issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 cauchy vol. 6 no. 3 pages: 100 – 161 malang november 2020 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 editorial board ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 table of contents matrix approach to the direct computation method for the solution of fredholm integro-differential equations of the second kind with degenerate kernels .................................................................................................................. 100 – 108 forecasting financial system stability using vector error correction model approach ...................................................................................................................................... 109 – 116 inclusion properties of the homogeneous herz-morrey ............................................... 117 – 121 local dynamics of an svir epidemic model with logistic growth ............................ 122 – 132 super total labeling (a,d)edge antimagic on the firecracker graph ..................... 133 – 139 the rule of hessenberg matrix for computing determinant of centrosymmetric matrices ........................................................................................................................................ 140 – 148 the metric dimension and local metric dimension of relative prime graph ....... 149 – 161 issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 cauchy vol. 7 no. 1 pages: 1 – 151 malang november 2021 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 editorial board ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 table of contents genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification ..................................... 1 – 12 a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units .................. 13 – 21 inclusion properties of the homogeneous herz-morrey spaces with variable exponent ....................................................................................................................................... 22 – 27 sentiment analysis on government performance in tourism during the covid19 pandemic period with lexicon based ........................................................................ 28 – 39 optimal prevention and treatment control on sveir type model spread of covid-19 ...................................................................................................................................... 40 – 48 analysis of landing airplane queue systems at juanda international airport surabaya ....................................................................................................................................... 49 – 63 on rainbow vertex antimagic coloring of graphs: a new notion .............................. 64 – 72 the confidence interval of the estimator of the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process .......................................................................................................................................................... 73 – 83 on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function ............................................................... 84 – 96 supplier selection analysis using minmax multi choice goal programming model .............................................................................................................................................. 97 – 104 spline nonparametric regression to analyze factors affecting gender empowerment measure (gem) in east java ................................................................... 105 – 117 modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression approach ................................................................................ 118 – 128 the ring homomorphisms of matrix rings over skew generalized power series rings ............................................................................................................................................... 129 – 135 cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 table of contents local hölder regularity of weak solutions for singular parabolic systems of plaplacian type ............................................................................................................................ 136 – 141 a study of count regression models for mortality rate .................................................. 142 – 151 issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 cauchy vol. 7 no. 2 pages: 152 – 331 malang may 2022 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines muhammad fakhruddin, department of mathematics, faculty of military mathematics and natural sciences, the republic of indonesia defense university, bogor, indonesia alfi yusrotis zakiyyah, universitas bina nusantara, indonesia bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia dr heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 editorial board subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 table of contents a note on generalized strongly p-convex functions of higher order ....................... 152 – 157 the generalized star modeling with heteroscedastic effects ..................................... 158 – 172 optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention ......................................................................... 173 – 185 an application of geographically weighted regression for assessing water polution in pontianak, indonesia ........................................................................................ 186 – 194 richards curve implementation for prediction of covid-19 spread in maluku province ........................................................................................................................................ 195 – 206 the properties of intuitionistic anti fuzzy module t-norm and t-conorm ............... 207 – 219 analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 220 – 230 average based-fts markov chain based on a modified frequency density partitioning to predict covid-19 in central java ......................................................... 231 – 239 spatial autoregressive model of tuberculosis cases in central java province 2019 ................................................................................................................................................ 240 – 248 goodwin model with clustering workers' skills in indonesian economic cycle ... 249 – 266 a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 ............................................................................................................................ 267 – 280 forecasting rice paddy production in aceh using arima and exponential smoothing models ..................................................................................................................... 281 – 292 multipolar intuitionistic fuzzy ideal in b-algebras ........................................................... 293 – 301 hybrid model of singular spectrum analysis and arima for seasonal time series data ................................................................................................................................................. 302 – 315 elliptical orbits mode application for approximation of fuel volume change ...... 316 – 331 cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the journal editor, the peer reviewer, the publisher and the society. as publisher of pure and applied mathematics journal, we take our duties to back up over all stages of publishing seriously and we recognize our ethical and other responsibilities. we are committed to ensuring that advertising, reprint or other commercial revenue has no impact or influence on editorial decisions. publication decisions the editor of cauchy is responsible for deciding which of the articles submitted to the journal should be published. the validation of the work in question and its importance to researchers and readers must always drive such decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 publication etics contribution to editorial decisions peer review assists the editor in making editorial decisions and through the editorial communications with the author may also assist the author in improving the paper. promptness any selected referee who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the editor and excuse himself from the review process. standards of 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been appropriately cited or quoted. cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 publication etics multiple, redundant or concurrent publication an author should not in general publish manuscripts describing essentially the same research in more than one journal or primary publication. submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. acknowledgement of sources proper acknowledgment of the work of others must always be given. authors should cite publications that have been influential in determining the nature of the reported work. authorship of the paper authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. all those who have made significant contributions should be listed as co-authors. where there are others who have participated in certain substantive aspects of the research project, they should be acknowledged or listed as contributors. the corresponding author should ensure that all appropriate co-authors and no inappropriate co-authors are included on the paper, and that all co-authors have seen and approved the final version of the paper and have agreed to its submission for publication. hazards and human or animal subjects if the work involves chemicals, procedures or equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia modeling plant stems using the deterministic lindenmayer system cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 286-295 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 03, 2021 reviewed: march 16, 2021 accepted: april 17, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11591 modeling plant stems using the deterministic lindenmayer system juhari1, muhammad zia alghar2 1,2 department of mathematics, faculty of science and technology universitas islam negeri maulana malik ibrahim email: juhari@uin-malang.ac.id, muhammadzia1904@gmail.com abstract plant morphology modeling can be done mathematically which includes roots, stems, leaves, to flower. modeling of plant stems using the lindenmayer system (l-system) method is a writing returns that are repeated to form a visualization of an object. deterministic l-system method is carried out by predicting the possible shape of a plant stem using its iterative writing rules based on the original object photo. the purpose of this study is to find a model of the plant stem with deterministic lindenmayer system method which will later be divided into two dimensional space three. the research was conducted by identifying objects in the form of pine tree trunks measured by the angle, thickness, and length of the stem. then a deterministic and parametric model is built with l-system components . the stage is continued by visualizing the model in two dimensions and three dimensions. the result of this research is a visualization of a plant stem model that is close to the original. addition color, thickness of the stem, as well as the parametric writing is done to get the results resembles the original. the iteration is limited to less than 20 iterations so that the simulation runs optimal. keywords: modeling; deterministic l-system; plant stems; visualization introduction growth is the process of increasing the size, volume and number of cells irrevisible (cannot return to original). on the stem of a plant, its growth includes the increase in size and volume on the trunk, branches, and branches. when the plant is still young its growth is fast and will slow down when have started to mature to age [1]. while the branching of the stem is a sign the growth of a plant. almost all plants branch. only monocot plants that do little branching on the stem. branching pattern the stem is generally divided into three, namely monopodial, sympodial, and dichotomous [2] [3] [4]. (a) (b) (c) http://dx.doi.org/10.18860/ca.v6i4.11591 mailto:juhari@uin-malang.ac.id mailto:muhammadzia1904@gmail.com modeling plant stems using the deterministic lindenmayer system juhari 287 figure 1. the branches of (a) monopodial, (b) sympodial, and (c) dichotomus the lindenmayer system or what is commonly referred to as the l-system is one the method used in mathematical studies developed by astrid lindenmayer. the lsystem uses a geometric aspect as its basis which is assisted in a manner computerization to produce a particular shape and model [5]. generally the lindenmayer system is a rewriting system with certain rules [6]. l-system is a branch of science in dynamic systems science that is applied to plant morphology, architectural design, to augmented reality (ar) components from video games and three-dimensional film. as of writing with the lindenmayer system used several main components, namely axioms, production, and letters [7]. the deterministic model is a mathematical model in which symptoms can be measured with certain certainty. in the deterministic model, the odds of each the incidence of subsequent events was not counted [6]. the l-system deterministic model can be formed in two dimensions or three dimensions. how to interpret the l-system basis a graphic in two dimensions is just a 2 x 2 dimensional matrix. however, in three dimensions using a rotation matrix measuring 3 x 3 [8]. the l-system also deals with the postulates of leonardo da vinci, namely on his notes at no. 394, which reads “all tree branches at any height if put together to have the same thickness as the stem below” [9]. an explanation of da vinci describes the condition of the diameter before and after branching, that is symbolized by d , d1 , and d2 . where d is the diameter of the stem, d1 and d2 are child stem diameter [10]. figure 2. measurement of stem thickness ratio from the da vinci postulate, the area in the parent stem (blue circle) will be obtained equals the area in the child stem (green circle). the implication of this is if the ratio of the two child stems add up to the thickness of the parent stem [10]. if the comparison of the parent stem to the stem of the child is done in the previously equation, the value of 0.707106 was obtained as the ratio of the thickness of the stems in the plant. the da vinci postulate ratio is used as a parameter in determining the thickness of the stem at l-system [6]. methods research data the research data used are some photos of the evergreen plant, results measurement of the angle, thickness of the trunk, and the length of the trunk on the pine tree to be modeled. these data are used as the basis for forming the l-system pattern that will be created. the data can be seen in the following table modeling plant stems using the deterministic lindenmayer system juhari 288 table 1. the results of the angle measurement in the xz plane angle measured (α) angle size first branch 0 o second branch 90 o third branch 180 o fourth branch 270 o table 2. the results of the angle measurement in the yz plane angle measured (β) left right first section 5o 5,2 o second section 6o 6,8 o third section 5,4o 5,2 o fourth section 5o 5,2 o table 3. the results of the angle measurement in the xy plane angle measured (θ) left right first branch 65o 62 o son of first branch 52o 54 o second branch 62o 63 o son of second branch 66o 64 o third branch 60o 62 o son of third branch 64o 61 o table 4. measurement results of plant stem length type of stem stem length mother stem (a) 17,20 cm daughter stem (b) 15,50 cm branching daughter stem (c) 9,30 cm modeling plant stems using the deterministic lindenmayer system juhari 289 table 5. results of measurements of plant stem thickness type of stem thickness trunk parent stem 9,20 cm right fork 6,40 cm left fork 6,20 cm daughter of the right fork 4,50 cm daughter of the right fork 4,60 cm daughter of the left fork 4,40 cm daughter of the left fork 4,50 cm research steps the steps taken to model plants using deterministic l-system are: (1) take picture of the object being modeled, namely in the form of an image photographed from various sides. (2) measurement of the angle, length and thickness of the stem for each branch on the object stem. (3) finding the average value and ratio of measurement results. (4) identify the various components of the l-system that build it, such as rules production, letters, axioms and other components. (5) performing a simulation by evaluating the results. research data the research data used are some photos of the evergreen plant, results measurement of the angle, thickness of the trunk, and the length of the trunk on the pine tree to be modeled. these data are used as the basis for forming the l-system pattern that will be created. result and discussion modeling results research modeling plants using the deterministic l-system method was carried out against three plants in a three-dimensional plane. as for the definition of the symbols used in this study can be observed in the following table table 6. definitions of symbols in the l-system f(l) : draw forward by l units, for l > 0 +(a) : rotates counterclockwise with rotation matrix r(α) of α degree -(a) : rotates clockwise with rotation matrix r(α) of a degree &(α) &(a) : rotates counterclockwise with rotation matrix r(β) of a degree modeling plant stems using the deterministic lindenmayer system juhari 290 ^(a) : rotates clockwise with rotation matrix r (β) of a degree /(a) : rotates counterclockwise with rotation matrix r (δ) of a degree \(a) : rotates counterclockwise with rotation matrix r (δ) of a degree |(a) : rotates with a rotation matrix r ( δ ) of 180o degree [ : saves the current location then moves according to the next command ] : returns to the original position stored in the symbol “[“ wr : specifies the thickness of the stem !(x) : determine the line thickness x the following are all production rules created for the three crop objects (pine plant) w = a(1,15) r1 = 0.9; r2 = 0.6; wr = 0,691 generation : 15 a0 = 61.25; a1 = 5.47; a2 = 90 p1 : a(l,w) --> !(w*wr)f(l)[e(l*r1,w*wr)] p2 : b(l,w):(l>=0.1) --> !(w*wr)f(l)[+(a0)^(a1)c(l*r2,w*wr)]f(wr) [-(a0)^(a1)c(l*r1,w*wr)] [^(a1)b(l*r2,w*wr2)f(l*r2,w*wr)] p3 : c(l,w):(l>=0.0) --> !(w*wr)f(l)[^(a1)d(l*r2,w*wr)] p4 : d(l,w):(l>=0.0) --> ! (w*wr)f(l)[^(a1)c(l*r2,w*wr)] p5 : e(l,w) --> !(w*wr)f(l)[[/(a2)&(l*a2)b(l*r1,w*wr)] [\(a2)&(l*a2)b(l*r1,w*wr)][/(2*a2)&(l*a2)b(l*r1,w*wr)] [\(0*a2)&(l*a2)b(l*r1,w*wr)]][[/(0.5*a2)&(l*a2)b(l*r1,w*wr)] [/(0.666*a2)fe(l*r1,w*wr)] p6 : s(l,w) --> tf p7 : t(l,w) --> f (ketapang kencana plant) v = {r1, r2, r3, wr, l, w, a, y, z, s, f, !, -, +, &, ^, /, \, (, ), [, ], *} modeling plant stems using the deterministic lindenmayer system juhari 291 axiom w = a(12,20) a(8,20) a(4,20) a(1,20) generations : 5 r1 = 0.8 r2 = 0.5 r3 = 0.3 wr = 0.707 p1 : a(l,w) --> !(w*0.5)sf(l)[-(70)y(l*r1,w*wr)][+(70)z(l*r1,w*wr)] [-(60)^(45)/(15)y(l*r1,w*wr)][+(60)&(45)/(15)z(l*r1,w*wr)] [+(120)&(135)\(15)y(l*r1,w*wr)][(120)&(225)\(15)z(l*r1,w*wr)] [/(90)-(70)y(l*r1,w*wr)][/(90)+(70)z(l*r1,w*wr)] p2 : y(l,w) --> !(w*0.3)sf(l)[[^(37.5)y(l*r3,w*wr)][&(37.5)y(l*r3,w*wr)]] sf(l)[[^(20)y(l*r3,w*wr)][&(20)y(l*r3,w*wr)]]sf(l) p3 : z(l,w) --> !(w*0.3)sf(l)[[^(37.5)z(l*r3,w*wr)][&(37.5)z(l*r3,w*wr)]] sf(l)[[^(20)z(l*r3,w*wr)][&(20)z(l*r3,w*wr)]]sf(l) p4 : s --> ssf (trembesi plant) v = {a0, a1, r1, r2, wr, l, w, a, b, c, s, f, !, -, +, ^, /, \, (, ), [, ], *} axiom w = a(1,90) generation : 10 r1 = 0.9 r2 = 0.6 a0 = 25 a1 = 10 wr = 0.707 p1 : a(l,w) --> !(w*0.4)-(10)sf(l*0.5)[+(a0)/(90)c(l*r2,w*wr)] [-(a1)\(90)a(l*r1,w*wr)][^(a0)\(90)c(l*r1,w*wr)] p2 : b(l,w) --> !(w*0.4)sf(l)[-(a0)c(l*r2,w*wr)][+(a1)c(l*r1,w*wr)] p3 : c(l,w) --> !(w*0.4)sf(l)[+(a0)a(l*r2,w*wr)][-(a0)a(l*r1,w*wr)] p4 : s --> sf modeling plant stems using the deterministic lindenmayer system juhari 292 visualization result the results of the visualization of the modeling of plant stems using the deterministic method system is done based on the measurement results of stem thickness, angle, and length stem of the object being modeled, which is then written in the lindenmayer rule system. furthermore, the l-system writing is visualized using computational applications namely l-studio. this application is specially designed for modeling plant growth developed at the university of cagliari [11]. the visualization results will be in a dimensional form three, so that the output can be viewed from various points of view. visual display on l-studio supports in visualization magnification, so that the output can be seen in form the details. the following is the output of the lindenmayer system program for three indoor plants three dimension. (pine plant) figure 3. visualization of pine trees from various iterations (trembesi plant) figure 4. visualization of trembesi from various perspectives modeling plant stems using the deterministic lindenmayer system juhari 293 ( ketapang kencana plant) figure 5. visualization of ketapang kencana plants from various perspectives after visualizing the l-studio program, it is followed by evaluating the results of the visualization. every detail of the visualization is enlarged and rotated in all directions. this is to ensure that there are no defects in the visualization. if there is defects, then changes are made to the components of the production rules. the iteration use on each plant is less than 20 iterations. it is intended to prevent programs that are not responding or errors when running on l-studio. the following is the comparison result of the visualization with the original object. (pine plant) figure 6. comparison of several visualization results of the l-system program on pine plant with the original object modeling plant stems using the deterministic lindenmayer system juhari 294 (ketapang kencana plant) figure 7. comparison of several visualization results of the l-system program on ketapang kencana plant with its original object (trembesi plant) figure 8. comparison of several visualization results of the l-system program on tamarind trees with the original object conclusion modeling plant stems using the deterministic lindenmayer system method, is a modeling that is more concise in level than using a method stochastic lindenmayer system , in the absence of a probability factor. the initial stages important in modeling modeling plant stems using the deterministic lindenmayer system juhari 295 plant stems is to determine the main component of l-system . in this modeling, researchers use three-dimensional visualization on the result. therefore, the visualization results are displayed from the front side, the side (round 90o x axis ), the top side (90o rotation of the z axis ), and the side slightly down (round 45o z axis ). the use of the l-studio application is very helpful in the process of visualizing the model plants, both in the iteration process, determine production rules, to deep loops do the visualization. the use of iterations needs to be considered in order for the running program to run smooth. the researcher uses less than 20 iterations so that it running optimally. references [1] a. shipunov, introduction of botany, usa: university of minot state, 2011. [2] r. mcgarry, "monopodial and sympodial branching architecture in cotton is differentially regulated by the gossypium hirsutum single flower truss and selfpruning orthologs," new phytologist, pp. 1-2, 26 april 2016. [3] j. power, "interactive arrangement of botanical l-system models," proceedings of the 1999 symposium on interactive 3d graphics si3d '99, pp. 175-182, 1999. [4] c. jacob, "genetic l-system programming: breeding and evolving artificial flowers with mathematica," in proceedings of the first international mathematica symposium, vol. 33976, pp. 215-222, 1995. [5] p. prusinkiewicz, "developmental models of herbaceous plants for computer imagery purposes," computer graphics, vol. 22, no. 0097-8930, pp. 141-150, 1988. [6] a. lindenmayer, the algorithmic beauty of plants, new york: spinger-verlag, 1990. [7] juhari, "pemodelan pertumbuhan tanaman zea mays l. menggunakan stochastic lsystem," jurnal cauchy, vol. 3, no. 2477-3344, p. 2, 2013. [8] c. h. iswanto, "penerapan stochastic l-systems pada pemodelan pertumbuhan batang tanaman," pp. 1-18, 1 janurary 2014. [9] j. ritcher, the notebooks of leonardo da vinci, new york: dover publications, 1970. [10] b. mandelbrot, the fractal geometry of nature, san fransisco: w.h. freeman, 1982. [11] suhartono, "pemodelan pertumbuhan tanaman zinnia menggunakan lindenmayer system dengan mathematica," jurnal cauchy, vol. 3, no. 2086-0382, pp. 33-37, 2013. [12] juhari, "pemodelan pertumbuhan batang tanaman menggunakan deterministic lsystems," p. 3, 25 november 2013. [13] m. kahfi, geometri transformasi, malang: ikip malang, 1997. inclusion properties of herz-morrey spaces with variable exponent cauchy –jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 22-27 p-issn: 2086-0382; e-issn: 2477-3344 submitted: may 02, 2021 reviewed: august 24, 2021 accepted: october 06, 2021 doi: https://doi.org/10.18860/ca.v7i1.12141 inclusion properties of herz-morrey spaces with variable exponent hairur rahman departement of mathematics, islamic state university of maulana malik ibrahim malang email: hairur@mat.uin-malang.ac.id abstract the inclusion properties in herz-morrey spaces has proved by rahman in 2020. this paper aims to discuss the inclusion of the homogeneous herz-morrey spaces and homogeneous weak herzmorrey spaces with variable exponent. we also investigated the inclusion between both spaces. this result will be useful to prove fractional integral on the homogeneous herz-morrey spaces with variable exponent. keywords: herz-morrey spaces; inclusion properties; variable exponent. introduction inclusion properties or inclusion relation between spaces has received a lot of attention from researchers. it seems that many authors have studied this issue in some spaces (see [1]-[5]). thus, this lead the author for discussing the inclusion properties especially in herz-morrey spaces. herz spaces can be traced back to the work of beurling. beurling [6] introduced a space 𝒜𝑝, which is the original version of non homogeneous herz spaces. lu et al [7] has given the inclusion properties in homogeneous herz spaces, as a proposition below. proposition 1.1. let 𝛼 ∈ ℝ, 𝑝 > 0, and 𝑞 ≤ ∞. the following inclusions are valid. a. if 𝑝1 ≤ 𝑝2, then 𝐾𝑞 𝛼,𝑝1 (ℝ𝑛) ⊂ 𝐾𝑞 𝛼,𝑝2 (ℝ𝑛) b. if 𝑞2 ≤ 𝑞1, then 𝐾𝑞1 𝛼,𝑝 (ℝ𝑛) ⊂ 𝐾𝑞2 𝛼−𝑛( 1 𝑞1 − 1 𝑞2 ),𝑝 (ℝ𝑛). this proposition can be proved by simply computation. in fact, if 0 < 𝑟 < 1, (a) is a consequence of the inequality (∑|𝑎𝑘 | ∞ 𝑘=1 ) 𝑟 ≤ ∑|𝑎𝑘 | 𝑟 ∞ 𝑘=1 . while, (b) can be deduced directly from the hölder inequality. in 2016, gunawan et al. (see [1] [2]) have proved the inclusion of morrey spaces and generalized morrey spaces. recently, rahman [8] also has proved the inclusion properties in herz-morrey spaces. these result have been motivated the author to study more about inclusion in homogenous herz-morrey spaces, but in this case the author uses variable exponent. since 1991, the research of kovacik and rakosnik [9] motivated many researchers to study about function spaces with variable exponent in several discussion. suppose that ω ⊂ ℝ𝑛 is an open set, 𝑝(⋅): ω → [1, ∞) is a measurable https://doi.org/10.18860/ca.v7i1.12141 mailto:hairur@mat.uin-malang.ac.id inclusion properties of herz-morrey spaces with variable exponent hairur rahman 23 function and 𝐿𝑝(⋅)(ω) is denoted the set of measurable functions 𝑓 on ω, such that for some positive 𝜆 satisfied ∫ ( | 𝑓(𝑥) | 𝜆 ) 𝑝( 𝑥 ) 𝑑𝑥 ω < ∞. if 𝐿𝑝(⋅)(ω) equipped by the luxemburg-nakano norm ‖ 𝑓 ‖ 𝐿 𝑝(⋅)(ω) = inf { 𝜆 > 0 ∶ ∫ ( | 𝑓(𝑥) | 𝜆 ) 𝑝(𝑥) 𝑑𝑥 ω ≤ 1}, then 𝐿𝑝(⋅)(ω) becomes a banach function spaces. since these spaces generalize the standard 𝐿𝑝 spaces, they are also referred to as variable 𝐿𝑝 spaces. 𝐿𝑝(⋅)(ω) is isometrically isomorphic to 𝐿𝑝(ω), when 𝑝(𝑥) = 𝑝 is a constant. in 2010, the boundedness of sublinear operators on herz-morrey space with variable exponent ℳ�̇�𝑝(⋅) 𝛼,𝑞 and ℳ�̇�𝑝(⋅) �̅�,𝑞 was proved by izuki [10]. then, xu and yang [11] developed the definition of herz-morrey spaces with variabel exponents. let 𝑝(⋅) ∈ 𝒫(ℝ𝑛), 0 < 𝑞 < ∞, 0 ≤ 𝜆 < ∞, and 𝛼(⋅) is a bounded real-valued measurable function on ℝ𝑛 , the homogeneous herz-morrey spaces with variable exponent ℳ�̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) consists all functions 𝑓 ∈ 𝐿𝑙𝑜𝑐 𝑞 ( ℝ𝑛 /{0} ) such that ‖𝑓‖ ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) = sup 𝐿∈ℤ 1 2𝐿𝜆 (∑ 2𝑘𝛼(⋅)𝑝‖ 𝑓𝜒𝑘 ‖𝐿𝑞 (ℝ𝑛) 𝑝𝐿 𝑘=−∞ ) 1 𝑝(⋅) < ∞, , where 𝐵𝑘 = { 𝑥 ∈ ℝ 𝑛 : |𝑥| ≤ 2𝑘 }, 𝐴𝑘 = 𝐵𝑘 /𝐵𝑘−1 and 𝜒𝑘 = 𝜒𝐴𝑘 is the characteristic function of the set 𝐴𝑘 for 𝑘 ∈ ℤ. as another spaces which have their weak type spaces, herz-morrey spaces also have their weak type spaces. for 𝛼(⋅) ∈ ℝ𝑛 , 𝑝(⋅) ∈ 𝒫(ℝ𝑛), 0 ≤ 𝜆 ≤ ∞ and 0 < 𝑞 ≤ ∞, the homogeneous weak herz-morrey spaces with variabel exponent ( 𝑊 ℳ�̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛)) is a set of measurable 𝑓 ∈ 𝐿𝑙𝑜𝑐 𝑞 (ℝ𝑛 /{0}) which is equipped with norm such that ‖ 𝑓 ‖ 𝑊 ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) = sup 𝛾>0 𝛾 sup 𝐿∈ℤ 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑝(⋅)𝑚𝑘 (𝛾, 𝑓) 𝑝(⋅) 𝑞 𝐿 𝑘=−∞ ) 1 𝑝(⋅) < ∞, where 𝑚𝑘 ( 𝛾, 𝑓 ) = |{ 𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾 }|. some authors have investigated those spaces in various terms of discussion (see [12] [15]). meanwhile, this article aims to discuss in terms inclusion properties and inclusion relation of the homogeneous herz-morrey spaces and homogeneous weak herz-morrey spaces with variable exponent. result and discussion our main results are the following: theorem 2.1. let 1 ≤ 𝑝1(⋅) ≤ 𝑝2(⋅) < 𝑞 < ∞, and 𝛼(⋅) is a bounded real-valued measurable fuction on ℝ𝑛 . then, the inclusion ℳ�̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ ℳ�̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛), is valid. inclusion properties of herz-morrey spaces with variable exponent hairur rahman 24 proof. we may take any 𝑓 ∈ ℳ�̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). then, by using hölder inequality and 𝑝1 ≤ 𝑝2 we have ‖𝑓‖ ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) = sup 𝐿∈𝑍 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑝1(⋅) 𝐿 𝑘=−∞ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1(⋅) ) 1 𝑝1(⋅) ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 (( ∑ (2𝑘𝛼(⋅)𝑝1(⋅)) 𝑝2(⋅) 𝑝1(⋅) 𝐿 𝑘=−∞ ) 𝑝1(⋅) 𝑝2(⋅) ( ∑ (‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1(⋅) ) 𝑝2(⋅) 𝑝2(⋅)−𝑝1(⋅) 𝐿 𝑘=−∞ ) 1− 𝑝1(⋅) 𝑝2(⋅) ) 1 𝑝1(⋅) ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 (( ∑ 2𝑘𝛼(⋅)𝑝2(⋅) 𝐿 𝑘=−∞ ) 𝑝1(⋅) 𝑝2(⋅) ( ∑ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1(⋅)𝑝2(⋅) 𝑝2(⋅)−𝑝1(⋅) 𝐿 𝑘=−∞ ) 1− 𝑝1(⋅) 𝑝2(⋅) ) 1 𝑝1(⋅) ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑝2(⋅) 𝐿 𝑘=−∞ ( ∑ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1(⋅)𝑝2(⋅) 𝑝2(⋅)−𝑝1(⋅) 𝐿 𝑘=−∞ ) 𝑝2(⋅)−𝑝1(⋅) 𝑝1(⋅) ) 1 𝑝2(⋅) ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑝2(⋅) 𝐿 𝑘=−∞ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝2(⋅) ) 1 𝑝2(⋅) ≤ ‖𝑓‖ ℳ�̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) . it is easy to know that 𝑓 ∈ ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛), where 𝛼(⋅) ∈ (ℝ𝑛) and 𝑝(⋅) ∈ 𝒫(ℝ𝑛). then, we have ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 ( ℝ𝑛). by the previous theorem, the author established the following inclusions. theorem 2.2. let 1 ≤ 𝑝1(⋅) ≤ 𝑝2(⋅) < 𝑞 < ∞, and 𝛼(⋅) is a bounded real-valued measurable fuction on ℝ𝑛 , then the following inclusion is valid. 𝐿𝑞 ( 𝑅𝑛) = ℳ �̇�𝑞,𝑞 𝛼(⋅),𝜆 ( ℝ𝑛 ) ⊆ ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 ( ℝ𝑛 ) ⊆ ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 ( ℝ𝑛). proof. theorem 2.1 has stated that ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). then, we only prove that 𝐿𝑞 (ℝ𝑛) = ℳ�̇� 𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ ℳ𝐾 ̇ 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). let 𝑓 ∈ 𝑀�̇�𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ), by using similar method as before, we get ‖ 𝑓 ‖ 𝑀�̇� 𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑞 𝐿 𝑘=−∞ ((∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 (∫ |𝜒𝑘 | 𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 ) 𝑞 ) 1 𝑞 ≤ sup 𝐿∈𝑍 1 2𝐿𝜆 ∑ 2 𝑘𝛼(⋅) 𝐿 𝑘=−∞ (∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 ( 2 𝑘𝑑 ) 1 𝑞 ≤ 𝐶 (∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 inclusion properties of herz-morrey spaces with variable exponent hairur rahman 25 ≤ ‖ 𝒇 ‖ 𝑳𝒒(ℝ𝒏). hence, 𝑓 ∈ 𝐿𝑞 (ℝ𝑛) and 𝐿𝑞 (ℝ𝑛) ⊆ ℳ �̇�𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ). in the other hand, for any 𝑓 ∈ 𝐿𝑞 (ℝ𝑛), there exist any constant 𝐶 such that 𝐶 = sup 𝐿∈𝑍 1 2𝐿𝜆 ∑ 2 𝑘𝛼(⋅) + 𝑘𝑑 𝑞𝐿 𝑘=−∞ . consequently, we have 𝑓 ∈ ℳ �̇� 𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) and ℳ�̇� 𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ 𝐿𝑞 (ℝ𝑛 ). it gives conclusion that 𝐿𝑞 (ℝ𝑛) = ℳ �̇� 𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ), where 𝛼(⋅) ∈ (ℝ𝑛 ). furthermore, we will prove that ℳ �̇�𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). by using similar method as the proof of theorem 2.1, we have ‖ 𝑓 ‖ ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ≤ ‖ 𝑓 ‖ ℳ �̇�𝑞,𝑞 𝛼(⋅),𝜆 (ℝ𝑛) , where 𝛼(⋅) ∈ (ℝ𝑛 ). the author also added the inclusion of the homogeneous weak herz-morrey spaces with variable exponent by the following theorem. theorem 2.3. let 1 ≤ 𝑝1(⋅) ≤ 𝑝2(⋅) ≤ 𝑞 < ∞, and 𝛼(⋅) is a bounded real-valued measurable fuction on ℝ𝑛 , the following inclusion holds: 𝑊 ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ 𝑊 ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). proof. let 𝑓 ∈ ‖𝑓‖ 𝑊ℳ�̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) , we have ‖ 𝑓 ‖ 𝑊 ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) = sup 𝛾>0 𝛾 sup 𝐿∈ℤ 1 2𝐿𝜆 ( ∑ 2𝑘𝛼(⋅)𝑝1(⋅)𝑚𝑘 (𝛾, 𝑓) 𝑝1(⋅) 𝑞 𝐿 𝑘=−∞ ) 1 𝑝1(⋅) ≤ sup 𝛾>0 𝛾 sup 𝐿∈ℤ 1 2𝐿𝜆 (∑ 2𝑘𝛼(⋅)𝑝2(⋅)𝑚𝑘 (𝛾, 𝑓) 𝑝2(⋅) 𝑞𝐿 𝑘=−∞ ) 1 𝑝2(⋅) ≤ ‖𝑓‖ 𝑊ℳ�̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) . the above inequality has shown that 𝑊 ℳ �̇� 𝑝2(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ⊆ 𝑊 ℳ �̇� 𝑝1(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). now, we state the inclusion relation between both spaces. theorem 2.4. let 1 ≤ 𝑝(⋅) ≤ 𝑞, and 𝛼(⋅) is a bounded real-valued measurable function on ℝ𝑛 . then, the inclusion ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ) ⊆ 𝑊 ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) is proper. proof. we use similar idea as before to prove this theorem. let 𝑓 ∈ ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ), 𝑎(⋅) ∈ ℝ𝑛 , 𝑝(⋅) ∈ 𝒫(ℝ𝑛) and 𝛾 > 0. we have observed that | {𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾} | 𝑝(⋅) 𝑞 ≤ (∫ |𝑓(𝑥)𝜒𝑘 | 𝑞 𝑑𝑥 𝐵(0,2𝑘) ) 𝑝(⋅) 𝑞 = ‖ 𝑓𝜒𝑘 ‖ 𝐿𝑞(ℝ𝑛) 𝑝(⋅) . multiplying both sides by ∑ 2𝑘𝛼(⋅)𝑝(⋅)𝐿𝑘=−∞ , we get inclusion properties of herz-morrey spaces with variable exponent hairur rahman 26 ∑ 2𝑘𝛼(⋅)𝑝(⋅)|{ 𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾 }| 𝑝(⋅) 𝑞 𝐿 𝑘=−∞ ≤ ∑ 2𝑘𝛼(⋅)𝑝(⋅) ‖ 𝑓𝜒𝑘 ‖ 𝐿𝑞(ℝ𝑛) 𝑝(⋅) 𝐿 𝑘=−∞ . clearly, we see that ‖ 𝑓 ‖ 𝑊ℳ�̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) ≤ ‖ 𝑓 ‖ ℳ𝐾 ̇ 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛) and 𝑓 ∈ 𝑊 ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛), which implies that ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛 ) ⊆ 𝑊 ℳ �̇� 𝑝(⋅),𝑞 𝛼(⋅),𝜆 (ℝ𝑛). conclusion by this result, the author can conclude that the homogeneous herz-morrey spaces with variable exponent have inclusion properties ... . this result will be useful to be used in proving fractional integral on the homogeneous herz-morrey spaces with variable exponent. acknowledgment this paper is partially supported by uin maulana malik ibrahim malang research and innovation program 2020. references [1] h. gunawan, d. i. hakim, k. m. limanta and a. a. masta, "inclusion property of generalized morrey spaces," math. nachr., pp. 1-9, 2016. 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[15] y. shi, x. tao and t. zheng, "multilinier riesz potential on morrey-herz spaces with non-doubling measure," journal of inequality and applications, vol. 10, 2010. multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 238-245 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 25, 2020 reviewed: february 19, 2021 accepted: april 11, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.10871 multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma1, juhari 2, ramadani a. rosa3 1,2,3 department of mathematics uin maulana malik ibrahim malang email: riadhea@uin-malang.ac.id, juhari@uin-malang.ac.id, ramadaniauiyanarosa@gmail.com abstract poverty population is one of the serious problems in indonesia. the percentage of population poverty used as a means for a statistical instrument to be guidelines to create standard policies and evaluations to reduce poverty. the aims of the research are to determine model population poverty using multivariate adaptive regression spline and bagging mars then to understand the most influence variable population poverty of central java province in 2018. the result of this research is the bagging mars model showed better accuracy than the mars model. since, gcv in the bagging mars model is 0,009798721 and gcv in the mars model is 6,985571. the most influence variable population poverty of central java province in 2018 based on mars model is the percentage of the old school expectation rate. then, the most influentce variable based on bagging mars model is the number of diarrhea disease. keywords: multivariate adaptive regression splines; bootstrap aggregating; generalized crossvalidation; poverty introduction poverty has concerned problem in the world even in indonesia. in indonesia, which is developing country, poverty has been affected in economics that it’s showed level of welfare. therefore, it has become a serious problem that must be resolved. the growth of economics is the fundamental factor to reduce poverty. based on bps data, indonesia has been able to deal with some economics global problem and succeeded in increasing economic growth. some programs realized such as credit procurement programs, agricultural development, equitable development, infrastructure improvement, to the procurement program inpress lagging village (idt) to help improve the community's living standards. the efforts considered significant because it reduced the experiencing of gaps between the rich and the underprivileged people and as an effort to realize the strategy of human quality development [1]. according to bps (badan pusat statistik) or indonesian statistics institution, level of poverty in indonesia has been reduced in recently. the percentage of privileged people reduced up to 0, 58% (in year-on-year) and at 2017 was the lowest poverty level rate. the http://dx.doi.org/10.18860/ca.v6i4.10871 mailto:ramadaniauiyanarosa@gmail.com multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 239 government succeeded in reducing poverty rate by 1.18 million people from an average. the government made a system for implement social protection based on a life cycle approach at 2018. however, in some areas poverty rate was slowed in reducing poverty [2]. mars were introduced assumption about the relationship between the dependent and independent variables to estimate general functions of high dimensional data. bagging mars is a method that improved performance of in mars method used bootstrap replicating. the past researches karisma & sri harini [3] used mars to find the classification of risk factors of ischemic and hemorrhagic patients by mars method, kilinc, b et al. [4] research to find models of metal concentrations to determine soil pollution by mars method, etc. mars model used combination from spline method and recursive partition. then, model in spline regression applied using a set of basis function to achieve q-order spline regression and estimated using least squares method. it has knot to find out the continuity basis function from one region in regression line to others. otherwise, bootstrap aggregating (bagging) used to minimize squared error value. the aimed of the research was the influenced poverty factor using mars and bagging mars then it can be used for guidelines standard policies and evaluation to reduce poverty. methods poverty is resident who have an average monthly expenditure per capita below the poverty line [5]. poverty influenced by some factors such as human resources, employment, inflation, unemployment, population density, health facilities, income, scarcity, transportation, education, business capital [6] . mars method used multivariate nonparametric approaches. it has recursive partition formed, high dimensional data, and discontinuity data. bagging mars is a method that used for improve performance on mars method with bootstrap replicating. mars developed by recursive partitioning regression (rpr) to estimate sub-region in each region continuous model in knots [7]. the advantage mars is unrequired standardization, produced accurate results, used in big data, and used for regression analysis and classification simultaneously. bagging mars recursive partitioning regression (rpr) unable to overcome the discontinuous data in knots. therefore, the rpr algorithm used to estimate and correlate data in subregions [8]. the basis function explained the relationship between the dependent and independent variables [9] . the regression model used basis functions (bf) as follows: 𝑦 = 𝛽0 ∑ 𝛽𝑚ℎ𝑚(𝑥) 𝑀 𝑚=1 (1) where ℎ𝑚 is a set of basis function, and 𝛽𝑚 is a coefficient of ℎ𝑚 in splines basis function defined as: ℎ𝑚 = ∏ [𝑆𝑘𝑚(𝑥𝑣(𝑘,𝑚) − 𝑡𝑘𝑚)] + 𝐾𝑚 𝑘=1 (2) after modified bf with the rpr model, the mars model obtained as follows: 𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚 𝑀 𝑚=1 ∏ [𝑆𝑘𝑚(𝑥𝑣(𝑘,𝑚) − 𝑡𝑘𝑚)]+ 𝐾𝑚 𝑘=1 (3) where 𝑎0 is a coefficient, 𝑎𝑚 is a coeefficient function basis-m m is a maximum basis, 𝐾𝑚 is an interction degree, 𝑥𝑣(𝑘,𝑚) is label of predictor variables, 𝑡𝑘𝑚 is knot of predictor variables 𝑥𝑣(𝑘,𝑚), and 𝑆𝑘𝑚 are variables that take values ± 1 [7]. multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 240 in matrix formed, mars model defined by (4) 𝑌 = 𝐵𝑎 + 𝜀 , 𝑌 = (𝑌1, … , 𝑌𝑛) 𝑇, 𝑎 = (𝑎0, … , 𝑎𝑀) 𝑇, 𝜀 = (𝜀0, … , 𝜀𝑛) 𝑇 (4) 𝐵 = [ 1 ∏ [𝑆1𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 … ∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 1 ∏ [𝑆2𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 ⋮ 1 ∏ [𝑆𝑛𝑚. (𝑥𝑣(1,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 …… … ∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 ⋮ ∏ [𝑆𝑀𝑚. (𝑥𝑣(𝑀,𝑚) − 𝑡1𝑚)] 𝐾𝑚 𝑘=1 ] (5) the gcv used to find the best model from mars method, which used smaller is better. it is determined value by trial and error combining the number of basis functions (bf), maximum interaction (mi), and minimum observation (mo) [4]. the gcv defined as: 𝐺𝐶𝑉 = 𝑀𝑆𝐸 [1− 𝐶(�̂�) 𝑛 ] 2 (6) where 𝑀𝑆𝐸 value defined as 1 𝑛 ∑ [𝑦𝑖 − 𝑓𝑀(𝑥𝑖)] 2𝑛 𝑖=1 , and c(m̂) defined as c(m̂) = c(m) + dm (7) where, c(m) is matrix trace [b(btb)−1bt] + 1 that is the number of parameters being fit and d represents a cost for each basis function optimization [7]. the research used data from sosial ekonomi nasional (susenas), bps (badan pusat statistik) or indonesian statistics institution for java province, and bps semarang regional. total data that used in this research was 350. it used mars and bagging mars to analyze, then the steps that employed are divided data into training and testing data. then, mars method resolved by determined data used mars method with a combination basis function (bf), maximum interaction (mi), and minimal observations (mo)[10]. besides, obtained minimum gcv value to determine the best model in mars and interpreted mars model. bagging mars method completed by determined bagging mars model using 50 replications. then, the best model in bagging mars method achieved. the last is determined variable that the most influenced of poverty in central java province in 2018. results and discussion statistics descriptive the descriptive analysis used to determine characteristic poverty in central java at 2018 (badan pusat statistik, 2019) multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 241 figure 1. descriptive analysis poverty population figure 1 showed the percentage of population poverty in central java at 2018. the histogram illustrated regency areas and the percentage of poverty population in those areas. the highest poverty in those areas was kabupaten wonosobo with 17,58%. the total of poverty population was almost fifth percent. the percentage of population poverty occurred by some factors such as social economic, technology, health care and others. then, the lowest population poverty was kota semarang from total population. it was under one in twenty percent. modeling poverty population mars and bagging mars methods the mars model showed in matrix pattern (see figure 3.2). the matrix plot discovered relationship between response variable, which is variable the percentage of population poverty (𝑌), and predictor variables, which is the number of diarrhea disease (𝑋1), the number of life expectancy (𝑋2), the percentage of human development index (hdi) (𝑋3), the percentage of expenditure per capita by non-food commodities (𝑋4), the percentage of open unemployement (𝑋5), the number of infant malnutrition (𝑋6), the percentage of family planning and birth control (𝑋7), the percentage of labor force participation rate (𝑋8), the percentage of expectation old school (𝑋9), the number of bpjs participants (𝑋10). 0 2 4 6 8 10 12 14 16 18 20 percentage of poverty population in central java province in 2018 multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 242 figure 2. matrix plot pattern of poverty population figure 2 illustrated that indicated unclear and difficult patterns of the relationship between variables. then, in each variable has different characteristics on those areas and predictor variable was not able to be explained. in addition, nonparametric method used in this research which is mars and bagging mars methods. the best model even in mars and bagging mars methods indicated by the gcv. the gcv in mars model was 6.985571 and the r-sq value was 75,7 %. then, it was five predictor variables that significant and affected population poverty. it was 𝑋1, 𝑋6, 𝑋9, 𝑋8, 𝑋10 using training data 85% and testing data 15%. the mars model obtained: f(x) = 12.8 − 0.000235 ∗ max(0, 𝑋1 − 19574) + 0.0107 ∗ max(0, 249 − 𝑋6) – 0.514 ∗ max(0, 𝑋8 − 67.5) + 7.35 ∗ max(0, 124 − 𝑋9) − 1.34e − 05 ∗ max(0, 597322 − 𝑋10) then, the interpretation of mars model is − 0.000235 ∗ max (0, 𝑋1 − 19574) when, the value of 𝑋1 was greater than 19574, for every increased number of diarrhea, it increased the percentage of the population poverty at 0.000235 in the central java province with an average number of cases of diarrhea less than 19574 people. 0. 0107 ∗ max (0, 249 −𝑋6) when, the value of 𝑋6 was smaller than 249, for every increased number of infant malnutrition, it increased the percentage of the population poverty at 0.0107 in the central java province with an average number of infant malnutrition less than 249 people. −0.514 ∗ max (0, 𝑋8 − 68) when, the value of 𝑋8 greater than 68, for every, increased in the percentage of labor force participation rate, it decreased the percentage of the population poverty by 0.514 in the central java province with an average percentage participation rate of a labor force more than 68 people. multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 243 7. 35 ∗ max (0, 12.4 −𝑋9) when, the value of 𝑋9 is smaller than 12.4, for every increased the percentage of old school expectancy, it decreased the percentage of the population poverty at 7.35 in the central java province with an average percentage of the old school expectancy is less than 12.4%. −1, 34𝑒−05∗ max (0, 597322 −𝑋10) when, the value of 𝑋10 was smaller than 597322, for every increased number of participants bpjs, it decreased the percentage of the population poverty of 0.0000134 in the central java province with an average number of participants bpjs less than 597322 people. in bagging mars method that used 50 times replicate the best model obtained at the 49th replication using minimum gcv. then, it was six predictor variables that have significant value affected population of poverty. it was 𝑋1, 𝑋4, 𝑋6, 𝑋7, 𝑋8, 𝑋10. the gcv was 0.009431298 and r-sq value 0.7955023. the model was: f̂(x) = 11.17643 – 0.0001232638 ∗ max(0, 13503 − 𝑋1) + 0.0001346581 ∗ max (0, 𝑋1 − 13503) + 1.637211 ∗ max(0, 48.96 − 𝑋4) – 0.6424541 ∗ max(0, 𝑋4 −48.96) – 0.0250127 ∗ max(0, 𝑋6 − 52) + 8.251765e − 05 ∗ max(0, 33664 −𝑋7) – 0.0001611239 ∗ max(0, 𝑋7 − 33664) – 0.07994066 ∗ max(0, 67.03 𝑋8) − 0,1345248 ∗ max(0, 𝑋8 − 67.03) + 1.335112e − 05 ∗ max(0, 𝑋10 −763837) (5) table 1. comparison mars and bagging mars model significance variables gcv mars 𝑋1, 𝑋6, 𝑋8, 𝑋9, 𝑋10 6.985571 bagging mars 𝑋1, 𝑋4, 𝑋6, 𝑋7, 𝑋8, 𝑋10 0.009798721 table 1 showed that the gcv of the bagging mars model was 0.009798721. then, mars model was 6.985571. gcv in the bagging mars model indicated a better accuracy than the mars model. since, bagging mars model has gcv minimum than mars model. best variable in mars and bagging mars methods the population poverty of central java using mars model affected by the number of diarrhea disease (𝑋1), the number of infant malnutrition (𝑋6), the percentage of labor force participation rate (𝑋8), the percentage of expectation old school (𝑋9), and the number of participants bpjs (𝑋10). table 2 is affected population poverty based on importance variables from mars method. table 2. importance variables mars model variable importance variables (%) 𝑋1 40.9 𝑋6 22.9 𝑋8 31.7 𝑋9 100 𝑋10 50.8 multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 244 moreover, bagging mars affected variable by importance variables that showed in table 3. the variables were the number of diarrhea disease (𝑋1), the percentage of expenditure per capita by non-food commodities (𝑋4), the percentage of family planning and birth control (𝑋7), the percentage of labor force participation rate (𝑋8), the percentage of old school expectancy (𝑋9), and the number of participants bpjs (𝑋10). table 3. importance variables bagging mars model variable importance variables 𝑋1 95.32921 𝑋4 0.000000 𝑋7 60.80385 𝑋8 0.000000 𝑋10 0.000000 mars and bagging mars method have distinction in importance variables. in mars method the best level of importance variable was 100% which is the percentage of old school expectancy (𝑋9) then in bagging mars method was 95.33% which is number of cases of diarrhea disease (𝑋1). conclusions bagging mars methods obtained better accuracy than the mars model. the most influenced variable population of poverty in central java at 2018 using mars method was the percentage of old school expectancy(𝑋9), then the bagging mars method is the variable number of cases of diarrhea disease(𝑋1). references [1] [2] [3] [4] [5] [6] [7] tjiptoherijanto, p. (1997). prospek perekonomian indonesia dalam rangka globalisasi. rineka cipta. badan pusat statistik. (2017). perhitungan dan analisis kemiskinan makro di indonesia. karisma & sri harini. (2019). multivariate adaptive regression spline in ishemic and hemorrhagic. journal aip converence proceedings of symposium on biomathematics, 1–8. kilinc, b. k., malkoc, s., koparal, a. s., & yazici, b. (2017). using multivariate adaptive regression splines to estimate pollution in soil. international journal of advanced and applied sciences. https://doi.org/10.21833/ijaas.2017.02.002 badan pusat statistik. (2019). kemiskinan dan ketimpangan. badan pusat statistik kemiskinan dan ketimpangan. https://www.bps.go.id/subject/23/kemiskinandan-ketimpangan.html kurniawan, m. d. (2017). analisis faktor-faktor penyebab kemiskinan di kabupaten musi banyuasin (studikasus di kecamatan sungai lilin). jurnal ilmiah ekonomi global masa kini. friedman, j. h. (1991). rejoinder: multivariate adaptive regression splines. the annals of statistics. https://doi.org/10.1214/aos/1176347973 rahmaniah, m. nanda dkk, (2016). bootstrap aggregating multivariate adaptive multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines of poverty in central java ria dhea ln karisma 245 [8] [9] [10] [11[ regression spline. jurnal eksponensial, 7(2), 163–170. jurnal eksponensial, 7(2), 163–170. breiman, l. (1996). bagging predictors. machine learning. https://doi.org/10.1007/bf00058655 shofa, b. & i. n. &. (2012). analisis survival dengan pendekatan multivariate adaptive regression spline pada kasus demam berdarah dengue (dbd). jurnal sains dan seni its, 1(1), 318–323. badan pusat statistik. (2019). https://semarangkab.bps.go.id. retrieved from https://semarangkab.bps.go.id/indicator/23/78/1/persentase-penduduk-miskinkabupaten-kota-di-jawa-tengah.html strongly summable vector valued sequence spaces defined by 2 modular cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 279-285 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 23, 2021 reviewed: march 29, 2021 accepted: april 07, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11484 strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho1, p.w. prasetyo2 1,2departement of mathematical education, universitas ahmad dahlan, yogyakarta, indonesia email: burhanudin@pmat.uad.ac.id abstract summability is an important concept in sequence spaces. one summability concept is strongly cesaro summable. in this paper, we study a subset of the set of all vector-valued sequence in 2modular space. some facts that we investigated in this paper include linearity, the existence of modular and completeness with respect to these modular. keywords: strongly; summable; sequence spaces; 2-modular introduction summability is an important concept in sequence spaces. the familiar example of sequence spaces that using the summability concept is ℓ𝑝 spaces. in [1], it is explained that kutner discusses spaces of strongly cesaro summable sequences, and furthermore, maddox generalizes this concept. if 𝜔 denote the set of all infinite sequence of real/complex numbers, then the set 𝑤 = {(𝑥𝑘 ) ∈ 𝜔: ∃𝐿, ∋ lim 𝑛→∞ 1 𝑛 ∑|𝑥𝑘 − 𝐿| = 0 𝑛 𝑘=1 }, denote the space of strongly cesaro summable sequence [2] [3]. let 𝑋 be a real linear space of dimension 𝑑 ≥ 2. a 2-norm on 𝑋 is a function ‖. , . ‖: 𝑋 × 𝑋 → ℝ , where for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, satisfy (i) ‖𝑥, 𝑦‖ = 0 if and only if 𝑥 and 𝑦 are linearly dependent (ii) ‖𝑥, 𝑦‖ = ‖𝑦, 𝑥‖ (iii) ‖𝛼𝑥, 𝑦‖ = |𝛼|‖𝑥, 𝑦‖, 𝛼 ∈ ℝ (iv) ‖𝑥 + 𝑦, 𝑧‖ ≤ ‖𝑥, 𝑧‖ + ‖𝑦, 𝑧‖. the pair (𝑋, ‖. , . ‖) is then called a 2-normed space [4]. the concept is initially introduced by gahler [5] in the middle of 1963. furthermore, in 1989, misiak generalized the 2normed concept to be n-normed [6]. since then, many kinds research on 2-normed (nnormed) spaces, include research on strongly cesaro summable vector-valued sequences or the generalize in 2-normed (n-normed) spaces [7] [8] [9] [10] [11]. in 1950, nakano developed modular function and it was generalized by musielak and orlicz [12] [13]. modular is the generalization of the norm. let 𝑌 be a real linear space, a functional 𝑔: 𝑌 → ℝ∗ is said tobe modular if it satisfies the following conditions: (i) 𝑔(𝑥) = 0 if and if 𝑥 = 0 (ii) 𝑔(−𝑥) = 𝑔(𝑥) (iii) 𝑔(𝛼𝑥 + 𝛽𝑦) ≤ 𝑔(𝑥) + 𝑔(𝑦), every 𝑥, 𝑦 ∈ 𝑌, 𝛼, 𝛽 ≥ 0, 𝛼 + 𝛽 = 1. http://dx.doi.org/10.18860/ca.v6i4.11484 mailto:burhanudin@pmat.uad.ac.id strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 280 the pair (𝑌, 𝑔) is then called a modular space. following the 2-norm (n-norm) concept, k. nourouzi and s. shabanian in 2009 initially introduced the n-modular concept [14] [15]. let 𝑋 be a real linear space of dimension 𝑑 ≥ 2. a 2-modular on 𝑋 is a function 𝜌(. , . ): 𝑋 × 𝑋 → ℝ∗ where for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, satisfy (i) 𝜌(𝑥, 𝑦) = 0 if and only if 𝑥 and 𝑦 are linearly dependent (ii) 𝜌(𝑥, 𝑦) = 𝜌(𝑦, 𝑥) (iii) 𝜌(−𝑥, 𝑦) = 𝜌(𝑥, 𝑦), (iv) 𝜌(𝛼𝑥 + 𝛽𝑦, 𝑧) ≤ 𝜌(𝑥, 𝑧) + 𝜌(𝑦, 𝑧), every 𝛼, 𝛽 ≥ 0, 𝛼 + 𝛽 = 1. the pair (𝑋, ‖. , . ‖) is then called a 2-modular space. the 2-modular space, with 𝜌 satisfies δ2-condition, if there exist 𝐿 > 0, such that 𝜌(2𝑥, 𝑦) ≤ 𝐿𝜌(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝑋. a sequence (𝑥𝑘 ) in 𝑋 is said to be 2-modular convergent to 𝑥0 ∈ 𝑋 if lim 𝑘→∞ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) = 0, ∀𝑦 ∈ 𝑋. it means that for every 𝜖 > 0, there exists an 𝑘0 ∈ ℕ, such that for any 𝑘 ∈ ℕ, 𝑘 ≥ 𝑘0, we have 𝜌(𝑥𝑘 − 𝑥0, 𝑦) < 𝜖, ∀𝑦 ∈ 𝑋. furthermore, a sequence (𝑥𝑘 ) in 𝑋 is called 2-modular cauchy sequence if, for all 𝑦 ∈ 𝑋, we have lim 𝑘,𝑙→∞ 𝜌(𝑥𝑘 − 𝑥𝑙 , 𝑦) = 0. the standard example of a 2-modular space is 𝑋 = ℝ2, with 2-modular on ℝ2 define by 𝜌(�̅�, �̅�) = √|det ( 𝑥1 𝑥2 𝑦1 𝑦2 )|, where �̅� = (𝑥1, 𝑥2), �̅� = (𝑦1, 𝑦2) ∈ ℝ 2. clearly that 𝜌 satisfies δ2-condition and the sequence (( 1 𝑛 , 0)) in ℝ2 is 2-modular convergent to (0,0) ∈ ℝ2. this paper will be constructed t spaces of strongly cesaro summable vector-valued sequences in 2-modular spaces based on the facts presented above. methods let (𝑋, 𝜌) be a 2-modular space, with 𝜌 satisfies δ2-condition and the dimension of 𝑋 greater than one. we define 𝑋𝜌 = {𝑥 ∈ 𝑋: 𝜌(𝑥, 𝑦) < ∞, ∀𝑦 ∈ 𝑋}. because 𝜌 satisfies δ2-condition, then there exists 𝐾 > 0, such that for all 𝑥, 𝑦 ∈ 𝑋𝜌, 𝑧 ∈ 𝑋 and 𝛼 ∈ ℝ, we have 𝜌(𝑥 + 𝑦, 𝑧) = 𝜌 ( 2𝑥 + 2𝑦 2 , 𝑧) ≤ 𝜌(2𝑥, 𝑧) + 𝜌(2𝑦, 𝑧) ≤ 𝐾𝜌(𝑥, 𝑧) + 𝐾𝜌(𝑦, 𝑧) < ∞ based on archimedean property, there exists 𝑛0 ∈ ℕ, such that 𝛼 ≤ 2 𝑛0 𝜌(𝛼𝑥, 𝑧) ≤ 𝜌(2𝑛0 𝑥, 𝑧) ≤ 𝐾𝑛0 𝜌(𝑥, 𝑧) < ∞. hence, we have that 𝑋𝜌 is a subspace linear of 𝑋. furthermore (𝑋𝜌, 𝜌) is a 2-modular space too. the notation 𝜔(𝑋𝜌) will donate as the set of all sequences in 𝑋𝜌 strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 281 𝜔(𝑋𝜌) = {(𝑥𝑘 ): 𝑥𝑘 ∈ 𝑋, 𝑘 ∈ ℕ} (1) where linear space operations are defined coordinatewise, (𝑥𝑘 ) + (𝑦𝑘 ) = (𝑥𝑘 + 𝑦𝑘 ), 𝛼(𝑥𝑘 ) = (𝛼𝑥𝑘 ) for all (𝑥𝑘 ), (𝑦𝑘 ) ∈ 𝜔(𝑋𝜌) and 𝛼 ∈ ℝ. the goal of this paper is that we want to extend the concept of strongly cesaro summable to 2-modular spaces valued sequences, defined as 𝑤0 𝜌 (𝑋𝜌) = {(𝑥𝑘 ) ∈ 𝜔(𝑋𝜌): 𝑙𝑖𝑚 𝑛→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) 𝑛 𝑘=1 = 0, ∀𝑦 ∈ 𝑋𝜌 } (2) 𝑤 𝜌(𝑋𝜌 ) = {(𝑥𝑘 ) ∈ 𝜔(𝑋𝜌): ∃𝑥0 ∈ 𝑋𝜌, 𝑙𝑖𝑚 𝑛→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛 𝑘=1 = 0, ∀𝑦 ∈ 𝑋𝜌 } (3) furthermore, we also studied the properties of 𝑤0 𝜌 (𝑋𝜌) and 𝑤 𝜌(𝑋𝜌). results and discussion henceforth, if not specified then 𝑋 is a 2-modular space with 2-modular 𝜌, that satisfies the δ2-conditions. first, we will prove that the mean cesaro theorem applies to 2-modular space. theorem 1. let sequence (𝑥𝑘 ) in 𝑋𝜌 2-modular convergent to 𝑥0 ∈ 𝑋𝜌, then lim 𝑛→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛 𝑘=1 = 0, ∀𝑦 ∈ 𝑋𝜌 proof. since the sequence (𝑥𝑘 ) in 𝑋𝜌 2-modular convergent to 𝑥0 ∈ 𝑋𝜌, then for all 𝜖 > 0, there exists 𝑛𝜖 ∈ ℕ, such that for all 𝑘 ≥ 𝑛𝜖 , we have 𝜌(𝑥𝑘 − 𝑥0, 𝑦) < 𝜖 2 , for all 𝑦 ∈ 𝑋. note that, for all 𝑛 ≥ 𝑛𝜖 , we have 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) = 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛𝜖 𝑘=1 + 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛 𝑘=𝑛𝜖+1 𝑛 𝑘=1 ≤ 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛𝜖 𝑘=1 + 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛 𝑚𝑎𝑥 𝑛 𝑘=𝑛𝜖+1 = 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛 ∑ 1 𝑛𝜖 𝑘=1 + 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛 𝑚𝑎𝑥 𝑛 ∑ 1 𝑛 𝑘=𝑛𝜖+1 = 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛𝜖 𝑛 + 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛 𝑚𝑎𝑥 𝑛 − 𝑛𝜖 𝑛 = 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛𝜖 𝑛 + 𝜌(𝑥𝑘 − 𝑥0, 𝑦)𝑛𝜖+1≤𝑘≤𝑛 𝑚𝑎𝑥 = 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛𝜖 𝑛 + 𝜖 2 . strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 282 by archimedean property, there exists 𝑛′ ≥ 𝑛𝜖 , such that for all 𝑛 ≥ 𝑛′, we have 𝜌(𝑥𝑘 − 𝑥0, 𝑦)1≤𝑘≤𝑛𝜖 𝑚𝑎𝑥 𝑛𝜖 𝑛 < 𝜖 2 . hence, for all 𝑛 ≥ 𝑛′, we have 1 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛 𝑘=1 < 𝜖. in other words, the proof is complete. ∎ based on theorem 1, we can say that for all 2-modular convergent sequence (𝑥𝑘 ) in 𝑋𝜌 is an element of 𝑤 𝜌(𝑋𝜌). theorem 2. the set 𝑤 𝜌(𝑋𝜌) is a linear subspace of 𝜔(𝑋𝜌). proof. note that for all (𝑥𝑘 ), (𝑦𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌) and 𝛼 ∈ ℝ, there exsist 𝑥0, 𝑦0 ∈ 𝑋𝜌 so that for all 𝑦 ∈ 𝑋𝜌, we have lim 𝑛→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 − x0, 𝑦) 𝑛 𝑘=1 = 0, and lim 𝑛→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 − y0, 𝑦) 𝑛 𝑘=1 = 0. therefore, 𝜌 satisfy δ2-condition, then there exists 𝐿 > 0 and 𝑛0 ∈ ℕ so that 0 ≤ 𝜌((𝑥𝑘 + 𝑦𝑘 ) − (𝑥0 + 𝑦0), 𝑦) = 𝜌((𝑥𝑘 − 𝑥0) + (𝑦𝑘 − 𝑦0), 𝑦) ≤ 𝜌(2(𝑥𝑘 − 𝑥0), 𝑦) + 𝜌(2(𝑦𝑘 − 𝑦0), 𝑦) ≤ 𝐿𝜌((𝑥𝑘 − 𝑥0), 𝑦) + 𝐿𝜌((𝑦𝑘 − 𝑦0), 𝑦) and 0 ≤ 𝜌(𝛼𝑥𝑘 − 𝛼𝑙, 𝑦) = 𝜌(𝛼(𝑥𝑘 − 𝑙), 𝑦) ≤ 𝜌(2𝑛0 (𝑥𝑘 − 𝑥0), 𝑦) ≤ 𝐿𝑛0 𝜌(𝑥𝑘 − 𝑥0, 𝑦). hence, we have lim 𝑛→∞ 1 𝑛 ∑ 𝜌((𝑥𝑘 + 𝑦𝑘 ) − (𝑥0 + 𝑦0), 𝑦) 𝑛 𝑘=1 = 0 and lim 𝑛→∞ 1 𝑛 ∑ 𝜌(𝛼𝑥𝑘 − 𝛼𝑥0, 𝑦) 𝑛 𝑘=1 = 0. in other words (𝑥𝑘 ) + (𝑦𝑘 ), 𝛼(𝑥𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌), and we proof that 𝑤 𝜌(𝑋𝜌) is a subspace linear of 𝜔(𝑋𝜌).∎ theorem 3. if (𝑥𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌), then for all 𝑦 ∈ 𝑋𝜌, ( 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) 𝑛 𝑘=1 ) is a bounded sequence of real numbers. proof. if (𝑥𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌), then there exist 𝑥0 ∈ 𝑋𝜌, such that for all 𝑦 ∈ 𝑋𝜌, we have lim n→∞ 1 n ∑ ρ(xk − 𝑥0, y) n k=1 = 0. hence, there exist 𝑛0 ∈ ℕ, such that for all 𝑛 ∈ ℕ, with 𝑛 ≥ 𝑛0 we have strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 283 1 n ∑ ρ(xk − 𝑥0, y) n k=1 ≤ 1. since 𝜌 satisfies the δ2-conditions, there exist 𝐿 > 0, for all 𝑦 ∈ 𝑋𝜌, we have 𝜌(𝑥𝑘 , 𝑦) = 𝜌 ( 2(𝑥𝑘 − 𝑥0) 2 + 2𝑥0 2 , 𝑦) ≤ 𝐿𝜌(𝑥𝑘 − 𝑥0, 𝑦) + 𝐿𝜌(𝑥0, 𝑦). it implies, 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) 𝑛 𝑘=1 ≤ 𝐿 𝑛 ∑ 𝜌(𝑥𝑘 − 𝑥0, 𝑦) 𝑛 𝑘=1 + 𝐿𝜌(𝑥0, 𝑦). if we set 𝑀 = sup {𝜌(𝑥1 − 𝑥0, 𝑦), 1 2 ∑ 𝜌(xk − 𝑥0, y), ⋯ , 1 n0 − 1 ∑ ρ(x1 − 𝑥0, y) n0−1 k=1 , 1 2 k=1 } then it follows that we have 𝐾 = 𝐿(𝑀 + 𝜌(𝑥0, 𝑦)), such that 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) ≤ 𝐾, 𝑛 𝑘=1 for all 𝑛 ∈ ℕ. this implies that for all 𝑦 ∈ 𝑋𝜌, ( 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) 𝑛 𝑘=1 ) is a bounded sequence. ∎ theorem 4. function 𝑔((𝑥𝑘 )) = sup { 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑧) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} (5) is a modular on 𝑤 𝜌(𝑋𝜌). proof. if (𝑥𝑘 ) = 𝟎 is the zero sequence. then it is clear that 𝑔((𝑥𝑘 )) = 0. conversely, if ((𝑥𝑘 )) = 0, then we have sup { 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑧) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} = 0. hence, it implies for all 𝑛 ∈ ℕ and 𝑧 ∈ 𝑋𝜌, we have 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦𝑘 ) 𝑛 𝑘=1 = 0 ⇔ 𝜌(𝑥𝑘 , 𝑧) = 0 ⇔ 𝑥𝑘 = 0, ∀𝑘 ∈ ℕ. thus, it is evident that (𝑥𝑘 ) = 𝟎. since 𝜌(−𝑥, 𝑦) = 𝜌(𝑥, 𝑦) applies, for all 𝑥, 𝑦 ∈ 𝑋𝜌, consequently, it is clear that 𝑔(−(𝑥𝑘 )) = 𝑔((𝑥𝑘 )). finally, for all 𝛼, 𝛽 ≥ 0 with 𝛼 + 𝛽 = 1, the for all (𝑥𝑘 ), (𝑦𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌) we have, 𝑔(𝛼(𝑥𝑘 ) + 𝛽(𝑦𝑘 )) = sup { 1 𝑛 ∑ 𝜌(𝛼𝑥𝑘 + 𝛽 𝑦𝑘 , 𝑧) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} = sup { 1 𝑛 ∑( 𝜌(𝑥𝑘 , z) + 𝜌(𝑦𝑘 , 𝑧)) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 284 ≤ sup { 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑧) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} + sup { 1 𝑛 ∑ 𝜌(𝑦𝑘 , 𝑧 ) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} = 𝑔((𝑥𝑘 )) + 𝑔((𝑦𝑘 )). this completes the proof. ∎ theorem 5. if 𝑋𝜌 2-modular complete, then (𝑤 𝜌(𝑋𝜌), 𝑔) is a modular complete. proof. let 𝑛 ∈ ℕ and (𝑥𝑖 ) be a 2-modular cauchy sequence in 𝑤 𝜌(𝑋𝜌), where 𝑥 𝑖 = (𝑥𝑘 𝑖 ), for all 𝑖 ∈ ℕ. hence, for all 𝜖 > 0, there exists 𝑛0 ∈ ℕ, such that for all 𝑖, 𝑗 ∈ ℕ, with 𝑖, 𝑗 ≥ 𝑛0, we have 𝑔(𝑥𝑖 − 𝑥 𝑗 ) = sup { 1 𝑛 ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 𝑗 , 𝑧) 𝑛 𝑘=1 , ∀𝑧 ∈ 𝑋𝜌} < 𝜖. it implies that, for all 𝑖, 𝑗 ≥ 𝑛0, we have 1 𝑛 ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 𝑗 , 𝑧) 𝑛 𝑘=1 < 𝜖, ∀𝑧 ∈ 𝑋𝜌, or ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 𝑗 , 𝑧) 𝑛 𝑘=1 < 𝑛𝜖, ∀𝑧 ∈ 𝑋𝜌, such that, 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 𝑗 , 𝑧) < 𝑛𝜖, ∀𝑧 ∈ 𝑋𝜌. hence, for all 𝑘 ∈ ℕ, (𝑥𝑘 𝑖 ) is a 𝜌-cauchy sequence in 𝑋𝜌. since 𝑋𝜌 complete 2-modular, then (𝑥𝑘 𝑖 ) is 2-modular convergent in 𝑋𝜌, for all 𝑘 ∈ ℕ. therefore, for 𝑘 ∈ ℕ, there exist 𝑥𝑘 ∈ 𝑋𝜌 , such that for all 𝑧 ∈ 𝑋𝜌, we have lim 𝑖→∞ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 , 𝑧) = 0. since, for all 𝑖, 𝑗 ≥ 𝑛0, we have 1 𝑛 ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 , 𝑧) 𝑛 𝑘=1 = lim 𝑗→∞ 1 𝑛 ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 𝑗 , 𝑧) 𝑛 𝑘=1 < 𝜖, ∀𝑧 ∈ 𝑋𝜌, then 𝑔 ((𝑥𝑘 𝑖 ) − (𝑥𝑘 )) = sup ( 1 𝑛 ∑ 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 , 𝑧) 𝑛 𝑘=1 ) < 𝜖, for all 𝑖 ≥ 𝑛0, such that 𝜌(𝑥𝑘 𝑖 − 𝑥𝑘 , 𝑧) < 𝑛𝜖, for all 𝑖 ≥ 𝑛0 therefore (𝑥𝑖 ) modular convergent to (𝑥𝑘 ), and (𝑥𝑘 𝑖 − 𝑥𝑘 ) ∈ 𝑤(𝑋𝜌). since (𝑥𝑘 𝑖 ) ∈ 𝑤(𝑋𝜌) and 𝑤(𝑋𝜌) is a linear spaces, so we have (𝑥𝑘 ) = (𝑥𝑘 𝑖 ) − (𝑥𝑘 𝑖 − 𝑥𝑘 ) ∈ 𝑤(𝑋𝜌). this complete the proof that (𝑤 𝜌(𝑋𝜌), 𝑔) is a complete modular (𝜌-complete). ∎ conclusions if (𝑋, 𝜌) is a 2-modular space, with 𝜌 satisfies δ2-condition, then we can construct 𝑤 𝜌(𝑋𝜌) ⊂ 𝑤(𝑋𝜌) is the space of strongly cesaro summable vector-valued sequences in 2-modular (𝑋𝜌, 𝜌). it certainly can be shown that 𝑤 𝜌(𝑋𝜌) is a linear space. furthermore, if (𝑥𝑘 ) ∈ 𝑤 𝜌(𝑋𝜌), then we can prove that for all 𝑦 ∈ 𝑋𝜌, ( 1 𝑛 ∑ 𝜌(𝑥𝑘 , 𝑦) 𝑛 𝑘=1 ) is a bounded strongly summable vector valued sequence spaces defined by 2 modular b. a. nurnugroho 285 sequence of real numbers. this fact provides a guarantee for us to be able to build a modular 𝑔 on 𝑤 𝜌(𝑋𝜌). finally, we proved that (𝑤 𝜌(𝑋𝜌), 𝑔) is modular complete, if (𝑋𝜌, 𝜌) is a 2-modular complete. acknowledgments the author would like to thank lppm uad for funding this research references [1] t. bilgin, "on strong a-summability defined by a modulus," chinese journal of mathematics, vol. 24, no. 2, pp. 159-166, 1996. 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[15] b. a. nurnugroho, s. and a. zulijanto, "2-linear operator on 2-modular spaces," far east journal of mathematical sciences , vol. 102, no. 12, pp. 3193-3210, 2017. supplier selection analysis using minmax multi choice goal programming model cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 97-104 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 16, 2021 reviewed: september 02, 2021 accepted: november 01, 2021 doi: https://doi.org/10.18860/ca.v7i1.12944 supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi1, eka susanti2, bambang suprihatin3, endro setyo cahyono4, anggun permata5, nurul fadhila yanita6 1,2,3,4,5,6 department of mathematics, universitas sriwijaya email: novirustiana@unsri.ac.id, eka_susanti@mipa.unsri.ac.id, endrosetyo_c@yahoo.co.id, bambangs@unsri.ac.id abstract production control, inventory and distribution is an important factor in trading activities. these three factors are discussed in a system called supply chain management (scm). procurement of goods from a company or trading business related to suppliers. in some cases, there are several suppliers that can be assessed by considering certain factors. in certain cases, the data from several factors that are considered are uncertainty, so the fuzzy approach can be used. the minmax multi choice goal programming model can be used to solve fuzzy supplier selection problems with linear membership function. it can be applied to selecting supplier of brastagi oranges. there are four suppliers, namely jaya, mako, baros. gina. there are three factor to consider, cost, quality and delivery. the decision maker selects the best supplier for ordering 17000 kg brastagi oranges. the results, the best supplier is gina with an order quantity of 10000 kg and mako with a total order of 7000 kg. keywords: fuzzy; minmax multi choice goal programming; supply chain management; supplier selection introduction supply chain management has three main components, namely the process of obtaining suppliers of raw materials, the process of changing raw materials into finished products and the product distribution process. the first stage in the supply chain is supplier selection. selection of suppliers aims to get products with good quality and competitive prices. supplier selection is related to the process of procuring goods to meet customer demands. price and quality, time of delivery is a consideration in supplier’s selection, especially for perishable products. fruit is a type of product that does not last long if not stored in the refrigerator. research related to supply chains with application in various fields and solutions have been carried out with several approaches. the application of fuzzy topsis in supplier selection was introduced by [1]. the fuzzy approach is also used by [2] in the selection of suppliers in manufacturing companies. the application of the supply chain concept to inventory control and supplier selection for planning new product production in several planning horizons was carried out by [3]. discussion of supply chain problems https://doi.org/10.18860/ca.v7i1.12944 mailto:novirustiana@unsri.ac.id mailto:eka_susanti@mipa.unsri.ac.id mailto:endrosetyo_c@yahoo.co.id mailto:bambangs@unsri.ac.id supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 98 by considering price, supply and demand factors is carried out by [4] and an efficient lagrangian relaxation algorithm is proposed to solve the model. a discussion of bioethanol supply chain network problems with a robust approach was introduced [5]. a deterministic approach to solving the supply chain problem of food product distribution is discussed by [6]. the application of the mix integer programming model to the distribution and supply chain problems of liquid helium is given by [7]. the research of the [8] is combines the concepts of siting, inventory and routing in the supply chain. there are two main studies related to the supplier selection model to be used, namely the concept of fuzzy and fuzzy goal programming. the goal programming (gp) model is used in problems with several objectives to be achieved simultaneously. the gp model with fuzzy numbers is called the fuzzy goal programming (fgp) model. the concept of fgp with random variables was introduced by [9]. fuzzy and probabilistic approaches to the fgp model are discussed by [10]. completion of the fgp model with a genetic algorithm is discussed by [11]. research [12] uses a multi-choice goal programming model to determine energy renewal facilities. [13] used the fgp model in production planning. the choice of waste transportation mode using the fgp model was introduced by [14]. the application of the weighted goal programming model in the urban planning process is given by [15]. the application of the gp model in capital management is given by [16]. the use of the fgp model in transportation problems with several modes of transportation is given by [17]. the research that has been mentioned is the implementation of the supply chain concept to supplier, inventory and distribution components. this research will discuss the problem of selecting suppliers of brastagi oranges using minmax multi choice goal programming models (minmax mcgp). the research focus is on component suppliers. this research is a basic research by developing the minmax multi choice goal programming introduced by [2]. in [2], the fuzzy number used is the trapezoid fuzzy number by considering the factors of price, quality and technology offered. in this study, price, quality and time of delivery are considering. linear membership function is used to define these tree factor. methods the steps for completing the supplier selection using the minmax mcgp method are: 1. data collection and description the data used in this study is primary, consist of data on the purchase with the parameters of cost, quality and delivery. the data collection period is from 18 february to 18 march 2020. 2. determine the fuzzy triangular membership value for the goal of price, quality and delivery. following are given fuzzy membership functions for the respective three goals, in order of price, quality and timeliness of delivery which are formulated based on the data in step 1. the restriction value of variable 𝑐, 𝑘, 𝑑 is determined based on the data in step 1. 𝜇(𝑐) = { 1, 𝑐 ≤ 7800 1 − [ (𝐶−𝑆𝐿1(𝑐)) 𝑆𝐿2(𝑐)−𝑆𝐿1(𝑐) ] , 7800 ≤ 𝑐 ≤ 10000 0, 𝑐 ≥ 10000 (1) supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 99 𝜇(𝑘) = { 1, 𝑘 ≥ 100. 𝑘 𝑆𝐿2(𝑘) , 0 < 𝑘 ≤ 100. 0, 𝑘 ≤ 0. (2) 𝜇(𝑑) = { 1, 𝑑 ≥ 100. 𝑑 𝑆𝐿2(𝑑) , 0 < 𝑑 ≤ 100. 0, 𝑑 ≤ 0. (3) where 𝜇(𝑐) is the membership function for the cost. 𝜇(𝑘) is the membership function for the quality. 𝜇(𝑑) is membership function for delivery 𝑘 is the percentage of average supplier quality. 𝑆𝐿1(𝑐) is satisfaction level lower bound for the unit cost. 𝑆𝐿2(𝑐) is satisfaction level upper bound for the unit cost. 𝑆𝐿2(𝑘) is satisfaction level upper bound for the unit quality. 𝑆𝐿2(𝑑)is satisfaction level upper bound for the unit delivery. 3. the minmax mcgp model formulation based on the membership function values defined in step 2. the following is the minmax mcgp model introduced by [2]. min 𝐷 subject to 𝐷 ≥ 𝛼𝑖 𝑑𝑖 + + 𝛽𝑖 𝑑𝑖 −, 𝑖 = 1,2, … 𝑚, 𝐷 ≥ 𝛿𝑖 (𝑒𝑖 + + 𝑒𝑖 −), 𝑖 = 1,2, … 𝑚, (4) 𝜇(𝑥𝑖 ) − 𝑑𝑖 + + 𝑑𝑖 − = 𝑦𝑖 , 𝑖 = 1,2, … 𝑚, 𝑦𝑖 − 𝑒𝑖 + + 𝑒𝑖 − = 𝑔𝑖,𝑚𝑎𝑥 , 𝑖 = 1,2, … , 𝑚, 𝑔𝑖,𝑚𝑖𝑛 ≤ 𝑦𝑖 ≤ 𝑔𝑖,𝑚𝑎𝑥 , 𝑖 = 1,2, … , 𝑚, 𝑑𝑖 +, 𝑑𝑖 −, 𝑒𝑖 +, 𝑒𝑖 − ≥ 0, 𝑖 = 1,2, … , 𝑚, where 𝐷 : the deviation variable of the objective function 𝛼𝑖 and 𝛽𝑖 : weight of the positive deviation penalty in the objective function 𝑑𝑖 + and 𝑑𝑖 − : positive and negative deviation of the objective function 𝛿𝑖 : the sum of the deviation in the objective function 𝑒𝑖 + and 𝑒𝑖 − : positive and negative deviation on |𝑦𝑖 − 𝑔𝑖,𝑚𝑎𝑥 |. 𝑦𝑖 : continuous variable with a range of interval value 𝑔𝑖,𝑚𝑖𝑛 and 𝑔𝑖,𝑚𝑎𝑥 : minimum and maximum 𝑦𝑖 value 𝜇(𝑥𝑖 ) : membership function for the supplier to i 4. completion of the model obtained in step (4) uses lingo 13.0 software 5. analyses and conclusion supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 100 results and discussion this research discusses supplier selection problem of citrus fruits for the type of brastagi oranges. the data used are primary data with a data collection period of 30 ordering periods. the research was conducted at a fruit shop in palembang . the following is given the research data. table 1. ordering the data for each supplier no supplie r name ordering delivery on time deliv ery price offered prece ntage of qualit y (%) date month date month cost (@kg) total 1 jaya 21 feb 21 feb √ 8500 45900000 80 2 mako 21 feb 21 feb √ 8000 43200000 85 3 baros 22 feb 24 feb √ 8500 45900000 80 4 gina 22 feb 22 feb √ 9000 48600000 95 5 jaya 23 feb 23 feb √ 8500 45900000 85 6 mako 24 feb 24 feb √ 8500 45900000 90 7 baros 25 feb 25 feb √ 8000 43200000 80 8 mako 25 feb 25 feb √ 9000 48600000 90 9 gina 26 feb 26 feb √ 9000 48600000 90 10 mako 27 feb 28 feb √ 8000 43200000 85 11 jaya 27 feb 27 feb √ 8500 45900000 85 12 baros 28 feb 28 feb √ 8000 43200000 85 13 mako 29 feb 1 maret √ 9000 48600000 85 14 gina 29 feb 29 feb √ 9500 51300000 90 15 jaya 1 maret 2 maret √ 8500 45900000 85 16 mako 1 maret 1 maret √ 9000 48600000 95 17 baros 2 maret 2 maret √ 9000 48600000 80 18 mako 3 maret 3 maret √ 9000 48600000 85 19 gina 4 maret 4 maret √ 9500 51300000 85 20 jaya 5 maret 6 maret √ 9000 48600000 85 21 baros 5 maret 5 maret √ 9000 48600000 80 22 mako 6 maret 6 maret √ 9000 48600000 90 23 gina 7 maret 7 maret √ 9500 51300000 95 24 jaya 8 maret 10 maret √ 9000 48600000 80 25 baros 8 maret 8 maret √ 9000 48600000 85 26 gina 9 maret 9 maret √ 9500 51300000 95 27 mako 9 maret 9 maret √ 9000 48600000 90 28 jaya 10 maret 12 maret √ 8500 45900000 85 29 baros 10 maret 10 maret √ 9000 48600000 80 30 gina 11 maret 11 maret √ 9500 51300000 85 31 jaya 12 maret 13 maret √ 9000 48600000 80 32 mako 13 maret 13 maret √ 9000 48600000 85 33 jaya 13 maret 14 maret √ 9000 48600000 90 supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 101 34 gina 14 maret 14 maret √ 9500 51300000 90 35 baros 15 maret 16 maret √ 8500 45900000 85 36 mako 15 maret 15 maret √ 9000 48600000 90 37 gina 16 maret 16 maret √ 9500 51300000 90 38 jaya 16 maret 18 maret √ 8500 45900000 85 39 mako 17 maret 17 maret √ 9000 48600000 90 40 gina 19 maret 18 maret √ 9000 48600000 95 41 baros 19 maret 19 maret √ 8500 45900000 90 42 jaya 19 maret 20 maret √ 8500 45900000 80 43 mako 19 maret 21 maret √ 8500 45900000 80 44 gina 20 maret 20 maret √ 9000 48600000 90 45 jaya 20 maret 22 maret √ 8000 43200000 80 46 baros 21 maret 21 maret √ 8500 45900000 90 47 mako 21 maret 22 maret √ 8500 45900000 85 (source : pd wibowo, 21 februari until maret 2020) table 1 can determine the percentage of on-time delivery, the variable price offered, and the varying percentage of quality citrus in good condition with the total of all oranges sent by the supplier. the price value of each supplier is obtained by adding up each price in purchases divided by the number of investments, determined the average value for each data cost, quality, and timeliness. the calculation results are given in table 2 below. table 2. value percentage criteria from four suppliers supplier 𝒙𝒊 cost (rp) quality (%) delivery (%) total order (kg) jaya 𝒙𝟏 8625 83,33 25,00 64800 mako 𝒙𝟐 8750 87,50 71,43 75600 baros 𝒙𝟑 8600 83,50 80,00 54000 gina 𝒙𝟒 9318 90,91 90,91 59400 determined the degree of membership for the level of satisfaction of the decision maker (dm) of each goal using (1), (2), (3). the calculation results are given in table 3 below: table 3. degree of membership for dm satisfaction level of each goal decision lowest highest 𝒄: cost > 10000 8465.4 8243.6 8021.8 7800 sl(c), satisfaction level c 0 0,7 0,8 0,9 1 𝒌 : kualitas 0 40 60 80 100 sl(k), satisfaction level k 0 0,4 0,6 0,8 1 𝒅 : ketepatan waktu 0 40 60 80 100 sl(d), satisfaction level d 0 0,4 0,6 0,8 1 the value of the level of satisfaction is in the interval [0,1]. based on table 3, it is known that for the lowest decision value, dm gives a satisfaction level value of 0. for the highest decision value, dm gives a satisfaction level value 1. the level of satisfaction for each goal of cost, quality, and time delivery is determined based on equations (1), (2), and (3). the results are given in table 4 below. supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 102 table 4. membership function value for each goal supplier amount of order cost quality delivery jaya 𝒙𝟏 0,625 0,83 0,25 mako 𝒙𝟐 0,568 0,88 0,71 baros 𝒙𝟑 0,636 0,84 0,8 gina 𝒙𝟒 0,31 0,91 0,91 average 0,53 0,865 0,6675 maximum value 0,636 0,91 0,91 the lower bound for the price goal is determined based on the average price value multiplied by the minimum order. the upper price is the product of the maximum value of the price times the maximum order. the same calculation is done for quality goals and on time delivery. we obtained a lower bound and an upper bound for the goal value of price, quality and on time delivery respectively 28876,5; 48081,6; 49013,2; 70308; 36045; 68796. the formulation of the minmax mcgp model (4) the problem of supplier’s selection of brastagi oranges with a maximum order quantity for each supplier of 10000 kg, minimum order of 15000 kg and a maximum of 17000 kg is given as follows. minimum d subject to 𝐷 ≥ 3𝑑1 + + 𝑑1 − 𝐷 ≥ 𝑒1 + + 𝑒1 − ; 𝐷 ≥ 𝑑2 + + 5𝑑2 − 𝐷 ≥ 𝑒2 + + 𝑒2 −; 𝐷 ≥ 𝑑3 + + 3𝑑3 − ; 𝐷 ≥ 𝑒3 + + 𝑒3 − (5) 0,625𝑥1 + 0,568𝑥2 + 0,636𝑥3 + 0,31𝑥4 − 𝑑1 + + 𝑑1 − = 𝑦1 𝑦1 − 𝑒1 + + 𝑒1 − = 48081,6 28876,5 ≤ 𝑦1 ≤ 48081,6 0,83𝑥1 + 0,88𝑥2 + 0,84𝑥3 + 0,91𝑥4 − 𝑑2 + + 𝑑2 − = 𝑦2 𝑦2 − 𝑒2 + + 𝑒2 − = 70308 49013,2 ≤ 𝑦2 ≤ 70308 0,25𝑥1 + 0,71𝑥2 + 0,8𝑥3 + 0,91𝑥4 − 𝑑3 + + 𝑑3 − = 𝑦3 𝑦3 − 𝑒3 + + 𝑒3 − = 68796 36045 ≤ 𝑦3 ≤ 68796 𝑥1 ≤ 10000; 𝑥2 ≤ 10000 ; 𝑥3 ≤ 10000 ; 𝑥4 ≤ 10000 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≥ 15000 ; 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 ≤ 17000 𝑑1 +, 𝑑1 −, 𝑒1 +, 𝑒1 −, 𝑦1, 𝑦2, 𝑦3 ≥ 0 solving the linear model (5) uses lingo 13 software and the solution is obtained in table 5 below. supplier selection analysis using minmax multi choice goal programming model novi rustiana dewi 103 table 5. minmax mcgp model solution for citrus fruit supplier selection no variable value 1. 𝑥1 0 2. 𝑥2 7000 3. 𝑥3 0 4. 𝑥4 10000 5. 𝑦1 39463.88 6. 𝑦2 28876.50 7. 𝑦3 36764.17 8. 𝐷1 + 0 9. 𝐷1 − 32387.88 10. 𝑒1 + 0 11. 𝑒1 − 8617.725 12. 𝐷2 + 0 13. 𝐷2 − 13616.5 14. 𝑒2 + 0 15. 𝑒2 − 39919.50 16. 𝐷3 + 0 17. 𝐷3 − 22694.17 18. 𝑒3 + 0 19. 𝑒3 − 32031.83 20. 𝐷 68082.50 in table 5, for a maximum total order of 17000 kg, an order is recommended for 𝑥2 (supplier mako) and 𝑥4 (supplier gina). the values of 𝑦1 (aspiration rate g1) = 39463.88, 𝑦2 (aspiration rate g2) = 28876.50, 𝑦3 (aspiration rate g3) = 36764.17, and other deviations are given in table 5. the values of 𝑥1, 𝑥2, 𝑥3, and 𝑥4 are 0, 7000, 0, 10000, respectively. it can be concluded that the order for selecting the best supplier is supplier gina with an order quantity of 10000 kg, supplier mako with an order quantity of 7000 kg. conclusions the results obtained the best supplier for orders of a maximum of 17000 kg are gina supplier with a total order of 1000 kg of brastagi oranges and mako supplier with a maximum order of 7000 kg. the best supplier order is obtained by looking at the difference in the value of the deviation from the target for each goal of price, quality and delivery. the difference in goal value results in a different order of supplier selection. acknowledgments this research is supported by universitas sriwijaya through sains teknologi dan seni (sateks) research scheme with the number of the research assignment contract number 0163.177/un9/sb3.lppm.pt/2020. references [1] r. kiani, m. goh, and n. kiani, “supplier selection with shannon entropy and fuzzy topsis in the context of supply chain risk management,” procedia soc. behav. sci., vol. 235, no. october, pp. 216–225, 2016. 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[17] l. chen, j. peng, and b. zhang, “uncertain goal programming models for bicriteria solid transportation problem,” appl. soft comput. j., 2016. super total labeling (a,d)-edge antimagic on the firecracker graph cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 133-139 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 18, 2020 reviewed: october 01, 2020 accepted: november 10, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.10145 super total labeling (a,d)-edge antimagic on the firecracker graph juhari mathematics department, universitas islam negeri maulana malik ibrahim malang email: juhari@uin-malang.ac.id abstract an an (a, d)-edge antimagic total labeling on (p, q)-graph g is a one-to-one map f from v (g) ∪ e(g) onto the integers 1, 2, . . ., p + q with the property that the edge-weights, w(uv) = f (u) + f(v) + f(uv) where uv ∈ e(g), form an arithmetic progression starting from a and having common difference d. such labeling is called super if the smallest possible labels appear on the vertices. in this paper, we investigate the existence of super (a, d)-edge antimagic total labeling of firecracker graph. keywords: super (a, d)-edge-antimagic total labeling; firecracker graph (fn, k) introduction the first label appears in the middle of the year 1960 it started by a ringel and rosa hypotheses [1]. in 1967 rosa called this label as liberation β-valuation from a graph with e side, if there is the function that mapping one to one from the set of points 𝑉(𝐺) to set of integers 0,1,2, … . , 𝑒, so every side xy in graph g gets a different label |𝑓(𝑥) − 𝑓(𝑦)| for every edges in graph g. one type of graf liberation is super total labeling (𝑎, 𝑑)-antimagic edge (seatl), where a smallest side dan d different value. this liberation introduced by simanjutak, bertault and miller in the year 2000 [1], [2], [3]. the entire release (𝑎, 𝑑)-edge antimagic is total labeling in some kinds of graf g that started by labeling all of the graphs first with consecutive original numbers, then proceed with buying all sides of the map such that the side weights form an arithmetic sequence with the first term and different d [4], [5]. the type of star graph, the super total labeling (𝑎, 𝑑)-edge antimagic (seatl) not yet found one of which is a graph firecracker that hasn't been labelled previously. this prompted the writer to examine how super total labeling (𝑎 𝑑)-edge antimagic (seatl) on the firecracker graph (fn, k). some of the problems formulated are as follows: (1) the upper limit d, so the firecracker graph has super total labeling (a, d)-edge antimagic? and (2) how bijective function of super total labeling (𝑎, 𝑑)-edge antimagic the firecracker graph? in order not to be widespread, this research needs to be done, and this research needs to be done on total labeling (𝑎, 𝑑)-edge antimagic the firecracker graph (fn, k) with n ≥ 2; k ≥ 3. in this session, n and k are a provision of the definition firecracker graph. super total labeling (𝒂, 𝒅)-edge antimagic. a graph is said to have total labeling (𝑎, 𝑑)-edge antimagic if there is a one-to-one mapping of one 𝑉(𝐺) ∪ 𝐸(𝐺) to integers. 1,2,3, … , 𝑝 + 𝑞 so the set of side weight 𝑤(𝑢𝑣) = 𝑓(𝑢) + http://dx.doi.org/10.18860/ca.v6i3.10145 mailto:juhari@uin-malang.ac.id super total labeling (a,d)-edge antimagic on the firecracker graph juhari 134 𝑓(𝑣) + 𝑓(𝑢𝑣) on all edge g is 𝑎, 𝑎 + 𝑑, … , 𝑎 + (𝑞 − 1)𝑑 for 𝑎 > 0 and 𝑑 >0 both integers [6]. the total labeling (𝑎, 𝑑)-edge antimagic called super total labeling (𝑎, 𝑑)edge antimagic if 𝑓(𝑉) = {1,2,3, . . , 𝑝} and 𝑓(𝐸) = {𝑝 + 1, 𝑝 + 2, 𝑝 + 3, … , 𝑝 + 𝑞}. to search upper limit different value d super total slowly (𝑎, 𝑑)-edge antimagic can certain by lemma [1], [7]: lemma 1 if a graph (p, q) is super total labeling (a,d)-edge antimagic so 𝑑 ≥ 2𝑝+𝑞−5 𝑞−1 prove. 𝑓(𝑉) = {1,2,3, . . , 𝑝} and 𝑓(𝐸) = {𝑝 + 1, 𝑝 + 2, 𝑝 + 3, … , 𝑝 + 𝑞} for example, graph (𝑝, 𝑞) is super total labeling (𝑎, 𝑑)-edge antimagic by mapping 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → {1,2,3, … , 𝑝 + 𝑞}. the minimum value that possible from the smallest weight side 𝛼(𝑢) + 𝛼(𝑢𝑣) + 𝛼(𝑣) = 1 + (𝑝 + 1) + 2 = 𝑝 + 4 and can be written: 𝑝 + 4 ≤ 𝛼. while on the other side, t h e maximum value that possible from the biggest weight side gained by the sum of 2 smallest labels and biggest label or can be written (𝑝 − 1) + (𝑝 + 𝑞) + 𝑝 = 3𝑝 + 𝑞 − 1. result: 𝑎 + (𝑞 − 1) 𝑑 ≤ 3𝑝 + 𝑞 − 1 𝑑 ≤ 3𝑝 + 𝑞 − 1 − (𝑝 + 4) 𝑞 − 1 𝑑 ≤ 2𝑝 + 𝑞 − 5 𝑞 − 1 the equation above has proved and got value𝑑 ≥ 2𝑝+𝑞−5 𝑞−1 from many kinds or graph family. firecracker graph firecracker graph is a graph that gets by star graph combination exactly one leaf of each graph is connected [8], [9], [10], usually symbolized 𝐹𝑛,𝑘 where n is the number of merged star graphs, while k is the number of points of each connected star graph. methods this research uses axiomatic descriptive method, which is by decreasing the existing axioms or theorems [11], [12], then applied in super total labeling (𝑎, 𝑑)-edge antimagic on the firecracker graph 𝐹𝑛,𝑘 . in addition, some systematic research techniques are as follows: (1) count the number of points v and side e on the firecracker graph 𝐹𝑛,𝑘 ; (2) determine the upper limit of the different d values in the firecracker graph 𝐹𝑛,𝑘 in accordance with the lemma 1; (3) determine or find eavl label (edge-antimagic vertex labeling) or labeling points (𝑎, 𝑑)edge antimagic of the firecracker graph 𝐹𝑛,𝑘 ; (4) determine the algorithm of the functional function eavl 𝑓(𝑥𝑖.𝑙 ) on the firecracker graph 𝐹𝑛,𝑘 by looking at the labeling pattern on the graph firecracker 𝐹𝑛,𝑘 which has been found then grouping the numbers on the label of points that form arithmetic rows ; (5) determine the algorithm for the functional function of side weights eavl (𝑤) on the firecracker graph 𝐹𝑛,𝑘 by looking at the firecracker graph labeling pattern;(6) label the sides of the firecracker graph 𝐹𝑛,𝑘 with seatl (super edge antimagic total labeling) or super total labeling (𝑎, 𝑑)-edge antimagic for each corresponding different value d;(7) determine the bijective function on the firecracker graph 𝐹𝑛,𝑘 ; and (8) write a conclusion. super total labeling (a,d)-edge antimagic on the firecracker graph juhari 135 results and discussion the first step in determining the labeling of super total (𝑎, 𝑑)-edge antimagic is to determine the number of points and the number of edges on the graph under study, in this case, firecracker graph. after that, select the value of d in the labeling that will be examined using lemma 1. the labeling pattern can be determined by detecting the way (pattern recognition) after labeling a specific firecracker graph. next, to determine patterns in general, the objective function is found by using the principle of an arithmetic sequence. the following will be presented lemma and theorems that have been found. lemma 2. there is a point labeling (𝑎, 1)edge antimagic on the firecracker graph 𝐹𝑛,𝑘 if n odd, n ≥ 2, and k ≥ 3. prove. first defined 𝑥𝑖.𝑙 is the point in the graph component of the firecracker 𝐹𝑛,𝑘 ,where 1 ≤ i ≤ n and 0 ≤ l ≤ k − 1. based on research results,if 𝛼: 𝑉(𝐹𝑛,𝑘 ) → {1,2, … , 𝑛𝑘} so 𝛼 labelation can be written as follow: 𝑖; if i odd (1 ≤ 𝑖 ≤ 𝑛), and l=0 (𝑛 + 𝑖); if i even (2 ≤ 𝑖 ≤ 𝑛 − 1), and l=0 𝑖; if i even (12 ≤ 𝑖 ≤ 𝑛 − 1), and l=1 (𝑛 + 𝑖); if i odd (1 ≤ 𝑖 ≤ 𝑛), and l=1 𝑛(𝑙 + 1) − 𝑖−1 2 ; if i odd (1 ≤ 𝑖 ≤ 𝑛), and (2 ≤ 𝑙 ≤ 𝑘 − 1) (𝑙𝑛 + 𝑛−𝑖−1 2 ) + 1; if i even (2 ≤ 𝑖 ≤ 𝑛 − 1), and (2 ≤ 𝑙 ≤ 𝑘 − 1) from the equation above 𝛼(𝑥𝑖,𝑙 ) is an objective function that maps 𝑉(𝐹𝑛,𝑘 ) = {𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑛𝑘 } to the set of integers {1,2, … , 𝑛𝑘}. if 𝑤𝛼 is defindes as the weight edge of the labeling point α, so 𝑤𝛼 is formulated: 𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑛 + 2𝑖) ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛 + 2𝑖 + 1) ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑙 + 1)𝑛 + (𝑖+1) 2 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑤𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑙 + 1)𝑛 + (𝑛+𝑖+1) 2 ; if i even, 1 ≤ 𝑖 ≤ 𝑛 − 1 and 2 ≤ 𝑙 ≤ 𝑘 − 1 theorem 1. there is super total labeling (2𝑛𝑘 + 𝑛 + 1,0)-edge antimagic on the firecracker graph 𝐹𝑛,𝑘 if n odd, n ≥ 2, and k ≥ 3. prove. first define the edge label 𝑓𝛼 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, . . . , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … , 2𝑛𝑘 − 1}, so the edge label 𝑓𝛼 for super total labeling (𝑎, 0)edge antimagic on the graph 𝐹𝑛,𝑘 can be formulated as follows: 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = 2(𝑛𝑘 − 𝑖) + 1 ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = 2(𝑛𝑘 − 𝑖) ; 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑙 = 1 𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + (1−𝑖) 2 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + (−2𝑛−𝑖+6) 2 ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 next, if there is wα even that defined as weight of super total labeling 𝛼(𝑥𝑖,𝑙 ) = = super total labeling (a,d)-edge antimagic on the firecracker graph juhari 136 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ), so wα can be obtained by adding up the edge weight formula eavl 𝑤𝛼 and the formula of edge label 𝑓𝛼 with the terms of boundaries i and l which correspond and can be stated as follows: 𝑊𝛼1 = 𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 𝑊𝛼2 = 𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 𝑊𝛼3 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑊𝛼4 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 by substituting the equation above is obtained: 𝑊𝛼1 = (𝑛 + 2𝑖) + 2(𝑛𝑘 − 𝑖) + 1 = 2𝑛𝑘 + 𝑛 + 1 𝑊𝛼2 = 2(𝑛𝑘 − 𝑖) + (𝑛 + 2𝑖 + 1) = 2𝑛𝑘 + 𝑛 + 1 𝑊𝛼3 = (2𝑘 − 𝑙)𝑛 + (1−𝑖) 2 + (𝑙 + 1)𝑛 + (𝑖+1) 2 = 2𝑛𝑘 + 𝑛 + 1 𝑊𝛼4 = (2𝑘 − 𝑙)𝑛 + (−2𝑛−𝑖+6) 2 + (𝑙 + 1)𝑛 + (𝑛+𝑖+1) 2 = 2𝑛𝑘 + 𝑛 + 1 based on the equation above, the set of total labeling edge weights can be written as 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. it can also be seen that 𝑊𝛼1 = 𝑊𝛼2 = ⋯ = 𝑊𝛼4 = 2𝑛𝑘 + 𝑛 + 1 or can be written as follows: ⋃ 4𝑡=1 𝑊𝛼𝑡 = {2𝑛𝑘 + 𝑛 + 1,2𝑛𝑘 + 𝑛 + 1, … , 2𝑛𝑘 + 𝑛 + 1}. from this it can be concluded that the firecracker graph 𝐹𝑛,𝑘 have super (2𝑛𝑘 + 𝑛 + 1,0)edge antimagic if n odd, n ≥ 2, and k ≥ 3. theorem 2. there are super total labeling ((𝑘 + 1)𝑛 + 3,2)-edge antimagic on the graph firecracker 𝐹𝑛,𝑘 if n odd (𝑛 ≥ 2), and 𝑘 ≥ 3. prove. first define the edge label 𝑓𝛼 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, . . . , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … , 2𝑛𝑘 − 1}, so edge label 𝑓𝛼 for super total labeling (𝑎, 2)edge antimagic on the graph 𝐹𝑛,𝑘 can be formulated as follows: 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑛𝑘 + 2𝑖 − 1) ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛𝑘 + 2𝑖) ; 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑙 = 1 𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑘 + 𝑙)𝑛 + (𝑖−1) 2 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑘 + 𝑙)𝑛 + (𝑛+𝑖−1) 2 ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 next, wα defined as weight of super total labeling edge 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) 𝑊𝛼1 = 𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 𝑊𝛼2 = 𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 𝑊𝛼3 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑊𝛼4 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 by substituting the equation above is obtained: 𝑊𝛼1 = (𝑛 + 2𝑖) + (𝑛𝑘 + 2𝑖 − 1) = (𝑘 + 1)𝑛 + 4𝑖 − 1 𝑊𝛼2 = (𝑛 + 2𝑖 + 1) + (𝑛𝑘 + 2𝑖) = (𝑘 + 1)𝑛 + 4𝑖 + 1 𝑊𝛼3 = (𝑙 + 1)𝑛 + (𝑖+1) 2 + (𝑘 + 𝑙)𝑛 + (𝑖−1) 2 super total labeling (a,d)-edge antimagic on the firecracker graph juhari 137 = (𝑘 + 2𝑙 + 1)𝑛 + 𝑖 𝑊𝛼4 = (𝑙 + 1)𝑛 + (𝑛+𝑖+1) 2 + (𝑘 + 𝑙)𝑛 + (𝑛+𝑖−1) 2 = (𝑘 + 2𝑙 + 2)𝑛 + 𝑖 based on the equation above, the set of total labeling edge weights can be written as 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. it can also be seen that 𝑊𝛼1 = 𝑊𝛼2 = ⋯ = 𝑊𝛼4 = 2𝑛𝑘 + 𝑛 + 1 or can be written as follows: ⋃ 4𝑡=1 𝑊𝛼𝑡 = {(𝑘 + 1)𝑛 + 3, (𝑘 + 1)𝑛 + 5, … , (3𝑘 + 1)𝑛 − 1}. from this it can be concluded that the firecracker graph 𝐹𝑛,𝑘 have super ((𝑘 + 1)𝑛 + 3,2)edge antimagic if n odd, n ≥ 2, and k ≥ 3. theorem 3. there is super total labeling (2𝑛𝑘 + 1,1)-edge antimagic on the graph firecracker 𝐹𝑛,𝑘 if n even (𝑛 ≥ 2), and 𝑘 ≥ 3. prove. to determine the total super labeling (𝑎, 1)–edge antimagic, defined first 𝑓𝑎 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, … , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … ,2𝑛𝑘 − 1}which is the labeling edge label and can be formulated: 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑛𝑘 − 4𝑖 + 2) ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (2𝑛𝑘 − 4𝑖) ; 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑙 = 1 𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 3𝑛 − 𝑖 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 2𝑛 − 1 ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 if 𝑊𝛼 defined as the total labeling edge weight 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ). 𝑊𝛼1 = 𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 𝑊𝛼2 = 𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 𝑊𝛼3 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑊𝛼4 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 by substituting the equation above is obtained: 𝑊𝛼1 = (𝑛 + 2𝑖) + (2𝑛𝑘 − 4𝑖 + 2) = ( 2𝑛𝑘 + 𝑛 − 2𝑖 + 2 ) 𝑊𝛼2 = (2𝑛𝑘 − 4𝑖) + (𝑛 + 2𝑖 + 1) = ( 2𝑛𝑘 + 𝑛 − 2𝑖 + 1 ) 𝑊𝛼3 = (2𝑘 − 𝑙)𝑛 + 3𝑛 − 𝑖 + (𝑙 + 1)𝑛 + (𝑖+1) 2 = ( 2𝑛𝑘 + 4𝑛 + (𝑖+1) 2 ) 𝑊𝛼4 = (2𝑘 − 𝑙)𝑛 + 2𝑛 − 𝑖 + (𝑙 + 1)𝑛 + (𝑛+𝑖+1) 2 = ( 2𝑛𝑘 + 7𝑛+1−𝑖 2 ) based on the equation above, the set of total labeling edge weights can be witten with 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. it can also be seen that the smalest edge lies in 𝑊𝛼2 and the biggest edge weights lies in 𝑊𝛼1, can be stated that 𝑊𝛼 forming arithmetic lines with initial term 2𝑛𝑘 + 1 and different 1 (one), or can be written ⋃ 4𝑡=1 𝑊𝛼𝑡 = {2𝑛𝑘 + 1,2𝑛𝑘 + 2,2𝑛𝑘 + 3, … , ((3𝑛𝑘 − 1) + 𝑖)}. so, can be conclude that the firecracker graph 𝐹𝑛,𝑘 have super (2𝑛𝑘 + 1,1)-eat; n even (𝑛 ≥ 2), and 𝑘 ≥ 3. super total labeling (a,d)-edge antimagic on the firecracker graph juhari 138 theorem 4. there is super total labeling {𝑛𝑘 + 𝑛 + 4,3} –edge antimagic on the combination of firecracker graph 𝑚𝐹𝑛𝑘 if 𝑚 ≥ 2, n even (𝑛 ≥ 2), and 𝑛 ≥ 3. prove. to determine the total super labeling (𝑎, 1)-edge antimagic, defined first 𝑓𝑎 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, … , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … ,2𝑛𝑘 − 1} which is the edge label and can be formulated: 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑛𝑘 + 4𝑖 − 2) ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛𝑘 + 4𝑖) ; 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑙 = 1 𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 𝑖 − 1 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛,and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 𝑛 + 𝑖 − 1 ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 if 𝑊𝛼 defined as the total edge labeling weights 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ). 𝑊𝛼1 = 𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 𝑊𝛼2 = 𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) + 𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 𝑊𝛼3 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 𝑊𝛼4 = 𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 by substituting the equation above is obtained: 𝑊𝛼1 = (𝑛 + 2𝑖) + (𝑛𝑘 + 4𝑖 + 2) = ( 𝑛𝑘 + 𝑛 + 6𝑖 + 2 ) 𝑊𝛼2 = (𝑛𝑘 + 2𝑖 + 1) + (𝑛𝑘 + 4𝑖) = ( 2𝑛𝑘 + 𝑛 + 6𝑖 + 1 ) 𝑊𝛼3 = (𝑙 + 1)𝑛 + (𝑖+1) 2 + (2𝑘 − 𝑙)𝑛 + 𝑖 − 1 = ( 2𝑛𝑘 + 𝑛 + (1−𝑖) 2 ) 𝑊𝛼4 = (2𝑘 − 𝑙)𝑛 + 𝑛 − 𝑖 + 1 + (𝑙 + 1)𝑛 + (𝑛+𝑖+1) 2 = ( 2𝑛𝑘 + 5𝑛+2𝑖−1 2 ) based on the equation above, the set of total labeling edge weights can be written with 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. it can also be seen that the smalest edge lies in 𝑊𝛼1 and the biggest edge weights lies in 𝑊𝛼2, can be stated that 𝑊𝛼 forming arithmetic lines with initial term 𝑛𝑘 + 𝑛 + 4 and different 3 (one), or can be written ⋃ 4𝑡=1 𝑊𝛼𝑡 = {𝑛𝑘 + 𝑛 + 4, 𝑛𝑘 + 𝑛 + 7, 𝑛𝑘 + 𝑛 + 10, … , ((3𝑛𝑘 + 1)𝑛 + 𝑖)}. so, can be conclude that the firecracker graph 𝐹𝑛,𝑘 have super (2𝑛𝑘 + 1,1)-eat; n even (𝑛 ≥ 2), and 𝑘 ≥ 3. conclusions based on the result, can be concluded that firecracker graph 𝐹𝑛,𝑘 have super total labeling (𝑎, 𝑑)-edge antimagic, with 𝑑 ∈ {0,1,2,3} and bijective function in some of lemma and theorem show about super complete labeling (𝑎, 𝑑)-edge antimagic on the firecracker graph. the bijective function for each liberation with d different value have shown in equation (1),(2),(3) until (10) above. open problem: super total labeling (𝑎, 𝑑)-edge antimagic on the firecracker graph 𝐹𝑛,𝑘 for d=0 and d=2 where n even and 𝑘 ≥ 3; super total labeling d=1 and d=3 where n odd and 𝑘 ≥ 3. super total labeling (a,d)-edge antimagic on the firecracker graph juhari 139 references [1] dafik, “structural properties and labeling of graphs statement of authorship,” no. november, pp. 1–139, 2007. 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[12] a. s. meliana deta anggraeni, mulyono, “pelabelan l(3,2,1) dan pembentukan graf middle pada beberapa graf khusus,” vol. 2, no. 1, pp. 76–83, 2013. comparisons between resampling techniques in linear regression: a simulation study cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 345-353 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 23, 2021 reviewed: may 25, 2022 accepted: july 24, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.14550 comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto1,*, punitha linganathan2 1,*department of statistics, ipb university, indonesia 2department of mathematics, universiti putra malaysia, malaysia email: anwarstat@gmail.com abstract parameter estimations in linear regression need to fulfill some assumptions. once the assumptions are not fulfilled, the conclusion is questionable. bootstraps and jackknife are resampling techniques that do not require assumptions in estimating the �̂�. the study aims to compare resampling techniques in linear regression. the data used in the study is clean, without any influential observations, outliers, or leverage points. the ordinary least square method was used as the primary method to estimate the parameters and then compared with resampling techniques. the variance, p-value, bias, and standard error are used as a scale to estimate the best method among random bootstrap, residual bootstrap and delete-one jackknife. after all the analysis, it was found that random bootstrap did not perform well while residual and delete-one jackknife works quite well. random bootstrap, residual bootstrap, and jackknife estimate better than ordinary least square. the study also found that residual bootstrap works well in estimating the parameter in the small sample. at the same time, it is suggested to use jackknife when the sample size is big because jackknife is more accessible to apply than residual bootstrap and jackknife works well when the sample size is large. keywords: jackknife; linear; regression; resampling introduction regression analysis is a statistical analysis that constructs relationships between dependent or response variables 𝑦 and independent or regressor variables (𝑥1,𝑥2, …, 𝑥𝑘). ordinary least square (ols) is a traditional way of finding parameter estimates, �̂� but it relies strongly on assumptions [1]. the reliability and validity of the conclusion in regression analysis are essential ([2], [3]), and they depend on how far the data follows the assumption and on the sample size of the data. it is easier to find the estimated regression coefficient, �̂� without any assumption or distribution. bootstrap and jackknife are resampling techniques that do not need any assumptions in estimating the �̂� ,([4]–[6]. sahinler and topuz [7] compared the bootstrap and jackknife methods. their research discussed strategies for building a regression model using the jackknife and bootstrap method. the four methods used in their research are bootstrap based on the resampling observations, bootstrap based on the resampling errors, delete-one jackknife regression and delete-d jackknife regression. these methods were used to find the parameter estimates, bias, standard errors, and confidence intervals. their research concluded that http://dx.doi.org/10.18860/ca.v7i3.14550 mailto:anwarstat@gmail.com comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 346 large bootstrap replicates ensure that the parameter is close to the true parameter. they also suggested that bootstrap replicate is sufficient for estimating the variance and 𝐵 = 1000 for estimating the standard errors. their research tests the accuracy of bootstrap and jackknife methods in estimating the distribution of regression parameters with various sample sizes and various bootstrap replicates. sahinler and topuz [7] and li et. al. [8] found that the bootstrap method is appropriate for linear regression and it is usable even when the error is not normally distributed. algamal and rasheed [9] further develop resampling in linear regression. the advantage of bootstrap approximations is that, in general, it needs a smaller sample than the ordinary least square for estimating the parameter. meanwhile, the disadvantages of bootstrap methods were discussed in ma et al., [10], wan et al., [11], [12], and phaladiganon et al., [13] a few of the disadvantages of the methods are as follows: a) bootstrap distribution of is not a good approximation of 𝐹, if the sample size is small and with the existence of an outlier, b) bootstrap is not suggested to use in dependence structure case like time series, and c) it is not preferable to use residual bootstrap when the assumptions are violated. algamal and rasheed, [10] concluded that jackknife method perform quite well when the sample size is large enough (𝑛 ≥ 50). meanwhile, recent studies by shao, j., & tu, d., [14] and beyaztas, u., & alin, a., [15] discussed bootstrap and jackknife in linear regression. based on that, the study is aimed to compare parameter estimates of multiple linear regression based on several resampling methods. there are several methods to estimate the �̂� in bootstrap and jackknife. the scope of this research is to investigate the bootstrap and jackknife method with different scenariosthis research considered random bootstrap, residual bootstrap, and jackknife delete-one observation. the study is limited to multiple linear regression model. first the sample size will be selected with different size and estimate the parameter. the bias and variance will be observed then the relationship between the bias and variance will be investigated. the distribution also will be observed by varying with the increase in the sample size. the value of bootstrap resampling with different bootstrap replicates and sample size gives less bias than ordinary least square. the jackknife coefficient is calculated by using, �̂�𝑗 = 1 𝑛 ∑ �̂�𝑗𝑖 𝑛 𝑖=𝑛 (1) where n is the sample size and �̂�𝑗𝑖 parameter estimate for each sample formed after deleting one of the observations. while the bootstrap coefficient is calculated from �̂�𝑏 = 1 𝐵 ∑�̂�𝑏𝑟 𝐵 𝑟=1 (2) �̂�𝑏𝑟 = �̂�𝑜𝑙𝑠 + (𝑥 ′𝑥) −1 𝑥′𝑒𝑏𝑟 (3) where 𝑟 = 1,2,…,𝐵 is bootstrap replicate, 𝑒𝑏𝑟 is error of the regression,𝑥 is the independent variable and �̂�𝑜𝑙𝑠 is the parameter estimate from ordinary least square method. comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 347 methods data the data used in this study is pressure-dropping data, which is available in montgomery et al., [16]. it has one dependent variable 𝑦, and four independent variables, that is 𝑥1,𝑥2,𝑥3 and 𝑥4. there are 62 observations in the data. the data was collected from research where the pressure drop was measured for two-phase flow through screen-plate bubble columns. the research was conducted to test the reason of the pressure drop through the bubble cap. a bubble column is used to observe the reaction between the gas and liquid. the first factor considered in that research is the superficial fluid velocity of the gas. the gas's speed and direction of motion are measured by flow in the column. the second factor is the kinematic viscosity. the friction caused by the thickness of gas when the gas moves through the liquid particles was calculated. then the distance across the space between two parallel threads was considered. the last factor used in research is the dimensionless number, which is not associated with the physical dimension. it is calculated to relate the gas's superficial fluid velocity and the liquid's superficial fluid velocity. for building the model, the dependent variable 𝑦 denotes the dimensionless factor for the pressure drop through a bubble cap. the independent variables are 𝑥1 (superficial fluid velocity of the gas (𝑐𝑚 𝑠⁄ ), 𝑥2 (kinematic viscosity), 𝑥3 (mesh opening, cm), and 𝑥4 (dimensionless number relating the gas's superficial fluid velocity to the liquid's superficial fluid velocity). simulation study scenarios the original data will be analyzed using ordinary least square regression data. then assumptions checkings will be conducted using the residuals of the model. then, using the sampe original data, resampling techniques using the residuals and random bootstrap resampling will be conducted with four different sample sizes, which are 20, 40,50 and 62. each sample will be used in three different bootstrap replicates, namely 100, 1000 and 10000. for the delete-one jackknife bootstrap, the resampling will be conducted at different sample sizes, namely 20, 40, 50 and 62. the bias, variance, standard error and p-value will be calculated for each method. the best method among this three methods will be chosen according to the value of bias, variance, standard error and p-value. results and discussion in this study, full model was used for the reference, which means all independent variables were included in the model regardless the significance of the variables. the fitted full regression model which was obtained based on ordinary least square using sas software is written as follows: �̂� = 5.88839 − 0.48460𝑥1 + 0.18263 𝑥2 + 35.39109𝑥3 + 5.92695𝑥4 random bootstrap approach random bootstrap technique was first used to analyze the data. the resampling was conducted at different sample size 20, 40, 50 and 62. the bootstrap replication were applied in every sample size, namely 100, 1000 and 1000. comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 348 table 1. summary statistics for multiple linear regression using random bootstrap at different bootstrap replicates and sample sizes for 𝛽0 and 𝛽3 parameter estimate bootstrap replicate sample size bias variance p-value standard error �̂�0 100 20 -2.2181 85.6307 0.0001 0.9254 40 3.3626 22.8120 <.0001 0.4776 50 1.5437 15.1000 <.0001 0.0469 62 -0.6707 19.9445 <.0001 0.4466 1000 20 -1.1044 83.9495 <.0001 0.2897 40 2.9549 34.4348 <.0001 0.0012 50 1.4503 18.4197 <.0001 0.1357 62 -0.8994 19.1731 <.0001 0.1385 10000 20 -1.2754 203.4686 <.0001 0.0108 40 2.6252 41.8398 <.0001 0.0647 50 1.3527 18.8707 <.0001 0.0434 62 -0.9042 4.9842 <.0001 0.0461 �̂�3 100 20 2.6410 574.9345 <.0001 2.3978 40 -5.7656 111.8724 <.0001 1.0577 50 -5.3883 61.2876 <.0001 0.7829 62 1.0310 125.2369 <.0001 1.1191 1000 20 2.4814 629.8633 <.0001 0.7936 40 -5.4017 211.8649 <.0001 0.4603 50 -4.5249 73.4070 <.0001 0.2709 62 1.6356 116.7295 <.0001 0.3417 10000 20 3.1247 634.0890 <.0001 0.2518 40 -4.5548 261.6325 <.0001 0.1618 50 -4.2045 87.0297 <.0001 0.0933 62 1.8947 37.2858 <.0001 0.1146 table 1 shows the changes in �̂�3 and �̂�0 at different sample sizes and bootstrap replicates. for each parameter estimate, as the sample size changes, the bias changes. more specifically, the bias is getting smaller as the sample size increases. the variance of �̂�3 decreases from 574.9345 when the sample is 20 to 61.2876 when the sample size is 50. but, the bias of �̂�3 increases when the sample is 62 . it can be observed that as the sample size increased from 20 to 62, the variance of parameter estimates decreased. meanwhile, the bias decreases as the bootstrap replicate increases. for b was set to 100, the intercept shows bias as 1.5437. this value decreases to 1.4503 when the number of bootstrap replicates, b, increases to 1000. when the number of bootstrap replicates was increased to 10000, the bias decreases again to 1.3527. from the results, it can be observed that the bias decreases as the replicate increases. when the bootstrap replicate, b increases from 100 to 1000, the variance decreases from 125.2369 to 116.7295. it decreases further to 37.2858 when b is equal to 10000, which shows 70.23% difference when we compare to 125.2369. comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 349 residual bootstrap approach the second resampling technique that has been used to analyze the data was residual bootstrap. this section displays some results such as parameter estimates, bias, and variances of the parameter estimates using residual bootstrap. the results of �̂�0 and �̂�1 are shown in table 2. in residual bootstrap, the results were more apparent than in random bootstrap. it shows a clear trend of parameter estimates, bias, and variance at different sample sizes and the number of bootstrap replicates. the bias decrease as the sample size increases. when 𝑛 = 20, the bias is 0.2307. then when the sample increased to 40 the bias became 0.2266 and bias is 0.0684 when the sample size is 50 and at last, when 𝑛 is 62 the bias became 0.01368. in general, there is a noticeable difference in bias when the sample size increases. table 2. summary statistics for multiple linear regression using residual bootstrap at different bootstrap replicates and sample sizes for 𝛽0 and 𝛽1 the resampling techniques in table 2 show a clear decrease of the variances when the sample size increases. let’s consider the changes in the variance of �̂�0 when the bootstrap replicate is 1000. when the sample size is 20 the variance is 28.6300, and the value parameter estimate bootstrap replicate sample size bias variance p-value standard error �̂�0 100 20 1.5277 30.9685 <.0001 0.5565 40 2.6535 27.0046 <.0001 0.5197 50 1.5324 19.8861 <.0001 0.4459 62 -0.3345 15.4073 <.0001 0.3925 1000 20 0.9635 28.6300 <.0001 0.1692 40 2.3838 22.7581 <.0001 0.1509 50 2.0622 20.0467 <.0001 0.1416 62 0.0035 15.4785 <.0001 0.1244 10000 20 0.6704 30.9949 <.0001 0.0557 40 2.2883 24.0894 <.0001 0.0491 50 2.2491 20.2725 <.0001 0.0450 62 -0.0193 17.0400 <.0001 0.0413 �̂�1 100 20 0.2307 0.2037 <.0001 0.0451 40 0.2266 0.1566 <.0001 0.0396 50 0.0684 0.1098 <.0001 0.0331 62 0.0137 0.0819 <.0001 0.0286 1000 20 0.1630 0.2196 <.0001 0.0148 40 0.2061 0.1612 <.0001 0.0127 50 0.0732 0.1322 <.0001 0.0115 62 -0.0066 0.1025 <.0001 0.0101 10000 20 0.1547 0.2103 <.0001 0.0046 40 0.2180 0.1579 <.0001 0.0040 50 0.0608 0.1338 <.0001 0.0037 62 -0.0024 0.1071 <.0001 0.0033 comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 350 becomes 22.7581 when the sample size is 40. then the variance decrease as the sample size increases to 50 and 62 where the bias become 19.8861 and 15.4785, respectively. now let’s observe the changes in bias caused by the bootstrap replicate, b, when it is increased from hundred to thousand then ten thousand. for the estimated constant, �̂�0, when the sample size is 40 the bias changes from 2.6535 to 2.3838, then 2.2883 when b increases from 100 to 1000 then 10000, respectively. the variance also decreases when the bootstrap replicate increases. delete-one jackknife approach the third technique that was used in this research is jackknife delete-one. the method was applied with different sample sizes , which are 20, 40, 50 and 62. table 3 and figure 1 display the changes in bias of all parameters for delete-one jackknife. the bias decreases as the sample size increases. but when sample size equal to the population size the bias shows an increasing state. using the population as sample size might show this type of result. plot of variance versus sample size for all parameters are shown in figure 2. from the plot, it can be seen that the variance also shows a decreased state from sample 20 to sample 62. small variances give a better estimation in linear regression. the bias and variance also not interrelated in delete-one jackknife. the p-value also shows that all parameter estimates are significant. the standard error also clearly shows that the increase in sample size will give a better estimation. table 3. summary statistics for multiple linear regression using delete-one jackknife at different sample size . parameter estimate sample size bias variance p-value standard error �̂�0 20 0.5586 2.9683 <.0001 0.3852 40 2.2937 0.7335 <.0001 0.1354 50 2.1648 0.3625 <.0001 0.0851 62 -3.1721 0.2212 <.0001 0.0597 �̂�1 20 0.1617 0.0161 <.0001 0.0284 40 0.2182 0.0054 <.0001 0.0117 50 0.0662 0.0046 <.0001 0.0096 62 0.6613 0.0029 <.0001 0.0069 �̂�2 20 0.0249 0.0001 <.0001 0.0017 40 0.0006 0.0000 <.0001 0.0007 50 -0.0045 0.0000 <.0001 0.0005 62 0.0054 0.0000 <.0001 0.0004 �̂�3 20 -2.0491 18.1473 <.0001 0.9526 40 -7.3628 3.7708 <.0001 0.3070 50 -5.6852 1.4059 <.0001 0.1677 62 -3.7014 0.9218 <.0001 0.1219 �̂�4 20 0.3589 1.9284 <.0001 0.3105 40 0.6624 0.4470 <.0001 0.1057 50 -0.0712 0.3889 <.0001 0.0882 62 0.7431 0.4339 <.0001 0.0837 comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 351 figure 1. changes of bias in all parameter estimation when sample size increases in delete-one jackknife. figure 2. changes of variance in all parameter estimation when sample size increases in delete-one jackknife. the difference between residual bootstrap estimation and random bootstrap estimation is obvious when the sample size is 20 (small). the residual bootstrap provided better parameter estimation than random bootstrap in bias and variance. this shows that residual has a big influence in linear regression. but, as the sample size increases, both residual and random bootstrap methods show similar results. the increase in bootstraps replicates and sample size gave better parameter estimation in both methods. jackknife delete-one gave a small variance, but the value of the bias was big when the sample size was small. the bias and variance decrease as the sample size increases. conclusions residual bootstrap, random bootstrap, and delete-one jackknife were compared. jackknife is not advisable to use when the sample size is small. however, when the sample -7 -6 -5 -4 -3 -2 -1 0 1 2 3 20 40 50 62 b ia s sample size β4 β3 β2 β1 β0 0 5 10 15 20 25 20 40 50 62 va ri a n ce sample size β4 β3 β2 β1 β0 comparisons between resampling techniques in linear regression: a simulation study anwar fitrianto 352 size is big enough which is near to population size, it will give better parameter estimation than random bootstrap and residual bootstrap. in a situation where the sample size is small due to cost consideration, it is better to use residual bootstrap than other methods in linear regression. in conclusion, it is advisable to use residual bootstrap when the sample is small. the bigger bootstrap replicates will give better parameter estimation. the jackknife can be used when the sample size is big enough. this method will be useful when the sample size is too big which may take time to process in both random and residual bootstrap. in the future, this research can be extended to observe how these methods react when there is an outlier, influential point or leverage point. moreover, the comparisons may involve other resampling techniques to compare which method works well in multiple linear regression. references [1] 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[16] d. c. montgomery, e. a. peck, and g. g. vining, introduction to linear regression analysis. john wiley & sons, 2021. cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 220-230 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 19, 2021 reviewed: december 10, 2021 accepted: january 07, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.13356 analysis of insurance customer factors to renewal using hybrid ahp-ftopsis kwardiniya andawaningtyas, evi ardiyani*, corina karim departement of mathemathics faculty of mathematics and natural science brawijaya university, 65145, malang, indonesia *corresponding author email: eviardiyani98@gmail.com* dina_math@ub.ac.id, corinaub@gmail.com abstract human life is full of uncertainties that have enormous risks. insurance is one way that can help humans reduce this risk. the human need for insurance causes competition among insurance companies in indonesia to be very competitive. competition between insurance companies is influenced by several factors, one of the factors is having customers who do insurance renewals. this study aims to determine the factors that influence customers to renew using the analytical hierarchy process (ahp) method and to rank customers' favorite insurance using the fuzzy technique for order preference by similarity to ideal solution (ftopsis) method. the results of the analysis using this method concluded that the main factors that influence customers in making renewals are features with sub-criteria for health protection needs. meanwhile, the customer's favorite insurance ratings for extending are takafullink salam cendikia with a closeness coefficient of 0.645, takaful al-khairat with a value of 0.563, takaful dana pendidikan with a value of 0.552, and takafullink salam with a value of 0.341. keywords: insurance; renewal; ahp; ftopsis introduction human life is full of elements of uncertainty that have enormous risks, such as accidents and death. humans need a guarantee or a method to reduce this risk which we usually call insurance. the human need for insurance causes the competition of insurance companies in indonesia to be very competitive. the biggest factor for an insurance company to be competitive is a customer who carries out a renewal. each customer has its own criteria which are the determining factors for a customer to renew. the decision support system (dss) is specific information that is intended to assist management in making decisions related to semi-structured issues. dss aims to assist decision makers in establishing an unstructured decision. unstructured decisions have vague problems, and it's difficult to find solutions. decision support systems are basically designed to support every stage of decision making, namely identifying problems, selecting relevant data, determining approaches, and evaluating alternative choices. in 2018, [1] conducted research on how to improve consumer satisfaction http://dx.doi.org/10.18860/ca.v7i1.13356 mailto:eviardiyani98@gmail.com* mailto:dina_math@ub.ac.id mailto:corinaub@gmail.com analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 221 learning (lbb) in malang using the danp-topsis method. identifying important human error factors in emergency departments in taiwan using hfacs, ahp, and ftopsis by [2]. [3] conducted research on the selection of favorite banks using the ahp and topsis methods. [4] discusses the selection of the best health applications and features that affect the ahp and ftopsis methods. comparasion on of anp and ahp methods studied by [5]. [6] conducted reseacrh comparison between topsis and saw. [7]discusses decision making using hybrid ahp-topsis. hybrid fuzzy ahp-topsis researched by [8]. [9] researched decision making using hybrid ahp-topsis. comparison beetwen saw, ahp, and topsis researched by [10]. [11] conducted research on the comparison between saw method and ahp method. integrated anp and topsis method for suplier performace assesment researched by [12]. [13] reasearched evaluation of smart and suistainable cities with anp and topsis method. [14] conducted research hybrid ahptopsis method under spherical fuzzy sets for system selection. hybrid ahp-topsis for selecting supplier in construction supply chain researched by [15]. this study aims to determine the factors that most influence insurance customers to renew and obtain favorite insurance alternatives by combining the ahp and ftopsis methods. the combination of the ahp and ftopsis methods is to obtain the criteria weights using the ahp method, then the ftopsis method uses the criteria weights that have been obtained by the ahp method to obtain the best alternative. methods analytical hierarchy process (ahp) method. ahp is a decision-making process with compilation of functional hierarchies with the main input being human [16]. ahp requires ideas from individuals and groups by obtaining their respective assumptions and obtaining the desired solutions. these ideas are used to determine criteria that can solve a problem. in this research, ahp method is used to determine criterion weight to be used in the ftopsis method. according [16] there are general measures of ahp method consists of seven steps. 1. defining the problem and determining the desired solution then arranging hierarchy of the problems by setting goals which are the overall system goals at the top level. 2. determine the priority of the elements. a. making pair comparasons by comparing elements in pairs according to given criteria. b. the pairwaise comparison matrix is filled using numbers to represent the relative importance of one element to another. the pairwise comparison matrix entry is the result of a questionnaire converted using table 1. 3. synthesis considerations for pairwise comparisons are synthesized to obtain overall priority. a. sum each column on the matrix. b. divide each value from the column by the total column obtain a normalized matrix. c. sum each row and divide by the number of elements to get the average value. 4. measure consistency. a. multiplies each value in the first column by the relative priority of the first element, the value in the second column by the relative priority if the second element, and so on. b. adding each row, the result divided by the corresponding relative priority element. analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 222 c. adding the results above for the elements that exist, called λmaks 5. calculating the consistency index (ci) (1) = number of elements 6. calculating the consistency ratio (cr) (2) where is index random consistency contained in table 2 7. check hierarchy consistency, the consistency ratio must be less or equal to 0.1. the calculation result can be declared correct. table 1. ahp rating scale difference -8 -7 -6 -5 -4 -3 -2 -1 0 ahp scale 9 8 7 6 5 4 3 2 1 source : [16] table 2. index random consistency matrix size ir value 1,2 0.00 3 0.58 4 0.90 5 1.12 6 1.24 7 1.32 8 1.41 9 1.45 10 1.49 11 1.51 source : [16] fuzzy technique for order preferences by similarity to ideal solution (ftopsis) method. the ftopsis method is a development of the topsis (technique for order preference by similarity to ideal solution). topsis method first introduced by yoon and hwang in 1981. the topsis method has a weakness, when the decision maker has difficulty determining a value. therefore, it is necessary to provide an assessment in the form of intervals such as applying fuzzy logic. fuzzy numbers, linguistic values and membership function shown in the figure 1 and table 3. in this research, ftopsis method is used to rank the alternatives. analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 223 figure 1. fuzzy numbers and linguistic table 3. the membership function of linguistic value linguistic value fuzzy number very low (vl) ( 0 , 0 , 0.2 ) low (l) ( 0, 0.2, 0.4 ) medium (m) (0.2, 0.4, 0.6) high (h) (0.4, 0.6, 0.8) very high (vh) ( 0.6, 0.8, 1 ) excellent (e) ( 0.8, 1, 1 ) source : [3] general measurer of ftopsis method consists of 9 steps. 1. assesing criteria and alternatives assumed that there are alternatives that will be evaluated against criteria . the weight of each criterion is denoted by . the ranking of the fuzzy criteria value of each decision for each alternative against the criterion denoted by with the membership function . 2. calculate the comparison value of each criterion and alternatives the fuzzy values for each decision maker are presented as fuzzy triangle the value of the fuzzy ratio is given by , with (3) fuzzy number linguistic value analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 224 fuzzy weight ratio with : (4) 3. make a decision matrix creating a decision matrix (dk) that is appropriate for the alternatives to be evaluated based on the following defined criteria : with states the performance of the calculation for i alternatives against the j criterion. 4. normalize the fuzzy decision matrix normalize the data using a linear scale transformation, the normalized matrix is defined by (5) with (6) (7) (6) benefit criteria, (7) cost criteria 5. calculate the normalized matrix weights the normalized matrix weight is calculated by multiplying the weight of the evaluated criterion by the normalized decision matrix (8) with 6. calculate the value of fuzzy positive ideal solution (fpis) and fuzzy negative ideal solution (fnis) (9) (10) with and is the set of benefit criteria. and is the set of cost criteria. 7. calculate the distance for each alternatives from fpis and fnis if there is and is two fuzzy triangular numbers, defined as and then the distance between and can be calculated by analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 225 (11) distance of each weighted alternative (i = 1, 2, 3, … , m) from fpis and fnis can be calculated by (12) (13) 8. calculate the closeness coefficient value the closeness coefficient ( ) represents the distance between fpis(a+) and fnis (a) simultaneously for each alternative, the closeness coefficient ( ) can be calculated by (14) with 9. sort alternative each alternative is sorted according to the decreasing closeness coefficient ( ) value. the best alternatives is the closeness coefficient ( ) value is close to fpis and far from fnis. results and discussion analytical hierarchy process (ahp) method. the first step in the ahp method is arrangement the hierarchichal structure. the hierarchical structure in this study consists of 4 levels, the first level is the goal, namely to determine the favorite type of insurance. the second level is the elaboration of the main aspects that influence the objectives, namely the criteria. the third level is the aspects that influence the criteria, namely sub-criteria. the fourth level or the lowest level is the level that consists of alternatives. the structure of the hierarchical system in this study can be seen in figure 2. then, we determine the priority of the elements by create formation of pairwise comparison matrix between sub-criteria and create weight matrix between sub-criteria based on the results of the questionnaire. next step is calculate value. the five criteria have a , it can be concluded that the pairwise comparison matrix between these subcriteria is consistent. the most influential criterion in choosing the customer’s favorite insurance for renewal in company a is the insurance feature with the subcriteria for the need for health protection having a weight value of 0.800. table 5 shows the evaluation result and final ranking of criterion. to determine the level of data consistency, we calculate the value. first, calculate the value of then calculate the using equation (1). is obtained by adding the results for the elements that exist and each number of subcriteria is the value of n used. table 5 shows the results. using the value that has been obtained, calculate the value using equation (2). if the the research can be continued. table 4 analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 226 shows the value. it is shown that the five criteria have a value of less than 0,1 which means that the data for the five criteria are consistent. so, research can be continued. table 4. value criteria value company image 0,068 agent 0,034 insurance features 0,064 claim 0,020 income 0,055 figure 2. hierarchical system table 5. ahp method result subcriteria (priority vector) honesty 0.639 achievment 0.087 tdp tls tlsc tak customer income (rp. 2,5 4,9 million /month) customer income (rp. 5,0 7,5 million /month) customer income (>rp. 7,5 million /month) goal favorite type of insurance and factors that influence customers to renew criterion sub-criterion alternative agent product mastery communication ease of contact honesty achievment track record insurance features health education investation claim ease of taking claims great claim company image time period for claiming income customer income (<rp. 2,5 million /month) analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 227 track record 0.274 product mastery 0.343 subcriteria (priority vector) communication 0.575 ease of contact 0.082 health 0.800 education 0.124 investation 0.075 ease of taking claims 0.123 great claim 0.557 time perios for claiming 0.320 customer income (<rp. 2,5 million /month) 0.055 customer income (rp. 2,5 – 4,9 million /month) 0.156 customer income (rp. 5,0 – 7,5 million/month) 0.545 customer income (>rp. 7,5 million/month) 0.244 afterward, we analyze the best alternative in ftopsis methods. the weights of criteria to be used in evalution process are calculated by using ahp method combined with the scores from the expert questionnaire. table 6 shows the data from the expert questionnaire. table 6. data from the expert questionnaire tdp tls tlsc tak honesty h vh m h achievment vh vh h m track record m h vh l product mastery l vh m m communication h l vh h ease of contact m l l m health h h vh e education e vh h h investation h vh e h ease of taking claims vh h h vh great claim m l vh h time period for claiming vh h vh h customer income (<rp. 2,5 million /month) h vh m m customer income (rp. 2,5 – 4,9 million /month) h h h m customer income (rp. 5,0 – 7,5 million/month) h m vh vh customer income (>rp. 7,5 million/month) h h m h then the next step is calculating the weight of the alternative matrix. table 7 shown the multiplication results of the expert questionnaire values that have been converted based on table 3 with the priority vector value for example, criteria honesty on alternative tdp is h then convert the value to fuzzy number based on table 3 which is (0.4, 0.6, 0.8). then do fuzzy multiplication with the value of the priority vector which is 0.639. after we get the multipclication of the priority vectors and the expert quiestionner, we calculate the fpis and fnis values, then we use these values to calculate the fpis and fnis distances using equation (12) and (13). table 8 shows the value of the fpis and fnis distance. after calculating the distance between fpis and fnis, we calculate value using equation (14). table 9 shows the results of calculating the value. analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 228 depends of the value in table 9 the alternatives ranking in ftopsis method, the first order is the tlsc alternative, the second is the tak alternative, the third is the tdp alternative, and the last order is the tls alternative. table 7. multiplication of the priority vectors by the results of the expert questionnaire tdp tls tlsc tak honesty (0.256,0.384,0.511) (0.384,0.511,0.639) (0.128,0.256,0.384) (0.256,0.384,0.511) achievment (0.052,0.070,0.087) (0.052,0.070,0.087) (0.035,0.052,0.070) (0.017,0.035,0.052) track record (0.055,0.109,0.164) (0.109,0.164,0.219) (0.164,0.219,0.274) (0,0.055,0.109) product mastery (0,0.069,0.137) (0.206,0.274,0.343) (0.069,0.137,0.206) (0.069,0.137,0.206) communication (0.230,0.345,0.460) (0,0.115,0.230) (0.345,0.460,0.575) (0.230,0.345,0.460) ease of contact (0.016,0.033,0.049) (0,0.016,0.033) (0,0.016,0.033) (0.016,0.033,0.049) health (0.320,0.480,0.640) (0.320,0.480,0.640) (0.480,0.640,0.800) (0.640,0.800,0.800) education (0.099,0.124,0.124) (0.075,0.099,0.124) (0.050,0.075,0.099) (0.050,0.075,0.099) investation (0.030,0.045,0.060) (0.045,0.060,0.075) (0.060,0.075,0.075) (0.030,0.045,0.060) ease of taking claims (0.074,0.098,0.123) (0.049,0.074,0.098) (0.049,0.074,0.098) (0.074,0.098,0.123) great claim (0.446,0.557,0.557) (0,0.111,0.223) (0.334,0.446,0.557) (0.223,0.334,0.446) time perios for claiming (0.192,0.256,0.320) (0.128,0.192,0.256) (0.192,0.256,0.320) (0.128,0.192,0.256) customer income (<rp. 2,5 million /month) (0.222,0.033,0.044) (0.033,0.044,0.055) (0.011,0.022,0.033) (0.011,0.022,0.033) customer income (rp. 2,5 – 4,9 million /month) (0.062,0.094,0.125) (0.062,0.094,0.125) (0.062,0.094,0.125) (0.031,0.062,0,094) customer income (rp. 5,0 – 7,5 million/month) (0.218,0.327,0.436) (0.109,0.218,0.327) (0.327,0.436,0.545) (0.327,0.436,0.545) customer income (>rp. 7,5 million/month) (0.098,0.146,0.195) (0.098,0.146,0.195) (0.049,0.098,0.146) (0.098,0.146,0.195) table 8. fpis and fnis distances. tdp tls tlsc tak 0.982 1.445 0.787 0.958 1.207 0.748 1.429 1.234 table 9. value alternative tdp 0.552 tls 0.341 tlsc 0.645 tak 0.563 conclusion the results of data analysis that has been carried out from the combination of the two methods indicate that the most influencing factor for insurance customers to renew at pt asuransi takaful keluarga is the insurance feature, namely the customer's need for health protection with weight value of 0.800. all health insurance companies must have health protection features. so, we see the next order of sub-criteria, honesty with a weighted value of 0.639, agent communication with a weighted value of 0.575, and good claims with a weighted value of 0.557. the sub-criteria that have the highest value analysis of insurance customer factors to renewal using hybrid ahp-ftopsis 229 weight are the criteria for company image, agent, and claims. the five criteria have a priority value that is quite close, indicating that the five criteria are mutually sustainable.the order of alternative choices for the customer's favorite insurance who renews at pt. family takaful insurance is takafulink salam scholar, takaful al-khairat, takaful fund education, and takafulink salam. references [1] k. andawaningtyas, e. w. handamari, and c. karim, “improving the guidance learning (lbb) consumer satisfaction in malang using danp topsis method,” cauchy, vol. 5, no. 3, p. 117, 2018, doi: 10.18860/ca.v5i3.5541. 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[16] kusrini, konsep dan aplikasi sistem pendukung keputusan. 2007. c-type ops transformation cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 401-410 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 05, 2022 reviewed: june 29, 2022 accepted: july 07, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.15749 c-type ops transformation ahmad lazwardi*, iin ariyanti, soraya djamilah universitas muhammadiyah banjarmasin, indonesia email: lazwardiahmad@gmail.com abstract the scope of the ops transformation is limited to the mclaurin series only. while there are still many cases in mathematical modeling which are modeled in the form of a more general series. the purpose of this study is to generalize the ops transformation into a more general form that can be used for any taylor series. this study uses a literature study method, namely by reviewing the ops transformation and then observing aspects that can be generalized. next is to construct a more general definition of ops transformation which is referred to as c-type ops transformation. at the end of this research, the ops transformation will be applied to solve ordinary differential equations with variable coefficients very briefly. keywords: c-type ops transformation; ops transformation; power series introduction taylor series is accessible to all students and it is a useful mathematical tool to nonlinear equations [1]. power series is an essensial method for solving many problems in mathematics such as algebra and differential equations [2]. many cases also appear in algebra which is involving power series such as solving polynomial homotopies equations [3]. as we know on algebraic geometry topics we talk about rings, ring extensions and ideals which recently appears as series forms. for example, is on commutative ring topics. if r be a commutative ring with identity. let r[x] and r[[x]] be the collection of polynomials and, respectively, of power series with coefficients in r. their multiplications are from a class of sequences 𝜆 = {𝜆𝑛} of positive integers which is related one-one correspondence to its power series [4]. another branch of mathematics which also involved by power series recently is differential equations[5]. the problem of finding formal power series solutions of differential equations has a long history and it has been extensively studied in literature. using power series method however, is a more systematic way and standard basic method for approximating the solutions of such differential equations analytically and thus studying the method is of greater importance [6]. ordinary power series has also appear as solutions for fractional differential equations as told by angstmann and henry in their publication namely generalized series expansions involving integer powers and fractional powers in the independent variable have recently been shown to provide solutions to certain linear fractional order differential equations [7]. this incredible discovery is related to i. area and j. nieto who discover the solution for fractional logistics equation which is appeared as power series form [8]. http://dx.doi.org/10.18860/ca.v7i3.15749 c-type ops transformation ahmad lazwardi 402 this research discovers new theory of ordinary power series expecially the alternating method to analyze ordinary power series by considering ordinary power series as a transformation which called ops transformation. this transformation will simplify the counting process of some equations involving sigma notation. reducing the use of sigma algebra, alternating it by algebra of linear transformation. the aim of this research is to generalize the concept of ops transformation for power series of form (1). we will name it as c-type ops transformation. the letter c is indicated the center of the power series. methods we will proceed this reseach through the following procedures. first we will define the more general form of ops transforamtion. after that we will find some basic results of our new definition. we will use such results to extend the theory of ops transformation. next we will make sure that our previous definition of ops transformation to become special case for our new definition. next we will find some theorems regarding our new transformation properties. we also will make sure that our new definition is able to be applied much wider than our previous one. results and discussion first we shall define the c-type of ops transformation. recall that power series centered at c is defined to be the real valued function of the form [9] .)( 0     n n n cxa the series has a value depending on what value of x we choose. some value of x will result the series tend to infinity. some other will result the series converges[10]. the set of x which result (1) converges is called convergence interval. the term “interval” makes sense because such set always forms an interval. the half-length of such interval is called convergence radius. some smooth function f(x) at point c is able to be approximated by power series which on some value of x0 lies on its convergence interval centered at c, the result will be same, i.e f(x0) = ∑ 𝑎𝑛(𝑥0 − 𝑐) 𝑛∞ 𝑛=0 . such functions are called “real analytic” functions. the method of resulting such series was given by taylor which is called taylor series as below [11] .)( ! )( 0 )(     n n n cx n cf special case of taylor series is when the value of c = 0, the series is called mclaurin series. lazwardi (2021) has already able to reformulate such series into more simple form called ops transformation. the ops transformation is defined as below .)})(({ 0     n n nn xaxaops therefore, for some mclaurin series of the form . ! )0( 0 )(   n n n x n f (1) (2) (4) c-type ops transformation ahmad lazwardi 403 its enough to write the series simply as 𝑂𝑝𝑠{ 𝑓(𝑛)(0) 𝑛! } . simplification of the form will make calculations and manipulations easier [12]. there are some properties regarding ops transformation as following: theorem 1. (shifting-entry) for each {𝑎𝑛} sequence, we have 𝑂𝑝𝑠{0,𝑎0,𝑎1,…} = 𝑥𝑂𝑝𝑠{𝑎𝑛}. theorem 2. for each {𝑎𝑛} sequence, we have 𝑂𝑝𝑠({𝑎𝑛})− 𝑎0 = 𝑥𝑂𝑝𝑠({𝑎𝑛+1}). beside two above theorems, ops transformation inherits linearity properties as well as sigma notations. theorem 3. for each {𝑎𝑛},{𝑏𝑛} sequences and any real numbers 𝛼,𝛽 , we have 𝑂𝑝𝑠(𝛼{𝑎𝑛} +𝛽{𝑏𝑛}) = 𝛼𝑂𝑝𝑠{𝑎𝑛} + 𝛽𝑂𝑝𝑠{𝑏𝑛}. the last theorem notices us that we can view ops transformation as linear transformation which mapping from the space of all real sequences to real numbers on its convergence radius. we shall use this necessary fact to simplify several calculations. besides that lazwardi was able to prove the formula regarding product of two ops transformations as following. theorem 4. for each {𝑎𝑛},{𝑏𝑛} sequences, we have .}{}{ 0           n k knknn baopsbopsaops this is just similiar with the product two power series but wihout involving double sigma notation. another important result of previous research is we can use the fact that the power series is always able to differentiate n-times, to construct the rule of differentiation for ops transformation. talking about differentiation of ops transformation meas we have to state the symbol for its derivative. we use 𝐷𝑥𝑂𝑝𝑠{𝑎𝑛} to notate the derivative of ops transformation on its radius convergence. therefore we have the following theorems. theorem 5. for each {𝑎𝑛} sequence, we have 𝐷𝑥𝑂𝑝𝑠({𝑎𝑛}) = 𝑂𝑝𝑠({(𝑛 + 1)𝑎𝑛+1}). here is some nice modification formula theorem 6. for each {𝑎𝑛} sequence, we have 𝑥𝐷𝑥𝑂𝑝𝑠({𝑎𝑛}) = 𝑂𝑝𝑠({𝑛𝑎𝑛}). if we pay more attention to the (1). there are some difference between (1) and (3) i.e the value of c will be varied and able to consider it as a variable. therefore we have at least 4 variable involved in calculations of (1) which is more complicated than (3) expecially special type of ordinary power series which called taylor series. as told by salwa in her research that many infinite series form are recently appear in sequence spaces expecially on 𝛽 − 𝑑𝑢𝑎𝑙 sequence spaces which is defined as infinite series form [13] one of popular application from taylor series is the iterative method of the (5) (6) (7) (8) (9) (10) c-type ops transformation ahmad lazwardi 404 differential transform methor has already been used for a while, by the ‘‘traditional’’ taylor series method users which have even better developed the method. suppose that we have power series of form (1) centered at c. we define definition 1. let c be a real number and suppose power series ∑𝑎𝑛(𝑥 − 𝑐) 𝑛 has positive convergence radius near c. define the c-type ops transformation as following 𝑂𝑝𝑠𝑐{𝑎𝑛}(𝑥) = ∑𝑎𝑛 ∞ 𝑛=0 (𝑥 − 𝑐)𝑛. it looks similiar to the previous form with additional superscript c. note that the additional superscript c on ops roles as index depending on value c on the right side. for some reason, we shall keep c to become upper index because we shall use lower index with another use on the next research. recall that one of the most suitable form which is similiar to our last definition is taylor series of analytic function on c. if 𝑓(𝑥) is an analytic function near c, then we can write f as taylor series on some neighborhood c as below 𝑓(𝑥) = ∑ 𝑓(𝑛)(𝑐) 𝑛! ∞ 𝑛=0 (𝑥 − 𝑐)𝑛. hence, taylor series of f can be written as c-type ops transformation as 𝑓(𝑥) = 𝑂𝑝𝑠𝑐 { 𝑓(𝑛)(𝑐) 𝑛! }(𝑥). for some reason, we just write 𝑓 = 𝑂𝑝𝑠𝑐 { 𝑓(𝑛)(𝑐) 𝑛! }. please pay more attention here. upper index c is viewed as variable (not necessary fixed). we can write 𝑂𝑝𝑠𝑎 { 𝑓(𝑛)(𝑐) 𝑛! } = ∑ 𝑓(𝑛)(𝑐) 𝑛! ∞ 𝑛=0 (𝑥 − 𝑎)𝑛. i.e when we change the value of upper index c by a, the value of 𝑓(𝑛)(𝑐) 𝑛! doesn’t change but the center of power series on the right side changes to a. its clear that ops transformation is a special case of c-type of ops transformation by taking value c = 0 [14], i.e 𝑂𝑝𝑠0{𝑎𝑛} = 𝑂𝑝𝑠{𝑎𝑛}. for more brief information. we shall discuss some more examples as following. (11) (12) (13) (14) (15) c-type ops transformation ahmad lazwardi 405 example 1. 𝑂𝑝𝑠𝑐{1} = 1 1−(𝑥−𝑐) . proof: observe that 𝑂𝑝𝑠𝑐{1} = ∑(𝑥 − 𝑐)𝑛. suppose that 𝑦 = 𝑥 − 𝑐 then the right side of equation become ∑𝑦𝑛 = 1 1−𝑦 for |𝑦| < 1. hence we have for |𝑥 − 𝑐| < 1 or 𝑐 − 1 < 𝑥 < 𝑐 + 1 we will get the series ∑(𝑥 − 𝑐)𝑛 will converge and we have 𝑂𝑝𝑠𝑐{1} = 1 1−(𝑥−𝑐) . here is another example example 2. 𝑂𝑝𝑠𝑐 { 1 𝑛! } = 𝑒𝑥−𝑐. now we shall analyze more properties of c-type ops transformation. first we success to preserve the “shifting index” properties as well as previous form [15] theorem 7. (shifting-entry) for each {𝑎𝑛} sequence, we have 𝑂𝑝𝑠𝑐{0,𝑎0,𝑎1,…} = (𝑥 − 𝑐)𝑂𝑝𝑠 𝑐{𝑎𝑛}. proof: observe that 𝑂𝑝𝑠𝑐{0,𝑎0,𝑎1,…} = 0 + ∑𝑎𝑛−1 ∞ 𝑛=1 (𝑥 − 𝑐)𝑛 = ∑𝑎𝑛(𝑥 − 𝑐) 𝑛+1 ∞ 𝑛=0 = (𝑥 − 𝑐)∑𝑎𝑛 ∞ 𝑛=0 (𝑥 − 𝑐)𝑛 = (𝑥 − 𝑐)𝑂𝑝𝑠𝑐{𝑎𝑛} hence by induction we can conclude as corollary below corollary 1. for each {𝑎𝑛} sequence, we have 𝑂𝑝𝑠𝑐 {0,0,0, . . ,0⏟ 𝑘−𝑒𝑛𝑡𝑟𝑖𝑒𝑠 𝑎0,𝑎1,…} = (𝑥 − 𝑐) 𝑘𝑂𝑝𝑠𝑐{𝑎𝑛}. proof: for n = 1 the statement is true due to theorem 7. lets assume for n = k, the statement is also true, i.e (17) holds. for n = k+1, we have 𝑂𝑝𝑠𝑐 { 0,0,0, . . ,0⏟ 𝑘+1−𝑒𝑛𝑡𝑟𝑖𝑒𝑠 𝑎0,𝑎1,…} = (𝑥 − 𝑐) 𝑂𝑝𝑠 𝑐 {0,0,0, . . ,0⏟ 𝑘−𝑒𝑛𝑡𝑟𝑖𝑒𝑠 𝑎0,𝑎1,…} = (𝑥 − 𝑐)((𝑥 − 𝑐)𝑘𝑂𝑝𝑠𝑐{𝑎𝑛}) = (𝑥 − 𝑐)𝑘+1𝑂𝑝𝑠𝑐{𝑎𝑛} theorem 8. for each {𝑎𝑛} sequence, we have (16) (17) (18) c-type ops transformation ahmad lazwardi 406 𝑂𝑝𝑠𝑐({𝑎𝑛})− 𝑎0 = (𝑥 − 𝑐)𝑂𝑝𝑠 𝑐{𝑎𝑛+1}. proof: observe that 𝑂𝑝𝑠𝑐({𝑎𝑛})− 𝑎0 = ∑𝑎𝑛(𝑥 − 𝑐) 𝑛 ∞ 𝑛=1 = ∑𝑎𝑛+1 ∞ 𝑛=0 (𝑥 − 𝑐)𝑛+1 = (𝑥 − 𝑐)∑𝑎𝑛+1 ∞ 𝑛=0 (𝑥 − 𝑐)𝑛 = (𝑥 − 𝑐)𝑂𝑝𝑠𝑐{𝑎𝑛+1} fortunately we also sucess to keep linearity properties of ops transformation[16]. theorem 9. for each {𝑎𝑛},{𝑏𝑛} sequences and any real numbers 𝛼, 𝛽, we have 𝑂𝑝𝑠𝑐(𝛼{𝑎𝑛} + 𝛽{𝑏𝑛}) = 𝛼𝑂𝑝𝑠 𝑐{𝑎𝑛}+ 𝛽𝑂𝑝𝑠 𝑐{𝑏𝑛}. proof: lets observe 𝑂𝑝𝑠𝑐(𝛼{𝑎𝑛}+ 𝛽{𝑏𝑛}) = ∑(𝛼𝑎𝑛 + 𝛽𝑏𝑛)(𝑥 − 𝑐) 𝑛 ∞ 𝑛=0 = 𝛼 ∑𝑎𝑛 ∞ 𝑛=0 (𝑥 −𝑐)𝑛 + 𝛽 ∑𝑎𝑛 ∞ 𝑛=0 (𝑥 − 𝑐)𝑛 = 𝛼𝑂𝑝𝑠𝑐{𝑎𝑛}+ 𝛽𝑂𝑝𝑠 𝑐{𝑏𝑛} although we success to prove linearity of ops transformation, but unfortunately that linearity of upper index, i.e 𝑂𝑝𝑠𝛼𝑐+𝛽𝑑{𝑎𝑛} ≠ 𝑂𝑝𝑠 𝛼𝑐{𝑎𝑛} +𝑂𝑝𝑠 𝛽𝑑{𝑏𝑛}. next we shall observe properties of c-type ops transformation for product of two power series. its still works similarily as previous result. theorem 10. for each {𝑎𝑛},{𝑏𝑛} sequences, we have 𝑂𝑝𝑠𝑐{𝑎𝑛}𝑂𝑝𝑠 𝑐{𝑏𝑛} = 𝑂𝑝𝑠 𝑐 {∑𝑎𝑘𝑏𝑛−𝑘 𝑛 𝑘=0 }. proof: let {𝑎𝑛},{𝑏𝑛} any two sequences, we have (19) (20) c-type ops transformation ahmad lazwardi 407 }{}{ n c n c bopsaops = ...))()(...)()()(( 2 210 2 210  cxbcxbbcxacxaa ......))()((...))((( 101100  cxbbcxacxbba ...)(.)()()( 2 11 2 20011000 cxbacxbacxbacxbaba  ...))(())(( 2 021120011000  cxbababacxbababa n n n k knk cxba )( 0 0                        n k knk c baops 0 from above theorem, we can conclude easily the following fact example 3. (𝑂𝑝𝑠𝑐{1})2 = 𝑂𝑝𝑠𝑐{𝑛 + 1}. translating the notation into sigma notation we get ∑(𝑛 +1)(𝑥 − 𝑐)𝑛 ∞ 𝑛=0 = ( 1 1 −(𝑥 − 𝑐) ) 2 from example 1 we can observe that how c-type ops transformations helps us to calculate, or manipulate some power series. here is another example example 4. (𝑂𝑝𝑠𝑐 { 𝟏 𝒏! }) 𝟐 = 𝑂𝑝𝑠𝒄 {∑ 𝟏 𝒌!(𝒏−𝒌)! 𝒏 𝒌=𝟎 }. therefore we concluce that ∑ ∑( 1 𝑘!(𝑛 − 𝑘)! )(𝑥 − 𝑐)𝑛 𝑛 𝑘=0 ∞ 𝑛=0 = 𝑒2𝑥−2𝑐 hence we can take another form of 𝑒𝑥 as following equation 𝑒𝑥 = (∑ ∑( 𝑒𝑐 𝑘!(𝑛 − 𝑘)! )(𝑥 − 𝑐)𝑛 𝑛 𝑘=0 ∞ 𝑛=0 ) 1 2 as for last discussion we shall observe how c-type ops transformation properties when we take its derivatives. we still use 𝐷𝑥𝑂𝑝𝑠 𝑐{𝑎𝑛} to notate the derivative of ops transformation on its radius convergence. consider the fact that the form 𝑥 − 𝑐 has the same derivative with x itself [17]. therefore we still able to adapt the formula for derivative of previous ops transformation as below c-type ops transformation ahmad lazwardi 408 theorem 11. for each {𝑎𝑛} sequence, we have 𝐷𝑥𝑂𝑝𝑠 𝑐({𝑎𝑛}) = 𝑂𝑝𝑠 𝑐({(𝑛 + 1)𝑎𝑛+1}). proof: observe that 𝐷𝑥𝑂𝑝𝑠 𝑐({𝑎𝑛}) = 𝑑 𝑑𝑥 ∑𝑎𝑛(𝑥 − 𝑐) 𝑛 ∞ 𝑛=0 = ∑𝑛𝑎𝑛(𝑥 −𝑐) 𝑛−1 ∞ 𝑛=1 = ∑(𝑛 + 1)𝑎𝑛+1(𝑥 − 𝑐) 𝑛 ∞ 𝑛=0 = 𝑂𝑝𝑠𝑐{(𝑛 + 1)𝑎𝑛+1} trivialy we also can conclude the next modification theorem theorem 12. for each {𝑎𝑛} sequence, we have (𝑥 − 𝑐)𝐷𝑥𝑂𝑝𝑠 𝑐({𝑎𝑛}) = 𝑂𝑝𝑠 𝑐{𝑛𝑎𝑛}. proof: observe that (𝑥 − 𝑐)𝐷𝑥𝑂𝑝𝑠 𝑐({𝑎𝑛}) })1{()( 1 n c anopscx      0 1 )()1()( n n n cxancx       0 1 1 )()1( n n n cxan     0 )( n n n cxna }{ n c naops from the last two theorems, we also can inductively conclude the following two corollaries. corollary 2. for each {𝑎𝑛} sequence, we have 𝐷𝑥 𝑘𝑂𝑝𝑠𝑐{𝑎𝑛} = 𝑂𝑝𝑠 𝑐 { (𝑛 + 𝑘)! 𝑛! 𝑎𝑛+𝑘}. where 𝐷𝑥 𝑘 is kth-derivative of ops transformation [15]. corollary 3. for each {𝑎𝑛} sequence, we have 𝑂𝑝𝑠𝑐{𝑛𝑘𝑎𝑛} = (𝑥 − 𝑐)𝐷𝑥((𝑥 − 𝑐)𝐷𝑥(…))(𝑥 −𝑐)𝐷𝑥𝑂𝑝𝑠 𝑐{𝑎𝑛}.⏟ 𝑘−𝑡𝑖𝑚𝑒𝑠 at the end of this research, we will try our transformation to solve some ordinary differential equation with variable coefficient. this solution must be exist due to [18] for example the equation (21) (22) (23) (24) c-type ops transformation ahmad lazwardi 409 02')1(2''  yyxy (25) we will find the solution of (25) near c = 1 as following: step 1: assume the solution has the form     0 )1( n n n xcy . step 2: transform (25) to ops transformation equation. }0{}{2}{)1(2}{ 11112 opscopscopsdxcopsd nnxnx  step 3: solve the equation }{2}{)1(2}{ 2 nnxn c x copscopsdxcopsd  }2{})1)(2{( 1 2 1 nn ncopscnnops   }2{ 1 n cops }22)1)(2{( 2 1 nnn cnccnnops   }0{ 1 ops step 4: remove ops transformation from the equation, we have (𝑛 + 2)(𝑛 + 1)𝐶𝑛+2 = (2𝑛 − 2)𝐶𝑛 for n = 0,1,2, ... step 5: by evaluating n one by one, we have the solution 𝑦 = 𝐶0 (1 − (𝑥 −1) 2 − 1 6 (𝑥 −1)4 + ⋯) +𝐶1(𝑥 −1) conclusions based on the discussion above, it can be concluded that c-type ops transformation is a generalization of the ordinary ops transformation with additional c as the upper index. all properties of ordinary ops transformations can still apply in c-type ops transformations. the c-type ops transformation can also be applied to solve ordinary differential equations for variable coefficients. references [1] h. ji he and f. yu ji, “taylor series solution for lane-emden equation,” j. math. chem., vol. 57, no. 1, pp. 1932–1934, 2019. 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[18] s. falkensteiner, y. zhang, and n. thieou, “on existence and uniqueness of formal power series solutions of algebraic ordinary differential equations,” mediterranian j. math., vol. 19, no. 2, pp. 95–114, 2022, doi: 10.3389/fphy.2021.795693. actuarial modeling of covid-19 insurance cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 362-369 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 04, 2022 reviewed: july 23, 2022 accepted: august 20, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.14999 actuarial modeling of covid-19 insurance mila kurniawaty*, maulana muhamad arifin, bagus kurniawan, sadam laksamana sukarno, muhammad teguh prayoga department of mathematics, faculty of mathematics and natural sciences, universitas brawijaya, malang, indonesia email: mila_n12@ub.ac.id abstract the coronavirus disease (covid-19) has spread to almost all countries in the world causing economic and financial crisis. many researchers are interested in studying infectious diseases especially in dynamical models of covid-19. peng et al in 2020 studied the generalized seir (susceptible-exposed-infected-recovered) of covid-19. we interested to develop their results to make financial arrangement. in this article, we provide an actuarial model of the covid-19 insurance based on the generalized seir model. we construct the dynamical models of premium and benefit based on generalized seir. based on its dynamical model, we formulate the premium and the premium reserves on hospitalization and death benefits of the covid-19 insurance by using equivalence principle. this actuarial model is expected to able to help financial arrangements to cover losses due to the outbreak of covid-19. keywords: premium; premium reserves; generalized seir; hospitalization benefit; death benefit introduction the novel coronavirus-caused pneumonia 2019 (covid-19) previously called 2019-ncov or sars-cov-2 (severe acute respiratory syndrome coronavirus 2) first appeared in wuhan in december 2019 and then spread rapidly throughout china [1]. based on data on woldometer (2021), this covid-19 case has spread to almost all countries in the world [2]. this condition had a huge impact on the world economy, financial institutions in crisis [3]. arfah et al. [4] studied a new strategy to solve the problem of the global financial crisis in the sharia aspect. from a financial point of view, a well-designed health care system that can reduce the financial impact of sudden outbreaks of a pandemic, such as soaring medical costs, hospital infrastructure, medical equipment, vaccination and quarantine. then the insurance program is expected to cover financial losses arising from disruptions in operation of regular businesses. by applying mathematical and actuarial techniques to model and measure financial risk, actuaries are expected to expand their expertise and tackle epidemics in the health care system. the mathematical modeling has been widely developed and analyzed as a consideration to determine the insurance premium (see [5], [6], [7]). feng and garrido [8] used the epidemic model to make financial arrangements. they used the sir (susceptible-infected-removed) model to study the infectious diseases. the class s http://dx.doi.org/10.18860/ca.v7i3.14999 mailto:mila_n12@ub.ac.id* actuarial modeling of covid-19 insurance mila kurniawaty 363 denoted a group of susceptible individuals to the certain diseases or virus. the class i denoted a group of individuals who are infected and capable of transmitting the disease. the individuals were excluded from the epidemic due to death or recovery through medical treatment are classified in class r. the dynamic model compartments as in [8] are given in figure 1 below. figure 1. dynamical model of the sir premium and benefit payment [8] the outbreak of covid-19 has attracted researchers’ interest in studying infectious diseases. there are several research which studied the sir model of the covid-19 epidemic (see [9] and [10]). however, in the development of the case, there is another factor influencing the spread of disease in covid-19 cases, namely exposed individuals as in [11] and [12]. peng et al. [13] and aldila et al. [14] added some classes influencing the covid-19 epidemic model. to characterize the outbreak of covid-19 in wuhan, peng et al. [13] generalized the classical seir (susceptible-exposed-infected-removed) model by introducing seven classes, that is {𝑆(𝑡), 𝑃(𝑡), 𝐸(𝑡), 𝐼(𝑡), 𝑄(𝑡), 𝑅(𝑡), 𝐷(𝑡)} which represent the number of the susceptible cases, insusceptible cases, exposed cases, infective cases, quarantined cases, recovered cases and death case, respectively, at time 𝑡. the epidemic model for covid-19 of [13] is given in figure 2. figure 2. dynamical model of the generalized seir for covid-19 [13] the total population is assumed constant, that is the summation of all classes, and the coefficient 𝛼, 𝛽, 𝛾 −1, 𝜃−1, 𝜆(𝑡), 𝜅(𝑡) represent the respective protection rate, infection rate, average latent time, average quarantine time, cure rate, and mortality rate. in this paper, the dynamical model of peng et al. [13] will be generalized to determine actuarial calculation of covid-19 insurance. in particular, our result improves the previous work due to feng and garrido [8]. the first one we construct the dynamical model of premium and benefit payment, and then we use the classical actuarial calculation actuarial modeling of covid-19 insurance mila kurniawaty 364 to determine actual present value of benefit payment and premium payment, and also the premium reserves (see [15]-[20]). methods in this research, we develop the research methods into some steps. the first one, the figure 2 is modified into actuarial concept, by adding the premium payment and benefit payment. the premium payment must be done by the population in class 𝑆, 𝐸, 𝐼, 𝑅, and 𝑃. the population in class 𝑄 have to get the hospitalization benefit, whereas the population in class 𝐷 have to get the death benefit. from the new figure will be construct the ordinary differential equation of dynamical model. based on the dynamical model will be constructed the premium rate and the premium reserve. the equivalence principle will be used to construct it. results and discussion dynamical model of premium and benefits in this section, the compartment model in peng et al. [13] will be generalized to dynamical model of premium payments for the covid-19 policyholders and benefit payments by insurance companies. the compartments of the dynamical model are given in figure 3. figure 3. the dynamical model of premium dan benefit payments on generalized seir in this case the policyholder is assumed to be out of insurance after recovery. by [13], the compartment of the generalized seir model is denoted by following system of ordinary differential equations: infective (i) 𝛾 𝛽 premium payment 𝜆(𝑡) 𝜅(𝑡) premium payment 𝛼 recovered (r) susceptible (s) exposed (e) insusceptible (p) quarantined (q) insurance death (d) 𝜃 premium payment hospitalization benefit premium payment death benefit actuarial modeling of covid-19 insurance mila kurniawaty 365 𝑑𝑆(𝑡) 𝑑𝑡 = −𝛼𝑆(𝑡) − 𝛽 𝑆(𝑡)𝐼(𝑡) 𝑁 (1) 𝑑𝐸(𝑡) 𝑑𝑡 = −𝛾𝐸(𝑡) + 𝛽 𝑆(𝑡)𝐼(𝑡) 𝑁 (2) 𝑑𝐼(𝑡) 𝑑𝑡 = 𝛾𝐸(𝑡) − 𝜃𝐼(𝑡) (3) 𝑑𝑄(𝑡) 𝑑𝑡 = 𝜃𝐼(𝑡) − 𝜆(𝑡)𝑄(𝑡) − 𝜅(𝑡)𝑄(𝑡) (4) 𝑑𝑅(𝑡) 𝑑𝑡 = 𝜆(𝑡)𝑄(𝑡) (5) 𝑑𝐷(𝑡) 𝑑𝑡 = 𝜅(𝑡)𝑄(𝑡) (6) 𝑑𝑃(𝑡) 𝑑𝑡 = 𝛼𝑆(𝑡) (7) with given initial value 𝑆(0) = 𝑆0, 𝐸(0) = 𝐸0, 𝐼(0) = 𝐼0, 𝑄(0) = 𝑄0, 𝑅(0) = 𝑅0, 𝐷(0) = 𝐷0, 𝑃(0) = 𝑃0, and 𝑆0 + 𝐸0 + 𝐼0 + 𝑄0 + 𝑅0 + 𝐷0 + 𝑃0 = 𝑁. in actuarial approach, the probability of each class is defined by rasio of each class to the total population, then we now introduce the deterministic functions 𝑠(𝑡), 𝑒(𝑡), 𝑖(𝑡), 𝑞(𝑡), 𝑟(𝑡), 𝑑(𝑡), and 𝑝(𝑡), represented as the fractions of the population in each of class 𝑆, 𝐸, 𝐼, 𝑄, 𝑅, 𝐷, and 𝑃, respectively. by dividing equations (1)-(7) by the constant total population size 𝑁, we have 𝑠′(𝑡) = −𝛼𝑠(𝑡) − 𝛽𝑠(𝑡)𝑖(𝑡), 𝑡 ≥ 0 (8) 𝑒′(𝑡) = −𝛾𝑒(𝑡) + 𝛽𝑠(𝑡)𝑖(𝑡), 𝑡 ≥ 0 (9) 𝑖′(𝑡) = 𝛾𝑒(𝑡) − 𝜃𝑖(𝑡), 𝑡 ≥ 0 (10) 𝑞′(𝑡) = 𝜃𝑖(𝑡) − 𝜆(𝑡)𝑞(𝑡) − 𝜅(𝑡)𝑞(𝑡), 𝑡 ≥ 0 (11) 𝑟′(𝑡) = 𝜆(𝑡)𝑞(𝑡), 𝑡 ≥ 0 (12) 𝑑′(𝑡) = 𝜅(𝑡)𝑞(𝑡), 𝑡 ≥ 0 (13) 𝑝′(𝑡) = 𝛼𝑠(𝑡), 𝑡 ≥ 0 (14) 𝑠(𝑡) + 𝑒(𝑡) + 𝑖(𝑡) + 𝑞(𝑡) + 𝑟(𝑡) + 𝑑(𝑡) + 𝑝(𝑡) = 1, 𝑡 ≥ 0 (15) with initial given value 𝑠(0) = 𝑠0, 𝑒(0) = 𝑒0, 𝑖(0) = 𝑖0, 𝑞(0) = 𝑞0, 𝑟(0) = 𝑟0, 𝑑(0) = 𝑑0, 𝑝(0) = 𝑝0, dan 𝑠0 + 𝑒0 + 𝑖0 + 𝑝0 = 1. premium and benefit payments we assume that the premium payment of an infectious disease insurance plan in the form of continuous annuities from the susceptibles, insuspectible, infected, and exposed. it means the policyholder are commited to pay the premiums continuously as long as they remain in susceptibles, insuspectible, infected, and exposed classes. otherwise, the insurance company will give the benefit if the policyholder are quarantined and death. once the individual dies or recovery after quarantined process in hospital, the plan terminates immediately. by using the principles of international actuarial notation as mention in [8], the actuarial present value (apv) of each class for a 𝑡-year period is denoted by �̅��̅�| 𝑠 , �̅��̅�| 𝑒 , �̅��̅�| 𝑖 , �̅� �̅�| 𝑞 , �̅��̅�| 𝑟 , �̅��̅�| 𝑑 , and �̅� �̅�| 𝑝 . to evaluate the annuity, we use the present value of payments due at time 𝑡, which is the discounted value of one monetary unit for a basic annuity. then it is multiplied by the probability of making those payments and then integrate these apv for all payment times 𝑡. the detailed evaluations of annuities can be found in ([17] and 18]). actuarial modeling of covid-19 insurance mila kurniawaty 366 by figure 3, there are 2 benefits, i.e, hospitalization benefit and death benefit. the hospitalization benefit will be given to quarantined individual and death benefit will be given to death individual. the medical and hospitalization expenses are continuously reimbursed for each quarantined policyholder during the whole period of treatment. hence, the total discounted value of a 𝑡-year annuity of hospitalization payments is can be descibed as follows �̅� �̅�| 𝑞 = ∫ 𝑒−𝛿𝑥 𝑡 0 . 𝑞(𝑥) 𝑑𝑥, 𝛿 > 0 (16) where 𝛿 is the discounting force of interest. when the policyholder is diagnosed with the infectious disease and hospitalized immediately, the medical expenses are to be paid immediately in a lump sum. since its obligation is fulfilled, the insurance plan terminates. in actuarial mathematics, the payment of a lump sum compensation can be analogized as whole life insurance. the apv of hospitalization benefit with lumpsum payment, denoted by �̅� �̅�| 𝑞 , is defined by �̅� �̅�| 𝑞 = ∫ 𝑒−𝛿𝑥 𝑡 0 . 𝜃𝑖(𝑥) 𝑑𝑥 = 𝜃�̅��̅�| 𝑖 (17) since 𝜃𝑖(𝑡) denotes the probability of being newly quarantined at time 𝑡. by the same concept of lumpsum payment of hospitalization benefit, the apv of death benefit is given by �̅��̅�| 𝑑 = ∫ 𝑒−𝛿𝑥 𝑡 0 . 𝜅(𝑥)𝑞(𝑥) 𝑑𝑥 (18) since the probability of being newly death at time 𝑡 is 𝜅(𝑡)𝑞(𝑡). as in the previous section, there are 4 classes must pay the premium, then the total discounted value of a 𝑡-year annuity premium of payments is given by �̅��̅�| 𝑠 + �̅�𝑡| 𝑒 + �̅�𝑡̅| 𝑖 + �̅� �̅�| 𝑝 = ∫ 𝑒−𝛿𝑥 𝑡 0 . (𝑠(𝑥) + 𝑒(𝑥) + 𝑖(𝑥) + 𝑝(𝑥)) 𝑑𝑥 (19) in this section, the compartment model in peng et al. [13] is generalized to dynamical model of premium payments for the covid-19 policyholders and benefit payments by insurance companies. premium rate and premium reserves the policy shall be analized with an infinite term for mathematical convenience. the premium based on an infinite term can be used to estimate the cost of insurance for relatively long policy. then the equation in the previous section must be applied for 𝑡 tend to infinity. proposition 1 in the generalized seir model (8)-(11), and (14), the following inequality holds �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅�∞̅| 𝑝 + �̅�∞̅| 𝑞 = 1 𝛿 (1 − ∫ 𝑒−𝛿𝑥 (𝜆(𝑥) + 𝜅(𝑥))𝑞(𝑥)𝑑𝑥 ∞ 0 ) (20) our study is based on one of three principles in [17] and almost used in ([15] [20]), i.e., the equivalence principle for determination of level premium is given by 𝐸[present value of benefits] = 𝐸[present value of benefit premium] (21) actuarial modeling of covid-19 insurance mila kurniawaty 367 therefore, by using the equations (16), (19), and equivalence principle with an infinite term, the level premium for a unit annuity claim payment plan of hospitalization benefit is given by �̅�(�̅�∞̅| 𝑞 ) = �̅�∞̅| 𝑞 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 by equation (20) we then have the following �̅�(�̅�∞̅| 𝑞 ) = �̅�∞̅| 𝑞 1 𝛿 (1 − ∫ 𝑒−𝛿𝑥 (𝜆(𝑥) + 𝜅(𝑥))𝑞(𝑥)𝑑𝑥 ∞ 0 ) − �̅� ∞̅| 𝑞 the net level premium of hospitalization benefit with lumpsum payment for the infinite term insurance plan is denoted by �̅�(�̅�∞̅| 𝑞 ). by equations (17), and (19)-(21) for infinite term we then have �̅�(�̅�∞̅| 𝑞 ) = �̅�∞̅| 𝑞 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 = 𝜃�̅�∞̅| 𝑖 1 𝛿 (1 − ∫ 𝑒−𝛿𝑥 (𝜆(𝑥) + 𝜅(𝑥))𝑞(𝑥)𝑑𝑥 ∞ 0 ) − �̅� ∞̅| 𝑞 in fact, the covid-19 insurance is a combination of the hospitalization and death insurances due to covid-19. then the net level premium of death benefit and hospitalization claim for the infinite term insurance plan is denoted by �̅�(�̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 ) = �̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 = �̅�∞̅| 𝑞 + ∫ 𝑒−𝛿𝑥 ∞ 0 . 𝜅(𝑥)𝑞(𝑥) 𝑑𝑥 1 𝛿 (1 − ∫ 𝑒−𝛿𝑥 (𝜆(𝑥) + 𝜅(𝑥))𝑞(𝑥)𝑑𝑥 ∞ 0 ) − �̅� ∞̅| 𝑞 meanwhile, the net level premium of death benefit and hospitalization claim with lumpsum payment for the plan of an infinite term insurance is given by �̅�(�̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 ) = �̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 = 𝜃�̅�∞̅| 𝑖 + + ∫ 𝑒−𝛿𝑥 ∞ 0 . 𝜅(𝑥)𝑞(𝑥) 𝑑𝑥 1 𝛿 (1 − ∫ 𝑒−𝛿𝑥 (𝜆(𝑥) + 𝜅(𝑥))𝑞(𝑥)𝑑𝑥 ∞ 0 ) − �̅� ∞̅| 𝑞 (22) we consider the net level premium, the total premium and the total benefit in order to obtained the premium reserves. in actuarial sciences, the premium reserve is very important to determine the ability of the insurance company to pay the claim of policyholders. there are some methods to determine the premium reserves. one of them is retrospective method, the detail of this method can be found in [19]. by ordinary differential equations in [4], where �̅�(𝑡) denotes accumulated premium reserves at time 𝑡 with lumpsum payment, we thus have �̅�′(𝑡) = �̅�(�̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 )(𝑠(𝑡) + 𝑒(𝑡) + 𝑝(𝑡) + 𝑖(𝑡)) − (𝜃𝑖(𝑡) + 𝜅(𝑡)𝑞(𝑡)) + 𝛿�̅�(𝑡) = ( �̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 ) (𝑠(𝑡) + 𝑒(𝑡) + 𝑝(𝑡) + 𝑖(𝑡)) − (𝜃𝑖(𝑡) + 𝜅(𝑡)𝑞(𝑡)) + 𝛿�̅�(𝑡) let us define 𝑓(𝑡) = ( �̅�∞̅̅̅| 𝑞 +�̅�∞̅̅̅| 𝑑 �̅�∞̅̅̅| 𝑠 +�̅�∞̅̅̅| 𝑒 +�̅�∞̅̅̅| 𝑖 +�̅� ∞̅̅̅| 𝑝 ) (𝑠(𝑡) + 𝑒(𝑡) + 𝑝(𝑡) + 𝑖(𝑡)) − (𝜃𝑖(𝑡) + 𝜅(𝑡)𝑞(𝑡)) we thus have �̅�′(𝑡) − 𝛿�̅�(𝑡) = 𝑓(𝑡) by multiplying both sides by 𝑒−𝛿𝑡 , yields �̅�(𝑡) = (∫ 𝑒−𝛿𝑥 𝑓(𝑥) 𝑡 0 𝑑𝑥) . 𝑒𝛿𝑡 + �̅�(0). 𝑒𝛿𝑡 or equivalently, actuarial modeling of covid-19 insurance mila kurniawaty 368 �̅�(𝑡) = (∫ 𝑒−𝛿𝑥 ( �̅�∞̅| 𝑞 + �̅�∞̅| 𝑑 �̅�∞̅| 𝑠 + �̅�∞̅| 𝑒 + �̅�∞̅| 𝑖 + �̅� ∞̅| 𝑝 ) (𝑠(𝑥) + 𝑒(𝑥) + 𝑝(𝑥) + 𝑖(𝑥)) 𝑡 0 − (𝜃𝑖(𝑥) + 𝜅(𝑥)𝑞(𝑥)) 𝑑𝑥) . 𝑒𝛿𝑡 + �̅�(0). 𝑒𝛿𝑡 (23) from equation (23), the amount of the premium reserve at time 𝑡 depends on the total premium, the death benefit and quarantined benefit, and also the initial value of the premium reserve. conclusions the covid-19 insurance by considering the generalized seir model was assumed that the recovery individuals not involved in premium payment since the policyholder who has been quarantined in the hospital and claims the benefit payment then the insurance plan terminates. therefore, the total premium of payment are depend on class 𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), and 𝑃(𝑡). meanwhile, the total benefit of payment are depend on class 𝑄(𝑡) and 𝐷(𝑡). by using equivalence principle, the net level premium is the ratio of actuarial present value of benefits to actuarial present value of premiums. hence we get the premium reserve by restrocpective approach. acknowledgments this research is supported by the doctoral grant no. 1631/un10.f09/pn/2021 at mathematics and natural sciences faculty, universitas brawijaya. references [1] c. huang, y. wang, x. li, l. ren, j. zhao, y. hu, li. zhang, g. fan, j. xu, x. gu, z. cheng, t. yu, j. xia, y. wei, w. wu, x. xie, w. yin, h. li, m. liu, y. xiao, h. gao, l. guo, j. xie, g. wang, r. jiang, z. gao, q. jin, j. wang, and b. cao. clinical features of patients infected 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[20] i. catarya. buku materi pokok asuransi ii. karunika universitas terbuka. jakarta, 1988. spatial autoregressive to model tuberculosis cases in central java province in 2019 cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 240-248 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 29, 2021 reviewed: december 09, 2021 accepted: december 14, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.13451 spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua1, i gede nyoman mindra jaya2* 1 post-graduate program in applied statistics, faculty of mathematics and natural sciences, universitas padjadjaran, indonesia 2 department of statistics, faculty of mathematics and natural sciences, universitas padjadjaran, indonesia *corresponding author email: mindra@unpad.ac.id*, hasrat20001@mail.unpad.ac.id abstract tuberculosis is an infectious disease caused by the bacterium mycobacterium tuberculosis. central java is one of the three provinces with the highest tuberculosis cases in indonesia. some of the risk factors used in this research are the spatial lag of the number of tuberculosis cases representing the agent component, the morbidity rate representing the host component, population density, proper sanitation, and proper drinking water which represent environmental components. this study aims to model the tuberculosis cases in central java province using the spatial autoregressive (sar) model. the sar model is a regression model where the response variable has a spatial correlation. the estimation method usually used in sar model is maximum likelihood. the moran's i on the number of tuberculosis cases in central java shows a positive spatial autocorrelation. the model was chosen based on the lm test and aic. the best model is the sar model. the results show that the greater the number of tuberculosis cases is influenced by the number of tuberculosis cases in the neighbouring areas. proper sanitation has a negative effect, on the contrary, the dense population has a positive effect on the number of tuberculosis cases in the province of central java. keywords: maximum likelihood; sar; spatial; tuberculosis introduction tuberculosis is an infectious disease caused by infection with the bacterium mycobacterium tuberculosis [1]. tuberculosis can be transmitted from human to human through the splash of the saliva of tuberculosis sufferers which spreads into the air when coughing or sneezing. single cough or sneeze can produce up to 3000 splashes of saliva [2]. an infection occurs when other people breathe air containing these droplets. this disease usually affects the lungs, but can also affect other parts of the body. according to who [3], in 2019 around 10 million people were infected with tuberculosis and 11,4 million of them died. in 2016, 45 percent of the estimated tuberculosis cases were in southeast asia http://dx.doi.org/10.18860/ca.v7i1.13451 mailto:mindra@unpad.ac.id mailto:email1@gmail.com spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 241 and indonesia is one of them. indonesia is the third country with the highest tuberculosis cases in the world after china and india with an estimated 845 thousand cases. during tuberculosis endemic in indonesia, the spread of tuberculosis is still very high. central java is one of the three provinces with the highest tuberculosis cases in indonesia, with 73,171 cases in 2019 [4]. tuberculosis prevention and control efforts have been carried out such as the bacille calmette guerin (bcg) vaccine in infants [5], increasing the number of case finding and treatment success in healthcare facilities. the biggest challenge in controlling tuberculosis is that there are many missing cases (unreported) which will further increase the transmission process due to patient ignorance. from an epidemiological perspective, the incidence of tuberculosis is an interaction between three components, namely the agent, host, and environment [1]. from the agent's side, intensive interactions between patients and other people can facilitate the transmission. the duration of contact time or the intensity of contact with people with tuberculosis can cause a person to be more easily exposed [6]. the host side (i.e., a person's susceptibility to tuberculosis) is strongly influenced by his body's resistance. as stated by pangaribuan et al [7] the factors that are related from the host side are age, gender, race, socioeconomic, living habits, marital status, occupation, heredity, nutrition, and immunity. in terms of the environment, arinil, et al [8] stated that environmental factors such as the physical environment of the house (occupancy density, ventilation, sanitation) and weather climate (temperature, humidity) were closely related to tuberculosis. prevention of tuberculosis transmission certainly requires control of these three components. identifying area with a high risk of tuberculosis transmission is important to know to see the inter-regional linkages. analytical tool for spatial data to model the number of cases of tuberculosis that occur is needed. in spatial epidemiology, the use of maps as a visualization method is needed to see the distribution of disease by geographic area [9]. several risk factors used in this research are the spatial lag of the number of tuberculosis cases representing the agent component, the morbidity rate representing the host component, population density, proper sanitation, and proper drinking water representing the environmental component. one of the spatial models that can be used is the spatial autoregressive (sar) model. the sar model is a regression model with a spatial correlation in the response variable. using cross-section data, this model combines a linear regression model with a spatial lag of the response variable [10]. the use of the sar model in infectious diseases, especially tuberculosis in central java, is due to agent factors that have high mobility from district to other districts, besides that there are other factors, namely host and environment that can be used as covariates. the estimation method usually used in the sar model is maximum likelihood. therefore, the purpose of this study was to model the sar of the number of tuberculosis cases in central java province using the maximum likelihood approach. in this case, we will use a standardized weighting matrix using the queen contiguity. methods data and variables this study used data from the publication of the health profile of the central java province [4] and the central bureau of statistics of the central java province [11]. the units of analysis are 35 districts/cities in central java. the variables used in this study are shown in table 1: spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 242 table 1. variables and data sources notation variable source 𝑦 number of confirmed cases of tuberculosis health profile of the central java province 𝑥1 proper sanitation 𝑥2 eligible drinking water facilities 𝑥3 population density central bureau of statistics of the central java province 𝑥4 morbidity rate moran’s i moran's i is the value of the test statistic used to determine whether there is spatial autocorrelation or spatial dependence in the data. moran's i has global and local measures. moran's i values range from -1 and 1. the global measure is used to measure the overall autocorrelation and the local is used to identify the autocorrelation on each unit. global moran's i can be seen in the following formula [12]: 𝐼 = 𝑛 ∑ ∑ 𝑤𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 ∑ ∑ (𝑦𝑖 − �̅�)(𝑦𝑗 − �̅�) 𝑛 𝑗=1 𝑛 𝑖=1 ∑ (𝑦𝑖 −�̅�) 2𝑛 𝑖=1 (1) with, n : number of spatial units �̅� : mean of n locations 𝑦𝑖 : observation variable at location i 𝑦𝑗 : observation variable at location j 𝑤𝑖𝑗 : elements of the spatial weight matrix w and the local moran's i can be seen in the following formula: 𝐼𝑖 = (𝑦𝑖 − �̅�) ∑ (𝑦𝑘 − �̅�) 2/𝑛𝑛𝑘=1 ∑ (𝑦𝑗 − �̅�) 𝑛 𝑗=1 (2) the null hypothesis for autocorrelation is 𝐼 = 𝐸(𝐼) no spatial dependence. the formula of test statistics can be written as follows: 𝑍(𝐼) = 𝐼 − 𝐸(𝐼) √𝑉𝐴𝑅(𝐼) ~𝑁(0,1) (3) with, 𝐸(𝐼) = − 1 𝑛 − 1 𝑉𝐴𝑅(𝐼) = 𝑛2𝑆1 − 𝑛𝑆2 + 3𝑆0 2 (𝑛2 − 1)𝑆0 2 − [𝐸(𝐼)]2 𝑆0 = ∑ ∑ 𝑤𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 ; 𝑆1 = 1 2 ∑ ∑ (𝑤𝑖𝑗 + 𝑤𝑗𝑖) 2𝑛 𝑗=1 𝑛 𝑖=1 ; 𝑆2 = ∑ (∑ 𝑤𝑖𝑗 𝑛 𝑗 𝑛 𝑖=1 + ∑ 𝑤𝑗𝑖 𝑛 𝑗 ) 2. lagrange multiplier (lm) test the lagrange multiplier (lm) test is used to determine whether there is a spatial dependence are not. there are two types of lm tests that have been developed, namely the spatial dependence of the dependent variable and the spatial error dependence. lm test statistics on the spatial dependence (𝐿𝑀𝐿𝐴𝐺) of the dependent variable are as follows [13]: 𝐿𝑀𝐿𝐴𝐺 = [(𝑒𝑇𝑊𝐴𝑦)/(𝑒 𝑇𝑒/𝑛)]2 [(𝑊𝐴𝑋�̂�) 2 𝑀(𝑊𝐴𝑋�̂�)/(𝑒 𝑇𝑒/𝑛)] + [𝑡𝑟(𝑊𝐴 𝑇𝑊𝐴 + 𝑊𝐴 2)] ~𝜒(1−𝛼);𝑑𝑓=1 2 (4) if the test statistic value is greater than the chi-square value (reject h0), then the model spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 243 made is the spatial autoregressive (sar) model. lm test statistics for the dependence of spatial error (𝐿𝑀𝐸𝑅𝑅) can be seen in the following formula: 𝐿𝑀𝐸𝑅𝑅 = [(𝑒𝑇𝑊𝐴𝑒)/(𝑒 𝑇𝑒/𝑛)]2 𝑡𝑟(𝑊𝐴 𝑇𝑊𝐴 + 𝑊𝐴 2) ~𝜒(1−𝛼);𝑑𝑓=1 2 (5) if the test statistic value is greater than the chi-square value (reject h0), then the model made is the spatial error model (sem). meanwhile, if the 𝐿𝑀𝐿𝐴𝐺 and 𝐿𝑀𝐸𝑅𝑅 values are both significant, the best model can be chosen by comparing the akaike information criterion (aic). model with the smaller aic value is the best model. spatial autoregressive (sar) model the sar model is a combination of a linear regression model with a spatial lag of the response variable using cross-section data. in general, the sar model can be written as follows [14]: 𝐲 = ρ𝐖𝐲 + 𝐗𝛃+ 𝛆; 𝛆~mvn(0,σ𝜀 2in) (6) with: 𝐲 : continuous response variable ρ : autoregressive coefficient 𝐖 : spatial weight matrix 𝛃 : intercept and regression coefficient 𝐗 : predictor variable 𝛆 : error this model assumes that the autoregressive process is only found in the response variable. maximum likelihood is one of the most commonly used estimators because it can provide the best linear unbiased estimation (blue) and overcome endogeneity in the sar model. the estimated parameters using the maximum likelihood method are as follows: �̂�𝑀𝐿 = (𝑿 𝑻𝑿)−1𝑿𝑻𝒚⏟ �̂�𝑂𝐿𝑆 − 𝜌(𝑿𝑻𝑿)−1𝑿𝑻𝑾𝝆𝒚⏟ �̂�𝐿 = �̂�𝑂𝐿𝑆 − 𝜌�̂�𝐿 (7) where �̂�𝐿 is an estimator of regression parameters that depends on the spatial autocorrelation ρ and the weight matrix (w). however, this form cannot be solved directly because the value of ρ is unknown. to be able to estimate the regression parameters, it can be done using a concentrated log-likelihood function (𝐿𝑐) which is a function of the mle residual which is defined as follows: ln𝐿𝑐(𝜌) = 𝐶 − 𝑛 2 ln[ 1 𝑛 (𝒆𝟎 − 𝜌𝒆𝑳) 𝑇(𝒆𝟎 − 𝜌𝒆𝑳)] + ln |𝑰 −𝜌𝑾𝜌| (8) the formula cannot be solved analytically so a numerical method is needed to find the estimated ρ parameter of the equation. results and discussion the case notification rate (cnr) of tuberculosis in central java province is 211, which means that there are 211 cases of tuberculosis being treated and reported among 100,000 residents in central java. judging from the number of cases of tuberculosis in central java there were as many as 73,171 cases during 2019. tuberculosis cases are a disease that is spread throughout the district in central java province. the lowest number of cases was in karanganyar regency with 514 cases and cilacap regency with the highest number of 4,703 cases. however, when viewed from the district/city cnr, spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 244 the highest cnr is tegal city at 832.5 per 100,000 population and the lowest cnr is temanggung regency at 45.72 per 100,000 [4]. the map of the distribution of tuberculosis cases in central java can be seen in figure 1. figure 1. map of tuberculosis case quantile in central java province in 2019 figure 1 shows that areas with a large number of cases (in solid red) tend to be close to areas with a large number of cases. areas with a small number of cases (in faded red) also tend to be adjacent to areas with a small number of cases. this indicates that there is a spatial dependence between regions. before conducting sar modeling, it is necessary to test the classical assumptions of multiple linear regression and moran's i tests. table 2. the statistic test result of classic assumptions and moran’s i statistic test p-value normality (shapiro-wilk) 0.9821 nonautocorrelation (durbin-watson) 0.4893 nonmulticolinierity (vif) 1.138817(x1); 1.049796(x2); 1.065710(x3); 1.074787(x4) homoscedasticity (breusch-pagan) 0.9749 moran’s i 0.000 (i= 0.4991) table 2 shows that all assumptions in multiple linear regression have been fulfilled. the results of the moran's i value using a spatial weight matrix based on queen contiguity on the number of tuberculosis cases in the province of central java is 0.499 with a p-value <0.05 (reject h0) which means there is a positive spatial autocorrelation. high tuberculosis cases areas will be surrounded by high areas as well, and low tuberculosis cases areas will be surrounded by low areas as well. for local moran's i values will produce different values for each location. there are several significant areas in local moran's i which is divided into two parts. first, the high-high areas include cilacap, banyumas, brebes, tegal, puralingga, and tegal city. this indicates that districts/cities in high-high areas have a high number of tuberculosis cases and the surrounding areas are also high, which is possible due to the transmission spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 245 of tuberculosis to the surrounding area. second, in the low-low area, there are magelang, boyolali, sragen, and karanganyar. this shows that in the low-low area the number of tuberculosis cases is low and the surrounding area is also low. for other regions, the local moran’s i is not significant. moran's i local results can be seen in figure 2: figure 2. local moran's i of tuberculosis case in central java province in 2019 the selection of the spatial model was carried out through the lagrange multiplier (lm) test as an initial identification. lm test is used to determine the spatial dependence more specifically whether the dependency on a response variable (lag), dependency on other variables that are not studied (error), or both (lag and error). the results of the lm test carried out can be seen in table 3. table 3. lm test results model lm-test value p-value aic sar 𝐿𝑀𝐿𝐴𝐺 12,635 0.000378 52.72579 sem 𝐿𝑀𝐸𝑅𝑅 8,973 0.002739 53.10059 from table 2 it can be seen that the spatial dependence in lag and error is significant because the p-value is smaller than alpha (0.05). the sar model will be applied in this study due to the aic value is smaller than the sem model. the sar model is a spatial regression model that involves spatial lag in the response variable. the estimation results of the sar model using the maximum likelihood method and using a spatial weight matrix based on queen contiguity can be seen in table 4. table 4. sar parameter estimation results estimate std. error z-value p-value (intercept) 3.383100 1.346100 2.51318 0.01196 proper sanitation (𝑥1) -0.010300 0.005600 -1.85760 0.06322 eligible drinking water facilities (𝑥2) 0.006170 0.005800 1.06111 0.28864 population density (𝑥3) 0.000094 0.000029 3.20521 0.00134 morbidity rate (𝑥4) -0.003020 0.020255 -0.14910 0.88147 spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 246 estimate std. error z-value p-value rho: 0.58787, lr test value: 11.396, p-value: 0.000 asymptotic standard error: 0.14246 z-value: 4.1267, p-value: 0.000 wald statistic: 17.03, p-value: 0.000 aic: 52.726 the spatial lag variable (rho) has a positive and significant coefficient in influencing the number of tuberculosis cases in central java province. this means that the greater the number of tuberculosis cases is influenced by a large number of tuberculosis cases in the surrounding area. this is in accordance with the research of mindra, et al [15] in the city of bandung. the variable of proper sanitation has a negative regression coefficient value and significantly influences the number of tuberculosis cases in central java province. this means that the more families that have access to proper sanitation (healthy latrines), the number of tuberculosis cases will decrease with the assumption that the other variables are constant. the population density variable has a positive and significant regression coefficient value in influencing the number of tuberculosis cases in central java province. this means that the denser the population of an area, the number of tuberculosis cases will increase with the assumption that the other variables are constant. variables eligible drinking water facilities and morbidity rates have no significant effect on tuberculosis cases in central java province. in the sar model, the covariate impact can be categorized in three types, namely direct impact, indirect impact, and total impact which can be seen in table 5. table 5. direct and indirect impact measure of sar model variable direct indirect total proper sanitation (𝑥1) -0.0116 -0.0135 -0.0251 eligible drinking water facilities (𝑥2) 0.0069 0.0080 0.0149 population density (𝑥3) 0.0001 0.0001 0.0002 morbidity rate (𝑥4) -0.0034 -0.0039 -0.0073 the direct impact is an impact that occurs locally in an area as a result of changes in predictor variables. the indirect impact is a spillover effect, which is the impact that occurs when the predictor variable is in the surrounding area. the total impact is a change that occurs in an area as a result of changes in the area and its surroundings. to find out whether the obtained sar model is good, it is necessary to carry out a diagnostic check, including assumptions of normality, non-autocorrelation, and homogeneity. the results of the diagnostic check performed in table 6 show that these assumptions have been fulfilled. table 6. diagnostic check of sar model statistic test p-value normality (shapiro-wilk) 0.2812 lm test for residual autocorrelation 0.4840 homoscedasticity (breusch-pagan) 0.7637 based on table 6, it can be seen that the p-value for the assumption of normality using the shapiro-wilk test is 0.2821 which is greater than alpha (0.05), which means the residuals are normally distributed. in the autocorrelation test, it was found that the p-value was also greater than alpha (0.05), which means that the residuals meet the non-autocorrelation assumption. likewise, in the homoscedasticity test using the spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 247 breusch-pagan test, it was found that the p-value was also greater than alpha (0.05) so that the assumption of homogeneity was also fulfilled. conclusions tuberculosis cases in central java province showed a positive spatial autocorrelation. it supports the hypothesis that tuberculosis cases are spatially dependent and that a spatial econometrics model should be considered. the best spatial econometrics model was chosen based on the lm test and the aic. the best model is the spatial autoregressive (sar) model. the estimation results of the sar show that the number of tuberculosis cases is influenced by a large number of tuberculosis cases in the neighbouring areas. proper sanitation (ownership of healthy latrines) has a negative effect on the number of tuberculosis cases, on the other hand, the dense population has a positive influence on the number of tuberculosis cases in the province of central java. references [1] k. ri, "pusat data dan informasi kementrian kesehatan ri tuberkolosis," kementrian kesehatan republik indonesia, jakarta, 2018. [2] k. ri, "pedoman nasional pengendalian tuberkolosis," jakarta, 2014. [3] w. h. organization, "global tuberculosis report 2020," geneva, 2020. [4] d. k. p. j. tengah, "profil kesehatan provinsi jawa tengah tahun 2019," semarang, 2020. [5] m. dara, c. d. acosta, v. rusovich, j. p. zellweger, r. centis and g. b. migliori, "bacille calmette–guérin vaccination: the current situation in europe," european respiratory journal, vol. 43, no. 1, pp. 24-35, 2014. [6] t. d. kristini and r. hamidah, "potensi penularan tuberculosis paru pada anggota keluarga penderita," jurnal kesehatan lingkungan indonesia, vol. 15, no. 1, 2020. [7] l. pangaribuan, kristina, d. pangaribuan, t. tejayanti and d. b. lolong, "faktorfaktor yang mempengaruhi kejadian tuberkulosis pada umur 15 tahun ke atas di indonesia (analisis data survei prevalensi tuberkulosis (sptb) di indonesia 20132014)," vol. 23, no. 1, 2020. [8] a. haq, u. f. achmadi and d. susanna, "analisis spasial (topografi) tuberkulosis paru di kota pariaman, bukittinggi, dan dumai tahun 2010-2016," jurnal ekologi kesehatan, vol. 18, no. 3, p. 149 – 158, 2020. [9] m. souris, epidemiology and geography principles, methods and tools of spatial analysis, great britain and the united states: iste ltd and john wiley & sons, inc, 2019. [10] j. lesage and r. k. pace, introduction to spatial econometrics, london: chapman and hall/crc, 2009. [11] b. p. s. p. j. tengah, "provinsi jawa tengah dalam angka," badan pusat statistik, semarang, 2020. [12] g. grekousis, spatial analysis methods and practice, cambridge: cambridge university press, 2020. [13] l. anselin, "lagrange multiplier test diagnostics for spatial dependence and spatial spatial autoregressive to model tuberculosis cases in central java province in 2019 hasrat ifolala zebua 248 heterogeneity," geographical analysis,, vol. 20, pp. 1-17, 2010. [14] i. g. n. m. jaya and y. andriyana, analisis data spasial perspektif bayesian, sumedang: alqaprint jatinangor, 2020. [15] i. g. n. jaya and e. al, "metode bayesian dalam penaksiran model spasial autoregressive (sar) (studi kasus pemodelan penyakit tb paru di kota bandung)," jurnal euclid, vol. 4, no. 2, 2017. bayesian hurdle poisson regression for assumption violation cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 384-393 p-issn: 2086-0382; e-issn: 2477-3344 submitted: march 05, 2022 reviewed: april 23, 2022 accepted: april 26, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.15549 bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah*, ani budi astuti, maria bernadetha t. mitakda department of statistics, faculty of mathematics and natural science, brawijaya university, indonesia email: nurkamilahs@student.ub.ac.id abstract violation of the poisson regression assumption can cause the model formed will produce an unbiased estimator. there is a good method for estimating parameters on small sample sizes and on all distributions, namely the bayesian method. the number of death due to chronic filariasis data violates the poisson regression assumption (overdispersion and response variable did not follow poisson distribution), so it is modeled with the bayesian hurdle poisson regression. with the bayesian method, convergence is fullfilled when 300000 iterations and 7 thin are performed. in addition to presenting an alternative method for estimating the hurdle poisson regression parameter, the model obtained can be used by the government in efforts to mitigate disease disasters through efforts to prevent, control, and handle cases of filariasis. the results showed that in the logit model only the percentage of households that have access to proper sanitation in 34 provinces in indonesia had a significant effect on the number of death due to chronic filariasis cases in 34 provinces in indonesia (𝑌). the truncated poisson model resulted in all predictor variables having a significant effect on the number of death due to chronic filariasis cases. keywords: bayesian; filariasis; hurdle; overdispersion; poisson introduction an important assumption in poisson regression analysis is that the response variable in the form of count distribute poisson, does not occur multicollinearity in the predictor variable, and occurs equidispersion (the mean of the data is equal to its variance). however, in certain cases, the assumption of conformity of poisson's distribution and equidispersion is not fullfilled. this can cause the model formed will produce an unbiased estimator [1]. equidispersion violations or often known as overdispersion (variance greater than the mean) can be overcome with zero inflated model and hurdle model. the handling of overdispersion in this study uses the hurdle poisson model because hurdle model better than the zero inflated model [2]. the parameter estimation method often used in the poisson hurdle model is maximum likelihood estimation (mle). however, mle cannot estimate parameters on small sample sizes and on certain distributions. there is a good method for estimating parameters on small sample sizes and on all distributions, namely the bayesian method. the advantage of the bayesian method is that it can estimate parameters for extremely small observations and can be used for all distributions [3]. the application of the bayesian method to overdispersion data has been carried out to analyze the number of filariasis sufferers in papua province, using the bayesian zero http://dx.doi.org/10.18860/ca.v7i3.15549 mailto:email1@gmail.com bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 385 inflated poisson model [4]. in this study will model data on the number of death from chronic filariasis cases in indonesia that violate the assumption of equidispersion and suitability of poisson distribution with bayesian hurdle poisson regression. filariasis or also known as elephant foot disease is believed to have existed since b.c. because in 1501-1480 bc found an ancient relief in a cemetery temple. queen hatshepsut in thebet, egypt who depicts the princess punt suffering from filariasis on her legs [5]. filariasis in indonesia is one of the endemic diseases (a disease that continues to infect certain regions) and was first reported by haga and van eecke in 1889 in jakarta caused by brugaria malayi [6]. acute clinical symptoms of filariasis disease include inflammation and swelling of the lymph canal accompanied by fever, headache, weak feeling and the onset of abscesses/ulcers while symptoms chronic clinical is the occurrence of enlargement that persists in the legs, arms, breasts and genitals of women and men [7]. one of the efforts to inhibit the transmission of filariasis disease is to mass preventive drug delivery (mpdd) filariasis implemented by endemic districts/cities of filariasis [5]. the success of the filariasis control program can be known by looking at the number of districts/cities that managed to reduce the number of microphilia to <1% [8]. this study discusses the influence of the number of chronic cases of filariasis in 34 provinces in indonesia (𝑋1), the number of districts/cities succeeded in reducing mikrophilia <1% in 34 provinces in indonesia (𝑋2), the number of districts/cities still carry out mass preventive drug delivery (mpdd) filariasis in 34 provinces in indonesia (𝑋3), population density in 34 provinces in indonesia (𝑋4), and the percentage of households that have access to proper sanitation in 34 provinces in indonesia (𝑋5) against the number of deaths from chronic filariasis in 34 provinces in indonesia (𝑌). the results of this study can be utilized for many things, namely (1) through the bayesian hurdle poisson regression model that is built can be identified factors that affect the number of cases of chronic filariasis death in indonesia, so that this information can be utilized for appropriate policy making for the central and local governments and related agencies in order to mitigate the disaster of chronic filariasis disease in indonesia through prevention efforts, control, and handling of the case. (2) by using bayesian parameter estimation approach, it is very useful and superior in various data challenge cases, namely for various sample sizes (any sample) small or large and various distributions (any distribution) with a data driven concept. methods this study uses secondary data from the indonesian health profile in 2020, namely the number of cases of chronic filariasis in 2020 with five predictor variables and one response variable [9]. the first step that must be done is testing the poisson regression assumption (poisson distribution suitability, non-multicollinearity, and overdispersion testing). the variables used in this study are the number of chronic cases of filariasis in 34 provinces in indonesia (𝑋1), the number of districts/cities succeeded in reducing mikrophilia <1% in 34 provinces in indonesia (𝑋2), the number of districts/cities still carry out mass preventive drug delivery (mpdd) filariasis in 34 provinces in indonesia (𝑋3), population density in 34 provinces in indonesia (𝑋4), and the percentage of households that have access to proper sanitation in 34 provinces in indonesia (𝑋5) against the number of deaths from chronic filariasis in 34 provinces in indonesia (𝑌). bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 386 poisson regression assumption poisson distribution suitability was tested with the kolmogorov-smirnov. kolmogorov-smirnov test statistics for testing the suitability of the poisson distribution are presented in equation (1)[10]. 𝐷 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚|𝐹𝑁 (𝑦(𝑖)) − 𝑃(𝑦(𝑖), 𝜆)| (1) if 𝐷 > 𝐷(𝑛,𝛼) or 𝑝𝑣𝑎𝑙𝑢𝑒 < 0.05, so we can conclude that response variable does not follow a poisson distribution. assumption of non-multicollinierity was tested with the 𝑉𝐼𝐹 criteria. if the 𝑉𝐼𝐹𝑗 exceeds 10, non-multicollinierity assumption is not fulfilled [11]. the third assumption test that must be done is the overdispersion test. the overdispersion test is carried out by calculating pearson chi square divided by the degrees of freedom of residual based on the formula (2). 𝜒𝑃𝑒𝑎𝑟𝑠𝑜𝑛 2 = ∑ (𝑦𝑖−�̂�𝑖) 2 �̂�𝑖 𝑛 𝑖=1 (2) where: �̂�𝑖 = �̂�𝑖 = 𝑒𝑥𝑝(�̂�0 + �̂�1𝑋𝑖1 + �̂�2𝑋𝑖2 + ⋯ + �̂�𝑘 𝑋𝑖𝑘 ) 𝑑𝑓 = 𝑛 − 𝑝 𝑛 : number of observations 𝑝 : number of parameters (𝑘 + 1) if (𝜒𝑃𝑒𝑎𝑟𝑠𝑜𝑛 2 𝑑𝑓⁄ ) > 1 then it can be said that observations contain overdispersion [12]. bayesian method suppose there are parameters 𝜃 to be estimated. in bayesian method, parameters 𝜃 treated as variable will have value in the domain 𝑓(𝜃). the prior distribution is the initial information to form the posterior. with prior information combined with data, calculating the posterior will be easier. based on the bayesian method, the posterior distribution is proportional (comparable) to the combination of the prior distribution and the likelihood function based on equation (3) [13]. 𝑓(𝜃|𝑦) ∝ 𝑓(𝑦|𝜃)𝑓(𝜃) (3) where: 𝑓(𝑦|𝜃) : likelihood function 𝑓(𝜃) : prior distribution function 𝑓(𝜃|𝑦) : posterior distribution function bayesian hurdle poisson regression there are three important components in bayesian method, namely (1) the likelihood function of the hpr model, (2) the prior distribution and (3) the posterior distribution. the likelihood function of the hpr model is as presented in equation (4). 𝑓(𝑌|𝛽, 𝛿) = ∏ 1 1+𝑒𝑥𝑝(𝑿𝑇𝜹) 𝑛 𝑖=1 𝑦𝑖=0 × ∏ [𝑒𝑥𝑝(−𝑒𝑥𝑝(𝑿𝑇𝜷))][𝑒𝑥𝑝(𝑿𝑇𝜷)] 𝑦𝑖 (1−[𝑒𝑥𝑝(−𝑒𝑥𝑝(𝑿𝑇𝜷))])𝑦𝑖! 𝑛 𝑖=1 𝑦𝑖>0 (4) the prior distribution for 𝛽 and 𝛿 is assumed to be normally distributed with the mean and variance 𝜎2 with the form as shown in equation (5). 𝑓(𝛽, 𝛿) = ∏ 1 𝜎𝛽 √2𝜋 𝑒𝑥𝑝 (− (𝛽−𝜇𝛽) 2 2𝜎𝛽 2 ) 𝑘 𝑗=0 × ∏ 1 𝜎𝛿√2𝜋 𝑒𝑥𝑝 (− (𝛿−𝜇𝛿) 2 2𝜎𝛿 2 ) 𝑘 𝑗=0 (5) the posterior distribution is obtained from the product of the likelihood function and the prior distribution in the form of an equation as presented in equation (6). bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 387 𝑓(𝛽, 𝛿|𝑌) ∝ ∏ 1 1+𝑒𝑥𝑝(𝑿𝑇𝜹) 𝑛 𝑖=1 𝑦𝑖=0 ∏ [𝑒𝑥𝑝(−𝑒𝑥𝑝(𝑿𝑇𝜷))][𝑒𝑥𝑝(𝑿𝑇𝜷)] 𝑦𝑖 (1−[𝑒𝑥𝑝(−𝑒𝑥𝑝(𝑿𝑇𝜷))])𝑦𝑖! 𝑛 𝑖=1 𝑦𝑖>0 × ∏ 1 𝜎𝛽√2𝜋 𝑒𝑥𝑝 (− (𝛽−𝜇𝛽) 2 2𝜎𝛽 2 ) 𝑘 𝑗=0 ∏ 1 𝜎𝛿√2𝜋 𝑒𝑥𝑝 (− (𝛿−𝜇𝛿) 2 2𝜎𝛿 2 ) 𝑘 𝑗=0 (6) the posterior distribution of the bayesian hurdle poisson regression model parameters has a complex function and requires a difficult integration process, so it is not easy to obtain analytically. therefore, a numerical approach is needed using the markov chain monte carlo (mcmc) simulation method [14]. bayesian model convergence test convergence test method consists of trace plot, autocorrelation plot, ergodic mean plot, and monte carlo error (mc error) [15]. convergence will be fullfilled if the trace plot does not form an ascending or descending pattern, the autocorrelation plot is close to one and the next lag is close to zero, after several iterations the ergodic mean plot is stable, or mc error is less than 5% of the standard deviation of each parameter. results and discussion the results of the analysis begin with testing the poisson regression assumption, then the parameter estimator of the bayesian hurdle poisson regression. result of poisson regression assumption test the first assumption in poisson regression is the response variable in the form of count with poisson distribution based on hypothesis. 𝐻0: the number of death due to chronic filariasis cases follows a poisson distribution versus 𝐻1: the number of death due to chronic filariasis cases does not follows a poisson distribution the results of the kolmogorov-smirnov test with software r showed that the 𝑝𝑣𝑎𝑙𝑢𝑒 less than 2.2 × 10−16. this suggests that the response variable did not follows a poisson distribution. then do fit distribution with easyfit software. poisson distribution ranked third after uniform and geometric distribution. since poisson regression is the most common regression model for modeling response variable in the form of count, then no one has researched related to uniform regression and geometric regression, the study still uses poisson's regression model, but uses the bayesian method to estimate the parameters because they have advantages that can be applied to all distribution. the next assumption is non-multocollinearity. the results of the multicollinearity test with the 𝑉𝐼𝐹𝑗 are presented in table 1. table 1. vif for all predictors variable 𝑉𝐼𝐹𝑗 𝑋1 4.782 𝑋2 1.530 𝑋3 3.872 𝑋4 1.173 𝑋5 3.162 bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 388 table 1 shows that the 𝑉𝐼𝐹 of all predictor variables is less than 10, so it can be concluded that the non-multicollinearity assumption is fullfilled. the last assumption in poisson regression is the occurrence of equidispersion. overdispersion testing was carried out with 𝜒𝑃𝑒𝑎𝑟𝑠𝑜𝑛 2 𝑑𝑓⁄ . data is said to contain overdispersion if (𝜒𝑃𝑒𝑎𝑟𝑠𝑜𝑛 2 𝑑𝑓⁄ ) > 1. the 𝜒𝑃𝑒𝑎𝑟𝑠𝑜𝑛 2 𝑑𝑓⁄ = 212.549, it can be concluded that the data contains overdispersion. because the two poisson regression assumptions are not fullfilled, then estimate the parameters with the bayesian hurdle poisson regression model. result of bayesian model convergence test in bayesian method, parameters are generated using the gibbs sampling algorithm with 300000 iterations and 7 thin. it is important to check the convergence of the model parameters to check the accuracy of the parameter estimation using the bayesian method. there are four methods for checking the convergence of parameters, namely (1) trace plot, (2) autocorrelation plot, and (3) ergodic mean plot (4) mc error. trace plots for each parameter are presented in figure 1. trace plot of 𝛿0 trace plot of 𝛿1 trace plot of 𝛿2 trace plot of 𝛿3 trace plot of 𝛿4 trace plot of 𝛿5 trace plot of 𝛽0 trace plot of 𝛽1 trace plot of 𝛽2 trace plot of 𝛽3 trace plot of 𝛽4 trace plot of 𝛽5 figure 1. trace plot for bayesian hurdle poisson regression parameters the figure 1 shows that the trace plot is random when 300000 iterations are carried out and 7 thin. it can be concluded that the parameters are convergent, so the iteration is stopped. the second method used to check the convergence is the autocorrelation plot. the figure 2 shows the autocorrelation plot for each parameter. bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 389 autocorrelation plot of 𝛿0 autocorrelation plot of 𝛿1 autocorrelation plot of 𝛿2 autocorrelation plot of 𝛿3 autocorrelation plot of 𝛿4 autocorrelation plot of 𝛿5 autocorrelation plot of 𝛽0 autocorrelation plot of 𝛽1 autocorrelation plot of 𝛽2 autocorrelation plot of 𝛽3 autocorrelation plot of 𝛽4 autocorrelation plot of 𝛽5 figure 2. autocorrelation plot for bayesian hurdle poisson regression parameters the figure 2 shows that the first lag in the autocorrelation plot is close to one and the next lag is close to zero, so the convergence of parameters is fulfilled. the third method used to check convergence is the ergodic mean plot. convergence will be fullfilled if after several iterations the ergodic mean plot is stable. the figure 3 shows the ergodic mean plot for each parameter. ergodic mean plot of 𝛿0 ergodic mean plot of 𝛿1 ergodic mean plot of 𝛿2 ergodic mean plot of 𝛿3 ergodic mean plot of 𝛿4 ergodic mean plot of 𝛿5 ergodic mean plot of 𝛽0 ergodic mean plot of 𝛽1 ergodic mean plot of 𝛽2 bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 390 ergodic mean plot of 𝛽3 ergodic mean plot of 𝛽4 ergodic mean plot of 𝛽5 figure 3. ergodic mean plot for bayesian hurdle poisson regression parameters the figure 3 shows that after 300000 iterations and 7 thin the ergodic mean plot is stable. it can be concluded that the parameters are convergent. in addition to using plots, convergence checks can also be done by comparing the mc error with 5% standard deviation for each parameter. the mc error for each parameter of the bayesian hurdle poisson regression model are presented in the table 2. table 2. mc error for bayesian hurdle poisson regression parameters model parameter estimator standard deviation 5% standard deviation mc error decision logit �̂�0 7.278704 0.363935 0.255295 convergence �̂�1 0.001624 8.12 × 10 −5 2.17 × 10−5 convergence �̂�2 0.175980 0.008799 0.003009 convergence �̂�3 0.242849 0.012142 0.002269 convergence �̂�4 0.000538 2.69 × 10 −5 3.59× 10−6 convergence �̂�5 0.084389 0.004219 0.002958 convergence truncated poisson �̂�0 0.918011 0.045901 0.040426 convergence �̂�1 0.000272 1.36 × 10 −5 3.33 × 10−6 convergence �̂�2 0.025134 0.001257 0.000488 convergence �̂�3 0.026595 0.00133 0.000515 convergence �̂�4 0.000131 6.54 × 10 −6 9.12× 10−7 convergence �̂�5 0.010116 0.000506 0.000445 convergence based on table 2, mc error on all parameters is less than 5% standard deviation, then the convergence is met. based on the four methods of checking the convergence, the results are the same, namely the convergence is fulfilled when 300000 and 7 thin amere performed. parameter estimation results of bayesian hurdle poisson regression model after the convergence is fullfilled, we can calculate the parameter estimator obtained from the sample generation using gibbs sampling. the parameter estimator is the average of the sample generation results for each parameter which is shown in table 3. testing bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 391 the bayesian model parameters using a confidence interval by looking at the lower limit of the 2.5% percentile and the upper limit of the 97.5% percentile. if it contains zero in that range, the decision to accept 𝐻0 or the 𝑗 th predictor variable has no significant effect to the response variable. table 3. parameter estimator of bayesian hurdle poisson regression model parameter parameter estimator percentile 2.5% percentile 97.5% decision logit 𝛿0 16.6551 5.0846 28.4970 reject 𝐻0 𝛿1 0.0004 -0.0031 0.0022 accept 𝐻0 𝛿2 −0.2270 -0.5155 0.0572 accept 𝐻0 𝛿3 −0.2974 -0.6936 0.0984 accept 𝐻0 𝛿4 0.0006 -0.0001 0.0014 accept 𝐻0 𝛿5 −0.1922 -0.3257 -0.0549 reject 𝐻0 truncated poisson 𝛽0 −4.2404 -5.7179 -2.7044 reject 𝐻0 𝛽1 −0.0027 -0.0032 -0.0023 reject 𝐻0 𝛽2 0.1500 0.1086 0.1911 reject 𝐻0 𝛽3 0.5121 0.4677 0.5551 reject 𝐻0 𝛽4 −0.0004 -0.0006 -0.0002 reject 𝐻0 𝛽5 0.0805 0.0636 0.0968 reject 𝐻0 based on table 3, the bayesian hurdle poisson regression model can be presented as follows 𝑙𝑜𝑔𝑖𝑡 �̂�𝑖 = 16,6551 − 0,1922𝑋5𝑖 (6) ln �̂�𝑖 = −4,2404 − 0,0027𝑋1𝑖 + 0,1500𝑋2𝑖 + 0,5121𝑋3𝑖 − 0,0004𝑋4𝑖 + 0,0805𝑋5𝑖 (7) the interpretation of the logit model in equation (6), that is, every 1% increase in the percentage of households that have access to proper sanitation in 34 provinces in indonesia will increase the probability of the number of cases of death due to chronic filariasis in 34 provinces in indonesia by exp(-0.1922) = 0.825 times of the original number of death from chronic filariasis cases. the interpretation of poisson's truncated model in equation (7) is: 1. every 1 person increase in the total number of chronic filariasis cases in 34 provinces in indonesia will increase the average number of deaths due to chronic filariasis in 34 provinces in indonesia by exp(-0.0027)=0.997≈1 person. 2. every increase in 1 district/city that succeeds in reducing microphilia <1% will increase the average number of cases of death due to chronic filariasis in 34 provinces in indonesia by exp(0.1500)=1.16≈1 person. 3. every increase in 1 district/city in indonesia that is still implementing mass preventive drug delivery (mpdd) will increase the average number of cases of death due to chronic filariasis in 34 provinces in indonesia by exp(0,5121)=1,669≈2 persons. 4. every 1 person/km2 increase in population density in 34 provinces in indonesia will increase the average number of cases of death due to chronic filariasis in 34 provinces in indonesia by exp(-0.0004)=0.9996≈1 person. bayesian hurdle poisson regression for assumption violation nur kamilah sa’diyah 392 5. every 1% increase in the percentage of households having access to proper sanitation in 34 provinces in indonesia will increase the average number of cases of death due to chronic filariasis in 34 provinces in indonesia by exp(0.0805)=1.08≈ 1 person. conclusions in the logit model, the percentage of households that have access to proper sanitation in 34 provinces in indonesia (𝑋5) has a significant effect on the number of cases of death due to chronic filariasis in 34 provinces in indonesia (𝑌). then in the truncated poisson model, all predictor variables, namely the number of all chronic cases of filariasis in 34 provinces in indonesia (𝑋1), the number of district/cities managed to reduce microphilia <1% in 34 provinces in indonesia (𝑋2), the number of district/cities that are still implementing the mass preventive drug delivery (mpdd) for filariasis in 34 provinces in indonesia (𝑋3), population density in 34 provinces in indonesia (𝑋4), as well as the percentage of households that have access to proper sanitation in 34 provinces in indonesia (𝑋5) have a significant effect on the number of deaths due to chronic filariasis in 34 provinces in indonesia (y). references [1] d. w. osgood, “poisson based regression analysis of aggregate crime rates,” quant. methods criminol., vol. 16, no. 1, pp. 577–599, 2017, doi: 10.4324/9781315089256-23. 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[15] i. ntzoufras, "bayesian modeling using winbugs", vol. 698. john wiley & sons, 2011. multipolar intuitionistic fuzzy ideal in b-algebras cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 293-301 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 17, 2021 reviewed: december 10, 2021 accepted: january 20, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.14003 multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo*, noor hidayat, vira hari krisnawati department of mathematics, university of brawijaya, malang, indonesia *corresponding author email: amigo.royyan@yahoo.com* abstract b-algebra is an algebraic structure which combine some properties from 𝐵𝐶𝐾-algebras and 𝐵𝐶𝐼-algebras. some researchers have investigated the concept of multipolar fuzzy ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras and multipolar intuitionistic fuzzy set in 𝐵-algebras. in this paper, we construct a new structure which is called a multipolar intuitionistic fuzzy ideal in 𝐵-algebras. this structure is a combination of three structures such as multipolar fuzzy ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras, fuzzy 𝐵-subalgebras in 𝐵-algebras, and multipolar intuitionistic fuzzy 𝐵-algebras. we investigated and proved some characterizes of the multipolar intuitionistic fuzzy ideal, such as a necessary condition and sufficient condition. keywords: b-algebras; multipolar fuzzy ideal; multipolar intuitionistic fuzzy set; multipolar intuitionistic fuzzy ideal introduction zadeh [1] introduced a new idea, namely a fuzzy set as a non-empty set with a degree of membership whose value in interval [0,1] in 1965. the degree of membership of each member of the set is determined by the membership function. that notion from zadeh became the basis for further researchers to develop fuzzy concepts in various fields such as graph theory, data analysis, decision making, and so on. a simple example of an algebraic structure is a group. not only groups, 𝐵𝐶𝐾-algebras, 𝐵𝐶𝐼-algebras and 𝐵-algebras are also other examples of algebraic structures. imai and iseki [2] proposed the notion a new algebraic structure called 𝐵𝐶𝐾-algebras in 1966. 𝐵𝐶𝐾-algebras is an important class of algebraic structure which is constructed from two different fragments, set theory and propositional calculus. in the same year, iseki [3] continued his research to propose the notion of 𝐵𝐶𝐼-algebras which is generalization from 𝐵𝐶𝐾-algebras. a new idea about algebraic structure is called 𝐵algebras which satisfies some properties from 𝐵𝐶𝐾-algebras and 𝐵𝐶𝐼-algebras was proposed by neggers and kim in [4]. they also investigated its properties. zhang [5] introduced the concepts of bipolar fuzzy sets which is the extension of fuzzy set. meng [6] studied about fuzzy implicative ideals in 𝐵𝐶𝐾-algebras in 1997. moreover, muhiuddin and al-kadi [7] introduced bipolar fuzzy implicative ideals in 𝐵𝐶𝐾-algebras. they discussed about the relationship between a bipolar fuzzy ideal and bipolar fuzzy implicative ideal. furthermore, chen et al. [8] introduced the concepts of multipolar fuzzy sets which is the extension of bipolar fuzzy set. kang et al. [9] proposed http://dx.doi.org/10.18860/ca.v7i1.14003 mailto:amigo.royyan@yahoo.com multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 294 the concepts about multipolar intuitionistic fuzzy set with finite degree and its application in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. in 1999, attanasov [10] introduced the new notion about intuitionistic fuzzy set. jun et al. [11] defined fuzzy 𝐵-algebras. then, al-masarwah and ahmad [12] discussed about multipolar fuzzy ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. ahn and bang [13] studied fuzzy 𝐵-subalgebras in 𝐵-algebras. recently, borzooei et al. [14] proposed the concept about multipolar intuitionistic fuzzy 𝐵-algebras and some properties. they constructed a simple multipolar fuzzy set. then, they also discussed about multipolar intuitionistic fuzzy subalgebras of 𝐵-algebras. in this paper, we construct a new structure which is called a multipolar intuitionistic fuzzy ideal in 𝐵-algebras. this structure is a combination of three structures which are the results of research by al-masarwah and ahmad [12], ahn and bang [13], and borzooei et al. [14]. next, we investigated and proved some necessary condition and sufficient condition of the multipolar intuitionistic fuzzy ideal. methods by using literary study and analogical related concepts from [12], [13] and [14], we propose the terminology of multipolar intuitionistic fuzzy ideal in 𝐵-algebras. we start to describe the structure of 𝐵-algebra, fuzzy 𝐵-algebra, and multipolar intuitionistic fuzzy sets. each structure is given its definition, examples, and some of its properties. definition 2.1 [15] 𝐵-algebra is a nonempty set 𝑋 with 0 as identity element (right) and a binary operation ∗ satisfying the following axioms for all 𝑥, 𝑦, 𝑧 ∈ 𝑋: i. 𝑥 ∗ 𝑥 = 0. ii. 𝑥 ∗ 0 = 𝑥. iii. (𝑥 ∗ 𝑦) ∗ 𝑧 = 𝑥 ∗ (𝑧 ∗ (0 ∗ 𝑦)). for all 𝑥, 𝑦 ∈ 𝑋, we define a partial ordering relation " ≤ " on 𝑋 by 𝑥 ≤ 𝑦 if and only if 𝑥 ∗ 𝑦 = 0 ([14]). example 2.2 [15] let 𝑋 = {0, 𝑎, 𝑏, 𝑐} be a set with cayley table as follows: table 1: cayley table for (𝑋;∗ ,0). ∗ 𝟎 𝒂 𝒃 𝒄 𝟎 0 0 𝑏 𝑏 𝒂 𝑎 0 𝑐 𝑏 𝒃 𝑏 𝑏 0 0 𝒄 𝑐 𝑏 𝑎 0 then, (𝑋;∗ ,0) is a 𝐵-algebra. example 2.3 [15] let (ℤ; −,0) with ′′ − ′′ be a substraction operation of integers ℤ. then, (ℤ; −,0) is a 𝐵-algebra. example 2.4 let (ℝ+ − {0};∗ ,1) with ′′ ∗ ′′ be a binary operation of ℝ+ − {0} defined by 𝑥 ∗ 𝑦 = 𝑥 𝑦 . multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 295 then, (ℝ+ − {0};∗ ,1) is a 𝐵-algebra. proposition 2.5 [16] if (𝑋;∗ ,0) is a 𝐵-algebra, then for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies the following conditions. i. (𝑥 ∗ 𝑦) ∗ (0 ∗ 𝑦) = 𝑥. ii. 𝑥 ∗ (𝑦 ∗ 𝑧) = (𝑥 ∗ (0 ∗ 𝑧)) ∗ 𝑦. iii. if 𝑥 ∗ 𝑦 = 0 then 𝑥 = 𝑦. iv. 0 ∗ (0 ∗ 𝑥) = 𝑥. v. (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) = 𝑥 ∗ 𝑦. vi. 0 ∗ (𝑥 ∗ 𝑦) = 𝑦 ∗ 𝑥. vii. 𝑥 ∗ 𝑦 = 0 if and only if 𝑦 ∗ 𝑥 = 0. viii. if 0 ∗ 𝑥 = 0 then 𝑋 contains only 0. definition 2.6 [16] a 𝐵-algebra (𝑋;∗ ,0) is called commutative 𝐵-algebra if for all 𝑥, 𝑦 ∈ 𝑋 satisfies: 𝑥 ∗ (0 ∗ 𝑦) = 𝑦 ∗ (0 ∗ 𝑥). example 2.7 let (ℤ; −,0) with ′′ − ′′ be a substraction operation of integers ℤ. then, (ℤ; −,0) is a commutative 𝐵-algebra. proposition 2.8 [16] if (𝑋;∗ ,0) is a commutative 𝐵-algebra, then for all 𝑥, 𝑦, 𝑧, 𝑡 ∈ 𝑋 satisfies the following rules. i. (0 ∗ 𝑥) ∗ (0 ∗ 𝑦) = 𝑦 ∗ 𝑥. ii. (𝑧 ∗ 𝑦) ∗ (𝑧 ∗ 𝑥) = 𝑥 ∗ 𝑦. iii. (𝑥 ∗ 𝑦) ∗ 𝑧 = (𝑥 ∗ 𝑧) ∗ 𝑦. iv. (𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ 𝑦 = 0. v. (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑡) = (𝑡 ∗ 𝑧) ∗ (𝑦 ∗ 𝑥). definition 2.9 [15] let (𝑋;∗ ,0) be a 𝐵-algebra. a nonempty subset 𝐼 of 𝑋 is called ideal of 𝑋 if it satisfies: i. 0 ∈ 𝐼, ii. for all 𝑥, 𝑦 ∈ 𝑋, if 𝑦 ∈ 𝐼 and 𝑥 ∗ 𝑦 ∈ 𝐼 then 𝑥 ∈ 𝐼. example 2.10 [15] let 𝐼 = ℤ+ ⋃ {0} be a subset of 𝐵-algebra (ℤ; −,0), then 𝐼 is ideal of ℤ. let (𝑋;∗ ,0) be a 𝐵-algebra. a non empty subset 𝐼 of 𝑋 is called subalgebras (𝐵-subalgebras) of 𝑋 if for all 𝑥, 𝑦 ∈ 𝐼 satisfies 0 ∈ 𝐼 and 𝑥 ∗ 𝑦 ∈ 𝐼 ([15]). definition 2.11 [11] let (𝑋;∗ ,0) be a 𝐵-algebra. a fuzzy set 𝐴 in 𝑋 is called fuzzy 𝐵-algebra if it satisfies the inequality for all 𝑥, 𝑦 ∈ 𝑋, 𝜇𝐴(𝑥 ∗ 𝑦) ≥ min{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}. let (𝑋;∗ ,0) be a 𝐵-algebra. a fuzzy set 𝐴 in 𝑋 is called fuzzy ideal 𝐵-algebra ([17]) if it satisfies for all 𝑥, 𝑦 ∈ 𝑋, 𝜇𝐴(0) ≥ 𝜇𝐴(𝑥), 𝜇𝐴(𝑥) ≥ min{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)}. multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 296 a 𝐵-algebra (𝑋;∗ ,0) in the example 2.2. if we define a fuzzy set 𝐴 in 𝑋 by 𝜇𝐴(0) = 𝜇𝐴(𝑏) = 1 and 𝜇𝐴(𝑎) = 𝜇𝐴(𝑐) = 0.5, then 𝐴 is fuzzy ideal of 𝑋. moreover, a 𝐵-algebra (ℝ+ − {0};∗ ,1) in the example 2.4, if we define a fuzzy set 𝐴 in ℝ+ − {0} by 𝜇𝐴(𝑥) = { 1 𝑖𝑓 𝑥 = 1, 0.5 𝑖𝑓 𝑥 ≠ 1, then 𝐴 is fuzzy ideal of ℝ+ − {0}. let (𝑋;∗ ,0) be a 𝐵-algebra. a multipolar intuitionistic fuzzy set over 𝑋 is a mapping (ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])𝑚 𝑥 ↦ (ℓ̂(𝑥), �̂�(𝑥)), where ℓ̂ ∶ 𝑋 → [0,1]𝑚 and �̂� ∶ 𝑋 → [0,1]𝑚 are multipolar fuzzy sets over 𝑋 which is satisfies the condition for all 𝑥 ∈ 𝑋, ℓ̂(𝑥) + �̂�(𝑥) ≤ 1 where 𝜋𝑖 ∶ [0,1] 𝑚 → [0,1] such that (𝜋𝑖 ∘ ℓ̂)(𝑥) + (𝜋𝑖 ∘ �̂�)(𝑥) ≤ 1 for 𝑖 = 1,2, … , 𝑚 (see [14]). results and discussion in this section, we will describe the structure of multipolar intuitionistic fuzzy ideal in 𝐵-algebras. the description begins with the definition of the new structure, then examples are given, and its properties are determined and proven. definition 3.1 let (𝑋;∗ ,0) be a 𝐵-algebra. a multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over 𝑋 is called multipolar intuitionistic fuzzy ideal in 𝑋 if it satisfies: i. (∀𝑥 ∈ 𝑋)(ℓ̂(0) ≥ ℓ̂(𝑥) and �̂�(0) ≤ �̂�(𝑥)) such that (𝜋𝑖 ∘ ℓ̂)(0) ≥ (𝜋𝑖 ∘ ℓ̂)(𝑥) and (𝜋𝑖 ∘ �̂�)(0) ≤ (𝜋𝑖 ∘ �̂�)(𝑥), ii. (∀𝑥, 𝑦 ∈ 𝑋)(ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)}) such that (𝜋𝑖 ∘ ℓ̂)(𝑥) ≥ inf{(𝜋𝑖 ∘ ℓ̂)(𝑥 ∗ 𝑦), (𝜋𝑖 ∘ ℓ̂)(𝑦)} and (𝜋𝑖 ∘ �̂�)(𝑥) ≤ sup{(𝜋𝑖 ∘ �̂�)(𝑥 ∗ 𝑦), (𝜋𝑖 ∘ �̂�)(𝑦)}, for 𝑖 = 1,2, … . , 𝑚. example 3.2 let (𝑋;∗ ,0) be a 𝐵-algebra in the example 2.2. given a multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over 𝑋 by (ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])5, multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 297 𝑥 ↦ { ((0.7,0.3), (0.6,0.25), (0.7,0.15), (0.63,0.2), (0.8,0.18)) 𝑖𝑓 𝑥 ∈ {0, 𝑏}, ((0.3,0.6), (0.4,0.5), (0.5,0.4), (0.2,0.7), (0.4,0.5)) 𝑖𝑓 𝑥 ∈ {𝑎, 𝑐}. then, (ℓ̂, �̂�) is 5-polar intuitionistic fuzzy ideal of 𝑋. example 3.3 let (ℝ+ − {0};∗ ,1) be a 𝐵-algebra in the example 2.4. given a multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over ℝ+ − {0} by (ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])5, 𝑥 ↦ { ((1,0), (1,0), (1,0), (1,0), (1,0)) 𝑖𝑓 𝑥 = 1, ((0.5,0.5), (0.4,0.4), (0.3,0.3), (0.2,0.2), (0.1,0.1)) 𝑖𝑓 𝑥 ≠ 1. then, (ℓ̂, �̂�) is 5-polar intuitionistic fuzzy ideal of ℝ+ − {0}. for any 𝜔 ∈ 𝑋 and multipolar intuitionistic fuzzy set (ℓ̂, �̂�) in 𝑋, we give the conditions for the set 𝐼(𝜔) to be an ideal of 𝑋 and its example. theorem 3.4 let (𝑋;∗ ,0) be a 𝐵-algebra and 𝑥 ∈ 𝑋. if (ℓ̂, �̂�) is a multipolar intuitionistic fuzzy ideal of 𝑋, then 𝐼(𝜔) is an ideal of 𝑋 where 𝐼(𝜔) = {𝑥 ∈ 𝑋|ℓ̂(𝑥) ≥ ℓ̂(𝜔) 𝑎𝑛𝑑 �̂�(𝑥) ≤ �̂�(𝜔)}. proof. let (ℓ̂, �̂�) be a multipolar intuitionistic fuzzy ideal of 𝑋 where 𝐼(𝜔) = {𝑥 ∈ 𝑋|ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ �̂�(𝜔)}. i. by using definition 3.1 (i) we have that ℓ̂(0) ≥ ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(0) ≤ �̂�(𝑥) ≤ �̂�(𝜔). hence, 0 ∈ 𝐼(𝜔). ii. let 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ∗ 𝑦 ∈ 𝐼(𝜔) and 𝑦 ∈ 𝐼(𝜔). then, ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂(𝜔) and �̂�(𝑥 ∗ 𝑦) ≤ �̂�(𝜔), ℓ̂(𝑦) ≥ ℓ̂(𝜔) and �̂�(𝑦) ≤ �̂�(𝜔). by using definition 3.1 (ii), we have ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} ≤ �̂�(𝜔), such that ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ �̂�(𝜔). hence, 𝑥 ∈ 𝐼(𝜔). therefore, 𝐼(𝜔) is an ideal of 𝑋. ∎ multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 298 example 3.5 let (𝑋;∗ ,0) be a 𝐵-algebra in the example 2.2. given a multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋 in the example 3.2 where 𝐼(𝑏) = {0, 𝑏|ℓ̂(0) ≥ ℓ̂(𝑏) and �̂�(0) ≤ �̂�(𝑏), ℓ̂(𝑏) ≥ ℓ̂(𝑏) and �̂�(𝑏) ≤ �̂�(𝑏)}. then, 𝐼(𝑏) is an ideal of 𝑋. next, we discuss some properties of multipolar intuitionistic fuzzy ideal in 𝐵-algebras. proposition 3.6 let (𝑋;∗ ,0) be a 𝐵-algebra. every multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋 satisfies the following implication for all 𝑥, 𝑦 ∈ 𝑋, if 𝑥 ≤ 𝑦 then ℓ̂(𝑥) ≥ ℓ̂(𝑦) and �̂�(𝑥) ≤ �̂�(𝑦). proof. let 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ≤ 𝑦. so, 𝑥 ∗ 𝑦 = 0. by using definition 3.1 (i) and (ii), we have that ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} = inf{ℓ̂(0), ℓ̂(𝑦)} = ℓ̂(𝑦) and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} = sup{�̂�(0), �̂�(𝑦)} = �̂�(𝑦). ∎ proposition 3.7 let (𝑋;∗ ,0) be a commutative 𝐵-algebra. for any multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋, if for all 𝑥, 𝑦 ∈ 𝑋 satisfies ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) 𝑎𝑛𝑑 �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦), then for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 𝑎𝑛𝑑 �̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). proof. let 𝑥, 𝑦, 𝑧 ∈ 𝑋 such that ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧 ≤ (𝑥 ∗ 𝑦) ∗ 𝑧. by using proposition 2.5 and 2.8, we have that ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧 = ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧 ≤ (𝑥 ∗ 𝑦) ∗ 𝑧. from proposition 3.6, we have ℓ̂ (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) and �̂� (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 299 so, from proposition 2.8, we get ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) = ℓ̂ ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ≥ ℓ̂ (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) and �̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) = �̂� ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ≤ �̂� (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). ∎ proposition 3.8 let (𝑋;∗ ,0) be a 𝐵-algebra. for any multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋, if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 𝑎𝑛𝑑 �̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧), then for all 𝑥, 𝑦 ∈ 𝑋 satisfies ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) 𝑎𝑛𝑑 �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦). proof. let 𝑥, 𝑦, 𝑧 ∈ 𝑋. if 𝑧 is replaced by 𝑦 on the assumption, then by using definition 2.1 (i) and (ii) we have ℓ̂(𝑥 ∗ 𝑦) = ℓ̂((𝑥 ∗ 𝑦) ∗ 0) = ℓ̂((𝑥 ∗ 𝑦) ∗ (𝑦 ∗ 𝑦)) = ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) and �̂�(𝑥 ∗ 𝑦) = �̂�((𝑥 ∗ 𝑦) ∗ 0) = �̂�((𝑥 ∗ 𝑦) ∗ (𝑦 ∗ 𝑦)) = �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦). ∎ based on proposition 3.7 and proposition 3.8, we get the following corollary. corollary if we assume that 𝑋 is a commutative 𝐵-algebra, then the statements in proposition 3.7 and proposition 3.8 are equivalent. furthermore, we also give another condition of multipolar intuitionistic fuzzy ideal in 𝐵-algebras such that make this following proposition. proposition 3.9 let (𝑋;∗ ,0) be a 𝐵-algebra. a multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over 𝑋 is a multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋 if and only if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝑥 ∗ 𝑦) ∗ 𝑧 = 0 implies ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑧)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑧)}. multipolar intuitionistic fuzzy ideal in b-algebras royyan amigo 300 proof. we assume that (ℓ̂, �̂�) is a multipolar intuitionistic fuzzy ideal over 𝑋. let 𝑥, 𝑦, 𝑧 ∈ 𝑋 such that (𝑥 ∗ 𝑦) ∗ 𝑧 = 0. so, 𝑥 ∗ 𝑦 ≤ 𝑧. by using definition 3.1 (i) and (ii), we have ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} ≥ inf{inf{ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧), ℓ̂(𝑧)} , ℓ̂(𝑦)} = inf{inf{ℓ̂(0), ℓ̂(𝑧)} , ℓ̂(𝑦)} = inf{ℓ̂(𝑦), ℓ̂(𝑧)} and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} ≤ sup{sup{�̂�((𝑥 ∗ 𝑦) ∗ 𝑧), �̂�(𝑧)} , �̂�(𝑦)} = sup{sup{�̂�(0), �̂�(𝑧)} , �̂�(𝑦)} = sup{�̂�(𝑦), �̂�(𝑧)}. conversely, we assume for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝑥 ∗ 𝑦) ∗ 𝑧 = 0. then ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑧)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑧)}. let 𝑥 ∈ 𝑋. by using definition 2.1 (ii) and definition 2.11, we have ℓ̂(0) = ℓ̂(𝑥 ∗ 𝑥) ≥ inf{ℓ̂(𝑥), ℓ̂(𝑥)} = ℓ̂(𝑥) and �̂�(0) = �̂�(𝑥 ∗ 𝑥) ≤ sup{�̂�(𝑥), �̂�(𝑥)} = �̂�(𝑥). then, let 𝑥, 𝑦 ∈ 𝑋. by using definition 2.1 (i), we have (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑦) = 0 such that ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑥 ∗ 𝑦)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑥 ∗ 𝑦)}. hence, (ℓ̂, �̂�) is a multipolar intuitionistic fuzzy ideal over 𝑋. ∎ conclusions in this paper, we apply the terminology of multipolar intuitionistic fuzzy ideal in 𝐵-algebras and investigate some properties. we also explain the conditions for a multipolar intuitionistic fuzzy set to be a multipolar intuitionistic fuzzy ideal and give some examples. these definitions and main results can be applied with similarly in other algebraic structure such as 𝐵𝐺-algebras, 𝐵𝐹-algebras and 𝐵𝐷-algebras. references [1] zadeh, l.a. 1965. fuzzy sets. information and control, vol. 8, no. 3, 338-353. 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[17] senapati, t.; bhowmik, m.; pal, m. 2011. fuzzy closed ideals of b-algebras. ijcset, vol. 1, no. 10, 669-673. on rainbow vertex antimagic coloring of graphs: a new notion cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 64-72 p-issn: 2086-0382; e-issn: 2477-3344 submitted: juni 30, 2021 reviewed: august 12, 2021 accepted: october 20. 2021 doi: https://doi.org/10.18860/ca.v7i1.12796 on rainbow vertex antimagic coloring of graphs: a new notion marsidi1, ika hesti agustin2, dafik3, elsa yuli kurniawati4 1department of mathematics education, universitas pgri argopuro jember, indonesia 2department of mathematics, university of jember, indonesia 3department of mathematics education, university of jember, indonesia 4cgant, university of jember, indonesia email: marsidiarin@gmail.com, ikahesti.fmipa@unej.ac.id, d.dafik@unej.ac.id, elsayuli@unej.ac.id abstract for a bijective function 𝑔: 𝐸(𝐺) → {1, 2,3, ⋯ , |𝐸(𝐺)|}, the associated weight of a vertex 𝑣 ∈ 𝑉(𝐺) under 𝑔 is 𝑤𝑔(𝑣) = σ𝑒∈𝐸(𝑣)𝑔(𝑒), where 𝐸(𝑣) is the set of vertices incident to 𝑣. the function 𝑔 is called a vertex-antimagic edge labeling if every vertex has distinct weight. a path 𝑃 in the edgelabeled graph 𝐺 is said to be a rainbow path if for any two vertices 𝑥 and 𝑥′, all internal vertices in the path 𝑥 − 𝑥′ have different weight. if for every two vertices 𝑥 and 𝑦 of 𝐺, there exists a rainbow 𝑥 − 𝑦 path, then 𝑔 is called a rainbow vertex antimagic labeling of 𝐺. when we assign each edge 𝑥𝑦 with the color of the vertex weight 𝑤𝑔(𝑣), thus we say the graph 𝐺 admits a rainbow vertex antimagic coloring. the smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of 𝐺 is called rainbow vertex antimagic connection number of 𝐺, denoted by 𝑟𝑣𝑎𝑐(𝐺). in this paper, we initiate to determine the rainbow vertex antimagic connection number of graphs, namely path (𝑃𝑛), wheel (𝑊𝑛 ), friendship (ℱ𝑛), and fan (𝐹𝑛). keywords: antimagic labeling; rainbow vertex coloring; rainbow vertex antimagic coloring; rainbow vertex antimagic connection number. introduction we consider a graph 𝐺(𝑉, 𝐸) in this paper are simple, connected and un-directed graph, where 𝑉 and 𝐸 are respectively a vertex set and edge set of 𝐺 [1]. the rainbow coloring problem has been studied by many researchers since many years ago. many good results has been published in some reputable journal [2]. thus, it has given many contributions in graph theory research of interest. there are many types of rainbow coloring, namely rainbow (edge) coloring, rainbow vertex coloring, strong rainbow edge/vertex coloring. the minimum number of colors for which an edge (vertex) coloring exists such that the graph 𝐺 is rainbow connected is called the rainbow connection number, denoted by 𝑟𝑐(𝐺) for edge coloring and the rainbow vertex connection number, denoted by 𝑟𝑣𝑐(𝐺) for vertex coloring, see [3]–[10] for detail. krivelevich and yuster [6] gave the lower bound for 𝑟𝑣𝑐(𝐺), namely 𝑟𝑣𝑐(𝐺) ≥ 𝑑𝑖𝑎𝑚(𝐺) – 1, where 𝑑𝑖𝑎𝑚(𝐺) is the diameter of graph 𝐺. an easy observation is that if 𝐺 has an order n, then 𝑟𝑣𝑐(𝐺) ≤ 𝑛 − 2 and 𝑟𝑣𝑐(𝐺) = 0 if and only if 𝐺 is a complete graph. notice that 𝑟𝑣𝑐(𝐺) ≥ 𝑑𝑖𝑎𝑚(𝐺) − 1 with equality if the diameter of 𝐺 is 1 or 2. https://doi.org/10.18860/ca.v7i1.12796 mailto:marsidiarin@gmail.com mailto:ikahesti.fmipa@unej.ac.id mailto:d.dafik@unej.ac.id mailto:elsayuli@unej.ac.id on rainbow vertex antimagic coloring of graphs: a new notion marsidi 65 meanwhile, in 2003, hartsfield and ringel [11] defined antimagic graphs. a graph 𝐺 is called antimagic if there exists a bijection 𝑓: 𝐸(𝐺) → {1,2, ⋯ , 𝑞} such that the weights of all vertices are distinct [12] . the vertex weight of a vertex 𝑣 under 𝑓, 𝑤𝑓 (𝑣), is the sum of labels of edges incident with 𝑣, that is, 𝑤𝑓 (𝑣) = ∑ 𝑓(𝑢𝑣)𝑢𝑣∈𝐸(𝐺) . in this case, 𝑓 is called an antimagic labeling. there many results were found for antimagicness of graph. there are extension types of vertex antimagic labeling, namely total vertex antimagic labeling, super total vertex antimagic labeling, (𝑎, 𝑑)-vertex antimagic labeling, super (𝑎, 𝑑)-vertex antimagic labeling. for detail, see galian dynamic survey of graph labeling [13] . in this study, we initiate to combine the two notion, namely rainbow coloring and antimagic labeling [14][15]. we name for this combination as rainbow vertex antimagic coloring. for a bijective function 𝑔: 𝐸(𝐺) → {1, 2,3, ⋯ , |𝐸(𝐺)|}, the associated weight of a vertex 𝑣 ∈ 𝑉(𝐺) under 𝑔 is 𝑤𝑔(𝑣) = σ𝑒∈𝐸(𝑣)𝑔(𝑒), where 𝐸(𝑣) is the set of vertices incident to 𝑣. the function 𝑔 is called a vertex-antimagic edge labeling if every vertex has distinct weight. a path 𝑃 in the edge-labeled graph 𝐺 is said to be a rainbow path if for any two vertices 𝑥 and 𝑥′, all internal vertices in the path 𝑥 − 𝑥′ have different weight. if for every two vertices 𝑥 and 𝑦 of 𝐺, there exists a rainbow 𝑥 − 𝑦 path, then 𝑔 is called a rainbow vertex antimagic labeling of 𝐺. when we assign each edge 𝑥𝑦 with the color of the vertex weight 𝑤𝑔(𝑣), thus we say the graph 𝐺 admits a rainbow vertex antimagic coloring. the rainbow vertex antimagic connection number of 𝐺, denoted by 𝑟𝑣𝑎𝑐(𝐺), is the smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of 𝐺. to determine the rainbow vertex antimagic connection number of any graph is considered to be hard problem. even, this study fall into np-hard problem. in this paper, we initiate to determine the rainbow vertex antimagic connection number of graphs, namely path (𝑃𝑛 ), wheel (𝑊𝑛), friendship (ℱ𝑛), and fan (𝐹𝑛 ) as well as fix the lower bound 𝑟𝑣𝑎𝑐(𝐺) of any graph. methods this research includes deductive analytic methods. the procedures to obtain the rainbow vertex antimagic connection number of are as follows. 1. define a graph 𝐺. 2. determine the cardinality of graph 𝐺 by obtaining the order and size of graph 𝐺. 3. determine the lower bound of 𝑟𝑣𝑎𝑐(𝐺) by using the obtained remark of sharpest lower bound. 4. determine the upper bound of 𝑟𝑣𝑎𝑐(𝐺) by constructing the bijective function, compute the vertex weight using 𝑤𝑔(𝑣) = σ𝑒∈𝐸(𝑣)𝑔(𝑒), and show that every two different vertices of 𝐺 satisfy the rainbow vertex antimagic coloring. 5. if the upper bound attains the lower bound, then we obtain the 𝑟𝑣𝑎𝑐(𝐺). if the upper bound does not attain the lower bound, then we return to determine the upper bound of 𝑟𝑣𝑎𝑐(𝐺). 6. finally we can construct a new theorem and its proof after we obtain the rainbow vertex antimagic connection number of graph 𝐺. on rainbow vertex antimagic coloring of graphs: a new notion marsidi 66 results and discussion in this section we have several theorems on the rainbow vertex antimagic coloring. we determine the minimum color taken to the graph such that it has rainbow vertex antimagic coloring. since we determine the minimum colors such that 𝐺 has rainbow vertex antimagic coloring, then the lower bound of rainbow vertex antimagic connection number of graph is at least and equal to rainbow vertex connection number. the lower bound of rainbow vertex antimagic connection number of any graph is mathematically written in the remark 1. remark 1 let 𝐺 be a connected graph, 𝑟𝑣𝑎𝑐(𝐺) ≥ 𝑟𝑣𝑐(𝐺). theorem 1 if 𝑃𝑛 be a path graph of order 𝑛 and 𝑛 ≥ 3, then 𝑟𝑣𝑎𝑐(𝑃𝑛) = { 3, 𝑛 = 3,4 𝑛 − 2, 𝑛 ≥ 5 proof. let 𝑃𝑛 be a path graph with vertex set 𝑉(𝑃𝑛) = {𝑣1, 𝑣2, 𝑣3, ⋯ , 𝑣𝑛 } and edge set 𝐸(𝑃𝑛 ) = {𝑣𝑖 𝑣{𝑖+1}: 1 ≤ 𝑖 ≤ 𝑛 − 1}. the diameter of 𝑃𝑛 is 𝑛 − 1. we divide into two cases to prove the rainbow vertex antimagic connection number as follows. case 1. for 𝑃𝑛 , 𝑛 = 3,4 path graph 𝑃𝑛 , 𝑛 = 3 have two edges. if we give labels on it, it gives three different weights on its edges exactly. it concludes that the rainbow vertex antimagic connection number of 𝑃3 is 3. furthermore for 𝑃4, we determine the all permutation of edge labeling on 𝑃4. let 𝑒1, 𝑒2, 𝑒3 are the edges of 𝑃4, thus there are six possibilities of edge labeling on 𝑃4 as follows. 1). if 𝑒1 = 1, 𝑒2 = 2, 𝑒3 = 3, then 𝑤𝑡(𝑣1) = 1, 𝑤𝑡(𝑣2) = 3, 𝑤𝑡(𝑣3) = 5, 𝑤𝑡(𝑣4) = 3. 2). if 𝑒1 = 1, 𝑒2 = 3, 𝑒3 = 2, then 𝑤𝑡(𝑣1) = 1, 𝑤𝑡(𝑣2) = 4, 𝑤𝑡(𝑣3) = 5, 𝑤𝑡(𝑣4) = 2. 3). if 𝑒1 = 2, 𝑒2 = 1, 𝑒3 = 3, then 𝑤𝑡(𝑣1) = 2, 𝑤𝑡(𝑣2) = 3, 𝑤𝑡(𝑣3) = 4, 𝑤𝑡(𝑣4) = 3. 4). if 𝑒1 = 2, 𝑒2 = 3, 𝑒3 = 1, then 𝑤𝑡(𝑣1) = 2, 𝑤𝑡(𝑣2) = 5, 𝑤𝑡(𝑣3) = 4, 𝑤𝑡(𝑣4) = 1. 5). if 𝑒1 = 3, 𝑒2 = 1, 𝑒3 = 2, then 𝑤𝑡(𝑣1) = 3, 𝑤𝑡(𝑣2) = 4, 𝑤𝑡(𝑣3) = 3, 𝑤𝑡(𝑣4) = 2. 6). if 𝑒1 = 3, 𝑒2 = 2, 𝑒3 = 1, then 𝑤𝑡(𝑣1) = 3, 𝑤𝑡(𝑣2) = 5, 𝑤𝑡(𝑣3) = 3, 𝑤𝑡(𝑣4) = 1. based on edge labelings and vertex weights above, it is easy to determine the rainbow vertex antimagic connection number of 𝑃4 at least 3. thus 𝑎𝑟𝑣𝑐(𝑃4) = 3. case 2. for 𝑃𝑛 , 𝑛 ≥ 5 based on remark 1, we have 𝑟𝑣𝑎𝑐(𝑃𝑛) ≥ 𝑟𝑣𝑐(𝑃𝑛 ) = 𝑑𝑖𝑎𝑚(𝑃𝑛 ) − 1 = 𝑛 − 1 − 1 = 𝑛 − 2. furthermore, to show the upper bound we construct the bijective function of edge labels. we have two conditions, namely for 𝑛 ≡ 1(mod 2) and 𝑛 ≡ 0(mod 2). for 𝑛 ≡ 1(mod 2), we have 𝑔(𝑣1𝑣2) = 3 𝑔(𝑣2𝑣3) = 1 𝑔(𝑣3𝑣4) = 2 𝑔(𝑣𝑛−1𝑣𝑛 ) = 4 𝑔(𝑣𝑖 𝑣𝑖+1) = 𝑖 + 1: 4 ≤ 𝑖 ≤ 𝑛 − 2 from the edge labels above, we have the vertex weight as follows. for 𝑃5, we have 𝑤(𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5) = (3,4,3,6,4). for 𝑃𝑛 : 𝑛 ≥ 6, we have on rainbow vertex antimagic coloring of graphs: a new notion marsidi 67 𝑤(𝑣1) = 3 𝑤(𝑣2) = 4 𝑤(𝑣3) = 3 𝑤(𝑣4) = 7 𝑤(𝑣𝑖 ) = 2𝑖 + 1: 5 ≤ 𝑖 ≤ 𝑛 − 2 𝑤(𝑣𝑛−1) = 𝑛 + 3 𝑤(𝑣𝑛) = 4 for 𝑛 ≡ 0(mod 2), we have 𝑔(𝑣1𝑣2) = 3 𝑔(𝑣2𝑣3) = 1 𝑔(𝑣3𝑣4) = 2 𝑔(𝑣𝑖 𝑣𝑖+1) = 𝑖: 4 ≤ 𝑖 ≤ 𝑛 − 1 from the edge labels above, we have the vertex weights in the following: 𝑤(𝑣1) = 3 𝑤(𝑣2) = 4 𝑤(𝑣3) = 3 𝑤(𝑣4) = 6 𝑤(𝑣𝑖 ) = 2𝑖 − 1: 5 ≤ 𝑖 ≤ 𝑛 − 1 𝑤(𝑣𝑛) = 𝑛 − 1 from the vertex weight above, it is easy to see that the different weight is 𝑛 − 2. it concludes that the rainbow vertex antimagic connection number of 𝑃𝑛 : 𝑛 = {3,4} is 3 and the rainbow vertex antimagic connection number of 𝑃𝑛 : 𝑛 ≥ 5 is 𝑛 − 2. furthermore, we show that every two different vertices of 𝑃𝑛 is rainbow vertex antimagic coloring. suppose that 𝑣 ∈ 𝑉(𝑃𝑛 ), refer to the vertex weight the rainbow vertex path is shown in table 1. table 1. the rainbow vertex path of 𝑃𝑛 case 𝒗 𝒗 rainbow vertex coloring 1 𝑣1 𝑣𝑛 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑖 , … , 𝑣𝑛−1 hence, the vertex coloring of 𝑃𝑛 is rainbow vertex antimagic coloring. thus, we obtain 𝑎𝑟𝑣𝑐(𝑃𝑛) is 3 for 𝑛 = 3,4 and 𝑎𝑟𝑣𝑐(𝑃𝑛 ) is 𝑛 − 2 for 𝑛 ≥ 5. ∎ theorem 2 if 𝑊𝑛 be a wheel graph of order 𝑛 + 1 and 𝑛 ≥ 3, then 𝑟𝑣𝑎𝑐(𝑊𝑛) = 2 if 𝑛 ≡ 1(mod 2) and 2 ≤ 𝑟𝑣𝑎𝑐(𝑊𝑛) ≤ 3 if 𝑛 ≡ 0(mod 2). proof. let 𝑊𝑛 be a wheel graph with vertex set 𝑉(𝑊𝑛) = {𝐴, 𝑥1, 𝑥2, 𝑥3, ⋯ , 𝑥𝑛} and edge set 𝐸(𝑊𝑛) = {𝐴𝑥𝑖 : 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖 𝑥{𝑖+1}: 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑛−1𝑥1}. the diameter of 𝑊𝑛 is 2. based on remark 1, we have 𝑟𝑣𝑎𝑐(𝑊𝑛) ≥ 𝑟𝑣𝑐(𝑊𝑛) = 𝑑𝑖𝑎𝑚(𝑊𝑛) − 1 = 2 − 1 = 1. since the vertex 𝐴 has degree of much greater than the others, it must have a different vertex weight than the others. the vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. from this condition, such that we have 𝑟𝑣𝑎𝑐(𝑊𝑛) ≥ 2. we divide into two cases to show the upper bound of the rainbow vertex antimagic connection number of 𝑊𝑛 as follows. on rainbow vertex antimagic coloring of graphs: a new notion marsidi 68 case 1. for 𝑊𝑛, 𝑛 ≡ 1(mod 2) to show the upper bound of (𝑊𝑛): 𝑛 ≡ 1(mod 2) , we construct the bijective function of edge labels. 𝑔(𝑥𝑖 𝑥𝑖+1) = { 𝑖 + 1 2 , if 𝑖 ≡ 1(mod 2) ⌈ 𝑛 2 ⌉ + 𝑖 2 , if 𝑖 ≡ 0(mod 2) 𝑔(𝐴𝑥𝑖 ) = 2𝑛 + 1 − 𝑖 from the edge labels above, we have the vertex weights in the following: 𝑤(𝑥𝑖 ) = 2𝑛 + 1 + ⌈ 𝑛 2 ⌉ 𝑤(𝐴) = 𝑛 2 (3𝑛 + 1) from the vertex weights above, it is easy to see that the different weight is 2. case 2. for 𝑊𝑛, 𝑛 ≡ 0(mod 2) to show the upper bound of 𝑟𝑣𝑎𝑐(𝑊𝑛): 𝑛 ≡ 0(mod 2), we construct the bijective function of edge labels. 𝑔(𝑥𝑖 𝑥𝑖+1) = { 𝑖 + 1 2 , if 𝑖 ≡ 1(mod 2) ⌈ 𝑛 2 ⌉ + 𝑖 2 , if 𝑖 ≡ 0(mod 2) 𝑔(𝐴𝑥𝑖 ) = 2𝑛 + 1 − 𝑖 from the edge labels above, we have the vertex weights in the following. 𝑤(𝑥1) = 3𝑛 + 1 𝑤(𝑥𝑖 ) = 2𝑛 + 1 + ⌈ 𝑛 2 ⌉ 𝑤(𝐴) = 𝑛 2 (3𝑛 + 1) from the vertex weight above, it is easy to see that the different weight is 3. furthermore, we show that every two different vertices of 𝑊𝑛 is rainbow vertex antimagic coloring. suppose that 𝑥, 𝑦 ∈ 𝑉(𝑊𝑛), refer to the vertex weight the rainbow vertex 𝑥 − 𝑦 path is shown in table 2. table 2. the rainbow vertex of 𝑥 − 𝑦 path of 𝑊𝑛 case 𝒙 𝒚 rainbow vertex coloring 𝒙 − 𝒚 1 𝑥𝑖 𝐴 𝑥𝑖 , 𝐴 2 𝑥𝑖 𝑥𝑖 𝑥𝑖 , 𝐴, 𝑥𝑖 hence, the vertex coloring of 𝑊𝑛 is rainbow vertex antimagic coloring. thus, we obtain 𝑟𝑣𝑎𝑐(𝑊𝑛) = 2 if 𝑛 ≡ 1(mod 2) and 2 ≤ 𝑟𝑣𝑎𝑐(𝑊𝑛) ≤ 3 if 𝑛 ≡ 0(mod 2). ∎ theorem 3 if ℱ𝑛 be a friendship graph of order 2𝑛 + 1 and 𝑛 ≥ 3, then 𝑟𝑣𝑎𝑐(ℱ𝑛) = 3. proof. let ℱ𝑛 be a friendship graph with vertex set 𝑉(ℱ𝑛) = {𝐴} ∪ {𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑛 } ∪ {𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑛 } and edge set 𝐸(ℱ𝑛) = {𝐴𝑥𝑖 ; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝐴𝑦𝑖 ; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖 𝑦𝑖 ; 1 ≤ 𝑖 ≤ 𝑛}. the diameter of ℱ𝑛 is 2. based on remark 1, we have 𝑟𝑣𝑎𝑐(ℱ𝑛) ≥ 𝑟𝑣𝑐(ℱ𝑛) = 𝑑𝑖𝑎𝑚(ℱ𝑛) − 1 = 2 − 1 = 1. since the vertex 𝐴 has degree of much greater than the on rainbow vertex antimagic coloring of graphs: a new notion marsidi 69 others, it must have a different vertex weight than the others. the vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. in the other hand, the vertex 𝑥𝑖 and 𝑦𝑖 are adjacent, such that based on the edge labeling it can not receive the same weight. from this condition, such that we have 𝑎𝑟𝑣𝑐(ℱ𝑛) ≥ 3. furthermore, to show the upper bound we construct the bijective function of edge labels. 𝑔(𝐴𝑥𝑖 ) = 𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 𝑔(𝑥𝑖 𝑦𝑖 ) = 2𝑛 + 1 − 𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 𝑔(𝐴𝑦𝑖 ) = 2𝑛 + 𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 from the edge labels above, we have the vertex weights in the following. 𝑤(𝑥𝑖 ) = 2𝑛 + 1 𝑤(𝑦𝑖 ) = 4𝑛 + 1 𝑤(𝐴) = 3𝑛2 + 𝑛 from the vertex weight above, it is easy to see that the different weight is 3. furthermore, we show that every two different vertices of ℱ𝑛is rainbow vertex antimagic coloring. suppose that 𝑥, 𝑦 ∈ 𝑉(ℱ𝑛), refer to the vertex weight the rainbow vertex 𝑥 − 𝑦 path is shown in table 3. table 3. the rainbow vertex of 𝑥 − 𝑦 path of ℱ𝑛 case 𝒙 𝒚 rainbow vertex coloring 𝒙 − 𝒚 1 𝑥𝑖 𝑥𝑖 𝑥𝑖 , 𝐴, 𝑥𝑖 2 𝑥𝑖 𝑦𝑖 𝑥𝑖 , 𝐴, 𝑦𝑖 3 𝑦𝑖 𝑦𝑖 𝑦𝑖 , 𝐴, 𝑦𝑖 4 𝑦𝑖 𝑥𝑖 𝑦𝑖 , 𝐴, 𝑥𝑖 hence, the vertex coloring of ℱ𝑛 is rainbow vertex antimagic coloring. thus, we obtain 𝑟𝑣𝑎𝑐(ℱ𝑛) is 3 . ∎ theorem 4 if 𝐹𝑛 be a fan graph 𝑛+1 and 𝑛 ≥ 3, then 𝑟𝑣𝑎𝑐(𝐹𝑛 ) = 2 if 𝑛 ≡ 1(mod 2) and 2 ≤ 𝑟𝑣𝑎𝑐(𝐹𝑛) ≤ 3 if 𝑛 ≡ 0(mod 2). proof. let 𝐹𝑛 be a fan graph with vertex set 𝑉(𝐹𝑛 ) = {𝐴, 𝑥1, 𝑥2, 𝑥3, ⋯ , 𝑥𝑛} and edge set 𝐸(𝐹𝑛 ) = {𝐴𝑥𝑖 : 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖 𝑥{𝑖+1}: 1 ≤ 𝑖 ≤ 𝑛 − 1}. the diameter of 𝐹𝑛 is 2. based on remark 1, we have 𝑟𝑣𝑎𝑐(𝐹𝑛) ≥ 𝑟𝑣𝑐(𝐹𝑛 ) = 𝑑𝑖𝑎𝑚(𝐹𝑛) − 1 = 2 − 1 = 1. since the vertex 𝐴 has degree of much greater than the others, it must have a different vertex weight than the others. the vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. from this condition, such that we have 𝑟𝑣𝑎𝑐(𝐹𝑛) ≥ 2. we divide into two cases to show the upper bound of the antimagic rainbow connection number of 𝐹𝑛 as follows. case 1. for 𝐹𝑛 , 𝑛 ≡ 1(mod 2) to show the upper bound of 𝑟𝑣𝑎𝑐(𝐹𝑛): 𝑛 ≡ 1(mod 2), we construct the bijective function of edge labels. 𝑔(𝑥𝑖 𝑥𝑖+1) = { 𝑖 2 , if 𝑖 ≡ 0(mod 2) 𝑛 + 𝑖 2 , if 𝑖 ≡ 1(mod 2) on rainbow vertex antimagic coloring of graphs: a new notion marsidi 70 𝑔(𝐴𝑥𝑖 ) = { 2𝑛 − 1, if 𝑖 = 𝑛 2𝑛 − 𝑖 − 1, if 1 ≤ 𝑖 ≤ 𝑛 − 1 from the edge labels above, we have the vertex weights in the following. 𝑤(𝑥𝑖 ) = 5𝑛 − 3 2 𝑤(𝐴) = 3𝑛2 − 𝑛 2 from the vertex weights above, it is easy to see that the different weight is 2. case 2. for 𝐹𝑛 , 𝑛 ≡ 0(mod 2) to show the upper bound of 𝑟𝑣𝑎𝑐(𝐹𝑛): 𝑛 ≡ 0(mod 2), we construct the bijective function of edge labels. 𝑔(𝑥𝑖 𝑥𝑖+1) = { 𝑖 2 , if 𝑖 ≡ 0(mod 2) 𝑛 + 𝑖 − 1 2 , if 𝑖 ≡ 1(mod 2) 𝑔(𝐴𝑥𝑖 ) = { 2𝑛 − 1, if 𝑖 = 𝑛 2𝑛 − 𝑖 − 1, if 1 ≤ 𝑖 ≤ 𝑛 − 1 from the edge labels above, we have the vertex weights in the following. 𝑤(𝑥𝑖 ) = { 3𝑛 − 2, if 𝑖 = 𝑛 5𝑛 2 − 2, if 1 ≤ 𝑖 ≤ 𝑛 − 1 𝑤(𝐴) = 3𝑛2 − 𝑛 2 from the vertex weight above, it is easy to see that the different weight is 3. furthermore, we show that every two different vertices of 𝐹𝑛 is rainbow vertex antimagic coloring. suppose that𝑥, 𝑦 ∈ 𝑉(𝐹𝑛 ), refer to the vertex weight the rainbow vertex 𝑥 − 𝑦 path is shown in table 4. table 4. the rainbow vertex of 𝑥 − 𝑦 path of 𝐹𝑛 case 𝒙 𝒚 rainbow vertex coloring 𝒙 − 𝒚 1 𝑥𝑖 𝐴 𝑥𝑖 , 𝐴 2 𝑥𝑖 𝑥𝑖 𝑥𝑖 , 𝐴, 𝑥𝑖 hence, the vertex coloring of 𝐹𝑛 is rainbow vertex antimagic coloring. thus, we obtain 𝑟𝑣𝑎𝑐(𝐹𝑛) = 2 if 𝑛 ≡ 1(mod 2) and 2 ≤ 𝑟𝑣𝑎𝑐(𝐹𝑛 ) ≤ 3 if 𝑛 ≡ 0(mod 2). ∎ the illustration of antimagic rainbow edge labeling can be seen in figure 1. based on the figure 1, we know that wheel graph 𝑊17 satisfy the rainbow vertex antimagic coloring and rainbow vertex antimagic connection number of 𝑊17 is 2. on rainbow vertex antimagic coloring of graphs: a new notion marsidi 71 figure 2. the illustration rainbow vertex antimagic coloring of 𝑊17 conclusions we have obtained the exact values of rainbow vertex antimagic connection number of some connected graphs, namely path (𝑃𝑛 ), wheel (𝑊𝑛), friendship (ℱ𝑛), and fan (𝐹𝑛 ). however, since obtaining rainbow vertex antimagic connection number of graph is considered to be np-complete 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[15] h. s. budi, dafik, i. m. tirta, i. h. agustin, and a. i. kristiana, “on rainbow antimagic coloring of graphs,” j. phys. conf. ser., vol. 1832, no. 1, 2021, doi: 10.1088/17426596/1832/1/012016. analysis of the rosenzweig-macarthur predator-prey model with anti-predator behavior cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 260-269 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 22, 2021 reviewed: march 17, 2021 accepted: april 14, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11472 analysis of the rosenzweig-macarthur predator-prey model with anti-predator behavior ismail djakaria1, muhammad bachtiar gaib2 , resmawan3 1,2,3department of mathematics, universitas negeri gorontalo, indonesia email: iskar@ung.ac.id, m.tiargaib@gmail.com, resmawan@ung.ac.id abstract this paper discusses the analysis of the rosenzweig-macarthur predator-prey model with antipredator behavior. the analysis is started by determining the equilibrium points, existence, and conditions of the stability. identifying the type of hopf bifurcation by using the divergence criterion. it has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (𝐸0), the non-predatory equilibrium point (𝐸1), and the co-existence equilibrium point (𝐸2). the existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. the divergence criterion indicates the existence of the supercritical hopf-bifurcation around the equilibrium point 𝐸2. finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order runge-kutta methods. keywords: rosenzweig-macarthur; predator-prey model; anti-predator behaviour; hopf bifurcation; divergence criterion; equilibrium point. introduction population dynamics are the most interesting research in mathematical biology which discusses the interactions that occur between prey and predator in a particular ecosystem [1]. this interaction has implemented to a simple mathematical model known as the lotka-volterra predator-prey model [2]. in a mathematical model, the predation process (interaction between prey and predator) is expressed in some form that is known as a functional response. this functional response has classified three functions, i.e. holling-type i, holling-type ii, and holling-type iii where each type determine the characteristic of the predator [3]. on the progress, rosenzweig and macarthur modifying the lotka-volterra predator-prey model with the assumption the attack rate of predator increases at a decreasing rate with prey density until it becomes constant due to satiation which is affected by holling-type ii functional response [4]. further, some modified of lotka-volterra predator-prey model by considering the infectious disease [5]-[7]. several research has discussed the modification of the rosenzweig-macarthur predator-prey model [8][9] is introduced predator foraging facilitation into holling-type ii functional response. furthermore, the rosenzweig-macarthur model has modified with various factors, e.g. the stage-structure [10][11], the refuge effect [12][13], the harvesting to one or more population [14][15]. from several studies described above, no one http://dx.doi.org/10.18860/ca.v6i4.11472 mailto:iskar@ung.ac.id mailto:m.tiargaib@gmail.com mailto:resmawan@ung.ac.id analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 261 (1) considering anti-predator behavior factors. in this article, the rosenzweig-macarthur predator-prey model by [6] modified considering anti-predator behavior factors [16]. these factors can be considered in the model because the dynamics of the model will be very complex when the prey population prefers to defending and provide resistance when the predation process is occurring. the structure of this paper is as follows. in the next section, the methods in our work are described. then, the analysis of the model has been discussed. finally, a brief conclusion of our work is given. methods the dynamics of the model is analyzed by carrying out the following steps: 1. modifying the rosenzweig-macarthur predator-prey model considering antipredator behavior factors. 2. simplifying the model by using non-dimensional to reduce the number of parameters and solving the equilibrium points of the model. 3. identifying the existence, local stability, and global stability of the equilibrium points. 4. identifying the hopf-bifurcation type by using the divergence criterion. 5. demonstrated the numerical simulations of the model to describe the analysis results by using the 4th-order runge-kutta method. results and discussion mathematical model in this article, the mathematical model is formulated based on the following assumptions: 1. the prey population is assumed to grow logistically with an intrinsic growth rate of 𝑟 and carrying capacity of the environment of 𝐾 and reduced due to the predation process. 2. the predator population is assumed to grow due to the predation process. 𝑐 is the conversion rate of the consumed prey into predator births. 3. the predation process follows holling-type ii functional response which is affected by the encounter rate function where there is foraging facilitation of predator (𝑤 = 0), 𝑎 is the saturated rate of the predator, 𝑏 is coefficient interaction on both population and ℎ is the predator time handling. 4. 𝑚 is the mortality of predators. 5. 𝜂 is the anti-predator behavior. from the following assumptions above, the dynamics of the model can be represented by the following set of differential equations: 𝑑𝑥 𝑑𝑡 = 𝑟𝑥(1− 𝑥 𝐾 )− (𝑎 −𝑏)𝑥𝑦 𝑦 +ℎ(𝑎 −𝑏)𝑥 𝑑𝑦 𝑑𝑡 = 𝑐(𝑎 −𝑏)𝑥𝑦 𝑦 +ℎ(𝑎 −𝑏)𝑥 −𝑚𝑦 −𝜂𝑥𝑦 where 𝑥 and 𝑦 are respectively the densities of prey and predator population at time 𝑡 and 𝑥(0),𝑦(0) > 0. analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 262 (2) (3) (4) to simplify our analysis, we reduce the number of parameters in system (1) by using the following parameter scales [17]: 𝑥 → 𝑥𝐾, 𝑦 → 𝑦(𝑎 −𝑏)𝐾ℎ, 𝑡 → 𝑡 𝑟 we obtain the following non-dimensional model 𝑑𝑥 𝑑𝑡 = 𝑥(1−𝑥)− 𝛼𝑥𝑦 𝑥 +𝑦 𝑑𝑦 𝑑𝑡 = 𝛽𝑥𝑦 𝑥 +𝑦 −𝛾𝑦 −𝛿𝑥𝑦 where 𝛼 = (𝑎 −𝑏) 𝑟 , 𝛽 = 𝑐 ℎ𝑟 , 𝛾 = 𝑚 𝑟 , 𝛿 = 𝜂𝐾 𝑟 existence and stability analysis of equilibrium points in this section, the equilibrium point of model (2) is obtained by solving [18]: 𝑥(1−𝑥)− 𝛼𝑥𝑦 𝑥 +𝑦 = 0 𝛽𝑥𝑦 𝑥 +𝑦 −𝛾𝑦 −𝛿𝑥𝑦 = 0 thus, from the system (3), we obtain the following equilibrium points, i.e.: 1. a trivial equilibrium point 𝐸0 = (0,0), always exists. 2. a non-predator equilibrium point 𝐸1 = (1,0), always exists too. 3. a co-existence equilibrium point 𝐸2 = (𝑥 ∗,𝑦∗), where 𝑥∗ = 𝛽 −𝛼𝛽 +𝛼𝛾 𝛽 −𝛼𝛿 , 𝑦∗ = (𝛽 −𝛼𝛽 +𝛼𝛾)(𝛽 −𝛾 −𝛿) (𝛽 −𝛼𝛿)(𝛾 +𝛿 −𝛼𝛿) which exists if 𝛽 > 𝛼(𝛽 −𝛾), 𝛾 +𝛿 < 𝛼𝛿 < 𝛽 now, study the local stability of the dynamics of the system (3) around each of equilibrium point. the jacobian matrix from the system (3) is determined as [19]: 𝐽(𝑥,𝑦) = ( 1−2𝑥 − 𝛼𝑦 𝑥 +𝑦 + 𝛼𝑥𝑦 (𝑥 +𝑦)2 − 𝛼𝑥 𝑥 +𝑦 + 𝛼𝑥𝑦 (𝑥 +𝑦)2 𝛽𝑦 𝑥 +𝑦 − 𝛽𝑥𝑦 (𝑥 +𝑦)2 −𝛿𝑦 𝛽𝑥 𝑥 +𝑦 − 𝛽𝑥𝑦 (𝑥 +𝑦)2 −𝛾 −𝛿𝑥 ) by evaluating this jacobian matrix (4) at each equilibrium point, we obtain the local stability properties of 𝐸0, 𝐸1, and 𝐸2 as follows. analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 263 theorem 1. the trivial equilibrium point 𝐸0 always unstable (saddle). proof: the jacobian matrix (4) evaluated in equilibrium point 𝐸0 is given by 𝐽(𝐸0) = ( 1 0 0 −𝛾 ) so, by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸0) is 𝜆1 = 1 and 𝜆2 = −𝛾. it means 𝜆1 > 0 and 𝜆2 < 0. therefore, stability of equilibrium point 𝐸0 is unstable (saddle).∎ theorem 2. if 𝛿 > 𝛽 −𝛾, then the non-predatory equilibrium point 𝐸1 of system (2) is locally asymptotically stable. proof: the jacobian matrix (4) evaluated in equilibrium point 𝐸1 is given by 𝐽(𝐸1) = ( −1 −𝛼 0 𝛽 −𝛾 −𝛿 ) so, by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸1) is 𝜆1 = −1 and 𝜆2 = 𝛽 −𝛾 −𝛿. it means 𝜆1 < 0. therefore, if 𝛿 > 𝛽 −𝛾 then each the eigenvalues of 𝐽(𝐸1) are negatif, and 𝐸1 is locally asymptotically stable.∎ theorem 3. the co-existence equilibrium point 𝐸2 is locally asymptotically stable if the conditions below are satisfied 𝛿2 < θ+υ ζ proof: the jacobian matrix (4) evaluated in equilibrium point 𝐸1 is given by 𝐽(𝐸2) = ( 𝑀11 𝑀12 𝑀21 𝑀22 ) where 𝑀11 = −𝛽2 +𝛼𝛽2 −𝛼𝛾2 −2𝛼𝛿(𝛼 −1)(𝛽 −𝛾)−𝛼𝛿2 +𝛼2𝛿2 (𝛽 −𝛼𝛿)2 𝑀12 = − 𝛼(𝛾 +𝛿 −𝛼𝛿)2 (𝛽 −𝛼𝛿)2 𝑀21 = (𝛽 −𝛾 −𝛿)(𝛽2𝛾 +𝛼2𝛾𝛿2 −𝛽(𝛾2 +2𝛾𝛿 +𝛿2(𝛼 −1)2)) (𝛽 −𝛼𝛿)2 𝑀22 = − 𝛽(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿) (𝛽 −𝛼𝛿)2 by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸2) is 𝜆1,2 = 1 2 . 1 (𝛽 −𝛼𝛿)2 (𝐴±𝐵) analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 264 where 𝐴 = ζ𝛿2 −θ−υ and 𝐵 = ψ2 −𝛼ω with ζ = (𝛼2 −𝛼 +𝛽 −𝛼𝛽) θ = 𝛿(𝛽(𝛽 −2𝛾)+2𝛼2(𝛽 −𝛾)−𝛼(𝛽2 +2(𝛽 −𝛾)−𝛽𝛾)) υ = 𝛽2(𝛾 −𝛼 +1)+𝛾2(𝛼 +𝛽) ψ = (𝛽2 −𝛼𝛽2 +𝛼𝛾2 +2𝛼𝛿(𝛼 −1)(𝛽 −𝛾)−𝛼𝛿2(𝛼 −1)−𝛽(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿)) ω = 4(𝛽 −𝛾 −𝛿)(𝛾 +𝛿 −𝛼𝛿)(𝛽2𝛾 +𝛼2𝛾𝛿2 −𝛽((𝛼 −1)2𝛿2 +𝛾2 +2𝛾𝛿)) according to (), the stability of equilibrium point 𝐸2 depending on the value of 𝐴. if 𝐴 < 0, we obtained: ζ𝛿2 −θ𝛿 −υ < 0 ζ𝛿2 < θ𝛿 +υ 𝛿2 < θ+υ ζ by the conditions above, the stability of equilibrium point 𝐸2 is locally asymptotically stable.∎ next, study the global stability of the dynamics of the system (3) around equilibrium point 𝐸2. we obtain the global stability properties of 𝐸2 by using the lyapunov function [20] as follows. theorem 4. the co-existence equilibrium 𝐸2 is globally asymptotically stable if the conditions below are satisfied: 𝑥∗ < (𝛼 −𝛽 +𝛾 +𝛿)(𝛾 +𝛿 −𝛼𝛿) 𝛼(𝛾 +𝛿 −𝛼𝛿)−(𝛽 −𝛾 −𝛿)2 proof: define a lyapunov function as follows 𝑉(𝑥,𝑦) = [𝑥 −𝑥∗ −𝑥∗ ln( 𝑥 𝑥∗ )]+[𝑦 −𝑦∗ −𝑦∗ ln( 𝑦 𝑦∗ )] by using the function �̇� < 0,∀ (𝑥,𝑦) ∈ ℝ2 +, we obtain: 𝜕𝑉 𝜕𝑥 . 𝜕𝑥 𝜕𝑡 + 𝜕𝑉 𝜕𝑦 . 𝜕𝑦 𝜕𝑡 ≤ 0 (1− 𝑥∗ 𝑥 )(𝑥(1−𝑥)− 𝛼𝑥𝑦 𝑥 +𝑦 )+(1− 𝑦∗ 𝑦 )( 𝛽𝑥𝑦 𝑥 +𝑦 −𝛾𝑦 −𝛿𝑥𝑦) ≤ 0 ( (1−𝑥)(𝑥 +𝑦)−𝛼𝑦 𝑥 +𝑦 )(𝑥 −𝑥∗)+( 𝛽𝑥 −𝛾(𝑥 +𝑦)−𝛿𝑥(𝑥 +𝑦) 𝑥 +𝑦 )(𝑦 −𝑦∗) ≤ 0 for (𝑥,𝑦) ∈ ℝ2 +, we obtain: analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 265 (5) −𝛼 +𝛼𝑥∗ +𝛽 −𝛾 −𝛿 −(𝛽 −𝛾 −𝛿)𝑦∗ < 0 −𝛼 +𝛼𝑥∗ +𝛽 −𝛾 −𝛿 −𝑥∗ (𝛽 −𝛾 −𝛿)2 (𝛾 +𝛿 −𝛼𝛿) < 0 𝑥∗ ( 𝛼(𝛾 +𝛿 −𝛼𝛿)−((𝛽 −𝛾 −𝛿)2) (𝛾 +𝛿 −𝛼𝛿) ) < 𝛼 −𝛽 +𝛾 +𝛿 𝑥∗ < (𝛾 +𝛿 −𝛼𝛿)(𝛼 −𝛽 +𝛾 +𝛿) 𝛼(𝛾 +𝛿 −𝛼𝛿)−((𝛽 −𝛾 −𝛿)2) by the conditions above, the stability of equilibrium point 𝐸2 is globally asymptotically stable.∎ analysis of hopf bifurcation type in this section, we’ll define the hopf-bifurcation type by using the divergence criterion [21]. system (3) underwent a hopf-bifurcation when it satisfies the following conditions: 𝛿2 < θ+υ ζ and 𝛼 > ψ2 ω to determine the hopf-bifurcation type of system (3) on equilibrium point 𝐸2, then we formed a new system. let 𝜙(𝑥,𝑦) is a divergence of (𝑎𝑓,𝑎𝑔). we obtain the coefficient value of 𝑎(𝑥,𝑦) of the system (3) when the parameter value 𝛼 = 2, 𝛽 = 0.79, 𝛾 = 0.5, and 𝛿 = 0.0186 with equilibrium point 𝐸2 ∗ = (0.279;0.157) as follows: 𝑎(𝑥,𝑦) = 1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2 so that a new system is obtained: 𝑧(𝑥,𝑦) = (1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2) (𝑥(1−𝑥)− 𝛼𝑥𝑦 𝑥 +𝑦 ) 𝑤(𝑥,𝑦) = (1+6.956𝑥 +13,386𝑦 −6.77𝑥2 +32.968𝑥𝑦 +55.507𝑦2) ( 𝛽𝑥𝑦 𝑥 +𝑦 −𝛾𝑦 −𝛿𝑥𝑦) by linearizing system (4), we obtained: 𝐽(𝐸2 ∗) = ( 1.337 −6.002 0.732 −1.337 ) by solving the characteristic equation, we obtained the eigenvalues of 𝐽(𝐸2 ∗) is 𝜆1,2 = ±1.615𝑖 for a system (5) to obtain the eigenvalues of conjugate complex numbers, then we can analyze the hopf-bifurcation of system (3) type by looking at the divergence value of system (3). we obtained: 𝜙𝑥𝑥(𝐸2 ∗) = −21.109 analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 266 based on the divergence value above, a stable limit cycle appears in the system (3). therefore, system (3) underwent a supercritical hopf-bifurcation. numerical simulations in this section, the numerical simulation is solved using the 4th-order runge-kutta method [22] with initial conditions and some values of the parameters. we choose the following set of parameter values: 𝛼 = 2, 𝛽 = 0.79, 𝛾 = 0.5 with different parameter control values as follows 𝛿1 = 0.011, 𝛿2 = 0.0186 and 𝛿3 = 0.026. we using the initial condition is 𝑥(0) = 0.3 and 𝑦(0) = 0.3. (a) (b) figure 1. (a) phase portrait of case 1 and (b) time-series portrait (a) (b) figure 2. (a) phase portrait of case 2 and (b) time-series portrait in case 1, we obtained the dynamics of the solution on the system (3) with parameter control values 𝛿1 = 0.011. based on figure 1(a), the trivial equilibrium point 𝐸0 = (0,0) is unstable (saddle) with eigenvalues 𝜆1 = 1 and 𝜆2 = −0.5. this coincides with theorem 1. the non-predator equilibrium point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 𝜆1 = −1 and 𝜆2 = 0.279. this coincides with theorem 2 on condition 𝛿 < 𝛽 −𝛾. the co-existence equilibrium point 𝐸2 = (0.273;0.156) is unstable (spiral) with eigenvalues 𝜆1,2 = 0.003±0.220𝑖. this analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 267 coincides with theorem 3 on condition 𝛿2 < θ+υ ζ . based on figure 1(b), the prey population and predator population have increased and decreased of total populations. the case continuously oscillates with a greater deviation value. hence, both population is unstable to a specific point. (a) (b) figure 3. (a) phase portrait of case 3 and (b) time-series portrait in case 2, we obtained the dynamics of the solution on the system (3) with parameter control values 𝛿1 = 0.0186. based on figure 2(a), the trivial equilibrium point 𝐸0 = (0,0) is unstable (saddle) with eigenvalues 𝜆1 = 1 and 𝜆2 = −0.5. this coincides with theorem 1. the non-predator equilibrium point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 𝜆1 = −1 and 𝜆2 = 0.271. this coincides with theorem 2 on condition 𝛿 < 𝛽 −𝛾. the coexistence equilibrium point 𝐸2 = (0.279;0.157) is center (spiral) with eigenvalues 𝜆1,2 = ±0.220𝑖. this coincides with theorem 3 on condition 𝛿2 = θ+υ ζ . based on figure 2(b), the oscillations that occur have a smaller deviation value. this condition explains that there is a stability transition from unstable to stable to a specific point. this stability transition has led to the appearance of hopf-bifurcation. in case 3, we obtained the dynamics of the solution on the system (3) with parameter control values 𝛿1 = 0.026. based on figure 3(a), the trivial equilibrium point 𝐸0 = (0,0) is unstable (saddle) with eigenvalues 𝜆1 = 1 and 𝜆2 = −0.5. this coincides with theorem 1. the non-predator equilibrium point 𝐸1 = (1,0) is unstable (saddle) with eigenvalues 𝜆1 = −1 and 𝜆2 = 0.264. this coincides with theorem 2 on condition 𝛿 < 𝛽 −𝛾. the coexistence equilibrium point 𝐸2 = (0.285;0.159) is stable (spiral) with eigenvalues 𝜆1,2 = −0.003±0.220𝑖. this coincides with theorem 3 on condition 𝛿2 > θ+υ ζ . based on figure 3(b), the dynamics between prey and predator begin to stabilize at 1500 days to a specific point. conclusions the rosenzweig-macarthur predator-prey model with anti-predator behavior has been studied. from the analysis of system (2), we obtain three equilibrium points, i.e., the trivial equilibrium point (𝐸0), the non-predatory equilibrium point (𝐸1), and the coexistence equilibrium point (𝐸2). the local stability conditions of each equilibrium point have been appointed, and the global stability conditions of the co-existence equilibrium analysis of the rosenzweig-macarthur predator-prey model with anti-predator behaviour ismail djakaria 268 point (𝐸2) have been obtained. our analysis also showed that the model occurs a supercritical hopf-bifurcation by using the divergence criterion. numerical analytic has been simulated to verify the theoretical results. no one extinction matters in any population. references [1] p. b. turchin, complex population dynamics: a theoretical/empirical synthesis, priceton university press, 2003. 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[22] a. suryanto, metode numerik untuk persamaan diferensial biasa dan aplikasinya dengan matlab, universitas negeri malang, 2017. richards curve implementation for prediction of covid-19 spread in maluku province cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 195-206 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 10, 2021 reviewed: december 22, 2021 accepted: january 05, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.13323 richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi, francis yunito rumlawang, yopi andry lesnussa* department of mathematics, faculty of mathematics and natural sciences, pattimura university, indonesia *corresponding author email: yopi_a_lesnussa@yahoo.com* rumlawang@yahoo.com, nanangondi21@gmail.com abstract the first case of covid-19 in maluku province, indonesia was reported at the end of march 2020 as many as 1 case and the total cumulative cases reported were 3.884 cases on november 4, 2020. the purpose of this study is to predict the spread of covid-19 cases in maluku province by estimating the richards function parameters are i is the population size, k is carrying capacity, k is the growth rate, a is the scaling parameter and mt is the turning point using the nonlinear least-squares (nls) method. the method use in this research is richards curve method. the results of this research found the estimation results, with rmse = 75,1057, the peak of the spread of covid-19 cases in the maluku province is predicted to occur on october 22, 2020, with a total of 3.623 cases and ends on may 25, 2023, with a total of 9.451 cases. this research can provide an overview of the results of predictions for the development of covid-19 for the government, making it easier for the government to make decisions in the future. keywords: carrying capacity; covid-19; prediction; richards curve; turning point introduction coronavirus is a group of viruses from the subfamily orthocronavirinae in the coronaviridae family and the order nidovirales. this group of viruses can cause disease in birds and mammals, including humans [1]. in 2002, the sars-cov coronavirus (sars coronavirus) caused severe acute respiratory syndrome (sars) in guangdong, china [2]. in 2012 the type of coronavirus mers-cov (mers coronavirus) caused middle eastern respiratory syndrome (mers) which occurred in saudi arabia and the middle east [3]. in early 2020, who (world health organization) received a report from china that there were 44 patients with severe pneumonia in wuhan city, hubei province, china [4]. subsequent research showed a close relationship with the coronavirus that caused sars in 2002 [5]. on february 11, 2020, who inaugurated the term covid-19 (coronavirus disease 2019) which is an infectious disease similar to influenza caused by severe acute respiratory syndrome 2 (sars-cov-2) [6], [7]. the first covid-19 was reported in indonesia on march 23, 2020, with two cases. data on march 31, 2020, showed that there were 1,528 confirmed cases and 136 deaths. http://dx.doi.org/10.18860/ca.v7i1.13323 mailto:yopi_a_lesnussa@yahoo.com* mailto:rumlawang@yahoo.com mailto:nanangondi21@gmail.com richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 196 in 1839 verhulst introduced the logistics equation to model population growth which became known as the verhulst equation and was rediscovered in 1912 [8], [9]. in 1959 in research entitled: a flexible growth function for empirical use, richards modified the verhulst equation and became known as the richards curve [10] or generalized logistic function [11] because it is an extension of the logistic model [12], [13] and in some literature, the richards curve is also called the theta logistic model [14], [15] with parameters namely k (carrying capacity), k (growth rate), mt (inflection point) and a (scaling parameter) the shape of the richards curve resembles the shape of the exponential curve [16]. richards curve is a model of a population growth curve in conditions where growth is not symmetrical with inflexion points [17], [18]. in 2004 the richards curve was used to predict the spread of sars in singapore, hong kong and beijing [19] after estimation with the richards curve, the results obtained are that the spread of sars in beijing is predicted to end on 27 june 2003 with a total of 2.595 cases, in hong kong it is predicted to end on 29 june 2003 with a total of 1.748 cases and in singapore it is predicted to end in may 28, 2003 with a total of 207 cases. the prediction results of the spread of sars in singapore, hong kong and beijing using the richards curve were considered quite successful, because based on the data obtained, singapore last reported cases of sars on may 18, 2003 with a total of 206 cases, hong kong on june 11, 2003 with a total of cases of 1.755 cases and beijing on june 11, 2003 with a total of 2.631 cases. besides that, the richards curve was widely used in other studies [20]–[22] and in 2020, the richards curve was used to predict the spread of covid-19 in the province of south sulawesi, indonesia, with the peak of the spread predicted to occur in mid-june 2020 july 2020 with a total of 10,000-12,000 cases and the end of the spread is predicted to occur at the end of november 2020. based on the above background, where the richards curve is considered quite good in predicting the spread of sars in singapore, hong kong and beijing in 2002, therefore in this study the richards curve will be used to predict the spread of covid-19 in maluku province. methods in general, the differential form of the richards curve is : [10], [23]   1 a di i i t ri dt k                    (1) where i is the population size, k is carrying capacity, k is the growth rate and a is the scaling parameter. to find a solution to equation 1, the integration technique can be written as: a a a k di r dt i k i                or it can be written: richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 197 based on the similarity of the two sides, the values of 1a  and 1a b i   are obtained so as to obtain : a a aa a k a b di di i k ii k i                      1 1 a a a i di di i k i                  so we get:   1 ln ln a a a a a a k di i k i i k i                        since we get r dt rt c  , we can write: :     1 ln ln a a a i k i rt c           or it can be written : so we get:   1 a a a rt c k i e i                to simplify the above form, both sides can be raised to the power of a so that we get: :   1 a a art ac k i e e           (2)       a a a a a k i b i di r dt i k i                  1 ln a a a k i rt c i              richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 198 from equation 2, since ,a r and c are constants, it is assumed that k is the product of ar and q is the product of  ac e  , so it can be written as:  1 a a kt k i q e          so we get:     1 1 kt a k i t q e               (3) since the inflection point of equation 3 is   1 1 a k a         [24] , let mt be the parameter of the inflection point of equation 3 then it can be written as : [25]     1 1 m k t t a k i t ae                 (4) where i is the population size or the total number of cases that occurred at the time of t , k is the carrying capacity or total of the latest cases, k is the rate of growth of cases, mt is the inflexion point or time of the peak of the spread of covid-19 cases where       1 1 1 1 m m m k t t a a k k i t aae                        . results and discussion covid-19 cases in maluku province have continued to increase since it was first reported on march 23, 2020, and as of november 4, 2020, the total cumulative cases of covid-19 in maluku province were reported as many as 3,884 cases, including 551 positive patient cases or with a percentage of 14.18%, 3,286 cases of patients cured or with a percentage of 84.6 and 47 cases of patients dying or with a percentage of 1.2%. cumulative case developments and the addition of daily cases of covid-19 in maluku province from march 23, 2020 – november 4, 2020, can be described as follows: table 1. cumulative and daily case data of covid-19 in maluku province date cumulative cases daily cases march 23, 2020 1 0 march 24, 2020 1 0 march 25, 2020 1 0 march 26, 2020 1 0 richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 199 date cumulative cases daily cases march 27, 2020 1 0 … … … october 30, 2020 3849 59 october 31, 2020 3851 2 november 1, 2020 3863 12 november 2, 2020 3863 0 november 3, 2020 3877 14 november 4, 2020 3884 7 the development of cumulative covid-19 cases in maluku province from 23 march – 4 november 2020 can be described as follows: figure 1. cumulative case development the first case of the spread of covid-19 in maluku province was reported on march 23, 2020, as many as 1 case and up to july 5, 2020 the total cumulative cases reported were 794 cases or with a growth rate of 755, 23%. on july 6 to october 22, 2020 the average daily addition of cases increased to 26 cases with the average growth rate increasing significantly as much as %1864, 953 from the previous one, which was 2620,183%, and from october 23 to november 4, 2020, the average increase in cases the daily rate of covid-19 in maluku province decreased by 18 cases. the graph of the daily increase in cases can be seen in figure 2, where the maluku province experienced the highest number of cases on october 2, 2020, which was 117 cases. figure 2. development of daily cases of covid-19 in maluku province 23 march – 4 november 2020 richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 200 parameter estimation results by using data on cumulative cases of covid-19 in maluku province from march 23 – november 4, 2020, an estimate was made with the richards function parameter using the nonlinear least square method in python with the following script: import scipy.optimize as optimize from scipy.optimize import curve_fit import numpy as np import pandas as pd def richardsfunction(t,k,a,k,tm): return k/(1 + a*(np.exp(-k*(t-tm))))**(1/a) df=pd.read_excel('coviddate_maluku.xlsx') data=df[0:227] y=data['cummulativecases'] t=np.arange(1,228,1) popt,pcov=optimize.curve_fit(richardsfunction,t,y,bounds=(0.01,np.inf)) the results obtained are: table 2. richards parameter estimation results (rmse: 75,1057) k a k mt 9.451,245 0,085 0,01 213,918 so by substituting the parameter values k , a , k and mt in equation (4) obtained the richards equation, namely :     1 20.01 0,08591 113, 8 9.451, 245 1 0, 085 t i t e                then it can be illustrated that the comparison between the cumulative covid19 case data from the richards function parameter estimation results with the actual data for [1, 227]t  is as follows: figure 3. comparison of the results of predictions of cumulative cases of covid-19 in maluku province with actual data richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 201 in figure 3. the cumulative comparison between the actual data and the predicted data using the richards function at the time of t = 1 to t = 227, we get rmse = 75.1057 while using the logistic function we get a larger rmse value of 85.1813. the comparison of the error values between the predicted results and the actual data can be seen in the following table 3: table 3. the error value of the predicted data with the actual data t actual predict error 1 1 16.65457518205870 15.65457518205870 2 1 17.48844921400830 16.48844921400830 3 1 18.35884011574600 17.35884011574600 4 1 19.26706628665680 18.26706628665680 5 1 20.21448005719650 19.21448005719650 … … … … 222 3849 3889.25714338151000 40.25714338151370 223 3851 3922.50525096264000 71.50525096263660 224 3863 3955.72658861666000 92.72658861666100 225 3863 3988.91835825547000 125.91835825547500 226 3877 4022.07779333937000 145.07779333936700 227 3884 4055.20215931066000 171.20215931065600 from figure 3, the richards curve can be described from the estimation results as follows: figure 4. richards curve of estimation results from figure 4, suppose that  ii t is the total cumulative cases on day i and  1ii t  is the total cumulative cases on day 1i  , then the total addition of daily cases can be formulated as follows : [26]      1 ; 1, 2, 3,...i i ij t i t i t i   (5) so the comparison between the predicted data and actual data from daily covid-19 cases in maluku province can be described as follows: richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 202 figure 5. comparison of the results of daily covid-19 case predictions in maluku province with actual data from figure 5, it can be seen that the results of daily case predictions for covid19 in maluku province are as follows: figure 6. daily cases of covid-19 in maluku province from estimated result turning point of case deployment from the results of richards parameter estimation with data on covid-19 cases in maluku province, the parameter mt value is 213.918, meaning that the time of the turning point for the spread of covid-19 in maluku province is predicted to occur on the 214th day, where the total cases on the 214th day are obtained. from the equation:     1 0.01 214 0,085213,91 18 9.451, 245 214 1 0, 085 i e                which is 3.622,654 means that the total cases at the inflection point are 3.623 cases or can be described as follows: figure 7. inflection point richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 203 for the addition of daily cases, the total addition of cases can be obtained at the inflexion point or when m t t namely :    214 213i i        1 1 0.01 214 0.01 2130,0851 0,081 5213,918 213,9 8 1 9.451, 245 9.451, 245 33, 358161 1 0, 085 1 0, 085e e                                  so the total addition of daily cases at the inflection point is 33 cases, so it can be concluded that the turning point of the covid-19 case in maluku province is based on the estimation results, namely     , 214, 33t i t  or can be described as follows : figure 8. turning point in figure 7, the point     , 214, 33t i t  which is the turning point of the curve is also the peak of the curve, namely when t = 214. end of case deployment from richards parameter estimation results with data on covid-19 cases in maluku province, the parameter k value is 9,451,245, meaning that the latest total cases for covid-19 cases in maluku province are predicted to be 9,451 cases. for example, if endt is the end time of covid-19 cases in maluku province, with a total of 9,450.5 cases or can be written as  endi t = 9.450,5 then the value of endt can be obtained from the equation:  . 213,918 1 0 01 0,0851 9.451, 245 9.450, 5 1 0, 085 end t e                 that is endt = 1.158,681 meaning that the time for the end of the covid-19 case in maluku province is predicted to occur on the 1.159th day. richards curve implementation for prediction of covid-19 spread in maluku province nanang ondi 204 figure 9. total case when 1.159t  from figure 9, when 1.159t  the population size will always be at number 1.159 and will only move towards the value of k or carrying capacity. conclusions from the estimation results of the richards function parameter with the cumulative case data of covid-19 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[26] m. höök, j. li, n. oba, and s. snowden, “descriptive and predictive growth curves in energy system analysis,” natural resources research. 2011, doi: 10.1007/s11053011-9139-z. forecasting population of madiun regency using arima method cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 420-431 p-issn: 2086-0382; e-issn: 2477-3344 submitted: may 26, 2022 reviewed: july 19, 2022 accepted: july 25, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.16156 forecasting population of madiun regency using arima method yuniar farida*, mayandah farmita, nurissaidah ulinnuha, dian yuliati department of mathematics, faculty of science and technology, sunan ampel state islamic university surabaya, indonesia email: yuniar_farida@uinsby.ac.id abstract the high population growth of the madiun regency can cause population density that can have implications for other problems, both in terms of social, economic, welfare, security, land availability, availability of clean water, and food needs. this study aims to predict the population growth of madiun regency using the arima method. the arima method is popular for forecasting time series data, which is reliable because the calculation process is done gradually. the arima method has three models, namely ar (autoregressive), ma (moving average), and arma (autoregressive moving average). this study uses annual population data of madiun regency from 1983 to 2021 and produces an arima forecasting model (0,2,1) with a mape value of 8.42%. this study also showed that from 2022 to 2024 is predicted to increase by 17947 people or 2.39%. the results of this study are expected to be used as information from the madiun regency government in anticipating the emergence of problems caused by the population level of madiun regency in the future. keywords: arima; forecasting; population; time series analysis introduction one of the most important problems globally is the high population growth in developing countries [1], [2]. indonesia is listed as one of the five most populous countries in the world. indonesia ranks fourth after china, india, and the united states and is the most populous asian continent [3]. according to the 2020 population census, conducted in 2020, the population of indonesia reached 270.20 million. indonesia has a land area of 1.9 million km2 and a population density of 141 people per km2, with an average annual population growth rate of 1.25% between 2010 and 2020 [4]. in indonesia, java ranks first as the most populous island, and east java is the second-most populous province after west java, with 40.67 million people. [4]. east java province consists of several cities and regencies, one of which is madiun regency, whose population growth rate ranks fifth in the 2020 period and experiences population growth every year. in 2015 the population was 676,087 people, while its development in 2019 was 749,070 [5]. population growth in madiun regency is affected by the high birth rate. in 2020, the birth rate in madiun regency will be 3928, with a population growth rate of 0.92% [5]. high population growth can cause various problems, such as regional spatial problems, housing, employment, education, economy, and security. in addition, it can also http://dx.doi.org/10.18860/ca.v7i3.16156 mailto:yuniar_farida@uinsby.ac.id population forecasting of madiun regency using arima method yuniar farida 421 cause problems in social aspects, welfare, availability of clean water, food needs, and can cause environmental damage [1], [6], [7]. population density, which can cause problems, needs to be anticipated by making predictions so that various handling strategies can be carried out. several statistical mathematical-based forecasting models can predict it, including exponential smoothing, moving average, and arima models (box-jenkins). other forecasting models are based on artificial intelligence, such as neural networks, genetic algorithmic, simulated annealing, and classification [8]. several previous studies by xu et al., predicted beijing's main area population using the long short term memory (lstm) model with mape 4.35% [9]. next, a study that indicated the population of residents in east kalimantan using exponential smoothing by pakpahan, basani, and hariani that yielded a mape of 14.81% [10]. on the other, there are studies related to the prediction of population prediction using the arima method conducted by mardiyah et al. in pasuruan city [11], nyoni, mutongi, and munyaradzi in the gambia [12], and nyoni in zimbabwe obtained mape < 3.94% [13]. based on the previous studies, the authors are interested in the arima method, which results in an excellent level of accuracy in some cases of population forecasting. the arima method is flexible and straightforward in an application, and accurate prediction results for the short term, but the forecasting accuracy for long-term forecasting is not good and will usually tend to be flat for a long time [14], [15]. in applying practice and forecasting the population, arima is also widely used in various case studies, including research by alabdulrazzaq related to predicting the spread of covid-19 with mape 4.2% [16]. other research by swaraj about covid-19 predictions in india with mape 4.7% [17]. additional research by guha and bandyopadhyay predicts the price of gold with a mape of 3.25% [18]. another study by banerjee related forecasting on the indian stock market with mape 3.33% [19]. then there was research by grigonytė and butkevičiūtė about predicting wind speed in latvia with mape 1% [20]. based on the explanation above, this study used the arima method to predict the population of the madiun regency. this research is expected to provide information for the madiun regency government to take policy steps to minimize and reduce risk due to the high rate of population growth of madiun regency. methods the data the data used in this study is data on residents of madiun regency from 1983 to 2021, which is taken from the central bureau of statistics of madiun regency, from the website https://bit.ly/pendudukkabmadiun [21]. table 1. the population of madiun regency no. year population 1. 1983 638586 2. 1984 627467 ⋮ ⋮ ⋮ 38 2020 744350 39 2021 750143 https://bit.ly/pendudukkabmadiun population forecasting of madiun regency using arima method yuniar farida 422 arima box and jenkins first developed the arima model in the 1970s [22]. the arima is one of the econometric methods used to predict univariate time-series data. box and jenkins state that this model does not use independent variables but instead utilizes the information in the circuit to generate pre-predicted values. therefore, the arima model requires an autocorrelation process in the series. autocorrelation is the correlation between two observations at different points in a time series. in other words, time series data is self-correlated. time series models in the arima method include autoregressive (ar), moving average (ma), and autoregressive moving average (arma) [23], [24]. the analysis with arima box jenkins begins by creating a series of plot periods and plotting the acf to determine whether the data is mean-stationary or variancestationary. differentiation must be done if the data are not stationary to the mean. otherwise, if the data is not stationary to the variance, a box-cox transformation is performed. repeat the process for the data stationer. after getting stationary data, the next step is to predict the data from arima based on the acf and pacf plots. then use ljung-box to test the parameters of the test model as well as test the residual hypothesis, which is residual white noise. it can be concluded, there are several stages of forecasting in arima, namely model identification, parameter estimation, diagnostic testing, and prediction. model identification when identifying the model on the arima method, the data used must meet the stationary or stability requirements. if the data does not meet the stationery requirements, the data must be stationary for the variance and average (mean) [25]. the transformation equation is as follows [26]. 𝑇(𝑍𝑡) ′ = 𝑍𝑡 𝜆 𝜆 (1) where: 𝑇(𝑍𝑡) : transformed data value 𝑍𝑡 : i th time data value 𝜆 : the estimated value of transformation parameters the transformed data is determined by the lambda value. for example, the following table shows some commonly used 𝜆 values and associated transformations. table 2. 𝜆 value and transformation 𝝀 value transformation -1.0 1 𝑍𝑡⁄ -0.5 1 √𝑍𝑡⁄ 0.0 ln𝑍𝑡 0.5 √𝑍𝑡 1.0 𝑍𝑡 for time-series data that have not satisfied the stationarity of the average, the data must be processed differentially to find the difference between one data and the previous data in sequence. the differencing equation is as follows. 𝑍𝑡 ′ = 𝑍𝑡 − 𝑍𝑡−1 (2) population forecasting of madiun regency using arima method yuniar farida 423 𝑍𝑡 ′ is the differentiated data value, where 𝑍𝑡 is the i th time data value. if the data is already stationary, a tentative model of arima (p, d, q) is obtained. annotation p is a lag that exceeds the significance limit on the partial autocorrelation function (pacf) plot graph, d is the level of differencing performed, q is the lag that crosses the significance limit of the autocorrelation function (acf) plot. autoregressive is a model in which a dependent variable is influenced by the value of the dependent variable itself because the data used is single. in general, ar is p ordo, with the form 𝐴𝑅(𝑝) as follows [27]. 𝑋𝑡 = ∅0 + ∅1𝑋𝑡−1 + ∅2𝑋𝑡−2 + ⋯+ ∅𝑖𝑋𝑡−𝑖 + 𝛼𝑡 (3) where: 𝑋𝑡 : time series t 𝑋𝑡−𝑖 : time series t-i 𝛼𝑡 : time error value t ∅0 : constant ∅𝑖 : coefficients of autoregressive moving average is a model that measures autocorrelation between the error or residual values. the ma is generally ornate q, with the following form of 𝑀𝐴(𝑞) [28]. 𝑋𝑡 = 𝑒𝑡 − 𝜃1𝛼𝑡−1 − 𝜃2𝛼𝑡−2 − ⋯− 𝜃𝑖𝛼𝑡−𝑖 (4) where: 𝛼𝑡−𝑖 : time error value t-i 𝜃𝑖 : coefficient of moving average autoregressive moving average or arma (p, q), with the following general equations [29]. 𝑋𝑡 = ∅0 + ∅1𝑋𝑡−1 + ⋯+ ∅𝑖𝑋𝑡−𝑖 + 𝛼𝑡 − 𝜃1𝛼𝑡−1 − ⋯− 𝜃𝑖𝛼𝑡−𝑖 (5) where: 𝑋𝑡 : stationary time series autoregressive integrated moving average data used must be stationary. arima's general statement is as follows [30]. ∅0(𝐵)(1 − 𝐵) 𝑑𝑍𝑡 = 𝜃0 + 𝜃𝑞(𝐵)𝑎𝑡 (6) where: ∅0 : autoregressive process 𝜃𝑞 : moving average process (1 − 𝐵)𝑑 : differentiating operator 𝑑 : differencing parameter 𝐵 : step-back operator 𝑍𝑡 : deviations from the average process parameter estimation tentative model determination requires several estimation stages through model feasibility tests to find the best model. the significance test hypothesis is as follows [29]. 𝐻0:∅ = 0 (indicates parameters are not yet significant) 𝐻1:∅ ≠ 0 (shows the parameters are significant) population forecasting of madiun regency using arima method yuniar farida 424 𝑡𝑐𝑜𝑢𝑛𝑡 = 𝜃 𝑆𝐸(𝜃𝑗) (7) where: 𝜃 : estimation of autoregressive model parameters and moving averages 𝑆𝐸(𝜃𝑗) : standard errors diagnostic test diagnostic tests are used to determine whether or not the model is the best. a good model, where the residual results of the white noise assumption test using the ljung-box test are as follows [6], [29]. 𝑄 = 𝑛(𝑛 + 2)∑ �̂�𝑘 2 (𝑛 − 𝑘) 𝑖 𝑘 (8) where: �̂�𝑘 : lag autocorrelation value k q : ljung-box test 𝑘 : lag time prediction accuracy value the results produced by the arima model are measured in terms of forecast accuracy. each method has a mape (mean absolute percentage error) error value that can be used to calculate the error value with the following formula [16], [31]. 𝑀𝐴𝑃𝐸 = ∑ |𝑃𝐸𝑡| 𝑛 𝑡=1 𝑛 (9) with 𝑃𝐸𝑡 = 𝑒𝑡 𝑍𝑡 × 100 (8) where: 𝑃𝐸𝑡 : percentage of errors at t 𝑒𝑡 : t-time error value 𝑍𝑡 : actual data of t-time the quality of the prediction can be shown by the mape value, which can be interpreted into four categories, namely excellent (mape < 10%), good (mape 11% 20%), good enough (mape 21% 50%), and not good (mape > 50%). results and discussion based on table 1, a time series plot is performed to determine the arima model and identify the stationarity of the data. population forecasting of madiun regency using arima method yuniar farida 425 figure 1. plot data on the population of madiun regency based on figure 1, the plot data shows an uptrend (positive). the data is not stationary because in 2019 there was an increase seen from the previous year's difference of 67,676 people. if there is no increase or decrease invariance and average, the data is stationary. figure 2. plot box-cox transformation figure 2 shows that the lambda value is equal to 1, the data can be said to be stationary in variance. stationary data on the average can be seen from the acf plot and time series plot. the data does not yet have a fixed pattern. population forecasting of madiun regency using arima method yuniar farida 426 figure 3. acf plot of madiun regency population from the plot figure 3, it appears that the lag-lag is falling slowly. the plot time series data also does not have a fixed pattern, so the data is not stationary against the average. as a result, it is necessary to do a further transformation process through differencing so that the data is stationary. figure 4. acf plot after differencing figure 4 shows that the data is stationary against the mean. if the data is stationary, the next step is to plot the autocorrelation function (pacf). population forecasting of madiun regency using arima method yuniar farida 427 figure 5. pacf plot after differencing based on figures 4 and 5 shows that the plot does not have an autocorrelation on the model, so the values 𝑀𝐴(𝑞) = 0 and 𝐴𝑅(𝑝) = 0, then obtained the tentative model arima (0,2,0). the model is a random walk where the autocorrelation coefficient is equal to 1, so the tentative models of arima are arima models (1,2,0), (0,2,1), and (1,2,1). a significance test and a residual white noise test were carried out to choose the model used in the prediction. test the significance of the parameters by knowing the pvalue. if the p-value is less than 0.05, then the model is significant. the results of the arima model's tentative significance test (1,2,0), (0,2,1), and (1,2,1) are as follows. table 3. significance test results model parameters coef se coef t-value p-value arima (1,2,0) ar (1) -0.580 0.139 -4.19 0.000 arima (0,2,1) ma (1) 0.950 0.131 7.26 0.000 arima (1,2,1) ar (1) -0.221 0.180 -1.23 0.228 ma (1) 0.946 0.122 7.73 0.000 table 3 shows that the arima models (1,2,1) are not significant because p-values are more than 0.05, arima models (1,2,0) and (0,2,1) are significant because p-values are less than 0.05. after conducting a parameter significance test, it is necessary to perform a residual white noise test to determine which model to use for prediction in performing residual tests using ljung-box. if the p-value is more than 0.05, the model meets the white noise requirement. ljung-box test results for arima models (1,2,0), (0,2,1), and arima models (1,2,1). population forecasting of madiun regency using arima method yuniar farida 428 table 4. ljung-box test results model lag chi-square df p-value arima (1,1,1) 12 5.34 10 0.867 24 11.17 22 0.972 36 23.15 34 0.920 arima (2,1,1) 12 1.83 10 0.998 24 7.23 24 0.999 36 11.94 36 1.000 arima (0,1,2) 12 0.53 9 1.000 24 6.15 21 0.999 36 11.41 33 1.000 table 4, after the ljung-box test, shows that the arima model (1,2,0), (0,2,1), and arima model (1,2,1) are white noise because the p-value is more than 0.05. after performing the white noise test, determine the mape value of the arima provisional model. table 5. mape value model mape arima (1,2,0) 12.45 arima (0,2,1) 8.42% arima (1,2,1) 8.92% based on table 3, table 4, and table 5 of the arima model, whose parameters are significant and meet the assumption of white noise, the arima model (0,2,1) has the smallest mape value. after choosing the best model, then predicted the number of residents of madiun regency. figure 6. actual data plot and forecasting figure 6 showed that there is not much difference between the actual data and the forecast and obtained mape < 10%, which means that the prediction model is already very good. the prediction of the number of residents of madiun regency for the next three years (2022 to 2024) is presented in table 6 below: population forecasting of madiun regency using arima method yuniar farida 429 table 6. population prediction year population 2022 758561 2023 767346 2024 776508 table 6 shows that the madiun regency population from 2022 to 2024 is predicted to increase by 17947 people or 2.39%. the results of these predictions show an uptrend every year. population growth, if followed by an increase in the quality of human resources, will become a regional potential for development. on the other hand, if the increase in population is not accompanied by good quality human resources, it will become a burden for regional development [32]. population data prediction is needed in the planning and evaluating of humanoriented development as the primary target because the population is both an object and a subject of the action. the object’s function means the population as a target, and the people carry out the development mark. the function of the issue means that the people are the sole actor in action. the two functions are expected to go hand in hand and line integrally [32]. conclusions based on the research results on the prediction of the population of madiun regency using the arima method, it can be concluded that the best model for predicting the number of residents of madiun regency is 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[32] p. k. madiun, data demografi, ekonomi dan sosial budaya kota madiun 2017. madiun: pemerintah kota madiun, 2017. optimal prevention and treatment control on sveir type model spread of covid-19 cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages pages 40-48 p-issn: 2086-0382; e-issn: 2477-3344 submitted: juni 21, 2021 reviewed: september 09, 2021 accepted: november 07, 2021 doi: https://doi.org/10.18860/ca.v7i1.12634 optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan department of mathematics, cenderawasih university jayapura indonesia email: jonner2766@gmail.com abstract covid-19 pandemic has disrupted the world's health and economy and has resulted in many deaths since the first case occurred in china at the end of 2019. moreover, the covid-19 disease spread throughout the world, including indonesia on march 2, 2020. coronavirus quickly spreads through droplets of phlegm through the throat to the lungs. researchers in the medical field and the exact science are jointly examined transmission, prevention, and optimal control of covid-19 disease. due to the prevention of covid-19, a vaccine has been found in early 2021, which at the time, the vaccination process was carried out worldwide against covid-19. this paper examines the spread model of sveir-type covid-19 by considering the vaccination subpopulation. in this study, control of prevention efforts (𝑢1 ∗ and 𝑢2 ∗ ) and healing efforts (𝑢3 ∗ ) are given and being analyzed with the fourth-order runge-kutta approach. based on numerical simulations, it can be seen that using the controls 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ can decrease the amount of infected people in the subpopulation compared to those without control. the 𝑢3 ∗ control can increase the number of recovered individual subpopulations. keywords: covid-19; sveir model; optimal control; treatment; vaccination. introduction coronavirus is a virus that attacks the respiratory system. corona virus interferes with mild respiratory and lung infections and can result in death [1]. corona virus is rapidly spreading to almost all countries in the world, and indonesia on march 2, 2020. to prevent the spread of covid-19, the government recommends frequent hand washing with soap/hand sanitizer and practicing cough etiquette. corona virus spreads by sprinkling phlegm from the throat of an infected person, especially in closed air circulation areas. be aware of covid-19, improve your health with healthy lifestyle, includes: balanced nutrition consumption, exercises, adequate rest, and frequent handwashing with soap. exact science and medicine experts are working together to prevent the spread of covid-19. consequently, mathematical researchers are also take part in assessing transmission, streamlining, and optimizing control of the disease. the optimal control strategy for the control of pandemic avian influenza along with the quarantine subpopulation [2]. studying the model of the spread of the corona virus type seir in wuhan, by controlling people's travel history to and from the city of wuhan [3]. then, compared to the pattern of the spread of covid-19 in wuhan, china, and internationally in january and february 2020. examine the mathematical modeling of the epidemic of covid-19 cases in nigeria from 29 https://doi.org/10.18860/ca.v7i1.12634 mailto:jonner2766@gmail.com optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 41 march to 12 june 2020 with the effect of public awareness programs in implementing health protocols according to government recommendations [4]. the model for the spread of covid19 by paying attention to symptomatically and asymptomatically infectious compartments [5]. studied the spread of the covid-19 outbreak by applying mathematical growth functions and analyzing cases caused by the disease [6]. examined a mathematical model on the sir type of covid-19 to predict its spread by considering the social distancing factors [7]. formulated a deterministic model on the spread of covid-19 by estimating the model parameters according to the pandemic data that occurred in india, then conducting a sensitivity analysis to identify the model parameters [8]. analyzed the covid-19 modeling based on morbidity data in anhui, china by taking into account increased morbidity and mitigation measures [9]. studied and analyzed corona virus disease spread system and the seir type cov-2 sar with the effectiveness of government intervention [10]. analyzed model and predicted the spread of covid-19, then examined the exposed subpopulation, with measurement to prevent and control the epidemic [11]. examined the growth of the logistic model of the covid-19 spread in china and compared with data globally, determined the parameter values with a least-squares approach [12]. studied a mathematical model for the covid-19 pandemic, type sir, with asymptomatic individual effects considered with the finite antibody duration and health policy [13]. studied the global dynamics of the seir type covid-19 under convex incidence rate, through a mathematical model [14]. other studies have been carried out to find the exponential growth rate of the epidemic and determine the basic reproduction number [15]. reviewed a nonlinear ordinary differential equation model for the spread of covid-19, the model studied predicts the total number of covid-19 cases in austria, france, and poland [16]. the model studied is based on the model studied by fang et al. (2020), figuring out the vaccinated subpopulation and providing optimal control of u1, u2, and u3. the objectives of this study includes: (1) to examine and analyze the sveir-type of covid-19 spread model, (2) determine the co-state function of the sveir-type of covid-19 spread model with control 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ . (3) determine numerical solution of the optimal control for prevention and treatment of the sveir-type of covid-19 spread model. optimal control of prevention and treatment in the model of the spread of covid19 is important to study, since the covid-19 disease is still spreading and has not been completely the treatment until now. this study will examine the sveir type of covid-19 spread model by considering vaccination subpopulations and provide preventive control measures (𝑢1 ∗ and 𝑢2 ∗ ), healing efforts (𝑢3 ∗ ) and analyzed simulation by numerical approach using runge-kutta fourth order. method the model used in this study comes from the development of the tian (2020) which contemplated the recruitment of suspected subpopulations denoted by s, namely individuals who have a history of travel to infected areas. vaccination subpopulation, namely susceptible individuals who are vaccinated to increase individual immunity to the covid-19 virus so as not to be infected with covid-19 disease. the exposed subpopulation, namely individuals who are positive for covid-19 from the results of the swab, but not severe, denoted e. infected subpopulations denoted by i, i.e. individuals who are positive for covid-19 from the swab results and experience severe illness due to covid-19. the recovered subpopulation is denoted by r, due to treatment in hospitals provided by the government or due to self-isolation at home by eating or taking vitamins to increase immunity so that they can recover from covid-19. the assumptions of the optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 42 model studied are as follows: (i) individuals who travel to areas infected with covid-19 or have been in contact with individuals infected with covid-19 enter the susceptible subpopulation. (ii) does not consider the natural mortality of each subpopulation. (iii) most of the susceptible individuals who want to be vaccinated yet entering the vaccination subpopulation. (iv) individuals who were successfully vaccinated entered the recovered subpopulation, and those who failed to enter the exposed subpopulation. (v) if the maximum number of individuals in the subpopulation is exposed or the virus is multiplying in the lungs, then the individual will enter the infected subpopulation. (vi) pay attention to deaths due to covid-19. (vi) individuals may experience recovery due to treatment or by self-isolation. the models of the spread of covid-19 studied are as follows: 𝑑𝑆 𝑑𝑡 = λ − 𝛽𝑆𝐼 𝑁 − 𝜃𝑆 (1) 𝑑𝑉 𝑑𝑡 = 𝜃𝑆 − (𝜎 + 𝑟)𝑉 (2) 𝑑𝐸 𝑑𝑡 = 𝛽𝑆𝐼 𝑁 + 𝜎𝑉 − 𝛾𝐸 (3) 𝑑𝐼 𝑑𝑡 = 𝛾𝐸 − (𝑑 +  + 𝜏)𝐼 (4) 𝑑𝑅 𝑑𝑡 = 𝑟𝑉 + ( + 𝜏)𝐼 (5) where n = s + v + e + i + r. description of the parameters as table 1 follows: table 1. parameter description and estimate value parameter description value reference = n recruitment rate entering the s subpopulation 0,047/day covid-19 go.id data  transmission rate from s to e 0,154/day assumed  vaccination rates from subpopulation s to v 0,04/day assumed  the transfer rate from subpopulation v to e, due to vaccine failure 0,005/day assumed r vaccination success rate 0,05/day assumed  transmission rate from subpopulation e to i 0,036/day assumed d death rate due to covid-19 0,002/day covid-19 go.id data  healing rate 0,036/day covid-19 go.id data  speed of healing with covid-19 self-isolation 0,04/day assumed equilibrium point non-endemic fixed point, to analyze equation (1)-(5) it is enough to use equation (1)-(4) because equation (5) is redundant to equation (1)-(4). based on equation (1)-(4), the non-endemic fixed point is obtained, namely: 𝐸0 = ( λ 𝜃 , λ 𝑟+𝜎 , 0,0), and the endemic point of system (1)-(4) is 𝐸1 = ( λ𝑁 𝛽𝐼 + 𝜃𝑁 , 𝜃λ𝑁 (𝑟 + 𝜎)(𝛽𝐼 + 𝜃𝑁) , 𝛽λ𝑁 𝛾(𝛽𝐼 + 𝜃𝑁) + 𝜎𝜃λ𝑁 𝛾(𝑟 + 𝜎)(𝛽𝐼 + 𝜃𝑁) , 𝐼∗∗) with 𝐼∗∗ = − 1 2 𝑐𝜃𝑁𝑇±√(𝑐𝜃𝑁𝑇)2+4𝑎𝛽𝑐2𝑇+4𝑏𝛽𝑐𝑇 𝑐𝛽𝑇 , 𝑎 = 𝛽λ𝑁, 𝑏 = 𝜎𝜃λ𝑁, 𝑇 = 𝑑 +  + 𝜏, 𝑐 = 𝑟 + 𝜎. optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 43 the reproduction number the reproduction number is a parameter that expresses the expectation of secondary infective individuals due to contracting primary infective individuals in the susceptible population. the standard parameters that need to be known to determine whether the disease is spreading or not are the basic reproduction number (r0), if r0 > 1 then the number of infected individuals increases, if r0 < 1 then the number of infected individuals does not increase or decrease. the basic reproduction number is calculated by using the next-generation method, the basic reproduction number of equations (1)– (5) are as follows: r0 = 𝜌(𝐺𝑈 −1) with  is radius of the matrix built by gu-1. the rate of new infections denoted with g, and u is individuals out. the jacobian matrix of g(x) and u(x), and denote g = [gi /xj] and u = [ui /xj], (i, j = 1, 2, 3, 4, 5). parameter the reproduction number, r0 as r0 = 𝑟𝛽λ 𝜃𝑁(𝑑+𝛿+𝜏)(𝑟+𝜎) . (6) result and discussion the handling of covid-19 carried out by the stakeholders are: control u1(t) for prevention with government appeals in government agencies, schools, and the general public by implementing physical distancing, namely maintaining a minimum distance of 1 meter from other people. then, implementing a mask when doing activities in public places or crowds. to continue, washing our hands regularly with soap and water or a hand sanitizer that contains at least 60% alcohol, especially after coming back from outdoor activities or in public places. the u2(t) control is vaccination counseling because many individuals are afraid to be vaccinated. this u2(t) control provides education to the public, how to prepare for vaccination, what to do after being vaccinated, and what are the effects after being vaccinated. because there is a lot of hoax information circulating, to scare people not to be vaccinated. u3(t) control is an effort to accelerate healing from covid-19 by individuals who are self-isolating in their own homes or places provided by the central government or local governments. the healing efforts given are: taking vitamins c, d, b, zinc, selenium, curcumin, echinacea. based on equations (1)-(5) after being given the control u1(t), u2(t) and u3(t) obtained a system of differential equations with optimal control as follows: 𝑑𝑆 𝑑𝑡 = λ − 𝛽(1−𝑢1(𝑡))𝑆𝐼 𝑁 − 𝜃(1 + 𝑢2(𝑡))𝑆 (7) 𝑑𝑉 𝑑𝑡 = 𝜃(1 + 𝑢2(𝑡))𝑆 − (𝜎 + 𝑟)𝑉 (8) 𝑑𝐸 𝑑𝑡 = 𝛽(1−𝑢1(𝑡))𝑆𝐼 𝑁 + 𝜎𝑉 − 𝛾𝐸 (9) 𝑑𝐼 𝑑𝑡 = 𝛾𝐸 − (𝑑 +  + 𝜏(1 + 𝑢3(𝑡)))𝐼 (10) 𝑑𝑅 𝑑𝑡 = 𝑟𝑉 + ( + 𝜏(1 + 𝑢3(𝑡)))𝐼 (11) the functional objective of optimal control studied in this paper are: 𝐽(𝑢1, 𝑢2, 𝑢3) = ∫ (𝐴𝐸(𝑡) + 𝐵𝐼(𝑡) + 𝐶1𝑢1 2(𝑡) + 𝐶2𝑢2 2(𝑡)+𝐶3𝑢3 2(𝑡))𝑑𝑡 𝑡𝑓 0 , (12) where the coefficients a, b is the balance weights of the individual compartments exposed and actively infected with covid-19, respectively. while the coefficient c1 is a parameter weight that corresponds to the control u1(t), c2 is a parameter weight that corresponds to the control u2(t), and c3 is a parameter weight corresponding to the control u3(t), and tf is optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 44 the end time of the period. let 𝑢1 ∗ (𝑡), 𝑢2 ∗ (𝑡), and 𝑢3 ∗ (𝑡), be the optimal control of the system (7)-(11) and (12), such that it satisfies 𝐽(𝑢1 ∗ , 𝑢2 ∗ , 𝑢3 ∗ ) = min 𝐽(𝑢1, 𝑢2, 𝑢3), (13) where the control set 𝑈 = {(𝑢1, 𝑢2, 𝑢3)|𝑢𝑖 : [0, 𝑡𝑓 ] → [0,1], lebesgue measurable, 𝑖 = 1, 2, 3} . the objective function (12), the optimal control 𝑢1 ∗ , 𝑢2 ∗ , 𝑢3 ∗ , obtained with provided that (13) with restriction system (7)-(11) by using matlab tools, the solution will be obtained [17]. by model (7)-(11), and minimum functional adjoint (12) obtained hamiltonian function h, that is 𝐻 = 𝐴𝐸 + 𝐵𝐼 + 𝐶1𝑢1 2 + 𝐶2𝑢2 2 + 𝐶3𝑢3 2 + 𝜆1 𝑑𝑆 𝑑𝑡 + 𝜆2 𝑑𝑉 𝑑𝑡 +𝜆3 𝑑𝐸 𝑑𝑡 +𝜆4 𝑑𝐼 𝑑𝑡 +𝜆5 𝑑𝑅 𝑑𝑡 (14) theorem 1 there exists a optimal control 𝑢1 ∗ (𝑡), 𝑢2 ∗ (𝑡) and 𝑢3 ∗ (𝑡) and associated solution 𝑆∗(𝑡), 𝑉 ∗(𝑡), 𝐸∗(𝑡), 𝐼∗(𝑡) , 𝑅∗(𝑡) from models (7)-(11) and (14). then there exist costate functions λi, i = 1, 2, 3, 4, 5 satisfying 𝑑𝜆1 𝑑𝑡 = (𝜆1 − 𝜆3) 𝛽(1−𝑢1)𝐼 𝑁 + (𝜆1 − 𝜆2)(1 + 𝑢2)𝜃 𝑑𝜆2 𝑑𝑡 = (𝜆2 − 𝜆3)𝜎 + (𝜆2 − 𝜆5)𝑟 𝑑𝜆3 𝑑𝑡 = −𝐴 + (𝜆3 − 𝜆4)𝛾 𝑑𝜆4 𝑑𝑡 = −𝐵 + (𝜆1 − 𝜆3) 𝛽(1−𝑢1)𝑆 𝑁 + (𝜆4 − 𝜆5)(𝛿 + 𝜏) + 𝜆4𝑑 𝑑𝜆5 𝑑𝑡 = 0, the transversality conditions are given by 𝜆𝑖 (𝑡𝑓 ) = 0, 𝑖 = 1, 2, 3, 4, 5. finally, from the optimality condition, we obtain the following optimal controls: 𝑢1 ∗ = min {𝑚𝑎𝑥 {0, (𝜆3−𝜆1)𝛽𝑆𝐼 2𝐶1𝑁 } , 1} 𝑢2 ∗ = min {𝑚𝑎𝑥 {0, (𝜆1−𝜆2)𝜃𝑆 2𝐶2 } , 1}. 𝑢3 ∗ = min {𝑚𝑎𝑥 {0, (𝜆4−𝜆5)𝜏𝐼 2𝐶3 } , 1}. proof: we use pontrygain’s maximum principle [17] on our model system (14), and the hamiltonian is given by, 𝐻 = 𝐴𝐼 + 𝐵1𝑢1 2 + 𝐵2𝑢2 2 + 𝜆1 (λ − 𝛽(1 − 𝑢1(𝑡))𝑆𝐼 𝑁 − 𝜃(1 + 𝑢2(𝑡))𝑆) +𝜆2(𝜃(1 + 𝑢2(𝑡))𝑆 − (𝜎 + 𝑟)𝑉) + 𝜆3 ( 𝛽(1−𝑢1(𝑡))𝑆𝐼 𝑁 + 𝜎𝑉 − 𝛾𝐸) + +𝜆4(𝛾𝐸 − (𝑑 +  + 𝜏(1 + 𝑢3(𝑡)))𝐼 ) + 𝜆5(𝑟𝑉 + ( + 𝜏(1 + 𝑢3(𝑡)))𝐼 ) 𝑑𝜆1 𝑑𝑡 = − 𝜕𝐻 𝜕𝑆 = (𝜆1 − 𝜆3) 𝛽(1−𝑢1)𝐼 𝑁 + (𝜆1 − 𝜆2)(1 + 𝑢2)𝜃 𝑑𝜆2 𝑑𝑡 = − 𝜕𝐻 𝜕𝑉 = (𝜆2 − 𝜆3)𝜎 + (𝜆2 − 𝜆5)𝑟 𝑑𝜆3 𝑑𝑡 = − 𝜕𝐻 𝜕𝐸 = −𝐴 + (𝜆3 − 𝜆4)𝛾 𝑑𝜆4 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼 optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 45 = −𝐵 + (𝜆1 − 𝜆3) 𝛽(1−𝑢1)𝑆 𝑁 + (𝜆4 − 𝜆5)(𝛿 + 𝜏) + 𝜆4𝑑 𝑑𝜆5 𝑑𝑡 = − 𝜕𝐻 𝜕𝑅 = 0. the optimality equations (14) and must satisfy transversality conditions λ(𝑡𝑓 ) = 0 for values i = 1, 2, 3, 4, 5. there exist unique optimal controls 𝑢1 ∗ (𝑡) and 𝑢2 ∗ (𝑡) which minimize j over u: the optimality necessary conditions that 𝜕𝐻 𝜕𝑢1 = 0, 𝜕𝐻 𝜕𝑢2 = 0 and 𝜕𝐻 𝜕𝑢3 = 0, then, by the bounds on the controls, it is easy to obtain and in the form 𝑢1 ∗ (𝑡) = (𝜆3−𝜆1)𝛽𝑆𝐼 2𝐶1𝑁 , 𝑢2 ∗ (𝑡) = (𝜆1−𝜆2)𝜃𝑆 2𝐶2 , and 𝑢3 ∗ (𝑡) = (𝜆4−𝜆5)𝜏𝐼 2𝐶3 . the optimal prevention control of disease, the reproduction numbers declared to be as follows: 𝑅0𝑝 ∗ = 𝑟𝛽(1−𝑢1)λ 𝜃𝑁(1+𝑢2)(𝑑+𝛿+𝜏)(𝑟+𝜎) . the optimal healing control of disease, the reproduction numbers declared to be as follows: 𝑅0ℎ ∗ = 𝑟𝛽λ 𝜃𝑁(𝑑+𝛿+𝜏(1+𝑢3))(𝑟+𝜎) . numerical method to solve the optimal control on the studied system of differential equations, it is solved by using a numerical method approach. the numerical solution used is the pontryagin maximum principle. completion of the optimal control system using the fourth-order runge–kutta procedure iterative method. the solution of the model (7)-(11) by guessing the initial and forward time from left to right with the same time co-state is solved from left to right by a forward runge–kutta fourth-order procedure in time with conditions of transversality [1]. suppose the initial number of subpopulations s0 = 61478 , v0 = 40000 (assumed), e0 = 20000 (assumed), i0 = 26940, r0 = 7637. figure 1. the dynamics of e with 𝑢1 ∗ and 𝑢2 ∗ controls figure 1, the optimal control on covid-19 prevention (𝑢1 ∗ ) and on efforts to increase the effectiveness of vaccination (𝑢2 ∗ ) are used equation (12). the results in figure 1 show that a significant difference in the e with the optimal control 𝑢1 ∗ and 𝑢2 ∗ compared to exposed subpopulation without control, using effective controls decreased the number of exposed covid-19 (e) than without control. optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 46 figure 2. the dynamics of i with 𝑢1 ∗ and 𝑢2 ∗ controls figure 2, the optimal control (𝑢1 ∗ ) and (𝑢2 ∗ ) on covid-19 are used to equation (12). the results in figure 2 show that there is difference in the i with control than i without control 𝑢1 ∗ and 𝑢2 ∗ , using effective controls decreased the number of active covid-19 (i) compared to without control. figure 3. the dynamics of i with 𝑢1 ∗ and 𝑢2 ∗ controls figure 4. the dynamics of r with 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ control figure 3, the optimal control on covid-19 prevention (𝑢1 ∗ ), the optimal control (𝑢2 ∗ ) and optimal control on covid-19 treatment (𝑢3 ∗ ) are application equation (12). the results in figure 3 show that there is difference in the i with control than i without control 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ compared to i without control, using effective controls decreased the number of active covid-19 (i) compared to with control strategy 𝑢1 ∗ and 𝑢2 ∗ and without control. optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 47 figure 4, the results in figure 4 show that a significant difference in the r with the optimal control strategy 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ compared to r without control. we see in figure 4 the number of recovered individuals in subpopulation of covid-19 increases rapidly with optimal control, while it is increase slowly without the control. figure 5. the profile of the optimal controls 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ figure 5: in this scenario, we consider the covid-19 optimal prevention and treatment control of covid-19 simultaneously. the profile of the optimal prevention control 𝑢1 ∗ , 𝑢2 ∗ and optimal treatment control 𝑢3 ∗ of this scenario in figure 5. conclusion until now, various efforts have been made by medical personnel in each country and who has been trying to find a cure and a vaccine for covid-19, but until now it has not been found so that the spread of covid-19 in the world has not been controlled. but in early 2021 a vaccine for covid-19 has been found and vaccinations have been carried out all over the world. having been vaccinated against covid-19 does not guarantee that you will not be infected with covid-19 again. the model studied in this paper discusses model covid-19 in indonesia concerning vaccinations. based on the numerical simulation obtained optimal control strategy 𝑢1 ∗ and 𝑢2 ∗ compared to e without control, using effective controls decreased the number of exposed covid-19 (e) compared to without control. optimal control strategy 𝑢1 ∗ , 𝑢2 ∗ and 𝑢3 ∗ compared to i without control, using effective controls decreased the number of active covid-19 (i) compared to with control strategy 𝑢1 ∗ and 𝑢2 ∗ and without control. the control result in a further increase in the number who recovered of covid-19 (r) compared optimal control strategy 𝑢1 ∗ and 𝑢2 ∗ and without control. acknowledgments the authors would like to thank kemenristek dikti for providing higher education grants for fiscal year 2020, through lppm universitas cenderawasih who sponsored the research. optimal prevention and treatment control on sveir type model spread of covid-19 jonner nainggolan 48 references [1] kemenkes ri, “dokumen resmi,” pedoman kesiapan menghadapi covid-19, pp. 0– 115, 2020. 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[17] s. lenhart and j. t. workman, optimal control applied to biological models, john chapman and hall, new york, 2007. spatio temporal modelling for government policy the covid-19 pandemic in east java cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 218-226 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 07, 2020 reviewed: december 03, 2020 accepted: april 12, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.10639 spatio temporal modelling for government policy the covid-19 pandemic in east java atiek iriany1, novi nur aini1, agus dwi sulistyono2 1 department of statistics faculty of mathematics and natural sciences, brawijaya university, indonesia 2 faculty of fisheries and marine science, brawijaya university, indonesia email: atiekiriany@ub.ac.id abstract covid-19 has cursorily spread globally. just in four months, its status altered into a pandemic. in indonesia, the virus epicenter is identified in java. the first positive case was identified in west java and later spread in all java. the large-scale social restrictions are seemingly inefficient as the sars-cov-2 transmission remains. as such, the government is struggling to find anticipatory policies and steps best to mitigate the transmission. in this particular article, we used a spatiotemporal model method for the total covid-19 cases in java and forecasted the total cases for the next 14 days, allowing the stakeholders to make more effective policies. the data we were using was the daily data of the cumulative number of covid-19 cases taken from www.covid19.go.id. data modeling was conducted using a generalized spatio-temporal autoregressive model. the model acquired to model the covid-19 cases in java was the gstar(1)(1,0,0) model. keywords: covid-19; forecasting, pandemic; spatio-temporal introduction as stipulated by who on 12 march 2020, covid-19 had become a pandemic [1]. the virus, firstly identified in wuhan in december 2019, rapidly spread throughout china and other 190 countries [2]. no research exactly explains how the sars-cov-2 was initially transmitted, but, in the meantime, it is believed that humans transmit this virus to humans. later research reveals that symptomatic patients transmit sars-cov-2 through droplets or sneezes [3]. moreover, another research mentions that sars-cov-2 can live in gas particles, e.g., air (generated through nebulizer) for approximately three hours [4]. due to its relatively rapid transmission and mortality rate which cannot be overlooked and no definitive therapy found, covid-19 is one of the diseases to which we should alert [5]. the coronavirus epicenter in indonesia is identified in java. the first positive case was identified in west java and later spread in all java. it indicates that adjacent locations closely pertain to the sars-cov-2 transmission. in response to the virus, china’s social distancing regulation is proven effective to stabilize the virus transmission, and hence the number declines [6]. indonesia, similar to china, issues the same regulation, namely the large-scale social restrictions (psbb). nevertheless, the regulation is seemingly inefficient as the sars-cov-2 transmission remains. as such, the government is struggling to find anticipatory policies and steps best to mitigate the transmission. http://dx.doi.org/10.18860/ca.v6i4.10639 http://www.covid19.go.id/ spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 219 many researchers, e.g., jia et al. [7], albana [8], and fajar [9] have studied the covid-19 transmission and aim to recommend some anticipatory efforts. meanwhile, we made covid-19 modeling using a spatio-temporal approach due to the sars-cov-2 transmission, which is mostly influenced by interplay, and numerous positive cases. several researchers used the spatio-temporal model [10] and [11]. one of the methods used to handle data attributed to time and location was generalized space-time autoregressive (gstar). some researchers, such as iriany [12], ruchjana [13], and prastyo [14], prefer this method. in this particular article, we used a spatio-temporal model for the total covid-19 cases in java and forecasted the total cases for the next 14 days, allowing the stakeholders to make more effective policies. methods data source the data we were using in this research were the daily data of the cumulative number of covid-19 cases taken from www.covid19.go.id. data stationarity according to the stationary time series data, neither a sharp decrease nor an increase in data value nor fluctuated data was found around the constant mean value [15]. stationary data had the mean 𝐸(𝑍𝑡) = µ and variance 𝑉𝑎𝑟(𝑍𝑡) = σ2. the mean value conditioned that data had to be stationary, so neither decrease nor an increase in data from time to time was allowed [16]. furthermore, the characteristic of a stationary time series was endlessly constant average and variance. there were two types of time series stationarity, namely stationarity to variance and the mean. a. stationarity to variance stationarity to variance was if 𝑉𝑎𝑟(𝑍𝑡) = 𝑉𝑎𝑟(𝑍𝑡−𝑘) for all t and k, the variance was constant from time to time [17]. to observe whether or not the data was stationary to variance, we used a box-cox plot. non-stationary data could be altered into stationary ones through transformation. b. stationarity to the mean stationarity to the mean was if 𝐸(𝑍𝑡) = 𝐸(𝑍𝑡−𝑘) for all t and k, the mean function remained constant from time to time. stationarity to the mean was observed using the acf (autocorrelation function) plot or the dickey-fuller test. non-stationary data could be altered into stationary ones through differencing. generalized space-time autoregressive integrated (gstar) the ar order was determined using the mpacf plot. correlation between zt and zt+k, after a dependence relationship, was linear. the variables zt+1, zt+2, …, and zt+k-1 were thus negated. the formula of correlation partial matrix function is as follows: ϕkk = 𝑐𝑜𝑣 [(𝑍𝑡−�̂�𝑡),(𝑍𝑡+𝑘−�̂�𝑡+𝑘)] √𝑣𝑎𝑟(𝑍𝑡−�̂�𝑡)√𝑣𝑎𝑟(𝑍𝑡+𝑘−�̂�𝑡+𝑘) (1) where ϕkk = partial correlation matrix coefficient at lag k 𝑍𝑡 = observation data at the time t �̂�𝑡 = predictor for 𝑍𝑡 𝑍𝑡+𝑘 = observation data at the time 𝑡 + 𝑘 �̂�𝑡+𝑘 = predictor for 𝑍𝑡+𝑘 http://www.covid19.go.id/ spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 220 the partial autoregression matrix at lag s became the last matrix coefficient when the data were leveraged for the vector autoregression process of the order s. the best model was selected among some models considered feasible for mpacf testing. model selection was conducted using aic. the less the aic value in a model, the better the model. the quantification of the aic value was as follows: 𝐴𝐼𝐶(𝑖) = ln (|𝑆(𝑝)| + 2𝑝𝑏2 𝑇 ) (2) where: b = the number of predicted parameters in the model t = the number of observations s(p) = residual sum of squares p = var model order the gstar model was introduced by borovkova, lopuha, and ruchjana in 2020 in wutsqa et al. [18]. it was more flexible and generalized than the star model and did not require the same parameter values at all locations. the gstar model (𝑝, 𝜆1, … . , 𝜆𝑙) is written as follows [19]: zt = ∑ [φ𝑘0 + 𝑝 𝑘=1 φ𝑘1𝑊] 𝑍𝑡−𝑝 + 𝑒𝑡 (3) where: φ𝑘0 = diag (𝜙𝑘0 1 , … , 𝜙𝑘0 𝑛 ), diagonal matrix of the parameter space-time lag spatial 0 and the parameter autoregressive lag at the time kth φ𝑘1 = diag (𝜙𝑘1 1 , … , 𝜙𝑘1 𝑛 ), diagonal matrix of the parameter space-time lag spatial 1 and the parameter autoregressive lag at the time kth w = weighing matrix (n×n) selected as such that 𝑊 𝑖𝑖 (𝑘) = 0 dan ∑ 𝑊 𝑖𝑗 (𝑘) = 1𝑖≠𝑗 e(t) = the white-nose vector in size of (n × 1) z(t) = the random vector in size of (n × 1) at the time t suhartono and subanar [20] introduced a new method for determining weight using the result of cross-correlation normalization between locations at a congruent time lag. �̂�𝑖𝑗(𝑘) = 𝑟𝑖𝑗 (𝑘) = ∑ [𝑍𝑖(𝑡)− 𝑍𝑙̅̅ ̅] 𝑛 𝑘+1 [[𝑍𝑗(𝑡−𝑘)− 𝑍𝑗] ̅̅ ̅̅ √(∑ [𝑍𝑖(𝑡)− 𝑍𝑙̅̅ ̅] 2𝑛 𝑡=1 )(∑ [𝑍𝑗(𝑡)− 𝑍𝑗 ̅̅ ̅]2𝑛𝑡=1 (4) the determination of location weight for the gstar model (1;p) is as follows: wij = 𝑟𝑖𝑗(1) ∑ |𝑟𝑖𝑘(1)|𝑘≠1 (5) with i ≠ j and the weight had fulfilled ∑ 𝑤𝑖𝑗𝑖≠𝑗 = 1. the weight of cross-correlation normalization represented the variance of correlation between locations occurring in the data. spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 221 results and discussion the covid-19 cases in indonesia were ever-increasing, and java was regarded as the transmission epicenter. the increase in the covid-19 cases is depicted in figure 1. figure 1. the plot of the time series of covid-19 cases in each province figure 1 indicates that as of 2 march-18 may 2020, the covid-19 cases increased in all provinces in java. on 18 may 2020, the highest number of cases, 5,555, was reportedly in jakarta, whereas the lowest one, 185, was in yogyakarta. using the data of the total covid-19 cases in six provinces in java, we identified the correlation between provinces and the covid-19 transmission in java. correlation between locations was identified using pearson’s correlation between provinces. the result of pearson correlation quantification is presented in table 1. table 1. the correlation value of the covid-19 cases between provinces in java banten jakarta west java central java yogyakarta east java banten 1 0.994 0.994 0.982 0.981 0.973 jakarta 0.994 1 0.995 0.989 0.975 0.967 west java 0.994 0.995 1 0.992 0.986 0.977 central java 0.982 0.989 0.992 1 0.983 0.980 yogyakarta 0.981 0.975 0.986 0.983 1 0.996 east java 0.973 0.967 0.977 0.980 0.996 1 in table 1, we can see that the data of the number of the covid-19 cases in six provinces in java had a high pearson’s correlation value which was higher than 0.9. it implies that the correlation of the covid-19 cases between provinces in java was strong. data stationarity test data stationarity testing was performed in two stages which were stationarity to variance and stationarity to the mean. stationarity to variance was tested using the boxcox transformation. data were regarded stationary if the lambda value was 1, signifying that var(zt) = var(zt-k). the result of the stationarity test to variance is shown in table 2. table 2. the result of box-cox transformation location λ transformation final transformation trans. λ trans. λ banten 0.20 zt0.20 1.00 zt0.20 jakarta 0.20 zt0.20 1.00 zt0.20 spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 222 location λ transformation final transformation trans. λ trans. λ west java 0.19 zt0.19 1.00 zt0.19 central java 0.00 ln(zt) 1.00 ln(zt) yogyakarta 0.00 ln(zt) 0.00 ln(zt) 1.00 ln(ln(zt)) east java 0.00 ln(zt) 0.50 zt0.50 1.00 ln(zt)0.50 as seen in table 2, the initial data had not fulfilled the stationarity to variance yet. several transformations were thus called for. after conducting the data stationarity test, we did the stationarity test to the mean. the test was conducted using an augmented dickey-fuller test. the result of the stationarity test to the mean is indicated in table 3. table 3. the result of the augmented dickey-fuller test location lag 0 1 2 banten π 98.73 60.83 34.96 p-value 0.001 0.001 0.001 jakarta π 107.89 32.98 21.15 p-value 0.001 0.001 0.001 west java π 100.74 40.22 28.78 p-value 0.001 0.001 0.001 central java π 122.94 55.84 29.66 p-value 0.001 0.001 0.001 yogyakarta π 75.55 51.84 42.09 p-value 0.001 0.001 0.001 east java π 155.97 57.49 25.85 p-value 0.001 0.001 0.001 from the augmented dickey-fuller test, we acquired predicted values less than the real ones (0.05). it indicates that the data had fulfilled the stationarity to variance. interpretation of the gstar model parameter model identification was aimed to find the autoregressive gstar model order. the order was elicited by identification using aic. the lag with the smallest aic value was regarded as the autoregressive gstar model order. table 4 lists the aic values. table 4. the aic value in model order selection lag ma 0 ma 1 ma 2 ma 3 ma 4 ma 5 ar 0 34.0876 35.2909 35.5546 36.0924 36.8882 36.1828 ar 1 31.9424 32.9755 33.3945 33.9363 33.9725 33.0619 ar 2 32.0815 33.1868 33.0648 33.8555 34.487 34.3527 ar 3 32.006 32.8989 33.096 35.128 35.9621 36.0125 ar 4 32.9289 34.1187 32.882 35.2756 35.429 42.0827 ar 5 34.6863 35.2259 35.61 35.7834 42.3544 table 4 shows the smallest aic value at the lag ar(1) and ma(0), hence the gstar(1)(1,0,0) model. spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 223 interpretation of the gstar model parameter the gstar model was a particular form of var engaging spatial elements. estimating the gstar(1) (1,0,0) spatial parameters with the ordinary least square method using cross-correlation normalization weight generated the following parameters. table 5. the parameters of the gstar(1)(1,0,0) model location parameter estimation banten ∅10 (1) 1.015 ∅11 (1) 0.793 jakarta ∅10 (2) 0.915 ∅11 (2) 0.984 west java ∅10 (3) 0.758 ∅11 (3) 1.031 central java ∅10 (4) -0.003 ∅11 (4) 0.256 yogyakarta ∅10 (5) 0.118 ∅11 (5) 0.061 east java ∅10 (6) 0.088 ∅11 (6) -0.013 referring to table 5, we generated the matrix equation of the gstar(1)(1,0,0) model, which is as follows: [ 𝑍1(𝑡) 𝑍2(𝑡) 𝑍3(𝑡) 𝑍4(𝑡) 𝑍5(𝑡) 𝑍6(𝑡)] = [ 1.015 0 0 0 0 0 0 0.915 0 0 0 0 0 0 0.758 0 0 0 0 0 0 −0.03 0 0 0 0 0 0 0.118 0 0 0 0 0 0 0.088] [ 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) 𝑍3(𝑡 − 1) 𝑍4(𝑡 − 1) 𝑍5(𝑡 − 1) 𝑍6(𝑡 − 1)] + [ 0.793 0 0 0 0 0 0 0.984 0 0 0 0 0 0 1.031 0 0 0 0 0 0 0.256 0 0 0 0 0 0 0.061 0 0 0 0 0 0 −0.013] [ 0 0.256 0.116 0.209 0.183 0.236 0.205 0 0.187 0.217 0.160 0.231 0.192 0.223 0 0.233 0.150 0.201 0.092 0.273 0.194 0 0.231 0.210 0.156 0.242 0.079 0.276 0 0.247 0.134 0.241 0.180 0.124 0.321 0 ] [ 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) 𝑍3(𝑡 − 1) 𝑍4(𝑡 − 1) 𝑍5(𝑡 − 1) 𝑍6(𝑡 − 1)] + [ 𝑒1(𝑡) 𝑒2(𝑡) 𝑒3(𝑡) 𝑒4(𝑡) 𝑒5(𝑡) 𝑒6(𝑡)] the following matrix equation was derived from the above equation. spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 224 [ 𝑍1(𝑡) 𝑍2(𝑡) 𝑍3(𝑡) 𝑍4(𝑡) 𝑍5(𝑡) 𝑍6(𝑡)] = [ 1.015 0 0 0 0 0 0 0.915 0 0 0 0 0 0 0.758 0 0 0 0 0 0 −0.03 0 0 0 0 0 0 0.118 0 0 0 0 0 0 0.088] [ 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) 𝑍3(𝑡 − 1) 𝑍4(𝑡 − 1) 𝑍5(𝑡 − 1) 𝑍6(𝑡 − 1)] + [ 0 0.259 0.118 0.212 0.186 0.239 0.202 0 0.184 0.213 0.157 0.227 0.198 0.229 0 0.240 0.155 0.207 0.024 0.069 0.049 0 0.059 0.054 0.009 0.015 0.005 0.017 0 0.015 −0.002 −0.004 −0.002 −0.002 −0.004 0 ] [ 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) 𝑍3(𝑡 − 1) 𝑍4(𝑡 − 1) 𝑍5(𝑡 − 1) 𝑍6(𝑡 − 1)] + [ 𝑒1(𝑡) 𝑒2(𝑡) 𝑒3(𝑡) 𝑒4(𝑡) 𝑒5(𝑡) 𝑒6(𝑡)] from the model generated, we made a comparison between the actual and predicted data, in which we acquired an rmse and mape value of 0.005 and 1.43, respectively. the two gave us a hint that the model generated was good. prediction result from the equation, we forecasted the total cases for the next 14 days, namely 19 may-1 june 2020, the result of which is presented in table 6. table 6. the predicted covid-19 cases on 19 may-1 june 2020 banten jakarta west java central java yogyakarta east java 1 628 5662 1689 1214 195 2321 2 651 5738 1746 1249 201 2438 3 674 5802 1804 1283 207 2561 4 699 5850 1862 1318 214 2689 5 725 5881 1920 1352 220 2822 6 754 5892 1978 1386 227 2961 7 784 5880 2035 1419 233 3105 8 817 5841 2092 1450 240 3255 9 852 5773 2148 1481 247 3411 10 891 5672 2202 1509 254 3573 11 933 5533 2254 1535 260 3741 12 979 5351 2305 1559 267 3916 13 1030 5122 2352 1579 274 4097 14 1086 4839 2396 1596 280 4284 the prediction stated that the total cases in all provinces in java would increase, except jakarta, in which there would be a declined total number of cases. the prediction was based on the assumption that there was no change in social interaction in the community. spatio-temporal modelling for government policy the covid-19 pandemic in java atiek iriany 225 conclusions the model acquired in modeling the covid-19 cases in java was the gstar(1)(1,0,0) model. our predicted covid-19 case data was close to the actual number of covid-19 cases in java. a spatio-temporal model could be used to predict the number of covid-19 cases in java. human-to-human transmission likely had a cross-location impact due to an interaction between individuals. our prediction indicates that all provinces in java, but jakarta, would likely have an increase in the total number of covid19 cases for the next 14 days. acknowledgments we would like to thank the university of brawijaya university for funding and support of this research. references [1] w. h. organization, “who director-general’s opening remarks at the media briefing on covid-19-11 march 2020,” geneva, switz., 2020. 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[20] s. suhartono and s. subanar, “the optimal determination of space weight in gstar model by using cross-correlation inference,” quant. methods, vol. 2, no. 2, pp. 45– 53, 2006. bühlmann's credibility model with claims of negative binomial and 2-poisson distribution cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 493-502 p-issn: 2086-0382; e-issn: 2477-3344 submitted: june 10, 2022 reviewed: february 22, 2023 accepted: march 09, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.16400 bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi1, taufiq iskandar2, nurbahari3, alim misbullah4, vina apriliani5* 1,2,3department of mathematics, syiah kuala university, banda aceh, indonesia 4department of informatics, syiah kuala university, banda aceh, indonesia 5department of mathematics education, uin ar-raniry, banda aceh, indonesia 1,5graduate school of natural science and technology, kanazawa university, kanazawa, japan email: vina.apriliani@ar-raniry.ac.id abstract one technique for determining the premium is using the credibility theory. in this study, a credibility premium determination model was derived with the best accuracy approach in the form of bühlmann’s credibility premium. the approach used was a parameteric approach where the claim data is assumed to have a negative binomial and 2-poisson distribution. the bühlmann's credibility premium formula is given explicitly for these two data distributions. the obtained model is also applied to the correct data following these distributions. from the simulation results, it is obtained that the premium values are very close in value so that both models can be applied to the data and have a high level of credibility because they have a high credibility factor value. the results of this study provide a basic contribution to the development of actuarial science, especially in the technique of determining insurance premiums. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc by-sa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: 2-poisson distribution; bühlmann's credibility; negative binomial distribution introduction in everyday life, humans are very vulnerable to risk. for example, the risk of accidents, property loss, illness, and even loss of life. this risk causes humans to lose their assets. therefore, insurance comes with the aim of minimizing loss if the risk does occur. insurance is an agreement between the insured (policyholder) and the insurer which requires the policyholder to pay a premium as compensation for the insurance benefits to be provided by the insurer in the event of a risk of failure to the policyholder [1]. one of the problems of insurance companies is how to determine the premium of a product. one of the techniques used is credibility theory. this theory predicts the amount of premium rates in the future based on experience data in the past. there are two predictive models that can be formed, namely a model for the number of claims and a model for the amount of claims (claim severity) that occur. this of course will be related to the type of data distribution used. the statistical approach that can be used in modeling the data is a parametric approach and a nonparametric approach. in this approach, we used a http://dx.doi.org/10.18860/ca.v7i4.16400 mailto:vina.apriliani@ar-raniry.ac.id https://creativecommons.org/licenses/by-sa/4.0/ bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 494 parametric approach where the claim data is assumed to follow a certain distribution [2]. one type of credibility that is widely used is the best accuracy credibility which consists of the bühlmann model and the bühlmann-straub model [3]. in the bühlmann model, policyholders are assumed to be the same number between time periods, while the bühlmann-straub model is a general form of the bühlmann model where the number of policyholders may differ between time periods. several studies related to the determination of premiums with bühlmann and bühlmann-straub credibility parametrically can be seen in the research conducted by [4]–[7]. the distributions that are commonly used to model many claims are the negative binomial distribution and the poisson distribution [8]. according to [9], mixed distribution is a distribution that can be considered in modeling the data. this is because, data modeling becomes more accurate. one of the mixed distributions used in this study is the 2-poisson distribution [10]. the use of 2-poisson distribution has not been found in previous studies. by using the assumption of a negative binomial and 2-poisson distribution on the data, the bühlmann’s credibility model will be determined on the data that satisfies this distribution. the equation for determining bühlmann's credibility parameter is given explicitly. in addition, the prediction results obtained through the application of the data are also compared. applications on nonparametric data using r can be seen in [11]. methods poisson distribution the poisson distribution is a distribution with one parameter (𝜆). the probability function for the poisson distribution is 𝑝𝑘 = 𝑒−𝜆𝜆𝑘 𝑘! , (1) with 𝑘 = 0,1,2, …. the expected value and variance of the poisson distribution are 𝐸(𝑋) = 𝜆, (2) 𝑉𝑎𝑟(𝑋) = 𝜆 (3) [12]. the other properties and the applications of this distribution can be seen in [13]. gamma distribution a random variable is said to have a gamma distribution with parameters 𝛼 and 𝛽, if it has a probability density function 𝑔𝑋 (𝑥; 𝛼, 𝛽) = 𝑥𝛼−1𝑒 −𝑥 𝛽⁄ 𝛤(𝛼)𝛽𝛼 , 𝑥 ∈ ℝ+ (4) with 𝛼 > 0, 𝛽 > 0, γ(𝛼) > 0, and γ(𝛼) = ∫ 𝑦𝛼−1 ∞ 0 𝑒−𝑦 𝑑𝑦. the parameter 𝛼 is called the shape parameter associated with the gamma distribution and the parameter 𝛽 is generally called the scale parameter because it multiplies the random variable with a gamma distribution by a positive constant. the expected value and variance of the gamma distribution are 𝐸(𝑋) = 𝛼 𝜏 , (5) 𝑉𝑎𝑟(𝑋) = 𝛼 𝜏2 . (6) the evidence can be found at [14]. bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 495 negative binomial distribution a negative binomial distribution is formed by an experiment that satisfies the following conditions: 1. an experiment consists of a series of independent experiments. 2. each experiment can only produce one of two possible outcomes, failure and success. 3. the experiment continues until a total number of 𝑥 successes. the negative binomial distribution can be formed from a mixed distribution of the poisson distribution and the gamma distribution. by using equation (1) and (4), it can be shown that this distribution has two parameters (𝛼 and 𝜏). the probability density function is 𝑝𝑋 (𝑥) = ( 𝑥 + 𝛼 − 1 𝛼 − 1 ) (1 − 𝜏)𝑥 𝜏𝛼 , (7) with 𝑥 = 0,1,2, … [15]. 2-poisson distribution the 2-poisson distribution is a distribution with three parameters (𝜆1, 𝜆2, and 𝑝). the probability function for the 2-poisson distribution is 𝑝𝑋 (𝑥) = 𝑒−𝜆1 𝜆1 𝑥 𝑥! 𝑝 + (1 − 𝑝) 𝑒−𝜆2 𝜆2 𝑥 𝑥! , (8) with 𝑥 = 0,1,2, … [16]. some of the applications of this distribution can be seen in [17]-[18]. bühlmann’s credibility bühlmann's credibility is a credibility model with the best accuracy approach. in this model, the number of policyholders observed is assumed to be the same every year. premiums are determined based on a linear model between past data and theoretical premiums. the parameters used in the bühlmann's credibility model are as follows: 1. the average value of individual claims or premiums and the expected values 𝝁(𝜽) = 𝑬(𝑿|𝚯), (𝟗) 𝝁 = 𝑬(𝝁(𝜽)), (𝟏𝟎) 2. the variance of the hypothetical mean 𝒂 = 𝑽𝒂𝒓(𝝁(𝜽)) = 𝑽𝒂𝒓(𝚯), (𝟏𝟏) 3. variance process and the expected value of variance 𝒗(𝜽) = 𝑽𝒂𝒓(𝑿|𝚯), (𝟏𝟐) 𝒗 = 𝑬(𝒗(𝜽)), (𝟏𝟑) 4. credibility coefficient 𝑲 = 𝒗 𝒂 , (𝟏𝟒) 5. credibility factor 𝒁 = 𝑵 𝑵 + 𝑲 , (𝟏𝟓) 6. credibility premium 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) (16) [19]. goodnes-of-fit test suppose 𝑚 is the largest value in the distribution of the observed data and 𝑟 is the number of parameters to be estimated, then with several predicted parameters, statistics bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 496 𝝌𝟐 = ∑ (𝒏𝒙 − 𝒏𝒑𝑿(𝒙)) 𝟐 𝒏𝒑𝒙(𝒙) 𝒎 𝒙=𝟎 (𝟏𝟕) asymptotically spread 𝝌𝟐 with degrees of freedom 𝒎 − 𝒓. hypothesis testing is one of the statistical tests carried out for testing the suitability of the parameter 𝛽𝑖 which is made with the following hypothesis: 𝐻0: 𝜃 = 𝛽𝑖 , (data has a distribution that matches the distribution of the test) 𝐻1: 𝜃 ≠ 𝛽𝑖 . (data does not have a distribution that matches the distribution of the test) by using the values of the calculated chi-square and the table chi-square, the following decision rules apply: if 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 ≤ 𝜒𝑡𝑎𝑏𝑙𝑒 2 then the null hypothesis is accepted and if 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 > 𝜒𝑡𝑎𝑏𝑙𝑒 2 then the null hypothesis is rejected, by setting the alpha value as well as the degrees of freedom of the chi-square distribution [20]. results and discussion estimating bühlmann’s credibility parameters using negative binomial distribution bühlmann's credibility model with claims of negative binomial distribution can be derived by providing an estimate of the credibility parameter assuming the frequency of claims with a negative binomial distribution. we know that the negative binomial distribution is a mixed distribution of the poisson distribution and the gamma distribution. suppose 𝑋|θ ∽ poisson(𝜃) and θ ∽ gamma(𝛼, 𝜏), it can be proven that 𝑋 has a negative binomial distribution with parameters 𝛼 and 𝜏 [21]. the following formula is given for the bühlmann’s credibility parameters using this distribution assumption. hypothetical mean and the expected value for negative binomial model the hypothetical mean for negative binomial model can be determined using equation (9). since 𝑋|θ ∽ poisson(𝜃) then 𝜇(θ) = 𝐸(𝑋|θ = θ) = 𝜃. the expected value of the hypothetical mean or known as the individual premium (𝜇) can be determined by equation (10) as follows: 𝜇 = 𝐸(𝜇(θ)) = 𝐸(θ). since θ has gamma distribution (𝛼, 𝜏), then according to equation (5), 𝜇 = 𝛼 𝜏 . (18) to determine the credibility coefficient, it is necessary to estimate the value of parameter 𝑎. the value of 𝑎 is the variance of the hypothetical mean. using equation (11), the formula for the variance value of the hypothetical mean can be determined as follows: 𝑎 = 𝑉𝑎𝑟(𝜇(𝜃)) = 𝑉𝑎𝑟(θ) = 𝐸(θ2) − (𝐸(θ)) 2 . since θ has gamma distribution, then according to equation (6), 𝑎 = 𝛼 𝜏2 . (19) variance process and the expected value for negative binomial model the variance process formula and the expected value can be determined using equation (12) and (13). since 𝑋|θ ∽ poisson(𝜃) then 𝑣(𝜃) = 𝑉𝑎𝑟(𝑋|θ = θ) = 𝜃. the expected value of the variance process (𝑣) ishypothetical𝑣 = 𝐸(𝑣(θ)) = 𝐸(θ) 𝑣 = 𝛼 𝜏 . (20) bühlmann’s credibility coefficient for negative binomial model the credibility coefficient is the ratio between the expected value of variance process bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 497 and variance of the hypothetical mean. by using equation (11), (19), and (20), it is obtained 𝐾 = 𝑣 𝑎 = 𝜏. (21) bühlmann's credibility premium for negative binomial model after obtaining the formula from the credibility parameter for the negative binomial model, it can be formulated a formula for determining the premium with the negative binomial model using equation (15) and (16) as follows: 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) μ, (22) with 𝑍 = 𝑁 𝑁 + 𝐾 = 𝑁 𝑁 + 𝜏 , �̅� = ∑ 𝑥𝑖 (𝑛𝑥𝑖 ) 𝑁 𝑖=1 (𝑛𝑥𝑖 ) . �̅� is the average of the observed data while μ is the individual premium which can be determined using equation (18). the 𝑍 variable is also called the bühlmann’s credibility factor for the frequency of claims with a negative binomial distribution, where 𝐾 is a credibility coefficient that satisfies equation (21). estimating bühlmann's credibility parameters using 2-poisson distribution the following formula is given for the bühlmann’s credibility parameters using this distribution assumption. hypothetical mean and the expected value for 2-poisson model as before, the hypothetical mean for 2-poisson model can be determined using equation (9). the 2-poisson distribution gives that 𝑋|θ ∽ poisson(𝜃) and θ ∽ 𝑢(𝜃) = { 𝑝 ∶ 𝜃 = 𝜆1 1 − 𝑝 ∶ 𝜃 = 𝜆2. (23) since 𝑋|θ ∽ poisson(𝜃), then according to equation (2), 𝜇(𝜃) = 𝐸(𝑋|θ = θ) = 𝜃. (24) the expected value of the hypothetical mean (𝜇) is 𝜇 = 𝐸(𝜇(θ)) = 𝐸(θ) = 𝑝𝜆1 + (1 − 𝑝)𝜆2 = 𝑝(𝜆1 − 𝜆2) + 𝜆2. (25) furthermore, the variance of the hypothetical mean can be determined using equation (11) and (23) as follows: 𝑎 = 𝑉𝑎𝑟(𝜇(𝜃)) = 𝑉𝑎𝑟(θ) = 𝐸(θ2) − (𝐸(θ)) 2 = 𝜆1 2 𝑝 + 𝜆2 2(1 − 𝑝) − (𝑝(𝜆1 − 𝜆2) + 𝜆2) 2 = 𝜆1 2 𝑝 + 𝜆2 2(1 − 𝑝) − (𝑝2(𝜆1 − 𝜆2) 2 + 2𝑝(𝜆1 − 𝜆2)𝜆2 + 𝜆2 2 ) = (𝜆1 2 − 𝜆2 2 )𝑝 + 𝜆2 2 − [(𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2) + 𝜆2 2 ] = (𝜆1 2 − 𝜆2 2 )𝑝 (𝜆1 − 𝜆2)(𝜆1 + 𝜆2) − [(𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2)] = [(𝜆1 + 𝜆2) − (𝑝(𝜆1 − 𝜆2) + 2)𝑝(𝜆1 − 𝜆2)] 𝑎 = [(1 − 𝑝)(𝜆1 + 𝜆2) − 2]𝑝(𝜆1 − 𝜆2). (26) variance process and the expected value for 2-poisson model the variance process and the expected value can be determined using equation (12) bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 498 and (13). since 𝑋|θ ∽ poisson(𝜃) then according to equation (3), we get 𝑣(𝜃) = 𝑉𝑎𝑟(𝑋|θ = θ) = 𝜃. the expected value of the variance process (𝑣) can be obtained using equation (23), which is as follows: 𝑣 = 𝐸(𝑣(θ)) = 𝐸(θ) 𝑣 = 𝑝𝜆1 + (1 − 𝑝)𝜆2 𝑣 = 𝑝(𝜆1 − 𝜆2) + 𝜆2. (27) bühlmann’s credibility coefficient for 2-poisson model the last part of determining bühlmann's credibility premium is the need to assign a credibility coefficient value. by using equation (14), (26), and (27), it is obtained 𝐾 = 𝑣 𝑎 𝐾 = 𝑝𝜆1 + (1 − 𝑝)𝜆2 [(1 − 𝑝)(𝜆1 + 𝜆2) − 2]𝑝(𝜆1 − 𝜆2) . (28) bühlmann's credibility premium for 2-poisson model bühlmann's credibility premium is obtained by using equation (15) and (28) so that the bühlmann credibility value for the 2-poisson model is 𝑃𝐶 = 𝑍�̅� + (1 − 𝑍) μ, (29) with 𝑍 = 𝑁 𝑁 + 𝐾 = 𝑁 𝑁 + 𝑝𝜆1+(1−𝑝)𝜆2 [(1−𝑝)(𝜆1+𝜆2)−2]𝑝(𝜆1−𝜆2) , �̅� = ∑ 𝑥𝑖 (𝑛𝑥𝑖 ) 𝑁 𝑖=1 (𝑛𝑥𝑖 ) . the 𝑍 variable is also known as the bühlmann’s credibility factor for the frequency of claims with 2-poisson distribution. the value of μ can be determined by equation (25) and �̅� is the average of the observed data. application on data the data used for the application of the model is data on the distribution of claims (𝑛𝑥 ) on the motor vehicle insurance portfolio in singapore [22]. the number of claims occurred from 1993 to 2001 which can be seen in table 1. table 1. portfolio of the number of claims from observation results (millions of dollars) 𝑥 𝑛𝑥 0 178.080 1 19.224 2 1.859 3 177 4 11 5 1 >5 0 total 𝑛𝑥 = 199.352 before applying to the model, it is first tested whether the claim frequency data in table 1 has a negative binomial and 2-poisson distribution or not. testing the distribution of data was carried out using the chi-square test. bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 499 negative binomial distribution test the negative binomial distribution has two estimators for the parameters (�̂� and �̂�) based on equation (7). parameter estimates can be obtained using the moment method. based on table 1, the average value of the number of claims �̅� = 0,1179923, �̅�2 = 0,01392218, and variance 𝑆2 = 0,12881027. the values of �̂� and �̂� are �̂� = �̅�2 𝑆2 − �̅� = 0,01392218 0,01081797 = 1,28694966, �̂� = �̅� 𝑆2 − �̅� = 0,1179923 0,01081797 = 10,9070648. the distribution test steps carried out are as follows: a. hypothesis formulation. 𝐻0 ∶ data has negative binomial distribution. 𝐻1 ∶ data is not distributed negative binomial. b. calculates the probability for each claim frequency and the expected value. the probability is calculated for each claim frequency (𝑝𝑥 ) for each 𝑥 in the table data and based on the parameter estimator values and the negative binomial distribution formula, then for 𝑥 = 0: 𝑝𝑥 = ( 𝛼 + 𝑥 − 1 𝑥 ) ( 𝜏 1 + 𝜏 ) 𝛼 ( 1 1 + 𝜏 ) 𝑥 𝑝0 = ( 1,28694966 + 0 − 1 0 ) ( 10,9070648 1 + 10,9070648 ) 1,28694966 ( 1 1 + 10,9070648 ) 0 𝑝0 = 0,89324646. for the expected value (𝑛𝑝𝑥 ): 𝑛𝑝0 = 199.352(0,89324646) = 178.070,47. in the same way, it will produce a portfolio in table 2. table 2. portfolio of the number of claims with negative binomial distribution 𝑥 𝑛𝑥 𝑛𝑝𝑋 (𝑥) 0 178.080 178.070,47 1 19.224 19.246,38 2 1.859 1.848,29 3 177 170,07 4 11 15,31 5 1 1,36 >5 0 0,12 total 199.352 199.352 c. determine the value of the chi-square test statistic. the chi-square test statistic determined by equation (17) is obtained 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 = 1,6912. with a 95% confidence interval, then 𝛼 = 0,05 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 with degrees of freedom 𝑚 − 𝑟 = 5 − 2 = 3 is 7,8147. based on the values of 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 , it can be concluded that 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 < 𝜒𝑡𝑎𝑏𝑙𝑒 2 and the null hypothesis is accepted. thus, the data used in this study has met the requirements for a negative binomial distribution. 2-poisson distribution test there are three parameters that are assumed to have 2-poisson distribution based on bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 500 equation (8). by using the moment method, it is obtained that 𝑝 ̂ = 0,77481, 𝜆1̂ = 0,06191, and 𝜆2̂ = 0,31092. the distribution test steps carried out are as follows: a. hypothesis formulation. 𝐻0 ∶ data has 2-poisson distribution. 𝐻1 ∶ data is not distributed 2-poisson. b. calculates the probability for each claim frequency and the expected value. the probability is calculated for each claim frequency (𝑝𝑥 ) for each 𝑥 in the table data and based on the parameter estimator values and the 2-poisson distribution formula, the portfolio is obtained in table 3. table 3. portfolio of the number of claims with 2-poisson distribution 𝑥 𝑛𝑥 𝑛𝑝𝑋 (𝑥) 0 178.080 178.081,52 1 19.224 19.217,82 2 1.859 1.868,36 3 177 170,53 4 11 12,89 5 1 0,79 >5 0 0,04 total 199.352 199.352 c. determine the value of the chi-square test statistic. the chi-square test statistic determined by equation (17) is obtained 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 = 0.6249634. with a 95% confidence interval, then 𝛼 = 0,05 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 with degrees of freedom 𝑚 − 𝑟 = 4 − 3 = 1 is 3,8414. based on the values of 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 and 𝜒𝑡𝑎𝑏𝑙𝑒 2 , it can be concluded that 𝜒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 2 < 𝜒𝑡𝑎𝑏𝑙𝑒 2 and the null hypothesis is accepted. thus, the data used has met the requirements for a 2-poisson distribution. parameter estimation of bühlmann's credibility premium based on the estimation of distribution parameter values that have been obtained, it can be determined the parameter estimation of bühlmann's credibility premium. determination of the estimated parameter value using equation (18)-(29). the alleged results are presented in the table 4. table 4. parameter estimation of bühlmann's credibility premium from the data used parameter estimation negative binomial 2-poisson �̅� 0,1179 0,1179 �̂� 0,1180 0,1179 �̂� 0,0108 0,3696 𝑣 0,1180 0,1179 �̂� 10,9071 0,3191 �̂� 0,9999 0,9494 𝑃𝐶 0,11790 0,11799 based on the results in table 4, it can be seen that the estimated frequency of claims in the next period assuming the data is negative binomial and 2-poisson distribution is 0.11790 and 0.11799, respectively. this means that in the negative binomial model it is estimated that there will be 11.79% of policyholders who will make insurance claims in the bühlmann's credibility model with claims of negative binomial and 2-poisson distribution ikhsan maulidi 501 next period, while in the 2-poisson model it is estimated that there will be 11.799% of policyholders who will make insurance claims. the two models give fairly close results, this is because the bühlmann's credibility factor for the two distribution models is quite large, namely each 0.9999 and 0.9494. conclusions bühlmann's credibility formula has been given for data with negative binomial and 2poisson distribution. both distributions are mixed distributions. mixed distributions are quite well used in determining premiums with credibility. this is because in a mixed distribution, the distribution of claim frequency often depends on the distribution of risk. the simulation on the data shows that the premium value obtained is very good with high credibility for both distributions modeled. references [1] t. futami, matematika asuransi jiwa. tokyo: kyoei life insurance, 1993. 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[22] s. m. simanjuntak, “beberapa model banyaknya klaim dalam sistem bonus malus,” ipb university, 2017. a study of count regression models for mortality rate cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 142-151 p-issn: 2086-0382; e-issn: 2477-3344 submitted: october 16, 2021 reviewed: november 04, 2021 accepted: november 05, 2021 doi: https://doi.org/10.18860/ca.v7i1.13642 a study of count regression models for mortality rate anwar fitrianto department of statistics, faculty of mathematics and natural sciences, ipb university email: anwarstat@gmail.com abstract in this study, poisson regression model, negative binomial 1 regression model (negbin 1) and negative binomial regression 2 (negbin 2) model were proposed to fit mortality rate data. the method used is comparing the values of akaike information criterion (aic) and bayesian information criterion (bic) to find out which method suits the data the most. the results show that the data indeed display higher variability. among the three models, the model preferred is negbin 1 model. keywords: mortality; poisson; regression; binomial; overdispersion introduction count data contain variables that count how many times something has happened, such as the number of cases with a particular disease in epidemiology [1]. linear regression models have often been applied to handle this kind of data, but the results are inefficient, inconsistent, and biased. this type of data is considered as count data with variable offset. mortality data is considered as the amount of data that contains the offset variable. a study of mortality for middle-aged men on ischemic heart disease (ihd) that affects mortality has been conducted by [2]. the results showed that there were 46 of 109 deaths around 11.4 years of follow-up due to ihd. in addition to studies on causes of death other than ihd, [3] has researched the global impact of hiv/aids. another study on mortality was conducted by [4] about the diarrheal disease. it has been found that diarrhea causes 1 in 9 child deaths worldwide, the second leading cause among children under 5 years of age. in addition, [5] examined the global causes of death due to disease in children under 5 years. in their study, diarrhea remained the second leading cause of death in children from infection in the last 30 years. in addition, malnutrition is said to be one of the world's worrisome problems. it affects about 6 million child deaths every year. [6] studied that poor nutrition during fetal development can cause severe physical damage, and malnutrition always increases susceptibility to disease. a study conducted by [7] stated that malnutrition (measured as poor anthropometric status) accounted for nearly 50% of childhood deaths. regarding the problem of mortality due to disease, [8] stated that the trend of injuries and deaths from road traffic accidents (rta) is becoming severe in countries such as india. not a day goes by without an rta in india; many people die or become disabled. in https://doi.org/10.18860/ca.v7i1.13642 mailto:anwarstat@gmail.com a study of count regression models for mortality rate anwar fitrianto 143 addition, suicide is one of the factors that contribute to the death rate. in a study by [9], suicidal behavior has always been a major health problem in many countries, both developed and developing countries. poisson regression model is one of the general linear models for data with offset variables. it is also the standard model for calculating data and contingency tables. in this model, the response variable is assumed to have a poisson distribution. in addition to poisson regression, negative binomial regression is also a generalized linear model where the dependent variable is the number of events. the negative binomial distribution is a two-parameter distribution that is generally more flexible than the poisson model [2]. this model can also model scattered quantities, which the poisson model cannot. the negative binomial model can be derived from the poisson distribution and the generalized poisson distribution. [10] has discussed several other specific mortality measures, such as age-specific crude death rates, cause-specific mortality rates, and infant and maternal mortality rates. in the data collection process, there may be biased and inaccurate data measurements. the inaccuracy of this data collection will cause overdispersion. this study aims to identify the most suitable method when dealing with mortality data which usually has overdispersion. methods data the data used in this study is mortality rate data which is available in [11]. the data consists of 163 observations (countries) with seven independent variables, which are the number of people dying per 100,000 live births due to ihd (𝑥1), diarrheal disease (𝑥2), hiv/aids (𝑥3), malaria (𝑥4), malnutrition (𝑥5), road accidents (𝑥6), and suicides (𝑥7). count regression models according to [12], the count regression model has been suggested to be used to model over-dispersed and zero-inflated count response variables. poisson regression is the standard model for modeling count data, while the negative binomial regression model is often introduced to solve count data with overdispersion. meanwhile, the zero-inflated poisson model (zip) and the zero-inflated negative binomial model (zinb) are introduced to solve a zero-inflated variable in which the data contains many zeros. moreover, [13] found that zip and zinb can be obtained by mixing a distribution degenerate at zero with a poisson regression and negative binomial regression, respectively. the probability mass function of the zip is, 𝑃(𝑌 = 𝑦𝑖) = { 𝜔𝑖 + (1 − 𝜔𝑖) 𝑒𝑥𝑝( 𝜆𝑖), 𝑦𝑖 = 0 (1 − 𝜔𝑖) 𝜆𝑖 𝑦𝑖! 𝑒𝑥𝑝( 𝜆𝑖), 𝑦𝑖 > 0 (1) meanwhile, the zinb's probability mass function can be formulated as: 𝑃(𝑌 = 𝑦𝑖) = { 𝜔𝑖 + (1 − 𝜔𝑖) ( 𝜃 𝜃+𝜆𝑖 ) 𝜃 , 𝑦𝑖 = 0 (1 − 𝜔𝑖) 𝛤(𝑦𝑖+𝜃) 𝑦𝑖!𝛤(𝜃) ( 𝜃 𝜃+𝜆𝑖 ) ( 𝜆𝑖 𝜃+𝜆𝑖 ) 𝑦𝑖 , 𝑦𝑖 > 0 (2) with 𝜆𝑖 = 𝑒𝑥𝑝( 𝑥𝑖𝛽). the 0's arise with probability 𝜔 from a second process. the function f that relates to the product 𝑥𝑖𝛾 to the probability 𝜔𝑖 is named as the zero-inflated link function, 𝜔𝑖 = 𝐹(𝑥𝑖𝛾). a study of count regression models for mortality rate anwar fitrianto 144 poisson regression model [14] studied about poisson regression model as the standard model for count data. a variable y is a count of events of poisson regression, and the marginal probability of poisson regression is written as: 𝑃(𝑌 = 𝑦𝑖) = 𝑒𝑥𝑝(−𝜆𝑖)𝜆𝑖 𝑦𝑖 𝛤(1+𝑦𝑖) ; (3) with 𝜆𝑖 = 𝑒𝑥𝑝( 𝛼 + 𝑥𝑖𝛽); 𝑦𝑖 = 0,1, . . . 𝑁. the rate parameter of poisson regression is 𝜆𝑖 and it is also known as its expected count is formulated as: 𝜆𝑖 = 𝑒 𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 . (4) based on equation (4), the log-linear model for mean rate is written as: 𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝, (5) with p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the regression, 𝛽𝑝 are the regression coefficients, and 𝑥𝑖 is the independent variable. [14] formulated maximum likelihood estimation (mle) of poisson regression. let y be a random variable with poisson distribution and with an unknown parameter value 𝜃. the probability mass function of y is obtained, which is 𝑃𝑦(𝑦; 𝜃) to emphasize the parameter 𝜃 and n is the independent trials in order to get the data 𝑦1, 𝑦2, 𝑦3, . . . , 𝑦𝑛. the joint probability mass function is as follows: 𝑃𝑌...𝑌𝑛(𝑦1, . . . , 𝑦𝑛; 𝜃) = 𝑃𝑦(𝑦1; 𝜃). . . . 𝑃𝑦(𝑦𝑛; 𝜃). (6) the likelihood of 𝜃 given data 𝑦1, . . . , 𝑦𝑛 can be obtained from equation (6) by applying the logarithm as follows: 𝐿(𝜃; 𝑦1, … , 𝑦1) = 𝑃𝑌...𝑌𝑛1 (𝑦1, . . . , 𝑦𝑛; 𝜃) = 𝑃𝑦(𝑦1; 𝜃). . . . 𝑃𝑦(𝑦𝑛; 𝜃). (7) the estimated value maximizes the maximum likelihood estimates 𝜃 = 𝜃. y follows a poisson distribution with unknown parameters, and the data is collected from the independent trials are of the form 𝑌1 = 𝑦1, 𝑌2 = 𝑦2, . . . , 𝑌𝑛 = 𝑦𝑛. on the other hand, the likelihood function of the poisson regression is written as: 𝐿 = ∏ 𝑒−𝜆𝑖𝜆 𝑖 −𝑦𝑖 𝑦𝑖! 𝑁 𝑖=1 . (8) the log-likelihood function of poisson regression is obtained by applying the logarithm of equation (8), l ∑ 𝑙𝑜𝑔 ( 𝑒−𝜆𝑖𝜆 𝑖 −𝑦𝑖 𝑦𝑖! )𝑁𝑖=1 . the standard negative binomial regression model according to [1], in most applications, the mean of the data is usually greater than the variance. if otherwise, it is called overdispersion in the particular data. but, based on the study of [15], the poisson regression model is inefficient when dealing with overdispersed data. while in a study by [16], the negative binomial distribution is more flexible than poisson distribution as it is a two-parameter when modeling the data with overdispersion. particularly, negative binomial regression can model overdispersed counts. the negative binomial model can be derived as a mixture of the gamma-poisson model. starting from the conditional mean of the poisson model, 𝐸(𝑦𝑖|𝑥𝑖. 𝜀𝑖) = 𝑒𝑥𝑝( 𝛼 + 𝑥𝑖𝛽 + 𝜀𝑖) = ℎ𝑖𝜆𝑖, (9) a study of count regression models for mortality rate anwar fitrianto 145 where ℎ𝑖 = 𝑒𝑥𝑝(𝜀𝑖). in the case of the poisson-gamma distribution, 𝑔(𝜃, 𝜃) is the poisson distribution while ℎ𝑖 = 𝑒𝑥𝑝(𝜀𝑖) follows gamma distribution. the ℎ𝑖 is assumed to follow a two-parameter gamma distribution, 𝑓(ℎ𝑖) = 𝜃𝜃 𝑒𝑥𝑝(−𝜃ℎ𝑖)ℎ𝑖 𝜃−1 𝛤(𝜃) . (10) once ℎ𝑖 has been integrated out from the joint distribution, then the marginal probability of negative binomial distribution is obtained as follows: 𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) = 𝛤(𝜃+𝑦𝑖) 𝑦𝑖!𝛤(𝜃) ( 𝜃 𝜃+𝜆𝑖 ) 𝜃 ( 𝜆𝑖 𝜃+𝜆𝑖 ) 𝑦𝑖 . (11) the mean of negative binomial is the same as poisson regression, which is written as 𝐸(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 = 𝑒 𝑥𝑖𝛽 and the variance of a negative binomial is written as: 𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 [1 + ( 1 𝜃 ) 𝜆𝑖] = 𝜆𝑖(1 + 𝑘𝜆𝑖), (12) where 𝑘 = 𝑉𝑎𝑟(ℎ𝑖). moreover, the rate parameter of negative regression 𝜆𝑖, which is also known as its expected counts, is written as: 𝜆𝑖 = 𝑒 𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 . (13) the log-linear model for the mean rate of negative binomial regression can be obtained by applying the logarithm of equation (13): 𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝, (14) where p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the regression, 𝛽𝑝 are the regression coefficients, and x's are the independent variables. [17] has discussed the mle of negative binomial regression in which random samples of n subjects are given. in a standard negative binomial model, the dependent variables 𝑦𝑖 and the predictor variables 𝑥1𝑖, 𝑥2𝑖, … , 𝑥𝑝𝑖 are included. predictor variables are combined to form the following matrix, 𝑿 = [ 1 𝑥11 … 𝑥1𝑝 1 𝑥12 ⋯ 𝑥2𝑝 ⋮ ⋮ ⋱ ⋮ 1 𝑥1𝑛 ⋯ 𝑥𝑛𝑝] . the 𝑖𝑡ℎ row of x is designated to be 𝑥𝑖 , from equation (11), the 𝜆𝑖 in which is replaced by 𝑒𝑥𝑖𝛽. the equation can be rewritten as, 𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) = 𝛤(𝜃+𝑦𝑖) 𝑦𝑖!𝛤(𝜃) ( 𝜃 𝜃+𝑒𝑥𝑖𝛽 ) 𝜃 ( 𝑒𝑥𝑖𝛽 𝜃+𝑒𝑥𝑖𝛽 ) 𝑦𝑖 . (15) the likelihood function of negative binomial is stated as below, 𝐿 = ∏ γ(𝜃+𝑦𝑖) 𝑦𝑖!γ(𝜃) 𝑁 𝑖=1 ( 𝜃 𝜃+𝑒𝑥𝑖𝛽 ) 𝜃 ( 𝑒𝑥𝑖𝛽 𝜃+𝑒𝑥𝑖𝛽 ) 𝑦𝑖 , (16) and the log-likelihood function of negative binomial regression is obtained by applying the logarithm to obtain the following equation:               ln1lnln1ln) 1 ln( 1                ii i iii n i yy x yxyyl e . (17) a study of count regression models for mortality rate anwar fitrianto 146 negative binomial 1 model and negative binomial p model [18] have shown the equation (9) is considered as a negative binomial 2 (negbin 2) model. they re-parameterized the negbin 2 model, and it is labeled as specification negative binomial 1 (negbin 1), which is written as, 𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖 + 𝑘𝜆𝑖 = 𝑉𝑎𝑟(𝑦𝑖|𝑥𝑖) = 𝜆𝑖(1 + 𝑘). (18) the marginal probability of negbin 1 is obtained by replacing 𝛳 with 𝛳𝜆𝑖 in equation (11), 𝑃(𝑌 = 𝑦𝑖|𝑥) = γ(𝜃𝜆𝑖+𝑦𝑖) 𝑦𝑖!γ(𝜃𝜆𝑖) ( 𝜃𝜆𝑖 𝜃𝜆𝑖+𝜆𝑖 ) 𝜃 ( 𝜆𝑖 𝜃𝜆𝑖+𝜆𝑖 ) 𝑦𝑖 . (19) by replacing 𝜃 with 𝜃𝜆𝑖 2−𝑃 in equation 19, the negative binomial p (negbin p) model is written as: 𝑃(𝑌 = 𝑦𝑖|𝑥𝑖) = 𝛤(𝜃𝜆𝑖 2−𝑃+𝑦𝑖) 𝑦𝑖!𝛤(𝜃𝜆𝑖 2−𝑃) ( 𝜃𝜆𝑖 2−𝑃 𝜃𝜆𝑖 2−𝑃+𝜆𝑖 ) 𝜃𝜆𝑖 2−𝑃 ( 𝜆𝑖 𝜃𝜆𝑖 2−𝑃+𝜆𝑖 ) 𝑦𝑖 . (20) overdispersion [19] have proposed that in almost the statistical study for the count data, it is always assumed that the dependent variable follows the poisson distribution. the mean is assumed to be equal to the variance. however, in real life, the variance is usually larger than the mean. [19] also stated that overdispersion indicates high variability around a model's fitted values in the poisson formulation. this case will lead to a negative binomial model as a proposal to correct this problem. when the data are over-dispersed, the variance is not the same as its mean, or 𝑉𝑎𝑟(𝑥𝑖) = 𝜑𝜆, where 𝜆 is the mean. if 𝜑 = 1, the poisson model is ordinary; if 𝜑 > 1, it means that the model is overdispersed model. consequently, [20] stated that a unique property of distributions in exponential families is the conditional variance equal the conditional mean. the dispersion parameter, 𝜑. in the poisson model, the dispersion parameter is set to constant value 𝜑 = 1. count data according to [16], count data indicates how many times or how frequent something happens. furthermore, [18] stated that an event outcome is the number of times an event occurs while an event count is a nonnegative random variable. the examples of count data included the number of patients hospitalized, the number of thieves arrested, and the number of natural disasters. in some cases of count, data have offset variables. [21] said that offset variable is always being analyzed by the generalized linear model (glm) and count regression model. the analysis is usually used whenever the data is recorded over an observed period. offset is used to denote the period observed in glm. other than that, offset is usually defined as a measure of exposure. the exposure can be the number of house years incurred, and the response will be the number of claims incurred. the log-linear mean rate for poisson regression and negative binomial model is, 𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝, (21) when applying poisson regression or negative binomial regression, the offset variable, 𝑙𝑜𝑔(𝑡) is added a study of count regression models for mortality rate anwar fitrianto 147 𝑙𝑜𝑔(𝜆𝑖) = 𝛽0 + 𝛽1𝑥1 + ⋯ + 𝛽𝑝𝑥𝑝 + 𝑙𝑜𝑔( 𝑡) (22) 𝑙𝑜𝑔(𝜆𝑖) − 𝑙𝑜𝑔( 𝑡) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝 𝑙𝑜𝑔 ( 𝜆𝑖 𝑡 ) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝. 𝑙𝑜𝑔(𝜆𝑖) − 𝑙𝑜𝑔( 𝑡) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝 𝑙𝑜𝑔 ( 𝜆𝑖 𝑡 ) = 𝛽0 + 𝛽1𝑥1+. . . +𝛽𝑝𝑥𝑝, where p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the regression, 𝛽𝑝 are the coefficients of the regression, 𝑥 is the independent variable, t is the period observed (exposure), log (t) is the offset variable and 𝜆𝑖 𝑡 is the rate. in this study, our interest is in modeling for the mortality data, which is count data. poisson regression and negative binomial regression are generally appropriate to deal with the count data. in this research, our interest is to find out which regression best fits the mortality data. modelling the mortality rate data poisson regression and negative binomial regression are the main study in this research in modeling the data. the model for poisson model and negative binomial model are written as equation (22), where p is the number of predictors or covariates in the model, 𝛽0 is the intercept of the regression, 𝛽𝑝 is the covariate coefficients, and 𝑥 is the independent variable. the 𝑙𝑜𝑔 ( 𝜆𝑖 𝑡 ) represents the number of people dying per time unit and the function βx is the relationship of death rate changes as a function of subject covariates. the null hypothesis states the slope is equal to zero, whereas the alternative hypothesis indicates the slope is not equal to zero. goodness-of-fit test deviance and person's chi-square will be carried out to check if the data has overdispersion or under-dispersion. the results of deviance and pearson's chi-square that are divided by the degree of freedom (df) should be approximately equal to one. if the values are more than one, this indicates that the data is overdispersion. goodness-of-fit is performed by using the proc genmod statement in sas. deviance for fitted poisson regression and negative binomial regression is written as: 𝐷 = 2 ∑ {𝑦𝑖𝑙𝑜𝑔 ( 𝑥𝑖 𝑦𝑖 ) − (𝑥𝑖 − 𝜆𝑖)} 𝑛 𝑖=1 . (23) and the pearson’s chi-square is defined as, 𝜒2 = ∑ (𝑥𝑖−𝜆𝑖) 2 𝑉𝑎𝑟(𝑥𝑖) 𝑛 𝑖=1 , (24) where 𝜆𝑖 = 𝑒 𝛽0+𝛽1𝑥1+...+𝛽𝑝𝑥𝑝 . a study of count regression models for mortality rate anwar fitrianto 148 results and discussion mortality rate data models proc genmod statement in sas version 9.4 was used to run the poisson regression analysis. at 5% level of significance, all independent variables contributed significantly to the mortality rate with the following estimated poisson regression model (table 1): ( 𝑙𝑜𝑔(𝜆𝑖) 𝑡 ) ̂ =6.5834 + 0.0008𝑥1+ 0.0039𝑥2+0.0010𝑥3+0.004𝑥4-0.003𝑥5– 0.0123𝑥6+0.0081𝑥7 table 1. analysis of maximum likelihood parameter estimates for poisson regression parameter degree of freedom estimate standard error chi-square pr > chisquare intercept 1 6.5834 0.0083 6255069.00 <.0001 𝑥1 1 0.0008 0.0000 507.71 <.0001 𝑥2 1 0.0039 0.0002 312.25 <.0001 𝑥3 1 0.0010 0.0000 1787.60 <.0001 𝑥4 1 0.0046 0.0002 529.74 <.0001 𝑥5 1 -0.0032 0.0003 95.57 <.0001 𝑥6 1 -0.0123 0.0004 1092.88 <.0001 𝑥7 1 0.0081 0.0004 463.89 <.0001 the estimated poisson model, along with the standard error of each estimated coefficient and p values, indicated that the ihd, diarrheal disease, aids/hiv, malaria, malnutrition, road accidents and suicides were significant predictors contributing to the mortality rate. as an alternative to the poisson regression model, the data were also analyzed using the negative binomial model. table 2 displays the result of the analysis based on maximum likelihood estimation for the negative binomial regression. table 2. analysis of maximum likelihood parameter estimates for negative binomial regression parameter degree of freedom estimate standard error chi-square pr > chi square intercept 1 6.5602 0.00875 5616.57 <.0001 𝑥1 1 0.0008 0.0004 4.02 0.0451 𝑥2 1 0.0046 0.0026 3.16 0.0755 𝑥3 1 0.0011 0.0003 13.13 0.0003 𝑥4 1 0.0054 0.0022 5.95 0.0147 𝑥5 1 -0.0041 0.0038 1.19 0.2750 𝑥6 1 -0.0138 0.0037 13.88 0.0002 𝑥7 1 0.0112 0.0045 6.13 0.0133 fitting the data using the negative binomial regression model found that all independent variables are except 𝑥2 (diarrheal disease) and 𝑥5(malnutrition) contribute significantly to the mortality rate. both variables have a more considerable p value (0.0755 for diarrheal diseases and 0.2750 for malnutrition). hence, diarrheal disease and malnutrition were not significant predictors, while the other variables ihd, aids/hiv, malaria, road accidents, and suicides, were the significant predictors. the predicted model using the negative binomial regression model for the mortality rate data is written as, ( 𝑙𝑜𝑔(𝜆𝑖) 𝑡 ) ̂ =6.5602+0.0008𝑥1+0.0046𝑥2+0.0011𝑥3+0.0054𝑥4-0.0041𝑥5-.0138𝑥6+0.0112𝑥7 a study of count regression models for mortality rate anwar fitrianto 149 descriptive statistics of the variables for checking overdispersion when the variance of a particular variable is higher than its mean, it indicates that the data has overdispersion. in this study, the dependent variable's mean and variance were 824.0061 and 105125.22, respectively, indicating overdispersion. table 3 displays the means and variances of all the variables in the study. all the variables were overdispersed and more considerable variability was given around a model's fitted values in poisson regression, 𝑉𝑎𝑟(𝑥𝑖) =𝜑𝜆, 𝜑 >1. as a consequence, the negative binomial regression was the better approach for modeling over-dispersed count data. table 3. the mean and the variance for each variable variable mean variance mortality 824.0061 105125.22 ihd 114.5747 6031.84 diarrhoel disease 22.0791 1118.02 aids/hiv 46.6652 11754.18 malaria 11.5688 481.2379 malnutrition 12.7489 477.6099 road accidents 17.2767 91.6093 suicides 10.0120 52.6231 goodness-of-fit test for poisson regression and negative binomial regression the main purpose of the goodness-of-fit test is to determine a more appropriate model. table 4 presents the deviance and pearson's chi-square to observe whether the deviance and pearson's chi-square obtained close to one. table 4. goodness-of-fit test for poisson regression and negative binomial regression model criterion df value value/df poisson regression deviance 155 15081.6228 97.3008 pearson's chi-square 155 14196.5148 91.5904 negative binomial regression deviance 155 167.1663 1.0785 pearson's chi-square 155 131.1002 0.8458 the value/df column of deviance and pearson's chi-square for the poisson model were 97.3008 and 91.5904, respectively, which were remarkably higher than one. the poisson model did not correctly describe the data. there was more significant variability among counts than will be expected for poisson distribution. this situation arises because repeated subjects may not be independent. one of the possible reasons for the overdispersion is that experimental conditions are not under control, hence 𝜆𝑖 varies with uncontrolled factors. the table shows that the negative binomial regression was the better alternative to model the mortality rate. the value/df of the deviance and pearson's chi-square were 1.0785 and 0.8458, respectively. both values were closer to one as compared to the corresponding values in the poisson regression model. comparison between poisson regression, negative binomial 1 and negative binomial 2. comparisons between all the three proposed models for the mortality data were given in table 5. the aic for poisson regression was larger compared to the other two. the aic value for negbin 1 was slightly smaller than the one for negbin 2. it indicated that negbin l was a better fit than poisson regression and negbin 2. on the other hand, the bic values for the three regressions were 16501, 2345, and 2347, respectively, for a study of count regression models for mortality rate anwar fitrianto 150 poisson, negbin 1, and negbin 2. the bic value for poisson regression was much higher when compared to the negative binomial regressions. thus, with lower aic and bic values, the negbin 1 was the better approach for the mortality rate data since it can explain more variation with the same number of independent variables.. table 5. aic and bic values between fifferent regression models regressions aic bic poisson 16476 16501 negbin 1 2317 2315 negbin 2 2319 2347 conclusions the analysis was conducted to compare the performance of three models: poisson regression, negbin 1 and negbin 2. the negbin 1 has been proven that it is the most appropriate model for overdispersed data. the mean and the variance were calculated to ensure that data has overdispersion. since the data were overdispersed, the results of deviance and pearson's chi-square showed that negative binomial was a better model for the data. then, the performance of aic and bic showed that negbin 1 is a better model, followed by negbin 2 and poisson regression. references [1] tutz, g. regression for categorical data, cambridge university press, new york, 2012. [2] kumpusalo, e., lakka, h.n., laaksomen, d.e., lakka, t.a., niskanen, l.k., salonen, j.t. and tuomilehto, j., "the metabolic syndrome and total and cardiovascular disease mortality in middle-age men", journal of american medical association, vol. 288, no. 3, pp. 2708-2716, 2002. [3] gayle, h.d. and hill, g.l. "global impact of human immunodeficiency virus and aids". clinical microbiology reviews, vol. 14, no. 2, pp. 327 – 335, 2001. [4] breman, j.g., jamison, d.t., and measam, a.r., disease control priorities in developing countries 2nd edition: worldbank, washington (dc), 2006. [5] claudio f. lanata, christa l. fischer-walker and ana c. olascoaga, carla x. torres, martin j. aryee, robert e. black, "global causes of diarrheal disease mortality in children <5 years of age: a systematic review", plos one. vol. 8, no.9, pp. 1-11, 2013. [6] bassett, l. and levinson, f.j. malnutrition is still a major contributor to child death. population reference bureau, boston, 2007. 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[18] cameron, c.a. and trivedi, p.k., regression aof count data, cambridge university press, cambridge, 2013. [19] berk, r and macdonald, j., "overdispersion and poisson regression", journal of quantitative criminology, vol. 24, no. 3, pp. 269-284. [20] turner, h., introduction to generalized linear model, esrc national centre for research method, 2008. [21] yan, j., guszcza, j., flynn, m., and wu, c.s., "applications of the offset in propertycasualty predictive modeling", casualty actuarial society e-forum, vol. 1, no. 1., pp. 366-385. http://www.worldlifeexpectancy.com/lifethe generalized star modeling with heteroscedastic effects cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 158-172 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 05, 2021 reviewed: august 20, 2021 accepted: december 20, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.13097 the generalized star modeling with heteroscedastic effects utriweni mukhaiyar1,*, syahri ramadhani2 1statistics research division, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia 2undergraduate programme in mathematics, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia *corresponding author email: utriweni@math.itb.ac.id abstract most of the generalized space time autoregressive (gstar) models assume the constant error variance. in fact, there are many space-time observations whose variability is changing over the times. in this study, a gstar model is built with an error variance that is not constant or has a heteroscedasticity effect, namely the combination of gstar–autoregressive conditional heteroscedasticity (arch). the parameters of the gstar–arch model are estimated using the generalized least square (gls) method to obtain the efficient parameter estimation. as a case study, the gstar–arch model is applied to the daily mean wind speed data of new orleans, florida and mississippi, in order to predict the occurrence of hurricane katrina that occurred in 2005. it is obtained that the heteroscedastic involvement in gstar modeling gives the better results in predictions, compared to the homoscedastics approach. furthermore, as the order of model is higher, the gstar model performances is better, which is shown by the least mean squared errors (mse) and mean absolute percentage error (mape). the obtained results show that the gstar model (3;0,0,1)–arch(1) predicts the hurricane katrina better than the gstar(3;0,0,1) and gstar(1;1)–arch(1) models. keywords: gstar; arch; conditional variance; generalized least squares; heteroskedasticity introduction a hurricane is a natural phenomenon in the form of wind gusts with a speed exceeding 119 km/hour. hurricanes are a type of tropical cyclone that usually forms on warm sea surfaces around the equator. the wind speed at one location is influenced by the wind speed of the previous time at that location and also influenced by the average wind speed at other locations. this means that the wind speed can be modeled with space-time models such as starma(p,q), star(p), stma(q), gstar(p,q) and starmag(p,q). the model used in this study is the gstar model which uses a certain weight matrix according to the location conditions. http://dx.doi.org/10.18860/ca.v7i1.13097 mailto:utriweni@math.itb.ac.id the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 159 the arrival of a storm which is usually predicted by weather satellites will be predicted using the gstar model. by using the daily average wind speed data one year before the storm, it is hoped that the arrival of the storm can be predicted earlier and reduce the number of victims, both property and life, due to the storm. however, a large increase in wind speed when a storm occur, causes the data to have heteroscedastic effect, so that the error generated by the gstar model has a non-constant variance. it makes the estimation of the initial model parameters being no longer efficient, thus a model that explains the variance of the error is not constant, namely the arch model is should be developed. the use of the arch is to model the variance of errors, such that it is expected to eliminate heteroscedastic elements and more efficient parameters can be produced. the development of gstar model in indonesia is very fast, both theoretically and in application. theoretically, it includes the stationary properties of the process using the inverse autocovariance matrix [1] as well as the kernel approach [2, 3] and minimum spanning tree approach [4], gstar with correlated errors [5, 6], gstar with outliers [7], gstar for discrete data [8], and invertibility of kernel gstar model [9]. the application of the gstar model has been carried out on economic data [10], tea plantations [11], palm oil production [12], red chili commodity prices [13], number of dengue fever cases [14], predictions of robbery cases in medan, north sumatra [15], the spread of covid-19 cases in java [16], and copper and gold grades vertical distribution [17]. on the other hand, the arch model, which accommodates the element of heteroscedasticity and exogenous variable which make the high volatility of process, is widely developed in economic problems. the impact of covid-19 as the exogenous factor to the economic sector be explored by [18]. sometimes the exogeneous factor cause a point of change happen and it should be detected [19]. however, its application has also been carried out to predict electric current [20], caterpillar pests in oil palm plantations [21], and rice prices [22]. the development of the gstar model with an error variance by considering the heteroscedasticity effect, has been investigated by [23] on the gstar(1,1) model with application to stock prices. the contruction of gstar model with the arch effect and estimate the parameters using the maximum likelihood method approach be explored by [24]. in this study, the gstar–arch model was developed by estimating the parameters using the generalized least square (gls) approach. methods generalized star model the gstar model is a generalization of the star model where the model parameters for each location that were initially considered homogeneous can be different. an observation at location i at time t are expressed as 𝑍𝑖,𝑡 . if the observations between locations are related, then these observations can be modeled using the gstar model. the general form of gstar is, 𝒁𝑡 = ∑ ∑ 𝚽𝑗𝑙 λ𝑘 ℓ=0 𝑘 𝑗=1 𝑾(𝓵)𝒁𝑡−𝑗 + 𝒆𝑡 the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 160 with 𝒁𝑡 is a n-dimensional column vector ( 𝑍1,𝑡 , 𝑍2,𝑡 , … , 𝑍𝑁,𝑡 )′, 𝑾 (𝓵) is a n-dimensional weight matrix for spatial lag-ℓ𝑡ℎ , 𝚽𝑗𝑙 is n-dimensional matrix of autoregressive parameters for spatial lag ℓ and time lag j, and 𝒆𝑡 is a n-dimensional vector of errors respective to the observations inside vector 𝒁𝑡 . for homoscedastic gstar, the 𝒆𝑡 is a white noise vector whose mean and variance are constant, and follows normal multivariate distribution, meanwhile for heteroscedastics case, the variance is not constant. the gstar modeling stage follows the box-jenkins iteration [1], consist of model identification, parameter estimation and diagnostic checking. before doing gstar modeling, the data of process must have stationary properties. if it is not stationary, a differentiation process should be performed on the data until the data is stationary. in estimating the parameters of the gstar model, it can be done using the ordinary least square (ols) method by constructing the gstar model into a linear form 𝒀 = 𝐗𝜷 + 𝒆, so that the ols estimator obtained is �̂� = (𝐗′𝐗)−1𝐗′𝒀 [1] arch(1) model consider a process {yt} which follows ar(p) model such that can be written as: 𝑌𝑡 = 𝜙0 + 𝜙1𝑌𝑡−1 + ⋯ + 𝜙𝑝𝑌𝑡−𝑝 + 𝑒𝑡 which 𝑒𝑡 is uncorrelated errors but has inconstant variance or depend on time. based on engle (1982), the error 𝑒𝑡 can be expressed as, 𝑒𝑡 = 𝑎𝑡 𝜎𝑡 (1) with 𝑎𝑡 is random sample which independent and has identical standard normal distribution, and 𝜎𝑡 2 = 𝛼0 + 𝛼1𝑒𝑡−1 2 + ⋯ + 𝛼𝑝𝑒𝑡−𝑝 2 (2) if the erros are known until (t-1) the the conditional variance of 𝑒𝑡 is stated as: 𝑉𝑎𝑟𝑡−1(𝑒𝑡 ) = 𝐸𝑡−1[𝑒𝑡 2] = 𝐸[𝑒𝑡 2|𝑒𝑡−1 2 , 𝑒𝑡−2 2 , … ] = 𝜎𝑡 2 from eq. (2), it can be said that the conditional variance of 𝑒𝑡 depends on squares of the past errors and inconstant. tthis condition is named as arch(p) model [25]. the simplest form of the arch(p) model and used in this study is the arch(1) model. in this model, the error variance at time t is affected by the square of the error of the previous one time lag. the arch(1) model is formulated as: 𝑒𝑡 = 𝑎𝑡 𝜎𝑡 and 𝜎𝑡 2 = 𝛼0 + 𝛼1𝑒𝑡−1 2 with 𝛼0 and 𝛼1 are non-negative parameters of arch(1) model. the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 161 the variance of arch(1) errors is, var(𝑒t) = 𝐸[𝑒𝑡 2] − (𝐸[𝑒𝑡 ]) 2 = 𝐸[𝑒𝑡 2] = 𝐸[𝑎𝑡 2]𝐸[𝜎𝑡 2] = 𝐸[𝛼0 + 𝛼1𝑒𝑡−1 2 ] = 𝐸[𝛼0] + 𝐸[𝛼1𝑒𝑡−1 2 ] = 𝛼0 + 𝛼1var(𝑒𝑡 ) then it is obtained, var(𝑒𝑡 ) = 𝛼0 1−𝛼1 (3) since the variance is positive then, based on eq. (3), 𝛼0 > 0 and 0 ≤ 𝛼1 < 1, be the stationary condition of arch(1) model. gstar(1;1) – arch (1) model consider 𝒁𝑡 = (𝑍1,𝑡 , 𝑍2,𝑡 , . . . , 𝑍𝑁,𝑡 )′ as a vector of observations in n location at time t, can be modeled as gstar(1;1)–arch(1), if it can be expressed as: 𝒁𝑡 = (𝜱𝟎 + 𝜱𝟏𝑾)𝒁𝑡−1 + 𝒆𝑡 (4) where 𝒆𝑡 ~ 𝑁(𝟎, ωt), is vector of errors which follows normal distribution with zero mean and inconstant variance over the time. the covariance matrix ωt is defined as ωt = diag(h1,t, h2,t, . . . , hn,t) and hi,t is a vector of erros variance on location i at time t, which can be modeled as arch(1), that is 𝒉𝑖,𝑡 = 𝜶0𝑖 + 𝜶1𝑖 𝒆𝑖,𝑡−1 𝟐 with 𝜶𝑘𝑖 is parameter of model for location i and k = 0, 1. the assumption used in this model is that the errors between locations are uncorrelated with each other so that the error variance at location i at time t is affected by the square of the errors on that location at time (t – 1) , but is not affected by the errors on the other locations. meanwhile, the observation value of location i is influenced by the observations on that location and also the neighbor locations. the method used to estimate this model is the generalized least square (gls) method. suppose that the matrix ω has eigenvalues, 𝜆1, 𝜆2, … , 𝜆𝑇 . by cholesky's decomposition, it can be written as 𝛀 = 𝐂𝚲𝐂′, with 𝚲 = diag(𝜆1, 𝜆2, … , 𝜆𝑇 ) is a diagonal matrix, and c is an orthogonal matrix. consider the matrix 𝐂 = 𝐏−1𝚲 − 1 2 , with 𝚲 1 2 = diag(√𝜆1, √𝜆2, … , √𝜆𝑇 ). thus , 𝛀−1 = 𝐏−1𝚲−1(𝐏′)−1 = 𝐏−1𝚲 − 1 2𝚲′ − 1 2(𝐏′)−1 = 𝐂𝐂′. let the transformed linear model, 𝒀∗ = 𝐗∗𝜷 + 𝒆∗ be defined with 𝒀∗ = 𝐂𝒀, 𝐗∗ = 𝐂𝐗, and 𝒆∗ = 𝐂𝒆. then, unbiased estimator of gstar–arch model parameters are presented as: �̂�gls = (𝐗 ′𝛀−1𝐗)−1𝐗′𝛀−1𝒀 (5) the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 162 with 𝛀 = diag(ℎ1(1), … , ℎ1(𝑇), … , ℎ𝑁 (1), … , ℎ𝑁 (𝑇)) is nt-dimentional of diagonal [26]. furthermore, the stages of gstar-arch modeling is illustrated in a flow chart as presented in fig. 1. figure 1. flowchart of the gstar-arch modeling stage. modeling is carried out to obtain a homoscedastic error. the equation for the mean is modeled by the gstar model while the variance is modeled by the arch model. results and discussion as a case study, the data used are the average daily wind speed in three states of the united states (n = 3) from september 1, 2004 to august 24, 2005 (t = 358) obtained from the national oceanic and atmospheric administration (noaa) that belongs to the united states. geographically, the united states is located around the equator, so some areas of the country are often hit by storms. hurricane katrina was one of the deadliest hurricanes that occurred in 2005. according to the united states department of oceanic and atmospheric research, the total loss caused by hurricane katrina was over a terdapat efek arch space-time data data plot data is stationary no yes gstar model identification difference parameter estimation arch effect test for errors no yes gstar model identify arch model parameter estimation of arch parameter estimation of gstar arch diagnostics tes of model homoscedastic of errors yes no short-time forecasting finish is there arch effect? the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 163 thousand million dollars and more than 1,800 people died. the states of interest are new orleans (louisiana), florida, and mississippi, each of which can be seen in fig. 2. the modeling is carried out with the help of the r application. the data will be modeled with the space-time model and must meet the stationary properties first. the stationary data can be seen from the plot of the row of observations at each location as in fig. 3(a). from the figure, it can be seen that there are several wind speed values that are higher than other observations. in addition, there is also a slight downward and rising pattern, which indicates a data pattern that is not stationary on average. therefore, the data differentiation is done first. the series plot after one-time differentiation can be seen in fig. 3(b). the stationary data is then centered so that it has a zero mean (centralized process). the process variability which occasionally increases, indicates that the variance is not constant. this will be accomodated in the modeling with heteroscedastic effect. in gstar modeling, one of the important elements that characterizes the relationship between locations is the presence of a weight matrix. the weight matrix has entries 𝑤𝑖𝑗 , which represents the weight of location-j to location-i. this matrix has zeros entries in the main diagonal and the total weight in one row is equal to one. in this study, the weight matrix used is uniform and binary weights. the spatial lag used is limited to only one spatial lag. for simplicity, the uniform and binary weight matrix be used, respectively, are 𝑾(𝟏) = ( 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 ) and 𝑾(𝟏) = ( 0 0 1 1 0 0 1 0 0 ) figure 2. map of the united states and the states where the observations are located (source: https://greatpersie.wordpress.com). new orleans (louisiana), florida, and mississippi are referred to as location 1, location 2, and location 3, respectively (1) (2) (3) the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 164 figure 3. plot series for each location, (a) before differentiation and (b) after differentiation by one time lag. after differentiation, the data become centered with a more stationary pattern in the mean the first stage in the modeling is model identification with the help of space-time acf and pacf plots, called stacf and stpacf. however, because the model has been determined at the beginning, namely the gstar(1;1) model, the model identification stage can be skipped. the stacf and stpacf plots obtained are used to see whether there is a relationship between time and location from the daily average wind rate data. the plots of stacf and stpacf can be seen in fig. 4. from fig. 4 it can be seen that the data have time and spatial dependence, although the gstar(1;1) model is not very appropriate to model this data. from the stpacf plot, the best possible model for the data is gstar(3;0,0,1). thus, the modeling be considered are gstar(1;1) and gstar(3;0,0,1) model. first, the obtained estimated parameters using ordinary least squares (ols) method for the gstar(1;1) model can be seen in table 1. table 1. the ols parameter estimation of gstar(1;1) model parameter 𝝓𝟎𝟏 𝝓𝟏𝟏 𝝓𝟎𝟐 𝝓𝟏𝟐 𝝓𝟎𝟑 𝝓𝟎𝟑 estimation -0.15 0.08 -0.08 -0.04 -0.28 0.20 (a) (b) the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 165 the next step is to test the presence of the arch effect on the error of each location. the existence of the arch effect can be detected from the plot of the squared error of each location which can be seen in fig. 5. figure 4. stacf (left) and stpacf (right) plot based on uniform weight matrix. the stacf has more patterned values than stpacf. thus the autoregressive model is more appropriate figure 5. the square errors plot of the gstar(1;1) model. it can be seen that the error variance is not constant, indicating the presence of the arch the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 166 to confirm the existence of the arch effect, the arch-lm test will be used (engle, 1982). the presence of arch effect on the error is indicated by the p-value which is smaller than 1% ≤ α ≤ 10%. the arch-lm test results for the first six time lags in table 2, show that the p-value is smaller in almost all locations and time lags. so it can be concluded that there is an arch effect on the gstar(1;1) model, means that the variance of errors is not constant over time. a slight difference found at location 2, florida. the p-values obtained are less than 9.2% until the third time lag, indicates that the wind speed value in this area tends to be more constant in average and variance than the other two observation locations. at a time lag of more than three, the wind speed in florida did not show any arch effect on the process. however, the presence of heteroscedasticity effects in two other locations, also in florida until the first three-time lags, be the reason to consider the inconstant variance in this case. table 2. the p-values of heteroscedastic effect existence using arch-lm test time lag new orleans (× 10−4) florida mississippi (× 10−2) 1 0.0019 0.0304 0.0300 2 0.0085 0.0731 0.0980 3 0.0067 0.0917 0.2000 4 0.0224 0.1550 0.4300 5 0.0666 0.2170 0.8000 6 0.1490 0.2950 0.9300 next, the inconstant erros variance of gstar(1;1) be modeled by arch(1). the obtained parameter estimation of model arch(1) model using maximum likelihood (ml) method, can be seen in tabel 3. table 3. the ml parameter estimation of arch(1) model. the i-th parameter of locationj is presented by 𝜶𝑖𝑗 for 𝑖 = 0,1 and 𝑗 = 1,2,3. parameter 𝜶𝟎𝟏 𝜶𝟏𝟏 𝜶𝟎𝟐 𝜶𝟏𝟐 𝜶𝟎𝟑 𝜶𝟏𝟑 estimation 1.01 0.25 0.66 0.18 0.60 0.30 thus, the variances of errors for each location are obtained as:: 𝜎1,𝑡 2 = 1.01 + 0.25𝑒1,𝑡−1 2 𝜎2,𝑡 2 = 0.66 + 0.18𝑒2,𝑡−1 2 𝜎3,𝑡 2 = 0.60 + 0.30𝑒3,𝑡−1 2 furthermore, the variances of errors of each location, for t=1,2,..., t be the entries of 𝑁𝑇 dimenstional of diagonal matrix, ω(t) = diag(𝜎1,𝑡 2 , 𝜎2,𝑡 2 , 𝜎3,𝑡 2 ). this matrix is used for parameter estimation of gstar(1;1) – arch (1) model by using the gls mehod. the estimated parameters are presented in table 4. the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 167 table 4. the gls parameter estimation of gstar(1;1) model with the errors variance follows arch(1) model parameter 𝝓𝟎𝟏 𝝓𝟏𝟏 𝝓𝟎𝟐 𝝓𝟏𝟐 𝝓𝟎𝟑 𝝓𝟎𝟑 estimation -0.11 0.08 -0.05 -0.05 -0.27 0.22 the estimated parameters of the model obtained using the gls method in table 4, are not much different from the estimated parameters obtained using the ols method in table 1. this is probably because the arch(1) model is not the right model to model the error variance of the gstar(1;1) model. for comparison, the modeling with the same steps was carried out again using a binary weight matrix and a weight matrix based on wind direction. determination of the best model is done by comparing the value of the mean squared error (mse) of each model. from table 5 it can be concluded that the use of a uniform weight matrix in the gstar(1;1)–arch (1) modeling is slightly better than the binary weight matrix. the three locations involved in modeling give the composition of the uniform and binary weight matrix are quite similar. table 5. the mse values of gstar(1;1) – arch(1) model with two types of thw weight matrix. the results using uniform weight matrix is slightly better than binary, although the difference is not significant weight matrix mse uniform 0.975 binary 0.978 the comparison between the original and estimated data using the gstar(1;1)– arch(1) model can be seen in fig. 6. from this figure, it can be seen that there is a big difference between the original data and estimated results. this is because the stpacf plot in fig. 4 indicates that the gstar(1;1) model is not appropriate for the data. from the stpacf plot, the possible next space-time model is the gstar (3;0,0,1) with a fixed error variance be modeled by the arch(1) model. this model explains that the condition of location i at t is influenced by its own condition at-(t – 1), (t – 2), (t – 3) and the conditions of other locations which are its closest neighbors at (t – 3). the modeling is carried out following the similar steps in modeling gstar(1;1) – arch (1) until the parameters are obtained before and after the arch element is calculated. the estimation parameter results can be seen in table 6. the estimation parameters obtained from the gstar(3;0,0,1) modeling are not much different from the gstar estimation parameters gstar (3;0,0,1)–arch(1) so that the estimation results obtained will also not far different. the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 168 figure 6. plot of original series (red) and estimation (blue, green) using gstar(1;1) – arch(1) model. the large errors indicate that the gstar(1;1) model is not suitable. table 6. the comparison of gstar(3;0,0,1) parameter estimation before and after the errors variance be modeled by arch(1). generally, there is no difference of both models. parameter 𝝓𝟏𝟎 𝟏 𝝓𝟐𝟎 𝟏 𝝓𝟑𝟎 𝟏 𝝓𝟑𝟏 𝟏 𝝓𝟏𝟎 𝟐 𝝓𝟐𝟎 𝟐 𝝓𝟑𝟎 𝟐 𝝓𝟑𝟏 𝟐 𝝓𝟏𝟎 𝟑 𝝓𝟐𝟎 𝟑 𝝓𝟑𝟎 𝟑 𝝓𝟑𝟏 𝟑 before -0.24 -0.40 -0.15 -0.07 -0.17 -0.32 -0.27 -0.02 -0.31 -0.37 -0.13 -0.07 after -0.21 -0.36 -0.13 -0.09 -0.13 -0.33 -0.24 -0.03 -0.30 -0.34 -0.10 -0.03 the comparison of the original and estimated data using the gstar(3;0,0,1)– arch(1) model can be seen in fig. 7. by comparing the plots in fig. 6 and fig. 7, it can be seen that the estimation results generated by this model are better than the gstar(1;1)–arch(1) model. the gstar(3;0,0,1)–arch(1) model can capture the pattern of process variability. however, this model is not the best model for the data. from fig. 7 it can be seen that the estimation results cannot reach the very high either low value of the original data. this may be due to the inaccurate selection of the arch(1) model as a model that explains the error variance for each location. the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 169 figure 7. plot of the original series (red) and estimation (green) gstar(3;0,0,1) – arch(1) model the mse value for the gstar(3;0,0,1)–arch(1) model is 0.875. this value is smaller than the mse value of the gstar(1;1)–arch (1) model, so the model that will be used for short-term prediction is the gstar(3;0,0,1)–arch(1) model. to make sure the selection of the gstar(3;0,0,1)–arch(1) model, a comparison was made with the gstar(3;0,0,1) model without the effect of heteroscedasticity. table 7 shows the comparison of mse, mad and mape values for the two models. although those values were not significant different, the model used for short-term prediction is the gstar(3;0,0,1)–arch(1) model, since it is slightly better. table 7. the comparison of gstar(3;0,1,1) and gstar(3;0,0,1)–arch(1) model from table 8, it can be concluded that the gstar (3;0,0,1)–arch (1) model is not very good for estimating the changes in the daily average wind speed, which are too large. in a relatively short period of time, august 25 – 29, 2005, there was an increase in the daily average wind speed. if this speed increase is assumed to be the beginning of hurricane katrina, then hurricane katrina is expected to hit new orleans and mississippi on september 1, 2005, which is three days later than the actual time of hurricane landfall in new orleans and mississippi. gstar(3;0,0,1) gstar(3;0,0,1)–arch(1) mse 0.88 0.86 mad 0.70 0.70 mape 7.48 6.86 the generalized star modeling with heteroscedastic effects utriweni mukhaiyar 170 table 8. the comparison of real data and its estimation using gstar(3;0,0,1)–arch(1) model conclusions the daily average wind speed data in new orleans, florida, and mississippi of the united states are not only influenced by the wind speed at the previous days in the same location, but the influenced of the wind speed in neighbor states can not be ignored. through the previous modeling, it was found that the gstar(3;0,0,1)–arch(1) model is a better model than the gstar(3;0,0,1) model. it means that, the wind speed in threeprevious days of the closest neighbors will influence today’s wind speed in the reference location. this model can capture the pattern of the wind speed volatilities compare to other observed models. however, the gstar(3;0,0,1)–arch(1) model is still not good enough to predict wind speeds that are extrme high nor low. this can be caused because the arch(1) model is not the right model to model the model error variance. further analysis of the error variance model such as garch(p,q) is also needed so that the error variance can be modeled better. as for the application 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regression cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 118-128 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 25, 2021 reviewed: november 05, 2021 accepted: november 08, 2021 doi: https://doi.org/10.18860/ca.v7i1.12995 modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar1, athifa salsabila deva2, maiyastri3, hazmira yozza4, aidinil zetra5 1,2,3,4mathematics department, faculty of mathematics and natural sciences, universitas andalas, padang 5political sciences department, faculty of social and political sciences, universitas andalas, padang email: ferrayanuar@sci.unand.ac.id, athifasalsabila4300@gmail.com, maiyastri@sci.unand.ac.id, hazmirayozza@sci.unand.ac.id, aidinil@soc.unand.ac.id abstract this study aims to construct the model for the length of hospital stay for patients with covid-19 using quantile regression and bayesian quantile approaches. the quantile regression models the relationship at any point of the conditional distribution of the dependent variable on several independent variables. the bayesian quantile regression combines the concept of quantile analysis into the bayesian approach. in the bayesian approach, the asymmetric laplace distribution (ald) distribution is used to form the likelihood function as the basis for formulating the posterior distribution. all 688 patients with covid-19 treated in m. djamil hospital and universitas andalas hospital in padang city between march-july 2020 were used in this study. this study found that the bayesian quantile regression method results in a smaller 95% confidence interval and higher value than the quantile regression method. it is concluded that the bayesian quantile regression method tends to yield a better model than the quantile method. based on the bayesian quantile regression method, it was found that the length of hospital stay for patients with covid-19 in west sumatra was significantly influenced by age, diagnoses, and discharge status. keywords: length of hospital stay; bayesian quantile regression; asymmetric laplace distribution (ald) introduction the problem of covid-19 has become the concern of the world community from every group. in cases of being infected with covid-19 in west sumatra province, not a few people have been declared cured, died, or are undergoing treatment at the hospital. people with criteria for severe symptoms of covid-19 must undergo treatment in a hospital [1]. certain factors influence the length of stay of covid-19 patients. an estimation of the regression model parameters is carried out using quantile regression and bayesian quantile regression methods to identify the factors that influence the length of stay of covid-19 patients. the estimated length of stay for covid-19 patients who are hospitalized can be used for specific purposes such as in health service activities. the need for health facilities at each level of health care. and the preparation of decisions related to mitigation scenarios and preparedness for covid-19 [2]–[4]. https://doi.org/10.18860/ca.v7i1.12995 mailto:ferrayanuar@sci.unand.ac.id mailto:athifasalsabila4300@gmail.com mailto:maiyastri@sci.unand.ac.id mailto:hazmirayozza@sci.unand.ac.id mailto:aidinil@soc.unand.ac.id modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 119 if linear model assumptions are fulfilled, such as no multicollinearity, homoscedasticity, and no autocorrelation, the ordinary least squares (ols) method is used to estimate the model parameters [5]. in the preliminary analysis, data on the length of stay of covid-19 patients in west sumatra province were not normally distributed. therefore, the use of ols was not efficient in estimating model parameters. for this reason, an analysis of the estimated parameters was carried out using quantile regression and bayesian quantile regression. quantile regression analysis was chosen because in estimating the parameters, it does not require any assumptions, including the assumption of normality, which only requires large data. the merging of quantile analysis into bayesian concepts is carried out so that the resulting estimator becomes more effective and natural so that it can produce a better predictive model that is closer to the actual value [6], [7]. research related to bayesian quantile regression was initiated by yu and mooyed [8]. research on this topic then developed rapidly, including research on numerical simulations in estimating the parameters of the bayesian quantile regression method using the gibbs sampling algorithm [9]. the application of the bayesian quantile regression method is also applied in the use of binary response data based on the asymmetric laplace distribution (ald) distribution [10]. subsequent research discussed the analysis of variable selection in quantile regression using the gibbs sampling concept [7]. further bayesian quantile regression analysis was also used to estimate the model by approximating the likelihood function [11], as well as the analysis of posterior inference with the likelihood of the ald distribution [12]. the application of bayesian quantile regression was also used in modeling the jeonse deposit in korea [13]. oh et al. do selecting variables using the bayesian quantile regression method using the savage– dickey density ratio [14]. furthermore, the application of bayesian quantile regression was also applied in constructing a low birth weight model using the gibbs sampling algorithm approach [15]. this study aims to construct a model of length of stay for covid-19 patients using quantile and bayesian quantile regression methods to then compare the results between two methods. this case is important to be investigated since the cases of covid-19 is increasing. as the results, rooms in hospitals become full. for this reason, this research needs to be carried out in an effort to find out what factors affect the length of stay of covid-19 patients. this research will give information on how to shorten the length of stay of covid-19 patients. methods material huskamp et al. and kaufman et al. have found that mortalities are higher for the old populace than young populace [16], [17]. yuki et al. recognized that older patients were more powerless to longer the length of hospital stay than younger patients [18]. this information implies that age could influence the length of hospital stay of a patient. many studies also investigated that the presence of hypertension, diabetes, and coronary artery disease were considered as hazard factors to covid-19 [19]. gebhard et al, demonstrated that covid-19 is deadlier for infected men than women [20]. the hypothesis model is constructed based on literatures to be then fitted to the data. the data used were 688 covid-19 patients treated at m. djamil hospital, padang city, and andalas university hospital in march-july 2020. in this study, the variables used are factors that are assumed to affect the length of stay of covid-19 patients in west sumatra modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 120 province, they are: age (𝑋1), gender (𝑋2) with male and female categories, diagnosis of covid-19 (𝑋3) by categories are asymptomatic person (asymp), person under supervision (perus), patients under supervision (paus), and positive, discharge status (𝑋4) with the categories are recovered, died, forced discharge, outpatient, referred to another hospital, and the number of comorbid (𝑋5). table 1 presents the frequency distribution for data of covid-19 patients by categorical independent variables, i.e., gender, diagnosis, and discharge status. table 1 shows that most diagnose of the respondents are paus (patients under supervision) with 87.7% of all respondents and 73.3% respondents were recovered. table 1. frequency distribution of covid-19 patients for categorical independent variables variable category frequency percentage gender (𝑋1) male 347 50.4 female 341 49.6 diagnose (𝑋3) asymp 1 0.1 perus 6 0.9 paus 604 87.8 positive 77 11.2 discharge status (𝑋4) recovered 504 73.3 died 141 20.5 forced discharge 30 4.4 outpatient 4 0.6 referred to another hospital 9 1.3 in figure 1 below, part (a) shows that the length of stay for covid-19 patients has a histogram that is skewed to the left, while part (b) shows that some data are not located around a linear line. based on both figures, these are informed that the data on the length of stay of covid-19 patients is not normally distributed. (a) (b) figure 1. data of length of hospital stay: (a) histogram and (b) qq-plot quantile regression method assummed that 𝒚 = (𝑦1, 𝑦2, ⋯ , 𝑦𝑛) ′ is response variable vector and 𝒙 = (𝑥1, 𝑥2, ⋯ , 𝑥𝑘 ) ′ is a covariate vector. in general, a linear regression equation model for the 𝜏-th quantile. modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 121 where 0 < 𝜏 < 1 with 𝑛 sample and 𝑘 predictor for 𝑖 = 1,2, … , 𝑛 written in the form: 𝑦𝑖 = 𝛽0𝜏 + 𝛽1𝜏 𝑥𝑖1 + 𝛽2𝜏 𝑥𝑖2 + ⋯ + 𝛽𝑘𝜏𝑥𝑖𝑘 + 𝜀𝑖 , (1) where 𝜷(𝜏) is parameter’s vector and 𝜺 is the leftover vector. the 𝜏-th conditional quantile function in the quantile regression method is defined as 𝑄𝑦𝑖 (𝜏|𝑥𝑖 ) = 𝑥 ′ 𝑖 𝜷(𝜏) then the estimated value of the parameter is �̂�(𝜏) obtained by minimizing [21]: ∑ 𝜌𝜏 (𝑦𝑖 − 𝑥𝑖 ′𝜷)𝑛𝑖=1 , (2) where 𝜌𝜏 (𝑢) = 𝑢(𝜏 − 𝐼(𝑢 < 0)) is a loss function which is equivalent to : 𝜌𝜏 (𝜀) = 𝜀(𝜏𝐼(𝜀 > 0) − (1 − 𝜏)𝐼(𝜀 < 0)), (3) 𝐼(. ) is an indicator function. with value 1 if 𝐼(. ) is true and zero rest. minimization of equation (2) was done by using the simplex method in linear programming. however, using the simplex method in estimating parameters is complicated to do. therefore, an approach with the bayes method is carried out so that the parameter estimation process becomes a little easier. bayesian quantile regression method yu and mooyed [8] found that minimizing the loss function of the quantile regression is equivalent to maximizing the likelihood function formed from the data assumed to be distributed in the asymmetric laplace distribution (ald). the ald is used in the likelihood distribution to make bayesian estimators more effective and natural. this estimation resulted in the ald distribution is a possible parametric relationship between the minimization problem of equation (2) and the maximum likelihood theory [7]. in addition, the quantile regression loss function is identical to the likelihood function of ald [22]. the ald distribution is one of the continuous probability distributions. a random variable 𝜀 has an ald distribution with probability density function 𝑓(𝜀) [7], [8]: 𝑓𝜏 (𝜀) = 𝜏(1 − 𝜏)𝑒𝑥𝑝(−𝜌𝜏 (𝜀)). (4) where 0 < 𝜏 < 1 and 𝜌𝜏 (𝜀) where defined in equation (3). the estimation of model parameters using the bayesian quantile regression method can be done for any data distribution by assuming the following [8]: 1. 𝑓(𝑦 ; 𝜇𝑖 ) has ald distribution. 2. 𝑔(𝜇𝑖 ) = 𝑥𝑖 ′𝜷(𝜏). the observation was given by 𝒚 = (𝑦1, 𝑦2, ⋯ , 𝑦𝑛 ). based on equation (4), to combine the quantile regression method into the bayesian method to estimate the parameter, 𝜷. ald was used to form the likelihood function. the ald has a combined representation of several distributions based on the exponential distribution and normal distribution [9]. a random variable 𝜀 can be expressed in: 𝜀 = 𝜃𝑧 + 𝑝𝑢√𝑧, (5) where 𝜃 = 1−2τ (1−τ)τ and 𝑝2 = 2 (1−τ)τ . the 𝜏-th quantile regression model can be written as: 𝑦𝑖 = 𝑥𝑖 ′𝜷𝜏 + 𝜎𝜃𝑧𝑖 + 𝜎𝑝𝑢𝑖 √𝑧𝑖 , (6) where 𝑧𝑖 ~𝑒𝑥𝑝(1) and 𝑢𝑖 ~𝑁(0,1), 𝑣𝑖 = 𝜎𝑧𝑖 , 𝒗 = (𝑣1, 𝑣2, ⋯ , 𝑣𝑛 ) ′. because of 𝑧𝑖 ~𝑒𝑥𝑝(1) then 𝑣𝑖 ~𝑒𝑥𝑝(𝜎), and 𝑖 = 1,2, ⋯ , 𝑛. so, we get the probability density function of 𝑦𝑖 : 𝑓(𝑦𝑖 ; 𝜷𝜏 , 𝑣𝑖 , 𝜎) = 1 𝑝√𝜎𝑣𝑖√2𝜋 𝑒𝑥𝑝 (− (𝑦𝑖−(𝑥𝑖 ′𝜷𝜏+𝜃𝑣𝑖)) 2 2𝑝2𝜎𝑣𝑖 ) , (7) and the likelihood function is obtained as follows: modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 122 𝐿(𝜷𝜏 , 𝒗, 𝜎) ∝ (∏ (𝜎𝑣𝑖 ) − 1 2 𝑛 𝑖=1 ) (𝑒𝑥𝑝 (− ∑ (𝑦𝑖−(𝑥𝑖 ′𝜷𝜏+𝜃𝑣𝑖)) 2 2𝑝2𝜎𝑣𝑖 𝑛 𝑖=1 )). (8) then, the prior distribution is selected for the parameter 𝜷𝜏 ~𝑁(𝑏0, 𝑩0). 𝑣𝑖 ~𝑒𝑥𝑝(𝜎), and 𝜎~𝐼𝐺(𝑎, 𝑏). the posterior distribution is obtained, i.e: (𝜷𝜏|𝒗, 𝜎, 𝒚)~𝑁 [(𝑩0 −1 + 𝑥𝑖 (𝑝 2𝜎𝒗)−𝟏𝑥𝑖 ′) −1 (𝑩0 −1𝑏𝟎 + 𝑥𝑖 (𝑝 2𝜎𝒗)−𝟏𝒚 − 𝑥𝑖 (𝑝 2𝜎𝒗)−𝟏𝜃𝒗), (𝑩0 −1 + 𝑥𝑖 (𝑝 2𝜎𝒗)−𝟏𝑥𝑖 ′) −1 ] ; (𝑣𝒊|𝜷𝜏 , 𝜎, 𝒚)~𝐺𝐼𝐺 ( 1 2 , ( (𝑦𝑖−𝑥𝑖 ′𝜷𝜏) 2 𝑝2𝜎 ) , ( 2 𝜎 + 𝜃𝟐 𝑝2𝜎 )) ; (𝜎|𝜷𝜏, 𝒗, 𝒚)~𝐼𝐺 ((𝑎 + 3𝑛 2 ) , (𝑏 + ∑ 𝑣𝑖 𝑛 𝑖=1 + ∑ (𝑦𝑖 − (𝑥𝑖 ′𝜷𝜏 + 𝜃𝑣𝑖 )) 2 2𝑝2𝑣𝑖 𝑛 𝑖=1 )). these posterior distribution then are used to estimate mean posterior and variance posterior as point estimate for unknown parameter using gibbs sampling iteration method [23], [24]. the goodness of fit for both methods is measured using 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 [25]. the formula for 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 is as follows: 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 = 1 − 𝑅𝐴𝑆𝑊𝜏 𝑇𝐴𝑆𝑊𝜏 , (9) where 𝑅𝐴𝑆𝑊𝜏 is the residual absolute sum of weighted differences between the observed dependent variable and the estimated quantile of conditional distribution in the more complex model. while, 𝑇𝐴𝑆𝑊𝜏 is the total absolute sum of weighted differences between the observed dependent variable and the estimated quantile of conditional distribution in the simplest model. the range values for 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 are between zero and one. the value of 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 indicates the goodness of fit of the proposed model in explaining the variance of the response variable. the higher the value of 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 the better the proposed model obtained. results and discussion data analysis begins with fitting the data to the hypothesis model using the ols method to select the significant variables involved for modeling in the quantile and bayesian analysis. based on ols analysis, the variables of age, diagnosis, and discharge status contributed significantly. furthermore, a model of the length of stay for covid-19 patients is constructed using the quantile regression method and the bayesian quantile regression method. the analysis results are then compared between both methods by looking at the width of the 95% confidence interval and 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 of the selected quantile. the quantile used are 0.10; 0.25; 0.50; 0.75; dan 0.90. r software was used to analyze the data. the results of the analysis from both methods are provided in table 3. modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 123 table 3. comparison between quantile and bayesian quantile method indicator variables quantile method bayesian quantile method estimate 95% ci estimate 95% ci 𝜏 = 0.10 intersep 2.0000 na -1.5006 14.6799 age (𝑋1) 0.0000 0.0000 0.0002 0.0041 diagnose (𝑋3) perus (𝑋3𝐷1) -2.0000 na 1.3864 15.3326 paus (𝑋3𝐷2) -1.0000 na 2.4139 14.6589 positive (𝑋3𝐷3) 0.0000 na 3.1337 14.8359 discharge status (𝑋4) recovered (𝑋4𝐷1) 2.0000* 0.0000 2.0446* 0.6021 died (𝑋4𝐷2) 0.0000 0.0000 0.0275 0.6264 outpatient (𝑋4𝐷3) 0.0000 na -0.2693 5.0697 referred to another hospital (𝑋4𝐷4) 0.0000 na -0.1649 2.0374 𝜏 = 0.25 intersep 1.0000 na -1.0118 14.2889 age (𝑋1) 0.0000 0.0000 0.0014 0.0094 diagnose (𝑋3) perus (𝑋3𝐷1) -1.0000 na 1.2671 14.8492 paus (𝑋3𝐷2) 0.0000 na 2.1446 14.2644 positive (𝑋3𝐷3) 3.0000 na 5.1174* 14.4329 discharge status (𝑋4) recovered (𝑋4𝐷1) 3.0000* 1.2544 2.6699* 1.2514 died (𝑋4𝐷2) 0.0000 0.9709 -0.2344 1.1873 outpatient (𝑋4𝐷3) 0.0000 na 1.5336 6.3011 referred to another hospital (𝑋4𝐷4) 0.0000 1.1155 0.1684 2.9599 𝜏 = 0.50 intersep 1.0000 na 1.4877 15.5613 age (𝑋1) 0.0000 0.0000 0.0002 0.0010 diagnose (𝑋3) perus (𝑋3𝐷1) 1.0000 na 0.9143 16.1529 paus (𝑋3𝐷2) 1.0000 na 0.9518 15.4168 positive (𝑋3𝐷3) 7.0000 na 6.8087 15.6258 discharge status (𝑋4) recovered (𝑋4𝐷1) 3.0000* 1.3265 2.4889* 2.1480 died (𝑋4𝐷2) -1.0000* 2.1006 -1.3962* 2.1555 outpatient (𝑋4𝐷3) 3.0000 6.1705 2.4900 7.1299 referred to another hospital (𝑋4𝐷4) 0.0000 2.9157 0.0397 4.2175 𝜏 = 0.75 intersep 2.0000 na 5.7668 22.7172 age (𝑋1) -1.05𝑥10 −7 0.0210 -0.0071* 0.0229 diagnose (𝑋3) perus (𝑋3𝐷1) 5.0000 na 0.1273 24.1500 paus (𝑋3𝐷2) 2.0000 na -1.4964 22.7243 positive (𝑋3𝐷3) 12.0000 na 9.2450 23.1280 modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 124 indicator variables quantile method bayesian quantile method estimate 95% ci estimate 95% ci discharge status (𝑋4) recovered (𝑋4𝐷1) 2.0000* 2.0233 2.2849* 2.2233 died (𝑋4𝐷2) -2.0000* 1.5733 -1.8134* 2.3198 outpatient (𝑋4𝐷3) 2.0000 na 2.8668 8.5570 referred to another hospital (𝑋4𝐷4) 0.0000 9.5010 0.5922 5.9888 𝜏 = 0.90 intersep -0.6596 na 8.3752 36.4268 age (𝑋1) -0.0213 0.0539 -0.0181 * 0.0339 diagnose (𝑋3) perus (𝑋3𝐷1) 8.7021 na 0.2688 38.0224 paus (𝑋3𝐷2) 5.6596 na -2.9586 36.2285 positive (𝑋3𝐷3) 27.5957 na 18.1682 37.1268 discharge status (𝑋4) recovered (𝑋4𝐷1) 5.9362* 2.7045 5.2751 * 2.7938 died (𝑋4𝐷2) -0.5957* 2.7368 -1.1434 * 2.8253 outpatient (𝑋4𝐷3) 2.1702 na 3.5015 10.9954 referred to another hospital (𝑋4𝐷4) 0.7660 na 1.2313 6.3550 * significant at 𝛼 = 0.05, na = not available. in table 3, it can be seen that for the quantile regression method, the 𝑋4𝐷1 variable (recovered) contributed significantly in each quantile, and the category died is significant in the quantile 0.50; 0.75; and 0.90. meanwhile, none were statistically significant for other categories in other quantiles in influencing the length of stay of covid-19 patients. meanwhile, by using the bayesian quantile regression method, the age contributed significantly at the quantile 0.75, and 0.90 in giving affects to the length of stay of covid19 patients. while, diagnose variable (only positive category) contributed significantly to the length of stay of covid-19 patients in quantile 0.25, discharge status (only recovered category) contributed significantly in all quantiles, discharge status (only died category) is significant in quantile 0.50; 0.75; and 0.90 to affect the length of stay of covid-19 patients. from the results of this estimation analysis, it is found that the bayesian quantile regression method as a whole has more significant parameter and smaller 95% confidence interval than the quantile regression method. in order to determine the best method including the best model, it could be based on the higher value of 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2. the 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 values for both methods for all selected quantiles are provided in table 4. table 4. the pseudo r2 values at all selected quantiles. quantile 𝝉𝒕𝒉 𝑷𝒔𝒆𝒖𝒅𝒐 𝑹𝟐 quantile bayesian quantile 0.10 0.27030 0.27235 0.25 0.57550 0.57843 0.50 0.87950 0.88262 0.75 0.93925 0.94244 0.90 0.67508 0.67787 modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 125 in table 4 above, it can be seen that for the quantile regression method, the model at quantile 0.75 is the best model because it has the highest value of 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2, that is 0.93925. this value informs that the proposed model can explain the variance of length of hospital stay for patients with covid-19 is 93.925%. this means that the proposed model at quantile 0.75 is acceptable and could be accepted. meanwhile for the bayesian quantile regression method, the quantile 0.75 is also as the best model because it has the highest value of 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2, that is 0.94244. this informs us that the model can explain the variance of the length of stay for covid-19 patients by 94.244%. since the 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 value obtained from bayesian quantile regression model is higher than quantile method at corresponding quantiles, we could conclude here that bayesian quantile method tends to result better model than quantile method. therefore, the best model for the length of stay of covid-19 patients in west sumatra is model at quantile 0.75 based on bayesian quantile regression method. this proposed model is formulated as follows: �̂� = 5.7668 − 0.0071𝑋1 + 0.1273𝑋3𝐷1 − 1.4964𝑋3𝐷2 + 9.2450𝑋3𝐷3 + 2.2849𝑋4𝐷1 − 1.8134𝑋4𝐷2 + 2.8668𝑋4𝐷3 + 0.5922𝑋4𝐷4. there were 75% of the length of stay for covid-19 patients diagnosed with perus (person under supervision) is 0.1273 days longer than patients diagnosed with asymp (asymptotic persons) assuming others constants. around 75% of the length of stay for covid-19 patients diagnosed with paus (patients under supervision) is 1.4964 days longer than patients diagnosed with asymp (asymptotic persons) assuming others constants. approximately, 75% of the length of stay of covid-19 patients diagnosed with positive was 9.2450 days longer than patients diagnosed with asymp (asymptotic persons) assuming other variables constant. the similar interpretation could be stated for other variables. furthermore, the convergence test of the proposed parameter model obtained was carried out. because of limited space, the selected results of these test are provided in figure 2 below. (a) (b) (c) figure 2. convergency test for category recovered at quantile 0.75 (a) trace-plot, (b) densityplot, dan (c) acf plot in figure 2 (a), it can be seen that the resulting trace-plot forms a pattern that converges to a value so that it can be stated that the model parameters have converged. modeling length of hospital stay for patients with covid-19 in west sumatra using quantile regression ferra yanuar 126 while in part (b), it can be seen that the resulting density plot resembles a normal distribution curve. it can be stated that the model parameters are normally distributed. then in part (c), the resulting acf plot shows a smaller autocorrelation value so that it can be stated that there is no autocorrelation between samples. based on these convergency test, it can be concluded that the model parameters have converged and proposed model could be accepted. conclusions this study found that the length of stay of covid-19 patients in west sumatra was influenced by age, diagnoses of covid-19 patients, and discharge status. from the analysis carried out, the bayesian quantile regression method is better in modeling the length of stay of covid-19 patients than quantile method. the 95% confidence interval based on bayesian quantile regression is smaller, and the 𝑃𝑠𝑒𝑢𝑑𝑜 𝑅2 value is greater than the quantile regression method. acknowledgments this research was funded by directorate of resources directorate general of higher education, ministry of education, culture, and research and technology of indonesia, in accordance with contract number 104/e4.1/ak.04.pt/2021. references [1] kemenkes ri, “kmk no. hk.01.07-menkes-413-2020 ttg pedoman pencegahan dan pengendalian covid-19.pdf.” 2020. 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[25] f. yanuar, h. yozza, f. firdawati, i. rahmi, and a. zetra, “applying bootstrap quantile regression for the construction of a low birth weight model,” makara journal of health research, vol. 23, no. 2, pp. 90–95, aug. 2019, doi: 10.7454/msk.v23i2.9886. on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 84-96 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 15, 2021 reviewed: august 19, 2021 accepted: october 06, 2021 doi: https://doi.org/10.18860/ca.v7i1.12934 on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari department of mathematics, faculty of science and technology, maulana malik ibrahim islamic state university of malang email: juhari@uin-malang.ac.id abstract this study discusses the construction of mathematical model modification of newton-secant method and solving nonlinear equations for multiple zeros by using a modified newton-secant method. a nonlinear equations for multiple zeros or multiplicity 𝑚 > 1 is an equation that has more than one root. the first step is to construct of mathematical model newton-secant method and its modification, namely to construct a mathematical model of the newton-secant method using the concept of the newton method and the concept of the secant method. the second step is to construct a modified mathematical model of the newton-secant method by adding the parameter 𝜃. after obtaining the formula for the modification newton-secant method, then applying the method to solve a nonlinear equations for multiple zeros. in this case, it is applied to the nonlinear equation trigonometric function 𝑓(𝑥) = (𝑐𝑜 𝑠2 𝑥 + 𝑥)5 which has a multiplicity of 𝑚 = 5. the solution is done by selecting four different initial guess, namely −2; −0,8; −0,2 and 2. furthermore, to determine the effectivity of this method, the researcher compared the result with the newton-raphson method, the secant method, and the newton-secant method that has not been modified. the obtained results from the construction of mathematical model newtonsecant method and its modification is an iteration formula modification of newton-secant method. and for the result of 𝑓(𝑥) using a modification of newton-secant method with four different initial guess, the root of 𝑥 is obtained approximately, namely −0.641714371 with fewer iterations if compared to using the newton method, the secant method, and the newton-secant method. based on the problem to find the root of the nonlinear equation 𝑓(𝑥) it can be concluded that the modification of newton-secant method is more effective than the newton method, the secant method, and the newton-secant method. keywords: modification; newton-secant method; nonlinear equation; multiple zeros; trigonometric function introduction in the fields of science, engineering and economics often involve problems mathematics. mathematical problems are often found in the form of nonlinear equations [1]. nonlinear equations in the form of functions 𝑓(𝑥) can be form of algebraic equations and transcendent equations. transcendent equations or non-algebraic equations are equations that cannot be expressed in algebraic operations. this equation consists of logarithmic functions, exponential functions, hyperbolic functions and trigonometric https://doi.org/10.18860/ca.v7i1.12934 mailto:juhari@uin-malang.ac.id on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 85 functions [2]. in finding a solution to a nonlinear equation means making that equation becomes zero, i.e. 𝑓(𝑥) = 0 [3]. in determining the solution of a complex nonlinear equation will be difficult if done using analytical methods so the numerical will be solution to this problem [4]. the solution obtained from the numerical method an approximate solution. the approximate solution is different from the exact solution, so there is the difference between exact solution and approximate solution. this difference is often referred to as an error [5]. in finding the roots of a nonlinear equation, it is not always singular or simple, sometimes nonlinear equations are in the form of multiple nonlinear equations, meaning the equation has a multiplicity 𝑚 > 1. the following is the definition of multiplicity: definition 1 (multiplicity) the root 𝛼 of 𝑓(𝑥) is said to have multiplicity (𝑚) if 𝑓(𝑥) = (𝑥 − 𝛼)𝑚ℎ(𝑥) for ℎ(𝑥) a continuous function with ℎ(𝑥) ≠ 0, and 𝑚 is a positive integer. if 𝑚 = 1 then 𝛼 is called a simple root. if 𝑚 ≥ 2 then 𝛼 is called a multiple root [6]. thus it can be said that a function has multiplicity if the multiplicity of a function is more than one [5]. the most famous numerical method for solving nonlinear equations is the newton-raphson method. in finding solutions to nonlinear equations, this method requires one initial guess and function derivative value. this method will fail if the initial guess selection gives the derivative value zero [7]. the following is the formula of newton-raphson method : 𝑥𝑛+1 = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) (1) with 𝑛 = 0,1,2, . . .. and 𝑓 ′(𝑥𝑛) ≠ 0. the numerical method that is no less famous is the secant method. this method able to overcome the weakness of the newton-raphson method. in the newton-raphson method is required first derivative of the function 𝑓(𝑥). the process of finding the derivative function 𝑓(𝑥) does not always easy, sometimes there are some functions that are difficult to find the derivative value. to overcome the weakness of newton's method then in the secant method derivative function is replaced by another equivalent form. so the secant method does not require another derivative of the function but requires two initial guesses [8]. the following is the formula of secant method : 𝑥𝑛+1 = 𝑥𝑛 − 𝑓(𝑥𝑛) ∙ (𝑥𝑛−1 − 𝑥𝑛 ) 𝑓(𝑥𝑛−1) − 𝑓(𝑥𝑛) (2) with 𝑛 = 1,2,3 . . .. in the calculation process using numerical method such as newton's method and the secant method are needed initial guess. determining the initial value will be easier if you pay attention the theory related the intermediate value theorem. the intermediate value theorem is a theorem that is used to determine the presence or absence of a solution at a certain interval limit. theorem 1 (intermediate value theorem) if 𝑓 ∈ 𝐶[𝑎, 𝑏] and 𝐾 is any number between 𝑓(𝑎) and 𝑓(𝑏), then there exists a number 𝑐 in (𝑎, 𝑏) for which 𝑓(𝑐) = 𝐾. [9] another theorem that is almost similar with the intermediate value theorem is bolzano's theorem. on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 86 theorem 2 (bolzano's theorem) if 𝑓: [𝑎, 𝑏] ⊂ ℝ → ℝ is a continuous function and if 𝑓(𝑎) ∙ 𝑓(𝑏) < 0, then there is at least one root 𝑥 ∈ (𝑎, 𝑏) such that 𝑓(𝑥) = 0. [10] each numerical method has a different order of conver gence. order of convergence is the speed of an iteration method in finding the roots simultaneously approximation of the equation of function 𝑓. the following is the definition of convergence : definition 2 (convergence) let the sequence 𝑥1, 𝑥2, . . . , 𝑥𝑛 convergence to 𝛼 and 𝑒𝑛 = 𝑥𝑛 − 𝛼 where 𝑛 ≥ 0. if the order of convergence 𝑝 > 0 and error constant 𝐶 ≠ 0, with lim 𝑛→∞ |𝑥𝑛+1 − 𝛼| |𝑥𝑛 − 𝛼| 𝑝 = lim 𝑛→∞ |𝑒𝑛+1| |𝑒𝑛| 𝑝 = 𝐶 then the sequence {𝑥𝑛} converges to 𝛼 with the order of convergence 𝑝 [11]. if 𝑝 = 1 then the iteration method has a linear convergence order. if 𝑝 = 2 then the iteration method has a quadratic order of convergence. if 1 < 𝑝 < 2 then iteration method has a superlinear order of convergence [12]. if 𝑝 = 3 then the iteration method has a cubic order of convergence [13]. numerical methods such as the newton-raphson method have the quadratic order of convergence. while the secant method has a superlinear order of convergence [12]. numerical methods often experience developments. the development aims to find methods that are considered more effective in solving existing problems [14]. based on this in 2002 kasturiarachi combine newton's method and the secant method become a new method, namely the leap-frogging newton‘s method or newton-secant method [13]. this method has a cubic convergence when used to solve simple nonlinear equations. whereas if used to solve multiple zeros of nonlinear equations the convergence to be linear. therefore, in [15] modified newton-secant method with the addition of parameter 𝜃. the purpose of this modification, namely to maintain the order of convergence newton-secant method to remain cubic, if used to find the roots of nonlinear equations [15]. the following theorem related to the parameter 𝜃 used to modify the newton-secant method : theorem 3 let 𝛼 ∈ 𝐷 be multiple root of a sufficiently differentiable function 𝑓 ∶ 𝐷 ⊂ 𝑹 → 𝑹 on an open interval 𝐷 with multiplicity 𝑚 > 1, which includes 𝑥0 as an initial approximation of 𝛼. then, the modification of newton-secant method has order three and 𝜃 = ( −1+𝑚 𝑚 ) −1+𝑚 , 𝑚 ∈ 𝑍+. proof: let 𝛼 is multiple zero of equation 𝑓(𝑥) = 0, then 𝑓(𝛼) = 0 and 𝑓′(𝛼) ≠ 0. next, suppose 𝑒𝑛 ≔ 𝑥𝑛 − 𝛼 𝑒𝑛,�̅� ≔ 𝑥𝑛̅̅ ̅ − 𝛼 𝑐𝑖 ≔ 𝑚! (𝑚 + 𝑖)! 𝑓 (𝑚+𝑖)(𝛼) 𝑓 (𝑚)(𝛼) using the taylor expansion of 𝑓(𝑥𝑛) around 𝑥𝑛 = 𝛼 we get 𝑓(𝑥𝑛) = 𝑐0𝑒𝑛 𝑚 + 𝑐1𝑒𝑛𝑒𝑛 𝑚 + 𝑐2𝑒𝑛 2𝑒𝑛 𝑚 + 𝑐3𝑒𝑛 3𝑒𝑛 𝑚 + 𝑂(𝑒𝑛 4) simplified to 𝑓(𝑥𝑛) = 𝑒𝑛 𝑚(𝑐0 + 𝑐1𝑒𝑛 + 𝑐2𝑒𝑛 2 + 𝑐3𝑒𝑛 3) + 𝑂(𝑒𝑛 4), (3) and on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 87 𝑓 ′(𝑥𝑛) = 𝑒𝑛 𝑚−1(𝑚 + (𝑚 + 1)𝑐1𝑒𝑛 + (𝑚 + 2)𝑐2𝑒𝑛 2 + (𝑚 + 3)𝑐3𝑒𝑛 3 + 𝑂(𝑒𝑛 4)). (4) if equation (3) is divided by equation (4), then we get 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) = 𝑒𝑛 ( (𝑐0 + 𝑐1𝑒𝑛 + 𝑐2𝑒𝑛 2 + 𝑐3𝑒𝑛 3) + 𝑂(𝑒𝑛 4) (𝑚 + (𝑚 + 1)𝑐1𝑒𝑛 + (𝑚 + 2)𝑐2𝑒𝑛 2 + (𝑚 + 3)𝑐3𝑒𝑛 3 + 𝑂(𝑒𝑛 4)) ) (5) furthermore, equation (5) can be written as 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) = 1 𝑚 𝑒𝑛 − 𝑐1 𝑚2𝑐0 𝑒𝑛 2 + −(1 + 𝑚)𝑐1 2 + 2𝑚𝑐0𝑐2 𝑚3𝑐0 2 𝑒𝑛 3 + 𝑂(𝑒𝑛 4), (6) and since 𝑒𝑛,�̅� = 𝑥𝑛̅̅ ̅ − 𝛼 = −1 + 𝑚 𝑚 𝑒𝑛 − 𝑐1 𝑚2𝑐0 𝑒𝑛 2 + −(1 + 𝑚)𝑐1 2 + 2𝑚𝑐0𝑐2 𝑚3𝑐0 2 𝑒𝑛 3 + 𝑂(𝑒𝑛 4). (7) for 𝑓(𝑥𝑛̅̅ ̅) we have 𝑓(𝑥𝑛̅̅ ̅) = 𝑒𝑛,�̅� 𝑚 (𝑐0 + 𝑐1𝑒𝑛,�̅� + 𝑐2𝑒𝑛,�̅� 2 + 𝑐3𝑒𝑛,�̅� 3 ) + 𝑂(𝑒𝑛,�̅� 4 ) (8) substituting (3)-(8) in modification of newton-secant method formula, which the method will be constructed in the research results section. so we get 𝑒𝑛+1 = 𝐷1𝑒𝑛 + 𝐷2𝑒𝑛 2 + 𝐷3𝑒𝑛 3 + 𝑂(𝑒𝑛 4), where 𝐷1 = 1 + 𝜃 𝑚 (−𝜃 + ( −1+𝑚 𝑚 ) 𝑚 ) , and 𝐷2 = 𝜃𝑚−2+𝑚(−𝑚(−1 + 𝑚)𝑚 + 𝜃𝑚𝑚(−1 + 𝑚))𝑐1 (−1 + 𝑚)((−1 + 𝑚)𝑚 − 𝜃𝑚𝑚)2𝑐0 , and 𝐷3 = 𝜃𝑚−3+𝑚𝐴 2(−1 + 𝑚)2((−1 + 𝑚)𝑚 − 𝜃𝑚𝑚)3𝑐0 2 , where 𝐴 = (−1 + 𝑚)2𝑚(−1 + 𝑚 + 2𝑚2)(𝑚𝑐1 2 − 2(−1 + 𝑚)𝑐0𝑐2) + 2𝜃2(−1 + 𝑚)2𝑚2𝑚((1 + 𝑚)𝑐1 2 − 2𝑚𝑐0𝑐2) − 𝜃(−1 + 𝑚)1+𝑚(𝑚(3 + 4𝑚)𝑐1 2 + 2(1 + 𝑚 − 4𝑚2)𝑐0𝑐2). therefore, to provide the three order of convergence, it is need to choose 𝐷𝑖 = 0 (𝑖 = 1, 2), so we have 𝜃 = ( −1 + 𝑚 𝑚 ) −1+𝑚 , and the error equation becomes 𝑒𝑛+1 = ( 𝑚𝑐1 2 − 2(−1 + 𝑚)𝑐0𝑐2 2𝑚2𝑐0 2 ) 𝑒𝑛 3 + 𝑂(𝑒𝑛 4) , and modification of newton-secant method has convergence order of three [15]. based on the description above the researcher intends to construct the modification of newton-secant method in solving nonlinear equations for multiple zeros. then apply the method to solve the nonlinear equations for multiple zeros. in this case, it is applied to a nonlinear trigonometric function. to determine the effectivity of modification of newton-secant method then the solution will also be compared with newton method, secant method, and newton-secant method. on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 88 methods research steps 1. construction of mathematical model newton-secant method and its modification with the following steps: a) construction of mathematical model newton-secant method.  analyzing the theory related to the origin of the newton-secant method based on kasturiarachi’s article 2002 entitled leapfrogging newton’s method.  performing analysis to obtain newton's approximation by using the equation of the tangent (𝑥0, 𝑓(𝑥0)) that intersects (𝑥0̅̅ ̅, 0).  create an equation of a secant connecting the points (𝑥0, 𝑓(𝑥0)) and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)) using the equation of the line that passes through two point and assumes a secant that satisfies the 𝑥-axis on point (𝑥1, 0).  substituting newton's approximation into the equation of secant connecting the points (𝑥0, 𝑓(𝑥0)) and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)).  write the iteration formula based on the process that has been done at the stages above. b) construction of mathematical model modification of newton-secant method with adding parameter 𝜃 to the second term of the newton-secant method. in this case, theorem 3 is used which is in the introduction section. 2. solving nonlinear equation that have a multiplicity 𝑚 > 1 using modification of newton-secant method. in this case the solution is done by selecting two different initial values. after that the researcher compared the result with the newton-raphson method, secant method, and newton-secant method that has not been modified. this comparison aims to determine the effectivity of modification of newton-secant method if when viewed from the iterations, convergence, and time needed to solve a nonlinear equation having a multiplicity of 𝑚 > 1. results and discussion 1. construction of mathematical model newton-secant method and its modification a. construction of mathematical model newton-secant method newton-secant method or also known as leap-frogging newton's method is a combination of newton method and secant method. based on this to do construction of mathematical model newton-secant method used the concept of newton's method and the concept of the secant method. suppose that the function 𝑓(𝑥) has the zero 𝛼 in the interval [𝑎, 𝑏] and 𝑓 ∈ 𝐶2[𝑎, 𝑏]. let 𝑥0 be the initial guess. if the equation of the tangent line at (𝑥0, 𝑓(𝑥0)) intersects (𝑥0̅̅ ̅, 0) then by using the concept of newton's method geometrically as follows : on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 89 figure 1. newton's approximation curve then we get newton's approximation : 𝑓 ′(𝑥𝑛) = δ𝑦 δ𝑥 𝑓 ′(𝑥0) = 𝑓(𝑥0) − 0 𝑥0 − 𝑥0̅̅ ̅ 𝑓 ′(𝑥0)(𝑥0 − 𝑥0̅̅ ̅) = 𝑓(𝑥0) 𝑥0 − 𝑥0̅̅ ̅ = 𝑓(𝑥0) 𝑓 ′(𝑥0) 𝑥0̅̅ ̅ = 𝑥0 − 𝑓(𝑥0) 𝑓 ′(𝑥0) (9) in this case used 𝑥0̅̅ ̅ instead of 𝑥1 because this is only used as an intermediate approximation. furthermore find the equation of the secant line connecting the points (𝑥0, 𝑓(𝑥0)) and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)). to find these equations is used the concept of the secant method geometrically. look at the following curve : figure 2. the concept of secant method to find the equation of the secant line based on figure 2 above, the following gradient is obtained: 𝑓 ′(𝑥0) = δ𝑦 δ𝑥 = [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] 𝑥0 − 𝑥0̅̅ ̅ tangent to the curve at 𝑥0 with gradient 𝑓 ′(𝑥0) on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 90 the gradient is used to get the equation of the line connecting the points (𝑥0, 𝑓(𝑥0)) and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)). based on the gradient formula to find the equation of the line that through two points, the following equation is obtained : 𝑦 − 𝑓(𝑥0) = 𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅) 𝑥0 − 𝑥0̅̅ ̅ (𝑥 − 𝑥0) (10) assume the secant line meets the x-axis at the point (𝑥1, 0). so with substituting a value of 𝑥1 in 𝑥 and value of 0 in 𝑦 in equation (10) then we get, 0 − 𝑓(𝑥0) = [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] 𝑥0 − 𝑥0̅̅ ̅ (𝑥1 − 𝑥0) 0 − 𝑓(𝑥0) 𝑥1 − 𝑥0 = [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] 𝑥0 − 𝑥0̅̅ ̅ −𝑓(𝑥0) 𝑥1 − 𝑥0 = [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] (𝑥0 − 𝑥0̅̅ ̅) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] ∙ (𝑥1 − 𝑥0) = −𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅) 𝑥1 − 𝑥0 = −𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] 𝑥1 = 𝑥0 − 𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] (11) then substitute newton's approximation (9) into equation (11) so that optained, 𝑥1 = 𝑥0 − 𝑓(𝑥0) ∙ (𝑥0 − (𝑥0 − 𝑓(𝑥0) 𝑓′(𝑥0) )) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] = 𝑥0 − 𝑓(𝑥0) ∙ ( 𝑓(𝑥0) 𝑓′(𝑥0) ) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] = 𝑥0 − [𝑓(𝑥0)] 2 𝑓′(𝑥0) [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] = 𝑥0 − [𝑓(𝑥0)] 2 𝑓 ′(𝑥0)[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] (12) repeating this above process, the iteration formula can be written as following : where 𝑥𝑛+1 = 𝑥𝑛 − [𝑓(𝑥𝑛)] 2 𝑓 ′(𝑥𝑛)[𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅)] , 𝑛 = 0, 1, 2, . . . . 𝑥𝑛̅̅ ̅ = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) (13) thus, we get equation (13) is the iteration formula for the newton-secant method. b. construction of mathematical model modification of newton-secant method in solving simple nonlinear equations the newton-secant method has cubic convergence. while to solve the nonlinear equations for multiple zeros or multiplicity 𝑚 > 1 the convergence is not cubic but becomes linear. therefore, to maintain the convergence of the newton-secant method to remain cubic it is necessary to modify the method. the process modification of newton-secant method is done by adding parameter 𝜃 to the second term of the newton-secant method formula. the addition of these parameter resulted in the convergence of the modification of newton-secant method is cubic. on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 91 the formula for the newton-secant method (13) can be written in the following form, 𝑥𝑛+1 = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅) ∙ 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) , 𝑛 = 0, 1, 2, . . . . refer to theorem 3 in the introduction section to the construction of mathematical model modification of newton-secant method is done by adding parameter 𝜃 to the second term newton-secant method. so that the iteration formula is obtained as following : where 𝑥𝑛+1 = 𝑥𝑛 − 𝜃𝑓(𝑥𝑛) 𝜃𝑓(𝑥𝑛 ) − 𝑓(𝑥𝑛̅̅ ̅) ∙ 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) , 𝑛 = 0, 1, 2, . . . . 𝑥𝑛̅̅ ̅ = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛) and 𝜃 = ( −1 + 𝑚 𝑚 ) −1+𝑚 (14) thus, we get equation (14) is modification of newton-secant method formula which can be used to find solution to nonlinear equations for multiple zeros. 2. solving nonlinear equation for multiple zeros (trigonometric function) use modification of newton-secant method in the previous explanation, the construction of mathematical model newtonsecant method and its modification has been explained. to know effectivity of modification of newton-secant method then it is necessary to apply a modified newtonsecant method to nonlinear equations for multiple zeros. in this case, it is taken an example of nonlinear equation of trigonometric function, namely 𝑓(𝑥) = (cos2 𝑥 + 𝑥)5 solving equation 𝑓(𝑥) use modification of newton-secant method will be applied with four different initial guess, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2. the following are the steps to find a solution to the equation 𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 using modification of newton-secant method: 1) determine the initial guess to be used the initial guess selection is based on theorem 1 (intermediate value theorem) in the introduction section. so that the initial guess are chosen, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2. 2) finding the derivative of 𝑓(𝑥) 𝑓(𝑥) = (cos2 𝑥 + 𝑥)5 𝑓 ′(𝑥) = 5(𝑐𝑜𝑠2𝑥 + 𝑥)4(−2 cos 𝑥 sin 𝑥 + 1). 3) determine the multiplicity of 𝑓(𝑥) based on the definition of multiplicity in the introduction section, it can be said that the multiplicity of 𝑓(𝑥) is 𝑚 = 5. 4) set the error to be used in this case the author sets the error used, namely 10−10. 5) calculating the value of 𝜃 on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 92 𝜃 = ( −1 + 𝑚 𝑚 ) −1+𝑚 = ( −1 + 5 5 ) −1+5 = ( 4 5 ) 4 = 0,4096 6) perform iteration using the modification of newton-secant method formula. 7) comparing the modification of newton-secant method with the newton’s method, the secant method, and the newton-secant method. it aims to know the effectiveness of the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function. the following is the result of solving 𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 using modification of newton-secant method, newton method, secant method, and newton-secant method which has not been modified. table 1. solution 𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 use modification of newton-secant method, newton method, secant method, and newton-secant method. 𝒇𝒊 𝒙𝟎 method n 𝒙𝒏 𝒇(𝒙𝒏) 𝜺 𝒇(𝒙) = (𝒄𝒐𝒔𝟐 𝒙 + 𝒙)𝟓 -2 mmns 5 -0,641714371 1,76869e-74 0,0000000000 mns 61 -0,641714371 8,67022e-49 0,0000000000 mn 93 -0,641714371 1,20449e-47 0,0000000000 ms 133 -0,641714371 -8,69088e-47 0,0000000000 -0,8 mmns 4 -0,641714371 -4,17656e-74 0,0000000000 mns 60 -0,641714371 -1,81522e-48 0,0000000000 mn 92 -0,641714371 -2,51216e-47 0,0000000000 ms 131 -0,641714371 -9,67797e-47 0,0000000000 -0,2 mmns 4 -0,641714371 -9,9601e-76 0,0000000000 mns 62 -0,641714371 3,45672e-48 0,0000000000 mn 96 -0,641714371 1,88037e-47 0,0000000000 ms 132 -0,641714371 -7,87677e-47 0,0000000000 2 mmns 5 -0,641714371 9,07176e-75 0,0000000000 mns 64 -0,641714371 -9,82514e-49 0,0000000000 mn 101 -0,641714371 1,84781e-47 0,0000000000 ms 133 -0,641714371 -8,69088e-47 0,0000000000 information : 𝑓𝑖 : nonlinear equation function with 𝑖 = 1, 2, 3, …. 𝑥0 : initial guess or initial value n : many iterations on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 93 𝑥𝑛 : root of function 𝑓(𝑥𝑛 ) : function of root 𝑥𝑛 𝜀 : error (𝑥𝑛+1 − 𝑥𝑛 ) mmns : modification of newton-secant method mns : newton-secant method mn : newton method ms : secant method based on table 1 above, the error is 0.0000000000, meaning that the error in the iteration value is less than the error tolerance constant used, namely 𝜀 < 10−10. therefore, the iteration process stops and the approximate root is −0.641714371 which is the solution of 𝑓(𝑥). in the table above, it can be seen that taking four different initial values, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2, if the search for a solution uses modification of newton-secant method, iterations are needed more little when compared to newton's method, the secant method, and the newton-secant method. based on the many iterations, it can be said that the modification of newton-secant method is more effective in solving the nonlinear equation 𝑓(𝑥) when compared to the newton method, the secant method, and the newton-secant method which have not been modified. the convergence of error values from the table of calculation results above can be seen in table 2 and table 3 below. table 2. convergence graph table of error values in the solution of 𝑓(𝑥) using modification of newtonsecant method and newton-secant method 𝒙𝟎 method modification of newton-secant method newton-secant method -2 -0,8 on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 94 -0,2 2 table 3. convergence graph table of error values in the solution of 𝑓(𝑥) using newton method and secant method. 𝒙𝟎 method newton method secant method -2 -0,8 on the modification of newton-secant method in solving nonlinear equations for multiple zeros of trigonometric function juhari 95 -0,2 2 conclusion combining two concepts of numerical method, namely the concept of newton's method and the concept of secant method result newton-secant method. then newtonsecant method modified by adding parameter 𝜃. so we get a new method, namely modification of newton-secant method with the following iteration formula: where 𝑥𝑛+1 = 𝑥𝑛 − 𝜃𝑓(𝑥𝑛) 𝜃𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅) ∙ 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛 ) , 𝑛 = 0, 1, 2, . . . . 𝑥𝑛̅̅ ̅ = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓 ′(𝑥𝑛 ) and 𝜃 = ( −1 + 𝑚 𝑚 ) −1+𝑚 solution of 𝑓(𝑥) = (cos2 𝑥 + 5)5 use modification of newton-secant method with four different initial guess, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2 is obtained the root of 𝑥 approximately, namely −0,641714371 with fewer iterations when compared to using the newton method, the secant method, and the newton-secant method. based on the problem of finding the root of the nonlinear equation trigonometric function 𝑓(𝑥) it can be concluded that the modification of newton-secant method is more effective than the newton method, the secant method, and the newtonsecant method that has not been modified. references [1] p. batarius, "perbandingan metode newton-raphson modifikasi dan metode secant modifikasi dalam penentuan akar persamaan," 2018. 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[8] s. c. chapra and r. p. canale, numerical methods for engineers sixth edition, new york: mcgraw-hill companies, inc, 2010. [9] r. l. burden and j. faires, numerical analysis ninth edition, usa: brooks/cole cengage learning, 2011. [10] m. n. vrahatis, "generalization of the bolzano theorem for simplices," elsevier, 2015. [11] j. h. mathews, numerical methods for mathematics, science, and engineering, new jersey: prentice-hall inc, 1992. [12] r. kumar and vipan, "comparative analysis of convergence of various numerical methods," journal of computer and mathematical sciences, vols. 6(6),290-297, 2015. [13] a. b. kasturiarachi, "leap-frogging newton's method," international journal of mathematical education in science and technology, vols. 33, no. 4, 521-527, 2002. [14] s. putra, d. ar and m. imran, "kombinasi metode newton dengan metode secant untuk menyelesaikan persamaan nonlinear," jurnal eksakta, vol. 2, 2011. [15] m. ferrara, s. sharifi and m. salimi, "computing multiple zeros by using a parameter in newton-secant method," sema journal, 2016. trace of positive integer power of squared special matrix cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 200-211 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 15, 2020 reviewed: january 04, 2021 accepted: january 27, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.10312 trace of positive integer power of squared special matrix rahmawati1, aryati citra2, fitry aryani3, corry corazon marzuki4, yuslenita muda5 1,2,3,4,5 department of mathematics, faculty of science and technology, state islamic university of sultan syarif kasim riau st. hr. soebrantas no. 155 simpang baru, panam, pekanbaru, 28293 email: 1rahmawati@uin-suska.ac.id , 2aryaticitra1@gmail.com, 3khodijah_fitri@uin-suska.ac.id, 4corry@uin-suska.ac.id, 5yuslenita.muda@uin-suska.ac.id abstract the rectangle matrix to be discussed in this research is a special matrix where each entry in each line has the same value which is notated by 𝐴𝑛. the main aim of this paper is to find the general form of the matrix trace of 𝐴𝑛 powered positive integer 𝑚 or notated by 𝑇𝑟(𝐴𝑛) 𝑚. to prove whether the general form of the matrix trace of 𝐴𝑛 powered positive integer can be confirmed, mathematics induction and direct proof are used. the main results present the general formula of (𝐴𝑛) 𝑚 and 𝑇𝑟(𝐴𝑛) 𝑚 with observing the pattern of power matrix for 2 ≤ 𝑚 ≤ 11,𝑛 ≥ 2, and 𝑚 ∈ ℤ+. keywords: direct proof; mathematics induction; matrix trace; squared matrix introduction the calculation of trace of power of square matrix has become attention. according to brezinski [1], trace of power of matrix is often used in some fields of mathematics, especially network analysis, number theory, dynamic systems, matrix theory, and differential equations. the discussion about trace matrix has been widely studied by several researchers before. datta et.al [2], has obtained algorithm of trace of power of squared matrix 𝑇𝑟(𝐴𝑘), with 𝑘 is an integer and 𝐴 is hassenberg matrix with a codiagonal unit. there is also discussion of trace in several applications in matrix theory and numerical linear algebra. for example in determining the eigenvalue of a symmetric matrix, the basic procedure in estimating a trace (𝐴𝑛) and trace (𝐴−𝑛) with 𝑛 integers, this is explained in pan [3]. chu. mt [4] discussed symbolic calculations on the power of squared tridiagonal of matrix trace. for example, 𝐴 a symmetric positive definite matrix, and for example {𝜆𝑘} notated its eigen value. for 𝑞 ∈ ℝ, 𝐴 𝑞 also symmetric definite matrices, and are listed in hignam [5] with formula 𝑇𝑟(𝐴𝑞) = ∑𝜆𝑘 𝑞 𝑘 . according to zarelua [6] in quantum and combinatorial theory, the trace matrix is a whole number in relation to the euler equations 𝑇𝑟(𝐴𝑝 𝑟 ) = 𝑇𝑟(𝐴𝑝 𝑟−1 ) 𝑚𝑜𝑑(𝑝𝑟) http://dx.doi.org/10.18860/ca.v6i4.10312 mailto:rahmawati@uin-suska.ac.id mailto:aryaticitra1@gmail.com mailto:khodijah_fitri@uin-suska.ac.id mailto:corry@uin-suska.ac.id mailto:yuslenita.muda@uin-suska.ac.id trace of positive integer power of squared special matrix rahmawati 201 for all matrix a integers, p is the prime number and r original number. then this article also discuss about invariant in dynamic system which is illustrate as form of trace of integer squared matrix, for example the number lefschetz. next, pahade and jha [7], discuss about the formation of general form of trace matrix ordo 2 × 2 square with powered positive integer. in that article there are two general forms of order trace 2× 2 with integer square n. first, the general form of order trace matrix trace 2 x 2 with even number square 𝑛, is 𝑇𝑟 (𝐴𝑛) = ∑   (−1)𝑟 𝑟 ! 𝑛 2⁄ 𝑟=0  𝑛 [𝑛  − (𝑟 + 1)] [𝑛  − (𝑟 + 2)] ⋯ [𝑛 − (𝑟 +(𝑟 − 1))] (det (𝐴)) 𝑟  (𝑇𝑟 (𝐴)) 𝑛−2𝑟 . second, the main form of trace matrix 2 × 2 with odd number square 𝑛, is 𝑇𝑟 (𝐴𝑛) = ∑   (−1)𝑟 𝑟 ! (𝑛−1) 2⁄ 𝑟=0  𝑛 [𝑛 − (𝑟 + 1)] [𝑛  − (𝑟 + 2)] ⋯ [𝑛 − (𝑟 + (𝑟 − 1))] (det (𝐴)) 𝑟  (𝑇𝑟 (𝐴)) 𝑛−2𝑟 . in the network analysis field, especially on triangle counting in a graph, based on avron [8], when analyzing a complex network, the important problem is calculating the total numbers of triangle on the simple connected graph. this number is equal to 𝑇𝑟(𝐴3) 6⁄ , where 𝐴 is adjacency matrix from the graph. then, in 2017, pahade and jha [9] discuss about trace of squared adjacency matrix on positive integers. in the paper, there is a symmetrical adjacency matrix on a complete simple graph with vertex n, for even number k is formulated 𝑇𝑟(𝐴𝑘) = ∑𝑠(𝑘,𝑟)𝑛(𝑛 − 1)𝑟(𝑛 − 2)𝑘−2𝑟 𝑛 2 𝑟=1 and for odd number k is formulated 𝑇𝑟(𝐴𝑘) = ∑ 𝑠(𝑘,𝑟)𝑛(𝑛 − 1)𝑟(𝑛 − 2)𝑘−2𝑟 𝑛−1 2 𝑟=1 with ),( rks is a number thats depend on k and r, and defined as 𝑠(𝑘,1) = 1,𝑠(𝑘, 𝑘 2 ) = 1,𝑠(𝑘, 𝑘−1 2 ) = 𝑘−1 2 , and 𝑠(𝑘,𝑟) = 𝑠(𝑘 − 1,𝑟)+ 𝑠(𝑘 − 2,𝑟 − 1). next, by this research, it will be decided the trace of rectangle matrix with the real number entries which for every entry row has an equal value. in this research, there are some related definitions and theorems. definition 1.1 (anton [10]) if 𝐴 is a rectangle matrix, then the definition of squared of powered non negative integers of 𝐴 is 𝐴0  =  𝐼 , 𝐴𝑛  = 𝐴𝐴…𝐴⏟ 𝑛 𝑓𝑎𝑘𝑡𝑜𝑟  (𝑛  >  0). next, if 𝐴 is invertible, then the definition of squared of powered negative integers of 𝐴 is 𝐴−𝑛  = (𝐴−1)𝑛  = 𝐴−1𝐴−1 …𝐴−1⏟ 𝑛 𝑓𝑎𝑘𝑡𝑜𝑟 . theorem 1.1 (andrilli, [11]) if 𝐴 is a rectangle matrix, and if 𝑟 and 𝑠 are nonnegative integers, then 1. 𝐴𝑟 𝐴𝑠  = 𝐴𝑟 + 𝑠 2. (𝐴𝑟)𝑠  = 𝐴𝑟𝑠 = (𝐴𝑠)𝑡 trace of positive integer power of squared special matrix rahmawati 202 definition 1.2 [10] if 𝐴 is a rectangle matrix, then the trace of 𝐴 which is stated as 𝑇𝑟(𝐴), is defined as the total entries on main diagonal of 𝐴. trace from 𝐴 cannot be defined when 𝐴 is not a rectangle matrix 𝑇𝑟 (𝐴) = 𝑎11 + 𝑎22 + ⋯+ 𝑎𝑛𝑛 = ∑  𝑎𝑖𝑖 𝑛 𝑖=1 . (1.1) theorem 1.2 [12] if 𝐴 and 𝐵 are rectangle matrix in the same order and 𝑐 is r scale, then apply: a. 𝑇𝑟(𝐴) =  𝑇𝑟(𝐴𝑇), b. 𝑇𝑟(𝑐𝐴) =  𝑐 𝑇𝑟(𝐴), (1.2) c. 𝑇𝑟(𝐴 +  𝐵) =  𝑇𝑟(𝐴) + 𝑇𝑟(𝐵), d. 𝑇𝑟(𝐴𝐵) =  𝑇𝑟(𝐵𝐴). methods the method used in order to reach the aim of this paper is using literature study or conceptual foundation by following steps.  finding the general formula of power matrix (𝐴𝑛) 𝑚 with 𝑚  ∈ ℤ+ and proof it using mathematical induction,  determining trace matrix (𝐴𝑛) 𝑚, notated by 𝑇𝑟 (𝐴𝑛) 𝑚, finding the general formula and using mathematical induction, we proof the formula obtained. results and discussion this research is going to discuss about positive integers squared trace of m from special matrix of 𝑛 𝑥 𝑛 order with the entries of real numbers where each entry has the same value in a row, which is noted with matrix 𝐴𝑛 𝑚. the research started by deciding the general form of matrix square of 𝐴𝑛 𝑚 by calculating matrix square in order of 2 x 2 to order of 5 x 5 squared by m positive integers. after the general matrix of 𝐴𝑛 𝑚 is formed, then this research is continued by looking for 𝑇𝑟(𝐴𝑛 𝑚). special matrix order of 𝒏 ×  𝒏 (𝒏  ≥  𝟐) squared by 𝒎 positive integers this part is going to explain about squaring of special matrix order of 𝑛  × 𝑛 ,𝑛 ≥ 2 with the real number entries where each entry has the same value in a row, this matrix is formulated as follows 𝐴𝑛  =  [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛]  , 𝑎𝑖  ∈  ℝ ;  𝑖  =  1, 2, … ,  𝑛. (2.1) it is special matrix in order of 22  to 55  which is formulated as follows. 𝐴2  = [ 𝑎1 𝑎1 𝑎2 𝑎2 ] , 𝐴3  = [ 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 ] , 𝐴4  = [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 ], 𝐴5  =  [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 𝑎4 𝑎5 𝑎5 𝑎5 𝑎5 𝑎5] trace of positive integer power of squared special matrix rahmawati 203 for 𝑛 = 2, it is decided the matrix squaring values of (𝐴2) 2 to (𝐴2) 11 which are presented in table 1 below. table 1. special matrix squaring values of (𝐴2) 2 to (𝐴2) 11 no special matrix squaring of 𝑨2 matrix squaring values 𝑨2 1. (𝐴2) 2 (𝑎1 + 𝑎2) 𝐴2 2. (𝐴2) 3 (𝑎1 + 𝑎2) 2 𝐴2 3. (𝐴2) 4 (𝑎1 + 𝑎2) 3 𝐴2 4. (𝐴2) 5 (𝑎1 + 𝑎2) 4 𝐴2 5. (𝐴2) 6 (𝑎1 + 𝑎2) 5 𝐴2 6. (𝐴2) 7 (𝑎1 + 𝑎2) 6 𝐴2 7. (𝐴2) 8 (𝑎1 + 𝑎2) 7 𝐴2 8. (𝐴2) 9 (𝑎1 + 𝑎2) 8 𝐴2 9. (𝐴2) 10 (𝑎1 + 𝑎2) 9 𝐴2 10. (𝐴2) 11 (𝑎1 + 𝑎2) 10 𝐴2 after getting the values of special matrix squaring of 𝐴2 which are in table 1, then it can be predicted that the general form of the special matrix squaring based on its recursive pattern is (𝐴2) 𝑚  = (𝑎1 + 𝑎2) 𝑚−1 𝐴2. according to the prediction, then the general form of matrix squaring of 𝐴2 is presented in theorem 2.1 below. theorem 2.1 if given the special matrix of 𝐴2  = [ 𝑎1 𝑎1 𝑎2 𝑎2 ] ; 𝑎1, 𝑎2  ∈  ℝ, then (𝐴2) 𝑚  = (𝑎1 + 𝑎2) 𝑚−1 𝐴2 with 𝑚  ∈ ℤ +. (2.2) proof: using mathematic induction. for example 𝑝(𝑚) : (𝐴2) 𝑚  = (𝑎1 + 𝑎2) 𝑚−1 𝐴2 1. for 𝑚 = 1 then 𝑝 (1) : (𝐴2) 1  = (𝑎1 + 𝑎2) 1−1 𝐴2 = (𝑎1 + 𝑎2) 0 𝐴2 = 𝐴2 2. for 𝑚 = 𝑘 then it is assumed that 𝑝 (𝑘) is correct, which is 𝑝 (𝑘) : (𝐴2) 𝑘  = (𝑎1 + 𝑎2) 𝑘−1 𝐴2 will be presented for 𝑚  =  𝑘 + 1 then 𝑝 (𝑘 + 1) is also correct, which is 𝑝 (𝑘 + 1) : (𝐴2) 𝑘+1  = (𝑎1 + 𝑎2) 𝑘 𝐴2. (2.3) then, (𝐴2) 𝑘+1  = (𝐴2) 𝑘 (𝐴2) = (𝑎1 + 𝑎2) 𝑘−1 𝐴2 𝐴2 = (𝑎1 + 𝑎2) 𝑘−1 (𝐴2) 2 = (𝑎1 + 𝑎2) 𝑘−1 (𝑎1 + 𝑎2) 𝐴2 = (𝑎1 + 𝑎2) (𝑘−1)+1 𝐴2 = (𝑎1 + 𝑎2) 𝑘 𝐴2 by giving attention to the equation (2.3) then 𝑝 (𝑘 + 1) is correct. due to step (1) and (2) are presented correctly, then theorem 2.1 is proven. ∎ trace of positive integer power of squared special matrix rahmawati 204 for 𝑛 = 3, it is decided the value of matrix squaring of (𝐴3) 2 to (𝐴3) 11 which are presented in table 2 below. table 2. the value of special matrix squaring of (𝐴3) 2 to (𝐴3) 11 no special matrix squaring of 𝐴3 the value matrix squaring of 𝐴3 1. (𝐴3) 2   3321 aaaa  2. (𝐴3) 3 (𝑎1 + 𝑎2 + 𝑎3) 2 𝐴3 3. (𝐴3) 4 (𝑎1 + 𝑎2 + 𝑎3) 3 𝐴3 4. (𝐴3) 5 (𝑎1 + 𝑎2 + 𝑎3) 4 𝐴3 5. (𝐴3) 6 (𝑎1 + 𝑎2 + 𝑎3) 5 𝐴3 6. (𝐴3) 7 (𝑎1 + 𝑎2 + 𝑎3) 6 𝐴3 7. (𝐴3) 8 (𝑎1 + 𝑎2 + 𝑎3) 7 𝐴3 8. (𝐴3) 9 (𝑎1 + 𝑎2 + 𝑎3) 8 𝐴3 9. (𝐴3) 10 (𝑎1 + 𝑎2 + 𝑎3) 9 𝐴3 10. (𝐴3) 11 (𝑎1 + 𝑎2 + 𝑎3) 10 𝐴3 after getting the values of the special matrix squaring of 𝐴3 which is in table 2, then it can be predicted the general form of the special matrix squaring is based on its recursive pattern which is (𝐴3) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3) 𝑚−1 𝐴3. according to the prediction, then the general form of matrix squaring of 3a is presented in theorem 2.2 below. theorem 2.2 if given the special matrix of 𝐴2  = [ 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 ] ; 𝑎1, 𝑎2, 𝑎3  ∈  ℝ, then (𝐴3) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3) 𝑚−1 𝐴3 with 𝑚  ∈ ℤ +. (2.4) proof: applying the same steps with theorem 2.1, then this theorem is proved. ∎ for 𝑛 = 4, it is decided the value of matrix squaring of (𝐴4) 2 to (𝐴4) 11 which are presented in table 3 below. table 3. the value of special matrix squaring of (𝐴4) 2 to (𝐴4) 11 no special matrix squaring of 𝐴4 matrix squaring value of𝐴4 1. (𝐴4) 2 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝐴4 2. (𝐴4) 3 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 2 𝐴4 3. (𝐴4) 4 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 3 𝐴4 4. (𝐴4) 5 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 4 𝐴4 5. (𝐴4) 6 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 5 𝐴4 6. (𝐴4) 7 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 6 𝐴4 7. (𝐴4) 8 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 7𝐴4 8. (𝐴4) 9 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 8𝐴4 9. (𝐴4) 10 (𝑎1 +𝑎2 +𝑎3 + 𝑎4) 9𝐴4 10. (𝐴4) 11 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 10𝐴4 after getting the values of special matrix squaring of 𝐴4 which is in table 3, then it can be predicted that the general form of the special matrix squaring is based on its recursive pattern which is (𝐴4) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝑚−1 𝐴4. according to the trace of positive integer power of squared special matrix rahmawati 205 prediction, then the general form of matrix squaring of 𝐴4 is presented in theorem 2.3 below. theorem 2.3 if given the special matrix of 𝐴4  = [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 ];  𝑎1, 𝑎2, 𝑎3, 𝑎4  ∈  ℝ, then (𝐴4) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝑚−1 𝐴4 with 𝑚  ∈ ℤ +. (2.5) proof: adopting the proof in theorem 2.1, then this theorem is proven as well. ∎ for 𝑛 = 5, it is decided the value of matrix squaring of (𝐴5) 2 to (𝐴5) 11 which is presented in the table 4 below. table 4. the value of special matrix squaring of (𝐴5) 2to (𝐴5) 11 no special matrix squaring of 𝐴5 matrix squaring value of 𝐴5 1. (𝐴5) 2 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 𝐴5 2. (𝐴5) 3 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 2𝐴5 3. (𝐴5) 4 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 3𝐴5 4. (𝐴5) 5 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 4𝐴5 5. (𝐴5) 6 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 5𝐴5 6. (𝐴5) 7 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 6𝐴5 7. (𝐴5) 8 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 7𝐴5 8. (𝐴5) 9 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 8𝐴5 9. (𝐴5) 10 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 9𝐴5 10. (𝐴5) 11 (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 10𝐴5 after getting the values of matrix squaring of 𝐴5 which is in table 3, then in can be predicted that the general form of the special matrix squaring is based on its recursive pattern which is (𝐴5) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 𝑚−1𝐴5. according to the prediction, then the general form of matrix squaring of 𝐴5 is presented in theorem 2.4 below. theorem 2.4 if given the special matrix of 𝐴5  =  [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 𝑎4 𝑎5 𝑎5 𝑎5 𝑎5 𝑎5] ; 𝑎1,𝑎2,𝑎3,𝑎4,𝑎5 ∈ ℝ then (𝐴5) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 𝑚−1𝐴5 with 𝑚  ∈ ℤ +. (2.6) proof: it is clear from above theorems. ∎ by giving attention to the recursive pattern of equation (2.2), equation (2.4), equation (2.5) and equation (2.6) which are trace of positive integer power of squared special matrix rahmawati 206 (𝐴2) 𝑚  = (𝑎1 + 𝑎2) 𝑚−1 𝐴2 (𝐴3) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3) 𝑚−1 𝐴3 (𝐴4) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝑚−1 𝐴4 (𝐴5) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 𝑚−1 𝐴5. it can be predicted that the general form of the special matrix squaring in order of 𝑛 ×𝑛, 𝑛 ≥ 2 is equal to the equation (2.1), which is (𝐴𝑛) 𝑚  = (𝑎1 + 𝑎2 + … + 𝑎𝑛) 𝑚−1 𝐴𝑛 = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1 𝐴𝑛 according to the prediction, then the general form of special matrix squaring in order of 𝑛  × 𝑛 ,𝑛 ≥ 2 is equal to equation (2.1) is presented in the theorem 2.5 below. theorem 2.5 if given the special matrix in order 𝑛 ×𝑛 , 𝑛 ≥ 2 which is 𝐴𝑛  =  [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛]  ; 𝑎𝑖  ∈  ℝ ,   𝑖  =  1,  2, … ,  𝑛 then (𝐴𝑛) 𝑚  = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1𝐴𝑛 ,  𝑑𝑒𝑛𝑔𝑎𝑛 𝑚  ∈ ℤ +. proof: again, by using mathematic induction, for example 𝑝 (𝑚) : (𝐴𝑛) 𝑚  =  (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1𝐴𝑛 , with 𝑚 ∈ ℤ + 1. for 𝑚  =  1 then 𝑝 (1) : (𝐴𝑛) 1  = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 1−1 𝐴𝑛 = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 0 𝐴𝑛 = 𝐴𝑛 2. for km  is assumed that 𝑝 (𝑘) is correct, which is 𝑝 (𝑘) : (𝐴𝑛) 𝑘  = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1𝐴𝑛, with 𝑚  ∈ ℤ +. will be presented for 𝑚  =  𝑘 + 1 then 𝑝 (𝑘  +  1) is also correct, which is 𝑝 (𝑘  +  1) : (𝐴𝑛) 𝑘+1  = (∑ 𝑎𝑖 𝑛 𝑖=1 ) (𝑘+1)−1 𝐴𝑛 =  (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘𝐴𝑛 (2.7) the proof is below (𝐴𝑛) 𝑘+1  = (𝐴𝑛) 𝑘 (𝐴𝑛) trace of positive integer power of squared special matrix rahmawati 207 = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1 𝐴𝑛𝐴𝑛 = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1 [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛] [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛]   = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1  [ 𝑎1 2 + 𝑎1𝑎2 + ⋯ + 𝑎1𝑎𝑖  + ⋯ + 𝑎1𝑎𝑛 𝑎1 2 + 𝑎1𝑎2 + ⋯ + 𝑎1𝑎𝑖  + ⋯ + 𝑎1𝑎𝑛 ⋯ 𝑎1 2 + 𝑎1𝑎2 + ⋯ + 𝑎1𝑎𝑖  + ⋯ + 𝑎1𝑎𝑛 𝑎1𝑎2 + 𝑎2 2 + ⋯ + 𝑎2𝑎𝑖  + ⋯ + 𝑎2𝑎𝑛 𝑎1𝑎2 + 𝑎2 2 + ⋯ + 𝑎2𝑎𝑖  + ⋯ + 𝑎2𝑎𝑛 ⋯ 𝑎1𝑎2 + 𝑎2 2 + ⋯ + 𝑎2𝑎𝑖  + ⋯ + 𝑎2𝑎𝑛 ⋮ ⋮   ⋮ 𝑎1𝑎𝑖  + 𝑎2𝑎𝑖  + ⋯ + 𝑎𝑖 2 + ⋯ + 𝑎𝑖𝑎𝑛 𝑎1𝑎𝑖  + 𝑎2𝑎𝑖  + ⋯ + 𝑎𝑖 2 + ⋯ + 𝑎𝑖𝑎𝑛 ⋯ 𝑎1𝑎𝑖  + 𝑎2𝑎𝑖  + ⋯ + 𝑎𝑖 2 + ⋯ + 𝑎𝑖𝑎𝑛 ⋮ ⋮ ⋮ ⋮ 𝑎1𝑎𝑛  + 𝑎2𝑎𝑛  + ⋯ + 𝑎𝑖𝑎𝑛  + ⋯ + 𝑎𝑛 2 𝑎1𝑎𝑛  + 𝑎2𝑎𝑛  + ⋯ + 𝑎𝑖𝑎𝑛  + ⋯ + 𝑎𝑛 2 ⋯ 𝑎1𝑎𝑛  + 𝑎2𝑎𝑛  + ⋯ + 𝑎𝑖𝑎𝑛  + ⋯ + 𝑎𝑛 2] = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1   [ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎1 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎1 ⋯ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎1 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎2 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎2 ⋯ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎2 ⋮ ⋮ ⋮ ⋮ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑖 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑖 ⋯ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑛 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑛 ⋯ (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛) 𝑎𝑛] = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1 (𝑎1 + 𝑎2 + ⋯ + 𝑎𝑖  + ⋯ + 𝑎𝑛)  [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛] = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘−1 (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝐴𝑛 = (∑  𝑎𝑖 𝑛 𝑖=1 ) (𝑘−1)+1 𝐴𝑛 = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑘 𝐴𝑛 by giving attention to the equation (2.7) then 𝑝 (𝑘 +  1) is correct. due to step (1) and (2) are presented correctly, then the theorem 2.5 is proven. ∎ trace of special matrix in order 𝒏 × 𝒏 (𝒏  ≥  𝟐) squared by positive integers in this part is going to be given the trace of special matrix of 𝐴2 𝑚,𝐴3 𝑚,𝐴4 𝑚, and 𝐴5 𝑚 which are contained in theorem 3.1 to theorem 3.4 as follows. theorem 3.1 if it is given the special matrix of 𝐴2 = [ 𝑎1 𝑎1 𝑎2 𝑎2 ] ; 𝑎1,𝑎2 ∈ ℝ then 𝑇𝑟 (𝐴2) 𝑚 = (𝑎1 + 𝑎2) 𝑚, with 𝑚 ∈ ℤ+. (3.1) proof. the proof of theorem uses direct proof. because of the known matrix of 𝐴2 then 𝑇𝑟(𝐴2) = 𝑎1 + 𝑎2. according to theorem 2.1, is got equation (2.2) which is (𝐴2) 𝑚 = (𝑎1 + 𝑎2) 𝑚−1𝐴2. by using the definition 1.2 and theorem 1.2 (b), it is formulated 𝑇𝑟 (𝐴2) 𝑚 = 𝑇𝑟((𝑎1 + 𝑎2) 𝑚−1𝐴2) = (𝑎1 + 𝑎2) 𝑚−1𝑇𝑟(𝐴2) = (𝑎1 + 𝑎2) 𝑚−1(𝑎1 + 𝑎2) = (𝑎1 + 𝑎2) (𝑚−1)+1 trace of positive integer power of squared special matrix rahmawati 208 = (𝑎1 + 𝑎2) 𝑚. according to the proof, then theorem 3.1 is proven. ∎ theorem 3.2 if it is given special matrix of 𝐴3 = [ 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 ] ;𝑎1,𝑎2,𝑎3 ∈ ℝ then 𝑇𝑟 (𝐴3) 𝑚 = (𝑎1 + 𝑎2 + 𝑎3) 𝑚, with 𝑚 ∈ ℤ+. (3.2) proof. it is clear from above theorem. ∎ theorem 3.3 if it is given the special matrix of 𝐴4  = [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 ]; 𝑎1, 𝑎2, 𝑎3, 𝑎4  ∈  ℝ, then 𝑇𝑟 (𝐴4) 𝑚 = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝑚, with 𝑚 ∈ ℤ+. (3.3) proof. the proof is clear. ∎ theorem 3.4 if given the special matrix of 𝐴5  =  [ 𝑎1 𝑎1 𝑎1 𝑎1 𝑎1 𝑎2 𝑎2 𝑎2 𝑎2 𝑎2 𝑎3 𝑎3 𝑎3 𝑎3 𝑎3 𝑎4 𝑎4 𝑎4 𝑎4 𝑎4 𝑎5 𝑎5 𝑎5 𝑎5 𝑎5] ; 𝑎1, 𝑎2, 𝑎3,𝑎4 and 𝑎5  ∈  ℝ then 𝑇𝑟 (𝐴5) 𝑚 = (𝑎1 +𝑎2 +𝑎3 +𝑎4 +𝑎5) 𝑚, with 𝑚 ∈ ℤ+. (3.4) proof. clearly proven by following theorem 3.1. ∎ by giving attention to the recursive pattern on equation (3.1), equation (3.2), equation (3.3) and equation (3.4) which are 𝑇𝑟 (𝐴2) 𝑚  = (𝑎1 + 𝑎2) 𝑚 𝑇𝑟 (𝐴3) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3) 𝑚 𝑇𝑟 (𝐴4) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 𝑚 𝑇𝑟 (𝐴5) 𝑚  = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 𝑚. it can be predicted that the general form of the trace of special matrix in order 𝑛 𝑥 𝑛,𝑛 ≥ 2 is equal to equation (2.1) squared by positive integer (nonnegative integer) which is 𝑇𝑟 (𝐴𝑛) 𝑚  = (𝑎1 + 𝑎2 + … + 𝑎𝑛) 𝑚 = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚 . according to the prediction, then the general form of trace of special matrix in order 𝑛 𝑥 𝑛 ,𝑛 ≥ 2 is presented in theorem 3.5 below. trace of positive integer power of squared special matrix rahmawati 209 theorem 3.5 if given special matrix in order 𝑛 𝑥 𝑛 ,𝑛 ≥ 2 which is 𝐴𝑛  =  [ 𝑎1 𝑎1 ⋯ 𝑎1 𝑎2 𝑎2 ⋯ 𝑎2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑖 𝑎𝑖 ⋯ 𝑎𝑖 ⋮ ⋮ ⋮ ⋮ 𝑎𝑛 𝑎𝑛 ⋯ 𝑎𝑛]  ,  𝑎𝑖  ∈  ℝ ;  𝑖  =  1, 2, … ,  𝑛. then 𝑇𝑟 (𝐴𝑛) 𝑚  = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑚, with 𝑚  ∈ ℤ+. proof: this theorem will be proven by direct proof. because matrix 𝐴𝑛 is known, then 𝑇𝑟 (𝐴𝑛) = ∑  𝑎𝑖 𝑛 𝑖=1 . from theorem 2.5, obtained (𝐴𝑛) 𝑚  = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1𝐴𝑛. so that by using definition 1.2 and theorem 1.2 (b) obtained 𝑇𝑟(𝐴𝑛) 𝑚  =  𝑇𝑟 ((∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1 𝐴𝑛)  = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1  𝑇𝑟 (𝐴𝑛) = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1  (∑ 𝑎𝑖 𝑛 𝑖=1 ) = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚 . based on the evidence, then theorem 3.5 is proven. ∎ the application of matrix 𝑨𝒏 𝒎 and 𝑻𝒓(𝑨𝒏 𝒎) in examples the following is given the example of question related to theorem 2.5 and theorem 3.5 as follows. example 1. consider matrix 𝐴4 as follows 𝐴4 = [ 3 3 3 3 12 12 12 12 25 25 25 25 10 10 10 10 ] determine (𝐴4) 80 and 𝑇𝑟(𝐴4) 80. solution: by giving attention to matrix 𝐴4, value of 𝑎1 = 3,𝑎2 = 12,𝑎3 = 25, and 𝑎4 = 10. based on theorem 2.5 obtained (𝐴4) 80 = (∑𝑎𝑖 4 𝑖=1 ) 80−1 𝐴4 = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 79𝐴4 = (3+ 12 + 25 + 10)79 [ 3 3 3 3 12 12 12 12 25 25 25 25 10 10 10 10 ] trace of positive integer power of squared special matrix rahmawati 210 = (50)79 [ 3 3 3 3 12 12 12 12 25 25 25 25 10 10 10 10 ] based on theorem 3.5 obtained 𝑇𝑟(𝐴4) 80 = (∑𝑎𝑖 4 𝑖=1 ) 80 = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4) 80 = (3+ 12 + 25 + 10)80 = (50)80 example 2. given matrix 𝐴5 as follows 𝐴5 = [ 8 8 8 8 8 5/16 5/16 5/16 5/16 5/16 −12 −12 −12 −12 −12 2/3 2/3 2/3 2/3 2/3 −5/12 −5/12 −5/12 −5/12 −5/12] determine (𝐴5) 27 and 𝑇𝑟(𝐴5) 27. solution : by giving attention to matrix ,5a value of 𝑎1 = 8,𝑎2 = 5/16,𝑎3 = −12,𝑎4 = 2/3 and 𝑎5 = −5/12. based on theorem 2.5 obtained (𝐴5) 27 = (∑𝑎𝑖 5 𝑖=1 ) 27−1 𝐴5 = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 26𝐴5 = (8 + (5/16) +(−12) +(2/3) + (−5/12))26 [ 8 8 8 8 8 5/16 5/16 5/16 5/16 5/16 −12 −12 −12 −12 −12 2/3 2/3 2/3 2/3 2/3 −5/12 −5/12 −5/12 −5/12 −5/12] = (−3 7 16 )26 [ 8 8 8 8 8 5/16 5/16 5/16 5/16 5/16 −12 −12 −12 −12 −12 2/3 2/3 2/3 2/3 2/3 −5/12 −5/12 −5/12 −5/12 −5/12] according to theorem 3.5 it is formulated 𝑇𝑟(𝐴5) 27 = (∑𝑎𝑖 5 𝑖=1 ) 27 = (𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5) 27 = (8+ (5/16) + (−12) + (2/3) +(−5/12))27 = (−3 7 16 )27. trace of positive integer power of squared special matrix rahmawati 211 conclusions based on elaboration and discussion in previous part, several conclusions can be drawn as follows. 1. the general form of integer of a special matrix form in order 𝑛 × 𝑛 ,𝑛 ≥ 2 in equation (2.1) is as follows. (𝐴𝑛) 𝑚  = (∑ 𝑎𝑖 𝑛 𝑖=1 ) 𝑚−1  𝐴𝑛,  with 𝑚  ∈ ℤ +. 2. general form of trace in a special matrix form in order 𝑛 × 𝑛 ,𝑛 ≥ 2 in equation (2.1) is as follows. 𝑇𝑟 (𝐴𝑛) 𝑚  = (∑  𝑎𝑖 𝑛 𝑖=1 ) 𝑚,  with 𝑚  ∈ ℤ+. acknowledgments we thank all authors (rr, ac, fa, ccm, and ym) for their responsibility to designed the research and approved the final manuscript. rr, ac wrote the manuscript, fa, ccm gave their suggestion and edited the manuscript and ym read, edited for the final content of the manuscript. none of the authors had a conflict of interest. references [1] c. brezinski, p. fika, and m. mitrouli, “estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments,” electron. trans. numer. anal., vol. 39, pp. 144–155, 2012. [2] b. n. datta and k. datta, “an algorithm for computing powers of a hessenberg matrix and its applications,” linear algebra appl., vol. 14, no. 3, pp. 273–284, 1976, doi: 10.1016/0024-3795(76)90072-0. [3] v. pan, “estimating the extremal eigenvalues of a symmetric matrix,” comput. math. with appl., vol. 20, no. 2, pp. 17–22, 1990, doi: 10.1016/08981221(90)90236-d. [4] m. t. chu, “symbolic calculation of the trace of the power of a tridiagonal matrix,” vol. 268, pp. 257–268, 1985. [5] n. higham, “functions of matrices: theory and computation, chapter 5 matrix sign function,” soc. ind. appl. math., no. march, p. 445, 2008. [6] a. v. zarelua, “on congruences for the traces of powers of some matrices,” proc. steklov inst. math., vol. 263, no. 1, pp. 78–98, 2008, doi: 10.1134/s008154380804007x. [7] j. pahade and m. jha, “trace of positive integer power of real 2 x 2 matrices,” adv. linear algebr. &amp; matrix theory, vol. 05, no. 04, pp. 150–155, 2015, doi: 10.4236/alamt.2015.54015. [8] h. avron, “counting triangles in large graphs using randomized matrix trace estimation categories and subject descriptors,” kdd-ldmta, 2010. [9] j. k. pahade and m. jha, “trace of positive integer power of adjacency matrix,” glob. j. pure appl. math., vol. 13, no. 6, pp. 2079–2087, 2017. [10] c. r. howard anton, elementary linear algebra: applications version, 11th edition 11. 2013. [11] j. asquith and b. kolman, elementary linear algebra, vol. 71, no. 457. 1987. [12] j. e. gentle, matrix algebra, theory, computations, and applications in statistics, vol. 102. 2009. cauchy jurnal matematika murni dan aplikasi volume 7, issue 1, november 2021 issn : 2086-0382 e-issn : 2477-3344 publication etics cauchy: jurnal matematika murni dan aplikasi is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions 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sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia subanar seno, gadjah mada university, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740595') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740557') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740556') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740541') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/736347') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5964') optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 173-185 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 23, 2021 reviewed: november 17, 2021 accepted: december 22, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.13184 optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi1,*, nursyiva irsalinda1, meksianis z. ndii2 1department of mathematics, faculty of applied science and technology, ahmad dahlan university, yogyakarta, indonesia 2department of mathematics, faculty of sciences and engineering, university of nusa cendana, indonesia *corresponding author email: yudi.adi@math.uad.ac.id* nursyiva.irsalinda@math.uad.ac.id, meksianis.ndii@staf.undana.ac.id abstract the existence of viral mutations in various infectious diseases can make it difficult to overcome outbreaks caused by these viruses. in this paper, we introduce an optimal control problem in a two-strain sir epidemic model with viral mutation and vaccine administration. the purpose of this study was to investigate the efficacy and cost-effectiveness of two disease prevention strategies, namely restriction of community mobility to prevent disease transmission and vaccine intervention. we consider the time-dependent control case, and we use pontryagin’s maximum principle to derive necessary conditions for the optimal control of the disease. we also calculate the average cost-effectiveness ratio (acer) and the incremental cost-effectiveness ratio (icer) to investigate the cost-effectiveness of all possible strategies of the control measures. the results of this study indicate that the most cost-effective disease control strategy is a combination of mobility restriction and vaccination. keywords: epidemic model; cost-effectiveness analysis; numerical simulation; optimal control; viral mutation introduction epidemiological modeling is a field of mathematical modeling that studies the causes, patterns, and effects of disease on health in a population. the sir (susceptible, infected, recovered) compartment model that kermack-mckendrick first introduced in 1927 became the basis for developing models of the spread of infectious diseases. according to the characteristics of the disease, different epidemic models by adding or modifying compartments have been developed and studied. among them by adding a compartment vaccination [1],[2],[3], treatment [4], quarantine [5], viruses or bacteria that cause disease[6], disease-carrying vectors [7], and others. in various types of infectious diseases caused by viruses, viruses mutations make the epidemic difficult to overcome immediately. the emergence of new variants of this virus increased the length of the epidemic period. such conditions are also currently happening in various parts of the world, namely the covid-19 pandemic. especially in http://dx.doi.org/10.18860/ca.v7i1.13184 mailto:yudi.adi@math.uad.ac.id* mailto:nursyiva.irsalinda@math.uad.ac.id mailto:meksianis.ndii@staf.undana.ac.id optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 174 indonesia, after experiencing a decline in cases for about nine months since the beginning of the pandemic in march 2020, the number of positive covid-19 cases again increased in mid-june 2021. the government has taken various policies to be able to end the spread of this covid-19 disease immediately. beside targeting vaccinations, the government is currently implementing community activity restrictions (ppkm) to control the spread of the covid-19 outbreak. many mathematical models of covid-19 have also been developed, as in [5], [8]–[12]. in the last few decade, optimal control theory has developed rapidly, and its diverse applications are widely used in various scientific and engineering fields. this theory has proven to be effective in mathematical epidemiology when it comes to determining how to remove or reduce the number of cases at the lowest possible cost. the optimal control theory has been utilized to capture intervention strategies in many research, see for example [5], [7], [10], [13]–[16] optimal control models involving vaccination strategies have also been developed, as in [3], [16], [17]. however, these models did not consider the presence of viral mutations that were presumed to be more virulent in the premutated viruses. as in 12 states across the united states, the more easily transmissible strain of sars-cov-2, b.1.1.7, has been found [18]. in this article, we will discuss the sir epidemic model by considering the presence of viral mutations. we are also considering vaccine intervention as one of prevention against diseases. motivated by this, in this article, we intend to modify the epidemic model with virus mutation and vaccine interventions studied in adi et al. [19]. instead of constant parameters of the intervention strategy, we use a control function to express the intervention strategy in this model. the goal is to find the best function for a given control measure by applying pontryagin’s maximum principle [20]. this study also observes which control strategy is the most cost-effective, which is determined through the average cost-effectiveness ratio (acer) and the incremental cost-effectiveness ratio (icer), as defined in [21]– [24]. besides being applied to the spread of covid-19, the model can also be used for other diseases involving viral mutations. this paper's structure is as follows. the methodologies used in our research are discussed in the following section. after then, the model's analysis was discussed. finally, we will provide a brief summary of our work. methods the optimal control problem is analyzed by performing the following steps: 1. we consider a modified sir epidemic model taking into account the presence of viral mutations and vaccine intervention. 2. considering a time-dependent constant case-control and using pontryagin's maximum principle to obtain the necessary conditions for optimal disease control. 3. demonstrating the numerical result of the existence of the optimal control by implementing the forward-backward fourth-order runge-kutta method. 4. computing the average cost-effectiveness ratio (acer) and additional costeffectiveness ratio (icer) to investigate the cost-effectiveness of all possible control action strategies. optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 175 results and discussion formulation of the optimal control problem modifying the standard sir model, adi et al. [19] have developed an epidemic model taking into account the presence of viral mutations and vaccine interventions. mutations are recorded in terms that transfer an individual infected with one strain to an individual infected with another strain. the populations subdivided into five classes, which are; susceptible (𝑆), infected by strain one (𝐼1), infected by strain two (𝐼2), vaccinated (𝑉), and recovered (𝑅). the model is given in (1) below. 𝑑𝑆 𝑑𝑡 = λ − 𝛽1𝑆𝐼1 − 𝛽2𝑆𝐼2 − 𝛾𝑆 − 𝜇𝑆, 𝑑𝐼1 𝑑𝑡 = 𝛽1𝑆𝐼1 − (𝜔 + 𝛼1 + 𝑐 + 𝜇)𝐼1, 𝑑𝐼2 𝑑𝑡 = 𝛽2𝑆𝐼2 + 𝜔𝐼1 + (1 − 𝜀)𝑉𝐼2 − (𝛼2 + 𝑑 + 𝜇)𝐼2, 𝑑𝑉 𝑑𝑡 = 𝛾𝑆 − (1 − 𝜀)𝑉𝐼2 − 𝜇𝑉, 𝑑𝑅 𝑑𝑡 = 𝛼1𝐼1 + 𝛼2𝐼2 − 𝜇𝑅. (1) the first four equations in the system (1) do not depend on 𝑅, so to analyze the dynamics of the model, the fifth equation is neglected. please refer to [19] for details. next, paying attention only to the first four equations, we introduce a time-dependent control in the system (1). the purpose is to control the spread of disease and study strategies to eradicate epidemics in a community. we introduce two control functions, 𝑢1(𝑡) and 𝑢2(𝑡), which represent attempts to prevent disease transmission from both viral strains and vaccinations, respectively. the corresponding state system is given by: 𝑑𝑆 𝑑𝑡 = λ − (1 − 𝑢1(𝑡))(𝛽1𝐼1 + 𝛽2𝐼2)𝑆 − 𝑢2(𝑡)𝑆 − 𝜇𝑆, 𝑑𝐼1 𝑑𝑡 = (1 − 𝑢1(𝑡))𝛽1𝑆𝐼1 − (𝜔 + 𝛼1 + 𝑐 + 𝜇)𝐼1, 𝑑𝐼2 𝑑𝑡 = (1 − 𝑢1(𝑡))𝛽2𝑆𝐼2 + 𝜔𝐼1 + (1 − 𝜀)𝑉𝐼2 − (𝛼2 + 𝑑 + 𝜇)𝐼2, 𝑑𝑉 𝑑𝑡 = 𝑢2(𝑡)𝑆 − (1 − 𝜀)𝑉𝐼2 − 𝜇𝑉, (2 ) where 𝑢1(𝑡) is a control strategy that maintains the state of the uninfected population in the susceptible class and reduces the rate at which individuals leave the susceptible class to the infected class, either by strain one or by strain two, and 𝑢2(𝑡) is a control strategy to increase the number of individuals vaccinated. medically, considering that both strategies have many limitations so that they are not fully effective, it is realistic to assume that 0 ≤ 𝑢𝑖 𝑚𝑎𝑥 < 1, 𝑖 = 1,2. hence, the bounded lebesgue measurable set of admissible control is represented as 𝛺 = {(𝑢1(𝑡), 𝑢2(𝑡))|0 ≤ 𝑢𝑖 (𝑡) ≤ 𝑢𝑖 𝑚𝑎𝑥 , 𝑖 = 1,2, 𝑡 ∈ [0, 𝑇]}. (3) the aim is to gain the optimal value 𝑢𝑖 ∗ of the control 𝑢𝑖 (𝑡) in the time interval [0, 𝑇], such that the associate state trajectories 𝑋∗ = (𝑆∗, 𝐼1 ∗, 𝐼2 ∗, 𝑉∗) are solutions of the system (2) in the interval [0, 𝑇] with the initial conditions: optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 176 𝑆(0) ≥ 0, 𝐼1(0) ≥ 0, 𝐼2(0) ≥ 0, 𝑉(0) ≥ 0, (4) and 𝑢𝑖 ∗ maximizes the objective function given by: 𝐽(𝑢1, 𝑢2) = ∫ [𝑤1𝑆(𝑡) + 𝑤2𝑉(𝑡) − 𝑤3𝐼1(𝑡) − 𝑤4𝐼2(𝑡) − 𝐶1𝑢1 2(𝑡) 2 − 𝐶2𝑢2 2(𝑡) 2 ] 𝑑𝑡 𝑇 0 , (5) with 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝐶1, 𝐶2 are positive weight constant where we want to maximize the susceptibles 𝑆(𝑡), and vaccinated individuals 𝑉(𝑡), and to minimize both infected individuals by strain one 𝐼1(𝑡) and by strain two 𝐼2(𝑡) (negative sign means maximizing) while keeping prevention cost 𝑢1(𝑡) and vaccination cost 𝑢2(𝑡) low. the cost of the prevention program could come from the implementation of the restriction of citizen mobilization, quarantine, or local lockdowns. at the same time, the cost of vaccination comes from everything needed to implement the vaccination program. our optimal control problem is to determining (𝑆∗, 𝐼1 ∗, 𝐼2 ∗, 𝑉∗) related to an admissible control 𝑢𝑖 ∗ on the time interval [0, 𝑇] satisfying equation (2) and the initial condition of (4) and maximizing the cost functional of equation (5) such that 𝐽(𝑢1 ∗ , 𝑢2 ∗ ) = max ω 𝐽(𝑢1, 𝑢2). (6) here, we consider that the objective function as a function of 𝑢1 and 𝑢2, so it is concave with respect to the control 𝑢𝑖 . from this property and noting that the control system also satisfies the lipschitz property corresponding to the state variables (𝑆, 𝐼1 , 𝐼2 , 𝑉), it is ensured that the optimal control u of the optimal control problem in equation (4) exists. hence, the maximum value can be obtained [25]–[27]. characteristic of the optimal controls in order to take advantage the pontryagin's maximal principle, the system (4) and the objective functional (5) need to be converted into a pointwise hamiltonian, ℋ with respect to (𝑢1, 𝑢2), and we get ℋ = 𝑤1𝑆(𝑡) + 𝑤2𝑉(𝑡) − 𝑤3𝐼1(𝑡) − 𝑤4𝐼2(𝑡) − 𝐶1𝑢1 2 2 − 𝐶2𝑢2 2 2 + 𝜆1[λ − (1 − 𝑢1)(𝛽1𝐼1 + 𝛽2𝐼2)𝑆 − 𝑢2𝑆 − 𝜇𝑆] + 𝜆2[(1 − 𝑢1)𝛽1𝑆𝐼1 − (𝜔 + 𝛼1 + 𝑐 + 𝜇)𝐼1] + 𝜆3[(1 − 𝑢1)𝛽2𝑆𝐼2 + 𝜔𝐼1 + (1 − 𝜀)𝑉𝐼2 − (𝛼2 + 𝑑 + 𝜇)𝐼2] + 𝜆4[𝑢2𝑆 − (1 − 𝜀)𝑉𝐼2 − 𝜇𝑉]. (7) where 𝜆1, 𝜆2, 𝜆3, 𝜆4 are the costate variables or adjoint variables associated with the state variables 𝑆, 𝐼1, 𝐼2, 𝑉. we summarize the necessary conditions for the optimal control 𝑢𝑖 ∗, 𝑖 = 1,2 in theorem 1 below. theorem 1. there is an optimal control 𝑢𝑖 ∗, 𝑖 = 1,2 corresponding to the optimal solution (𝑆∗, 𝐼1 ∗, 𝐼2 ∗, 𝑉∗) that maximizes the objective functional 𝐽(𝑢1, 𝑢2) over ω. moreover, there exist costate variables or adjoint variables, 𝜆𝑗 , 𝑗 = 1,2,3,4 that satisfies 𝑑𝜆𝑗 𝑑𝑡 = − 𝜕ℋ 𝜕𝑋 with transversality condition 𝜆𝑗 (𝑇) = 0, 𝑗 = 1,2,3,4. furthermore, the associated optimal control 𝑢𝑖 ∗, 𝑖 = 1,2 are given by optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 177 𝑢1 ∗ = min {max {0, (𝜆1 − 𝜆2)𝛽1𝐼1 ∗𝑆∗ + (𝜆1 − 𝜆3)𝛽2𝐼2 ∗𝑆∗ 𝐶1 } , 𝑢1 𝑚𝑎𝑥 }, 𝑢2 ∗ = min {max {0, (𝜆4 − 𝜆1)𝑆 ∗ 𝐶2 } , 𝑢2 𝑚𝑎𝑥 }. (8) proof. the adjoint system is derived by taking the partial derivative of the hamiltonian ℋ with respect to the associated state variables so that 𝑑𝜆1 𝑑𝑡 = − 𝜕ℋ 𝜕𝑆 = −𝑤1 + (𝜆1 − 𝜆2)(1 − 𝑢1)𝛽1𝐼1 + (𝜆1 − 𝜆3)(1 − 𝑢1)𝛽2𝐼2 +(𝑢2 + 𝜇)𝜆2 − 𝜔𝜆3, 𝑑𝜆2 𝑑𝑡 = − 𝜕ℋ 𝜕𝐼1 = 𝑤3 + (𝜆1 − 𝜆2)(1 − 𝑢1)𝛽1𝑆 + (𝜔 + 𝛼1 + 𝑐 + 𝜇)𝜆2 − 𝜔𝜆3, 𝑑𝜆3 𝑑𝑡 = − 𝜕ℋ 𝜕𝐼2 = 𝑤4 + (𝜆1 − 𝜆3)(1 − 𝑢1)𝛽2𝑆 + (𝛼2 + 𝑑 + 𝜇)𝜆3 +(𝜆4 − 𝜆3)(1 − 𝜀)𝑉, 𝑑𝜆4 𝑑𝑡 = − 𝜕ℋ 𝜕𝑉 = −𝑤2 + (𝜆4 − 𝜆3)(1 − 𝜀)𝐼2 + 𝜇𝜆4, (9) along with the transversality conditions 𝜆𝑗 (𝑇) = 0, 𝑗 = 1,2,3,4. then, the optimal control 𝑢𝑖 ∗ are defined by solving 𝜕ℋ 𝜕𝑢𝑖 = 0. this lead to the condition of optimal controls 𝜕ℋ 𝜕𝑢1 = −𝐶1𝑢1 + (𝜆1 − 𝜆2)𝛽1𝐼1 ∗𝑆∗ + (𝜆1 − 𝜆3)𝛽2𝐼2 ∗𝑆∗ = 0, 𝜕ℋ 𝜕𝑢2 = −𝐶2𝑢2 + (𝜆4 − 𝜆1)𝑆 ∗ = 0. hence, we have 𝑢1 = (𝜆1 − 𝜆2)𝛽1𝐼1 ∗𝑆∗ + (𝜆1 − 𝜆3)𝛽2𝐼2 ∗𝑆∗ 𝐶1 , 𝑢2 = (𝜆4 − 𝜆1)𝑆 ∗ 𝐶2 . (10) since 𝑢𝑖 ∗, 𝑖 = 1,2 must belong to ω, we get 𝑢1 ∗ = { 0 (𝜆1 − 𝜆2)𝛽1𝐼1 ∗𝑆∗ + (𝜆1 − 𝜆3)𝛽2𝐼2 ∗𝑆∗ 𝐶1 𝑢1 𝑚𝑎𝑥 , if 𝑢1 ≤ 0 , if 0 < 𝑢1 < 𝑢1 𝑚𝑎𝑥 , if 𝑢1 ≥ 𝑢1 𝑚𝑎𝑥 , 𝑢2 ∗ = { (𝜆4 − 𝜆1)𝑆 ∗ 𝐶2 , if 𝑢2 ≤ 0 , if 0 < 𝑢2 < 𝑢2 𝑚𝑎𝑥 , if 𝑢2 ≥ 𝑢2 𝑚𝑎𝑥 . which can also be characterized by 𝑢1 ∗ = min {max {0, (𝜆1 − 𝜆2)𝛽1𝐼1 ∗𝑆∗ + (𝜆1 − 𝜆3)𝛽2𝐼2 ∗𝑆∗ 𝐶1 } , 𝑢1 𝑚𝑎𝑥 }, (11) optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 178 𝑢2 ∗ = min {max {0, (𝜆4 − 𝜆1)𝑆 ∗ 𝐶2 } , 𝑢2 𝑚𝑎𝑥 }. this completes the proof. the following section provides numerical simulations of the optimality system, the control profile, and discussions. numerical results and discussion we observe the optimal trajectories of the optimal system through some numerical simulations. we applied the forward-backward sweep method described in [20], which is very commonly used in the literature of optimal control problems, as in the literature [9], [14], [23]. for numerical simulation, we use a set of parameter values as in [19] and take the weight factor 𝑤1, 𝑤2, 𝑤3, 𝑤4, equal to one 𝐶1 = 2, and 𝐶2 = 2 due to the lack of the available literature and data. it should be noted that the weight values selected for the simulation are only for the theoretical sense to describe the control strategy proposed in this model. for the maximum control, we set 𝑢1, 𝑢1 𝑚𝑎𝑥 = 0.5 under the assumption that it is difficult to maintain community discipline in implementing prevention of disease transmissions such as restrictions on community interaction/mobilization, local lockdown, and quarantine. as for the control with vaccination, 𝑢1 𝑚𝑎𝑥 = 0.7 was taken based on the assumption that the vaccine was not yet fully effective and the lack of awareness of the individual to be vaccinated. we will focus on comparing the three control strategies.  strategy i: combination of prevention of disease transmission and vaccination. in this case 𝑢1 and 𝑢2 are defined as control variables.  strategy ii: use restrictions on community interaction/mobilization as a control. in this case, only 𝑢1 is taken as a control variable.  strategy iii: vaccine intervention as the only control, so only 𝑢2 as the control variable. figure 1 shows the impact of implementing various strategies on the population size of 𝑆(𝑡) (fig. 1a) and 𝑉(𝑡) (fig. 1b) for 50 days. it can be seen that without implementing the control strategy, the number of susceptible individuals and vaccinated individuals is lower than if the control strategy is applied. with optimal control strategies, most susceptible individuals will be protected or vaccinated against the virus, thus leading to higher individuals in the vaccinated class (fig. 1b) and ultimately resulting in fewer individuals being infected by either strain one or strain two see figure 2. optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 179 figure 1. simulation results without and with the implementation of various control strategies. (a) susceptible individuals, (b) vaccinated individuals. in figures 2(a) 2(d), we show the impact of using optimal control strategies on the number of individuals infected by strain one and strain two. this suggests that disease in infectious populations can be reduced more rapidly when both controls are applied (strategy i) compared to the situation without control or by using a single control, i.e., prevention of transmission only (strategy ii) or vaccination only (strategy iii). from the simulation results, the trajectories of optimal control show that the combination of two control strategies can lead to desired disease control. fig. 2(a) – 2(b) show a comparison of the number of individuals infected by strains one and by strain two using strategy i and strategy iii. figures 2(c) 2(d) show the situation of individuals infected by strain one and strain two by implementing strategy ii and without control strategy. based on the number of infected individuals, it appears that strategy i is the best strategy that can be applied to end the spread of the disease immediately. the corresponding timedependent controls 𝑢1(𝑡) and 𝑢2(𝑡) are depicted in figure 3. optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 180 figure 2. simulation results for individuals infected by strain one (a), (c) and infected individuals by strain two (b), (d) without and with the implementation of various control strategies. figure 3(a) tells us that strategy i can be implemented by maintaining preventive transmission control 𝑢1(𝑡) and vaccination 𝑢2(𝑡) at their upper bounds for about 30 days and 35 days, respectively, and gradually decreasing to their lower bounds. figure 3(b) illustrates the implementation of strategy ii, which shows that the control 𝑢1(𝑡) is kept at its upper bound over time. while figure 3(c) shows that if strategy iii is implemented, then the control 𝑢2(𝑡) should be maintained at its upper bound most of the time. when these controls are implemented on a broad scale, it is also critical to adopt an approach that provides optimal cost, i.e., less cost. as a result, we will look at the cost-effectiveness of these controls in the next section. optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 181 figure 3. control profile for each strategy. (a) strategy i, (b) strategy ii, (c) strategy iii. cost-effectiveness analysis in this section, we use the average cost-effectiveness ratio (acer) and the incremental cost-effectiveness ratio (icer) to carry out the cost-effectiveness analysis. the average cost-effective ratio (acer) is calculated as follows [21]: acer = the total cost (tc) total number of infections averted (ta) . (12) the total number of individuals infected averted during the intervention period t is obtained by using ta = ∫(𝐼1 ∗ + 𝐼2 ∗)𝑑𝑡 − 𝑇 0 ∫(𝐼1 + 𝐼2)𝑑𝑡, 𝑇 0 (13) where 𝐼1 ∗, 𝐼2 ∗ are the solution of infected classes by strain one and the infected classes by strain two without controls and 𝐼1, 𝐼2 are the optimal solution with controls. the total optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 182 cost implemented during the period t is calculated as follows: t𝑐 = ∫ 1 2 (𝐶1𝑢1 2 + 𝐶2𝑢2 2)𝑑𝑡. 𝑇 0 based on this cost analysis, the most cost-effective strategy is the one with the smallest acer value [23]. now, we calculate the total cost invested and total infected averted in each strategy to analyze the cost-effectiveness. using the formula (12), we find that strategy i has the smallest acer value and strategy ii has the largest acer value, as seen in figure 4. the results are also given in table 1. thus, according to the acer value, the most effective intervention strategy is strategy i. figure 4. average cost-effectiveness ratio (acer) results for strategy i – iii the icer, on the other hand, is calculated by dividing the cost difference between two feasible interventions by the difference in their effects. mathematically, it is expressed as [22], [24]: icer = difference in costs produced by strategies i and j difference in the total number of infection averted in strategies i and j . (14) the difference between the total number of infected individuals without controls and the total number of infected individuals with controls is used to compute the total number of averted infections. furthermore, we employed the cost functions 𝐶1 2 𝑢1 2 and 𝐶2 2 𝑢2 2 across time to calculate the total cost of the implemented strategies. we also used the parameter values from the preceding section to calculate the total cost and total infections averted, as shown in table 1, with total averted infections are ranked according to their increasing in order. then, the icer is calculated using the formula in (14). first, we computed for the competing strategies ii and iii as follows: icer (ii) = 989,582.93 − 0 102,599,77 − 0 = 9.6451, icer (iii) = 1,026,524.16 − 989,582.93 112,334.16 − 102,599,77 = 3.7949. the results of the icer computation (as shown in table 1) show that strategy ii has a 8,8 8,9 9 9,1 9,2 9,3 9,4 9,5 9,6 9,7 average cost-effective ratio (acer) strategi ii strategi iii strategi i optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 183 higher icer value than strategy iii. as a result, implementing prevention transmission control 𝑢1 alone is more expensive and ineffective than using vaccine intervention control 𝑢2. as a result, strategy ii is removed from the list of possible control strategies. the icer for strategies iii and i now need to be recalculated. the calculation is as follows: icer (iii) = 1,026,524.16 112,334.16 = 9.1381, icer (i) = 1,029,506.04 − 1,026,524.16 112,886.09 − 112,334.16 = 5.4026. table 2 summarizes the results of the calculations. table 1. strategies i – iii in order of increasing number of averted infected strategy total infected averted total cost acer icer strategy ii 102,599.77 989,582.93 9.6451 9.6451 strategy iii 112,334.16 1,026,524.16 9.1381 3.7949 strategy i 112,886.09 1,029,506.04 9.1199 table 2. comparison between strategies iii and i strategy total infected averted total cost icer strategy iii 112,334.16 1,026,524.16 9.1381 strategy i 112,886.09 1,029,506.04 5.4026 it is clearly shown from table 2 that strategy iii has an icer value greater than strategy i. therefore, due to its cost-effectiveness and health benefits, strategy i, that combination of prevention of disease transmission and vaccination, is the best of all possible options. conclusions this paper has presented and analyzed a modified sir epidemic model considering a time-dependent constant control that includes two control variables. the two control variables considered in this model are prevention of disease transmission, such as by restricting community interactions and administering vaccines. numerical simulation of the optimal control problem was carried out using three strategies. strategy i, a combination of prevention of disease transmission and vaccination, strategy ii, only prevention of disease transmission by restriction community interaction is taken as a control variable, and strategy iii, if the vaccine intervention is the only intervention carried out. all strategies show control profiles adjusted for the number of infected individuals in the community. stronger interventions are needed to substantially reduce the number of infected individuals and the cost of implementing the strategy. furthermore, analysis to determine the most cost-effective strategy was carried out using acer and icer. based on calculating acer and icer, we found that using both controls simultaneously was the most cost-effective method and vaccination was the most cost-effective method in a single intervention. when only one intervention is applied, our simulations reveal that vaccination is the best single intervention strategy. however, the combination of vaccination and the restriction of community interactions, i.e., strategy i, gave the best results in reducing the number of infected individuals with the cheapest cost compared to a single intervention strategy. we think that our work optimal control and cost-effectiveness analysis in an epidemic model with viral mutation and vaccine intervention yudi ari adi 184 will serve as a foundation for mathematical models that examine cost-effectiveness analyses using real-world data, especially on an epidemic model which considers viral mutation and vaccination. acknowledgments we thank ahmad dahlan university for supporting this work through fundamental research grant (no. pd-273/sp3/lppm-uad/vi/2021). references [1] s. ullah and m. a. khan, “modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study,” chaos, solitons and fractals, vol. 139, 2020, doi: 10.1016/j.chaos.2020.110075. 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[23] d. aldila, “cost-effectiveness and backward bifurcation analysis on covid-19 transmission model considering direct and indirect transmission,” commun. math. biol. neurosci., pp. 1–28, 2020. [24] fatmawati, u. dyah purwati, f. riyudha, and h. tasman, “optimal control of a discrete age-structured model for tuberculosis transmission,” heliyon, 2020, doi: 10.1016/j.heliyon.2019.e03030. [25] l. cesari, optimization-theory and applications. new york, ny, usa: springer, 1983. [26] t. burden, j. ernstberger, and k. r. fister, “optimal control applied to immunotherapy,” discret. contin. dyn. syst. ser. b, vol. 4, no. 1, pp. 135–146, 2004, doi: 10.3934/dcdsb.2004.4.135. [27] c. campos, c. j. silva, and d. f. m. torres, “numerical optimal control of hiv transmission in octave/matlab,” math. comput. appl., 2020, doi: 10.3390/mca25010001. cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of 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of the author. cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 publication etics contribution to editorial decisions peer review assists the editor in making editorial decisions and through the editorial communications with the author may also assist the author in improving the paper. promptness any selected referee who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the editor and excuse himself from the review process. standards of objectivity reviews should be conducted objectively. personal criticism of the author is inappropriate. referees should express their views clearly with supporting arguments. acknowledgement of sources reviewers should identify relevant published work that has not been cited by the authors. any statement that an observation, derivation, or argument had been previously reported should be accompanied by the 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statements constitute unethical behavior and are unacceptable. data access and retention authors are asked to provide the raw data in connection with a paper for editorial review and should be prepared to provide public access to such data (consistent with the alpspstm statement on data and databases), if practicable, and should in any event be prepared to retain such data for a reasonable time after publication. originality and plagiarism the authors should ensure that they have written entirely original works, and if the authors have used the work and/or words of others that this has been appropriately cited or quoted. cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 publication etics multiple, redundant or concurrent publication an author should not in general publish manuscripts describing essentially the same research in more than one journal or primary publication. submitting the same manuscript to more than one 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if the work involves chemicals, procedures or equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 6, issue 3, november 2020 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated kusno kusno, university of jember, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia dr riswan efendi, uin sultan syarif kasim riau, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity abdussakir abdussakir, (scopus id:57202352728), universitas islam negeri maulana malik ibrahim malang, indonesia cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 publication etics cauchy: jurnal matematika murni dan aplikasi is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the journal editor, the peer reviewer, the publisher and the society. as publisher of pure and applied mathematics journal, we take our duties to back up over all stages of publishing seriously and we recognize our ethical and other responsibilities. we are committed to ensuring that advertising, reprint or other commercial revenue has no impact or influence on editorial decisions. publication decisions the editor of cauchy is responsible for deciding which of the articles submitted to the journal should be published. the validation of the work in question and its importance to researchers and readers must always drive such decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 publication etics contribution to editorial decisions peer review assists the editor in making editorial decisions and through the editorial communications with the author may also assist the author in improving the paper. promptness any selected referee who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the editor and excuse himself from the review process. standards of objectivity reviews should be conducted objectively. personal criticism of the author is inappropriate. referees should express their views clearly with supporting arguments. acknowledgement of sources reviewers should identify relevant published work that has not been cited by the authors. any statement that an observation, derivation, or argument had been previously reported should be accompanied by the relevant citation. a reviewer should also call to the editor's attention any substantial similarity or overlap between the manuscript under consideration and any other published paper of which they have personal knowledge. disclosure and conflict of interest privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers. reporting standards authors of reports of original research should present an accurate account of the work performed as well as an objective discussion of its significance. underlying data should be represented accurately in the paper. a paper should contain sufficient detail and references to permit others to replicate the work. fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. data access and retention authors are asked to provide the raw data in connection with a paper for editorial review and should be prepared to provide public access to such data (consistent with the alpspstm statement on data and databases), if practicable, and should in any event be prepared to retain such data for a reasonable time after publication. originality and plagiarism the authors should ensure that they have written entirely original works, and if the authors have used the work and/or words of others that this has been appropriately cited or quoted. cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 publication etics multiple, redundant or concurrent publication an author should not in general publish manuscripts describing essentially the same research in more than one journal or primary publication. submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. acknowledgement of sources proper acknowledgment of the work of others must always be given. authors should cite publications that have been influential in determining the nature of the reported work. authorship of the paper authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. all those who have made significant contributions should be listed as co-authors. where there are others who have participated in certain substantive aspects of the research project, they should be acknowledged or listed as contributors. the corresponding author should ensure that all appropriate co-authors and no inappropriate co-authors are included on the paper, and that all co-authors have seen and approved the final version of the paper and have agreed to its submission for publication. hazards and human or animal subjects if the work involves chemicals, procedures or equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 7, issue 2, may 2022 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia subanar seno, gadjah mada university, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740595') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740557') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740556') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740541') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/736347') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5964') elliptical orbits mode application for approximation of fuel volume change cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 316-331 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 19, 2021 reviewed: january 09, 2022 accepted: january 10, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.14407 elliptical orbits mode application for approximation of fuel volume change jovian dian pratama*, ratna herdiana*, susilo hariyanto department of mathematics, diponegoro university *corresponding author email: joviandianpratama@yahoo.com*, herdiana.math@gmail.com*, sus2_hariyanto@yahoo.com abstract at a 45.507.21 candirejo tuntang gas station, it is difficult to ensure the stock of fuel supplies because there is always a difference between calculations using dipsticks and fuel dispensers. because the calculation method used by gas stations throughout indonesia is linear interpolation which is not smooth, then by using the pertalite (pertamina fuel products) measuring book data a smooth volume change approximation function will be formed. this article presents the elliptical orbits mode (eom) as a proposed method in approximating the function that describes the volume change of fuel with respect to fuel height in underground tank (ut). since the calculation by the gas station is not smooth, it is necessary for a smoother data fitting by considering residual square error (rss) and mean square error (mse). the results of the elliptical orbits mode approximation will be compared with the circle orbits mode and least square data fitting. the result show that eom(θ) method with elliptical height control produces smaller rss and mse compared to using com, eom, least square degree two and three. in next research, the approximation results will be applied to the fuel dispenser data. keywords: orbits mode; data fitting; ellipse; fuel; approximation introduction based on the assumptions given in [1] and [2] the previous orbits mode data fitting research which was used to calibrate the dipstick measuring instrument that converts the height to the volume of fuel in the buried tank, it is explained that the approximation function of the change in fuel volume in the tank is only based on height. in [1] and [2] it is explained that the orbits mode data fitting-based calibration is limited by several assumptions and field conditions, including the following: 1. the resulting approximation function is the change in the volume of fuel in the ut which only depends on the variable height of the fuel in the ut. 2. orbits mode data fitting proposed by the author is used only for the distribution of data that forms a semicircle or ellipse in the first quadrant. 3. the data to be approximated is the fuel measurement manual in the ut from the semarang regency metrology agency. http://dx.doi.org/10.18860/ca.v7i1.14407 mailto:joviandianpratama@yahoo.com elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 317 4. it is assumed that ut is not tilted or flat during measurement, including tank trucks that deliver fuel to gas stations or are filling supplies at gas stations. 5. ut for the right and left have the same shape alias symmetrical, due to the second assumption. in [3] data fitting is applied to approximate the shape of an island on the map, then in [4] the curve fitting which is used to detect enlarged and shrinking eye retina, research in [3] and [4] has their additional algorithm to approach the desired result, which will also be applied to the orbits mode data fitting starting from the proposed method and the object applied to calibration is something new. in [5] hyper least square or hyperls was also introduced and [6] also calibrated data in the form of curves but using an orthogonal matrix where the more data the more complicated, so the method will be difficult for large data. from [7] there is a design drawing of a buried tank where the tank is in the form of a capsule tube with a cross section that is not flat or protruding so that according to [8] also, changes in volume in the tank tend to form a semicircle or half an ellipse or a parabola. the approximation function used by gas stations throughout indonesia is linear interpolation which is not smooth, then by using the pertalite (pertamina fuel products) measuring book data a smooth volume change approximation function with elliptical orbits mode will be formed, and then will be any improvement on ellips height control to minimize residual sum of square (rss) and mean square error (mse), where the data used is the change in the volume of fuel in the tank based on changes in the height of the fuel in the ut in units (cm) and will be converted to fuel volume (liter). therefore, the author proposes method because the calculation is simpler for small and large data and is smoother, although only for data that tends to be semicircular or elliptical, to approximate the fuel volume with minimized errors. orbits mode data fitting is a method proposed by the author in approximating the function of the data which tends to be in the form of a semi-circle or half an ellipse. in [1] and [2] the author introduced the orbits mode data fitting method only in a circle shape, then compared it with cubic spline interpolation and least square data fitting, but this time the authors made the orbits mode data fitting method in the shape of an ellipse too, because in the value approach there is a volume of fuel which has not been detected in the function. definition 1 (ellipse equation) in [9] the ellipse equation is presented in equation (1) which (𝑝,𝑞) is the center point of the ellipse with the major and minor axes adjusting 𝑎 and 𝑏, (𝑥 −𝑝)2 𝑎2 + (𝑦 −𝑞)2 𝑏2 = 1 (1) definition 2 (𝑨𝒊 set for ellipse mode) a set 𝐴𝑖 of points formed from two ellipse equations is defined as follows, 𝐴𝑖 = {(𝑥,𝑦) | 𝑑𝑖1 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑𝑖2 }, (2) with 𝑖 = 1,2,…,𝑘. the set (2) it can be visualized as follows, elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 318 figure 1. set visualization 𝐴𝑖 (2) with 𝑥𝑤 2 = (𝑥 − 𝑤 2 ) and 𝑤 = height of ut and 𝑙 = a half of maximum fuel change in ut defined 𝑤 2 (√𝑑𝑖2 −√𝑑𝑖1) = 𝑙 2 (√𝑑𝑖2 −√𝑑𝑖1) = 𝑡𝑖 the thickness of the ellipse from the partition interval taken from the maximum and minimum values of the volume change [δ𝑉(ℎ)𝑚𝑖𝑛,δ𝑉(ℎ)𝑚𝑎𝑥] will be divided by several partitions where the thickness with the most points is sought, then [δ𝑉(ℎ)𝑚𝑖𝑛,δ𝑉(ℎ)𝑚𝑎𝑥] = [𝑑11,𝑑12]∪ [𝑑21,𝑑22]∪ [𝑑31,𝑑32] ∪…∪[𝑑𝑘1,𝑑𝑘2] (3) with 𝑑𝑖2 = 𝑑(𝑖+1)1 and the intersection of the respective sub-blankets of the minimum and maximum volume intervals in equation (3) is denoted for each [𝑑𝑖1,𝑑𝑖2]∩ [𝑑(𝑖+1)1,𝑑(𝑖+1)2] is equal to 𝑑𝑖2 or 𝑑(𝑖+1)1, where 𝑖 = 1,2,…,𝑘 with 𝑘 is the number of blankets dividing the maximum and minimum intervals of the volume change. definition 3 (partition of 𝑨𝒊 set) partition of 𝐴𝑖 set that divide 𝐴𝑖 set to become partitions or sets of points between 2 ellipses equations, defined as follows, 𝐴1 = {(𝑥,𝑦) | 𝑑11 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑12 } 𝐴2 = {(𝑥,𝑦) | 𝑑21 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑22 } 𝐴3 = {(𝑥,𝑦) | 𝑑31 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑32 } … (4) elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 319 𝐴𝑘 = {(𝑥,𝑦) | 𝑑𝑘1 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑𝑘2 } set of partitions (4) can be described as follows, figure 2. the visualization of partitions set 𝐴1,𝐴2,𝐴3 up to 𝐴𝑘 (4) definition 4 (elliptical orbits mode) elliptical orbits mode was choose partition has the most points, defined as follows: 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑀𝑎𝑥(𝑛(𝐴1),𝑛(𝐴2),𝑛(𝐴3),…,𝑛(𝐴𝑘)) (5) with 𝑖 = 1,2,…,𝑘. if there is a condition where 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴𝑘1) = ⋯ = 𝑛(𝐴𝑘𝑚), then the average ellipses scale 𝐴𝑘1,𝐴𝑘2,…,𝐴𝑘𝑚 is taken so 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴𝑚 ̅̅ ̅̅ ), therefore, the inequality whose ellipse will change is defined 𝐴𝑚̅̅ ̅̅ as follows, 𝐴𝑚̅̅ ̅̅ = {(𝑥,𝑦) |( 𝑑𝑘11 +⋯+𝑑𝑘𝑚1 𝑚 ) < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < ( 𝑑𝑘12 +⋯+𝑑𝑘𝑚2 𝑚 ) } the next step, because we have obtained 𝐴𝑖 or 𝐴𝑚̅̅ ̅̅ , then we approximate ellipse equation, divided which can be devide into two cases: case 1, [𝑴𝒂𝒙(𝒏(𝑨𝒊)) = 𝒏(𝑨𝒎)] based on the set with the maximum number of points between 2 ellipses, choose 𝐴𝑚 = {(𝑥,𝑦) | 𝑑𝑚1 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑𝑚2 }, so that we get: 𝑑𝑚1 < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < 𝑑𝑚2 ⟹ ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 = ( 𝑑𝑚1 +𝑑𝑚2 2 ) ⟹ 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 (6) elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 320 the steps in (6) can be visualized as follows, figure 3. visualization steps in (6) therefore, from (6) the result of the elliptical orbital mode is a semicircular function by substituting 𝑥𝑤 2 = 𝑥𝑑𝑚, as follows: 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑥𝑑𝑚 ( 𝑤 2 ) ) 2 (7) with 𝑥 = the fuel level in ut and 𝑥𝑑𝑚 = 𝑥 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 . theorem 1 (defined intervals for elliptical orbits mode approximation function) if 𝑥𝑤 2 = 𝑥𝑑𝑚 is substituted to 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1+𝑑𝑚2 2 ) −( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 then 𝑦 is defined in ℝ in the interval 0 ≤ 𝑥 ≤ 𝑤 2 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 . proof: substitute 𝑥𝑤 2 = 𝑥𝑑𝑚 to 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1+𝑑𝑚2 2 ) −( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 so that it is obtained: 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑥𝑑𝑚 ( 𝑤 2 ) ) 2 ⟺ 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) − ( 𝑥 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 ( 𝑤 2 ) ) 2 ⟺ 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) − ( 𝑥2 −𝑤𝑥√ 𝑑𝑚1+𝑑𝑚2 2 +( 𝑤 2 ) 2 ( 𝑑𝑚1+𝑑𝑚2 2 ) ( 𝑤 2 ) 2 ) elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 321 ⟺ 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑑𝑚1 +𝑑𝑚2 2 ) − ( 𝑥2 −𝑤𝑥√ 𝑑𝑚1+𝑑𝑚2 2 ( 𝑤 2 ) 2 ) ⟺ 𝑦 = ( 𝑙 2 )√ 𝑥(𝑤√ 𝑑𝑚1+𝑑𝑚2 2 −𝑥) ( 𝑤 2 ) 2 ⟺ 𝑦 = ( 𝑙 𝑤 )√𝑥(𝑤√ 𝑑𝑚1 +𝑑𝑚2 2 −𝑥) with 𝑤,𝑑𝑚1,𝑑𝑚2 ∈ ℝ +. obviously 𝑦 is defined in ℝ since 𝑥(𝑤√ 𝑑𝑚1+𝑑𝑚2 2 −𝑥) ≥ 0, then 𝑥 must be on both interval 0 ≤ 𝑥 ≤ 𝑤√ 𝑑𝑚1+𝑑𝑚2 2 and 0 ≤ 𝑥 ≤ 𝑤 2 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 , because 𝑤√ 𝑑𝑚1+𝑑𝑚2 2 > 𝑤 2 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 . ∎ substitute 𝑥𝑤 2 = 𝑥𝑑𝑚 so that the value 𝑦 is defined in 0 ≤ 𝑥 ≤ 𝑤 2 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 , so that (7) the function of volume change to fuel level in the ut can be visualized as follows, figure 4. visualization of steps in (6) to (7) the function is formed from the half ellipse selected for the calibration of the buried tank whose cross-sectional area is circular but convex, so that the volume of ut will be calculated as a function of the change in volume with respect to height, which then the coordinates are taken from the change in units (cm) that will be converted to volume per centimeter (liter/cm). elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 322 case 2, [𝑴𝒂𝒙(𝒏(𝑨𝒊)) = 𝒏(𝑨𝒌𝟏) = ⋯ = 𝒏(𝑨𝒌𝒎)] based on the set with the maximum number of points between two ellipses equations, 𝐴𝑚̅̅ ̅̅ = {(𝑥,𝑦) |( 𝑑𝑘11+⋯+𝑑𝑘𝑚1 𝑚 ) < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < ( 𝑑𝑘12+⋯+𝑑𝑘𝑚2 𝑚 ) }, then: ( 𝑑𝑘11 +⋯+𝑑𝑘𝑚1 𝑚 ) < ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 < ( 𝑑𝑘12 +⋯+𝑑𝑘𝑚2 𝑚 ) ⟹ ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 = ( ( 𝑑𝑘11+⋯+𝑑𝑘𝑚1 𝑚 )+( 𝑑𝑘12+⋯+𝑑𝑘𝑚2 𝑚 ) 2 ) ⟹ ( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 +( 𝑦 ( 𝑙 2 ) ) 2 = ( 𝑑𝑚1 +𝑑𝑚2 2 ) . (8) the steps in (8) are visualized similarly in figure 11 with 𝑑𝑚1 = ( 𝑑𝑘11+⋯+𝑑𝑘𝑚1 𝑚 ) and 𝑑𝑚2 = ( 𝑑𝑘12+⋯+𝑑𝑘𝑚2 𝑚 ), then ⟹ 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑥𝑤 2 ( 𝑤 2 ) ) 2 . (9) therefore, the result of the orbital mode ellipse is a half-ellipse function, as follows: 𝑦 = ( 𝑙 2 )√( 𝑑𝑚1 +𝑑𝑚2 2 ) −( 𝑥𝑑𝑚 ( 𝑤 2 ) ) 2 , (10) with 𝑥 = is the fuel level in ut and 𝑥𝑑𝑚 = 𝑥 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 . as stated theorem 1 about defined interval for (10) substituted 𝑥𝑤 2 = 𝑥𝑑𝑚 so that 𝑦 is defined values since 0 ≤ 𝑥 ≤ 𝑤 2 −( 𝑤 2 )√ 𝑑𝑚1+𝑑𝑚2 2 , we get (10) the function of the change in volume to the fuel level in the ut which is visualized similarly in figure 4. methods research steps 1. construction of mathematical model elliptical orbits mode methods with the following steps: a) construction of mathematical model elliptical orbits mode.  review for ellipse equation  define 𝐴𝑖 set for ellipse mode  make partition for 𝐴𝑖 set (definition)  choose partitions of set 𝐴𝑖 with 𝑛(𝐴𝑖) is the maximum value  divide onto two cases singular and plural maximum value elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 323  create function 𝑦 = 𝑓(𝑥) from chosen 𝐴𝑖 sets  translate 𝑦 = 𝑓(𝑥) so that 𝑦 is defined on 0 and so on, and  controlling height to find ellipse’s height or vertical axis that minimized residual sum of square (rss) and mean square error (mse). 2. construction of mathematical model elliptical orbits mode will be applied on data from candirejo gas station, measuring book from government metrology agency specific for pertalite (fuel product of pertamina) only and visualize it. 3. measuring performance using rss [10] and mse [11,12] and compare with circle orbits mode from [1], least square with n = 2 and n = 3 from [13], and elliptical with height control. results and discussion pertalite measuring book data a calculation of gas station 45.507.21 candirejo using a measuring book from government metrology agency to determine the volume of fuel in the buried tank, so the authors are just obtained the following data which is not proceed by authors, we need this data from metrologi agency as constructor of approximation function, as follows: table 1. fuel volume measuring book data for pertalite tanks from metrology agency height (x) volume diff (y) height (x) volume diff (y) height (x) volume diff (y) 0 0.0 0.0 75 7085.9 117.7 150 16124.4 111.1 1 237.1 237.1 76 7203.5 117.6 151 16235.6 111.2 2 294.3 57.2 77 7321.2 117.7 152 16346.7 111.1 3 351.4 57.1 78 7438.8 117.6 153 16457.8 111.1 4 409.4 58.0 79 7556.5 117.7 154 16568.9 111.1 5 468.2 58.8 80 7674.1 117.6 155 16680.0 111.1 6 527.1 58.9 81 7791.8 117.7 156 16791.1 111.1 7 586.1 59.0 82 7909.4 117.6 157 16902.2 111.1 8 646.7 60.6 83 8027.1 117.7 158 17013.3 111.1 9 707.3 60.6 84 8144.7 117.6 159 17124.4 111.1 10 767.9 60.6 85 8262.4 117.7 160 17232.6 108.2 11 831.6 63.7 86 8380.0 117.6 161 17337.9 105.3 12 896.1 64.5 87 8497.6 117.6 162 17443.2 105.3 13 960.6 64.5 88 8615.3 117.7 163 17548.4 105.2 14 1026.7 66.1 89 8732.9 117.6 164 17653.7 105.3 15 1093.3 66.6 90 8855.0 122.1 165 17758.9 105.2 16 1160.0 66.7 91 8980.0 125.0 166 17864.2 105.3 17 1228.3 68.3 92 9105.0 125.0 167 17969.5 105.3 18 1297,2 68.9 93 9230.0 125.0 168 18065.7 96.2 19 1366.2 69.0 94 9355.0 125.0 169 18161.0 95.3 20 1439.3 73.1 95 9480.0 125.0 170 18256.2 95.2 21 1513.3 74.0 96 9605.0 125.0 171 18351.4 95.2 22 1588,0 74.7 97 9730.0 125.0 172 18446.7 95.3 23 1668.0 80.0 98 9855.0 125.0 173 18541.9 95.2 elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 324 height (x) volume diff (y) height (x) volume diff (y) height (x) volume diff (y) 24 1748.0 80.0 99 9980.0 125.0 174 18637.1 95.2 25 1830.0 82.0 100 10105.0 125.0 175 18732.4 95.3 26 1913,3 83.3 101 10230.0 125.0 176 18825.5 93.1 27 1997,4 84.1 102 10355.0 125.0 177 18916.4 90.9 28 2084,3 86.9 103 10480.0 125.0 178 19007.3 90.9 29 2171.3 87.0 104 10605.0 125.0 179 19098.2 90.9 30 2261.8 90.5 105 10730.0 125.0 180 19189.1 90.9 31 2352.7 90.9 106 10855.0 125.0 181 19280.0 90.9 32 2446.7 94.0 107 10980.0 125.0 182 19370.9 90.9 33 2541.9 95.2 108 11105.0 125.0 183 19458.3 87.4 34 2637.1 95.2 109 11230.0 125.0 184 19545.2 86.9 35 2732.4 95.3 110 11355.0 125.0 185 19632.2 87.0 36 2830.0 97.6 111 11480.0 125.0 186 19719.1 86.9 37 2930.0 100.0 112 11605.0 125.0 187 19804.0 84.9 38 3030.0 100.0 113 11730.0 125.0 188 19884.0 80.0 39 3130.0 100.0 114 11855.0 125.0 189 19964.0 80.0 40 3230.0 100.0 115 11980.0 125.0 190 20044.0 80.0 41 3330.0 100.0 116 12105.0 125.0 191 20124.0 80.0 42 3430.0 100.0 117 12230.0 125.0 192 20202.2 78.2 43 3530.0 100.0 118 12355.0 125.0 193 20276.3 74.1 44 3630.0 100.0 119 12480.0 125.0 194 20350.4 74.1 45 3730.0 100.0 120 12605.0 125.0 195 20420.0 69.6 46 3832.6 102.6 121 12730.0 125.0 196 20486.7 66.7 47 3937.9 105.3 122 12855.0 125.0 197 20553.3 66.6 48 4043,2 105.3 123 12980.0 125.0 198 20617.5 64.2 49 4148.4 105.2 124 13105.0 125.0 199 20680.0 62.5 50 4257.8 109.4 125 13227.1 122.1 200 20742.5 62.5 51 4368.9 111.1 126 13344.7 117.6 201 20802.9 60.4 52 4480.0 111.1 127 13462.4 117.7 202 20860.0 57.1 53 4591.1 111.1 128 13580.0 117.6 203 20917.1 57.1 54 4702.2 111.1 129 13697.6 117.6 204 20974.3 57.2 55 4813.3 111.1 130 13815.3 117.7 205 21028.6 54.3 56 4924.4 111.1 131 13932.9 117.6 206 21082.7 54.1 57 5035,6 111.2 132 14050.6 117.7 207 21136.8 54.1 58 5146.7 111.1 133 14168,2 117.6 208 21187.1 50.3 59 5257.8 111.1 134 14285.9 117.7 209 21222.9 35.8 60 5368.9 111.1 135 14403.5 117.6 210 21258.6 35.7 61 5480.0 111.1 136 14521.2 117.7 211 21294.3 35.7 62 5591.1 111.1 137 14638.8 117.6 212 21330.0 35.7 63 5702.2 111.1 138 14756.5 117.7 213 21365,7 35.7 64 5813.3 111.1 139 14874.1 117.6 214 21387.1 21.4 65 5924.4 111.1 140 14991.8 117.7 215 21399.1 12.0 66 6035.6 111.2 141 15109.4 117.6 216 21411.0 11.9 67 6146.7 111.1 142 15227.1 117.7 217 21422.9 11.9 68 6262.4 115.7 143 15344.7 117.6 218 21434.8 11.9 elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 325 69 6380.0 117.6 144 15457.8 113.1 219 21446.7 11.9 70 6497.6 117.6 145 15568.9 111.1 220 21458.6 11.9 71 6615.3 117.7 146 15680.0 111.1 221 21470.5 11.9 72 6732.9 117.6 147 15791.1 111.1 222 21482.4 11.9 73 6850.6 117.7 148 15902.2 111.1 222.3 21486.0 3.6 74 6968.2 117.6 149 16013.3 111.1 approximation using elliptical orbits mode (eom) table 1 is used as sample data; we derive elliptical orbits mode approach as an approximation to the change of fuel volume. to obtain a half-ellipse function from the smallest to the largest abscissa, choose, 𝑤 2 = maximum height on underground tank 2 = 222.3 2 = 111.15 𝑙 2 = maximum volume change on undergorund tank = 125 after that, select the difference 𝑑𝑖1 and 𝑑𝑖2 for the prefix of the 𝐴𝑖 set which is 𝑡𝑖 = 0.2 defined as follows, 𝐴𝑖 = {(𝑥,𝑦) |0.8 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 1} with step-size = 0.02 then the number of partitions is obtained 𝑡𝑖 𝑠𝑡𝑒𝑝−𝑠𝑖𝑧𝑒 = 10, so that the partitions are obtained from the 𝐴𝑖set with 𝑖 = 1,2,…,10, 𝐴1 = {(𝑥,𝑦) | 0.98 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 1} 𝐴2 = {(𝑥,𝑦) | 0.96 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.98} 𝐴3 = {(𝑥,𝑦) | 0.94 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.96} 𝐴4 = {(𝑥,𝑦) | 0.92 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.94} 𝐴5 = {(𝑥,𝑦) | 0.90 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.92} 𝐴6 = {(𝑥,𝑦) | 0.88 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.90} 𝐴7 = {(𝑥,𝑦) | 0.86 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.88} 𝐴8 = {(𝑥,𝑦) | 0.84 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.86} 𝐴9 = {(𝑥,𝑦) | 0.82 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.84} 𝐴10 = {(𝑥,𝑦) | 0.80 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.82} elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 326 accordingly, the value of 𝑛(𝐴𝑖) for 𝑖 = 1,2,…,10 is provided in table 2. table 2. calculation of elliptical orbits mode𝑛(𝐴𝑖) 𝑨𝒊 𝒏(𝑨𝒊) 𝐴1 11 𝐴2 15 𝐴3 16 𝐴4 26 𝐴5 25 𝐴6 19 𝐴7 9 𝐴8 3 𝐴9 0 𝐴10 0 according to table 2 it is obtained that 𝑀𝑎𝑥(𝑛(𝐴𝑖)) = 𝑛(𝐴4), with 𝑖 = 1,2,…,10 the selected 𝐴4 set , after that from the 𝐴4 set the following functions 𝑦 = 𝑓(𝑥) will be formed, 0.92 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.94 ⇒ (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 = ( 0.92+0.94 2 ) 0.92 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.94 ⇒ (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 = 0.93 0.92 < (𝑥 −111.15)2 (111.15)2 + 𝑦2 (125)2 < 0.94 ⇒ 𝑦 = 125√0.93− (𝑥 −111.15)2 (111.15)2 . then the translation 𝑦 = 𝑓(𝑥) to be defined at 0 ≤ 𝑥 ≤ 111.15−111.15√0.93 or around 0 ≤ 𝑥 ≤ 3,96, substitution 𝑥𝑤 2 = (𝑥 −111.5) with 𝑥𝑑𝑚 = (𝑥 −111.15√0.93), so that we get: 𝑦 = 125√0.93− (𝑥 −111.15√0.93) 2 (111.15)2 (11) with 𝑦 = change in the volume of fuel with respect to fuel height, x., for visualization of the graph of changes in the volume of fuel obtained: figure 5. graph of change in fuel volume by eom elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 327 based on figure 5, the eom results produce a function that is fit to the pertalite volume change data, the volume as function of height (h) is then obtained by the following integration: 𝑉(ℎ) = ∫125√0.93− (𝑥 −111.15√0.93) 2 (111.15)2 ℎ 0 𝑑𝑥 using the help of maple 2015 the integral results are obtained as follows, 𝑉(ℎ) = 535945891 800000000000000000 √18792746045928961 + 116250000 17040701 √896879arcsin( 10182971929 93000000000000 √896879√93) + 1 160000000000 (−8094332975000000000000 ℎ2 +1735249799334822290000000 ℎ +18792746045928961) 1 2 ℎ − 535945891 800000000000000000 (−8094332975000000000000 ℎ2 +1735249799334822290000000 ℎ +18792746045928961) 1 2 + 116250000 17040701 √896879arcsin( 19 93000000000000 √896879√93(5000000 ℎ −535945891)); (12) where 𝑉(ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3√0.93. the eom version of the fuel volume calculation uses (12) with 𝑉𝑚𝑎𝑥 = 20.296.55 liters. elliptical orbits mode with elliptical height control on data pertalite based on figure 5, it can be seen that the volume change function according to eom will regress more pertalite data if the ellipse height is higher, so it is necessary to adjust the ellipse height. the eom result at (11) has an elliptical height 125 which represents the equation so that it has a volume change function with respect to the fuel level in the ut, with the general form of (11): 𝐸𝑂𝑀(𝜃,𝑥) = 𝜃 ∙√0.93− (𝑥 −111.15√0.93) 2 (111.15)2 (13) with 𝜃 is the height of the ellipse, defined on the interval 0 ≤ 𝑥 ≤ 222.3√0.93. choose 𝜃 the one that minimizes the residual sum square 𝐸 = ∑ (𝑦𝑖 −𝐸𝑂𝑀(𝜃,𝑥𝑖)) 2 ⌊222.3√0.93⌋ 𝑖=1 𝐸 = ∑(𝑦𝑖 2 −2𝐸𝑂𝑀(𝜃,𝑥𝑖)𝑦𝑖 +𝐸𝑂𝑀 2(𝜃,𝑥𝑖)) 214 𝑖=1 elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 328 𝐸 = ∑𝑦𝑖 2 214 𝑖=1 −2∑𝐸𝑂𝑀(𝜃,𝑥𝑖)𝑦𝑖 214 𝑖=1 +∑(𝐸𝑂𝑀(𝜃,𝑥𝑖)) 2 214 𝑖=1 𝐸 = ∑𝑦𝑖 2 214 𝑖=1 −2∑𝑦𝑖 ∙ 𝜃 √0.93− (𝑥 −111.15√0.93) 2 (111.15)2 214 𝑖=1 +∑(𝜃√0.93− (𝑥 −111.15√0.93) 2 (111.15)2 ) 2 214 𝑖=1 𝐸 = ∑𝑦𝑖 2 214 𝑖=1 −2∑𝑦𝑖 ∙ 𝜃 √0.93− (𝑥 −111.15√0.93) 2 (111.15)2 214 𝑖=1 +∑𝜃2 (0.93− (𝑥 −111.15√0.93) 2 (111.15)2 ) 214 𝑖=1 then, to find 𝜃 the minimization of 𝐸, find the solution of the equation 𝜕𝐸 𝜕𝜃 = 0, we get: ⟺ −2∑𝑦𝑖 √0.93− (𝑥 −111.15√0.93) 2 (111.15)2 214 𝑖=1 +2𝜃∑(0.93− (𝑥 −111.15√0.93) 2 (111.15)2 ) 214 𝑖=1 = 0 ⟺ −2∑ 𝑦𝑖√0.93− (𝑥−111.15√0.93) 2 (111.15)2 214 𝑖=1 +2𝜃∑ (0.93− (𝑥−111.15√0.93) 2 (111.15)2 )214𝑖=1 2√0.93− (𝑥−111.15√0.93) 2 (111.15)2 = 0 ⟺ −∑𝑦𝑖 214 𝑖=1 +𝜃∑√0.93− (𝑥 −111.15√0.93) 2 (111.15)2 214 𝑖=1 = 0 ⟺ 𝜃 = ∑ 𝑦𝑖 214 𝑖=1 ∑ √0.93− (𝑥−111.15√0.93) 2 (111.15)2 214 𝑖=1 by using the data in table 1, it is obtained 𝜃 ≈ 130,37 that from (27) it is obtained: 𝐸𝑂𝑀(130,37,𝑥) = 130,37 ∙√0.93 − (𝑥 − 111.15√0.93) 2 (111.15)2 and visualized the graph of the function of the change in fuel volume and the actual fuel volume change in the reservoir as follows: figure 6. graph of changes in fuel volume by 𝐸𝑂𝑀(𝜃) elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 329 based on figure 6 𝑦 = 𝑓(𝑥) the 𝐸𝑂𝑀(𝜃)results produce a function that is more fit to the pertalite data than the eom results in figure 5, then we calculate the volume function 𝑉(ℎ) with integral and with (24) obtained: 𝑉𝜃(ℎ) = 130,37 125 ∙𝑉(ℎ) where .𝑉𝜃(ℎ) is defined on the interval 0 ≤ ℎ ≤ 222.3. calculation of the volume of the fuel 𝐸𝑂𝑀(𝜃) version has 𝑉𝑚𝑎𝑥 = 21.166,71 liters. comparison of approximate visualization results data visualization results from circle orbits mode, elliptical orbits mode, 𝐸𝑂𝑀(𝜃), least square data fitting 𝑛 = 2, and least square data fitting 𝑛 = 3, as follows: figure 7. comparison graph of approximation method results the results of the calculation of changes in the volume of fuel based on the height of the fuel in the ut will be applied to the pertalite data to search for rss and mse from each result. pertalite data approximation rss and mse calculation comparison of the results between the proposed method and other methods can be seen from the calculation of rss and mse with liter unit in table 3 below: table 3. pertalite data approximation rss and mse calculation method rss mse 𝐶𝑂𝑀 40.390,49 185,28 𝐸𝑂𝑀 8.529,37 39,67 𝐿𝑆(𝑛 = 2) 8.980,63 40,09 𝐿𝑆(𝑛 = 3) 7.574,51 33,81 𝐸𝑂𝑀(𝜃) 6.415,32 29,84 based on table 3 the calculation of com which has the largest rss and mse, for eom has rss and mse which is slightly below 𝐿𝑆(𝑛 = 2), but still above 𝐿𝑆(𝑛 = 3), then by controlling the height of the ellipse to find 𝜃 that minimizes the rss we obtain the minimum i.e. 6.415,32 and its mse 29,84. the smallest value rss and mse are obtained elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 330 when using 𝐸𝑂𝑀(𝜃). pertalite data approximation is not compared to the calculation of gas stations and cubic spline interpolation because the rss and mse are definitely 0 and have unsmooth approximation. defined domain interval and maximum volume in the orbits mode data fitting construction, there is a reduction in the bbm altitude domain in the ut, so that it is only defined at a certain height. the results of the comparison of the defined domain height and the maximum volume of each approximation method are as follows: table 4. domain height intervals and maximum volume approximation method approximation method domain height (cm) 𝑽𝒎𝒂𝒙 maximum volume (liter) gas station calculation 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.486.00 liter. circle orbits mode 0 ≤ ℎ ≤ 217,1 𝑉𝑚𝑎𝑥 = 18.508,85 liter. elliptical orbits mode 0 ≤ ℎ ≤ 222,3√0,93 𝑉𝑚𝑎𝑥 = 20.296,55 liter. 𝐸𝑂𝑀(𝜃) 0 ≤ ℎ ≤ 222,3√0,93 𝑉𝑚𝑎𝑥 = 21.166,04 liter. least square 𝑛 = 2 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.248,90 liter. least square 𝑛 = 3 0 ≤ ℎ ≤ 222,3 𝑉𝑚𝑎𝑥 = 21.256,66 liter. the calculation result of circle orbits mode is only defined to altitude 217,1 cm there is a reduction of 5,2 cm and elliptical orbits mode is only defined to a height of 222,3√0,93 cm or about 214,382 cm, there is a reduction of (222,3−222,3√0,93) cm or about 7,92 cm. for further research, this has no effect on the application of daily sales data (according to dispenser) if the maximum height of fuel data is below 214,38 cm. so that the value is defined for all data as well as each approximation method as well. approximation results will be validated by measuring mean average deviation (mad) based on [14] and then mean absolute percentage error (mape) based on [15]. if aproximation results has mape below on 10% then aproximation methods is very feasible. conclusions based on the results and discussion, it can be concluded that the method of approximating the pertalite data with the smallest rss and mse is 𝐸𝑂𝑀(𝜃) by 𝜃 ≈ 130,37, resulting in rss and mse respectively are 6.415,32 and 29,81. 𝐸𝑂𝑀(𝜃) also produces a more fit half-ellipse function than other approximation methods. the results of the comparison of the approximation of the pertalite data are compared with 𝐶𝑂𝑀, 𝐸𝑂𝑀, 𝐿𝑆(𝑛 = 2), and 𝐿𝑆(𝑛 = 3) although 𝐸𝑂𝑀(𝜃) produces rss and mse, which are smaller than other methods, there is a reduction in the altitude domain and has a different maximum volume compared to the calculation of gas stations. according to the gas station metrology measurement book, the height of the ut is 222,3 cm and has a maximum volume of 21.486 liters, but 𝐸𝑂𝑀(𝜃) only detects the volume of fuel up to a height of about 214,1 cm and the maximum volume is below the calculation of the gas station. the author hopes for the development of this research, applied to different types of fuel such as pertamax and dexlite. as well as for a more real problem under study, use data on changes in the height and volume of bbm based on daily sales according to the bbm dispenser which must first be tested for the accuracy of the bbm dispenser used. as well as calculating errors using mape, mad, and other error calculations. elliptical orbits mode application for approximation of fuel volume change jovian dian pratama, ratna herdiana, susilo hariyanto 331 references [1] pratama, j.d., herdiana, r., hariyanto, s., application of orbits mode data fitting for dipstick calibration of altitude measuring instruments into volume of fuel oil in the tank compared with cubic spline interpolation, snast – national seminar on application of science and technology, akprind – yogyakarta, 138 146, 2021 [2] pratama, j.d., herdiana, r., hariyanto, s., application of orbits mode data fitting for dipstick calibration. journal of mathematics education: judika education, 4(2), 107118., 2021. 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"mean absolute percentage error for regression models", neurocomputing 2016 arxiv:1605.02541. 2015. issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 cauchy vol. 6 no. 4 pages: 162 – 308 malang may 2021 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 editorial board ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 6, issue 4, may 2021 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. 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2086-0382 e-issn : 2477-3344 table of contents selection of specialization class using support vector machine (svm) method in sekolah menengah atas negeri 1 ambon ........................................................................ 162 – 168 optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm ....................... 169 – 180 learning interest of poliwangi students to learn mathematics engineering through moocs using dummy regression .................................................................... 181 – 187 stability analysis of hiv/aids model with educated subpopulation ......................... 188 – 199 trace of positive integer power of squared special matrix ............................................ 200 – 211 distance and areas weighting of gwr kriging for stunting cases in east java ..... 212 – 217 spatio temporal modelling for government policy the covid-19 pandemic in east java ........................................................................................................................................ 218 – 226 dynamical of ratio-dependent eco-epidemical model with prey refuge................. 227 – 237 poverty in central java using multivariate adaptive regression splines and bootstrap aggregating multivariate adaptive regression splines ........................ 238 – 245 invertibility of generalized space-time autoregressive model with random weight ............................................................................................................................................ 246 – 259 analysis of the rosenzweig-macarthur predator-prey model with anti-predator behavior ........................................................................................................................................ 260 – 269 bayesian generalized self method to estimate scale parameter of invers rayleigh distribution ............................................................................................................... 270 – 278 strongly summable vector-valued sequence spaces defined by 2-modular .......... 279 – 285 modeling plant stems using the deterministic lindenmayer system ........................ 286 – 295 regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) .......................................................................... 296 – 308 the metric dimension and local metric dimension of relative prime graph cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 149-161 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 06, 2020 reviewed: october 07, 2020 accepted: 10 november 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.10103 the metric dimension and local metric dimension of relative prime graph inna kuswandari1, fatmawati2, mohammad imam utoyo3 mathematics department, faculty of science and technology, universitas airlangga, mulyorejo, surabaya 60115, indonesia. email: ikuswandari94@gmail.com, fatma47unair@gmail.com, m.i.utoyo@fst.unair.ac.id abstract this study aims to determine the metric dimensions and local metric dimensions of relative prime graphs formed from modulo 𝑛 integer rings, namely 𝐺ℤ𝑛 with 𝑛 ≥ 2. as a vertex set is ℤ𝑛 ∖{0} and 𝑢𝑣 ∈ 𝐺ℤ𝑛 if 𝑢 and 𝑣 are relatively prime. by finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of 𝐺ℤ𝑛 are 𝑛 − 2+ 𝑘 and |𝑃0|− 1+ 𝑘 respectively, where 𝑘 is the number of vertices groups that formed multiple 2,3, … , 𝑘 and |𝑃0| is the cardinality of set 𝑃0. this research can be developed by determining the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings. key words : metric dimension; modulo 𝑛; relative prime graph; resolving set; rings. introduction graph 𝐺 is defined as a non-empty and finite set of 𝑉(𝐺) whose elements are called vertices and set 𝐸(𝐺) (maybe empty) whose elements are called edges which are the unordered pair of different vertices in 𝑉(𝐺) [1]. any problems whose objects can be described as vertices and edges can be solved by the concept of graph theory, this has become one of the supporting factors in the field of graph theory research developing very rapidly today. the notion of metric dimensions was introduced firstly by [2] and independently by [3] (in [4]). in [3], the concepts of bases and dimensions have been built on the graph. a bases on a graph is a set of vertices with minimal cardinality that implies in each vertex of graph having a different representation (resolving set) to the bases, while the dimensions are number elements of the bases. meanwhile, the local metric dimension was introduced by [5] by considering different representations of two adjacent vertices so that the local metric dimension of a graph is obtained. the research related to the metric dimensions and local metric dimensions of a graph has been carried out by many researchers no exception to the graph of the results of operations (comb, corona, joint, etc.). the development of research in the field of graph theory is also supported by the expansion of research objects in algebraic systems, namely groups or rings. in [6], the zero divisor graph is introduced from any commutative ring by defining the vertices on the graph are elements of the ring, while the two vertices are adjacent if the product is zero. by using http://dx.doi.org/10.18860/ca.v6i3.10103 mailto:ikuswandari94@gmail.com mailto:fatma47unair@gmail.com mailto:utoyo@fst.unair.ac.id the metric dimension and local metric dimension of relative prime graph fatmawati 150 the definition in [6], the research was also carried out in [7,8,9] and succeeded in finding several properties related to the diameter, girth, isomorphism of the graph, radius, and domination set of zero divisor graphs constructed from the commutative and non commutative rings. in addition to the zero divisor of graph, research has also been developed on the jacobson graph formed from commutative rings that have nonzero unit elements [10,11]. the definition of vertices and edges is built from the radical jacobson and the unit of the ring. specifically, in [11], jacobson graph was formed from the commutative ring 𝑍3𝑛 and obtained several properties that related to the graph with its forming ring. some of the research above inspired the author to examine the metric dimensions and local metric dimensions on graphs constructed from commutative rings. as an object of research, we choose a ring of modulo 𝑛 integer with the adjacency between two vertices was chosen based on the relative prime properties between the two vertices. the following is given the definition of the greatest common divisor and the relative prime of two positive integers. definition 1 [12]. let 𝑎 and 𝑏 be two positive integers. the positive integer 𝑑 that satisfies 𝑑 = 𝑝𝑎 + 𝑞𝑏 for some integer 𝑝 and 𝑞 is called greatest common divisor (abbreviated gcd) of 𝑎 and 𝑏, denoted by 𝑑 = gcd (𝑎,𝑏). definition 2 [12]. two positive integers 𝑎 and 𝑏 are relatively prime if gcd(𝑎,𝑏) = 1. the purpose of this study is to determine the metric dimensions and local metric dimensions of relative prime graphs. the definition of metric dimensions and local metric dimensions which refer to [13] and [5] is given as follows. definition 3 [13]. suppose that 𝐺 is a connected graph of order 𝑛 and 𝑊 = {𝑤1,𝑤2,…,𝑤𝑗} ⊆ 𝑉(𝐺),1 ≤ 𝑗 ≤ 𝑛 is an ordered set of j-tuples of vertices in 𝐺. representation of vertice 𝑣 ∈ 𝑉(𝐺) with respect to 𝑊 is an ordered pair j-tuples 𝑟(𝑣|𝑊) = (𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2),…,𝑑(𝑣,𝑤𝑗)), where 𝑑(𝑣,𝑤𝑖) representing the distance between vertex 𝑣 and vertex 𝑤𝑖, 1 ≤ 𝑖 ≤ 𝑗. the set 𝑊 is called the resolving set of 𝐺 if each vertex in 𝐺 has different representation with respect to 𝑊. the resolving set with a minimum cardinality is called a bases, whereas number of elements on the bases are called dimensions of the graph 𝐺, denoted by 𝑑𝑖𝑚(𝐺). since the calculation of dimensions in a graph is built using the concept of distance (metric), it is called the metric dimension. definition 4 [5]. let g be the connected graph and 𝑊 ⊆ 𝑉(𝐺). the set 𝑊 is called the local resolving set of a graph 𝐺 if each of two adjacent vertices in 𝐺 has a different representation with respect to 𝑊, i.e. 𝑢𝑣 ∈ 𝐸(𝐺) implies 𝑟(𝑢|𝑊) ≠ 𝑟(𝑣|𝑊). the set of local resolving set with minimal cardinality is called a local bases, while number of elements on a bases are called the local metric dimensions of graph 𝐺, denoted by 𝑑𝑖𝑚𝑙(𝐺). methods this study aims to determine the metric dimensions and local metric dimensions of relative prime graph 𝐺𝑍𝑛. the most important step in determining the metric dimensions and local metric dimensions is to determine the pattern of resolving set and local resolving set that have a minimum cardinality as a bases set. the main difference between them is that the metric dimension and local metric dimension of relative prime graph fatmawati 151 1 1 1 2 3 2 3 3 4 4 5 2 6 2 on the local metric dimension the different representation are only required at adjacent vertices. results and discussion in this section, we will discuss the definition of a relative prime graph built from a ring of modulo 𝑛 integer and an exploration of the basic properties of a relative prime graph related to the characteristics of a graph. the discussion begins with the definition of a relative prime graph. definition 5. let ℤ𝑛 be a rings of modulo 𝑛 integer ℤ𝑛, where 𝑛 positive integer and 𝑛 ≠ 1. we defined a new graph 𝐺 with 𝑉(𝐺) = ℤ𝑛 ∖{0} and 𝐸(𝐺) = {𝑢𝑣;𝑢 relatively prime with 𝑣}. graph 𝐺 which formed from ring of modulo 𝑛 integer with add relatively prime as condition for adjacency two vertices are called relative prime graph, denoted by 𝐺ℤ𝑛. the number of vertices of 𝐺𝑍𝑛is denoted |𝑉(𝐺ℤ𝑛)| and the number of edges is denoted |𝐸(𝐺ℤ𝑛)|. example: a b c figure 1. some relative prime graphs a: 𝐺ℤ4; b: 𝐺ℤ5; c: 𝐺ℤ7 the graph on figure 1: a: 𝐺ℤ4 consists of three vertices, namely 1,2,3 which are mutually relative prime. hence vertex 1 adjacent to 2, vertex 2 adjacent to 3, and vertex 3 adjacent to 1. b: 𝐺ℤ5 consists of four vertices, namely 1,2,3,4. the adjacency of vertices 1,2,3 similar with condition of 𝐺ℤ4. furthermore, vertex 4 is relatively prime to 1 and 3, while vertex 2 is not relatively prime to 4. hence vertex 2 is not adjacent to 4. c: 𝐺ℤ7 consists of six vertices, namely 1,2,3,4,5,6. vertex 1 is relatively prime to vertices 2,3,4,5,6; vertex 2 is relatively prime to vertices 1,3,5; vertex 3 is relatively prime to vertices 1,2,4,5; vertex 4 is relatively prime to vertices 1,3,5; vertex 5 is relatively prime to vertices 1,2,3,4,6; and vertex 6 is relatively prime to vertices 1,3,5. since the three vertices (i.e. 2,4,6) are not relatively prime to each other, then they are not adjacent to each other. likewise, vertex 3 is not adjacent to 6 because 3 is not relatively prime to 6. the results of this study begin with an exploration the basic properties of 𝐺ℤ𝑛, furthermore determine the metric dimensions and local metric dimensions of 𝐺ℤ𝑛. based on definition 5 above, there are some basic properties of 𝐺ℤ𝑛 for 𝑛 ≥ 2, as follows: a. the set of vertices in 𝐺ℤ𝑛is 𝑉(𝐺ℤ𝑛) = {1,2,3,…,𝑛 − 1}, hence |𝐸(𝐺ℤ𝑛)| = 𝑛 − 1. b. 𝐺ℤ𝑛 is not an empty graph. c. 𝐺ℤ𝑛 is a trivial graph for 𝑛 = 2 because it consist only one vertex. the metric dimension and local metric dimension of relative prime graph fatmawati 152 d. 𝐺ℤ𝑛 is a connected graph. e. 𝐺ℤ𝑛 is a complete graph for 𝑛 = 2,3,4. specifically for 𝑛 = 3 it is a path graph and for 𝑛 = 4 it is a sikel graph. f. 𝐺ℤ𝑛 is a regular graph for 𝑛 = 3,4 because every vertex has the same degree. g. there is a vertex 1 that adjacent to every vertex in 𝐺ℤ𝑛 for 𝑛 ≥ 3. based on definition 5, a vertex 𝑢 is adjacent to vertex 𝑣 if 𝑢 is relatively prime with 𝑣. in 𝐺ℤ𝑛, there are vertices that are adjacent to each vertex in 𝐺ℤ𝑛, included in this category are vertex 1 and several vertices which are prime numbers. based on the definition of two adjacent vertices on 𝐺ℤ𝑛, the adjacency of vertices are divided into two groups, namely (1) vertices that are adjacent to every vertex, and (2) vertices that are not adjacent to each other. the vertices that not adjacent are the vertices that not relatively prime to each other, i.e. the vertices that have a common divisor other than 1. furthermore, the vertices that have a common divisor other than 1 can be divided into multiple of 2, multiple of 3, multiple of 5, ..., multiple of 7, ... . suppose 𝐴 = {2,3,5,7,…} = {𝑝1,𝑝2,…,𝑝𝑘} represent a set of ordered prime numbers, then the non adjacent vertices are grouped together in multiples of 2, multiples of 3, ... to multiples of prime numbers 𝑝𝑘, where 2𝑝𝑘 ≤ 𝑛 − 1. in this case, 𝑝𝑘 is the largest prime numbers and 𝑘 represents number of groups (multiples 2, multiples 3, etc.). the vertices that adjacent to every vertices in 𝐺ℤ𝑛 are categorized into group 𝑝0. example: in 𝐺ℤ15, 𝑉(𝐺ℤ15) = {1,2,3,…, 14}. the grouping of vertices in 𝐺ℤ15 are: the vertices in group 𝑝0 are 1,11,13. the vertices in group of multiple 𝑝1= 2 are 2,4,6,8,10,12,14. the vertices in group of multiple 𝑝2= 3 are 3,6,9,12. the vertices in group of multiple 𝑝3= 5 are 5,10. the vertices in group of multiple 𝑝4= 7 are 7,14. in this case, there are four groups of multiples, where 7 is the largest prime number that satisfy 2x7 ≤ 14. the vertices 1,11,13 are adjacent to every vertex in 𝐺ℤ15 so that it is categorized in group 𝑝0. it appears that there are vertices that are in more than one group of multiples, for example vertices 6 and 12 are in groups of multiple 2 and multiple 3. likewise, vertex 15 is in groups of multiple 3 and multiple 5. furthermore, if 𝑃𝑖 is the set of vertices in the group of multiples 𝑝𝑖, where 𝑖 = 1,2,…,𝑘 and ⌊𝑥⌋ represent the largest integer that same or less than 𝑥, then |𝑃𝑖| = ⌊ 𝑛−1 𝑝𝑖 ⌋. on 𝐺ℤ15, |𝑃1| = ⌊ 14 2 ⌋ = 7; |𝑃2| = ⌊ 14 3 ⌋ = ⌊4,67⌋ = 4; |𝑃3| = ⌊ 14 5 ⌋ = ⌊2,8⌋ = 2; |𝑃4| = ⌊ 14 7 ⌋ = 2. so, in general in 𝐺ℤ𝑛 the number of vertices in the group of multiples 𝑝𝑖 is ⌊ 𝑛−1 𝑝𝑖 ⌋, while the number of vertices in the group of multiples 𝑝𝑖 and 𝑝𝑗 is ⌊ 𝑛−1 𝑝𝑖.𝑝𝑗 ⌋ where 1 ≤ 𝑖,𝑗 ≤ 𝑘. the lemma that expressed the number of edge of 𝐺ℤ𝑛 is given as follows. lemma 1. |𝐸(𝐺ℤ𝑛)| = (𝑛−1)(𝑛−2) 2 − ∑ ( ⌊ 𝑛−1 𝑝𝑖 ⌋ 2 )𝑘𝑖=1 , where ⌊ 𝑛−1 𝑝𝑖 ⌋ is the number of vertices in groups of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘 and 2𝑝𝑘 ≤ 𝑛 − 1. proof. as explained above, the vertices on 𝐺ℤ𝑛 are divided into 2 major groups, namely the group 𝑝0 and the groups of multiples 𝑝𝑖. due to the specific properties of each group, the number of edges on graph 𝐺ℤ𝑛 can be calculated by asumption the number of edges of the complete graph with 𝑛 −1 vertices, then edge reduction is done based on the non relatively prime properties between any two vertices. as we know, the number of edge of the complete graph with 𝑛 − 1 vertices is (𝑛−1)(𝑛−2) 2 . since an edge is formed by two different vertices, the metric dimension and local metric dimension of relative prime graph fatmawati 153 then there is a reduction in the edge for each group of multiples 𝑝𝑖 as many as ( ⌊ 𝑛−1 𝑝𝑖 ⌋ 2 ). thus, |𝐸(𝐺ℤ𝑛)| = (𝑛−1)(𝑛−2) 2 − ∑ ( ⌊ 𝑛−1 𝑝𝑖 ⌋ 2 )𝑘𝑖=1  table 1. number of edges on graph 𝐺ℤ𝑛 𝑛 |𝑉(𝐺ℤ𝑛)| groups of vertices reduction of edge |𝐸(𝐺ℤ𝑛)| 2 1 0 3 2 𝑝0 : 1,2 2(2− 1) 2 = 1 4 3 𝑝0 : 1,2,3 3(3− 1) 2 = 3 5 4 𝑝0 : 1,3 4(4− 1) 2 − 1 = 5 𝑝1 : 2,4 ( 2 2 ) = 1 6 5 𝑝0 : 1,3,5 5(5− 1) 2 − 1 = 9 𝑝1 : 2,4 ( 2 2 ) = 1 7 6 𝑝0 : 1,5 6(6−1) 2 −3 −1 = 11 𝑝1 : 2,4,6 ( 3 2 ) = 3 𝑝2 : 3,6 ( 2 2 ) = 1 8 7 𝑝0 : 1,5,7 7(7−1) 2 −3 −1 = 17 𝑝1 : 2,4,6 ( 3 2 ) = 3 𝑝2 : 3,6 ( 2 2 ) = 1 9 8 𝑝0 : 1,5,7 8(8−1) 2 −6 −1 = 21 𝑝1 : 2,4,6,8 ( 4 2 ) = 6 𝑝2 : 3,6 ( 2 2 ) = 1 the degree of each vertex on any graph is the number of the vertices that adjacent to that vertex. thus, the degree of vertex 𝑢 ∈ 𝑉(𝐺ℤ𝑛) is determined based on their adjacency to (𝑛 − 2) other vertices as in observation 2. furthermore, lemma 3 states the minimum and maximum degree of the vertex on 𝐺ℤ𝑛. observation 2. if deg𝐺ℤ𝑛 (𝑢) denoted the degree of any vertex 𝑢 ∈ 𝑉(𝐺ℤ𝑛), then deg𝐺ℤ𝑛 (𝑢) = |{𝑣 ∈ 𝑉(𝐺ℤ𝑛):𝑣 relatively prime with 𝑢}|. lemma 3. if 𝑢 ∈ 𝑉(𝐺ℤ𝑛), then deg𝐺ℤ𝑛 (𝑢) = { 0, 1, 2 ≤ deg𝐺ℤ𝑛 (𝑢) ≤ 𝑛 − 2, 𝑛 = 2 𝑛 = 3 𝑛 ≥ 4 proof. let 𝑢 ∈ 𝑉(𝐺ℤ𝑛). the metric dimension and local metric dimension of relative prime graph fatmawati 154 for 𝑛 = 2, then 𝑢 = 1 and 𝐺ℤ𝑛 is the trivial graph, hence deg𝐺ℤ𝑛 (𝑢) = 0. for 𝑛 = 3, then 𝑢 ∈ {1,2}. the number 1 is relatively prime with 2, hence vertex 1 adjacent to vertex 2. thus, deg𝐺ℤ𝑛 (𝑢) = 1. for 𝑛 ≥ 4, it means |𝑉(𝐺ℤ𝑛)| ≥ 3. to show the minimum degree of any vertex is 2, we use claim: every vertex in 𝐺ℤ𝑛 is adjacent to at least two other vertices. proof of claim: since the criterion for two adjacency vertices are relative prime and the vertices in 𝐺ℤ𝑛 are natural numbers, it is sufficient to show that any natural number other than 1 must be relative prime to the natural number before and after it. suppose any natural number 𝑎 where 𝑎 ≠ 1, it will be shown that 𝑎 relative prime with 𝑎 − 1 and 𝑎 also relative prime with 𝑎 + 1. based on definition 2, the natural number 𝑎 is relative prime with 𝑎 − 1 if there are integers 𝑝 and 𝑞 such that 1 = 𝑝𝑎 + 𝑞(𝑎 − 1). by choosing 𝑝 = 1 and 𝑞 = −1, equality hold. hence 𝑎 relative prime with 𝑎 − 1. in a similar way, it can be shown that 𝑎 relative prime with 𝑎 + 1. thus, 𝑎 adjacent to 𝑎 − 1 and also 𝑎 adjacent to 𝑎 + 1. therefore, for any 𝑢 ∈ 𝑉(𝐺ℤ𝑛), there are at least two vertices that adjacent to 𝑢, so deg𝐺ℤ𝑛 (𝑢) ≥ 2. on the other hand, based on the property of the group 𝑝0, where each vertex is adjacent to every vertex in 𝐺ℤ𝑛, so for any 𝑢 ∈ 𝑃0, deg𝐺ℤ𝑛 (𝑢) = |𝑉(𝐺ℤ𝑛)|− 1 = 𝑛 − 1 − 1 = 𝑛 −2 and this is the maximum degree of any vertex in 𝐺ℤ𝑛. thus, it is proven that 2 ≤ deg𝐺ℤ𝑛 (𝑢) ≤ 𝑛 − 2, for 𝑛 ≥ 4 and the whole lemma is proven.  for example, on graph 𝐺ℤ7 we have 𝑉(𝐺ℤ7) = {1,2,3,4,5,6}. the degree of each vertex is: deg𝐺ℤ7 (1) = |{2,3,4,5,6}| = 5; deg𝐺ℤ7 (2) = |{1,3,5}| = 3; deg𝐺ℤ7 (3) = |{1,2,4,5}| = 4; deg𝐺ℤ7 (4) = |{1,3,5}| = 3; deg𝐺ℤ7 (5) = |{1,2,3,4,6}| = 5; deg𝐺ℤ7 (6) = |{1,5}| = 2. it can be seen that 2 ≤ deg𝐺ℤ7 (𝑢) ≤ 5, for every 𝑢 ∈ 𝑉(𝐺ℤ7). in 𝐺ℤ𝑛, there is vertex 1 which is adjacent to each vertex in 𝐺ℤ𝑛. several other vertices are also similar. vertices of this property will form a complete subgraph of 𝐺ℤ𝑛 as shown in theorem 4. theorem 4. there is a complete subgraph formed by vertices of 𝐺ℤ𝑛. proof. a complete graph is a graph where every two vertices are adjacent. based on the property of the vertices on 𝐺ℤ𝑛, there are vertices that are adjacent to each vertex, namely the vertices in the group 𝑝0. these vertices will form the complete subgraph 𝐾𝑚, where 𝑚 = |{𝑢 ∈ 𝑉(𝐺ℤ𝑛):gcd(𝑢,𝑣) = 1,∀𝑣 ∈ 𝑉(𝐺ℤ𝑛)}|.  from theorem 4, the vertices in 𝐺ℤ𝑛 that form the complete subgraph are vertices in group 𝑝0, where 𝑚 = |{𝑢 ∈ 𝑉(𝐺ℤ𝑛):gcd(𝑢,𝑣) = 1,∀𝑣 ∈ 𝑉(𝐺ℤ𝑛)}| = |𝑃0|. on the other hand, the vertices in the group of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘 have the same property that they are not adjacent to each other. this inspires that the vertices in the group of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘 form the multipartite subgraph of 𝐺ℤ𝑛 which is stated in the theorem 5. theorem 5. for 𝑛 ≥ 5, 𝐺ℤ𝑛 is isomorphic with a 𝐾𝑚 + 𝐻 graph where 𝐻 is the 𝑘-partite graph, 𝐾𝑚 is a complete graph with 𝑚 vertices and 𝑚 = |𝑃0|. proof. based on theorem 4, the vertices in the group 𝑝0 form the complete subgraph 𝐾𝑚 of 𝐺ℤ𝑛. meanwhile, the vertices in the group of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘 are not adjacent to the metric dimension and local metric dimension of relative prime graph fatmawati 155 each other for each 𝑖, so the vertices in the group of multiples 𝑝𝑖 can be partitioned into 𝑘 partitions with each partition consisting of vertices in the same group of multiples. on the same partition, there are no adjacent vertices according to the properties of the vertices in the group of multiples 𝑝𝑖. the connectedness of the vertices between partitions is based on the relatively prime properties of these vertices. since every vertex in 𝐾𝑚 is adjacent to every vertex in the group of multiples 𝑝𝑖, this means that every vertex in 𝐾𝑚 is adjacent to every vertex on all partitions (as many as 𝑘 partitions) that are formed. suppose the 𝑘 partitions formed along with adjacency their vertices are called graph 𝐻, then 𝐻 is a 𝑘-partite graph formed from vertices in the group of multiples 𝑝𝑖. every vertex in the group 𝑝0 is adjacent to every vertex in 𝐺ℤ𝑛, especially in the group of multiples 𝑝𝑖. this means that every vertex in 𝐾𝑚 is adjacent to every vertex in 𝐻. therefore, there is a bijective function 𝑓 from 𝑉(𝐺ℤ𝑛) to 𝑉(𝐾𝑚 + 𝐻) which preserves the adjacency between vertices in 𝐺ℤ𝑛. thus, 𝐺ℤ𝑛 ≅ 𝐾𝑚 + 𝐻.  theorem 5 applies for 𝑛 ≥ 5, spesifically for 𝑛 = 2,3,4 then 𝐺ℤ𝑛 ≅ 𝐾𝑛−1 (complete graph with 𝑛 −1 vertices). for example: on 𝐺ℤ8 where 𝑉(𝐺ℤ8) = {1,2,3,4,5,6,7}. ≅ + 𝐺ℤ8 𝐾3 + 𝐻2,2 figure 2. isomorphism 𝐺ℤ8 with 𝐾3 + 𝐻2,2 figure 2 illustrates the isomorphism 𝐺ℤ8 with 𝐾3 + 𝐻2,2 where 𝐻2,2 is 2-partite graph. in 𝐺ℤ8, vertices 1,5,7 is adjacent to each vertex such that 𝑚 = 3. meanwhile, the vertices in the group of multiples 2 are 2,4,6; the vertices in the group of multiples 3 are 3,6 and no vertex are multiples of 5, so there are 2 partitions. in the example above, the vertices on the first partition are 2 and 4, while the vertices on the second partition are 3 and 6. since the number of vertices on the first partition is 2 and the number of vertices on the second partition is 2, it is written 𝐻2,2. the existence of graph 𝐻2,2 as a 2-partite graph is not unique. another alternative is that if three vertices are selected on the first partition (i.e. vertices 2,4,6) and on the second partition one vertex is chosen (i.e. vertex 3), then the 2-partite graph in this case is written 𝐻3,1. thus, 𝐺ℤ8 ≅ 𝐾3 + 𝐻2,2 ≅ 𝐾3 + 𝐻3,1. next, we will be determined the value of the metric dimension and the local metric dimension of 𝐺ℤ𝑛 for 𝑛 ≥ 2. the determination of metric dimensions is calculated using the concept of the distance between two vertices, namely the length of the shortest path connecting the two vertices. theorem 6. if 𝑢,𝑣 ∈ 𝑉(𝐺ℤ𝑛), then 𝑑(𝑢,𝑣) ≤ 2. proof. let 𝑢,𝑣 ∈ 𝑉(𝐺ℤ𝑛). based on theorem 5, there are three possibilities for 𝑢 and 𝑣, namely (i) 𝑢,𝑣 ∈ 𝑉(𝐾𝑚), (ii) 𝑢,𝑣 ∈ 𝑉(𝐻), dan (iii) 𝑢 ∈ 𝑉(𝐾𝑚) ,𝑣 ∈ 𝑉(𝐻). (i) suppose 𝑢,𝑣 ∈ 𝑉(𝐾𝑚). since 𝐾𝑚 is a complete graph, then distance between two different vertices is 1, so that 𝑑(𝑢,𝑣) = { 0, 𝑢 = 𝑣 1, 𝑢 ≠ 𝑣 . 1 2 3 5 4 7 6 1 5 7 3 2 4 6 the metric dimension and local metric dimension of relative prime graph fatmawati 156 (ii) suppose 𝑢,𝑣 ∈ 𝑉(𝐻), there are two condition, i.e. 𝑢 and 𝑣 on the same partitions or 𝑢 and 𝑣 on the different partitions. a. if 𝑢 and 𝑣 are vertex on the same partition, then there is 𝑧 ∈ 𝑉(𝐾𝑚) so that 𝑑(𝑢,𝑧) = 1 and 𝑑(𝑧,𝑣) = 1. therefore, 𝑑(𝑢,𝑣) = 2. so 𝑑(𝑢,𝑣) = { 0, 2, 𝑢 = 𝑣 𝑢 ≠ 𝑣 . b. if 𝑢 and 𝑣 are vertices on the different partition, then the distance is 1 if 𝑢 adjacent to 𝑣. for vertex 𝑢 which is not adjacent to 𝑣, there is 𝑥 ∈ 𝑉(𝐾𝑚) so that 𝑑(𝑢,𝑥) = 1 and 𝑑(𝑥,𝑣) = 1. therefore 𝑑(𝑢,𝑣) = 2. thus 𝑑(𝑢,𝑣) = { 1, 2, gcd(𝑢,𝑣) = 1 gcd(𝑢,𝑣) ≠ 1 . (iii) suppose 𝑢 ∈ 𝑉(𝐾𝑚) ,𝑣 ∈ 𝑉(𝐻), then 𝑑(𝑢,𝑣) = 1 because every vertex in 𝐾𝑚 is adjacent to all vertices in 𝐻. from all possibilities (i), (ii), and (iii) it is proven that 𝑑(𝑢,𝑣) ≤ 2.  corollary 7. let 𝑢,𝑣 ∈ 𝑉(𝐺ℤ𝑛), 𝑑(𝑢,𝑣) = 2 if and only if 𝑢 is not relatively prime to 𝑣. proof. it is clear, is a direct result of theorem 6. the vertices in the group 𝑝0 apart from forming a complete subgraph of 𝐺ℤ𝑛 are also adjacent to all vertices in 𝐺ℤ𝑛. due to this specific property, in determining the elements of the resolving set, only one vertex is allowed. on the other hand, the vertices in the group of multiples 𝑝𝑖 also have a specific property. two groupings of vertices in 𝐺ℤ𝑛, each group have specific properties, so it is impossible for the metric bases of 𝐺ℤ𝑛 consist of vertices in the group 𝑝0 or in the group of multiples 𝑝𝑖 only. this condition is illustrated in theorem 8 below by observing the vertices in the group 𝑝0. theorem 8. every subset of 𝐾𝑚 is not a resolving set. proof. referring to theorem 4, the vertices in group 𝑝0 form a complete subgraph with 𝑚 vertices and the metric dimension is 𝑚 − 1. that is, the representation of other vertices in the group 𝑝0 against the 𝑚 − 1 vertices is the same as the vertex representation outside the group 𝑝0 for the 𝑚 − 1 verices. as a result, the set consisting of 𝑚 − 1 vertices is not a resolving set. the same condition also applies to sets whose element are less than 𝑚 − 1 vertices. based on theorem 5 which states 𝐺ℤ𝑛 ≅ 𝐾𝑚 +𝐻, it is obtained that each vertex in 𝐾𝑚 is adjacent to every vertex in 𝐻, it means that the distance is 1. for the same reason, any vertex taken from 𝐾𝑚 is not a resolving set. evidently, every subset of 𝐾𝑚 is not a resolving set.  based on the proof of theorem 8, the resolving set can not contain only vertices in 𝐾𝑚. on the other hand, the vertices in groups of multiples 𝑝1,𝑝2, ..., 𝑝𝑘 has a similar property that are not adjacent to each other. representation of two different vertices of a certain group of multiple to another vertices in the same group of multiple and to different groups of multiple are presented in the following lemma 9 and lemma 10. lemma 9. if 𝑢,𝑣 ∈ 𝑃𝑖, where 𝑢 ≠ 𝑣, then 𝑟(𝑢|𝑃𝑖 ∖ {𝑢,𝑣}) = 𝑟(𝑣|𝑃𝑖 ∖{𝑢,𝑣}). proof. the vertices in the group of multiples 𝑝𝑖 have a similar characteristic, namely they are not adjacent. based on the proof of theorem 6 (ii) a, the distance between different vertices is 2. thus, 𝑟(𝑢|𝑃𝑖 ∖{𝑢,𝑣}) = (2,2,…,2) and 𝑟(𝑣|𝑃𝑖 ∖{𝑢,𝑣}) = (2,2,…,2). therefore, 𝑟(𝑢|𝑃𝑖 ∖ {𝑢,𝑣}) = 𝑟(𝑣|𝑃𝑖 ∖{𝑢,𝑣}).  lemma 10. if 𝑢,𝑣 ∈ 𝑃𝑗1, then 𝑟(𝑢|𝑃𝑗2) = 𝑟(𝑣|𝑃𝑗2), where 1 ≤ 𝑗1, 𝑗2 ≤ 𝑘 and 𝑢,𝑣 ∉ 𝑃𝑗1 ∩𝑃𝑗2. the metric dimension and local metric dimension of relative prime graph fatmawati 157 proof. based on the proof of theorem 6 (ii) b, the distance between two vertices on different partitions is 1 or 2. furthermore, 𝑟(𝑢|𝑃𝑗2) = (𝑑(𝑢,𝑝𝑗2),𝑑(𝑢,2𝑝𝑗2),𝑑(𝑢,3𝑝𝑗2),…,𝑑(𝑢,𝑡𝑝𝑗2)) and 𝑟(𝑣|𝑃𝑗2) = (𝑑(𝑣,𝑝𝑗2),𝑑(𝑣,2𝑝𝑗2),𝑑(𝑣,3𝑝𝑗2),…,𝑑(𝑣,𝑡𝑝𝑗2)), where 𝑡𝑝𝑗2 ≤ 𝑛 − 1. the distance between two vertices in different groups of multiples is 1 if the two vertices are relatively prime and the distance is 2 if they are not relatively prime. since the properties of the group of multiples 𝑝𝑗1 and multiples 𝑝𝑗2 are similar, namely that the vertices are not adjacent in each group, while 𝑢 and 𝑣 are vertices in the same multiple group, then 𝑟(𝑢|𝑃𝑗2) = 𝑟(𝑣|𝑃𝑗2).  based on the results of lemma 9 and lemma 10, the vertices in the group of multiple 𝑝𝑖 become element of the resolving set leaving only one vertex in each group. this also applies to the vertices in the group 𝑝0, which leaves only one vertex. suppose that the vertex that must be left in the group of multiples 𝑝𝑖 are 𝑎𝑖 where 1 ≤ 𝑖 ≤ 𝑘 and the vertex that must be left in the group 𝑝0 are 𝑎0, the following is given the metric dimension of 𝐺ℤ𝑛. theorem 11. 𝐷𝑖𝑚(𝐺ℤ𝑛) = 𝑛 − 𝑘 −2, where 𝑘 represents number of groups of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘. proof. suppose 𝑊 = 𝑄0 ∪ 𝑄1 ∪ 𝑄2 ∪…∪𝑄𝑘, where 𝑄0 = 𝑃0 ∖ {𝑎0}, 𝑎0 is an arbitrary vertex left from the group 𝑝0, 𝑄1 = 𝑃1 ∖ {𝑎1}, 𝑎1 is an arbitrary vertex left from the group of multiple 𝑝1, 𝑄2 = 𝑃2 ∖ {𝑎2}, 𝑎2 is an arbitrary vertex left from the group of multiple 𝑝2, ⋮ 𝑄𝑘 = 𝑃𝑘 ∖ {𝑎𝑘}, 𝑎𝑘 is an arbitrary vertex left from the group of multiple 𝑝𝑘. the representation of every vertex in 𝑉(𝐺ℤ𝑛 ∖𝑊) with respect to 𝑊 is: 𝑟(𝑎0|𝑊) = ( 1,1,1,…,1⏟ as many as |𝑃0|−1 , 1,1,1,…,1⏟ as many as |𝑃1|−1 , 1,1,1,…,1⏟ as many as |𝑃2|−1 ,…, 1,1,1,…,1⏟ as many as |𝑃𝑘|−1 ) 𝑟(𝑎1|𝑊) = ( 1,1,1,…,1⏟ as many as |𝑃0|−1 , 2,2,2,…,2⏟ as many as |𝑃1|−1 ,𝑑(𝑎1,𝑄2),…,𝑑(𝑎1,𝑄𝑘)) 𝑟(𝑎2|𝑊) = ( 1,1,1,…,1⏟ as many as |𝑃0|−1 ,𝑑(𝑎2,𝑄1), 2,2,2,…,2⏟ as many as |𝑃2|−1 ,…,𝑑(𝑎2,𝑄𝑘)) ⋮ 𝑟(𝑎𝑘|𝑊) = ( 1,1,1,…,1⏟ as many as |𝑃0|−1 , 𝑑(𝑎𝑘,𝑄1), 𝑑(𝑎𝑘,𝑄2),…, 2,2,2,…,2⏟ as many as |𝑃𝑘|−1 ) where 𝑑(𝑎𝑖,𝑄𝑗) = { 1,if 𝑎𝑖 ∉ 𝑃𝑗 2, if 𝑎𝑖 ∈ 𝑃𝑗 , 1 ≤ 𝑖,𝑗 ≤ 𝑘. it appears that the representation of every vertex with respect to 𝑊 is different, so that 𝑊 is a resolving set. next, it will be shown that the cardinality of 𝑊 is minimal. suppose that any set 𝑋 is taken whose cardinality is one less than 𝑊, that is, |𝑋| = |𝑊|− 1. there are three possibilities for elements of set 𝑋, namely (i) all vertices in group 𝑝0 are element of 𝑋; (ii) all vertices in the group of multiples 𝑝𝑖 are element of 𝑋; and (iii) all vertices of the group of multiple 𝑝𝑗 (certain 𝑗), 1 ≤ 𝑗 ≤ 𝑘 being element of 𝑋. (i) if all vertices in the group 𝑝0 are element of 𝑋, then 𝑘 + 1 of the vertices in 𝐺ℤ𝑛 must be left in the group of multiples 𝑝𝑖 with the number of groups being 𝑘. a. suppose that as many as 𝑘 − 1 groups leave each one vertex, then there are two vertices that must be left by the group of multiple 𝑝𝑗 (certain 𝑗), 1 ≤ 𝑗 ≤ 𝑘. due to the metric dimension and local metric dimension of relative prime graph fatmawati 158 the similar properties of vertices in the group of multiple 𝑝𝑗, the representation of the two vertices with respect to 𝑋 will be the same. thus 𝑋 is not being a resolving set. b. suppose that all vertices in the 𝑘 − 1 group are element of 𝑋, then there is one particular group, name the group of multiple 𝑝𝑗 (certain 𝑗), 1 ≤ 𝑗 ≤ 𝑘 leaving 𝑘 + 1 vertex. for the same reason as (i)a, these 𝑘 − 1 vertices will have the same representation of 𝑋 because the vertices in the group of multiples 𝑝𝑗 are nonmutually adjacent. so 𝑋 is not a resolving set. (ii) if all the vertices in the group of multiples 𝑝𝑖 are element of 𝑋, then |𝑋| = |𝑊| − 1+ 𝑘 = (𝑛 − 2 − 𝑘) − 1+ 𝑘 = 𝑛 − 3 ≥ 𝑛 − 2− 𝑘 for 𝑘 ≥ 1. the equity hold only to 𝑘 = 1. so, for 𝑘 ≥ 2 then |𝑋| > |𝑊|. this contradicts the cardinality of the set 𝑋. (iii) if all vertices in a group of multiples 𝑝𝑖, namely 𝑝𝑗 dengan 1 ≤ 𝑗 ≤ 𝑘 as element of 𝑋, then from the vertices in 𝐺ℤ𝑛 must be left 𝑘 + 1 vertices in 𝑘 − 1 group of multiples 𝑝𝑖 (other than 𝑝𝑗) and in the group 𝑝0. a. suppose that all vertices in the group 𝑝0 are element of 𝑋, then from 𝑘 − 1 group of multiples 𝑝𝑖 must be left 𝑘 vertices. this is similar to case (i). b. suppose all vertices in the group of multiples 𝑝𝑖 (other than 𝑝𝑗) are element of 𝑋, it means that all vertices in the group of multiples 𝑝𝑖 are element of 𝑋. this is similar to the case (ii). from all of the above possibilities it can be concluded that 𝑋 is not a resolving set. if all vertices in one group are selected as element of set 𝑋, then 𝑋 is not a resolving set. likewise, if there is a group 𝑝0 or a group of multiples 𝑝𝑖 that leaves more than one vertex, it will result that 𝑋 not being a resolving set. since |𝑋| = |𝑊| − 1 and 𝑊 are resolving set, it means that 𝑊 is the resolving set with minimal cardinality or the metric bases of 𝐺ℤ𝑛. since each group leaves one vertex and the number of groups are 𝑘 +1, the metric dimension of 𝐺ℤ𝑛 is (𝑛 − 1) − (𝑘 + 1) = 𝑛 − 𝑘 − 2. it is proven that 𝑑𝑖𝑚(𝐺ℤ𝑛) = 𝑛 −𝑘 − 2.  based on theorem 11, the metric bases of 𝐺ℤ𝑛 consists of a combination of vertices in the group 𝑝0 and the group of multiples 𝑝𝑖 with the conditions each leaving only one vertex. the final part of this research is to determine the value of the local metric dimension of 𝐺ℤ𝑛 by first determining the local resolving set. in this case the vertex representation may be the same as long as the vertices are not adjacent. theorem 12. 𝐷𝑖𝑚𝑙(𝐺ℤ𝑛) = |𝑃0| + 𝑘 − 1, where 𝑘 representing number of groups of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘 and |𝑃0| is the cardinality of set 𝑃0. proof. based on the property of each group of multiples 𝑝𝑖, 𝑖 = 1,2,…,𝑘, which the vertices are not mutually adjacent, the property of the group 𝑝0 that all elements are adjacent to every vertex, and according to the definition of the local metric dimension, then we choose the set 𝑊𝑙 = {1,𝑝01,…,𝑝𝑜(𝑚−2),𝑝11,𝑝21,…,𝑝𝑘1}, where 1,𝑝01,…,𝑝𝑜(𝑚−2) are 𝑚 − 1 vertices in the group 𝑝0, 𝑝11 is one vertex in the group of multiples 𝑝1, 𝑝21 is one vertex in the group of multiples 𝑝2, 𝑝𝑘1 is one vertex in the group of multiples 𝑝𝑘. the representation of every vertex in 𝐺ℤ𝑛 with respect to 𝑊𝑙 is: 𝑟(1|𝑊𝑙) = (0,1,…,1,1,1,…,1); 𝑟(𝑝01|𝑊𝑙) = (1,0,…,1,1,1,…,1); 𝑟(𝑝𝑜(𝑚−2)|𝑊𝑙) = (1,1,…,0,1,1,…,1); 𝑟(𝑝11|𝑊𝑙) = (1,1,…,1,0,1,…,1); 𝑟(𝑝21|𝑊𝑙) = (1,1,…,1,1,0,…,1); 𝑟(𝑝𝑘1|𝑊𝑙) = (1,1,…,1,1,1,…,0); the metric dimension and local metric dimension of relative prime graph fatmawati 159 𝑟(𝑝02|𝑊𝑙) = (1,1,…,1,1,1,…,1); 𝑟(𝑝03|𝑊𝑙) = (1,1,…,1,1,1,…,1); 𝑟(𝑝12|𝑊𝑙) = (1,1,…,1,2,2,…,1); 𝑟(𝑝13|𝑊𝑙) = (1,1,…,1,2,2,…,1); 𝑟(𝑝14|𝑊𝑙) = (1,1,…,1,2,2,…,1); 𝑟(𝑝1𝑠|𝑊𝑙) = (1,1,…,1,2,2,…,1). it appears that 𝑟(𝑝13|𝑊𝑙) = 𝑟(𝑝13|𝑊𝑙) = 𝑟(𝑝14|𝑊𝑙) = 𝑟(𝑝1𝑠|𝑊𝑙), but 𝑝12,𝑝13,𝑝14,𝑝1𝑠 are vertices in the group of multiples 𝑝1 which is not mutually adjacent. in the concept of local metric dimensions, the above still meets the criteria, meaning that 𝑊𝑙 is a local resolving set. next, it will be shown that the cardinality of 𝑊𝑙 is minimal. suppose any set 𝑋𝑙 whose cardinality is reduced to one from the set 𝑊𝑙, i.e. |𝑋𝑙| = |𝑊𝑙| − 1. there are three possibilities for element of 𝑋𝑙, namely (i) all vertices in group 𝑝0 become element of 𝑋𝑙; (ii) at least one vertex from each group of multiples 𝑝𝑖 become element of 𝑋𝑙; and (iii) the group 𝑝0 leaves more than one vertex. (i) suppose that all vertices in the group 𝑝0 are members of 𝑋𝑙, then there are 𝑘 − 2 vertices from the group of multiple 𝑝𝑖 as many as 𝑘 that can be element of 𝑋𝑙. assuming at least one vertex in each group of multiple 𝑝𝑖 becomes a element of 𝑋𝑙, it means that there are at least two groups whose vertices are not represented as element of 𝑋𝑙. it is sufficient to show that there are two vertices from two different groups, these two vertices are adjacent and have the same representation respect to 𝑋𝑙. suppose that the two groups not represented in the element of the set 𝑋𝑙 are the group of multiples 𝑝𝑗1 and 𝑝𝑗2 where 1 ≤ 𝑗1, 𝑗2 ≤ 𝑘. vertex 𝑝𝑗1 is adjacent to 𝑝𝑗2 because both are prime numbers and are the first element in their respective group of multiples. in addition, vertices 𝑝𝑗1 and 𝑝𝑗2 are adjacent to all vertices in the group 𝑝0, so that 𝑟(𝑝𝑗1|𝑋𝑙) = 𝑟(𝑝𝑗2|𝑋𝑙). consequently, 𝑋𝑙 is not a resolving set. (ii) suppose that at least one vertex from each group of multiples 𝑝𝑖 is a element of 𝑋𝑙, then there are at least two vertices in the group 𝑝0 that are not element of 𝑋𝑙. the two vertices in the group 𝑝0 will have the same representation respect to 𝑋𝑙, because the two vertices in the group 𝑝0 are adjacent to every vertex in 𝐺ℤ𝑛. so 𝑋𝑙 is not a resolving set. (iii) suppose that the group 𝑝0 leaves more than one vertex, then a. if the group 𝑝0 leaves two vertices, then whatever the condition for selecting vertices in the group of multiples 𝑝𝑖, the remaining two vertices in the group 𝑝0 will have the same representation respect to 𝑋𝑙. consequently, 𝑋𝑙 is not a resolving set. b. if the group 𝑝0 leaves more than two vertices, then there is a group of multiples 𝑝𝑖 where all the vertices are not 𝑋𝑙 element. in this case, the remaining vertices in the group 𝑝0 will have the same representation of 𝑋𝑙 regardless of the vertex conditions in the group of multiples 𝑝𝑖. as a result, 𝑋𝑙 is not a resolving set. from all of the above possibilities, it can be concluded that 𝑋𝑙 is not a local resolving set. if there are more than one vertex left in group 𝑝0, then the representation of the remaining vertices respect to 𝑋𝑙 will be the same. likewise, if there is one or more groups of multiples 𝑝𝑖 that are not represented in the element of the set 𝑋𝑙, it can always be found the vertices of the group of multiples 𝑝𝑖 which have the same representation respect to set 𝑋𝑙 eventhough they are adjacent. since |𝑋𝑙| = |𝑊𝑙| − 1 and 𝑊𝑙 are local resolving set, it can be concluded that 𝑊𝑙 is a local resolving set with minimal cardinality or local metric bases of 𝐺ℤ𝑛. the cardinality of the set 𝑊𝑙 is the local metric dimension of 𝐺ℤ𝑛. it is proven that 𝑑𝑖𝑚𝑙(𝐺𝑍𝑛) = |𝑃0|− 1 + 𝑘.  based on theorem 12, the local metric bases consists of vertices in the group 𝑝0 and the group of multiples 𝑝𝑖, provided that the group 𝑝0 leaves only one vertex while in each group of multiples 𝑝𝑖 are represented by only one vertex. the metric dimension and local metric dimension of relative prime graph fatmawati 160 example: suppose given 𝐺ℤ17, where 𝑉(𝐺ℤ17) = {1,2,3,…, 16}. it will determine the metric dimension and the local metric dimension of 𝐺ℤ17 and some metric bases and local metric bases. firstly, the vertices of 𝐺ℤ17 were grouped as follows: group 𝑝0:1,11,13 then |𝑃0| = 3. group of multiple 2 are 2,4,6,8,10,12,14,16. group of multiple 3 are 3,6,9,12,15. group of multiple 5 are 5,10,15. group of multiple 7 are 7,14. there are four groups of multiples, namely multiples of 2, multiples of 4, multiples of 5, and multiples of 7, so 𝑘 = 4. (1) 𝑑𝑖𝑚(𝐺ℤ17) = 17 − 4 − 2 = 11. some of the metric bases of 𝐺ℤ17 are: 𝑊1 = {1,11,2,3,4,6,8,10,12,14,15}, then 𝑟(5|𝑊1) = (1,1,1,1,1,1,1,2,1,1,2); 𝑟(7|𝑊1) = (1,1,1,1,1,1,1,1,1,2,1); 𝑟(9|𝑊1) = (1,1,1,2,1,2,1,1,2,1,2); 𝑟(13|𝑊1) = (1,1,1,1,1,1,1,1,1,1,1); 𝑟(16|𝑊1) = (1,1,2,1,2,2,2,2,2,2,1). 𝑊2 = {1,13,2,3,4,6,10,12,14,15,16}, then 𝑟(5|𝑊2) = (1,1,1,1,1,1,2,1,2,2,1); 𝑟(7|𝑊2) = (1,1,1,1,1,1,1,1,2,1,1); 𝑟(8|𝑊2) = (1,1,2,1,2,2,2,2,2,1,2); 𝑟(9|𝑊2) = (1,1,1,2,1,2,1,2,1,2,1); 𝑟(11|𝑊2) = (1,1,1,1,1,1,1,1,1,1,1). 𝑊3 = {11,13,2,4,6,9,10,12,14,15,16}, then 𝑟(1|𝑊3) = (1,1,1,1,1,1,1,1,1,1,1); 𝑟(3|𝑊3) = (1,1,1,1,1,2,2,1,2,2,1); 𝑟(5|𝑊3) = (1,1,1,1,1,1,2,1,1,2,1); 𝑟(7|𝑊3) = (1,1,1,1,1,1,1,1,2,1,1); 𝑟(8|𝑊3) = (1,1,2,1,2,1,2,2,2,1,2). (2) 𝑑𝑖𝑚𝑙(𝐺ℤ17) = 3 − 1+ 4 = 6. some of the local metric bases of 𝐺ℤ17 are: 𝑊𝑙1 = {1,11,2,3,5,7}, then 𝑟(4|𝑊𝑙1) = (1,1,2,1,1,1); 𝑟(6|𝑊𝑙1) = (1,1,2,2,1,1); 𝑟(8|𝑊𝑙1) = (1,1,2,1,1,1); 𝑟(9|𝑊𝑙1) = (1,1,1,2,1,1); 𝑟(10|𝑊𝑙1) = (1,1,2,1,2,1); 𝑟(12|𝑊𝑙1) = (1,1,2,2,1,1); 𝑟(13|𝑊𝑙1) = (1,1,1,1,1,1); 𝑟(14|𝑊𝑙1) = (1,1,2,1,1,2); 𝑟(15|𝑊𝑙1) = (1,1,1,2,2,1); 𝑟(16|𝑊𝑙1) = (1,1,2,1,1,1). it appears that 𝑟(4|𝑊𝑙1) = 𝑟(8|𝑊𝑙1) and 𝑟(6|𝑊𝑙1) = 𝑟(12|𝑊𝑙1), but vertex 4 is not adjacent to 8 and vertex 6 is not adjacent to 12. 𝑊𝑙2 = {1,13,4,9,5,7}, then 𝑟(2|𝑊𝑙2) = (1,1,2,1,1,1); 𝑟(3|𝑊𝑙2) = (1,1,1,1,1,1); 𝑟(6|𝑊𝑙2) = (1,1,2,2,1,1); 𝑟(8|𝑊𝑙2) = (1,1,2,1,1,1); 𝑟(10|𝑊𝑙2) = (1,1,2,1,2,1); 𝑟(11|𝑊𝑙2) = (1,1,1,1,1,1); 𝑟(12|𝑊𝑙2) = (1,1,2,2,1,1); 𝑟(14|𝑊𝑙2) = (1,1,2,1,1,2); 𝑟(15|𝑊𝑙2) = (1,1,1,2,2,1); 𝑟(16|𝑊𝑙2) = (1,1,2,1,1,1). it appears that 𝑟(2|𝑊𝑙1) = 𝑟(8|𝑊𝑙1) and 𝑟(6|𝑊𝑙1) = 𝑟(12|𝑊𝑙1), but vertex 2 is not adjacent to 8 and vertex 6 is not adjacent to 12. 𝑊𝑙3 = {1,11,8,12,5,14}, then 𝑟(2|𝑊𝑙3) = (1,1,2,2,1,2); 𝑟(3|𝑊𝑙3) = (1,1,1,2,2,1); 𝑟(4|𝑊𝑙3) = (1,1,2,2,1,2); 𝑟(6|𝑊𝑙3) = (1,1,2,2,1,2); 𝑟(7|𝑊𝑙3) = (1,1,1,1,1,1); 𝑟(9|𝑊𝑙3) = (1,1,1,2,1,1); 𝑟(10|𝑊𝑙3) = (1,1,2,2,2,2); 𝑟(13|𝑊𝑙3) = (1,1,1,1,1,1); 𝑟(15|𝑊𝑙3) = (1,1,1,2,2,1); 𝑟(16|𝑊𝑙3) = (1,1,2,2,1,2). it appears that 𝑟(2|𝑊𝑙3) = 𝑟(4|𝑊𝑙3) = 𝑟(6|𝑊𝑙3) = 𝑟(16|𝑊𝑙3), but the vertices 2,4,6,16 are not adjacent to each other. the metric dimension and local metric dimension of relative prime graph fatmawati 161 conclusions this research focuses on determining the metric dimension and local metric dimension of relative prime graphs 𝐺𝑍𝑛. based on the previous discussion, it can be concluded that: (1) 𝑑𝑖𝑚(𝐺𝑍𝑛) = 𝑛 −𝑘 − 2; (2) 𝑑𝑖𝑚𝑙(𝐺𝑍𝑛) = |𝑃0| − 1 + 𝑘, where 𝑘 is the number of group multiples 𝑝1,𝑝2,…,𝑝𝑘, and |𝑃0| is the cardinality of set 𝑃0. in the future, the research can be extended to other topics such as fractional metric dimensions, local fractional metric dimensions, domination numbers/set, graph coloring, and graph labeling as well as expansion of research objects in special rings. references [1] g. chartrand and l. lesniak, graphs and digraphs, third edition, chapman & hall/crc, florida, 2000. 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[13] g. chartrand, l. eroh, m. a. johnson and o. r. oellermann, "resolvability in graphs and the metric dimension of a graph", discrete applied mathematics 105, pp. 99-113, 2000. regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) cauchy –jurnal matematika murni dan aplikasi volume 6 (4) (2021), pages 296-304 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 27, 2021 reviewed: april 29, 2021 accepted: may 04, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11758 regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing1, yudhie andriyana2, bertho tantular3 1statistics indonesia, jakarta, indonesia 1,2,3department of statistics, padjadjaran university, bandung, indonesia email: robinson@bps.go.id abstract generally, modeling poverty aims to obtain the best criteria for assessing poverty status. there are two approaches to model the factors that affect poverty, namely consumption approach and discrete choice model. the advantage of the discrete choice model compared to the consumption approach is that the discrete choice model provides a probabilistic estimate for classifying samples into different poverty categories. the aim of this study is to determine the factors that impact poverty in yogyakarta through regularized ordinal regression used elastic net approach both for parallel, non-parallel, and semi-parallel models. the data used in this study is susenas march 2018 for yogyakarta provinces. the result of this study shows that the best discrete choice model for yogyakarta’s modelling is the parallel model. households that live in villages, have a large number of household members, are headed by women, have elderly household heads, have low education, and work in the primary sector tend to be more vulnerable to poverty. therefore, a simultaneous policy with inclusive economic development is needed to reduce cross-border, cross-gender, and cross-sector inequality. keywords: elastic net; ordinal regression; parallel; poverty introduction poverty is one of the problems in economic development. every country tries to alleviate poverty with various programs. as an institution that released the official poverty rate in indonesia, bps [1] defines poverty as the inability to meet basic needs from an economic perspective, both food and non-food, which is measured in terms of expenditure. generally, modeling poverty aims to obtain the best criteria for assessing poverty status. rouband & razafindrakoto [2] assert that there is a correlation between objective and subjective poverty measures and further argue that various forms of poverty cannot be reduced to one another. a poverty approach is generally a monetary approach, but there is a growing literature that tries to bring up an index of multidimensional aspects of poverty. the impact factor in poverty approaches with two models. the first uses a regression approach between consumption expenditure per adult equivalent to several potential explanatory variables called the consumption approach. the second model is discrete choice model. the discrete approach is to categorize poverty into three categories based on household consumption expenditure compared to a region's poverty http://dx.doi.org/10.18860/ca.v6i4.11758 regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 297 line. the advantages of the discrete model are the influence of independent variables to vary across poverty categories. one of the most common regression models for ordinal data types is the cumulative logit model [3], also known as the proportional odds model or the ordinal logistic regression model. to improve the prediction accuracy in ordinal regression model with different regression coefficients for each response category, the dogev model was introduced to improve prediction accuracy [4]. the dogev model has a requirement which is the data that used has extreme values. moreover, the model has parallel and non-parallel models yet. wurm, rathouz, & hanlon [5] introduced a regression model using different regression coefficients for each response category known as regulized ordinal regression. the data that has ordinal response or dependent should be explained using parallel or non-parallel. when, the number of household observation used maximum likelihood then it is the proper model [5]. after that, both the nonparallel model that includes the parallel model as a particular case and the parallel model will provide an inconsistent estimation coefficient if there are errors in the modeling. the number of explanatory variables increases, we need a variable selection technique that will reduce some variables. this step is needed because it is impossible to estimate each coefficient with a high degree of accuracy. then more realistic modeling goal is built a model for out-of-sample prediction and determine the most important explanatory variables. two variable selection methods that are often used are the lasso and ridge methods. lasso and ridge regression are techniques that minimize the penalized likelihood objective function. lasso regression uses the l1 penalty, while ridge regression uses the l2 penalty. both penalties produce coefficient estimates that are closer to zero than the maximum likelihood estimator; for example, the estimate is "close" to zero yet. the estimation results in an estimation bias towards zero, but a trade-off occurs in terms of reducing the variance, which often reduces the overall mean squared error. lasso has properties with some approximate coefficients close to zero. this method provided a natural way to select variables because only the most relevant predictor of the response variable will have a non-zero coefficient. however, it is a group of variables that are highly correlated then, the lasso tends to choose one variable from the correlated group and ignores the others. the elastic net penalty was introduced to overcome those limitations [5]. the elastic net penalty method is the weighted average between lasso and ridge, by dividing the lasso properties and shrinking some coefficients to zero so that it has a unique solution in most cases. based on the previous description, the main problem to be examined is how the factors that affect poverty in yogyakarta through regularized ordinal regression with elastic net approach both for parallel, non-parallel, and semi-parallel models. methods data the data that used in this study is susenas consumption module in march 2018 by bps. then used as the response variable and explanatory variables. base on theoretical studies, residential, community, household and individual characteristics influenced differences in household expenditure. this study uses household data and the variables that related to household characteristics only. the variation in household characteristics will affect the households’ expenditure. regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 298 methodology the aim of poverty modelling is to know factor that influence such as poverty. the focus of study is the family that has characteristics in specific poverty status. the framework of poverty study would be assumed that the real poverty status in household unable observed or unconsidered by well-being ratio, with general mode: 𝑙𝑜𝑔𝑖𝑡 (𝒑𝑖) = 𝐗𝑖 𝑇𝜷 i=1,2,...,n (1) where: 𝐗𝑖 = covariate matrix 𝜷 = vector of regression coefficients 𝒑𝑖 = category probability the logit model that used is one of the generalized linear model (glm) models. the glm model has three components, namely random component, systematic component and link function. the form of the glm depends on the three components that related to each other. a. random components 𝑌 is a random variable of ordinal response with three categories, which is poor, almost poor, not poor 𝑌 ~ multinomial (𝑛; 𝑝1,𝑝2,𝑝3) 𝑓(𝑦;𝑝1,𝑝2,𝑝3,𝑛) = 𝑛! 𝑦1!𝑦2!𝑦3! 𝑝1 𝑦1𝑝2 𝑦2𝑝3 𝑦3 b. systematic component the systematic component of the model is a set of 𝜷 parameters and a covariate 𝐗 that forms a linear combination of 𝐗𝑖 𝑇𝜷 the general form of the linear predictor is : 𝜼𝑖 = 𝐗i t𝜷 (2) according to wurm, rathouz, & hanlon [5], the linear form of the predictor consists of:: i. parallel model 𝐗𝑖 = (𝐈𝐾×𝐾 | 𝒙𝑖 𝑇 ⋮ 𝒙𝑖 𝑇 ) 𝐾×(𝑃+𝐾) , 𝜷 = ( 𝒃𝟎 𝒃 ) (𝑃+𝐾)×1 ii. nonparallel model 𝐗𝑖 = (𝐈𝐾×𝐾 | 𝒙𝑖 𝑇 0 0 0 0 𝒙𝑖 𝑇 0 0 0 0 ⋱ ⋯ 0 0 ⋮ 𝒙𝑖 𝑇 ) 𝐾×(𝑃𝐾+𝐾) , 𝜷 = ( 𝒃𝟎 𝐁1 𝐁2 ⋮ 𝐁𝑘) (𝑃𝐾+𝐾)×1 iii. semi-parallel model 𝐗𝑖 = ( 𝐈𝐾×𝐾 || 𝒙𝑖 𝑇 𝒙𝑖 𝑇 ⋮ 𝒙𝑖 𝑇 𝒙𝑖 𝑇 0 ⋮ 0 0 𝒙𝑖 𝑇 ⋮ 0 ⋯ … ⋱ ⋯ 0 0 ⋮ 𝒙𝑖 𝑇 ) 𝐾×(𝑃(𝐾+1)+𝐾) , 𝜷 = ( 𝒃0 𝒃 𝐁1 𝐁2 ⋮ 𝐁𝑘) (𝑃(𝐾+1)+𝐾)×1 𝒃0 = vector intercept, 𝒃 = vector slope for parallel model 𝐁𝑖= matrix slope for nonparallel model regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 299 𝐈𝐾×𝐾= matrix identity 𝒙𝒊 = vektor covariat without intercept c. link function the link function is a function that connect systematic components and the expected (average) value the random component explained the relationship working e(𝒚) = n𝒑 with explanatory variable in linier predictor. we have model 𝒑 directly or model a monotonous function. 𝐸 (𝒚𝑖|𝒙𝑖) = 𝑔(𝒑𝑖) = 𝜼𝑖 = 𝐗𝑖 𝑇𝜷 (3) elastic net penalty suppose 𝜷 has the length q and 𝛽𝑗 shows the j element. wurm, rathouz, & hanlon [5] wrote the objective elastic net function as: 𝑀(𝜷;𝛼,𝜆,𝑐1,…,𝑐𝑄) = − 1 𝑁∗ ℓ(𝜷)+𝜆∑ 𝑐𝑗 (𝛼|𝛽𝑗|+ 1 2 (1−𝛼)𝛽𝑗 2) 𝑄 𝑗=1 (4) where: ℓ(𝜷) = ∑ ℓ𝑖(𝜷) 𝑁 𝑖=1 and ℓ𝑖(𝜷) = 𝐿𝑖(ℎ(𝐗𝑖 𝑇𝜷)) in model ℓ(𝜷) is loglikelihood function, 𝜆 > 0 and 0 ≤ 𝛼 ≤ 1. wurm, rathouz, & hanlon [5] wrote elastic net objective function for each model shape derived from equation 4 as follows: objective function for parallel model is: 𝑀(𝒃0,𝒃;𝛼,𝜆) = − 1 𝑁∗ ℓ(𝒃0,𝒃)+𝜆∑(𝛼|𝑏𝑗|+ 1 2 (1−𝛼)𝑏𝑗 2) 𝑃 𝑗=1 objective function for nonparallel model is: 𝑀(𝒃0,𝐁;𝛼,𝜆) = − 1 𝑁∗ ℓ(𝒃0,𝐁)+𝜆∑∑(𝛼|𝐵𝑗|+ 1 2 (1−𝛼)𝐵𝑗𝑘 2 ) 𝐾 𝑘=1 𝑃 𝑗=1 objective function for semiparallel model is: 𝑀(𝑏0,𝑏,𝐵;𝛼,𝜆,𝜌) = − 1 𝑁∗ ℓ(𝑏0,𝑏,𝐵) +𝜆(𝜌∑(𝛼|𝑏𝑗|+ 1 2 (1−𝛼)𝑏𝑗 2) 𝑃 𝑗=1 +∑∑(𝛼|𝐵𝑗|+ 1 2 (1−𝛼)𝐵𝑗𝑘 2 ) 𝐾 𝑘=1 𝑃 𝑗=1 ) when 𝜆 ≥ 0 and 𝛼 ∈ [0,1] are tuning parameters and 𝜌 ≥ 0 is tuning parameters which determines the extent to eliminated the parallel term results and discussion firstly, we discuss characteristics of the socioeconomic variables of the household as a general overview of the respondents that used in the study. we use pie charts for the descriptive characteristics of the respondents. it is used to illustrate the frequency of each category in the research variables. table 1. characteristic of responden variable category poverty status total poor almost poor not poor region type rural 5,99 4,49 54,53 65,01 urban 5,56 3,57 25,86 34,99 the total number single 0,39 0,25 7,95 8,59 regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 300 household member living with a couple 1,93 1,03 14,05 17,01 living with a couple with other family members. 9,24 6,78 58,38 74,39 marital status never marriage 0,25 0,11 4,17 4,53 marriage 10,16 7,35 65,58 83,10 divorce 0,07 0,11 2,85 3,03 divorce by death 1,07 0,50 7,77 9,34 gender male 10,34 7,49 70,01 87,84 female 1,21 0,57 10,38 12,16 age 15-64 years old 8,92 6,63 70,86 86,41 65+ years old 2,64 1,43 9,52 13,59 education primary and junior 8,84 5,67 39,16 53,67 senior high school 2,64 2,32 26,64 31,60 collage 0,07 0,07 14,59 14,73 sector economy primary 6,13 3,53 19,54 29,21 secondary 2,92 2,14 17,40 22,47 tertiary 2,50 2,39 43,44 48,32 total 11.05 8,06 80,39 the first step is to test chi-square independence. chi-square independence analysis use when it has a relationship between categorical variables. this method has done at first step. then seeing whether the independent variable/predictor used has a relationship (dependent) with the dependent/response variable. the null hypothesis formulation there is no dependency between poor status and variables the explanation, while the alternative hypothesis there is a dependency between poor status with the explanatory variable. table 2 is the probability value of the results less than 0.05 then it means all independent variables have a dependent relationship with the dependent variable/response. table 2. independent test of category variables on poor status in this study, used the ordinalnetcv function on the ordinalnet software r version 3.61 package. in this study, we compare the results of parallel, non-parallel and semiparallel models with the aic, bic and loglik. in general, parallel and semi-parallel models have similar performance, but non-parallel models are much worse. this model might be due to the unidentified out-of-sample log-likelihood non-parallel model (nonmonotonous cumulative probability) in the first few values of λ. table 3. comparison of the aic, bic and loglik values of the three ordinal regression models model aic bic loglik parallel 3192.01 3269.21 -319.1258 non-parallel 3403.23 3468.56 -339.732 semi-parallel 3209.88 3358.35 -321.276 table 3 shows that the values of aic, bic, and loglik have the smallest on the parallel model, moreover, the lambda parameters obtained in all three models for each fold, in table 4. the variability of lambda values is the lowest in the parallel model. category variables value chi square df p.value region type 41,086 2 0.000 the total number household member 30,327 4 0.000 marital status 27,660 6 0.000 gender 7,491 2 0.024 age 181,148 4 0.000 education 32,699 2 0.000 sector economy 154,836 4 0.000 regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 301 table 4. comparison of lambda values for the three regression models fold parallel model nonparallel model semiparallel model fold1 0,0015 0,0010 0,0019 fold2 0,0012 0,0229 0,0031 fold3 0,0019 0,0292 0,0009 fold4 0,0025 0,0180 0,0040 fold5 0,0012 0,0372 0,0051 table 5 shows that five dummy variables have a positive coefficient, six dummy variables that have a negative coefficient and one dummy variable with a zero coefficient. a positive coefficient value means the chance of the understudy category to be poor is higher compared to the reference category, furthermore the negative coefficient means the chance of the category understudy is smaller for the poor status compared to the reference category. the zero coefficient means that the opportunity for the category studied is not significantly different for poor status compared to the reference category. table 5. ordinal regression variabel category logit(p[y<=1]) logit(p[y<=2]) intercept -2.389 -1.706 region type (*rural) region type (urban) 0.042 0.042 the total number household member *single living with a couple 1.249 1.249 living with a couple with other family members. 0.539 0.539 marital status * never marriage marriage 0.000 0.000 divorce -1.157 -1.157 divorce by death -0.367 -0.367 gender (*male) gender (female) 0.319 0.319 age (*15-64 years old) age (non produktif 65+) 0.390 0.390 education *primary&junior senior high school -0.455 -0.455 collage -2.912 -2.912 sector economy *primer secondary sector -0.408 -0.408 tertiary sector -1.057 -1.057 *baseline category discussion of the results a. residential type regional type is a category of respondent's residential area; there are two categories: urban and rural areas. the location of the household is one of the factors which is often associated with poverty status. the regional type is due to differences in access to primary facilities such as education and health. the results of this study show that the status of the area of residence significantly affected the poverty status of a household. rural households have a higher tendency to become poorer than urban households. this result is in line with some previous studies such as [6] and [7] that suggest that rural households are more vulnerable to poverty due to limited access. b. household size the size of a household indicates the number of people who live in that household. the more people live in a household; then the more resources are needed to keep the household members prosperous. the results of this study show that compared to households consisting of only one person, households of 2 or more people had a higher inclination to live in poverty. the results of this study are in developing countries which regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 302 show that as the number of households increases, the average per capita consumption decreases, indicating that households are approaching poor status. the results of the studies that conducted in developing countries such as [6] and [8] show that the larger household’s size, the lower the average consumption per capita. it indicates that these households are getting closer to poor status. the problem is even though they live in one household, about 20 percent of the items used together [8]. therefore, they must allocate limited income to more needs. c. marital status marital status is related to responsibility for household expenses. someone who has a never married status tends to have income that and use it for personal needs. where, the income generated becomes cumulative from household members, in the results, there is equal opportunities between those who are never married and those who are married. compared to someone who are never married, household with divorcee-household head have less probability to be poor. according to [9], divorcee usually has economic planning and economic adaptation strategies to align with the amount of income a family needs every day of their life. it proves that from the way a divorcee to save, set aside in part piecemeal revenue that could be used to meet the needs of their child's education and are used for urgent needs. d. household gender there are characteristic differences between households dominated by men and women. in general, households headed by women are often identified with households with higher chances of poverty. the research results in some regions, both in developed and developing countries, showed that households led by women are more prone to poverty, because female heads of households generally generate lower incomes and generally have more dependencies ([7], [10], [11]). the yogyakarta data also shows alike. based on the data collected in this study, female heads of households tend to bear a large number of household members. e. head of household age one of the factors that influence a person's level of productivity is age. a person who has at a productive age is likely to have a higher income than someone has at an unproductive age. therefore, it is a common misconception that households with lowincome households are less likely to become poorer. the results of this study support these general assumptions. the result of this research is with the research that has done in [7] and have shown that as a productive age passes, one's income tends to decline, and the risk of becoming poor can higher. f. head of household education education is one of the crucial factors that determine one's well-being. educational attainment increases potential income of individuals, and as a result, increasing income definitely helped them to out from poverty [12]. in line with previous research, this study showed consistent results. households headed by a person with a high school education have a higher tendency to be poor compared to those with a lower middle school. g. economic sector of head of household the field of work in which household heads work has an impact on household poverty status. this is due to differences in income levels in each industry sector. the primary regularized ordinal regression with elastic net approach (case study: poverty modeling in yogyakarta province 2018) pardomuan robinson sihombing 303 sectors comprising agriculture and mining generally have lower income levels than other sectors. the results of this study show that households with households working in the secondary and tertiary sectors have a lower tendency to be poor compared to those with primary income from the primary sector. besides that, the results of this study show that households who work in the tertiary sector have a higher chance of being poor than those who work in other sectors. the result of this research is in line with [13] and [14] that state that the shift from the agricultural sector is effective in alleviating poverty. conclusions some factors determined poverty such as household size, marital status, the gender of household head, age of head of household, level of education of the head of household, and occupation of the head of household. based on aic and bic criteria, the best model to yogyakarta poverty data is parallel model. households that live in villages, have a large number of household members, are headed by women, have elderly household heads, have low education, and work in the primary sector tend to be more vulnerable to poverty. therefore, a simultaneous policy with inclusive 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[14] f. fahar, “kemiskinan dan ketenagakerjaan di kepulauan riau 2014: permasalahan dan implikasi kebijakan,” no. february, 2015. a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units cauchy –jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 13-21 p-issn: 2086-0382; e-issn: 2477-3344 submitted: pebruary 14, 2021 reviewed: april 20, 2021 accepted: october 12, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.11575 a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata1,2, khairil anwar notodiputro2*, bagus sartono2 1mathematics education study program, raja ali haji maritime university, tanjungpinang 2department of statistics, ipb university, bogor email: alonadwinata@umrah.ac.id, khairil@apps.ipb.ac.id*, bagusco@apps.ipb.ac.id *corresponding author abstract generalized linear mixed models (glmm) combined with the l1 penalty (least absolute shrinkage and selection operator/lasso) is called lasso glmm. lasso glmm reduces overfitting and selects predictor variables in modeling. the aim of this study is to evaluate the performance model for predicting covid-19 patients with certain congenital disease that require icu based on the results of blood tests laboratory and patient’s vital signs. this study used binary response variables, 1 if the patient was admitted to the icu and 0 if the patient was not admitted to the icu. the fixed effect predictor variables are the results of blood tests laboratory and patient’s vital signs. the random effect predictor variable is patient's congenital disease. the result showed that the average of accuracy and auc from lasso glmm is more than the average of accuracy and auc from lasso glm by using 5% level of significance. respiratory rate and lactate show a significance effect to predict the icu needs of covid-19 patients. the random effects patient's congenital disease has significance effect at 5% level of significance. it means that the icu needs for covid-19 patients varies among patient's congenital disease. we can conclude that glmm lasso with the random effect of patient’s congenital diseases has better modeling performance to predict the icu needs of covid-19 patients based on the results of blood tests laboratory and patient’s vital signs. the results of this modeling can quickly detect covid-19 patients who need the icu and can help medical staff use icu resources optimally. keywords: covid 19; glmm; glmmlasso; lasso introduction generalized linear model (glm) is an approach that can be used to model the effect of predictor variables on response variables derived from exponential family distribution. for observations in certain groups there is usually a correlation between observations then the glm study is expanded to include random effects on linear predictors. when the glm model added a random effect, the model called generalized linear mixed models (glmm) [1]. glmm modeling has a problem with the number of predictor variables used in relation to complexity in modeling. the more predictor variables used in modeling, the estimation is very unstable [2]. the existence of predictor variables that are not related http://dx.doi.org/10.18860/ca.v7i1.11575 mailto:alonadwinata@umrah.ac.id mailto:khairil@apps.ipb.ac.id mailto:bagusco@apps.ipb.ac.id a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 14 to the response variables in the model will cause overfitting problems. to improve the accuracy of the model prediction, a penalty is added in modeling [3]. the addition of penalty function in modeling was carried out by tibshirani (1996) using the l1 penalty, namely 𝜆 ∑ |𝛽𝑗| 𝑝 𝑗=1 which is called least absolute shrinkage and selection operator (lasso). lambda (λ) in the l1 penalty function is a shrinkage parameter (λ) that determines the amount of shrinkage regression coefficient. lasso reduces overfitting and selects predictor variables in modeling [4]. modeling with a combination of glm and glmm with lasso techniques in this study are called lasso glm and lasso glmm. researchers have discussed various problems on lasso glm, such as arnold and tibshirani (2016) [5], hossain et al. (2015) [6], zhang and zou (2014) [7], simon et al. (2013) [8], friedman et al. (2010) [9]. the lasso glm optimizes the objective function by using coordinate descent optimization. this algorithm is available in the r programming language, namely glmnet package [9]. some researchers have discussed variable selection procedures in glmm using the l1 penalty, including thomson and hossain (2018) [10], groll and tutz (2014) [2], schelldorfer et al. (2011) [11], ibrahim et al. (2010) [12]. the lasso glmm produces stable estimations because penalty l1 can select the important predictor variables used in glmm [2]. the glmms using the l1 penalty are useful whenever there is a grouping structure among high dimensional observations [11]. previous studies also have found an algorithm for estimating the maximum likelihood in the glmm model with the addition of the l1 penalty function. the penalized loglikelihood function maximize using gradient ascent algorithm, this algorithm is called glmmlasso [13]. the glmmlasso algorithm in the r programming language is included in the glmmlasso package [14]. in this study, researchers apply lasso glm and lasso glmm to predict the icu needs for covid-19 patients. the surge in covid-19 cases is putting enormous pressure on the health care system. intensive care units (icu) is one of the health facilities needed by patients with covid-19 confirmation. the study examines the prediction of icu for covid19 patients. the icu needs for covid-19 patients were analyzed using the results of blood tests laboratory, vital signs and the patient's congenital disease. the predictor variables for blood test laboratory results and patient's vital signs were fixed effect, whereas predictor variables for patient's congenital disease were assumed to be fixed effect for lasso glm and random effect for lasso glmm. previous researchers have discussed the performance of lasso glm and lasso glmm modeling on rainfall data, the results showed that modeling with lasso glmm has better performance than lasso glm [15]. to predict the icu needs for covid-19 patients based on laboratory results of blood tests, patient’s vital signs and congenital disease, researchers conducted modeling with lasso glm and lasso glmm. the aim of this study is to evaluate the model's performance in predicting covid-19 patients with certain congenital disease groups that require icu based on the results of blood tests laboratory and patient’s vital signs. methods data the study used data from patients confirmed by covid-19 at the sírio-libanês hospital, são paulo, brasilia. data were collected after 12 hours of confirmed covid-19 patients undergoing treatment in the hospital. total data were 98 patients, with 52 icu patients and 46 non-icu patients. the study used binary response variables, 1 if the patient was admitted to the icu and 0 if the patient was not admitted to the icu. the fixed effect predictor variables for a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 15 modeling totally used 32 variables, 26 variables from the results of blood tests laboratory and 6 variables patient’s vital signs. the fixed effect predictor variables used in modeling can be seen in table 1. researchers assumed patient’s congenital disease as fixed effect predictor variables in modeling using lasso glm and a random effect predictor variable in modeling using lasso glmm. table 1. research variables variable variable name type information y the covid-19 patient's status binary 1 = icu patient, 0 = non-icu patient x1 albumin numeric fixed effect x2 be_venous numeric fixed effect x3 bic_venous numeric fixed effect x4 billirubin numeric fixed effect x5 calcium numeric fixed effect x6 creatinin numeric fixed effect x7 ffa numeric fixed effect x8 ggt numeric fixed effect x9 glucose numeric fixed effect x10 hematocrite numeric fixed effect x11 hemoglobin numeric fixed effect x12 lactate numeric fixed effect x13 leukocytes numeric fixed effect x14 linfocitos numeric fixed effect x15 neutrophiles numeric fixed effect x16 p02_venous numeric fixed effect x17 pc02_venous numeric fixed effect x18 pcr numeric fixed effect x19 ph_venous numeric fixed effect x20 platelets numeric fixed effect x21 potassium numeric fixed effect x22 sat02_venous numeric fixed effect x23 sodium numeric fixed effect x24 ttpa numeric fixed effect x25 urea numeric fixed effect x26 dimer numeric fixed effect x27 bloodpressure_diastolic numeric fixed effect x28 bloodpressure_sistolic numeric fixed effect x29 heart_rate numeric fixed effect x30 respiratory_rate numeric fixed effect x31 temperature numeric fixed effect x32 oxygen_saturation numeric fixed effect research methods modeling was carried out to predict the icu needs for covid-19 patients based on the results of blood tests laboratory, vital signs and congenital diseases. there are many predictor variables used in modeling. we select the variables to determine the important predictor variables, then a simpler model is obtained by adding the l1 penalty function to the model. the algorithms of this research were as follows: a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 16 1. lasso glm modeling predicted the icu needs for covid-19 patients a. determine the optimum lambda value b. lasso glm modeling used the r package glmnet c. analyze the parameters from the modeling results d. determine the model accuracy. 2. lasso glmm modeling predicted the icu needs for covid-19 patients a. determine the optimum lambda value b. lasso glmm modeling used the r package glmmlasso c. analyze the parameters from the modeling results the random effects in modeling used hypothesis 𝐻0:𝜎 2 = 0. this hypothesis was tested by using likelihood ratio, 𝐺2 = 2(loglik𝐿𝐴𝑆𝑆𝑂𝐺𝐿𝑀𝑀 − loglik𝐿𝐴𝑆𝑆𝑂𝐺𝐿𝑀). if 𝐺 2 > 𝜒(𝑑𝑏=1,𝛼=0.05) 2 then 𝐻0 is rejected. d. determine the model accuracy. to evaluate the performance of lasso glm and lasso glmm, researchers have chosen the best model to predict the hospitalization needs of a patient with covid-19. the best model was selected based on accuracy and auc. the steps for selecting the best model were as follows: a. partition data with a composition of 80% modeling data and 20% validation data. data partitioning was performed 30 times b. modeling the lasso glm and lasso glmm used modeling data for each replication c. assessing model performance based on auc and accuracy values using validation data for each replication d. statistically perform a performance difference of lasso glm and lasso glmm used paired sample t-test. results and discussion 1. lasso glm modeling lasso glm selects variables based on λ. the λ optimum is obtained when the binomial deviance value is minimum. cross validation plot to optimize lasso glm shrinkage parameters is shown in figure 1. figure 1. cross validation plot to optimize glm lasso shrinkage parameters based on figure 1, the optimum λ was 0.024. the predictor variables included in the modeling are fixed effect predictor variables. there are 26 features laboratory blood test results and 6 patient's vital signs, and a patient's congenital disease as dummy variable. a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 17 lasso glm modeling used the r package glmnet. the plot of the lasso glm coefficient for each log λ value can be seen in figure 2. the regression coefficient with non-zero values results from the lasso glm modeling is shown in table 2. figure 2. plot of lasso glm coefficients for each shrinkage parameter value table 2. lasso glm coefficient variables coefficient lactate -0.45 p02_venous -1.82 sodium -0.03 bloodpressure_sistolic 0.83 respiratory_rate 3.78 oxygen_saturation 4.26 htn 0.23 disease group 1 -0.52 2. lasso glmm modeling the same as lasso glm, lasso glmm also required optimum λ in modeling. figure 3 shows the binomial deviance value for each value of λ. the optimum λ is 19.6 that obtained when the smallest deviance. figure 3. cross validation plot for optimizing lasso glmm shrinkage parameters there are 26 features laboratory blood test results, 6 patient's vital signs and a patient's congenital disease as random effect in lasso glmm. the modeling use r a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 18 package glmmlasso. the plot of the lasso glmm coefficient spread for each λ can be seen in figure 4. the regression coefficients go to zero along to the increasing λ. the regression coefficient of the lasso glmm modeling with λ = 19.6 result 4 non-zero predictor variables that is shown in table 3. figure 4. plot of lasso glmm coefficients for each shrinkage parameter table 3. lasso-penalized logistic mixed effects regression model (glmm-lasso) fixed effects coefficient standard error z p(>|z|) (intercept) -6.88 0.54 -12.636 0.000 lactate -0.65 0.38 -1.70 0.08 bloodpressure_systolic 1.55 1.88 0.82 0.41 respiratory_rate 5.11 1.25 4.09 0.04 oxygen_saturation 8.64 5.36 1.61 0.11 the patient’s congenital disease as random effect had standard deviation 0.8262 with 𝐺2 = 4.12 dan 𝜒(𝑑𝑏=1,𝛼=0.05) 2 =3.84. then, 𝐻0 is rejected. it means that the random effects for patient's congenital disease was significant at 5% level of significance. 3. selection of the best model data were divided randomly with a composition of 80% modeling data and 20% validation data. furthermore, there are 79 patients as modeling data and 19 patients as validation data. data partitioning was carried out in 30 replications. the optimum λ is obtained based on the modeling data taken for each replication. a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 19 figure 5. auc of lasso glm and lasso glmm for 30 replications furthermore, the lasso glm and lasso glmm modeling was carried out for each replication. assessment of modeling performance use the accuracy and auc in the validation data. comparison of the accuracy and auc of 30 replications for each model is shown in figure 5 and figure 6. figure 6. accuracy of lasso glm and lasso glmm for 30 replications the performance differences of lasso glm and lasso glmm can be statistically stated by paired sample t-test of auc and accuracy. the results of the paired sample t-test for these two models can be seen in table 4. the hypothesis about accuracy or auc of the two models is as follows:  h0: average accuracy of lasso glm is less than or equal to average accuracy of lasso glmm  h0: average auc of lasso glm is less than or equal to average auc of lasso glmm table 4. the paired sample t-test of accuracy and auc criteria t-stat p-value accuracy 5.5746 0.0000 auc 2.2058 0.0178 the t-test results in table 4 showed the p-value for accuracy and auc less than 0.05. it means the average of accuracy and auc from lasso glmm is more than the average of accuracy and auc from lasso glm by using 5% level of significance. discussion the ability to identify patients who need the icu is needed. the solution to this problem can be done by identifying the most important variables that affect the icu needs for covid-19 patients. the paired sample t-test of accuracy and auc in table 4 showed that modeling with lasso glmm has better performance than lasso glm. figure 4 shows the effect of the predictor variables for each lambda value. by using lambda 19.6, this model produced four non-zero fixed effect predictor variables which are the focus of attention to predict the icu needs of covid-19 patients, namely lactate, blood pressure systolic, respiratory rate and oxygen saturation. among these four predictors, only respiratory rate had a significant effect at the 5% level of significance and lactate had a significant effect at the 10% level of significance. meanwhile, blood pressure systolic and oxygen saturation had no significant effect. the odds ratio of respiratory rate was 165.67. it meant that the odds of covid-19 a combination of generalized linear mixed model and lasso methods for estimating number of patients covid 19 in the intensive care units alona dwinata 20 patient required the icu was 165.67 higher given an increase of a unit respiratory rate (respirations per minute/rpm) than before the increase. covid-19 damages the respiratory system. respiratory rate is one measure used to identify respiratory tract infections immediately before and during the first days of symptoms. the normal respiratory rate for adults at rest is 12 to 20 rpm [16]. the findings of a study suggest that the stability of nightly respiratory rate measurements in healthy individuals at night rest is a useful metric for tracking changes in health [16]. the odds ratio of lactate was 0.52. it meant that the odds of a covid-19 patient required the icu was 0.52 lower given an increase of a unit lactate (mmol/l) than before the increase. arterial lactatemia higher than central vein (a reversed delta a-cv lactate) indicates a disturbance in the mitochondrial metabolism of lung cells caused by severe inflammation [17]. an increase in one unit of venous blood lactate reduces reversed delta a-cv lactate. lasso glmm produced an auc of 0.96. this means that glmm lasso has good predictive performance in predicting the icu needs of covid-19 patients. the random effects patient's congenital disease was significant at 5% level of significance. it means that the icu needs for covid-19 patients varies among patient's congenital disease. we can conclude that glmm lasso with the random effect of patient’s congenital diseases has better modeling performance to predict the icu needs of covid-19 patients. conclusions in this study, modeling with lasso glmm has better performance to predict the icu needs of covid-19 patients than lasso glm. lasso glmm has good predictive performance in predicting the icu needs of covid-19 patients with an auc 0.96. respiratory rate has a significant effect at 5% level of significance and lactate has a significant effect at 10% level of significance in lasso glmm. respiratory rate shows the largest significance effect to predict the icu needs of covid-19 patients. random effects of patient congenital disease had a significant effect on covid-19 patients requiring icu at 5% level of significance. it means that the icu needs for covid-19 patients varies among patient's congenital disease. we can conclude that glmm lasso with the random effect of patient’s congenital diseases has better modeling performance to predict the icu needs of covid-19 patients based on the results of blood tests laboratory and patient‘s vital signs. acknowledgments the authors would like to thank to all persons who contributed to the improvement of this paper. references [1] j. jiang, linear and generalized linear mixed models and their applications. 2007. 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[17] g. nardi et al., “lactate arterial-central venous gradient among covid-19 patients in icu: a potential tool in the clinical practice,” crit. care res. pract., 2020, doi: 10.1155/2020/4743904. learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 181-187 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 29, 2020 reviewed: march 17, 2021 accepted: april 11, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.10212 learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati1, aprilia divi yustita2, siska aprilia hardiyanti3, i wayan suardinata4 1,2,3,4politeknik negeri banyuwangi email: ika@poliwangi.ac.id abstract moocs is a learning system in the form of online courses that is massive and open to allow participants to enjoy unlimited content and can be accessed via the web. mathematical techniques taught using moocs which will be developed in the following year are expected to be liked by students. the purpose of this study was to determine student interest in studying moocs. this study uses a dummy regression model on learning hours in each category. dummy regression is considered a suitable model because dummy regression can quantify qualitative data. qualitative data were obtained from a questionnaire distributed to 240 students. the questionnaire contains indicators of student moocs interest, including cognitive, affective, and psychomotor interests. the result of this study is the amount of time studying mathematics influenced by students' interest in learning mathematics through moocs by 60.7%, and the rest 39.3% is influenced by other factors. the model is yi = 1,562 + 4,729 d1 + 1,461 d2 + 𝜀𝑖 . so it can be concluded that the interest of students who want to study mathematics through moocs is the highest with an average student learning hours of 4,729 minus 1,562 equal to 3,647 hours. keywords: dummy regression model; learning interest; moocs introduction massive open online courses (moocs) can be qualified as revolution education has begun to grow and become popular today. individuals can get training in the areas needed and developed with educational training that is open to all students throughout the world [1]. moocs caters to a large number of students and provides a combination of open online courses, short video lectures, automated conversations, quizzes, peer and self-talk, and student collaboration through discussion forums. there are various kinds of moocs designed according to the level of thinking since 2016 [2]. massive open online courses (moocs) contribute significantly to individual empowerment because they can help people learn about various topics [3]. the goal of moocs is the best learning resource and new ways of learning in the classroom. learning can help students learn fully, work together, and is also supported by expert guidance [4]. the use of moocs in learning already exists in various countries. singapore moocs can reduce university and university level tuition fees improve community access to such courses. they also provide skills and job training for community members [5]. this contradicts the results of research which state that moocs can provide new forms of learning through technology and save significant costs for education [6]. in portugal, there is a study that states there is a relationship between interest in educational success [7]. other studies add learning competencies that can influence participation, perseverance, and sustainability [8]. a good formal education system supports the students involved. factors that affect this performance according to http://dx.doi.org/10.18860/ca.v6i4.10212 mailto:ika@poliwangi.ac.id learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati 182 a survey conducted at the nigerian private university (redeemer university) namely hours of study [9]. while the learning domain that must be learned in learning can be categorized as the cognitive domain (knowledge), the psychomotor domain (skills), and the affective domain (attitude) according to bloom [10]. because this research analyzed interests published in the domain which is an indicator that shows interest in moocs. the analysis used dummy analysis. dummy analysis can be used to predict interest. the research that has been done is the students' interest in soap [11]. the dummy regression model is also used for the performance of students majoring in mathematics fmipa [12]. dummy variables have often been used in strategy research to study the effects of categorical variables [13,14]. the advantage of using these puppet variables, variables 1 and 0 is that they can be questioned and interpreted as the resulting regression estimates [15]. methods multiple linear regression analysis the dummy regression analysis is a double linear regression analysis whose variable is qualitative. so before the use of dummy regression analysis, it must first be understood the analysis of a double linear regression [12] following on each observation, represented the ith bservation, applies the equation 𝑌𝑖 = 𝛽0 + 𝛽1𝑋1𝑖 + 𝛽2𝑋2𝑖 + ⋯ + 𝛽𝑝𝑋𝑝𝑖 + 𝜀𝑖 (1) system equations (1) can be written in the form of a matrix, by defining each matrix into the following matrix: 𝑌 = [ 𝑌1 𝑌2 ⋮ 𝑌𝑛 ] ; 𝑋 = [ 1 𝑋11 𝑋12 ⋯ 𝑋1𝑘 1 𝑋21 𝑋22 ⋯ 𝑋2𝑘 ⋮ 1 ⋮ 𝑋𝑛1 ⋮ ⋯ ⋮ 𝑋𝑛2 ⋯ 𝑋𝑛𝑘 ] ; 𝛽 = [ 𝛽0 𝛽1 ⋮ 𝛽𝑛 ] ; 𝜀 = [ 𝜀1 𝜀2 ⋮ 𝜀𝑛 ] (2) or equations (2) can be written in the form of another matrix as follows : 𝐘 = 𝐗𝛃 + ϵ (3) based on the assumptions above𝜀𝑖 ~𝑁(0, 𝜎 2), then the equation (1) can be written in the form of expectation value: 𝐸(𝑌𝑖 ) = 𝛽0 + 𝛽1𝑋1𝑖 + 𝛽2𝑋2𝑖 + ⋯ + 𝛽𝑘 𝑋𝑖𝑘 (4) estimation parameters the estimation of the parameters can be obtained using the smallest quadratic method so that the equation (4) can be written in a matrix form : �̂� = (x′x)−1x′y (5) hypothesis testing was conducted to test the overall regression parameter. overall test of the regression parameters as follows: h0: β0 = β1 … = βk = 0 h1: at least one βj ≠ 0 learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati 183 the sum of squares total (sst) is the total of sum squared regression (ssr) and the sum of squared error (sse), or it can be written: sst=ssr+sse (6) test statistics which used are f test statistics: f = ssr/dfr sse/dfe = msr mse explanation dfr = degree of freedom in regression dfe = degree of freedom in error msr = mean sum of squares regression mse = mean sum of squares error h0 rejected if 𝐹0 > 𝐹(𝛼.𝑘,𝑛−𝑘−1) by minimizing the number of squared errors, it is obtained: sse = ∑ (yi − ŷi) 2n i=1 (7) sst = ∑ (yi − y̅i) 2n i=1 (8) from equation (6), (7), and (8) obtained 𝑆𝑆𝑅 = �̂�′𝑋′𝑌′ − (∑ 𝑌𝑖 𝑛 𝑖=1 ) 2 𝑛 (9) when variable a free variable is inserted one by one gradually into a regression equation, it is performed a sequential f test [16] hypothesis testing for partial regression coefficient parameters the f test is used to determine the effect of the independent variable on the dependent variable simultaneously. after the f test is carried out, the t-test or partial regression test is carried out. this test is used to determine the effect of each independent variable on the dependent variable. partial regression hypothesized will be testing h0 : βj = 0 h1 : βj ≠ 0 test statistics: thit = β̂j se(βj) (10) h0 rejected if |thit| > 𝑡(𝛼 2 ;𝑛−𝑘−1) coefficient of determination after knowing the effect simultaneously and the effect of each independent variable. the next step of analysis is to find the percentage of the independent variables as a whole to the dependent variable. multiple coefficients of determination r2 measures the proportion of total diversity in the y-free variables that can be explained by the regression equation model together. size of regression coefficient determined by the formula: r2 = ssr sst (11) then from the independent variable and the dependent variable, the model is determined. there are many ways to build a regression model whose free variables contain. variable a qualitative variable, one of which is using a variable doll or commonly called a dummy variable. dummy variables are used as an attempt to see how learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati 184 the classifications in the sample affect the estimation parameters. variables dummy also tries to make the quantification of qualitative variables. for example, if you want to estimate variable. the value of a variable y is influenced by one variable quantitative variable (x) and one variable qualitative free variable that has two categories, such as category 1 and category 2. the dummy model of the example is a. y = a + bx + cd1 (dummy intercept model) b. y = a + bx + c(d1x) (dummy slope model) c. y = a + bx + c(d1x) + dd1 (dummy intercept and slope model) this research used dummy intercept model or sympel dummy regression model by modupe by the formula [9] yi =bo+b1zi+ei bo = intercept b1 = regression coefficient zi = 1 if the unit to i is a group that is = 0 if the unit to i as the reference group results and discussion data presentation respondents, in this case, were 240 students. these students have been given knowledge about the moocs that was developed. students are given a questionnaire totaling 20 questions. the questionnaire was designed to contain indicators of interest in the cognitive, affective, and psychomotor domains. after completing the questionnaire then make groups of them. it is namely students into students who are interested in learning mathematics through moocs (a), students who do not like to learn mathematics through moocs (b), and students who do not want to learn mathematics through moocs (c). the grouping scores can be seen in table 1. table 1. student interest grouping score category score interval a 60-80 b 40-59 c 20-39 the data in the research has been categorized as in table 1 then it was changed to dummy variables. the reference category is chosen. it is c category. so that category c becomes d0, category a becomes d1, and category c becomes d2. test of significance and coefficient of determination the significance test is divided into 2 tests, namely the f-test to find out the significance simultaneously and the t-test to find out the partial significance. f-test is used to test a regression that is by testing hypotheses that involve more than one coefficient. the f-test can also be used to test the linearity of a regression equation. it can also be used to see the effect between independent and dependent variables. f-test on the results of this study can be seen in table 2. learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati 185 tabel 2. result of f test f-statistic f p-value 183,313 2 0.000 table 2 indicate that the value of the f-test is 183.313 with a p-value of 0,000, it can be stated that the model or independent variable factors, in this case, the variable of interest in studying engineering mathematics through moocs affects the dependent variable, the amount of time studied the t-test can be used to test hypotheses about individual coefficients, the ttest is also often called a partial test. the results of the t-test can be seen in table 3. tabel 3. result of t test coefficient t-value sig d0 6.282 0.000 d1 16.040 0.000 d2 5.257 0.000 based on table 3 above, it is known that all predictor variables significantly influence the model. it can be seen sig value which has a value less than 0,05. so, the regression coefficient in each category produced on the variable significantly influences the number of learning hours. the coefficient of determination is used to find out how much influence is dependent on the independent variable. it can be seen in table 4. tabel 4. table of effect from the amount of time studying in mathematics towards students interest to moocs model r r2 1 0.779a 0.607 a. predictors: (constant), d2, d1 b. dependent variable: the amount of time studying mathematics table 4 reflected the correlation between the amount of time studying mathematics and students’s interest in learning mathematics through moocs. r-value is 0,779. this value shows that their correlation is strong. besides, r2 shows a value of 60.7%, this gives the meaning of the amount of time studying mathematics influenced by students' interest in learning mathematics through moocs by 60.7%, and the rest 39.3% is influenced by other factors. interpretation of dummy regression model the data in this model uses the results of the data which are categorized into 3 dependent variables. it can be seen in table 5. tabel 5. the estimated learning interest through the amount of time studying mathematics in 3 categories model unstandardized coefficient sig. b std. error 1 (constat) 1,562 0,249 0,000 d1 4,729 0,295 0,000 d2 1,461 0,278 0,000 learning interest modelling of poliwangi students to learn mathematics engineering through moocs using dummy regression ika yuniwati 186 table 5 shows that they are students who do not want to learn mathematics through moocs (d0), students who are interested in learning through moocs (d1), students who do not like to learn through moocs (d2) and 1 independent variable the average hours of study (y) are dummy and a linear regression model is obtained yi = 1,562 + 4,729 d1 + 1,461 d2 + 𝜀𝑖 the interpretation of the regression model above is the average study time for students who do not want to learn engineering mathematics by 1,562 hours. the average difference in hours of study for students interested in learning engineering mathematics through moocs with students who do not want to learn engineering mathematics through moocs is 4,729 hours or in other words the interest of students who do not want to study engineering mathematics through moocs is lower than the interest of students interested in learning engineering mathematics through moocs. the average study time for students who interested in learning engineering mathematics by 3,467 hours. the average difference in hours of study for students who do not like to study engineering mathematics through moocs with students who do not want to study engineering mathematics through moocs is 1,461 hours or in other words the interest of students who do not likes to study engineering mathematics through moocs is lower than that of students who do not want to study engineering mathematics through moocs. the average study time for students who do not like study engineering mathematics by 0,101 hours or equals to 6 minutes. conclusions the results of this study are interest of students who do not want to study engineering mathematics through moocs is lower than the interest of students who are interested in learning engineering mathematics through moocs. moreover, the interest of students who do not want to study engineering mathematics through moocs is lower than the interest of students who do not like to study engineering mathematics through moocs. references [1] h. bicen, “determining the effect of using social media as a moocs tool,” procedia comput. sci., vol. 120, pp. 172–176, 2017, doi: 10.1016/j.procs.2017.11.225. 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[16] n. . draper and s. h., applied regression analysis, no. 48. 1981. sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 28-39 p-issn: 2086-0382; e-issn: 2477-3344 submitted: juni 09, 2021 reviewed: september 03, 2021 accepted: october 25, 2021 doi: https://doi.org/10.18860/ca.v7i1.12488 sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana1, ahmad ashril rizal2 1center of data analytic research and services, universitas jenderal achmad yani yogyakarta 2information technology, universitas islam negeri mataram email: adripriadana3202@gmail.com, ashril.rizal@gmail.com abstract the covid-19 pandemic impact has affected all industries in indonesia and even the world, including the tourism industry. the government has conducted many programs to answer the needs of the tourism industry, especially in making tourism and business destination management programs and carrying out activities oriented, especially during the covid-19 pandemic. meanwhile, the government has a role in making policies, especially in the roadmap, for developing the tourism industry. however, the government also needs a way to figure out public sentiments towards the policies that have been implemented. this study aimed to track trending topics and analyze the sentiment of public opinion in instagram to figure out government performance in tourism during the covid-19 pandemic period. the results of trending topics will be classified by sentiment analysis using a lexicon-based and naive bayes classifier. instagram data taken since january 2020 showed the five highest topics in the tourism sector, namely health protocols, hotels, homes, streets, and beaches. of the five topics, sentiment analysis was carried out with the lexicon-based and naive bayes classifier, showing that beaches get an incredibly positive sentiment, namely 80.87%, and hotels provide the highest negative sentiment, 57.89%. the accuracy of the confusion matrix's sentiment results shows that the accuracy, precision, and recall are 82.53%, 86.99%, and 83.43%, respectively. keywords: sentiment analysis; government performance in tourism; covid-19 pandemic period; lexicon based introduction the covid-19 pandemic impact has affected all industries in indonesia and even the world, including the tourism industry, and spreads to various other sectors. the tourism industry in indonesia has links with other sectors such as hotels, restaurants, transportation, and small micro medium enterprises. furthermore, it impacts on souvenir and culinary entrepreneurs, travel agents, and tour guides. the value of the decline in state revenue in the tourism sector due to covid-19 on a national scale is, of course, tremendous. the government should not count and study the impact but pay attention to concrete steps in saving the tourism industry in indonesia. a careful strategy and planning are needed to save the tourism industry in indonesia after covid-19, which can be obtained from social media. one of the social https://doi.org/10.18860/ca.v7i1.12488 sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 29 media that is widely used by indonesian people is instagram. from andi link1 shows that instagram is one of the most-used social media platforms in 2020. in indonesia, there are 63 million instagram users in 2020. data sources from social media are indeed instrumental in research. one of the studies sourced from instagram data analyzed human selfies by determining several hashtags as a basis. we have shown how image data is detected as a human face using the haar cascade method. image analysis to detect human faces using the haar cascade method shows that the applied method produces an accuracy value of 71.48% [1]. collecting data on instagram can be done using the web scraping method. simple additive weighting is successfully applied to the decision support system for selecting endorsement accounts on instagram. this study’s instagram account parameter include the number of followers, the number of likes, the number of comets, and posts that are always updated. the determination of the best parameters for selecting the endorse account on instagram has been successfully carried out, as shown by the system accuracy of 75% [2]. the instagram platform plays an increasingly central role in social media, essential for users to interact or communicate. instagram contributes to developing tourism destinations, which it is clear that instagram and its users are transforming into a new form [3]. previously, yadav conducted research related to trip mode's effect on opinions on hotel aspects using a social media analysis approach. knowing the exact customer expectations will allow service providers to focus more on those aspects that are important. it will help hoteliers prioritize their efforts, allocate resources according to customer needs, and provide tailor-made offers to customers to increase customer satisfaction and optimize resource utilization. with social media being an open source of information, positive or negative sentiments of a hotel's opinion or related aspects can affect a hotel's business. it also provides an opportunity to identify the essential features as perceived by the customer and ascertain what the main reasons for customer dissatisfaction are. the data is taken from hotel visitor reviews based on travel mode on tripadvisor. tripadvisor reviews are divided into five modes of travel with very few single travelers (4% of overall reviews), and the majority are family travelers (44%), couples giving reviews 22%, business travelers 19%, and friends 11% [4]. there is another opinion mining platform for extracting and classifying hotel reviews posted by users on tourism websites. the system visits web pages starting at the given url, extracts reviews from page content, then uses opinion mining to process the content and classifies reviews as positive, negative, and neutral. the proposed process has acceptable accuracy and has the advantage that it does not depend on domains and does not require expensive resources to operate. according to the review, it can be concluded that in the tourism domain, the analysis is made aspect oriented. it is because of the many aspects expressed by users about opinions and mixed sentiments present in a review [5]. the sentiment analysis method is also carried out to research halal tourism in europe. halal tourism has recently received significant attention from academics and practitioners. this study analyzes tweets related to halal tourism. this study's findings can help various stakeholders, such as marketers who want to target the halal tourism market. these studies have also contributed to existing knowledge about tourism in general and halal tourism in particular [6]. the government has conducted many programs to answer the needs of the tourism industry, especially in making tourism and business destination management programs and carrying out activities oriented, especially during the covid-19 pandemic. meanwhile, the government has a role in making policies, especially in the roadmap, for 1https://andi.link/hootsuite-we-are-social-indonesian-digital-report-2020/ sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 30 developing the tourism industry. however, the government also needs a way to figure out public sentiments towards the policies that have been implemented. this study aimed to track trending topics and analyze the sentiment of public opinion in instagram to figure out government performance in tourism during the covid-19 pandemic period. some studies analyzed the sentiment of public opinion in social media that could help the government, or the relevant authorities develop responses or programs to address existing problems in the community, especially during the covid-19 pandemic. shofiya and abidi in 2021 [7] analyzed the sentiment of public opinion about social distancing programs in canada using twitter data during the covid-19 pandemic period. they used the support vector machine (svm) technique to classify sentiment. obiedat et al. in 2021 [8] analyzed the sentiment of public opinion to enhance the government decisions in jordan during covid-19 pandemic based on facebook data. they used whale optimization algorithm & support vector machines (woa-svm) methods compared with some other methods to classify sentiment. habibi et al. in 2021 [9] analyzed the sentiment and modeled the topic of public opinion about covid-19 epidemics in indonesia based on twitter data. they used latent dirichlet allocation (lda) to model topics and naive bayes methods compared with other methods to classify sentiment. prastyo et al. in 2020 [10] analyzed the sentiment of public opinion on twitter about the indonesian government’s handling of covid-19. they used svm with normalized poly kernel to classify sentiment. in this study, the results of trending topics will be classified by sentiment analysis using a lexicon-based and naive bayes classifier. there are some studies that use lexicon-based to analyze sentiment. an arabic sentiment lexicon built through automatic lexicon expansion (moarlex) is a lexicon for a sentiment on a large scale. the lexicon is intended to provide accessible arabic resources that can be used in sentiment analysis tasks. one of the advantages of using the proposed lexicon and the techniques used in constructing it is that it can include terms commonly used in social media [11]. meanwhile, indonesian can use the sastrawi library, which can be accessed openly. a literature review needs to be carried out to provide information about sentiment analysis studies on social media. the researchers introduced various methods, but the most common methods used in the lexicon-based method are sentiwordnet and tf-idf. at the same time, those for machine learning are naïve bayes and svm. choosing the right sentiment analysis method depends on the data itself [12]. different preprocessing methods affect the polarity classification of sentiments on twitter. removing urls, deleting stop words, and deleting numbers affects classifier performance minimally. meanwhile, removing stop words, numbers, and urls is appropriate for reducing noise but not affecting performance [13]. methods data collection first, we did data extraction to collect instagram data by using a web data extraction method. it is used to routinely extract data from a web data source [14]. we got the instagram post data by using hashtags that are near related to tourism in indonesia. the data is used as a dataset is data with captions based on predetermined hashtags in various languages. sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 31 data cleaning process the second step in this research is the data cleaning process. figure 1 shows the data cleaning process, in which there is a processing stage inside of it. in the process, several steps are carried out namely lowercase, removing symbol, stemming, tokenizing, and bag of words. the lowercase process is the process of converting all the letters in the instagram caption to lowercase. it functions to facilitate the next process in determining sentiment. table 1 shows examples of the lowercase process. table 1. lowercase process no. real caption after lowercase process 1. trust and maximize your vacation with @ giliketapang001 guaranteed satisfying fun trust and maximize your vacation with @ giliketapang001 guaranteed satisfying fun 2. are you paid for being sent home? what if we open a small business? are you paid for being sent home? what if we open a small business? 3. just bored at home just bored at home 4. the holiday has ended. hopefully, this feeling of happiness can last even though the vacation time is over? the feeling that i want a year off the holiday has ended. hopefully, this feeling of happiness can last even though the vacation time is over? the feeling that i want a year off figure 1. data cleaning process sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 32 the removing symbol process will delete the symbol in the caption. the deleted symbols is a period (.), comma (,), exclamation (!), ask (?), (@), (#), (https: //), and several other unique symbols and emoticons. this process is done because it is assumed that the caption's scraping results do not determine the sentiment results. table 2 shows examples of captions after removing symbols. the stemming process aimed to replace affix words into root words by removing all affixes, whether it's an affix at the beginning, in the middle, or at the end of the word. at this stage, we used the sastrawi library to replace the affixed word with the root word. tokenizing process is used to collect the number of words in the data set. the data can be a single word. it means that if there are two words or more than two words in the data set, we only can use one word. bag of words is a concept from text analysis. this concept represents the document as an essential information pocket without sorting its words. this method works by counting the total of words’ frequency who appeared in a document dataset. so the output of the bag of words model is a frequency vector. table 2. removing symbol process. no. caption after removing symbol process 1. trust and maximize your vacation with @ giliketapang001 guaranteed satisfying fun. trust and maximize your vacation with giliketapang001 guaranteed satisfying fun. 2. are you paid for being sent home? what if we open a small business? are you paid for being sent home what if we open a small business 3. just bored at home. just bored at home 4. the holiday has ended. hopefully, this feeling of happiness can last even though the vacation time is over? the feeling that i want a year off. the holiday has ended. hopefully, this feeling of happiness can last even though the vacation time is over the feeling that i want a year off determination of trending topics trending topics are determined by counting ten captions with the highest frequency. this determination is done by counting the n-grams after getting the word tokenizing results. sentiment analysis using lexicon base and naive bayes classifier in general, the sentiment analysis process is carried out with the steps shown in figure 2. the lexicon-based process is carried out after cleaning the previously obtained dataset. the number of words in the data will be calculated based on the rules in the data dictionary. the data dictionary is a collection of words in the great dictionary of the indonesian language or kbbi and has been classified into negative and positive classes. naive bayes classifier is a classification with the concept of likelihood when referring to the bayes hypothesis. bayes' hypothesis scientifically determines the relationship between the probability of two events a and b, p (a) and p (b) and the likelihood that event a is formed by b and event b is adapted by a, p (a | b) and p (b | a). so the bayes equation is shown in equation 1. 𝑃(𝐴 𝐵⁄ ) = 𝑃 (𝐵 𝐴)𝑃(𝐴)⁄ 𝑃(𝐵) (1) in this case, the calculated classification is p (a | b), which is the probability that the sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 33 hypothesis is correct (valid) for the observed sample b data. the b data is sample data with an unknown class (label), and a is a hypothesis that b is data with a known class (label). p (a) is the probability of hypothesis a, p (b) is the probability of the observed sample data, p (a | b) is the probability of sample data b if it is assumed that the hypothesis is correct (valid) [15]. figure 2. sentiment analysis proses performance evaluation in this study, performance evaluation was carried out using the confusion matrix as well as precision, recall, and accuracy. precision and recall values are calculated using equation 2 and equation 3. the calculation of predictive accuracy is done using equation 4 [16], 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 𝑇𝑃 𝑇𝑃 + 𝐹𝑃 x 100% (2) 𝑟𝑒𝑐𝑎𝑙𝑙 = 𝑇𝑃 𝑇𝑃 + 𝐹𝑁 x 100% (3) 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 x 100% (4) results and discussion this paper's research results are in the form of trending topics in the tourism sector since the covid-19 pandemic hit. the evaluation results are based on data obtained from instagram. the data distribution used is shown in table 3. data collection was carried out by scraping method using python. the libraries used in natural language processing are the natural language toolkit (nltk) and the sastrawi library as cleaning data in indonesian. trending topics in the tourism sector in this study, the total dataset used is 195136 data. for the first time, the similarity process of each caption is carried out. this function is used because one caption uses a different hashtag, and that hashtag is used as a parameter in this paper. so that one caption will be taken as a dataset if there are the same captions. the search results for trending topics on instagram are the accumulated result of bigram and tri-gram using nltk. there is a word order from each caption that is part of a sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 34 sentence with the same pattern. examples like the caption in english translation (1) “follow health protocols for holidays, it seems complicated” and caption (2) “to the beach you have to obey health protocols, prefer #dirumahaja”. if caption (1) and caption (2) go through the cleaning stage, caption (1) will become “follow health protocols for very complicated holidays” and caption (2) will become “beaches must comply with health protocols, at home”. after going through the tokenizing stage using bi-gram, there will be a caption (1) ∩ caption (2) become health protocol. the number of slices becomes a parameter for determining the trending topic, which is a projection of the cumulative frequency of words that appear. table 3. frequency hashtag no. hashtag frequency 1 #pariwisata 12397 2 #indonesia 13396 3 #liburan 13997 4 #pesonaindonesia 14175 5 #jalanjalan 14196 6 #wonderfulindonesia 14114 7 #wisataindonesia 14248 8 #kuliner 13873 9 #visitindonesia 14113 10 #wisata 14115 11 #exploreindonesia 14141 12 #wisataalam 14269 13 #pantai 14157 14 #dirumahaja 13945 figure 3. wordcloud instagram caption sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 35 figure 4. top 5 instagram trending topics figure 3 and figure 4 show the results of the trending topic of the data dataset. figure 3 is a wordcloud which describes the distribution of text in the dataset. the size of the text is directly proportional to the frequency with which it appears in the dataset. graphically, figure 4 shows the top five trending topics based on the dataset used. the trending sequences that emerged were health protocols (1301), hotels (1227), homes (739), roads (772), and beaches (544). sentiment analysis at this stage, the sentiment results are based on a predefined data dictionary. the lexion-based method used refers to the data dictionary. in addition to calculating bayes's probabilities, the number of words in the caption that fall into the negative and positive class is one of the classification references. for example in english translation, “follow the health protocol for the holidays, it seems like it's very complicated” as the caption (1), will be ”follow the health protocol for a very complicated holiday” after the cleaning process. in the caption, the words, follow/protocol/health/for/holiday/very are neutral words, and the word /complicated/is negative word. figure 5 shows the sentiment results in trending tourism topics on instagram. health protocol is the highest topic among instagram users. however, most wrote positive captions to the health protocol. it is evident from the sentiment calculation results that 73.12% of captions are writing positive things related to health protocols. not a few instagram users have complained about health protocols while on vacation or traveling during a pandemic. still, many instagram users invite and emphasize complying with health protocols to reduce the number of corona cases in indonesia. besides, many hotel complaints not only from tourists but from the hotel and its employees. with the closure of hotel access and the slow pace of hotel openings, many instagram users have complained about being laid off. regarding the topic of beaches, many positive captions have appeared. for example, access to the beach is still open even though health protocols are applied. besides, a caption discusses beaches in indonesia that are getting prettier and cleaner since the covid-19 pandemic hit. we can see that beach sentiment is 80.87% positive and only 19.13% with the negative caption. sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 36 figure 5. sentiment analysis of trending topics algorithm performance the sentiment results in this paper were evaluated using a confusion matrix. the results of true positive (tp), true negative (tn), false positive (fp), and false negative (fn) are shown in table 4. table 4. confusion matrix of sentiment classification class classified as positive classified asnegative positive 2911 435 negative 578 1874 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛: 𝑇𝑃 𝑇𝑃+𝐹𝑃 ∗ 100% = 2911 2911+435 ∗ 100% = 86.99% recall: 𝑇𝑃 𝑇𝑃+𝐹𝑁 ∗ 100% = 2911 2911+578 ∗ 100% = 83.43% accuracy: 𝑇𝑃+𝑇𝑁 𝑇𝑃+𝑇𝑁+𝐹𝑃+𝐹𝑁 ∗ 100% = 2911+1874 2911+1874+435+578 ∗ 100% = 82.53% in this study, the lexicon-based method worked by creating a dictionary of opinion words (lexicon) compiled beforehand. the words contained in the data dictionary were divided into two classes, namely classes containing positive words and classes containing negative words. the dictionary was used to identify whether a sentence contains a certain opinion or not. in the text segmented based on word order, the lexicon method would perform a search process on the data dictionary, which words in the text contained positive or negative words. in the naive bayes classifier process, the results of the search for word classes in the data dictionary in the lexicon process would count the number of words in the dataset that fall into the positive class or the words that fall into the negative class. the naive sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 37 bayes classifier used a prior probability, a true probability value, before carrying out experiments on each label which was the frequency of each label in the training set. the purpose of this process was to classify the caption results on the data in each dataset. the naive bayes classifier method performs the text classification process based on the previously stored training data. in the implementation, there are three stages: making a list of n-grams, then making a list of word classes that have been carried out in the lexicon process and making a classifier. classifiers need to be trained and to do so requires a list of captions classified into positive and negative classes manually. in its implementation, around 250 positive captions and 250 negative tweet captions were used to train the classifier. based on the calculated confusion matrix results, we can see that the accuracy of the sentiment results is 82.53%. precision describes the level of accuracy between the requested data and the predictive results given by the model. the precision result from the sentiment is 86.99%. meanwhile, the recall which describes the success of the model in recovering considerable information is 83.43%. some captions should be positive in the negative class or negative captions classified as positive because the lexicon-based method does not use a learning method. sometimes in indonesian captions, there are ambiguous sentences that are positive, but the words chosen to make the caption classified as negative we can see on an example in the caption (3) of the dataset in english translation like “for those of you who are nagging about wanting to go on a walk, there are more victims, it's better to stay safe with #dirumahaja”. of course, the caption should fall into the positive class. however, words like nagging and victims are included in the negative data dictionary, and words like dirumahaja are classified as positive. thus, the caption contains more negative words so that it will enter the negative class. conclusions this study aims to track trending topics in social media instagram since covid-19 hit. the results of trending topics will be classified by sentiment analysis using a lexicon-based and naive bayes classifier. according to analysis results using instagram data taken since january 2020, it shows the five highest tourism sector topics, namely health protocols, hotels, homes, streets, and beaches. of the five topics, sentiment analysis was carried out with the lexicon-based and naive bayes classifier, showing that beaches get a very positive sentiment, namely 80.87%, and hotels provide the highest negative sentiment 57.89%. the accuracy of the confusion matrix's sentiment results shows that the accuracy, precision, and recall are 82.53%, 86.99%, and 83.43%, respectively. in the future, further investigations will be carried out on how the public perceives the government's performance in the tourism sector during the new normal. moreover, the lexicon-based method is indeed high-speed in classifying text data. however, the sentiment results are very dependent on the data dictionary used and the caption context. in indonesia, netizens often write this article with a specific purpose. captions written using this figure can make the caption fall into the wrong classification. there is a need for a learning method to understand captions with specific figures to fall into the correct classification. sentiment analysis on government performance in tourism during the covid-19 pandemic period with lexicon based adri priadana 38 acknowledgments the authors would like to thank the center of data analytic research and services, universitas jenderal achmad yani yogyakarta, for supporting the data of this research. references [1] a. priadana and m. habibi, “face detection using haar cascades to filter selfie face image on instagram,” in 2019 international conference of artificial intelligence and information technology (icaiit), 2019, pp. 6–9, doi: 10.1109/icaiit.2019.8834526. 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[16] e. m. martín and á. p. del pobil, robust motion detection in real-life scenarios, 1st ed. springer-verlag london, 2012. a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 474-482 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 10, 2022 reviewed: august 27, 2022 accepted: september 13, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.15398 a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi*, nina salsabila sulistiani, pringgo widyo laksono industrial engineering department universitas sebelas maret jalan ir. sutami 36 a surakarta, indonesia 57126 email: cucuknur@staff.uns.ac.id* abstract in an intensed business competition, a company has to improve its competitiveness by focus more on supply chain management. one of the crucial problems in supply chain deals with the allocation of orders. more and more companies are starting to adopt mass customization logistics service (mcls) mode to determine the optimal allocations of order both from suppliers and to customized logistics services at the possible lowest cost. for this purpose, logistics service integrator (lsi) is needed to integrate the logistics tasks which operationally done by functional logistics service provider (flsp). this research aims at developing an optimization model to determine optimal decisions concerning order allocations of the needed items from the manufacturer to the respective suppliers and logistics tasks from lsi to flsps. the problems were formulated using mixed integer linear programming (milp). the results of the analysis show that the demand becomes the only sensitive parameter towards both decision variables and objective function, while the purchasing cost only impact significantly to the objective function. keywords: mixed integer linear programming; order allocation; mass customization logistics services. introduction in an intensed business competition, many companies made their best effort to improve their competitiveness through highly product adjustments, increase product quality, and reduce product costs with timely distribution. hence, supply chain management has become more important in increasing the company competitiveness [1]. the company has to manage its supply chain efficiently to cope with increasing customer variety and demand, the advances of communication technology and information systems, and high competition in the era of globalization, and environmental awareness [2]. the short-term goal of supply chain management is to increase productivity while in the same time reduce total inventory costs and total cycle time. in the long-term, the goal of supply chain management is to increase customer satisfaction, market share, and profits for all parties involved in the supply chain, namely suppliers, manufacturers, distribution centers, and customers [3]. to achieve this goal, it is necessary to have good coordination of each element in the supply chain. several important decisions should be made by the decision makers in the supply http://dx.doi.org/10.18860/ca.v7i3.15398 mailto:cucuknur@staff.uns.ac.id a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 475 chain such as supplier selection, order allocation, and third-party logistics selection. supplier selection activities are important for the company and involving multi criteria decision making techniques due to its problem nature [4-6]. order allocation problem is usually solved using constrained mathematical programming approach which formulated in single or multi-objective formulation. the common objective of the model is to minimize cost or maximizing profit and maximizing total value purchasing [7]. with optimal order allocation the company can run the entire supply chain with its best performance [8]. supplier selection and order allocation have attracted many researchers. for example, the model in [9] proposed multi attribute utility theory determine the optimal order allocation with two stages. in the first stage, supplier selection was performed using and in the second stage the optimal order allocation was found using multi-objective integer linear programming involving social and environmental objectives. other recent model in supplier selection and order allocation were developed by [10]. in the research, the sustainable criteria were used to determine the weight of criteria using best-worst method (bwm). afterwards, the results of the weight were used to determine the suppliers rating and rating were found by the measurement alternatives and ranking according to compromise solution (marcos) method. research [11] only developed an optimization model to determine the optimal order allocation. research [12] considered the transportation alternatives and lateral transhipment in order allocation problem. the model was used to determine the optimal order allocation and transportation alternative for three echelon supply chain consisting of supplier, manufacturer, and retailer. supply chain transportation has to be managed efficiently. hence, according to [1315], more and more companies adopted mass customization logistic service (mcls) mode to make the oprations more efficient. in mcls, customized logistics services are provided where the order allocation of logistics tasks are conducted by logistic service integrator (lsi). the lsi allocates the logistics tasks to functional logistics service providers (flsp). the research to solve mcls problems has been conducted by many researchers. for example, the scheduling problems of the mcls have been solved by [13] for deterministic and by [16] for uncertain flsp’s time. the optimization models have been developed to solve the order allocation problems of mcls such as in [17, 18]. both researches only considered the order allocation of logistics tasks from lsi to flsps. in fact, the manufacturer that uses the lsi services need to determine the optimal allocation of the needed items from the suppliers. hence, an optimization model needs to develop in order to integrating the decision making of order allocations of needed items and logistics tasks. the problem is formulated using mixed integer linear programming (milp) method to determine the allocation of orders to suppliers and the allocation of logistics tasks to flsp to minimize the total supply chain costs. methods the model is formulated using milp method. the objective function of the model is to minimize manufacturer’s costs which comprise of supplier cost, outsourcing services cost, and transportation cost. there are two decision variables in the model, namely the allocation of order from each supplier and the assignment of logistics tasks to the respective flsp. several assumptions that involved in the modeling process are: (1) each supplier can supply more than one product, (2) the quantity of orders to each supplier is assumed to be constant for each period, (3) the budget for purchasing of orders is assumed to be constant for each period, and (4) each has different outsourcing price and a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 476 service capacity. model notations index i : supplier index (i=1…i) f : flsp index (f=1…f) j : procedure index (j=1…j) k : product index (k=1…k) decision variables 𝑋𝐶𝑘𝑖 : the order quantity of product k from supplier i 𝑄𝑓𝑗𝑘 : the number of logistics tasks assigned by the lsi to the flsp f for procedure j for product k 𝑋𝑓𝑗𝑘 { 1, if flsp 𝑓 of procedure 𝑗 is selected for product 𝑘 0, otherwise parameters cki : unit cost of product k from supplier i ($) tc : unit transportation cost per kg ($) wcfjk : the mass of product k in procedure j processed by flsp f (kg) ocki : unit order cost of product k from supplier i ($) b : total budget for procurement ($) dck : demand for product k (unit) capcki : product k capacity from supplier i (unit) pfjk : unit service price of product k processed by flsp f for procedure j ($) afjk : maximum service capacity of flsp f for procedure j and for product k varfjk : variable for linearization m : big positive number (assumption of m value = 1000000) model formulation the formulation of the cost components is shown in equations (1)-(3). equation (1) expresses the supplier cost which determines by multiplying the order allocation with the summation of unit product cost and order cost. equation (2) calculates the total cost of outsourcing services incurred by the company for flsp and lsi services. the total cost was calculated by multiplying the order quantity with service price and the number of logistics tasks performed by flsp. equation (3) calculate total transportation cost which expressed as the function of the mass of product. tbp = (∑ ∑ 𝐶𝑖𝑘 + 𝑂𝐶𝑖𝑘 𝑁𝑘 𝑘 𝑁𝑖 𝑖 ) 𝑥 𝑋𝐶𝑖𝑘 (1) tblo = ∑ ∑ ∑ 𝑋𝑓𝑗𝑘 𝑁𝑘 𝑘 𝑥 𝑃𝑓𝑗𝑘 𝑥 𝑄𝑓𝑗𝑘 𝑁𝑗 𝑗 𝑁𝑖 𝑓 (2) tbt = ∑ ∑ ∑ 𝑇𝐶 . 𝑁𝑗 𝑘 𝑊𝐶𝑓𝑗𝑘 𝑁𝑗 𝑗 𝑁𝑖 𝑓 (3) the constraints of the model are expressed in equations (4)-(12). equation (4) ensures the expenditure to cover all the costs is not over budget. equations (5) and (6) ensure the order quantity covers all the demand and does not exceed the supplier capacity. equation (7) ensures at least one flsp is selected to prevent the service delays. equation (8) is needed to ensure all the demand are processed by flsp. equation (9) is a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 477 needed to ensure the number of logistics tasks assigned to the flsp for each procedure does not exceed the capacity of each flsp for each procedure. equations (10) and (11) express the non-negative and integer values of the decision variables. equation (12) defines the binary decision variable. (∑ ∑ 𝐶𝑖𝑘 + 𝑂𝐶𝑖𝑘 ) ․ 𝑋𝐶𝑖𝑘 + (∑ ∑ ∑ 𝑋𝑓𝑗𝑘 𝑁𝑘 𝑘 . 𝐶1𝑓𝑗𝑘 . 𝑄𝑓𝑗𝑘 . 𝑇𝐶 . 𝑊𝐶𝑓𝑗𝑘 ) 𝑁𝑗 𝑗 𝑁𝑖 𝑓 𝑁𝑘 𝑘 𝑁𝑖 𝑖 ≤ 𝐵 (4) ∑ 𝑋𝐶𝑖𝑘 ≥ 𝐷𝐶𝑘 𝑁𝑖 𝑖 (5) 𝑋𝐶𝑖𝑘 ≤ 𝐶𝐴𝑃𝐶𝑖𝑘 (6) ∑ 𝑋𝑓𝑗𝑘 ≥ 1 𝑁𝑖 𝑓 (7) ∑ 𝑄𝑓𝑗𝑘 = 𝐷𝐶𝑘 𝑁𝑖 𝑓 (8) 𝑄𝑓𝑗𝑘 ≤ 𝐴𝑓𝑗𝑘 (9) 𝑋𝐶𝑖𝑘 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (10) 𝑄𝑓𝑗𝑘 ≥ 0 (11) 𝑋𝑓𝑗𝑘 ∈ {0,1} (12) in equation (4), there is a non-linear function as the result of multiplication of two decision variables. hence, we have to conduct linearization by adding a surrogate variable. equation (13) and (14) show the lower bound and upper bounds of the surrogate variable. in this case, the surrogate variable should not be greater than the integer decision variable to ensure the consistency of the model. 𝑉𝑎𝑟𝑓𝑗𝑘 ≥ 𝑄𝑓𝑗𝑘 − (1 − 𝑋𝑓𝑗𝑘 ) 𝑥 𝑀 (13) 𝑉𝑎𝑟𝑓𝑗𝑘 ≤ 𝑀 𝑥 𝑋𝑓𝑗𝑘 (14) 𝑉𝑎𝑟𝑓𝑗𝑘 ≤ 𝑄𝑓𝑗𝑘 (15) results and discussion optimization results in this section, we give a numerical example and sensitivity analysis to show the implementation of the model and how sensitive the model to the change of some input parameters. in the numerical example, a single manufacturer has to order three kinds of raw materials form three suppliers. all raw materials can be supplied by all suppliers except for raw material a which only supplied by supplier 1 and 2. after determines the order allocation for each supplier, the delivery of the orders is done by single lsi which then order the logistics services to three flsp with eight procedures. unit product cost and unit order cost are shown in table 1. the demand for each product is set at 9,500 units; the maximum budget for procurement expenditure is $5,000 and transportation cost per kg is $0.00023. the other parameters which deal with the flsp activities are shown in table 2. a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 478 table 1. unit product and order cost raw material supplier unit product cost ($) unit order cost ($) supplier capacity (unit) a 1 7.75 0.210 10,000 2 6.15 0.210 5,000 3 b 1 320 0.013 15,000 2 290 0.014 1,000 3 278.6 0.007 8,000 c 1 151.33 0.021 8,000 2 34 0.004 10,000 3 39.65 0.005 1,000 table 2. service cost and fls capacity. flsp procedure raw material service cost ($) flsp capacity (unit) a b c a b c 1 2.2 2.2 2.2 3350 3369 2576 2 2.8 2.8 2.8 4557 3148 5560 3 4.3 4.3 4.3 5847 5288 1170 4 4.3 4.3 4.3 3573 3150 4734 1 5 4.4 4.4 4.4 2311 3395 2048 6 5.5 5.5 5.5 4500 4561 6751 7 5.1 5.1 5.1 3457 2390 4286 8 6.5 6.5 6.5 7890 6732 7865 1 2.3 2.3 2.3 2576 1096 3711 2 3.2 3.2 3.2 2278 5589 3402 3 4.5 4.5 4.5 3553 2911 2999 4 5.2 5.2 5.2 4428 2152 3350 2 5 4.8 4.8 4.8 4113 4866 5821 6 4.6 4.6 4.6 3235 2387 4598 7 4.8 4.8 4.8 2389 8798 6851 8 5.6 5.6 5.6 6541 4531 3452 1 3.4 3.4 3.4 3711 5241 3350 2 3.5 3.5 3.5 3947 4335 4756 3 3.5 3.5 3.5 1320 3350 5899 4 4.8 4.8 4.8 2987 4850 1887 3 5 5.1 5.1 5.1 3541 1741 2435 6 5.0 5.0 5.0 3576 5768 1578 7 5.2 5.2 5.2 8976 2566 1897 8 5.8 5.8 5.8 5456 3459 1765 lingo 18.0 was used to solve the model using the embedded branch and bound method. the global optimum was found in the sixth iteration resulted a minimum cost at $1676.58. the optimal order for each supplier and the order for each flsp is shown in a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 479 table 3 and table 4 respectively. from table 3, the manufacturer should order raw material a from supplier 1 and 2, raw material b from supplier 1 and 3, and raw material c only from supplier 2. as shown in table 4, all flsp are assigned to process the procedure for all materials. table 3. optimal raw material order allocation raw material supplier order allocation (unit) a 1 4500 2 5000 3 0 b 1 1500 2 0 3 8000 c 1 0 2 9500 3 0 table 4. order allocation for each flsp flsp procedure order allocation (unit) a b c 1 1 3350 3369 2576 2 4557 3148 5560 3 5847 5288 1170 4 3573 3150 4734 5 2311 3395 2048 6 4500 4561 6751 7 3457 2390 4286 8 7890 6732 7865 2 1 2439 890 3574 2 996 2017 1 3 2333 862 2431 4 2940 1500 2879 5 3648 4364 5017 6 1424 1 1171 7 1 4544 3317 8 1 1 1 3 1 3711 5241 3350 2 3947 4335 3939 3 1320 3350 5899 4 2987 4850 1887 5 3541 1741 2435 6 3576 4938 1578 7 6042 2566 1897 8 1609 2767 1634 sensitivity analysis the scenario for sensitivity analysis is shown in table 5. six parameters are studied to determine how sensitive the model towards the change of those parameters. for each parameter, we set four values each with the decrease and increase of 15% and 30% from the base line. resume of the results of sensitivity analysis are shown in table 6. from the table we can see that the change of all parameters value has the same effect on the decision variables, both the order allocation and outsourcing decisions. all the parameters value change are insensitive to both decision variables except for the demand. the increase of a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 480 demand by 15% made the model infeasible. this result indicates that when the demand increases by 15% the manufacturer should find other suppliers to fulfill the demand or otherwise requires some suppliers to increase their capacities. two parameters are sensitive towards the objective function, namely the purchasing cost and demand. this becomes an indication for the manufacturer to have high awareness to those parameters especially when their value of those parameters increases. table 5. sensitivity analysis scenarios parameter value changes (%) c -30 -15 0 15 30 tc -30 -15 0 15 30 oc -30 -15 0 15 30 b -30 -15 0 15 30 dc -30 -15 0 15 30 p -30 -15 0 15 30 table 6. resume of the results of sensitivity analysis parameter order allocation objective function suppliers flsp purchasing cost insensitive insensitive sensitive unit transportation cost insensitive insensitive insensitive order cost insensitive insensitive insensitive maximum expenditure cost insensitive insensitive insensitive demand sensitive sensitive sensitive unit outsourcing cost insensitive insensitive insensitive conclusions in this research, we developed a milp model to solve order allocation problem in a supply chain consists of multi supplier, single manufacturer considering mcls to minimize total supply chain costs. mcls was represented by single lsi which responsible to process the delivery of the raw material through a serial procedure done by several flsps. the costs of the supply chain comprise of purchasing cost, transportation cost, order cost, and outsourcing service cost. based on the results of sensitivity analysis, among six parameters there are only one parameter 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[17] x. hu, g. wang, x. li, y. zhang, s. feng, and a. yang, “joint decision model of supplier selection and order allocation for the mass customization of logistics services”, transportation research part e: logistics and transportation review, vol. 120, pp. 76-95, 2018. a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) cucuk nur rosyidi 482 [18] g. wang, x. hu, x. li, y. zhang, s. feng, and a. yang, “multi-objective decisions for provider selection and order allocation considering the position of the codp in a logistics service supply chain”, computers & industrial engineering, vol. 140, 2020. confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 411-419 p-issn: 2086-0382; e-issn: 2477-3344 submitted: may 07, 2022 reviewed: may 15, 2022 accepted: june 23, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.15989 confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad*, i wayan mangku, bib paruhum silalahi department of mathematics, ipb university, bogor, indonesia email: idipbfaisalmuhammad@apps.ipb.ac.id abstract asymtotic normality of an estimator for the mean function of a compound cyclic poisson process in the present of power function trend which introduced by safitri in 2002. to provided information on parameters guarantees (mean function) covered in an interval, it is necessary to find a convidence interval for the mean function of a compound cyclic poisson process in the presence of power function trend. the objectives of this paper are: (i) to construct confidence interval for the mean function of a compound cyclic poisson process with significance level 0 < 𝛼 < 1, (ii) to prove that the probability that the mean function contained in the confidence interval converges to 1 − 𝛼, and (iii) to observe, using simulation study, that the probabilities of the mean function contained in the confidence intervals for bounded length of observation interval. this paper showed that a confidence interval for the mean function and a theorem about convergence of the probability that the mean function contained in confidence interval. the simulation study shows that the probability that the mean function contained in the confidence interval is in accordance with the theorem. the contribution of this study is to provide information for users regarding confidence interval for the mean function of a compound cyclic poisson process in the presence of power function trend. keywords: compound cyclic poisson process; power function tren; mean function; confidence interval; poisson process. introduction there are many events in everyday life that are uncertain, such as the birth and death process [1] the queue process [2] and the estimation of total insurance claims [3], which can be modeled using a stochastic process. a stochastic process is process that describes series of random events at certain time intervals [4]. a special form of stochastic process is the compound poisson process. a compound poisson process is a process of adding sequencess random variables of independent and identically distributed (i.i.d) with certain distribution as many as poisson random variables, and independent of the poisson process. based on the time aspect, stochastic process can be classified in two categories, namely discrete time stochastic process and continuous time stochastic process. a special form of continuous time stochastic process is the poisson process. the poisson process is a counting a process in which the number of events in a poisson distribution time interval. http://dx.doi.org/10.18860/ca.v7i3.15989 mailto:idipbfaisalmuhammad@apps.ipb.ac.id confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 412 based on the intensity aspect, the poisson process can be classified in two categories, that is homogeneous poisson process with constant intensity function (not dependent on time) and the non-homogeneous poisson process with intensity function depends on time. one type of non-homogeneous poisson process is the cyclic or periodic poisson process [5]. the period can be daily, weekly, yearly or in other forms [6]. this non-homogeneous poisson process is widely applied to real phenomena, such as th phenomenon of earthquakes [7], healthcare [8], radio burst rates [9], and traffic accidents [10]. the study of the compound periodic poisson process is widely. this research begins with the estimation of the expected value function on the compound periodic poisson process [11] [12], then it was developed with a power trend [13] [14]. the compound cyclic poisson does not follow the usual distribution patern. one aspect which can be estimated is the mean value. since this value depends on the time of observation, it is called the mean function. in [15], an estimator for the mean function of a compound cyclic poisson process has been constructed and studied. the asymptotic normality of this estimator also has been proven. furthemore, to give assurance information that the mean function is included in an interval, it is necessary to construct a confidence interval for mean function of the compound cyclic poisson process in the presence of power function trend. as an application of the asymptotic normality, it can be determined the confidence interval of the estimator for the periodic component. in [16] studied the confidence intervals for the mean and variance functions of compound poisson process with power function intensity have been studied, while in [17] confidence intervals for the mean and variance functions of compound poisson process with exponential of linear function intensity have been studied. specifically, this research was conducted to (i) to construct confidence interval for the mean function of a compound cyclic poisson process in the presence of power function trend, (ii) to prove convergence to 1 − 𝛼 of probability that the mean function included in the confidence interval, and (iii) to check using simulation study that the probabilities of the mean function contained in the confidence intervals for bounded length of observation interval. the contribution of this study is to provide information for users regarding confidence interval for the mean function of a compound cyclic poisson process in the presence of power function trend. methods the estimator for the mean function suppose that {𝑁(𝑡), 𝑡 ≥ 0} is a nonhomogeneous poisson process with (unknown) intensity function 𝜆 which is assumed to be locally integrable. suppose that 𝜆 has two components, namely a cyclic component (𝜆𝑐 ) with (known) period 𝜏 > 0 and a power function trend component (𝑎𝑠𝑏 ). in other words, for all 𝑎 ≥ 0 and 𝑠 ≥ 0, the intensity function 𝜆(𝑠) can be expressed as 𝜆(𝑠) = 𝜆𝑐 (𝑠) + 𝑎𝑠 𝑏 . (1) the value of b is assumed to be known real number and 0 < 𝑏 < 1 2 . we do not assumed any parametric form for the cyclic component c  , except that it is cyclic or periodic, which satisfies confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 413 𝜆𝑐 (𝑠) = 𝜆𝑐 (𝑠 + 𝑘𝜏) (2) for all 𝑠 ≥ 0 and all k∈ ℕ. suppose that {𝑌(𝑡), 𝑡 ≥ 0} is a process where 𝑌(𝑡) = ∑ 𝑋𝑖 𝑁(𝑡) 𝑖=1 (3) with {𝑋𝑖 , 𝑖 ≥ 1} is a sequence of independent and identically distributed random variables which having mean 𝜇 < ∞ and variance 𝜎2 < ∞, and also independent of {𝑁(𝑡), 𝑡 ≥ 0}. the process {𝑌(𝑡), 𝑡 ≥ 0} is called a compound cyclic poisson process with power function trend [7]. suppose that 𝜓(𝑡) is notation of the mean function of 𝑌(𝑡), that is 𝜓(𝑡) = 𝐸(𝑌(𝑡)) = 𝐸[𝑁(𝑡)]𝐸[𝑋1] = 𝛬(𝑡)μ (4) with 𝜇 = 𝐸(𝑋𝑖) and 𝛬(𝑡) = ∫ 𝜆(𝑠) 𝑑𝑠 𝑡 0 . (5) let 𝑡𝑟 = 𝑡 − ⌊ 𝑡 𝜏 ⌋ 𝜏, where ⌊ 𝑡 𝜏 ⌋ represents the largest integer less than or equal to 𝑡 𝜏 , 𝑡 𝜏 ∈ ℝ and 𝑘𝑡,𝜏 = ⌊ 𝑡 𝜏 ⌋. then, for any real number 𝑡 ≥ 0, 𝑡 can be expressed as 𝑡 = 𝑘𝑡,𝜏 𝜏 + 𝑡𝑟 with 0 ≤ 𝑡𝑟 ≤ 𝜏. let 𝜃 = 1 𝜏 ∫ λ𝑐 (𝑠) 𝑑𝑠 𝜏 0 denotes the global intensity of the periodic component in the process {n(t ), t ≥ 0} and it is assumed that 𝜃 > 0. this 𝜃 can be written as 𝛬𝑐 (𝑡𝑟) + 𝛬𝑐 𝑐 (𝑡𝑟 ) with 𝛬𝑐 (𝑡𝑟 ) = ∫ λ𝑐 (𝑠) 𝑑𝑠 𝑡𝑟 0 (6) and 𝛬𝑐 𝑐 (𝑡𝑟) = ∫ λ𝑐 (𝑠) 𝑑𝑠. 𝜏 𝑡𝑟 (7) by using (6) and (7) and substituting (1) into (5), then for any t ≥ 0, 𝛬(𝑡) can be written as 𝛬(𝑡) = (𝑘𝑡,𝜏 + 1)𝛬𝑐 (𝑡𝑟)+𝑘𝑡,𝜏 𝛬𝑐 𝑐 (𝑡𝑟 ) + 𝑎 𝑏+1 𝑡𝑏+1. (8) by substituting (8) into (4), we have 𝜓(𝑡) = ((𝑘𝑡,𝜏 + 1)𝛬𝑐 (𝑡𝑟)+𝑘𝑡,𝜏 𝛬𝑐 𝑐 (𝑡𝑟 ) + 𝑎 𝑏+1 𝑡𝑏+1) 𝜇. (9) estimation of mean function in [11] an estimator of the mean function 𝜓(𝑡) has been formulated as follows �̂�𝑛,𝑏 (𝑡) = ((𝑘𝑡,𝜏 + 1)�̂�𝑐,𝑛,𝑏 (𝑡𝑟 ) + 𝑘𝑡,𝜏 �̂�𝑐,𝑛,𝑏 𝑐 (𝑡𝑟 ) + �̂�𝑚,𝑏 𝑏 + 1 𝑡𝑏+1) �̂�𝑛 (10) where �̂�𝑐,𝑛,𝑏 (𝑡𝑟 ) = (1 − 𝑏)𝜏1−𝑏 𝑛1−𝑏 ∑ 1 𝑘𝑏 𝑘𝑛,𝜏 𝑘=1 𝑁([𝑘𝜏, 𝑘𝜏 + 𝑡𝑟 ]) − �̂�𝑚,𝑏 (1 − 𝑏)𝑛 𝑏 𝑡𝑟 , (11) �̂�𝑐,𝑛,𝑏 𝑐 (𝑡𝑟) = (1 − 𝑏)𝜏1−𝑏 𝑛1−𝑏 ∑ 1 𝑘𝑏 𝑘𝑛,𝜏 𝑘=1 𝑁([𝑘𝜏 + 𝑡𝑟 , 𝑘𝜏 + 𝜏]) + �̂�𝑚,𝑏(1 − 𝑏)𝑛 𝑏 (𝑡𝑟 − 𝜏), (12) confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 414 �̂�𝑚,𝑏 = (1 + 𝑏)𝑁([0, 𝑚]) 𝑚(1+𝑏) − (1 + 𝑏) 𝑚𝑏 �̃�𝑛 , (13) �̃�𝑛 = (1 − 𝑏) 𝑛1−𝑏 𝜏𝑏𝑏2 ∑ 1 𝑘𝑏 𝑘𝑛,𝜏 𝑘=1 𝑁([𝑘𝜏, 𝑘𝜏 + 𝜏]) − (1 + 𝑏)(1 − 𝑏)𝑛𝑏 𝑁([0, 𝑛]) 𝑛(1+𝑏)𝑏2 , (14) �̂�𝑛 = 1 𝑁[0, 𝑛] ∑ 𝑋𝑖 𝑁([0,𝑛]) 𝑖=1 . (15) with �̂�𝑛 = 0 when 𝑁([0, 𝑛]) = 0. asymptotic normally of the estimator for the mean function theorem 1 (the asymptotic normally of the estimator for the mean function) suppose that the intensity 𝜆 statisfies (1) and locally integrable. if 𝑌(𝑡) statisfies (2), then √𝑛1−𝑏 (�̂�𝑛,𝑏(𝑡) − 𝜓(𝑡)) 𝑑 → normal (0, (𝑘𝑡,𝜏 + 1) 2 𝑎(1 − 𝑏)𝜏𝑡𝑟 𝜇 2 + 𝑘𝑡,𝜏 2 𝑎(1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )𝜇 2) (16) as 𝑛 → ∞. the proofs of theorem 1 can be proved through a rough analysis [11]. results and discussion our main results are a confidence interval for the mean function 𝝍(𝒕) and a theorem about convergence of the probability that 𝝍(𝒕) contained in the confidence interval. corollary 1 (the confidence interval for 𝝍(𝒕)) for given a significant level 𝛼, where 0 < 𝛼 < 1, the confidence interval for 𝜓(𝑡) in the case 0 < 𝑏 < 1 2 is given by 𝐼𝜓,𝑛 = [�̂�𝑛,𝑏 (𝑡) −  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 , �̂�𝑛,𝑏(𝑡) +  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ] with 𝑉𝑛,𝑏 = (𝑘𝑡,𝜏 + 1) 2 �̂�𝑚,𝑏 (1 − 𝑏)𝜏𝑡𝑟 �̂�𝑛 2 + 𝑘𝑡,𝜏 2 �̂�𝑚,𝑏 (1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )�̂�𝑛 2 𝑛1−𝑏 , where  denotes the standard normal distribution and 𝑉𝑛,𝑏 denotes the studenize version of (16). theorem 2 (convergence of probability that 𝝍(𝒕) ∈ 𝑰𝝍,𝒏) for confidence interval 𝐼𝜓,𝑛 of 𝜓(t) given in corollary 1, we have that 𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) → 1 − 𝛼 as 𝑛 → ∞. confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 415 proof of theorem 2: the probability that 𝜓(𝑡) contained in the confidence interval 𝐼𝜓,𝑛 can be computed as follows. . 𝑃 (�̂�𝑛,𝑏 (𝑡) −  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ≤ 𝜓(t) ≤ �̂�𝑛,𝑏 (𝑡) +  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ) = 𝑃 (−−1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ≤ −�̂�𝑛,𝑏 (𝑡) + 𝜓(t) ≤  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ) = 𝑃 (−1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ≥ �̂�𝑛,𝑏(𝑡) − 𝜓(t) ≥ − −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ) = 𝑃 (−−1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ≤ �̂�𝑛,𝑏 (𝑡) − 𝜓(t) ≤  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ) = 𝑃 (−−1 (1 − 𝛼 2 ) ≤ �̂�𝑛,𝑏(𝑡)−𝜓(t) √𝑉𝑛,𝑏 ≤ −1 (1 − 𝛼 2 )). by the studentize version of (16), we have that �̂�𝑛,𝑏(𝑡)−𝜓(t) √𝑉𝑛,𝑏 𝑑 → normal(0,1), as 𝑛 → ∞. therefore 𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) converges to 𝑃 (−−1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ −1 (1 − 𝛼 2 )) as 𝑛 → ∞, where 𝑍 is the standard normal random variable. further we can simplify the above probability as follows. 𝑃 (−−1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ −1 (1 − 𝛼 2 )) = 𝑃 (𝑍 ≤ −1 (1 − 𝛼 2 )) − 𝑃 (𝑍 ≤ −1 (1 − 𝛼 2 )) = 𝑃 (𝑍 ≤ −1 (1 − 𝛼 2 )) − 𝑃 (𝑍 ≥ −1 (1 − 𝛼 2 )) = 𝑃 (𝑍 ≤ −1 (1 − 𝛼 2 )) − (1 − 𝑃 (𝑍 ≤ −1 (1 − 𝛼 2 ))) =  (−1 (1 − 𝛼 2 )) − (1 −  (−1 (1 − 𝛼 2 ))) = (1 − 𝛼 2 ) − (1 − (1 − 𝛼 2 )) = 1 − 𝛼. this completes the proof of theorem 2. simulation of the confidence interval for the mean function the purpose of this simulation is to check the probability that the mean function 𝜓(𝑡) is contained in the confidence intervals for some different significant levels, period, and length of observation interval, using generated data. this simulation was carried out with the help of r software and scilab software for illustration the results. confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 416 the programing stage is carried out by generating the realization of compound periodic poisson process with a power function trend with the formulation of the intensity function: 𝜆(𝑠) = sin 2𝜋𝑠 𝜏 + 1 + 𝑎𝑠𝑏 . in this simulation, we choose significant levels 𝛼 = 1%, 5% and 10%, 𝜏 = 1, 𝑠 = 2.5, 𝑎 = 0.1, 𝑏 = 0.4, 𝑛 = 20, 50 and 100 with 1000 repetitions. table 1. simulation results of confidence interval for the mean function 𝜓(𝑡) 𝛼 𝑛 a b c d e 1% 20 985 15 98.5% 1.5% 0.5% 50 988 12 98.8% 1.2% 0.2% 100 990 10 99.0% 1.0% 0.0% 5% 20 941 59 94.1% 5.9%% 0.9% 50 950 50 95.0% 5.0% 0.0% 100 952 48 95.2% 4.8% 0.2% 10% 20 891 109 89,1% 10.9% 0.9% 50 907 93 90.7% 9.3% 0.7% 100 917 83 91.7% 8.3% 1.7% (a= the number confidence interval containing the parameter, b= the number confidence interval that do not contain the parameter, c= percentage of confidence interval containing the parameter, d= percentage of confidence interval that does not contain the parameter, e= absolute error between 𝛼 and percentage of confidence interval that does not contain the parameters) based on simulation results, percentage of confidence interval that does not contain parameter at 𝑠 = 2.5 and 𝜏 = 1 with 𝛼 = 1%, 5% and 10% fir observation interval [0, 𝑛] with 𝑛 = 20, 50 and 100 respectively from 0.0% − 0.5%, 0.0% − 0.9% and from 0.7% − 1.7%. the error that obtained between 𝛼 and percentage of confidence interval that does not contain parameters also tend to be small, between 0% and 1.7%. this shows that the result of the simulation of the confidence interval for the mean function 𝜓(𝑡) for the compound poisson process with different significant levels is in accordance with the theory obtained. the simulation results based on the first 200 estimators can be seen in figure 1. confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 417 figure 1. confidence interval for the mean function 𝜓(𝑡) based on the first 200 estimators with 𝑠 = 2.5, 𝜏 = 1, 𝛼 = 1% and 𝑛 = 100 the illustration in figure 1 shows some of the results of the confidence interval simulation for the mean function 𝜓(𝑡) at 𝑠 = 2.5 and 𝜏 = 1 based on the first 200 estimators with significance level of 𝛼 = 1% and 𝑛 = 100. it can be seen in the figure that the horizontal line is the true value of the mean function 𝜓(𝑡) and vertical lines are the confidence intervals of the mean function 𝜓(𝑡). if the horizontal and vertical lines do not intersect each other, this indicates that the value of the mean function 𝜓(𝑡) is not in that interval. in figure 1, there are three non intersecting lines which indicates there are three confidence intervals based on the first 200 estimators do not contain the value of the mean function 𝜓(𝑡). since in table 1 there are 10 confidence intervals that do not contain the mean function 𝜓(𝑡), this shows that there are seven confidence intervals based on the 201-st to 1000-th estimators do not contain the value of the mean function 𝜓(𝑡). the illustration results in figure 1 show that the probability of the mean function 𝜓(𝑡) is contained in the confidence interval already close to 1 − 𝛼 for 𝜏 = 1 and 5, 𝛼 = 1%, 5% and 10% for bounded time interval observation. conclusions according to the main results, it can be concluded that confidence interval for the mean function 𝜓(𝑡) of compound cyclic poisson process in the presence of power function trend is 𝐼𝜓,𝑛 = [�̂�𝑛,𝑏 (𝑡) −  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 , �̂�𝑛,𝑏 (𝑡) +  −1 (1 − 𝛼 2 ) √𝑉𝑛,𝑏 ], where  denotes the standard normal distribution and 𝑉𝑛,𝑏 = (𝑘𝑡,𝜏 + 1) 2 �̂�𝑚,𝑏 (1 − 𝑏)𝜏𝑡𝑟 �̂�𝑛 2 + 𝑘𝑡,𝜏 2 �̂�𝑚,𝑏 (1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )�̂�𝑛 2 𝑛1−𝑏 . convergence of the probability that the mean function 𝜓(𝑡) contained in the confidence interval is confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 418 𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) → 1 − 𝛼, as 𝑛 → ∞. the simulation results show that the probability of the mean function 𝜓(𝑡) included in the confidence interval already close to 1 − 𝛼 for a finite length observation interval. a recommendation for futher research can be to use different intensity functiom and different observation function from this study at thr simulation stage, so that they can show more diverse simulation results. refrences [1] e. roflin, "analysis of time series with calendar effects," management science, vol. 26, pp. 106-112, 2000. [2] s. udayabaskaran and v. t. dora pravina, "transient analysis of an m/m/1 queueing system with server operating in three models," far east journal of mathematical science, vol. 101, pp. 1395-1418, 2017. [3] a. chadidjah, "proses poisson dalam estimasi total klaim," in prosiding seminar nasional matematika dan pendidikan matematika dengan tema peran matematika dan pendidikan matematika dalam menghadapi isu-isu global, 2015, pp. 325-336. [4] s. m. ross, introduction to probaility models, ninth ed. florida: academic press inc, 2010. [5] i. w. mangku, "estimating the intensity function of a cyclic posson process," univ of amsterdam, 2001. [6] t. a. walls and j. l. schafer, "models for intensice longitudinal data," oxford univ pr, 2006. [7] j. geng, w. shi, and g. hu, "bayesian nonparametric nonhomogeneous poisson process with applications to usgs earthquake data," eslevier, vol. 41, p. 100495, march 2021. 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[12] f. i. makhmudah, i. w. mangku, and h. sumarno, "estimating the variance function of a compound cyclic poisson process," far east journal of mathematical science (fjms), vol. 100, no. 6, pp. 911-922, sep 2016. [13] i. f sari, i. w. mangku, and h. sumarno, "estimating the mean function of a compound cyclic poissom process in the presence od power function trend," far east j. math. sci, vol. 100, no. 11, pp. 1825-1840, 2016. confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend faisal muhamad 419 [14] a. fajri, "pendugaan ragam pada proses poisson periodik majemuk dengan tren fungsi pangkat," ipb university, 2018. [15] n. i. safitri, "sebaran asimtotik penduga fungsi nilai harapan proses poisson periodik majemuk dengan tren fungsi pangkat," ipb university, 2022. [16] a. fajri, "selang kepercayaan fungsi nilai harapan dan fungsi ragam proses poisson majemuk dengan intensitas fungsi pangkat," ipb university, 2017. [17] s. utami, "interval kepercayaan fungsi nilai harapan dan fungsi ragam proses poisson majemuk degan intensitas eksponensial fungsi linear," ipb university, 2018. 11254-yundari fix cauchy –jurnal matematika murni dan aplikasi volume 6(4)(2021), pages 246-259 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 05, 2021 reviewed: march 16, 2021 accepted: march 30, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11254 invertibility of generalized space-time autoregressive model with random weight yundari1, setyo wira rizki2 1mathematics department, faculty of mathematics and natural science, universitas tanjungpura pontianak, indonesia 2statistics department, faculty of mathematics and natural science, universitas tanjungpura pontianak, indonesia email: yundari@math.untan.ac.id abstract the generalized linear process accomplishes stationarity and invertibility properties. the invertibility property must be having a series of convergence conditions of the process parameter. the generalized space-time autoregressive (gstar) model is one of the stationary linear models therefore it is necessary to reveal the invertibility through the convergence of the parameter series. this article studies the invertibility of model gstar(1;1) with kernel random weight. the result shows that the model gstar(1;1) under kernel random weight fulfills the invertibility property and obtains a finite order of generalized space-time moving average (gstma) process. the other result obtained is the time order of the finite orde 7 � � � 30 . on the triangular kernel resulted in the relatively great value n, so that it does not apply to the kernel with a finite value n. the gstar(1;1) model with random kernel weight is applied to the data of tea production in six plantantion area in west java. the rmse value of data estimation obtained is quite small. it follows the original data pattern at each research location respectively. keywords: autoregressive process; generalized linear process; invertibility; stationarity introduction theoretically, the first order of the autoregressive model, ar(1), of the univariate time series is equivalent to the moving average model with infinity order, ma ( )∞ [1]. it happens to the multivariate model that the vector autoregressive model, var(1), is comparable with the model vma( )∞ [2]. these properties are known as the invertibility property of the autoregression model orde 1. the gstar(1;1) model is a member of the autoregression model family [3]. the question of research, is the gstar(1;1) model also equivalent to the gstma(∞;1) model? the theoretical study of the gstar model has be done in [4] about the model stationarity using the inverse of the autocorrelation matrix. furthermore, [5] tells about the estimation property of the parameter gstar using the least square method, observes the error assumption of the model gstar [6] and the gstar containing outlier [7]. also, the development of the gstar model has been carried out by several researchers such as gstar-garch [8], gstar-sur [9], gstar-kriging [10] and others. invertibility of generalized space-time autoregressive model with random weight yundari 247 several researchers have developed the spatial weight matrix determination, such as [11] using a uniform spatial weight matrix namely the closest neighbors are given the same weight. [5] uses a binary weight matrix considering the uniform weight as a comparison. the weight matrix determination using cross-correlation have also be done by [12]. all of the researchers use distance as the basis of the weight matrix determination. [13] proposes a fuzzy set approach based on observational data in determining the weight matrix, but the approach still produces the weights assigned is not random. determining random weight matrix have be be done by author by using some kernel functions approach [3]. furthermore, the spatial weight effect of the random kernel is also examined for its stationary properties [14]. some of the space-time data applied using the gstar model are the tourist number data at several tourist attractions [15], the tea production data [5], the gdp data in the countries in europe [11], the chili prices prediction [16], the data of log gamma ray [3], the rainfall data [10] and so on. this paper discusses the gstar model with a random weight using the kernel function. the kernel function used is uniform, triangular, epanechnikov, cosine dan gaussian. the kernel functions present the constant function, linear, square, cosine, dan exponential. moreover, the research talks about the weight matrix effect of kernel spatial to its invertibility. in notation, the weight matrix using the kernel function is denoted by � ( )ijw=w ɶ and the parameter matrix gstar(1;1) is represented by �φ . besides, the study discloses the convergence of each kernel function to its invertibility. the article begins with the invertibility theory of the ar(1) model and var(1). the next section explains the kernel function and continued with the study of the gstar(1;1) model under the kernel weight. both results and discussion will be conferred in the next section about the invertibility of the gstar model under both the kernel weight and the convergence to determine the order of the gstma model. in the last section, the paper implements the gstar(1;1) model with the gaussian kernel weight on the tea production data in the six plantation area in west java. methods this section will discuss the theories underlying the research namely the ar(1) model and var(1) which is the basis of the gstar(1;1) model formation. after that, the research studies the properties of each invertibility. the last, it will have conversed about the kernel function used to form the spatial weight matrix of the gstar(1;1) model. invertibility of ar(1) and var(1) process the autoregression process (ar(p)) is defined as below [1]: 1 1 ...t t p t p ty y y aφ φ− −= + + + with 1 2, , ..., pφ φ φ are the autoregression parameters and ta is white noise process with mean is zero and variance is 2aσ . if 1p = then it will be known as process ar(1), which is formulated as: 1t t ty y aφ −= + . besides, the process ar(1) is one of stationer linear models under a stationarity condition 1φ < , model ar(1) has the invertibility property such as: 1 1 2 ( 1) 2 1 t t t k k t t k t k t t t y y a y a a a a a φ φ φ φ φ − − − − − − − = + = + + + + + +⋯ ⋯ invertibility of generalized space-time autoregressive model with random weight yundari 248 the last model obtained is the ma( ∞ )model and the ma model surely stationer with 1φ < and over convergence process. it results in the ar(1) is invertible to ar(1) ≈ ma( ∞ ). on the process ar(1), it considers one random variable with some times. if the observation is worked by using several random variables which each of them through the process ar(1) so it is known as the first order of the vector of autoregressive (var(1)). the model var(1) can be framed as follows: 1 tt t aφ −= +y y � the necessary and sufficient condition of stationarity of the var(1) is a solution of 1 0ki bφ− = less than one. the process var(1) can also be represented in the vector of moving average (vma) or in the other words satisfied the invertibility property [2]. the kernel function the continuous real function is denoted as the kernel function if satisfied the sum of integral is one, symmetrically for each , the mean equal to zero and the finite variance. an example of the kernel function along with its efficiency properties which learned in table 1. the notation { }2(k) ( )r k x dx= ∫ states “roughness” of the kernel function k. the notation 2 2 ( )k x k x dxσ = ∫ is a variance of the kernel function, while the efficiency of the kernel function is obtained from { }5 / 4( *) / (k) 1c k c = , where ( ){ }1/ 524 2( ) ( ) kc k r k σ= . table 1. the shape of kernel function and its properties. the bound of its domain between -1 and 1 (and 0 for outside the domain), except for the gaussian kernel is applicable for all the real numbers [17]. the kernel the form of function ( )r k 2 kσ the efficiency the domain uniform (seragam kernel) ( ) 1 / 2k x = 1 2 1 3 0.9295 (-1,1) triangular ( ) 1k x x= − 2 3 1 6 0.9859 (-1,1) epanechnikov ( )23( ) 1 4 k x x= − 3 5 1 5 1 (-1,1) cosinus ( ) cos 4 2 k x π π =       2 16 π 2 8 1 π − 0.9897 (-1,1) gaussian 21 ( ) e x p 22 x k x π = −       1 2 π 1 0.9512 ℝ the approximation of the kernel function as the weight function is generally used to estimate density and regression function. the procedure of the kernel function is the sum of some kernel function to each point corresponded to every surrounding point (see figure 1). in general, the kernel function of a point linked to it’s the nearest point, for invertibility of generalized space-time autoregressive model with random weight yundari 249 example, x and y are x y k h −      . the notation h is a bandwidth controlling smoothness of the kernel function. figure 1. the plot of kernel density estimation. if there are a lot of observations which is close to point x then f(x) has great value. on other hand, if there are less �� closed to the point x then f(x) has a small value. the gstar(1;1) model with the kernel weight the novel method to determine the spatial weight matrix of the model gstar recommended is by using the kernel function. kernel location weight is attained by adopting the kernel estimator of nadaraya-watson [18] and using an average value of the observation on every single location .iy average value selection of observation in each location is intended to find overall data property (data centering) by ignoring outlier of an observation data. centralization process { }( )iy t following a model gstar (1;1) kernel weight is written as: � 0 1 1 ( ) ( 1) ( 1) ( ), 1,..., , 1,..., n iji i j ii i j wy t y t y t t t t i nφ φ ε = = − + − + = =∑ (1) this model has a spatial weight 1 i j ij n i i y y k h w y y k h= ≠             − = − ∑ ℓ ℓ ℓ ɶ , with notation (.)k is the kernel function, h represents a smoother parameter of the kernel function k and ( )iy t declares an observation on-time t at location i. the term weight matrix can be written as, invertibility of generalized space-time autoregressive model with random weight yundari 250 � � � � � � � 12 1 21 2 1 2 0 0 0 n n n n w w w w w w =              w ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ the result of the weight matrix w , by the kernel function approach, appears to satisfy the properties of the random weight matrix. it is caused by the weight that originated from the random variable data. it is the observation data and fulfilled the property 1 1 n ij j w = =∑ , 1n > . after obtaining the matrix of kernel spatial weight, the parameter estimation of the gstar (1;1) model is carried out using the least squares method followed by validating model. the model validating is held by doing 2 steps namely the parameter significance test and the residual test. the parameter significance test uses the parameter matrix eigen value of the gstar(1;1) model and the residual test using the plot of data error (error randomness) and the qq plot of error (normality). results and discussion the symbol writing of parameter matrix for the gstar(1;1) model based on equation (1) to 0 01 0( , , )ndiag φ φ=φ ⋯ , 1 11 1( , , )ndiag φ φ=φ ⋯ dan =(wij), so that the model gstar(1;1) can be expressed in the matrix as follows � �( ) ( ) ( 1) ( 1) ( ) ( ) ( 1) ( ). t t t t t t t = − + − + = − + 0 1 0 1 y φ y φ wy ε y φ +φ w y ε representation of the gstma model from the gstar(1;1) model as below, ( ) ( ) ( ){ } ( ) ( 1) ( ) = ( 2) ( 1) ( ) t t t t t t = − + − + − + 0 1 0 1 0 1 y φ +φ w y ε φ +φ w φ +φ w y ε ε ( ) ( ) ( ) ( ) 2 2 = ( 2) ( 1) ( ) = ( ) ( 1) ( 2) t t t t t t − + − + + − + − 0 1 0 1 0 1 0 1 φ +φ w y φ +φ w ε ε ε φ +φ w ε φ +φ w y ⋮ ( ) ( ) � 2 0 = ( ) ( 1) ( 2) = ( ) i i t t t t i ∞ = + − + − + −∑ 0 1 0 1ε φ +φ w ε φ +φ w ε φε ⋯ with � 0 1φ=(φ +φ w) . for the gstar(1;1) model which is stationer, all of the eigenvalues �φ are between -1 and 1 so that � 0 for i i→ → ∞φ . this is stated in theorem 1. invertibility of generalized space-time autoregressive model with random weight yundari 251 theorem 1. if � 0 1φ= (φ +φ w), and eigenvalue of �φ is between -1 dan 1 so �lim , n n→∞ =φ 0 for n =0,1,2,…. proof: the matrix � �'φ φ is positive definite so that the matrix � ck×∈φ ℓ can be stated by the singular value decomposition (svd), i.e a diagonal matrix r , min{ , }r r r k×∈ ≤d ℓ and matrix c , ck k× ×∈ ∈u v ℓ ℓ , so that � � ⇔ n n φ = u d v φ = u d v because of matrix elements, d is a root of the eigenvalue of matrix �φ and eigenvalue of � φ is between -1 dan 1 so lim n n→∞ =d 0 . it resulted �lim n n→∞ =φ 0. this caused the invertibility property of the gstar(1;1) model satisfied because of the coefficient of process { }( )t i−y limits to zero. it confirms that gstar (1;1) gstma( ;1)∞≃ . the orde determination of gstma is theoretically done by considering the convergence level of every kernel function used. if it is reviewed from every viewpoint of the kernel function, the limit value approaching zero (table 2). the invertibility property stated that gstar(1;1) gstma( ;1)∞≃ . in statistics, the orde of time gstma on the gstma( ;1)∞ model does not mean infinite, but it can be determined by a finite order such as n. it is stated in theorem 2. table 2. the limit result of each kernel function. it seems that the overall kernel function having a limit value is zero. the kernel the limit result of the function the uniform 1 lim 0 2 n n→∞   =    the triangular ( )lim (1 ) 0n n x →∞ − = the epanechnikov 23lim (1 ) 0 4 n n x → ∞  − =    the cosinus lim cos( ) 0 4 2 n n x π π →∞   =    the gaussian 21 lim exp 0 22 n n x π→∞    − =      theorem 2. if given a process { }( )iy t following the model gstar(1;1) with a weight matrix of kernel spatial and satisfied the invertibility property so that gstar (1;1) gstma( ;1)∞≃ then � � 0 ( 1 ) ( ) ( ) 0 n i i t t t i = − + − − →∑φ y ε φ ε , for n →∞ . invertibility of generalized space-time autoregressive model with random weight yundari 252 proof: given the gstar(1;1) model and the gstma(∞,1) model, with the help of matrix norm from the difference of both equivalent models, obtained 0 1 1 ( 1) ( ) ( ) ( 1) ( ) = ( 1) ( ( ) ( 1)) = ( 1) ( n ni i i i n i i t t t i t t i t t i t i t t = = = − + − − = − − − − − − − − − − − − ∑ ∑ ∑ φy ε φε φy φε φy φ y φy φy φy 2 2 2 2 3 1) ( 2) ( ( ) ( 1)) = ( 2) ( 2) ( ( ) ( 1)) n i i n i i t t i t i t t t i t i = = + − − − − − − − − − − − − − − ∑ ∑ φy φ y φy φ y φ y φ y φy ⋮ = ( ) n t n−φ y furthermore, by using the property of matrix norm, it can be written as, ɶ ɶ ɶ ɶ( ) ( ) n n n n t n t n c c− ≤ − ≤ ≤φ y φ y φ φ , for a constant c ∈ℝ . it is defined previously that � � 0 1 = φ + φφ w so ɶ( ) ɶ0 1 0 1 n n n n n = φ + φ ≤ φ + φφ w wɶ (2) the value of diagonal matrix elements iφ is between -1 and 1, so that ( )0 0 0 0 n n nn i i maks aφ≤ = =     ∑φ φ for 00 1a< < . similar to the parameter ar concerning location that is ( )1 1 1 0 n n nn i i m a k s bφ ≤ = =    ∑φ φ for 0 0 1b< < . it results in equation (2) being: � 0 0 nn n na b≤ +φ w (3) the matrix is a spatial weight matrix and obtained through the kernel function. for each, the kernel function converges to zero (table 2) and 0lim 0 n n a →∞ = 0limb 0 n n → ∞ = . by using norm ∞ℓ on matrix [19] is � � 1 , m a k s ij i j n w ∞ ≤ ≤ =w , therefore that can be attained the norm of rank n of the spatial weight matrix towards zero. in other words, the gstar(1;1) model is equivalent to the gstma(n;1) model.□ by using theorem 2 of every kernel function forming the weight matrix will produce a convergence rate differently. the convergence rate resulted in the discovery of a finite value n which is an orde of the gstma model. the value n for each kernel function with an error of 0.001 can be seen in figure 2. from figure 2, it can be classified into 3 groups based on the size of n, such as the group 1 15n≤ ≤ , 16 30n≤ ≤ , and 30n > . on the group1 15n≤ ≤ applies to the uniform kernel function, n = 11 and the gaussian, n = 9. the functions describe that the convergence reached is relatively fast to head zero. the next group is16 30n≤ ≤ satisfied by the epanechnikov kernel function, and the cosinus, . both groups can be categorized into finite n, but the triangular kernel function, n invertibility of generalized space-time autoregressive model with random weight yundari 253 = 688, can be said .n →∞ it caused by the triangular function containing a differentiable absolute value function, therefore that it takes time to get convergence. each kernel function raised to the power of n forms a geometric sequence. as a result, the ratio of the gaussian function becomes the smallest, therefore, that the convergence is also faster. the next smallest ratio in a row is the kernel function of the uniform, the epanechnikov, the cosine, and the triangular. it proved that the invertibility property of the model gstar(1;1) can approached by using the gstma(n;1) model with .n < ∞ some error values can be seen in table 3. the kerne l the convergency of the parameter matrix ( )( )( ) 0nnn x m aks k xφ ≤ ≈ →w the convergency plot of error 0.001 1 ( ) 2 k x = , 1x < . ( ) (1 )k x x= − , 1x < . 23( ) (1 ) 4 k x x= − , 1x < . ( ) 4 2 k x cos x π π = , 1x < . ( ) 2 1/ 2 exp 2 ( ) 2 x k x π   −   = , rx ∈ . 0 20 40 60 80 100 0 .0 0 .5 1 .0 1 .5 2 .0 n y 0 20 40 60 80 100 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 n y 0 20 40 60 80 100 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n y 0 20 40 60 80 100 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n y 0 20 40 60 80 100 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n y t h e u n if o rm t h e t ri a n g u la r t h e e p a n e c h n ik o v t h e c o s in u s t h e g a u s s ia n n=11 n=688 n=25 n=30 n=9 invertibility of generalized space-time autoregressive model with random weight yundari 254 note: for i jy y x h −= figure 2. the order value of the gstma model is equivalent to the gstar(1;1) model for every single kernel function and the error 0.001. table 3. the finite order of the gstma model satisfied the invertibility property of the model gstar(1;1) the kernel the order of the model gstma on some errors 0.01 0.001 0.0001 the uniform 8 11 14 the triangular 459 688 917 the epanechnikov 17 25 33 the cosinus 20 29 39 the gaussian 6 8 11 case study the gstar(1;1) model with the kernel weight will be applied to the tea production data in the 6 plantation field in west java. the data plot can be seen on figure 3. the figure 3 presents the modelling data with time t=200 by 6 observation locations. the plot of each location shows the data stationary has not been fulfilled (weak stationary), so it is necessary to doing differencing of the data. figure 3. the data plot of each location. the plot illustrates the data is under weak stationary condition and needs differencing so that the data plot is stationary invertibility of generalized space-time autoregressive model with random weight yundari 255 the data is stationary to the mean dan variance after passing once differencing process. the modelling carried out in this paper is the gstar(1;1) model. it does not need to identify model. the next step is to determine the spatial weight matrix using the gaussian kernel (table 1.) with the optimum bandwith value. this spatial weight matrix is random because it uses the function of tea plantation random variable from each the plantation area. the spatial weight matrix with the gaussian kernel is represented as following. � = � � � � � 0 0.25 0.27 0.21 0 0.24 0.23 0.24 0 0.10 0.27 0.11 0.16 0.23 0.16 0.14 0.25 0.14 0.12 0.22 0.18 0.24 0.24 0.25 0.12 0.22 0.17 0 0.17 0.32 0.13 0 0.13 0.32 0.17 0 � � � � � � the next step is to determine the parameter value through the least square estimation method and the parameter significance by considering the eigen value obtained of the parameter matrix of the gstar(1;1) model. the value of parameter estimation with its validation can be seen on table 4. table 4. the result of paremeter estimation using least square method and its validation the parameter of each location the value of parameter estimation confidence interval 95% validation (the eigen value of parameter matrix < 1) phi01; phi 11 -0.32; 0.01 (-0.37;-0.27); (-0.03;0.06) valid phi02; phi12 -0.48; 0.34 (-0.52;-0.44); (0.27;0.40) valid phi03; phi13 -0.56; 0.59 (-0.60;-0.51); (0.53;0.66) valid phi04; phi 14 -0.30; 0.37 (-0.34;-0.27); (0.33;0.42) valid phi05; phi15 -0.49; 0.25 (-0.54;-0.45); (0.19;0.30) valid phi06; phi16 -0.37; 0.18 (-0.40;-0.34); (0.13;0.22) valid the result of the parameter estimation of the gstar(1;1) model with the gaussian kernel weight is shown as below, � � � �( )( ) ( 1)t t= −0 1y + w yφ φ with � 0.32 0 0 0 0 0 0 0.48 0 0 0 0 0 0 0.56 0 0 0 0 0 0 0.30 0 0 0 0 0 0 0.49 0 0 0 0 0 0 0.37 −   −    − =  −   −   −  0φ and � 1 0.01 0 0 0 0 0 0 0.34 0 0 0 0 0 0 0.59 0 0 0 0 0 0 0.37 0 0 0 0 0 0 0.25 0 0 0 0 0 0 0.18         =           φ , invertibility of generalized space-time autoregressive model with random weight yundari 256 the data plot estimated from each location with the gstar(1;1) model can be viewed in figure 4 with its rmse value respectively. it concludes that the estimation value to follow the original value pattern and the rmse value is quite small. figure 4. the plot of the both estimation and original value of the gstar(1;1) model with the gaussian kernel weight. the black line presents the original value and the red line is the estimation value. the residual test of this model can be viewed in figure 5. the residual scatter plot and qq plot show that the assumptions of randomness and normality are met. invertibility of generalized space-time autoregressive model with random weight yundari 257 (a) (b) figure 5. the results of the residual test plot that meet the assumption of randomness (a) and normality (b) conclusion the use of the kernel weight matrix also affects the invertibility property of the model gstar(1;1) to the order of the gstma( ∞ ,1). the result obtained is the time order of the finite orde 7 < � < 30 . on the triangular kernel resulted in the relatively great value n, so that it does not apply to the kernel with a finite value n. the model implementation of the tea production data over 6 plantation area in west java can be applied to this research. invertibility of generalized space-time autoregressive model with random weight yundari 258 this is because the spatial weight to use production data applied to the gaussian kernel function according to the data description from each research location. references [1] g. box, g. jenkins and g.reinsel, time series analysis, forecasting and control, 3rd edition, new jersey: prentice hall, 1994. 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[14] yundari, n. huda, u. pasaribu, u. mukhaiyar and k. sari, "stationary process in gstar(1;1) through kernel function approach," in aip conference proceeding, pontianak, 2020. invertibility of generalized space-time autoregressive model with random weight yundari 259 [15] d. u. wutsqa and suhartono, "seasonal multivariate time series forecasting on tourism data by using var-gstar model," jurnal ilmu dasar, vol. 11, no. 1, pp. 101109, 2010. [16] n. fadlilah, u. mukhiayar and f. fahmi, "the generalized star(1;1) modeling with time correlated galats to red chili weekly prices of some traditional markets in bandung," in aip conference proceeding, bandung, 2015. [17] m. wand and m. jones, kernel smoothing, new york: springerscience+business media b.v, 1995. [18] m. michalack, "time series pediction with periodic kernels," pattern analitic application, vol. 14, no. 0, pp. 283-293, 2011. [19] r. horn and c. johnson, matrix analysis, 2nd, new york: cambridge univesity press, 2013. 𝜼−𝐈𝐧𝐭𝐮𝐢𝐭𝐢𝐨𝐧𝐢𝐬𝐭𝐢𝐜 𝐅𝐮𝐳𝐳𝐲 𝐒𝐨𝐟𝐭 𝐆𝐫𝐨𝐮𝐩𝐬 cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 354-361 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 23, 2021 reviewed: may 25, 2022 accepted: july 17, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.14555 𝜼−𝐈𝐧𝐭𝐮𝐢𝐭𝐢𝐨𝐧𝐢𝐬𝐭𝐢𝐜 𝐅𝐮𝐳𝐳𝐲 𝐒𝐨𝐟𝐭 𝐆𝐫𝐨𝐮𝐩𝐬 mustika ana kurfia*, noor hidayat, corina karim mathematics department, universitas brawijaya, malang, indonesia email: muzematika@gmail.com abstract the fuzzy set theory was introduced by zadeh in 1965 and the soft set theory was introduced by molodtsov in 1999. recently, many researchers have developed these two theories and combined the theory of fuzzy set and soft set became the fuzzy soft set. in this research, we present the idea of the 𝜂 −intuitionistic fuzzy soft group defined on the 𝜂 −intuitionistic fuzzy soft set. the main purpose of this research is to create a new concept, which is an 𝜂 −intuitionistic fuzzy group. to achieve this, we combine the concept of 𝜂 −intuitionistic fuzzy group and intuitionistic fuzzy soft group. as the main result, we prove the correlation between intuitionistic fuzzy soft group and 𝜂 −intuitionistic fuzzy soft group along with some properties of 𝜂 −intuitionistic fuzzy soft group. also, we prove some properties of subgroup of an 𝜂 −intuitionistic fuzzy soft group. an 𝜂 −intuitionistic fuzzy soft homomorphism is also proved. keywords: intuitionistic fuzzy group; intuitionistic fuzzy soft group; 𝜂 −intuitionistic fuzzy group; 𝜂 −intuitionistic fuzzy soft group introduction the theory of fuzzy has been studied by many researchers in various fields. zadeh introduced the fuzzy set theory in [1] by defining a membership function that maps each member of a set to a closed interval of 0 and 1. then atanassov formed the intuitionistic fuzzy set that consist of membership function and nonmembership function in [2]. the theory of fuzzy set and intuitionistic fuzzy set was developed into group theory became fuzzy subgroup in [3] and intuitionistic fuzzy subgroup in [4]. the intuitionistic fuzzy subgroup was studied in various types. for example, the intuitionistic l-fuzzy subgroups formed in [5], the (𝑠, 𝑡] −intuitionistic fuzzy subgroups defined in [6], definition of (𝛼, 𝛽)cut of intuitionistic fuzzy subgroups in [7], and t-intuitionistic fuzzy subgroups in [8]. doda and sharma studied the finite groups of different orders and gave the idea of recording the count of intuitionistic fuzzy subgroups in [9]. zhou and xu extended the intuitionistic fuzzy sets based on the hesitant fuzzy membership in [10]. the concept of the (𝜆, 𝜇) −intuitionistic fuzzy subgroups and normal subgroups were defined in [11]. then the fundamental properties of t-intuitionistic fuzzy abelian subgroup along with the homomorphism of t-intuitionistic fuzzy abelian subgroup were studied in [12]. latif, et al. studied the fundamental theorems of t-intuitionistic fuzzy isomorphism of t-intuitionistic fuzzy subgroup in [13]. moreover, the concept of 𝜉 −intuitionistic fuzzy subgroup, 𝜉 −intuitionistic fuzzy cosets, and 𝜉 −intuitionistic fuzzy normal subgroup were characterized in [14]. based on those research, shuaib, et al. in [15] formed a concept http://dx.doi.org/10.18860/ca.v7i3.14555 mailto:muzematika@gmail.com 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 355 called an 𝜂 −intuitionistic fuzzy subgroup on an 𝜂 −intuitionistic fuzzy set along with 𝜂 −intuitionistic fuzzy homomorphism. while the theory of soft set is introduced in [16] which is an ordered pair of parameter and function that maps each member of parameter to the power set of an empty set. the soft sets constructed in the form of membership function became fuzzy soft sets defined in [17]. maji, et al. introduced the concept of intuitionistic fuzzy soft set which is a generalization of intuitionistic fuzzy set and soft set in [18]. furthermore, the operation properties and algebraic structure of intuitionistic fuzzy soft set were discussed in [19]. soft group on the soft set is defined in [16]. then aygünoglu and aygun in [20] developed the concept of soft group in the form of membership function became fuzzy soft group. the concept of intuitionistic fuzzy soft set to semigroup was applied in [21] and intuitionistic fuzzy soft ideals over ordered ternary semigroup was defined in [22]. the concept of soft group is developed into an intersection called soft int-group in [23]. moreover, karaaslan, et al. in [24] applied the concept of soft int-group into intuitionistic fuzzy soft set by forming the intuitionistic fuzzy soft group and investigate some properties of intuitionistic fuzzy soft group. based on [15] and [24], we introduce the notion of the 𝜂 −intuitionistic fuzzy soft group on the 𝜂 −intuitionistic fuzzy soft set and give some basic properties. moreover, we define the notion of the 𝜂 −intuitionistic fuzzy subgroup and investigate the properties of homomorphism of an 𝜂 −intuitionistic fuzzy soft group. methods the method of this research is literature review, data collecting techniques by conducting review studies of books, notes, and other scientific research results related to the object of the problem. in this paper, we begin by forming the definition of 𝜂 −intuitionistic fuzzy soft set based on the definition of intuitionistic fuzzy soft set and 𝜂 −intuitionistic fuzzy set. then, we form the definition of 𝜂 −intuitionistic fuzzy soft group based on the definition of intuitionistic fuzzy soft group. we continue to prove some properties of the 𝜂 −intuitionistic fuzzy soft group along with subgroup of an 𝜂 −intuitionistic fuzzy soft group. moreover, we continue to define the definition of image and pre-image of an 𝜂 −intuitionistic fuzzy soft group and prove the homomorphism of an 𝜂 −intuitionistic fuzzy soft group. here is shown the definition of intuitionistic fuzzy soft set, 𝜂 −intuitionistic fuzzy set, and intuitionistic fuzzy soft group. definition 1[18]. let 𝑋 be a non empty set and 𝐸 be a set of parameter with 𝐴 ⊆ 𝐸. let ℐℱ(𝑋) be a set of all intuitionistic fuzzy set of 𝑋. an intuitionistic fuzzy soft set of 𝐴 over 𝑋 is defined by γ𝐴 = {(𝑎, 𝛾𝐴(𝑎)): 𝑎 ∈ 𝐴}, where 𝛾𝐴(𝑎): 𝐸 → ℐℱ(𝑋) such as 𝛾𝐴(𝑎) = ∅ if 𝑎 ∉ 𝐴. therefore, for all 𝑎 ∈ 𝐸, 𝛾𝐴(𝑎) is called intuitionistic fuzzy value set of 𝑎. 𝛾𝐴(𝑎) can be written as 𝛾𝐴(𝑎) = {(𝑥, 𝜇𝛾𝐴(𝑎)(𝑥), 𝜇𝛾𝐴(𝑎)(𝑥)) : 𝑥 ∈ 𝑋}, for all 𝑎 ∈ 𝐸. definition 2 [15]. suppose 𝐴 is an intuitionistic fuzzy set of a non empty set 𝑋 where 𝜇𝐴 be a membership function and 𝜈𝐴 be a nonmembership function in 𝐴. let 𝜂 ∈ [0, 1]. an 𝜂 −intuitionistic fuzzy set is defined by 𝐴𝜂 = {(𝑥, 𝜇𝐴𝜂 (𝑥), 𝜈𝐴𝜂 (𝑥)): 𝑥 ∈ 𝑋}, where 𝜇𝐴𝜂 (𝑥) = 𝜓[𝜇𝐴(𝑥), 𝜂] = √𝜇𝐴(𝑥) ⋅ 𝜂 and 𝜈𝐴𝜂 (𝑥) = 𝜓 ′[𝜈𝐴(𝑥), 1 − 𝜂] = √𝜈𝐴(𝑥) ⋅ (1 − 𝜂). 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 356 definition 3 [24]. let 𝐺 be an arbitrary group and γ𝐺 be an intuitionistic fuzzy soft set over universe 𝑋, then γ𝐺 is called intuitionistic fuzzy soft group on 𝐺 over 𝑋 if 1. 𝛾𝐺 (𝑥𝑦) ⊇ 𝛾𝐺 (𝑥) ∩ 𝛾𝐺 (𝑦), 2. 𝛾𝐺 (𝑥 −1) = 𝛾𝐺 (𝑥), for all 𝑥 ∈ 𝐺. results and discussion in this section, we introduce the notion of 𝜂 −intuitionistic fuzzy soft group on 𝜂 −intuitionistic fuzzy soft set. inspiring from definition 1 and definition 2, we define the notion of 𝜂 −intuitionistic fuzzy soft set. definition 4. suppose γ𝐴 be an intuitionistic fuzzy soft set of parameter 𝐴 over universe 𝑋 and 𝜂 ∈ [0, 1]. an 𝜂 −intuitionistic fuzzy soft set of 𝐴 over 𝑋 is defined by γ𝐴 𝜂 = {(𝑎, 𝛾𝐴 𝜂 (𝑎)): 𝑎 ∈ 𝐴}, where 𝛾𝐴 𝜂(𝑎) is an 𝜂 −intuitionistic fuzzy set of 𝑋 defined as 𝛾𝐴 𝜂 (𝑎) = ψ[𝛾𝐴(𝑎), 𝜂], for all 𝑎 ∈ 𝐴, where ψ[𝛾𝐴(𝑎), 𝜂] = {(𝑥, 𝜓[𝜇𝛾𝐴(𝑎)(𝑥), 𝜂], 𝜓[𝜈𝛾𝐴(𝑎)(𝑥), 1 − 𝜂]): 𝑥 ∈ 𝑋}. the value of 𝜓[𝜇𝛾𝐴(𝑎)(𝑥), 𝜂] = √𝜇𝛾𝐴(𝑎)(𝑥) ⋅ 𝜂 and 𝜓[𝜈𝛾𝐴(𝑎)(𝑥), 1 − 𝜂] = √𝜇𝛾𝐴(𝑎)(𝑥) ⋅ (1 − 𝜂). some basic properties such as intersection and union of 𝜂 −intuitionistic fuzzy soft set is proved by the following proposition. proposition 1. the intersection of any two 𝜂 −intuitionistic fuzzy soft sets is an 𝜂 −intuitionistic fuzzy soft set. proof. let γ𝐴 𝜂 = {(𝑎, 𝛾𝐴 𝜂 (𝑎)): 𝑎 ∈ 𝐴} and γ𝐵 𝜂 = {(𝑏, 𝛾𝐵 𝜂(𝑏)): 𝑏 ∈ 𝐵}, respectively be two 𝜂 −intuitionistic fuzzy soft sets of 𝐴 and 𝐵 over 𝑋. for any 𝑐 ∈ 𝐴 ∩ 𝐵, then 𝛾(𝐴∩̌𝐵)𝜂 (𝑐) = ψ[𝛾𝐴∩̌𝐵 (𝑐), 𝜂] = {( 𝑥, 𝜓[min[𝜇𝛾𝐴(𝑐)(𝑥), 𝜇𝛾𝐵(𝑐)(𝑥)] , 𝜂], 𝜓[max[𝜈𝛾𝐴(𝑐)(𝑥), 𝜈𝛾𝐵(𝑐)(𝑥)] , 1 − 𝜂] ) : 𝑥 ∈ 𝑋} = {( 𝑥, 𝜓[min[𝜇𝛾𝐴(𝑐)(𝑥), 𝜂] , min[ 𝜇𝛾𝐵(𝑐)(𝑥), 𝜂]], 𝜓[max[𝜈𝛾𝐴(𝑐)(𝑥), 1 − 𝜂] , max[𝜈𝛾𝐵(𝑐)(𝑥), 1 − 𝜂]] ) : 𝑥 ∈ 𝑋} = {( 𝑥, min[𝜓[𝜇𝛾𝐴(𝑐)(𝑥), 𝜂] , 𝜓[ 𝜇𝛾𝐵(𝑐)(𝑥), 𝜂]], max[𝜓[𝜈𝛾𝐴(𝑐)(𝑥), 1 − 𝜂] , 𝜓[ 𝜈𝛾𝐵(𝑐)(𝑥), 1 − 𝜂]] ) : 𝑥 ∈ 𝑋} = ψ[𝛾𝐴(𝑐), 𝜂] ∩ ψ[𝛾𝐵 (𝑐), 𝜂] = 𝛾𝐴𝜂∩̃𝐵𝜂 (𝑐). hence γ𝐴 𝜂 ∩̃ γ𝐵 𝜂 is an 𝜂 −intuitionistic fuzzy soft set. ∎ remark 1. the union of any two 𝜂 −intuitionistic fuzzy soft set is an 𝜂 −intuitionistic fuzzy soft set. the notion of 𝜂 −intuitionistic fuzzy soft group is defined based on definition 3. an 𝜂 −intuitionistic fuzzy soft group must satisfy 2 axioms to be an 𝜂 −intuitionistic fuzzy soft group. definition 5. let 𝐺 be a group and γ𝐺 𝜂 be an 𝜂 −intuitionistic fuzzy soft set over universe 𝑈, then γ𝐺 𝜂 is called an 𝜂 −intuitionistic fuzzy soft group over 𝑈 if 1. 𝛾𝐺 𝜂 (𝑥𝑦) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦), 2. 𝛾𝐺 𝜂 (𝑥−1) = 𝛾𝐺 𝜂 (𝑥), for all 𝑥, 𝑦 ∈ 𝐺. 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 357 the property of the identity element of the group on an 𝜂 −intuitionistic fuzzy soft group is proved by the following proposition. proposition 2. let 𝐺 be a group and γ𝐺 𝜂 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈, then 𝛾𝐺 𝜂 (𝑒) ⊇ 𝛾𝐺 𝜂 (𝑥) for all 𝑥 ∈ 𝐺. proof. let γ𝐺 𝜂 = {(𝑥, 𝛾𝐺 𝜂 (𝑥)): 𝑥 ∈ 𝐺} be an 𝜂 −intuitionistic fuzzy soft group and 𝑒 be an identity element of 𝐺. for 𝑒 ∈ 𝐺, then 𝛾𝐺 𝜂 (𝑒) = 𝛾𝐺 𝜂 (𝑥𝑥 −1). since γ𝐺 𝜂 is an 𝜂 −intuitionistic fuzzy soft group, then 𝛾𝐺 𝜂 (𝑒) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑥−1) = 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑥) = 𝛾𝐺 𝜂 (𝑥). so, 𝛾𝐺 𝜂 (𝑒) ⊇ 𝛾𝐺 𝜂 (𝑥), for all 𝑥 ∈ 𝐺. ∎ an 𝜂 −intuitionistic fuzzy soft set is called an 𝜂 −intuitionistic fuzzy soft group if it satisfies 2 axioms in definition 5. here we prove the alternative way of 𝜂 −intuitionistic fuzzy soft group. theorem 1. an 𝜂 −intuitionistic fuzzy soft set is called 𝜂 −intuitionistic fuzzy soft group if and only if 𝛾𝐺 𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺. proof. (⇒) let γ𝐺 𝜂 = {(𝑥, 𝛾𝐺 𝜂 (𝑥)): 𝑥 ∈ 𝐺} be an 𝜂 −intuitionistic fuzzy soft group. from definition 5, then for all 𝑥, 𝑦 ∈ 𝐺 we have 𝛾𝐺 𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦−1) = 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦). (⇐) since 𝛾𝐺 𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺, then 𝛾𝐺 𝜂 (𝑥−1) = 𝛾𝐺 𝜂 (𝑒𝑥) ⊇ 𝛾𝐺 𝜂 (𝑒) ∩ 𝛾𝐺 𝜂 (𝑥) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑥) = 𝛾𝐺 𝜂 (𝑥), and 𝛾𝐺 𝜂 (𝑥) = 𝛾𝐺 𝜂 (𝑒(𝑥−1)−1) ⊇ 𝛾𝐺 𝜂 (𝑒) ∩ 𝛾𝐺 𝜂 (𝑥−1) ⊇ 𝛾𝐺 𝜂 (𝑥)−1 ∩ 𝛾𝐺 𝜂 (𝑥−1) = 𝛾𝐺 𝜂 (𝑥−1). hence 𝛾𝐺 𝜂 (𝑥−1) = 𝛾𝐺 𝜂 (𝑥) for all 𝑥 ∈ 𝐺. then for all 𝑥, 𝑦 ∈ 𝐺 we have 𝛾𝐺 𝜂 (𝑥𝑦) = 𝛾𝐺 𝜂 (𝑥(𝑦−1)−1) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦−1) = 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦). hence 𝛾𝐺 𝜂 (𝑥𝑦) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦) for all 𝑥, 𝑦 ∈ 𝐺. therefore, γ𝐺 𝜂 is an 𝜂 −intuitionistic fuzzy soft group. ∎ since 𝜂 −intuitionistic fuzzy soft group is defined based on intuitionistic fuzzy soft group, so there is a correlation between those concepts. the following theorem, we prove that every intuitionistic fuzzy soft group is an 𝜂 −intuitionistic fuzzy soft group. theorem 2. if γ𝐺 is an intuitionistic fuzzy soft group, then γ𝐺 𝜂 is an 𝜂 −intuitionistic fuzzy soft group for all 𝜂 ∈ [0, 1]. proof. let γ𝐺 = {(𝑥, 𝛾𝐺 (𝑥)): 𝑥 ∈ 𝐺} be an intuitionistic fuzzy soft group over the universe 𝑈 where 𝛾𝐺 (𝑥) = {(𝑢, 𝜇𝛾𝐺(𝑥)(𝑢), 𝜈𝛾𝐺(𝑥)(𝑢)) : 𝑢 ∈ 𝑈}. for any 𝑥, 𝑦 ∈ 𝐺 and 𝜂 ∈ [0, 1], then 𝛾𝐺 𝜂 (𝑥𝑦−1) = ψ[𝛾𝐺 (𝑥𝑦 −1), 𝜂]. since γ𝐺 is an intuitionistic fuzzy soft group, then 𝛾𝐺 𝜂 (𝑥𝑦−1) ⊇ ψ[(𝛾𝐺 (𝑥) ∩ 𝛾𝐺 (𝑦)), 𝜂] = {( 𝑢, 𝜓[min[𝜇𝛾𝐺(𝑥)(𝑢), 𝜇𝛾𝐺(𝑦)(𝑢)], 𝜂], 𝜓[max[𝜈𝛾𝐺(𝑥)(𝑢), 𝜈𝛾𝐺(𝑦)(𝑢)], 1 − 𝜂] ) : 𝑢 ∈ 𝑈} = {( 𝑥, 𝜓[min[𝜇𝛾𝐺(𝑥)(𝑢), 𝜂] , min[ 𝜇𝛾𝐺(𝑦)(𝑢), 𝜂]], 𝜓[max[𝜈𝛾𝐺(𝑥)(𝑢), 1 − 𝜂] , max[𝜈𝛾𝐺(𝑦)(𝑢), 1 − 𝜂]] ) : 𝑢 ∈ 𝑈} 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 358 = {( 𝑥, min[𝜓[𝜇𝛾𝐺(𝑥)(𝑢), 𝜂] , 𝜓[ 𝜇𝛾𝐺(𝑦)(𝑢), 𝜂]], max[𝜓[𝜈𝛾𝐺(𝑥)(𝑢), 1 − 𝜂] , 𝜓[ 𝜈𝛾𝐺(𝑦)(𝑢), 1 − 𝜂]] ) : 𝑢 ∈ 𝑈} = ψ[𝛾𝐺 (𝑥), 𝜂] ∩ ψ[𝛾𝐺 (𝑦), 𝜂] = 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦). therefore, γ𝐺 𝜂 is an 𝜂 −intuitionistic fuzzy soft group. ∎ here we define the 𝜂 −intuitionistic fuzzy subgroup of an 𝜂 −intuitionistic fuzzy group. the properties of 𝜂 −intuitionistic fuzzy subgroup are proved in the following theorem. definition 6. suppose 𝐺 be a group and 𝐻 be a subgroup of 𝐺. let γ𝐺 𝜂 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈, then γ𝐻 𝜂 is called an 𝜂 −intuitionistic fuzzy soft subgroup of γ𝐺 𝜂 if γ𝐻 𝜂 is an 𝜂 −intuitionistic fuzzy soft group over 𝑈. theorem 3. let γ𝐺 𝜂 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈. suppose γ𝐻 𝜂 and γ𝑁 𝜂 be two 𝜂 −intuitionistic fuzzy soft subgroups of γ𝐺 𝜂 , then γ𝐻 𝜂 ∩̃ γ𝑁 𝜂 is an 𝜂 −intuitionistic fuzzy soft subgroup of γ𝐺 𝜂 . proof. defined γ𝐻 𝜂 ∩̃ γ𝑁 𝜂 = {(𝑥, 𝛾𝐻∩̃𝑁 𝜂 (𝑥)): 𝑥 ∈ 𝐻 ∩ 𝑁}. for any 𝑥, 𝑦 ∈ 𝐺, then 𝛾𝐻∩̃𝑁 𝜂 (𝑥𝑦−1) = 𝛾𝐻 𝜂 (𝑥𝑦−1) ∩ 𝛾𝑁 𝜂 (𝑥𝑦−1) ⊇ (𝛾𝐻 𝜂 (𝑥) ∩ 𝛾𝐻 𝜂 (𝑦)) ∩ (𝛾𝑁 𝜂 (𝑥) ∩ 𝛾𝑁 𝜂 (𝑦)) = (𝛾𝐻 𝜂 (𝑥) ∩ 𝛾𝑁 𝜂 (𝑥)) ∩ (𝛾𝐻 𝜂 (𝑦) ∩ 𝛾𝑁 𝜂 (𝑦)) = 𝛾𝐻∩̃𝑁 𝜂 (𝑥) ∩ 𝛾𝐻∩̃𝑁 𝜂 (𝑦). hence γ𝐻 𝜂 ∩̃ γ𝑁 𝜂 is an 𝜂 −intuitionistic fuzzy soft subgroup of γ𝐺 𝜂 . ∎ theorem 4. let γ𝐺 𝜂 be an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 𝑒 be an identity elemen of 𝐺. then γ𝐺 𝜂 |𝑒 = {(𝑥, 𝛾𝐺 𝜂 (𝑥)): 𝛾𝐺 𝜂 (𝑥) = 𝛾𝐺 𝜂 (𝑒), 𝑥 ∈ 𝐺} is an 𝜂 −intuitionistic fuzzy subgroup of 𝐺. proof. since (𝑒, 𝛾𝐺 𝜂 (𝑒)) ∈ γ𝐺 𝜂 |𝑒, then γ𝐺 𝜂 |𝑒 ≠ ∅. let (𝑥, 𝛾𝐺 𝜂 (𝑥)), (𝑦, 𝛾𝐺 𝜂 (𝑦)) ∈ γ𝐺 𝜂 |𝑒 , we have 𝛾𝐺 𝜂 (𝑥) = 𝛾𝐺 𝜂 (𝑦) = 𝛾𝐺 𝜂 (𝑒) for 𝑥, 𝑦 ∈ 𝐺. then 𝛾𝐺 𝜂 (𝑥𝑦−1) ⊇ 𝛾𝐺 𝜂 (𝑥) ∩ 𝛾𝐺 𝜂 (𝑦) = 𝛾𝐺 𝜂 (𝑒) ∩ 𝛾𝐺 𝜂 (𝑒) = 𝛾𝐺 𝜂 (𝑒). since 𝛾𝐺 𝜂 (𝑒) ⊇ 𝛾𝐺 𝜂 (𝑥𝑦−1), then 𝛾𝐺 𝜂 (𝑥𝑦−1) = 𝛾𝐺 𝜂 (𝑒). thus (𝑥𝑦−1, 𝛾𝐺 𝜂 (𝑥𝑦−1)) ∈ γ𝐺 𝜂 |𝑒 . therefore γ𝐺 𝜂 |𝑒 is an 𝜂 −intuitionistic fuzzy subgroup of 𝐺. ∎ here notioned the definition of image and pre-image of an 𝜂 −intuitionistic fuzzy soft group. definition 7. let γ𝐴 𝜂 and γ𝐵 𝜂 be two 𝜂 −intuitionistic fuzzy soft sets over universe 𝑈, and let 𝜙: 𝐴 → 𝐵, then 1. image of γ𝐴 𝜂 under the map 𝜙, denoted by 𝜙(γ𝐴 𝜂 ) defined by 𝜙(γ𝐴 𝜂 ) = {(𝑏, 𝜙(𝛾𝐴 𝜂 )(𝑏)): 𝑏 ∈ 𝐵}, where for all 𝑏 ∈ 𝐵, then 𝜙(𝛾𝐴 𝜂 )(𝑏) = { ∩ {𝛾𝐴 𝜂 (𝑎): 𝑎 ∈ 𝐴, 𝑓(𝑎) = 𝑏}, if 𝜙(𝑎) ∈ 𝜙(𝐴), 𝛾∅ 𝜂 , others. 2. pre-image of γ𝐵 𝜂 under 𝜙, denoted by 𝜙−1(γ𝐵 𝜂 ) defined by 𝜙−1(γ𝐵 𝜂 ) = {(𝑎, 𝜙−1(𝛾𝐵 𝜂)(𝑎)): 𝑎 ∈ 𝐴}, where for all 𝑎 ∈ 𝐴, then 𝜙−1(𝛾𝐵 𝜂)(𝑎) = 𝛾𝐵 𝜂 (𝜙(𝑎)). lemma 1. let γ𝐴 𝜂 and γ𝐵 𝜂 be two 𝜂 −intuitionistic fuzzy soft sets over 𝑈, then 𝜙(γ𝐴 𝜂 ) and 𝜙−1(γ𝐵 𝜂 ) are 𝜂 −intuitionistic fuzzy soft sets over universe 𝑈. 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 359 proof. let γ𝐴 𝜂 = {(𝑎, 𝛾𝐴 𝜂 (𝑎)): 𝑎 ∈ 𝐴} and γ𝐵 𝜂 = {(𝑏, 𝛾𝐵 𝜂(𝑏)): 𝑏 ∈ 𝐵} respectively be two 𝜂 −intuitionistic fuzzy soft sets of 𝐴 and 𝐵 over 𝑈. let 𝜙: 𝐴 → 𝐵. from definition 7, there is 𝑏 ∈ 𝜙(𝐴) such that 𝜙(𝑎) = 𝑏, then 𝜙(𝛾𝐴 𝜂 )(𝑏) =∩ {𝛾𝐴 𝜂 (𝑎): 𝑎 ∈ 𝐴, 𝜙(𝑎) = 𝑏}. from proposition 1, we have 𝜙(𝛾𝐴 𝜂)(𝑏) is an 𝜂 −intuitionistic fuzzy set. if 𝜙(𝑎) ∉ 𝜙(𝐴), then 𝜙(𝛾𝐴 𝜂 )(𝑏) = 𝛾∅ 𝜂 is an 𝜂 −intuitionistic fuzzy set. then ∀𝑎 ∈ 𝐴, we have 𝜙−1(𝛾𝐵 𝜂 )(𝑎) = 𝛾𝐵 𝜂 (𝜙(𝑎)) is an 𝜂 −intuitionistic fuzzy set. therefore, 𝜙(γ𝐴 𝜂) and 𝜙−1(γ𝐵 𝜂 ) are 𝜂 −intuitionistic fuzzy soft sets over 𝑈. ∎ the map 𝑓: 𝐺1 → 𝐺2 of a group 𝐺1 to a group 𝐺2 is called homomorphism if for all 𝑥, 𝑦 ∈ 𝐺1 then 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑓(𝑦). the following theorems, we prove that every image and preimage of the 𝜂 −intuitionistic fuzzy group under the homomorphism function are the 𝜂 −intuitionistic fuzzy soft group. theorem 5. let γ𝐺1 𝜂 is an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 𝜙hom: 𝐺1 − 𝐺2, then 𝜙(γ𝐺1 𝜂 ) is an 𝜂 −intuitionistic fuzzy soft group. proof. let 𝐺1 and 𝐺2 are two groups, 𝜙hom: 𝐺1 − 𝐺2, and γ𝐺1 𝜂 is an 𝜂 −intuitionistic fuzzy soft group over 𝑈. let 𝑢, 𝑣 ∈ 𝐺2, 𝑢 ∉ 𝜙(𝐺1) or 𝑣 ∉ 𝜙(𝐺1), then 𝜙(𝛾𝐺1 𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1 𝜂 )(𝑣) = 𝛾∅ 𝜂, means 𝜙(𝛾𝐺1 𝜂 )(𝑢𝑣) ⊇ 𝜙(𝛾𝐺1 𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1 𝜂 )(𝑣). since 𝑢 ∉ 𝜙(𝐺1), then 𝑢 −1 ∉ 𝜙(𝐺1), so 𝜙(𝛾𝐺1 𝜂 )(𝑢−1) = 𝜙(𝛾𝐺1 𝜂 )(𝑢) = 𝛾∅ 𝜂 . suppose 𝜙(𝑥) = 𝑢 and 𝜙(𝑦) = 𝑣 for 𝑥, 𝑦 ∈ 𝐺1. let 𝑧 = 𝑥𝑦, then 1. 𝜙(𝛾𝐺1 𝜂 )(𝑢𝑣) =∩ {𝛾𝐺1 𝜂 (𝑧): 𝑧 ∈ 𝐺1, 𝜙(𝑧) = 𝑢𝑣} =∩ {𝛾𝐺1 𝜂 (𝑥𝑦): 𝑥, 𝑦 ∈ 𝐺1, 𝜙(𝑥𝑦) = 𝑢𝑣} ⊇∩ {𝛾𝐺1 𝜂 (𝑥) ∩ 𝛾𝐺1 𝜂 (𝑦): 𝑥, 𝑦 ∈ 𝐺1, 𝜙(𝑥) = 𝑢, 𝜙(𝑦) = 𝑣} = (∩ {𝛾𝐺1 𝜂 (𝑥): 𝑥 ∈ 𝐺1, 𝜙(𝑥) = 𝑢}) ∩ (∩ {𝛾𝐺1 𝜂 (𝑦): 𝑦 ∈ 𝐺1, 𝜙(𝑦) = 𝑣}) = 𝜙(𝛾𝐺1 𝜂 )(𝑢) ∩ 𝜙(𝛾𝐺1 𝜂 )(𝑣). 2. 𝜙(𝛾𝐺1 𝜂 )(𝑢) =∩ {𝛾𝐺1 𝜂 (𝑥): 𝑥 ∈ 𝐺1, 𝜙(𝑥) = 𝑢} =∩ {𝛾𝐺1 𝜂 (𝑥−1): 𝑥 ∈ 𝐺1, 𝜙(𝑥 −1) = 𝑢−1} = 𝜙(𝛾𝐺1 𝜂 )(𝑢−1). therefore, 𝜙(γ𝐺1 𝜂 ) is an 𝜂 −intuitionistic fuzzy soft group over 𝑈. ∎ theorem 6. let γ𝐺2 𝜂 is an 𝜂 −intuitionistic fuzzy soft group over universe 𝑈 and 𝜙hom: 𝐺1 − 𝐺2, then 𝜙 −1(γ𝐺2 𝜂 ) is an 𝜂 −intuitionistic fuzzy soft group. proof. let 𝐺1 and 𝐺2 are two groups, 𝜙hom: 𝐺1 − 𝐺2, and γ𝐺2 𝜂 is an 𝜂 −intuitionistic fuzzy soft group over 𝑈. for all 𝑥, 𝑦 ∈ 𝐺1 we have 1. 𝜙−1(𝛾𝐺2 𝜂)(𝑥𝑦) = 𝛾𝐺2 𝜂 (𝜙(𝑥𝑦)) = 𝛾𝐺2 𝜂 (𝜙(𝑥)𝜙(𝑦)) ⊇ 𝛾𝐺2 𝜂 (𝜙(𝑥)) ∩ 𝛾𝐺2 𝜂 (𝜙(𝑥)) = 𝜙−1(𝛾𝐺2 𝜂 )(𝑥) ∩ 𝜙−1(𝛾𝐺2 𝜂 )(𝑦). 2. 𝜙−1(𝛾𝐺2 𝜂)(𝑥−1) = 𝛾𝐺2 𝜂 (𝜙(𝑥−1)) = 𝛾𝐺2 𝜂 ((𝜙(𝑥)) −1 ) = 𝛾𝐺2 𝜂 (𝜙(𝑥)) = 𝜙−1(𝛾𝐺2 𝜂 )(𝑥). 𝜂 −intuitionistic fuzzy soft groups mustika ana kurfia 360 therefore, 𝜙−1(γ𝐺2 𝜂 ) is an 𝜂 −intuitionistic fuzzy soft group over 𝑈. ∎ based on theorem 5 and theorem 6, we know that image and pre-image of an 𝜂 −intuitionistic fuzzy soft group under 𝜂 −intuitionistic fuzzy soft homomorphism are also an 𝜂 −intuitionistic fuzzy group. conclusions based on the result, it is concluded that 𝜂 −intuitionistic fuzzy soft group depends on intuitionistic fuzzy soft group. it is proved that every intuitionistic fuzzy soft group is an 𝜂 −intuitionistic fuzzy soft group. the definition of 𝜂 −intuitionistic 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[24] f. karaaslan, k. kaygisiz, and n. cagman, “on intuitionistic fuzzy soft groups,” j. new results sci., vol. 2, no. 3, pp. 72–86, 2013. average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 231-239 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 18, 2021 reviewed: december 08, 2021 accepted: december 21, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.13371 average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto*, zaenurrohman, titi udjiani srrm department of mathematics, faculty science and mathematics, diponegoro university, indonesia *corresponding author email: susilohariyanto@lecturer.undip.ac.id*, zaenurrohman8.zr@gmail.com, udjianititi@yahoo.com abstract every day, new covid-19 positive cases are discovered in central java. many research has used various methodologies to try to forecast new positive instances. the fuzzy time series (fts) approach is one of them. many fts are now in development, including the fts markov chain. the duration of the gap in the fts must be determined carefully because it will affect the flr, which will be used to estimate the forecast value. the average-based method can be used to determine the optimum interval length; however, other research use frequency density partitioning to determine the optimal interval length in order to produce superior predicting values. the goal of this research is to improve the accuracy of forecasting values by modifying the frequency density partition on the average based-fts markov chain. the approach utilized is average-based, with the length of the interval determined by the average, the forecast value determined by the fts markov chain, and the frequency density partition modified to provide the ideal interval. the average-based fts markov chain approach with adjustments to the frequency density partition achieves an accuracy rate of 89.3 percent, according to the findings of this study. because changes to the frequency density partition can produce a good level of accuracy in forecasting new positive cases of covid-19 in central java, it is hoped that this modification of the frequency density partition on the average-based fts markov chain can be used as a model for forecasting in fields other than new positive cases. covid-19. keywords: average based; fts markov chain; modified frequency density partitioning; covid19; mape introduction covid-19 first appeared in the city of wuhan, hubei province, china, which spread almost all over the world, including indonesia. at the beginning of 2020, indonesia experienced a covid-19 pandemic, which, to this day, new positive cases are still being found[1]. the government is still thinking about how to make indonesia free from covid-19. many experts have estimated the amount of new covid-19 positive cases in indonesia. forecasting is the process of predicting what will happen in the future over a lengthy period of time[2]. however, no approach for accurately forecasting anything, http://dx.doi.org/10.18860/ca.v7i1.13371 mailto:susilohariyanto@lecturer.undip.ac.id* mailto:zaenurrohman8.zr@gmail.com mailto:udjianititi@yahoo.com average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 232 including new covid-19 positive cases, has been developed to yet. the fuzzy time series (fts) is one of the forecasting approaches for determining the number of new positive cases in indonesia. the concept of fuzzy logic is used in the forecasting of fts. song and chissom introduced the fts in 1993[3], and it has since been widely developed, including the markov method[4], chen’s method[5], chen and hsu’s method[6], the weighted method[7], the multiple-attribute fuzzy time series method[8], the percentage change method[9], and markov chain method[10]. ruey chyn tsaur, developing fuzzy time series by merging the fuzzy time series method with the markov chain concept [10]. markov chain is a stochastic process in which future events only depend on today's events. markov chain is used in the defuzzification process [11]. the determination of the length of the interval in the fuzzy time series does not have a definite formula, but the determination of the length of the interval in the fuzzy time series is based on the researcher. as a result, even if each researcher is utilizing the same data, the length of the interval will vary[3]. even though the determination of the length of the interval is a very influential part in the formation of a fuzzy logical relationship (flr).[12]. one method that can be used to determine the length of the interval is the average based. this average based uses an average-based method in determining the length of the interval [12]. chen and hsu in 2004 also developed a fuzzy time series. chen and hsu developed fuzzy time series by repartitioning based on frequency density. chen and hsu's frequency density repartitioning algorithm divides the interval with the highest frequency density into four sub-intervals, the interval with the second highest density into three sub-intervals, the interval with the third highest density into two subintervals, and the interval with the lowest density into one sub-interval.[6][13]. based on the description above, the researcher is interested in modifying the frequency density partitioning algorithm used by chen and hsu, namely by exchanging the partition between the interval with the first densest frequency with the third densest interval, which was originally the first densest interval partitioned into 4 subintervals, the researcher changed the first densest was partitioned into 2 sub-intervals, and for the interval with the third densest frequency initially partitioned into 2 subintervals the researcher changed it to 4 sub-intervals. furthermore, the researcher will use the average-based method to determine the interval length in the fuzzy time series type markov chain, and apply it to forecasting new positive cases of covid-19 in central java. methods fuzzy time series the definition of fuzzy time series was first introduced by song dan chisom (1993). let 𝑈 universe of discourse, with 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛} on a fuzzy set 𝐴𝑖 , defined as[3]: 𝐴𝑖 = 𝑓𝐴(𝑢1) 𝑢1 + 𝑓𝐴(𝑢2) 𝑢2 + ⋯ + 𝑓𝐴(𝑢𝑛) 𝑢𝑛 (1) where 𝑓𝐴 is theemembership ofkthe fuzzypset 𝐴𝑖 and 𝑢𝑘 is anxelementuof the fuzzy set 𝐴𝑖 and 𝑓𝐴(𝑢𝑘 ) shows the degreebof membership of 𝑢𝑘 in 𝐴𝑖 where 𝑘 = 1,2,3, … , 𝑛. definition. if 𝐹(𝑡) is causeddby 𝐹(𝑡 − 1), then thecrelation in the first orderrmodel 𝐹(𝑡) cantbe stated assfollows: [5]. average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 233 𝐹(𝑡) = 𝐹(𝑡 − 1) ○ 𝑅(𝑡, 𝑡 − 1) (2) where “○” is max-min composition operator, and 𝑅(𝑡, 𝑡 − 1) is a relation matrix to describe the fuzzy relationship between 𝐹(𝑡 − 1) dan 𝐹(𝑡). average-based average based is an algorithm that can be used to set the interval length that is determined at the initial stage of forecasting when using fuzzy time series. the steps of the average based algorithm are as follows [12], [14]: a. determine the absolute difference (lag) between data 𝑛 + 1 and data 𝑛 with the formula: 𝑙𝑎𝑔𝐷𝑛 = |(𝐷𝑎𝑡𝑎 𝑛 + 1) − (𝐷𝑎𝑡𝑎 𝑛)| (3) b. determine the length of the interval 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = ( 𝑡𝑜𝑡𝑎𝑙 𝑙𝑎𝑔 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑑𝑎𝑡𝑎 ) : 2 (4) c. determine the basis value of the interval length according to table 1. following: table 1. basis mapping table range basis 0,1 – 1,0 0,1 1,1 -10 1 11-100 10 101-1000 100 d. the length of the interval is then rounded up according to the interval basis table. modification frequency density partition in this study, modifications to the frequency density partition were used, the algorithm used is as follows: a. the interval with the first densest frequency is divided into 2 subintervals. b. the interval with the second densest frequency is divided into 3 subintervals. c. the interval with the third densest frequency is divided into 4 subintervals d. eliminates intervals that have no frequency. fuzzy time series markov chain markov chain's fuzzy time series forecasting procedure is as follows[10]: step 1. collecting historical data (𝑌𝑡). step 2. defines the u universe set of data, with d1 and d2 being the corresponding positive numbers. 𝑈 = [𝐷𝑚𝑖𝑛 − d1, 𝐷𝑚𝑎𝑥 + d2] (5) step 3. specify the number of fuzzy intervals. step 4. defining the fuzzy set in the universe of discourse u, the fuzzy ai set declares the linguistic variable of the share price by 1 ≤ 𝑖 ≤ 𝑛. step 5. fuzzification of historical data. if a time series data is included in the 𝑢𝑖 interval, then that data is fuzzification into 𝐴𝑖 . step 6. specifies fuzzy logical relationship (flr) and fuzzy logical relationships group (flrg). step 7. calculate forecasting results for time series data, using flrg, a probability can be obtained from a state heading to the next state. in order to calculate the predicting value, a markov probability transition matrix with a dimension of 𝑛 𝑥 𝑛 was used. if average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 234 state 𝐴𝑖 transition to a state 𝐴𝑗 and pass the state 𝐴𝑘 , 𝑖, 𝑗 = 1, 2, . . . , 𝑛, then we can obtain flrg. the transition probability formula is as follows: 𝑃𝑖𝑗 = 𝑀𝑖𝑗 𝑀𝑖 , 𝑖, 𝑗 = 1, 2, … , 𝑛 (6) with: 𝑃𝑖𝑗 = probability of transition from state 𝐴𝑖 to state 𝐴𝑗 one step 𝑀𝑖𝑗 = number of transitions from state 𝐴𝑖 to state 𝐴𝑗 one step 𝑀𝑖 = the amount of data included in the 𝐴𝑖 the probability matrix r of all states can be written as follows: 𝑅 = [ 𝑃11 ⋯ 𝑃1𝑛 ⋮ ⋱ ⋮ 𝑃𝑛1 ⋯ 𝑃𝑛𝑛 ] (7) matrix r reflects the transition of the entire system. if 𝐹(𝑡 − 1) = 𝐴𝑖 , then the process will be defined in the 𝐴𝑖 at the time of (𝑡 − 1), then the forecasting results 𝐹(𝑡) will be calculated using the [𝑃𝑖1, 𝑃𝑖2, … , 𝑃𝑖𝑛 ] on the matrix r. forecasting results 𝐹(𝑡) is the weighted average value of the 𝑚1, 𝑚2, ..., 𝑚𝑛 (midpoint of 𝑢1, 𝑢2, ..., 𝑢𝑛 ). the forecasting output result value on 𝐹(𝑡) can be determined by using the following rules: rule 1: if fuzzy logical relationship group 𝐴𝑖 is one-to-one (suppose 𝐴𝑖 → 𝐴𝑘 where 𝑃𝑖𝑘 = 1 and 𝑃𝑖𝑗 = 0, 𝑗 ≠ 𝑘) then the forecasting value of 𝐹(𝑡) is 𝑚𝑘 the middle value of the 𝑢𝑘 . 𝐹(𝑡) = 𝑚𝑘 𝑃𝑖𝑘 = 𝑚𝑘 (8) rulet2: if the flrg 𝐴𝑖 is one-to-many (e.g. 𝐴𝑗 → 𝐴1, 𝐴2, . . . , 𝐴𝑛 . 𝑗 = 1, 2, . . . , 𝑛), when 𝑌(𝑡 − 1) at time (𝑡 − 1) is included in state 𝐴𝑗 then the forecasting 𝐹(𝑡), is: 𝐹(𝑡) = 𝑚1𝑃𝑗1 + 𝑚2𝑃𝑗2 + … + 𝑚𝑗−1𝑃𝑗(𝑗−1) + 𝑌(𝑡 − 1)𝑃𝑗𝑗 + 𝑚𝑗+1𝑃𝑗(𝑗+1) + … + 𝑚𝑛𝑃𝑗𝑛 (9) where 𝑚1, 𝑚2, . . . , 𝑚𝑗−1, 𝑚𝑗+1, … , 𝑚𝑛 is the middle value 𝑢1, 𝑢2, . . . , 𝑢𝑗−1, 𝑢𝑗+1, . . . , 𝑢𝑛 , and 𝑌(𝑡 − 1) are state values 𝐴𝑗 at time 𝑡 − 1. rule 3: if the flrg 𝐴𝑖 is empty (𝐴𝑖 → ∅) forecast value 𝐹(𝑡) is 𝑚𝑖 which is the middle value of 𝑢𝑖 with the following equation: 𝐹(𝑡) = 𝑚𝑖 (10) step 8. adjusting the trend of forecasting values with the following rules:  if state 𝐴𝑖 communicates with 𝐴𝑖 , startingffrom state 𝐴𝑖 at time 𝑡 − 1 expressed as 𝐹(𝑡 − 1) = 𝐴𝑖 , and undergoing an increasingatransition to state 𝐴𝑗 at the time 𝑡 where (𝑖 < 𝑗), thendthe adjustment value is: 𝐷𝑡1 = ( 𝑙 2 ) (11) where 𝑙 is the basis interval.  ifestate 𝐴𝑖 communicatesswith 𝐴𝑖 , startinggfrom state 𝐴𝑖 atpthestime 𝑡 − 1 expressed as 𝐹(𝑡 − 1) = 𝐴𝑖 , anddexperiencing a decreasingmtransitionxto state 𝐴𝑗 atzthevtime 𝑡 where (𝑖 > 𝑗) the adjustment valueeis: 𝐷𝑡1 = − ( 𝑙 2 ) (12) average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 235  ifzstate 𝐴𝑖 at the time 𝑡 − 1 is expressed 𝐹(𝑡 − 1) = 𝐴𝑖 , and undergoes a jumpbforward transitionrtoqstate 𝐴𝑖+𝑠 atythegtime 𝑡 where (1 ≤ 𝑠 ≤ 𝑛 − 𝑖) then the adjustment value is: 𝐷𝑡2 = ( 𝑙 2 )𝑠 (13) where 𝑠 is the number of forward jumps.  if state 𝐴𝑖 atpthemtime 𝑡 − 1 is as 𝐹(𝑡 − 1) = 𝐴𝑖 , andhundergoes anjumpbackwardxtransition topstate 𝐴𝑖−𝑣 at thertime 𝑡 where (1 ≤ 𝑣 ≤ 𝑖) thenlthe adjustment value is: 𝐷𝑡2 = − ( 𝑙 2 )𝑣 (14) where 𝑣 is the number of jumps backward. step 9. determine the final forecast value based on the adjustment of the trend of the forecasting value if flrg 𝐴𝑖 is one-to-many and state 𝐴𝑖+1 can be accessed from state 𝐴𝑖 where state 𝐴𝑖 is related to 𝐴𝑖 then the forecasting result becomes ’(𝑡) = 𝐹(𝑡) + 𝐷𝑡1 + 𝐷𝑡2 = 𝐹(𝑡) + ( 𝑙 2 ) + ( 𝑙 2 ) . ifqflrg 𝐴𝑖 isoone-too-many andastate 𝐴𝑖+1 can be accessed from 𝐴𝑖 wherewstate 𝐴𝑖 is not related to 𝐴𝑖 then the forecasting values becomes 𝐹’(𝑡) = 𝐹(𝑡) + 𝐷𝑡2 = 𝐹(𝑡) + ( 𝑙 2 ). if flrg 𝐴𝑖 is one to many and state 𝐴𝑖−2 can be accessed from state 𝐴𝑖 where 𝐴𝑖 is not related to 𝐴𝑖 then the forecasting result is 𝐹’(𝑡) = 𝐹(𝑡) − 𝐷𝑡2 = 𝐹(𝑡) − ( 𝑙 2 ) 𝑥 2 = 𝐹(𝑡)– 𝑙. if 𝑣 is jumpxstep, theggeneralmform ofpthe forecast is: 𝐹’(𝑡) = 𝐹(𝑡) ± 𝐷𝑡1 ± 𝐷𝑡2 = 𝐹(𝑡) ± ( 𝑙 2 ) ± ( 𝑙 2 ) 𝑣. (15) forecasting error measurement the reliability of a forecast can be determined by looking at mean average percentage error (mape), this mape formulas[15]: 𝑀𝐴𝑃𝐸 = 1 𝑛 ∑ |𝑌(𝑡) − 𝐹′(𝑡)| 𝑌(𝑡) 𝑥 100% 𝑛 𝑡=1 (16) with 𝑌𝑡 : actual data period 𝑡, 𝐹′𝑡 : 𝑡 period forecasting value, and 𝑛 : the predictable amount of data. results and discussion forecasting with an average based-fts markov chain with modified frequency density partitioning, the first step is to collect covid-19 in the central java period june 25, 2021 until august 20, 2021 as a universe discourse (u). next, determine the greatest value (𝐷𝑚𝑎𝑥 = 5655) and smallest values (𝐷𝑚𝑖𝑛 = 1428), and the value of d1 = 8 and d2 = 5, so it can be defined 𝑈 = [1428 − 8, 5655 + 5] = [1420, 5660]. then, calculate the absolute difference from historical data, the average absolute difference from 57 data points is 658,554, which is then divided by 2 to yield 329,277. the value of 329,277 is then determined using table 1. the basis of the length of the interval is 100, so that u can be partitioned into the same interval length, namely 𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5, … , 𝑢39, 𝑢40, 𝑢41, 𝑢42, 𝑢43 successively the value for each interval is average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 236 table 2. universe discourse of new positive cases of covid-19 𝑢1 = [1420, 1520] ⋮ 𝑢2 = [1520, 1620] 𝑢41 = [5420, 5520] 𝑢3 = [1620, 1720] 𝑢42 = [5520,5620] 𝑢4 = [1720, 1820] 𝑢43 = [5620, 5720] the next step is to distribute the data to each interval and determine the frequency density, resulting in the densest interval, which is then repartitioned using a modified method. the following outcomes were achieved: table 3. frequency density and repartition interval interval frequency redivided interval 𝑢29 = [4220, 4320] 5 𝑢29,1 = [4220, 4270], 𝑢29,2 = [4270, 4320] 𝑢28 = [4120, 4220] 4 𝑢28,1 = [4120, 4153.33] 𝑢28,2 = [4153.33, 4186.67] 𝑢28,3 = [4186.67, 4220] 𝑢16 = [2920, 3020] 3 𝑢16,1 = [2920, 2945], 𝑢16,2 = [2945, 2970], 𝑢16,3 = [2970, 2995] 𝑢16,3 = [2995, 3020] 𝑢17 = [3020, 3120] 3 𝑢17,1 = [3020, 3045], 𝑢17,2 = [3045, 3070], 𝑢17,3 = [3070, 3095], 𝑢17,4 = [3095, 3120] 𝑢19 = [3220, 3320] 3 𝑢19,1 = [3220, 3245], 𝑢19,2 = [3245, 3270], 𝑢19,3 = [3270, 3295], 𝑢19,4 = [3295, 3320] 𝑢27 = [4020, 4120] 3 𝑢27,1 = [4020, 4045], 𝑢27,2 = [4045, 4070], 𝑢27,3 = [4070, 4095], 𝑢27,4 = [4095, 4120] 𝑢32 = [4520, 4620] 3 𝑢32,1 = [4520, 4545], 𝑢32,2 = [4545, 4570], 𝑢32,3 = [4570, 4590], 𝑢32,4 = [4590, 4620] 𝑢33 = [4620, 4720] 3 𝑢33,1 = [4620, 4645], 𝑢33,2 = [4645, 4670], 𝑢33,3 = [4670, 4695], 𝑢33,4 = [4695, 4720] 𝑢2, 𝑢3, 𝑢4, 𝑢5, 𝑢6, 𝑢8, 𝑢11, 𝑢14, 𝑢20, 𝑢23, 𝑢26, 𝑢34, 𝑢42 0 removed next, look for the middle value (𝑚1) for each interval, we get: table 4. middle value 𝒖𝒊 𝒎𝒊 𝒖𝒊 𝒎𝒊 𝑢1 1470 ⋮ ⋮ 𝑢7 2070 𝑢39 5270 𝑢9 2270 𝑢40 5370 𝑢10 2370 𝑢41 5470 𝑢12 2570 𝑢43 5670 average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 237 furthermore, defining fuzzy sets, fuzzy sets that can be formed from the universe conversation are 44 fuzzy sets. the fuzzy sets formed is as follows: 𝐴1 = { 1 𝑢1⁄ + 0,5 𝑢7 ⁄ + 0 𝑢9⁄ + 0 𝑢10⁄ + ⋯ + 0 𝑢40⁄ + 0 𝑢41⁄ + 0 𝑢43⁄ } 𝐴2 = { 0,5 𝑢1 ⁄ + 1 𝑢7⁄ + 0,5 𝑢9 ⁄ + 0 𝑢10⁄ + ⋯ + 0 𝑢40⁄ + 0 𝑢41⁄ + 0 𝑢43⁄ } 𝐴3 = { 0 𝑢1⁄ + 0,5 𝑢7 ⁄ + 1 𝑢9⁄ + 0,5 𝑢10 ⁄ + ⋯ + 0 𝑢40⁄ + 0 𝑢41⁄ + 0 𝑢43⁄ } ⋮ 𝐴49 = { 0 𝑢1⁄ + 0 𝑢7⁄ + 0 𝑢9⁄ + 0 𝑢10⁄ + ⋯ + 1 𝑢40⁄ + 0,5 𝑢41 ⁄ + 0 𝑢43⁄ } 𝐴50 = { 0 𝑢1⁄ + 0 𝑢7⁄ + 0 𝑢9⁄ + 0 𝑢10⁄ + ⋯ + 0,5 𝑢40 ⁄ + 1 𝑢41⁄ + 0,5 𝑢43 ⁄ } 𝐴51 = { 0 𝑢1⁄ + 0 𝑢7⁄ + 0 𝑢9⁄ + 0 𝑢10⁄ + ⋯ + 0 𝑢40⁄ + 0,5 𝑢41 ⁄ + 1 𝑢43⁄ } the next step is to perform fuzzification, the data from the fuzzification results are presented in the following table: table 5. fuzzification results t actual data fuzzy data t actual data fuzzy data 1 2311 𝐴3 ⋮ ⋮ ⋮ 2 2064 𝐴2 54 3263 𝐴18 3 3079 𝐴14 55 3078 𝐴14 4 2702 𝐴6 56 1428 𝐴1 5 2932 𝐴8 57 1432 𝐴1 the next step, determine the flr and flrg, as shown in table 6. and table 7: table 6. flr data order flr data order flr 1-2 𝐴3 → 𝐴2 ⋮ ⋮ 2-3 𝐴2 → 𝐴14 54-55 𝐴18 → 𝐴14 3-4 𝐴14 → 𝐴6 55-56 𝐴14 → 𝐴1 4-5 𝐴6 → 𝐴8 56-57 𝐴1 → 𝐴1 table 7. flrg current state next state current state next state 𝐴1 (1)𝐴1 ⋮ ⋮ 𝐴2 (1)𝐴14 𝐴49 (1)𝐴35, (1)𝐴42 𝐴3 (1)𝐴2 𝐴50 (1)𝐴48 𝐴4 (1)𝐴6 𝐴51 (1)𝐴43 the initial forecast will be calculated next. for example, for 𝑡 = 2, june 26, 2021, the forecast computation based on formulas (8), (9), and (10) is 𝐹(2) = 𝑚𝑘 𝑃𝑖𝑘 = 𝑚𝑘 = 2070. the summary of the initial forecasting results is as follows: table 8. initial forecasting results (𝐹(𝑡)) period actual data 𝑭(𝒕) period actual data 𝑭(𝒕) 6/25/21 2311 na ⋮ ⋮ ⋮ 6/26/21 2064 2070 8/19/21 1428 2070 6/27/21 3078 3082,5 8/20/21 1432 1428 average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 238 after we get the initial forecasting, the next step is to adjust the forecasting trend. for example, adjustment value for june 26, 2021, the next step is 𝐴2 and the current state is 𝐴3 then the adjustment calculation uses the forecast adjustment rule (14) we get 𝐷𝑡2 = − ( 𝑙 2 ) 𝑣 = − ( 100 2 ) 1 = −(50). dor the calculation of other forecasting value adjustment using equations (11), (12), (13), and (14). the following is a forecast adjustment table. table 8. forecasting trend adjustment value period flr 𝐷𝑡𝑛 period flr 𝐷𝑡2 6/25/21 𝐴3 → 𝐴2 na ⋮ ⋮ ⋮ 6/26/21 𝐴2 → 𝐴14 -50 8/19/21 𝐴14 → 𝐴1 -650 6/27/21 𝐴14 → 𝐴9 600 8/20/21 𝐴1 → 𝐴1 0 calculate the final forecast value. the final forecast is the sum of the initial forecast value with the forecast adjustment value by following the equation (15). for example, the final forecast value for june 26, 2021 data is 𝐹′2 = 𝐹2 ± 𝐷𝑡2 = 2070 + (−50) = 2020. by doing the same way, the summary of the final forecasting result is as follows: table. 9. final forecast value period y(t) 𝐹′𝑡 period y(t) 𝐹′𝑡 6/25/21 2311 na ⋮ ⋮ ⋮ 6/26/21 2064 2020 8/19/21 1428 1907.5 6/27/21 3078 3682,5 8/20/21 1432 1428 furthermore, the average based-fts markov chain is based on the modified frequency density partition to forecast data for new positive cases on august 21, 2021, the current state is 𝐴1, from flrg it is known that the next state of 𝐴1 is 𝐴1, then based on equations (8), (9), and (10) the forecasting result is 1 times the data of the previous new positive case 𝑌(𝑡 − 1) is 1432. the last step is to calculate the forecast accuracy value using mape. the mape values of average based-fts based on a modified frequency density partitioning is 10,7%. for forecasting results using average based-fts based on a modified frequency density partitioning, are presented in the following figure: figure 1. graph of forecasting results using average based-fts markov chain based on modified frequency density partitioning 0 1000 2000 3000 4000 5000 6000 7000 actual data forecasting value average based-fts markov chain with modifications to the frequency density partition to predict covid-19 in central java susilo hariyanto 239 conclusions forecasting new positive cases of covid-19 in central java using average based-fts markov chain based on modified frequency density partitioning has a good level of accuracy, this can be seen from the mape value obtained which is 10.7%. and for the predicted value of new positive cases on august 21, 2021, it is 1432 new positive cases of covid-19 in central java. references [1] d. handayani, “penyakit virus corona 2019,” j. respirologi indones., vol. 40, no. 2, 2020. [2] d. c. montgomery, c. l. jennings, and m. kulahci, introduction time series analysis and forecasting, willey, 2015. [3] q. song and b. s. chissom, “forecasting enrollments with fuzzy time series part i,” fuzzy sets syst., vol. 54, no. 1, pp. 1–9, 1993. [4] j. sullivan and w. h. woodall, “a comparison of fuzzy forecasting and markov modeling,” fuzzy sets syst., vol. 64, no. 3, pp. 279–293, 1994. [5] s. m. chen, “forecasting enrollments based on fuzzy time series,” fuzzy sets syst., vol. 81, no. 3, 1996. [6] s. m. chen and c.-c. hsu, “a new method to forecast enrollments using fuzzy time series,” int. j. appl. sci. eng., vol. 2, no. 3, pp. 234–244, 2004. [7] h. k. yu, “weighted fuzzy time series models for taiex forecasting,” phys. a stat. mech. its appl., vol. 349, no. 3–4, 2005. [8] c. h. cheng, g. w. cheng, and j. w. wang, “multi-attribute fuzzy time series method based on fuzzy clustering,” expert syst. appl., vol. 34, no. 2, 2008. [9] m. stevenson and j. porter, “fuzzy time series forecasting using percentage change as the universe of discourse,” change, vol. 1971, no. 3.89, 1972. [10] r. c. tsaur, “a fuzzy time series-markov chain model with an application to forecast the exchange rate between the taiwan and us dollar,” int. j. innov. comput. inf. control, vol. 8, no. 7 b, 2012. [11] y. a. r. langi, “penentuan klasifikasi state pada rantai markov dengan menggunakan nilai eigen dari matriks peluang transisi,” j. ilm. sains, vol. 11, no. 1, 2011. [12] s. xihao and l. yimin, “average-based fuzzy time series models for forecasting shanghai compound index *,”, vol. 4, no. 2, pp. 104-111, 2008. [13] t. a. jilani and s. m. a. burney, “a refined fuzzy time series model for stock market forecasting,” phys. a stat. mech. its appl., vol. 387, no. 12, 2008. [14] j. noh, w. wijono, and e. yudaningtiyas, “model average based fts markov chain untuk peramalan penggunaan bandwidth jaringan komputer,” j. eeccis, vol. 9, no. 1, 2015. [15] s. makridakis, s. wheelwright c, and v. e. mcgee, metode dan aplikasi peramalan. binarupa aksara, 1999. levi decomposition of frobenius lie algebra of dimension 6 cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 394-400 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 03, 2022 reviewed: april 11, 2022 accepted: april 14, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.15656 levi decomposition of frobenius lie algebra of dimension 6 henti*, edi kurniadi, ema carnia departement of mathematics of fmipa, universitas padjadjaran, indonesia email: henti17001@mail.unpad.ac.id abstract in this paper, we study notion of the frobenius lie algebra m2,1(ℝ) ⋊ 𝔤𝔩2(ℝ) of dimension 6. the finite dimensional lie algebra can be expressed in terms of decomposition between levi subalgebra and the radical (maximal solvable ideal). this form of decomposition is called levi decomposition. our main object is further denoted by 𝔞𝔣𝔣(2) ≔ m2,1(ℝ) ⋊ 𝔤𝔩2(ℝ). the work aims to obtain levi decomposition of frobenius lie algebra 𝔞𝔣𝔣(2) of dimension 6. to obtained levi subalgebra and the radical, we apply literature reviews about lie algebra and decomposition levi in dagli result. the main result of this paper is frobenius lie algebra 𝔞𝔣𝔣(2) can be decomposition be semisimple levi subalgebra 𝔥 of dimension 4 and radical solvable rad(𝔤) of dimension 2. thus, the levi decomposition form of the frobenius lie algebra is given. keywords: frobenius lie algebra; levi decomposition; lie algebra; radical introduction a vector space over a field that is equipped by lie brackets which is neither commutative nor associative is called lie algebra [1]. any finite dimensional lie algebra can be expressed as semidirect sum between levi subalgebra (lie subalgebra) and its radical (maximal solvable ideal) and this form is called a levi decomposition [1]. we denote a finite dimensional lie algebra by 𝔤. on the other hand, for finite dimensional case, the lie algebra 𝔤 can be written in levi decomposition form which is given in the following form 𝔤 = 𝔥 ⋉ rad(𝔤) (1) where 𝔥 is a levi subalgebra of 𝔤 and rad(𝔤) is a radical or solvable maximal ideal of 𝔤. let 𝑆 = {𝑒1, 𝑒2, … , 𝑒𝑛} be a basis of 𝔤 and we define 𝐶(𝔤) = (𝐶(𝔤)𝑖,𝑗) be a matrix whose lie brackets entries of 𝔤 are given by 𝐶(𝔤)𝑖,𝑗 ≔ [𝑒𝑖, 𝑒𝑗]𝔤 , 1 ≤ 𝑖, 𝑗 ≤ 𝑛. (2) this matrix 𝐶(𝔤) ∈ 𝑀𝑎𝑡(𝑛 × 𝑛, 𝑆(𝔤)) is called a structure matrix of 𝔤 where 𝑆(𝔤) denotes as symmetric algebra of 𝔤 [2]. the notion of lie algebras has been widely studied. one of which is the investigation of lie algebra with dimension 8 which can be carried out by levi's decomposition [3]. rais introduced the lie algebra notion 𝑀𝑛,𝑝(ℝ) ⋊ 𝔤𝔩𝑛(ℝ) where 𝑀𝑛,𝑝(ℝ) is a vector space of matrices of size 𝑛 × 𝑝 with real number entries and 𝔤𝔩𝑛(ℝ) is the lie algebra of a vector space of matrices of size 𝑛 × 𝑛 equipped with lie brackets [4]. furthermore, we can see the notions of lie algebra in [5] and [6]. http://dx.doi.org/10.18860/ca.v7i3.15656 mailto:henti17001@mail.unpad.ac.id* levi decomposition of frobenius lie algebra of dimension 6 henti 395 let 𝔤 be a lie algebra with 𝔤∗ is a dual vector space of 𝔤 where 𝔤∗ consisting of real valued all linear functional on 𝔤. the lie algebra 𝔤 is said to be a frobenius lie algebra if there exists a linear functional 𝜑 ∈ 𝔤∗ so that the skew-symmetric bilinear form 𝐵𝜑 (𝑥, 𝑦) ≔ 𝜑([𝑥, 𝑦]) is non degenerate. many studies of frobenius lie algebras have been carried out over the years. for instance, the properties of principal elements on frobenius lie algebra one of them is frobenius lie algebra cannot be unimodular [7]. an example of frobenius lie algebra is the affine lie algebra 𝔞𝔣𝔣(2) can be seen in the classification of frobenius lie algebra with dimension less than or equal to 6 [8]. kurniadi have constructed frobenius lie algebra with dimension less than or equal to 6 from non-commutative nilpotent lie algebra with dimension less than or equal to 4 [9]. other example of frobenius lie algebra, notation lie algebra 𝑀𝑛,𝑝(ℝ) ⋊ 𝔤𝔩𝑛(ℝ) where 𝑛 = 𝑝 = 3 is frobenius lie algebra of dimension 18 [10]. moreover, the lie algebra 𝑀3(ℝ) ⋊ 𝔤𝔩3(ℝ) has quasi-associative algebra structure [11]. the frobenius lie algebra m2(ℝ) ⋊ 𝔤𝔩2(ℝ) is the left-symmetric algebra [12]. it has been proven that the affine lie algebra that is denoted by 𝔞𝔣𝔣(𝑛) ≔ ℝ𝑛 ⋊ 𝔤𝔩𝑛(ℝ) is frobenius lie algebra where ℝ 𝑛 is another form of 𝑀𝑛,1(ℝ) [13]. readers can study more about frobenius lie algebra in the following articles: [14], [15], [16], and [17]. in this paper, we study about decompose frobenius lie algebra for special case 𝑛 = 2 of the affine lie algebra 𝔞𝔣𝔣(𝑛). the notion m2,1(ℝ) ⋊ 𝔤𝔩2(ℝ) can be written in simpler formulas as ℝ2 ⋊ 𝔤𝔩2(ℝ) and we can denote it by 𝔞𝔣𝔣(2) which is known as the affine lie algebra. in the nice formula, the affine lie algebra 𝔞𝔣𝔣(2) ≔ ℝ2 ⋊ 𝔤𝔩2(ℝ) can be expressed in the form of a matrix 𝔞𝔣𝔣(2) ≔ {( 𝑋 𝑌 0 0 ) ; 𝑋 ∈ 𝔤𝔩2(ℝ), 𝑌 ∈ ℝ 2} ⊆ 𝔤𝔩3(ℝ) (3) where 𝔤𝔩3(ℝ) is 3 × 3 real matrix. the purpose of this research is to give decompose this lie algebra into levi subalgebra and radical. methods we used literature study for the research method, especially the study of frobenius lie algebra 𝔞𝔣𝔣(2) and about levi decomposition of lie algebra in [18]. first, we given an affine lie algebra 𝔞𝔣𝔣(2). we proved the affine lie algebra 𝔞𝔣𝔣(2) not solvable. then, it is proved that lie algebra 𝔞𝔣𝔣(2) can be decomposed into its subalgebra and radical. before going into the discussion, we would like to introduce the theoretical foundations used in this study as follows: definition 1 [19] let 𝔤 be a vector space and a bilinear form [. , . ]: 𝔤 × 𝔤 ∋ (𝑥, 𝑦) ↦ [𝑥, 𝑦] ∈ 𝔤. the bilinear form [. , . ] is called a lie bracket for 𝔤 if the following condisitions are satisfied: 1. [𝑥, 𝑦] = −[𝑦, 𝑥]; ∀ 𝑥, 𝑦 ∈ 𝔤 2. [𝑥, [𝑦, 𝑧]] + [𝑦, [𝑧, 𝑥]] + [𝑧, [𝑥, 𝑦]] = 0; ∀ 𝑥, 𝑦, 𝑧 ∈ 𝔤. the vector space 𝔤 equipped by lie brackets is called lie algebra. definition 2 [19] a linear subspace 𝔥 of 𝔤 is called a lie sub-algebra if [𝔥, 𝔥] ⊆ 𝔥, we denote by 𝔥 < 𝔤. if we have [𝔤, 𝔥 ] ⊆ 𝔥, we call 𝔥 as an ideal of 𝔤 and then write 𝔥 ⊴ 𝔤. definition 3 [19] let 𝔤 be a lie algebra. the derived series of 𝔤 is defined by 𝐷0(𝔤) = 𝔤 and 𝐷𝑛(𝔤) = [𝐷𝑛−1(𝔤), 𝐷𝑛−1(𝔤)] ∀𝑛 ∈ ℕ (4) the lie algebra 𝔤 is said to be solvable, if there exists an 𝑛 ∈ ℕ with 𝐷𝑛(𝔤) = {0}. theorem 1 let 𝔤 be lie algebra then, levi decomposition of frobenius lie algebra of dimension 6 henti 396 i. if 𝔤 is solvable then the subalgebras and homomorphic images of 𝔤 are solvable. ii. if 𝔥 is a solvable ideal of 𝔤 and 𝔤/𝔥 is solvable, then 𝔤 is solvable iii. if 𝔥 and 𝔦 are solvable ideals of 𝔤 then 𝔥 + 𝔦 is also a solvable ideal of 𝔤. this theorem shows that the sum of all solvable ideals of a lie algebra is a solvable ideal. so, in every finite-dimensional lie algebra 𝔤, there exists a maximal solvable ideal. this ideal is called the radical of 𝔤 and denoted by 𝑅𝑎𝑑(𝔤). theorem 2 [18] let 𝑉 be vector space over a field and let 𝔤 be a subalgebra of 𝔤𝔩(𝑉), the 𝔤 is solvable if tr(𝑥𝑦) = 0 for all 𝑥 ∈ 𝔤 and 𝑦 ∈ [𝔤, 𝔤]. theorem 3 [18] let 𝔤 be lie algebra over a field 𝔽, then 𝑅𝑎𝑑(𝔤) = {𝑥 ∈ 𝔤 | 𝑇𝑟(ad 𝑥 ⋅ ad 𝑦) = 0} (5) for all 𝑦 ∈ [𝔤, 𝔤]. definition 4 [19] let 𝔤 be a lie algebra. if its radical is trivial i.e 𝑅𝑎𝑑(𝔤) = {0} then 𝔤 is called semisimple. the lie algebra 𝔤 is said to be simple if it is not abelian and if it contains no ideal other than 𝔤 and {0}. definition 5 [1] let 𝑉 be a space vector. a linear map 𝜌: 𝑉 → 𝑉 is said to be endomorphism on 𝑉 if the following condition satisfied: 1. 𝜌(𝑥 + 𝑦) = 𝜌(𝑥) + 𝜌(𝑦) 2. 𝜌(𝑥𝑦) = (𝜌(𝑥))𝑦 = 𝑥(𝜌(𝑦)) for all 𝑥, 𝑦 ∈ 𝑉. the set of all endomorphism on 𝑉 is denoted by 𝐸𝑛𝑑(𝑉). furthermore, the endomorphism 𝐸𝑛𝑑(𝑉) equipped by lie bracket [𝑥, 𝑦] = 𝑥𝑦 − 𝑦𝑥 for all 𝑥, 𝑦 ∈ 𝐸𝑛𝑑(𝑉) is lie algebra and it is called a general linear algebra, we denoted by 𝔤𝔩(v). definition 6 [19] let 𝔤 be a lie algebra and 𝑥 ∈ 𝔤. the map 𝑎𝑑: 𝔤 → 𝔤 defined by ad 𝑥 ∶ 𝔤 ∈ 𝑦 ↦ ad 𝑥(𝑦) = [𝑥, 𝑦] ∈ 𝔤 (6) is a derivation. the map ad: 𝔤 → 𝔤𝔩(𝔤) is called an adjoint representation. let a representation of lie algebra 𝔤 in the dual vector space 𝔤∗ is denoted by ad∗ whose value on 𝔤 is defined by 〈ad∗(𝑥)𝜑, 𝑦〉 = 〈𝜑, ad∗(−𝑥)𝑦〉 = 〈𝜑, [𝑦, 𝑥]〉 (7) for 𝜑 ∈ 𝔤∗, for all 𝑥, 𝑦 ∈ 𝔤. a stabilizer of lie algebra 𝔤 at the point 𝜑 ∈ 𝔤∗ is given in the following form: 𝔤𝜑 = {𝑥 ∈ 𝔤 | ad∗(𝑥)𝜑 = 0} (8) definition 7 [20] let 𝔤 be a lie algebra whose 𝔤∗ be a dual vector space of 𝔤. a lie algebra 𝔤 is said to be frobenius lie algebra if there exist linear functional 𝜑 ∈ 𝔤∗ such that the stabilizer of 𝔤 on 𝜑 is equal to 0. furthermore, we review briefly some basic notations needed in levi decomposition. we explain levi’s theorem which states that a finite dimensional lie algebra can be expressed as the semidirect sum of the levi subalgebra and the radical. theorem 4 [18] let 𝔤 be a lie algebra and let 𝔤 be not solvable, then 𝔤/𝑅𝑎𝑑(𝔤) is a semisimple lie subalgebra. theorem 5 [18] let 𝔤 be a finite dimensional lie algebra. if 𝔤 is not solvable, then there is a semisimple subalgebra 𝔰 of 𝔤 such that 𝔤 = 𝔰 ⊕ 𝑅𝑎𝑑(𝔤). (9) in this decomposition, 𝔰 ≅ 𝔤/𝑅𝑎𝑑(𝔤) and we have commutation relations as follows [𝔰, 𝔰] = 𝔰, [𝔰, 𝑅𝑎𝑑(𝔤)] ⊆ 𝑅𝑎𝑑(𝔤), [𝑅𝑎𝑑(𝔤), 𝑅𝑎𝑑(𝔤)] ⊆ 𝑅𝑎𝑑(𝔤). (10) the example of levi decomposition can be seen in the work of [18], one of all example its levi decomposition as follows example 1 [18] let 𝔤 be a lie algebra spanned by levi decomposition of frobenius lie algebra of dimension 6 henti 397 {𝑥1 = ( 0 1 0 1 0 0 0 0 1 ) , 𝑥2 = ( 1 0 0 0 −1 0 0 0 1 ) , 𝑥3 = ( 0 1 0 0 0 0 0 0 1 ) , 𝑥4 = ( 0 1 0 1 0 0 0 0 0 )} (11) where lie bracket non-zero is [𝑥1, 𝑥2] = 4𝑥1 − 4𝑥3 − 2𝑥4, [𝑥1, 𝑥3] = 𝑥1 − 𝑥2 − 𝑥4, [𝑥2, 𝑥3] = −2𝑥1 + 2𝑥3 + 2𝑥4, [𝑥2, 𝑥4] = −4𝑥1 + 4𝑥3 + 2𝑥4, [𝑥3, 𝑥4] = −𝑥1 + 𝑥2 + 𝑥4. the lie algebra 𝔤 can be express in 𝔤 = 𝑅𝑎𝑑(𝔤) ⋊ 𝔥 where radical 𝑅𝑎𝑑(𝔤) = 𝑠𝑝𝑎𝑛 {( 0 0 0 0 0 0 0 0 1 )} and levi subalgebra 𝔥 is spanned by {𝑧1 = ( 0 1 0 1 0 0 0 0 0 ) , 𝑧2 = ( 1 0 0 0 −1 0 0 0 0 ) , 𝑧3 = ( 0 1 0 0 0 0 0 0 0 )}. results and discussion in this section, let 𝔞𝔣𝔣(2) be the affine lie algebra and let 𝔞𝔣𝔣(2) be realized in the following matrix form 𝔞𝔣𝔣(2) = {( 𝑎 𝑏 𝑥 𝑐 𝑑 𝑦 0 0 0 ) | 𝑎, 𝑏, 𝑐, 𝑑, 𝑥, 𝑦 ∈ ℝ} ⊆ 𝔤𝔩3(ℝ), (12) with the standard basis for 𝔞𝔣𝔣(2), we have 𝑆 = {𝑥1 = ( 1 0 0 0 0 0 0 0 0 ) , 𝑥2 = ( 0 1 0 0 0 0 0 0 0 ) , 𝑥3 = ( 0 0 0 1 0 0 0 0 0 ) , 𝑥4 = ( 0 0 0 0 1 0 0 0 0 ) , 𝑥5 = ( 0 0 1 0 0 0 0 0 0 ) , 𝑥6 = ( 0 0 0 0 0 1 0 0 0 )}. (13) the lie brackets for the affine lie algebra 𝔞𝔣𝔣(2) is defined by [𝑎, 𝑏] = 𝑎𝑏 − 𝑏𝑎, ∀ 𝑎, 𝑏 ∈ 𝔞𝔣𝔣(2) such that the non-zero lie brackets for the affine lie algebra 𝔞𝔣𝔣(2) as follows (14) theorem 5 [8] let 𝔞𝔣𝔣(2) be a lie algebra of dimension 6 with basis in the equation (13). let 𝔞𝔣𝔣(2)∗ be its dual vector space of 𝔞𝔣𝔣(2). then there exist a linear functional 𝜑 = 𝑥2 ∗ + 𝑥6 ∗ ∈ 𝔞𝔣𝔣(2)∗ such that 𝔤𝜑 = {0}. therefore, the affine lie algebra 𝔞𝔣𝔣(2) is frobenius. in this section of the discussion is our main result, we will prove the proposition 1 and the proposition 2 as follows. proposition 1. the affine lie algebra 𝔞𝔣𝔣(2) is not solvable. proof. we have that 𝐷1(𝔞𝔣𝔣(2)) = [𝐷(𝔞𝔣𝔣(2)), 𝐷(𝔞𝔣𝔣(2))] = 𝑠𝑝𝑎𝑛{𝑥1 − 𝑥4, 𝑥2, 𝑥3, 𝑥5, 𝑥6}. next, we compute 𝐷2(𝔞𝔣𝔣(2)) also obtained 𝐷2(𝔞𝔣𝔣(2)) = [𝐷1(𝔞𝔣𝔣(2)), 𝐷1(𝔞𝔣𝔣(2))] = 𝑠𝑝𝑎𝑛{𝑥1 − 𝑥4, 𝑥2, 𝑥3, 𝑥5, 𝑥6}. therefore, there not exist 𝑛 > 0 that causes 𝐷𝑛(𝔞𝔣𝔣(2)) = {0}. thus, the affine lie algebra 𝔞𝔣𝔣(2) is not solvable. ∎ proposition 2. let 𝔞𝔣𝔣(2) be frobenius affine lie algebra whose basis 𝑆 = {𝑥𝑖}𝑖=1 6 where the non-zero brackets for 𝔞𝔣𝔣(2) in the equation (14). the affine lie algebra 𝔞𝔣𝔣(2) is not [𝑥1, 𝑥2] = 𝑥2, [𝑥1, 𝑥3] = −𝑥3, [𝑥1, 𝑥5] = 𝑥5, [𝑥2, 𝑥3] = 𝑥1 − 𝑥4, [𝑥2, 𝑥4] = 𝑥2, [𝑥2, 𝑥6] = 𝑥5, [𝑥3, 𝑥4] = −𝑥3, [𝑥3, 𝑥5] = 𝑥6, [𝑥4, 𝑥6] = 𝑥6. levi decomposition of frobenius lie algebra of dimension 6 henti 398 solvable then there exist 𝔥 = 𝑠𝑝𝑎𝑛{𝑥1, 𝑥2 + 𝑥5 + 𝑥6, 𝑥3 + 𝑥5 + 𝑥6, 𝑥4} is the semisimple lie subalgebra of 𝔞𝔣𝔣(2) and 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} is the radical of 𝔞𝔣𝔣(2) such that 𝔞𝔣𝔣(2) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} ⋊ 𝑠𝑝𝑎𝑛{𝑥1, 𝑥2 + 𝑥5 + 𝑥6, 𝑥3 + 𝑥5 + 𝑥6, 𝑥4}. (15) proof. firstly, we have the structure matrix of 𝔞𝔣𝔣(2) is 𝐶(𝔞𝔣𝔣(2)) = ( 0 𝑥2 −𝑥3 0 𝑥5 0 −𝑥2 0 𝑥1 − 𝑥4 𝑥2 0 𝑥5 𝑥3 𝑥4 − 𝑥1 0 −𝑥3 𝑥6 0 0 −𝑥2 𝑥3 0 0 𝑥6 −𝑥5 0 −𝑥6 0 0 0 0 −𝑥5 0 −𝑥6 0 0 ) . (16) then, we find the maximal linearly independent set in the structure matrix such that basis 𝐵 = {𝑦1 = 𝑥1 − 𝑥4, 𝑦2 = 𝑥2, 𝑦3 = 𝑥3, 𝑦4 = 𝑥5, 𝑦5 = 𝑥6} of the product space [𝔞𝔣𝔣(2), 𝔞𝔣𝔣(2)]. next, calculate ad 𝑥𝑖 and ad 𝑦𝑗 for 1 ≤ 𝑖 ≤ 6, 1 ≤ 𝑗 ≤ 5, we get ad 𝑥1 = [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] , ad 𝑥2 = [ 0 0 1 0 0 0 −1 0 0 1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] , ad 𝑥3 = [ 0 −1 0 0 0 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] , ad 𝑥4 = [ 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] , ad 𝑥5 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0] , ad 𝑥6 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0] , ad 𝑦1 = ad 𝑥1 − 𝑥4 = [ 0 0 0 0 0 0 0 2 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1] . (17) furthermore, compute the radical of 𝔞𝔣𝔣(2) where 𝑥 = ∑ 𝛼𝑖𝑥𝑖 6 𝑖=1 ∈ 𝑅𝑎𝑑(𝔞𝔣𝔣(2)), then find value 𝛼𝑖 using equations (17), we have ∑ 𝛼𝑖 6 𝑖=1 𝑇𝑟 (𝑎𝑑 𝑥𝑖 ⋅ 𝑎𝑑 𝑦𝑗) = 0 ; 1 < 𝑗 ≤ 5 𝛼1 [ 5 0 0 0 0] + 𝛼2 [ 0 5 5 0 0] + 𝛼3 [ 0 5 0 0 0] + 𝛼4 [ −5 0 0 0 0 ] + 𝛼5 [ 0 0 0 0 0] + 𝛼6 [ 0 0 0 0 0] = [ 0 0 0 0 0] . (18) levi decomposition of frobenius lie algebra of dimension 6 henti 399 next, we solve linear equations (18) such we find that 𝛼1 − 𝛼4 = 0, 𝛼2 = 0, 𝛼3 = 0, 𝛼5 = 𝑠, 𝛼6 = 𝑡, and with 𝛼1 = 0, then we get 𝑥 = ∑ 𝛼𝑖𝑥𝑖 6 𝑖=1 = 0𝑥1 + 0𝑥2 + 0𝑥3 + 0𝑥4 + 𝑠𝑥5 + 𝑡𝑥6 = 𝑠𝑥5 + 𝑡𝑥6. therefore, we obtain the radical of 𝔞𝔣𝔣(2) is 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} = 𝑠𝑝𝑎𝑛{𝑟1, 𝑟2}. (19) next, we find basis levi subalgebra of 𝔞𝔣𝔣(2). in this cases, 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) is abelian because [𝑟𝑖, 𝑟𝑗] = 0 for all 1 ≤ 𝑖, 𝑗 ≤ 2. complement on 𝔞𝔣𝔣(2) respect to 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) spanned by {𝑥1, 𝑥2, 𝑥3, 𝑥4}. the quotient algebra 𝔞𝔣𝔣(2)/𝑅𝑎𝑑(𝔞𝔣𝔣(2)) is spanned by �̅�1, �̅�2, �̅�3, �̅�4 and we have its brackets as follows [�̅�1, �̅�2] = �̅�2, [�̅�1, �̅�3] = −�̅�3, [�̅�2, �̅�3] = �̅�1 − �̅�4, [�̅�2, �̅�4] = �̅�2, [�̅�3, �̅�4] = −�̅�3. (20) we set levi subalgebra spanned by 𝑧1 = 𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 𝑧2 = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, 𝑧3 = 𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2, 𝑧4 = 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2. (21) next, we calculate to determine the four unknown 𝛼, 𝛽, 𝛾, 𝛿 such that 𝑧1, 𝑧2, 𝑧3, 𝑧4 span a semisimple lie algebra that is isomorphic to 𝔞𝔣𝔣(2)/𝑅𝑎𝑑(𝔞𝔣𝔣(2)). since 𝑧1, 𝑧2, 𝑧3, 𝑧4 have the same commutation relations as �̅�𝑖, 1 ≤ 𝑖 ≤ 4 written in equations (20), we then get [𝑧1, 𝑧2] = 𝑧2, [𝑧1, 𝑧3] = −𝑧3, [𝑧2, 𝑧3] = 𝑧1 − 𝑧4, [𝑧2, 𝑧4] = 𝑧2, [𝑧3, 𝑧4] = −𝑧3. (22) we substitution the equation (22) onto (23) such that equation can be written as [𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2] = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, (23) [𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2] = −(𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2), (24) [𝑥2 + ∑ 𝛽𝑗𝑟𝑗 2 𝑗=1 , 𝑥3 + ∑ 𝛾𝑗𝑟𝑗 2 𝑗=1 ] = (𝑥1 + ∑ 𝛼𝑗𝑟𝑗 2 𝑗=1 ) − (𝑥4 + ∑ 𝛿𝑗𝑟𝑗 2 𝑗=1 )(25) [𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2] = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, (26) [𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2, 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2] = −(𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2). (27) then, we apply the equations (23), (24), (25), (26), and (27) to compute 𝛼𝑖, 𝛽𝑖, 𝛾𝑖, 𝛿𝑖, 1 ≤ 𝑖 ≤ 2. from equations (23) and (26) obtained that 𝛽1 = 𝛽2 = 1. from equations (24) and (27) obtained that 𝛾1 = 𝛾2 = 1. for equations (25), we obtained that 𝛼1 − 𝛿1 = 𝛼2 − 𝛿2 = 0 and let 𝛼𝑖 = 0, such that we have 𝑧1 = 𝑥1, 𝑧2 = 𝑥2 + 𝑟1 + 𝑟2, 𝑧3 = 𝑥3 + 𝑟1 + 𝑟2, 𝑧4 = 𝑥4. thus, the levi subalgebra spanned by {𝑧1 = ( 1 0 0 0 0 0 0 0 0 ) , 𝑧2 = ( 0 1 1 0 0 1 0 0 0 ) , 𝑧3 = ( 0 0 1 1 0 1 0 0 0 ) , 𝑧4 = ( 0 0 0 0 1 0 0 0 0 )}. (28) ∎ conclusions it has been proven in proposition 2 that the affine lie algebra 𝔞𝔣𝔣(2) can be decomposed into its subalgebra and radicals which written in the equations (15). from our result of this paper, other research can study about decomposition of the general formula affine lie algebra 𝔞𝔣𝔣(n) of dimension 𝑛(𝑛 + 1). for future research, the decomposition process can be expanded from the decomposition result of 𝔞𝔣𝔣(𝑛) in its radical and levi subalgebra form such that we can find structure frobenius lie algebra 𝔞𝔣𝔣(𝑛) of its decomposition. levi decomposition of frobenius lie algebra of dimension 6 henti 400 references [1] j. e. humphreys, introduction to lie algebras and representation theory. springer, 1972. [2] m. a. alvarez, m. c. rodríguez-vallarte, and g. salgado, “contact and frobenius solvable lie algebras with abelian nilradical,” commun. algebr., vol. 46, no. 10, pp. 4344–4354, 2018, doi: 10.1080/00927872.2018.1439048. 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[14] diatta, a., b. manga, and a. mbaye, “on systems of commuting matrices, frobenius lie algebras and gerstenhaber’s theorem,” arxiv:2002.08737., 2020. [15] d. n. pham, “$\frak{g}$-quasi-frobenius lie algebras,” arch. math., vol. 52, pp. 233–262, 2016, [online]. available: http://arxiv.org/abs/1701.01680. [16] m. goze and e. remm, “contact and frobenius forms on lie groups,” differ. geom. its appl., vol. 35, pp. 74–94, 2014, doi: 10.1016/j.difgeo.2014.05.008. [17] t. barajas, e. roque, and g. salgado, “principal derivations and codimension one ideals in contact and frobenius lie algebras,” commun. algebr., vol. 47, no. 12, pp. 5380–5391, 2019, doi: 10.1080/00927872.2019.1623238. [18] m. dagli, “levi decomposition of lie algebras; algorithms for its computation,” iowa state university, 2004. [19] j. hilgert and k.-h. neeb, structure and geometry of lie groups. new york: springer monographs in mathematics, 2012. [20] a. i. ooms, on frobenius lie algebras, vol. 8, no. 1. 1980. spline nonparametric regression to identify factors affecting gender empowerment measure (gem) in east java cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 105-117 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 25, 2021 reviewed: october 10, 2021 accepted: november 05, 2021 doi: https://doi.org/10.18860/ca.v7i1.12993 spline nonparametric regression to identify factors affecting gender empowerment measure (gem) in east java luluk mahfiroh1, yuniar farida2* 1,2 department of mathematics, faculty of science and technology, sunan ampel state islamic university surabaya, indonesia email: mahfirohluluk@gmail.com, yuniar_farida@uinsby.ac.id* *corresponding author abstract gender is a multidimensional issue that's not limited to gender discrimination, but also includes the economic, educational, and health aspects, which then become the focus of almost all the sustainable development goals (sdgs). evaluation of the development devoted to the perspective of the gender using several indicators, gender development index (gdi) and gender empowerment measure (gem). gem describes the role of women in the economic sphere and is measured by equality in political participation. gem of east java for 5 consecutive years (2014 – 2018) is lower than the average national gem. this study aims to identify factors affecting gem in east java using nonparametric regression spline quadratic. the result of the regression model shows that all variables selected are affecting gem in east java, they are the labor force participation rate (lfpr) population of women (𝑥1), school participation rate (spr) high school population of women (𝑥2), percentage of population female that working in the formal sector (𝑥3), sex ratio (𝑥4), percentage of population female that working as members of people’s representative council (𝑥5), percentage of population female that working as civil servants (𝑥6), and rate of women's income donations (𝑥7). the model generates 𝑅 2 value of 93.74% and mape of 3.22%. this research contributes to the implementation of non-parametric spline regression in identifying various factors that influence social phenomena. keywords: gender empowerment measure (gem); gender development index (gdi); nonparametric regression; spline; generalized cross-validation (gcv) introduction discussion about gender is indeed inseparable from the concept of gender equality and justice. the goal is to kill the patriarchal culture that overrides the role of women. at the beginning of the world's development, history records that very few female figures played important events. information about women's roles in social, economic, and political movements is also minimal [1]. the male domination of women influences it. such dominance is inseparable from the patriarchal ideology adopted by most of the world's people since a long time ago [2]. gender is a multidimensional issue that's not limited to gender discrimination, but also includes the economic, educational, and health aspects, which then become the focus of almost all the sustainable development goals (sdgs) [3]. in indonesia, evaluation of https://doi.org/10.18860/ca.v7i1.12993 mailto:mahfirohluluk@gmail.com mailto:yuniar_farida@uinsby.ac.id* spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 106 development outcomes that are devoted to a gender perspective uses several indicators, namely the gender development index (gdi) and the gender empowerment measure (gem). gdi explains the gap in human development between men and women. meanwhile, gem describes women's role in the economic sphere and is measured by equality in political participation [4]. gem is the percentage of women who work as professionals, managers, technicians, and all forms of leadership [5]. gem east java's achievements in 2017 and 2018 were 69.37 and 69.71. it is lower than the national gem average of 71.74 in 2017 and 72.1 in 2018 [6]. gem's calculations indicate the proportion of managers, administrative staff, professional workers, and technicians; the balance of representation in parliament; and non-farm workers' wages. in addition to these indicators, other factors that may influence it, including the labor force participation rate (lfpr), school participation rate (spr), the percentage of the population working in the formal sector, the gender ratio, the percentage of income contributions, the percentage of the population female that working as members of people’s representative council and civil servants. to find out the relationship of these factors to gem values can be done by mathematical analysis. one of the analytical methods used to solve the problem is mathematical modeling [7]. mathematical modeling can be done using regression analysis to determine the causal relationship with other variables. [8]. regression analysis is grouped into 3, namely parametric, nonparametric, and semiparametric [9]. research conducted by adawiyah [10], it is obtained that the spline estimator is strongly influenced by the number of knots, the number of orders, the location of knot points, and the best spline regression model using two-knot points. conducted by fadhilah's research [11], the best spline model relies heavily on determining the optimal knot point with a minimum generalized cross-validation (gcv) value. the best-truncated spline regression model lies in order 2 [12]. the regression function estimation with a nonparametric approach is done with the spline technique to adjust effectively to data [13]. this research used modeling spline nonparametric regression to determine the factors that affect the gender empowerment measure in east java. nonparametric regression is used because the data is not in a particular pattern or the regression curve's shape is limited. methods data and data processing the data used in this study is secondary data in 2013-2018 sourced from bps (badan pusat statistik) of east java province both through the official website of www.bps.go.id and publication books available at the service center. in this study, there are eight (8) variables, which consist of one dependent variable (𝑦) dan seven independent variables (𝑥). they are gender empowerment measure (gem) (𝑦), labor force participation rate (lfpr) of the female population (𝑥1), school participation rate (spr) of female population in the high school level (𝑥2), percentage of female population in the formal sector (𝑥3), sex ratio (𝑥4), percentage of female that working as members of people’s representative council (𝑥5), percentage of female population that working as civil servants (𝑥6), and percentage of women's income donations (𝑥7). the data obtained in this study were analyzed using the nonparametric spline regression method for longitudinal data with the rstudio program. the steps to achieve the objectives of this research are as follows: spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 107 a) create a descriptive statistical analysis and scatter plot for each variable a scatter plot is used to detect relationship patterns between the y variable with each x variable, which is predicted to be a factor in its influence. scatter plots provide information on the pattern of regression curve shapes used in modeling [14]. b) create a model of gem with nonparametric spline regression. nonparametric regression models, in general, can be presented as follows: 𝑦𝑖 = 𝑓(𝑥𝑖) + 𝜀𝑖 ; 𝑖 = 1, 2, 3, … , 𝑛 (1) with 𝑦𝑖 is the response variable; 𝑥𝑖 is predictor variable; 𝑓(𝑥𝑖) is a regression function that doesn't follow a particular pattern; 𝜀𝑖 = (𝜀1, 𝜀2, … , 𝜀𝑛) 𝑇 is a free mutual error vector with zero alignments and diversity 𝜎2 [15]. estimation of function 𝑓(𝑥𝑖) in nonparametric regression is performed with the spline estimator [16]. if a regression curve is 𝑓 is an additive model and approached with the spline function, the regression model is: 𝑦𝑖𝑗 = ∑ 𝛽ℎ𝑖𝑥𝑖𝑗𝑝 ℎ + ∑ 𝛼𝑙𝑖(𝑥𝑖𝑗𝑝 − 𝐾𝑙𝑖)+ 𝑞 + 𝜀𝑖𝑗 𝑚 𝑙=1 𝑞 ℎ=0 (2) with 𝑦𝑖𝑗 is response variables in the i-subject and j-time observations; 𝑥𝑖𝑗𝑝 is ppredictor variables on the i-subject and j-time observations; 𝜀𝑖𝑗 is a random error on isubject and j-time observation; 𝑞 is polynomial degrees. in this study, the value of 𝑖 is 1, 2, ..., 38, that represent the subject/region in east java (38 cities/regencies); 𝑗 is the number of observations (2013-2018) with a value of 1, 2, ..., 6; 𝐾𝑙𝑖 is the number of knots. then, the function of (𝑥𝑖𝑗𝑝 − 𝐾𝑙𝑖)+ 𝑞 is given by [17]: (𝑥𝑖𝑗𝑝 − 𝐾𝑙𝑖)+ 𝑞 = { (𝑥 − 𝐾𝑙𝑖) 𝑞 ; 𝑥𝑖𝑗𝑝 ≥ 𝐾𝑙𝑖 0 ; 𝑥𝑖𝑗𝑝 < 𝐾𝑙𝑖 this study used a nonparametric regression spline with a second-degree polynomial curve or quadratic for longitudinal data. the quadratic curves are selected because they can give a smaller error than linear curves and are suitable for data with more complex patterns. in this study, the ordinary least square (ols) method is used to estimate the parameter value of 𝛽 in equation [2]. by using the spline regression model as a smooth curve estimation 𝑓(𝑥), the estimation equation is as follows [18]: �̂� = (𝑋𝑇𝑋)−1𝑋𝑇𝑦 (3) with matrix x as follows: 𝑋 = [ 1 𝑥111 … 𝑥111 𝑞 (𝑥111 − 𝐾11)+ 𝑞 … (𝑥111 − 𝐾𝑚1)+ 𝑞 … 𝑥11𝑝 … 𝑥11𝑝 𝑞 (𝑥11𝑝 − 𝐾𝑚11)+ 𝑞 … (𝑥11𝑝 − 𝐾𝑚𝑝1) + 𝑞 1 𝑥211 … 𝑥211 𝑞 (𝑥211 − 𝐾22)+ 𝑞 … (𝑥211 − 𝐾𝑚2)+ 𝑞 … ⋮⋱⋮ 𝑥21𝑝 … 𝑥21𝑝 𝑞 (𝑥21𝑝 − 𝐾𝑚12)+ 𝑞 … (𝑥21𝑝 − 𝐾𝑚𝑝2) + 𝑞 1 𝑥𝑛𝑡1 … 𝑥𝑛𝑡1 𝑞 (𝑥𝑛𝑡1 − 𝐾𝑚𝑛)+ 𝑞 … (𝑥𝑛𝑡1 − 𝐾𝑚𝑛)+ 𝑞 … 𝑥𝑛𝑡𝑝 … 𝑥𝑛𝑡𝑝 𝑞 (𝑥𝑛𝑡𝑝 − 𝐾𝑚1𝑛)+ 𝑞 … (𝑥𝑛𝑡𝑝 − 𝐾𝑚𝑝𝑛) + 𝑞 ] spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 108 c) select the optimal knot point using the gcv method. the selection of optimal knot points is critical in nonparametric regression models. knot points are joint fusion points that have behavior changes in the data. one appropriate method for selecting the optimal knot points is generalized crossvalidation (gcv) [19]. it is said to be the optimal knot point when obtained the lowest or minimum gcv value. the gcv function, according to eubank [20], is as follows: 𝐺𝐶𝑉(𝐾𝑙𝑖) = 𝑀𝑆𝐸(𝐾𝑙𝑖) (𝑛−1𝑡𝑟𝑎𝑐𝑒[𝐼 − 𝐻𝐾𝑙𝑖]) 2 (4) with 𝑛 is the amount of data; i is an identity matrix, and mse is mean square error. then, 𝐻 = 𝑋(𝑋𝑇𝑋)−1𝑋𝑇 and 𝑀𝑆𝐸(𝐾𝑙𝑖) is defined as: 𝑀𝑆𝐸(𝐾𝑙𝑖) = 𝑛 −1 ∑ ∑(𝑦𝑖𝑗 − 𝑓(𝑥𝑖𝑗𝑝)) 2 𝑡 𝑗=1 𝑛 𝑖=1 d) modeling gem using spline with optimal knot points. e) interpret the model and conclude. results and discussion characteristics of gem in east java year 2013-2018 an overview of the data can be seen in the following table. table 1. the descriptive statistics of gem variable (y) and various variable affecting the gem (𝑥1 ….. 𝑥7) variable average variance maximum value minimum value 𝑦 65.82 75.82 83.29 42.09 𝑥1 55.19 33.55 72.80 43.56 𝑥2 71.55 190.07 100 25.30 𝑥3 31.87 269.49 67.61 5.98 𝑥4 97.13 6.40 101.64 90.64 𝑥5 17.13 87.80 51.52 0 𝑥6 48.11 33.96 83.88 29.93 𝑥7 32.73 18.88 40.34 22.81 table 1 shows that the average gem in east java in 2013-2018 was 65.82 with a variance of 75.82. high variance data shows the data used is spread far from the average caused by the outlier value. this happens because of differences in women's participation in areas with city and district status. in districts, the majority of women are only housewives, while women's participation in the public sector is minimal. the maximum value is 83.29, which is the gem value of surabaya city in 2018, while the minimum value is 42.09, which is the gem value of the sampang regency in 2013. some variables that have an average score below 50% indicate a gender gap where women's involvement in the variable is lower than that of men. characteristics of variable x5 or percentage of women as members of parliament in east java in 2013-2018 obtained the minimum value is 0 this is because in bangkalan regency in the period 2014 – 2018 there are no members of the dprd of the female sex. spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 109 relationship patterns between 𝒚 (gem) variable with 𝒙 variables relationship patterns between 𝑦 variables with 𝑥 variables do not form a specific pattern for each year. it can be seen from the random plot spread. so that all variables can be used as nonparametric components. the scatter plot of their relationship is shown in figures 1 to 7 as follows: 2016 2017 2018 figure 1. scatter plots between 𝑦 variable and𝑥1 (labor force participation rate (lfpr) of the female population) variable 2016 2017 2018 figure 2. scatter plots between 𝑦 variable and 𝑥2 (school participation rate (spr) high school level female population) variable 2016 2017 2018 figure 3. scatter plots between 𝑦 variable and 𝑥3 (percentage of female population in the formal sector) variable spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 110 2016 2017 2018 figure 4. scatter plots between 𝑦 variables and 𝑥4 (sex ratio) variable 2016 2017 2018 figure 5. scatter plots between 𝑦 variables and 𝑥5 (percentage of females that working as members of people’s representative council) variable 2016 2017 2018 figure 6. scatter plots between 𝑦 variables and 𝑥6 (percentage of female population that working as civil servants) variable 2016 2017 2018 figure 7. scatter plots between 𝑦 variables and 𝑥7 (percentage of women's income donations) variables spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 111 scatter plot between variable y (gem) and all variables x (𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥7) are random spread and do not follow a specific pattern. so those variables are included in nonparametric components. spline quadratic nonparametric regression modeling with one-knot points the nonparametric regression model of a quadratic spline with the one-knot point in the gem data of east java province is as follows: 𝑦𝑖𝑗 = 𝛽0𝑖 + 𝛽1𝑖𝑥1𝑖𝑗 + 𝛽1𝑖(𝑥1𝑖𝑗) 2 + 𝛼1𝑖(𝑥1𝑖𝑗 − 𝐾1𝑖) 2 + 𝛽2𝑖𝑥2𝑖𝑗 + 𝛽2𝑖(𝑥2𝑖𝑗) 2 + 𝛼1𝑖(𝑥2𝑖𝑗 − 𝐾1𝑖) 2 + ⋯ + 𝛽7𝑖𝑥7𝑖𝑗 + 𝛽7𝑖(𝑥7𝑖𝑗) 2 + 𝛼1𝑖(𝑥7𝑖𝑗 − 𝐾1𝑖) 2 + 𝜀𝑖𝑗 here is the gcv value for a one-knot point. table 2. smallest gcv value with one-knot point order regency/city 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 𝒙𝟓 𝒙𝟔 𝒙𝟕 gcv 23 pacitan 70.89 65.98 13.3 95.37 16.31 45.61 38.8 1.07e-26 ponorogo 59.57 74.24 18.08 99.84 12.47 45.67 34.65 trenggalek 62.56 64.25 14.78 98.71 15.28 47.22 36.75 tulungagung 57.63 74.18 28.49 95.12 6.69 50.7 37.7 blitar 53.75 64.25 27.71 100.24 15.98 53.95 40.09 kediri 52.1 78.59 34.61 100.56 29.23 52.29 30.65 malang 51.15 58.56 31.93 101.01 17.04 50.46 36.5 lumajang 46.99 55.7 22.74 95.22 14.55 45.91 23.16 ⋮ ⋮ malang city 51.57 80.62 60.92 97.35 23.44 50.3 34.22 probolinggo city 50.17 75.5 54.51 97.15 24.47 49.55 30.83 pasuruan city 53.74 72.88 50.9 97.98 6.98 50.45 31.02 mojokerto city 56.47 81.99 50.69 96.52 28.46 50.7 36.47 madiun city 54.1 81.2 48.4 93.71 34.27 54.13 38.42 surabaya city 52.06 71.02 64.72 97.59 39.04 68.19 35.1 batu city 55.93 83.4 41.47 101.35 25.63 53.31 29.97 the table above shows that the minimum gcv is 1.07e-26 in the 23rd order segment with optimal knot points on each 𝑥 variable; it only has a one-knot point. spline quadratic nonparametric regression modeling with two-knot points the next step is to create quadratic spline regression with two-knot points. the nonparametric regression model of the quadratic spline with two-knot points in gem data of east java province is as follows: 𝑦𝑖𝑗 = 𝛽0𝑖 + 𝛽1𝑖𝑥1𝑖𝑗 + 𝛽1𝑖(𝑥1𝑖𝑗) 2 + 𝛼1𝑖(𝑥1𝑖𝑗 − 𝐾1𝑖) 2 + 𝛼2𝑖(𝑥1𝑖𝑗 − 𝐾2𝑖) 2 + 𝛽2𝑖𝑥2𝑖𝑗 + 𝛽2𝑖(𝑥2𝑖𝑗) 2 + 𝛼1𝑖(𝑥2𝑖𝑗 − 𝐾1𝑖) 2 + 𝛼2𝑖(𝑥2𝑖𝑗 − 𝐾2𝑖) 2 + ⋯ + 𝛽7𝑖𝑥7𝑖𝑗 + 𝛽7𝑖(𝑥7𝑖𝑗) 2 + 𝛼1𝑖(𝑥7𝑖𝑗 − 𝐾1𝑖) 2 + 𝛼2𝑖(𝑥7𝑖𝑗 − 𝐾2𝑖) 2 + 𝜀𝑖𝑗 here are the resulting gcv values. (5) (6) spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 112 table 3. smallest gcv value with two-knot points order regency/city 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 𝒙𝟓 𝒙𝟔 𝒙𝟕 gcv 32 pacitan 43.79 53.46 10.44 95.32 15.06 69.87 38.36 2.83e-28 51.91 82.78 19.65 94.84 6.8 57.55 36.63 ponorogo 46.44 71.67 14.59 95.39 16.89 71.36 38.99 54.06 94.26 23.4 94.91 11.13 66.47 38.05 trenggalek 44.47 64.44 14.36 99.75 11.17 57.8 34.05 47.04 71.52 14.08 95.04 20.17 48.66 30.65 tulungagung 46.22 78.7 19.77 99.88 13.06 60.38 34.92 50.69 80.54 18.43 95.47 25.77 59.11 31.54 blitar 45.52 50.21 10.05 98.67 11.29 58.95 35.92 44.61 56.7 14.03 97.08 12.12 47.62 24.97 kediri 47.99 70.63 16.92 98.73 17.1 64.2 37.13 47.28 74.39 24.3 97.57 16.04 53.8 25.84 ⋮ ⋮ malang city 52.8 74.94 39.9 99.98 29.54 56.49 35.2 51.94 85.71 53.21 96.66 35.61 58 36.69 probolinggo city 48.65 66.19 26.26 98.9 8.41 46.53 26.58 52.56 67.42 45.15 93.57 27 52.35 37.84 pasuruan city 51.72 80.73 32.06 99.17 21.61 51.94 27.22 54.85 87.47 49.88 93.78 37.57 54.89 38.68 mojokerto city 49.55 61.96 19.55 98.69 16.33 44.11 24 54.91 60.59 62.27 97.54 28.48 50.53 34.7 madiun city 51.86 79.87 24.99 98.82 27.01 51.1 25.1 74.22 75.76 65.83 97.61 43.84 52.76 35.29 surabaya city 50.23 78.87 20.1 97.33 11.2 48.99 28.95 49.82 76.73 35.53 101.1 20.24 53.59 29.45 batu city 53.02 86.7 24.82 97.42 14.11 54.47 30.02 54.9 86.42 44.16 101.46 28.08 57 30.21 optimal knot point selection the selection of the best model is based on the selection of optimal knot points with the minimum gcv grades. gcv with one-knot point produces 48 alternate knot points with each gcv value and obtained a minimum value of 1.07e-26, while gcv with two-knot points also has 48 alternate knot points gcv value obtained a minimum value of 2.83e-28. so, selected nonparametric regression modeling spline quadratic using 2 knot points with the gem model in each city or regency in east java is: 𝑦𝑖𝑗 = 𝛽01 + 𝛽11𝑥1𝑖𝑗 + 𝛽12(𝑥1𝑖𝑗) 2 + 𝛼11𝑗(𝑥1𝑖𝑗 − 𝐾11𝑖) 2 + 𝛼21𝑖(𝑥1𝑖𝑗 − 𝐾21𝑖) 2 + 𝛽21𝑥2𝑖𝑗 + 𝛽22(𝑥2𝑖𝑗) 2 + 𝛼12𝑖(𝑥2𝑖𝑗 − 𝐾12𝑖) 2 + 𝛼22𝑖(𝑥2𝑖𝑗 − 𝐾22𝑖) 2 + 𝛽31𝑥3𝑖𝑗 + 𝛽32(𝑥3𝑖𝑗) 2 + 𝛼13𝑖(𝑥3𝑖𝑗 − 𝐾13𝑖) 2 + 𝛼23𝑖(𝑥3𝑖𝑗 − 𝐾23𝑖) 2 + 𝛽41𝑥4𝑖𝑗 + 𝛽42(𝑥4𝑖𝑗) 2 + 𝛼14𝑖(𝑥4𝑖𝑗 − 𝐾14𝑖) 2 + 𝛼24𝑖(𝑥4𝑖𝑗 − 𝐾24𝑖) 2 + 𝛽51𝑥5𝑖𝑗 + 𝛽52(𝑥5𝑖𝑗) 2 + 𝛼15𝑖(𝑥5𝑖𝑗 − 𝐾15𝑖) 2 + 𝛼25𝑖(𝑥5𝑖𝑗 − 𝐾25𝑖) 2 spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 113 + 𝛽61𝑥6𝑖𝑗 + 𝛽62(𝑥6𝑖𝑗) 2 + 𝛼16𝑖(𝑥6𝑖𝑗 − 𝐾16𝑖) 2 + 𝛼26𝑖(𝑥6𝑖𝑗 − 𝐾26𝑖) 2 + 𝛽71𝑥7𝑖𝑗 + 𝛽72(𝑥7𝑖𝑗) 2 + 𝛼17𝑖(𝑥7𝑖𝑗 − 𝐾17𝑖) 2 + 𝛼27𝑖(𝑥7𝑖𝑗 − 𝐾27𝑖) 2 overall gem model in each city or regency in east java obtains 𝑅2 value is 93.74%, which means that the model can explain the diversity of gem variable values of east java province by 93.74%; while the residual (6,26%) is explained by other variables that are not in the regression model. this model has a mean absolute percentage error (mape) value of 3.22%. parameter testing of quadratic spline nonparametric regression model the test results are displayed in the following anova table 4. table 4. anova results on model source df ss ms fcount p-value decision regresi 6 16134.355 2689.089 512.053 1.96e-120 failed to reject h0 error 205 1076.755 5.2516 total 227 17211.111 the decision obtained is reject 𝐻0, this means there is at least one significant parameter in the quadratic spline nonparametric regression model. the considerable parameter is an entire estimator 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 in each year from 2013-2018. interpretation of results of the quadratic spline nonparametric regression model for example, the implementation of the quadratic spline nonparametric regression model in longitudinal data is used in surabaya. the estimated coefficient of parameters and knot points obtained is substituted in equation (7), so it obtained a model of nonparametric regression of quadratic spline with longitudinal data for the city of surabaya as follows. in gem data, surabaya city is 37th out of a total of 38 cities/regencies, hence the value 𝑖 = 37 𝑦37𝑗 = 0.0008 − 0.0134𝑥1.37.𝑗 − 0.0363(𝑥1.37.𝑗) 2 + 0.0049(𝑥1.37.𝑗 − 50.32) 2 + 0.3059(𝑥1.37.𝑗 − 49.82) 2 − 0.0191𝑥2.37.𝑗 − 0.0391(𝑥2.37.𝑗) 2 + 0.0007(𝑥2.37.𝑗 − 78.87) 2 − 0.0132(𝑥2.37.𝑗 − 76.73) 2 − 0.0302𝑥3.37.𝑗 + 0.0048(𝑥3.37.𝑗) 2 + 0.2975(𝑥3.37.𝑗 − 20.10) 2 + 0.002(𝑥3.37.𝑗 − 35.53) 2 + 0.0457𝑥437𝑗 + 0.0458(𝑥437𝑗) 2 + 0.1919(𝑥437𝑗 − 97.33) 2 − 0.0056(𝑥4.37.𝑗 − 101.1) 2 + 0.111𝑥5.37.𝑗 + 0.0843(𝑥5.37.𝑗) 2 + 0.4336(𝑥5.37.𝑗 − 11.2) 2 − 0.0274(𝑥5.37.𝑗 − 20.24) 2 + 0.0028𝑥6.37.𝑗 + 0.0011(𝑥6.37.𝑗) 2 − 0.0042(𝑥6.37.𝑗 − 48.99) 2 + 0.0255(𝑥6.37.𝑗 − 53.59) 2 + 0.009𝑥7.37.𝑗 + 0.3078(𝑥7.37.𝑗) 2 − 0.0122(𝑥7.37.𝑗 − 28.95) 2 − 0.0009(𝑥7.37.𝑗 − 29.45) 2 from the model above, if the 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the influence of lfpr female population (𝑥1) on gem is: (7) spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 114 �̂� = − 0.0134𝑥1 − 0.0363(𝑥1) 2 + 0.0049(𝑥1 − 50.32) 2 + 0.3059(𝑥1 − 49.82) 2 if the lfpr female population variable (𝑥1) is below 50.32 and then there is an increase of 1 unit, then the gem value tends to decrease in value by 0.0363. at the time of lfpr female population (𝑥1) between the moderate intervals of 49.82 and 50.32 and there is an increase of 1 unit, the gem value will decrease by 0.0314. lastly, if the lfpr female population (𝑥1) is greater than or equal to 49.82 then there is an increase of 1 unit, then gem has an increase of 0.2745. overall, the higher the lfpr female population score (𝑥1) then the gem value will also be higher, and vice versa. this is because the lfpr is a representation of the large number of people participating in the economy, in this case is counted by the participation of working women. the model's interpretation about spr high school level female population variable (𝑥2) is if the 𝑥1, 𝑥3, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the influence of spr on gem is: �̂� = − 0.0191𝑥2 − 0.0391(𝑥2) 2 + 0.0007(𝑥2 − 78.87) 2 − 0.0132(𝑥2 − 76.73) 2 if the spr high school level female population (𝑥2) is below 78.87 and then there is an increase of 1 unit, then the gem value will decrease by 0.0391 units. when the aps is between the moderate intervals of 76.73 and 78.87 and there is an increase of 1 unit, the gem value will tend to decrease by 0.0384. lastly, if the spr high school level female population (𝑥2) is greater than or equal to 76.73 then there is an increase of 1 unit, then gem also decreases by about 0.0516 units. overall, the higher the spr high school female population (𝑥2) score, the higher the gem score, and vice versa. this is because the higher the level of education that a person finishes, the more it will encourage his participation in the job market. a person's chances of getting a job also tend to be in line with their level of education, especially the share of the current job market usually increases their education. the model's interpretation about percentage of female population working age working in the formal sector variable (𝑥3) is if the 𝑥1, 𝑥2, 𝑥4, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the influence of formal sector on gem is: �̂� = − 0.0302𝑥3 + 0.0048(𝑥3) 2 + 0.2975(𝑥3 − 20.10) 2 + 0.0020(𝑥3 − 35.53) 2 if the percentage of female population working age working in the formal sector (𝑥3) is below 20.10 and then there is an increase of 1 unit, then the gem value will increase by 0.0048 units. when the percentage of female population working age working in the formal sector (𝑥3) was between the moderate interval segments of 20.10 and 35.53 and there was an increase of 1 unit, the gem value also increased by 0.3023. lastly, if the percentage of the female population working age working in the formal sector (𝑥3) is greater than or equal to 35.53, there is an increase of 1 unit, the gem also increases by about 0.3043 units. so overall, if the percentage of the female population working age working in the formal sector (𝑥3) increases, gem value will also increase, and vice versa. this is because, the role of women in the economic field as measured in gem is women who work as professional workers, leadership, technicians, and technical or skilled workers. spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 115 the model's interpretation about the sex ratio variable is if the 𝑥1, 𝑥2, 𝑥3, 𝑥5, 𝑥6 and 𝑥7 assumed to be constant, the influence of sex ratio on gem is: �̂� = 0.0457𝑥4 + 0.0458(𝑥4) 2 + 0.1919(𝑥4 − 97.33) 2 − 0.0056(𝑥4 − 101.1) 2 if the sex ratio (𝑥4)is below the segment of 97.33 then if the value increases by one unit, gem will increase by 0.0458. whereas if the value of the sex ratio (𝑥4)is located between the moderate intervals of 97.33 and 101.1 then if the ratio value increases by one unit, gem tends to increase by 0.2377. if the sex ratio (𝑥4) is worth more than or equal to 101.1 then if there is an increase of one unit will affect the increase in gem by 0.2321. so overall, if the value of the sex ratio (𝑥4) increases then the gem value will also increase, and vice versa. this is because the sex ratio is a comparison of the number of male population per 100 female population in a given region and time. the model's interpretation about percentage female that working as members of people’s representative council variable (𝑥5) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥6 and 𝑥7 assumed to be constant, the influence of the people’s representative council on gem is: �̂� = 0.111𝑥5 + 0.0843(𝑥5) 2 + 0.4336(𝑥5 − 11.20) 2 − 0.0274(𝑥5 − 20.24) 2 if the percentage of females that work as members of the people's representative council(𝑥5) is below the segment of 11.20 then if the value increases by one unit, gem will experience an increase in value of 0.0843. meanwhile, if the percentage value is located between the moderate intervals of 11.20 and 20.24 then if the ratio value increases by one unit, gem also tends to increase by 0.5179 if the percentage value is more than or equal to 20.24 then if there is an increase of one unit will affect the increase in gem by 0.49. so overall, if the value of percentage female that working as members of the people's representative council (𝑥5) increases, then the gem value will also increase, and vice versa. this is because, gem also highlights the decision of women to participate in politics, so the role of women in the field of political decision-making is measured by the membership of the people’s representative council. the model's interpretation about percentage of population female that working as civil servants variable (𝑥6) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5 and 𝑥7 assumed to be constant, the influence of civil servants on gem is: �̂� = 0.0028𝑥6 + 0.0011(𝑥6) 2 − 0.0042(𝑥6 − 48.99) 2 + 0.0255(𝑥6 − 53.59) 2 if the percentage of women working as civil servants(𝑥6) is below the segment of 48.99 then if the value increases by one unit, gem will experience an increase in value of 0.0011. whereas if the percentage value is located between the moderate intervals of 48.99 and 53.59 then if the percentage value increases by one unit, gem also tends to decrease by 0.0031. if the percentage of women working as civil servants(𝑥6)is worth more than or equal to 53.59 then if there is an increase of one unit will affect the increase in gem value by 0.0224. so overall, if the percentage of population female that working as civil servants (𝑥6) increases, gem value will also increase, and vice versa. this is because, the employment status of civil servants is included in the status of formal sector employment but within the scope of statehood, the percentage of population female that working as civil servants is considered to affect gem. spline nonparametric regression to identify factors affecting gender empowerment measure (gem) luluk mahfiroh 116 the model's interpretation about the percentage of women's income donations variable (𝑥7) is if the 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5 and 𝑥6 assumed to be constant, the influence of income on gem is: �̂� = 0.0090𝑥7 + 0.3078(𝑥7) 2 − 0.0122(𝑥7 − 28.95) 2 − 0.0009(𝑥7 − 29.45) 2 if the percentage of women's income donations (𝑥7) is below 28.95 and there is an increase of 1 unit, then the gem value will increase by 0.3078 units. when the percentage value is between the medium interval segments of 28.95 and 29.45 and there is an increase of 1 unit, the gem value also increases by 0.296. lastly, if the percentage of women's income donations (𝑥7) is greater than or equal to 29.45, there is an increase of 1 unit, then gem also continues to increase by about 0.295 units. so overall, if the percentage value of women's income donations (𝑥7) increases, then the value of gem will also increase, and vice versa. this is because the income contribution is a contribution of the value of the proceeds received in return from members of households who work in this case women. conclusions identification of factors that affect gem using spline nonparametric regression results in a highly optimized model with mape at 3.22%. all factors (variable x) studied have a significant influence on gem value (y) so that the value 𝑅2 is 93.74% gem (y). this value indicates that the resulting model is already excellent. all research variables showed significant results on the model so that the factors that influenced gender empowerment measure (gem) east java province is labor force participation rate (lfpr) population of women (𝑥1), school participation rate (spr) high school level female population (𝑥2), percentage of female population that working in the formal sector (𝑥3), sex ratio (𝑥4), percentage of female population that working as members of people’s representative council (𝑥5), percentage of female population that working as civil servants (𝑥6), and percentage of female's income donations (𝑥7). areas below the interval are dominated by regions with district status, and madura islands have the most areas below the interval. references [1] gadis arifia and nur iman 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[20] r. l. eubank, nonparametric regression and spline smoothing, 2nd editio. new york: mercel dekker, 1988. forecasting rice paddy production in aceh using arima and exponential smoothing models cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 281-292 p-issn: 2086-0382; e-issn: 2477-3344 submitted: october 19, 2021 reviewed: december 09, 2021 accepted: january 05, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.13701 forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana1,*, amelia1, riezkypurnama sari1, ulyanabilla1, taufan talib2 1mathematics department, universitas samudra, indonesia 2mathematics department, universitas pattimura, indonesia *corresponding author email: nurviana@unsam.ac.id*, ameliamat@unsam.ac.id, riezkypurnamasari@unsam.ac.id, ulya.nabilla@unsam.ac.id, taufan.talib@fkip.unpatti.ac.id abstract indonesia targets aceh to be one of the rice paddy production centers and be able to carry out self-sufficient production in rice paddy and become a national granary. however, in reality, aceh's rice paddy production in its province is not consistent from year to year. this province has not been able to meet the food needs of rice paddy independently, so that it supplies rice paddy from other regions due to the difficulty of detecting the presence of a surplus of rice paddy.the purpose of this research is to forecast the yield ofrice paddy production in aceh for the future. the mathematical model that can be used is a time series model namely autoregressive integrated moving average (arima) and exponential smoothing. the forecasting results of rice paddy production in the next 5 years using the arima (3,1,1) model are 2453401; 2154784; 2111594; 1615171; and 2062436. while the estimation results using the winter exponential smoothing model are 1625925; 1645196; 1687667; 1605530; and 1555213. arima model (3,1,1) produces an mse/mad value of 3,34041 × 1010, while the winter exponential smoothing model produces an mse/mad value of 3,08616 × 1010. therefore, it can be concluded that the winter exponential smoothing model.by obtaining this resultsanalysis, the aceh government can make the right policies in planning for the provision of rice paddy food in the future. keywords: arima; forecasting; exponential smoothing; rice paddy introduction aceh is one of the provinces in indonesia which has large agricultural land. the majority of the population in aceh rely their life on the agricultural sector for their livelihood. according to the central bureau of statistics, the highest economic source of aceh is in the agricultural sector, where agriculture has a good contribution to the economy and fulfills the basic needs of the society. the agriculture of aceh excels in various commodities such as rice paddy, corn, soybeans, and chilies. the rice paddy plants are stapled food crop commodities whose needs continue to increase from year to year following population growth. rice paddy cultivation is the main activity and main source of income for more than 100 million households in developing countries in asia, africa, and latin america. in asiapacific more than 90 percent of the world's rice paddy has been produced and consumed[1]. the rice paddy plant is an ancient agricultural crop that until now is considered a staple crop in most tropical countries, especially in asia and africa. rice http://dx.doi.org/10.18860/ca.v7i1.13701 mailto:nurviana@unsam.ac.id mailto:ameliamath@unsam.ac.id mailto:riezkypurnamasari@unsam.ac.id mailto:ulya.nabilla@unsam.ac.id mailto:taufan.talib@fkip.unpatti.ac.id forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 282 paddy is the most important food crop in aceh because almost all people use rice paddy as a staple food and rice paddy is also a strategic food commodity that has a considerable influence on economic stability, especially inflation, social and political stability. indonesia targets aceh as one of the centers of rice paddy production and can carry out self-sufficient production in rice paddy and become a national granary [2]. aceh has many genotypes or local rice paddy varieties. according to [3]there are 50 aceh rice paddy genotypes that have been collected through exploration activities, but only a few local rice varieties are often planted.the varieties are ramos, dewi, sigupai, tinggong, and siputeh. based on bps [4], local rice paddy production of aceh is not consistent from year to year or in other words, it increases and decreases every year. in 2020, the rice paddy plants production reached 1.75 million tons, an increase from the previous year of 1.71 million tons. if it is converted to rice paddy, in 2020 rice paddy production in aceh will reach 1 million tons. this increase was due to an increase in the harvested area of rice paddy plants from 310.01 thousand hectares to 320.75 thousand hectares [4]. therefore, the province of aceh should have been able to meet the food needs of rice paddy independently. however, aceh still supplies rice paddy from other regions due to the difficulty of detecting the presence of a surplus of rice paddy. a mathematical model is needed to make plans related to food commodities, especially rice paddy. one of the mathematical models that can be used is the time series model. this model is used to estimate production results in the future period based on previous data. the time series model that will be used in this research is autoregressive integrated moving average (arima) and exponential smoothing. previous research related to the study of rice paddy production has been carried out by [5]which discussed the forecast of rice paddy production in gorontalo province using the double moving average method.the forecasting results for the next 5 years were obtained, in 2019 of 326318.5 tons, in 2020 of 32094.5 tons, and so on until 2023 of 304826.5 tons. other research [6]using double exponential smoothing model to assess the estimated production value for the next year. the application of this model obtained predictions of rice paddy harvests in the kudus regency in 2019 of 163,435.90 tons. other rice paddy production research [7]using the fuzzy time series model for forecasting the amount of rice paddy production in southeast sulawesi, the results of forecasting rice paddy production in 2015 were 657768.25191 tons. based on the explanation above, researchers are interested in analyzing the rice paddy production results using the arima and exponential smoothing models. the purpose of this study is to predict the local rice paddy production of aceh result in the future period and to see which of the two models is the best in estimating aceh's local rice paddy production. therefore, hopes that the government can make more precise planning in the provision of rice paddy food and can make aceh a national rice paddy production center. methods arima process arima is a time series model that can predict data for a certain period of time based on past data [8]. this model has very good accuracy when used for short-term forecasting. meanwhile, for long-term forecasting, the accuracy of the forecast is not good and usually, it will tend to be flat (level or constant) for a fairly long period. the arima process developed by box and jenkins in 1976 was a model that does not assume certain patterns in the historical data that was forecasted and was a model that completely ignores the independent variables in making forecasts that were used in forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 283 model formation[9]. the arima process is a combined model between autoregressive (ar) and moving average (ma). this model can represent stationary and non-stationary time series [10] [11]. the general form of arima model (p,d,q) is defined as[12]: 𝑌𝑡 − 𝑌𝑡−𝑑 = 𝛾0 + ∑ 𝛼𝑖 (𝑌𝑡−𝑖 − 𝑌𝑡−𝑖−𝑑) 𝑝 𝑖=1 + ∑ 𝛽𝑖 𝜀𝑡−𝑖 𝑞 𝑖=1 + 𝑒𝑡 (1) in practice, the data is commonly non-stationary so that modifications need to be made, by using differencing, to produce stationary data. exponential smoothing model exponential smoothing is an analytical time series model that is quite good and convenient in low ease of operation. the exponential smoothing model continuously makes improvements related to forecasting by taking the smoothing average of past values from a time series data by decreasing exponentially[13]. in general, the exponential smoothing model is divided into 3 models, namely single, double, and triple exponential smoothing (holt-winter's model). this research concentrates on triple exponential smoothing. this model is used when the data pattern shows very large differences, trends, and seasonal behavior. to deal with seasonality, a third equation parameter has been developed called the “holt-winters” model after the name of the inventor. the holt-winters method is based on three equations, namely stationary, trend, and seasonal elements[14]. the basic equation for the holt-winters method is as follows: [15] overall smoothing: st = α xt it−l + (1 − α)(st−1 + bt−1) trend smoothing: bt = γ(st − st−1) + (1 − γ)bt−1 seasonal smoothing: it = β xt st + (1 − β)it−l forecast: 𝐹𝑡+𝑚 = (𝑆𝑡 + 𝑏𝑡 𝑚)𝐼𝑡−𝐿+𝑚 results and discussion analysis of the data on the data on the amount of rice paddy production using the arima box-jenkins method and the exponential smoothing method. the data processed is data on rice paddy production in aceh from 1993 to 2020 and analyzed with usingminitab 19. data characteristics total rice paddy production result data plot rice paddy production from 1993 to 2020. figure 1. time series plot of total rice paddy production in aceh from 1993 to 2020 forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 284 descriptive statistics show the least quantity of rice paddy production in 2001 was 1,246,614 tons. meanwhile, the highest number of productions occurred in 2017, which was 2.49613 tons and the average total rice paddy production from 1993 to 2020 is around 1.61315 tons. based on figure 1, the amount of rice paddy production tends to go up and down. the fluctuation of the data on the amount of rice paddy production is not at a constant average value so that there is an indication that the data is not stationary. forecasting rice paddy production using the arima model a. model identification the initial step in identifying the data is to know whether the data is stationary in the mean and variance. identification has been done by determining: the time series plot, acf plot, pacf plot, and box-cox transformation. the identification process starts from determining whether the rice paddy production data is stationary to the variance or not. figure 2. box-cox plot of total rice paddy production based on figure 2, the rice paddy production data is not stationary to the variance because the rounded value is less than 1, where the rounded value is said to be good if the value is 1. figure 3. box-cox of total rice paddy production after transformation based on figure 3, a rounded value of 1.00 is obtained so that the data can be said to have been stationary in variance. furthermore, the differencing stage is carried out so that the data is stationary to the mean. stationary data to the mean can be seen visually through the acf plot. the following is an acf plot of total rice paddy production. forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 285 figure 4. acf plot of total rice paddy production before differencing figure 4 shows that the quantity of rice paddy production is not stationary in the mean, because the lags in the acf plot are still decreasing slowly. therefore, it is necessary to do differencing. here is the time series plot after differencing. figure 5. time series plot of rice paddy production after differencing figure 5 shows that the pattern of the amount of rice paddy production is stationary in the mean after the differencing process is carried out once. the next step is to identify the model to get the arima conjecture model. the identification of the arima model has been known based on the acf and pacf plots. the following is a plot of acf and pacf for the amount of rice paddy production after the differencing process at lag 1. forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 286 figure 6. acf plot of total rice paddy production data after differencing figure 7. pacf plot of total rice paddy production data after differencing based on figure 6 and 7, it is showing that there are no lags that come out or all lags are in a significant line in the acf plot, while the pacf plot is cut off at the 3rd lag. so it can be concluded that the suggested arima (p,d,q) models for rice paddy production were (1,1,0), arima (1,1,1), arima (3,1,0) and arima (3,1,1) b. estimation of parameters and diagnostic tests of residual the next step is the estimation of the model parameters for the tentative models that have been selected. the best model was selected based on the minimum values of means square error (mse). table 2. suggested arima (p,d,q) models for rice paddy production suspected model estimates of parameters mean square error value type coef se coef (1,1,0) 𝜙1 -0,354 0,183 4,32646e+10 (1,1,1) 𝜙1 -1,028 0,276 3,95604e+10 𝜃1 -0,853 0,465 (3,1,0) 𝜙1 -0,171 0,179 3,36785e+10 𝜙2 -0,091 0,189 𝜙3 -0,800 0,268 (3,1,1) 𝜙1 -0,396 0,339 3,34041e+10 𝜙2 -0,187 0,231 𝜙3 -0,812 0,290 𝜃1 -0,351 0,411 forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 287 based on table 2, the best selected arima model for rice paddy production is arima (3,1,1) with mse =3,34041e+10. so, it can be analyzed in the next step, namely by testing the assumption of residual white noise and normal distribution. testing the assumption of the residual white noise is carried out using the boxljung test statistic with the following hypothesis formulation: 𝐻0 : residual white noise 𝐻1 : residual is not white noise if the significance level is set at 5%, then the rejection area is rejected h0if q < xα,df−k−p−q 2 or p-value> α. table 3. results of the box-ljung statistic for residuals of arima (3,1,1) model lag q df p-value (3,1,1) 12 3,51 8 0.898 24 9.41 20 0.978 36 * * * 48 * * * based on table 3, the model of arima (3,1,1) since p-value is greater than the level of significance we do not reject the null hypothesis. indicated that the residuals for the model was white noise at the 5% level of significance. testing the assumption of normal distribution of residuals is carried out using the kolmogorov smirnov test. the following are the results of testing the normal distribution of the residual assumption. hypothesis: 𝐻0 : residual-normally distributed 𝐻1 : residual-not normally distributed figure 8. probability plot of residual of arima (3,1,1) base on figure, the results of the residual of arima (3,1,1) was 0,141 with an observed significance level of 0.05 indicating that residual for the model was normally distributed at the 5 % level of significance. hence, the diagnostic test that arima (3,1,1) model was appropriate for rice paddy production. table 6. rice paddy production forecasting results using model arima (3,1,1) year rice paddy production forecasting results 2021 2453401 2022 2154784 forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 288 year rice paddy production forecasting results 2023 2111594 2024 1615171 2025 2062436 estimation using exponential smoothing model the plot of rice paddy data shows that rice paddy production fluctuates every year. the graph also shows that start from 1993s, rice paddy production continued to experience a very significant increase until 2013. however, in 2018 rice paddy production decreased very rapidly, which is equal to 1,697,756 tons. the existence of a trend element in rice paddy production (tons) can be seen from the results of the acf (autocorrelation function) as it shown below. figure 9. acf plot of rice paddy production based on the picture above, it can conclude that the data has an element of trend because the lag movement is slowly decreasing towards 0. the trend element can also spot from the results of trend analysis in the image below: figure 10. trend analysis plot of rice paddy production trend analysis plot in the figure 10, it is obvious that the data has an element of trend. it can be shown from the fits line that has increased linearly. when the fits line increases or decreases linearly, then the data have an element of trend. the presence of seasonal elements in rice paddy production (tons) can spot from the pacf (partial autocorrelation function) plot shown in the figure 12. forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 289 figure 11. pacf plot of rice paddy production in the figure 11, the data has a seasonal element because the lag movement is repeating. in lag 2 which increases as well as in lag 4 and the lag that decreases from the previous lag is spotted in lag 3 and lag 5 and so does lag 6. based on the results of testing the trend and seasonal elements on rice paddy production data, the selection of the appropriate exponential smoothing solution method is winter exponential smoothing. a. determining the smoothing constant value 𝜶, 𝜸, and 𝜷 the smoothing constant used in the winter exponential smoothing model is 𝛼, 𝛾,and 𝛽. the optimal constant value is selected based on the smallest mape value. the following is the mape value of some of the best smoothing constant values. table 7. value of constants and mape no 𝜶 𝜸 𝛃 mape(%) 1 0.8 0.9 0.1 8.25 2 0.8 0.8 0.1 8.26 3 0.7 0.7 0.1 8.31 4 0.7 0.8 0.1 8.23 5 0.7 0.7 0.1 8.27 6 0.7 0.9 0.1 8.20 7 0.7 0.6 0.1 8.31 8 0.8 0.6 0.1 8.36 9 0.7 0.5 0.1 8.34 10 0.9 0.1 0.1 8.27 in the table 7, it can be seen that the smoothing constant value which has the smallest mape value is α = 0,7,γ = 0,9 ,β = 0,1 and the smallest mape value is 8.20%. the smoothing constant value will be used in the mathematical model of winter exponential smoothing in order to obtain forecasting results for the future period. the forecasting results for rice paddy production can be seen in the table 8. forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 290 table 8. estimation of rice paddy production yield in the next 5 years using winter exponential smoothing method year rice paddy production forecasting results 2021 1625925 2022 1645196 2023 1687667 2024 1605530 2025 1555213 the estimation results show that rice paddy production has an average increase of 1.88% from 2021 to 2023. however, rice paddy production has decreased by an average of 4% from 2023 to 2025. figure 12 shows the plot results between the actual data and forecasting data using the winter exponential smoothing method. based on the figure, it has been shown that the mape value generated for forecasting rice paddy production is 7.74% and less than 10%. this means that the average percentage error between the actual value and the forecast value using the winter exponential smoothing method is very small. therefore, it is fair to say that the winter exponential smoothing method provides better forecasting value for forecasting the rice paddy production in aceh. figure 12. plot between actual data and estimated data using winter method comparison of estimated results of rice paddy production using the arima and winter exponential smoothing models comparison of the estimation results of rice paddy production in the next 5 years using the arima and winter exponential smoothing models can be seen in the following table: table 9. comparison of estimated results of rice paddy production using arima (3,1,1) and winter exponential smoothing model year estimated results arima (3,1,1) winter exponential smoothing 2021 2453401 1625925 2022 2154784 1645196 2023 2111594 1687667 2024 1615171 1605530 2025 2062436 1555213 mse/mad 3,34041e+10 3.08616e+10 forecasting rice paddy production in aceh using arima and exponential smoothing models nurviana 291 figure 13. comparison graph of rice paddy production estimation results using arima(3,1,1) and winter exponential smoothing models based on table 9 and figure 14, the estimation results of rice paddy production using the arima model have increased from 2021 to 2022 with an increased rate of 0.84%. however, the estimated production results from 2022 to 2023 have decreased with a decline rate of 5.64%. furthermore, the results of the estimated rice paddy production have increased again by 2.14% from 2023 to 2024. in 2025, the estimated results of rice paddy have decreased by 0.75%. while the estimation results using the winter exponential smoothing model show that rice paddy production has an average rate of increase of 1.88% from 2021 to 2023 and has decreased with an average decline of 4% from 2023 to 2025. meanwhile, for testing the estimation results, the arima model (1,1,3) produces an mse/mad value of 3,34041 × 1010, while the winter exponential smoothing model produces an mse/mad value of 3.08616 × 1010. the best model is the model that has the smallest mse/mad value. therefore, it can conclude that the winter exponential smoothing model is the best model that can explain the actual data pattern and is used to estimate rice paddy production in aceh. conclusions the estimation results of rice paddy production of aceh in the next five years by using arima (3,1,1) model, respectively 2453401; 2154784; 2111594; 1615171; and 2062436 with the mse/mad 3,34041 × 1010. while the estimation results using the winter exponential smoothing model, respectively 1625925; 1645196; 1687667; 1605530; and 1555213 with the mse/mad 3.08616 × 1010.therefore, it can conclude that the winter exponential smoothing model is the 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[15] r. j. hyndman, forecasting: principles & practice. australia: university of western australia, 2014. bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 270-278 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 22, 2021 reviewed: april 24, 20201 accepted: april 30 , 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.11482 bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar1, rahmi febriyuni2, izzati rahmi hg3 1,2,3mathematics department, faculty of mathematics and natural sciences, universitas andalas, padang email: ferrayanuar@sci.unand.ac.id, raahmifebriyuni@gmail.com, izzatirahmihg@sci.unand.ac.id abstract the purposes of this study are to estimate the scale parameter of invers rayleigh distribution under mle and bayesian generalized square error loss function (self). the posterior distribution is considered to use two types of prior, namely jeffrey’s prior and exponential distribution. the proposed methods are then employed in the real data. several criteria for the selection model are considered in order to identify the method which results in a suitable value of parameter estimated. this study found that bayesian generalized self under jeffrey’s prior yielded better estimation values than mle and bayesian generalized self under exponential distribution. keywords: bayesian generalized self; exponential distribution; inverse rayleigh; jeffrey’s prio; mle. introduction rayleigh distribution is a special form of weibull distribution, meanwhile, inverse rayleigh distribution is a special form of inverse weibull distribution. the inverse rayleigh distribution is very useful lifetime model that can be used for analyzing infant mortality, survival analysis, reliability and quality control. the probability density function (pdf) of the inverse rayleigh distribution with scale parameter 𝜃 is defined as follows [1]: 𝑓(𝑥; 𝜃) = 2𝜃 𝑥3 exp (− 𝜃 𝑥2 ) , 𝜃 > 0, 𝑥 > 0. (1) the cumulative distribution function (cdf) of the inverse rayleigh distribution is given by 𝐹(𝑥; 𝜃) = exp (− 𝜃 𝑥2 ) , 𝜃 > 0, 𝑥 > 0 . (2) here 𝜃 is the scale parameter. the behavior of instantaneous failure rate of the inverse rayleigh distribution has been increasing and decreasing failure rate patterns for lifetime data. a significant amount of work has been done related to the inverse rayleigh distribution model in the classical framework but not much in a bayesian setup, especially in bayesian generalized self (squared error loss function). several studies http://dx.doi.org/10.18860/ca.v6i4.11482 mailto:ferrayanuar@sci.unand.ac.id mailto:raahmifebriyuni@gmail.com mailto:izzatirahmihg@sci.unand.ac.id bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 271 have used inverse rayleigh distribution for several cases. soliman and al-aboud [2] used classical method and bayesian for parameter estimation based on a set of upper record values from the rayleigh distribution. aslam and jun [3] derived an acceptance sampling plan from a truncated life test where multiple items in a group could be tested simultaneously by a tester when the lifetime of an item followed either an inverse rayleigh or a log-logistic distribution. soliman et al. [4] discussed the parameter estimation for an inverse rayleigh distribution based on lower record values. they implemented a maximum likelihood (ml) estimator of the unknown parameter and bayesian analysis with informative prior used to derive these estimators and the predictive intervals. ali [5] explored the modeling of the heterogeneity existing in the lifetime processes using the mixture of the inverse rayleigh distribution, and the spotlight is the bayesian inference of the mixture model using non-informative (the jeffreys and the uniform) and informative (gamma) priors. studied by dey & dey [6] derived bayesian estimation of the scale parameter and reliability function of an inverse rayleigh distribution. yousef & lafta [7] explored how to estimate the scale parameter for distribution of inverse rayleigh using different methods, such as the method of maximum likelihood estimator and moment method. dey [8] obtained bayesian estimates of an inverse rayleigh distribution using squared error and linex loss functions. meanwhile, rasheed [9] designed some bayesian estimators for the parameter scale and reliability function of the inverse rayleigh distribution under the generalized squared error loss function (self). in the present study, we consider the estimation of unknown parameters in an inverse rayleigh distribution. the aim of this study is to estimate the scale parameter of inverse rayleigh distributions using frequentist method (mle) and the bayesian approach which are employed to empirical data. the bayesian approach here is bayesian generalized self under two types of priors, namely jeffrey’s prior and exponential’s prior. the criteria to determine better performance of estimation method are based on the smallest value of akaike information criteria (aic), akaike information criteria correction (aicc) and bayesian information criteria (bic). the remainder of this study is organized as follows: the maximum likelihood estimation (mle), bayesian generalized self, jeffrey’s method, exponential distribution, criteria for the goodness of fit of parameter estimation method are derived in section 2. estimation method using bayesian generalized self under two types of priors and implementation of the proposed method to the real data are discussed in section 3. finally, section 4 as the last section provides some concluding remarks. methods in this section, we explore all methods which are implemented in this present study. the maximumm likelihood estimation method in the beginning and then followed by bayesian approach and criteria for model selection. maximum likelihood estimation in this section, we derived the classical estimator of the scale parameter for the inverse rayleigh distribution represented by the maximum likelihood estimator. let 𝑋1, 𝑋2, … 𝑋𝑛 be a sequence of i.i.d random variables from invers rayleigh distribution with scale parameter θ, written as 𝑋 ~ 𝐼𝑅𝐷 (𝜃), with probability density function of 𝑋 is 𝑓(𝑥 ; 𝜃) as presented by eq. (1). thus, the maximum likelihood estimation is formulated as follows [10]: bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 272 𝐿(𝜃) = ∏ 𝑓(𝑥𝑖 , 𝜃) 𝑛 𝑖=1 = 2𝑛 𝜃𝑛 (∏ 1 𝑥𝑖 3 𝑛 𝑖=1 ) 𝑒𝑥𝑝 (−𝜃 ∑ 1 𝑥𝑖 2 𝑛 𝑖=1 ) (3) to obtained the estimate for θ is derived by maximizing eq. (3) until we have: 𝜃𝑀�̂� = 𝑛 ∑ 1 𝑥 𝑖 2 𝑛 𝑖=1 = 𝑛 𝑇 , 𝑤ℎ𝑒𝑟𝑒 𝑇 = ∑ 1 𝑥𝑖 2 𝑛 𝑖=1 (4) bayesian generalized self method this section deals with the problem of obtaining bayesian estimators for the scale parameter θ from the inverse rayleigh distribution the bayes method is a parameter estimation method based on the bayes theorem. the basic concepts of the bayes method are as follows. suppose that 𝑋1, 𝑋2, … . , 𝑋𝑛 is a random example of the distribution 𝑓(𝒙 ; 𝜃) where 𝜃 is the parameter of the distribution. estimation of parameter 𝜃 will be based on random example 𝑋1, 𝑋2, … . , 𝑋𝑛. the bayes method is an estimation method based on combining information obtained from samples (objective knowledge), known as the likelihood function, with prior information regarding the distribution of estimated parameters [11], [12]. multiplying the likelihood function by the prior distribution gives the posterior distribution. in other words, the posterior distribution is a conditional probability density function of a parameter θ which is given the observation 𝐱 = (𝑋1, 𝑋2, … . , 𝑋𝑛). the formula for defining the posterior distribution is stated by the following formula [13]: 𝑓(𝜃|𝒙) = 𝑓(𝒙|𝜃)𝑓(𝜃) ∫ 𝑓(𝒙|𝜃)𝑓(𝜃)𝑑𝜃 = 𝑓(𝒙,𝜃) 𝑓(𝒙) (5) meanwhile, the estimator for the scale parameter (𝜃) using the bayes generalized self method will be described as follows [9]: 𝐿(𝜃 ; 𝜃) = ∑ 𝛼𝑗 𝜃 𝑗 (𝜃 − 𝜃) 𝑛 𝑖=1 = (𝛼0 + 𝛼1𝜃 + ⋯ + 𝛼𝑘 𝜃 𝑘 ) (𝜃 − 𝜃) (6) the estimator for parameter 𝜃 is obtained by minimizing the expectation for 𝜃, which is denoted by 𝐿(𝜃 ; 𝜃). the expected value of this function can be found by combining 𝐿(𝜃 ; 𝜃) and the probability density function of 𝜃, here denoted by ℎ(𝜃 |𝑥). thus, the expectation of 𝜃 using the bayes generalized self is as follows: 𝐸[𝐿(𝜃 ; 𝜃)] = ∫ 𝐿(𝜃 ; 𝜃) ℎ(𝜃 |𝑥) 𝑑𝜃 ∞ 0 𝐸[𝐿(𝜃 ; 𝜃)] = 𝛼0 𝜃 2 − 2 𝛼0 𝜃 𝐸(𝜃 | 𝑥) + 𝛼0 𝐸(𝜃 2 | 𝑥) + 𝛼1 𝜃 2𝐸(𝜃 | 𝑥) − 2 𝛼1 𝜃 𝐸(𝜃 2 | 𝑥) + 𝛼1 𝐸(𝜃 3 | 𝑥) + ⋯ + 𝛼𝑘 𝜃 2𝐸(𝜃𝑘 | 𝑥) − 2 𝛼𝑘 𝜃 𝐸(𝜃 𝑘+1 | 𝑥) + 𝛼𝑘 𝐸(𝜃 𝑘+2 | 𝑥). (7) to obtain the estimated value for 𝜃 with the bayesian generalized self method, the eq. (7) is derived on 𝜃 , so that: 𝜃𝐵�̂� = 𝛼0 𝐸(𝜃 | 𝑥) + 𝛼1 𝐸(𝜃 2 | 𝑥) + ⋯ + 𝛼𝑘 𝐸(𝜃 𝑘+1 | 𝑥) 𝛼0 + 𝛼1 𝐸(𝜃 | 𝑥) + ⋯ + 𝛼𝑘 𝐸(𝜃 𝑘 | 𝑥) (8) bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 273 jeffreys’ prior as non-informative prior the most widely used noninformative priors in bayesian analysis is jeffreys’ prior. this method is also attractive because it is proper under mild conditions and requires no elicitation of hyperparameters. jeffreys’ rule is derived from likelihood function then take the prior distribution to be the determinant of the square root of the fisher information matrix, denoted by 𝑓(𝜃) ∝ √𝐼(𝜃). fisher's information for the parameter 𝜃, defined as [13], [14] 𝐼(𝜃) = −𝑛 𝐸 ( 𝜕2 ln(𝑓(𝑥𝑖 ; 𝜃)) 𝜕𝜃2 ) let b is constant, thus : 𝑓(𝜃) ∝ √𝐼(𝜃) = 𝑏 √−𝑛 𝐸 ( 𝜕2 ln(𝑓(𝑥𝑖 ; 𝜃)) 𝜕𝜃2 ) (9) for inverse rayleigh distribution, it’s found that 𝜕2 ln(𝑓(𝑥𝑖 ; 𝜃)) 𝜕𝜃2 = − 1 𝜃2 . thus, it’s also obtained that 𝐸 ( 𝜕2 ln(𝑓(𝑥𝑖 ; 𝜃)) 𝜕𝜃2 ) = − 1 𝜃2 (10) by substituting eq. (10) into eq. (9), then it results 𝑓(𝜃) = 𝑏 𝜃 √𝑛 , 𝜃 > 0 (11) by combining this jeffrey’s prior and likelihood function, it yields the following posterior distribution : ℎ1(𝜃 | 𝑥1, 𝑥2, … , 𝑥𝑛 ) = 𝑇𝑛 𝜃𝑛−1 exp(−𝜃𝑇) г(𝑛) (12) the posterior distribution in eq. (12) has identic form with gamma distribution with scale parameter 1 𝑇 and shape parameter n, written as 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑇 , 𝑛). exponential distribution as conjugate prior we also derive the parameter estimation for 𝜃 based on bayesian generalized self with exponential distribution as prior. the probability distribution function for random variable 𝜃 which has exponential distribution with scale parameter 𝜆, written as 𝜃~𝐸𝑥𝑝(𝜆), is formulated as follows: 𝑔(𝜃) = 1 𝜆 exp ( −𝜃 𝜆 ) , 𝜃 > 0, 𝜆 > 0 (13) eq. (13) then is combined with likelihood function in eq. (3) until we have the posterior distribution as follows: ℎ2(𝜃 | 𝑥1, 𝑥2, … , 𝑥𝑛 ) = (𝑇 + 1 𝜆 ) 𝑛+1 𝜃𝑛 exp (−𝜃 (𝑇 + 1 𝜆 )) г(𝑛 + 1) (14) bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 274 this posterior distribution has a similar form with gamma distribution with scale parameter is 1 𝑇+ 1 𝜆 and shape parameter is n+1, written as 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑇+ 1 𝜆 , 𝑛 + 1) or 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑃 , 𝑛 + 1) with 𝑃 = 𝑇 + 1 𝜆 . criteria model selection the akaike information criterion (aic) which is widely used for statistical inference, is an estimator of out-of-sample prediction error and thereby the relative quality of statistical models for a given set of data [15]. given several models for the data, aic estimates the quality of each model, relative to each of the other models. this method provides a means for each model. when a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. aic estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model. in estimating the amount of information lost by a model, aic deals with the trade-off between the goodness of fit of the model and the simplicity of the model. in other words, aic deals with both the risk of overfitting and the risk of underfitting. suppose that we have a statistical model of some data. let k be the number of estimated parameters in the model. let �̂� be the maximum value of the likelihood function for the model. then the aic value of the model is the following [15]: 𝐴𝐼𝐶 = 2𝑘 − 2𝑙𝑛(�̂�) for condition 𝑛 𝑘 < 40 with n represent the amount of data, it’s suggested to use aicc (akaike information criteria correction): 𝐴𝐼𝐶𝑐 = 𝐴𝐼𝐶 + 2𝑘(𝑘 + 1) 𝑛 − 𝑘 − 1 another method to estimate the quality of each model relative to each of the other models is bayesian information criteria (bic), which is represented by following: 𝐵𝐼𝐶 = 𝑘𝑙𝑛(𝑛) − 2𝑙𝑛 (𝐿(𝜃)). given a set of candidate models for the data, the preferred model is the one with the minimum aic, aicc and bic value. results and discussion in this current study, we then employ the bayesian generalized self under non informative prior that is jeffreys prior to estimate the scale parameter of invers rayleight distribution. we also consider the bayesian self under informative prior namely an exponential distribution. both methods as well as mle are employed to the empirical data then. the most suitable method to be implemented is determined based on the smallest values of aic, aicc and bic. https://en.wikipedia.org/wiki/statistical_inference https://en.wikipedia.org/wiki/estimator https://en.wikipedia.org/wiki/out-of-sample https://en.wikipedia.org/wiki/statistical_model https://en.wikipedia.org/wiki/goodness_of_fit https://en.wikipedia.org/wiki/overfitting https://en.wikipedia.org/wiki/statistical_model https://en.wikipedia.org/wiki/statistical_parameter https://en.wikipedia.org/wiki/likelihood_function bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 275 bayesian generalized self under jeffrey’s prior based on eq. (12) is obtained that 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑇 , 𝑛). it can be derived that 𝐸 (𝜃 | 𝑥) = 𝑛 𝑇 and in general we also obtain 𝐸 (𝜃𝑚 | 𝑥) = г (𝑛+𝑚) г(𝑛) 𝑇𝑚 . these expected values then be substituted into eq. (8) to derive the formula to estimate 𝜃, under jeffrey’s prior, denoted here as 𝜃�̂� : 𝜃�̂� = 𝛼0 ( 𝑛 𝑇 ) + 𝛼1 ( (𝑛+1)𝑛 𝑇2 ) + ⋯ + 𝛼𝑘 ( (𝑛+𝑘)(𝑛+𝑘−1)…(𝑛+1)𝑛 𝑇𝑘+1 ) 𝛼0 + 𝛼1 ( 𝑛 𝑇 ) + ⋯ + 𝛼𝑘 ( (𝑛+𝑘−1)(𝑛+𝑘−2)…(𝑛+1)𝑛 𝑇𝑘 ) in this study, we choose first polynomial until fourth polynomial to be applied to estimate 𝜃: 𝜃𝐽1̂ = 𝛼0 ( 𝑛 𝑇 ) + 𝛼1 ( (𝑛+1)𝑛 𝑇2 ) 𝛼0 + 𝛼1 ( 𝑛 𝑇 ) (15) 𝜃𝐽2̂ = 𝛼0 ( 𝑛 𝑇 ) + 𝛼1 ( (𝑛+1)𝑛 𝑇2 ) + 𝛼2 ( (𝑛+2)(𝑛+1)𝑛 𝑇3 ) 𝛼0 + 𝛼1 ( 𝑛 𝑇 ) + 𝛼2 ( (𝑛+1)𝑛 𝑇2 ) (16) 𝜃𝐽3̂ = 𝛼0 ( 𝑛 𝑇 ) + 𝛼1 ( (𝑛+1)𝑛 𝑇2 ) + ⋯ + 𝛼3 ( (𝑛+3)(𝑛+2)(𝑛+1)𝑛 𝑇4 ) 𝛼0 + 𝛼1 ( 𝑛 𝑇 ) + ⋯ + 𝛼3 ( (𝑛+2)(𝑛+1)𝑛 𝑇3 ) (17) 𝜃𝐽4̂ = 𝛼0 ( 𝑛 𝑇 ) + 𝛼1 ( (𝑛+1)𝑛 𝑇2 ) + ⋯ + 𝛼4 ( (𝑛+4)(𝑛+3)…(𝑛+1)𝑛 𝑇5 ) 𝛼0 + 𝛼1 ( 𝑛 𝑇 ) + ⋯ + 𝛼4 ( (𝑛+3)(𝑛+2)(𝑛+1)𝑛 𝑇4 ) (18) bayesian generalized self under exponential distribution it has been proved that 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑇+ 1 𝜆 , 𝑛 + 1) or 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 ( 1 𝑃 , 𝑛 + 1) with 𝑃 = 𝑇 + 1 𝜆 . then, it can be proved that 𝐸 (𝜃𝑚 | 𝑥) = г (𝑛 + 1 + 𝑚) г(𝑛 + 1) 𝑃𝑚 (19) by substituting 𝑚 = 1,2, . . . , 𝑘 to eq. (19), we then derive the estimate formula for scale parameter, 𝜃 under exponential prior, denoted here as 𝜃�̂� : 𝜃�̂� = 𝛼0 ( 𝑛+1 𝑃 ) + 𝛼1 ( (𝑛+2)(𝑛+1) 𝑃2 ) + ⋯ + 𝛼𝑘 ( (𝑛+𝑘+1)…(𝑛+1) 𝑃𝑘+1 ) 𝛼0 + 𝛼1 ( 𝑛+1 𝑃 ) + ⋯ + 𝛼𝑘 ( (𝑛+𝑘)(𝑛+𝑘−1)…(𝑛+1) 𝑃𝑘 ) (20) in this study, we choose first polinomial until fourth polinomial based eq. (20) to be used to estimate 𝜃�̂� . bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 276 𝜃𝐸1̂ = 𝛼0 ( 𝑛+1 𝑃 ) + 𝛼1 ( (𝑛+2)(𝑛+1) 𝑃2 ) 𝛼0 + 𝛼1 ( 𝑛+1 𝑃 ) (21) 𝜃𝐸2̂ = 𝛼0 ( 𝑛+1 𝑃 ) + 𝛼1 ( (𝑛+2)(𝑛+1) 𝑃2 ) + 𝛼2 ( (𝑛+3)(𝑛+2)(𝑛+1) 𝑃3 ) 𝛼0 + 𝛼1 ( 𝑛+1 𝑃 ) + 𝛼2 ( (𝑛+2)(𝑛+1) 𝑃2 ) (22) 𝜃𝐸3̂ = 𝛼0 ( 𝑛+1 𝑃 ) + 𝛼1 ( (𝑛+2)(𝑛+1) 𝑃2 ) + ⋯ + 𝛼3 ( (𝑛+4)…(𝑛+1) 𝑃4 ) 𝛼0 + 𝛼1 ( 𝑛+1 𝑃 ) + ⋯ + 𝛼3 ( (𝑛+3)(𝑛+2)(𝑛+1) 𝑃3 ) (23) 𝜃𝐸4̂ = 𝛼0 ( 𝑛+1 𝑃 ) + 𝛼1 ( (𝑛+2)(𝑛+1) 𝑃2 ) + ⋯ + 𝛼4 ( (𝑛+5)…(𝑛+1) 𝑃5 ) 𝛼0 + 𝛼1 ( 𝑛+1 𝑃 ) + ⋯ + 𝛼4 ( (𝑛+4)…(𝑛+1) 𝑃4 ) (24) implementation of proposed method to real data the result of analytical study above then implemented to real data. the real data set represents the 72 exceedances for the years 1958–1984 (rounded to one decimal place) of flood peaks (in m3/s) of the wheaton river near carcross in yukon territory, canada [16]. the data are as follows: 1.7 2.2 14.4 1.1 0.4 20.6 5.3 0.7 1.9 13.0 12.0 9.3 1.4 18.7 8.5 25.5 11.6 14.1 22.1 1.1 2.5 14.4 1.7 37.6 0.6 2.2 39.0 0.3 15.0 11.0 7.3 22.9 1.7 0.1 1.1 0.6 9.0 1.7 7.0 20.1 0.4 2.8 14.1 9.9 10.4 10.7 30.0 3.6 5.6 30.8 13.3 4.2 25.5 3.4 11.9 21.5 27.6 36.4 2.7 64.0 1.5 2.5 27.4 1.0 27.1 20.2 16.8 5.3 9.7 27.5 2.5 27.0 in this present study, we fix several values for 𝛼0=100, 𝛼1=50, 𝛼2=10, 𝛼3=8, 𝛼4=7 and 𝜆=0.8 to be applied to estimate the scale parameter 𝜃. based on this real data, we calculate that: 𝑇 = ∑ 1 𝑥𝑖 2 = 138,99 ∑ 𝑙𝑛 ( 1 𝑥1 3 ) 72 𝑖=1 72 𝑖=1 = −388,38 𝑃 = ∑ 1 𝑥𝑖 2 72 𝑖=1 + 1 𝜆 = 140,24 we then employ both proposed methods and mle to this empirical data. the comparison of criteria for model selection based on three methods are provided in table 1. table 1. estimated values for aic, aicc, and bic prior mean criteria model selection aic aicc bic mle 0.5180 917.6820 917.7391 919.9587 jeffrey’s prior 𝜃𝐽1̂ 0.5195 917.6778 917.7349 919.9545 𝜃𝐽2̂ 0.5198 917.6748 917.7319 919.9515 bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 277 𝜃𝐽3̂ 0.5200 917.6728 917.7299 919.9495 𝜃𝐽4̂ 0.5201 917.6718 917.7289 919.9485 exponential’s prior 𝜃𝐸1̂ 0.5220 917.6816 917.7387 919.9583 𝜃𝐸2̂ 0.5223 917.6786 917.7357 919.9553 𝜃𝐸3̂ 0.5225 917.6766 917.7337 919.9533 𝜃𝐸4̂ 0.5226 917.6756 917.7327 919.9523 table 1 informs us that this present study yielded almost similar values for estimated mean for all three methods (in the third column). based on the criteria model selection, this study found that jeffrey’s prior as noninformation prior, tends to result smaller values than mle and exponential ‘s prior for all four polynomials. the smallest values for these criteria are at jeffrey’s prior at fourth polynomial (𝜃𝐽4̂). these results inform us that the method to estimate scale parameter of invers rayleigh distribution using bayesian generalized self under jeffrey’s prior tends to result better values than mle and bayesian generalized self under exponential’s prior. this present study proved it by employing all proposed method to real data with size sample is relatively moderate, n = 72. conclusions this study employed the mle, bayesian generalized self under jeffrey’s prior and bayesian generalized self under exponential’s prior to estimate the scale parameter of invers rayleight distribution of a real data. all 72 sample of flood peaks data in canada are involved in this study. this real data has invers rayleigh distribution. this study found that estimation mean of scale parameter from invers rayleigh distribution based on mle, bayesian generalized self under jeffrey’s prios and bayesian generalized self under exponential’s prior tend to result similar values. based on criteria of selection model, this study proved that bayesian generalized self under jeffrey’s prior tend to result the smallest value of aic, aicc and bic. acknowledgments this research was partly funded by drpm, the deputy for strengthening research and development of the ministry of research and technology / national research and innovation agency of indonesia, in accordance with contract number 123/sp2h/amd/lt/drpm/2020. bayesian generalized self method to estimate scale parameter of inverse rayleigh distribution ferra yanuar 278 references [1] a. rasheed, “reliability estimation in inverse rayleigh distribution using precautionary loss function,” mathematics and statistics journal, vol. 2, no. 3, pp. 9–15, 2016. 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[16] k. fatima and s. p. ahmad, “weighted inverse rayleigh distribution,” international journal of statistics and systems, vol. 12, no. 1, pp. 119–137, 2017. the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process cauchy –jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 73-83 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 05, 2021 reviewed: october 15, 2021 accepted: ocotober 30, 2021 doi: https://doi.org/10.18860/ca.v7i1.12848 the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi1, bonno andri wibowo2, nina valentika3, muhammad syazali4, vina apriliani5 1department of mathematics, syiah kuala university, banda aceh, indonesia 2department of mathematics, ipb university, bogor, indonesia 3department of mathematics, pamulang university, tangerang selatan, indonesia 4department of mathematics, universitas pertahanan, bogor, indonesia 5department of mathematics education, uin ar-raniry, banda aceh, indonesia email: ikhsanmaulidi@unsyiah.ac.id bonno1818@gmail.com , dosen02339@unpam.ac.id, muhamad.syazali@idu.ac.id, vina.apriliani@ar-raniry.ac.id abstract the nonhomogeneous poisson process is one of the most widely applied stochastic processes. in this article, we provide a confidence interval of the intensity estimator in the presence of a periodic multiplied by trend power function. this estimator's confidence interval is an application of the formulation of the estimator asymptotic distribution that has been given in previous studies. by using the asymptotic theorem, the distribution was derived in the form of a confidence interval for the intensity function. in addition, constructive proof of the convergent in probability has been provided for all power functions. the results of this study contribute to the study of statistical analysis of the estimators that have been formulated previously. keywords: asymptotic distribution; interval confidence; intensity function; poisson process. introduction there are many events in nature can be modeled by stochastic modeling processes. the stochastic process is a set of random variables that map the sample space to a state space. one of the stochastic processes is a counting process which states the number of events at a time interval. the counting process assuming the number of events has a poisson distribution is called the poisson process. some basic theories related poisson process can be seen in [1]–[3]. due to the intensity function, the poisson process is divided into two categories, namely the homogeneous poisson process and the nonhomogeneous poisson process. a homogeneous poisson process has a constant intensity function (independent of time), while a nonhomogeneous poisson process has a time-dependent intensity function. this nonhomogeneous poisson process is widely applied to real phenomena, such as the phenomenon of earthquakes [4], traffic accidents [5], and radio burst rates [6]. on the other hand, the study of the nonhomogeneous poisson process in the form a periodic intensity function also has been conducted in recent years. [7] studied the https://doi.org/10.18860/ca.v7i1.12848 mailto:ikhsanmaulidi@unsyiah.ac.id mailto:bonno1818@gmail.com mailto:dosen02339@unpam.ac.id mailto:muhamad.syazali@idu.ac.id mailto:vina.apriliani@ar-raniry.ac.id the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 74 estimation of the intensity function in a nonhomogeneous poisson process by including a trend component in the periodic intensity function. the trend component began with a trend in the form of an additive linear function. then the study was continued with a trend in the form of a multiplicative linear function [8]. the estimation of the intensity function was carried out using the general kernel function approach, and [9] examined this poisson process with a uniform kernel approach. other related studies can be seen in [10]–[13]. [14] studied the estimation of the periodic poisson process intensity function with the power function trend using a general kernel. furthermore, the statistical properties of these estimators have also been proven. [15] has given strong consistency of these estimators. in addition, the asymptotic normality of the estimator has also been formulated and given a numerical simulation of the consistency of the estimator [16]. the results obtained in that study are the estimator of the periodic component which converges to the normal distribution by providing certain conditions. as an application of the asymptotic normality, it can be determined the confidence interval of the estimator for the periodic component. this study provides the theorems for the confidence interval for the intensity function parameters and their proofs. the contribution of this study is to provide the characteristics of the estimator, especially in terms of accuracy. with a certain number of samples (interval length), it can be determined how accurately the estimator predicts the value of the parameter in the form of a confidence interval. methods the estimator for periodic component of the intensity function suppose that {𝑁(𝑡), 𝑡 ≥ 0} is a nonhomogeneous poisson process with intensity function 𝜆 which locally integrable and unknown. suppose also that 𝜆 is a periodic function with the trend of the power function, then λ which depends on the time variable 𝑠 can be expressed as 𝜆(𝑠) = 𝜆𝑐 ∗ (𝑠). 𝑎𝑠𝑏 . (1) the values of the 𝑎 and 𝑏 constants are assumed to be known, so that what is not known is the function of the periodic component of the intensity function, namely 𝜆𝑐 ∗ . equation (1) can also be stated as follows 𝜆(𝑠) = 𝜆𝑐 (𝑠). 𝑠 𝑏 , (2) with 𝜆𝑐 (𝑠) = 𝑎𝜆𝑐 ∗ (𝑠). [14] has been given the kernel type estimator for 𝜆𝑐 (𝑠) by using general kernel functions. the estimator for periodic component of the intensity is �̂�𝑐,𝑛,𝑘 (𝑠) = 𝜏 𝑛 ∑ 1 ℎ𝑛 (𝑠 + 𝑘𝜏) 𝑏 ∞ 𝑘=0 ∫ 𝐾 ( 𝑥 − (𝑠 + 𝑘𝜏) ℎ𝑛 ) 𝑁(𝑑𝑥). (3) 𝑛 0 on equation (3), the constant 𝜏 is a period of the intensity function which satisfies 𝜆𝑐 (𝑠 + 𝑘𝜏) = 𝜆𝑐 (𝑠), for 𝑘 ∈ 𝑍. with 𝑛 is the length of the time interval used. in this case, since the poisson process is a discrete stochastic process, it is clear that 𝑛 is a natural number. the function 𝐾 called a kernel function if it satisfies the following properties: (k1) 𝐾 is a probability density function, (k2) 𝐾 is bounded, and (k3) 𝐾 is defined in [-1,1] [17]. the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 75 the asymptotic normality of the estimator theorem 1 (the asymptotic normal distribution for �̂�𝒄,𝒏,𝒌(𝒔), 𝟎 < 𝒃 < 𝟏) suppose that the intensity 𝜆 satisfies (1) and locally integrable. the kernel function 𝐾 satisfies (k1), (k2), (k3), 𝜆𝑐 has a bounded second derivative around of 𝑠, 0 < 𝑏 < 1, 𝑛1−𝑏 ℎ𝑛 → 0, 𝑛 𝑏+1ℎ𝑛 → ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, a) if (𝑛1+𝑏 ℎ𝑛 5 ) 1 2 → 0, then (𝑛1+𝑏 ℎ𝑛 ) 1 2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(0, 𝜎2) (4) as 𝑛 → ∞, with 𝜎2 = 𝜏𝜆𝑐(𝑠) (1−𝑏) ∫ 𝐾2(𝑧)𝑑𝑧. 1 −1 b) if (𝑛1+𝑏 ℎ𝑛 5 ) 1 2 → 1, then (𝑛1+𝑏 ℎ𝑛 ) 1 2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(𝜇, 𝜎2) (5) as 𝑛 → ∞, with 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 and 𝜎2 = 𝜏𝜆𝑐(𝑠) (1−𝑏) ∫ 𝐾2(𝑧)𝑑𝑧. 1 −1 theorem 2 (the asymptotic normal distribution for �̂�𝒄,𝒏,𝒌(𝒔), 𝒃 = 𝟏) suppose that the intensity 𝜆 satisfies (1) and locally integrable. the kernel function 𝐾 satisfies (k1), (k2), (k3), 𝜆𝑐 has a bounded second derivative around of 𝑠, 𝑏 = 1, ln (𝑛)ℎ𝑛 → 0, 𝑛2ℎ𝑛 𝑙𝑛(𝑛) → ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, a) if ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 0, then ( 𝑛2ℎ𝑛 ln (𝑛) ) 1 2 (�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(0, 𝜎2) (6) as 𝑛 → ∞, with 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾 2(𝑧)𝑑𝑧 1 −1 . b) if ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 1, then ( 𝑛2ℎ𝑛 ln (𝑛) ) 1 2 (�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(𝜇, 𝜎2) (7) as 𝑛 → ∞, with 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 and 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾 2(𝑧)𝑑𝑧. 1 −1 theorem 3 (the asymptotic normal distribution for �̂�𝒄,𝒏,𝒌(𝒔), 𝒃 > 𝟏) suppose that the intensity 𝜆 satisfies (1) and locally integrable. the kernel function 𝐾 satisfies (k1), (k2), (k3), and 𝜆𝑐 has a bounded second derivative around of 𝑠, 𝑏 > 1, 𝑛2ℎ𝑛 → ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, a) if (𝑛2ℎ𝑛 5 ) 1 2 → 0, then (𝑛2ℎ𝑛) 1 2 (�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(0, 𝜎2) (8) as 𝑛 → ∞, with 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾 2(𝑧)𝑑𝑧 1 −1 , and 𝜁(𝑏) = lim 𝑛→∞ (∑ 1 𝑘𝑏 ∞ 𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 76 b) if (𝑛2ℎ𝑛 5 ) 1 2 → 1, then (𝑛2ℎ𝑛 ) 1 2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠)) 𝑑 → normal(𝜇, 𝜎2) (9) as 𝑛 → ∞, with 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 , 𝜎2 = 𝜏2−𝑏 𝜆𝑐(𝑠)𝜁(𝑏) ∫ 𝐾 2(𝑧)𝑑𝑧 1 −1 , and 𝜁(𝑏) = lim 𝑛→∞ (∑ 1 𝑘𝑏 ∞ 𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . the proofs of theorem 1, theorem 2, and theorem 3 above can be proved through a rough analysis, [18]. it is recommended to study the basic theory to proof these theorems in [19]–[21]. results and discussion suppose that ф denotes the standard normal distribution with ф−1 is the inverse. based on theorem 1, theorem 2, and theorem 3 above, it can be given some confidence interval for 𝜆𝑐 with significant level 1 − 𝛼 as follows: corollary 1 (the confidence interval for 𝝀𝒄 for 𝟎 < 𝒃 < 𝟏) suppose that all conditions on theorem 1 are satisfied, the for a significant level α where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐 for 0 < 𝑏 < 1 has been given in the following conditions: a) if (𝑛1+𝑏 ℎ𝑛 5 ) 1 2 → 0 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏ℎ𝑛 , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏ℎ𝑛 ), where 𝜎2 = 𝜏𝜆𝑐(𝑠) (1−𝑏) ∫ 𝐾2(𝑧)𝑑𝑧. 1 −1 b) if (𝑛1+𝑏 ℎ𝑛 5 ) 1 2 → 1 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛1+𝑏 ℎ𝑛 , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛1+𝑏 ℎ𝑛 ), where 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 and 𝜎2 = 𝜏𝜆𝑐(𝑠) (1−𝑏) ∫ 𝐾2(𝑧)𝑑𝑧. 1 −1 corollary 2 (the confidence interval for 𝝀𝒄 for 𝒃 = 𝟏) suppose that all conditions on theorem 2 are satisfied, the for a significant level α where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐 for 𝑏 = 1 has been given in the following conditions: a) if ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 0 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 ) , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 ) ), where 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾 2(𝑧)𝑑𝑧. 1 −1 the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 77 b) if ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 1 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − (𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 , �̂�𝑐,𝑛,𝑘 (𝑠) + (𝜎ф −1 (1 − 𝛼 2 ) − 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ), where 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 and 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾 2(𝑧)𝑑𝑧. 1 −1 corollary 3 (the confidence interval for 𝝀𝒄 for 𝒃 > 𝟏) suppose that all conditions on theorem 3 are satisfied, the for a significant level α where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐 for 𝑏 > 1 has been given in the following conditions: a) if (𝑛2ℎ𝑛 5 ) 1 2 → 0 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ), where 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾 2(𝑧)𝑑𝑧 1 −1 b) if (𝑛1+𝑏 ℎ𝑛 5 ) 1 2 → 1 then 𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛2ℎ𝑛 , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛2ℎ𝑛 ), where 𝜇 = 𝜆𝑐 ′′(𝑠) 2 ∫ 𝑧2𝐾(𝑧)𝑑𝑧 1 −1 , 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾 2(𝑧)𝑑𝑧 1 −1 , and 𝜁(𝑏) = lim 𝑛→∞ (∑ 1 𝑘𝑏 ∞ 𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . to strengthen the reasons for the above confidence intervals, it is given the probability convergence theorems for these confidence interval. theorem 4. convergence in probability of the confidence interval for 𝝀𝒄 and 𝟎 < 𝒃 < 𝟏 if λ̂c,n,k is the estimator for periodic component of the intensity function that is given in equation (3). also, iλc,n is a confidence interval that is given in corollary 1, then for the value 0 < b < 1 satisfies p(λc,n(s)ϵiλc,n ) → 1 − α + o(1), provided n → ∞. the proof of theorem 4: case (a) assumption (𝒏𝟏+𝒃𝒉𝒏 𝟓 ) 𝟏 𝟐 → 𝟎 p(λc(s)ϵiλc ) = 𝑃 (λ̂c,n,k − 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏 ℎ𝑛 ≤ λc,n(s) ≤ λ̂c,n,k + 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏 ℎ𝑛 ) the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 78 p(λc,n(s)ϵiλc,n ) = p (− 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏 ℎ𝑛 ≤ λc,n(s) − λ̂c,n,k ≤ 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏ℎ𝑛 ) p(λc,n(s)ϵiλc,n ) = 𝑃 (− 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏 ℎ𝑛 ≤ λ̂c,n,k − λc,n(s) ≤ 𝜎ф−1 (1 − 𝛼 2 ) √𝑛1+𝑏 ℎ𝑛 ) p(λc,n(s)ϵiλc,n ) = 𝑃 (−𝜎ф −1 (1 − 𝛼 2 ) ≤ √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc,n(s)) ≤ 𝜎ф −1 (1 − 𝛼 2 )), let 𝑌 = √𝑛1+𝑏ℎ𝑛 (λ̂c,n,k − λc(s)), then based on theorem 1 𝑌~normal(0, 𝜎 2), by using central limit theorem 𝑍 = 𝑌 𝜎 ~normal(0,1). therefore p(λc(s)ϵiλc ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − 𝑃 (𝑍 < −ф−1 (1 − 𝛼 2 )). since the normal distribution has a symmetricity property, 𝑃 (𝑍 < −ф−1 (1 − 𝛼 2 )) = 𝑃 (𝑍 ≥ −ф−1 (1 − 𝛼 2 )), then p(λc(s)ϵiλc ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − (𝑍 ≥ −ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − 1 + p (𝑍 ≤ ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = ф (ф −1 (1 − 𝛼 2 )) − 1 + ф (ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = 1 − 𝛼 2 − 1 + 1 − 𝛼 2 = 1 − 𝛼, provided 𝑛 → ∞. case (b). assumption (𝒏𝟏+𝒃𝒉𝒏 𝟓 ) 𝟏 𝟐 → 𝟏 p(λc(s)ϵiλc ) = 𝑃 (λ̂c,n,k − 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛1+𝑏 ℎ𝑛 ≤ λc,n(s) ≤ λ̂c,n,k + 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛1+𝑏 ℎ𝑛 ) = p (− 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛1+𝑏 ℎ𝑛 ≤ λc,n(s) − λ̂c,n,k ≤ 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛1+𝑏 ℎ𝑛 ) = 𝑃 (− 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛1+𝑏ℎ𝑛 ≤ λ̂c,n,k − λc,n(s) ≤ 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛1+𝑏 ℎ𝑛 ) the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 79 = 𝑃 (− (𝜎ф−1 (1 − 𝛼 2 ) − 𝜇) ≤ √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc,n(s)) ≤ (𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)) suppose that 𝑌 = √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc(s)) then based on theorem 1b 𝑌~normal(𝜇, 𝜎 2), and 𝑍 = 𝑌 − 𝜇 𝜎 ~normal(0,1). therefore p(λc(s)ϵiλc ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − 𝑃 (𝑍 < −ф−1 (1 − 𝛼 2 )). since the normal distribution has a symmetricity property 𝑃 (𝑍 < −ф−1 (1 − 𝛼 2 )) = 𝑃 (𝑍 ≥ −ф−1 (1 − 𝛼 2 )), so p(λc(s)ϵiλc ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − (𝑍 ≥ −ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = p (𝑍 ≤ ф −1 (1 − 𝛼 2 )) − 1 + p (𝑍 ≤ ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = ф (ф −1 (1 − 𝛼 2 )) − 1 + ф (ф−1 (1 − 𝛼 2 )) p(λc,n(s)ϵiλc,n ) = 1 − 𝛼 2 − 1 + 1 − 𝛼 2 = 1 − 𝛼, provided 𝑛 → ∞. theorem 5. convergence in probability of the confidence interval for 𝝀𝒄 and 𝒃 = 𝟏 if λ̂c,n,k is the estimator for periodic component of the intensity function that is given in equation (3). also, iλc, is an confidence interval that is given in corollary 2, then for the value b = 1 satisfies p(λc(s)ϵiλc ) → 1 − α + o(1), provided n → ∞. the proof of theorem 5 case (a) assumption ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 𝟎 p(λc(s)ϵiλc ) = 𝑃 (λ̂c,n,k − 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 ) ≤ λc,n(s) ≤ λ̂c,n,k + 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 )) = p (−𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 ) ≤ λc,n(s) − λ̂c,n,k ≤ 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 )) the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 80 = 𝑃 (−𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 ) ≤ λ̂c,n,k − λc,n(s) ≤ 𝜎√ ln(𝑛) 𝑛2ℎ𝑛 ф−1 (1 − 𝛼 2 )) = 𝑃 (−𝜎ф−1 (1 − 𝛼 2 ) ≤ √ 𝑛2ℎ𝑛 ln(𝑛) (λ̂c,n,k − λc,n(s)) ≤ 𝜎ф −1 (1 − 𝛼 2 )). let = √ 𝑛2ℎ𝑛 ln(𝑛) (λ̂c,n,k − λc(s)), then based on theorem 2 𝑌~normal(0, 𝜎 2), by using central limit theorem 𝑍 = 𝑌 𝜎 ~normal(0,1). therefore p(λc(s)ϵiλc ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )) by using the same arguments before, it is obtained p(λc(s) ϵ iλc ) = 1 − 𝛼, provided 𝑛 → ∞. case (b). assumption ( 𝑛2ℎ𝑛 5 ln (𝑛) ) 1 2 → 𝟏 p(λc(s)ϵiλc ) = 𝑃 (λ̂c,n,k − (𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ≤ λc(s) ≤ λ̂c,n,k + (𝜎ф −1 (1 − 𝛼 2 ) − 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ) = p (−𝜎ф−1 (1 − 𝛼 2 ) + 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ≤ λc(s) − λ̂c,n,k ≤ 𝜎ф −1 (1 − 𝛼 2 ) − 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ) = 𝑃 (−ф−1 (1 − 𝛼 2 ) − 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ≤ λ̂c,n,k − λc(s) ≤ 𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)√ ln(𝑛) 𝑛2ℎ𝑛 ) = 𝑃 (− (𝜎ф−1 (1 − 𝛼 2 ) − 𝜇) ≤ √ 𝑛2ℎ𝑛 ln(𝑛) (λ̂c,n,k − λc(s)) ≤ (𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)), suppose that 𝑌 = √ 𝑛2ℎ𝑛 ln(𝑛) (λ̂c,n,k − λc(s)), then according to theorem 1b 𝑌~normal(𝜇, 𝜎2) and 𝑍 = 𝑌 − 𝜇 𝜎 ~normal(0,1). therefore p(λc,n(s) ϵ iλc,n ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )). the same arguments gave us p(λc(s) ϵ iλc ) = 1 − 𝛼, provided 𝑛 → ∞. the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 81 theorem 6. convergence in probability of the confidence interval for 𝝀𝒄,𝒏 and 𝒃 > 𝟏 if λ̂c,n,k is the estimator for periodic component of the intensity function that is given in equation (3). also, iλc,n is a confidence interval that is given in corollary 3, then for the value b > 1 satisfies p(λc(s)ϵiλc ) → 1 − α + o(1), provided n → ∞. the proof of theorem 6 case a. assumption (𝑛2ℎ𝑛 5 ) 1 2 → 0 p(λc(s)ϵiλc ) = 𝑃 (λ̂c,n,k − 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ≤ λc(s) ≤ λ̂c,n,k + 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ) = p (− 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ≤ λc(s) − λ̂c,n,k ≤ 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ) = 𝑃 (−𝜎 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ≤ λ̂c,n,k − λc(s) ≤ 𝜎 𝜎ф−1 (1 − 𝛼 2 ) √𝑛2ℎ𝑛 ) = 𝑃 (−𝜎ф−1 (1 − 𝛼 2 ) ≤ √𝑛2ℎ𝑛 (λ̂c,n,k − λc(s)) ≤ 𝜎ф −1 (1 − 𝛼 2 )). let 𝑌 = √𝑛2ℎ𝑛 (λ̂c,n,k − λc(s), then according to theorem 3a 𝑌~normal(0, 𝜎 2) and 𝑍 = 𝑌 𝜎 ~normal(0,1). therefore p(λc(s)ϵiλc ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )). the same arguments gave us p(λc(s) ϵ iλc ) = 1 − 𝛼, provided 𝑛 → ∞. case b. assumption (𝑛2ℎ𝑛 5 ) 1 2 → 𝟏 p(λc(s)ϵ iλc ) = 𝑃 (λ̂c,n,k − 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛2ℎ𝑛 ≤ λc(s) ≤ λ̂c,n,k + 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛2ℎ𝑛 ) = p (− 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛2ℎ𝑛 ≤ λc(s) − λ̂c,n,k ≤ 𝜎 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛2ℎ𝑛 ) the confidence interval for the periodic intensity function in the presence of power function trend on the nonhomogeneous poisson process ikhsan maulidi 82 = 𝑃 (− 𝜎ф−1 (1 − 𝛼 2 ) − 𝜇 √𝑛2ℎ𝑛 ≤ λ̂c,n,k − λc(s) ≤ 𝜎ф−1 (1 − 𝛼 2 ) + 𝜇 √𝑛2ℎ𝑛 ) = 𝑃 (− (𝜎ф−1 (1 − 𝛼 2 ) − 𝜇) ≤ √𝑛2ℎ𝑛 (λ̂c,n,k − λc(s)) ≤ (𝜎ф −1 (1 − 𝛼 2 ) + 𝜇)). suppose that 𝑌 = √𝑛2ℎ𝑛 (λ̂c,n,k − λc(s), then based on theorem 3b 𝑌~normal(𝜇, 𝜎 2) and 𝑍 = 𝑌 − 𝜇 𝜎 ~normal(0,1). therefore p(λc(s) ϵ iλc ) = 𝑃 (−ф −1 (1 − 𝛼 2 ) ≤ 𝑍 ≤ ф−1 (1 − 𝛼 2 )). by using the same arguments, it is obtained that p(λc(s) ϵ iλc ) = 1 − 𝛼, provided 𝑛 → ∞. conclusions from the results that have been studied, the formula to determine the confidence interval for parameter of the periodic component of the nonhomogeneous poisson process with the intensity in the form of periodic function has been obtained. these confidence intervals have been given for each case of the values of b, this is because the results of previous studies show that the variance of the estimator is given in a different function for each case of the values of b. these confidence intervals have been proved to converge in probability 1 − 𝛼. the recommendation for further research that can be done is providing numerical simulations for each confidence interval case, there are 6 cases. the simulation can be started by determining the bandwidth function ℎ𝑛 which satisfies all the conditions in the given case and determining the probability of the estimator being in the confidence 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[21] r. m. dudley, real analysis and probability. wardswort & brooks, 1989. dynamical of ratio-dependent eco-epidemical model with prey refuge cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 227-237 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 17, 2020 reviewed: february 11, 2021 accepted: march 16, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.10827 dynamical of ratio-dependent eco-epidemical model with prey refuge adin lazuardy firdiansyah department of tadris matematika, stai muhammadiyah probolinggo jl. soekarno hatta 94b, sukabumi, mayangan, probolinggo, indonesia email: adin.lazuardy@gmail.com abstract this paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. the prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. here, we get the boundedness and three equilibrium points where are existence under certain conditions. in the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. we find that the system undergoes hopf bifurcation around the interior equilibrium point by choosing prey refuge as a bifurcation parameter. we also find a condition for uniform persistence. finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge. keywords: eco-epidemiology model; global stable; hopf bifurcation; local stable; persistence introduction one of the natural phenomena that described the interaction between one species and another individual is the prey-predator interaction. this interaction depends on whether the effects are profitable or detrimental. in the real world, prey-predator interaction also can be influenced by infectious diseases. these diseases can affect population size in the predator-prey interaction. since then, the combination of epidemiological and ecological becomes important issues that are often discussed by many researchers. mathematical studies have considered this issue into an ecoepidemiology model that contains the class of susceptible and infective populations. currently, several studies have focused on the spread of disease in prey only, e.g. [1]–[3]. it is well known that predators prefer to capture infected prey because they are easier to catch than susceptible prey. however, the predator can become infected after eating them. therefore, several researchers are interested to investigate the spread of disease not only in prey but also in predators, e.g. [4][5]. moreover, some studies have reviewed the spread of disease in both populations, e.g. [6]. base on several experiments, the spread of infectious disease becomes an important factor to know the regulation of population density [7]. in this paper, we focus on the situation where predators can eat infected prey only. it appropriates to the fact that infected prey tends to change its behavior. infected prey shall live in an area that is accessible to predators [8]. moreover, infected prey is less agile than healthy prey and can be predated by predators easily [1]. hudson et al. [9] have http://dx.doi.org/10.18860/ca.v6i4.10827 mailto:adin.lazuardy@gmail.com dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 228 observed that predators selectively capture heavily infected red grouse. predators that consumed infected prey populations are described by the response functions. it is well known that response functions are an important component in the eco-epidemiology model. generally, several researchers use holling types as a response function in their model. according to [10], holling types are divide into three types, namely holling type i, holling type ii, and holling type iii. holling type i means that the consumption rate of predator increases linearly with the density of prey but it achieves a constant value if predators are surfeit, e.g. [6][5]. holling type ii means that the consumption rate of predators increases if the density of prey is low, e.g. [1]–[4]. meanwhile, holling type iii means that the consumption rate of predator increases when the density of prey is large but it decreases when the density of prey is low, e.g. [11]. in holling type iii, predators easily switch to eat one prey to another or they focus to eat prey in a location where it is most abundant [12]. the response function for the holling type depends on the density of prey. this is unrealistic because it ignores the effects of predator interference. base on the experiment, the density of predators can influence the consumption rate. in the modeling, the consumption rate that depended on the density of prey can’t describe the dynamics behavior when the density of predator influences the system [12]. currently, several researchers have considered the density of both populations as a ratio-dependent function. this function depends on the ratio of prey to predator density [13]. according to [14], the ratio-dependent model is a more reasonable dynamic than the previous model. one of the phenomena that reduce predation risk is prey refuge. it can avoid the extinction of prey and influence the stability of the dynamic behavior [15]. according to [3], the prey refuge that is incorporated in the model is divided into two types, namely the refuge for a constant-number of prey and the refuge for a constant-proportion of prey. it is well known that the refuge for a constant-number of prey has a stronger stabilizing effect than the refuge for a constant-proportion of prey [14]. therefore, the model is more realistic by incorporating prey refuge and it gives an accessible factor to the predator. in this study, we modify the eco-epidemiology model from [1] by changing holling type ii into a ratio-dependent function. here, we also observe the effect of prey refuge in the system. further, this article presents the results in the form of model analysis. moreover, it is well observed that there are hopf bifurcations around the positive equilibrium point and the condition for uniform persistence. finally, several simulations are performed to illustrate the analytical results. methods we use several methods to modify the eco-epidemiology model from [1]. the method is presented as follows. 1. reviewing and studying the eco-epidemiology model from previous literature. 2. modifying the eco-epidemiology model by changing the holling ii type into the ratiodependent function. 3. investigating the boundedness, equilibrium points, and dynamical behavior in the modified model. 4. investigating hopf bifurcation and persistence in the modified model. 5. performing numerical simulation by using the 5th-order predictor-corrector method as a numerical method to support the analytical results. dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 229 results and discussion the mathematical model in this paper, the eco-epidemiology model consists of three populations, namely susceptible prey, infected prey, and predator. let 𝑋𝑆(𝑡), 𝑋𝐼(𝑡), and 𝑌(𝑡) is respectively defined as the density of susceptible prey, infected prey, and predator at the time 𝑡. according to [1], the general eco-epidemiology model is presented as follows. 𝑋�̇� = 𝑅 − 𝛽𝑋𝑆𝑋𝐼 − 𝛿𝑋𝑆, 𝑋𝐼̇ = 𝛽𝑋𝑆𝑋𝐼 − 𝑓(𝑋𝐼, 𝑌)𝑌 − 𝜂𝑋𝐼, �̇� = 𝑒𝑓(𝑋𝐼, 𝑌)𝑌 − 𝛾𝑌, (1) with 𝑋𝑆 ≥ 0, 𝑋𝐼 ≥ 0, 𝑌 ≥ 0. the first equation of system (1) expresses that in the absence of disease, the prey population grows by following the equation as below. 𝑋�̇� = 𝑅 − 𝛿𝑋𝑆, where 𝑅 is expressed as the level of recruitment in prey populations such as immigrants and new-born and 𝛿 is defined as the natural death rate of susceptible prey. here, the growth of the prey population is affected by factors such as disease. the spread of disease is denoted by bilinear incidence rate 𝛽𝑋𝑆𝑋𝐼 with 𝛽 is the transmission rate. we assume that the prey is not infected because of inherited disease but other sources. moreover, the disease spreads on susceptible prey only. we also assume that the infected prey populations do not recover nor reproduce. the second equation of system (1) describes the development of the infected prey population. they will be erased by the natural death rate 𝜂 and the predation of the predator. here, predation is denoted by the response function 𝑓(𝑋𝐼, 𝑌). we assume that infected prey can hide. hiding behavior gives protection for infected prey which can protect from predation. the protection of infected prey is denoted by constant 𝑚. predators can capture infected prey by following a ratio-dependent function as in [14]. 𝑓(𝑋𝐼, 𝑌) = { 0 , if 0 ≤ 𝑋𝐼 ≤ 𝑚, 𝑎(𝑋𝐼 − 𝑚) 𝑋𝐼 − 𝑚 + 𝜉𝑌 , if 𝑋𝐼 > 𝑚, with 𝑎 is expressed as the predation rate and 𝜉 is defined as half capturing saturation constant. according to [1], if the density of infected prey populations is below the constant 𝑚, then predators cannot eat them and will die exponentially. meanwhile, if the density of infected prey populations is above the constant 𝑚, then predators can eat them. thus, when 𝑋𝐼 > 𝑚, then system (1) shall become as follows. 𝑋�̇� = 𝑅 − 𝛽𝑋𝑆𝑋𝐼 − 𝛿𝑋𝑆, 𝑋𝐼̇ = 𝛽𝑋𝑆𝑋𝐼 − 𝑎(𝑋𝐼 − 𝑚)𝑌 𝑋𝐼 − 𝑚 + 𝜉𝑌 − 𝜂𝑋𝐼, �̇� = 𝑎𝑒(𝑋𝐼 − 𝑚)𝑌 𝑋𝐼 − 𝑚 + 𝜉𝑌 − 𝛾𝑌, (2) with 𝑋𝑆(0) ≥ 0, 𝑋𝐼(0) ≥ 0, 𝑌(0) ≥ 0. the last equation represents the behavior of the predator population. predators only consume the infected prey and don’t consume the susceptible prey. when the infected prey population is absent, then the predator experiences a natural death rate 𝛾. here, it is assumed that disease does not spread from infected prey to predator. in the model, all parameters are positive values. the meaning of parameter and their units are summarized in table 1. dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 230 table 1. units and the meaning of parameters in system (2) parameters biological meaning units 𝑋𝑆 the number of susceptible prey number 𝑋𝐼 the number of infected prey number 𝑌 the number of predators number 𝑅 the level of recruitment in prey time-1 𝛽 the level of infection of disease mass-1 time-1 𝛿 the level of natural mortality in susceptible prey time-1 𝜂 the level of natural mortality in infected prey time-1 𝛾 the level of natural mortality in predators time-1 𝜉 half saturation constant number 𝑎 the level of predation in predators mass-1 time-1 𝑒 the level of alteration from prey into predators time-1 𝑚 the measure of prey in the refuge number the boundedness to show the biological validity, we shall prove the boundedness of the system (2) as follows. theorem 1. all solutions of the system (2) are uniformly bounded. proof: we define 𝑊 = 𝑋𝑆 + 𝑋𝐼 + 1 𝑒 𝑌. by differentiating 𝑊 to 𝑡, we get 𝑑𝑊 𝑑𝑡 = 𝑑𝑋𝑆 𝑑𝑡 + 𝑑𝑋𝐼 𝑑𝑡 + 1 𝑒 𝑑𝑌 𝑑𝑡 . by substituting system (3), we get, 𝑑𝑊 𝑑𝑡 = 𝑅 − (𝛿𝑋𝑆 + 𝜂𝑋𝐼 + 𝛾 𝑌 𝑒 ). next, we choose 𝑞 = min{𝛿, 𝜂, 𝛾}. thus, we get 𝑑𝑊 𝑑𝑡 ≤ 𝑅 − 𝑞𝑊, by using the theory of differential equation, we get 𝑊(𝑡) ≤ 𝑅 𝑞 + 𝐶𝑒−𝑞𝑡, where 𝐶 is the arbitrary positive constant. for 𝑡 → ∞, we get lim 𝑡→∞ sup 𝑊(𝑡) ≤ 𝑅 𝑞 . thus, all solutions of system (2) enter into 𝛺 = {(𝑋𝑆, 𝑋𝐼, 𝑌) ∈ ℝ+ 3 : 𝑊(𝑡) ≤ 𝑅 𝑞 }. the equilibrium points by setting 𝑋�̇� = 𝑋𝐼̇ = �̇� = 0, we get the possible equilibrium points as below. 1. axial equilibrium 𝐸0 ( 𝑅 𝛿 , 0,0) always existent. dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 231 2. planar equilibrium 𝐸1 ( 𝜂 𝛽 , 𝛽𝑅−𝛿𝜂 𝜂𝛽 , 0) exists when 𝛽𝑅 > 𝛿𝜂. 3. interior equilibrium 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗), where 𝑋𝑆 ∗ = 𝑅 𝛽𝑋𝐼 ∗+𝛿 , 𝑌∗ = (𝑎𝑒−𝛾)(𝑋𝐼 ∗−𝑚) 𝛾𝜉 , and 𝑋𝐼 ∗ is the positive root of the quadratic equation (3) as follows. 𝐴1(𝑋𝐼 ∗)2 + 𝐴2𝑋𝐼 ∗ + 𝐴3 = 0, (3) with 𝐴1 = −𝛽(𝑎𝑒 − 𝛾 + 𝑒𝜂𝜉), 𝐴2 = 𝛽𝑚(𝑎𝑒 − 𝛾) − 𝛿(𝑎𝑒 − 𝛾 + 𝑒𝜂𝜉) + 𝑒𝑅𝛽𝜉, 𝐴3 = 𝛿𝑚(𝑎𝑒 − 𝛾). this point 𝐸∗ exists when it satisfies the condition 𝑎𝑒 > 𝛾 and 𝑋𝐼 ∗ > 𝑚. it is clear to show that 𝐴1 < 0 and 𝐴3 > 0. thus, the determinant of the equation (3) is 𝐷 = (𝐴2) 2 − 𝐴1𝐴3 ≥ 0. therefore, to get the explicit form of the root of the equation (3), we have to check the following two cases: a. for 𝐷 = 0, the equation (3) has a twin positive root where 𝑋𝐼 ∗ = − 𝐴2 2𝐴1 with 𝐴2 > 0. b. for 𝐷 > 0, the probability that equation (3) has positive roots is as follows.  if 𝐴2 > 0, then the equation (3) has two positive roots where 𝑋𝐼1,2 ∗ = −𝐴2±√𝐷 2𝐴1 .  if 𝐴2 < 0, then the equation (3) has a single positive root where 𝑋𝐼 ∗ = −𝐴2−√𝐷 2𝐴1 . dynamical behaviour to investigate the stability in system (2), we have to determine the eigenvalues of the jacobian matrix. here, we identify the jacobian matrix at 𝐸(𝑋𝑆, 𝑋𝐼, 𝑌) as follows. 𝐽(𝐸) = [ −𝛽𝑋𝐼 − 𝛿 −𝛽𝑋𝑆 0 𝛽𝑋𝐼 𝛽𝑋𝑆 − 𝑎𝜉𝑌2 (𝑋𝐼 − 𝑚 + 𝜉𝑌) 2 − 𝜂 − 𝑎(𝑋𝐼 − 𝑚) 2 (𝑋𝐼 − 𝑚 + 𝜉𝑌) 2 0 𝑎𝑒𝜉𝑌2 (𝑋𝐼 − 𝑚 + 𝜉𝑌) 2 𝑎𝑒(𝑋𝐼 − 𝑚) 2 (𝑋𝐼 − 𝑚 + 𝜉𝑌) 2 − 𝛾 ] . (4) to check the stability of 𝐸0 ( 𝑅 𝛿 , 0,0), we get the jacobian matrix by replacing 𝐸(𝑋𝑆, 𝑋𝐼, 𝑌) in the equation (4) with 𝐸0 ( 𝑅 𝛿 , 0,0). hence, we obtain the eigenvalues of the jacobian matrix, namely 𝜆1 = −𝛿, 𝜆2 = 𝛽𝑅 𝛿 − 𝜂, and 𝜆3 = 𝑎𝑒 − 𝛾. the point 𝐸0 is locally asymptotically stable when 𝑎𝑒 < 𝛾 and 𝛽𝑅 < 𝛿𝜂. to investigate the stability of 𝐸1 ( 𝜂 𝛽 , 𝛽𝑅−𝛿𝜂 𝜂𝛽 , 0), we have to identify the jacobian matrix by replacing 𝐸(𝑋𝑆, 𝑋𝐼, 𝑌) in the equation (4) with 𝐸1 ( 𝜂 𝛽 , 𝛽𝑅−𝛿𝜂 𝜂𝛽 , 0). thus, we get the eigenvalues 𝜆1 = 𝑎𝑒 − 𝛾 and other eigenvalues are the roots of the quadratic equation 𝜆2 + 𝜑1𝜆 + 𝜑2 = 0 with 𝜑1 = 𝛽𝑅−𝛿𝜂 𝜂 + 𝛿 and 𝜑2 = 𝛽𝑅 − 𝛿𝜂. by using the routh-hurwitz criteria, the eigenvalues have negative real roots when 𝛽𝑅 > 𝛿𝜂. hence, the point 𝐸1 is locally asymptotically stable when 𝑎𝑒 < 𝛾 and 𝛽𝑅 > 𝛿𝜂. from the above discussion, we get the following theorem as follows. theorem 2. the axial equilibrium 𝐸0 is locally asymptotically stable when 𝑎𝑒 < 𝛾 and 𝛽𝑅 < 𝛿𝜂 and the planar equilibrium 𝐸1 is locally asymptotically stable when 𝑎𝑒 < 𝛾 and 𝛽𝑅 > 𝛿𝜂. theorem 2 means that if the level of natural mortality in predator is lower than a certain value and the level of infection is lower than a certain value, then the point 𝐸0 dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 232 becomes stable. meanwhile, if the level of infection is greater than a certain value, then the point 𝐸1 becomes stable. now, we shall investigate the stability of 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗). by replacing 𝐸(𝑋𝑆, 𝑋𝐼, 𝑌) in the equation (4) with 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗), we get the jacobian matrix where 𝑢𝑖𝑗 is the entry of matrix with row 𝑖 and column 𝑗 as follows. 𝐽(𝐸∗) = [ 𝑢11 𝑢12 0 𝑢21 𝑢22 𝑢23 0 𝑢32 𝑢33 ], with 𝑢11 = −𝛽𝑋𝐼 ∗ − 𝛿; 𝑢12 = −𝛽𝑋𝑆 ∗; 𝑢21 = 𝛽𝑋𝐼 ∗, 𝑢22 = 𝛽𝑋𝑆 ∗ − 𝑎𝜉(𝑌∗)2 (𝑋𝐼 ∗ − 𝑚 + 𝜉𝑌∗)2 − 𝜂; 𝑢23 = − 𝑎(𝑋𝐼 ∗ − 𝑚)2 (𝑋𝐼 ∗ − 𝑚 + 𝜉𝑌∗)2 , 𝑢32 = 𝑎𝑒𝜉(𝑌∗)2 (𝑋𝐼 ∗ − 𝑚 + 𝜉𝑌∗)2 ; 𝑢33 = − 𝑎𝑒𝜉𝑌∗(𝑋𝐼 ∗ − 𝑚) (𝑋𝐼 ∗ − 𝑚 + 𝜉𝑌∗)2 . hence, we obtain the characteristic equation of 𝐸∗, namely 𝜆3 + 𝜇1𝜆 2 + 𝜇2𝜆 + 𝜇3 = 0 (5) with 𝜇1 = −(𝑢11 + 𝑢22 + 𝑢33), 𝜇2 = 𝑢11𝑢22 + 𝑢11𝑢33 + 𝑢22𝑢33 − 𝑢12𝑢21 − 𝑢23𝑢32, 𝜇3 = −𝑢11𝑢22𝑢33 + 𝑢11𝑢23𝑢32 + 𝑢12𝑢21𝑢33. by using the routh-hurwitz criteria, the eigenvalues have negative real roots when 𝜇1 > 0, 𝜇3 > 0, 𝜇1𝜇2 > 𝜇3. thus, the point 𝐸 ∗ is locally asymptotically stable when 𝜇1 > 0, 𝜇3 > 0, 𝜇1𝜇2 > 𝜇3. therefore, we get the following theorem. theorem 3. the interior equilibrium 𝐸∗ is locally asymptotically stable when it satisfies the condition 𝜇1 > 0, 𝜇3 > 0, 𝜇1𝜇2 > 𝜇3. in the next theorem, we shall prove that the point 𝐸1 is globally asymptotically stable under a certain condition. theorem 4. the point 𝐸1 is globally asymptotically stable in the 𝑋𝑆 − 𝑋𝐼 plane. proof: by applying the dulac function as 𝐻(𝑋𝑆, 𝑋𝐼) = 1 𝑋𝐼 , we have 𝐻(𝑋𝑆, 𝑋𝐼) = 1 𝑋𝐼 , ℎ1(𝑋𝑆, 𝑋𝐼) = 𝑅 − 𝛽𝑋𝑆𝑋𝐼 − 𝛿𝑋𝑆, ℎ2(𝑋𝑆, 𝑋𝐼) = 𝛽𝑋𝑆𝑋𝐼 − 𝜂𝑋𝐼, where 𝐻(𝑋𝑆, 𝑋𝐼) > 0 in the 𝑋𝑆 − 𝑋𝐼 plane. thus, we get ∆(𝑋𝑆, 𝑋𝐼) = 𝜕 𝜕𝑋𝑆 (ℎ1𝐻) + 𝜕 𝜕𝑋𝐼 (ℎ2𝐻) = −𝛽 − 𝛿 𝑋𝐼 < 0. base on bendixson-dulac criteria, there is no limit cycle in the 𝑋𝑆 − 𝑋𝐼 plane. thus, the point 𝐸1 is globally asymptotically stable in the 𝑋𝑆 − 𝑋𝐼 plane. hopf bifurcation in this section, we shall investigate hopf bifurcation around 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗). hopf bifurcation guarantees that all solutions of system (2) enter a limit cycle around 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗). here, we choose the constant 𝑚 as a bifurcation parameter. hopf bifurcation around 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗) is presented in the following theorem. dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 233 theorem 5. the system (2) undergoes a hopf bifurcation around 𝐸∗(𝑋𝑆 ∗, 𝑋𝐼 ∗, 𝑌∗) when 𝑚 passes through a critical value 𝑚 = 𝑚𝑐. proof: we consider equation (5) as the characteristic equation at 𝐸∗. next, we choose 𝑚 = 𝑚𝑐 such that 𝜇1𝜇2 = 𝜇3 where 𝜇1, 𝜇2, 𝜇3 > 0. therefore, we get (𝜆2 + 𝜇2)(𝜆 + 𝜇1) = 0 (6) with the roots are 𝜆1,2 = ±𝑖√𝜇2 and 𝜆3 = −𝜇1. for all 𝑚, the roots become 𝜆1,2 = 𝑣1(𝑚) ± 𝑖𝑣2(𝑚) and 𝜆3 = −𝜇1(𝑚) where 𝑣1(𝑚) and 𝑣2(𝑚) are real. next, we shall prove the transversality condition as follows. 𝑑 (𝑅𝑒 𝜆𝑗(𝑚)) 𝑑𝑚 | 𝑚=𝑚𝑐 ≠ 0, 𝑗 = 1,2. by substituting 𝜆1 = 𝑣1(𝑚) + 𝑖𝑣2(𝑚) into equation (6) and differentiating to 𝑚, we obtain 𝐾𝑣1̇ − 𝐿𝑣2̇ + 𝑀 = 0, 𝐿𝑣1̇ + 𝐾𝑣2̇ + 𝑁 = 0, (7) where 𝐾 = 3(𝑣1 2 − 𝑣2 2) + 𝜇2 + 2𝑣1𝜇1, 𝐿 = 6𝑣1𝑣2 + 2𝑣2𝜇1, 𝑀 = 𝜇1̇(𝑣1 2 − 𝑣2 2) + 𝑣1𝜇2̇ + 𝜇3̇, and 𝑁 = 2𝑣1𝑣2𝜇1̇ + 𝑣2𝜇2̇. next, we solve equation (7). thus, we have 𝑣1̇ = − 𝐾𝑀 + 𝐿𝑁 𝐾2 + 𝐿2 . since 𝐾𝑀 + 𝐿𝑁 ≠ 0 and 𝜇1̇ ≠ 0, we obtain 𝑑 (𝑅𝑒 𝜆𝑗(𝑚)) 𝑑𝑚 | 𝑚=𝑚𝑐 = − 𝐾𝑀 + 𝐿𝑁 𝐾2 + 𝐿2 | 𝑚=𝑚𝑐 ≠ 0, 𝑗 = 1,2, and 𝜆3(𝑚𝑐) = −𝜇1(𝑚𝑐) < 0. thus, the system (2) occurs hopf bifurcation when 𝑚 passes through a critical value 𝑚 = 𝑚𝑐. persistence to show that all species are present and are not extinct in the future time, we shall prove that system (2) is uniform persistence. theorem 6. let the assumption of theorem 4 holds. if the inequalities 𝑎𝑒 > 𝛾 and 𝛽𝑅 > 𝛿𝜂 hold, then the system (2) is uniform persistence. proof: we consider average lyapunov function 𝜎(𝑋) = 𝑋𝑆 𝑟1𝑋𝐼 𝑟2𝑌𝑟3 with 𝑟1, 𝑟2, 𝑟3 > 0. here, 𝜎(𝑋) is nonnegative 𝐶1 in ℝ+ 3 . thus, we have 𝜗(𝑋) = 1 𝜎 𝑑𝜎 𝑑𝑡 = 𝑟1 𝑋�̇� 𝑋𝑆 + 𝑟2 𝑋𝐼̇ 𝑋𝐼 + 𝑟3 �̇� 𝑌 , = 𝑟1 ( 𝑅 𝑋𝑆 − 𝛽𝑋𝐼 − 𝛿) + 𝑟2 (𝛽𝑋𝑆 − 𝑎(𝑋𝐼 − 𝑚)𝑌 𝑋𝐼(𝑋𝐼 − 𝑚 + 𝜉𝑌) − 𝜂) + 𝑟3 ( 𝑎𝑒(𝑋𝐼 − 𝑚) 𝑋𝐼 − 𝑚 + 𝜉𝑌 − 𝛾). in the system, the point 𝐸1 is the only equilibrium point which is no limit cycle around the equilibrium point. therefore, theorem 4 holds. hence, it is enough to prove that 𝜗(𝑋) > 0 for all equilibrium point 𝑋 ∈ 𝑏𝑑 ℝ+ 3 . thus, we get 𝜗(𝐸0) = 𝑟2 ( 𝛽𝑅 − 𝛿𝜂 𝛿 ) + 𝑟3(𝑎𝑒 − 𝛾) > 0, 𝜗(𝐸1) = 𝑟3(𝑎𝑒 − 𝛾) > 0. dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 234 we note that if the inequalities 𝑎𝑒 > 𝛾 and 𝛽𝑅 > 𝛿𝜂 hold, then 𝜗(𝐸0) > 0 holds. meanwhile, if the inequalities 𝑎𝑒 > 𝛾 holds, then 𝜗(𝐸1) > 0 holds. therefore, theorem 6 expresses the instability of the point 𝐸0 and 𝐸1. numerical solutions the analytical results obtained are incomplete without numerical investigation. in this section, we present the numerical solution by using the 5th-order predictor-corrector method at ∆𝑡 = 0.01. here, we will give four simulations to verify our analytical results and also to demonstrate the effect of the prey refuge. we choose several parameters as in (8). their units are given as in table 1. 𝑅 = 2, 𝛽 = 1, 𝛿 = 1, 𝜂 = 0.5, 𝛾 = 0.5, 𝜉 = 1, 𝑎 = 2, 𝑒 = 0.75, 𝑚 = 0.5. (8) simulation 1. it confirms that system (2) has an equilibrium 𝐸∗(1.0685,0.8717,0.7434). on simulation 1, theorem 3 and theorem 6 are satisfied. we observe that all solutions converge to the point 𝐸∗, see figure 1(a). thus, the point is locally asymptotically stable, see figure 1(b), and the system (2) is uniform persistence. (a) (b) figure 1. the dynamics of system (2) with 𝑚 = 0.5 and other parameters as in (8) simulation 2. we replace 𝑚 = 0.5 into 𝑚 = 0.0002. here, we investigate that system (2) has an equilibrium point 𝐸∗(1.8305,0.0926,0.1848). on simulation 2, theorem 3 is not satisfied but theorem 6 is satisfied. therefore, system (2) is uniform persistence but the equilibrium point 𝐸∗ is unstable, see figure 2. (a) (b) figure 2. this figure shows the instability of system (2) dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 235 simulation 3. to see hopf bifurcation in the system (2), we choose 𝑚 = 𝑚𝑐 = 0.0004 as a bifurcation parameter. thus, we identify that the equilibrium point of system (2) is 𝐸∗(1.8277,0.0943,0.1878). if we choose 𝑚 = 0.005 > 𝑚𝑐 = 0.0004, then all solutions of the system (2) convergent to 𝐸∗(1.7795,0.1239,0.2378), see figure 3(a). meanwhile, figure 3(b) shows the phase portrait of the system (2) with 𝑚 = 0.005 which means that the point 𝐸∗ is stable. furthermore, if we choose 𝑚 = 0.0001 < 𝑚𝑐 = 0.0004, then the point 𝐸∗(1.8319,0.0918,0.1833) is unstable, see figure 4(a). meanwhile, the phase portrait in the system (2) with 𝑚 = 0.0001 that is presented in figure 4(b) means that the solution of the system (2) enter a limit cycle around 𝐸∗. thus, theorem 5 is satisfied. simulation 4. to see the effect of prey refuge in the system (2), we use several values of prey refuge, namely 𝑚1 = 0.1, 𝑚2 = 0.65, and 𝑚3 = 1.3. figure (5) shows the time graph of the system (2) by using 𝑚 makes different values. here, we obtain that all populations exist no matter how large 𝑚 with 𝑚 < 𝑋𝐼. this prey refuge creates the system (2) to become stable rapidly and no extinction occurs. here, it is worthy to attention that when we choose the constant 𝑚 with 𝑚 < 𝑋𝐼, then the measure of prey refuge doesn’t lead to predator extinction. (a) (b) figure 3. the dynamics of system (2) with 𝑚 = 0.005 and other parameters as in (8) (a) (b) figure 4. the dynamics of system (2) with 𝑚 = 0.0001 and other parameters as in (8) dynamical of ratio-dependent eco-epidemiology model with prey refuge adin lazuardy firdiansyah 236 figure 5. the effect of prey refuge by using 𝑚 makes different values conclusions in this study, we have observed three species in the eco-epidemiology model with the ratio-dependent function incorporating prey refuge. we obtain three equilibrium points where all points, i.e. axial equilibrium, planar equilibrium, and interior equilibrium, are locally asymptotically stable under certain conditions. moreover, the planar equilibrium point in the system (2) is globally asymptotically stable. next, we find that hopf bifurcation occurs around the interior equilibrium by choosing a bifurcation parameter in the constant 𝑚. when 𝑚 > 𝑚𝑐 = 0.0004, the system (2) is stable. however, when 𝑚 < 𝑚𝑐 = 0.0004, the system (2) is unstable. furthermore, we also find a condition for uniform persistence. if the level of natural mortality in predators is lower than a certain value and the level of infection is greater than a certain value, then all species exist in the future time. next, by applying the prey refuge, all populations exist no matter how large 𝑚 with 𝑚 < 𝑋𝐼. we conclude that system (2) is stable faster and there is no extinction. references [1] c. maji, d. kesh, and d. mukherjee, “bifurcation and global stability in an ecoepidemic model with refuge,” energy, ecol. environ., vol. 4, no. 3, pp. 103–115, 2019, doi: 10.1007/s40974-019-00117-6. 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[15] s. wang, z. ma, and w. wang, “dynamical behavior of a generalized ecoepidemiological system with prey refuge,” adv. differ. equations, vol. 2018, no. 1, pp. 1–20, 2018, doi: 10.1186/s13662-018-1704-x. a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 267-280 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 30, 2021 reviewed: january 22, 2022 accepted: january 26, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.13462 a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi*, nurul gusriani , betty subartini department of mathematics of fmipa of universitas padjadjaran, indonesia *corresponding author email: edi.kurniadi@unpad.ac.id*, nurul.gusriani@unpad.ac.id, betty.subartini@unpad.ac.id abstract this paper relates quasi-associative algebras or koszul algebras of matrix lie algebras of finite dimension to finite dimensional frobenius lie algebras which is written as a semi direct sum. let m2(ℝ)⋊𝔤𝔩2(ℝ) be the lie algebra of the semi-direct sum of the real vector space m2(ℝ) and the lie algebra 𝔤𝔩2(ℝ) of the sets of all 2×2 real matrices. the research aims to give explicit formulas of left-symmetric algebra structures on m2(ℝ)⋊𝔤𝔩2(ℝ). in this paper, a frobenius functional is constructed in order to show that the lie algebra m2(ℝ)⋊𝔤𝔩2(ℝ) is the real frobenius lie algebra of dimension 8. moreover, a bilinear form corresponding to this frobenius functional is symplectic. then the obtained symplectic bilinear form induces the leftsymmetric algebra structures on m2(ℝ)⋊𝔤𝔩2(ℝ). in other words, we show that the lie algebra m2(ℝ)⋊𝔤𝔩2(ℝ) is the left-symmetric algebra. thus, we give the formulas of its left-symmetric algebra structure explicitely. the left-symmetric algebra structures for case of higher dimension of this lie algebra type are still an open problem to be investigated. keywords: left-symmetric algebra; frobenius lie algebra; frobenius functional; semi-direct sum; symplectic form. introduction let ℝ be the field of real numbers of characteristic zero and m2(ℝ) be the ℝ-vector space of all 2×2 real matrices with entries contained in ℝ. the space m2(ℝ) is considered as the ℝ-algebra structure given by usual addition and multiplication of matrices. in addition, the space m2(ℝ) is the lie algebra equipped by the lie brackets in the form of the matrix commutator [𝑥,𝑦]:= 𝑥𝑦 −𝑦𝑥 for all matrices 𝑥,𝑦 ∈ m2(ℝ). we denote by 𝔤𝔩2(ℝ) the lie algebra for m2(ℝ) with respect to the mentioned lie bracket above. moreover, we shall let 𝔥2 ≔ m2(ℝ)⋊𝔤𝔩2(ℝ) represent for the semi-direct sum lie algebra of the vector space m2(ℝ) and the lie algebra 𝔤𝔩2(ℝ) of dimension 8. furthermore, the lie algebra 𝔥2 has the following matrix realization: http://dx.doi.org/10.18860/ca.v7i1.13462 mailto:edi.kurniadi@unpad.ac.id* mailto:nurul.gusriani@unpad.ac.id mailto:betty.subartini@unpad.ac.id a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 268 𝜎 ≔ 𝜎(𝑈,𝑋) = ( 𝑋 𝑈 0 0 ) ∈ 𝔥2 ⊂ m4(ℝ) (1) with 𝑈 ∈ m2(ℝ) and 𝑋 ∈ 𝔤𝔩2(ℝ). let 𝔥2 ∗ be a dual vector space of 𝔥2. we also realize in the following matrix 𝜎∗ ≔ 𝜎∗(𝛼,𝛽) = ( 𝛽 0 𝛼 0 ) ∈ 𝔥2 ∗ ⊂ m4(ℝ) (2) with 𝛼 ∈ m2(ℝ) and 𝛽 ∈ 𝔤𝔩2(ℝ). a value of a linear functional 𝜎 ∗ ∈ 𝔥2 ∗ on a matrix 𝜎 ∈ 𝔥2 is denoted by 〈𝜎∗,𝜎〉 = tr(𝑈𝛼)+tr(𝑋𝛽) = 〈𝛼,𝑈〉+〈𝛽,𝑋〉 (3) where tr is denoted the matrix trace. the notion of the lie algebra 𝔥2 comes originally from the lie algebra 𝔥𝑛,𝑝 ≔ m𝑛,𝑝(𝕂)⋊𝔤𝔩𝑛(𝕂) where m𝑛,𝑝(k) is the real vector space of the set 𝑛 ×𝑝 matrices and 𝔤𝔩𝑛(𝕂) is the lie algebra of the set 𝑛 ×𝑛 matrices over a field 𝕂 of characteristic 0. this lie algebra was introduced by [1] in the context of coadjoint representations of the affine lie group g𝑛,𝑝 ≔ m𝑛,𝑝(𝕂)⋊gl𝑛(𝕂) where gl𝑛(𝕂) is the lie group of all invertible matrices of the lie algebra 𝔤𝔩𝑛(𝕂). in our case, we take 𝑛 = 𝑝 = 2 and 𝕂 = ℝ. we denote by m2(ℝ) ≔ m2,2(ℝ) and we determine the lie algebra 𝔥2 of the lie group g2. in addition, the case 𝑝 = 1 is considered as the affine lie algebra, denoted by 𝔞𝔣𝔣(𝑛), of dimension 𝑛(𝑛 +1). more precise, the realization of this affine lie algebra 𝔞𝔣𝔣(𝑛) is of the form m𝑛,1(ℝ)⋊𝔤𝔩𝑛(ℝ) where m𝑛,1(ℝ) is isomorphic to the space ℝ 𝑛. interestingly, the lie algebra 𝔞𝔣𝔣(𝑛) is frobenius since it has a trivial stabilizer on a certain linear functional. in the other words, the lie group aff(𝑛) of 𝔞𝔣𝔣(𝑛) has an open coadjoint orbit [2]. moreover, the lie algebra 𝔞𝔣𝔣(𝑛) is non-solvable. regarding the notion of a frobenius lie algebra, this lie algebra type was introduced by ooms in the context to answer professor jacobson’s question on properties of finite dimensional lie algebras having an exact simple module over the universal enveloping algebras([3],[4]). the answer that the universal enveloping algebra has an exact simple module if its finite dimensional lie algebra is frobenius. furthermore, if a lie algebra 𝔤 is frobenius then there exists a linear functional 𝜓 defined on 𝔤 such that a bilinear alternating form is non-degenerate. namely, it is a symplectic linear form and such the linear functional 𝜓 is called a frobenius functional [5]. the notion of the frobenius functional gives some important implications. among them, firstly, since the alternating bilinear form corresponding to the frobenius functional is the symplectic linear form, then the dimension of frobenius lie algebra is always even. secondly, the stabilizer of frobenius lie algebras corresponding to the frobenius functional is trivial, then we have a fact that the frobenius lie algebra is not nilpotent [6]. appearing in many different areas of lie algebras, for instance, in the study of bounded homogeneous domain, simple hypersurface singularities, and solutions of yang-baxter equation [7], frobenius lie algebras bring great significance in many areas of lie algebras. in ([5],[8]), a notion of a left-symmetric algebra structure on a lie algebra was introduced where the bilinear product was defined. it was proved that an 𝑛-dimensional lie algebra 𝔤 has left-symmetric algebra structures if there exists a 𝔤-module 𝐾 of dimension 𝑛 such that the 1-cocyle space contains a bijective 1-cocyle. indeed, not every type of lie algebras has affine structure. these structures are important because it arise a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 269 in many areas of mathematics such as convex homogeneous cones, affine manifolds, and vertex algebras. associating to the frobenius functional, the left-symmetric algebras can be induced by the symplectic structure corresponding to a frobenius functional [9]. in the present work, we are interested in studying the frobenius lie algebra and leftsymmetric algebra for the lie algebra 𝔥2 of the lie group g2. in this paper, we calculate a frobenius functional of 𝔥2 corresponding to its pfaffian and we prove that 𝔥2 is 8-dimensional frobenius lie algebra. the obtained frobenius functional of 𝔥2 is applied to contruct a symplectic structure on 𝔥2 which induces a leftsymmetric algebra structure on 𝔥2. the research aims to give explicit formulas of leftsymmetric algebra structures on m2(ℝ)⋊𝔤𝔩2(ℝ). furthermore, with respect to a basis 𝔥2, we calculte the left-symmetric algebra structures explicitely. let 𝜀𝑖𝑗, 1 ≤ 𝑖,𝑗 ≤ 4 be element of 𝔥2 ⊂ m4(ℝ) of the form in the equation (1). we organize the paper as follows. section 1 is explained the background of this research, state of art, the aim of research, statement of the main results, and some basic notions . section 2 is devoted to research method. section 3 discusses our main results to prove that 𝔥2 ≔ m2(ℝ)⋊𝔤𝔩2(ℝ) is 8-dimensional frobenius lie algebra and 𝔥2 has left-symmectric structures. in this section, we also give explicit formulas for leftsymmetric structures of 𝔥2 and discussion for future research related to our main results. in the end of this paper, the conclusion is given. methods our main object was considered from the notion of the affine lie group g𝑛,𝑝 ≔ m𝑛,𝑝(𝕂)⋊gl𝑛(𝕂) which corresponds to the notion of coadjoint representations. regarding this affine lie group, particularly, we consider the special case for 𝑛 = 𝑝 = 2 and we obtained the lie algebra 𝔥2 ≔ m2(ℝ)⋊𝔤𝔩2(ℝ) of the lie group g2 ≔ m2(ℝ)⋊ gl2(ℝ). we studied some related papers corresponding to a frobenius lie algebra and we offered another alternative to show whether it was a frobenius lie algebra or not coresponding to its pfaffian. particularly on some oom’s work [4]. on the other hand, burde introduced the notation of left-symmetric algebra structures on a lie group and a lie algebra [8]. we proved the existence of left-symmetric algebra structures on 𝔥2 and we listed its structures explicitely. our constructions based on symplectic form which induces its left-symmetric algebra structures. in the next section, we shall briefly review some basic notions of lie algebras, frobenius lie algebras and their properties, pfaffian of a frobenius lie algebras, and left-symmetric structures on a lie algebra. in section results and discussion, we shall complete the proof of our main results. 1.1 frobenius lie algebra we shall first introduce the notion of a lie algebra as follows: definition 1[10]. let 𝔤 be a vector space. a lie bracket on 𝔤 is a bilinear form on 𝔤×𝔤, usually denoted by [ , ], which satisfies: 1. [𝑥,𝑥] = 0 for 𝑥 ∈ 𝔤, a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 270 2. jacobi identity, that is [𝑥,[𝑦,𝑧]] = [[𝑧,𝑥],𝑦]+[[𝑥,𝑦],𝑧]. (4) a vector space 𝔤 together with a lie bracket [ , ] is called a lie algebra. let 𝔤 be a lie algebra with 𝔤∗ is dual vector space consisting of linear functionals on 𝔤. we denote by 𝔤𝔩(𝔤) the lie algebra of endomorphism of 𝔤 to itself. then a representation of the lie algebra 𝔤 on himself is given by the following map ad ∶ 𝔤 ∋ 𝑥 ↦ ad(𝑥) ∈ 𝔤𝔩(𝔤). (5) furthermore, the corresponding representation of the lie algebra 𝔤 on the space 𝔤∗ is considered as the dual map to ad and is written in the following formula 〈ad∗(𝑥)𝜎∗,𝑦〉 = 〈𝜎∗,−ad(𝑥)𝑦〉 = 〈𝜎∗,−[𝑥,𝑦]〉 (6) where 𝑥,𝑦 ∈ 𝔤and 𝜎∗ ∈ 𝔤∗. let 𝜎∗ be an element of 𝔤∗ and 𝐵𝜎∗ be alternating bilinear form corresponding to 𝜎 ∗ which is defined by 𝐵𝜎∗ ∶ 𝔤×𝔤 ∋ (𝑥,𝑦) ↦ 〈𝜎 ∗, [𝑥,𝑦]〉 ≔ 𝐵𝜎∗(𝑥,𝑦) ∈ ℝ. (7) the kernel of 𝐵𝜎∗ is given by the following formula ker(𝐵𝜎∗) = {(𝑥,𝑦) ∈ 𝔤×𝔤 ; 〈𝜎 ∗, [𝑥,𝑦]〉 = 0 } = {𝑥 ∈ 𝔤 ; 〈𝜎∗, [𝑥,𝑦]〉 = 0,∀ 𝑦 ∈ 𝔤 } = {𝑥 ∈ 𝔤 ; 〈ad∗(𝑥)𝜎∗,𝑦〉 = 0,∀ 𝑦 ∈ 𝔤 } = {𝑥 ∈ 𝔤 ; ad∗(𝑥)𝜎∗ = 0} the latter formula is nothing but a stabilizer of 𝔤 correponding to 𝜎∗ which is denoted by 𝔤𝜎 ∗ . definition 2[11]. let 𝔤 be a lie algebra. the lie algeba 𝔤 is said to be frobenius if there exists a linear functional 𝜎∗ ∈ 𝔤∗ such that the alternating bilinear form in the equation (7) is non-degenerate. such 𝜎∗ satisfying the eqution (7) is called a frobenius functional. indeed, the non-commutative lie algebra of dimension 2 is always frobenius. it is well known that affine lie algebra 𝔞𝔣𝔣(1) ≔ ℝ⋊ℝ is 2-dimensional frobenius lie algebras. those frobenius lie algebras form a large class of lie algebras. for example, some parabolic and seaweed subalgebras of semi-simple lie algebras, 𝑗-algebras, and borel subalgebras of simple lie algebras are frobenius lie algebras [7]. now, let 𝑇 ≔ {𝑥1,𝑥2,…,𝑥𝑛} be a basis for a frobenius lie algebra 𝔤 of dimension 𝑛 with 𝑛 is even. for 𝑥𝑗,𝑥𝑘 ∈ 𝑇, we denote by 𝑀𝔤(ℝ) an 𝑛 ×𝑛 matrix whose entries are defined by 𝑀𝔤(ℝ) ≔ ([𝑥𝑗,𝑥𝑘])𝑗,𝑘=1 𝑛 . moreover, for 𝜎∗ ∈ 𝔤∗ we define a matrix 𝑀𝔤(ℝ)(𝜎 ∗) stand for an 𝑛 ×𝑛 matrix whose entries are defined by 𝑀𝔤(ℝ)(𝜎 ∗) ≔ (〈𝜎∗, [𝑥𝑗,𝑥𝑘]〉)𝑗,𝑘=1 𝑛 . (8) we get the following theorem : a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 271 theorem 1[4]. let 𝔤 be a lie algebra of even dimensional with a basis 𝑇 ≔ {𝑥1,𝑥2,…,𝑥𝑛}. the lie algebra 𝔤 is frobenius if one of the following equivalent conditions is satisfied: 1. the stabilizer 𝔤𝜎 ∗ = {0} for some 𝜎∗ ∈ 𝔤∗. 2. the determinant of the matrix 𝑀𝔤(ℝ) is not equal to zero. 3. the determinant of the matrix 𝑀𝔤(ℝ)(𝜎 ∗) is not equal to zero for some frobenius functionals 𝜎∗ ∈ 𝔤∗. the relation of alternating bilinear form and a linear functional on 𝔤 in the term of a frobenius lie algebra can be explained as follows: in general, let 𝜓 be alternating bilinear form in a lie algebra 𝔤. if 𝜓 is non-degenerate closed 2-form then 𝔤 is said to be a quasi-frobenius lie algebra. furthermore, if there exists a linear functional 𝜎∗ ∈ 𝔤∗ such that 𝜓(𝛼,𝛽) = d𝜎∗(𝛼,𝛽) = −〈𝜎∗, [𝛼,𝛽]〉 (𝛼,𝛽 ∈ 𝔤) (9) with d𝜎∗ denotes the differential of 𝜎∗, then 𝔤 is also called a frobenius lie algebra [5]. indeed, this statement is equivalent to definition of the frobenius lie algebra. example 1. the nice examples of low dimensional frobenus lie algebras are 4dimensional frobenius lie algebras over a field with characteristic ≠ 2 and 6dimensional frobenius lie algebras over an algebraically closed field classified in [6]. in the other hand, the (2𝑛 +1)-dimensional heisenberg lie algebra is not frobenius lie algebra. since we shall relate a frobenius functional to pfaffian, we need to recall the notion of pfaffian for the square matrix 𝐴 ≔ (𝐴𝑗𝑘)𝑗,𝑘=1 2𝑛 where 𝐴 is an alternating matrix [12]. the formula of pfaffian is given by pf(𝐴) ≔ 1 2𝑛𝑛! ∑ sgn(𝜏)∏ 𝐴𝜏(2𝑖−1)𝜏(2𝑖) 𝑛 𝑘=1𝜏∈𝑆(2𝑛) . (10) the pfaffian for the lie algebra 𝔤 is defined as the pfaffian of matrix 𝑀𝔤(ℝ). remark 1 [12]. let 𝐴 be an even dimensional square-alternating matrix. then det(𝐴) = pf(𝐴)2. example 2. let 𝔞𝔣𝔣(1) be 2-dimensional affine lie algebra with non-zero bracket is [𝑎,𝑏] = 𝑏. we obtain pf(𝔞𝔣𝔣(1)) = 𝑏. moreover, the 4-dimensional frobenius lie algebra σ with non-zero brackets in the following formulas [6] [𝑑,𝑎] = 𝑎, [𝑐,𝑏] = 𝑎 [𝑑,𝑏] = 1 2 𝑏, [𝑑,𝑐] = 1 2 𝑐 has pfaffian of the form pf(σ) = 𝑎2 [13]. a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 272 the notion of pfaffian is very important in representation theory of lie groups, for instance, in notion of square-integrable representation, the duflo-moore operator can be associated to pfaffian [13]. in this paper, we shall relate pfaffian to a frobenius functional contruction. 1.2 left-symmetric algebra structure the notion of left-symmetric algebra first have been arisen in the theory of lie groups 𝐺 endowed a left-invariant affine structure [14]. let 𝐴,𝐵, and 𝐶 left-invariant vector fields of 𝐺 that are the section of the map 𝜓 ∶ 𝑇𝐺 → 𝐺 with 𝑇𝐺 is a tangent bundle of 𝐺. let ∇ be a connection in 𝑇𝐺 with both curvature and torsion are zero, namely : [𝐴,𝐵] = ∇a𝐵 −∇𝐵𝐴, ∇[a,b]𝐶 = ∇𝑋∇𝑌𝑍 −∇𝑌∇𝑋𝑍. (11) let 𝑎,𝑏, and 𝑐 be elements of a lie algebra 𝔤 of 𝐺. firstly, we define 𝑎𝑏 ≔ 𝑎 ∗𝑏 = ∇a𝐵. let ∆(𝑎,𝑏,𝑐) be associator for 𝑎,𝑏, and 𝑐 given by ∆(𝑎,𝑏,𝑐) ≔ (𝑎𝑏)𝑐 −𝑎(𝑏𝑐). (12) in addition, a left-symmectric or a non-associative algebra structure on the lie algebra 𝔤 is given by the following formula ∆(𝑎,𝑏,𝑐) = ∆(𝑏,𝑎,𝑐). (13) we start with some basic definitions of left-symmetric algebra structure which shall be needed for the next section. definition 3[8]. a non-associative algebra 𝐿 is called a left-symmetric algebra (lsa) if a bilinear product defined by 𝐿 ×𝐿 ∋ (𝑎,𝑏) ↦ 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝐿 satisfies the equation (15). a given lie bracket by the following commutator [𝑎,𝑏] = 𝑎𝑏 −𝑏𝑎 (14) for all 𝑎,𝑏,𝑐 ∈ 𝐿 defines a lie algebra 𝔤 of 𝐿. in other words, the algebra 𝐿 is a lie algebra denoted by 𝔤 = 𝔤(𝐿) with respect to the lie bracket in the equation (14). we call a left-symmetric algebra by a lie admissible lie algebra or vinberg algebra [8]. example 3[5]. let 𝔞𝔣𝔣(1) be a 2-dimensional affine lie algebra with basis 𝐵 ≔ {𝑎,𝑏} and non-zero brackets [𝑎,𝑏] = 𝑏. the left-symmetric algebra structures on 𝔞𝔣𝔣(1) are given by 𝑎2 = −𝑎, 𝑎𝑏 = 0, 𝑏𝑎 = −𝑏, 𝑏2 = 0. (15) indeed, 𝔞𝔣𝔣(1) is lsa. furthermore, we can see that the equation (15) satisfies the equation (14). a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 273 remark 2[8]. a lie algebra 𝔤 has structure affines if it satisfies the equations (13) and (14). we mention here that not every lie algebra has left-symmetric algebra structure. for examples, filiform lie algebras 𝔤 of dimension 10 ≤ dim𝔤 ≤ 13 do not have leftsymmetric algebra structure ( [8], [14] ). let 𝔤 be a frobenius lie algebra whose frobenius functional is 𝜎∗. we define an alternating bilinear form 𝐵𝜎∗ as in the equation (7) whose a representation matrix is in the equation (8). for all 𝑎,𝑏,𝑐 ∈ 𝔤, using a product 𝑎𝑏 ≔ 𝑎 ∗𝑏, then we have that the symplectic form 𝐵𝜎∗ induces a left-symmetric algebra structure( [5], [9] ) defined by 𝐵𝜎∗(𝑎𝑏,𝑐):= −𝐵𝜎∗(𝑏,[𝑎,𝑐]) = −〈𝜎 ∗, [𝑏, [𝑎,𝑐]]〉. (16) the non-degeneracy of 𝐵𝜎∗ and the jacobi identity in the equation (4) guarantee that the equation (16) satisfies the equations (13) and (14). in other words, we find that the symplectic form 𝐵𝜎∗ induces a left-symplectic algebra structure. we also observe that the determinant of a representation matrix of 𝐵𝜎∗ is not equal to zero since 𝔤 is the frobenius lie algebra. furthermore, we shall apply the equation (16) to find leftsymmetric algebra structure explicitely. results and discussion in this section, we shall prove our main result as we summarize as follows : proposition 1. let 𝑆 = {𝜀11,𝜀12,𝜀21,𝜀22,𝜀13,𝜀14, 𝜀23, 𝜀24} be a basis for the lie algebra 𝔥2. then 𝔥2 is 8-dimensional frobenius lie algebra whose the frobenius functional 𝜎0 ∗ ≔ 𝜀14 ∗ + 𝜀23 ∗ ∈ 𝔥2 ∗ is considered with respect to the pfaffian of 𝔥2, denoted by pf(𝔥2). its pfaffian pf(𝔥2) is written in the following form : pf(𝔥2) = (𝜀13𝜀24 −𝜀14𝜀23) 2 ∈ s(𝔥2), (17) where s(𝔥2) is a symmetric algebra of degree 4. furthermore, the lie algebra 𝔥2 is leftsymmetric algebra whose formulas are in the following products 𝜀11 2 = −𝜀11 𝜀11𝜀12 = 0 𝜀11𝜀21 = −𝜀21 𝜀11𝜀22 = 0 𝜀11𝜀13 = 𝜀13 𝜀11𝜀14 = 0 𝜀11𝜀23 = 0 𝜀11𝜀24 = −𝜀24 𝜀12𝜀21 = −𝜀22 𝜀12𝜀11 = −𝜀12 𝜀12 2 = 0 𝜀12𝜀14 = −𝜀13 𝜀12𝜀22 = 0 𝜀12𝜀13 = 0 𝜀21 2 = 0 𝜀12𝜀23 = 𝜀13 𝜀12𝜀24 = 𝜀14 −𝜀23 𝜀21𝜀24 = 0 𝜀21𝜀11 = 0 𝜀21𝜀12 = −𝜀11 𝜀22𝜀21 = 0 a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 274 𝜀21𝜀22 = −𝜀21 𝜀21𝜀13 = 𝜀23 −𝜀14 𝜀22𝜀14 = 0 𝜀21𝜀14 = 𝜀24 𝜀21𝜀23 = −𝜀24 𝜀21𝜀24 = 0 𝜀22𝜀11 = 0 𝜀22𝜀12 = −𝜀12 𝜀22𝜀21 = 0 𝜀22 2 = −𝜀22 𝜀22𝜀13 = −𝜀13 𝜀13𝜀21 = −𝜀14 𝜀22𝜀23 = 0 𝜀22𝜀24 = 𝜀24 𝜀13𝜀14 = 0 𝜀13𝜀11 = 0 𝜀13𝜀12 = 0 𝜀14𝜀21 = 0 𝜀13𝜀22 = −𝜀13 𝜀13 2 = 0 𝜀14 2 = 0 𝜀13𝜀23 = 0 𝜀13𝜀24 = 0 𝜀23𝜀21 = −𝜀24 𝜀14𝜀11 = −𝜀14 𝜀14𝜀12 = 0 𝜀23𝜀14 = 0 𝜀14𝜀22 = 0 𝜀14𝜀13 = 0 𝜀24𝜀21 = 0 𝜀14𝜀23 = 0 𝜀14𝜀24 = 0 𝜀24𝜀14 = 0 𝜀23𝜀11 = 0 𝜀23𝜀12 = 0 𝜀23𝜀22 = −𝜀23 𝜀23𝜀13 = 0 𝜀23 2 = 0 𝜀23𝜀24 = 0 𝜀24𝜀11 = −𝜀24 𝜀24𝜀12 = −𝜀23 𝜀24𝜀22 = 0 𝜀24𝜀13 = 0 𝜀24𝜀23 = 0 𝜀24 2 = 0 where the products on 𝔥2 is defined by 𝔥2 ×𝔥2 ∋ (𝜀𝑖𝑗,𝜀𝑘𝑙) ↦ 𝜀𝑖𝑗𝜀𝑘𝑙 ≔ 𝜀𝑖𝑗 ∗𝜀𝑘𝑙 ∈ 𝔥2 . (18) proof. let 𝜀𝑗𝑘 be elements of 𝔥2 ⊂ m4(ℝ) with 1 ≤ 𝑗,𝑘 ≤ 4 corresponding to the form in the equation (1). we let the set 𝑆 = {𝜀11,𝜀12,𝜀21,𝜀22, 𝜀13, 𝜀14,𝜀23,𝜀24} be a basis for 𝔥2. the lie algebra 𝔥2 is endowed with the lie brackets written as the matrix commutator with respect to the basis 𝑆 as follows: [𝜀𝑗𝑘,𝜀𝑖𝑙] = 𝜀𝑗𝑘𝜀𝑖𝑙 −𝜀𝑖𝑙𝜀𝑗𝑘 (19) where 1 ≤ 𝑗,𝑘, 𝑖, 𝑙 ≤ 4. in addition, we give the non-zero brackets for 𝔥2 in the following formulas : [𝜀12,𝜀21] = 𝜀11 −𝜀22 [𝜀11,𝜀21] = −𝜀21, a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 275 [𝜀11,𝜀12] = 𝜀12 [𝜀21,𝜀22] = −𝜀21, [𝜀12,𝜀22] = 𝜀12 [𝜀11,𝜀13] = 𝜀13, [𝜀11,𝜀14] = 𝜀14 [𝜀12,𝜀23] = 𝜀13, [𝜀12,𝜀24] = 𝜀14 [𝜀21,𝜀13] = 𝜀23, [𝜀21,𝜀14] = 𝜀24 [𝜀22,𝜀23] = 𝜀23, [𝜀22,𝜀24] = 𝜀24. (20) firstly, in order to show that 𝔥2 is the frobenius lie algebra, we just compute the pfaffian of the matrix 𝑀𝔥2(ℝ) ≔ ([𝜀𝑘𝑗,𝜀𝑖𝑙])𝑗,𝑘,𝑖,𝑙=1 4 defined before with respect to the basis 𝑆. since we have that the determinant for 𝑀𝔥2(ℝ) in the following form det𝑀𝔥2 (ℝ) = (𝜀13𝜀24 −𝜀14𝜀23) 4, (21) is not equal to zero, then 𝔥2 is the frobenius lie algebra as desired. in the next step, using the theorem 5, we can also see that 𝔥2 is the frobenius lie algebra by constructing a frobenius functional such that a stabilizer of 𝔥2 ata that point is trivial. secondly, to prove that 𝔥2 is left-symmetric algebra, we should find a frobenius functional corresponding the pfaffian for 𝔥2. we observe, using remark 7, we have that the pfaffian of 𝔥2 can be written of the form pf(𝔥2 ) = (𝜀13𝜀24 −𝜀14𝜀23) 2, (22) which is contained in the symmetric algebra 𝑆(𝔥2) of degree 4. since 𝔥2 is frobenius lie algebra, the existence of some frobenius functionals are guaranteed. from the equation (22), we first claim that a frobenius functional is 𝜎𝑚𝑛 ∗ ≔ 𝜀14 ∗ +𝜀23 ∗ which is contained in the dual space 𝔥2 ∗ of vector space 𝔥2. we recalll the value of a linear functional 𝜎𝑚𝑛 ∗ on 𝔥2 is defined by 〈𝜎𝑚𝑛 ∗ , 𝜀𝑗𝑘〉 = 1 for 𝑚 = 𝑗,𝑛 = 𝑘 and 0 otherwise. let 𝑀𝔥2(ℝ)(𝜎 ∗) be the matrix defined in the equation (10). namely, we have the 8×8 matrix 𝑀𝔥2(ℝ)(𝜎 ∗) ≔ (〈𝜎𝑚𝑛 ∗ , [𝜀𝑘𝑗,𝜀𝑖𝑙]〉)𝑗,𝑘,𝑖,𝑙=1 4 which can be seen in the following form 𝑀𝔥2(ℝ)(𝜎𝑚𝑛 ∗ ) = ( 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0) . (23) since the determinant of matrix 𝑀𝔥2(ℝ)(𝜎𝑚𝑛 ∗ ) is equal to 1, then 𝔥2 is again the frobenius lie algebra. in other words, we have shown that our claim is true. thus, 𝜎𝑚𝑛 ∗ is the frobenius functional. indeed, one can also check that the stabilizer of 𝔥2 at this frobenius functional 𝜎𝑚𝑛 ∗ is trivial and this again show that 𝜎𝑚𝑛 ∗ is the frobenius functional. a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 276 corresponding to the frobenius functional 𝜎𝑚𝑛 ∗ , we construct the symplectic linear form 𝐵𝜎𝑚𝑛∗ as defined in the equation (7). we shall show that 𝐵𝜎𝑚𝑛∗ induces leftsymmetric algebra structure in the equations (13) and (14) with respect to the the product 𝔥2 ×𝔥2 ∋ (𝑎,𝑏) ↦ 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝔥2. let 𝑎,𝑏,𝑐, and 𝛼 be elements of 𝔥2. firstly, we shall show that the equation (13). namely, we shall show that (𝑎𝑏)𝑐 −(𝑏𝑎)𝑐 = 𝑎(𝑏𝑐)−𝑏(𝑎𝑐). (24) let us observe that 𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐,𝛼) = 𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐,𝛼)−𝐵𝜎𝑚𝑛∗ ((𝑎𝑏)𝑐,𝛼), = 𝐵𝜎𝑚𝑛∗ (𝑐, [𝑎𝑏,𝛼])−𝐵𝜎𝑚𝑛∗ (𝑐,[𝑏𝑎,𝛼]), = 〈𝜎𝑚𝑛 ∗ , [𝑐, [𝑎𝑏,𝛼]]〉−〈𝜎𝑚𝑛 ∗ , [𝑐, [𝑏𝑎,𝛼]]〉, = 〈𝜎𝑚𝑛 ∗ , [𝑐, [𝑎𝑏,𝛼]]−[𝑐,[𝑏𝑎,𝛼]]〉, = 〈𝜎𝑚𝑛 ∗ , [𝑐, [𝑎𝑏,𝛼] −[𝑏𝑎,𝛼]]〉, = 〈𝜎𝑚𝑛 ∗ , [𝑐, [𝑎𝑏 −𝑏𝑎,𝛼]]〉, = 〈𝜎𝑚𝑛 ∗ , [𝑐, [[𝑎,𝑏],𝛼]]〉, = 𝐵𝜎𝑚𝑛∗ (𝑐,[[𝑎,𝑏],𝛼]). therefore, we get the nice formula as follows : 𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐,𝛼) = 𝐵𝜎𝑚𝑛∗ (𝑐,[[𝑎,𝑏],𝛼]). (25) in the similar way, we have 𝐵𝜎𝑚𝑛∗ (𝑏(𝑎𝑐)−𝑎(𝑏𝑐),𝛼) = 𝐵𝜎𝑚𝑛∗ (𝑐, [[𝑎,𝑏],𝛼]). (26) by the equality of equations (25) and (26), then we have 𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐 +𝑎(𝑏𝑐)−𝑏(𝑎𝑐),𝛼) = 0. (27) but the non-degeneracy of the bilinear form 𝐵𝜎𝑚𝑛∗ guarantees that (𝑏𝑎)𝑐 −(𝑎𝑏)𝑐 + 𝑎(𝑏𝑐)−𝑏(𝑎𝑐) = 0. in other words, the equation (13) holds for all 𝑎,𝑏,𝑐 ∈ 𝔥2. therefore, the symplectic bilinear form 𝐵𝜎𝑚𝑛∗ corresponding to 𝜎𝑚𝑛 ∗ induces the left-symmetric on the frobenius lie algebra 𝔥2. secondly, we shall show that the equation (14) holds. let us observe that 𝐵𝜎𝑚𝑛∗ (𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏],𝛼) = 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝛼)−𝐵𝜎𝑚𝑛∗ (𝑏𝑎,𝛼)−𝐵𝜎𝑚𝑛∗ ([𝑎,𝑏],𝛼), = 𝐵𝜎𝑚𝑛∗ (𝑎, [𝑏,𝛼])−𝐵𝜎𝑚𝑛∗ (𝑏,[𝑎,𝛼])−𝐵𝜎𝑚𝑛∗ ([𝑎,𝑏],𝛼), = 𝐵𝜎𝑚𝑛∗ (𝑎, [𝑏,𝛼])+𝐵𝜎𝑚𝑛∗ (𝑏,[𝛼,𝑎])+𝐵𝜎𝑚𝑛∗ (𝛼,[𝑎,𝑏]), = 〈𝜎𝑚𝑛 ∗ , [𝑎,[𝑏,𝛼]]〉+〈𝜎𝑚𝑛 ∗ , [𝑏, [𝛼,𝑎]]〉+〈𝜎𝑚𝑛 ∗ , [𝛼,[𝑎,𝑏]]〉, = 〈𝜎𝑚𝑛 ∗ , [𝑎,[𝑏,𝛼]]+[𝑏,[𝛼,𝑎]]+[𝛼,[𝑎,𝑏]]〉, a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 277 = 〈𝜎𝑚𝑛 ∗ ,0〉 = 0. thus, we obtain 𝐵𝜎𝑚𝑛∗ (𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏],𝛼) = 0. (28) by the same argument of non-degeneracy of 𝐵𝜎𝑚𝑛∗ , we get 𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏] = 0. thus, the equation (14) holds. we proved that, the alternating bilinear form 𝐵𝜎𝑚𝑛∗ induces the leftsymmetric structure on 𝔥2. more specific, for 𝑎,𝑏 ∈ 𝔥2, there exist scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙 where 1 ≤ 𝑗,𝑘, 𝑖, 𝑙 ≤ 4 such that 𝑎 ≔ ∑ 𝛼𝑘𝑗𝜀𝑘𝑗1≤𝑗,𝑘≤4 , 𝑏 ≔ ∑ 𝛽𝑖𝑙𝜀𝑖𝑙1≤𝑖,𝑙≤4 . (29) moreover, since 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝔥2, then there also exists scalars 𝜑𝑠𝑡 where 1 ≤ 𝑠,𝑡 ≤ 4 such that 𝑎𝑏 ≔ ∑ φ𝑠𝑡𝜀𝑠𝑡1≤𝑠,𝑡≤4 . (30) we shall calculate scalar 𝜑𝑠𝑡 written as products of scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙 where 1 ≤ 𝑗,𝑘, 𝑖, 𝑙,𝑠,𝑡 ≤ 4 satisfying the equationss (13) and (14). applying induced left-symmetric structure from the symplectic form 𝐵𝜎𝑚𝑛∗ corresponding to the frobenius fuctional 𝜎𝑚𝑛 ∗ ≔ 𝜀14 ∗ +𝜀23 ∗ then we get the following products 𝜑𝑠𝑡 = ∑ 𝛼𝑘𝑗𝛽𝑖𝑙.1≤𝑗,𝑘,𝑖,𝑙≤4 (31) therefore, by choosing suitable scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙, we have the products 𝜀𝑗𝑘𝜀𝑖𝑙. indeed, these product satisfies the equationss (13) and (14) because they are induced by simplectic form for 𝔥2. thus, 𝔥2 is left-symmetric algebra. we apply formulas in the equations (16), (29), (30), and (31) to find the explicit left-symmetric structures on 𝔥2. we compute these left-symmetric structures with respect to the basis 𝑆 as mentioned above. then we obtain : 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝜀𝑘𝑗) = −𝐵𝜎𝑚𝑛∗ (𝑏,[𝑎,𝜀𝑘𝑗]), = −〈𝜎𝑚𝑛 ∗ , [𝑏,[𝑎,𝜀𝑘𝑗]]〉, = −〈𝜎𝑚𝑛 ∗ , [∑ 𝛽𝑖𝑙𝜀𝑖𝑙1≤𝑖,𝑙≤4 , [∑ 𝛼𝑘𝑗𝜀𝑘𝑗1≤𝑗,𝑘≤4 , 𝜀𝑘𝑗]]〉, (32) where 𝜀𝑘𝑗 ∈ 𝑆 ⊂ 𝔥2 for 1 ≤ 𝑗,𝑘 ≤ 4. on the other hand, we have the following formulas : 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝜀𝑘𝑗) = 𝐵𝜎𝑚𝑛∗ ( ∑ φ𝑠𝑡𝜀𝑠𝑡 1≤𝑠,𝑡≤4 ,𝜀𝑘𝑗), = 〈𝜎𝑚𝑛 ∗ , [ ∑ φ𝑠𝑡𝜀𝑠𝑡 1≤𝑠,𝑡≤4 , 𝜀𝑘𝑗 ]〉, (33) where 𝜀𝑘𝑗 ∈ 𝑆 ⊂ 𝔥2 for 1 ≤ 𝑗,𝑘 ≤ 4. let 𝜀𝑘𝑗 = 𝜀11, then the equation (32) can be computed with respect the lie brackets in the equation (20) as follows : a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 278 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝜀11) = −〈𝜎𝑚𝑛 ∗ , [𝑏,[𝑎,𝜀𝑘𝑗]]〉, = −𝛽8𝛼2 +𝛽5𝛼3 +𝛽3𝛼5 +𝛽1𝛼6. (34) in the other hand, we obtain from the equation (33) that 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝜀11) = 〈𝜎𝑚𝑛 ∗ , [ ∑ φ𝑠𝑡𝜀𝑠𝑡 1≤𝑠,𝑡≤4 , 𝜀𝑘𝑗 ]〉, = −φ14. (35) therefore, we get φ14 = 𝛽8𝛼2 −𝛽5𝛼3 −𝛽3𝛼5 −𝛽1𝛼6. (36) in the similar way, we obtain the following formulas using the equations (20), (32), and (33) simultaneously : φ11 = −(𝛽1𝛼1 +𝛽2𝛼3), φ12 = −(𝛽1𝛼2 +𝛽2𝛼4), φ21 = −(𝛽3𝛼1 +𝛽4𝛼3), φ22 = −(𝛽3𝛼2 +𝛽4𝛼4), φ13 = 𝛽5𝛼1 −𝛽6𝛼2 +𝛽7𝛼2 −𝛽5𝛼4 −𝛽4𝛼5 −𝛽2𝛼6, φ23 = 𝛽5𝛼3 −𝛽8𝛼2 −𝛽4𝛼7 −𝛽2𝛼8, φ24 = 𝛽6𝛼3 −𝛽8𝛼1 −𝛽7𝛼3 +𝛽8𝛼4 −𝛽3𝛼7 −𝛽1𝛼8. (34) therefore, the following formula 𝑎𝑏 = ( ∑ 𝛼𝑘𝑗𝜀𝑘𝑗 1≤𝑗,𝑘≤4 )( ∑ 𝛽𝑖𝑙𝜀𝑖𝑙 1≤𝑖,𝑙≤4 ) = ∑ φ𝑠𝑡𝜀𝑠𝑡 1≤𝑠,𝑡≤4 is determined by the equation (34). by choosing suitable 𝛼𝑘𝑗 and 𝛽𝑖𝑙 then we have the left-symmetric structure as stated in proposition 1. for example if we fix 𝛼1 = 1 and 𝛽1 = 1, then we have the product 𝜀11 ∗𝜀11 = 𝜀11 2 = −𝜀11. ∎ as discussion, there is a still open problem for a generalization of left-symmetric structure of 𝔥2 to left-symmetric structure of 𝔥𝑛,𝑝:= m𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ). the lie algebra 𝔥𝑛,𝑝 is frobenius lie algebra whenever 𝑝 is factor of 𝑛 [1]. we offered an alternative proof to show 𝔥𝑛,𝑝:= m𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) is frobenius for 𝑛 = 𝑝 = 2 as explained before. since 𝔥𝑛,𝑝 is frobenius lie algebra, then there exists a frobenius functional. therefore, we can construct a symplectic linear form which induces left-symmetric structure for 𝔥𝑛,𝑝. thus, we consider the following conjecture: a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 279 conjecture 1. the lie algebra 𝔥𝑛,𝑝:= m𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) where 𝑝 devides 𝑛 has leftsymmetric structures. these structures are induced by a symplectic form of 𝔥𝑛,𝑝. one of the interesting problems is how to find a frobenius functional for 𝔥𝑛,𝑝 corresponding to the pfaffian of 𝔥𝑛,𝑝 in order to give explicit formulas for left-symmetric structure for 𝔥𝑛,𝑝. furthermore, if the conjecture 12 is true, then the next question is how about leftsymmetric structures for general frobenius lie algebras. we also proved that that all 4dimensional frobenius lie algebra are left-symmetric algebras. as mentioned before that not all lie algebras have left-symmetric structure. but we guess that lie algebras of frobenius types in general have left-symmetric structures. in other words, in more general case we consider the following conjecture conjecture 2. let 𝔤 be a finite dimensional frobenius lie algebra. then 𝔤 is left-symmetric algebra whose left-symmetric structures are induced by a symplectic form corresponding to frobenius functional of 𝔤. it would be interesting to study the completeness of left-symmetric algebra of 𝔥𝑛,𝑝 whenever a frobenius lie algebra is equal to its radical and in general, the completeness of any finite dimensional frobenius lie algebra. conclusions we showed that 𝔥2 ≔ m2(ℝ)⋊𝔤𝔩2(ℝ) is the 8-dimensional frobenius lie algebra. furthermore, we proved the existence of left-symmetric structures on the frobenius lie algebra 𝔥2 and we listed the explicit formulas of left-symmetric structures. therefore, 𝔥2 is left-symmetric algebra. our construction based on the symplectic form corresponding to a frobenius functional of 𝔥2 which induced the left-symmetric structures on 𝔥2. our result can motivate the left-symmetric structure for 𝔥𝑛,𝑝:= m𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) and for general case of frobenius lie algebras. for future research, we stated conjecture 12 and conjecture 13 which are still open problem to be investigated. this is very interesting if conjecture 12 and conjecture 13 are true because we can study the radical of 𝔥𝑛,𝑝, denoted by rad( 𝔥𝑛,𝑝), and if we have 𝔥𝑛,𝑝 = rad( 𝔥𝑛,𝑝), then we come to the notion of a completeness of left-symmetric algebra of 𝔥𝑛,𝑝. moreover, left-symmetric algebras relate affine structures and it is interesting to investigate for the lie groups case[15] . acknowledgments we thank universitas padjadjaran who has funded the work through riset percepatan lektor kepala (rplk) in the year 2021 with the contract number 1959/un6.3.1/pt.00/2021. a left-symmetric structure on the semi-direct sum real frobenius lie algebra of dimension 8 edi kurniadi 280 references [1] m. rais, “la representation du groupe affine,” ann.inst.fourier,grenoble, vol. 26, pp. 207--237, 1978. 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[15] ayala,v, a. da silva, and m. ferreira, “affine and bilinear systems on lie groups,” syst. &control lett., vol. 117, pp. 23--29, 2018. hybrid model of singular spectrum analysis and arima for seasonal time series data cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 302-315 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 01, 2021 reviewed: december 10, 2021 accepted: december 23, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.14136 hybrid model of singular spectrum analysis and arima for seasonal time series data gumgum darmawan1,2,*, dedi rosadi1, budi n ruchjana2 1gadjah mada university, yogyakarta, indonesia 2padjadjaran university, bandung, indonesia *corresponding author email: gumgum.darmawan@gmail.ugm.ac.id*, dedirosadi@gadjahmada.edu, budi.nurani@unpad.ac.id abstract hybrid models between singular spectrum analysis (ssa) and autoregressive integrated moving average (arima) have been developed by several researchers. in the ssa-arima hybrid model, ssa is used in the decomposition and reconstruction process, while forecasting is done through the arima model. in this paper, hybrid ssa-arima uses two auto grouping models. the purpose of this paper is to analyze seasonal data using the ssa-arima hybrid by auto grouping. the first model namely the alexandrov method and the second method is alternative auto grouping with long memory approach. the two hybrid models were tested for two types of seasonal pattern, multiplicative and additive seasonal time series data. the analysis results using both methods give accurate result; as seen from the mape generated the 12 observations for future, the value is below 5%. for additive seasonal pattern, the hybrid ssa-arima method with alexandrov auto grouping is more accurate (mape= 0.13%) than the hybrid ssa-arima method with alternative method but for multiplicative seasonal pattern the hybrid ssa-arima with alternative auto grouping is more accurate (mape = 3.63%) than the hybrid ssa-arima method with alexandrov method. keywords: arima; automatic grouping; long memory effect; seasonal pattern, singular spectrum analysis introduction singular spectrum analysis (ssa) is a relatively new non-parametric method that has proved its capability in various time series types. solving all these problems correspond to the so-called basic capabilities of ssa. besides, the method has several extensions. first, the multivariate version of the method permits the simultaneous expansion of several time series data; see, for example, [1]. second, the ssa ideas lead to several forecasting procedures for time series; see [2]. third, ssa has been utilized for change-point detection in time series. the ssa technique has been used as a filtering method in [3]. fifth, a family of the causality test based on the multivariate ssa technique has been introduced in [4]. sixth, ssa can be applied for missing value imputation [5]. ssa can be applied in various disciplines, from mathematics and physics to economics and financial mathematics, meteorology and oceanography, to social sciences. http://dx.doi.org/10.18860/ca.v7i1.14136 mailto:gumgum.darmawan@gmail.ugm.ac.id mailto:dedirosadi@gadjahmada.edu mailto:budi.nurani@unpad.com hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 303 for instance, in climatology ([6], [7], [8]) and biomedical data time series analysis [9]. hybrid modeling of ssa in time series data has been carried out by many researchers. the hybrid model is carried out so that the advantages of two or more models make a positive contribution to the forecasting results. ssa hybrid model with other time series models includes arima, neural network, arimax, par, varimax, and others. [10], performed hybrid ssa with neural network.[11] perform the hybrid ssa-algorithm firefly-bp neural network process. [12] carried out a hybrid ssa model with armax. [13] combining the ssa model with par(p), this model was applied to wind speed data.[14], built the ssa-varimax hybrid model and used it for climate data. the arima model is often used as a comparison for the ssa model, such as [15], comparing ssa, arima, and other time series models for tourism cases in various countries in europe. the result has indicated that there is no good time series model for all tourism data. [16] compared ssa and arima for predicting ambulance demand. the ssa-arima hybrid model studied by [17] was applied to the annual runoff data. [18], the ssa-arima hybrid model was compared with the basic ssa and arima models. the result showed that the ssa-arima hybrid model was the most accurate. however, many of these papers do not discuss specific data forms (e.g., seasonal patterns), so we consider it necessary to examine this hybrid model for seasonal data. in this study, the ssa and the arima were employed collectively to forecast two types of time series data. both models run to get fast and accurate computation. in ssa, there are two methods of automatic grouping (alexandrov and alternative). the forecasting performance of the hybrid ssa-arima model was compared between the two methods (alternative vs. alexandrov). this paper contributes to the analysis of the seasonal patterns (additive and multiplicative) by the ssa-arima hybrid. the purpose of this paper was to analyze seasonal data using the ssa-arima hybrid by auto grouping for two types of seasonal patterns. this paper was organized as follows: the current section was an introduction where we briefly outlined the use of ssa and introduced our study. in the next section, the methods section, we described the detailed methodology of ssa and arima, briefly outlined forecasting using a linear recurrent formula, identification of fractional differencing parameter, identification of hidden periodicities based on periodogram and automatic grouping on alexandrov method ([19], [20]) also alternative automatic grouping [21]. this section also included a proposed algorithm for automatic hybrid ssaarima. in the results and discussion section, we demonstrated the abilities of hybrid ssaarima in real-time series data. in this part, we also investigated three types of time series data: seasonal with no trend, multiplicative seasonal with the trend, and additive seasonal with the trend. this section also discussed the comparison result between hybrid ssa-arima with the alexandrov method and hybrid ssa-arima with an alternative method for real data analysis. methods singular spectrum analysis the (non-parametric) ssa method has received a fair amount of attention in the literature. the first phase of ssa is the decomposition, where the time series are broken down into four components: trend, seasonal, cyclical, and noise. this phase consists of the embedding and singular value decomposition steps. the second phase, namely the reconstruction phase, consists of grouping and diagonal average process. the hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 304 forecasting process can be done once the four stages have been completed. for the completeness of presentation of our method, we presented the complete phase of the ssa algorithm in the following section. embedding the embedding step will transform one-dimensional time series  1 2 tx = x , x , ....., x into multi-dimensional series 1 2 kx , x , ..., x with vectors 𝑋 = (𝑋𝑖,𝑋𝑖+1,𝑋𝑖+2, . . ,𝑋𝑖+𝐿−1) 𝑇 ∈ 𝑅𝐿 , where 𝑖 = 1,2,…,𝐾, 𝐾 = 𝑇 − 𝐿 + 1. the parameter window length l defines the embedding process, where 2 ≤ 𝐿 ≤ 𝑇 − 1 [22]. if we need to emphasize the size (dimension) of the vectors xi, then we shall call them l-lagged vectors. the l-trajectory matrix (or simply the trajectory matrix) of the series x is defined as 𝑋 = [ 𝑥1 𝑥2 ⋯ 𝑥𝐾 𝑥2 ⋮ 𝑥3 ⋮ … ⋯ 𝑥𝐾+1 ⋮ 𝑥𝐿 𝑥𝐿+1 … 𝑥𝑇 ] (1) the lagged vectors xi are the columns of the trajectory matrix x. both the rows and column of x are sub-series of the original series. the (i,j) element of matrix x is 𝑥𝑖𝑗 = 𝑥𝑖+𝑗−1 which yields that x has equal elements on the ‘antidiagonals’ i+j=const. hence the trajectory matrix is a hankel matrix. singular value decomposition the second step, the svd step, makes the singular value decomposition of the trajectory matrix x and represents it as a sum of rank-one bi-orthogonal elementary matrices. set 𝑆 = 𝑋𝑋𝑇 and denoted by 𝜆1,𝜆2,…,𝜆𝐿the eigenvalues of s taken in the decreasing order of magnitude (𝜆1 ≥ 𝜆2 ≥ ⋯ ≥ 𝜆𝐿 ≥ 0)and by u1, u2,…., ul the orthonormal system of the eigenvectors of the matrix s corresponding to these eigenvalues. 𝑑 = 𝑚𝑎𝑥{𝑖,𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝜆𝑖 > 0} = 𝑟𝑎𝑛𝑘 𝑋 if we denote 𝑉𝑖 = 𝑋𝑇𝑈𝑖 √𝜆𝑖 , then the svd of the trajectory matrix can be written as 𝑋 = 𝑋1 + 𝑋2 + ⋯+ 𝑋𝑑, where eigenvector ui, eigenvalues iλ form matrix 𝑉𝑖 𝑇𝑋. the three elements of svd forming are called eigen triple. grouping the purpose of this step is to appropriately identify the trend, the oscillatory components with different periods and noise. this step can be skipped if one does not want to extract hidden information by regrouping and filtering components precisely. the grouping procedure partitions the set of indices 1,2,…., l into m disjoint subsets 𝐼 = 𝐼1, 𝐼2,…,𝐼𝑚, so the elementary matrix in equation (2) is regrouped into m groups. let 𝐼 = {𝑖1, 𝑖2,…, 𝑖𝑝}. then the resultant matrix xi corresponding to the group i is defined as 𝑋𝑖 = 𝑋𝑖1 + 𝑋𝑖2+.. .+𝑋𝑖𝑝. the matrices are computed for i1, i2,…im, and substituted into equation (2) to obtain the new expansion. the grouping process is the phase when the lxk matrix is grouped into several sub-groups, namely trend patterns, seasonal or periodic, and noise patterns. here, in this paper, the patterns are identified by fourier series analysis and long-memory analysis. fourier series analysis is hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 305 used to identify a seasonal pattern, and long memory series analysis is used to identify the differencing parameter of data. we use the gph method [23] to identify the differencing parameter of time series. diagonal averaging the next step in basic ssa transforms each resultant matrix of the grouped decomposition (3) into a new one-dimensional series of length n and is called diagonal averaging. let y denote a matrix with orde (lxk), with the elements    ijy ,1 i l,1 j k , and define l* = min(l, k), k*=max(l, k), and t=l+k-1. let 𝑦𝑖𝑗 ∗ = 𝑦𝑖𝑗 if l<k and * ij ji y y otherwise. diagonal averaging transfers matrix y into a series 1 2 ny , y , ...., y by the formula. 𝑦𝑘 = { 1 𝑘 ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝑘 𝑚=1 𝑓𝑜𝑟 1 ≤ 𝑘 < 𝐿 1 𝐿 ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝐿∗ 𝑚=1 𝑓𝑜𝑟 𝐿∗ ≤ 𝑘 < 𝐾∗ 1 𝑇 − 𝑘 + 1 ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝑇−𝐾∗+1 𝑚=𝑘−𝐾∗+1 𝑓𝑜𝑟 𝐾∗ < 𝑘 ≤ 𝑇 (2) using equation (3), the resultant matrix ii x , i = 1,2, ..., m will form the series         k k k kty y y y1 2, ,..., ; therefore, the original sequence will be decomposed into the sum of the m series. forecasting using linear recurrent formula the forecasting method used in this paper is the linear recurrent formula (lrf). in the above notation, the recurrent forecasting algorithm (briefly, r-forecasting) can be formulated as follows. 1. the time series 𝑌𝑇+𝑀 = (𝑦1,𝑦2,…𝑦𝑇+𝑀)is defined by 𝑦𝑖 = { �̃�𝑖 𝑓𝑜𝑟 𝑖 = 1,2,…. ,𝑇 ∑ 𝑎𝑗𝑦𝑖−𝑗 𝐿−1 𝑗=1 𝑓𝑜𝑟 𝑖 = 𝑇 + 1,…,𝑇 + 𝑀 (3) 2. the numbers 𝑦𝑇+1,𝑦𝑇+2,…𝑦𝑇+𝑀 form the m terms of the recurrent forecast thus, the r-forecasting is formed by the direct use of the lrf with coefficients {𝑎𝑗, 𝑗 = 1,…,𝐿 − 1}. autoregressive integrating moving average a time series  tx is said to follow the autoregressive integrated moving average model if the differenced that is 𝑊𝑡 = ∇ 𝑑𝑥𝑡then this process is arma processes. if 𝑊𝑡 tw is arma(p,q), then {𝑋𝑡} is arima(p,d,q) process. the mathematic model of arima(p,d,q) is as follow; hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 306 ∅(𝐵)(1 − 𝐵)𝑑(𝑋𝑡 − 𝜇) = 𝜃(𝐵)𝑎𝑡 (4) where t = index of observation d = differencing parameter (1,2,3,..)  = mean of observation 𝑎𝑡~𝐼𝐼𝐷(0,𝜎𝑎 2) ∅(𝐵) = 1 − ∅1𝐵 − ∅2𝐵 2 − ⋯− ∅𝑝𝐵 𝑝 is a polynomial of ar(p) 𝜃(𝐵) = 1 − 𝜃1𝐵 − 𝜃2𝐵 2 − ⋯− 𝜃𝑝𝐵 𝑝 is polynomial of ma(q) identification of fraction differencing parameter the fractional differencing operator is described as a limitless binomial series growth in powers of the backward-shift operator [24]. one of the methods to estimate the value of the fractional differencing parameter is the geweke and porter-hudak method (gph). according to the gph method, the differencing parameter of time series data can be estimated by the least square method of the arfima (autoregressive fractionally integrated moving average) model. the gph procedure can be described as follows. for the arfima model given in (4), let 𝑊𝑡 = (1 − 𝐵) 𝛿𝑍𝑡 and let 𝑓𝑊(𝜔)and 𝑓𝑍(𝜔)be the spectral density function of {𝑊𝑡} and {𝑍𝑡}, respectively. then, 𝑓𝑧(𝜔𝑗) = 𝑓𝑊(𝜔𝑗){2𝑠𝑖𝑛( 𝜔𝑗 2 ⁄ )} −2𝛿 ,𝜔𝑗 ∈ (−𝜋,𝜋) (5) where; 𝑓𝑧(𝜔𝑗) = 𝜎𝑎 2 2𝜋 | 𝜃𝑞(𝑒 −𝑖𝜔𝑗) 𝜙𝑝(𝑒 −𝑖𝜔𝑗) | 2 , 𝜔𝑗 ∈ (−𝜋,𝜋) is the spectral density of a regular arma(p,q) model. ln[𝑓𝑧(𝜔𝑗)] = 𝛿ln{2𝑠𝑖𝑛( 𝜔𝑗 2 ⁄ )} −2 + 𝑙𝑛[𝑓𝑊(𝜔𝑗)], ln[𝑓𝑧(𝜔𝑗)] = 𝑙𝑛[𝑓𝑊(0)] + 𝛿 ln{2𝑠𝑖𝑛( 𝜔𝑗 2 ⁄ )} −2 + 𝑙𝑛[ 𝑓𝑊(𝜔𝑗) 𝑓𝑊(0) ]. replacing 𝜔𝑗 by the fourier frequencies 𝜔𝑗 = 2𝜋𝑗 𝑇 ⁄ and adding 𝑙𝑛[𝐼𝑍(𝜔𝑗)] to both sides, the periodogram{𝑍𝑡} gives ln[𝐼𝑍(𝜔𝑗)] = 𝑙𝑛[𝑓𝑊(0)] + 𝛿ln{2𝑠𝑖𝑛( 𝜔𝑗 2 ⁄ )} −2 + 𝑙𝑛[ 𝐼𝑍(𝜔𝑗) 𝑓𝑍(𝜔𝑗) ] + 𝑙𝑛[ 𝑓𝑊(𝜔𝑗) 𝑓𝑊(0) ]. for 𝜔𝑗 closes to zero, i.e. for 𝑗 = 1,2,…, 𝑇 2⁄ such that 𝑚 𝑇⁄ → ∞ we have 𝑙𝑛[ 𝑓𝑊(𝜔𝑗) 𝑓𝑊(0) ] ≈ 0, thus, 𝑌𝑗 = 𝛽0 + 𝛽1𝑋𝑗 + 𝑒𝑗, 𝑗 = 1,2,…,𝑚 (6) 𝑌𝑗 = 𝑙𝑛[𝐼𝑍(𝜔𝑗)], 𝑋𝑗 = 𝑙𝑛( 1 4𝑠𝑖𝑛2 ( 𝜔𝑗 2 ⁄ ) ) hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 307 �̂�0 = 𝑙𝑛[𝑓𝑊(0)],𝐼𝑍 = 1 2𝜋 [𝛾0 + ∑𝛾𝑡𝑐𝑜𝑠(𝜔𝑗. 𝑡) 𝑇−1 𝑡=0 ] the least-square estimator of long memory coefficient is given by �̂�1 = 𝛿 = ∑ (𝑋𝑗 − �̅�) 𝑚 𝑗=1 (𝑌𝑗 − �̅�) ∑ (𝑋𝑗 − �̅�) 2𝑚 𝑗=1 ;𝑚 = 𝑔(𝑇) = 𝑇𝛼,0 < 𝛼 < 1 (7) identification of hidden periodicities based on periodogram identification of hidden periodicities based on periodogram analysis can be described as follows; 1. fit the data to equation (7): 𝑋𝑡 = ∑(𝑎𝑘𝑐𝑜𝑠𝜔𝑘𝑡 + 𝑏𝑘𝑠𝑖𝑛𝜔𝑘𝑡), [𝑇 2⁄ ] 𝑘=0 (8) where k = 0,1,…,[t/2] and k is fourier frequency determined by 𝜔𝑘 = 2𝜋𝑘 𝑇⁄ . 2. compute ka and kb by (8) and (9) : 𝑎𝑘 = { 1 𝑇 ∑𝑥𝑡𝑐𝑜𝑠𝜔𝑘𝑡 𝑇 𝑡=1 𝑓𝑜𝑟 𝑘 = 0 𝑎𝑛𝑑 𝑘 = 𝑇 2 𝑖𝑓 𝑇 𝑒𝑣𝑒𝑛 2 𝑇 ∑𝑥𝑡𝑐𝑜𝑠𝜔𝑘𝑡 𝑇 𝑡=1 𝑓𝑜𝑟 𝑘 = 1,2,…, (𝑇 − 1) 2 (9) 𝑏𝑡 = 2 𝑇 ∑𝑥𝑡𝑠𝑖𝑛𝜔𝑘𝑡, 𝑇 𝑡=1 𝑘 = 1,2,… (𝑇 − 1) 2 (10) 3. compute the values of ordinate ( )ki  by formula (10) ; 𝐼(𝜔𝑘) = { 𝑇𝑎0 2 𝑘 = 0 𝑇 2 (𝑎𝑘 2 + 𝑏𝑘 2) 𝑘 = 1,2,…,[ 𝑇 − 1 2 ] 𝑇𝑎𝑇 2⁄ 2 𝑘 = [ 𝑇 2 ] (11) 4. test for significance of every fourier frequency as follow; 0 h : 0   (data do not have a seasonal pattern) 1 h : 0  or 0  (data have a seasonal pattern) statistic test: 𝑇 = 𝐼(1)(𝜔(1)) ∑ 𝐼(𝜔𝑘) [𝑇 2⁄ ] 𝑘=1 (12) hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 308 where, (1) (1)i (ω ) : maximum ordinate of periodogram of fourier frequency ki(ω ) : value of periodogram ordinate at k-th’ fourier frequency. test criteria: reject 0h if t >gα with α = significant level. the value of gα can be seen in table fisher [25]. automatic grouping in ssa in this section, we explain two methods ssa automatic grouping based on trend extraction based namely alexandrov, and ssa automatic grouping based on long memory approach, namely alternative method. first, we introduce the periodogram of a time series. let us consider the fourier representation of the elements of a time series 𝑋 = (𝑥1,𝑥2,…,𝑥𝑇)of length t. 𝑋𝑡 = 𝐶0 + ∑ (𝑐𝑘𝑐𝑜𝑠( 2𝜋𝑡𝑘 𝑇⁄ )) + 𝑠𝑘𝑐𝑜𝑠( 2𝜋𝑡𝑘 𝑇⁄ ) + (−1)𝑡𝐶𝑇 2⁄ , 1≤𝑘≤ 𝑁−1 2 (13) where ,  k 0 t t 1 and t 2c = 0if t is an odd number. the periodogram of x at the frequencies   n 2 k=0 ω k n is defined as 𝐼𝑋 𝑁(𝑘 𝑇⁄ ) = 𝑇 2 { 2𝑐0 2 𝑘 = 0 (𝑐𝑘 2 + 𝑠𝑘 2) 0 < 𝑘 < 𝑇 2⁄ 2𝑐𝑇 2⁄ 2 𝑘 = 𝑇 2⁄ ,𝑇 = 𝑒𝑣𝑒𝑛 (14) define the kth element of the discrete fourier transform of x as 𝐹𝑘(𝑋) = ∑ 𝑒 −𝑖2𝜋𝑡𝑘 𝑇⁄ 𝑥𝑡 𝑇−1 𝑡=0 . the periodogram  txi  at the frequencies   2 0 t k k t      can be simplified as 𝐼𝑋 𝑇(𝑘 𝑇⁄ ) = 1 𝑇 { 2|𝐹𝑘(𝑋)| 2 𝑓𝑜𝑟 0 < 𝑘 < 𝑇/2 |𝐹𝑘(𝑋)| 2 𝑓𝑜𝑟 𝑘 = 0 𝑜𝑟 𝑇 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑘 = 𝑇/2 all frequency in the interval (0,0.5) is multiplied by two. this is done to ensure the following property; ‖𝑋‖2 2 = ∑ 𝑥𝑡 2𝑇−1 𝑡=0 = ∑ 𝐼𝑋 𝑁(𝑘 𝑇⁄ ) ⌊𝑇 2⁄ ⌋ 𝑘=0 where 𝜋𝑋 𝑇(𝜔) = ∑ 𝐼𝑇 𝑋(𝑘 𝑇⁄ ),𝜔 ∈ [0,0.5]𝑘:0≤𝑘≤𝜔 is the cumulative contribution of the frequencies 0, ? then, for given  0 0, 0.5  , [19] define the contribution of low frequencies from the interval  00, to t x r as 𝐶(𝑋,𝜔0) = 𝜋𝑋 𝑇(𝜔0) 𝜋𝑋 𝑇(0.5)⁄ (15) then, given parameters  0ω 0,0.5 and  0ω 0,1 ,[19] proposed to select those svd components whose eigenvectors satisfy the following criteria: 𝐶(𝑈𝑗,𝜔0) ≤ 𝐶0 (16) hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 309 where uj is a corresponding jth eigenvector. the second procedure (alternative method) of the automatic grouping algorithm can be described using the following steps: 1. denote the original series as 𝑋 = (𝑥1,𝑥2,…,𝑥𝑇), the univariate time series data with length t 2. define window length (l) 2 ≤ 𝐿 ≤ 𝑇 2⁄ .. 3. determine the shape of the trajectory matrix based on l as in equation (3). 4. find the eigen triple  i i iλ ,v ,u , i = 1,2,….,d (d is the number of positive eigenvalues) of the xxt matrix, where x is the trajectory matrix. 5. identify the ̂ (order difference) of each series ui by equation (6) in section 2.3. 6. make a differencing for all ui based on the value of ̂ (step 5) to obtain the udi series. using this step, then we obtain data with the stationary pattern. 7. specify the maximum value of seasonal order of pi of udi; here, the pi values are between 0 and 24. 8. group udi series based on pi values obtained from the previous step 7, a grouping of udi is based on periods, 3,6,12, and 24, udi with pi=0 is grouped as non-seasonal data, where when pi>24 is considered as cycle pattern. 9. create diagonal averaging for grouping components from step 8; see equation (3). 10. forecasting (linear recurrent method) with equation (4). 11. determine mape values based on the number of our sample data. the mape of the hstep ahead forecast is defined as 𝑀𝐴𝑃𝐸 = ( 1 ℎ ∑| 𝑦𝑛+ℎ − �̂�𝑛+ℎ �̂�𝑛+ℎ | ℎ 𝑖=1 ) × 100% (17) where n denotes the last data used in the in-sample data and ˆ n h y  denotes the forecast values. 12. go to step 2 until all of the l choices have been calculated. the best model will give the smallest mape value. algorithm for hybrid ssa-arima in this section, we proposed our new automatic algorithm on singular spectrum analysis. the procedure of our automatic grouping algorithm can be described using the following steps: 1. denote the original series as 𝑋 = (𝑥1,𝑥2,…,𝑥𝑇), the univariate time series data with length t. 2. define window length (l) 2 ≤ 𝐿 ≤ 𝑇 2⁄ . 3. determine the shape of the trajectory matrix based on l as in equation (3). 4. find the eigen triple  i i iλ ,v ,u , i = 1,2,….,d (d is the number of positive eigenvalues) of the xxt matrix, where x is the trajectory matrix. hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 310 5. use one method of automatic grouping procedures (section 2.4). this procedure takes the best parameter (window length (l)). 6. build model arima of all grouping (rc’s) by auto-arima procedures. 7. forecast a new series of all groups by model arima. 8. sum all results of forecast to get predicted of new time series. 9. determine mape values based on the number of our sample data (17). this procedure is described in figure 1. figure 1. flowchart of hybrid ssa-arima results and discussion for our empirical study, we analyzed airpassengers, and co2 data, available in r package datasets. here, we consider two seasonal data, namely airpassengers, which has a trend and multiplicative seasonal pattern; and co2, which contains an additive seasonal pattern with the trend. hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 311 figure 2. airpassengers figure 3. co2 figure 2 reveals that there has been a gradual increase in several passengers with a multiplicative seasonal pattern from 1949 to 1960. figure 3 shows that there has been a sharp increase in mauna loa atmospheric co2 concentration with an additive seasonal pattern. table 1. grouping and contribution of data co2 rc hybrid-ssa-arima (alexandrov grouping) hybrid-ssa-arima (alternative grouping) principle component (pc) contribution principle component (pc) contribution rc1 pc1,pc2,pc3,pc6 pc7,pc8 99.49% pc1,pc8,pc9,pc16,pc17 pc21,pc23,pc25 98.40% rc2 pc4,pc5,pc12,pc15,pc16 pc17 0.28% pc2,pc3,pc10,pc12,pc13 pc14,pc15,pc18,pc19,pc20 0.93% rc3 pc10,pc11,pc18,pc19 pc22,pc28,pc29 0.07% pc24 0.01% rc4 pc13,pc14,pc26,pc30,pc31 0.06% pc4,pc5,pc6,pc22,pc28 pc31,pc32,pc33,pc34,pc35 0.40% rc5 pc20,pc21,pc23,pc24,pc25 pc27 0.06% pc11,pc26,pc27,pc37,pc39 pc40,pc41 0.06% co2 data had been successfully decomposed into five groups through ssa, each group consisting of 5 or 6 principle components. pc1 to pc31 were pcs that had successfully identified the pattern, while pc32 and so on were pcs with random patterns. rc1 was the largest contribution of 99.49%; this showed that the trend pattern was very dominant in this data. on the right side of this table, co2 data had been successfully decomposed into five groups through ssa; the number of pcs for each rc is uneven. rc3 only consists of 1 pc. pc1 to pc41 were pcs that had successfully identified the pattern, while pc42 and so on were pcs with random patterns. rc1 had the largest contribution of 98.49%; this showed that the trend pattern was very dominant in this data. table 2. separability of co2 rc hybrid-ssa-arima (alexandrov grouping) hybrid-ssa-arima (alternative grouping) rc1 rc2 rc3 rc4 rc5 rc1 rc2 rc3 rc4 rc5 rc1 1 0.021 0.023 0.000 0.006 1 0.010 0.006 0.351* 0.000 rc2 0.021 1 0,000 0,001 0,001 0.010 1 0.000 0.000 0.001 rc3 0.023 0,000 1 0,001 0.001 0.006 0.000 1 0.000 0.002 rc4 0.000 0,001 0,001 1 0.001 0.351* 0.000 0.000 1 -0.002 rc5 0.00 0,001 0.001 0.001 1 0.000 0.001 0.002 -0.002 1 time co 2 1960 1970 1980 1990 3 2 0 3 3 0 3 4 0 3 5 0 3 6 0 time a ir p a s s e n g e rs 1950 1952 1954 1956 1958 1960 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 312 this table shows the correlation values between groups from rc1 to rc5. this best model turned out to not have a very good separability which can be seen from the correlation between rc1 and rc4, which was quite large. however, the correlation values for the others were very small. the right side of this table shows the correlation values between groups from rc1 to rc5. this best model turned out to not have a very good separability. it can be seen from the correlation between rc1 and rc4, which is quite large. however, the correlation values for the others were very small. table 3. grouping and contribution of airpassengers rc hybrid-ssa-arima (alexandrov grouping) hybrid-ssa-arima (alternative grouping) principle component (pc) contribution principle component (pc) contribution rc1 pc1,pc2,pc3,pc6,pc11 pc14,pc15 83.58% pc15,pc19 0.80% rc2 pc4,pc5,pc16,pc17,pc23 pc24,pc29 7.20% pc1,pc2,pc3,pc16,pc17,pc18 pc21,pc22,pc23,pc28,pc29 81.51% rc3 pc9,pc10,pc19,pc20 pc33,pc35,pc36 3.26% pc4,pc5,pc6,pc20,pc24,pc26, pc30 8.54% rc4 pc7,pc8,pc25,pc26,pc30 pc31 3.13% pc7,pc8,pc9,pc10,pc11 pc33,pc34,pc36,pc37 5.98% rc5 pc12,pc13,pc18,pc21,pc22 pc27,pc28,pc34 2.91% pc31,pc32 0.31% airpassengers data had been successfully decomposed into five groups through ssa with each group consisted of 6 to 8 principle components. pc1 to pc36 were pcs that had successfully identified the pattern, while pc37 and so on were pcs with random patterns. rc1 had the largest contribution of 83.589% which indicated that the trend pattern dominated in this data. airpassengers data had been successfully decomposed into five groups via ssa, rc1 and rc5 consisted of 2 pcs, rc2 consists of 11 pcs, rc3 consisted of 7 pcs, and rc4 9pcs. pc1 to pc36 were pcs whose patterns had been identified, while pc38 and so on were pcs with random patterns. rc2 had the most considerable contribution of 81, 51%, which indicated that seasonal patterns dominated in this data. table 4. separability of airpassengers rc hybrid-ssa-arima (alexandrov grouping) hybrid-ssa-arima (alternative grouping) rc1 rc2 rc3 rc4 rc5 rc1 rc2 rc3 rc4 rc5 rc1 1 -0.053 -0.003 0.008 -0.015 1 -0.017 0.000 0.000 0.000 rc2 -0.053 1 0.005 0.001 0.000 -0.017 1 -0.133 0.033 -0.001 rc3 -0.003 0.005 1 -0.003 -0.002 0.000 -0.133 1 0.000 -0.002 rc4 0.008 0.001 -0.003 1 -0.001 0.000 0.033 0.000 1 0.000 rc5 -0.015 0.000 -0.002 -0.001 1 0.000 -0.001 -0.002 0.000 1 this table showed the correlation values between groups from rc1 to rc5. this best model displayed good separability, which could be seen from rc's very small correlation value. the hybrid-ssa-arima (alternative grouping) model showed the correlation value between groups from rc1 to rc5. this best model showed good separability which could be seen from the very small correlation value between rc. hybrid model of singular spectrum analysis and arima for seasonal time series data g.darmawan 313 table 5. mape of two methods from airpassengers dan co2 data hybrid-ssa-arima (alternative grouping) hybrid-ssa-arima (alexandrov-grouping) airpassengers 3.63% 4.64% co2 0.48% 0.13% this table showed that the airpassengers data using the hybrid-ssa-arima (alternative grouping) method was more accurate than the hybrid-ssa-arima (alexandrov grouping) method. the mape value of the hybrid-ssa-arima (alternative grouping) method was 3.63%, while the the value of hybrid-ssa-arima (alexandrov grouping) method was 4.64%. on the co2 data, on the other hand, the hybrid ssa-arima (alexandrov grouping) method was more accurate than the hybrid ssa-arima (alternative grouping) method because the mape value for co2 data using the hybrid-ssa-arima (alexandrov grouping) method was 0.13%, while for the hybrid-ssa-arima (alternative grouping) method, the value was 0.48%. conclusions based on the analysis results, it appeared that the ssa-arima method with alexandrov auto grouping was better for data with additive seasonal patterns such as co2 data. the mape value for this method was 0.13% for co2 data while the ssa-arima method with alternative auto grouping was 0.48% for co2 data. on the other hand, for data with multiplicative seasonal patterns such as airpassengers data, the ssa-arima hybrid method with alternative auto grouping produced more accurate forecasts with the mape value of 3.63%. in comparison, the ssa-arima hybrid method with alexandrov auto grouping containing the same data gave a mape value of 4.64%. in general, the separability value had good separability as the correlation value was close to zero only one significant correlation value is 0.351. the correlation value between rc1 and rc4 for co2 data generated from the ssa-arima hybrid method with alternative auto grouping. as for the contribution, in general, it is dominated by rc1, except for airpassengers data which was analyzed through the ssa-arima auto hybrid method with alternative auto grouping; the contribution is dominated by rc2, amounting to 81.51%. in general, both the ssa-arima with alexadarov auto grouping and the ssaarima with alternative auto grouping provide satisfactory forecast results. the mape value of the two methods was below 5% 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[25] w. wei, time series analysis univariate and multivariate methods. california: addison-wesley publishing company, 2006 the properties of intuitionistic anti fuzzy module t-norm and t-conorm cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 207-219 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 15, 2021 reviewed: december 10, 2021 accepted: february 20, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.13351 the properties of intuitionistic anti fuzzy module t-norm and tconorm ongky denny wijaya, abdul rouf alghofari, noor hidayat, mohamad muslikh* mathematics department brawijaya university, malang, indonesia *corresponding author email: mslk@ub.ac.id* ongkydenny@gmail.com, abdul_rouf@ub.ac.id, noorh@ub.ac.id abstract zadeh have introduced fuzzy set in 1965 and atanassov have introduced intuitionistic fuzzy set in 1986 in theirs paper. now, many of researcher connecting intuitionistic fuzzy set with algebra theory. we interested to combine some concepts in references over intuitionistic fuzzy set, module of a ring, t-norm, t-conorm, and intuitionistic anti fuzzy. in this paper, we discuss about intuitionistic anti fuzzy module t-norm and t-conorm (iafmtc) and their properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. we have investigated and prove all general properties of iafmtc and properties related to module homomorphism, maps, pre-image, and anti-image. keywords: intuitionistic fuzzy set; t-norm; t-conorm; module introduction in 1965, zadeh [1] introduced new concept about fuzzy set. this concept highlighting the membership status of an indeterminate or fuzzy set. in this concept, membership status is defined as a function whose value is in the interval [0, 1]. in 1986, atanassov [2] introduced the notion of intuitionistic fuzzy sets as generalization of fuzzy sets, i.e. highlight membership function and non membership function which the value sum of both functions lies in [0,1]. today, much of the research in the field of algebra was utilised with intuitionistic fuzzy sets on their works. as on the article which written by isaac and john [3] they introduce intuitionistic fuzzy module and their properties. next, rahman and saikia [4] was introduced a concept of t-norms with respect to the intuitionistic fuzzy submodule, then he describes some properties. in other side, rasuli [5] was introduced the concepts and properties of intuitionistic fuzzy vector space with respect to t-norm and t-conorm. in contrast to the results above, sharma in his article [6] introduces a submodule of the intuitionistic anti-fuzzy module. sharma also discusses the concept of intuitionistic antifuzzy modules and the properties it gives rise to. in [7] sharma was discussed about several properties in maps, pre-image, and anti-image in intuitionistic fuzzy module. a function called t-norm is studied on statistical metric [8] and both of t-norm and its dual, i.e. t-conorm is studied on probabilistic metric spaces [9]. before many research in http://dx.doi.org/10.18860/ca.v7i1.13351 mailto:mslk@ub.ac.id mailto:ongkydenny@gmail.com mailto:abdul_rouf@ mailto:noorh@ the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 208 intuitionistic fuzzy sets related with algebra, there are also many research in fuzzy sets related with algebra. some of them are fuzzy modules over a t-norm by rasuli in [10] and anti-fuzzy submodule of a module by sharma in [11]. some properties about homomorphism in group algebraic structure related by intuitionistic fuzzy was discussed by sharma in [12]. motivated by the results of [3-12] in this article, we introduced some relatively new concepts on intuitionistic anti fuzzy module with respect to t-norm and t-conorm (iafmtc) as opposite of intuitionistic fuzzy submodule related by t-norm and t-conorm. this concept combines some concepts from the results mentioned above as the gap of previous research. we investigate general properties on iafmtc and properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. we give also some examples to illustrate the main results in this article. methods we make literature review from [3-12]. then we construct a new structure named intuitionistic anti fuzzy module t-norm and t-conorm by modifying definition intuitionistic anti fuzzy module (iafm) from [6] or intuitionistic fuzzy module w.r.t tnorm and t-conorm (ifmtc) from [4]. we have modified definition of iafm by change minimum value with t-norm and maximum value using t-conorm. also, we have modified definition of ifmtc by change “less than or equal” sign with “greater than or equal” sign. results and discussion in this paper let 𝑅 denotes commutative ring with unity 1. definition of ring see [13] or [14]. now, this is definition of module over 𝑅. definition 1. [14] let 𝑅 be a ring with identity 1. an abelian group 𝑀 said to be a module over 𝑅 (denoted by 𝑅-module) if the map ⋅: 𝑅 × 𝑀 → 𝑀 (𝑟, 𝑚) ↦⋅ (𝑟, 𝑚) = 𝑟𝑚 satisfying the four conditions: (1) 𝑟(𝑚1 + 𝑚2) = 𝑟𝑚1 + 𝑟𝑚2, (2) (𝑟1 + 𝑟2)𝑚 = 𝑟1𝑚 + 𝑟2𝑚, (3) (𝑟1𝑟2)𝑚 = 𝑟1(𝑟2𝑚), (4) 1𝑚 = 𝑚, for all 𝑟, 𝑟1, 𝑟2 ∈ 𝑅 and 𝑚, 𝑚1, 𝑚2 ∈ 𝑀. for the future, module over a ring 𝑅 will be denoted by 𝑅-module. for a submodule of a module have simple criteria as defined as below. definition 2. [14] let 𝑀 be a 𝑅-module and 𝑁 be a non empty subset of 𝑀. 𝑁 is said to be submodule of 𝑀 if two conditions below is satisfied: (1) 𝑁 is a subgroup of 𝑀, i.e. for all 𝑎, 𝑏 ∈ 𝑁 then 𝑎 − 𝑏 ∈ 𝑁. (2) for all 𝑛 ∈ 𝑁 and 𝑟 ∈ 𝑅, 𝑟𝑛 ∈ 𝑁. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 209 the homomorphism maps on the module not much different on the other algebraic structures. for instance, homomorphism in vector spaces (we called it by linear maps, see [15]) is relatively similar with homomorphism maps in module. now, definition of homomorphism maps on the module given in definition 3. definition 3. [14] let 𝑀 and 𝑁 be a 𝑅-module and 𝑓: 𝑀 → 𝑁 is a map. 𝑓 is said to be 𝑅-module homomorphism, if for all 𝑎, 𝑏 ∈ 𝑀 and 𝑟 ∈ 𝑅 satisfy two conditions as follows. (1) 𝑓(𝑎 + 𝑏) = 𝑓(𝑎) + 𝑓(𝑏), (2) 𝑓(𝑟𝑎) = 𝑟𝑓(𝑎). homomorphism 𝑓 is said an epimorphism if 𝑓 surjective, monomorphism if 𝑓 injective, isomorphism if 𝑓 bijective. if domain and codomain of a homomorphism 𝑓 are equal, then we said 𝑓 endomorphism. we called 𝑓 automorphisma, if 𝑓 bijective and endomorphism. zadeh have introduced fuzzy sets as follows. definition 4. [1] let 𝑋 be non empty set. a fuzzy set 𝐴 in 𝑋 is characterized by a membership function 𝜇𝐴 which associates each point in 𝑋 with a real number in the interval [0,1]. in other words, we say the fuzzy sets is a set 𝐴 = {(𝑥, 𝜇𝐴(𝑥))|𝑥 ∈ 𝑋} which 𝜇𝐴: 𝑋 → [0,1]. the value of 𝜇𝐴 at 𝑥, i.e. 𝜇𝐴(𝑥) representing the degree of membership of 𝑥 in 𝐴. atanassov [2] on 1986 have introduced the concept of intuitionistic fuzzy set as follows. this concept is considered as generalization of fuzzy sets. definition 5. [2] let 𝑋 be a non empty set. an intuitionistic fuzzy set (ifs) 𝐴 in 𝑋 is defined as an set of the form 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥))|𝑥 ∈ 𝑋} which 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥) are the functions defined by 𝜇𝐴: 𝑋 → [0,1] and 𝜈𝐴: 𝑋 → [0,1], which 0 ≤ 𝜇𝐴(𝑥) + 𝜈𝐴(𝑥) ≤ 1 for every 𝑥 ∈ 𝑋. the functions 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥) define the degree of membership and degree of nonmembership respectively, for every 𝑥 ∈ 𝑋. the definition of triangular norm was introduced on [8] in 1942. in the beginning, triangular norm used in the study of probabilistic metric spaces. in [9], triangular norm and triangular conorm was studied more at probabilistic metric spaces. in the following, triangular norm and triangular conorm are defined as follows. definition 6. [4, 5] let 𝑇 be a function which defined as 𝑇: [0,1] × [0,1] → [0,1]. 𝑇 is said to be a triangular norm (denoted by t-norm) if for all 𝑥, 𝑦, 𝑧 ∈ [0,1], four axioms are hold: (1) neutral element, i.e. 𝑇(𝑥, 1) = 𝑥, (2) monotonicity, i.e. if 𝑦 ≤ 𝑧 then 𝑇(𝑥, 𝑦) ≤ 𝑇(𝑥, 𝑧), (3) commutativity, i.e. 𝑇(𝑥, 𝑦) = 𝑇(𝑦, 𝑥), (4) associativity, i.e. 𝑇(𝑥, 𝑇(𝑦, 𝑧)) = 𝑇(𝑇(𝑥, 𝑦), 𝑧). the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 210 definition 7. [4, 5] let 𝐶 be a function which defined as 𝐶: [0,1] × [0,1] → [0,1]. 𝑇 is said to be a triangular conorm (denoted by t-conorm) if for all 𝑥, 𝑦, 𝑧 ∈ [0,1], four axioms are hold: (1) neutral element, i.e. 𝐶(𝑥, 0) = 𝑥, (2) monotonicity, i.e. if 𝑦 ≤ 𝑧 then 𝐶(𝑥, 𝑦) ≤ 𝐶(𝑥, 𝑧), (3) commutativity, i.e. 𝐶(𝑥, 𝑦) = 𝐶(𝑦, 𝑥), (4) associativity, i.e. 𝐶(𝑥, 𝐶(𝑦, 𝑧)) = 𝐶(𝐶(𝑥, 𝑦), 𝑧). example 1. [5] the examples of t-norm and t-conorm is given as follows. table 1. the examples of t-norm and t-conorm name t-norm t-conorm standard intersection/ standard union 𝑇𝑚 (𝑥, 𝑦) = min{𝑥, 𝑦} 𝐶𝑚 (𝑥, 𝑦) = max{𝑥, 𝑦} bounded sum 𝑇𝑏 (𝑥, 𝑦) = max{0, 𝑥 + 𝑦 − 1} 𝐶𝑏 (𝑥, 𝑦) = min{1, 𝑥 + 𝑦} algebraic product/ algebraic sum 𝑇𝑝(𝑥, 𝑦) = 𝑥𝑦 𝐶𝑝(𝑥, 𝑦) = 𝑥 + 𝑦 − 𝑥𝑦 drastic 𝑇𝐷 (𝑥, 𝑦) = { 𝑦 if 𝑥 = 1 𝑥 if 𝑦 = 1 0 otherwise 𝐶𝐷(𝑥, 𝑦) = { 𝑦 if 𝑥 = 0 𝑥 if 𝑦 = 0 1 otherwise nilpotent minimum/ nilpotent maximum 𝑇𝑛𝑀 (𝑥, 𝑦) = { min{𝑥, 𝑦} if 𝑥 + 𝑦 > 1 0 otherwise 𝐶𝑛𝑀 (𝑥, 𝑦) = { max{𝑥, 𝑦} if 𝑥 + 𝑦 < 1 1 otherwise hamacher product/ einstein sum 𝑇𝐻0 (𝑥, 𝑦) = { 0 if 𝑥 = 𝑦 = 0 𝑥𝑦 𝑥 + 𝑦 − 𝑥𝑦 otherwise 𝐶𝐻2 (𝑥, 𝑦) = 𝑥 + 𝑦 1 + 𝑥𝑦 definition 8. [5] let 𝑇 be a t-norm and 𝐶 be a t-conorm. 𝑇 is said to be idempotent t-norm and 𝐶 is said to be idempotent t-conorm if for all 𝑥 ∈ [0,1] satisfy 𝑇(𝑥, 𝑥) = 𝑥 and 𝐶(𝑥, 𝑥) = 𝑥 respectively. in [6], there are several corollaries of t-norm and t-conorm as follows. corollary 1. [5] let 𝑇 be a t-norm, then for all 𝑥 ∈ [0,1], (1) 𝑇(𝑥, 0) = 0, (2) 𝑇(0,0) = 0. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 211 corollary 2. [5] let 𝐶 be a t-conorm, then for all 𝑥 ∈ [0,1], (1) 𝐶(𝑥, 1) = 1, (2) 𝐶(0,0) = 0. definition 9. [7] let 𝑋 and 𝑌 are non empty sets, 𝑓: 𝑋 → 𝑌 a maps, 𝐴 = (𝜇𝐴, 𝜈𝐴) and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) are intuitionistic fuzzy sets on 𝑋 and 𝑌 respectively. (1) for all 𝑦 ∈ 𝑌, intuitionistic fuzzy sets 𝑓(𝐴) = {(𝑦, 𝜇𝑓(𝐴)(𝑦), 𝜇𝜈(𝐴)(𝑦)) |𝑦 ∈ 𝑌} which 𝜇𝑓(𝐴)(𝑦) = { max 𝑥∈𝑓−1(𝑦) 𝜇𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 0 𝑓 −1(𝑦) = 0 and 𝜈𝑓(𝐴)(𝑦) = { min 𝑥∈𝑓−1(𝑦) 𝜈𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 1 𝑓 −1(𝑦) = 0 , is called by image of 𝐴 by 𝑓. (2) for all 𝑥 ∈ 𝑋, intuitionistic fuzzy sets 𝑓 −1(𝐵)(𝑥) = (𝜇𝑓−1(𝐵)(𝑥), 𝜈𝑓−1(𝐵)(𝑥)) which 𝜇𝑓−1(𝐵)(𝑥) = 𝜇𝐵 (𝑓(𝑥)) and 𝜈𝑓−1(𝐵)(𝑥) = 𝜈𝐵 (𝑓(𝑥)), is called by pre-image 𝐴 by 𝑓. (3) for all 𝑦 ∈ 𝑌, intuitionistic fuzzy sets 𝑓(𝐴) = {(𝑦, 𝜇�̂�(𝐴)(𝑦), 𝜈�̂�(𝐴)(𝑦)) |𝑦 ∈ 𝑌} which 𝜇�̂�(𝐴)(𝑦) = { min 𝑥∈𝑓−1(𝑦) 𝜇𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 1 𝑓 −1(𝑦) = 0 and 𝜈�̂�(𝐴)(𝑦) = { max 𝑥∈𝑓−1(𝑦) 𝜈𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 0 𝑓 −1(𝑦) = 0 , is called by anti-image of 𝐴 by 𝑓. intuitionistic fuzzy module has discussed before in [3] by isaac and john on 2011. let 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑀, then 𝐴 is called by intuitionistic fuzzy module if satisfy six axioms, i.e. (1) 𝜇𝐴(0) = 1, (2) 𝜇𝐴(𝑥 + 𝑦) ≥ min{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}, (3) 𝜇𝐴(𝑎𝑥) ≥ 𝜇𝐴(𝑥), (4) 𝜈𝐴(0) = 0, (5) 𝜈𝐴(𝑥 + 𝑦) ≤ max{𝜈𝐴(𝑥), 𝜈𝐴(𝑦)}, (6) 𝜈𝐴(𝑎𝑥) ≤ 𝜈𝐴(𝑥), for all 𝑥 ∈ 𝑀 and 𝑎 ∈ 𝑅. intuitionistic anti fuzzy module (abbreviated by iafm) was discussed by sharma in [6]. intuitionistic anti fuzzy module obtained by modifying 𝜇𝐴(0) = 1 by 𝜇𝐴(0) = 0, min by max on second and fifth axiom, sign ≥ by ≤ and vice versa. in this section, we introduce intuitionistic anti fuzzy module t-norm and tconorm (abbreviated by iafmtc), the example, some corollaries, and the properties related by module homomorphism, image, pre-image, and anti-image. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 212 definition 10. let 𝑀 be 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) be intuitionistic fuzzy set on 𝑀. 𝐴 is called by intuitionistic anti fuzzy module t-norm and t-conorm (denoted by 𝐴 ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀)) if six axioms below are holds: (1) 𝜇𝐴(0) = 0, (2) 𝜇𝐴(𝑥 + 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)), (3) 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥), (4) 𝜈𝐴(0) = 1, (5) 𝜈𝐴(𝑥 + 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)), (6) 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥), for all 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. in following, we give example of intuitionistic anti fuzzy module t-norm and tconorm. example 2. let 𝑌 is a set 𝑌 = {1,2}. the power set of 𝑌 is 𝑃(𝑌) = {∅, {1}, {2}, {1,2}}. it is easy to check 𝑃(𝑌) form a ring under the operations ⊕ (symmetric difference) and ∩ (intersection). now, also easy to check that 𝑃(𝑌) is a 𝑃(𝑌)-module. if 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑃(𝑌) which 𝜇𝐴: 𝑃(𝑌) → [0,1] 𝐵 ↦ 𝜇𝐴(𝐵) = { 0 𝐵 = ∅ 0.4 𝐵 = {1} or {2} 0.5 𝐵 = {1,2} and 𝜈𝐴: 𝑃(𝑌) → [0,1] 𝐵 ↦ 𝜈𝐴(𝐵) = { 1 𝐵 = ∅ 0.3 𝐵 = {1} or {2} 0.25 𝐵 = {1,2} then 𝐴 is intuitionistic anti fuzzy module: (a) bounded sum t-norm and bounded sum t-conorm. (b) algebraic product t-norm and algebraic sum t-conorm. proof. to prove it, we must check all conditions on definition 10. ◼ corollary 3. if 𝑀 is 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥), for all 𝑥 ∈ 𝑀. proof. since 𝑀 is 𝑅-module then for all 𝑥 ∈ 𝑀 is satisfy −𝑥 ∈ 𝑀. based on axiom on definition 10, we have 𝜇𝐴(𝑎(−𝑥)) = 𝜇𝐴((−𝑎)𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎(−𝑥)) = 𝜈𝐴((−𝑎)𝑥) ≥ 𝜈𝐴(𝑥). now, choose 𝑎 as multiplication identity on 𝑅, i.e. 𝑎 = 1. we have 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥). ◼ corollary 4. if 𝑀 is 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝐴(𝑥) = 𝐴(−𝑥), for all 𝑥 ∈ 𝑀. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 213 proof. we prove based on axiom on definition 10. consider that 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) = 𝜇𝐴(−(−𝑥)) ≤ 𝜇𝐴(−𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥) = 𝜈𝐴(−(−𝑥)) ≥ 𝜈𝐴(−𝑥). this imply 𝜇𝐴(−𝑥) = 𝜇𝐴(𝑥) dan 𝜈𝐴(−𝑥) = 𝜈𝐴(𝑥). now, we have 𝐴(−𝑥) = (𝜇𝐴(−𝑥), 𝜈𝐴(−𝑥)) = (𝜇𝐴(𝑥), 𝜈𝐴(𝑥)) = 𝐴(𝑥). ◼ next, we give lemma about idempotent t-norm and t-conorm. lemma 1. 𝑇 is a idempotent t-norm if and only if 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. proof. (⇒) take any 𝑥, 𝑦 ∈ [0,1]. if 𝑥 ≤ 𝑦 then 𝑇(𝑥, 𝑦) ≤ 𝑇(𝑥, 1) = 𝑥 = 𝑇(𝑥, 𝑥) ≤ 𝑇(𝑥, 𝑦) such that 𝑇(𝑥, 𝑦) = 𝑥. if 𝑥 ≥ 𝑦 then 𝑇(𝑥, 𝑦) ≥ 𝑇(𝑦, 𝑦) = 𝑦 = 𝑇(1, 𝑦) ≥ 𝑇(𝑥, 𝑦) such that 𝑇(𝑥, 𝑦) = 𝑦. therefore, we have 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦}. (⇐) given that 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. let 𝑥 = 𝑦, we have 𝑇(𝑥, 𝑥) = min{𝑥, 𝑥} = 𝑥. therefore, min{𝑥, 𝑦} is idempotent t-norm. ◼ lemma 2. 𝐶 is a idempotent t-norm if and only if 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. proof. (⇒) take any 𝑥, 𝑦 ∈ [0,1]. if 𝑥 ≥ 𝑦 then 𝐶(𝑥, 𝑦) ≥ 𝐶(𝑥, 0) = 𝑥 = 𝐶(𝑥, 𝑥) ≤ 𝐶(𝑥, 𝑦) such that 𝐶(𝑥, 𝑦) = 𝑥. if 𝑥 ≤ 𝑦 then 𝐶(𝑥, 𝑦) ≤ 𝐶(𝑦, 𝑦) = 𝑦 = 𝐶(0, 𝑦) ≤ 𝐶(𝑥, 𝑦) such that 𝐶(𝑥, 𝑦) = 𝑦. therefore, we have 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦}. (⇐) given that 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. let 𝑥 = 𝑦, we have 𝑇(𝑥, 𝑥) = max{𝑥, 𝑥} = 𝑥. therefore, max{𝑥, 𝑦} is idempotent t-conorm. ◼ corollary 5. if 𝑀 be a 𝑅-module, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀), 𝑇 and 𝐶 are idempotent t-norm and tconorm respectively then 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀(𝑀). (iafm denote intuitionistic anti fuzzy module in [6]). proof. according to lemma 1 and 2, if 𝑇 and 𝐶 are idempotent t-norm and t-conorm respectively then 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} and 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. therefore, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀(𝑀). ◼ remark 1. every intuitionistic anti fuzzy module with respect to idempotent t-norm and idempotent t-conorm is intuitionistic anti fuzzy module. the next main result is about some properties with respect to intuitionistic anti fuzzy module t-norm and t-conorm. theorem 1. if 𝑀 be a 𝑅-module, 𝑎 is a unit on 𝑅, and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝜇𝐴(𝑎𝑥) = 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎𝑥) = 𝜈𝐴(𝑥) for all 𝑥 ∈ 𝑀. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 214 proof. take any 𝑥 ∈ 𝑀 and let 𝑎 is a unit on 𝑅. since 𝑎 is a unit, then there exist 𝑏 ∈ 𝑅 such that 𝑎𝑏 = 𝑏𝑎 = 1. consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 𝜇𝐴(1𝑥) = 𝜇𝐴((𝑎𝑏)𝑥) = 𝜇𝐴((𝑏𝑎)𝑥) = 𝜇𝐴(𝑏(𝑎𝑥)) ≤ 𝜇𝐴(𝑎𝑥) and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 𝜈𝐴(1𝑥) = 𝜈𝐴((𝑎𝑏)𝑥) = 𝜈𝐴((𝑏𝑎)𝑥) = 𝜈𝐴(𝑏(𝑎𝑥)) ≥ 𝜈𝐴(𝑎𝑥). therefore, 𝜇𝐴(𝑎𝑥) = 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎𝑥) = 𝜈𝐴(𝑥) for all 𝑥 ∈ 𝑀. ◼ theorem 2. let 𝑀 be a 𝑅-module, 𝑁 is a submodule of 𝑀, and 𝐴 = (𝜇𝐴, 𝜈𝐴) with degree of membership 𝜇𝐴 and degree of non membership 𝜈𝐴 which defined as 𝜇𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 0 if 𝑥 ∉ 𝑁 and 𝜈𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 0 if 𝑥 ∉ 𝑁 . if 𝐴′ = (𝜇𝐴𝐶 , 𝜈𝐴) is intuitionistic fuzzy set, where 𝜇𝐴𝐶 : 𝑀 → {0,1} is defined by 𝜇𝐴𝐶 (𝑥) = { 1 if 𝑥 ∉ 𝑁 0 if 𝑥 ∈ 𝑁 then 𝐴′ ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). proof. take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. the proof is divided into three cases as follows. (1) first case. if 𝑥, 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝑎𝑥 ∈ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) = 0 ≤ 0 = 𝐶(0,0) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 0 ≤ 0 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) = 1 ≥ 1 = 𝑇(1,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 1 ≥ 1 = 𝜈𝐴(𝑥). (2) second case. if 𝑥 ∉ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁 and 𝑎𝑥 ∉ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) = 1 ≤ 1 = 𝐶(1,0) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 1 ≤ 1 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) = 0 ≥ 0 = 𝑇(0,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 0 ≥ 0 = 𝜈𝐴(𝑥). (3) third case. 𝑥, 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁 or 𝑥 + 𝑦 ∈ 𝑁 and 𝑎𝑥 ∉ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) ≤ 1 = 𝐶(1,1) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 1 ≤ 1 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) ≥ 0 = 𝑇(0,0) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 0 ≥ 0 = 𝜈𝐴(𝑥). therefore, based on three cases proof above, we conclude that 𝐴′ ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 215 theorem 3. let 𝑀 be a 𝑅-module, 𝑁 is a non empty set of 𝑀, and 𝐴 = (𝜇𝐴, 𝜈𝐴) is an intuitionistic fuzzy set which degree of membership 𝜇𝐴: 𝑁 → [0,1] and degree of non membership 𝜈𝐴: 𝑁 → [0,1] defined as 𝜇𝐴(𝑥) = { 0 if 𝑥 ∈ 𝑁 𝑐1 if 𝑥 ∉ 𝑁 and 𝜈𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 𝑐2 if 𝑥 ∉ 𝑁 , for 0 ≤ 𝑐1 ≤ 1, 0 ≤ 𝑐2 ≤ 1, and , 0 ≤ 𝑐1 + 𝑐2 ≤ 1. 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) if and only if 𝑁 is a submodule of 𝑀. proof. (⇒) take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅 which means 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. based on given 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥), we have 𝜇𝐴(0) = 0 and 𝜈𝐴(0) = 1 imply 0 ∈ 𝑁. now, consider that 𝜇𝐴(𝑥 − 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(−𝑦)) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)) = 𝐶(0,0) = 0 and 𝜈𝐴(𝑥 − 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(−𝑦)) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)) = 𝑇(1,1) = 1, which mean 𝑥 − 𝑦 ∈ 𝑁. consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 0 and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 1, which mean 𝑎𝑥 ∈ 𝑁. based on definition 2, 𝑁 is a submodule of 𝑀. (⇐) the proof of 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) directly from definition 10 as follows. take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. (1) obviously, 0 ∈ 𝑁 imply 𝜇𝐴(0) = 0. (2) the proof is divided into three cases as follows. (a) if 𝑥 ∈ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0. thus, 𝜇𝐴(𝑥 + 𝑦) = 0 ≤ 0 = 𝐶(0,0) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (b) if 𝑥 ∈ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁, 𝜇𝐴(𝑥) = 0, dan 𝜇𝐴(𝑦) = 𝑐1. thus, 𝜇𝐴(𝑥 + 𝑦) = 𝑐1 ≤ 𝑐1 = 𝐶(0, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (c) if 𝑥 ∉ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 or 𝑥 + 𝑦 ∉ 𝑁, 𝜇𝐴(𝑥) = 𝑐1, and 𝜇𝐴(𝑦) = 𝑐1. if 𝑥 + 𝑦 ∈ 𝑁 then 𝜇𝐴(𝑥 + 𝑦) = 0 = 𝐶(0,0) ≤ 𝐶(𝑐1, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). if 𝑥 + 𝑦 ∉ 𝑁 then 𝜇𝐴(𝑥 + 𝑦) = 𝑐1 = 𝐶(𝑐1, 0) ≤ 𝐶(𝑐1, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (3) the proof is divided into two cases as follows. (a) if 𝑥 ∈ 𝑁 then 𝜇𝐴(𝑥) = 0 and 𝑎𝑥 ∈ 𝑁. thus, 𝜇𝐴(𝑎𝑥) = 0 ≤ 0 = 𝜇𝐴(𝑥). (b) if 𝑥 ∉ 𝑁 then 𝜇𝐴(𝑥) = 𝑐1 and 𝑎𝑥 ∈ 𝑁 or 𝑎𝑥 ∉ 𝑁. if 𝑎𝑥 ∈ 𝑁 then 𝜇𝐴(𝑎𝑥) = 0 ≤ 𝑐1 = 𝜇𝐴(𝑥). if 𝑎𝑥 ∉ 𝑁 then 𝜇𝐴(𝑎𝑥) = 𝑐1 ≤ 𝑐1 = 𝜇𝐴(𝑥). (4) obviously, 0 ∈ 𝑁 imply 𝜈𝐴(0) = 1. (5) the proof is divided into three cases as follows. (a) if 𝑥 ∈ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. thus, 𝜈𝐴(𝑥 + 𝑦) = 1 ≥ 1 = 𝑇(1,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (b) if 𝑥 ∈ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁, 𝜈𝐴(𝑥) = 1, dan 𝜈𝐴(𝑦) = 𝑐2. thus, 𝜈𝐴(𝑥 + 𝑦) = 𝑐2 ≥ 𝑐2 = 𝑇(1, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 216 (c) if 𝑥 ∉ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 or 𝑥 + 𝑦 ∉ 𝑁, 𝜈𝐴(𝑥) = 𝑐2, and 𝜈𝐴(𝑦) = 𝑐2. if 𝑥 + 𝑦 ∈ 𝑁 then 𝜈𝐴(𝑥 + 𝑦) = 1 = 𝑇(1,1) ≥ 𝑇(𝑐2, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). if 𝑥 + 𝑦 ∉ 𝑁 then 𝜈𝐴(𝑥 + 𝑦) = 𝑐2 = 𝑇(𝑐2, 1) ≥ 𝑇(𝑐2, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (6) the proof is divided into two cases as follows. (a) if 𝑥 ∈ 𝑁 then 𝜈𝐴(𝑥) = 1 and 𝑎𝑥 ∈ 𝑁. thus, 𝜈𝐴(𝑎𝑥) = 1 ≥ 1 = 𝜈𝐴(𝑥). (b) if 𝑥 ∉ 𝑁 then 𝜈𝐴(𝑥) = 𝑐2 and 𝑎𝑥 ∈ 𝑁 or 𝑎𝑥 ∉ 𝑁. if 𝑎𝑥 ∈ 𝑁 then 𝜈𝐴(𝑎𝑥) = 1 ≥ 𝑐2 = 𝜈𝐴(𝑥). if 𝑎𝑥 ∉ 𝑁 then 𝜈𝐴(𝑎𝑥) = 𝑐2 ≥ 𝑐2 = 𝜈𝐴(𝑥). therefore, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ theorem 4. if 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then the set 𝑁 = {𝑥 ∈ 𝑀|𝐴(𝑥) = (0,1)} is a submodule of 𝑀. proof. take any 𝑥, 𝑦 ∈ 𝑁 and 𝑎 ∈ 𝑅. this means 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. now, we want to show 𝑁 is a submodule of 𝑀 as follows. (1) consider that 𝜇𝐴(𝑥 − 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(−𝑦)) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)) = 𝐶(0,0) = 0, 𝜈𝐴(𝑥 − 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(−𝑦)) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)) = 𝑇(1,1) = 1. now, we have 𝜇𝐴(𝑥 − 𝑦) ≤ 0 and 𝜈𝐴(𝑥 − 𝑦) ≥ 1 which means 𝜇𝐴(𝑥 − 𝑦) = 0 and 𝜈𝐴(𝑥 − 𝑦) = 1. thus, 𝐴(𝑥 − 𝑦) = (𝜇𝐴(𝑥 − 𝑦), 𝜈𝐴(𝑥 − 𝑦)) = (0,1) which imply 𝑥 − 𝑦 ∈ 𝑁. (2) consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 0 and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 1 which 𝜇𝐴(𝑎𝑥) = 0 and 𝜈𝐴(𝑎𝑥) = 1 must hold. thus, 𝐴(𝑎𝑥) = (𝜇𝐴(𝑎𝑥), 𝜈𝐴(𝑎𝑥)) = (0,1), which implies 𝑎𝑥 ∈ 𝑁. therefore, 𝑁 is a submodule of 𝑀. ◼ theorem 5. let 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). if 𝐴(𝑥 − 𝑦) = (0,1) then a(x) = 𝐴(𝑦) for all 𝑥, 𝑦 ∈ 𝑀. proof. take any 𝑥, 𝑦 ∈ 𝑀. given 𝜇𝐴(𝑥 − 𝑦) = 0 and 𝜈𝐴(𝑥 − 𝑦) = 1. we will show 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) directly from axiom on the definition 10. consider that 𝜇𝐴(𝑥) = 𝜇𝐴(𝑥 − 𝑦 + 𝑦) ≤ 𝐶(𝜇𝐴(𝑥 − 𝑦), 𝜇𝐴(𝑦)) = 𝐶(0, 𝜇𝐴(𝑦)) = 𝜇𝐴(𝑦) and the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 217 𝜇𝐴(𝑦) = 𝜇𝐴(𝑥 − 𝑥 + 𝑦) = 𝜇𝐴(𝑥 − (𝑥 − 𝑦)) ≤ 𝐶 (𝜇𝐴(𝑥), 𝜇𝐴(−(𝑥 − 𝑦))) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑥 − 𝑦)) = 𝐶(𝜇𝐴(𝑥), 0) = 𝜇𝐴(𝑥). thus, we have 𝜇𝐴(𝑥) ≤ 𝜇𝐴(𝑦) and 𝜇𝐴(𝑦) ≤ 𝜇𝐴(𝑥) or equivalently 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦). now, consider that 𝜈𝐴(𝑥) = 𝜈𝐴(𝑥 − 𝑦 + 𝑦) ≥ 𝑇(𝜈𝐴(𝑥 − 𝑦), 𝜈𝐴(𝑦)) = 𝑇(1, 𝜈𝐴(𝑦)) = 𝜈𝐴(𝑦) and 𝜈𝐴(𝑦) = 𝜈𝐴(𝑥 − 𝑥 + 𝑦) = 𝜈𝐴(𝑥 − (𝑥 − 𝑦)) ≥ 𝑇 (𝜈𝐴(𝑥), 𝜈𝐴(−(𝑥 − 𝑦))) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑥 − 𝑦)) = 𝑇(𝜈𝐴(𝑥), 1) = 𝜈𝐴(𝑥). thus, we have 𝜈𝐴(𝑥) ≤ 𝜈𝐴(𝑦) and 𝜈𝐴(𝑦) ≤ 𝜈𝐴(𝑥) or equivalently 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦). therefore, 𝐴(𝑥) = (𝜇𝐴(𝑥), 𝜈𝐴(𝑥)) = (𝜇𝐴(𝑦), 𝜈𝐴(𝑦)) = 𝐴(𝑦). ◼ in the next theorem, we discuss about properties of intuitionistic anti fuzzy module t-norm and t-conorm with respect to module homomorphism, image, pre-image, and anti-image. theorem 6. let 𝑀 and 𝑁 be a 𝑅-module, 𝛼: 𝑀 → 𝑁 is 𝑅-module homomorphism, and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) is intuitionistic fuzzy set on 𝑁. if 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁) then 𝛼 −1(𝐵) = (𝜇𝛼−1(𝐵), 𝜈𝛼−1(𝐵)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). proof. let identity element of addition on 𝑀 and 𝑁 is denoted by 0𝑀 and 0𝑁 respectively. given 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). we will prove 𝛼 −1(𝐵) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). take any 𝑥1, 𝑥2 ∈ 𝑀 and 𝑎 ∈ 𝑅. consider that (1) 𝜇𝛼−1(𝐵)(0𝑀 ) = 𝜇𝐵 (𝛼(0𝑀)) = 𝜇𝐵 (0𝑁 ) = 0. (2) 𝜇𝛼−1(𝐵)(𝑥1 + 𝑥2) = 𝜇𝐵 (𝛼(𝑥1 + 𝑥2)) = 𝜇𝐵 (𝛼(𝑥1) + 𝛼(𝑥2)) ≤ 𝐶 (𝜇𝐵 (𝛼(𝑥1)), 𝜇𝐵 (𝛼(𝑥2))) = 𝐶 (𝜇𝛼−1(𝐵)(𝑥1), 𝜇𝛼−1(𝐵)(𝑥2)). (3) 𝜇𝛼−1(𝐵)(𝑎𝑥1) = 𝜇𝐵 (𝛼(𝑎𝑥1)) = 𝜇𝐵 (𝑎𝛼(𝑥1)) ≤ 𝜇𝐵 (𝛼(𝑥1)) = 𝜇𝛼−1(𝐵)(𝑥1). (4) 𝜈𝛼−1(𝐵)(0𝑀) = 𝜈𝐵 (𝛼(0𝑀)) = 𝜈𝐵 (0𝑁 ) = 1. (5) 𝜈𝛼−1(𝐵)(𝑥1 + 𝑥2) = 𝜈𝐵 (𝛼(𝑥1 + 𝑥2)) = 𝜈𝐵 (𝛼(𝑥1) + 𝛼(𝑥2)) ≥ 𝑇 (𝜈𝐵 (𝛼(𝑥1)), 𝜈𝐵 (𝛼(𝑥2))) = 𝑇 (𝜈𝛼−1(𝐵)(𝑥1), 𝜈𝛼−1(𝐵)(𝑥2)). (6) 𝜈𝛼−1(𝐵)(𝑎𝑥1) = 𝜈𝐵 (𝛼(𝑎𝑥1)) = 𝜈𝐵 (𝑎𝛼(𝑥1)) ≥ 𝜈𝐵 (𝛼(𝑥1)) = 𝜈𝛼−1(𝐵)(𝑥1). therefore, 𝛼−1(𝐵) = (𝜇𝛼−1(𝐵), 𝜈𝛼−1(𝐵)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ theorem 7. let 𝑀 and 𝑁 be a 𝑅-module, 𝛼: 𝑀 → 𝑁 is 𝑅-module epimorphism, and 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑀. if 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then �̂�(𝐴) = (𝜇�̂�(𝐴), 𝜈�̂�(𝐴)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 218 proof. take any 𝑦1, 𝑦2 ∈ 𝑁 and 𝑎 ∈ 𝑅. given 𝛼 is 𝑅-module epimorphism, i.e. surjective homomorphism, obviously 𝛼−1(𝑦1 + 𝑦2) ≠ ∅. let 𝑦1 = 𝛼(𝑥1) and 𝑦2 = 𝛼(𝑥2), for all 𝑥1, 𝑥2 ∈ 𝑀. consider that (1) 𝜇�̂�(𝐴)(0𝑁 ) = min 𝑥∈𝛼−1(0𝑁) 𝜇𝐴(𝑥) = 𝜇𝐴(0𝑀) = 0. (2) 𝜇�̂�(𝐴)(𝑦1 + 𝑦2) = 𝜇�̂�(𝐴)(𝛼(𝑥1) + 𝛼(𝑥2)) = 𝜇�̂�(𝐴)(𝛼(𝑥1 + 𝑥2)) = min 𝑥∈𝛼−1(𝛼(𝑥1+𝑥2)) 𝜇𝐴(𝑥) = 𝜇𝐴(𝑥1 + 𝑥2) ≤ 𝐶(𝜇𝐴(𝑥1), 𝜇𝐴(𝑥2)) = 𝐶 ( min 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜇𝐴(𝑥), min 𝑥∈𝛼−1(𝛼(𝑥2)) 𝜇𝐴(𝑥) ) = 𝐶 (𝜇�̂�(𝐴)(𝛼(𝑥1)), 𝜇�̂�(𝐴)(𝛼(𝑥2))) = 𝐶 (𝜇�̂�(𝐴)(𝑦1), 𝜇�̂�(𝐴)(𝑦2)). (3) 𝜇�̂�(𝐴)(𝑎𝑦1) = 𝜇�̂�(𝐴)(𝑎𝛼(𝑥1)) = 𝜇�̂�(𝐴)(𝛼(𝑎𝑥1)) = min 𝑥∈𝛼−1(𝛼(𝑎𝑥1)) 𝜇𝐴(𝑥) = 𝜇𝐴(𝑎𝑥1) ≤ 𝜇𝐴(𝑥1) = min 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜇𝐴(𝑥) = 𝜇�̂�(𝐴)(𝛼(𝑥1)) = 𝜇�̂�(𝐴)(𝑦1). (4) 𝜈�̂�(𝐴)(0𝑁 ) = max 𝑥∈𝛼−1(0𝑁) 𝜈𝐴(𝑥) = 𝜈𝐴(0𝑀) = 1. (5) 𝜈�̂�(𝐴)(𝑦1 + 𝑦2) = 𝜈�̂�(𝐴)(𝛼(𝑥1) + 𝛼(𝑥2)) = 𝜈�̂�(𝐴)(𝛼(𝑥1 + 𝑥2)) = max 𝑥∈𝛼−1(𝛼(𝑥1+𝑥2)) 𝜈𝐴(𝑥) = 𝜈𝐴(𝑥1 + 𝑥2) ≥ 𝑇(𝜈𝐴(𝑥1), 𝜈𝐴(𝑥2)) = 𝑇 ( max 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜈𝐴(𝑥), max 𝑥∈𝛼−1(𝛼(𝑥2)) 𝜈𝐴(𝑥) ) = 𝑇 (𝜈�̂�(𝐴)(𝛼(𝑥1)), 𝜈�̂�(𝐴)(𝛼(𝑥2))) = 𝑇 (𝜈�̂�(𝐴)(𝑦1), 𝜈�̂�(𝐴)(𝑦2)). (6) 𝜈�̂�(𝐴)(𝑎𝑦1) = 𝜈�̂�(𝐴)(𝑎𝛼(𝑥1)) = 𝜈�̂�(𝐴)(𝛼(𝑎𝑥1)) = max 𝑥∈𝛼−1(𝛼(𝑎𝑥1)) 𝜈𝐴(𝑥) = 𝜈𝐴(𝑎𝑥1) ≥ 𝜈𝐴(𝑥1) = max 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜈𝐴(𝑥) = 𝜈�̂�(𝐴)(𝛼(𝑥1)) = 𝜈�̂�(𝐴)(𝑦1). therefore, �̂�(𝐴) = (𝜇�̂�(𝐴), 𝜈�̂�(𝐴)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). ◼ conclusions we have define the intuitionistic anti fuzzy module t-norm and t-conorm and investigate their general properties and properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. we give some example, corollary, and the properties. several properties which proved are iafmtc with respect to idempotent t-norm and t-conorm, iafmtc with respect to characteristic function, submodule of a module iafmtc, module homomorphism, maps, pre-image, and antiimage from intuitionistic fuzzy sets. the properties of intuitionistic anti fuzzy module t-norm and t-conorm ongky denny wijaya 219 references [1] l. a. zadeh, "fuzzy sets," journal of information and control, vol. 8, pp. 338-353, 1965 [2] k. t. atanassov, "intuitionistic fuzzy sets," journal of fuzzy sets and systems, vol. 20, pp. 87-96, 1986 [3] p. isaac and p. p. john, on intuitionistic fuzzy submodules of a module, "journal of mathematical sciences and applications", vol. 1, no. 3, pp. 1447-1454, 2011 [4] s. rahman and h. k. saikia, "atanassov's intuitionistic fuzzy submodules with respect to a t-norm", journal of soft computing, vol. 17, pp. 1253-1262, 2013 [5] r. rasuli, "norms over intuitionistic fuzzy vector space", journal of algebra letters, pp. 1-19, 2019 [6] p. k. sharma, "on intuitionistic anti-fuzzy submodule of a module", international journal of fuzzy mathematics and systems, vol. 2, pp. 127-132, 2012 [7] p. k. sharma, "(α,β)-cut of intuitionistic fuzzy modules", international journal of mathematical sciences and applications, vol. 1, no. 3, 2011 [8] k. menger, "statistical metrics", journal of mathematics", vol. 28, pp. 535-537, 1942 [9] b. schweizer and a. sklar, probabilistic metric spaces, dover publication, 2005 [10] r. rasuli, “fuzzy modules over a t-norm”, int. j. open problems compt. math., vol. 9, no. 3, pp. 12-18, 2016 [11] p. k. sharma, "anti fuzzy submodule of a module", advances in fuzzy sets and systems, pp. 1-9, 2012 [12] p. k. sharma, "homomorphism of intuitionistic fuzzy groups", international mathematical forum, vol. 6, no. 64, 3169 3178, 2011 [13] d. s. dummit and r. m. foote, abstract algebra, prentice-hall international, 1990 [14] b. hartley and t. o. hawkes, rings, modules, and linear algebra, chapman and hall, 1983 [15] s. lang, linear algebra second edition, addison-wesley, 1972 analysis of landing airplane queue systems at juanda international airport surabaya cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 49-63 p-issn: 2086-0382; e-issn: 2477-3344 submitted: juni 29, 2021 reviewed: september 08, 2021 accepted: november 05, 2021 doi: https://doi.org/10.18860/ca.v7i1.12772 analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida1, fadilah akbar2, moh. hafiyusholeh3, moh. hartono4 1,2,3department of mathematics, faculty of science and technology, sunan ampel state islamic university surabaya 4department of mechanical engineering, politeknik negeri malang email: yuniar_farida@uinsby.ac.id, fadilakbar783@gmail.com, hafiyusholeh@uinsby.ac.id, moh.hartono@polinema.ac.id abstract before the covid-19 pandemic hit all over the world, juanda international airport was preparing to realize the construction of terminal 3. this construction project impression that juanda airport is experiencing an overload, including in the airplane landing queue. this study aims to analyze the queuing system at the juanda international airport apron before the pandemic, whether effective, quite effective, or less effective in serving the number of existing flights with two terminals. an analysis of the queuing system was conducted in several scenarios. they are in normal/regular condition, a scenario if there is an increase in flight frequency, and a scenario if there is a reduction in aprons’ number because of certain exceptional situations. to analyze the airplane’s landing queue at juanda airport apron, the queuing model (m/m/51): (fcfs/∞/∞) is used. from this model, the results show that in normal conditions, the estimated waiting time for each airplane in the system is 0.18 hours with a queue of 2 up to 3 planes/hour, categorized as effective. in one apron reduction scenario, each airplane’s estimated waiting time in the system is 0.7 hours, with a queue of 6 up to 7 planes categorized as less effective. in the scenario of additional flights, only 9 other flights are allowed every day to keep the service performance still quite effective. by obtaining this results analysis, the decision of pt. angkasa pura 1 (persero) to build terminal 3 is suitable to reduce queuing time and improve juanda international airport services to be more effective. keywords: queueing system model; juanda internasional airport; landing airplane queue; waiting time; multi-channel single-phase introduction juanda international airport surabaya has been named the 3rd busiest airport in indonesia after the first position is soekarno – hatta international airport jakarta. the second position is i gusti ngurah rai international airport denpasar bali. surabaya juanda international airport has also been called the 6th largest airport after padang minangkabau international airport. [1]. due to the high activity of juanda airport every year, as seen in 2020, the average movement of its aircraft has increased by 97.2%, or 13,504 aircraft. it has grown in average passengers by 175% or 1,133,855 passengers, causing juanda international airport surabaya to realize terminal 3. this project aims to increase the number of facilities from existing services. terminal 3 construction is carried https://doi.org/10.18860/ca.v7i1.12772 mailto:yuniar_farida@uinsby.ac.id mailto:fadilakbar783@gmail.com mailto:hafiyusholeh@uinsby.ac.id mailto:moh.hartono@polinema.ac.id analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 50 out to accommodate passengers’ number, which continues to grow every year. if the linkages are connected, the airplane’s more passengers will also increase [2]. an increase in airplane passengers’ number usually increases each route’s flight frequency and opens new destination routes [2]. the increase in flight frequency can cause queuing problems that are getting longer. the impact of long airplane queues will make customers feel less satisfied and the potential for passengers to switch to other modes of transportation so that the airport can decrease the number of passengers for the domestic flight, but for the international flight will give a negative impression to foreign tourists and lead to a reduction in the assessment of airport services ratings in the world [3]. reducing the number of passengers will also be followed by reducing the number of airplanes used, so finally, it impacts decreasing income for pt. angkasa pura 1 (persero) as an organization that manages the juanda international airport. one of the airports’ performances is determined by the airplane’s queue that will park at the apron. with the number of aprons available, if the plane’s number continues to increase, the waiting time for the plane in the queue will also be longer [4]. a situation like this can endanger the plane that will make a landing at the airport, the amount of fuel used by the plane is always adjusted to the flight route, the distance traveled, and the total capacity of the plane filled, so that the plane is only filled with sufficient fuel [5]. the danger of a very long queue in a landing airplane can cause the plane to run out of fuel to make an emergency landing or land at the nearest airport. the flight schedule will also be delayed [6], [7]. the landing plane’s waiting time in the juanda international airport apron can be analyzed using the queuing system analysis concept, a statistical method used to solve a queue problem. a. k. erlang invented this method in 1909 with the research title “probability theory of telephone conversations” [8]. queuing occurs when customers who come to be served to exceed the facilities, resulting in customers waiting for service and a queue appearing [9]. in some cases, services can be improved to avoid growing queues [10]. however, the cost of improving services can cause significant losses because it requires enormous costs if it is not right on target. so that in this study, the performance analysis of the juanda airport apron was carried out using a queuing system model. in queuing at the airport, the queuing system is modeled by assuming aprons are servants while airplanes are considered customers [11], [12]. the apron is used as a parking space for an airplane, where the plane does dropping and picks up passengers, so the apron is considered a servant [13]. meanwhile, an airplane is regarded as a customer because it lands to enter the airport and leave it [14], [15]. the queuing process on an airplane occurs when the plane comes to the airport to make a landing, but with the complete apron condition so that the plane carries out a holding process or rotates in the sky before making a final approach and then landing [16]. holding is when the plane delays carrying out the final approach process and takearound flight near the airport. the plane will land until it arrives at the time the plane lands to land. in contrast, the final approach is when the aircraft has made the final approach to the runway for landing [17], [18]. the final procedure can be carried out when the airport has allowed the plane to land, provided there is an apron available for the plane to park and carry out the process of dropping and raising passengers [19]. if all the apron conditions are still used, the plane that will make the landing will hold it until there is one apron available and land according to the plane’s order [20], [21]. several studies related to the airport’s queuing system, including research conducted by thiagaraj and seshaiah investigated the queuing system at the bengaluru international airport devanahalli, india [22]. in this study, airport facilities considered analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 51 servants are in the form of a runway used as several servers, focusing on the aircraft queuing system. still, this research also models the passenger queuing system at the terminal. therefore, this study aims to model the queuing system to analyze the impact or effect of delays in-service performance and cost losses. however, this study does not explain whether the apron airport’s service performance as a whole is effective, quite effective, or ineffective. the following research was conducted by afsah novita sari, its investigation regarding the plane queuing system model at yogyakarta adi sutjipto international airport [23]. in this study, the data were obtained from apron movement control for 21 days for commercial aircraft landing and taking off. the number of aprons is 11 pieces. that research only makes queuing models without analyzing the queuing system’s performance. the following research was conducted by anggit ratnakusuma et al., related to the queuing system analysis at semarang ahmad yani international airport [24]. the data was obtained from apron movement control for seven days for commercial airplane landing and taking off. the number of aprons used at the airport is six (6) aprons. the data obtained has a poisson distribution. this research only makes queuing models without analyzing the queuing system’s performance. referring to the various studies above, this study will model the queuing system and analyze the existing queuing system’s performance of apron at juanda international airport. an analysis of the queuing system was conducted in several scenarios; they are normal conditions, a scenario if there is an increase in flight frequency, and a scenario if there is a reduction in the number of aprons because of certain exceptional conditions. it is hoped that this research will provide an objective assessment of apron performance as one of the key performance indicators for the service of juanda international airport. finally, the analysis results can be used as input to evaluate the terminal 3 construction project’s feasibility at juanda international airport. methods data collection this research was conducted at juanda international airport by observing apron movement control (amc) from aircraft movement activities. this observation is also carried out based on monitoring from the flightradar 24 application. it aims to observe aircraft holding in the sky near the airport—international juanda surabaya and data correction. the plane holding location is carried out in the sky over the gresik area if it lands on the runway direction 10, as in figure 1. the sky above the eastern sea between java and madura islands lands on the runway at 28, as in figure 2. research time for data collection was conducted for seven consecutive days starting from monday, december 23, 2019, to sunday, december 29, 2019. analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 52 (a) (b) figure 1. flight holding area of (a) runway 10 and (b) runway 28 (source: google maps) the decision to land on runway direction 10 or direction 28 is determined from the wind’s direction that blows in the airport area. aircraft are required to land and take off against the wind blowing. when the wind blows from the east, the plane will land and take off from the runway direction 10, and this event often occurs during the dry season in indonesia. when the wind blows from the west, the plane will land and take off from the runway at 28, and this event often occurs during indonesia’s dry season. data collection was carried out in the apron, amounting to 51 units, observing the queue system, recording the aircraft that landed, then parked at the apron, then took off. the number of planes in and out of the apron is recorded every time with an interval of 1 minute, where each point is observed for 1 hour from 04.30 wib to 22.30 wib. furthermore, the researcher interviewed several air traffic controller employees of airnav indonesia about the airport’s queuing system in general. this interview aims to add references or general knowledge about the queuing system at the studied airport because each airport’s queuing system is different. this difference depends on the condition of the service facilities owned by each airport. queueing system analysis this research was conducted to measure the performance of the queue time for aircraft landing under normal conditions. assuming that security procedures are met, no obstacles can hinder the landing process, such as bad weather, technical airport operations, wind direction changes, and other factors. this aims to limit the research problem so that the discussion can be more focused on time. in carrying out this research, steps are used to run systematically and precisely according to the desired target. the research steps are presented in the flow chart of figure 3. figure 2. research flowchart and data processing from the flowchart in figure 3, some of the equations needed in the calculations to process the data that have been obtained are described as follows: analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 53 1. average rate amount of airplane calculate the average number of the plane landing (𝜆) and the number of the plane taking off (𝜇), by equations (1) and (2) [25]. 𝜆 = average landing airplane time (1) 𝜇 = average takeoff plane time (2) 2. steady-state check performing steady-state checks with equation (3). 𝜌 = 𝜆 𝑆𝜇 < 1 (3) by the number of servers or aprons (𝑆) in use, the steady-state condition (𝜌) is said to be achieved if the conditions of the average number of the plane landing are more significant than the average number of the plane taking off, or it can be written as 𝜆 > 𝜇 or also 𝜌 < 1 [26]. 3. kolmogorov-smirnov distribution fit test perform a distribution fittest. form a distribution on the landing data or take-off data with the kolmogorov-smirnov test using equation (4) with a significant level (α) of 5%. 𝐷 = 𝑆𝑢𝑝|𝑆(𝑥) − 𝐹𝑜 (𝑥)| (4) with, 𝑆𝑢𝑝 : the largest value limit used 𝑆(𝑥) : cumulative distribution of sample data 𝐹0 (𝑥) : the cumulative distribution of the hypothesized data data can be said to be poisson distributed when 𝐷 < 𝐷 ∗ (𝛼). 𝐷 ∗ (𝛼) is obtained from the kolmogorov smirnov table [27]. 4. determining the queue structure there are four types of queuing structures; the selection of these structures can be determined based on the system’s queuing system to be studied. below is an explanation of the four queuing forms [28], [29]. (a) single-channel single-phase this structure is used when there is only one server serving and only one incoming and outgoing path, as shown in figure 4 [30]. figure 3. single-channel single-phase structure analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 54 (b) single-channel multi-phase this structure is used when multiple servers serve consecutively and only have one incoming and outgoing path, as shown in figure 5 [30]. figure 4. single-channel multi-phase structure (c) multi-channel single-phase this structure is used when two or more servers serve simultaneously and only have two or more incoming and outgoing paths, as shown in figure 6 [31]. figure 5. multi-channel single-phase structure this queue structure is suitable for this study. the airplane landing queue in the apron uses a multi-channel single phase, where there are 51 aprons (which represent multichannel), and there is only one queue phase, namely the runaway. (d) multi-channel multi-phase this structure is used when two or more servers serve simultaneously and sequentially and have two or more incoming and outgoing paths, as shown in figure 7 [31]. figure 6. multi-channel multi-phase structure 5. the use of kendall notation kendal notation is a notation that is now the global standard for queuing system models. a series of symbols and slashes describe the queuing process with notation as follows [32], [33]: (𝑎/𝑏/𝑐) ∶ (𝑑/𝑒/𝑓) the symbols 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑓 are the queue model’s essential elements with characteristics shown in table 1. analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 55 table 1. kendall distribution description characteristics notation explanation arrival time distribution (a) and service time distribution (b) 𝑀 markovian (or poisson arrivals or equivalently exponential interarrival or service time distribution) 𝐷 deterministic 𝐸𝑘 erlang 𝑘 type 𝐺𝐼 the general distribution of interarrival time 𝐺 the general distribution of service time number of parallel server (𝑐) 1, 2, … , ∞ finite or infinite queue discipline (𝑑) fcfs the first one comes served first lcfs the last one comes served first siro random selection of services pq priority queue ps processor sharing gd general discipline maximum system capacity (𝑒) 1, 2, … , ∞ finite or infinite size of the calling source (𝑓) 1, 2, … , ∞ finite or infinite this study illustrates the airplane landing queueing system model using (m/m/51): (fcfs/∞/∞) uses poisson arrival with exponential service time distribution. there are 51 aprons as servers, and the queuing discipline is first come first service (fcfs). there are no limited customers on the entire system, and the size of the source from which customers arrive is infinite, too. 6. modeling and calculating the queueing system because this study uses a queuing system model (m/m/51): (fcfs/∞/∞), then to measure the performance of the queuing system using the formula for the model (m/m/s):(fcfs/∞/∞). this model helps determine how to analyze the system appropriately. in most queues found, usually, the data used is poisson distributed; this is because, in the queue, there is a period. here is the explanation [34].  (𝑴/𝑴/𝑺): (𝑭𝑪𝑭𝑺/∞/∞) queue model in this model, the first sign (m) shows the arrival rate with a poisson distribution. the second sign (m) shows the exponential distribution of the service time [35]. queuing model is a poisson distributed service with a server capacity limit of (s) [36]. a reasonably simple model but has several assumptions that need to be fulfilled. this model has more than one number of aprons or servers so that the maximum number of individuals can be served simultaneously by as many servers (s) [37]. in this model, the mean plane landing rate (𝜆) and the mean take-off rate (𝜇) follow the poisson distribution, and the service time ( 1 𝜇 ) follow the exponential distribution. by supposing 𝑟 = 𝜆 / 𝜇 and 𝜌 = 𝑟 / 𝑆, the queuing system model for (m/m/s): (fcfs/∞/∞) is, using equation (5) to equation (9) [38]. 𝑃0 = {[∑ ( 𝜆 𝜇 ) 𝑛 𝑛! 𝑆−1 𝑛=0 ] + ( 𝜆 𝜇 ) 𝑆 𝑆! (1 − 𝜌) } −1 (5) analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 56 𝐿𝑞 = ( 𝜌 ( 𝜆 𝜇 ) 𝑆 𝑆! (1 − 𝜌)2 ) 𝑃0 (6) 𝐿𝑠 = 𝐿𝑞 + 𝜆 𝜇 (7) 𝑊𝑞 = 𝐿𝑞 𝜆 (8) 𝑊𝑠 = 𝑊𝑞 + 1 𝜇 (9) with, 𝑃0 : chance of not queuing 𝐿𝑞 : estimated airplane in the queue (planes) 𝐿𝑠 : estimated airplane in the system (planes) 𝑊𝑞 : estimated waiting time in the queue (hour) 𝑊𝑠 : estimated waiting time in the system (hour) results and discussion an overview of the airplane queuing system model at juanda international airport surabaya in general, the queuing system at juanda international airport surabaya only consists of a single-phase (because the airport only has one runway) with multi-channels (there are 51 active aprons). the illustration of the queuing system will be explained in figure 7. figure 7. simulation of aircraft queueing system model at juanda international airport surabaya a plane can be said to land if the airplane has successfully landed on the runway, then parked on the apron, then dropped the passengers. meanwhile, an airplane can take off when a passenger has boarded the plane, then exits the apron to the runway, and then the plane manages to fly after the wheels take off from the runway. steady-state condition check from the data that has been obtained after making observations for one week, the data will be looked at for the average rate with time intervals of one hour. after obtaining the average rate of the plane landing (𝜆) and taking off (𝜇), then the steady-state value (𝜌) is obtained as follows. the average rate of landing airplanes: 𝜆 = 220 17 = 12.94 airplanes/hour analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 57 the average rate of take-off airplanes: 𝜇 = 188.85 17 = 11.10 airplanes/hour with steady-state size: 𝜌 = 𝜆 𝑆𝜇 = 12.94 51 × 11.10 = 0.02 because 𝜌 = 0.02 < 1, it can be said that the steady-state conditions are met for the queuing system at juanda airport. distribution suitability test with data after the queuing system at juanda international airport can be proven that the steady-state conditions are met, the next step is to test the data distribution’s suitability. the data used will be checked whether the data is included in the poisson distribution or general distribution. test the distribution of the amount landing airplane: 𝐷 = 𝑆𝑢𝑝|𝑆(𝑥) − 𝐹𝑜 (𝑥)| = 0.304 𝐷 ∗ (𝛼) = 0.803 because 𝐷 < 𝐷 ∗ (𝛼) is fulfilled, the data for the number of planes that land is a poisson distribution. test the distribution of the amount take-off airplane: 𝐷 = 𝑆𝑢𝑝|𝑆(𝑥) − 𝐹0(𝑥)| = 0.604 𝐷 ∗ (𝛼) = 1.598 because 𝐷 < 𝐷 ∗ (𝛼) is fulfilled, the data for the number of planes that take off is an exponential distribution. model of the regular queuing system for juanda international airport surabaya following the minister of transportation of the republic of indonesia number pm 89 years 2015, concerning flight delays as described in chapter ii section 3 and chapter v section 9 subsection 1. we can assume the queue time for aircraft at the airport apron is categorized as effective, quite effective, and less effective if it meets the following conditions as shown in table 2 as follows [39], [40]: table 2. performance category category 𝑾𝒒, 𝑾𝒔 (hour) 𝑳𝒒, 𝑳𝒔 (planes) effective < 0,5 < 3 quite effective < 0,5 > 3 0.5 − 1 ≤ 3 less effective 0.5 − 1 > 3 > 1 ≥ 1 after checking the steady-state conditions and testing the distribution of data on the airplane number that landed and the number of airplanes that take-off using the kolmogorov-smirnov compatibility test, it is known that the queuing system model for juanda international airport follows (m/m/51) : (fcfs/∞/∞). the queuing discipline in analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 58 this model includes first come first service (fcfs), where first come first will be served by the number of planes that land and the source of calls has no limit. the distribution test found that the data were poisson distributed with 51 aprons in use. by using the formula that has been presented, the results are obtained as follows. table 3. normal queue system no performance measure result 1 average rate amount of landing airplane (𝜆) 12.94 planes/hour 2 average rate amount of takeoff airplane (𝜇) 11.10 planes/hour 3 chance of not queueing (𝑃0) 31.16 % 4 estimated airplane in queue (𝐿𝑞 ) 1.2 planes 5 estimated airplane in system (𝐿𝑠 ) 2.36 planes 6 estimated waiting time in queue (𝑊𝑞 ) 0.09 hour 7 estimated waiting time in system (𝑊𝑠 ) 0.18 hour the queuing system analysis in table 3, indicates that under normal conditions, the apron queuing system performance in juanda international airport surabaya is categorized as effective. queuing system scenario when there is a reduction in the number of aprons it was assumed that when there is a reduction in the number of aprons used. the duration of time spent serving aircraft landing and taking off was extended, from operating 17 hours a day to 24 hours a day. this is because to serve all flights with a reduced number of aprons, it is not enough to operate for the usual duration. it will take longer to service all scheduled flights. after running the scenario, the reduction of one apron, of which only 50 aprons were left in operation. the simulation results can be used as evaluation material for the management of juanda surabaya international airport if they want to use other aprons for purposes other than commercial plane operational activities. one example is when essential state guest planes, logistics, military, and emergencies land emergency at juanda international airport. table 4. queuing system simulation when there is a reduction in the number of aprons no performance measure result 1 average rate amount of landing airplane (𝜆) 9.16 planes/hour 2 average rate amount of takeoff airplane (𝜇) 7.86 planes/hour 3 chance of not queueing (𝑃0) 31.16 % 4 estimated airplane in queue (𝐿𝑞 ) 5.28 planes 5 estimated airplane in system (𝐿𝑠 ) 6.44 planes 6 estimated waiting time in queue (𝑊𝑞 ) 0.58 hour 7 estimated waiting time in system (𝑊𝑠 ) 0.70 hour the queuing system analysis that simulates the reduction of one apron in table 4 indicates that in the reduced condition of one apron used at juanda international airport surabaya, the queuing system service performance is categorized as less effective. queuing system scenario when there is an increase in flight frequency the scenario of adding a landing or take-off flight schedule every day is carried out as an evaluation material for the management of juanda international airport in serving airlines that want to add new flight route schedules, both domestic and international analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 59 routes. this scenario is carried out to determine the extent to which the tolerance limit for the maximum addition of juanda international airport’s performance in surabaya is still sufficiently effective to turn out to be less effective. thus, the average number of planes landing and taking off within 17 hours of airport operational time is the estimate obtained. therefore, the performance of increasing the number of flights will be measured as follows. increasing the number of flights one by one using software pom qm to obtain the maximum limit that juanda international airport surabaya to maintain reasonably effective service performance. the increase in the number of flights that can be done is only 9 flights. table 5. queuing system when there is increase 9 flight frequency no performance measure result 1 average rate amount of landing airplane (𝜆) 12.41 planes/hour 2 average rate amount of takeoff airplane (𝜇) 10.57 planes/hour 3 chance of not queueing (𝑃0) 30.91 % 4 estimated airplane in queue (𝐿𝑞 ) 4.33 planes 5 estimated airplane in system (𝐿𝑠 ) 5.50 planes 6 estimated waiting time in queue (𝑊𝑞 ) 0.35 hour 7 estimated waiting time in system (𝑊𝑠 ) 0.44 hour the queuing system analysis in table 5 indicates that with the addition of 9 flight frequencies at juanda international airport surabaya, the queuing system service performance is still considered quite effective. to measure service performance, we added back the number of flight frequencies one by one. after the addition of more than 9 flights, the queue system service performance becomes overloaded. the queueing system performance indicated this changed from quite effective to less effective. the time for the aircraft to land becomes more delayed than before. overload at juanda international airport can occur when the additional 10 flight frequencies are added. this overload is obtained from the performance measurement results shown in table 6 as follows. table 6. queueing system when there is increase 10 flights frequency no performance measure result 1 average rate amount of landing airplane (𝜇) 12.35 planes/hour 2 average rate amount of takeoff airplane (𝜇) 10.51 planes/hour 3 chance of not queueing (𝑃0) 30.88 % 4 estimated airplane in queue (𝐿𝑞 ) 7.97 planes 5 estimated airplane in system (𝐿𝑠 ) 9.14 planes 6 estimated waiting time in queue (𝑊𝑞 ) 0.65 hour 7 estimated waiting time in system (𝑊𝑠 ) 0.74 hour the queuing system analysis in table 6 indicates that with 10 flight frequencies at juanda international airport surabaya, the queuing system service performance is categorized as less effective. discussion this research was conducted through observation at juanda international airport surabaya by the end of 2019 when the covid-19 pandemic virus had not yet hit analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 60 indonesia. the flight schedule was running normally; unlike today (2021), there was a pandemic covid-19 virus that made the government make a social restriction policy. this policy caused a decrease drastically in the number of flight frequencies so that the airport was very rarely even had almost no queue. in the data collection during the week conducted before the pandemic, the busy conditions of domestic and international flights at juanda international airport surabaya usually went on average the number of domestic flight arrivals as many as 200.86 flights/day or 201 flights/day and international flights as many as 19.14 flights/day or 20 flights/day. while the number of domestic flights departures is as many as 169.71 flights/day or 170 flights/day, the number of international flights departures is as many as 19.14 flights/day. the number of domestic and international airlines is 14 airlines. the performance of the queuing system in three scenarios was presented in the graph as shown as in figure 9 (the y-axis represents the number of planes) and figure 10 (the y-axis represents the waiting time in an hour): figure 8. comparison graph of queueing system waiting performance figure 9. comparison graph of queueing system time performance from figure 9 and figure 10, we can conclude that under normal conditions, the performance of the queue system is effective but is prone to turning less effective in the following two scenarios:  in the emergency or specific conditions, which must reduce an apron, although just only one apron is reduced, it will potentially decrease juanda international 0 2 4 6 8 10 normal condition reduction one apron increase 9 frequency increase 10 frequency queueing system waiting performance lq ls 0 0,2 0,4 0,6 0,8 normal condition reduction one apron increase 9 frequency increase 10 frequency queueing system time performance wq ws analysis of landing airplane queue systems at juanda international airport surabaya yuniar farida 61 airport’s service performance to be less effective because it will cause long queues.  in the scenario if there were an additional 10 flights. in this scenario, if there is an increase in the number of airplane passengers, then the tolerance limit for the number of different flight routes allowed is only 9 flights. this number is not comparable with the many domestic and international airlines, including 14 airlines. it does not satisfy the airline because, of course, not all additional airline flight routes can be allowed. there will be a strict review to determine which airlines are allowed to add their flight routes. from the analysis above, so the decision of pt. angkasa pura 1 (perero) to build terminal 3 is suitable to improve airport performance quality in serving passengers and airlines. also, the number of passengers can be accommodated a lot so that the airlines will also have more frequent flights, which will increase income for pt. angkasa pura 1 (persero). in the future, research can be developed by examining airports that have more than one runway. because at the airport, other variables can affect the aircraft queuing system, namely the number of runways, so that the servants in the queuing system at the airport are the apron and the runway. conclusions the queuing system model used at juanda international airport is (m/m/51): (fcfs/∞/∞). in normal conditions, the queuing system’s performance at juanda international airport surabaya can be considered effective. but, it was prone to turning into less effective in reducing aprons, although only one apron. the tolerance limit for the number of additional flight routes allowed is only 9 flights. this tolerance limits relatively small compared to the potential increase in the number of airplane passengers every year. so the decision of pt. angkasa pura 1 (persero) to build terminal 3 is suitable to improve airport performance quality in serving passengers and airlines. the development of terminal 3 will make the queuing system’s service performance will be more effective. acknowledgments we would like to express our gratitude and many thanks to pt. angkasa pura 1 (persero) surabaya juanda international airport and airnav indonesia have supported data, information, and references for this research’s convenience. references [1] e. herlambang, “jumlah bandara di indonesia, 20 paling sibuk!,” pariwisata indonesia, 2020. 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[40] a. susetyadi, “waktu tunggu take off dan landing pesawat udara pada runway bandar udara soekarno hatta,” war. penelit. perhub., vol. 24, no. 6, pp. 559–566, 2012. on graceful chromatic number of vertex amalgamation of tree graph family cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 432-444 p-issn: 2086-0382; e-issn: 2477-3344 submitted: june 07, 2022 reviewed: july 24, 2022 accepted: august 13, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.16334 on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana*, ahmad aji , edy wihardjo, deddy setyawan mathematics education department, jember university of jember, indonesia email: arika.fkip@unej.ac.id abstract definition graceful k-coloring of graph 𝐺 = (𝑉,𝐸) is proper vertex coloring 𝑐:𝑉(𝐺) → {1,2,…,𝑘);𝑘 ≥ 2, which induces a proper edge coloring 𝑐′:𝐸(𝐺) → {1,2,…,𝑘 − 1} defined 𝑐′ (𝑢𝑣) = |𝑐(𝑢) − 𝑐(𝑣)|. the minimum vertex coloring from graph 𝐺 can be colored with graceful coloring called a graceful chromatic number with notation 𝜒𝑔 (𝐺). the method used in this research is the axiomatic deductive method and pattern recognition. the detection method finds coloring patterns and graceful chromatic numbers based on the amalgamation operation of point family tree graphs. in this paper, we have investigated the graceful chromatic number of vertex amalgamation of tree graph family: path graph, centipede graph, broom, and e graph. the results of this study are expected to be used as a basis for studies in developing science and applications related to graceful chromatic numbers on the results of point amalgamation operations in tree family graphs. keywords: graceful coloring; tree graph family; graceful chromatic number; vertex amalgamation. introduction graceful coloring is the topic of vertex coloring used in this research. vertex coloring is the assignment of color to a vertex where each neighboring vertex is assigned a different color [1][2][3]. graceful coloring is the coloring of each point that induces the coloring of each sedge by calculating the difference between two neighboring points [4]. the minimum number of colors used in the vertex coloring of graph g is referred to as the chromatic numbers in the graceful coloring of g, which is denoted by 𝜒𝑔(𝐺). research on the chromatic number on graceful coloring has been done before. in 2016, zhang [4] investigated the graceful coloring of path, wheel, complete bipartite, and circle graphs. in 2017, english et. al [5] studied graceful chromatic numbers in tree graphs. furthermore, in 2019, mincu et al [6] investigated the graceful coloring of 13 graphs goldner-harary, fritsch, friendship, fan, wagner, petersen, house prism, octahedron, jellyfish, umbrella, spindle, and diamond graphs. in the same year, alfarisi et al [7] studied graceful coloring on pan, tadpole, and sun graphs. in 2020, sania et al [8] studied graceful staining on a family of uncyclic graphs consisting of bull, cricket, caveman, peach, and flowerpot graphs. furthermore, in 2021, khoirunnisa et al [9] continued their research with the same topic, but the graph used was the graph of the results of the comb operation. http://dx.doi.org/10.18860/ca.v7i3.16334 mailto:arika.fkip@unej.ac.id on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 433 the graph that results from the comb operation is a ladder graph with a first-degree path and a ladder graph with a circle. based on this description, research was conducted to find a particular graph's graceful chromatic number. the graph that will be studied is the result of vertex amalgamation operations in the tree graph family. the family of tree graphs used is path graphs, centipedes, brooms, and 𝐸. this research will use a point amalgamation operation. the vertex amalgamation operation is the acquisition of a new graph from several graphs shown by attaching it to the point that has been selected, which is called a terminal point [10][11][12][13]. the reason for choosing the vertex amalgamation operation is that the basic graph arrangement of the vertex amalgamation cannot change when it is operated compared to the edge amalgamation operation, so the colouring pattern used is no different from the base graph [14]. this article aims to obtain the of a vertex amalgamation of a tree graph family. a tree graph family is a graph that has the same properties as a tree graph. for example, a connected graph with no circuits is called a tree graph [15]. if graph g has the same starting and ending points, then the graph is a circuit graph. path graph, sweep graph, centipede graph, and graph e are some examples of tree graphs [16][17][18]. in order to achieve the objectives of the research in this article, here are some definitions, lemmas, and propositions related to graceful coloring. definition 1. [4] graceful 𝑘-coloring of graph g is proper vertex coloring 𝑐:𝑉(𝐺) → {1,2,…,𝑘}, where 𝑘 ≥ 2 induces proper edge coloring 𝑐′:𝐸(𝐺) → {1,2,…,𝑘 − 1 defined 𝑐′(𝑢𝑣) = |𝑐(𝑢) − 𝑐(𝑣)|. proper vertex coloring c of graph g is graceful coloring if c is graceful k-coloring for 𝑘 ∈ 𝛮. definition 2. [4] the graceful chromatic number of a graph 𝐺, denoted 𝜒𝑔(𝐺), is the minimum 𝑚 where 𝐺 has graceful 𝑚-coloring. definition 3. [10] the amalgamation operation 𝑎𝑚𝑎𝑙(𝐺𝑖,𝑣0𝑖, 𝑡) is a graph operation used to obtain a new graph by gluing all graphs 𝐺𝑖 as much as t at the terminal point (𝑣0𝑖). lemma 1. [4] if 𝐻 is a subgraph of a graph 𝐺, then 𝜒𝑔(𝐺) ≥ 𝜒𝑔(𝐻). lemma 2. [4] it follows that if 𝐺 is a nontrivial connected graph, then 𝜒𝑔(𝐺) ≥ δ(𝐺) + 1, where δ(𝐺) is maximum degree in 𝐺. proposition 1. [4] for each integer 𝑛 ≥ 5, 𝜒𝑔(𝑃𝑛) = 4 methods the method used in this research is the axiomatic deductive method and pattern recognition. the detection method is used to find coloring patterns and graceful chromatic numbers on the result of the amalgamation operation of point family tree graphs. results and discussion in this article, we determine the graceful chromatic numbers in graphs resulting from the amalgamation vertex of family tree graphs. the following are the results of the research presented in four theorems. theorem 1. based on definition 3, given a graph of 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) for 𝑛 ≥ 3, then the graceful coloring chromatic number is 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = { 4, 𝑚 + 1, for 𝑚 = 2 for 𝑚 ≥ 3 . on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 434 proof. the graph 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) has a vertex set 𝑉(𝑎𝑚𝑎𝑙(𝑝𝑛,𝑣,𝑚)) = {𝑥𝑗 𝑖│1 ≤ 𝑖 ≤ 𝑚,1 ≤ 𝑗 ≤ 𝑛 − 1} ∪ {𝑣} and an edge set 𝐸(𝑎𝑚𝑎𝑙(𝑝𝑛,𝑣,𝑚)) = {𝑣𝑥1 𝑖 ,𝑥𝑗𝑥𝑗+1 |1 ≤ 𝑖 ≤ 𝑚,1 ≤ 𝑗 ≤ 𝑛 − 2}. proving graceful chromatic numbers on a graph 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) will be divided into two cases as follows. case 1. 𝑚 = 2 we know that graph 𝑃𝑛 is a subgraph of graph 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚), such that based on lemma 1 and proposition 1 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≥ 𝜒𝑔(𝑃𝑛) = 4 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≥ 4. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≤ 4, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) → {1,2,…,4} as follows 𝑓(𝑣) = { 1, 2, 3, 4, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 1, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑥𝑖 2, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1} for 𝑣 ∈ {𝑥𝑖 1, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑥𝑖 2, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) → {1,2,3}coloring as follows. 𝑓(𝑒) = { 1, 2, for 𝑒 ∈ {𝑉𝑥1 1},{𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 0 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 1 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2} for 𝑒 ∈ {𝑣𝑥1 2},{𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 1 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 0 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2} 𝑓 is a graceful 4−coloring of 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≤ 4, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = 4. case 2. 𝑚 ≥ 3 we know that ∆(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) = 𝑚, such that based on lemma 2 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≥ ∆ (𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) + 1 = 𝑚 + 1 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≥ m + 1. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≤ m + 1, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) → {1,2,…,m + 1} as follows. 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1 for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} , for 𝑣 ∈ {𝑥𝑖 1, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1},{𝑥𝑖 2, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 1, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1},{𝑥𝑖 2, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 𝑚} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) → {1,2,…,m} coloring as follows. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 435 𝑓(𝑒) = { 1, 2, 𝑧, 𝑧 − 1, for 𝑒 ∈ {𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 0 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 1 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2;3 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 1 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 0 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 2);1 ≤ 𝑖 ≤ 𝑛 − 2;3 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,3 ≤ 𝑧 ≤ 𝑚} 𝑓 is a graceful 𝑚 + 1−coloring of 𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) ≤ m + 1, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = m + 1. theorem 2. based on definition 3, given a graph of 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) for 𝑛 ≥ 5, then the graceful coloring chromatic number is 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = { 5, 𝑚 + 1, for 𝑚 ≤ 3 for 𝑚 ≥ 4. proof. the graph 𝑎𝑚𝑎𝑙(c𝑝𝑛,𝑣,𝑚) has a vertex set 𝑉(𝑎𝑚𝑎𝑙(c𝑝𝑛,𝑣,𝑚)) = {𝑥𝑖 𝑧|1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑦𝑖 𝑧|1 ≤ 𝑖 ≤ 𝑛 − 1; 1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑣} and an edge set 𝐸(𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) = {𝑣𝑥1 𝑧|1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 |1 ≤ 𝑧 ≤ 𝑚;1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧|1 ≤ 𝑖 ≤ 𝑛 − 1; 1 ≤ 𝑧 ≤ 𝑚}. proving graceful chromatic numbers on a graph 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) will be divided into two cases as follows. case 1. 𝑚 ≤ 3 we know that graph 𝑃𝑛 is a subgraph of graph 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚), such that based on lemma 1 and proposition 1 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≥ 𝜒𝑔(𝑃𝑛) = 4 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≥ 4. the proper vertex coloring will induce supposed 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) = 4, and it is known that the centipede is a subgraph of the 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚). then graceful coloring can be done on the centipede graph first. the proper side coloring is as follows. given a graph 𝐶𝑝𝑛, according to the definition of graceful coloring of edges 𝑐′(𝑥2𝑥3) ≠ 𝑐′(𝑥3𝑥4) ≠ 𝑐′(𝑥3𝑦2). based on the several possibilities that have been obtained, we can see that the color of edge 3 that is formed can only be formed using a combination colors of vertex 1 and 4. while coloring the color of the next edge 3, we cannot do it. so it can be concluded if 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≥ 5. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≤ 5, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) → {1,2,…,5} as follows 𝑓(𝑣) = { 1, 2, 3, 4, 5, 𝑧 + 1, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛} , for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛},{𝑦1 2} for 𝑣 ∈ {𝑦𝑖 1,1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑦𝑖 2,1 < 𝑖 ≤ 𝑛 − 1},{𝑦𝑖 3,1 ≤ 𝑖 ≤ 𝑛 − 1} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛} for 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 3} on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 436 the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) → {1,2,…,4} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 4, 𝑧, for 𝑒 ∈ {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑥1 2𝑥2 2},{𝑥1 3𝑥2 3},{𝑥2 1𝑦1 1} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3𝑥𝑖+1 3 , 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1},{𝑥2 2𝑦1 2},{𝑥1 1𝑥2 1} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 3}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3𝑥𝑖+1 3 , 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 3} there is graceful m+1−coloring of 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≤ m + 1, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) = m + 1. case 2. for 𝑚 ≥ 4 we know that ∆(𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) = m, such that based on lemma 2 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≥ ∆ (𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) + 1 = 𝑚 + 1 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≥ m + 1. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≤ m + 1, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚)) → {1,2,…,m + 1} as follows. 𝑓(𝑣) = { 1, 2, 3, 4, 5, 𝑧 + 1, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;4 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, , {𝑥𝑖 3, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛},{𝑦1 2} for 𝑣 ∈ {𝑦𝑖 1,1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑦𝑖 2,1 < 𝑖 ≤ 𝑛 − 1}, {𝑦𝑖 𝑧,1 ≤ 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛}, {𝑥𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;4 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛}, {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛;4 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 𝑚} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚)) → {1,2,…,m} coloring as follows. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 437 𝑓(𝑒) = { 1, 2, 3, 4, 𝑧 − 1, 𝑧, for 𝑒 ∈ {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1},{𝑥1 2𝑥2 2},{𝑥1 3𝑥2 3},{𝑥2 1𝑦1 1}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3𝑥𝑖+1 3 , 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1}, {𝑥𝑖+1 3 𝑦𝑖 3, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1},{𝑥2 2𝑦1 2},{𝑥1 1𝑥2 1} {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑖+1 𝑧 𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 ≤ 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 3}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 3}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 3 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 3𝑥𝑖+1 3 , 𝑖 ≡ 1 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1} {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 4);1 < 𝑖 ≤ 𝑛 − 1;4 ≤ 𝑧 ≤ 𝑚}, for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,4 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} there is graceful 𝑚 + 1−coloring of 𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) ≤ m + 1, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝐶𝑝𝑛,𝑣,𝑚) = m + 1. theorem 3. based on definition 3, given a graph of 𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) for 𝑛 ≥ 4 and 𝑘 − 𝑛 ≥ 3, then the graceful coloring chromatic number is 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) = { 𝑘 − 𝑛 + 2, 𝑚 + 1, for 𝑚 + 𝑛 < 𝑘 + 1 for 𝑚 + 𝑛 ≥ 𝑘 + 1. proof. the graph 𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) has a vertex set 𝑉(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) = {𝑥𝑖 𝑧,𝑦𝑗 𝑧|1 ≤ 𝑖 ≤ 𝑛 − 1,1 ≤ 𝑗 ≤ 𝑘 − 𝑛,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑣} and an edge set 𝐸(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) = {𝑣𝑥1 𝑧,𝑥𝑖 𝑧𝑥𝑖+1 𝑧 ,𝑥𝑖+1 𝑧 𝑦𝑗 𝑧|1 ≤ 𝑖 ≤ 𝑛 − 2,1 ≤ 𝑗 ≤ 𝑘 − 𝑛,1 ≤ 𝑧 ≤ 𝑚}. proving graceful chromatic numbers on a graph 𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) will be divided into two cases as follows. case 1. for 𝑚 + 𝑛 < 𝑘 + 1 we know that ∆(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) = 𝑘 − 𝑛 + 1, such that based on lemma 2 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≥ ∆ (𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) + 1 = 𝑘 − 𝑛 + 2 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≥ 𝑘 − n + 2. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≤ k − n + 2, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) → {1,2,…,k − n + 2} as follows. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 438 subcase 1. 𝑛 ≡ 1 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, 𝑗 + 2, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚} untuk 𝑣 ∈ {𝑦2 𝑧,1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;3 ≤ 𝑧 ≤ 𝑚}, untuk 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑦𝑗 𝑧,3 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 2}, {𝑦𝑗 𝑧,1 ≤ 𝑗 ≤ 𝑘 − 𝑛;3 ≤ 𝑧 ≤ 𝑚} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) → {1,2,…,k − n + 1} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 1, 𝑗 + 1, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;3 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑛−1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑛−1 2 𝑦2 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥1 1𝑥2 1} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;3 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−1 𝑧 𝑦𝑗 𝑧,3 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 2} {𝑥𝑛−1 𝑧 𝑦𝑗 𝑧,1 ≤ 𝑗 ≤ 𝑘 − 𝑛;3 ≤ 𝑧 ≤ 𝑚} subcase 2. for 𝑛 ≡ 2 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, 𝑗 + 1, 𝑘 − 𝑛 + 2, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥2 3},{𝑥2 𝑧,5 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 2 and 𝑧 = 4}, {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 𝑛 − 2;𝑧 = 3 dan 5 ≤ 𝑧 ≤ 𝑚}, for 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑦𝑗 𝑧,1 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑛−1 𝑧 ,1 ≤ 𝑧 ≤ 𝑚} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) → {1,2,…,k − n + 1} coloring as follows. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 439 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 2 𝑘 − 𝑛 + 1, 𝑘 − 𝑛 − 𝑗 + 1, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 4} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥2 𝑧𝑥3 𝑧,5 ≤ 𝑧 ≤ 𝑚},{𝑥2 3𝑥3 3},{𝑥1 1𝑥2 1} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥2 1𝑥3 1},{𝑥2 2𝑥3 2},{𝑥2 4𝑥3 4} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,5 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−2 𝑧 𝑥𝑛−1 𝑧 ,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−1 𝑧 𝑦𝑗 𝑧,1 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 𝑚} subcase 3. for 𝑛 ≡ 0 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, 𝑗 + 1, 𝑘 − 𝑛 + 2, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥2 3},{𝑥2 𝑧,5 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 2;1 ≤ 𝑧 ≤ 2 dan 𝑧 = 4}, {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 𝑛 − 2;𝑧 = 3 dan 5 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑦𝑗 𝑧,2 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑛−1 𝑧 ,1 ≤ 𝑧 ≤ 𝑚} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) → {1,2,…,k − n + 1} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 2 𝑘 − 𝑛 𝑘 − 𝑛 + 1, 𝑘 − 𝑛 − 𝑗 + 1, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 4} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥2 𝑧𝑥3 𝑧,5 ≤ 𝑧 ≤ 𝑚},{𝑥2 3𝑥3 3},{𝑥1 1𝑥2 1} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 𝑛 − 3;1 ≤ 𝑧 ≤ 𝑚}, {𝑥2 1𝑥3 1},{𝑥2 2𝑥3 2},{𝑥2 4𝑥3 4} for 𝑒 ∈ {𝑉𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,5 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−2 𝑧 𝑥𝑛−1 𝑧 ,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑛−1 𝑧 𝑦𝑗 𝑧,2 ≤ 𝑗 ≤ 𝑘 − 𝑛;1 ≤ 𝑧 ≤ 𝑚} there is graceful 𝑘 − 𝑛 − 2−coloring of 𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≤ 𝑘 − 𝑛 + 2, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) = 𝑘 − 𝑛 + 2. case 2. for 𝑚 + 𝑛 ≥ 𝑘 + 1 we know that ∆(𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) = 𝑚, such that based on lemma 2 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≥ ∆ (𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚)) + 1 = m + 1 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≥ 𝑚 + 1. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≤ m + 1 as same case 1. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 440 theorem 4. based on definition 3, given a graph of 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) for 𝑛 ≥ 2, then the graceful coloring chromatic number is 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) = { 4, 𝑚 + 1, for 𝑚 = 2 for 𝑚 ≥ 3 proof. the graph 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) has a vertex set 𝑉(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) = {𝑥𝑖 𝑧|1 ≤ 𝑖 ≤ 2𝑛 + 2,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑦𝑖 𝑧│1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑣} and an edge set 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) = {𝑣𝑥1 𝑧|1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 |1 ≤ 𝑖 ≤ 2𝑛 + 1,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 |1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑧 ≤ 𝑚} ∪ {𝑥𝑛+1 𝑧 𝑦1 𝑧|1 ≤ 𝑧 ≤ 𝑚}. proving graceful chromatic numbers on a graph 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) will be divided into two cases as follows. case 1. for 𝑚 = 2 we know that graph 𝑃𝑛 is a subgraph of graph 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚), such that based on lemma 1 and proposition 1 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≥ 𝜒𝑔(𝑃𝑛) = 4 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≥ 4. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) ≤ k − n + 2, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,…,4} as follows. subcase 1. for 𝑛 ≡ 1 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);3 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2}, {𝑥3 2},{𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 2},{𝑥1 1} for 𝑣 ∈ {𝑥3 1},{𝑦1 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥1 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2 }, {𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 2} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);3 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑛+1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥2 1𝑥3 1},{𝑥1 2𝑥2 2},{𝑥3 2𝑥4 2},{𝑣𝑥1 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑦1 𝑧𝑦2 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥2 2𝑥3 2},{𝑥1 1𝑥2 1},{𝑥3 1𝑥4 1},{𝑣𝑥1 2} {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} subcase 2. for 𝑛 ≡ 2 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2}, {𝑥1 1},{𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 2} for 𝑣 ∈ {𝑥1 2},{𝑦1 𝑧,1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2 }, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 2} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3} coloring as follows. on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 441 𝑓(𝑒) = { 1, 2, 3, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑦1 𝑧𝑦2 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥1 2𝑥2 2},{𝑣𝑥1 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0(𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑛+1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 2},{𝑥1 1𝑥2 1},{𝑣𝑥1 2} {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} subcase 3. 𝑛 ≡ 0 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 2, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2}, {𝑥1 1},{𝑦1 1},{𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 2} for 𝑣 ∈ {𝑥𝑖 1, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2},{𝑦1 2},{𝑥1 2} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2 }, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 2} the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3} coloring as follows. (𝑒) = { 1, 2, 3, for 𝑒 ∈ {𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1}, {𝑥1 2𝑥2 2},{𝑦1 2𝑦1 2},{𝑣𝑥1 1},{𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0(𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 2} for 𝑒 ∈ {𝑥𝑖 1𝑥𝑖+1 1 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1}, {𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1},{𝑥𝑛+1 2 𝑦1 2},{𝑣𝑥1 2}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2},{𝑦2 1𝑦2 2} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 2} there is graceful 4−coloring of 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≤ 4, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) = 4. case 2. for 𝑚 ≥ 3 we know that ∆(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) = m, such that based on lemma 2 we get 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≥ ∆ (𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) + 1 = 𝑚 + 1 or 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≥ m + 1. furthermore, we prove that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≤ m + 1, define a proper vertex coloring 𝑐 ∶ 𝑉(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,…,m + 1} as follows. subcase 1. for 𝑛 ≡ 1 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);3 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑥3 2},{𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 𝑚},{𝑥2 𝑧;3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥3 1},{𝑥3 𝑧;3 ≤ 𝑧 ≤ 𝑚},{𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚 }, {𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 𝑚},{𝑥2 1},𝑥2 2} for 𝑣 ∈ {𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚 } on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 442 the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3,…,m} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 1, untuk 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);3 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑛+1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚},{𝑥2 1𝑥3 1},{𝑥3 2𝑥4 2},{𝑥2 𝑧𝑥3 𝑧,3 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚} untuk 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);2 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑦1 𝑧𝑦2 𝑧,1 ≤ 𝑧 ≤ 𝑚},{𝑥2 2𝑥3 2},{𝑥3 𝑧𝑥4 𝑧,𝑧 = 1 dan 3 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑀},{𝑥1 1𝑥2 1} untuk 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚} untuk 𝑒 ∈ {𝑣𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} untuk 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 𝑚} subcase 2. for 𝑛 ≡ 2 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2} {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;3 ≤ 𝑧 ≤ 𝑚} , {𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 2 }, {𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;3 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚 } the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3,…,m} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 1, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2} {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;3 ≤ 𝑧 ≤ 𝑚} , {𝑦1 𝑧𝑦2 𝑧,1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0(𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑛+1 𝑧 𝑦1 𝑧,1 ≤ 𝑧 ≤ 𝑚},{𝑥1 1𝑥2 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 2}, {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;3 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑣𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 𝑚} on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 443 subcase 3. for 𝑛 ≡ 0 (mod 3) 𝑓(𝑣) = { 1, 2, 3, 4, 𝑧 + 1, for 𝑣 ∈ {𝑉},{𝑥𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 2, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2}, {𝑥2 𝑧,3 ≤ 𝑧 ≤ 𝑚},{𝑦𝑖 𝑧, 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 < 𝑧 ≤ 𝑚}, {𝑦1 𝑧,𝑧 = 1 dan 3 ≤ 𝑧 ≤ 𝑚} for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;𝑧 = 1 and 3 ≤ 𝑧 ≤ 𝑚} {𝑦1 2},{𝑥1 2} , for 𝑣 ∈ {𝑥𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 2;1 ≤ 𝑧 ≤ 𝑚 }, {𝑦𝑖 𝑧, 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛; 1 ≤ 𝑧 ≤ 𝑚},{𝑥2 1},{𝑥2 2} for 𝑣 ∈ {𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚 } the proper vertex coloring will induce the proper edge of 𝑐′ ∶ 𝐸(𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚)) → {1,2,3,…,m} coloring as follows. 𝑓(𝑒) = { 1, 2, 3, 𝑧, 𝑧 − 1, for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1,𝑧 = 1, and 3 ≤ 𝑧 ≤ 𝑚,{𝑦1 2𝑦1 2},{𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 0(𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛;1 ≤ 𝑧 ≤ 𝑚}, {𝑥𝑛+1 𝑧 𝑦1 𝑧,𝑧 = 1 dan 3 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 0 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;𝑧 = 1 and 3 ≤ 𝑧 ≤ 𝑚},{𝑥𝑖 2𝑥𝑖+1 2 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑦2 𝑧𝑦2 𝑧,𝑧 = 1 and 3 ≤ 𝑧 ≤ 𝑚},{𝑥1 1𝑥2 1},{𝑥𝑛+1 2 𝑦1 2} , for 𝑒 ∈ {𝑥𝑖 𝑧𝑥𝑖+1 𝑧 , 𝑖 ≡ 1 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 2𝑛 + 1;1 ≤ 𝑧 ≤ 𝑚}, {𝑦𝑖 𝑧𝑦𝑖+1 𝑧 , 𝑖 ≡ 2 (𝑚𝑜𝑑 3);1 < 𝑖 ≤ 𝑛 − 1;1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑣𝑥1 𝑧,1 ≤ 𝑧 ≤ 𝑚} for 𝑒 ∈ {𝑥1 𝑧𝑥2 𝑧,2 ≤ 𝑧 ≤ 𝑚} there is graceful 𝑚 + 1−coloring of 𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚). therefore, it obtained that 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) ≤ m + 1, hence 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) = 𝑚 + 1. conclusions based on the discussion described in chapter four, four new theorems of graceful coloring are obtained from the amalgamation operation of vertex family tree graphs. the graceful chromatic number obtained is as follows. 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = { 4, 𝑚 + 1, for 𝑚 = 2 for 𝑚 ≥ 3 . 𝜒𝑔𝑎𝑚𝑎𝑙(𝑃𝑛,𝑣,𝑚) = { 5, 𝑚 + 1, for 𝑚 ≤ 3 for 𝑚 ≥ 4. 𝜒𝑔𝑎𝑚𝑎𝑙(𝐵𝑘,𝑛,𝑣,𝑚) = { 𝑘 − 𝑛 + 2, 𝑚 + 1, for 𝑚 + 𝑛 < 𝑘 + 1 for 𝑚 + 𝑛 ≥ 𝑘 + 1. 𝜒𝑔𝑎𝑚𝑎𝑙(𝐸3,𝑛,𝑣,𝑚) = { 4, 𝑚 + 1, for 𝑚 = 2 for 𝑚 ≥ 3 on graceful chromatic number of vertex amalgamation of tree graph family arika indah kristiana 444 references [1] o. levin, discrete mathematics. greeley: university of northern colorado, 2019. 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[18] m. ajmal, w. nazeer, w. khalid, and s. m. kang, “zagreb polynomials and multiple zagreb indices for the line graphs of banana tree graph , firecracker graph,” glob. j. pure appl. math., vol. 13, no. 6, pp. 2659–2672, 2017. local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type cauchy –jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 136-141 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 07, 2021 reviewed: october 11, 2021 accepted: october 21, 2021 doi: https://doi.org/10.18860/ca.v7i1.13105 local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type khoirunisa khoirunisa1, corina karim2, m. muslikh3 1,2,3department of mathematics, universitas brawijaya email: khoir.n97@gmail.com, co_mathub@ub.ac.id, mslk@ub.ac.id abstract local hölder regularity of weak solutions for degenerate parabolic systems of p-laplacian type was proved by corina, k. in this paper, we aim to show the local hölder regularity of weak solutions in the singular case with 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2, so that the local hölder regularity of weak solutions for parabolic systems of p-laplacian type can hold for both cases. by applying poincaré inequality, we show that its weak solutions within hölder space, or we can tell that the local hölder regularity for the singular case is valid. keywords: hölder regularity; singular case; weak solutions introduction hölder regularity of weak solutions for parabolic equations in singular and degenerate case was introduced in [1] dan [2], where the coefficients are measurable and satisfy elliptic condition. however, the result in [2] is not showing the boundary estimates. after that, bögelein was interested to prove for the boundary regularity from the previous result in [2], see [3]. then, [4] investigated the same problem for nonlinear parabolic equations in the degenerate case. in 2002, dibenedetto et al discussed about the regularity of weak solutions for quasilinear parabolic equations in singular and degenerate case to proving their hölder character in [5]. the result in [2] was extended by misawa to a larger class of right-hand side terms, but the singular case was ecluded here, see [6]. meanwhile, for singular case, we can see the hölder regularity in [7]. in addition, the holder regularity of gradient solutions for all cases was proved in [8]. the study about hölder regularity of gradients solution was continued in [9] for evolutionary p-laplacian systems where the coefficients is hölder continuous. based on [10], the global weak solutions for similar case is exist. they used variational method to prove the existence of weak solutions globally. in 2018, karim started to investigated about simpler p-laplacian type in singular parabolic systems and showed that the weak solutions is bounded [11]. to prove the local boundedness in [11], we can adopt the method to prove energy estimates of singular parabolic equations in [12]. on the other hand, the existence of weak solutions in singular case was proved in [13] by using the galerkin method. furthermore, the intrinsic scaling method was used to treat the weak solutions to prove the hölder regularity for degenerate case [14]. the intrinsic scaling https://doi.org/10.18860/ca.v7i1.13105 mailto:khoir.n97@gmail.com mailto:co_mathub@ub.ac.id local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type khoirunisa khoirunisa 137 method is based on [15], which used the intrinsic scaling method to approach the regularity in degenerate and singular partial differential equations. motivated by the last result in[14], we would investigated the hölder regularity of weak solutions in singular case. let ω ⊂ ℝ𝑛 be a bounded domain, 𝑛 ≥ 2, and 𝜕ω is smooth boundary. the unknown function 𝑢: (0, 𝑇) × ω → ℝ𝑚is vector valued function, 𝑢 = (𝑢1(𝑧), 𝑢2(𝑧), … , 𝑢𝑚(𝑧)), where 𝑧 = (𝑡, 𝑥1, 𝑥2, … , 𝑥𝑛 ). let 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2, consider the parabolic systems { 𝜕𝑡 𝑢 − div (|𝐷𝑢| 𝑝−2𝐷𝑢) = 0 in(0, 𝑇) × ω, 𝑢(0, 𝑥) = 𝑢0(𝑥) on 𝜕𝑝(0, 𝑇) × ω, (1) where 𝑢0(𝑥) ∈ 𝑊 1,𝑝(ω, ℝ𝑚). the result in [13] shows that for any initial condition, there exists weak solutions of (1) from ω into ℝ𝑚. on the other hand, the local boundedness of the weak solution of (1) was proved by karim in [11]. they modified the intrinsic scaling from the original work by dibenedetto [12]. they used the intrinsic scaling for singular case. their main theorem established that intrinsic scaling well-worked to prove the local boundedness of weak solution of (1). we now turn to notion of hölder continuous functions. for any 𝑥, 𝑦 ∈ ω̅, if 𝑢 ∈ 𝐶0(ω̅) satisfy sup 𝑥,𝑦∈ ω̅,𝑥≠𝑦 |𝑢(𝑥) − 𝑢(𝑦)| |𝑥 − 𝑦|𝛼 < ∞, then 𝑢 ∈ 𝐶0,𝛼 (ω̅). karim in [14] shows that the weak solutions of (1) in degenerate case satisfy the following theorem. theorem 1.[14] let 𝑝 ≥ 2 and u is weak solutions of (1) in 𝑄(𝜆2−𝑝𝜌2, 𝜌)(𝑧0). then, 𝑢 is locally hölder continuous with some 0 < 𝛼 < 1. furthermore, for any 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 with 𝑧0 ′ ∈ 𝑄 and 0 < 𝜌0 < 1, there exist 𝐶 > 0 such that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, holds for any 𝑧, 𝑧′ ∈ 𝑄𝜌0 where 𝑄𝑟 (𝑧0) ⊂ 𝑄(𝜆 2−𝑝𝜌2, 𝜌)(𝑧0) ⊂ 𝑄𝜌0 (𝑧0 ′ ) ⊂ 𝑄3𝜌0 (𝑧0 ′ ). theorem 1 implies that the local hölder regularity of weak solutions (1) in degenerate case is proved. here, we aim to prove for the singular case, so that the local hölder regularity of weak solutions (1) can be proven for both cases or 2𝑚 𝑚+2 < 𝑝 < ∞. methods we can prove the local hölder regularity of weak solutions for parabolic systems of plaplacian type in singular case by showing that the weak solutions are elements of hölder space. our method is using poincaré inequality to show that the weak solutions within campanato space, then by isomorphism between campanato and hölder space, it is easy to see that its weak solutions are elements of hölder space. the poincaré inequality that we use is in the following theorem. theorem 2. (poincaré inequality).[16] let ω ⊂ ℝ𝑛 be a bounded open set and 1 ≤ 𝑝 ≤ ∞. then, there exists positive constant 𝐶 = 𝐶(ω, 𝑝) so that ||𝑢 − 𝑢ω||𝐿𝑝 ≤ 𝐶||𝐷𝑢|| 𝐿𝑝 , (2) local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type khoirunisa khoirunisa 138 where 𝑢ω = 1 |ω| ∫ 𝑢 ω 𝑑𝑥. results and discussion for singular equation (1) with 1 < 𝑝 < 2, we have the cylinder of the type 𝑄(𝜌)(𝑧0)(𝑡0 − 𝜌 2, 𝑡0 + 𝜌 2) × 𝐵 (𝜆 𝑝−2 2 𝜌, 𝑥0). by switching the cylinder 𝑄(𝜌)(𝑧0) to 𝑄(1), we have 𝑣(𝑠, 𝑦) = 𝑢(𝑡0 + 𝜆 2−𝑝𝜌𝑠, 𝑥0 + 𝜌𝑦) 𝜆𝜌 ; (𝑠, 𝑦) ∈ 𝐵(1) × (−1,1) ≡ 𝑄(1). suppose that on such a certain cylinder the relations 1 |𝑄(𝜌)(𝑧0)| ∫ |𝐷𝑢|𝑝𝑑𝑧 ≈ 𝜆𝑝. (3) 𝑄(𝜌)(𝑧0) hence, we have 𝜕𝑡 𝑢 − 𝜆 𝑝−2div(𝐷𝑢) = 0, in 𝑄(𝜌)(𝑧0). let 𝑢 ∈ 𝐿∞(0 , 𝑇; 𝑊1,𝑝(ω, ℝ𝑚)) be a weak solutions of (1) in cylinder 𝑄 = (0, 𝑇) × ω where 𝑧0 = (𝑡0, 𝑥0) ∈ 𝑄, 𝜆 ≥ 1and 0 < 𝜌 < 1 such that 𝑄 (𝜌 2, 𝜆 𝑝−2 2 𝜌) ⊂ 𝑄. while, our main theorem is the following. theorem 3.let 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2 and u is a weak solution of (1) in 𝑄(𝜌2, 𝜆 𝑝−2 2 𝜌)(𝑧0). then 𝑢 is locally 𝛼-hölder continuous with some 0 < 𝛼 < 1. furthermore, for any 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 with 𝑧0 ′ ∈ 𝑄 and 0 < 𝜌0 < 1, there 𝐶 > 0 such that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, holds for any 𝑧, 𝑧′ ∈ 𝑄𝜌0 (𝑧0 ′ ) where 𝑄𝑟 (𝑧0) ⊂ 𝑄(𝜌 2, 𝜆 𝑝−2 2 𝜌)(𝑧0) ⊂ 𝑄𝜌0 (𝑧0 ′ ) ⊂ 𝑄3𝜌0 (𝑧0 ′ ) proof. figure 1. any cylinder in 𝑄 local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type khoirunisa khoirunisa 139 let 𝑄3𝜌0 (𝑧0 ′ ) ≔ 𝑄 (3(𝜌0) 2, 3(𝜌0) 2 𝑝) ⊂ 𝑄 is any cylinder centered in 𝑧0 ′ = (𝑡0 ′ , 𝑥0 ′ ) ∈ 𝑄, and 0 < 𝜌0 < 1. let 𝑧0 = (𝑡0, 𝑥0) ∈ 𝑄𝜌0 (𝑧0 ′ ) ≔ 𝑄 ((𝜌0) 2, (𝜌0) 2 𝑝) (𝑧0 ′ ), where 𝜆 ≥ 1. it is easy to see that ∫ |𝐷𝑢|𝑝 𝑑𝑧 ≤ |𝑄𝑟 (𝑧0)| 1 |𝑄(𝜌2, 𝜆 𝑝−2 2 𝜌)(𝑧0)| ∫ |𝐷𝑢|𝑝 𝑑𝑧 𝑄(𝜌2,𝜆 𝑝−2 2 𝜌)(𝑧0)𝑄𝑟(𝑧0) by using relation (3) we have 1 |𝑄𝑟(𝑧0)| ∫ |𝐷𝑢|𝑝 𝑑𝑧 𝑄𝑟(𝑧0)) ≤ 𝐶𝑟𝑚+𝑝𝛼 (4) where0 < 𝑟 < 1, 0 < 𝛼 < 1, and 𝐶 = 𝜋𝑟2−𝑚. next, recall the poincaré inequality. then, substitute (4) to (2) we have ∫ |𝑢 − 𝑢𝑟 | 𝑝 𝑑𝑧 ≤ 𝐶𝑟𝑝 ∫ |𝐷𝑢|𝑝 𝑑𝑧, 𝑄𝑟(𝑧0)𝑄𝑟(𝑧0) ≤ 𝐶𝑟𝑚+𝑝+𝑝𝛼 , holds for any 𝑧0 ∈ 𝑄𝜌0 (𝑧0) and 0 < 𝑟 < 2(𝜌0) 2 𝑝. we apply the characterization between hölder continuous functions and campanato space, which implies that for 𝑢 ∈ 𝐿𝑝 (𝑄𝜌0 (𝑧0 ′ )) and sup 𝑄𝜌0 (𝑧0 ′ ) 1 𝑟𝑚+𝑝+𝑝𝛼 ∫ |𝑢 − 𝑢𝑟 | 𝑝 𝑑𝑧 ≤ 𝐶, (5) 𝑄𝑟(𝑧0) then 𝑢 ∈ 𝐶0,𝛼 (𝑄𝜌0 (𝑧0 ′ )). thus, we have for any two points (𝑡1, 𝑥1), (𝑡2, 𝑥2) ∈ 𝑄𝜌0 (𝑧0 ′ )with |𝑡1 − 𝑡2| = 𝑟 𝑝, |𝑢(𝑡1, 𝑥1) − 𝑢(𝑡2, 𝑥1)| ≤ 𝐶|𝑡1 − 𝑡2| 𝛼 𝑝 , (6) and let |𝑥1 − 𝑥2| = 𝑟, then |𝑢(𝑡2, 𝑥2) − 𝑢(𝑡2, 𝑥1)| ≤ 𝐶|𝑥1 − 𝑥2| 𝛼 . (7) moreover, we can conclude from (6),(7) and triangle inequality that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, in 𝑄𝜌0 (𝑧0 ′ ) or we have local hölder regularity of weak solutions for singular parabolic systems of p-laplacian type khoirunisa khoirunisa 140 𝑢 ∈ 𝐶 , 𝛼 𝑝 ,𝛼 (𝑄𝜌0 (𝑧0 ′ )). since, 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 is arbitrary, then 𝑢 ∈ 𝐶 𝑙𝑜𝑐 , 𝛼 𝑝 ,𝛼 (𝑄, ℝ𝑛 ). by this result, we can tell that the local hölder regularity of weak solutions for parabolic systems of p-laplacian type in singular case is proved. conclusions based on the previous results and discussion, it can be concluded that in singular case the weak solutions for parabolic systems of p-laplacian type are elements of hölder space, 𝑢 ∈ 𝐶 , 𝛼 𝑝 ,𝛼 (𝑄𝜌0 (𝑧0 ′ )). since 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 is arbitrary, then 𝑢 ∈ 𝐶 𝑙𝑜𝑐 , 𝛼 𝑝 ,𝛼 (𝑄, ℝ𝑛 ) or we can say that the local hölder regularity of weak solutions is proved. acknowledgement this paper partially supported by the doctoral grant no. 44/un10.f09/pn/2020 at mathematics and natural sciences faculty, universitas brawijaya. references [1] c. ya-zhe and e. dibenedetto, “hölder estimates of solutions of singular parabolic equations with measurable coefficients,” archive for rational mechanics and analysis, vol. 118, no. 3, pp. 257–271, 1992. [2] e. dibenedetto, “on the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,” annali della scuola normale superiore di pisa-classe di scienze, vol. 13, no. 3, pp. 487–535, 1986. [3] v. bögelein, “global gradient bounds for the parabolic p-laplacian system,” proceedings of the london mathematical society, vol. 111, no. 3, pp. 633–680, 2015. [4] m. m. porzio and v. vespri, “holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,” journal of differential equations, vol. 103, no. 1, pp. 146–178, 1993. [5] e. dibenedetto, j. m. urbano, and v. vespri, “current issues on singular and degenerate evolution equations,” in handbook of differential equations: evolutionary equations, vol. 1, elsevier, 2002, pp. 169–286. [6] m. misawa, “a hölder estimate for nonlinear parabolic systems of p-laplacian type,” journal of differential equations, vol. 254, no. 2, pp. 847–878, 2013. 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[16] b. dacorogna, introduction to the calculus of variations. world scientific publishing company, 2014. inclusion properties of the homogeneous herz-morrey y aces cauchy – jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 117-121 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 18, 2020 reviewed: october 05, 2020 accepted: november 05, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.10114 hairur rahman department of mathematics, islamic state university of maulana malik ibrahim malang email: hairur@mat.uin-malang.ac.id abstract in this paper, we have discussed the inclusion properties of the homogeneous herz-morrey spaces and the homogeneous weak homogeneous spaces. we also studied the inclusion relation between those spaces. keywords: homogeneous herz-morrey spaces; homogeneous weak herz-morrey spaces; inclusion properties. introduction the subject discussion about inclusion properties of any spaces or inclusion relation between spaces has interested to study. some authors have studied about inclusion relation in some spaces (see [1], [2], [3], [4] and [5]). it guided us to discuss the inclusion properties of other spaces. regarding c.b. morrey in [6] who introduced morrey spaces, many authors have defined the generalization of morrey spaces and combined with other spaces. lu and xu [7] introduce one of the homogeneous herz-morrey spaces. these spaces are the generalization of morrey spaces and herz spaces. let 𝛼 ∈ ℝ, 0 < 𝑝 ≤ ∞, 0 < 𝑞 < ∞, and 0 ≤ 𝜆 < ∞, the homogeneous herz-morrey spaces ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) are defined by ℳ�̇�𝑝,𝑞 𝛼,𝜆(𝑅𝑛) ∶= {𝑓 ∈ 𝐿𝑙𝑜𝑐 𝑞 (ℝ𝑛/{0}) ∶ ‖𝑓‖ ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) < ∞}, where ‖𝑓‖ ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) = sup 𝐿∈ℤ 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝 𝐿 𝑘=−∞ ) 1 𝑝 with 𝐵𝑘 = 𝐵(0,2 𝑘 ) = {𝑥 ∈ ℝ𝑛: |𝑥| ≤ 2𝑘 }, 𝐴𝑘 = 𝐵𝑘 /𝐵𝑘−1 for 𝑘 ∈ ℤ and 𝜒𝑘 = 𝜒𝐴𝑘 for 𝑘 ∈ ℤ be the characteristic function of the set 𝐴𝑘 . lu and xu also defined the homogeneous weak herz-morrey spaces. for 𝛼 ∈ ℝ, let 0 < 𝑝 ≤ ∞, 𝜆 ≥ 0 and 0 < 𝑞 < ∞, the homogeneous weak herz-morrey spaces (𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛)) is a set of measurable 𝑓 ∈ 𝐿𝑙𝑜𝑐 𝑞 (ℝ𝑛/{0}) which completed by norm such that ‖𝑓‖ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) = sup 𝛾>0 𝛾 sup 𝐿∈ℤ 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝𝑚𝑘 (𝛾, 𝑓) 𝑝 𝑞 𝐿 𝑘=−∞ ) 1 𝑝 < ∞, where 𝑚𝑘 (𝛾, 𝑓) = |{𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾}|. some authors have studied those spaces in different term of discussion (see [7], [8], [9], [10]). meanwhile, in this article, the authors would like to discuss the inclusion properties and inclusion relation of the homogeneous herz-morrey spaces and the homogeneous weak herz-morrey spaces. inclusion properties of the homogeneous herz-morrey y aces http://dx.doi.org/10.18860/ca.v6i3.10114 mailto:hairur@mat.uin-malang.ac.id inclusion properties of the homogeneous herz-morrey hairur rahman 118 results and discussion now, we formulate our main results of this paper as follows: theorem 1.1. let 1 ≤ 𝑝1 ≤ 𝑝2 < 𝑞 < ∞, then the following inclusion holds: ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛). generally, by theorem 1.1, we have the following inclusions of the homogeneous herz-morrey spaces. theorem 1.2. let 1 ≤ 𝑝1 ≤ 𝑝2 < 𝑞 < ∞, then the following inclusion holds: 𝐿𝑞 (𝑅𝑛) = ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) ⊆ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛). besides, we have the inclusion property of the homogeneous weak herz-morrey spaces, also the inclusion relation of the homogeneous herz-morrey spaces. theorem 1.3. let 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞, the following inclusion holds: 𝑊ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (ℝ𝑛) ⊆ 𝑊ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (ℝ𝑛). theorem 1.4. let 1 ≤ 𝑝 ≤ 𝑞. then the inclusion ℳ�̇�𝑝,𝑞 𝛼,𝜆 (ℝ𝑛) ⊆ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(𝑅𝑛) is proper. the proof of each theorem will be described in the following section. the proof of theorem 1.1. for proofing theorem 1.1., we shall show that ‖𝑓‖ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛) ≤ ‖𝑓‖ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) by applying hölder inequality. proof of theorem 1.1. let we first take for any 𝑓 ∈ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛), then by using hölder inequality and 𝑝1 ≤ 𝑝2 we obtain that ‖𝑓‖ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛) = sup 𝐿∈𝑍 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝1 𝐿 𝑘=−∞ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1 ) 1 𝑝1 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 (( ∑ (2𝑘𝛼𝑝1 ) 𝑝2 𝑝1 𝐿 𝑘=−∞ ) 𝑝1 𝑝2 ( ∑ (‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1 ) 𝑝2 𝑝2−𝑝1 𝐿 𝑘=−∞ ) 1− 𝑝1 𝑝2 ) 1 𝑝1 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 (( ∑ 2𝑘𝛼𝑝2 𝐿 𝑘=−∞ ) 𝑝1 𝑝2 ( ∑ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1𝑝2 𝑝2−𝑝1 𝐿 𝑘=−∞ ) 1− 𝑝1 𝑝2 ) 1 𝑝1 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝2 𝐿 𝑘=−∞ ( ∑ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝1𝑝2 𝑝2−𝑝1 𝐿 𝑘=−∞ ) 𝑝2−𝑝1 𝑝1 ) 1 𝑝2 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝2 𝐿 𝑘=−∞ ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝2 ) 1 𝑝2 ≤ ‖𝑓‖ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) . by this observation, we know that 𝑓 ∈ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛). hence it concludes that ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛). inclusion properties of the homogeneous herz-morrey hairur rahman 119 the proof of theorem 1.2. since it has been stated in theorem 1.1 that ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛), therefore in proving theorem 1.2, we need to prove that 𝐿𝑞 (𝑅𝑛) = ℳ�̇�𝑞,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛). proof of theorem 1.2. to prove that 𝐿𝑞 (𝑅𝑛) = ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛), we need to show that ‖𝑓‖𝑳𝒒(𝑹𝒏) = ‖𝑓‖𝑀�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) . let take for any 𝑓 ∈ 𝑀�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛), by applying hölder inequality for the norm. we obtain that ‖𝑓‖ 𝑀�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) ≤ sup 𝐿∈𝑍 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑞 𝐿 𝑘=−∞ ((∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 (∫ |𝜒 𝑘 | 𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 ) 𝑞 ) 1 𝑞 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 ∑ 2𝑘𝛼 𝐿 𝑘=−∞ (∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 (2𝑘𝑑 ) 1 𝑞 ≤ sup 𝐿∈𝑍 2−𝐿𝜆 ∑ 2 𝑘𝛼+ 𝑘𝑑 𝑞 𝐿 𝑘=−∞ (∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 ≤ 𝐶 (∫ |𝑓(𝑥)|𝑞 𝑑𝑦 𝐵(0,2𝑘) ) 1 𝑞 ≤ ‖𝒇‖𝑳𝒒(𝑹𝒏), it means that 𝑓 ∈ 𝐿𝑞 (𝑅𝑛). then 𝐿𝑞 (𝑅𝑛) ⊆ ℳ�̇�𝑞,𝑞 𝛼,𝜆 (𝑅𝑛). meanwhile, for any 𝑓 ∈ 𝐿𝑞 (𝑅𝑛), we can find any constant 𝐶 such that 𝐶 = sup 𝐿∈𝑍 2−𝐿𝜆 ∑ 2 𝑘𝛼+ 𝑘𝑑 𝑞𝐿 𝑘=−∞ , then it shows that 𝑓 ∈ ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) which means ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) ⊆ 𝐿𝑞 (𝑅𝑛). hence, it concludes that 𝐿𝑞 (𝑅𝑛) = ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛). next, we have to prove that ℳ�̇�𝑞,𝑞 𝛼,𝜆 (𝑅𝑛) ⊆ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) by showing ‖𝑓‖ ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ≤ ‖𝑓‖ ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) . by using a similar way for proving theorem 1.1., and since 𝑞 > 𝑝2, it is clear that ‖𝑓‖ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) ≤ ‖𝑓‖ ℳ�̇�𝑞,𝑞 𝛼,𝜆(𝑅𝑛) . therefore, the proof is complete. the proof of theorem 1.3. one way for proving theorem 1.3. is showed that ‖𝑓‖ 𝑊ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛) ≤ ‖𝑓‖ 𝑊ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) . proof of theorem 1.3. let take for any 𝑓 ∈ ‖𝑓‖ 𝑊ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (𝑅𝑛) , then by observing the norm of 𝑓 we obtain that ‖𝑓‖ 𝑊ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (ℝ𝑛) = sup 𝛾>0 𝛾 sup 𝐿∈ℤ 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝1 𝑚𝑘 (𝛾, 𝑓) 𝑝1 𝑞 𝐿 𝑘=−∞ ) 1 𝑝1 ≤ sup 𝛾>0 𝛾 sup 𝐿∈ℤ 2−𝐿𝜆 ( ∑ 2𝑘𝛼𝑝2 𝑚𝑘 (𝛾, 𝑓) 𝑝2 𝑞 𝐿 𝑘=−∞ ) 1 𝑝2 ≤ ‖𝑓‖ 𝑊ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (𝑅𝑛) . by the observation, it concludes that 𝑊ℳ�̇�𝑝2,𝑞 𝛼,𝜆 (ℝ𝑛) ⊆ 𝑊ℳ�̇�𝑝1,𝑞 𝛼,𝜆 (ℝ𝑛). inclusion properties of the homogeneous herz-morrey hairur rahman 120 the proof of theorem 1.4. proving theorem 1.4 is used a similar idea as previous theorems which shall show that ‖𝑓‖ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) ≤ ‖𝑓‖ ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) . proof of theorem 1.4. let 𝑓 ∈ ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛 ), 𝑎 ∈ ℝ𝑛, and 𝛾 > 0. we observe that |{𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾}| 𝑝 𝑞 ≤ (∫ |𝑓(𝑥)𝜒𝑘 | 𝑞 𝑑𝑥 𝐵(0,2𝑘) ) 𝑝 𝑞 = ‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝 multiplying both side by ∑ 2𝑘𝛼𝑝𝐿𝑘=−∞ , then we obtain that ∑ 2𝑘𝛼𝑝|{𝑥 ∈ 𝐴𝑘 : |𝑓(𝑥)| > 𝛾}| 𝑝 𝑞 𝐿 𝑘=−∞ ≤ ∑ 2𝑘𝛼𝑝‖𝑓𝜒𝑘 ‖𝐿𝑞(ℝ𝑛) 𝑝 𝐿 𝑘=−∞ . it says merely that ‖𝑓‖ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) ≤ ‖𝑓‖ ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) , therefore 𝑓 ∈ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛). hence, it is proved that ℳ�̇�𝑝,𝑞 𝛼,𝜆(ℝ𝑛) ⊆ 𝑊ℳ�̇�𝑝,𝑞 𝛼,𝜆(𝑅𝑛). conclusions by this result, the author can conclude that the homogeneous herz-morrey spaces have inclusion properties as stated above. this result will be useful to be used in proving fractional integral on the homogeneous herz-morrey spaces. acknowledgemet this article research is partially supported by uin maliki malang research and innovation program 2020. references [1] h. gunawan, d. i. hakim, k. m. limanta and a. a. masta, "inclusion property of generalized morrey spaces," math. nachr., pp. 1-9, 2016. 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[10] y. mizuta and t. ohno, "herz-morrey spaces of variable exponent, riesz potential operator and duality," complex variable and elliptic equations, vol. 60, no. 2, pp. 211240, 2015. microsoft word 1 sampul depan.doc 31  perambatan gelombang optik pada grating sinusoidal dengan chirp dan taper isnani darti jurusan matematika, fakultas mipa universitas brawijaya, jl. veteran malang 65145 email: isnanidarti@yahoo.com abstrak artikel ini membahas model perambatan gelombang optik pada grating sinusoidal takhomogen. model tersebut diturunkan dengan mereduksi secara eksak persamaan helmholtz menjadi sistem persamaan diferensial orde satu dengan syarat awal yang dapat diselesaikan dengan metode runge-kutta orde empat. metode ini disebut metode integrasi langsung (mil). formulasi mil sangat sederhana baik dalam hal penurunannya maupun implementasinya karena tidak memerlukan prosedur iterasi maupun optimasi. dengan menggunakan mil, dipelajari perubahan respon optik pada grating sinusoidal akibat variasi amplitudo modulasi indeks (taper) dan variasi frekuensi spasial grating (chirp). hasil simulasi menunjukkan bahwa taper menyebabkan adanya fenomena penghilangan side-lobe pada spektrum transmitansi. adanya chirp menyebabkan penghalusan side-lobe pada spektrum transmitansi dengan semakin besar parameter chirp menyebabkan peningkatan transmitansi di sekitar pusat band-gap dari grating homogen. selain implementasi integrasi numerik (runge-kutta), mil merupakan metode eksak sehingga dapat digunakan untuk mengevaluasi validitas metode yang sering digunakan yaitu persamaan moda tergandeng (pmt). dari hasil perbandingan dapat disimpulkan bahwa secara umum pmt kurang akurat dalam menganalisis struktur grating sinusoidal baik homogen maupun tak-homogen. kata kunci: grating sinusoidal tak-homogen (taper, chirp), metode integrasi langsung, persamaan moda tergandeng, metode runge-kutta orde empat. 1. pendahuluan salah satu blok komponen penting dalam peralatan optik terpadu adalah struktur grating, yaitu suatu sistem yang terbuat dari beberapa lapisan medium dielektrik yang disusun secara bergantian (periodik). struktur ini memiliki sifat dasar bahwa gelombang dengan frekuensi yang termasuk dalam suatu interval tertentu akan dipantulkan secara sempurna oleh struktur grating. interval frekuensi ini disebut dengan band gap; lihat (joannopoulos et al. 1995, soukoulis 1993, soukoulis 1996). salah satu struktur grating yang sering dikaji adalah grating dengan indeks bias berbentuk sinusoidal (disebut grating sinusoidal). struktur grating tersebut telah diaplikasikan dalam berbagai peralatan optik yang menarik seperti filter (lei et al. 1997), cermin sempurna (joannopoulos et al. 1995), optical limiter dan switching (scalora et al., 1994, tran, 1997) dan sensor (mandal et al., 2005). sebelum mendesain peralatan yang menggunakan struktur grating, adalah sangat penting untuk mengetahui perilaku transmisi gelombang optik yang melewati struktur tersebut. untuk mempelajari perilaku transmisi secara eksperimen dibutuhkan fasilitas yang sangat mahal seperti “clean room”. oleh karena itu pengertian secara teoritis dan prinsip-prinsip dasar sangat diperlukan untuk efisiensi waktu dan biaya. untuk itu perlu dikembangkan metode analisis yang sangat akurat dan efisien sebagai dasar untuk simulasi komputer. metode yang sering banyak diaplikasikan untuk mempelajari fenomena pada grating sinusoidal adalah persamaan moda tergandeng (pmt) (de sterke at at., 1991). secara umum pmt dikembangkan dengan mengasumsikan bahwa amplitudo gelombang bervariasi secara lambat (slowly varying envelope isnani darti  32 volume 1 no. 1 november 2009 approximation, disingkat svea) sehingga untuk kasus-kasus tertentu keakuratannya diragukan. untuk mengatasi hal tersebut, suryanto dan darti (2005) telah mengembangkan metode integrasi langsung (mil). dalam metode ini, persamaan maxwell diselesaikan dalam domain frekuensi dan ditransformasikan ke dalam sistem persamaan diferensial yang dapat diintegrasikan secara langsung. metode tersebut telah diaplikasikan untuk mempelajari respon optik baik dari grating step-index dengan beberapa lapisan cacat (suryanto dan darti, 2005; suryanto, 2006) maupun grating sinusoidal dengan pergeseran fasa (suryanto dan darti, 2008; suryanto, 2009). dalam artikel ini metode integrasi langsung akan diaplikasikan untuk mempelajari perubahan respon optik pada grating sinusoidal akibat variasi amplitudo modulasi indeks (taper) dan variasi frekuensi spasial (chirp). 2. grating sinusoidal tak-homogen dan metode numerik struktur grating sinusoidal yang akan dikaji dalam artikel ini adalah struktur grating dengan indeks bias berbentuk ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ λ += z z znzn π δ 2 sin0 (1) dengan ,0n ( )zδ dan ( )zλ masing-masing adalah rata-rata indeks bias medium, kedalaman modulasi indeks sinusoidal dan panjang periode grating. secara skematik, struktur grating yang dikaji dapat dilihat pada gambar 1. gambar 1. gambar skematik variasi indeks grating terhadap z. 2.1. persamaan gelombang dan metode integrasi langsung perambatan gelombang elektromagnetik secara umum dimodelkan oleh persamaan maxwell. pada medium linear satu dimensi, persamaan maxwell direduksi menjadi persamaan helmholtz (suryanto et al., 2003a, suryanto et al., 2003b): ( ) 0222 2 =+ eznk dz ed (2) dimana e adalah gelombang elektrik, ck /ω= adalah konstanta propagasi di ruang hampa dengan c adalah kecepatan cahaya. persamaan (2) dapat dinyatakan dalam bentuk sistem persamaan diferensial: ( ) ( ) ( ) ( ) ( ).22 zeznk dz zdv zv dz zde −= = (3) perhatikan bahwa sistem persamaan (3) diturunkan dari persamaan helmholtz (2) tanpa melakukan aproksimasi. untuk menyelesaikan persamaan (3), diperlukan syarat awal. syarat awal ditentukan dengan mengasumsikan bahwa gelombang elektromagnetik dengan frekuensi ω dan amplitudo dtga diiluminasikan dengan sudut normal hanya dari perambatan gelombang optik pada grating sinusoidal dengan chiper dan taper  volume 1 no. 1 november 2009 33 sebelah kiri grating. secara umum, gelombang datang tersebut akan dipantulkan (misalkan gelombang pantul mempunyai amplitudo ptla ) oleh struktur grating dan sebagian diteruskan oleh grating (misal dengan amplitudo tra ). jika grating diasumsikan terletak di antara z = 0 dan z = maxz dan medium di luar grating mempunyai indeks bias konstan 0nn = , maka medan elektrik di luar grating secara umum adalah ( ) 0;00 ≤+= − zeaeaze ziknptl zikn dtg (4.a) ( ) max)( ;0 zzeaze lzikntr ≥= −− . (4.b) dengan mengimplementasikan kondisi antar-muka pada z = 0 dan z = maxz , didapat bahwa ( ) ( ) tr tr aiknlv ale 0−= = (5) dan amplitudo gelombang datang dan gelombang pantul ditentukan oleh ( ) ( ) ( ) ( ) .00 2 1 00 2 1 0 0 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += v kn i ea v kn i ea ptl dtl (6) penyelesaian persamaan (3) dengan kondisi awal (5) dapat dicari dengan menggunakan metode integrasi numerik mulai dari z = maxz sampai z = 0. dalam artikel ini, metode yang digunakan adalah metode runge-kutta orde empat. setelah seluruh medan elektrik terhitung, amplitudo gelombang datang dan gelombang pantul dapat dihitung dengan mengaplikasikan persamaan (6) dan reflektansi dan transmitansi dapat ditentukan dengan persamaan: 2 dtg ptl a a r = dan 2 dtg tr a a t = . (7) 2.2. persamaan moda-tergandeng (pmt) untuk menurunkan pmt, pertama diasumsikan bahwa variasi modulasi amplitudo grating ( )zδ bernilai cukup kecil, sehingga kuadrat indeks bias dalam persamaan (1) dapat diaproksimasi menjadi ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ λ += z z znnzn π δ 2 sin2 0 2 0 2 . (8) selanjutnya, medan elektrik diasumsikan sebagai superposisi dari moda normal dari medium konstan (yaitu jika ( ) 0=zδ ) tetapi dengan amplitudo tergantung pada posisi z, yaitu ( ) ( ) ( )zikzbzikzaze 00 exp)(exp)( +−= (9) dengan 00 knk = . dengan mengasumsikan svea dan menganggap bahwa moda harmonik spasial order tiga sangat kecil, persamaan helmholtz (2) direduksi menjadi pmt: ( )( ) ( )( )zkkiak z b zkkibk z a −−= ∂ ∂ −= ∂ ∂ 0 0 2exp 2 1 2exp 2 1 δ δ (10) isnani darti  34 volume 1 no. 1 november 2009 dengan λ = π2 k . untuk grating homogen, penyelesaian analitik pmt dapat ditemukan dalam literatur (yeh, 1988), yaitu: ( ) ( )[ ] ( )[ ] ( ) [ ] ( ) ( )[ ] ( ) [ ] ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ δ− δ+ − −= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ δ δ+ −δ+− = kzia szkiszs zzsi zb kzia szkiszs zzskizzss za 2 1 exp sinh 2 1 cosh sinh 2 1 exp sinh 2 1 cosh sinh 2 1 cosh 0 maxmax max 0 maxmax maxmax κ (11) dimana kkk −=δ 02 , δκ k2 1 = , 2 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ δ−= k s κ and ( )00 aa = . tetapi untuk grating tak-homogen, penyelesaian eksak relatif sulit didapatkan. dalam artikel ini pmt pada persamaan (1) diintegralkan secara numerik seperti pada mil, yaitu menggunakan metode runge-kutta order empat dengan syarat awal ( ) .0 )( max max = = zb aza tr (12) reflektansi dan transmitansi dapat ditentukan dengan cara yang identik dengan mil, yaitu ( ) ( ) 2 0 0 a b r = dan ( ) ( ) 2 max 0a za t = . (13) 3. simulasi numerik perambatan gelombang optik pada grating sinusoidal 3.1. grating homogen pada bagian ini akan ditunjukkan hasil simulasi numerik gelombang optik pada grating homogen. khususnya akan dibahas sifat-sifat transmisi grating sinusoidal homogen, yaitu struktur dengan indeks bias seperti pada persamaan (1) dengan 20 =n , ( ) 4.00 == δδ z dan ( ) 02 1 n z b =λ=λ . (a) (b) gambar 2. spektrum transmitansi grating sinusoidal homogen dengan (a) n = 10 dan (b) n = 20; dihitung dengan mil dan pmt. pada gambar 2 ditunjukkan pengaruh jumlah perioda grating terhadap karakteristik transmisi. terlihat bahwa semakin banyak perioda, semakin banyak pula jumlah frekuensi resonan (frekuensi dengan transmitansi t = 1) yang muncul. terlebih perambatan gelombang optik pada grating sinusoidal dengan chiper dan taper  volume 1 no. 1 november 2009 35 lagi, pada ujung-ujung band-gap untuk jumlah perioda yang lebih besar, resonan memiliki lebar spektrum yang lebih sempit. perbandingan hasil mil dengan pmt untuk grating sinusoidal homogen juga dapat dilihat pada gambar 2. secara umum pmt memberikan spektrum transmitansi dengan jumlah resonan yang sama dengan mil tetapi dengan side-lobe lebih dangkal untuk spektrum di sebelah kiri band-gap dan lebih dalam untuk spektrum di sebelah kanan band-gap. lebih lanjut, spektrum transmitansi hasil pmt tergeser ke kanan jika dibandingkan spektrum hasil mil. hasil ini menunjukkan bahwa pmt kurang akurat dalam menganalisis grating sinusoidal. perbedaan antara spektrum hasil pmt dan mil semakin berkurang dengan semakin kecilnya nilai 0δ . 3.2. grating dengan taper linear untuk melihat pengaruh taper, diasumsikan bahwa amplitudo modulasi indeks bias mengalami peningkatan (untuk taper positif) atau penurunan (untuk taper negatif) secara linear, yaitu grating mempunyai modulasi amplitudo secara linear: ( ) ( )znzn δ+= 10δδ (14) dengan panjang perioda dan rata-rata modulasi indeks bias bernilai konstan yaitu ( ) 02 1 n z =λ dan 4.00 =δ . (a) (b) gambar 3. spektrum transmitansi grating (n = 10) dengan taper linear: (a) taper positif ; (b) taper negatif. gambar 4. spektrum transmitansi grating dengan taper positif dan n = 20. transmitansi dari grating dengan jumlah perioda n = 10 dapat dilihat pada gambar 3.a untuk taper positif dan gambar 3.b untuk taper negatif. adanya taper pada grating, baik positif ataupun negatif menunjukkan adanya fenomena penghilangan side-lobe. perbedaannya, taper positif mengakibatkan interval band-gap semakin lebar dan sebaliknya taper negatif mengakibatkan penghilangan interval band-gap. fenomena isnani darti  36 volume 1 no. 1 november 2009 penghilangan side-lobe dan pelebaran/penghilangan interval band-gap semakin terlihat apabila parameter taper δn diperbesar atau jika jumlah perioda grating n ditingkatkan. sebagai contoh, fenomena tersebut dapat dilihat dari spektrum transmitansi grating sinusoidal dengan taper linear dan jumlah perioda n = 20, lihat gambar 4. pada gambar 3 dan 4 juga dibandingkan spektrum transmitansi grating dengan taper hasil mil dengan hasil pmt. pada gambar-gambar tersebut dapat dilihat bahwa perbedaan antara hasil mil dan pmt serupa dengan masalah grating homogen. hasilhasil simulasi menunjukkan bahwa pmt semakin tidak akurat jika parameter taper atau jumlah perioda grating diperbesar. 3.3. grating dengan chirp chirp pada grating dibuat dengan mengubah periodasitas dari grating homogen, yaitu panjang perioda masing-masing grating diberikan oleh ( ) ( ) negatif untuk ;1 positif untuk;1 minmax max minmax min chirpi n chirpi n i i −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ−λ −λ=λ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ−λ+λ=λ (15) dengan ( ) ( ) ( ) .4.0;1;1 0minmax ==λ−=λλ+=λ δδσσ zbb gambar 5 menunjukkan spektrum transmitansi dari grating dengan chirp dan jumlah perioda n = 10. terlihat pada gambar tersebut bahwa chirp menghaluskan side-lobe dan apabila parameter chirp diperbesar maka transmitansi meningkat di sekitar pusat band-gap dari grating homogen. pada gambar 5 terlihat bahwa pmt untuk masalah grating dengan chirp sangat tidak akurat. hal ini ditunjukkan oleh perbedaan spektrum transmitansi hasil mil dan pmt. hasil simulasi lain menunjukkan bahwa semakin besar parameter chirp ataupun semakin besar jumlah perioda grating menyebabkan pmt semakin tidak akurat. (a) (b) gambar 5. spektrum transmitansi grating dengan jumlah perioda n = 10: (a) chirp negatif dan (b) chirp positif. 4. kesimpulan dalam artikel ini telah dikembangkan mil untuk analisis grating sinusoidal takhomogen. dalam metode tersebut persamaan helmholtz secara langsung diubah menjadi sistem persamaan diferensial orde satu dengan syarat awal yang dapat diselesaikan secara langsung oleh metode runge-kutta order empat. jika dibandingkan dengan mil, pmt secara umum kurang akurat dalam menganalisis grating sinusoidal baik homogen maupun tak-homogen. dengan menggunakan mil, telah dipelajari pengaruh taper dan chirp terhadap respon optik pada grating sinusoidal. ditunjukkan juga bahwa taper dan chirp masingperambatan gelombang optik pada grating sinusoidal dengan chiper dan taper  volume 1 no. 1 november 2009 37 masing menyebabkan adanya fenomena penghilangan dan penghalusan side-lobe pada spektrum transmitansi. parameter chirp yang besar mengakibatkan peningkatan transmitansi di sekitar pusat band-gap (grating homogen). ucapan terima kasih artikel ini merupakan bagian dari hasil penelitian fundamental yang dibiayai oleh direktorat penelitian dan pengabdian kepada masyarakat (dp2m) dengan surat perjanjian pelaksanaan hibah penugasan penelitian desentralisasi nomor: 320/sp2h/pp/dp2m/iii/2008, direktorat jenderal pendidikan tinggi, departemen pendidikan nasional. penulis mengucapkan terima kasih kepada a. suryanto (jurusan matematika, universitas brawijaya) atas diskusi dan saran yang sangat bermanfaat pada penulisan artikel ini. daftar pustaka de sterke, c.m., k.r. jackson, b.d. robert, 1991, nonlinear coupled-mode equations on a finite interval: a numerical procedure, j. opt. soc. am. b. 8: 403. joannopoulos, j.d., r.d. meade dan j.n. winn, 1995, photonics crystals, princeton university press, princeton, nj. lei, x-y., h. li, f. din, w. zhang dan n-b. ming, 1997, novel application of a perturbed photonic crystal: high-quality filter, appl. phys. lett. 71: 2889. mandal, j., y. shen, s. pal, t. sun, k.t.v. grattan dan a.t. augousti, 2005, bragg grating tuned fiber laser system for measurement of wider range temperature and strain, opt. comm. 244: 111. scalora, m., j.p. dowling, c.m. bowden dan m.j. bloemer,1994, optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials, phys. rev. lett. 73: 1368. soukoulis, c.m., 1993, photonics band gaps and localization, plenum, new york. soukoulis, c.m., 1996, photonics band gaps materials, kluwer academic, dordrecht. suryanto, a., e. van groesen, m. hammer dan h.j.w.m. hoekstra, 2003a, a finite element scheme to study the nonlinear optical response of a finite grating without and with defect, opt. and quant. electr. 35: 313. suryanto, a., e. van groesen dan m. hammer, 2003b, finite element analysis of optical bistability in one-dimensional nonlinear photonics band gap structures with a defect, j. nonl. opt. phys. & mater. 12:187. suryanto, a. dan i. darti, 2005, perubahan sifat-sifat transmisi gelombang optik pada struktur grating satu dimensi akibat adanya beberapa lapisan cacat, laporan penelitian dasar dp2m-dikti. suryanto, a. 2006, transmission characteristics of one-dimensional photonics bandgap structures with some defects, j. nonl. opt. phys. & mater. 15: 331. suryanto, a. dan i. darti, 2008, pengaruh variasi frekuensi spatial dan pergeseran fasa terhadap karakteristik optical bistability pada fiber bragg grating nonlinear berstruktur sinusoidal, laporan penelitian fundamental dp2m-dikti. suryanto, a., 2009, on the numerical modelling of optical switching in nonlinear phase-shifted grating, j. nonl. opt. phys. & mater. 18: 129. tran, p., 1997, optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect, j. opt. soc. am. b14: 2589. yeh, p., 1988, optical waves in layered media, john wiley & sons, new york. 1a sampul depan identifikasi struktur dependensi dengan copula (aplikasi pada data klimatologi) mutiah salamah1 dan heri kuswanto2 1,2jurusan statistika, institut teknologi sepuluh nopember (its) surabaya e-mail: 1mutiah_s@statistika.its.ac.id, 2heri_k@statistika.its.ac.id abstract paper ini mendiskusikan peranan copula dalam memodelkan struktur dependensi antara dua variabel iklim, yaitu kecepatan angin dan rata-rata tekanan udara permukaan laut. kedua variabel iklim tersebut mempunyai distribusi marginal yang berbeda yang mana kecepatan angin tidak berdistribusi normal dengan kecenderungan heavy-tailed, sedangkan tekanan laut berdistribusi normal. oleh karena itu dependensi keduanya tidak bisa diwakili dengan menggunakan korelasi pearson. masalah ini bisa diatasi dengan copula. hasil analisis menunjukkan bahwa copula-t adalah model terbaik untuk menjelaskan struktur dependensi ataran kedua variabel yang dibahas. keywords: copula, struktur dependensi , heavy-tailed pendahuluan analisis dependensi antara dua atau lebih variabel merupakan subjek penting dalam aplikasi sehari-hari misalnya marketing, percobaan klinis, ekonomi dan bisnis. analisis standar biasanya ditampilkan dalam bentuk satu nilai tingkat korelasi, yang kemudian memberikan petunjuk kepada praktisi mengenai kebijakan yang harus diambil. karena kebijakan yang salah akan mengakibatkan kerugian yang signifikan, maka analisis harus dilakukan secara benar dan teliti. paper ini menyajikan suatu metode yang mendapatkan banyak perhatian dalam beberapa tahun terakhir dalam ilmu statistika yaitu pemodelan copula. ini merupakan suatu metode statistika yang diperuntukkan untuk memodelkan struktir dependensi antara dua atau lebih variabel, dan mempunyai kemampuan untuk mengeksplorasi lebih banyak informasi tentang dependensi daripada korelasi pearson, kendall atau spearman. dalam prakteknya, seringkali prakstisi mengabaikan atau bahkan tidak mengetahui distribusi marginal dari variabel yang dianalisis. korelasio pearson seringkali enjadi pilihan paling mudah dan sederhana untuk mengukur dependensinya. peranan copula menjdi penting ketika satu atau kedua variabel mempunyai distribusi marginal yang tidak normal atau mempunyai tail dependensi. sebagaimana yang kita ketahui, korelasi pearson diperkenalkan dengan mengasusmsikan bahwa variablenya berdistribusi normal. banyak distribusi bivariat atau multivariat yang dikembangkan sebagai alternativ untuk mengatasi kasus ketidaknormalan pada distribusi marginal seperti bivariate gamma (moran (1969), nadarajah dan gupta (2006)), farlie-gumbel-morgenster (conway (1979), bivariate exponential distribution (gumbel (1960)), dan sebagainya. namun, distribusi-distribusi tersebut terbatas pada marginal yang sama, dan mempunyai struktur yang komplek pada fungsi densitas pribabilitas-nya (termasuk struktur dependensiya) yang tentunya tidak disukai dalam praktek. copula datang dengan kefleksibilitasannya, dimana distribusi marginal dari variabel bisa berbeda atau bahkan tidak diketahui. copula telah dapat diaplikasikan dengan baik pada banyak bidang seperti hidrologi (favre et. al. (2004), salvadori dan de michele (2007), genest et. al. (2007)), finance dan asuransi (cerubini et. al .(2004), denuit et. al. (2005), mcneil et.al. (2005)) dan masih banyak lagi. akhir-akhir ini, copula telah dikembangkan pada bidang ekonometrik dan time series seperti pada patton (2002;2009) dan lainnya. paper ini membahas aplikasi copula dalam memodelkan struktur dependensi antara dua variabel iklim yaitu kecepatan angin dan tekanan udara permukaan laut. aplikasi coopula dalam bidang meteorologi masih sangat sedikit dan tergolong masih baru. beberapa diantaranya adalah de michele dan salvadori (2003), vannitsem (2007), vrac et. al. (2005) and schölzel dan friedrich (2008). identifikasi struktur dependensi dengan copula jurnal cauchy – issn: 2086-0382 79 konsep dasar copula bagian ini menjelaskan secara singkat tentang konsep dasar copula. deskripsi yang diberikan terbatas pada dua variabel random atau kasus bivariat. theori yang digambarkan disini tentu saja dapat dikembangkanuntuk kasus multivaiat dengan duaatau lebih variabel random. misalkan kita punya 2 vektor random x dengan kumulatif distribusi marginal ��� dan ���, maka berdasarkan teorema sklar (1959), distribusi bersama dari vektorl random dapat dituliskan sebagai fungsi dari marginal distribusinya sebagai berikut ����� � � �����1�,�����2�� dimana �: �0,1� � �0,1� � �0,1� adalah fungsi distribusi bersama dari variabel random yang ditransformasi �� � ������� untuk � � 1,2. transformasi ini menghasilkan distribusi marginal uniform ��. � adalah fungsi copula dengan densitasnya adalah ��. teorema sklar dia atas mempunyai dua implikasi penting, yaitu probabilitas desitas bersama dapat di ekspresikan sebagai perkalian antara densitas marginal dan densitas copula sebagai berikut: ����� � �����1������2�.�����,��� berbagai fungsi copula telah dikembangkan sejauh ini. namun, kita batasi diskusi dalam paper ini pada tiga copula yang terkait dengan kasus yang kita bahas. mengenai teori copula lebih detail bisa ditemukan pada genest dan favre (2007), nielsen (1999) atau joe (1997). a. copula gumbel dan clayton copula gumbel dan clayton termasuk dalam copula archimedian, yang memungkinkan mempunyai struktur dependensi yang lebih luas. dengan copula archimedian, beberapa copula dapat di bangiktan dengan fungsi generator. dalam bentuk umum, copula archimedian mempunyai bentuk sebagai berikut: ����,��� � � ��������,������ dimana ���� adalah fungsi generator. ���� adalah fungsi tidak turun yang memetakan �0,1� ke dalam sehingga ��0� � ∞ and ��1� � 0. copula gumbel mempunya fungsi generator sebagai berikut: ���� � � log����$ ,%≥1 sedangkan copula clayton ���� � ��$ 1 % ,% & 0. kedua copula di atas memepunyai perilaku tail yang berbeda. copula clayton mempunyai tail dependensi bawah sedangkan copula gumbel mempunyai tail dependensi atas. b. copula-t copula-t termasuk dalam copula elips sebagaimana copula gaussian (copula dengan marginal normal). namun tidak seperti copula normal yang simetri, copula-t mempunyai potensi untuk membangkitkan nilai ekstrim karena t adalah distribusi ynag skew. copula-t didefinisikan sebagai berikut: ����,��� � �'�(,σ� �'�(� �� ����,�'�(� �� ����� dimana �' mendefinisikan cdf dari distribusi t dengan derajat bebas v. copula digunakan secara bersamaan dengan korelasi tau kendall dan spreamann correlation. ini karena kedua korelasi tersebut diurunkan dari rank, yang sesuai dengan teori copula. copula sendiri tidak berubah dengan adanya transformasi. deskripsi data dan pentingnya mempelajari variabel yang dibahas dalam paper ini kita tertarik untuk mengaplikasikan metode copula pada kasus klimatologi. kita tertarik untuk mempelajari struktur dependensi antara kecepatan angin dengan rata-rata tekanan udara permukaan laut (mslp). data diperoleh dari stasiun surabaya / juanda. kita menganalisa data harian selama 9 tahun antara tahun 2000 sampai 2009. ada beberapa pengamatan yang missing dan mereka dihilangkan dari analisis. jumlah data keseluruhan sebanyak 1400 data berpasangan. kecepatan angin diukur dengan km/jam sedangkan mslp diukur dalam mbar. sekarang kita mendiskusikan pentingnya menganalisis kedua variabel tersebut. kecepetan angin (bersamaan dengan arah angin) merupakan variabel penting d alam beberapa disiplin ilmu seperti meteorologi, penerbangan, kelautan, konstruksi dan lain sebagainya. kecepatan angin yang tinggi mempunyai hubungan dengan petir yang hebat serta angin puting beliung yang bis amenyebabkan kerusakan fatal dan kerugian dalam hidup. ini juga merupakan input utama dari pembangkit tenaga angin. berbagai penelitian dalam klimatologi dan meteorologi ditujukan untuk memprediksi kecepatan angin. mempelajari mslp sangat penting terutama untuk bidang yang sensitif terhadap cuaca, seperti pilot, petani, pelaut, dan sebagainya. kecepatan angin dan mutiah salamah dan heri kuswanto 80 volume 1 no. 2 mei 2010 mslp merupakan suatu indikator penting dalam perubahan iklim (harrison dan larkin (1997). mempelajari struktur dependensi antara kedua variabel tersebut juga subjek yang penting. mengembangkan sistem peramalan (model) untuk kecepatan angin membutuhkan mslp sebagai salah satu input. selanjutnya, kedua variabel ini juga merupakan input untuk peramalan variabel iklim yang lain seperti curah hujan, presipitasidan petir sebagaimana diuraikan dalam qian et al. (2000), murphree and van den dool (1988), kutiel (2004). dengan mempertimbangkan struktur dependensi antar variabel tersebut akan dapat memperoleh peramalan yangg lebih reliabel. sebaliknya, mengabaikannya akan menghasilkan analisis yang salah. aplikasi dan pembahasan gambar berikut menampilkan scatter plot antara variabel yang dibahas dalam skala alsinya. kita tidak melihat adanya trend atau tendensi tertentu dari plot. untuk menyimpulkan apakah kedua variabel berkorelasi atau tidak sangat sulit jika kita hany amenggunakan scatter plot. salah satu informasi penting yang bisa dibaca dari gambar 1 adalah, bahwa plot-plotnya terkonsentrasi pada interval tertentu, yang mengindikasikan dependensi atara kedua variabel tersebut. gambar 1. scatter plot antara kecepatan angin dan mslp dalam skala asli gambar 2. histogram (distribusi marginal) dari masing-masing variabel 1004 1006 1008 1010 1012 1014 0 5 1 0 1 5 2 0 2 5 mslp w in d histogram of mslp mslp f re q u e n cy 1004 1006 1008 1010 1012 1014 1016 0 2 0 0 4 0 0 6 0 0 8 0 0 histogram of wind wind f re q u e n cy 0 5 10 15 20 25 30 0 2 0 0 4 0 0 6 0 0 8 0 0 identifikasi struktur dependensi dengan copula jurnal cauchy – issn: 2086-0382 81 meskipun pengetahuan tentang distribusi marginal tidak disyaratkan dalam pemodelan copula, informasi tentang distribusi marginal bermanfaat untuk menjamin bahwa copula memang diperlukan. jika kedua variabel mempunyai marginal normal, maka korelasi pearson dapat diaplikasikan dengan mudah. kita dapat mengduga distribusi marginal secara empiris dengan menggunakan histogram. (gambar 2). histogram sebelah kiri menggambarkan histogram dari mslp dan sebelah kanan mewakili histogram dari kecepatan angin. dari gambar, terlihat bahwa mslp sepertinya simetri dan berdistribusi normal, sedangkan kecepatan angin tidak simetri dan mempunyai tail yang panjang. kenyataan bahwa salah satu variabel tidak berdistribusi normal menyebabkan struktur dependensi dari kedua variabel itu tidak dapat diwakili oleh korelasi pearson. sebagaimana yang dijelaskan sebelumnya, copula mempunyai kemampuan untuk mendeskripsikan struktur dependensi antara variabel dengan marginal yang berbeda dan memodelkan dependensi tailnya. langkah pertama dalam analisis dengan copula adalah mentransformasi variable ke dalam distribusi marginal yang uniform. scatter plot pada gambar 1 kemudian di transformasi ke domain [0,1], seperti yang bisa dilihat pada gambar berikut: gambar 3. scatter plot antara kecepatan angin dan mslp dalam skala transformasi (uniform �0,1�) plot dalam gambar mengkonfirmasi bahwa kedua variabel saling berhubungan, meskipun derajat hubungannya sangat rendah. ini tampak jelas dari plot-plot yang terpusat pada pojok, namun tidak jelas apakan pojok bawah atau atas. bagaimanapun juga, ini suatu indikasi dari dependensi tail. seperti yang didiskusikan sebelumnya, paling tidak ada 3 copula yang mempunyai karakteristik seperti ini yaitu copulat, clayton dan gumbel. gambar 4 mengilustrasikan suatu kasus ideal dengan plot yang dibangkitkan dari 5000 pengamatan. dari plot tersebut, nampak jelas bahwa copula mempunyai karakteristik sendiri terkait dengan dependensi tail. untuk meyakinkan copula mana yang cocok dengan kasus kita, seleksi model perlu dilakukan. 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 u v mutiah salamah dan heri kuswanto 82 volume 1 no. 2 mei 2010 (a) (b) (c) gambar 4. sampel plot untuk (a)copula-t, (b) clayton, dan (c) gumbel 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 t-copula u v 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 clayton-copula u v 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 gumbel-copula u v identifikasi struktur dependensi dengan copula jurnal cauchy – issn: 2086-0382 83 sebelum kita melangkah pada penaksiran model copula, perlu ditampilkan koefisien korelasi hasil estimasi dengan 3 metode yaitu pearson, kendall and spearman. tabel 1. koefisien korelasi pearson kendall spearmann correlation 0.09399 0.1228 0.1793 ketiga ukuran tersebut memberikan nilai yang sangat kecil. khususnya korelasi pearson, nilai korelasiny apaling kecil diantara yang lain. menggunakan ukuran korelasi ini akan mengakibatkan kesimpulan yang bias, yaitu kedua variabel yang dianalisis tidak berhubungan. dalam praktek, kondisi ini akan menyarankan untuk mengabaikan korelasi antara keduanya (jika digunakan sebagai model input dalam peramalan klimatologi) atau mengeluarkan mslp sebagai variabel independen dari kecepatan angin (jika kita ingin meramalkan kecepatan angin). perlu dicatat bahwa korelasi yang lain dihitung dari data yang sudah ditransformasi, dan memberikan nilai yang lebih besar secraa signifikan daripada korelasi pearson. untuk meyakinkan apakah nilai korelasi tersebut signifikan atau tidak, perlu dilakukan pengujian hipotesa. hasil pengujian menujukkan bahwa kedua variabel tersebut berkorelasi secar asignifikan (dengan 95% confidence interval) dengan p-value < 2.2e-16 untuk kendall dan spearmann. oleh karena itu, kita harus mempertimbangkan korelasi tersebut. yang belum teridentifikasi adalah tentang peranan copula dan copula mana yang mewakili struktur dependensi dengan baik. berikut hasil estimasi parameter copula untuk tiga macam copula:. tabel 2. estimasi model copula copula estimasi z log likelihood t 0.1585 9.514 41.76 gumbel 1.0795 93.56 21.48 clayton 0.1644 8.4185 29.754 statistik z menyarankan bahwa parameter dari ketiga copula tersebut signifikan. estimasinya ditampilkan dengan tau kendall. model terbaik akan diseleksi berdasarkan nilai log-likelihood. karena nilai ini mewakili kemungkinan maksimum bahwa model yang diestimasi akan cocok dengan model sebenarnya, maka semakin besar log likelihood akan semakin baik modelnya. dari tabel, copula-t memeberikan nilai paling tinggi. oleh karena itu, kita menyimpulkan bahwa struktur dependensi antara kecepatan angin dan mslp lebih bagus diwakili oleh copula-t dengan parameter 0.1585 bersesuaian dengan 0.1228. hasil ini menyarankan bahw akita harus memperhatikan nilai ekstrim (atas atau bawah). kedua variabel secra asignifikan saling bergantung pada kondisi ekstrim rendah atau tinggi. pengujian kesesuaian model tidak dibahas dalam paper ini. penutup kita telah menujukkan aplikasi copula untuk memodelkan struktur dependensi antara kecepatan angin dengan mslp. korelasi pearson tidak lagi dapat dipakai karena distribusi marginal dari keduanya berbeda. copula dapat menangkap dependensi tail antara kedua variabel dan menyarankan copula-t sebagai model terbaik. hasil yang ditampilkan dalam paper ini adalah kasus khusus per stasiun. oleh sebab itu, stasiun lain mungkin akan mempunyai struktur dependensi yang berbeda. daftar pustaka [1] cherubini, u., luciano, e. and vecchiato, w. (2004).copula methods in finance, wiley. [2] conway, d.a. (1979). farlie-gumbelmorgenstern distributions. in: encyclopedia of statistical sciences, 3, s.kotz and n.l. johnson, editors (john wiley & sons, new york), 28-31. [3] de michele, c. and salvadori, g .(2003). a generalized pareto intensity duration model of storm rainfall exploiting 2-copulas, j. geophys. res., 108(d2); 4067. [4] denuit, m., dhaene, j., goovaerts, m. and kaas, r. (2005). actuarial theory for dependent risks. measures, orders and models. wiley, chichester. [5] embrechts, p., lindskog, f., and mcneil, a.(2001). modelling dependence with copulas and applications to risk management, in: handbook of heavy tailed distributions in finance, edited by: rachev, s., 329–384. [6] genest, c. and favre, a.-c.(2007). everything you always wanted to know about copula modeling but were afraid to ask, j. hydrol.eng.; 12;347–368. [7] genest, c., favre, a.-c., beliveau, j., and jacques, c. (2007). metaelliptical copulas and their use in frequency analysis of multivariate hydrological data, water resour. res., 43; w09401 mutiah salamah dan heri kuswanto 84 volume 1 no. 2 mei 2010 [8] gumbel, e.j. (1960). bivariate exponential distributions. journal of the american statistical association; 55; 698-707. [9] harrison, d. e. and larkin, n. k. (1997). darwin sea level pressure: 1876-1996 : evidence for climate change? geophysical research letters, 24(14); 1779-1782. [10] joe, h. (1997), multivariate models and dependence concepts. chapman and hall, london. [11] kutiel, t. (1991). recent spatial and temporal variations in mean sea level pressure over europe and the middle east, and their influence on the rainfall regime in the galilee, israel. theoretical and applied climatology, 44(3-4). [12] mcneil, a.j., frey, r. and embrechts, p. (2005). quantitative risk management: concepts,techniques, tools. princeton university press, princeton. [13] moran, p. a. p. (1969). statistical inference with bivariate gamma distributions, biometrika; 54; 385–394. [14] murphree, t., and van den dool, h., (1988). calculating tropical winds from time mean sea level pressure fields. journal of the atmospheric sciences, 45; 3269-3282 [15] nadarajah, s. and gupta, a. k. (2006). some bivariate gamma distributions, applied mathematics letters;19(8); 767-774. [16] patton a. j. (2002). applications of copula theory in financial econometrics, june ph.d. dissertation department of economics, university of california, san diego [17] patton, a. j. (2009) . copula-based models for financial time series, 2009, in t.g. andersen, r.a. davis, j.-p. kreiss and t. mikosch (eds.) handbook of financial time series, springer verlag [18] nelsen, r.b. (1999), an introduction to copulas. lecture notes in statistics 139, springer-verlag, new york. [19] qian, b., cortereal, j. and xu, h. (2000). nonseasonal variability of monthly mean level pressure and precipitation variability over europe. physics and chemistry of the earth, part b: hydrology, oceans and atmosphere; 25(2);177-181. [20] salvadori , g. and de michele, c. (2007), on the use of copulas in hydrology: theory and practice. journal of hydrologic engineering, 12(4); 369–380 [21] schölzel, c. and friedrich, p. (2008). multivariate non-normally distributed randomvariable in climate research – introduction to the copula approach. nonlinear process geophys.;25;761-772. [22] sklar, a. (1959). fonctions de repartition an dimensions et leurs marges. publ. inst. statist. univ. paris 8, 229-231. [23] vannitsem, s. (2007). statistical properties of the temperature maxima in an intermediate order quasi-geostrophic model, tellus a; 59; 80–95. [24] vrac, m., chedin, a., and diday, e.(2005). clustering a global field of atmospheric profiles by mixture decomposition of copulas, j. atmos. ocean. tech.; 22; 1445– 1459. ummu habibah kajian model longand shortterm runoff _5_ kajian model longand shortterm runoff (lst) dan implementasinya untuk menghitung debit banjir ummu habibah1 dan suharmadi2 1jurusan matematika, universitas brawijaya, malang email: ummu915@gmail.com 2jurusan matematika, institut teknologi sepuluh nopember, surabaya abstrak air hujan merupakan salah satu aspek dari siklus hidrologi yang berperan penting dalam ketersediaan air di dalam bumi. akan tetapi apabila terjadi hujan lebat dalam durasi waktu yang cukup lama maka air hujan tersebut dapat mengakibatkan terjadinya aliran permukaan (surface runoff) yang berpotensi menimbulkan banjir. untuk mengetahui jumlah potensi air yang ada pada suatu daerah pengaliran, diperlukan perhitungan hidrologi dari data-data curah hujan. untuk menghitung jumlah air atau debit sungai pada waktu banjir digunakan formulasi model longand short-term runoff (lst). formulasi model lst diperoleh dari model fisisnya. pada penelitian ini dikaji proses terbentuknya formulasi model lst dari perilaku sistem berdasarkan fenomena siklus hidrologi. selanjutnya formulasi model lst tersebut akan diimplementasikan untuk menghitung debit banjir pada suatu daerah pengaliran. hasil penelitian menunjukkan bahwa formulasi model lst dapat digunakan untuk menghitung debit banjir dan merupakan model yang baik karena pada saat implementasi, error yang dihasikan antara debit banjir pengamatan dan debit banjir perhitungan adalah kecil. kata kunci: siklus hidrologi, surface runoff , model lst pendahuluan air hujan merupakan salah satu aspek dari siklus hidrologi yang berperan penting dalam ketersediaan air di dalam bumi. akan tetapi apabila terjadi hujan lebat dalam durasi waktu yang cukup lama maka air hujan tersebut dapat mengakibatkan terjadinya aliran permukaan (surface runoff) yang berpotensi banjir. untuk mengetahui jumlah potensi air yang ada pada suatu daerah pengaliran diperlukan perhitungan hidrologi dari data-data curah hujan. untuk menghitung jumlah air atau debit sungai pada waktu banjir digunakan model longand short-term runoff (lst). model ini digunakan untuk menganalisa aliran long-term dan shortterm (banjir) dan dapat juga digunakan untuk meramalkan banjir real time. berdasarkan uraian di atas, maka permasalahan yang akan dibahas pada penelitian ini adalah bagaimana kajian formulasi model lst dan implementasinya untuk menghitung debit sungai pada waktu banjir. sedangkan batasan masalah adalah data yang digunakan untuk mengimplementasikan model lst adalah data sekunder yang didapatkan dari perum jasa tirta yaitu berupa data curah hujan di stasiun dampit dan data tinggi muka air di sungai lesti, sedangkan untuk nilai parameterparameter dan data lainnya menggunakan data artifisial. kajian teori siklus hidrologi air di bumi ini mengulangi terus menerus sirkulasi yang berupa penguapan, presipitasi dan pengaliran keluar (outflow). air menguap ke udara dari permukaan tanah dan laut, berubah menjadi awan sesudah melalui beberapa proses dan kemudian jatuh sebagai hujan atau salju ke permukaan laut atau daratan. sebelum tiba ke permukaan bumi sebagian langsung menguap ke udara dan sebagian tiba ke permukaan tanah. sebagian akan tertahan oleh tumbuh-tumbuhan di mana sebagian akan menguap dan sebagian lagi akan jatuh atau mengalir melalui dahandahan ke permukaan tanah. air hujan yang tiba ke permukaan tanah akan masuk ke dalam tanah (infiltrasi). bagian lain yang merupakan kelebihan akan mengisi lekuk-lekuk permukaan tanah, kemudian mengalir ke daerah-daerah yang rendah, masuk ke sungai-sungai dan akhirnya ke laut. tidak semua butir air yang mengalir akan tiba ke laut. dalam perjalanan ke laut sebagian akan menguap dan kembali ke udara. sebagian air yang masuk ke dalam tanah, keluar kembali segera ke sungaisungai (disebut aliran intra = interflow). tetapi sebagian besar akan tersimpan sebagai air tanah (groundwater) yang akan keluar sedikit demi sedikit dalam jangka waktu yang lama ke permukaan tanah di daerah-daerah yang rendah ummu habibah dan suharmadi 188 volume 1 no. 4 mei 2011 (disebut groundwater runoff = limpasan air tanah). sirkulasi yang kontinu antara air laut dan air daratan berlangsung terus. sirkulasi air ini disebut siklus hidrologi (hydrological cycle). infiltrasi horton konsep infiltrasi horton adalah limpasan permukaan dimulai pada tempat dan saat intensitas curah hujan melampaui suatu tingkat di mana air dapat memasuki tanah. persamaan infiltrasi horton adalah sebagai berikut: ( ) ( )0 expc cf f f f kt= + − − (1) dimana : f : kapasitas infiltrasi / daya serap 0f : kapasitas infiltrasi maksimum (pada awal hujan) cf : kapasitas infiltrasi rendah k : parameter kapasitas infiltrasi t : waktu dari mulainya hujan aliran long-term dan short-term (banjir) model lst adalah model yang terdiri dari tangki-tangki penyimpanan. pada model lst terdapat dua aliran yaitu aliran long-term dan short-term. secara umum yang disebut aliran longterm adalah aliran yang mengalir di suatu daerah pengaliran atau sungai. dalam model lst, yang disebut aliran long-term adalah aliran yang keluar dari tangki penyimpanan dan mengalir menuju ke suatu daerah pengaliran atau sungai. banjir disebut juga dengan aliran shortterm karena aliran banjir terjadi secara langsung ketika hujan turun dan berlangsung dengan cepat. aliran short-term dapat juga mengakibatkan aliran long-term apabila banjir yang terjadi sangat besar sehingga air tersebut akan mengalir ke suatu daerah pengaliran. hukum manning hukum manning dapat digunakan untuk menghitung kecepatan aliran dalam saluran yang didefiniskan dengan : 2 1 3 2 1 = ⋅ ⋅ s v j i k (2) jadi, 2 1 3 2 1 = ⋅ = ⋅ ⋅ ⋅ s q v csa j i csa k 2 3= ⋅ ⋅p j csa (3) dimana : sk : koefisien kekasaran v : kecepatan aliran rata-rata ( / )m s i : gradien/kemiringan permukaan air j : jari-jari hidrolisis ( )m q : debit csa : luas penampang melintang air 2( )m jari-jari hidrolisis (ketinggian sungai) dapat dihitung dengan membandingkan antara luas penampang melintang air ( )csa dengan keliling basah ( )wp . csa j wp = (4) persamaan kontinuitas persamaan kontinuitas pada fluida didefinisikan dengan ( ) ( ) ( ) 0 u v w t x y z ρ ρ ρ ρ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ (5) dimana : ( , , , )x y z tρ ρ= : rapat massa (kg/m3) u : kecepatan pada arah sumbu x (m/s2) v : kecepatan pada arah sumbu y (m/s2) w : kecepatan pada arah sumbu z (m/s2) t : waktu (s) kajian formulasi model lst model lst terdiri dari 3 tangki penyimpanan. pada tangki paling atas dibagi menjadi 2 lapis sehingga jumlah tangki penyimpanan menjadi 4. bentuk model fisis dari model lst adalah seperti gambar di bawah ini: gambar 1. model lst 1q 2q 3q 4s 3s 2s 1s f 1b 1g 4q 2b 2g 3b 2a 3a 4a 5a ( )3z ( )1z 1a 5q ( )2z kajian model long-and short-term runoff (lst) dan implementasinya… jurnal cauchy – issn: 2086-0382 189 dimana : js : simpanan air dalam tangki penyimpanan r : rata-rata curah hujan f : kapasitas infiltrasi (daya serap) je : evapotranspirasi pada permukaan tangki penyimpanan 1q : aliran permukaan (surface runoff) jz : ketinggian batas aliran yang keluar tangki penyimpanan (height of runoff outlet) 2q dan 3q : aliran bawah permukaan (subsurface runoff) yang keluar tangki penyimpanan 4q dan 5q : aliran air tanah (groundwater runoff) yang keluar tangki penyimpanan ia dan jb : parameter-parameter aliran dengan memperhatikan perilaku sistem dan pengetahuan tentang siklus hidrologi maka diperoleh persamaan kontinuitas tiap tangki penyimpanan sebagai berikut : 1 2 1 1 2 3 1 3 4 1 1 4 2 2 3 5 , , ds ds r e f q q f q g dt dt ds ds g e q g g e q dt dt = − − − − = − − = − − − = − − (6) aliran jq dan perembesan jg dihitung sebagai berikut : ( ) ( ) 1 1 1 1 2 2 1 3 3 2 3 1 2 2 4 4 3 5 5 4, 2 3 3 5 , 3 , , , m q a s z m q a s q a s z g b s q a s q a s g b s = − = = = − = = = = (7) dengan mengasumsikan bahwa hukum manning dapat diaplikasikan untuk aliran permukaan, maka 3 5=m dapat digunakan dalam persamaan 1 1 1 1( ) mq a s z= − . tingkat infiltrasi f dari lapisan atas ke lapisan bawah pada tangki atas adalah: ( )1 2 3 2f b z z s= + − (8) ketika terdapat cukup banyak air di lapisan atas, persamaan infiltrasi horton ditunjukkan dalam bentuk parameter-parameter model. dalam kasus 32 zs > , diperoleh hubungan-hubungan dibawah ini: ( ) ( ) ( ) 0 1 2 2 3 3 1 2 3 1 2 exp / = + − −  = + +   = + + c c c f f f f kt f b b z z a b z k k a b b (9) metode dan teknik analisis implementasi model lst untuk menghitung debit banjir langkah-langkah dalam implementasi model lst dapat dilakukan sebagai berikut : gambar 2. bagan alir implementasi formulasi model lst sedangkan langkah-langkah untuk menghitung debit banjir sq menggunakan model lst dapat dilakukan pada bagan alir sebagai berikut: mulai inisialisasi parameter-parameter menggunakan data artifisial menghitung debit banjir sq menggunakan model lst menghitung error 0q dengan sq selesai input: 1. data curah hujan (r) 2. data tinggi muka air ( 0q ) output: plot 0q , sq , 0q dengan sq , error : permukaan tangki penyimpanan mulai 1 simpanan 2 dan aliran 3q infiltrasi horton input: 1. data curah hujan (r) 2. data tinggi muka air ( 0q ) ummu habibah dan suharmadi 190 volume 1 no. 4 mei 2011 gambar 3. bagan alir perhitungan debit banjir menggunakan model lst pada model lst, debit banjir perhitungan sq diestimasi dari debit banjir aktual (pengamatan) yaitu )( 5430 qqqqqs ++−= (10) dimana : sq : debit banjir perhitungan menggunakan formulasi model lst 0q : debit banjir pengamatan (aktual) 3q : aliran bawah permukaan yang keluar dari tangki penyimpanan dua 4q : aliran air tanah yang keluar dari tangki penyimpanan tiga 5q : aliran air tanah yang keluar dari tangki penyimpanan empat hasil dan pembahasan berdasarkan bagan alir pada gambar 2., diperoleh hasil dari implementasi model lst untuk menghitung debit banjir menggunakan script m-file yang ditunjukkan pada tabel 1. dari tabel 1 diperoleh error antara data pengamatan dan perhitungan menggunakan formulasi model lst adalah kecil. tabel 1. perbandingan debit banjir pengamatan dan perhitungan beserta error tanggal debit banjir pengamatan )(m debit banjir dengan model lst )(m error perhitungan dan pengamatan 1 316.1700 316.2600 0.09000 2 316.1700 316.1780 0.01020 3 316.4500 316.4195 0.01988 4 316.0900 316.0944 0.00236 5 316.1200 316.1403 0.02190 6 316.1230 316.1260 0.00166 7 316.1100 316.1310 0.02217 8 316.0600 316.0627 0.00155 9 316.5030 316.5241 0.02221 10 316.4550 316.4576 0.00154 11 316.4300 316.4511 0.02222 12 316.4200 316.4226 0.00153 ... ... … … ... ... … … ... ... … … 31 316.7000 316.7211 0.02222 berikut ini adalah grafik yang menunjukkan debit banjir pengamatan pada bulan januari 2004. gambar 4. hidrograf debit banjir pengamatan untuk debit banjir perhitungan menggunakan formulasi model lst ditunjukkan pada grafik dibawah ini. gambar 5. hidrograf debit banjir perhitungan simpanan 1, aliran 1q dan 2q simpanan 3 dan aliran 4q simpanan 4 dan aliran 5q rembesan 1g rembesan 2g selesai output : sq 1 kajian model long-and short-term runoff (lst) dan implementasinya… jurnal cauchy – issn: 2086-0382 191 selanjutnya perbandingan debit banjir pengamatan dan perhitungan ditunjukkan pada grafik dibawah ini gambar 6. hidrograf debit banjir pengamatan dan perhitungan kemudian untuk error antara debit pengamatan dan debit banjir perhitungan ditunjukkan pada grafik berikut. gambar 7. grafik error antara pengamatan dan perhitungan dari perhitungan debit banjir menggunakan model lst pada tabel 1 dan gambar 7 di atas terlihat bahwa error antara data aktual dengan data perhitungan adalah kecil sehingga dapat disimpulkan bahwa model lst adalah model yang baik untuk menghitung debit banjir. penutup hasil penelitian menunjukkan bahwa formulasi model lst dapat digunakan untuk menghitung debit banjir dan merupakan model yang baik karena pada saat implementasi, error yang dihasikan antara debit banjir pengamatan dan debit banjir perhitungan adalah kecil yaitu antara 0.00153 sampai dengan 0.09000. daftar pustaka [1] bellomo, n. dan preziosi, l., (1994), modelling mathematical methods and scientific computation, politecnico di torino, torino. [2] de smedt, f., (1988), introduction to river water quality management, interuniversity post-graduate programme in hidrology, vrije universiteit brussel. [3] direktorat jenderal pengairan, (1974), analisa run-off dengan metode storage function, seminar pengairan rainfall & run off relation and design flood (dpma bandung), 27-30 agustus, 6, jilid i. [4] direktorat jenderal pengairan, (1974), analisa run-off dengan metode storage function, seminar pengairan rainfall & run off relation and design flood (dpma bandung), 27-30 agustus, 6, jilid ii. [5] hanselman, d., dan littlefield, b., (2001), “mastering matlab 6 a comprehensive tutorial and reference”, prentice hall, new jersey. [6] linsley, r., kohler, m., dan paulus, j., (1982), ” hydrologi for engineers”, in: hidrologi untuk insinyur, ed: sianipar, t. dan haryadi, e., penerbit erlangga, jakarta. [7] nagai, a, (2002), hydrologic modeling of rainfall-runoff process and its application to real-time flood forescating. [8] penny, j., lindfield, g., (2000), “numerical methods using matlab”, second edition, prentice hall, new jersey. [9] http://watercycle.gsfc.nasa.gov/images/ watergraphic_low.jpg the first zagreb index, the wiener index, and the gutman index of the power of dihedral group cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 513-520 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 07, 2022 reviewed: february 02, 2023 accepted: february 25, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.16991 the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani1, sahin two lestari1, dara purnamasari1, abdul gazir syarifudin2, salwa1, i gede adhitya wisnu wardhana1* 1 department of mathematics, faculty of mathematics and natural sciences, university of mataram, indonesia 2department of magister mathematics, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia. email: adhitya.wardhana@unram.ac.id abstract research on graphs combined with groups is an interesting topic in the field of combinatoric algebra where graphs are used to represent a group. one type of graph representation of a group is a power graph. a power graph of the group 𝐺 is defined as a graph whose vertex set is all elements of 𝐺 and two distinct vertices 𝑎 and 𝑏 are adjacent if and only one vertice is the power of other vertice. in addition to mathematics, graph theory can be applied to various fields of science, one of which is chemistry, which is related to topological indices. in this study, the topological indexes will be discussed, namely the zagreb index, the wiener index, and the gutman index of the power graph of the dihedral group where 𝑛 is a prime power. the method used in this research is a literature review. for the main result, we gives the first zagreb index, wiener index, and gutman index of the power graph of the dihedral group. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: first zagreb index; wiener index; gutman index; power graph; dihedral group introduction in mathematics, graph theory has many uses, especially in algebraic structures where graphs are used to represent a group. many types of graphs are developed from a group, one of which is a power graph. the first power graph introduced by kalarev in 2013 [1] is to define a directed power graph of a semigroup. and motivated by this, askin and buyukkose discuss the undirected power graph of semigroups and groups [2]. recently, there have been many studies discussing the power graph of a group, one of which is the study by asmarani et al. which deals with the power graph of a dihedral group when 𝑛 = 𝑝𝑚 where 𝑝 is a prime number and an 𝑚 is a natural number [3]–[5]. besides mathematics, graph theory has benefits in other fields, one of which is chemistry, which is related to topological indices. topological indices represent chemical structures and are useful for predicting the chemical and physical properties of molecular structures. not only researching graphs related to chemical structures but over time research on topological indices has developed to examine graphs in general. several types http://dx.doi.org/10.18860/ca.v7i4.16991 mailto:adhitya.wardhana@unram.ac.id https://creativecommons.org/licenses/by-sa/4.0/ the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 514 of topological indexes are interesting to discuss. some of them are the first zagreb index, the wiener index, and the gutman index [6]. research on the topological index of a graph, especially graphs related to groups is interesting to do. several previous research results discuss the topological index of graphs related to groups, namely the topological index of the non-commuting graph of a dihedral group, the szeged and wiener indices for the coprime graph of the dihedral group [7], and connectivity indices of the coprime graph of generalized quaternion group [8]. for other graph representations of a group, see [8]–[14]. recently, not many studies have investigated the topological indices of graphs associated with groups, especially the power graphs of dihedral groups. therefore, in this study, topological indices will be discussed, namely the first zagreb index, wiener index, and the gutman index of the power graph of a dihedral group when 𝑛 = 𝑝𝑚 where 𝑝 is a prime number and an 𝑚 is a natural number. methods this study uses a deductive proof method to find new knowledge from an algebraic structure from a previous study. we start by studying the algebraic structure for several cases looking for a pattern. and with the foundation of the pattern, we stated the conjecture for a general case, if the conjecture is proven by deductive proof, the conjecture is stated as a theorem. results and discussion preliminaries in this section, we present some definitions and theorems that are needed in this research. definition 1 [15] group 𝐺 is said to be a dihedral group of order 2𝑛, 𝑛 ≥ 3, and 𝑛 ∈ ℕ, is a group composed of two elements 𝑎,𝑏 ∈ 𝐺 with the property 𝐺 = 〈𝑎,𝑏|𝑎𝑛 = 𝑒,𝑏2 = 𝑒,𝑏𝑎𝑏−1 = 𝑎−1〉 the dihedral group of order 2n is denoted by 𝐷2𝑛. definition 2 [5] power graph of group g denoted by 𝒢(𝐺) is an undirected graph whose vertex set is g and two vertices 𝑎,𝑏 ∈ 𝐺 are adjacent if and only if 𝑎 ≠ 𝑏 and 𝑎𝑚 = 𝑏 or 𝑏𝑚 = 𝑎 for some positive integer 𝑚. we will give some topological indices of the power graph such as the zagreb index, wiener index, and gutman index. the definitions are as follows definition 3 [16] let 𝒢 be a simple connected graph. the first zagreb index of 𝒢, denoted by 𝑀1(𝒢), is defined as 𝑀1(𝒢) = ∑ (deg(𝑣)) 2 𝑣∈𝑉(𝒢) where 𝑑𝑒𝑔(𝑣) is the number of edges that incident to 𝑣. definition 4 [2] let 𝒢 be a simple connected graph. the wiener index of 𝒢, denoted by 𝑊(𝒢), is defined as the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 515 𝑊(𝒢) = ∑ 𝑑(𝑢, 𝑣) 𝑢,𝑣∈𝑉(𝒢) where 𝑑(𝑢,𝑣) is the shortest distance between vertex 𝑢 and 𝑣. definition 5 [17] let 𝒢 be a simple connected graph. the gutman index of 𝒢, denoted by 𝐺𝑢𝑡(𝒢), is defined as 𝐺𝑢𝑡(𝒢) = ∑ deg(𝑢)deg(𝑣)𝑑(𝑢,𝑣) 𝑢,𝑣∈𝑉(𝒢) where 𝑑𝑒𝑔(𝑢) and deg⁡(𝑣) are the number of edges that incident to 𝑢 and 𝑣 and 𝑑(𝑢,𝑣) are the shortest distance between vertex 𝑢 and 𝑣. theorem 1 [3] if 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers, then the power graph of a dihedral group is a graph that has two non-disjoint subgraphs, namely a complete subgraph and a star subgraph. example 1 power graph of the dihedral group 𝐷2.3 as shown in the following figure figure 1. power graph of the dihedral group 𝐷2.3 theorem 2 [3] the vertex degree of the power graph of a dihedral group when 𝑛 = 𝑝𝑚 where 𝑝 prime numbers and an 𝑚 natural numbers are a. deg(𝑒) = 2𝑛 − 1 b. deg(𝑎𝑖) = 𝑛 − 1 for every 𝑖 ∈ ℤ,1 ≤ 𝑖 ≤ 𝑛 − 1 c. deg(𝑎𝑗𝑏) = 1 for every 𝑗 ∈ ℤ, 0 ≤ 𝑗 ≤ 𝑛 − 1 main result if 𝑛 = 𝑝𝑚 where 𝑝 is prime and an 𝑚 natural number then the first zagreb index, wiener index, and gutman index of the power graph of a dihedral group, respectively is 𝑛2(𝑛 + 1), 7𝑛2 2 − 5𝑛 2 , 1 2 ⁡(𝑛4 + 𝑛) + 3 2 (𝑛3 − 𝑛2) as shown in the following theorem. theorem 3 if 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers, then the zagreb index of the power graph of the dihedral group 𝐷2𝑛 is 𝑛 2(𝑛 − 1). proof. 𝑀1(𝒢(𝐷2𝑛))⁡⁡⁡⁡⁡⁡⁡⁡= ∑ deg(𝑢) 2 𝑢∈𝑉(𝒢(𝐷2𝑛)) the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 516 = deg(𝑒)2 .1 + ∑ deg(𝑎𝑖) 2 𝑛−1 𝑖=1 + ∑ deg(𝑎𝑖𝑏) 2 𝑛−1 𝑖=0 = (2𝑛 − 1)2. 1 + (𝑛 − 1)2(𝑛 − 1) + 12.𝑛 = 4𝑛2 − 4𝑛 + 1 + (𝑛 − 1)3 + 𝑛 = 4𝑛2 − 4𝑛 + 1 + 𝑛3 − 3𝑛2 + 3𝑛 − 1 + 𝑛 = 𝑛3 + 𝑛2 = 𝑛2(𝑛 + 1) theorem 4 if 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers then the wiener index of the power graph of the dihedral group 𝐷2𝑛 is 7𝑛2 2 − 5𝑛 2 . proof. let 𝐷2𝑛 = {𝑒, 𝑎,𝑎 2,…,𝑎𝑛−1,𝑏,𝑎𝑏,…,𝑎𝑛−1𝑏}, a dihedral group with 𝑛 = 𝑝𝑚 where 𝑝 is a number prime and 𝑚 natural number then the dihedral group can be partitioned into 3 partitions namely 𝑉1 = {𝑒}, 𝑉2 = {𝑎,𝑎 2,…,𝑎𝑛−1⁡} and 𝑉3 = {𝑏,𝑎𝑏,𝑎 2⁡𝑏, …, 𝑎𝑛−1𝑏}. to prove the wiener index of the power graph of a dihedral group can be divided into 4 cases. case 1. for 𝑒 ∈ 𝑉1 and 𝑥 ∈ 𝑉(𝒢(𝐷2𝑛) where 𝑒 ≠ 𝑣, obtained ∑ 𝑑(𝑒,𝑥) 𝑥∈𝐷2𝑛 ∗ = (2𝑛 − 1).1 = 2𝑛 − 1 case 2 for 𝑎𝑖,𝑎𝑗 ∈ 𝑉2 where 1 ≤ 𝑖 ≤ 𝑛 − 1, 1 ≤ 𝑗 ≤ 𝑛 − 1, and 𝑖 ≠ 𝑗 obtained ∑ 𝑑(𝑎𝑖,𝑎𝑗)⁡= ( 𝑛 − 1 2 ).1 1≤𝑖≤𝑛−1 ⁡1≤𝑗≤𝑛−1 𝑖≠𝑗 = (𝑛 − 1)! 2!(𝑛 − 3)! = (𝑛 − 1)(𝑛 − 2) 2 = 𝑛2 2 − 3𝑛 2 + 1 case 3 for 𝑎𝑐𝑏,𝑎𝑑𝑏 ∈ 𝑉3 where 0 ≤ 𝑐 ≤ 𝑛 − 1, 0 ≤ 𝑑 ≤ 𝑛 − 1, and 𝑐 ≠ 𝑑 obtained ∑ 𝑑(𝑎𝑐𝑏, 𝑎𝑑𝑏) = ( 𝑛 2 ).2 0≤𝑐≤𝑛−1 0≤𝑑≤𝑛−1 𝑐≠𝑑 = 𝑛! 2! (𝑛 − 2)! .2 = 𝑛(𝑛 − 1) = 𝑛2 − 𝑛 case 4 for 𝑎𝑒 ∈ 𝑉2 and 𝑎 𝑓𝑏 ∈ 𝑉3 where 1 ≤ 𝑒 ≤ 𝑛 − 1, 0 ≤ 𝑓 ≤ 𝑛 − 1 obtained the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 517 ∑ 𝑑(𝑎𝑒,𝑎𝑓𝑏)⁡⁡⁡⁡⁡⁡= (𝑛 − 1)𝑛.2 1≤𝑒≤𝑛−1 0≤𝑓≤𝑛−1 = 2𝑛2 − 2𝑛 based on definition 4 and the four cases, the wiener index of the power graph of the dihedral group 𝐷2𝑛 when 𝑛 = 𝑝 𝑚 for a prime number 𝑝 and 𝑚 natural numbers is: 𝑊((𝒢(𝐷2𝑛) = ∑ 𝑑(𝑢, 𝑣) 𝑢,𝑣∈𝑉(𝒢(𝐷2𝑛) =⁡ ∑ 𝑑(𝑒,𝑥) 𝑥∈𝐷2𝑛 ∗ + ∑ 𝑑(𝑎𝑖,𝑎𝑗) 1≤𝑖≤𝑛−1 ⁡1≤𝑗≤𝑛−1 𝑖≠𝑗 +⁡ ∑ 𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏) 0≤𝑐≤𝑛−1 0≤𝑑≤𝑛−1 𝑐≠𝑑 ⁡ + ∑ 𝑑(𝑎𝑒,𝑎𝑓𝑏) 1≤𝑒≤𝑛−1 0≤𝑓≤𝑛−1 = (2𝑛 − 1) + ( 𝑛2 2 − 3𝑛 2 + 1) + (𝑛2 − 𝑛) + (2𝑛2 − 2𝑛) = 7𝑛2 2 − 5𝑛 2 teorema 5 if 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers then the gutman index of the power graph of the dihedral group 𝐷2𝑛 is 1 2 (𝑛4 + 𝑛) + 3 2 (𝑛3 − 𝑛2). proof. let 𝐷2𝑛 = {𝑒, 𝑎,𝑎 2,…,𝑎𝑛−1,𝑏,𝑎𝑏,…,𝑎𝑛−1𝑏}, a dihedral group with 𝑛 = 𝑝𝑚 where 𝑝 is a number prime and an 𝑚 natural number then the dihedral group can be partitioned into 3 partitions namely 𝑉1 = {𝑒}, 𝑉2 = {𝑎,𝑎 2,…,𝑎𝑛−1⁡} and 𝑉3 = {𝑏,𝑎𝑏,𝑎 2⁡𝑏, …, 𝑎𝑛−1𝑏}. to prove the gutman index of the power graph of a dihedral group can be divided into 5 cases. case 1 for 𝑒 ∈ 𝑉1 and 𝑎 𝑖 ∈ 𝑉2 where 1 ≤ 𝑖 ≤ 𝑛 − 1, obtained ∑ (deg(𝑒) .deg(𝑎𝑖)).𝑑(𝑒,𝑎𝑖) = (((2𝑛 − 1)(𝑛 − 1))1)(𝑛 − 1) 1≤𝑖≤𝑛−1 = 2𝑛3 − 5𝑛2 + 4𝑛 − 1 case 2 for 𝑒 ∈ 𝑉1 and 𝑎 𝑗𝑏 ∈ 𝑉3 where 0 ≤ 𝑗 ≤ 𝑛 − 1, obtained ∑ (deg(𝑒).deg(𝑎𝑗𝑏)).𝑑(𝑒,𝑎𝑗𝑏) = (((2𝑛 − 1)1)1)𝑛 0≤𝑗≤𝑛−1 = 2𝑛2 − 𝑛 case 3 for 𝑎𝑘, 𝑎𝑙 ⁡∈ 𝑉2 where 1 ≤ 𝑘 ≤ 𝑛 − 1, 1 ≤ 𝑙 ≤ 𝑛 − 1, and 𝑘 ≠ 𝑙, obtained ∑ (deg(𝑎𝑘) .deg(𝑎𝑙)).𝑑(𝑎𝑘,𝑎𝑙) = (((𝑛 − 1)(𝑛 − 1))1) 1≤𝑘≤𝑛−1 1≤𝑙≤𝑛−1 𝑘≠𝑙 ( 𝑛 − 1 2 ) = (𝑛 − 1)(𝑛 − 1) (𝑛 − 1)! 2!(𝑛 − 3)! = (𝑛 − 1)3(𝑛 − 2) 2 = 𝑛4 2 − 5𝑛3 2 + 9𝑛2 2 − 7𝑛 2 + 1 case 4 the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 518 for 𝑎𝑐𝑏,𝑎𝑑𝑏⁡ ∈ 𝑉3 where 0 ≤ 𝑐 ≤ 𝑛 − 1,⁡0 ≤ 𝑑 ≤ 𝑛 − 1, and 𝑐 ≠ 𝑑, obtained ∑ (deg(𝑎𝑐𝑏).deg(𝑎𝑑𝑏)).𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏) = (1.1.2) ⁡0≤𝑐≤𝑛−1 0≤𝑑≤𝑛−1 𝑐≠𝑑 ( 𝑛 2 ) = 2( 𝑛 2 ) = 2 𝑛! 2! (𝑛 − 2)! = 𝑛(𝑛 − 1) = 𝑛2 − 𝑛 case 5 for 𝑎𝑒 ∈ 𝑉2 and 𝑎 𝑓𝑏 ∈ 𝑉3 where 1 ≤ 𝑒 ≤ 𝑛 − 1,⁡0 ≤ 𝑓 ≤ 𝑛 − 1, obtained ∑ (deg(𝑎𝑒). deg(𝑎𝑓𝑏)).𝑑(𝑎𝑒,𝑎𝑓𝑏)⁡= (((𝑛 − 1)1)2) 1≤𝑒≤𝑛−1 0≤𝑓≤𝑛−1 (𝑛 − 1)𝑛 = 2𝑛(𝑛 − 1)(𝑛 − 1) = 2𝑛(𝑛2 − 2𝑛 + 1) = 2𝑛3 − 4𝑛2 + 2𝑛 based on definition 5 and the five cases, the gutman index of the power graph of the dihedral group 𝐷2𝑛 when 𝑛 = 𝑝 𝑚 for a prime number p and m natural numbers is: 𝐺𝑢𝑡(𝒢(𝐷2𝑛)) = ∑ (deg(𝑢).deg(𝑣)).𝑑(𝑢,𝑣) 𝑢,𝑣∈𝑉(𝒢(𝐷2𝑛) = ∑ (deg(𝑒). deg(𝑎𝑖)).𝑑(𝑒,𝑎𝑖) 1≤𝑖≤𝑛−1 + ∑ (deg(𝑒).deg(𝑎𝑗𝑏)).𝑑(𝑒,𝑎𝑗𝑏)⁡ 0≤𝑗≤𝑛−1 + ∑ (deg(𝑎𝑘) .deg(𝑎𝑙)).𝑑(𝑎𝑘,𝑎𝑙) 1≤𝑘≤𝑛−1 1≤𝑙≤𝑛−1 𝑘≠𝑙 + ∑ (deg(𝑎𝑐𝑏). deg(𝑎𝑑𝑏)).𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏) ⁡0≤𝑐≤𝑛−1 0≤𝑑≤𝑛−1 𝑐≠𝑑 + ∑ (deg(𝑎𝑒) .deg(𝑎𝑓𝑏)).𝑑(𝑎𝑒,𝑎𝑓𝑏) 1≤𝑒≤𝑛−1 0≤𝑓≤𝑛−1 = (2𝑛3 − 5𝑛2 + 4𝑛 − 1) + (2𝑛2 − 𝑛) + ( 𝑛4 2 − 5𝑛3 2 + 9𝑛2 2 − 7𝑛 2 + 1) + (𝑛2 − 𝑛) + (2𝑛3 − 4𝑛2 + 2𝑛) = 𝑛4 2 + 3𝑛3 2 − 3𝑛2 2 + 𝑛 2 = 1 2 (𝑛4 + 𝑛) + 3 2 (𝑛3 − 𝑛2) the first zagreb index, the wiener index, and the gutman index of the power of dihedral group evi yuniartika asmarani 519 conclusions the results obtained from this study are the first zagreb index, wiener index, and gutman index of the power graph for the dihedral group 𝐷2𝑛 where 𝑛 = 𝑝 𝑚, 𝑝 is prime and 𝑚 a natural number respectively is 𝑛2(𝑛 + 1), 7𝑛2 2 − 5𝑛 2 , 1 2 (𝑛4 + 𝑛) + 3 2 (𝑛3 − 𝑛2). references [1] j. abawajy, a. kelarev, and m. chowdhury, “power graphs: a survey,” 2013. 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[16] t. mansour, m. a. rostami, e. suresh, and g. b. a. xavier, “on the bounds of the first reformulated zagreb index,” turkish journal of analysis and number theory, vol. 4, no. 1, pp. 8–15, jan. 2016, doi: 10.12691/tjant-4-1-2. [17] j. p. mazorodze, s. mukwembi, and t. vetrík, “the gutman index and the edgewiener index of graphs with given vertex-connectivity,” discussiones mathematicae graph theory, vol. 36, no. 4, pp. 867–876, 2016, doi: 10.7151/dmgt.1900. the ring homomorphisms of matrix rings over skew generalized power series rings cauchy –jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 129-135 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 25, 2021 reviewed: august 26, 2021 accepted: october 11, 2021 doi: https://doi.org/10.18860/ca.v7i1.13001 the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol1, fitriani2 1,2department of mathematics, faculty of mathematics and natural sciences universitas lampung, bandar lampung email: ahmadfaisol@fmipa.unila.ac.id, fitriani.1984@fmipa.unila.ac.id abstract let 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) be a matrix rings over skew generalized power series rings, where 𝑅1, 𝑅2 are commutative rings with an identity element, (𝑆1, ≤1), (𝑆2, ≤2) are strictly ordered monoids, 𝜔1: 𝑆1 → 𝐸𝑛𝑑(𝑅1), 𝜔2: 𝑆2 → 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. in this research, we define a mapping 𝜏 from 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) to 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≤1) → (𝑆2, ≤2), and ring homomorphisms 𝜇: 𝑅1 → 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≤1, 𝜔1]] → 𝑅2[[𝑆2, ≤2, 𝜔2]]. furthermore, we prove that 𝜏 is a ring homomorphism, and also we give the sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism. keywords: matrix rings; homomorphisms; skew generalized power series rings. introduction in [1], it has been explained that a matrix is an arrangement of mathematical objects in rectangular rows and columns enclosed by square brackets or regular brackets. these mathematical objects are commonly called entries. if the matrix entries are members of a ring, the matrix is called the matrix over the ring [2]. a ring is a nonempty set with two binary operations and satisfies several axioms [3]. the skew generalized power series rings (sgpsr) 𝑅[[𝑆, ≤, 𝜔]] is one example of a ring [4]. this ring is defined as the set of all functions 𝑓 from a strictly ordered monoid (𝑆, ≤) to a ring 𝑅 with an identity element, that supp(𝑓) is artinian and narrow, with pointwise addition operation and convolution multiplication operation using a monoid homomorphism 𝜔: 𝑆 → end(𝑅). some research related to the properties of sgpsr 𝑅[[𝑆, ≤, 𝜔]], can be seen in mazurek et al. [5]-[10] and faisol et al. [11]-[16]. a set of matrices over a ring that forms a ring under matrix addition and matrix multiplication is called a matrix ring [17]. furthermore, the set of all 𝑛 × 𝑛 matrices with entries in ring 𝑅 is a matrix ring denoted by 𝑀𝑛 (𝑅). in 2021, rugayah et al. [18] have constructed the set of all matrices over sgpsr 𝑅[[𝑆, ≤, 𝜔]], denoted by 𝑀𝑛 (𝑅[[𝑆, ≤, 𝜔]]). moreover, they have defined the ideal of matrix ring over sgpsr 𝑅[[𝑆, ≤, 𝜔]] and studied its ideal properties. one of the essential concepts in the ring structure is a ring homomorphism, a mapping from ring to ring that preserves binary operations on these rings. in [19], the matrix ring homomorphism from 𝑀𝑛(𝑅1) to 𝑀𝑛 (𝑅2) defined by 𝜎([𝑎𝑖𝑗 ]) = [𝜇(𝑎𝑖𝑗 )] for https://doi.org/10.18860/ca.v7i1.13001 mailto:ahmadfaisol@fmipa.unila.ac.id mailto:fitriani.1984@fmipa.unila.ac.id the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol 130 every 𝑎𝑖𝑗 ∈ 𝑅1 where 𝜇: 𝑅1 → 𝑅2 is a ring homomorphism has constructed. several studies related to matrix ring homomorphism can be seen in [20],[21]. this construction motivates us to study the ring homomorphism on the ring matrix over sgpsr 𝑅[[𝑆, ≤ , 𝜔]]. therefore, in this research, matrix rings over the sgpsr 𝑅[[𝑆, ≤, 𝜔]] were constructed, i.e., 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) where 𝑅1, 𝑅2 are rings, (𝑆1, ≤1), (𝑆2, ≤2) are strictly ordered monoids, and 𝜔1: 𝑆1 → 𝐸𝑛𝑑(𝑅1), 𝜔2: 𝑆2 → 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. next, the maping 𝜏 from 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) to 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) is defined by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≤1) → (𝑆2, ≤2), and ring homomorphisms 𝜇: 𝑅1 → 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≤1, 𝜔1]] → 𝑅2[[𝑆2, ≤2, 𝜔2]]. furthermore, it is proved that 𝜏 is a matrix ring homomorphism, and the sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism are also given. methods the method used in this research is a literature study from books and scientific journals. the following steps can be obtained in the results. we construct the matrix rings over sgpsr 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]), where 𝑅1, 𝑅2 are given rings, strictly ordered monoid (𝑆1, ≤1), (𝑆2, ≤2), strictly ordered monoid homomorphism 𝛿: (𝑆1, ≤1) → (𝑆2, ≤2), and monoid homomorphisms 𝜔1: 𝑆1 → end(𝑅1), 𝜔2: 𝑆2 → end(𝑅2). next, we define a mapping τ from mn(r1[[s1, ≤1, ω1]]) to mn(r2[[s2, ≤2, ω2]]), by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≤1) → (𝑆2, ≤2), ring homomorphisms 𝜇: 𝑅1 → 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≤1, 𝜔1]] → 𝑅2[[𝑆2, ≤2, 𝜔2]]. furthermore, we prove that τ is a ring homomorphism. finally, we give sufficient conditions for τ to be a monomorphism, epimorphism, and isomorphism. results and discussion mazurek and ziembowski [4] give the structure of skew generalized power series rings (sgpsr) as follows. let 𝑅1, 𝑅2 are rings, (𝑆1, ≤1), (𝑆2, ≤2) are strictly ordered monoids, and 𝜔1: 𝑆1 → 𝐸𝑛𝑑(𝑅1), 𝜔2: 𝑆2 → 𝐸𝑛𝑑(𝑅2) are monoid homomorphisms. homomorphic image of 𝜔1 and 𝜔2 are denoted by 𝜔1 𝑠 and 𝜔2 𝑢 for all 𝑠 ∈ 𝑆1 and ∈ 𝑆2 . therefore, 𝜔1 𝑠+𝑡 = 𝜔1(𝑠 + 𝑡) = 𝜔1(𝑠) + 𝜔1(𝑡) = 𝜔1 𝑠 + 𝜔1 𝑡 , (1) and 𝜔2 𝑢+𝑣 = 𝜔2(𝑢 + 𝑣) = 𝜔2(𝑢) + 𝜔2(𝑣) = 𝜔2 𝑢 + 𝜔2 𝑣 , (2) for every all 𝑠, 𝑡 ∈ 𝑆1 and 𝑢, 𝑣 ∈ 𝑆2. next, let 𝑅1[[𝑆1, ≤1, 𝜔1]] = {𝑓: 𝑆1 → 𝑅1|supp(𝑓) artinian and narrow} and 𝑅2[[𝑆2, ≤2, 𝜔2]] = {𝛼: 𝑆2 → 𝑅2|supp(𝛼) artinian and narrow}, where supp(𝑓) = {𝑠 ∈ 𝑆1|𝑓(𝑠) ≠ 0} and supp(𝛼) = {𝑢 ∈ 𝑆2|𝛼(𝑢) ≠ 0}. under pointwise addition and convolution multiplication defined by (𝑓 + 𝑔)(𝑠) = 𝑓(𝑠) + 𝑔(𝑠), (3) (𝛼 + 𝛽)(𝑢) = 𝛼(𝑢) + 𝛽(𝑢), (4) and (𝑓𝑔)(𝑠) = ∑ 𝑓(𝑥)𝜔1 𝑥 (𝑔(𝑦))𝑥+𝑦=𝑠 , (5) (𝛼𝛽)(𝑢) = ∑ 𝛼(𝑝)𝜔2 𝑝 (𝛽(𝑞))𝑝+𝑞=𝑢 , (6) 𝑅1[[𝑆1, ≤1, 𝜔1]] and 𝑅2[[𝑆2, ≤2, 𝜔2]] be a skew generalized power series rings, for every 𝑠 ∈ 𝑆1, 𝑓, 𝑔 ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]], and 𝑢 ∈ 𝑆2, 𝛼, 𝛽 ∈ 𝑅2[[𝑆2, ≤2, 𝜔2]]. the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol 131 according to [22], a strictly ordered monoid homomorphism δ: (s1, ≤1) → (s2, ≤2) is a monoid homomorphism such that if s <1 t, then δ(s) <2 δ(t) for every s, t ∈ s1. now, let δ be a monomorphism such that for any artinian and narrow subset 𝑇 of s1, δ(𝑇) is an artinian and narrow subset of s2, and μ: r1 → r2 is a ring homomorphism such that for every s ∈ s1 the following diagram is commutative: 𝑆1 𝑓 ↓ 𝑅1 𝛿 → 𝜇 → 𝑆2 ↓ 𝛼 𝑅2 𝜔1 𝑠 ↓ ↻ ↓ 𝜔2 𝛿(𝑠) 𝑅1 𝜇 → 𝑅2 figure 1. commutative diagram 𝜔2 𝛿(𝑠) ∘ 𝜇 = 𝜇 ∘ 𝜔1 𝑠 for 𝑓 ∈ r1[[s1, ≤1, ω1]], let 𝛼: 𝑆2 → 𝑅2 be the map defined as follows: 𝛼(𝑡) = { 𝜇 ∘ 𝑓 ∘ 𝛿 −1(𝑡) if 𝑡 ∈ 𝛿(𝑆1) 0 otherwise. (7) since supp(𝛼) ⊆ 𝛿(supp(𝑓)), based on [23](1.(a)), 𝛼 ∈ 𝑅2[[𝑆2, ≤2, 𝜔2]]. therefore, we can define a map 𝜎: 𝑅1[[𝑆1, ≤1, 𝜔1]] → 𝑅2[[𝑆2, ≤2, 𝜔2]] by putting 𝜎(𝑓) = 𝛼 in (7). according to [24](lemma 8.1.6), the map σ: r1[[s1, ≤1, ω1]] → r2[[s2, ≤2, ω2]] is a ring homomorphism, and ker(σ) = (ker(μ))[[s1, ≤1, ω1]]. now, we construct the matrix rings 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]), that are the sets of all matrices over sgpsr 𝑅1[[𝑆1, ≤1, 𝜔1]] and 𝑅2[[𝑆2, ≤2, 𝜔2]] defined by 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) = {[𝑓𝑖𝑗 ]|𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]]; 𝑖, 𝑗 = 1, 2, ⋯ , 𝑛}, (8) and 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) = {[𝛼𝑖𝑗 ]|𝛼𝑖𝑗 ∈ 𝑅2[[𝑆2, ≤2, 𝜔2]]; 𝑖, 𝑗 = 1, 2, ⋯ , 𝑛}, (9) with addition matrix operation [𝑓𝑖𝑗 ] + [𝑔𝑖𝑗 ] = [𝑓𝑖𝑗 + 𝑔𝑖𝑗 ] , (10) [𝛼𝑖𝑗 ] + [𝛽𝑖𝑗 ] = [𝛼𝑖𝑗 + 𝛽𝑖𝑗 ] , (11) and multiplication matrix operation [𝑓𝑖𝑗 ][𝑔𝑖𝑗 ] = [ℎ𝑖𝑗 ] , (12) [𝛼𝑖𝑗 ][𝛽𝑖𝑗 ] = [𝛾𝑖𝑗 ] , (13) where ℎ𝑖𝑗 = ∑ 𝑓𝑖𝑘 𝑔𝑘𝑗 𝑛 𝑘=1 dan 𝛾𝑖𝑗 = ∑ 𝛼𝑖𝑘 𝛽𝑘𝑗 𝑛 𝑘=1 , for every [𝑓𝑖𝑗 ], [𝑔𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) and [𝛼𝑖𝑗 ], [𝛽𝑖𝑗 ] ∈ 𝑀𝑛(𝑅2𝑅2[[𝑆2, ≤2, 𝜔2]]). for every [𝑓𝑖𝑗 ] ∈ 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]), we define the map 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) → 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) by 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )], (14) for every [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]). the following theorem shows that 𝜏 is a ring homomorphism. the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol 132 proposition 1 let 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]) be matrix rings over sgpsr. the mapping 𝜏 that is defined in (14) is a ring homomorphism. proof based on [24](lemma 8.1.6), for i, j = 1, 2, ⋯ , n, there is 𝛼𝑖𝑗 = 𝜎(𝑓𝑖𝑗 ) ∈ r2[[𝑆2, ≤2, 𝜔2]] for every 𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]]. therefore, 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )] = [𝛼𝑖𝑗 ] is well-defined. for any 𝑡 ∈ 𝑆2, 𝑓𝑖𝑗 , 𝑔𝑖𝑗 ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]], we have 𝜇 ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 −1(𝑡) = 𝜇 ((𝑓𝑖𝑗 + 𝑔𝑖𝑗 )(𝛿 −1(𝑡))) = 𝜇 (𝑓𝑖𝑗 (𝛿 −1(𝑡)) + 𝑔𝑖𝑗 (𝛿 −1(𝑡))) = 𝜇 (𝑓𝑖𝑗 (𝛿 −1(𝑡))) + 𝜇 (𝑔𝑖𝑗 (𝛿 −1(𝑡))) = (𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1)(𝑡) + (𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1)(𝑡), and 𝜇 ∘ (𝑓𝑖𝑗 𝑔𝑖𝑗 ) ∘ 𝛿 −1(𝑡) = 𝜇 ((𝑓𝑖𝑗 𝑔𝑖𝑗 )(𝛿 −1(𝑡))) = 𝜇 ( ∑ 𝑓𝑖𝑗 (𝑥)𝜔1 𝑥 (𝑔𝑖𝑗 (𝑦)) 𝑥+𝑦=𝛿−1(𝑡) ) = ∑ 𝜇 (𝑓𝑖𝑗 (𝑥)𝜔1 𝑥 (𝑔𝑖𝑗 (𝑦))) 𝑥+𝑦=𝛿−1(𝑡) = ∑ 𝜇 (𝑓𝑖𝑗 (𝑥)) 𝜇 (𝜔1 𝑥 (𝑔𝑖𝑗 (𝑦))) 𝑥+𝑦=𝛿−1(𝑡) = ∑ 𝜇 (𝑓𝑖𝑗 (𝑥)) 𝜔2 𝛿(𝑥) (𝜇 (𝑔𝑖𝑗 (𝑦))) 𝑥+𝑦=𝛿−1(𝑡) 𝛿(𝑥)+𝛿(𝑦)=𝑡 = ∑ 𝜇 (𝑓𝑖𝑗 (𝛿 −1(𝑢))) 𝜔2 𝑢 (𝜇 (𝑔𝑖𝑗 (𝛿 −1(𝑣)))) 𝑢+𝑣=𝑡 = ∑ (𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1)(𝑢)𝜔2 𝑢 ((𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1)(𝑣)) 𝑢+𝑣=𝑡 = (𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1)(𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1)(𝑡). in other words, 𝜇 ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 −1 = (𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1) + (𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1) and 𝜇 ∘ (𝑓𝑖𝑗 𝑔𝑖𝑗 ) ∘ 𝛿 −1 = (𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1)(𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1) for every 𝑓ij, 𝑔ij ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]]. now, we prove that τ is a ring homomorphism. for any [𝑓𝑖𝑗 ], [𝑔𝑖𝑗 ] ∈ mn(𝑅1[[𝑆1, ≤1, 𝜔1]]), we obtain: (i) τ([𝑓𝑖𝑗 ] + [𝑔𝑖𝑗 ]) = τ([𝑓𝑖𝑗 + 𝑔𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 + 𝑔𝑖𝑗 )] = [𝜇 ∘ (𝑓𝑖𝑗 + 𝑔𝑖𝑗 ) ∘ 𝛿 −1] = [(μ ∘ 𝑓𝑖𝑗 ∘ δ −1) + (μ ∘ 𝑔𝑖𝑗 ∘ δ −1)] = [μ ∘ 𝑓𝑖𝑗 ∘ δ −1] + [μ ∘ 𝑔𝑖𝑗 ∘ δ −1] = [𝜎(𝑓𝑖𝑗 )] + [𝜎(𝑔𝑖𝑗 )] = 𝜏([𝑓𝑖𝑗 ]) + 𝜏([𝑔𝑖𝑗 ]). the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol 133 (ii) 𝜏([𝑓𝑖𝑗 ][𝑔𝑖𝑗 ]) = 𝜏([∑ 𝑓𝑖𝑘 𝑔𝑘𝑗 𝑛 𝑘=1 ]) = [𝜎 (∑ 𝑓𝑖𝑘 𝑔𝑘𝑗 𝑛 𝑘=1 )] = [𝜇 ∘ (∑ 𝑓𝑖𝑘 𝑔𝑘𝑗 𝑛 𝑘=1 ) ∘ 𝛿 −1] = [∑ 𝜇 ∘ (𝑓𝑖𝑘 𝑔𝑘𝑗 ) ∘ 𝛿 −1 𝑛 𝑘=1 ] = [∑(𝜇 ∘ 𝑓𝑖𝑘 ∘ 𝛿 −1)(𝜇 ∘ 𝑔𝑘𝑗 ∘ 𝛿 −1) 𝑛 𝑘=1 ] = [𝜇 ∘ 𝑓𝑖𝑗 ∘ 𝛿 −1][𝜇 ∘ 𝑔𝑖𝑗 ∘ 𝛿 −1] = [𝜎(𝑓𝑖𝑗 )][𝜎(𝑔𝑖𝑗 )] =𝜏([𝑓𝑖𝑗 ])𝜏([𝑔𝑖𝑗 ]) according to (i) and (ii), it is proved that τ is a ring homomorphism. ∎ the following proposition shows that ker(𝜏) = 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]). proposition 2 let 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]) be matrix rings over sgpsr. let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) → 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) is the map that is defined in (14). then, ker(𝜏) = 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]). proof for any [𝑓𝑖𝑗 ] ∈ ker(𝜏), we have 𝜏([𝑓𝑖𝑗 ]) = [𝜎(𝑓𝑖𝑗 )] = [0]. therefore, for i, j = 1, 2, ⋯ , n , 𝜎(𝑓𝑖𝑗 ) = 0. so, 𝑓𝑖𝑗 ∈ ker(σ). based on [24](lemma 8.1.6), ker(𝜎) = (ker(𝜇))[[𝑆1, ≤1, 𝜔1]]. therefore, 𝑓𝑖𝑗 ∈ (ker(𝜇))[[𝑆1, ≤1, 𝜔1]] for all i, j = 1, 2, ⋯ , n. so, [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]). then, we get ker(𝜏) ⊂ 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]). on the other side, for any [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]), we have 𝑓𝑖𝑗 ∈ (ker(𝜇))[[𝑆1, ≤1, 𝜔1]] for all i, j = 1, 2, ⋯ , n. according to [24](lemma 8.1.6), ker(𝜎) = (ker(𝜇))[[𝑆1, ≤1, 𝜔1]]. therefore, 𝑓𝑖𝑗 ∈ ker(𝜎). then, 𝜎(𝑓𝑖𝑗 ) = 0 for all i, j = 1, 2, ⋯ , n. so, we get [𝜎(𝑓𝑖𝑗 )] = [0] = 𝜏([𝑓𝑖𝑗 ]). in other words, [𝑓𝑖𝑗 ] ∈ ker(𝜏). hence, 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]) ⊂ ker(𝜏). so, it is proved that ker(𝜏) = 𝑀𝑛 ((ker(𝜇))[[𝑆1, ≤1, 𝜔1]]) ∎ next, we give sufficient conditions for 𝜏 to be a monomorphism, epimorphism, and isomorphism. proposition 3 let 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]) be matrix rings over sgpsr. let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) → 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) is the map that is defined in (14). if 𝛿 is an isomorphism and 𝜇 is a monomorphism, then 𝜏 is a monomorphism. proof based on proposition 1, it is clear that 𝜏 is a ring homomorphism. so, we only have to show that 𝜏 is injective. if τ([𝑓𝑖𝑗 ]) = τ([𝑔𝑖𝑗 ]), then [𝜎(𝑓𝑖𝑗 )] = [𝜎(𝑔𝑖𝑗 )]. hence, [μ ∘ 𝑓𝑖𝑗 ∘ δ −1] = [μ ∘ 𝑔𝑖𝑗 ∘ δ −1]. therefore, we get μ ∘ 𝑓𝑖𝑗 ∘ δ −1 = μ ∘ 𝑔𝑖𝑗 ∘ δ −1 for all i, j = the ring homomorphisms of matrix rings over skew generalized power series rings ahmad faisol 134 1, 2, ⋯ , n. in other words, for any t ∈ s2, we have μ (𝑓𝑖𝑗 (δ −1(t))) = μ (𝑔𝑖𝑗 (δ −1(t))). since μ is a monomorphism, 𝑓𝑖𝑗 (δ −1(t)) = 𝑔𝑖𝑗 (δ −1(t)). since δ is an isomorphism, 𝑓𝑖𝑗 (s) = 𝑔𝑖𝑗 (s) for every s ∈ s1. so, 𝑓𝑖𝑗 = 𝑔𝑖𝑗 for all 𝑖, 𝑗 = 1, 2, ⋯ , n. therefore [𝑓𝑖𝑗 ] = [𝑔𝑖𝑗 ]. so, it is proved that if τ([𝑓𝑖𝑗 ]) = τ([𝑔𝑖𝑗 ]), then [𝑓𝑖𝑗 ] = [𝑔𝑖𝑗 ]. hence, τ is injective. ∎ proposition 4 let 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]) be matrix rings over sgpsr. let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) → 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) is the map that is defined in (14). if 𝜎 is an epimorphism, then 𝜏 is an epimorphism. proof based on proposition 1, 𝜏 is a ring homomorphism. so, we only have to show that 𝜏 is surjective. in other words, we have to prove that im(𝜏) = 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]). it is clear that im(𝜏) ⊂ 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]), so it suffices to show that 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) ⊂ im(𝜏). for any [αij] ∈ mn(𝑅2[[𝑆2, ≤2, 𝜔2]]), then α𝑖𝑗 ∈ 𝑅2[[𝑆2, ≤2, 𝜔2]] for all 𝑖, 𝑗 = 1, 2, ⋯ , 𝑛. since 𝜎 is an epimorphism, there is 𝑓𝑖𝑗 ∈ 𝑅1[[𝑆1, ≤1, 𝜔1]] such that 𝜎(𝑓𝑖𝑗 ) = α𝑖𝑗 . therefore, there is [𝑓𝑖𝑗 ] ∈ 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) such that [α𝑖𝑗 ] = [𝜎(𝑓𝑖𝑗 )] = τ([𝑓𝑖𝑗 ]) for every [α𝑖𝑗 ] ∈ 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]). so, [α𝑖𝑗 ] ∈ im(𝜏). in other words, 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) ⊂ im(𝜏). hence, that 𝜏 is surjective. ∎ corollary 5 let 𝑀𝑛(𝑅1[[𝑆1, ≤1, 𝜔1]]) and 𝑀𝑛(𝑅2[[𝑆2, ≤2, 𝜔2]]) be matrix rings over sgpsr. let 𝜏: 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) → 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) is the map that is defined in (14). if 𝛿 is an isomorphism, 𝜇 is a monomorphism, and 𝜎 is an epimorphism, then 𝜏 is an isomorphism. conclusions a ring homomorphism 𝜏 from the matrix ring 𝑀𝑛 (𝑅1[[𝑆1, ≤1, 𝜔1]]) to the matrix ring 𝑀𝑛 (𝑅2[[𝑆2, ≤2, 𝜔2]]) can be constructed by using a strictly ordered monoid homomorphism 𝛿: (𝑆1, ≤1) → (𝑆2, ≤2), and ring homomorphisms 𝜇: 𝑅1 → 𝑅2 and 𝜎: 𝑅1[[𝑆1, ≤1, 𝜔1]] → 𝑅2[[𝑆2, ≤2, 𝜔2]]. furthermore, it also proves that ker(𝜏) is equal to the 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[24] m. ziembowski, “right gaussian rings and related topics,” university of edinburgh, 2010. a note on generalized strongly p-convex functions of higher order cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 152-157 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 16, 2021 reviewed: october 12, 2021 accepted: october 18, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.12938 a note on generalized strongly p-convex functions of higher order corina karim*, ekadion maulana mathematics department, faculty of mathematics and natural sciences, brawijaya university, malang, indonesia *corresponding author email: co_mathub@ub.ac.id* rm.ekadion.m@gmail.com abstract generalized strongly 𝑝-convex functions of higher order is a new concept of convex functions which introduced by saleem et al. in 2020. the schur type inequality for generalized strongly 𝑝convex functions of higher order also studied by them. this paper aims to revise schur type inequality for generalized strongly 𝑝-convex functions of higher order in their paper. in order to revise it, we show that the contradiction was true. this paper showed that schur type inequality for generalized strongly 𝑝-convex functions of higher order previously is not valid and we give the correct schur type inequality for generalized strongly 𝑝-convex functions of higher order. keywords: schur type inequality; 𝑝-convex functions; strongly convex of higher order introduction convexity is a basic notion in many branches of applied mathematics. convexity is an important thing on functional analysis, geometry, mathematical programming, probability, and statistics. on functional analysis, convexity has intended to ensure existence and uniqueness of solutions of problems of calculus of variations and optimal control. on mathematical programming, convexity has intended to ensure convergence of optimization algorithms [1]. convexity appears in ancient greek geometry. archimedes (ca. 250 bc) used convexity on study of the area and arch length. archimedes has been the first person who gave a definition of convexity, similar to the geometric definition which used till today, a set is said to be convex if it contains all line segments between each of its points [1]. some geometric properties of convex sets and functions have studied before 1960 by great mathematicians hermann minkowski and werner fenchel. in 1891, minkowski proved that, in euclidean space ℝ𝑛 , every compact convex set with center at the origin and volume greater than 2𝑛 contains at least one point with integer coordinates different from the origin [2]. afterwards, in 1951, werner fenchel’s monograph stimulated the development of convexity theory. several researchers have been considered for classical convexity such that some of these new concepts are based on an extension of the domain of convex functions or http://dx.doi.org/10.18860/ca.v7i1.12938 mailto:co_mathub@ub.ac.id mailto:rm.ekadion.m@gmail.com a note on generalized strongly p-convex functions of higher order corina karim 153 sets to a generalized form[3-9]. some examples of these new concepts are quasiconvexity [10], exponential convexity [11, 12], logarithmical convexity [13], ℎ-convexity [14, 15], and 𝑝-convexity [3, 4, 16, 17]. 𝑝-convex funtions and their properties was introduced by zhang and wan [16] in 2007. in 2018 maden et al. [17] discussed strongly 𝑝-convex funtions and hermite-hadamard inequality for it. in 2020, saleem et al. [3, 4] discussed about generalized 𝑝-convex funtions and generalized strongly 𝑝-convex funtions of higher order. in [3] discussed definition and properties of generalized strongly 𝑝-convex functions of higher order also some of type inequalities which are hermite-hadamard, fejér, and schur. in schur type inequality for generalized strongly 𝑝-convex funtions of higher order in [3] indirectly mentioned that 𝜙(𝑥3 𝑝 − 𝑥2 𝑝 ) ≤ (𝑥3 𝑝 − 𝑥1 𝑝 )𝜙 ( 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝) for any 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥3 𝑝 − 𝑥1 𝑝 ∈ (0,1) and 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . so, this paper revises the correction of schur type inequality in [3]. methods in this section, we discussed about some definitions which are implemented with generalized strongly 𝑝-convex functions of higher order. we also give examples of strongly 𝑝-convex funtions of higher order and its generalized. definition 1. (see [1]) (convex set) let 𝑋 ⊂ ℝ𝑛. 𝑋 is called to be convex if 𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝑋, (1) for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1]. definition 2. (see [3]) ( 𝒑-convex set) let 𝑋 ⊂ ℝ𝑛. 𝑋 is called to be 𝑝-convex if [𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝] 1 𝑝 ∈ 𝑋, (2) for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1]. definition 3. (see [18]) (q-convex uniform) let 𝑋 be a banach space and real number 𝑞 ≥ 2. defined 𝛿𝑋 (𝜖) = inf {1 − ‖ 𝑓+𝑔 2 ‖ ; 𝑓, 𝑔 ∈ 𝑋, ‖𝑓‖ ≤ 1, ‖𝑔‖ ≤ 1, ‖𝑓 − 𝑔‖ ≥ 𝜖}. 𝑋 is called to be 𝑝-convex uniform if there exists a constant 𝑐 > 0 such that 𝛿𝑋 (𝜖) ≥ 𝑐𝜖 𝑞 , for 0 < 𝜖 ≤ 2. lemma 1. (see [19]) let 𝑋 be a 𝑞-convex uniform with 𝑞 ≥ 2, then there exists a constant 𝜇 > 0 such that ‖𝑡𝑥 + (1 − 𝑡)𝑦‖𝑞 ≤ 𝑡‖𝑥‖𝑞 + (1 − 𝑡)‖𝑦‖𝑞 − 𝜇𝜙(𝑡)‖𝑥 − 𝑦‖𝑞 for every 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ (0,1), where 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . lemma 1 used to prove an example of strongly 𝑝-convex funtions of higher order and generalized strongly 𝑝-convex funtions of higer order. it has been proved in [19]. a note on generalized strongly p-convex functions of higher order corina karim 154 definition 4. (see [3]) (strongly 𝒑-convex functions of higher order) let 𝑋 be a 𝑝-convex set. a function 𝑓: 𝑋 → ℝ is called to be strongly 𝑝-convex functions of higher order if for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1], then 𝑓 ([𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝] 1 𝑝) ≤ 𝑡𝑓(𝑥) + (1 − 𝑡)𝑓(𝑦) − 𝜇𝜙(𝑡)‖𝑥𝑝 − 𝑦𝑝‖𝑞 , (3) with 𝜇 ≥ 0, 𝑞 > 0, and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞. example: let 𝑞 ≥ 2 and 𝑝 > 1. defined 𝑋 = {𝑥 ∈ ℝ; |𝑥|𝑝 ∈ 𝑙𝑞 } and 𝜓: 𝑋 → ℝ, where 𝜓(𝑥) = ‖|𝑥|𝑝‖𝑞 𝑞 , then 𝜓 is strongly 𝑝–convex functions of higher order. definition 5. (see [3]) (generalized strongly 𝒑-convex funtions of higher order) let 𝑋 be a 𝑝-convex set. a function 𝑓: 𝑋 → ℝ is called to be strongly 𝑝-convex functions of higher order with respect to 𝜂: 𝐴 × 𝐴 → 𝐵, with 𝐴, 𝐵 ⊆ ℝ if for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1], then 𝑓 ([𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝] 1 𝑝) ≤ 𝑓(𝑦) + 𝜂(𝑓(𝑥), 𝑓(𝑦)) − 𝜇𝜙(𝑡)‖𝑥𝑝 − 𝑦𝑝‖𝑞 , (4) with 𝜇 ≥ 0, 𝑞 > 0, and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞. example: function 𝜓 in example of strongly 𝑝-convex functions of higher order with respect to 𝜂(𝑥, 𝑦) = 𝑥 − 𝑦 is generalized strongly 𝑝-convex functions of higher order with rescpect to 𝜂. results and discussion in this section, we discussed about revised schur type inequality for generalized strongly 𝑝-convex functions of higher order. we showed that 𝜙(𝑥3 𝑝 − 𝑥2 𝑝 ) ≤ (𝑥3 𝑝 − 𝑥1 𝑝 )𝜙 ( 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝) is not valid for any 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥3 𝑝 − 𝑥1 𝑝 ∈ (0,1) and 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . theorem 1. (schur type inequality) let 𝑋 is 𝑝-convex set and 𝐴, 𝐵 ⊂ ℝ. defined a generalized strongly 𝑝-convex function of higher order with respect to 𝜂(∙,∙): 𝐴 × 𝐴 → 𝐵 with 𝜇 ≥ 0 and 𝑝 > 0, then for every 𝑥1, 𝑥2, 𝑥3 ∈ 𝑋 such that 𝑥1 < 𝑥2 < 𝑥3 and 𝑥3 𝑝 − 𝑥1 𝑝 , 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥2 𝑝 − 𝑥1 𝑝 ∈ (0,1), then the following inequality hold 𝑓(𝑥3)(𝑥3 𝑝 − 𝑥1 𝑝 ) − 𝑓(𝑥2)(𝑥3 𝑝 − 𝑥1 𝑝 ) + (𝑥3 𝑝 − 𝑥2 𝑝 )𝜂(𝑓(𝑥1), 𝑓(𝑥3)) − (𝑥3 𝑝 − 𝑥1 𝑝 )𝜇𝜙 ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) ‖𝑥3 𝑝 − 𝑥1 𝑝 ‖ 𝑞 ≥ 0. (5) proof: let 𝑥1, 𝑥2, 𝑥3 ∈ 𝑋 with some criterion which are 𝑥1 < 𝑥2 < 𝑥3 and 𝑥3 𝑝 − 𝑥1 𝑝 , 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥2 𝑝 − 𝑥1 𝑝 ∈ (0,1). based on those criteria, then we have 0 < 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 and 0 < 𝑥2 𝑝 − 𝑥1 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 . (6) based on (6), then we have a note on generalized strongly p-convex functions of higher order corina karim 155 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 , 𝑥2 𝑝 − 𝑥1 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 ∈ (0,1). (7) it’s clear that 𝑥3 𝑝 − 𝑥1 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1 ⟺ 𝑥3 𝑝 − 𝑥1 𝑝 − 𝑥2 𝑝 + 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1 ⇔ (𝑥3 𝑝 − 𝑥2 𝑝 ) + (𝑥2 𝑝 − 𝑥1 𝑝 ) 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1 ⇔ 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 + 𝑥2 𝑝 − 𝑥1 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1. (8) choose 𝑡 = 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝, so from (8) can get 𝑡 + 𝑥2 𝑝 − 𝑥1 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1 ⇔ 𝑥2 𝑝 − 𝑥1 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 = 1 − 𝑡 ⇔ 𝑥2 𝑝 − 𝑥1 𝑝 = (1 − 𝑡)(𝑥3 𝑝 − 𝑥1 𝑝 ) ⇔ 𝑥2 𝑝 − 𝑥1 𝑝 = (1 − 𝑡)𝑥3 𝑝 − (1 − 𝑡)𝑥1 𝑝 ⇔ 𝑥2 𝑝 − 𝑥1 𝑝 = (1 − 𝑡)𝑥3 𝑝 − 𝑥1 𝑝 + 𝑡𝑥1 𝑝 ⇔ 𝑥2 𝑝 = 𝑡𝑥1 𝑝 + (1 − 𝑡)𝑥3 𝑝 ⇔ 𝑥2 = (𝑡𝑥1 𝑝 + (1 − 𝑡)𝑥3 𝑝 ) 1 𝑝. (9) from (9), we can write 𝑓(𝑥2) = 𝑓 ([𝑡𝑥1 𝑝 + (1 − 𝑡)𝑥3 𝑝 ] 1 𝑝). (10) after that, because 𝑓 is generalized strongly 𝑝-convex functions of higher order, then from (10) and 𝑡 = 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝 we can get 𝑓(𝑥2) ≤ 𝑓(𝑥3) + ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) 𝜂(𝑓(𝑥1), 𝑓(𝑥3)) − 𝜇𝜙 ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) ‖𝑥3 𝑝 − 𝑥1 𝑝 ‖ 𝑞 (11) if all segments on (11) times by 𝑥3 𝑝 − 𝑥1 𝑝 , then we have 𝑓(𝑥2)(𝑥3 𝑝 − 𝑥1 𝑝 ) ≤ 𝑓(𝑥3)(𝑥3 𝑝 − 𝑥1 𝑝 ) + ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) 𝜂(𝑓(𝑥1), 𝑓(𝑥3))(𝑥3 𝑝 − 𝑥1 𝑝 ) − 𝜇𝜙 ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) ‖𝑥3 𝑝 − 𝑥1 𝑝 ‖ 𝑞 (𝑥3 𝑝 − 𝑥1 𝑝 ) ⟺ 𝑓(𝑥3)(𝑥3 𝑝 − 𝑥1 𝑝 ) − 𝑓(𝑥2)(𝑥3 𝑝 − 𝑥1 𝑝 ) + (𝑥3 𝑝 − 𝑥2 𝑝 )𝜂(𝑓(𝑥1), 𝑓(𝑥3)) − (𝑥3 𝑝 − 𝑥1 𝑝 )𝜇𝜙 ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝 ) ‖𝑥3 𝑝 − 𝑥1 𝑝 ‖ 𝑞 ≥ 0, this completes the proof of theorem 1. to show that 𝜙(𝑥3 𝑝 − 𝑥2 𝑝 ) ≤ (𝑥3 𝑝 − 𝑥1 𝑝 )𝜙 ( 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝) is not valid for any 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥3 𝑝 − 𝑥1 𝑝 ∈ (0,1) and 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 , first we take 𝑞 = 2, then we have 𝜙(𝑡) = 𝑡2(1 − 𝑡) + 𝑡(1 − 𝑡)2 = 𝑡2 − 𝑡3 + 𝑡(1 − 2𝑡 + 𝑡2) = 𝑡2 − 𝑡3 + 𝑡 − 2𝑡2 + 𝑡3 = 𝑡 − 𝑡2 = 𝑡(1 − 𝑡). (12) after that, we have to take 𝑥 = 𝑥3 𝑝 − 𝑥1 𝑝 = 1 2 and 𝑦 = 𝑥3 𝑝 − 𝑥2 𝑝 = 1 4 , then its clear that 𝑥3 𝑝 − 𝑥2 𝑝 = 1 4 < 1 2 = 𝑥3 𝑝 − 𝑥1 𝑝 . so, we can get 𝜙 ( 1 4 ) = 1 4 (1 − 1 4 ) = 3 16 , a note on generalized strongly p-convex functions of higher order corina karim 156 1 2 𝜙 ( 1 4⁄ 1 2⁄ ) = 1 2 𝜙 ( 1 2 ) = 1 2 [ 1 2 (1 − 1 2 )] = 1 8 = 2 16 . and 𝜙 ( 1 4 ) = 3 16 > 2 16 = 1 2 𝜙 ( 1 4⁄ 1 2⁄ ). so, there exists 𝑥3 𝑝 − 𝑥2 𝑝 = 1 4 , 𝑥3 𝑝 − 𝑥1 𝑝 = 1 2 and 𝑥3 𝑝 − 𝑥2 𝑝 = 1 4 < 𝑥3 𝑝 − 𝑥1 𝑝 = 1 2 with 𝑞 = 2 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 but (𝑥3 𝑝 − 𝑥2 𝑝 ) > (𝑥3 𝑝 − 𝑥1 𝑝 )𝜙 ( 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝) . in other words, 𝜙(𝑥3 𝑝 − 𝑥2 𝑝 ) > (𝑥3 𝑝 − 𝑥1 𝑝 )𝜙 ( 𝑥3 𝑝 −𝑥2 𝑝 𝑥3 𝑝 −𝑥1 𝑝) is not valid for 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥3 𝑝 − 𝑥1 𝑝 ∈ (0,1) and 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . remarks: if 𝜙 satisfies 𝑥𝜙(𝑦) ≥ 𝜙(𝑥𝑦) for any 𝑥, 𝑦 ∈ (0,1) and 𝑞 > 0, then (5) can be written as 𝑓(𝑥3)(𝑥3 𝑝 − 𝑥1 𝑝 ) − 𝑓(𝑥2)(𝑥3 𝑝 − 𝑥1 𝑝 ) + (𝑥3 𝑝 − 𝑥2 𝑝 )𝜂(𝑓(𝑥1), 𝑓(𝑥3)) − 𝜇𝜙(𝑥3 𝑝 − 𝑥2 𝑝 )‖𝑥3 𝑝 − 𝑥1 𝑝 ‖ 𝑞 ≥ 0. conclusion schur type inequality for generalized strongly 𝑝-convex functions of higher order on [3] has a correction. (𝑥3 𝑝 − 𝑥1 𝑝 )𝜇𝜙 ( 𝑥3 𝑝 − 𝑥2 𝑝 𝑥 3 𝑝 − 𝑥 1 𝑝) ≥ 𝜙(𝑥3 𝑝 − 𝑥1 𝑝 ), is not valid for any 𝑥3 𝑝 − 𝑥2 𝑝 , 𝑥3 𝑝 − 𝑥1 𝑝 ∈ (0,1) and 𝑥3 𝑝 − 𝑥2 𝑝 < 𝑥3 𝑝 − 𝑥1 𝑝 with 𝑝, 𝑞 > 0, 𝑡 ∈ [0,1], and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . so, (5) is the correct schur type inequality for generalized strongly 𝑝-convex funtions of higher order. acknowledgments this paper supported by the dpp/spp grant no. 1519/un10.f09/pn/2021 at mathematics and natural sciences faculty, universitas brawijaya. references [1] d. henrion, "convexity", the princeton companion to applied mathematics, part ii, vol. 8, pp. 89-90, 2015. [2] g. g. magaril-il'yaev and v. m. tikhomirov, convex analysis: theory and applications, 2003. [3] m. s. saleem, y. m. chu, n. jahangir, h. akhtar, and c. y. jung, "on generalized strongly p-convex functions of higher order", journal of mathematics, vol. 2020, hindawi, 2020. 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[10] b. de finneti, "sulle stratificazioni covesse", ann. mat. pura appl, vol. 30, pp. 173183, 1949. [11] m. a. noor and k. i. noor, "on expopentially convex funtions", journal of orissa mathematical society, vol. 38, pp. 33-51, 2019. [12] m. a. noor and k. i. noor, "strongly exponentially convex funtions", u.p.b. sci. bull, vol. 81, 2019. [13] j. e. pečarić, f. proschan, y. l. tong, convex funtions, partial orderings, and statistical applications, academic press, boston, 1992. [14] s. varošanec, "on h-convexity", math. anal. appl, vol. 326, pp. 303-311, 2007. [15] h. angulo, j. giménez, a. m. moros, and k. nikodem, "on strongly h-convex funtions", ann. funct. anal, vol. 2, pp. 85-91, 2011. [16] k. zhang and j. wan, "p-convex funtions and their properties", pure and applied mathematics, vol. 23, 2007. [17] s. maden, s. turhan, and i̇. i̇scan, hermite-hadamard inequality for strongly p-convex funtions, springer, berlin, 2018. [18] h. k. xu, "fixed point theorem for uniformly lipschitzian semigroups in uniformly convex spaces", journal of mathematical analysis and applications, vol. 152, pp. 391398, 1990. 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issue was very appreciated bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia subanar seno, gadjah mada university, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740595') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740557') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740556') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740541') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/736347') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5964') microsoft word 1 sampul depan.doc 38  pemodelan copula: studi banding kuantifikasi autokorelasi fachrur rozi jurusan matematika, fakultas sains dan teknologi universitas islam negeri (uin) maulana malik ibrahim malang e-mail: fachrurkibar@yahoo.com abstrak dalam tulisan ini akan dijelaskan beberapa metode kuantifikasi dependensi antara dua variabel acak (bivariat) dan perbandingan antara metode kuantifikasi dependensi tersebut. selain itu akan diperkenalkan teori copula dalam kaitannya dengan kuantifikasi dependensi pada data time series, yang biasa disebut autokorelasi, khususnya kuantifikasi autokorelasi kendall’s tau melalui copula. kata kunci: autokorelasi, copula, kendall’s tau. 1. pendahuluan dalam kehidupan sehari-hari, sering kita dipertemukan dengan fenomena hubungan antara beberapa karakteristik yang diduga mempunyai keterkaitan antara karakteristik yang satu dengan karakteristik yang lain. dalam ilmu statistika keterkaitan ini sering disebut dependensi (keterhubungan) antara variabel yang satu dengan variabel yang lain. jogdeo (1982) mengatakan: “hubungan ketergantungan (dependensi) antara beberapa variabel acak adalah salah satu persoalan yang sangat banyak dipelajari dalam ilmu probabilitas dan statistika. …..…..” dalam tulisan ini akan dijelaskan beberapa metode kuantifikasi dependensi antara dua variabel acak dan perbandingan antara metode kuantifikasi dependensi tersebut. salah satu hal yang bisa dikatakan baru adalah memperkenalkan teori copula kaitannya dengan dependensi antara dua variabel acak, khususnya dependensi pada data time series, yang biasa disebut autokorelasi. dalam hal ini copula hadir sebagai perluasan metode dalam memodelkan dependensi antar variabel acak, copula akhir-akhir ini banyak dikembangkan dalam bidang biostatistika, ilmu aktuaria, dan keuangan. pembahasan pada tulisan ini hanya dilakukan dilakukan untuk dependensi antara dua variabel, khususnya pada kasus dependensi data time series, yang biasa disebut autokorelasi lag-1,. dengan demikian, teori-teori yang dijelaskan lebih ditekankan pada dependensi antara dua variabel (bivariat). dalam membandingkan kuantifikasi dependensi dilakukan dengan membandingkan hasil simulasi dari kuantifikasi dependensi yang yang diperoleh. 2. copula dan sifat-sifatnya pada bagian ini, akan dijelaskan mengenai definisi copula dan sifat-sifat dasarnya sebagai teori dasar yang akan digunakan dalam pembahasan selanjutnya. definisi 1. copula dua dimensi (2-copula) adalah fungsi 2:c i i→ yang memenuhi sifat-sifat: a. untuk setiap 2( , )tu x y i= ∈ , maka berlaku ( , 0) 0 (0, )c x c y= = dan ( ,1)c x x= ; (1, )c y y= . (2.1) pemodelan copula: studi banding kuantifikasi autokorelasi  volume 1 no. 1 november 2009 39 b. untuk setiap 21 1 2 2( , ) , ( , ) t tu x y v x y i= = ∈ sedemikian sehingga u v≤ , maka berlaku 2 2 2 1 1 2 1 1( , ) ( , ) ( , ) ( , ) 0c x y c x y c x y c x y− − + ≥ . (2.2) himpunan dari semua copula dua dimensi didefinisikan sebagai c2. mengingat, [ ] [ ]( ) ( , ) ( , ) ( , 0) (0, ) (0, 0) 0, 0,c c u v c u v c u c v c v u v = − − + = × (2.3) maka hal ini akan menunjukkan bahwa ( , )c u v sebagai pengaitan suatu bilangan di i terhadap persegi panjang [ ] [ ]0, 0,u v× . 2.1. copula dan variabel acak teorema 1. (teorema sklar) misalkan h adalah fungsi distribusi gabungan dari variable x dan y, dengan f dan g masing-masing adalah fungsi distribusi marginal dari x dan y. maka terdapat sebuah copula c sedemikian sehingga untuk setiap ,x y r∈ berlaku ( )( , ) ( ), ( ) ( , )h x y c f x g y c u v= = , (2.4) dengan ( )u f x= dan ( )v g y= . jika f dan g kontinu, maka copula c tunggal, jika f dan g tidak kontinu, maka copula c tunggal pada ( ) ( )range f range g× . sebaliknya, misalkan c adalah sebuah copula, f dan g masing-masing adalah fungsi distribusi marginal dari x dan y. maka terdapat fungsi distribusi gabungan h sedemikian sehingga untuk setiap ,x y r∈ berlaku ( )( , ) ( ), ( ) ( , )h x y c f x g y c u v= = . sebagai konsekuensi dari teorema sklar, jika x dan y adalah variabel acak dengan fungsi distribusi gabungan h dan mempunyai fungsi distribusi marginal masing-masing adalah f dan g, maka untuk setiap ,x y r∈ berlaku ( ) ( )max ( ) ( ) 1, 0 ( , ) min ( ), ( )f x g y h x y f x g y+ − ≤ ≤ . (2.5) 2.2. copula empiris definisi 2. misalkan ( ){ } 1 , n k k k x y = adalah sampel berukuran n dari distribusi bivariat yang kontinu. copula empiris adalah fungsi cn yang didefinisikan sebagai ( ) ( ) ( ) , , i jn banyaknya pasangan data x y dalam sampel sehingga x x dan y yi j c n n n ≤ ≤⎛ ⎞ =⎜ ⎟ ⎝ ⎠ di mana ( )ix dan ( )jy , 1 ,i j n≤ ≤ , adalah statistik urutan dari sampel. copula frekuensi empiris adalah fungsi cn yang didefinisikan sebagai ( ) ( ){ }( ) ( ) 11 / , , , ,, 0 , . n i j k k k n n jika x y a d a la h elem en d a ri x yi j c n n la in n ya = ⎧⎛ ⎞ = ⎨⎜ ⎟ ⎝ ⎠ ⎩ (2.6) ilustrasi: misalkan diberikan sampel acak {( , )}k k k nx y ∈ dari variabel acak ( , )x y , tabel 1. data sampel acak k 1 2 3 4 5 6 x 1.3 2.4 3.2 1.7 4.3 0.8 y 2.5 3.5 2.6 2.1 3.2 1.8 fachrur rozi  40 volume 1 no. 1 november 2009 berdasarkan definisi 2, maka grafik dari copula frekuensi empiris , i j c n n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ dan , i j c n n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ , , 1, 2,..., 6i j = dari variabel x dan y adalah: (a) (b) gambar 1. (a) grafik copula frekuensi empiris , i j c n n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ; (b) grafik copula frekuensi empiris , i j c n n ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 3. dependensi autokorelasi berdasarkan pembatasan masalah pada pendahuluan, pembahasan akan difokuskan dalam mempelajari perbandingan kuantifikasi dependensi dalam data deret waktu (time series), yang biasanya disebut dengan istilah autokorelasi, dan lebih khusus lagi untuk kuantifikasi autokorelasi lag-1. bentuk klasik yang umum digunakan dalam kuantifikasi autokorelasi ini adalah koefisien autokorelasi dari fungsi autokorelasi (autocorrelation coefficient function/acf). bentuk kuantifikasi dependensi yang lain adalah bentuk modern yang menggunakan konsep konkordan, yang salah satunya kuantifikasi autokorelasi dengan pendekatan copula. 3.1. autokorelasi suatu himpunan hasil pengamatan yang dilakukan berdasarkan urutan waktu, biasa disebut time series. data time series yang dibahas dalam tulisan ini adalah berkaitan dengan data time series diskrit. adapun suatu fenomena statistika yang berkembang dalam kaitannya dengan runtutan waktu yang sesuai dengan aturan probabilistik di sebut proses stokastik. jadi analisa dari time series, dianggap sebagai realisasi dari proses stokastik. salah satu proses yang khusus dalam proses stokastik, disebut proses stasioner (box & jenkins, 1976). definisi 3. misalkan ( )t t tz z ∈= adalah proses stokastik pada ruang probabilistik ( , , )f pω . maka x dikatakan stasioner kuat, jika untuk setiap m n∈ , { }1 2, ,..., mt t t t⊂ dan setiap 0h > dengan { }1 2, ,..., mt h t h t h t+ + + ⊂ , kita punya ( ) ( )1 11 1,..., ,...,m mt h t h m t t mp z z z z p z z z z+ +< < = < < (3.1) untuk setiap 1 2, ,..., mz z z r∈ . jika m = 1, asumsi kestasioneran berakibat bahwa fungsi distribusi peluang ( )tf z adalah sama untuk setiap t t∈ , dan cukup menuliskan ( )f z . oleh karena itu proses stasioner mempunyai mean konstan: [ ] [ ] ( ).te z e z zf z dz ∞ −∞ μ = = = ∫ (3.2) pemodelan copula: studi banding kuantifikasi autokorelasi  volume 1 no. 1 november 2009 41 dan variansi konstan: ( ) ( )2 22 2( ) ( )z te z e z z f z dz ∞ −∞ ⎡ ⎤ ⎡ ⎤σ = −μ = −μ = −μ⎢ ⎥ ⎣ ⎦⎣ ⎦ ∫ . (3.3) dalam praktek, mean dan variansi dari proses stationer ditaksir dengan 1 1ˆ n t t z z n = μ = = ∑ (3.4) dan ( )22 1 1ˆ n z t t z z n = σ = −∑ (3.5) di mana 1 2, ,..., nz z z adalah pengamatan/sampel time series. asumsi kestasioneran juga berakibat bahwa fungsi distribusi peluang gabungan 1 2 ( , )t tf z z adalah sama untuk setiap 1 2,t t yang mana merupakan interval konstan yang terpisah. sehingga, karakteristik distribusi gabungan ini dapat diduga dengan memplot diagram pencar dari data pasangan ( ) 1 2 ( , ) ,t t t t kz z z z += yang merupakan bagian dari data time series yang dipisahkan oleh interval konstan atau lag k. gambar 2. contoh diagram pencar untuk data time series dengan lag-1 dalam hal ini kita dapat berbicara mengenai kovariansi dari tz dan t kz + , yang dipisahkan oleh k interval waktu diskrit, yang disebut autokovariansi pada lag k yang didefinisikan oleh: [ ] ( )( )cov ,k t t k t t kz z e z z+ +⎡ ⎤γ = = −μ −μ⎣ ⎦ (3.6) sehingga, dari definisi autokovariansi, kita bisa definisikan kuantifikasi autokorelasi pada lag k, didefinisikan oleh: ( )( ) ( ) ( ) ( )( ) 2 2 2 t t k k t t k t t k z e z z e z e z e z z + + + ⎡ ⎤−μ −μ⎣ ⎦ρ = ⎡ ⎤ ⎡ ⎤−μ −μ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎡ ⎤−μ −μ⎣ ⎦= σ (3.7) dalam praktek, kuantifikasi autokorelasi di atas ditaksir oleh: ( )( ) ( ) 1 2 1 ˆ n k t t k t k n t t z z z z z z − + = = − − ρ = − ∑ ∑ (3.8) 0.9 0.95 1 1.05 1.1 1.15 0.9 1 1.1 1.2zi zi+1 fachrur rozi  42 volume 1 no. 1 november 2009 bentuk terakhir pada persamaan (3.8) merupakan bentuk klasik dari dependensi time series yang disebut koefisien autokorelasi (autocorrelation function). 3.2. kendall’s tau dan copula pada bagian ini akan dijelaskan bentuk kuantifikasi dependensi lain selain koefisien autokorelasi, salah satunya adalah statistik kendall’s τ, yaitu kuantifikasi dependensi yang didasarkan atas data rangking. definisi 4. misalkan 1 1( , )x y dan 2 2( , )x y dua vektor acak yang independen dan berdistribusi identik pada ruang peluang ( , , )a pω . kendall’s τ didefinisikan sebagai. ( ) ( ). 1 2 1 2 1 2 1 2( )( ) 0 ( )( ) 0x y p x x y y p x x y yτ ≡ τ = − − > − − − < (3.9) selanjutnya kita menyebut bentuk di atas adalah definisi kendall’s τ untuk populasi. jadi, kendall’s τ adalah perbedaan antara peluang dari konkordan dan peluang dari diskordan. dalam prakteknya, kita dapat mendefinisikan ukuran dependensi kendall’s τ berdasarkan sampel. misalkan { }1 1( , ),..., ( , )n nx y x y , 2n ≥ adalah sampel berukuran n dari vaktor acak kontinu ( , )x y . setiap pasang sampel { }( , ), ( , )i i j jx y x y , , {1,..., }, i j n i j∈ ≠ merupakan suatu diskordan atau konkordan. maka jelas terdapat 2 n⎛ ⎞ ⎜ ⎟ ⎝ ⎠ pasangan berbeda dari sampel yang ada. misalkan k menyatakan banyaknya pasangan konkordan, dan d menyatakan banyaknya pasangan diskordan. maka kendall’s τ untuk sampel didefinisikan menjadi ˆ 2 k d k d nk d − − τ = = + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (3.10) dengan definisi kendall’s τ di atas, kita dapat menunjukkan bahwa copula mempunyai hubungan dengan kendall’s τ, untuk menunjukkan hubungan tersebut, sebelumnya perlu didefinisikan terlebih dahulu suatu fungsi konkordan q, yang menyatakan perbedaan peluang dari konkordan dan peluang diskordan antara dua vektor 1 1( , )x y dan 2 2( , )x y dari variabel acak kontinu dengan fungsi distribusi gabungan (yang mungkin) berbeda 1h dan 2h , tetapi dengan fungsi distribusi marginal yang sama f dan g. kemudian akan ditunjukkan bahwa fungsi konkordan ini bergantung pada distribusi dari 1 1( , )x y dan 2 2( , )x y melalui copula mereka. teorema 2. misalkan 1 1( , )x y dan 2 2( , )x y adalah dua vektor random dengan fungsi distribusi gabungan masing-masing 1h dan 2h , di mana ~ix f dan ~iy g , 1, 2i = . lebih lanjut, misalkan 1c dan 2c menyatakan copula dari 1 1( , )x y dan 2 2( , )x y , sedemikian sehingga ( )1 1( , ) ( ), ( )h x y c f x g y= dan ( )2 2( , ) ( ), ( )h x y c f x g y= . jika q menyatakan perbedaan antara peluang dari konkordan dan peluang diskordan dari 1 1( , )x y dan 2 2( , )x y , yang didefinisikan sebagai ( ) ( )1 2 1 2 1 2 1 2( )( ) 0 ( )( ) 0q p x x y y p x x y y= − − > − − − < , (3.11) maka kita peroleh: 21 2 2 1( , ) 4. ( , ) ( , ) 1iq q c c c u v dc u v= = −∫∫ . (3.12) pemodelan copula: studi banding kuantifikasi autokorelasi  volume 1 no. 1 november 2009 43 berdasarkan definisi fungsi konkordan pada teorema 2, maka kita dapat mendefinisikan kendall’s τ untuk x dan y melalui copula dengan teorema berikut (nelsen, 1999): teorema 3. misalkan x dan y variabel acak kontinu dengan copula c. maka kendall’s τ untuk x dan y diberikan oleh 2 . ( , ) 4. ( , ) ( , ) 1x y c i q c c c u v dc u vτ ≡ τ = = −∫∫ . (3.20) perhatikan bahwa bentuk integral yang ada pada persamaan (3.20) dapat diinterpretasikan sebagai ekspektasi dari fungsi ( , )c u v , di mana u dan v variabel acak yang berdistribusi (0,1)u , atau dengan kata lain [ ]4. ( , ) 1c ce c u vτ = − . (3.21) teorema 4. misalkan x dan y variabel acak kontinu dengan fungsi distribusi gabungan h, dan misalkan [ ] ' ' ( , ) ( ', ') ( , ') ( , ') ' ' y x t h x y h x y h x y h x y dxdydx dy ∞ ∞ −∞ −∞ −∞ −∞ = −∫ ∫ ∫ ∫ . (3.22) maka kendall’s τ untuk x dan y diberikan oleh . 2x y tτ = . berdasarkan definisi copula empiris, dan dengan memperhatikan kembali definisi kendall’s τ untuk populasi untuk suatu variabel acak kontinu x dan y dengan copula c seperti pada persamaan 3.22 dengan modifikasi yaitu [ ] 1 1 ' ' 0 0 0 0 2 ( , ) ( ', ') ( ', ) ( , ') ' ' v u c u v c u v c u v c u v d u d vd u d vτ = −∫ ∫ ∫ ∫ , (3.23) maka teorema berikut (nelsen, 1999) akan menjelaskan bentuk turunan koefisien kendall’s τ untuk sampel. teorema 5. misalkan cn dan cn masing-masing adalah fungsi copula empirik dan fungsi frekuensi copula empirik untuk sampel ( ){ } 1 , n k k k x y = . jika t adalah koefisien kendall’s τ untuk sampel, maka 11 2 2 1 1 2 , , , , 1 jn n i n n n n i j p q n i j p j i q p q t c c c c n n n n n n n n n −− = = = = ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ∑ ∑ ∑ ∑ . (3.24) 3.3. kendall’s tau untuk autokorelasi pada subbab ini, akan dijelaskan pendekatan kendall’s τ untuk kuantifikasi autokorelasi dari data time series, dan dikhususkan untuk autokorelasi lag-1 (first-order serial dependence). misalkan diberikan barisan variabel acak 1 2, ,..., nx x x , 3n ≥ yang merupakan data time series dan 1 2, ,..., nr r r adalah barisan ranking yang bersesuaian dengan barisan variabel acak, maka ukuran autokorelasi lag-1 secara khusus didasarkan pada data pasangan 1 2 2 3 1( , ), ( , ),..., ( , )n nr r r r r r− , (3.25) dan mungkin menambah dengan 1( , )nr r , dalam kasus barisan variabel acak tersebut bersiklus. misalkan diberikan barisan variabel acak 1 2, ,..., nx x x , 3n ≥ yang bersesuaian dengan barisan ranking 1 2, ,..., nr r r dari barisan acak. maka kuantifikasi kendall’s τ untuk autokorelasi lag-1 untuk kasus barisan bersiklus dapat didefinisikan sebagai: 4 1 2 1 2 .( 1)n n d n n n ⎛ ⎞⎛ ⎞ τ = − = −⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠ , (3.26) fachrur rozi  44 volume 1 no. 1 november 2009 di mana d menyatakan banyaknya diskordan, atau { } ( ) 1 1 1 1 1 1 1 1 1 1 1 ( , ) ( , ) ( , ) n n i j i j i j i j i j i n n i j i j i j d i r r r r i r r r r i r r r r − + + + + = = + + + = = = < > + > < = < > ∑ ∑ ∑∑ (3.27) di mana ( )i a menyatakan fungsi indikator dari himpunan a. untuk kasus barisan yang tidak bersiklus, dengan mensubstitusikan n-1 untuk n pada persamaan (3.26) dan (3.27). 3.4. pengujian keberartian autokorelasi setelah melakukan kuantifikasi autorkorelasi, untuk dapat membandingkan hasil yang telah diperoleh dari masing-masing ukuran dependensi yang telah dijelaskan pada subbab sebelumnya, perlu dilakukan pengujian keberartian dependensi dari masingmasing ukuran dependensi, artinya hasil kuantifikasi autokorelasi harus diuji keberartiannya. 3.5. pengujian untuk koefisien autokorelasi klasik misalkan 1 2( , ,..., ), nz z z z n n= ∈ adalah proses stationer dengan waktu diskrit berukuran n, dan 1ρ̂ adalah taksiran untuk koefisien autokorelasi lag-1, maka arcana (2005) mengatakan bahwa pada tingkat keberartian α pada tingkat keberartian α , nilai 1ρ̂ dikatakan berarti untuk pengujian dua arah jika / 21ˆ z n αρ > , (3.28a) sedangkan, untuk pengujian satu arah, jika 11ˆ z n −αρ > atau 1ˆ z n αρ < , (3.28b) di mana zα adalah suatu nilai sehingga ( )p z zα< = α dengan ~ (0,1)z n . 3.6. pengujian untuk koefisien autokorelasi kendall’s τ misalkan 1 2( , ,..., ), nz z z z n n= ∈ adalah proses stationer dengan waktu diskrit berukuran n, dan ˆ nτ adalah taksiran kendall’s τ untuk koefisien autokorelasi lag-1, maka genest & fergusen (1999) mendefinisikan [ ]ˆ ˆ ˆ( ) n n n n e t var τ − τ = τ , (3.29) di mana [ ]ˆ 0ne τ = adalah mean dari ˆ nτ yang didefinisikan, [ ] 2ˆ 3( 1)n e n τ = − , 3n ≥ , (3.30) untuk kasus proses stasioner yang bersiklus maupun tidak bersiklus. dan ˆ( )nvar τ adalah variansi dari ˆ nτ yang berbeda untuk kasus proses stasioner yang bersiklus dan tidak bersiklus. untuk kasus proses stasioner yang bersiklus ˆ( ) 0nvar τ = , untuk 3n = , dan 3 2 2 2 20 14 98ˆ( ) 45 ( 1) n n n var n n − − τ = − , untuk 4n ≥ . (3.31) sedangkan untuk kasus proses stasioner yang tidak bersiklus ˆ( ) 8 / 9nvar τ = , untuk 3n = , dan pemodelan copula: studi banding kuantifikasi autokorelasi  volume 1 no. 1 november 2009 45 3 2 2 2 20 74 54 148ˆ( ) 45( 1) ( 2) n n n n var n n − + + τ = − − , untuk 4n ≥ . (3.32) selanjutnya untuk pengujian keberartian nilai ˆ nτ , genest & fergusen (1999) mengatakan pada tingkat keberartian α , nilai ˆ nτ dikatakan berarti untuk pengujian dua arah, jika ,n nt tα> , (3.33a) sedangkan, untuk pengujian satu arah, jika tn > tα,n atau tn < – tα,n (3.33b) di mana ,ntα adalah suatu nilai sehingga ,( )n np t tα> = α dengan ~n nt t . 4. simulasi dan hasil perbandingan kuantifikasi autokorelasi 4.1. desain simulasi simulasi ini dilakukan untuk membandingkan hasil kuantifikasi autokorelasi klasik dan kuantifikasi autokorelasi kendall’s tau melalui copula empirik. adapun desain dari simulasi yang dilakukan adalah membangun barisan data yang mengikuti model proses stasioner, simulasi ini dilakukan untuk barisan data berukuran 10, 15, dan 20n = . cara membangun barisan data dalam simulasi ini adalah sebagai berikut: misalkan ie , 1, 2,...,i n= adalah data yang dibangkitkan secara acak dari distribusi normal (0,1), maka desain 2 1/ 21 1(1 )x e −= + θ dan 1.i i ix x e−= θ + , 2,...,i n= (4.1) di mana nilai θ disimulasikan untuk beberapa nilai (1/ 2) jθ = , 1, 2,..., 5j = dan 0θ = . selanjutnya, kuantifikasi autokorelasi dari barisan data yang ada dilakukan dengan metode yang telah dijelaskan sebelumnya. percobaan ini lakukan sebanyak 500 kali untuk setiap ukuran n, kemudian dilakukan pengujian keberartian autokorelasi berdasarkan hipotesis nol yang menyatakan bahwa tidak terdapat autokorelasi yang berarti pada barisan data simulasi. perbandingan terhadap hasil kuantifikasi autokorelasi untuk masing-masing metode dilakukan dengan menghitung prosentase penolakan hipotesis nol. sehingga dalam simulasi ini diperlukan data acak berdistribusi normal (0,1) sebanyak 500 x 6 x (10+15+20) = 135000 data. 4.2. hasil simulasi setelah dilakukan simulasi berdasarkan desain di atas, maka hasilnya dapat dilihat pada tabel 2. dari tabel tersebut, dapat dikatakan secara keseluruhan, jika nilai θ semakin mendekati nol, maka prosentase penolakan hipotesis nol yang menyatakan bahwa autokorelasi pada model proses stasioner cenderung menurun. dalam batas nilai θ yang digunakan dalam simulasi ini, perbandingan antara kuantifikasi autokorelasi klasik dengan kuantifikasi autokorelasi kendall’s tau, dapat dikatakan mengenai beberapa hal, yaitu: 1. untuk nilai 1 1 1, ,2 4 8θ = , prosentase penolakan hipotesis nol berdasarkan kuantifikasi autokorelasi kendall’s tau melalui copula lebih tinggi dibanding prosentase penolakan hipotesis nol berdasarkan kuantifikasi autokorelasi klasik. fachrur rozi  46 volume 1 no. 1 november 2009 2. sebaliknya untuk nilai 1 1, , 016 32θ = , prosentase penolakan hipotesis nol berdasarkan kuantifikasi autokorelasi klasik lebih tinggi dibanding prosentase penolakan hipotesis nol berdasarkan kuantifikasi autokorelasi kendall’s tau melalui copula. 3. hal ini menunjukkan bahwa untuk nilai θ tertentu kuantifikasi autokorelasi kendall’s tau melalui copula dikatakan lebih baik dibanding dengan kuantifikasi autokorelasi klasik, sedangkan untuk θ yang mendekati nol, kuantifikasi autokorelasi klasik dikatakan lebih baik dibanding dengan kuantifikasi autokorelasi kendall’s tau melalui copula. tabel 2. prosentase penolakan hipotesis non pada simulasi pengujian autokorelasi statistic n theta ½ ¼ 1/8 1/16 1/32 0 autocorrelation function 10 15.4% 3.0% 2.0% 0.8% 1.6% 0.2% kendall' tau dg copula 10 22.6% 8.4% 6.2% 3.6% 3.4% 2.8% autocorrelation function 15 32.6% 11.8% 4.0% 4.0% 3.2% 0.6% kendall' tau dg copula 15 38.8% 18.4% 8.0% 7.2% 6.6% 4.6% autocorrelation function 20 49.4% 15.4% 6.0% 3.2% 4.0% 1.2% kendall' tau dg copula 20 54.8% 20.8% 10.6% 7.4% 5.6% 5.2% 5. kesimpulan dan saran berdasarkan penjelasan teori dan hasil simulasi yang dilakukan pada bagian sebelumnya, penulis mencoba mengambil beberapa kesimpulan sebagai berikut: 1. kendall’s tau melalui copula adalah metode alternatif yang dapat digunakan dalam kuantifikasi dependensi antara dua variabel acak, lebih khusus dapat digunakan dalam kuantifikasi autokorelasi lag-1. 2. pada model 1.i i ix x e−= θ + , i = 1,2,…,n , untuk batas θ tertentu, kuantifikasi autokorelasi kendall’s tau melalui copula dikatakan lebih baik dibanding dengan kuantifikasi autokorelasi klasik, sedangkan untuk θ yang mendekati nol, kuantifikasi autokorelasi klasik dikatakan lebih baik dibanding dengan kuantifikasi autokorelasi kendall’s tau melalui copula. 3. untuk simulasi, perlu adanya penelitian lebih lanjut mengenai pemilihan model time series yang lain serta penentuan batas nilai θ yang digunakan dalam model. pemodelan copula: studi banding kuantifikasi autokorelasi  volume 1 no. 1 november 2009 47 daftar pustaka arcana, i nyoman. (2005), batch process, how to measure a process capability with a better way: a case study at soft drink factory, j. int. conf. of app. math. (icams), bandung, indonesia, pp 5-6. box, g.e., dan jenkins, g.m. (1976), time series analysis: forecasting and control, holden day, san francisco, pp 23-28. genest, c., dan fergusen, t., (1999), kendall’s tau for autocorrelation, departement of statistics papers of university of california, los angeles, pp 1-9, 11, 12. jogde, k., (1982), concepts of dependence, in encyclopedia of statistical sciences, vol.1, s. kotz dan n.l, johnson, editor, john wiley & sons, new york. nelsen, b. roger, (1999), an introduction to copula, spinger-verlag, new york. schmitz, v., (2003), copulas and stochastic processes, disertasi program doktor, shaker verlag, aachen. an application of geographically weighted regression for assessing water polution in pontianak, indonesia cauchy –jurnal matematika murni dan aplikasi volume 7(2) (2022), pages 186-194 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 01, 2021 reviewed: november 11, 2021 accepted: january 05, 2022 doi: http://dx.doi.org/10.18860/ca.v7i1.13266 an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar1, naomi nessyana debataraja1,*, and rossie wiedya nusantara2 1department of statistics, universitas tanjungpura, indonesia 2department of soil sciences, universitas tanjungpura, indonesia *corresponding author email: dkusnand@untan.ac.id, naominessyana@untan.ac.id*, rwiedyanusantara@gmail.com abstract geographically weighted regression (gwr) is an exploratory analytical tool that creates a set of location-specific parameter estimates. the estimates can be analyzed and represented on a map to provide information on spatial relationships between the dependent and the independent variables. a problem that is faced by the gwr users is how best to map these parameter estimates. this paper introduces a simple mapping technique that plots local t-values of the parameters on one map. this study employed gwr to evaluate chemical parameters of water in pontianak city. the chemical oxygen demand (cod) was used as the dependent variable as an indicator of water pollution. factors used for assessing water pollution were ph (𝑋1), iron (𝑋2), fluoride (𝑋3), water hardness (𝑋4), nitrate (𝑋5), nitrite (𝑋6), detergents (𝑋7) and dissolved oxygen, do, (𝑋8). samples were taken from 42 locations. chemical properties were measured in the laboratory. the parameters of the gwr model from each site were estimated and transformed using geographic information systems (gis). the results of the analysis show that 𝑋1, 𝑋2, 𝑋3, 𝑋5 and 𝑋7 influence the amount of cod in water. the resulting map can assist the exploration and interpretation of data. keywords: chemical parameters; geographically weighted regression; modelling; t-value mapping introduction in a residential area, human activities are one of the critical aspects that affect the quality of water resources. the more activities in the area, the higher the waste discharged into the environment. as the capital city of the west kalimantan province, the level of land use in pontianak city has increased every year. this increase has resulted in a decrease in the carrying capacity of the city. one form of land use that has experienced a very rapid rise is the land for settlements. this condition is closely related to population growth. the total population of pontianak city is estimated at 646,661 people, with a population density of 5,998 people/km2 and a population growth rate of 1.95% per year [1]. the quantity and quality of water in a region significantly affects the life of living things. changes in the quality and quantity of water are strongly influenced by the patterns of land management that exist in the area. waste generated from human http://dx.doi.org/10.18860/ca.v7i1.13266 mailto:dkusnand@untan.ac.idm mailto:naominessyana@untan.ac.idi mailto:rwiedyanusantara@gmail.com an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 187 activities in daily life can cause deterioration in water quality. discharged waste has different characteristics that determine the degree of water quality around it. waste produced from the activities of human life is diverse both in type and content. the waste can be in the form of organic compounds degraded by microorganisms as well as inorganic compounds such as soap, detergent, shampoo, and other cleaning agents that can contaminate water [2]. parameters in measuring water quality include physical, biological and chemical parameters. these parameters are essential variables for measuring the water quality [3]. however, this paper focused on chemical variables. water pollution can also be seen from the amount of oxygen content dissolved in water, namely through the measurement of chemical oxygen demand [4]. chemical oxygen demand (cod) is the total amount of oxygen needed to oxidize organic matter chemically. household waste is the primary source of organic waste and is one of the leading causes of high cod concentrations. this condition has an impact on humans and the environment, one of which is that many aquatic biotas die because the level of oxygen dissolved in water is small. the criteria for proper water use are increasingly difficult to obtain. this research aimed to investigate the relationships among many variables and the cod in the research area. the samples were collected from several locations having different characteristics and types of land and environment. this sampling procedure could create a dependence between data measurements and their locations. hence, it generates spatial data. techniques of spatial analysis can then be applied to the collected data. this research utilized the geographically weighted regression (gwr) to investigate the relationship between the dependent variable and the corresponding independent variables. as an exploratory method, gwr provides extra information for any spatial data set and should be useful across all disciplines in which spatial data are utilized. applications of gwr include studies in a wide variety of demographic fields including but not limited to the analysis of health and disease (see for example [5, 6, 7, 8]), environmental equity [9], housing markets [10, 11], population density and housing [12], poverty mapping in malawi [13], urban poverty [14], demography and religion [2], as well as environmental conditions [15]. brown et al. [16] used gwr to investigate the relationships between land cover, rainfall, and surface water habitat in predominately agriculture regions in southeast australia. it was found that gwr provided a better estimate than the ols method. in this study, gwr was applied to investigate the relationship among variables of chemical contained in the water samples. methods the research was carried out in pontianak city (lat. 0°02' n – 0°01' s, long. 109°16' – 109°23 e). pontianak is the capital city of the west kalimantan province, indonesia. it covers approximately 107.82 km2. soil conditions in the city of pontianak consist of soil types of organosol, gley, humus, and alluvial, each of which has different characteristics. the sampling method was carried out by stratified random sampling. subsequent sub-populations called strata were formed based on the criteria of the area flowed by the same tributary. it is assumed that the level of water pollution is homogeneous. the sample units studied were rivers/ditches, with a total sample of 42 water samples from different locations, representing the six districts in the city of pontianak. the sites were plotted into a map of pontianak city in figure 1 [17]. the samples were taken in the same an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 188 conditions, namely when the water receded. in this study, the response variable is cod, while the independent variables used include ph (𝑋1), iron (𝑋2), fluoride (𝑋3), hardness (𝑋4), nitrate (𝑋5), nitrite (𝑋6), detergent (𝑋7), and dissolved oxygen, do, (𝑋8). figure 1. map of sample locations suppose we have a set of observations ijx for i = 1, 2, …, n cases and j = 1, 2, …, k independent variables, and a set of dependent variables  iy for each case. this notation is standard data set for a global regression model. now suppose that in addition to this, we have a set of location coordinates   ,i iu v for each case. the underlying model for gwr is as follows [18]:    0 1 , , k i i i j i i ij i j y u v u v x       (1) where       0 1, , , , ,ku v u v u v   are k + 1 continuous functions of the location (u, v) in the geographical study area, and 𝜀𝑖 ∼ 𝑁(0, 𝜎 2). the log-likelihood for any particular set of estimates of the functions may be written as follows (see, for example [18]):            2 0 1 02 1 1 1 , , , , , | , , 2 n k k i i i ij j i i i j l u v u v u v d y u v x u v                   (2) where d is the union of the set  ijx ,  iy and   ,i iu v . rather than attempting to maximize equation (2) globally, we consider the local likelihood. we consider the problem of estimating       0 1, , , , ,ku v u v u v   on a pointwise basis. that is, given a specific point in geographical space  0 0,u v , we attempt to estimate       0 1, , , , ,ku v u v u v   . the point  0 0,u v may or may not correspond to one of the observed  ,i iu v . if these functions are reasonably smooth, we can assume that a simple regression model 0 1 k i ij j i j y x       (3) holds close to the point  0 0,u v , where each j is a constant valued approximation of the an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 189 corresponding  ,j u v in eq. (1). we can calibrate a model of this sort by considering observations close to the point  0 0,u v . an obvious way to do this is to use weighted least squares; this is to choose  0 1, , , k   to minimize   2 0 0 1 1 n k i i ij j i j w d y x             (4) where d0i is the distance between the points  0 0,u v and  ,i iu v . this result gives us the standard gwr approach. we simply set  0 0ˆ ,j u v as ˆ j to obtain the familiar gwr estimates. at this stage, it is worth noting that eq. (4) maybe multiplied by 2 1   and be considered as a local log-likelihood expression:      0 0 0 1 | | n k i k i wl d w d l d       (5) where  0 |kwl d  is an empirical estimate of the expected log-likelihood at the point of estimation gwr employs a weighted distance decay function for model calibration. the gwr assumes that observations closer together will have more impact on each other than on observations further apart. the weighting function for including related samples can be calculated using the exponential distance decay function: 2 2 exp ij ij d b           (6) where ij is the weight of observation j for observation i, dij is the distance between observation i and j and b are the kernel bandwidth when the distance between observations is greater than the kernel bandwidth, the weight rapidly approaches zero. fixed bandwidth kernel calculates a bandwidth that is held constant over space, whereas the adaptive bandwidth kernel can adapt bandwidth distance in relation to variabledensity; bandwidths are smaller where data are dense and more abundant when data are sparse. in this study, all gwr models used the adaptive kernel bandwidth as sample densities varied spatially. the optimal bandwidth distance was determined automatically in gwr using the akaike information criterion (aic). results and discussion tests on spatial aspects consist of two stages, namely the analysis of spatial dependency and spatial heterogeneity test. the spatial dependency test performed using the moran's i test, whereas the spatial heterogeneity test used the breusch-pagan test. the test results are presented in table 1. table 1. test on spatial aspects tests p-value decision moran’s i 9,763e-08 reject h0 breusch-pagan 0,1375 do not reject h0 moran’s i test indicated the existence of dependency spatial in the model. whereas the results of breush-pagan showed there are no differences in characteristics among an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 190 observation points. the next step is the selection of bandwidth that will be used in gwr modeling. bandwidth selection can be made by examining the cross-validation (cv) value between weighting functions. the weighting function used were fixed gaussian, fixed bisquare, and fixed tricube (table 2). table 2. cross-validation values and the bandwidth for each model. model cv bandwidth gaussian 19,587.93 0.020963 bisquare 20,381.45 0.07239735 tricube 20,260.00 0.07239735 table 2 shows that the fixed gaussian model gives a minimum value of the cv with the bandwidth of 0.020963. the fixed gaussian model was then used to obtain the weighting matrix for each location. models obtained by the gwr was compared to that of the ordinary least squares (ols) method in term of their sse. the results showed that gwr performed better than that of the ols (table 3). table 3. comparison between the gwr and the ols methods. sse df f p-value ols 17,330.621 33,000 gwr 5,195.428 16,602 3.3357 0.005773 models for all 42 locations of the cod on the eight dependent variables are presented in table 4. the coefficient of determination (r2) also showed in the table. table 4. regression model for the 42 locations no location r2 model 1 s. raya dalam 1 .78 y=190-8.9x1-13.4x2+70.3x3-+0.02x4-16.3x5-11.6x6-8682x7+0.9x8 2 s. raya dalam 2 .82 y=174-9.0x1-12.2x2+76x3+0.34x4-19.3x5-15.2x6-9011x7+0.1x8 3 jl. sepakat 2 .75 y=249-12.2x1-14.0x2-73x3-0.72x4-11.1x5-11.5x6+10665x7-3.9x8 4 jl. parit h. husin 2 .95 y=181-12.2x1-6.1x2-8.2x3+0.92x4-20.4x5+29.3x6+19033x7+3.6x8 5 jl. parit h. husin 1 .82 y=173-9.6x1-11.6x2-84x3+0.35x4-19.5x5-15.4x6+9350x7-0.2x8 6 jl. imam bonjol .83 y=174-10.1x1-10.9x2-92x3+0.31x4-19.4x5-14.8x6-9693x7-0.7x8 7 jl. media .83 y=190-11.6x1-10.2x2-97x3+0.14x4-18.2x5-12.9x6-10117x7-1.9x8 8 jl. perdana .81 y=241-14.3x1-11.5x2+88x3-0.38x4-14.2x5-12.2x6-11432x7-3.4x8 9 jl. purnama .84 y=275-17.2x1-10.1x2+81x3-0.65x4-13.2x5-10.9x6-12776x7-3.2x8 10 jl. tebu .89 y=172-13.6x1-6.3x2+69x3-0.17x4-10.1x5-11.8x6-3942x7+3.0x8 11 jl. karet .93 y=222-21.5x1-5.9x2+53x3-0.01x4-7.0x5-6.9x6-3448x7+3.2x8 12 jl. ampera .90 y=340-37.1x1-5.0x2-27x3+0.07x4-3.7x5+22.8x6-4795x7-3.9x8 13 jl. wahidin .88 y=283-27.8x1-5.6x2+56x3-0.03x4-8.2x5+4.4x6-5345x7+0.3x8 14 jl. purnama jaya .84 y=264-17.3x1-9.7x2+86x3-0.45x4-14.0x5-10.9x6-12128x7-3.7x8 15 jl. uray bawadi .86 y=260-19.5x1-7.8x2+82x3-0.25x4-13.6x5-6.3x6-10159x7-3.3x8 16 jl. hm suwignyo .86 y=259-21.5x1+6.8x2+74x3-0.17x4-12.0x5-2.6x6-8077x7-2.2x8 17 ujung suwignyo .85 y=143-6.6x1-8.8x2+91x3-0.10x4-16.3x5-11.9x6-7250x7-0.9x8 18 jl. gst hamzah .86 y=223-17.1x1-7.2x2+78x3-0.13x4-13.0x5-5.4x6-7484x7-2.0x8 19 jl. hasanuddin .85 y=122-4.1x1-9.2x2+94x3-0.11x4-17.0x5-13.8x6-6912x7-0.2x8 an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 191 no location r2 model 20 jl. k yos sudarso .86 y=141-8.4x1-6.9x2+79x3-0.21x4-12.3x5-14.1x6-4619x7+2.3x8 21 jl. wr. supratman .85 y=171-9.5x1-8.8x2+95x3-0.03x4-17.1x5-11.3x6-8755x7-2.3x8 22 pasar flamboyan .84 y=180-10.6x1-9.3x2+98x3+0.09x4-18.2x5-12.5x6-9767x7-2.4x8 23 depan ramayana .84 y=163-10.6x1-9.4x2+100x3+0.15x4-19.1x5-13.7x6-9544x7-1.9x8 24 jl. flora siantan .95 y=178-15.8x1-6.5x2+65x3-0.17x4-8.3x5-16.3x6-2715x7+5.7x8 25 jl. teluk selamat .84 y=82+0.4x1-8.9x2+97x3-0.21x4-17.4x5-19.7x6-5497x7+2.4x8 26 jl. parit makmur .84 y=91+0.1x1-10.0x2+101x3-0.09x4-19.7x5-17.9x6-6875x7+0.8x8 27 jl. puring 1 .85 y=105-1.5x1-9.8x2+101x3-0.04x4-19.6x5-16.5x6-7557x7-0.2x8 28 jl. selat sumba 2 .84 y=97-0.4x1-9.9x2+102x3+0.01x4-20.8x5-18.4x6-7735x7+0.9x8 29 parit pangeran .84 y=85+1.2x1-9.9x2+99x3+0.01x4-22.0x5-21.1x6-7593x7+0.9x8 30 jl. keb. nasional .83 y=85+1.3x1-9.7x2+91x3+0.12x4-23.1x5-23x6-8678x7+1.7x8 31 jl. selat panjang .84 y=86+1.1x1-9.7x2+95x3+0.11x4-22.9x5-22.5x6-8431x7+1.2x8 32 jl. tritura .84 y=110-2.1x1-9.7x2+102x3+0.15x4-21.4x5-18.4x6-8714x7-0.5x8 33 tanjung hilir .84 y=116-3x1-9.6x2+102x3+0.14x4-21x5-17.4x6-8751x7-0.8x8 34 tanjung raya 1 .84 y=128-4.8x1-9.7x2+100x3+0.27x4-21.4x5-17.5x6-9334x7-0.6x8 35 tanjung raya 2 .84 y=132-5.3x1-9.9x2+99x3+0.32x4-21.4x5-17.6x6-9445x7-0.5x8 36 jl. panglima aim .84 y=120-4x1-10.1x2+95x3+0.39x4-22.4x5-19.7x6-9562x7+0.4x8 37 jl. ya’ m sabran .83 y=96-0.6x1-10.2x2+86x3+0.37x4-23.6x5-22.3x6-9998x7+2.2x8 38 jl. tani .83 y=120-3.9x1-10.6x2+82x3+0.53x4-23.1x5-21.2x6-10021x7+1.7x8 39 tj. raya 2 ujung .83 y=163-7.9x1-11.8x2+59x3+0.54x4-21.6x5-16.8x6-9212x7+1.2x8 40 jl. tabrani ahmad .87 y=216-19.0x1-6.2x2+64x3-0.08x4-9.5x5-4.7x6-4688x7+1.0x8 41 jl. harapan jaya .85 y=281-17.1x1-10.0x2+65x3-0.86x4-11.3x5-3.0x6-13148x7+0.1x8 42 jl. merdeka .85 y=162-9.0x1-8.5x2+89x3-0.10x4-15.6x5-10.4x6-7406x7-1.4x8 y = cod; x1 = ph; x2 = iron; x3 = fluoride; x4 = water hardness; x5 = nitrate; x6 = nitrite; x7 = detergents; x8 = do the coefficient of determination of the model varied between 75% (location 3) to 95% (location 4 and 24). partial significance tests were carried out to examine which parameters are significant. the t-statistic was used for the tests. the values of the tstatistics for each parameter are presented in figure 2 (a) to (h). (a) t value for 1 (b) t value for 2 an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 192 (c) t value for 3 (d) t value for 4 (e) t value for 5 (f) t value for 6 (g) t value for 7 (h) t value for 8 notes: ●: |t| < 1.64; : 1.64 ≤ |t| < 1.96; +: 1.96 ≤ |t| < 2.33; : |t| > 2.33 figure 2. (a) to (h) are the plot of t values for each 𝛽 in every location the sample locations marked with ● indicate that the corresponding coefficient of 𝛽i is not significant (the |t| value is smaller than 1.64). the symbol  is used to indicate that the coefficient of 𝛽 i is almost significant (1.64 ≤ |t| < 1.96). whereas the symbols + and  are used to indicate that the coefficient of 𝛽 i are significant (|t| ≥ 1.96). the values of t for 𝛽1 are generally small for the samples located close to the rivers (fig. 2 (a)). the variable ph (x1) for those locations is not significant, hence it does not contribute to modeling the cod. however, variable ph appeared to be significant for the sample located at some distance from the river (there were 18 sample locations). in general, the t values for 𝛽2, 𝛽3, 𝛽5, and 𝛽7 are significant (fig. 2 (b), (c), (e), (g)). these results indicate that the contents of the variable of iron, fluoride, nitrate, and detergent have a great influence in an application of geographically weighted regression for assessing water polution in pontianak, indonesia dadan kusnandar 193 modeling the cod. high nitrogen compounds in the form of oxidized nitrogen, such as nitrate, tend to reduce the level of dissolved oxygen in water through the oxidation of ammonia [13, 10]. likewise, detergent, where one of the integral ingredients is phosphate compound, has a major role in the occurrence of eutrophication in the water body [22]. the t value for 𝛽4, 𝛽6, and 𝛽8 (fig.2 (d), (f), and (h)) are generally small, indicating that water hardness (𝑋4), nitrite (𝑋6), and do (𝑋8) are not significant in the whole samples. the three variables are, therefore, not important in predicting the cod of the samples. plotting the t value of the regression coefficients of the sample location in a map enables the researcher to identify the importance of the variables to the regression models in each sample location. the application of gwr allows getting a different model in each location. conclusions this study has demonstrated the superiority of gwr to model the spatially varying relationship between variables over ols regression. gwr 42 samples point can result in 42 regression models that accommodate the characteristics of the location. the variable ph (x1) generally small for the samples located close to the rivers, however it is significant for the samples located at some distances from the river. the variable of iron, fluoride, nitrate and detergent significant in regression modeling of cod. these results could provide the researcher with a more accurate prediction of chemical oxygen demand in each location. acknowledgments the authors would like to acknowledge the financial support no. 098/sp2h/lt/drpm/2018 from the ministry of research, technology, and higher education of the republic of indonesia to conduct the research. references [1] bps kota pontianak, pontianak municipality in figure, bps, pontianak, 2019. 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[17] d. kusnandar. n.n. debataraja, s.w. rizki, and e. saputri, "water quality mapping in pontianak city using multiple discriminant analysis," the 4th indoms international conference on mathematics and its applications, aip conference proceedings, vol 2268, pp. 020006-1020006-2, 2020 [18] a. s. fotheringham, c. brunsdon, and m. charlton, geographically weighted regression: the analysis of spatially varying relationships', john wiley & sons, chichester, 2002. 3 desy norma spectrum detour graf n-partisi komplit spectrum detour graf n-partisi komplit desy norma puspita dewi jurusan matematika uin maulana malik ibrahim malang e-mail:phyta_23@yahoo.co.id abstrak matriks detour dari graf g adalah matriks yang elemen ke-(i,j) merupakan panjang lintasan terpanjang antara titik �� ke titik �� di g. himpunan nilai eigen matriks detour dari graf terhubung langsung g adalah spectrum detour. spectrum detour dari graf g biasanya dinotasikan dengan ������ �.. dalam artikel ini, hanya menentukan spectrum detour graf n-partisi komplit ��,���,���,…,����, dan graf 3partisi komplit ��,�,��. dalam menentukan spectrum detour graf tersebut dengan cara menggambar pola grafnya, mencari matriks detournya, setelah itu dicari nilai eigen dan vektor eigen dari matriks tersebut, sehingga diperoleh pola (konjektur) spectrum detour, kemudian merumuskan konjektur sebagai teorema yang dilengkapi dengan bukti-bukti. kata kunci: graf n-partisi komplit, matriks detour, dan spectrum, abstract the detour matrices of a graph is for its (i,j) entry the length of the longest path between vertices �� to �� of g. set of detour matrices eigenvalues of a connected graph g are detour spectrum. detour spectrum of g, denoted by ������ �. in this article, only determination of detour spectrum of complete npartition graph ��,���,���,…,����, and complete 3-partition graph ��,�,��. determination of spectrum detour graph are picturing model graph, then finding detour matrices, after that finded eigenvalues and eigenvectors from that matrices, with the result that detour spectrum obtained model of detour spectrum, end then formulate the model as theorem with its prove. keywords: complete npartitions graph, detour matrices, and spectrum pendahuluan graf g adalah pasangan (v(g),e(g)) dengan v(g) adalah himpunan tidak kosong dan berhingga dari objek-objek yang disebut titik (vertex), dan e(g) adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di v(g) yang disebut sisi (edge). misalkan terdapat suatu graf g, dari suatu graf tersebut dibentuk matriks adjacency atau matriks keterhubungan. matriks adjacency merupakan matriks simetri. matriks adjacency dapat dirubah menjadi matriks detour, yang unsur-unsur ke--(i,j) merupakan panjang lintasan terpanjang antara titik i dan j. setelah dibentuk menjadi matriks detour, maka dapat dicari nilai eigen dan vektor eigen dari matriks tersebut. biasanya spectrum graf dibentuk dari nilai eigen dari matriks terhubung langsung. dalam pengertian, nilai eigen dari graf g dinotasikan dengan �� , i = 1,2,…,n dan spectrum ditulis dengan spec(g). matriks detour didefinisikan dd=dd(g) dari g sehingga unsur ke (i,j) adalah panjang lintasan terpanjang antara titik i dan j. nilai eigen dari dd(g) disebut dd-nilai eigen dari g dan membentuk dd-spectrum dari g, dinotasikan dengan spec�� g�. karena matriks detour simetris, semua nilai eigen µ , i = 1,2,…,n adalah real dan dapat diberi label µ� ! µ� ! " !µ#. jika µ $ ! µ % ! " ! µ & adalah nilai eigen dari matriks detour, maka dd-spectrum dapat ditulis sebagai ������ � ' (µ $)� µ %)� …… µ &)*+ di mana mmenyatakan banyaknya basis untuk ruang vektor eigen m . dan m1 / m2 / " / m� 'n. (ayyaswamy dan balachandran, 2010:250). kajian teori 1. graf definisi 1 graf g adalah pasangan himpunan (v,e) dengan v adalah himpunan tidak kosong dan berhingga dari obyek-obyek yang disebut sebagai titik dan e adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di g yang disebut sebagai sisi (chartrand dan lesniak, 1986:4). desy norma puspita dewi 14 volume 2 no. 1 november 2011 sehingga jika ' 1 �, 2 ��, maka 1 � ' 3�1, �2, … , ��4 dan 2 � ' 3�1, �2, … ,��4, dimana �� 5 1 �, 6 ' 1,2, … , 9 disebut titik (vertex) dan �� 5 2 �, : ' 1,2, … , ) disebut sisi (edge). 2. adjacent dan incident sisi � ' ;� dikatakan menghubungkan titik ; dan �. jika � ' ;� adalah sisi di graf , maka ; dan � disebut terhubung langsung (adjacent). � dan � serta ; dan � disebut terkait langsung (incident). titik ; dan � disebut ujung dari �. dua sisi berbeda �1 dan �2 disebut terhubung langsung jika terkait langsung pada titik yang sama. untuk selanjutnya sisi � ' ;, �� ditulis � ' ;� (abdussakir, dkk, 2009:6). 3. graf komplit definisi 2 graf komplit (complete graph) adalah graf dengan dua titik yang berbeda saling terhubung langsung (adjacent). graf komplit dengan 9 titik dinyatakan dengan �� (chartrand dan lesniak, 1986:9). definisi 3 graf g dikatakan partisi n-komplit jika g adalah graf partisi-n dengan himpunan partisi 11, 12, … , 1� sehingga jika ; < 1� dan � < 1�, 6 < :, maka ;� < 2 �. maka graf ini dinotasikan dengan �=1,=2,…,=> (abdussakir, dkk. 2009:23). 4. graf terhubung definisi 4 misalkan graf. misalkan ; dan � adalah titik pada . jalan (trail) ;� pada yang dinotasikan ? adalah barisan berhingga yang berganti ?: ; ' �0, �1, �1, �2, �2, … ,��, �� ' � antara titik dan sisi yang diawali dan diakhiri dengan titik dengan �� ' ��a1�� adalah sisi di untuk 6 ' 1,2, … , 9. �0 disebut titik awal dan �� disebut titik akhir. titik �0, �1, �2, … , �� disebut titik internal, dan 9 menyatakan panjang dari ?. jika �0 < ��, maka w disebut jalan terbuka. jika �0 ' �� maka w disebut jalan tertutup. jalan yang tidak mempunyai sisi disebut jalan trivial (abdussakir, dkk. 2009:49). 5. nilai eigen dan vector eigen definisi 5 misalkan a sebuah matrik n × n. bilangan � disebut nilai eigen (eigenvalue) dari a jika terdapat vektor tidak nol � 5 b� sedemikian sehingga ax = �x . kemudian vektor x disebut vektor eigen (eigenvector) dari a yang berpasangan ke nilai eigen � (jain & gunawardena, 2004:151). 6. spectrum graf misalkan terdapat suatu graf g, dari suatu graf tersebut dibentuk matriks adjacency atau matriks keterhubungan. matriks adjacency merupakan matriks simetri. matriks adjacency dapat dirubah menjadi matriks detour, yang unsur-unsur ke-(i,j) merupakan panjang lintasan terpanjang antara titik i dan j. setelah dibentuk menjadi matriks detour, maka dapat dicari nilai eigen dan vektor eigen dari matriks tersebut. misalkan g graf berorder p dan a matriks keterhubungan dari graf g. suatu vektor tak nol x disebut vektor eigen (eigen vector) dari a jika cd adalah suatu kelipatan skalar dari x, yakni cd ' �d, untuk sebarang skalar �. skalar � disebut nilai eigen (eigen value) dari a, dan x disebut sebagai vektor eigen dari a yang bersesuaian dengan �. untuk menentukan nilai eigen dari matriks a, persamaan cd ' �ed�d ditulis kembali dalam bentuk c f �e�d ' 0, dengan i matriks identitas berordo p. persamaan ini akan mempunyai solusi tak nol jika dan hanya jika g�h c f �e�d ' 0 persamaan g�h c f �e� ' 0 akan menghasilkan persamaan polinomial dalam variabel � dan disebut persamaan karakteristik dari matriks a. skalar-skalar � yang memenuhi persamaan karakteristik ini tidak lain adalah nilai-nilai eigen dari matriks a. misalkan �1, �2, … , �� adalah nilai eigen berbeda dari a, dengan �1, �2, … , ��, dan misalkan )�1, )�2, … , )�� adalah banyaknya basis untuk ruang vektor eigen masingmasing i� , maka matriks berordo 2 j 9� yang memuat �1, �2, … , �� pada baris pertama dan )�1, )�2, … , )�� pada baris kedua disebut spectrum graf g, dan dinotasikan dengan ���� �. jadi spectrum graf g dapat ditulis dengan ���� � ' ( ��)�� ��)�� …… ��)��+ (abdussakir, dkk, 2009:82-83). 7. graf dalam matriks detour matriks detour didefinisikan dd = dd(g) dari g sehingga unsur atau entry (i,j) adalah panjang lintasan terpanjang antara titik i dan j. nilai eigen dari dd(g) disebut ddnilai eigen dari g dan membentuk ddspectrum dari g, yang dinotasikan dengan ������ �. selama matriks detour simetris, semua nilai eigen k� , i = 1, 2, …, n adalah real dan dapat diberi label k1 ! k2 ! " ! k�. jika k�1 ! k�2 ! " ! k�l adalah nilai eigen dari matriks detour, maka dd-spectrum dapat ditulis sebagai spectrum detour graf n-partisi komplit jurnal cauchy – issn: 2086-0382 15 ������ � ' (µ $)� µ %)� …… µ &)*+ di mana )� menunjukkan banyaknya basis untuk ruang vektor eigen dalam k�m dan tentunya )1 / )2 / … / )* ' 9. (ayyaswamy dan balachandran, 2010) pembahasan 1. spectrum detour dari graf n-partisi komplit ��,���,���,…,���� pembahasan spectrum detour dari graf npartisi komplit ��,���,���,…,����, dibatasi pada 9 ! 2, ) ! 1 dan 9, ) 5 n. 1.1 spectrum detour graf 3-partisi komplit op,q,r ������ ��,s,t� ' u64 f81 8 y 1.2 spectrum detour graf 4-partisi komplit op,q,r,z ������ ��,s,t,[� ' u169 f131 13 y 1.3 spectrum detour graf 5-partisi komplit op,q,r,z,^ ������ ��,s,t,[� ' u361 f191 19 y 1.4 spectrum detour graf 6-partisi komplit op,q,r,z,^,_ ������ ��,s,t,[� ' u676 f261 26 y teorema 1: jika ��,��1,��2,…���� adalah graf n-partisi komplit dengan 9 ! 2, ) ! 1; 9, ) 5 n dan � ' 9 / 9 / 1� / 9 / 2� / " / 9 / )�, maka: ������ ��,��1,��2,…����' ( � f 1�2 f � f 1� 1 � f 1� + dimana ������ ��,��1,��2,…���� adalah spectrum detour dari graf n-partisi komplit dan 9 bilangan asli. bukti: misalkan bb ��,��1,��2,…���� adalah matrik detour adjacent dari ��,��1,��2,…����, maka bb ��,��1,��2,…���� ' cd dd e 0 � f 1 � f 1 … � f 1� f 1 0 � f 1 … � f 1� f 1 � f 1 0 … � f 1f f f g f� f 1 � f 1 � f 1 … 0 hi ii j dari matriks detour adjacent di atas, maka akan dicari nilai eigennya dengan menentukan det k�e f bb ��,��1,��2,…����l ' 0 mmλ cdd de1 0 0 … 00 1 0 … 00 0 1 … 0f f f g f0 0 0 … 1hi iij f cd dd e 0 � f 1 � f 1 … � f 1� f 1 0 � f 1 … � f 1� f 1 � f 1 0 … � f 1f f f g f� f 1 � f 1 � f 1 … 0 hi ii j mm ' 0 mm λ f � f 1� f � f 1� … f � f 1�f � f 1� λ f � f 1� … f � f 1�f � f 1� f � f 1� λ … f � f 1�f f f g ff � f 1� f � f 1� f � f 1� … λ m m ' 0 kita kalikan matriks di atas dengan 1a =a1�, sehingga diperoleh m m m λf � f 1� 1 1 … 1 1 λf � f 1� 1 … 1 1 1 λf � f 1� … 1f f f g f1 1 1 … λf � f 1�m m m ' 0 dimisalkan �p ' k q=a1l, maka mm fλp 1 1 … 11 fλp 1 … 11 1 fλp … 1f f f g f1 1 1 … fλp mm ' 0 melalui operasi basis elementer, matriks det k�e f bb ��,��1,��2,…����l direduksi menjadi matriks segitiga atas, sehingga diperoleh m m mfλ p 1 1 … 10 akrs%a�lrs 1 … 10 0 akrs%a�l rsa��rsa� … 1f f f g f0 0 1 … akrs%a ta�� ta�� rsa ta���%lrsa ta�� ta�� m m m ' 0 sehingga det k�e f bb ��,��1,��2,…����l tidak lain adalah hasil perkalian diagonal matriks segitiga atas tersebut, sehingga diperoleh det kλi f dd k#,#��,#��,…#�z�l' λp f p f 1�� λp / 1�ta� karena det kλe f bb ��,��1,��2,…����l ' 0, maka λp f p f 1�� λp / 1�ta� ' 0 sehingga didapat nilai eigen �p ' � f 1� atau λp ' f1, karena λp ' ra ta�� maka nilai eigennya diperoleh � ' � f 1� atau � ' f1 r ta�� ' p f 1� r ta�� ' f1 λ ' p f 1�� λ ' f p f 1� sedangkan untuk vektor eigennya, yaitu cd ' �d ' 0 desy norma puspita dewi 16 volume 2 no. 1 november 2011 ( ) ' 1 ' 2 ' 1 ' 01 1 1 01 1 1 01 1 1 01 1 1 p p x x x x λ λ λ λ −   −       −          =−               −      ⋯ ⋯ ⋯ ⋮⋮⋮ ⋮ ⋮ ⋱ ⋮ ⋯ kemudian, akan dibuktikan bahwa untuk λ ' n f 1�� akan didapatkan banyaknya basis vektor eigen adalah 1. untuk λ ' f p f 1� akan didapatkan ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 2 1 2 1 1 1 1 1 1 0 1 1 1 1 0 01 1 1 1 1 0 1 1 1 1 1 p p p p xp xp xp p x p p −  −   − −     −        − −          =−       − −             −    − −  ⋯ ⋯ ⋯ ⋮⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋯ ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 p p xp xp xp p x −   − −       − −          − − =               − −      ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮⋮ ⋯ dengan mereduksi matriks di atas menjadi bentuk eselon tereduksi baris, aka didapatkan ( ) 1 2 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 p p x x x x −  −        −          =−                    ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮⋮ ⋯ kemudian didapat d1 ' d=, d2 ' d=, … , d=a1 ' d= sehingga diperoleh d1 ' d2 ' " ' d=a1 ' d=. misal d= ' � maka vektor eigennya adalah ( ) 1 2 1 1 1 1 1 1 p p x s x s s s x s sx −                        = = =                       ⋮ ⋮ ⋮ jadi didapatkan banyaknya basis ruang eigen untuk λ ' p f 1�� adalah 1. untuk λ ' f p f 1� akan didapatkan ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 1 1 0 1 1 1 1 0 01 1 1 1 1 0 1 1 1 1 1 p p p p xp xp xp p x p p −  − −  − −     − −       − −          =− −      − −                − −   − −  ⋯ ⋯ ⋯ ⋮⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋯ ( ) 1 2 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 p p x x x x −                    =                    ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮⋮ ⋯ dengan mereduksi matriks di atas menjadi bentuk eselon tereduksi baris, maka didapatkan ( ) 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p p x x x x −                    =                    ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮⋮ ⋯ kemudian didapat d1 / d2 / " / d =a1� / d= ' 0 sehingga diperoleh d1 ' fd2 f " f d =a1� f d=. maka vektor eigennya adalah ( ) ( ) ( ) 2 11 2 2 1 1 2 pp p p p p x x xx x x s x x x x − − − − − −           = =               ⋯ ⋮ ⋮ jadi didapatkan banyaknya basis ruang eigen untuk � ' f � f 1� adalah � f 1�. jadi terbukti bahwa ������ ��,��1,��2,…���� ' ( � f 1�2 f � f 1� 1 � f 1� +. 2. spectrum detour dari graf 3-partisi komplit 2,2,nk pembahasan spectrum detour dari graf 3partisi komplit 2,2,nk dibatasi pada 9 ! 5. 2.1 spectrum detour graf 3-partisi komplit op,p,z ( )2,2,5 25 1029 25 1029 6 8 1 1 3 4 ddspec k  + − − − =     2.2 spectrum detour graf 3-partisi komplit op,p,^ ( )2 ,2 ,6 29 129 7 29 1297 6 8 1 1 3 5 ddspec k  + − − − =     2.3 spectrum detour graf 3-partisi komplit op,p,_ ( )2,2,7 33 1597 33 1597 6 8 1 1 3 6 ddspec k  + − − − =     2.4 spectrum detour graf 3-partisi komplit op,p,{ ( )2,2,8 37 1929 33 1929 6 8 1 1 3 7 ddspec k  + − − − =     2.5 spectrum detour graf 3-partisi komplit op,p,| ( )2,2,9 41 2293 41 2293 6 8 1 1 3 8 ddspec k  + − − − =     2.6 spectrum detour graf 3-partisi komplit op,p,}~ ( )2,2,10 45 2689 45 2689 6 8 1 1 3 9 ddspec k  + − − − =     spectrum detour graf n-partisi komplit jurnal cauchy – issn: 2086-0382 17 berdasarkan hasil spectrum detour dari graf 3-partisi komplit ��,�,�� di atas, dapat diperoleh dugaan sementara bahwa bentuk umum dari spectrum detour adalah: ( ) ( ) ( ) 2 2 2,2, (2 2 2 1) (2 2 2 1)16 220 793 16 220 793 6 8 1 1 3 1 dd n n n n n spec k n n n n n+ + + + + + + − + + + + − −=  −   dengan 5n ≥ dan n adalah bilangan asli. penutup kesimpulan berdasarkan pembahasan mengenai spectrum detour dari graf n-partisi komplit, diperoleh kesimpulan: (a). untuk graf n-partisi komplit ��,���,���,…,���� dengan 9 ! 2, ) !1; 9, ) 5 n dan � ' 9 / 9 / 1� / 9 / 2� / " / 9 / )�, maka ( ) ( ) ( ) ( ) 2 , 1 , 2 , 1 1 1 1 d d n n n n m p p s p e c k p + + … +  − − − =   −   , (b). untuk graf 3-partisi komplit ��,�,�� dengan 9 ! 2, 9 5 n ( ) ( ) ( ) 2 2 2,2, (2 2 2 1) (2 2 2 1)16 220 793 16 220 793 6 8 1 1 3 1 dd n n n n n spec k n n n n n+ + + + + + + − + + + + − −=  −   saran pada artikel ini, penulis hanya memfokuskan pada spectrum detour yang digambarkan oleh dua bentuk graf n-partisi komplit yaitu graf n-partisi komplit ��,���,���,…,���� dan graf 3-partisi komplit ��,�,��. pada bentuk graf 3-partisi komplit ��,�,�� masih merupakan konjektur, sehingga perlu diselidiki lebih lanjut. karena masih banyaknya bentuk dari graf ini, maka untuk penulisan skripsi selanjutnya diteliti pada graf lain. daftar pustaka [1] abdussakir, dkk. 2009. teory graf : topik dasar untuk tugas akhir/skripsi. malang: uin-malang press. [2] chartrand, g and lesniak, l. 1986. graph and digraph: second edition california: a division wadsworth [3] ayyaswamy, s.k. dan balachandran, s. (2010). “on detour spectra of some graphs”. world academy of science, enggineering and technology. (www.waset.org/journals/waset/v67/v6788.pdf. diakses 2 februari 2011). [4] jain, s. k. 2004. linear algebra; an interactive approach. australia: thomson learning. on irregular colorings of unicyclic graph family cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 503-512 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 01, 2022 reviewed: july 24, 2022 accepted: august 15, 2022 doi: http://dx.doi.org/10.18860/ca.v7i4.16917 on irregular colorings of unicyclic graph family arika indah kristiana*, dafik, qurrotul a’yun, robiatul adawiyah, ridho alfarisi mathematics education study program, faculty of teacher training and education university of jember, indonesia email: arika.fkip@unej.ac.id abstract an irregular coloring is a proper coloring where each vertex on a graph must have a different code. the color code of a vertex v is 𝑐𝑜𝑑𝑒(𝑣) = (𝑎0,𝑎1,𝑎2, . . . ,𝑎𝑘) where 𝑎0 = 𝑐(𝑣) and 𝑎𝑖 = 𝑖,1 ≤ 𝑖 ≤ 𝑘 is the number of vertices that are adjacent to 𝑣 and colored 𝑖. the minimum k-color used in irregular coloring is called the irregular chromatic number and denoted by 𝜒𝑖𝑟. the type of research used in this research is exploratory research. in this paper, we discuss the irregular chromatic number of the bull graph, pan graph, sun graph, peach graph, and caveman graph. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: irregular coloring; irregular chromatic number; unicyclic graph introduction a graph 𝐺 = (𝑉,𝐸) is a pair of a vertex set denoted by 𝑉(𝐺) and an edge set (may be empty) denoted by 𝐸(𝐺). more detail definitions and some properties can be seen in [1]. in that year, a four-color theorem which discusses maps coloring was discovered. this theorem states that there is a separation of a plane into adjacent areas resulting in an image called a map. the maximum colors which is needed to color the area on the map so that two neighboring areas do not have the same color is four [2]. 𝑐𝑜𝑑𝑒(𝑣) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘) where 𝑎0 = 𝑐(𝑣) and 𝑎𝑖 = 𝑖,(1 ≤ 𝑖 ≤ 𝑘) is the number of vertices that are adjacent to 𝑣, and colored 𝑣, where 𝑐(𝑣) is the color at vertex 𝑣. the minimum color used in irregular coloring is called the irregular chromatic number and denoted by 𝜒𝑖𝑟 [2]. an irregular coloring is included proper coloring of g, it follows that [2] 𝜒(𝐺) ≤ 𝜒𝑖𝑟(𝐺). the following are the theorems, corollaries, and observations that will be used in this paper. lemma 1. [2] for every pair 𝑎,𝑏 of integers with 2 ≤ 𝑎 ≤ 𝑏, there is a connected graph 𝐺 with 𝜒(𝐺) = 𝑎 and 𝜒𝑖𝑟(𝐺) = 𝑏. corollary 1. [2] for every graph 𝐺, 𝜔(𝐺) ≤ 𝜒(𝐺) the clique number 𝜔(𝐺) of a graph 𝐺 is the maximum order of a complete subgraph of 𝐺. observation 1.[2] let 𝑐 be a (proper) vertex coloring of a nontrivial graph 𝐺 and let 𝑢 and 𝑣 be two distinct vertices of 𝐺. a. if 𝑐(𝑢) ≠ 𝑐(𝑣), then 𝑐𝑜𝑑𝑒(𝑢) ≠ 𝑐𝑜𝑑𝑒 (𝑣) b. if 𝑑𝑒𝑔𝐺𝑢 ≠ 𝑑𝑒𝑔𝐺𝑣, then 𝑐𝑜𝑑𝑒(𝑢) ≠ 𝑐𝑜𝑑𝑒 (𝑣) c. if 𝑐 is irregular coloring and 𝑁(𝑢) = 𝑁(𝑣), 𝑐(𝑢) ≠ 𝑐(𝑣) http://dx.doi.org/10.18860/ca.v7i4.16917 mailto:arika.fkip@unej.ac.id https://creativecommons.org/licenses/by-sa/4.0/ on irregular colorings of unicyclic graph family arika indah kristiana 504 lemma 2. [3] let 𝑐 be an irregular k-coloring of the vertices of a graph 𝐺. the number of different possible color codes of the vertices of degree r in g is 𝑘( 𝑟 + (𝑘 + 1) − 1 𝑟 ) = 𝑘( 𝑟 + 𝑘 − 2 𝑟 ) corollary 2. [3] if c is an irregular k-coloring of a nontrivial connected graph g, then g contains at most 𝑘( 𝑟 + 𝑘 − 2 𝑟 ) vertices of degree r. corollary 3. [3] if 𝜒𝑖𝑟(𝐺) = 𝑘 where 𝑛 ≥ 3, then 𝑛 ≤ (𝑘)( 𝑘 2 ) = 𝑘2(𝑘−1) 2 corollary 4.[3] let 𝑘 ≥ 3 be an integer, 𝜒𝑖𝑟(𝐶𝑛) ≥ 𝑘 for all integers n. such that (𝑘 − 1)2(𝑘 −2) +2 2 ≤ 𝑛 ≤ 𝑘2(𝑘 − 1) 2 propotition 1.[3] 𝜒𝑖𝑟(𝐶𝑛) = 4 if n is even and 𝜒𝑖𝑟(𝐶𝑛) = 3 if n is odd for 3 ≤ 𝑛 ≤ 9. lemma 3. [3] let 𝑘 ≥ 3, if 𝑛 = 𝑘2(𝑘−1) 2 , then 𝜒𝑖𝑟(𝐶𝑛−1) ≥ 𝑘 + 1 a. rohoni et. al have discussed irregular coloring in the double wheel graph family [4], fan graphs families [5], and wheel related graphs [6]. avudainayaki et. al [7] has discussed irregular coloring on central and middle graphs of double star graphs, anderson et. al [8] discussed irregular coloring on regular graphs, and kristiana et. al [9] has discussed local irregularity coloring of some family graphs. there are some previous results of irregular coloring of bipartite graph dan tree graph family [10], star families [11], some generalized graph [12], and mycielskian graphs [13]. in this paper, we observed the irregular coloring of bull graph, pan graph, sun graph, peach graph, and caveman graph. the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges [14]. the pan pan graph is a graph obtained by combining a cycle graph cn with a singleton star graph k1[15]. the sun graph of order 2𝑛 is a cycle 𝐶𝑛 with an edge terminating in a pendent vertex attached to each vertex[16]. a peach graph is a circular graph cm which has 𝑛 pendants at one vertex, which is at vertex 𝑥1 contained in the cycle graph[17]. a caveman graph arises by modifying a set of fully connected clusters (caves) by removing one edge from each cluster and using it to connect to a neighboring one such that the clusters form a single loop [18]. method the type of research used in this research is exploratory research. this research is research conducted to dig up data and find new things that you want to know so that the results obtained can be used as knowledge development. the following is a description of the steps taken to determine irregular chromatic numbers: (1) determining the graph that will be researched; (2) determining the cardinality of the graph that is used as research; (3) do vertex coloring of the researched graph; (4) create a code from each vertex in the graph. specifically, 𝑐𝑜𝑑𝑒 (𝑣) = (𝑎0,𝑎1,…,𝑎𝑘) =, where 𝑎0 = 𝑐(𝑣) and 𝑎𝑖 (1 ≤ 𝑖 ≤ 𝑘) are the number of vertices of color 𝑖 adjacent to 𝑣; (5) if the codes are the same for each vertex, proceed to step 3. (6) determining irregular chromatic numbers. result and discussion this research is focused on finding irregular chromatic numbers in the unicyclic graph family, including bull graph, pan graph, sun graph, peach graph, and caveman graph. on irregular colorings of unicyclic graph family arika indah kristiana 505 theorem 1. the irregular chromatic number of bull graph 𝑩𝟑,𝒏 is 𝒏 + 𝟏. proof: this graph has the vertex set 𝑉(𝐵3,𝑛) = {𝑥𝑖| 1 ≤ 𝑖 ≤ 3} ∪ {𝑦2𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑦3𝑖|1 ≤ 𝑖 ≤ 𝑛} and edge set is 𝐸(𝐵3,𝑛) = {𝑥1𝑥𝑖| 2 ≤ 𝑖 ≤ 3} ∪{𝑥2𝑥3} ∪{𝑥2𝑦2𝑖| 1 ≤ 𝑖 ≤ 𝑛} ∪{𝑥3𝑦3𝑖| 1 ≤ 𝑖 ≤ 𝑛}.|𝑉(𝐵3,𝑛)| = 2𝑛 + 3 and |𝐸(𝐵3,𝑛)| = 2𝑛 + 3. we need to prove the upper bound 𝜒𝑖𝑟(𝐵3,𝑛) ≤ 𝑛 + 1 and the lower bound 𝜒𝑖𝑟(𝐵3,𝑛) ≥ 𝑛 + 1. first, we prove that the lower bound of the irregular chromatic number of bull graph is 𝜒𝑖𝑟(𝐵3,𝑛) ≥ 𝑛 + 1. assume that 𝜒𝑖𝑟(𝐵3,𝑛) < 𝑛 + 1. we have some condition as follows. (1) 𝑐(𝑥𝑖) = 𝑐(𝑦2𝑖) and 𝑐(𝑥𝑖) 𝑐(𝑦2𝑖) ∈ 𝐸(𝐵3,𝑛). it's contradicting. (2) 𝑐(𝑥𝑖) = 𝑐(𝑦3𝑖) and 𝑐(𝑥𝑖) 𝑐(𝑦3𝑖) ∈ 𝐸(𝐵3,𝑛). it's contradicting. (3) 𝑐(𝑥2𝑘) = 𝑐(𝑦2𝑙) and 𝑘 ≠ 𝑙. it's contradicting. (4) 𝑐(𝑥3𝑘) = 𝑐(𝑦3𝑙) and 𝑘 ≠ 𝑙. it's contradicting. based on (1), (2), (3), and (4), we get the lower bound of the irregular coloring on a bull graph is 𝜒𝑖𝑟(𝐵3,𝑛) ≥ 𝑛 + 1. furthermore, we prove that the upper bound of the irregular chromatic number of the bull graph is 𝜒𝑖𝑟(𝐵3,𝑛) ≤ 𝑛 + 1. the color function 𝑐 on this graph is defined as follows. 𝑐(𝑥𝑖) = 𝑖,1 ≤ 𝑖 ≤ 3 𝑐(𝑦2𝑖) = { 𝑖 𝑖 = 1 𝑖 + 1 2 ≤ 𝑖 ≤ 𝑛 𝑐(𝑦3𝑖) = { 𝑖 1 ≤ 𝑖 ≤ 2 𝑖 + 1 3 ≤ 𝑖 ≤ 𝑛 based on the color function, the code for each vertex in the bull graph is obtained as follows. 𝑐𝑜𝑑𝑒(𝑥1) = (𝑐(𝑥1),0,1,1,0,0,…,0⏟ 𝑛+1 ) 𝑐𝑜𝑑𝑒(𝑥2) = (𝑐(𝑥2),2,0,2,1,1,…,1⏟ 𝑛+1 ) 𝑐𝑜𝑑𝑒(𝑥3) = (𝑐(𝑥3),2,1,0,1,1,…,1⏟ 𝑛+1 ) 𝑐𝑜𝑑𝑒(𝑦2𝑖) = (𝑐(𝑦2𝑖),0,1,0,0,0,…,0⏟ 𝑛+1 ) 𝑐𝑜𝑑𝑒(𝑦3𝑖) = (𝑐(𝑦3𝑖),0,0,1,0,0,…,0⏟ 𝑛+1 ) based on the color function and vertex code obtained, each neighboring vertex has a different color and each vertex in the graph has a different color code. therefore, the upper bound of the irregular chromatic number on the bull graph is 𝜒𝑖𝑟(𝐵3,𝑛) ≤ 𝑛 + 1. the lower and upper bound of the irregular chromatic number of bull graph is 𝑛 + 1 ≤ 𝜒𝑖𝑟(𝐵3,𝑛) ≤ 𝑛 + 1. so, 𝜒𝑖𝑟(𝐵3,𝑛) = 𝑛 + 1. theorem 2. let (𝒌−𝟏)( 𝒌 − 𝟏 𝟐 ) + 𝟏 ≤ 𝒏 ≤ 𝒌( 𝒌 𝟐 ) and 𝒌 ≥ 𝟒. irregular chromatic number of pan graph 𝑷𝒂𝒏 is 𝜒𝑖𝑟(𝑃𝑎𝑛) = { 3 𝑓𝑜𝑟 𝑛 𝑜𝑑𝑑,3 ≤ 𝑛 ≤ 9 3 𝑓𝑜𝑟 𝑛 = 4 4 𝑓𝑜𝑟 𝑛 𝑒𝑣𝑒𝑛,6 ≤ 𝑛 ≤ 9 on irregular colorings of unicyclic graph family arika indah kristiana 506 proof: this graph has the vertex set 𝑉(𝑃𝑎𝑛) = {𝑥𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪{𝑦} and the edge set is 𝐸(𝑃𝑎𝑛) = {𝑥𝑖𝑥𝑖+1|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪{𝑥1𝑥𝑛} ∪ {𝑥𝑖𝑦}. |𝑉(𝑃𝑎𝑛)| = 𝑛 + 1 and |𝐸(𝑃𝑎𝑛)| = 𝑛 + 1. the pan graph contains the 𝐶𝑛 subgraph where 𝐶𝑛 is a cycle graph. in this case, the 𝐶𝑛 subgraph is labeled according to the 𝐶𝑛 graph. there are three cases in this proof. case 1. 𝜒𝑖𝑟(𝑃𝑎𝑛) = 3,𝑓𝑜𝑟 𝑛 𝑜𝑑𝑑,3 ≤ 𝑛 ≤ 9 we need to prove the upper bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3 and the lower bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. first, we prove that the lower bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. assume that 𝜒𝑖𝑟(𝑃𝑎𝑛) < 3, for example 𝜒𝑖𝑟(𝑃𝑎𝑛) = 2, we get the color on 𝑥𝑖 is 1,2,1,2, . . . ,1 or 2,1,2, . . . ,2 periodically. since 𝑐(𝑥1) = 𝑐(𝑥𝑛) and 𝑥1𝑥𝑛 ∈ 𝐸(𝑃𝑎𝑛). this contradicts the definition of proper coloring where each pair of adjacent vertices must have different colors. therefore, 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3 where 3 ≤ 𝑛 ≤ 9. define the color function 𝑐:𝑣 → {1,2,3,…,𝜒𝑖𝑟},𝑆 = {1,2,3,…,𝜒𝑖𝑟} and 𝑝,𝑞 ∈ 𝑁. 𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1 ∈ 𝑉(𝑃𝑎𝑛), with 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1), 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ∈ 𝑆. if 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ≠ 𝑐(𝑥(𝑞(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑞+2(𝑚𝑜𝑑 𝑛))+1), then 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1) = 𝑐(𝑥(𝑞+1(𝑚𝑜𝑑 𝑛))+1). the color of y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦), 𝑥𝑖𝑦 ∈ 𝐸(𝑃𝑎𝑛). based on the color function, it can be written, if it is known that 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). we get the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. the lower and upper bound of the irregular chromatic number of pan graph is 3 ≤ 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) = 3. case 2. 𝜒𝑖𝑟(𝑃𝑎𝑛) = 3,𝑛 = 4 we need to prove the upper bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3 and the lower bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. first, we prove that the lower bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. assume that 𝜒𝑖𝑟(𝑃𝑎𝑛) < 3, for example 𝜒𝑖𝑟(𝑃𝑎𝑛) = 2. define the color function 𝑐:𝑣 → {1,2} is 𝑐(𝑥1) = 1,𝑐(𝑥2) = 2,𝑐(𝑥3) = 1,𝑐(𝑥4) = 2,𝑐(𝑦) = 2. the color code obtained is 𝑐𝑜𝑑𝑒(𝑥1) = (1,0,3), 𝑐𝑜𝑑𝑒(𝑥2) = (2,2,0),𝑐𝑜𝑑𝑒(𝑥3) = (1,0,2),𝑐𝑜𝑑𝑒(𝑥4) = (2,2,0),𝑐𝑜𝑑𝑒(𝑦) = (2,1,0). based on the code obtained, there is the same code, namely 𝑐𝑜𝑑𝑒(𝑥2) = 𝑐𝑜𝑑𝑒(𝑥4). this contradicts the definition of irregular colorin that each vertex in a graph must have a different color code. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 3. furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. define the color function 𝑐:𝑣 → {1,2,3} is 𝑐(𝑥1) = 1,𝑐(𝑥2) = 3,𝑐(𝑥3) = 1,𝑐(𝑥4) = 2,𝑐(𝑦) = 2. the color code obtained is 𝑐𝑜𝑑𝑒(𝑥1) = (1,0,2,1), 𝑐𝑜𝑑𝑒(𝑥2) = (3,2,0,0),𝑐𝑜𝑑𝑒(𝑥3) = (1,0,1,1),𝑐𝑜𝑑𝑒(𝑥4) = (2,2,0,0),𝑐𝑜𝑑𝑒(𝑦) = (2,1,0,0). so, it can be concluded that 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. we get the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. the lower and upper bound of the irregular chromatic number of pan graph is 3 ≤ 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 3. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) = 3. case 3. 𝜒𝑖𝑟(𝑃𝑎𝑛) = 4,𝑓𝑜𝑟 𝑛 𝑒𝑣𝑒𝑛,6 ≤ 𝑛 ≤ 9 we need to prove upper bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 4 and lower bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 4. first, we prove that the lower bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 4. according to lemma 3, if 𝑘 = 3, then 𝜒𝑖𝑟(𝐶𝑛−1) ≥ 4. we get an odd n, so on irregular colorings of unicyclic graph family arika indah kristiana 507 𝑛 − 1 is even. since pan has the subgraph cn, where cn is a cycle graph labeled as cn. so, based on lemma 3, we can conclude that 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 4. next, 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 4. furthermore, we prove that the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 4 where 3 ≤ 𝑛 ≤ 9. define the color function 𝑐:𝑣 → {1,2,3,…,𝜒𝑖𝑟},𝑆 = {1,2,3,…,𝜒𝑖𝑟} and 𝑝,𝑞 ∈ 𝑁. 𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1 ∈ 𝑉(𝑃𝑎𝑛), with 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ∈ 𝑆. if 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ≠ 𝑐(𝑥(𝑞(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑞+2(𝑚𝑜𝑑 𝑛))+1), then 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1) = 𝑐(𝑥(𝑞+1(𝑚𝑜𝑑 𝑛))+1). the color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦), 𝑥𝑖𝑦 ∈ 𝐸(𝑃𝑎𝑛). based on the color function, it can be written, if it is known that 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). so, we get the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 4. the lower and upper bound of the irregular chromatic number of pan graph is 4 ≤ 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 4. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) = 4. figure 1. 𝜒𝑖𝑟(𝑃𝑎7) = 3 theorem 3. let (𝒌−𝟏)( 𝒌 − 𝟏 𝟐 ) + 𝟏 ≤ 𝒏 ≤ 𝒌( 𝒌 𝟐 ) irregular chromatic number of pan graph 𝑷𝒂𝒏 for 𝒏 ≥ 𝟑,𝒌 ≥ 𝟒 is 𝜒𝑖𝑟(𝑃𝑎𝑛) = { 𝑘 𝑓𝑜𝑟 𝑛 ≠ 𝑘( 𝑘 2 ) −1 𝑘 + 1 𝑓𝑜𝑟 𝑛 = 𝑘( 𝑘 2 ) −1 proof: this graph has the vertex set 𝑉(𝑃𝑎𝑛) = {𝑥𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪{𝑦} and the edge set is 𝐸(𝑃𝑎𝑛) = {𝑥𝑖𝑥𝑖+1|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪{𝑥1𝑥𝑛} ∪ {𝑥𝑖𝑦}. |𝑉(𝑃𝑎𝑛)| = 𝑛 + 1 and |𝐸(𝑃𝑎𝑛)| = 𝑛 + 1. the pan graph contains the 𝐶𝑛 subgraph where 𝐶𝑛 is a cycle graph. in this case, the 𝐶𝑛 subgraph is labeled according to the 𝐶𝑛 graph. there are two cases in this proof. case 1. 𝜒𝑖𝑟(𝑃𝑎𝑛) = 𝑘,𝑓𝑜𝑟 𝑛 ≠ 𝑘( 𝑘 2 ) − 1 we need to prove upper bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘 and lower bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘. first, we prove that the lower bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘. according to corollary 4, if 𝑘 ≥ 3 is an integer, then 𝜒𝑖𝑟(𝐶𝑛) ≥ 𝑘. since 𝑃𝑎𝑛 has the subgraph 𝐶𝑛, where 𝐶𝑛 is a cycle graph labeled as 𝐶𝑛. therefore, we get 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘. next, we prove that the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘. color function 𝑐:𝑣 → {1,2,3,…,𝜒𝑖𝑟},𝑆 = {1,2,3,…,𝜒𝑖𝑟} and 𝑝,𝑞 ∈ 𝑁. 𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1 ∈ 𝑉(𝑃𝑎𝑛), with 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ∈ 𝑆. on irregular colorings of unicyclic graph family arika indah kristiana 508 if 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ≠ 𝑐(𝑥(𝑞(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑞+2(𝑚𝑜𝑑 𝑛))+1), then 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1) = 𝑐(𝑥(𝑞+1(𝑚𝑜𝑑 𝑛))+1). the color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦), 𝑥𝑖𝑦 ∈ 𝐸(𝑃𝑎𝑛). based on the color function, it can be written, if it is known that 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). so, we get the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘. the lower and upper bound of the irregular chromatic number of pan graph is 𝑘 ≤ 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) = 𝑘. case 2. 𝜒𝑖𝑟(𝑃𝑎𝑛) = 𝑘 + 1,𝑓𝑜𝑟 𝑛 = 𝑘( 𝑘 2 ) − 1 we need to prove upper bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘 + 1 and lower bound 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘 + 1. first, we prove that the lower bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘 +1. according to lemma 3, if 𝑛 = 𝑘2(𝑘+1) 2 = 𝑘( 𝑘 2 ), then 𝜒𝑖𝑟(𝐶𝑛−1) ≥ 𝑘 + 1. since 𝑃𝑎𝑛 has the subgraph 𝐶𝑛, where 𝐶𝑛 is a cycle graph labeled as 𝐶𝑛. therefore, if 𝑛 − 1, then 𝑘( 𝑘 2 ) − 1. so, if 𝑛 = 𝑘( 𝑘 2 ) −1, then 𝜒𝑖𝑟(𝑃𝑎𝑛) ≥ 𝑘 + 1. next, we prove that the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘 +1. color function 𝑐:𝑣 → {1,2,3,…,𝜒𝑖𝑟},𝑆 = {1,2,3,…,𝜒𝑖𝑟} and 𝑝,𝑞 ∈ 𝑁. 𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1 ∈ 𝑉(𝑃𝑎𝑛),with 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1), 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ∈ 𝑆. if 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ≠ 𝑐(𝑥(𝑞(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑞+2(𝑚𝑜𝑑 𝑛))+1), then 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1) = 𝑐(𝑥(𝑞+1(𝑚𝑜𝑑 𝑛))+1). the color on y is 𝑐(𝑦), for 𝑐(𝑦) ∈ 𝑆, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦), 𝑥𝑖𝑦 ∈ 𝐸(𝑃𝑎𝑛). based on the color function, it can be written, if it is known that 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). so, we get the upper bound of the irregular chromatic number on the pan graph is 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘 + 1. the lower and upper bound of the irregular chromatic number of pan graph is 𝑘 +1 ≤ 𝜒𝑖𝑟(𝑃𝑎𝑛) ≤ 𝑘 + 1. so, 𝜒𝑖𝑟(𝑃𝑎𝑛) = 𝑘 + 1. theorem 4. the irregular chromatic number of sun graph 𝑺𝒏 for 𝒌 𝟐 + 𝒌+ 𝟏 ≤ 𝒏 ≤ 𝒌𝟐 + 𝟑𝒌+ 𝟐 and 𝒌 ∈ 𝑵 is 𝒌+ 𝟐. proof: this graph has the vertex set 𝑉(𝑆𝑛) = {𝑥𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑦𝑖|1 ≤ 𝑖 ≤ 𝑛} and the edge set is 𝐸(𝑆𝑛) = {𝑥𝑖𝑥𝑖+1|1 ≤ 𝑖 ≤ 𝑛 −1} ∪{𝑥1𝑥𝑛} ∪ {𝑥𝑖𝑦𝑗|1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑛}. |𝑉(𝑆𝑛)| = 2𝑛 and |𝐸(𝑆𝑛)| = 2𝑛. we need to prove upper bound 𝜒𝑖𝑟(𝑆𝑛) ≤ 𝑘 + 2 and lower bound 𝜒𝑖𝑟(𝑆𝑛) ≥ 𝑘 + 2. first, we prove that the lower bound of the irregular chromatic number on the sun graph is 𝜒𝑖𝑟(𝑆𝑛) ≥ 𝑘 + 2. assume that 𝜒𝑖𝑟(𝑆𝑛) < 𝑘 + 2. if 𝑥𝑖 colored with k+1-color, then at most k + 1 vertex of the same color. because of 𝑥𝑘𝑦𝑘 ∈ 𝐸(𝑆𝑛) and 𝑦𝑘 are colored with k+1-color, then there are two neighboring vertices having the same color, namely 𝑐(𝑥𝑘) = 𝑐(𝑦𝑘). this contradicts the definition of proper coloring, where each neighboring vertex must have a different color. thus, it can be concluded that the upper bound of the irregular chromatic number of a sun graph is 𝜒𝑖𝑟(𝑆𝑛) ≥ 𝑘 + 2. next, we prove that the upper bound of the irregular chromatic number on the sun graph is 𝜒𝑖𝑟(𝑆𝑛) ≤ 𝑘 + 2. the color function c on this graph is defined as follows. on irregular colorings of unicyclic graph family arika indah kristiana 509  for 𝑥𝑖 vertices, where 𝑛 ≡ 1(𝑚𝑜𝑑 𝑘 + 2) the color of the vertices 1,2,3,…,𝑘 + 2,1,2,3,…,𝑘 + 2 periodically.  for 𝑥𝑖 vertices, where n not equivalent 1(𝑚𝑜𝑑 𝑘 + 2) the color of the vertices 1,2,3,…,𝑘 + 2,1,2,3,…,𝑘 + 2,…,1,2,3,…,𝑘 + 2,2 periodically.  for 𝑦𝑖 vertices, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦𝑗) where 𝑥𝑖𝑦𝑗 ∈ 𝐸(𝑆𝑛), and 𝑐(𝑥𝑖) ≠ 𝑐(𝑥𝑗) for 𝑥𝑖𝑦𝑘,𝑥𝑗𝑦𝑙 ∈ 𝐸(𝑆𝑛), with 𝑐(𝑦𝑘) = 𝑐(𝑦𝑙),𝑘 ≠ 𝑙. we know that the 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘) and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘), then based on the color function obtained, 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). thus, the upper bound of the irregular chromatic number on a sun graph is 𝜒𝑖𝑟(𝑆𝑛) ≤ 𝑘 + 2. the ilustration of irregular coloring of sun graph can be seen in figure 2. the lower and upper bound of the irregular chromatic number of caveman graph is 𝑘 + 2 ≤ 𝜒𝑖𝑟(𝑆𝑛) ≤ 𝑘 + 2. so, 𝜒𝑖𝑟(𝑆𝑛) = 𝑘 + 2. figure 2. 𝜒𝑖𝑟(𝑆𝑛) = 4 theorem 5. irregular chromatic number of peach graph 𝑪𝒎 𝒎 for 𝒎 ≥ 𝟑 is 𝒎 + 𝟏. proof: this graph has the vertex set 𝑉(𝐶𝑚 𝑚) = {𝑥𝑖|1 ≤ 𝑖 ≤ 𝑚}∪{𝑦𝑖|1 ≤ 𝑖 ≤ 𝑚} and the edge set is 𝐸(𝐶𝑚 𝑚) = {𝑥𝑖𝑥𝑖+1|1 ≤ 𝑖 ≤ 𝑚 − 1} ∪{𝑥1𝑥𝑚} ∪{𝑥1𝑦𝑖|1 ≤ 𝑖 ≤ 𝑚}. |𝑉(𝐶𝑚 𝑚)| = 2𝑚 and |𝐸(𝐶𝑚 𝑚)| = 2𝑚. we need to prove upper bound 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≤ 𝑚 + 1 and lower bound 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≥ 𝑚 +1. first, we prove that the lower bound of the irregular chromatic number on the peach graph is 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≥ 𝑚 + 1. assume that 𝜒𝑖𝑟(𝐶𝑚 𝑚) < 𝑚 + 1. we have the following cases. case 1. 𝑐(𝑥𝑖) = 𝑐(𝑦𝑗), 𝑥𝑖𝑦𝑗 ∈ 𝐸(𝐶𝑚 𝑚). it’s contradicts. case 2. 𝑐(𝑥𝑘) = 𝑐(𝑦𝑙), 𝑘 ≠ 𝑙,𝑦𝑘𝑥1,𝑦𝑗𝑥1 ∈ 𝐸(𝐶𝑚 𝑚). it’s contradicts. based on case 1 and case 2 above, the lower bound is obtained from the irregular chromatic number on the peach graph is 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≥ 𝑚 + 1. next, we prove that the upper bound of the irregular chromatic number on the peach graph is 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≤ 𝑚 + 1. the color function c on this graph is defined as follows. 𝑐(𝑥𝑖) = 𝑖,1 ≤ 𝑖 ≤ 𝑚 𝑐(𝑦𝑗) = { 𝑗 𝑓𝑜𝑟 2 ≤ 𝑗 ≤ 𝑚 𝑚 + 1 𝑓𝑜𝑟 𝑗 = 1 on irregular colorings of unicyclic graph family arika indah kristiana 510 based on the color function above, the code for each vertex in the peach graph is obtained as follows. 𝑐𝑜𝑑𝑒 (𝑥𝑖) = { (𝑐(𝑥𝑖),0,2,1,1,1,1,0,0,…,1,1,2⏟ 𝑚+1 𝑓𝑜𝑟 𝑖 = 1 (𝑐(𝑥𝑖),0,…,0⏟ , 𝑖−2 1,0,1,0,…,0⏟ , 𝑚−1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑚 − 1 (𝑐(𝑥𝑖),1,0,0,0,0,0,0,…,0,0,1,0⏟ 𝑚+1 𝑓𝑜𝑟 𝑖 = 𝑚 𝑐𝑜𝑑𝑒(𝑦𝑖) = (𝑐(𝑦𝑖),1,0,0,0,…,0,0,0)⏟ 𝑚+1 based on the color function and vertex code obtained, each neighboring vertex has a different color and each vertex in the graph has a different color code. therefore, the upper bound of the irregular chromatic number on the peach graph is 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≤ 𝑚 + 1. the illustration of irregular coloring of sun graph can be seen in figure 3. the lower and upper bound of the irregular chromatic number of peach graph is 𝑚 + 1 ≤ 𝜒𝑖𝑟(𝐶𝑚 𝑚) ≤ 𝑚 + 1. so, 𝜒𝑖𝑟(𝐶𝑚 𝑚) = 𝑚 + 1. figure 3. 𝜒𝑖𝑟(𝐶8 8) = 9 theorem 6. irregular chromatic number of caveman graph 𝑲𝒎,𝟑 for 𝒌 𝟐 + 𝒌 + 𝟏 ≤ 𝒎 ≤ 𝒌𝟐 + 𝟑𝒌+ 𝟐 and 𝒌 ∈ 𝑵 is 𝒌+ 𝟐. proof: this graph has the vertex set 𝑉(𝐾𝑚,3) = {𝑥𝑖|1 ≤ 𝑖 ≤ 2𝑚} ∪{𝑦𝑗|1 ≤ 𝑗 ≤ 𝑚} and the edge set is 𝐸(𝐾𝑚,3) = {𝑥𝑖𝑥𝑖+1|1 ≤ 𝑖 ≤ 2𝑚 − 1} ∪{𝑥1𝑥2𝑚}∪ {𝑥𝑖𝑦𝑖|1 ≤ 𝑖 ≤ 2𝑚,𝑖 = 𝑜𝑑𝑑,1 ≤ 𝑗 ≤ 𝑚}. |𝑉(𝐾𝑚,3)| = 3𝑚 + 6 and |𝐸(𝐾𝑚,3)| = 3𝑚 + 6. we need to prove upper bound 𝜒𝑖𝑟(𝐾𝑚,3) ≤ 𝑘 + 2 and lower bound 𝜒𝑖𝑟(𝐾𝑚,3) ≥ 𝑘 + 2. first, we prove that the lower bound of the irregular chromatic number on the caveman graph is 𝜒𝑖𝑟(𝐾𝑚,3) ≥ 𝑘 + 2. assume that 𝜒𝑖𝑟(𝐾𝑚,3) ≤ 𝑘 + 2, for example is 𝜒𝑖𝑟(𝐾𝑚,3) = 𝑘 + 1. if 𝑥𝑖, where deg(𝑥𝑖) = 3 is colored with k+1-color, then at most k+1 vertex of the same color. because of 𝑥𝑖𝑦𝑗 ∈ 𝐸(𝐾𝑚,3) and 𝑦𝑗 are colored with k+1-color, then there are two neighboring vertices having the same color, namely 𝑐(𝑥𝑖) = 𝑐(𝑦𝑗). this contradicts the definition of proper coloring, where each neighboring vertex must have a different color. on irregular colorings of unicyclic graph family arika indah kristiana 511 thus, it can be concluded that the lower bound of the irregular chromatic number of a caveman graph is 𝜒𝑖𝑟(𝐾𝑚,3) ≥ 𝑘 + 2. next, we prove that the upper bound of the irregular chromatic number on the caveman graph is 𝜒𝑖𝑟(𝐾𝑚,3) ≤ 𝑘 + 2. the color function c on this graph is defined as follows.  for 𝑥𝑖 vertices, where deg(𝑥𝑖) = 3, the color of the vertices 1,2,3,…,𝑘 + 2,1,2,3,…,𝑘 + 2 periodically.  for 𝑥𝑖 and 𝑥𝑗 vertices, with deg(𝑥𝑖) = deg(𝑥𝑗) = 2, 𝑥𝑖 = 𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1and 𝑥𝑗 = 𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1. if 𝑆 = {1,2,3,…,𝑘 + 2}, 𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1,𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1 ∈ 𝑉(𝐾𝑚,3) ≥ 𝑘 + 1, with 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ∈ 𝑆. if 𝑐(𝑥(𝑝(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑝+2(𝑚𝑜𝑑 𝑛))+1) ≠ 𝑐(𝑥(𝑞(𝑚𝑜𝑑 𝑛))+1),𝑐(𝑥(𝑞+2(𝑚𝑜𝑑 𝑛))+1), then 𝑐(𝑥(𝑝+1(𝑚𝑜𝑑 𝑛))+1) = 𝑐(𝑥(𝑞+1(𝑚𝑜𝑑 𝑛))+1).  for 𝑦𝑗 vertices, 𝑐(𝑥𝑖) ≠ 𝑐(𝑦𝑗) for 𝑥𝑖𝑦𝑗 ∈ 𝐸(𝐾𝑚,3)  for 𝑦𝑗 vertices, 𝑐(𝑦𝑘) = 𝑐(𝑦𝑙) when 𝑥𝑖𝑦𝑗,𝑥𝑗𝑦𝑙 ∈ 𝐸(𝐾𝑚,3) and 𝑐(𝑥𝑖) ≠ 𝑐(𝑦𝑗) we know that the 𝑐𝑜𝑑𝑒(𝑥𝑝) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘) and 𝑐𝑜𝑑𝑒(𝑥𝑞) = (𝑎0,𝑎1,𝑎2,…,𝑎𝑘) then based on the color function obtained, 𝑐𝑜𝑑𝑒(𝑥𝑝) ≠ 𝑐𝑜𝑑𝑒(𝑥𝑞). thus, the upper bound of the irregular chromatic number on a caveman graph is 𝜒𝑖𝑟(𝐾𝑚,3) ≤ 𝑘 + 2. the illustration of irregular coloring of caveman graph can be seen in figure 4. the lower and upper bound of the irregular chromatic number of peach graph is 𝑘 + 2 ≤ 𝜒𝑖𝑟(𝐾𝑚,3) ≤ 𝑘 + 2. so, 𝜒𝑖𝑟(𝐾𝑚,3) = 𝑘 + 2. figure 4. 𝜒𝑖𝑟(𝐾4,3) = 3 conclusions in this paper, we get the irregular chromatic number of the unicyclic graph family, namely bull graph, pan graph, sun graph, peach graph, caveman graph. on irregular colorings of unicyclic graph family arika indah kristiana 512 references [1] k. anitha, b. selvam, and k. thirusangu. new irregular colourings of graphs. journal of applied science and computations. vol. 6, no.5, pp.269-279,2019. 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[18] d. koutra. exploring and making sense of large graphs. pittsburgh. computer science department. carnegie mellon university, 2015. optimalization route to tourism places in west java using a-star algorithm cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 464-473 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 11, 2022 reviewed: july 13, 2022 accepted: july 28, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.17032 optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha*, sudradjat supian, herlina napitupulu departement of mathematics, faculty of mathematics and natural science, universitas padjadjaran, sumedang, indonesia email: muhammad16088@mail.unpad.ac.id abstract indonesia has many tourism places, one of which is west java province. tourist places in west java province are scattered in various regions, then many roads can be passed by tourists to reach these tourist places. west java also has a complex route; therefore, an optimal route is needed to pass the complex route to tourism places in west java province. based on these problems, the algorithm used is the a-star algorithm. the purpose of this research is to use the a-star algorithm to find the optimal route to tourism places which is influenced by distance and traffic density. the steps taken are to describe all the main lines in west java into a graph, then process the data that has been obtained from several government agencies in the province of west java so that it can be solved using the a-star algorithm. the results of the calculation of the a-star algorithm can produce an optimal route. by using the help of python, the optimal route to tourism in the province of west java can be obtained. this optimal route can help tourists to go to tourism places with the shortest distance in the province of west java. keywords: a-star algorithm; optimal route; road density; graph. introduction tourism is a variety of tourism activities and is supported by various facilities and services provided by the community, businessmen, government, and local governments. each region throughout indonesia has a variety of diversity and uniqueness of each, the province of west java is one of them. west java province has tourism spots scattered in each district and city. this province has all the potential of mountains, seas, beaches, rice fields, valleys, waterfalls, and local wisdom. west java province has a complex route one of them is traffic density, where the route is one of the tools used to reach tourism places. optimal route search has been widely applied to various navigation applications. the process of finding the optimal route finding the smallest cost or value of a route from the starting nodes to the destination nodes. optimal route search techniques that are often used are; blind search and heuristic search. blind search is easier to understand than heuristic search, but the solution search time is faster and the results obtained are more varied than the heuristic search [1]. one of the optimal route search algorithms using heuristic search is the a-star algorithm. this algorithm is a planning method that can be applied to situations where information about the global http://dx.doi.org/10.18860/ca.v7i3.17032 mailto:muhammad16088@mail.unpad.ac.id optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 465 environment has been obtained and also has a heuristic value that can be used as a basis for consideration [2], [3]. several previous studies have discussed finding the optimal route using the astar algorithm. research and testing on the application of finding the closest culinary route in bandar lampung, it was concluded that the tests carried out on the algorithm manually and the application got valid results with the same distance [4]. determining the best route can be done with the a-star algorithm, this simulation determines the starting nodes to the end nodes with the obstacles given in each route and gets the best route compared to other routes [1]. the search for the closest route between hospitals in samarinda is obtained using the a-star algorithm [2]. the a-star algorithm generates the shortest path for making roguelike games in the process of pursuing enemies against the player character. [5]. semi-optimized crane placement and configuration in a modular construction was successful in resulting in significant total cost reductions compared to lift planning algorithms [6]. based on the description above, the motivation for this research is to use the astar algorithm to find the optimal route which is affected by the distance and road density to tourist places in west java province. the parameter that is used to obtain the optimal route is the distance unit, with the use of four-wheeled vehicles. the road used in the study is a two-way road. methods the method used in this research is the a-star algorithm. in finding the shortest route with this algorithm, the route is drawn using a graph. in this graph, each edge has two values, namely the weight value, and the heuristic value. then the a-star algorithm calculates the weight value at the 𝑛th node plus the heuristic value at the 𝑛th node. after that, the algorithm calculates the weight value of the 𝑛 − 1 node to the 𝑛th node plus the heuristic value of the 𝑛th node until it reaches the destination node. when the optimal value has been obtained, the algorithm performs a callback to call the route that has been passed or can be referred to as the optimal route. graph a graph is a configuration of edge pairs and vertices. many things can be represented as a graph, for example, a highway as an edge and a bus station as a node [7] a linear graph or graph 𝐺 = (𝑉, 𝐸) consists of a non-empty set of 𝑉 = {𝑣1, 𝑣2, 𝑣3, … } called vertices or nodes, and the set 𝐸 = {𝑒1, 𝑒2, 𝑒3, … } called edge that connects a pair of vertices [8]. a graph 𝐺 is called connected if every node of the graph has a path connecting the two vertices, or it can be said to be connected if a node 𝑣𝑖 and 𝑣𝑗 there is at least one edge [9]. if e is an edge, then 𝑒 = (𝑣𝑖 , 𝑣𝑗 ) where 𝑣𝑖 and 𝑣𝑗 are different elements of 𝑉. edge 𝑒 = (𝑣𝑖 , 𝑣𝑗 ) is a pair of vertices 𝑣𝑖 and vertices 𝑣𝑗 , for an undirected graph, can also be denoted as 𝑣𝑖 𝑣𝑗 or 𝑣𝑗 𝑣𝑖 [1]. graphs can be implemented for optimal route finding. intersections, starting nodes, and ending nodes are represented as vertices. then, the length of the road segment is represented as an edge. shortest path problem the shortest route problem is a problem to find a path between two or more vertices in a weighted graph, the weight sought is the weight of the smallest edge that traversed [2], [10]. the shortest route problem is an optimization problem that uses a optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 466 weighted graph, where the weights can represent the distance between cities, delivery times, travel costs, and so on [10]. the shortest route obtained will optimize a special linear function of the route in the form of distance, time, and or costs that will be faced during the trip [9]. the shortest route problem can be applied to both directed and undirected graphs [11]. several kinds of shortest path problems, among others; search for the shortest path between two nodes, search for the shortest path between all pairs of nodes, search for the shortest path from a certain node to all other nodes, and search for the shortest path between two nodes that pass through certain nodes [8]. decision tree a tree in discrete mathematics means an undirected graph that is connected and does not contain circuits [12]. a tree has many applications such as; expression tree, binary search tree, prefix code, huffman code, and decision tree [12]. decision trees have many advantages in modeling optimization problems, including; being easy to interpret, easy to understand, allowing for exploring all possibilities, more accurate in terms of modeling, and complex decision-making can be changed to be simpler [12]. a-star algorithm optimal route-finding algorithms have played many important roles in various scientific disciplines such as cybernetics, satellite navigation systems for vehicles on highways, vehicle path planning, and robotic path planning [6]. the a-star algorithm is the best first search algorithm because it modifies the heuristic value, this algorithm minimizes the total cost of the path and under the right conditions can provide the best solution with optimal time [13]. in 1968 peter hart, nils nilsson, and bertram raphael of the stanford research institute described the a-star algorithm for the first time. the algorithm is an extension of dijkstra's algorithm which produces better time performance by using heuristic values [14]. the heuristic function contained in the a-star algorithm to calculate the estimated value of a node that has been traversed is [15]. 𝐹(𝑛) = 𝐺(𝑛) + 𝐻(𝑛) (1) where 𝐹(𝑛): total estimated cost 𝐺(𝑛): cost from starting node to 𝑛th node 𝐻(𝑛): estimated cost to arrive at a destination from the n-th node the a-star algorithm uses two list, namely open and closed. open is a list that is used to store nodes/vertices that have been calculated and their heuristic values have also been calculated but have not been selected as the best node (𝐹(𝑛)). in other words, the list contains nodes that still have a chance to be selected as the best node. closed is a list to store the nodes that have been calculated and have been selected as the best node (𝐹(𝑛)). in other words, the opportunity to be selected from the nodes in closed does not exist (dalem, 2018). in the a-star algorithm, the value of each iteration of the open list is compared and the smallest value is selected. then the value of the next open list is calculated and then compared with other 𝐹(𝑛) values. when the value of 𝐹(𝑛) is the smallest and there is a destination node, then the iteration stops. optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 467 result and discussion case study the a-star algorithm can be used when the processed data has a heuristic value which is a benchmark for calculations in the algorithm. in this study, the data were secondary data obtained from several related departement in the province of west java. the data used are as follows: 2016 track distance data from the department of spatial planning; 2016 provincial traffic density data from the department of transportation; and 2019 tourism places data from the department of tourism and culture. the form of data obtained is data on distances between provincial boundaries, the road ends, and intersections. the data can be seen at the link https://bit.ly/3oiddgc. figure 1. provincial, national toll road, national non-toll road, and west java province tourism places figure 1 is a combination of the provincial road (blue), national non-toll road (green), and national toll road (red) in 2016. then, on the provincial road, the density value is obtained from the value of the volume of the road divided by the capacity of the road with a value range of 0.00 – 1.00 from the department of transportation of west java province which was managed in 2016. non-toll national route with the distance value between each node obtained with the help of google maps. the department of transportation of west java province does not have density data from the national nontoll road because the route is managed by the ministry of transportation, then the lane is considered to have 0 density value. national toll road with the value of the distance https://bit.ly/3oiddgc optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 468 between each node obtained with the help of google maps. if the toll road is considered unimpeded, then the lane is considered to have 0 density value. figure 1 contains 239 nodes stating: provincial boundaries; road boundaries; intersections; and tourism in west java province a total of 208 nodes are: provincial boundaries; road boundaries; and intersections. furthermore, there are 43 tourism places adjacent to the main route in west java province. then the tourism place is drawn by passing the nearest route (regency/city road or village road) so that the node that states the tourism place is located on the main route. however, among the 43 nodes resulting from the withdrawal of the tourism sites, there are 11 nodes adjacent to the provincial boundary nodes, road boundaries, and intersections. thus, 11 of the 208 nodes were converted into tourism nodes, and 32 other tourism nodes were added. apart from the 43 tourism places nodes, the rest of the nodes are used as starting nodes. complete data related to road names, calculating distances and density values of the 239 interrelated nodes can be seen in full at the link https://bit.ly/3oiddgc. table 1. tourist places from each city/regency city/regency node tourism places kabupaten kota sukabumi a1 curug bibijilan a2 bukit sabak kabupaten cianjur a3 taman wisata alam sevillage a4 wana wisata pokland kabupaten kota bogor a5 wana wisata rusa a6 taman buah mekarsari kabupaten kota bekasi a7 gedung juang a8 taman buaya indonesia jaya kota depok a9 taman herbal insani a10 taman rekreasi wiladatika kabupaten karawang a11 candi jiwa a12 sian djin ku poh temple kabupaten purwakarta a13 green valley waterpark a14 bale panyawangan diorama nusantara kabupaten bandung barat a15 gua pawon stone garden citatah kabupaten bandung a16 situ cileunca a17 farm house susu lembang kota bandung a18 alun alun kota bandung a19 bukit moko kota cimahi a20 alam wisata cimahi a21 curug tilu leuwi opat kabupaten garut a22 pucak guha a23 curug sanghyang taraje kabupaten kota tasikmalaya a24 taman wisata curug cikoja a25 wisata alam pasir kirisik kabupaten ciamis a26 wisata alam ciung wanara a27 objek wisata situ wangi kota banjar a28 curug panganten a29 ecopark kota banjar kabupaten pangandaran a30 batu karas a31 pantai karapyak kabupaten kuningan a32 wisata bukit panembongan a33 telaga remis pasawahan kabupaten kota cirebon a34 pantai kejawanan a35 keraton kasepuhan cirebon kabupaten indramayu a36 museum bandar cimanuk a37 pantai tirtamaya https://bit.ly/3oiddgc optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 469 kabupaten subang a38 wisata buaya blanakan a39 pemandian air panas ciater kabupaten majalengka a40 curug cilutung a41 taman wisata alam cadas gantung kabupaten sumedang a42 cadas pangeran a43 curug cipongkor table 1 describes two names of tourism places given from each city/regency in west java province. numeric result how to calculate the a-star algorithm is done as in the example below: figure 2. graph for a-star algorithm example given the value of 𝐻(𝑛) at nodes ab, az, bc, bd, ce, de, dz, and ez sequentially, namely 5, 3, 4, 2, 6, 3, 1, and 2. based on figure 2, the route from the starting node to the destination node can be modeled as follows 1. 𝐴 → 𝐵 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝐵 + 𝐻(𝑛)𝐴𝐵 𝐴 → 𝑍 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝑍 + 𝐻(𝑛)𝐴𝑍 2. 𝐴 → 𝐵 → 𝐶 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝐵𝐶 + 𝐻(𝑛)𝐵𝐶 𝐴 → 𝐵 → 𝐷 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝐵𝐷 + 𝐻(𝑛)𝐵𝐷 3. 𝐴 → 𝐵 → 𝐷 → 𝐸 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝐵𝐷𝐸 + 𝐻(𝑛)𝐷𝐸 𝐴 → 𝐵 → 𝐷 → 𝑍 = 𝐹(𝑛) = 𝐺(𝑛)𝐴𝐵𝐷𝑍 + 𝐻(𝑛)𝐷𝑍 table 2. decision tree drawing iteration open list closed list hold decision tree 0 𝐵 − 𝑍 𝐴 − 1 𝐶 − 𝐷 − 𝑍 𝐴 − 𝐵 𝑍 2 𝐶 − 𝐸 − 𝑍 𝐴 − 𝐵 − 𝐷 𝐶 − 𝑍 optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 470 iteration open list closed list hold decision tree 3 − 𝐴 − 𝐵 − 𝐷 − 𝑍 − the flow used by a-star algorithm starts with the initial node initiation into the open list then the current node (the current node) is made the best node (𝐹 score). as long as the current node is not the destination node, then move the current node into the closed list. then calculate and compare the 𝐹 score values of all open lists. after the current node with the smallest 𝐹 score value is the destination node, then the iteration is stopped and performs a backtrack to display the route. python is used to help find the optimal route by using equation (1) where 𝐹(𝑛) is the total distance value, 𝐺(𝑛) is the distance from the starting node to the nth node, and 𝐻(𝑛) is a heuristic estimate of traffic density between nodes. numerical is done by finding the optimal route which is the optimal distance and the route traversed by the a-star algorithm. the initial node used is the southwest node of west java province in sukabumi regency (node a), with the destination node being 43 tourism places nodes. the route column in table 2 is depicted with symbols representing provincial boundary nodes; road boundaries; and intersections. between these nodes is the name of the road traversed by the optimal route generated from the a-star algorithm. full data on the road can be seen at the link https://bit.ly/3oiddgc. table 3. distance and route from node a to tourist places destination node tourism places distances route a1 curug bibijilan 128.36 a, b, c, a1 a2 bukit sabak 141.24 a, b, d, f, i, j, q, p, o, n, a2 a3 taman wisata alam sevillage 180.84 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, a3 a4 wana wisata pokland 179.04 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4 a5 wana wisata rusa 199.84 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a5 a6 taman buah mekarsari 213.71 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, aq, ao, ar, a6 a7 gedung juang 239.67 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, aq, ao, ar, ho, hp, a7 a8 taman buaya indonesia jaya 242.91 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, aq, ao, ar, a6, as, a8 a9 taman herbal insani 191.81 a, b, d, f, i, h, s, t, az, hq, ay, aw, ax, a9 a10 taman rekreasi wiladatika 197.01 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, am, a10 a11 candi jiwa 310.50 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, aq, ao, ar, a6, as, a8, bd, be, bf, bh, a12, a11 a12 sian djin ku poh temple 284.31 a, b, d, f, i, h, s, t, az, hq, ay, av, au, ap, aq, ao, ar, a6, as, a8, bd, be, bf, bh, a12 https://bit.ly/3oiddgc optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 471 destination node tourism places distances route a13 green valley waterpark 244.34 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, ca, bz, a14, bx, a13 a14 bale panyawangan diorama nusantara 238.54 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, ca, bz, a14 a15 gua pawon 195.64 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, a15 stone garden citatah a16 situ cileunca 230.98 a, b, c, y, z, hz, dg, cx, cy, df, a16 a17 farm house susu lembang 219.32 a, b, c, y, z, hz, dg, cx, cw, cq, co, cl, a18, a17 a18 alun alun kota bandung 214.31 a, b, c, y, z, hz, dg, cx, cw, cq, co, cl, a18 a19 bukit moko 232.12 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, a15, cc, cd, cg, a21, ch, ci, a19 a20 alam wisata cimahi 214.54 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, a15, cc, ce, cf, cm, a20 a21 curug tilu leuwi opat 215.94 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, a15, cc, cd, cg, a21 a22 pucak guha 167.90 a, b, c, y, aa, dr, a22 a23 curug sanghyang taraje 220.73 a, b, c, y, aa, dr, dq, a23 a24 taman wisata curug cikoja 270.60 a, b, c, y, aa, dr, a22, hk, ds, du, a24 a25 wisata alam pasir kirisik 293.51 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, dz, a25 a26 wisata alam ciung wanara 341.01 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, dz, dy, ea, eb, a26 a27 objek wisata situ wangi 316.91 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, dz, a25, ec, a27 a28 curug panganten 355.71 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, dz, dy, ea, eb, a26, ed, a28 a29 ecopark kota banjar 350.11 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, dz, dy, ea, eb, a26, ed, a29 a30 batu karas 292.25 a, b, c, y, aa, dr, a22, hk, ds, du, do, a30 a31 pantai karapyak 342.20 a, b, c, y, aa, dr, a22, hk, ds, du, do, eg, ei, a31 a32 wisata bukit panembongan 363.67 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, a40, gk, el, a32 a33 telaga remis pasawahan 335.67 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, gm, gn, a33 a34 pantai kejawanan 350.61 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, fn, fh, fg, ff, ez, a34 a35 keraton kasepuhan cirebon 349.81 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, fn, fh, fg, ff, fe, a35 a36 museum bandar cimanuk 359.31 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, gz, fs, ft, fv, hf, a36 a37 pantai tirtamaya 362.15 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, fn, fm, fk, fj, fp, fo, a37 a38 wisata buaya blanakan 285.94 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, ca, bz, a14, bx, a13, br, bp, bo, gd, a38 a39 pemandian air panas ciater 253.08 a, b, d, f, i, j, q, p, o, n, a2, ab, ac, ad, a4, cb, a15, cc, cd, cg, a21, ch, ci, a39 a40 curug cilutung 326.31 a, b, c, y, z, hz, dg, cx, cy, cz, db, gw, dh, dk, dl, a40 optimalization route to tourism places in west java using a-star algorithm muhammad herlambang prakasa yudha 472 destination node tourism places distances route a41 taman wisata alam cadas gantung 323.81 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43, gp, go, fn, a41 a42 cadas pangeran 251.11 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42 a43 curug cipongkor 254.51 a, b, c, y, z, hz, dg, cx, cy, cz, db, hi, gs, gq, a42, a43 based on table 3 for the tourism destination of curug bibijilan (node a1), the optimal route chosen is a, b, c, a1. in other words, the route it takes is jl. raya ciracap – jl. surade – tegalbuleud – jl. sagaranten – cidolog curug bibijilan with a route distance of 128.36 km. all search results for the optimal route along with the road names from each point can be seen at the link https://bit.ly/3oiddgc. based on these numerical results, the conclusion that can be drawn is that optimizing the route to tourism in west java using the a-star algorithm can produce an optimal route. in particular, it can be concluded that the search for optimal routes that are affected by distance and traffic density can be solved by the a-star algorithm. conclusions based on the research, the a-star algorithm can be used when information about the distance and traffic density has been obtained. in this case, traffic density becomes a heuristic value that can be used as a basis for consideration. it can be concluded that the optimal route which is affected by the distance and traffic density can be found using the a-star algorithm. this research can also help tourists to get through the most optimal route to tourism places in west java province. references [1] i. b. g. w. a. dalem, “penerapan algoritma a*(star) menggunakan graph untuk menghitung jarak terpendek,” jurnal resistor (rekayasa sistem komputer), vol. 1, no. 1, pp. 41–47, 2018. 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[15] h. i̇. şahın and a. r. kavsaoğlu, “indoor path finding and simulation for smart wheelchairs,” in 2021 29th signal processing and communications applications conference (siu), 2021, pp. 1–4. m farhan q misklasifikasi mahasiswa baru dengan analisis regresi logistik _7_ misklasifikasi mahasiswa baru f saintek uin sunan kalijaga jalur tes tulis… jurnal cauchy – issn: 2086-0382 175 misklasifikasi mahasiswa baru f saintek uin sunan kalijaga jalur tes tulis dengan analisis regresi logistik mohammad farhan qudratullah program studi matematika fakultas sains dan teknologi universitas islam negeri sunan kalijaga yogyakarta, indonesia e-mail: aching_lo@yahoo.com abstrak pada tahun ajaran 2008/ 2009, uin sunan kalijaga membuka 2 (dua) jalur penerimaan mahasiswa baru, yaitu jalur reguler yang meliputi tes tulis dan seleksi (seleksi khusus dan mahasiswa berprestasi), serta jalur snmptn. selama ini, penerimaan mahasiswa baru melalui jalur tes tulis hanya berdasarkan pilihan saat mendaftar dan hasil ujian tulis tanpa mempertimbangkan variabel lain seperti nilai uan/ uas. penelitian ini bertujuan untuk mengetahui variabel apa saja (nilai tes tulis dan nilai uan/uas) yang mampu membedakan karakteristik mahasiswa program studi yang satu dengan yang lainnya, sehingga memungkinkan untuk mengetahui besar tingkat misklasifikasi yang terjadi pada program studi cluster sains fakultas saintek. adapun alat analisis yang digunakan adalah analisis regresi logistik multinomial. pada tingkat kepercayaan 90% diperoleh bahwa dari 7 (tujuh) variabel independen yang digunakan, terdapat 5 (lima) variabel yang mampu membedakan karakteristik mahasiswa baru program studi yang satu dengan yang lainnya, yaitu nilai tes numerik (nt_numerik), nilai tes spasial (nt_spasial), nilai uan matematika (uanmat), nilai uas fisika (uanfis), dan nilai uas kimia (uaskim), sedangkan 2 (dua) variabel lainnya yaitu: nilai tes verbal (nt_verbal) dan nilai uas biologi (uasbio) tidak signifikan. misklasifikasi mahasiswa baru jalur tes cukup tinggi, yaitu mancapai 35,1%. misklasifikasi dari yang paling rendah berturut-turut adalah program studi matematika 17,7%, program studi kimia 33,3%, program studi biologi 47,7%, dan yang paling tinggi program studi fisika mencapai 50%. sehingga proses penerimaan mahasiswa baru pada keempat program studi pada umumnya perlu mempertimbahkan nilai uan/ uas. kata kunci: analisis regresi logistik multinomial, mahasiswa baru, misklasifikasi pendahuluan di era globalisasi yang bercirikan high competition ini, tuntutan terhadap perguruan tinggi bukan hanya sebatas kemampuan untuk menghasilkan lulusan yang diukur secara akademik, melainkan keseluruhan program dari lembaga-lembaga perguruan tinggi tersebut harus mampu membuktikan kualitas yang tinggi demi terciptanya manusia indonesia seutuhnya, yaitu menguasai ilmu pengetahuan dan teknologi (iptek), estetika (seni), moral dan etika. di indonesia, tingkat human development index (hdi) yang mengukur pembandingan antara life expectancy, literacy, education, dan standard of living belum beranjak naik secara signifikan, rangking ini tidak berbeda jauh dengan negara vietnam yang baru merdeka (tabel 1), sementara akses pendidikan bagi usia 19-24 tahun yang terdaftar sebagai mahasiswa di indonesia juga belum menggembirakan yaitu sebesar 14 persen. kondisi partisipasi untuk melanjutkan di perguruan tinggi ini masih rendah apabila dibandingkan dengan negara malaysia (38 persen) atau mesir (30 persen). universitas islam negeri (uin) sunan kalijaga hadir untuk memenuhi tuntutan masyarakat dan dunia kerja terhadap lembaga pendidikan tinggi yang dapat mengintegrasikan keislaman dan keilmuan serta bermanfaat bagi peradaban. uin sunan kalijaga diharapkan dapat menghasilkan pekerja yang profesional, intelektual yang agamis, dan pemimpin bangsa yang moralis. tabel 1. posisi kualitas sumberdaya manusia tahun 1995, 2000, 2002, 2006 berdasarkan hdi. negara tahun 1995 2000 2002 2006 china 111 99 96 81 thailand 58 76 70 74 philipina 100 77 77 84 malaysia 59 61 59 61 indonesia 104 109 110 108 vietnam 120 108 109 109 sumber: undp berbagai edisi kehadiran uin sunan kalijaga merupakan perjuangan panjang umat islam indonesia, yang dimulai sejak tahun 1951. tranformasi iain sunan kalijaga menjadi uin sunan kalijaga secara de jure, ditandai dengan terbitnya keputusan presiden ri nomor 50 tahun 2004 tertanggal 21 juni 2004. implikasinya dalam aspek akademik, uin sunan kalijaga mohammad farhan qudratullah 176 volume 1 no. 4 mei 2011 mendapatkan ijin penyelenggaraan program studi ’umum’ di luar ilmu-ilmu keislaman yang ditandai dengan berdirinya 2 (dua) fakultas baru, yaitu fakultas sains dan teknologi (f saintek) yang terdiri atas 10 program studi, yaitu matematika, biologi, kimia, fisika, pendidikan matematika, pendidikan biologi, pendidikan kimia, pendidikan fisika, teknologi informasi, dan teknologi industri.dan fakultas ilmu sosial dan himanora (f isoshum) yang terdiri atas 3 program studi yaitu psikologi, sosiologi, dan ilmu komunikasi. demi percepatan mewujudkan visi, misi, dan tujuan uin sunan kalijaga sebagai center for excellence dalam bidang pengembangan keilmuaan dan keislaman yang integratifinterkonektif, uin sunan kalijaga terus melakukan pembenahan diberbagai aspek. salah satunya adalah sistem penerimaan mahasiswa baru dengan melakukan revisi dan penambahan jalur penerimaan mahasiswa baru yang mampu menjaring mahasiswa baru yang berkualitas dan mampu mengakomodir semua program studi yang semakin beragam. pada tahun ajaran 2008/ 2009, uin sunan kalijaga membuka 2 (dua) jalur penerimaan mahasiswa baru, yaitu jalur reguler yang meliputi tes tulis dan seleksi (seleksi khusus dan mahasiswa berprestasi), serta jalur snmptn. jalur tes tulis diadakan sendiri oleh uin sunan kalijaga yang merupakan tes potensi akademik yang meliputi tes verbal, tes numerik, dan tes spasial. mahasiswa baru yang diterima adalah sejumlah mahasiswa yang dirangking berdasarkan hasil penjumlahan nilai ketiga tes tersebut yang disesuaikan dengan pilihannya. dengan kata lain, penerimaan mahasiswa baru hanya berdasarkan hasil ujian tulis tanpa mempertimbangkan prestasi akademik lainnya seperti nilai uan/ uas. serta belum diketahui apakah mahasiswa yang diterima telah masuk program studi yang sesuai atau tidak. penelitian ini akan mengidentifikasi faktor-faktor apasaja yang dapat dijadikan kriteria bahwa seorang mahasiswa telah masuk program studi yang sesuai dengan pilihannya atau tidak. penelitian ini bertujuan untuk mengetahui nilai-nilai apa saja (nilai tes tulis dan nilai uan/uas) yang membedakan karakteristik mahasiswa program studi yang satu dengan yang lainnya pada f saintek, khususnya program studi cluster sains (matematika, biologi, kimia, dan fisika), dan mendeteksi mahasiswamahasiswa yang kemungkinan memiliki karakteristik yang berbeda atau kemungkinan salah memilih program studi (misklasifikasi). sehingga hasil penelitian ini dapat dijadikan bahan evaluasi dalam sistem penerimaan mahasiswa baru uin sunan kalijaga, khususnya yang akan masuk pada program studi-program studi di f saintek yang jelas memiliki karakteristik berbeda dengan program studi yang lainnya. alat statistika yang akan digunakan adalah analisis regresi logistik dan data yang digunakan adalah data mahasiswa baru 4 (empat) program studi sains (matematika, biologi, kimia, dan fisika) angkatan 2008/ 2009 yang diterima melalui jalur tes tulis yang meliputi nilai tes potensi akademik (tes verbal, tes numerik, dan tes spasial) dan nilai uan/ uas untuk matapelajaran mipa (matematika, biologi, kimia, dan fisika). analisis regresi logistik analisis regresi logistik merupakan salah satu alat analisis dalam statistika yang merupakan bentuk khusus dari analisis regresi, yaitu variabel dependennya merupakan data skala nominal atau ordinal, sedangkan variabel independennya dapat berbentuk nominal, ordinal, skala, ataupun rasio. dinamakan regresi logistik, karena analisis regresi ini pembentukan modelnya didasarkan atas kurva logistik. nilai yang dihasilkan persamaan regresi logistik merupakan peluang kejadian yang digunakan sebagai ukuran untuk pengklasifikasian. jika variabel dependen terdiri atas 2 (dua) klasifikasi, maka disebut analisis regresi logistik biner. dan jika variabel dependen terdiri atas 3 (tiga) klasifikasi atau lebih, maka disebut analisis regresi logistik multinomial. model regresi logistik misalkan terdapat n hasil pengamatan ���;���;���;…;�� , dengan �� adalah variabel dependen yang dikotomi/biner subjek ke-i yang diberi kode 0 atau 1 dan ��� adalah variabel independen subjek ke-i sejumlah p variabel, i = 1, 2, …, n. maka model regresi logistik dapat ditulis: � �� � ������ �∑ ����� ��� �1������� �∑ ����� ��� � (1) dimana � �� � � �|�� dengan melakukan transformasi logit, � �� � ��� � ��1 � ��! diperoleh suatu fungsi penghubung untuk model regresi logistik berikut: � �� � ��" � ��1 � ��# misklasifikasi mahasiswa baru f saintek uin sunan kalijaga jalur tes tulis… jurnal cauchy – issn: 2086-0382 177 � �� � �� $ %%& ������ �∑ ����� ��� �1������� �∑ ����� ��� �1�1������� �∑ ����� ��� �' (() � �� � *����� ��� (2) model regresi logistik multinomial merupakan perluasan dari model regresi logistik biner, misalkan variabel dependen terdiri atas 3 (tiga) kategori maka variabel dependen y dapat diberi kode 0, 1, atau 2, dimana y = 0 merupakan kategori acuan. pada model regresi logistik biner terdapat 1 (satu) fungsi logit y = 1 terhadap y = 0, maka pada model regresi logistik multinomial 3 (tiga) kategori terdapat 2 (dua) fungsi logit, yaitu fungsi logit y = 1 terhadap y = 0 dan fungsi logit y = 2 terhadap y = 0. misalkan �+ �� � , � � -|�� dimana j = 0, 1, 2 dan mengikuti kaidah pada model regresi logistik biner, maka kedua fungsi logit dapat dirumuskan berikut: �� �� � ��"�� ���� ��# � ��� , � � 1|��, � � 0|�� ! � �� $ %%& ������� �∑ ������ ��� �1�������� �∑ ������ ��� �1�1�������� �∑ ������ ��� �' (() � ��� � *������ ��� (3) �� �� � ��"�� ���� ��# � ��� , � � 2|��, � � 0|�� ! � �� $ %%& ������� �∑ ������ ��� �1�������� �∑ ������ ��� �1�1�������� �∑ ������ ��� �' (() � ��� � *������ ��� (4) sehingga diperoleh: , � � 1|�� � �� �� � , � � 0|��.���2��� � *������ ��� 3 , � � 2|�� � �� �� � , � � 0|��.���2��� � *������ ��� 3 karena �� ����� ����� �� � 1, maka dapat ditulis �� �� � 11 �������� � ∑ ������ ��� � ������� �∑ ������ ��� (5) �� �� � ������� � ∑ ������ ��� 1 �������� � ∑ ������ ��� � ������� �∑ ������ ��� (6) �� �� � ������� � ∑ ������ ��� 1 �������� � ∑ ������ ��� � ������� �∑ ������ ��� (7) berdasarkan hasil di atas, dapat digeneralisasikan (j-1) model regresi logistik multinomial untuk variabel independen terdiri atas j kategori adalah: �+ �� � ������+ �∑ ��+��� ��� 1�∑ ������+ �∑ ��+��� ��� 45�+�� (8) dan fungsi logitnya adalah: �+ �� � ��"����0 ��# � ��+ � *��+��� ��� (9) dimana j = 1, 2, ... , j-1 metode dan teknik analisis estimasi parameter penaksiran parameter pada model regresi logistik dapat mengunakan metode maksimum likelihood estimator (mle). dengan menetapkan asumsi distribusi binomial dan setiap objek pengamatan saling independen, fungsi likelihood metode mle merupakan fungsi linear maka untuk memperoleh taksiran parameter dilakukan proses iterasi dengan metode newton-raphson, dengan menentukan nilai awal dari β, yaitu β0. estimasi maksimum likelihood merupakan pendekatan dari estimasi weighted least square, dimana matrik pembobotnya berubah setiap iterasi. proses menghitung estimasi maksimum likelihood ini disebut juga sebagai iteratif reweighted least square. uji serentak dalam uji serentak ini, digunakan likelihood-rasio test, metode ini merupakan metode pengujian model dengan membandingkan likelihood untuk model lengkap (l1) dan likelihood untuk model yang semua parameternya sama dengan nol (l0). h0 : �� � �� � 6 � � � 0 h1 : minimal ada satu �+ 7 0, j =1, 2, …, p mohammad farhan qudratullah 178 volume 1 no. 4 mei 2011 statistik uji yang digunakan adalah: 8� � 2��"9091# (10) pada tingkat kepercayaan 1 :�%, h0 ditolak jika 8� < χ =,?@�� atau p-value (sig.) a :. uji parsial uji parsial untuk menguji signifikasi setiap parameter dalam model yang dapat dilakukan dengan wald test. uji ini dilakukan untuk mengetahui apakah setiap variabel independen dapat diandalkan membangun model atau tidak dalam proses pengklasifikasian. h0 : �+ � 0 h1 : �+ 7 0 untuk j = 1, 2, …, p statistik uji yang digunakan adalah: b� � �+��c���+ �� (11) pada tingkat kepercayaan 1 :�%, h0 ditolak jika b� < χ =,?@�� atau p-value (sig.) a :. uji kesesuaian model uji kesesuaian model regresi logistik yang digunakan adalah chi-square test. h0 : model sesuai h1 : model tidak sesuai statistik uji yang digunakan adalah: χ� � *�d+ �+�e+ ��+�e+�1 �e+ f +�� (12) pada tingkat kepercayaan 1 :�%, h0 ditolak jika χ� < χ =,?@�� atau p-value (sig.) a :. evaluasi model evaluasi fungsi klasifikasi (fungsi logistik) dilakukan dengan membuat tabulasi antara actual group dan predicted group yang diperoleh dari fungsi logistik. selanjutnya dihitung proporsi pengamatan yang benar klasifikasinya, diharapkan proporsi pengamatan yang benar diklasifikasikan tersebut sebesar mungkin atau proporsi pengamatan yang salah sekecil mungkin. data dan variabel data yang digunakan adalah data mahasiswa baru jalur tes tulis tahun ajaran 2008/ 2009 fakultas sains dan teknologi uin sunan kalijaga yang bersumber pada bagian akademik fakultas sains dan teknologi dan dapic teknologi uin sunan kalijaga yang terdiri atas 58 mahasiswa dengan perincian 12 mahasiswa program studi matematika, 6 mahasiswa program studi fisika, 22 mahasiswa program studi kimia, dan 15 mahasiswa program studi biologi. sesuai tujuan penelitian ini, maka variabel dependennya adalah variabel kelompok yaitu variabel yang berupa kategori program studi yang terdiri dari matematika (mat) dengan kode 1, fisika (fis) dengan kode 2, kimia(kim) dengan kode 3, dan biologi(bio) dengan kode 4. selanjutnya variabel independen yang digunakan dalam penelitian ini ada 7 (tujuh) variabel, yaitu: 1. nt_verbal : nilai hasil ujian tulis yakni tes verbal 2. nt_numerik : nilai hasil ujian tulis yakni tes numerik 3. nt_spasial : nilai hasil ujian tulis yakni tes spasial 4. uanmat : nilai uan bidang studi matematika 5. uasbio : nilai uas bidang studi biologi 6. uaskim : nilai uas bidang studi kimia 7. uasfis : nilai uas bidang studi fisika hasil dan pembahasan berikut adalah hasil analisis data yang proses analisisnya mengunakan bantuan software spss.15.0: tabel 1. hasil likelihood rasio test tahap awal model chi-kuadrat df sig keterangan keseluruhan 57,902 21 0,000 signifikan pervariabel 1. nt_verbal 0,836 3 0,841 tidak signifikan 2. nt_numerik 19,869 3 0,000 signifikan 3. nt_spasial 9,444 3 0,024 signifikan 4. uanmat 12,950 3 0,005 signifikan 5. uasfis 11,133 3 0,011 signifikan 6. uaskim 8,372 3 0,039 signifikan 7. uasbio 3,015 3 0,389 tidak signifikan misklasifikasi mahasiswa baru f saintek uin sunan kalijaga jalur tes tulis… jurnal cauchy – issn: 2086-0382 179 tabel 2. hasil likelihood rasio test tahap akhir model chi-kuadrat df sig keterangan keseluruhan 61,690 15 0,000 signifikan pervariabel 1. nt_numerik 3 0,000 signifikan 2. nt_spasial 3 0,010 signifikan 3. uanmat 3 0,001 signifikan 4. uasfis 3 0,012 signifikan 5. uaskim 3 0,078 signifikan langkah awal dalam proses ini adalah memasukan semua variabel independen kedalam model dan ringkasan hasilnya disajikan dalam tabel 1. pada tabel 1 disajikan hasil uji serentak atau uji keseluruhan dari model, tampak bahwa nilai sig. = 0,000 < 0,10 yang berarti ho ditolak, yaitu minimal ada satu � 7 0. kemudian untuk pervariabel, tampak bahwa pada tingkat kepercayaan 90% terdapat 2 (dua) variabel independen tidak signifikan yaitu nt_verbal dan uasfis, maka perlu dilakukan analisis ulang dengan mengeluarkan variabel independen yang tidak signifikan (sig. < 0,10) satu demi satu dari model, mulai dari nt_verbal dan diperoleh bahwa uasfis masih belum signifikan sehingga harus dikeluarkan dari model dan ringkasannya disajikan dalam tabel 2, tampak bahwa semua variabel independen telah signifikan sehingga proses analisis selanjutnya, yaitu uji parsial mengunakan wald test. tabel 3. hasil estimasi parameter tahap akhir variabel b wald df sig exp(b) keterangan mat 0. intercept -13,061 2,861 1 0,091 1. nt_numerik 0,542 5,763 1 0,016 1,720 signifikan 2. nt_spasial -0,184 0,236 1 0,627 0,832 tidak signifikan 3. uanmat 1,589 7,202 1 0,007 4,900 signifikan 4. uasfis -0,371 0,299 1 0,585 0,690 tidak signifikan 5. uaskim -1,273 3,636 1 0,057 0,280 signifikan fis 0. intercept -2,561 0,167 1 0,683 1. nt_numerik -0,376 4,670 1 0,031 0,687 signifikan 2. nt_spasial 1,168 5,357 1 0,021 3,217 signifikan 3. uanmat 2,274 5,969 1 0,015 9,714 signifikan 4. uasfis -1,796 5,258 1 0,022 0,166 signifikan 5. uaskim -1,021 3,58 1 0,076 0,360 signifikan kim 0. intercept 5,103 2,342 1 0,126 1. nt_numerik -0,047 0,407 1 0,524 0,955 tidak signifikan 2. nt_spasial 0,269 1,732 1 0,188 1,308 tidak signifikan 3. uanmat 0,584 2,052 1 0,152 1,793 tidak signifikan 4. uasfis -1,084 6,101 1 0,014 0,338 signifikan 5. uaskim -0,402 1,249 1 0,264 0,669 tidak signifikan bio sebagai kelompok acuan berdasarkan nilai koefisien dari tabel 3 di atas, dapat disusun 3 (tiga) fungsi logit, yaitu: 1. fungsi logit program studi matematika dengan biologi sebagai acuan �� �� � 13,061 �0,542.nt_numerik* 0,184.nt_spasial � 1,589.uanmat* 0,371.uasfis 1,273.uaskim* (13) 2. fungsi logit program studi fisika dengan biologi sebagai acuan �� �� � 2,561 0,376.nt_numerik* � 1,168.nt_spasial* � 2,274.uanmat* 1,796.uasfis* 1,021.uaskim* (14) 3. fungsi logit program studi kimia dengan biologi sebagai acuan �n �� � 5,103 0,047.nt_numerik � 0,296.nt_spasial � 0,584.uanmat 1,084.uasfis* 0,402.uaskim (15) ket: * signifikan pada tingkat kepercayaan 90% mohammad farhan qudratullah 180 volume 1 no. 4 mei 2011 tampak bahwa pada tingkat kepercayaan 90% terdapat 3 (tiga) variabel independen yang membedakan mahasiswa yang memilih program studi matematika dan biologi, yaitu nt_numerik, uanmat, dan uaskim. nilai tes numerik dan nilai uas matematika mahasiswa program studi matematika cenderung lebih tinggi, sedangkan nilai uas kimianya cenderung lebih rendah dibanding program studi biologi. kemudian yang membedakan program studi fisika dan biologi adalah 5 (lima) variabel independen (nt_numerik, nt_spasial, uanmat, uasfis, dan uaskim). nilai ujian tes spatial, uan matematika mahasiswa program studi fisika cenderung lebih tinggi, sedangkan nilai tes numerik uas fisika dan uas kimia cenderung lebih rendah dibanding program studi biologi. selanjutnya yang membedakan mahasiswa program studi kimia dan program studi biologi hanya satu variabel independen, yaitu uasfis. nilai uas fisika mahasiswa program studi kimia cenderung lebih rendah dibanding program studi biologi. dari uraian di atas dapat dikembangkan dan disusun variabel-variabel independen yang membedakan karakteristik mahasiswa antara program studi yang selengkapnya disajikan dalam tabel 4 berikut: tabel 4. karakteristik 4 (empat) program studi program studi fis kim bio mat nt_numerik (+/-) nt_spasial (-/+) uasfis (+/-) nt_numerik (+/-) nt_spasial (-/+) uanmat (+/-) uasfis (+/-) uaskim (-/+) nt_numerik (+/-) uanmat (+/-) uaskim (-/+) fis nt_numerik (-/+) nt_spasial (+/-) uanmat (+/-) uaskim (-/+) nt_numerik (-/+) nt_spasial (+/-) uanmat (+/-) uasfis (-/+) uaskim (-/+) kim uasfis (-/+) +/: program studi pada baris nilainya > program studi pada kolom -/+ : program studi pada baris nilainya < program studi pada kolom tabel 5 merupakan ringkasan uji kesesuaian model mengunakan uji chi-kuadrat, tampak bahwa nilai sig. = 0,995 > 0,10, artinya pada tingkat kepercayaan 90% ho tidak ditolak atau dengan kata lain model sesuai dan dapat digunakan tabel 5. hasil uji kesesuaian model chikuadrat df sig keterangan pearson 111,564 153 0,995 model sesuai deviance 84,291 153 1,000 model sesuai sedangkan untuk mengevaluasi fungsi logistik, dari tabel 6 dapat dilihat bahwa 2 atau 16,7% mahasiswa matematika diprediksi masuk dalam program studi lain dalam hal ini kimia, 3 atau 50% mahasiswa fisika diprediksi masuk program studi lain, yaitu kimia, 8 atau 33,3% mahasiswa program studi kimia diprediksi masuk program studi lain, yaitu 2 program studi matematika, 1 program studi fisika, dan 5 program studi biologi, terakhir 7 atau 47,7% mahasiswa program studi biologi diprediksi masuk program studi lain, yaitu 1 program studi matematika, 2 program studi fisika, dan 4 program studi kimia. secara keseluruhan dari 58 mahasiswa baru f saintek yang diterima melalui ujian tulis terdapat 20 mahasiswa atau 35,1% yang misklasifikasi dalam pemilihan program studi, misklasifikasi terbesar adalah program studi fisika dan diikuti program studi biologi. tabel 6. hasil pengklasifikasian model prediksi % kebenaran data aktual mat fis kim bio prediksi mat 10 0 2 0 83,3 fis 0 3 3 0 50,0 kim 2 1 16 5 66,7 bio 1 2 4 8 53,3 % kebenaran prediksi (keseluruhan) 64,9 penutup dari uraian analisis data dan pembahasan di atas dapat ditarik beberapa kesimpulan sebagai berikut: 1. pada tingkat kepercayaan 90% dari 7 (tujuh) variabel independen yang digunakan, terdapat 5 (lima) variabel yang mampu membedakan karakteristik mahasiswa baru program studi yang satu dengan yang lainnya pada program studi cluster sains fakultas saintek, yaitu nilai tes numerik (nt_numerik), nilai tes spasial (nt_spasial), nilai uan matematika (uanmat), nilai uas misklasifikasi mahasiswa baru f saintek uin sunan kalijaga jalur tes tulis… jurnal cauchy – issn: 2086-0382 181 fisika (uanfis), dan nilai uas kimia (uaskim), sedangkan 2 (dua) variabel lainnya yaitu: nilai tes verbal (nt_verbal) dan nilai uas biologi (uasbio) tidak signifikan. 2. misklasifikasi mahasiswa baru jalur tes tulis 2008/ 2009 pada program studi cluster sains fakultas saintek cukup tinggi, yaitu mancapai 35,1%. misklasifikasi dari yang paling rendah berturut-turut adalah program studi matematika 17,7%, program studi kimia 33,3%, program studi biologi 47,7%, dan yang paling tinggi program studi fisika mencapai 50%. sehingga proses penerimaan mahasiswa baru pada keempat program studi pada umumnya perlu mempertimbahkan nilai uan/ uas. adapun beberapa saran untuk penelitian selanjutnya, yaitu: 1. pada penelitian ini program studi yang diteliti adalah program studi yang masuk dalam cluster sains di fakultas saintek. penelitian ini dapat diperluas dengan melibatkan semua program studi di fakultas saintek maupun semua fakultas serta penelitian bukan hanya melibatkan satu angkatan tapi beberapa angkatan. 2. variabel independen yang digunakan adalah nilai-nilai sebelum mahasiswa ikut perkuliahan, yaitu nilai tes tulis tulis (nilai tes verbal, nilai tes numerik, nilai tes spasial) dan nilai uan/ uas (matematika, fisika, kimia, biologi). variabel independen ini dapat diperluas dengan mempertimbangkan nilai matakuliah dasar pada semester awal, ipk, jalur masuk, dan lain sebagainya. 3. untuk menganalisis misklasifikasi mahasiswa baru dapat juga mengunakan alat statistika lain seperti analisis diskriminan, struktur equation modeling (sem), maupun mengunakan jaringan syaraf tiruan (jst) daftar pustaka [1] agresti, a. (2007). an introduction to categorical data analysis, second edition. new jersey: john wiley & sons, inc. [2] chatterjee, s. and hadi, a.s. (2006). regreeion analysis by example, fourth edition. new jersey: john willey & sons, inc. [3] hosmer, d.w. and lamenshow (1989). applied logistic regression. new york: willey and sons. [4] kleinbaum, d.g. (1994). logistic regression. springer-verlag, new york. [5] uin sunan kalijaga (2004). kerangka dasar keilmuan & pengembangan kurikulum uin. yogyakarta: pokja akademik uin sunan kalijaga [6] uin sunan kalijaga (2007). sistem penerimaan mahasiswa baru tahun ajaran 2008/ 2009 uin sunan kalijaga. yogyakarta: pokja akademik uin sunan kalijaga. cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 publication etics cauchy: jurnal matematika murni dan aplikasi is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewed and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an 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acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia subanar seno, gadjah mada university, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740595') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740557') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740556') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/740541') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/736347') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5964') genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification cauchy – jurnal matematika murni dan aplikasi volume 7(1) (2021), pages 1-12 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 18, 2020 reviewed: may 21, 2021 accepted: november 02, 2021 doi: http://dx.doi.org/10.18860/ca.v7i1.10337 genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah1, elok faiqah2, heri kuswanto3, nlp satyaning pradnya paramita4 1,3department of statistics, faculty of mathematics, computing, and data sciences institut teknologi sepuluh nopember, 60111 surabaya, indonesia email: irhamah@statistika.its.ac.id, elokfaiqoh212@gmail.com, heri_k@statistika.its.ac.id, sat.pradnyaparamita@gmail.com abstract colon cancer is the second leading cause of cancer-related deaths globally; hence, research on that topic, such as disease diagnosis, needs to be undertaken with improvement. microarray has an essential role in biomedical research as a tool for identifying and classifying diseases, especially cancer. this study aims to develop a classification model using fuzzy support vector machines (fsvm) hybridized with genetic algorithm (ga) for classifying individuals based on gene expression into two classes, i.e., normal and cancer, and compare classification performance based on the use of selection method. fuzzy memberships were used in svm to deal with the case of imbalanced microarray data. meanwhile, the role of the genetic algorithm is, firstly, to select the relevant genes as the features and, secondly, to optimize the parameter of fsvm as ga can handle the problem of nonlinear optimization that has a high dimension, adaptable, and easily combined with other methods. the results show that classification using fcbf selection has a higher accuracy value than those without the selection. fsvm optimized using ga has the highest accuracy value compared to other classification methods used in this study. keywords: feature selection; fuzzy svm; genetic algorithm; parameter optimization; svm introduction cancer is a leading cause of death globally: an estimated 7.6 million people died of cancer in 2005, and 84 million people will die in the next ten years if action is not taken. more than 70% of all cancer deaths occur in lowand middle-income countries, where cancer prevention, diagnosis, and treatment resources are limited or non-existent [1]. the cancer diagnosis can be done based on its morphological structure but has difficulties due to very thin differences in morphological structures between different types of cancer [2]. in 1999, alon researched gene expression related to colon cancer which contained data microarray data. microarray data is a technology in molecular and medical biology that can see differences in gene expression. microarray data is a type of data with very high dimensions used in bioinformatics. the characteristics of data microarray are the small amount of data and a large number of features. microarray data consists of thousands of spots (attributes), and from each spot consists of millions of http://dx.doi.org/10.18860/ca.v7i1.10337 mailto:irhamah@statistika.its.ac.id mailto:elokfaiqoh212@gmail.com mailto:heri_k@statistika.its.ac.id mailto:sat.pradnyaparamita@gmail.com genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 2 copies of dna molecules that respond to a gene. the collection of genes will be used to classify a class of diseases [3]. therefore, the study was conducted using microarray data from colon cancer, which will be carried out by typing genes, features, or variables using the support vector machine (svm). the svm is one of the classification methods which gives the best results with accuracy values reaching 80% to 90% and can be used to deal with high dimensional data. [4] implemented svm on the microarray data and showed promising results. svm has a better performance in classification than other methods [5], but the effect of imbalanced data on svm will be a drawback in the paradigm of maximizing margins. some researchers propose fsvm, which applies fuzzy membership to each sample and formulates svm so that different sample inputs have other contributions and can handle imbalanced data. in [6], svm and nb are used to classify microarray data where the methodologies involve dimension reduction of microarray data using ica, followed by the feature selection using fbfe. still, there was no specific method used to deal with imbalanced data. therefore, this study compares the classification performance of fsvm and svm analysis on colon cancer microarray imbalanced data. the biggest problem in setting the svm model is in determining the hyperparameter values of the svm [7]. setting the parameter values will increase the classification accuracy of the svm model [8]. thus, in this study, ga is used to optimize the value of parameters on the svm model to increase the classification performance. ga can handle high-dimensional nonlinear optimization problems [9]. this study also compares the effect of fast correlation based filter (fcbf) in classification performance with variable selection. the fcbf algorithm is based on the idea that good features are a feature that is relevant to the class but not redundant to pertinent other features. the methods implemented in this research are svm without variable selection, svm with fcbf variable selection, svm ga without variable selection, svm ga with fcbf variable selection, fsvm without variable selection, fsvm with fcbf variable selection, fsvm ga without variable selection, and fsvm ga with fcbf variable selection. the best method is obtained from the highest classification performance value. methods fast correlation based filter (fcbf) fast correlation based filter or fcbf is a variable selection method developed by [10]. in general, a feature is good if it is relevant to the class concept and not redundant to any of the other relevant characteristics. suppose we adopt the correlation between two variables as a goodness of fit measure. in that case, the above definition means that a feature is good if it is highly correlated to the class but not correlated to any other components. in other words, if the correlation between an element and the class is high enough to make it relevant to (or predictive of) the class and the correlation between it and any other suitable features does not reach a level, then it can be predicted by any of the other relevant features, it will be regarded as a good feature for the classification task. in this sense, the problem of feature selection boils down to find a suitable measure of correlations between components and a sound procedure to select features based on this measure. there exist broadly two approaches to measure the correlation between two random variables. one is based on classical linear correlation, and the other is based on information theory. this study used a second approach: choosing a correlation degree based on the entropy theory information [10]. decresing of entropy in x reflects genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 3 additional information of x by y called information gain [11]. to measure information gain, symmetrical uncertainty is used. information retrieval is symmetrical from two random variables x and y. symmetrical uncertainty values are carried out in the range 0 to 1 with a value of 1 which indicates another value or x and y dependent and 0, which shows that x and y are independent. support vector machines (svm) svm was first introduced by vapnik in 1992. svm is a technique for finding a separation function in classifications that can separate two or more different data groups. svm can find the best hyperplane function between unlimited functions to separate objects. the best hyperplane is located right in the middle between two object sets of two classes. finding the best hyperplane is equivalent to maximizing the margin or distance between two sets of objects from two different classes. svm works to find a separation function with maximum margins [12]. svm is a classification technique with a training process (supervised learning) to find the legitimate line of the best hyperplane with f(x), ( ) ( ( ))f sign gx x (1) where ( ) t g  x w x b ;𝒙, 𝑤 ∈ 𝑅𝑛 and 𝑏 ∈ 𝑅 if the value of g (x) is negative then the observation goes into the negative class, so if g (x) is positive then obstruction will enter the positive class. the concept of hyperplane on svm is as follows, figure 1. the concept of hyperplane in svm in figure 1. the margin is equal to d+ d+. the classification function is a hyperplane plus a margin zone. this separates points from the two classes with the highest distance (margin) between the two classes. 0t b  i x w is a separator hyperplane, d+ d+ will be the shortest distance on the closest object from class +1 (-1). because separation can be solved without error, all observations i = 1, 2, ..., n is measured as, 1 for 1 1 for 1 t t b yi b yi           x w i x w i where 1 0, 1, 2,.., n i i i i y i n    w x in general, in the real-world domain (real world problem) rarely are linearly separable, mostly nonlinear. the method for classifying data that cannot be separated from linear functions is by transforming data into feature space dimensions so that they (2) genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 4 can be separated linearly in feature space. input space with 2 dimensions cannot separate data into two classeslinearly. therefore, it is necessary to map the input vector by function ( )x into a new vector space with a higher dimension (3 dimensions). the way to classify data is to use the transformation function ( ) i i x x into the feature space so that there is a separate field that can separate data according to its category. by using the ( ) i i x x transformation function, the value is generated, 1 ( ) ( ) ( ) n i i i j i f x y x x b      feature space in practice usually has a higher dimension than input vectors. this causes the computation of feature space to be very large, because there is a possibility that feature space can have an unlimited number of features. in addition, it is difficult to know the right transformation function. the transformation function in svm is to use kernel tricks [13]. the kernel trick is to calculate the scalar product in the form of a kernel function.    ( ) ti jk x x i jx , x then the transformation function in the above equation can be used without the need to know the transformation function explicitly. thus the resulting function is,   1 ( ) n i i i f x y k b    i jx , x where 0 ; 1, 2, ....,c i ni   the requirement for a function to function as a kernel is to fulfill the mercer theorem, which is a necessary and sufficient condition for symmetric functions ( )k i j x , x . the most commonly used kernel functions are as follows, a. kernel gaussian (rbf) :     2 expk         i j i j x , x x x b. kernel linear :   tjk i i jx , x x x c. kernel polynomial :     p k r  t i j i j x , x x x fuzzy support vector machines fuzzy support vector machine (fsvm) is a development of support vector machine for multiclass problems. by using the decision function obtained from svm for a class pair, for each class a polyhedral pyramidal membership function is defined. fsvm uses membership functions to classify variable that cannot be classified by decision functions [14]. there are several applications that only want to focus on accuracy for class classification. for this purpose fuzzy membership can be determined as a function of each class. suppose given a series of training [15]: 1 1 1 ( , , ),....., ( , , ) n n n y x s y x s fuzzy membership becomes a function in the class si = 1 if yi = 1 and si = 0.1 if yi = -1 by using lagrangian, the decision function for fsvm is stated as follows ( ) 1 n f x y b i i i    k(x , x ) i j when 0 ; 1, 2,...,s c i ni i   (3) (4) (5) (6) genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 5 pre-processing data pre-processing datais a process to improve the quality of raw data, so that it can improve accuracy and efficiency for the next data mining process. in this study, preprocessing of data was carried out using transformations. one method of transformation is scaling where one of the advantages of scaling is that it can avoid features with a greater range of values dominating features with a smaller range of values. each data is linearly transformed into a range [0, 1] using the following equation, ' min max min a a a v v    wherev'is the transformed value, v is the initial value, maxa is maximum value on variable, and mina is minimum value on the variable. genetic algorithm genetic algorithm (ga) was first discovered by john holand in 1975. the ga concept is based on the theory of evolution with the principle of natural selection developed by darwin. ga is a technique for identifying solutions to optimization problems. the steps taken in the ga method are as follows: step 1 : define, which defines the operator in ga that corresponds to the problem. at this research, the selection and optimization variables were carried out using genetic algorithm step 2 : initialize, which is to form an initial population consisting of n chromosomes. set n = 100 step 3 : fitness, which evaluates the fitness of each chromosome in the population step 4 : selection, which applies the roulette wheel selection method which gives a set of m mating populations with size n step 5 : crossover, which is the recombine process. this process randomly pairs all m chromosomes to form n / 2 pairs. if a random number [0.1] is less than pc, then crossovers occur step 6 : mutation, which is using mutation probability (pm) to carry out the process of inheritance mutation step 7 : replace, which is replacing the old population with the new population. the new population is obtained by selecting the best n chromosomes obtained by evaluating the fitness value of parents and new breeds step 8: test, that is, if the criteria has been met, then the process is stopped and return to the best solution of the current population. if the criteria have not been met, then go back to step 2. furthermore, elitism is one of the techniques that is done to maintain the best individual who has the highest fitness value to survive for the next generation microarray data microarray data is one of the technologies used to measure the level of expression of thousands of genes simultaneously in one observation and appears as a device of the microarray which is usually summarized in the list of genes and expressed in two conditions or classifications based on the phenotype. microarray data is a type of high dimensional data because it has a number of genes (variables) hundreds or even thousands, while the number of observations that usually do not reach 100 or far smaller than the number of variables. two common methods for analyzing data microarray are clustering and classification [16]. based on the information they possess, (7) genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 6 microarray has an important role in biomedical research as a tool for identifying and classifying diseases, especially cancer. the colon cancer data stores information in the form of gene expressions obtained from patient of tumor colon tissues and normal colon tissues. colon data consists of 62 patients, 40 of which are patients of the tumor class and the other 22 are normal classpatients. the number of variables in colon cancer data is 2000 variables. the steps of microarray experiment to get a colon cancer microarray data are as follows [17]. 1. obtaining mrna from the observed cell (for example in the case of a tumor, the sample observed is a cell that has a tumor. 2. mrna is converted to cdna using reverse tranciptase enzyme. 3. marking cdma from tumor cells in red and cdna from normal cells in green. 4. the sample is hybridized, iecdna binds to dna. 5. the sample is scanned to measure the expression of each gene through fluorescence contained (fluorescence is related to the amount of cdna in the sample for the gene). 6. a bright red glowing point is a gene that is highly expressed in tumor cells, while a bright green glow is a gene that is highly expressed in normal cells. if the gene is expressed in both samples (tumors and normal), the color produced is bright yellow. from the process, the final data is obtained which consists of thousands of points that have different colors and need to be interpreted. color dots must be changed to a certain value to be analyzed later. here is an illustration of the image to get microarray data, figure 2. microarray data process (source: gibson & muse, 2002) results and discussion description of data the data used in this study is the microarray type colon cancer data by u. alon, n. barkai, d.a. notterman, k. gish, s. ybarra, d. mack, and a.j. levine in 1999 [18]. data were taken from human intestinal tissue. the expression of these genes is stored in 2000 variables. this data consists of 62 patients, where 40 patients are having tumor colon tissue (tumor/tissue) while 22 are normal patients (normal colon tissue). the research variable used are shown in table 1, table 1. research variables variable information measurement scale dependent (y) nominal colon cancer class: 0 = normal 1 = tumor independent (x) gene expression ratio genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 7 characteristics of colon cancer data the data used are observations made on 62 samples taken from human intestinal tissue. the data is divided into two classes, namely the tumor class and normal class. the tumor class is described as negative by coding 1 and the normal class is described as positive by coding 0. the description of the two colon cancer classes is shown in figure 3. figure 3. percentage of normal class and tumor in colon cancer data in this observation, a positive sample is stated as a normal sample with code 0 and a negative class is stated as a tumor sample with code 1. figure 3. shows that of the 62 patients taken there were 64.5% of the total samples stated as tumors and the remaining 35.5% were declared normal . based on the proportion of samples of normal and tumor class classes, it is known that colon cancer data is imbalanced data so that the appropriate method to classify colon cancer class is the fuzzy support vector machine (fsvm) method. colon cancer data has a number of genes or variables as many as 2000 variables so that if it will complicate the classification because the data distribution pattern will become very complex. following is the pattern of data distribution shown in figure 4. figure 4. distribution patterns of several variables in colon cancer data the pattern of colon cancer data distribution shown in figure 4. in red is the tumor class and the green color is normal class is spread evenly so that it is difficult in classification. the function separator or hyperplane is expected to be able to help overcome the classification problems in colon cancer data so that in this study an analysis will be conducted using fuzzy support vector machine (fsvm) classification method. 0 (normal) 1 (tumor) c ategory 64.5% 35.5% 10000 5000 0 2001000 16008000 2001000 1600 800 0 2000 1000 0 200 100 0 1000050000 200 100 0 200010000 v9 v418 v897 v1493 v1940 negativ e (tumor) positiv e (normal) kelas : matrix plot of v9; v418; v897; v1493; v1940 genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 8 pre-processing data data pre-processing is a process carried out to improve the quality of raw data by producing precise accuracy values and efficient classification results. in this study, preprocessing is done using variable transformation and selection. one method of transformation is scaling where one of the advantages of scaling is that it can avoid features with a greater range of values dominating features with a smaller range of values. table 2. results of transformation of colon cancer data scaling variable variable name before transformation after transformation mean varians mean varians x1 h55933 7016 9566467 0.3936 0.0569 x2 r39465 4967 4791241 0.4087 0.0623 x3 r39465_1 4095 3305418 0.3851 0.0614 x4 r85482 3988 4076712 0.2784 0.0403 x5 u14973 2937 1841267 0.2556 0.0384 ... ... ... ... ... ... ... ... ... ... ... ... x1999 r77780 53.25 1479.39 0.2477 0.0404 x2000 t49647 42.97 806.28 0.3070 0.0551 table 2 shows that after transformation using scaling, the average value of each variable becomes smaller and ranges between [0,1]. the variance value of the transformed variable is also small which indicates that the observation value with small data distribution or diversity between observations of one with other observations is quite small.in this study, the selection of variables used is fcbf (fast correlation baser filter) and the variables obtained after selecting variables with fcbf are 15 variables from 2000 variables. the 15 variables obtained were further analyzed using svm and fsvm classifications. colon cancer classification performance the svm classification consists of svm classifications with or without fcbf variable selection. the results of classification methods performance are shown in table 3. in svm without fcbf selection, the optimum valueof cost and gamma parameter are 128 and 0.001953125. based on the auc, the svm without selection was good because the auc value was 86.42%. the accuracy value obtained is also quite high, that is 83.57%. whereas for svm classification with fcbf variable selection, the optimal value of cost and gamma parameter are 2048 and 0.03125. the auc value obtained in the svm classification with fcbf selection is very good because it is located in a range of 90100%. it means that the svm model with fcbf selection has been very good in classifying colon cancer data. the accuracy obtained is 91.90% which means that by using a cost value 2048and gamma 0.03125, svm models can classify 91.90% of observations correctly. in addition, the model can also classify positive or normal classes correctly at 88.33% which can be seen from the sensitivity value and 95.50% of the model can classify negative classes or tumors with a specificity value. genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 9 table 3. performance of colon cancer classification method variable selection performance classification (%) optimal parameters accuracy sensitivity specificity g-means auc cost gamma svm without variable selection 83.57 83.33 89.5 85.31 86.42 128 0.001953125 with fcbf variable selection 91.9 88.33 95.5 91.92 91.4 2048 0.03125 svm ga without variable selection 83.57 83.33 89.5 85.31 86.42 1272.99 0.0011941 with fcbf variable selection 95 91.67 97.5 94.29 94.58 18652 0.0161011 fsvm without variable selection 75.71 40 95 47.13 67.5 32768 0.0000610 with fcbf variable selection 90.24 81.67 95 87.09 88.33 2048 0.125 fsvm ga without variable selection 87.38 86.25 43011.2 0.000088722 with fcbf variable selection 90.03 97.5 2968.8 0.15275 on svm-ga without selection, the optimal cost parameter is 1272.99 and gamma optimal parameters is 0.0011941 with the highest fitness value of 94.64%.the model is also classified as good in classifying based on auc value in the range of 80 to 90%. the cost and gamma optimal parameters from svm-ga-fcbf is 18652.00 and 0.0161011. the model can classify the tumor class and normal correctly by 95% which can be known from the constellation value. all classification performance values which include sensitivity, specificity, g-means, and auc values have values above 90%. before conducting fsvm classification, firstlywe have to determine the optimal value of cost and gamma parameter from training data as has been done in svm analysis from the optimal parameter range. the determination of optimal parameters is based on auc value and accuracy. this study uses 10 fold and the highest average for the auc value and the accuracy value in the training data was obtained from the cost value 32768 and gammavalue 0.000061035 with the accuracy value obtained is 98.74% and the auc value is 98.22%. these parameters will be usied in fsvm analysis using data testing. the following classification performance is obtained using the optimal parameter values in training data that is cost value 32768 and gamma value 0.000061035.according to gorunescu 2011, the auc value in the range 60-70 shows a poor classification. this is also supported by the gmeans value of 47.13% which means that the stability between the performance of the classification of minority classes and the majority class is 47.13%. meanwhile for fsvm with fcbf selection, the cost and gamma optimal values are 2048 and 0.125 with an average value of 98.39% accuracy and an average auc value of 97.89%. optimal parameters will be used to find performance testing data classification. classification performance on testing data will be calculated using the optimal parameter value in training data: cost 2048 and gamma 0.125. the auc value obtained is 88.33%, which means that the model is good in classifying. the model can also classify positive or normal classes and negative or tumor classes correctly by sensiticity value is 81.67% and specificity value is 95%. in the classification using fsvm without selection and with ga optimization, optimization will be performed on each fold with a range of costs 215-216 and the range of gamma is 2-3-2-2. the average fitness value value generated for the range cost 215-216 genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 10 and the range gamma is 2-3 2-2 in the training data is 98.22%. average fitness value is used to get the highest auc value from the k-fold cross validation (kcv) method which is a reliable method for predicting errors in a classification. this method is usually used by researchers to reduce the bias that occurs because of sampling data to be used. the average fitness value value for testing data is 86.25%. there are 7 folds that have a fitness value of up to 100% then the cost and gamma optimal parameters of ga optimization results can be taken from one of the seven cost values and gamma values which has a fitness value of 100%. the cost and gamma optimal parameters obtained from fsvm fcbf selection will be optimized using ga optimization. the range used in the fsvm ga optimization fcbf selection is 211-212 for the cost parameter and 2-3-2-2 for the gamma parameters. by using this range of parameters, the fitness value value generated in the training data reaches 98.35% so that the value range 211212 for the cost parameter and 2-3-2-2 for gamma parameters can be used to find the optimal para-meter using data testing. the average fitness value value generated using a range of cost 211-212 and the range of gamma 2-3-2-2 is 97.5%. the value of fitness-value obtained to fold 1 to fold 9 is 100% so that for cost and gamma optimal can be taken from one of the folds that have a fitness value of 100%. the results of svm grid search method and ga for svm optimization without selection have the same accuracy and auc values. this can occur because the variables have not been selected. when compared to svm grid search and svm-ga for optimization, it can be seen that with ga optimization can increase the accuracy and auc value. ga for svm optimization has an auc value of 94.58% and an accuracy value of 95%. based on auc value, the svm-ga optimization classification performance is very good.classification method using fcbf selection has a value higher accuracy than without selection, both for for svm method or fsvm according to a comparison table of classification in table 3. in addition, it is known that the best method that can be used to classify the class of colon cancer is fsvm using fcbf with ga optimization, since it yields the highest auc compared to other methods (that is 97.50%). auc is considered first than accuracy because this study deals with imbalance data. conclusions classification analysis has been carried out using svm and fsvm on colon cancer data thatconsist of 35.5% normal class and 64.5% tumor class. after selecting variables using fcbf, from about 2000 variables are then reduced to 15 variables. the results of svm classification with variable selection give higher accuracy values than the svm without variable selection. in addition, svm ga optimization with variable selection also produces better results than svm without selection with ga optimization. fsvm classification method also produces the same results, where using variable selection produces higher accuracy values than without variable selection. among the four methods used, ga-fsvm classification method produces the highest fitness value compared to other classification methods investigated by this study. the fitness value for fcbf selection is 97.50% and the one without selection is 86.25%. genetic algorithm for variable selection and parameter optimization in svm and fuzzy svm for colon cancer microarray classification irhamah 11 acknowledgments the authors thank to the ministry of research, technology, and higher education, republic of indonesia and institut teknologi sepuluh nopember surabaya indonesia for the financial support under “penelitian dasar unggulan perguruan tinggi” references [1] who, world cancer day : global action to avert 8 million cancer-realted deaths by 2015. retrieved from https://www.who.int/mediacentre/news/releases/2006/pr06/en/, 2002 [2] t. r. golub, d.k. slonim, p. tamayo, c. huard, m. gaasenbeek, & j.p. mesirov, "molecular classification of cancer: class discovery and class prediction by gene expression monitoring", science, 286, 531-537, 1999 [3] m. m. babu, introduction to micoarray data analysis, u.k : horizon press, 2013 [4] t.s. furey, n. cristianini, n. duffy, d. w. bednarski, m. schummer, & d. haussler, "support vector machine classification and validation of cancer tissue samples using 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its, 2018 [18] u. alon, n. barkai, d. a. notterman, k. gish, s. ybarra, d. mack, and a. j. levine, "broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays", procnatlaadsci u s a. jun 8;96(12):6745-6750, 1999 1a sampul depan pendekatan cart untuk mendapatkan faktor yang mempengaruhi terjangkitnya penyakit demam tifoid di aceh utara muhammad sjahid akbar1, dina yuanita2, dan sri harini3 1,2jurusan statistika its 3jurusan matematika, uin malulana malik ibrahim malang e-mail: sri_harini21@yahoo.com abstract typhoid fever is a disease caused by salmonella typhi bacteria. it is attack the digestive tract. typhoid fever caused by poor sanitation and personal hygiene is not good. according to the basic health research in 2007 showed that the prevalence of typhoid fever in indonesia of 1.6%. nad province is hight typhoid fever prevalence(2,96 %). because having traced the biggest contributor was derived from nad. therefore, the research conducted to find factors that influence the outbreak of typhoid fever in nad. research using the cart method. the results of the analysis indicate that the main factor causing typhoid fever was drinking water reservoirs. the other factors are waste water reservoirs, the physical quality of drinking water, a habit washing hands with soap before eating, the bowel, the dump, gender, socioeconomic status, habits of washing hands with soap after defecation and health education. keywords: cart, typhoid fever pendahuluan penyakit demam tifoid seringkali menjadi sebab seseorang harus menjalani rawat inap. demam tifoid atau typhoid fever yang biasa juga disebut typhus atau types oleh orang awam, merupakan penyakit yang disebabkan oleh bakteri salmonella typhi (s. typhi). bakteri s. typhi menyerang bagian saluran pencernaan. puslitbang sistem dan kebijakan kesehatan menyatakan demam tifoid disebabkan pencemaran air minum dan sanitasi yang buruk. demam tifoid adalah penyakit infeksi akut yang menyerang mulai dari usia balita, anak-anak dan dewasa. data world health organization (who) tahun 2003 memperkirakan terdapat sekitar 17 juta kasus demam tifoid di seluruh dunia dengan kejadian 600.000 kasus kematian tiap tahun (anonim, 2008). angka kejadian demam tifoid diketahui lebih tinggi pada negara berkembang khususnya di daerah tropis. sehingga tak heran jika demam tifoid banyak ditemukan di indonesia. hasil riset dasar kesehatan tahun 2007 menunjukkan bahwa persentase penduduk yang terjangkit demam tifoid dibandingkan dengan seluruh penduduk (prevalensi) di indonesia sebesar 1,6%. provinsi nad merupakan prevalensi tifoid tertinggi yaitu sebesar 2,96%. setelah ditelusuri ternyata penyumbang terbesar berasal dari kabupaten aceh utara. oleh karena itu penelitian dilakukan di wilayah aceh utara untuk mendapatkan faktor-faktor yang menyebabkan terjangkitnya penyakit demam tifoid. ada tiga penelitian yang digunakan untuk dasar penelitian ini. tugas akhir nunik hidayati mahasiswa s1 jurusan statistika fmipa its, thesis rahayu lubis mahasiswa pasca sarjana jurusan kesehatan masyarakat di universitas sumatera utara, dan penelitian bambang wasito tjipto peneliti dari puslitbang system dan kebijakan kesehatan. hidayati (2001) memodelkan kasus penyakit demam tifoid di jawa timur dengan menggunakan regresi poisson. asumsi yang harus dipenuhi apabila menggunakan metode regresi poisson adalah variabel dependen harus diskrit dan berdistribusi poisson. ada beberapa faktor resiko yang diduga mempengaruhi terjangkitnya penyakit demam tifoid antara lain kepadatan penduduk, prosentase cakupan penduduk pemakai air bersih, prosentase cakupan penduduk pemakai jamban keluarga, prosentase kondisi rumah yang memenuhi syarat, prosentase cakupan pembuangan sampah sementara yang memenuhi syarat, prosentase cakupan tempat pengolahan makanan yang memenuhi syarat, dan prosentase cakupan penduduk pemakai sarana pembuangan air limbah. hasil dari penelitian hidayati variabel yang mempengaruhi terjangkitnya demam tifoid adalah kepadatan penduduk, prosentase cakupan penduduk pemakai air bersih, prosentase cakupan pembuangan sampah sementara yang memenuhi syarat, prosentase cakupan tempat pengolahan makanan yang memenuhi syarat, dan prosentase cakupan penduduk pemakai sarana pembuangan air limbah. sedangkan penelitian lubis (2007) mempelajari faktor risiko yang muhammad sjahid akbar, dina yuanita, dan sri harini 72 volume 1 no. 2 mei 2010 berhubungan dengan kejadian penyakit demam tifoid pada penderita yang dirawat di rsud dr. soetomo surabaya dengan menggunakan regresi logistik. variabel yang digunakan seperti tingkat pengetahuan, higiene perorangan, kebiasaan makan/minum diluar rumah dan sanitasi lingkungan. hasilnya faktor yang mempengaruhi kejadian penyakit demam tifoid adalah hygiene perorangan dan kualitas air minum. selain itu, tjipto (2009) meneliti faktor-faktor yang berpengaruh terhadap kejadian penyakit demam tifoid pada balita di indonesia dengan analisis multivariate logistik biner. tjipto (2009) menyatakan bahwa demam tifoid erat kaitannya dengan higiene perorangan dan sanitasi lingkungan. hasil penelitian menunjukkan bahwa faktor-faktor yang berpengaruh agar tidak terjadi penyakit infeksi tifoid adalah buang air besar ditempat yang baik (jamban), dan mencuci tangan dengan benar (memakai sabun). tujuan penelitian ini adalah untuk mendapatkan faktor yang mempengaruhi terjangkitnya demam tifoid menggunakan metode classification and regression trees (cart). alasan menggunakan metode cart adalah cart merupakan salah satu metode non parametrik dengan hasil analisis berupa topologi pohon atau berupa grafis sehingga hasil analisis lebih mudah diinterpretasi (lewis dan roger, 2000). data yang digunakan merupakan data sekunder yang diambil dari riskesdas tahun 2007 dan susenas tahun 2007. data dihimpun oleh badan litbangkes departemen kesehatan ri. total sampel art di aceh utara adalah sebanyak 2.491 art. pada penelitian ini data yang digunakan 1816 data art dengan batasan art minimal berusia 10 tahun. variabel respon yang digunakan berskala biner yaitu , 1 untuk anggota rumah tangga terinfeksi demam tifoid dan 2 untuk anggota rumah tangga yang tidak terinfeksi demam tifoid. sedangkan variabel prediktor yang digunakan dalam penelitian ini adalah. asal daerah (x1), jenis kelamin (x2), status sosial ekonomi (x3), kualitas fisik air minum (x4), tempat penampungan air minum(x5), tempat pembuangan sampah(x6), tempat penampungan air limbah (x7), tempat buang air besar (x8), kebiasaan cuci tangan pakai sabun setelah buang air besar (x9), kebiasaan cuci tangan pakai sabun sebelum makan (x10), dan penyuluhan kesehatan (x11). classification and regression trees (cart) classification and regression trees (cart) adalah suatu metode teknik pohon keputusan (breiman et al., 1993). cart menghasilkan suatu pohon klasifikasi jika variabel responnya kategorik, dan menghasilkan pohon regresi jika variabel responnya kontinu. tujuan utama cart adalah untuk mendapatkan suatu kelompok data yang akurat sebagai penciri dari suatu pengklasifikasian. klasifikasi pohon dalam cart melibatkan 4 komponen, yaitu variabel respon, variabel prediktor, data learning, dan data testing. data learning untuk verifikasi model dan data testing untuk validasi model. sebagai ilustrasi struktur pohon klasifikasi dapat dilihat pada gambar 1. simpul utama dinotasikan dengan t1 sedangkan internal nodes (simpul dalam) dinotasikan dengan t2, t3, t4, t7, t9 dan t13. simpul akhir atau simpul terminal adalah t5, t6, t8, t10, t11, t12, t14 dan t15 . penghitungan depth (kedalaman) pohon dimulai dari simpul utama t1 yang berada pada kedalaman 1, sedangkan t2 dan t3 berada pada kedalaman 2 begitu seterusnya sampai pada t14 dan t15 yang berada pada kedalaman 6. pembentukan pohon klasifikasi terdiri atas 3 tahap yang memerlukan learning sample l. tahap pertama adalah pemilihan pemilah. menurut breiman et al. (1993) setiap pemilahan hanya bergantung pada nilai yang berasal dari satu variabel independen. rumus kemungkinan pemilah disajikan sebagai berikut. variabel independen kontinu = 1−n pemilahan variabel independen kategori nominal = 12 1 −−l pemilahan (1) variabel independen kategori ordinal = l 1 pemilahan setelah semua kemungkinan pemilah didapatkan, masing-masing pemilah dicari nilai goodness of split. goodness of split merupakan suatu evaluasi pemilahan oleh pemilah s pada simpul t. goodness of split ���, �� didefinisikan sebagai penurunan keheterogenan. sehingga semakin besar nilai goodness of split semakin homogen simpul anak yang dihasilkan. ���, �� � ∆ ��, �� � ��� �� ���� � �� � (2) pengembangan pohon dilakukan dengan mencari semua kemungkinan pemilah pada simpul �� sehingga ditemukan pemilah s* yang memberikan nilai penurunan keheterogenan tertinggi yaitu, ∆ ���, ��� � ������ ��, ��� (3) dengan ��� adalah fungsi keheterogenan indeks gini, ���, �� adalah kriteria goodness of split, �� ���� adalah proporsi pengamatan dari simpul t menuju simpul kiri, dan � �� � adalah proporsi pengamatan dari simpul t menuju simpul kanan. pendekatan cart untuk mendapatkan faktor yang mempengaruhi… jurnal cauchy – issn: 2086-0382 73 indeks gini sebagai metode pemilahan yang digunakan mempunyai fungsi sebagai berikut. ��� � ∑ �� |����� ���|�� (4) dengan, �� |�� adalah proporsi kelas i pada simpul t, dan ���|�� adalah proporsi kelas j pada simpul t. gambar 1 struktur klasifikasi pohon tahap kedua adalah penentuan simpul terminal (penghentian pembentukan pohon). simpul t dapat dijadikan simpul terminal jika (1) tidak terdapat penurunan keheterogenan yang berarti. (2) hanya terdapat satu pengamatan (n=1) pada tiap simpul anak. (3) adanya batasan minimum n/pengamatan pada simpul anak. dan (4) adanya batasan jumlah level atau tingkat kedalaman pohon maksimal (lewis, 2000). tahap ketiga adalah penandaan label tiap simpul terminal berdasarkan aturan jumlah anggota kelas terbanyak, yaitu: ����|�� � ���� ���|�� � ���� � �!� ��!� (5) dengan ���|�� adalah proporsi kelas j pada simpul t, "���� adalah jumlah pengamatan kelas j pada simpul t, dan "��� adalah jumlah pengamatan pada simpul t. label kelas simpul terminal t adalah �� yang memberi nilai dugaan kesalahan pengklasifikasian simpul t terbesar. setelah terbentuk pohon maksimal tahap selanjutnya adalah pemangkasan pohon untuk mencegah terbentuknya pohon klasifikasi yang berukuran sangat besar dan kompleks. sehingga diperoleh ukuran pohon yang layak berdasarkan cost complexity prunning. besarnya resubtitution estimate pohon t pada parameter kompleksitas # yaitu : $%�&� � $�&� ' %|&(| (6) dengan rα(t) adalah resubtitution suatu pohon t pada kompleksitas α, r(t) adalah resubstitution estimate, α adalah parameter cost complexity bagi penambahan satu simpul akhir pada pohon t, dan |&(| adalah banyaknya simpul terminal pohon t. cost complexity prunning menentukan pohon bagian t(α) yang meminimumkan rα(t) pada seluruh pohon bagian untuk setiap nilai α. nilai parameter kompleksitas α akan secara perlahan meningkat selama proses pemangkasan. selanjutnya pencarian pohon bagian t(α) < tmax yang dapat meminimumkan rα(t) yaitu : $% �&�%�� � )*+&,-./0 $%�&� (7) setelah dilakukan pemangkasan diperoleh pohon klasifikasi optimal yang berukuran sederhana namun memberikan nilai pengganti yang cukup kecil. penduga pengganti yang sering digunakan adalah penduga sampel uji (test sample estimate) dan validasi silang lipat v (cross validation v-fold estimate). menurut breiman et al. (1993) jika jumlah sampel yang digunakan lebih kecil dari 3000 pengamatan penduga pengganti yang digunakan adalah cross validation v-fold estimate. penduga validasi silang lipat v sering digunakan apabila amatan yang ada tidak cukup besar. amatan dalam l dibagi secara acak menjadi v bagian yang saling lepas dengan ukuran kurang lebih sama besar untuk setiap kelasnya. pohon t(v) dibentuk dari l-lv dengan v = 1, 2, ..., v. misalkan d(v)(x) adalah hasil pengklasifikasian. penduga sampel uji untuk r(t1(v)) yaitu 12�342 �5�6 � � 78 ∑ 93:�5���;� < �;6�0=,�=�>�? (8) dengan "@ � "/b adalah jumlah amatan dalam lv. kemudian dilakukan prosedur yang sama kedalaman 1 kedalaman 6 kedalaman 2 2 pemilah 6 t15 t14 1 3 4 pemilah 7 t13 t12 2 3 t8 t9 pemilah 4 t7 t4 t10 t11 3 4 pemilah 5 pemilah 2 pemilah 3 pemilah 1 t1 t3 t2 t5 t6 muhammad sjahid akbar, dina yuanita, dan sri harini 74 volume 1 no. 2 mei 2010 menggunakan seluruh l, maka penduga validasi silang lipat v untuk 42 �5� adalah : 1c8�42� � � 5 ∑ 12�34�8�658�� (9) pohon klasifikasi optimum dipilih t* dengan 1c8�4�� � � d2 1 c8�42� (10) aplikasi dan pembahasan penelitian menggunakan variabel respon kategorik berskala biner. bernilai 1 untuk anggota rumah tangga terinfeksi demam tifoid dan 0 untuk anggota rumah tangga yang tidak terinfeksi demam tifoid, sehingga didapatkan pohon klasifikasi untuk menjelaskan keterkaitan 11 variabel prediktor yang diduga mempengaruhi terjangkitnya penyakit demam tifoid. pada klasifikasi pohon data sampel anggota rumah tangga terjangkit dan tidak terjangkit demam tifoid di aceh utara dibagi menjadi dua kelompok yaitu data learning dan data testing. penelitian ini menggunakan perbandingan data learning 75% (1.362 data) dan testing 25% (454 data). tahap pertama pembentukan pohon klasifikasi maksimal adalah pemilah-pemilah. perhitungan pemilah pada setiap variabel prediktor menggunakan persamaan (1). hasil yang diperoleh adalah variabel asal daerah, variabel jenis kelamin, variabel status sosial ekonomi, variabel kualitas air minum, variabel tempat buang air besar, variabel kebiasaan cuci tangan pakai sabun setelah buang air, variabel kebiasaan cuci tangan pakai sabun sebelum makan, dan variabel keikutsertaan penyuluhan dengan 1 kemungkinan pemilahan. variabel kondisi penampungan air minum dan variabel kondisi tempat pembuangan sampah dengan 3 kemungkinan pemilahan. dan variabel kondisi penampungan air limbah dengan 15 kemungkinan pemilahan. penelitian ini menggunakan metode pemilahan indeks gini sesuai persamaan (4). pemilah terbaik adalah pemilah yang menghasilkan nilai penurunan keheterogenan tertinggi (kriteria pemilahan goodness of split pada persamaan (3)). pemilah terbaik pada simpul 1 (pemilah utama) pada penelitian ini adalah variabel tempat penampungan air (x5). variabel tempat penampungan air terpilih sebagai pemilah utama karena menghasilkan nilai penurunan keheterogenan tertinggi pada simpul 1 (gambar 2). informasi hasil perhitungan penurunan keheterogenan pada setiap pemilah di simpul 1 disajikan pada table 1. tahap kedua yaitu penentuan simpul terminal. simpul t dikatakan sebagai simpul terminal jika tidak terdapat penurunan keheterogenan yang berarti sehingga tidak akan dipilah lagi. simpul terminal adalah simpul yang berwarna merah, biru dan putih. pohon klasifikasi maksimal (maximal tree) dari data anggota rumah tangga yang terjangkit maupun tidak terjangkit demam tifoid ditunjukkan pada gambar 2. tabel 1. nilai penurunan keheterogenan variabel pemilah pada simpul 1 pemilah split ∆ ��, �� x1 1 0,000418599 x2 1 0,000126786 x3 1 0,000714858 x4 1 0,000867109 x5 1,2 0,001200313 x5 1,3 2,08527e-05 x5 2,3 0,000896645 x6 1,2 9,47417e-05 . . . x7 2,3 0,000400087 . . . . . . x8 1 0,000343279 x9 1 0,000118855 x10 1 0,001019946 x11 1 0,00041325 pohon klasifikasi maksimal terdiri dari 89 simpul terminal dengan 15 kedalaman. kedalaman adalah jumlah level atau tingkatan dalam pohon maksimal dimana tiap level terdiri atas beberapa simpul. kedalaman dihitung dari simpul utama sampai simpul terminal (simpul akhir). pohon klasifikasi akan semakin besar jika kedalaman pohon juga semakin besar. tahap ketiga adalah penandaan label kelas. pemberian label kelas untuk setiap simpul terminal berdasarkan rumus pada persamaan (5). perbedaan warna pada tiap simpul terminal menunjukkan adanya perbedaan label kelas. simpul terminal dengan warna biru menunjukkan pada simpul tersebut ditandai dengan label kelas 1 yang berarti anggota rumah tangga terjangkit demam tifoid, dengan persentase jumlah pengamatan yang terjangkit demam tifoid mendekati 100%. warna biru akan berubah secara perlahan menjadi warna putih jika persentase jumlah pengamatan yang terjangkit demam tifoid pada simpul terminal tersebut berkisar 50%. sedangkan untuk simpul terminal berwarna merah menunjukkan label kelas 2 yang berarti anggota rumah tangga tidak pendekatan cart untuk mendapatkan faktor yang mempengaruhi… jurnal cauchy – issn: 2086-0382 75 terjangkit demam tifoid, dimana persentase jumlah pengamatan kelas yang tidak terjangkit demam tifoid pada simpul tersebut mendekati 100%. gambar 2. pohon klasifikasi maksimal tabel 2. kesalahan klasifikasi data learning pada pohon maksimal kelas aktual kelas prediksi total aktual 1 2 1 68 0 68 2 167 1.127 1.294 total prediksi 235 1.127 1.362 benar 1 0,871 total benar 0,877 tabel 2 menunjukkan hasil klasifikasi pohon maksimal untuk data learning. kesalahan klasifikasi terjadi bila data pada kelas aktual 1 (terjangkit demam tifoid) masuk ke dalam kelas prediksi 2 (tidak terjangkit demam tifoid) begitupun sebaliknya. tidak terjadi kesalahan pengklasifikasian pada kelas 1 yang merupakan kelas bagi anggota rumah tangga yang terjangkit demam tifoid. pada kelas 2 (kelas bagi anggota rumah tangga yang tidak terjangkit demam tifoid) terjadi kesalahan pengklasifikasian sebanyak 167 pengamatan. ketepatan klasifikasi untuk data learning pada pohon klasifikasi maksimal adalah sebesar 68 ' 1.127 1.362 � 100% � 87,7% selanjutnya dilakukan pemangkasan pohon klasifikasi maksimal. breiman, et al (1993) menyatakan pemangkasan pohon klasifikasi dilakukan apabila pohon klasifikasi yang terbentuk berukuran sangat besar dan kompleks dalam penggambaran struktur data. sehingga pada akhirnya diperoleh ukuran pohon yang layak dan berdasarkan cost complexity minimum. gambar 3 memberikan informasi bahwa nilai relative cost pohon klasifikasi maksimal lebih besar dibandingkan relative cost pohon klasifikasi optimal. oleh karena itu perlu dilakukan pemangkasan pohon maksimal agar didapatkan nilai relative cost yang paling kecil. garis hijau menunjukkan nilai relative cost minimum pada pohon optimal sebesar 0,599 (persamaan 10). gambar 3. plot relative cost setelah dilakukan pemangkasan terhadap pohon klasifikasi maksimal maka dihasilkan pohon klasifikasi optimal yang memiliki relative costi terkecil dengan 9 kedalaman dan 16 simpul terminal yang disajikan dalam gambar 4 dan spilters pada pohon klasifikasi optimal disajikan pada gambar 5. gambar 4. pohon klasifikasi optimal gambar 5. spilters pohon klasifikasi optimal variabel prediktor yang menjadi pemilah utama pada pohon klasifikasi optimal adalah muhammad sjahid akbar, dina yuanita, dan sri harini 76 volume 1 no. 2 mei 2010 tempat penampungan air minum (x5) dengan skor variabel penting 100. dengan kata lain penampungan air minum merupakan faktor utama yang mempengaruhi anggota rumah tangga terjangkit atau tidak terjangkit demam tifoid. keterangan dari dr. satinta febrianti yang berdinas di rumah sakit yasmin banyuwangi, penyebab seseorang terjangkit demam tifoid adalah bakteri salmonella thypi.penularannya melalui makanan dan minuman yang telah tercemari oleh bakteri salmonella thypi. orang yang kelelahan lebih mudah terjangkit penyakit demam tifoid karena daya tahan tubuhnya menurun. apabila seseorang dengan daya tahan tubuh menurun mengkonsumsi makanan atau minuman yang tercemar oleh bakteri s.thypi maka orang tersebut mudah terjangkit penyakit demam tifoid. hal ini sesuai dengan hasil penelitian ini yang mendghasilkan tempat penampungan air minum sebagai faktor utama yang mempengaruhi terjangkitnya demam tifoid. karena dengan tidak mempunyai tempat penampungn air minum atau tempat penampungan air minum terbuka maka mudah sekali bakteri salmonella thypi mencemari air yang merupakan bahan pokok untuk keperluan sehari-hari. sehingga orang yang tidak mempunyai tempat penampungan air minum atau tempat penampungan air minumnya terbuka lebih rentan terjangkit demam tifoid. selain tempat penampungan air minum variabel yang juga berkontribusi dalam pembentukkan pohon optimal adalah variabel tempat penampungan air limbah (x7) dengan skor 70.61, variabel kualitas fisik air minum (x4) dengan skor 55.23, variabel kebiasaan cuci tangan pakai sabun sebelum makan (x10) dengan skor 48.12, dan variabel tempat buang air besar (x8) dengan skor 40.60. variabel tempat pembuangan sampah(x6), variabel jenis kelamin (x2), dan variabel status sosial ekonomi (x3) juga berkontribusi dalam pembentukan pohon optimal dengan skor variabel penting masingmasing adalah 37.50, 33.80, 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar (x9) dan penyuluhan kesehatan (x11) memiliki skor variabel penting dibawah 20 . simpul utama (simpul 1) dipilah oleh variabel penampungan air minum dengan mengelom-pokkan 931 anggota rumah tangga yang tidak memiliki tempat penampungan air minum dan anggota rumah tangga yang penampungan air minumnya terbuka pada simpul kiri menjadi simpul 2. sisannya yaitu 431 anggota rumah tangga yang tempat penampungan air minumnya terbuka dikelompokkan pada simpul kanan menjadi simpul terminal 16. simpul 2 terdapat 62 anggota rumah tangga yang terjangkit demam tifoid (6,7%) dan 869 anggota rumah tangga yang tidak terjangkit demam tifoid (93,3%). sedangkan simpul terminal 16 terdapat 6 anggota rumah tangga yang terjangkit demam tifoid (1,4%) dan 425 anggota rumah tangga yang tidak terjangkit demam tifoid (98,6%). karena proporsi terbesar pada simpul terminal 16 adalah tidak terjangkit demam tifoid, maka pada simpul terminal 6 diberi label kelas tidak terjangkit demam tifoid (persamaan 5). terjadi kesalahan pengklasifikasian pada simpul terminal 16 dengan label kelas tidak terjangkit demam tifoid, karena terdapat 6 anggota rumah tangga yang dinyatakan terjangkit demam tifoid. proses pemilahan akan terjadi lagi pada simpul 2 namun pada simpul terminal 16 tidak akan terjadi pemilahan. simpul 2 dipilah variabel kebiasaan cuci tangan pakai sabun sebelum makan. sebanyak 663 anggota rumah tangga yang mencuci tangan pakai sabun sebelum makan dipilah pada simpul kiri menjadi simpul 3 dan 268 anggota rumah tangga yang tidak mencuci tangan pakai sabun sebelum makan dipilah pada simpul kanan menjadi simpul 13. pada simpul 3 terdapat 56 anggota rumah tangga yang dinyatakan terjangkit demam tifoid (8,4%) dan 607 anggota rumah tangga yang tidak terjangkit demam tifoid (91,6%). sedangkan pada simpul 13 terdapat 6 anggota rumah tangga yang terjangkit demam tifoid (2,2%) dan 262 anggota rumah tangga yang tidak terjangkit demam tifoid (97,8). pemilahan akan dilakukan terus-menerus sampai simpul terminal. tabel 3 menunjukkan hasil klasifikasi pohon maksimal untuk data learning. kesalahan klasifikasi terjadi bila data pada kelas aktual 1 (terjangkit demam tifoid) masuk ke dalam kelas prediksi 2 (tidak terjangkit demam tifoid) begitupun sebaliknya. jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 17 dari 68 jumlah amatan. jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 199 dari 1.294 jumlah amatan. dengan demikian diperoleh ketepatan pengklasifikasian sebesar 51 ' 1.095 1.362 � 100% � 84,1% tabel 3. ketepatan pohon klasifikasi optimal dari data learning kelas aktual prediksi kelas total aktual 1 2 1 51 17 68 2 199 1.095 1.294 total prediksi 250 1.112 1.362 benar 0,750 0,846 total benar 0,841 jurnal cauchy selanjutnya dilakukan uji validasi. dilakukan validasi adalah untuk mengetahui layak atau tidak model pohon klasifikasi dalam pengklasifikasian data baru. caranya yaitu data testing klasifikasi yang telah terbentuk sebelumnya dari data learning sebesar 25% dari total data keseluruhan yaitu 454 data tabel 4 menunjukkan bahwa data sebanyak 454 pengamatan menghasilkan ketepatan pengklasifikasian sebesar jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 18 dari 37 jumlah amatan. sedangkan jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 47 dari 417 jumlah amatan. karen pada data testing sudah tinggi yaitu 85,7% maka model pohon klasifikasi optimal yang dihasilkan sudah baik. tabel 4. kelas aktual 1 2 total prediksi benar total benar penutup metode klasifikasi optimal dengan ketepatan klasifikasi data learning ketepatan klasifikasi data 85,7%. variabel yang berpengaruh terhadap terjangkitnya penyakit demam tifoid di aceh utara pada pohon optimal adalah variabel tempat penampungan air minum sebagai faktor utama dengan skor tertinggi sebesar 100, tempat penampungan air limba kualitas fisik air minum dengan skor 55.23, kebiasaan cuci tangan pakai sabun sebelum makan dengan skor 48.12, variabel tempat buang air besar dengan skor 40.60, tempat pembuangan sampah dengan skor 37.50, jenis kelamin dengan skor 33.80 dan status sosial ekonomi dengan skor jurnal cauchy – issn: 2086 selanjutnya dilakukan uji validasi. dilakukan validasi adalah untuk mengetahui layak atau tidak model pohon klasifikasi dalam pengklasifikasian data baru. caranya yaitu data dimasukkan kedalam model pohon klasifikasi yang telah terbentuk sebelumnya dari learning. data sebesar 25% dari total data keseluruhan yaitu 454 data. tabel 4 menunjukkan bahwa data sebanyak 454 pengamatan menghasilkan ketepatan pengklasifikasian sebesar jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 18 dari 37 jumlah amatan. sedangkan jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 47 dari 417 jumlah amatan. karen pada data testing sudah tinggi yaitu 85,7% maka model pohon klasifikasi optimal yang dihasilkan sudah baik. tabel 4. ketepatan pohon klasifikasi optimal dari data kelas aktual prediksi kelas 1 19 2 47 total prediksi 66 benar 0,514 total benar 0,857 penutup metode cart klasifikasi optimal dengan ketepatan klasifikasi learning sebesar 84,1%, sedangkan ketepatan klasifikasi data 85,7%. variabel yang berpengaruh terhadap terjangkitnya penyakit demam tifoid di aceh utara pada pohon optimal adalah variabel tempat penampungan air minum sebagai faktor utama dengan skor tertinggi sebesar 100, tempat penampungan air limba kualitas fisik air minum dengan skor 55.23, kebiasaan cuci tangan pakai sabun sebelum makan dengan skor 48.12, variabel tempat buang air besar dengan skor 40.60, tempat pembuangan sampah dengan skor 37.50, jenis kelamin dengan .80 dan status sosial ekonomi dengan skor issn: 2086-0382 selanjutnya dilakukan uji validasi. dilakukan validasi adalah untuk mengetahui layak atau tidak model pohon klasifikasi dalam pengklasifikasian data baru. caranya yaitu data dimasukkan kedalam model pohon klasifikasi yang telah terbentuk sebelumnya dari data testing yang digunakan sebesar 25% dari total data keseluruhan yaitu tabel 4 menunjukkan bahwa data sebanyak 454 pengamatan menghasilkan ketepatan pengklasifikasian sebesar 85,7%. jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 18 dari 37 jumlah amatan. sedangkan jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 47 dari 417 jumlah amatan. karena ketepatan klasifikasi pada data testing sudah tinggi yaitu 85,7% maka model pohon klasifikasi optimal yang dihasilkan ketepatan pohon klasifikasi optimal dari data testing prediksi kelas 1 2 19 18 47 370 66 388 0,514 0,887 0,857 cart menghasilkan pohon klasifikasi optimal dengan ketepatan klasifikasi sebesar 84,1%, sedangkan ketepatan klasifikasi data testing 85,7%. variabel yang berpengaruh terhadap terjangkitnya penyakit demam tifoid di aceh utara pada pohon optimal adalah variabel tempat penampungan air minum sebagai faktor utama dengan skor tertinggi sebesar 100, tempat penampungan air limbah dengan skor 70.61, kualitas fisik air minum dengan skor 55.23, kebiasaan cuci tangan pakai sabun sebelum makan dengan skor 48.12, variabel tempat buang air besar dengan skor 40.60, tempat pembuangan sampah dengan skor 37.50, jenis kelamin dengan .80 dan status sosial ekonomi dengan skor pendekatan cart untuk mendapatkan faktor yang mempengaruhi… 0382 selanjutnya dilakukan uji validasi. tujuan dilakukan validasi adalah untuk mengetahui layak atau tidak model pohon klasifikasi dalam pengklasifikasian data baru. caranya yaitu data dimasukkan kedalam model pohon klasifikasi yang telah terbentuk sebelumnya dari yang digunakan sebesar 25% dari total data keseluruhan yaitu tabel 4 menunjukkan bahwa data testing sebanyak 454 pengamatan menghasilkan ketepatan pengklasifikasian sebesar jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 18 dari 37 jumlah amatan. sedangkan jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 47 dari a ketepatan klasifikasi pada data testing sudah tinggi yaitu 85,7% maka model pohon klasifikasi optimal yang dihasilkan ketepatan pohon klasifikasi optimal total aktual 37 417 454 0,887 menghasilkan pohon klasifikasi optimal dengan ketepatan klasifikasi sebesar 84,1%, sedangkan adalah sebesar 85,7%. variabel yang berpengaruh terhadap terjangkitnya penyakit demam tifoid di aceh utara pada pohon optimal adalah variabel tempat penampungan air minum sebagai faktor utama dengan skor tertinggi sebesar 100, tempat h dengan skor 70.61, kualitas fisik air minum dengan skor 55.23, kebiasaan cuci tangan pakai sabun sebelum makan dengan skor 48.12, variabel tempat buang air besar dengan skor 40.60, tempat pembuangan sampah dengan skor 37.50, jenis kelamin dengan .80 dan status sosial ekonomi dengan skor pendekatan cart untuk mendapatkan faktor yang mempengaruhi… tujuan dilakukan validasi adalah untuk mengetahui layak atau tidak model pohon klasifikasi dalam pengklasifikasian data baru. caranya yaitu data dimasukkan kedalam model pohon klasifikasi yang telah terbentuk sebelumnya dari yang digunakan sebesar 25% dari total data keseluruhan yaitu testing sebanyak 454 pengamatan menghasilkan jumlah kesalahan pengklasifikasian untuk kelas 1 (terjangkit demam tifoid) adalah sebanyak 18 dari 37 jumlah amatan. sedangkan jumlah kesalahan pengklasifikasian untuk kelas 2 (tidak terjangkit demam tifoid) adalah sebanyak 47 dari a ketepatan klasifikasi pada data testing sudah tinggi yaitu 85,7% maka model pohon klasifikasi optimal yang dihasilkan menghasilkan pohon klasifikasi optimal dengan ketepatan klasifikasi sebesar 84,1%, sedangkan adalah sebesar 85,7%. variabel yang berpengaruh terhadap terjangkitnya penyakit demam tifoid di aceh utara pada pohon optimal adalah variabel tempat penampungan air minum sebagai faktor utama dengan skor tertinggi sebesar 100, tempat h dengan skor 70.61, kualitas fisik air minum dengan skor 55.23, kebiasaan cuci tangan pakai sabun sebelum makan dengan skor 48.12, variabel tempat buang air besar dengan skor 40.60, tempat pembuangan sampah dengan skor 37.50, jenis kelamin dengan .80 dan status sosial ekonomi dengan skor 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar dan penyuluhan kesehatan dengan skor variabel penting dibawah 20. daftar pustaka [1] [2] [3] [4] [5] [6] [7] [8] [9] [10 pendekatan cart untuk mendapatkan faktor yang mempengaruhi… 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar dan penyuluhan kesehatan dengan skor variabel penting dibawah 20. daftar pustaka ] anonim. 2007. <http://ummusalma.wordpress.com/2007/ 01/22/helloworld/ september 2009>. ] breiman l, friedman j.h, olshen r.a, dan stone c.j. 1993. trees. chapman and hall. ] departemen kesehatan ri. 2008. kesehatan dasar (laporan nasional 2007) jakarta. ] hidayati, n terhadap faktor mempengaruhi penyakit demam typhoid di provinsi jawa timur”. jurusan statistika fmipa its ] jevuska. 2008. fever), <http://www.jevuska.com/2008/05/ /demam-tifoid 26 september 2009 ] kompas. 2005. waspadai penyakit tipus <http://www.kompas.com/ kompas tanggal akses: 28 agustus2009 ] lewis dan roger j. 2000. to classification and regression trees (cart) analysis ] lubis, r. (200 penyakit demam tifoid penderita yang dirawat di rsud dr. thesis, mahasiswa jurusan ilmu kesehatan masyarakat universitas sumatera utara ] salma, u. 2007. <http//ummusalma.word ,tanggal akses 23 agustus 2009>. 10] steinberg d. dan phillip c. 2005. classification and regression trees salford system, san diego. pendekatan cart untuk mendapatkan faktor yang mempengaruhi… 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar dan penyuluhan kesehatan dengan skor variabel penting dibawah 20. daftar pustaka anonim. 2007. demam tifoid http://ummusalma.wordpress.com/2007/ 01/22/helloworld/, tanggal akses: 27 september 2009>. breiman l, friedman j.h, olshen r.a, dan stone c.j. 1993. classification and regression . chapman and hall. departemen kesehatan ri. 2008. kesehatan dasar (laporan nasional 2007) n. 2001. “analisis regresi poisson terhadap faktor mempengaruhi penyakit demam typhoid di provinsi jawa timur”. jurusan statistika fmipa its jevuska. 2008. demam tifoid (typhoid http://www.jevuska.com/2008/05/ tifoid-typhoid 26 september 2009>. kompas. 2005. masyarakat diminta waspadai penyakit tipus http://www.kompas.com/ kompas tanggal akses: 28 agustus2009 lewis dan roger j. 2000. to classification and regression trees (cart) analysis. presented at the 2000. . (2007). “faktor resiko kejadian penyakit demam tifoid penderita yang dirawat di rsud dr. thesis, mahasiswa jurusan ilmu kesehatan masyarakat universitas sumatera utara. sumatera utara. salma, u. 2007. demam tifoid <http//ummusalma.word ,tanggal akses 23 agustus 2009>. steinberg d. dan phillip c. 2005. fication and regression trees salford system, san diego. pendekatan cart untuk mendapatkan faktor yang mempengaruhi… 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar dan penyuluhan kesehatan dengan skor variabel demam tifoid, http://ummusalma.wordpress.com/2007/ tanggal akses: 27 breiman l, friedman j.h, olshen r.a, dan classification and regression . chapman and hall. new york. departemen kesehatan ri. 2008. kesehatan dasar (laporan nasional 2007) “analisis regresi poisson terhadap faktor-faktor yang mempengaruhi penyakit demam typhoid di provinsi jawa timur”. skripsi, mahasiswa jurusan statistika fmipa its. surabaya. demam tifoid (typhoid http://www.jevuska.com/2008/05/ typhoid-fever, tanggal akses: masyarakat diminta waspadai penyakit tipus http://www.kompas.com/ kompas tanggal akses: 28 agustus2009>. lewis dan roger j. 2000. an introduction to classification and regression trees . presented at the 2000. faktor resiko kejadian penyakit demam tifoid penderita yang dirawat di rsud dr. soetomo surabaya thesis, mahasiswa jurusan ilmu kesehatan masyarakat universitas . sumatera utara. demam tifoid, <http//ummusalma.word-press.com ,tanggal akses 23 agustus 2009>. steinberg d. dan phillip c. 2005. fication and regression trees salford system, san diego. pendekatan cart untuk mendapatkan faktor yang mempengaruhi… 77 22.09. sedangkan variabel kebiasaan cuci tangan pakai sabun setelah buang air besar dan penyuluhan kesehatan dengan skor variabel http://ummusalma.wordpress.com/2007/ tanggal akses: 27 breiman l, friedman j.h, olshen r.a, dan classification and regression new york. departemen kesehatan ri. 2008. riset kesehatan dasar (laporan nasional 2007). “analisis regresi poisson faktor yang mempengaruhi penyakit demam typhoid di skripsi, mahasiswa . surabaya. demam tifoid (typhoid http://www.jevuska.com/2008/05/10fever, tanggal akses: masyarakat diminta waspadai penyakit tipus, http://www.kompas.com/ kompas-cetak/, an introduction to classification and regression trees . presented at the 2000. faktor resiko kejadian penyakit demam tifoid penderita yang soetomo surabaya”. thesis, mahasiswa jurusan ilmu kesehatan masyarakat universitas press.com steinberg d. dan phillip c. 2005. cart– fication and regression trees. ca: 4. imam fahcruddin positifitas dan ketercapaian positifitas dan ketercapaian sistem linier fractional waktu kontinu imam fahcruddin mahasiswa progam studi s2 matematika fmipa universitas gadjah mada yogyakarta e-mail: fahrudinuin@gmail.com abstract this paper studies a solution of the fractional continuous-time linier system. necessary and sufficient condition were established for the internal and external positivity of fractional systems. sufficient conditions are given for the reachability of fractional positive systems. keywords: fractional systems, positive systems, reachability pendahuluan sudah menjadi familiar, ketika fenomena dalam kehidupan sehari-hari sering dimodelkan dalam persamaan differensial. beberapa model matematika yang telah dipublikasikan, berupa sistem persamaan differensial dengan orde n, � � �. oleh sebab itu, sebagian besar pemerhati matematika sudah familiar dengan notasi-notasi ����� atau ��� � , � ��� atau ���� �� , bahkan ����� atau ���� �� ,� � �. lantas, bagaimana saat orde dari persamaan differensial tersebut bukan bilangan bulat, tetapi real bahkan kompleks, seperti ����� ��� atau ������ ? lebih lanjut, bagaimana aplikasi dari persamaan differensial tersebut, apakah hanya sekedar fantasi matematika saja. fractional calculus merupakan salah satu topik dalam matematika yang mengkaji integral atau differensial dengan sebarang orde (podlubny, 1999 : 42). pembahasan mengenai fractional calculus mulai dibahas secara rinci oleh leibnitz pada abad ke-18, kemudian dikembangkan lagi oleh l’hospital, euler, lagrange, laplace, riemann, fourier, liouville, dan caputo. seiring dengan perkembangan sains dan teknologi, kompleksitas dari kejadian alam yang dialami tak bisa dipungkiri lagi. hal ini menyebabkan adanya sub-sub sistem dalam sebuah model perlu dikaji lebih dalam. dalam berbagai aplikasi pemodelan, sering dihadapkan pada permasalahan positifitas. yang berarti variabel input, output, bahkan variabel state diinginkan bernilai tidak negatif. seperti dalam model ekonomi, misalnya kuantitas dari investasi dan pajak selalu bernilai positif. selain itu pemodelan dari masalah transpor polutan dan populasi juga memerlukan variabel-variabel yang bernilai positif. sehingga penelitian mengenai positifitas pada sistem matematika berkembang pesat dewasa ini. pada paper ini, dibicarakan dua aspek penting terkait sistem linier fractional waktu kontinu, yaitu dari sisi kuantitatif maupun kualitatif. mengenai aspek kuantitatif akan dibicarakan solusi dari sistem, sedangkan aspek kualitatif akan dikaji positifitas dan ketercapaian pada sistem tersebut. dalam pembahasannya, disertai beberapa contoh aplikatif tentang sirkuit elektronika, yang didasarkan pada hasil penelitian dzieliński. beberapa notasi yang digunakan dalam paper ini, diantaranya himpunan matriks nxm atas bilangan riil dinotasikan ����, dan �� � ����. himpunan matriks mxn atas bilangan riil yang entri-entrinya nonnegatif dinotasikan �����, dan ��� � �����. suatu matriks a dengan entri nonnegatif dinotasikan dengan � � 0, dan matriks identitas nxn dinotasikan dengan ��. kajian teori pada subbab ini, terlebih dahulu akan dipaparkan beberapa alat-alat yang digunakan untuk menyelesaikan solusi dari sistem linier fractional waktu kontinu. definisi 1 (chen, 1984 : 56) diberikan fungsi f terdefinisi untuk 0 � � � ∞. transformasi laplace dari f, dinotasikan dengan atau !"��� #, didefinisikan oleh �$ % !"��� # % & '()*��� +�,∞, (1) asalkan integral (1) ada untuk setiap s yang lebih besar atau sama dengan suatu nilai $,. dari definisi diatas, dapat dihasilkan beberapa sifat transformasi laplace, seperti sifat konvolusi, yaitu: jika ��$ % !"���� #, �$ % !"� �� # maka ! -& ���� . / � �/ +/*, 0 % ��$ �$ . positifitas dan ketercapaian sistem linier fractional waktu kontinu jurnal cauchy – issn: 2086-0382 19 kemudian, berikut ini dipaparkan beberapa fungsi yang berperan penting dalam pembahasan sistem linier fractional, yaitu fungsi gamma yang menggeneralisasi n! ke bilangan tidak bulat, bahkan rasional dan fungsi mittag-leffler yang merupakan generalisasi dari fungsi eksponensial ')1*, yang berguna untuk mendapatkan solusi dari sistem tersebut. definisi 2 (podlubny, 1999 : 1) fungsi gamma yang dinotasikan dengan 2�� didefinisikan oleh integral 2�� % 3 ��(�'(*+�, � 4 0.∞, definisi 3 (podlubny, 1999 : 16) suatu fungsi kompleks z yang didefinisikan 56�7 % 8 792�:; < 1 , ∞ 9>, disebut sebagai fungsi mittag-leffler dengan satu parameter. dalam paper ini, definisi fractional derivative yang digunakan adalah definisi yang dikembangkan oleh caputo, yaitu: definisi 4 (kaczorek, 2011 : 30) diberikan ?'�: � �� . 1,� dengan � � �, dan ��� merupakan fungsi differensiabel sampai ke-n. derivative fractional caputo dengan orde : � �, dinotasikan dengan �6, didefinisikan sebagai �6��� % +6+�6 ��� % 12�� . : 3 ��� �/ +/�� . / 6��(� ,*, untuk � . 1 � : � �, dengan 2�� % & ��(�'(*+�∞, , ?'�� 4 0 adalah fungsi gamma dan ��� �/ % ���@ @� merupakan persamaan differensial dengan orde �. dari definisi 1, dapat diperoleh hasil transformasi laplace dari derivative fractional caputo dengan orde : � �, yang disajikan dalam teorema dibawah ini: teorema 5 (kaczorek, 2011 : 31) transformasi laplace dari derivative fractional caputo diberikan oleh !"�6��� # % $6 �$ . 8 $6(9��9(� �0� �9>� . bukti dengan menggunakan sifat konvolusi pada transformasi laplace, diperoleh !"�6��� # % ! a 12�� . : 3 ��� �/ +/�� . / 6��(�*, b % 12�� . : !"��(6(�# !"���� # % 12�� . : 2�� . : $�(6 c$� �$ . 8 $�(9��9(� �0� �9>� d % $6 �$ . 8 $6(9��9(� �0� �9>� . karena terkait dengan sistem, maka ada beberapa jenis matriks yang mempunyai karakter khusus dan menjadi syarat positifitas dan ketercapaian pada sistem linier fractional waktu kontinu, diantaranya adalah matriks metzler dan matriks monomial. definisi 6 (kaczorek, 2008 : 225) suatu matriks persegi atas bilangan rill � % efghi dikatakan matriks metsler jika entrientri yang bukan pada diagonal utamanya nonnegatif, i.e. fgh � 0 untuk j k l. definisi 7 (kaczorek, 2008 : 226) suatu matriks persegi atas bilangan riil dikatakan monomial jika dan hanya jika setiap baris dan kolom dari entri matriks tersebut hanya memuat satu entri positif dan entri yang lain bernilai 0. pembahasan solusi sistem linier fractional waktu kontinu pandang sistem linier fractional waktu kontinu yang diberikan dengan sistem linier berikut ini: �6��� % ���� < mn�� , 0 � : � 1, (8a) o�� % p��� < �n�� , (8b) dengan ��� � �� merupakan variabel state,n�� � �� variabel input, o�� � �q variabel output dan � � ����, m � �r��,p ��q�r,� � �q��. teorema 9 (kaczorek, 2008 : 224) solusi dari sistem (8a) diberikan oleh ��� % φ,�� �, < 3 φ�� . / mn�τ +/,*, ��0 % �,, (10) dengan φ,�� % 56���6 % 8 �9�962�;: < 1 , ∞ 9>, �11 φ�� % 8 �9��9�� 6(�2"�; < 1 :#∞9>, , �12 imam fahcruddin 20 volume 2 no. 1 november 2011 dimana 56���6 adalah fungsi matriks mittage-leffler, 2�� adalah fungsi gamma. bukti dengan menerapkan transformasi laplace pada (8a), diperoleh !"�6��� # = !"���� < mn�� #= !"���� # < !"mn�� # (13) berdasarkan definisi 1 dan teorema 5, hasil transformasi laplace pada (8a) adalah !"�6��� # % $6t�$ . $6(��,, (14) t�$ % !"��� # % & ��� '()*+�,∞, (15) hasil subsitusi (14) ke (13), diperoleh $6t�$ . $6(��, % �t�$ < mu�$ , (16) t�$ % "�r$6 . �#(�v$6(��, < mu�$ w, (17) karena "�r$6 . �#v∑ �9$(�9�� 6∞9>, w % �r, sehingga "�r$6 . �#(� % v∑ �9$(�9�� 6∞9>, w. kemudian disubsitusikan ke (17), diperoleh t�$ % y8 �9$(�9�� 6∞9>, zv$6(��, < mu�$ w % 8 �9$(�96�� �, < 8 �9$(�9�� 6mu�$ ∞9>, ∞ 9>, kemudian hasil konvolusi dan invers transformasi laplace pada persamaan diatas adalah ��� % !(�"t�$ # % 8 �9!(�e$(�9�� 6i∞9>, �, <8 �9!(�e$(�9�� 6mu�$ i∞9>, % 8 �9�962�;: < 1 [ 9>, �, <8 �9!(�e$(�9�� 6i!(�"mu�$ #[9>, % φ,�� �, < 3 φ�� . / mn�τ +/*, dengan φ,�� % 56���6 % 8 �9�962�;: < 1 , ∞ 9>, φ�� % !(�"�r$6 . �# % 8 �9!(�e$(�9�� 6i∞9>, % 8 �9��9�� 6(�2"�; < 1 :#∞9>, . catatan 1. sama halnya dengan derivative orde 1, dari (11) dan (12), untuk : % 1, diperoleh φ,�� % φ�� % 8 ��� 92�; < 1 % '\*. ∞ 9>, perhatikan sirkuit elektronik yang terdiri dari resistor, superkondensator, kumparan, dan sumber tegangan (arus). dipilih variabel state yang diinginkan adalah jarak lintas (across) sumber tegangan dengan superkondensator dan arus dalam kumparan. diketahui bahwa hubungan antara arus j]�� dalam superkondensator dengan sumber tegangan n^�� (dzieliński, 2009) adalah j]�� % p _`a�* *_ untuk � 0: � 1, (18) dengan c merupakan kapasitas dari superkondensator. sama halnya, hubungan tegangan nb�� pada kumparan dengan arus jb�� adalah nb�� % c d1e�* *d untuk 0 � f � 1, (19) dengan l merupakan inductance dari kumparan. dengan menggunakan hubungan (18) dan (19) dan hukum kirchhoff`s maka sirkuit linier fractional memenuhi persamaan state: gh hi+ 6�]+�6+j�b+�j kl lm % n��� �� � � � o-�]�b0 < nm�m o', dengan komponen �] � ?�� merupakan jarak lintas tegangan dengan superkondensator, kemudian �b � ?�� merupakan arus dalam kumparan, dan ' � ?� adalah tegangan dari sirkuit. contoh. perhatikan sirkuit linier elektronika pada figur a1 yang terdiri dari resistor ?�, ? , ?p, kapasitor p�,p , induktor c�,c dan sumber tegangan '�,' . menurut (18) dan (19), dan hukum kirchhoff, sirkuit linier fractional dari figur a1 memenuhi persamaan state: j� % p� +6�n�+�6� , j % p +6�n +�6� '� % �?� < ? j� < c� +j�j�+�j� < n� . ?pj , ' % �? < ?p j < c +j�j +�j� < n . ?pj�. gambar 1. contoh sirkui elektronik positifitas dan ketercapaian sistem linier fractional waktu kontinu jurnal cauchy – issn: 2086-0382 21 persamaan state di atas dapat dibentuk: gh hh hh i _�`� *_� _�`� *_� d�g� *d� d�g� *d� kl ll ll m % �q n�n j�j r < m '�' 0, dengan � % gh hh hh i 0 0 �]� 00 0 0 �]�. �b� 0 . s��stb� stb�0 . �b� stb� . s��stb� kl ll ll m , m % gh hh i0 00 0�b� 00 �b�kl ll m . menurut teorema 9, solusi dari sistem linier fractional pada sirkuit elektronik diatas dapat diselesaikan dengan mudah. positifitas sistem linier fractional waktu kontinu diawal tadi telah disinggung tentang konsep dasar dari sistem positif. pembahasan mengenai positifitas dari sistem linier fractional waktu kontinu dibagi menjadi dua aspek, yaitu positif internal dan positif eksternal. definisi 19 (kaczorek, 2008 : 225) sistem linier fractional (8) dikatakan positif internal (internally positive) jika dan hanya jika ��� � ��r dan o�� � ��u untuk � � 0 untuk sebarang initial state �, � ��r dan semua input n�� � ���, � � 0. dari definsi 19, mengindikasikan bahwa variabel state dan output dari sistem (8) diharapkan selalu bernilai nonnegatif untuk � � 0 . syarat ini diperlukan agar terjadi kesesuaian antara model yang diteliti dengan fakta riil yang terjadi. oleh karena itu, perlu adanya dilakukan karakterisasi sehingga dapat dihasilkan suatu kondisi yang dibutuhkan agar ��� � ��r untuk sebarang initial state �, � ��r dan semua input n�� � ���, � � 0. lemma 20 diberikan � � ?r�r, � � 0 dan 0 � : � 1. maka φ,�� % 8 �9�962�;: < 1 � ��r�r ∞ 9>, dan φ�� % 8 �9��9�� 6(�2"�; < 1 :#∞9>, � ��r�r jika dan hanya jika a merupakan matriks metsler. bukti �y . dari ekspansi φ,�� % �r < �2�: < 1 < z , φ�� % �r ��6(� 2�: < � � 6(�2�2: < z jika a matriks metzler , jelas bahwa φ,�� � ��r�r dan φ�� � ��r�r untuk bilangan kecil � 4 0. �{ . telah diketahui bahwa '\* � ��r�r untuk � � 0 (21) jika dan hanya jika a merupakan matriks metzler (kaczorek, 2002). sehingga φ,�� % 8 | ���6 92�;: < 1 . ���6 9;! ~ ∞ 9>, % 8 |;! . 2�;: < 1 2�;: < 1 ~|���6 9;! ~ � 0 ∞ 9>, untuk � � 0. (22) karena ;! � 2�;: < 1 untuk 0 � : � 1. kemudian dari (22) dan (21) diperoleh φ,�� � '\*_ � 0 untuk � � 0. begitupun sebaliknya untuk kasus φ�� . berikut ini akan disajikan syarat cukup dan perlu suatu sistem linier fractional waktu kontinu dapat dikatakan positif internal: teorema 23 sistem fractional derivative waktu kontinu (2) merupakan positif internal jika dan hanya jika matriks a adalah matriks metzler dan m � ��r��, p � ��q�r, � � ��q��. bukti (y). dari teorema 1, diketahui bahwa solusi dari (8a) dinyatakan dalam bentuk (10). karena matriks a pada sistem (8a) adalah matriks metzler, dari lemma 20 disimpulkan ��� � ��r. lebih lanjut, apabila m � ��r��, p � ��q�r, � � ��q�� dari (28), disimpulkan o�� � ��u untuk semua n�� � ���, � � 0. ({). dipilih n�� % 0,� � 0 dan �, % 'g (kolom ke-i dari matriks identitas ��). trayektori dari sistem (8a) akan berada pada orthan ��� hanya jika �6��0 % �'g � 0, yang berakibat fgh � 0 untuk j k l. yang berarti a merupakan matriks metzler. dengan analogi yang sama, untuk �, % 0 diperoleh �6��0 % mn�0 � 0, yang berakibat m � ��r�� untuk sebarang n�0 � ���. dari (8b), untuk n�� % 0,� � 0 diperoleh o�0 % p�, � 0 yang berakibat p � ��q�r untuk sebarang �, � ��r. kemudian asumsikan �, % 0, dari (8b) diperoleh o�0 % �n�0 � 0 yang berakibat � � ��q�� untuk sebarang n�0 � ���. dalam menganalisa suatu sistem, terkadang cukup diteliti variabel input dan kondisi awalnya saja, untuk menjustifikasi positivitas dari varibel outputnya. berikut ini disajikan definisi dari positif eksternal, serta matriks impulse respon. definisi 24 (kaczorek, 2008 : 226) sistem linier fractional waktu kontinu (8) dikatakan positif eksternal (externally imam fahcruddin 22 volume 2 no. 1 november 2011 positive) jika untuk semua input n�� ����, � � 0 dan �, % 0 maka variabel output o�� � ��u , � � 0. definisi 25 (kaczorek, 2011 : 36) output dari sistem fractional single-input single-output (siso) dengan kondisi awal 0, untuk dirac impulse n�� % ��� disebut impulse respon dari sistem. lemma 26 matriks impulse respon dari sistem (8) dinotasikan dengan ��� � �q�� adalah ��� % pφ�� m < ���� , untuk � � 0 . (27) bukti subsitusikan (3) ke (2b) sehingga untuk �, % 0, n�� % ��� , dan o�� % ��� diperoleh o�� % 3pφ�� . / mn�/ +/ < �n�� * , % pφ�� m < ���� ,untuk � � 0. syarat cukup dan perlu agar sistem linier fractional waktu kontinu dikatakan positif eksternal disajikan dalam teorema dibawah ini: teorema 28 (kaczorek, 2008 : 226) sistem linier fractional waktu kontinu (8) adalah positif eksternal (externally positive) jika dan hanya jika matriks impulse respon (27) nonnegative, i.e. ��� � ��q��, � � 0. bukti karena matriks impulse respon pada sistem (8) telah diketahui, maka untuk sebarang input n�� , variabel output o�� dapat dinyatakan dalam bentuk o�� % 3���,/ n�/ +/* , , �29 dengan perhitungan langsung ataupun metode grafik. dari lemma 26, variabel output o�� dari sistem (8) dapat dinyatakan dalam bentuk (29), yaitu: o�� % 3��� . / n�/ +/. * , apabila kondisi ��� � ��q�� untuk � � 0 dipenuhi, maka dari (29) disimpulkan o�� � ��u , � � 0 untuk setiap input n�� ����, � � 0. reachability sistem linier fractional waktu kontinu terkadang untuk mendesain suatu sistem, diperlukan suatu varibel input yang mengendalikan state awal, menuju state yang dikehendaki. hal ini perlu dilakukan, terkait sistem yang telah terbentuk, apakah sudah benarbenar optimal dalam aspek tertentu. oleh karena itu, berikut ini disajikan definisi sistem yang reachable. definisi 29 (kaczorek, 2008 : 226) state �� � ?�r dari sistem linier fractional (8) dikatakan reachable (dapat dicapai) pada waktu �� jika terdapat suatu input n�� � ���, � � e0,��i yang mengendalikan state pada sistem (8), mulai dari state awal yang bernilai 0, i.e. �, % 0 ke ��. jika setiap state �� � ?�r reachable pada waktu ��, maka sistem dikatakan tercapai pada waktu ��. jika untuk setiap state �� � ?�r terdapat waktu �� sedemikian hingga state dalam keadaan reachable, maka sistem (8) dikatakan reachable. mengenai syarat cukup dan perlu sistem (8) dikatakan reachable dipaparkan dalam beberapa teorema dibawah ini: teorema 30 (kaczorek, 2008 : 226) sistem linier fractional dengan waktu kontinu (8) adalah reachable pada waktu �� jika matriks ?v��w % 3 φ�/ *, mm�φ��/ +/ �31 adalah matriks monomial. lebih lanjut, variabel input yang mengendalikan state pada sistem (8) dari �, % 0 menuju ��, yaitu n�� % m�φ�v�� . �w?(�v��w��. (32) bukti diketahui bahwa matriks (31) merupakan matriks monomial, maka ?(�v��w � ��r�r dan variabel input yang didefinisikan (32) merupakan vektor non-negatif, i.e. n�� � ?��, � � 0. dari (10) untuk �, % 0, � % ��, (31) dan (32) diperoleh �v��w % 3 φv�� . /wmm�φ�v�� . /w+/?(�v��w�� *� , % 3 φ�/ mm�φ��/ +/?(�v��w�� *� , % ?v��w?(�v��w�� % ��. terlihat variabel input (32) mengendalikan state dari sistem (8) dari �, % 0 menuju ��. teorema 33 (kaczorek, 2008 : 227) jika � % +jf�"f�, f ,… ,fr# � ��r�r, m � ��r�� adalah matriks monomial, maka sistem fractional derivative (8) adalah reachable. bukti dari persamaan (12), apabila matriks a adalah matriks diagonal, jelas bahwa matriks φ�� dan matriks φ�� m adalah monomial. persamaan (31) juga dapat ditulis dalam bentuk ?v��w % 3 φ�/ *, m�φ�/ m �+/, �34 positifitas dan ketercapaian sistem linier fractional waktu kontinu jurnal cauchy – issn: 2086-0382 23 dan juga merupakan matriks monomial. dengan demikian, dari teorema 30, dapat disimpulkan bahwa sistem linier fractional (8) adalah reachable. penutup berdasarkan pembahasan, diperoleh kesimpulan bahwa bentuk umum solusi dari sistem linier fractional waktu kontinu diperoleh dari transformasi laplace (teorema 9). syarat cukup dan perlu suatu sistem fractional dapat dikatakan internal positif atau eksternal positif telah diberikan pada teorema 23 dan 28. syarat cukup sistem linier fractional waktu kontinu dikatakan reachable disajikan pada teorema 33. daftar pustaka [1] a. dzieliński, d. sierociuk, and g. sarwas,. (2009). ultracapacitor parameters identification based on fractional order model. proc ecc’09 1. cd-rom (2009). [2] chen, c. t. (1984). linier system theory and design. new york: cbs college publishing. [3] kaczorek, t. (2002). positive 1d and 2d systems. london: springer-verlag. [4] kaczorek, t. (2008). fractional positive continuous-time linier systems and their reachability. int. j. appl. math. comput. sci., 2008, vol. 18, no. 2, 223-228. doi: 10.2478/v10006-008-0020-0. [5] kaczorek, t. (2011). selected problems of fractional systems theory. berlin heidelberg : springer-verlag. [6] podlubny, i. (1999). fractional differential equations. london: academic pres 1a sampul depan membatasi k-ketenggaan simpul dalam pembangkitan random graph metode erdos royi untuk meningkatkan kinerja komputasi zainal abidin1 dan agus zainal arifin2 1 jurusan teknik informatika fakultas sains dan teknologi uin maulana malik ibrahim malang. 2 jurusan informatika fakultas teknologi informatika institut teknologi sepuluh nopember keputih, sukolilo, surabaya, indonesia e-mail : 1 br52s@cs.its.ac.id, 2agusza@cs.its.ac.id abstract edges generation by random graph erdos-royi methods was needed high computation, it’s caused low performance. in fact, edge generation was used frequently with many nodes. this paper is described a node restriction by k-nearest neighbour on edge generation of random graph erdos royi method. result of node restriction by k-nearest neighbour can be reduced computation time. keywords: random graph, erdos royi, k-nearest neighbour, computation time. pendahuluan graph adalah himpunan simpul dan busur yang menghubungkan semua atau beberapa simpul (diestel, 2000). graph dengan anggota simpul-simpul tidak saling terhubung antar satu dan simpul lain disebut graph terisolasi. pembangkitan busur dari graph terisolasi sering digunakan sebagai sarana untuk simulasi. simulasi graph sering digunakan untuk mengetahui hubungan antar penulis dan pembimbing dalam melakukan aktifitas penelitian (newman, 2001a). graph yang terbentuk bisa diketahui seberapa banyak seorang peneliti melakukan pembimbingan. banyaknya bimbingan dapat diketahui dari jumlah busur yang terhubung keluar dari simpul peneliti dan bobotnya. graph dipakai untuk melihat karya peneliti terbaik (newman, 2001b). karya tulis dianggap sebauh simpul kemudian dibangkitkan busur kearahnya jika ada karya tulis yang mengacu pada karya tulis tersebut. graph digunakan untuk mengetahui seberan dari sel tumor otak (gunduz, 2004; demir 2005). sel-sel dari otak dianggap sebagai suatu simpul. simpul-simpul yang diperoleh dibangkitkan busur-busur menggunakan suatu probabilitas. probilitas digunakan untuk menentukan batas apakah suatu simpul terhubung atau tidak terhubung dengan simpul yang lain. graph dipakai mengukur dan menganalisa kerapatan (abidin dan arifin, 2008; abidin dan arifin, 2009). simpul-simpul yang digunakan untuk membangkitkan busur berjumlah ribuan. simpul digunakan dalam deteksi sebaran sel tumor sejumlah sekitar 4000 buah (gunduz, 2004; demir 2005). simpul digunakan dalam analisa kerapatan berjumlah antara 2000 sampai dengan sekitar 7000 buah (abidin dan arifin, 2008; abidin dan arifin, 2009). random graph dengan metode erdos royi adalah membangkitkan graph dari graph terisolasi dengan membangkitkan busur dari setiap simpul dengan semua simpul dengan suatu batasan sebuah probabilitas (watts dan strogatz, 1998). menghubungkan setiap simpul ke semua simpul yang lain memerlukan komputasi yang besar. persamaan 1 merupakan jumlah busur b yang mungkin terbentuk dengan jumlah simpul n. )1(21 −= nnb . (1) komputasi besar membuat kinerja komputer jadi rendah. prosesor dari komputer jadi sibuk. pembangkitan random graph metode erdos royi masih dapat ditingkatkan kinerjanya. pembatasan dalam bentuk probalitas antar simpul dapat digunakan sebagai dasar peningkatan kinerja. jika pembangkitan dibatasi dengan probablitas, maka tidak perlu setiap simpul dicoba dibangkitan busur dan dihitung probablitasnya, tetapi mungkin hanya perlu dicoba pada simpul-simpul terdekatnya saja. sejumlah n simpul terdekat atau ketetanggaan diberi notasi n-ketetangggan. dalam penelitian ini menjelaskan tentang pembatasan simpul sejumlah n-ketetanggan untuk peningkatan kinerja pada pembangkitan graph dengan metode erdos dan royi. zainal abidin dan agus zainal arifin 98 volume 1 no. 2 mei 2010 kajian graph a. dasar-dasar graph graph (g) adalah himpunan simpul (vertex, v) dan busur (edge, e), ditulis dengan g=(v, e) (reinhard diestel, 2000). graph ditampilkan dalam titik dan garis. titik adalah lambang dari simpul. garis merupakan lambang dari busur yang menghubungkan antara dua simpul. ukuran (size, order) graph g, |g| adalah jumlah simpul yang menjadi anggota himpunan dari graph, walaupun simpul tersebut tidak dihubungkan oleh suatu busur. jumlah busur dituliskan dengan ||g||. gambar 1, contoh graph dengan ukuran 7. simpul nomor enam tidak terhubung ke simpul yang lain. graph pada gambar 1 dapat ditulis dalam bentuk himpunan simpul v dan busur e, misal, v = {1, 2, 3, 4, 5, 6, 7}, e={{1,2}, {1,5}, {2,5},{3,4}, {5,7}}. busur {x,y} dapat tulis dengan busur xy atau yx. simpul x dan y dari graph g dikatakan saling berketetanggaan, jika xy adalah busur graph g. jika semua simpul dalam graph g saling berpasangan satu sama lain, maka graph g disebut komplet (complete) dituliskan dengan kn, dimana n adalah jumlah dari simpul. k-ketetanggaan terdekat (k-nn) dari simpul i bisa diperoleh dengan menarik sebuah lingkaran dengan berpusat pada simpul i sampai diperoleh k simpul lain yang berada dalam lingkaran. gambar 2, 3-ketetanggaan terdekat dari simpul a adalah tiga simpul, yaitu simpul b, c, dan d. 7-ketetanggaan terdekat dari simpul a diperoleh dengan memperpanjang jari-jari lingkaran sampai diperoleh 7 simpul yang berada dalam lingkaran, yaitu simpul b, c, d, e, f, g, dan h. dua simpul (i dan j) bukan anggota dari 7ketetanggaan terdekat dari simpul a, karena berada diluar lingkaran. graph terdiri dari berbagai jenis (newman, 2003), yaitu graph berarah, tak berarah, berbobot, dan tak berbobot. graph tak berarah (undirected graph) adalah pasangan busur xy sama dengan pasangan busur yx. busur tersebut dikatakan sebagai busur tak berarah. simpul x dan y disebut sebagai titik akhir (endpoint). sebuah graph g disebut graph tak berarah jika setiap busurnya terhubung tak berarah (levitin, 2005). jika pasangan busur xy tidak sama dengan busur yx, maka busur tersebut disebut sebagai busur berarah. busur xy meninggalkan x menuju y disebut juga x sebagai ekor, dan y sebagai kepala. graph g disebut sebagai graph berarah jika semua busur terhubung secara berarah (levitin, 2005). gambar 3a adalah contoh penggambaran dari graph tak berarah. busur yang menghubungkan antar simpul tidak mempunyai tanda arah. gambar 3b adalah contoh graph berarah. busur antar simpul pada gambar 3b mempunyai simbol anak panah yang menunjukkan arah hubungan dari simpul ekor ke simpul kepala. simpul c dan f mempunyai dua busur, yaitu cf dan fc. gambar 1. graph dengan tujuh simpul dan empat busur (diestel, 2000). gambar 2. ilustrasi k-ketetanggaan terdekat a b gambar 3. graph tidak berarah dan berarah. (a) graph tak berarah, (b) graph berarah (levitin, 2005) untuk kepentingan suatu komputasi atau sebuah algoritma, secara umum graph dapat digambarkan dengan bentuk matriks, disebut sebagai matriks adjacency. matriks adjacency dari graph dengan ukuran n adalah matriks n x n. setiap elemen dari matriks mewakili satu busur dari graph. elemen baris ke i dan kolom ke j bernilai satu jika simpul ke i terhubung dengan simpul ke j. elemen baris ke i dan kolom ke j bernilai nol jika simpul ke i tidak terhubung dengan simpul ke j (levitin, 2005). gambar 4 membatasi k-ketetanggaan simpul dalam pembangkitan random graph… jurnal cauchy – issn: 2086-0382 99 merupakan adjacency matriks dari graph tak berarah pada gambar 3a. graph tak berarah mempunyai matriks adjacency yang simetris. graph berbobot (weigthed graph) merupakan graph dengan suatu nilai pada simpul atau busur. graph tak berbobot (unweigthed graph) adalah graph dengan simpul dan busurnya tidak mempunyai nilai. nilai bisa hanya dimiliki oleh salah satu elemen dari graph, simpul saja atau hanya busur. tetapi yang sering, nilai dimiliki oleh busur. nilai pada simpul digunakan untuk mewakili jumlah keanggotaan, luasan, atau besaran suatu simpul. nilai pada busur digunakan untuk mewakili jumlah busur yang terhubung dengan sepasang simpul, jarak dua simpul, atau biaya yang dibutuhkan untuk melewati busur. gambar 5 adalah contoh graph berbobot dengan matriks adjacency. a b c d e f a 0 0 1 1 0 0 b 0 0 1 0 0 1 c 1 1 0 0 1 0 d 1 0 0 0 1 0 e 0 0 1 1 0 1 f 0 1 0 0 1 0 gambar 4. graph dalam adjacency matriks (levitin, 2005) (a) (b) gambar 5. (a) graph berbobot, (b) matriks dari graph berbobot. ditinjau dari jumlah penghubung dalam setiap simpul, terdapat pasangan simpul dengan busur jamak (multi edge) dan pasangan simpul dengan busur secara tunggal (single edge). graph dengan busur jamak, simpul x dan y dihubungkan dengan lebih dari satu simpul. gambar 3b merupakan graph berbusur jamak. peng-gambaran dalam matriks adjacency berupa matriks berbobot. bobot dalam graph busur jamak mewakili dari jumlah busur yang menghubungkan antara simpul x dan simpul y. b. random graph random graph pertama kali dikenalkan oleh erdős dan rényi tahun 1959 (diestel, 2000). random graph adalah graph dengan simpul sejumlah n dan setiap pasang simpulnya terhubung atau tidak terhubung dengan suatu probabilitas p atau (1-p) (newman, 2003). random graph di atas dinotasikan sebagai gn,p. secara teknis, graph g dengan m adalah jumlah busur yang muncul, maka probabilitas kemunculan busur dinyatakan dalam persamaan 8. gambar 6. graph dibangun dengan model erdos dan renyi, n = 16 dan p = 1/7 (newman, 2001c) mmm pp −− )1( , (8) dimana )1(21 −= nnm . (9) dengan m adalah maksimum jumlah busur yang mungkin terjadi, persamaan 9. dari persamaan 8 dan 9 di atas, sering muncul random graph yang dinotasikan dengan gn,m. gambar 6 merupakan contoh random graph yang dibangun dengan model erdos dan renyi dengan probabilitas, p = 1/7. bahan dan metode a. bahan bahan untuk uji coba pembatasan kketetanggaan simpul menggunakan citra tiruan berwarna hitam. citra tiruan hitam penuh dikenakan pengotoran dengan derau putih. ukuran citra tiruan untuk uji coba adalah 200x500 piksel. pengotoran citra hitam penuh dengan derau menggunakan teknik pengkotoran salt and paper (gonzales, 2002). tingkat kepadatan derau pada citra tiruan mulai 0,003 dan 0,201. citra tiruan yang dipakai untuk bahan uji coba sejumlah 100 buah. citra tiruan yang telah dikotori dengan derau putih digunakan sebagai bahan model dari zainal abidin dan agus zainal arifin 100 volume 1 no. 2 mei 2010 graph. satu piksel putih pada citra tiruan dianggap sebagai sebuah simpul pada graph. model menghasilkan graph dengan simpulsimpul yang tersebar secara acak. graph yang dihasilkan berupa graph dengan simpul-simpul yang tidak saling terhubung atau disebut sebagai graph terisolasi. simpul-simpul dalam graph terisolasidigunakan sebagai bahan untuk membangkitkan busur-busur dengan metode erdos royi. gambar 7 potongan dari citra dengan ukuran 20x50 piksel yang telah diperbesar 400%. data jumlah simpul pada 100 citra tiruan hasil pengkotoran citra hitam penuh dengan menggunakan teknik pengkotoran salt and paper terdapat di dalam tabel 1. pada tabel 1 tercantum jumlah simpul atau piksel putih dan tingkat kepadatan simpul dalam citra tiruan. jumlah simpul terkecil 61 buah dan terbesar adalah 9951. variasi jumlah simpul untuk mengetahui kelebihan dan kelemahan random graph dengan metode erdos royi dengan k-nn. b. metode metode untuk menurunkan komputasi pembentukan graph dengan metode random graph erdos dan royi terdiri dari tiga tahapan. tiga tahapan itu adalah : inisialisasi graph, mencari k ketetanggaan simpul, dan menghubungkan setiap simpul dengan k tetangga dengan probabilitas p. gambar 8 diagram alir metode random graph erdos royi dengan k-nn. 1) inisialisasi graph pada tahapan inisialisasi graph digunakan untuk menentukan nilai-nilai parameter awal yang dipakai untuk membangun random graph metode erdos royi yang diintegrasikan dengan k-nn. dua paramater awal adalah jumlah ketanggaan dari setiap simpul, k dan jarak terjauh dari semua pasangan simpul, l. a b c gambar 7. citra sampel yang telah dikotori dengan salt and paper. (a) potongan dari citra dengan jumlah simpul 937. (b) potongan dari citra yang telah terkotori dengan jumlah simpul 2041. (c) potongan dari citra yang telah terkotori dengan jumlah simpul 2933. ( ) lvudeuvp ⋅−⋅= βα ,),( . (2) nilai l digunakan untuk menentukan nilai probabilitas antar dua simpul , p(u,v), dengan metode waxman (gunduz, 2004), persamaan 2, dimana α dan β adalah bilangan konstan dengan besar antara nol sampai dengan satu. d(u,v) adalah jarak euclidean antara simpul u dan v. dalam graph, l bisa diperoleh dengan persamaan 3, dimana skala adalah lebar dimensi dari graph. dalam penelitian ini, graph dibangun berdasarkan citra, maka l diperoleh dari panjang diagonal citra, persamaan 4, dimana p dan l merupakan panjang dan lebar citra sampel. skalal ⋅= 2 , (3) 22 lpl += , (4) random graph metode erdos royi menghubungkan setiap simpul (n) ke semua simpul lain (n-1). keterhubungan pasangan simpul memperhatikan probabilitas waxman, seperti dalam persamaan 2. dengan kata lain, random graph metode erdos dan royi mencoba memeriksa keterhubungan semua kemungkinan pasangan simpul, n(n-1). di sisi lain, jika graph g dengan setiap simpul dihubungkan dengan k ketetanggaan dan k jauh lebih besar dari ln(n), maka graph g dijamin menjadi graph terhubung (watts dan strogatz, 1998, distel, 2000), sesuai persamaan 5. syarat agar random graph menjadi terhubung, seperti pada persamaan 6 (watts dan strogatz, 1998, distel,2000). )ln(nk >> . (5) 1)ln( >>>>>> nkn . (6) dalam penelitian ini, untuk mendapatkan nilai k jauh lebih besar dari ln(n), k diperoleh dengan dua pangkat pembulatan ke bawah ln(n), seperti persamaan 7.  )ln(2 nk = . (7) random graph metode erdos dan royi dengan k-nn menghubungkan setiap simpul (n) dengan k ketetanggaannya. sehingga graph dapat dihasilkan dengan random graph metode erdos dan royi dengan k-nn. komputasi dari random graph metode erdos dan royi dengan k-nn adalah n(k+k2). 2) mencari k ketetanggaan simpul. setelah diperoleh jumlah tetangga dari setiap simpul adalah k, tahapan selanjutnya membatasi k-ketetanggaan simpul dalam pembangkitan random graph… jurnal cauchy – issn: 2086-0382 101 adalah mencari simpul yang menjadi tetangga dari setiap simpul dengan jumlah k. k tetangga terdekat dapat diperoleh dengam membuat jendela persegi panjang. ukuran panjang sisi adalah pembulatan ke atas akar k, seperti persamaan 8. gambar 8. diagram alir metode random graph erdos dan royi dengan k-nn tabel 1.data jumlah simpul dalam citra tiruan untuk uji coba nama file kerapatan jumlah simpul ug001.tif 0.003 61 ug002.tif 0.005 153 ug003.tif 0.007 257 ug004.tif 0.009 351 ug005.tif 0.011 459 ug006.tif 0.013 559 ug007.tif 0.015 610 ug008.tif 0.017 789 ug009.tif 0.019 919 ug010.tif 0.021 937 ug011.tif 0.023 1080 ug012.tif 0.025 1101 ug013.tif 0.027 1258 ug014.tif 0.029 1338 ug015.tif 0.031 1482 ug016.tif 0.033 1627 ug017.tif 0.035 1695 ug018.tif 0.037 1781 ug019.tif 0.039 1861 ug020.tif 0.041 2041 ug021.tif 0.043 2156 ug022.tif 0.045 2192 ug023.tif 0.047 2237 ug024.tif 0.049 2289 ug025.tif 0.051 2482 ug026.tif 0.053 2481 ug027.tif 0.055 2713 ug028.tif 0.057 2836 ug029.tif 0.059 2889 ug030.tif 0.061 2933 ug031.tif 0.063 3046 ug032.tif 0.065 3167 ug033.tif 0.067 3384 ug034.tif 0.069 3397 ug035.tif 0.071 3504 ug036.tif 0.073 3540 ug037.tif 0.075 3572 ug038.tif 0.077 3673 ug039.tif 0.079 3837 ug040.tif 0.081 4004 ug041.tif 0.083 3970 ug042.tif 0.085 4082 ug043.tif 0.087 4221 ug044.tif 0.089 4424 ug045.tif 0.091 4417 ug046.tif 0.093 4491 ug047.tif 0.095 4631 ug048.tif 0.097 4778 ug049.tif 0.099 4696 ug050.tif 0.101 4897 ug051.tif 0.103 5036 ug052.tif 0.105 5150 ug053.tif 0.107 5249 ug054.tif 0.109 5336 ug055.tif 0.111 5428 ug056.tif 0.113 5546 ug057.tif 0.115 5685 ug058.tif 0.117 5773 ug059.tif 0.119 5925 ug060.tif 0.121 5952 ug061.tif 0.123 6041 ug062.tif 0.125 6088 ug063.tif 0.127 6072 ug064.tif 0.129 6255 ug065.tif 0.131 6515 ug066.tif 0.133 6409 ug067.tif 0.135 6542 ug068.tif 0.137 6736 ug069.tif 0.139 6794 ug070.tif 0.141 6991 ug071.tif 0.143 7017 ug072.tif 0.145 7217 ug073.tif 0.147 7297 ug074.tif 0.149 7281 ug075.tif 0.151 7502 ug076.tif 0.153 7543 ug077.tif 0.155 7611 ug078.tif 0.157 7843 ug079.tif 0.159 7906 penambahan ukuran jendela pembangkitan random graph mulai inisialisasi graph l, k keluaran matriks adjacency selesai penentuan lebar jendela hitung tetangga dalam jendela tetangga < k ya tidak mencari k ketetanggaan zainal abidin dan agus zainal arifin 102 volume 1 no. 2 mei 2010 ug080.tif 0.161 7995 ug081.tif 0.163 7986 ug082.tif 0.165 8135 ug083.tif 0.167 8362 ug084.tif 0.169 8276 ug085.tif 0.171 8564 ug086.tif 0.173 8592 ug087.tif 0.175 8787 ug088.tif 0.177 8709 ug089.tif 0.179 8724 ug090.tif 0.181 9087 ug091.tif 0.183 9044 ug092.tif 0.185 9092 ug093.tif 0.187 9170 ug094.tif 0.189 9537 ug095.tif 0.191 9380 ug096.tif 0.193 9504 ug097.tif 0.195 9693 ug098.tif 0.197 9667 ug099.tif 0.199 9886 ug100.tif 0.201 9951 gambar 9 : ilustrasi pembuatan jendela k-nn dengan n=34 dan k=8,  k . (8) diasumsikan bahwa semua bagian dalam jendela terisi dengan simpul secara penuh. jika semua bagian jendela terisi penuh, maka terdapat jumlah tetangga >= k. jika dalam jendela terdapat tetangga terdekat kurang dari k, maka lebar sisi jendela ditambah sebesar k/2. gambar 9 merupakan ilustrasi pembuatan jendela k-nn. ukuran graph 34, diperoleh ln(n) = 3,5 sehingga k = 8. pada ilustrasi dicari 8-ketetanggaan dari titik berlabel angka satu. proses inisialisasi diperoleh lebar jendela tiga dan titik yang menjadi tetangganya delapan. pada jendela ukuran 3x3 hanya terdapat empat simpul tetangga, jumlah simpul masih kurang dari k. lebar jendela ditambah untuk mendapatkan jumlah tetangga lebih besar atau sama dengan k. hasil dari penambahan lebar jendela diperoleh simpul tetangga sejumlah sembilan. 3) pembangkitan random graph setiap memperoleh k tetangga terdekat, suatu simpul dibuat busur yang menghubungkan simpul dengan semua k tetangganya. dalam pembentukan graph ditetapkan suatu nilai probabilitas setiap pasang simpul, p, dengan persamaan 2 dengan kondisi di atas bisa di hasilkan random graph gn,p. tabel 2. simulasi perhitungan probalitas waxman d p 1 0.2309609 2 0.0561505 3 0.0136511 4 0.0033188 5 0.0008069 6 0.0001962 7 0.0000477 8 0.0000116 9 0.0000028 10 0.0000007 persamaan 2 menghasilkan probabilitas yang mendekati satu sampai mendekati nol. jika jarak euclidean antara dua buah titik semakin dekat maka probabilitasnya semakin besar. demikian pula sebaliknya, jika jaraknya jauh maka probabilitasnya semakin kecil, cenderung nol. tabel 2 uji coba rumus 21 dalam simulasi sepuluh angka dengan lebar skala 10x10. 4) pembangunan graph hasil dari proses penentuan obyek berupa citra biner, nilai nol dan satu di citra biner dijadikan acuan pembentukan graph. piksel bernilai satu dianggap sebagai sebuah simpul. asumsi bahwa simpul-simpul dalam sampel sebagai simpul terasing yang tidak terhubung dengan busur. gambar 10 merupakan citra tiruan simpul-simpul yang tersimpan dalam citra biner. dalam citra tiruan terdapat 144 piksel putih yang berarti terdapat 144 simpul terasing. satu kotak kecil berisikan 9 piksel putih. pembentukan graph diawali dengan penghubungan setiap simpul dari citra sampel dengan busur. pembangkitan busur-busur dalam pembentukan graph menggunakan random graph metode erdos dan royi dengan k-nn. simpul yang saling terhubung dihitung probabilitasnya menggunakan metode waxman, persamaan 2. nilai probabilitas rendah menunjukan bahwa dua simpul mempunyai jarak yang jauh (panjang). sebaliknya probabilitas tinggi menunjukkan bahwa dua simpul mempunyai jarak yang dekat (pendek), lihat tabel 2. membatasi k-ketetanggaan simpul dalam pembangkitan random graph… jurnal cauchy – issn: 2086-0382 103 keterhubungan antar dua simpul dibatasi dengan nilai ambang. jika nilai probabilitas dua simpul lebih kecil dari nilai ambang, maka busur yang menghubungkan dua simpul tersebut dihapus. tujuan dari pemotongan garis penghubung yang mempunyai probabilitas rendah adalah untuk menghapus hubungan dua simpul yang jauh . manfaatnya, simpul hanya terhubung dengan simpul-simpul yang dekat, sehingga dapat diketahui nilai-nilai karakter dari setiap simpul terhadap simpul tetangga terdekatnya. gambar 10. citra tiruan simpul gambar 11. graph dari citra tiruan. gambar 10 merupakan graph hasil dari citra tiruan. setiap simpul pada citra tiruan dihubungkan dengan nilai ambang probabilitas 0.2 dari citra berukuran 50x50. tampak graph dengan simpul-simpul yan saling berdekatan mempunyai busur lebih banyak dibandingkan dengan yang letaknya berjauhan. jumlah busur dari sebuah simpul dipengaruhi jumlah simpul tetangga yang dekat dan jarak antar simpul tetangga. tampak pada gambar 11, simpul-simpul yang mempunyai jarak dekat dengan tetangganya mempunyai jumlah busur lebih banyak dibandingkan dengan simpul yang jarak antar tetangganya jauh. jumlah busur pada setiap simpul merupakan degree dari simpul tersebut. simpul yang mempunyai degree tinggi adalah simpul yang dikelilingi oleh simpul lain dengan jarak yang dekat. sebaliknya simpul dengan degree rendah adalah simpul yang di sekitarnya terdapat simpul-simpul dengan jarak yang cenderung jauh. simpul dengan jumlah tetangga sedikit cenderung mempunyai degree rendah. simpul dengan tetangga sedikit biasanya berada dipinggir suatu area. aplikasi dan pembahasan 1) lingkungan uji coba uji coba lakukan di komputer acer aspire 3610. perangkat keras pendukungnya adalah prosesor 1.8 ghz, kapasitas memori 2 gbyte, kapasitas harddisk 40 gbyte. perangkat lunak pendukung adalah i windows xp service pack 1, bahasa pemrograman menggunakan matlab versi 7.1 2) skenario uji coba uji coba random graph metode erdos royi dengan k-nn. uji coba akan membandingkan random graph metode erdos royi murni dengan random graph metode erdos royi dengan k-nn. uji coba ini untuk melihat kebutuhan waktu komputasi dari masing-masing metode. uji coba dilaksanakan guna mengetahui keberhasilan random graph metode erdos dan royi dengan k-nn dalam menurunkan waktu komputasi pada saat membangkitkan suatu graph. metode diujicobakan dengan 100 data (tabel 4.1). data berupa citra yang dikotori dengan derau menggunakan teknik pengkotoran salt and paper (gonzalez, 2002). semua piksel putih dalam citra dianggap sebagai simpulsimpul terasing pada graph. semua data digunakan untuk masukan dalam pembangkitan random graph menggunakan metode erdos dan royi. kemudian hasil coba pertama dibandingkan dengan hasil uji coba pembangkitan random graph yang dilaksanakan dengan metode erdos dan royi dengan k-nn. 3) pelaksanaan dan evaluasi uji coba uji coba dilaksanakan sesuai dengan skenario yang telah dibahas pada subbab skenario uji coba. uji coba dilaksanakan untuk mengevaluasi hasil-hasil uji coba.hasil metode erdos royi murni dibandingkan dengan erdos royi dengan k-nn. kedua metode diujicobakan pada citra tiruan diperoleh dari citra hitam penuh yang dikotori dengan derau. pengkotoran menggunakan metode pengkotoran salt and zainal abidin dan agus zainal arifin 104 volume 1 no. 2 mei 2010 paper (gonzalez, 2002). jumlah citra 100 dengan kerapatan mulai 0.002 sampai dengan 0.2. jumlah piksel putih lebih lengkap dapat dilihat pada tabel 1. perlakuan uji coba kedua metode dikenakan pada lingkungan uji coba tertera pada tabel 3. gambar 12 perbandingan waktu komputasi dari dua metode. pada gambar 12 waktu komputasi erdos royi lebih besar jika banding dengan erdos royi dengan k-nn. data lebih lengkap tercantum dalam tabel 4. pada jumlah simpul kecil di bawah 1258, waktu komputasi random graph metode erdos dan royi lebih cepat. tetapi waktu komputasi random graph metode erdos dan royi dengan k-nn tidak terlalu lambat dibanding dengan komputasi random graph metode erdos dan royi murni. tabel 3 : parameter uji coba random graph. parameter nilai probabilitas 0,5 l 538,516 alfa 0,95 beta 0,05 grafik perbandingan waktu komputasi 0 10 20 30 40 50 60 70 80 90 100 6 1 3 5 1 6 1 0 9 3 7 1 2 5 8 1 6 2 7 1 8 6 1 2 1 9 2 2 4 8 1 2 8 3 6 3 0 4 6 3 3 9 7 3 5 7 2 3 9 7 0 4 2 2 1 4 4 9 1 4 7 7 8 5 1 5 0 5 4 2 8 5 7 7 3 6 0 4 1 6 2 5 5 6 5 4 2 6 9 9 1 7 2 8 1 7 5 4 3 7 9 0 6 8 1 3 5 8 5 6 4 8 7 2 4 9 0 8 7 9 3 8 0 9 6 6 7 9 9 5 1 jumlah simpul w a k tu k o m p u ta s i erdos royi erdos royi dengan k-nn gambar 12 : grafik komputasi random graph erdos royi dan erdos royi dengan k-nn. tabel 4 : perbandingan waktu komputasi random graph erdos dan royi dengan random graph erdos royi k-nn. jumlah simpul kerapatan waktu komputasi er k-nn er 61 0.003 0.57294 0.00697 153 0.005 0.62569 0.02100 257 0.007 0.54349 0.05890 351 0.009 0.47827 0.10926 459 0.011 0.88367 0.18505 559 0.013 0.86169 0.27676 610 0.015 0.86304 0.32878 789 0.017 0.82398 0.55010 919 0.019 0.83146 0.74452 937 0.021 0.80041 0.77365 1080 0.023 0.82323 1.02799 1101 0.025 1.45573 1.06906 1258 0.027 1.46713 1.39193 1338 0.029 1.44778 1.57528 1482 0.031 1.44725 1.93332 1627 0.033 1.49927 2.32916 1695 0.035 1.49051 2.52808 1781 0.037 1.50575 2.79336 1861 0.039 1.53051 3.04980 2041 0.041 1.54944 3.66728 2156 0.043 1.60874 4.09628 2192 0.045 1.62677 4.23201 2237 0.047 1.64248 4.40775 2289 0.049 1.63693 4.61314 2481 0.051 1.67972 5.42537 2482 0.053 1.67597 5.42627 2713 0.055 1.74183 6.47728 2836 0.057 1.81065 7.08220 2889 0.059 1.83860 7.34796 2933 0.061 1.84444 7.57942 3046 0.063 3.15187 8.16770 3167 0.065 3.23873 8.83156 3384 0.067 3.54660 10.08787 3397 0.069 3.46173 10.20142 3504 0.071 3.50742 10.81216 3540 0.073 3.65244 11.03016 3572 0.075 3.64004 11.22606 3673 0.077 3.67181 11.87704 3837 0.079 3.76098 12.96722 3970 0.081 3.73449 13.87264 4004 0.083 3.73913 14.11632 4082 0.085 3.75886 14.67194 4221 0.087 3.70121 15.68447 4417 0.089 3.80541 17.17473 4424 0.091 3.83085 17.21913 4491 0.093 3.91290 17.75805 4631 0.095 4.01942 18.87249 4696 0.097 4.10087 19.40843 4778 0.099 4.19061 20.10800 4897 0.101 4.29645 21.10801 5036 0.103 4.47701 22.30385 5150 0.105 4.63355 23.35821 membatasi k-ketetanggaan simpul dalam pembangkitan random graph… jurnal cauchy – issn: 2086-0382 105 5249 0.107 4.73978 24.23739 5336 0.109 4.80595 25.05935 5428 0.111 4.92789 25.94123 5546 0.113 5.00948 27.06993 5685 0.115 5.16376 28.43455 5773 0.117 5.19992 29.34578 5925 0.119 5.17961 30.91552 5952 0.121 5.20977 31.16242 6041 0.123 5.20647 32.10460 6072 0.125 5.29950 32.45810 6088 0.127 5.24221 32.61672 6255 0.129 5.20859 34.42880 6409 0.131 5.20513 36.13983 6515 0.133 5.22762 37.36209 6542 0.135 5.26608 37.65784 6736 0.137 5.32041 39.93466 6794 0.139 5.40414 40.61880 6991 0.141 5.62366 43.02693 7017 0.143 5.63420 43.31324 7217 0.145 5.84581 45.84301 7281 0.147 5.90240 46.63678 7297 0.149 5.93010 46.84052 7502 0.151 6.24529 49.53723 7543 0.153 6.22858 50.08511 7611 0.155 6.30604 50.96423 7843 0.157 6.65871 54.11894 7906 0.159 6.72723 54.98794 7986 0.161 6.76583 56.11708 7995 0.163 6.84309 56.23221 8135 0.165 11.69431 58.21586 8276 0.167 12.07412 60.26962 8362 0.169 12.30286 61.56285 8564 0.171 12.70018 64.59212 8592 0.173 12.79342 64.96993 8709 0.175 13.00610 66.77934 8724 0.177 13.08824 66.93141 8787 0.179 13.16771 67.93681 9044 0.181 14.28052 71.92699 9087 0.183 13.72087 72.75217 9092 0.185 13.75461 72.71987 9170 0.187 13.69513 74.00391 9380 0.189 14.59682 79.08805 9504 0.191 16.60406 79.52803 9537 0.193 18.77744 80.03217 9667 0.195 17.57210 82.66109 9693 0.197 18.70974 83.11653 9886 0.199 19.80950 86.60223 9951 0.201 18.38680 87.59296 terlihat pada tabel 4 bahwa random graph metode erdos royi dengan k-nn lebih unggul pada posisi jumlah simpul lebih dari 1258. penambahan jumlah simpul tidak menambah waktu komputasi secara signifikan. penambahan waktu komputasi pada random graph metode erdos dan royi dengan k-nn bertambah secara linier terhadap jumlah simpul. waktu komputasi random graph metode erdos dan royi bertambah secara eksponensial terhadap jumlah simpul. kelemahan erdos royi dengan k-nn adalah pada waktu komputasi untuk mencari k ketetanggaan. pada data dengan tingkat kepadatan simpul rendah, jarak antar simpul relatif berjauhan, sehingga pencarian simpul tetangga memerlukan waktu relatih lama. data dengan tingkatan kepadatan simpul tinggi, waktu untuk pencarian simpul tetangga relatif cepat, karena dengan ukuran jendela yang tidak lebar tetangga simpul sejumlah k dapat ditemukan. pencarian ketetanggan pada area dengan kerapatan tinggi hanya memerlukan proses memperlebar jendela dengan waktu yang pendek. pada area dengan jumlah simpul 1338 waktu komputasi dari random graph metode erdos dan royi dengan knn. gambar 13b graph hasil random graph dari erdos royi. gambar 13c graph hasil random graph dengan metode erdos royi dengan k-nn. kedua graph dibangkitkan dari citra biner (gambar 13a) yang berisi 257 piksel putih dengan tingkat kepadatan 0.007. graph pada gambar 13 dibangkitkan dengan probabilitas 0,4. tampak dalam gambar bahwa dua graph yang dihasilkan tidak memiliki perbedaan, jumlah simpul yang terhubung oleh busur adalah sama, yaitu 252. a b c gambar 13 : contoh dari hasil uji coba random graph, (a) citra biner, (b) erdos royi, (c) erdos royi dengan k-nn. simpul-simpul pada gambar 13a yang tidak terhubung oleh busur merupakan simpul yang mempunyai tetangga sangat jauh, atau probabilitasnya lebih kecil dari 0,5. pasangan simpul yang probabilitasnya lebih kecil dari nilai ambang, maka busurnya tidak dihubungkan. simpul-simpul dengan jarak yang saling berdekatan mempunyai busur yang banyak. simzainal abidin dan agus zainal arifin 106 volume 1 no. 2 mei 2010 pul-simpul dengan jarak yang saling berjauhan mempunyai busur yang sedikit. dari graph dapat diketahui jumlah simpul yang saling berdekatan dengan melihat jumlah busur yang terhubung dengan simpul. simpul dengan jumlah busur (degree) besar berarti simpul mempunyai tetangga yang banyak. dengan kata lain, simpul berada di area yang rapat. penutup a. simpulan waktu komputasi pada random graph metode erdos dan royi dapat dikurangi dengan mengintegrasikan metode k-nn. k-nn digunakan untuk menghubungkan busur dari setiap simpul dengan k tetangga terdekat. random graph metode erdos dan royi yang semula menghubungkan n(n-1) simpul, setelah diintegrasikan dengan knn menjadi hanya menghubungkan n(k) simpul. waktu komputasi random graph metode erdos dan royi n(n-1), sedangkan random graph metode erdos dan royi dengan k-nn menjadi n(k+k2), dimana k adalah waktu untuk menghubungkan busur dari simpul ke k jumlah tetangga, dan k2 adalah waktu untuk mencari tetangga dalam jendela. b. saran pengurangan waktu komputasi pada random graph erdos royi dengan k-nn masih terdapat waktu komputasi untuk mencari k ketetanggaan yang relatif besar. waktu komputasi mencari k ketetanggaan masih perlu dikurangi. metode pembuatan jendela bisa ditingkatkan ketepatannya agar diperoleh k ketetanggaan lebih tepat dan waktu pencarian k ketetanggaan lebih cepat. daftar pustaka [1] abidin, zainal dan arifin agus z. 2008. generation graph with random graph erdos royi method by medical image to help diagnoses of osteoporosis. 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[4] andre, m., ijaz, k., tillinghast, j. d., krebs, v. e., diem, l. a., metchock, b., crisp, t., mcelroy, p. d.(2006) “transmission network analysis to complement routine tuberculosis contact investigations” american journal of public healt. volume 96 nomor 11. [5] diestel, reinhard (2000). graph theory electronic edition. springer-verlag: new york. [6] demir, c., yener, b., dan gultekin, s. h. (2005). “augmented cell-graph for automated cancer diagnosis”. bioinformatic vol 21 suplemen 2. ii7-ii12 [7] gonzalez, r. c., woods, r. e., eddins, s. l.. (2002). digital image processing. prentice hall. new jersey. [8] guesebroek, jan-mark., smeulders, a.w .m., dan cornellisen f (1999). “segmentation of tissue by distance graph matching”. cytometry 35:11-22 [9] gunduz, c., yener, b., dan gultekin, s. h. (2004). “the cell graphs of cancer”. bioinformatic vol. 20 suplemen 1. i145-i151. [10] levitin, anany. (2007). intoduction to the design and analysis of algoritms second edition. pearson education, inc. [11] newman, m. e. j.. (2001a). “scientific collaboration network. ii. shostest paths, weighted network, and centrality”. physical review vol. e64 016132 [12] newman, m. e. j.. (2001b), “who is the best connected scientist ? a study of scientific coauthorship network”. physical review vol. e64 0011144. [13] newman, m. e. j., (2001c), “random graph models of sosial network”. proc. natl. acad. sci. usa 99. 2566-2572. [14] newman, m. e. j., 2003, “the structure and function of complex networks”. siam review vol. 45 number 2 : 167-256 [15] watts, d. j., strogatz, s. h.. (1998). “collective dynamics of ‘small-word’ models” nature. volume 393 halaman 440442. a monte carlo simulation study to assess estimation methods in confimatory factor analysis (cfa) on ordinal data cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 332-344 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 20, 2021 reviewed: march 07, 2022 accepted: march 28, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.14434 a monte carlo simulation study to assess estimation methods in confimatory factor analysis (cfa) on ordinal data nina fitriyati*, and madona yunita wijaya universitas islam negeri syarif hidayatullah jakarta, indonesia email: nina.fitriyati@uinjkt.ac.id abstract likert-type scale data are ordinal data and are commonly used to measure latent constructs in the educational, social, and behavioral sciences. the ordinal observed variables are often treated as continuous variables in factor analysis, which may cause misleading statistical inferences. two robust estimators, i.e., unweighted least squares (uls) and diagonally weighted least squares (dwls) have been developed to deal with ordinal data in confirmatory factor analysis (cfa). in this research, we conduct an extensive simulation study to examine the impact of uls and dwls estimations in the accuracy of parameter estimates as well as the model fit measures by varying the number of likert scale points. we use synthetic data generated in a monte carlo experiment to explore the behavior of these methods (dwls and uls) and compare their performance with normal theory-based ml and gls (generalized least squares) under different levels of experimental conditions. the simulation results indicate that both dwls and uls yield consistently accurate parameter estimates across all conditions considered. the likert data can be treated as a continuous variable under ml or gls when using at least five likert scale points to produce trivial bias. however, these methods generally fail to provide a satisfactory fit. furthermore, we provide empirical studies in the field of psychological measurement data to present how theoretical and statistical instances have to be taken into consideration when ordinal data are used in the cfa model. keywords: confirmatory factor analysis; diagonally weighted least squares; generalized least squares; likert data; maximum likelihood introduction factor analysis such as cfa is often used in various fields, including social science, economics, and psychology to validate survey instruments. cfa is developed from the concept of exploratory factor analysis (efa) where researchers have the hypothesized model based on theory or previous analytical research. the model should be specified before analysis regarding the number of factors in the model, the number of indicators reflecting each factor, and whether a relationship exists between these factors. factor analysis plays an important role to provide evidence of construct validity and information about the internal structure of the measurement instrument [1] [2]. thompson [3] stated that cfa is also known as a measurement model which is an important component of the structural equation model (sem) class, describing how the measured indicators can reflect a latent variable/construct. the measurement model describes how the latent variable depends on the indication of the observed variable which also explains the measurement properties such as the validity and reliability of the observed variable [4]. http://dx.doi.org/10.18860/ca.v7i3.14434 mailto:nina.fitriyati@uinjkt.ac.id a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 333 a questionnaire is one of the tools in quantitative research that is generally used to study latent constructs through their observed indicators. the indicators are collected through a series of questions in the form of a questionnaire measured on a likert-type scale. this scale offers a series of fixed-choice options where there is a rank order between the choices given [5]. the results of measuring data using a likert scale are also known as ordinal data. when collecting data with a likert scale for analysis, researchers tend to ignore the actual data structure and treat it like a continuous variable since the methods designed for continuous variables tend to be easy to apply and easy to use. in this case, the maximum likelihood (ml) method is used to estimate the model parameters [6]. the ml method is the most popular since it gives asymptotic unbiased results and consistent parameter estimates [7] [8] [9] [10]. however, this method uses pearson correlation which requires the assumption of continuous measurement for both latent and observed variables. pearson correlation also tends to underestimate the true correlation between ordinal variables. however, in principle, this method cannot be directly applied to ordinal data because the characteristics of the data itself tend to be different. several theories and estimation methods are available which are designed to handle ordinal data, such as dwls (diagonally weighted least squares) and uls (unweighted least squares). several studies have addressed the comparison between uls and dwls performance in finite samples [11] [12] [13] [14]. all these studies have certain limitations in that they did not compare with standard estimation procedures such as ml or gls. as of today, many researchers are still in their comfort zone using the standard estimation, knowing that the observed variables violated the normal assumption. in maydeu-olivers [11] study only considered a non-standard model, whereas tate [12] considered one replication per condition. the other two studies only considered a certain condition in terms of the number of categories used per indicator. to conduct further research in this area, an extensive simulation study was conducted to examine the impact of uls and dwls estimations in the accuracy of parameter estimates as well as the model fit measures by varying the number of likert scale points. in addition, ml and gls were also compared to give the readers insight into the risk of treating ordinal data as a continuous variable in the factor analysis. methods this research was implemented as a monte carlo simulation study, where it involves creating data by pseudo-random sampling to study the behavior of statistical methods under various conditions [15]. in this study, we evaluate the performance of cfa estimation methods under different experimental conditions. in addition, numerical examples using secondary data are also considered to support the findings. estimation methods the most common method used for parameter estimation in the cfa model is the maximum likelihood (ml). it is used more frequently in many research as it has many desirable statistical properties, including asymptotic unbiasedness, asymptotic consistency, asymptotic normality, and asymptotic efficiency. the ml estimation method assumes that the observed indicators are multivariate normally distributed and measured on continuous scales [16]. to estimate the model parameters (𝜃), ml maximizes the likelihood function of the observed data, or equivalently minimizes the following function [7]: a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 334 𝐹𝑀𝐿 = log|∑(𝜃)| − log|𝐒| + tr(𝐒∑(𝜃) −1) − 𝑘, (1) where 𝑆 is the sample variance-covariance matrix, ∑(𝜃) is the model implied covariance matrix, and 𝑘 is the number of observed indicators. generalized least squares (gls) is another estimation method that assumes the multivariate normality of the data similar to ml. the model parameters are estimated by minimizing the following function: 𝐹𝐺𝐿𝑆 = 1 2 𝑡𝑟(𝐒 − ∑(𝜃)𝐒−1)2. (2) the main difference between the two estimators is that the sample covariance matrix is used in the weight matrix instead of the model-implied covariance matrix. however, this estimator provides a better empirical fit than ml when the models are misspecified [17] [18] [19]. the assumptions of normality and continuous scales are often violated since many research, such as psychology, social sciences, and educational research, use likert scales to measure observable variables. likert-type scales are categorical (ordinal) data where the response categories have a rank order, i.e., one score can be said to be higher than another but not the distance between the two scores. if the assumption is not satisfied, then the resulting cfa model may not be reliable and the conclusion obtained can be misleading. alternative estimators available are dwls and uls, which have been suggested to deal with ordinal data [20] [6]. when the data are treated as categorical (ordinal), correlation structure is involved instead of covariance structure and the model is fitted to polychoric correlations. both methods share a similar fit function, i.e., least-square function to estimate the model parameters involving minimizing the following objective function: 𝐹 = (𝐒 − 𝝈(𝜃)) ′ 𝐖(𝐒 − 𝝈(𝜃)), (3) where 𝐖 is the weight matrix and (𝐒 − 𝝈(𝜃)) denotes the difference between the population threshold and polychoric correlations and those implied by the model. the weight matrix 𝐖 determines the minimization procedure. when the choice of 𝐖 is the identity matrix, the method is known as uls (muthen, 1993). a second choice is 𝐖 = (𝑑𝑖𝑎𝑔(𝚪)) −1/2 known as dwls. it uses the elements in the diagonal matrix of the estimated polychoric correlations (the estimated variances) as the weights [21]. simulation design monte carlo simulation studies were carried out to compare the impact of various estimation methods (ml, gls, uls, and dwls) on the cfa model by manipulating four experimental factors, i.e., the number of latent variables or factors, number of items, the number of response categories, and sample sizes. more specifically, we are interested in evaluating four popular model fit indices (i.e., chi-square test, rmsea, cfi, and tli) and also bias in factor loadings across different experimental conditions. different cfa models were generated under different settings of dimensionality, i.e., a one-factor model and a correlated two-factor model. each factor was measured by four and eight ordinal observed indicators. at least four indicators are necessary for a one-factor model to be (over-)identified. additionally, it was found to have optimal a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 335 accuracy of parameter estimates and marginally improved by adding more indicators [22]. the coefficients of factor loadings on the same factor were varied from low to high loadings, i.e., 0.5 to 0.9, to mimic real-world data as it is very rarely happened to have constant factor loadings on the same factor. it was also suggested from many empirical research and simulation studies were reported standardized factor loadings range from 0.4 to 0.9 [23] [24] [25]. for a two-factor model, the inter-factor correlation was set to have a moderate correlation of 0.4. the factor variances were all set equal to 1 in the population. four different sample sizes are considered from low to large samples: n = 100, 200, 500, and 1000 [26]. all generated models assumed symmetric observed distributions generated from normal latent response distribution. to investigate the influence of categorization or the number of likert scale points (c), each indicator was generated by considering 𝑐 = 2,3, … ,10 categories. a total of 144 experimental conditions (2 × 2 × 9 × 4) was created in this study by considering the combination of the four experimental factors, i.e., number of factors (𝑓 = 1,2), number of items (𝑖 = 4,8) number of categories (𝑐 = 2, … ,10), and sample sizes (𝑁 = 100, 200, 500, 1000) [26]. each experimental condition was evaluated by generating 500 datasets. all data were generated using mplus while cfa model evaluations were carried out using the 'lavaan’ package in r [27] [28]. outcome variables in this study, a model assessment was done on parameter estimates and model fit. for each experimental condition, parameter estimates including factor loadings and interfactor correlation were examined. the bias, i.e., the difference between the estimated (𝜃 ) and true parameter (𝜃), were computed to evaluate the performance of the four estimation methods. the average absolute relative bias (arb) was considered to take into account the magnitude of the true parameter value. let 𝜃𝑖𝑗 be the parameter estimates of the 𝑗th parameter (1,2, … , 𝑝) in the 𝑖th replicate ( 𝑖 = 1,2, . . , 𝑅). the arb can be expressed as follows [29]: 𝐴𝑅𝐵 = 1 𝑅 ∑ ( 1 𝑝 ∑ (�̂�𝑖𝑗−𝜃𝑗) 𝜃𝑗 𝑝 𝑗=1 ) 𝑅 𝑖=1 . (4) a trivial bias is indicated with an arb value less than 5%, a moderate bias is indicated with an arb value between 5% and 10%, and a substantial bias is indicated with an arb value above 10% [30] [31]. to assess model fit, the chi-square test and rmsea were evaluated in terms of the rate of rejection (type i error) of the proposed model. a rate of rejection greater than 5% indicates inflated type i error rates, showing that the test statistics may have been underestimated. other fit indices such as tli and cfi were evaluated by averaging across r replications. they were also computed as the proportion of satisfactory fit across r replications (tli / cli > 0.9). results and discussion parameter estimation the results of average relative bias (arb%) values for factor loadings for each estimation method depending on the number of likert scale points are presented in figure 1 and 2. when ordinal indicators were treated as continuous, the number of likert a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 336 scale points had an impact on the accuracy of the estimated factor loadings, whether ml or gls estimations were used. the estimated factor loadings suffered from a downward bias with a fewer number of likert scale points (𝑐 < 5) and the bias increased dramatically when only considering two categories in the observed indicator variable. increasing sample size did not seem to reduce the bias. using at least five response categories produced moderate bias since the arb values were less than 10%. the accuracy of parameter estimates improved with the increased number of response categories. the factor loadings were essentially unbiased with nine or ten response categories under ml and gls. the dwls and uls estimation methods were superior to the other two methods. both methods consistently provided more accurate factor loadings, evidenced by their smaller arb values. the resulting bias was trivial even in the condition of a small sample size (𝑁 = 100). unlike ml and gls, the number of likert scale points did not influence the accuracy. the bias remained stable below 5% across a different number of response categories. in general, both methods appeared to perform well under various conditions. the impact of the number of indicators to measure a latent construct on the accuracy of factor loading estimates can be notably seen in the one-factor model. the bias decreased as more indicators were used based on dwls and uls methods. in contrast, the arb values were generally larger when using more indicators under ml and gls. in a two-factor model, the bias was relatively similar whether 4or 8indicators per factor were used. figure 1. plot of arb(%) for factor loadings by varying the number of likert scale points, the number of indicators per factor, and sample size, for a one-factor model. a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 337 figure 2. plot of arb(%) for factor loadings by varying the number of likert scale points, the number of indicators per factor, and sample size, for a two-factor model. the arb values for inter-factor correlations in two-factor models are presented in figure 3 by varying the number of indicators, the number of likert scale points, and sample size. the plots also show a comparison of the performance between the four different estimation methods. similar to factor loadings, the number of likert scale points appeared to have an impact on the estimated inter-factor correlations. ml and gls methods performed equally worst in estimating the correlation between the two factors in the condition when two-response categories were used since they substantially underestimated the true correlation parameter (arb < −10%). the arb values can be improved by increasing sample size; however, the resulting estimates were still biased. in the case of five or more response categories, both methods were able to achieve accuracy in the inter-factor correlation estimates with trivial bias. higher accuracy was obtained as the sample size increased. it is interesting to see that the ml estimator generally produced negative bias while in contrast, gls produced positive bias. meanwhile, the inter-factor correlation bias was consistently lower under dwls and uls and essentially unbiased across different scenarios. a larger sample size reduced relative bias, particularly in a two-factor model with four indicators per factor. a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 338 figure 3. plot of arb(%) for inter-factor correlations by varying the number of likert scale points and sample sizes for a two-factor model with four and eight indicators per factor. model fit figure 4 and 5 present rejection rates associated with rmsea and chi-square test (note that not all experimental condition's results are shown here for brevity). the empirical type i error for the chi-square test under ml and gls estimation methods were well within the range of 0.03 and 0.10, which is very near to the nominal value of type i error (𝛼 = 0.05) when considering four indicators per factor. the rate was the highest for a larger sample size, particularly for two response categories. increasing the number of likert scale points, the type i error was able to be controlled at a 5% level. in contrast, the type i error inflated along with the increase in the use of several indicators per factor, particularly for small sample sizes and under the ml method. gls performed generally better than ml within this similar condition. overall, dwls worked well in maintaining chi-square type i error rates across most experimental conditions. in contrast, the rmsea fit indices were found to be insensitive to most conditions of the study. the rejection rates were consistently below 0.05, regardless of the estimation methods used. except for the uls method, which was found with poor performance in the condition of small sample size (𝑁 = 100) and two response categories. the proportions of tli and cli values greater than 0.9 across 500 data generations obtained from fitting a one-factor model is displayed in figure 6. other models are not shown here since the resulting tli and cli was pretty much similar. the estimation methods did not influence both cfi and tli values in larger sample sizes (𝑁 ≥ 500), since they performed consistently well with a proportion above 0.9. in smaller sample sizes, gls was the most conservative one compared to other methods concerning the tli index. this suggests that the model fit worse based on gls when the models were correctly specified. as the level of response categories increased, gls was able to improve its tli performance, but still below a satisfactory level. again, both dwls and uls were able to maintain their consistency concerning tli and cfi indices above 0.9 across different experimental conditions. a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 339 figure 4. plot of rejection rate from chi-square test and rmsea for a one-factor model with 4 indicators per factor. figure 5. plot of rejection rate from chi-square test and rmsea for a two-factor model with 8 indicators per factor. figure 6. plot of a proportion of tli/cli values above 0.9 for a one-factor model with 4 indicators per factor. a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 340 empirical studies to support the findings from the simulation studies, empirical studies were carried out using available data from open-source psychometrics projects (https://openpsychometrics.org/rawdata/) [32]. three datasets were considered. the first dataset was collected from the taylor manifest anxiety scale (tmas). it consisted of 50 true-false statements to identify the eligibility criteria for individuals for enrolling in the stress studies and other related psychological phenomena [33]. the second dataset was obtained from the nature relatedness scale (nr-6) to measure how an individual’s trait level of feeling emotionally connected to nature [34]. the test consisted of six statements of opinions rated on a five-point scale of how many people agree with each of the statements. the third dataset was collected from the right-wing authoritarianism scale (rwas) to understand the psychologies of fascist regimes and their followers [35] [36]. a total of 22 statements of opinions was used in the test rated on a nine-point scale from very strongly disagree (1) to very strongly agree (9). not all observations in the datasets are used in building the cfa models, but instead, the samples were randomly selected from the main datasets, thus each study using 750-1500 observations. table 1 summarizes model fit indices from fitting a one-factor model to the three studies. the chi-square tests were not able to detect a good fit model across all studies. this can be expected since the test is highly sensitive to sample size. alternatively, rmsea statistics can be used to assess model fit. in the tmas study, the estimated rmsea value based on uls was above 0.05 or associated with a p-value < 0.001, indicating a poor fit. this is in line with the findings in figure 5, that uls performed the worst when two response categories are used. concerning cfi and tli indices, both ml and gls were not able to reach satisfactory fit since both values were below 0.9, particularly the tli index obtained from the gls method showed a strong underestimation. this is in agreement with the results shown in figure 6. dwls outperformed the other methods in the study with two likert scale points. meanwhile, in the nr-6 and rwas studies, both dwls and uls performed almost equally well. in the case of the rwas study, only uls indicated a good fit based on rmsea criteria. the associated factor loading estimates are reported in table 2. all loadings were statistically significant at a 5% level, irrespective of the estimation methods used. however, all coefficients based on ml and gls were consistently lower than dwls and uls. table 1. summary of model fit indices for tmas, nr-6, and rwas studies (cases in boldface indicated a good model fit) study estimation method chi-square test rmsea cfi tli statistic df p-value statistic p-value tmas (50 indicators, 2-point scale) ml 14494.4 1175 <0.001 0.050 0.203 0.718 0.706 gls 8369.3 1175 <0.001 0.037 1.000 0.210 0.176 dwls 13559.6 1175 <0.001 0.049 0.999 0.945 0.943 uls 0.070 <0.001 0.952 0.950 nr-6 (6 indicators, 5-point scale) ml 133.5 9 <0.001 0.095 <0.001 0.964 0.939 gls 117.9 9 <0.001 0.089 <0.001 0.842 0.737 dwls 62.2 9 <0.001 0.062 0.074 0.996 0.993 uls 0.043 0.183 0.993 0.989 rwas (22 indicators, 9-point scale) ml 1923.6 209 <0.001 0.105 <0.001 0.855 0.840 gls 1017.0 209 <0.001 0.072 <0.001 0.289 0.214 dwls 1361.8 209 <0.001 0.087 <0.001 0.992 0.991 uls 0.050 0.507 0.994 0.993 https://openpsychometrics.org/rawdata/ a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 341 table 2. estimated standardized factor loadings for tmas, nr-6, and rwas studies (only the first five-factor loadings are presented) study indicator ml gls dwls uls tmas tmas1 0.320 0.334 0.417 0.419 tmas2 0.324 0.353 0.451 0.444 tmas3 0.258 0.266 0.337 0.344 tmas4 0.339 0.353 0.450 0.449 tmas5 0.277 0.285 0.358 0.361 nr-6 nr-61 0.503 0.527 0.557 0.560 nr-62 0.540 0.546 0.586 0.588 nr-63 0.779 0.793 0.837 0.805 nr-64 0.598 0.622 0.726 0.733 nr-65 0.848 0.846 0.891 0.899 rwas rwas1 0.559 0.660 0.638 0.612 rwas2 0.715 0.792 0.798 0.791 rwas3 0.806 0.857 0.859 0.834 rwas4 0.757 0.811 0.831 0.820 rwas5 0.672 0.771 0.737 0.715 discussion the comparison of cfa model performance using different estimation methods (ml, gls, dwls, and uls) was discussed in this study under various experimental conditions such as factor dimensionality, number of indicators, number of likert scale points, and sample size. it is worth highlighting that each method has its strengths and shortcomings. this study gives clear evidence that dwls and uls outperform the ml method in all conditions. meanwhile, gls presents the worst performance, particularly related tli index to measure model fit. uls and dwls consistently provide accurate factor loading estimates even in smaller sample sizes. this study also reveals that using at least a 5-point likert scale, the data can be treated as continuous and use the ml method to estimate the model parameters since the resulting parameter values are found to have trivial bias. however, the drawback is that it is very unlikely to achieve a satisfactory fit based on the chi-square test, and it is more profound when using more indicators in a two-factor model. this, of course, has implications in empirical studies. the estimated parameters can only be interpreted and be trusted once the model shows no lack of fit. the choice of estimators (ml, dwls, and uls) when evaluating model fit based on rmsea do not seem to have an impact on the rate of rejection, since all methods are able to control the type i error at a 5% level. with an exception of uls, which performs poorly under small sample size and two-response categories. the result from the empirical study also supports these findings. the model fit based on tli and cfi performs almost equally well, except ml that only works best when the number of response categories increases. the tli index is rather conservative under gls and it is very unlikely to achieve a satisfactory fit as compared to cfi, particularly with 3 or fewer response categories in the condition of a small sample size. a monte carlo simulation studi to assess estimation methods in cfa on ordinal data nina fitriyati 342 it is important to note that any working recommendations provided in this study are based on the current model configurations. this research did not take into account the possible violation such as non-normality in latent distribution and asymmetric threshold. these types of violations would be interesting to investigate and see how the impact of different estimation methods on model performance. a future study should also address the estimators' behavior when the cfa model is not correctly specified, such as omitting important factor loadings and ignoring significant inter-correlation between two or more factors. conclusions the present findings suggest the importance to select appropriate estimation methods based on how the data are measured. the study focuses on a type of data collected using a survey instrument measured in a likert-type scale, which is ordinal in nature. maximum likelihood as the most common estimation method used, which has several nice properties, failed to give accurate parameter estimation when the data are ordinal, particularly with fewer numbers of likert scale points. in addition, the goodness of fit test, particularly the 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[36] b. altemeyer, the authoritarians, university of manitoba, 2007. on the study of rainbow antimagic coloring of special graphs cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 585-596 p-issn: 2086-0382; e-issn: 2477-3344 submitted: october 19, 2022 reviewed: february 20, 2023 accepted: march 20, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.17836 on the study of rainbow antimagic coloring of special graphs dafik1,2,*, riniatul nur wahidah 1,2, ermita rizki albirri 1,2 , sharifah kartini said husain 3 1mathematics edu. depart. university of jember, indonesia 2pui-pt combinatorics and graph, cgant, university of jember, indonesia 3institute for mathematical research, university putra malaysia, malaysia email: d.dafik@unej.ac.id abstract let 𝐺 be a connected graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). the bijective function 𝑓:𝑉(𝐺) → {1,2,…,|𝑉(𝐺)|} is said to be a labeling of graph where 𝑤(𝑥𝑦) = 𝑓(𝑥) + 𝑓(𝑦) is the associated weight for edge 𝑥𝑦 ∈ 𝐸(𝐺). if every edge has different weight, the function 𝑓 is called an edge antimagic vertex labeling. a path 𝑃 in the vertex-labeled graph 𝐺, with every two edges 𝑥𝑦,𝑥′𝑦′ ∈ 𝐸(𝑃) satisfies 𝑤(𝑥𝑦) ≠ 𝑤(𝑥′𝑦′) is said to be a rainbow path. the function 𝑓 is called a rainbow antimagic labeling of 𝐺, if for every two vertices 𝑥,𝑦 ∈ 𝑉(𝐺), there exists a rainbow 𝑥 − 𝑦 path. graph 𝐺 admits the rainbow antimagic coloring, if we assign each edge 𝑥𝑦 with the color of the edge weight 𝑤(𝑥𝑦). the smallest number of colors induced from all edge weights of edge antimagic vertex labeling is called a rainbow antimagic connection number of 𝐺, denoted by 𝑟𝑎𝑐(𝐺). in this paper, we study rainbow antimagic connection numbers of octopus graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sun flower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi jahangir graph 𝐽𝑛. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc by-sa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: antimagic labeling; rainbow coloring; rainbow antimagic connection number; special graphs introduction the definition of graph used in this paper follows from chartrand and zhang [9]. in the latest days, graph theory has many applications, one of them is graph coloring. the application of graph coloring can be found in many area, such as data mining, image segmentation, clustering, image capturing, networking. chartrand, et al. [10] extended the graph coloring concept into a rainbow coloring of graph. let 𝑐:𝐸(𝐺) → {1,2, . . . ,𝑘},𝑘 ∈ ℕ be the edge coloring of a connected graph where the two adjacent edges may have the same color. if for every two vertices 𝑥,𝑦 ∈ 𝑉(𝐺), there exists a rainbow 𝑥 − 𝑦 path, if no two edges of the 𝑥 − 𝑦 path are the same color, then the path is called a rainbow path. a coloring of graph 𝐺 is said to be rainbow connection, if for every two vertices 𝑥,𝑦 ∈ 𝑉(𝐺) have a rainbow 𝑥 − 𝑦 path. the edge colored 𝐺 which every two different vertices have a rainbow connection is called rainbow coloring of graph, see [10]. some results in regards to the concept of rainbow coloring of graphs can been found by nabila, et al [21] and ma, et al. [19]. some other type of rainbow coloring are rainbow vertex coloring and rainbow total coloring. some relevant results of rainbow vertex coloring can be found in lie. h, et al. [15], http://dx.doi.org/10.18860/ca.v7i4.17836 https://creativecommons.org/licenses/by-sa/4.0/ on the study of rainbow antimagic coloring of special graphs dafik 586 bustan et al. [8] and li. x et al. [17], while some results of total rainbow coloring can be found results in lie. h et al. [16] and ma. y et al.[20]. furthermore, the other concepts in graph theory is graph labeling, one of the concept of graph labeling is an antimagic labeling of graph 𝐺, defined by hartsfield and ringel [13]. baca et al. has found some antimagic labeling results in [4], [5], [6]. moreover, some results on antimagic labeling have been contributed by dafik 𝑒𝑡 𝑎𝑙. in [11]. in addition, the research on antimagic labeling can also be found in several papers [2], [22], [25]. arumugam et al. [3], defined a new concept by combining graph coloring and graph labeling. the bijective function 𝑓 ∶ 𝐸(𝐺) → {1,2, . . . , |𝐸(𝐺)|}, the vertex weight of the vertex 𝑥 is 𝑤(𝑥) = ∑ 𝑓(𝑥𝑦) 𝑥𝑦∈𝐸(𝑥) and 𝐸(𝑥) is the set of edges incident to 𝑥 for every 𝑥 ∈ 𝑉(𝐺). if for every two adjacent vertices 𝑥,𝑦 ∈ 𝑉 (𝐺),𝑤(𝑥) ≠ 𝑤(𝑦), then the bijective function 𝑓 is called a local antimagic labeling. so, each local antimagic label is a vertex coloring in 𝐺 with vertex 𝑥 colored with 𝑤(𝑥). based on the definition of arumugam [3], dafik 𝑒𝑡 𝑎𝑙. [12] defined the combination of the concepts of antimagic labeling and rainbow coloring into a new concept called rainbow antimagic coloring. in this study, we will study the combination of rainbow coloring and antimagic labeling, and it tends to the new notion, namely a rainbow antimagic coloring. the lower bound of the rainbow antimagic connection number has been determined in septory et al. stated in the following lemma. lemma 1. let 𝐺 be any connected graph. let 𝑟𝑐(𝐺) and δ(𝐺) be the rainbow connection number of 𝐺 and the maximum degree of 𝐺, 𝑟𝑎𝑐(𝐺) ≥ 𝑚𝑎𝑥 {𝑟𝑐(𝐺),δ(𝐺)}. while dafik et al. also characterised the existence of rainbow 𝑢 − 𝑣 path of any graph of 𝑑𝑖𝑎𝑚(𝐺) ≤ 2 in the following theorem. theorem 1. let 𝐺 be a connected graph of diameter 𝑑𝑖𝑎𝑚(𝐺) ≤ 2. let 𝑓 be any bijective function from 𝑉(𝐺) to the set {1,2,…, |𝑉(𝐺)| }, there exists a rainbow 𝑥 − 𝑦 path. some other results in regards on this notion can be read on [1], [7], [12], [14], [23] and [24]. in this paper, we will study the rainbow antimagic connection number of octopus graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sun flower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi jahangir graph 𝑆𝐽𝑛. method to determine the number of rainbow antimagic coloring of graph, we use the following steps: 1. for any graph 𝐺, identify the set of vertices 𝑉(𝐺) and set of edges 𝐸(𝐺). 2. analyze the lower bound of rainbow antimagic connection number (𝑟𝑎𝑐) based on lemma: 𝑟𝑎𝑐(𝐺) ≥ max {𝑟𝑐(𝐺),δ(𝐺)}. 3. label the vertices of the graph 𝐺 with the function: 𝑉(𝐺) → {1,2,3, . . . , |𝑉(𝐺)|}. 4. determine the edge weight based on the sum of vertex label which incident with the edge. to calculate edge weight we give the function, 𝑤(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) for 𝑢,𝑣 𝜖 𝑉(𝐺). 5. verify that every two vertex in the graph 𝐺 have rainbow paths. if not, repeat the step 3. 6. determine the upper bound of 𝑟𝑎𝑐(𝐺) from the number of different edge weight. 7. the exact value of rainbow antimagic connection number can be determined if lower on the study of rainbow antimagic coloring of special graphs dafik 587 bound is the same with upper bound of rainbow antimagic connection number. on the study of rainbow antimagic coloring of special graphs dafik 588 results and discussion in this section, we will show our new results on those graph above stated in a theorem. we start to write the theorem, provide the cardinality of the graph, obtain lower and upper bound, establish the rainbow antimagic connection number and show the existence of rainbow path for any to vertices and finally conclude the proof. theorem 2. for 𝑛 ≥ 3 , 𝑟𝑎𝑐(𝑂𝑛) = 2𝑛 . 𝑃𝑟𝑜𝑜𝑓. the octopus graph 𝑂𝑛 is a graph with vertex set 𝑉( 𝑂𝑛 ) = {𝑥} ∪ { 𝑦𝑖, 𝑧𝑖,1 ≤ 𝑗 ≤ 𝑛}, and edge set 𝐸(𝑂𝑛) = {𝑥𝑦𝑖,𝑥𝑧𝑖,1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑦𝑖𝑦𝑖+1, 1 ≤ 𝑖 ≤ 𝑛 − 1}. the cardinality of vertex set is |𝑉(𝑂𝑛)| = 2𝑛 + 1 and the cardinality of edge set is |𝐸(𝑂𝑛)| = 3𝑛 − 1. based on definition of octopus graph, the graph 𝑂𝑛 has maximum degree of δ (𝑂𝑛) = 2𝑛. to prove the rainbow antimagic connection number of 𝑂𝑛, the first step is to determine the lower bound of 𝑟𝑎𝑐(𝑂𝑛). based on lemma 1. we have 𝑟𝑎𝑐(𝑂𝑛) ≥ δ(𝑂𝑛). since, the labels of the vertices with the bijection 𝑓:𝑉(𝑂𝑛) → {1,2,…,|𝑉(𝑂𝑛)|}, we have 𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). it implies for each edge 𝑢𝑥,𝑣𝑥 ∈ 𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). thus 𝑟𝑎𝑐 (𝑂𝑛) ≥ 2𝑛. the second step is to determine the upper bound of 𝑟𝑎𝑐(𝑂𝑛). define the vertex labeling 𝑓 ∶ 𝑉(𝑂𝑛) → {1,2, . . . ,2𝑛 + 1 } as follows. 𝑓(𝑥) = 1 𝑓(𝑦𝑖) = { 3+𝑖 2 , for 𝑖 is odd 3+𝑛+𝑖 2 , for 𝑖 is even,𝑛 is odd 2+𝑛+𝑖 2 , for 𝑖 is even,𝑛 is even 𝑓(𝑧𝑖) = 𝑛 + 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 the edge weight 𝑓 can be expressed as 𝑤(𝑥𝑧𝑖) = 2 + 𝑛 + 𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑥𝑦𝑖) = { 5+𝑖 2 , for 𝑖 is odd 5+𝑛+𝑖 2 , for 𝑖 is even,𝑛 is odd 4+𝑛+𝑖 2 , for 𝑖 is even,𝑛 is even 𝑤(𝑦𝑖𝑦𝑖+1) = { 7+2𝑖+𝑛 2 , for 1 ≤ 𝑖 ≤ 𝑛,𝑛 is odd 6+2𝑖+𝑛 2 , for 1 ≤ 𝑖 ≤ 𝑛,𝑛 is even the next step is to count the number of different edge weights inducing the rainbow antimagic coloring on the graph 𝑂𝑛. the edge weights are included in the sets 𝑤(𝑥𝑦𝑖) = {3,4,5,…,𝑛 + 2} and 𝑤(𝑥𝑧𝑖) = {𝑛 + 3,𝑛 + 4,𝑛 + 5,…,2𝑛 + 2}. the number of distinct colors of 𝑤(𝑥𝑦𝑖) ∪ 𝑤(𝑥𝑧𝑖) is 2𝑛. to prove this number, we use the formula of an arithmetic sequence formula. the following is an illustration of determining the number of distinct colors. 𝑈𝑠 = 𝑎 + (𝑠 − 1)𝑑 2𝑛 + 2 = 3 + (𝑠 − 1)1 2𝑛 + 2 = 3 + 𝑠 − 1 𝑠 = 2𝑛 on the study of rainbow antimagic coloring of special graphs dafik 589 it implies that the edge weight 𝑓 ∶ 𝑉(𝑂𝑛) → {1,2, . . . ,2𝑛 + 1} induces a rainbow antimagic coloring of 2𝑛 colors. therefore 𝑟𝑎𝑐 (𝑂𝑛 ) ≤ 2𝑛. combining two bounds, we have the exact value of 𝑟𝑎𝑐 (𝑂𝑛) = 2𝑛. the last is to show the existence of the rainbow 𝑥 − 𝑦 path of 𝑂𝑛. according to the theorem 2, since 𝑑𝑖𝑎𝑚(𝑂𝑛) = 2, for every two vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. it completes the proof. the illustration of a rainbow antimagic coloring of octopus graph 𝑂𝑛 can be seen in figure 1. figure 1. the illustration of rainbow antimagic coloring of octopus graph 𝑂7 theorem 3. for 𝑛 ≥ 3, 𝑟𝑎𝑐(𝑆𝑡𝑛) = 3𝑛 . 𝑃𝑟𝑜𝑜𝑓. the sandat graph 𝑆𝑡𝑛 is a graph with vertex set 𝑉( 𝑆𝑡𝑛 ) = {𝑎} ∪ { 𝑥 𝑖,𝑦𝑖,𝑧𝑖,1 ≤ 𝑖 ≤ 𝑛} and edge set 𝐸(𝑆𝑡𝑛) = {𝑎𝑥𝑖,𝑎𝑦𝑖,𝑎𝑧𝑖,𝑥𝑖𝑦𝑖,𝑦𝑖𝑧𝑖 1 ≤ 𝑖 ≤ 𝑛}. the cardinality of vertex set is |𝑉(𝑆𝑡𝑛)| = 3𝑛 + 1 and the cardinality of edge set is |𝐸(𝑆𝑡𝑛)| = 5𝑛. based on definition of sandat graph, the graph 𝑆𝑡𝑛 has maximum degree of δ (𝑆𝑡𝑛) = 3𝑛. to prove the rainbow antimagic connection number of 𝑆𝑡𝑛 , the first step is to determine the lower bound of 𝑟𝑎𝑐(𝑆𝑡𝑛). based on lemma 1. we have 𝑟𝑎𝑐(𝑆𝑡𝑛) ≥ δ(𝑆𝑡𝑛). since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝑡𝑛) → {1,2,…,|𝑉(𝑆𝑡𝑛)|}, we have 𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). it implies for each edge 𝑢𝑥,𝑣𝑥 ∈ 𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). thus 𝑟𝑎𝑐 (𝑆𝑡𝑛) ≥ 3𝑛. the second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝑡𝑛). define the vertex labeling 𝑓 ∶ 𝑉(𝑆𝑡𝑛) → {1,2, . . . ,3𝑛 + 1 } as follows. 𝑓(𝑎) = 2 𝑓(𝑥𝑖) = 3𝑛 + 3 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑦𝑖) = { 1 , for 𝑖 = 𝑛 𝑖 + 1 , for ≤ 𝑖 ≤ 𝑛 𝑓(𝑧𝑖) = 3𝑛 + 2 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 the edge weight 𝑓 can be expressed as 𝑤(𝑎𝑥𝑖) = 3𝑛 + 5 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 on the study of rainbow antimagic coloring of special graphs dafik 590 𝑤(𝑎𝑦𝑖) = { 3 , for i = 1 𝑖 + 3 , for 2 ≤ 𝑖 ≤ 𝑛 𝑤(𝑎𝑧𝑖) = 3𝑛 + 4 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑥𝑖𝑦𝑖) = { 3𝑛 + 2 , for 𝑖 = 1 3𝑛 + 4 − 𝑖 , for 2 ≤ 𝑖 ≤ 𝑛 𝑤(𝑦𝑖𝑧𝑖) = { 3𝑛 + 1 , for 𝑖 = 1 3𝑛 + 3 − 𝑖 , for 2 ≤ 𝑖 ≤ 𝑛 the next step is to count the number of different edge weights inducing the rainbow antimagic coloring on the graph 𝑆𝑡𝑛. the edge weights are included in the sets 𝑤(𝑎𝑥𝑖) ∪ 𝑤(𝑎𝑦𝑖) ∪ 𝑤(𝑎𝑧𝑖) ∪ 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑧𝑖) = {5,6,7,…,3𝑛 + 3 }. the number of distinct colors of 𝑤(𝑎𝑥𝑖) ∪ 𝑤(𝑎𝑦𝑖) ∪ 𝑤(𝑎𝑧𝑖) ∪ 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑧𝑖) is 3𝑛. based on edge weights the number of edge wights is determined in the same way in theorem 2. it implies that the edge weight 𝑓 ∶ 𝑉(𝑆𝑡𝑛) → {1,2, . . . ,3𝑛 + 1} induces a rainbow antimagic coloring of 3𝑛 colors. therefore 𝑟𝑎𝑐 (𝑆𝑡𝑛 ) ≤ 3𝑛. combining two bounds, we have the exact value of 𝑟𝑎𝑐 (𝑆𝑡𝑛) = 3𝑛. the last is to show the existence of the rainbow 𝑥 − 𝑦 path of 𝑆𝑡𝑛. according to the theorem 1, since 𝑑𝑖𝑎𝑚(𝑆𝑡𝑛) = 2, for every two vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. it completes the proof. the illustration of a rainbow antimagic coloring of sandat graph 𝑆𝑡𝑛 can be seen in figure 2. figure 2. the illustration of rainbow antimagic coloring of sandat graph 𝑆𝑡6. theorem 4. for 𝑛 ≥ 4, 𝑟𝑎𝑐(𝑆𝑓𝑛) = 3𝑛 . 𝑃𝑟𝑜𝑜𝑓. the sunflower graph 𝑆𝑓𝑛 is a graph with vertex set 𝑉( 𝑆𝑓𝑛 ) = {𝑐} ∪ { 𝑥 𝑖,𝑦𝑖,𝑧𝑖,1 ≤ 𝑖 ≤ 𝑛} and edge set 𝐸(𝑆𝑓𝑛) = {𝑐𝑥𝑖,𝑐𝑦𝑖,𝑐𝑧𝑖,𝑦𝑖𝑧𝑖,𝑧𝑖𝑧𝑖+1,1 ≤ 𝑖 ≤ 𝑛}. the cardinality of vertex set is |𝑉(𝑆𝑓𝑛)| = 3𝑛 + 1 and the cardinality of edge set is |𝐸(𝑆𝑓𝑛)| = 5𝑛. based on definition of sunflower graph, the graph 𝑆𝑓𝑛 has maximum degree of δ (𝑆𝑓𝑛) = 3𝑛. to prove the rainbow antimagic connection number of 𝑆𝑓𝑛 , the first step is to determine the lower bound of 𝑟𝑎𝑐(𝑆𝑓𝑛). based on lemma 1. we have 𝑟𝑎𝑐(𝑆𝑓𝑛) ≥ δ(𝑆𝑓𝑛). since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝑓𝑛) → {1,2,…,|𝑉(𝑆𝑓𝑛)|}, we have on the study of rainbow antimagic coloring of special graphs dafik 591 𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). it implies for each edge 𝑢𝑥,𝑣𝑥 ∈ 𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). thus 𝑟𝑎𝑐 (𝑆𝑓𝑛) ≥ 3𝑛. the second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝑓𝑛). define the vertex labeling 𝑓 ∶ 𝑉(𝑆𝑓𝑛) → {1,2, . . . ,3𝑛 + 1 } as follows. 𝑓(𝑐) = 1 𝑓(𝑥𝑖) = 2𝑛 + 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑦𝑖) = 2𝑛 − 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑧𝑖) = 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 the edge weight 𝑓 can be expressed as 𝑤(𝑐𝑥𝑖) = 2𝑛 + 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑐𝑦𝑖) = 2𝑛 − 𝑖 + 3 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑐𝑧𝑖) = 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑦𝑖𝑧𝑖) = 2𝑛+3 , for 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑧𝑖𝑧𝑖+1) = { 2𝑖 + 3 , for 1 ≤ 𝑖 ≤ 𝑛 − 1 𝑛 + 3 , for 𝑖 = 𝑛 the next step is to count the number of different edge weights inducing the rainbow antimagic coloring on the graph 𝑆𝑓𝑛. the edge weights are included in the sets 𝑤(𝑐𝑥𝑖) ∪ 𝑤(𝑐𝑦𝑖) ∪ 𝑤(𝑐𝑧𝑖) = {3,4,5,…,3𝑛 + 2 },𝑤(𝑦𝑖𝑧𝑖) = {2𝑛 + 3} and 𝑤(𝑧𝑖𝑧𝑖+1) = {𝑛 + 3} ∪ {5,7,9,…2𝑛 + 1}. the number of distinct colors of 𝑤(𝑐𝑥𝑖) ∪ 𝑤(𝑐𝑦𝑖) ∪ 𝑤(𝑐𝑧𝑖) ∪ 𝑤(𝑦𝑖𝑧𝑖) ∪ 𝑤(𝑧𝑖𝑧𝑖+1) is 3𝑛. based on edge weights the number of edge wights is determined in the same way in theorem 2. it implies that the edge weight 𝑓 ∶ 𝑉(𝑆𝑓𝑛) → {1,2, . . . ,3𝑛 + 1} induces a rainbow antimagic coloring of 3𝑛 colors. therefore 𝑟𝑎𝑐 (𝑆𝑓𝑛 ) ≤ 3𝑛. combining two bounds, we have the exact value of 𝑟𝑎𝑐 (𝑆𝑓𝑛) = 3𝑛. the last is to show the existence of the rainbow 𝑥 − 𝑦 path of 𝑆𝑓𝑛. according to the theorem 1, since 𝑑𝑖𝑎𝑚(𝑆𝑓𝑛) = 2, for every two vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. it completes the proof. the illustration of a rainbow antimagic coloring of sunflower graph 𝑆𝑓𝑛 can be seen in figure 3. on the study of rainbow antimagic coloring of special graphs dafik 592 figure 3. the illustration of rainbow antimagic coloring of sunflower graph 𝑆𝑓6. theorem 5. for 𝑛 ≥ 3, 𝑟𝑎𝑐(𝑉𝑛) = 𝑛 + 2 . 𝑃𝑟𝑜𝑜𝑓. the volcano 𝑉𝑛 is a graph with vertex set 𝑉( 𝑉𝑛 ) = {𝑥1,𝑥2,𝑥3} ∪ {𝑦𝑖,1 ≤ 𝑖 ≤ 𝑛} and edge set 𝐸(𝑉𝑛) = {𝑥1𝑥2,𝑥2𝑥3,𝑥3𝑥1} ∪ {𝑥𝑖𝑦𝑖,1 ≤ 𝑖 ≤ 𝑛}. the cardinality of vertex set is |𝑉(𝑉𝑛)| = 𝑛 + 3 and the cardinality of edge set is |𝐸(𝑉𝑛)| = 𝑛 + 3. based on definition of volcano graph, the graph 𝑉𝑛 has maximum degree of δ (𝑉𝑛) = 𝑛 + 2. to prove the rainbow antimagic connection number of 𝑉𝑛 , the first step is to determine the lower bound of 𝑟𝑎𝑐(𝑉𝑛). based on lemma 1. we have 𝑟𝑎𝑐(𝑉𝑛) ≥ δ(𝑉𝑛). since, the labels of the vertices with the bijection 𝑓:𝑉(𝑉𝑛) → {1,2,…, |𝑉(𝑉𝑛)|}, we have 𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). it implies for each edge 𝑢𝑥,𝑣𝑥 ∈ 𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). thus 𝑟𝑎𝑐 (𝑉𝑛) ≥ 𝑛 + 2. the second step is to determine the upper bound of 𝑟𝑎𝑐(𝑉𝑛). define the vertex labeling 𝑓 ∶ 𝑉(𝑉𝑛) → {1,2, . . . ,𝑛 + 3 } as follows. 𝑓(𝑥1) = 1 𝑓(𝑥2) = 2 𝑓(𝑥3) = 3 𝑓(𝑦𝑖) = 𝑖 + 3 the edge weight 𝑓 can be expressed as 𝑤(𝑥1𝑥2) = 3 𝑤(𝑥2𝑥3) = 5 𝑤(𝑥1𝑥3) = 4 𝑤(𝑥𝑖𝑦𝑖) = 𝑖 + 4 the next step is to count the number of different edge weights inducing the rainbow antimagic coloring on the graph 𝑉𝑛. the edge weights are included in the sets on the study of rainbow antimagic coloring of special graphs dafik 593 𝑤(𝑥1𝑥2) ∪ 𝑤(𝑥2𝑥3) ∪ 𝑤(𝑥1𝑥3) = {3,4,5 } and 𝑤(𝑥𝑖𝑦𝑖) = {5,6,7,…,𝑛 + 4}. the number of distinct colors of 𝑤(𝑥1𝑥2) ∪ 𝑤(𝑥2𝑥3) ∪ 𝑤(𝑥1𝑥3) ∪ 𝑤(𝑥𝑖𝑦𝑖) is 𝑛 + 2. based on edge weights the number of edge wights is determined in the same way in theorem 2. it implies that the edge weight 𝑓 ∶ 𝑉(𝑉𝑛) → {1,2, . . . ,𝑛 + 3} induces a rainbow antimagic coloring of 𝑛 + 2 colors. therefore 𝑟𝑎𝑐 (𝑉𝑛 ) ≤ 𝑛 + 2. combining two bounds, we have the exact value of 𝑟𝑎𝑐 (𝑉𝑛) = 𝑛 + 2. the last is to show the existence of the rainbow 𝑥 − 𝑦 path of 𝑉𝑛. according to the theorem 1, since 𝑑𝑖𝑎𝑚(𝑉𝑛) = 2, for every two vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. it completes the proof. the illustration of a rainbow antimagic coloring of volcano graph 𝑉𝑛 can be seen in figure 4. figure 4. the illustration of rainbow antimagic coloring of volcano graph 𝑉7. theorem 6. for 𝑛 ≥ 3, 𝑟𝑎𝑐(𝑆𝐽𝑛) = 𝑛 . 𝑃𝑟𝑜𝑜𝑓. the semi jahangir graph 𝑆𝐽𝑛 is a graph with vertex set 𝑉( 𝑆𝐽𝑛 ) = {𝑎} ∪ { 𝑥𝑖,1 ≤ 𝑖 ≤ 𝑛} ∪ { 𝑦𝑖,1 ≤ 𝑖 ≤ 𝑛 − 1} and edge set (𝑆𝐽𝑛) = {𝑎𝑥𝑖,1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑦𝑖,𝑦𝑖𝑥𝑖+1,1 ≤ 𝑖 ≤ 𝑛 − 1}. the cardinality of vertex set is |𝑉(𝑆𝐽𝑛)| = 2𝑛 and the cardinality of edge set is |𝐸(𝑆𝐽𝑛)| = 3𝑛 − 2. based on definition of semi jahangir graph, the graph 𝑆𝐽𝑛 has maximum degree of δ (𝑆𝐽𝑛) = 𝑛. to prove the rainbow antimagic connection number of 𝑆𝐽𝑛 , the first step is to determine the lower bound of 𝑟𝑎𝑐(𝑆𝐽𝑛). based on lemma 1. we have 𝑟𝑎𝑐(𝑆𝐽𝑛) ≥ δ(𝑆𝐽𝑛). since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝐽𝑛) → {1,2,…,|𝑉(𝑆𝐽𝑛)|}, we have 𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). it implies for each edge 𝑢𝑥,𝑣𝑥 ∈ 𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). thus 𝑟𝑎𝑐 (𝑆𝐽𝑛) ≥ 𝑛. the second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝐽𝑛). define the vertex labeling 𝑓 ∶ 𝑉(𝑆𝐽𝑛) → {1,2, . . . ,2𝑛} as follows. 𝑓(𝑎) = { 𝑛 , for 𝑛 is odd 𝑛 + 1 , for 𝑛 is even 𝑓(𝑥𝑖) = { 2 , for 𝑖 = 1 2𝑖 + 2 , for 2 ≤ 𝑖 ≤ 𝑛 − 1 4 , for 𝑖 = 𝑛 𝑓(𝑦𝑖) = { 2𝑛 − 2𝑖 + 1 , for 1 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉ − 1 2𝑛 − 2𝑖 − 3 , for ⌈ 𝑛 2 ⌉ ≤ 𝑖 ≤ 𝑛 − 2 on the study of rainbow antimagic coloring of special graphs dafik 594 𝑓(𝑦𝑛−1) = { 𝑖 − 1 , for 𝑛 is odd 𝑖 , for 𝑛 is even the edge weight 𝑓 can be expressed as 𝑤(𝑎𝑥𝑖) = { 𝑛 + 2 , for 𝑛 is odd 𝑛 + 3 , for 𝑛 is even 𝑤(𝑎𝑥𝑖) = { 𝑛 + 2𝑖 + 2 , for 𝑛 is odd,2 ≤ 𝑖 ≤ 𝑛 − 1 𝑛 + 2𝑖 + 3 , for 𝑛 is even,2 ≤ 𝑖 ≤ 𝑛 − 1 𝑤(𝑎𝑥𝑛) = { 𝑛 + 4 , for 𝑛 is odd 𝑛 + 5 , for 𝑛 is even 𝑤(𝑥𝑖𝑦𝑖) = { 2𝑛 + 1 , for 𝑖 = 1 2𝑛 + 3 , for 2 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉ − 1 2𝑛 − 1 , for ⌈ 𝑛 2 ⌉ ≤ 𝑖 ≤ 𝑛 − 2 𝑤(𝑎𝑥𝑛−1𝑦𝑛−1) = { 3𝑛 − 2 , for 𝑛 is odd 3𝑛 − 1 , for 𝑛 is even 𝑤(𝑦𝑖𝑥𝑖+1) = { 2𝑛 + 5 , for 2 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉ − 1 2𝑛 + 1 , for ⌈ 𝑛 2 ⌉ ≤ 𝑖 ≤ 𝑛 − 2 2𝑛 − 5 , for 𝑖 = 𝑛 − 1 the next step is to count the number of different edge weights inducing the rainbow antimagic coloring on the graph 𝑆𝐽𝑛. the edge weights are included in the sets 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑥𝑖+1) = {2𝑛 + 1,2𝑛 + 3,2𝑛 + 5 } and 𝑤(𝑎𝑥𝑖) = {𝑛 + 3,𝑛 + 4,𝑛 + 5,…,3𝑛 + 1}. the number of distinct colors of 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑥𝑖+1) ∪ 𝑤(𝑎𝑥𝑖) is 𝑛. based on edge weights the number of edge wights is determined in the same way in theorem 2. it implies that the edge weight 𝑓 ∶ 𝑉(𝑆𝐽𝑛) → {1,2, . . . ,3𝑛 − 2} induces a rainbow antimagic coloring of 𝑛 colors. therefore 𝑟𝑎𝑐 (𝑆𝐽𝑛 ) ≤ 𝑛. combining two bounds, we have the exact value of 𝑟𝑎𝑐 (𝑆𝐽𝑛) = 𝑛. the last is to show the existence of the rainbow 𝑥 − 𝑦 path of 𝑆𝐽𝑛. suppose we take any 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛), there are two possibilities for 𝑥,𝑦, namely: 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≤ 2 or 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≥ 3. suppose 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≤ 2, based on theorem 1, we must have the rainbow 𝑥 − 𝑦 path. for 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≥ 3, we have two case: first case for path 𝑥𝑖 − 𝑦𝑗 we use the path 𝑥𝑖,𝑎,𝑥𝑗,𝑦𝑗 or 𝑥𝑖,𝑎,𝑥𝑗+1,𝑦𝑗. second case for path 𝑦𝑖 − 𝑦𝑗 we use the path 𝑦𝑖,𝑥𝑖,𝑎,𝑥𝑗,𝑦𝑗 or 𝑦𝑖,𝑥𝑖,𝑎,𝑥𝑗+1,𝑦𝑗 or 𝑦𝑖,𝑥𝑖+1,𝑎,𝑥𝑗,𝑦𝑗 or 𝑦𝑖,𝑥𝑖+1,𝑎,𝑥𝑗+1,𝑦𝑗. thus, for 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) there is a rainbow 𝑥 − 𝑦 path. it completes the proof.the illustration of a rainbow antimagic coloring of semi jahangir graph 𝑆𝐽𝑛 can be seen in figure 5. on the study of rainbow antimagic coloring of special graphs dafik 595 figure 5. the illustration of rainbow antimagic coloring of semi jahangir graph 𝑆𝐽6. concluding remarks based on these results, the authors get the results of the rainbow antimagic connection number on several graphs. the authors finds the exact value of the octopus graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sunflower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi jahangir graph 𝑆𝐽𝑛. based on the results of this study, this study raises an open problem. determine the exact value of the rainbow antimagic connection number of operation of graphs. references [1] al jabbar z l, dafik adawiyah r, albirri e r, agustin i h, on rainbow antimagic coloring of some special graph, journal of physics: conference series. 1465 (1) (2020). 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[25] wang t m, zhang g h, on antimagic labeling of regular graphs with particular factors, journal of discrete algorithms, 23 (2013), 76–82. a fractional-order leslie-gower model with fear and allee effect cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 521-534 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 18, 2022 reviewed: february 23, 2023 accepted: march 05, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.17336 a fractional-order leslie-gower model with fear and allee effect adin lazuardy firdiansyah*, dewi rosikhoh state islamic institute of madura, pamekasan, indonesia email: adin.lazuardy@iainmadura.ac.id abstract to describe the interaction of prey and predator, we consider a predator-prey model based on the lesliegower model. the model is formed by assuming fear effect in the prey and allee effect in predators. in order to account for the memory effect, we apply the caputo fractional-order derivative. the model has four possible equilibrium points, namely the origin, the predator extinction point, the prey extinction point, and all population exist point. here, we show that two local stable points and two unstable points. furthermore, we also investigate the stability changing caused by hopf bifurcation when the order of fractional derivative changes. finally, we perform several simulations to support our analysis results. we observe numerically by using the predictor-corrector method for the local stability, the existence of hopf bifurcation, and the influence of fear factor and allee effect to prey and predator. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: hopf bifurcation; leslie-gower model; local stability introduction the relationship between predators and prey is still a special thing in the development of ecological modeling. the main concern of this situation is how to maintain the availability of the ecosystem resources. in ecology, the presence of prey depends on how they can protect themselves, and the presence of predators depends on the availability of prey [1]. these relationships make researchers interested in forming predator-prey interactions into mathematical models. several modifications have been made by researchers to build models that are more suitable for biological behavior. for example, the interaction between prey and predator considering the fear factor in prey [2]–[5], the influence of allee effect on the availability of prey and predator [6]–[9], the impact of refuge in prey to the existence of predator [10]–[12], and the exploitation of prey and predator by harvesting [6], [12], [13]. in the last decade, the allee effect has received great attention for the population dynamics. the allee effect is divided into two types, namely demographic allee effect and component allee effect. according to [14], [15], the component allee effect is a scenario at the lower population density affected by the positive interaction between growth rate and population density so that can increase their extinction. this effect can make a demographic allee effect to a small population density. to be specific, if the population has a high density, then the competition for food will increase and the growth rate of population will decrease. therefore, the demographic allee effect does not hold for large population density [16]. the scenario occurs as a result of several conditions, such as http://dx.doi.org/10.18860/ca.v7i4.17336 mailto:adin.lazuardy@iainmadura.ac.id https://creativecommons.org/licenses/by-sa/4.0/ a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 522 obligate cooperators, mate finding problem, and anti-predator strategies in prey. we give an example, namely if a population reproduces sexually, then it can increase its density and an individual can find a mate easily. thus, it can reduce the risk of inbreeding. in contrast to the large population density, the small population density has the risk of inbreeding. in the ecological model, the allee effect can be constructed on prey population [6], [7], [17], predator population [9], [16], [18], or both population [8]. according to [16], for the literature on predator-prey model with the allee effect, predators are more prone than their prey because predators are usually smaller than prey population. for example, the spotted owls (strix occidentalis caurina) lost their habitat causing them to be unable to find their mates [19]. in addition, several experimental studies have shown that the presence of predators can change prey behavior even more strongly than the direct predation effect [2]–[4]. fear of prey can affect the physiological state of juvenile prey (e.g., reduce prey reproduction) and be harmful to their survival as adults [3], [5], [20], [21]. for example, sparrows (melospiza melodia) during their breeding season without direct predation using electric fences, and it is found that there is a 40% reduction in the density of offspring due to predation risk [21]. recently, we study the dynamical behavior of predator-prey interaction by assuming that (i) the allee effect occurs in predators and (ii) prey is afraid of predators because prey is always alert to possible predator attacks. biologically, the growth rate of individual should involve previous and current conditions [22]. that is, all current conditions of population density depend on all previous conditions [23]. therefore, we also consider the memory effect which means the effect of all previous biological conditions to the present condition by replacing the first-order derivative with the fractional-order derivative [1], [9], [17], [24]. the memory effect shows that the population dynamics of present condition depend on all previous conditions stored in their memory system such as the experience in foraging, the best place to take shelter, the perfect time to migrate, and so on [1]. to play the superlative form of model, ordinary calculus is less effective in describing complex phenomenon involving memory effect and hereditary biological properties [25]. thus, the fractional calculus is applied to solve the problem. because, it has the ability to describe biological conditions related to the memory effect [26]. there are several well-known fractional-order derivatives that are used as operators in the predator-prey model. by considering the availability of analytical tools, we choose the caputo fractional-order for our model as done by [1], [9], [17], [24]. according to [22], the caputo fractional-order can be used on the classic initial condition as in the integer order equations. it has rich analytical tools in observing the dynamic of predator-prey systems. in this manuscript, we organize several contents as follows. the section 1 presents several methods to solve the model. in the section 2, the mathematical model is formulated to obtain the first-order model and replacing it with the fractional-order derivative operator. in the section 3 and 4, the model is solved to explore the dynamical behaviors by investigating the equilibrium points, local stability, and hopf bifurcation. in the section 4, the analytical results are demonstrated through several numerical simulations. we end this discussion by giving the conclusion in the section 5. a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 523 method in observing the interaction of the predator-prey population, we perform several steps to modify a modified leslie-gower predator-prey model. the steps are presented as follows. 1) reviewing and studying the previous literature related to the problems taken. 2) formulating the modified leslie-gower model by adding the fear factor and allee effect. next, we transform it into a fractional-order model. 3) identifying equilibrium points and local stability in the model. 4) investigating hopf bifurcation in the model. 5) performing several numerical simulations to observe dynamical behavior in the model. the numerical method used in this paper is the predictor-corrector method for fractional-order equations. results and discussion mathematical model the interaction between prey and predator population is presented in a modified leslie-gower model proposed by aziz-alaoui and okiye [27] and yu [28]. for the next research, yu [29] considers a modified leslie-gower model incorporating the beddington-deangelis functional response. the modified leslie-gower model with beddington-deangelis function response can be written as follows. 𝑑𝑁 𝑑𝑇 = 𝑟𝑁 (1 − 𝑁 𝐾 ) − 𝑎𝑁𝑃 1 + 𝑏𝑁 + 𝑐𝑃 , 𝑑𝑃 𝑑𝑇 = 𝑠𝑃 (1 − 𝑒𝑃 𝑘 + 𝑁 ), (1) where 𝑁 = 𝑁(𝑇) and 𝑃 = 𝑃(𝑇) are the density of prey and predator population at time 𝑡. the parameters 𝑟, 𝐾, 𝑏, 𝑐, 𝑤, 𝑠, 𝑒, 𝑘 are positive values. in the particular, the biological meaning of parameters can be shown in table 1. table 1. the biological meaning of parameters in system (1) parameter biological meaning 𝑟 the intrinsic growth rate of prey 𝑠 the intrinsic growth rate of predator 𝑎 the capture rate by predator against prey 𝑏 the measure of handling time by predator against prey 𝑐 the amount of disturbance among predator 𝑒 the reproduction rate of predator 𝐾 the carrying capacity of prey 𝑘 the environmental protection of predator according to [30], the predation of the predator can influence the behavior of prey indirectly resulting in fear. consequently, the protection of frightened prey diminishes and leaves their newborn [20]. therefore, we consider the fear factor multiplying the intrinsic growth rate of prey with 𝑓(𝑢, 𝑃) = 1 1+𝑢𝑃 , where the parameter 𝑢 is the fear rate of prey. biologically, the fear factor satisfies several conditions as follows. a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 524 𝑓(0, 𝑃) = 1, 𝑓(𝑢, 0) = 1, lim 𝑢→∞ 𝑓(𝑢, 𝑃) = 0, lim 𝑢→∞ 𝑓(𝑢, 𝑃) = 0, 𝜕𝑓(𝑢, 𝑃) 𝜕𝑢 < 0, 𝜕𝑓(𝑢, 𝑃) 𝜕𝑃 < 0. the biological meaning of conditions can be shown in [31]. in this article, we are interested to observe the allee effect as done by feng and kang [8] and assume that the allee effect occurs only in predators. therefore, our system becomes the following system. 𝑑𝑁 𝑑𝑇 = 𝑟𝑁 1 + 𝑢𝑃 − 𝑟𝑁2 𝐾 − 𝑎𝑁𝑃 1 + 𝑏𝑁 + 𝑐𝑃 , 𝑑𝑃 𝑑𝑇 = 𝑠𝑃 ( 𝑃 𝑃 + 𝑛 − 𝑒𝑃 𝑘 + 𝑁 ), (2) where the parameter 𝑛 is the measure of the allee effect in predator. for simplicity, system (2) is formed into a non-dimensional system by using parameters (𝑥, 𝑦, 𝑡) → ( 𝑁 𝐾 , 𝑒𝑃 𝐾 , 𝑟𝑇). thus, we obtain the following system. 𝑑𝑥 𝑑𝑡 = 𝑥 1 + 𝜌𝑦 − 𝑥2 − 𝑥𝑦 𝛿 + 𝛽𝑥 + 𝛾𝑦 , 𝑑𝑦 𝑑𝑡 = 𝜃𝑦 ( 𝑦 𝑦 + 𝜇 − 𝑦 𝜈 + 𝑥 ), (3) where 𝜌 = 𝑢𝐾 𝑒 , 𝛿 = 𝑟𝑒 𝑎𝐾 , 𝛽 = 𝑏𝑒𝑟 𝑎 , 𝛾 = 𝑐𝑟 𝑎 , 𝜃 = 𝑠 𝑟 , 𝜇 = 𝑛𝑒 𝐾 , 𝜈 = 𝑘 𝐾 . from system (3), we can see that the system only has 7 dimensionless parameters, that is 𝜌, 𝛿, 𝛽, 𝛾, 𝜃, 𝜇, and 𝜈. nondimensionalization can reduce the number of parameters by grouping them in a meaningful way. in general, the groupings can provide a relative measure of the influence of dimension parameters [32]. for example, 𝜃 is the ratio of growth rates of predator to prey such that 𝜃 > 1 and 𝜃 < 1 have definite ecological significance. that is, prey can reproduce more rapidly than predator. model with caputo operator to form system (3) into a fractional-order model, we use the similar manner as in [1], [22], [24], [33]. by replacing the left-hand sides of the system (3) with the caputo fractional-order derivative, we obtain the following model with 𝛼 is the memory effect. 𝐷∗ 𝛼𝑥 = 𝑥 1 + 𝜌𝑦 − 𝑥2 − 𝑥𝑦 𝛿 + 𝛽𝑥 + 𝛾𝑦 , 𝐷∗ 𝛼𝑦 = 𝜃𝑦 ( 𝑦 𝑦 + 𝜇 − 𝑦 𝜈 + 𝑥 ), (4) where 𝐷∗ 𝛼 shows the caputo fractional-order derivative for a real-valued function 𝑓 defined as follows. 𝐷∗ 𝛼𝑓(𝑡) = 1 𝛤(1 − 𝛼) ∫ (𝑡 − 𝜏)−𝛼𝑓′(𝜏) 𝑡 0 𝑑𝜏, a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 525 with 𝑡 ≥ 0, 𝛤(∙) is euler’s gamma function, 𝑓 ∈ 𝐶𝑛([0, +∞), ℝ), and 𝛼 ∈ [0,1) [34]. when we replace the operator with the caputo fractional-order, our model has an inconsistency time’s dimension between the left-hand sides of model with its right-hand side. to overcome this condition, we can adjust by rescaling all favorable parameters. thus, let 𝜃 = 𝜃𝛼, �̂� = 𝜇𝛼, �̂� = 𝜈, �̂� = 𝜌𝛼, 𝛿 = 𝛿𝛼, �̂� = 𝛽𝛼, and 𝛾 = 𝛾𝛼, we have 𝐷∗ 𝛼𝑥 = 𝑥 1 + �̂�𝑦 − 𝑥2 − 𝑥𝑦 𝛿 + �̂�𝑥 + 𝛾𝑦 , 𝐷∗ 𝛼𝑦 = 𝜃𝑦 ( 𝑦 𝑦 + �̂� − 𝑦 �̂� + 𝑥 ). (5) form model (5), we can simplify by re-denoting all parameters. we remove the hat . ̂ on each parameters. thus, we have the final model as follows. 𝐷∗ 𝛼𝑥 = 𝑥 1 + 𝜌𝑦 − 𝑥2 − 𝑥𝑦 𝛿 + 𝛽𝑥 + 𝛾𝑦 , 𝐷∗ 𝛼𝑦 = 𝜃𝑦 ( 𝑦 𝑦 + 𝜇 − 𝑦 𝜈 + 𝑥 ). (6) equilibrium points and local stability to investigate the existence and stability of the equilibrium points, we use the magtinon condition presented in the following condition. lemma 1. (see [35]) consider the caputo fractional-order system with initial conditions as follows. 𝐷∗ 𝛼�⃗�(𝑡) = 𝑓(𝑡, �⃗�), �⃗�(0) = �⃗�0, where 𝑥 ∈ ℝ𝑛, 𝑛 ∈ ℕ, and 𝛼 = (0,1]. the point �⃗�∗ can be called an equilibrium point when �⃗�∗ satisfies 𝑓(𝑡, �⃗�∗) = 0. moreover, the point �⃗�∗ is locally asymptotically stable when the eigenvalues 𝜆𝑖, 𝑖 = 1,2, … , 𝑛 of the jacobian matrix 𝐽(�⃗� ∗) satisfies |arg(𝜆𝑖)| > 𝛼𝜋 2 . from lemma 1, we identify the equilibrium points of the system (6) by setting 𝐷∗ 𝛼𝑥 = 𝐷∗ 𝛼𝑦 = 0. therefore, we obtain the equilibrium points and their existence condition as follows. 1) the point 𝐸0(0,0) is always exists. it means that both populations are extinct. 2) the point 𝐸1(1,0) is always exists. it explains that the predator is extinct. 3) the point 𝐸2(0, 𝜈 − 𝜇) exists when 𝜈 > 𝜇. it shows that prey is extinct. 4) the interior point 𝐸∗(𝑥∗, 𝑦∗), where 𝑥∗ = 𝑦∗ + 𝜇 − 𝜈 and exists if 𝑦∗ > 𝜈 − 𝜇 with 𝑦∗ is positive roots of the cubic equations as follows. (𝑦∗)3 + 𝜔1(𝑦 ∗)2 + 𝜔2𝑦 ∗ + 𝜔3 = 0, (7) where 𝜔1 = 𝜌(𝜇 − 𝜈)(2𝛽 + 𝛾) + 𝛿𝜌 + 𝛽 + 𝛾 + 𝜌 3𝜌(𝛾 + 𝛽) , a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 526 𝜔2 = 𝛽𝜌(𝜇2 − 2𝜇𝜈 + 𝜈2) + (𝜇 − 𝜈)(𝛿𝜌 + 2𝛽 + 𝛾) − 𝛽 + 𝛿 − 𝛾 + 1 3𝜌(𝛾 + 𝛽) , 𝜔3 = (𝜇 − 𝜈 − 1)(𝛽𝜇 − 𝛽𝜈 + 𝛿) 𝜌(𝛾 + 𝛽) . let 𝑠 = 𝑦∗ + 𝜔1, we obtain 𝑔(𝑠) = 𝑠3 + 3𝑝𝑠 + 𝑞 = 0, (8) where 𝑝 = 𝜔2 − 𝜔1 2 and 𝑞 = 𝜔3 − 3𝜔1𝜔2 + 2𝜔1 3. by using cardan’s method as in [7], we have the existence condition of an equilibrium point as follows. lemma 2. let 𝑦∗ > 𝜈 − 𝜇. the point 𝐸∗(𝑥∗, 𝑦∗) is a positive equilibrium point with 𝑦∗ is positive roots of (7) when it satisfies one of the following conditions. 1) if 𝑞 < 0, then (8) has a single positive root. thus, system (6) has a unique equilibrium point 𝐸∗(𝑠1 − 𝜔1 + 𝜇 − 𝜈, 𝑠1 − 𝜔1) with 𝑠1 > 𝜔1. 2) if 𝑞 > 0 and 𝑝 < 0, then a) if 𝑞2 + 4𝑝3 = 0, then (8) has a positive root of multiplicity two. thus, system (4) has a unique equilibrium point 𝐸∗(√−𝑝 + 𝜇 − 𝜈, √−𝑝). b) if 𝑞2 + 4𝑝3 < 0, then (8) has two positive points. thus, system (6) has two possible equilibrium points, that is, 𝐸1 ∗(𝑠1 − 𝜔1 + 𝜇 − 𝜈, 𝑠1 − 𝜔1) and/or 𝐸2 ∗(𝑠2 − 𝜔1 + 𝜇 − 𝜈, 𝑠2 − 𝜔1) with 𝑠1,2 > 𝜔1. 3) if 𝑞 = 0 and 𝑝 < 0, then (8) has an unique positive root. thus, system (6) has an unique equilibrium point 𝐸∗(√−3𝑝 + 𝜇 − 𝜈, √−3𝑝). it is known that if (8) has two positive roots, then their positive roots are 𝑠1 = (−4𝑞+4√4𝑝3+𝑞2) 2 3 −4𝑝 2(−4𝑞+4√4𝑝3+𝑞2) 1 3 and 𝑠2 = − 𝑠1 2 + √𝑠1 3+4𝑞 2√𝑠1 . meanwhile, if (8) has a positive root, then the positive root is 𝑠1 = (−4𝑞+4√4𝑝3+𝑞2) 2 3 −4𝑝 2(−4𝑞+4√4𝑝3+𝑞2) 1 3 . the local stability analysis is done by identifying the eigenvalue of the jacobian matrix. the jacobian matrix 𝐽(𝐸) for all equilibrium points (𝑥, 𝑦) can be presented as follows. 𝐽(𝐸) = [ 1 1 + 𝜌𝑦 − 2𝑥 − 𝑦(𝛿 + 𝛾𝑦) (𝛿 + 𝛽𝑥 + 𝛾𝑦)2 − 𝑥𝜌 (1 + 𝜌𝑦)2 − 𝑥(𝛿 + 𝛽𝑥) (𝛿 + 𝛽𝑥 + 𝛾𝑦)2 𝜃 ( 𝑦 𝑥 + 𝜈 ) 2 𝜃𝑦(𝑦 + 2𝜇) (𝑦 + 𝜇)2 − 2𝜃𝑦 𝑥 + 𝜈 ] . (9) the local stability of any point is ensured by the following theorems. theorem 1. 𝐸0(0,0) is unstable and 𝐸1(1,0) is a non-hyperbolic point. proof: the jacobian matrix for 𝐸0(0,0) and 𝐸1(1,0) is presented as follows. 𝐽(𝐸0) = [ 1 0 0 0 ], a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 527 𝐽(𝐸1) = [ −1 −𝜌 − 1 𝛽 + 𝛿 0 0 ]. based on the jacobian matrix 𝐽(𝐸0), we know that the eigenvalues of 𝐽(𝐸0) are 𝜆1 = 1 and 𝜆2 = 0. thus, the point 𝐸0 is unstable because |arg(𝜆1)| = 0 < 𝛼𝜋 2 . moreover, for 𝐸1(1,0), the eigenvalues of 𝐽(𝐸1) are 𝜆1 = −1 and 𝜆2 = 0. it is shown that |arg(𝜆2)| = 𝛼𝜋 2 . thus, the point 𝐸1 is a non-hyperbolic point. theorem 2. 𝐸2(0, 𝜈 − 𝜇) is locally asymptotically stable. proof: by substituting 𝐸2 into (9), we obtain 𝐽(𝐸2) = [ 1 1 + 𝜌(𝜈 − 𝜇) − (𝜈 − 𝜇) 𝛿 + (𝜈 − 𝜇)𝛾 0 𝜃 ( 𝜈 − 𝜇 𝜈 ) 2 −𝜃 ( 𝜈 − 𝜇 𝜈 ) 2 ] . from jacobian matrix 𝐽(𝐸2), we obtain the eigenvalues, that is 𝜆1 = 1 1+𝜌(𝜈−𝜇) − (𝜈−𝜇) 𝛿+(𝜈−𝜇)𝛾 and 𝜆2 = −𝜃 ( 𝜈−𝜇 𝜈 ) 2 . by using the existence condition of 𝐸2, it is clear that 𝜆1 < 0 when 1 1+𝜌(𝜈−𝜇) < (𝜈−𝜇) 𝛿+(𝜈−𝜇)𝛾 and 𝜆2 < 0. thus, |arg(𝜆1,2)| = 𝜋 > 𝛼𝜋 2 . in other words, the point 𝐸2 is locally asymptotically stable. theorem 3. suppose that 𝜑1 = 𝑦∗(𝛿 + 2𝛽𝑥∗ + 𝛾𝑦∗) (𝛿 + 𝛽𝑥∗ + 𝛾𝑦∗)2 − 𝜃 ( 𝑦∗ 𝑦∗ + 𝜇 ) 2 − 1 1 + 𝜌𝑦∗ , 𝜑2 = 𝜃 ( 𝑦∗ 𝑦∗ + 𝜇 ) 2 ( 𝑥∗(𝛿 + 𝛽𝑥∗) − 𝑦∗(𝛿 + 2𝛽𝑥∗ + 𝛾𝑦∗) (𝛿 + 𝛽𝑥∗ + 𝛾𝑦∗)2 + 1 + 𝜌(𝑥∗ + 𝑦∗) (1 + 𝜌𝑦∗)2 ) , 𝛼∗ = 2 𝜋 |tan−1 ( √4𝜑2 − 𝜑1 2 𝜑1 )|. 𝐸∗(𝑥∗, 𝑦∗) is locally asymptotically stable if it satisfies one of the following conditions. a) 𝜑1 2 ≥ 4𝜑2, 𝜑1 < 0, and 𝜑2 > 0. b) 𝜑1 2 < 4𝜑2, and if 𝜑1 < 0, or 𝜑1 > 0 and 𝛼 < 𝛼 ∗. proof: the jacobian matrix at 𝐸∗(𝑥∗, 𝑦∗) is presented as follows. 𝐽(𝐸∗) = [ − 1 1 + 𝜌𝑦∗ + 𝑦∗(𝛿 + 2𝛽𝑥∗ + 𝛾𝑦∗) (𝛿 + 𝛽𝑥∗ + 𝛾𝑦∗)2 −𝑥∗ ( 𝜌 (1 + 𝜌𝑦∗)2 + (𝛿 + 𝛽𝑥∗) (𝛿 + 𝛽𝑥∗ + 𝛾𝑦∗)2 ) 𝜃 ( 𝑦∗ 𝑦∗ + 𝜇 ) 2 −𝜃 ( 𝑦∗ 𝑦∗ + 𝜇 ) 2 ] , from jacobian matrix 𝐽(𝐸∗), we have the characteristic equation 𝜆2 − 𝜑1𝜆 + 𝜑2 = 0. therefore, we obtain the eigenvalues 𝜆1,2 = 𝜑1±√𝛬 2 with 𝛬 = 𝜑1 2 − 4𝜑2. suppose 𝜑1 2 ≥ 4𝜑2, a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 528 we have 𝛬 ≥ 0. in this condition, we observe that 𝜆1,2 < 0 when 𝜑1 < 0 and 𝜑2 > 0. consequently, |arg(𝜆1,2)| > 𝛼𝜋 2 . thus, the point 𝐸∗ is locally asymptoticaly stable. however, if 𝜑1 2 < 4𝜑2, then 𝛬 < 0. the eigenvalues 𝜆1,2 are a pair of complex conjugate. by using lemma 1, |arg(𝜆1,2)| > 𝛼𝜋 2 when 𝛼 < 𝛼∗ and one of two conditions, that is 𝜑1 < 0 or 𝜑1 > 0. thus, 𝐸 ∗ is locally asymptoticaly stable. finally, the stability condition of 𝐸∗(𝑥∗, 𝑦∗) is proven. hopf bifurcation hopf bifurcation for a fractional-order system is a change in the stability of systems that enters a limit cycle when there is a pair of complex eigenvalues. based on theorem 3, the order of derivatives can affect the stability of the interior equilibrium point with 𝜑1 2 < 4𝜑2 and 𝜑1 > 0. in this article, we take the order of derivative as the bifurcation parameter. therefore, to ensure the existence of hopf bifurcation, we present the following theorem. theorem 4. (existence of hopf bifurcation) suppose that 𝜑1 2 < 4𝜑2 and 𝜑1 > 0. the point 𝐸∗ enters a hopf bifurcation when 𝛼 passes through 𝛼∗. proof: based on theorem 3, the roots of the characteristic equation for 𝐸∗ are a pair of complex conjugate eigenvalues with positive real parts. it is easy to confirm that 𝑚(𝛼∗) = 0 and 𝑑𝑚(𝛼∗) 𝑑𝛼 | 𝛼=𝛼∗ ≠ 0 with 𝑚(𝛼) = 𝛼𝜋 2 − min 1≤𝑖≤2 |arg(𝜆𝑖)|. according to [36], the point 𝐸 ∗ undergoes a hopf bifurcation when 𝛼 crosses 𝛼∗. numerical simulations to demonstrate the dynamical analysis results, we perform several numerical simulations. in this article, we use the predictor-corrector approximation for fractionalorder equations developed by diethelm, et. al. [37]. because the field data is not available, we choose the parameter values that satisfy the results of the stability conditions obtained from the previous discussion. for the first simulation, we choose several parameter values as follows: 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜇 = 0.3, 𝜈 = 0.5, 𝛼 = 0.9. in this work, we obtain that 𝐸2(0,0.2) is a unique equilibrium point because there are two unstable points, that is 𝐸0(0,0) and 𝐸1(1,0), and a local stable point 𝐸2(0,0.2) which is shown by all solutions converged to 𝐸2. we can see its phase portrait in figure 1. it means that the predator can live for a long time even though the prey has become extinct. it is caused because the great fear of prey causes prey to become extinct. meanwhile, the predator can survive against the allee effect because the environmental protection of predator is very good. a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 529 figure 1. phase portrait of system (6) using the parameter values as follows: 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜇 = 0.3, 𝜈 = 0.5, 𝛼 = 0.9. for the second simulation, we set 𝜇 = 0.4 and the other parameter values are made the same as in the previous simulation. based on lemma 2, the point 𝐸∗ lies in the interior plane. therefore, we can interpret that all populations can live together for a long time because some predator populations die due to increased the allee effect. thus, the population density of predator will descrease. consequently, prey population can survive. now, if we take 𝛼 = 0.6, the solution converges to the equilibrium point 𝐸∗ as in figure 2(a). in other words, the point 𝐸∗ is a local stable point. meanwhile, when we take 𝛼 = 0.9, the solution loses stability and oscillates as in figure 2(b). it shows that the solution enters limit cycles or undergoes a hopf bifurcation which corresponds to theorem 4. therefore, the population density fluctuates when the memory effect is large, that is 𝛼 > 0.9. (a) 𝛼 = 0.6 a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 530 (b) 𝛼 = 0.9 figure 2. time series of system (6) using the parameter values as follows: 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜇 = 0.4, 𝜈 = 0.5 figure 3. time series of system (6) using the parameter values as follows: 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜇 = 0.6, 𝜈 = 0.5 in this work, we want to observe the effect of the order derivative on the fractionalorder system. by using 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜇 = 0.6, 𝜈 = 0.5, we obtain the simulation as in figure 3 with 𝛼 = {0.6,0.7,0.8,0.9,1}. it is obseved that all solutions oscillate and converge to the interior point 𝐸∗. moreover, if we assign the order of derivative with larger values, then we can observe that the solution of system converges more quickly to the equilibrium point. therefore, the order of derivative affects the rate of convergence. it means that the when memory effect is large, then in this case, the prey population can increase for a long time. moreover, the predator populations tend to remain constant. from the previous experiment, we observe that the allee effect greatly affects the dynamical behaviour. therefore, we want to observe the impact of allee effect on the fractional-order system. by using 𝛼 = 0.6 and setting the parameter 𝜇 with varying values, we get the simulation as in figure 4. in this work, it is shown that the number of prey increases when the parameter 𝜇 is large. but, when we take a large parameter 𝜇, the number of predator decreases. thus, the allee effect constant is inversely proportional to the number of predator and directly proportional to the number of prey. this result is the same as the conslusion in [9]. now, we observe the influence of fear effect on the system. by assigning 𝜇 = 0.4 and using the varying parameters 𝜌, we have the simulation as in figure 5. it is obeserved that all populations decrease when the parameter 𝜌 is huge. this a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 531 means that the fear effect constant is inversely proportional to the density of two populations. figure 4. the time series of system (6) using the parameter values as follows: 𝜌 = 1.2, 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜈 = 0.5, 𝛼 = 0.6 figure 5. the time series of system (6) using the parameter values as follows: 𝛿 = 0.1, 𝛽 = 0.8, 𝛾 = 0.3, 𝜃 = 0.6, 𝜈 = 0.5, 𝜇 = 0.4, 𝛼 = 0.6 conclusions this manuscript has studied the predator-prey interaction formed into a fractionalorder leslie-gower model with allee and fear effect. we find two local stable points and two unstable points with certain conditions of existence and stability. we also present the existence of hopf bifurcation by assigning the order of derivative as the bifurcation parameter both analytically and numerically. from the results obtained, when the order of fractional derivative is large, then the solution converges faster. here, we observe that when 𝛼 < 0.9, then the solutions of system are locally asymptotically stable. however, when 𝛼 > 0.9, then the solutions of system are isolated by limit cycle via hopf bifurcation. we also conclude that the allee effect constant is directly proportional to the number of prey and inversely proportional to the number of predator. furthermore, the fear effect constant is inversely proportional to the number of the two populations. a fractional-order lelie-gower model with fear and allee effect adin lazuardy firdiansyah 532 references [1] h. s. panigoro, r. resmawan, a. t. r. sidik, n. walangadi, a. ismail, and c. husuna, “a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey,” jambura j. math., vol. 4, no. 2, pp. 355–366, 2022, doi: 10.34312/jjom.v4i2.15220. 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[37] k. a. i. diethelm, n. j. ford, and a. d. freed, “a predictor-corrector approach for the numerical solution of fractional differential equation,” nonlinear dyn., vol. 29, no. 1–4, pp. 3–22, 2002, doi: https://doi.org/https://doi.org/10.1023/a:1016592219341. spatial analysis of dengue disease in jakarta province cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 535-547 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 30, 2022 reviewed: february 25, 2023 accepted: march 04, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.17423 spatial analysis of dengue disease in jakarta province muhamad sobari1,4*, i gede nyoman mindra jaya2, budi nurani ruchjana3 1post-graduate program in applied statistics, universitas padjadjaran, bandung 2department of statistics, universitas padjadjaran, bandung 3department of mathematics, universitas padjadjaran, bandung 4central bureau of statistics of tasikmalaya regency, west java email: muhamad21059@mail.unpad.ac.id abstract dengue disease is a virus-borne illness spread by the bite of the female aedes aegypti mosquito. jakarta is at risk for dengue disease because it has a lot of people who live in urban slums. the objective of this study is to identify the risk factors that affect the number of dengue disease cases in jakarta by considering spatial dependencies. this study used a spatial autoregressive (sar) model with the queen contiguity spatial weight matrix to account for the spatial dependence. the number of dengue disease cases in jakarta is strongly affected by the number of flood-prone areas, the number of slum neighborhood associations, the population density, the number of hospitals and the number of public health centers per 1,000 people also the spatial lag. when dengue disease cases increase in one sub-district, the number of dengue disease cases in the sub-districts around it will increase as well because of the positive and significant spatial lag coefficient. based on the direct impact, each additional one percent of flood-prone points in one sub-district will increase the number of dengue disease cases in that sub-district by four cases. this research contributes that flood control is important in jakarta to control dengue disease. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: dengue disease; spatial autoregressive; queen contiguity introduction the female aedes aegypti mosquito bite is the primary method of transmission for the dengue virus, which causes dengue disease. these mosquitoes can breed in slum places such as there are pools of water that are not taken care of, dark, and damp [1]. dengue disease is still one of the main health problems that threaten the indonesian people because indonesia is a tropical country that is vulnerable to vector-borne diseases, that is diseases that increase the likelihood and risk of occurrence due to changes in the weather cycle [2]. dengue virus is highly sensitive to changes in average temperature, humidity, and increased rainfall which can affect the life cycle and reproduction of the aedes aegypti mosquito that carries the virus [2], [3]. and indonesia is a tropical country with changes in two seasonal cycles, that is the dry season and the rainy season, so that dengue disease will increase during the rainy season because the weather conditions become humid and waterlogging often occurs due to water channels that do not flow or post-floods that are http://dx.doi.org/10.18860/ca.v7i4.17423 mailto:muhamad21059@mail.unpad.ac.id https://creativecommons.org/licenses/by-sa/4.0/ spatial analysis of dengue disease in jakarta province muhamad sobari 536 not cleaned immediately. based on research [1], slum areas have more potential to become a breeding ground for aedes aegypti mosquitoes, so it is necessary to have a program to eradicate mosquito nests in slum areas to eradicate mosquitoes that carry the dengue virus to reduce dengue disease cases. meanwhile, according to research [4], areas that often flood due to high rainfall have a vulnerability in the health sector, that is dengue disease because of climate change. so, it can be concluded that flooding has an indirect effect on dengue disease through climate change. several studies show that dengue disease is related to mobility and population density and people's living behavior. as done by [5], [6] shows that there will be a rise in dengue disease cases because of increased population density. this shows that dengue disease spreads more easily in areas with a high population density. mosquitoes live in the tropics with warm temperatures, in areas below 1,000 meters sea level in indonesia [7]. jakarta, in particular, is located in a region with ideal conditions for mosquito breeding. in 2019, the province with the highest population density in indonesia which reached 15,328 people per km2 was jakarta [8]. jakarta province is also the highest percentage of urban slum households (the lowest 40 percent of the population) which reached 42.73 percent. urban slum households are defined as households that: do not have access to safe drinking water sources; do not have access to proper sanitation; do not have access to floor area >= 7.2 m2 per capita; and do not have access to proper roof, floor, and wall conditions [9]. jakarta province is vulnerable to the transmission of dengue disease due to high population density, percentage of urban slum households, some characteristics that may flood if the rainfall is high, and an optimum temperature for the breeding of the aedes aegypti mosquito. this is evident in 2019 the dengue disease morbidity rate per 100,000 population jakarta province is the top 10 provinces in indonesia which reached 82.45 [10]. research on the spread of dengue disease with a spatial approach has been carried out by [11] using the moran’s i method and the local indicators of spatial association (lisa) which shows that dengue disease cases, population, population density, temperature, rainfall, and wind speed have positive spatial autocorrelation between villages in padang municipality on the six variables. furthermore, research [12] using the moran and geary's c index method shows that dengue disease transmission in semarang municipality exhibits spatial autocorrelation. both studies have the same conclusion, that is there is a positive spatial autocorrelation in the number of dengue disease cases. another study was conducted by [5] using the spatial autoregressive (sar) model, and the study's findings show that factors that significantly influence the number of dengue disease cases in central java province are number of protected spring facilities, population percentage access to sustainable drinking water, population density, number of village polyclinics per 1,000 population, number of public health centers per 1,000 population, and percentage of clean water quality free of bacteria, fungi, and chemicals. in addition, research by [13] comparing spatial durbin model (sdm) and sar model revealed that sar performed better than sdm in predicting the factors that influence the transmission of dengue disease in central java province. in general, the number of residents and the average length of schooling are factors that affect the spread of dengue disease in central java province. the two studies have something in common, that is the unit observation and analysis is the regency and municipality in central java province. this study looks at how various environmental and social issues in jakarta influence the spatial analysis of dengue disease in jakarta province muhamad sobari 537 spread of dengue disease by accounting for spatial dependencies. relying on this background, the objective of this study is to identify risk factors that significantly affect the number of dengue disease cases in jakarta province, using subdistricts as observation and analysis units. methods data and variables this study relied on secondary data from the central bureau of statistics of jakarta province (bps jakarta province) and the provincial government of jakarta's office of communication, information, and statistics (diskominfotik jakarta province). the data were obtained from publications published by bps jakarta [14] and from websites managed by diskominfotik jakarta province [15]. the unit observation and analysis in this study covers 42 out of 44 sub-districts jakarta province in 2017. the data used are not all sub-districts in jakarta province and 2017 dataset is due to the limited data availability. the two sub-districts not included in this study are all sub-districts in kepulauan seribu regency. the following variables were used in this study: table 1. data source and variable name notation variable name data source y number of dengue disease cases diskominfotik jakarta province x1 number of flood-prone points diskominfotik jakarta province x2 number of slum neighborhood associations (rt) bps jakarta province x3 population density (people per km2) diskominfotik jakarta province x4 number of hospitals per 1,000 population diskominfotik jakarta province x5 number of public health centers per 1,000 population diskominfotik jakarta province data analysis steps data processing was carried out using r software version 4.1.2 and thematic map creation using q.gis desktop 3.16.15. steps of data analysis were carried out as follows: 1. exploring data for all variables using thematic maps so that the pattern of distribution of data between sub-districts can be known; 2. before modeling with spatial regression, the classical assumptions of multiple linear regression models must be tested. [16]; 3. before performing moran’s i test, it is necessary to create a spatial weight matrix. the most common way to represent spatial data relationships is through the concept of contiguity. that is, areas will be considered related if their boundaries have the same points. in the concept of queen contiguity every region that touches the boundary of another region, either a side or a corner, is considered a neighbor [17]. queen contiguity is the spatial weight matrix used in this study; 4. checking whether there is an autocorrelation between sub-districts by conducting the moran's i test (moran index) [18]. the hypothesis of moran's i test is as follows: 𝐻0 : no spatial autocorrelation under given w (spatial weight matrix) 𝐻1 : there is a spatial autocorrelation under the given w moran's i is defined: spatial analysis of dengue disease in jakarta province muhamad sobari 538 𝐼 = 𝑁 𝑆0 ∑ ∑ 𝑤𝑖𝑗 (𝑦𝑖 − �̅�)(𝑦𝑗 − �̅�) 𝑁 𝑗=1 𝑁 𝑖=1 ∑ (𝑦𝑖 − �̅�) 2𝑁 𝑖=1 (1) where 𝑁 denotes the number of observations, �̅� denotes the average value 𝑦𝑖 from 𝑁 locations, 𝑦𝑖 denotes the value at locations i, 𝑦𝑗 denotes the value at locations j, 𝑤𝑖𝑗 denotes the spatial weight matrix element and moran's i test statistic is defined: 𝑍𝐼 = 𝐼 − 𝐸(𝐼) √𝑉𝑎𝑟(𝐼) (2) where 𝑍𝐼 denotes the moran's i test statistic value, 𝐸(𝐼) denotes the expected value of moran's i and 𝑉𝑎𝑟(𝐼) denotes variance of moran's i which defined: 𝑉𝑎𝑟(𝐼) = 𝑁2. 𝑆1 − 𝑁. 𝑆2 + 3. 𝑆0 2 (𝑁 − 1)(𝑁 + 1)𝑆0 2 − [𝐸(𝐼)]2 (3) 𝐸(𝐼) = − 1 𝑛 − 1 (4) 𝑆2 = ∑ 𝑁 𝑖=1 (∑ 𝑤𝑖𝑗 𝑁 𝑗=1 + ∑ 𝑤𝑗𝑖 𝑁 𝑗=1 ) 2 (5) 𝑆1 = 1 2 ∑ ∑(𝑤𝑖𝑗 + 𝑤𝑗𝑖 ) 2 𝑁 𝑗=1 𝑁 𝑖=1 (6) 𝑆0 = ∑ ∑ 𝑤𝑖𝑗 𝑁 𝑗=1 𝑁 𝑖=1 (7) the decision criteria in making conclusions are to reject 𝐻0 if 𝑍𝐼 > 𝑍𝛼 2 ; 5. checking whether there is a spatial dependence on lag or error by using the lagrange multiplier (lm) test. the lagrange multiplier (lm) test is used to determine the type of spatial analysis that is appropriate to use [19], [20]. the general form of the sar model is defined as follows [20]–[22]: 𝒚 = 𝜌𝑾𝒚 + 𝑿𝜷 + 𝜺; 𝜺~𝑁(𝟎, 𝜎𝜺 𝟐𝑰) (8) where 𝒚 denotes the response variable, 𝑿 denotes the predictor variable, 𝜌 denotes the spatial autocorrelation coefficient on the response variable, 𝑾 denotes the spatial weight matrix, 𝜷 denotes the intercept and regression coefficient and 𝜺 denotes the error. the hypotheses for the spatial dependence on lag are as follows: 𝐻0 ∶ 𝜌 = 0 (no spatial dependence on lag) 𝐻1 ∶ 𝜌 ≠ 0 (lag has a spatial dependence) the test statistic for the spatial dependence on lag are as follows: 𝐿𝑀𝐿𝐴𝐺 = [(𝑒𝑇 𝑊𝐴𝑦)/(𝑒 𝑇 𝑒/𝑁)]2 [(𝑊𝐴𝑋�̂�) 2 𝑀(𝑊𝐴𝑋�̂�)/(𝑒 𝑇 𝑒/𝑁)] + [𝑡𝑟(𝑊𝐴 𝑇 𝑊𝐴 + 𝑊𝐴 2)] ~𝜒(1−𝛼);𝑑𝑓=1 2 (9) the decision criteria in making conclusions are to reject 𝐻0 if 𝐿𝑀𝐿𝐴𝐺 > 𝜒(1−𝛼);𝑑𝑓=1 2 . if 𝐻0 is rejected, the spatial autoregressive (sar) model is used. and here are the test statistics for spatial dependence on error: 𝐿𝑀𝐸𝑅𝑅 = [(𝑒𝑇 𝑊𝐴𝑒)/(𝑒 𝑇 𝑒/𝑁)]2 [𝑡𝑟(𝑊𝐴 𝑇 𝑊𝐴 + 𝑊𝐴 2)] ~𝜒(1−𝛼);𝑑𝑓=1 2 (10) spatial analysis of dengue disease in jakarta province muhamad sobari 539 the decision criteria in making conclusions are to reject 𝐻0 if 𝐿𝑀𝐸𝑅𝑅 > 𝜒(1−𝛼);𝑑𝑓=1 2 . if 𝐻0 is rejected, the spatial error model (sem) is used. if both 𝐿𝑀𝐿𝐴𝐺 and 𝐿𝑀𝐸𝑅𝑅 are significant, comparing the akaike information criterion (aic) values allows one of the best models to be chosen. the best model is the one with the lowest aic value [16]. the aic calculated using the maximum likelihood estimation (mle) method is as follows [23]: 𝐴𝐼𝐶 = −2𝐿𝑚 + 2𝑚 (11) where 𝐿𝑚 denotes the maximum log-likelihood and 𝑚 denotes the number of model parameters; 6. estimate the parameters of the sar model. the sar model is a model whose dependent variables are spatially correlated. parameter estimation using the maximum likelihood method is defined as follows [24], [25]: �̂�𝑀𝐿 = (𝑿 𝑻𝑿)−1𝑿𝑻𝒚 − 𝜌(𝑿𝑻𝑿)−1𝑿𝑻𝑾𝒑𝒚 = �̂�𝑂𝐿𝑆 − 𝜌�̂�𝐿 (12) with �̂�𝐿 is a regression parameter estimator based on weight matrix (w) and spatial autocorrelation 𝜌. equation (12), however, cannot be directly solved because the value 𝜌 is unavailable. as a result, the log-likelihood concentrated function (𝐿𝑐 ) is employed, as defined below [18]: ln 𝐿𝑐 (𝜌) = 𝐶 − 𝑛 2 𝑙𝑛 [ 1 𝑛 (𝒆𝟎 − 𝜌𝒆𝑳) 𝑇 (𝒆𝟎 − 𝜌𝒆𝑳)] + 𝑙𝑛|𝑰 − 𝜌𝑾𝜌| (13) with c is a constant. equation (13) is a non-linear function in one parameter and is maximized using a numerical technique with direct search; 7. interpretation of the obtained sar model, including the direct impact of covariates; 8. diagnostic testing of the sar model. results and discussion descriptive analysis figure 1 depicts the distribution of dengue disease cases number in the jakarta province. considering figure 1 the sub-districts with a high dengue disease cases number are shown in dark red, with the majority located in the municipalities of east jakarta and north jakarta and a tiny portion in the municipalities of west jakarta and south jakarta. the sub-districts with a moderate number of dengue disease cases are denoted in light red, with the majority found in the municipalities of west jakarta and south jakarta and a tiny portion in the municipalities of east jakarta and north jakarta. and the sub-districts with a low number of dengue disease cases are marked in white, with the majority of these sub-districts being in the municipality of central jakarta and a tiny portion in the municipalities of south jakarta and north jakarta. this indicates that sub-districts with high dengue disease cases likely to be located near sub-districts with moderate dengue disease cases. spatial analysis of dengue disease in jakarta province muhamad sobari 540 figure 2 shows that cilincing, koja, tanjung priok, cengkareng, and pulo gadung are the sub-districts with the highest number of flood-prone points. in comparison to dengue disease cases number in these sub-districts, the number of dengue disease cases is also high. in addition, it can be noticed that pademangan, taman sari, gambir, senen, menteng, johar baru, and cilandak are the sub-districts with the lowest flood-prone points. moreover, as compared to dengue disease cases number in these sub-districts, the number of dengue disease cases is comparatively low. this indicates that there is a positive correlation between the number of flood-prone points and the dengue disease cases number in jakarta province. figure 2 also shows that cilincing, koja, cengkareng, and jatinegara are the subdistricts with the highest number of slum neighborhood associations. in comparison to dengue disease cases number in these sub-districts, the number of dengue disease cases is also high. cempaka putih and pancoran can be noted to have the fewest slum neighborhood associations. comparatively to dengue disease cases number in these subdistricts, the number of dengue disease cases is comparatively low. this indicates a substantial correlation between the number of slum neighborhood associations and dengue disease cases number in jakarta province. we can see in figure 2 that koja, kramat jati, and jatinegara are the sub-districts with a high population density, and these sub-districts also have a significant prevalence of dengue disease cases. penjaringan and cilandak are the sub-districts with the lowest population density, and both sub-districts also have the lowest number of dengue disease. this indicates a positive correlation between population density and the number of cases of dengue disease in jakarta province. we can also see in figure 2 that cilincing, cakung, cengkareng, and cipayung are sub-districts with a low number of hospitals per 1,000 populations. however, when compared with dengue disease cases number, these sub-districts are classified as subfigure 1. map of the distribution of dengue disease cases in jakarta province spatial analysis of dengue disease in jakarta province muhamad sobari 541 districts with a high dengue disease cases number, indicating a negative relationship between the number of hospitals per 1,000 populations and the number of cases of dengue disease in jakarta province. figure 2. map of distribution of predictor variable spatial analysis of dengue disease in jakarta province muhamad sobari 542 and we can also notice from figure 2 the sub-districts with the lowest public health centers per 1,000 people number are cakung, koja, cengkareng, and ciracas. however, when compared to the number of dengue disease cases, these sub-districts are classified as sub-districts with a high number of dengue disease cases. this indicates a negative correlation between the number of public health centers per 1,000 people and dengue disease case number in jakarta province. relying on figures 1 and 2, we can conclude that descriptively there is a relationship between the number of cases of dengue disease and all predictor variables used in this study. however, in order to be more convincing, a spatial regression analysis must be performed. spatial analysis table 2 shows the results of the linear regression model's classical assumption test. the p-values of the normality assumption test and the homoscedasticity assumption test are greater than 0.05, implying that the normality and homoscedasticity assumptions are fulfilled. table 2 also shows that the durbin watson (𝑑) value is between 4−𝑑𝑈<𝑑<4−𝑑𝐿 which can be concluded that the non-autocorrelation assumption is fulfilled, and the vif value lower than 5 for all variables can be concluded that the non-multicollinearity assumption is fulfilled. table 2. the results of the classical assumption of linear regression model statistic test result normality (shapiro-wilk) p-value = 0.3081 non-autocorrelation (durbin-watson) 2.0914 (dl=1.2546 du=1.7814) nonmulticollinearity (vif) (x1) 1.4457, (x2) 1.4091, (x3) 1.2851, (x4) 1.1829, (x5) 1.0831 homoscedasticity (breusch-pagan) p-value = 0.3219 even though the linear regression model has all of the assumptions fulfilled, it is still necessary to investigate whether there is an autocorrelation between sub-districts by performing the moran's i test. before carrying out the moran's i test, a spatial weight matrix is needed. and the spatial weight matrix used in this study is the neighbor (contiguity) spatial weight matrix with the neighboring type is (queen). the following table shows the results of the spatial autocorrelation test using moran's i test. table 3. spatial autocorrelation test results with moran's i variable statistic moran’s i p-value (α=5%) number of dengue disease cases 0.4140 1.893x10-6* number of flood-prone points 0.2926 0.0004* number of slum neighborhood associations (rt) 0.1591 0.05191 population density (people per km2) 0.1596 0.0363* number of hospitals per 1,000 population 0.1217 0.0877 number of public health centers per 1,000 population 0.0919 0.2044 *) significant according to table 3, there is a spatial autocorrelation in dengue disease cases number because the p-value is less than 0.05. two of the five predictor variables also spatial analysis of dengue disease in jakarta province muhamad sobari 543 showed that there was a spatial autocorrelation. moran's i statistical values are all between 0 and 1, indicating that the closer an area is, the more similar the variable values are. table 4. lr dan lm test results and aic value test value p-value (α=5%) aic lr 9.9310 0.0016 451.7152 lmlag 7.6763 0.0056 443.7842 lmerr 5.1326 0.0235 446.3620 the likelihood ratio (lr) test is used to determine which model performs better, spatial regression or linear regression. and the lagrange multiplier (lm) test is employed to find whether the spatial dependence is on the dependent variables (lag), on unresearched variables (error), or both (error and lag). table 4 shows the output of the lr and lm tests that were performed. according to the results in table 4, the lr test is significant because the p-value is less than 0.05, indicating that there is a significant difference between the spatial regression model and the linear regression model. it can also be seen that the aic of the linear regression model is the largest when compared to the two spatial regression models. and it is clear that both the spatial dependence in lag and the spatial dependence in error are significant, as indicated by the p-value less than 0.05. however, this study use the sar model because its aic value is lower than the aic value in the sem model. table 5 below shows the estimation results of the sar model parameters. table 5. estimation of sar model parameters estimate std error z-value pr (>|z|) (intercept) 20.839 29.102 0.7161 0.4739 number of flood-prone points 3.5296 0.9152 3.8568 0.0001* number of slum neighborhood associations (rt) 0.1752 0.1336 1.3116 0.1896 population density (people per km2) -0,0003 0.0006 -0.4603 0.6453 number of hospitals per 1,000 population 2.8001 315.00 0.0089 0.9929 number of public health centers per 1,000 population -792.52 539.49 -1.4690 0.1418 spasial lag (rho) 0.56618 0.12682 4.4644 0.000008* statistic wald: 19.931; p-value: 0.000008 *) significant according to table 5, the spatial lag variable (rho) has a statistically significant and positive coefficient, indicating that as dengue disease cases number in one sub-district increases, so will dengue disease cases number in neighboring sub-districts. and the wald test p-value is less than 0.05, indicating a significant relationship between the number of dengue disease cases in jakarta province and all predictor factors. dengue disease cases number in jakarta province is significantly affected by the number of flood-prone points, the number of slum neighborhood associations, the population density, the number of hospitals and the number of public health centers per 1,000 populations, and the spatial lag. at a significance level of 5%, only the number of flood-prone points and spatial lag have a significant impact on the number of cases of spatial analysis of dengue disease in jakarta province muhamad sobari 544 dengue disease in jakarta province. this is consistent with lilis wijaya's 2018 research, which found that floods led to post-flood ailments such as diarrhea, dengue disease, leptospirosis, acute respiratory infection (ari), intestinal worms, skin problems, and many more [26]. wang et al. (2016) reported that the aedes aegypti mosquito will live longer if the humidity level is high, such as during the rainy season, particularly in areas prone to floods, where the huge volume of standing water will make the disease more likely to spread [2]. the number of slum neighborhood associations and the number of public health centers per 1,000 people has a same coefficient sign with previous studies [1], [5]. the sign of the coefficient indicates the direction of the relationship between the predictor variable and the number of cases of dengue disease. although in this study the effect is almost significant with the p-value still below 0.2, which means the error rate is still below 20%. meanwhile, the population density variable and the number of hospitals per 1,000 population have different coefficient signs from previous studies [5], [6]. however, the pvalue shows that the effect is highly insignificant because the p-value exceeds 0.5, which means the error rate is above 50%. based on the parameter estimation results in table 5, the form of the sar model in this study is as follows: �̂�𝑖 = 0.56618 ∑ 𝑤𝑖𝑗 𝑦𝑗 𝑛 𝑗=1,𝑖≠𝑗 + 20.839 + 3.5296 𝑋1𝑖 + 0.1752 𝑋2𝑖 − 0,0003𝑋3𝑖 +2.8001𝑋4𝑖 − 792.52𝑋5𝑖 (14) the impact of covariates in the sar model could be classified into three categories: total impact; indirect impact; and direct impact; [16]. total impact refers to the changes that occur in one sub-district as a consequence of changes in that sub-district and its surroundings. indirect impact refers to the effect that takes place when the predictor factors in the bordering sub-district change. and impacts that occur locally in an area, which in this study is a sub-district, as a consequence of changes in predictor factors in that sub-district are referred to as direct impacts. the magnitude of direct and indirect impact of table 6 shows the sar model used in this study. table 6. measures of direct, indirect and total impact of the sar model variable direct impact p-value direct impact indirect impact p-value indirect impact total impact p-value total impact number of flood-prone points 3.8601 0.0002* 4.2760 0.0859 8.1361 0.0110* number of slum neighborhood associations (rt) 0.1916 0.2004 0.2123 0.3637 0.4039 0.2751 population density (people per km2) -0.0003 0.6310 -0.0004 0.7318 -0.0007 0.6816 number of hospitals per 1,000 population 3.0623 0.9833 3.3923 0.9584 6.4546 0.9671 number of public health centers per 1,000 population -866.74 0.1388 -960.11 0.2705 -1826.8 0.1908 *) significant table 6 shows the number of flood-prone points in jakarta province has a substantial direct effect on the number of dengue disease cases. no variable has a significant indirect effect on dengue disease in jakarta province, based on an examination of indirect effects. the number of flood-prone points in jakarta province has a significant impact on the total number of dengue disease cases. the growth in the number of floodspatial analysis of dengue disease in jakarta province muhamad sobari 545 prone points will have a direct impact on dengue disease cases number in jakarta province. each one percent increase in flood-prone points in a sub-district will result in a rise of 3.86 cases of dengue disease in that sub-district. table 7. sar model diagnostic test statistic test p-value conclusion normality (shapiro-wilk) 0.8740 residuals are normally distributed lm test for residual autocorrelation 0.7339 there is no autocorrelation between residuals homogeneity (breusch-pagan) 0.6246 the homogeneity assumption is met it is important to conduct a diagnostic test that involves the assumptions of homogeneity, normality and non-autocorrelation to determine the quality of the sar model. the sar model satisfies all the assumptions, as shown in table 7. conclusions moran's i test showed the dengue disease cases number, the number of flood-prone points, and the population density have spatial correlation whereas the number of slum neighborhood associations, the number of hospitals per 1,000 populations, and the number of public health centers per 1,000 population have no geographical correlation. relying on the lagrange multiplier test, the best spatial model is the spatial autoregressive (sar) model. the growth in the flood-prone points number in jakarta province will have a direct effect on the number of instances of dengue disease. each additional one percent of flood-prone points in a sub-district will result in 3.86 additional cases of dengue disease in that sub-district. the derived sar model is valid since it satisfies the assumptions of normality, absence of autocorrelation, and homogeneity. recommendations are made for the jakarta provincial government to enhance flood management regulations in order to lower the incidence of dengue disease cases. additional variables with a substantial association to the frequency of dengue disease cases can be included in future investigations. in addition, the spatial durbin model (sdm) technique can be utilized for more research on the number of dengue disease cases if all predictor variables exhibit a spatial lag. or geographically weighted poisson regression (gwpr) can also be utilized, which overcomes the presence of spatial heterogeneity in the response data in the form of count data (amount). or the conditional autoregressivebessag york mollie (car-bym) can also be utilized, which can accommodate geographical and non-spatial features induced by the heterogeneity of cases between regions. acknowledgments the authors would like to thank the rector of universitas padjadjaran, who offered financial help to disseminate studies reports. this research is part of pdkn contract dikti: 094/e5/pg.02.00.pt/2022 and drpm: 1318/un6.3.1/pt.00/2022 and also it is part of academic leadership grant (alg) contract 2023/un6.3.1/pt.00/2022. we are also thankful for the discussion on social media analytics through the rise_sma project funded through the european union year 2019-2024 and also 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[26] m. g. saputra and f. ummah, “kesiapan masyarakat dalam menghadapi penyakit pasca banjir di dusun lohgawe desa gawerejo kecamatan karangbinangun kabupaten lamongan,” jurnal asri (administrasi rumah sakit indonesia), vol. 7, no. 2, pp. 54–62, 2021. 6. reni damayanti automorfisme graf bintang dan graf lintasan reni tri damayanti mahasiswa pascasarjana jurusan matematika universitas brawijaya email: si_cerdazzz@rocketmail.com abstrak salah satu topik yang menarik untuk dikaji pada teori graf adalah tentang automorfisme graf. automorfisme pada suatu graf g adalah isomorfisme dari graf g ke g sendiri. dengan kata lain, automorfisme graf g merupakan suatu permutasi dari himpunan titik-titik v(g) atau sisi-sisi dari graf g, e(g) yang menghasilkan graf yang isomorfik dengan dirinya sendiri. jika ϕ adalah suatu automorfisme dari g ke g dan v� v(g) maka ������ ������ ntuk mencari automorfisme pada suatu graf, biasanya dilakukan dengan menentukan semua kemungkinan fungsi yang satu-satu, onto, dan isomorfisme dari himpunan titik pada graf tersebut. sehingga berdasarkan hal itu dapat diketahui dan diuraikan automorfisme graf bintang dan graf lintasan serta penjabarannya. berdasarkan hasil pembahasan, dapat diperoleh: (1) graf bintang-� ( �,�) memiliki �+1 titik, banyaknya automorfisme dari graf tersebut adalah �!. permutasinya α adalah automorfisme yang harus mengawetkan derajat titik-titiknya, oleh karena itu permutasinya harus berbentuk ���� �� dan ���� �� untuk setiap ��,��,�� � �� �,��. jika α� �,��= (������…��) fungsi bijektif maka α� �,�� merupakan automorfisme; (2) dari graf lintasan �� maka banyaknya automorfisme hanya ada 2 fungsi permutasi yang berbentuk: (a) untuk n genap: �� ����� ������� ������� … ������� �!, untuk n ganjil: �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % dan (b) �� =��� ��� ��� …��� . kata kunci: graf bintang � �,� , graf lintasan ��, isomorfisme graf, automorfisme graf, dan grup simetri pendahuluan graf didefinisikan sebagai himpunan titik (vertek) yang tidak kosong dan himpunan garis atau sisi (edge) yang mungkin kosong. himpunan titik dari suatu graf g dinyatakan dengan v(g) dan himpunan sisi dinyatakan dengan e(g). salah satu topik yang menarik untuk dikaji pada teori graf adalah tentang automorfisme graf. automorfisme dari suatu graf g merupakan suatu permutasi dari himpunan titik-titik v(g) atau himpunan sisi-sisi dari graf g (e(g)). dengan kata lain, automorfisme dari suatu graf g adalah isomorfisme dari graf g ke dirinya sendiri, yaitu fungsi yang memetakan ke dirinya sendiri. pada bab ini akan dibahas mengenai automorfisme suatu graf pada graf bintang dan graf lintasan. pada graf bintang, teorema yang dibangun adalah (1) banyaknya fungsi permutasi yang automorfisme, (2) bentuk fungsi permutasi yang automorfisme yaitu titik v1� &� �,�� yang selalu dipetakan ke dirinya sendiri sedangkan titik lainnya dapat dipetakan ke sebarang titik kecuali v1. sedangkan pada graf lintasan, teorema yang dibangun adalah banyaknya fungsi permutasi yang automorfisme dari graf �� yaitu hanya 2 fungsi yang dibedakan berdasarkan banyak titik genap dan ganjil. kemudian mengetahui bahwa grup automorfisme dari graf bintang �,� isomorfik dengan grup simetri '� dan grup automorfisme dari graf lintasan �� isomorfik dengan grup siklik orde-2 �(� . graf lintasan dan graf bintang definisi 1. graf lintasan adalah graf yang terdiri dari sebuah lintasan tunggal. graf lintasan dengan � verteks dilambangkan oleh ��. perhatikan bahwa �� memiliki �-tepi, dan dapat diperoleh dari graf siklus )� dengan menghapus sebuah sisi. contoh 1: 2p 3p 4p1p gambar 1. graf lintasan dari gambar 1. di atas, graf p1 hanya mempunyai satu titik, maka p1 tidak mempunyai sisi. pada graf p2 mempunyai dua titik dan satu sisi. pada graf p3 mempunyai tiga titik dan dua sisi. sedangkan, pada graf p4 mempunyai empat titik dan tiga sisi. jadi, penulis dapat menentukan beberapa ciri khusus dari graf lintasan ��adalah setiap titik ujung dan titik pangkal selalu berderajat 1 dan titik selain titik ujung dan titik pangkal selalu berderajat 2. reni damayanti 36 volume 2 no. 1 november 2011 definisi 2. suatu graf g lengkap partisi-� adalah graf partisi-� dengan himpunan-himpunan partisi &�,&�,…,&� yang memiliki sifat tambahan yaitu jika * � &+ dan � � &,, . / maka *� � ��0 . jika |&+| �+, kemudian graf ini dinotasikan dengan �2�,2�,…,2� . (order pada bilangan 2�,2�,… ,2� tidak penting.) ingat bahwa graf lengkap partisi-� adalah lengkap jika dan hanya jika 2+ 1 untuk semua -, dalam hal ini adalah �. jika 2+ 4 untuk semua -, kemudian graf lengkap partisi-� adalah tetap dan dinotasikan dengan ��� . maka, ��� 5 �. suatu graf bipartisi lengkap dengan himpunan partisi &� dan &�, dimana |&�| 6 dan |&�| �, kemudian dinotasikan dengan �6,� . graf �1,� disebut graf bintang. contoh 2: gambar 2 graf bipartisi definisi 3. dua buah graf g1 dan g2 dikatakan isomorfik jika terdapat pemetaan satu-satu ϕ antara v(g1) pada v(g2) sedemikian hingjga misal *� � ��0� jika dan hanya jika (ϕ(u)ϕ(v)) � e(g2). jika g1 isomorfis terhadap g2 dapat dikatakan bahwa g1 dan g2 saling isomorfik dan dapat ditulis g1 ≅ g2. contoh 3: gambar 3. g1 isomorfik dengan g2 tetapi tidak isomorfik dengan g3 pemetaan ϕ: v(g1) → v(g2) didefinisikan oleh: gambar 4. pemetaan satu-satu ���� *�, ���� *�, ���� *�, ���7 *7 akan dibuktikan bahwa (v1,v2), (v1,v3), (v1,v4), (v2,v3), (v2,v4), (v3,v4)� e(g1) jika dan hanya jika (ϕ(v1),ϕ(v2)), (ϕ(v1),ϕ(v3)), (ϕ(v1),ϕ(v4)), (ϕ(v2),ϕ(v3)), (ϕ(v2),ϕ(v4)), (ϕ(v3),ϕ(v4)) � e(g2). ���,�� � ��0� dan ����� ,���� � �*�,*� � ��0� ���,�� � ��0� dan ����� ,���� � �*�,*� � ��0� ���,�7 � ��0� dan ����� ,���7 � �*�,*7 � ��0� ���,�� � ��0� dan ����� ,���� � �*�,*� � ��0� ���,�7 � ��0� dan ����� ,���7 � �*�,*7 � ��0� ���,�7 � ��0� dan ����� ,���7 � �*�,*7 � ��0� berdasarkan uraian diatas terbukti bahwa g1 ≅ g2. definisi 4 automorfisme pada suatu graf g adalah isomorfisme dari graf g ke g sendiri. dengan kata lain automorfisme graf g merupakan suatu permutasi dari himpunan titik-titik v(g). jika ϕ adalah suatu automorfisme dari g dan v�v(g) maka ������� ������ . contoh 4: misal diberikan graf g seperti di bawah ini: gambar 5. graf g diberikan pemetaan �:&�0 9 &�0 , maka automorfisme yang mungkin dari graf g di atas adalah: g1 v5 v1 v3 v4 v2 v6 v7 v5 v1 v3 v4 v2 v6 v7 g2 g3 v3 v4 v2 v1 g2 u3 u4 u2 u1 g1 ϕ v(g1) v(g2) v3 v4 v2 v1 v3 v4 v2 v1 2 1 3 4 automorfisme graf bintang dan graf lintasan jurnal cauchy – issn: 2086-0382 37 ��1 1 ��2 2 ��3 3 ��1 2 ��2 3 ��3 1 ��1 3 ��2 1 ��3 2 ��1 1 ��2 3 ��3 2 ��1 1 ��2 3 ��3 2 ��1 1 ��2 3 ��3 2 hasil dan pembahasan berikut ini ditentukan banyaknya automorfisme dari masing-masing graf ke dirinya sendiri berdasarkan bentuk-bentuk permutasi yang mengacu pada pemetaan titiknya yang memenuhi fungsi tersebut, dengan pola sebagai berikut: graf bintang pada graf bintang banyaknya automorfisme yang mengacu pada pemetaan titiknya sebagai berikut: dari penjelasan tentang automorfisme pada graf bintang berdasarkan bentuk-bentuk permutasi yang mengacu pada pemetaan titiknya yang memenuhi fungsi tersebut, maka dapat dibuat tabel banyaknya automorfisme dari kemungkinan banyak fungsi tersebut melalui tempat kedudukan titik-titiknya dengan bentuk fungsi �1 �……… <=>=? � : tabel 2 banyaknya automorfisme melalui bentuk permutasi titiknya dari �,� graf bintang ( �,�) σ fungsi berbentuk �……… <=>=?� �,� 1 1 1! �,� 2 2 · 1 2! �,7 6 3 · 2 · 1 3! �,a 24 4 · 3 · 2 · 1 4! �,c 120 5 · 4 · 3 · 2 · 1 5! … … … �,� ��� e 1 �� e2 …2· 1 �! tabel 1. banyaknya automorfisme dari g(k�,g) → g(k�,g) graf bintang automorfisme banyaknya automorfisme �,� α = (1)(2)(3) 1 2 α = (1)(. .) 1 �,� α = (1)(2)(3)(4) 1 6 α = (1)( . . . ) 2 α = (1)( . )( . . ) 3 �,7 α = (1)(2)(3)(4)(5) 1 24 α = (1)( . . . . ) 6 α = (1) (.) (. . .) 8 α = (1)( . )( . )( . . ) 6 α = (1)( . . )( . . ) 3 �,a α = (1)(2)(3)(4)(5)(6) 1 120 α = (1)( . . . . . ) 24 α = (1)( . )( . . . . ) 30 α = (1)( . )( . )( . . . ) 20 α = (1)( . )( . )( . )( . . ) 10 α = (1)( . )( . . )( . . ) 15 α = (1)( . . )( . . . ) 20 �,c α = (1)( . . . . . .) 120 120 atau dapat ditulis φ= (1) (2) (3) atau dapat ditulis φ= (1 2 3) atau dapat ditulis φ= (1 3 2) atau dapat ditulis φ= (1) (2 3) atau dapat ditulis φ= (2) (1 3) atau dapat ditulis φ= (3) (1 2) reni damayanti 38 volume 2 no. 1 november 2011 dari uraian automorfisme graf bintang di atas maka berdasarkan banyak titik dapat dibuat teorema tentang banyak automorfisme dari graf �,� untuk � bilangan asli yang fungsinya berbentuk (1) �……… <=>=? � , yaitu sebagai berikut: teorema 1 graf bintang-� ( �,�) memiliki �+1 titik. banyaknya automorfisme dari graf tersebut adalah �!. diketahui: misalkan � adalah automorfisme dari �,� ke dirinya sendiri. akan dibuktikan: ���� �� dan ���� �� dengan i . 1 dan 4 . 1. bukti graf �,� memiliki � m 1 titik. misalkan titik-titiknya adalah &� �,�� ���,��,��,… ,�� � . misalkan � adalah automorfisme dari �,� ke dirinya sendiri. karena α(��) = ��, yang berarti mengawetkan derajat �� itu sendiri, sehingga banyaknya titik yang dapat dipermutasikan adalah sebanyak � titik, maka permutasinya dapat dirumuskan sebanyak �!. selanjutnya, akan ditunjukkan bahwa ���� �� dan ���� ��. karena derajat titik �� 1 sehingga ���� �� juga mengawetkan derajat titiknya. andaikan ���� �� dengan i . 1 dan �� �n �i . 4 . 6 . gambar 6. bintang-� ( �,�) ���,�� � �� �,� �����,�� ����� ,���� ���,�n o �� �,� jadi, ���� �� maka � bukan automorfisme. sehingga, pengandaian i . 1 salah. jadi, seharusnya pengandaian menjadi i 1 atau ���� ��. dari teorema 1 di atas, maka dapat diturunkan teorema sebagai berikut: teorema 2 grup automorfisme dari graf bintang �,�isomorfik dengan grup simetri '� atau �'�,p 5 �q� �,��,p�. bukti akan ditunjukkan ada korespondensi satusatu dari anggota �'�,p pada �q� �,��,p�. buat pemetaan r dari '� ke q� �,�� dengan aturan sebagai berikut (contoh dapat dilihat pada lampiran): jika� � '�dengan � ��� �� �� … �s ,1 t u t � maka ↓ ↓ ↓ ↓ r�� ��v$ � �v� � �vw � … �vx �� karena � dan r�� korespondensinya sama, maka bentuk permutasinya sama. dapat ditunjukkan bahwa r bersifat bijektif dan homomorfisme. jika y,z � '� dan r�y ,r�z � q� �,�� maka y 9 r�y z 9 r�z jadi, r�yz r�y r�z dengan demikian r adalah isomorfisme. jadi, �'�,p 5 �q� �,��,p� terbukti. graf lintasan pada graf lintasan banyaknya automorfisme yang mengacu pada pemetaan titiknya sebagai berikut: tabel 3 banyaknya automorfisme dari g(pn) → g(pn) graf lintasan automorfisme banyaknya automorfisme p2 β = (1)(2) 1 2 β =(1 2) 1 p3 β =(1)(2)(3) 1 2 β =(1 3)(2) 1 p4 β =(1)(2)(3)(4) 1 2 β =(1 4)(2 3) 1 p5 β =(1)(2)(3)(4)(5) 1 2 β =(1 5)(2 4)(3) 1 p6 β =(1)(2)(3)(4)(5)(6) 1 2 β = (1 6)(2 5)(34) 1 dari tabel 3 di atas, maka dapat dibuat bentuk umum dari banyaknya fungsi permutasi yang automorfisme sebagai berikut: 1 �� automorfisme graf bintang dan graf lintasan jurnal cauchy – issn: 2086-0382 39 tabel 4. bentuk umum automorfisme dari g(pn) → g(pn) berdasarkan banyak titik genap dan ganjil �genap �� ��� ��� ��� …��� sebanyak 1 �� ����� ������� ������� … "������ �% �ganjil �� ��� ��� ��� …��� sebanyak 1 �� ����� ������� ������� … "�� �� ���� �� �%"�� �� % dari uraian automorfisme graf lintasan di atas maka berdasarkan banyak titik dapat dibuat teorema tentang banyak automorfisme dari graf pn, yaitu sebagai berikut: teorema 3 dari graf lintasan �� maka banyaknya automorfisme hanya ada 2 fungsi yang berbentuk: a. untuk n genap, permutasinya berbentuk: �� ����� ������� ������� … ������� �! dan �� = ��� ��� ��� …��� b. untuk n ganjil, permutasinya berbentuk: �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % dan �� = ��� ��� ��� …��� bukti a. untuk n genap, permutasinya berbentuk: �� ����� ������� ������� … "������ �% sehingga, ���� �� → mengawetkan derajat titik 1 ���� ���� → mengawetkan derajat titik 2 ���� ���� → mengawetkan derajat titik 2 ↓ ↓ �����! ����� �! → mengawetkan derajat titik 2 ↓ ↓ ������ 2 → mengawetkan derajat titik 2 ���� 1 → mengawetkan derajat titik 1 karena graf lintasan (��) ini jumlah titiknya genap, maka ���� ,��� � � ���� sehingga, ����� ,��� � ����� ,���� � ���,���� � ���� ���� ,��� � � ���� sehingga, ����� ,��� � ����� ,���� � �����,���� � ���� [����!,���� �!\ � ���� sehingga, �[����,��� �!\ [�����!,����� �!\ ���� �,���! ����,��� �! � ���� begitu pula untuk ������ ,��� � � ���� sehingga, ������� ,��� � ������� ,���� � ���,�� � ���� jadi,�� ����� ������� ������� … ������� �! terbukti automorfisme. selanjutya, untuk fungsi identitas tidak perlu ditunjukkan karena sudah jelas automorfisme. b. untuk n ganjil, permutasinya berbentuk: �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % sehingga, ���� �� → mengawetkan derajat titik 1 ���� ���� → mengawetkan derajat titik 2 ���� ���� → mengawetkan derajat titik 2 ↓ ↓ �"��#$� ����#$� �% �"��#$� % → mengawetkan derajat titik 2 ↓ ↓ ������ 2→ mengawetkan derajat titik 2 ���� 1 → mengawetkan derajat titik 1 maka, ���� ,��� � � ���� sehingga, ����� ,��� � ����� ,���� � ���,���� � ���� ���� ,��� � � ���� sehingga,����� ,��� � ����� ,���� � �����,���� � ���� ["��#$� ��,��#$� �%\ � ���� sehingga, �["�� �� ��,�� �� �%\ reni damayanti 40 volume 2 no. 1 november 2011 ]�[�� �� ��,�"�� �� �%\^ "��#$� ��,��#$� �% � ���� begitu pula untuk ������ ,��� � � ���� sehingga, ������� ,��� � ������� ,���� � ���,�� � ���� jadi, �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % adalah automorfisme. jadi, berdasarkan pada bagian a dan b maka teorema terbukti benar. setelah mengetahui banyaknya automorfisme graf lintasan (��) hanya ada 2 fungsi yaitu yang berbentuk: �� ��� ��� ��� …��� dan �� ����� ������� ������� … ������� �! untuk � genap �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % untuk � ganjil, dari teorema 3 di atas, maka dapat diturunkan teorema sebagai berikut: teorema 4 grup automorfisme dari graf lintasan ��isomorfik dengan grup siklik orde-2 �(� atau �(�,p 5 ���,p . bukti misalkan �(�,p 5 ���,p . akan ditunjukkan ada korespondensi satusatu dari anggota �(�,p pada �q��� ,p . misalkan (� = {τ�,τ�} dan q��� `a�,a�b. selanjutnya, anggota (� dikorespondensikan satu-satu pada titik-titik dari �� sebagai berikut: τ�~a� τ�~a� untuk � genap maupun � ganjil karena dari teorema 3 grup automorfisme graf lintasan �� dan grup siklik orde-2 �(� adalah 2, jadi ���,p 5 �(�,p . selanjutnya, untuk fungsi identitas tidak perlu ditunjukkan karena sudah jelas automorfisme. penutup dari hasil dan pembahasan, secara umum dapat disimpulkan bahwa: 1. graf bintang-�( �,�) memiliki �+1 titik. banyaknya automorfisme dari graf tersebut adalah �!. permutasinya α adalah automorfisme yang harus mengawetkan derajat titik-titiknya, oleh karena itu permutasinya harus berbentuk ���� �� dan ���� �� untuk setiap ��,��,�� � �� �,� . 2. dari graf lintasan �� maka banyaknya automorfisme hanya ada 2 fungsi yang berbentuk: a. untuk n genap, permutasinya berbentuk: �� ����� ������� … ������� �! dan �� = ��� ��� ��� …��� b. untuk n ganjil, permutasinya berbentuk: �� ����� ������� ������� … "��#$� ����#$� �%"��#$� % dan �� = ��� ��� ��� …��� 3. grup automorfisme dari graf bintang �,� isomorfik dengan grup simetri '� atau �'�,p 5 �q� �,��,p�. 4. grup automorfisme dari graf lintasan �� isomorfik dengan grup siklik orde-2 �(� atau �(�,p 5 ���,p . daftar pustaka [1] wilson, r.j. dan watkins, j. j. 1990, graphs an introductory approach, canada: john wiley and sons, inc. [2] chartrand, gery dan lesniak, linda, 1986, graphs and digraphs second edition, hal. 5, 10, 250, california: a division of wadsworth, inc. yusron rijal filter halaman web pornografi _7_ filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit yusron rijal1 dan awalia nofitasari2 1email: yusronrijal@yahoo.com 2pt. elpro solusitama, 2email: awalia.nofitasari@gmail.com abstract this paper presents an effort to detect pornographic webpages. it was stated that a positive relationship exists between percentage of human skin color in an image and the image itself (jones et.al., 1998). based on the statement, rather than using the traditional method of text-filtering, this paper propose a new approach to detect pornographic images by using skin color detection. the skin color detection performed by using rgb, hsi, and ycbcr color model. using algorithm stated by ap-apid (ap-apid, 2005), the system will classify nude and not-nude images. if one or more nude images are found, the system will block the webpage. keywords: webpage filtering, image processing, pornography, nudity, skin color, nude images pendahuluan perkembangan teknologi di bidang penangkapan dan pengolahan citra serta penyimpanan data, seiring dengan akses internet yang semakin luas, secara signifikan meningkatkan aliran informasi di masyarakat. salah satunya adalah penyebaran informasi yang berkaitan dengan pornografi. internet adalah media yang mudah diakses, sehingga tiap orang yang mengerti cara membuka web browser, menggunakan situs pencarian kemudian masuk ke situs tertentu, dapat menampilkan halaman web yang mengandung pornografi hanya dengan mengetikkan beberapa kata kunci. (lin, et. al, 2003:1) kebanyakan sistem komersial yang didesain untuk mencegah akses terhadap hal-hal yang mengandung pornografi seperti netnanny, cybersitter, cyberpatrol dan childwebguardian memblokir situs web dengan membandingkan suatu alamat ip, alamat url, atau teks dalam suatu halaman web dengan alamat ip, alamat url, atau kata-kata tertentu yang ada dalam basis data sistem tersebut. pendekatan tersebut efektif untuk memblokir situs web porno yang populer dan halaman web yang memiliki link ke situs web porno, tetapi kurang efektif dalam memblokir halaman web yang berisi koleksi citra pornografi karena halaman-halaman tersebut sering tidak berisi link ke halaman web lain yang mengandung pornografi atau tidak berisi katakata yang mengandung pornografi. (liang, et. al, 2004:1) data statistik terhadap 4.000.000 halaman html yang dihimpun oleh starykevitch dan daoudi (starykevitch, 2002) menunjukkan bahwa 70% dari halaman-halaman tersebut berisi citra digital, dan sebuah halaman html rata-rata berisi 18,8 citra digital. tinjauan statistik terhadap 1.232 halaman web porno dan 6.967 halaman web non-porno menunjukkan bahwa 72% halaman web porno berisi lebih dari 5 citra digital dan 60% dari halaman web porno berisi lebih dari 10 citra digital. 40% dari halaman web porno tersebut berisi lebih dari 5 link menuju file video dan citra digital. karenanya, kemampuan untuk mendeteksi citra pornografi dapat menjadi alat yang berguna untuk menyaring hal-hal yang mengandung pornografi di internet. (abadpour, 2005:1) pornografi adalah konsep yang hingga kini belum memiliki definisi yang cukup saksama untuk membedakan citra mana yang mengandung pornografi dan citra mana yang tidak (chan, et. al, 1999:1). berdasarkan definisi pornografi oleh lin (lin et. al, 2003:2), halaman web yang mengandung citra pornografi didefinisikan sebagai halaman web yang didalamnya terdapat citra yang mengandung obyek manusia telanjang, manusia menunjukkan organ seksual, atau manusia sedang melakukan hubungan seksual. sebuah citra dapat digolongkan sebagai citra pornografi bila didalamnya terdapat obyek manusia yang telanjang, manusia yang menunjukkan organ seksual, atau manusia yang sedang melakukan hubungan seksual. citra tersebut umumnya memiliki banyak warna kulit, sehingga warna kulit merupakan salah satu perhatian utama dalam pendeteksian citra pornografi. lebih jauh lagi, lin (lin et al, 2003) menyebutkan bahwa situs web pornografi umumnya mengandung kata-kata yang mengindikasikan pornografi. yusron rijal dan awalia nofitasri 208 volume 1 no. 4 mei 2011 metodologi perancangan sistem tujuan utama penelitian yang dilakukan penulis adalah untuk menyaring halaman web yang mengandung citra pornografi. penyaringan halaman web yang mengandung citra pornografi diimplementasikan dengan membuat script penyaringan halaman web yang mengandung citra pornografi untuk kemudian ditambahkan kedalam script situs web tertentu, misalnya sebagai penyaring masukan pada layanan buku tamu online (umumnya pada situs web pribadi atau blog) atau layanan pendaftaran promosi iklan online (umumnya pada situs web iklan baris). bahasa pemrograman yang digunakan adalah php dengan apache sebagai web server. penyaringan halaman web yang mengandung unsur pornografi dilakukan menggunakan kombinasi dua metode, yaitu pencocokan kata (text matching) dan identifikasi citra pornografi. secara umum, langkah-langkah penyaringan halaman web yang mengandung unsur pornografi dapat digambarkan pada blok diagram berikut: gambar 1. blok diagram penyaringan halaman web berdasarkan blok diagram tersebut, proses identifikasi citra dibagi menjadi 4 tahap, yaitu: 1. cari url dalam komentar. 2. cari kata-kata yang mengandung unsur pornografi. 3. identifikasi citra pornografi pada url tersebut. 4. klasifikasikan halaman web, apakah termasuk halaman web yang mengandung unsur pornografi atau tidak. identifikasi url dalam komentar url dalam komentar diidentifikasi berdasarkan pernyataan-pernyataan berikut: 1. kebanyakan url menggunakan protokol http (boutell, 2009). 2. protokol yang digunakan untuk mengakses file tertentu dalam sebuah web server dan menampilkannya dalam web browser adalah protokol http (university of illinois at urbanachampaign, 2009). 3. protokol dan nama resource dipisahkan dengan tanda titik dua dan dua garis miring (sun, 2009). 4. untuk dapat diakses, sebuah url tidak dapat mengandung spasi (blogger, 2009). sehingga, sebuah rangkaian karakter yang memiliki rangkaian karakter “http://” akan dianggap url oleh sistem dan rangkaian karakter tersebut akan disimpan hingga menemukan karakter spasi (“ ”). url dapat pula disebutkan dalam sebuah hyperlink. untuk mengantisipasi hal tersebut, sistem menghilangkan semua tag html dalam isi komentar sebelum melakukan pencarian rangkaian karakter “http://”. contoh isi komentar disampaikan pada gambar 2. gambar 2. contoh komentar url pada gambar 2 adalah http://www. break.com/pictures_nsfw/hottie-on-thebed6641 10.html. identifikasi kata-kata yang mengandung unsur pornografi teknik yang digunakan adalah text matching, dimana isi html sebuah halaman web dicocokkan dengan daftar kata yang dimiliki sistem. bila ditemukan adanya kata yang sama, sistem akan melakukan analisa lebih lanjut. bila tidak ketemu, sistem akan menganggap bahwa komentar aman untuk disimpan. daftar kata yang mengandung unsur pornografi disusun berdasarkan hasil pengamatan terhadap 24 sampel halaman web. kata yang dipilih adalah kata yang sering muncul dalam sampel halaman web pornografi tapi tidak muncul dalam sampel bukan halaman web pornografi. contoh hasil text matching disampaikan pada gambar 3. gambar 3. contoh hasil text matching isi komentar identifikasi url dalam komentar identifikasi citra pornografi identifikasi kata-kata yang mengandung unsur pornografi status komentar (diblokir atau disimpan) penarikan kesimpulan filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit jurnal cauchy – issn: 2086-0382 209 identifikasi citra pornografi jika proses pencarian kata yang mengindikasikan pornografi menghasilkan nilai true, sistem mencari alamat citra pada isi html halaman web. citra yang diidentifikasi adalah citra jenis jpeg (ekstensi .jpg dan .jpeg). dengan demikian, alamat citra adalah alamat yang diawali dengan kata “http://” dan diakhiri dengan kata “.jpg” atau “.jpeg”. alamat citra dicari pada: a. alamat yang disebutkan dalam properti src sebuah image. contoh: <img src=” http://localhost/samples/article2231447.ece _files/siren_280_710789b.jpg” . . . >. . .</img> b. alamat yang disebutkan dalam properti href sebuah hyperlink. contoh: <a href=” http://2.bp.blogspot.com/_64iclf2k6xq/sr5 wnduusji/aaaaaaaaanu/p2tsto8zv5q/s1 600-h/gacktcalen1.jpg”>. . .</a> bila jumlah alamat citra yang dihasilkan dari proses pencarian berjumlah lebih dari satu, sistem melakukan proses penyalinan file citra. penyalinan file citra digital merupakan proses menyalin (copy) citra digital kedalam komputer server. bila alamat citra tidak ditemukan, sistem menganggap bahwa halaman web tersebut tidak mengandung citra pornografi, sehingga sistem kemudian menyimpan isi komentar. contoh hasil pencarian alamat citra digital disampaikan pada gambar 4 dan gambar 5. gambar 4. alamat file citra digital pada properti src sebuah image gambar 5. hasil penyalinan file citra digital secara umum, langkah-langkah identifikasi citra pornografi adalah (ap-apid, 2005:3): 1. deteksi piksel warna kulit dalam citra. 2. cari atau bentuk area-area kulit berdasarkan hasil pendeteksian piksel. 3. analisa tiap area kulit untuk mendapatkan petunjuk pornografi. 4. klasifikasikan citra, apakah termasuk citra pornografi atau bukan. untuk mendeteksi piksel warna kulit, digunakan model warna tertentu. model warna yang digunakan harus dapat membedakan piksel yang menyerupai warna kulit dengan piksel yang tidak menyerupai warna kulit. dari piksel-piksel yang menyerupai warna kulit, diperkirakan daerah-daerah yang dapat dibentuk oleh pikselpiksel tersebut. tiap daerah kulit kemudian dihitung luasnya. luas daerah kulit digunakan sebagai bahan penarikan kesimpulan (berdasarkan kriteria-kriteria yang telah ditetapkan) untuk mengambil kesimpulan apakah suatu citra tergolong citra pornografi atau tidak. pemilihan model warna untuk mendeteksi piksel warna kulit, digunakan tiga macam model warna: rgb, hsi, dan ycbcr. lin (lin et al, 2003:4) menyebutkan bahwa model warna default untuk file citra digital jenis jpeg dan gif adalah rgb. model warna hsi mewakili penerimaan mata manusia terhadap warna. ycbcr dapat digunakan untuk mengidentifikasi warna kulit manusia tanpa memperhatikan faktor perbedaan ras (chai and bouzerdoum, 1999). model warna rgb tidak sesuai untuk mengidentifikasi piksel warna kulit karena model warna rgb tidak hanya menggambarkan warna tapi juga kecerahan warna. hal ini dapat diatasi dengan penggunaan model warna hsi dan ycbcr. model warna hsi dan ycbcr dapat memisahkan tingkat kecerahan piksel dengan kromatisitas warna (chai & bouzerdoum, 1999). persamaan yang digunakan untuk mendapatkan nilai hsi sebagai berikut (abadpour, 2005:15): )))(()((2 2 cos 2 1 bgbrgr bgr h −−+− −− = − (1) ),,min( 3 1 bgr bgr s ++ −= (2) )( 3 1 bgri ++= (3) persamaan yang digunakan untuk mendapatkan nilai ycbcr sebagai berikut (blogspot, 2009): )*114,0()*587,0()*299,0( bgry ++= (4) )*5,0()*331,0()*169,0( bgrcb +−+−= (5) )*081,0()*419,0()*5,0( bgrcr −+−+= (6) penetapan batas ambang untuk memisahkan piksel warna kulit dengan piksel bukan warna kulit, diperlukan penentuan batas ambang untuk tiap model warna. batas ambang ditentukan berdasarkan hasil eksperimen terhadap 85 sampel citra kulit manusia dan 58 sampel citra bukan kulit manusia. pendeteksian warna kulit pada dasarnya, deteksi kulit pada citra digital melibatkan dua proses, yaitu pembacaan piksel dan segmentasi citra. pembacaan piksel yusron rijal dan awalia nofitasri 210 volume 1 no. 4 mei 2011 dilakukan untuk mengetahui nilai rgb tiap piksel. dari nilai rgb tersebut, dilakukan perhitungan nilai hsi dan ycbcr. dari nilai rgb, hsi, dan ycbcr yang diperoleh, dilakukan segmentasi citra. segmentasi citra merupakan proses pemisahan piksel bukan warna kulit dengan piksel warna kulit menggunakan batas ambang tertentu untuk tiap model warna. piksel yang dianggap bukan piksel warna kulit akan diubah warnanya menjadi hitam, sedangkan piksel warna kulit akan diubah warnanya menjadi putih. dengan demikian, citra yang dihasilkan dari proses segmentasi citra adalah sebuah citra biner. yang dimaksud citra biner adalah citra hitam putih, dimana piksel warna kulit direpresentasikan dengan warna putih, dan piksel bukan warna kulit direpresentasikan dengan warna hitam. segmentasi citra dilakukan untuk mempermudah proses perhitungan luas area kulit. hasil segmentasi citra dari gambar 5 disampaikan pada gambar 6. gambar 6. citra biner hasil segmentasi batas ambang untuk model warna rgb ditetapkan berdasarkan pernyataan-pernyataan berikut: 1. kulit manusia terdiri atas darah yang berwarna merah dan melanin yang berwarna kuning atau coklat. dengan demikian, kulit manusia (pada citra selain hitam putih dan keabuan) tidak mungkin berwarna hitam, putih, atau abu-abu. 2. piksel (pada model warna rgb) dapat digolongkan sebagai piksel warna kulit jika memiliki nilai r > g atau r > b atau keduanya. pernyataan tersebut bila digunakan memerlukan analisa lebih lanjut, karena pernyataan r > g atau r > b hanya berlaku jika citra tidak memiliki piksel (selain warna kulit) berwarna merah (brown et.al., 2001). dari pernyataan-pernyataan tersebut, batas ambang untuk model warna rgb ditetapkan sebagai berikut: a. piksel dianggap bukan warna kulit jika berwarna putih (255, 255, 255), berwarna hitam (0, 0, 0), ataupun abu-abu (r = g = b). b. piksel dianggap bukan warna kulit jika r <= g atau r <= b. batas ambang hsi dan ycbcr diperoleh dari hasil eksperimen terhadap 85 sampel citra warna kulit dan 58 sampel citra bukan warna kulit. batas ambang untuk model warna hsi sebagai berikut: a. hue lebih besar dari -1 dan lebih kecil dari 15 (-1 < h < 15), dengan nilai saturation antara 3 dan 65 (3 < s < 65) b. hue lebih besar sama dengan 15 dan lebih kecil dari 45 (15 <= h < 45), dengan nilai saturation antara 3 dan 70 (3 < s < 70) c. hue lebih besar sama dengan 45 dan lebih kecil dari 75 (45 <= h < 75), dengan nilai saturation antara 3 dan 65 (3 < s < 65) batas ambang ycbcr ditetapkan sebagai berikut: a. kroma biru lebih besar dari -61 dan lebih kecil dari 7 (-61 < cb < 7) b. kroma merah lebih besar dari 10 dan lebih kecil dari 101 (10 < cr < 101) segmentasi citra akan membentuk areaarea kulit. perhitungan luas area kulit dilakukan menggunakan metode freeman’s chain code yang telah dimodifikasi (rijal et.al, 2006). model chain code yang digunakan penulis adalah chain code dengan delapan arah mata angin (gambar 7). tiap arah pergeseran direpresentasikan dengan angka tertentu, dimana angka terkecil adalah arah permulaan pergeseran. untuk menyederhanakan algoritma perhitungan luas, pergerakan chain code ditetapkan ke kanan searah jarum jam. angka terkecil menunjukkan arah permulaan penelusuran piksel. sehingga, penelusuran piksel dimulai dari kanan koordinat, kemudian kanan bawah, bawah, kiri bawah, kiri, kiri atas, atas, kanan atas, demikian seterusnya hingga semua piksel telah ditelusuri. area yang dikenai proses perhitungan luas adalah area berwarna putih. satuan luas yang digunakan adalah piksel. contoh hasil perhitungan luas disampaikan pada gambar 8. gambar 7. chain code yang dimodifikasi gambar 8. contoh hasil perhitungan luas dari informasi luas tiap area kulit, dihitung luas piksel kulit. luas piksel kulit kemudian dibandingkan dengan luas piksel bukan kulit. jika 1 7 5 3 2 86 4 jurnal cauchy persentase jumlah piksel warna kulit dibandingkan dengan jumlah piksel bukan warna kulit lebih besar dari 50 perse citra pornografi. bila kurang dari sama dengan 50 persen, citra dianggap bukan citra pornografi. batas ambang piksel kulit terhadap piksel bukan warna kulit ditentukan berdasarkan hasil eksperimen terhadap 120 pornografi. penarikan kesimpulan penarikan kesimpulan merupakan proses penentuan apakah suatu halaman web tergolong halaman web digunakan untuk penarikan kesimpulan adalah adanya kata yang mengindikasikan unsur pornografi dan jumlah citra pornografi. jika sistem tidak menemukan kata yang mengindikasikan pornografi, sistem akan menyimpan komentar. jika sis kata yang mengindikasikan pornografi tapi tidak metemukan satu pun citra pornografi dalam halaman web, sistem akan menyimpan komentar. bila sistem menemukan adanya kata yang mengindikasikan pornografi dan metemukan setidaknya satu citra porno web, komentar tidak akan disimpan dan sistem akan menampilkan halaman pemblokiran komentar ( gambar 9. hasil dan pembahasan untuk menguji akurasi penyaringan halaman web, digunakan 10 sampel url terdiri atas 6 halaman web pornografi dan 4 halaman web bukan pornografi. untuk menguji akurasi identifikasi citra pornografi, digunakan 178 sampel citra. sampel citra terdiri atas 118 citra pornografi dan 60 citra bukan pornografi. uji aku merupakan uji kuantitatif, sehingga hasil pengujian dibagi dalam empat kelompok, yaitu kelompok negative kelompok false negative uji akurasi penya filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit jurnal cauchy – issn: 2086 persentase jumlah piksel warna kulit dibandingkan dengan jumlah piksel bukan warna kulit lebih besar dari 50 perse citra pornografi. bila kurang dari sama dengan 50 persen, citra dianggap bukan citra pornografi. batas ambang prosentase piksel kulit terhadap piksel bukan warna kulit ditentukan berdasarkan hasil eksperimen terhadap 120 citra pornografi dan 32 citra bukan pornografi. penarikan kesimpulan penarikan kesimpulan merupakan proses penentuan apakah suatu halaman web tergolong halaman web pornografi digunakan untuk penarikan kesimpulan adalah adanya kata yang mengindikasikan unsur pornografi dan jumlah citra pornografi. jika sistem tidak menemukan kata yang mengindikasikan pornografi, sistem akan menyimpan komentar. jika sis kata yang mengindikasikan pornografi tapi tidak metemukan satu pun citra pornografi dalam halaman web, sistem akan menyimpan komentar. bila sistem menemukan adanya kata yang mengindikasikan pornografi dan metemukan setidaknya satu citra porno web, komentar tidak akan disimpan dan sistem akan menampilkan halaman pemblokiran komentar (gambar 9). gambar 9. halaman pemblokiran komentar hasil dan pembahasan untuk menguji akurasi penyaringan halaman web, digunakan 10 sampel url terdiri atas 6 halaman web pornografi dan 4 halaman web bukan pornografi. untuk menguji akurasi identifikasi citra pornografi, digunakan 178 sampel citra. sampel citra terdiri atas 118 citra pornografi dan 60 citra bukan pornografi. uji akurasi penyaringan halaman web merupakan uji kuantitatif, sehingga hasil pengujian dibagi dalam empat kelompok, yaitu kelompok true positive (tn), kelompok kelompok false negative uji akurasi penyaringan halaman web filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit issn: 2086-0382 persentase jumlah piksel warna kulit dibandingkan dengan jumlah piksel bukan warna kulit lebih besar dari 50 persen, citra dianggap citra pornografi. bila kurang dari sama dengan 50 persen, citra dianggap bukan citra pornografi. prosentase perbandingan luas piksel kulit terhadap piksel bukan warna kulit ditentukan berdasarkan hasil eksperimen citra pornografi dan 32 citra bukan penarikan kesimpulan penarikan kesimpulan merupakan proses penentuan apakah suatu halaman web tergolong pornografi atau tidak. kriteria yang digunakan untuk penarikan kesimpulan adalah adanya kata yang mengindikasikan unsur pornografi dan jumlah citra pornografi. jika sistem tidak menemukan kata yang mengindikasikan pornografi, sistem akan menyimpan komentar. jika sistem menemukan kata yang mengindikasikan pornografi tapi tidak metemukan satu pun citra pornografi dalam halaman web, sistem akan menyimpan komentar. bila sistem menemukan adanya kata yang mengindikasikan pornografi dan metemukan setidaknya satu citra pornografi dalam halaman web, komentar tidak akan disimpan dan sistem akan menampilkan halaman pemblokiran ambar 9). halaman pemblokiran komentar hasil dan pembahasan untuk menguji akurasi penyaringan halaman web, digunakan 10 sampel url terdiri atas 6 halaman web pornografi dan 4 halaman web bukan pornografi. untuk menguji akurasi identifikasi citra pornografi, digunakan 178 sampel citra. sampel citra terdiri atas 118 citra pornografi dan 60 citra rasi penyaringan halaman web merupakan uji kuantitatif, sehingga hasil pengujian dibagi dalam empat kelompok, yaitu true positive (tp), kelompok (tn), kelompok false positive kelompok false negative (fn). ringan halaman web filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit 0382 persentase jumlah piksel warna kulit dibandingkan dengan jumlah piksel bukan warna n, citra dianggap citra pornografi. bila kurang dari sama dengan 50 persen, citra dianggap bukan citra pornografi. perbandingan luas piksel kulit terhadap piksel bukan warna kulit ditentukan berdasarkan hasil eksperimen citra pornografi dan 32 citra bukan penarikan kesimpulan merupakan proses penentuan apakah suatu halaman web tergolong atau tidak. kriteria yang digunakan untuk penarikan kesimpulan adalah adanya kata yang mengindikasikan unsur pornografi dan jumlah citra pornografi. jika sistem tidak menemukan kata yang mengindikasikan pornografi, sistem akan tem menemukan kata yang mengindikasikan pornografi tapi tidak metemukan satu pun citra pornografi dalam halaman web, sistem akan menyimpan komentar. bila sistem menemukan adanya kata yang mengindikasikan pornografi dan metemukan grafi dalam halaman web, komentar tidak akan disimpan dan sistem akan menampilkan halaman pemblokiran halaman pemblokiran komentar untuk menguji akurasi penyaringan halaman web, digunakan 10 sampel url. sampel url terdiri atas 6 halaman web pornografi dan 4 untuk menguji akurasi identifikasi citra pornografi, digunakan 178 sampel citra. sampel citra terdiri atas 118 citra pornografi dan 60 citra rasi penyaringan halaman web merupakan uji kuantitatif, sehingga hasil pengujian dibagi dalam empat kelompok, yaitu (tp), kelompok true false positive (fp), dan ringan halaman web filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit persentase jumlah piksel warna kulit dibandingkan dengan jumlah piksel bukan warna n, citra dianggap citra pornografi. bila kurang dari sama dengan 50 persen, citra dianggap bukan citra pornografi. perbandingan luas piksel kulit terhadap piksel bukan warna kulit ditentukan berdasarkan hasil eksperimen citra pornografi dan 32 citra bukan penarikan kesimpulan merupakan proses penentuan apakah suatu halaman web tergolong atau tidak. kriteria yang digunakan untuk penarikan kesimpulan adalah adanya kata yang mengindikasikan unsur pornografi dan jumlah citra pornografi. jika sistem tidak menemukan kata yang mengindikasikan pornografi, sistem akan tem menemukan kata yang mengindikasikan pornografi tapi tidak metemukan satu pun citra pornografi dalam halaman web, sistem akan menyimpan komentar. bila sistem menemukan adanya kata yang mengindikasikan pornografi dan metemukan grafi dalam halaman web, komentar tidak akan disimpan dan sistem akan menampilkan halaman pemblokiran untuk menguji akurasi penyaringan url. sampel url terdiri atas 6 halaman web pornografi dan 4 untuk menguji akurasi identifikasi citra pornografi, digunakan 178 sampel citra. sampel citra terdiri atas 118 citra pornografi dan 60 citra rasi penyaringan halaman web merupakan uji kuantitatif, sehingga hasil pengujian dibagi dalam empat kelompok, yaitu true (fp), dan menghasilkan 6 halaman web pornografi dideteksi sebagai halaman web pornografi (tp = 60%), 0 halaman web pornografi deteksi sebagai bukan halaman web pornografi (tn = 0%), 2 halaman web bukan pornografi d halaman web pornografi (fp = 20%), 2 halaman web bukan pornografi dideteksi sebagai bukan halaman web pornografi (fn = 20%). hasil pengujian disampaikan pada tabel 1. adanya citra bukan pornografi yang didentifikasi sebagai citra pornografi. tabel 1. uji akurasi identifikasi citra pornografi hasil sebagai berikut: 1. 2. 3. 4. beberapa hasil pengujian disampaikan pada tabel 2 dan tabel 3. dipengaruhi oleh kualitas citra, tingkat pencahayaan, serta keragaman warna kulit antar ras manusia. munculn dipengaruhi oleh antara lain adanya warna latar belakang obyek manusia pada citra menyerupai warna kulit manusia, serta adanya obyek manusia pada citra yang memenuhi sebagian besar luasan citra, misalnya pada citra foto manusia. no. 1 filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit pengujian terhadap 10 sampel url menghasilkan 6 halaman web pornografi dideteksi sebagai halaman web pornografi (tp = 60%), 0 halaman web pornografi deteksi sebagai bukan halaman web pornografi (tn = 0%), 2 halaman web bukan pornografi d halaman web pornografi (fp = 20%), 2 halaman web bukan pornografi dideteksi sebagai bukan halaman web pornografi (fn = 20%). hasil pengujian disampaikan pada tabel 1. munculnya adanya citra bukan pornografi yang didentifikasi sebagai citra pornografi. tabel 1. hasil uji akurasi penyaringan halaman web uji akurasi identifikasi citra pornografi pengujian terhadap sampel memberikan hasil sebagai berikut: 112 sampel citra pornografi dideteksi sebagai citra pornografi (tp = 62,9%). 6 sampel citra pornografi dideteksi sebagai citra pornografi (tn = 3,37%) 51 sampel citra bukan pornografi dideteksi sebagai citra bukan pornografi (fn = 28,7%) 9 sampel citra bukan p sebagai citra pornografi (fp = 5%) beberapa hasil pengujian disampaikan pada tabel 2 dan tabel 3. munculnya dipengaruhi oleh kualitas citra, tingkat pencahayaan, serta keragaman warna kulit antar ras manusia. munculn dipengaruhi oleh antara lain adanya warna latar belakang obyek manusia pada citra menyerupai warna kulit manusia, serta adanya obyek manusia pada citra yang memenuhi sebagian besar luasan citra, misalnya pada citra foto manusia. tabel 2. beberapa hasil pengujian terhadap citra no. citra sampel filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit pengujian terhadap 10 sampel url menghasilkan 6 halaman web pornografi dideteksi sebagai halaman web pornografi (tp = 60%), 0 halaman web pornografi deteksi sebagai bukan halaman web pornografi (tn = 0%), 2 halaman web bukan pornografi d halaman web pornografi (fp = 20%), 2 halaman web bukan pornografi dideteksi sebagai bukan halaman web pornografi (fn = 20%). hasil pengujian disampaikan pada tabel 1. munculnya false positive adanya citra bukan pornografi yang didentifikasi sebagai citra pornografi. hasil uji akurasi penyaringan halaman web uji akurasi identifikasi citra pornografi pengujian terhadap sampel memberikan hasil sebagai berikut: sampel citra pornografi dideteksi sebagai citra pornografi (tp = 62,9%). 6 sampel citra pornografi dideteksi sebagai citra pornografi (tn = 3,37%) 51 sampel citra bukan pornografi dideteksi sebagai citra bukan pornografi (fn = 28,7%) 9 sampel citra bukan p sebagai citra pornografi (fp = 5%) beberapa hasil pengujian disampaikan pada tabel munculnya true negative dipengaruhi oleh kualitas citra, tingkat pencahayaan, serta keragaman warna kulit antar ras manusia. munculnya dipengaruhi oleh antara lain adanya warna latar belakang obyek manusia pada citra menyerupai warna kulit manusia, serta adanya obyek manusia pada citra yang memenuhi sebagian besar luasan citra, misalnya pada citra foto beberapa hasil pengujian terhadap citra pornografi citra sampel hasil pengolahan citra filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit pengujian terhadap 10 sampel url menghasilkan 6 halaman web pornografi dideteksi sebagai halaman web pornografi (tp = 60%), 0 halaman web pornografi deteksi sebagai bukan halaman web pornografi (tn = 0%), 2 halaman web bukan pornografi dideteksi sebagai halaman web pornografi (fp = 20%), 2 halaman web bukan pornografi dideteksi sebagai bukan halaman web pornografi (fn = 20%). hasil pengujian disampaikan pada tabel 1. false positive dipengaruhi oleh adanya citra bukan pornografi yang didentifikasi hasil uji akurasi penyaringan halaman web uji akurasi identifikasi citra pornografi pengujian terhadap sampel memberikan sampel citra pornografi dideteksi sebagai citra pornografi (tp = 62,9%). 6 sampel citra pornografi dideteksi sebagai citra pornografi (tn = 3,37%) 51 sampel citra bukan pornografi dideteksi sebagai citra bukan pornografi (fn = 28,7%) 9 sampel citra bukan pornografi dideteksi sebagai citra pornografi (fp = 5%) beberapa hasil pengujian disampaikan pada tabel true negative dipengaruhi oleh kualitas citra, tingkat pencahayaan, serta keragaman warna kulit antar ya false positive dipengaruhi oleh antara lain adanya warna latar belakang obyek manusia pada citra menyerupai warna kulit manusia, serta adanya obyek manusia pada citra yang memenuhi sebagian besar luasan citra, misalnya pada citra foto beberapa hasil pengujian terhadap citra pornografi hasil pengolahan citra identifikasi ? filter halaman web pornografi menggunakan kecocokan kata dan deteksi warna kulit 211 pengujian terhadap 10 sampel url menghasilkan 6 halaman web pornografi dideteksi sebagai halaman web pornografi (tp = 60%), 0 halaman web pornografi deteksi sebagai bukan halaman web pornografi (tn = 0%), 2 ideteksi sebagai halaman web pornografi (fp = 20%), 2 halaman web bukan pornografi dideteksi sebagai bukan halaman web pornografi (fn = 20%). hasil dipengaruhi oleh adanya citra bukan pornografi yang didentifikasi hasil uji akurasi penyaringan halaman web uji akurasi identifikasi citra pornografi pengujian terhadap sampel memberikan sampel citra pornografi dideteksi sebagai citra pornografi (tp = 62,9%). 6 sampel citra pornografi dideteksi sebagai 51 sampel citra bukan pornografi dideteksi sebagai citra bukan pornografi (fn = 28,7%) ornografi dideteksi beberapa hasil pengujian disampaikan pada tabel dapat dipengaruhi oleh kualitas citra, tingkat pencahayaan, serta keragaman warna kulit antar positive dapat dipengaruhi oleh antara lain adanya warna latar belakang obyek manusia pada citra menyerupai warna kulit manusia, serta adanya obyek manusia pada citra yang memenuhi sebagian besar luasan citra, misalnya pada citra foto beberapa hasil pengujian terhadap citra ter identifikasi ? ya yusron rijal dan awalia nofitasri 212 volume 1 no. 4 mei 2011 2 ya 3 ya 4 ya 5 ya tabel 3. beberapa hasil pengujian terhadap citra bukan pornografi no. citra sampel hasil pengolahan citra ter identifikasi ? 1 tidak 2 ya 3 tidak 4 ya 5 tidak 6 tidak 7 tidak 8 tidak 9 tidak 10 ya penutup hasil evaluasi menunjukkan bahwa penyaringan halaman web pornografi mengggunakan pendekatan deteksi warna kulit dapat dilakukan. keberhasilan penyaringan halaman web pornografi sangat dipengaruhi oleh tingkat keberhasilan pendeteksian warna kulit. akurasi penyaringan halaman web dapat ditingkatkan dengan mengkombinasikan pendeteksian citra dengan sistem ip filtering dan text filtering. beberapa metode yang dapat dilakukan untuk meningkatkan akurasi pendeteksian citra antara lain: 1. melakukan penggolongan warna kulit berdasarkan kecerahan warna kulit. 2. mengkombinasikan metode deteksi warna kulit dengan pendeteksian tepi. 3. mengkombinasikan metode deteksi warna kulit dengan pengenalan fitur. 4. mengkombinasikan metode deteksi warna kulit dengan identifikasi kontur. daftar pustaka [1] abadpour, a. & kasaei, s. 2005. pixel-based skin detection for pornography filtering, (online), (http://ce.sharif.edu/~ipl/papers/ ijeee-05-aa.pdf, diakses 15 pebruari 2009) [2] achmad, u. 2005. pengolahan citra digital dan teknik pemrogramannya. jogjakarta: graha ilmu [3] anderson, d.l, shapiro, l. 2006. introduction to chain codes, (online), (http://www. mind. ilstu.edu/curriculum/chain_codes_intro/cha in_codes_intro.php?modgui=162&compgui =1704&itemgui=2966, diakses 10 maret 2009) [4] ap-apid, r. 2005. an algorithm for nudity detection, (online), (http://www.math.admu .edu.ph/~raf/pcsc05/proceedings/ai4.pdf, diakses 15 pebruari 2009) [5] arifianto, r.d. 2008. deteksi wajah menggunakan metode segmentasi berbasis model warna pada objek bergerak, (skripsi). stikomp surabaya [6] 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(online), (http://www.poesiafilter.org/pdf/ deliverable_3_1.pdf, diakses 18 pebruari 2009) [25] sun. 2009. what is a url?, (online), (http://java.sun.com/docs/books/tutorial/ networking/urls/definition.html, diakses 18 maret 2009) [26] university of illinois at urbana-champaign. 2009. what’s a url?, (online), (http:// www.cites.illinois.edu/101/url101.html, diakses 18 maret 2009) [27] wang, j., li, j., wiederhold, g., & firschein, o. 1998. system for screening objectionable images, (online), (http://infolab.stanford. edu/image/jcc98/wang.pdf, diakses 17 pebruari 2009) [28] webdevelopernotes. 2009. urls – what is an url?, (online), (http://www.webdevelopersnotes.com/basics/urls.php3, diakses 18 maret 2009) [29] wikipedia. 2007. rgb color model, (online), (http://www.wikipedia.org/wiki/rgb color model.htm, diakses 18 pebruari 2009) [30] wilson, brian. 2009. url encoding, (online), (http://www.blooberry.com/indexdot/html /topics/urlencoding.htm, diakses 18 maret 2009) [31] wordpress. 2009. color management, (online), (http://warto.files.wordpress.com/ 2008/11/g-color-management.ppt, diakses 10 maret 2009) on the dominant local resolving set of vertex amalgamation graphs cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 597-607 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 19, 2022 reviewed: december 20, 2022 accepted: march 16, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.18891 on the dominant local resolving set of vertex amalgamation graphs reni umilasari1,2, liliek susilowati1,*, slamin3, savari prabhu4 1,2 department of mathematics, airlangga university, surabaya 60115, indonesia 2department of informatics engineering, university of muhammadiyah jember, jember 69121 indonesia 3study program of informatics, university of jember, jember 68121 indonesia 4mathematics department, rajalakshmi engineering college, chennai 602105, tamil nadu, india email: liliek-s@fst.unair.ac.id abstract in graph theory, there is a new topic of the dominant local metric dimension which be symbolized by 𝐷𝑑𝑖𝑚𝑙 (𝐻) and it was a combination of local metric dimension and dominating set. there are some terms in this topic that is dominant local resolving set and dominant local basis. an ordered subset 𝑊𝑙 is said a dominant local resolving set of 𝐺 if 𝑊𝑙 is dominating set and also local resolving set of 𝐺. while dominant local basis is a dominant local resolving set with minimum cardinality. this study uses literature study method by observing the local metric dimension and dominating number before detecting the dominant local metric dimension of the graphs. after obtaining some new results, the purpose of this research is how the dominant local metric dimension of vertex amalgamation product graphs. some special graphs that be used are star, friendship, complete graph and complete bipartite graph. based on all observation results, it can be said that the dominant local metric dimension for any vertex amalgamation product graph depends on the dominant local metric dimension of the copied graphs and how the terminal vertex is constructed. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: dominant local metric dimension; vertex amalgamation; star; friendship; complete bipartite introduction metric dimension and dominating set are graph topics with numerous variations. for metric dimensions, there are more than five development concepts, such as partition dimension, local metric dimension, complement metric dimension, central metric dimension, fractional metric dimension, and star metric dimension. more results about metric dimension and its expansion can be seen at [1] about the simultaneous local metric dimension, the local metric dimension of amalgamation [2] and corona product of star and path graph [3] , complement metric dimension [4], fractional metric dimension [5], and the new one is star metrc dimension [6]. let 𝐺 be a connected graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). the distance between any two 𝑎 and 𝑏 of 𝑉(𝐺) is denoted by 𝑑(𝑎, 𝑏). it be defined as shortest path from 𝑎 to 𝑏. the resolving set of 𝐺 is an ordered set which can be written as 𝑊, where 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑘 } ⊆ 𝑉(𝐺) and 𝑟(𝑣|𝑊) = (𝑑(𝑣, 𝑤1), 𝑑(𝑣, 𝑤2), … . , 𝑑(𝑣, 𝑤𝑘 )) is defined as http://dx.doi.org/10.18860/ca.v7i4.18891 mailto:liliek-s@fst.unair.ac.id https://creativecommons.org/licenses/by-sa/4.0/ on the dominant local resolving set of vertex amalgamation graphs reni umilasari 598 representation of 𝑣 ⊆ 𝑉(𝐺) to 𝑊 by using the concept of distance. the rule to select resolving set of 𝐺 is every vertex of 𝑉(𝐺) should have different representation to 𝑊. the minimum cardinality of 𝑊 is called the basis of 𝐺 [7]. the number of basis is referred to the metric dimension because the concept of this topic is based on distance. next, we introduce the differences between metric dimension and local metric dimension. in the metric dimension, all vertices must have different representations to the resolving set, whereas in the local metric dimension, only any two adjacent vertices must be different. it also can be said that a vertex's representation can be the same as another vertices even though they are not adjacent. [8]. some examples of the local metric dimension have been published at [9], [10] and [11]. more clearly, there is a paper which describe the similarity between metric dimension and the local metric dimension [12]. in 2021, umilasari et al introduced the new concept, which is a combination of dominating set and local metric dimension. they defined that an ordered subset 𝑊𝑙 is said a dominant local resolving set of 𝐺 if 𝑊𝑙 is dominating set and also local resolving set of 𝐺. for a clearer understanding of this term, you can see the illustration in figure 1. all the vertices in the graph of figure 1 (a) can be dominated by 𝑥4. but the vertices which adjacent {𝑉(𝐺)\𝑥4} have same representation to 𝑥4. while {𝑥2, 𝑥3} in figure 1 (b) is a local resolving set. as can be seen, each pair of adjacent vertices has a different representation to {𝑥2, 𝑥3}. the problem is 𝑥5 and 𝑥6 cannot be dominated by 𝑥2 or 𝑥3. if we take two vertices like in figure 1(c), {𝑥2, 𝑥4} can dominate all vertices of the graph, the representation of any two vertices to {𝑥2, 𝑥4} is different. therefore, {𝑥2, 𝑥4} are elements of a dominant local metric dimension of the graph. figure 1. the illustration of dominant local metric dimension of graphs after obtaining some new results, in this paper we continue to expand on how the dominant local metric dimension of a vertex amalgamation product graphs. the vertex amalgamation product of a graph 𝐻, denoted by 𝑎𝑚𝑎𝑙(𝐻, 𝑣, 𝑘), is defined as creating a new graph by gluing 𝑘-copies of 𝐻 in a terminal vertex 𝑣 [13]. in this paper, we determine the dominant local metric dimension of the vertex amalgamation for some special graphs, which are star, complete graph, complete bipartite graph, and friendship graph. to make it easier to present each of the theorems produced, several theorems are given below, which can be seen in [14]. lemma 1. let 𝐺 be a connected graph. if there is no local dominant resolving set with cardinality 𝑝, then for every 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < 𝑝 is not a local dominant resolving set. lemma 2. let 𝐺 be a connected graph and 𝑊𝑙 ⊆ 𝑉(𝐺). for every 𝑣𝑖 , 𝑣𝑗 ∈ 𝑊𝑙 then 𝑟(𝑣𝑖 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑗 |𝑊𝑙 ). some new results about the dominant local metric dimension of star, complete on the dominant local resolving set of vertex amalgamation graphs reni umilasari 599 graph, complete bipartite graph, and friendship graph are given in table 1. on the dominant local resolving set of vertex amalgamation graphs reni umilasari 600 table 1. dominant local metric dimension of special graphs [14][15][16] graphs dominating number (𝜸(𝑮)) local metric dimension (𝐝𝐢𝐦𝒍(𝑮)) dominan local metric dimension (𝐝𝐢𝐦𝒍(𝑮)) star (𝑆𝑛) 𝛾 (𝑆𝑛) = 1 dim𝑙 (𝑆𝑛) = 1 ddim𝑙 (𝑆𝑛) = 1 complete (𝐾𝑛) 𝛾 (𝐾𝑛) = 1 dim𝑙 (𝐾𝑛) = 𝑛 − 1 ddim𝑙 (𝐾𝑛) = 𝑛 − 1 complete bipartite (𝐾𝑚,𝑛) 𝛾(𝐾𝑚,𝑛) = 2 dim𝑙 (𝐾𝑚,𝑛) = 2 ddim𝑙 (𝐾𝑚,𝑛) = 2 friendship (𝐹𝑛 ) 𝛾(𝐹𝑛 ) = 1 dim𝑙 (𝐹𝑛) = 𝑛 ddim𝑙 (𝐹𝑛) = 𝑛 methods in this research, there are several procedures. we start by determining the special graphs to be operated by the vertex amalgamation product and observing the local metric dimensions and dominating number of the graphs. then, we construct the vertex amalgamation product graphs from the special graphs that we have chosen. we continue by labeling the vertex and attempting to find the least dominant local basis. this is accomplished by observing and recording the representation of each vertex which can be dominated and has different representation from the local resolving set (two nonneighbouring vertex can have the same representation). the minimum local dominant basis is then determined. in summary, the procedures of the research can be seen in the following flowchart in figure 2. we also give some examples of each step below. a) let 𝐺 = 𝑃4 b) 𝑑𝑖𝑚𝑙 (𝑃4) = 1 and 𝛾(𝑃4) = 2 , it can be seen at [16] c) let |𝑊𝑙 | = 1, 𝑊𝑙 = {𝑣1} illustration: based on the illustration above, 𝑣1 can’t dominate 𝑣3 and 𝑣4. when we choose 𝑊𝑙 = {𝑣2}, 𝑊𝑙 = {𝑣3}, 𝑊𝑙 = {𝑣4} the condition remains the same. minimally, there exist one vertex that can’t be dominated. d) let |𝑊𝑙 | = 2, 𝑊𝑙 = {𝑣1, 𝑣2} illustration: we can see that 𝑣4 can’t be dominated by 𝑣1 or 𝑣2 . e) let |𝑊𝑙 | = 2, 𝑊𝑙 = {𝑣2, 𝑣3} illustration: since ∀𝑣𝑖 𝑣𝑗 ∈ 𝐸(𝑃4), 𝑟(𝑣𝑖 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑗 |𝑊𝑙 ) then 𝑊𝑙 is basis local of 𝑃4. all vertices of 𝑉(𝑃4) can be dominated by 𝑣2 and 𝑣3. therefore, 𝑊𝑙 is dominant local basis of 𝑃4 or 𝐷𝑑𝑖𝑚𝑙 (𝑃4) = 2. to more clearly understand this research method, we can see the flowchart in figure 2. on the dominant local resolving set of vertex amalgamation graphs reni umilasari 601 figure 2. flowchart for determining the minimum dominant local resolving set of graphs results and discussion in this section, we determine the dominant local metric dimension of the vertex amalgamation product for some special graphs, which are star, complete graph, complete bipartite graph, and friendship graph. theorem 1. let 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) is a vertex amalgamation of star with the order of star is 𝑛 ≥ 3, then 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = { 1, 𝑣 𝑖𝑠 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝑆𝑛 𝑘, 𝑣 𝑖𝑠 𝑝𝑒𝑛𝑑𝑎𝑛𝑡 𝑜𝑓 𝑆𝑛 on the dominant local resolving set of vertex amalgamation graphs reni umilasari 602 proof. case 1. 𝑣 is center vertex of star it is very clearly to see that 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) ≅ 𝑆𝑛, then by the table 1 we can conclude that 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = 1. ∎ case 2. 𝑣 is pendant vertex of star let the vertex set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) is 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = {𝑣, 𝑣𝑗 , 𝑢1𝑗 , 𝑢2𝑗 , … , 𝑢𝑖𝑗 |𝑣, 𝑢𝑖 ∈ 𝑉(𝑆𝑛), 𝑖 = 1,2, … , 𝑛 − 2, 𝑗 = 1,2, … , 𝑘} and the edge set is 𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = {𝑣𝑣𝑗 , 𝑣𝑗 𝑢𝑖𝑗 |𝑖 = 1,2,3, … , 𝑛 − 2, 𝑗 = 1,2,3, … , 𝑘}. choose 𝑊𝑙 = {𝑣𝑗 } is the local basis of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) for every 𝑗 = 1,2,3, … , 𝑘, |𝑊𝑙 | = 𝑘. we can show below that the representation of every two adjacent vertices of 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) is different. i. for 𝑢𝑖𝑗 𝑣𝑗 ∈ 𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) since 𝑣𝑗 is element of 𝑊𝑙 , then there exist 0 on 𝑖 th element in 𝑟(𝑣𝑗 |𝑊𝑙 ), while for 𝑟(𝑢𝑖𝑗 |𝑊𝑙 ) there are no zero elements, hence 𝑟(𝑣𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑢𝑖𝑗 |𝑊𝑙 ). ii. for 𝑣𝑣𝑗 ∈ 𝐸(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) since 𝑣𝑗 is element of 𝑊𝑙 , then there exist 0 on 𝑖 th element in 𝑟(𝑣𝑗 |𝑊𝑙 ), while for 𝑟(𝑣|𝑊𝑙 ) there are no zero elements, hence 𝑟(𝑣𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑣|𝑊𝑙 ). by i and ii therefore 𝑊𝑙 is a local resolving set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). further, because 𝑣𝑗 is adjacent to 𝑣 and 𝑢𝑖𝑗 , so we can said that 𝑊𝑙 is a dominant local resolving set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). next, take any 𝑆 ⊆ 𝑉(𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) with |𝑆| < |𝑊𝑙 |. without loss of generality, let |𝑆| = |𝑊𝑙 | − 1 with 𝑊𝑙 = {𝑣𝑗 |𝑗 = 1,2,3, … , 𝑘 − 1}, so 𝑢𝑖𝑘 are not adjacent to 𝑆. so 𝑆 is not a dominant local resolving set of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). based on lemma 1, any set 𝑇 with |𝑇| < |𝑆| is not a dominant local resolving set of 𝐺. therefore, 𝑊𝑙 = {𝑣𝑗 } is a dominant local basis of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘). then its is proven that 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘)) = 𝑘 for 𝑣 is pendant vertex of star of 𝑆𝑛. ∎ figure 3. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆4, 𝑣, 3)) = 1. figure 4. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝑆6, 𝑣, 5)) = 5 on the dominant local resolving set of vertex amalgamation graphs reni umilasari 603 figure 3 gives the illustration of dominant local metric dimension of 𝑎𝑚𝑎𝑙(𝑆𝑛, 𝑣, 𝑘) for 𝑣 is the center vertex of 𝑆𝑛. while in figure 4, 𝑣 is the pendant of star. the next theorem, we will show the dominan local metric dimension of complete graph. because the graphs are regular, then we can select any vertex of complete graph as the linkage vertex. theorem 2. let 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is a vertex amalgamation of complete graph with the order of complete graph is 𝑛 ≥ 3, then 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = 𝑝 × (𝐷𝑑𝑖𝑚𝑙 (𝐾𝑛) − 1). proof. let 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = {𝑣, 𝑣𝑖𝑗 |𝑖 = 1,2,3, … , 𝑛 − 1, 𝑗 = 1,2,3, … , 𝑝} and the edge set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is 𝐸(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = {𝑣𝑣𝑖𝑗 , 𝑣𝑥𝑗 𝑣𝑦𝑗 |𝑖 = 1,2,3, … , 𝑛 − 1, 𝑗 = 1,2,3, … , 𝑝, 𝑣𝑥 𝑣𝑦 ∈ 𝐸(𝐾𝑛), 𝑥 ≠ 𝑦 }. the 𝑗-th copy of 𝐾𝑛 with 𝑗 = 1,2,3, … , 𝑝 is called (𝐾𝑛)𝑗 . let 𝐵 be a local dominant basis of 𝐾𝑛, 𝐵𝑗 is a local dominant basis of (𝐾𝑛)𝑗 , so that for every 𝑗 = 1,2,3, … , 𝑝, |𝐵𝑖 | = |𝐵|. select 𝑊𝑙 = ⋃ 𝐵𝑗 𝑚 𝑗=1 , with 𝐵𝑗 = {𝑣𝑖𝑗 |𝑗 = 1,2,3, … , 𝑛 − 2} for every 𝑗 = 1,2,3, … , 𝑝, then |𝑊𝑙 | = 𝑝(𝑛 − 2). by lemma 2 for every 𝑣𝑖𝑗 , 𝑣𝑘𝑙 ∈ 𝐵𝑖 then 𝑟(𝑣𝑖𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑘𝑙 |𝑊𝑙 ). next, we take any two adjacent vertices in 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 . let 𝑥, 𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 , then for 𝑥𝑦 = 𝑣, 𝑣𝑛𝑗 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝))\𝑊𝑙 with 𝑗 = 1,2,3, … , 𝑝. since 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝) is a connected graph, 𝑑(𝑣𝑛𝑗 , 𝑧) = 𝑑(𝑣𝑛𝑗 , 𝑣) + 𝑑(𝑣, 𝑧)for every 𝑧 ∈ 𝐵𝑖 so that 𝑑(𝑧, 𝑣𝑛𝑗 ) ≠ 𝑑(𝑧, 𝑣) caused 𝑟(𝑣𝑛𝑗 |𝐵𝑖) ≠ 𝑟(𝑣|𝐵𝑖). because of 𝐵𝑖 ⊆ 𝑊𝑙 then 𝑟(𝑣𝑛𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑧|𝑊𝑙 ). based on the explanation above, 𝑊𝑙 = ⋃ 𝐵𝑖 𝑚 𝑖=1 is a local resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). since, every 𝑣𝑖𝑗 ∈ 𝑊𝑙 with 𝑖 = 1,2,3, … , 𝑛 − 2 and 𝑗 = 1,2,3, … , 𝑝 is adjacent to 𝑣 and 𝑣𝑛𝑗 , then 𝑊𝑙 is a dominating set. so that, 𝑊𝑙 = ⋃ 𝐵𝑗 𝑝 𝑗=1 is a local dominant resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). take any 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < |𝑊𝑙 |. let |𝑆| = |𝑊𝑙 | − 1, then there exists 𝑗 such as 𝑆 contains maximal |𝐵𝑗 | − 1 elements of (𝐾𝑛)𝑗 . since 𝐵𝑗 is a local dominant basis of (𝐾𝑛)𝑗 then there exist two vertices in (𝐾𝑛)𝑗 that have the same representation toward 𝑆, so that 𝑆 is not a local dominant resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). based on lemma 1 then 𝑊𝑙 = ⋃ 𝐵𝑗 𝑝 𝑗=1 is a local dominant basis of 𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝). by table 1 we know that |𝐵𝑖 | = 𝐷𝑑𝑖𝑚𝑙 ((𝐾𝑛)𝑖 ) − 1, then it has been proven that 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾𝑛, 𝑣, 𝑝)) = 𝑝 × (𝐷𝑑𝑖𝑚𝑙 (𝐾𝑛) − 1). ∎ the example of dominant local metric dimension of vertex amalgamation complete graph be given in figure 5. the graph show that 𝑎𝑚𝑎𝑙(𝐾4, 𝑣, 3) has the dominant local metric dimension equals six. figure 5. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾4, 𝑣, 3)) = 6. theorem 3. let 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝) is a vertex amalgamation of complete bipartite graph with the order is 𝑚, 𝑛 ≥ 2, then 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = 𝑝 + 1. on the dominant local resolving set of vertex amalgamation graphs reni umilasari 604 proof. let the vertex set of 𝐾𝑚,𝑛 is 𝑉(𝐾𝑚,𝑛) = {𝑎𝑖 |𝑖 = 1,2, … , 𝑚} ∪ {𝑏𝑗 |𝑗 = 1,2, … , 𝑛}, and the edge set is 𝐸(𝐾𝑚,𝑛) = {𝑎𝑖 𝑏𝑗 |𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛}. 𝑉 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = {𝑣, 𝑎𝑖𝑘 , 𝑏𝑗𝑘 |𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑘 = 1,2, … , 𝑝} and the edge set is 𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = {𝑣𝑏𝑗𝑘 , 𝑎𝑖𝑘 𝑏𝑗𝑘 |𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑘 = 1,2, … , 𝑝}. choose, 𝑊𝑙 = {𝑣, 𝑏1𝑘 } for every 𝑘 = 1,2,3, … , 𝑝, then |𝑊𝑙 | = 𝑝 + 1. we can show below that the representation of every two adjacent vertices of𝑉 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) is different. i. for 𝑣𝑏𝑗𝑘 ∈ 𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) since 𝑣 is element of 𝑊𝑙 , then there exist 0 on 1 𝑠𝑡 element in 𝑟(𝑣|𝑊𝑙 ), while for 𝑟(𝑏𝑗𝑘 |𝑊𝑙 ) there are no zero elements except 𝑏1𝑘 the representation to 𝑊𝑙 is 𝑟(𝑏1𝑘 |𝑊𝑙 ) = (1,0), hence 𝑟(𝑣|𝑊𝑙 ) ≠ 𝑟(𝑏𝑗𝑘 |𝑊𝑙 ). ii. for 𝑎𝑖𝑘 𝑏𝑗𝑘 ∈ 𝐸 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) since for 𝑖 = 2,3, … , 𝑚, 𝑗 = 1,2, … , 𝑛, 𝑑(𝑎𝑖𝑘 , 𝑣) = 𝑑(𝑎𝑖𝑘 , 𝑏𝑗𝑘 ) + 𝑑(𝑏𝑗𝑘 , 𝑣) hence 𝑟(𝑎𝑖𝑘|𝑊𝑙 ) ≠ 𝑟(𝑏𝑗𝑘 |𝑊𝑙 ). from the two explanations above we know that 𝑊𝑙 is the local resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). since 𝑣 is adjacent to 𝑏𝑗𝑘 for 𝑗 = 1,2, … , 𝑛 and 𝑘 = 1,2, … , 𝑝. the vertex 𝑏1𝑘 is adjacent to 𝑎𝑖𝑘 for 𝑖 = 2,3, … , 𝑚 and 𝑘 = 1,2, … , 𝑝, thus 𝑊𝑙 is dominant local resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). take any 𝑆 ⊆ 𝑉(𝐺) with |𝑆| < |𝑊𝑙 |. let |𝑆| = |𝑊𝑙 | − 1 the two possibilities below: a. if 𝑣 ∉ 𝑊𝑙 𝑣 ∉ 𝑊𝑙 , then all vertices 𝑏𝑗𝑘 with 𝑗 = 2,3, … , 𝑛 and 𝑘 = 1,2, … , 𝑝 cannot be dominated by 𝑊𝑙 . b. if 𝑣 ∈ 𝑊𝑙 𝑣 ∈ 𝑊𝑙 , then there exist 𝑏1𝑘 ∉ 𝑊𝑙 for 𝑘 = 1,2, … , 𝑝. without loss of generality suppose that 𝑏11 ∉ 𝑊𝑙 . it means that 𝑎𝑖1 cannot be dominated by 𝑊𝑙 for 𝑖 = 2,3, … , 𝑚. therefore, from two possibilities above 𝑆 is not a local dominant resolving set of 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝) or we can conclude that 𝑊𝑙 = 𝑝 + 1 is dominant local basis of 𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝). hence, we get 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾𝑚,𝑛, 𝑣, 𝑝)) = 𝑝 + 1. ∎ the example of a dominant local basis for vertex amalgamation of a complete bipartite graph is depicted as red vertices in figure 5, where 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾3,3, 𝑣, 3)) = 4. figure 6. 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐾3,3, 𝑣, 3)) = 4. on the dominant local resolving set of vertex amalgamation graphs reni umilasari 605 theorem 4. let 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) is a vertex amalgamation of friendship graph with the order is 𝑛 ≥ 3 then 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) = { 𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛), 𝑣 𝑖𝑠 𝑎 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐹𝑛 1 + 𝑝(𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛) − 1), 𝑣 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑐𝑒𝑛𝑡𝑒𝑟 𝑣𝑒𝑟𝑡𝑒𝑥 𝑜𝑓 𝐹𝑛 proof. case 1. 𝑣 is a center vertex of 𝐹𝑛 it is very clearly to see that 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) ≅ 𝐹𝑛 , then by the table 1 we can conclude that 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = 𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛). ∎ case 2. 𝑣 is not a center vertex of 𝐹𝑛 let 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) = {𝑣, 𝑣𝑖 , 𝑥𝑖𝑘 , 𝑦𝑖𝑗 |𝑖 = 1,2,3, … , 𝑝; 𝑗 = 1,2,3, … , 𝑛; 𝑘 = 2,3,4, … , 𝑛} and 𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = {𝑣𝑖 𝑥𝑖𝑘 , 𝑣𝑖 𝑦𝑖𝑗 , 𝑥𝑖𝑘 𝑦𝑖𝑗 , 𝑣𝑦𝑖1|𝑖 = 1,2, … , 𝑝; 𝑗 = 1,2, … , 𝑛; 𝑘 = 2,3,4, … , 𝑛 }. the 𝑖-th copy of 𝐹𝑛 with 𝑖 = 1,2,3, … , 𝑝 is called (𝐹𝑛)𝑖 . let 𝐵 be a local dominant basis of 𝐹𝑛 , 𝐵𝑖 is a local dominant basis of (𝐹𝑛 )𝑖, so that for every 𝑖 = 1,2,3, … , 𝑝, |𝐵𝑖 | = |𝐵| = 𝑛. select 𝑊𝑙 = {𝑣} ⋃ (𝐵𝑖 − 1) 𝑝 𝑖=1 , suppose 𝐵𝑖 − 1 = {𝑥𝑖𝑘 |𝑘 = 2,3, … , 𝑛} for every 𝑖 = 1,2,3, … , 𝑝, then |𝑊𝑙 | = 1 + 𝑝(𝑛 − 1). by lemma 2 for every 𝑥𝑎𝑏 , 𝑥𝑐𝑑 ∈ 𝐵𝑖 then 𝑟(𝑥𝑎𝑏 |𝑊𝑙 ) ≠ 𝑟(𝑥𝑐𝑑 |𝑊𝑙 ). next, we take any two adjacent vertices in 𝑉(𝐺)\𝑊𝑙. let 𝑣𝑖 , 𝑦𝑖𝑗 ∈ 𝑉(𝐺)\𝑊𝑙 with 𝑖 = 1,2,3, … , 𝑝 and 𝑗 = 2,3, … , 𝑛. since 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) is a connected graph, for 𝑑(𝑦𝑖𝑗 , 𝑣) = 𝑑(𝑦𝑖𝑗 , 𝑣𝑖 ) + 𝑑(𝑣𝑖 , 𝑣) for 𝑗 ≠ 1, 𝑣 ∈ 𝑊𝑙 so that 𝑑(𝑣, 𝑦𝑖𝑗 ) ≠ 𝑑(𝑣, 𝑣𝑖 ) caused 𝑟(𝑦𝑖𝑗 |𝑊𝑙 ) ≠ 𝑟(𝑣𝑖 |𝑊𝑙 ). then, 𝑊𝑙 is local resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝). moreover, since 𝑣 adjacent to 𝑣𝑖 and 𝑥𝑖𝑘 adjacent to 𝑦𝑖𝑗 , hence 𝑊𝑙 is dominant local resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝). take any 𝑆 ⊆ 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝)) with |𝑆| < |𝑊𝑙 |. let |𝑆| = |𝑊𝑙 | − 1 the two possibilities below. a) if 𝑣 ∉ 𝑊𝑙 𝑣 ∉ 𝑊𝑙 , then all vertices 𝑦𝑖1 with 𝑖 = 1,2, … , 𝑝 cannot be dominated by 𝑊𝑙 . b) if 𝑣 ∈ 𝑊𝑙 𝑣 ∈ 𝑊𝑙 , then there exist 𝑥𝑖𝑘 ∉ 𝑊𝑙 for 𝑘 = 2,3 … , 𝑛. without loss of generality suppose that 𝑥1𝑛 ∉ 𝑊𝑙 . it means that 𝑦1𝑛 cannot be dominated by 𝑊𝑙 . therefore, from two possibilities above 𝑆 is not a local dominant resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝) or we can conclude that |𝑊𝑙 | = 1 + 𝑝(𝑛 − 1) is dominant local basis of 𝑎𝑚𝑎𝑙(𝐹𝑛, 𝑣, 𝑝). hence, we get 𝐷𝑑𝑖𝑚𝑙 (𝑎𝑚𝑎𝑙(𝐹𝑛 , 𝑣, 𝑝)) = 1 + 𝑝(𝐷𝑑𝑖𝑚𝑙 (𝐹𝑛) − 1) . ∎ figure 6 gives an axample of 𝑎𝑚𝑎𝑙(𝐹3, 𝑣, 3), where 𝑣 is not the center vertex of friendship. those graph has dominant local resolving set equals seven. figure 7. 𝐷𝑑𝑖𝑚𝑙 (𝐹3, 𝑣, 3) = 7 on the dominant local resolving set of vertex amalgamation graphs reni umilasari 606 conclusion based on the findings of this study, it is possible to conclude that the dominant local metric dimension for any vertex amalgamation product graph is determined by the dominant local metric dimension of the copied graphs and how the terminal vertex is chosen. this topic can be expanded by observing the dominant local metric dimension for the vertex amalgamation product with the special graphs that will be glued are different graphs. next, we can determined the dominant local metric dimension for another product of graphs. moreover, the program application of this concept can be generated for any connected graph. acknowledgments this work was supported by drpm, kemenristek of indonesia, dcree 672/un3/2022, contract no. 75/un3.15/pt/2022 and 010/e5/pg.02.00.pt/2022, year 2022. references [1] g. a. barragán-ramírez, a. estrada-moreno, y. ramírez-cruz, and j. a. rodríguezvelázquez, “the simultaneous local metric dimension of graph families,” symmetry (basel)., vol. 9, no. 8, pp. 1–22, 2017, doi: 10.3390/sym9080132. 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[16] r. umilasari, l. susilowati, and slamin, “dominant local metric dimension of wheel related graphs,” iop conf. ser. mater. sci. eng., vol. 1115, no. 1, p. 012029, 2021, doi: 10.1088/1757-899x/1115/1/012029. the distance irregular reflexive k-labeling of graphs cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 622-629 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 18, 2023 reviewed: february 28, 2023 accepted: march 16, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.19747 the distance irregular reflexive k-labeling of graphs ika hesti agustin1,2, dafik1,3, n. mohanapriya 4 , marsidi 5 , ismail naci cangul 6 1pui-pt combinatorics and graph, university of jember, indonesia 2 department of mathematics, university of jember, indonesia 3 department of mathematics education, university of jember, indonesia 4department of mathematics, kongunadu arts and science college, india 5department of mathematics education, university of pgri argopuro jember, indonesia 6uludag university, mathematics department, gorukle 16059 bursa, turkey email: ikahesti.fmipa@unej.ac.id abstract a total 𝑘-labeling is a function 𝑓𝑒 from the edge set to the set {1,2,…,𝑘𝑒} and a function 𝑓𝑣 from the vertex set to the set {0,2,4,…,2𝑘𝑣}, where 𝑘 = max{ ke,2kv}. a distance irregular reflexive k-labeling of graph g is total k-labeling. if for every two different vertices 𝑢 and 𝑢′ of 𝐺, 𝑤(𝑢) ≠ w(u′), where w(u) = σui∈ n(u) fv(ui) + σuv∈ e(g) fe(uv). the minimum k for graph 𝐺 which has a distance irregular reflexive klabeling is called distance reflexive strength of the graph 𝐺, denoted by 𝐷𝑟𝑒𝑓(g). in this paper, we determine the lower bound of distance reflexive strength of any graph and the exact value of distance reflexive strength of path, star, and friendship graph. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc by-sa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: distance irregular reflexive k-labeling; distance reflexive strength; path; star; friendship. introduction a simple graph 𝐺 is a pair (𝑉(𝐺),𝐸(𝐺)), where 𝑉(𝐺) denotes a non-empty vertex set, whereas 𝐸(𝐺) denotes an unordered pair of two distinct vertices 𝑢,𝑣 ∈ 𝑉(𝐺). if there is a path between every pair of distinct vertices in 𝐺, the graph is said to be connected. among the various types of problems that arise while studying graph theory, one that has grown in popularity in recent years is graph labeling. throughout this paper all graphs are simple, connected and undirected 𝐺(𝑉,𝐸). the vertex set and edge set are denoted by v(g) and e(g), respectively [1]. a graph labeling is a map that connects graph elements to numbers (usually to the positive or non-negative integers). gallian [2] describes a recent concept of graph labeling. the most common choices of domain are the set of labels to the vertex set (called vertex labeling), the edge set (called edge labeling), or to both vertices and edges (called total labeling). in many cases it is quite interesting to consider the sum of all labels associated with a graph element called the weight of the element which is denoted by 𝑤. in this research, we concentrate on studying a novel kind of labeling called irregular labeling. if each vertex has a different weight (label sum), when we assign positive integer labels to the edges or vertices of a connected graph with at least three vertices, http://dx.doi.org/10.18860/ca.v7i4.19747 https://creativecommons.org/licenses/by-sa/4.0/ the distance irregular reflexive k-labeling of graphs ika hesti agustin 623 they become irregular. the irregularity strength of a graph g is the minimum of all such irregular assignments' maximum weights. on [3-13], you can view the results of a graph's irregularity strength. the overall vertex irregularity strength on graph g was evaluated by baca et al.; for more information, see [15]. we define a 𝑘-labeling for a graph 𝐺 and assign the numbers {1,2,…,𝑘} to the graph's elements by using a vertex irregular reflexive labeling. let 𝑘 be a natural number; total 𝑘-irregular labeling is a function𝑓 ∶ 𝑉(𝐺) ∪ 𝐸(𝐺) → { 1,2,3, . . . ,𝑘}. a vertex irregular reflexive total 𝑘-labeling is one of the many types of irregular 𝑘-labeling. vertex irregular reflexive labeling is described in detail by tanna et al. in their article [14]. total 𝑘-labeling is a function 𝑓𝑒 from 𝐸(𝐺) to the first natural number up to 𝑘𝑒 and a function 𝑓𝑣 from 𝑉(𝐺) to the non-negative even number up to 2𝑘𝑣, where 𝑘 = 𝑚𝑎𝑥{𝑘𝑒,2𝑘𝑣}. a vertex irregular reflexive 𝑘-labeling of the graph 𝐺 is the total 𝑘-labeling, if for every two different vertices 𝑥 and 𝑥′ of 𝐺, 𝑤𝑡(𝑥) ≠ 𝑤𝑡(𝑥′), where 𝑤𝑡(𝑥) = 𝑓𝑣(𝑥) + σ𝑥𝑦∈ 𝐸(𝐺) 𝑓𝑒(𝑥𝑦). the reflexive vertex strength of the graph g, denoted by 𝑟𝑣𝑠(𝐺), is the minimum 𝑘 for a graph 𝐺 with a vertex irregular reflexive 𝑘-labeling. the minimum number of edges between two vertices 𝑢 and 𝑣 is called the distance between these vertices and is denoted as 𝑑(𝑢,𝑣). motivated by the definition of distance irregular labeling [16] and irregular reflexive labeling of graphs, we develop a new concept namely distance irregular reflexive labeling. we have given a formal definition as follows: a total 𝑘-labeling is a function 𝑓𝑒 from 𝐸(𝐺) to the set {1,2,…,𝑘𝑒} and a function 𝑓𝑣 from 𝑉(𝐺) to the set {0,2,4,…,2𝑘𝑣}, where 𝑘 = 𝑚𝑎𝑥{𝑘𝑒,2𝑘𝑣}. a distance irregular reflexive k-labeling of the graph 𝐺 is the total 𝑘-labeling, if for every 𝑢,𝑢′ ∈ 𝑉(𝐺),𝑢 ≠ 𝑢′, 𝑤(𝑢) ≠ 𝑤(𝑢′), where 𝑤(𝑢) = σ𝑢𝑖∈ 𝑁(𝑢)𝑓𝑣(𝑢𝑖) + σ𝑢𝑣∈ 𝐸(𝐺)𝑓𝑒(𝑢𝑣) and 𝑁(𝑢) is the neighbourhood of distance 1. the minimum 𝑘 for graph 𝐺 which has a distance irregular reflexive 𝑘labeling is called distance reflexive strength of the graph 𝐺, denoted by ɒ𝑟𝑒𝑓(𝐺). in this paper, we determine the lower bound of distance reflexive strength of any graph and the exact value of distance reflexive strength for three types of connected graphs: path, star, and friendship graph. method the method used in determining the distance reflexive strength of a graph is as follows: 1. defines a graph as data or a research object. 2. determine the vertex set and edge set of the graph. 3. determine the lower bound distance reflexive strength of a graph with the following lemma and corollary. lemma: ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ and corollary: ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ + 1, if 𝑝 + 𝛿 − 1 > δ(2𝑡 − 1). 4. construct vertex and edge labels based on the definition of distance irregular reflexive labeling. 5. determine the upper bound of the distance reflexive strength of a graph with the obtained function in point 4. 6. if the upper bound of distance reflexive strength of a graph is the same as the lower bound of distance reflexive strength, then the value of distance reflexive strength can be determined from the graph. the distance irregular reflexive k-labeling of graphs ika hesti agustin 624 7. if the upper bound of distance reflexive strength of the graph is not the same as the lower bound of distance reflexive strength, then point 4 is repeated until the upper bound of distance reflexive strength is equal to the lower bound of distance reflexive strength. results and discussion before determining the distance reflexive strength of a graph, we need to determine the lower bound which is used as a basic reference and proof in the theorem. in addition, the lower bound is used to guarantee the obtained distance reflexive strength is the smallest or minimum value. the lower bound of distance reflexive strength is presented in the following lemma. lemma 1. let 𝐺 be a graph which has 𝑝,𝛿, and δ. for any graph g, ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝 + 𝛿 − 1 2δ ⌉ where 𝑝,𝛿, and δ are order, minimum degree, and maximum degree, respectively. proof. assume that 𝐺 is a graph of order 𝑝, the minimum degree 𝛿, and the maximum degree δ. the greatest label on the graph for every distance irregular reflexive 𝑘-labeling is 𝑘′ = ɒ𝑟𝑒𝑓(𝐺). we choose 𝑢 as a vertex of degree 𝛿, in such a case that the vertex weight of 𝑢 is at least: 𝑤(𝑢) = σ𝑢𝑖∈ 𝑁(𝑢) 𝑓𝑣(𝑢𝑖) + σ𝑢𝑣∈ 𝐸(𝐺) 𝑓𝑒(𝑢𝑣) = 0 + 1(𝛿) = 𝛿. moreover, to get the minimum 𝑘′ the vertex weight must form an arithmetic sequence, namely 𝛿,𝛿 + 1,𝛿 + 2,…,𝛿 + 𝑝 − 1. in other hand, we choose 𝑢′ as a vertex of degree δ, such that 𝑤(𝑢′) = σ𝑢𝑖 ′∈ 𝑁(𝑢′) 𝑓𝑣(𝑢𝑖 ′) + σ𝑢′𝑣′∈ 𝐸(𝐺) 𝑓𝑒(𝑢′𝑣′) = 2𝑘𝑣(δ)+ 𝑘𝑒(δ). since 𝑘′ = 𝑚𝑎𝑥{𝑘𝑒,2𝑘𝑣}, then 𝑤(𝑢′) = 2𝑘𝑣(δ)+ 𝑘𝑒(δ) ≤ 𝑘′(2δ). it is obtained the largest vertex weight is 𝑝 + 𝛿 − 1. since 𝑝 + 𝛿 − 1 ≤ 𝑤𝑡(𝑢′), then 𝑘′(2δ) ≥ 𝑝 + 𝛿 − 1, such that 𝑘′ ≥ 𝑝+𝛿−1 2δ . since ɒ𝑟𝑒𝑓(𝐺) should be integer and we need a sharpest lower bound, it implies 𝑘′ ≥ ⌈ 𝑝+𝛿−1 2𝛥 ⌉ or ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉. ∎ from the lemma 1, we establish the corollary that can be used to calculate the exact value of distance reflexive strength as follows. corollary 1. let 𝐺 be a graph which has 𝑝,𝛿, and δ. if 𝑝 + 𝛿 − 1 > δ(2𝑡 − 1), then ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉+ 1, where 𝑝,𝛿,δ, and 𝑡 are order, minimum degree, maximum degree, and an odd integer, respectively. proof. let 𝐺 be a graph which has 𝑝,𝛿, and δ. let 𝑝 + 𝛿 − 1 = 𝑡(2δ). according to lemma 1, we have ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ > ⌈ δ(2𝑡−1) 2δ ⌉ = ⌈ 2𝑡−1 2 ⌉ = 𝑡. moreover, suppose 𝑡 is ɒ𝑟𝑒𝑓(𝐺), 𝑡 will be the label with the largest size under any distance irregular reflexive k-labeling. on the vertex with the maximum degree, the greatest weight can be attained. let 𝑢′ be the vertex of greatest degree, with u′ is vertex weight being the distance irregular reflexive k-labeling of graphs ika hesti agustin 625 𝑤(𝑢′) = σ𝑢𝑖 ′∈ 𝑁(𝑢′) 𝑓𝑣(𝑢𝑖 ′) + σ{𝑢′𝑣′∈ 𝐸(𝐺)} ≤ (𝑡 − 1)(δ) + 𝑡(δ) = δ(2𝑡 − 1). based on lemma 1, we obtain 𝑝 + 𝛿 − 1 ≤ δ(2𝑡 − 1). it contradicts with 𝑝 + 𝛿 − 1 > δ(2𝑡 − 1). hence 𝑡 ≠ ɒ𝑟𝑒𝑓(𝐺), such that ɒ𝑟𝑒𝑓(𝐺) > ⌈ 𝑝+𝛿−1 2δ ⌉ > ⌈ δ(2𝑡−1) 2δ ⌉ = ⌈ 2𝑡−1 2 ⌉ = 𝑡. thus, it concludes that ɒ𝑟𝑒𝑓(𝐺) ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ + 1. ∎ we use those lemma and corollary above to determine the distance reflexive strength of path, star, and friendship graph which presented in the following theorems. theorem 1. let 𝐺𝑛 be a graph isomorphic with a path graph. for ∀𝑛 ≥ 3, ɒ𝑟𝑒𝑓(𝐺𝑛) = { ⌈ 𝑛 4 ⌉ + 1, if 𝑛 ≡ 3,4(𝑚𝑜𝑑 8) ⌈ 𝑛 4 ⌉, if 𝑛 ≡ 0,5,6,7(𝑚𝑜𝑑 8) where 𝑛 is natural number. proof. the graph 𝐺𝑛, 𝑛 ≥ 3 has a vertex set is 𝑉(g𝑛) = {𝑎𝑖; 1 ≤ 𝑖 ≤ 𝑛 } and an edge set 𝐸(g𝑛) = { 𝑎𝑖𝑎𝑖+1; 1 ≤ 𝑖 ≤ 𝑛 − 1}. as such 𝛿(𝑃𝑛) = 1 and δ(𝑃𝑛) = 2, for 𝑛 ≡ 0,5,6,7 (𝑚𝑜𝑑 8) base on lemma 1 we have ɒ𝑟𝑒𝑓(g𝑛) ≥ ⌈ 𝑝 + 𝛿 − 1 2δ ⌉ = ⌈ 𝑛 + 1 − 1 2(2) ⌉ = ⌈ 𝑛 4 ⌉. for 𝑛 ≡ 3 (𝑚𝑜𝑑 8), it gives ɒ𝑟𝑒𝑓(g𝑛) ≥ ⌈ 𝑛 4 ⌉ = ⌈ 8𝑏+3 4 ⌉ = 2𝑏 + 1 and for 𝑛 ≡ 4 (𝑚𝑜𝑑 8), it gives ɒ𝑟𝑒𝑓(g𝑛) ≥ ⌈ 𝑛 4 ⌉ = ⌈ 8𝑏+4 4 ⌉ = 2𝑏 + 1, where 𝑏 is natural number. since ⌈ 𝑛 4 ⌉ = 2𝑏 + 1 is odd number, 2(2⌈ 𝑛 4 ⌉ − 1) will be the biggest vertex label on path. in other hand, 𝑛 > 2(2⌈ 𝑛 4 ⌉ − 1), this condition satisfies with corollary 1, such that we have ɒ𝑟𝑒𝑓(g𝑛) ≥ ⌈ 𝑛 4 ⌉ + 1. furthermore, we show the upper bound of ɒ𝑟𝑒𝑓(g𝑛) by constructing and defining the function as follows. we give an illustration in figure 1 to show the upper bound of 𝑃3,𝑃4,𝑃5,𝑃6, and 𝑃7. now we give the construction of vertex and edge labels in the following steps: (i) give the label on vertices with the function as follows. for 1 ≤ 𝑖 ≤ 2𝑘 + 1, 𝑓𝑣(𝑎𝑖) = 𝑘 where 𝑘 = { ⌈ 𝑛 4 ⌉ + 1, if 𝑛 ≡ 3,4 (𝑚𝑜𝑑 8) ⌈ 𝑛 4 ⌉ , if 𝑛 ≡ 0,5,6,7 (𝑚𝑜𝑑 8) the distance irregular reflexive k-labeling of graphs ika hesti agustin 626 figure 1. the construction of vertex and edge labels on 𝑃3,𝑃4,𝑃5,𝑃6, and 𝑃7. (ii) give the label on edges with the function as follows. for 1 ≤ 𝑖 ≤ 2𝑘, 𝑓𝑒(𝑎𝑖𝑎𝑖+1) = 𝑘 + 1 − ⌈ 𝑖 2 ⌉. (iii) the vertex weight, 𝑤(𝑎1) = 2𝑘 and give the vertex weight 𝑤(𝑎𝑖) = 4𝑘 + 2 − 𝑖, for 2 ≤ 𝑖 ≤ 2𝑘. (iv) determine 𝑤(𝑎2𝑘+1) = 2𝑘 + 1, and for 2𝑘 + 2 ≤ 𝑖 ≤ 𝑛 − 1,𝑤(𝑎𝑖) = 4𝑘 + 1 − 𝑖. (v) for 2𝑘 + 2 ≤ 𝑖 ≤ 𝑛, label each vertex 𝑎𝑖 with 𝑓𝑣(𝑎𝑖−1) and label edge 𝑎𝑖−1𝑎𝑖 with 𝑤(𝑎𝑖−1) − 𝑓𝑣(𝑎𝑖−2) − 𝑓𝑒(𝑎𝑖−2𝑎𝑖−1) − 𝑓𝑣(𝑎𝑖). (vi) if the edge label of point [4] is 0, then label each vertex 𝑎𝑖 with 𝑓𝑣(𝑎𝑖−1) − 2 and label edge 𝑎𝑖−1𝑎𝑖 with 𝑤(𝑎𝑖−1) − 𝑓𝑣(𝑎𝑖−2) − 𝑓𝑒(𝑎𝑖−2𝑎𝑖−1) − 𝑓𝑣(𝑎𝑖). (vii) the vertex weight of 𝑎𝑛 as follows. 𝑤(𝑎𝑛) = 𝑓𝑣(𝑎𝑛−1) + 𝑓𝑒(𝑎𝑛−1𝑎𝑛), since 𝑤(𝑎𝑛) is the sum of two labels (vertex and edge), then 𝑤(𝑎𝑛) must be different from the others. since the all vertices are distinct, ɒ𝑟𝑒𝑓(g𝑛) ≥ 𝑘 and ɒ𝑟𝑒𝑓(g𝑛) ≤ 𝑘, it concludes that ɒ𝑟𝑒𝑓(g𝑛) = 𝑘 for 𝑛 ≥ 3. ∎ the distance irregular reflexive k-labeling of graphs ika hesti agustin 627 theorem 2. let h𝑛 be a graph isomorphic with a star graph. for ∀𝑛 ≥ 3, ɒ𝑟𝑒𝑓(h𝑛) = 𝑛. proof. the graph 𝐻𝑛,𝑛 ≥ 3 has a vertex set 𝑉(h𝑛) = {𝛼,𝑥𝑖; 1 ≤ 𝑖 ≤ 𝑛} and an edge set 𝐸(h𝑛) = {𝛼 𝑥𝑖; 1 ≤ 𝑖 ≤ 𝑛}. the star graph has n vertices of degree 1 and one dominant vertex. since 𝛿(h𝑛) = 1 and the dominant vertex always contributes the constant label to other vertices, the smallest vertex weight is at least 1 and the largest vertex weight on vertices of degree 1 is at least n. since 𝛼 is the dominant vertex and we can label it with 0, the vertex weight on vertices of degree 1 depends on 1 label (only edge), such that ɒ𝑟𝑒𝑓 (h𝑛) ≥ 𝑛. since δ(h𝑛) = 𝑛, then ɒ𝑟𝑒𝑓(h𝑛) ≥ 𝑛 ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ and satisfies the lemma 1. furthermore, we show the upper bound of ɒ𝑟𝑒𝑓(h𝑛) by defining the following function: 𝑓𝑣(𝛼) = 𝑓𝑣(𝑥𝑖) = 0 𝑓𝑒(𝛼 𝑥𝑖) = 𝑖: 1 ≤ 𝑖 ≤ 𝑛 by those vertex and edge labels, we have the vertex weight as follows. 𝑤(𝛼) = ∑[𝑓𝑣(𝑥𝑖) + 𝑓𝑒(𝛼 𝑥𝑖)] 𝑛 𝑖=1 = 𝑛(𝑛 + 1) 2 𝑤(𝑥𝑖) = 𝑖: 1 ≤ 𝑖 ≤ 𝑛 we know that all vertex weights of graph h𝑛 are distinct and it gives a distance reflexive strength on the star graph. since ɒ𝑟𝑒𝑓(h𝑛) ≤ 𝑛 and ɒ𝑟𝑒𝑓(h𝑛) ≥ 𝑛, it concludes that ɒ𝑟𝑒𝑓(h𝑛) = 𝑛 for 𝑛 ≥ 3. ∎ theorem 3. let m𝑛 be a graph isomorphic with friendship graph (𝐹𝑛). for ∀𝑛 ≥ 3, ɒ𝑟𝑒𝑓(m𝑛) = { ⌈ 2n + 1 3 ⌉ + 1, if 𝑛 ≡ 1(𝑚𝑜𝑑 3) ⌈ 2𝑛 + 1 3 ⌉ , otherwise where 𝑛 is natural number. proof. the graph m𝑛,𝑛 ≥ 3 has a vertex set 𝑉(m𝑛) = {𝑎,𝑢𝑖,𝑣𝑖; 1 ≤ 𝑖 ≤ 𝑛} and an edge set 𝐸(m𝑛) = {𝑎𝑢𝑖; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑎𝑣𝑖; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢𝑖𝑣𝑖; 1 ≤ 𝑖 ≤ 𝑛}. the graph m𝑛 has 2𝑛 vertices of degree 2 and one dominant vertex. since 𝛿(m𝑛) = 2 and the dominant vertex always contributes the constant label to other vertices, the smallest vertex weight is at least 2 and the biggest vertex weight on vertices of degree 2 is at least 2𝑛 + 1. since a is the dominant vertex and we can label it with 0, the vertex weight on vertices of degree 2 depends on 3 labels (2 edges and 1 adjacent vertex). since label of vertex must be an even number, in such a case that ɒ𝑟𝑒𝑓(m𝑛) ≥ ⌈ 2𝑛+1 3 ⌉. since δ(m𝑛) = 2𝑛, then ɒ𝑟𝑒𝑓(m𝑛) ≥ ⌈ 2𝑛+1 3 ⌉ ≥ ⌈ 𝑝+𝛿−1 2δ ⌉ and satisfies the lemma 1. for 𝑛 ≡ 1(𝑚𝑜𝑑 3), there is an integer s such that 2𝑛 = 2(3𝑠 + 1) = 6𝑠 + 2, then ⌈ 2𝑛+1 3 ⌉ = ⌈ 6𝑠+3 3 ⌉ = 2𝑠 + 1. the vertex weight 2𝑛 + 1 = 6𝑠 + 3 is the sum of 3 labels where each label is not more than 2𝑠 + 1, such that 2𝑛 + 1 = 6𝑠 + 3 = (2𝑠 + 1) + (2𝑠 + 1) + (2𝑠 + 1). as such the vertex label should be an even number, for 𝑛 ≡ 1(𝑚𝑜𝑑 3) we get ɒ𝑟𝑒𝑓(m𝑛) ≥ ⌈ 2𝑛+1 3 ⌉ + 1. furthermore, we show the upper bound of ɒ𝑟𝑒𝑓(m𝑛) by defining the following function: the distance irregular reflexive k-labeling of graphs ika hesti agustin 628 𝑓𝑣(𝑎) = 0 𝑓(𝑢𝑖) = { 0, if 𝑖 = 1 2⌈ 𝑖 − 1 3 ⌉, if 2 ≤ 𝑖 ≤ 𝑛 𝑓𝑣(𝑢𝑖) = 𝑓𝑣(𝑣𝑖) 𝑓𝑒(𝑢𝑖𝑣𝑖) = { 2 3 𝑖, if 𝑖 ≡ 0 (𝑚𝑜𝑑 3) 1 3 (2𝑖 + 1), if 𝑖 ≡ 1 (𝑚𝑜𝑑 3) 1 3 (2𝑖 − 1), if 𝑖 ≡ 2 (𝑚𝑜𝑑 3) 𝑓𝑒(𝑎𝑢𝑖) = 𝑓𝑒(𝑢𝑖𝑣𝑖) 𝑓𝑒(𝑎𝑣𝑖) = { 2 3 𝑖 + 1, if 𝑖 ≡ 0 (𝑚𝑜𝑑 3) 1 3 (2𝑖 + 1) + 1, if 𝑖 ≡ 1 (𝑚𝑜𝑑 3) 1 3 (2𝑖 − 1) + 1, if 𝑖 ≡ 2 (𝑚𝑜𝑑 3) by those vertex and edge labels, we have the vertex weight as follows. 𝑤(𝑎) = ∑[𝑓𝑣(𝑢𝑖) + 𝑓𝑣(𝑣𝑖) + 𝑓𝑒(𝑎𝑢𝑖) + 𝑓𝑒(𝑎𝑣𝑖)] 𝑛 𝑖=1 𝑤(𝑢𝑖) = 2𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 𝑤(𝑣𝑖) = 2𝑖 + 1 ∶ 1 ≤ 𝑖 ≤ 𝑛 we know that the all vertex weights of graph m𝑛 are distinct. thus, it gives a distance reflexive strength on m𝑛 for 𝑛 ≥ 3. ∎ conclusions we determined the lower bound and ɒ𝑟𝑒𝑓(𝐺), where 𝐺 be several graphs which isomorphic with path, star, and friendship graphs. since every graph has different characteristics, the distance reflexive strength varies. based on the characteristics of these graphs, we can characterize the distance reflexive strength. characterizing the distance reflexive strength is a difficult problem that requires extensive research. as a result, we present the following open problems. open problem 1. find out the exact value of ɒ𝑟𝑒𝑓(𝑃𝑛) for 𝑛 ≡ 1,2 (𝑚𝑜𝑑 8) and any graphs apart from those families. open problem 2. characterize the distance reflexive strength of graph. the distance irregular reflexive k-labeling of graphs ika hesti agustin 629 references [1] g. chartrand, l. lesniak, p. zhang, graphs & digraphs, sixth ed., taylor & francis group, boca raton, new york, 2016. 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[16] slamin. 2017. on distance irregular labeling of graphs. far east journal of mathematical sciences (fjms). volume 102, number 5, 2017, pages 919-932. http://dx.doi.org/10.17654/ms102050919. comparing several missing data estimation methods in linear regression;real data example and a simulation study cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 641-648 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 27, 2023 reviewed: april 01, 2023 accepted: april 23, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.20548 comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto1,2*, yap ee jia3, budi susetyo1 , la ode abdul rahman1 1department of statistics, ipb university, bogor, west java, indonesia 2institute of engineering mathematics, universiti malaysia perlis, malaysia 3department of mathematics and statistics, universiti putra malaysia, serdang, malaysia email: anwarstat@gmail.com abstract one of the causes of bias in parameter estimation is incomplete data when analyzed using standard statistical procedures. in addition, if the analysis is performed on missing data, the researcher may not have sufficient observations necessary for the analysis. for this reason, a method is needed to estimate the missing data. until now, there are many methods for estimating lost data.. the main objective of the study is to compare the performance between listwise deletion (ld), mean substitution (ms) and multiple imputation (mi) methods in estimating parameters. the performance will be measured through bias, standard error and 95% confidence interval of interested estimates for handling missing data with 10% missing observations. a complete empirical data set was used and assumed as population data. ten percent of total observations in the population ere set as missing arbitrarily by generating random numbers from a uniform distribution, . then, bias of parameter estimates and confidence interval of parameter estimates are calculated to compare the three methods. a monte carlo simulation was carried out to know the properties of missing data estimation methods. simulation of 1000 sampled data with 20, 50, and 100 observations and each sample is set to have 10% missing observations. standard statistical analyses are run for each missing data and get the average of parameter estimates to calculate the bias and standard error of parameter estimates for every missing data method. the analysis was conducted by using sas version 9.2. it was found that the mi method provided the smallest bias and standard error of parameter estimates and a narrower confidence interval compared to the ld and ms methods. meanwhile, the ld method gives a smaller bias of parameter estimates and standard error for small sample size of missing data. and, ms method is strongly recommended not to use for handling missing data because it will result in large bias and standard error of parameter estimates. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: incomplete, mcar, missing, regression, simulation introduction many researchers try to solve the problems of missing data analysis by using last observation carried forward (locf), complete case, available case, and single imputation method [1]. the locf is a method on substituting the last available measurement whenever there is a missing value [2]. this method was popular for treating data with monotone and non-monotone missing patterns and is valid for longitudinal studies. beside that, many people use complete case methods such as the listwise deletion (ld) http://dx.doi.org/10.18860/ca.v7i4.20548 mailto:anwarstat@gmail.com https://creativecommons.org/licenses/by-sa/4.0/ comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 642 method to solve the missing data problem, [3]. ld is the simplest and easiest method among conventional methods of dealing with missing data, [4]. although ld is the simplest method, the number of deleted cases will increase as the values are missing arbitrarily. the major advantage of listwise deletion is that it produces a complete data set, which in turn allows for the use of standard analysis techniques, [5]. however, there is a loss in power using ld method even though the data is mcar, especially since a large number of subjects have been deleted. furthermore, pairwise deletion (pd) is another case-based method. it uses all available information in statistical analysis. at the same time, the mean substitution (ms) method is also popular since it is one of the single imputation methods. a study by [6] found that overall mean computed from mean substitution (ms) is equal to the complete case values but the variance of the same variable is underestimated. meanwhile, [7] defined the ms method as an approach to substitute the missing items with the mean of the non-missing items. it could be problematic when applying to categorical data. although the ms method can be used to analyze a complete case, it will reduce the variability and weaken covariances and correlation estimates in the data, [8]. another method is multiple imputation (mi) proposed by [3]. it is conducted by replacing each missing value with a set of plausible values. in the mi method, missing values for any variable are predicted using existing values from other variables. the predicted values, called “imputes”, are substituted for the missing values, resulting in a full data set called an imputed data set, [9]. [10] provided three phases for mi inference. conventional statistical methods and analysis tools presume that all variables in a specified model are measured for all cases. the default method for all statistical software is simply deleting the cases with missing value on the interesting variables such as ld method. [11] supported that if the standard approach analysis is used to analyze incomplete data, the estimation of such analysis can be biased. complete case analysis by ignoring the missing data will lead to an inefficient, biased, unreliable result. missing data results in information loss and statistical power, [12]. meanwhile, for a small data set that contains a relatively large number of missing observations, many cases will be simply deleted by the default method. based on that fact, the research aims to compare the performance of listwise deletion (ld) and mean substitution (ms) methods on bias, standard error and 95% confidence interval. method data the data set involved in this study is about human resources obtained from human development reports of united nationals development programme, [13]. several variables are involved, such as the human development index, life expectancy at birth and gross national income variable. human development enlarges people’s choices. the most critical of these wide-ranging choices are to live a long and healthy life, be educated and access the resources needed for a decent living standard. the measure includes life expectancy, literacy, and a modified measure of income, [14]. meanwhile, life expectancy at birth reflects the overall mortality level of a population. it summarizes the mortality pattern across all age groups from children to the elderly, [15]. life expectancy relates to the value people attach to living long, healthy, and well. the life expectancy at birth component in the hdi data is calculated using a minimum value of 20 years and a maximum value of 85 years. the other variable included in the study is gross national income. according to [16], the gross national income is the total comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 643 domestic and foreign output claimed by residents of a country, consisting of the gross domestic product plus factor incomes earned by foreign residents minus income earned in the domestic economy by non-residents. in the data, the standard of living dimension is measured by gross national income per capita. the human development index is expected to be strongly affected by life expectancy at birth and gross national income. thus, human development index is response variable, denoted by , life expectancy at birth and gross national income per capita are the predictors. simulation designs monte carlo simulation was conducted to compare performance between missing data estimation methods in estimating missing values. according to [17], the researcher begins the simulation by creating a model with known population parameters that the researcher sets the values. in the monte carlo simulation, b samples of n size are drawn from the population and, for each sample, estimates the interested parameter. next, a sampling distribution is estimated for each population parameter by collecting the parameter estimates from all the samples. the properties of that sampling distribution, such as its mean and variance, come from this estimated sampling distribution. sas version 9.2 was used for the simulation. there are few steps involved in the simulation: step 1: develop a simple linear regression model with known parameters. the simulation is begun with a simple linear regression model 𝑦 = 𝛽0 + 𝛽1𝑥 + 𝜀 with arbitrary known parameters 𝛽0=5 and 𝛽1=10. the independent variable 𝑋 is generated as 2 × observation number. in this simulation, the number of observations was set equal to 20. then random errors were generated from a normal distribution 𝜀~𝑁(0, 𝜎 = 2), step 2: generate the missing data set let’s d is the number of variable observations in the generated data set that are missing arbitrarily and 𝑑 < 𝑛. the missing measurements are selected by generating a random number from a uniform distribution 𝑈(0,1). the measurements of the variable 𝑋1 with the first d smallest random numbers are made to be missing by denoted as dot “.” . now, a missing data set are created, step 3: repeat b samples from a population repeat step 2 for 1000 times to generate 1000 missing data sets with 20 observations, step 4: missing data analysis using ld, ms and mi method for each missing data set, statistical analysis are carried out to calculate the parameter estimates of the linear regression model by using ld, ms and mi method, step 5: calculate the standard error (se) of parameter estimates and confidence intervals after obtaining 1000 parameter estimates for the linear regression model based on each missing data estimation method, bias, standard errors and confidence intervals of the parameters are calculated as follows: 𝐵𝑖𝑎𝑠 = �̅�𝑖 − 𝛽𝑖 = 𝑖 = 1, 2, ...., 1000. (1) 𝑆𝐸 = √∑ �̄�𝑖 2 − (∑ �̄�𝑖) 2 𝑛 𝑛 − 1 (2) comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 644 where 𝛽0 = 5 and 𝛽0 = 10. meanwhile, the 95% confidence interval of 𝛽𝑖 is calculated as: �̄�𝑖 ± 𝑡0.05/2;𝑑𝑓 √ ∑ �̄�𝑖 2 − (∑ �̄�𝑖) 2 𝑛 𝑛 − 1 , (3) step 6: repeat step 1 to step 5 for 50 and 100 observations, respectively. mechanism of missing data [3] had developed a taxonomy and terminology of missing data mechanisms. the missing data mechanisms include missing completely at random (mcar), missing at random (mar), and missing not at random (mnar). it is essential to understand the mechanism because the problems caused by the missing data and the solutions to these problems are different. regarding tests for missing data, [18] proposed an mcar test that is distributed as a 𝜒2 under the 𝐻0 and called 𝑑 2. the little’s mcar test was conducted using a custom sas macro. little’s mcar test is a chi-square test to determine whether data is mcar, [19]. most researchers use little’s mcar test to test mcar. the 𝑑2 sums the squared standardized mean differences across the j missing data patterns. the 𝑑2 is mahalanobis distances which is written as: 𝑑2 = ∑ 𝑚𝑗 (�̄�𝑜𝑏𝑠.𝑗 − �̂�𝑜𝑏𝑠.𝑗 ) 𝐽 𝑗=1 ∑̂𝑜𝑏𝑠.𝑗 −1 (�̄�𝑜𝑏𝑠.𝑗 − �̂�𝑜𝑏𝑠.𝑗 ), (4) where 𝑚𝑗 is the number of cases in pattern 𝑗, 𝑑𝑓 =∑𝑝𝑗 − 𝑝, where 𝑝𝑗 is the number of complete variables for pattern j. the estimates �̂� and ∑̂ shown in equation (4) are obtained via maximum likelihood estimation using expectation maximization (em) algorithm. it is an iterative procedure that produces maximum likelihood (ml) estimates of a covariance matrix and mean vector, assuming mar. the data is considered as mcar if the p value from little’s mcar test is insignificant. unfortunately, no standard statistical test determines if the missing data is mar, [20]. the three common missing data mechanisms are as follows: missing completely at random according to [11], mcar mechanism indicates that the probability of a missing value is unrelated to any observed and unobserved values. [21] explained the mcar by denoting a single variable with missing data as 𝑊. suppose a set of variables is represented by vector 𝑍. now, let 𝑅𝑊be a dummy variable with a value 1 if 𝑊is missing and 0 if 𝑊is observed. the expression of mcar mechanism is written as: 𝑃𝑟(𝑅𝑊 = 1|𝑊, 𝑍) = 𝑃𝑟(𝑅𝑊 = 1) (5) the probability that 𝑊 is missing depends neither on the observed variables in 𝑍 vector nor on the missing values 𝑊itself. complete case analysis will only give unbiased result if the missing data is assumes mcar. comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 645 missing at random mcar mechanism is a special case of mar mechanism. for the data to be mar, the probability of 𝑊is missing depends on observed variables but independent on 𝑊itself. [22] claimed that the data is mar when the missingness is correlated with other observed variables in the analysis. the mar mechanism is expressed as follows: 𝑃𝑟(𝑅𝑊 = 1|𝑊, 𝑍) = 𝑃𝑟(𝑅𝑊 = 1|𝑍) (6) since missing 𝑊is arising from the same distribution as the observed 𝑍, thus missing 𝑊can be predicted by using the observed 𝑍 distribution, [11]. mcar or mar data are sometimes considered ignorable missingness, [23]. this is because for those data still can produce unbiased estimates without needing a model to explain the missingness. missing not at random if the data are not mcar or mar, then the data would be mnar, where the probability of 𝑊is missing depends on both observed and missing variables. the data for mnar is a case of non-ignorable missingness. the missing data mechanism must be modeled as part of the estimation process for a valid estimation. unfortunately, mcar mechanism is fraught with difficulty. the model for the missing data mechanism must be carefully tailored with each situation because every mnar situation is different, [21]. she also stated that there is no information in data to help choose the appropriate model and no statistic to tell how well a chosen model fits the data. results and discussion bias of parameter estimates as shown in table 1, mi results in the least bias in parameter estimates. for 𝑏0, there is only 0.0047 bias from the true parameter estimates from the original data. the 𝑏1 can be said to be unbiased parameter estimates since the bias value is small enough, which is only -0.0001. while 𝑏2 have biased downward with 0.0038. the three-parameter estimates using the mi method yield good parameter estimates where their biases are nearly zero. table 1. bias of parameter estimates of ld, ms and mi method methods bias of parameter estimates 𝒃𝟎 𝒃𝟏 𝒃𝟐 ld -0.0170 0.0002 0.0202 ms 0.2716 -0.0020 -0.4992 mi 0.0047 -0.0001 -0.0038 the mi was expected to have the smallest bias compared to the other two methods. although the missing data is mcar, since the human development index and life expectancy at birth are highly correlated, it helps predict the missing value nearest to the original value. in the ms method, each missing value is substituted by the mean of the observed variable. it means that all missing values are replaced with same value. it causes the substituted values are not similar or nearer to the original value since a single mean value only substitutes every missing value. that is the reason ms method yields the most extensive bias among these three methods. comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 646 table 2. 95% confidence interval and corresponding length of parameter estimates of ld, ms and mi methods estimator 𝒃𝟎 𝒃𝟏 𝒃𝟐 true value 0.8252 0.00736 -2.5594 method lci uci length lci uci length lci uci length ld 0.6739 0.9426 0.2687 0.0065 0.0086 0.0021 -2.7980 -2.2802 0.5178 ms 0.9738 1.2199 0.2461 0.0044 0.0064 0.0020 -3.2985 -2.8186 0.4800 mi 0.7015 0.9583 0.2567 0.0063 0.0083 0.0020 -2.8077 -2.3187 0.4889 meanwhile, the performance of each estimation method of missing data about 95% confidence interval is displayed in table 2. the ms method gives the smallest confidence interval length based on the table. but, true parameter values from ms method did not fall in the interval. hence, the ms method was not suitable for estimating the true value. but, the ld and mi methods can help estimate true value since true value falls in the confidence interval. mi method was preferable than the ld method in estimating parameters since it can produce a narrower confidence interval. results of the simulation study when there are 10% missing observations in a small data set (n=20), for after 1000 simulation runs, the ms method provides the least bias of parameter estimates. however, the bias of parameter estimates for ms method increases as the sample size increases. in other words, the ms method is not good enough to handle medium or large numbers of missing data since it will produce large bias. table 3. bias of parameter estimates from 1000 simulation runs for ld, ms and mi with 10% missing data in 20, 50, and 100 observations, respectively number of simulation number of observations number of missing observations methods bias of parameter estimates 𝑏0 𝑏1 1000 20 2 ld -1.0013 0.0125 ms 0.0969 0.0070 mi -1.0950 0.0141 50 5 ld -0.8805 0.0093 ms 2.0675 -0.0024 mi -0.7981 0.0071 100 10 ld -0.8420 0.0083 ms 6.3112 -0.0095 mi -0.7075 0.0062 meanwhile, the ld and mi methods are better for handling many missing data. bias of 𝑏0 produced by ld method for small number of missing data is negative with -1.0013 and 𝑏1 is 0.0125. when 𝑛 is increases to 50, bias of parameter estimates reduces. table 3 shows that the bias for 𝑏0 and 𝑏1 in ld method decrease as the number of observations increases. as we see in table 3, mi method follows the trends as ld method that the bias of parameter estimates is reduced as 𝑛 increases. however, we notice that the bias from mi method reduces more drastically compared to ld method as 𝑛 increases. the results from the simulation study follow the results from the empirical data where the mi method gives the least bias for a large number of missing data. comparing several missing data estimation methods in linear regression;real data example and a simulation study anwar fitrianto 647 table 4. standard error of parameter estimates from 1000 simulation runs for ld, ms and mi with 10% missing data in 20, 50, and 100 observations, respectively. number of simulation number of observations number of missing observations methods standard error of parameter estimates 𝑏0 𝑏1 1000 20 2 ld 0.2368 0.0109 ms 10.3413 0.0763 mi 0.2411 0.0110 50 5 ld 0.2305 0.0086 ms 14.6124 0.0744 mi 0.1466 0.0045 100 10 ld 0.1838 0.0062 ms 20.4287 0.0647 mi 0.1110 0.0010 table 4 shows the standard error of parameter estimates for ld, ms and mi method. the standard error of 𝑏0 from ms method is increase as 𝑛 increases. and, although the standard error of 𝑏1 decreases as 𝑛 increases, but the standard error of 𝑏1 is the largest compared to the standard error of parameter estimates from ld and mi methods. overall, the ms method is not suggested to handle missing data because it will result in the largest standard error of parameter estimates if we compare to the ld and mi methods. on the other hand, the ld and mi method provide a small standard error of parameter estimates. the ld method provides the smallest standard error of parameter estimates when n =20. however, the ms method gives the smallest standard error of parameter estimates, increasing to 50 and 100. conclusions and future works missing data may result in biased parameter estimates. however, a different sample size of missing data is recommended by various methods. the ld method is more appropriate for treating small missing data because it gives negligible bias and standard error of parameter estimates. however, the ms method is strongly recommended not to use for handling missing data due to large bias and standard error of parameter estimates. from the study result, mi method can reduce the bias of parameter estimates of missing data. based on that fact, it is strongly recommended to analyze missing data using mi method rather than ld method and ms method because the mi method results in smaller bias and standard error of parameter estimates. in the future, analyses of data with missing values in two or more variables are suggested. in the statistical analysis, other than standard error and bias of parameter estimates, analysts can also compare the mean and efficiency of different missing data estimation methods at different missing percentages. furthermore, analyzing missing data with different mechanisms or patterns can also be a research topic. references [1] s. ghosh and p. pahwa, “assessing bias associated with missing data from joint canada,” in united states survey 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[23] t. d. pigott, “a review of methods for missing data,” educational research and evaluation, vol. 7, no. 4, pp. 353–383, 2001 mathematical model of iteroparous and semelparous species interaction cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 445-463 p-issn: 2086-0382; e-issn: 2477-3344 submitted: june 16, 2022 reviewed: june 22, 2022 accepted: june 27, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.16447 mathematical model of iteroparous and semelparous species interaction arjun hasibuan*, asep kuswandi supriatna, and ema carnia department of mathematics, faculty of mathematics and natural sciences, universitas padjadjaran, jl. raya bandung-sumedang km 21, jatinangor, sumedang, west java province, 45363, indonesia email: arjun17001@mail.unpad.ac.id abstract to study the survival of a species in an ecosystem it is very important to consider the dynamics of the species. a species can be categorized based on its reproductive strategy either semelparous or iteroparous. in this paper, we examine the dynamics involving both categories of species in an ecosystem. we focus on one semelparous and one iteroparous species influenced by densitydependent and also by harvesting factors in which there are two age classes for each species. we study two different models, i.e competitive and non-competitive models. we also consider two type of competition, i.e intraspecific and interspecific competition. the approach that we use in this research is the multispecies leslie matrix model. in addition, we use m-matrix theory to obtain the locally stable asymptotically of the model. our results show that the level of competition both intraspecific and interspecific competition affect the co-existence equilibrium point and the stability of the equilibrium point. we also present explicitly the conditions for all equilibrium points to exist and to be locally stable asymptotically. this theory can be applied to study the dynamics of natural resource models including the effects of different management to the growth of the resources, such as in fisheries. keywords: density-dependent; harvesting; multispecies; leslie matrix; age-structured model introduction in an ecosystem, the survival of a species is an important thing to study. species in the same ecosystem have reciprocal relationships between one species and other. the survival of each species can be affected by density-dependent, harvesting, competition, predator-prey, and so on. of course, the important thing to do is to ensure the survival of these species to survive. the survival of a species can be studied with a system dynamics approach. in some studies, species in an ecosystem can be categorized based on their reproductive strategy, including species with semelparous and iteroparous strategy. research on semelparous species can be seen in [1]–[3]. then, research on iteroparous species can be seen in [4]–[6]. semelparous species are species that reproduce only once in their lifetime shortly before dying. then, iteroparous species are species that reproduce more than once in the lifetime of the species. both species allow to live together and interact in the same ecosystem. in this research, we focus on studying the growth of multispecies cases consisting of one semelparous species and one iteroparous species http://dx.doi.org/10.18860/ca.v7i3.16447 mailto:arjun17001@mail.unpad.ac.id mathematical model of iteroparous and semelparous species interaction arjun hasibuan 446 using a dynamic system approach, especially using the leslie matrix model. this model is a population growth model based on age class which was introduced in 1945 by leslie in [7]. research on studying the dynamics of population growth using the leslie matrix model has been carried out by several researchers. these studies can be in the form of single species and multispecies cases. several studies on single species cases include in [3], [8], [9], and many more. then, several multispecies studies examine the effect of density-dependent on the leslie matrix model which is one of the nonlinear models of the leslie matrix model. in 1968, pennycuick et al. [10] focused on simulating the case of single species and multispecies interacting in the form of competition and predator-prey using the leslie matrix. in 1980, travis et al. [11] reviewed two competing species and provided a case study on semelparous. in 2011, kon [12] studied two semelparous species with one species containing two age classes while the other species amounting to one age class. in 2012, kon [13] conducted a study on two semelparous species that have a predator-prey relationship and observed the effect of coprime traits from the number of age classes in both species. coprime is a condition where two numbers have the greatest common factor of one, in which case the number is the number of age classes of each species. then in 2017, kon [1] examined the leslie multispecies semelparous matrix model which has an arbitrary number of age classes. then, there are also studies on multispecies but with other methods using the rosenzweig-macarthur model (see [14], [15]), the leslie-gower model (see [16], [17]), and the lotka-volterra model (see [18]– [20]). our aim in this paper is to study the growth dynamics of an ecosystem consisting of one semelparous species and an iteroparous species with two age classes in each species. in addition, we combined the density-dependent effect of the first age class for the two species. then, we consider the effect of harvesting that occurs in the second age class in each species on the growth of each species. next, we divide the case into two models consisting of without competition and with competition. both models were analyzed and seen the influence of the level of intraspecific and interspecific competition on the equilibrium point and locally stable asymptotically for each equilibrium point. methods leslie's matrix model with one iteroparous species and one semelparous species without competition in this section, we present one of the models that we studied, namely the multispecies leslie matrix model with the case of one iteroparous species (𝑥) and one semelparous species (𝑦) in this case each species has two age classes. in this first model, we assume that the growth of both species is influenced by density-dependent occurrence in the first age class and harvesting is carried out in the second age class. in this case, the densitydependent problem used in the model uses the classical beverton-holt function which is also used in wikan's research [21]. this problem is presented in the following model and we refer to as model 1: 𝑥1(𝑡 + 1) = 𝑓𝑥1 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡) 𝑥2(𝑡 + 1) = 𝑠𝑥1(1 − ℎ𝑥2) 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑥1(𝑡) (1) mathematical model of iteroparous and semelparous species interaction arjun hasibuan 447 𝑦1(𝑡 + 1) = 𝑓𝑦2𝑦2(𝑡) 𝑦2(𝑡 + 1) = 𝑠𝑦1(1 − ℎ𝑦2) 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑦1(𝑡) there are several parameters in the model 1. first, 𝑓𝑥1 > 0 and 𝑓𝑥2 > 0 are the birth rates of the 1st and 2nd age classes of species 𝑥, respectively. second, 𝑓𝑦2 > 0 is the birth rate of the 2nd age classes of species 𝑦. third, 0 < 𝑠𝑥1, 𝑠𝑦1 < 1 are the survival rates of the 1st age classes of species 𝑥 and 𝑦, respectively. fourth, 0 < ℎ𝑥2, ℎ𝑦2 < 1 are the harvesting rates of the 2nd age classes of species 𝑥 and 𝑦, respectively. in addition, the variables 𝑥𝑖(𝑡), and 𝑦𝑖(𝑡) represent the total population of each species 𝑥 and 𝑦 for the age class 𝑖 = {1,2}. simply put, equation 1 in (1) means that the population of the first age class of species 𝑥 at time 𝑡 + 1 is obtained by adding the number of newborn from the first age class and the second class at time 𝑡. the newborn of first age class is affected by density-dependent while the newborn of the second age class is not affected by density-dependent. then, equation 2 in (1) means that the number of population of the second age class of species 𝑥 at time 𝑡 + 1 is obtained from the number of surviving populations which is influenced by density-dependent of the first age class at time 𝑡. equations 3 and 4 in (1) have the same meaning as equations 1 and 2 in (1) but in species 𝑦 there is no birth in the first age class. model 1 is constructed based on research conducted by leslie [7], travis et al. [11], and wikan [21]. in addition, model 1 is adjusted based on the assumptions and simplifications in this research. leslie matrix model with one iteroparous species and one semelparous species with competition effect in this section, we present a model which is an extension of the previous model. the problems raised in this section involve the effect of competition between the same species, also known as intraspecific competition, and competition between different species, also known as interspecific competition. the following is an extension of the model 1 and we refer to it as model 2. 𝑥1(𝑡 + 1) = 𝑓𝑥1 1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡) 𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡) 𝑥2(𝑡 + 1) = 𝑠𝑥1(1 − ℎ𝑥2) 1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡) 𝑥1(𝑡) (2) 𝑦1(𝑡 + 1) = 𝑓𝑦2𝑦2(𝑡) 𝑦2(𝑡 + 1) = 𝑠𝑦1(1 − ℎ𝑦2) 1 + 𝑏𝑥1(𝑡) + 𝑎𝑦1(𝑡) 𝑦1(𝑡) the description of the parameters and variables in the model 2 is the same as in model 1 where the difference is only in parameters 𝑎 > 0 and 𝑏 > 0. in model 2, to simplify the problem, we assume that the level of competition between the first age class in species 𝑥 and species 𝑦 has the same value, namely 𝑎. then, the level of competition between the first age class in species 𝑦 against species 𝑥 and vice versa has the same value, namely 𝑏. the level of competition within the same species is referred to as intraspecific competition (𝑎) and the level of competition between different species is referred to as interspecific competition (𝑏). details of model 2 are the same as model 1 but there are differences in the first age class birth rate, first age class survival rate and second age class mathematical model of iteroparous and semelparous species interaction arjun hasibuan 448 harvesting rate in species 𝑥 which are influenced by density-dependent and competition. then, species 𝑦 is only affected by density-dependent and competition on the survival rate of the first age class and the level of harvesting of the second age class. model 2 is constructed based on research conducted by leslie [7], travis et al. [11], wikan [21], and cushing [22]. in addition, model 2 is adjusted based on the assumptions and simplifications in this research. local stability criteria using m-matrix in this section, we present the definition of m-matrix and the theorem that ensures a matrix has absolute eigenvalues less than one to determine the locally stable asymptotically of the model. definition 1. (see [11] or [23]) (m-matrix) a square matrix of size 𝑛, for example, 𝑀 = (𝑚𝑖𝑗) (1 ≤ 𝑖, 𝑗 ≤ 𝑛) is called an m-matrix if it is satisfied that 𝑚𝑖𝑗 ≤ 0 ∀𝑖 ≠ 𝑗 and if any of the following things are true: 1. all minor principals of the 𝑀 matrix are positive 2. all eigenvalues of the 𝑀 matrix have a positive real part 3. the matrix 𝑀 is a non-singular matrix and 𝑀−1 is positive 4. there is a vector 𝑣 > 0 so that it meets 𝑀𝑣 > 0 or 5. there is a vector 𝑢 > 0 s so that it satisfies 𝑀𝑇𝑢 > 0 theorem 1. (see [11]) suppose a matrix 𝐽 has the following form 𝐽 = [ 𝐴𝑚×𝑚 𝐵𝑚×𝑛 𝐶𝑛×𝑚 𝐷𝑛×𝑛 ] and the matrix 𝐺 = 𝐼 − 𝑆𝐽𝑆−1 is an m-matrix with 𝑆 = 𝐼 if 𝐵 and 𝐶 ≥ 0 or 𝑆 = [ 𝐼𝑚 0 0 −𝐼𝑛 ] if 𝐵 and 𝐶 ≤ 0, where 𝐼, 𝐼𝑚, and 𝐼𝑛 are identity matrices with sizes 𝑚 + 𝑛, 𝑚, and 𝑛, respectively, then matrix 𝐽 has a spectral radius of less than one. the spectral radius is the largest modulus of all the eigenvalues. theorem 1 and definition 1 are used to determine the locally stable asymptotically of model 1 and model 2 in the results and discussion section. results and discussion in the previous section, we have presented model 1, model 2, definition 1 and theorem 1. in this section, the two models, definition, and theorem will then be used to analyze the equilibrium point and the locally stable asymptotically of each equilibrium point. equilibrium point of model 1 the equilibrium point of model 1 can be obtained by expressing the variables 𝑥 and 𝑦 to the left of the model 1 depending on time 𝑡. the equilibrium model 1 is obtained as follows: mathematical model of iteroparous and semelparous species interaction arjun hasibuan 449 𝑥1(𝑡) = 𝑓𝑥1 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡) 𝑥2(𝑡) = 𝑠𝑥1(1 − ℎ𝑥2) 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑥1(𝑡) (3) 𝑦1(𝑡) = 𝑓𝑦2𝑦2(𝑡) 𝑦2(𝑡) = 𝑠𝑦1(1 − ℎ𝑦2) 1 + 𝑥1(𝑡) + 𝑦1(𝑡) 𝑦1(𝑡) then by determining the solution of equation (3), the equilibrium point of the model 1 is obtained as follows: 1. the equilibrium point with both species going extinct is 𝐸0 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 0 0 0 0 ] . 2. the equilibrium point with species 𝑥 exists while species 𝑦 is extinct, i.e 𝐸𝑥 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 𝑅𝑥 − 1 (𝑅𝑥 − 1)𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑥 0 0 ] . the condition 𝐸𝑥 exists if it is fulfilled 𝑅𝑥 = 𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 > 1. 𝑅𝑥 is referred to as the expected number of offspring per individual per lifetime when densitydependent effects are neglected on harvest-influenced growth of species 𝑥. 3. the equilibrium point with species 𝑦 exists while species 𝑥 is extinct, i.e 𝐸𝑦 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 0 0 𝑅𝑦 − 1 𝑅𝑦 − 1 𝑓𝑦2 ] . the condition 𝐸𝑦 exists if it is fulfilled 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) > 1. 𝑅𝑦 is referred to as the expected number of offspring per individual per lifetime when densitydependent effects are neglected on harvest-influenced growth of species 𝑦. it can be seen that in the model (1) there is no equilibrium point where the two species survive or co-existence equilibrium point. locally stable asymptotically at the equilibrium point of model 1 in this section, we perform a locally stable asymptotically analysis of the model 1. the equilibrium point is said to be asymptotically stable if it is satisfied that the spectral radius of the jacobian matrix at the equilibrium point is less than one. the locally stable asymptotically of each equilibrium point of the model 1 is stated in the following theorem. mathematical model of iteroparous and semelparous species interaction arjun hasibuan 450 theorem 2. (locally stable asymptotically at the equilibrium point model 1) for the leslie multispecies matrix model with the case of one iteroparous species and one semelparous species in which there are two classes each whose growth is influenced by density-dependent, harvesting and without the influence of competition described in the model 1, among others: 1. the equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1 and 𝑅𝑦 < 1. 2. the equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. 3. the equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 𝑅𝑥 and 𝑅𝑦 > 1. proof : in determining the stability of all equilibrium points of the model 1, it can be obtained by determining the linearization of the model 1, namely 𝐽(𝐸) = 𝐽 ([ 𝑥1 𝑥2 𝑦1 𝑦2 ]) = [ 𝑓𝑥1(1 + 𝑦1) (1 + 𝑥1 + 𝑦1) 2 𝑓𝑥2 − 𝑓𝑥1𝑥1 (1 + 𝑥1 + 𝑦1) 2 0 𝑃𝑥(1 + 𝑦1) 0 −𝑃𝑥𝑥1 0 0 0 0 𝑓𝑦2 −𝑃𝑦𝑦1 0 𝑃𝑦(1 + 𝑥1) 0 ] (4) where 𝑃𝑥 = 𝑠𝑥1(1 − ℎ𝑥2) (1 + 𝑥1 + 𝑦1) 2 and 𝑃𝑦 = 𝑠𝑦1(1 − ℎ𝑦2) (1 + 𝑥1 + 𝑦1) 2 . the reason for using the m-matrix theory in this study is due to the complexity of determining the spectral radius matrix 𝐽(𝐸) at the corresponding equilibrium point 𝐸. the use of definition 1 and theorem 1 guarantee that the spectral radius value of the 𝐽(𝐸) matrix at the corresponding equilibrium is less than one. based on the 𝐽(𝐸) matrix in (4) the elements of rows 1-2 columns 3-4 and rows 3-4 columns 1-2 are non-positive because the elements of the equilibrium point are guaranteed to be zero or positive. theorem 1 says matrix 𝑆 for matrix 𝐽(𝐸) is 𝑆 = [ 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 ] . the next step is to substitute all the equilibrium points in (4) and analyze their stability one by one. 1. the jacobian matrix for the equilibrium point 𝐸0 is 𝐽(𝐸0) = [ 𝑓𝑥1 𝑓𝑥2 0 0 𝑠𝑥1(1 − ℎ𝑥2) 0 0 0 0 0 0 𝑓𝑦2 0 0 𝑠𝑦1(1 − ℎ𝑦2) 0 ] . next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸0))𝑆 −1 and examine the element 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 that is non-positive and that all minor principals of 𝐺 are positive. if these conditions are met, then the 𝐺 matrix is an m-matrix so that the spectral radius 𝐽(𝐸0) is less than one. as a result, the equilibrium point 𝐸0 is locally stable asymptotically. here we present the obtained matrix 𝐺 = [ 1 − 𝑓𝑥1 −𝑓𝑥2 0 0 −𝑠𝑥1(1 − ℎ𝑥2) 1 0 0 0 0 1 −𝑓𝑦2 0 0 −𝑠𝑦1(1 − ℎ𝑦2) 1 ] and all minor principals of 𝐺 obtained are mathematical model of iteroparous and semelparous species interaction arjun hasibuan 451 𝑃𝑀1 = |𝑔11| = 1 − 𝑓𝑥1, 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = 1 − 𝑅𝑥, 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = 1 − 𝑅𝑥, and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑥)(1 − 𝑅𝑦). in this paper, we define 𝑃𝑀𝑖 (𝑖 = 1,2,3,4) as the 𝑖-th minor principal of the matrix 𝐺 for each equilibrium point under consideration. it is clear that the element 𝑔𝑖𝑗 < 0 for 𝑖 ≠ 𝑗 in the 𝐺 matrix is nonpositive by recalling the previously defined parameters. then, 𝑃𝑀2 and 𝑃𝑀4 will be positive if met 𝑅𝑥 < 1. in addition, 𝑅𝑥 < 1 implicitly results in 𝑓𝑥1 < 1 so that 𝑃𝑀1 > 0. furthermore, 𝑃𝑀4 is positive if 𝑅𝑦 < 1 because it must be fulfilled that 𝑅𝑥 < 1. therefore, 𝐺 is an m-matrix, that is if it is filled with 𝑅𝑥 < 1 and 𝑅𝑦 < 1. hence, according to theorem 1, the equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1 and 𝑅𝑦 < 1. 2. the jacobian matrix for the equilibrium point 𝐸𝑥 is 𝐽(𝐸𝑥) = [ 𝑓𝑥1 𝑅𝑥 2 𝑓𝑥2 𝑓1(1 − 𝑅𝑥) 𝑅𝑥 2 0 𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑥 2 0 𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥) 𝑅𝑥 2 0 0 0 0 𝑓𝑦2 0 0 𝑠𝑦1(1 − ℎ𝑦2) 𝑅𝑥 0 ] next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸𝑥))𝑆 −1 and make sure the matrix 𝐺 is an m-matrix. here we present the obtained matrix 𝐺 = [ 𝑅𝑥 2−𝑓𝑥1 𝑅𝑥 2 −𝑓𝑥2 𝑓1(1−𝑅𝑥) 𝑅𝑥 2 0 −𝑠𝑥1(1−ℎ𝑥2) 𝑅𝑥 2 1 𝑠𝑥1(1−ℎ𝑥2)(1−𝑅𝑥) 𝑅𝑥 2 0 0 0 1 −𝑓𝑦2 0 0 − 𝑠𝑦1(1−ℎ𝑦2) 𝑅𝑥 1 ] and all minor principals of 𝐺 obtained are 𝑃𝑀1 = |𝑔11| = 𝑅𝑥 2 − 𝑓𝑥1 𝑅𝑥 2 , 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = − 1 − 𝑅𝑥 𝑅𝑥 , 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = − 1 − 𝑅𝑥 𝑅𝑥 , and 𝑃𝑀4 = |𝐺| = − (1 − 𝑅𝑥)(𝑅𝑥 − 𝑅𝑦) 𝑅𝑥 . based on the previously defined parameters, the element 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 will be nonpositive if 𝑅𝑥 > 1 is satisfied. because of 𝑅𝑥 > 1, consequently 𝑃𝑀2 and 𝑃𝑀3 are positive. besides that, 𝑃𝑀4 is also positive but with the additional condition that is 𝑅𝑥 > 𝑅𝑦. then, it is clear that 𝑓𝑥1 < 𝑅𝑥 2 = (𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1) 2 so that 𝑃𝑀1 > 0. therefore, 𝐺 is an m-matrix, if it is fulfilled 𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. hence, according to theorem 1, the equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. mathematical model of iteroparous and semelparous species interaction arjun hasibuan 452 3. the jacobian matrix for the equilibrium point 𝐸𝑦 is 𝐽(𝐸𝑦) = [ 𝑓𝑥1 𝑅𝑦 𝑓𝑥2 0 0 𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑦 0 0 0 0 0 0 𝑓𝑦2 − 𝑠𝑦1(1 − ℎ𝑦2)(𝑅𝑦 − 1) 𝑅𝑦 2 0 1 𝑓4𝑅𝑦 0 ] next, determine the matrix 𝐺 = 𝐼 − 𝑆 (𝐽(𝐸𝑦)) 𝑆 −1 and make sure the matrix 𝐺 is an m-matrix. here we present the obtained matrix 𝐺 = [ 𝑅𝑦 − 𝑓𝑥1 𝑅𝑦 −𝑓𝑥2 0 0 −𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑦 1 0 0 0 0 1 −𝑓𝑦2 𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦) 𝑅𝑦 2 0 − 1 𝑓4𝑅𝑦 1 ] and all minor principals of 𝐺 obtained are 𝑃𝑀1 = |𝑔11| = 𝑅𝑦 − 𝑓𝑥1 𝑅𝑦 , 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = 𝑅𝑦 − 𝑅𝑥 𝑅𝑦 , 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = 𝑅𝑦 − 𝑅𝑥 𝑅𝑦 , and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑦)(𝑅𝑥 − 𝑅𝑦) 𝑅𝑦 . note that all elements 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are nonpositive except for 𝑔41. then, 𝑔41 will be negative if 𝑅𝑦 > 1. next, focus on the minor principal terms of the 𝐺 matrix. 𝑃𝑀2 and 𝑃𝑀3 are positive if 𝑅𝑦 > 𝑅𝑥. because of 𝑅𝑦 > 1 and 𝑅𝑦 > 𝑅𝑥, consequently 𝑃𝑀4 are positive. then, since 𝑅𝑦 > 𝑅𝑥 where 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) and 𝑅𝑥 = 𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 it follows that 𝑃𝑀1 is positive because 𝑅𝑦 > 𝑓𝑥1. hence, 𝐺 is an 𝑀-matrix and the equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 𝑅𝑥 and 𝑅𝑦 > 1. ∎ figure 1. population growth graph for each age class of each species 𝑥 and 𝑦 in case i model 1 mathematical model of iteroparous and semelparous species interaction arjun hasibuan 453 equilibrium point of model 2 with the same treatment as determining the equilibrium point in the model 1, the equilibrium model 2 is obtained as follows: 𝑥1(𝑡) = 𝑓𝑥1 1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡) 𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡) 𝑥2(𝑡) = 𝑠𝑥1(1 − ℎ𝑥2) 1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡) 𝑥1(𝑡) (5) 𝑦1(𝑡) = 𝑓𝑦2𝑦2(𝑡) 𝑦2(𝑡) = 𝑠𝑦1(1 − ℎ𝑦2) 1 + 𝑏𝑥1(𝑡) + 𝑎𝑦1(𝑡) 𝑦1(𝑡) then, there are four equilibrium points from the model 2, namely 1. the equilibrium point with both species going extinct is 𝐸0 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 0 0 0 0 ] . 2. the equilibrium point with species 𝑥 exists while species 𝑦 is extinct, i.e 𝐸𝑥 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 𝑅𝑥 − 1 𝑎 (𝑅𝑥 − 1)𝑠𝑥1(1 − ℎ𝑥2) 𝑎𝑅𝑥 0 0 ] . the condition 𝐸𝑥 exists if it is fulfilled 𝑅𝑥 = 𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 > 1. 3. the equilibrium point with species 𝑦 exists while species 𝑥 is extinct, i.e 𝐸𝑦 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 0 0 𝑅𝑦 − 1 𝑎 𝑅𝑦 − 1 𝑎𝑓𝑦2 ] . the condition 𝐸𝑦 exists if it is fulfilled 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) > 1. 4. the equilibrium point with species 𝑥 and 𝑦 exists if one of them is satisfied, namely 𝑎2 > 𝑏2 or 𝑎 > 𝑏, 𝑎(𝑅𝑦 − 1) > 𝑏(𝑅𝑥 − 1), and 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), or 𝑎 2 < 𝑏2 or 𝑎 < 𝑏, 𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1), and 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1) with mathematical model of iteroparous and semelparous species interaction arjun hasibuan 454 𝐸𝑥𝑦 = [ 𝑥1 𝑥2 𝑦1 𝑦2 ] = [ 𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1) (𝑎2 − 𝑏2) 𝑠𝑥1(1 − ℎ𝑥2) (𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)) (𝑎2 − 𝑏2)𝑅𝑥 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1) (𝑎2 − 𝑏2) 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1) (𝑎2 − 𝑏2)𝑓𝑦2 ] . in this second model, we obtain four equilibrium points where an equilibrium point appears with all species existing or a co-existence equilibrium point. the level of competition in both species affects the existence of a co-existence equilibrium point. figure 2. population growth graph for each age class of each species 𝑥 and 𝑦 in case ii model 1 locally stable asymptotically at the equilibrium point of model 2 this section discusses the locally stable asymptotically of model 2 which is presented in theorem 3 below. theorem 3. (locally stable asymptotically at the equilibrium point of model 2) for the system in the case of one iteroparous species and one semelparous species, each of which consists of two classes whose growth is affected by density-dependent, harvesting and competition which is specifically described in the model 2, among others: 1. the equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1, and 𝑅𝑦 < 1. 2. the equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 1 and 𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1). 3. the equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 1 and 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1). 4. the equilibrium point 𝐸𝑥𝑦 is locally stable asymptotically if 𝑎 > 𝑏, 𝑎(𝑅𝑦 − 1) > mathematical model of iteroparous and semelparous species interaction arjun hasibuan 455 𝑏(𝑅𝑥 − 1), 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), and 𝑓𝑥1(𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑥 − 𝑎𝑅𝑦)) < (𝑎 2 − 𝑏2)𝑅𝑥 2. proof: the steps to determine the stability of all equilibrium points of the model 2 can be carried out as in model 1. linearization of the model 2, namely 𝐽(𝐸) = 𝐽 ([ 𝑥1 𝑥2 𝑦1 𝑦2 ]) = [ 𝑓𝑥1(1 + 𝑏𝑦1) (1 + 𝑎𝑥1 + 𝑏𝑦1) 2 𝑓𝑥2 − 𝑓𝑥1𝑥1𝑏 (1 + 𝑎𝑥1 + 𝑏𝑦1) 2 0 𝑃𝑥(1 + 𝑏𝑦1) 0 −𝑃𝑥𝑏𝑥1 0 0 0 0 𝑓𝑦2 −𝑃𝑦𝑏𝑦1 0 𝑃𝑦(1 + 𝑏𝑥1) 0 ] (6) with 𝑃𝑥 = 𝑠𝑥1(1 − ℎ𝑥2) (1 + 𝑎𝑥1 + 𝑏𝑦1) 2 and 𝑃𝑦 = 𝑠𝑦1(1 − ℎ𝑦2) (1 + 𝑏𝑥1 + 𝑎𝑦1) 2 . the elements of rows 1-2 columns 3-4 and rows 3-4 columns 1-2 in (6) are non-positive because the elements of the equilibrium point are guaranteed to be zero or positive, so theorem 1 says matrix 𝑆 for matrix 𝐽(𝐸) is 𝑆 = [ 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 ] . the next step is to substitute all the equilibrium points in (6) and analyze its stability one by one. 1. the jacobian matrix for the equilibrium point 𝐸0 is 𝐽(𝐸0) = [ 𝑓𝑥1 𝑓𝑥2 0 0 𝑠𝑥1(1 − ℎ𝑥2) 0 0 0 0 0 0 𝑓𝑦2 0 0 𝑠𝑦1(1 − ℎ𝑦2) 0 ] . next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸0))𝑆 −1 and make sure the matrix 𝐺 is an m-matrix. here we present the obtained matrix 𝐺 = [ 1 − 𝑓𝑥1 −𝑓𝑥2 0 0 −𝑠𝑥1(1 − ℎ𝑥2) 1 0 0 0 0 1 −𝑓𝑦2 0 0 −𝑠𝑦1(1 − ℎ𝑦2) 1 ] and all minor principals of 𝐺 obtained are 𝑃𝑀1 = |𝑔11| = 1 − 𝑓𝑥1, 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = 1 − 𝑅𝑥, 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = 1 − 𝑅𝑥, and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑥)(1 − 𝑅𝑦). by considering the matrix 𝐺, it is clear that the values of all 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are nonpositive. next is focus on determining the conditions for 𝑃𝑀𝑖 > 0 (𝑖 = 1,2,3,4). 𝑃𝑀2 and 𝑃𝑀3 are positive if 𝑅𝑥 < 1 is satisfied. because of 𝑅𝑥 < 1 so that 𝑃𝑀1 is positive and an additional condition for 𝑃𝑀4 to be positive is 𝑅𝑦 < 1. therefore, 𝐺 is an m-matrix, and the equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1 and 𝑅𝑦 < 1. mathematical model of iteroparous and semelparous species interaction arjun hasibuan 456 2. the jacobian matrix for the equilibrium point 𝐸𝑥 is 𝐽(𝐸𝑥) = [ 𝑓𝑥1 𝑅𝑥 2 𝑓𝑥2 𝑏𝑓1(1 − 𝑅𝑥) 𝑎𝑅𝑥 2 0 𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑥 2 0 𝑏𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥) 𝑎𝑅𝑥 2 0 0 0 0 𝑓𝑦2 0 0 𝑎𝑠𝑦1(1 − ℎ𝑦2) 𝑎 + 𝑏(𝑅𝑥 − 1) 0 ] . next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸𝑥))𝑆 −1 and make sure the matrix 𝐺 is an m-matrix. here we present the obtained matrix 𝐺 = [ 𝑅𝑥 2 − 𝑓𝑥1 𝑅𝑥 2 −𝑓𝑥2 𝑏𝑓1(1 − 𝑅𝑥) 𝑎𝑅𝑥 2 0 −𝑠𝑥1(1 − ℎ𝑥2) 𝑅𝑥 2 1 𝑏𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥) 𝑎𝑅𝑥 2 0 0 0 1 −𝑓𝑦2 0 0 − 𝑎𝑠𝑦1(1 − ℎ𝑦2) 𝑎 + 𝑏(𝑅𝑥 − 1) 1 ] and all minor principals of 𝐺 obtained are 𝑃𝑀1 = |𝑔11| = 𝑅𝑥 2 − 𝑓𝑥1 𝑅𝑥 2 , 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = − 1 − 𝑅𝑥 𝑅𝑥 , 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = − 1 − 𝑅𝑥 𝑅𝑥 , and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑥) (𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1)) 𝑎 + 𝑏(𝑅𝑥 − 1) . in the matrix 𝐺, it can be seen that all 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are non-positive because 𝑅𝑥 > 1 which is a condition for 𝐸𝑥 to exist. therefore, the next step is to focus on determining the positive terms of the minor principal of the 𝐺 matrix. it is clear that 𝑃𝑀2 and 𝑃𝑀3 are positive because 𝑅𝑥 > 1. then, it is clear that 𝑃𝑀1 is positive because in fact 𝑓𝑥1 < 𝑅𝑥 2 = (𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1) 2. since 𝑅𝑥 > 1, 𝑃𝑀4 is positive if 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1) < 0 or 𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1). therefore, 𝐺 is an mmatrix, and the equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 1 and 𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1). 3. the jacobian matrix for the equilibrium point 𝐸𝑦 is 𝐽(𝐸𝑦) = [ 𝑎𝑓𝑥1 𝑎 + 𝑏(𝑅𝑦 − 1) 𝑓𝑥2 0 0 𝑎𝑠𝑥1(1 − ℎ𝑥2) 𝑎 + 𝑏(𝑅𝑦 − 1) 0 0 0 0 0 0 𝑓𝑦2 𝑏𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦) 𝑎𝑅𝑦 2 0 1 𝑓4𝑅𝑦 0 ] . next, determine the matrix 𝐺 = 𝐼 − 𝑆 (𝐽(𝐸𝑦)) 𝑆 −1 and make sure the matrix 𝐺 is an m-matrix. here we present the obtained matrix mathematical model of iteroparous and semelparous species interaction arjun hasibuan 457 𝐺 = [ − 𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1) 𝑎 + 𝑏(𝑅𝑦 − 1) −𝑓𝑥2 0 0 −𝑎𝑠𝑥1(1 − ℎ𝑥2) 𝑎 + 𝑏(𝑅𝑦 − 1) 1 0 0 0 0 1 −𝑓𝑦2 𝑏𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦) 𝑎𝑅𝑦 2 0 − 1 𝑓4𝑅𝑦 1 ] . and all minor principals of 𝐺 obtained are 𝑃𝑀1 = |𝑔11| = − 𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1) 𝑎 + 𝑏(𝑅𝑦 − 1) , 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = − 𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1) 𝑎 + 𝑏(𝑅𝑦 − 1) , 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = − 𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1) 𝑎 + 𝑏(𝑅𝑦 − 1) , and 𝑃𝑀4 = |𝐺| = (𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)) (1 − 𝑅𝑦) 𝑎 + 𝑏(𝑅𝑦 − 1) . the equilibrium point of 𝐸𝑦 is exist if 𝑅𝑦 > 1 consequently all elements of 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are non-positive. next is the focus on determining the conditions so that all the principal minor matrices 𝐺 are positive. since 𝑅𝑦 > 1, so we have 𝑃𝑀2, 𝑃𝑀3, and 𝑃𝑀4 are positive if 𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1) < 0 or 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1). in addition, 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1) the result is satisfied 𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1) < 0 or 𝑎(𝑓𝑥1 − 1) < 𝑏(𝑅𝑦 − 1). then, because of 𝑅𝑦 > 1 and 𝑎(𝑓𝑥1 − 1) < 𝑏(𝑅𝑦 − 1) so that 𝑃𝑀1 is fulfilled with a positive value. therefore, 𝐺 is an m-matrix and the equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 1 and 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1). 4. the jacobian matrix for the equilibrium point 𝐸𝑥𝑦 is 𝐽(𝐸𝑥𝑦) = [ 𝐴𝑥𝑓𝑥1 𝐶𝑅𝑥 2 𝑓𝑥2 − 𝐵𝑥𝑏𝑓𝑥1 𝐶𝑅𝑥 2 0 𝑠𝑥1(1 − ℎ𝑥2)𝐴𝑥 𝐶𝑅𝑥 2 0 − 𝑏𝑠𝑥1(1 − ℎ𝑥2)𝐵𝑥 𝐶𝑅𝑥 2 0 0 0 0 𝑓𝑦2 − 𝑏𝐵𝑦 𝐶𝑅𝑦𝑓𝑦2 0 𝐴𝑦 𝐶𝑓𝑦2𝑅𝑦 0 ] . next is the 𝐺 matrix for the equilibrium point 𝐸𝑥𝑦 is 𝐺 = [ 1 − 𝐴𝑥𝑓𝑥1 𝐶𝑅𝑥 2 −𝑓𝑥2 − 𝐵𝑥𝑏𝑓𝑥1 𝐶𝑅𝑥 2 0 𝐴𝑥𝑠𝑥1(ℎ𝑥2 − 1) 𝐶𝑅𝑥 2 1 − 𝑏𝑠𝑥1(1 − ℎ𝑥2)𝐵𝑥 𝐶𝑅𝑥 2 0 0 0 1 −𝑓𝑦2 − 𝑏𝐵𝑦 𝐶𝑅𝑦𝑓𝑦2 0 − 𝐴𝑦 𝐶𝑅𝑦𝑓𝑦2 1 ] and the principal minor of the matrix 𝐺 are mathematical model of iteroparous and semelparous species interaction arjun hasibuan 458 𝑃𝑀1 = |𝑔11| = 1 − 𝐴𝑥𝑓𝑥1 𝐶𝑅𝑥 2 , 𝑃𝑀2 = | 𝑔11 𝑔12 𝑔21 𝑔22 | = 𝑎𝐵𝑥 𝐶𝑅𝑥 𝑃𝑀3 = | 𝑔11 𝑔12 𝑔13 𝑔21 𝑔22 𝑔23 𝑔31 𝑔32 𝑔33 | = 𝑎𝐵𝑥 𝐶𝑅𝑥 , and 𝑃𝑀4 = |𝐺| = 𝐵𝑥𝐵𝑦 𝐶𝑅𝑥𝑅𝑦 where 𝐴𝑥 = (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑥 − 𝑎𝑅𝑦)), 𝐴𝑦 = (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑦 − 𝑎𝑅𝑥)) 𝐵𝑥 = (𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)) , 𝐵𝑦 = (𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1)) , and 𝐶 = (𝑎 2 − 𝑏 2). in this case, the conditions that meet the requirements will be determined so that 𝐺 is called the m-matrix. first focus on 𝑃𝑀4 is positive. because of 𝐸𝑥𝑦 exists if 𝐵𝑥, 𝐵𝑦, 𝐶 > 0 or 𝐵𝑥, 𝐵𝑦, 𝐶 < 0. however, 𝑃𝑀4 is positive if it is fulfilled 𝐵𝑥, 𝐵𝑦, 𝐶 > 0. because 𝐵𝑥, 𝐵𝑦, 𝐶 > 0 consequently fulfilled 𝑔13, 𝑔23, 𝑔31 < 0, and 𝑃𝑀2, 𝑃𝑀3 > 0. then, 𝑃𝑀1 is positive if 𝑓𝑥1𝐴𝑥 < 𝐶𝑅𝑥 2. finally, all the conditions for 𝐺 to be called an m-matrix have been fulfilled. therefore, 𝐺 is an m-matrix, and the equilibrium point 𝐸𝑥𝑦 is locally stable asymptotically if 𝑓𝑥1 (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑦 − 𝑎𝑅𝑥)) < (𝑎2 − 𝑏2)𝑅𝑥 2, 𝑎(𝑅𝑦 − 1) > 𝑏(𝑅𝑥 − 1), 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), and 𝑎 2 > 𝑏2 or 𝑎 > 𝑏. ∎ numerical simulations of model 1 and model 2 in the previous subsection, an analysis of the existing condition and local stability asymptotically from each equilibrium point has been carried out on model 1 and model 2. in this section, we perform a numerical simulation of the results from the analysis of model 1 and model 2. in this case, we divide the two models into two cases and each case is divided into as many subcases as the asymptotically local stability conditions of theorem 2 and theorem 3. in the simulation of model 1, we assume for all subcases of the model 1 case, including: 1. 𝑠𝑥1 = 0.6 and 𝑠𝑦1 = 0.4 respectively that if there are 10 individuals in the first age class of species 𝑥 and 𝑦 then only 6 individuals and 4 individuals are able to survive from species 𝑥 and 𝑦. 2. ℎ𝑥2 = 0.5 and ℎ𝑦2 = 0.3 respectively that if there are 10 individuals in the first age class of species 𝑥 and 𝑦 then there are only 5 individuals and 3 individuals harvested from species 𝑥 and 𝑦. figure 3. population growth graph for each age class of each species 𝑥 and 𝑦 in case iii model 1 mathematical model of iteroparous and semelparous species interaction arjun hasibuan 459 then, the birth rate for the simulation in the model 1 case is divided into 3 subcases based on theorem 1, including: i. 𝑓𝑥1 = 0.7, 𝑓𝑥2 = 0.9, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 0.97 < 1 and 𝑅𝑦 = 0.84 < 1. ii. 𝑓𝑥1 = 1, 𝑓𝑥2 = 5, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 2.5 > 1 and 𝑅𝑦 = 0.84 < 𝑅𝑥. iii. 𝑓𝑥1 = 0.7, 𝑓𝑥2 = 0.9, and 𝑓𝑦2 = 10 consequently 𝑅𝑥 = 0.97 < 1 and 𝑅𝑦 = 2.8 > 𝑅𝑥. the results of the model 1 simulation for each subcase i-iii are presented in figure 13. figures 1-3 respectively for the parameters given in each subcase i-iii of the model 1 simulation show that the locally stable asymptotically towards the equilibrium point 𝐸0 = [0,0,0,0]𝑇, 𝐸𝑥 = [1.5,0.18,0,0] 𝑇, and 𝐸𝑦 = [0,0,1.8,0.18] 𝑇. in the simulation of model 2, we assume for all subcases of model 2 for survival and harvesting rates are equal to model 1. then, the levels of intraspecific and interspecific competition are 𝑎 = 0.2 and 𝑏 = 0.1, respectively. then, the birth rate for the simulation in the model 1 case is divided into 4 subcases based on theorem 2, including: i. 𝑓𝑥1 = 0.5, 𝑓𝑥2 = 1, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 0.8 and 𝑅𝑦 = 0.84. ii. 𝑓𝑥1 = 30, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 60 consequently 𝑅𝑥 = 36 and 𝑅𝑦 = 16.8. iii. 𝑓𝑥1 = 5, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 100 consequently 𝑅𝑥 = 11 and 𝑅𝑦 = 28. iv. 𝑓𝑥1 = 20, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 60 consequently 𝑅𝑥 = 26 and 𝑅𝑦 = 16.8. figure 4. population growth graph for each age class of each species 𝑥 and 𝑦 in case i model 2 mathematical model of iteroparous and semelparous species interaction arjun hasibuan 460 figure 5. population growth graph for each age class of each species 𝑥 and 𝑦 in case ii model 2 figure 4-7 is the simulation result of model 2 for each subcase i-iv. figure 4-7 for each parameter that satisfies theorem 2 conditions in subcases i-iv of the model 2 simulation that sequentially locally stable asymptotically towards the equilibrium point 𝐸0 = [0,0,0,0]𝑇, 𝐸𝑥 = [175,1.46,0,0] 𝑇, 𝐸𝑦 = [0,0,135,1.35] 𝑇 dan 𝐸𝑥𝑦 = [114,1.32,22,0.36] 𝑇. figure 6. population growth graph for each age class of each species 𝑥 and 𝑦 in case iii model 2 mathematical model of iteroparous and semelparous species interaction arjun hasibuan 461 figure 7. population growth graph for each age class of each species 𝑥 and 𝑦 in case iv model 2 conclusions in this paper, we compare two different models: model 1 and model 2. our focus is to compare the presence and absence of the influence of intraspesific and interspecific competition in the equilibrium point and its local stability of model 1 and model 2. mathematically the conditions under which the positive/non-trivial equilibrium point exists and the local stability of this equilibrium of the model is easy to interpret. however, biologically only some conditions can be interpreted because of the complexity of conditions. simply put, the results of our study show that the level of competition has a role in the equilibrium point and its local stability of the model 1 and model 2. model 1 shows that there is no coexistence equilibrium point so model 1 is never locally stable at the point where both species exist. in model 2, one of the conditions that is easily interpreted is that the coexistence equilibrium point occurs when 𝑎 > 𝑏 which means the intensity of the intraspecific competition level is greater than the intensity of the interspecific level competition. the inequality of 𝑎 > 𝑏 is one of the locally stable asymptotically conditions of the co-existence equilibrium point in model 2. the results of this study can be applied to problems that have similarities mathematical structure to this case. there still some limitation in this model to fit in a realistic real case, and hence we think that this research should be further developed, for example by increasing. the number of species, the number of classes, and so on, which is mathematically interesting and realistically important. this theory can be applied to study the dynamics of natural resource models including the effects of different management to the growth of the resources, such as in fisheries. mathematical model of iteroparous and semelparous species interaction arjun hasibuan 462 acknowledgments the work is part of the research done by first author in the master of mathematics study program at padjadjaran university. the first author would like to thank for the tuition fee during the candidature covered by alg-unpad research grant with contract number 1959/un6.3.1/pt.00/2021. the preparation and the apc (article processing charge) for manuscript publication is funded by “penelitian tesis magister (ptm)” from the indonesian ministry of education, culture, research and technology in 2022 with contract number 1318/un6.3.1/pt.00/2022. references [1] r. kon, “stable bifurcations in multi-species semelparous population models,” 2017, doi: 10.1007/978-981-10-6409-8_1. 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[12] r. kon, “age-structured lotka-volterra equations for multiple semelparous populations,” siam j. appl. math., vol. 71, no. 3, pp. 694–713, 2011, doi: 10.1137/100794262. [13] r. kon, “permanence induced by life-cycle resonances: the periodical cicada problem,” j. biol. dyn., vol. 6, no. 2, 2012, doi: 10.1080/17513758.2011.594098. [14] l. k. beay, a. suryanto, i. darti, and trisilowati, “stability of a stage-structure rosenzweig-macarthur model incorporating holling type-ii functional response,” 2019, doi: 10.1088/1757-899x/546/5/052017. [15] j. li, x. liu, and c. wei, “the impact of role reversal on the dynamics of predatormathematical model of iteroparous and semelparous species interaction arjun hasibuan 463 prey model with stage structure,” appl. math. model., vol. 104, 2022. [16] j. wang, y. cai, s. fu, and w. wang, “the effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge,” chaos, vol. 29, no. 8, 2019. [17] e. rahmi, i. darti, a. suryanto, and trisilowati, “a modified leslie–gower model incorporating beddington–deangelis functional response, double allee effect and memory effect,” fractal fract., vol. 5, no. 3, 2021. [18] t. hidayati and w. kurniawan, “stability analysis of lotka-volterra model in the case of interaction of local religion and official religion,” int. j. educ. res. soc. sci., vol. 2, no. 3, 2021. [19] w. w. mohammed, e. s. aly, a. e. matouk, s. albosaily, and e. m. elabbasy, “an analytical study of the dynamic behavior of lotka-volterra based models of covid19,” results phys., vol. 26, 2021. [20] y. lin, q. din, m. rafaqat, a. a. elsadany, and y. zeng, “dynamics and chaos control for a discrete-time lotka-volterra model,” ieee access, vol. 8, 2020. [21] a. wikan, “dynamical consequences of harvest in discrete age-structured population models,” j. math. biol., vol. 49, no. 1, 2004, doi: 10.1007/s00285-0030251-5. [22] j. m. cushing, “three stage semelparous leslie models,” j. math. biol., vol. 59, no. 1, 2009, doi: 10.1007/s00285-008-0208-9. [23] c. c. travis and w. m. post, “dynamics and comparative statics of mutualistic communities,” j. theor. biol., vol. 78, no. 4, 1979, doi: 10.1016/00225193(79)90190-5. microsoft word 1 sampul depan.doc 48  penyelesaian persamaan nonlinier orde-tinggi untuk akar berganda mohammad jamhuri jurusan matematika, fakultas sains dan teknologi, universitas islam negeri maulana malik ibrahim malang j4m3sh@gmail.com abstrak dalam paper ini dikembangkan sebuah metode orde-empat untuk mencari akar berganda dari persamaan nonlinier. metode tersebut di dasarkan pada metode orde-lima dari jarrat (untuk akar-akar sederhana) yang hanya memerlukan satu perhitungan fungsi dan tiga kali perhitungan turunan. efisiensi informasi dari metode tersebut sama dengan metode-metode dengan orde yang lebih rendah. untuk kasus-kasus akar berganda, telah ditemukan metode-metode yang hanya memerlukan satu kali perhitungan turunan. sehingga metode-metode tersebut lebih efisien jika dibandingkan dengan metode-metode lainnya. kata kunci: akar berganda, orde-tinggi, persamaan nonlinier. 1. pendahuluan ada berbagai macam literature untuk masalah penyelesaian persamaan nonlinier dan sistem persamaan nonlinier. lihat pada contoh ostrowski (1960), traub (1964), neta (1983) dan pada referensi-referensi yang lainnya. dalam penelitian ini akan dikembangkan sebuah metode titik-tetap orde-tinggi untuk penyelesaian akar berganda. sebenarnya terdapat banyak metode yang dapat digunakan untuk mencari akar dari persamaan nonlinier 0, lihat neta (1983). metode newton hanya salah satu metode orde-satu kecuali yang dimodifikasi sehingga menjadi orde-dua tingkat konvergensinya, lihat rall (1996) atau schroder (1996). untuk memodifikasi diperlukan pengetahuan tentang multiplicity. traub (1964) telah menyarankan penggunaan sebuah metode untuk atau , beberapa metode tersebut memerlukan turunan yang libih tinggi dari pada yang digunakan untuk masalah akar sederhana yang hanya memiliki satu akar. sehingga yang pertama dari metode-metode tersebut adalah mengetahui multiplicity . dalam beberapa hal, terdapat metode orde-tinggi yang dikembangkan oleh hamsen dan patrick (1977), victory dan neta (1987), dan dong (1987). karena secara umum tidak dapat diketahui multiplicity-nya, traub (1964) menyarankan sebuah jalan untuk mengaproksimasinya pada saat iterasi. sebagai contoh, metode newton yang dimodifikasi dengan tingkat konvergensi kuadratik adalah 1 dan metode halley (1964) dengan tingkat konvergensi kubik adalah 1 2 2 2 dimana adalah kependekan dari . metode orde-tiga lainnya telah dikembangkan oleh victory dan neta (1987) yang didasarkan metode orde-empat-nya king (1973) untuk akar-akar sederhana.  penyelesaian persamaan nonlinear orde‐tinggi untuk akar berganda  volume 1 no. 1 november 2009 49     3 dimana 2 1 1 1   4 dan 1 5 sebelumnya, dua metode orde-tiga telah dikembangkan oleh dong (1987), keduanya memerlukan informasi yang sama dan keduanya juga didasarkan pada keluarga metodemetode orde-empat (untuk akar-akar sederhana) oleh jarrat (1966): 1 1 1 6 1 1 1 1 1 7 dimana . langkah awal dari metode yang akan dibentuk disini adalah metode jarrat (1996) yang diberikan 8 dimana seperti diatas dan 9 jarrat (1996) telah menunjukkan bahwa metode ini (untuk akar sederhana) adalah dari orde-lima jika parameter-parameter yang dipilih adalah sebagai berikut: 1, 1 8 , 3 8 , 1 6 , 2 3 10 metode tersebut memerlukan satu fungsi-dan tiga turunan-untuk setiap langkah perhitungan. sehingga efisiensi informasi adalah 1.25 (traub, 1964). karena jarrat (1996) tidak memberikan konstanta error asymptotic-nya, maka digunakan redfern (1994) untuk memperolehnya, 1 24 1 2 1 4 1 8 dimana diberikan oleh 14 dengan 1. 2. skema baru untuk orde-tinggi untuk memaksimalkan orde-konvergensi untuk sebuah akar dengan perkalian harus ditentukan enam parameter , , , , , _3. misalkan , ̂ , adalah error pada untuk iterasi ke, yaitu: ̂ mohammad jamhuri  50 volume 1 no. 1 november 2009 jika dan di expansi menggunakan deret taylor (setelah dipotong sampai orde ke, ) diperoleh ! 12 atau ! 1 13 dimana ! ! , 14 1 ! 1 15 untuk mengekspansi dan digunakan manipulasi symbolic, seperti redfern (1994), diperoleh 1 ! ̂ 1 1 ̂ 2 ̂ 16 ̂ 1 1 1 2 1 2 1 1 2 17 dimana untuk memudahkan dipilih 2 18 sehingga 1 ! 19 dimana 2 3 1 2 4 2 3 3 2 2 2 25 21 4 21 34 2 12 13 48 6 2 20 error-nya diberikan oleh  penyelesaian persamaan nonlinear orde‐tinggi untuk akar berganda  volume 1 no. 1 november 2009 51 2 ̂ 1 2      8 5 6 ̂ 4 4 1 3 7 ̂ 4 21 dimana ̂ 2 1 ̂ 22 berikutnya ekspansi dalam bentuk 1 ! 1 1 2 ! 23 dimana 1 3 1 ̂ 2 1 6 3 2 ̂ 32 ̂ ̂ ̂ ̂ ̂ ̂ ̂ 24 dimana 16 2 1 7 2 1 8 6 1 3 15 1 1 2 7 4 8 1 1 6 3 16 4 1 16 1 32 2 4 2 4 2 1 2 5 2 8 16 2 48 2 5 51 98 5 27 26 8 24 2 48 2 5 35 42 128 2 1 32 2 berikutnya substitusikan 13 , 15 , 19 dan 23 kedalam 8 dan expansi kuasi menggunakan deret taylor, sehingga diperoleh 26 mohammad jamhuri  52 volume 1 no. 1 november 2009 dimana koefisien tergantung pada parameter-parameter , , , , . kelima parameter tersebut dapat digunakan untuk menghilangkan koefisien , , dan . sehingga orde dari metode tersebut adalah 4. sebenarnya, kecuali untuk 2, digunakan dan sehingga hanya 4 parameter yang di gunakan. ini merupakan syarat perlu untuk memperoleh metode orde-empat. tabel 1. hasil dari contoh 2 0 0.8 0.1296 0.6 0.4096 1 1.00074058 0.21954564 5 1.02772227 0.31600247 2 2 1.00000014 0.750396 13 karena sangat kompleknya persamaan di atas, parameter-parameter yang digunakan untuk 2,3,4,5 dan diberikan dalam tabel 2 berikut ini. metode-metode tersebut semuanya memiliki orde-empat. tabel 2. parameter-parameter hasil contoh 2 2 2 3 4 5 6 1 4 3 3 2 2 5 2 3 2 5 2 3 1 3 3 5 4 0.064783 0.021737 0.008212 1 2 1 2 2 25 108 43 72 0.437458 0.430345 0.368149 2 3 1 4 25 72 7.904129 18.815436 39.687683 0 2 125 72 5.912818 15.894083 35.699379 1 2 2 9 13 18 5 1296 37 108 0.236261 0.164791 0.120179 3 8 7 8 2 25 81 5 972 0.154675 0.101387 0.073031 1 8 1 8 2 25 0.083527 0.069672 0.057025 batas kesalahan diberikan oleh 27 dimana , , dan adalah yang diberikan dalam table diatas untuk setiap . untuk 3, dapat dipilih dengan parameter dengan bebas untuk menyamakn . ringkasnya, dalam penelitian ini telah dihasilkan metode orde-empat yang menggunakan satu fungsi dan tiga turunan dalam setiap iterasi. efisiensi informasi pada metode-metode tersebut adalah 1, seperti metode-metode yang telah disebutkan diatas untuk akar-akar yang banyak. indeks efisiensinya adalah 1.4142 yang lebih rendah daripada metode-metode orde-tiga. dalam hal 2 diperoleh sebuah metode yang hanya memerlukan dua perhitungan turunan 0 sehingga efisiensi informasinya adalah 4/3 dan indeks efisiensinya adalah 1.5874.  penyelesaian persamaan nonlinear orde‐tinggi untuk akar berganda  volume 1 no. 1 november 2009 53 3. simulasi numerik dalam contoh pertama ini digunakan sebuah polynomial kuadratik yang mempunyai dua akar pada 1. 2 1 28 dalam contoh ini, dimulai dengan 0, dan kekonvergenannya diperoleh dalam 1 iterasi. dalam contoh kedua, diambil polynomial yang mempunyai dua akar pada 1. 2 1 29 dimulai pada 0.8, metode ini konvergen dalam 1 iterasi. jika dimulai dengan 0.6, metode ini memerlukan 2 kali iterasi. hasil perhitungannya diberikan dalam table 1. hasil yang sama juga diperoleh jika dimulai dengan 0.8 dan 0.6 untuk konvergen pada 1. contoh berikutnya adalah polinomial dengan 3 akar pada 1. 8 24 34 23 6 30 iterasinya dimulai dengan 0 dan hasilnya diringkas dalam tabel 3. contoh lainnya dengan 2 akar pada 0 adalah 31 dimulai pada 0.1 metode ini konvergen dalam 1 iterasi, tetapi jika nilai awalnya dimulai pada 0.2, metode ini konvergen dalam 1 iterasi. hasil perhitungannya diberikan dalam tabel 4. contoh terakhir adalah polinomial yang mempunyai akar ganda pada 1 3 8 6 24 19 32 tabel 3. hasil dari contoh 3 0 0 6 1 0.95239072 0.23148417 3 2 0.99999683 0.63 16 tabel 4. hasil dari contoh 4 0 0.1 0.11051709 1 0.2 0.48856110 1 1 0.12654311 4 0.16013361 9 0.17709827 3 0.31369352 7 2 0.3739 20 0 0.14341725 15 0 tabel 5. hasil dari contoh 5 0 0 19 1 1.46056319 9.725126111 2 1.00101187 0.368806435 4 3 1 0 mohammad jamhuri  54 volume 1 no. 1 november 2009 daftar pustaka dong, c., (1987), a family of multipoint iterative function for finding multiple zeros of nonlinear equations, int. j. comput. math., 21, pp 363-367 halley, e., (1964), a new, exact and easy method of finding the roots of equations generally and that without any previous reduction., phil. trans. r. soc. london, 18, pp 136-148. hansen, e., patrick, m., (1977), a family of root finding methods, numer. math., 27, pp 257-269. jarrat, p., (1966), some fourth order multipoint methods for solving equations, math. comp., 20, pp 434-437. jarrat, p., (1996), multipoints iterative methods for solving certain equations, comput. j., 8, pp 398-400. king, r.f., (1973), a family of fourth order methods for nonlinear equations, siam j. numer. anal., 10, pp 876-879. neta, b., (1983), numerical methods for the solution of equations, net-a-sof, california. ostrowski, a.m., (1960), solution of equations and system of equations, academic press, new york. rall, l.b., (1996), convergence of the newton process to multiple solutions, numer. math., 9, pp 23-37. redfern, d., (1994), the maple handbook, springer-verlag, new york. schroder, e., (1996), uber unendlich viele algorithm zur auflosung der gleichungen, math. ann., 2, pp 23-37. traub, j.f., (1964), iterative methods for the soslution of equations, prentice hall, new jersey. victory, h.d., neta, b., (1987), a higher order method for multiple roots of equations, int. j. comput. math., 21, pp 363-367. aplikasi dua segitiga sebangun pada studi venus transit di matahari tanggal 8 juni 2004 dari bpd lapan watukosek nanang widodo balai pengamatan dirgantara lapan watukosek email: nang_widodo@yahoo.com abstrak transit planet venus di cakram matahari (jari-jari = 696000 km) merupakan peristiwa alam yang dapat dilihat secara berkala. planet venus merupakan planet kedua dalam sistem tata surya yang mempunyai orbit lebih dekat ke matahari (= 0,723 astronomical unit) dibanding jarak bumi-matahari (= 149.600.000 km = 1 au). sehingga pada suatu waktu tertentu ada peluang berada tepat di depan bumi, saat menghadap matahari atau dikenal dengan transit venus. proses pengamatan fenomena transit venus di cakram matahari tersebut dapat diimplimentasikan sebagai aplikasi dua segitiga sebangun, dimana jari-jari planet venus (jari-jari = 6051,8 km) dinyatakan sebagai tinggi benda dan jari-jari tinggi bayangan venus sebesar 20880 km (= 3,65 mm pada cakram matahari). dimana diameter matahari 1.392.000 km (= 240 mm pada lembar sket). dengan pengukuran jarak tempuh venus transit 72,4 mm (419 920 km di cakram matahari) terhadap waktu kontak pertama bayangan venus pada jam 05.28 ut (12.28 wib) di tepi timur hingga akhir transit pada 17.50 ut (14.50 wib) diperoleh kecepatan bayangan venus sebesar 49,286 km/detik. kata kunci: orbit, segitiga sebangun,venus transit. abstract transit of the planet venus in the sun disk (radius = 696.000 km) is a natural event that can be seen on a regularly. venus is the second planet in the solar system have orbits closer to the sun (=0.723 astronomical units) compared to the earth-sun distance (= 149,600,000 km = 1 au). so that at any given time there are opportunities right in front of the earth, while facing the sun or known by the venus transit. the observation of the venus transit on the sun disc is implemented of application two congruent triangles, where the radius of venus (6051.8 km) is expressed as object height and radius the shadow height of venus is 20,880 km (= 3.65 mm in the sun disk). where the diameter of the sun 1392 million km (= 240 mm on the sketch). during a transit, venus cover distance 72.4 mm or 419,920 km on the sun disk. the first contact venus shadow on the sun disk at 05:28 ut (12:28 pm) on the east limb until the end the transit at 17:50 ut (14:50 pm) obtained orbital speed of venus shadow is 49.286 km/s. keywords: orbit, triangle congruent, venus transit pendahuluan henry chamberlain russel adalah pengamat pertama fenomena venus transit tahun 1874. dimana cahaya matahari dipantulkan oleh aerosol dan dibiaskan oleh atmosfer planet venus, glenn s (2004). planet venus disebut juga bintang pagi dan bintang sore oleh william, (1978). hal terjadi karena planet venus memantulkan cahaya dari matahari, selain itu posisi orbitnya berdekatan dengan bumi. venus mengitari matahari dalam 224,701 hari bumi (+ 0,615 tahun bumi). karena dari perbedaan laju orbit bumi dan venus, maka venus mengorbit 2,6 kali sedangkan bumi mengorbit 1,6 kali, sebelum kedua planet segaris. periode ini (583,92 hari bumi) disebut earthvenus synodic cycle. peristiwa transit venus ini serupa dengan fenomena bulan melintas di depan matahari atau peristiwa gerhana matahari. tabel 1. ukuran fisik bumi dan venus ukuran fisika bumi venus diameter 12.760 km 12.100 km jarak rata-rata orbit 149,6 juta km 108,2 juta km periode orbit 365,25 hari 224,7 hari sudut inklinasi orbit 0o 3,4o peristiwa venus akan transit kembali di tempat yang sama dalam waktu 243 tahun atau 121,5 + 8 tahun di tempat berbeda, nick a.f. (2004) mailto:nang_widodo@yahoo.com aplikasi dua segitiga sebangun pada studi venus transit di matahari tanggal 8 juni 2004… jurnal cauchy – issn: 2086-0382 39 contoh foto planet venus yang diambil dari proyek magelan nasa, nick . (2012) gambar 1. sebagian permukaan venus, (sumber: venus unveiled*image credit: nasa magelan project). gambar 2. siklus-siklus sinodik berdasarkan perhitungan dari metoda eso vt-2004 pada saat kontak pertaman transit venus masuk cakram matahari, jarak bumi matahari 149.596.765 km (taraf kesalahan + 3534 km = 0,0237 %) dengan solar parallax (ii): 8,79440”. perbedaan waktu kontak transit venus tergantung pada posisi bujur dari stasiun pengamat (http://www.isthe.com/chongo/tech/ astro/venus2004.html). fenomena transit venus ini dapat digunakan untuk menjelaskan aplikasi teori dua segitiga sebangun. konsep kesebangunan segitiga telah banyak dibahas dalam materi matematika geometri, dalam menguji kesebangunan dua segitiga ∆ abc dan ∆ def terdapat syarat-syarat antara lain: 1. perbandingan panjang sisi-sisi yang bersesuaian pada kedua segitiga adalah sama besar 2. dua pasang sudut yang bersesuaian adalah sama besar, atau dinyatakan dalam persamaan berikut: 𝐴𝐵 𝐴𝐷 = 𝐴𝐶 𝐴𝐸 = 𝐵𝐶 𝐷𝐸 (1) penjelasan persamaan (1) dapat digambarkan pada gambar 3 gambar 3. dua segitiga sebangun abc dan ade dengan asumsi indeks bias atmosfer bumi dan atmosfer venus diabaikan, maka dalam penelitian ini akan dilakukan perbandingan bayangan venus dari persamaan (1) dan bayangan venus (hasil sket) di cakram matahari. gambar 4. jalur venus transit matahari (dari kiri ke kanan) (sumber: http://en.wikipedia.org/w/index.php? title=file2004 venus_transit.svg&p) metode penelitian dalam penelitian ini akan digunakan data hasil pengamatan transit venus tanggal 8 juni 2004 antara pukul 05.28 ut sampai 07.50 ut (12.28 – 14.50 wib) dari balai pengamatan dirgantara lapan watukosek dengan teleskop sunspot. teleskop sunspot secara rutin digunakan untuk pengamatan aktivitas daerah aktif (sunspot = bintik matahari). sehingga pada lembar sket tersebut tampak tiga grup sunspot di samping venus transit. terdapat perbedaan arah timur-barat matahari pada gambar 4, disebabkan gambar 5. adalah hasil proyeksi cahaya matahari pada bidang proyeksi dalam hal ini lembar sket adalah bayangan fotosfer matahari. http://www.isthe.com/chongo/tech/%20astro/venus2004.html http://www.isthe.com/chongo/tech/%20astro/venus2004.html http://en.wikipedia.org/w/index.php?%20title=file2004%20venus_transit.svg&p http://en.wikipedia.org/w/index.php?%20title=file2004%20venus_transit.svg&p http://en.wikipedia.org/w/index.php?title=file:2004_venus_transit.svg&page=1 nanang widodo 40 volume 3 no. 1 november 2013 gambar 5. lintasan venus di matahari (dari kanan ke kiri), (sumber: pengamatan fotosfer matahari lapan bpd watukosek). perbandingan diameter sket cakram matahari (24 cm) : diameter matahari (1.392.000 km) = 1 : 5.800.000.000. jika terjadi kesalahan 1mm dalam sket = 5800 km dalam ukuran sebenarnya di matahari. dalam penelitian ini akan dilakukan perbandingan beberapa besaran dari hasil pengamatan dan perhitungan, antara lain: 1. perbandingan tinggi bayangan hasil perhitungan dengan hasil pengamatan (sket lintasan bayangan venus). 2. perbandingan kecepatan orbit (dalam teori) sesungguhnya dengan kecepatan bayangan venus hasil pengamatan. 3. mengetahui tingkat kesalahan pengukuran yang dipengaruhi oleh beberapa faktor fisis. segitiga abc segitiga abc terdiri dari titik a adalah titik pengamatan di stasiun bumi, b titik pusat venus dan c adalah titik singgung permukaan planet venus. karena perbandingan jari-jari venus terhadap jarak bumi-venus sangat kecil sebesar 1 : 6843, maka titik f dapat didekati dengan titik c, seperti dalam perhitungan berikut. di mana jari-jari venus = 6050 km dan jarak bumi-venus = 41.400.000 km, sehingga perbandingan jari-jari venus terhadap jarak bumi-venus = 1 : 6843 gambar 6. titik singgung f pada cakram venus. 𝐴𝐶 = √𝐴𝐵2 + 𝐵𝐶 2 (2) 𝐴𝐹 = √𝐴𝐵2 − 𝐵𝐹2 (3) di mana bf = bc = jari-jari venus. sehingga jarak b – f = 41400000,442 – 41399999,558 = 1 km, dapat diabaikan karena ketebalan atmosfer venus. asumsi ini digunakan untuk mencari besar bayangan venus di cakram matahari. hasil dan pembahasan segitiga ade terdiri dari titik d adalah titik pusat bayangan planet venus di cakram matahari dan e adalah titik terluar dari jari-jari bayangan. diameter bayangan venus dari perhitungan persamaan (1), diperoleh 𝐷𝐸ℎ𝑖𝑡 = 𝐵𝐶 ( 𝐴𝐷 𝐴𝐵 ) = 6050 ∗ ( 149600000 41439200 ) = 43695 𝑘𝑚 diameter bayangan venus dalam sket, de(sket) = 7,3 mm 𝐷𝐸𝑠𝑘𝑒𝑡 = ( 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑏𝑎𝑦𝑎𝑛𝑔𝑎𝑛 𝑉𝑒𝑛𝑢𝑠 (𝑚𝑚) 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑀𝑎𝑡𝑎ℎ𝑎𝑟𝑖 (𝑚𝑚) ) ∗ 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑀𝑎𝑡𝑎ℎ𝑎𝑟𝑖 𝐷𝐸𝑠𝑘𝑒𝑡 = ( 7.3 240 ) ∗ 1392000 = 42340 𝑘𝑚 selisih diameter = 43695 – 42340 = 1355 km atau dalam ukuran sket = + 0,2337 mm. gambar 7. aplikasi ∆ abc dan ∆ ade pada transit venus di cakram matahari. tabel 2.posisi dan waktu pengamatan venus transit 8 juni 2004, lapan watukosek no waktu obserasi (ut) titik tepi timur venus, mm 1 05.28 0 2 05.44 8,6 3 06.02 18,0 4 06.16 26,8 5 06.48 41,7 6 07.08 51,5 7 07.30 62,9 8 07.50 72,4 lintasan bayangan venus transit di cakram matahari menempuh jarak 72,4 mm (419.920 km), dimana diameter matahari (sket) = 240 mm (1392000 km). lintasan bayangan venus sejauh 72.4 mm ditempuh dalam waktu 2 aplikasi dua segitiga sebangun pada studi venus transit di matahari tanggal 8 juni 2004… jurnal cauchy – issn: 2086-0382 41 jam 22 menit. maka, kecepatan bayangan venus = 49,286 km/dtk. sedangkan kecepatan orbit venus sebenarnya, 𝑣 = ( 22 7 ) (𝑅𝑝𝑒𝑛𝑑𝑒𝑘 + 𝑅𝑝𝑎𝑛𝑗𝑎𝑛𝑔 ) 224.7 𝑑𝑎𝑦𝑠 = (0,5)(3.14286)(108.942.109 + 107.476.259) (224.7)(24)(3600) 𝑘𝑚 𝑑𝑒𝑡𝑖𝑘 = 35,03 𝑘𝑚 𝑑𝑒𝑡𝑖𝑘 selisih kecepatan orbit venus dengan kecepatan bayangan saat transit sebesar 14,25 km/dtk ~ 0,0024mm/detik (dalam sket). perbedaan kecepatan venus transit disebabkan beberapa faktor antara lain; 1. perekaman bayangan venus dengan menggambar pada kertas sket (cakram matahari), sehingga tingkat kesalahan masih besar. 2. mengeliminasi pengaruh indek bias atmosfer bumi dan venus. 3. faktor koreksi ukuran dalam lembar sket cakram matahari yaitu 1 mm (sket) = 5800 km ukuran sebenarnya di cakram matahari. 4. mengeliminasi kondisi pergerakan aerosol dan awan di atmosfer bumi yang mengakibatkan bayangan semu batas tepi venus. setelah kejadian venus transit pada 8 juni 2004 ini disusul dengan siklus pendek 8 tahunan, venus kembali transit di cakram matahari pada tanggal 6 juni 2012. terdapat 3 jangka waktu siklus venus transit yaitu siklus pendek 8 tahun, siklus sedang 113,5 tahun dan 129,5 tahun. pada masa yang akan datang, peristiwa venus transit kembali terjadi pada 11 desember 2117 (atau siklus 113,5 tahun) setelah peristiwa venus transit tanggal 8 juni 2004. penutup peristiwa transit venus di cakram matahari ini dapat membantu pemahaman aplikasi dua segitiga sebangun yang diterapkan pada kejadian alam, dimana cakram matahari sebagai bidang proyeksi yang digunakan untuk merekam lintasan bayangan venus selama transit. jari-jari venus sebesar 6050 km setelah diproyeksikan di cakram matahari diperoleh tinggi bayangan 21460 km. berdasarkan hitungan dua segitiga sebangun diperoleh tinggi bayangan venus 21841 km, sehingga terdapat koreksi 381 km (+ 1,75 %). kecepatan bayangan venus transit 49,286 km/detik relatif lebih cepat dibandingkan kecepatan orbit venus sebenarnya yaitu 35,05 km/detik. selisih kecepatan bayangan venus transit sebesar 14,25 km/dtk atau 0,0024 mm/dtk (dalam sket) disebabkan beberapa faktor fisis di alam dan ketepatan perekaman bayangan venus di cakram matahari. daftar pustaka [1] kesebangunan segitiga, www.crayonpedia.org/mw, diunduh tanggal 11 pebruari 2012, [2] william j.k, (1978),”exploration of the solar system”, macmillan publising co.inc, printed in the united states of america, newyork [3] glenn schneider (2004), “transit of venus 08 june 2004 trace white light ingress/egress imaging” steward observatory. university of arizona.gschneider@as.arizona.edu, http://nicmosis.as.arizona.edu:8000. how far is the sun ? transit of venus 8 june 2004 observations, diunduh tanggal 20 nopember 2012. http://www.isthe.com/chongo/tech/astro/ venus2004.html [4] nick anthony fiorenza (2012), “the venus transits the pentagonal cycle of venus”, cycles of the heart june 8, 2004 and june 56, 2012. [5] transit of venus, 2004, diunduh tanggal 24 oktober 2012 http://en.wikipedia.org/w/index.php?title= file2004_venus_transit.svg&p [6] synodic cycles and planetary retrogrades to learn more about synodic cycles and synodic astrology, www.lunarplanner.com/hcpages/venus.h tml , diunduh 2 pebruari 2013 [7] jack a stone and jay h. zimmerman “index of refraction air”, diunduh 2 desember 2012, http://emtodbox.nist.gov/wavelength/doc umentation.nsp#indexof http://www.crayonpedia.org/mw mailto:arizona.gschneider@as.arizona.edu http://nicmosis.as.arizona.edu:8000/ http://www.isthe.com/chongo/tech/astro/venus2004.html http://www.isthe.com/chongo/tech/astro/venus2004.html http://en.wikipedia.org/w/index.php?title=file2004_venus_transit.svg&p http://en.wikipedia.org/w/index.php?title=file2004_venus_transit.svg&p http://www.lunarplanner.com/hcpages/venus.html http://www.lunarplanner.com/hcpages/venus.html http://emtodbox.nist.gov/wavelength/documentation.nsp#indexof http://emtodbox.nist.gov/wavelength/documentation.nsp#indexof mila kurniawaty estimasi harga multi-state european call option _5_ estimasi harga multi-state european call option menggunakan model binomial mila kurniawaty1 dan endah rokhmati2 1jurusan matematika, universitas brawijaya, malang. email: mila akuwni@yahoo.com 2jurusan matematika, institut teknologi sepuluh nopember, surabaya abstrak option merupakan kontrak yang memberikan hak kepada pemiliknya untuk membeli (call option) atau menjual (put option) sejumlah aset dasar tertentu (underlying asset) dengan harga tertentu (strike price) dalam jangka waktu tertentu (sebelum atau saat expiration date). perkembangan option belakangan ini memunculkan banyak model pricing untuk mengestimasi harga option, salah satu model yang digunakan adalah formula black-scholes. multi-state option merupakan sebuah option yang payoff-nya didasarkan pada dua atau lebih aset dasar. ada beberapa metode yang dapat digunakan dalam mengestimasi harga call option, salah satunya masyarakat finance sering menggunakan model binomial untuk estimasi berbagai model option yang lebih luas seperti multi-state call option. selanjutnya, dari hasil estimasi call option dengan model binomial didapatkan formula terbaik berdasarkan penghitungan eror dengan mean square error. dari penghitungan eror didapatkan eror rata-rata dari masing-masing formula pada model binomial. hasil eror rata-rata menunjukkan bahwa estimasi menggunakan formula 5 titik lebih baik dari pada estimasi menggunakan formula 4 titik. kata kunci: option, formula black-scholes, multi-state option, model binomial. pendahuluan perkembangan dunia perekonomian sekarang ini semakin pesat, seiring dengan kebutuhan masyarakat yang terus meningkat sehingga mendorong para pelaku ekonomi termasuk para investor untuk bersaing memperoleh keuntungan semaksimal mungkin. dengan membayar sejumlah uang tertentu untuk investasi awal, investor dapat menguasai saham yang nilainya berlipat ganda dari investasi awal. oleh karena itu, diperlukan alat investasi yang berupa option. pada dasarnya, option merupakan kontrak yang memberikan hak kepada pemiliknya untuk membeli atau menjual sejumlah aset dasar tertentu dengan harga tertentu (strike price) dalam jangka waktu tertentu. aset dasarnya dapat berupa saham, kurs, indeks, atau komoditas. menurut waktu exercise-nya, ada beberapa jenis option. salah satunya adalah european option, yang hanya dapat di-exercise pada saat jatuh tempo. jika sebuah option yang payoff-nya berdasarkan pada dua atau lebih aset dasar maka dinamakan multi-state option. model binomial paling banyak digunakan dalam komunitas finance untuk estimasi berbagai model option yang lebih luas seperti pada multi-state european option dalam mengkaji mengenai estimasi harga multi-state european call option menggunakan model binomial, akan diperoleh suatu pemahaman yang mendalam mengenai estimasi tersebut dan menjadi alternatif lain bagi komunitas finance, khususnya para investor untuk mengestimasi harga option di samping modelmodel lain yang telah ada agar dapat memaksimumkan keuntungan dan meminimumkan kerugian. pendekatan proses return dalam mengestrimasi harga multi-state european call option dengan model binomial digunakan suatu pendekatan proses return. dalam melakukan pendekatan proses return diperlukan pemahaman mengenai fungsi payoff dan formula black-scholes. fungsi payoff berdasarkan waktu exercise-nya, ada beberapa jenis option. salah satunya adalah european option, yang hanya dapat di-exercise pada saat jatuh tempo. fungsi payoff dari european call option adalah sebagai berikut ������ � max �� �,0� atau dapat dijabarkan sebagai berikut ������ � ��� �,���� �� � �0 , ���� �� � �� dimana �� adalah stock price pada saat jatuh tempo dan � adalah strike price. estimasi harga multi-state european call option menggunakan model binomial jurnal cauchy – issn: 2086-0382 183 formula black-scholes untuk european option persamaan black-scholes dari option pricing adalah sebagai berikut ���� � �� � ����!� � "� ���! "# � 0 (1) berdasarkan asumsi model black-scholes pada persamaaan (1), persamaan black-scholes untuk european vanilla call option dapat dibentuk sebagai berikut ���� � �� � ����!� � "� ���! "#, ∞ $ � $ ∞,% � 0 (2) dimana # � #&�,%' merupakan nilai european call option, � adalah harga aset dasar, % adalah jatuh tempo, " adalah suku bunga bebas resiko dan ( adalah volatilitas. kondisi awal (payoff saat jatuh tempo) dapat dinyatakan sebagai berikut #&�,0' � max � �,0� dengan � adalah strike price. solusi dari persamaan (2) diperoleh sebagai berikut #&�,%' � )*+� , max&�� �' ∞ 1 ��(√21% exp4 56 �ln�� 9ln� � :" ( 2 ;%<= 2( % > ?@a�� operasi ekspektasi #&�,%' � b&max&�� �,0'')*+&�*c' dapat dinyatakan sebagai berikut #&�,%' � )*+&�*c' d max&�� �'e∞&��;�'a�� (3) dengan % � g h. e&��;�' merupakan transition density function dari harga aset �� yang dapat dinyatakan sebagai berikut e&��;�' � j!k�√ l� expm n&op !q*op!'*r+* s�� t�u� ��� v (4) dari persamaan tersebut terlihat bahwa fungsi densitas berdistribusi normal dengan variabel ln !q! , yang mempunyai mean w" s�� x% dan varians ( %. model yang digunakan adalah two-state option dan densitas bersama dari harga dua aset dasar �j dan � adalah lognormal bivariate. dalam dunia netral resiko, return aset dasar diberikan sebagai berikut ln !y∆[!y � \] , i = 1,2 dimana �] merupakan harga dari aset dasar i pada saat ini, �]∆c merupakan harga aset dasar i pada suatu periode ∆h berikutnya dan \] adalah return aset dasar. sesuai perhitungan pada (4), variabel acak normal \] mempunyai mean r" �y� t∆h dan varians (] ∆h. dimana " adalah tingkat bunga bebas resiko dan ( adalah varians dari proses lognormal. ^ merupakan koefisien korelasi antara \j dan \ dan (] adalah volatilitas harga aset dasar �], � � 1,2. proses normal bivariate bersama \j,\ � didekati oleh sepasang variabel acak diskrit \j_,\ _� dengan mengikuti distribusi berikut tabel 1. distribusi variabel diskrit \j_,\ _� \j_ \ _ probabilitas `j `j `j `j 0 ` ` ` ` 0 �j � �a �b �c dimana `] � d](]√δh , � � 1,2. hasil dan pembahasan nilai probabilitas untuk mendapatkan nilai probabilitas 4321 ,,, pppp dan 5p , mean, varian, dan covarian dari \j_,\ _� disamakan dengan \j,\ �. sehingga didapatkan persamaan yang bersesuaian sebagai berikut trppppve a ∆      −=−−+= 2 )()( 2 1 432111 σ ζ trppppve a ∆      −=+−−= 2 )()( 2 2 432122 σ ζ tppppva ∆=+++= 214321 2 11 )()var( σζ (5) tppppva ∆=+++= 224321 2 22 )()var( σζ (6) tppppvvaa ∆=−+−= ρσσζζ 2143212121 )(),cov( agar persamaan (5) dan (6) konsisten, harus ditentukan 21 λλλ == sehingga diperoleh empat persamaaan independen untuk lima nilai probabilitas sebagai berikut 1 2 1 4321 2 λσ σ tr pppp ∆      − =−−+ 2 2 2 4321 2 λσ σ tr pppp ∆      − =+−− 24321 1 λ =+++ pppp 24321 λ ρ =−+− pppp karena jumlahan probabilitas harus sama dengan satu, maka diberikan kondisi berikut 154321 =++++ ppppp mila kurniawaty dan endah rokhmati 184 volume 1 no. 4 mei 2011 sehingga didapatkan penyelesaian persamaan tersebut dan diperoleh nilai masing-masing probabilitas yang terjadi             +             − + −∆ += 2 2 2 2 1 2 1 21 221 4 1 λ ρ σ σ σ σ λλ rr t p             −             − − −∆ += 2 2 2 2 1 2 1 22 221 4 1 λ ρ σ σ σ σ λλ rr t p             +             − − − − ∆ += 2 2 2 2 1 2 1 23 221 4 1 λ ρ σ σ σ σ λλ rr t p             −             − + − − ∆ += 2 2 2 2 1 2 1 24 221 4 1 λ ρ σ σ σ σ λλ rr t p 25 1 1 λ −=p , dengan λ ≥ 1 adalah parameter bebas. estimasi harga multi-state european call option menggunakan model binomial. dalam model binomial, perubahan harga saham tiap periode dari interval waktu δh diasumsikan mempunyai dua kemungkinan hasil, yaitu e dengan probabilitas � dan a dengan probabilitas 1 � &e � a' seperti pada gambar berikut gambar 1. kontruksi pohon binomial jika harga saham (stock option) saat ini �-, maka harga saham pada saat h adalah sebagai berikut �c � �-efac*f, h � 1,2,…,g dimana 0 $ a $ 1 $ e dan h � 0,1,…,h. harga call saat ini dinotasikan dengan #, dan #i∆c dan #j∆c menunjukkan harga call setelah satu periode (waktu jatuh tempo dalam konteks sekarang) yang bersesuaian dengan pergerakan naik dan turunnya harga aset, misal � menyatakan strike price dari call, maka payoff dari call pada saat jatuh tempo adalah sebagai berikut k#i∆c � max&e� �,0'dengan probabilitas � #j∆c � max&a� �,0'dengan probabilitas 1 �� (7) nilai terkini dari call diberikan sebagai berikut # � t�u∆[v&j*t'�w∆[x dengan � � x*ji*j dan y � )+∆c. jika � dan �∆c menyatakan harga aset saat ini dan harga aset satu periode setelah ∆h, maka mean dan varians dari !∆[! adalah �e � &1 �'a dan �e � &1 �'a z�e � &1 �'a{ . dengan derajat akurasi |&∆h' nilai dari parameter e dan a adalah e � )�√∆c dan a � )*�√∆c. karena terdapat dua aset dasar yang mendasari harga european call option yaitu �j dan � , maka kemungkinan yang terjadi dapat digambarkan sebagai berikut gambar 2. kontruksi pohon binomial dua aset dasar jika �], � � 1,2,…,h menyatakan harga aset dasar � dan }&�j,� ,…,�f,g' menyatakan nilai dari sekuritas derivatif, maka secara umum fungsi payoff-nya dinyatakan sebagai fungsi linier berikut }&�,0' � max&∑ �]�] � �,0f]�j ' (8) dimana � dan �], � � 1,2,…,h adalah konstanta. dengan menggunakan fungsi payoff untuk multi-state option pada (8) dan persamaan (7), maka dapat dijabarkan sebagai berikut ��� � ��� #i�i�δc � max&ej�j � e � �,0',dengan peluang �j#i�j�δc � max&ej�j � a � �,0',dengan peluang � #j�j�δc � max&aj�j � a � �,0',dengan peluang �a#j�i�δc � max&aj�j � e � �,0',dengan peluang �b#-,-δc � max&�j � � �,0',dengan peluang �c � dengan e] � )�y,a] � )*�y, � � 1,2. sehingga menurut formula binomial, diperoleh harga dari multi-state call option sebagai berikut r cpcpcpcpcp c tt ud t dd t du t uu ∆∆∆∆∆ ++++ = 0,054321 21212121 (9) jika ,1=λ maka 05 =p dan formula 5 titik berkurang menjadi 4 titik sebagai berikut r cpcpcpcp c t ud t dd t du t uu ∆∆∆∆ +++ = 21212121 4321 (10) dimana r = )+∆c. simulasi untuk mengestimasi harga option hasil estimasi call option dengan model binomial dengan σ1=0.2 dan σ2=0.3, untuk λ bernilai 1.0 sampai 2.0 dapat digambarkan sebagai berikut: �j ej�j aj�j � e � a � � �-e �-a probabilitas � probabilitas 1 � estimasi harga multi-state european call option menggunakan model binomial jurnal cauchy – issn: 2086-0382 185 gambar 3. grafik call option model binomial untuk beberapa nilai lamda dari grafik terlihat bahwa pertambahan nilai λ menurunkan harga call option. hal ini berarti, penggunaan formula 5 titik dapat menurunkan harga call option dari pada formula 4 titik. mean square error (mse) penghitungan eror menggunakan mean square error dituliskan sebagai berikut: m rmse m n n∑ == 1 2η dimana =η market option price – model option price. error dari model binomial berdasarkan mean square error sesuai penghitungan apabila ditampilkan dalam grafik sebagai berikut gambar 4. grafik harga call dan mean square error model binomial formula 4 titik dan 5 titik dari hasil penghitungan eror menggunakan mean square error didapatkan rata-rata eror sebagai berikut: tabel 2. tabel hasil average mse dari model binomial 4 titik dan 5 titik model binomial average mse formula 4 titik 0.244817 formula 5 titik 0.2271071 berdasarkan hasil penghitungan eror dari dua formula pada model binomial dapat disimpulkan bahwa formula 5 titik mempunyai eror yang lebih kecil daripada formula 4 titik . oleh karena itu formula 5 titik pada model binomial dapat dinyatakan sebagai formula yang yang lebih baik daripada formula 4 titik. penutup berdasarkan hasil pembahasan mengenai estimasi harga multi-state european call option dengan model binomial dapat disimpulkan bahwa estimasi harga multi-state european call option dengan model binomial diperoleh dengan pendekatan proses return yang mempunyai variabel acak normal didekati dengan sepasang variabel acak diskrit untuk mendapatkan nilai probabilitas yang terjadi pada masing-masing aset dasar sehingga dapat diestimasi menggunakan model binomial. estimasi harga multi-state european call option menggunakan model binomial diperoleh dua formula, yaitu: 1. formula 5 titik # � �j#i�i�δc � � #i�j�δc � �a#j�j�δc � �b#j�i�δc � �c#-,-δcy 2. formula 4 titik # � �j#i�i�δc � � #i�j�δc � �a#j�j�δc � �b#j�i�δcy formula terbaik dalam model binomial pada multi-state option adalah formula 5 titik karena mempunyai eror rata-rata yang lebih kecil daripada formula 4 titik. dari hasil simulasi dapat dilihat bahwa estimasi harga multi-state european call option menggunakan model binomial sangat dipengaruhi oleh besarnya suatu parameter bebas λ. semakin tinggi nilai λ maka harga dari call option akan semakin turun. daftar pustaka [1] kwok, yue-kuen. 1998. mathematical models of financial derivatives. singapore: springer. [2] pliska, r. stanley. 1997. introduction tomathematical finance: discrete time model. oxford: blackwell. [3] bodie, kane, marcus. 2005. investment. sixth edition. mcgraw-hill, international edition. [4] sembel, roy, dan fardiansyah, tedy. 2002. sekuritas derivatif: madu atau racun. jakarta: salemba empat. grafik harga call option untuk 6 nilai lamda 34 34.2 34.4 34.6 34.8 35 35.2 35.4 35.6 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 data ke h ar g a ca ll o p ti o n lamda=1.0 lamda=1.2 lamda=1.4 lamda=1.6 lamda=1.8 lamda=2.0 grafik call option formula 4 titik 34 34.5 35 35.5 36 1 9 17 25 33 41 49 57 65 73 81 89 97 data keh a rg a c a ll o p ti o n c_m arket c_form ula 4 titik grafik call option formula 5 titik 34 34.5 35 35.5 36 1 9 17 25 33 41 49 57 65 73 81 89 97 data keh a rg a c a ll o p ti o n c_market c_formula 5 titik mse form ula 4 titik dan 5 titik 0 0.2 0.4 0.6 0.8 1 1 10 19 28 37 46 55 64 73 82 91 100 d at a keformula 4 t it ik formula 5 t it ik mila kurniawaty dan endah rokhmati 186 volume 1 no. 4 mei 2011 [5] rudiger, seydel. 2002. tools for computational finance. koln: springer. [6] higham, desmond j. 2004. an introduction to financial option valuation: mathematics, stochastics, and computation. cambridge: cambridge university press. [7] rahayu, s.k.t. 2006. estimasi nilai european call option menggunakan metode historical data dan filter kalman. skripsi its surabaya. [8] hull, john c. 2002. option future and other derivatives. new jersey: prentice hall. [9] ross, sheldon m. 2004. an introduction to mathematical finance: options and other topics. cambridge: cambridge university press. pemanfaatan suhu udara dan kelembapan udara dalam persamaan regresi untuk simulasi prediksi total hujan bulanan di pangkalpinang akhmad fadholi stasiun meteorologi depati amir pangkalpinang email: akhmad.fadholi@bmkg.go.id abstrak simulasi prediksi curah hujan bulanan (rr) dengan menggunakan prediktor suhu udara (t) dan kelembapan udara (rh) telah dicoba dilakukan di stasiun meteorologi depati amir pangkalpinang. evaluasi dilakukan dengan membandingkan dan menghitung besarnya penyimpangan prediksi total hujan bulanan terhadap total hujan aktualnya. simulasi prediksi total hujan bulanan ini digunakan dua metode regresi, yaitu regresi linier sederhana dan regresi linier berganda. hasil pengolahan data menunjukkan bahwa simulasi prediksi total hujan bulanan tahun 2011 di daerah studi didapatkan rerata rmse = 14.8 mm/bulan menggunakan prediktor suhu udara, rmse = 14.5 mm/bulan menggunakan prediktor kelembapan udara, dan rmse = 14.9 mm/bulan menggunakan prediktor suhu udara dan kelembapan udara sekaligus. kata kunci: hujan bulanan, kelembaban udara, linear, regresi, suhu udara abstract monthly rainfall prediction simulation (rr) using air temperature (t) and relative humidity (rh) as predictors, has been done at meteorological station depati amir pangkalpinang. evaluation of prediction was examined by comparing and computing between the prediction output and observation values. both linear and multi-linear regression methods were used in data processing. results show the monthly rainfall prediction simulation of 2011 having the mean of rmse = 14.8 mm/month using air temperature data as predictor, rmse = 14.5 mm/month using relative humidity data as predictor, and rmse = 14.9 mm/month usi ng both air temperature and relative humidity data as predictors at onces. key words: monthly rainfall, relative humidity, linear, regression, air temperature pendahuluan kota pangkalpinang berada di wilayah belahan bumi selatan. secara geografis (sandy, 1995) kota pangkalpinang terletak pada 02° 04' 02° 10' ls dan 106° 04' 106° 07' bt dengan lingkungan fisik sekitar stasiun pada umumnya adalah perumahan, hutan, dan pantai. di wilayah tropis, curah hujan merupakan salah satu unsur iklim yang paling tinggi keragamannya (wirjohamidjojo dan swarinoto, 2010). karakteristik curah hujan di berbagai daerah tentunya tidak sama. kondisi ini diakibatkan oleh beberapa faktor (nieuwolt, 1977), yakni: geografis, topografis, dan orografis. belum lagi ditambah dengan struktur dan orientasi kepulauan (kartasapoetra, 2006). akibatnya pola sebaran curah hujan cenderung tidak merata antara daerah yang satu dengan daerah yang lain dalam ruang lingkup yang luas. mengingat bahwa hujan di wilayah ropis banyak berpengaruh terhadap kehidupan manusia dalam segala aspeknya (wirjohamidjojo dan swarinoto, 2007), maka penulis berusaha mengumpulkan dan melakukan pengolahan data curah hujan dimaksud. selain itu disertai juga dengan pengolahan data suhu udara dan kelembapan udara selama 21 tahun di stasiun meteorologi depati amir pangkalpinang. sementara itu untuk pembuatan simulasi prediksi total hujan bulanan digunakan metode regresi linier sederhana dan regresi linier berganda. bertalian dengan adanya tipe-tipe total hujan bulanan, maka presisi prediksi total hujan bulanan akan berbeda-beda dari tempat yang satu dengan tempat yang lain. prediksi total hujan bulanan dengan metode tertentu sangat sesuai dengan tempat yang satu, tetapi dapat juga tidak sesuai pada tempat yang lain. untuk itu evaluasi prediksi total hujan bulanan sangat diperlukan sehingga hasil kajian dapat digunakan sebagai masukan dalam menyiapkan prediksi total hujan bulanan pada bulan-bulan berikutnya. a. curah hujan curah hujan adalah butir-butir air atau kristal es yang jatuh atau keluar dari awan atau akhmad fadholi 2 volume 3 no. 1 november 2013 kelompok awan. jika curahan dimaksud dapat mencapai permukaan bumi disebut sebagai hujan (tjasyono, 1999). jika setelah keluar dari dasar awan tetapi tidak jatuh sampai ke permukaan bumi disebut sebagai virga (soepangkat, 1994). butir air yang dapat keluar dari awan dan mampu mencapai permukaan bumi harus memiliki garis tengah paling tidak sebesar 200 mikrometer (1 mikrometer = 0,001 cm). kurang dari ukuran diameter tersebut, butir-butir air dimaksud akan habis menguap di atmosfer sebelum mampu mencapai permukaan bumi5). banyaknya curah hujan yang mencapai permukaan bumi atau tanah selama selang waktu tertentu dapat diukur dengan jalan mengukur tinggi air hujan dengan cara tertentu. hasil dari pengukurannya dinamakan curah hujan, yaitu tanpa mengingat macam atau bentuknya pada saat mencapai permukaan bumi dan tidak memperhitungkan endapan yang meresap ke dalam tanah, hilang karena penguapan, atau pun mengalir. dari bentuk dan sifatnya, hujan ada yang disebut dengan shower atau hujan tiba-tiba. hujan tersebut ditandai dengan permulaan dan akhir yang mendadak dengan variasi intensitas yang umumnya cepat, dengan titik-titik air atau partikelpartikel yang lebih besar daripada hujan biasa dan jatuhnya dari awan-awan cumulus (cu) ataupun cumulonimbus (cb) yang pertumbuhannya bersifat konvektif. hujan kontinu yang permulaan dan akhirnya tidak secara mendadak dan tidak tampak terjadi pengurangan perawanan sejak permulaan sampai pada akhirnya aktifitas tersebut. hujan ini jatuhnya dari awan-awan yang pada umumnya berbentuk merata seperti awanawan stratus (st), altostratus (as), maupun nimbustratus (ns). b. suhu udara untuk keperluan operasional klimatologi di indonesia, khususnya bagi stasiun yang beroperasi kurang dari 24 jam sehari, maka suhu udara permukaan rata-rata harian dapat dihitung dengan persamaan berikut: 𝑇𝑚𝑒𝑎𝑛 = 2∗𝑇7 + 𝑇13 + 𝑇18 4 (1) dengan: tmean = suhu udara permukaan ratarata harian (°), t7 = suhu udara pengamatan jam 07.00 lt; t13 = suhu udara pengamatan jam 13.00 lt; t18 = suhu udara pengamatan jam8.00 lt. keadaan suhu udara pada suatu tempat di permukaan bumi akan ditentukan oleh faktor faktor (tanudidjaja, 1993) sebagai berikut : a) lamanya penyinaran matahari semakin lama matahari memancarkan sinarnya disuatu daerah, makin banyak panas yang diterima. keadaan atmosfer yang cerah sepanjang hari akan lebih panas daripada jika hari itu berawan sejak pagi. b) kemiringan sinar matahari suatu tempat yang posisi matahari berada tegak lurus di atasnya, maka radiasi matahari yang diberikan akan lebih besar dan suhu ditempat tersebut akan tinggi, dibandingkan dengan tempat yang posisi mataharinya lebih miring. c) keadaan awan adaya awan di atmosfer akan menyebabkan berkurangnya radiasi matahari yang diterima di permukaan bumi. karena radiasi yang mengenai awan, oleh uap air yang ada di dalam awan akan dipencarkan, dipantulkan, dan diserap. d) keadaan permukaan bumi perbedaan sifat darat dan laut akan mempengaruhi penyerapan dan pemantulan radiasi matahari. permukaan darat akan lebih cepat menerima dan melepaskan panas energy radiasi matahari yang diterima dipermukaan bumi dan akibatnya menyebabkan perbedaan suhu udara di atasnya. c. kelembapan udara kelembapan udara adalah banyaknya uap air yang terkandung dalam udara atau atmosfer. besarnya tergantung dari masuknya uap air ke dalam atmosfer karena adanya penguapan dari air yang ada di lautan, danau, dan sungai, maupun dari air tanah. disamping itu terjadi pula dari proses transpirasi, yaitu penguapan dari tumbuhtumbuhan. sedangkan banyaknya air di dalam udara bergantung kepada banyak faktor, antara lain adalah ketersediaan air, sumber uap, suhu udara, tekanan udara, dan angin5). uap air dalam atmosfer dapat berubah bentuk menjadi cair atau padat yang akhirnya dapat jatuh ke bumi antara lain sebagai hujan. kelembapan udara yang cukup besar memberi petunjuk langsung bahwa udara banyak mengandung uap air atau udara dalam keadaan basah. berbagai ukuran dapat digunakan untuk menyatakan nilai kelembapan udara. salah satunya adalah kelembapan udara relative (nisbi). kelembapan udara nisbi (wirjohamidjojo, 2006) memiliki pengertian sebagai nilai perbandingan antara tekanan uap air yang ada pada saat pengukuran (e) dengan nilai tekanan uap air maksimum (em) yang dapat dicapai pada suhu udara dan tekanan udara saat pemanfaatan suhu udara dan kelembapan udara dalam persamaan regresi … jurnal cauchy – issn: 2086-0382 3 pengukuran. persamaan untuk kelembapan udara relatif adalah seperti berikut: 𝑅𝐻 = 𝑒 𝑒𝑚 𝑥 100 (2) dengan: 𝑅𝐻 = kelembapan udara relatif (%), 𝑒 = tekanan uap air pada saat pengukuran (mb), 𝑒𝑚 = tekanan uap air maksimum yang dapat dicapai pada suhu udara dan tekanan udara saat pengukuran (mb). metode penelitian a. data data yang digunakan dalam tulisan ini adalah data iklim yang diperoleh dari stasiun meteorologi depati amir pangkalpinang, yang terdiri atas: (1). data total curah hujan bulanan (2). data rerata suhu udara bulanan (3). data rerata kelembapan udara bulanan data suhu udara dan kelembapan udara bulanan merupakan rata-rata bulanan hasil dari jumlah data rata-rata harian selama satu bulan kemudian dibagi dengan banyaknya data pada bulan yang bersangkutan. panjang data yang digunakan adalah 22 tahun dari tahun 1990 2011. data total hujan, suhu udara, dan kelembapan udara bulanan selama 21 tahun (1990-2010), digunakan untuk membentuk persamaan regresi. data suhu udara dan kelembapan udara bulanan tahun 2011 digunakan untuk memprediksi total hujan. sedangkan data total hujan bulanan pada tahun 2011 digunakan sebagai pembanding dalam melakukan verifikasi hasil prediksi total hujan bulanan. b. metode a) regresi linier sederhana metode prediksi regresi linier sederhana dilakukan dengan cara membentuk persamaan regresi agar dapat melakukan simulasi memprediksi total hujan bulanan di stasiun meteorologi depati amir pangkalpinang. adapun persamaan yang digunakan (nazir, 2003) adalah sebagai berikut: 𝑌 = 𝐴 + 𝐵𝑋 (3) dengan: y = variabel yang diduga predictant/dependent), a = konstanta, b = koefisien regresi, dan x = variabel penduga (predictor /independent). koefisien a dan b pada persamaan di atas dapat dihitung dengan cara sebagai berikut: 𝐵 = 𝑛 ∑ 𝑋𝑌 − (∑ 𝑋)(∑ 𝑌) 𝑛 ∑ 𝑋2 − (∑ 𝑋)2 (4) sedangkan: 𝐴 = 𝑌𝑚𝑒𝑎𝑛 − 𝐵 ∗ 𝑋𝑚𝑒𝑎𝑛 (5) dengan: x = data suhu udara (kelembapan udara); y = data total hujan (mm); dan n = banyak data. b) regresi linier berganda metode prediksi regresi linier berganda ini dilakukan dengan cara membentuk persamaan regresi yang digunakan untuk melakukan simulasi prediksi total hujan bulanan menggunakan lebih dari dari satu variable independen. hasil prediksi total hujan bulanan menggunakan metode ini dibandingkan dengan prediksi total hujan bulanan menggunakan regresi linier sederhana sehingga dapat terlihat hasil prediksi yang lebih baik setelah dicocokkan dengan data observasi. adapun persamaan umum (usman dan akbar, 2000) metode ini adalah sebagai berikut: 𝑌 = 𝐵0 + 𝐵1𝑋1 + 𝐵2 𝑋2 + … . . + 𝐵𝑘 𝑋𝑘 (6) dengan: bo = konstanta; b1, b2, ….. bk = koefisien variabel x1, x2,...., xk; y = variable yang diduga (variabel dependent); dan xi = variabel penduga (variabel independent). untuk analisis dengan metode regresi dibedakan dua jenis variabel ialah variable bebas (independent) atau variabel prediktor dan variabel tidak bebas (dependent) atau variable respon. variabel bebas merupakan variable yang dapat mempengaruhi varibel tidak bebas atau variabel yang dapat memprediksi harga variabel tidak bebas. variabel ini dinyatakan dengan x1, x2, ..., xk. sedangkan variabel tidak bebas merupakan variabel yang terjadi karena variabel bebas atau variabel yang mencerminkan respon dari variabel bebas, dinyatakan dengan y (sudjana, 1995). dalam tulisan ini variabel bebas (independent) atau prediktor adalah suhu udara dan kelembapan udara, sedangkan variabel tidak bebas (dependent) atau variabel respon adalah total hujan. proses pembuatan prakiraan ada dua tahap, tahap pertama membuat persamaan regresi untuk tiap bulan berdasarkan bulan yang sama selama 21 tahun dari tahun 1990 2010 dan tahap kedua memprediksi total hujan bulan dengan memberikan nilai variabel penduga (prediktor) pada persamaan regresi yang dibuat. dalam penginputan data prediktor pada masingmasing persamaan regresi digunakan perbedaan waktu (timelag) 1 (satu) bulan dengan prediktan. akhmad fadholi 4 volume 3 no. 1 november 2013 c) root mean square error metode ini digunakan untuk mengetahui besarnya penyimpangan yang terjadi antara nilai prediksi total hujan dibandingkan dengan nilai total hujan aktualnya yang terjadi selama satu tahun. dari nilai ini dapat dilakukan analisa prediksi total hujan dengan prediktor mana diantara suhu maupun kelembapan udara atau suhu dan kelembapan udara yang memiliki nilai penyimpangan yang besar atau kecil (wilks, 1995). perlu diketahui bahwa untuk validasi hasil prakiraan semakin besar nilai rmse, maka semakin jauh nilai data total hujan bulanan prakiraan terhadap total hujan aktualnya dan semakin kecil nilai rmse maka semakin baik prediksi total hujannya. karena tingkat kesalahan yang dapat diminimalisir dapat meningkatkan tingkat akurasi prakiraan (soetamto dan maria, 2010). d) koefisien korelasi nilai koefisien korelasi pearson (trihendradi, 2005) di gunakan untuk menentukan besarnya hubungan atau kedekatan antara total hujan yang telah diprediksi dengan total hujan aktual yang terjadi. dalam hal ini kedekatan yang dicari adalah besarnya nilai prediksi dengan menggunakan prediktor mana diantara suhu atau kelembapan udara atau suhu dan kelembapan udara yang paling baik. kuat tidaknya hubungan (prihatini dkk, 2000) antara prediksi total hujan bulanan dengan total hujan observasinya dapat diukur dengan suatu nilai yang disebut dengan koefisien korelasi. nilai koefisien korelasi ini paling sedikit -1 dan paling besar 1. jadi r = koefisien korelasi, dapat di nyatakan sebagai berikut: a) jika harga r mendekati +1, berarti hubungan antara total hujan bulanan b) yang diprediksi dengan total hujan bulanan observasinya sangat kuat dan positif. c) jika harga r mendekati -1, berarti hubungan antara total hujan bulanan yang diprediksi dengan total hujan bulanan observasinya sangat kuat dan negatif. d) jika harga r mendekati +0.5 atau -0.5 , berarti hubungan antara total hujan bulanan yang diprediksi dengan total hujan bulanan observasinya dianggap cukup kuat. e) jika harga r lebih kecil dari +0.5 atau lebih besar dari -0.5, berarti hubungan antara total hujan bulanan yang diprediksi dengan total hujan bulanan observasinya dianggap lemah. untuk validasi hasil prakiraan dengan menggunakan koefisien korelasi, semakin kuat korelasi maka semakin baik hasil validasi berarti semakin tinggi tingkat akurasi prakiraan. hasil dan pembahasan a. hasil bentuk persamaan regresi linier sederhana hasil pengolahan menggunakan data suhu (t) dan data kelembapan udara (rh) disajikan pada tabel 1. prediksi curah hujan tahun 2011 menggunakan variabel penduga suhu udara dan kelembapan udara dapat dilihat pada tabel 2. bentuk persamaan regresi linier berganda hasil pengolahan menggunakan data suhu udara, kelembapan udara dan total hujan bulanan dicantumkan pada tabel 3. tabel 1. persamaan regresi linier sederhana untuk prediksi total hujan bulanan dengan prediktor t dan rh (sumber: pengolahan data). no bulan persamaan regresi linier sederhana menggunakan prediktor menggunakan prediktor suhu udara (t) kelembaban udara (rh) 1 januari y = 2889.6 99.5x y = -1022.6 + 15.4x 2 februari y = 1496.3 49.5x y = -1936.2 + 25x 3 maret y = 1316.8 40.4x y = 183.9 + 0.7x 4 april y = 3610.3 125.5x y = -1231.9 + 17.5x 5 mei y = 1850.9 60.5x y = -1160.9 + 16.3x 6 juni y = 1021.6 32.5x y = -555.7 + 8.5x 7 juli y = 2066.4 71.2x y = -1170.1 + 16.3x 8 agustus y = 2381.2 83x y = -1378.5 + 19.3x 9 september y = 2081.4 72.8x y = -795.8 + 11.4x 10 oktober y = 2788.7 96.4x y = -994 + 14.5x 11 nopember y = 3691.3 130.2x y = -1421.7 + 19.7x 12 desember y = 178.1 + 5x y = -1071.8 + 15.9x pemanfaatan suhu udara dan kelembapan udara dalam persamaan regresi … jurnal cauchy – issn: 2086-0382 5 tabel 2. prediksi total hujan bulanan tahun 2011 menggunakan prediktor t dan rh di stasiun meteorologi depati amir pangkalpinang (sumber: pengolahan data). no bulan prediksi cht prediksi chrh tahun 2011 (mm) tahun 2011 (mm) 1 januari 301.7 300.8 2 februari 206.7 155.1 3 maret 256.1 245.5 4 april 331.9 253.4 5 mei 244.1 229.4 6 juni 136.0 154.8 7 juli 123.0 145.8 8 agustus 125.3 121.2 9 september 62.4 51.1 10 oktober 83.9 62.8 11 nopember 164.1 164.8 12 desember 311.5 259.1 tabel 3. persamaan regresi linier berganda menggunakan prediktor t dan rh (sumber: pengolahan data). no bulan persamaan regresi linire berganda 1 januari y = 689.7 51.1x1 + 10.9x2 2 februari y = 73 + 10.9x1 1.9x2 3 maret y = 80.7 + 4.5x1 + 0.6x2 4 april y = 93.5 + 0.8x1 + 1.6x2 5 mei y = 54.7 + 4x1 +0.3x2 6 juni y = -21.6 + 6.4x1 0.2x2 7 juli y = -77.6 + 8.5x1 0.2x2 8 agustus y = -58.3 +6.9x1 0.2x2 9 september y = -49 + 5.5x1 0.2x2 10 oktober y = -48.7 + 9.8x1 0.6x2 11 nopember y = -18.7 + 19.9x1 3.3x2 12 desember y = 508.5 6.9x1 0.2x2 perlu diketahui bahwa jika total hujan hasil observasi mengalami kenaikan dan total hujan hasil prediksi juga mengalami kenaikan atau sebaliknya, maka dapat dikatakan bahwa prediksi tersebut mendekati total hujan yang sebenarnya dan hasil prediksi tersebut adalah cukup baik. hasil perhitungan menunjukan bahwa pada tahun 2011 di dapat nilai koefisien korelasi pearson antara simulasi prediksi total hujan bulanan menggunakan prediktor suhu udara adalah r = 0,63; menggunakan prediktor kelembapan udara diperoleh sebesar r = 0,44; menggunakan prediktor suhu udara dan kelembapan udara sekaligus diperoleh sebesar r = 0,63. b. pembahasan a) bulan januari. pada bulan januari 2011 diperoleh nilai prediksi total hujan bulanan sebesar 301.7 mm menggunakan prediktor suhu udara. penyimpangan terhadap total hujan observasi sebesar 46.8 mm. sementara itu menggunakan prediktor kelembapan udara didapat nilai total hujan bulanan prediksi sebesar 300.8 mm. penyimpangan terhadap data observasi diperoleh 47.7 mm. selanjutnya menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 298.6 mm. akhmad fadholi 6 volume 3 no. 1 november 2013 penyimpangan terhadap total hujan aktualnya adalah 45.5 mm. b) bulan februari pada bulan februari 2011 diperoleh nilai prediksi total hujan bulanan dengan menggunakan prediktor suhu udara sebesar 206.7 mm. penyimpangan terhadap data total hujan aktualnya sebesar 103.2 mm. menggunakan prediktor kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 155.1 mm. penyimpangan terhadap data observasi didapat 154.8 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan sebesar 202.0 mm. penyimpangan terhadap data aktualnya sebesar 45.5 mm. grafik perbandingan antara total hujan bulanan menggunakan prediktor t dengan data aktualnya tahun 2011 dapat dilihat pada gambar 1, sedangkan prediksi menggunakan prediktor rh dapat dilihat pada gambar 2. gambar 1. perbandingan antara prediksi total hujan bulanan dengan prediktor t terhadap data aktualnya di stasiun meteorologi depati amir pangkalpinang tahun 2011 (sumber: pengolahan data) gambar 2. perbandingan antara prediksi total hujan bulanan dengan prediktor rh dengan hasil observasinya di stasiun meteorologi depati amir pangkalpinang tahun 2011 (sumber: pengolahan data) 0 50 100 150 200 250 300 350 400 c h ( m m ) bulan rr bulanan terhadap rh 2011 data obs 2011 0 50 100 150 200 250 300 350 400 c h ( m m ) bulan rr bulanan terhadap t dan rh 2011 data obs 2011 pemanfaatan suhu udara dan kelembapan udara dalam persamaan regresi … jurnal cauchy – issn: 2086-0382 7 gambar 3. perbandingan antara prediksi total hujan bulanan menggunakan prediktor (t dan rh) dengan total hujan bulanan observasi observasi di stasiun meteorologi depati amir pangkalpinang 2011 (sumber: pengolahan data) c) bulan maret pada bulan maret 2011 menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 256.1 mm. penyimpangan terhadap data total hujan aktualnya sebesar 27.6 mm. menggunakan prediktor kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 245.5 mm. penyimpangan terhadap data observasi didapat 17.0 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan sebesar 245.1 mm. penyimpangan terhadap data aktualnya sebesar 16.6 mm. d) bulan april pada bulan april 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 331.9 mm. penyimpangan terhadap data total hujan aktualnya sebesar 24.3 mm. menggunakan prediktor kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 253.4 mm. penyimpangan terhadap data observasi sebesar 102.8 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 245.9 mm. penyimpangan terhadap data aktualnya sebesar 110.3 mm. e) bulan mei pada bulan mei 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 244.1 mm. penyimpangan terhadap data aktualnya sebesar 99.8 mm. sementara itu menggunakan prediktor kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 229.4 mm. penyimpangan terhadap data observasi sebesar 114.5 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara didapat nilai diprediksi total hujan bulanan sebesar 190.0 mm. penyimpangan terhadap data aktualnya sebesar 153.9 mm. f) bulan juni pada bulan juni 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 136.0 mm. penyimpangan terhadap data aktualnya sebesar 135.6 mm. menggunakan predictor kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 154.8 mm. penyimpangan terhadap data observasi sebesar 116.8 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 134.4 mm. penyimpangan terhadap data observasi sebesar 137.2 mm. g) bulan juli pada bulan juli 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 123.0 mm. penyimpangan terhadap data total hujan aktualnya sebesar 31.9 mm. menggunakan prediktor kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 145.8 mm. penyimpangan terhadap data observasi sebesar 54.7 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara didapat nilai 0 50 100 150 200 250 300 350 400 c h ( m m ) bulan rr bulanan terhadap t dan rh 2011 data obs 2011 akhmad fadholi 8 volume 3 no. 1 november 2013 prediksi total hujan bulanan sebesar 138.6 mm. penyimpangan terhadap data aktualnya sebesar 47.5 mm. h) bulan agustus pada bulan agustus 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 125.3 mm. penyimpangan terhadap data total hujan aktualnya sebesar 81.7 mm. menggunakan prediktor kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 121.2 mm. penyimpangan terhadap data actual sebesar 77.6 mm. sedangkan menggunakan predictor suhu udara dan kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 116.2 mm. penyimpangan terhadap data observasi sebesar 72.6 mm. i) bulan september pada bulan september 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 62.4 mm. penyimpangan terhadap data total hujan aktualnya sebesar 116.2 mm. menggunakan prediktor kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 51.1 mm. penyimpangan terhadap data observasi didapat 27.5 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan sebesar 87.7 mm. penyimpangan terhadap data aktualnya sebesar 9.1 mm. j) bulan oktober pada bulan oktober 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 83.9 mm. penyimpangan terhadap data actual total hujan bulanan sebesar 218.0 mm. menggunakan prediktor kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 62.8 mm. penyimpangan terhadap data aktualnya sebesar 239.1 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 181.1 mm. penyimpangan terhadap data aktualnya sebesar 120.8 mm. k) bulan nopember pada bulan nopember 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 164.1 mm. penyimpangan terhadap data total hujan aktualnya sebesar 187.8 mm. menggunakan prediktor kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 164.8 mm. besar penyimpangan terhadap data actual sebesar 187.1 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara didapat nilai prediksi total hujan bulanan sebesar 253.7 mm. penyimpangan terhadap data observasi sebesar 98.2 mm. l) bulan desember pada bulan desember 2011 dengan menggunakan prediktor suhu udara diperoleh nilai prediksi total hujan bulanan sebesar 311.5 mm. penyimpangan terhadap data actual total hujan bulanan sebesar 43.0 mm. menggunakan prediktor kelembapan udara dihasilkan nilai prediksi total hujan bulanan sebesar 259.1 mm. besar penyimpangan terhadap data actual sebesar 9.4 mm. sedangkan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai prediksi total hujan bulanan sebesar 306.5 mm. penyimpangan terhadap data observasi sebesar 38.0 mm. dengan menggunakan nilai rmse rerata terlihat bahwa secara keseluruhan bahwa prediksi total hujan bulanan 2011 di daerah studi menghasilkan nilai rmse rerata sebesar 14.8 mm dengan menggunakan prediktor suhu udara. rmse rerata sebesar 14.5 mm menggunakan predictor kelembapan udara. sedangkan untuk prediksi total hujan bulanan dengan menggunakan prediktor suhu udara dan kelembapan udara diperoleh nilai rerata rmse sebesar 14.9 mm. dari ketiga prediktor ini nampak bahwa prediksi total hujan bulanan dengan menggunakan prediktor suhu dan kelembapan udara memiliki nilai penyimpangan yang relatif lebih besar daripada menggunakan prediktor suhu atau kelembapan udara. nilai penyimpangan luaran prediksi menunjukkan nilai penyimpangan yang cukup besar. sedangkan untuk prediksi total hujan bulanan menggunakan prediktor kelembapan udara mempunyai nilai penyimpangan yang relatif lebih rendah dari yang lainnya. hal tersebut di atas menunjukkan bahwa prediksi total hujan bulanan 2011 dengan menggunakan prediktor suhu dan kelembapan udara cukup baik dibandingkan dengan prakiraan menggunakan prediktor suhu maupun kelembapan udara. meskipun nilai rmse terhitung lebih besar yaitu 0.1 mm dari suhu udara dan 0.4 mm dari kelembapan, namun nilai korelasinya merupakan tertinggi di antara ketiga model regresi. grafik prediksi total hujan bulanan menggunakan prediktor suhu udara dengan grafik total hujan hasil observasi secara umum nampak pola yang terjadi adalah cenderung teratur, dan terlihat prediksi total hujan cenderung mengikuti total hujan aktualnya. namun, ada selisih curah hujan hasil prediksi dengan data observasi tidak konsisten. pada pemanfaatan suhu udara dan kelembapan udara dalam persamaan regresi … jurnal cauchy – issn: 2086-0382 9 model regresi dengan predictor suhu udara atau kelembapan, hasil prediksi hujan lebih rendah dari data observasi terjadi pada bulan februari, april, mei, juni, september, oktober dan november. sedangkan dengan menggunakan predictor suhu dan kelembapan udara secara bersamaan, hasil prediksi bulan september mempunyai nilai lebih tinggi disbanding data observasi. pada bulan februari, hasil prediksi dari ketiga model regresi menunjukkan kesamaan dimana cuah hujan meningkat setelah januari, padahal pada data observasi curah hujan bulan februari mengalami penurunan. penutup berdasarkan uraian yang telah dipaparkan oleh penulis dalam bab-bab tersebut di atas, maka dapat ditarik beberapa kesimpulan sebagai berikut: 1. prediksi total hujan bulanan tahun 2011 di stasiun meteorologi depati amir pangkalpinang dengan menggunakan prediktor suhu udara (t) dan menggunakan suhu udara dengan kelembapan udara (t dan rh) menunjukkan nilai prediksi yang cukup baik pada bulan september, sedangkan menggunakan kelembapan udara (rh) nilai prediksi cukup baik tampak pada bulan desember. 2. prediksi total hujan bulanan menggunakan dua prediktor suhu udara dan kelembapan udara sekaligus (t dan rh) menggunakan persamaan regresi linier berganda menghasilkan luaran yang relatif lebih baik dibandingkan dengan menggunakan satu prediktor. daftar pustaka [1] sandy, i.m. (1995). atlas republik indonesia. depok: pt indograf bakti & jurusan geografi fmipa-ui. [2] wirjohamidjojo, s. & swarinoto, y.s (2010). iklim kawasan indonesia. jakarta: badan meteorologi klimatologi dan geofisika, jakarta. [3] nieuwolt, s. (1977). tropical climatology: an introduction to the climates of low latitudes. toronto: john wiley & sons. [4] kartasapoetra, a.g. (2006). klimatologi: pengaruh iklim terhadap tanah dan tanaman. jakarta: bumi aksara. [5] wirjohamidjojo, s. & y.s. swarinoto. (2007). praktek meteorologi pertanian. jakarta: badan meteorologi dan geofisika. [6] tjasyono, b. (1999). klimatologi umum. bandung: itb. [7] soepangkat. (1994). pengantar meteorologi. jakarta: akademi meteorologi dan geofisika. [8] tanudidjaja, (1993). ilmu pengetahuan bumi dan antariksa. jakarta : penerbit departemen pendidikan dan kebudayaan. [9] wirjohamidjojo, s. (2006). kamus istilah meteorologi aeronautika. jakarta : penerbit badan meteorologi dan geofisika. [10] nazir, m. (2003). metode penelitian. jakarta: pt ghalia indonesia. [11] usman, h. & akbar, r.p.s. (2000). pengantar statistik. jakarta: penerbit bumi aksara. [12] sudjana. (1995). metoda statistika. bandung: penerbit tarsito. [13] wilks, d.s. (1995). statistical methods in the atmospheric science, san diego: academic press. [14] soetamto & maria, u.a. (2010). modul pelatihan peningkatan akurasi prakiraan musim. jakarta: bmkg. [15] trihendradi, c. (2005). step by step spss 13, analisis data statistik. yogyakarta: penerbit andi. [16] prihatini, djatmiko, h.t., & swarinoto, y.s. (2000). kaitan southern oscillation index dengan total hujan bulanan di pontianak. jurnal meterologi & geologi, 1(1). 1a sampul depan linieritas integral henstock-pettis pada ruang euclide rn hairur rahman jurusan matematika fakultas sains dan teknologi universitas islam negeri maulana malik ibrahim malang abstract in this paper we study henstock-pettis integral on the euclidean space ℜn. we discuss some properties linear integrable. keywords: henstock integral, euclidean space ℜn, properties linear integrable, henstock-pettis integral. pendahuluan pada tahun 1914, perron mengembangkan perluasan lain integral lebesgue dan menunjukkan bahwa integralnya mempunyai sifat bahwa setiap derivatifnya terintegral pada garis lurus. selanjutnya henstock dan kurzweil secara terpisah mengitlakkan integral riemann dengan mengubah konstanta positif δ menjadi fungsi positif δ dan ternyata integral yang disusun keduanya ekuivalen. oleh karena itu, integral yang mereka susun terkenal dengan nama henstock-kurzweil (lee dan vborn, 2000). dari kajian tentang integral henstock banyak sifat-sifat yang telah diungkapkan baik dalam ℜ maupun ruang ℜn. menurut penelitian, masalah mengenai sifat-sifat pada integral henstock kemungkinan dapat dikembangkan menjadi masalah yang lebih luas dalam integral henstock-pettis, khususnya sejauh mana sifatsifat integral henstock dari fungsi bernilai real dapat dikembangkan ke dalam integral henstockpettis pada ruang euclide ℜn. dari kajian tentang integral henstock-kurzweil banyak sifat-sifat yang telah diungkapkan baik dalam ℜ maupun ruang ℜn. menurut penelitian sebelumnya, masalah mengenai sifat-sifat pada integral henstock-kurzweil yang telah dilakukan oleh guoju dan tianqing, 2001, guoju dan swabik, 2001, indrati, 2002, serta dharmawidjaya, 2003, kemungkinan dapat dikembangkan menjadi masalah yang lebih luas dalam integral henstockkurzweil-pettis, khususnya sejauh mana sifatsifat integral henstock dari fungsi bernilai real dapat dikembangkan ke dalam integral henstockkurzweil-pettis pada ruang euclide ℜn. definisi 1.1. diberikan fungsi volume α pada ℜn, sel � � ℜ�dan e ruang banach. fungsi f : e → x dikatakan terintegral-α henstock pada e terhadap α, ditulis singkat f ∈ ℋ(e, α, x) jika terdapat vektor a ∈ x sehingga untuk setiap bilangan � > 0 terdapat fungsi positif pada e dan untuk setiap partisi perron -fine = �� �, ����, � �, ����, � , � � , ��� �� pada e berlaku �� � � � � ������ �� � �� � � � � ����� ��� � � � ��� � � �. pembahasan akan dibahas sifat-sifat lanjut dari integral henstock-pettis pada ruang euclide ℜ�, yakni memuat pembahasan yang berkaitan dengan sifat-sifat linearitas fungsi terintegral henstockpettis pada ruang euclide ℜ� . definisi 2.1. diberikan x ruang banach dan !" ruang dualnya, volume α pada ℜn, dan se � � ℜ�. fungsi f : e → x dikatakan terintegral-α henstockpettis pada e, ditulis singkat dengan f ∈ ℋ#(e, α), jika untuk setiap � " ∈ !" dan sel � � $, fungsi � "� terintegral-� henstock pada a dan terdapat vektor ��%,&,'� ∈ ! sehingga � "(��%,&,'�) � �ℋ� * � "� +�& . selanjutnya vektor ��%,&,'� di atas disebut nilai integral henstock-pettis fungsi f pada a dan ditulis ��%,&,'� � �ℋ#� * � +�,& khususnya ��%,,,'� � �ℋ#� * � +�,, jadi � " -�ℋ#� * � +�,& . � �ℋ� * � "� +�.& teorema 2.2. jika f ∈ hp(e, α), maka vektor ��%,&,'� ∈ ! yang dimaksud di dalam definisi 2.1 adalah tunggal bukti: jika terdapat vektor �� �%,&,'� dan �� �%,&,'� seperti pada definisi 2.1 maka � " /�� �%,&,'�0 � �ℋ� * � "� +�& hairur rahman 66 volume 1 no. 2 mei 2010 dan � " /�� �%,&,'�0 � �ℋ� * � "� +�& oleh karena itu, � " /�� �%,&,'�0 � � " /�� �%,&,'�0 � �ℋ� * � "� +�& � �ℋ� * � "� +�& atau � " /�� �%,&,'�0 � � " /�� �%,&,'�0 untuk setiap x ∗∈x∗ jadi �� �%,&,'� � �� �%,&,'� ◙ contoh 2.3 jika � "���� � � 1 untuk setiap �� di dalam sel e ⊂ ℜn, maka f ∈ hp(e,α) dan � "(��%,&,'�) � ����1. bukti : diberikan sebarang bilangan ε > 0 dan � " ∈ !" maka dapat ditemukan fungsi positif δ pada e sehingga jika a ⊂ e sebarang sel dan d � �� , �� �� sebarang partisi perron δ-fine pada sel a, maka |� d � ∑ �"���4��� � � �"(���,�,��)| � |� d � ∑ 1�� � � ����1| � | ����1 � ����1| � 0 � � jadi � "� terintegral henstock pada e dan untuk sel a ⊂ e di atas, maka terdapat vektor ��%,&,'� ∈ ! sehingga � "(��%,&,'�) � �ℋ� * � "� +�& � �ℋ� * 1 +�& � ����1 jadi f terintegral henstock-pettis pada e. teorema 2.4 diberikan x ruang banach dan !" dualnya, volume α pada ℜn dan sel e ⊂ ℜn. (i) jika f,g ∈ ℋ#(e,α) maka f + g ∈ ℋ#(e,α) dan ��%67,&,'� � ��%,&,'� 8 ��7,&,'� (ii) jika f ∈ ℋ#(e,α) dan c∈ ℜ maka cf ∈ ℋ#(e,α) dan ��9%,&,'� � 1. ��%,&,'� bukti : (i). fungsi f ∈ ℋ#(e,α), maka untuk setiap � " ∈ !" fungsi � "� terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��%,&,'� ∈ ! sehingga � "(��%,&,'�) � �ℋ� * � "� +�& dan fungsi g ∈ ℋ#(e,α), maka untuk setiap � " ∈ !" fungsi � "g terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��7,&,'� ∈ ! sehingga � "(��7,&,'�) � �ℋ� * � ": +�& karena a ⊂ e sebarang sel, maka berlaku untuk fungsi f dan fungsi g. jadi untuk setiap � " ∈ !" fungsi � "(f+g) terintegral henstock pada e. oleh karena itu terdapat vektor ��%67,&,'� ∈ ! sehingga untuk sel a ⊂ e di atas berlaku � "(��%67,&,'�) � �ℋ� * � "�� 8 :� +�& � �ℋ� * � "� +�& 8 �ℋ� * � ": +�& � � "(��%,&,'�) 8 � "(��7,&,'�) jadi f + g ∈ ℋ#(e,α) dan ��%67,&,'� � ��%,&,'� 8 ��7,&,'� (ii). fungsi f ∈ ℋ#(e,α), maka untuk setiap � " ∈ !" fungsi � "f terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��%,&,'� ∈ ! sehingga � "(��%,&,'�) � �ℋ� * � "� +�& oleh karena itu, untuk setiap � " ∈ !" dan c ∈ ℜ maka � "cf terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��9%,&,'� ∈ !. sehingga � "(��9%,&,'�) � �ℋ� * � "1� +�& � 1�ℋ� * � "� +�& � 1� "(��%,&,'�) jadi cf ∈ ℋ#(e,α) dan ��9%,&,'� � 1. ��%,&,'� ◙ teorema 2.5 diberikan x ruang banach dan !" dualnya, volume α dan β di dalamnya ℜn dan sel e⊂ ℜn. (i). jika f ∈ ℋ#(e,α) dan f ∈ ℋ#(e, β), maka f ∈ ℋ#(e, α +β) dan ��%,&,'6;� � ��%,&,'� 8 ��%,&,;� (ii). jika f ∈ hp (e,α) dan e∈ ℜ dengan e> 0 maka f ∈ hp (e,eα) dan ��%,&,<'� � =. ��%,&,'� bukti : (i). fungsi f ∈ ℋ#(e,α), maka untuk setiap � " ∈ !" fungsi � "f terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��%,&,'� ∈ ! sehingga linieritas integral henstock-pettis pada ruang euclide rn volume 1 no. 2 mei 2010 67 � "(��%,&,'�) � �ℋ� * � "� +�& dan fungsi f ∈ ℋ#(e,β), maka untuk setiap � " ∈ !" fungsi � "f terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��%,&,;� ∈ ! sehingga � "(��%,&,;�) � �ℋ� * � "� +>& karena a ⊂ e sebarang sel, maka berlaku untuk fungsi f yang terintegral henstockpettis pada e terhadap α dan β. jadi untuk setiap � " ∈ !" fungsi � "f terintegral henstock pada e. oleh karena itu terdapat vektor ��%,&,'6;� ∈ ! sehingga sel a ⊂ e di atas berlaku ��%,&,'6;� � �ℋ� * � "� +�� 8 >�& � �ℋ� * � "� +�& 8 �ℋ� * � "� +>& � ��%,&,'� 8 ��%,&,;� jadi f ∈ ℋ#(e,α+β), dan ��%,&,'6;� � ��%,&,'� 8 ��%,&,;� (ii). fungsi f ∈ ℋ#(e,α), maka untuk setiap � " ∈ !" fungsi � "f terintegral henstock pada e dan untuk setiap sel a ⊂ e terdapat vektor ��%,&,'� ∈ ! sehingga � "(��%,&,'�) � �ℋ� * � "� +�& oleh karena itu, jika e ∈ ℜ dengan e > 0 maka untuk sel a ⊂ e di atas terdapat vektor ��%,&,<'� ∈ ! sehingga � "(��%,&,<'�) � �ℋ� * � "� +=�& � =�ℋ� * � "� +�& � =� "(��%,&,'�) jadi f ∈ ℋ#(e,eα) dan ��%,&,<'� � =. ��%,&,'� ◙ teorema 2.6 jika e1, e2 ⊂ ℜn sel-sel tak saling tumpang tindih, f ∈ ℋ#(e1,α) dan, f ∈ ℋ#(e2,α) maka, f ∈ ℋ#(e1 ∪ e2,α) dan jika a⊂ e1 sel dan b⊂ e2 sel maka ��%,&?@,'� � ��%,&,'� 8 ��%,@,'� bukti : diberikan sebarang bilangan ε > 0 dan � " ∈ !", maka terdapat fungsi positif δ1 pada e1 dan δ2 pada e2 sehingga untuk a ⊂ e1 sebarang sel dan b ⊂ e2 sebarang sel dan d1 � �� , �� �� dan d2 � �� , �� �� berturut-turut partisi perron δ-fine pada a dan b berlaku |� d1 � ∑ �"���4��� � � �"(���,�,��)| � �2 dan |� d2 � ∑ �"���4��� � � �"(���,�,��)| � �2 � |� d � ∑ 1�� � � ����1| � | ����1 � ����1| � 0 � � diambil fungsi δ : ��� � � b ���� � , cdef �� ∈ � +fg �� h i ���� � , cdef �� ∈ i +fg �� h �min( ���� � , ���� �), cdef �� ∈ � m i n diperoleh δ fungsi positif pada e1 ∪ e2 dan untuk sel-sel a ⊂ e1, dan b ⊂ e2 di atas dan d 1 ∪ d2 partisi perron δ-fine pada a∪b. oleh karena itu diperoleh |�d1 ? d2 � ∑ �"���4��� � � /� "(��%,&,'�) 8 � "(��%,&,;�)0 | � |��d1 �∑ � "���� ��� � 8 �d2�∑ � "���� ��� �� � /� "(��%,&,'�) 8 � "(��%,&,;�)0 | � |��d1 �∑ � "���� ��� � 8 �d2�∑ � "���� ��� �� � /� "(��%,&,'�) 8 � "(��%,&,;�)0 | o |��d1 �∑ � "���� ��� � 8 � "(��%,&,'�)| 8 |��d2�∑ � "���� ��� � 8 � "(��%,&,;�)| � �2 8 �2 � � jadi � "f terintegral henstock pada e1 ∪ e2 dan untuk sel-sel a ⊂ e1, b⊂ e2 di atas terdapat vektor (��%,&?@,'�) ∈ ! sehingga � "(��%,&?@,'�) � �ℋ� * � "� +�&?@ � �ℋ� * � "� +�& 8 �ℋ� * � "� +�@ � � "(��%,&,'�) 8 � "(��%,@,'�) jadi f ∈ ℋ#(e1∪e2, α) dan ��%,&?@,'� � ��%,&,'� 8 ��%,@,'� ◙ selanjutnya teorema di bawah ini merupakan akibat teorema 2.6. teorema 2.7 diberikan sel e ⊂ ℜn. jika a⊂ e sel dan d = {(d1, d2, …, dm)} divisi pada a serta f ∈ ℋ#(di,α) untuk setiap i, maka f∈ ℋ#(e,α) dan ��%,&,'� � � ��%,pq,'� r ��� bukti : diberikan sebarang bilangan ε > 0 dan � " ∈ !", maka terdapat fungsi positif δ dan di, sehingga jika ��is , ��s �: e � 1,2, … , w� partisi perron δi-fine pada di berlaku hairur rahman 68 volume 1 no. 2 mei 2010 x� "(��%,pq,'�) � � � "����s ���is � y s�� x � �z untuk setiap i = 1,2,..,m. dibentuk fungsi positif δ : ��� � � [ � ��� � , �� ∈ dg\� � �, d � 1,2, … , z min� � ��� �� , �� ∈ ]� � � ^\e __`w �� ∈ � n diperoleh δ fungsi positif pada e dan ��� � o � ��� � untuk setiap ��∈di, i = 1,2,…,m. diambil sebarang partisi perron δ-fine pi pada di untuk setiap i = 1,2,…,m. menurut teorema 2.6.2., a � b a�r��� merupakan partisi perron δ-fine pada e. akibatnya, untuk a ⊂ e sebarang sel p merupakan partisi perron δ-fine pada a. oleh karena itu pi merupakan partisi perron δ-fine di untuk setiap i =1,2,…,m, maka diperoleh x� � "(��%,pq,'�) r ��� � # � � "��c4���a�r ��� x o x� � "(��%,pq,'�) r ��� � � #� � "��c4���a� r ��� x ( )( )( ) ( ) ( )∑ ∑∑ = = ∗ −≤ m i m i idf pyfxxx 1 * 1 ,, 1 αα p ( )( )( ) ( ) ( )∑ ∑ = ∗ −≤ m i idf pyfxxx 1 * ,, 1 αα p ε ε =< ∑ = m i m1 jadi untuk setiap � " ∈ !" fungsi � "� terintegral henstock pada e dan untuk setiap sel a ⊂ e di atas terdapat vektor ( ) xx af ∈α,, sehingga jadi f ∈hp (e,α) dan ( )( ) ( )∑ = ∗ = m i dfaf i xxx 1 ,,,, αα ◙ teorema 2.8. jika f=0 (fungsi nol) h,d pada e maka f∈hp(e,α) dan jika a⊂e sel maka ( ) θα =,, afx bukti : karena f = 0 h.d pada e, maka terdapat himpunan a⊂ e dengan µα(a) = 0 sehingga jika ∗∗ ∈ xx berakibat ( )      ∈≠ −∈= ∗ ax aex xfx jika,0 jika,0 dibentuk a = ∪ ∞ =1i ia dimana ai = ( ){ } aiixfiax ⊂=≤≤−∈ ,....2,1,1: dengan µα (ai) = 0 diberikan bilangan ε > 0. untuk setiap i terdapat himpunan terbuka gi dengan ukuran kurang dari ii2 ε sehingga gi ⊃ai. didefinisikan fungsi positif δ pada e sedemikian sehingga n( x ,δ( x )) ⊂gi, untuk x ∈ ai, i=1,2,…, dan sebarang fungsi positif untuk x yang lain . oleh karena itu untuk sebarang a ⊂ e sel dan sebarang partisi perron δ-fine p={(b, x )} pada a berlaku ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε ε α αα α = < = += − ∑ ∑ ∑∑ ∑ ∞ = ∗ ∈ ∗ ∉ ∗ ∈ ∗ ∗ i ix axfx axfxaxfx axfx i i ax axax i ii .2 0 1 p untuk setiap ∗∗ ∈ xx dan 1≤∗x . hal ini menunjukkan bahwa untuk setiap ∗∗ ∈ xx fungsi fx ∗ terintegral henstock pada e. dan untuk sel a ⊂ e di atas terdapat vektor ( ) xx af ∈α,, sehingga ( )( ) ( ) ( ) ( )∫ ∫ === =∗ 000 * ,, ad fdxxx a af αα αα h h jadi f ∈hp (e,α) dan ( ) θα =,, afx berikut ini akibat teorema 3.1.8 di atas. teorema 3.1.9. jika f ∈ hp(e,α) dan g = f h.d pada e maka g ∈hp(e,α) dan jika a⊂ e sel maka ( ) ( )αα ,,,, afag xx = ( )( ) ( ) ( )( )∑ ∑ ∫ = ∗ = ∗ = = m i df m i d af i i xx fdxxx 1 ,, 1 * ,, α α αh linieritas integral henstock-pettis pada ruang euclide rn volume 1 no. 2 mei 2010 69 bukti : ambil fungsi h = g f. diperoleh h = 0 (fungsi nol) h.d pada e. menurut teorema 3.1.8, h ∈hp (e,α) dan ( ) θα =,, ahx untuk setiap sel a⊂ e. karena h ∈hp(e,α), f ∈hp(e,α) dan g = h + f, maka berdasarkan teorema 3.1.4 (i) diperoleh, g ∈hp (e,α) dan untuk setiap ∗∗ ∈ xx ( )( ) ( ) ( )∫ += ∗∗ a ag dfhxxx αα h,, ( ) ( )∫∫ ∗∗ += aa fdxhdx αα hh ( )∫ ∗+= a fdx αh0 ( )∫ ∗= a fdx αh ( )( )α∗= ,a,fxx jadi ( ) ( )αα ,,,, afag xx = ◙ teorema 3.1.10. jika f ∈ hp (e,α) dan ∗x f ≥ 0 h.d pada e dan jika a⊂ e sel maka ( )( ) 0,, ≥∗ αafxx bukti : karena ∗x f ≥ 0 h.d pada e maka tidak mengurangi arti jika dianggap ∗x f ( x ) ≥ 0 untuk setiap x ∈ e dan ∗∗ ∈ xx . oleh karena itu, karena f ∈ hp(e,α), jadi untuk setiap ∗∗ ∈ xx fungsi ∗x f terintegral henstock pada e dan untuk setiap sel a⊂ e terdapat vektor ( ) xx af ∈α,, sehingga berakibat ( )( ) ( ) 0,, ≥= ∫ ∗∗ a af fdxxx αα h ◙ di bawah ini merupakan akibat teorema 3.1.10 teorema 2.11. jika f ∈ hp (e,α) dan g ∈ hp (e,α) dan fx ∗ ≤ gx ∗ h.d pada e dan jika a ⊂ e sel maka ( )( ) ( )( )αα ,,,, agaf xxxx ∗∗ ≤ bukti : karena fx ∗ ≤ gx ∗ h.d pada e maka ∗x g – ∗x f ≥ 0 h.d. pada e. oleh karena itu, berdasarkan teorema 3.1.10 untuk setiap ∗∗ ∈ xx fungsi ∗x (f-g) terintegral henstock pada e dan untuk setiap a ⊂ e sel terdapat vektor ( ) xx afg ∈− α,, sehingga ( )( ) ( ) ( ) 0,, ≥−= ∫ ∗−∗ αα dfgxxx a afg h ( ) ( ) 0≥−= ∫∫ ∗∗ αα dfxdgx aa hh ( )( ) ( )( ) 0,,,, ≥−= ∗∗ αα afag xxxx jadi ( )( ) ( )( )αα ,,,, agaf xxxx ∗∗ ≤ ◙ daftar pustaka [1]. dharmawidjaya, s., 2003, on the bounded interval functions, proceedings of the international conference 2003 on mathematics and its application, seamsgadjah mada university, universitas gadjah mada, indonesia. [2]. indrati, ch. r., 2002, integral henstockkurzweil pada ruang euclide ℜn berdimensi-n, disertasi, universitas gadjah mada, indonesia. [3]. gordon, r.a., 1994, the integral of lebesgue, denjoy, perron and henstock, american mathematical society, usa. [4]. guoju, ye and tianqing, 2001.on henstockdunford and henstock-pettis integral, ijmms, 25:7, hindawi publishing corp. pp 467-478. [5]. guoju, ye, and ,swabik. s, 2001. the macshane and the weak. macshane intergal of banach space value function defined on ℜn. mathematical notes, miscole, vol, 2., no, 2., pp127-136. [6]. guoju, ye, and swabik. s, 2004, topics in banach space integration, manuscript in preparation. [7]. guoju, ye, lee, p.y.,wu, congxin, 1999, convergence theorem of the denjoybochner, denjoy-pettis and denjoy-dunford integral, southeast asian bulletin of mathematics, springer-verlag, vol 23; 135143. [8]. rahman, hairur, 2005, integral henstockpettis pada ruang euclide ℜn , tesis, universitas gadjah mada, indonesia. hairur rahman 70 volume 1 no. 2 mei 2010 [9]. rahman, hairur, 2005, kekonvergenan integral henstock-pettis pada ruang euclide ℜn, universitas gadjah mada, indonesia. [10]. lee, p.y., 1989, lanzhou lectures on henstock integration, word scientific, singapore. [11]. lee, p.y. dan vborn, r., 2000, integral: an easy approach after kurzweil and henstock, cambridge university press. [12]. pfeffer,w.f.,1993, the riemann approach to integration, cambridge university press, new york, usa. [13]. royden, h.l., 1989, real analysis, third edition, macmillan publishing company, new york, usa. [14]. swabik. s. and ye, guoju.,1991.the macshne and the pettis intergal of banach space value function defined on ℜn, chzech, math journal. microsoft word 1 sampul depan.doc 7  pendeteksian outlier dengan metode regresi ridge sri harini jurusan matematika, fakultas sains dan teknologi universitas islam negeri maulana malik ibrahim malang e-mail: sriharini21@yahoo.co.id abstrak dalam analisis regresi linier berganda adanya satu atau lebih pengamatan pencilan (outlier) akan menimbulkan dilema bagi para peneliti. keputusan untuk menghilangkan pencilan tersebut harus dilandasi alasan yang kuat, karena kadang-kadang pencilan dapat memberikan informasi penting yang diperlukan. masalah outlier ini dapat diatasi dengan berbagai metode, diantaranya metode regresi ridge (ridge regression). untuk mengetahui kekekaran regresi ridge perlu melihat nilai-nilai r2, press, serta leverage (hii), untuk metode regresi ridge dengan berbagai nilai tetapan bias k yang dipilih. kata kunci: outlier, press, regresi ridge, r2, leverage (hii) 1. pendahuluan pada analisis regresi berganda sering ditemui satu atau lebih pengamatan tidak sesuai dengan model yang digunakan pada sebagian besar pengamatan lainnya. hal ini dapat terjadi karena kesalahan dalam pencatatan pengamatan-pengamatan tersebut, kesalahan alat ukur, atau karena ketidakcocokan model yang digunakan. pengamatan semacam itu disebut pencilan (outlier). pencilan bisa dihilangkan bila ada penjelasan tentang kasus pencilan yang menunjukkan situasi khusus yang tercakup dalam model. pencilan dalam data regresi berganda dapat berpengaruh pada hasil analisis statistik. pengamatan pencilan mungkin menghasilkan residual yang besar dan sering berpengaruh terhadap fungsi regresi yang dihasilkannya. untuk itu perlu dilakukan identifikasi terhadap pencilan ini guna melihat kesalahan sampel observasi. (walker dan birch, 1988). 2. kajian pustaka pendeteksian pencilan (outlier) seringkali model regresi dibentuk dari data yang banyak mengandung kekurangan, diantaranya adalah adanya pencilan yaitu pengamatan dengan residual yang besar. pencilan sering menyebabkan kesalahan dalam pemilihan model, dan biasanya dihilangkan. kenyataannya, beberapa pencilan dapat memberi informasi yang berarti, misalnya pencilan timbul dari kombinasi keadaan yang tidak biasa yang mungkin penting dan perlu diselidiki lebih lanjut. oleh karena itu adanya pencilan dalam data perlu diselidiki secara seksama, barangkali dapat diketahui ada alasan dibalik keganjilan itu. pencilan dapat disebabkan oleh kesalahan dalam data atau status fisik yang ganjil dari obyek yang dianalisis. kesalahan dalam data berupa gangguan, penyimpangan instrumen, kesalahan operator, atau kesalahan pencetakan (retnaningsih, 2001). pendeteksian pencilan terhadap nilai-nilai variabel x, dapat menggunakan matrik topi yang didefinisikan sebagai h=x(x'x)-1x'. unsur ke-i pada diagonal utama matrik topi disebut leverage (hii). unsur hii dapat diperoleh dari hii=xi(x'x)-1xi'. nilai diagonal hii terletak antara 0 dan 1 dan jumlahnya sama dengan p, yaitu banyak parameter regresi di dalam fungsi regresi termasuk suku intersep (neter, wasserman, dan kutner, 1990). sri harini  8 volume 1 no. 1 november 2009 nilai leverage yang besar menunjukkan pencilan dari nilai-nilai variabel x untuk pengamatan ke-i. hal ini disebabkan, bahwa hii adalah ukuran jarak antara nilai x untuk pengamatan ke-i dengan rata-rata nilai x untuk semua pengamatan. sehingga, nilai hii yang besar menunjukkan pengamatan ke-i berada jauh dari pusat semua pengamatan variabel x. suatu nilai hii dianggap besar apabila nilainya lebih besar dari 2p/n dan dapat berpotensi sebagai pengamatan yang berpengaruh. regresi ridge (ridge regression) regresi ridge merupakan salah satu metode yang dianjurkan untuk memper-baiki masalah multikolinearitas dengan cara memodifikasi metode kuadrat terkecil, sehingga dihasilkan penduga koefisien regresi lain yang bias (neter, et al., 1990). modifikasi metode kuadrat terkecil tersebut dilakukan dengan cara menambah tetapan bias, k, yang relatif kecil pada diagonal matriks x'x, sehingga penduga koefisien regresi dipengaruhi oleh besarnya tetapan bias, k. pada umumnya nilai k berkisar antara 0 dan 1. untuk menentukan penduga ridge, dimulai dari asumsi model linier secara umum, yaitu : y = x β + ε …………………………………… (2.1) dimana : y adalah vektor pengamatan pada variabel respon yang berukuran (nx1) x adalah matrik yang berukuran nx(p+1) dari p variabel bebas β adalah vektor berukuran (p+1)x1 dari koefisien regresi ε adalah vektor berukuran (nx1) dari error dalam regresi ridge variabel bebas x dan variabel tak bebas y ditransformasikan dalam bentuk variabel baku z dan y*, dimana transformasi variabel bebas dan tak bebas ke bentuk variabel baku diperoleh dari z= xs xx − dan y*= ys yy − . selanjutnya z'z= ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ − xs xx . ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − xs xx dan z'y= ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − xs xx ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ys yy . sementara itu rumus dari korelasi rxx= ( ) xx ss xxxx )( −− . sehingga persamaan normal kuadrat terkecil (x'x)b=x'y akan berbentuk (rxx)b=rxy, dengan rxx adalah matrik korelasi variabel x dan rxy adalah vektor korelasi antara y dan masing-masing variabel x. akibat dari transformasi matrik x ke z dan vektor y ke y*, maka akan menjadikan persamaan normal regresi ridge berbentuk : (rxx+ki) *b̂ =rxy. penduga koefisien regresi ridge menjadi : *b̂ =(rxx+ k i)-1 rxy ………………………………………………………………… (2.2) dimana : *b̂ adalah vektor koefisien regresi ridge rxx adalah matrik korelasi variabel x yang berukuran pxp rxy adalah vektor korelasi antara variabel x dan y berukuran px1 k adalah tetapan bias i adalah matrik identitas berukuran pxp. masalah yang dihadapi dalam regresi ridge adalah penentuan nilai dari k. prosedur yang cukup baik untuk menentukan nilai k ini adalah dengan menggunakan nilai statistik cp-mallows, yaitu ck. statistik cp-mallows adalah suatu kriteria yang berkaitan dengan rata-rata kuadrat error (mean square error) dari nilai kesesuaian model. nilai k yang terpilih adalah yang meminimumkan nilai ck (myers, 1990). nilai ck dapat dirumuskan sebagai berikut : 2σ̂ k k jkr c = n + 2 + 2 tr[hk] ……………………………… (2.3) dimana : jkrk adalah jumlah kuadrat residual dari regresi ridge n adalah banyak pengamatan pendeteksian outlier dengan metode regresi ridge   volume 1 no. 1 november 2009 9 hk = [z(z'z+ki)-1z'] dengan i adalah matrik identitas tr [hk] adalah teras dari matrik hk 2σ̂ adalah penduga varian metode kuadrat terkecil acuan lain yang digunakan untuk memilih nilai k adalah dengan melihat nilai vif (myers, 1990). nilai vif untuk koefisien regresi ridge *b̂ didefinisikan sebagai diagonal dari matrik (rxx+ki)-1rxx(rxx+ki)-1 . rumusan ini didapat dengan serangkaian proses sebagai berikut : jika di metode kuadrat terkecil diketahui nilai koefisien penduga b̂ dan varian( b̂ ): b̂ = (x'x)-1 x'y dengan y=x b̂ varian( b̂ ) = 2σ (x'x)-1 dalam regresi ridge harga *b̂ dan varian( *b̂ ) diketahui sebagai : *b̂ = (x'x+ki)-1 x'y = (x'x+ki)-1 x' xb varian( *b̂ ) = 2σ (x'x+ki)-1 (x'x) (x'x)-1 (x'x) (x'x+ki)-1 = 2σ (x'x+ki)-1 (x'x) (x'x+ki)-1 sehingga vif merupakan diagonal matrik (x'x+ki)-1 (x'x) (x'x+ki)-1. bila x dibakukan, maka vif dari regresi ridge adalah diagonal dari matrik (rxx+ki)-1rxx(rxx+k i)-1. leverage dalam regresi ridge ketika teknik bias digunakan pada regresi ridge, untuk mengurangi efek dari multikolinearitas, maka rumus pencilan di dalam data tersebut dapat dimodifikasi. seperti halnya di dalam metode regresi kuadrat terkecil, maka pencilan dalam regresi ridge dapat diukur dengan nilai leverage (hii). untuk itu nilai hii pada regresi kuadrat terkecil berubah sebagai fungsi dari k, guna mendapatkan nilai hii pada regresi ridge (retnaningsih,2001). dengan memakai penduga (2.2), maka nilai-nilai vektor dugaan y adalah : *ˆiy = z b* = z(z'z+ki*)-1z'y oleh karena itu, matrik h untuk regresi ridge menjadi h*=z(z'z+ki*)-1z', dan unsur ke-i pada diagonal utama matrik h* adalah hii* = zi (z'z+ki*)-1 zi'. matrik h* berperan sama seperti matrik h pada metode kuadrat terkecil. sehingga, nilai dugaan ke-i dapat ditulis dalam bentuk elemen h* sebagai berikut (walker dan birch, 1988): *ˆiy = ∑ = n j jij yh 1 * unsur diagonal matrik topi ridge hii* dapat diinterpretasikan sama sebagai leverage pada diagonal matrik topi pada metode kuadrat terkecil. lichtenstein dan velleman (1983) dalam walker dan birch (1988) mengungkapkan beberapa fakta penting dari sifat unsur diagonal matrik h*. pertama, untuk k>0, maka nilai hii* < hii dengan i = 1, 2, …, n. dengan demikian, untuk setiap pengamatan, nilai leverage regresi ridge lebih kecil dari leverage regresi kuadrat terkecil. kedua, leverage menurun secara monoton sejalan dengan kenaikan k. ketiga, laju penurunan leverage tergantung pada posisi baris tertentu dari z sepanjang sumbu utama. artinya, leverage dari baris yang terletak di sumbu utama yang berpadanan dengan akar karakteristik besar akan berkurang lebih sedikit dari pada leverage dari baris yang terletak di sumbu utama yang berpadanan dengan akar karakteristik kecil. sri harini  10 volume 1 no. 1 november 2009 3. pembahasan penurunan rumus dari regresi ridge jika *β̂ adalah penduga dari vektor β , maka jumlah kuadrat residual dapat ditulis (hoerl dan kennard, 1970) sebagai berikut : φ = (y-x *β̂ )'(y-x *β̂ ) = (y-x β̂ )'(y-x β̂ )+( β̂ *β̂ )'x'x( β̂ *β̂ ) = minφ + φ ( *β̂ ) dimana β̂ adalah penduga kuadrat terkecil dari β . untuk φ tetap, maka dipilih nilai *β̂ dan dibuat meminimumkan *β̂ ' *β̂ dengan kendala ( β̂ *β̂ )'x'x( β̂ *β̂ )= 0φ , sehingga masalah lagrange (hoerl dan kennard, 1970) menjadi : =f *β̂ ' *β̂ + k 1 [( *β̂ β̂ )'x'x( *β̂ β̂ )0φ ] *β̂∂ ∂f = 2 *β̂ + k 1 [2(x'x) *β̂ -2(x'x) β̂ ] = 0 *β̂ [1+ k 1 (x'x)] k 1 (x'x) β̂ ] *β̂ = [ki+ (x'x)]-1 (x'x) β̂ jadi penduga regresi ridge adalah : *β̂ = [(x'x + ki)]-1 x'y adapun sifat-sifat regresi ridge sebagai berikut (marquardt, 1970): 1. penduga *β̂ adalah transformasi linier dari β̂ , dan transformasi hanya tergantung pada x dan k. *β̂ = [ki+ (x'x)]-1 x'y, tetapi x'y = (x'x) β̂ maka, *β̂ = [(x'x+ki)]-1 (x'x) β̂ = zk β̂ e( *β̂ ) = zk β̂ sehingga *β̂ adalah penduga bias dari β̂ 2. varian *β̂ adalah : v( *β̂ ) = 2σ [ki+ (x'x)]-1 (x'x) [ki+ (x'x)]-1 v( *β̂ ) = var [ki+ (x'x)]-1 x'y = [ki+ (x'x)]-1 x' 2σ ix[ki+ (x'x)]-1 v( *β̂ ) = 2σ [ki+ (x'x)]-1 (x'x) [ki+ (x'x)]-1 3. mse (mean square error) dari *β̂ : e(l2) = tr[v( *β̂ )]+ β̂ '(zk-i)'(zk-i) β̂ = varian + (bias)2 e(l2) = e[( *β̂ β̂ )'( *β̂ β̂ )] = 2σ ∑ = + p j j i k1 2)(λ λ + k2 β '(x'x+ki)-2 β = v( *β̂ )+k2 β '(x'x+ki)-2 β 4. jika k≥ 0, dan misal *β̂ memenuhi persamaan *β̂ = [(x'x + ki)]-1 x'y, maka *β̂ meminimumkan jumlah kuadrat residual : *)ˆ(*)'ˆ(*)ˆ( βββ xyxy −−=φ . 5. jika *β̂ adalah solusi dari [ki+ (x'x)] *β̂ = x'y untuk nilai k yang diberikan, maka *β̂ adalah fungsi monoton turun kontinyu dari k, sedemikian hingga pada saat k ∞→ , *β̂ → 0. pendeteksian outlier dengan metode regresi ridge   volume 1 no. 1 november 2009 11 6. jika ββ ' terbatas, maka ada tetapan k>0 sedemikian hingga mse dari *β̂ kurang dari mse penduga kuadrat terkecil 7. dalam persamaan (x'x+ki) *β̂ = x'y, g = x'y adalah vektor gradien dari )(βφ . misal kγ adalah sudut antara *β̂ dan g, maka kγ adalah fungsi monoton turun kontinyu dari k, sedemikian hingga k .0, →∞→ kγ pemilihan nilai tetapan bias k merupakan sesuatu yang tidak dipisahkan dalam regresi ridge. untuk itu perlu dirunut dari mana asal nilai tetapan bias k tersebut. untuk melihat *β̂ dari sudut pandang mse, maka hoerl dan kennard (1970) mengekspresikan hal tersebut ke dalam bentuk e(l2), dimana : e(l2) = 2σ ∑ = + p j j i k1 2)(λ λ + k2 β '(x'x+ki)-2 β = )()( 2)1( kk γγ + elemen kedua, yaitu )(2 kγ , adalah jarak kuadrat dari z β ke β . elemen )(2 kγ akan bernilai nol, jika k=0, sehingga )(2 kγ dapat dipandang sebagai bias kuadrat. elemen pertama, yaitu )(1 kγ , merupakan total varian dari dugaan parameter. total varian dari semua *β̂ j adalah jumlah diagonal elemen 2σ z(x'x)-1 z'. total varian turun seiring dengan kenaikan k, sementara bias kuadrat naik seiring dengan kenaikan k. total varian )(1 kγ adalah kontinyu, merupakan fungsi monoton turun dari k. 1.)(2 321 −+−= ∑ kdk d jj λλσ γ = -2 ∑ + 3 2 )( kj j λ λ σ bias kuadrat )(2 kγ adalah kontinyu, merupakan fungsi monoton naik dari k. = dk d 2γ ∑ + +−+ 4 2222 )( ))(2()(2 k kkkk j jjjj λ λαλα = ∑ + −+ 3 22222 )( 222 k kkk j jjjj λ ααλα = ∑ + 3 2 )( 2 k k j jj λ λα ∑ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − + =+ p j j j j jj kk k dk d dk d 1 3 2 3 2 21 )()( 2 λ λ σ λ αλγγ = 0 022 =− jjjk λσαλ 2 2 j k α σ = sedangkan untuk rumusan ck yang digunakan sebagai alternatif pemilihan nilai tetapan bias k dapat diturunkan sebagai berikut (myers, 1990): ∑ ∑ = = + n i i n i i ybiasyvar 1 2* 1 * ]ˆ[ˆ 2 2 * 1 ][ ˆ k i n i atr yvar = ∑ = σ sri harini  12 volume 1 no. 1 november 2009 222* 1 )()ˆ( kki n i aitrjkrybias −−=∑ = σ * 2 2 2 1 2 2 2 2 ˆ( ) ( ) ( ) σ σ σ σ = − −= = − − ∑ n i i k k k k biasy jkr tr i a jkr tr i a jadi penduga dari ∑ ∑ = = + n i i n i i ybiasyvar 1 2* 1 * ]ˆ[ˆ diberikan oleh : ck = e 2)](ˆ[ ii yey − = [ )ˆ()]ˆ()( 2 iii yvyeye +− ck = 2 1 2* 1 * ]ˆ[ˆ σ ∑ ∑ = = + n i i n i i ybiasyvar = +2)( katr 2 2 )( k k aitr jkr −− σ = )(22 k k atrn jkr +− σ bila *')**'(* 1 xkixxxh k −+= dan 1)()( += kk htratr ck = ]1)([2 ˆ 2 ++− k k htrn jkr σ ck = )(22 ˆ 2 k k htrn jkr ++− σ identifikasi pencilan leverage (hii) adalah elemen-elemen diagonal dari matrik proyeksi least squares yang disebut matrik topi, h = x(x'x)-1x', yang menjelaskan pendugaan atau nilai-nilai dugaan, karena : hyxby =≡ˆ . elemen-elemen diagonal h merupakan jarak antara xi dan .x oleh karena h adalah matrik proyeksi, maka dia simetris dan idempoten (h2=h). elemen-elemen dari matrik topi yang dipusatkan adalah : )/1( ~ nhh ijij −= . hal ini berimplikasi, bahwa 1)/1( ≤≤ ihn . jumlah akar karakteristik dari matrik proyeksi tidak nol sama dengan rank dari matrik. dalam hal ini rank(h) = rank(x)=p dan trace h=p, atau karena x rank penuh, maka ∑ = n i ih 1 = p. ukuran rata-rata elemen diagonal adalah p/n. data yang diinginkan adalah yang jauh dari pengamatan berpengaruh, dimana masing-masing pengamatan mempunyai hi dekat dengan rata-rata p/n. untuk itu perlu beberapa kriteria untuk memutuskan kapan nilai hi cukup besar atau cukup jauh dari rata-ratanya. jika variabel-variabel bebas didistribusikan secara independen, maka dapat dicari distribusi eksak dari fungsi-fungsi tertentu dari hi. belsley, kuh, dan welsch (1980) mengambil teori distribusi untuk mencari batas kritis dari nilai leverage sebagai berikut : statistik λ wilks untuk dua grup, dimana 1 grup terdiri dari titik tunggal : )' ~ det( )~'~)(~)('~)1( ~ ' ~ det( )~( xx xxixixnxx x iii −−− =λ pendeteksian outlier dengan metode regresi ridge   volume 1 no. 1 november 2009 13 = ) ~ ' ~ det( ) ~ ' ~ det()'~) ~ ' ~ (~ 1 1 1 xx xxxxxx n n ii − − − = 1~ 1 i h n n − = 1) 1 ( 1 n h n n i −− = )1( 1 i h n n − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ λ λ− − − )~( )~(1 1 i i x x p pn ~ fp-1,n-p sehingga dapat ditulis : ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − )1( ) 1 ( 1 i i h n h p pn ~ fp-1,n-p untuk p besar (lebih dari 10) dan n-p besar (lebih dari 50), maka pada tabel f nilai-nilainya kurang dari 2 sehingga nilai 2(p/n) merupakan batas yang cukup bagus. selanjutnya, pengamatan ke-i adalah titik leverage ketika hi melebihi 2(p/n). penurunan rumus press rumus press umumnya adalah iii yy −− ,ˆ . rumus ini bisa ditulis sebagai berikut (myers, 1990) : ei,-i = yi xi'b-i ei,-i = ii ii ii ii yxh xxxxxx xxxy −− −− − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − +− ' 11 1 1 )'(')'( )'(' = ii iiiii iiii h yxxxxh yxxxxy − −− −− − −− − 1 )'( )'( '1' '1' = ii iiiiiiiiiiiii h yxxxxhyxxxxhyh − −−− −− − −− − 1 )'()'()1()1( '1''1' = ii iiiiii h yxxxxyh − −− −− − 1 )'()1( '1' = ii iiiiii h yxyxxxxyh − −−− − 1 )'()'()1( 1' = = ii i h yy − − 1 ˆ = ii i h e −1 4. kesimpulan masalah pencilan ini dapat diatasi degan berbagai metode, diantaranya metode regresi ridge (ridge regression). hal ini ditinjau dari ketepatan model, dimana metode regresi ridge memberikan hasil yang relatif lebih baik dibandingkan dengan metode kuadrat terkecil. ii iiiiiii h yhyyh − +−− 1 ˆ)1( sri harini  14 volume 1 no. 1 november 2009 daftar pustaka belsley, d.a., kuh, e., dan welsch, r.e, (1980), regression diagnostics. john wiley. new york. hoerl, a.e dan kennard, r.w., (1970), ridge regression: biased estimation for nonorthogonal problems. technometrics, vol. 12, no. 1. marquardt, d.w., (1970), generalized inverses, ridge regression, biased linearestimation, and nonlinier estimation. technometrics, vol. 12, no. 3. mason, r.l. dan gunst, r.f., (1985), outlier-induced collinearities. technometrics, vol. 2, no. 4. myers, r.h, (1990), classical and modern regression with applications. 2nd edition. pws-kent, boston. neter, j., wasserman, w. dan kutner, m.h., (1990), applied linear statistical models, regression, analysis of variance & experimental design, richard d. irwin inc. illinois. toppan company. ltd, tokyo. retnaninsih, e., (2001), studi perbandingan metode regresi ridge dengan kuadrat trekecil parsial pada struktur ekonomi dan tingkat kesra penduduk indonesia. tidak dipublikasikan, tesis program master, its, surabaya. walker, e. dan birch, j.b., (1988). influence measure in ridge regression. technometrics, 25: 221-227. microsoft word 1 sampul depan.doc 15  empat model aproksimasi binomial harga saham model black-scholes abdul aziz jurusan matematika, fakultas sains dan teknologi universitas islam negeri (uin) maulana malik ibrahim malang e-mail: abdulaziz_uinmlg@yahoo.com abstrak kami akan menyajikan empat bentuk nilai parameter-parameter u, d, dan p dalam model binomial harga saham, yang dihasilkan dengan menggunakan penyamaan ekspektasi dan variansi model diskrit dengan kontinu. metode pertama menggunakan asumsi u . d = 1, yang mana metode ini dapat menghasilkan tiga bentuk solusi untuk parameter-parameter u, d, dan p dalam model binomial harga saham. metode kedua menggunakan asumsi p = 0,5. dari kedua metode ini ternyata dapat dihasilkan empat bentuk solusi u, d, dan p yang berbeda dan akan dibandingkan hasilnya dalam pendekatan nilai option dalam model binomial dengan model black-scholes. kata kunci: aproksimasi, binomial, black-scholes, harga saham, parameter. 1. pendahuluan call option pada sebuah saham merupakan sebuah perjanjian hak, tetapi bukan obligasi, untuk membeli saham tersebut pada suatu hari tertentu, t yang akan datang, yang diistilahkan sebagai strike date atau jatuh tempo (date of expiration), dengan harga tertentu k, yang diistilahkan sebagai strike price atau exercise price. sebaliknya, put option pada sebuah saham merupakan sebuah perjanjian hak untuk menjual saham pada suatu hari tertentu yang akan datang dengan harga tertentu pula (stampfli, j., goodman, v., 2001) tentu saja, pemegang option (holder) akan menggunakan atau mengabaikan hak pilihnya pada option tersebut, yang diistilahkan sebagai exercise, tergantung pada harga saham di pasar bebas pada waktu t tersebut. pemegang call option akan menggunakan haknya dengan membeli saham itu pada waktu t dengan harga k pada penulis option (writer), jika harga saham pada pasar bebas pada waktu t, st, lebih tinggi dibandingkan dengan k, harga saham pada option, sehingga menguntungkan bagi pemegang option. dan, penulis option wajib untuk menjual sahamnya pada pemegang option dengan harga dan waktu sesuai perjanjian option. sebaliknya, jika harga saham pada pasar bebas lebih rendah dibandingkan dengan strike price maka pemegang option dapat mengabaikan haknya, dan ia lebih baik membeli saham pada pasar bebas dengan harga yang lebih menguntungkan. serupa untuk put option, pemegang option akan menjual sahamnya pada penulis option jika harga saham tersebut pada pasar bebas lebih rendah dari pada strike price. dan, penulis option wajib membeli saham tersebut dari pemegang option. sebaliknya, lebih baik menjual pada pasar bebas jika harga saham di pasar bebas lebih tinggi dari pada strike price. harga saham di pasar bebas pada waktu tertentu yang akan datang tidak dapat dipastikan oleh seseorang. harga saham dapat mengalami perubahan turun naik setiap detiknya. padahal, harga saham tersebut pada waktu tertentu sangat diperlukan oleh dua pihak, penulis option dan pemegang option, dalam pembuatan perjanjian option, sebagai perjanjian transaksi jual beli saham pada waktu yang akan datang. banyak pendekatan numerik yang dilakukan oleh para ilmuan untuk memperkirakan harga saham di pasar bebas pada waktu tertentu dengan memodelkan gerakan fluktuasi harga saham, sehingga mereka dapat menentukan harga option yang mungkin menguntungkan abdul aziz   16 volume 1 no. 1 november 2009 bagi kedua pihak tersebut. pemegang option akan memperoleh keuntungan, jika menggunakan hak optionnya, dari nilai option (option value) yang diperoleh dari selisih harga saham pada pasar bebas dengan harga saham pada option, yang diistilahkan dengan payoff, v, setelah dikurangi dengan harga option (option price), yang diistilahkan dengan profit atau keuntungan (return). sedangkan penulis option hanya memperoleh keuntungan sebesar harga atau biaya option, baik jika pemegang option menggunakan atau mengabaikannya. jadi keuntungan atau kerugian yang diperoleh oleh pemegang call option pada waktu t (hull, john c., 2003): profit = payoff – biaya option = vc – c = max(k-st, 0) – c sebaliknya, bagi pemegang put option akan mendapatkan keuntungan atau kerugian: profit = payoff – biaya option = vp – c = max(st-k, 0) – p artinya, jika profit bernilai positif maka pemegang option mendapatkan keutungan, dan sebaliknya jika negatif merupakan kerugian yang maksimal sebesar biaya option. 2. model binomial harga saham harga saham pada pasar bebas kenyataannya akan selalu berubah naik atau turun dengan perubahan waktu. kemungkinan dua arah perubahan inilah yang digunakan sebagai dasar model binomial. misalkan harga saham pada saat t = 0, saat pembuatan option, adalah s0 dan pada saat t = t akan naik dengan peluang p menjadi su atau akan turun dengan peluang 1-p menjadi sd, sehingga nilai option pada saat t = 0, saat pembuatan option, adalah v0 dan pada saat t = t akan naik menjadi u atau akan turun menjadi d. gambar 1. grafik perubahan harga saham dan harga option permodelan matematika diharapkan dapat membantu kita untuk memahami keadaan sekarang dan prediksinya pada waktu yang akan datang. oleh karena itu, agar model binomial ini dapat berhasil dengan lebih baik maka harus sesuai dengan keadaan dunia nyata. masalah yang dihadapi sekarang adalah bagaimana kita memilih p, u, dan d sedemikian hingga model binomial ini mendekati pada keadaan dunia nyata. kita mulai dengan diskritisasi, yaitu menjadikan waktu kontinu t menjadi diskrit dengan menggantikan t oleh waktu yang sama lamanya katakanlah ti. misalkan kita gunakan notasi berikut: m : banyaknya selang waktu, m t t =δ : ti : i . ∆t, i = 0, 1, …, m si : s(ti) selanjutnya bidang (s,t) diwakili oleh garis-garis lurus paralel dengan jarak ∆t. dan kita ganti nilai-nilai kontinu si sepanjang paralel t = ti dengan nilai-nilai diskrit sji, untuk semua i dan j yang sesuai. untuk lebih memahami lihat gambar 2. gambar ini s0 su = u s0 v0 d u sd = d s0 p 1-p  empat model aproksimasi binomial harga saham model black‐scholes   volume 1 no. 1 november 2009 17 menunjukkan sebuah hubungan grid, katakanlah perubahan dari t ke t+∆t, atau dari ti ke ti+1. gambar 2. prinsip metode binomial sehingga asumsi-asumsi yang digunakan dalam permodelan ini adalah (figlewski, stephen, 1990): (a1) harga s, sebagai harga awal, selama setiap periode waktu ∆t hanya dapat berubah dalam dua kemungkinan yaitu naik menjadi su atau turun menjadi sd dengan 0 < d < u. di sini u dan d masing-masing merupakan faktor perubahan naik dan turun yang konstan untuk setiap ∆t. (a2) peluang perubahan naik adalah p, p(naik) = p. sehingga p(turun) = 1 – p. (a3) ekspektasi harga saham secara acak kontinu, dengan suku bunga bebas resiko r, dari si pada waktu ti menjadi si+1 pada waktu ti+1 adalah: tr ii esse δ + = .)( 1 . asumsi selanjutnya adalah tidak ada pembayaran dividen selama periode waktu tersebut. jika ada pembayaran dividen, q, maka persamaan (2.1) menjadi tqrii esse δ− + = )( 1 .)( dengan model binomial kita bisa membangun skema (tree) untuk fluktuasi harga saham secara diskrit. dari gambar 3, kita misalkan harga saham pada saat t = t0 adalah s0 = s00 = s, dan harga saham pada saat t = t1 adalah s01 = sd dan s11 = su. sehingga secara umum harga saham pada saat t = ti terdapat i+1 kemungkinan dengan rumus umum jijji duss −= 0 , i = 0,1,…,m dan j = 0,1,…i. (1) sehingga diperoleh nilai-nilai option, untuk european call option )0,max( ksv jmjm −= , dan untuk european put option )0,max( jmjm skv −= . pada american option, kita bisa meng-exercise sebelum jatuh tempo, t ≤ t, sehingga perlu juga untuk menghitung nilai-nilai option untuk ti dimana i = m-1, m-2,…,0, karena ada kemungkinan nilai-nilai option di waktu-waktu tersebut lebih baik dari pada pada waktu jatuh temponya. (ross, sheldon m., 1999) s su = u . s sd = d . s p 1-p si+1 si ti+1= t + ∆t ti = t s t abdul aziz   18 volume 1 no. 1 november 2009 persamaan (1) adalah tidak rekursif, artinya perhitungan yang memerlukan waktu relatif lama, sehingga perlu adanya bentuk rekursif yang diperoleh sebagai berikut, dengan bantuan persamaan ( ) trii esse δ+ =1 . (2) gambar 3. skema fluktuasi harga saham secara binomial sedangkan ( ) 1,1,11, )1()1( ++++δ −+=−+== ijijjijiijtrji sppsdspupssees . (3) sehingga bentuk rekursif untuk nilai option, v, ( ) ( ) ( )1,1,11, )1( +++δ−δδ−+δ− −+=== ijijtrtrjitrijtrji vppveeveveev . (4) jadi, nilai-nilai option untuk european call option ( )0,max ksv jmjm −= , dan ( )1,1,1 )1( +++δ− −+= ijijtrji vppvev dan untuk european put option ( )0,max jmjm skv −= , dan ( )1,1,1 )1( +++δ− −+= ijijtrji vppvev , sedangkan untuk american call option ( )0,max ksv jmjm −= , dan ( ) ( ){ }1,1,1 )1(,0,maxmax +++δ− −+−= ijijtrjiji vppveksv dan untuk american put option ( )0,max jmjm skv −= , dan ( ) ( ){ }1,1,1 )1(,0,maxmax +++δ− −+−= ijijtrjiji vppveskv untuk i = 0,1,…,m dan j=0,1,…,i . 3. metode i, dengan asumsi u.d = 1 untuk menentukan tiga parameter yang belum diketahui, u, d, dan p, diperlukan tiga persamaan, yaitu (stampfli, j., goodman, v., 2001): s su sd su2 sud sd2 su3 su2d sud2 sd3 t0 t1 t2 t3  empat model aproksimasi binomial harga saham model black‐scholes   volume 1 no. 1 november 2009 19 (p.1) menyamakan ekspektasi harga saham model diskrit dengan model kontinu. (p.2) menyamakan variansi model diskrit dengan model kontinu. (p.3) menyamakan u . d = 1. konsekuensi dari asumsi (a1) dan (a2) untuk model diskrit ini adalah ( )dppusdspupsse iiii )1()1()( 1 −+=−+=+ . di sini si adalah sebuah nilai sebarang untuk ti, yang berubah secara acak menjadi si+1, sehingga sesuai kedua asumsi tersebut persamaan (2) dan (3) memberikan dppue tr )1( −+=δ . ini merupakan persamaan pertama yang diperlukan untuk menentukan u, d, p. selanjutnya perhatikan bahwa dengan menyelesaikan persamaan (4) untuk p akan diperoleh: dduppddpudppue tr +−=−+=−+=δ )()1( du de p tr − − = δ . (5) karena p merupakan peluang yang harus memenuhi 0 ≤ p ≤ 1 maka haruslah er∆t – d ≤ u – d atau er∆t ≤ u dan u – d > 0 atau d ≤ u, sehingga diperoleh d ≤ er∆t ≤ u. pertidaksamaan-pertidaksamaan ini berhubungan dengan gerakan naik dan turunnya harga aset terhadap suku bunga bebas resiko r. pertidaksamaan terakhir ini bukanlah merupakan asumsi baru tetapi merupakan prinsip no-arbitrage bahwa 0 < d < u. selanjutnya kita menghitung variansi. dari model kontinu kita terapkan hubungan tr ii esse δ+ + = )2(22 1 2 )( σ . (6) persamaan (2) dan (6) menghasilkan variansi ( ) )1()()()( 22 2222)2(2212 11 −=−=−= δδδδ++++ ttritritriiii eesesessesesvar σσ . di sisi lain, dengan menggunakan persamaan (3) dan (4), varian untuk model diskrit memenuhi ( ) ( ) ( ) ( )( )222212 11 )1()1()()()( dppusdspuspsesesvar iiiiii −+−−+=−= +++ ( ) ( ) ( )tritrii edppusesdppus δδ −−+=−−+= 2222 2222 )1()1( (7) sehingga dengan menyamakan hasil kedua variansi tersebut, persamaan (6) dan (7), menghasilkan ( ) ( )trittri edppusees δδδ −−+=− 222222 )1(12σ ( ) trttr edppuee δδδ −−+=− 2222 )1(12σ ( ) 222 )1( 2 dppue tr −+=δ+σ ( ) ( ) 2222 2 ddupe tr +−=δ+σ ( ) 22 22 2 du de p tr − − = δ+σ (8) selanjutnya, dengan menyamakan persamaan (5) dan (8) serta misalkan kita memilih menyamakan u . d = 1 akan dihasilkan ( ) 22 22 2 du de du de trtr − − = − − δ+δ σ ( ) ( )( )dudu de du de trtr +− − = − − δ+δ 22 2σ ( )( ) ( ) 22 2 dededu trtr −=−+ δ+δ σ ( ) 222 2 deduddeue trtrtr −=−−+ δ+δδ σ abdul aziz   20 volume 1 no. 1 november 2009 ( ) ( ) 222 21 dededu trtr −=−−+ δ+δ σ ( ) ( ) trtr eedu δ+δ =−+ 221 σ ( ) ( ) trtrtr eeedu δδ+δ =−+ 21 σ ( ) ( ) trtrtrtr eeeedu δδ+δδ− =−+ 2σ ( ) trtr eedu δ+δ− =−+ 2σ ( ) trtr ee u u δ+δ− =−+ 21 σ ( ) trtr ueueu δ+δ− =−+ 2 12 σ ( ) 01 22 =−−+ δ+δ− trtr ueueu σ ( )( ) 0122 =++− δ+δ− trtr eeuu σ ( )( ) 0122 =++− δ+δ− trtr eeuu σ (9) dengan memisalkan ( )( )trtr ee δ+δ− += 221 σβ persamaan (9) menjadi persamaan kuadrat yang lebih sederhana yaitu 0122 =+− uu β dengan akar-akar 12 −±= ββu dimana 012 >−β . karena d < u maka kita pilih 12 −+= ββu sehingga diperoleh nilai untuk u, d dan p yaitu 12 −+= ββu , d=1/u, du de p tr − − = δ dengan ( )( )trtr ee δ+δ− += 221 σβ selanjutnya, dengan aproksimasi bilangan eksponensial xe x +≈ 1 akan diperoleh nilai untuk β ( )( ) ( ) tttrtr δ+=δ+=δ+++δ−= 221221221 1211 σσσβ sehingga untuk nilai u ( ) ( ) 111111 44122212221221 −δ+δ++δ+=−δ++δ+= tttttu σσσσσ ttttttt δ+δ+=δ+δ+≈δ+δ+δ+= σσσσσσσ 221 22 2 14 4 122 2 1 111 tet δ≈δ+≈ σσ1 sehingga kita memperoleh nilai u, d dan p sebagai ,, tt edeu δ−δ == σσ dan. du de p tr − − = δ . nilai-nilai dari parameter-parameter terakhir ini telah diperkenalkan oleh cox, ross, dan rubinstein. 7 atau jika diinginkan juga untuk nilai p 1 1 2 − − = − − = − − = − − = δ δ+δ δ δ δ−δ δ−δ δ−δ δ−δδ t ttr t t tt ttr tt ttrtr e e e e ee ee ee ee du de p σ σ σ σ σσ σ σσ σ ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +δ= δ δ+δ = δ δ+δ = −δ+ −δ+δ+ ≈ 1 2 1 2 1 2121 11 t r t tt t ttr t ttr r σσ σ σ σ σ σ σ . sehingga kita memperoleh nilai u, d dan p yang lain sebagai ,, tt edeu δ−δ == σσ dan ( )121 +δ= tp r σ .  empat model aproksimasi binomial harga saham model black‐scholes   volume 1 no. 1 november 2009 21 4. metode ii, dengan asumsi p = 0.5 sekarang, kita coba membandingkan metode di atas dengan memilih p = 0.5 pada (p.3) untuk menghitung ulang dalam menentukan nilai u dan d. dan tetap dengan menyamakan ekspektasi dan rata-rata pada model kontinu dan diskritnya sebagaimana metode sebelumnya. dengan mensubstitusikan nilai p dada persamaan (5) diperoleh )(5.05.05.0)1( dududppue tr +=+=−+=δ sehingga tredu δ=+ 2 dan pada persamaan (8) diperoleh ( ) 2 1 22 22 2 = − − = δ+ du de p tr σ sehingga ( ) tredu δ+=+ 2222 2 σ . misalkan u = b + c dan d = b – c maka diperoleh trebcbcbdu δ==−++=+ 22 atau treb δ= dan ( ) trecbcbcbcbcbdu δ+=+=+−+++=+ 2222222222 22222 σ atau ( ) 22222 2 cecbe trtr +=+= δδ+σ sehingga ( ) ( )122 2222 −=−= δδδδ+ ttrtrtr eeeec σσ atau 1 2 −= δδ ttr eec σ . sehingga diperoleh ⎟ ⎠ ⎞⎜ ⎝ ⎛ −+=−+= δδδδδ 111 22 ttrttrtr eeeeeu σσ dan ⎟ ⎠ ⎞⎜ ⎝ ⎛ −−=−−= δδδδδ 111 22 ttrttrtr eeeeed σσ . jadi, dengan metode ini diperoleh 2 1 ,11,11 22 =⎟ ⎠ ⎞⎜ ⎝ ⎛ −−=⎟ ⎠ ⎞⎜ ⎝ ⎛ −+= δδδδ peedeeu ttrttr σσ . 5. perbandingan hasil numerik dari kedua metode di atas ternyata dapat diperoleh empat bentuk solusi nilai-nilai untuk parameter-parameter u ,d, dan p dalam model binomial, yaitu 12 −+= ββu , d=1/u, du de p tr − − = δ dengan ( )( )trtr ee δ+δ− += 221 σβ ,, tt edeu δ−δ == σσ dan. du de p tr − − = δ . ,, tt edeu δ−δ == σσ dan ( )121 +δ= tp r σ . 2 1 ,11,11 22 =⎟ ⎠ ⎞⎜ ⎝ ⎛ −−=⎟ ⎠ ⎞⎜ ⎝ ⎛ −+= δδδδ peedeeu ttrttr σσ . abdul aziz   22 volume 1 no. 1 november 2009 berikut ini adalah tabel dan grafik hasil komputasi numerik untuk menghitung nilai european option dengan keempat model binomial solusi aproksimasi parameterparameter di atas yang juga diperbandingkan dengan nilai option dengan model blackscholes. pada contoh di sini menggunakan data-data: s = 5, k = 10, r = 0.06. σ = 0.3, t = 1. tabel 1. hasil numerik nilai european put option m 8 16 32 64 128 256 512 model 1 4.4251 4.4292 4.4299 4.4299 4.4300 4.4304 4.4304 model 2 4.4248 4.4289 4.4297 4.4298 4.4300 4.4304 4.4304 model 3 4.2010 4.2057 4.2065 4.2065 4.2067 4.2071 4.2071 model 4 4.4247 4.4293 4.4298 4.4296 4.4302 4.4303 4.4304 bs 4.4305 4.4305 4.4305 4.4305 4.4305 4.4305 4.4305 tabel 2. hasil numerik nilai european call option periode 8 16 32 64 128 256 512 model 1 0.0074 0.0116 0.0122 0.0123 0.0124 0.0127 0.0127 model 2 0.0071 0.0112 0.0120 0.0122 0.0124 0.0127 0.0127 model 3 0.107 0.0168 0.0183 0.0187 0.0190 0.0195 0.0195 model 4 0.0071 0.0117 0.0121 0.0120 0.0126 0.0126 0.0128 bs 0.0128 0.0128 0.0128 0.0128 0.0128 0.0128 0.0128 gambar 4. perbandingan numerik european put option model 1, 2,3 dan 4 gambar 5. perbandingan numerik european put option model 1, 2, dan 4 4 4.2 4.4 4.6 8 16 32 64 128 256 512 perbandingan numerik european put option model 1 model 2 model 3 model 4 black scholes 4.424 4.426 4.428 4.43 4.432 8 16 32 64 128 256 512 perbandingan numerik european put option model 1 model 2 model 4 black scholes  empat model aproksimasi binomial harga saham model black‐scholes   volume 1 no. 1 november 2009 23 gambar 6. perbandingan numerik european call option model 1, 2,3 dan 4 gambar 7. perbandingan numerik european call option model 1, 2, dan 4 dari tabel dan garfik perbandingan di atas dapat dilihat bahwa aproksimasi binomial model 3 sangat tidak sesuai atau paling lemah dan besar galatnya dibandingkan dengan ketiga model lainnya karena dihasilkan dengan melibatkan banyak aproksimasi dalam menghasilkan perumusannnya. untuk kasus european put option, model 3 under estimate, sedangkan untuk kasus european call option, model 3 over estimate. 6. kesimpulan model binomial, kecuali model 3, dapat digunakan sebagai pendekatan diskritisasi dalam menentukan nilai option. semakin besar banyaknya grid, m, maka metode ini akan semakin mendekati pada nilai option dengan model black-scholes. aproksimasi binomial model 3 sangat tidak sesuai atau paling lemah dan besar galatnya dibandingkan dengan ketiga model lainnya karena dihasilkan dengan melibatkan banyak aproksimasi dalam menghasilkan perumusannnya. untuk kasus european put option, model 3 under estimate, sedangkan untuk kasus european call option, model 3 over estimate. 0 0.05 0.1 0.15 8 16 32 64 128 256 512 perbandingan numerik european call option model 1 model 2 model 3 model 4 black scholes 0.006 0.008 0.01 0.012 0.014 8 16 32 64 128 256 512 perbandingan numerik european call option model 1 model 2 model 4 black scholes abdul aziz   24 volume 1 no. 1 november 2009 daftar pustaka figlewski, stephen, (1990), theoretical valuation models, dalam: financial options from theory to practice, salomon brothers center for the study of financial institutions, new york university. hull, john c., (2003), options, futures, and other derivatives, fifth edition, prentice hall, new jersey. ross, sheldon m., (1999), an introduction to mathematical finance, option and other topics, cambridge university press. stampfli, j., goodman, v., (2001), the mathematics of finance, brooks/cole, usa. solusi numerik model reaksi-difusi (turing) dengan metode beda hingga implisit 1junik rahayu, 2usman pagalay, dan 3ari kusumastuti 1,2,3jurusan matematika uin maulana malik ibrahim malang e-mail: rahayujunik@yahoo.com abstrak alan turing (1952) mengemukakan bahwa sistem interaksi bahan kimia dipengaruhi oleh difusi yang tidak stabil yang kemudian berkembang menjadi pola spasial. hasil dari penelitian ini disebut dengan model reaksi-difusi (turing). pada umumnya model difusi mempunyai difusifitas berupa konstanta. barras dkk. (2006) mengganti mekanisme murray (2003) dalam menganalisis model ini, sehingga terbentuklah model dengan rasio pertumbuhan domain yang tumbuh secara eksponensial sebagai difusifitasnya. hal inilah yang membuat model ini lebih menarik dibandingkan dengan model difusi yang lain. berbagai model matematika dipastikan mempunyai solusi, begitu juga dengan model ini. paper ini membahas penyelesaian numerik pada contoh model. digunakan metode beda hingga implisit sebagai metode dasar menyelesaikan model. dalam model terdapat dua konsentrasi yang bereaksi untuk mencapai suatu kesetimbangan. dari konsentrasi ini akan diperiksa keterikatan pada pertumbuhan domain dengan adanya dinamika gangguan kecil serta pengaruh pertumbuhan domain terhadap penyelesaian numerik model. dari penyelesaian numerik diperoleh bahwa pertumbuhan domain mempengaruhi dua konsentrasi dalam model dan penyelesaian numerik. kata kunci: metode beda hingga implisit, model reaksi-difusi (turing), pertumbuhan domain. abstract alan turing (1952) telling that chemicals interaction system influenced by unstable diffusion which later;then round into pattern of spasial. result of from this research is referred as with of reaction-diffusion (turing) model’s. in general diffusion model’s have coeficient diffuson in the form of constanta. barras et. al (2006) changing mechanism of murray (2003) in analysing this model, is so that formed by model with domain growth ratio which grow by eksponensial as its. this matter make this model more is interesting compared to other diffusion model. various mathematics model ascertained to have solution, so also with this model. this paper study the solving of numeric at model of example.used by different method till implisit as basic method finish model. in model there are two concentration reacting to reach an is balance. than this concentration will be checked by binding at domain growth with existence of small trouble dynamics and also influence of domain growth to solving of model numerik. from solving of numeric obtained that domain growth influence two concentration in model and numerical solution. keywords: reaction-diffusion (turing) model’s, finite difference methods, implicit scheme,domain growth rate. pendahuluan alan turing (1952) mengemukakan bahwa sistem interaksi bahan kimia dipengaruhi oleh difusi yang tidak stabil yang kemudian berkembang menjadi pola spasial. dalam era integrasi biologi, model hasil penelitian alan turing merupakan salah satu contoh pertama bagaimana mengintegrasikan proses sederhana yang dapat memberikan hasil yang kompleks, dalam hal ini, kombinasi dari proses penyetabilan yang menghasilkan sistem yang tidak stabil. pada model tersebut, diasumsikan bahwa sel tidak bergerak tetapi hanya menanggapi pembedaan isyarat kimia. hasil dari penelitian ini disebut dengan model reaksi-difusi (turing). barras dkk (2006) mengganti mekanisme model murray (2003) dalam menganalisis model ini dengan domain pertumbuhan menggunakan kinetika schnakenberg, yang timbul dari suatu penerapan hukum aksi massa untuk skema trimolecular. menurut barras dkk (2006) model terdiri dari 2 persamaan diferensial parsial dan 1 persamaan diferensial biasa, sehingga membentuk sistem. dalam jurnalnya barras dkk (2006) mengungkap bahwa proses transisi dalam mailto:rahayujunik@yahoo.com solusi numerik model reaksi-difusi (turing) dengan metode beda hingga implisit jurnal cauchy – issn: 2086-0382 19 model mencapai puncak didorong oleh pertumbuhan domain sehingga menghasilkan urutan pola. model matematis yang kompleks seperti model ini, sukar mendapatkan solusinya dengan solusi analitis. walaupun metode numerik dalam pencarian solusi dari suatu sistem juga jarang digunakan akhir-akhir ini, akan tetapi metode ini merupakan alternatif dalam menyelesaikan persoalan matematik. metode ini dapat digunakan untuk mendekati solusi secara eksak. salah satu metode numerik untuk menyelesaikan persamaan diferensial parsial seperti model ini adalah metode beda hingga. dalam metode beda hingga terdapat bermacam skema, salah satunya skema implisit yang stabil tanpa syarat. kajian teori a. analisis persamaan diferensial parsial pada model reaksi-difusi (turing) suatu persamaan yang di dalamnya terdapat turunan parsial dan terdapat dua atau lebih variabel bebas maka persamaan tersebut disebut persamaan diferensial parsial (partial differential equation/pde) (ayres, 1992). bentuk umum persamaan diferensial parsial linear orde 2 dalam 2 variabel bebas adalah: 𝐴𝑓𝑥𝑥 + 𝐵𝑓𝑥𝑦 + 𝐶𝑓𝑦𝑦 + 𝐷𝑓𝑥 + 𝐸𝑓𝑦 + 𝐹𝑓 = 𝐺 menurut sasongko (2010) persamaan di atas dapat dinyatakan sebagai kondisi-kondisi berikut: 1. apabila koefisien , , , , , ,a b c d e f g adalah konstanta atau fungsi yang terdiri dari variabel bebas saja, maka persamaan tersebut disebut linier. 2. apabila koefisien , , , , , ,a b c d e f g adalah fungsi dari variabel tak bebas dan atau merupakan turunan dengan orde yang lebih rendah daripada persamaan diferensialnya , , u u x t         maka persamaan tersebut disebut kuasilinier. 3. apabila koefisien , , , , , ,a b c d e f g adalah fungsi dengan orde turunan yang sama dengan orde persamaan diferensialnya 2 2 2 2 2 , , , u u u x tx t           maka persamaan tersebut disebut persamaan non-linier. tipe dari persamaan diferensial orde dua ditentukan oleh determinan (𝐷) jika: a. 2 4 0,d b ac   maka bertipe eliptik. b. 2 4 0,d b ac   maka bertipe parabolik. c. 2 4 0,d b ac   maka bertipe hiperbolik. model turing menurut barras dkk (2006) berbentuk: 2 2 2 2 2 2 2 2 1u u a uv u t l x v d v b uv v v t l x dl l dt                      dengan mengganti persamaan 𝑑𝐿 𝑑𝑡 = 𝜌𝐿 menjadi persamaan diferensial biasa, diasumsikan 𝐿(0) = 1, maka model menjadi bentuk berikut: 2 2 2 2 1 ( ) ( ( ) ) t xx t x t x u u a uv u l t d v v b uv v v t e l t l              dari derinisi yang telah diuraikan di atas, maka model dapat diklasifikasikan menjadi persamaan diferensial parsial kuasilinier orde dua tipe parabolik. solusi model adalah fungsi ( , )u x t dan ( , )v x t yang memenuhi persamaan di atas. solusi tersebut merupakan solusi umum, sehingga diperlukan subtitusi kondisi batas dan kondisi awal agar didapatkan solusi khusus. kondisi batas yang digunakan pada model adalah dirichlet boundary conditions. untuk interval 0 0.002t  dan 0 1x  . nilai batas (0, ) 0.9u t  ; (0, 0.002) 0.9u  ; (0, ) 1v t  dan (0, 0.002) 1v  untuk semua .t sedangkan kondisi awal yang digunakan untuk model adalah ( )l t yang dirumuskan sebagai berikut: 𝑢(𝑥,0) = 𝑣(𝑥,0) = 𝐿(𝑡) = 𝑒𝜌𝑡 (1) persamaan tersebut akan digunakan untuk membuat iterasi numerik pada pembahasan. b. analisis model reaksi-difusi (turing) pemodelan matematika mengenai model reaksi-difusi dikemukakan oleh alan turing (1952) yang mengidentifikasi perkembangan embrio menjadi dewasa. dalam penelitiannya alan turing mengasumsikan bahwa sistem interaksi bahan kimia dipengaruhi oleh difusi yang tidak stabil yang kemudian berkembang menjadi pola spasial. barras dkk (2006) mengganti mekanisme model murray (2003) dalam menganalisis model junik rahayu, usman pagalay, dan ari kusumastuti 20 volume 3 no. 1 november 2013 dengan domain pertumbuhan menggunakan kinetika schnakenberg, yang timbul dari suatu penerapan hukum aksi massa untuk skema trimolecular, sehingga terbentuklah model. pada model diasumsikan proses difusi dalam kasus pertumbuhan domain yang tumbuh secara eksponensial. selanjutnya brownian motion untuk persamaan 𝑢𝑡 = 1 𝐿(𝑡)2 𝑢𝑥𝑥 + 𝑎 − 𝑢𝑣 2 − 𝜌𝑢 dapat dituliskan sebagai, 𝑢𝑡 − 1 𝐿(𝑡)2 𝑢𝑥𝑥 − 𝑎 + 𝑢𝑣 2 + 𝜌𝑢 = 0 (2) menurut zauderer (1998:2-5), untuk menyelesaikan persamaan  ,u x t di atas, digunakan asumsi-asumsi sebagai berikut: 1. ekspektasi dari variabel acak 𝑥 atau disebut juga sebagai lokasi perpindahan partikel dalam gelombang yang didefinisikan: 𝐸(𝑥) = 𝑥 = (𝑝 − 𝑞)𝛿 dengan c adalah kecepatan difusi, dan dalam masalah ini kecepatan difusi dianggap sama dengan nol. 2. varian dari suatu variabel acak x atau disebut juga dengan besarnya perpindahan yang terjadi dari suatu proses difusi, didefinisikan sebagai berikut: 𝑉(𝑥) = 4𝑝𝛿2 dengan 𝐷 adalah konstanta atau koefisien difusi yang dalam hal ini diasumsikan besarnya sama dengan 2/𝐿(𝑡)2. 3. asumsi dasar difusi yang digunakan adalah  ,u x t yang merupakan distribusi peluang. dimana distribusi peluang dari suatu partikel pada langkah x dan pada waktu yang ke t  sama dengan peluang ketika berada pada titik x  pada waktu t dikalikan dengan peluang perpindahan partikel ke arah kanan (p) pada suatu langkah ditambah dengan peluang partikel pada saat berada di titik x  pada waktu t dikalikan dengan probabilitas perpindahan ke arah kiri (q) pada suatu langkah, dimana 1,p q  yang dapat dituliskan dalam bentuk berikut: 𝑢(𝑥,𝑡 + 𝜏) = 𝑝𝑢(𝑥 − 𝛿,𝑡) + 𝑞𝑢(𝑥 + 𝛿,𝑡) (3) dimana  merupakan partisi waktu. 4. p adalah peluang perpindahan partikel ke arah kanan, sedangkan q adalah peluang perpindahan partikel ke arah kiri, dimana , .p q r untuk menyelesaikan brownian motion persamaan  ,u x t di atas, digunakan deret taylor sebagai berikut: a. untuk 𝑢(𝑥,𝑡 + 𝜏) = 𝑢(𝑥,𝑡) + 𝜏𝑢𝑡(𝑥,𝑡). b. untuk 𝑢(𝑥 − 𝛿,𝑡) = 𝑢(𝑥,𝑡) − 𝛿𝑢𝑥(𝑥,𝑡) + 1 2 𝛿2𝑢𝑥𝑥(𝑥,𝑡). c. untuk 𝑢(𝑥 + 𝛿,𝑡) = 𝑢(𝑥,𝑡) + 𝛿𝑢𝑥(𝑥,𝑡) + 1 2 𝛿2𝑢𝑥𝑥(𝑥,𝑡) sedangkan untuk persamaan 𝑣𝑡 = 𝑑 𝐿(𝑡)2 𝑣𝑥𝑥 + 𝑏 + 𝑢𝑣2 − 𝑣 − 𝜌𝑣 dapat ditulis sebagai: 𝑣𝑡 − 𝑑 𝐿(𝑡)2 𝑣𝑥𝑥 − 𝑏 − 𝑢𝑣 2 + 𝑣 + 𝜌𝑣 = 0. menurut zauderer (1998:2-5), untuk menyelesaikan persamaan  ,v x t di atas, digunakan asumsi-asumsi sebagai berikut: 1. ekspektasi dari variabel acak 𝑥 atau disebut juga sebagai lokasi perpindahan partikel dalam gelombang yang didefinisikan:     ,e x x p q    dengan c adalah kecepatan difusi, dan dalam masalah ini kecepatan difusi dianggap sama dengan nol. 2. varian dari suatu variabel acak x atau disebut juga dengan besarnya perpindahan yang terjadi dari suatu proses difusi, didefinisikan sebagai berikut:   24 ,v x p dengan 𝐷 adalah konstanta atau koefisien difusi yang dalam hal ini diasumsikan besarnya sama dengan 2𝑑/𝐿(𝑡)2. 3. asumsi dasar difusi yang digunakan adalah  ,v x t yang merupakan distribusi peluang. dimana distribusi peluang dari suatu partikel pada langkah x dan pada waktu yang ke t  sama dengan peluang ketika berada pada titik x  pada waktu t dikalikan dengan peluang perpindahan partikel ke arah kanan  p pada suatu langkah ditambah dengan peluang partikel pada saat berada di titik x  pada waktu t dikalikan dengan probabilitas perpindahan ke arah kiri  q pada suatu langkah, dimana 𝑝 + 𝑞 = 1, yang dapat dituliskan dalam bentuk berikut: 𝑣(𝑥,𝑡 + 𝜏) = 𝑝𝑣(𝑥 − 𝛿,𝑡) + 𝑞𝑣(𝑥 + 𝛿,𝑡) (4) dimana  merupakan partisi waktu. 4. p adalah peluang perpindahan partikel ke arah kanan, sedangkan q adalah peluang solusi numerik model reaksi-difusi (turing) dengan metode beda hingga implisit jurnal cauchy – issn: 2086-0382 21 perpindahan partikel ke arah kiri, dimana , .p q r untuk menyelesaikan brownian motion persamaan  ,v x t di atas, digunakan deret taylor sebagai berikut: a. untuk 𝑣(𝑥,𝑡 + 𝜏) = 𝑣(𝑥,𝑡) + 𝜏𝑣𝑡(𝑥,𝑡). b. untuk 𝑣(𝑥 − 𝛿,𝑡) = 𝑣(𝑥,𝑡) − 𝛿𝑢𝑥(𝑥,𝑡) + 1 2 𝛿2𝑣𝑥𝑥(𝑥,𝑡). c. untuk 𝑣(𝑥 + 𝛿,𝑡) = 𝑢(𝑥,𝑡) + 𝛿𝑣𝑥(𝑥,𝑡) + 1 2 𝛿2𝑣𝑥𝑥(𝑥,𝑡). nilai parameter, kondisi awal dan kondisi batas mengacu pada keterangan barras dkk (2006) dengan  merupakan rasio domain pertumbuhan, u dan v adalah dilution effect, energi kinetik 0.9a  dan 0.1b  dan koefisien difusi 0.06.d  beberapa nilai  yang sesuai dengan keterangan barras dkk (2006) yaitu 0.001,  0.05  dan 0.01.  c. metode beda hingga skema implisit untuk model reaksi-difusi (turing) dibentuk skema beda hingga untuk turunan parsial fungsi u dan v yang terdiri dari dua variabel bebas x dan .t berikut merupakan deret taylor:(causaon dan mingham, 2010) 2 0 0 0 ( , ) ( , ) ( , ) ... 2! xx x u x x t u x t u x t             1 01 ( , ) , 1 ! n n n x u x t o x n        (5) dalam skema implisit, untuk menghitung variabel di suatu titik perlu dibuat suatu sistem persamaan yang mengandung variabel di titik tersebut dan titik-titik sekitarnya pada waktu yang sama (triatmodjo, 2002). berikut merupakan langkah iterasi pada skema implisit: gambar 1. gambar skema implisit gambar skema di atas digunakan untuk menghitung turunan petama dan kedua model reaksi-difusi (turing). pembahasan a. metode beda hingga skema implisit model reaksi-difusi (turing) berikut merupakan model reaksi-difusi (turing): { 𝑢𝑡 = 1 𝐿(𝑡)2 𝑢𝑥𝑥 + 𝑎 − 𝑢𝑣 2 − 𝜌𝑢 𝑣𝑡 = 𝑑 𝐿(𝑡)2 𝑣𝑥𝑥 + 𝑏 + 𝑢𝑣 2 − 𝑣 − 𝜌𝑣 𝐿(𝑡) = 𝑒𝜌𝑡 pada persamaan 𝑢𝑡 = 1 𝐿(𝑡)2 𝑢𝑥𝑥 + 𝑎 − 𝑢𝑣 2 − 𝜌𝑢, maka dapat dinyatakan bentuk diskritnya sebagai berikut: −𝛼𝑢𝑖−1 𝑛+1 + (1 + 2𝛼)𝑢𝑖 𝑛+1 − 𝛼𝑢𝑖+1 𝑛+1 = 𝑢𝑖 𝑛 + δ𝑡(𝑎 − 𝑢𝑖 𝑛(𝑣𝑖 𝑛)2 − 𝜌𝑢𝑖 𝑛 dengan 𝛼 = δ𝑡 𝐿(𝑡)2δ𝑥2 . adapun stensilnya dapat digambarkan sebagai berikut: gambar 2. stensil untuk persamaan  ,u x t selanjutnya pada persamaan 2 2 , ( ) t xx d v v b uv v v l t      maka dapat dinyatakan bentuk diskritnya sebagai berikut:   2 1 1 1 1 1 1 2 ( ). n n n n n n n n i i i i i i i i v v v v t b u v v v                  dengan 2 2 . ( ) t d d l t x       stensilnya dapat di lihat pada gambar 3 berikut ini. gambar 3. stensil untuk persamaan  ,v x t jaringan titik hitung beda hingga implisit untuk model pada daerah 𝑥0 ≤ 𝑥 ≤ 𝑅 dan 𝑡0 ≤ junik rahayu, usman pagalay, dan ari kusumastuti 22 volume 3 no. 1 november 2013 𝑡 ≤ 𝑇 adalah sebagai berikut dapat dilihat pada gambar 4. gambar 4. jaringan titik hitung beda hingga implisit untuk model. digunakan kondisi awal sebagai berikut: 𝑢(𝑥,0) = 0.9 + 𝑏𝑖𝑙𝑎𝑛𝑔𝑎𝑛 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣(𝑥,0) = 1 + 𝑏𝑖𝑙𝑎𝑛𝑔𝑎𝑛 𝑟𝑎𝑛𝑑𝑜𝑚 setelah didapatkan nilai awal dan nilai batas, iterasi dilakukan pada hasil diskritisasi dengan sesuai jaringan titik hitung pada gambar 4. b. penyelesaian numerik model reaksidifusi (turing) diselesaikan contoh model reaksi-difusi (turing) pada daerah batas 0 < 𝑥 < 1 dan 0 < 𝑡 < 0.002, rasio pertumbuhan domain 𝜌 = 0.001, energi kinetik 𝑎 = 0.9 dan 𝑎 = 0.1 serta rasio koefisien difusi 𝑑 = 0.06 sehingga model dapat dituliskan sebagai berikut: { 𝑢𝑡 = 1 𝐿(𝑡)2 𝑢𝑥𝑥 + 0.9 − 𝑢𝑣 2 − 0.001𝑢 𝑣𝑡 = 0.06 𝐿(𝑡)2 𝑣𝑥𝑥 + 0.1 + 𝑢𝑣 2 − 𝑣 − 0.001𝑣 𝐿(𝑡) = 𝑒𝛼𝑡 (6) dipilih nilai δ𝑡 = 0.00002 dan δ𝑥 = 0.01. selanjutnya dilakukan iterasi dengan kondisi batas sebagai berikut: 𝑢(𝑥0, 𝑡) = 𝑢(0,𝑡) = 0.9,𝑢(𝑅,𝑡) = 𝑢(1,𝑡) = 0.9 𝑣(𝑥0, 𝑡) = 𝑣(0,𝑡) = 1,𝑣(𝑅,𝑡) = 𝑣(1,𝑡) = 1 sehingga diperoleh: 𝑢𝑖 𝑛 = 0.9,∀𝑛 = 0,1,2,…,100,∀𝑖 = 0,1,2,…,100 𝑣𝑖 𝑛 = 1, ∀𝑛 = 0,1,2,…,100, ∀𝑖 = 0,1,2,…,100 langkah berikutnya yaitu dilakukan iterasi kondisi awal sebagai berikut: 𝑢𝑖 𝑛 = 𝑓(𝑡𝑖) = 0.9 + 𝑏𝑖𝑙𝑎𝑛𝑔𝑎𝑛 𝑟𝑎𝑛𝑑𝑜𝑚, ∀𝑛 = 0, ∀𝑖 = 1,2,…,99. 𝑣𝑖 𝑛 = 𝑔(𝑡𝑖) = 1 + 𝑏𝑖𝑙𝑎𝑛𝑔𝑎𝑛 𝑟𝑎𝑛𝑑𝑜𝑚, ∀𝑛 = 0, ∀𝑖 = 1,2,…,99. setelah didapatkan nilai awal dan nilai batas, iterasi dilakukan pada hasil diskritisasi sesuai jaringan titik hitung pada gambar 5. gambar 5. jaringan titik hitung skema beda hingga implisit untuk model dengan parameter x dan t gambar 6. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.001. gambar 7. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap waktu (𝑡) dengan 𝜌 = 0.001. gambar 8. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.001. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.91 0.92 0.93 0.94 0.95 0.96 jarak (x) k o n s e n tr a s i u 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 0.9 0.91 0.92 0.93 0.94 0.95 0.96 waktu (t) k o n s e n tr a s i u 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 0.9 0.91 0.92 0.93 0.94 0.95 0.96 waktu (t) k o n s e n tr a s i u solusi numerik model reaksi-difusi (turing) dengan metode beda hingga implisit jurnal cauchy – issn: 2086-0382 23 gambar 9. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.001. gambar 10. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.05. gambar 11. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap waktu (𝑡) dengan 𝜌 = 0.05. gambar 12. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.05 gambar 13. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap waktu (𝑡) dengan 𝜌 = 0.05. gambar 14. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.01. gambar 15. grafik solusi numerik untuk 𝑢(𝑥,𝑡) terhadap waktu (𝑡) dengan 𝜌 = 0.01. gambar 16. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap jarak (𝑥) dengan 𝜌 = 0.01. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 waktu (t) k o n s e n tr a s i v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.91 0.92 0.93 0.94 0.95 0.96 jarak (x) k o n s e n tr a s i u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 jarak (x) k o n s e n tr a s i v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 jarak (x) k o n s e n tr a s i v 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 waktu (t) k o n s e n tr a s i v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.91 0.92 0.93 0.94 0.95 0.96 jarak (x) k o n s e n tr a s i u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 jarak (x) k o n s e n tr a s i v 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 0.9 0.91 0.92 0.93 0.94 0.95 0.96 waktu (t) k o n s e n tr a s i u junik rahayu, usman pagalay, dan ari kusumastuti 24 volume 3 no. 1 november 2013 gambar 17. grafik solusi numerik untuk 𝑣(𝑥,𝑡) terhadap waktu (𝑡) dengan 𝜌 = 0.01. c. interpretasi penyelesaian numerik model reaksi-difusi (turing) kondisi batas yang digunakan dalam bahasan ini adalah    0 , 0, 0.9,u x t u t     , 1, 0.9.u r t u t     0 , 0, 1,v x t v t     , 1, 1.v r t v t  hal tersebut diinterpretasi bahwa 0 x dan r merupakan batas domain yang diselesaikan sehingga efek dilusi sebelum 0 x dan r diabaikan. nilai batas 0.9 dapat dimaknai bahwa energi kinetik non-dimensional pada titik 0 0x  sebesar 0.9 dan nilai batas 1 dapat dimaknai bahwa energi kinetik non-dimensional pada titik 𝑥𝑛 = 𝑅 sebesar 1 pada masing-masing konsentrasi untuk semua waktu 𝑡. dengan adanya kondisi batas yang diberikan, maka dapat memberikan batasan daerah yang akan diselesaikan. parameter-parameter yang digunakan di dalam model yaitu  merupakan rasio pertumbuhan domain, u dan v adalah dilution effect, energi kinetik pada 0.9a  dan 0.1b  dan koefisien difusi 0.06.d  kondisi awal yang digunakan dalam pembahasan contoh model adalah sebagai berikut: ( ) 0, 9 , 0, 1, 2,..., 99 n i i u f t bilangan random n i       ( ) 1 , 0, 1, 2,..., 99 n i i v g t bilangan random n i       kondisi tersebut dapat dimaknai bahwa energi kinetik non-dimensional pada titik 𝑥0 pada waktu 𝑡𝑖 untuk masing-masing konsentrasi dipengaruhi oleh adanya penambahan bilangan random di belakang suatu konstanta. dengan membandingkan gambar 1, 2, 3, 4, 5 dan 6 dapat diketahui bahwa nilai  mempengaruhi perubahan konsentrasi ( , )u x t dan ( , ).v x t sehingga dapat disimpulkan nilai  mempengaruhi penyelesaian numerik pada model reaksi-difusi (turing). penutup berdasarkan pembahasan, dapat diperoleh bahwa untuk menyelesaikan model dengan menttransformasikan dalam bentuk skema beda hingga implisit menggunakan beda maju untuk turunan pertama terhadap waktu dan beda simetrik untuk tururnan kedua terhadap ruang, sehingga diperoleh bentuk diskrit model reaksidifusi (turing) sebagai berikut:     2 1 1 1 1 1 1 2 ( ) n n n n n n n i i i i i i i u u u u t a u v u                   2 1 1 1 1 1 1 2 ( ). n n n n n n n n i i i i i i i i v v v v t b u v v v                  selanjutnya dilakukan iterasi dengan parameter, kondisi batas dan kondisi awal pada daerah batas yang telah ditentukan pada hasil diskritisasi di atas. untuk menghitung solusi numerik digunakan program yang tertera pada lampiran. berdasar hasil perhitungan diperoleh solusi numerik untuk model reaksi-difusi (turing) berupa matriks ukuran 101x101. simulasi gambar, menunjukkan rasio domain pertumbuhan ( ) mempengaruhi dua konsentrasi pada proses difusi serta mempengaruhi penyelesaian numerik model. peneliti lain diharapkan dapat mengembangkan penelitian ini dalam kasus dua dimensi ataupun dengan menurunkan model yang berupa persamaan diferensial parsial menjadi persamaan diferensial biasa sehingga dapat dibandingkan hasilnya dengan penelitian ini. daftar pustaka [1] ayres, f. 1992. persamaan diferensial. jakarta: erlangga. [2] causon, d.m dan mingham, c.g.. 2010. introductory finite difference methods for pdes. united kingdom: ventus publishing aps. [3] barras, i., crampin e. j., dan maini p. k.. 2006. mode transitions in a model reactiondiffusion system driven by domain growth and noise. bulletin of mathematical biology (2006) 68: 981-995. [4] murray, j.d.. 2003. mathematical biology 3rd edition in 2 volumes: spatial models and biomedical applications. new york: springer. [5] sasongko, s. b. 2010. metode numerik dengan scilab. yogyakarta: c.v andi offset. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 -3 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 waktu (t) k o n s e n tr a s i v solusi numerik model reaksi-difusi (turing) dengan metode beda hingga implisit jurnal cauchy – issn: 2086-0382 25 [6] triatmodjo, b. 2002. metode numerik. yogyakarta: beta offset. [7] turing, a.m.. 1952. the chemical basis of morphogenesis. london: phil. trans. r. soc. [8] zauderer, e.. 1998. partial differential equations of applied mathematics, second edition. new york: john wiley. pemodelan pertumbuhan tanaman zea mays l.menggunakan stochastic l-system juhari jurusan matematika uin maulana malik ibrahim malang e-mail : jo_alkanderi57@yahoo.co.id abstrak l-systems memiliki fleksibilitas dalam mensimulasikan struktur dan proses pengembangan pertumbuhan tanaman secara visual dan realistik. penelitian ini bertujuan untuk memodelkan pertumbuhan tanaman jagung menggunakan l-systems dan memvisualisasikan model pertumbuhan tanaman jagung tersebut dari kecil hingga dewasa dalam ruang dimensi tiga. penelitian dilakukan dalam tiga tahap yang diawali dari identifikasi kebutuhan data tehadap pertumbuhan tanaman jagung (zea mays l.). tahap kedua, membangun model secara manual yang meliputi identifikasi dan penentuan komponen lsystems (huruf, aksioma, dan aturan produksi). tahap ketiga, melakukan simulasi dan visualisasi model pertumbuhan tanaman jagung yang telah didapat menggunakan processing dengan bahasa java dalam ruang dimensi tiga. ketiga tahapan tersebut menghasilkan model stochastic l-systems dari pertumbuhan tanaman jagung dalam ruang dimensi tiga. visualisasi model tanaman jagung yang telah dihasilkan pada penelitian ini lebih menekankan pada penyempurnaan model yang dilakukan pada penelitian sebelumnya terutama pada pewarnaan, pembentukan batang, dan adanya tulang daun pada tanaman jagung setiap iterasinya. model tanaman jagung divisualisasikan mulai dari kecil hingga dewasa (fase vegetatif) yang memiliki tulang daun dan kelengkungan daun berbeda dari daun bawah sampai pada daun atas. tanaman jagung yang divisualisasikan hanya terbatas sampai 8 iterasi saja yang sudah mampu mewakili pertumbuhan tanaman jagung pada fase vegetatif. kata kunci: tanaman jagung (zea mays l.), pemodelan, stochastic l-systems abstract l-systems have the flexibility in structure and development process of simulating plant growth and visually realistic. this research aims to model the growth of corn plants using l-systems and visualize the corn crop growth model from small to mature in three dimensional space. the research was conducted in three stages starting from the identification of data needs by taking action against plant growth of corn (zea mays l.). the second stage, building on the model manually which include the identification and determination of components of l-systems (letters, axioms, and rules of production). the third stage, perform simulations and visualizations of the corn plant growth model has been obtained using processing with java in a three dimensional space. the third stage of the stochastic model generate l-systems of plant growth of corn in three dimensional space. visualization model a corn plant has produced on this research is emphasized on consummation model performed on previous study especially in staining, the formation of stem, and the bone leaves on the corn plant any iterasinya. model a corn plant divisualisasikan ranging from childhood to manhood ( vegetative phase ) that have a curvature leaves and leaves different from lower leaves until the leaves upon. a corn plant divisualisasikan confined to 8 iterating which are able to represent plant growth corn on vegetative phase. keywords: zea mays l., modeling, stochastic l-systems pendahuluan program simulasi berbasis pendekatan metoda l-systems memiliki fleksibiltas dalam mensimulasikan struktur dan proses pengembangan pertumbuhan tanaman secara visual dan realistik beserta faktor-faktor lingkungan yang mempengaruhinya. tahun 1998 fournier and andrieu mengembangkan model pertumbuhan tanaman jagung dalam 3d mulai dari proses tanaman kecil hingga tumbuh menjadi dewasa. batang dan daun tanaman jagung pada gambar 1 menggunakan aproksimasi segitiga (gambar 1.a), dimana batang menggunakan delapan segitiga dan setiap pembentukan bidang pada daun menggunakan dua segitiga, sehingga total seluruh bidang segitiga yang digunakan untuk memodelkan tanaman jagung adalah 29 segitiga. mailto:jo_alkanderi57@yahoo.co.id juhari 50 volume 3 no. 1 november 2013 gambar 1 (a) representasi batang dan daun tanaman jagung; (b) hasil simulasi dan visualisasi tanaman jagung. representasi pemodelan tanaman jagung dengan menggunakan segitiga belum cukup untuk menggambarkan keadaan tanaman yang sesungguhnya karena jika diperhatikan bentuk batang pada hasil model tanaman jagung pada gambar 1 tidak menyerupai batang tanaman jagung sebenarnya yang pada keyataannya lebih cenderung berbentuk silinder/tabung. pewarnaan pada tanaman jagung hasil simulasi pada gambar 1 (b) masih monoton pada satu warna. penelitian lainnya yang sudah dilaksanakan menggunakan l-systems adalah memodelkan bentuk daun menggunakan lsystems dan algoritma genetik [6], memodelkan bentuk–bentuk batang dimensi tiga menggunakan l-systems [3], dan memodelkan morfologi batang tanaman pada dimensi dua dengan l-systems [2]. penelitian tersebut masih memerlukan pengembangan pada tanaman yang lebih kompleks pada dimensi tiga. pengembangan yang akan dilakukan memodelkan tanaman jagung (zea mays l.) tiga dimensi menggunakan sthochastic l-systems. penelitian diawali dari identifikasi kebutuhan data terhadap pemodelan pertumbuhan tanaman jagung. data yang dibutuhkan berupa perkiraan sudut, panjang daun dan jumlah daun dalam satu tanaman jagung. data yang diperoleh digunakan untuk mendesain model tanaman jagung dalam bidang tiga dimensi. desain pemodelan pertumbuhan tanaman jagung, selanjutnya dilakukan pengujian dan validasi simulasi hasil program yang telah didapat. penelitian bertujuan untuk menyusun model dan memvisualisasi tanaman jagung dalam ruang dimensi tiga menggunakan sthochastic l-systems. kajian pustaka a. l-systems beberapa istilah yang menjadi komponen utama pada l-systems adalah: a) huruf huruf adalah himpunan hingga dan simbol-simbol formal, misalnya dalam bentuk dan seterusnya, atau mungkin beberapa huruf lainnya. b) aksioma aksioma (inisiator) adalah suatu string dari simbol-simbol pada . himpunan string dari dinotasikan . jika diberikan , maka beberapa contoh string yang dapat dibentuk yaitu: dan seterusnya. panjang dari suatu string adalah jumlah simbol dalam string. c) produksi produksi (aturan penulisan kembali) adalah suatu pemetaan simbol ke string . ini diberi label dan ditulis dengan notasi: . jika suatu simbol tidak memiliki aturan produksi, maka dapat diasumsikan bahwa simbol tersebut dipetakan pada dirinya sendiri sehingga menjadi konstanta l-systems. [4] tabel 1. generasi l-systems contextsensitive b. penafsiran grafis pada l-systems pada l-systems terdapat simbol-simbol yang dapat ditafsirkan secara grafis. jika diasumsikan suatu satuan panjang dan perputaran sudut , maka perintah-perintah dari simbol-simbol pada l-systems adalah sebagai berikut: : menggambar ke depan satu satuan sepanjang ; : bergerak ke depan satu satuan sepanjang tanpa harus menggambar; : berputar berlawanan arah jarum jam dengan sudut ; : berputar searah jarum jam dengan sudut ; dan | : berputar 180o atau berbalik arah v ,,, cba w v v *v },,{ cbav  ,,,,,, bbccaabaabcacbba w w va  * vw wap : va  a 0 g baaaaaaaa 1 g abaaaaaaa 2 g aabaaaaaa 3 g aaabaaaaa h  f h g h    )a( )b( pemodelan pertumbuhan zea mays l. menggunakan sthochastic l-system jurnal cauchy – issn: 2086-0382 51 penafsirkan l-systems secara grafis dapat diartikan menggambar secara grafis barisan generasi yang dihasilkan dari aksioma dan aturan produksi yang diberikan. contohnya, jika diberikan aksioma dan aturan produksi dengan , dan , maka dimulai dengan aksioma akan diperoleh produksi generasi pertama dengan string: jika diasumsikan bahwa satu satuan sudut adalah 𝜋 3 radian, maka penafsiran grafis dari generasi pertama dapat dilihat pada gambar berikut ini: gambar 2. penafsiran grafis dari l-systems c. percabangan l-systems pada dimensi tiga pada dimensi dua, simbol l-systems yang digunakan hanya berkisar pada f, +, dan -. namun, simbol l-systems pada dimensi dua tidak cukup untuk memvisulaisasikan grafis l-systems pada dimensi tiga. sehingga memerlukan simbol tambahan. bentuk gerakan grafika turtle pada dimensi tiga bergerak dengan arah dan . arah sudut tersusun atas 3 arah bagian yang bertumpuh pada sumbu dan , sedangkan konstanta dari dan dengan cara penambahan dan pengurangan untuk inisialisasi gerakan. penggambaran gerakan dinyatakan dalam sistem koordinat dinotasikan menggunakan 6 notasi yaitu ),,,,,( zyxzyx  . koordinat baru dan dari gerakan dihitung dari perkalian koordinat dari gerakan saat itu dengan rotasi matrik dan . penafsiran grafis l-systems dimensi tiga dapat dilihat pada gambar berikut: gambar 3. penafsiran grafis l-systems 3d matrik rotasi dari gambar 3 memenuhi persamaan sebagai berikut: rzyxzyx ],,[]',','[   dimana r adalah matriks rotasi 33 . secara khusus, rotasi dengan sudut  tentang vektor  yx , dan  z mengikuti aturan:            100 0cossin 0sincos )(    z r                cos0sin 010 sin0cos )( y r              cossin0 sincos0 001 )( x r contoh string percabangan, jika diberikan aksioma dan aturan produksi dengan ]}[,/,\,,^,,,{  fv , dan ffffffffffp ][_][^][]\[:  , maka dimulai dengan aksioma akan diperoleh produksi generasi pertama dengan string: fffffffff ][_][^][]\[  asumsikan bahwa satu satuan sudut rotasi  dan sudut kemiringan cabang  adalah 𝜋 4 radian, maka penafsiran grafis generasi pertama dapat dilihat pada gambar berikut ini : gambar 3. percabangan l-systems pada dimensi tiga objek yang dijadikan penelitian adalah foto tanaman jagung jenis hibrida. lokasi penelitian adalah areal pertanian politeknik negeri jember. pemodelan tanaman jagung dilakukan dalam tiga },,{  fv fw  fffffp : f 1 g ffff   yx, z yx, z yx  , z yx, z ryrx, rz fw  f 1 g 1 g  x  z  y ^   \ /_ 1 2 3 1 2 3 4 1 6 5 3 4 2 0 x 4 \   ^ y z _ juhari 52 volume 3 no. 1 november 2013 tahapan. tahap pertama, pengambilan data penelitian yang berupa pengukuran sudut, panjang daun, dan jumlah daun dalam satu tanaman. tahap kedua, pembuatan model secara manual untuk menentukan dan menafsirkan komponen-komponen l-systems penyusun tanaman jagung. penentuan komponen l-systems meliputi pemilihan huruf yang dipilih mulai dari huruf a sampai z beserta huruf kecil a sampai z serta pembuatan aturan produksi berdasar pada huruf yang telah ditentukan. tahap ketiga, simulasi dan visualisasi tanaman jagung. penentuan simulasi tanaman jagung dilakukan dengan mengubah jumlah iterasi. perubahan jumlah iterasi berfungsi untuk mengatur pertumbuhan batang tanaman sekaligus besar dan banyaknya daun jagung dalam model tersebut. hasil penelitian a. hasil model penelitian pemodelan tanaman jagung dimensi tiga dibangun dengan menggunakan jenis l-systems context free dan stochastic. adapun definisi dari simbol-simbol l-systems dimensi tiga pada tanaman jagung dilihat pada tabel 2 sebagai berikut: tabel 2. simbol – simbol l-systems dimensi tiga pada tanaman jagung huruf arti f maju membuat batang dengan panjang 5 satuan dan tebal 5 satuan dalam sistem koordinat + berputar ke kiri dengan sudut 10o pada bidang xy terhadap sumbu z menggunakan matriks rotasi )( z r berputar ke kanan dengan sudut 10o pada bidang xy terhadap sumbu z menggunakan matriks rotasi )(  z r _ berputar ke kiri dengan sudut 10o pada bidang xz terhadap sumbu y menggunakan matriks rotasi )( y r ^ berputar ke kanan dengan sudut 10o pada bidang xz terhadap sumbu y menggunakan matriks rotasi )(  y r \ berputar ke kiri dengan sudut 10o pada bidang yz terhadap sumbu x menggunakan matriks rotasi )( x r / berputar ke kanan dengan sudut 10o pada bidang yz terhadap sumbu x menggunakan matriks rotasi )(  x r [ menyimpan posisi saat ini dan bergerak sesuai perintah selanjutnya ] kembali ke posisi semula yang disimpan oleh simbol ”[” t bergeser tanpa menggambar (berpindah) berikut semua aturan produksi yang telah dibuat untuk memodelkan tanaman jagung. y k:p xj:p wi:p u g:p rf:p qe:p oc:p nb:p ma:p ks:p j r:p ip:p gn:p fm:p el:p cj:p bi:pac:p1       232221 191817 151413 11109 765 32 zl:p vh:p pd:p lt:p ho:p dk:p       24 20 16 12 8 4 k__l_m],_j__j__j__c_i_j_j_j_^^^[/////////x:p 25 _m]j__j__k__lc_i_j_j_j_^^\^\\\\\\\[\y:p 26 p_r_s_t],n_o_p_p_p_^^^[/////////u:p 27 p_r_s_t]n_o_p_p_p_^^\^\\\\\\\[\q:p 28 _m]j__j__k__lc_i_j_j_j_^^\^\\\\\\\[\w:p 29 p_r_s_t]n_o_p_p_p_^^\^\\\\\\\[\z:p 30 ded:p 31 z]dgw][d[v:p 32 +x]dv+++\\q][\----\d[\b:p 33 y]db---u][-d[a:p 34 +x]da+++-y][//---d[//h:p 35 q]dh---u][-+++d[+g:p 36 [g]---------ttttt [g]---------ttttt-[g]-------ttttt____ttttt_____^^^^^^ ^^^ttttt---------[g] ttttt---------[g]-ttttt-------____[g]ttttt_____^^^^^^^^[g]^--------ttttt[g]---------ttttt [g]---------ttttt ____ttttt_____^^^^^^^^^ttttt ---------[g]ttttt--------[g]ttttt---------[g]f :p37           pemodelan pertumbuhan zea mays l. menggunakan sthochastic l-system jurnal cauchy – issn: 2086-0382 53 b. hasil pemrograman desain visualisasi pertumbuhan tanaman menggunakan l-systems didasarkan pada kerangka prototype hasil pertumbuhan tanaman jagung selama pengukuran. berdasarkan visual grafis pertumbuhan tanaman jagung yang dihasilkan, selanjutnya format grafis digunakan sebagai data visual pertumbuhan tanaman dalam desain antar muka permodelan pertumbuhan tanamaan yang dikembangkan. berikut output hasil program visualisasi tanaman jagung tiga dimensi. gambar 4. hasil output program l-systems tanaman jagung program dijalankan sehingga menghasilkan visualisasi tanaman jagung tiga dimensi baik tanaman individu maupun landscape. berikut hasil visualisasi tanaman jagung individu mulai umur kurang dari 5 hst, 7 hst, 14 hst, 24 hst, 34 hst, 44 hst, dan 54 hst. (hst = hari setelah tanam) gambar 5. hasil visualisasi pertumbuhan tanaman jagung dari kecil hingga dewasa untuk memvisualisasikan tanaman jagung secara landscape, maka tidak perlu mengubah script program, cukup dengan memasukkan semua aturan produksi yang telah dibuat dan jumlah iterasi sesuai dengan yang diinginkan. gambar 6. hasil visualisasi landscape tanaman jagung kesimpulan pertumbuhan tanaman jagung disusun oleh model pertumbuhan tanaman menggunakan lsystems context free dan stochastic. model pertumbuhan tanaman jagung terbentuk dari gabungan aturan produksi, aksioma, serta sudut percabangan daun. komponen tersebut diperoleh dengan melakukan serangkaian proses identifikasi secara manual yang meliputi pengukuran sudut, tinggi tanaman dan jumlah daun pada 5 sampel tanaman jagung yang kemudian dirata-rata; model tanaman jagung divisualisasi dalam ruang dimensi tiga dengan memperhatikan sudut kelengkungan daun setiap iterasinya. hasil visualisasi tanaman jagung ditampilkan secara individu dan landscape. tanaman jagung landscape hasil visualisasi model dibuat 12 tanaman yang mewakili areal tanaman jagung sebenarnya. referensi [1] chuai-aree, s., siripant, s., and lursinsap, c. 2000. animating plant growth in l-system by parametric functional symbols. thailand : department of mathematics. [2] ashlock d., k. m. bryden, and s. p. gent. 2004. simultaneous evolution of bracketed lsystem rules and interpretation. canada : mathematics and statistics university of guelph. [3] luis, d., ding, y., and jingyi, y. 2010. modeling complex unfoliaged trees from a sparse set of images. usa : university of delaware. [4] mishra, j., dan mishra, s. 2007. l-system fractal. netherland : elsevier [5] prusinkiewicz, p. and lindenmayer, a. 1990. the algorithmic beauty of plants. new york : springer-verlag juhari 54 volume 3 no. 1 november 2013 [6] yodthong, r., siripant, s., lursinsap, c. 2005. modeling leaf shapes using l-systems and genetic algorithms. thailand : faculty of engineering, chulalongkorn university. [7] viruchpintu, r and khiripet, n. 2006. realtime 3d plant structure modeling by l-system with actual measurement parameters. thailand : national electronics and computer technology center, pathumthani microsoft word 1 sampul depan.doc 1  super edge-magic labeling pada graph ulat dengan himpunan derajat {1, 4} dan n titik berderajat 4 abdussakir jurusan matematika, fakultas sains dan teknologi universitas islam negeri maulana malik ibrahim malang, indonesia e-mail: abdussakir1975@yahoo.co.id abstrak pelabelan total sisi ajaib super (edge magic total labeling) pada suatu graph (v, e) dengan order p dan ukuran q adalah fungsi bijektif f dari v ∪ e ke himpunan {1, 2, 3, …, p + q} sehingga untuk masing-masing sisi xy di g berlaku f(x) + f(xy) + f(y) = k, dengan k konstanta. pelabelan total sisi ajaib yang memetakan v ke {1, 2, …, p} disebut pelabelan sisi ajaib super (super edge-magic labeling). graph yang dapat dikenakan pelabelan sisi ajaib super disebut graph sisi ajaib super. pada artikel ini akan dijelaskan bahwa graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4, untuk n bilangan asli, adalah sisi ajaib super. kata kunci: graph, pelabelan, total sisi ajaib. 1. pendahuluan graph g adalah pasangan (v, e) dengan v adalah himpunan tidak kosong dan berhingga dari objek-objek yang disebut titik, dan e adalah himpunan (mungkin kosong) pasangan takberurutan dari titik-titik berbeda di v yang disebut sisi. banyaknya unsur di v disebut order dari g dan dilambangkan dengan p(g), dan banyaknya unsur di e disebut ukuran dari g dan dilambangkan dengan q(g). jika yang dibicarakan hanya satu graph, maka order dan ukuran masing-masing akan ditulis p dan q. sisi e = (u, v) dikatakan menghubungkan titik u dan v. jika e = (u, v) adalah sisi di graph g, maka u dan v disebut terhubung langsung, v dan e serta u dan e disebut terkait langsung, dan u, v disebut ujung dari e. derajat dari titik v di graph g, ditulis degg v, adalah banyaknya sisi di g yang terkait langsung dengan v. titik yang berderajat satu disebut titik ujung. untuk selanjutnya, sisi e = (u, v) akan ditulis e = uv. himpunan derajat dari graph g, ditulis dg, adalah himpunan yang memuat derajat semua titik di g. pada graph g berikut, g : diperoleh bahwa degg w = 2, degg x = 2, degg y = 3, dan degg y = 1. jadi himpunan derajat dari graph g adalah dg = {1, 2, 3}. jika yang dibicarakan hanya satu graph, maka himpunan derajat akan ditulis d. jalan u-v dalam graph g adalah barisan berhingga yang berselang-seling w : u = vo, e1, v1, e2, v2, …, en, vn = v antara titik dan sisi, yang dimulai dari titik dan diakhiri dengan titik, dengan ei = vi-1vi adalah sisi di g. v0 disebut titik awal, vn disebut titik akhir, v1, v2, …, vn-1 disebut titik internal, dan n menyatakan panjang w. jalan yang tidak mempunyai sisi disebut jalan trivial. jika v0 = vn, maka w disebut jalan tertutup. jika semua sisi di w berbeda, maka w disebut trail. jika semua titik di w berbeda, maka w disebut lintasan. graph berbentuk lintasan disebut graph lintasan. • • • x z y w • abdussakir  2 volume 1 no. 1 november 2009 2. graph ulat graph ulat (caterpillar) adalah graph yang jika semua titik ujungnya dibuang akan menghasilkan lintasan. perlu diingat kembali bahwa titik ujung adalah titik yang berderajat satu. berikut ini adalah beberapa contoh graph ulat. (a) (b) (c) himpunan derajat pada (a), (b), dan (c) masing-masing adalah {1, 3, 4}, {1, 3, 4}, dan {1, 4}. pada (c) terdapat 3 titik berderajat 4. pada artikel ini, yang akan dibahas adalah graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4, untuk n bilangan asli. graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4 akan memuat sebanyak 2n titik ujung, untuk n bilangan asli. pelabelan total sisi ajaib (edge-magic total labeling) pada suatu graph (v, e) dengan order p dan ukuran q adalah fungsi bijektif f dari v ∪ e ke {1,2,3,…, p+q} sehingga untuk masing-masing sisi xy di g berlaku f(x) + f(xy) + f(y) = k, dengan k konstanta. konstanta k disebut bilangan ajaib. pelabelan total sisi ajaib dapat dimaknai bahwa jumlah label suatu sisi dan label titik yang terkait langsung dengan sisi tersebut adalah sama, untuk semua sisi. graph yang dapat dikenakan pelabelan total sisi ajaib disebut graph total sisi ajaib. pelabelan total sisi ajaib yang memetakan himpunan titik v ke {1,2,…, p} disebut pelabelan sisi ajaib super (super edge-magic labeling). graph yang dapat dikenakan pelabelan sisi ajaib super disebut graph sisi ajaib super. pada contoh berikut, gambar (a) adalah pelabelan total sisi ajaib dan gambar (b) adalah pelabelan sisi ajaib super. konstanta pada gambar (a) adalah 12, sedangkan pada gambar (b) adalah 9. perhatikan pada gambar (b), titik dipetakan pada himpunan {1, 2, 3}. pada artikel ini akan dijelaskan bahwa graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4 adalah sisi ajaib super, untuk n bilangan asli. dengan tujuan mempermudah penulisan, dalam artikel ini graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4 akan disimbolkan dengan un. untuk membuktikan bahwa un adalah sisi ajaib super, perlu ditunjukkan adanya suatu fungsi bijektif f dari v(un) ∪ e(un) ke {1, 2, 3, …, ⏐v(un)⏐+ ⏐e(un)⏐} sehingga untuk masing-masing sisi xy di g berlaku f(x) + f(xy) + f(y) = k, dengan k adalah konstanta. selain itu, perlu ditunjukkan bahwa f memetakan v(un) ke {1, 2, 3, …, ⏐v(un)⏐}. 3. pembahasan pembahasan bahwa graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4 adalah sisi ajaib super, untuk n bilangan asli, disajikan dalam teorema berikut beserta buktinya, dan beberapa contoh sebagai aplikasi fungsi/pelabelan yang disajikan dalam teorema. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 4 6 5 2 1 3 • • • 1 3 2 5 4 6 (a) (b)  super edge­magic labelling pada graph ulat dengan…  volume 1 no. 1 november 2009 3 teorema 1. graph ulat dengan himpunan derajat d = {1, 4} dan n titik berderajat 4 adalah sisi ajaib super, untuk n bilangan asli. bukti: graph ulat un dengan himpunan derajat d = {1, 4} dan n titik berderajat 4, n bilangan asli, dapat digambar sebagai berikut. un : berdasarkan gambar, himpunan titik pada un adalah v(un) = { x1, x2, x3,…, xn, y1, y2, y3, …, yn + 2, z1, z2, z3, …, zn} dan himpunan sisi pada un adalah e(un) = { x1y2, x2y3, …, xnyn + 1, y1y2, y2y3, …, yn + 1yn + 2, z1y2, z2y3, …, znyn + 1}. jadi, order dari un adalah p(un) = ⏐v(un)⏐= 3n + 2 dan ukuran dari un adalah q(un) = ⏐e(un)⏐= 3n + 1. jadi, p(un) + q(un) = 6n + 3. dalam hal ini terdapat dua kasus, yaitu untuk n bilangan asli ganjil dan n bilangan asli genap. a. pelabelan pada un, dengan n bilangan asli ganjil definisikan fungsi f dari v(un) ∪ e(un) ke {1, 2, 3, …, 6n + 3} sebagai berikut. f(xi) = 2 13 +i , untuk i ganjil dan 1 ≤ i ≤ n. f(xi) = 2 333 ++ in , untuk i genap dan 1 ≤ i ≤ n. f(yi) = 2 13 −i , untuk i ganjil dan 1 ≤ i ≤ n + 2. f(yi) = 2 133 ++ in , untuk i genap dan 1 ≤ i ≤ n + 2. f(zi) = 2 33 +i , untuk i ganjil dan 1 ≤ i ≤ n. f(zi) = 2 533 ++ in , untuk i genap dan 1 ≤ i ≤ n. f(yiyi + 1) = 6n – 3i + 6, 1 ≤ i ≤ n + 1. f(xiyi + 1) = 6n – 3i + 5, 1 ≤ i ≤ n. f(ziyi + 1) = 6n – 3i + 4, 1 ≤ i ≤ n. dengan mengecek pada masing-masing interval indeks titik (genap atau ganjil) akan dapat ditunjukkan bahwa f adalah fungsi injektif. karena f fungsi injektif dengan domain dan kodomain yang mempunyai banyak anggota sama dan berhingga, maka f adalah fungsi surjektif. jadi f adalah fungsi bijektif. (1) untuk 1 ≤ i ≤ n + 1 dan i ganjil diperoleh f(yi) + f(yiyi + 1) + f(yi + 1) = ( 2 13 −i ) + (6n – 3i + 6) + ( 2 1)1(33 +++ in ) = 2 1515 +n • • • • • • • • • • • • • • … x1 x2 x3 … xn z1 z2 z3 … zn y1 y2 y3 y4 yn +1 yn + 2 abdussakir  4 volume 1 no. 1 november 2009 (2) untuk 1 ≤ i ≤ n + 1 dan i genap diperoleh f(yi) + f(yiyi + 1) + f(yi + 1) = ( 2 133 ++ in ) + (6n – 3i + 6) + ( 2 1)1(3 −+i ) = 2 1515 +n (3) untuk 1 ≤ i ≤ n dan i ganjil diperoleh f(xi) + f(xiyi + 1) + f(yi + 1) = ( 2 13 +i ) + (6n – 3i + 5) + ( 2 1)1(33 +++ in ) = 2 1515 +n (4) untuk 1 ≤ i ≤ n dan i genap diperoleh f(xi) + f(xiyi + 1) + f(yi + 1) = ( 2 333 ++ in ) + (6n – 3i + 5) + ( 2 1)1(3 −+i ) = 2 1515 +n (5) untuk 1 ≤ i ≤ n dan i ganjil diperoleh f(zi) + f(ziyi + 1) + f(yi + 1) = ( 2 33 +i ) + (6n – 3i + 4) + ( 2 1)1(33 +++ in ) = 2 1515 +n (6) untuk 1 ≤ i ≤ n dan i genap diperoleh f(zi) + f(ziyi + 1) + f(yi + 1) = ( 2 533 ++ in ) + (6n – 3i + 4) + ( 2 1)1(3 −+i ) = 2 1515 +n dengan demikian diperoleh bahwa un adalah total sisi ajaib dengan bilangan ajaib k = 2 1515 +n . selanjutnya dapat ditunjukkan bahwa f memetakan v(un) ke {1, 2, 3, …, 3n + 2}. jadi f adalah pelabelan sisi ajaib super pada un dengan n bilangan asli ganjil. b. pelabelan pada un, dengan n bilangan asli genap definisikan fungsi f dari v(un) ∪ e(un) ke {1, 2, 3, …, 6n + 3} sebagai berikut. f(xi) = 2 13 +i , untuk i ganjil dan 1 ≤ i ≤ n. f(xi) = 2 33 in + , untuk i genap dan 1 ≤ i ≤ n. f(yi) = 2 13 −i , untuk i ganjil dan 1 ≤ i ≤ n + 2. f(yi) = 2 233 −+ in , untuk i genap dan 1 ≤ i ≤ n + 2. f(zi) = 2 33 +i , untuk i ganjil dan 1 ≤ i ≤ n. f(zi) = 2 233 ++ in , untuk i genap dan 1 ≤ i ≤ n. f(yiyi + 1) = 6n – 3i + 6, 1 ≤ i ≤ n + 1. f(xiyi + 1) = 6n – 3i + 5, 1 ≤ i ≤ n.  super edge­magic labelling pada graph ulat dengan…  volume 1 no. 1 november 2009 5 f(ziyi + 1) = 6n – 3i + 4, 1 ≤ i ≤ n. dengan mengecek pada masing-masing interval indeks titik (genap atau ganjil) akan dapat ditunjukkan bahwa f adalah fungsi injektif. karena f fungsi injektif dengan domain dan kodomain yang mempunyai banyak anggota sama dan berhingga, maka f adalah fungsi surjektif. jadi f adalah fungsi bijektif. (1) untuk 1 ≤ i ≤ n + 1 dan i ganjil diperoleh f(yi) + f(yiyi + 1) + f(yi + 1) = ( 2 13 −i ) + (6n – 3i + 6) + ( 2 2)1(33 −++ in ) = 2 1215 +n (2) untuk 1 ≤ i ≤ n + 1 dan i genap diperoleh f(yi) + f(yiyi + 1) + f(yi + 1) = ( 2 233 −+ in ) + (6n – 3i + 6) + ( 2 1)1(3 −+i ) = 2 1215 +n (3) untuk 1 ≤ i ≤ n dan i ganjil diperoleh f(xi) + f(xiyi + 1) + f(yi + 1) = ( 2 13 +i ) + (6n – 3i + 5) + ( 2 2)1(33 −++ in ) = 2 1215 +n (4) untuk 1 ≤ i ≤ n dan i genap diperoleh f(xi) + f(xiyi + 1) + f(yi + 1) = ( 2 33 in + ) + (6n – 3i + 5) + ( 2 1)1(3 −+i ) = 2 1215 +n (5) untuk 1 ≤ i ≤ n dan i ganjil diperoleh f(zi) + f(ziyi + 1) + f(yi + 1) = ( 2 33 +i ) + (6n – 3i + 4) + ( 2 2)1(33 −++ in ) = 2 1215 +n (6) untuk 1 ≤ i ≤ n dan i genap diperoleh f(zi) + f(ziyi + 1) + f(yi + 1) = ( 2 233 ++ in ) + (6n – 3i + 4) + ( 2 1)1(3 −+i ) = 2 1215 +n dengan demikian diperoleh bahwa un adalah total sisi ajaib dengan bilangan ajaib k = 2 1215 +n . selanjutnya dapat ditunjukkan bahwa f memetakan v(un) ke {1, 2, 3, …, 3n + 2}. jadi f adalah pelabelan sisi ajaib super pada un dengan n bilangan asli genap. berdasarkan dua kasus pada n, maka diperoleh bahwa graph ulat un dengan himpunan derajat {1, 4} dan n titik berderajat empat adalah sisi ajaib super, untuk semua n bilangan asli. berikut ini contoh pelabelan sis ajaib super pada graph ulat u3 dan u4 menggunakan teorema di atas. perhatikan bahwa label titik dan sisi sesuai dengan rumus pada teorema. abdussakir  6 volume 1 no. 1 november 2009 u3 : pada u3, diperoleh bahwa bilangan ajaib adalah 30 = 2 15)3(15 + . u4 : pada u4, diperoleh bahwa bilangan ajaib adalah 36 = 2 12)4(15 + . 4. kesimpulan berdasarkan pembahasan dapat disimpulkan bahwa bahwa graph ulat un dengan himpunan derajat {1, 4} dan n titik berderajat empat adalah sisi ajaib super, untuk semua n bilangan asli. untuk n bilangan asli ganjil dan genap masing-masing bilangan ajaib adalah k = = 2 1515 +n dan k = 2 1215 +n . pelabelan seperti yang dijelaskan dalam teorema dimungkinkan bukan satu-satu pelabelan sisi ajaib super pada graph ulat un. dengan demikian, disarankan kepada pembaca untuk mencari rumus pelabelan yang berbeda pada graph ulat un. selain itu, karena graph ulat banyak jenisnya, maka disarankan kepada pembaca untuk melakukan penelitian pada beberapa jenis graph ulat yang lain. daftar pustaka bondy, j.a. & murty, u.s.r., (1976), graph theory with applications, the macmillan press ltd, london. chartrand, g. & lesniak, l., (1986), graph and digraph, 2nd edition, california: wadsworth, inc. miller, mirka., (2000), open problems in graph theory: labelings and extremal graphs, prosiding konferensi nasional x matematika di bandung, juli, 17-20. • • • • • • • • • • • 2 9 5 3 10 6 1 21 8 18 4 15 11 12 7 20 17 14 19 16 13 • • • • • • • • • • • • • • 2 9 5 12 3 10 6 13 1 27 8 24 4 21 11 18 7 15 14 26 23 20 17 25 22 19 16 issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 cauchy vol. 7 no. 3 pages: 332 – 492 malang november 2022 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines muhammad fakhruddin, department of mathematics, faculty of military mathematics and natural sciences, the republic of indonesia defense university, bogor, indonesia alfi yusrotis zakiyyah, universitas bina nusantara, indonesia bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia dr heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 issn : 2086-0382 e-issn : 2477-3344 editorial board subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 7, issue 3, november 2022 issn : 2086-0382 e-issn : 2477-3344 table of contents a monte carlo simulation study to assess estimation methods in cfa on ordinal data ................................................................................................................................................. 332 – 344 comparisons between resampling techniques in linear regression: a simulation study ........................................................................................................................ 345 – 353 η-intuitionistic fuzzy soft groups ............................................................................................ 354 – 361 actuarial modeling of covid-19 insurance .......................................................................... 362 – 369 mathematics model of covid-19 with two-stage vaccination, symptomatic, asymptomatic, and quarantine individuals ................................................................... 370 – 383 bayesian hurdle poisson regression for assumption violation ................................... 384 – 393 levi decomposition of frobenius lie algebra of dimension 6 394 – 400 c-type ops transformation ........................................................................................................ 401 – 410 confidence intervals for the mean function of a compound cyclic poisson process in the presence of power function trend ....................................................... 411 – 419 forecasting population of madiun regency using arima method ............................. 420 – 431 on graceful chromatic number of vertex amalgamation of tree graph family 432 444 mathematical model of iteroparous and semelparous species interaction ............. 445 – 463 optimalization route to tourism places in west java using a-star algorithm ... 464 – 473 a mixed integer linear programming model of order allocation involving mass customization logistic service (mcls) ............................................................................ 474 – 482 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages .......................... 483 – 492 2 ari kusumastuti pengenalan pola pengenalan pola gelombang khas dengan interpolasi ari kusumastuti dosen jurusan matematika fakultas sains dan teknologi universitas islam negeri (uin) maulana malik ibrahim malang e-mail: arikusumastuti@gmail.com abstrak pengenalan bentuk khas gelombang merupakan masalah yang penting dalam pencitraan suatu bentuk objek yang bervibrasi. prosedur pengenalan bentuk khas gelombang teridentifikasi dengan suatu fourier transform infra red. turunan kedua ftir merupakan pengenalan bentuk gelombang di daerah sidik jari objek. permasalahan yang muncul adalah belum teridentifikasi secara detail bentuk khas gelombang tersebut secara visual pada turunan kedua ftir. penelitian ini berupaya memberikan jawaban terhadap pencitraan secara detai bentuk gelombang hasil turunan kedua ftir dengan pendekatan interpolasi. prosedur interpolasi akan membaca kembali data berpasangan pada turunan kedua ftir sehingga terbaca bentuk khas gelombang objek. data berpasangan yang dimaksud adalah bilangan gelombang dan penyerapan. studi kasus penelitian ini menggunakan data spektra objek yang selanjutnya akan terbaca bentuk khas gelombangnya secara unik dengan interpolasi. keywords: interpolasi, bentuk khas gelombang. pendahuluan kajian pada bidang matematika terapan sangat bermanfaat dalam menjawab permasalahan yang muncul di luar bidang matematika. banyak permasalahan yang muncul dari berbagai latar belakang disiplin ilmu lain yang penting untuk dianalisis. permasalahan identifikasi bentuk vibrasi molekul, misalnya merupakan topik yang sangat membutuhkan peran matematika pada analisis lanjutan. penelitian pada bidang bioteknologi menggunakan fourier transform infra red untuk mendeteksi vibrasi molekuler sampai di tingkat sidik jari. hasil pembacaan ftir ini menghasilkan data bilangan gelombang (wave number) dan penyerapan (absorbansi). kelemahan ftir adalah tidak teridentifikasi visual secara detail bentuk gelombang khas suatu objek sampai di tingkat sidik jari. interpolasi merupakan teknik peramalan fungsi dari suatu data berpasangan. pada penelitian ini masalah interpolasi digunakan sebagai alat untuk mempertajam pengenalan pola gelombang di level sidik jari dari data ftir. prosedur interpolasi ini dapat mengenali bentuk khas gelombang secara detail sehingga setiap objek teridentifikasi secara unik bentuk gelombang khasnya. prosedur yang digunakan pada penelitian ini adalah; 1. pengujian secara statistik data turunan kedua ftir. uji data dilakukan untuk mendapatkan data berdistribusi normal dan uji pengaruh untuk mendapatkan data yang tidak dipengaruhi oleh waktu pengambilan sampel. 2. meramalkan data dengan interpolasi. penelitian ini ditujukan untuk mendapatkan ramalan secara detail bentuk khas gelombang pada suatu objek secara unik. kajian pustaka interpolasi interpolasi memainkan peranan yang sangat penting dalam metode numerik. fungsi yang tampak rumit menjadi lebih sederhana bila dinyatakan dalam polinom interpolasi. interpolasi berguna untuk menaksir harga-harga tengah antara titik data yang sudah tepat. interpolasi mempunyai orde atau derajat. interpolasi ada beberapa macam yaitu, interpolasi beda terbagi newton, interpolasi lagrange, interpolasi spline (munir, 2006). triatmodjo (2002) menambahkan, dalam interpolasi dicari suatu nilai yang berada diantara beberapa titik data yang telah diketahui nilainya. untuk dapat memperkirakan nilai tersebut, pertama kali dibuat suatu fungsi atau persamaan yang melalui titik-titik data. setelah persamaan kurva terbentuk, kemudian dihitung nilai fungsi yang berada diantara titik-titik data. interpolasi lagrange digunakan untuk mencari titik-titik antara dari n buah titik p1(x1,y1), p2(x2,y2), p3(x3,y3), …, pn(xn,yn) dengan menggunakan pendekatan fungsi polynomial yang disusun dalam kombinasi deret dan didefinisikan dengan: ari kusumastuti 8 volume 2 no. 1 november 2011 daerah penolakan h0 ( ); 1; 1k k nfα − − daerah penerimaan h0 kesulitan utama yang muncul dari proses interpolasi adalah teknis komputasi. oleh karena itu perlu suatu mekanisme pendukung. software matlab dapat digunakan untuk mempermudah pelaksanaan perhitungan interpolasi, bahkan sampai dengan penyusunan fungsi dan penggambaran grafiknya (djojodihardjo, 2000). deret taylor deret taylor merupakan dasar untuk menyelesaikan masalah dalam metode numerik, terutama penyelesaian persamaan diferensial. deret taylor akan memberikan suatu fungsi dengan benar jika semua suku dari deret tersebut diperhitungkan. persamaan deret taylor (triatmojo, 2002): analysis of variance analisis of variance atau sering dikenal dengan anova digunakan untuk menyelidiki hubungan antara variabel respons (dependen) dengan 1 atau beberapa variabel prediktor (independen) (irawan dan astuti, 2006). anova pada dasarnya terdiri dari dua kelompok. pengelompokan ditentukan dari jumlah variabel bebasnya. bila variabel yang akan dianalisis terdiri dari satu variabel terikat dan satu variabel bebas disebut anova satu arah (one way anova). bila variabel yang akan dianalisis terdiri dari satu variabel terikat dan lebih dari variabel bebas disebut dengan anova dua arah (two way anova) (hartono, 2004). one way anova digunakan untuk mengetahui apakah data dari sampel yang ada sudah cukup kuat untuk menggambarkan populasinya, atau apakah bisa suatu dilakukan generalisasi tentang populasi berdasarkan hasil sampel (harini, 2010). irawan dan astuti (2006) menambahkan bahwa, jika hasil analisa diperoleh p-value < α, maka variabel prediktor tersebut mempunyai hubungan yang kuat, tetapi jika nilai p-value yang diperoleh > α, maka variabel prediktor tersebut tidak ada hubungan dengan variabel respons. output analisis anova ditampilkan dalam window session dengan hipotesis: h0 : sampel tiap perlakuan sama (µ1 = µ2) h1 : ada perlakuan yang tidak sama hipotesis awal akan ditolak apabila nilai f hitung melebihi fα, k-1, k(n-1), dimana α adalah tingkat kesalahan, k adalah banyak replikasi dan n adalah banyaknya perlakuan. nilai nya dapat dilihat pada table. selain menggunakan nilai f, dapat juga dilihat dari nilai p-value yang diperoleh. hipotesis awal akan ditolak apabila nilai p-value kurang dari α (irawan dan astuti, 2006). gambar 1. grafik daerah penolakan untuk fα, k-1, k(n-1) (irawan dan astuti, 2006) analisis korelasi korelasi adalah hubungan, begitu pula dengan analisis korelasi yaitu suatu analisis yang digunakan untuk melihat hubungan antara dua variabel atau lebih (odi, 2008). analisis korelasi ada beberapa jenis, salah satunya adalah korelasi pearson product moment (riduwan dan sunarto, 2009). irawan dan astuti (2006) menambahkan bahwa, koefisien korelasi pearson berguna untuk mengukur tingkat keeratan hubungan linear antara 2 variabel. korelasi pearson product moment (ppm) dilambangkan “r” dengan ketentuan nilai r tidak lebih dari harga (-1 ≤ r ≤ +1). tabel interpretasi nilai r sebagai berikut: tabel 1. interpretasi koefisien nilai r interval koefisien tingkat hubungan 0,80 1,000 sangat kuat 0,60 0,799 kuat 0,40 0,599 cukup kuat 0,20 0,399 rendah 0,00 0,199 sangat rendah sumber: riduwan dan sunarto, 2009 nilai korelasi berkisar antara -1 sampai +1. nilai korelasi negative berarti hubungan antara 2 variabel adalah negatif. artinya, apabila salah satu variabel menurun, maka variabel lainnya akan meningkat. sebaliknya, nilai korelasi positif berarti hubungan antara kedua variabel adalah positif. artinya, apabila salah satu variabel meningkat, maka variabel lainnya akan meningkat pula dan apabila nilai korelasi bernilai pengenalan pola gelombang khas dengan interpolasi jurnal cauchy – issn: 2086-0382 9 0, artinya tidak ada korelasi (irawan dan astuti, 2006). second derivative (2d) program menghitung turunan numerik sangat sederhana. rumus-rumus turunan dinyatakan sebagai fungsi (munir, 2006). derivative dapat digunakan untuk mengumpulkan informasi tentang grafik fungsi. karena derivative menunjukkan tingkat perubahan dari suatu fungsi, untuk menentukan dimana suatu fungsi naik, maka hanya memeriksa dimana derivativenya positif. dengan cara yang sama, untuk menemukan dimana suatu fungsi turun, maka hanya memeriksa dimana derivativenya negatif. titik dimana derivative sama dengan 0 disebut titik kritis. pada titik-titik ini, fungsi itu konstan dan grafiknya horizontal (barroroh, 2009). dengan membuat turunan spektra, visualisasi dari pantulan spektra dapat ditingkatkan, sehingga pengujian yang lebih baik dapat dimungkinkan. analisis pada turunan pertama, sangat bermanfaat untuk menempatkan posisi dari puncak, lembah, dan red-edge inflection point (r-eip). turunan kedua dimaksudkan untuk menentukan posisi dari r-eip. r-eip adalah spektral region pada batas antara panjang gelombang merah dan infra merah di mana nilai spektral vegetasi meningkat tajam (ustin et al., 2000 dalam hartini, 2001). perbedaan dari posisi puncak, lembah dan r-eip digunakan untuk menjelaskan sifat dari vegetasi. gambar 2. posisi dari puncak (p), lembah (t) dan red-edge inflection point (r-eip) pada plot pantulan spektral vegetasi (hartini, 2001) pengujian derivative pertama dan kedua secara esensial memberlakukan logika yang sama, yaitu menjelaskan apa yang terjadi pada derivative f’(x) didekat suatu titik kritis x0. pengujian derivative pertama mengatakan bahwa maksima dan minima itu berpasangan denfan f’ melintasi nol dari satu arah ke arah yang lain, yang ditunjukkan oleh tanda dari f’ dekat x0. ari kusumastuti 10 volume 2 no. 1 november 2011 pengujian derivative kedua hanyalah pengamatan dengan informasi yang sama ditunjukkan pada kemiringan dari garis singgung f’(x) dititik x0 (barroroh, 2009). spektroskopi infra merah spektroskopi infra merah merupakan salah satu alat yang banyak dipakai untuk mengidentifikasi senyawa baik alami maupun buatan. bila sinar infra merah dilewatkan melalui cuplikan senyawa organik, maka sejumlah frekuensi akan diserap sedang frekuensi yang lain diteruskan atau ditransmisikan tanpa diserap. gambaran antara persen absorbansi atau persen transmitansi lawan frekuensi akan menghasilkan suatu spektrum infra merah. transisi yang terjadi didalam serapan infra merah berkaitan dengan perubahan-perubahan vibrasi dalam molekul (sastrohamidjojo, 2001). daerah radiasi spektroskopi infra merah berkisar pada bilangan gelombang 1280-10 cm-1 atau pada panjang gelombang 0,78-1000 μm (khopkar 1990). hayati (2007) menambahkan bahwa, dilihat dari segi aplikasi dan instrumentasi spektroskopi infra merah dibagi ke dalam tiga jenis radiasi yaitu infra merah dekat, infra merah pertengahan, dan infra merah jauh. daerah spektroskopi infra merah dapat dilihat pada tabel 2. tabel 2. daerah spektroskopi infra merah daerah panjang gelombang μm bilangan gelombang cm-1 dekat 0.78-2.5 12800-4000 pertengahan 2.5-50 4000-200 jauh 50-100 200-10 sumber: hayati (2007) energi dalam spektroskopi infra merah dibutuhkan untuk transisi vibrasi, maka radiasi infra merah hanya terbatas pada perubahan energi setingkat molekul. untuk tingkat molekul, perbedaan dalam keadaan vibrasi dan rotasi digunakan untuk mengadsorbsi sinar infra merah. jadi untuk dapat mengadsorbsi, molekul harus memiliki perubahan momen dipol sebagai akibat dari vibrasi. radiasi medan listrik yang berubah-ubah akan berinteraksi dengan molekul dan akan menyebabkan amplitudo salah satu gerakan molekul (khopkar, 1990). ada 2 jenis instrmentasi untuk absorbsi infra merah yaitu, instrumentasi dispersi (konvensional) yang hanya digunakan untuk analisis kualitatif dan instrumentasi yang menggunakan fourier transform (ftir) dapat digunakan untk analisis kuantitatif dan kualitatif (hayati, 2007). spektroskopi ftir (fourier transform infrared) merupakan salah satu teknik analitik yang sangat baik dalam proses identifikasi struktur molekul suatu senyawa. komponen utama spektroskopi ftir adalah interferometer michelson yang mempunyai fungsi menguraikan (mendispersi) radiasi infra merah menjadi komponen-komponen frekuensi. penggunaan interferometer michelson tersebut memberikan keunggulan metode ftir dibandingkan metode spektroskopi infra merah konvensional maupun metode spektroskopi yang lain. diantaranya adalah informasi struktur molekul dapat diperoleh secara tepat dan akurat (memiliki resolusi yang tinggi). keuntungan yang lain dari metode ini adalah dapat digunakan untuk mengidentifikasi sampel dalam berbagai fase (gas, padat atau cair). kesulitan-kesulitan yang ditemukan dalam identifikasi dengan spektroskopi ftir dapat ditunjang dengan data yang diperoleh dengan menggunakan metode spektroskopi yang lain (harmita, 2006). delwiche, et al (2007) telah berhasil mengukur jumlah protein glicinin dan βconglicinin yang terdapat pada biji kedelai menggunakan near-infrared spectroscopy (nir) sampai pada batas screening. sebelumnya protein ini biasa dipisahkan melalui metode ultrasentrifugasi dan elektroforesis. mossoba, et al (2007) juga telah melakukan penelitian tentang pengukuran kuantitatif asam lemak trans menggunakan spektroskopi infra merah. metode yang digunakan yaitu melalui pengukuran ketinggian pita absorbsi asam lemak trans pada 966 cm-1 menggunakan metode second derivative (2d). metode ini berhasil mengidentifikasi dan memisahkan adanya interferensi pita pada 962956 cm-1 yang dimilki lemak jenuh pada pita asam lemak trans pada 966 cm-1. keberhasilan pemisahan pita interferensi ini dapat meningkatkan sensitivitas dan akurasi penentuan asam lemak trans pada konsentrasi rendah (≤ 0.5% dari lemak total) (barroroh, 2009). pembagian daerah spektra infra merah daerah spektra infra merah dapat dibagi menjadi 2, yaitu (mudasir dan candra, 2008): 1. daerah frekuensi gugus fungsional terletak pada daerah radiasi 4000–1400 cm-1. pita-pita absorpsi pada daerah ini utamanya disebabkan oleh vibrasi dua atom, sedangkan frekuensinya karakteristik terhadap massa atom yang berikatan dan konstanta gaya ikatan. 2. daerah sidik jari (fingerprint) yaitu daerah yang terletak pada 1400–400 cm-1. pita-pita absorpsi pada daerah ini pengenalan pola gelombang khas dengan interpolasi jurnal cauchy – issn: 2086-0382 11 berhubungan dengan vibrasi molekul secara keseluruhan. setiap atom dalam molekul akan saling mempengaruhi sehingga dihasilkan pita-pita absorpsi yang khas untuk setiap molekul. menurut hayati (2007), spektroskopi infra merah mengandung banyak serapan yang berhubungan dengan sistem vibrasi yang berinteraksi dalam suatu molekul akan memberikan puncak-puncak yang sangat karakteristik dalam spektra. corak puncak ini dikenal sebagai “sidik jari” molekul yang merupakan daerah yang mengandung sejumlah besar vibrasi yang tidak dapat dimengerti. dengan membandingkan spektra infra merah dari dua senyawa yang diperkirakan identik maka dapat dinyatakan kedua senyawa tersebut identik atau tidak. akan jauh lebih sulit untuk membedakan ikatan-ikatan tertentu dalam area sidik jari daripada dalam area yang lebih ‘bersih’ yang berada dalam area dengan bilangan gelombang yang lebih besar. hal penting dalam area sidik jari ini adalah setiap senyawa yang berbeda menghasilkan pola lembah yang berbeda-beda pada spektrum bagian ini. hasil dan pembahasan data ftir gambar 3. data second derivation ftir gambar 4. hasil pengenalan pola dengan interpolasi -0.001 -0.0008 -0.0006 -0.0004 -0.0002 0 0.0002 0.0004 0.0006 0.0008 0 500 1000 1500 2000 2500 3000 3500 4000 4500 ari kusumastuti 12 volume 2 no. 1 november 2011 (a). interpolasi (b). perbesaran second derivative gambar 5. a) hasil analisis interpolasi dan (b) perbesaran second derivative. ekspansi deret taylor dari data second derivation ftir: y = 4.7907e-12 x8 – 3.7484e-8 x7 + 1.2831e-4 x6 – 0.2510 x5 + 306.8267 x4 – 2.4006e+4 x3 + 1.1739e+8 x2 – 3.2802e+10 x + 4.0100e+12. penutup pada akhir penelitian ini prosedur interpolasi mampu membaca secara lebih detail dibandingkan dengan second derivation ftir. selanjutnya analisis syaraf tiruan dapat di kerjakan sehingga bentuk spectra gelombang objek mampu dikenali untuk mengganti ftir yang relatif mahal. prosedur pemberian bobot yang efektif pada langkah jst mampu mengenali pola gelombang suatu objek dengan baik. daftar pustaka [1] mathews, j.h., (1999). numerical methods using matlab [2] chopra, (2002) numerical methods for engineering, mc graw hill [3] naseem (2010), fundamental numerical analysis and error estimation, anamaya publisher, new delhi [4] barroroh (2009). identifikasi pola khas spektra inframerah dengan second derivative. penelitian [5] hayati (2007). dasar-dasar analisis spektroskopi [6] harini(2010). praktikum statistik elementer [7] riduan (2009). pengantar statistik untuk penelitian. alfabeta, bandung [8] sastrohadimidjojo(2001). spektroskopi. liberty, yogyakarta -0.00001 -0.00001 -0.00000 0 0.000005 0.00001 965 970 975 980 985 990 s ec on d d er iv at iv e (2 d ) bilangan gelombang 1 abdul aziz analisis critical value analisis critical root value pada data nonstationer abdul aziz dosen jurusan matematika fakultas sains dan teknologi universitas islam negeri (uin) maulana malik ibrahim malang e-mail : abdulaziz_uinmlg@yahoo.com abstract a stationery process can be done t-test, on the contrary at non stationery process t-test cannot be done again because critical value of this process isn’t t-distribution. at this research, we will do simulation of time series ar(1) data in four non stationery models and doing unit root test to know critical value at ttest of non stationery process. from the research is yielded that distribution of critical point for t-test of non stationery process comes near to normal with restating simulation of random walk process which ever greater. result of acquirement of this critical point has come near to result of dickey-fuller test. from this research has been obtained critical point for third case which has not available at tables result of dickey-fuller test. key words: non stationery, unit root test, critical value, distribution, simulation abstrak pada sebuah data stationer dapat dilakukan t-test, sebaliknya pada data nonstationer t-test tidak dapat dilakukan lagi karena critical root value (titik akar kritis) untuk proses ini tidak berdistribusi t. pada penelitian ini, kami akan melakukan simulasi data time series ar(1) dalam empat model nonstationer dan dilakukan unit root test untuk mengetahui critical value pada t-test proses nonstationer. dari penelitian dihasilkan bahwa distribusi titik kritis untuk t-test proses nonstationer mendekati normal dengan perulangan simulasi proses random walk yang semakin besar. hasil perolehan titik kritis ini sudah mendekati dari hasil dickey-fuller test. dari penelitian ini telah diperoleh titik akar kritis untuk kasus ketiga yang belum ada di tabel hasil dickey-fuller test. kata kunci: nonstationer, unit root test, critical root value, distribusi, simulasi pendahuluan multivariate time series banyak dipakai dalam permodelan ekonomi. dengan beberapa time series yang saling berpengaruh sehingga membentuk suatu model time series baru yang dinamakan sebagai vector autoregression (var). pada proses stationer dapat dilakukan t-test, sebaliknya pada proses nonstationer t-test tidak dapat dilakukan lagi karena critical value untuk proses ini tidak berdistribusi t. pada penelitian ini, kami akan melakukan simulasi data time series ar(1) nonstationer dan dilakukan unit root test untuk mengetahui critcal value pada t-test proses nonstationer. berdasarkan latar belakang tersebut, maka pada penelitian ini kami merumuskan permasalahan yaitu bagaimana critical value dan distribusinya untuk t-test pada proses nonstationer secara simulasi dengan kasus: a. true process: 1t t ty y e−= + dengan estimated regression: 1t t ty y eβ −= + b. true process: 1t t ty y e−= + dengan estimated regression: 0 1 1t t ty y eβ β −= + + c. true process: 0 1t t ty y eβ −= + + dengan estimated regression: 0 1 1t t ty y eβ β −= + + d. true process: 0 1t t ty y eβ −= + + dengan estimated regression: 0 1 1 2t t ty y t eβ β β−= + + + masing-masing model dengan siginifikansi kepercayaan 5% dan 10%. metode yang akan digunakan pada penelitian ini adalah metoda simulasi. unit root test dilakukan dengan perulangan data simulasi menggunakan software matlab versi 6.5., hingga diperoleh suatu nilai yang konvergen. kajian pustaka misalkan suatu proses time series ar(1), 1t t ty y eβ −= + (1) dengan et white noise. jika |β| < 1 maka ma(∞) representasinya adalah 1 i t t i i y eβ ∞ − = = ∑ (2) abdul aziz 2 volume 2 no. 1 november 2011 dengan [ ] 0te y = (3) dan ( ) 2 21 tvar y σ β = − (4) yang bebas dari variabel t, sehingga dikatakan sebagai stationary process. sebaliknya, jika β = 1 maka ma(∞) representasinya adalah 1 0 t t i i i i y e e ∞ ∞ − = = = =∑ ∑ (5) dengan [ ] 0te y = (6) dan ( ) 2tvar y tσ= (7) yang merupakan fungsi dari variabel t, sehingga dikatakan sebagai nonstationary process. jika {yt} stationary process maka hipotesis 0 1: 0, : 0h hβ β= ≠ (8) adalah t-test valid. sebaliknya, hipotesis 0 1: 1, : 1h hβ β= < (9) adalah t-test yang tidak valid, karena {yt} adalah proses nonstasioner di bawah h0. untuk melakukan uji t dengan hipotesis kedua di atas perlu dibuat critical value tersendiri agar menjadi valid. critical value t-test untuk proses nonstasioner dapat diperoleh dengan dua cara: 1. menggunakan estimasi koefisien autoregressive �( )1t t β= − (10) 2. menggunakan estimasi ols terhadap residual variance � �( ) 1 testt var β β − = (11) dengan � ( ) 1' 'x x x yβ −= (12) dan �( ) � ( ) ( )1 12 '' ' 1 e e cov x x x x t β σ − −= = − ɵ ɵ (13) yang distribusinya dapat diketahui secara simulasi dengan siginifikansi kepercayaan tertentu dengan aturan tolak hipotesis null jika nilai statistik (t-test) kurang dari nilai kritis (critical value). metode penelitian dalam melakukan penelitian ini, kami menyusun beberapa langkah prosedur yang dilakukan dari awal hinga akhir penelitian dengan bantuan bahasa pemrograman komputer, yaitu: 1. membangkitkan dua true process dengan model random walk tanpa drift (konstanta), 1t t t y y e−= + (14) dan dengan drift (konstanta), 1 1t t ty y e−= + + (15) masing-masing berukuran 50 x 1. 2. melakukan perhitungan t-test untuk kasus pertama: kasus 1, time series yang dibangkitkan dengan model true process tanpa drift, 1t t ty y e−= + , yang akan diestimasi dengan model regresi tanpa konstanta ataupun time trend, 1t t ty y eβ −= + dengan hipotesis: 0 1: 1, : 1h hβ β= < (14) dan ( ) 1 testt var β β − = (15) dimana: (16) (17) (18) (19) (20) (21) 3. melakukan perhitungan t-test untuk kasus kedua: kasus 2, time series yang dibangkitkan dengan model true process tanpa drift, 1t t ty y e−= + , yang akan diestimasi dengan model regresi dengan konstanta tanpa time trend, 0 1 1t t ty y eβ β −= + + dengan hipotesis: 0 1 1 1: 1, : 1h hβ β= < (22) dan (23) dimana : ( ) [ ]1 0 1' ' 'x x x yβ β β − = = (24) (25) (26) (27) ( ) 1' 'x x x yβ −= 1tx y −= ty y= ( ) ( ) 12 'var x xβ σ −= 2 ' 1 e e t σ = − e y x β= − ( ) 1 1 1 testt var β β − = [ ]11 tx y −= ty y= ( ) ( ) 12 'covar x xβ σ −= analisis critical root value pada data nonstationer jurnal cauchy – issn: 2086-0382 3 (28) (29) 4. melakukan perhitungan t-test untuk kasus ketiga: kasus 3, time series yang dibangkitkan dengan model true process dengan drift, 11t t ty y e−= + + , yang akan diestimasi dengan model regresi dengan konstanta tanpa time trend, 0 1 1t t ty y eβ β −= + + dengan hipotesis: 0 1 1 1: 1, : 1h hβ β= < (30) dan (31) dimana : ( ) [ ]1 0 1' ' 'x x x yβ β β − = = (32) (33) (34) (35) (36) (37) 5. melakukan perhitungan t-test untuk kasus keempat: kasus 4, time series yang dibangkitkan dengan model true process dengan drift, 11t t ty y e−= + + , yang akan diestimasi dengan model regresi dengan konstanta dan time trend, 0 1 1 2t t ty y t eβ β β−= + + + dengan hipotesis: 0 1 1 1: 1, : 1h hβ β= < (38) dan (39) dimana : ( ) [ ]1 0 1 2' ' 'x x x yβ β β β − = = (40) (41) (42) (43) (44) (45) 6. mengulangi langkah (1) sampai dengan (5) hingga 50.000 kali. 7. menentukan titik kritis pada signifikansi 0.05 (dengan mengambil data persentil ke 5) dan 0.10 (dengan mengambil data persentil ke 10) dari data t-test (berukuran 5.000) untuk masing-masing kasus. 8. melakukan time plot terhadap dua true process yang dibangkitkan terakhir kali. 9. melakukan histogram untuk distribusi t-test pada keempat kasus. 10. menganalisis hasil ouput program. 11. mengambil kesimpulan. hasil dan pembahasan berikut ini merupakan hasil output simulasi komputer untuk mengetahui distribusi dan nilai kritis t-test pada proses non stationer dengan menggunakan estimasi ols terhadap residual variance dengan menggunakan data simulasi berukuran 50 yang dilakukan perulangan hingga 50.000 perulangan dengan dua model true process, persamaan (14) dan (15). gambar 1: time plot data terakhir 2 ' 2 e e t σ = − e y x β= − ( ) 1 1 1 testt var β β − = [ ]11 tx y −= ty y= ( ) ( ) 12 'covar x xβ σ −= 2 ' 2 e e t σ = − e y x β= − ( ) 1 1 1 testt var β β − = [ ]11 tx y t−= ty y= ( ) ( ) 12 'covar x xβ σ −= 2 ' 3 e e t σ = − e y x β= − abdul aziz 4 volume 2 no. 1 november 2011 perolehan data t-test untuk masing-masing kasus berukuran 50.000 adalah sebagai berikut: tabel 1. statistitik deskriptif critical value mean median minimum maximum kasus 1 -0.4146 -0.4934 -4.4895 3.8730 kasus 2 -1.5403 -1.5587 -6.2006 2.1809 kasus 3 -0.2422 -0.2413 -5.1322 4.7800 kasus 4 -2.2031 -2.1837 -5.9714 1.4687 dari 50.000 data tersebut diambil data percentil ke-5 dan ke-10, sehingga diperoleh critical value masing-masing sebagai berikut: tabel 2. tabel critical value α = 0.05 α = 0.10 kasus 1 -1.96 -1.62 kasus 2 -2.95 -2.62 kasus 3 -1.92 -1.55 kasus 4 -3.52 -3.20 hasil ini dapat berbeda untuk setiap kali dilakukan penelitian simulasi. namun dengan besarnya ukuran data simulasi (50.000) maka dapat dijamin untuk diterima sesuai dengan hukum bilangan besar. sedangkan dari tabel dickey-fuller test pada tabel b.6 hamilton j.d.(1994) diperoleh tabel 3. tabel dickey-fuller test α = 0.05 α = 0.10 kasus 1 -1.95 -1.62 kasus 2 -2.86 -2.57 kasus 3 kasus 4 -3.41 -3.12 distribusi titik kritis untuk masing-masing kasus tampak seperti empat gambar berikut: gambar 2: distribusi critical value kasus 1 gambar 3: distribusi critical value kasus 2 analisis critical root value pada data nonstationer jurnal cauchy – issn: 2086-0382 5 gambar 4: distribusi critical value kasus 3 gambar 5: distribusi critical value kasus 4 penutup distribusi titik kritis untuk t-test proses nonstationer mendekati normal dengan perulangan simulasi proses random walk yang semakin besar. hasil perolehan titik kritis ini sudah mendekati dari hasil dickey-fuller test. dari penelitian ini telah diperoleh titik kritis untuk kasus ketiga yang belum ada di tabel hasil dickey-fuller test. untuk pengembangan penelitian selanjutnya dapat dilakukan untuk metode selain simulasi atau distribusi critical value pada proses dengan model yang lain. daftar pustaka [1]. bibby, john, prediction and improved estimation in linear models, john wiley & sons, 1979. [2]. gujarati, d., basic econometrics, mcgrawhill, inc., 1978 [3]. greene, william.h, econometrics analysis, macmillan, inc., 1995 [4]. hamilton, d.j., time series analysis, princeton university press, new jersey, 1994. [5]. hogg & craig, introduction to mathematical statistics, macmillan, inc., 1978. [6]. judge, g.g., et.al., the theory and practice of econometrics, john wiley & sons, inc., 1985. abdul aziz 6 volume 2 no. 1 november 2011 [7]. judge, g.g., et.al., introduction to theory and practice of econometrics, john wiley & sons, inc., 1988. [8]. netter, j., et.al., applied linear statistical models, richard d.irwin, inc., 1990. [9]. wonnacot, j.r. & thomas wonnacott, econometrics, john wiley and sons, inc., 1979. [10]. walpole & myers, probability and statistics for engineers and scientists, macmillan inc., 1989. issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 cauchy vol. 7 no. 4 pages: 493 – 650 malang may 2023 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia managing editor : mohammad jamhuri, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia juhari, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia 1. editorial board : prof hadi susanto, department of mathematical sciences, university of 2. essex and department of mathematics of khalifa university, united kingdom mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines muhammad fakhruddin, department of mathematics, faculty of military mathematics and natural sciences, the republic of indonesia defense university, bogor, indonesia alfi yusrotis zakiyyah, universitas bina nusantara, indonesia bety hayat susanti, politeknik siber dan sandi negara, indonesia dian savitri, universitas negeri surabaya, indonesia meta kallista, universitas telkom, indonesia dani suandi, universitas bina nusantara, bandung, indonesia anwar fitrianto, department of statistics, ipb university, indonesia sri harini, universitas islam negeri maulana malik ibrahim malang, indonesia dr heni widayani, faculty of mathematics and natural sciences, institut teknologi bandung, indonesia corina karim, brawijaya uiversity javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/272712') cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 editorial board subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, universitas islam negeri maulana malik ibrahim malang, indonesia abdussakir, universitas islam negeri maulana malik ibrahim malang, indonesia ari kusumastuti, universitas islam negeri maulana malik ibrahim malang, indonesia fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 7, issue 4, may 2023 issn : 2086-0382 e-issn : 2477-3344 table of contents bühlmann's credibility model with claims of negative binomial and 2-poisson distribution .................................................................................................................................. 493 – 502 on irregular colorings of unicyclic graph family .............................................................. 503 – 512 the first zagreb index, the wiener index, and the gutman index of the power of dihedral group ...................................................................................................................... 513 – 520 a fractional-order leslie-gower model with fear and allee effect............................ 521 – 534 spatial analysis of dengue disease in jakarta province ................................................... 535 – 547 spatial analysis of dengue disease in jakarta province ................................................... 548 – 558 characteristic of quaternion algebra over fields .............................................................. 559 – 567 systematic literature review on adjustable robust shortest path problem ......... 568 – 584 on the study of rainbow antimagic coloring of special graphs ................................... 585 – 596 on the dominant local resolving set of vertex amalgamation graphs .................... 597 – 607 covid-19 data analysis in tarakan with poisson regression and spatial poisson process ........................................................................................................................................... 608 621 the distance irregular reflexive k-labeling of graphs .................................................... 622 – 630 determination of term life insurance premiums with varying interest rates following the cir model and varying benefits value ................................................ 631 – 641 comparing several missing data estimation methods in linear regression;real data example and a simulation study .............................................................................. 642 – 650 corina karim splitting field dan ketunggalannya _4_ splitting field dan ketunggalannya atas polinomial field corina karim1 dan ari andari2 1,2jurusan matematikam universitas brawijaya, malang abstrak suatu field e disebut extension field f, jika �⊂ � di mana f merupakan field. dengan kata lain e disebut extension field f, jika f subfield dari field e. sedangkan splitting field merupakan generalisasi dari extension field yang memenuhi beberapa aksioma. field yang digunakan pada splitting field adalah field finite extension, dimana field finite extension adalah extension field yang mempunyai basis berhingga n. kata kunci: extension field, finite extension dan splitting field. pendahuluan r ring, suatu barisan tak hingga ���,��,� ,…� � dikatakan suatu polinomial atas ring r jika terdapat suatu bilangan bulat tak negatif n sedemikian sehingga terdapat �� � 0 � ,�� � �. polinomial tersebut dinotasikan sebagai berikut: ���� � �� � ��� � � � � �� ���� � � � ����� ∞ ��� dimana x disebut indeterminate atas ring r. jika f field dan f[x] ring polinomial x atas f maka f[x] daerah integral dengan elemen kesatuan dan memuat subring sejati f. polinomial ���� � ���� disebut irreducible jika degree ���� � 1, dan jika ���� � ���."��� dimana ���,"��� � ����, maka ��� � � atau "��� � �. jika polinomial ���� tidak irreducible maka disebut reducible (bhattacharya, dkk, 1986). selanjutnya fraleigh (1994) menyebutkan bahwa suatu field e disebut extension field f, jika �⊂ � di mana f merupakan field. dengan kata lain e disebut extension field f, jika f subfield dari field e. dan jika e extension field f yang mempunyai basis berhingga n di mana e adalah ruang vektor atas f, maka e adalah suatu finite extension berderajat n. selanjutnya dinotasikan [e : f] = n, artinya e extension field f berdimensi n. di lain pihak, menurut gallian (1990), e extension field f dan # � �, α disebut algebraic atas f, jika α akar dari suatu polinomial f[x]. ����� � ����,��#� � 0�. jika α bukan algebraic atas f, maka α disebut transendental atas f. dan menurut fraleigh (1994) suatu e extension field dari field f disebut algebraic extension dari f jika setiap elemen di e algebraic atas f. jika e bukan algebraic extension maka e disebut transcenddental extension. termotivasi dari pengertian extension field dan finite extension, maka dalam makalah ini akan dibahas tentang splitting field atas polinomial di f(x) dan membuktikan ketunggalan dari splitting field tersebut. pembahasan definisi 1 (bhattacharya, dkk, 1986). jika ���� polinomial di f[x] dengan degree ≥ 1 maka k extension field f disebut splitting field ���� atas f jika: i) faktor ���� dapat ditulis menjadi faktor linier $��� % ���� � &�� ' #���� ' # �…�� ' #��, #� � $, dengan c sebarang skalar. ii) $ � ��#�,# ,…,#�� % $ dibangun oleh f dengan akar-akar #�,# ,…,#� � ���� dan ���� � $. contoh 1: field (�√2� � +� � ,√2|�,, � (. adalah splitting field dari � ' 2 � (��� atas (. bukti: jelas ( ⊆ (�√2�. akan dibuktikan: (i). (�√2� extension field (. (ii). (�√2� splitting field ( [x] atas(. bukti (i). ( ⊂ (�√2�. klaim / � +1,√2. akan ditunjukkan: s adalah basis untuk ruang vektor ( (√2) a) ambil sebarang y∈( �√2�. maka y dapat dinyatakan sebagai 0 � � � ,√2, �,, � ( � �.1 � ,.√2, corina karim dan ari andari 162 volume 1 no. 4 mei 2011 jadi s merentang ( (√2). ▲ b) untuk suatu �,, ∈(, maka 0 � � � ,√2 ⇒ 0 � �.1 � ,.√2 akan terpenuhi jika � � 0 dan , � 0 jadi s bebas linier. ▲ dari a) dan b) terbukti s basis untuk ( (√2) dan �((√2):( � � 2. jadi ((√2) finite extension (. karena ((√2) finite extension (, maka ((√2) extension field (. ▲ bukti (ii). akan ditunjukkan: a) ��� � &�� ' #���� ' # �…�� ' #��, #� � (�√2� pilih ��� � � ' 2 � (��� ��� � � ' 2 � 1� ' √221� � √22 � 1� ' √223� ' 1'√224 di mana 5√2 � (1√22 ▲ b) (1√22 � (�#�,# ,…,#�� % (1√22 dibangun oleh ( dengan akar-akar #�,# ,…,#� � ��� atas (. karena (1√22 finite extension atas ( maka (1√22 algebraic extension atas (. sehingga ∃ ��� � (���, ��� 6 0 di mana �#� � 0. ambil sebarang # � (1√22 maka # � � � √2, # � 1� � √2,2 � � � 2√2�, � 2, # ' � � 2√2�, � 2, � 0 sehingga ∃ ��� � (��� di mana ��� � � ' � � 2√2�, � 2, jadi (1√22 dibangun oleh ( dengan # � (��� ▲ jadi (1√22 splitting field ���atas (. ▲ contoh 2 splitting field � � 1 � 7��� atas 7 adalah field 7. teorema 2 (bhattacharya, dkk,1986) jika k splitting field ���� � ���� atas f maka k finite extension f dan k algebraic extension atas f. bukti : karena menurut definisi 1. bagian ii) k = ��#�,# ,…,#�� % k dibangun oleh f dengan #�,# ,…,#� � $. maka k algebraic atas f. sehingga k finite extension f. karena k finite extension maka k juga algebraic extension f. ▲ teorema 3 (dummit and foote, 1991 ) misal 8 9 � →~ �: isomorfisma field. ���� � ���� polinomial dan �:��� � �:��� polinomial yang didapat dari 8 terhadap koefisien ����. jika e splitting field ����atas f dan e’ splitting field �:��� atas f’. maka isomorfisma 8 “extended to” isomorfisma ; 9 � →~ �:. ; yang dibatasi (restricted to) f adalah isomorfisma 8 : ; : � →~ �: 8 : � →~ �: bukti: 8 : f →~ f ’ isomorfisma field. ���� ֏ �:��� #� � #�� ֏ #�: � #�:� e splitting field ���� atas f �:splitting field �:���atas �: akan dibuktikan: a) ; : � →~ �: 8 : � →~ �: b) σ isomorfisma bukti: a) dengan menggunakan induksi matematika terhadap derajat n pada e field extension f. i) untuk n = 1. jika n = 1 maka e = f padahal ; : � →~ �: 8 : � →~ �: maka σ = ϕ . ▲ ii) andai untuk n = 2 benar maka akan dibuktikan n > 2 benar untuk sebarang field extension. bukti: definisikan : φ : ���� < ��#� = ֏ =�#� ambil sebarang # � � splitting field ����,���� > �. misal ?��� polinomial terkecil yang memuat #. maka ?���| ����, karena # � ����. splitting field dan ketunggalannya atas polinomial field jurnal cauchy – issn: 2086-0382 163 ambil sebarang ?@�a� � �@��� % ?��� � ?@���. karena �@��� splitting field atas �@ maka ada elemen b � �: dan b � ?@���. jadi ada homomorfisma ring c extending 8 % c : ��#� →~ �@�b� (1) 8 : � → ~ �: karena ���#�:�� � 1 maka: ���#�:�� � 2 ��:��#�����#�:�� � ��:��#��.2 ��:�� � ��:��#��.2 (karena ��:�� � 2 benar) maka ��:��#�� e 1 jadi ��:��#�� f 1 dan ��:��#�� � 1. jika ��:��#�� f 1 maka ada σ extending c % ; : � → ~ �: (2) c : ��#� →~ �:�b� karena σ extends c dan c extends 8 maka σ extends 8. dari (1) dan (2) diperoleh: ; : � → ~ �: c : ��#� →~ �:�b� 8 : � → ~ �: jadi ; : � →~ �: 8 : � →~ �: ▲ jika ��:��#�� � 1 maka ada σ extending c % ; : � → ~ �: di mana ; � c (3) c : ��#� →~ �:�b� dari (1) dan (3) diperoleh ; : � →~ �: 8 : � →~ �: ▲ b) andai �: splitting field �@��� atas �: dan 8 isomorfisma. akan dibuktikan: σ isomorfisma ⇒ i) σ homomorfisma ii) σ onto iii) σ satu-satu bukti: i) didefinisikan: ; : � →~ �: ;��� � �: % 8 : � →~ �: 8��� � �: ambil sebarang #,b � � splitting field ���� atas f dan g � h, maka: • ;�#b� � �#b�: � #:b: � ;�#�.;�b� • ;�g#� � �g#�: � g.#: � g.;�#� jadi σ homomorfisma. ▲ ii) ambil sebarang ���� � ����, maka: ���� � #� � #�� � �� #���, #� � � himpunan �i��� � 8�#� � #�� � �� #���� karena 8 homomorfisma, maka: �i��� � 8�#�� � 8�#��� � �� 8�#���� � 8�#�� � 8�#��� � �� 8�#���� padahal �: splitting field �@��� � �@���, maka: �@��� � &�� ' #���� ' # �…�� ' #�� di mana #� > �: sehingga �@��� � &1� ' 8����21� ' 8�� �2… 1� ' 8����2 (4) disisi lain, jika ada b� � �:, maka: �@��� � &�� ' b���� ' b �…�� ' b�� (5) dari (4) dan (5) maka himpunan +8����,8�� �,…,8����. � +b�,b ,…,b�. karena �: splitting field �@��� � �@���maka �@ � �:�b�,b ,…,b�� � �:18����,8�� �,…,8����2 � �:1;����,;�� �,…,;����2 � 8���1;����,;�� �,…,;����2 � ;���18����,8�� �,…,8����2 � ;1��#�,# ,…,#��2 corina karim dan ari andari 164 volume 1 no. 4 mei 2011 �@ � ;��� jadi σ onto. ▲ iii) σ satu-satu. ambil sebarang ��,�� � �@ % ;���� � ;�� � ;���� � ;�� � ;���� ' ;�� � � 0 ;��� ' � � � 0 (karena ; homomorfisma) jadi �� ' � � ker ; ;���� � ;�� � ;���� ' ;�� � � 0 8���� ' 8�� � � 0 (karena 8 � ;) 8��� ' � � � 0 (karena 8 homomorfisma) jadi �� ' � � ker 8 jadi ; satu-satu. ▲ akibat 4 (ketunggalan splitting field) sebarang dua splitting field polinomial ���� � ���� atas field f adalah isomorfik. bukti: ambil sebarang splitting field ���� � ����. misal: e splitting field ���� atas f dan �: splitting field �:���atas �:. e splitting field ���� � ���� isomorfik dengan �: splitting field �:��� � �:��� jika ada isomorfisma ring ; : � < �: dan 8 : � < �: ; : � �: 8 : � �: komutatif (; � 8). jelas. dari teorema 3, di mana � � �: dan ; m 8 . ▲ kesimpulan dari hasil pembahasan dapat ditarik kesimpulan bahwa jika e dan e’ splitting field atas polinomial-polinomial f(x)∈ f[x] maka kedua splitting field tersebut isomorfik. dan untuk selanjutnya disebut ketunggalan splitting field. daftar pustaka [1] anton,h. 1987. aljabar linier elementer. erlangga. jakarta. 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[12] leon, steven j. 2001. aljabar linear dan aplikasinya. edisi kelima. erlangga. jakarta. [13] raisinghania, m.d., and anggarwal, r.s. 1980. modern algebra. s.chard and company ltd. new delhi. [14] roman, s. 1991. advanced linear algebra. springer verlag. new york. [15] sims, charles c. 1984. abstract a computational approach. john wiley and sons, inc. new york. [16] spindler, karlheins. 1994. abstract algebra with application. volume ii. marcell dekker, inc. usa. [17] stewart, ian. 1989. galois theory. second ed., chapman and hall. london. [18] whitelaw, thomas a. 1995. introduction to abstract algebra. third ed., chapman and hall. new york. →~ →~ generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier lukman hakim1) dan ari kusumastuti2) 1)mahasiswa pascasarjana jurusan matematika universitas brawijaya malang 2)jurusan matematika uin maulana malik ibrahim malang e-mail: bedek_me@yahoo.com abstrak persamaan schrodinger adalah persamaan diferensial parsial nonlinier yang menginterpretasikan pergerakan suatu partikel atau atom. penelitian ini berupaya untuk memperoleh analisis konstruksi bentuk umum solusi ananalitik persamaan schrodinger nonlinier dengan fungsi airy. fungsi airy adalah solusi persamaan diferensial airy, adapun langkah pertama adalah manipulasi bentuk persamaan schrodinger nonlinier menjadi bentuk persamaan airy dengan menerapkan transformasi fourier. dengan demikian didapatkan solusia nanalitik persamaan airy dengan generalisasi fungsi airy. dan langkah selanjutnya adalah menerapkan invers dari transformasi fourier yang digunakan untuk memdapatkan solusi analitik bagi persamaan schrodinger nonlinier, dalam hal ini diberikan kondisi awal bilangan kompleks pada invers transformasi fourier, yaitu ���,�� � �� �. adapun hasil dari penelitian ini adalah solusi bagi persamaan schrodinger nonlinier ketika pangkat dari modulusnya di analisis dengan bentuk ganjil dan genap memberikan bentuk solusi yang sama yaitu: ���,�� � 1�� ��� ���3 � ��� ∞ � �� sedangkan ketika di analisis hingga dimensi � maka didapatkan generalisasi fungsi airy sebagai berikut: ����,��,��,… ,��, �� � 1�� ��� � !��3 "� 1 #�$ � $%� & � �� "� �� 1 #�$ㅦ$%' & � ù� ) "� #�$�*�$%� 1& � ù� � "� #�$�$%� & � ù� �+" 1�2ð�� � � � … . *. . *. � /0ù1∑ 345467 89:. *. . *. ���& ; <<< <<< = . � �ù kata kunci: persamaan schrodinger nonlinier, persamaan airy, fungsi airy, transformasi fourier dan invers transformasi fourier. pendahuluan persamaan diferensial parsial (pdp) adalah kajian ilmu matematika yang berkaitan langsung dengan kehidupan manusia karena persamaan diferensial parsial dapat digunakan untuk menterjemahkan fenomena alam menjadi suatu persamaan yang sistematis dan logis (purwanto, 2003). secara definitif, persamaan diferensial adalah persamaan yang mengandung fungsifungsi turunan, baik turunan parsial maupun turunan biasa. dan dalam menyelesaikan persamaan diferensial secara analitik, biasanya terbatas pada persamaan-persamaan diferensial dengan bentuk tertentu dan biasanyahanya digunakanuntuk menyelesaikan persamaanpersamaan yang linier. sedangkan persamaan diferensial nonlinier tidak mudah diselesaikan secara langsung, akan tetapi harus melalui transformasi-transformasi yang menjadikan persamaan nonlinier menjadi persamaan yang linier (ross, 1984). selain sebagai alat untuk memodelkan masalah, maka persamaan diferensial parsial juga menjadi alat yang digunakan untuk menyelesaikan permasalahan matematika fisika, matematika kimia dan fenomena yang ada di alam (purwanto, 2003). misalnya, fenomena alam yang terjadi di laut yaitu adanya gerakan partikel di bawah laut yang menimbulkan gelombang yang disebut dengan gelombang internal. gelombang ini terjadi karena terdapat perbedaan rapat massa pada setiap lapisan laut dan perbedaan rapat massa disebabkan oleh perbedaan kadar garam dan temperatur dari setiap lapisan laut, selain di laut pergerakan partikel dapat juga terjadi pada medan kuantum. dalam hal ini persamaan diferensial parsial yang generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier jurnal cauchy – issn: 2086-0382 87 menggambarkan pergerakan elektron pada medan kuantum adalah model persamaan schrodinger (sudirham dan utari, 2010). persamaan schrodinger adalah persamaan diferensial parsial yang menggambarkan bagaimana pergerakan suatu partikel khususnya partikel elektron. dalam ilmu fisika, persamaan schrodinger diperkenalkan oleh fisikawan erwin schrodinger pada tahun 1925 dan dijelaskan juga bagaimana hubungan antara ruang dan waktu pada sistem mekanika kuantum. persamaan ini sangat penting seperti halnya persamaan newton yang menjadi dasar berkembangnya keilmuan fisika, dan sedangkan persamaan schrodinger menjadi dasar berkembangnya ilmu fisika modern yang berkenaan dengan mekanika kuantum (sudirham dan utari, 2010). fungsi airy adalah suatu fungsi yang menjadi solusi bagi persamaan diferensial airy. adapun bentuk persamaan airy adalah >?? � �> � 0 dan solusinya disebut dengan fungsi airy. selanjutnya diberikan penjelasan bahwa fungsi airy adalah salah satu bentuk model penyelesaian persamaan diferensial schrodinger. dengan demikian asumsi-asumsi yang harus dipenuhi persamaan schrodinger agar dapat diselesaikan dengan fungsi airy adalah persamaan schrodinger harus dikontruks menjadi model persamaan airy, kemudian diselesaikan dan didapatkan fungsi airy (valle dan manuael, 2004). berdasarkan pemaparan di atas peneliti ini bertujuan untuk memaparkan analisis bentuk generalisasi fungsi airy sebagai solusi analitik bagi persamaan schrodinger nonlinier. dan dalam penelitian ini dibatasi pada generalisasi bentuk umum solusi ketika pangkat dari modulus persamaan schrodinger nonlinier dianalisis dengan bentuk pangkat ganjil dan genap, dan generalisasi bentuk umum solusi ketika dimensi dari persamaan schrodinger nonlinier dinaikkan hingga dimensi �. adapun metode penelitian yang digunakan pada penelitian ini adalah analisis brownian motion persamaan schrodinger nonlinier, transformasi persamaan schrodingernonlinier ke dalam bentuk persamaan airy dengan transformasi fourier, menyelesaikan persamaan airy sehingga solusi yang disebut dengan fungsi airy, menerapkan invers transformasi fourier dan memberikan kondisi awal sehingga didapatkan solusi persamaan schrodinger nonlinier. kajian pustaka persamaan diferensial parsial persamaan diferensial adalah persamaan yang memuat turunan satu atau lebih dari variabel tak bebas terhadap satu atau lebih variabel bebas.berdasarkan jumlah variabel bebasnya, persamaan diferensial dikelompokkan menjadi persamaan diferensial biasa (pdb) atau ordinarydifferential equation (ode) dan persamaan diferensial parsial (pdp) atau partialdifferential equation (pde) (ross, 1984). persamaan diferensial parsial (pdp) adalah persamaan diferensial yang memuat turunan parsial satu atau lebih dari variabel tak bebas terhadap satu atau lebih variabel bebas (ross, 1984). orde persamaan diferential parsial ordo/orde suatu persamaan diferensial adalah pangkat turunan tertinggi yang muncul dalam persamaan diferensial (stewart, 2003). sedangkan tingkat derivatif parsial tertinggi merupakan tingkat dari persamaan differensial parsial tersebut dan pangkat tertinggi dari orde tertinggi merupakan derajat dari persamaan differensial tersebut (soeharjo,1996). persamaan schrodinger persamaan schrodinger diajukan pada tahun 1925 oleh ahli fisika yaitu erwin schrodinger (1887-1961).persamaan ini pada awalnya merupakan jawaban dari dualitas partikel gelombang yang lahir dari gagasan de broglie yang menggunakan persamaan kuantisasi cahaya planck dan prinsip fotolistrik einstein dalam menentukan kuantisasi pada orbit elektron. selain erwin schrodinger ada dua orang fisikawan lainnya yang mengajukan teorinya yaitu werner heisenberg dengan mekanika matriks dan paul dirac dengan aljabar kuantum.ketiga teori ini merupakan teori kuantum lengkap yang berbeda dan dikerjakan secara terpisah namun ketiganya setara.teori erwin schrodinger kemudian lebih sering digunakan dengan alasan rumusan matematisnya yang relatif sederhana dan lebih aplikatif (sudirham dan utari, 2010). persamaan schrodinger banyak kegunaannya dan karena penerapannya mencapai ketelitian sangat tinggi dan akurat.penggunaan persamaan schrodinger pada sistem fisis memungkinkan untuk memberikan ketelitian yang sangat tinggi dan berdasarkan penelitian terbaru bahwa persamaan schrodinger nonlinier mempunyai peluang hingga tingkatan nano.sehingga penerapan ini menghasilkan ramalan-ramalan baru misalnya, penemuan positron yang merupakan anti materi dari elektron. selanjutnya persamaan schrodinger menjadi landasan berkembangnya keilmuan di bidang mekanika kuantum dan dewasa ini persamaan schrodinger telah diterapkan di berbagai bidang lukman hakim dan ari kusumastuti 88 volume 2 no. 2 mei 2012 fisika yaitu fisika matematika, optik tidak linier, sistem kuantum partikel banyak, fisika plasma dan superkonduktivitas. dan pada penelitian ini dikhususkan untuk mempelajari solusi analitik persamaan schrodinger nonlinear menggunakan generalisasi fungsi airy, adapun bentuk umum persamaan schrodinger nonlinier adalah (polyanin dan zaitsev, 2004): a���,��a� a����,��a�� +|���,��|����,�� � 0 (1) dimana: ���,��: fungsi bernilai kompleks � : waktu + : bilangan bernilai riil |�|� : ��c �c : konjugat fungsi ���,�� integral fourier misalkan didefinisikan fungsi de��� suatu fungsi dengan periode 2f maka fungsi ini dapat diinterpretasikan ke dalam bentuk deret fourier (agarwal dan o’regan, 2009), yaitu: de��� � g�2 #�g� cosù�� k�� � ù��� . �%� (2) dimana ù� � �ðf g� � 1f �de������ ù�� ��, � l 0 e *e k� � 1f �de���� � ù�� ��, � l 1 e *m dengan demikian didapatkan nilai dari $n� adalah g�2 � 12f �de��� � �� e *e (3) nilai g� ��� ù�� adalah g� ��� ù�� � ��� ù�� o1f �de������ ù�� �� e *e p (4) nilai k�� � ù�� adalah k�� � ù�� � � � ù�� o1f �de���� � ù�� �� e *e p (5) substitusi persamaan (3), (4) dan (5) ke persamaan (2), dengan f q ∞ maka didapatkan de��� � 12f �de��� � �� e *e (6) 1f# st ttt tu cosù�� �de������ ù���� e *e � � ù�� �de���� � ù���� e *e vw www wx. �%� kemudian didefinisikan ∆ù sebagai perubahan frekuensi sudut, yaitu: ∆ù � ù�8� � ù� � �� 1�ðf ��ðf � ðf (7) maka diperoleh dari persamaan (7) adalah 1f � ∆ùð (8) dengan menerapkan persamaan (8) pada persamaan (6) maka didapatkan de��� � 12f �de��� � �� e *e 1ð# st ttt tu���� ù���∆ù �de������ ù���� e *e � � ù���∆ù �de���� � ù���� e *e vw www wx. �%� (9) karena f q ∞ maka z [eq. de��� � d��� sehingga nilai dari �e q 0 dan mengakibatkan nilai dari ��e\ de��� � ��e*e ] 0, begitu juga nilai ∆ù � ðe q 0. suatu deret adalah jumlah dari suku demi suku dan dikarenakan suku menuju tak hingga maka didekati dengan limit sehingga berupa suatu luasan kurva dari persamaan. berdasarkan definisi integral reimann bahwa limit dari deret infinit ekuivalen dengan integral yang mempunyai batas bawah 0 dan batas atas ∞ (purcell dan varberg, 1999). dengan demikian persamaan (9) yang semula berupa deret infinit menjadi definisi suatu integral (agarwal dan o’regan, 2009), d��� � 1ð� st tt tu cosù� � d������ ù� �� . *. � � ù� � d���� � ù� ��. *. vw ww wx�ù. � (10) persamaan (10) disebut dengan integral fourier. transformasi fourier pandang persamaan (10), yaitu d��� � 1ð� st tt tu cosù� � d������ ù� �� . *. � � ù� � d���� � ù� ��. *. vw ww wx�ù. � generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier jurnal cauchy – issn: 2086-0382 89 � 1ð� ^ � d������ �ù� � ù���� . *. _�ù. � (11) persamaan (11) dapat dinyatakan dalam bentuk yang ekuivalen sebagai berikut: d��� � 12ð � ^ � d������ �ù� �ù���� . *. _�ù. *. (12) analog dengan persamaan (12) maka dapat dibentuk persamaan baru yang valid yaitu 12ð � ^ � d���� �� � � �ù� �ù���� . *. _�ù. *. � 0 (13) karena fungsi � � � adalah fungsi ganjil maka nilai dari persamaan (13) adalah nol, dikarenakan integral dari �∞ sampai ∞ pada fungsi ganjil menghasilkan nilai nol maka persamaan (13) benar (agarwal dan o’regan, 2009). selanjutnya kombinasi dari persamaan (12) dan (13) menghasilkan d��� � 12ð � ^ � d������ �ù� �ù���� . *. _�ù. *. 12ð � ^ � d��� � � �ù� �ù���� . *. _�ù. *. � 12ð � ^ � d���` cos�ù� � ù�� � � �ù� � ù��a�� . *. _�ù. *. (14) berdasarkan rumus euler maka persamaan (14) menjadi d��� � 12ð � � d���/0�ù3*ù9����ù . *. . *. � 12ð � � d���/0ù�3*9����ù . *. . *. (15) persamaan (15) dapat diuraikan menjadi d��� � 12ð � ^ � d���/*0ù9�� . *. _/0ù3�ù. *. (16) selanjutnya integral yang di dalam kurung dari persamaan (16) disimbolkan dengan b�ù� yang disebut dengan transformasi fourier dari d���, yaitu: b�ù� � � d���/0妾9�ù. *. (17) fungsi airy pandang persamaan diferensial orde dua berikut: ��>������ � �>��� � 0 (18) selanjutnya persamaan (18) disebut persamaan airy dan solusi persamaan airy disebut dengan fungsi airy. solusi persamaan (18) akan mudah dicari dengan menerapkan transformasi fourier, jika didefinisikan c�ù� sebagai transformasi fourier dari >���, yaitu: c�ù� � � /*0ù3>�����. *. (19) jika persamaan (19) diturunkan dua kali terhadap � maka didapatkan ��c�ù���� � ����� " � /*0ù3>����� . *. & � � /*0ù3>dd�����. *. � /*0ù3>?���|*.. � ��� ù�/*0ù3>?�����.*. � ù � /*0ù3>?�����. *. � ùe / *0ù3>���|*.. � ��� ù�/*0ù3>�����. *. f � �ù�c1ù���: maka diperoleh ��c�ù���� � �ù�c1ù���: (20) selanjutnya persamaan (19) diturukan terhadap ù dengan langkah berikut: �c�ù��ù � ��ù" � /*0ù3>����� . *. & � � � � /*0ù3>�����. *. � � �c1ù���: maka didapatkan �c�ù��ù � � �c1ù���: (21) berdasarkan persamaan (21) maka didapatkan maka diperoleh �c�ù��ù � �c�ù� (22) berdasarkan persamaan (20) dan (22) maka didapatkan modifikasi persamaan (18), adalah �ù�c�ù� � �c�ù��ù ù�c�ù� � �c�ù��ù ù��ù � �c�ù�c�ù� ù�3 � z� c�ù� c�ù� � /ùg� 0 (23) lukman hakim dan ari kusumastuti 90 volume 2 no. 2 mei 2012 langkah selanjutnya adalah menggunakan invers transformasi fourier untuk mendapatkan solusi dalam bentuk >���, yaitu: >��� � 12ð � /0ù3c�ù��ù . *. (24) substitusi persamaan (23) pada persamaan (24) maka didapatkan >��� � 12ð � /0`ù38ù g� a�ù. *. (25) berdasarkan rumus euler, maka didapatkan /0`ù38ùg� a � cos�ù� ù�3 � � � �ù� ù�3 � (26) dengan demikian diperoleh >��� � 12ð stt ttu � cos�ù� ù�3 ��ù � � � �ù� ù�3 � . *. �ù . *. vww wwx (27) karena fungsi � � � adalah fungsi ganjil maka persamaan (27) menjadi >��� � 12ð � ��� �ù� ù�3 ��ù . *. (28) karena fungsi ���� adalah fungsi genap maka persamaan (28) menjadi >��� � 1ð� ��� �ù� ù�3 ��ù . � (29) karena persamaan (29) adalah solusi bagi persamaan airy (18) maka disebut dengan fungsi airy dan kemudian fungsi airy disimbolkan dengan + ���, yaitu: + ��� � 1ð� ��� �ù� ù�3 ��ù . � (30) (oliver dan manuael, 2004). pembahasan analisis brownion motion persamaan schrodinger nonlinier suatu gelombang pada dasarnya adalah pergerakan bebas partikel-partikel yang merambat ke segala arah dan pergerakannya dipengaruhi oleh energi masing-masing partikel, akan tetapi setiap partikel berpeluang sama untuk bergerak ke segala arah. persamaan schrodinger adalah salah satu model gelombang yang interpretasinya banyak diterapkan pada mekanika kuantum. adapun asumsi yang mendasari persamaan schrodinger adalah adanya pergerakan acak pertikel dan setiap partikel mempunyai peluang gerak yang sama baik ke kanan maupun ke kiri karena pergerakan gelombangnya diasumsikan bergerak ke kanan dan ke kiri. dalam buku yang berjudul “partial differential equations of applied mathematics” erich zauderer menyebutkan bahwa pergerakan suatu partikel dapat diinterpretasikan dalam bentuk distribusi probabilitas yang menyatakan bahwa probabilitas partikel di � pada saat � ô sama dengan probabilitas partikel di � � ä pada saat � dikalikan dengan probabilitas ç yang berpindah ke kanan ditambah dengan probabilitas partikel di � ä pada saat � dikalikan dengan probabilitasî yang berpindah ke kiri, sehingga pergerakan partikel dapat dinyatakan dengan persamaan matematis, yaitu: ���,� ô� � ç��� � ä, �� î��� ä, �� (31) berdasarkan ekspansi deret taylor (2. 22) maka didapatkan sistem persamaan dari persamaan (3.1) adalah klm ln ���,� ô� � ���,�� ô�9��,����� � ä, ô� � ���,��� ä�3��,�� 12 ä��33��,����� ä, ô� � ���,�� ä�3��,�� 12 ä��33��,�� o (32) selanjutnya substitusi sistem persamaan (32) pada persamaan (31) dan didapatkan ���,�� ô�9���,�� � ç ^���,��� ä�3��,�� 12 ä��33��,�� _ î p���,�� ä�3��,�� 12 ä��33��,��q (33) karena pergerakan partikel adalah kejadian peluang maka nilai dari pergerakan peluang ke kanan dan ke kiri yaitu: �ç î� ] 1 maka persamaan (33) menjadi ô�9��,�� � ��ç î�ä�3��,�� 12 ä��33��,�� (34) jika masing-masing dari ruas persamaan (34) dibagi dengan ô maka didapatkan �9��,�� � ��ç î� ô ä�3��,�� 12가�ô �33��,�� (35) jika ruas kanan dipindah ke ruas kiri maka persamaan (35) menjadi �9��,��� ��ç î�ä ô �3��,��� ä�2ô �33��,�� � 0 (36) kemudian persamaan (36) dikenal dengan persamaaan difusi satu dimensi, dengan pergerakan gelombangnya ke kanan dan ke kiri. jika diasusmsikan nilai ��ç î�ä ô ] 0 maka didapatkan persamaan difusi satu dimensi dengan mengabaikan kecepatan pergerakan partikel, yaitu: a���,��a� � �ä�2ô�a����,��a�� � 0 (37) jika ruas kiri dari persamaan (37) ditambah suatu fungsi dengan bentuk +|���,��|����,�� maka didapatkan generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier jurnal cauchy – issn: 2086-0382 91 a���,��a� � �ä�2ô�a����,��a�� +|���,��|����,�� � 0 (38) jika diasumsikan nilai dari ä 7 �ô ] �1 maka persamaan (38) menjadi a���,��a� � ����,��a�� +|���,��|����,�� � 0 (39) dimana + adalah konstanta riil dan bentuk persamaan (39) disebut dengan persamaan schrodinger nonlinier satu dimensi. solusi analitik persamaan schrodinger nonlinier dengan generalisasi fungsi airy meninjau persamaan schrodinger nonlinier (39), yaitu: a���,��a� a����,��a�� +|���,��|����,�� � 0 (40) dimana + adalah konstanta real dan |�|� � ��c, untuk �c adalah konjugat dari � suatu fungsi kompleks. kasus i: jika � genap maka persamaan (40) dapat dinyatakan sebagai berikut: a���,��a� a����,��a�� +|���,��|�r���,�� � 0 (41) selanjutnya, jika [ � 1 maka persamaan (41) menjadi a���,��a� a����,��a�� +|���,��|����,��� 0 (42) jika persamaan (42) ditransformasi dengan transformasi fourier, dan transformasi fourier yang digunakan adalah ��ù��,��,�� � � /*0ù9���,����∞ *. (43) maka invers trasformasi fourier dari ��ù, �� adalah ���,�� dan dapat dinyatakan sebagai berikut: ���,�� � 12ð � /*0ù9��ù, ���ù ∞ *. (44) persamaan (43) memberikan makna bahwa bentuk ���,�� ditranformasi menjadi bentuk ��ù��,��,�� yang secara fisis mentransformasi domain dari bentuk spasial ke dalam bentuk frekuensi, dan kemudian penulisan ��ù��,��,�� akan disingkat dengan ��ù, ��. kemudian, jika persamaan (43) diturunkan terhadap � maka didapatkan a��ù,��a� � s /*0ù尷 a���,��a� �� . *. � /*0ù9���,��|*.. � ��� ù�/*0ù9���,����. *. � ù��ù,�� (45) memandang transformasi untuk |���,��|� adalah |��ù,��|� maka berdasarkan definisi pada analisis kompleks didapatkan bahwa |��ù,��|� � ��ù,�� ·�u�ù,�� (46) kemudian dengan memandang persamaan (43) maka transformasi |��ù,��|� pada persamaan (46) dapat dinyatakan |��ù,��|� � " � /*0ù9���,����. *. &" � /*0ù9�c��,����. *. & (47) berdasarkan persamaan (45) maka didapatkan modifikasi dari persamaan (42) adalah ù��ù,�� a���ù,��a�� +|��ù,��|���ù,�� � 0 (48) bentuk sederhana persamaan (48) adalah a���ù,��a�� � �� ù � +|��ù,��|����ù,�� (49) misal � ù � +|��ù,��|� � v, maka persamaan (49) menjadi a���ù,��a�� � v��ù,�� (50) analog dengan persamaan (18) maka solusi persamaan (50) adalah ��ù,�� � 1ð� ��� �ù�3 vù� . � �ù substitusi v dengan � ù � +|��ù,��|� maka didapatkan ��ù,�� � 1ð� ��� �ù�3 ` � ù�+|��ù,��|�aù� . � �ù (51) substitusi |��ù,��|� dengan persamaan (47), maka persamaan (51) menjadi ��ù,�� � 1ð� ��� � ! ù �3 � ù� � + � !" � /*0ù9���,���� . *. & " � /*0ù9�c��,����. *. &; << =ù ; <<< << =. � �ù (52) dengan demikian persamaan (52) adalah solusi persamaan airy (50) dan untuk mendapatkan solusi persamaan schrodinger (41) maka transformasi fourier yang terdapat pada persamaan (43) harus diinverskan. selanjutnya menggunakan invers transformasi fourier (44) maka diperoleh solusi persamaan schrodinger nonlinier adalah lukman hakim dan ari kusumastuti 92 volume 2 no. 2 mei 2012 ���,�� � 1ð� ��� � ! ù �3 � ù� � + � !" 12ð � /0ù9��ù,���ù . *. & " 12ð � /0ù9�u�ù,���ù . *. &; << =ù ; <<< << =. � �ù (53) selanjutnya pandang faktor berikut 12ð � /0ù9��ù,���ù . *. diberikan sebarang kondisi awal dan diasumsikan kondisi awalnya adalah fungsi komplek, maka dalam penelitian ini penulis mengambil kondisi awal sebagai berikut: ��ù,�� � ù� � (54) sehingga saat � � 0 didapatkan ��ù,0� � ù� 0 � ù� (55) karena �u adalah konjugat dari � maka didapatkan �u�ù,�� � ù� � � (56) sehingga saat � � 0 didapatkan ��ù,0� � ù� � 0 � ù� (57) berdasarkan persamaan (55) dan (57) maka persamaan (53) menjadi ���,�� � 1ð� ��� � ! ù �3 � ù� � + � !" 12ð � /0ù9ù��ù . *. & " 12ð � /0ù9ù� �ù . *. &; << =ù ; <<< << =. � �ù (58) kemudian pandang faktor berikut 12ð � /0ù9ù��ù . *. (59) berdasarkan sifat integral dan perhitungan integral biasa maka persamaan (59) menjadi 12ð � /0ù9ù��ù . *. � 1ð �� � 1ð �� � 0 dengan demikian persamaan (58) menjadi ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù (60) persamaan (60) adalah fungsi airy yang menjadi solusi bagi persamaan schrodinger nonlinier (42). analog dengan proses pada persamaan (42) maka ketika mengambil nilai [ hingga �, maka solusi bagi persamaan berikut a�a� ��,�� a��a�� ��,�� +|���,��|�r���,�� � 0 adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù (61) dengan demikian dapat disimpulkan bahwa bentuk umum fungsi airy sebagai solusi persamaan schrodinger nonlinier (42) adalah persamaan (61). kasus ii: jika � ganjil maka persamaan (40) dapat dinyatakan sebagai berikut: a���,��a� a����,��a�� +|���,��|�r8����,�� � 0 (62) jika [ � 0 maka dari persamaan (62) didapatkan a���,��a� a����,��a�� +|���,��|���,�� � 0 (63) kemudian memandang transformasi untuk |���,��| adalah |��ù,��|dan berdasarkan transformasi fourier yaitu persamaan (43) maka didapatkan |��ù,��| � w � ‱*0ù9���,����. *. w (64) berdasarkan persamaan (45) maka didapatkan modifikasi dari persamaan (63) adalah ù��ù,�� a���ù,��a�� +|��ù,��|��ù,�� � 0 persamaan ini dapat disederhanakan menjadi a���ù,��a�� � �� ù � +|��ù,��|���ù,�� misal � ù � |��ù,��| � v, maka persamaan menjadi a���ù,��a�� � v��ù,�� (65) selanjutnya analog dengan persamaan (18) maka persamaan (65) mempunyai solusi fungsi airy dengan bentuk berikut: ��ù,�� � 1ð� ��� �ù�3 vù� . � �ù substitusi v dengan � ù � +|��ù,��| maka didapatkan ��ù,�� � 1ð� ��� �ù�3 ` � ù�+|��ù,��|aù� . � �ù (66) substitusi |��ù,��| dengan persamaan (64) pada persamaan (66) maka didapatkan ��ù,�� � 1ð� ��� � ! ù�3 � ù� � +w � /*0ù9���,����. *. wù; <=.� �ù (67) dengan demikian persamaan (67) adalah solusi persamaan airy (65) dan untuk mendapatkan solusi persamaan schrodinger (63) maka transformasi fourier yang terdapat pada persamaan (67) harus diinverskan dan analog generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier jurnal cauchy – issn: 2086-0382 93 dengan persamaan (44) maka persamaan (67) menjadi ���,�� � 1ð� ��� � ! ù�3 � ù� � +w 12ð � /0ù9��ù,���� . *. wù; <=.� �ù diberikan kondisi awal adalah persamaan (55) dan (57) maka didapatkan ���,�� � 1ð� ��� � ! ù�3 � ù� � +w 12ð � /0ù9ù��� . *. wù; <=.� �ù (68) analog dengan persamaan (59) maka didapatkan hasil dari persamaan (68) adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù (69) dengan demikian persamaan (69) adalah solusi persamaan schrodinger (63). analog dengan proses kasus ii maka solusi bagi persamaan berikut a�a� ��,�� a��a�� ��,�� +|���,��|�r8����,�� � 0 adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù bentuk solusi analitik persamaan schrodinger nonlinier dimensi tinggi dengan generalisasi fungsi airy pada paparan sebelumnya didapatkan generalisasi bahwa pangkat � dari modulus suku +|���,��|����,�� untuk setiap � genap maupun ganjil menghasilkan solusi dengan bentuk yang sama dan hal ini memberikan kesimpulan bahwa solusi analitik persamaan shrodinger nonlinier satu dimensi a�a� ��,�� a��a�� ��,�� +|���,��|����,�� � 0 adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù selanjutnya, analog dengan persamaan (3.12) maka persamaan schrodinger nonlinier dua dimensi dapat dinyatakan sebagai berikut: a�a� a��a��� a ��a��� +|�|�� � 0 (70) dimana � � ����,��, �� dan memandang ��ù,�� sebagai transformasi fourier dari ����,��, �� dengan bentuk berikut: ��ù,�� � � � /*0ù�9837��. *. �����.*. (71) jika persamaan (71) diturunkan terhadap � maka didapatkan a��ù,��a� � aa�" � � /*0ù�9837�� . *. �����.*. & � � � /*0ù�9837��9.*. ����� . *. � � � ! x/*0ù�9837��y*.. � ��� ù�1 ����/*0ù�9837�� ��.*. ; <=���.*. � � �� ù�1 ����/*0ù�9837�� _����.*. . *. � ù�1 ��� � � /*0ù�9837�� �����.*. . *. � ù�1 �����ù,�� (72) untuk mendapatkan tranformasi fourier dari {7|�ù,9�{377 maka analog dengan proses persamaan (72) maka didapatkan a���ù,��a��� � �ù��� 1����ù,�� (73) selanjutnya memandang transformasi fourier dari |����,��, ��|� adalah |��ù,��|� maka dapat dinyatakan |��ù,��|� � ��ù,���u�ù,�� (74) berdasarkan persamaan (71) maka persamaan (74) menjadi |��ù,��|� � " � � /*0ù�9837��. *. �����.*. & " � � /*0ù�9837��c. *. �����.*. & (75) dengan persamaan (73) dan (74) maka persamaan (70) menjadi ù�1 �����ù,�� a���ù,��a�� � ù��� 1����ù,�� +|��ù,��|���ù,�� � 0 persamaan ini dapat disederhanakan menjadi a���ù,��a�� � `ù��� 1�� � ù�1 ����+|��ù,��|� a��ù,�� (76) misal ù��� 1�� � ù�1 ���� +|��ù,��|� � v maka persamaan (76) menjadi a���ù,��a�� � v��ù,�� (71) berdaskan persamaan (18) maka solusi bagi persamaan (71) adalah ��ù,�� � 1ð� ����ù�3 vù��ù . � substitusi v dengan ù��� 1�� � ù�1 ����+|��ù,��|� maka didapatkan lukman hakim dan ari kusumastuti 94 volume 2 no. 2 mei 2012 ��ù,�� � 1ð� ����ù�3 ` ù��� 1�� � ù�1 ����+|��ù,��|�aù��ù . � kemudian substitusi |��ù,��|� dengan persamaan (75) maka didapatkan ��ù,�� � 1ð� ��� � ! ù �3 �� 1��ù� � �1 ���ù� �+ � !" � � /*0ù�98}�� . *. ����� . *. & " � � /*0ù�98}��c.*. ����� . *. &; << =ù ; <<< << = �ù.� (72) persamaan (72) adalah solusi bagi persamaan airy (71), sehingga untuk mendapatkan solusi persamaan schrodinger (70) maka transformasi fourier yang terdapat pada persamaan (72) harus diinverskan. selanjutnya pandang persamaan (71) sebagai transformasi fourier maka invers dari transformasi fourier tersebut adalah 1�2ð�� � � /0ù�9837� . *. . *. ��ù,���㈹��� (73) dengan menerapkan invers transformasi fourier (73) pada persamaan (72) maka didapatkan solusi bagi persamaan schrodinger (70) adalah ����,��, �� � 1ð� ��� � ! ù �3 �� 1��ù� � �1 ���ù� �+ � ! 1�2ð�' e � � /0ù�9837� . *. . *.��ù,������� f e � � /0ù�9837� . *. . *.�u�ù,������� f ; <<< << = ù ; << << << << = �ù.� (74) berdasarkan persamaan (74) maka generalisasi fungsi airy sebagai solusi persamaan schrodinger nonlinier dimensi � yaitu ����,��,��,…,��, �� � 1ð� ��� � ! ù�3 "� 1 #�$ � $%� & �ù� "� �� 1 #�$�$%' & �ù� ) "� #�$�*�$%� 1& � 2� � "� #�$�$%� & � ù� � + � ! 1�2ð�� � � � …. *. . *. � /0ù�∑ 345467 89�. *. . *. ��� ; << = ; << << << << << << << << = . � �ù penutup dari paparan pembahasan di atas dapat disimpulkan bahwa bentuk generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier, yaitu: bentuk generalisasi fungsi airy ketika pangkat � dari modulus persamaan schrodinger nonlinier a. jika � genap yaitu 2[, maka didapatkan bentuk umum fungsi airy untuk persamaan schrodinger nonlinier adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù b. jika � ganjil yaitu 2[ 1, maka didapatkan bentuk umum fungsi airy untuk persamaan schrodinger nonlinier adalah ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù dengan demikian dapat diambil kesimpulan bahwa ketika pangkat dari modulus dianalisis dengan bentuk genap dan ganjil menghasilkan penyelesaian yang sama, yaitu: ���,�� � 1ð� ��� �ù�3 � ù�� . � �ù sedangkan bentuk umum ketika di analisis pada persamaan schrodinger nonlinier dimensi tinggi maka didapatkan generalisasi fungsi airy sebagai berikut: ����,��,��,…,��, �� � 1ð� ��� � ! ù�3 "� 1 #�$ � $%� & �ù� "� �� 1 #�$�$%' & � ù� ) "� #�$�*�$%� 1& � ù� � "� #�$�$%� & �ù� � + � ! 1�2ð�� � � � …. *. . *. � /0ù�∑ 345467 89�. *. . *. ��� ; << = ; << << << << << << << << = . � �ù daftar pustaka [1] agarwal, ravi p. dan o’regan, donal. 2009. ordinary and partial diffreential equations. new york: springer. [2] polyanain, a. d. dan zaitsev. 2004. handbook of nonlinear partial differential equatiuons. new york: chapman & hall. generalisasi fungsi airy sebagai solusi analitik persamaan schrodinger nonlinier jurnal cauchy – issn: 2086-0382 95 [3] purcell, edwin j. dan dale varberg. 1999. kalkulus dan geometri analitik. jilid 2. jakarta: erlangga. [4] purwanto, agus. 2003. fisika matematika 1&2. surabaya: its press. [5] ross, shepley l. 1984. differential equations. third edition. new york: john wiley&sons. inc. [6] sudirham, sudaryatno dan ning utari. 2010. mengenal sifat-sifat materi. bandung:: darpublic. [7] valle, oliver dan manuael, soares. 2004. airy functions and applications to physics. london: imperial college press. lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf 4. daftar isi.pdf 5.1. elva ravita sari studi tentang persamaan fuzzy _54-65.pdf analisis matematik terhadap azimat numerik jurusan matematika uin maulana malik ibrahim malang matematika merupakan ilmu pengetahuan dasar yang tidak terlepas dari alam dan agama. masyarakat saat ini masih banyak yang percaya tentang benda mempercayai dan menggunakan jimat. jimat atau azimat atau ta terdiri dari tulisan yang ditulis pada kertas, kain, kayu dan lain sebagainya yang dianggap mempunyai kesaktian dan dapat mengobati segala macam penyakit. adalah konsep matematika apakah yang direpresentasikan dalam azimat numerik yaitu bagaimana penjelasan mengenai azimat numerik berdasarkan klasifikasi pada persegi ajaib langkah dalam menganalisis azimat numerik yaitu mengumpulkan azimat y mengidentifikasi pola-pola perhitungan persegi ajaib kesimpulan. berdasarkan hasil penelitian diperoleh azimat yang dikhususkan dalam bentuk tabel menggunakan konsep persegi ajaib ajaib yang terdapat dalam azimat numerik yaitu persegi semi sempurna (perfect magic square), (bondered), persegi ajaib penjumlahan persegi ajaib. berdasarkan hasil analisis tersebut, azimat numerik yang berbentuk tabel n x n secara matematika tidak mempunyai kekuatan ap (magic square). kata kunci: bilangan, azimat, persegi ajaib pendahuluan matematika merupakan ilmu pengetahuan dasar yang dibutuhkan semua manusia dalam kehidupan sehari-hari baik secara langsung maupun tidak langsung. matematika sendiri banyak membantu dan mempermudah untuk menyelesaikan permasalahan dalam kajian ilmu yang lainnya, seperti halnya pada ilmu kedokteran, ilmu hukum, ilmu perbintangan, ilmu agama dan masih banyak ilmu-ilmu yang lainnya, bahkan dalam sebuah buku karangan hairur rahman (2007) dituliskan bahwa m merupakan ilmu yang tidak terlepas dari alam dan agama yang kebenarannya dapat dilihat dalam al-qur’an. oleh karena itu, h antara matematika dengan ilmu agama memang sangat erat, sama seperti hubungan matematika dengan ilmu-ilmu yang lain. suatu ilmu dapat berkembang karena berbaur dengan ilmu-ilmu yang lain. kebanyakan masyarakat kurang mengetahui bahwa perkembangan ilmu yang semakin pesat ini, dapat merubah hal yang tidak mungkin untuk dijelaskan menjadi suatu hal yang dapat dijelaskan secara rinci. sebuah buku karangan abil ( dituliskan bahwa menurut konteks agama islam, azimat dalam bentuk tulisan atau sering juga disebut wifiq adalah tulisan yang menggunakan simbol-simbol dalam bahasa arab, baik berupa nalisis matematik terhadap azimat numerik rosy aliviana dan abdussakir jurusan matematika uin maulana malik ibrahim malang e-mail: rosy_aliviana@yahoo.com abstrak matematika merupakan ilmu pengetahuan dasar yang tidak terlepas dari alam dan agama. masyarakat saat ini masih banyak yang percaya tentang benda-benda yang membawa pada kesyirikan. seperti halnya menggunakan jimat. jimat atau azimat atau tamimah merupakan terdiri dari tulisan yang ditulis pada kertas, kain, kayu dan lain sebagainya yang dianggap mempunyai kesaktian dan dapat mengobati segala macam penyakit. permasalahan yang diangkat dalam penelitian ini konsep matematika apakah yang direpresentasikan dalam azimat numerik yaitu bagaimana penjelasan mengenai azimat numerik berdasarkan klasifikasi pada persegi ajaib (magic square) langkah dalam menganalisis azimat numerik yaitu mengumpulkan azimat yang berbentuk kotak, pola perhitungan persegi ajaib (magic square), mengklasifikasikannya dan menarik kesimpulan. berdasarkan hasil penelitian diperoleh azimat yang dikhususkan dalam bentuk tabel menggunakan konsep persegi ajaib (magic square) dan diperoleh lima klasifikasi jenis dari konsep persegi ajaib yang terdapat dalam azimat numerik yaitu persegi semi-ajaib (semimagic square) (perfect magic square), persegi ajaib simetris (symmetric magic square), persegi persegi ajaib penjumlahan-perkalian (addition-multiplication magic square), . berdasarkan hasil analisis tersebut, azimat numerik yang berbentuk tabel n x n secara matematika tidak mempunyai kekuatan apa-apa dan merupakan susunan bilangan dari persegi ajaib bilangan, azimat, persegi ajaib (magic square) matematika merupakan ilmu pengetahuan dasar yang dibutuhkan semua manusia dalam hari baik secara langsung atematika sendiri banyak membantu dan mempermudah untuk menyelesaikan permasalahan dalam kajian ilmuilmu yang lainnya, seperti halnya pada ilmu kedokteran, ilmu hukum, ilmu perbintangan, ilmu ilmu yang lainnya, bahkan dalam sebuah buku karangan hairur rahman (2007) dituliskan bahwa matematika lepas dari alam dan agama yang kebenarannya dapat dilihat . oleh karena itu, hubungan antara matematika dengan ilmu agama memang sangat erat, sama seperti hubungan matematika suatu ilmu dapat berkembang karena ilmu yang lain. kebanyakan masyarakat kurang mengetahui bahwa perkembangan ilmu yang semakin pesat ini, dapat merubah hal yang tidak mungkin untuk dijelaskan menjadi suatu hal yang dapat sebuah buku karangan abil (1994) dituliskan bahwa menurut konteks agama islam, azimat dalam bentuk tulisan atau sering juga adalah tulisan yang menggunakan simbol dalam bahasa arab, baik berupa huruf, angka, gambar, maupun kombinasi ketiganya dan diyakini oleh k tradisional mempunyai khasiat atau kekuatan tertentu. azimat terdiri dari tiga macam yaitu azimat numerik, azimat alfabetik dan azimat pictorial. azimat dengan tulisan berupa angka angka arab disebut azimat azimat dengan tulisan berupa huruf atau huruf hijaiyah disebut azimat dengan tulisan berupa gambar tertentu disebut azimat pictorial azimat numerik dapat matematika karena menggunakan angka atau simbol bilangan. pada ini merupakan salah satu contoh yang diambil dari kitab al gozali: gambar 1. azimat anak cerdas (sumber: al gozali, azimat tersebut digunakan agar anak menjadi cerdas dan memperoleh ilmu yang tinggi. dengan demikian, analisis matematika terhadap azimat numerik sangat mungkin untuk nalisis matematik terhadap azimat numerik matematika merupakan ilmu pengetahuan dasar yang tidak terlepas dari alam dan agama. masyarakat benda yang membawa pada kesyirikan. seperti halnya mimah merupakan sebuah bahan yang terdiri dari tulisan yang ditulis pada kertas, kain, kayu dan lain sebagainya yang dianggap mempunyai permasalahan yang diangkat dalam penelitian ini konsep matematika apakah yang direpresentasikan dalam azimat numerik yaitu bagaimana (magic square). langkahang berbentuk kotak, , mengklasifikasikannya dan menarik kesimpulan. berdasarkan hasil penelitian diperoleh azimat yang dikhususkan dalam bentuk tabel dan diperoleh lima klasifikasi jenis dari konsep persegi (semimagic square), persegi ajaib persegi ajaib kosentrik multiplication magic square), dan variasi . berdasarkan hasil analisis tersebut, azimat numerik yang berbentuk tabel n x n secara apa dan merupakan susunan bilangan dari persegi ajaib huruf, angka, gambar, maupun kombinasi ketiganya dan diyakini oleh kalangan islam tradisional mempunyai khasiat atau kekuatan azimat terdiri dari tiga macam yaitu azimat numerik, azimat alfabetik dan azimat azimat dengan tulisan berupa angkaazimat numeric, sedangkan lisan berupa huruf-huruf arab atau huruf hijaiyah disebut azimat alfabetik. azimat dengan tulisan berupa gambar-gambar azimat pictorial. dapat dianalisis secara matematika karena menggunakan angka-angka pada gambar 1.1 di bawah salah satu contoh azimat numerik al-aufaq karangan al azimat anak cerdas halaman 15) digunakan agar anak menjadi cerdas dan memperoleh ilmu yang dengan demikian, analisis matematika terhadap azimat numerik sangat mungkin untuk rosy aliviana dan abdussakir 106 dilakukan agar dapat mengetahui rahasia angka yang ada di dalam azimat tersebut selama ini, masyarakat pengguna azimat tidak mengetahui secara rinci yang dipakai atau digunakannya. mereka hanya meyakini bahwa azimat tersebut memang mempunyai kekuatan tanpa mengetahui apa sebenarnya yang tertulis dalam azimat tersebut. kalangan masyarakat muslim pedesaan pada umumnya dan sebagian masyarakat perkotaan, keberadaan azimat menjadi suatu yang sangat penting untuk membantu mengatasi berbagai permasalahan hidup, bahkan tidak sedikit orang yang sudah dianggap berpengetahuan luas dalam bidang aga memberikan azimat kepada masyarakat sebagai media untuk mengatasi masalah atau untuk kesaktian. tidak jarang ada orang yang menjadikan azimat sebagai barang dagangan untuk diperjualbelikan. keberadaan azimat ini oleh masyarakat diyakini tidak seked media, tetapi azimat itulah yang kemudian dianggap dapat membantu masyarakat dengan kekuatan yang dimilikinya. adapun batasan masalah pada penelitan ini adalah penulis hanya membatasi pada numerik yang berbentuk tabel dengan konsep matematika yaitu persegi ajaib (magic square) tujuan dari penelitian ini adalah menjelaskan secara rinci mengenai azimat numerik berdasarkan klasifikasi matematikanya kajian teori numerik dan sejarah bilangan numerik dalam kamus besar bahasa indonesia memiliki arti yang berwujud nomor (angka); yang bersifat angka atau sistem angka sehingga dalam menyatakan bilangan, manusia menggunakan lambang atau simbol bilangan yang disebut angka (numeral). melihat perkembangan sejarahnya, simbol bilangan berbeda-beda berdasarkan kemajuan budaya suatu bangsa. simbol bilangan suatu bangsa kadang merupakan hasil adopsi dan adaptasi dari simbol bilangan bangsa lainnya. sejarah bilangan dimulai dari bangsa india abad ke-2 sebelum masehi (sm) sampai abad ke 6 m. kemudian dikembangkan oleh bangsa arab pada abad ke-7 sampai ke-9 m, bangsa mesir kuno, bangsa sumeria, bangsa babilonia dan kemudian bangsa yunani. sedangkan dalam matematika, untuk memudahkan uraian, penjelasan, atau keterangan, orang memerlukan seperangkat kesepakatan atau perjanjian tentang makna penggunakan lambang-lambang tertentu (muhsetyo, 1997:5). metode paling awal yang digunakan oleh manusia merupakan sistem tallis gambar. sistem tallis mungkin merupakan volume agar dapat mengetahui rahasia-rahasia angka yang ada di dalam azimat tersebut. arakat pengguna azimat tentang azimat yang dipakai atau digunakannya. mereka hanya meyakini bahwa azimat tersebut memang mempunyai kekuatan tanpa mengetahui apa sebenarnya yang tertulis dalam azimat tersebut. at muslim pedesaan pada umumnya dan sebagian masyarakat perkotaan, keberadaan azimat menjadi suatu yang sangat penting untuk membantu mengatasi berbagai permasalahan hidup, bahkan tidak sedikit orang yang sudah dianggap berpengetahuan luas dalam bidang agama juga memberikan azimat kepada masyarakat sebagai media untuk mengatasi masalah atau untuk kesaktian. tidak jarang ada orang yang menjadikan azimat sebagai barang dagangan untuk diperjualbelikan. keberadaan azimat ini oleh masyarakat diyakini tidak sekedar sebagai media, tetapi azimat itulah yang kemudian dianggap dapat membantu masyarakat dengan dapun batasan masalah pada penelitan ini adalah penulis hanya membatasi pada azimat numerik yang berbentuk tabel dengan konsep (magic square). tujuan dari penelitian ini adalah mengenai azimat numerik berdasarkan klasifikasi matematikanya. numerik dalam kamus besar bahasa g berwujud nomor g bersifat angka atau sistem angka. sehingga dalam menyatakan bilangan, manusia menggunakan lambang atau simbol bilangan melihat perkembangan sejarahnya, simbol beda berdasarkan kemajuan simbol bilangan suatu bangsa kadang merupakan hasil adopsi dan adaptasi dari simbol bilangan bangsa lainnya. angsa india pada 2 sebelum masehi (sm) sampai abad ke. kemudian dikembangkan oleh bangsa arab 9 m, bangsa mesir kuno, bangsa sumeria, bangsa babilonia dan sedangkan dalam matematika, untuk memudahkan uraian, penjelasan, atau keterangan, orang memerlukan seperangkat kesepakatan atau perjanjian tentang lambang tertentu metode paling awal yang digunakan oleh istem tallis dan sistem mungkin merupakan metode yang paling awal digunakan manusia untuk melakukan pengecekan terhadap kuantitas-kuantitas tertentu. gambar, bilangan diwujudkan dengan pengulangan simbol-simbol yang mewakili objek tertentu. contoh dari sistem gambar adalah sebagai berikut: gambar 2. lambang bilangan mesir kuno (sumber: www-history.mcs.st andrew.ac.uk/histtopics/egyptian_numerals.html kemudian dikembangkan oleh bangsa arab pada abad ke-7 sampai ke mesir kuno, bangsa sumeria, bangsa babilonia dan kemudian bangsa yunani sebelum angka india masuk ke arab sekitar abad ketujuh masehi, bangsa arab menggunakan huruf untuk melambangkan bilangan. sistem ini dikenal dengan nama al-jumal atau dikenal juga dengan nama yang diambil dari empat huruf pertama. berikut ini adalah tabel huruf al-jumal tabel 1. huruf huruf lambang desimal 1 ا 2 ب 3 ج 4 د 5 ه 6 و 7 ز 8 ح 9 ط 10 ي 20 ك 30 ل 40 م 50 ن bangsa mesir kuno mempunyai tiga macam sistem numerasi, yaitu sistem hieratic, dan demotic. sistem merupakan sistem yang sangat kompleks untuk digunakan dalam kehidupan sehari biasanya dituliskan pada batu. sistem kemudian dikembangkan menjadi sistem yang lebih sederhana yang dikenal dengan sistem hieratic. sistem hieratic digunaka di kuil-kuil dan ditulis di daun papyrus sehingga dikenal pula dengan sistem kuil. sistem dikembangkan dari sistem sistem numerasi yang banyak digunakan dalam kehidupan sehari-hari. volume 2 no. 2 mei 2012 metode yang paling awal digunakan manusia untuk melakukan pengecekan terhadap kuantitas tertentu. dan pada sistem , bilangan diwujudkan dengan simbol yang mewakili objek dari sistem gambar adalah sebagai lambang bilangan mesir kuno history.mcs.standrew.ac.uk/histtopics/egyptian_numerals.html) kemudian dikembangkan oleh bangsa 7 sampai ke-9 m, bangsa eria, bangsa babilonia dan kemudian bangsa yunani. sebelum angka india masuk ke arab sekitar abad ketujuh masehi, bangsa arab menggunakan huruf untuk melambangkan bilangan. sistem ini dikenal dengan nama huruf atau dikenal juga dengan nama abjad, yang diambil dari empat huruf pertama. berikut jumal bangsa arab: huruf al-jumal huruf lambang desimal 60 س 70 ع 80 ف 90 ص 100 ق 200 ر 300 ش 400 ت 500 ث 600 خ 700 ذ 800 ض 900 ظ 1000 غ bangsa mesir kuno mempunyai tiga macam sistem numerasi, yaitu sistem hieroglyph, . sistem hieroglyph merupakan sistem yang sangat kompleks untuk digunakan dalam kehidupan sehari-hari dan biasanya dituliskan pada batu. sistem hieroglyph kemudian dikembangkan menjadi sistem yang lebih sederhana yang dikenal dengan sistem digunakan oleh pendeta kuil dan ditulis di daun papyrus sehingga dikenal pula dengan sistem kuil. sistem demotic dikembangkan dari sistem hieratic dan menjadi sistem numerasi yang banyak digunakan dalam jurnal cauchy – issn: 2086-0382 berikut disajikan tabel perubahan secara bertahap angka brahma menjadi angka desimal di eropa (abdussakir, 2009:54-57) gambar 3. perubahan angka india ke angka desimal (sumber: http://www.archimedes-lab.org/numeral.html persegi ajaib (magic square) a. sejarah munculnya persegi ajaib persegi ajaib ditemukan di yu sekitar tahun 2200 sm. terdapat legenda bahwa dahulu kala terdapat bencana banjir di sekitar sungai kuning. saat raja yu berusaha untuk menyalurkan air ke laut, terlihat kura-kura dengan pola titik titik bulat bilangan yang diatur dalam suatu corak petak sembilan tiga aneh pada tempurung. persegi ajaib yang ditemukan raja yu tersebut disebut lo shu. bilangan dalam lo s ditunjukkan sebagai titik-titik atau noktah pada suatu tali. lo shu sebenarnya adalah persegi ajaib berukuran 3 x 3 (alex. 2004:4). b. pengertian persegi ajaib sebuah buku karangan w. s andrews (1960) dijelaskan bahwa persegi ajaib persegi magis (magic square) bilangan dalam kotak-kotak yang berbentuk persegi dengan sifat jumlah bilangan menurut masing-masing baris, kolom, ataupun diagonalnya adalah sama. persegi ajaib berukuran � � �, sebanyak � disusun dalam kotak-kotak persegi dengan syarat tidak ada bilangan yang ditulis berulang dan jumlah bilangan-bilangan menurut masing masing baris, kolom, ataupun diagonal adalah sama. berikut ini adalah contoh persegi ajaib berukuran 3 � 3, 4 � 4 dan 5 � bahwa jumlah bilangan pada masing baris, kolom, dan diagonal adalah sama. gambar 4. magic square 3 × 3, 4 × 4 dan 5 × 5 c. klasifikasi magic square terdapat tujuh klasifikasi dalam square yaitu sebagai berikut: analisis matematika terhadap azimut numerik 0382 perubahan secara menjadi angka desimal 57). perubahan angka india ke angka desimal lab.org/numeral.html) sejarah munculnya persegi ajaib ersegi ajaib ditemukan di cina oleh raja u sekitar tahun 2200 sm. terdapat legenda bahwa dahulu kala terdapat bencana banjir di u berusaha untuk menyalurkan kura dengan pola titiktitik bulat bilangan yang diatur dalam suatu corak ilan tiga aneh pada tempurung. segi ajaib yang ditemukan raja yu tersebut disebut lo shu. bilangan dalam lo shu tau noktah pada hu sebenarnya adalah persegi ajaib sebuah buku karangan w. s andrews persegi ajaib atau adalah susunan kotak yang berbentuk persegi dengan sifat jumlah bilangan-bilangan masing baris, kolom, ataupun diagonalnya adalah sama. persegi ajaib � � � bilangan segi dengan syarat tidak ada bilangan yang ditulis berulang dan bilangan menurut masingmasing baris, kolom, ataupun diagonal adalah berikut ini adalah contoh persegi ajaib � 5. perhatikan pada masing-masing baris, kolom, dan diagonal adalah sama. magic square 3 × 3, 4 × 4 dan 5 × 5 terdapat tujuh klasifikasi dalam magic 1. persegi semi-ajaib ( adalah sebuah matriks yang berukuran jika dijumlahkan dari elemen setiap baris dan kolom adalah sama. dengan mengabaikan jumlah kedua diagonal. contoh persegi semi ajaib berukuran 5 � 5 adalah �� �� 17131925 172341011 81420212 jumlah bilangan pada adalah sama, yaitu 65. 2. diabolik, pandiagonal sempurna (perfect magic square) persegi ajaib jika ditambahkan maka jumlah dari setiap baris, kolom, diagonal utama dan diagonal kedua adalah sama atau konstan. contoh jumlah pada persegi ajaib berukuran 3 � 3 yaitu 15 adalah �4 93 58 1 3. persegi ajaib simetris square) adalah persegi ajaib yang mempunyai jumlah dari dua sel dari setiap dua sel yang simetris dan dua sel ditengah maka jumlahnya sama. persegi ajaib simetris juga disebut dengan associative magic square contoh persegi ajaib simetris ini adalah sebagai berikut: tabel 2. persegi ajaib 4 x 4 1 15 14 12 6 7 8 10 11 13 3 2 jumlah setiap baris, kolom dan diagonalnya adalah 34, serta jumlah dari masing 2 sel pojok atas-bawah, kanan adalah 34. 4. persegi ajaib konsentrik adalah persegi ajaib yang menghilangkan bagian atas, bawah, kiri dan kanan kolom akan menghasilkan persegi ajaib lain. contohnya adalah �� �� �� 44948478910 5153736181945 6162229243444 4333272523177 jumlah dari setiap kolom, bari diagonal adalah 125. 5. sebuah persegi ajaib nol atau ( square) adalah persegi ajaib yang jika dijumlahkan memiliki urutan baris, kolom, dan diaogonal adalah bilangan 0. persegi ajaib normal ini mengandung bilangan negatif. analisis matematika terhadap azimut numerik 107 ajaib (semimagic square) adalah sebuah matriks yang berukuran n x n, dari elemen setiap baris dan kolom adalah sama. dengan mengabaikan jumlah kedua diagonal. contoh persegi semi adalah 24561218 15162239 �� �� � pada baris dan kolom atau persegi ajaib (perfect magic square) adalah persegi ajaib jika ditambahkan maka jumlah dari setiap baris, kolom, diagonal utama dan diagonal kedua adalah sama atau konstan. persegi ajaib berukuran 276� persegi ajaib simetris (symmetric magic adalah persegi ajaib yang mempunyai jumlah dari dua sel dari setiap dua sel yang simetris dan dua sel ditengah maka jumlahnya sama. persegi ajaib simetris associative magic square. contoh persegi ajaib simetris ini adalah persegi ajaib 4 x 4 14 4 7 9 11 5 2 16 umlah setiap baris, kolom dan diagonalnya adalah 34, serta jumlah dari masing 2 sel bawah, kanan-kiri dan tengah konsentrik atau bordered adalah persegi ajaib yang menghilangkan bagian atas, bawah, kiri dan kanan kolom akan menghasilkan persegi ajaib lain. 39302621282011 38311314323512 40123424142�� �� �� � umlah dari setiap kolom, baris dan kedua sebuah persegi ajaib nol atau (zero magic adalah persegi ajaib yang jika dijumlahkan memiliki urutan baris, kolom, dan diaogonal adalah bilangan 0. persegi ajaib normal ini mengandung bilangan rosy aliviana dan abdussakir 108 �� �� 410�9�3�2 11�8�7�1 5 �12�6 0 6 12 �5 1 7 8�11 6. persegi ajaib perkalian (geometric) matriks persegi dari bilangan yang hasil setiap elemen baris, kolom, diagonal utama dan diagonal kedua adalah konstan. contohnya adalah sebagai berikut: � 432 6 18 164 72 24 108854 3648 12144 2162 jumlah setiap baris, kolom dan kedua diagonal adalah 746496. 7. persegi ajaib penjumlahan (addition-multiplication magic square) adalah persegi ajaib dimana jika dijumlahkan dan dikalikan dalam setiap baris kedua diagonal memiliki jumlah yang sama. contohnya adalah �� �� �� � 162 207 51105 152 10092 27 91 26 133 12029 138 243 136 45 3857 30 17458 75 17113200153 6820378 1841554 225 108 23 90 17 52 1897669 50117232 87102175 jika setiap baris, kolom dan kedua diagonal dikalikan dan dijumlahkan hasilnya adala2,05 � 10�� (stephens, 1993:5 pada perkembangan selanjutnya, terdapat berbagai variasi persegi ajaib yang muncul. berikut ini juga merupakan persegi ajaib yang disebut ixohoxi. masing-masing baris, kolom, dan diagonal mempunyai jumlah 19998. hal ya menarik dari persegi ajaib ini adalah jika dibalik (yang atas di bawah dan yang bawah di atas), tetap terbaca sebagai bujur sangkar ajaib. tabel 3. persegi ajaib ixohoxi 1118 8181 1888 8888 1811 8118 8111 1188 8881 1881 8818 1111 pembahasan klasifikasi azimat numerik berdasarkan konsep persegi ajaib (magic square) azimat numerik merupakan azimat yang terdiri dari angka-angka atau simbol bilangan yang menggunakan bahasa arab dan diyakini oleh kalangan islam tradisional memiliki kh atau kekuatan tertentu. analisis matematik tentang azimat numerik pada batasan masalah dikhususkan pada azimat berbentuk tabel yang memiliki konsep matematika yaitu persegi ajaib (magic square). volume 2 3 9�10 �4�� �� � (geometric) adalah matriks persegi dari bilangan yang hasil setiap elemen baris, kolom, diagonal utama dan diagonal kedua adalah konstan. contohnya adalah sebagai berikut: 161082162 � umlah setiap baris, kolom dan kedua diagonal adalah 746496. penjumlahan-perkalian multiplication magic square) dimana jika dijumlahkan dalam setiap baris, kolom, dan kedua diagonal memiliki jumlah yang sama. 120 116 25243 39 3438 150 26123 119 10452 216 16187102175 1354619 1148160 �� �� �� �� olom dan kedua diagonal dikalikan dan dijumlahkan hasilnya adalah :5-7). pada perkembangan selanjutnya, terdapat berbagai variasi persegi ajaib yang muncul. berikut ini juga merupakan persegi ajaib yang masing baris, kolom, dan diagonal mempunyai jumlah 19998. hal yang menarik dari persegi ajaib ini adalah jika dibalik (yang atas di bawah dan yang bawah di atas), tetap terbaca sebagai bujur sangkar ajaib. persegi ajaib ixohoxi 8811 1181 1818 8188 klasifikasi azimat numerik berdasarkan (magic square) azimat numerik merupakan azimat yang angka atau simbol bilangan yang menggunakan bahasa arab dan diyakini oleh kalangan islam tradisional memiliki khasiat atau kekuatan tertentu. analisis matematik tentang azimat numerik pada batasan masalah dikhususkan pada azimat berbentuk tabel yang memiliki konsep matematika yaitu persegi ajaib berdasarkan jenis klasifikasi dari persegi ajaib (magic square), maka di bawah ini merupakan hasil analisis dari berbagai macam azimat numerik yang telah dikelompokkan. a. persegi semi-ajaib (semimagic square) i. azimat hidup sejahtera azimat ini digunakan agar hidup seseorang bisa sejahtera dan dilindungi dari murka allah, serta diberikan panjang umur dan kekayaan yang melimpah. azimat ini terbentuk dari surat al ikhlas yang setiap huruf al jumal dijumlahkan, yang terdiri dari: د , ح, ا, �, ل, ل, ا, و, �, ل, .1 ,ل ل, ا .2 ا,� , ,م ص, ل, jumlahnya 231 د, , ل .3 ي ,م ,ل jumlahnya 114 د, د, ل, و, ي, م, ل,و .4 jumlahnya 126 ل,و .5 , ي, م ,ك , ل, ن ,� jumlahnya د, ح, ا, ا, و, ق, ك .6 jumlahnya bentuk azimatnya adalah sebagai gambar 5. azimat hidup sejahtera (sumber: as syadhli halaman 21) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 4. lambang desimal dari azimat hidup sejahtera 120 191 126 114 220 120 191 126 231 220 120 191 114 231 220 120 126 114 231 220 191 126 114 231 berdasarkan hasil analisis numerik pada azimat diatas, jumlah setiap baris dan kolomnya adalah 1002, tetapi jumlah kedua diagonalnya berbeda yaitu 730 dan 1002. maka azimat ini termasuk dalam persegi semi-ajaib berukuran 6 x 6. b. persegi ajaib sempurna square) i. azimat penambah barokah azimat ini digunakan apabila seseorang ingin menambah barokah dari tuhan. bentuk azimatnya dapat dilihat pada gambar 6. azimat tersebut jika diterjemahkan ke dalam lambang desimal akan seperti tabel 5. volume 2 no. 2 mei 2012 berdasarkan jenis klasifikasi dari persegi , maka di bawah ini merupakan hasil analisis dari berbagai macam azimat numerik yang telah dikelompokkan. (semimagic square) azimat ini digunakan agar hidup seseorang bisa sejahtera dan dilindungi dari murka allah, serta diberikan panjang umur dan kekayaan yang melimpah. azimat ini terbentuk dari surat al huruf al jumal-nya dijumlahkan, yang terdiri dari: , ق jumlahnya 220 jumlahnya 231 jumlahnya 114 jumlahnya 126 jumlahnya 191 jumlahnya 120. bentuk azimatnya adalah sebagai berikut: azimat hidup sejahtera (sumber: as syadhli halaman 21) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat hidup sejahtera 114 231 220 126 114 231 191 126 114 120 191 126 220 120 191 231 220 120 berdasarkan hasil analisis numerik pada azimat diatas, jumlah setiap baris dan kolomnya adalah 1002, tetapi jumlah kedua diagonalnya berbeda yaitu 730 dan 1002. maka azimat ini termasuk ajaib (semimagic square) persegi ajaib sempurna (perfect magic azimat penambah barokah azimat ini digunakan apabila seseorang ingin menambah barokah dari tuhan. bentuk dapat dilihat pada gambar 6. jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil jurnal cauchy – issn: 2086-0382 gambar 6. azimat penambah barokah (sumber: al gozali halaman 5) tabel 5. lambang desimal dari azimat penambah barokah 4 9 2 3 5 7 8 1 6 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 15. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau perfect magic square berukuran 3 � 3. ii. azimat mendapatkan belas kasih azimat ini digunakan agar mendapat belas kasih, pengayoman dari tuhan dan diberi kecukupan dalam hidupnya. azimat ini terbentuk dari: ي, ف, ا, ك .1 (maha mencukupi) jumlahnya 111 ي, ي, ل, و .2 (maha mengayomi) jumlahnya 56 د, و, د, و .3 (maha belas kasih) jumlahnya 10 (terdapat 2 huruf yang sama yaitu dalam kitab dituliskan hanya dihitung 2 huruf saja). jumlah keseluruhannya 177. bentuk azimatnya adalah sebagai berikut: gambar 7. azimat mendapatkan belas kasih (sumber: al gozali halaman 4) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 6. lambang desimal dari azimat mendapatkan belas kasih 44 47 50 36 49 37 43 48 38 52 45 42 46 41 39 51 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 177. maka azimat tersebut termasuk analisis matematika terhadap azimut numerik 0382 azimat penambah barokah (sumber: al gozali halaman 5) lambang desimal dari azimat penambah berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 15. maka azimat tersebut termasuk dalam perfect magic square mendapatkan belas kasih azimat ini digunakan agar mendapat belas kasih, pengayoman dari tuhan dan diberi kecukupan dalam hidupnya. azimat ini terbentuk (maha mencukupi) jumlahnya 111 (maha mengayomi) jumlahnya 56 a belas kasih) jumlahnya 10 (terdapat 2 huruf yang sama yaitu و dan د, maka dalam kitab dituliskan hanya dihitung 2 huruf jumlah keseluruhannya 177. bentuk azimatnya adalah sebagai berikut: azimat mendapatkan belas kasih gozali halaman 4) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat mendapatkan 36 48 42 51 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 177. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau scuare berukuran 3 x 3. iii. azimat bermacam-macam khasiat azimat ini digunakan untuk melindungi diri, menjaga diri dari musuh, gangguan jin dan dari tipu daya manusia. bentuk azimatnya adalah sebagai berikut: gambar 8. azimat bermacam (sumber: kitab al gozali halaman 7) azimat di atas jika diterjemahkan kedalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 7. lambang desimal dari azimat bermacam macam khasiat 549 542 544 546 545 550 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 1638. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau scuare berukuran 3 x 3. iv. azimat mempercepat jodoh azimat ini digunakan jika keluarga laki laki atau perempuan uang belum bertemu dengan jodohnya walaupun sudah lanjut usia akan segera bertemu. bentuk azimatnya adalah sebagai berikut: gambar 9. azimat mempercepat jodoh (sumber: halfeasy halaman 78 azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 8. lambang desimal dari azimat mempercepat jodoh 209 214 208 210 213 306 berdasarkan hasil analisis pada azimat di atas, terdapat 1 kolom yang penulisannya kurang tepat. jumlah dari elemen pada baris horisontal dan vertikal yang bersinggungan dengan kolom analisis matematika terhadap azimut numerik 109 dalam persegi ajaib sempurna atau perfect magic macam khasiat azimat ini digunakan untuk melindungi diri, menjaga diri dari musuh, gangguan jin dan dari tipu daya manusia. bentuk azimatnya adalah azimat bermacam-macam khasiat (sumber: kitab al gozali halaman 7) atas jika diterjemahkan kedalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat bermacammacam khasiat 547 548 543 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 1638. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau perfect magic epat jodoh azimat ini digunakan jika keluarga lakilaki atau perempuan uang belum bertemu dengan jodohnya walaupun sudah lanjut usia akan segera bertemu. bentuk azimatnya adalah azimat mempercepat jodoh halaman 78-79) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat mempercepat 207 212 211 berdasarkan hasil analisis pada azimat di atas, terdapat 1 kolom yang penulisannya kurang tepat. jumlah dari elemen pada baris horisontal dan vertikal yang bersinggungan dengan kolom rosy aliviana dan abdussakir 110 yang berwarna merah adalah 730. sehingga, agar memenuhi syarat-syarat pada klasifikasi square angka 306 tersebut diganti dengan angka 206 seperti pada tabel di bawah ini: tabel 9. lambang desimal dari azimat mempercepat jodoh 209 214 207 208 210 212 213 206 211 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 630. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau scuare berukuran 3 x 3. c. persegi ajaib kosentrik atau i. azimat memperlancar berbicara azimat ini digunakan apabila seseorang ingin lancar saat berbicara atau setiap perkataan yang keluar dari isi hati. bentuk azimatnya adalah sebagai berikut: gambar 10. azimat memperlancar berbicara (sumber: al gozali halaman 14) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 10. lambang desimal dari azimat memperlancar berbicara 15 22 9 16 2 14 21 8 19 1 13 25 6 18 5 12 23 10 17 4 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 39. maka azimat tersebut termasuk dalam persegi ajaib kosentrik atau bordered jika bagian atas, bawah dan kiri, dihilangkan akan menghasilkan persegi ajaib (magic square) lain. d. persegi ajaib simetris (symmetric magic square) i. azimat menyiksa musuh azimat ini digunakan jika seseorang ingin menyiksa musuhnya. bentuk azimatnya adalah sebagai berikut: volume yang berwarna merah adalah 730. sehingga, agar a klasifikasi magic angka 306 tersebut diganti dengan angka 206 seperti pada tabel di bawah ini: lambang desimal dari azimat mempercepat berdasarkan hasil analisis pada azimat di setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 630. maka azimat tersebut termasuk dalam persegi ajaib sempurna atau perfect magic persegi ajaib kosentrik atau bordered azimat memperlancar berbicara azimat ini digunakan apabila seseorang ingin lancar saat berbicara atau setiap perkataan yang keluar dari isi hati. bentuk azimatnya azimat memperlancar berbicara (sumber: al gozali halaman 14) jemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat memperlancar berbicara 3 20 7 24 11 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 39. maka azimat tersebut termasuk dalam bordered yang mana jika bagian atas, bawah dan kiri, kanan kolom dihilangkan akan menghasilkan persegi ajaib (symmetric magic azimat ini digunakan jika seseorang ingin menyiksa musuhnya. bentuk azimatnya adalah gambar 11. azimat menyiksa musuh (sumber: as-syadli halaman 50) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 11. lambang desimal dari azimat menyiksa musuh 62 65 68 67 56 61 57 70 63 64 59 58 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 250. maka azimat tersebut termasuk dalam persegi ajaib simetris atau magic square. ii. azimat pengobat rasa takut azimat ini digunakan untuk menghilang kan rasa takut dihati dan memberi rasa aman dan keselamatan pada diri. bentuk azimatnya adalah sebagai berikut: gambar 12. azimat pengobat rasa takut (sumber: as-syadhli halaman 50) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 12. lambang desimal dari azimat pengobat rasa takut 119 122 125 124 113 118 114 127 120 121 116 115 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 478. maka azimat tersebut termasuk dalam persegi ajaib simetris atau magic square. iii. azimat bermacam-macam manfaa azimat ini digunakan untuk melindungi diri dari gangguan orang jahat selama perjalanan, volume 2 no. 2 mei 2012 azimat menyiksa musuh syadli halaman 50) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat menyiksa musuh 68 55 61 66 63 60 58 69 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 250. maka azimat tersebut termasuk dalam persegi ajaib simetris atau symmetric pengobat rasa takut azimat ini digunakan untuk menghilangkan rasa takut dihati dan memberi rasa aman dan keselamatan pada diri. bentuk azimatnya adalah azimat pengobat rasa takut syadhli halaman 50) atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat pengobat rasa takut 125 112 118 123 120 117 115 126 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 478. maka azimat tersebut termasuk dalam persegi ajaib simetris atau symmetric macam manfaat azimat ini digunakan untuk melindungi diri dari gangguan orang jahat selama perjalanan, jurnal cauchy – issn: 2086-0382 keluar rumah, dan dimudahkan rizkinya. bentuk azimatnya adalah sebagai berikut: gambar 13. azimat bermacam-macam manfaat (sumber: as-syadhli halaman 53) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 13. lambang desimal dari azimat bermacam macam manfaat 27 30 33 20 32 21 26 31 22 35 28 25 29 24 23 34 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 110. maka azimat tersebut termasuk dalam persegi ajaib simetris atau magic square. iv. azimat memikat orang azimat ini digunakan apabila seseorang ingin siapa saja yang memandangnya akan hanyut atau terpikat. bentuk azimatnya adalah sebagai berikut: gambar 14. azimat memikat orang (sumber: abbas halaman 79) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 14. lambang desimal dari azimat memikat orang 40 10 900 70 899 71 39 11 72 902 8 38 9 37 73 901 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 1020. maka azimat tersebut termasuk dalam persegi ajaib simetris atau magic square. e. persegi ajaib penjumlahan (addition-multiplication magic square) i. azimat kekuatan azimat ini digunakan agar semua orang tunduk kepadanya dan apapun yang diinginkan analisis matematika terhadap azimut numerik 0382 keluar rumah, dan dimudahkan rizkinya. bentuk azimatnya adalah sebagai berikut: macam manfaat syadhli halaman 53) atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat bermacam20 31 25 34 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 110. maka azimat tersebut termasuk dalam persegi ajaib simetris atau symmetric ni digunakan apabila seseorang ingin siapa saja yang memandangnya akan hanyut atau terpikat. bentuk azimatnya adalah azimat memikat orang (sumber: abbas halaman 79) azimat di atas jika diterjemahkan ke dalam an diperoleh hasil seperti lambang desimal dari azimat memikat 70 11 38 901 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 1020. maka azimat tersebut termasuk dalam persegi ajaib simetris atau symmetric persegi ajaib penjumlahan-perkalian multiplication magic square) azimat ini digunakan agar semua orang tunduk kepadanya dan apapun yang diinginkan akan tercapai. bentuk azimatnya adalah sebagai berikut: gambar 15. azimat kekuatan (sumber: al gozali azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 15. lambang desimal dari azimat kekuatan 9 7 5 5 3 1 1 9 7 7 5 3 3 1 9 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 25. maka azimat tersebut termasuk dalam persegi ajaib penjumlahan multiplication magic square). ii. azimat keselamatan azimat ini digunakan agar tidak terjerumus ketempat yang nista dan apabila dibenci orang, maka orang tersebut tidak akan membenci lagi, diibaratkan besi yang leleh dengan bara api. bentuk azimatnya adalah sebagai berikut: gambar 16. azimat (sumber: al gozali halaman 37) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 16. lambang desimal dari azimat keselamatan 289 329 66 66 102 289 102 66 329 329 289 102 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 786. maka azimat tersebut termasuk dalam persegi ajaib penjumlahan-perkalian multiplication magic square) analisis matematika terhadap azimut numerik 111 akan tercapai. bentuk azimatnya adalah sebagai azimat kekuatan (sumber: al gozali halaman 12) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat kekuatan 3 1 9 7 5 3 1 9 7 5 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 25. maka azimat tersebut termasuk dalam persegi ajaib penjumlahan-perkalian (additionmultiplication magic square). azimat ini digunakan agar tidak terjerumus ketempat yang nista dan apabila dibenci orang, maka orang tersebut tidak akan membenci lagi, diibaratkan besi yang leleh dengan bara api. bentuk azimatnya adalah azimat keselamatan (sumber: al gozali halaman 37) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat keselamatan 66 102 289 329 329 289 102 66 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya maka azimat tersebut termasuk dalam persegi perkalian (additionsquare). rosy aliviana dan abdussakir 112 f. variasi persegi ajaib i. azimat memerangi hawa nafsu azimat ini digunakan untuk memerangi hawa nafsu yang dimiliki manusia. bentuk azimatnya adalah sebagai berikut: gambar 17. azimat memerangi hawa nafsu (sumber: as syadhli halaman 43) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 17. lambang desimal dari azimat memerangi hawa nafsu 16 10 186 12 9 21 19 13 15 196 14 5 6 23 209 205 17 22 2 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 250. maka azimat tersebut termasuk dalam variasi dari persegi ajaib (magic square) ii. azimat memperbanyak rizki azimat ini digunakan untuk memperbanyak rizki dan mendapatkan barokah dari allah. bentuk azimatnya adalah sebagai berikut: gambar 18. azimat memperbanyak rizki (sumber: as syadhli jalil halaman 44) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 18. lambang desimal dari azimat memperbanyak rizki 16 10 1 12 9 21 19 13 15 11 14 5 6 23 24 20 17 22 2 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masing volume azimat memerangi hawa nafsu azimat ini digunakan untuk memerangi hawa nafsu yang dimiliki manusia. bentuk azimatnya adalah sebagai berikut: azimat memerangi hawa nafsu (sumber: as syadhli halaman 43) atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat memerangi 26 188 25 7 4 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 250. maka azimat tersebut termasuk (magic square). digunakan untuk memperbanyak rizki dan mendapatkan barokah dari allah. bentuk azimatnya adalah sebagai azimat memperbanyak rizki (sumber: as syadhli jalil halaman 44) azimat di atas jika diterjemahkan ke dalam diperoleh hasil seperti lambang desimal dari azimat memperbanyak rizki 12 26 13 3 14 25 24 7 2 4 berdasarkan hasil analisis pada azimat diatas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 65. maka azimat tersebut termasuk dalam variasi dari persegi ajaib (magic square) iii. azimat memerangi hawa nafsu azimat ini digunakan untuk memerangi hawa nafsu dan tampak berwibawa dihadapan orang lain. bentuk azimatnya adalah sebagai berikut: gambar 19. azimat memerangi hawa nafsu (sumber: as syadhli halaman 44) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel di bawah ini: tabel 19. lambang desimal dari azimat memerangi hawa nafsu 16 10 2203 9 21 19 15 2213 5 6 23 2222 17 22 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masing masing baris, kolom dan kedua diagonalnya adalah 2267. maka azimat tersebut termasuk dalam variasi dari persegi ajaib iv. azimat balas dendam azimat ini digunakan untuk membalas dendam kepada orang yang telah menganiaya kita sampai orang tersebut meninggal dunia dan jika orang tersebut hidup maka diberikan hidup yang paling hina. bentuk azimatnya adalah sebagai berikut: gambar 20. azimat balas dendam (sumber: as syadhli halaman 45) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti tabel 20. berdasarkan hasil analisis pada azimat pada tabel 20. jumlah dari setiap elemen pada masing-masing baris, kolom dan kedua diagonalnya adalah 585. maka azimat tersebut volume 2 no. 2 mei 2012 masing baris, kolom dan kedua diagonalnya adalah 65. maka azimat tersebut termasuk dalam (magic square). azimat memerangi hawa nafsu azimat ini digunakan untuk memerangi hawa nafsu dan tampak berwibawa dihadapan k azimatnya adalah sebagai azimat memerangi hawa nafsu (sumber: as syadhli halaman 44) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti lambang desimal dari azimat memerangi hawa nafsu 12 26 13 2205 14 25 2226 7 2 4 berdasarkan hasil analisis pada azimat di atas, jumlah dari setiap elemen pada masingmasing baris, kolom dan kedua diagonalnya adalah 2267. maka azimat tersebut termasuk dalam variasi dari persegi ajaib (magic square). digunakan untuk membalas dendam kepada orang yang telah menganiaya kita sampai orang tersebut meninggal dunia dan jika orang tersebut hidup maka diberikan hidup yang paling hina. bentuk azimatnya adalah azimat balas dendam (sumber: as syadhli halaman 45) azimat di atas jika diterjemahkan ke dalam lambang desimal akan diperoleh hasil seperti berdasarkan hasil analisis pada azimat umlah dari setiap elemen pada masing baris, kolom dan kedua diagonalnya adalah 585. maka azimat tersebut analisis matematika terhadap azimut numerik jurnal cauchy – issn: 2086-0382 113 termasuk dalam variasi dari persegi ajaib (magic square). tabel 20. lambang desimal dari azimat balas dendam 16 10 521 12 26 9 21 19 13 523 15 531 14 25 5 6 23 544 7 540 17 22 2 4 penjelasan mengenai azimat numerik berdasarkan klasifikasi persegi ajaib (magic square) wifiq atau wafaq bermakna selaras, serasi dan sistematis. menurut istilah wafaq berarti tulisan yang berupa huruf, angka, simbol yang membentuk pola yang sistematis dan serasi yang diyakini akan membentuk pola energi yang dapat difungsikan sesuai dengan niat sang pembuat wafaq (umar, 2010). keserasian dari wafaq adalah bila angkaangka dalam kotak persegi wafaq saling dijumlahkan secara vertikal, horisontal dan diagonal maka akan menghasilkan jumlah yang sama (dijumlahkan kebawah, kesamping ataupun miring, semua jumlahnya sama). dengan modal tersebutlah dari sisi matematika dapat dijelaskan dengan konsep persegi ajaib (magic square). azimat numerik yang telah dikumpulkan dan dianalisis berdasarkan klasifikasi persegi ajaib (magic square) diatas terdapat enam klasifikasi, yaitu persegi semi-ajaib (semimagic square), persegi ajaib sempurna (perfect magic square), persegi ajaib simetris (syimetric magic square), persegi ajaib kosentrik atau bondered, persegi ajaib penjumlahan-perkalian (additionmultiplication magic square), dan variasi persegi ajaib. menurut analisis diatas maka secara matematik salah satu jenis azimat numerik yang berbentuk kotak merupakan azimat yang terdiri dari angka-angka atau simbol bilangan yang menggunakan bahasa arab dan tidak memiliki khasiat atau kekuatan tertentu, melainkan pengembangan dari salah satu konsep matematika yaitu persegi ajaib (magic square). selain itu, jika dilihat dari segi penulisan, tidak semua penulisan azimat tersebut benar. terbukti pada saat penulis melakukan perhitungan, terdapat beberapa azimat yang kurang tepat dalam segi penulisannya. kasus seperti ini dimungkinkan pada saat penulis buku tersebut menyalin dari buku-buku atau kitab-kitab sebelumnya terdapat suatu kemiripan angka (arab) dan tidak dapat dilihat dengan jelas. al-qur’an surat yunus ayat 106 yang berbunyi: ÿω uρ äí ô‰s?  ïβ èβρßš «!$# $ tβ ÿω y7ãè x ζtƒ ÿω uρ x8•�ûøtƒ ( βî*sù |m ù=yè sù y7 ‾ρî*sù # ]œî) zïiβ tïϑî=≈ ©à9 $# ∩⊇⊃∉∪ artinya: “dan janganlah kamu menyembah apa-apa yang tidak memberi manfaat dan tidak (pula) memberi mudharat kepadamu selain allah; sebab jika kamu berbuat (yang demikian), itu, maka sesungguhnya kamu kalau begitu termasuk orang-orang yang zalim”. dari potongan ayat diatas menjelaskan bahwa, zalim yang dimaksudkan adalah zalim terhadap diri sendiri dengan mempercayai kekuatan pada azimat tersebut. seseorang yang mempercayai sesuatu hal tanpa melihat dan mengetahui apa yang terkandung di dalamnya merupakan salah satu wujud perbuatan syirik. penutup hasil dari analisis matematik terhadap azimat numerik yang berbentuk tabel � � � terkonstruk dari konsep matematika yaitu persegi ajaib (magic square) dengan klasifikasi yaitu persegi semi-ajaib (semimagic square), persegi ajaib sempurna (perfect magic square), persegi ajaib simetris (symmetric magic square), persegi ajaib kosentrik (bondered), persegi ajaib penjumlahan-perkalian (addition-multiplication magic square), dan variasi persegi ajaib. azimat numerik pada bentuk tabel yang telah diteliti dalam pandangan matematika bukan merupakan sesuatu hal yang luar biasa serta tidak memiliki suatu kegunaan yang khusus, melainkan hanyalah sebuah tabel � � � yang angka didalamnya terkonstruk dari sebuah konsep matematika. saat azimat disalin dari buku ke buku atau dari kitab ke kitab terdapat juga kesalahan tulisan karena kemiripan ataupun kesalahan percetakan. pada penelitian ini, penulis menganalisis azimat numerik yang berbentuk tabel. selain azimat numerik terdapat juga azimat alfabetik dan azimat pictorial. oleh karena azimat alfabetik telah diteliti, maka penulis memberikan saran kepada pembaca yang tertarik dengan permasalahan ini untuk mengembangkannya dengan meneliti dan menganalisis azimat numerik yang berbentuk selain tabel atau dengan meneliti dan menganalisis azimat pictorial. daftar pustaka [1] alex, persi. 2004. the electronic journal of combinatorics 11(2). journal: stanford university [2] al-ghozali. al-aufaq. diambil dari ponpes. islam as-salafi rosy aliviana dan abdussakir 114 volume 2 no. 2 mei 2012 [3] al-mashiri, imam abil abbas ahmad bin alil bauni. 1994. sakti mandra guna. pekalongan: t.b bahagia [4] andrews, w. s. 1960. magic squares and cubes. new york: dover publications, inc. [5] as-syahdi, abu hasan. 1249 h. as-sirul jalil. surabaya: mahtabah al-hidayah [6] beck, matthias dan andrew van herick. 2010. american mathematical society: enumeration of 4 x 4 magic squares. mathematics of computation: article electronically published on march 29. [7] departemen agama ri. 2007. al-qur’an dan terjemahannya 30 juz. solo: pt qomari prima publisher. [8] departemen pendidikan dan kebudayaan. 1989. kamus besar bahasa indonesia. jakarta: balai pustaka [9] hanafi, muhammad. 2000. asma’ul husna. surabaya: sinar terang [10] rahman, hairur. 2007. indahnya matematika dalam al-qur’an. malang: uin-malang press. [11] samsuri, muhammad. 1998. mujarrobat besar. surabaya: apollo [12] stephens, daryl lynn. 1993. matrix properties of magic squares: denton, texas lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf 4. daftar isi.pdf 5.1. elva ravita sari studi tentang persamaan fuzzy _54-65.pdf model pergerakan tumpahan minyak di perairan selat sunda agus salim 1) , taufik edi sutanto 2) 1) mathematic department, faculty of science and technology uin jakarta 2) mathematic department ,faculty of science and technology uin jakarta abstrak selat sunda merupakan daerah yang memiliki potensi kekayaan alam yang sangat besar. akan tetapi potensi tersebut terancam akibat tumpahan minyak yang terjadi karena adanya aktivitas transportasi dan penyimpanan minyak mentah maupun olahan di dan sekitar selat sunda. sebuah penelitian pengkajian resiko ekologis tumpahan minyak dilakukan untuk mengukur seberapa besar ancaman tumpahan minyak tersebut dan menyiapkan strategi preemptive untuk meminimalisir dampak yang ditimbulkan jika tumpahan minyak berskala besar terjadi di masa depan. dari analisa trajectory yang dilakukan dapat disimpulkan bahwa daerah yang rawan terkena dampak tumpahan minyak adalah daerah-daerah pantai panimbang, pulau sumur, pantai cigeulis, cimanggu, selat panaitan, cilegon, anyer, penengahan, dan jabung sragi. penggunaaan data ahir trajectory menjadi acuan dalam penentuan tingkat resiko dari partikel minyak yang mengeksposure ekosistem di sekitar selat sunda tersebut dan terpenting adalah dapat digunakan untuk meminimalisir dampak akibat tumpahan minyak dimasa depan. kata kunci: tumpahan minyak, selat sunda, tra-jectory. pendahuluan pada saat terjadi sebuah bencana tumpahan minyak di laut, hal utama yang menjadi perhatian adalah, akan kemanakah minyak yang tumpah tersebut akan mengalir atau dampak apa yang akan ditimbulkannya. mengetahui lintasan (trajectory) dari tumpahan merupakan informasi penting (crucial) untuk menentukan kebijakan lanjutan dalam menangani bencana tersebut, terutama untuk menyelamatkan lingkungan dan melakukan pembersihan. tanpa penanganan yang tepat maka kerugian kerusakan lingkungan akan berdampak luas dan berdampak hingga jangka waktu lama. namun demikian untuk dapat memperkirakan pergerakan tumpahan minyak dilaut secara tepat, biasanya sangatlah sulit untuk dilakukan. hal ini disebabkan cukup banyak proses kimiawi dan fisik yang terlibat, serta pada saat bencana terjadi informasi yang ada biasanya tidak lengkap. untuk mengatasi hal tersebut perlu adanya suatu pemodelan trajectory sedini mungkin dengan data yang senantiasa diperbaharui (up to date) sehingga jika terjadi tumpahan minyak, penanggulangan yang efektif dan efisien dapat dilakukan dengan baik. sebuah pemodelan trajectory tumpahan minyak akan menganalisa berbagai akibat dan kemungkinan dari pergerakan minyak. analisa harapan (likelihood) dari berbagai kemungkinan tersebut adalah suatu simulasi yang biasa disebut sebagai analisa trajectory. hasil akhirnya berupa sebuah peta prediksi dari pergerakan minyak yang dapat dimanfaatkan untuk membantu penanganan bencana tumpahan minyak. pada saat awal-awal tumpahan minyak terjadi prediksi jalur tumpahan minyak biasanya terhambat oleh input data yang tidak lengkap. detail tumpahan minyak (lokasi, volume, tipe minyak) seringnya masih berupa perkiraan. lebih jauh lagi data lingkungan sekitar (angin, arus laut, dan kedalaman) terkadang tidak terhimpun atau bahkan tidak tersedia. meskipun demikian pemodelan harus dilakukan dengan mengamati data dan mencoba untuk mengerti sifat fisik dan kimiawi yang akan mempengaruhi pergerakan minyak dan kelanjutan dari tumpahan tersebut. dengan pemahaman proses-proses fisik, ilmuwan dapat menganalisa situasi yang ada dan melakukan sebuah prediksi (forecast). bila prediksi awal tidak akurat, tim model tumpahan minyak mengkaji ulang informasi yang ada (termasuk memperbaharui) dan memperbaiki prediksi. sebuah prediksi yang tidak akurat biasanya disebabkan oleh kesalahan informasi tentang tumpahan (misal lokasi dan volume) atau ketidak tepatan informasi ingkungan agus salim, taufik edi susanto 100 volume 3 no. 2 mei 2014 (cuaca, angin, dan arus). seiring berjalannya waktu prediksi akan menjadi semakin akurat, karena kondisi awal menjadi semakin tidak signifikan untuk prediksi. sebagai ilustrasi, bagan alur analisis trajectory . untuk mendapatkan sebuah analisa trajectory yang baik suatu faktor ketidakpastian (uncertainty) perlu untuk diikutsertakan dalam pemodelannya. ketidak pastian tersebut dipengaruhi oleh berbagai faktor, diantaranya detail tumpahan, kondisi minyak, angin, dan arus. lebih jelasnya tabel 1 merupakan informasi yang diambil dari [1] yang dapat menjelaskan lebih rinci akan besarnya pengaruh ketidakpastian. salah satu proses oil weathering adalah evaporasi, yaitu perubahan dari suatu cairan menjadi bentuk gas. pada evaporasi minyak, bagian dari minyak yang lebih ringan akan terevaporasi terlebih dahulu. pada minyak jenis tertentu, terutama minyak jenis ringan, sebagian besar akan terevaporasi. pada suhu 15°c bensin akan terevaporasi sempurna setelah dua hari, 80% solar akan terevaporasi, 40% minyak light crude, 20% minyak heavy crude, dan hanya sekitar 5% untuk tipe minyak bunker c. secara umum seluruh proses evaporasi yang dapat terjadi akan memakan waku kurang dari lima hari. proses oil weathering lainnya adalah dispersi atau dalam hal ini pemecahan partikel minyak kecil (droplet) oleh ombak (gelombang/waves) dapat terjadi bila diameter droplet minyak relative kecil (50-70 micron). partikel minyak yang terpecah menjadi penting, karena hal tersebut merupakan salah satu faktor yang dapat digunakan untuk meghilangkan pratikel minyak dari permukaan laut. banyaknya partikel minyak yang terpecah berantung pada sifat minyak yang tertumpah. diantara sifat yang mempengaruhi tersebut adalah kekentalan (viscosity) minyak dan keadaan air laut. metodologi data dan lokasi penelitan penelitian dilakukan di selat sunda, lokasi studi trajectory berada diantara koordinat 7o24’ – 7o45’ lintang selatan dan 108o25’ – 108 o45’ bujur timur (gambar 1). waktu yang digunakan dalam studi kasus analisa trajectory dalam penelitian ini terbagi menjadi dua kasus. kasus pertama adalah kasus ketika tumpahan terjadi pada awal tahun (maret) dan akhir tahun (oktober). gambar 1. lokasi studi kasus trajectory. kedua kasus ini dipilih dikarenakan karakteristik arus laut sepanjang tahun menurut data satelit ocean surface currents (oscar) berbeda pada bulan-bulan tersebut. pada bulan maret, arus cenderung bergerak dari arah utara ke selatan, dimana pada bulan oktober (akhir tahun) terdapat kecenderungan dari tahun ke tahun arus akan bergerak dari samudera hindia masuk ke samudera pasifik bagian barat. data kecepatan arus mingguan di selat sunda dari tahun 1992 hingga 2011 awal (gambar 2) model pergerakan tumpahan minyak di perairan selat sunda cauchy – issn: 2086-0382 101 gambar 2. graph rangkuman data kecepatan arus di selat sunda (mingguan) dari tahun 19922011. dalam bentuk tabel, sebagian data tersebut dapat dilihat pada lampiran 1. dari kedua data tersebut dapat terlihat bahwa terdapat kecenderungan data kecepatan arus di selat sunda cenderung stasioner dengan beberapa lompatan yang biasa terjadi di awal tahun. fakta ini dapat membantu analisa trajectory menjadi lebih akurat, mengingat kecepatan arus yang cenderung tidak terlalu variatif (variable). rata-rata kecepatan angin di selat sunda dari tahun 1992-2011 adalah 0.386 m/s dengan variansi rata-rata sebesar 0.046. dari data tersebut dapat ditarik kesimpulan bahwa dengan kepercayaan 95% rata-rata angin pada suatu hari di selat sunda akan terletak pada interval 0.818 m/s sampai dengan 0.04 m/s. data penting lain yang diperlukan untuk analisa trajectory adalah data angin di sekitar selat sunda. menggunakan data pada periode waktu yang sama, terlihat data dari satelit quikscat memperlihatkan bahwa angin bergerak cenderung berlawanan arah dengan arus, yang mengakibatkan pergerakan minyak cenderung tidak terlalu jauh. rangkuman graph data kecepatan angin mingguan selama beberapa tahun (gambar 12) menunjukkan sifat yang sama dengan arus, yaitu stasioner dengan beberapa pengecualian lompatan dibeberapa kasus waktu (dalam bentuk tabel, sebagian data tersebut dapat dilihat pada lampiran 1). rata-rata kecepatan angin total dari seluruh 3750 data adalah 6.191 m/s dan variansi sebesar 0.767. selang kepercayaan untuk rata-rata kecepatan angin adalah antara 7.94 hingga 4.43 m/s. gambar 3. data kecepatan angin di selat sunda berdasarkan satelit quickscat. data penting lain dari analisa trajectory, tentu saja data tumpahan minyak itu sendiri. di selat sunda terdapat beberapa tangki kilang minyak dan distribusi minyak produk level tinggi (bensin, solar, minyak tanah, dll), namun tidak menutup kemungkinan terjadinya tumpahan produk minyak dengan kekentalan tinggi seperti oli mesin. berikut adalah karakteristik dari ketiga tipe minyak tersebut yang akan digunakan sebagai input data analisa trajectory. data terakhir yang dibutuhkan untuk melakukan analisa trajectory adalah data peta polygon selat sunda. gambar 13 menunjukkan polygon dari wilayah perairan selat sunda, sebagian koordinat bujur dan lintang verteksverteks dari polygon tersebut dapat dilihat pada lampiran 2 yang terdiri dari 287 verteks yang membangun 2 kepulauan besar jawa dan sumatera, serta beberapa pulau kecil diantaranya. agus salim, taufik edi susanto 102 volume 3 no. 2 mei 2014 model matematis model tumpahan minyak biasanya terdiri dalam tiga tahapan model [2]. prediksi trajectory tumpahan minyak menggunakan data actual atau percobaan. model ini disebut juga sebagai model deterministik dan menghasilkan sebuah ekspektasi prediksi dari tumpahan minyak. model kedua, backtrack, menjalankan model deterministik secara runtun waktu terbalik dan biasanya digunakan untuk menentukan sumber minyak yang didapatkan di laut atau pantai. model ini terkadang disebut juga sebagai “model misteri” dan digunakan sebagai alat untuk mengidentifikasi adanya tumpahan minyak yang dilakukan secara sengaja. membangun sebuah model matematika yang akan mencerminkan mdel yang sesungguhnya di dunia nyata bukanlah suatu proses yang mudah. model yang dibuat haruslah dapat diterima karena kemiripan sifat dengan model aslinya. oleh karena itu beberapa asumsi harus dibuat terlebih dahulu sebelum pemodelan dapat dilakukan. asumsi yang digunakan dalam pemodelan penyebaran tumpahan minyak yang digunakan dalam penelitian ini adalah: 1) penyebaran searah dengan arah arus air. 2) ukuran maksimum tumpahan (droplet) adalah ukuran partikel minyak sedemikian sehingga ia tidak akan mengambang. 3) arus diasumsikan konstan dan tenang (placid). 4) efek suhu, angin, dan tekanan diabaikan. menggunakan asumsi-asumsi tersebut diharapkan model yang terbentuk dapat mewakili penyebaran minyak yang sesungguhnya. model tersebut akan dikombinasikan dengan model system persamaan differensial hidrodinamika fluida seperti yang dikemukakan di [3], yaitu: 𝛿𝑢 𝛿𝑡 + 𝑢 𝛿𝑢 𝛿𝑥 + 𝑣 𝛿𝑢 𝛿𝑦 + 𝑔 𝛿𝑤 𝛿𝑥 = −𝑔 𝑢√𝑢2 + 𝑣 2 (𝑤 + ℎ)𝑐2 + 𝑓𝑣 𝛿𝑣 𝛿𝑡 + 𝑢 𝛿𝑣 𝛿𝑥 + 𝑣 𝛿𝑣 𝛿𝑦 + 𝑔 𝛿𝑤 𝛿𝑦 = −𝑔 𝑣√𝑢2 + 𝑣 2 (𝑤 + ℎ)𝑐2 + 𝑓𝑢 𝛿𝑤 𝛿𝑡 + 𝛿[(𝑤 + ℎ)𝑢] 𝛿𝑥 + 𝛿[(𝑤 + ℎ)𝑣] 𝛿𝑦 = 0 (1) 𝑢, 𝑣 adalah kecepatan arus terhadap komponen timur dan utara (bujur dan lintang), w adalah kedalaman laut, t waktu, h kedalaman dimana air diasumsikan tidak dipengaruhi arus atas, dan c adalah suatu ketetapan koefisien tertentu. pada permasalahan nyata tentu saja terdapat sebuah batasan [pantai] yang merupakan sebuah boundary condition (bc) bagi gelombang. bc yang dipergunakan dalam pemodelan penelitian ini menggunakan formula 𝜁(𝑖,𝑗)(𝑘) = 𝐻(𝑖,𝑗) cos(𝜎𝑘 − 𝑔(𝑖,𝑗)) , 𝑘 = 0,1, … , 𝑁 − 1 (2) dimana 𝐻 adalah amplitudo, 𝑔 jeda fase (phase lag), 𝑁 banyaknya langkah dalam satu periode gelombang, dan  adalah frekuensi angular dari komponen gelombang. dari model hidrodinamik yang telah ditetapkan diatas, kemudian model gelombang laut dikembangkan dengan memperhatikan gravitasi permukaan gelombang akibat angin, atau yang biasa disebut sebagai irregular long crested (ilc). ilc ditambahkan dengan cara menambahkan sejumlah gelombang regular long crested (rlc) dengan amplitudo dan panjang gelombang yang berbeda-beda, sehingga pada saat tertentu tinggi gelombang dimodelkan sebagai berikut: 𝜁 = 𝐻𝑡𝑖𝑑𝑒 + ∑ 𝜁𝑎𝑡 cos(𝑘𝑡 (𝑥 sin 𝜃 + 𝑦 cos 𝜃) + 𝜔𝑡 𝑡 + 𝜑𝑡 ) 𝑛 𝑡=1 (3) model pergerakan tumpahan minyak di perairan selat sunda cauchy – issn: 2086-0382 103 htide adalah tinggi gelombang, g adalah percepatan gravitasi,  arah propagasi gelombang,  ∈ [0,2], t adalah waktu, t adalah sudut fase, kt adalah gelombang yang didapat dari persamaan dispersi. model untuk difusi minyak terdiri dari tiga bagian: difusi gravitasi, difusi glutinosity, dan difusi tekanan permukaan. ketiga bagian model tersebut diberikan oleh persamaan berikut: 𝑟(𝑡) = 𝐾1(δ𝑔𝑉𝑡 2) 1 4 𝑟(𝑡) = 𝐾2 ( δ𝑔𝑉𝑡 2 3 𝑣 1 2 ) 1 6 𝑟(𝑡) = 𝐾3 ( 𝜎2𝑡 3 𝜌2𝑣 ) 1 4 dimana δ𝑔 = (𝜌 − 𝜌0) ∗ 𝑔/𝜌, o dan  adalah adalah kepadatan minyak dan air. 𝑟(𝑡) adalah skala difusi minyak, 𝜎 = 𝛿𝑤𝑎 − 𝛿𝑎𝑜 − 𝛿𝑜𝑤, dimana 𝛿𝑤𝑎 , 𝛿𝑎𝑜 , dan 𝛿𝑜𝑤 adalah tekanan permukaan antara air dengan udara, minyak dengan udara, dan minyak dengan air. v adalah koefisien kekentalan kinematik air, t menyatakan waktu, v sebagai volume minyak, dan k1, k2, dan k3 adalah koefisien percobaan dengan nilai tertentu (biasanya antara 1-1,5). pemodelan diatas memperhitungkan setiap satuan tumpahan (droplet) dari tumpahan minyak yang terjadi. hal tersebut mengakibatkan penyelesaian model diatas menjadi sangat kompleks dan membutuhkan komputasi yang sangat tinggi. selanjutnya hasil simulasi tersebut dilakukan dengan bantuan software aplikasi gnome dan spill tool untuk melihat hasil simulasi terbaik. analisis trajectory gnome setelah data kelautan, geografis, dan karakteristik minyak telah didapatkan selanjutnya analisa tajectory dilakukan menurut standar internasional noaa (national oceanic and atmospheric administration). badan khusus internasional penanganan bencana di laut emergency response division (erd) , dibawah noaa mendesain suatu software khusus untuk melakukan analisa trajectori tumpahan minyak yang diberi nama gnome (general noaa operational modeling environment). perangkat lunak gnome akan digunakan untuk menganalisa simulasi trajectory dari tumpahan minyak di selat sunda. gnome (gambar 4) memiliki dua fungsi utama, standar dan diagnostic mode. pada mode standar digunakan loc file, yang berupa informasi peta polygon, arus, angin, karakteristik minyak, kedalaman laut, dan data lainnya. pada mode tersebut pengguna hanya memvariasikan berbagai parameter tambahan seperti jumlah tumpahan dan pengaruh tambahan untuk melihat dampaknya terhadap perubahan trajectory minyak. walaupun menggunakan mode standar sangatlah mudah, namun loc file yang tersedia sangat terbatas, yang sebagian besar adalah loc file laut disekitar amerika. gambar 4. perangkat lunak analisa trajectory erd-noaa gnome. agus salim, taufik edi susanto 104 volume 3 no. 2 mei 2014 dalam penelitian ini digunakan mode diagnostic, dikarenakan loc file untuk selat sunda tidak tersedia. untuk mengawali penggunaan mode diagnostic, pertama-tama diperlukan peta polygon dari daerah yang akan diteliti. peta polygon tersebut menggunakan format bna (atlas boundary file). bna diawali dengan tiga parameter awal, sebagai contoh “234”,”1”,33 yang berarti feature 234, polygon daratan, lalu terdiri dari 33 verteks. setelah baris ini diikuti dengan 33 vertex lokasi pulau yang akan diteliti. parameter berikutnya dalam peta polygon adalah batas peta, format batas peta adalah "map bounds","1",7 yang berarti terdapat 7 verteks yang akan membatasi perhitungan pergerakan droplet minyak. daerah yang dapat atau mungkin terjadi tumpahan minyak dinyatakan dengan "spillablearea", "1", 15. sama seperti sebelumnya 15 menyatakan daerah tersebut dikarakterisasi oleh 15 verteks pembatas daerah tumpahan. data arus laut di gnome menggunakan format cats finite element – velocities on triangles, steady state. gambar 15 memperlihatkan salah satu contoh penggunaan segitiga pada finite element dalam data arus gnome. arus yang dapat digunakan tidak hanya arus permukaan, namun juga kecepatan arus pada berbagai kedalaman. data angin dalam diagnostic mode gnome menggunakan format wnd. dalam format ini arah, besaran, dan waktu akan kecepatan angin diberikan dalam bentuk seperti yang nampak pada gambar 5. gambar 5. format wnd untuk data angin laut pada gnome. hasil dan pembahasan hasil simulasi & analisa ketidakpastian simulasi tumpahan minyak jenis medium crude oil sebesar 3000 barrel dilakukan dengan tumpahan berawal dari jalur distribusi minyak. simulasi memanfaatkan data angin dan arus seperti yang sudah dijabarkan sebelumnya dan menggunakan data musim awal tahun, dalam hal ini bulan maret. setelah menjalankan simulasi selama beberapa hari, didapatkan hasil seperti pada gambar 6. gambar 6. simulasi pergerakan minyak di selat sunda setelah 10 hari. model pergerakan tumpahan minyak di perairan selat sunda cauchy – issn: 2086-0382 105 dari simulasi tersebut dapat disimpulkan bahwa daerah yang terkena dampak, bila terjadi tumpahan pada awal tahun adalah sumatera bagian selatan (penengahan), pantai cigeulis, pulau sumur, dan selat panaitan, dengan dampak terbesar terjadi di pulau sumur 1. simulasi berikutnya kemudian dilakukan dengan menggunakan data musiman untuk akhir tahun. jumlah tumpahan dan jenis minyak sama seperti simulasi sebelumnya. hasil simulasi dalam jangka waktu empat hari setelah tumpahan diberikan dalam gambar 7. gambar 7. simulasi jika tumpahan minyak terjadi pada akhir tahun. hasil simulasi selama empat hari untuk data simulasi akhir tahun (gambar 18) menunjukkan bahwa selat sunda minimal terkena dampak tumpahan minyak. dalam minggu pertama daerah di sekitar bojonegara, anyer, dan penengahan terkena dampak yang cukup besar. dalam waktu lebih dari dua minggu jika jumlah tumpahan cukup banyak, maka ada kemungkinan pencemaran akan menjangkau kalimantan dan negara tetangga singapura.permasalahan berikutnya yang cukup penting untuk dianalisa adalah seberapa besar dampak pencemaran pada lokasi-lokasi yang telah disebutkan diatas. untuk menganalisa tingkat pencemaran tersebut digunakan roc (response options calculator). dengan menggunakan parameter yang sama dengan trajectory gnome, roc dapat menghitung berapa bagian minyak yang telah melalui proses oil weathring (evaporasi, dispersi, dan residue) (gambar 8). gambar 8. hasil perhitungan presentasi kadar pencemaran di selat sunda. kesimpulan dan saran dari analisa trajectory yang dilakukan dapat disimpulkan bahwa daerah yang rawan terkena dampak tumpahan minyak adalah daerah-daerah sebagai berikut (gambar 20) pantai panimbang, pulau sumur, pantai cigeulis, cimanggu, selat panaitan, cilegon, anyer, penengahan, jabung sragi. saran dari penelitian ini adalah penelitian ini masih dalam skala laboratorium (simulasi). akan lebih lengkap bila agus salim, taufik edi susanto 106 volume 3 no. 2 mei 2014 dilakukan proses validasi terhadap hasil simulasi pergerakan partikel minyak di akhir pengamatan dengan waktu,bukan dan kondisi sebenarnya di lapang. sehingga akan dapat terlihat validitas dan keakuratan model yang dibangun. penelitian ke depan adalah melakukan proses validasi hasil akhir tersebut pada kondisi,iklim sebenarnya. bibliography [1] a. jimoh and m. alhassan, “modelling and simulation of crude oil dispersion,” leonardo electron. j. pract. technol., vol. 5, no. 8, pp. 17– 28, 2006. [2] m. l. spaulding, v. s. kolluru, e. anderson, and e. howlett, “application of three-dimensional oil spill model (wosm/oilmap) to hindcast the< i> braer</i> spill,” spill sci. technol. bull., vol. 1, no. 1, pp. 23–35, 1994. [3] f. yu and y. yin, “oil spill visualization based on the numeric simulation of tidal current,” in proceedings of the 7th acm siggraph international conference on virtual-reality continuum and its applications in industry, 2008, p. 28. model pergerakan tumpahan minyak di perairan selat sunda cauchy – issn: 2086-0382 107 lampiran-lampiran lampiran 1: data kecepatan angin & arus di selat sunda selama beberapa tahun tanggal rerata angin variansi date rerata arus variansi 7/29/1999 4.328433 1.7936478 12/11/1992 0.526463 0.369623 7/30/1999 5.2825038 1.8516024 12/16/1992 1.00257 0.984601 7/31/1999 5.8838556 0.95648555 12/21/1992 0.635397 0.669616 8/1/1999 5.5123405 1.6233164 12/26/1992 0.460883 0.482043 8/2/1999 7.4288562 1.0785921 1/1/1993 0.387529 0.33567 8/3/1999 7.3761565 1.8431835 1/6/1993 0.476694 0.454172 8/4/1999 8.485566 1.1580675 1/11/1993 0.359789 0.333094 8/5/1999 7.839028 0.99348769 1/16/1993 0.534405 0.514174 8/6/1999 7.5892326 1.4727978 1/21/1993 0.832444 0.818649 8/7/1999 7.0998994 1.7550339 1/26/1993 0.615157 0.932113 8/8/1999 6.9291045 1.1770411 1/31/1993 0.664582 0.796478 8/9/1999 5.4590789 1.4110885 2/5/1993 0.19502 0.244655 8/10/1999 6.7429213 1.3620754 2/10/1993 0.309104 0.251402 8/11/1999 7.3185649 1.6466063 2/15/1993 0.163413 0.150595 8/12/1999 10.005049 1.3535395 2/20/1993 0.297904 0.307282 8/13/1999 8.0106097 0.47421375 2/25/1993 0.458922 0.360722 … … … … … … 3/6/2003 6.3151514 3.6660507 1/1/2011 0.404298 0.369638 3/7/2003 8.1872091 3.1166491 1/6/2011 0.307276 0.333587 3/8/2003 6.1029508 2.2004977 1/11/2011 0.315274 0.319965 3/9/2003 4.3183583 0.93758907 1/16/2011 0.334781 0.401904 3/10/2003 3.5829664 2.3982565 1/21/2011 0.229897 0.285396 3/11/2003 2.4757332 1.3811929 1/26/2011 0.59927 0.566984 3/12/2003 5.6816771 3.7906258 1/31/2011 0.583347 0.519379 3/13/2003 3.9427958 1.1597723 2/5/2011 0.325766 0.421876 3/14/2003 5.6493751 1.3974251 2/10/2011 0.274698 0.375643 3/15/2003 5.2963544 3.9728989 2/15/2011 0.359002 0.37683 3/16/2003 3.1243345 0.63200567 2/20/2011 0.385987 0.412617 3/17/2003 2.5974114 1.3625091 2/25/2011 0.319986 0.323262 3/18/2003 8.294951 3.0138123 3/2/2011 0.136407 0.163611 lampiran 2: format peta polygon analisis trajectory "2068","1",26 104.529419,-5.826419 104.529419,-5.544913 104.545898,-5.555848 104.559631,-5.599585 104.61731,-5.659719 104.650269,-5.717113 104.636536,-5.741708 104.655762,-5.758105 104.655762,-5.771769 104.677734,-5.793629 104.702454,-5.831884 104.694214,-5.867403 104.724426,-5.911117 104.724426,-5.927508 104.685974,-5.932972 104.669495,-5.924777 104.669495,-5.938436 104.606323,-5.9439 104.60083,-5.93024 pendahuluan metodologi data dan lokasi penelitan model matematis analisis trajectory gnome hasil dan pembahasan hasil simulasi & analisa ketidakpastian kesimpulan dan saran lampiran-lampiran lampiran 1: data kecepatan angin & arus di selat sunda selama beberapa tahun lampiran 2: 1a sampul depan sebuah telaah elips dan lingkaran melalui sebuah pendekatan aljabar matriks rahmat sagara sekolah tinggi keguruan dan ilmu pendidikan kebangkitan nasional sampoerna school of education building jl. kapten tendean no. 88 c. 4th floor, jakarta selatan 12710, indonesia phone: +62 21 577 2275 ext. 7243, fax: +62 21 577 2276 e-mail: rahmat.sagara@sampoernaeducation.ac.id abstract in this article, ellipse and circle will be learnt in depth via matrix algebra approach. the discussion of the both is started from their classic definition continued by surveying ellipse in matrix form. during the survey, some properties about ellipse will be explained and also, the procedure in drawing the figures can be obtained geometrically using some aspect in geometry: rotation and translation. at the end of the discussion, the new definition of the figures is deduced. both of them are defined as” a set of points in a plane that are the same distance from a fixed point” but in different point of view about the ‘distance’. the ‘distance’ in the definition is derived from different norm definition. the difference lies on the positive definite matrix used in the norm definition. base on the new definition, we’ll have the conclusion that circle is a special type of ellipse. keywords: circle, distance, ellipse, norm, positive definite matrix pendahuluan elips dan lingkaran memiliki definisi yang berbeda. masing-masing didefinisikan sebagai berikut: lingkaran didefinisikan sebagai himpunan semua titik-titik pada bidang datar yang berjarak sama terhadap sebuah titik tertentu. sebuah titik ini disebut titik pusat dari lingkaran. elips didefinisikan sebagai himpunan semua titik-titik pada bidang datar yang jumlah jaraknya terhadap dua titik-titik tertentu tetap. kedua titik-titik ini disebut titik-titik fokus dari elips. berdasarkan kedua definisi di atas, sebuah lingkaran bisa disebut elips, yakni elips dengan titik-titik fokusnya berimpit dengan titik pusatnya. misalkan r adalah jarak yang dinyatakan pada definisi yang pertama (dengan kata lain r adalah panjang jari-jari dari lingkaran), t adalah jumlah jarak yang dinyatakan pada definisi kedua, titik-titik fokus dari elips adalah f�c, 0� and f�� c, 0� dimana c adalah sebuah bilangan real non-negatif, dan tanpa mengurangi keumuman misalkan titik asal �0,0� adalah titik pusat bagi kedua bangun datar tersebut. maka bisa ditunjukkan bahwa lingkaran dan elips masing-masing bisa dirumuskan sebagai berikut: � � ��, �� � ��|�� � �� � ��� (1.1) dan � � ���, �� � ��� ���� � ������ � � !�". (1.2) pada bidang kartesius, jika sembarang titik pada kedua bangun datar itu dinamai dengan p, titik potong dengan sumbu-x dinamai dengan a dan a�, dan titik potong dengan sumbuy dinamai dengan b dan b� maka kedua bangun datar tersebut masing-masing digambarkan seperti gambar 1 dan gambar 2. gambar 1. lingkaran dengan titik pusat �0,0� dan panjang jari-jari r gambar 2. elips dengan titik pusat �0,0� titik-titik fokus f dan &� serta jumlah jarak titik pada elips ke titik-titik focus ' � �! � ��. rahmat sagara 86 volume 1 no. 2 mei 2010 pandang gambar 2 dan misalkan bahwa koordinat dari (, (�, ) dan )* berturut-turut adalah �+, 0�, � +, 0�, �,, 0�, dan � ,, 0�, dimana + dan , keduanya adalah bilangan real positif. jika berimpit dengan ( atau (�, maka akan didapatkan ' � 2+ (1.3) dan jika berimpit dengan ) atau )*, maka akan didapatkan !� '� � ,� � /�. (1.4) dari persamaan 1.3 dan 1.4 didapatkan +� � ,� � /�. (1.5) dengan demikian, persamaan elips bisa dituliskan sebagai berikut: � � ���, �� � ��� ��0� � ��1� � 1" (1.6) atau � � ��, �� � ��|,��� � +��� � +�,�� (1.7) dimana + dan , keduanya adalah bilangan real positif sebagai mana dipaparkan di atas. bilangan 2+ dan 2, masing-masing adalah panjang sumbu mayor dan panjang sumbu minor dari elips. pembahasan 1. elips dalam pendekatan aljabar matriks misalkan p�x, y� merupakan sebuah titik pada elips seperti pada persamaan 1.7. jelas bahwa akan memenuhi persamaan ,��� � +��� � +�,� (2.1) atau dalam bentuk persamaan matriks, akan memenuhi persamaan: 5� �6 7,� 00 +�8 7��8 � +�,� . (2.2) pandang persamaan 2.2. matriks bujur sangkar yang digunakan pada persamaan itu bersifat definit positif karena simetri dan (sesuai dengan diasumsikan sebelumnya bahwa + 9 0 dan , 9 0 yang mengakibatkan +� dan ,� yang tiada lain adalah) nilai-nilai eigen dari matriks itu bernilai positif. kemudian karena +� dan ,� adalah nilai-nilai eigen dari matriks itu, jelas bahwa +�,� adalah nilai determinan dari matriks itu. lebih jauh, bahwa secara umum, matriks bujursangkar yang bisa digunakan dalam persamaan elips tidak hanya matriks diagonal, tetapi semua matriks definit positif. tentunya dua matriks definit positif yang berbeda akan menghasilkan dua buah elips yang berbeda pula. dengan demikian, elips bisa dirumuskan sebagai berikut: �! � ���, �� � ���5� �6: 7��8 � |:|" (2.3) dimana m adalah sebuah matriks definit positif berukuran 2x2 dan |:| adalah determinan dari matriks m. perumusan ini bisa dijelaskan sebagai berikut: karena m adalah sebuah matriks simetri maka terdapat matriks orthogonal p dan matriks diagonal d sedemikian sehingga m=pdpt, dimana pt adalah matriks transpos dari p. semua komponen diagonal dari d adalah nilai eigen dari m dan karena m definit positif maka semua komponen diagonal ini positif. vektor kolom ke-i dari matriks p adalah vektor eigen satuan dari m yang bersesuaian dengan nilai eigen ke-i dari m, yakni komponen diagonal ke-i dari d. dengan demikian, ekspresi: 5� �6: 7��8 bisa dituliskan sebagai: 5; <6= 7;<8 dimana 7;<8 � >? 7��8 dan karena p merupakan matriks orthogonal, didapatkan juga 7��8 � > 7;<8. dengan demikian persamaan 2.3 bisa dituliskan sebagai: �! � ���, �� � ��� 7��8 � > 7;<8 @�;, <� � �� "(2.4) dimana �� � ��;, <� � ���5; <6= 7;<8 � |=|" (2.5) dan p dan d masing masing matriks orthogonal dan matriks diagonal sedemikian sehingga : � >=>? seperti yang telah dipaparkan sebelumnya. dari persamaan 2.4 dan 2.5 terlihat bahwa elips yang dirumuskan pada persamaan 2.3 adalah hasil transformasi a dari elips ��, atau �! � a���� dibawah definisi: a b7;<8c � > 7;<8. (2.6) lebih dalam, peranan matrks d dan p dalam pembentukan elips dipaparkan sebagai berikut: pandang matriks d dan tulis = � de� 00 e!f. tanpa mengurangi keumuman asumsikan bahwa e! g e�, maka elips �� seperti 2.5 adalah elips sebuah telaah elips dan lingkaran melalui sebuah pendekatan aljabar matriks jurnal cauchy – issn: 2086-0382 87 yang berpusat di �0,0� mempunyai panjang sumbu mayor dan sumbu minor masing-masing adalah 2he! dan 2he�. hal ini didapat dari analogi persamaan 2.5 dan persamaan 1.6 dimana he! dan he� pada persamaan 2.5 bersesuaian dengan + dan , pada persamaan 1.6. sekarang pandang matriks p dan tulis > � 7i!! i!�i�! i��8. karena p adalah matriks ortognal maka >>? � >?> � jk yaitu matriks identitas orde 2. dari persamaan >>? � jk didapat hubungan: d i!!� � i!�� i!!i�! � i!�i��i�!i!! � i��i!� i�!� � i��� f � 71 00 18 atau didapat sistem persamaan: i!!� � i!�� � 1 (2.7) i!!i�! � i!�i�� � 0 (2.8) i�!� � i��� � 1 (2.9) dan dari persamaan >?> � jk ini didapat hubungan: d i!!� � i�!� i!!i!� � i�!i��i!�i!! � i��i�! i!�� � i��� f � 71 00 18 atau didapat sistem persamaan: i!!� � i�!� � 1 (2.10) i!!i!� � i�!i�� � 0 (2.11) i!�� � i��� � 1 (2.12) dari persamaan 2.8 dan 2.11 didapatkan �i!! i����i!� i�!� � 0. (2.13) pandang persamaan 2.13 untuk kedua kasus berikut: 1. kasus i!� l i�! jika i!� l i�! maka i!! � i��; dan jika i!! � i�� maka berdasarkan persamaan 2.7 dan 2.9 (atau persamaan 2.10 dan 2.12) didapat bahwa i!�� � i�!� akan tetapi karena i!� l i�! maka i!� � i�!. dengan demikian bisa dituliskan bahwa > � dm nn m f (2.13) dimana m dan n adalah bilagan real yang memenuhi sifat m� � n� � 1. 2. kasus i!! l i�� jika i!! l i�� maka i!� � i�!; dan jika i!� � i�! maka berdasarkan persamaan 2.7 dan 2.9 (atau persamaan 2.10 dan 2.12) didapat bahwa i!!� � i��� akan tetapi karena i!! l i�� maka i!! � i��. dengan demikian bisa dituliskan bahwa > � dm nn mf (2.14) dimana m dan n adalah bilagan real yang memenuhi sifat m� � n� � 1. matriks p pada persamaan 2.13 dan 2.14 keduanya memberikan elips yang sama. hal ini dijamin oleh fakta berikut: 1. dengan menggunakan matriks p pada persamaan 2.13, hasil perkalian dari pdpt: dm nn m f de! 00 e�f d m n n mf� de!m� � e�n� �e! e��mn�e! e��mn e!n� � e�m�f 2. dengan menggunakan matriks p pada persamaan 2.14, hasil perkalian dari pdpt: dm nn mf de! 00 e�f dm nn mf� de!m� � e�n� �e! e��mn�e! e��mn e!n� � e�m�f pada kedua kasus di atas bisa dilihat bahwa keduanya memberikan matriks perkalian pdpt yang sama. selain itu, fenomena ini juga bisa dijelaskan dengan fakta bahwa kelipatan dari suatu vektor eigen yang bersesuaian dengan suatu nilai eigen tertentu merupakan sebuah vektor eigen yang bersesuaian dengan nilai eigen itu. dengan demikian, apapun bentuk matriks p yang dipakai, baik pada persamaan 2.13 maupun pada persamaan 2.14, tidak akan memberikan elips yang berbeda. tranformasi yang didefinisikan pada persamaan 2.6 jika menggunakan matriks p seperti pada persamaan 2.13 secara geometri bisa ditafsirkan sebagai rotasi sebesar p (diukur dalam derajat) dengan sumbu rotasi titik asal �0,0� sedemikian sehingga m � cos�p� dan n � sin�p�. dengan menulis kembali elips pada persamaan 2.3 dan ganti |:| dengan bilangan real positif u, maka akan didapatkan elips baru sebagai berikut: �v � ���, �� � ��|5� �6: 7��8 � u" (2.15) atau �v � ���, �� � ��� 7��8 � > 7;<8 @�;, <� � �� " (2.16) dimana �� � ��;, <� � ���5; <6= 7;<8 � u" (2.17) dan > � dm nn m f dan = � dλ� 00 λ!f masing-masing adalah matriks orthogonal dan matriks diagonal sehingga : � >=>?. rahmat sagara 88 volume 1 no. 2 mei 2010 sembarang titik -��;, <� pada elips �� akan memenuhi: 5; <6 dλ� 00 λ!f 7;<8 � u yang ekivalen dengan e�;� � e!<� � u atau w�xy� � z �xy[ � 1. (2.18) bisa dilihat bahwa persamaan 2.18 serupa dengan ekspresi yang dipenuhi titik-titik pada elips � seperti pada persamaan 1.6. dengan demikian, analogi dengan elips e pada persamaan 1.6 (dimana + dan , pada persamaan 1.6 masingmasing bersesuaian dengan \ ]̂� dan \ ]̂[ pada persamaan 2.6), �� adalah elips yang berpusat di �0,0�, panjang sumbu mayor 2\ ]̂�, panjang sumbu minor 2\ ]̂[ dan titik-titik fokus _\u ^[�^�^[^� , 0` and _ \u ^[�^�^[^� , 0`. hal khusus pada persamaan 2.18, jika nilai eigen the eigen e! � e� � e, maka persamaan 2.18 akan menjadi: ;� � <� � ]̂ (2.19) dan persamaan ini serupa dengan ekspresi yang dipenuhi oleh titik-titik pada lingkaran seperti pada persamaan 1.1. dengan demikian, pada hal khusus tersebut, elips �� pada persamaan 2.17 merupakan sebuah lingkaran dengan pusat di �0,0� dan panjang jari-jari � � \]̂ , begitu pula dengan persamaan 2.15 karena matriks orthogonal p akan berupa matriks identitas yang artinya bahwa lingkaran tersebut dirotasi dengan besar sudut 0a, artinya bahwa transformasi ini tidak merubah bentuk awal. 2. elips dalam pendekatan jarak untuk sembarang vektor �b � 7�!��8 � ��, definisikan sebuah norm dari vektor itu sebagai berikut: c�bc: � √�b�:�b (3.1) dimana m adalah sebuah matriks definit positif. dengan norm ini, definisikan pula jarak antara dua titik -��, �� dan e�;, <� sebagai norm dari selisih dua vektor 7��8 dan 7;<8 sebagai berikut: f:�-, e� � g7��8 7;<8g: � 5� ; � <6: 7� ;� <8 (3.2) dengan demikian jarak -��, �� dan titik asal �0,0� adalah f:�-, � � g7��8g: � 5� �6: 7��8 (3.3) berdasarkan definisi jarak pada persamaan 3.3, bisa dikatakan bahwa elips �v � ���, �� � ��|5� �6: 7��8 � u" adalah merupakan himpunan semua titik-titik yang berjarak sama terhadap titik asal, yakni sebesar √u. pusat dari elips bisa diperluas menjadi sembarang titik dengan menggunakan aspek lain dari geometri yaitu translasi. dengan demikian jika diinginkan bahwa pusat dari elips itu adalah h�i, i�, maka translasi dari elips �v harus dilakukan dengan vektor arah translasi 7ii8 sehingga didapat elips ���, �� � ��|5� i � i6: 7� i� i8 � u". penutup misalkan h�i, i� sebuah titik tertentu dan -��, �� titik sembarang di dalam bidang kartesius. misalkan pula m adalah sebuah matriks definit positif berukuran 2x2 dan u sebuah bilangan real positif. pandang definisi jarak pada persamaan 3.2 dan sebuah himpunan yang diformulasikan sebagai berikut: j � � ��|f:�-, h� � u� maka: 1. jika > � dm nn m f dan = � dλ� 00 λ!f masing-masing adalah matriks orthogonal dan matriks diagonal sehingga : � >=>?, maka j adalah sebuah elips dengan: a. titik pusat h, b. merupakan hasil transformasi a��� � �a! k a����� dimana a� b7��8c � dm nn m f 7��8 adalah rotasi sebesar p (diukur dalam derajat) dengan sumbu rotasi titik asal sebuah telaah elips dan lingkaran melalui sebuah pendekatan aljabar matriks jurnal cauchy – issn: 2086-0382 89 �0,0� sedemikian sehingga m � cos�p� dan n � sin�p�, a b7��8c � 7� � i� � i8 adalah translasi dengan arah vektor translasi 7ii8 dan � � l��, �� � ��m ��ue� � ��ue! � 1n adalah elips yang berpusat di �0,0�, panjang sumbu mayor 2\ ]̂�, panjang sumbu minor 2\ ]̂[ dan titik-titik fokus _\u ^[�^�^[^� , 0` and _ \u ^[�^�^[^� , 0`. 2. jika : � 7λ 00 λ8, maka j adalah sebuah lingkaran dengan: a. titik pusat h, b. panjang jari-jari \]̂, 3. jika : � 71 00 18, maka j adalah sebuah lingkaran dengan: a. titik pusat h, b. panjang jari-jari √u akhirnya, dari penelaahan ini bisa dikemukakan definisi dari elips dan lingkaran sebagai berikut: elips adalah himpunan semua titik-titik pada bidang datar yang berjarak sama terhadap sebuah titik tertentu, dimana jarak yang dimaksud didefinisikan seperti pada persamaan 3.2 dan lingkaran adalah hal khusus dari elips yang mana matriks definit positif yang dipakai dalam definisi jarak itu berupa matriks identitas atau kelipatannya. daftar pustaka [1] hogg, r. v., and craig, a. t., (1978), introduction to statistical mathematics, 5th ed., prentice-hall, inc. [2] mardia, k. v., kent, j. t., and bibby, j. m., (1979) multivariate analysis, academic press. [3] meyer, c. d., (2000), matrix analysis and applied linear algebra, society for industrial and applied mathematics. [4] rich, b., and thomas, c., (2009), geometry scauhm’s outlines, 4th ed., mcgraw-hill companies, inc. [5] http://en.wikipedia.org/wiki/ellipse penerapan logika fuzzy dan sem untuk mengukur kedalaman spiritual dosen, karyawan, dan mahasiswa any tsalasatul fitriyah mahasiswa s2 matematika, fakultas mipa universitas brawijaya malang telp. 08990444003; email: any.tsalasatul@yahoo.com abstract measuring the depth of spiritual used to knowing rank of depth of spiritual. need a measurement to measure of depth of spiritual. in this reseach study case at the state islamic university of maulana malik ibrahim malang. the depth of spiritual is vegue, therefore in this study fuzzy logic used to measure of depth of spiritual. in this reseach used fuzzy multi attribute desicion making (fmadm) with topsis method. in addtion, relationships between variables that affect the depth of spiritual lecture, staff, and student will be investigated using structural equation modeling (sem). form calculation scores of depth of spiritual using topsis method at interval 0 to 1. topsis method can be applied to measure the depth of spiritual of lecture, staff, and student. furthermore the relationship between variables will be analyzed using sem, the results of the analysis data of lecture and staff show that variables have great influence on the variables that are more spiritual depth than behavioral variables. the same conclusion was also obtained based on the analysis of student data. keyword: depth of spiritual, fmadm, sem, topsis abstrak pengukuran kedalaman spiriual digunakan untuk mengetahui tingkat kedalaman spiritual seseorang. untuk mengukur kedalaman spiritual diperlukan patokan atau ukuran yang sesuai. penelitian ini dilakukan di universitas islam negeri maulana malik ibrahim malang. kedalaman spiritual merupakan sesuatu yang kabur, oleh karena itu logika fuzzy dapat membantu untuk pengukuran kedalaman spiritual. logika fuzzy yang digunakan dalam penelitian ini adalah fuzzy multi atributte desicion making (fmadm) dengan metode topsis. dalam penelitian ini digunakan pula structural equation modeling (sem) untuk menguji hubungan antar variabel yang mempengaruhi kedalaman spiritual dosen, karyawan dan mahasiswa. hasil perhitungan skor kedalaman spiritual menggunakan metode topsis berupa skor yang berada pada selang interval antara 0 sampai 1. metode topsis ternyata dapat diterapkan untuk mengukur kedalaman spiritual dosen , karyawan, dan mahasiswa. analisis hubungan antar variabel dengan sem dari data dosen dan karyawan menunjukkan bahwa variabel ibadah memiliki pengaruh yang lebih besar terhadap variabel kedalaman spiritual dibandingkan variabel perilaku. kesimpulan yang sama juga diperoleh berdasarkan analisa pada data mahasiswa. pendahuluan kedalaman spiritual merupakan tingkat perilaku untuk mendekatkan diri kepada tuhannya secara rasa. kedalaman spiritual tidak hanya bisa dinilai dengan ibadah sehari-hari, tapi juga dinilai dengan perilaku yang dilakukan dalam kehidupannya. ibadah adalah mendekatkan diri kepada tuhan, dengan jalan menaati segala perintah-nya, menjauhi larangan-nya dan mengamalkan segala yang diizinkan-nya sebagai tanda mengabdikan/ memperhambakan diri kepada tuhan. demikian pula ibadah juga bermakna untuk mewujudkan keimanan dengan amal-amal sholeh yang merupakan pengembangan ke arah yang positif atau baik dari fitrah manusia. adapun fungsi dasar ibadah bagi manusia untuk menjaga keselamatan akidah, menjaga hubungan antara manusia dengan tuhannya dan berfungsi untuk mendisiplinkan sikap dan prilaku [1]. sayangnya kedalaman spiritual merupakan sesuatu yang kabur untuk diukur, tetapi logika fuzzy dapat membantu dalam pengukurannya karena logika fuzzy memungkinkan nilai keanggotaan antara 0 dan 1. berbagai teori dalam perkembangan logika fuzzy menunjukkan bahwa pada dasarnya logika fuzzy dapat digunakan untuk memodelkan berbagai sistem [2]. salah satu metode dalam logika fuzzy untuk mendukung keputusan adalah fuzzy multi attribute decision making (fmadm), fmadm merupakan suatu metode pengambilan keputusan dari sejumlah alternatif keputusan berdasarkan kriteria tertentu. salah satu metode dalam fmadm adalah metode topsis. metode topsis merupakan metode pengambilan keputusan yang didasarkan pada kriteria-kriteria tertentu. metode ini sering digunakan karena penerapan logika fuzzy dan sem untuk mengukur kedalaman spiritual dosen, karyawan, dan mahasiswa cauchy – issn: 2086-0382 85 alternatif terpilih yang terbaik tidak hanya memiliki jarak terpendek dari solusi ideal positif, namun juga memiliki jarak terpanjang dari solusi ideal negatif [3]. beberapa peneliti menggunakan fmadm dalam studi kasus menentukan lokasi suatu gudang [4]. juliyanti dkk [5] juga menerapkan fmadm untuk menentukan guru berprestasi dengan metode ahp dan topsis. cabang ilmu matematika yang lain adalah ilmu statistika. dalam statistika terdapat suatu metode, yaitu structural equation modeling (sem) yang merupakan suatu metode statistika yang digunakan untuk menguji serangkaian hubungan antar variabel yang terbentuk dari variabel laten dan variabel manifest. penelitian yang dilakukan renganathan [6] menerapkan metode sem untuk mengukur persepsi pelanggan dalam sektor perbankan. oleh karena itu, pada penelitian ini akan dibahas sistem yang berbasis logika fuzzy dalam hal ini fmadm yang dapat diterapkan untuk mengukur kedalaman spiritual dosen, karyawan, dan mahasiswa serta akan diteliti tentang hubungan antar variabel yang mempengaruhi nilai kedalaman spiritual dosen, karyawan, dan mahasiswa dengan menggunakan sem. kajian teori 1. fuzzy multi attribute desicion making (fmadm) definisi 1: misalkan 𝐴 = {𝑎𝑖 |𝑖 = 1, 2, … , 𝑛} adalah himpunan alternatif-alternatif keputusan dan 𝐶 = {𝑐𝑗|𝑗 = 1, 2, … , 𝑚} a dalah himpunan tujuan yang diharapkan, maka akan ditentukan alternatif 𝑥 0 yang memiliki derajat harapan tertinggi terhadap tujuan-tujuan yang relevan cj [3]. salah satu metode dalam fmadm adalah topsis. menurut kusumadewi dkk [3] prosedur topsis mengikuti langkah-langkah sebagai berikut: 1. membuat matriks keputusan yang ternormalisasi 2. membuat matriks keputusan yang ternormalisasi terbobot 3. menentukan matriks solusi ideal positif dan matriks solusi ideal negatif 4. menentukan jarak antara nilai setiap alternatif dengan matriks solusi ideal positif dan negatif. 5. menentukan nilai preferensi untuk setiap alternatif. 2. weighted least square (wls) estimator menurut wijanto [7], dalam wls, fungsi 𝐹(𝑆, σ(𝜃)) yang diminimumkan adalah sebagai berikut: 𝐹𝑊𝐿𝑆(𝜃) = (𝑠 − 𝜎) ′𝑊−1 (𝑠 − 𝜎) dimana s’= (s11, s21, s22, s31, ... , skk) adalah suatu vektor dari elemen-elemen pada separuh bagian bawah, termasuk diagonal matrik kovarian s yang berdimensi k x k, yang digunakan untuk mencocokkan model dengan data. σ’= (σ11, σ21, σ22, σ31, ... , σkk) adalah suatu vektor dari elemen-elemen yang berkaitan pada ∑(θ) yang dihasilkan kembali dari parameterparameter model θ. w-1 adalah suatu matrik definit positif w-1 yang berdimensi u x u, dimana u= k(k+1)/2 dan elemen-elemennya ditandai dengan wgh,ij. 3. structural equation modeling (sem) definisi 2: sem merupakan teknik analisis multivariat yang dikembangkan guna menutupi keterbatasan yang dimiliki oleh model-model analisis sebelumnya yang telah digunakan secara luas dalam penelitian statistika. model-model yang dimaksud diantaranya adalah regression analysis (analisis regresi), path analysis (analisis jalur), dan confirmatory factor analysis (analisis faktor konfirmatori)[8]. ada beberapa tahapan dalam prosedur sem menurut wijanto (2008), yaitu : 1. spesifikasi model (model specification) 2. identifikasi (identification) 3. estimasi (estimation) 4. uji kecocokan (testing fit) 5. respesifikasi (respecification) metode penelitian studi kasus pada penelitian ini dilakukan di universitas islam negeri maulana malik ibrahim malang. data pada penelitian ini adalah data primer yang diperoleh dari survey kuesioner. kuesioner disebarkan pada dosen, karyawan, dan mahasiswa. sebanyak 405 dosen dan karyawan menjadi resonden sedangkan kuesioner untuk mahasiswa sebanyak 409 responden. analisa data pada penelitian ini menggunakan software matlab untuk mengukur kedalaman spiritual dosen, karyawan dan mahasiswa dengan menggunakan fmadm metode topsis. software lisrel 8.80 digunakan untuk menganalisa guna mengetahui faktor yang mempengaruhi skor kedalaman spiritual dosen, karyawan, dan mahasiswa. any tsalasatul fitriyah 86 volume 3 no. 2 mei 2014 hasil dan pembahasan 1. aplikasi fmadm metode topsis untuk mengukur kedalaman spiritual data yang diperoleh dari hasil survey kuesioner kemudian dianalisis menggunankan fmadm metode topsis. tahapan metode topsis yang pertama adalah membuat matriks ternormalisasi dan matriks ternormalisasi berbobot. matriks ternormalisasi merupakan bentuk perbandingan berpasangan setiap alternatif di setiap kriteria sedangkan matriks ternormalisasi berbobot merupakan matriks yang berisi matriks ternormalisasi yang telah dikalikan dengan bobot preferensi. pada penelitian ini, peneliti memberikan bobot preferensi sebagai berikut: 𝑊 = [5 5 4 4 3 3 3 2 2 3 3 5 5 ] tahap berikutnya adalah menentukan solusi ideal positif dan solusi ideal negatif. solusi ideal positif didefinisikan sebagai jumlah dari seluruh nilai terbaik yang dapat dicapai untuk setiap atribut, sedangkan solusi negatif-ideal terdiri dari seluruh nilai terburuk yang dicapai untuk setiap atribut. hasil perhitungan solusi ideal positif dan negatif untuk data dosen dan karyawan diperoleh solusi ideal positif (a+) adalah sebagai berikut:0 𝐴+ = {0,0628; 0,2956; 0.1213; 0.3247; 0.0245; 0.1694; 0.0240; 0.1184; 0.0239; 0.1226; 0.0891; 0.1082} solusi ideal negatif (a-) adalah sebagai berikut: 𝐴− = {0.3139; 0.1182; 0.3033; 0.0649; 0.1225; 0.0678; 0.1202; 0.0474; 0.1193; 0.0245; 0.2227; 0.0433} untuk data mahasiswa diperoleh solusi ideal positif (a+) adalah sebagai berikut: 𝐴+ = {0.0778; 0.3384; 0.0708; 0.4148; 0.0573; 0.2559; 0.0431; 0.2011; 0.0444; 0.1256; 0.0232; 0.1829; 0.0359} solusi ideal negatif (a-) adalah sebagai berikut: 𝐴− = {0.3889; 0.1354; 0.3541; 0.0830; 0.2867; 0.0512; 0.2157; 0.0402; 0.2220; 0.0251; 0.1159; 0.0366; 0.1795} setelah menentukan solusi ideal positif dan negatif, langkah selanjutnya adalah menentukan jarak antara nilai setiap alternatif dengan matriks solusi ideal positif dan negatif serta menentukan nilai preferensi untuk setiap alternatif. pada penelitian ini diberikan kriteria pada skor kedalaman spiritual yang diperoleh dari hasil perhitungan dengan metode topsis. kriteria tersebut adalah sebagai berikut: skor makna interval 0,00 – 0,20 kedalaman spiritual kurang sekali interval 0,21 – 0,40 kedalaman spiritual kurang interval 0,41 – 0,60 kedalaman spiritual cukup interval 0,61 – 0,80 kedalaman spiritual baik interval 0,81 – 1,00 kedalaman spiritual baik sekali hasil perhitungan kedalaman spiritual dosen dan karyawan akan ditunjukkan dalam diagram lingkaran sebagai berikut: gambar 2. kedalaman spiritual dosen dan karyawan hasil perhitungan kedalaman spiritual dosen dan karyawan menunjukkan bahwa sebanyak 33% memiliki skor kedalaman spiritual tertinggi yang berarti kedalaman spiritual baik sekali. skor kedalaman spiritual terendah yang berarti kedalaman spiritual kurang sekali dimiliki sebanyak 7% responden. untuk hasil perhitungan kedalaman spiritual mahasiswa, ditunjukkan dalam diagram lingkaran berikut: pada responden mahasiswa, paling banyak responden memiliki skor pada interval 0,41-0,60 yang berarti kedalaman spiritual cukup dan interval 0,61-0,80 yang bermakna kedalaman spiritual baik yaitu sebanyak 33% dan 45%. 2. aplikasi sem setelah analisis dengan menggunakan fmadm metode topsis, selanjutnya data akan di analisis dengan sem untuk mengetahui hubungan antar variabel yang mempengaruhi variabel kedalaman spiritual. penerapan logika fuzzy dan sem untuk mengukur kedalaman spiritual dosen, karyawan, dan mahasiswa cauchy – issn: 2086-0382 87 gambar 3. kedalaman spiritual mahasiswa 1. spesifikasi model spesifikasi model pada penelitian ini: a. persamaan struktural 𝐾𝑆 = 𝛾11𝐼𝑏𝑎𝑑𝑎ℎ + 𝛾12𝑃𝑒𝑟𝑖𝑙𝑎𝑘𝑢 + 1 b. persamaan pengukuran variabel eksogen 𝐼𝐵𝐷. 1 = 𝜆11𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿1 𝐼𝐵𝐷. 2 = 𝜆21𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿2 𝐼𝐵𝐷. 3 = 𝜆31𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿3 𝐼𝐵𝐷. 4 = 𝜆41𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿4 𝐼𝐵𝐷. 5 = 𝜆51𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿5 𝐼𝐵𝐷. 6 = 𝜆61𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿6 𝐼𝐵𝐷. 8 = 𝜆81𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿8 𝑃𝑅. 1 = 𝜆92𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿9 𝑃𝑅. 2 = 𝜆102𝐼𝑏𝑎𝑑𝑎ℎ + 𝛿10 c. persamaan pengukuran variabel endogen 𝐾𝑆. 1 = 𝜆11𝐾𝑆 + 1 𝐾𝑆. 2 = 𝜆21𝐾𝑆 + 2 𝐾𝑆. 3 = 𝜆31𝐾𝑆 + 3 2. identifikasi identifikasi dimaksudkan untuk menjaga agar model yang dispesifikasi bukan merupakan model yang under-identified. pada data dosen dan karyawan jumlah sampel n=405. jumlah parameter yang akan diestimasi adalah 29 dengan banyaknya indikator variabel laten endogen 3 dan indikator variabel laten eksogen 10, maka degree of freedom yang dihasilkan adalah 62. pada data mahasiswa degree of freedom adalah 75. karena degree of freedom 62 dan 75 sehingga model tersebut overidentified jadi model tersebut dapat diidentifikasikan estimasinya. 3. estimasi pada penelitian ini estimasi parameter menggunakan weigthed least square (wls) estimator karena estimasi parameter wls penggunaannya tidak bergantung pada jenis distribusi datanya serta jenis data pada penelitian ini adalah ordinal [9]. dengan data sebesar 405 dan 409 sudah memenuhi syarat untuk menggunakan estimasi parameter wls. penggunaan estimasi parameter wls disertai dengan penggunaan matriks korelasi sebagai input data. 4. uji kecocokan hasil uji kecocokan menunjukkan bahwa model pada penelitian ini belum fit, karena probabilitas chi-square menunjukkan p=0,00. selain itu nilai ecvi, aic dan caic masih menunjukkan model belum fit. uji kecocokan pada data mahasiswa menunjukkan bahwa model pada penelitian ini belum fit, karena probabilitas chi-square menunjukkan p=0,00. selain itu nilai ecvi, aic dan caic masih menunjukkan model belum fit. karena uji kecocokan pada data dosen dan karyawan maupun pada data mahasiswa menunjukkan hasil yang belum fit, maka perlu dilakukan respeksifikasi agar mendapatkan model yang fit. 5. respeksifikasi untuk mendapatkan model yang cocok, maka peneliti perlu melakukan modifikasi. modifikasi dilakukan dengan menambahkan hubungan path dari beberapa variabel manifest dengan variabel laten, mengkorelasikan error pada beberapa variabel juga dilakukan untuk memodifikasi model. respeksifikasi pada data dosen dan karyawan menghasilkan path diagram estimasi sebagai berikut: dari hasil estimasi pada data dosen dan karyawan setelah respeksifikasi, diketahui bahwa indikator ibd.7 dan ibd.8 yang paling dominan mempengaruhi variabel ibadah, yaitu sebesar 0,8 dan 0,9. pada variabel perilaku, indikator pr.2 yang paling berpengaruh yaitu sebesar 1,41. pada variabel kedalaman spiritual (ks), indikator ks.1 memiliki korelasi dengan ks.2 dan ks.3 sebesar 0,11 dan 0,17 sedangkan faktor yang paling berpengaruh dominan terhadap kedalaman spiritual adalah indikator ks.2 yaitu penghayatan. any tsalasatul fitriyah 88 volume 3 no. 2 mei 2014 gambar 4. hasil estimasi data dosen dan karyawan hasil estimasi model struktural ditunjukkan pada gambar 5 berikut: gambar 5. model struktural data dosen dan karyawan pada estimasi model struktural, variabel ibadah mempunyai korelasi dengan variabel ks sebesar 0,76 dan variabel perilaku memiliki korelasi dengan variabel ks sebesar 0,38 dengan error sebesar 0,23. sedangkan korelasi antara variabel ibadah dengan perilaku sebesar 0,44. setelah memperoleh hasil estimasi, uji kecocokan perlu dilakukan untuk menguji kesesuaian model dengan data. uji kecocokan pada model yang telah respeksifikasi menunjukkan bahwa model telah fit. probabilitas chi-square sebesar 0,0013 menunjukkan model telah fit. ecvi pada model ini adalah sebesar 0,42 namun ecvi for saturated model adalah 0,45 sedangkan ecvi for independence model adalah sebesar 16,55. sehingga dapat dikatakan model sudah fit. nilai aic dan caicjuga telah menunjukkan bahwa model telah fit karena model aic/caic lebih kecil dari independence aic/caic dan saturated aic/caic. nilai nfi, nnfi, pnfi, cfi, ifi, dan rfi. pada model ini dapat dikatakan fit karena nilainya telah mendekati 1. nilai critical n (cn) sebesar 216,86 menunjukkan model ini cukup mewakili data. nilai gfi sebesar 0,82 mengindikasikan bahwa model ini marginal fit dan nilai agfi sebesar 0,91 menyatakan bahwa model pada kasus ini adalah fit. selain pada penilaian goodness of fit, uji kecocokan dilakukan dengan menguji validitas data yang digunakan, pada output lisrel pada path diagram t-value. analisa data menunjukkan t-value pada seluruh indikator yang mempengaruhi variabel laten adalah valid karena nilai t-value tidak ada yang berada pada rentang -2 < t-value < 2. setelah menguji validitas data, uji kecocokan berikutnya adalah menguji reabilitas data. untuk menguji reabilitas data menurut ghozali dan fuad (2005) dapat digunakan rumus construct reability (cr) dan variance extracted (ve). untuk menghitung cr digunakan rumus : 𝐶𝑅 = (∑ 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑙𝑜𝑎𝑑𝑖𝑛𝑔)2 (∑ 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑙𝑜𝑎𝑑𝑖𝑛𝑔)2 + ∑ 𝑒𝑗 dan untuk menghitung ve digunakan rumus: 𝑉𝐸 = ∑ 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 2 ∑ 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 2 + ∑ 𝑒𝑗 perhitungan cr dan ve pada data ini diperoleh cr ibadah sebesar 0,89; cr perilaku sebesar 0,82; dan cr ks sebesar 0,73. dan nilai ve ibadah sebesar 0,5028; ve perilaku sebesar 0,6252; dan ve ks sebesar 0,52. hasil analisa dengan sem pada data dosen dan karyawan dapat dikatakan bahwa semakin tinggi derajat ibadah dan perilaku maka akan semakin tinggi pula kedalaman spiritual yang dimiliki dosen dan karyawan. dengan kata lain terdapat pengaruh positif dan signifikan antara variabel ibadah dan perilaku terhadap variabel kedalaman spiritual. selain itu, penelitian ini juga menemukan adanya variabel ibadah dan penerapan logika fuzzy dan sem untuk mengukur kedalaman spiritual dosen, karyawan, dan mahasiswa cauchy – issn: 2086-0382 89 variabel perilaku. hasil analisa menunjukkan adanya pengaruh yang positif dan signifikan antara variabel ibadah dengan variabel perilaku. hasil modifikasi pada data mahasiswa, menghasilkan path diagram estimasi sebagai berikut: gambar 6. hasil estimasi data mahasiswa pada data mahasiswa, indikator yang paling berpengaruh terhadap adalah ibd.2, ibd.3, ibd.4, ibd.6, dan ibd.7, yaitu sebesar 0,67, 0,66, 0,64, 0,65, dan 0,68. ibd.6 memiliki korelasi terhadap ibd.1 dan ibd.7 sebesar 0,18 dan 0,11 sedangkan ibd.4 dan ibd.5 berkorelasi sebesar 0,30. pada variabel perilaku, pr.2 dan pr.3 yang memiliki pengaruh paling dominan, yaitu sebesar 0,45 dan 0,51. pada variabel kedalaman spiritual, indikator ks.2 yang paling berpengaruh yaitu sebesar 1,35. indikator ks.1 memiliki korelasi dengan ks.2 dan ks.3 sebesar 0,29 dan -0,60. model struktural yang diperoleh dari hasil estimasi digambarkan pada gambar 7 berikut ini: gambar 7. model struktural data mahasiswa variabel ibadah berpengaruh sebesar 0,56 terhadap variabel ks, sedangkan variabel perilaku berpengaruh sebesar 0,48 dengan error seberar 0,31. korelasi antara variabel ibadah dan perilaku yaitu sebesar 0,26. uji kecocokan goodness of fit yang dihasilkan oleh output lisrel pada data mahasiswa, menunjukkan bahwa model telah fit dengan probabilitas chi-square sebesar 0,0065. ecvi pada model ini adalah sebesar 0,44 dan ecvi for saturated model adalah 0,51 sedangkan ecvi for independence model adalah sebesar 6,54 sehingga dapat dikatakan model sudah fit. nilai aic dan caic juga telah menunjukkan bahwa model telah fit karena model aic/caic lebih kecil dari independence aic/caic dan saturated aic/caic. uji kecocokan dengan nfi, nnfi, pnfi, cfi, ifi, dan rfi pada model ini dapat dikatakan fit karena nilainya telah mendekati 1. nilai critical n (cn) sebesar 291,27 menunjukkan model ini cukup mewakili data. nilai gfi sebesar 0,93 mengindikasikan bahwa model ini good fit dan nilai agfi sebesar 0,88 menyatakan bahwa model pada kasus ini adalah marginal fit. berdasarkan analisa dengan lisrel, seluruh indikator pada model ini adalah valid dengan ditandainya tidak ada nilai t-value yang berada pada rentang antara -2 sampai 2 serta diperoleh cr ibadah sebesar 0,89; cr perilaku sebesar 0,788; dan cr ks sebesar 0,854. untuk nilai variance extracted (ve), ve ibadah sebesar 0,5011; ve perilaku sebesar 0,54; dan ve ks sebesar 0,6625. karena nilai cr pada setiap variabel lebih dari 0,7 dan nilai ve lebih dari 0,5 maka dapat dikatakan bahwa data telah reliabel. hasil analisa dengan sem pada data mahasiswa dapat dikatakan bahwa semakin tinggi derajat ibadah dan perilaku maka akan semakin tinggi pula kedalaman spiritual yang dimiliki oleh mahasiswa. atau dengan kata lain terdapat pengaruh positif dan signifikan antara variabel ibadah dan perilaku terhadap variabel kedalaman spiritual. selain itu, penelitian ini juga menemukan adanya variabel ibadah dan variabel perilaku. hasil analisa menunjukkan any tsalasatul fitriyah 90 volume 3 no. 2 mei 2014 adanya pengaruh yang positif dan signifikan antara variabel ibadah dengan variabel perilaku. kesimpulan berdasakan hasil analisa data responden di lokasi penelitian dengan metode topsis dan sem maka dapat disimpulkan sebagai berikut: 1. fuzzy multi atributte decision making (fmadm) metode topsis dapat diterapkan untuk mengukur kedalaman spiritual dosen, karyawan dan mahasiswa. 2. stuctural equation modeling (sem) dapat digunakan untuk mengetahui hubungan antar variabel yang mempengaruhi kedalaman spiritual dosen, karyawan, serta mahasiswa. berdasarkan analisis data hasil penelitian diperoleh bahwa variabel ibadah yang lebih berpengaruh terhadap variabel kedalaman spiritual dibandingkan variabel perilaku dengan model struktural yang dihasilkan sem pada data dosen dan karyawan adalah: 𝐾𝑆 = 0,76 𝐼𝑏𝑎𝑑𝑎ℎ + 0,38 𝑃𝑒𝑟𝑖𝑙𝑎𝑘𝑢 +0,23. pada data mahasiswa, model struktural yang dihasilkan sem adalah: 𝐾𝑆 = 0,56 𝐼𝑏𝑎𝑑𝑎ℎ + 0,48 𝑃𝑒𝑟𝑖𝑙𝑎𝑘𝑢 +0,31. daftar pustaka [1] pradana, “ibadah dan syari’at,” 2012. . [2] m. djunaidi, e. setiawan, and f. w. andista, “penentuan jumlah produksi dengan aplikasi metode fuzzymamdani,” j. ilm. tek. ind., vol. 4, no. 2, pp. 90–98, 2005. [3] r. kusumadewi, s., hartati, s., harjoko, a., wardoyo, fuzzy multi-attribute decision making (fmadm). yogyakarta: grahailmu, 2006. [4] m. ashrafzadeh, f. m. rafiei, n. m. isfahani, and z. zare, “application of fuzzy topsis method for the selection of warehouse location: a case study,” interdiscip. j. contemp. res. bus., vol. 3, no. 9, pp. 655–667, 2012. [5] j. juliyanti, i. mohammad isa, and m. imam, “pemilihan guru berprestasi menggunakan metode ahp dan topsis,” pemantapan keprofesionalan peneliti, pendidik, dan prakt. mipa untuk mendukung pembang. karakter bangsa, 2011. [6] r. renganathan, s. balach, and k. govindarajan, “customer perception towards banking sector: structural equation modeling approach,” african j. bus. manag., vol. 6, no. 46, pp. 11426– 11436, 2012. [7] s. h. wijanto, “structural equation modeling dengan lisrel 8.8: konsep dan tutorial,” 2008. [8] t. m. hox, j.j dan bechger, “an introduction to structural equation modeling, family science review,” vol. 11, pp. 354–373, 1998. [9] i. dan f. ghozali, struktural equation modeling teori, konsep, dan aplikasi dengan program lisrel lisrel 8.80 edisi 3. semarang: badan penerbit universitas diponegoro, 2005. kajian teori 1. fuzzy multi attribute desicion making (fmadm) 2. weighted least square (wls) estimator 3. structural equation modeling (sem) metode penelitian hasil dan pembahasan 1. aplikasi fmadm metode topsis untuk mengukur kedalaman spiritual 2. aplikasi sem kesimpulan daftar pustaka 1a sampul depan aplikasi program dinamik pada metode pelaksanaan pengecoran plat lantai agung sedayu jurusan teknik arsitektur fakultas sains dan teknologi uin maulana malik ibrahim malang mahasiswa program doktor teknik sipil universitas brawijaya malang abstract the development of world construction projects at this time that established by widespread methods of physical implementation which more excellent and sophisticated. the contractors are competing to obtain optimum results, and reduce losses more lower. the results obtained should not give benefit one person only, the other one should not to lose and suffer. islam guides human how they work give benefit to their own and others. many methods, equipment, and expertise deployed by the contractor for their work more optimal, one method for optimizing the physical implementation of the project is dynamic programming. dynamic programming to solve the problem optimization is not only with one step, but the events that took place linked to each other. case study in this paper is the application of dynamic programming of implementation method of concrete casting on floor slab of green house building, it’s belongs to department of biology, uin maliki malang with backward induction phase. in modern century as this era has many fast-growing method of implementation that are categorized as advanced, but the problem that the methods actually optimal in terms of cost and time, due to cost and time parameters are very important in the progress of construction projects. the methods is very sophisticated so make project to be done very fast but it is very expensive financing, or the method is still conventional and traditional so that project costs will be reduced as cheaply, but the project need long time to be finished. where is the decision that one will be expected to demand that we use what method. mathematical formulas of dynamic programming as recursive equation with change big problem into smaller sub-problems, so the completion of smaller problems will be found solving of the original problem. optimization process in this case that slab casting which solved in two methods, manually calculations with a recursive formula and use software ds win 2.0. optimization result of the two method are compared, and the two method produce similar results. optimal or not parameter of the two method is optimum time and cost. keywords: dynamic programming, project implementation method, floor slab casting pendahuluan sabda rasululloh saw, ”sesungguhnya allah sangat mencintai orang yang jika melakukan sesuatu pekerjaan dilakukan secara itqan (tepat, terarah, jelas, dan tuntas).” (hr thabrani) dari hadits di atas kita dianjurkan dalam melakukan tugas dan pekerjaan sehari-hari hendaknya dilakukan dengan itqan yakni sistematis, rapi, tepat sasaran, jelas, selesai tepat waktu, dan dapat memberikan manfaat, artinya manfaat bukan sesuatu pekerjaan yang sia-sia (hafidhuddin & tanjung, 2005:24). di dalam pekerjaan implementasi fisik proyek konstruksi, indikator itqan ini sangatlah penting dan benarbenar ditegakkan, agar mendapat ridlo allah swt dan memberi manfaat bagi kemaslahatan bagi diri sendiri dan orang lain. di jaman modern sekarang ini, banyak sekali metode pelaksanaan yang digunakan, salah satunya adalah dengan rumus matematika program dinamik. pemrograman dinamik adalah cara pengambilan keputusan dengan melibatkan tahapan-tahapan secara berurutan (aminudin, 2005:22). pemrograman dinamik menyelesaikan permasalahan optimasi tidak dengan sekali langkah, namun dengan mengubah masalah yang cukup besar ke dalam sub masalah yang lebih kecil, sehingga dari rangkaian penyelesaian masalah yang lebih kecil akan ditemukan penyelesaian masalah aslinya (taha, 1997:79). pada contoh kasus kali ini adalah penerapan progam dinamik pada metode pelaksanaan pengecoran plat lantai gedung green house jurusan biologi uin maliki malang (gambar 1 – 4) dengan metode induksi mundur. di era modern saat ini telah banyak berkembang pesat metode pelaksanaan yang tergolong mutakhir, namun persoalannya adalah apakah metode tersebut benar-benar optimal dari segi biaya dan waktu, karena parameter biaya dan waktu sangat penting dalam kemajuan proyek konstruksi (soeharto, 1999:102). metodenya canggih sehingga proyek cepat selesai namun secara pembiayaan sangat mahal, atau sebaliknya metodenya masih konvensional atau tradisional agung sedayu 56 volume 1 no. 2 mei 2010 yang berdampak biaya ditekan semurah mungkin, namun proyek selesainya lama. gambar 1. denah lantai 1 green house gambar 2. denah lantai 2 green house gambar 3. tampak depan green house gambar 4. tampak samping kanan green house program dinamik program dinamik (dynamic programming) adalah suatu teknik untuk menyelesaikan masalah yang melibatkan sekumpulan keputusan yang saling berhubungan dalam tujuan agar secara keseluruhan mencapai keaktifannya (mulyono, 2004:77). beberapa istilah yang digunakan dalam penyelesaian masalah program dinamik adalah : 1. stage, langkah pertama adalah mengidentifikasikan stage atau tahapan dalam proses keputusan. misalnya, terdapat (n) tahapan yang diberi label 1, 2, 3, 4, …, n-1, n. 2. states, adalah tahapan yang menggambarkan status semua informasi dalam membuat keputusan. misalnya, jika dalam stage 3 terdapat state 5, maka ditulis dengan s3 = 5 3. decision, variable keputusan atas stage i ditulis dengan di . misalnya dalam stage 3 terdapat states 5, maka d3 mungkin sama dengan 7 atau 8 ( menuju states 7 atau 8). 4. return function. untuk menghitung efektivitas digunakan fungsi dengan notasi f yang dapat berupa biaya, laba, jarak, atau beberapa hitungan yang lain. fungsi f tersebut dinamakan return function, misalnya, fi (si,di). apabila nilai fi optimum ditulis dengan fi*(si) atau dikenal dengan istilah optimum return function. misalnya, dalam stage 2 terdapat stages 2 atau i = 2, dan si = 2, maka fungsinya ditulis : f2 = min f2(2,d2) d2 = min {f2(2,4),f2(2,5),f2(2,6)} 5. recursions, fi*(si) = min {d(si,di) + fi+1*(di)} analisis biaya pelaksanaan untuk menunjang dan memperjelas optimasi metode pelaksanaan yang mana yang lebih menguntungkan, maka terlebih dahulu dilakukan analisis biaya pelaksanaan pengecoran plat lantai pada setiap stages dan states, sebagai berikut : 1. stage 3 (n=3), merupakan tahap persiapan pengecoran. tahap ini merupakan tahap persiapan yang mencakup kordinasi dan perencanaan, dengan biaya pelaksanaan masuk biaya tak langsung (overhead) dengan satuan lump sum (ls) (soeharto,2001:50) sebesar rp. 500.000,00 dengan durasi waktu 1 hari. 2. stage 2 (n=2), merupakan tahap pembuatan dan pemasangan bekesting tahap ini merupakan tahap pemasangan dan pembuatan bekesting, yang terdiri dari 2 pilihan langkah, meliputi : aplikasi program dinamik pada metode pelaksanaan pengecoran plat lantai jurnal cauchy – issn: 2086-0382 57 a. pemasangan bekesting dengan perancah bambu. pekerjaan pembuatan bekesting ini ditunjang oleh perancah bambu. analisis biaya pekerjaan ini ditunjukkan pada tabel 1 di bawah. tabel 1. analisis biaya pekerjaan stage 2 (n=2) kegiatan pemasangan bekesting dengan perancah bambu no item pekerjaan volume satuan koefisien harga satuan jumlah 1 bambu 140 btg rp. 17,500.00 rp2,450,000.00 2 kayu meranti reng 2cmx3cmx4m 0.048 m3 rp. 3,505,000.00 rp 168,240.00 3 kayu kasau 4cmx6cmx4m 0.288 m3 rp . 4,250,000.00 rp 1,224,000.00 4 paku 7,5 cm 7.5 kg rp. 12,500.00 rp 93,750.00 5 paku 4 cm 5 kg rp. 9,750.00 rp 48,750.00 6 upah mandor 1 oh 2.1125 rp. 65,000.00 rp 137,312.50 7 upah tukang 2 oh 3.25 rp. 55,000.00 rp 357,500.00 8 upah pekerja 5 oh 3.25 rp. 40,000.00 rp 650,000.00 9 upah bongkaran 3 oh 1.45 rp. 40,000.00 rp 174,000.00 total biaya rp5,303,552.50 b.pemasangan bekesting dengan scaffolding (perancah besi). pekerjaan pembuatan bekesting ini ditunjang oleh pekerjaan scafolding. analisis biaya pekerjaan ini ditunjukkan pada tabel 2 di bawah. tabel 2. analisis biaya pekerjaan stage 2 (n=2) kegiatan pemasangan bekesting dengan scafolding no item pekerjaan volume satuan koefisien harga satuan jumlah 1 sewa scafolding 130.8 m2 12 rp 3,850.00 rp6,042,960.00 2 upah pasang 130.8 m2 1.85 rp 1,300.00 rp 314,574.00 3 upah bongkaran 130.8 m2 1.25 rp 900.00 rp 147,150.00 4 kayu kasau 4cmx6cmx4m 0.186 m3 rp4,250,000.00 rp 790,500.00 total biaya rp7,295,184.00 dua pekerjaan di atas dapat dipilih, namun pekerjaan pembuatan bekesting untuk dua teknik di atas memiliki kesamaan dalam analisis biayanya. pekerjaan bekesting yang dilakukan ditunjukkan dalam analisis biaya sebagaimana tabel 3 di bawah. tabel 3. analisis biaya pekerjaan pembuatan bekesting no item pekerjaan volume satuan koefisien harga satuan jumlah 1 papan kayu 2cmx20cmx4m 175.3 bh rp 4,000.00 rp 701,200.00 2 kayu meranti reng 2cmx3cmx4m 0.147 m3 rp3,505,000.00 rp 515,235.00 3 kayu kasau 4cmx6cmx4m 0.285 m3 rp4,250,000.00 rp1,211,250.00 4 paku 7,5 cm 12.5 kg rp 12,500.00 rp 156,250.00 5 paku 4 cm 10 kg rp 9,750.00 rp 97,500.00 6 upah mandor 1 oh 2.5025 rp 65,000.00 rp 162,662.50 7 upah tukang 2 oh 3.85 rp 55,000.00 rp 423,500.00 8 upah pekerja 5 oh 3.85 rp 40,000.00 rp 770,000.00 9 upah bongkaran 3 oh 1.71 rp 40,000.00 rp 205,200.00 total biaya rp4,242,797.50 dari analisis biaya yang dilakukan dapat dibuat tabel besar biaya dan durasi pelaksanaan pembuatan bekesting dengan perbandingan dua metode di atas, sebagai berikut : tabel 4. perbandingan analisis biaya dan waktu pekerjaan pembuatan bekesting pekerjaan bekesting + perancah bamboo pekerjaan bekesting + scafolding total biaya rp.4.242.797,50 + rp.5,303,552.50 = rp. 9.546.350,00 rp.4.242.797,50 + rp.7,295,184.00 = rp.11.537.981,50 durasi 5,56 hr + 4,7 hr = 10,26 hr 5,56 hr + 3,1 hr = 8,66 hr agung sedayu 58 volume 1 no. 2 mei 2010 3. stage 1 (n=1), merupakan tahap pengecoran beton (19,62 m3) tahap ini merupakan tahap pengecoran beton, yang terdiri dari 3 option pilihan, meliputi: a. pengecoran dengan cara manual pekerjaan pengecoran ini dilakukan dengan cara manual (murni tenaga manusia). analisis biaya pekerjaan ini ditunjukkan pada tabel 5 di bawah. tabel 5. analisis biaya pekerjaan stage 1 (n=1) kegiatan pengecoran beton cara manual no item pekerjaan volume satuan koefisien harga satuan jumlah 1 semen 6278.4 kg rp 1,200.00 rp 7,534,080.00 2 pasir 10.5948 m3 rp123,500.00 rp 1,308,457.80 3 kerikil 16.0884 m3 rp 61,900.00 rp 995,871.96 4 mandor 1 oh 2.4336 rp 65,000.00 rp 158,184.00 5 tukang 2 oh 5.07 rp 55,000.00 rp 557,700.00 6 pekerja 5 oh 5.07 rp 40,000.00 rp 1,014,000.00 total biaya rp11,568,293.76 b. pengecoran dengan menggunakan small concrete mixer pekerjaan pengecoran ini dilakukan dengan menggunakan small concrete mixer. analisis biaya pekerjaan ini ditunjukkan pada tabel 6 di bawah. tabel 6. analisis biaya pekerjaan stage 1 (n=1) kegiatan pengecoran beton dengan menggunakan small concrete mixer no item pekerjaan volume satuan koefisien harga satuan jumlah 1 sewa peralatan concrete mixer 3.72 oh rp 270,000.00 rp 1,004,400.00 2 semen 6278.4 kg rp 1,200.00 rp 7,534,080.00 3 pasir 10.5948 m3 rp 123,500.00 rp 1,308,457.80 4 kerikil 16.0884 m3 rp 61,900.00 rp 995,871.96 5 mandor 1 oh 1.7376 rp 65,000.00 rp 112,944.00 6 tukang 2 oh 3.62 rp 55,000.00 rp 398,200.00 7 pekerja 5 oh 3.62 rp 40,000.00 rp 724,000.00 8 bahan bakar 26.04 ltr rp 4,500.00 rp 117,180.00 total biaya rp12,195,133.76 c. pengecoran dengan dengan ready mix dan concrete pump pekerjaan pengecoran ini dilakukan dengan ready mix dan concrete pump. analisis biaya pekerjaan ini ditunjukkan pada tabel 7 di bawah. tabel 7. analisis biaya pekerjaan stage 1 (n=1) kegiatan pengecoran beton dengan menggunakan ready mix dan concrete pump item pekerjaan volume satuan koefisien harga satuan jumlah 1 harga beton segar 19.62 m3 rp 500,000.00 rp 9,810,000.00 2 sewa concrete pump 2.5 jam rp1,250,000.00 rp 3,125,000.00 3 mandor 1 oh 0.232143 rp 65,000.00 rp 15,089.29 4 tukang 1 oh 0.357143 rp 55,000.00 rp 19,642.86 5 pekerja 2 oh 0.357143 rp 40,000.00 rp 28,571.43 total biaya rp12,998,303.57 tabel 8. analisis biaya pekerjaan pembesian item pekerjaan volume satuan koefisien harga satuan jumlah 1 besi sa ø = 12 106.875 m rp 77,875.00 rp 8,322,890.63 2 besi sa ø = 10 42.375 m rp 67,750.00 rp 2,870,906.25 3 kawat bendrat 25 kg rp 15,100.00 rp 377,500.00 4 upah mandor 1 oh 2.0605 rp 65,000.00 rp 133,932.50 5 upah tukang 2 oh 3.17 rp 55,000.00 rp 348,700.00 6 upah pekerja 4 oh 3.17 rp 40,000.00 rp 507,200.00 total biaya rp12,561,129.38 aplikasi program dinamik pada metode pelaksanaan pengecoran plat lantai jurnal cauchy – issn: 2086-0382 59 selain tahapan teknik pengecoran di atas, pekerjaan pengecoran didahului dengan pekerjaan pembesian. dimana pekerjaan pembesian ini juga tercakup untuk setiap teknik pelaksanaan dari ketiga teknik tersebut di atas (tabel 8) dari analisis biaya yang dilakukan dapat dibuat tabel besar biaya dan durasi pelaksanaan pengecoran dengan perbandingan tiga metode di atas, sebagai berikut : tabel 9. perbandingan analisis biaya dan waktu pekerjaan pengecoran pekerjaan pembesian + pengecoran manual pekerjaan pembesian + pengecoran dengan small concrete mixer pekerjaan pembesian + pengecoran dengan ready mix total biaya rp. 12.561.129,38 + rp.11.568.293,76 = rp. 24.129.423,14 rp. 12.561.129,38 + rp 12.195.133,76 = rp. 24.756.263,14 rp. 12.561.129,38 + rp. 12.998.303,57 = rp. 25.559.432,95 durasi 3,17 hr + 5,87 hr = 9,04 hr 3,17 hr + 4,12 hr = 7,29 hr 3,17 hr + 0,357 hr = 3,527 hr 4. stage 0 (n=0), merupakan tahap pengerasan beton (curing beton). pada tahap ini pekerjaan pengecoran plat selesai, dan menunggu masa pengerasan beton (curing beton). pemodelan dan optimasi dengan program dinamik 1. pemodelan program dinamik dari beberapa langkah analisis biaya dan waktu pelaksanaan proyek di atas, maka dapat dibuat model jaringan langkah atau tahapan pekerjaan. jaringan tersebut dibentuk dengan 4 stages (0, 1, 2, dan 3) dan 7 states, dengan state g pada stage 0, state d,e,dan f pada stage 1, state b dan c pada stage 2, dan state a pada stage 3. penjelasan model jaringan tersebut ditunjukkan pada gambar 5 di bawah. gambar 5. bagan jaringan tahapan pelaksanaan pengecoran plat lantai greenhouse keterangan: a : tahap persiapan pengecoran b : tahap pemasangan bekesting dengan perancah bamboo c : tahap pemasangan bekesting dengan scaffolding (perancah besi) d : pengecoran secara manual e : pengecoran dengan small concrete mixer f : pengeceron dengan ready mix dan concrete pump g : pengecoran selesai / curing beton dari hasil analisa biaya dikaitkan dengan bagan jaringan pada gambar 5, didapatkan penjelasan sebagai berikut 1 = biaya rp. 500.000,dan durasi 1 hari 2 = biaya rp. 9.546.350,dan durasi 10,26 hari 3 = biaya rp 11.537981,50 dan durasi 8,66 hari 4 = biaya rp. 24.129.423,14 dan durasi 9,04 hari a b c e d f g 1 1 2 3 2 3 4 5 6 2 3 agung sedayu 60 volume 1 no. 2 mei 2010 5 = biaya rp. 24.756.263,14 dan durasi 7,29 hari 6 = biaya rp 25.559.432,95 dan durasi 3,527 hari biaya dan lamanya waktu antara dua metode disebut dengan jmn(biaya) dan pmn (waktu), dimana m adalah tahap awal dan n tahap berikutnya. biaya dari a ke b dengan besar jab = rp 500.000,dan waktu proyek pab = 1 hari. adapun besar biaya dan lama waktu dari bagan tersebut dijelaskan pada tabel 10 – 11 di bawah sebagai berikut: tabel 10. biaya proyek jab=rp.500.000, jbd=rp.9.546.350, jce=rp.11.537.981,5 jdg=rp.24.129.423,14 jac=rp.500.000, jbe=rp.9.546.350, jcf=rp.11.537.981,5 jeg=rp.24.756263,14 jbf = rp.9.546.350, jcd=rp.11.537.981,5 jfg=rp.25.559.432,95 tabel 11. lama (waktu) proyek pab = 2 hari pbd = 10,26 hari pce = 8,66 hari pdg = 9,04 hari pac = 2 hari pbe = 10,26 hari pcf = 8,66 hari peg = 7,29 hari pbf = 10,26 hari pcd= 8,66 hari pfg = 3,57 hari 2. proses optimasi dengan program dinamik proses optimasi ini dilakukan dengan metode induksi mundur, yang dimulai dengan tahapan akhir atau tujuan, kemudian diurutkan secara mundur sampai ke awal permasalahan. proses optimasi yang dicari adalah meminimumkan biaya dan waktu dilakukan pada setiap stage, sebagai berikut : a. stage 0 (tahap pengerasan beton/ curing beton) dengan fungsi (n) = 0, didapat f0(g) = 0; k0 = stop. tahap ini merupakan akhir dari proses pelaksanaan pengecoran plat lantai. jadi pada stage 0 ini pekerjaan berhenti. b. stage 1 (tahap pengecoran beton) tahap stage 1 difungsikan sebagai n = 1, f1(s) = jn-1+ f0(x) dan g1(s) = jn-1+ g0(x). tahap ini adalah pengecoran beton dengan urutan langkah metode sebagai berikut, • untuk tahap dg f1(d) = {jdg + f0(g)} = rp.24.129.423,14 + 0 = rp. 24.129.423,14, k1(d) = go to g g1(d) = {pdg + g0(g)} = 9,04 hari + 0 = 9,04 hari, k1(d) = go to g • untuk tahap eg f1(e) = {jeg + f0(g)} = rp.24.756.263,14 + 0 = rp. 24.756.263,14, k1(e) = go to g g1(e) = {peg + g0(g)} = rp.7,29 hari + 0 = 7,29 hari, k1(g) = go to g • untuk tahap fg f1(f) = {jfg + f0(g)} = rp.25.559.432,95 + 0 = rp. 24.756.263,14, k1(f) = go to g g1(f) = {pfg + g0(g)} = rp.3,527 hari + 0 = 3,527 hari, k1(g) = go to g besar biaya yang termurah pada f1(d) = rp. 24.129.423,14, dan lama waktu yang paling singkat adalah g1(f) dengan 3,527 hari. keputusan pada biaya termurah dengan cara pengecoran secara manual melibatkan tenaga manusia. sedangkan apabila kita memilih waktu yang tercepat dan paling singkat maka kita gunakan tahap f menuju g dengan menggunakan ready mix. kondisi pada stage 1 ini dapat ditabelkan seperti di bawah ini: tabel 12. stage 1 state s keputusan = k k1(s) f1(s) d biaya = rp.24.129.423,14+ 0 = rp.24.129.423,14 waktu = rp.3,527 hari + 0 = 3,527 hari g rp.24.129.423,14 3,527 hari e biaya = rp.24.756.263,14 + 0 = rp. 24.756.263,14 waktu = rp.7,29 hari + 0 = 7,29 hari g rp. 24.756.263,14 7,29 hari f biaya = rp.25.559.432,95 + 0 = rp. 24.756.263,14 waktu = rp.3,527 hari + 0 = 3,527 hari g rp. 24.756.263,14 3,527 hari c. stage 2 (tahap pemasangan dan pembuatan bekesting) stage 2 dengan fungsi n = 2, f2(s) = jn-1+ f0(x) dan g2(s) = jn-1+ g0(x). pada tahap ini dilakukan pemasangan dan pembuatan bekesting, dengan langkah-langkah berikut : • untuk tahap bd f2(b) = {jbd+f1(d)} = rp.9.546.350 + rp.24.129.423,14 = rp.33.675.773,14, k2(b) = go to d g2(b) = {pbd+ g1(d)} = 10,26 hari + 9,04 hari = 19,30 hari, k2(b) = go to d • untuk tahap be f2(b) = {jbe+f1(e)} = rp.9.546.350 + rp.24.756.263,14 = rp.34.302.613,14, k2(b) = go to e g2(b) = {pbe+ g1(e)} = 10,26 hari + 7,29 hari = 18,05 hari, k2(b) = go to e aplikasi program dinamik pada metode pelaksanaan pengecoran plat lantai jurnal cauchy – issn: 2086-0382 61 • untuk tahap bf f2(b) = {jbf+f1(f)} = rp.9.546.350 + rp.25.559.432,95 = rp.35.105.782,95, k2(b) = go to f g2(b) = {pbf+ g1(f)} = 10,26 hari + 3,527 hari = 13,787hari, k2(b) = go tof • untuk tahap cd f2(c) = {jcd+f1(d)} = rp.11.537.981,5 + rp.24.129.423,14 = rp.35.667.404,64, k2(c) = go to d g2(c) = {pcd+ g1(d)} = 8,66 hari + 9,04 hari = 17,70 hari, k2(c) = go to d • untuk tahap ce f2(c) = {jce+f1(e)} = rp.11.537.981,5 + rp.24.756.263,14 = rp.36.294.244,64, k2(c) = go to e g2(c) = {pce+ g1(e)} = 8,66 hari + 7,29 hari = 15,95 hari, k2(c) = go to e • untuk tahap cf f2(c) = {jcf+f1(f)} = rp.11.537.981,5 + rp.25.559.432,95 = rp.37.097.414,45, k2(c) = go to f g2(c) = {pcf+ g1(f)} = 8,66 hari + 3,527 hari = 12,187 hari, k2(c) = go to f besar biaya yang termurah pada f2(b) = rp.33.675.773,14, dan lama waktu yang paling singkat adalah g2(c) sebesar 12,187 hari. keputusan yang diambil untuk biaya termurah dengan cara pembuatan perancah dari bambu. sedangkan apabila kita memilih waktu yang tercepat dan paling singkat adalah dengan menggunakan scafolding. biaya termurah dan waktu paling singkat yang ditempuh pada stage 2 ini dapat ditabelkan sebagai berikut : tabel 13. stage 2 state s keputusan k1(s) f1(s) x = d x = e x = f b biaya = rp.9.546.350 + rp.24.129.423,14 = rp.33.675.773,14 waktu = 10,26 hari + 9,04 hari = 19,30 hari biaya = rp.9.546.350 + rp.24.756.263,14 = rp.34.302.613,14 waktu = 10,26 hari + 7,29 hari = 18,05 hari, biaya= rp.9.546.350 + rp.25.559.432,95 = rp.35.105.782,95 waktu = 10,26 hari + 3,527 hari = 13,787hari biaya : d waktu: f rp.33.675.773,14, 13,787 hari c biaya = rp.11.537.981,5 + rp.24.129.423,14 = rp.35.667.404,64 waktu = 8,66 hari + 9,04 hari = 17,70 hari, biaya = rp.11.537.981,5 + rp.24.756.263,14 = rp.36.294.244,64 waktu = 8,66 hari + 7,29 hari = 15,95 hari biaya = rp.11.537.981,5 + rp.25.559.432,95 = rp.37.097.414,45, waktu = 8,66 hari + 3,527 hari = 12,187 hari, biaya : d waktu: f rp.35.667.404,64 12,187 hari d. stage 3 (tahap persiapan) stage ini difungsikan dengan, n = 3, f3(s) = jn-1 + f2(x) dan g3(s) = jn-1+ g2(x). langkah pengerjaan pada stage 3 ini adalah sebagai berikut, • untuk tahap ab f3(a) = {jab + f2(b)} = rp.500.000 + rp.33.675.773,14 = rp.34.175.773,14, k3(a) = go to b g3(a) = {pab+ g2(b)} = 1 hari + 13,787 hari = 14.787 hari, k3(a) = go to b • tahap ac f3(a) = {jac + f2(c)} = rp.500.000 + rp.35.667.404,64 = rp.36.167.404,64, k3(a) =go to c g3(a) = {pac+ g2(c)} = 1 hari + 12,187 hari = 13.187 hari, k3(a) = go to c besar biaya yang termurah pada f3(a) = rp.34.175.773,14, dan lama waktu yang paling singkat adalah g2(c) sebesar 13,187 hari. keputusan untuk biaya termurah kita pakai tahap a menuju b. sedangkan apabila kita memilih waktu yang tercepat maka kita gunakan tahap a menuju c. kondisi pada stage 2 ini dapat ditabelkan seperti di bawah ini : agung sedayu 62 volume 1 no. 2 mei 2010 tabel 14. stage 3 state s keputusan k3(s) f3(s) x = b x = c a biaya = rp.500.000 + rp.33.675.773,14 = rp.34.175.773,14 waktu = 1 hari + 13,787 hari = 14.787 hari biaya = rp.500.000 + rp.35.667.404,64 =rp.36.167.404,64 waktu = 1 hari + 12,187 hari = 13.187 hari biaya: e waktu: c rp.34.175.773,14 13,187 hari e. perbandingan hasil analisis untuk biaya dan waktu proyek dari proses perhitungan secara manual seperti sebelumnya di dapatkan metode pelaksanaan yang optimum sebagai berikut, 1. biaya paling minimum didapatkan dengan urutan dengan urutan tahapan a-b-d-g dengan biaya rp.34.175.773,14. deskripsi tahapan tersebut dimulai : tahapan persiapan – pemasangan bekesting + perancah bambu – pengecoran beton secara manual – curing beton. namun dari segi waktu durasi proyek membutuhkan waktu 20,3 hari. 2. waktu tercepat didapatkan dengan urutan langkah dengan urutan a-c-f-g yang memakan waktu 13,187 hari, dengan penjelasan urutan tahapan pelaksanaan : tahap persiapan pengecoran – pemasangan bekesting dengan scaffolding (perancah besi) – pengeceron dengan ready mix dan concrete pump – curing beton. sementara biaya yang dibutuhkan untuk tahapan ini adalah sebesar rp. 37.597.414,45. network analysis dengan program ds windows 2.0 proses ini dilakukan dengan memakai program bantu ds win 2.0 untuk melakukan evaluasi dan perbandingan terhadap perhitungan yang dilakukan secara manual dengan program dinamik. analisis yang dilakukan adalah analisis jaringan (network) yang menghubungkan notenote yang berisi metode pelaksanaan pengecoran. hasil yang hendak dicapai adalah yang paling minimum. hasil pengerjaannya adalah berikut ini, 1. minimasi biaya pelaksanaan pengecoran beton proses perhitungan diawali dengan input data setiap cabang yang merupakan jalur network biaya pekerjaan yang terdiri dari 11 cabang sebagaimana pada tabel 15. setelah dibuat jaringan dan diinput data seperti pada tabel 15, selanjutnya dilakukan running solver yang menghasilkan output perhitungan sebagaimana pada tabel 16 di bawah. tabel 15. input data cabang network biaya pelaksanaan tabel 16. ouput running solver optimasi biaya minimum aplikasi program dinamik pada metode pelaksanaan pengecoran plat lantai jurnal cauchy – issn: 2086-0382 63 dari tabel di atas tampak bahwa hasil urutan tahapan pekerjaan pengecoran adalah cabang 1 – 3 – 9 atau kalau menurut gambar 5, tahap a-b-d-g, dengan besar biaya rp 34.175.780, hal ini menunjukkan persamaan dengan hasil perhitungan pemrograman dinamik, hanya terdapat perbedaan pada besar biaya, karena program ds win 2.0 membaca angka pembulatan terdekat dari data angka yang dimasukkan (input). 2. minimasi waktu pelaksanaan pengecoran beton langkah ini sama dengan minimasi biaya, hanya saja item yang dioptimasi adalah waktu pengerjaan proyek yang terdiri dari 11 cabang network. input data sebagaimana tabel 17. hasil input pada tabel 17 di proses melalui solver, sehingga didapatkan hasil running seperti pada tabel 18 yang merupakan waktu minimum atau yang paling cepat. tabel 17. input data cabang network waktu pelaksanaan tabel 18. ouput running solver optimasi waktu minimum dari tabel di atas tampak bahwa hasil urutan tahapan pekerjaan pengecoran adalah cabang 2 – 8 – 11 atau tahap a-c-f-g, dengan lama waktu 13,187 hari, hal ini menunjukkan persamaan persis dengan hasil perhitungan pemrograman dinamik, karena digit angka yang sedikit. penutup di dalam manajemen proyek konstruksi faktor biaya dan waktu menjadi variabel yang sangat penting. visi apa yang ingin dicapai oleh sebuah kontraktor dalam setiap pelaksanaan, apakah biaya yang minimum atau waktu yang minimum. secara teori kedua nilai harapan ini tidak dapat dicapai kedua-duanya, sebab waktu yang cepat identik membutuhkan biaya yang mahal, sebaliknya biaya yang murah akan membuat proyek selesai dalam waktu yang lama. maka tentang keputusan tersebut, kontraktorlah yang menentukan pilihan. namun yang jelas, aspek manfaat jangan sampai terabaikan, artinya bermanfaat untuk diri sendiri dan orang lain, agar usaha dan segala aktivitas kita diridloi allah swt. ada banyak metode telah dikembangkan dan digunakan oleh para kontraktor dalam melaksanakan proyek fisiknya agar berhasil dengan optimum. optimum maksudnya biaya yang minimum atau waktu yang pendek. aplikasi formula matematika dapat diterapkan di berbagai persoalan dan mampu diterjemahkan dalam berbagai bidang, termasuk urusan optimasi metode pelaksanaan proyek. dari sekian banyak pemodelan matematika, salah satunya adalah pemrograman dinamik. program ini menyelesaikan persoalan secara berurutan saling terkait, baik proses maju maupun mundur. dari analisis metode yang dilakukan dengan program dinamik, ternyata optimasi berjalan sangat efektif dan relatif singkat, sehingga sang kontraktor cepat dalam mengambil keputusan, dengan tingkat resiko yang kecil. hal ini tampak jelas pada optimasi metode pengecoran plat pada gedung green house jurusan biologi uin maliki malang. walau sederhana studi kasus ini mencoba mempraktikkan program dinamik agung sedayu 64 volume 1 no. 2 mei 2010 dalam metode pelaksanaan proyek konstruksi. dan tidak menutup kemungkinan program dinamik tersebut dapat diterapkan pada contoh kasus lain di bidang konstruksi yang rumit dan kompleks. daftar pustaka [1] aminudin. 2005. prinsip-prinsip riset operasi. jakarta : erlangga [2] hafidhuddin, didin, dan tanjung, hendri. 2005. manajemen syariah dalam praktik. jakarta : gema insani [3] mulyono, sri. 2004. riset operasi edisi revisi. jakarta : penerbit fakultas ekonomi ui. [4] soeharto, iman. 1999. manajemen proyek (dari konseptual sampai operasional) jilid 1. jakarta : erlangga. [5] soeharto, iman. 2001. manajemen proyek (dari konseptual sampai operasional) jilid 2. jakarta : erlangga. [6] taha, a hamdy. 1997. riset operasi : suatu pengantar alih bahasa daniel wirajaya. jakarta : binarupa. on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages cauchy –jurnal matematika murni dan aplikasi volume 7(3) (2022), pages 483-492 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 20, 2021 reviewed: march 07, 2022 accepted: march 28, 2022 doi: http://dx.doi.org/10.18860/ca.v7i3.14434 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari*, mohamad febry andrean mathematics study program, faculty of science and technology, maulana malik ibrahim state islamic university malang, indonesia email: juhari@uin-malang.ac.id* abstract elliptic curve cryptography includes symmetric key cryptography systems that base their security on mathematical problems of elliptic curves. there are several ways that can be used to define the elliptic curve equation that depends on the infinite field used, one of which is the infinite field prima (fp where p > 3). elliptic curve cryptography can be used for multiple protocol purposes, digital signatures, and encryption schemes. the purpose of this study is to determine the process of hiding encrypted messages using the noiseless steganography method as well as the generation of private keys and public keys and the process of verifying the validity of the elliptic curves cryptography digital signature algorithm (ecdsa). the result of this thesis is that a line graph is obtained that store or hides a message using the steganography method and a message authenticity from the process of key generation and verification of validity using the ecdsa method. by selecting three samples consisting of one test sample and two differentiating samples, a line graph, an md5 hash value, and a value at the point are obtained m(r, s) different. successively obtained values m(r, s) to message “matematika 2018”, “matematika 2018”, and “2018 matematika” are m(94,67), m(15,17), and m(9,16). the discussion in this thesis only covers the elliptic curves on prime finite field. so, for the next thesis, the next researcher can do a discussion about the elliptic curve on the finite field (f2m ) or the application of elliptic curve cryptography and other steganography methods. keywords: cryptography; steganography; algorithm; elliptic curve; digital signature introduction the form of communication in this era has gone through several stages of development. this can be clearly seen from the way people use various digital devices as a means of communication. thanks to digital communication devices, people can communicate remotely through voice, text, images, and video. different types of messages sent only to http://dx.doi.org/10.18860/ca.v7i3.14434 mailto:juhari@uin-malang.ac.id* on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 484 certain parties are confidential. therefore, in sending a message or information, it is necessary to pay attention to its authenticity or authenticity so that the message you want to convey is received by the right person an algorithm is needed to maintain the authenticity of the message or information, namely elliptic curves cryptography digital signature algorithm [1], [2]. a cryptographic value that depends on the message body and the sender of the message is called a digital signature or digital signature. digital signatures generate different signatures on each document. it is a digital signature taken from the document itself [3]. basically, the use of digital signature functions the same as signatures in printed documents, namely as a process for authentication [4]. the use of a digital signature combines two cryptographic algorithms at once, namely the first one-way hash function algorithm that will produce a message digest, and the second algorithm is the public key algorithm used to encrypt the message digest. a hash function is a function that accepts an arbitrary-length string input and then compresses it into a fixed-sized message digest [5]. one of the one-way hash functions used is md5 (message digest 5) which is an improvement over md4 [6]. broadly speaking, md5 manufacturing has four steps, namely the addition of bits of the blocker, the addition of the original message length value, initialization of the md buffer, and message processing [7]. the public key algorithm in its implementation uses a pair of keys that are a public key that can be deployed, and a private key known to the owner only. elliptic curve cryptography (ecc) is cryptography that operates on elliptic curve domains. in the process of working on cryptographic algorithms, elliptic curves require mathematical concepts, namely abstract algebra including group theory, rings, and fields [8]. in addition to abstract algebraic concepts, there is a theory of numbers, especially in the modular concept of arithmetic. in the application of cryptography, one of them is the elliptic curve digital signature algorithm or elliptic curve digital signature algorithm which is based on the elgamal signature algorithm. the result of this algorithm is in the form of the authenticity of a message m. the method for keeping a message confidential is not just by using cryptography [9]. another technique that can be used besides cryptography is steganography [10]. steganography can be viewed as a complement to cryptography because they complement each other. the security of a message can be improved by combining cryptography and steganography [11]. in general, the technique used is to encrypt messages first with cryptographic algorithms, then encrypted messages are hidden in other media (voice, text, video, and images) by steganography methods [12]. if the concealment of the message on conventional steganography can degrade the quality of the cover, then the concealment of the message on the noiseless steganography or nostega methods does not cause damage. some research on elliptic curve cryptography has been done by annisa hardiningsih hr, creation and verification of digital signatures using the md5 hash function and the rsa algorithm cryptography on a document. the results show that each electronic document produces a different signature, even though it is signed by the same person and electronic documents that have not changed their contents result in the decryption value of the digital signature and message digest modulo n of the same value [13]. in previous studies has been discussed related elliptic curve cryptography and is given a simple example of the use of elliptic curve cryptography in the elgamal encoding process to make it easier to understand [14]. the study only discussed the application of elliptic curve cryptography to the elgamal encoding process only and was limited to finite fields fp. on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 485 in previous studies discusses the level of security and performance of cryptographic algorithms elliptic curves cryptography digital signature algorithm (ecdsa) or algorithms rivest-shamir-adleman (rsa) [15]. so that this research will be carried out a combination of cryptography and steganography methods without the need for a cover to hide the message. the steganography method used is noiseless steganography (nostega) and the cryptographic method used is the elliptic curves cryptography digital signature algorithm (ecdsa). method the stages of this study consist of five steps. the steps are as follows: 1. represents message 𝑀 into 8-bit ascii code. 2. perform a message concealment using the steganography method. 3. determining the equation of an elliptic curve on a primed finite field 𝐹𝑝. 4. define elements of an ellipse group a. calculating the modulo squared residual value 𝑝. b. comparing it with the value of 𝑦2 = 𝑥3 + 𝑎𝑥 + 𝑏(𝑚𝑜𝑑 𝑝). c. specifying values 𝑃(𝑥, 𝑦) on an elliptic curve as a generator of the elliptic group. d. determining a base point 𝐵(𝑥, 𝑦) selected from the ellipse group. 5. determining the elliptic curve algorithm using the elliptic curves cryptography digital signature algorithm a. performs the process of generating the public key and private key of the elliptic curves cryptography digital signature algorithm. b. perform the signature generation process of elliptic curves cryptography digital signature algorithm. c. verify the validity of the elliptic curves cryptography digital signature algorithm signature. results and discussion application of steganography method to a message change a message 𝑀 = ” matematika 2018” to a binary representation (each character is converted into an 8-bit ascii code) to 01001101 01100001 01110100 01100101 01101101 01100001 01110100 01101001 01101011 01100001 00100000 00110010 00110000 00110001 00111000 next, it will be converted back the above bit groups to decimal values 77 97 116 101 109 97 116 105 107 97 32 50 48 49 56 then a graph will be made using the decimal value above, for example, the value states the number of covid-19 cases in pasuruan regency. on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 486 picture 1. line graph that hides messages “matematika 2018” elliptic curve equation in primed finite field 𝑭𝒑 for example, given 𝐺𝐹(97) and selected 𝑎 = 1 and 𝑏 = 3 with 𝑎 and 𝑏 fulfill 4(1)3 + 27(3)2 = 247 ≢ 0 (𝑚𝑜𝑑 97), so that the elliptic curve equation is obtained [16]: 𝐺𝐹(97): 𝑦2 = 𝑥3 + 𝑥 + 3 to determine the points in an elliptic curve 𝐺𝐹(97), using the method of searching the set of modulo quadratic residues. that is by looking for all elements of the modulo 97 quadratic residual set annotated with 𝑄𝑅97, using all the elements of the set 𝐺𝐹(97) as a 𝑦point that is then squared, and the result of the square of the 𝑦 point is in modulo with 97, then the set is obtained 𝑄𝑅(97). modulo prima elliptic group elements 𝑮𝑭(𝟗𝟕) determined all points 𝑃(𝑥, 𝑦) on an elliptic curve 𝑦2 ≡ 𝑥3 + 𝑥 + 3 (𝑚𝑜𝑑 97) by 𝑥 and 𝑦 equation side 𝐺𝐹(97). then it is known the element inside 𝐺𝐹(97) is {0, 1, 2, … , 96}. performed calculations for all points on the curve 𝑦2 by substituting elements 𝐺𝐹(97) elliptic curve equation. a. find for modulo quadratic residues 𝟗𝟕(𝑸𝑹𝟗𝟕) table 1. modulo quadratic residue 97 𝒚 ∈ 𝑮𝑭𝟗𝟕 𝒚 𝟐(𝒎𝒐𝒅 𝟗𝟕) 𝑸𝑹𝟗𝟕 0 y2(mod 97) 0 1 y2(mod 97) 1 2 y2(mod 97) 4 3 y2(mod 97) 9 ⋮ ⋮ ⋮ 93 y2(mod 97) 16 94 y2(mod 97) 9 95 y2(mod 97) 4 96 y2(mod 97) 1 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 487 b. determining the value of 𝒚𝟐 ≡ 𝒙𝟑 + 𝒙 + 𝟑 (𝒎𝒐𝒅 𝟗𝟕) 𝑦2 is the value of the predetermined elliptic curve equation in table 1. by substituting each value 𝑥 ∈ 𝐺𝐹97 to equations 𝑦 2 ≡ 𝑥3 + 𝑥 + 3 (𝑚𝑜𝑑 97) then the results are obtained in table 2. table 2. value y2 ≡ x3 + x + 3 (mod 97) 𝒙 ∈ 𝑮𝑭𝟗𝟕 𝒚 𝟐 0 3 1 5 2 13 ⋮ ⋮ 94 70 95 90 96 1 c. determining sequential pairs (𝒙, 𝒚) ⊂ 𝑬𝟗𝟕 based on table 2, for 𝑥 = 1 obtained value 𝑦2 = 13 + 1 + 3 (𝑚𝑜𝑑 97) = 5. once equalized against the modulo quadratic residual value of 97 in the table 2, apparently 𝑦2 = 5 also found in 𝑄𝑅97 hence for value 𝑦1 = 11 and 𝑦2 = 18. then get a pair of dots (𝑥, 𝑦) = (1,11) dan (𝑥, 𝑦) = (1,18) which are the elements of the ellipse group 𝐸97(1,3). not all 𝑥 ∈ 𝐺𝐹97 will generate a value 𝑦 2 of elements 𝑄𝑅97. for example, for 𝑥 = 0 obtained value 𝑦2 = 03 + 0 + 3 (𝑚𝑜𝑑 97) = 3, while 𝑦2 not contained on 𝑄𝑅97. so, for 𝑥 = 0 no value 𝑦 that filled. picture 2. elliptic curve point 𝐺𝐹(97) so, the points contained on the elliptic curve are 96 points, if coupled with the o point in the infinity, then the points on the elliptic curve form a group with element 𝑛 = 97. on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 488 elliptical group generator 𝑮𝑭(𝟗𝟕) let’s 𝑃 ∈ 𝐺𝐹(97), then 𝑃 called generator or generator of 𝐺𝐹(97) if each element 𝐺𝐹(97) can be written as a rank of 𝑃 or 𝐺𝐹(97) = {𝑃𝑛 |𝑛 ∈ 𝐺𝐹(97)} where 𝐺𝐹(97) is a prime number with elements in the galois field {0,1,2, . . . ,36}. in the previous discussion, 96 points have been obtained 𝑃(𝑥, 𝑦) so that the generator of the elliptic group 𝐺𝐹(97) can be searched by summing and doubling the points of the ellipse curve with the following formula: a. addition elliptic curve point let’s 𝑃(𝑥1, 𝑦1) ∈ 𝐸(𝐹𝑝), 𝑄(𝑥2, 𝑦2) ∈ 𝐸(𝐹𝑝), and 𝑃 ≠ 𝑄, then 𝑃 + 𝑄 = (𝑥3, 𝑦3) where 𝑥3 = 𝜆2 − 𝑥1 − 𝑥2, 𝑦3 = 𝜆(𝑥1 − 𝑥3) − 𝑦1, and 𝜆 = 𝑦2−𝑦1 𝑥2−𝑥1 b. doubling a point let’s 𝑃 = (𝑥1, 𝑦1) ∈ 𝐸(𝐹𝑝) then 𝑃 + 𝑃 = 2𝑃 = (𝑥3, 𝑦3) where 𝑥3 = 𝜆 2 − 2𝑥1, 𝑦3 = 𝜆(𝑥1 − 𝑥3) − 𝑦1, and 𝜆 = 3𝑥1 2+𝑎 2𝑦1 of the 96 points of the curve that exist, it turns out that all of these points are generators of the elliptic group 𝐺𝐹(97). elliptic curves cryptography digital signature algorithm there are three digital signature elliptic curve algorithms used: 1. generation of public key and private key elliptic curves cryptography digital signature known equations of elliptic curves over infinity fields 𝐺𝐹(𝑝) that is 𝑦2 = 𝑥3 + 𝑥 + 3 (𝑚𝑜𝑑 97). from the equation obtained pairs of points of the elliptical curve of 96 points and one infinite point which can be seen in the appendix in the table. for the generation of public and private keys a value is required 𝑃𝐴 and 𝑃𝐵 for each of the two sides. sender generates its public and private keys as follows: a. select an integer 𝑥 = 3 b. count 𝑃𝐴 = 𝑥 ∙ 𝐵 𝑃𝐴 = 3 ∙ (0,10) 𝑃𝐴 = 2(0,10) + (0,10) by using the formula for doubling the points of the elliptic curve described earlier. the following will be shown the calculation process for the values of 2𝑃 and 3𝑃: a. let’s 𝑃(𝑥1 = 0, 𝑦1 = 10) ∈ 𝐺𝐹(97), then 𝑃 + 𝑃 = 2𝑃 = (𝑥3, 𝑦3) where: 𝑥3 = ( 3𝑥1 2+𝑎 2𝑦1 ) 2 − 2𝑥1 = ( 3 ∙ 02 + 1 2 ∙ 10 ) 2 − 2 ∙ 0 = ( 1 20 ) 2 − 0 = (1 ∙ 20−1)2 − 0 = (1 ∙ 34)2 − 0 = 342(𝑚𝑜𝑑97) = 89 𝑦3 = ( 3𝑥1 2+𝑎 2𝑦1 ) (𝑥1 − 𝑥3) − 𝑦1 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 489 = ( 1 20 ) (0 − 89) − 10 = 34 ∙ (−89) − 10 = 34 ∙ 8 − 10 = 78 − 10 = 68 (𝑚𝑜𝑑97) = 68 so 2𝑃 = (89,68) b. let’s 𝑃(𝑥1 = 0, 𝑦1 = 10) ∈ 𝐺𝐹(97), 𝑄(𝑥2 = 89, 𝑦2 = 68) ∈ 𝐺𝐹(97), and 𝑃 ≠ 𝑄, then 𝑃 + 𝑄 = (𝑥3, 𝑦3) where: 𝑥3 = ( 𝑦2−𝑦1 𝑥2−𝑥1 ) 2 − 𝑥1 − 𝑥2 = ( 68−10 89−0 ) 2 − 0 − 89 = ( 58 89 ) 2 − 89 = (58 ∙ 89−1)2 − 89 = (58 ∙ 12)2 − 89 = 172 − 89 = 6(𝑚𝑜𝑑 97) = 6 𝑦3 = ( 𝑦2−𝑦1 𝑥2−𝑥1 ) (𝑥1 − 𝑥3) − 𝑦1 = 17 ∙ (0 − 6) − 10 = 17 ∙ (−6) − 10 = (17 ∙ 91) − 10 = 82 𝑚𝑜𝑑 97 = 82 so 3𝑃 = (6,82) then obtained value 𝑃𝐴 = (6,82). 𝑃𝐴 = (6,82) is the sender public key and 𝑥 = 3 the private key. recipient generates her private key and public key as follows: a. select any integer 𝑦 = 2 b. count 𝑃𝐵 = 𝑦 ∙ 𝐵 𝑃𝐵 = 2 ∙ (0,10) by using the formula of doubling the points of the ellipse curve. let’s 𝑃(𝑥1 = 0, 𝑦1 = 10) ∈ 𝐺𝐹(97), then 𝑃 + 𝑃 = 2𝑃 = (𝑥3, 𝑦3) where: 𝑥3 = ( 3𝑥1 2+𝑎 2𝑦1 ) 2 − 2𝑥1 = ( 3 ∙ 02 + 1 2 ∙ 10 ) 2 − 2 ∙ 0 = ( 1 20 ) 2 − 0 = (1 ∙ 20−1)2 − 0 = (1 ∙ 34)2 − 0 = 342(𝑚𝑜𝑑97) = 89 𝑦3 = ( 3𝑥1 2+𝑎 2𝑦1 ) (𝑥1 − 𝑥3) − 𝑦1 = ( 1 20 ) (0 − 89) − 10 = 34 ∙ (−89) − 10 = 34 ∙ 8 − 10 = 78 − 10 = 68 (𝑚𝑜𝑑97) = 68 then obtained value 𝑃𝐵 = (89,68). so, 𝑃𝐵 = (89,68) is recipient's public key and 𝑦 = 2 the private key. 2. elliptic curves cryptography digital signature generate procedure sender generates a digital signature for a message m = “matematika 2018” as follows: a. choose a random integer 𝑘, whose value lies in the hose [1, 𝑝 − 1], will be selected 𝑘 = 10 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 490 b. count 𝑘 ∙ 𝐵 = (𝑥1, 𝑦1) and 𝑟 = 𝑥1 𝑚𝑜𝑑 𝑝. if 𝑟 = 0 then back to the stage 1. 𝑘 ∙ 𝐵 = 10 ∙ (0,10) = 5 ∙ (0,10) + 5 ∙ (0,10) = (3,18) + (3,18) = (94,19) so, 𝑟 = 𝑥1 = 94 𝑚𝑜𝑑 97 = 94 c. count 𝑘−1𝑚𝑜𝑑 𝑝 𝑘−1 𝑚𝑜𝑑 𝑝 = 10−1 𝑚𝑜𝑑 97 = 68 d. calculate the hash value of 𝑀, that is 𝑒 = 𝐻(𝑀). with the message conveyed “matematika 2018” then obtained 𝑒 = 𝑎6𝑏𝑑3𝑑71𝑒𝑐𝑓𝑑38391462𝑑𝑏𝑒𝑑𝑒𝑒65𝑏𝑎02 (hexadecimal) 𝑒 = 22163443765967189024324661321080961280 (decimal) e. count 𝑠 = 𝑘−1(𝑒 + 𝑥 ∙ 𝑟)𝑚𝑜𝑑 𝑝. if 𝑠 = 0, then repeat to the stage 1. 𝑠 = 10−1(221634437659671890243246613210809612802 + 3 ∙ 94) 𝑚𝑜𝑑 97 𝑠 = 67 then obtained a message 𝑀 be (94,67) from the generation of digital signatures. 3. elliptic curves cryptography digital signature verification procedure recipient will verify the digital signature (𝑟, 𝑠) from sender as follows: a. verify that 𝑟 and 𝑠 located inside the hose [1, 𝑝 − 1]. b. retrieve the sender public key, which is 3𝑃𝐴. c. recipient calculates the hash value of 𝑀, that is 𝑒 = 𝐻(𝑀). 𝑒 = 𝑎6𝑏𝑑3𝑑71𝑒𝑐𝑓𝑑38391462𝑑𝑏𝑒𝑑𝑒𝑒65𝑏𝑎02 (hexadecimal) 𝑒 = 221634437659671890243246613210809612802(desimal) d. count 𝑤 = 𝑠−1 𝑚𝑜𝑑 𝑝 𝑤 = 67−1 𝑚𝑜𝑑 97 = 42 e. count 𝑢1 = 𝑒 ∙ 𝑤 𝑚𝑜𝑑 𝑝 and 𝑢2 = 𝑟 ∙ 𝑤 𝑚𝑜𝑑 𝑝 𝑢1 = 𝑒 ∙ 𝑤 𝑚𝑜𝑑 𝑝 𝑢1 = (221634437659671890243246613210809612802 ∙ 42)𝑚𝑜𝑑 97 𝑢1 = 0 thus, the value of the value is obtained 𝑢1 = 0 𝑢2 = 𝑟 ∙ 𝑤 𝑚𝑜𝑑 𝑝 𝑢2 = 94 ∙ 42 𝑚𝑜𝑑 97 = 68 thus, the value of the value is obtained 𝑢2 = 68 f. count (𝑥1, 𝑦1) = 𝑢1 ∙ 𝐵 + 𝑢2 ∙ 𝑃𝐴 (𝑥1, 𝑦1) = 𝑢1 ∙ 𝐵 + 𝑢2 ∙ 𝑃𝐴 (𝑥1, 𝑦1) = 0 ∙ (0,10) + 68 ∙ (6,82) (𝑥1, 𝑦1) = 0 + (30(6,82) + 30(6,82) + 8(6,82)) (𝑥1, 𝑦1) = 0 + ((93,41) + (93,41) + (39,71)) (𝑥1, 𝑦1) = 0 + (26,57) + (39,71) (𝑥1, 𝑦1) = 0 + (94,19) (𝑥1, 𝑦1) = (94,19) g. count 𝑣 = 𝑥1 𝑚𝑜𝑑 𝑝 on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 491 𝑣 = 𝑥1 𝑚𝑜𝑑 𝑝 𝑣 = 94 𝑚𝑜𝑑 97 = 94 from the calculation of the value 𝑣 in the procedure of verifying the validity of obtaining the value 𝑣 = 94. in the previous procedure obtained the value 𝑟 = 94 on point 𝑀(94,67). if, 𝑣 = 𝑟 = 94 then the valid signature or authenticity of a message received is correct. conclusion the steganography method can be used to hide a secret message in another message as a cover so that the existence of the secret message cannot be detected. as well as the use of the elliptic curves digital signature algorithm method on a message to identify or authenticate the authenticity of a message sent and received correctly from the real sender and recipient. references [1] r. munir, kriptografi, bandung: informatika bandung, 2019. [2] r. a. m. m.-k. mojtaba bisheh-niasar, "cryptographic accelerators for digital signature based on ed25519," ieee transactions on very large scale integration (vlsi) systems, vol. 29, no. 7, pp. 1297-1305, 2021. [3] p. kitsos, n. sklavos and o. koufopavlou, "an efficient implementation of the digital signature algorithm," 2016. [4] j. tian, g. xiong, z. li and g. gou, "a survey of key technologies for constructing network covert channel. security and communication networks," pp. 1-20, 2020. [5] l. k. a. c. m. &. l. c. demir, "the pitfalls of hashing for privacy," ieee communications surveys & tutorials, vol. 20(1), pp. 551-565, 2017. [6] z. e. s. b. w. g. &. a. e. rasjid, " a review of collisions in cryptographic hash function used in digital forensic tools," procedia computer science, pp. 381-392, 2017. [7] n. hayati, a. budiman and m. sharif, "implementasi algoritma rc4a dan md5 untuk menjamin confidentiality dan integrity pada file teks," vol. 1, p. 7, 2017. [8] w. l. t. m. c. p. l. &. r. j. castryck, "csidh: an efficient post-quantum commutative group action," pp. (pp. 395-427), 2018. [9] p. g. a. k. t. m. c. &. y. v. k. dixit, "traditional and hybrid encryption techniques: a survey. in networking communication and data knowledge engineering," pp. (pp. 239248), 2018. [10] m. s. r. m. s. m. l. s. a. h. m. m. &. a. h. m. taha, "combination of steganography and cryptography: a short survey," vol. 518, 2019. [11] h. t. s. &. h. h. t. alrikabi, "enhanced data security of communication system using combined encryption and steganography," vol. 145, p. 15(16), 2021. [12] i. j. p. p. v. p. j. &. h. b. kadhim, "comprehensive survey of image steganography: techniques, evaluations, and trends in future research. neurocomputing," vol. 335, pp. 299-326, 2019. on the application of noiseless steganography and elliptic curves cryptography digital signature algorithm methods in securing text messages juhari 492 [13] h. a, "implementasi fungsi hash md5 dan kriptografi algoritma rsa pada pembuatan tanda tangan digital," 2021. [14] x. g. d. l. n. l. b. g. m. &. q. c. duan, "a new high capacity image steganography method combined with image elliptic curve cryptography and deep neural network," no. ieee access, 2020. [15] d. p. &. s. a. k. timothy, "a hybrid cryptography algorithm for cloud computing security," in 2017 international conference on microelectronic devices, circuits and systems (icmdcs), no. ieee, pp. (pp. 1-5), 2017. [16] w. stallings, criptography and net securitywork principles and practice, pearson education limited, 2017. characteristic of quaternion algebra over fields cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 559-567 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 21, 2022 reviewed: october 21, 2022 accepted: november 18, 2022 doi: http://dx.doi.org/10.18860/ca.v7i4.17625 characteristic of quaternion algebra over fields muhammad faldiyan, ema carnia*, asep k. supriatna department of mathematics, faculty of mathematics and natural sciences, padjadjaran university, indonesia email: ema.carnia@unpad.ac.id abstract quaternion is an extension of the complex number system. quaternion are discovered by formulating 4 points in 4-dimensional vector space using the cross product between two standard vectors. quaternion algebra over a field is a 4-dimensional vector space with bases {1, 𝑖, 𝑗, 𝑘}. quaternion algebra is an algebra that is not commutative but has an identity element and an inverse element in each element, so it is often referred to as a skew field. skew field is an interesting field structure to be studied further. the purpose of this study is to obtain the characteristics of split quaternion algebra and determine how it interacts with central simple algebra. the research method used in this paper is literature study on quaternion algebra, field and central simple algebra. the results of this study establish the equivalence of split quaternion algebra as well as the theorem relating central simple algebra and quaternion algebra. the conclusion obtained from this study is that split quaternion algebra has five different characteristics and quaternion algebra is a central simple algebra with dimensions less than equal to four. this research can be further applied to modeling movements in ℝ3 such as robots, 3d carboard film animation, and drone control. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc by-sa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: quaternion algebra; characteristics; split; central simple algebra introduction quaternion is an extension of the complex number system. in order to define 3 points in the vector space ℝ3 in the same way as 2 points defined in complex field ℂ, william rowan hamilton sought a solution in that space [1]. the results of multiplication at 3 points in ℝ3 did not work because hamilton could not find the multiplication value of 𝑖𝑗 and 𝑗𝑖. by defining 4 points in the vector space ℝ4 in the same way, hamilton expanded his investigation. then he found the solution by using the cross product between the standard vectors 𝑖, 𝑗, and 𝑘. it is essential to compute the product of 𝑖𝑗, 𝑗𝑘, 𝑖𝑘, 𝑗𝑖, 𝑘𝑗, and 𝑘𝑖 with 𝑖2 = 𝑗2 = 𝑘2 = −1. to form the quaternion number, these products must be noncommutative, in other word, 𝑖𝑗 = −𝑗𝑖 = 𝑘, 𝑗𝑘 = −𝑘𝑗 = 𝑖, and 𝑘𝑖 = −𝑖𝑘 = 𝑗 [2]. astronautics, robotics, computer graphics and visualization, animation, special effects in movies, navigation, and many other fields can all benefit from the use of quaternion numbers. let 𝐾 represent a field with characteristic not equal to 2. if 𝛼, 𝛽 ∈ 𝐾∗ = 𝐾\{0}, then exists a four-dimensional 𝐾-algebra with basis {1, 𝑖, 𝑗, 𝑘}. the symbol for this 𝐾-algebra is written as 𝐻𝐾 (𝛼, 𝛽). 𝐾-algebra is an algebra over a field 𝐾 and isomorphic with it for some 𝛼, 𝛽 ∈ 𝐾∗. an example for quaternion algebra is 𝐻𝐾 (−1, −1) = ℍ which is hamilton’s http://dx.doi.org/10.18860/ca.v7i4.17625 mailto:ema.carnia@unpad.ac.idm https://creativecommons.org/licenses/by-sa/4.0/ characteristic of quaternion algebra over fields muhammad faldiyan 560 quaternion. with the group center at 𝐾, quaternion algebra is a central simple algebra over field 𝐾 that is associative, non-commutative, and devoid of two-sided ideals [3]. for parameterizing smooth rotation and spatial transformations in vector space, quaternion representations are greatly favored. they eliminate the issue of gimbal locks and are typically regarded as being resilient to sheer/scaling noise and perturbations (i.e., numerically stable rotations). the model has additional degrees of freedom because quaternion rotations operate on two planes of rotation, whereas complex rotations only operate on one [4]. tarnauceanu’s research [5] discusses the characteristics of quaternion numbers. this research studies about the properties of quaternion numbers in general, but has not discussed further about the characteristics of quaternion numbers over the field. savin’s research [6] discusses quaternion algebra which is split over a certain field. this research learns more about the property of split in quaternion algebra over field in general. savin’s research [7] discusses about the class of division quaternion algebras over the field. this research studies the special properties of division algebra in quaternion algebra. acciaro’s research [3] discusses about the characteristics of quaternion algebras over quadratic field. this research is a development of research results from previous studies. this study discusses the characteristics of quaternion algebra over a field, the characteristics of split quaternion algebra, and the use of division algebra to prove the theorems discussed in this study. the author’s research discusses the development of acciaro’s research by adding and completing the proof of the theorem used in acciaro’s research. according to study in [8], several characteristics of quaternion are described which are not a division algebra and have isotropic elements. according to study [3], additional characteristics of quaternion algebra, including its relationship to the hilbert symbol and isomorphic with square matrix over a field, are discussed in this paper. in both studies, the theorem which discusses the characteristics of quaternion algebra is not shown to be proven directly. in this study, we will discuss about the characteristics of quaternion algebra, its full proof, and how it relates to central simple algebra. the main goals of this study were obtained the characteristics of split quaternion algebra and obtained the connection between quaternion algebra and central simple algebra. this research is expected to provide benefits for the development of mathematics about split quaternion algebra over a field and central simple algebra. methods in this section, we will discuss the materials and methods used and discuss about the definitions used in this study related to quaternion algebra. the research used in this study is a literature study on quaternion algebra over a field and its properties. the following are definitions and reference theories used in this study to answer the existing problems. definition 1. (see [9]) (quaternion) quaternion can be expressed mathematically as follows. ℍ = {𝑎 + 𝑏𝑖 + 𝑐𝑗 + 𝑑𝑘; 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ} where the following multiplication rule conditions apply: 1. 𝑖2 = 𝑗2 = 𝑘2 = −1. 2. 𝑖𝑗 = 𝑘, 𝑗𝑖 = −𝑘, 𝑗𝑘 = 𝑖, 𝑘𝑗 = −𝑖, 𝑘𝑖 = 𝑗, 𝑖𝑘 = −𝑗. 3. for any element 𝑎 ∈ ℝ is commutative with 𝑖, 𝑗, and 𝑘. characteristic of quaternion algebra over fields muhammad faldiyan 561 definition 2 (see [10]) (algebra over field) suppose that 𝐾 is a field and 𝐴 is a vector space over field 𝐾 equipped with binary addition operations from 𝐴 × 𝐴 to 𝐴. if 𝐴 meets the axioms below, it is an algebra over a field 𝐾, or so-called 𝐾-algebra: (𝜆𝑎)𝑏 = 𝑎(𝜆𝑏) = 𝜆(𝑎𝑏), ∀𝜆 ∈ 𝐾 𝑎𝑛𝑑 ∀𝑎, 𝑏 ∈ 𝐴 definition 3 (see [3]) (zero divisor) the associative algebra 𝐴 over a field is a division algebra if and only if it contains a multiplication identity element 1 ≠ 0 and every non-zero element in 𝐴 has a left and right multiplication inverse. if 𝐴 is an algebra which has finite dimensions, it can only be a division algebra if it does not have a zero divisor. definition 4 (see [11]) (division ring) division ring is a ring 𝐾 where each non-zero element has an inverse (two sides) i.e. 𝐾\{0}. definition 5 (see [11]) (division algebra) division algebra is an algebra over division ring 𝐾 (every non-zero element has an inverse). definition 6 (see [3]) (isotropic) vector is said to be isotropic if norm of a non-zero vector 𝑞 ∈ 𝐻𝐾 (𝛼, 𝛽) is zero (𝑁(𝑞) = 0). definition 7 (see [12]) (central simple algebra) let 𝐾 be a field, 𝐿 be its field extension, and 𝐴 be a 𝐾-algebra with finite dimensions. 𝐴 is said to be central simple algebra over 𝐾 if it satisfies one of the following equivalent conditions: 1. 𝐴 is a simple ring with center 𝐾. 2. there exists an isomorphism 𝐴 ≅ 𝑀𝑛(𝐷) where 𝐷 is a division algebra with center 𝐾. 3. 𝐴𝐿 isomorphic for some 𝑑 with ring matrix 𝑀𝑑 (𝐿). definition 8 (see [13]) (center group) let 𝐴 represent 𝐾-algebra. the center of a group 𝐴 is defined by the set 𝑍(𝐴) = {𝑧 ∈ 𝐴| 𝑎𝑧 = 𝑧𝑎, ∀𝑎 ∈ 𝐴} 𝐴 is said to be center or central if 𝑍(𝐴) = 𝐾. definition 9 (see [11]) (simple algebra) let 𝐴 represent 𝐾-algebra. 𝐴 is said to be simple if it does not have a non-trivial two-sided ideal, i.e. 𝐴 only has {0} and 𝐴 which is a two-sided ideal. definition 10 (see [13]) (central simple algebra) central simple algebra is 𝐾-algebra which is central and simple. characteristic of quaternion algebra over fields muhammad faldiyan 562 theorem 11 (see [8]) let 𝐴 be a finite dimensional 𝐾-algebra and 𝐾 be a field. if and only if there are integers 𝑛 > 0 and a finite field extension 𝐾|𝑘 such that 𝐴 ⊗𝑘 𝐾 is isomorphic with a ring matrix 𝑀𝑛 (𝐾), then 𝐴 is a central simple algebra. definition 12 (see [14]) (field extension) consider that 𝐸 and 𝐹 are fields. 𝐸 is a field extension of 𝐹 if and only if 𝐹 is a subfield of 𝐸. the vector space 𝐸 over a field 𝐹 is identified by the symbols 𝐸/𝐹 and its dimensions are denoted by [𝐸: 𝐹]. definition 13 (see [3]) (split algebra) let 𝐴 represent a central simple algebra over a field 𝐾. if 𝐴 is isomorphic to the square matrix 𝑀2(𝐾) over 𝐾, then 𝐴 is said to be split over field 𝐾. definition 14 (see [15]) (hilbert equation) for any field 𝐹 and any element 𝑎, 𝑏 ∈ 𝔽∗ = 𝔽\{0}. we say that 𝑎𝑋2 + 𝑏𝑌2 = 𝑍2 is the hilbert equation, and the solution (𝑥, 𝑦, 𝑧) ∈ 𝔽3 is trivial if and only if 𝑥 = 𝑦 = 𝑧 = 0, and non-trivial otherwise. definition 15 (see [15]) (hilbert symbol) let 𝑅 ⊆ 𝔽 be a subring and 𝔽 be a field. define the mapping that resolves the quadratic diagonal equation in three variables with coefficients in 𝔽 with solvency 𝑅 non-trivial: ℎ𝑅,𝔽(∙,∙) ∶ 𝔽 ∗ × 𝔽∗ → {−1,1} (𝑎, 𝑏) ⟼ { 1, 𝑖𝑓 ∃(𝑥, 𝑦, 𝑧) ∈ 𝑅3\{0} ∶ 𝑎𝑥2 + 𝑏𝑦2 = 𝑧2 −1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 the definitions and theorems written above will be used in the results and discussion section to answer the research problems asked in the introduction section. the steps in this research are as follows. 1. the literature study is searched using publish or perish software by entering keywords, year of publication, and maximum article limit. the keyword entered is quaternion algebra and quadratic field. the year of publication included is the last five years of research on quaternion algebra and quadratic fields. the maximum article limit is 500 articles. 2. the literature obtained from publish or perish software was selected by eliminating literature in the form of books, doctoral thesis, proceedings, and other topics that were not relevant to this research. 3. screening articles through abstracts that are irrelevant to the topic who want to discuss. 4. read the entire article and choose one article that you want to cover further. 5. one article that has been selected, understood in depth and developed the result of the research. characteristic of quaternion algebra over fields muhammad faldiyan 563 6. write the progress of the research results of previous paper in this research and complete the proof of the theorems in the previous paper. results and discussion a quaternion known as a ring which has an inverse for each non-zero element and is not commutative with regard to multiplication. it is well known that quaternion algebra possesses central and simple features since it is a central simple algebra over a field. quaternion algebra has special characteristics in its properties and in this study, the properties of quaternion algebra over fields with characteristics other than two will be studied in more detail. the characteristics of split quaternion algebra over a field and its equivalents are demonstrated in the following theorem. theorem 16 [3] let 𝐾 represent a field with characteristic is not equal to 2 (𝑐ℎ𝑎𝑟(𝐾) ≠ 2) and let 𝛼, 𝛽 ∈ 𝐾\{0} and 𝐴 is an quaternion algebra (𝐴 = 𝐻𝐾 (𝛼, 𝛽)). the following statements are equivalent. 1. 𝐴 ≅ 𝐻𝐾 (1, −1) ≅ 𝑀2(𝐾); 2. 𝐴 is not a division algebra; 3. 𝐴 contains an isotropic element; 4. the equation 𝛼𝑥2 + 𝛽𝑦2 = 1 has a solution (𝑥, 𝑦) ∈ 𝐾 × 𝐾; 5. 𝛼 is a norm of 𝐾(√𝛽); if any of the statements above apply to 𝐴, then 𝐴 said to be split over a field 𝐾. proof: the equivalence of this theorem will be demonstrated by searching for the implications of the statements (1) ⟹ (2), (2) ⟹ (3), (3) ⟹ (4), (4) ⟹ (5), and (5) ⟹ (1). we reviewed 5 cases of implications for solving the equivalence of the theorem. 1. (𝟏) ⟹ (𝟐) first, it is known that the quaternion algebra 𝐻𝐾 (1, −1) can be defined as follows. 𝐻𝐾 (1, −1) = {𝑎 + 𝑏𝑖 + 𝑐𝑗 + 𝑑𝑘 │𝑖 2 = 1, 𝑗2 = −1, 𝑘2 = 1} with 𝑖𝑗 = 𝑘, 𝑗𝑖 = −𝑘. then, it is known that 𝐻𝐾 (1, −1) is isomorphic with 𝑀2(𝐾) where the base is {1, 𝑖, 𝑗, 𝑘} and the square matrix of the base can be defined as follows. 1 = ( 1 0 0 1 ) , 𝑖 = ( 0 1 1 0 ) , 𝑗 = ( 0 1 −1 0 ) , 𝑘 = ( 1 0 0 −1 ) for instance, choose two non-zero elements in 𝐻𝐾 (1, −1), namely 1 + 𝑖 = ( 1 1 1 1 ) and 1 − 𝑖 = ( 1 −1 −1 1 ). multiply the two arbitrary elements so that (1 + 𝑖)(1 − 𝑖) = ( 1 1 1 1 ) ( 1 −1 −1 1 ) = ( 1 − 1 −1 + 1 1 − 1 −1 + 1 ) = ( 0 0 0 0 ) the fact that zero results from the product of two non-zero elements means that 𝐻𝐾 (1, −1) has a zero divisor. since ∃(1 + 𝑖), (1 − 𝑖) ≠ 0 ∈ 𝐻𝐾 (1, −1) ∋ (1 + 𝑖)(1 − 𝑖) = 0, it may be deduced from definition 3 that a quaternion algebra is not a division algebra. characteristic of quaternion algebra over fields muhammad faldiyan 564 based on the evidence above, it can be concluded that 𝐻𝐾 (1, −1) ≅ 𝑀2(𝐾) is not a division algebra. 2. (𝟐) ⟹ (𝟑) first, it is known that 𝐻𝐾 (𝛼, 𝛽) is not a division algebra which means that 𝐻𝐾 (𝛼, 𝛽) has a zero divisor. an element 𝑎 ∈ 𝐻𝐾 (𝛼, 𝛽) is said to have an inverse if its norm is not equal to zero (𝑁(𝑎) ≠ 0). 𝐻𝐾 (𝛼, 𝛽) is not a division algebra if it is connected to the concept of norm, which means that some of its member lack an inverse or in other word, it has a zero divisors element. consider choosing the element 𝑏 ≠ 0 ∈ 𝐻𝐾 (𝛼, 𝛽) in such a way that 𝐻𝐾 (𝛼, 𝛽) is not a division algebra. the norm of element 𝑏 is zero (𝑁(𝑏) = 0) because the element 𝑏 lacks an inverse. based on the evidence above, it can be concluded that there are isotropic elements in 𝐻𝐾 (𝛼, 𝛽). 3. (𝟑) ⟹ (𝟒) first, it is well known that quaternion algebra has isotropic elements. suppose we choose 𝑔 = 1 + 𝑖 ∈ 𝐻𝐾 (𝛼, 𝛽) which is an isotropic element. so according to definition 6, the norm of isotropic element is as follows. 𝑁(𝑔) = 𝑔�̅� = 0 (1 + 𝑖)(1 − 𝑖) = 0 12 − 𝑖2 = 0 1 − 𝛼 = 0 → 𝛼 = 1 the equation 𝛼𝑥2 + 𝛽𝑦2 = 1 can be changed to 𝑥2 + 𝛽𝑦2 = 1. there are two cases of this quadratic equation, namely  if 𝛽 = 0, then the value of 𝑥 = √1 = ±1  if 𝛽 ≠ 0, then the value of 𝑦 = ± √1−𝑥2 √𝛽 we have reviewed these two cases and concluded that for any value of 𝛽, the equation 𝑥2 + 𝛽𝑦2 = 1 must have a solution (𝑥, 𝑦) ∈ 𝐾 × 𝐾 so that based on the proof above, the equation 𝛼𝑥2 + 𝛽𝑦2 = 1 must have a solution on the field 𝐾 × 𝐾 for algebra having isotropic elements. 4. (𝟒) ⟹ (𝟓) first, it is known that the equation 𝛼𝑥2 + 𝛽𝑦2 = 1 has a solution (𝑥, 𝑦) ∈ 𝐾 × 𝐾. assume that 𝛽 is not a square in 𝐾\{0}. since 𝛽 is not a square in 𝐾\{0} then 𝑥 ≠ 0. then the equation can be formed as 𝛼𝑥2 + 𝛽𝑦2 = 1 𝛼 + 𝛽 ( 𝑦 𝑥 ) 2 = 1 𝑥2 𝛼 = ( 1 𝑥 ) 2 − 𝛽 ( 𝑦 𝑥 ) 2 ( 1 𝑥 ) 2 − 𝛽 ( 𝑦 𝑥 ) 2 = 𝑁 ( 1 𝑥 + ( 𝑦 𝑥 ) √𝛽) the aforementioned evidence leads to the conclusion that 𝛼 is the norm of field 𝐾(√𝛽). 5. (𝟓) ⟹ (𝟏) characteristic of quaternion algebra over fields muhammad faldiyan 565 first, it is known that 𝛼 = 𝑁 (𝐾(√𝛽)) such that 𝛼 = 1 𝑥2 − 𝛽 ( 𝑦2 𝑥2 ) 𝛼 + 𝛽 ( 𝑦2 𝑥2 ) = 1 𝑥2 𝛼𝑥2 + 𝛽𝑦2 = 1 if 𝛼, 𝛽 are not both negative values, then 𝛼, 𝛽 have a solution in this case. let’s say we select 𝛼 = 1, 𝛽 = −1 such that we obtain 𝐻𝐾 (1, −1). based on the evidence above, it can be concluded that the norm of the quadratic field contains the quaternion algebra 𝐻𝐾 (1, −1). based on the evidence from the five cases and based on the nature of the implications, it can be concluded that statements (1) − (5) are equivalent. another characteristic of quaternion algebra is that it discusses its relationship to central simple algebra. it is known that quaternion algebra is a central simple algebra. this theorem explains the structure of central simple algebra and establishes a connection between quaternion algebra and central simple algebra. theorem 17 if 𝐴 is a central simple algebra over a field 𝐹 with dimension less than 4, then 𝐴 is isomorphic to the quaternion algebra over a field 𝐹. proof: let 𝐴 represent a central simple algebra with dimension less than 4. since 𝐴 is the central simple algebra, 𝐴 ≅ 𝑀𝑛(𝐷), where 𝐷 is a division algebra over field 𝐹, according to definition 7, since 𝐴 ≅ 𝑀𝑛(𝐷), the dimension of 𝐴 is 𝑛 2. 𝐴 has dimension less than 4 so that the possible values of 𝑛 are 1 or 2. consider the following two scenarios: 𝑛 = 1 and 𝑛 = 2.  if 𝑛 = 1, then 𝐴 ≅ 𝐷 so that 𝐴 is a division algebra over field 𝐹.  if 𝑛 = 2, then 𝐴 ≅ 𝑀2(𝐷) so that = 𝐹. 𝐴 ≅ 𝑀2(𝐹). assume that 𝐴 is a division algebra. take a field 𝐾 = 𝐹(𝑖) ⊆ 𝐴 and the non-central element 𝑖 ∈ 𝐴. since 𝐹 ⊆ 𝐾 ⊆ 𝐴 so that dim𝐹 𝐾 = 2 dan dimf 𝐴 = 4. thus, 𝐾 is a quadratic field extension of 𝐹 and assume that 𝑖 ∈ 𝐾 has been chosen such that 𝑖2 = 𝑎 ∈ 𝐹. let 𝑓 be the inner automorphism in 𝐴 defined by 𝑖. following that, 𝑓 2 = identity and eigenspace decomposition 𝐴 = 𝐴+ ⊕ 𝐴−, where 𝐴+ = {𝑥 ∈ 𝐴: 𝑓(𝑥) = 𝑥} = {𝑥 ∈ 𝐴: 𝑖𝑥 = 𝑥𝑖}, 𝑎𝑛𝑑 𝐴− = {𝑥 ∈ 𝐴: 𝑓(𝑥) = −𝑥} = {𝑥 ∈ 𝐴: 𝑖𝑥 = −𝑥𝑖} consider a non-zero element 𝑗 ∈ 𝐴−. since 𝐾 ⊆ 𝐴+ and 𝐾 ∙ 𝑗 ∈ 𝐴−, we get 𝐾 = 𝐴+ dan 𝐾 ∙ 𝑗 = 𝐴−. we have 𝑗2𝑖 = 𝑖𝑗2 because of 𝑖𝑗 = −𝑗𝑖 which means that 𝑗2 ∈ 𝐴+ = 𝐾. on other hand, 𝐹(𝑗) is also an extension of the quadratic field of 𝐹, so that 𝑗 contains quadratic equation 𝑗2 + 𝑐𝑗 − 𝑏 = 0, for 𝑏, 𝑐 ∈ 𝐹. the equation 𝑐𝑗 = 𝑏 − 𝑗2 ∈ 𝐾 suggests that 𝑐 = 0 and 𝑗2 = 𝑏 ∈ 𝐹 from these equations, thus 𝐴 = 𝐾 ⊕ 𝐾 ∙ 𝑗 = 𝐹 ⊕ 𝐹𝑖 ⊕ 𝐹𝑗 ⊕ 𝐹𝑖𝑗 𝐴 isomorphic with quaternion algebra ( 𝑎,𝑏 𝐹 ). based on the above evidence, it can be concluded that 𝐴 is isomorphic with quaternion algebra over a field 𝐹. characteristic of quaternion algebra over fields muhammad faldiyan 566 conclusions the conclusion of this study is that there are five characteristics of quaternion algebra that are split over a field, which are equivalent to a square matrix with element of field, is not a division algebra, has isotropic elements, is equivalent to the hilbert symbol which has a solution to the field, and is equivalent to the norm of a quadratic field. besides that, quaternion algebra over a field isomorphic with a central simple algebra with dimensions less than four. acknowledgements the author would like to thank to the indonesian ministry of education, culture, research, and technology for funding this research under master thesis research project in 2022 entitled “aljabar quaternion atas komposit lapangan kuadratik”, grant number 1318/un6.3.1/pt.00/2022. references [1] t. y. lam, “hamilton’s quaternions,” handbook of algebra, vol. 3, pp. 429–454, 2003. [2] j. c. familton, “quaternions: a history of complex noncommutative rotation groups in theoretical physics,” columbia university, new york, 2015. [3] v. acciaro, d. savin, m. taous, and a. zekhnini, “on quaternion algebras that split over specific quadratic number fields,” italian journal of pure and applied mathematics-n, vol. 47, no. 2022, pp. 91–107, jun. 2019. [4] s. zhang, y. tay, l. yao, and q. liu, “quaternion knowledge graph embeddings,” in proceedings of the 33rd international conference on neural information processing systems, 2019, pp. 2735–2745. doi: https://doi.org/10.48550/arxiv.1904.10281. [5] m. tărnăuceanu, “a characterization of the quaternion group,” analele universitatii ovidius constanta seria matematica, vol. 21, no. 1, 2013, doi: https://doi.org/10.2478/auom-2013-0013. [6] d. savin, “about some split central simple algebras,” mar. 2014, [online]. available: http://arxiv.org/abs/1403.3443 [7] d. savin, “about division quaternion algebras and division symbol algebras,” nov. 2014, [online]. available: http://arxiv.org/abs/1411.2145 [8] p. gille and t. szamuely, central simple algebras and galois cohomology. cambridge university press, 2009. [9] k. conrad, “quaternion algebras,” storrs, 2018. [10] m. hazewinkel, n. gubareni, and v. v. kirichenko, algebras, rings and modules, vol. 1. springer, 2004. [11] j. voight, quaternion algebras. springer, 2021. [online]. available: http://www.springer.com/series/136 [12] e. tengan, “central simple algebras and the brauer group,” xviii latin american algebra colloquium, aug. 2009. [13] g. berhuy and f. oggier, an introduction to central simple algebras and their applications to wireless communication, vol. 191. american mathematical society, 2013. [online]. available: www.ams.org/bookpages/surv-191 [14] c. schwarzweller, “field extensions and kronecker’s construction,” formalized mathematics, vol. 27, no. 3, pp. 229–235, oct. 2019, doi: 10.2478/forma-2019-0022. characteristic of quaternion algebra over fields muhammad faldiyan 567 [15] l. a. wallenborn, “berechnung des hilbert symbols, quadratische form-¨ aquivalenz und faktorisierung ganzer zahlen (computing the hilbert symbol, quadratic form equivalence and integer factoring),” 2013. spatial durbin model (sdm) for identified influence dengue hemorrhagic fever factors in kabupaten malang indah resti ayuni suri1), henny pramoedyo2), endang wahyu handamari3) 1magister students of mathematics department, brawijaya university, malang 2,3lecturer of mathematics department, brawijaya university, malang 1email: ayuni_suri@yahoo.com abstrak dengue hemorrhagic fever or usually populer call dbd (demam berdarah degue) is the cronic desease that caused by virus infection who carry by aedes aegypti mousquito. the observation act by dbd descriptioning and some factors territorial view that influence them, also dbd’s modeling use spatial durbin model (sdm). sdm is the particullary case from spatial autoregresive model (sar), it means modeling with spatial lag at dependen variable and independen variable. this observation use ratio dbd invectors amount with population amount of citizenry at kabupaten malang in 2009. some variable was used, those are the precentation of existention free number embrio, ratio of civil amount between family, procentation of healthy clinic between invectors and procentase of the invectors who taking care by medical help with amount of invectors. the fourth variables are independen variable to ratio of dbd invector amount with population of citizenry amount, as dependen variable trough spatial sdm modelling. the result of sdm parameter modelling, the significant influence variable in session % is the procentation of free amount embrio existention from their own district, the procentation of healthy clinic amount with the dbd invector amount from their own district, the ratio of the population of citizenry with the family from their neighborhood district, and the procentation of healthy clinic amount with the dbd invector amount from their neighborhood district. kata kunci: dbd modelling , spatial durbin model. introduction territory indonesia has natural resources and biodiversity, but indonesia is also rich in disasters often hit several regions in indonesia. disaster hit several regions in indonesia in recent years, ranging from natural disasters of floods, landslides, forest fires, volcanic eruptions, droughts, earthquakes and tsunamis. the disaster caused fatalities and property no small value. according to who, the definition of a disaster is any occurrence that causes damage, ecological disruption, loss of human life or worsening of health status or specific health services on a scale that requires a response from outside the affected community or region. from the above definition it can be concluded that the so-called catastrophic not only natural disasters, but the extraordinary incident (klb) can also be categorized as a disaster. extraordinary events that have occurred in indonesia include dengue fever, diarrhea, polio (who, 2003). given the importance of effort and awareness of the community to not provide a means or a place for the proliferation of extraordinary events, all people should have an early alert to the threat of pandemic disaster. one is dengue fever, people should have a concern for the environmental hygiene (sanitation) so that the mosquito aedes aegypti is not able to grow. especially for malang regency consists of 39 centers from 33 districts. from each district in malang, is expected to prevent the breeding of mosquitoes that will improve public health in particular and attempt to degrade and reduce morbidity and mortality due to infectious diseases. it is really very difficult to stop the aedes aegypti mosquito population. actually, to this day is still considered effective activities to reduce aedes aegypti mosquito populations is to break the chain development. dengue hemorrhagic fever (dengue hemorrhagic fever), known as dengue, an acute disease caused by a viral infection carried by the mosquito aedes aegypti. during this time a lot of research in the fields of health, environment, economics, and other areas not involving spatial effects as being influential in the model so often lead to conclusions that are less precise. spatial effects exist be explained by a model that can capture these phenomena. anselin (1988) distinguish the spatial effects into two parts, namely the spatial dependency and spatial heterogeneity. spatial dependencies are shown with similar properties for locations that are close together, while the spatial heterogeneity shown by the difference in mailto:ayuni_suri@yahoo.com sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 75 75 nature between the various locations. spatial dependencies occur in areas which have close proximity resulting in spatial interaction in the area. spatial data analysis needs to be done with attention to the linkages antarlokasi (spatial autocorrelation). spatial linkages can be seen from a distance antarlokasi so as to analyze the spatial relationship is often used as a weighting antarlokasi distance. weighting matrix is expressed in used to calculate the autocorrelation coefficient. matrix is a matrix whose entries are weighted value assigned to a specific interregional comparisons. weighting was based on the spatial relationships between regions. in some cases the spatial model, weighting only be assessed on the dependent variable, while durbin model (sdm) is a model using spatial weighting on independent and dependent variables. the condition of a region generally associated with conditions in other areas, especially areas adjacent. is known as spatial relationships. the first law of geography proposed by tobler, stating that everything is inter-related to each other, but something close to have more influence than anything away (anselin, 1988). the law is the basis of assessment of the problem based on the effect of location or spatial methods. there are several studies that use spatial models spatial durbin model (sdm) of which is the research conducted by kissling and carl (2007) applied research in the fields of ecology and biogeography that models the effect of rain forest relation to the spread of the organism in auckland, new zealand. stating that spatial autocorrelation can affect the independent and dependent variables so w_1 x added in the modeling. bekti (2011) apply epidemiological research in modeling the relationship influences the availability of infrastructure, sanitation, clean water, and health facilities as independent variables on the incidence of diarrhea in tuban, east java as the dependent variable using the test moran's i. in the study, stated that the lag of the dependent and independent variables plays an important role in modeling the incidence of diarrhea and the factors that influence it. in the second study, the type of matrix used is the weighted contiguity matrix and is recommended for future studies using different methods in order to obtain a more precise modeling. in the study indrayati (2012) modeled the effect of human spatial density urban, bare soil, and density industrial area, as the independent variables on the dependent variable vegetation density in the city of surabaya. this research will be carried out modeling of human resources in health. the model used for spatial modeling approach using influence spatial area of the dependent and independent variables. the method used is the spatial durbin model (sdm), which is one of the model using spatial weighting on the independent variables and the dependent variable. in this analysis will want to obtain a regression model of the factors that significantly influence the number of dengue fever patients in every district in malang. by exploiting the spatial information, the model predicted results can be presented in a map to know dengue fever in malang. literature review spatial data spatial data is measurement data containing location information. spatial data collected from different spatial locations indicating dependence between measurement data by location (cressie, 1993). the word is derived from the spatial space, which means space is spatial and spatial means. spatial data is measurement data containing an information site. spatial data has a certain coordinate system as the basis of reference so as to have two essential parts that make it different from other data, location information (spatial) and descriptive information (attributes) are described as follows (gis consortium aceh nias, 2007): 1. location information (spatial), corresponds to a coordinate both geographic coordinates (latitude and longitude) and the xyz coordinates, including the datum and projection information. 2. descriptive information (attributes) or non spatial information is information relating to the location (spatial), for example: type of vegetation, population, area, zip code, and so on. purwaamijaya (2008) describes one of the sources for spatial data is to map analog. analog maps are maps in printed form, such as topographic maps, soil maps, and so on. in general, analog maps made with cartographic techniques that are likely to have a spatial reference, such as coordinates, scale, wind direction, and so on. spatial data becomes an important medium for development planning and management of natural resources in the coverage area of sustainable national, regional, and local levels. the use of spatial data increased as digital mapping technology and its use in geographic information systems (gis). mapping is a process of presentation of information the actual earth, good shape earth's surface and its natural axis, based on the map scale, map projection systems, as well as symbols indah resti ayuni, henny pramoedyo, endang wahyu handamari 76 76 volume 2 no. 2 mei 2012 of the elements of the earth are presented. advances in technology, particularly in the field of computer makes the map not only in tangible form (on paper, real folders, or hardcopy), but can also be stored in digital form that can be presented on the screen, known as virtual maps (maps virtual or softcopy). digital mapping is a process of making maps in a digital format that can be saved and printed as desired manufacturer. model spatial the model is a functional relationship, including variables, parameters, and equations. a variable is a symbol used to represent an item that can be numeric or value whatever. the independent variable (independent variable) is a variable whose value does not depend on anything in the equation, while the dependent variable (the dependent variable) is a variable whose value depends on the independent variable. the parameter is a constant value which is usually the coefficients of the variables in an equation. the value of the parameters obtained from the data (taylor, 1999). according prahasta (2004), in general, a model is a representation of reality. therefore the aim of developing a model is to help the user to understand, describe, or make predictions about how things work in the real world the use of gis in spatial modeling will make the output (output) into a spatial information that is both two-dimensional and three-dimensional so that the information will become more clear and useful for businesses utilization and processing of natural resources. as a simplification of a system, the spatial model using gis also has four major components of a system, the hardware (hardware), software (software), data (dataware), and human resource (brainware). spatial method is a method to obtain observations that influenced the effects of space or location. models often use spatial relations in the form of dependency relationship with the model autoregressive covariance structure (wall, 2004). lesage and pace (2009) states that the autoregressive process is shown through correlation dependence (autocorrelation) between a set of observations or location. spatial modeling during this time a lot of research in the fields of health, environment, economics, and other areas not involving spatial effects as being influential in the model so often lead to conclusions that are less precise. existing spatial effects should be explained by a model that can anselin (1988) distinguish the spatial effects into two parts, namely the spatial dependency and spatial heterogeneity. spatial dependencies are shown with similar properties for locations that are close together, while the spatial heterogeneity shown by the difference in nature between the various locations. spatial dependencies occur in areas which have close proximity resulting in spatial interaction in the area. the first law of geography proposed by tobler, stating that everything is inter-related to each other, but something close to have more influence than anything away (anselin, 1988). the law is the basis of assessment of the problem based on the effect of location or spatial methods. based on the data types, spatial modeling can be divided into modeling approaches and point area. 1. this type of approach point geographically weighted regression them (gwr), geographically weighted poisson regression (gwpr), space-time autoregressive (star), and the generalized space-time autoregressive (gstar). 2. this type of approach among area spatial autoregressive model (sar), spatial error model (sem), spatial durbin model (sdm), conditional autoregressive model (car), and the spatial autoregressive moving average (sarma). spatial autoregressive models are models that follow the autoregressive process, which is indicated by the dependence relationship between a set of observations or location. the relationship is shown by the lag on the dependent and independent variables. type spatial autoregressive models such as sar, sem, hr, and car (anselin, 1988). spatial durbin model (sdm) according to anselin (1988) general model of spatial autoregressive between the independent variable (x) with the dependent variable (y) is expressed in the form (2.1) (2.2) (2.3) which is: : vector variable depends : matrix variable independent : the vector parameter regression coefficients : layer spatial variable coefficient depends : parameter coefficients team the spatial error : vector error sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 77 77 : error vector, which is normally distributed capture these phenomena. with mean zero and variance : weighting matrix : identity matrix, indah resti ayuni, henny pramoedyo, endang wahyu handamari 78 78 volume 2 no. 2 mei 2012 : number of observations or location : number of variable independent. when ρ = 0 and λ = 0 it will be a simple linear regression model parameter estimation can be done via ordinary least squares (ols) as the following equation: (2.4) with . in this model there is no effect or no spatial autocorrelation. matrix and in equation (2.1) and (2.2) is a weighting that shows relationship antarlokasi contiguity or distance function. diagonal is zero or w_ij = 0 for i = j and w_ij ≠ 0 for i ≠ j, where i = 1,2, ..., n and j = 1,2,...,n is an observation or location. spatial durbin model (sdm) is a model with the effect of spatial autoregressive lag on the dependent and independent variables. sdm model form is as follows: development in a region other than the independent variable affected is also affected by other variables, the spatial relationships. representation of the spatial durbin location factor model in the form of the so-called adjacency matrix weighting matrix (lesage, 1999). spatial data analysis needs to be done with attention to the linkages between locations (spatial autocorrelation). spatial linkages can be seen from the distance between locations so as to analyze the spatial relationship is often used as a weighting distances between locations. weighting matrix is expressed in w is used to calculate the autocorrelation coefficient. the matrix w is a matrix whose entries are given weighted values for comparisons between specific areas. weighting is based on the spatial $ relationships between regions. in some cases the ! " # # () ! ('* +*, spatial model, weighting only be assessed on the %&' $ *& ' dependent variable, whereas hr is one of the model using spatial weighting on independent ! ( * ! .,% +*% , /0 and dependent variables. *&' %& ' assumptions spatial durbin model (sdm) where k = i = a lot of variables and a lot of scrutiny. equation (2.5) can be expressed in matrix form as shown in equation (2.6) where the coefficient of the spatial lag parameter vector of independent variables declared in () (2.6) atau 1 2 3 1 2 3 1 2 4' 1 2 4' 2 4' 1 2 4' 2 4' 5 this is 1 dan () (' ( 5 sdm model was developed on the grounds that in some cases the spatial dependency relationship does not only occur on the dependent variable, but also on the independent variables. therefore, it should be added spatial lag model (lesage and pace, 2009). the basic principle of spatial durbin model is almost the same as the weighted regression (weighted regression), with a variable weighting factor is location. proximity and linkages antarlokasi this phenomenon led to the emergence of spatial autocorrelation. spatial durbin model is the development of a simple regression model that has to accommodate spatial autocorrelation phenomena, both in the dependent variable and the independent variable. for example, to determine the level of sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 79 79 spatial autocorrelation according to lembo (2006) spatial autocorrelation is the correlation between a variable with itself by space. spatial autocorrelation can also mean a measure of the similarity of objects in space (distance, time, and region). if there are systematic patterns in the spread of a variable, then there is spatial autocorrelation. the existence of spatial autocorrelation indicates that the value of an attribute in a specific area related to the value of these attributes in other areas adjacent (neighboring). another definition is the correlation between a variable with itself by the space, in the spatial domain means that the correlation between the value in location-i with the value in location-j (anselin, 1988). spatial autocorrelation can measure the occurrence of an event in the area of the adjacent unit. the existence of spatial autocorrelation indicates that the value of a variable in a particular area is affected by the value of the variable in other adjacent areas. the important issue in spatial autocorrelation is: a) the value of a variable in the region r is determined by variables measured in several areas around and b) variable region r affect the value of the variable in the surrounding area (ngudiantoro, 2004). if from a variable found in the form of a systematic spatial autocorrelation, then the variable is said to be spatially berautokorelasi. if the areas adjacent or similar areas, the spatial region berautokorelasi positive. conversely, if the area or adjacent areas are not similar or opposite indah resti ayuni, henny pramoedyo, endang wahyu handamari 80 80 volume 2 no. 2 mei 2012 $o' : a ) ,& ' < < < the spatial berautokorelasi negative territory, and decrypted form indicates there is no spatial autocorrelation. one common statistic used in spatial autocorrelation is moran index and geary index. test methods geary spatial patterns can be described into three parts, namely clustered (gangs), dispersed (like a chess board), and random (random). spatial autocorrelation is positive if in an area adjacent to each other have similar values. if described will form cluster spatial autocorrelation will be negative if in an area adjacent to each other have different values or do not like. if described will form a pattern like a chessboard. meanwhile, if n : number of data +, : value of the variable at location i +% : value of the variable at location j lee and wong (2001) stated that geary autocorrelation coefficient ranges from 0, which indicates a positive spatial autocorrelation, up to 2 which indicates negative spatial autocorrelati on. the absence of spatial autocorrelation indicated by the expected value of c (e(c)) by 1. geary underlying hypothesis test can be written as follows: g) h i (spatial autocorrelation), g' h i j (no spatial autocorrelation) varians geary autocorrelation coefficient i is given by the equation: ' r s there is a form that randomly showed no spatial autocorrelation. klm i n p /f' f 2 2 qf ; (2.9) sawada (2009) describes the measurement f) <$ $ %& ' .,% of spatial autocorrelation can be given by a statistic (normalized) cross-product that shows f' b >c d b ?c d t>? ot?> a the degree of relationship between the f <$ .,u .u, relationship of the entries of the two matrices, one matrix determines the spatial relationships among n locations (weight matrix) and others to explain the definition of matrix similarity between a collection of values on a variable at the location n. all measurements of spatial autocorrelation have a common source, namely cross-product matrix. cross-product matrix is also called a general statistical cross-product, which is expressed by the equation: ,& ' if g) true, significance testing geary autocorrelation coefficient c using the test statistic z as follows: v i wx 4y w (2.10) z[\] w value of z(c) compared with the critical points of the distribution n (0,1) to test g) stating there is no spatial autocorrelation. criteria for decision reject g) if the value of v i ^ v_` where v_` is a critical point z test with level α error. it can be concluded that there is autocorrelation $ $ spatial. criteria for decision making can also be 6 ! ! 7,% 8,% /09 ,&' %&' with 6 is a general cross-product statistic, .,% are elements of the matrix of spatial proximity done by comparing the p-value and α. p-value can be obtained from the following calculation: a 2 blcde fpv i g v g 2v i r / h f v i v i (2.11) measurement point i to j places commonly called the spatial weighting matrix, and 8,% is a measure of the relationship between i and j in some other dimension (brittle, 2010 ). one test of spatial if the p-value < j then g) spatial autocorrelation. spatial weighting matrix rejected, so there is autocorrelation that include general cross product in measuring spatial autocorrelation is geary test. sawada (2009) explains that geary autocorrelation coefficient is denoted by c, which measures the difference between the values of a variable at a nearby location. geary autocorrelation coefficient c is given by the following equation: b b a sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 81 81 < ,& ' < weighting matrix can be used to determine the proximity of spatial data or spatial relationships. weighting matrix is used to calculate the autocorrelation coefficient. weighting matrix is a matrix whose entries are given weighted values for comparisons between specific areas. weighting is based on the spatial relationships between regions (ngudiantoro, 2004). lee and wong (2001) states that if there are c $4' + <>cd <?cd =>? @> 4@? (2.8) :; b >c d @> 4@ a n units in the observation area, it can be used spatial weighting matrix of size n × n for with value f) <$ which is: c : geary index $ %& ' 7, % determining the closeness of the relationship between the unit area. each unit area described as rows and columns. each value in the matrix describes the spatial relationship between 7,% : index/size analysis of the spatial proximity of the location states i and j geographic characteristics with rows and indah resti ayuni, henny pramoedyo, endang wahyu handamari 82 82 volume 2 no. 2 mei 2012 ˆ columns. a value of 1 and 0 are used as a matrix to describe the closeness between regions. paradis (2009) stated that the study of spatial patterns and spatial processes, is expected to close observations are more likely to be similar than observations that are far apart. it is usually related to the weights for each pair (+, +% ). in its simplest form, the weight is given a value of 1 for the neighbor who is near and given a value of 0 for any other. lee and wong (2001) called the binary matrix denoted by w. the matrix w has some interesting characteristics. first, all .,, diagonal elements are 0, because it is assumed that a unit area is not adjacent to itself. second, the matrix w is symmetric where .,% .%, . symmetry owned by w matrix essentially describes the interrelationship of spatial relationships. the three, rows in the matrix w shows how a spatial region associated with other regions. in general, there are two concepts to determine the elements of the matrix of weights, respectively found in the proximity principle (contiguty matrix) and the principle of distance (distance matrix). therefore the number of values in a row i is the number of neighbors owned by the i-th region. notation is the sum of } € { ~ •! a' 2 • ' ‚ a} 2 • } /0 q ,&' maximum likelihood estimation maximum likelihood estimation (mle) is a classical estimation method of the most popular used in the parameter estimation process. (mendenhall, scheaffer and wackerly, 1981) states that this method is used to determine the estimator for ƒ, where the expected value is the value that makes the data most likely observation (the most likely) occurs. based on this principle, if the data observed are more likely (more likely) has a value of ƒ ƒ' than ƒ ƒ have value, it will be selected as the alleged ƒ' . likelihood function is the joint density function of n random variables „' „ | „$ evaluated at x_1, x_2, ..., x_n: f +' + | +$ … † +' + | +$ ƒ or values +' + | +$ certain likelihood function is a function of θ and is denoted by ‡ ƒ . if „' „ | „$ is a random sample of independent, identical, and certain distributed (iid) of the population is distributed † + ƒ' | ƒ* has a likelihood function: ‡ ƒ + ‡ ƒ' | ƒ* +' | +$ (2.15) the line: $ $ ,& ' † +, ƒ' | ƒ* which is: ., ! .,% /0 / %&' the properties of parameter estimators can be seen from unbiased, consistent, and efficient. .,0 : the total value of the i-th row .,% : the value of the i-th row j-th column one way to determine the weighting matrix is to use distance bands. anselin and rey (2008) explains that building a simple weighting matrix is based on the measurement of distance. i and j are considered neighbors if j falls within a specified critical distance (distance bands) of i. distance bands are denoted by k the elements of the matrix .,% defined as: mnop q,% g p r j s estimation of parameters of sdm anselin (1988) has written estimate model parameters using maximum likelihood hr estimation (mle) and also wrote some subset of the statistics hypothesis testing and mle-based tests such as the following: 1 2 2 1 2 2 1 (2.16) differential equation (2.16) with respect to will .,% l tuvpnww xp /0 z form jacobian functions as follows: ‰ š ‹œ š 2 distance euclid ‹ • ‡ ţ in geostatistical models, structural correlation between adjacent response depends only on the distance between locations not on the locations of the value of the response. the n ' s $ř •• a ' $ř e+a ř2 ' ŗ ŗ • a ‡ ( • ' ŗ sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 83 83 distance between the location can be determined using euclidean distance. euclidean distance (euclidean distance) is defined as the shortest straight line between two points. this distance is n s • • a ‰ e+a ř2 ŗ • a ‡ ( • ř /ŕ $ř ŗ 5 2 the sum of the square of the distance of the two vector values. suppose there are two vectors p and q, where { a' a | a} and e+a l2 ŕp 2 2 1 r / 2 1 r˜™ p 2 ~ • ' • | • } , where m is the number of coordinates, the euclidean distance between vectors p and q are determined by the equation: operation natural logarithm likelihood in equation (2.17) indah resti ayuni, henny pramoedyo, endang wahyu handamari 80 80 volume 2 no. 2 mei 2012 vw ‡ $ c n ' s c 2 7 2 ' ŕp 2 the mosquito aedes aegypti. early symptoms of ••a 5 ' •a vims transmission, ie fever or history of acute 2 1 r p 2 2 1 r˜ fever lasts 2-7 days and the bleeding tendency. 2 $ vw /ŕ 2 $ vw c 2 2 ' 5 dhf patients will also decrease levels of thrombosit. in dhf patients will experience acute •a ŕp 2 2 1 r p 2 2 1 rš (2.17) seizures and can lead to death. obtained parameters ( and . best model selection criteria for selecting the best model for this study using the information coefficient of determination r2 and akaike's information criterion (aic). coefficient of determination r2 according to gujarati (2003), the coefficient of determination stating the size of the accuracy or suitability of a regression line is applied to a group of data and research results are used to determine the proportion of the total diversity of dependent variables diterang by several independent variables simultaneously. determination coefficient ranges from 0 to 1. the greater the deviation r2 indicates smaller, it indicates that the model approaches the actual data and the model is said to more accurately. the coefficient of determination in these equations are susceptible to the addition of the independent variables. the more independent variables are involved, then the value will be higher. to compare two or more r2 should be taken into account that there are many independent variebel in the model. criteria for selecting the best model to use the criteria of r2 by the following formula: › = 1 – ::y œ4œ• ′ œ4œ• where ţ• is the average of y. alternative criteria for selecting the best model. aic (akaike's information criterion) akaike develop methods for selecting the best model. aic is defined as follows: ¢ ¢ method types and sources of data the data used in this study is secondary data obtained from the district health office in malang. the data is malang district health publications in 2009 covering 33 districts this data includes the data of dengue hemorrhagic fever (dhf) and factors affecting dengue hemorrhagic fever (dhf) in 2009. district location coordinates in appendix 1 and appendix 2 there are research data used. research methods in this research, spatial autocorrelation test using test geary. significant spatial data are performed modeling using spatial autocorrelation spatial durbin model (sdm). in this study required two processing stages, namely preparation stage and the stage of data analysis. here's a full explanation of the methods of research: 1. preparation at this stage, prepared maps that have been digitized and made input attribute data into the map. variables used consists of a dependent variable (y) and four independent variable (x). data analysis step-by-step analysis of the data is as follows: a. determine the nearest neighbor distance of each point as seen from the distance matrix has been formed, each point has one nearest neighbor. b. determine the distance the band is the greatest value of the nearest neighbor distance. aic = 2/ ÿ$ ¡ / ¡ $ $ c. determining geary autocorrelation coefficient where n is the number of observations. according to winarno (2009) as a measure of the accuracy of the model, selected models with the lowest aic value. additionally akaike develop methods for selecting the best model, and formulate it as follows: aic = -2£¤ /¥ where: £¤ = maximum loglikelihood m = number of parameters in the model scarlet fever dengue disease is an infectious disease caused by the dengue virus and transmitted by sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 81 81 by equation (2.8) and the manifold based on equations (2.9) and then tested for significance by equation (2.10). d. calculate the spatial weighting matrix based on equation (2.13). e. determine the spatial proximity point using distance matrices, corresponding to equation (2.14). the size of the distance matrix based on the number of points. results and discussion test of spatial autocorrelation indah resti ayuni, henny pramoedyo, endang wahyu handamari 82 82 volume 2 no. 2 mei 2012 no variable geary test geary coefficien t value (¦ statistic test v i test conclusion result (§ ¨© 1 ratio of number of patients with dengue population (y) 0,66395 -1,9747 there spatial autocorrela tion 2 percentage of non existence figures flick („' 0,64338 -2,0955 there spatial autocorrela tion 3 ratio of total population by household („ 0,59818 -2,5379 there spatial autocorrela tion 4 percentage of health centers with number of patients („ª 0,29459 -4,1450 there spatial autocorrela tion 5 percentage of patients treated patients with medical assistance („« 0,63819 -2,1260 there spatial autocorrela tion no parame ter variable modelling 1 (', j ( , j j r / z q estimasi pƒ-¡ r wald 7 1 ( ) intersep 0.326 3.928* 2 ('' percentage of non existence figures flick („' -0.000 6.429* 3 (' ratio of total population by household („ 0.014 3.408 4 (' ª percentage of health centers with number of patients („ª -0.002 11.57* 5 (' « percentage of patients treated patients with medical assistance („« 0.0002 1.600 6 ( ' percentage of non existence figures flick neighbor (7' „' -0.000 3.642 the test results by the method of geary's spatial autocorrelation data on the number of dengue patients in malang regency briefly shown in table 1 as follows: table 1. spatial autocorrelation test results with geary method. 5%, then the variables are positive spatial autocorrelation or mengerombol data patterns, i.e the closer the area the more it will have similar characteristics or value of the variables will be similar (relatively similar). test results spatial weighting matrix weighting matrix can be used to determine the proximity of spatial data or spatial relationships. weighting matrix is used to calculate the autocorrelation coefficient. determine the weighting matrix is to use distance bands. the distance between the location can be determined using euclidean distance. euclidean distance (euclidean distance) is defined as the shortest straight line between two points. parameter estimation in the first model, there is a lag on the dependent variable dependencies and lag on independent variables. parameter estimation model called the spatial durbin model (sdm) are presented in table 2 and the modeling of spatial autoregressive (sar) are presented in table 3. table 2. modeling 1 or spatial durbin model (sdm): lag on the dependent variable and lag on independent variables (', j ( , j j . based on table 1, geary spatial autocorrelation test results showed all variables are significant spatial autocorrelation in the ratio of number of patients with hemorrhagic fever with a population (y), where the number persentese free larvae („' , the ratio of number of residents with household („ , percentage of health centers the number of patients („ª , the percentage of patients who handled medical assistance „« 0 it can be seen from the absolute value of the test statistic geary on that variable is greater than the critical value, v)0)¬ (1,96). the absolute value of the test statistic for all variables geary row is -1.9748, 2.0955, 2.5379, -4.1450 and -2.1260. spatial autocorrelation found on all variables are positive spatial autocorrelation is seen from 0.66395, 0.64338, 0.59818, 0.29459, and 0.63819. geary coefficient is a value in the range of 0 to 1. this shows, with a significant rate of sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 83 83 5 (' « percentage of patients treated patients with medical assistance („« 0,0002 1.032 6 ( ' percentage of non existence figures flick neighbor (7' „' 7 ( rasio jumlah ratio of total population by household neighbor (7 „ 8 ( ª percentage of health centers with number of patients neighbor (7ª „ª ) 9 ( « percentage of patients treated patients with medical assistance neighbor (7 „ 10 ratio of number of patients with dengue population neighbor 0.0102 0.042 aic = -67.047 rsqr = 0.343 no parame ter variable modelling 2 (', j ( , j r / z q estimasipƒ ¡ r wald 7 1 ( ) intersep 0.0248 0,424 2 ('' percentage of non existence figures flick („' -0.0003 1,281 3 (' ratio of total population by household („ 0.0118 1,697 4 (' ª percentage of health centers with number of patients („ª -0.0021 10.853* 7 ( rasio jumlah ratio of total population by household neighbor (7 „ 0.012 9,0* 8 ( ª percentage of health centers with number of patients neighbor (7ª „ª ) -0.001 10.09* 9 ( « percentage of patients treated patients with medical assistance neighbor (7« „« -0.000 1.953 10 ratio of number of patients with dengue population neighbor 0.127 3.905* aic = -78.9384 rsqr = 0.5204 when without any lag independent variables, there are no independent variables without weighting or spatial autoregressive (sar). table 3. modelling 2 or spatial autoregressive (sar): modeling without lag independent variables (', j ( , j « « in the second modeling spatial autoregressive (sar), when without any lag independent variables, there are no independent variables without weighting are significant. r2 value of 34.3% and aic values of -67.047. dengue map and factors affecting in malang from the results of modeling spatial durbin model (sdm), the ratio of the number of patients with dengue fever with a population as the dependent variable and the percentage of larvae being free numbers, the ratio of the population of the household, the percentage of health centers with a number of patients, and the percentage of indah resti ayuni, henny pramoedyo, endang wahyu handamari 84 84 volume 2 no. 2 mei 2012 patients treated with the amount of medical assistance patients, as independent variables in each sub-district malang. the following shows some of the results of the mapping that maps the results of observations on the dependent variable and maps the results of observations on the independent variables and map the results predicted ratio of the number of patients with dengue fever resulting from the modeling results predicted using the best models are modeling spatial durbin model (sdm). mapping as follows: some images figure 4. map of results with observations percentage of total health center total patient figure 1. map results observations ratio of number of patients with dengue population figure 2. percentage of presence map results observations numbers free flick figure 5. map results percentage of patients treated observations medical assistance by number of patients figure 6. map results predicted ratio of number of patients with dengue fever in malang conclusions and recommendations conclusions 1. beberapa independent variable percentage existence free numbers larvae, the ratio of the number of resident households, the percentage of health centers with a number of people, the percentage of patients treated figure 3. map results observations ratio of total population by household with a number of medical assistance patients sdm for identified influence dengue hemorrhagic fever factors… jurnal cauchy – issn: 2086-0382 85 85 showed clustered patterns. the best model of the number of patients with dengue fever was first modeling, namely spatial durbin model (sdm). model of dengue fever in malang using the spatial durbin model (sdm) through the process of estimation parameters as follows: • 0 /9q7' • ¢ 0z/®¯ 2 0 ¯„' 0 q¯„ 2 0 / „ª 0 /„« 2 0 z7' „' 0 /¯7' „ 2 0 q7' „ª 2 0 7' „« . 2. estimates of the parameters of the best model first modeling spatial durbin model (sdm) factors that significantly influence the level 5% is the percentage of non-existence of numbers of larvae district itself, the percentage of health centers with a number of people from the district itself, the ratio of number of residents with home stairs from the neighboring districts, and the percentage of health centers with the number of patients from neighboring districts. suggestions advice can be given of the results of this research using human spatial modeling by adding a variable which contains spatial autocorrelation and the factors that affect the ratio of the number of patients with dengue fever. references anselin, l. 1988. spatial econometrics: methods and models. kluwer academic publishers. netherlands. anselin, l and s. j. rey. 2008. spatial econometrics: foundations. http://toae.org/serge/pub/wvu08/foundati ons.pdf. diakses tanggal 1 november 2011. anselin, l. 1999. spatial autocorrelation. university of illinois. urbana. arlinghaus, s. l. 1996. practical handbook of spatial statistics. united states. crc. press. inc. bekti, r. d. 2011. spatial durbin model untuk mengidentifikasi faktor-faktor yang berpengaruh terhadap kejadian diare di kabupaten tuban. its. surabaya. cressie, n.a.c. 1993. statistics for spatial data. john willey and sons, inc. new york. d'agustino, m. and v. dardanoni. 2007. what's so special about euclid distance? a characterization with application to mobility and spatial voting. http://www 3.unipv.it/websiep/wp/597.pdf. diakses tanggal 6 november 2011. dinas kesehatan. 2011. laporan tahunan seksi p2b2. malang dinas kesehatan dan keluarga berencana. 2006. pencapaian 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(2009). ekonometrika: teori aplikasi. penerbit ekonisia, fakultas ekonomi u2: yogyakarta. http://bse.ictcenter-/ http://www.lpc.uottawa.ca/publications/m http://www.stat.ufl.edu/~mdaniels/spatial hidden markov model application to transfer the trader online forex brokers farida suharleni1), agus widodo2), endang wahyu h.3) 1magister student of mathematics department, brawijaya university, malang 2,3lecturer of mathematics department, brawijaya university, malang 1email: farida_jelita@yahoo.co.id abstract hidden markov model is elaboration of markov chain, which is applicable to cases that can’t directly observe. in this research, hidden markov model is used to know trader’s transition to broker forex online. in hidden markov model, observed state is observable part and hidden state is hidden part. hidden markov model allows modeling system that contains interrelated observed state and hidden state. as observed state in trader’s transition to broker forex online is category 1, category 2, category 3, category 4, category 5 by condition of every broker forex online, whereas as hidden state is broker forex online marketiva, masterforex, instaforex, fbs and others. first step on application of hidden markov model in this research is making construction model � � � �, �, � by making a probability of transition matrix (a) from every broker forex online. next step is making a probability of observation matrix (b) by making conditional probability of five categories, that is category 1, category 2, category 3, category 4, category 5 by condition of every broker forex online and also need to determine an initial state probability (π) from every broker forex online. the last step is using viterbi algorithm to find hidden state sequences that is broker forex online sequences which is the most possible based on model � � � �, �, � and observed state that is the five categories. application of hidden markov model is done by making program with viterbi algorithm using delphi 7.0 software with observed state based on simulation data. example: by the number of observation t = 5 and observed state sequences o = (2,4,3,5,1) is found hidden state sequences which the most possible with observed state o as following : � � � � 4, � 2, � � 1, � � 5, � � 3 where x1 = fbs, x2 = masterforex, x3 = marketiva, x4 = others, and x5 = instaforex. keywords: hidden markov model, trader, broker forex online introduction electronic technology-based life was intense, vigorous today. investment world to feel the impact of the development of electronic technology. for example: investing through forex (foreign exchange) or foreign currency exchange is taking advantage of the electronic technology better known as forex online. this online forex trading brokerage services utilizing the so-called online forex broker. by definition, online forex broker is a party that can be a company, individual or agency in which he is established or function as an intermediary to bridge or reconcile between the buyer and seller. the presence of online forex broker is easier for traders to trade forex online. however, along with the increasing number of online forex brokers, the greater the opportunity for traders to move from one broker to another broker. limitation of the research problem extent of the problem in this research is: 1. the data used are primary data, the survey data using questionnaires. respondents in this study are online forex traders in indonesia. 2. five kinds of online forex brokers that will be examined in this study is marketiva, masterforex, instaforex, fbs, and the fifth one is a collection of other online forex brokers (others). the purpose of research along with the formulation of the problem, the objectives of this study were: 1. knowing the hidden markov model construction for traders shift towards online forex broker. 2. implementing a hidden markov model construction is on the software using delphi 7.0 software. literature review chance opportunity is a value between 0 and 1 that describes the size of the opportunity for the emergence of a specific event on a specific condition. other terms of the probability opportunities. odds of an event that must happen or opportunities certainty is 1, denoted p (s) = 1, while odds improbable events or opportunities impossibility is 0, denoted. ��� � 0 in full, the probability of an event a, denoted p (a), are: 0 � ��� � 1. (nugroho, 2008) hidden markov model application to transfer the trader online forex brokers jurnal cauchy – issn: 2086-0382 67 conditional genesis conditional genesis is an event that will occur with other conditions incident has occurred. provided a and b are two events in the sample space, chances conditional b if event a is known or has been denoted by ���|� . thus, it can be denoted: : ���|� , and read: "the odds of a b, if a has occurred" or "chance b, if a is known to opportunity", where ���|� � ��� � � ��� with ��� � 0. (walpole, 1995) stochastic processes the term stochastic processes are widely used for events related to the time-oriented observation. stochastic process is a family of random variables � � , ! "# where t is the parameter "time" of the set t. t is a parameter of "time" that has discrete or continuous nature. that is, if, then the stochastic process is parameterized "timing" discrete and usually denoted. if " � �0,1,2,3 … # then the stochastic process is parameterized "time" is continuous and is usually denoted by � %, & � 0,1,2,3 … . #. all possible prices happened t random variable is called the state space "state" which is denoted by q. state space "state" is called discrete if finite or infinite countable, otherwise state space "state" is called continuous if the interval of the real line containing. for example, the number of messages that arrive in the time period from 0 to t is �'� , ( 0# parameterized "time" and continuous state space "state" discrete. one of the stochastic process is markov chain models which have the property that the stochastic behavior in the future depends only on the time or the current situation and do not depend on the state of the past. (lester, 2009) markov chain basic concepts of markov chain markov chain model was developed by a russian expert aa markov in 1906. markov chain is a stochastic process that has the property that if the value ) is known, then * where + � , is not affected by ,. where � -.the definition means that the phenomenon of the future is only affected by the phenomenon of the present, and not influenced by the past. mathematically, the markov chain can are expressed as follows: �� ).� � -| �, , / , ) � 0 � �� ).� � -| ) � 0 , where 0, ! 1 and 1 � 23�,/, 345 are the set of possible state and ! ". (lester, 2009) early opportunity matrix (�) value of early chances are �� 6 � 37 � 87, so that for every i, 87 ( 0 , n n is the number of state and ∑ 87 � 1.47:� initially opportunities outlined in the matrix is called a row vector as the contents of the existing opportunities in the beginning it was a chance that symbolized 87 so, � � ;8� 8 8� / 84< (hidayanto, 2009) opportunity transition matrix (a) opportunities outlined in the transition matrix of transitional opportunities dimension n x n. the matrix a is formed by =7, the transition from state i opportunity to state j in one step can be denoted: =7, � �� ).� � -| ) � 0 a matrix formed by the above is called the transition matrix of the markov chain opportunities. the shape of the transition matrix a the following opportunities: new state � � > =�� =� / =�4= � = / /=4� =4 / = 4? // =44 @ new state initial state for i = 1,2, ..., n and j = 1,2, ... n. in the transition matrix of the markov chain opportunities has traits traits that entry on each line numbered one. in other words every i holds: ∑ =7, � 14,:� , =7, ( 0. (sumarminingsih, 2006) hidden markov model (hmm) hidden markov models or hidden markov model (hmm) is a statistical model of a system that assumed a markov process with unknown parameters. the problem is to determine the hidden parameters (hidden) of the parameters that can be observed (observed). specified parameters can then be used for further analysis. a hidden markov model can be considered as a dynamic bayesian network simplest. in the general markov model, its state can be observed directly, therefore the transition state the opportunity to be the only parameter. in the hidden markov model, statenya can’t be observed directly, but can be observed are the parameters that are affected by the state. each state has a probability distribution that may arise, therefore the output sequence produced by the hidden markov model provide some more information about the sequence of states. (http://www.wikipedia.com) definition of hmm hidden markov model (hmm) is a statistical tool that is used to model the rows that have a time span between the data with each other data. the data will be analyzed using the hmm data with markov process. (evans, 2003) farida suharleni, agus widodo, endang wahyu h. 68 volume 2 no. 2 mei 2012 elements of hmm hmm can be defined as accumulated's five elements (n, m, �, a, b). if considered � � � �, �, � the hmm has a certain element of n and m. the explanation will these elements are: 1. the matrix of transition opportunities (a) � � 2=7,5 , =7, � � a ).� � 3,|||| ) � 37b3 for each 1< i, j < n. 2. matrix chance observation (b) � � �c7�de # an output matrix opportunity with c7�de a chance observation at time t categorized de the condition state at time t is the state 37. can be written: c7�de � � �f) � de|||| ) � 37 , 1 � 0 � ', 1 � g � h where the observation matrix mem-possess opportunity features features that entry on each line of a. in other words: i c7�de � 1 j e:� , i = 1,2,3……,n 3. matrix opportunities early (π) � � �87# , 87 � � � k � 37 for i = 1,2,3……, n , 87 ( 0 4. n, the number of hidden state. denote the set is limited to state that it probably is. l � �3�, 3 , … , 34#. 5. m, the number of observed states can be observed. denote the set is limited to observation is probably m � �d�, d , … , dj#. (rabiner, 1989) notation hmm notation used in the hmm are: n = number of observations made, usually related to time, o = number of hidden state, p = number of observed state, l = �3�, 3 , / , 34# � the set of hidden state, m = �d�, d , / , dj# � the set of states that may be observed, � = matrix of transition opportunities, � = matrix of observation opportunities, � = matrix of early chances, g = �f�, f , / , fh � line of observation, i = � �, , / , h � rows hidden state, (stamp, 2004) viterbi algorithm viterbi algorithm is used to select the rows of x are willing hidden state-customized with a maximum o observations. viterbi algorithm is a dynamic programming algorithm to find the hidden state sequence dihasil her under observation, especially in the scope of hmm. procedure find hidden state ranks up with the viterbi algorithm formulated as follows 1. inisialisasi j��0 � 87c7�f� , 1 � 0 � ' dan k��0 � 0 2. rekursif j)� � max�l7l4mj)n��0 . =7,o c,�f) k)� � arg max�l7l4mj)n��0 . =7,o , 2 � � " , 1 � � ' optimal measures currently taken from the previous optimal step. the focus of his is to find rows that have a value of likelihood (probability) maximum. using likelihood value at the previous step is j)n��0 multiplied by the transition opportunities will be selected maximum value. the maximum value selected multiplied by the chance observation by observation to state j t namely (f)) thus obtained likelihood current (current likelihood) are denoted as j)� the best measure of the state sequence obtained by mj)n��0 , =7,o the maximum argument of which is denoted by k)�. 3. terminasi �q � max�l7l4 jh�0 3hq � arg max�l7l4 jh�0 4. backtracking for � " r 1, " r 2, … , 1 , 3)q � k).� �3).�q (rabiner, 1989) the definition of forex trading online forex (foreign exchange) is a type of transaction that traded currency in the currency of a country other state. forex is a trade involving major financial markets in the world for 24 hours continuously and it is an investment instrument that involves analytical support in achieving a level of profit to the movement of major currencies in the world. the definition of online forex trading is the trading or foreign currency exchange foreign currency contracts with a single unit called a lot and done online. in the trading market of sellers and buyers are both commonly called the trader (trader). online broker understanding and duties of the broker is very simple, namely, a party can be a company, individual or agency where he stands or function as an intermediary to bridge or reconcile between the buyer and seller. forex, gold, silver, index instruments are generally traded. forex trading in the common world among the biggest banks in the world. if a trader wants to get into the trade, must have access to or contact with these banks that are in fact very difficult in because of the huge capital requirements. therefore, the emerging online retail broker that connects traders to banks so they can keep trading with small capital. online brokers are usually chosen according to kebutuhan individual trader. in addition, the presence of hidden markov model application to transfer the trader online forex brokers jurnal cauchy – issn: 2086-0382 69 these online brokers, may facilitate the traders to transact online. (jaya, 2012) methods type of research related to the nature of the data and data analysis techniques are used, this type of research in this study is qualitative and quantitative. data collection the data used to build the initial odds matrix (), the matrix of transition opportunities (a) and a chance observation matrix (b) is the quantitative data, in the form of numerical data which is then processed mathematically. to establish state observed in this case used qualitative data, through interviews with respondents and data collection through the media. the data used in this study in the form of primary data. primary data is survey data using questionnaires to online forex trader in indonesia. population and sample prior to sampling, preliminary survey first conducted by distributing questionnaires to 30 online forex trader in indonesia as a respondent. of the 30 respondents, only 28 respondents who filled out questionnaires correctly and completely, the proportion of subjects success in completing the questionnaire was thus obtained p = 0.93 q = 0.07. sampling process done by using purposive sampling technique (based on the consideration in accordance with the purpose of research). samples taken are online forex traders in indonesia. determination of the number of respondents was calculated by using the formula: &6 � stu vw x y3 & � &6z1 { &6' | description: n = total population members &6 = the estimated number of samples n = number of samples taken α = 0.05 level of confidence z = the value of the normal distribution (for α = 0.05 then = 1.96) d = a tolerable margin of error in determining the sample mean = 0.05 p = proportion of questionnaires were filled in correctly and completely q = proportion of questionnaires filled in incorrect and / or incomplete (cochran, 1991) estimated number of samples are obtained &6 � }1,960,05� 0,93 � 0,07 &6 � 100,0353 assumed number of online forex trader in indonesia is 10% of the number of internet users in indonesia in 2012, that is 80 million people. thus, the number of online forex trader population is 10% of 80 million people, which is 8 million people. to obtain the following calculation: & � 100,0353z1 { 100,03538000000 | & � 100,0341 � 100 responden from the above calculations, the number of samples taken was 100 respondents. interview and questionnaire before researchers distributed questionnaires to the respondents, the researchers are looking for information to make observations through interviews with respondents or through the media to determine the parameters of observation (observed state). furthermore, the researchers spread forex online questionnaires to respondents in indonesia. data results of this questionnaire in the form of data transfer broker online forex trader to the period october 2011 to march 2012. data obtained from the questionnaire will then be processed to get the early opportunity matrix (), the matrix of transition opportunities (a) and a chance observation matrix (b). data analysis observed state and hidden state 1. based on research conducted before making the questionnaire, the characteristics of online forex brokers are divided into 5 categories: leverage up to 1:1000, often to invent a unique competition, a 30% deposit, using the trading platform metatrader 4 and 5, the availability of pamm, profit restrictions during the news. 2. 25% deposit bonus, up to 1:500 leverage, rarely entered the competition, using the trading platform metatrader 4, representatives / ib is still a bit in indonesia. 3. trading platform is java based, each opening an account will get $ 5, 1:500 leverage, no deposit bonus, there is a chat service on the trading platform, the process of withdrawal of time (up to 1 week). 4. there are many offices in indonesia, just leverage up to 1:500, rarely hold trading competitions, using the trading platform farida suharleni, agus widodo, endang wahyu h. 70 volume 2 no. 2 mei 2012 metatrader 4 and 5 (for metatrader demo version 5 only accounts only). 5. leverage up to 1:200 only, rarely hold trading competitions, using the trading platform metatrader 4 and 5, is listed / regulated in some countries like the u.s. and the uk, the service payment system slightly so that the withdrawal process takes 2-3 days (using a wire transfer) , spread currency pairs traded little changed at any time yet. further 5 categories above are components of hidden markov models (hmm) is a state observed in this study. while the hidden state is 5 online forex brokers, that marketiva, masterforex, instaforex, fbs and others. initial construction opportunity matrix (�) the data was obtained through a questionnaire grouped by online forex broker is selected in october 2011. the number of respondents who chose each forex broker online divided by the total respondents in october 2011, in order to obtain the initial value, where i {marketiva, masterforex, instaforex, fbs and others}. construction opportunities transition matrix (a) the data have been obtained through questionnaires, calculated how much displacement of respondents in the period october 2011 to march 2012 and who continue to use the same online forex broker. this data is set out in tabular form. each displacement divided by the number of respondents forex broker online in october 2011, found that the transition opportunities later written down in the form of a 5x5 matrix a. if the matrix a has been obtained, it can be made schemes hidden markov models but first do the construction for the matrix observation opportunities. opportunity matrix construction observation (b) data obtained through the questionnaire grouped by category selected, then calculated how the number of respondents for each online forex broker in each category, and the number is divided by the number of respondents online forex broker in 2012, found that observation opportunities later written in matrix form b. construction program creating software design using delphi 7.0 software to search for a maximal series of hidden states using the viterbi algorithm. discussion online forex broker is selected respondents based on responses to questionnaires obtained by the number of respondents who use online forex broker in october 2011 as follows: table 1. number of respondents forex broker online in october 2011 no broker forex online responden proporsi 1 marketiva 16 0,132 2 masterforex 57 0,471 3 instaforex 30 0,248 4 fbs 12 0,099 5 others 6 0,050 total 121 1 column proportions in table 1 is obtained by dividing the number of respondents of each forex broker online by the total number of respondents. for example: there are 16 respondents marketiva divided the total number of respondents 121 then the result obtained is 0.132. while the data of respondents who use online forex broker in march 2012 are shown in table 2 below: table 2. number of respondents forex broker online in march 2012 no broker forex online responden proporsi 1 marketiva 20 0,165 2 masterforex 49 0,405 3 instaforex 34 0,281 4 fbs 8 0,066 5 others 10 0,083 total 121 1 the number of respondents in each forex broker online is changing. for example: the number of respondents marketiva in october 2011 there were 16 respondents, in march 2012 changed to 20 respondents. this change also implies a change in the proportion of each respondent online forex broker. for example: the proportion of respondents marketiva in october 2011 is 0.132, but in march 2012 the proportion of respondents marketiva be 0.165. the data in table 1 column matrix of proportions used to construct early chances � � �87# that the results presented in table 3 below: table 3. matrix of opportunity beginning �8 : no (i) broker forex online (37) peluang awal �87 1 marketiva 0,132 2 masterforex 0,471 3 instaforex 0,248 4 fbs 0,099 5 others 0,050 total 1 hidden markov model application to transfer the trader online forex brokers jurnal cauchy – issn: 2086-0382 71 early chances matrix �can be written: � � ;0,132 0,471 0,248 0,099 0,050< displacement trader in choosing online forex broker processed responses to questionnaires to describe the displacement trader in choosing a forex broker online from october 2011 to march 2012, as shown in table 4 below: table 4. displacement trader in choosing a forex broker online maret 2012 oktober 2011 m a rk e ti v a m a s te rf o re x in s ta fo re x f b s o th e rs marketiva 9 2 2 2 1 16 masterforex 5 38 10 3 1 57 instaforex 3 3 18 2 4 30 fbs 3 4 3 1 1 12 others 0 2 1 0 3 6 total 20 49 34 8 10 121 table 4 can be read: the first line shows traders using marketiva broker in october 2011 amounted to 16 traders. then in march 2012, from 16 traders marketiva is there 9 traders who continue to use the broker marketiva, 2 traders move to broker masterforex; 2 traders move to broker instaforex, 2 traders move to broker fbs, and one trader to move to broker others. thus, the data in table 4 show that the displacement occurs trader of online forex broker online forex broker one to the other of the period from october 2011 to march 2012. for example: the number of traders using marketiva broker in october 2011 there were 16 traders, while in march 2012 increased to 20 traders. opportunity transition matrix (a) referring to table 4 can be formed matrix a transition opportunities below: broker forex online bulan maret 2012 � � �� �� �0,563 0,125 0,125 0,125 0,0620,088 0,667 0,175 0,053 0,0170,100 0,100 0,600 0,067 0,1330,250 0,334 0,250 0,083 0,0830,088 0,334 0,166 0,013 0,500�� �� � value entry in the matrix a is obtained by dividing the number of respondents each forex broker online in october 2011 by the total number of respondents each forex broker online in october 2011. for example: the value of 0.563 is obtained by dividing the number of respondents broker marketiva in october 2011 that as many as nine respondents to the total number of respondents broker marketiva in october 2011 that as many as 16. the matrix a is the matrix of transition opportunities to transfer trader online forex broker. for example, a trader the opportunity to move from masterforex marketiva is equal to 0.088, mathematically written ��h=�g� 0d= | h=+ ������� = 0,088. opportunity observation matrix (b) options traders on online forex broker in each category based on the data obtained are presented in table 5 below: table 5. option traders on forex brokers online by category kategori broker forex online maret 2012 1 2 3 4 5 total marketiva 1 3 13 2 1 20 masterforex 6 5 2 33 3 49 instaforex 26 1 3 2 2 34 fbs 1 6 1 0 0 8 others 1 1 1 1 6 10 total 35 16 20 38 12 121 table 5 shows the options trader on online forex broker based category, for example: traders who choose a broker marketiva by category 1 by 1 person, based on category 2 as many as 3 people, based on category 3 as many as 13 people, according to category 4 by 2 people and based on category 5 by 1 person. based on table 5 can be generated matrix observation opportunities (b) the following: kategori � � �� �� �0,050 0,150 0,650 0,100 0,0500,123 0,102 0,041 0,673 0,0610,765 0,029 0,088 0,059 0,0590,125 0,750 0,125 0,000 0,0000,100 0,100 0,100 0,100 0,600�� �� � value entry in the matrix b is obtained by dividing the number of respondents of each forex broker online in march 2012 for each category selected by the total number of respondents each forex broker online in march 2012. for example: the value of 0.050 is obtained by dividing the number of respondents broker marketiva in october 2012 that choosing a category 1sebanyak one respondent to the total number of respondents broker marketiva in march 2012 that as many as 20. the matrix b is a matrix which shows the trader the opportunity to choose a forex broker online by category. for example: if you use a broker instaforex trader in march 2012, then the chances of traders choose a category 1 is 0.765, mathematically written = 0.765. broker forex online bulan oktober 2011 broker forex online bulan maret 2012 farida suharleni, agus widodo, endang wahyu h. 72 volume 2 no. 2 mei 2012 viterbi algorithm viterbi algorithm is a dynamic programming algorithm to find the hidden state sequence the maximum of a sequence of observations. source code of the viterbi algorithm is shown in table 6 below: table 6. source code algoritma viterbi source code algoritma viterbi 1. inisialisasi for i := 1 to n do begin delta[1,i] := phi[i]*b[i,o[1]]; psi[1,i] := 0; end; 2. rekursif for i := 2 to t do begin for j := 1 to m do begin for k := 1 to n do p[k] := delta[i-1,k]*a[k,j]; argmax := 1; for k := 2 to n do if p[argmax] < p[k] then argmax := k; delta[i,j] := p[argmax]*b[j,o[i]]; psi[i,j] := argmax; end; end; 3. terminasi for i := t downto 1 do begin argmax := 1; for j := 2 to n do if delta[i,argmax] < delta[i,j] then argmax := j; q[i] := argmax; end; implementation of hidden markov model in the line forex broker online figure 1. implementation of hidden markov models for sequence search online forex broker. figure 1 shows the output of a program that has been run with the data desired by the transition matrix a and matrix opportunity observation opportunities b and table 3 for the matrix early chances. for example, the observation sequence used simulated data with the number of observations t = 5, f � �f� � 2, f � 4, f� � 3, f� � 5, f� � 1 . rows of hidden states that best fits the observations o are: � � � � 4, � 2, � � 1, � � 5, � � 3 thus the ranks of online forex brokers are: (x1 = fbs, x2 = masterforex, x3 = marketiva, x4 = others, x5 = instaforex). conclusion the conclusions obtained are: 1. hidden markov models can be used to find the transfer trader of online forex broker to make the construction of the model through model construction , � � � �, �, � earned-right hidden state line that best fits the observation state o. for example: the number of observations t = 5 with the line observed state o = (2,4,3,5,1), be the line most likely hidden state with observed state o: � � � � 4, � 2, � � 1, � � 5, � � 3 where x1 = fbs, x2 = masterforex, x3 = marketiva, x4 = others, x5 = instaforex. 2. applications of hidden markov models has been done by making a program with the viterbi algorithm in delphi 7.0 software with state observed form the simulation data. suggestion further research on online forex brokers can be developed by using the baum-welch algorithm that aims to optimize the model parameters in a way to renew (re-estimation) model � � � �, �, � . references [1]. anonim. 2011.hmm/model markov tersembunyi. http://www.wikipedia.com [2]. cochran, w. g. 1991. teknik penarikan sampel. penerjemah: rudiansyah. universitas indonesia press. jakarta. [3]. evans, l. c. 2003. an introduction to stochastic differential equations. versi 1.2. departement of mathematics, uc berkeley. [4]. hidayanto, a. 2009. teori umum rantai markov. jurnal. achmad.blog.undip.ac.id. [5]. jaya, i. g. t. 2012. aplikasi analytic hierarchy process untuk memilih broker forex online. skripsi. jurusan matematika universitas brawijaya, malang. hidden markov model application to transfer the trader online forex brokers jurnal cauchy – issn: 2086-0382 73 [6]. lestari, y. d. 2009. penerapan model hidden markov pada peramalan harga premium. penelitian. jurusan matematika institut teknologi sepuluh november, surabaya. [7]. nugroho, s. 2008. dasar-dasar metode statistika. pt. grasindo. jakarta [8]. rabiner, l. r. 1989. a tutorial on hidden markov models and selected applications in speech recognation. proceedings of the ieee, 77 (2), 257-286. [9]. stamp, m. 2004. a revealing introduction to hidden markov models. [10]. www.cs.sjsu.edu/faculty/stamp/rua/hmm. pdf [11]. sumarminingsih, e. 2006. modul proses stokastik. jurusan matematika fmipa universitas brawijaya, malang. [12]. walpole, r. e. 1995. pengantar statistika edisi ke-3. pt gramedia pustaka utama, jakarta lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf 4. daftar isi.pdf 5.1. elva ravita sari studi tentang persamaan fuzzy _54-65.pdf teorema titik tetap di ruang banach amanatul husnia, hairur rahman jurusan matematika universitas islam negeri maulana malik ibrahim malang e-mail: niaja10@yahoo.com abstrak ruang banach merupakan suatu konsep penting dalam analisis fungsional. pada tahun 1992, seorang ahli matematika berasal dari polandia membuktikan teorema yang menyatakan ketunggalan titik tetap. teorema tersebut disebut juga dengan teorema titik tetap banach. teorema titik tetap banach (teorema kontraksi) merupakan teorema ketunggalan dari suatu titik tetap pada suatu pemetaan yang disebut kontraksi dari ruang metrik lengkap ke dalam dirinya sendiri. pengertian ruang banach sendiri adalah ruang norm yang lengkap, dikatakan lengkap jika barisan cauchy tersebut konvergen. penelitian ini bertujuan untuk mengetahui pembuktian titik tetap di ruang banach dengan kondisi yang diberikan yaitu pada pemetaan kannan dan pemetaan fisher. berdasarkan hasil pembahasan, diperoleh bahwa pemetaan kannan dan pemetaan fisher mempunyai titik tetap yang tunggal 𝑇(𝑥) = 𝑥 dan pemetaan tersebut merupakan pemetaan titik tetap terhadap dirinya sendiri di ruang metrik lengkap. kata kunci: titik tetap, pemetaan kontraksi, ruang metrik lengkap, ruang banach abstract banach space is an important concept in functional analysis. in 1992, a mathematician from poland proved the uniqueness of fixed point. the theorem is also called banach fixed point theorem. banach fixed point theorem (contraction theorem) is a unique fixed point theorem on a mapping called the contraction of a complete metric space into itself. the definition of banach space itself is a complete norm space, to be said complete if the cauchy sequence is convergent. this study aims to determine the evidence of fixed point in banach space with the given conditions, namely kannan mapping and fisher mapping. based on the results of the discussion, it is obtained that kannan mapping and fisher mapping has a single fixed point 𝑇(𝑥) = 𝑥 and the mapping is a fixed point mapping to itself in a complete metric space. keywords: fixed point, contraction mapping, complete metric space, banach space pendahuluan matematika merupakan abstraksi dari dunia nyata. abstraksi secara bahasa berarti proses pengabstrakan. abstraksi sendiri dapat diartikan sebagai upaya untuk menciptakan definisi dengan jalan memusatkan perhatian pada sifat yang umum dari berbagai objek dan mengabaikan sifat-sifat yang berlainan. untuk menyatakan hasil abstraksi, diperlukan suatu media komunikasi atau bahasa. bahasa yang digunakan dalam matematika adalah bahasa simbol. penggunaan bahasa simbol mempunyai dua keuntungan yaitu sederhana dan universal. sederhana di sini berarti sangat singkat dan universal berarti bahwa ahli matematika di belahan bumi manapun akan dapat memahaminya (abdussakir, 2009). menurut kreyzig (1978:1-2) misalnya dalam analisis fungsional memusatkan perhatian pada “ruang”. hal ini merupakan dasar penting untuk mengkaji ruang banach, ruang norma, ruang metrik, dan ruang hilbert dengan sangat rinci. dalam hubungan ini konsep “ruang” yang digunakan dalam ruang banach mempunyai arti yang sangat luas. ruang banach adalah ruang norma yang lengkap, artinya bahwa ruang banach adalah ruang norma, ruang yang memenuhi sifat-sifat ruang norma, dikatakan lengkap bahwa barisan cauchy tersebut konvergen (wilde, 2003:84). ruang banach merupakan suatu konsep penting dalam analisis fungsional. pada tahun 1992, seorang ahli matematika berasal dari polandia membuktikan teorema yang menyatakan keberadaan dan ketunggalan suatu titik tetap. teorema tersebut disebut juga dengan teorema titik tetap banach atau prinsip kontraksi banach. teorema ini menyediakan teknik untuk memecahkan berbagai masalah yang diterapkan dalam matematika sains (ilmu matematika) dan ilmu teknik. teorema titik tetap banach (teorema kontraksi) merupakan teorema ketunggalan dari teorema titik tetap di ruang banach cauchy – issn: 2086-0382 117 suatu titik tetap pada suatu pemetaan yang disebut kontraksi dari ruang metrik lengkap ke dalam dirinya sendiri. kajian teori 1. ruang metrik ruang metrik memperjelas konsep jarak. definisi dari metrik bermanfaat untuk mengetahui aplikasi yang lebih umum dari konsep jarak. di dalam kalkulus dipelajari tentang fungsi-fungsi yang terdefinisi dalam garis bilangan real ℝ. di dalam bilangan real ℝ terdefinisi fungsi jarak, yaitu memasangankan 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| dengan setiap pasangan titik 𝑥, 𝑦 ∈ ℝ, jadi ℝ mempunyai fungsi jarak atau disebut dengan 𝑑, dimana jarak 𝑑 (𝑥, 𝑦) = |𝑥 − 𝑦| dengan setiap pasangan titik 𝑥, 𝑦 ∈ ℝ (kreyszig, 1978:2-3). definisi 2.1.1 ruang metrik (𝑋, 𝑑), dimana 𝑋 merupakan himpunan dan 𝑑 merupakan metrik di 𝑋 (fungsi jarak 𝑋) yaitu fungsi yang didefinisikan pada 𝑋 × 𝑋 untuk setiap 𝑥, 𝑦, 𝑧 ∈ 𝑋, sehingga diperoleh 1. 𝑑(𝑥, 𝑦) ≥ 0 (𝑑 adalah bernilai real, terbatas, dan tidak negatif) 2. 𝑑(𝑥, 𝑦) = 0 ⇔ 𝑥 = 𝑦 3. 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) (simetri) 4. 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦) (ketaksamaan segitiga) (kreyszig, 1978:3). contoh didefinisikan fungsi 𝑑: ℝ2 × ℝ2 → ℝ yaitu 𝑑(𝑎, 𝑏) = √(𝑥1 − 𝑦1) 2 + (𝑥2 − 𝑦2) 2 dengan 𝑎 = (𝑥1, 𝑥2) dan 𝑏 = (𝑦1, 𝑦2). tunjukkan bahwa fungsi 𝑑 adalah metrik! definisi 2.1.2 (persekitaran) misalkan (𝑋, 𝑑) adalah ruang metrik, maka untuk suatu > 0, persekitaran titik di 𝑥0 ∈ 𝑋 merupakan himpunan 𝑣𝜀 (𝑥0) = {𝑥 ∈ 𝑋: 𝑑(𝑥0, 𝑥) < } (sherbert dan bartle, 2000:329). 2. himpunan terbuka dan himpunan tertutup definisi 2.2.1 misalkan (𝑋, 𝑑) adalah ruang metrik, untuk sebarang 𝑥 ∈ 𝑋 dan setiap 𝑟 > 0, himpunan-himpunan 1. 𝐵𝑥 (𝑟) = (𝑦 ∈ 𝑋 ⃓ 𝑑(𝑥, 𝑦) < 𝑟) disebut bola terbuka 2. 𝐵𝑥 (𝑟) = (𝑦 ∈ 𝑋 ⃓ 𝑑(𝑥, 𝑦) ≤ 𝑟) disebut bola tertutup (rynne dan youngson, 2008:13). contoh a. diketahui ruang metrik (𝑋, 𝑑) dengan metrik 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| b. 𝐵(0,1) = (𝑦 ∈ 𝑋 ⃓ − 1 < 𝑦 < 1) disebut bola terbuka berpusat di 0 dengan jari-jari 1 pada ruang metrik (𝑋, 𝑑). c. diketahui ruang metrik (𝑋, 𝑑) dengan metrik 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| d. 𝐵(0,1) = (𝑦 ∈ 𝑋 ⃓ − 1 ≤ 𝑦 ≤ 1) disebut bola tertutup berpusat di 0 dengan jari-jari 1 pada ruang metrik (𝑋, 𝑑). definisi 2.2.2 (titik interior) titik 𝑝 disebut suatu titik interior himpunan 𝐸 jika terdapat suatu persekitaran dari 𝑝 yang merupakan subset dari 𝐸 (soemantri, 1988). definisi 2.2.3 (himpunan terbuka) himpunan 𝐸 disebut himpunan terbuka jika setiap anggotanya merupakan titik interior himpunan 𝐸 (soemantri, 1988). definisi 2.2.4 (titik limit) misalkan 𝐴 ⊆ ℝ, suatu titik 𝑐 ∈ ℝ disebut titik limit jika untuk setiap 𝛿 > 0 terdapat paling sedikit satu titik 𝑥 ∈ 𝐴, 𝑥 ≠ 𝑐 sedemikian sehingga |𝑥 − 𝑐| < 𝛿 (bartle dan sherbert, 2000:97). menurut soemantri (1988) titik 𝑝 ∈ 𝑋 disebut titik limit himpunan 𝐸 subset 𝑋, bila setiap sekitar titik 𝑝 memuat paling sedikit satu titik 𝑞 ∈ 𝑋 dan 𝑞 ≠ 𝑝. definisi 2.2.5 (himpunan tertutup) himpunan 𝐸 disebut tertutup jika semua titik limitnya termuat di dalam 𝐸 (soemantri, 1988). definisi 2.2.6 (himpunan terbatas) himpunan 𝐸 dalam ruang metrik (𝑋) disebut terbatas jika terdapat titik 𝑝 ∈ 𝑋 dan bilangan 𝑀 > 0 sehingga untuk setiap 𝑥 ∈ 𝑋, maka jarak 𝑑(𝑝, 𝑥) ≤ 𝑀 (soemantri, 1988). 3. kekonvergenan dan kelengkapan definisi 2.3.1 (barisan terbatas) barisan 〈𝑥𝑛〉 di ruang metrik 𝑋 = (𝑋, 𝑑) disebut barisan terbatas jika daerah jangkauan (range) dari barisan tersebut merupakan himpunan bagian terbatas di 𝑋 (soemantri, 1988). amanatul husnia, hairur rahman 118 volume 3 no. 2 mei 2014 definisi 2.3.2 (barisan konvergen) barisan 〈𝑥𝑛 〉 di ruang metrik 𝑋 = (𝑋, 𝑑) dikatakan konvergen jika ada 𝑥 ∈ 𝑋, maka lim 𝑛→∞ 𝑑(𝑥𝑛 , 𝑥) = 0 𝑥 disebut limit dari 〈𝑥𝑛〉 dapat juga ditulis dengan lim 𝑛→∞ 𝑥𝑛 = 𝑥 atau 𝑥𝑛 → 𝑥 barisan 〈𝑥𝑛 〉 yang tidak konvergen disebut divergen (kreyszig, 1978:25). teorema 2.3.3 jika barisan 〈𝑥𝑛〉 konvergen di dalam ruang metrik (𝑋, 𝑑), maka barisan 〈𝑥𝑛 〉 tersebut terbatas dan limit barisan 〈𝑥𝑛〉 tunggal. definisi 2.3.4 (barisan cauchy) barisan 〈𝑥𝑛 〉 di dalam ruang metrik (𝑋, 𝑑) dikatakan barisan cauchy jika untuk setiap > 0, terdapat 𝑁 ∈ ℕ sedemikian sehingga untuk semua 𝑚, 𝑛 > 𝑁 berlaku 𝑑(𝑥𝑚 , 𝑥𝑛 ) < (ghozali, 2010:12). teorema 2.3.5 setiap barisan yang konvergen dalam suatu metrik (𝑋, 𝑑) merupakan barisan cauchy definisi 2.3.6 (ruang metrik lengkap) ruang metrik (𝑋, 𝑑) dikatakan lengkap jika setiap barisan cauchy konvergen di dalam 𝑋 (sherbet dan bartle, 2000:330). 4. ruang vektor bernorma definisi 2.4.1 (ruang vektor bernorma) ruang vektor bernorma adalah ruang vektor 𝑋 dengan pemetaan ∥ ∥: 𝑋 → 𝑅+, dengan sifat-sifat 1. ∥ 𝑥 ∥= 0 jika dan hanya jika 𝑥 = 0 (𝑥 ∈ 𝑋) 2. ‖𝛼𝑥‖ = |𝛼|‖𝑥‖ untuk setiap 𝑥 ∈ 𝑋 dan skalar 𝛼 3. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖ untuk setiap 𝑥, 𝑦 ∈ 𝑋 ruang vektor bernorma ini dinotasikan dengan (𝑥, ∥ ∥) dan pemetaan ini ∥ ∥ disebut “norma” pada ruang (cohen, 2003:174). contoh misalkan 𝑋 merupakan ruang vektor berdimensi hingga di 𝔽 dengan basis {𝑒1, 𝑒2, 𝑒3, … , 𝑒𝑛} yang mana 𝑥 ∈ 𝑋 dapat juga ditulis dengan 𝑥 = ∑ 𝜆𝑗𝑒𝑗 𝑛 𝑗=1 dengan 𝜆1, 𝜆2, 𝜆3, … , 𝜆𝑛 ∈ 𝔽. maka fungsi ‖𝑥‖: 𝑋 → ℝ didefinisikan dengan ‖𝑥‖ = (∑|𝜆𝑗 | 2 ) 1 2 𝑛 𝑗=1 merupakan norma di 𝑋 5. kekonvergenan dalam ruang bernorma menurut cohen (2003:178) dalam mempertimbangkan ruang bernorma menjadi ruang metrik dapat diketahui dengan satu cara. kemudian gagasan yang terkait pada kekonvergenan barisan di ruang metrik dapat dipindahkan ke ruang bernorma. oleh sebab itu, dapat disimpulkan dengan barisan 〈𝑥𝑛〉 di ruang norma konvergen jika terdapat bilangan > 0 dan terdapat elemen 𝑥 ∈ 𝑋 serta terdapat bilangan bulat positif 𝑁 seperti ‖𝑥𝑛 − 𝑥‖ < dimana 𝑛 > 𝑁 dapat ditulis dengan 𝑥𝑛 → 𝑥 atau 𝑙𝑖𝑚𝑥𝑛 = 𝑥 dan 𝑥 disebut limit pada barisan. definisi 2.5.1 (ruang banach) setiap ruang vektor bernorma yang lengkap disebut ruang banach (cohen, 2003:178). 6. teorema titik tetap definisi 2.6.1 misalkan 𝑇 merupakan pemetaan dari ruang metrik (𝑋, 𝑑) ke dalam dirinya sendiri a. sebuah titik 𝑥 ∈ 𝑋 sedemikian sehingga 𝑇(𝑥) = 𝑥 maka 𝑥 disebut titik tetap pada pemetaan 𝑇 b. jika ada 𝛼, dengan 0 < 𝛼 < 1, maka untuk setiap pasangan dari titik 𝑥, 𝑦 ∈ 𝑋 diperoleh 𝑑(𝑇𝑥 , 𝑇𝑦 ) ≤ 𝛼 𝑑(𝑥, 𝑦) kemudian 𝑇 disebut pemetaan kontraksi atau kontraksi sederhana, sedangkan 𝛼 disebut kontraksi konstan di 𝑇 (cohen, 2003:116). teorema 2.6.2 jika 𝑇 adalah pemetaan kontraksi di ruang metrik 𝑋 maka 𝑇 kontinu di 𝑋. teorema 2.6.3 (teorema titik tetap/ titik tetap banach) setiap pemetaan kontraksi di ruang metrik lengkap hanya mempunyai titik tetap tunggal. 7. pemetaan definisi 2.7.1 (pemetaan) misalkan 𝑋 dan 𝑦 adalah ruang metrik. pemetaan 𝑇 dari himpunan 𝑋 ke himpunan 𝑌 dinotasikan dengan 𝑇: 𝑋 → 𝑌 adalah suatu teorema titik tetap di ruang banach cauchy – issn: 2086-0382 119 pengawanan setiap 𝑥 ∈ 𝑋 dikawankan secara tunggal dengan 𝑦 ∈ 𝑌 dan ditulis 𝑦 = 𝑇(𝑥). definisi 2.7.2 (pemetaan kontinu) misalkan 𝑋 = (𝑋, 𝑑1) dan 𝑌 = (𝑌, 𝑑2) adalah ruang metrik. pemetaan 𝑇: 𝑋 → 𝑌 dikatakan kontinu di titik 𝑥0 ∈ 𝑋 jika untuk setiap > 0 terdapat 𝛿 > 0 sedemikian sehingga untuk setiap 𝑥 ∈ 𝑋 dengan 𝑑1(𝑥, 𝑥0) < 𝛿 maka berlaku 𝑑2(𝑇(𝑥), 𝑇(𝑥0)) < pemetaan 𝑇 dikatakan kontinu pada 𝑋 jika 𝑇 kontinu di setiap titik anggota 𝑋. definisi 2.7.3 (komposisi pemetaan) misalkan 𝐴, 𝐵 dan 𝐶 adalah ruang metrik. jika 𝑓: 𝐴 → 𝐵 dan 𝑔: 𝐵 → 𝐶 maka komposisi pemetaan 𝑔𝑜𝑓 merupakan pemetaan dari 𝐴 → 𝐶 yang didefinisikan (𝑔𝑜𝑓)(𝑥) = 𝑔(𝑓(𝑥)) untuk setiap 𝑥 ∈ 𝑋 komposisi (𝑓𝑜𝑓)(𝑥) = 𝑓(𝑓(𝑥)) = 𝑓2(𝑥) dan jika komposisi sebanyak 𝑛 suku, maka (𝑓𝑜𝑓𝑜𝑓𝑜 … 𝑜𝑓) = 𝑓𝑛 (𝑥) definisi 2.7.4 (pemetaan kontraksi) misalkan (𝑋, 𝑑) merupakan ruang metrik. pemetaan 𝑇: 𝑋 → 𝑋 dikatakan pemetaan kontraksi, jika ada konstanta 𝑐 dengan 0 ≤ 𝑐 ≤ 1 berlaku 𝑑(𝑇(𝑥)𝑇(𝑦)) ≤ 𝑐𝑑(𝑥, 𝑦) untuk setiap 𝑥, 𝑦 ∈ 𝑋 pembahasan dalam matematika teorema titik tetap banach juga dikenal sebagai teorema pemetaan kontraksi yang merupakan alat penting dalam teori ruang metrik, untuk menjamin keberadaan dan ketunggalan titik tetap pemetaan diri pada ruang metrik, dan menyediakan metode kontraksi untuk menemukan titik tetap (banach, 1992:133). teorema 3.1.2 misalkan 𝑇 adalah pemetaan kontraksi pada ruang metrik (𝑋, 𝑑) ke dalam dirinya sendiri. maka 𝑇 𝑛 adalah pemetaan kannan, untuk setiap 𝑛 adalah bilangan bulat positif (kannan, 1969:71-78). bukti menurut definisi pemetaan kontraksi (definisi 2.7.4) bahwa misalkan (𝑋, 𝑑) merupakan ruang metrik. pemetaan 𝑇: 𝑋 → 𝑋 dikatakan pemetaan kontraksi, jika ada konstanta 𝑐 dengan 0 ≤ 𝑐 ≤ 1 sehingga 𝑑(𝑇(𝑥)𝑇(𝑦)) ≤ 𝑐𝑑(𝑥, 𝑦) untuk setiap 𝑥, 𝑦 ∈ 𝑋. sekarang akan ditunjukkan bahwa 𝑇 𝑛 adalah pemetaan kannan, jika ada konstanta 𝑘 dengan 0 ≤ 𝑘 ≤ 1 sehingga 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) ≤ 𝑘[𝑑(𝑇(𝑥), 𝑥) + 𝑑(𝑇(𝑦), 𝑦)] untuk setiap 𝑥, 𝑦 ∈ 𝑋. 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) = 𝑑(𝑇𝑇 𝑛−1(𝑥), 𝑇𝑇 𝑛−1(𝑦)) ≤ 𝑐𝑑(𝑇 𝑛−1(𝑥), 𝑇 𝑛−1(𝑦)) = 𝑐𝑑(𝑇𝑇 𝑛−2(𝑥), 𝑇𝑇 𝑛−2(𝑦)) ≤ 𝑐 2𝑑(𝑇 𝑛−2 (𝑥), 𝑇 𝑛−2(𝑦)) sehingga diperoleh 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) ≤ 𝑐 𝑛 𝑑(𝑥, 𝑦) (3.1) untuk setiap 𝑥, 𝑦 ∈ 𝑋. karena 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑇 𝑛 (𝑥), 𝑥) + 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) + 𝑑(𝑇 𝑛 (𝑦), 𝑦) dengan menggunakan ketaksamaan (3.1), maka diperoleh 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) ≤ 𝑐 𝑛 𝑑(𝑥, 𝑦) ≤ 𝑐 𝑛 [𝑑(𝑇 𝑛(𝑥), 𝑥) + 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) + 𝑑(𝑇 𝑛(𝑦), 𝑦)] = 𝑐 𝑛 𝑑(𝑇 𝑛 (𝑥), 𝑥) + 𝑐 𝑛 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) + 𝑐 𝑛 𝑑(𝑇 𝑛(𝑦), 𝑦) sehingga mengakibatkan (1 − 𝑐 𝑛 )𝑑(𝑇 𝑛(𝑥), 𝑇 𝑛(𝑦)) ≤ 𝑐 𝑛[𝑑(𝑇 𝑛 (𝑥), 𝑥) + 𝑑(𝑇 𝑛 (𝑦), 𝑦)] 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) ≤ 𝑐 𝑛 (1−𝑐 𝑛) [𝑑(𝑇 𝑛 (𝑥), 𝑥) + 𝑑(𝑇 𝑛(𝑦), 𝑦)] (3.2) untuk setiap 𝑥, 𝑦 ∈ 𝑋. karena 𝑐 < 1, maka dapat diambil 𝑛 sebarang dengan 𝑐 𝑛 < 1 3 , sehingga (1 − 𝑐 𝑛 ) > 1 − 1 3 (1 − 𝑐 𝑛 ) > 2 3 atau 1 (1−𝑐 𝑛) < 3 2 oleh karena itu 𝑐 𝑛 (1−𝑐 𝑛) < 1 2 dimana 𝑐 𝑛 (1−𝑐 𝑛) = 𝑘, dengan menggunakan ketaksamaan (3.2) diperoleh 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛 (𝑦)) ≤ [𝑑(𝑇 𝑛 (𝑥), 𝑥) + 𝑑(𝑇 𝑛(𝑦), 𝑦)] untuk setiap 𝑥, 𝑦 ∈ 𝑋, dimana 0 ≤ 𝑘 ≤ 1 2 . sehingga terbukti bahwa 𝑇 𝑛 adalah pemetaan kannan. contoh amanatul husnia, hairur rahman 120 volume 3 no. 2 mei 2014 misalkan 𝑋 adalah himpunan bilangan real dengan −2 < 𝑥 < 2 dan didefinisikan metrik dengan 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| 𝑇 adalah pemetaan pada ruang metrik (𝑋, 𝑑) ke dalam dirinya sendiri dengan 𝑇𝑥 = { − 𝑥 4 , |𝑥| ≤ 1 𝑥 4 , 1 < |𝑥| < 2 maka 𝑑(𝑇𝑥, 𝑇𝑦) = |𝑇𝑥 − 𝑇𝑦| ≤ |𝑇𝑥| + |𝑇𝑦| = | 𝑥 4 | + | 𝑥 4 | sehingga mengakibatkan 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 1 4 |𝑥| + |𝑦| (3.3) untuk setiap 𝑥, 𝑦 ∈ 𝑋. 𝑑(𝑥, 𝑇𝑥) = |𝑥 − 𝑇𝑥| ≥ |𝑥| − |𝑇𝑥| dan 𝑑(𝑦, 𝑇𝑦) = |𝑦 − 𝑇𝑦| ≥ |𝑦| − |𝑇𝑦| sehingga 𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦) ≥ |𝑥| − |𝑇𝑥| + |𝑦| − |𝑇𝑦| = |𝑥| − | 𝑥 4 | + |𝑦| − | 𝑦 4 | = |𝑥| (1 − 1 4 ) + |𝑦|(1 − 1 4 ) = |𝑥| ( 3 4 ) + |𝑦|( 3 4 ) = 3 4 (|𝑥| + |𝑦|) maka 𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦) ≥ 3 4 (|𝑥| + |𝑦|) (3.4) oleh karena itu, dengan menggunakan ketaksamaan (3.3) dan (3.4) diperoleh 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 1 3 [𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)] untuk setiap 𝑥, 𝑦 ∈ 𝑋. jadi, terbukti bahwa 𝑇 adalah pemetaan kannan. dari teorema dan contoh di atas akan ditunjukkan bahwa pemetaan kannan mempunyai titik tetap yang tunggal. pertama akan ditunjukkan bahwa ruang metrik (𝑋, 𝑑) adalah lengkap, dapat diketahui bahwa kondisi pemetaan kannan adalah 𝑑(𝑇(𝑥), 𝑇(𝑦)) ≤ 𝑘[𝑑(𝑇(𝑥), 𝑥) + 𝑑(𝑇(𝑦), 𝑦)] (3.5) untuk setiap 𝑥, 𝑦 ∈ 𝑋, dimana 0 ≤ 𝑘 < 1 2 . 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+1(𝑥)) = 𝑑(𝑇𝑇 𝑛−1(𝑥), 𝑇𝑇 𝑛(𝑥)) dengan menggunakan ketaksamaan (3.5), diperoleh 𝑑(𝑇 𝑛(𝑥), 𝑇 𝑛+1(𝑥)) ≤ 𝑘[𝑑(𝑇 𝑛−1(𝑥), 𝑇 𝑛 (𝑥)) + 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+1(𝑥))] mengakibatkan (1 − 𝑘)𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+1(𝑥)) ≤ 𝑘𝑑(𝑇 𝑛−1(𝑥), 𝑇 𝑛 (𝑥)) 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+1(𝑥)) ≤ 𝑘 (1−𝑘) 𝑑(𝑇 𝑛−1(𝑥), 𝑇 𝑛 (𝑥)) (3.6) dengan mengubah 𝑛 menjadi 𝑛 − 1 dari persamaan di atas, diperoleh 𝑑(𝑇 𝑛−1(𝑥), 𝑇 𝑛 (𝑥)) ≤ 𝑘 (1 − 𝑘) 𝑑(𝑇 𝑛−2(𝑥), 𝑇 𝑛−1(𝑥)) maka dari ketaksamaan (3.6), diperoleh 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+1(𝑥)) ≤ 𝑘 (1−𝑘) 𝑘 (1−𝑘) 𝑑(𝑇 𝑛−2(𝑥), 𝑇 𝑛−1(𝑥)) = ( 𝑘 (1−𝑘) )2𝑑(𝑇 𝑛−2(𝑥), 𝑇 𝑛−1(𝑥)) ≤ ( 𝑘 (1−𝑘) )𝑛 𝑑((𝑥), 𝑇(𝑥)) untuk setiap 𝑥, 𝑦 ∈ 𝑋, dimana 0 ≤ 𝑘 < 1 2 . 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+𝑟 (𝑥)) ≤ 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+(𝑥)) + 𝑑(𝑇 𝑛+1(𝑥), 𝑇 𝑛+2(𝑥)) + ⋯ + 𝑑(𝑇 𝑛+𝑟−1(𝑥), 𝑇 𝑛+𝑟 (𝑥)) dengan menggunakan ketaksamaan segitiga, diperoleh 𝑑(𝑇 𝑛 (𝑥), 𝑇 𝑛+𝑟 (𝑥)) ≤ ( 𝑘 (1−𝑘) ) 𝑛 𝑑((𝑥), 𝑇(𝑥))+≤ ( 𝑘 (1−𝑘) ) 𝑛+1 𝑑((𝑥), 𝑇(𝑥)) + … +≤ ( 𝑘 (1−𝑘) )𝑛+𝑟−1𝑑((𝑥), 𝑇(𝑥)) = ( 𝑘 (1−𝑘) ) 𝑛 [1 + ( 𝑘 (1−𝑘) ) + ( 𝑘 (1−𝑘) ) 2 + ⋯ + ( 𝑘 (1−𝑘) ) 𝑛+𝑟−1 ] 𝑑((𝑥), 𝑇(𝑥)) ≤ ( 𝑘 (1−𝑘) ) 𝑛 [1 + ( 𝑘 (1−𝑘) ) + ( 𝑘 (1−𝑘) ) 2 + ⋯ + ∞] 𝑑((𝑥), 𝑇(𝑥)) dengan rasio 𝑘 (1−𝑘) < 1, maka barisan tersebut konvergen yaitu konvergen terhadap 1 1−( 𝑘 (1−𝑘) ) . maka 1 + ( 𝑘 (1 − 𝑘) ) + ( 𝑘 (1 − 𝑘) ) 2 + ⋯ + ∞ = 1 1 − ( 𝑘 (1 − 𝑘) ) = 1 − 𝑘 1 − 2𝑘 dari persamaan di atas diperoleh teorema titik tetap di ruang banach cauchy – issn: 2086-0382 121 𝑑(𝑇 𝑛(𝑥), 𝑇 𝑛+𝑟 (𝑥)) ≤ ( 𝑘 (1 − 𝑘) ) 𝑛 ( 1 − 𝑘 1 − 2𝑘 ) 𝑑((𝑥), 𝑇(𝑥)) karena barisan 〈𝑇 𝑛 (𝑥)〉 adalah barisan cauchy yang konvergen maka 𝑥 mempunyai titik tetap tunggal yaitu 𝑇(𝑥) = 𝑥. teorema 3.1.4 misalkan 𝑥, 𝑦 ∈ 𝑋 dan 𝛼, 𝛽 ∈ [0, 1 2 ] dengan 𝑑(𝑇(𝑥), 𝑇(𝑦)) ≤ 𝛼[𝑑(𝑇(𝑥), 𝑥) + 𝑑(𝑇(𝑦), 𝑦) + 𝛽𝑑(𝑥, 𝑦)] maka 𝑇 mempunyai titik tetap tunggal di 𝑋 (fisher, 1976:193-194). bukti ambil titik 𝑥0 ∈ 𝑋 dan 〈𝑥𝑛 〉 barisan di 𝑋 didefinisikan dengan 𝑥𝑛 = 𝑇𝑥𝑛−1, 𝑛 ∈ 𝑁 maka 𝑥0, 𝑥1 = 𝑇(𝑥0), 𝑥2 = 𝑇(𝑥1), 𝑥3 = 𝑇(𝑥2), … , 𝑥𝑛−1 = 𝑇(𝑥𝑛−2), 𝑥𝑛 = 𝑇(𝑥𝑛−1) akan ditunjukkan bahwa 〈𝑥𝑛〉 adalah barisan cauchy 𝑑(𝑥1, 𝑥2) = 𝑑(𝑇(𝑥0), 𝑇(𝑥1)) ≤ 𝛼[𝑑(𝑇(𝑥0), 𝑥0) + 𝑑(𝑇(𝑥1), 𝑥1)] + 𝛽𝑑(𝑥0, 𝑥1) = 𝛼[𝑑(𝑥1, 𝑥0) + 𝑑(𝑥2, 𝑥1)] + 𝛽𝑑(𝑥0, 𝑥1) = 𝛼[𝑑(𝑥1, 𝑥2) + 𝑑(𝑥0, 𝑥1)] + 𝛽𝑑(𝑥0, 𝑥1) ≤ 𝛼 1−𝛼 𝑑(𝑥0, 𝑥1) + 𝛽𝑑(𝑥0, 𝑥1) 𝑑(𝑥2, 𝑥3) = 𝑑(𝑇(𝑥1), 𝑇(𝑥2)) ≤ 𝛼[𝑑(𝑇(𝑥1), 𝑥1) + 𝑑(𝑇(𝑥2), 𝑥2)] + 𝛽𝑑(𝑥1, 𝑥2) = 𝛼[𝑑(𝑥2, 𝑥1) + 𝑑(𝑥3, 𝑥2)] + 𝛽𝑑(𝑥1, 𝑥2) = 𝛼[𝑑(𝑥2, 𝑥3) + 𝑑(𝑥1, 𝑥2)] + 𝛽𝑑(𝑥1, 𝑥2) ≤ 𝛼 1−𝛼 𝑑(𝑥1, 𝑥2) + 𝛽𝑑(𝑥1, 𝑥2) ≤ ( 𝛼 1−𝛼 )2𝑑(𝑥0, 𝑥1) + 𝛽 2𝑑(𝑥0, 𝑥1) 𝑑(𝑥3, 𝑥4) = 𝑑(𝑇(𝑥2), 𝑇(𝑥3)) ≤ 𝛼[𝑑(𝑇(𝑥2), 𝑥2) + 𝑑(𝑇(𝑥3), 𝑥3)] + 𝛽𝑑(𝑥2, 𝑥3) = 𝛼[𝑑(𝑥3, 𝑥2) + 𝑑(𝑥4, 𝑥3)] + 𝛽𝑑(𝑥2, 𝑥3) = 𝛼[𝑑(𝑥3, 𝑥4) + 𝑑(𝑥2, 𝑥3)] + 𝛽𝑑(𝑥2, 𝑥3) ≤ 𝛼 1−𝛼 𝑑(𝑥2, 𝑥3) + 𝛽𝑑(𝑥2, 𝑥3) ≤ ( 𝛼 1−𝛼 )3𝑑(𝑥0, 𝑥1) + 𝛽 3𝑑(𝑥0, 𝑥1) . . . 𝑑(𝑥𝑛 , 𝑥𝑛−1) = 𝑑(𝑇(𝑥𝑛), 𝑇(𝑥𝑛−1)) ≤ 𝛼[𝑑(𝑇(𝑥𝑛 ), 𝑥𝑛 ) + 𝑑(𝑇(𝑥𝑛−1), 𝑥𝑛−1)] + 𝛽𝑑(𝑥𝑛, 𝑥𝑛−1) = 𝛼[𝑑(𝑥𝑛, 𝑥𝑛−1) + 𝑑(𝑥𝑛+1, 𝑥𝑛 )] + 𝛽𝑑(𝑥𝑛 , 𝑥𝑛−1) = 𝛼[𝑑(𝑥𝑛−1 , 𝑥𝑛+1) + 𝑑(𝑥𝑛 , 𝑥𝑛−1)] + 𝛽𝑑(𝑥𝑛 , 𝑥𝑛−1) ≤ 𝛼 1−𝛼 𝑑(𝑥𝑛 , 𝑥𝑛−1) + 𝛽𝑑(𝑥𝑛 , 𝑥𝑛−1) ≤ ( 𝛼 1−𝛼 )𝑛 𝑑(𝑥𝑛, 𝑥𝑛−1) + 𝛽 𝑛 𝑑(𝑥𝑛, 𝑥𝑛−1) secara umum diperoleh jika 𝑚 merupakan bilangan bulat positif maka berlaku 𝑑(𝑥𝑚 , 𝑥𝑚+1) ≤ ( 𝛼 1−𝛼 )𝑚 𝑑(𝑥0, 𝑥1) + 𝛽 𝑚 𝑑(𝑥0, 𝑥1) = (𝑞)𝑚 𝑑(𝑥0, 𝑥1) + 𝑟 𝑚 𝑑(𝑥0, 𝑥1) (3.7) dengan 𝑞 = 𝛼 1−𝛼 dan 𝑟 = 𝛽. karena 𝛼, 𝛽 ∈ [0, 1 2 ], jelas bahwa 0 < 𝑞 < 1 dan 0 < 𝑟 < 1 ambil > 0 dan ambil bilangan 𝑛, 𝑚 ∈ ℕ dengan sifat ketaksamaan segitiga pada metrik dan jumlah dari barisan geometri, didapatkan untuk 𝑛 > 𝑚 𝑑(𝑥𝑚 , 𝑥𝑛 ) ≤ 𝑑(𝑥𝑚 , 𝑥𝑚+1) + 𝑑(𝑥𝑚+1, 𝑥𝑚+2) + 𝑑(𝑥𝑚+2, 𝑥𝑚+3) + ⋯ + 𝑑(𝑥𝑛−1, 𝑥𝑛) ≤ (𝑞𝑚 + 𝑞𝑚+1 + 𝑞𝑚+2 + 𝑞𝑚+3 + ⋯ + 𝑞𝑛−1)𝑑(𝑥0, 𝑥1) + (𝑟 𝑚 + 𝑟𝑚+1 + 𝑟𝑚+2 + 𝑟𝑚+3 + ⋯ + 𝑟𝑛−1)𝑑(𝑥0, 𝑥1) ≤ 𝑞𝑚 (1 + 𝑞 + 𝑞2 + 𝑞3 + ⋯ + 𝑞𝑛−𝑚−1)𝑑(𝑥0, 𝑥1) + 𝑟𝑚 (1 + 𝑟 + 𝑟2 + 𝑟3 + ⋯ + 𝑟𝑛−𝑚−1) 𝑑(𝑥0, 𝑥1) = 𝑞𝑚 ∑ 𝑞𝑖𝑛−𝑚−1𝑖=0 𝑑(𝑥0, 𝑥1) + 𝑟 𝑚 ∑ 𝑟𝑖𝑛−𝑚−1𝑖=0 𝑑(𝑥0, 𝑥1) = 𝑞𝑚 ∑ 𝑞𝑖∞𝑖=0 𝑑(𝑥0, 𝑥1) + 𝑟 𝑚 ∑ 𝑟𝑖∞𝑖=0 𝑑(𝑥0, 𝑥1) (3.8) karena 0 < 𝑞 < 1 dan 0 < 𝑟 < 1, maka deret ∑ 𝑞𝑖∞𝑖=0 pada ketaksamaan (3.8) konvergen ke 1 1−𝑞 dan deret ∑ 𝑟𝑖∞𝑖=0 konvergen ke 1 1−𝑟 sehingga diperoleh 𝑑(𝑥𝑚 , 𝑥𝑛 ) ≤ 𝑞𝑚 1 − 𝑞 𝑑(𝑥0, 𝑥1) + 𝑟𝑚 1 − 𝑟 𝑑(𝑥0, 𝑥1) untuk 𝑛 > 𝑚 > 𝑁. karena 𝑙𝑖𝑚𝑚→∞𝑞 𝑚 = 0 dan 𝑙𝑖𝑚𝑚→∞𝑟 𝑚 = 0, maka 𝑙𝑖𝑚𝑚→∞𝑑(𝑥𝑚, 𝑥𝑛) = 0 maka 〈𝑥𝑛〉 adalah barisan cauchy. karena 𝑋 lengkap, maka 〈𝑥𝑛〉 konvergen. katakan 𝑥𝑛 → 𝑥. artinya sedemikian sehingga jika 𝑁 ∈ ℕ, untuk setiap 𝑛 ≥ 𝑁 berlaku 𝑑(𝑥𝑛 , 𝑥) < 𝜀 2 akan ditunjukkan bahwa 𝑥 adalah titik tetap dari pemetaan 𝑇. dari sifat ketaksamaan segitiga dan prinsip fisher, didapatkan 𝑑(𝑥, 𝑇(𝑥)) ≤ 𝑑(𝑥, 𝑥𝑛) + 𝑑(𝑥𝑛 , 𝑇(𝑥)) = 𝑑(𝑥, 𝑥𝑛) + (𝑇(𝑥𝑛−1), 𝑇(𝑥)) ≤ 𝑑(𝑥, 𝑥𝑛) + 𝛼[𝑑(𝑇(𝑥𝑛−1), 𝑥𝑛−1) + 𝑑(𝑇(𝑥), 𝑥)] + 𝛽𝑑(𝑥𝑛−1, 𝑥) karena 𝑥𝑛 → 𝑥 diperoleh ketaksamaan 𝑑(𝑥, 𝑇(𝑥)) < 𝜀 2 + 𝛼[𝑑(𝑇(𝑥𝑛−1), 𝑥𝑛−1) + 𝑑(𝑇(𝑥), 𝑥)] + 𝛽𝑑(𝑥𝑛−1, 𝑥) amanatul husnia, hairur rahman 122 volume 3 no. 2 mei 2014 < 𝜀 2(1−𝛼) + ( 𝛼 1−𝛼 ) 𝑑(𝑇(𝑥𝑛−1), 𝑥𝑛−1) + 𝛽𝑑(𝑥𝑛−1, 𝑥) = 𝜀 2(1−𝛼) + ( 𝛼 1−𝛼 ) 𝑑(𝑥𝑛−1, 𝑥𝑛) + 𝛽𝑑(𝑥𝑛−1, 𝑥) menurut ketaksamaan (3.8) 𝑑(𝑥𝑛−1, 𝑥𝑛) ≤ ( 𝛼 1 − 𝛼 ) 𝑛−1 𝑑(𝑥0, 𝑥1) + (𝛽)𝑛−1 𝑑(𝑥0, 𝑥1) sehingga diperoleh 𝑑 ≤ 𝜀 2(1−𝛼) + ( 𝛼 1−𝛼 ) ( 𝛼 1−𝛼 ) 𝑛−1 𝑑(𝑥0, 𝑥1) + ( 𝛽 1−𝛽 ) 𝑛−1 𝑑(𝑥0, 𝑥1) + 𝛽𝑑(𝑥𝑛−1, 𝑥) = 𝜀 2(1−𝛼) + ( 𝛼 1−𝛼 ) 𝑛 𝑑(𝑥0, 𝑥1) + ( 𝛽 1−𝛽 ) 𝑛 𝑑(𝑥0, 𝑥1) = 𝜀 2(1−𝛼) + 𝑞𝑛 𝑑(𝑥0, 𝑥1) + (𝑟) 𝑛−1 𝑑(𝑥0, 𝑥1) untuk 𝑛 → ∞, maka 𝑞𝑛 → 0 dan 𝑟𝑛 → 0, sehingga 𝑑(𝑥, 𝑇(𝑥)) < 2(1 − 𝛼) karena sebarang, maka 𝑑(𝑥, 𝑇(𝑥)) = 0 atau 𝑇(𝑥) = 𝑥, dengan demikian terbukti bahwa pemetaan fisher pada 𝑋 yang lengkap mempunyai titik tetap tunggal. contoh misalkan 〈𝑥𝑛〉 adalah barisan cauchy di 𝑙2 akan ditunjukkan bahwa barisan di 𝑙2 tersebut konvergen, untuk setiap 𝑛 ∈ 𝑁 dan 𝑥𝑛 = (𝑥𝑛1, 𝑥𝑛2, 𝑥𝑛3, … ) yang telah didefinisikan pada ruang 𝑙2 atau ∑ |𝑥𝑛𝑘 | 2∞ 𝑘=1 konvergen terhadap 𝑛. dimana 〈𝑥𝑛〉 adalah barisan cauchy, untuk setiap > 0 terdapat bilangan bulat positif 𝑁, sehingga √∑|𝑥𝑛𝑘 − 𝑥𝑚𝑘 | 2 ∞ 𝑘=1 < dengan 𝑚, 𝑛 > 𝑁. dengan menggunakan definisi 𝑙2, diperoleh ∑ |𝑥𝑛𝑘 − 𝑥𝑚𝑘 | 2∞ 𝑘=1 < 2, 𝑚, 𝑛 > 𝑁 sehingga |𝑥𝑛𝑘 − 𝑥𝑚𝑘 | < , 𝑚, 𝑛 > 𝑁 untuk setiap 𝑘 ∈ 𝑁. maka untuk setiap 𝑘, (𝑥𝑛𝑘 ) adalah barisan cauchy di 𝐶 sehingga 𝑙𝑖𝑚𝑛→∞𝑥𝑛𝑘 ada ketika 𝐶 merupakan ruang metrik lengkap. 𝑙𝑖𝑚𝑛→∞𝑥𝑛𝑘 dimana 𝑥 = (𝑥1, 𝑥2, 𝑥3, … ) akan ditunjukkan bahwa 𝑥 ∈ 𝑙2 dan 〈𝑥𝑛 〉 konvergen terhadap 𝑥. maka 𝑙2 dapat dikatakan lengkap ∑ |𝑥𝑛𝑘 − 𝑥𝑚𝑘 | 2𝑟 𝑘=1 < 2, 𝑚, 𝑛 > 𝑁 dapat diperhatikan bahwa 𝑟 = 1,2,3, … maka 𝑛 merupakan titik dan 𝑙𝑖𝑚𝑚→∞𝑥𝑚𝑘 = 𝑥 · 𝑘 ∑ |𝑥𝑛𝑘 − 𝑥 · 𝑘| 2𝑟 𝑘=1 < 2, 𝑛 > 𝑁 ambil titik (𝑎1, 𝑎2, 𝑎3, … , 𝑎𝑟 ), (𝑏1, 𝑏2, 𝑏3, … , 𝑏𝑟 ), (𝑐1, 𝑐2, 𝑐3, … , 𝑐𝑟 ) ∈ 𝐶 𝑟 dengan menggunakan ketaksamaan segitiga di 𝐶 𝑟, diperoleh √∑|𝑎𝑘 − 𝑐𝑘 | 2 𝑟 𝑘=1 ≤ √∑|𝑎𝑘 − 𝑏𝑘 | 2 𝑟 𝑘=1 + √∑|𝑏𝑘 − 𝑐𝑘 | 2 𝑟 𝑘=1 misalkan 𝑎𝑘 = 𝑥 · 𝑘, 𝑏𝑘 = 𝑥𝑛𝑘 dan 𝑐𝑘 = 0, sehingga diperoleh √∑|𝑥 · 𝑘|2 𝑟 𝑘=1 ≤ √∑|𝑥 · 𝑘 − 𝑥𝑛𝑘 | 2 𝑟 𝑘=1 + √∑|𝑥𝑛𝑘 | 2 𝑟 𝑘=1 ≤ + √∑ |𝑥𝑛𝑘 | 2𝑟 𝑘=1 ≤ + √∑ |𝑥𝑛𝑘 | 2∞ 𝑘=1 jika 𝑛 > 𝑁 dan konvergen terhadap ∑ |𝑥𝑛𝑘 | 2∞ 𝑘=1 , tentu 𝑥 ∈ 𝑙2. oleh karena itu, dapat dilihat pada ketaksamaan segitiga sebelumnya √∑|𝑥𝑛𝑘 − 𝑥 · 𝑘| 2 ∞ 𝑘=1 < , 𝑛 > 𝑁 yang mengakibatkan bahwa barisan 〈𝑥𝑛〉 konvergen terhadap 𝑥 dan 𝑙2 adalah lengkap. penutup dari pembahasan pada bab sebelumnya, dapat ditarik kesimpulan bahwa teorema titik tetap banach juga dikenal sebagai teorema pemetaan kontraksi, sebelum mencari ketunggalan titik tetap dapat dicari kelengkapan ruang metrik, dikatakan lengkap jika suatu barisan cauchy tersebut konvergen, sehingga dapat dibuktikan bahwa teorema titik tetap di ruang banach mempunyai titik tetap yang tunggal. dalam membuktikan teorema titik tetap di ruang banach, diperlukan suatu teorema yaitu: pemetaan kannan 𝑑(𝑇(𝑥), 𝑇(𝑦)) ≤ 𝑘[𝑑(𝑇(𝑥), 𝑥) + 𝑑(𝑇(𝑦), 𝑦)] untuk setiap 𝑥, 𝑦 ∈ 𝑋 dan 𝛼 ∈ [0, 1 2 ], maka 𝑇 mempunyai titik tetap tunggal di 𝑋. dan pemetaan fisher 𝑑(𝑇(𝑥), 𝑇(𝑦)) ≤ 𝛼[𝑑(𝑇(𝑥), 𝑥) + 𝑑(𝑇(𝑦), 𝑦) + 𝛽𝑑(𝑥, 𝑦)] untuk setiap 𝑥, 𝑦 ∈ 𝑋 dan 𝛼, 𝛽 ∈ [0, 1 2 ], maka 𝑇 mempunyai titik tetap tunggal di 𝑋. teorema titik tetap di ruang banach cauchy – issn: 2086-0382 123 1. saran dalam penulisan skripsi ini, penulis menggunakan pemetaan kannan dan pemetaan fisher untuk membuktikan titik tetap di ruang banach. oleh karena itu penulis memberikan saran kepada pembaca yang tertarik pada permasalahan ini supaya mengembangkannya dengan menggunakan pada fungsi ruang yang lainnya bibliography [1] abdussakir. 2009. pentingnya matematika dalam pemikiran islam: state islamic university of malang. (online: http://abdussakir.wordpress.com/artikel/ diakses 20 desember 2013). [2] al-mahali, m.j.a. dan as-suyuthi, a.j.a.. 2010. tafsir jalalain 1. surabaya: bina ilmu surabaya [3] al-maraghi, m.a.. 1993. tafsir al-maraghi 2. mesir: musthafa al-babi al-halabi [4] banach, s.. 1992. sur les operations dans les ensembles abstraits et leur application aux equations integrales, fund. math. 133-181. [5] bartle, r.g. and sherbert, d.r.. (2000). introduction to real analysis, third edition. new york: john wiley and sons. [6] cohen, g.. 2003. a course in modern analysis and its applications. united states of america: cambridge university press. [7] fisher. 1976. a fixed point theorem for compact metric space. publ. inst. math. 25, 193-194. [8] ghozali, m.s.. 2010. analisis real 1. bandung. pusat perbukuan departemen pendidikan [9] kreyzig, e.. 1989. introductory functional analysis with application. united states of america. [10] kannan, r.. 1969. some result on fixed point theorems, bull. calcutta. math. soc, vol. 60, 71-78. [11] nasoetion, a.h.. 1980. landasan matematika. jakarta: pt. bharatara karya aksara. [12] quth, s.. 2002. tafsir fi dzilalil qur’an jilid 24. jakarta: bina insani [13] rynne, b.p. and youngson, m.a.. 2008. linear functional analysis. new york: springgerverlag. [14] soemantri. 1988. analisis real 1. jakarta: universitas terbuka [15] wijaya, b.t.. 2011. spectrum detour graf mpartisi komplit. skripsi s1. malang: uin maulana malik ibrahim malang. [16] wilde, f.i.. 2003. topological vector space. london. http://abdussakir.wordpress.com/artikel/ pendahuluan kajian teori 1. ruang metrik 2. himpunan terbuka dan himpunan tertutup 3. kekonvergenan dan kelengkapan 4. ruang vektor bernorma 5. kekonvergenan dalam ruang bernorma 6. teorema titik tetap 7. pemetaan pembahasan penutup 1. saran bibliography determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 630-640 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 06, 2023 reviewed: april 09, 2023 accepted: april 16, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.20542 determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita*, i gusti putu purnaba, donny citra lesmana department of mathematics, ipb university, bogor, indonesia email : dianpuspita@apps.ipb.ac.id abstract in life will always be surrounded by various things that can happen, which can lead to the risk of death and financial loss. one way to overcome is insurance. one type of insurance is term life insurance. term life insurance is an insurance that provides protection for a certain period that has been agreed in the policy. insurance premiums and benefits in the long term will be affected by interest rates . one of the models that can be used is the cox ingersoll ross model (cir). this research purposes to simulate the cir model that will be carried out to determine interest rates for calculating term life insurance premiums for five years, with premiums paid at the beginning of the 1 m interval and benefits paid at the end of the 1 m interval when the participant dies. the method to estimate parameters in cir model is ordinary least square. the results of this research is the cir model can be applied to calculate the term life insurance premiums for five years and the premium calculation results show that the amount of the premium increase every year with varying benefits. the contribution of this research as information to insurance companies regarding the amount of premiums paid monthly. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc by-sa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: premium; term life insurance; cir model; varying interest rates introduction insurance is a risk transfer from a financial loss caused by the deaths of insurance participants to an insurance company based on a policy agreement. one type of insurance is life insurance. life insurance is insurance to cover the financial costs of death. life insurance consists of four types namely whole life insurance, term life insurance, pure endowment insurance and endowment insurance [1]. in insurance, there is an agreement called a policy between insurance participants and insurance companies that include participants' obligation to pay premium contributions to insurance companies and insurance liability to pay the benefits if something happens to insurance participants has been compromised in policy, and other agreements associated with insurance. premiums and benefits are affected by interest rates. normally the interest rates used are constant even though the payment of the premiums for long-term payments on which interest rates will change over time. the cox ingersoll ross model is one of the http://dx.doi.org/10.18860/ca.v7i4.20542 mailto:dianpuspita@apps.ipb.ac.id https://creativecommons.org/licenses/by-sa/4.0/ determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 631 stochastic models used for fluctuating rates in period of time. the cir model has a characteristic that the movement of interest rates would lead to the average interest rate and cir model ensures that interest rates will be positive [2]. research relating to the cir model has been done, such as the moment for solution of the cir model [3],[4] then installment joint life insurance actuarial models with the stochastic interest rates [5] , premiums calculation for life insurance [6],[7] and the cox ingersoll ross estimate model at the indonesian bank using the maximum likelihood estimation method [8]. the cir model involving life insurance premiums has been carried out, by determining long-term life insurance premiums with interest rates following vasicek and the cox ingersoll ross [9] and calculating joint life insurance premiums with vasicek and cir [10]. research related to varying benefits and varying rates of interest has been researched regarding the annual net premium of term life insurance for cases of multiple decrements with varying interest rates [11] and single net premium of units linked term life insurance using the point to point method with varying benefits [12] . none of the literature states any limitations in using cir model in research. in addition to the financial research related to insurance and interest rates, cir model research is also in mathematic computation to solve problems related to method and formula in the cir model such as research about applied the role of adaptivity in a numerical method for the cox – ingersoll – ross model [13] and formula and full parameter in cir model [14], [15]. research that has been done before still calculates annual insurance premium payments and annual single premium payments paid at the beginning of the year. therefore, this research aims to determine varying interest rates in the future using cir model simulation to calculate term life insurance premiums 5 years with premiums paid at the beginning of 1 𝑚 year intervals with 𝑚 = 12 months or premiums payment every month and benefits paid at the end of 1 𝑚 year interval of death or at the end of each month of death with the amount varying benefits each year. methods data this research used secondary data such as monthly history data bi interest rates period january 2015 until december 2020 which was accesed from website https://www.bps.go.id/indicator/13/379/12/bi rates.html and indonesian life table 2019. the data analysis process was done using octave software and microsoft excel. estimation cir model the cox ingersoll ross model is one of the stochastic models used for fluctuating interest rates in period of time. the cir model can be used to estimate interest rates that changes in period of time in the future. the algorithm in the process of determining cir model following the steps: a. input monthly history data bi interest rates period januari 2015 until december 2019. b. estimate parameters cir model using algorithm as following the equation [9] . 𝜅 = ∑ 𝑟𝑖 𝑛−1 𝑖=1 ∑ ( 1 𝑟𝑖 )𝑛−1𝑖=1 − (𝑛 − 1) 2 − ∑ ( 1 𝑟𝑖 ) ∑ 𝑟𝑖+∆𝑡 + 𝑛−1 𝑖=1 𝑛−1 𝑖=1 (𝑛 − 1) ∑ 𝑟𝑖+∆𝑡 𝑟𝑖 𝑛−1 𝑖=1 (∑ 𝑟𝑖 𝑛−1 𝑖=1 ∑ ( 1 𝑟𝑖 )𝑛−1𝑖=1 − (𝑛 − 1) 2) ∆𝑡 (1) https://www.bps.go.id/indicator/13/379/12/bi%20rate.html determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 632 𝜽 = ∑ 𝒓𝒊+∆𝒕 𝒓𝒊 𝒏−𝟏 𝒊=𝟏 (∑ 𝒓𝒊 𝒏−𝟏 𝒊=𝟏 ) − (𝒏 − 𝟏)(∑ 𝒓𝒊+∆𝒕 𝒏−𝟏 𝒊=𝟏 ) ∑ 𝒓𝒊 𝒏−𝟏 𝒊=𝟏 ∑ ( 𝟏 𝒓𝒊 )𝒏−𝟏𝒊=𝟏 − (𝒏 − 𝟏) 𝟐 − ∑ ( 𝟏 𝒓𝒊 ) ∑ 𝒓𝒊+∆𝒕 + 𝒏−𝟏 𝒊=𝟏 𝒏−𝟏 𝒊=𝟏 (𝒏 − 𝟏) ∑ 𝒓𝒊+∆𝒕 𝒓𝒊 𝒏−𝟏 𝒊=𝟏 (𝟐) 𝝈 = √ 𝟏 𝒏 − 𝟐 ∑ ( 𝒓𝒊+∆𝒕 √𝒓𝒊 − ( 𝜿𝜽∆𝒕 √𝒓𝒊 + 𝟏 − 𝜿∆𝒕𝒓𝒊 √𝒓𝒊 )) 𝟐 𝒏−𝟏 𝒊=𝟏 (𝟑) where 𝜿 is the speed of mean reversion, 𝜽 is the long-term value, 𝝈 is the volatility interest rate, 𝒓𝒊 is the interest rates period of time, 𝒏 is a lot of data used, and ∆𝒕 is the interval of time. c. fit test cir model using the equation mape is one of the statistics method used for the level of forecasting accuracy. the better model can be used when mape has the lower value. mape is calculated following the equation: 𝑀𝐴𝑃𝐸 = 100 𝑛 ∑ | 𝐴𝑡 − 𝐹𝑡 𝐴𝑡 | 𝑛 𝑡=1 % (4) where 𝐴𝑡 is an actual data by bi interest rates, 𝐹𝑡 is a data estimate by the cir model, and 𝑛 is a lot of data used [13]. simulation the future interest rates the cox ingersoll ross model is one of the stochastic models used for fluctuating interest rates in period of time. the cir model can be used to simulate changes in interest rates in the future. the algorithm in the interest rates simulation process for the future using the cir model following the steps: a. input parameter values such as r0, κ, θ, σ, ∆t, m dan 𝑡. where r0 is interest rates in bi rates, κ is the speed of mean reversion, θ is the long-term value, σ is the volatility interest rate, m is a lot of simulations, and ∆t is the interval of time, and 𝑡 is the due date monthly. b. generate random variable εt~ n (0,1) used to calculate interest rates r (t + ∆t). where εt is a random number which is normally distributed n (0,1), r (t + ∆t) an interest rates period of time. c. calculate future interest rates with octave software as follow the equation 𝒓𝒕+∆𝒕 = 𝒓𝒕 + 𝜿 (𝜽 − 𝒓𝒕)∆𝒕 + 𝛔√𝒓𝒕 √∆𝒕 𝜺𝒕 (5) where 𝐫𝐭 is an interest rates in the cir model d. monte carlo simulation for interest rates which is calculated following the equation �̅�𝒊 = 𝟏 𝒎 ∑ 𝒓𝒊𝒋 𝒎 𝒋=𝟏 (𝟔) determination term life insurance premium this research to determine term life insurance premium paid at the beginning of each month. the steps of research are carried out as follow: a. calculate monthly interest rates in the cir model using the equation determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 633 𝒋𝒏 = (𝟏 + 𝒊𝒏) 𝟏/𝟏𝟐 − 𝟏 (𝟕) where 𝑗𝑛 is monthly interest rates, 𝑖𝑛 is an interest rates in the cir model b. calculate the discount factor of interest rates in the cir model using the equation (𝒗∗)𝒌 = 𝟏 (𝟏 + 𝒋𝟏) × (𝟏 + 𝒋𝟐) × … × (𝟏 + 𝒋𝒌) 𝐰𝐢𝐭𝐡 𝟏 ≤ 𝐤 ≤ 𝟔𝟎 (𝟖) where (𝑣∗)𝑘 is the discount factor in 𝑘 month c. calculate the probability of death for age fraction in indonesian life table 2019 𝑞𝑥+𝑡𝑦 = 𝑦𝑞𝑥 1 − 𝑡 𝑞𝑥 (9) where 𝑞𝑥+𝑡𝑦 is probability of death at fractional age with y = 1 12 and 𝑡 = 1 12 , . . , 11 12 d. calculate the actuarial present value of the annuity of term life insurance using the equation �̈� 𝑥∶𝑛| ∗(𝑚) = 1 𝑚 ∑ (𝑣∗)𝑘 𝑚𝑛−1 𝑘=0 𝑝𝑥𝑘 𝑚 (10) where �̈� 𝑥∶𝑛| ∗(𝑚) is the actuarial present value of the annuity monthly, (𝑣∗)𝑘 is the discount factor, 𝑝𝑥𝑘 𝑚 is the probability of survival at fractional age with 𝑘 m = 1 12 , . . , 60 12 e. calculate the actuarial present value of the varying benefits of term life insurance using the equation 𝐴 𝑥∶𝑛| 1∗(𝑚) = ∑ 𝑏𝑛 (𝑣 ∗)𝑘+1 𝑚𝑛−1 𝑘=0 𝑝𝑥𝑘 𝑚 𝑞 𝑥+ 𝑘 𝑚 1 𝑚 (11) where 𝐴 𝑥∶𝑛| 1∗(𝑚) is the actuarial present value of the varying benefits, 𝑏𝑛 is the varying benefit each year, (𝑣∗)𝑘 is the discount factor, 𝑝𝑥𝑘 𝑚 is the probability of survival at fractional age with 𝑘 m = 1 12 , . . , 60 12 , 𝑞 𝑥+ 𝑘 𝑚 1 𝑚 is the probability of death at fractional age. f. calculate the value of monthly net premiums term life insurance using the equation 𝑃 𝑥∶ 𝑛| 1(𝑚) = 𝐴∗𝑥:𝑛| 1(𝑚) �̈� 𝑥∶𝑛| ∗(𝑚) (12) where 𝑃 𝑥∶ 𝑛| 1(𝑚) is the net premiums monthly , 𝐴 𝑥∶𝑛| 1∗(𝑚) is the actuarial present value of the varying benefits, and �̈�𝑥∶𝑛| ∗(𝑚) is the actuarial present value of the annuity monthly. results and discussion data this research used data monthly interest rates from the bi 7-day repo rate from january 2015 until december 2020 which was accessed from website https://www.bps.go.id/indicator/13/379/12/bi rate.html. the data monthly interest rates from bi as shown in figure 1. https://www.bps.go.id/indicator/13/379/12/bi%20rate.html determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 634 figure 1. interest rates bi 7-day repo rate estimation parameter cir model estimation parameters in the cir model calculates using equation (1) to (3) and solved with octave software. estimate parameters cir model as shown in table 1. table 1. parameters cir model parameter value 𝜅 0,5309 𝜃 0,047218 𝜎 0,7679 based on table 1, the parameter values are κ is the speed of reversion as 0,5309, θ is the long-term value as 0,047218 and σ is the volatility as 0,7679. the error value of the parameter values in the cir model using mape in equation 4 is 3,9014%. error mape <10% indicates that the estimated data is excellent for describing the actual data in bi interest rates. the comparison between data interest rates in bi and estimated data interest rates in the cir model is shown in figure 2. figure 2. interest rates bi and estimate interest rates cir model simulation the future interest rates interest rates in the cir model for january 2022 to december 2026 with numeric variable 𝑟0 = 0,035, ∆𝑡 = 1 12 , 𝑛 = 60, simulated 100 times and calculated using equation (5) with octave software. the results to estimate future interest rates as shown in table 2. 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 in te re st r a te s b i 7 -d a y r e p o r a te time (month) 0 0,01 0,02 0,03 0,04 0,05 0,06 1 2 3 4 5 6 7 8 9 10 11 12 v a lu e o f in te re st r a te s time (month) bi interest rates interest rates cir model determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 635 table 2. interest rates in the cir model month interest rates (%) month interest rates (%) month interest rates (%) month interest rates (%) month interest rates (%) 1 0,025255 13 0,050327 25 0,053169 37 0,052515 49 0,036882 2 0,027668 14 0,056575 26 0,051294 38 0,047182 50 0,029854 3 0,025565 15 0,056146 27 0,044594 39 0,057194 51 0,040682 4 0,018886 16 0,055893 28 0,041811 40 0,053692 52 0,027046 5 0,016036 17 0,053001 29 0,037798 41 0,056603 53 0,048051 6 0,023475 18 0,045527 30 0,032737 42 0,061089 54 0,050728 7 0,028245 19 0,049414 31 0,038453 43 0,051467 55 0,048738 8 0,034898 20 0,052352 32 0,036794 44 0,031064 56 0,050071 9 0,029248 21 0,056978 33 0,044810 45 0,036524 57 0,043225 10 0,035791 22 0,047014 34 0,062456 46 0,038016 58 0,053388 11 0,040769 23 0,052004 35 0,072659 47 0,036822 59 0,065475 12 0,056432 24 0,052581 36 0,068307 48 0,032409 60 0,062451 the effective interest rates monthly cir model this research will calculate the value of term life insurance premiums for 5 years with premiums paid at the beginning of 1 𝑚 with 1 𝑚 = 1 12 years or premiums paid every month, so the effective interest rates in the cir model are required every month. calculate the effective interest rates every month using equation (7). the results of interest rate every month in the cir model are shown in table 3. table. 3 interest rates monthly in the cir model month interest rates (%) month interest rates (%) month interest rates (%) month interest rates (%) month interest rates (%) 1 0,002081 13 0,004100 25 0,004326 37 0,004274 49 0,003023 2 0,002277 14 0,004597 26 0,004177 38 0,003849 50 0,002454 3 0,002106 15 0,004563 27 0,003642 39 0,004646 51 0,003329 4 0,001560 16 0,004543 28 0,003419 40 0,004368 52 0,002226 5 0,001327 17 0,004313 29 0,003097 41 0,004599 53 0,003919 6 0,001936 18 0,003717 30 0,002688 42 0,004954 54 0,004132 7 0,002324 19 0,004027 31 0,003149 43 0,004191 55 0,003974 8 0,002863 20 0,004261 32 0,003016 44 0,002553 56 0,004080 9 0,002405 21 0,004629 33 0,003660 45 0,002994 57 0,003533 10 0,002935 22 0,003836 34 0,005061 46 0,003114 58 0,004344 11 0,003336 23 0,004234 35 0,005862 47 0,003018 59 0,005299 12 0,004585 24 0,004280 36 0,005521 48 0,002661 60 0,005061 from the monthly effective interest rates in the cir model, the value of the discount factor is calculated using equation (8). the results of calculating the discount factor with the monthly effective interest rate in the cir model are shown in table 4. determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 636 table 4. the value of discount factor in the cir model mont h discoun t factor (%) month discoun t factor (%) mont h discoun t factor (%) month discoun t factor (%) month discoun t factor (%) 1 0,99792 4 13 0,96678 0 25 0,91851 4 37 0,87593 3 49 0,83832 5 2 0,99565 7 14 0,96235 7 26 0,91469 3 38 0,87257 4 50 0,83627 2 3 0,99356 4 15 0,95798 6 27 0,91137 3 39 0,86853 9 51 0,83349 8 4 0,99201 6 16 0,95365 4 28 0,90826 8 40 0,86476 2 52 0,83164 6 5 0,99070 2 17 0,94955 8 29 0,90546 4 41 0,86080 4 53 0,82840 0 6 0,98878 8 18 0,94604 2 30 0,90303 7 42 0,85656 1 54 0,82499 1 7 0,98649 6 19 0,94224 7 31 0,90020 2 43 0,85298 6 55 0,82172 6 8 0,98368 0 20 0,93824 9 32 0,89749 5 44 0,85081 4 56 0,81838 7 9 0,98132 0 21 0,93392 6 33 0,89422 3 45 0,84827 4 57 0,81550 6 10 0,97844 8 22 0,93035 8 34 0,88971 9 46 0,84564 1 58 0,81197 9 11 0,97519 5 23 0,92643 5 35 0,88453 4 47 0,84309 7 59 0,80769 9 12 0,97074 4 24 0,92248 8 36 0,87967 7 48 0,84085 9 60 0,80363 2 the actuarial present value of the initial annuity term life insurance the actuarial present value of the initial annuity of term life insurance is used to obtain 5 years term life insurance premium will be calculated the actuarial present value of the annuity using equation (10) as follow. �̈� 30∶5| ∗(12) = 1 12 ∑ (𝑣∗)𝑘 59𝑘=0 𝑝𝑥𝑘 12 = 1 12 (1 + 1 (1 + 0,002081) 𝑝301 12 + 1 (1 + 0,002081)(1 + 0,002277) 𝑝302 12 + 1 (1 + 0,002081)(1 + 0,002277) … (1 + 0,005299) 𝑝3048 12 𝑝3411 12 ) = 1 12 (1 + (0,9999375)(0,997924) + (0,999875)(0,995657) + ⋯ + (0,807699)(0,9990700)(0,9990925) �̈� 30∶5| ∗(12) = 4,5220188 the results of calculating the initial annuity of 5 years term life insurance with varying interest rates in the cir model for ages 30 to 75 years are shown in figure 3. determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 637 figure 3. the actuarial present value of the initial annuity term life insurance the initial annuity value for a 30 years man is 4,522019, and the initial annuity value for a woman is 4,523243. the annuity value for a 31 years man is 4,521676, and the annuity value for a 31 years woman is 4,522986. the annuity value decreases every year because the older individual, the annuity value will decrease. the actuarial present value of the varying benefit term life insurance it is assumed that the number of benefits varies each year as in the first year is 𝑏1 = 1 unit, in the second year is 𝑏2 = 1,5 unit, in three years is 𝑏3 = 2 unit, in four years is 𝑏4 = 2,5 unit, and in five years is 𝑏5 = 3 unit. calculate the actuarial present value of the varying benefits paid at the end of 1 12 years of death for an individual aged 30 years using equation (11) as follow. 𝐴 30∶5| 1∗(12) = ∑ 𝑏𝑛 (𝑣 ∗)𝑘+159𝑘=0 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 𝐴 30∶5| 1∗(12) = ∑ 𝑏1 (𝑣 ∗)𝑘+111𝑘=0 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 + ∑ 𝑏2 (𝑣 ∗)𝑘+123𝑘=12 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 + ∑ 𝑏3 (𝑣 ∗)𝑘+135𝑘=24 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 + ∑ 𝑏4 (𝑣 ∗)𝑘+147𝑘=36 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 + ∑ 𝑏5 (𝑣 ∗)𝑘+159𝑘=48 𝑝30𝑘 12 𝑞 30+ 𝑘 12 1 12 = 1(0,00073966) + 1,5(0,00075942) + 2(0,00077835) + 2,5(0,00079175) + 3(0,0080956) 𝐴 30∶5| 1∗(12) = 0,00784355 the results of calculating the actuarial present value of 5 years term life insurance with varying benefits for ages 30 to 75 years are shown in figure 4. figure 4. the actuarial present value of the varying benefits term life insurance 4,300000 4,350000 4,400000 4,450000 4,500000 4,550000 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 v a lu e o f in it ia l a n n u ty initial annuity in man initial annuity in woman 0,0000000 0,1000000 0,2000000 0,3000000 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 b e n e fi t v a lu e age (year) benefit value for man benefit value for woman determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 638 figure 4 shows the value of the varying benefits of term life insurance with various interest rates in the cir model from ages 30 to 75 years for man and woman. the benefits value for a man aged 30 is 0,0078436, and the benefit value for a woman aged 30 is 0,0058254. the value of benefits for a man aged 31 is 0,0084167, and the value of benefits for a woman aged 31 is 0,0062626. if one unit of benefits value is rp. 10.000.000 so benefit value for man aged 30 is 0,0078436 × rp. 10.000.000 = rp 78.436. the results of the varying benefits in term life insurance for men and women increase every year because the older a person is the greater the possibility of someone's death, so the varying benefits obtained will increase yearly. net premium term life insurance the monthly net premium value of term life insurance with interest rates in the cir model and varying benefits using the equivalence principle in equation (12). the results of calculating the monthly net premium for five years term life insurance paid at the beginning of each month for individuals aged 30 years are as follows 𝑃 30: 5| 1(12) = 𝐴∗ 30:5| 1(12) �̈� 30∶5| ∗(12) = 0,0078436 4,522019 = 0,00173452 the results of calculating the net premium term life insurance with various interest rates in the cir model for ages 30 to 75 years are shown in figure 5. figure 5. the net premium term life insurance figure 5 shows that the value of term life insurance premiums monthly for a man aged 30 is 0,0017345 and for a woman aged 30 is 0,0012879. the premium for a man aged 31 is 0,0018614 and for a woman aged 31 is 0,0013846. the unit of premium value is rupiah, if one unit premium value is rp. 10.000.000 so premium value for man aged 30 is 0,0017345 × rp. 10.000.000 = rp 17.345. the value of the premium paid to men and women aged 30 years to 75 years increases every year. because of the greater chance of someone’s death so the premium payments will increase. conslusions the cox ingersoll ross model simulation can be applied to obtain variable interest rates in the future with parameter values κ=0,5309, θ=0,047218, σ=0,7679 and an error value in mape of 3,9014%, indicating that the estimated cir model value is very good to describe the actual data. the results of the premiums obtained from varying interest rates in the cir model show that the older individual is the greater premiums paid with the varying benefits obtained increasing every year. for further research can be suggested to use another stochastic model as a simulation to obtain interest rates that will be used to calculate premiums in whole life insurance or other life insurance. 0,0000000 0,0200000 0,0400000 0,0600000 0,0800000 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 p re m iu m v a lu e ages (year) man premium woman premium determination of term life insurance premiums with varying interest rates following cox ingersoll ross model and varying benefits value dian puspita 639 references [1] d. p. setiawati, f. agustiani, and r. marwati, “penentuan premi asuransi jiwa berjangka, asuransi tabungan berjangka, asuransi dwiguna berjangka dengan program aplikasinya,” j. eurekamatika, vol. 7, no. 2, pp. 100–114, 2019. 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[26] r. m. soffan, m. vasicek, m. reverting, m. gompertz, and m. vasicek, “menggunakan model stokastik,” vol. 5, no. 1, pp. 1–10, 2011. 1a sampul depan optimalisasi reduksi noise menggunakan chain-code termodifikasi pada pendeteksian wajah yusron rijal e-mail: yusronrijal@yahoo.com abstract digital image face detection had been developing so much in this last decade. the searching for the best method is still walking until today. this face detection research is using color model based segmentation method combined with template matching. there are 4 main process in this research. first, motion detection is used to minimize the size of the image by subtracting image n with image n+m. then, by using ycbcr color model based segmentation, the image is separated by skin color area and non-skin color area. third, is to reduce the noise by using the blurring way and by filtering its wide. the last one is matching the area that last with template that has been prepared to detect which area is face. the result of the research, from 217 images and 10 videos , shows that this method reach up to 70,5% face detection accuration procentage. keywords : face detection, skin color segmentation, noise reducing, template matching pendahuluan pengolahan citra merupakan suatu sistem dimana proses dilakukan dengan masukkan berupa citra dan hasilnya juga berupa citra. pada awalnya pengolahan citra ini dilakukan untuk memperbaiki kualitas citra, namun dengan berkembangnya dunia komputasi yang ditandai dengan semakin meningkatnya kapasitas dan kecepatan proses komputer, serta munculnya ilmu-ilmu komputasi yang memungkinkan manusia dapat mengambil informasi dari suatu citra, maka pengolahan citra tidak dapat dilepaskan dengan bidang computer vision. dalam perkembangan lebih lanjut pengolahan citra dan computer vision digunakan sebagai pengganti mata manusia, dengan perangkat input image capture seperti kamera dan scanner dijadikan sebagai mata dan mesin komputer dijadikan sebagai otak yang mengolah informasi. salah satu bidang yang menggunakan pengolahan citra yang saat ini banyak dikembangkan orang adalah biometric, yaitu bidang yang mempelajari bagaimana dapat mengindentifikasi ciri unik yang ada pada tubuh manusia otak bisa menentukan apakah objek yang dilihat oleh mata itu adalah manusia atau bukan, wajah atau tangan dan sebagainya. oleh karena itu, sangat penting penelitian yang bertujuan agar komputer dapat memiliki fungsi mendeteksi objek tersebut layaknya manusia diperlukan suatu cara dengan metode tertentu. dalam hal ini adalah bagaimana komputer mampu mendeteksi suatu objek bergerak dan menentukan bagianbagian wajah manusia. apabila komputer dapat didesain agar memiliki kecerdasan mendekati kecerdasan yang dimiliki manusia, tentu hal tersebut dapat memudahkan kerja manusia. dengan begitu banyak kegiatan manusia yang bisa didampingi oleh sistem komputer, seperti pengenalan wajah, pendeteksian wajah, object tracking, image registration, deteksi jalur kendaraan, dan lain sebagainya. metode penelitian 1. perancangan sistem dalam perancangan sistem, secara garis besar dapat digambarkan pada blok diagram berikut : gambar 1. blok diagram berdasarkan blok diagram diatas, pendeteksian wajah ini dapat dibagi menjadi 5 tahap, yaitu tahap citra awal, tahap deteksi kulit, tahap pengurangan derau, tahap template matching dan tahap hasil deteksi. tiap-tiap tahap memiliki proses-proses pengolahan citra di dalamnya. atau secara umum dapat dilihat pada flowchart sebagai berikut: 2. motion detection tujuan tahapan ini adalah untuk mengurangi proses representasi pola warna kulit terhadap citra yang akan diolah. citra yang digunakan sebagai masukan adalah citra hasil capture webcam dengan format webcam motion detection computer noise reduction result template matching skin represntation jurnal cauchy bmp dan dengan resolusi 320 yang penulis gunakan memiliki kecepatan 15 (frame per second kelipatan detik ke 0,1 dan ke 0,15 dilakukan capture posisi wajah relatif menghadap lurus seperti pada gambar setiap citra masukan akan diakses tiap tiap pikselnya untuk mengambil nilai hijau dan biru (rgb). serta dihitung nilai abu abu, y, cb dan cr proses pengaburan dalam setiap pendeteksian wajah, citra masukan yang dibutuhkan adalah dua. sebagaiman citra dilakukan dua kali dalam setiap pendeteksian. kedua citra masuka saling dikurang mendeteksi gerak. sebelum dilakukan pendeteksian gerak, kedua citra masukan akan dilakukan proses pengaburan terlebih dulu. mendapatkan hasil pendeteksian gerak yang lebih baik. pada dasarnya proses pengaburan adalah perkalian matriks antara matriks kernel dengan matriks citra berpengaruh pada hasil proses. semakin besar ukuran dihasilkan. pada sistem ini penulis menggunakan matriks kernel rata proses pengaburan dapat dilihat pada gambar gambar optimalisasi reduksi noise menggunakan chain jurnal cauchy – issn: 2086 dan dengan resolusi 320 yang penulis gunakan memiliki kecepatan 15 frame per second) atau 15 citra per detik. kelipatan detik ke 0,1 dan ke 0,15 dilakukan citra yang akan diproses selanjutnya. posisi wajah relatif menghadap lurus seperti pada gambar 2 gambar setiap citra masukan akan diakses tiap pikselnya untuk mengambil nilai hijau dan biru (rgb). serta dihitung nilai abu abu, y, cb dan cr-nya. proses pengaburan dalam setiap pendeteksian wajah, citra masukan yang dibutuhkan adalah dua. sebagaimana telah disebutkan di atas, citra dilakukan dua kali dalam setiap pendeteksian. kedua citra masuka saling dikurangkan ( mendeteksi gerak. sebelum dilakukan pendeteksian gerak, kedua citra masukan akan dilakukan proses pengaburan terlebih dulu. mendapatkan hasil pendeteksian gerak yang ih baik. pada dasarnya proses pengaburan adalah perkalian matriks antara matriks kernel dengan matriks citra [3]. ukuran matriks yang digunakan berpengaruh pada hasil proses. semakin besar matriks, semakin kabur citra yang dihasilkan. pada sistem ini penulis menggunakan matriks kernel rata-rata berukuran 3x3. proses pengaburan dapat dilihat pada gambar gambar 3. hasil proses pengaburan optimalisasi reduksi noise menggunakan chain issn: 2086-0382 dan dengan resolusi 320 x yang penulis gunakan memiliki kecepatan 15 ) atau 15 citra per detik. kelipatan detik ke 0,1 dan ke 0,15 dilakukan citra yang akan diproses selanjutnya. posisi wajah relatif menghadap lurus 2. gambar 2. citra masukan setiap citra masukan akan diakses tiap pikselnya untuk mengambil nilai hijau dan biru (rgb). serta dihitung nilai abu . dalam setiap pendeteksian wajah, citra masukan yang dibutuhkan adalah dua. telah disebutkan di atas, citra dilakukan dua kali dalam setiap pendeteksian. kedua citra masuka kan (image subtraction sebelum dilakukan pendeteksian gerak, kedua citra masukan akan dilakukan proses pengaburan terlebih dulu. tujuannya untuk mendapatkan hasil pendeteksian gerak yang pada dasarnya proses pengaburan adalah perkalian matriks antara matriks kernel dengan . ukuran matriks yang digunakan berpengaruh pada hasil proses. semakin besar matriks, semakin kabur citra yang dihasilkan. pada sistem ini penulis menggunakan rata berukuran 3x3. proses pengaburan dapat dilihat pada gambar hasil proses pengaburan optimalisasi reduksi noise menggunakan chain 0382 x 240. webcam yang penulis gunakan memiliki kecepatan 15 ) atau 15 citra per detik. setiap kelipatan detik ke 0,1 dan ke 0,15 dilakukan citra yang akan diproses selanjutnya. posisi wajah relatif menghadap lurus ke depan, citra masukan setiap citra masukan akan diakses tiap pikselnya untuk mengambil nilai-nilai merah, hijau dan biru (rgb). serta dihitung nilai abu dalam setiap pendeteksian wajah, citra masukan yang dibutuhkan adalah dua. telah disebutkan di atas, capturing citra dilakukan dua kali dalam setiap pendeteksian. kedua citra masukan tersebut akan image subtraction) untuk sebelum dilakukan pendeteksian gerak, kedua citra masukan akan dilakukan proses tujuannya untuk mendapatkan hasil pendeteksian gerak yang pada dasarnya proses pengaburan adalah perkalian matriks antara matriks kernel dengan . ukuran matriks yang digunakan berpengaruh pada hasil proses. semakin besar matriks, semakin kabur citra yang dihasilkan. pada sistem ini penulis menggunakan rata berukuran 3x3. keluaran proses pengaburan dapat dilihat pada gambar hasil proses pengaburan optimalisasi reduksi noise menggunakan chain webcam yang penulis gunakan memiliki kecepatan 15 fps setiap kelipatan detik ke 0,1 dan ke 0,15 dilakukan citra yang akan diproses selanjutnya. ke depan, setiap citra masukan akan diakses tiapnilai merah, hijau dan biru (rgb). serta dihitung nilai abudalam setiap pendeteksian wajah, citra masukan yang dibutuhkan adalah dua. capturing citra dilakukan dua kali dalam setiap n tersebut akan ) untuk sebelum dilakukan pendeteksian gerak, kedua citra masukan akan dilakukan proses tujuannya untuk mendapatkan hasil pendeteksian gerak yang pada dasarnya proses pengaburan adalah perkalian matriks antara matriks kernel dengan . ukuran matriks yang digunakan berpengaruh pada hasil proses. semakin besar matriks, semakin kabur citra yang dihasilkan. pada sistem ini penulis menggunakan keluaran proses pengaburan dapat dilihat pada gambar 3. proses tersebut akan mendeteksi pergerakan yang terjadi citra manapun yang bertindak sebagai pengurang tidak menjadi masalah. karena setiap hasil pengurangan dijadikan angka mutlak atau selalu indikator pergerakan. hasil dari proses deteksi gerak ini akan menjadi dasar bagi proses cropping gerak dapat dilihat pada gambar proses mempersempit area pendeteksian wajah sehingga diharapkan dapat mempercepat proses pendeteksian wajah dilakukan berdasarkan pada koordinat piksel putih terdekat dengan sumbu (0,0) atau xmin dan ymin. dan terjauh dari sumbu (0,0) atau xmax dan ymax. seperti pada gambar menjadi masukan bagi tahap selanjutnya, yaitu tahap deteksi kulit. optimalisasi reduksi noise menggunakan chain-code termodifikasi pada proses motion detection kedua citra tersebut akan mendeteksi pergerakan yang terjadi citra manapun yang bertindak sebagai pengurang tidak menjadi masalah. karena setiap hasil pengurangan dijadikan angka mutlak atau selalu dijadikan positif. perbedaan nilai yang dihasilkan menjadi indikator pergerakan. hasil dari proses deteksi gerak ini akan menjadi dasar bagi proses cropping selanjutnya. hasil dari proses deteksi gerak dapat dilihat pada gambar gambar proses cropping proses mempersempit area pendeteksian wajah sehingga diharapkan dapat mempercepat proses pendeteksian wajah dilakukan berdasarkan pada koordinat piksel putih terdekat dengan sumbu (0,0) atau xmin dan ymin. dan terjauh dari sumbu (0,0) atau xmax dan ymax. seperti pada gambar gambar hasil dari proses menjadi masukan bagi tahap selanjutnya, yaitu tahap deteksi kulit. code termodifikasi pada motion detection kedua citra hasil proses pengaburan tersebut akan saling dikurangankan untuk mendeteksi pergerakan yang terjadi citra manapun yang bertindak sebagai pengurang tidak menjadi masalah. karena setiap hasil pengurangan dijadikan angka mutlak atau dijadikan positif. perbedaan nilai yang dihasilkan menjadi indikator pergerakan. hasil dari proses deteksi gerak ini akan menjadi dasar bagi proses selanjutnya. hasil dari proses deteksi gerak dapat dilihat pada gambar gambar 4. hasil proses deteksi gerak cropping proses cropping mempersempit area pendeteksian wajah sehingga diharapkan dapat mempercepat proses pendeteksian wajah [1,2 dilakukan berdasarkan pada koordinat piksel putih terdekat dengan sumbu (0,0) atau xmin dan ymin. dan terjauh dari sumbu (0,0) atau xmax dan ymax. seperti pada gambar gambar 5. hasil proses hasil dari proses cropping menjadi masukan bagi tahap selanjutnya, yaitu tahap deteksi kulit. code termodifikasi pada pendeteksian wajah hasil proses pengaburan dikurangankan untuk mendeteksi pergerakan yang terjadi citra manapun yang bertindak sebagai pengurang tidak menjadi masalah. karena setiap hasil pengurangan dijadikan angka mutlak atau perbedaan nilai yang dihasilkan menjadi indikator pergerakan. hasil dari proses deteksi gerak ini akan menjadi dasar bagi proses selanjutnya. hasil dari proses deteksi gerak dapat dilihat pada gambar 4. hasil proses deteksi gerak dilakukan untuk mempersempit area pendeteksian wajah sehingga diharapkan dapat mempercepat proses ,2]. proses dilakukan berdasarkan pada koordinat piksel putih terdekat dengan sumbu (0,0) atau xmin dan ymin. dan terjauh dari sumbu (0,0) atau xmax dan ymax. seperti pada gambar 5. hasil proses croppping cropping ini yang kan menjadi masukan bagi tahap selanjutnya, yaitu pendeteksian wajah 91 hasil proses pengaburan dikurangankan untuk [1,2,11]. citra manapun yang bertindak sebagai pengurang tidak menjadi masalah. karena setiap hasil pengurangan dijadikan angka mutlak atau perbedaan nilai yang dihasilkan menjadi indikator pergerakan. hasil dari proses deteksi gerak ini akan menjadi dasar bagi proses selanjutnya. hasil dari proses deteksi hasil proses deteksi gerak dilakukan untuk mempersempit area pendeteksian wajah sehingga diharapkan dapat mempercepat proses . proses cropping dilakukan berdasarkan pada koordinat piksel putih terdekat dengan sumbu (0,0) atau xmin dan ymin. dan terjauh dari sumbu (0,0) atau croppping ini yang kan menjadi masukan bagi tahap selanjutnya, yaitu yusron rijal 92 3. skin representation pada tahap bertujuan untuk memisahkan area berwarna kulit dengan area berwarna non lebih mengoptimalkan hasil pen penulis melakukan dua kali segmentasi, yaitu berdasarkan rgb dan y pencarian range nilai yang digunakan pada kedua proses segmentasi dilakukan dengan metode iterasi, yaitu dilakukan berulang sampai m [1,2,7,9,1 tersebut didasarkan pada data hasil sampling. segmentasi untuk mengeliminasi area dipastikan bukan warna kulit, seperti putih mutlak, hitam mutlak, hijau mutlak dan lain gambar 6 gambar hasil dari proses segmentasi rgb tersebut dilanjutkan dengan proses segmentasi y segmentasi ini bertujuan area berwarna kulit. nilai y yang menggunakan nilai rgb yang telah didapatkan sebelumnya. persamaan y = cb = cr = gambar 4. noise reduction seperti yang dapat d bahwa ter menyertai area tahap pengurangan derau bertujuan untuk mengurangi sebanyak mungkin tersebut. pengurangan derau ini memiliki dua proses, yaitu pengaburan derau dan filter luas. pengaburan derau bertujuan untuk mengurangi yusron rijal skin representation pada tahap representasi warna kulit bertujuan untuk memisahkan area berwarna kulit dengan area berwarna non lebih mengoptimalkan hasil pen penulis melakukan dua kali segmentasi, yaitu berdasarkan rgb dan y pencarian range nilai yang digunakan pada kedua proses segmentasi dilakukan dengan metode iterasi, yaitu dilakukan berulang sampai menemukan range nilai yang tepat 10,11,12]. tetapi pengulangan range nilai tersebut didasarkan pada data hasil sampling. segmentasi berdasarkan rgb untuk mengeliminasi area dipastikan bukan warna kulit, seperti putih k, hitam mutlak, hijau mutlak dan lain-lain, seperti yang dapat dilihat pada 6. gambar 6. hasil proses segmentasi rgb hasil dari proses segmentasi rgb tersebut dilanjutkan dengan proses segmentasi y segmentasi ini bertujuan area berwarna kulit. nilai ycbcr didapatkan dari perhitungan yang menggunakan nilai rgb yang telah didapatkan sebelumnya. persamaan sebagai berikut: )*222.0( r += )*159.0( r−= (*)*5.0(( r −= gambar 7. hasil proses segmentasi ycbcr noise reduction seperti yang dapat d bahwa terdapat banyak derau menyertai area-area kulit hasil segmentasi. tahap pengurangan derau bertujuan untuk mengurangi sebanyak mungkin tersebut. pengurangan derau ini memiliki dua proses, yaitu pengaburan derau dan filter luas. pengaburan derau bertujuan untuk mengurangi skin representation representasi warna kulit bertujuan untuk memisahkan area berwarna kulit dengan area berwarna non lebih mengoptimalkan hasil pen penulis melakukan dua kali segmentasi, yaitu berdasarkan rgb dan ycbcr. pencarian range nilai yang digunakan pada kedua proses segmentasi dilakukan dengan metode iterasi, yaitu dilakukan berulang enemukan range nilai yang tepat tetapi pengulangan range nilai tersebut didasarkan pada data hasil sampling. berdasarkan rgb untuk mengeliminasi area-area yang dapat dipastikan bukan warna kulit, seperti putih k, hitam mutlak, hijau mutlak lain, seperti yang dapat dilihat pada . hasil proses segmentasi rgb hasil dari proses segmentasi rgb tersebut dilanjutkan dengan proses segmentasi y segmentasi ini bertujuan untuk me bcr didapatkan dari perhitungan yang menggunakan nilai rgb yang telah didapatkan sebelumnya. dengan persamaan sebagai berikut: )*707.0( g ++ )*332.0( g−+ ())*419.0 g +− hasil proses segmentasi ycbcr seperti yang dapat dilihat pada gambar dapat banyak derau area kulit hasil segmentasi. tahap pengurangan derau bertujuan untuk mengurangi sebanyak mungkin pengurangan derau ini memiliki dua proses, yaitu pengaburan derau dan filter luas. pengaburan derau bertujuan untuk mengurangi representasi warna kulit bertujuan untuk memisahkan area berwarna kulit dengan area berwarna non-kulit. untuk lebih mengoptimalkan hasil pendeteksian kulit, penulis melakukan dua kali segmentasi, yaitu pencarian range nilai yang digunakan pada kedua proses segmentasi dilakukan dengan metode iterasi, yaitu dilakukan berulang-ulang enemukan range nilai yang tepat tetapi pengulangan range nilai tersebut didasarkan pada data hasil sampling. berdasarkan rgb dilakukan area yang dapat dipastikan bukan warna kulit, seperti putih k, hitam mutlak, hijau mutlak, biru mutlak lain, seperti yang dapat dilihat pada . hasil proses segmentasi rgb hasil dari proses segmentasi rgb tersebut dilanjutkan dengan proses segmentasi ycbcr. untuk mencari area bcr didapatkan dari perhitungan yang menggunakan nilai rgb yang telah engan persamaan )*07.0( b+ )*05.0() b+ )*081.0( b− hasil proses segmentasi ycbcr ilihat pada gambar dapat banyak derau-derau yang area kulit hasil segmentasi. tahap pengurangan derau bertujuan untuk mengurangi sebanyak mungkin derau-derau pengurangan derau ini memiliki dua proses, yaitu pengaburan derau dan filter luas. pengaburan derau bertujuan untuk mengurangi representasi warna kulit ini bertujuan untuk memisahkan area berwarna kulit. untuk deteksian kulit, penulis melakukan dua kali segmentasi, yaitu pencarian range nilai yang digunakan pada kedua proses segmentasi dilakukan dengan ulang enemukan range nilai yang tepat tetapi pengulangan range nilai dilakukan area yang dapat dipastikan bukan warna kulit, seperti putih , biru mutlak lain, seperti yang dapat dilihat pada hasil dari proses segmentasi rgb tersebut bcr. ncari areabcr didapatkan dari perhitungan yang menggunakan nilai rgb yang telah engan persamaan ilihat pada gambar 7, derau yang area kulit hasil segmentasi. tahap pengurangan derau bertujuan untuk derau pengurangan derau ini memiliki dua proses, yaitu pengaburan derau dan filter luas. pengaburan derau bertujuan untuk mengurangi derau yang besar. proses pengaburan derau ini pada dasarnya sama tahap citra awal. proses filter luas kecil, berikutnya adalah mengurangi area yang cukup besar dinyatakan sebagai wajah. sebelumnya kita perlu untuk melakukan penghitungan luas tiap menghitung luas area, penulis memanfaatkan metode disebut dengan menghitung banyaknya piksel berwarna putih pada tiap area. sebagai penentu arah prioritas. derau-derau kecil dan mengahaluskan area yang besar. proses pengaburan derau ini pada dasarnya sama tahap citra awal. gambar 8. gambar 9 proses filter luas setelah mengurangi derau kecil, berikutnya adalah mengurangi area yang cukup besar dinyatakan sebagai wajah. untuk dapat melakukan filter luas, sebelumnya kita perlu untuk melakukan penghitungan luas tiap menghitung luas area, penulis memanfaatkan metode freeman’s chain code isebut chain code gambar 10 proses penghitungan luas dilakukan dengan menghitung banyaknya piksel berwarna putih pada tiap area. sebagai penentu arah prioritas. gambar 1 5 6 volume 1 no. 2 mei 2010 derau kecil dan mengahaluskan area yang besar. proses pengaburan derau ini pada dasarnya sama dengan proses pengaburan pada tahap citra awal. citra sebelum pengaburan derau . hasil proses pengaburan proses filter luas setelah mengurangi derau kecil, berikutnya adalah mengurangi area yang cukup besar tetapi kurang besar untuk dinyatakan sebagai wajah. untuk dapat melakukan filter luas, sebelumnya kita perlu untuk melakukan penghitungan luas tiap menghitung luas area, penulis memanfaatkan freeman’s chain code chain code [5,6,8]. 0. modifikasi freeman’s chain code proses penghitungan luas dilakukan dengan menghitung banyaknya piksel berwarna putih pada tiap area. chain code sebagai penentu arah prioritas. gambar 11. ilustrasi penghitungan luas 3 4 6 7 volume 1 no. 2 mei 2010 derau kecil dan mengahaluskan area yang besar. proses pengaburan derau ini pada dengan proses pengaburan pada citra sebelum pengaburan derau pengaburan derau setelah mengurangi derau-derau yang kecil, berikutnya adalah mengurangi area tetapi kurang besar untuk untuk dapat melakukan filter luas, sebelumnya kita perlu untuk melakukan penghitungan luas tiap-tiap area. untuk menghitung luas area, penulis memanfaatkan freeman’s chain code, selanjutnya cukup freeman’s chain code proses penghitungan luas dilakukan dengan menghitung banyaknya piksel berwarna chain code digunakan sebagai penentu arah prioritas. ilustrasi penghitungan luas 1 2 8 volume 1 no. 2 mei 2010 derau kecil dan mengahaluskan area-area yang besar. proses pengaburan derau ini pada dengan proses pengaburan pada citra sebelum pengaburan derau derau derau yang kecil, berikutnya adalah mengurangi area-area tetapi kurang besar untuk untuk dapat melakukan filter luas, sebelumnya kita perlu untuk melakukan tiap area. untuk menghitung luas area, penulis memanfaatkan , selanjutnya cukup freeman’s chain code proses penghitungan luas dilakukan dengan menghitung banyaknya piksel berwarna digunakan ilustrasi penghitungan luas jurnal cauchy berdasarkan gambar 1 didapat luas sebesar 1 piksel yang telah bertanda merah. selanjutnya ilustrasi dapat digambarkan seperti gambar gambar setiap perpindahan posisi pointer, maka akan dihitung luasnya sehingga didapatkan luas sebesar 16 piksel. ini dilakukan berulang sebanyak jumlah area pada citra. setelah didapatkan luas tiap barulah dapat dilakukan proses filter luas. jika luas suatu area tidak memenuhi luas maka area tersebut akan dihapus. 5. template matching tahap untuk menentukan area yang merupakan wajah dari tahap deteksi kulit akan di menjadi citra kulit tersebut yang akan dicocokan dengan template terlihat pada gambar 1 citra masukan sebelum dicocokan dengan template ukuran citra masukan tersebut disamakan optimalisasi reduksi noise menggunakan chain jurnal cauchy – issn: 2086 berdasarkan gambar 1 didapat luas sebesar 1 piksel yang telah bertanda merah. selanjutnya ilustrasi dapat digambarkan seperti gambar berikut gambar 12. ilustrasi proses hitung luas setiap perpindahan posisi pointer, maka akan dihitung luasnya sehingga didapatkan luas sebesar 16 piksel. ini dilakukan berulang sebanyak jumlah area pada citra. setelah didapatkan luas tiap barulah dapat dilakukan proses filter luas. jika luas suatu area tidak memenuhi luas maka area tersebut akan dihapus. gambar 13. hasil proses filter luas template matching tahap template matching untuk menentukan area yang merupakan wajah dari tahap deteksi kulit akan di menjadi citra-citra baru. kulit tersebut yang akan dicocokan dengan template yang telah disiapkan. terlihat pada gambar 1 citra masukan sebelum dicocokan dengan template, dilakukan penskalaan terlebih dahulu. ukuran citra masukan tersebut disamakan optimalisasi reduksi noise menggunakan chain issn: 2086-0382 berdasarkan gambar 11, berarti telah didapat luas sebesar 1 piksel yang telah bertanda merah. selanjutnya ilustrasi dapat digambarkan berikut ini : ilustrasi proses hitung luas setiap perpindahan posisi pointer, maka akan dihitung luasnya dan diberi tanda warna sehingga didapatkan luas sebesar 16 piksel. ini dilakukan berulang sebanyak jumlah area setelah didapatkan luas tiap barulah dapat dilakukan proses filter luas. jika luas suatu area tidak memenuhi luas maka area tersebut akan dihapus. hasil proses filter luas template matching template matching untuk menentukan area-area warna kulit mana yang merupakan wajah [14,15]. selanjutnya dari tahap deteksi kulit akan di baru. citra dengan kulit tersebut yang akan dicocokan dengan yang telah disiapkan. terlihat pada gambar 14. citra masukan sebelum dicocokan dengan , dilakukan penskalaan terlebih dahulu. ukuran citra masukan tersebut disamakan optimalisasi reduksi noise menggunakan chain 0382 , berarti telah didapat luas sebesar 1 piksel yang telah bertanda merah. selanjutnya ilustrasi dapat digambarkan ilustrasi proses hitung luas setiap perpindahan posisi pointer, maka dan diberi tanda warna sehingga didapatkan luas sebesar 16 piksel. ini dilakukan berulang sebanyak jumlah area setelah didapatkan luas tiap-tiap area, barulah dapat dilakukan proses filter luas. jika luas suatu area tidak memenuhi luas threshold hasil proses filter luas template matching ini berfungsi area warna kulit mana selanjutnya, hasil dari tahap deteksi kulit akan di-crop otomatis dengan area warna kulit tersebut yang akan dicocokan dengan yang telah disiapkan. seperti yang citra masukan sebelum dicocokan dengan , dilakukan penskalaan terlebih dahulu. ukuran citra masukan tersebut disamakan optimalisasi reduksi noise menggunakan chain , berarti telah didapat luas sebesar 1 piksel yang telah bertanda merah. selanjutnya ilustrasi dapat digambarkan setiap perpindahan posisi pointer, maka dan diberi tanda warna. hal ini dilakukan berulang sebanyak jumlah area tiap area, barulah dapat dilakukan proses filter luas. jika threshold, ini berfungsi area warna kulit mana , hasil otomatis area warna kulit tersebut yang akan dicocokan dengan seperti yang citra masukan sebelum dicocokan dengan , dilakukan penskalaan terlebih dahulu. ukuran citra masukan tersebut disamakan dengan ukuran citra untuk mempermudah proses pencocokan. menjadi citra hitam sebelum disiapkan. yaitu 70 gambar 1 tiap kecocokan yan sehingga indikator wajah atau bukan. 6. keseluruhan proses. tahap ini akan menggambar persegi empat berwarna merah pada area yang dideteksi sebagai wajah. bergantung pada tahap suatu area dinyatakan wajah, meskipun bukan wajah, maka area tersebut akan dikelilingi oleh persegi tersebut dideteksi sebagai wajah. dideteksi sebagai wajah, maka area tersebut tidak dikelilingi oleh segi empat merah. aplikasi citra dengan berbagai latar belakang. kondisi optimalisasi reduksi noise menggunakan chain-code termodifikasi pada dengan ukuran citra untuk mempermudah proses pencocokan. gambar 1 citra-citra baru tersebut akan menjadi citra hitam sebelum dicocokan dengan disiapkan. template yaitu template dengan sudut kemiringan 70°, −20°, −45° gambar 15. template setiap citra masukan dicocokan dengan tiap-tiap template kecocokan yan sehingga hasil rata indikator wajah atau bukan. tahap hasil deteksi tahap ini berfungsi sebagai keluaran dari keseluruhan proses. tahap ini akan menggambar persegi empat berwarna merah pada area yang dideteksi sebagai wajah. oleh karena itu, tahap ini sepenuhnya bergantung pada tahap suatu area dinyatakan wajah, meskipun bukan wajah, maka area tersebut akan dikelilingi oleh persegi empat merah sebagai tanda bahwa area tersebut dideteksi sebagai wajah. sebaliknya, jika suatu area wajah tidak dideteksi sebagai wajah, maka area tersebut tidak dikelilingi oleh segi empat merah. gambar 1 aplikasi dan uji coba dilakukan dengan menggunakan citra dengan berbagai latar belakang. kondisi code termodifikasi pada dengan ukuran citra template untuk mempermudah proses pencocokan. gambar 14. hasil proses citra baru tersebut akan menjadi citra hitam-putih terlebih dahulu dicocokan dengan template yang disiapkan ada 7 buah. dengan sudut kemiringan °, dan −70°. template 70° (paling kiri) sampai (paling kanan) setiap citra masukan dicocokan dengan template. tiap-tiap prosentase kecocokan yang didapatkan akan dirata hasil rata-rata tersebut yang menjadi indikator wajah atau bukan. tahap hasil deteksi tahap ini berfungsi sebagai keluaran dari keseluruhan proses. tahap ini akan menggambar persegi empat berwarna merah pada area yang dideteksi sebagai wajah. oleh karena itu, tahap ini sepenuhnya bergantung pada tahap-tahap sebelumnya. jika suatu area dinyatakan wajah, meskipun bukan wajah, maka area tersebut akan dikelilingi oleh empat merah sebagai tanda bahwa area tersebut dideteksi sebagai wajah. sebaliknya, jika suatu area wajah tidak dideteksi sebagai wajah, maka area tersebut tidak dikelilingi oleh segi empat merah. gambar 16. hasil deteksi wajah dan pembahasan uji coba dilakukan dengan menggunakan citra dengan berbagai latar belakang. kondisi code termodifikasi pada pendeteksian wajah template dengan tujuan untuk mempermudah proses pencocokan. hasil proses cropping citra baru tersebut akan putih terlebih dahulu dicocokan dengan template yang telah yang disiapkan ada 7 buah. dengan sudut kemiringan (paling kiri) sampai (paling kanan) setiap citra masukan dicocokan dengan tiap prosentase g didapatkan akan dirata rata tersebut yang menjadi indikator wajah atau bukan. tahap ini berfungsi sebagai keluaran dari keseluruhan proses. tahap ini akan menggambar persegi empat berwarna merah pada area yang oleh karena itu, tahap ini sepenuhnya tahap sebelumnya. jika suatu area dinyatakan wajah, meskipun bukan wajah, maka area tersebut akan dikelilingi oleh empat merah sebagai tanda bahwa area tersebut dideteksi sebagai wajah. sebaliknya, jika suatu area wajah tidak dideteksi sebagai wajah, maka area tersebut tidak dikelilingi oleh segi empat merah. hasil deteksi wajah pembahasan uji coba dilakukan dengan menggunakan citra dengan berbagai latar belakang. kondisi pendeteksian wajah 93 dengan tujuan untuk mempermudah proses pencocokan. cropping citra baru tersebut akan diubah putih terlebih dahulu yang telah yang disiapkan ada 7 buah. dengan sudut kemiringan 20°, 45°, (paling kiri) sampai −70° setiap citra masukan dicocokan dengan tiap prosentase hasil g didapatkan akan dirata-rata, rata tersebut yang menjadi tahap ini berfungsi sebagai keluaran dari keseluruhan proses. tahap ini akan menggambar persegi empat berwarna merah pada area yang oleh karena itu, tahap ini sepenuhnya tahap sebelumnya. jika suatu area dinyatakan wajah, meskipun bukan wajah, maka area tersebut akan dikelilingi oleh empat merah sebagai tanda bahwa area sebaliknya, jika suatu area wajah tidak dideteksi sebagai wajah, maka area tersebut tidak uji coba dilakukan dengan menggunakan citra dengan berbagai latar belakang. kondisi yusron rijal 94 pencahayaan harus terang dan merata. pada area wajah tidak dihalangi atau tertutupi oleh objek lainnya. pertama yang perlu menjadi perhatian utama adalah hal ini penting karena dengan didapatkan hasil deteksi kulit yang optimal, prosentase akurasi pendeteksian wajah akan meningkat. selanjutnya adalah proses matching secara efisien dan efektif untuk menghasilkan prosentase akurasi pendeteksian wajah yang baik. pengujian dilakukan untuk mengetahui berapa nilai kecocokkan dan rata kecocokkan citra wajah terhadap tabel 1. no 1 2 3 4 5 6 7 8 yusron rijal pencahayaan harus terang dan merata. pada area wajah tidak dihalangi atau tertutupi oleh objek pertama yang perlu menjadi perhatian utama adalah optimal hal ini penting karena dengan didapatkan hasil deteksi kulit yang optimal, prosentase akurasi pendeteksian wajah akan meningkat. elanjutnya adalah proses matching. bagaimana metode ini dapat secara efisien dan efektif untuk menghasilkan prosentase akurasi pendeteksian wajah yang pengujian dilakukan untuk mengetahui berapa nilai kecocokkan dan rata kecocokkan citra wajah terhadap uji sampel te citra pencahayaan harus terang dan merata. pada area wajah tidak dihalangi atau tertutupi oleh objek pertama yang perlu menjadi perhatian optimalnya hasil dari deteksi kulit. hal ini penting karena untuk proses pencocokan dengan didapatkan hasil deteksi kulit yang optimal, prosentase akurasi pendeteksian wajah elanjutnya adalah proses . bagaimana metode ini dapat secara efisien dan efektif untuk menghasilkan prosentase akurasi pendeteksian wajah yang pengujian dilakukan untuk mengetahui berapa nilai kecocokkan dan ratakecocokkan citra wajah terhadap template sampel terhadap template kecocokan (pixel) 611 577 360 545 424 319 526 564 pencahayaan harus terang dan merata. pada area wajah tidak dihalangi atau tertutupi oleh objek pertama yang perlu menjadi perhatian hasil dari deteksi kulit. untuk proses pencocokan dengan didapatkan hasil deteksi kulit yang optimal, prosentase akurasi pendeteksian wajah elanjutnya adalah proses template . bagaimana metode ini dapat dilakukan secara efisien dan efektif untuk menghasilkan prosentase akurasi pendeteksian wajah yang pengujian dilakukan untuk mengetahui -rata prosentase template. template kecocokan prosentase kecocokan 99% 93% 58% 88% 69% 52% 85% 92% pencahayaan harus terang dan merata. pada area wajah tidak dihalangi atau tertutupi oleh objek pertama yang perlu menjadi perhatian hasil dari deteksi kulit. untuk proses pencocokan. dengan didapatkan hasil deteksi kulit yang optimal, prosentase akurasi pendeteksian wajah template dilakukan secara efisien dan efektif untuk menghasilkan prosentase akurasi pendeteksian wajah yang pengujian dilakukan untuk mengetahui rata prosentase prosentase kecocokan dinyatakan bahwa suatu area tersebut adalah wajah jika memiliki nilai kecocokan 500 dan n pendeteksian wajah dilakukan untuk mengetahui tingka kondisi uji coba, yaitu pada satu orang pengguna normal (tidak beraksesoris dan bertingkah), pada pengguna lebih dari satu, pada pengguna beraksesoris pengguna berekspresi/be tabel no 1 2 3 4 akurasi pendeteksian baik pengujian terhadap citra diam maupun citra bergerak. belakang dan sumber pencahayaan beragam sesungguhnya dari tabel 3. kondisi pencahayaan sistem ini. faktor latar belakang mempengaruhi proses segmentasi yang dapat mengakibatkan kesalahan pendeteksian, terutama pada latar belakang yang sewarna dengan kulit. 9 berdasarkan data dari tabel 1, dapat dinyatakan bahwa suatu area tersebut adalah wajah jika memiliki nilai kecocokan 500 dan nilai prosentasenya berkisar selanjutnya, proses p pendeteksian wajah dilakukan untuk mengetahui tingkat akurasi dari sistem ini kondisi uji coba, yaitu pada satu orang pengguna normal (tidak beraksesoris dan bertingkah), pada pengguna lebih dari satu, pada pengguna beraksesoris (topi, kaca mata, jilbab dll) pengguna berekspresi/be tabel 2. uji coba pendeteksian wajah no 1 satu pengguna 2 lebih dari satu pengguna 3 pengguna beraksesoris 4 pengguna berekspresi berdasarkan data uji, s akurasi pendeteksian baik pengujian terhadap citra diam maupun citra bergerak. proses pengujian juga belakang dan sumber pencahayaan beragam. diharapkan akan diketahui kinerja sesungguhnya dari tabel 3. uji coba no sumber cahaya 1 sinar matahari 2 sinar matahari 3 melawan sinar matahari 4 lampu 5 sinar matahari dari tabel 3 dapat dilihat bahwa faktor kondisi pencahayaan sistem ini. faktor latar belakang mempengaruhi proses segmentasi yang dapat mengakibatkan kesalahan pendeteksian, terutama pada latar belakang yang sewarna dengan kulit. volume 1 no. 2 mei 2010 311 berdasarkan data dari tabel 1, dapat dinyatakan bahwa suatu area tersebut adalah wajah jika memiliki nilai kecocokan ilai prosentasenya berkisar selanjutnya, proses p pendeteksian wajah dilakukan untuk mengetahui t akurasi dari sistem ini kondisi uji coba, yaitu pada satu orang pengguna normal (tidak beraksesoris dan bertingkah), pada pengguna lebih dari satu, pada pengguna (topi, kaca mata, jilbab dll) pengguna berekspresi/bertingkah. coba pendeteksian wajah kondisi satu pengguna lebih dari satu pengguna pengguna beraksesoris pengguna berekspresi berdasarkan data uji, s akurasi pendeteksian wajah ini baik pengujian terhadap citra diam maupun citra proses pengujian juga belakang dan sumber pencahayaan . diharapkan akan diketahui kinerja sesungguhnya dari algoritma coba terhadap sumber cahaya sumber cahaya sinar matahari sinar matahari melawan sinar matahari lampu sinar matahari dari tabel 3 dapat dilihat bahwa faktor kondisi pencahayaan sangat berpengaruh pada sistem ini. faktor latar belakang mempengaruhi proses segmentasi yang dapat mengakibatkan kesalahan pendeteksian, terutama pada latar belakang yang sewarna dengan kulit. volume 1 no. 2 mei 2010 311 berdasarkan data dari tabel 1, dapat dinyatakan bahwa suatu area tersebut adalah wajah jika memiliki nilai kecocokan lebih dari ilai prosentasenya berkisar 76 selanjutnya, proses p pendeteksian wajah dilakukan untuk mengetahui t akurasi dari sistem ini melalui kondisi uji coba, yaitu pada satu orang pengguna normal (tidak beraksesoris dan bertingkah), pada pengguna lebih dari satu, pada pengguna (topi, kaca mata, jilbab dll) rtingkah. coba pendeteksian wajah prosentase akurasi lebih dari satu pengguna pengguna beraksesoris pengguna berekspresi berdasarkan data uji, secara keseluruhan wajah ini sebesar 70,5% baik pengujian terhadap citra diam maupun citra proses pengujian juga dilakukan pada latar belakang dan sumber pencahayaan . diharapkan akan diketahui kinerja algoritma ini. terhadap sumber cahaya akurasi (%) waktu (detik) 80 33 46 86 80 dari tabel 3 dapat dilihat bahwa faktor sangat berpengaruh pada sistem ini. faktor latar belakang mempengaruhi proses segmentasi yang dapat mengakibatkan kesalahan pendeteksian, terutama pada latar belakang yang sewarna dengan kulit. volume 1 no. 2 mei 2010 50% berdasarkan data dari tabel 1, dapat dinyatakan bahwa suatu area tersebut adalah lebih dari -80 %. selanjutnya, proses pengujian pendeteksian wajah dilakukan untuk mengetahui melalui beberapa kondisi uji coba, yaitu pada satu orang pengguna normal (tidak beraksesoris dan bertingkah), pada pengguna lebih dari satu, pada pengguna dan pada prosentase akurasi 80% 72% 80% 50% ecara keseluruhan sebesar 70,5% baik pengujian terhadap citra diam maupun citra dilakukan pada latar belakang dan sumber pencahayaan yang . diharapkan akan diketahui kinerja terhadap sumber cahaya waktu ratarata (detik) 4 5 5 4 4 dari tabel 3 dapat dilihat bahwa faktor sangat berpengaruh pada sistem ini. faktor latar belakang mempengaruhi proses segmentasi yang dapat mengakibatkan kesalahan pendeteksian, terutama pada latar optimalisasi reduksi noise menggunakan chain-code termodifikasi pada pendeteksian wajah jurnal cauchy – issn: 2086-0382 95 sehingga prosentase keberhasilan dari sistem ini rata-rata keseluruhan adalah 65%. prosentase tersebut didapatkan dari uji coba pada berbagai latar belakang dan kondisi pencahayaan. penutup a. dari pengujian sistem secara keseluruhan, prosentase akurasi dari sistem pendeteksian wajah ini sebesar 65% dengan kecepatan rata-rata proses sebesar 4 detik. b. proses segmentasi sangat bergantung pada kondisi pencahayaan. akibatnya, nilai ambang pada suatu kondisi pencahayaan dengan kondisi pencahayaan yang lain bisa jadi berbeda. c. keuntungan dari penggunaan model warna ycbcr sebagai dasar segmentasi deteksi warna kulit adalah pengaruh luminasi dapat dipisahkan. pada model warna ycbcr, semua informasi tentang tingkat kecerahan diberikan oleh komponen y (luminasi), karena komponen cb (biru) dan komponen cr (merah) tidak tergantung dari luminasi. d. proses penskalaan pada proses template matching bisa menjadi kelemahan dalam akurasi pendeteksian wajah. perubahan ukuran panjang dan lebar dari citra menyerupai ukuran template mengakibatkan beberapa area citra berubah bentuk. sehingga perubahan bentuk tersebut menyerupai bentuk wajah. e. prosentase akurasi penggunaan template matching sebagai penentu wajah mencapai 70,5%. tetapi cara ini memiliki kelemahan karena proses dilakukan pada bidang 2 dimensi. akibatnya, apabila terdapat area bukan wajah yang memiliki bentuk seperti template akan dinyatakan sebagai wajah. setelah melakukan penelitian ini, penulis mendapatkan beberapa hal yang perlu untuk dipelajari lebih lagi, yaitu: a. untuk mendapatkan proses pendeteksian wajah yang lebih cepat, hendaknya citra masukan melalui proses pengecilan terlebih dahulu. setelah ukuran citra masukan diperkecil, barulah proses-proses pendeteksian wajah dilakukan. setelah proses pendeteksian wajah selesai, barulah ukuran citra tersebut dikembalikan seperti semula. b. untuk memperoleh akurasi pendeteksian wajah yang lebih baik, maka penentuan area wajah dilakukan berdasarkan ekstraksi fitur area wajah, seperti mata, hidung atau mulut, bukan berdasarkan template matching. daftar pustaka [1] achmad, b. & firdausy, k. 2005. teknik pengolahan citra digital menggunakan delphi. jogjakarta: ardi publishing. [2] ahmad, u. 2005. pengolahan citra digital teknik pemrogramannya. jogjakarta: graha ilmu. [3] basuki, a. & palandi, j.f.f. 2005. pengolahan citra digital menggunakan visual basic. jogjakarta: graha ilmu. [4] castleman, k.r. 1996. digital image processing. new jersey: prentice-hall. [5] lensu, l. 1998. freeman chain code, (online), (www.it.lut.fi/kurssit/9900/ 010588000/exercises/11/freeman/solution .html, diakses 23 maret 2008). [6] munir, r. 2004. pengolahan citra digital dengan pendekatan algoritmik. bandung: informatika. [7] padilla, michael and fan, zhong, 2003, ee368 digital image processing project – automatic face detection using color based segmentation and template/energy thresholding, departement of electrical engineering, jurnal ee386. [8] parker, p.m. 2005. chain code, (online), (www.websters-onlinedictionary.org/definition/image, diakses 23 maret 2008). [9] rayleigh, j.w.s. 1899. additive primaries, (online), (http://en.wikipedia.org/ wiki/additive_primaries#additive_primarie s, diakses 23 maret 2008). [10] rgbworld. 2007. color, (online), (www.rgbworld.com.color.html, diakses 23 maret 2008). [11] rijal. yusron, & mardi supeno & mauridy h. purnomo. 2006. deteksi wajah pada obyek bergerak dengan menggunakan kombinasi gabor filter dan gaussian low pass filter. seminar nasional aplikasi teknologi informasi (snati) 2006. universitas islam indonesia [12] t.s. caetano and d.a.c. barone. 2000. a probabilistic model for the human skin color. universidade federal do rio grande do sul – instituto de informáticaav. bento golçalves, bloco iv – porto alegre – rs – brazil [13] wikipedia. 2007. rgb color model, (online), (www.wikipedia.org/wiki/rgb color model.htm, diakses 23 maret 2008). yusron rijal 96 volume 1 no. 2 mei 2010 [14] wikipedia. 2008. webcam, (online), (http://en.wikipedia.org/ wiki/webcam, diakses 24 maret 2008). [15] wikipedia. 2007. ycbcr color model, (online), (www.wikipedia.org/wiki/ ycbcr color model.htm, diakses 15 februari 2008). suma’inna dan gugun gumilar kestabilan persamaan fungsional jensen 1hilwin nisa’, 2hairur rahman, 3imam sujarwo 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang 3jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: azzami09hadza@gmail.com abstrak persamaan fungsional jensen merupakan variasi dari persamaan fungsional cauchy additive yang paling sederhana dan paling bagus. persamaan fungsional jensen dapat diaplikasikan sebagai model dari suatu proses fisik ketika persamaan fungsional jensen tersebut stabil. oleh karena itu, dengan diketahuinya kestabilan dari persamaan fungsional tersebut, dapat menambah referensi para peneliti lain yang akan mengaplikasikan persamaan fungsional jensen. pada artikel ini akan ditunjukkan kestabilan persamaan fungsional jensen. untuk mengetahui kestabilannya digunakan teorema kestabilan hyers-ulam-rassias. berdasarkan hasil analisis, telah dibuktikan bahwa persamaan fungsional jensen telah memenuhi teorema kestabilan hyers-ulam-rassias, sehingga dapat dikatakan bahwa persamaan fungsional jensen tersebut stabil. kata kunci: persamaan fungsional cauchy additive, persamaan fungsional jensen, kestabilan hyers-ulamrassias abstract jensen functional equation is the simplest and the most elegant one among variations of cauchy additive functional equation. jensen functional equation can be applied as a model of physical process when it is stable. this article showed stability of jensen functional equation. to know the stability, we used hyers-ulam-rassias stability theorem. based on the analysis, we proved that jensen functional equation satisfy hyers-ulam-rassias stability theorem, so it can be said that jensen functional equation is stable. keywords: additive cauchy functional equation, jensen functional equation, hyers-ulam-rassiass stability pendahuluan persamaan fungsional merupakan salah satu pembahasan dari matematika modern. di antara berbagai macam persamaan fungsional, persamaan fungsional yang paling penting adalah persamaan fungsional cauchy additive. hal ini dikarenakan sifat-sifat dari persamaan fungsional cauchy additive dapat digunakan untuk mengembangkan teorema-teorema dari persamaan fungsional lain. ada banyak variasi dari persamaan fungsional cauchy additive, akan tetapi diantara variasi persamaan fungsional cauchy additive tersebut yang paling bagus dan paling sederhana adalah persamaan fungsional jensen. suatu fungsi 𝑓: ℝ → ℝ disebut sebagai persamaan fungsional jensen jika memenuhi 𝑓 ( 𝑥+𝑦 2 ) = 𝑓(𝑥)+𝑓(𝑦) 2 , ∀𝑥, 𝑦 ∈ ℝ. persamaan fungsional jensen dapat diaplikasikan sebagai model dari suatu proses fisik ketika persamaan fungsional jensen tersebut stabil. dengan diketahuinya kestabilan dari persamaan fungsional jensen, akan dapat dijadikan referensi para peneliti lain yang bermaksud untuk mengaplikasikan persamaan fungsional jensen tersebut. adapun konsep kestabilan yang digunakan untuk meneliti kestabilan persamaan fungsional jensen pada artikel ini adalah konsep kestabilan hyersulam-rassias. dalam teorema kestabilan hyersulam-rassias, persamaan fungsional yang dijadikan acuan adalah persamaan fungsional cauchy additive. jika yang diteliti kestabilannya adalah persamaan fungsional lain, maka persamaan fungsional cauchy additive dalam teorema kestabilan hyers-ulam-rassias diganti dengan persamaan fungsional tersebut. oleh karena itu, untuk meneliti kestabilan persamaan fungsional jesnen, persamaan fungsional cauchy additive dalam teorema kestabilan hyerss-ulam-rassias tersebut diganti dengan persamaan fungsional jensen. artikel ini merupakan upaya ilmiah untuk meneliti kestabilan dari persamaan fungsional jensen dalam rangka penelitian pengembangan. pada peneltian ini dilakukan pembuktian terhadap teorema kestabilan hyers-ulam-rassiass pada persamaan fungsional jensen. jika persamaan fungsional jensen mailto:azzami09hadza@gmail.com hilwin nisa’, hairur rahman, imam sujarwo 24 volume 3 no. 4 mei 2015 memenuhi teorema kestabilan hyers-ulamrassiass, maka dapat dikatakan bahwa persamaan fungsional jensen tersebut stabil. kajian teori 1. persamaan fungsional persamaan fungsional adalah persamaan fungsi yang belum diketahui fungsinya [1]. 2. persamaan fungsional cauchy additive suatu fungsi 𝑓: ℝ → ℝ dikatakan suatu fungsi additive jika fungsi tersebut memenuhi persamaan fungsional cauchy additive 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) + 𝑓(𝑦) untuk setiap 𝑥, 𝑦 ∈ ℝ [1]. 3. persamaann fungsional jensen suatu fungsi 𝑓: ℝ → ℝ disebut persamaan jensen, jika persamaan tersebut memenuhi 𝑓 ( 𝑥 + 𝑦 2 ) = 𝑓(𝑥) + 𝑓(𝑦) 2 ∀𝑥, 𝑦 ∈ ℝ [1]. 4. ruang bernorma misalkan 𝐸 suatu ruang vektor. suatu pemetaan ‖. ‖: 𝐸 → ℝ disebut norm, jika ∀𝑥, 𝑦 ∈ 𝐸 dan 𝜆 ∈ ℝ berlaku 1. ‖𝑥‖ ≥ 0; 2. ‖𝑥‖ = 0 ↔ 𝑥 = 0; 3. ‖𝜆𝑥‖ = |𝜆|‖𝑥‖; 4. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖. (e, ‖ . ‖) disebut ruang vektor bernorma dan ‖𝑥‖ disebut norm dari 𝑥 [2]. setiap ruang bernorma (𝐾, ‖. ‖) merupakan ruang metrik terhadap 𝑑: 𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖ untuk setiap 𝑥, 𝑦 ∈ 𝐾 [3]. 5. barisan konvergen suatu barisan 𝑋 = {𝑥𝑛} ∈ ℝ dikatakan konvergen ke 𝑥 ∈ ℝ atau 𝑥 dikatakan suatu limit dari {𝑥𝑛}, jika untuk setiap 𝜀 > 0 terdapat suatu bilangan asli 𝐾(𝜀) sedemikian hingga untuk setiap 𝑛 ≥ 𝐾(𝜀), 𝑥𝑛 memenuhi |𝑥 − 𝑥𝑛| < 𝜀 [4]. 6. barisan cauchy suatu barisan 𝑋 = {𝑥𝑛} ∈ ℝ dikatakan suatu barisan cauchy jika untuk setiap 𝜀 > 0 terdapat suatu bilangan asli 𝐻(𝜀) sedemikian hingga untuk setiap bilangan asli 𝑛, 𝑚 ≥ 𝐻(𝜀), 𝑥𝑛 dan 𝑥𝑚 memenuhi |𝑥𝑛 − 𝑥𝑚| < 𝜀 [4]. di dalam sembarang ruang metrik (𝑋, 𝑑) setiap barisan konvergen merupakan barisan cauchy [5]. 7. ruang banach ruang banach merupakan ruang bernorma yang lengkap [6]. ruang bernorma dikatakan lengkap jika setiap barisan cauchy-nya konvergen [2]. 8. kestabilan hyers-ulam-rassiass misalkan 𝐸1 dan 𝐸2 merupakan ruang banach, dan misalkan 𝑓: 𝐸1 → 𝐸2 suatu fungsi yang memenuhi pertidaksamaan ‖𝑓(𝑥 + 𝑦) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝜃(‖𝑥‖𝑝 + ‖𝑦‖𝑝 ) untuk 𝜃 > 0, 𝑝 ∈ [0,1), dan ∀𝑥, 𝑦 ∈ 𝐸1. maka ada 𝐴: 𝐸1 → 𝐸2 suatu fungsi additive yang tunggal, sedemikian hingga ‖𝑓(𝑥) − 𝐴(𝑥)‖ ≤ 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 (2.16) ∀𝑥 ∈ 𝐸1 [7]. pembahasan untuk membuktikan kestabilan persamaan fungsional jensen dengan menggunakan konsep kestabilan hyers-ulam-rassias, maka persamaan fungsional cauchy additive dalam teorema kestabilan hyers-ulam-rassias tersebut diganti dengan persamaan fungsional jensen. oleh karena itu, untuk meneliti kestabilan persamaan fungsional jensen adalah dengan membuktikan teorema berikut ini: misalkan 𝑓: 𝐸1 → 𝐸2 merupakan suatu fungsi antara ruang banach. jika f memenuhi pertidaksamaan fungsional ‖2𝑓 ( 𝑥+𝑦 2 ) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝜃(‖𝑥‖𝑝 + ‖𝑦‖𝑝) ... (1) (3.2) ∀𝜃 ≥ 0, dengan 0 ≤ 𝑝 < 1 dan ∀𝑥, 𝑦 ∈ 𝐸1 ada 𝐴: 𝐸1 → 𝐸2, suatu fungsi additive yang tunggal, sedemikian hingga ‖𝑓(𝑥) − 𝐴(𝑥)‖ ≤ 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 ∀𝑥 ∈ 𝐸1. bukti: karena fungsi tersebut terjadi di antara ruang banach (𝐸1 𝑑𝑎𝑛 𝐸2) dan ∀𝑥, 𝑦 ∈ 𝐸1, dan karena ruang banach merupakan ruang bernorma yang lengkap, maka akan berlaku sifat dari ruang bernorma sebagi berikut: 1. ‖𝑥‖ ≥ 0; 2. ‖𝑥‖ = 0 ↔ 𝑥 = 0; kestabilan persamaan fungsional jensen cauchy – issn: 2086-0382 25 3. ‖𝜆𝑥‖ = |𝜆|‖𝑥‖, dimana 𝜆 ∈ ℝ; 4. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖. untuk membuktikan teorema hyers-ulam-rassias, maka harus ditunjukkan bahwa: 1. { 𝑓(2𝑛𝑥) 2𝑛 } 𝑛=1 ∞ merupakan suatu barisan cauchy untuk setiap 𝑥 ∈ 𝐸1. 2. jika 𝐴(𝑥) = lim 𝑛→∞ 𝑓(2𝑛𝑥) 2𝑛 , maka 𝐴 merupakan fungsi additive. 3. 𝐴 memenuhi ‖𝑓(𝑥) − 𝐴(𝑥)‖ ≤ 2𝜃 2−2𝑝 ‖𝑥‖𝑝 , ∀𝑥 ∈ 𝐸1. 4. 𝐴 merupakan fungsi yang tunggal. asumsikan 𝑓(0) = 0, ‖2𝑓 ( 𝑥 + 𝑦 2 ) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝜃(‖𝑥‖𝑝 + ‖𝑦‖𝑝 ). jika diambil y = 0 dan f(0) = 0, maka ‖2𝑓 ( 𝑥 + 0 2 ) − 𝑓(𝑥) − 𝑓(0)‖ ≤ 𝜃(‖𝑥‖𝑝 + ‖0‖𝑝 ) ‖2𝑓 ( 𝑥 2 ) − 𝑓(𝑥)‖ ≤ 𝜃(‖𝑥‖𝑝 ). dengan mengganti 𝑥 = 2𝑥 dan kedua sisi dibagi dengan 2, maka akan diperoleh ‖𝑓(𝑥) − 1 2 𝑓(2𝑥)‖ ≤ 𝜃 2 (‖2𝑥‖𝑝), ∀𝑥 ∈ 𝐸1. misalkan 𝑛, 𝑚 bilangan bulat nonnegatif dengan 𝑛 < 𝑚, maka ‖ 1 2𝑛 𝑓(2𝑛 𝑥) − 1 2𝑚 𝑓(2𝑚 𝑥)‖ = ‖ ‖ ‖ 1 2𝑛 𝑓(2𝑛 𝑥) − 1 2𝑛+1 𝑓(2𝑛+1𝑥) + 1 2𝑛+1 𝑓(2𝑛+1𝑥) − 1 2𝑛+2 𝑓(2𝑛+2𝑥) + ⋯ + 1 2𝑚−1 𝑓(2𝑚−1𝑥) − 1 2𝑚 𝑓(2𝑚 𝑥) ‖ ‖ ‖ . karena 𝑓 suatu fungsi di antara ruang banach, dengan menggunakan sifat keempat pada ruang bernorma (ketaksamaan segitiga), maka diperoleh ‖ 1 2𝑛 𝑓(2𝑛 𝑥) − 1 2𝑚 𝑓(2𝑚𝑥)‖ ≤ ‖ 1 2𝑛 𝑓(2𝑛 𝑥) − 1 2𝑛+1 𝑓(2𝑛+1𝑥)‖ + ‖ 1 2𝑛+1 𝑓(2𝑛+1𝑥) − 1 2𝑛+2 𝑓(2𝑛+2𝑥)‖ + ⋯ + ‖ 1 2𝑚−1 𝑓(2𝑚−1𝑥) − 1 2𝑚 𝑓(2𝑚𝑥)‖ = 1 2𝑛 ‖𝑓(2𝑛 𝑥) − 1 2𝑛 𝑓(2𝑛+1𝑥)‖ + 1 2𝑛+1 ‖𝑓(2𝑛+1𝑥) − 1 2𝑛 𝑓(2𝑛+2𝑥)‖ + ⋯ + 1 2𝑚−1 ‖𝑓(2𝑚−1𝑥) − 1 2𝑚 𝑓(2𝑚𝑥)‖ ≤ 1 2𝑛 𝜃 2 (‖2𝑛+1𝑥‖𝑝) + 1 2𝑛+1 𝜃 2 (‖2𝑛+2𝑥‖𝑝) + ⋯ + 1 2𝑚−1 𝜃 2 (‖2𝑚 𝑥‖𝑝) = 2𝑝 2𝑛 𝜃 2 (‖2𝑛 𝑥‖𝑝) + 2𝑝 2𝑛+1 𝜃 2 (‖2𝑛+1𝑥‖𝑝) + ⋯ + 2𝑝 2𝑚−1 𝜃 2 (‖2𝑚−1𝑥‖𝑝) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) ( 2𝑝𝑛 2𝑛 + 2𝑝(𝑛+1) 2𝑛+1 + ⋯ + 2𝑝(𝑚−1) 2𝑚−1 ) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) ∑ 2𝑘𝑝 2𝑘 𝑚−1 𝑘=𝑛 . jadi, ‖ 1 2𝑛 𝑓(2𝑛 𝑥) − 1 2𝑚 𝑓(2𝑚𝑥)‖ ≤ 2𝑝 𝜃 2 (‖𝑥‖𝑝 ) ∑ 2𝑘𝑝 2𝑘 . 𝑚−1 𝑘=𝑛 berdasarkan definisi barisan cauchy, barisan {𝑥𝑛 } dikatakan cauchy jika berlaku |𝑥𝑛 − 𝑥𝑚 | < 𝜀 untuk setiap 𝜀 > 0. jika 𝑛 → ∞ dan karena 0 ≤ 𝑝 < 1, maka 2𝑝𝜃 2 (‖𝑥‖𝑝 ) ∑ 2𝑘𝑝 2𝑘 𝑚−1 𝑘=𝑛 = 0. oleh karena itu, lim 𝑛→∞ ‖ 1 2𝑛 𝑓(2𝑛𝑥) − 1 2𝑚 𝑓(2𝑚 𝑥)‖ = 0, sehingga dapat dikatakan bahwa { 𝑓(2𝑛𝑥) 2𝑛 } 𝑛=1 ∞ merupakan barisan cauchy untuk setiap 𝑥 ∈ 𝐸1. karena 𝐸1 merupakan ruang banach, dimana setiap barisan cauchy-nya konvergen, maka terdapat fungsi 𝐴: 𝐸1 → 𝐸2 yang didefinisikan dengan 𝐴(𝑥) = lim 𝑛→∞ 𝑓(2𝑛𝑥) 2𝑛 untuk setiap 𝑥 ∈ 𝐸1 . selanjutnya akan ditunjukkan bahwa fungsi 𝐴: 𝐸1 → 𝐸2 merupakan fungsi additive. pandang bahwa ‖𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦)‖ = ‖ lim 𝑛→∞ { 2𝑓 ( 2𝑛(𝑥+𝑦) 2 ) 2𝑛 − 𝑓(2𝑛 𝑥) 2𝑛 − 𝑓(2𝑛𝑦) 2𝑛 }‖ = ‖ lim 𝑛→∞ 1 2𝑛 {2𝑓 ( 2𝑛 (𝑥 + 𝑦) 2 ) − 𝑓(2𝑛 𝑥) − 𝑓(2𝑛 𝑦)}‖ = lim 𝑛→∞ 1 2𝑛 ‖{2𝑓 ( 2𝑛 (𝑥 + 𝑦) 2 ) − 𝑓(2𝑛 𝑥) − 𝑓(2𝑛 𝑦)}‖. dengan menggunakan pertidaksamaan (1), maka didapatkan ‖𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦)‖ ≤ lim 𝑛→∞ 𝜃(‖𝑥‖𝑝 + ‖𝑦‖𝑝) 2𝑛 = 0. jadi ‖𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦)‖ ≤ 0. berdasarkan sifat pertama pada ruang bernorma dan ‖𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦)‖ ≤ 0, maka didapatkan ‖𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦)‖ = 0. berdasarkan sifat kedua pada ruang bernorma, maka didapatkan hilwin nisa’, hairur rahman, imam sujarwo 26 volume 3 no. 4 mei 2015 𝐴(𝑥 + 𝑦) − 𝐴(𝑥) − 𝐴(𝑦) = 0 𝐴(𝑥 + 𝑦) = 𝐴(𝑥) + 𝐴(𝑦). dari definisi fungsi addditive, maka dapat ditunjukkan bahwa 𝐴: 𝐸1 → 𝐸2 merupakan fungsi additive. selanjutnya akan ditunjukkan bahwa a memenuhi ||𝑓(𝑥) − 𝐴(𝑥)|| ≤ 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 , ∀𝑥 ∈ 𝐸1. ||𝑓(𝑥) − 𝐴(𝑥)|| = lim 𝑛→∞ ‖ 1 20 𝑓(20𝑥) − 1 2𝑛 𝑓(2𝑛𝑥)‖ ≤ lim 𝑛→∞ 2𝑝𝜃 2 (‖𝑥‖𝑝) ∑ 2𝑘𝑝 2𝑘 𝑛−1 𝑘=0 = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ ∑ 2(𝑝−1)𝑘 𝑛−1 𝑘=0 = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ ∑ 1 2(1−𝑝)𝑘 𝑛−1 𝑘=0 ... karena 𝑝 < 1 = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ (1 + ∑ 1 2(1−𝑝)𝑘 𝑛−1 𝑘=1 ) ... karena 𝑝 < 1 = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ (1 + 1 2(1−𝑝) 1 − 1 2(1−𝑝) ) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ (1 + 1 2(1−𝑝) 2(1−𝑝)−1 2(1−𝑝) ) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ (1 + 1 2(1−𝑝) − 1 ) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) lim 𝑛→∞ ( 2(1−𝑝) − 1 + 1 2(1−𝑝) − 1 ) = 2𝑝𝜃 2 (‖𝑥‖𝑝 ) ( 2(1−𝑝) 2(1−𝑝) − 1 ) = 2𝜃 2(2(1−𝑝) − 1) (‖𝑥‖𝑝) = 𝜃 ( 2 2𝑝 − 1) (‖𝑥‖𝑝) = 𝜃 ( 2−2𝑝 2𝑝 ) (‖𝑥‖𝑝 ) = 2𝑝𝜃 2 − 2𝑝 (‖𝑥‖𝑝). jadi, ||𝑓(𝑥) − 𝐴(𝑥)|| ≤ 2𝑝 𝜃 2 − 2𝑝 (‖𝑥‖𝑝 ). karena 2𝑝 < 2, ∀𝑝 ∈ [0,1), maka ||𝑓(𝑥) − 𝐴(𝑥)|| ≤ 2𝜃 2 − 2𝑝 (‖𝑥‖𝑝). terbukti ||𝑓(𝑥) − 𝐴. (𝑥)|| ≤ 2𝜃 2−2𝑝 ‖𝑥‖𝑝 , ∀𝑥 ∈ 𝐸1. selanjutnya akan ditunjukkan bahwa 𝐴 tunggal. andaikan 𝐴 tidak tunggal, maka akan ada fungsi additive yang lain 𝐵: 𝐸1 → 𝐸2 sedemikian hingga ||𝑓(𝑥) − 𝐵(𝑥)|| ≤ ||𝑓(𝑥) − 𝐴(𝑥)|| ≤ 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 ∀𝑥 ∈ 𝐸1. ||𝐴(𝑥) − 𝐵(𝑥)|| = ||𝐴(𝑥) − 𝑓(𝑥) + 𝑓(𝑥 − 𝐵(𝑥)|| ≤ ||𝐴(𝑥) − 𝑓(𝑥)|| + ||𝑓(𝑥) − 𝐵(𝑥)|| = ||𝑓(𝑥) − 𝐴(𝑥)|| + ||𝑓(𝑥) − 𝐵(𝑥)|| ≤ 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 + 2𝜃 2 − 2𝑝 ‖𝑥‖𝑝 . jadi, ||𝐴(𝑥) − 𝐵(𝑥)|| ≤ 4𝜃 2 − 2𝑝 ‖𝑥‖𝑝 . ||𝐴(𝑥) − 𝐵(𝑥)|| = lim 𝑛→∞ ‖ 1 2𝑛 𝐴(2𝑛 𝑥) − 1 2𝑛 𝐵(2𝑛 𝑥)‖ = lim 𝑛→∞ 1 2𝑛 ‖𝐴(2𝑛𝑥) − 𝐵(2𝑛 𝑥)‖ ≤ lim 𝑛→∞ 1 2𝑛 4𝜃 2 − 2𝑝 ‖𝑥‖𝑝 = 4𝜃 2 − 2𝑝 ‖𝑥‖𝑝 lim 𝑛→∞ 1 2𝑛 = 0 dimana 𝑛 ∈ ℝ. karena ||𝐴(𝑥) − 𝐵(𝑥)|| ≤ 0 dan berdasarkan sifat pertama pada ruang bernorma, maka 𝐴(𝑥) = 𝐵(𝑥), ∀𝑥 ∈ 𝐸1. jadi terbukti bahwa 𝐴 tunggal dan pembuktian teorema hyers-ulam-rassias di atas telah lengkap, sehingga terbukti bahwa persamaan jensen tersebut stabil. kesimpulan pada artikel ini telah ditunjukkan bahwa persamaan fungsional jensen telah memenuhiteorema kestabilan hyers-ulam-rassias. oleh karena itu, dapat dikatakan bahwa persamaan fungsional jensen tersebut stabil. dengan demikian, persamaan fungsional jensen ini dapat diaplikasikan sebagai model dari suatu proses fisik. daftar pustaka [1] p. k. sahoo and p. kannappan, "introduction to functional equations," in introduction to functional equations, new york, crc press, 2011. [2] r. coleman, "calculus on normed vector spaces," london, springer science+business media new york, 2012. [3] s. darmawijaya, "pengantar analisis abstrak," yogyakarta, jurusan matematika fakultas matematika dan ilmu pengetahuan alam ugm, 2007. [4] r. g. bartle and d. r. sherbert, "introduction to real analysis," new york, john wiley & sons, inc., 2000. kestabilan persamaan fungsional jensen cauchy – issn: 2086-0382 27 [5] m. muslikh, "analisis real," malang, universitas brawijaya press, 2012. [6] r. s. al-mosadder, "on stability of some types of functional equations," gaza, islamic university of gaza, 2012. [7] s. m. jung, "hyers-ulam-rassias stability of functional equations in nonlinear analysis," london, springer science+business media, 2011. [8] a. g. parlos, "linearization of nonlinear dynamics," 2014. [online]. available: http://parlos.tamu.edu/meen651/lineariz ation.pdf. [accessed 17 agustus 2014]. [9] r. c. robinson, an introduction dynamical system continuous and discrete, new jersey: pearson education, 2004. [10] w. e. boyce and r. c. diprima, elementary differential equations and boundary value problems, new york: john wiley & sons, inc, 2009. [11] r. o. kwofie, "a mathematical model of a suspension bridge – case study: adomi bridge, atimpoku, ghana," global advanced research journal of engineering, technology, and innovation, vol. 1(3), pp. 047-062, 2012. [12] g. vries, t. hillen, m. lewis, j. müller and m. schönfisch, a course in mathematical biology: quantitative modeling with mathematical and computational methods, alberta: siam, 2006. [13] p. k. s. d. p. kannappan, "introduction to functional equations," in introduction to functional equations, new york, crc press, 2011, p. 2. pendahuluan kajian teori 1. persamaan fungsional 2. persamaan fungsional cauchy additive 3. persamaann fungsional jensen 4. ruang bernorma 5. barisan konvergen 6. barisan cauchy 7. ruang banach 8. kestabilan hyers-ulam-rassiass pembahasan kesimpulan daftar pustaka 59 optimasi parameter neural network pada data time series untuk memprediksi rata-rata kekuatan gempa per periode (studi kasus gempa bumi di maluku utara) optimization parameter of neural network for time series data to predict the magnitude of periodics earthquake (study case earthquake in north maluku) muzakir hi sultan fakultas matematika dan ilmu pengetahuan alam, program studi magister matematika, universitas brawijaya jl. veteran malang, telp. (0341)554403. email: zhasya91@yahoo.com abstrak gempa bumi merupakan suatu pergerakan tanah yang terjadi secara tiba-tiba hingga menimbulkan getaran, besarnya kekuatan gempa dapat mengakibatkan bencana baik kerusakan maupun korban jiwa. untuk mengantisipasi bencana yang akan datang maka diperlukan suatu model khususnya untuk meramalkan besarnya kekuatan gempa. pada penelitian ini, digunakan model arima dan model kombinasi dari neural network-algoritma genetik (nn-ga) untuk memprediksi rata-rata kekuatan gempa bumi setiap bulan khususnya yang terjadi di wilayah maluku utara. data yang digunakan adalah data kekuatan gempa berdasarkan skala richter yang diperoleh dari badan meteorologi, klimatologi dan geofisika (bmkg) kota ternate. sebagai input pada model arima dan nn-ga digunakan rata-rata kekuatan gempa bumi 36 bulan dan rata-rata kekuatan gempa 36 bulan berikutnya digunakan sebagai target untuk prediksi. untuk mengupdate parameter (bobot) dari neural network digunakan metode gradient descent dan untuk mendapatkan parameter yang lebih optimal pada layer output, maka di diterapkan algoritma genetik. hasil peramalan dari kedua model kemudian dibandingkan dan model terbaik ditentukan dari nilai mean square error (mse) yang terkecil. dari hasil peramalan dengan model arima diperoleh mse sebesar 1.0125, sedangkan pada model nn-ga diperoleh mse sebesar 0.9196. nilai tersebut, menunjukkan bahwa model nn-ga lebih baik dari model arima untuk peramalan rata-rata kekuatan gempa bumi beberapa bulan ke depan. kata kunci : metode peramalan time series arima, neural network (nn), gradient descent dan algoritma genetik (ga). abstract earthquake is a ground motion that occurs suddenly to cause vibration. the strength of earthquake magnitude can lead to serious damage and fatalities. to anticipate the impending disaster, then we need a model specifically for earthquake forecasting. this research, uses arima model and a combination of neural network-genetic algorithm (nn-ga) to predict the average magnitude of earthquake in each periods. particularly, in the area of north maluku. the data used in this research is the average of earthquake measured by richter scale (sr) obtained from the meteorology, climatology and geophysics (bmkg) ternate city. the average magnitude of 36 months earthquake is used for input of the arima model and the nn-ga models, whereas the next 36 months earthquake is used as target for the prediction. to update the parameters (weights) of neural networks the gradient descent method is used and to get an optimal parameters in the output layer, the genetic algorithm is applieds. the forecasting results of the two models are then compared and the best model is determined from the smallest mean square error (mse). the result show that the mse of forecasting based on arima model is 1.0125. while the mse of the nn-ga models is 0.9196. these values, indicate that the nn-ga models is better than arima model to forecast the average of earthquake magnitude for the next several months. key words : time series arima forecasting, neural network (nn), gradient descent, genetic algorithm (ga) pendahuluan secara geografis, kepulauan indonesia terletak pada perbenturan tiga lempeng kerak bumi yaitu lempeng eruasia, lempeng pasifik, dan lempeng india australia. ditinjau secara geologis, kepulauan indonesia berada pada pertemuan dua jalur gempa utama, yaitu jalur gempa sirkum mailto:zhasya91@yahoo.com muzakir hi sultan 60 volume 3 no. 2 mei 2014 pasifik dan jalur gempa alpide transasiatik sehingga menyebabkan wilayah indonesia sangat rawan bahaya goncangan gempa bumi. catatan direktorat vulkanologi dan mitigasi bencana geologi (dvmbg) departemen energi dan sumber daya mineral menunjukan bahwa ada 28 wilayah di indonesia yang dinyatakan rawan gempa dan tsunami salah satunya adalah wilayah maluku utara [1]. gempa bumi adalah pergerakan tanah secara tiba-tiba yang terjadi di bumi hingga menimbulkan getaran yang disebut gelombang seismik. besarnya kekuatan gempa yang terjadi pada hiposentrum (pusat gempa) akan mengakibatkan terjadinya besaran getaran serta pengaruhnya yang berbeda pada daerah yang berbeda pula di permukaan bumi (epiosentrum) [2]. menurut [3] gempa bumi merupakan suatu kejadian yang tidak sepenuhnya tunggal, tetapi kejadian-kejadian tersebut saling berhubungan baik kekuatan, waktu dan tempat kejadian yang satu dengan yang lainnya. [4]menyatakan bahwa untuk menghitung atau memprediksi kondisi gempa pada masa yang akan datang dapat dilakukan dengan menggunakan pemodelan ann (artificial neural network). [5] mengaplikasikan ann pada data time series untuk memprediksi sifat dari gempa bumi. pengembangan model ann telah banyak dilakukan dengan banyak cara seperti penggabungan dan optimalisasi struktur jaringan [6]. neural network (nn) telah digunakan secara luas dalam menyelesaikan permasalahan di bidang geofisika, kedokteran, meteorologi, sosial dan industri. [7] melakukan penelitian yang paling awal dalam penggunaan jaringan syaraf (neural network) untuk peramalan data time series. kelebihan lain penggunaan neural network jika diterapkan pada data time series adalah bahwa data tidak perlu memenuhi asumsi sebaran seperti pada metode-metode statistika [8]. terdapat dua tipe neural network, yaitu pelatihan dengan pengarahan (supevised training) dan pelatihan tanpa pengarahan (unsupevised training). salah satu metode neural network yang menggunakan algoritma pelatihan dengan pengarahan adalah neural network propagasi balik (backpropagation) [9]. untuk penentuan bobot pada hidden layer selama proses pelatihan menggunakan metode gradient descent [10]. kemudian bobot di optimasi menggunakan algoritma genetik (ga) untuk menurunkan kesalahan yang terjadi pada hasil peramalan. tujuan penelitian ini adalah mengkaji cara kerja dari arima dan neural network algoritma genetik, memperoleh suatu output yang optimal pada neural network setelah parameternya (bobot) di optimasi menggunakan metode algortima genetik, dan membandingkan tingkat akurasi yang dihasilkan oleh arima dan neural network algoritma genetik pada data time series dalam memprediksi rata-rata kekuatan terjadinya gempa bumi di wilayah maluku utara untuk nperiode kedepan. dasar teori 1. analisis time series time series adalah serangkaian data pengamatan yang disusun menurut waktu, di mana data pengamatan tersebut bersifat acak dan saling berhubungan secara statistika. analisis data deret waktu pada dasarnya digunakan untuk melakukan analisis data yang mempertimbangkan pengaruh waktu [11]. 2. pengujian stasioneritas time series suatu data dapat dikatakan stasioner apabila pola data tersebut berada pada kesetimbangan disekitar nilai rata-rata (mean) yang konstan dan variansi disekitar rata-rata tersebut konstan selama waktu tertentu [12]. a. stasioner pada varians bagi data yang tidak statisioner pada varians maka ditransformasikan terlebih dahulu. transformasi data dapat dilakukan dengan metode transformasi box-cox [13], dengan bentuk sebagai berikut: 𝑇(𝑋𝑡 ) = 𝑋𝑡 () = { 𝑋𝑡  − 1  ,  ≠ 0 ln 𝑋𝑡 ,  = 0 (1) di mana : λ : parameter transformasi 𝑇(𝑋𝑡) : data transformasi xt : pengamatan pada waktu t b. stasioner pada mean jika data tidak stasioner terhadap mean maka perlu dilakukan pembedaan (differencing). operator shift mundur (backward shift) sangat tepat untuk menggambarkan proses differencing [12]. penggunaan backshift adalah sebagai berikut: 𝐵𝑋𝑡 = 𝑋𝑡−1 (2) di mana: 𝑋𝑡 : nilai pengamatan pada waktu ke-t 𝑋𝑡−1 : nilai pengamatan pada waktu ke-t-1 𝐵 : backshift secara umum jika terdapat differencing sampai orde ke-d ditulis sebagai 𝑋𝑡 𝑑 = (1 − 𝐵)𝑑 𝑋𝑡 , 𝑑 ≥ 1 (3) optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 61 3. koefisien fungsi autokorelasi (acf) koefisien autokorelasi merupakan suatu alat penentu dari indentifikasi pola dasar yang menggambarkan data [14]. menurut [15] proses stasioner suatu time series (𝑋𝑡 ) diperoleh dari 𝐸(𝑋𝑡 ) = 𝑋 dan 𝑉𝑎𝑟(𝑋𝑡 ) = (𝑋𝑡 − 𝑋) 2 yang konstan dan kovariansi 𝐶𝑜𝑣(𝑋𝑡 , 𝑋𝑡+1). dari koefisien autokorelasi antara 𝑋𝑡 dan 𝑋𝑡+1 untuk lag-k sebagai berikut: 𝜌𝑘 = 𝐶𝑜𝑣(𝑋𝑡 , 𝑋𝑡+1) √𝑉𝑎𝑟(𝑋𝑡 )√𝑉𝑎𝑟(𝑋𝑡+1) = 𝐸(𝑋𝑡 − �̅�)𝐸(𝑋𝑡+1 − �̅�) √𝑉𝑎𝑟(𝑋𝑡 )√𝑉𝑎𝑟(𝑋𝑡+1) (4) sedangkan untuk fungsi autokovariansi antara 𝑋𝑡 dan 𝑋𝑡+1 untuk lag-k sebagai berikut: 𝛾𝑘 = 𝐶𝑜𝑣(𝑋𝑡 , 𝑋𝑡+1) = 𝐸(𝑋𝑡 − �̅�)𝐸(𝑋𝑡+1 − �̅�) (5) dimana 𝛾𝑘 disebut fungsi autokovariansi, untuk keadaan stasioner 𝑉𝑎𝑟(𝑋𝑡 ) = 𝑉𝑎𝑟(𝑋𝑡+1) = 𝛾0 sehingga persamaan (4) dan (5) menjadi: 𝜌𝑘 = 𝛾𝑘 𝛾0 (6) koefisien fungsi autokorelasi 𝜌𝑘 diduga dengan koefisien autokorelasi data. 𝜌𝑘 = ∑ (𝑋𝑡 − �̅�)(𝑋𝑡+𝑘 − �̅�) 𝑛−𝑘 𝑡=1 ∑ (𝑋𝑡 − �̅�) 2𝑛−𝑘 𝑡=1 (7) 4. koefisien fungsi autokorelasi parsial (pacf) autokorelasi parsial digunakan untuk mengetahui korelasi atau hubungan antara 𝑋𝑡 dan 𝑋𝑡+𝑘 , apabila pengaruh dari time lag 1,2,3,…,k-j dianggap terpisah [12]. menurut [11] taksiran dari pacf adalah berdasarkan koefisien autokorelasi pada persamaan yule-walker untuk time lag-k secara rekursif sebagai berikut: 𝜙𝑘𝑘 = 𝜌𝑘𝑘 − ∑ 𝜙𝑘−1, 𝜌𝑗 𝑘−𝑗 𝑘−1 𝑗=1 1 − ∑ 𝜙𝑘−1, 𝜌𝑗 𝑗 𝑘−1 𝑗=1 (8) 5. pembentukan model arima model ini merupakan kombinasi dari model autoregressive dan moving average ditambah dengan integrated (pembedaan). model autoregressive (ar): model autogressive dengan ar (p). bentuk umum model ar (p) adalah: 𝑋𝑡 = 𝜙1 𝑋𝑡−1 + 𝜙2𝑋𝑡−2 + ⋯ + 𝜙𝑝 𝑋𝑡−𝑝 + 𝑒𝑡 (9) persamaaan (9) menggunakan operator b (backshift): 𝜙𝑝(𝐵)𝑋𝑡 = 𝑒𝑡 (10) dengan 𝜙𝑝(𝐵) = 1 − 𝜙1𝐵 − 𝜙2𝐵 2−. . . −𝜙𝑝𝐵 𝑝. model moving avarage (ma): bentuk umum model ma(q) dapat ditulis sebagai berikut: 𝑋𝑡 = 𝑒𝑡 − 𝜃1𝑒𝑡−1 − 𝜃2𝑒𝑡−2−. . . . −𝜃𝑞 𝑒𝑡−𝑞 (11) persamaaan (11) menggunakan operator b (backshift): 𝑋𝑡 = 𝜃𝑞(𝐵)𝑒𝑡 (12) dengan 𝜙𝑞 (𝐵) = 1 − 𝜙1𝐵 − 𝜙2𝐵 2−. . . −𝜙𝑞 𝐵 𝑞. model autoregresivve moving avarage (arma): model merupakan suatu gabungan dari model ar (p) dan ma (q). bentuk umum model arma (p,q), yaitu 𝑋𝑡 = 𝜙1𝑋𝑡−1 + ⋯ + 𝜙𝑝 𝑋𝑡−𝑝 + 𝑒𝑡 − 𝜃1𝑒𝑡−1 − ⋯ − 𝜃𝑞 𝑒𝑡−𝑞 (13) persamaaan (13) menggunakan operator b (backshift): 𝜙𝑝(𝐵)𝑋𝑡 = 𝜃𝑞 (𝐵)𝑒𝑡 (14) model autoregresivve moving avarage integreted (arima): model ini merupakan gabungan dari model arma (p,q) dan differencing (d), secara umum model arima (p,d,q) untuk suatu data time series 𝑋𝑡 adalah sebagai berikut [12] 𝜙𝑝 (𝐵)(1 − 𝐵) 𝑑𝑋𝑡 = 𝜃𝑞 (𝐵)𝑒𝑡 (15) muzakir hi sultan 62 volume 3 no. 2 mei 2014 dengan 𝜙𝑝(𝐵) = 1 − 𝜙1𝐵 − 𝜙2𝐵 2−. . . −𝜙𝑝𝐵 𝑝 𝜃𝑞 (𝐵) = 1 − 𝜃1 𝐵 − 𝜃2𝐵 2−. . . . −𝜃𝑞 𝐵 𝑞 di mana: 𝜙𝑝(𝐵) : faktor ar (p) 𝜃𝑞 (𝐵) : faktor ma (q) (1 − 𝐵)𝑑 : order differencing 6. tahapan pemodelan autoregressive integrated moving average (arima) tahapan pemodelan metode arima adalah: 1. identifikasi model langkah pertama dalam pembentukan model arima adalah pembentukan plot data time series. pembuatan plot data time series bertujuan untuk mendeteksi stasioneritas data time series. 2. menentukan orde autoregressive (ar) dan moving average (ma) setelah data terbukti stasioner langkah selanjutnya adalah mengidentifikasi order autoregressive (ar) dan moving average (ma) dapat dilakukan dengan cara melihat plot acf dan pacf dari data tersebut. 3. estimasi parameter metode yang digunakan untuk mengestimasi parameter arima 𝜙 dan 𝜃 yaitu metode kuadrat terkecil (least square method). untuk mempermudah proses estimasi biasanya dilakukan dengan program minitab. 4. pemeriksaan diagnosis setelah berhasil mengestimasi nilai-nilai parameter dari model arima yang ditetapkan sementara, selanjutnya perlu dilakukan pemeriksaan diagnostik untuk membuktikan bahwa model tersebut cukup memadai dan menentukan model mana yang terbaik digunakan untuk peramalan [12]. a. uji signifikansi parameter uji signifikansi parameter bertujuan untuk membuktikan bahwa model arima tersebut cukup memadai untuk digunakan. uji signifikasi parameter model pada parameter yaitu dengan uji t. b. uji sisa white noise suatu proses {𝑒𝑡} disebut white noise jika datanya terdiri dari variabel random yang tidak berkorelasi dan berdistribusi normal. menguji sampel residual dari acf tidak signifikan memakai statistik uji ljung box [14] 1. hipotesis 𝐻0 ∶ 𝜌1 = 𝜌2 =. . . = 𝜌𝐾 = 0 (white niose) 𝐻1 ∶ ∃ 𝜌𝑘 ≠ 0 , 𝑘 = 1,2, . . . , 𝐾 (tidak white niose) 2. statistik uji yaitu uji ljung box 𝒬𝐾 = 𝑛(𝑛 + 2) ∑ 𝜌𝑘 2 (𝑛 − 𝑘) 𝐾 𝑘=1 (15) 3. kriteria keputusan tolak 𝐻0 jika 𝒬𝐾 > 𝜒(𝛼,𝑑𝑓) 2 ; 𝑑𝑓 = 𝐾 − 𝑚, di mana 𝐾 banyaknya lag dan 𝑚 banyaknya parameter atau p-value < 𝛼0.05, artinya 𝑒𝑡 barisan yang tidak memiliki korelasi. 7. konsep dasar pemodelan artificial neural networks (ann) neural networks (nn) salah satu metode yang digunakan untuk pengenalan pola, signal processing dan peramalan, sebagaimana metodemetode lain yang terdapat dalam nn [16]. model dari neural networks terdiri atas 3 layer yaitu layer input, hidden layer/kerner layer dan layer output [17]. 8. metode neural network backpropagation algoritma bacpropagation menggunakan error output untuk menggubah nilai bobotbobotnya dalam arah mundur (backward). untuk mendapatkan error tahap perambatan maju (forward propagation) harus dikerjakan terlebih dahulu [16]. misalkan diberikan n inputan data yang berbeda 𝑥𝑖 (𝑖 = 1,2, … , 𝑛) yang dihubungkan dengan output data �̂�𝑘 (𝑘 = 1,2, … , 𝑝), maka interpolasi dari neural networks adalah: �̂�𝑘 = 𝑓 0 [∑(𝑤𝑗𝑓𝑗 ℎ 𝑚 𝑗=1 (∑ 𝑣𝑗𝑖 𝑛 𝑖=1 𝑥𝑖 (𝑘) + 𝑣0𝑗) + 𝑤0)] (16) di mana: 𝑥𝑖 : variabel input sebanyak 𝑛 �̂�𝑘 : nilai output layer 𝑘 : indeks banyak pasangan input-output 𝑣𝑗𝑖 : bobot dari neuron ke-𝑖 pada layer input yang menuju pada hidden layer 𝑣0𝑗 : bias dari neuron ke-𝑗 pada hidden layer 𝑓𝑗 ℎ : fungsi aktifasi neuron ke-𝑗 pada hidden layer 𝑤𝑗 : bobot dari neuron ke-𝑗 pada hidden layer yang menuju pada output layer 𝑤0 : bias pada neuron output layer 𝑓0 : fungsi aktifasi pada hidden layer fungsi aktivasi yang sering digunakan neural networks, salah satunya fungsi sigmoid biner dengan interval di antara 0 dan 1 [10], sebagai berikut: f(x) = 1 1 + 𝑒 −x (17) dengan turunannya optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 63 f(x)′ = 𝑒 −x (1 + 𝑒 −x)2 = 1 1 + 𝑒−x (1 − 1 1 + 𝑒 −x ) = f(x)(1 − f(x)) (18) langkah-langkah algoritma pelatihan untuk neural network adalah sebagai berikut: 1. inisialisasi bobot (diberi nilai kecil secara acak) 2. ulangi langkah 2 hingga 9 sampai kondisi iterasi terpenuhi. 3. untuk masing-masing pasangan data pelatihan, lakukan langkah 3-8 fase i : propagasi maju (feedforward) 4. masing-masing unit input 𝑥𝑖(𝑖 = 1,2, … , 𝑛) menerima sinyal masukan 𝑥𝑖 dan sinyal tersebut diteruskan ke unit-unit bagian berikutnya. 5. hitung semua output pada hidden layer 𝑎𝑗(𝑘) ℎ (k = 1,2, … , n) 𝑔𝑗(𝑘) ℎ = ∑ 𝑣𝑗𝑖 𝑛 𝑖=1 𝑥𝑖(𝑘) + 𝑣0𝑗 (19) kemudian menghitung fungsi aktifasi 𝑓𝑗 ℎ pada hidden layer 𝑎𝑗(𝑘) ℎ = 𝑓𝑗 ℎ (𝑔𝑗(𝑘) ℎ ) = 𝑓𝑗 ℎ (∑ 𝑣𝑗𝑖 𝑛 𝑖=1 𝑥𝑖(𝑘) + 𝑣0𝑗 ) (20) bila fungsi aktifasi sigmoid maka: 𝑓𝑗 ℎ = 1 1 + 𝑒−𝑔𝑗(𝑘) ℎ (21) 6. hitung semua output pada layer output 𝑎(𝑘) 0 (k = 1,2, … , m) 𝑔(𝑘) 0 = ∑ 𝑤𝑗 𝑎𝑗(𝑘) ℎ 𝑚 𝑗=1 + 𝑤0 (22) kemudian menghitung fungsi aktifasi 𝑓0 pada layer output 𝑎(𝑘) 0 = 𝑓0(𝑔(𝑘) 0 ) = 𝑓0 (∑ 𝑤𝑗𝑎𝑗(𝑘) ℎ 𝑚 𝑗=1 + 𝑤0 ) (23) dengan 𝑓0 = 1 1+𝑒 −𝑔(𝑘) 0 fase ii : propogasi mundur (backpropagation) 7. masing-masing layer output �̂�𝑘 (𝑘 = 1,2, … , p) menerima pola target (𝑦𝑘 ) sesuai dengan pola input dan hitung nilai galatnya (𝛿𝑘 0). 𝛿𝑘 0 = (𝑦𝑘 − �̂�𝑘 ) 𝑓 0 ′(𝑔(𝑘) 0 ) (24) karena fungsi 𝑓0(𝑔(𝑘) 0 ) = �̂�𝑘 menggunakan fungsi sigmoid maka: 𝑓0 ′ (𝑔(𝑘) 0 ) = 𝑓0(𝑔(𝑘) 0 ) (1 − 𝑓0(𝑔(𝑘) 0 )) = �̂�𝑘 (1 − �̂�𝑘 ) (25) sehingga persamaan (24) di subtitusikan ke persamaan (25)diperoleh: 𝛿𝑘 0 = (𝑦𝑘 − �̂�𝑘 )�̂�𝑘 (1 − �̂�𝑘 ) (26) hitung perubahan bobot 𝑤𝑘𝑗 dengan laju pembelajaran 𝛼. ∆𝑤𝑗𝑘 = 𝛼𝛿𝑘 0𝑎𝑗(𝑘) ℎ ; 𝑘 = 1,2, … , 𝑝 ; 𝑗 = 1,2, … , 𝑚 dan perubahan bobot bias 𝑤0𝑘 ∆𝑤0𝑘 = 𝛼𝛿𝑘 0 8. hitung faktor 𝛿 berdasarkan galat di setiap hidden layer 𝑎𝑗(𝑘) ℎ (k = 1,2, … , n) δ𝑗 = ∑ 𝛿𝑘 0 𝑝 𝑘=1 𝑤𝑗 (27) faktor 𝛿 unit tersembunyi 𝛿𝑗(𝑘) ℎ = δ𝑗𝑓𝑗 ℎ ′(𝑔𝑗(𝑘) ℎ ) = δ𝑗𝑎𝑗(𝑘) ℎ (1 − 𝑎𝑗(𝑘) ℎ ) (28) maka perubahan bobot 𝑣𝑗𝑖 dengan laju pembelajaran 𝛼. ∆𝑣𝑗𝑖 = 𝛼𝛿𝑗(𝑘) ℎ 𝑥𝑖(𝑘) ; 𝑗 = 1,2, … , 𝑚 ; 𝑖 = 1,2, … , 𝑛 dan perubahan bobot bias 𝑣0𝑗 ∆𝑣0𝑗 = 𝛼𝛿𝑗(𝑘) ℎ fase iii : perubahan bobot 9. masing-masing output layer �̂�𝑘 (𝑘 = 1,2, … , p) di update bobotnya. setelah menurunkan algoritma gradient descent, diperoleh dua persamaan yang digunakan untuk mengupdate bobot 𝑤𝑗 , 𝑤0, 𝑣𝑗𝑖 dan 𝑣0𝑗 sebagai berikut: a. untuk updating bobot dan bias pada output layer 𝑤𝑗(𝑘 + 1) = 𝑤𝑗 (𝑘) + 𝛼 ∑ 𝛿𝑘 0𝑎𝑗(𝑘) ℎ 𝑝 𝑘=1 (29) 𝑤0(𝑘 + 1) = 𝑤0(𝑘) + 𝛼 ∑ 𝛿𝑘 0 𝑝 𝑘=1 (30) b. untuk updating bobot dan bias pada hidden layer 𝑣𝑗𝑖(𝑘 + 1) = 𝑣𝑗𝑖 (𝑘) + 𝛼 ∑ 𝛿𝑗(𝑘) ℎ 𝑥𝑖(𝑘) 𝑝 𝑘=1 (31) 𝑣0𝑗(𝑘 + 1) = 𝑣0𝑗 (𝑘) + 𝛼 ∑ 𝛿𝑗(𝑘) ℎ 𝑝 𝑘=1 (32) dengan 𝛼 > 0 yang merupakan laju pembelajaran (learning rate) 9. metode algoritma genetik (ga) algoritma genetik (ga) merupakan suatu konsep komputasi yang pertama kali diutarakan oleh john holland dari universitas michigan pada tahun 1975. algoritma genetik adalah algoritma yang dikembangkan dari proses pencarian solusi optimasi menggunakan pencarian acak, ini muzakir hi sultan 64 volume 3 no. 2 mei 2014 terlihat pada proses pembangkitan populasi awal yang menyatakan sekumpulan solusi yang dipilh secara acak. algoritma ini memanfaatkan proses seleksi secara alamiah yang dikenal dengan proses evolusi dan operasi genetika atas kromosom [18]. 1. inisialisasi langkah awal untuk memperoleh nilai parameter yang optimum dari neural networks adalah dengan inisialisasi. kromosom memiliki panjang variabel yang didefinisikan sebagai berikut: 𝑐ℎ𝑟𝑜𝑚 = {𝑥𝑛1, 𝑥𝑛2, … , 𝑥𝑛𝑝} (33) di mana 𝑥𝑛1, 𝑥𝑛2, … , 𝑥𝑛𝑝 adalah kemungkinan calon–calon solusi sebanyak 𝑝 dan 𝑐ℎ𝑟𝑜𝑚 dinamakan populasi. dalam algoritma genetika, 𝑥𝑛1 dinamakan kromosom pertama dengan panjang 𝑛. 2. fungsi evaluasi fungsi evaluasi adalah fungsi yang menghitung nilai fitness di setiap kromosom, fungsi fitness adalah error antara target output dan saat output. 3. fungsi seleksi untuk mendapatkan solusi yang terbaik, maka program harus menyeleksi solusi yang memiliki nilai fitness yang baik (fitness terbesar) dan mampu bertahan dalam populasi. suatu metode seleksi yang umum digunakan adalah metode roulette-wheel (roda reulatte) [17] 4. crossover dan mutasi crossover adalah salah satu komponen paling penting dalam ga. sebuah kromosom yang mengarah pada solusi yang bagus dapat diperoleh dari proses memindah-silangkan dua buah kromosom [18]. persamaan crossover didefenisikan sebagai: 𝑋′ = 𝑟𝑋 + (1 − 𝑟)�̅� �̅�′ = (1 − 𝑟)�̅� + 𝑟�̅� (33) di mana �̅� dan �̅� adalah dua vektor berdimensi-k yang dinotasikan individu (keluarga) dari populasi dengan 𝑟 ∈ [0,1]. dari persamaan ini, dapat disajikan dalam bentuk: gambar 1. proses hitung crossover tiga titik dalam dua neutron mutasi berfungsi untuk menggantikan kromosom yang hilang dari populasi akibat proses seleksi yang memungkinkan munculnya kembali kromosom yang tidak muncul pada inisialisasi populasi, salah satu metode mutasi yang digunakan adalah mutasi seragam sebagai berikut: 𝑥𝑖 = { 𝑈(𝑎𝑖 , 𝑏𝑖 ) 𝑗𝑖𝑘𝑎 𝑖 = 𝑗 𝑥𝑖 𝑦𝑎𝑛𝑔 𝑙𝑎𝑖𝑛 (34) di mana 𝑎𝑖 batas bawah dan 𝑏𝑖 batas atas untuk setiap variabel i. berikut adalah gambar yang menyajikan proses mutasi yang muncul di antara parameter dari nn. gambar 2. mutasi seragam dari dua titik dalam dua neutron metode penelitian metode penelitian yang digunakan dalam penelitian ini adalah: 1. pengumpulan data data yang digunakan dalam penelitian ini diperoleh dari badan meteorologi, klimatologi dan geofisika (bmkg) kota ternate berupa kekuatan gempa dalam skala richter dari setiap kejadian gempa dengan kurun waktu januari 2007 sampai dengan desember 2012. 2. variabel penelitian berikut ini adalah variabel yang digunakan dalam penelitian ini. 1. variabel input (𝑥𝑖) adalah data rata-rata kekuatan gempa per periode pada waktu sebelumnya (𝑦𝑡−1, 𝑦𝑡−2, … , 𝑦𝑡−𝑛 ). 2. variabel output (𝑦𝑡 ) adalah data rata-rata kekuatan gempa per periode yang dijadikan data aktual/target. 3. metode analisis peramalan dengan arima adapun tahapan pengujian yang harus dilakukan untuk mendapatkan model arima terbaik adalah:  membuat plot data yang ada secara grafis.  melakukan pengujian kestasioneran, apabila diketahui bahwa data tidak stasioner dalam mean maka perlu distasionerkan dengan cara pembedaan (differencing). sedangkan apabila data tidak stasioner dalam varians maka dilakukan transformasi data. jika data sudah stasioner maka dicari nilai acf dan pacf dan merumuskan model umum arima (p,d,q).  estimasi parameter model. optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 65  diagnosis checking terhadap model:  uji signifikasi model dengan uji t  uji kesesuaian model dengan asumsi kecukupan white noise dengan ljung box  jika model belum sesuai maka diulangi sampai kondisi terpenuhi.  menetapkan model terbaik untuk peramalan berdasarkan nilai mse yang terkecil. setelah model yang tepat telah ditemukan maka peramalan untuk beberapa periode yang akan datang dapat ditentukan. gambar 3. flowchart proses peramalan model arima peramalan dengan neural network (nn)algoritma genetik (ga) untuk mendapatkan model peramalan menggunakan model neural network backpropagation-algoritma genetik (ga) dilakukan pelatihan (training) dan peramalan (testing) dengan tahapan-tahapan sebagai berikut:  input yang digunakan adalah semua data ratarata kekuatan gempa per periode.  algoritma backpropagation (propagasi balik)  inisialisasi bobot  feedforward pola data input hingga mencapai lapisan output.  hasil pada lapisan output akan dibandingkan dengan nilai target dan di hitung nilai mean square error (mse). jika sudah terpenuhi sesuai dengan toleransi kesalahan, maka proses perhitungan dihentikan. namun sebaliknya, jika kriteria belum terpenuhi maka dilanjutkan ke langkah berikutnya.  bergerak dari lapisan output kembali ke lapisan hidden dan melakukan penyesuaian bobot serta mengulangi.  optimasi parameter neural network backpropagation dengan agoritma genetik, sebagai berikut:  inisalisasi parameter inisialisasi populasi kromosom berdasarkan model peramalan.  fungsi evaluasi evaluasi populasi kromosom untuk menentukan nilai kecocokan (fitness) suatu kromosom (solusi).  seleksi seleksi kromosom solusi sesuai dengan nilai fitnessnya menggunakan metode roulette-wheel (roda roulatte).  crossover dan mutasi crossover kromosom induk untuk meningkatkan populasi menggunakan metode one-cut point. mutasi gen untuk mendapatkan kromosom baru menggunakan metode mutasi seragam.  hingga maksimum generasi tercapai dengan toleransi kesalahan sebesar 𝛼 = 0.001.  hitung mean square error (mse) sebagai pengujian neural network backpropagation yang merupakan selisih antara output jaringan dengan target dan simpan semua hasil pelatihan yang nantinya digunakan untuk peramalan.  membandingkan nilai mean square error (mse) hasil peramalan dari neural network backpropagationalgoritma genetik dengan metode arima dan neural network backpropagation secara individu.  hasil peramalan terbaik. muzakir hi sultan 66 volume 3 no. 2 mei 2014 gambar 4. flowchart proses peramalan model nnga hasil dan pembahasan 1. peramalan menggunakan arima untuk 36 bulan/periode kedepan 1. identifikasi model langkah pertama yang harus dilakukan untuk menganalisis data time series adalah membuat plot data rata-rata kekuatan gempa seiring waktu. plot ini dibuat dengan tujuan untuk mendeteksi kestasioneran data baik dalam mean maupun varians secara grafis. juga dapat digunakan untuk mengetahui kecenderungan rata-rata kekuatan gempa yang terjadi pada beberapa periode sebelumnya. 3632282420161284 8 7 6 5 4 index s r rata-rata kekuatan gempa 36 bulan/ periode (a) 5,02,50,0-2,5-5,0 0,037 0,036 0,035 0,034 0,033 0,032 0,031 0,030 0,029 0,028 lambda s td e v lower c l upper c l limit estimate 1,01 lower c l -1,42 upper c l 3,50 rounded value 1,00 (using 95,0% confidence) lambda box-cox plot of c2 (b) gambar 5. (a) plot time series kekuatan gempa (b). plot box-cox pada gambar 5 (a) terlihat bahwa terdapat perbedaan fluktuasi data rata-rata kekuatan gempa pada beberapa periode waktu, sehingga dicurigai bahwa terdapat sifat ketidakstasioneran terhadap varians pada data tersebut. akan tetapi nilai 𝜆 = 1 pada transformasi box-cox gambar 5 (b), menunjukan bahwa data masih memiliki sifat stasioneritas pada varians. karena secara grafis terdapat perubahan ratarata dari waktu ke waktu, maka perlu dilakukan differencing (pembedaan) untuk memenuhi sifat stasioneritas terhadap mean seperti pada gambar 6 (a). 3632282420161284 3 2 1 0 -1 -2 -3 -4 index c 4 rata-rata kekuatan gempa 36 bulan/ periode (a) 35302520151051 1,0 0,8 0,6 0,4 0,2 0,0 -0,2 -0,4 -0,6 -0,8 -1,0 lag a u to c o rr e la ti o n acf rata-rata kekuatan gempa 36 bulan/periode (b) 35302520151051 1,0 0,8 0,6 0,4 0,2 0,0 -0,2 -0,4 -0,6 -0,8 -1,0 lag p a rt ia l a u to co rr e la ti o n pacf rata-rata kekuatan gempa 36 bulan/ periode (c) gambar 6. (a) plot time series kekuatan gempa satu kali differencing (b). plot acf satu kali differencing (c) plot acf satu kali differencing optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 67 berdasarkan plot autokorelasi function (acf) dan plot partial function autokorelasi (pacf) pada gambar 6 (b) dan 6 (c) terlihat bahwa nilai autokorelasi dari keduanya mempunyai satu lag yang signifikan yaitu pada lag ke-1. sehingga dugaan sementara model arima (p,d,q) berdasarkan plot acf dan pacf adalah arima (1,1,0), arima (0,1,1) dan diperoleh juga model yang lain yang merupakan gabungan dari kedua model tersebut yaitu arima (1,1,1). 2. uji signifikansi parameter uji signifikasi parameter harus memenuhi syarat untuk mendukung ketepatan dalam pemilihan model, sehingga dilakukan pengujian sebagai berikut: tabel 1. uji parameter untuk model arima (1,1,0), arima (0,1,1) dan arima (1,1,1) berdasarkan uji signifikasi pada tabel 1 diketahui bahwa semua nilai parameter |𝑡𝐻𝑖𝑡𝑢𝑛𝑔 | > 𝑡𝛼 2 atau p-value < 𝛼 = 0.05, maka 𝐻0 ditolak (parameter tidak signifikan). sehingga semua model memiliki nilai parameter yang signifikan. 3. diagnostic checking model suatu model dikatakan sesuai apabila deret residualnya bersifat white niose. pengujian white noise menggunakan statistik uji ljung-box dengan hipotesis 𝐻0 adalah residual memenuhi syarat white noise. tabel 2. uji white noise untuk model arima (1,1,0), arima (0,1,1) dan arima (1,1,1) berdasarkan tabel 2 dapat di lihat bahwa nilai 𝒬𝐾 < 𝜒(𝛼,𝑑𝑓) 2 atau p-value > 𝛼 = 0.05 untuk setiap lag, maka 𝐻0 di terima. sehingga di simpulkan bahwa residual memenuhi syarat bersifat white noise. 4. pemilihan model terbaik berdasarkan pengujian syarat-syarat di atas, dapat dijelaskan bahwa model dugaan arima (1,1,0), arima (0,1,1), dan arima (1,1,1) adalah model yang sesuai. langkah selanjutnya adalah membandingkan nilai mean square error (mse) dari ketiga model tersebut. tabel 3. uji mse untuk model arima (1,1,0), arima (0,1,1) dan arima (1,1,1) model arima mse arima (1,1,0) 1,2320 arima (0,1,1) 1,0607 arima (1,1,1) 0,6505 berdasarkan tabel 3 maka dapat disimpulkan bahwa model arima (1,1,1) adalah model yang terbaik untuk meramalkan nilai data beberapa bulan/periode ke depan dengan nilai mse (mean square error) terkecil yaitu 0,6505. 5. peramalan arima pada rata-rata kekuatan gempa bumi setelah diperoleh model terbaik yaitu model arima (1,1,1), selanjutnya akan dilakukan peramalan rata-rata kekuatan gempa bumi untuk 36 bulan/periode ke depan di wilayah maluku utara. model arima (1,1,1) secara matematis dapat ditulis sebagai berikut: (1 − 𝜙1𝐵)(1 − 𝐵)𝑋𝑡 = (1 − 𝜃1𝐵)𝑒𝑡 (1 − 𝜙1𝐵)(1 − 𝐵)𝑋𝑡 = (1 − 𝜃1𝐵)𝑒𝑡 (1 − 𝜙1𝐵)(𝑋𝑡 − 𝐵𝑋𝑡 ) = 𝑒𝑡 − 𝜃1𝐵𝑒𝑡 (1 − 𝜙1𝐵)(𝑋𝑡 − 𝐵𝑋𝑡 ) = 𝑒𝑡 − 𝜃1𝐵𝑒𝑡 𝑋𝑡 − 𝐵𝑋𝑡 − 𝜙1𝐵𝑋𝑡 + 𝜙1𝐵 2𝑋𝑡 = 𝑒𝑡 − 𝜃1𝐵𝑒𝑡 𝑋𝑡 − 𝑋𝑡−1 − 𝜙1𝑋𝑡−1 + 𝜙1𝑋𝑡−2 = 𝑒𝑡 − 𝜃1𝑒𝑡−1 𝑋𝑡 = (1 + 0,1176)𝑋𝑡−1 − 0,1176𝑋𝑡−2 + 𝑒𝑡 −(0,9638𝑒𝑡−1) 2. peramalan menggunakan nn-ga untuk 36 bulan/periode kedepan 1. data training menggunakan nn dan nn-ga untuk rata-rata gempa bumi 36 bulan/periode data training bertujuan untuk mempelajari pola data yang akan digunakan selanjutnya pada data testing (peramalan). untuk menggambarkan model koefisien |𝒕𝑯𝒊𝒕𝒖𝒏𝒈 | 𝒕( 𝜶 𝟐 ,𝒅𝒇) pvalue ket arima (1,1,0) 𝜙1 = 0,5611 2,94 2,021 0,008 signifikan arima (0,1,1) 𝜃1 = 0,8940 5,16 2,021 0,000 signifikan arima (1,1,1) 𝜙1 = 0,1176 2,47 2,021 0,043 signifikan 𝜃1 = 0,9638 4,43 2,021 0,000 signifikan model lag 𝒅𝒇 𝝌𝒉𝒊𝒕𝒖𝒏𝒈 𝟐 (𝓠𝑲) 𝝌(𝜶,𝒅𝒇) 𝟐 pvalue ket arima (1,1,0) 12 9 6,7 16,919 0,666 white noise 24 21 7,0 32,671 0,998 white noise arima (0,1,1) 12 9 7,1 16,919 0,627 white noise 24 21 13,1 32,671 0,905 white noise arima (0,1,1) 12 7 17,4 14,017 0,115 white noise 24 19 26,3 30,144 0,122 white noise muzakir hi sultan 68 volume 3 no. 2 mei 2014 hasil simulasi data training di atas akan ditampilkan grafik antara data eksak dengan data hasil training dari nn dan nn-ga serta lamanya iterasi. berikut ini adalah grafik hasil data traning rata-rata gempa bumi 36 periode kedepan menggunakan metode nn secara individu. (a) (b) gambar 7. (a) perbandingan hasil data training rata–rata gempa 36 bulan/periode menggunakan model nn (b). nilai mse dari model nn pada setiap iterasi berikut ini adalah grafik hasil data training rata-rata gempa bumi untuk 36 periode menggunakan metode kombinasi nn-ga. (a) (b) gambar 8. (a) perbandingan hasil data training rata–rata gempa 36 bulan/periode menggunakan model nn-ga (b). nilai mse dari model nn-ga pada setiap iterasi. gambar 7 (a) dan gambar (b) menampilkan perbandingan kesalahan dari model nn dan nn-ga. nilai kesalahan sendiri diperoleh dari nilai mutlak selisih prediksi yang dihasilkan oleh ke dua metode dengan nilai eksak. tujuannya untuk melihat kesalahan hasil data prediksi dari ke dua metode tersebut, yang sebenarnya bisa juga dilihat melalui tabel 4 karena lebih mudah untuk melihat kedekatan hasil prediksi yang dihasilkan dari ke dua metode dengan nilai eksak. semakin kecil nilai kesalahan dari kedua model maka semakin dekat dengan nilai eksak yang diperoleh untuk 36 bulan/periode berikutnya. berdasarkan hasil pada tabel 5.4 diperlihatkan bahwa untuk data training rata-rata kekuatan gempa 36 bulan/periode, kombinasi model nn-ga lebih baik dibandingkan dengan hasil model nn secara individu. hal ini dapat di lihat nilai mse pada nn-ga sebesar 0.00016 yang secara keseluruhan lebih kecil dibandingkan dengan nilai mse dari nn sebesar 0.000093, serta jumlah iterasi yang berpengaruh terhadap lamanya waktu komputasi, perbandingan iterasi dari kedua model tersebut yaitu nn-ga hanya membutuhkan 9 kali iterasi sedangkan nn membutuhkan 12 kali iterasi. 2. data testing menggunakan metode arima, nn dan nn-ga untuk peramalan rata-rata gempa bumi 36 bulan/periode kedepan untuk menggambarkan hasil simulasi di atas akan ditampilkan grafik perbandingan antara data aktual dengan data hasil prediksi dari arima, nn dan nn-ga. berikut ini adalah grafik data hasil peramalan rata-rata gempa bumi 36 bulan /periode kedepan tanpa menggunakan ga. 0 2 4 6 8 10 12 0 5 10 15 20 25 30 grafik kesalahan setiap iterasi tanpa ga jumlah iterasi n il a i e r r o r optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 69 tabel 4. perbandingan hasil prediksi dan nilai kesalahan dari model nn dan nn-ga untuk data training 36 bulan/periode hasi training 12 bulan/ periode per akt nn nn-ga per akt nn nn-ga 1 6.3 6.28 6.29 19 5.4 5.38 5.39 2 5.1 5.08 5.091 20 4.8 4.78 4.79 3 5.5 5.48 5.489 21 7.6 7.58 7.588 4 5.0 4.99 4.992 22 4.4 4.39 4.390 5 4.8 4.78 4.789 23 5.2 5.18 5.192 6 5.1 5.08 5.094 24 6.6 6.58 6.588 7 4.7 4.69 4.686 25 5.7 5.68 5.685 8 6.6 6.58 6.590 26 5.2 5.18 5.192 9 6.7 5.68 5.690 27 4.5 4.48 4.493 10 5.5 5.48 5.489 28 6.5 6.48 6.485 11 6.0 5.98 5.986 29 4.7 4.68 4.692 12 4.6 4.58 4.592 30 4.1 4.08 4.093 13 5.8 5.78 5.789 31 4.7 4.68 4.691 14 4.2 4.19 4.195 32 5.2 5.19 5.189 15 5.2 5.18 5.192 33 6.0 5.98 5.989 16 5.5 5.48 5.493 34 6.4 6.81 6.387 17 4.9 4.88 4.888 35 5.2 5.18 5.188 18 4.8 4.78 4.793 36 5.6 5.58 5.591 mse nn nn-ga 0.00016 0.000093 (a) (b) gambar 9. (a) perbandingan data hasil peramalan rata–rata gempa 36 bulan/periode kedepan menggunakan model arima dan nn (b). nilai mse dari model arima dan nn pada setiap iterasi berikut ini adalah grafik data hasil peramalan rata – rata gempa untuk 36 bulan/periode kedepan menggunakan ga (a) muzakir hi sultan 70 volume 3 no. 2 mei 2014 gambar 9 (a) dan gambar 10 (a) menampilkan perbandingan kesalahan dari model yaitu arima dan nn-ga. nilai kesalahan sendiri diperoleh dari nilai mutlak selisih prediksi yang dihasilkan oleh kedua metode dengan nilai eksak. (b) gambar 10. (a) perbandingan hasil peramalan rata– rata gempa 36 bulan/periode kedepan menggunakan model arima dan nn-ga (b). nilai mse dari model arima dan nn-ga pada setiap iterasi. tujuannya untuk melihat kesalahan hasil prediksi dari kedua metode tersebut, yang sebenarnya bisa juga dilihat melalui tabel 5 karena lebih mudah untuk melihat kedekatan hasil prediksi yang dihasilkan dari ke tiga metode dengan nilai eksak. semakin kecil nilai kesalahan dari kedua model maka semakin dekat dengan prediksi yang diperoleh untuk 36 periode berikutnya. gambar 9 (b) dan 10 (b) menampilkan perbandingan rata rata waktu komputasi oleh masing – masing algoritma untuk menyelesaikan tiap – tiap data uji coba. dapat dilihat bahwa algoritma nn-ga mendapatkan waktu komputasi yang sama dengan nn-ga, namun menghasilkan prediksi yang berbeda. hal ini disebabkan karena solusi yang didapat dengan menggunakan nn-ga lebih baik dibadingkan dengan nn secara individu. tabel 5 perbandingan hasil prediksi dan kesalahan dari model arima dan nn-ga untuk data testing (peramalan) 36 bulan/periode kedepan hasi peramalan 36 bulan/periode per akt arima nn-ga per akt arima nn-ga 37 5.4 5.58 6.28 55 4.9 4.62 5.38 38 5.6 4.05 5.08 56 5.4 5.10 4.78 39 7.0 5.41 5.48 57 5.2 5.94 5.75 40 4.9 4.86 4.98 58 4.6 6.25 4.38 41 5.8 4.80 4.78 59 5.3 5.11 5.18 42 5.4 5.01 5.08 60 4.9 5.52 6.58 43 4.5 5.26 4.68 61 5.3 5.78 5.68 44 6.4 5.16 6.57 62 5.0 3.98 5.19 45 4.8 7.34 5.68 63 5.5 5.30 4.48 46 4.9 3.98 5.48 64 4.8 4.81 6.47 47 6.4 5.35 5.98 65 5.4 4.71 4.69 48 5.6 6.11 4.58 66 5.4 4.90 4.08 49 4.1 5.61 5.78 67 5.3 5.16 4.68 50 5.3 5.08 4.19 68 6.8 5.07 5.18 51 4.9 4.43 5.18 69 7.6 7.22 5.98 52 4.7 6.36 5.48 70 5.3 3.95 6.38 53 4.6 4.61 4.88 71 5.5 5.26 5.18 54 5.5 4.03 4.78 72 5.0 6.00 5.58 mse arima nn-ga 1.0125 0.9196 optimasi parameter neural networks pada data time series untuk memprediksi rata… cauchy – issn: 2086-0382 71 berdasarkan hasil pada tabel 5.5 diperlihatkan bahwa untuk data peramalan ratarata kekuatan gempa bumi 36 periode berikutnya, kombinasi model nn-ga lebih baik dibandingkan dengan hasil model arima dan nn secara individu. hal ini dapat di lihat nilai mse pada nnga sebesar 0.9196 yang secara keseluruhan lebih kecil dibandingkan dengan nilai mse dari arima sebesar 1.0125 dan nn sebesar 0.9664 secara individu serta jumlah iterasi yang sama yaitu 12 kali iterasi. kesimpulan kesimpulan yang didapatkan pada penelitian ini adalah model arima yang terbaik adalah model arima (1,1,1) dan berdasarkan hasil training model kombinasi neural network algoritma genetik lebih baik daripada model neural network untuk peramalan. pemilihan parameter-parameter dan jumlah iterasi dapat berpengaruh terhadap performa dari neural network algoritma genetik. performa yang dimaksud adalah ketepatan prediksi berdasarkan nilai mse yang kecil dan lamanya waktu komputasi. hasil yang diperoleh menunjukan bahwa kombinasi dari metode neural network algoritma genetik dalam memprediksi rata-rata gempa bumi di wilayah maluku utara pada bulan/periode berikutnya secara keseluruhan lebih baik dari pada metode arima secara individu berdasarkan kriteria nilai mse terkecil. bibliography [1] “badan meteorologi, klimatologi dan geofisika.” . [2] n. morris and s. handoko, gempa bumi. elex media komputindo, 2002. [3] y. altinok and d. kolcak, “an application of the semi-markov model for earthquake occurrences in north anatolia, turkey,” j. balk. geophys. soc., vol. 2, no. 4, pp. 90–99, 1999. [4] a. c. c. ronen, s. schultz, p.s.hattori m., seismic-guide estimation of log properties. 1994, p. part 1, 2, and 3. [5] b. h. russell, l. r. lines, and d. p. hampson, “application of the radial basis function neural network to the prediction of log properties from seismic attributes,” explor. geophys., vol. 34, no. 1/2, pp. 15–23, 2003. [6] y. zhangang, c. yanbo, and k. w. e. cheng, “genetic algorithm-based rbf neural network load forecasting model,” in power engineering society general meeting, 2007, pp. 1–6. [7] a. lapedes and r. farber, “nonlinear signal processing using neural networks,” 1987. [8] c. m. bishop, “neural networks for pattern recognition,” 1995. [9] e. yoan, pengembangan metode jaringan syaraf tiruan dengan fungsi radial basis (fbr) fuzzy dan aplikasinya. depok: universitas indonesia. [10] d. r. sari, “analisa algoritma back propagation dengan metode gradien descent untuk beban listrik jangka pendek.” [11] j. d. cryer, time series analysis. wadsworth publ. co., 1986. [12] s. makridakis, s. c. wheelwright, and v. e. mcgee, “metode dan aplikasi peramalan,” bin. aksara. jakarta, 1999. [13] g. e. p. box, g. m. jenkins, and g. c. reinsel, time series analysis: forecasting and control. john wiley & sons, 2013. [14] l. arsyad, “peramalan bisnis,” ghalia indones. jakarta, 1995. [15] w. w.-s. wei, time series analysis. addisonwesley publ, 1994. [16] j. j. siang, “jaringan syaraf tiruan dan pemrogramannya menggunakan matlab,” penerbit andi, yogyakarta, 2005. [17] suhartono, feedforward neural network untuk peramalan time series. yogyakarta: penerbit andi, 2007. [18] p. andrey, “genetic algorithm for optimization. programs for matlab.,” ugm yogyakarta. pendahuluan dasar teori 1. analisis time series 2. pengujian stasioneritas time series 3. koefisien fungsi autokorelasi (acf) 4. koefisien fungsi autokorelasi parsial (pacf) 5. pembentukan model arima 6. tahapan pemodelan autoregressive integrated moving average (arima) 7. konsep dasar pemodelan artificial neural networks (ann) 8. metode neural network backpropagation 9. metode algoritma genetik (ga) metode penelitian 1. pengumpulan data 2. variabel penelitian 3. metode analisis peramalan dengan arima peramalan dengan neural network (nn)-algoritma genetik (ga) hasil dan pembahasan 1. peramalan menggunakan arima untuk 36 bulan/periode kedepan 2. peramalan menggunakan nn-ga untuk 36 bulan/periode kedepan kesimpulan bibliography covid-19 data analysis in tarakan with poisson regression and spatial poisson process cauchy –jurnal matematika murni dan aplikasi volume 7(4) (2023), pages 608-621 p-issn: 2086-0382; e-issn: 2477-3344 submitted: january 13, 2023 reviewed: april 06, 2023 accepted: may 18, 2023 doi: http://dx.doi.org/10.18860/ca.v7i4.19653 covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro*, ika nurwanitantya wardani, khusnia nurul khikmah mathematics department, faculty of mathematics and natural sciences, universitas negeri surabaya, surabaya, east java, indonesia 60231 email: ayuninsofro@unesa.ac.id abstract coronavirus is included in the virus family that causes diseases ranging from mild to severe symptoms entered indonesia in march 2020 and included north kalimantan province, tarakan. that caused many implications in every aspect, but the outspread and the patterns still needed to be discovered. one approach was to use generalized linear models. the two methods are poisson regression and stochastic with spatial poisson process. the variables used were rainfall, population density, and temperature in each village in tarakan. the poisson regression analysis founds that only one factor affected temperature. then, the results were refined with the spatial poisson process, where in addition to the influencing factors also, the distribution patterns are obtained. the analysis showed that the pattern of case distribution was included in the non-homogeneous poisson process criteria. then the case density intensity model was obtained using regression from the model known that the covariate variables significantly influence rainfall and temperature. compared with general poisson regression analysis results, only the average temperature variables had a significant effect. thus, a better method was used, namely the spatial poisson process. the two models' aic values show it. the aic value of the spatial poisson process model is 89.742, and it was smaller than the poisson regression. copyright © 2023 by authors, published by cauchy group. this is an open access article under the cc bysa license (https://creativecommons.org/licenses/by-sa/4.0/) keywords: covid-19; deterministic; generalized linear models; poisson regression; spatial poisson process; stochastic. introduction coronavirus (cov) is included in the virus family that causes diseases ranging from mild to severe symptoms. previously, there were two types of coronaviruses that were known to cause illness with severe symptoms, such as middle east respiratory syndrome (mers) and severe acute respiratory syndrome (sars). research said sars is transmitted from civet cats to humans and mers from camels to humans. however, in the last few years, around the end of 2019, the international community was shocked by a virus outbreak originating from wuhan city, china, called coronavirus disease or better known as covid-19 [1]. covid-19 is a new type of virus that has never been identified before in humans. coronavirus is a zoonosis, which means it is transmitted between animals and humans. on december 31, 2019, the who china country office reported an unknown aetiology pneumonia case in wuhan city, hubei province, china. on january 7, 2020, http://dx.doi.org/10.18860/ca.v7i4.19653 mailto:ayuninsofro@unesa.ac.id https://creativecommons.org/licenses/by-sa/4.0/ covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 609 china identified this case as a new type of coronavirus (covid-19) [2]. on january 30 2020, who designated the public health emergency of international concern (pheic). the increase in the number of covid-19 cases took place quite quickly, and there has been a spread outside the wuhan region and other countries. as of june 8 2020, globally, there were 6,931,000 confirmed cases in 214 countries with 400,857 deaths (cfr 5.78%). with no exception, indonesia included. as of june 8, 2020, the number of confirmed cases was 32,033 positive cases with 1,883 deaths. according to the official website covid19.go.id owned by the national disaster management agency (bnpb), on the map of the spread of covid-19 in indonesia until june 8, 2020, north kalimantan province is currently ranked the 24th and 4th most in kalimantan after south kalimantan, central kalimantan and east kalimantan. the map mentioned positive cases of covid-19 patients in north kalimantan accumulated as many as 165 cases with 2 cases of death. tarakan, as of june 8, 2020, became the third contributor to this number, with a total of 46 cases. the information was obtained from the official website of tarakankota.go.id from a source: press release task force for the acceleration of covid-19 countermeasures of tarakan, where the case has spread in 14 out of 20 villages in the tarakan. the increase in the number of cases does not necessarily reveal the pattern of its spread. reason various factors cause the distribution pattern of covid-19 cases to be random. tarakan is elected to be the study location because, in this city, it is known that there has been fairly high population mobility in recent times, which has an effect on population density in some areas. add to this the erratic weather from march to the present based on the forecast from tarakan bmkg. these two things are predicted to be able to influence the addition of covid-19 cases based on previous research. therefore, researchers are interested in implementing it in tarakan. in this article, first, the covid-19 data will be analyzed with the simplest approach, which is poisson regression. only influencing factors that are obtained through this approach [3]–[5]. thus, the research will be continued with a more relevant method, namely spatial poisson process. the spatial poisson process in this study is used as an approach from spatial point pattern data. a spatial point pattern is a statistical method for random patterns at points in a dimensional space, where the points represent the location of the research object [6]. covid-19 outspread, and the variables at risk will be known with this method. the location point data used in this study is one type of spatial data included in the spatial point pattern, and this study uses a geographical point consisting of each location coordinates where covid-19 patients live. the location is a random point on the map. this spatial method is used to obtain information from data that is influenced by space or location. covid-19 cases’ outspread patterns can be found by testing clarified data homogeneity with contour plots. the outspread pattern will be tested again using the poisson regression method by taking research variables, namely temperature, rainfall, and population density [7]. however, data transformation will be performed before testing with poisson regression [6]. the goal is for the data used in regression to have a range that does not differ much, so the analysis can be more effective. after that, a regression analysis can be done to find out what factors can influence covid19 outspread in tarakan. thus, two results will be obtained, outspread pattern and influential variables. previous research related to the spatial poisson process method has been conducted [8] regarding the analysis of the mammary gland tumours development in female rats, analysis of the medical centre distribution pattern in surabaya, the analysis of the gas stations distribution patterns by [9]. mixed poisson regression models have also been carried out [10] regarding predictions on heart disease, and also by [11] regarding covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 610 ames salmonella assay data. meanwhile, research on spatial analysis in covid-19 cases has been carried out by several researchers, such as [12], which analyze factors of population density, elderly population, urbanization, average temperature, and annual rainfall in each province in iran using the spatial weight estimation and spatial autocorrelation methods. other studies were also conducted by [13], who analyzed the number of cases of death and recovery in wuhan, china using the spatial data panel model, and [14], who analyzed humidity and temperature. also, regarding the analysis of covid-19 outspread with local transmission factors and transportation flows in brazil. based on that background, this study will analyze the covid-19 data in tarakan using poisson regression and the spatial poisson process, which is an approach of generalized linear models (glm). it also adjusts to the distribution of covid-19 patients in tarakan who, after being tested, were distributed to poisson. this research is expected to provide advice to the government, especially in tarakan, in optimizing the handling of covid-19 cases. methods poisson distribution the poisson distribution belongs to an exponential family. poisson distribution is one type of distribution of the number of events at certain time intervals. simeon first announced this distribution denis poisson (1781-1840) published his probability theory in 1838 in his recherches “sur la probabilit𝑒 des jugements en mati𝑒re criminelle et en 𝑚𝑎𝑡𝑖𝑒𝑟𝑒 civile work” (examples of criminal and civil law opportunities). his work focuses on random variables that count among other discrete amounts. when the expected value occurs at an interval is 𝜆, then the probability of occurring 𝑥 times is: 𝑝(𝑦; 𝜆) ≔ 𝑒−𝜆𝜆𝑦 𝑦! , 𝑦 = 0,1,2, … (1) where 𝑦 is a non-negative integer stating the number of successes that occur, and 𝜆 is a positive actual number which is the average number of successes that occur per unit of time, distance, area, or volume; and 𝑒 = 2.71823 [15], [16]. the following will be discussed regarding an approach used in this study. generalized linier model (glm) generalized linear model (glm) is a general form of the linear model. glm is also a procedure that provides regression analysis and variance analysis for one response variable with one or more explanatory variables, which can also be called a covariate. glm allows researchers to examine the interactions between responses and each response influence, covariates influence, and the covariate's interaction with responses. for example, it is known that the vector 𝑦 has 𝑛 components, which is the realization of a 𝑌 response matrix. each component is independent and is distributed with the mean or (𝑌) = 𝜆. if the model formed has a predictor of 𝑥, with some unknown parameter 𝛽𝑖 , … , 𝛽𝑛, then the model is a linear combination 𝜆 = ∑ 𝑥𝑖 𝛽𝑖 𝑝 𝑖=1 . as a transition from the linear model to the generalized linear model, the form is described through three components, namely [17]: 1. random component, i.e. values of observing the response of 𝑌 that is independent of specific distributions. 2. systematic component, i.e. a linear combination of variable 𝑥 with parameter 𝛽. 3. link between random and systematic/link function, which is a function that explains the expected value of the response variable (𝑌) that connects with explanatory variables through linear equations. covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 611 poisson regression poisson regression can describe the relationship between response variables, where the response variable is the poisson distribution with predictor variables. poisson regression uses glm to use the model in observations where the response variable does not require normal distribution. the poisson regression model is a standard model for data whose response variable is abnormally distributed and discrete type, with poisson distribution as its primary condition. the poisson distribution provides a realistic model for various random phenomena as long as the value of poisson's independent variable is a non-negative number. here is the poisson regression model: 𝑦~𝑝𝑜𝑖𝑠𝑠𝑜𝑛(𝜆) 𝜆𝑖 = 𝑒 𝑥𝑖 ′𝛽 (2) where 𝜆 is the average value of the event numbers that occur in a specific time interval or area, is a predictor variable notated as follows: 𝑥 = [1 𝑥1 𝑥2 … 𝑥𝑘 ] ′ furthermore, 𝛽 is the poisson regression parameter that is noted as follows [18]: 𝛽 = [𝛽0 𝛽1 𝛽2 … 𝛽𝑘 ] poisson regression analysis can be directly done without doing data transformation first. however, poisson regression cannot make a covid-19 outspread pattern, which is this study's goal. because poisson regression can be done using data count. meanwhile, spatial data is needed to analyze the distribution patterns, such as the location of patients infected with covid-19. so that researchers conduct further analysis using the poisson process method, which will be explained in the following sub. parameter estimation in poisson regression uses the likelihood function and the likelihood equation based on the poisson distribution. the equation is as follows [19]: 𝐿(𝑦𝑖 ; �̂�) = ∏ 𝑒−[𝜆(𝑥𝑖;�̂�)][𝜆(𝑥𝑖 ; �̂�) 𝑦𝑖 ] 𝑦𝑖 ! 𝑛 𝑖=1 (3) spatial point process the spatial point process is stochastic (a collection of random variables indexed by space or time) in each realization consisting of a limited and infinite set of points in space, for example, 𝑆, where 𝑆 ⊂ ℝ2 [20], [21]. the spatial point process is used as a statistical model to analyze the pattern of point distribution, where the point represents the location of an object of research [22]. spatial point process definitions include processes that are limited and can be calculated. in this study, the discussion will be limited to the point process. for example, 𝑈, whose realization is 𝑢 = {𝑢1, … , 𝑢𝑛 }, 𝑛 ≥ 0, is a finite set of parts locally in space or region 𝑆. it is said that the point process is defined at 𝑆 to determine the distribution of 𝑈. we can determine the distribution of the number of points (𝑈). for each 𝑛 = 1 depending on 𝑛(𝑈) = 𝑛, the combined distribution of 𝑛 points on 𝑈. then, the approximation equivalent determines the distribution of a variable. for example, 𝑁(𝐴) = (𝑈λ) for the subset 𝐴 ⊆ 𝑆, where 𝑈λ = 𝑈 ∩ 𝐴. in this study, a spatial point process 𝑈 in ℝ2 is defined as a set of finite parts locally in ℝ2, i.e. (𝐴) is a finite random variable each time 𝐴 ⊆ ℝ2 is a restricted region. next, the poisson regression, which will be the method of approach in this research, will be discussed regarding the covid-19 case. spatial poisson process in this section, a specific spatial point process model will be discussed by a random intensity function with an analogy using a generalized linear model and a random effect model called the poisson process. the poisson process is one of the most widely studied covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 612 stochastic processes in the field of point processes and the random phenomena applied science because of their easy-to-use properties as a mathematical model and also mathematically interesting. the poisson process has two main properties including [23]: 1. the number of poisson points distribution. in all forms, the poisson point process is related to the poisson distribution, which implies that the probability of a random variable poisson 𝑋 is equal to 𝑥 given in equation (1), where 𝜆 is a single poisson parameter used to define the poisson distribution. 2. complete independence. another significant trait is that for collections of separate (or non-overlapping) sub-regions of the underlying space, the number of points in each restricted sub-region will be completely independent of the others. the poisson process is modelled for complete spatial randomness. usually, a linear log model of the intensity function is considered as follows: log 𝜆(𝑢) = 𝒛(𝑢)′𝛽, 𝑢 ∈ 𝑆 (4) where 𝛽 = (𝛽1, … , 𝛽𝑛) is an unknown parameter and 𝒛(𝑢) ≡ (𝑧1(𝑢), … , 𝑧𝑝(𝑢)) ′ is a covariate function, the covariate function can be obtained from the pixel image transformation function explained in the following subsection. the poisson process is divided into two types, homogenous and non-homogeneous [6], [21], [24]. the poisson process is homogenous or stationary when it has a constant or single parameter, for example, 𝜆. this parameter is called levels or intensities related to the number of poisson points expected in some restricted areas. the characteristics of the homogeneous poisson process are as follows: 1. number of 𝑁(𝑈 ∩ 𝐴) from the point inside region 𝐵 has the poisson distribution. 2. expectation value from a point inside region 𝐵 is 𝐸(𝑈 ∩ 𝐴) = 𝜆 ∙ area (𝐴). 3. if 𝐵1 and 𝐵2 disjoin or are different regions from space, then 𝑁(𝑈 ∩ 𝐴1), and 𝑁(𝑈 ∩ 𝐴2), are random variables independent of each other. 4. if 𝑁(𝑈 ∩ 𝐴) = 𝑛, 𝑛 points were independent and distributed evenly in region 𝐴. while the poisson process is said to be non-homogeneous if the intensity function (𝜆) is not constant and varies according to changes in time or area. the non-homogeneous poisson process with the intensity function (𝑢) depends on the u parameter. the characteristics of the non-homogeneous poisson process are as follows: 1. number of 𝑁(𝑈 ∩ 𝐴) from the point inside region 𝐵 has the poisson distribution. 2. expectation value from a point inside region 𝐵 is 𝐸[𝑁(𝑈 ∩ 𝐴)] = ∫ 𝜆(𝑢)𝑑𝑢 𝐵 . if 𝐵1, 𝐵2 disjoin or are different regions from space, then 𝑁(𝑈 ∩ 𝐴1), and 𝑁(𝑈 ∩ 𝐴2), are random variables which are independent to each other. 3. if 𝑁(𝑈 ∩ 𝐴) = 𝑛, 𝑛 points were independent and distributed evenly in region 𝐵, with probability as follows: 𝑓(𝑢) = 𝜆(𝑢) 𝐼 where 𝐼 = ∫ 𝜆(𝑢)𝑑𝑢 𝐵 . a homogeneity test on the poisson process will be carried out to determine whether the observed intensity of the point pattern is included in the homogeneous point pattern or non-homogeneous point pattern. thus, when estimating parameters, obtain a model that matches the characteristics or characteristics of the observed point patterns. the homogeneity test of the poisson process can be done using the chi-square test. the test hypotheses are as follows: 𝐻0: the intensity of homogeneous covid-19 cases 𝐻1: the intensity of the covid-19 case is not/ non-homogeneous the intensity referred to in this study is a large number of confirmed cases in each covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 613 location grid. where the rejection of the hypothesis can be done using the following test statistics: 𝜒2 = ∑ (𝑛𝑗 − �̅�𝑡𝑗 ) 2 �̅�𝑡𝑗 𝑟 𝑗=1 where �̅� = 𝑛 𝑡 , 𝑛 is the total number of points, and 𝑡 is the total number of regions. the test result will reject 𝐻0 if the p-value< 𝛼, which is 𝛼 = 0.05 [6]. spatial poisson process intensity function estimation in this study, data used is the outspread cases of covid-19 in tarakan. mle (maximum likelihood estimator) is used to find intensity function 𝜆 that maximizes data occurring possibilities. suppose you have an 𝑛 sample size, 𝑢 = {𝑢1, … , 𝑢𝑛 } from spatial poisson process as described above. then the likelihood function 𝐿(𝜆; 𝑢 = {𝑢1, … , 𝑢𝑛 }) = 𝑓(𝑢 = {𝑢1, … , 𝑢𝑛 }; 𝜆) is the probability obtained from sample 𝑢. so the intensity function 𝜆 becomes: λ = ∫ 𝜆(𝑢)𝑑𝑢 𝑆 (5) the observed point 𝑛 probability is 𝑒−λ λn 𝑛! and the observation probability density is 𝜆(𝑢𝑖) λ . because the observations are independent of each other, then, 𝑃(𝑢 = {𝑢1, … , 𝑢𝑛 }) = ∏ 𝜆(𝑢𝑖) λ 𝑛 𝑖=1 . the likelihood function to obtain a sample 𝑢 = {𝑢1, … , 𝑢𝑛 } = 𝑃(𝑛) ∙ 𝑃(𝑢 = {𝑢1, … , 𝑢𝑛 }|𝑛) is: 𝐿(𝜆; 𝑢 = {𝑢1, … , 𝑢𝑛}) = 𝑒 −λ λn 𝑛! ∏ 𝜆(𝑢𝑖 ) λ 𝑛 𝑖=1 (6) suppose the sample is ordered 𝑢1 < 𝑢2 < ⋯ < 𝑢𝑛 . so, likelihood function from an ordered sample is a sample of the likelihood function unordered multiple by 𝑛!. the likelihood function from an ordered sample is: 𝐿(𝜆) = 𝑒−λ λn 𝑛! ⋅ ∏ 𝜆(𝑢𝑖 ) λ 𝑛 𝑖=1 ⋅ 𝑛! 𝐿(𝜆) = 𝑒−λ ⋅ ∏ 𝜆(𝑢𝑖 ) λ 𝑛 𝑖=1 = 𝑒∫ 𝜆 (𝑢)𝑑𝑢 𝑆 ⋅ ∏ 𝜆(𝑢𝑖 ) 𝑛 𝑖=1 (7) so, its likelihood function is as follows [22], [25], [26]: ℓ(𝛽) = log(𝐿(𝜆)) = ∑ log 𝜆𝛽 (𝑢𝑖 ) − ∫ 𝜆𝛽 (𝑢)𝑑𝑢 𝑆 𝑛 𝑖=1 (8) in general, numeric integration is needed to calculate integrals [26]. in this study, covariate function data 𝑍(𝑢) that definite all spatial locations, shown in pixel image or contour plot form, which will be explained in the following sub. poisson regression in spatial poisson process poisson regression analysis was carried out in the spatial poisson process because general poisson regression could not include spatial analysis to find covid-19 distribution patterns in tarakan. regression analysis on the poisson process is carried out with the same steps as in general poisson regression. however, before doing so, the data transformation is performed first. thus, the model obtained is also different. the following is a poisson process model with covariate functions [10], [24]. covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 614 𝜆𝑖 (𝑢) = 𝑒 (𝛽0+𝛽𝑖𝑍(𝑢) ′), 𝑖 = 1,2, … , 𝑛 (9) where 𝛽 = (𝛽1, … , 𝛽𝑛) is an unknown parameter and 𝑍(𝑢) ′ ≡ (𝑧1(𝑢), … , 𝑧𝑝(𝑢)) ′ covariate function. in this section, covariate variables are displayed as pixel images. nadaraya-watson smoother was used for spatial smoothing of the mark value at the point distribution. spatial functions can be written as follows: �̃�(𝑞) = ∑ 𝑚𝑖 𝑘(𝑞 − 𝑟𝑖 ) 𝑛 𝑗=1 ∑ 𝑘(𝑞 − 𝑟𝑖 ) 𝑛 𝑗=1 where 𝑘 is a gaussian kernel function that can be formulated as 𝑘(𝑛) = 1 √2𝜋 𝑒 ( 1 2 (−𝑛2)) , 𝑛 ∈ ℝ; 𝑞 is the spatial location. 𝑟𝑖 is the 𝑖 data location. 𝑚𝑖 is the 𝑖 data mark value, and �̃�(𝑞) is the mean value for the spatial location of 𝑢 [6]. the value range of each covariate variable in this study is different. so, after pixel image transformation, there will be standardization variables to overcome differences in data that are too far away, then standardization to reduce the massive range of covariate variables used in the study. within the scope of regression, standardization is used to reduce colinearity due to interactions in the regression model [27]. after the data transformation is obtained, the next step is to conduct a regression analysis as usual. then we conduct the hypothesis testing. hypothesis testing is a test conducted on all parameter coefficients of the independent variable on the response variable. this test is carried out simultaneously and partially. the test statistic used in the simultaneous test to test the effect of the overall parameter coefficient in the model is the g or likelihood ratio test. concurrent test hypotheses are as follows: 𝐻0: 𝛽1 = 𝛽2 = ⋯ = 𝛽𝑝 = 0 and 𝐻1: at least there is 𝛽𝑝 ≠ 0. concurrent test statistics used for the model are [24]: 𝐺 = 2 log 𝐿(�̂�) 𝐿(�̂�) (10) furthermore, 𝐿(�̂�) and 𝐿(�̂�) values are maximum likelihood values for each model. statistical 𝐺 is the approach of the chi-square distribution, so the test criteria are rejected 𝐻0 if 𝐺 > 𝜒(𝛼,𝑛) 2 where 𝑛 is the number of predictor variables or the number of parameters in the model or p-value< 𝛼, where the level of significance is 𝛼 = 0.05. in comparison, a partial test is performed to determine the effect of the independent variable parameters individually by comparing the standard error. the partial test hypothesis for 𝑖 = 1,2, … , 𝑘 is as follows: 𝐻0: 𝛽𝑖 = 0 and 𝐻1: 𝛽𝑖 ≠ 0 for 𝑖 = 1,2, … , 𝑘. the partial test statistics used for the model are: 𝑊 = [ �̂�𝑖 𝑆𝐸�̂�𝑖 ] 2 (11) where �̂�𝑖 is estimator 𝛽𝑖 . 𝑆𝐸�̂�𝑖 is the standard error estimator 𝛽, 𝑊 is the wald test following the distribution of 𝜒2 with a freedom degree one. 𝐻0 is rejected if 𝑊 > 𝜒1,𝛼 2 or pvalue< 𝛼, so it can be concluded that the independent variable influences the response variable. akaike information criteria (aic) the aic method is a method that can be used to choose the best regression model found by akaike and schwarz. the method is based on the maximum likelihood estimator (mle) method. let 𝐿 be the maximum value of the likelihood function of a model, and k is covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 615 the number of parameters estimated in the model. then to calculate the value of aic can be used the following formula: 𝐴𝐼𝐶 = 2𝑘 − 2 ln 𝐿(�̂�) (12) where 𝐿(�̂�) is the likelihood value, and 𝑘 is the number of estimated parameters. if multiple models are given for a data set, a better model is a model with a smaller aic value [28], [29]. data analysis sources and techniques the data used in this study are secondary data on covid-19 cases confirmed in tarakan from 30 march to 8 june 2020 obtained from the official website of the city government of tarakan, namely tarakankota.go.id. the data consisted of 46 cases of covid-19 patients spread across 12 out of 20 villages in tarakan. the data will be modelled using the poisson regression model to determine the predictor variables that affect the response variable, where the response variable (𝑌) is the number of sufferers in each village, and the predictor variable is rainfall (𝑋1) in each village with millimetres, population density (𝑋2) in each village with the unit of person/km, and temperature (𝑋3) in each village with the unit °c, where the rainfall and temperature data are obtained from the bmkg website in tarakan, while the population density data is obtained from the tarakan central statistics agency website. poisson regression analysis has not met the objectives of this study, so the analysis was continued with the spatial poisson process method using data on the number of patients with covid-19, as many as 46 cases where the location of each patient will be displayed in a plot using latitude and longitude data on google maps. after that, it is divided into 20 grids which are restricted to the number of villages in tarakan. then, a homogeneity test can be performed and also displays a contour plot to reinforce the results obtained. then, performing pixel image transformation and poisson regression testing can also be done. the covariates used were the same as the predictor variable data in poisson regression. results and discussion this section will explain how to find out the distribution patterns of covid-19 and also the variables that influence it. the first method used is poisson regression, then proceed with the spatial poisson process. it aims to compare the results of variables that significantly influence the two methods. modeling and testing poisson regression parameters the data used for the application of poisson regression is data on the number of covid-19 cases per village in tarakan with predictor variables such as rainfall, population density and temperature. estimated parameters used in poisson regression using the maximum likelihood method. the results obtained from the parameter estimation using the maximum likelihood method in the following table: table 1. result of poisson regression parameter estimates. parameter estimation standard error z-value p-value 𝛽0 −5,331 3,868 −1,378 0,578 𝛽1 1,738 1,386 1,254 0,210 𝛽2 −6,245 × 10 −6 1,718 × 10−5 −0,363 0,716 𝛽3 0,015 6,484 × 10 −3 2,305 0,021 based on table 1, an estimation result is obtained from the covariate. the estimated covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 616 value of the average rainfall is −1.738, the population density is −0.000006245, and the average temperature is 0.01495. after obtaining the estimated parameters, the next step is testing the parameters simultaneously to determine whether there is an influence of the predictor variables on the response variable by testing the hypothesis as follows: 𝑯𝟎: 𝛽1 = 𝛽2 = 𝛽3 and 𝑯𝟏: at least one 𝛽𝑖 ≠ 0, 𝑖 = 1,2,3. the 𝐺 value in this analysis is 65.306, and the degree of freedom is 19, while 𝜒(3)(0.05) 2 = 7.815. because the value of 𝐺 > 𝜒(3)(0.05) 2 , reject 𝐻0, which means at least one predictor variable that influences the response variable, namely the number of cases of covid-19. to find out the effect is by each of these predictor variables, a partial parameter test is performed using the following hypothesis: 𝑯𝟎: 𝛽𝑖 = 0 and 𝑯𝟏: at least one 𝛽𝑖 ≠ 0, 𝑖 = 1,2,3. based on table 1, shows that the test results failed to reject 𝐻0, which means that the parameter 𝛽0 is not significant because the p-value of 0.578 is greater than 𝛼 = 0.05. tests on parameter 𝛽1 obtained the test results failed to reject 𝐻0, that the p-value of 0.21 is greater than 𝛼 = 0.05, which means that the variable 𝑋1 does not affect the response variable. tests on parameter 𝛽2 obtained the test results failed to reject 𝐻0, that the pvalue of 0.716 is greater than 𝛼 = 0.05, which means that the variable 𝑋2 does not affect the response variable. tests on parameter 𝛽3 obtained the test results reject 𝐻0 that the p-value of 0.021 is smaller than 𝛼 = 0.05, which means that the 𝑋3 variable influences the response variable. from the results of hypothesis testing that has been done, it is found that the predictor variable that significantly influences temperature. so, the poisson regression model is obtained as follows: 𝜆 = 𝑒(0.01495𝑋3) spatial poisson process modelling in this discussion, the distribution pattern analysis will be conducted first. however, before that, the data is pre-processed. the data used to model the poisson process cannot be used directly, so pre-processing of the data is needed first the next step of data preprocessing is to group the plots above into 20 grids of the same size. the number of grids was chosen to approach the many urban villages in tarakan. with the r program's help, the results of the grid division are presented in figure 1. below. figure 1. result of data pre-processing. exploration of location data and covariate variables in this discussion, the distribution pattern analysis will be conducted first. however, before that, the data is pre-processed. the data used to model the poisson process cannot be used directly, so pre-processing of the data is needed first. in this study, the coordinates of each infected village were plotted and plotted with the following display. in the covid-19 data, there are 46 observation points obtained from the official website of the city government of tarakan. the covariate variables used in the study are covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 617 average rainfall, population density magnitude, and the average temperature in each village. in this section, the distribution pattern analysis will be carried out by doing a homogeneity test. the results of r software show that the homogeneity test of the data fulfils the non-homogeneous poisson process, where the p-value < 0.05. it will be clarified again with the results of the contour plots shown in figure 2. with the r program's help, the resulting drawings where the contour plots formed show the peak density of the spread of covid-19 in tarakan. figure 2. the contour of covid-19 distribution sites. based on the picture above, it is known visually that the data distribution is uneven. the centre of distribution is seen at one point, which is known to be the east tarakan, mamburungan village. this is estimated because the level of population mobility in the east tarakan region is higher than in other tarakan areas. thus, this initial assumption is that the homogeneity of covid-19 cases in tarakan is not homogeneous. the characteristics of each covariate variable are illustrated below in figures 3 (a) to 3 (c). figure 3(a). rainfall mark point pattern for each region based on figure 3 (a), the level of rainfall in each village infected by the virus is almost the same. it can be seen that many circles accumulate, and some adjacent intersects patient locations. figure 3(b). population density mark point pattern for each region based on figure 3(b), it is known that the level of population density in each village infected by the virus is different. it is seen that several variations of the circle formed. covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 618 however, some are similar, so several circles are seen piling up, and some adjacent intersect patient locations. figure 3(c). mark point pattern average temperature of each region figure 3(c) shows that the temperature level in each village infected by the virus is almost similar. it is seen that many circles accumulate, and some adjacent patient locations intersect. the covariate variables in this study were also pre-processed by transforming them into pixel images, as presented in the following image. (a) (b) (c) figure 4. variable covariate (a) rainfall, (b) population density, and (c) temperature in the form of pixel image the data analysis step is carried out, beginning with the visualization of density and contour with the help of r software. then, the homogeneity test of the data is carried out to determine the intensity of the pattern of distribution of data to meet the homogeneous poisson process or non-homogeneous poisson process. then the poisson process model parameter estimation is performed using the maximum likelihood method. the results obtained are then analyzed and interpreted. poisson regression model in the spatial poisson process data distribution of covid-19, which has been explored visually, then performed poisson regression modelling. estimated parameters used in poisson regression using the maximum likelihood method. the results obtained from the estimated parameters using the maximum likelihood method with the help of the r program are presented in the following table: table 2. results of estimated regression parameters in the poisson process parameter estimation standard error z-value p-value 𝛽0 −0.023 0.248 −0.093 8.215 × 10 −3 𝛽1 −4.589 × 10 −3 2.339 × 10−3 −1.962 0.0498 𝛽2 1.294 × 10 −3 8.394 × 10−4 1.542 0.123 𝛽3 3.459 × 10 −3 1.535 × 10−3 2.253 0.024 based on table 2, the estimation results from the covariate are obtained. the average value of average rainfall is −0.004589, the estimated value of the population is 0.001294, and the average estimated value is 0.003459. after obtaining the estimated parameters, covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 619 then the simultaneous parameter testing is to determine whether there is a covariate effect on the response variable by conducting the following hypothesis test: 𝑯𝟎: 𝛽1 = 𝛽2 = 𝛽3 and 𝑯𝟏: at least one 𝛽𝑖 ≠ 0, 𝑖 = 1,2,3. the 𝐺 value in this analysis is 86.669, and the degree of freedom is 19, while 𝑋𝜒(3)(0.05) 2 = 7.815. because the value is 𝐺 > 𝜒(3)(0.05) 2 , it rejects 𝐻0, which means that at least one free variable is against the covid-19 case. to determine the effect of each covariate, partial parameters were tested using the following hypotheses: 𝑯𝟎: 𝛽𝑖 = 0 and 𝑯𝟏: at least one 𝛽𝑖 ≠ 0, 𝑖 = 1,2,3. based on table 2, it was shown that the test results were rejected by 𝐻0, which means that the parameter 𝛽0 is significant because the p-value of 0.008215 is smaller than 𝛼 = 0.05. tests on parameter 𝛽1 obtained the rejected 𝐻0 test results, namely the p-value of 0.0498 is smaller than 𝛼 = 0.05, which means that the variable 𝑋1 affects the response variable. tests on parameter 𝛽2 obtained the rejected 𝐻0 test results. the p-value of 0.1231 is greater than 𝛼 = 0.05, which means that the 𝑋2 variable does not conflict with the response variable. tests on parameter 𝛽3 obtained the rejected 𝐻0 test results. the pvalue of 0.0242 is smaller than 𝛼 = 0.05, which means that the 𝑋3 variable applies to the response variable. from the results of hypothesis testing that has been done, the significant covariates needed are rainfall and temperature. so, the following model is obtained: 𝜆 = 𝑒(−0,022987−0,00459𝑋1−0,00346𝑋3) selection of the best model based on the data processing results of several variables used in this study, aic values will be obtained to find the best model. the aic value is obtained based on the loglikelihood value previously described. aic calculation results can be seen in the following table: table 3. models obtained from both methods. method aic poisson regression 101.38 regression on poisson process 89.742 in table 3, the aic values for each model are obtained with the help of r software. furthermore, from this table, it can be seen that the spatial poisson process model is better than the usual poisson regression model, namely with the aic value is 89.742. so, the best model is obtained from the regression method in the spatial poisson process. from the results obtained in the distribution analysis of covid-19 cases in tarakan, differences were found from previous studies. in [12] study, the results showed that the factors that determine the high cases of covid-19 in iraq, namely the high rate of urbanization and the high cases of the elderly, the equation of the average ambient temperature could also determine the level of additional cases. the results of this study support previous research in shokouhi [14], which found eight countries, such as china, japan, south korea, and others. covid-19 cases are affected by temperature, humidity, and latitude. therefore in this study, improvisation can be done is that researchers can prove that weather factors that can affect covid-19 cases, not only environmental temperature but the level of rainfall, can also affect the increase in covid-19 cases in an area. however, everything is inseparable from the characteristics of each region is different. each country has its environmental conditions. meanwhile, temperature and rainfall can be tested, and many other environmental factors are also predicted to influence the increasing cases of covid-19 in an area. as in covid-19 data analysis in tarakan with poisson regression and spatial poisson process a’yunin sofro 620 the [14] study, the results show that the humidity and temperature factors significantly influence the increasing cases of covid-19 in each country. as for this study, it did not use the humidity factor because the data obtained were the same for each village, so further analysis could not be done. conclusions based on the conducted analysis results, direct modelling using poisson regression obtained the results of only temperatures that have a significant effect on the response variable. then, after further analysis using the spatial poisson process, another result was obtained from the distribution pattern of covid-19 in tarakan. where the distribution pattern is visually not homogeneous or included in the non-homogeneous poisson process criteria, this can be seen from the results of the contour plot, where there is a peak spread of cases in the east tarakan area. the average rainfall and temperature significantly affect the covid-19 intensity model in tarakan using the regression method in the spatial poisson process. the aic test results also showed that the spatial poisson process model is better than the regular poisson regression. all the analysis results are expected to be information for the city government regarding handling covid-19 cases in the city of tarakan going forward. the research proves that temperature and rain significantly affect the spread of covid-19 in tarakan. the results of this discovery can be used to provide directions to the community in related seasons to be more vigilant. in future studies, please combine covid-19 case data with a map of the area under study. this will give better and more accurate results than just using a grid because each region has a different 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[29] j. e. cavanaugh and a. a. neath, “the akaike information criterion: background, derivation, properties, application, interpretation, and refinements,” wiley interdiscip rev comput stat, vol. 11, no. 3, p. e1460, 2019. evaluasi premi polis last survivor pasangan suami istri menggunakan metode copula frank irma fauziah1 1program matematika fakultas sains dan teknologi universitas islam negeri syarif hidayatullah jakarta e-mail : irma_s2mathugm@yahoo.com atau irmaristianto08@gmail.com abstrak tulisan ini mengkaji tentang sebuah produk asuransi, ketahanan hidup, yang didasarkan pada dua kelompok umur: pasangan menikah berusia kurang dari 55 tahun dan menikah berusia lebih dari 55 tahun. penilaian premi untuk pasangan menikah yang kurang dari 55 tahun diperoleh dengan metode frasier dengan memperhitungkan probabilitas kematian setelah kematian terjadi dari salah satu tertanggung dengan asumsi kematian pasangan menikah adalah independen. sedangkan untuk pasangan yang sudah menikah lebih dari 55 tahun, premi penilaian diperoleh dengan metode frasier untuk menghitung probabilitas kematian dengan menganggap kematian pasangan menikah adalah dependen, asumsi ini dimodelkan oleh frank kopula, dimana kopula ini adalah salah satu dari keluarga kopula archimedes. kata kunci: metode frasier, frank copula, hidup bersama dan status terakhir abstract this paper introduces an insurance product, survivorship, based on two age groups : married couples less than 55 years old and married couples over 55 years old. premiums valuation for married couples who less than 55 years are obtained using frasier method with take into account the lapse rate after death occurs from one of the insured with mortality assumption of married couples are independent. while for married couples over 55 years, premiums valuation are obtained using frasier method for calculating the probability of death by assuming mortality of married couples are dependent, this assumption is modeled by frank copula. this copula is one of family archimedean copula. keywords : frasier methods, frank copula, joint life and last survivor status pendahuluan di indonesia, polis asuransi single life lebih mendominasi produk-produk asuransi di setiap perusahaan asuransi, sedangkan polis asuransi last survivor belum begitu banyak mendapat perhatian. di kanada, industri asuransi menetapkan harga polis asuransi last survivor menggunakan metode frasier untuk menaksir probabilitas kematian tertanggung dengan mengasumsikan mortalitas pasangan suami istri adalah saling bebas. metode ini menaksir probabilitas kematian dari tertanggung terakhir pada polis last survivor pada tahun tertentu yang telah dihitung pada awal tahun berlakunya polis last survivor. polis last survivor adalah polis asuransi jiwa yang menanggung minimal dua orang dimana benefitnya dibayarkan jika semua tertanggung meninggal dunia. jika tertanggungnya merupakan pasangan suami istri maka ada beberapa hal yang harus dianalisis terkait valuasi premi last survivor yaitu : 1. resiko yang dialami pasangan suami istri pasangan suami istri kurang lebih mempunyai resiko yang sama diantaranya : a. penyakit menular (margus p, 2002) seperti tubelkolosis (gusti arlina, 2003) dan penyakit kelamin seperti hiv, aids, hepatitis b (dr.m. atoillah i, 2007) b. kecelakaan yang dialami bersama (martikeinan p, valkonen t, 1996) c. stress cardiomyopathy (parkes, 1969 ; martikeinan p, 1996; dan heekyung youn, arkady shemyakin, 1999). menurut parkes, stress cardiomyopathy terjadi pada pasangan suami istri yang berusia 55 tahun ke atas. 2. faktor lapse lapse adalah kejadian dimana pemegang polis tidak melanjutkan kontraknya disebabkan tidak sanggup membayar premi. mailto:irma_s2mathugm@yahoo.com mailto:irmaristianto08@gmail.com evaluasi premi last survivor pasangan suami istri menggunakan metode copula frank jurnal cauchy – issn: 2086-0382 43 berdasarkan pertimbangan dua faktor tersebut maka dapat disimpulkan produk asuransi jiwa last survivor dengan asumsi mortalitas pasangan suami istri saling dependen yang keduanya hanya dikhususkan untuk pasangan suami istri yang keduanya berusia 55 tahun keatas. sedangkan faktor lapse setelah terjadi kematian dari salah satu pasangan suami istri dan asmsi bahwa mortalitas pasangan suamiistri adalah independen diperhitungkan untuk pemegang polis last survivor dengan usia pasangan suami istri kurang dari 55 tahun. tulisan ini hanya terbatas pada perhitungan premi tunggal bersih asuransi jiwa last survivor berjangka 10 tahun yang dihitung menggunakan tabel multilife yang memuat angka klaim frasier yang diturunkan dari tabel mortalitas single life us female and male tahun 2004. landasan teori a. bunga majemuk semua polis asuransi jiwa baik single life maupun multilife mengharuskan premi dibayar di muka sebelum asuransi efektif sedangkan benefit baru akan dibayarkan pada masa yang akan datang sehingga premi itu dikenakan bunga. oleh karena itu, perhitungan bunga pada asuransi menggunakan teknik perhitungan bunga majemuk dan diskonto. akumulasi bunga majemuk dari dana sebesar rp 1,. adalah 𝑎(𝑡) = (1 + 𝑖)𝑡 , 𝑢𝑛𝑡𝑢𝑘 𝑡 ≥ 0 (1) dengan i menyatakan suku bunga dalam satu periode. faktor diskonto dinotasikan dengan vt yang diberikan oleh, 𝑎−1(𝑡) = 𝑣 𝑡 = 1 (1 + 𝑖)𝑡 𝑢𝑛𝑡𝑢𝑘 𝑡 ≥ 0 (2) b. status single life a) fungsi survival misalkan x adalah variabel random kontinu yang menyatakan jatah usia yang akan dijalani oleh seorang bayi yang baru lahir, maka fungsi distribusi dari x adalah 𝐹𝑋(𝑥) = pr(𝑋 ≤ 𝑥) , 𝑥 ≥ 0 (3) dan fungsi survival yang menyatakan probabilitas seorang bayi yang baru lahir akan mencapai usia x tahun adalah 𝑠(𝑥) = 1 − 𝐹𝑋(𝑥) = pr(𝑋 > 𝑥) , 𝑥 ≥ 0 (4) b) sisa usia bagi (x) notasi t(x) menyatakan sisa usia dari (x) dinyatakan dengan t(x) = x – x. notasi aktuaria yang berhubungan dengan probabilitas tentang t(x) dinyatakan sebagai, 1. probabilitas (x) akan meninggal t tahun kemudian dinyatakan sebagai berikut: 𝑞𝑥 𝑥 = pr(𝑇(𝑥) ≤ 𝑡) = 1 − 𝑠(𝑥 + 𝑡) 𝑠(𝑥) , 𝑡 ≥ 0 (5) 2. probabilitas (x) akan hidup mencapai usia x+t tahun dinyatakan sebagai berikut : 𝑝𝑡 𝑥 = 1 − 𝑞𝑡 𝑥 = pr(𝑇(𝑋) > 𝑡) = 𝑠(𝑥 + 𝑡) 𝑠(𝑥) , 𝑡 ≥ 0 (6) c) sisa usia bulat bagi (x) variabel random yang berhubungan dengan sisa usia bulat untuk (x) sampai terjadi kematian dinotasikan dengan k(x). fungsi distribusi dari k(x) didefinisikan sebagai : 𝐹𝐾(𝑥)(𝑦) = ∑ 𝑞ℎ 𝑥 𝑘 ℎ=0 = 𝑞𝑘+1 𝑥 , 𝑦 ≥ 0 (7) dan k bilangan bilat terbesar dari y. d) hubungan tabel mortalitas dan fungsi survival misalkan £(x) merupakan variabel random yang menyatakan jumlah orang yang masih hidup sampai usia x tahun yang dinyatakan dengan £(𝑥) = ∑ 𝐼𝑗 (𝑥) 𝑙0 𝑗=1 1. £(𝑥)~𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙(𝑙0, 𝑠(𝑥)) 2. 𝐼𝑗 ~𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 dengan 𝐼𝑗 = { 1, 𝑗 𝑚𝑎𝑠𝑖ℎ ℎ𝑖𝑑𝑢𝑝 𝑠𝑎𝑚𝑝𝑎𝑖 𝑢𝑠𝑖𝑎 𝑥 0, 𝑗 𝑚𝑒𝑛𝑖𝑛𝑔𝑔𝑎𝑙 𝑠𝑒𝑏𝑒𝑙𝑢𝑚 𝑢𝑠𝑖𝑎 𝑥 karena 𝐸[𝐼𝑗 ] = 𝑠(𝑥) maka 𝑙𝑥 = 𝐸[£(𝑥)] = 𝑙0𝑠(𝑥). probabilitas seseorang yang berusia x akan hidup paling sedikit k tahun yaitu 𝑝𝑘 𝑥 = 𝑙𝑥+𝑘 𝑙𝑥 maka probabilitas seseorang berusia x akan meninggal sebelum mancapai usia x+k tahun adalah 𝑞𝑘 𝑥 = 1 − 𝑝𝑘 𝑥 = 𝑙𝑥+𝑘 − 𝑙𝑥 𝑙𝑥 = 𝑑𝑥 𝑙𝑥 (8) tabel mortalitas yang digunakan dalam tulisan ini adalah life table us male&female 2004. irma fauziah 44 volume 3 no. 1 november 2013 e) sistem pembayaran benefit asuransi single life berjangka diskrit misalkan bk+1 menyatakan fungsi benefit dan vk+1 menyatakan fungsi diskonto maka asuransi berjangka n tahun yang menyediakan benefit pada akhir tahun kematian tertanggung adalah 𝑏𝑘+1 = { 1, 𝑘 = 0,1, … , 𝑛 − 1 0, 𝑦𝑎𝑛𝑔 𝑙𝑎𝑖𝑛 (9) untuk mempermudah perhitungan, asumsikan benefitnya sebesar rp 1,. present value pembayaran benefit adalah sebesar zk+1 = bk+1. vk+1 variabel random dari zk+1 dinotasikan dengan z. 𝑍 = { 𝑣𝑘+1, 𝐾 = 0,1, … , 𝑛 − 1 0, 𝑦𝑎𝑛𝑔 𝑙𝑎𝑖𝑛 (10) actuarial presesnt value (apv) dari asuransi berjangka n tahun adalah                 1 0 1 :| 1 0 11 pr n k nxxk n k kk aq kxkvveze (11) f) anuitas jiwa single life berjangka n tahun diskrit present value variabel random dari anuitas jiwa awal berjangka n tahun sebesar 1 pertahun adalah 𝑌 = { �̈�𝐾+1|̅̅ ̅̅ ̅̅ ̅ 0 ≤ 𝐾 ≤ 𝑛 �̈�𝑛|̅̅ ̅ 𝐾 ≥ 𝑛 (12) dan actuarial present value (apv) nya adalah   :: 1 1.. .. .. 1 0 0 x n n n k k nk x x k n x k x k k e y a p q a p v p a            (13) g) status last survivor kelangsungan status dari m individu yang berusia x1, x2,…, xm masih berlangsung jika paling sedikit satu dari (x1), (x2),…,(xm) hidup dan status ini akan berkahir jika semua individu meninggal dunia dinamakan status last survivor. status ini dinotasikan dengan (𝑥1𝑥2 … 𝑥𝑚̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅). distribusi sisa usia yang lengkap yang akan dijalani oleh status last survivor yaitu:    1 2 1 2... max ( ), ( ),..., ( )m mk x x x k x k x k x (14) dengan k(xi) adalah sisa usia lengkap yang dijalani oleh individu ke-i. misalkan status last survivor pasangan suami istri dengan x menyatakan usia suami dan y menyatakan usia istri maka fungsi distribusi dari 𝐾(𝑥𝑦̅̅ ̅) dengan asumsi sisa usia pasangan suami istri adalah saling bebas dinyatakan sebagai,    kykxyxk kykykyk kxkxkyk xyk qqp qpq qpq kfkxyk        )(pr )( (15) probabilitas salah seorang pasangan suami istri akan hidup k tahun dinotasikan dengan 𝑝𝑘 𝑥𝑦̅̅ ̅̅ = 𝑝𝑘 𝑥𝑦 + 𝑝𝑘 𝑥 𝑞𝑘 𝑥 + 𝑝𝑘 𝑦 𝑞𝑘 𝑥 (16) c. sistem pembayaran benefit asuransi last survivors berjangka diskrit jika k menyatakan sisa usia lengkap dari status (𝑥𝑦̅̅ ̅) maka 1. waktu pembayaran adalah k+1 2. present value pada pembayaran benefit ini pada saat polis dikeluarkan sebesar z = vk+1. 3. actuarial present value (apv) yaitu :     1 1 1 1 : 0 pr ( ) n k k xy n k e z e v v k xy k a            (17) d. anuitas jiwa last survivors berjangka n tahun diskrit actuarial present value (apv) dari anuitas last survivor berjangka n tahun adalah :     1 .. .. .. 1 : 0 pr ( ) n k n xy nn xy k e y a k xy k a p a        (18) e. copula frank copula frank merupakan salah satu keluarga copula archimedean dan pertama kali dikenalkan oleh frank pada tahun 1979 yang mempunyai bentuk :      1 11 , ln 1 1 u v e e c u v e                  ,     (19) α adalah parameter yang mengukur penyimpangan dari asumsi saling bebas. copula ini dibangun dari fungsi pembangkit (generator) yang dinotasikan dengan ( )u yaitu : 𝜑(𝑢) = ln 𝑒 𝛼𝑢 − 1 𝑒 𝛼 − 1 (20) evaluasi premi last survivor pasangan suami istri menggunakan metode copula frank jurnal cauchy – issn: 2086-0382 45 dengan fungsi invers yaitu :      1 ln 1 1 u e e u        (21) copula ini mempunyai kelebihan jika dibandingkan dengan keluarga copula archimedean yang lain yaitu : 1. dependensi lengkap, copula frank mengijinkan dependensi negative antar marginal,     2. copula ini digunakan untuk memodelkan data yang menunjukkan dependensi yang lemah 3. mempunyai batas-batas frechet-hoeffding yang termuat dalam interval parameter dependensi. pembahasan a. status polis last survivor dalam polis last survivor yang menanggung pasangan suami yang berusia x tahun dan istri yang berusia y tahun maka terdapat empat status yang bergantung pada status hidup keduanya, misalkan a, b, c dan d adalah empat status tersebut dengan status a menyatakan status pasutri keduanya masih hidup, status b menyatakan status bahwa istri masih hidup dan suami sudah meninggal, status c menyatakan status istri sudah meninggal dan suami masih hidup dan status d menyatakan status bahwa keduanya sudah meninggal. jadi asumsi mortalitas pada keempat status merupakan bobot rata-rata angka mortalitas yang ditunjukkan pada tabel 1. tabel 1. populasi dan status status tp (x) tk (y) pspat (k+1) pif a h h 𝑝𝑘 𝐴 𝑥𝑦 = 𝑝𝑘 𝑥 𝑝𝑘 𝑦 ya b m h 𝑝𝑘 𝐵 𝑥𝑦 = 𝑞𝑘 𝑥 𝑝𝑘 𝑦 ya c h m 𝑝𝑘 𝐶 𝑥𝑦 = 𝑝𝑘 𝑥 𝑞𝑘 𝑦 ya d m m 𝑝𝑘 𝐷 𝑥𝑦 = 𝑞𝑘 𝑥 𝑞𝑘 𝑦 tidak keterangan. h : hidup m : meninggal tp : tertanggung pertama tk : tertanggung kedua pspat : probabilitas status pada awal tahun pif : policy in force b. angka klaim last survivor frasier bill frasier pada tahun 1978 mengenalkan metode pendekatan untuk menghitung probabilitas kematian pada tahun tertentu dari tertanggung terakhir, yang dihitung pada awal berlakunya polis, angka klaim tersebut dinyatakan dengan,               k x k y x k y k k x k y x k k x k y y kjls k k x k y k x k y k x k y p p q q p q q q p q q p p p q q p             1 1 k kxy xy k xy q q q     (22) dengan 𝑞𝑘 𝑥𝑦̅̅ ̅̅ menyatakan probabilitas bahwa kedua tertanggung meninggal dalam k tahun pertama. persamaan (22) dapat dinyatakan sebagai probabilitas bersyarat, yaitu : | |: a b c k xy k xy y k xmeninggal k xy x k ymeninggalx k y kjls k a b c k xy k xy k xy p q p q p q q p p p            (23) populasi status a akan berkurang karena terjadi kematian salah satu tertanggung maka polis bermigrasi ke status b dan c dan jika keduanya meninggal maka polis bermigrasi ke status d. secara rekursif dinyatakan sebagai : ktbn mtbn 𝑝𝑘+1 𝐴 𝑥𝑦 = 𝑝𝑘 𝐴 𝑥𝑦 − 𝑝𝑘 𝐴 𝑥𝑦 × 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ − 𝑝𝑘 𝐴 𝑥𝑦 × (𝑞𝑦+𝑘|𝑥ℎ − 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ) 𝑝𝑘+1 𝐵 𝑥𝑦 = 𝑝𝑘 𝐵 𝑥𝑦 − 𝑝𝑘 𝐵 𝑥𝑦 × 𝑞𝑥+𝑘|𝑦𝑚 − 𝑝𝑘 𝐴 𝑥𝑦 × (𝑞𝑦+𝑘|𝑦ℎ − 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅) 𝑝𝑘+1 𝐶 𝑥𝑦 = 𝑝𝑘 𝐶 𝑥𝑦 − 𝑝𝑘 𝐶 𝑥𝑦 × 𝑞𝑦+𝑘|𝑥𝑚 − 𝑝𝑘 𝐴 𝑥𝑦 × (𝑞𝑦+𝑘|𝑥ℎ − 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ) 𝑝𝑘+1 𝐷 𝑥𝑦 = 𝑝𝑘 𝐷 𝑥𝑦 − 𝑝𝑘 𝐴 𝑥𝑦 × 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ − 𝑝𝑘 𝐴 𝑥𝑦 × (𝑞𝑦+𝑘|𝑦𝑚 − 𝑞𝑥+𝑘,𝑦+𝑘̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ) + − 𝑝𝑘 𝐵 𝑥𝑦 × 𝑞𝑥+𝑘|𝑦𝑚 + − 𝑝𝑘 𝐶 𝑥𝑦 × 𝑞𝑦+𝑘|𝑥𝑚 keterangan. ktbn : klaim total bersih nol mtbn : migrasi total bersih nol c. pengaruh tingkat lapse pada setiap status jika data tidak tersedia maka seorang aktuaris boleh mengasumsikan tingkat lapse yang lebih rendah diantara tertanggung yang masih hidup (margus, 2002). untuk menetapkan jumlah populasi pada tahun berikutnya berdasarkan asumsi tingkat lapse yang diberikan, secara terpisah pada setiap status yang berlaku maka, status a,  . .1 11 a a e o y a k xy k xy k xy p p r      status b,  . .1 11 b b e o y b k xy k xy k xy p p r      status c,  . .1 11 c c e o y c k xy k xy k xy p p r      dimana 1 a k xy r  menyatakan tingkat lapse pada akhir tahun ke k+1 pada status a, 1 b k xy r  menyatakan tingkat lapse pada akhir tahun ke irma fauziah 46 volume 3 no. 1 november 2013 k+1 pada status b, 1 c k xy r  menyatakan tingkat lapse pada akhir tahun ke k+1 pada status c, dan notasi . .a e o y k xy p menyatakan probabilitas hidup atau populasi akhir tahun ke-k pada status a yang besarnya dinyatakan dengan . .a e o y a k xy k xy p p : a k xy x k y k p q      | : a k xy y k xhidup x k y k p q q        | : a k xy x k yhidup x k y k p q q       (25) persamaan ini diperoleh dari persamaan (24), dimana persamaan (24) merupakan populasi awal tahun. sedangkan status d adalah status dimana keduanya meninggal maka populasi status d tidak berubah. d. model copula frank untuk anuitas dan asuransi berjangka last survivor notasi (𝑥𝑦̅̅ ̅) menyatakan status dari dua tertanggung yang berusia x dan y tahun masih berlangsung jika salah satu dari (x) atau (y) masih hidup dan status tersebut berakhir jika keduanya meninggal dunia disebut status last survivor. probabilitas salah satu tertanggung dari suami ataupun istri akan hidup mencapai usia x+k dan y+k tahun dinotasikan dengan k xyp , berdasarkan copula frank dinyatakan sebagai   1 11 1 log 1 1 k yk x qq k exy e e p e                     (26) anuitas awal berjangka n tahun untuk status last survivor dinotasikan dengan �̈�𝑥𝑦̅̅ ̅̅ :�̅� yang besarnya adalah,   1.. : 0 1 11 1 log 1 1 k yk x qq n k xy n e k e e a v e                             (27) nilai tunai asuransi atau premi tunggal bersih asuransi last survivor sebesar rp 1,. selama n tahun adalah 1 1 1 : 0 n k jls k kxyxy n k a v p q               11 .. 1 1 .. 0 1 1 11 log 1 1 1 k yk x k yk x qq n k e qq k e e e v e e e                          (28) e. model copula frank untuk premi last survivor misalkan premi tunggal bersih untuk asuransi last survivor berjangka n tahun dengan benefit sebesar rp 1,. bagi suami yang berusia x tahun dan istri yang berusia y tahun dinotasikan dengan 1 :xy n p . fungsi kerugian: .. .. 1 1 1 : k k n n xyxy n l v p a a p           k = 0,1,2,…,n-1   .. .. 1 1 1 : 0 0 k k n n xyxy n e l e v p e a a p                                 11 .. 1 1 .. 0 1 : .. 1 0 1 1 11 log 1 1 1 1 11 1 log 1 1 k yk x k yk x k yk x qq n k e qq k xy n qq n k e k e e e v e e e p e e v e                                                      (29) contoh 1. karena ketidak tersediaan data di lapangan, maka sebagai contoh andaikan pasangan suami istri yang mempunyai usia lebih dari 55 tahun mengalihkan resiko ketidakpastian hidup dengan mengikuti asuransi berjangka last survivor berjangka dengan masa asuransi 10 tahun, pasangan suami istri tersebut berturut-turut berusia 60 tahun dan 55 tahun, dengan uang pertanggungan sebesar rp. 100 000 000,. dan suku bunga 6% pertahun, parameter dependensi mortalitas pasutri dihitung menggunakan copula frank dengan mengambil nilai parameter α = -3,367, -3, -2,5, -2, -1,5, -1 dan 1(valdez, frees, dan carriere, 1999) maka premi tahunan netto nya adalah : tabel 2. premi netto tahunan berjangka 10 tahun untuk suami yang berusia 60 tahun dan istri yang berusia 55 tahun upsi p (𝛼) ptn (rp) x = 60 -3.367 292 506.01 y = 55 -3 273 782.33 -2.5 247 579.07 -2 220 892.55 -1.5 194 163.79 -1 167 914.84 1 78 471.49 keterangan. upsi : usia pasangan suami-istri p : parameter ptn : premi tahunan netto dengan perhitungan sebagai berikut: x=60 (1) (2) (3) (4) y=55 mortalitas single life k qx+k qy+k kqx kqy 0 0.01176 0.00472 0.00000 0.00000 1 0.01293 0.00514 0.01176 0.00472 2 0.01416 0.00559 0.02454 0.00984 3 0.01536 0.00611 0.03836 0.01538 4 0.01656 0.00670 0.05313 0.02139 5 0.01785 0.00739 0.06881 0.02795 6 0.01933 0.00817 0.08543 0.03513 7 0.02099 0.00898 0.10311 0.04301 evaluasi premi last survivor pasangan suami istri menggunakan metode copula frank jurnal cauchy – issn: 2086-0382 47 8 0.02286 0.00978 0.12193 0.05160 9 0.02492 0.01058 0.14200 0.06087 10 0.02706 0.01147 0.16339 0.07081 (5) (6) (7) (8) (9) α = -3.3 α = -3 α = -2.5 α = -2 α = -1.5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00019 0.00017 0.00015 0.00013 0.00011 0.00080 0.00073 0.00063 0.00054 0.00045 0.00189 0.00172 0.00151 0.00129 0.00109 0.00352 0.00323 0.00283 0.00245 0.00208 0.00577 0.00530 0.00467 0.00406 0.00346 0.00870 0.00803 0.00711 0.00620 0.00532 0.01242 0.01149 0.01022 0.00896 0.00773 0.01700 0.01578 0.01410 0.01242 0.01077 0.02250 0.02096 0.01882 0.01666 0.01452 0.02898 0.02708 0.02444 0.02175 0.01907 (10) (11) (12) (13) (14) α = -1 α = 1 α = -3.3 α = -3 α = -2.5 0.00000 0.00000 0.00019 0.00017 0.00015 0.00009 0.00003 0.00061 0.00055 0.00048 0.00038 0.00014 0.00109 0.00100 0.00088 0.00091 0.00035 0.00164 0.00151 0.00133 0.00173 0.00069 0.00225 0.00208 0.00185 0.00290 0.00117 0.00295 0.00274 0.00245 0.00448 0.00185 0.00375 0.00349 0.00314 0.00655 0.00277 0.00463 0.00434 0.00392 0.00917 0.00399 0.00560 0.00526 0.00478 0.01244 0.00556 0.00663 0.00626 0.00573 0.01644 0.00755 0.00775 0.00734 0.00676 (15) (16) (17) (18) (19) α = -2 α = -1.5 α = -1 α = 1 vk 0.00013 0.00011 0.00009 0.00003 1.00000 0.00041 0.00035 0.00029 0.00011 0.94340 0.00076 0.00064 0.00053 0.00021 0.89000 0.00115 0.00099 0.00083 0.00033 0.83962 0.00161 0.00139 0.00117 0.00049 0.79209 0.00215 0.00186 0.00158 0.00068 0.74726 0.00278 0.00242 0.00207 0.00092 0.70496 0.00349 0.00307 0.00265 0.00122 0.66506 0.00429 0.00379 0.00330 0.00158 0.62741 0.00517 0.00461 0.00404 0.00200 0.59190 0.00615 0.00552 0.00488 0.00252 0.55839 (20) (21) (22) (23) (24) α = -3.3 α = -3 α = -2.5 α = -2 α = -1.5 1.00000 1.00000 1.00000 1.00000 1.00000 0.94322 0.94323 0.94326 0.94328 0.94330 0.88929 0.88935 0.88944 0.88952 0.88959 0.83804 0.83817 0.83836 0.83853 0.83870 0.78930 0.78954 0.78985 0.79016 0.79045 0.74295 0.74329 0.74377 0.74423 0.74467 0.69882 0.69930 0.69995 0.70059 0.70121 0.65680 0.65741 0.65826 0.65910 0.65992 0.61675 0.61751 0.61856 0.61962 0.62066 0.57858 0.57949 0.58076 0.58204 0.58330 0.54221 0.54327 0.54475 0.54625 0.54775 (25) (26) (27) (28) (29) α = -1 α = 1 α = -3.3 α = -3 α = -2.5 1.00000 1.00000 0.00018 0.00016 0.00014 0.94331 0.94337 0.00054 0.00049 0.00043 0.88966 0.88987 0.00092 0.00084 0.00073 0.83886 0.83932 0.00130 0.00119 0.00105 0.79072 0.79155 0.00168 0.00155 0.00138 0.74509 0.74638 0.00207 0.00192 0.00172 0.70180 0.70365 0.00247 0.00230 0.00207 0.66070 0.66321 0.00287 0.00269 0.00243 0.62166 0.62491 0.00326 0.00306 0.00279 0.58453 0.58861 0.00362 0.00342 0.00314 0.54922 0.55418 0.00396 0.00376 0.00348 (30) (31) (32) (33) α = -2 α = -1.5 α = -1 α = 1 0.00012 0.00010 0.00008 0.00003 0.00037 0.00031 0.00026 0.00010 0.00063 0.00054 0.00045 0.00018 0.00091 0.00078 0.00065 0.00026 0.00120 0.00103 0.00087 0.00036 0.00151 0.00131 0.00111 0.00048 0.00184 0.00160 0.00137 0.00061 0.00217 0.00191 0.00165 0.00076 0.00251 0.00222 0.00194 0.00093 0.00284 0.00254 0.00223 0.00111 0.00317 0.00285 0.00253 0.00132 nilai parameter dependensi α = -3,367 menunjukkan angka mortalitas yang sangat tinggi sehingga premi yang dihasilkan paling mahal jika dibandingkan dengan parameter dependensi yang lain. nilai parameter α = 1 menunjukkan angka mortalitas last survivor yang menurun sehingga premi yang dihasilkan paling murah di antara parameter dependensi yang lain. nilai parameter dependensi α yang mendekati nol menunjukkan independensi mortalitas pasangan suami-istri. 2. andaikan pasangan suami-istri yang berturut-turut berusia 54 tahun dan 43 tahun mengasuransikan diri mereka dengan mengambil polis last survivor berjangka 10 tahun dengan benefit sebesar rp. 100 000 000,. dan suku bunga 6%. karena usia pasangan suami-istri dibawah 55 tahun maka premi tunggal bersih dihitung dengan mengasumsikan mortalitas pasangan suami istri yang independen dan tingkat lapse 5% sampai terjadi kematian pertama dan 1% setelahnya. besar premi adalah     9 1 1 54:43 :101 0 .. 954:34:10 :10 54:43 0 1, 06 1 100000000. 100000000. 63033, 53 1, 06 1 k jls k k xy k k xy k k q q a p a q           dengan perhitungan sebagai berikut : x=54 (1) (2) (3) y=43 probabilitas terjadi lapse pada akhir tahun ke-k+1 k 1 a k xy r  1 b k xy r  1 c k xy r  𝑥ℎ dan 𝑦ℎ 𝑥ℎ 𝑦ℎ 0 5.00% 1.00% 1.00% 1 5.00% 1.00% 1.00% 2 5.00% 1.00% 1.00% 3 5.00% 1.00% 1.00% 4 5.00% 1.00% 1.00% 5 5.00% 1.00% 1.00% 6 5.00% 1.00% 1.00% 7 5.00% 1.00% 1.00% 8 5.00% 1.00% 1.00% irma fauziah 48 volume 3 no. 1 november 2013 9 5.00% 1.00% 1.00% 10 5.00% 1.00% 1.00% (4) (5) (6) (7) (8)=(4)x(5) keduanya meninggal x k q  y k q  k x q k yq :x k y kq   0.00749 0.00188 0.00000 0.00000 0.00001 0.00795 0.00206 0.00749 0.00188 0.00002 0.00846 0.00224 0.01537 0.00393 0.00002 0.00906 0.00244 0.02370 0.00617 0.00002 0.00981 0.00263 0.03255 0.00859 0.00003 0.01071 0.00282 0.04204 0.01120 0.00003 0.01176 0.00301 0.05230 0.01399 0.00004 0.01293 0.00320 0.06345 0.01695 0.00004 0.01416 0.00343 0.07556 0.02010 0.00005 0.01536 0.00370 0.08865 0.02347 0.00006 0.01656 0.00400 0.10265 0.02707 0.00007 kesimpulan 1. produk asuransi last survivor berjangka untuk pasangan suami-istri yang berusia 55 tahun keatas, premi dimodelkan dengan asumsi dependensi mortalitas pasangan suami istri menggunakan copula frank, asumsi ini diciptakan sebagai perlindungan terhadap resiko finansial perusahaan asuransi, 2. produk asuransi last survivor berjangka untuk pasangan suami-istri yang berusia di bawah 55 tahun ke bawah, premi dimodelkan menggunakan asumsi independensi mortalitas pasangan suami-istri dan asumsi lapse pada setiap status polis last survivor. referensi [1] bowers, newton l, jr.,gerber, hans u,.etc.1997. actuarial mathematics, second ed.the society of actuaries. schaunburg illinois [2] frasier, w. m. 1978. second to die joint life cash value and reverses. the actuary (april) : 4 [3] frees, e. w., carriere, and e. valdez. 1995. annuity valuation with dependent mortality. actuarial research clearing house 2 : 31-80 [4] gusti arlina. 2003. kekerapan tuberkolosis paru pada pasangan suami-istri penderita tuberkolosis paru yang berobat di bagian paru rsup adam malik. universitas sumatera utara [5] heekyung youn, shemyakin arkady.1999. statistical aspect of joint life insurance pricing. american stat. association : 34-38 [6] kellison, stephen g. 1991. the theory of interest. edisi kedua. irwin mcgraw-hill : united state of america [7] margus, paul. 2002. generalized frasier claim rates under survivorship life insurance policies. naaj [8] m. atoillah i, dr. 2007. epidemiologi hepatitis b [9] national vital statistics report, vol 56, no. 9, 28 desember 2007 [10] nelsen, b. roger. 2005. an introduction to copula. springer-verlag, new york [11] p feferman a. 2005. broken heart syndrome. universidade federal de sao paulo : brazil [12] sembiring, r.k.ph.d.2001. buku materi pokok asuransi jiwa i dan asuransi jiwa ii. karunika jakarta : universitas terbuka keakuratan solusi pada persamaan difusi menggunakan skema crank-nicolson afidah karimatul laili1, ari kusumastuti2 1mahasiswa jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang e-mail: aphid.laili@gmail.com, arikususmastuti@gmail.com abstrak persamaan difusi adalah persamaan diferensial parsial linier yang merupakan representasi berpindahnya suatu zat dalam pelarut dari bagian berkonsentrasi tinggi ke bagian yang berkonsentrasi rendah. penelitian ini bertujuan untuk menentukan distribusi temperatur persamaan difusi dengan menggunakan skema crank-nicolson. pertama, mendiskritisasikan persamaan difusi menggunakan skema crank-nicolson. diskritisasi akan menghasilkan matriks. selanjutnya menentukan kestabilan dan konsistensi. kestabilan dan konsistensi untuk menunjukkan bahwa metode yang digunakan tersebut memiliki solusi yang dapat mendekati solusi analitiknya sehingga diketahui bahwa solusi tersebut akurat. matriks hasil diskritisasi akan disimulasikan dalam program. hasil simulasi menunjukkan bahwa distribusi temperatur menurun terhadap waktu karena adanya perpindahan panas. kata kunci: solusi akurat, persamaan difusi, perpindahan panas balik, skema crank-nicolson. abstract diffusion equation is a linear differential equation that represents the transfer of substance from the high concentration part to the lower concentration part. this research is determine the temperature distribution of diffusion equation using crank-nicholson scheme. first, discretization diffusion equation using crank-nicholson scheme. obtained from the discretization is matrix. next, determining stability and consistency. the stability and consistency to indicate that the method used have a solution that can be approximating analytical solution so known regularization. matrix discretization results will be simulated in the program. the simulation results show that the temperature distribution decreases with time to heat transfer. keywords: regularization, diffusion equation, backward heat equation, crank-nicholson scheme. pendahuluan estimasi error adalah suatu proses yang bertujuan untuk mencari solusi terbaik dengan mempertimbangkan besarnya nilai error yang dihasilkan dengan metode numerik. dalam prosesnya, estimasi error didapatkan dari ekspansi daret taylor yang dipotong setelah suku turunan yang diinginkan. dengan pemotongan order yang ke n, maka hasil perhitungan akan mendekati solusi. jadi dalam estimasi error akan dihasilkan suatu solusi yang akurat. solusi akurat yaitu dekatnya suatu solusi pendekatan terhadap nilai sebenarnya. dalam prosesnya, dibutuhkan suatu metode numerik yang akan menghasilkan solusi pendekatan terbaik. solusi pendekatan salah satunya adalah skema crank-nicolson. skema crank-nicolson adalah pengembangan dari metode beda hingga skema eksplisit dengan metode beda hingga maju skema implisit. namun bentuk dari skema crank-nicolson adalah skema implisit. kelebihan metode ini dibandingkan dengan metode beda hingga yang lain adalah stabil tanpa syarat. penelitian terdahulu oleh (durmin, 2013) telah meneliti tentang perbandingan solusi dari skema implisit dan skema crank-nicolson untuk model perpindahan panas. fokus penelitian (durmin, 2013) adalah membandingkan solusi dari skema implisit dan skema crank-nicolson dengan cara simulasi. penelitian terdahulu oleh (le, q.h., & nguyen, 2013) meneliti tentang keakuratan solusi pada persamaan perpindahan panas balik dengan menggunakan ketaksamaan. pada hasil mailto:aphid.laili@gmail.com mailto:arikususmastuti@gmail.com keakuratan solusi pada persamaan difusi menggunakan skema crank-nicolson cauchy – issn: 2086-0382 148 diperoleh dengan error yang relatif kecil dan mendekati solusi sesungguhnya. dengan telah diketahuinya bahwa telah didapatkan error yang relatif kecil, penulis ingin mengetahui estimasi error pada persamaan yang sama dengan metode yang berbeda pada penentuan solusi pendekatannya. pada penelitian ini diselesaikan persamaan difusi menggunakan skema crank-nicolson, dalam penyelesaiaannya dilakukan diskritisasi menggunakan metode beda hingga skema cranknicolson, kemudian menentukan syarat kestabilan dan menentukan syarat konsistensi untuk mengetahui bahwa hasil diskritisasi tersebut akurat. selanjutnya melakukan simulasi dari skema yang digunakan dan interpretasi hasil. kajian pustaka 1. persamaan difusi persamaan difusi yang dipakai adalah persamaan perpindahan panas balik 𝜕𝑢 𝜕𝑡 (𝑥, 𝑡) − 𝑎(𝑡) 𝜕2𝑢 𝜕𝑥2 (𝑥, 𝑡) = 𝑓(𝑥, 𝑡), 𝑢(0, 𝑡) = 𝑢(𝜋, 𝑡) = 0, 𝑢(𝑥, 1) = 𝑔(𝑥) = cos(1) sin(𝑥) exp(12 + 1) , (1) dengan domain 𝑡 ∈ [0,1], 𝑥 ∈ [0, 𝜋], 𝑎(𝑡) adalah fungsi 2𝑡 + 1, dengan solusi eksak 𝑢(𝑥, 𝑡) = cos(𝑡) sin(𝑥) exp(𝑡2+𝑡) , serta 𝑓(𝑥, 𝑡) = − sin(𝑡) sin(𝑥) exp(𝑡2+𝑡) . 𝑢(𝑥, 𝑡) adalah fungsi distribusi temperatur dan 𝑢(𝑥, 1) adalah distribusi temperatur awal, 𝑢𝑡 adalah variabel panas yang bergantung pada 𝑡, 𝑢𝑥𝑥 adalah variabel panas yang bergantung pada 𝑥, dan 𝑎(𝑡) adalah konstanta panas (le, q.h., & nguyen, 2013). 2. skema crank-nicolson skema crank-nicolson merupakan salah satu skema pengembangan dari skema eksplisit dan implisit, yaitu merupakan nilai rerata darai kedua metode tersebut. pada skema crank-nicolson diferensial terhadap waktu 𝑡 dituliskan dalam bentuk beda maju, yaitu (triatmodjo, 2002) 𝜕𝑢(𝑥,𝑡) 𝜕𝑡 = 𝑢𝑗 𝑛+1 −𝑢𝑗 𝑛 ∆𝑡 (2) sedangkan, diferensial terhadap ruang 𝑥 merupakan rerata dari skema eksplisit dam implisit dengan menggunakan beda pusat 𝜕𝑢(𝑥,𝑡) 𝜕𝑥 ≈ 1 2 ( 𝑢𝑗+1 𝑛 −𝑢𝑗−1 𝑛 ∆𝑥 + 𝑢𝑗+1 𝑛+1 −𝑢𝑗−1 𝑛+1 ∆𝑥 ) (3) untuk diferensial orde 2 terhadap waktu dapat dituliskan sebagai berikut 𝜕 2 𝑢(𝑥,𝑡) 𝜕𝑥2 = 1 2 ( 𝑢𝑗−1 𝑛+1−2𝑢𝑗 𝑛+1+𝑢𝑗+1 𝑛+1 ∆𝑥2 ) + 1 2 ( 𝑢𝑗−1 𝑛 −2𝑢𝑗 𝑛+𝑢𝑗+1 𝑛 ∆𝑥2 ), (4) 3. keakuratan solusi keakuratan solusi numerik diukur berdasarkan kriteria konvergensi, konsistensi serta stabilitas. konvergensi berhubungan dengan besarnya penyimpangan solusi pendekatan oleh metode beda hingga terhadap solusi eksak. “aproksimasi solusi pasti konvergen ke solusi analitiknya, jika konsistensi dari persamaan beda dan stabilitas dari skema yang diberikan terpenuhi (zauderer, 2006)”. kriteria stabilitas dan konsitensi merupakan kondisi perlu dan cukup agar diperoleh solusi konvergen. analisis kestabilan dari skema yang digunakan dapat dicari menggunakan stabilitas von neumann dengan mensubstitusikan 𝑢𝑗 𝑛 = 𝜌𝑛𝑒𝑖𝑎𝑗 ke dalam persamaan beda yang digunakan, sedangkan untuk analisis konsistensi dapat dicari dengan menggunakan ekspansi deret taylor. syarat perlu dan cukup stabilitas von neumann yaitu |𝜌| ≤ 1 dan kriteria konsistensi akan terpenuhi jika ∆𝑥 → 0 dan ∆𝑡 → 0. jika syarat kestabilan dan konsistensi terpenuhi maka solusi numerik tersebut akan mendekati solusi analitik (zauderer, 2006). pembahasan 1. solusi persamaan difusi dengan skema crank-nicolson persamaan difusi yang digunakan adalah persamaan (1) yang akan dianalisis dengan skema crank-nicolson (durmin, 2013). mengacu pada persamaan (4), maka bentuk diskrit dari persamaan (1) adalah sebagai berikut: 𝑢𝑗 𝑛+1 − 𝑢𝑗 𝑛 ∆𝑡 − 1 2 (𝑎𝑛 𝑢𝑗−1 𝑛 − 2𝑢𝑗 𝑛 + 𝑢𝑗+1 𝑛 ∆𝑥2 + 𝑎𝑛 𝑢𝑗−1 𝑛+1 − 2𝑢𝑗 𝑛+1 + 𝑢𝑗+1 𝑛+1 ∆𝑥2 ) = 𝑓𝑗 𝑛 (5) kemudian untuk semua variabel dengan superskrip 𝑛 dikelompokkan ke ruas kanan, sehingga afidah karimatul laili 149 volume 3 no. 3 november 2014 − [ 𝑎𝑛 2∆𝑥2 ] 𝑢𝑗−1 𝑛+1 + [ 1 ∆𝑡 + 𝑎𝑛 ∆𝑥2 ] 𝑢𝑗 𝑛+1 − [ 𝑎𝑛 2∆𝑥2 ] 𝑢𝑗+1 𝑛+1 = [ 𝑎𝑛 2∆𝑥2 ] 𝑢𝑗−1 𝑛 + [ 1 ∆𝑡 − 𝑎𝑛 ∆𝑥2 ] 𝑢𝑗 𝑛 + [ 𝑎𝑛 2∆𝑥2 ] 𝑢𝑗+1 𝑛 + 𝑓𝑗 𝑛 (6) diasumsikan sebagai: 𝐴𝑗 = 𝐶𝑗 = 𝐷𝑗 = 𝐹𝑗 = 𝑎𝑛 2∆𝑥2 ; 𝐵𝑗 = 1 ∆𝑡 + 𝑎𝑛 ∆𝑥2 , 𝐸𝑗 = 1 ∆𝑡 − 𝑎𝑛 ∆𝑥2 , sehingga persamaan di atas dapat ditulis kembali sebagai: −𝐴𝑗𝑢𝑗−1 𝑛+1 + 𝐵𝑗𝑢𝑗 𝑛+1 − 𝐶𝑗𝑢𝑗+1 𝑛+1 = 𝐷𝑗𝑢𝑗−1 𝑛 + 𝐸𝑗𝑢𝑗 𝑛 + 𝐹𝑗𝑢𝑗+1 𝑛 + 𝑓𝑗 𝑛 (7) kemudian untuk 𝑛 = 1,2, … , 𝑀 − 1 dan 𝑗 = 1,2, … , 𝑀. misalkan 𝑀 = 5, 𝑀 adalah banyaknya iterasi, maka pada persamaan (7) akan diperoleh suatu matriks, [ 𝐵1 𝐴2 𝐶1 𝐵2 … … 0 0 0 0 ⋮ ⋮ ⋮ ⋮ 0 0 0 0 … … 𝐵𝑀−2 𝐶𝑀−2 𝐴𝑀−1 𝐵𝑀−1] [ 𝑢1 𝑛+1 𝑢2 𝑛+1 ⋮ 𝑢𝑀−2 𝑛+1 𝑢𝑀−1 𝑛+1 ] = [ 𝐷1 𝐷2 ⋮ 𝐷4 𝐷5] (8) maka matriksnya 𝐴𝑢𝑗 𝑛+1 = 𝐷𝑗 , dimana 𝐴 dan 𝐵 adalah matriks tridiagonal dengan ukuran (𝑀 − 1) × (𝑀 − 1) dan unsur 𝑢𝑗 𝑛 dan 𝑓𝑗 𝑛 diketahui dan selesaiannya adalah 𝑢𝑗 𝑛+1 = 𝐴−1(𝐷𝑗) yang berukuran (𝑀 − 1) × 1. 2. keakuratan solusi hasil skema cranknicolson untuk menunjukan bahwa persamaan (5) bernilai benar dan memiliki solusi yang dapat mendekati solusi analitik, maka cukup dengan menunjukan bahwa persamaan beda yang digunakan tersebut stabil dan konsisten. mengetahui apakah metode yang digunakan untuk mendekati persamaan difusi tersebut stabil atau tidak, maka uji kestabilan dapat dilakukan menggunakan analisa stabilitas van neumann, dengan cara mensubstitusikan 𝑢𝑗 𝑛 = 𝜌𝑛𝑒𝑖𝑎𝑗 , ∀ 𝑖 = √−1 ke dalam persamaan (5) yang terlebih dahulu dikalikan dengan ∆t, sehingga diperoleh persamaan sebagai berikut: 𝑢𝑗 𝑛+1 − 𝑢𝑗 𝑛 = (𝑎𝑛∆𝑡 𝑢𝑗−1 𝑛 − 2𝑢𝑗 𝑛 + 𝑢𝑗+1 𝑛 2∆𝑥2 + 𝑎𝑛∆𝑡 𝑢𝑗−1 𝑛+1 − 2𝑢𝑗 𝑛+1 + 𝑢𝑗+1 𝑛+1 2∆𝑥2 ) + ∆𝑡𝑓𝑗 𝑛 (9) kemudian dapat dicari dengan cara mensubstitusikan 𝑢𝑗 𝑛 = 𝜌𝑛𝑒𝑖𝑎𝑗, ∀ 𝑖 = √−1 ke dalam persamaan tersebut dan ∆𝑡𝑓𝑗 𝑛 dianggap kecil , sehingga: 𝜌𝑛+1𝑒𝑖𝑎𝑗 − 𝜌𝑛𝑒𝑖𝑎𝑗 = 𝑎𝑛∆𝑡 2∆𝑥2 (𝜌𝑛𝑒𝑖𝑎(𝑗−1) − 2𝜌𝑛𝑒𝑖𝑎𝑗 + 𝜌𝑛𝑒𝑖𝑎(𝑗+1)) + 𝑎𝑛∆𝑡 2∆𝑥2 (𝜌(𝑛+1)𝑒𝑖𝑎(𝑗−1) − 2𝜌(𝑛+1)𝑒𝑖𝑎𝑗 + 𝜌(𝑛+1)𝑒𝑖𝑎(𝑗+1)) (10) untuk penyederhanaan, persamaan (10) dibagi dengan 𝜌𝑛𝑒𝑖𝑎𝑗, misalkan 𝑎𝑛 diasumsikan sebagai 𝑘 sehingga diperoleh: 𝜌 = 1 + 𝑘∆𝑡 2∆𝑥2 (𝑒−𝑖𝑎 − 2 + 𝑒𝑖𝑎) [1 − 𝑘∆𝑡 2∆𝑥2 (𝑒−𝑖𝑎 − 2 + 𝑒𝑖𝑎)] (11) karena 𝑒±𝑖𝑎 = cos 𝑎 ± 𝑖 sin 𝑎, maka persamaan (11) dapat ditulis: 𝝆 = 𝟏+ 𝒌∆𝒕 𝟐∆𝒙𝟐 (𝐜𝐨𝐬 𝒂−𝒊 𝐬𝐢𝐧 𝒂−𝟐+𝐜𝐨𝐬 𝒂+𝒊 𝐬𝐢𝐧 𝒂) [𝟏− 𝒌∆𝒕 𝟐∆𝒙𝟐 (𝐜𝐨𝐬 𝒂−𝒊 𝐬𝐢𝐧 𝒂−𝟐+𝐜𝐨𝐬 𝒂+𝒊 𝐬𝐢𝐧 𝒂)] (12) sehingga diperoleh: 𝝆 = 𝟏 + 𝒌∆𝒕 𝟐∆𝒙𝟐 (𝟐 𝐜𝐨𝐬 𝒂 − 𝟐) [𝟏 − 𝒌∆𝒕 𝟐∆𝒙𝟐 (𝟐 𝐜𝐨𝐬 𝒂 − 𝟐)] (13) misalkan 𝒌∆𝒕 𝟐∆𝒙𝟐 = 𝑺 |𝝆| = √[ 𝟏 + 𝑺(𝟐 𝐜𝐨𝐬 𝒂 − 𝟐) [𝟏 − 𝑺(𝟐 𝐜𝐨𝐬 𝒂 − 𝟐)] ] 𝟐 (13) persamaan stabil jika dan hanya jika |𝝆| < 𝟏 atau keakuratan solusi pada persamaan difusi menggunakan skema crank-nicolson cauchy – issn: 2086-0382 150 𝟏 + 𝟒𝑺(𝒄𝒐𝒔 𝒂 − 𝟏) 𝟏 − 𝟒𝑺(𝒄𝒐𝒔 𝒂 − 𝟏) ≤ 𝟏 (14) karena −2 ≤ cos 𝑎 − 1 ≤ 0, maka persamaan (14) terpenuhi untuk setiap 𝑆 ∈ 𝑅. sehingga didapatkan kestabilan dari persamaan difusi menggunakan skema crank-nicolson adalah stabil tanpa syarat. setelah diperoleh syarat kestabilan maka selanjutnya syarat konsistensi, untuk mengetahui skema yang digunakan konsisten atau tidak, dapat dilakukan dengan ekspansi deret taylor yang disubstitusikan kedalam persamaan (5). ekspansi deret taylor yang digunakan adalah sebagai berikut: 𝑢𝑗 𝑛±1 = 𝑢𝑗 𝑛 ± ∆𝑡𝑢𝑡|𝑗 𝑛 + 1 2 ∆𝑥2𝑢𝑡𝑡|𝑗 𝑛 ± 1 6 ∆𝑡3𝑡𝑡𝑡|𝑗 𝑛 + ⋯ (15) 𝑢𝑗±1 𝑛±1 = 𝑢𝑗 𝑛 ± ∆𝑡𝑢𝑡|𝑗 𝑛 ± ∆𝑥𝑢𝑥|𝑗 𝑛 + 1 2 (∆𝑡2𝑢𝑡𝑡|𝑗 𝑛 + 2∆𝑡∆𝑥𝑢𝑡𝑥|𝑗 𝑛 + ∆𝑥2𝑢𝑥𝑥|𝑗 𝑛) + 1 6 (∆𝑡3𝑢𝑡𝑡|𝑗 𝑛 + 3∆𝑡2∆𝑥𝑢𝑡𝑡𝑥|𝑗 𝑛 + 3∆𝑡∆𝑥2𝑢𝑡𝑥𝑥|𝑗 𝑛 + ∆𝑥3𝑢𝑥𝑥𝑥|𝑗 𝑛) + ⋯ (16) 𝑢𝑗±1 𝑛 = 𝑢𝑗 𝑛 ± ∆𝑥𝑢𝑥|𝑗 𝑛 + 1 2 ∆𝑥2𝑢𝑥𝑥|𝑗 𝑛 ± 1 6 ∆𝑥3𝑢𝑥𝑥𝑥|𝑗 𝑛 + ⋯ (17) selanjutnya substitusikan persamaan (15), (16) dan (17) kedalam persamaan (5), dengan sedikit manipulasi aljabar maka diperoleh persamaan berikut: (𝑢𝑡 − 𝑎𝑛 2 𝑢𝑥𝑥 − 𝑎𝑛 2 𝑢𝑥𝑥 − 𝑓)| 𝑗 𝑛 + ( 1 2 𝑢𝑡𝑡 − 𝑎𝑛 ∆𝑥 − 8𝑎𝑛∆𝑥3𝑢𝑡𝑥𝑥𝑥) ∆𝑡| 𝑗 𝑛 − 𝑎𝑛 6 𝑢𝑥𝑥𝑥∆𝑥|𝑗 𝑛 + ( 1 6 𝑢𝑡𝑡𝑡 − 12𝑎 𝑛∆𝑥2𝑢𝑡𝑡𝑥𝑥) ∆𝑡 2| 𝑗 𝑛 − 𝑎𝑛 12 𝑢𝑥𝑥𝑥𝑥∆𝑥 2 | 𝑗 𝑛 + ⋯ = 0 (18) suku pertama pada persamaan (18) adalah persamaan difusi yang telah diselesaikan. suku kedua dan seterusnya adalah suku tambahan yang didapatkan dari penyelesaian menggunakan persamaan beda hingga dan disebut truncation error. truncation error atau galat pemangkasan yang didapatkan adalah ( 1 2 𝑢𝑡𝑡 − 𝑎𝑛 ∆𝑥 − 8𝑎𝑛∆𝑥3𝑢𝑡𝑥𝑥𝑥) ∆𝑡| 𝑗 𝑛 − 𝑎𝑛 6 𝑢𝑥𝑥𝑥∆𝑥|𝑗 𝑛 + ( 1 6 𝑢𝑡𝑡𝑡 − 12𝑎𝑛∆𝑥2𝑢𝑡𝑡𝑥𝑥) ∆𝑡 2| 𝑗 𝑛 − 𝑎𝑛 12 𝑢𝑥𝑥𝑥𝑥∆𝑥 2|𝑗 𝑛 + ⋯ (19) karena ∆𝑥 dan ∆𝑡 sangat kecil maka jumlah dari limit tersebut akan semakin kecil, karena berapapun nilai 𝑢𝑡𝑡, 𝑢𝑡𝑥𝑥𝑥 dan 𝑢𝑥𝑥𝑥 jika dikalikan dengan nilai dari ∆𝑡 dan ∆𝑥 akan semakin kecil. error pemotongan yang dihasilkan akan menuju nol untuk ∆𝑥 → 0 dan ∆t → 0. jadi skema cranknicolson konsisten terhadap persamaan difusi. 3. simulasi dan interpretasi hasil persamaan yang digunakan dalam simulasi adalah persamaan (7) yang merupakan bentuk diskrit dari persamaan difusi. dalam simulasi digunakan ∆𝑥 = 0,0698 dan ∆𝑡 = 0,0222, sehingga simulasi persamaan difusi dapat dilihat pada gambar (1) berikut: afidah karimatul laili 151 volume 3 no. 3 november 2014 gambar 1. solusi numerik persamaan difusi menggunakan skema crank-nicolson gambar 2. solusi analitik persamaan difusi pada gambar 1 solusi numerik di atas perubahan temperatur berjalan dari 𝑥 = 0 di 𝑡 berapapun berada pada temperatur 𝑢(𝑥, 𝑡) = 0 kemudian berjalan naik sampai pada temperatur tebesar yaitu pada 𝑥 = 1,827 dan 𝑡 = 0 dengan temperatur 𝑢(𝑥, 𝑡) = 0,4855 kemudian berjalan turun sampai pada 𝑥 = 𝜋 di 𝑡 = 0 dengan temperatur 𝑢(𝑥, 𝑡) = 0. pada gambar 2 solusi analitik di atas perubahan temperatur berjalan secara sama yaitu dari 𝑥 = 0 di 𝑡 berapapun berada pada temperatur 𝑢(𝑥, 𝑡) = 0 kemudian berjalan naik sampai pada temperatur tebesar yaitu pada 𝑥 = 1,536 dan 𝑡 = 0 dengan temperatur 𝑢(𝑥, 𝑡) = 0,977 kemudian berjalan turun sampai pada 𝑥 = 𝜋 di 𝑡 = 0 dengan temperatur 𝑢(𝑥, 𝑡) = 0. perubahan temperatur pada solusi numerik dan solusi analitik bergerak secara sama. perubahan temperatur terjadi secara signifikan yaitu pada ruang 𝑥 = 0 dengan temperatur yang awal nya kecil 𝑢(𝑥, 𝑡) = 0 kemudian perlahan mengalami kenaikan sampai pada ruang tengah 𝑥. kemudian temperatur 𝑢(𝑥, 𝑡) mengalami penurunan secara terus menerus sampai pada ruang 𝑥 maksimal. perubahan temperatur tersebut berjalan secara sama di 𝑡 berapapaun kesimpulan berdasarkan hasil pembahasan, dapat diperoleh kesimpulan antara lain: 1. hasil diskritisasi skema crank-nicolson pada persamaan difusi stabil pada saat ∆𝑡 dan ∆𝑥 berapapun, karena skema crank-nicolson. hasil diskritisasi memenuhi syarat konsistensi karena error pemotongannya menuju nol untuk ∆𝑥 → 0 dan ∆t → 0. jadi, hasil diskritisasi mendekati solusi analitik. 2. pada simulasi dan interpretasi yang dilakukan pada solusi analitik dan solusi numerik menunjukkan bahwa solusi numerik merupakan solusi pendekatan dari solusi analitik. perubahan temperatur terjadi secara sama pada solusi analitik dan solusi numerik. daftar pustaka [1]. durmin. (2013). studi perbandingan perpindahan panas menggunakan metode beda hingga dan cranh-nicholson. surabaya: tidak diterbitkan. [2]. le, t. p., q.h., d. t., & nguyen, t. (2013). a backward parabolic equation with timedependent coefficient: regulation and error estimates. journal of computational and applied mathematics, 237 , 432-441. [3]. triatmodjo, b. (2002). metode numerik dilengkapi dengan program komputer. yogyakarta: beta offset. [4]. zauderer, e. (2006). partial differential equations of applied mathematics. canada: wiley. pendahuluan kajian pustaka 1. persamaan difusi 2. skema crank-nicolson 3. keakuratan solusi pembahasan 1. solusi persamaan difusi dengan skema crank-nicolson 2. keakuratan solusi hasil skema crank-nicolson 3. simulasi dan interpretasi hasil kesimpulan daftar pustaka skema numerik persamaan leslie gower dengan pemanenan cauchy – issn: 2086-0382 1 skema numerik persamaan leslie gower dengan pemanenan trija fayeldi jurusan pendidikan matematika universitas kanjuruhan malang email: trija_fayeldi@yahoo.com abstrak model predator-prey merupakan salah satu model yang sangat banyak diteliti dan dimodifikasi pada penelitian ini, akan dibangun suatu skema numerik untuk menyelesaikan suatu model predatorprey. model yang dipilih pada penelitian ini adalah model kontinu leslie-gower yang telah dimodifikasi pada pemanenan. model kontinu tersebut akan didiskritisasi dengan menggunakan metode euler. selanjutnya, akan diamati perilaku model hasil diskritisasi tersebut dan pada akhirnya akan diamati apakah perubahan parameter dapat mempengaruhi kestabilan model dengan menggunakan perangkat lunak matlab. hasil penelitian menujukkan bahwa perilaku model hasil diskritisasi konsisten dengan model kontinunya. pada simulasi numerik dengan matlab, ditunjukkan bahwa perubahan parameter akan mempengaruhi kestabilan model. kata kunci: predator-prey, leslie-gower, stabil, saddle, source, pemanenan. pendahuluan leslie-gower memperkenalkan model dengan populasi predator tumbuh mengikuti model logistik [1]. pada model ini, diasumsikan bahwa bahwa carrying capacity (kapasitas pendukung) predator sebanding dengan banyaknya populasi prey. model ini kemudian diteliti dan dikembangkan oleh beberapa peneliti. aguirre mengembangkan penelitian model predator-prey leslie-gower dengan penambahan efek [2]. hasil penelitian ini menunjukkan bahwa model leslie-gower dengan penambahan efek allee dapat memiliki dua limit cycle. aziz mengkaji model leslie-gower dengan mengasumsikan fungsi respon yang menyatakan besarnya pemangsaan predator terhadap prey mengikuti fungsi respon holling tipe ii [3]. mereka meneliti kestabilan global dan keterbatasan sistem tersebut. model yang dikembangakan oleh aziz, dkk. kemudian dimodifikasi dengan penambahan waktu tunda [4]. dari hasil penelitian, diperoleh bahwa waktu tunda mempengaruhi perilaku dinamik model tersebut. penelitian ini dikembangkan dengan mengamati bifurkasi dari [5]. chen memperkenalkan perlindungan terhadap prey dan menunjukkan bahwa perlindungan tidak mempengaruhi perilaku model secara signifikan [6]. selain itu, pengaruh efek impulsif pada model diteliti oleh song [7]. zhu melakukan penelitian model leslie-gower dengan fungsi respon holling tipe ii tanpa adanya kontrol impulsif [8] dan mengkaji eksistensi dan kestabilan global solusi periodik positif model dengan mengkonstruksi suatu fungsi lyapunov.. tinjauan pustaka 1. sistem dinamik diskrit bentuk umum dari suatu sistem dinamik diskrit dapat dilihat pada definisi berikut. definisi 1 misalkan�⃗� (n) = [x1(n), x2(n), …, xk(n)]t dan �⃗�= [g1, g2, …, gk]t . bentuk umum dari suatu sistem dinamik diskrit adalah �⃗� (n + 1) =�⃗� (�⃗� (n)) (1) bentuk (1) disebut sebagai sistem dinamik diskrit nonlinear jika gi memuat perkalian antarvariabel tak bebas, ∀i = 1, 2, …, k. definisi 2 misalkan g adalah fungsi dari r ke r. titik p* dikatakan sebagai titik kesetimbangan dari g jika memenuhi g(p*) = p*. 2. sistem dinamik diskrit linear bentuk umum persamaan beda orde satu dengan variabel bebas adalah �⃗� (n + 1) = a�⃗� (n) (2) dengan a = (aij), ∀i = 1, 2, …, k adalah matriks yang bernilai real, dan �⃗� (n) = [x1(n), x2(n), … , xk(n)]t , t menyatakan transpose dari vektor a. jika diberikan nilai awal �⃗� (n) =�⃗�0 maka persamaan (2) memiliki solusi umum �⃗� (n) = an �⃗� (0) (3) jika matriks a dapat didiagonalkan maka solusi umum persamaan (2) adalah �⃗� (n) = c1𝜆1 𝑛�⃗�1+ … + c3𝜆𝑘 𝑛�⃗�𝑘 (4) skema numerik persamaan leslie gower dengan pemanenan cauchy – issn: 2086-0382 35 dengan ci adalah sembarang konstanta dan 𝑣𝑖⃗⃗⃗⃗ adalah vektor eigen yang bersesuaian dengan nilai eigen λi. teorema 1 misalkan f(λ) = λ2 ─ pλ + q maka kondisi-kondisi berikut: 1. f(1) = 1 ─ p + q > 0 2. f(─1) = 1 + p + q > 0 3. f(0) = q < 1 merupakan syarat perlu dan cukup agar persamaan tersebut memiliki akar-akar karakteristik |λi|, i =1, 2. lemma 1 jika f(λ) = λ2 ─ pλ + q dengan 𝜆1 dan 𝜆2 adalah dua akar dari f(λ) = 0 maka 1. |λ1| < 1 dan |λ2| < 1 jika dan hanya jika f(1) > 0 , f(−1) > 0, dan f(0) < 1. 2. |λ1| < 1 dan |λ2| > 1 atau |λ1| > 1 dan |λ2| < 1 jika dan hanya jika f(1) > 0 , f(−1) < 0. 3. |λ1| > 1 dan |λ2| > 1 jika dan hanya jika f(1) > 0 , f(−1) > 0, dan f(0) > 1. 4. λ1 dan λ2 adalah kompleks dan |λ1| = |λ2| = 1 jika dan hanya jika f(1) > 0 , f(1) > 0, f(0) = 1, dan p2 – 4q < 0. akibat 1 1. titik kesetimbangan bersifat stabil asimtotik (sink) f(1) > 0 , f(−1) > 0, dan f(0) < 1, 2. titik kesetimbangan bersifat tak stabil pelana (saddle) jika f(1) > 0 , f(−1) < 0, 3. titik kesetimbangan bersifat tak stabil (source) jika f(1) > 0 , f(−1) > 0, dan f(0) > 1, 4. titik kesetimbangan merupakan titik kesetimbangan non-hyperbolic jika f(1) > 0, p2 – 4q < 0, dan q = 1. 3. model predatorprey leslie-gower pada tahun 1948, leslie dan gower memperkenalkan model dengan populasi prey tumbuh mengikuti model logistik sehingga pertumbuhan predator juga terbatas disebabkan oleh predator memangsa prey. selain itu, carrying capacity predator sebanding dengan banyaknya populasi prey, sehingga model leslie-gower dinyatakan sebagai sistem persamaan diferensial nonlinear berikut. 𝑑𝑥 𝑑𝑡 =(r1 ─ a1y ─ b1x) x (5a) 𝑑𝑦 𝑑𝑡 = (𝑟2 − 𝑎2 𝑦 𝑥 )y (5b) dengan semua parameter bernilai positif. kedua parameter, yaitu r1 dan r2 berturut-turut menunjukkan pertumbuhan intrinsik prey dan predator. parameter a1 merupakan parameter interaksi pemangsaan predator terhadap prey. parameter b1 adalah parameter antraksi antar prey, sedangkan b2 adalah parameter interaksi antar predator [9].. model predator prey yang digunakan pada penelitian ini adalah sebagai berikut. 𝑑𝐻 𝑑𝑡 = (r1 ─ a1p ─ b1h) h ─ c1h (6a) 𝑑𝑃 𝑑𝑡 =(𝑟2 − 𝑎2 𝑃 𝐻 )p ─ c2p (6b) dengan h dan p berturut-turut adalah kepadatan spesies prey dan predator pada waktu t. pada model ini, diasumsikan 0 < ci < ri, i = 1, 2 untuk mengontrol kepadatan prey dan predator. model ini mempunyai dua titik kesetimbangan, yaitu e1 dan e2 berturut turut e1 =( 𝑟1−𝑐1 𝑏1 ,0)dan titik kesetimbangan e2 = (h2, p2) dengan h2 = (𝑟1−𝑐1)𝑎2 𝑎1(𝑟2−𝑎2)+𝑎2𝑏1 dan p2 = (𝑟1−𝑐1)(𝑟2−𝑐2) 𝑎1(𝑟2−𝑎2)+𝑎2𝑏1 . menurut zhu [8], titik kesetimbangan (h*, p*) bersifat stabil global. metodologi penelitian metodologi penelitian yang dilakukan pada penelitian ini adalah studi literatur. pada penelitian ini, persamaan model leslie-gower kontinu (6a) dan (6b) didiskritisasi dengan menggunakan metode euler. kemudian, dilakukan pencarian titik kesetimbangan dan analisis kestabilan pada titik kesetimbangan tersebut secara lokal. akhirnya, dilakukan simulasi numerik untuk beberapa kasus parameter model. hasil dan pembahasan 1. diskritisasi model pandang kembali (6a) dan (6b). dengan menggunakan beda maju, persamaan (6a) dan (6b) dapat didiskritisasi menjadi seperti berikut. hn+1 = hn + (k1 – a1pn – b1hn) hn (7a) pn+1 = pn +(𝑘2 − 𝑎2 𝑃𝑛 𝐻𝑛 )pn (7b) dengan k1 = (r1 – c1) dan k2 = (r2 – c2). 2. titik kesetimbangan model misalkan titik kesetimbangan dari (7a) dan (7b) adalah (h*, p*) maka haruslah berlaku h* = h* + (k1 – a1 p* – b1 h*) h* (8a) p* = p* +(𝑘2 − 𝑎2 𝑃∗ 𝐻∗ ) p* (8b) trija fayeldi 36 volume 3 no. 4 mei 2015 berdasarkan (8a) dan (8b), diperoleh h* = 0 atau k1 – a1 p* – b1 h* = 0 (9a) dan p* = 0 atau (𝑘2 − 𝑎2 𝑃∗ 𝐻∗ )= 0. (9b) berdasarkan (8b), jelas h* ≠ 0. substitusikan p* = 0 ke (9a) k1 − a1 (0) – b1 h* = 0 k1 – b1 h* = 0 k1 = b1h* h* = 𝑟1−𝑐1 𝑏1 (10) sehingga diperoleh e1 =( 𝑟1−𝑐1 𝑏1 ,0). untuk k1 – a1 p* – b1 h* = 0 diperoleh p* = 𝑘1−𝑏1𝐻 ∗ 𝑎1 . (11) substitusikan (11) ke (9b), diperoleh (𝑘2 − 𝑎2𝑃 ∗ 𝐻∗ )= 0 (𝑘2 − 𝑎2(𝑘1−𝑏1𝐻 ∗) 𝑎1𝐻 ∗ )= 0 𝑎1𝑘2𝐻 ∗−𝑎2(𝑘1−𝑏1𝐻 ∗) 𝑎1𝐻 ∗ = 0 a1k2h* − a2k1 + a2b1h* = 0 h* (a1k2 + a2b1) = a2k1 h* = 𝑎2𝑘1 𝑎1𝑘2+𝑎2𝑏1 (12) substitusikan (12) ke (11), diperoleh p* = 𝑘1−𝑏1𝐻 ∗ 𝑎1 p* = 𝑘1−𝑏1( 𝑎2𝑘1 𝑎1𝑘2+𝑎2𝑏1 ) 𝑎1 . p* = 𝑘1(𝑎1𝑘2+𝑎2𝑏1)−𝑎2𝑏1𝑘1 𝑎1(𝑎1𝑘2+𝑎2𝑏1) p* = 𝑘1(𝑎1𝑘2+𝑎2𝑏1−𝑎2𝑏1) 𝑎1(𝑎1𝑘2+𝑎2𝑏1) p* = 𝑎1𝑘1𝑘2 𝑎1(𝑎1𝑘2+𝑎2𝑏1) p* = 𝑘1𝑘2 𝑎1𝑘2+𝑎2𝑏1 . sehingga diperleh e2 =( 𝑎2𝑘1 𝑎1𝑘2+𝑎2𝑏1 , 𝑘1𝑘2 𝑎1𝑘2+𝑎2𝑏1 ). 3. analisis kestabilan model analisis kestabilan model dilakukan dengan mengamati perilaku model secara lokal di sekitar titik kesetimbangannya. misalkan, f(h, p) = h + (k1 – a1p – b1h) h (13a) g(h, p) = p +(𝑘2 − 𝑎2 𝑃 𝐻 )p (13b) matriks jacobi dari sistem (13a) dan (13b) adalah j =( 𝜕𝐹 𝜕𝐻 𝜕𝐹 𝜕𝑃 𝜕𝐺 𝜕𝐻 𝜕𝐺 𝜕𝑃 ) =( 1 + (𝑘1 − 𝑎1𝑃 − 2𝑏1𝐻) −𝑎1𝑃 𝑎2𝑃 2 𝐻2 1 + (𝑘2 − 2𝑎2 𝑃 𝐻 ) )(14) . untuk e1 diperoleh: je1 =( 1 − 𝑘1 − 𝑎1𝑘1 𝑏1 0 1 + 𝑘2 ). nilai-nilai eigen dari je1 adalah λ1 = 1 – k1 dan λ2 = 1 + k2. jelas |λ2| = |1 + k2| = 1 + k2 > 1. lemma jika 0 < k1 < 2 maka |λ1| < 1. bukti: 0 < k1 < 2 −2 < −k1 < 0 −2 + 1 < −k1 + 1 < 0 + 1 −1 < 1 – k1 < 1 |1 – k1| < 1 (terbukti) berdasarkan lemma tersebut, dapat dituliskan teorema berikut. teorema 2 jika k1, k2 > 0 maka 1. e1 saddle jika k1 < 2 2. e1 source jika k1 > 2 3. e1 nonhiperbolik jika k1 = 2 adapun matriks jacobi untuk e2 adalah je2 =( 1 − 𝑏1𝐻 ∗ −𝑎1𝐻 ∗ 𝑘2 2 1 − 𝑘2 ). misalkan, p dan q berturut-turut adalah trace dan determinan je2 maka dapat ditulis p = 2 – (b1h* + k2) q = 1 – (b1h* + k2) + k1k2 lemma 2 jika k1, k2 > 0 maka 1 – p + q > 0 bukti: misalkan k1, k2 > 0. 1 – p + q = 1 – [2 – (b1h* + k2)] + 1 – (k2 + b1h*) + k1k2 = b1h* + k2 + – (b1h* + k2) + k1k2 = k1k2 > 0 (terbukti) lemma 3 jika h* > 𝑘2 𝑏1 (k1 – 1) maka q < 1. bukti: misalkan, h* > 𝑘2 𝑏1 (k1 – 1) maka skema numerik persamaan leslie gower dengan pemanenan cauchy – issn: 2086-0382 37 b1h* > k2(k1 – 1) b1h* > k1k2 – k2 b1h* + k2 > k1k2 b1h* + k2 − k1k2 > 0 −( b1h* + k2) + k1k2 < 0 1 −( b1h* + k2) + k1k2 < 1 q < 1 (terbukti) lemma 4 jika h* < 1 2𝑏1 [4 + k2 (k1 – 2)] maka 1 + p + q >0. bukti: misalkan h* < 1 2𝑏1 [4 + k2 (k1 – 2)] h* < 1 𝑏1 [2 + 𝑘2 2 (k1 – 2)] b1h* + k2 < 2 + 𝑘1𝑘2 2 2 + 𝑘1𝑘2 2 > b1h* + k2 4 + k1k2 > 2(k2 + b1h*) 4 + k1k2 − 2(b1h* + k2) > 0 1 + [2 – (b1h* + k2)] + [1 – (b1h* + k2) + k1k2] > 0 1 + p + q > 0 (terbukti). berdasarkan lemma 2, lemma 3, dan lemma 4 dapat dibuat teorema berikut. teorema 3 jika 𝑘2 𝑏1 (k1 – 1) < h* < 1 2𝑏1 [4 + k2(k1 – 2)] maka titik kesetimbangan e2 akan stabil. 4. simulasi numerik berikut akan disajikan simulasi numerik dari sistem (7a) dan (7b) dengan c1 = 0,1; r1 = 0.3; c2= 0.1; r2 = 0.2; a1 = 0.1; b1 = 0.1; a2 = 0.1. pada simulasi tersebut, diperoleh nilai eigen pada titik kesetimbangan e1 = (3, 0) adalah |λ1| = 0,7 dan |λ2| = 1,1 yang berarti titik kesetimbangan e1 saddle. adapun nilai eigen yang diperoleh pada titik kesetimbangan e2 = (1.5, 1.5) adalah |λ1| = 0,8832 dan |λ2| = 0,8832 yang berarti titik kesetimbangan e2 stabil. gambar 1 plot sistem (7a) dan (7b) dengan c1 = 0,1; r1 = 0.3; c2= 0.1; r2 = 0.2; a1 = 0.1; b1 = 0.1; a2 = 0.1. simulasi numerik berikutnya menggunakan parameter c1 = 1; r1 = 3.5; c2= 0.1; r2 = 0.2; a1 = 0.3; b1 = 0.2; dan a2 = 0.5. pada simulasi tersebut, diperoleh nilai eigen pada titik kesetimbangan e1 = (12.5, 0) adalah |λ1| = 1,5 dan |λ2| = 1,1 yang berarti titik kesetimbangan e1 source. adapun nilai eigen yang diperoleh pada titik kesetimbangan e2 = (9.62, 1.92) adalah |λ1| = 0,89 dan |λ2| = 0,87 yang berarti titik kesetimbangan e2 stabil. gambar 2 plot sistem (7a) dan (7b) dengan c1 = 1; r1 = 3.5; c2= 0.1; r2 = 0.2; a1 = 0.3; b1 = 0.2; dan a2 = 0.5. trija fayeldi 38 volume 3 no. 4 mei 2015 kesimpulan berdasarkan uraian pada hasil dan pembahasan, maka dapat diambil kesimpulan bahwa perilaku model hasil diskritisasi konsisten dengan model kontinunya. selain itu, pada simulasi numerik dengan matlab, ditunjukkan bahwa perubahan parameter akan mempengaruhi kestabilan model. bibliography [1] p. h. leslie, "some furhter notes on the use of matrices in population mathematics," biometrika, vol. 35, pp. 213245, 1948. [2] e. p. aguirre and e. saez, "two limits cycles in a leslie-gower predator-prey model with additive allee effect," nonlinear analysis: real world applications, vol. 10, no. 3, pp. 1401-1416, 2009. [3] m. a. aziz-alaoui and m. d. okiye, "boundedness and global stability for a predator-prey model with modified lesliegower and holling-type ii," applied math, vol. 16, pp. 1069-1075, 2003. [4] a. f. nindjin, m. aziz-alaoui and m. cadivel, "analysis of a predator-prey model with modified leslie-gower and holling-type ii schemes with time delay," nonlinear analysis: real world applications, vol. 7, pp. 1104-1118, 2006. [5] r. yafia and f. e. adnani, "limit cycle and numerical simuations for small and large delays in a predator-prey with modified leslie-gower and holling-type ii schemes," nonlinear analysis: real world applications, vol. 9, no. 5, pp. 2055-2067, 2008. [6] l. chen and l. chen, "permanence of a discrete periodic volterra model with mutual interference," discrete dynamics in nature and society, 2009. [7] x. song and y. li, "dynamic behaviors of the periodic predator-prey model with modified leslie-gower holling-type ii schemes and impulsive effect," nonlinear analysis: real world applications, vol. 9, no. 1, pp. 64-79, 2008. [8] y. zhu and k. wang, "existence and global attractivity of positive periodic solutions for a predator-prey model with modified leslie-gower holling-type ii schemes," journal of mathematical analysis and applications, vol. 384, no. 2, pp. 400-408, 2011. [9] h. f. huo, x. wang and c. c. chavez, "dynamics of a stage-structured lesliegower predator-prey odel," the mathematical and theoretical biology institute, 2011. pendahuluan tinjauan pustaka 1. sistem dinamik diskrit 2. sistem dinamik diskrit linear 3. model predatorprey leslie-gower metodologi penelitian hasil dan pembahasan 1. diskritisasi model 2. titik kesetimbangan model 3. analisis kestabilan model 4. simulasi numerik kesimpulan bibliography cauchy – issn: 2086-0382 55 aplikasi quasigroup dalam pembentukan kunci rahasia pada algoritma hibrida (rsa-quasigroup cipher) muhammad khudzaifah jurusan matematika, f.mipa, universitas brawijaya, malang, indonesia email : m_khudzaifah@yahoo.com abstrak pada artikel ini dibahas penerapan quasigrup di bidang kriptografi. didefinisikan suatu operasi quasigroup order 𝑝 − 1 sehingga bisa membentuk suatu algoritma kriptografi yang disebut sebagai quasigrup cipher, quasigrup cipher merupakan algoritma kriptografi simetris. algoritma kriptografi simetris memiliki sistem keamanan lemah karena kunci yang digunakan untuk proses enciphering sama dengan kunci yang digunakan untuk proses deciphering. sehingga pada artikel ini algoritma quasigroup cipher dimodifikasi dengan menggabungkannya dengan algoritma rsa menjadi suatu algoritma hibrida yang memiki dua tingkatan kunci. kata kunci : quasigrup, kriptografi, algoritma hibrida. abstract in this article discusses application of quasigroup in the field of cryptography, a quasigroup order 𝑝 − 1 operation is defined so that it can form a cryptographic algorithm called a cipher quasigrup, quasigrup cipher is a symmetric cryptographic algorithms, symmetric cryptographic algorithms have weak security systems which are used as the key for enciphering process same as the key used for deciphering process. so at this article quasigroup cipher algorithm is modified by combining the rsa algorithm into a hybrid algorithm that is thinking about the two key levels. keywords: quasigrup, cryptography, hybrid algorithm pendahuluan kriptografi memegang peranan penting seiring dengan perkembangan teknologi informasi dan komunikasi. masalah keamanan merupakan salah satu aspek penting dari sebuah sistem informasi. salah satu hal yang penting dalam komunikasi menggunakan komputer dan dalam jaringan komputer untuk menjamin keamanan pesan, data, ataupun informasi adalah enkripsi. disini enkripsi dapat diartikan sebagai kode atau cipher. di sisi lain, beberapa teori dalam aljabar abstrak khususnya teori mengenai quasigroup telah dikembangkan secara luas mulai dari bentuk integrasinya dengan disiplin ilmu lain hingga aplikasinya dalam berbagai bidang. banyak teori quasigroup yang telah diterapkan dalam steganografi, teori pengkodean, dan kriptografi. penggunaan quasigroup pada kriptografi adalah quasigroup cipher, yang merupakan kriptografi kunci simetri yang keamanannya kurang baik. hal ini dikarenakan proses dekripsinya dan enkripsinya menggunakan kunci yang sama. oleh karena itu akan dikombinasikan dengan algoritma kunci asimetris, yang proses enkripsi dan dekripsinya menggunakan kunci yang berbeda. pada tahun 2004 gligoroski mengembangkan quasigroup cipher dengan mendefinisikan quasigroup order p − 1, dan dikombinasikan dengan algoritma elgamal. dalam tesis ini dibahas penggunaan quasigroup pada kriptografi yang dikombinasikan dengan algoritma rsa. alasan pemilihan algoritma rsa ini adalah karena proses enkripsi dan dekripsi rsa lebih cepat daripada algoritma elgamal. teori dasar 1. mengkontruksi kunci rahasia dalam bentuk quasigroup cipher pada sub-bab ini akan diperkenalkan definisi dan teori quasigroup yang dikutip dari (markovski, et al., 1997) teorema 1.1 misalkan (𝐴,∗) adalah quasigroup yang menetapkan sebuah operasi biner \ pada a sedemikian sehingga untuk semua 𝑥, 𝑦 ∈ 𝐴 berlaku, 𝑥\𝑦 = 𝑧 ↔ 𝑥 ∗ 𝑧 = 𝑦, maka groupoid (𝐴,\) adalah quasigroup. mailto:m_khudzaifah@yahoo.com muhammad khudzaifah 56 volume 3 no. 2 mei 2014 definisi 1.2 operasi ∖ adalah dual dari ∗ sehingga quasigroup (𝐴,∖) adalah dual dari quasigroup (𝐴,∗).(𝐴,∗,∖) adalah quasigroup yang diperoleh dari hasil perluasan quasigroup (𝐴,∗). teorema 2.3 quasigroup (𝐴,∗,∖) memenuhi persamaan identitas : 𝑥 ∖ (𝑥 ∗ 𝑦) = 𝑦, 𝑥 ∗ (𝑥 ∖ 𝑦) = 𝑦 , untuk semua 𝑥, 𝑦 ∈ 𝐴. definisi 2.4 misalkan 𝑢𝑖 ∈ 𝐴 +, 𝑘 ∈ ℕ, 𝑘 ≥ 1, dan 𝑎1 ∈ 𝐴 maka, 𝑓∗(𝑢1𝑢2 … 𝑢𝑘 ) = 𝑣1𝑣2 … 𝑣𝑘 ⇔ 𝑣1 = 𝑎1 ∗ 𝑢1, 𝑣2 = 𝑣2 ∗ 𝑢2, 𝑣3 = 𝑣3 ∗ 𝑢3, ⋮ 𝑣𝑖+1 = 𝑣𝑖 ∗ 𝑢𝑖+1, 𝑖 = 1,2, … , 𝑘 − 1, 𝑓∖(𝑢1𝑢2 … 𝑢𝑘 ) = 𝑣1𝑣2 … 𝑣𝑘 ⇔ 𝑣1 = 𝑎1 ∖ 𝑢1, 𝑣2 = 𝑣1 ∖ 𝑢2, 𝑣3 = 𝑣2 ∖ 𝑢2, ⋮ 𝑣𝑖+1 = 𝑣𝑖 ∖ 𝑢𝑖+1, 𝑖 = 1,2, … , 𝑘 − 1. sixtuple (𝐴,∗,∖, 𝑎1, 𝑓∗, 𝑓∖) disebut quasigroup cipher atas alfabet 𝐴. 2. mengkontruksi kunci rahasia dalam bentuk quasigroup cipher menggunakan quasigroup atas order p-1 misalkan himpunan huruf alfabet adalah himpunan berhingga 𝑄 dan dinotasikan 𝑄+ adalah himpunan semua kata tak kosong atau string berhingga yang terdiri atas anggota dari q. anggota dari 𝑄+ akan dinotasikan sebagai 𝑎1𝑎2, … 𝑎𝑛 . 𝑎𝑖 ∈ 𝑄. misalkan ∗ adalah operasi pada quasigroup pada himpunan 𝑄 dan (𝑄,∗) adalah quasigroup, untuk 𝑎 ∈ 𝑄 didefinisikan dua fungsi 𝑒𝑎 , 𝑑𝑎 : 𝑄 + ⟶ 𝑄+ sebagai berikut misal 𝑎𝑖 ∈ 𝑄, 𝛼 = 𝑎1𝑎2 … 𝑎𝑛 . maka 𝑒𝑎 (𝛼) = 𝑏1𝑏2 … 𝑏𝑛 ⟺ 𝑏1 = 𝑎 ∗ 𝑎1, 𝑏2 = 𝑏1 ∗ 𝑎2, … , 𝑏𝑛 = 𝑏𝑛−1 ∗ 𝑎𝑛 . sehingga 𝑏𝑖+1 = 𝑏𝑖 ∗ 𝑎𝑖+1 untuk setiap 𝑖 = 0,1, … , 𝑛 − 1, dimana 𝑏0 = 𝑎, dan 𝑑𝑎 (𝛼) = 𝑐1𝑐2 … 𝑐𝑛 ⟺ 𝑐1 = 𝑎 ∗ 𝑎1, 𝑐2 = 𝑎1 ∗ 𝑎2, … , 𝑐𝑛 = 𝑎𝑛−1 ∗ 𝑎𝑛 . sehingga 𝑐𝑖+1 = 𝑎𝑖 ∗ 𝑎𝑖+1 untuk setiap 𝑖 = 0,1, … , 𝑛 − 1, dimana 𝑎0 = 𝑎. definisi dan teori quasigroup pada sub bab ini dikutip dari [1] definisi 2.1 fungsi 𝑒𝑎 dan 𝑑𝑎 disebut sebagai 𝑒 transformasi string dan 𝑑-transformasi string dari 𝑄+ berdasarkan operasi ∗ dengan leader 𝑎. definisi 2.2 jika dipilih sebanyak 𝑘 leaders 𝑎1, 𝑎2, … , 𝑎𝑘 ∈ 𝑄 maka didefinisikan pemetaan fungsi komposisi 𝐸𝑘 = 𝐸𝑎1…𝑎𝑘 = 𝑒𝑎1 ∘ 𝑒𝑎2 ∘ ∙∙∙ ∘ 𝑒𝑎𝑘 dan 𝐷𝑘 = 𝐷𝑎1…𝑎𝑘 = 𝑑𝑎1 ∘ 𝑑𝑎2 ∘ ∙∙∙ ∘ 𝑑𝑎𝑘 disebut sebagai 𝐸dan 𝐷quasigroup transformasi string pada 𝑄+. lemma 2.3 fungsi 𝐸𝑘 dan 𝐷𝑘 adalah permutasi pada 𝑄 +. lemma 2.4 pada quasigroup (𝑄,∗) dengan diberikan himpunan dari 𝑘 𝑙𝑒𝑎𝑑𝑒𝑟𝑠 {𝑎1, 𝑎2, … , 𝑎𝑘 } maka invers dari 𝐸𝑘 = 𝐸𝑎1…𝑎𝑘 = 𝑒𝑎1 ∘ 𝑒𝑎2 ∘ ∙∙∙ ∘ 𝑒𝑎𝑘 adalah 𝐸𝑘 −1 = 𝐷𝑘 = 𝐷𝑎𝑘…𝑎1 = 𝑑𝑎𝑘 ∘ ∙∙∙ ∘ 𝑑𝑎1 . definisi 2.5 quasigroup (𝑄,∗) dan k-tuple (𝑎1, 𝑎2, … , 𝑎𝑘 ) dari leader 𝑎𝑖 ∈ 𝑄, sistem ((𝑄,∗), (𝑎1, 𝑎2, … , 𝑎𝑘 ), 𝐸𝑎1…𝑎𝑘 , 𝐷𝑎𝑘…𝑎1 ) terdefinisi sebagai quasigroup stream cipher atas string di 𝑄+. lemma 2.6 untuk suatu 𝑝 bilangan prima dan bilangan 𝐾 yang memenuhi 1 ≤ 𝐾 ≤ 𝑝 − 2, fungsi 𝑓𝐾 (𝑗) = 1 1+(𝐾+𝑗)𝑚𝑜𝑑(𝑝−1) 𝑚𝑜𝑑 𝑝 adalah permutasi dari element di ℤ𝑝 ∗ lemma 2.7 operasi biner ∗ pada himpunan 𝑄 = {1,2, … , 𝑝 − 1} didefinisikan sebagai 𝑖 ∗ 𝑗 = 𝑖 × 𝑓𝐾 (𝑗)𝑚𝑜𝑑 𝑝 membentuk quasigroup (𝑄,∗). akibat 2.8 jika didefinisikan fungsi 𝑔(𝑖, 𝑗, 𝐾) = ((𝑖 × 𝑗−1𝑚𝑜𝑑 𝑝) − 1 − 𝐾) 𝑚𝑜𝑑 (𝑝 − 1) yang mengambil parameter 𝑖, 𝑗, 𝐾 dari himpunan 𝑄 = {1,2, … , 𝑝 − 1}, yang memetakan himpunan 𝐴 = {1,2, … , 𝑝 − 1}3 ke himpunan 𝐵 = {1,2, … , 𝑝 − 2} maka pembagi kiri (𝑄,\) dari quasigroup (𝑄,∗) aplikasi quasigroup dalam pembentukan kunci rahasia pada algoritma hibrida cauchy – issn: 2086-0382 57 yang didefinisikan pada lemma 3.2.7 didefinisikan sebagai 𝑖 \ 𝑗 = { 𝑔(𝑖, 𝑗, 𝐾), jika 𝑔(𝑖, 𝑗, 𝐾) ≠ 0 𝑝 − 1, jika 𝑔(𝑖, 𝑗, 𝐾) = 0 pembahasan 1. mengkontruksi algoritma hibrida (rsaquasigroup cipher) ilustrasi proses encipher dan decipher 1) a membangkitkan kunci publik dan kunci privat dengan algoritma rsa yang mana kunci publik akan dikirimkan ke b. 2) b mengenkripsi pesan dengan algoritma quasigroup cipher, dan mengenkripsi session key dari quasigroup cipher dengan kunci publik yang diberikan oleh a dengan menggunakan algoritma rsa. pesan dan key yang telah terenkripsi dikirim ke a. 3) a mendekripsi session key dari b dengan menggunakan kunci privat algoritma rsa, lalu mendekripsi pesan dari b dengan menggunakan session key yang sudah terdekripsi dengan algoritma quasigroup cipher. 2. algoritma hibrida (rsa-quasigroup cipher) algoritma enkripsi 1) pilih sebarang bilangan bulat 𝐾, 1 ≤ 𝐾 ≤ 𝑝 − 1 yang mana quasigroup (𝑄,∗) terdefinisi untuk elemen {1,2, … , 𝑝 − 1} dengan persamaan pada lemma 2.7, dengan p adalah sebarang bilangan prima yang dipilih 2) enkripsi 𝐾 dengan algoritma rsa 3) pilih 𝑘 ≥ 3 bilangan bulat acak 𝑎𝑖, 𝑖 = 1,2, … , 𝑘, 1 ≤ 𝑎𝑖 ≤ 𝑝 − 2 untuk menjadi leader untuk quasigroup cipher dan enkripsikan dengan algoritma rsa 4) ubah setiap karakter pada pesan 𝑚𝜇 menjadi bilangan bulat pada range {1,2, … , 𝑝 − 1}, dengan 𝜇 adalah indeks setiap karakter dari pesan 5) secara berulang hitung 𝑚𝜇 𝑖 = 𝑎𝑖 ∗ 𝑚𝜇 𝑖−1, dimana 𝑚𝜇 0 = 𝑚𝜇, 𝑖 = 1, … , 𝑘 dan ∗ adalah operasi quasigroup yang terdefinisi pada lemma 2.7 6) 𝑐𝜇 = 𝑚𝜇 𝑘 dan update nilai leader dengan 𝑎𝑖 = 𝑚𝜇 𝑖 , 𝑖 = 1, … , 𝑘 − 1 dan 𝑎𝑘 = 1 + (∑ 𝑚𝜇 𝑖𝑘 𝑖=1 𝑚𝑜𝑑 (𝑝 − 1)). 7) didapatkan session key terenkripsi dan pesan yang terenkripsi 𝑐𝜇 (ciphertext) algoritma dekripsi 1) dekripsi session key dengan algoritma rsa, maka didapatkan 𝐾 untuk membuat (𝑄,\) dan didapatkan sejumlah 𝑘 leader. 2) secara berulang hitung 𝑐𝜇 𝑘 = 𝑎𝑘 \𝑐𝜇, 𝑐𝜇 𝑖 = 𝑎𝑖 \𝑐𝜇 𝑖+1, 𝑖 = 𝑘 − 1, … , 1 dan \ adalah operasi 𝑞𝑢𝑎𝑠𝑖𝑔𝑟𝑜𝑢𝑝 yang terdefinisi pada akibat 2.8 3) 𝑚𝜇 = 𝑐𝜇 1 dan update nilai leader dengan 𝑎𝑖 = 𝑐𝜇 𝑖+1, 𝑖 = 1, … , 𝑘 − 1 dan 𝑎𝑘 = 1 + (𝑐𝜇 + ∑ 𝑐𝜇 𝑖𝑘 𝑖=2 𝑚𝑜𝑑 (𝑝 − 1)). 4) didapatkan plaintext 𝑚𝜇 kesimpulan algoritma kriptografi yang didasarkan dari quasigroup yaitu quasigroup cipher memiliki keamanan cukup baik, hal ini dibuktikan pada contoh ketika kriptanalis salah mendekripsi satu huruf saja maka pesan tidak bisa terbaca. kelemahan algoritma quasigroup adalah karena kuncinya adalah kunci simetris, sehingga bila kuncinya bocor kepada orang lain, maka pesan bisa dibaca orang lain. sehingga diperkuat dengan algoritma rsa yang memiliki kunci asimetris menjadi algoritma hibrida yang tingkat keamanannya lebih tinggi karena memiliki 2 tingkatan kunci. bibliography [1] d. gligoroski, "stream cipher based on quasigroup string transformation in zp*.," faculty of natural sciences institute of informatics, skopje, 2004. [2] d. ariyus, pengantar ilmu kriptografi, yogyakarta: andi press, 2008. [3] m. a. al-turky, "on the number and equivalent latin squares," journal of al-anbar university for pure science, pp. 71-75, 2007. [4] p. b. bhattacharya, s. k. jain and s. r. nagpaul, basic abstrack algebra, new york: cambridge university press, 1990. [5] j. bell, "an introduction to sdr’s and latin squares," more-head electonic journal of applicable mathematics, pp. 1-8, 2005. [6] koscielny, "generating quasigroups for cryptograpic applications," international muhammad khudzaifah 58 volume 3 no. 2 mei 2014 journal of applied mathematic and computer science, pp. 559-569, 2002. [7] e. ochodkova. and v. snasel, "using quasigroups for secure encoding of file sistem," in proceedings of the confrence for security and protection of information, 2001. [8] s. markovski, d. glikoroski and s. andonova., "using quasigroups for one-one secure encoding," in proceeding of vii-th confrence for logic and computing-lira’97, 1997. pendahuluan teori dasar 1. mengkontruksi kunci rahasia dalam bentuk quasigroup cipher 2. mengkontruksi kunci rahasia dalam bentuk quasigroup cipher menggunakan quasigroup atas order p-1 pembahasan 1. mengkontruksi algoritma hibrida (rsa-quasigroup cipher) 2. algoritma hibrida (rsa-quasigroup cipher) algoritma enkripsi algoritma dekripsi kesimpulan bibliography 5.nurul_henstock-kurzweil henstockkurzweil integral on [a,b] siti nurul afiyah dosen stmik stie asia malang e-mail: noeroel_afy@yahoo.com abstract the theory of the riemann integral was not fully satisfactory. many important functions do not have a riemann integral. so, henstock and kurzweil make the new theory of integral. from the background, the writer will be research about henstock-kurzweil integral and also theorems of henstockkurzweil integral. henstockkurzweil integral is generalized from riemann integral. in this case the writer uses research methods literature or literature study carried out by way explore, observe, examine and identify the existing knowledge in the literature. in this thesis explain about partition which used in henstock kurzweil integral, definition and some property of henstockkurzweil integral. and some properties of henstockkurzweil integral as follows: value of the henstockkurzweil integral is unique, linearity of the henstock-kurzweil integral, additivity of the henstock-kurzweil integral, cauchy criteria, nonnegativity of henstock-kurzweil integral and primitive function. keywords: riemann integral, � � ���� partition, henstock-kurzweil integral. introduction we have already mentioned the developments, during the 1630’s, by fermat and descrates leading to analytic geometry and the theory of the derivatives. however, the subject we know as calculus did not begin to take shape until the late 1660’s when issac newton (16421727) created his theory of fluxions and invented the method of inverse tangents to find areas under curves. the reversal of the process for finding tangent lines to find areas was also discovered in the 1680’s by leibniz (1646-1716), who was unaware of newton unpublished work and who arrived at the discovery by a very different route. leibniz introduced the terminology calculus differential and calculus integral, since finding tangents lines involved differences and finding areas involved summations. thus they had discovered that integration, being a process of summation, was inverse to the operation of differentiation. during a century and a half of development and refinement of techniques, calculus consisted of these paired operations and their applications, primarily to physical problems. in the 1850s, bernhard riemann (1826-1866) adopted a new and different viewpoint. he separated the concept of integration from its companion, differentiation, and examined the motivating summation and limit process of finding areas by itself. he broadened the scope by considering all functions on an interval for which this process of integration could be defined: the class of integrable functions. the fundamental theorem of calculus became a result that held only for a restricted set of integrable functions. the viewpoint of riemann led others to invent other integration theories, the most significant being lebesgue’s theory of integration. the theory of the riemann integral was not fully satisfactory. many important functions do not have a riemann integral even after we extend the class of integrable functions slightly by allowing "improper" riemann integrals. for example characteristic function. in 1957, the czech mathematician jaroslav kurzweil discovered a new definition of this integral elegantly similar in nature to riemann's original definition which he named the gauge integral; the theory was developed by ralph henstock. due to these two important mathematicians, it is now commonly known as henstock-kurzweil integral. the simplicity of kurzweil's definition made some educators advocate that this integral should replace the riemann integral in introductory calculus courses, but this idea has not gained traction. concerning the background of the study, the writer formulates the statement of the problems as follows: 1. how does the concept δ-fine partition of henstock-kurzweil integral? 2. how does the definition of henstockkurzweil integral? 3. how does the fundamental properties of henstock-kurzweil integral? review of the related literature 1. supremum and infimum we start with a straightforward definition similar to many others in this course. read the henstock-kurzweil integral on [a,b] jurnal cauchy – issn: 2086-0382 25 definitions carefully, and note the use of ≤ and ≥ here rather than < and > . definition 1 let s be a subset of ℜ 1. a number ℜ∈u is said to be an upper bound of s if us ≤ for all ss ∈ 2. a number ℜ∈w is said to be a lower bound of s if sw ≤ for all ss ∈ definition 2 let s be a subset of ℜ . 1. if s is bounded above, then an upper bound u is said to be supremum (or a least upper bound) of s if no number smaller than u is an upper bound of s. 2. if s is bounded below, then a lower bound w is said to be infimum (or a greatest lower bound) of s if no number greater than w is a lower bound of s. 2. limit of function the essence of the concept of limit for real valued functions of a real variable is this: if l is a real number, then ( ) lxf xx = → 0 lim means that the value f(x) can be made as close to l as we wish by taking x sufficiently close to x0. this is made precise in the following definition. figure 1. the limit of f at x0 is l definition 3 we say that f(x) approaches the limit l as x approaches 0x , and write ( ) lxf xx = → 0 lim if f is defined on some deleted neighborhood of 0x , and for every 0>ε , there is a 0>δ such that ( ) ε<− lxf if δ<−< 00 xx 3. compact sets definition 4 a subset k of r is said to be compact if every open cover of k has a finite sub cover. in other words, a set k is compact if, whenever it is contained in the union of a collection g { }αg= of open sets in r , then it is contained in the union of some finite number of sets in g. theorem 1 if k is a compact subset of r, then k is closed and bounded. proof: we shall first show that k is bounded. for each � � , let �: ���,��. since each � is open and since � � � ����� �, we see that the collection � �:� � � is an open cover of k. since k is compact, this collection has a finite sub cover , so there exists � � such that, � � � � � ��� � ���,�� therefore k is bounded, since it is contained in the bounded interval ���,��. we show that k is bounded, by showing that its complement � ���� is open. to do so, let � ���� be arbitrary and for each � � , we let � ! "# � �:|# � �| % 1 �' (. since � ) �, we have � � � � ~ �� . since k is compact, there exists � � such that � � �� ~ �� �� now it follows from this that � + ,� � 1 �' ,� 1 �' . /, so that ,� � 1 �' ,� 1 �' . � ���� . but since u was an arbitrary point in ����, we infer that ���� is open. 4. continuity definition 5 a) we say that f is continuous at 0x if f is defined on an open interval ( )ba, containing 0x and ( ).)(lim 0 0 xfxf xx = → b) we say that f is continuous from the left at 0x if f is defined on an open interval ( )0, xa and ( ).)( 00 xfxf =− c) we say that f is continuous from the right at 0x if f is defined on an open interval ( )bx ,0 and ( )00 )( xfxf =+ siti nurul afiyah 26 volume 2 no. 1 november 2011 theorem 2 a) a function f is continuous at 0x if and only if f is defined on an open interval ( )ba, containing 0x and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf , whenever δ<− 0xx b) a function f is continuous from the right at 0x if and only if f is defined on an open interval [ )bx ,0 and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf holds whenever δ+≤≤ 00 xxx c) a function f is continuous from the left at 0x if and only if f is defined on an open interval ( ]0, xa and for each 0>ε , there is a 0>δ such that ε<− )()( 0xfxf holds whenever 00 xxx ≤≤− δ definition 6 a function f is continuous on an open interval ( )ba, if it is continuous at every point in ( )ba, . if, in addition. ( ) ( )bfbf =− or ( ) ( )afaf =+ then f is continuous on ( ] [ )baorba ,, , respectively, if f continuous on ( )ba, and ( ) ( )bfbf =− or ( ) ( )afaf =+ both hold, then f is continuous on [ ]ba, . definition 7 a function f is piecewise continuous on [ ]ba, if a) ( ).0 +xf exist for all 0x in [ )ba, ; b) ( ).0 −xf exist for all 0x in ( ]ba, ; c) ( ) ( ) ( )000 xfxfxf =−=+ for all but finitely many points 0x in ( )ba, . if c) fails to hold at some 0x in ( )ba, , f has a jump discontinuity at 0x . also, f has a jump discontinuity at a if ( ) ( )afaf ≠=+ or at b if ( ) ( )bfbf ≠− . 5. uniform continuity definition 8 let ra ⊆ , let raf →: ,we say that f is uniformly continuous on a if for each 0>ε there is a ( ) 0>εδ such that if x, u a∈ are any number satisfying , then ( ) ( ) ε<− ufxf . theorem 3 if f is continuous on a closed interval [ ]ba, , then f is uniformly continuous on [ ]ba, . proof: suppose that 0>ε . since f is continuous on [ ]ba, , for each t in [ ]ba, there is a positive number tδ such that ( ) ( ) 2 ε <− tfxf if ttx δ2<− and [ ]bax ,∈ if ( )ttt tti δδ +−= , , the collection [ ]{ }batih t ,∈= is an open covering of [ ]ba, . since [ ]ba, is compact, the heine-borel theorem implies that there are finitely many points nttt ,...,, 21 an [ ]ba, such that nttt iii ,...,, 21 cover [ ]ba, . now define { } nttt δδδδ ,...,,min 21 = we will show that if δ<− 'xx and [ ]baxx ,', ∈ then ( ) ( ) ε<− 'xfxf from the triangle inequality. ( ) ( ) ( ) ( )( ) ( ) ( )( )'' xftftfxfxfxf rr −+−=− ( ) ( )( ) ( ) ( )( )'xftftfxf rr −+−≤ since nttt iii ,...,, 21 cover [ ]ba, , x must be in one of these intervals. suppose that rt ix ∈ ; that is, rtr tx δ<− with rtt = , ( ) ( ) . 2 ε <− rtfxf henstock-kurzweil integral on [a,b] jurnal cauchy – issn: 2086-0382 27 such that, ( ) ( ) .2''' rr ttrrr txxxtxxxtx δδγ ≤+<−+−≤−+−=− therefore with rtt = and x replaced by x’ implies that ( ) ( ) . 2 ' ε <− rtfxf this imply that ( ) ( ) ε<− 'xfxf . definition 9 (lipschitz functions) let ra ⊆ , let raf →: . if there exist a constant 0>k such that ( ) ( ) uxkufxf −≤− for all aux ∈, , then f is said to be a lipschitz functions on a theorem 4 if raf →: is a lipschitz functions, then f is uniform continuous on a. proof: if the a lipschitz conditions satisfied with constant k, then given 0>ε , we can take k εδ =: . if aux ∈, satisfy δ<− ux , then ( ) ( ) εε =⋅<− k kufxf therefore f is uniformly continuous on a. 6. upper and lower integral definition 10 if f is bounded on [ ]ba, and { }nxxxp ,...,, 10 is a partition of [ ]ba, , let ( )xfm jj xxx j ≤≤− = 1 sup and ( )xfm jj xxx j ≤≤− = 1 inf the upper sum of f over p is ( ) ( )∑ = −−= n j jjj xxmps 1 1 and the upper integral of f over [ ]ba, , denoted by ( )dxxfb a∫ −−−− is the infimum of all upper sums. the lower sum of f over p is ( ) ( )∑ = −−= n j jjj xxmps 1 1 and the lower integral of f over [ ]ba, , denoted by ( )dxxfb a∫ −−− is the supremum off all lower sums. 7. riemann integral riemann integral, defined in 1854, was the first of the modern theories of integration and enjoys many of the desirable properties of an integration theory. the groundwork for the riemann integral of a function f over the interval [ ]ba, begins with dividing the interval into smaller subintervals. with infimum and suprimum taken include all partitions p on [ ]ba, , if the upper integral and lower integral same, then f can be said integrable on [ ]ba, . and called riemann function f on [ ]ba, and denoted by [ ]baf ,∈ this same value is called the riemann integral function f on [ ]baf ,∈ and written ( ) ( )dxxfr b a ∫ definition 11 let [ ] ℜ⊂ba, . a partition of [ ]ba, is a finite set of numbers { }nxxxp ,...,, 10= such that bxax n == ,0 and ii xx <−1 for ni ,...,2,1= . for each subinterval [ ]ii xx ,1− , define its length to be [ ]( ) 11 , −− −= iiii xxxxℓ . the mesh of the partition is then the length of the largest subinterval, [ ]ii xx ,1− : ( ) { }nixxp ii ,...,2,1:max 1 =−= −µ thus the point { }nxxx ,...,, 10 form an increasing sequence of numbers in [ ]ba, that divides the interval [ ]ba, into contiguous subintervals. let [ ] ℜ→baf ,: , { }nxxxp ,...,, 10= be a partition of [ ]ba, , and [ ]iii xxt ,1−∈ for each i. riemann began by considering the approximating (riemann) sums { }( ) ( )( ),,, 1 1 1 − = = −= ∑ ii n i i n ii xxtftpfs siti nurul afiyah 28 volume 2 no. 1 november 2011 defined with respect to the partition p and the set of sampling points { }n ii t 1= . riemann considered the integral of f over [ ]ba, to be a “limit” of the sums { }( ),,, 1 n ii tpfs = in the following sense. definition 12 a function [ ] ℜ→baf ,: is riemann integrable over [ ]ba, if there is an ℜ∈a such that for all 0>ε there is a 0>δ so that if p is any partition of [ ]ba, with ( ) δµ <p and [ ]iii xxt ,1−∈ for all i then { }( ) ε<−= atpfs nii 1,, we write ( )∫∫ == b a b a dttffa or, if we set [ ] ∫= i fbai .,, this definition defines the integral as a limit of sums as the mesh of the partition approaches 0. discussion 1. concept δδδδ-fine partition of henstockkurzweil integral let [ ]ba, be a compact interval in ℜ . let d be a finite collection of interval-point pairs [ ]( ){ }n iiii vu 1 ,, =ξ , where [ ]( ){ } n iii vu 1 , = are nonoverlapping subintervals of [ ]ba, . let ( )ξδ be a positive function on [ ]ba, , ( ) [ ] +ℜ→baei ,:.. ξδ . we say [ ]( ){ }n iiii vud 1 ,, == ξ is δ-fine henstockkurzweil partition of [ ]ba, if [ ] ( )( ) ( ) ( )( )iiiiiiiii bvu ξδξξδξξδξξ +−=⊂∈ ,,, for all ni ,...,3,2,1= given an δ-fine henstock-kurzweil partition [ ]( ){ }n iiii vud 1 ,, == ξ we write ( ) ( )( )ii n i i uvfdfs −= ∑ =1 , ξ for integral sum over d , whenever [ ] ℜ→baf ,: example 1 let ( ) xxf = consider a division bxxxa n =<<<= ...10 and { }nξξξ ,...,, 21 . and this time choose the points ( )1 2 1 −+= iii xxξ , clearly [ ]iii xx ,1−∈ξ for ni ,,2,1 ⋅⋅⋅= then, ( ) ( )( ) ( )( ) ( ) ( ) ( )22 0 2 1 2 1 2 1 1 11 1 2 1 2 1 2 1 2 1 , ab xx xx xxxxxxfdfs n n n i ii ii n i iiii n i i −= −= −= −+=−= ∑ ∑∑ = − − = −− = ξ then, ( ) ( )( ) ( )221 1 2 1 , abxxfdfs ii n i i −=−= − = ∑ ξ 2. definition of henstock-kurzweil integral definition 13 a function [ ] ℜ→baf ,: is said to be henstock-kurzweil integrable on [ ]ba, if there exists a real number a such that for every 0>ε there exists [ ] +ℜ→ba,:δ such that for every δ-fine henstock-kurzweil partition [ ]( ){ }n iiii vud 1 ,, == ξ of [ ]ba, , we have ( )( ) εξ <−−∑ = auvf ii n i i 1 we denote the henstock-kurzweilintegral (also write as hk-integral) a by ( ) ( )dxxfhk b a∫ . example 2 define [ ] ℜ→1,0:f the dirichlet’s function (= the characteristic function of the rational numbers in [ ]1,0 ), by ( )    ∉ ∈ = qxif qxif xf 0 1 then ( )xf is henstock-kurzweil integrable on [ ]1,0 . and ( ) 0 1 0 =∫ xf to prove this assertion, we will define an appropriate gauge εδ , first we enumerate these rational numbers as ,..., 21 rr . we define henstock-kurzweil integral on [a,b] jurnal cauchy – issn: 2086-0382 29 ( ) ,...2,12 1 == −− iforr ii εδ , and if [ ]1,0∈x is irrational we define ( ) 1=ξδ ; clearly εδ is a gauge on [ ]1,0 . if p is a δ-fine tagged partition, there can be at most two subintervals in p that have the number ir as tag, and the length of each of those subintervals is 12 −−≤ iε . hence the contribution to ( )pfs , from subintervals with tag ir is i−≤ 2ε . since the terms in ( )pfs , with tags at irrational points contribute 0, we readily see that ( ) εε =<≤ ∑ ∞ =1 2 ,0 i ipfs since 0>ε is arbitrary, this shows that ( )xf is henstock-kurzweil integrable on [ ]1,0 . and ( ) 0 1 0 =∫ xf 3. fundamental properties of henstockkurzweil integral theorem 5 (unique property) if f is henstock-kurzweil integrable over [ ]ba, , then the value of the integral is unique. proof: suppose that f is henstock-kurzweil integrable on [ ]ba, and both real number a and b satisfy definition 3.2.1. fix 0>ε choose aδ and bδ corresponding to a and b, respectively, in the definition with 2 ' ε ε = . let ( )ba δδδ ,min= and suppose for every δ-fine henstock-kurzweil partition [ ]( ){ }n iiii vud 1 ,, == ξ of [ ]ba, , then ( )( ) ( )( ) ( )( ) ( )( ) εεε ξξ ξξ =+< −−+−−≤       −−−      −−=− ∑∑ ∑∑ == == '' 11 11 buvfuvfa auvfbuvfba n i iii n i iii n i iii n i iii since ε was arbitrary, it follows that ba = . thus, the value of the integral is unique. theorem 6 (linearity of the henstockkurzweil integral) if f and g are henstock-kurzweil integrable on [ ]ba, , then so are gf + and fα where α is real. furthermore, ( ) ∫ ∫∫ +=+ b a b a b a gfgf and ( ) ∫∫ = b a b a ff αα proof: let a and b denote respectively the integrals of f and g on [ ]ba, . given 0>ε , there is a ( ) 01 >ξδ such that for any δ1-fine division [ ]( )ξ;, vud = we have ( )( ) 2 ε ξ <−−∑ auvf similarly, there is a ( ) 02 >ξδ such that for any δ2fine division [ ]( )ξ;, vud = we have ( )( ) 2 ε ξ <−−∑ buvg now put ( ) ( ) ( )( )ξδξδξδ 21 ,min= . note that any δ-fine division is also δ1-fine and δ2-fine. therefore for any δ-fine division [ ]( )ξ;, vud = we have ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) f g v u a b f v u a g v u b ξ ξ ξ ξ ε + − − + ≤ − − + − − < ∑ ∑ ∑ the proof is complete. example 3 valuate ∫ + 1 0 gf where ( ) 2xxf = and ( ) xxg = . solution: by theorem 3.3.3, dxxdxxdxxxgf ∫∫∫∫ +=+=+ 1 0 1 0 2 1 0 2 1 0 now, we evaluate value dxx∫ 1 0 2 consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . and this time choose the intermediate points ( ) 2 1 2 11 2 3 1       ++= −− iiiii xxxxξ , then ( ) ( ) ( ) iiiiiiii xxxxxxxx =<      ++<=≤ −−−− 2 1 2 2 1 2 11 22 1 2 11 3 1 0 for ni ,,2,1 ⋅⋅⋅= ; that is ( )iii xx ,1−∈ξ for each i . so 3 1 1 0 2 =∫ dxx and then , we evaluate value dxx∫ 1 0 consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . and this time choose the points siti nurul afiyah 30 volume 2 no. 1 november 2011 ( )1 2 1 −+= iii xxξ , clearly [ ]iii xx ,1−∈ξ for ni ,,2,1 ⋅⋅⋅= now ( ) ( )( ) ( )( ) ( ) ( ) 2 1 1 2 1 1 2 1 2 1 2 1 2 1 , 2 0 2 1 2 1 2 1 1 11 1 =⋅= ⋅= −= −= −+=−= ∑ ∑∑ = − − = −− = n n n i ii ii n i iiii n i i xx xx xxxxxxfdfs ξ so, 2 1 1 0 =∫ dxx that, 6 5 2 1 3 1 1 0 1 0 2 1 0 2 1 0 =+=+=+=+ ∫∫∫∫ dxxdxxdxxxgf theorem 7 (additivity of the henstockkurzweil integral) let bca << . if f is henstock-kurzweil integrable on [ ]ca, and on [ ]bc, and ∫∫∫ += b c c a b a fff proof: let a denote the integral of f on [ ]ca, and b that of f on [ ]bc, . given 0>ε , there is a ( ) 01 >ξδ , defined on [ ]ca, , such that for any δ1-fine division [ ]( )ξ;, vud = of [ ]ca, we have ( )( ) 2 ε ξ <−−∑ auvf similarly, there is a ( ) 02 >ξδ defined on [ ]bc, such that for any δ2-fine division 0 �1�,23;5� of [ ]bc, we have ( )( ) 2 ε ξ <−−∑ buvf define ( ) ( )( )ξξδξδ −= c,min 1 when [ )ca,∈ξ , ( )( )c−ξξδ ,min 2 when ( ]bc,∈ξ , and ( ) ( )( )cc 21 ,min δδ when c−ξ . note for any δ-fine division d of 16,73, c is always a division point of d. therefore for any δ-fine division 0 �1�,23;5� of 16,73 with σ over d, writing 21 σ+σ=σ where 1 σ is the partial sum over [ ]ca, and 2 σ over [ ]bc, we have ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 f v u a b f v u a f v u b ξ ξ ξ ε − − + ≤ − − + − − < ∑ ∑ ∑ hence f is henstock-kurzweil integrable to a+b on [ ]ba, . alternatively, let [ ]ca,χ denote the characteristic function of [ ]ca, and [ ]caff ,1 χ= . similarly, let [ ]bcff ,2 χ= . then it follows from theorem 3.3.1 that ( ) ∫ ∫∫∫ +=+= c a b c b a b a fffff 21 example 7 let ( ) xxf = and let 1 2 1 0 << .and, if f is henstock-kurzweil integrable on [ ]      = 2 1 ,0, ca and on [ ]      = 1, 2 1 , bc and ∫∫∫ += b c c a b a fff solution: consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 . and this time choose the points ( )1 2 1 −+= iii xxξ , clearly [ ]iii xx ,1−∈ξ for ni ,,2,1 ⋅⋅⋅= now ( ) ( )( ) ( )( ) ( ) ( ) ( )22 2 0 2 1 2 1 2 1 1 11 1 2 1 2 1 2 1 2 1 , ab xx xx xxxxxxfdfs n n i ii ii n i iiii n i i −= −= −= −+=−= ∑ ∑∑ = − − = −− = ξ with same procedure we get; on [ ]      = 2 1 ,0, ca henstock-kurzweil integral on [a,b] jurnal cauchy – issn: 2086-0382 31 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 2 1 1 2 2 0 2 2 1 , 2 1 2 1 2 1 2 n n i i i i i i i i i n i i i n s f d f x x x x x x x x x x c a ξ − − − = = − = = − = + − = − = − = − ∑ ∑ ∑ so 0>ε , there is a ( ) 01 >ξδ , defined on 10, � : 3, such that for any δ1-fine division 0 �1�,23;5� of 10, �: 3 we have ( ) 28 1 2 1 22 ε<−− ac and on 1;,73 1�: ,13, we get ( ) ( )( ) ( )( ) ( ) ( ) ( )22 2 0 2 1 2 1 2 1 1 11 1 2 1 2 1 2 1 2 1 , cb xx xx xxxxxxfdfs n n i ii ii n i iiii n i i −= −= −= −+=−= ∑ ∑∑ = − − = −− = ξ 0>ε , there is a ( ) 02 >ξδ , defined on 1�: ,13, such that for any δ1-fine division 0 �1�,23;5� of 1�: ,13 we have ( ) 28 3 2 1 22 ε<−− cb therefore for any δ-fine division 0 �1�,23;5� of 10,13 with σ over d, writing 21 σ+σ=σ where 1σ is the partial sum over [ ]ca, and 2 σ over 1�: ,13 we have ( ) ( ) ( ) ( ) ( ) 2 2 1 2 1 1 3 2 8 8 1 3 8 8 b a f v u f v uξ ξ ε  − − + ≤    − − + − − <∑ ∑ hence f is henstock-kurzweil integrable to � : on 10,13 lemma 8 (cauchy criteria) a function is henstock-kurzweil integrable on [ ]ba, if and only if for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = and [ ]( )';','' ξvud = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf where the first sum is over d and the second over d’. proof ( )⇒ we will prove that if a function is henstockkurzweil integrable on [ ]ba, then for every 0>ε , there is a ( ) 0>ξδ such that for any δfine division [ ]( )ξ;, vud = and [ ]( )';','' ξvud = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf a function is henstock-kurzweil integrable on [ ]ba, then for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = and [ ]( )';','' ξvud = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' ' ' ' ' 'f v u f v u f v u a a f v u f v u a a f v u ξ ξ ξ ξ ξ ξ ε − − − = − − + − − ≤ − − + − − < ∑ ∑ ∑ ∑ ∑ ∑ analogous to the situation for real-valued sequences, the condition that ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf ( )⇐ we have already proved that the integrability of f implies the cauchy criterion. so, assume the cauchy criterion holds. we will prove that f is henstock-kurzweil integrable . if for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = and [ ]( )';','' ξvud = we have ( )( ) ( )( ) εξξ <−−− ∑∑ ''' uvfuvf then a function is henstock-kurzweil integrable on [a,b]. for each ν∈k , choose a 0>kδ so that for any two division [ ]( )ξ;, vud = and [ ]( )';','' ξvud = , and corresponding sampling points, we have ( )( ) ( )( ) k uvfuvf 1 ''' <−−− ∑∑ ξξ replacing k δ by { }kδδδ ,...,,min 21 , we may assume that 1+ ≥ kk δδ . next for each k, fix a partition [ ]( )kkkk vud ξ;,= and set of sampling point { }n ii 1=ξ . note for kj > thus ( )( ) ( )( ) { }kj uvfuvf jjjkkk ,min 1 <−−− ∑∑ ξξ , siti nurul afiyah 32 volume 2 no. 1 november 2011 which implies that sequence ( )( )∑ ∞ = − 1k kkk uvf ξ is a cauchy sequence in r, and hence converges. let a be a limit of this sequence. it follows from the previous inequality that ( )( ) k auvf 1 <−−∑ ξ it remains to show that a satisfies definition 3.2.1 fix 0>ε and let division [ ]( )ξ;, vud = . then ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ε ξξξ ξξξ ξ =+< −−+−−−≤ −−+−−−= −− ∑∑∑ ∑∑∑ ∑ === === kk auvfuvfuvf auvfuvfuvf auvf n k kkk n k kkk n i iii n k kkk n k kkk n i iii 11 111 111 it now follows that f is henstock-kurzweil integrable on [ ]ba, theorem 9 if f is henstock-kurzweil integrable on [ ]ba, , then so it is on a subinterval [ ]dc, of [ ]ba, . proof : since f is henstock-kurzweil integrable on [ ]ba, , the cauchy condition holds. take any two δ-fine divisions of [ ]dc, . say d1 and d2, and denote by s1 and s2 respectively the riemann sums of f over d1 and d2. similarly, take another δ-fine division d3 of [ ] [ ]bdca ,, ∪ and denote by s3 the corresponding riemann sums. then the union 31 dd ∪ forms a δ-fine division of [ ]ba, . here the division points and associated points of 31 dd ∪ are the union of those from d1 and d3. the riemann sum of f over 31 dd ∪ is 31 ss + . and similarly that over 32 dd ∪ is 32 ss + .therefore by the cauchy condition we have ( ) ( ) ε<+−+≤− 323121 ssssss hence the result follows from lemma 3.3.7 with [ ]ba, replaced by [ ]dc, theorem 10 (nonnegativity of the henstockkurzweil integral) if f and g are henstock-kurzweil integrable on [ ]ba, and if ( ) ( )xgxf ≤ for almost all in x in [ ]ba, , then ∫∫ ≤ b a b a gf proof: in view of theorem 3.3.9 we may assume that ( ) ( )xgxf ≤ for all x. given 0>ε , as in the proof of theorem 3.2.1, there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = we have ( )( ) εξ <−− ∫∑ b a fuvf , ( )( ) εξ <−− ∫∑ b a guvg it follows that ( )( ) ( )( ) εξξε +<−<−<− ∫∑∑∫ b a b a guvguvff since ε is arbitrary, we have the required inequality. theorem 11 if f is henstock-kurzweil integrable on [ ]ba, with the primitive f, then for every 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = we have ( ) ( ) ( )( ) εξ <−−−∑ uvfufvf we shall make a few remarks. before proof, from the computational point of view, we may regard ( )( )uvf −ξ as an approximation of ( ) ( )ufvf − . then the difference ( ) ( ) ( )( )uvfufvf −−− ξ is an error. the definition of the henstock-kurzweil integral says that the absolute error is also small, whereas henstock’lemma. in fact, the two are equivalent by theorem 3.3.8. another way of putting it is that taking any partial sum 1σ of σ we still have ( ) ( ) ( ) ( )1f v f u f v uξ ες − − − < that is to say, the selected error is again small, and indeed it is equivalent to the above two. proof: given 0>ε , there is a ( ) 0>ξδ such that for any δ-fine division [ ]( )ξ;, vud = we have ( ) ( ) ( )( ) 4 εξ <−−−σ uvfufvf let 1σ be a partial sum of σ and e1 the union of [ ]vu, from 1σ . suppose e2. thus we can choose a δ-fine division [ ]( )ξ;,2 vud = of e2 such that henstock-kurzweil integral on [a,b] jurnal cauchy – issn: 2086-0382 33 ( ) ( ) ( )( ) 42 εξ <−−−σ uvfufvf where 2σ is over d2. now writing 213 σ+σ=σ we have ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 1 3 2 2 f v f u f v u f v f u f v u f v f u f v u ξ εξ ξ σ − − − ≤ σ − − − +σ − − − < consequently the result follows. example 8 let ( ) x xf 1= for 10 ≤< x . given 0>ε , we shall construct ( )ξδ so that f is henstock kurzweil integrable on [ ]1,0 . consider a division 1...0 10 =<<<= nxxx and { }nξξξ ,...,, 21 with 01 =ξ and iii xx ≤≤− ξ1 for ni ,...,2= . note that the primitive of x 1 is x2 . then we can write ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 1 . n i i i i n i i i x i n i i i i i f x x x dx x x x x x x x x ξ ξ − = − = − − = − − ≤ − − + − − ≤ + − − ∑ ∑∫ ∑ we shall prove that above is less than ε for suitable fine−δ divisions. suppose ( ) ξξδ c− for 10 ≤< ξ and 210 << c so that 01 =ξ always. if the above division is fine−δ and [ ]vu, is a typical interval [ ]ii xx ,1− in the division with 0≠u and vu ≤≤ ξ , then ( ) cvuv 220 ≤<−< ξδ , re-arranging we get ( )cuv 211 −≤ , and finally ( ) ( ).2122 ccucvuvuv −≤<− now choose c so that 210 << c and ( ) 2212 ε≤− cc . in addition, put ( ) 160 2εδ ≤ . then for the given fine−δ division the above inequality is less than ( ) ( ) εδ <− − + ∑ = − n i ii xx c c 2 1 21 2 02 for example, when 10 ≤< ε we may choose 6 ε =c . hence the function is henstock-kurzweil integrable on [ ]1,0 . conclusion from the discussion we get conclusion that: 1. δ-fine henstock-kurzweil partition [ ]( ){ }n iiii vud 1 ,, == ξ we write ( ) ( )( )ii n i i uvfdfs −= ∑ =1 , ξ where d be a finite collection of interval-point pairs [ ]( ){ }n iiii vu 1 ,, =ξ , where [ ]( ){ } n iii vu 1 , = are non-overlapping subintervals of [ ]ba, . let ( )ξδ be a positive function on [ ]ba, , ( ) [ ] +ℜ→baei ,:.. ξδ . and if [ ] ( )( ) ( ) ( )( )iiiiiiiii bvu ξδξξδξξδξξ +−=⊂∈ ,,, for all ni ,...,3,2,1= . 2. a function [ ] ℜ→baf ,: is said to be henstock-kurzweil integrable on [ ]ba, if there exists a real number fs such that for every 0>ε there exists [ ] +ℜ→ba,:δ such that for every δ-fine henstock-kurzweil partition [ ]( ){ }n iiii vud 1 ,, == ξ of [ ]ba, , we have ( ) ., ε<− fsdfs 3. and the fundamental properties of henstock kurzweil integral as follows: value of the henstockkurzweil integral is unique, linearity of the henstock-kurzweil integral, additivity of the henstock-kurzweil integral, cauchy criteria, nonnegativity of henstockkurzweil integral, and primitive function. bibliography [1] ding and g.j. ye., (2009). generalized gronwall-bellman inequalities using the henstock-kurzweil integral, southeast asian bulletin of mathematic(2009)33: 103-713. [2] douglas s kurtz, charles w swartz. (2004). theories of integration; the integrals of riemann, lebesgue, henstock-kurzweil and mc. shane. london: world scientific publishing co.pc.ltd. [3] g. bartle, robert. and donald r. sherbert. (1994). introduction to real analysis, 2nd edition. new york: wiley. [4] gordon, russel a. (1955). the integrals of lebesgue, denjoy, perron, and henstock, lyrary of congress cataloging in publication data. american mathematical society. siti nurul afiyah 34 volume 2 no. 1 november 2011 [5] hutahean, effendi. (1989). fungsi riil. itb: bandung [6] lee, p.-y. (1989). lanzhou lectures on henstock . singapore: world scientific. [7] mark bridger, real analysis, a constructive approach, northeastern university. department of mathematics boston, m.a, a john willey and sons, inc., publication [8] przynski, william and philip. (1987). introduction to mathematical analysis. mcgraww hill international. [9] slavik, antonin and stefan schwabik, praha. (2006). henstock-kurzweil and mcshane product integration; descriptive definitions. october 23, 2006 [10] rahman, hairur. (2008). pengantar analisis real. malang: uin malang. pemodelan pertumbuhan tanaman zinnia menggunakan lindenmayer system dengan mathematica suhartono jurusan teknik informatika uin maulana malik ibrahim malang email : galipek@gmail.com abstrak pendekatan dalam mempelajari pemodelan pertumbuhan tanaman saat ini adalah dengan menggunakan metoda l-system yaitu sistem penulisan berulang (rewriting system) yang dilakukan secara paralel dengan menggunakan aturan gramatikal. dengan menggunakan software mathematica telah diidentifikasi pemodelan pertumbuhan tanaman zinnia sebanyak 6 tahap pertumbuhan selama 25 hari dan dapat divisualisasikan. kata kunci : identifikasi, pemodelan pertumbuhan tanaman zinnia , l-system abstract the approaches in the study of plant modelling growth at this time is using the method of l-system, that is a system of repetitive writing repetitive performed in parallel by using grammatical rules. by using mathematica this modeling has been identified as 6 zinnia plant growth during the growth phase of 25 days and can be visualized. keywords: identification, modeling plant growth zinnia, l-system pendahuluan pendekatan tentang arsitektur tanaman telah dapat dikategorikan sebagai model tanaman. arsitektur tanaman dapat direpresentasikan dalam bentuk grafika 3d. aristid lindenmayer menjelaskan teori pertumbuhan sel anabaena catenula dengan menggunakan sistem penulisan berulang (rewriting system) dan dilakukan secara paralel yang dinyatakan dengan menggunakan aturan gramatikal yang disebut sebagai metoda lsystem (prusinkiewics, dkk, 1990). kemudian metoda l-system banyak diterapkan untuk pemodelan berbagai jenis tanaman (prusinkiewics, dkk, 2003; pachepsky, dkk, 2004; hirafuji, 1991; somporn, 2004; suhartono, dkk, 2011). pengembangan metoda l-system untuk menggambarkan pertumbuhan tanaman berdasarkan karakteristik lingkungan menjadi pemodelan tanaman secara realistik ( mech dkk, 1996; chuai-aree dkk, 2000; suyanto, dkk, 2010, suhartono, dkk, 2011, suhartono, dkk, 2013). hasil penelitian yang dihasilkan adalah identifikasi pemodelan pertumbuhan tanaman zinnia dengan metoda l-system yang mendekati tanaman secara real. pemodelan pertumbuhan tanaman yang dihasilkan dapat mengidentifikasi 6 tahap proses pertumbuhan tanaman zinnia. data percobaan penelitian dilakukan di laboratorium jaringan komputer jurusan teknik informatika uin maliki malang. data pertumbuhan tanaman dipakai tanaman kembang kertas (zinnia elegane jacq), penanaman dilakukan dengan polibag sejumlah 3 tanaman zinnia sebagai data untuk digunakan menyusun pemodelan pertumbuhan tanaman. sedangkan faktor-faktor lingkungan yang meliputi suhu dan kelembaban lingkungan, intensitas cahaya, pemupukan dan faktor lingkungan dalam penelitian ini lebih bersifat sebagai data pendukung terhadap pengamatan pertumbuhan tanaman zinnia. pemupukan dilakukan dengan cara mencampur pupuk dengan air sebanyak 5 liter untuk setiap kelompok tanaman. pengukuran dilakukan pengamatan hingga hari yang ke 25 dengan interval pengamatan selama 5 hari terhadap indikator pertumbuhan struktur model tanaman meliputi panjang batang, diameter batang, tinggi daun, lebar daun, diameter bunga, dan tinggi tanaman. pemodelan pertumbuhan tanaman perkembangan penelitian pemodelan pertumbuhan tanaman dimulai saat aristid lindenmayer memperkenalkan teori suhartono 34 volume 3 no. 1 november 2013 pertumbuhan cell anabaena catenula dengan menggunakan rewriting string telah diidentifikasi sebagai metoda lindenmayer system (l-system) ( prusinkiewics, dkk, 1990). kemudian diperbaiki dan dirancang menjadi pemodelan pertumbuhan secara realistik pada tanaman tinggi (prusinkiewics, dkk, 2003). penerapan metoda lsystem telah ditingkatkan sebagai alat pemodelan pada berbagai jenis tanaman (pachepsky, dkk, 2004). metoda l-system yang disebut teori rewriting string adalah pola aturan (grammar) yang dijumpai pada tanaman yang bersifat sebagai pola yang memiliki kesamaan dan berulang. sintak terhadap pertumbuhan tanaman dalam metoda l-system akan dinotasikan dan disusun dalam kaidah aturan (rules) pertumbuhan tanaman. pada kondisi sesungguhnya tanaman yang mengalami pertumbuhan akan diawali dari bibit, kemudian pertumbuhan tanaman dilanjutkan dengan tumbuhnya tunas muda yang dinotasikan sebagai axiom pada metoda l-system. aturan reproduksi tanaman diberikan pada tahap tunas muda yang tumbuh dan berkembang, kemudian disusul oleh batang yang kemudian diikuti oleh daun muda, kesemuanya berkembang secara paralel, terakhir adalah bunga. pengembangan pemodelan pertumbuhan tanaman untuk mendapatkan variasi model pertumbuhan yang sesuai dengan keinginan dengan memodifikasi sintak pada pertumbuhan tanaman pada metoda l-system dilakukan dengan metoda genetic l-system programming. piranti lunak untuk memodelkan pertumbuhan tanaman, pertama piranti lunak lstudio (karwowski.r dkk, 2006) merupakan perangkat lunak pemodelan pertumbuhan tanaman yang dikembangkan di universitas calgary tahun 1999 bersifat komersial. bahasa yang dikembangkan adalah bahasa c++ yang ditambahkan format l-system dengan notasi format bahasa l+c. piranti lunak plant vr (somporn.c.a dkk, 2004) merupakan piranti lunak pertumbuhan tanaman untuk parametric lsystem sebagai contoh untuk tanaman kedelai, bahasa pemrograman yang dikembangkan adalah bahasa pascal. piranti lunak mathematica dengan aplikasi klsystems dan evolvica (jacob.c, 1995) merupakan piranti lunak pemodelan pertumbuhan tanaman dan evolusi pemodelan pertumbuhan tanaman sebagai contoh untuk tanaman bunga lychnis coronaria. metoda l-system metoda lindenmayer system (l-system) adalah aturan formal yang disusun sebagai gramatika dalam bentuk axioma, dimana simbolsimbol yang digunakan merepresentasikan pertumbuhan tanaman, terjadi pergantian simbol secara paralel dan simultan pada masing-masing tahap. disini perbedaan penting gramatika chomsky dan l-system terletak pada hal produksi. di gramatika chomsky produksi dipakai sebagai urutan (sequentiallly) sedangkan pada gramatika l-system produksi dipakai sebagai paralel dan simultan untuk mengganti komponen. ini akibat dari refleksi motivasi biologi, dimana produksi adalah pertumbuhan, deferensiasi sel dan morfogenesis. rewriting system konsep utama dari lindenmayer system adalah penulisan berulang. penulisan berulang adalah teknik untuk mendifinisikan objek secara kompleks dengan cara mengganti bagian dari objek dengan cara rewriting rule atau production (prusinkiewics, dkk, 1990). contoh dari objek grafika yang didefinisikan secara aturan rewriting rule adalah snowflake curve, pada tahun 1905 oleh von koch (prusinkiewics, dkk, 1990). proses dari rewriting rule terdapat dua bagian pembentukan yaitu initiator dan generator. dimana menerapkan generator pada initiator, kemudian menerapkan generator pada hasil yang terakhir, dan seterusnya. dimana menerapkan generator pada initiator, kemudian menerapkan generator pada hasil yang terakhir, dan seterusnya. jika digunakan sebagai initiator dan sebagai generator setelah satu iterasi didapat , dimana mengganti setiap baris dengan generator, dapat di lihat pada gambar 1. gambar 1. konstruksi dari kurva snowflake (prusinkiewics.p, dkk, 1990) struktur pemodelan pertumbuhan tanaman zinnia data biologi tanaman didapat dari pengamatan pertumbuhan tanaman zinnia untuk membangun simulasi pertumbuhan tanaman pemodelan pertumbuhan tanaman zinnia menggunakan lindenmayer system dengan mathematica jurnal cauchy – issn: 2086-0382 35 zinnia. diagram alir untuk model tanaman zinnia dan visualisasi pertumbuhan tanaman zinnia dapat dilihat pada gambar 2. gambar 2. diagram dari model pertumbuhan dan visualisasi tanaman zinnia gambar 2 menjelaskan tentang prototipe dari model pertumbuhan dan visualisasi tanaman zinnia menggunakan metoda l-system dan menggabungkan model matematika dengan data bilogi dari struktur arsitektur tanaman zinnia. metoda l-system digunakan untuk model kualitatif yang dibangun untuk merepresentasikan bangunan dan topologi bagi tanaman zinnia. diagram ini memiliki enam blok, (1) mendifinisikan model kualitatif yang dibangun dari pengamatan pertumbuhan tanaman zinnia dalam siklus hidup tanaman zinnia, (2) mengukur karakteristik tanaman zinnia yang dikumpulkan dari data tanaman dilapangan, (3) mengubah data mentah menjadi fungsi pertumbuhan tanaman berdasarkan perkiraan fungsi pada setiap struktur tanaman zinnia, (4) mendefinisikan model kuantitatif yang terdiri dari model kualitatif dan fungsi pertumbuhan tanaman, (5) memvisualisasikan model kuantitatif , dan (6) mengevaluasi model kuantitatif. model kualitatif proses pemodelan dimulai dengan menggunakan model kualitatif. dimana mengambil data dari pertumbuhan tanaman dan melakukan pengamatan, pengamatan ini akan mengetahui topologi dan urutan pertumbuhan (prusinkiewicz, dkk, 1990) dari modul pada arsitektur tanaman zinnia. untuk memproduksi daun tanaman zinnia dimulai setelah tunas, kemudian batang sebagai dasar dari munculnya sepasang daun, kemudian muncul sepasang daun, kemudian muncul daun yang dikuti oleh kuncup bunga jalur ini adalah sebagai batang utama, kemudian muncul dua tunas pada ketiak daun pada order ke dua sebelum daun terakhir yang dekat dengan kuncup bunga. pada notasi p3 menjelaskan pertumbuhan sepasang daun pada setiap batang sampai pucuk tunas sebelum berbunga. maka model tanaman zinnia dapat dinotasikan ω : a4 p1 : a4→ i[a2][a2]ik0 p2 : ai→ ai+1, 0 ≤ i < 4 p3 : li→ li+1, 0 ≤ i < 4 p4 : ki→ ki+1, i≥ 0 model tanaman zinnia tersebut akan dibangun model tanaman dengan menggunakan metoda dol-system, yaitu komponen tanaman zinnia adalah himpunan dari struktur komponen, hubungan komponen dan proses pertumbuhan komponen yang dapat dinotasikan sebagai gzinnia=(σ,p={p1,…,p9},α), dimana ∑ adalah alphabet dimana ∑=(δ1,…, δn), setiap simbol alphabet mewakili unit morfologi seperti sprout,stalk,leaf, bloom, α adalah string awal, disebut sebagai aksioma dan p=(p1,…, pn), a set of productions or rewrite rules. produksi pertama adalah menjelaskan tentang hubungan antar komponen yang mendasari dari model kualitatif dimana notasi tunas adalah sprout, kemudian noasi batang adalah stalk, kemudian noasi daun adalah leaf dan kemudian notasi bunga adalah bloom p1 = sprout(4) → f stalk(3) [leaf(1)] f stalk(2) [leaf(1)] [sprout(2)] [sprout(2)] f stalk(1) bloom(0) notasi lengkap dengan metoda dol-system untuk model kualitatif dapat dilihat pada gambar 3. gambar 3. struktur model kualitatif tanaman zinnia dengan metoda l-system. model kuantitatif model kuantitatif pada penelitian ini adalah mengkombinasikan string l-system pertumbuhan tanaman zinnia dari model kualitatif dengan pendekatan fungsi pertumbuhan dari struktur tanaman zinnia sehingga didapat model tanaman zinnia secara keseluruhan yaitu berumur 25 hari dalam bentuk virtual plant tanaman zinnia seperti dapat dilihat pada gambar 6. fungsi pertumbuhan dihasilkan dari data pengamatan lapangan pada pertumbuhan struktur tanaman zinnia sesuai dengan waktu dalam siklus hidup tanaman, grafik pertumbuhan untuk batang pada gambar 4. data lapangan dirubah ke fungsi pertumbuhan g(t) suhartono 36 volume 3 no. 1 november 2013 (somporn.c.a,dkk, 2004) sesuai dengan waktu pengamatan yang dilakukan dengan persamaan 1. 𝐺(𝑡) = 𝐿 + 𝑈 − 𝐿 1 + 𝑒𝑚(𝑇−𝑡) (1) di mana l : nilai minimum panjang atau lebar. u : nilai maksimum panjang atau lebar. m : nilai kemiringan diperkirakan dari data mentah. t : waktu di (u-l)/2. t : waktu independen variabel. gambar 4. fungsi pertumbuhan struktur batang tanaman zinnia nilai minimum dari variabel l, nilai maksimum dari variabel u, nilai kemiringan grafik dari variabel m dan waktu t didapat pada setiap komponen arsitektur tanaman zinnia seperti pada tabel 1. tabel 1. nilai u, t, m dan t untuk simbol stalk, leaf dan bloom simbol nilai l nilai u nilai m nilai t stalk 0 10.3 0.9 4.4 leaf 0 6.6 0.9 4.1 bloom 0 7.2 0.8 4 setiap komponen dari arsitektur tanaman zinnia seperti stalk, leaf dan bloom akan dikontrol oleh fungsi pertumbuhan yang ditunjukkan pada produksi p1, p2, p3, p4 untuk produksi pada notasi p1 menjelaskan perubahan tunas, produksi pada notasi p2 menjelaskan perubahan perpanjangan batang, produksi pada notasi p3 menjelaskan perubahan ukuran daun, produksi pada notasi p4 menjelaskan perubahan ukuran bunga seperti gambar 5. visualisasi model kuantitatif pertumbuhan tanaman template mathevolvica (jacob, 1995) dapat mengimplemetasikan metoda l-system pada pemodelan pertumbuhan tanaman, template ini terdiri dari program klsystem.m dan turtleinterpretation.m sebagai inisialisasi implementasi metoda l-system. pemodelan pertumbuhan tanaman zinnia yang dikodekan ke dalam program mathematica sesuai dengan metoda l-system dapat dilihat pada gambar 9, kemudian tiap kode dari metoda l-system akan dikonversi ke turtle bentuk grafik komputer, visualisasi model tanaman zinnia kalau sudah cocok berhenti jika diperlukan perubahan maka dilakukan perubahan model kuantitatif dan model kualitatif tanaman zinnia sesuai dengan metoda lsystem. gambar 5. kode program mathematica untuk pemodelan pertumbuhan tanaman zinnia dengan menggunakan template math evolvica pada program mathematica didapat visualisasi pertumbuhan tanaman zinnia pada gambar 6 dimulai dengan munculnya tunas yang dinotasikan sebagai sprout(4) dimana notasi ini tidak merepresentasikan bentuk grafik, produksi pertama untuk rule grammar adalah mengganti sprout dengan empat struktur stalks yang di ikuti dengan tiga struktur leaf, tiap struktur leaf terdiri dari dua daun leaf(0), untuk stalks(2) akan muncul cabang diatas dua daun leaf(0) antara dua struktur leaf akan muncul dua tunas sebagai cabang tanaman dengan ditandai sprout(2). dan pada ujung paling tinggi pada stalks direpresentasikan sebagai bloom(0), yang direpresentasikan sebagai bunga zinnia gambar 6.1, proses berikutnya adalah produksi kedua yaitu melanjutkan aturan rule grammar seperti proses pada produksi pertama yang dikatakan sebagai metoda rewriting gambar 6.2. gambar 6. visualisasi pertumbuhan pemodelan pertumbuhan tanaman zinnia 5 10 15 20 25 t waktu 2 4 6 8 10 g t fungsi p ertumbuhan fungsi pertumbuhan batang terhadap waktu t real p endekatan 10.3 6.4 6.1 6.2 6.5 6.3 6.6 pemodelan pertumbuhan tanaman zinnia menggunakan lindenmayer system dengan mathematica jurnal cauchy – issn: 2086-0382 37 visualisasi pertumbuhan tanaman zinnia dilakukan sampai proses pertumbuhan tanaman pada tahap keenam pada gambar 6. kesimpulan berdasar penerapan data pertumbuhan tanaman zinnnia, penerapan metoda l-system pada pemodelan pertumbuhan tanaman zinnia dapat mengidentifikasi pertumbuhan tanaman zinnia selama 25 hari dalam 6 tahap, dimana pada setiap tahap pertumbuhan tanaman dapat divisualisasikan. daftar pustaka [1] brooks dan peter, 2006, metric for it service management (jan van bon, ed), zaltbommel : van haren publishing. [2] chuai-aree.s, siripant.s dan lursinsap.c, 2000, animating plant growth in l-system by parametric functional symbols, proc. of intern. conf. on intelligent technology 2000, pp 135-143, december 13-15, university bangkok, thailand [3] hirafuji.m, 1991, a plant growth model by neural networks and l-system, proc.9.th ifac symp. identification and system parameter estimation, vol 3 fukuoka, japan, pp.997-1022 [4] jacob.c, 2001, illustrating evolutionary computation with mathematica. morgan kaufmann publishers, san fancisco. [5] james a. freeman, 1994, fuzzy systems for control applications: the truck backerupper, the mathematica journal, miller freeman publications [6] karwowski.r dan prusinkiewics.p, 2006, the l-system based plant modeling environment l-studio 4.0. in proceeding of the 4th international workshop on functional structural plant models, pp. 403405 [7] mech. r dan prusinkiewics.p, 1996, visual model of plants interacting with their environment, proceedings of siggraph 96. in computer graphics proceedings, annual conferenceseries, 1996, acm siggrapp, pp.397-410. [8] pachepsky.l.b, m.kaul, c.walthall, j.lydon, h.hong, c.s.t daughtry, 2004, soybean growth and development visualized with l-systems simulation: effect of temperature, international jurnal of biotronic vol 33, 31-47 [9] prusinkiewics.p, and lindenmayer, 1990, the algoritmic beauty of plant, springerverlag, new york. [10] prusinkiewics.p, jim hanan, mark hammel dan mech. r, 2003, l-system : from the theory to visual models of plants, siggraph l-system and beyond, page 2.1-2.12 [11] somporn.c.a , suchada siripant, chidchanok lursinsap, 2004, animating plant growth in l-system by parametric functional symbols, 4th international workshop on functional structural plant models. [12] suhartono, mochamad hariadi & mauridhi hery purnomo, 2011, integration of fuzzy system into genetic l-system programming based plant modeling environment with mathematica, australian journal of basic and applied, vol. 5(11). pp. 1760-1765 [13] suhartono, mochamad hariadi & mauridhi hery purnomo, 2011, integration of artificial neural network into genetic lsystem programming based plant modeling environment with mathematica, international journal of academic research, vol. 3. no. 6, i part [14] suhartono, mochamad hariadi & mauridhi hery purnomo, 2010, hybrid genetic lsystem method for representing indentification of plant growth visualization, proceding seminar nasional teknologi industri (snti), isbn : 978-97918265-2-5 statistik uji parsial pada model mixed geographically weighted regression statistik uji parsial pada model mixed geographically weighted regression (studi kasus jumlah kematian bayi di jawa timur tahun 2012) mahmuda1, sri harini2 1mahasiswa jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang e-mail: ammyima@gmail.com1, sriharini21@yahoo.co.id2 abstrak mixed geographically weigted regression (mgwr) adalah pengembangan dari model regresi dimana terdapat parameter yang dihitung pada setiap lokasi pengamatan, sehingga setiap lokasi pengamatan mempunyai nilai parameter yang berbeda-beda, regresi yang demikian dinamakan geographically weighted regression (gwr). penaksir parameter untuk gwr ini menggunakan wls, karena dalam gwr terdapat matriks pembobot yang dibutuhkan. dengan mgwr akan menghasilkan penaksir parameter yang sebagian bersifat global dan sebagian lain bersifat lokal sesuai dengan lokasi pengamatan. penaksir pada model mgwr dapat dilakukan dengan metode wls. selanjutnya diperlukan serangkaian prosedur untuk melakukan uji hipotesis terhadap parameter model yang dihasilkan. uji hipotesis ini digunakan untuk kesesuaian model dan juga menentukan variabel bebas mana yang berpengaruh signifikan terhadap model. dari hasil penelitian didapatkan model statistik uji dari model mgwr adalah statistik uji f dan uji t. pada aplikasi gwr4 didapatkan bahwa dari ke tujuh variabel, ternyata kesehatan ibu dan kesehatan bayi yang signifikan mempengaruhi jumlah kematian bayi di jawa timur tahun 2012. sehingga model yang didapatkan dari pengujian signifikansi model gwr pada data jumlah kematian bayi di jawa timur tahun 2012 menggunakan aplikasi gwr4 adalah: �̂� = 33,930030 + 465,611379 𝑋6 − 469,063181 𝑋7 selain menggunakan statistik uji f dan uji t, dapat juga digunakan metode lainnya pada pengujian model mgwr. estimasi parameter juga dapat menggunakan metode selain wls, serta data yang sesuai dengan kebutuhan peneliti. kata kunci: geographically weighted regression (gwr), mixed geographically weighted regression (mgwr), statistik uji t. abstract mixed geographically weigted regression (mgwr) is the development of a regression model where there are parameters calculated at each observation location, so that each location has a different parameters value, such regression is called geographically weighted regression (gwr). the parameter estimator for this gwr use wls, because it requires weighting matrix gwr. mgwr will produce parameter estimator that some are global and some others are local according to the observation location. estimator at are mgwr model can be done using wls method. furthermore, a series of procedures required to test the hypothesis on the resulted model parameters .this hypothesis test is used for the suitability of the model and also determine which independent variable that influence the model significantly. from the study we obtain that a test statistical model of mgwr model is f test and t test. according to the application of gwr4, we obtained that from all of seven variables the maternal health and infant health are significantly affect the number of infant death in east java in 2012. the model obtained from testing the significance of the gwr models on data of the number of infant deaths in east java in 2012 using gwr4 application are: �̂� = 33,930030 + 465,611379 𝑋6 − 469,063181 𝑋7 in addition to the statistic of f test and t test, we can also use other methods on the mgwr model testing. parameter estimation can also use methods other than wls, and use data according to the needs of researcher. keywords: geographically weighted regression (gwr), mixed geographically weighted regression (mgwr), statistics t test. mailto:ammyima@gmail.com1 mailto:sriharini21@yahoo.co.id2 mahmuda 175 volume 3 no. 3 november 2014 pendahuluan dalam matematika terdapat beberapa cabang yang sangat bemanfaat untuk kehidupan manusia, salah satu cabang diantaranya adalah ilmu statistika. dalam statistika terdapat suatu metode yang disebut dengan regresi, regresi merupakan metode yang memodelkan hubungan antara variabel terikat dan variabel bebas. di dalam regresi terdapat statistik uji yang terbagi menjadi statistik uji simultan dan statistik uji parsial. dalam regresi terdapat parameter yang dihitung pada setiap lokasi pengamatan, sehingga setiap lokasi pengamatan mempunyai nilai parameter yang berbeda-beda, regresi yang demikian dinamakan geographically weighted regression (gwr). dalam regresi juga terdapat gabungan antara regresi linier global dan model gwr, dalam kasus ini disebut mixed geographically weigted regression (mgwr), dengan mgwr akan menghasilkan penaksir parameter yang sebagian bersifat global dan sebagian lain bersifat lokal sesuai dengan lokasi pengamatan. untuk mengetahui signifikansi dalam model mgwr, maka menggunakan statistik uji simultan dan statistik uji parsial. dalam jurnal hasbi yasin (2013) sudah diteliti statistik uji simultan, maka dari itu peneliti melanjutkan penelitian dengan menggunakan statistik uji parsial dalam model mgwr. oleh karena itu dalam penelitian ini akan dilakukan statistik uji parsial pada model mixed geographically weighted regression. kajian teori 1. geographically weighted regression (gwr) [1] menyatakan bahwa ”model gwr merupakan pengembangan dari model regresi global dimana ide dasarnya diambil dari regersi nonparametrik”. model gwr adalah pengembangan dari model regresi global di mana setiap parameter dihitung pada setiap lokasi pengamatan, sehingga setiap lokasi pengamatan mempunyai nilai parameter yang berbeda-beda. model ini merupakan model regresi linier lokal yang menghasilkan penaksir parameter yang bersifat lokal untuk setiap titik di mana data tersebut dikumpulkan. model gwr dapat ditulis sebagai berikut: 𝑦𝑖 = 𝛽0(𝑢0, 𝑣0) + ∑ 𝛽𝑘(𝑢𝑖, 𝑣𝑖) 𝑝 𝑘=1 𝑥𝑖𝑘 + 𝑖; 𝑖 = 1,2, … , 𝑛 dengan 𝑦𝑖 : nilai observasi variabel respon ke-i 𝑥𝑖𝑘 : nilai observasi variabel prediktor k pada pengamatan ke-i 𝛽0(𝑢0, 𝑣0) : nilai intercept model regresi gwr 𝛽𝑘 : koefisien regresi (𝑢𝑖, 𝑣𝑖) : menyatakan titik koordinat (lintang, bujur) lokasi i 𝑖 : error ke-i penaksir parameter model gwr penaksir parameter model gwr menggunakan metode wls yaitu dengan memberikan pembobot yang berbeda pada setiap lokasi pengamatan. sehingga penaksir parameter model untuk setiap lokasinya adalah [2]: �̂�(𝑢𝑖, 𝑣𝑖) = (𝑋 𝑇𝑊(𝑢𝑖, 𝑣𝑖)𝑋) −1𝑋𝑇𝑊(𝑢𝑖, 𝑣𝑖)𝑌 dengan 𝑋 : matriks peubah prediktor 𝑌 : matriks peubah respon �̂� : vektor penduga parameter gwr untuk pengamatan ke-i (𝑢𝑖, 𝑣𝑖): koordinat spasial (longitude, latitude) untuk pengamatan ke-i 2. statistik uji model gwr pengujian hipotesis pada model gwr terdiri dari pengujian kesesuaian model gwr dan pengujian parameter model. pengujian kesesuaian model gwr (goodness of fit) dilakukan dengan hipotesis sebagai berikut [3]: h0 = 𝛽𝑘(𝑢𝑖, 𝑣𝑖) = 𝛽𝑘 (tidak ada perbedaan yang signifikan antara model regresi global dan gwr) h1: paling tidak ada satu 𝛽𝑘(𝑢𝑖, 𝑣𝑖) ≠ 𝛽𝑘 untuk setiap 𝑘 = 1,2, … , 𝑞 (ada perbedaan yang signifikan antara model regresi global dan gwr). 𝐹 = (𝑅𝑆𝑆(𝐻0) − 𝑅𝑆𝑆(𝐻1)) 𝜏1 𝑅𝑆𝑆(𝐻1) 𝛿1 = 𝑦𝑇[(𝑰−𝑯)−(𝑰−𝑳)𝑡(𝑰−𝑳)]𝑦 𝜏1 𝑦𝑇(𝑰−𝑳)𝑡(𝑰−𝑳)𝑦 𝛿1 adapun pengujian signifikansi parameter model pada setiap lokasi dilakukan dengan menguji parameter secara parsial. pengujian ini dilakukan untuk mengetahui parameter mana saja yang signifikan mempengaruhi variabel terikatnya. adapun bentuk hipotesisnya adalah [4]: h0 = 𝛽𝑘(𝑢𝑖, 𝑣𝑖) = 0 h1 = 𝛽𝑘(𝑢𝑖, 𝑣𝑖) ≠ 0, dengan k=1,2,…,q statistik uji parsial pada model mixed geographically weighted regression cauchy – issn: 2086-0382 176 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = �̂� (𝑢𝑖, 𝑣𝑖) ( �̂� √𝐶𝑘𝑘 ) 3. mixed geographically weighted regression (mgwr) mixed geographically weighted regression merupakan gabungan dari model regresi linier global dengan model gwr, sehingga dengan model mgwr akan dihasilkan penaksir parameter yang sebagian bersifat global dan sebagian yang lain bersifat lokal sesuai dengan pengamatan data [1]. model mgwr dengan p variabel bebas dan q variabel bebas diantaranya bersifat lokal, dengan mengasumsikan bahwa intersep model bersifat lokal dapat dituliskan sebagai berikut: 𝑦𝑖 = 𝛽0(𝑢0, 𝑣0) + ∑ 𝛽𝑘(𝑢𝑖, 𝑣𝑖) 𝑞 𝑘=1 𝑥𝑖𝑘 + ∑ 𝛽𝑘 𝑞 𝑘=𝑞+1 𝑥𝑖𝑘 + 𝑖; 𝑖 = 1,2, … , 𝑛 dengan: 𝑦𝑖 : nilai observasi variabel respon ke-i xik : nilai observasi variabel prediktor k pada pengamatan ke-i β0(u0, v0) : nilai intercept model regresi βk : koefisien regresi (ui, vi) : menyatakan titik koordinat (lintang, bujur) lokasi i εi : error ke-i 4. penaksir parameter model mgwr cara mencari penaksir parameter pada model mgwr dapat dilakukan dengan menggunakan metode wls seperti halnya pada model gwr. penaksir parameter mgwr dilakukan dengan mengidentifikasikan terlebih dahulu variabel global dan variabel lokal pada model mgwr. dalam bentuk matriks dapat dituliskan sebagai berikut: 𝑦 = 𝑋𝑙𝛽𝑙(𝑢𝑖, 𝑣𝑖) + 𝑋𝑔𝛽𝑔 + dengan: 𝑋𝑔 : matriks variabel bebas global 𝑋𝑙 : matriks variabel bebas lokal 𝛽𝑔 : vektor parameter variabel bebas global 𝛽𝑙(𝑢𝑖, 𝑣𝑖) : matriks parameter variabel bebas lokal langkah selanjutnya adalah menuliskan model mgwr menjadi gwr untuk mencari penaksir parameter model mgwr: �̂� = 𝑦−𝑋𝑔𝛽𝑔 = 𝑋𝑙𝛽𝑙(𝑢𝑖, 𝑣𝑖) + sehingga penaksir parameter model gwr yang pertama adalah: �̂�𝑙(𝑢𝑖, 𝑣𝑖) = [𝑋𝑙 𝑇𝑊(𝑢𝑖, 𝑣𝑖)𝑋𝑔] −1 𝑋𝑙 𝑇𝑊(𝑢𝑖, 𝑣𝑖)�̃� misalkan 𝑥𝑙 𝑇 = 1, 𝑥𝑖1, 𝑥𝑖2, . . . , 𝑥𝑖𝑞 adalah elemen baris ke-i dari matriks 𝑋𝑙. maka nilai prediksi untuk �̃� pada (𝑢𝑖, 𝑣𝑖) untuk seluruh pengamatan dapat dituliskan sebagai berikut: �̃� = (�̃�1, �̃�2, … , �̃�𝑛) 𝑇 = 𝑆𝑙�̃� dengan 𝑆𝑙 = [ 𝑥𝑖1 𝑇 [𝑋𝑙 𝑇𝑊(𝑢1, 𝑣1)𝑋𝑔] −1 𝑋𝑙 𝑇𝑊(𝑢1, 𝑣1) 𝑥𝑖2 𝑇 [𝑋𝑙 𝑇𝑊(𝑢2, 𝑣2)𝑋𝑔] −1 𝑋𝑙 𝑇𝑊(𝑢2, 𝑣2) ⋮ 𝑥𝑖𝑛 𝑇 [𝑋𝑙 𝑇𝑊(𝑢𝑛, 𝑣𝑛)𝑋𝑔] −1 𝑋𝑙 𝑇𝑊(𝑢𝑛, 𝑣𝑛)] sehingga fitted-value dari respon pada koefisien lokal dari n lokasi pengamatan adalah �̃� = 𝑆𝑙(𝑦 − 𝑋𝑔𝛽𝑔) dengan 𝑆𝑔 = 𝑋𝑔(𝑋𝑔 𝑇𝑋𝑔) −1 𝑋𝑔 𝑇 oleh karena itu, nilai fitted-value dari respon untuk n lokasi pengamatan adalah �̂� = (�̂�1, �̂�2, … , �̂�𝑛) 𝑇 = �̂� + 𝑋𝑔𝛽𝑔 = 𝑆𝑙(𝑦 − 𝑋𝑔𝛽𝑔) + 𝑋𝑔𝛽𝑔 = 𝑆𝑙𝑦 + (𝐼 − 𝑆𝑙)𝑋𝑔[𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)𝑋𝑔] −1 𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)𝑦 = [𝑆𝑙 + (𝐼 − 𝑆𝑙)𝑋𝑔𝑆𝑙𝑦 + (𝐼 − 𝑆𝑙)𝑋𝑔[𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)𝑋𝑔] −1 𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)] 𝑦 = 𝑆𝑦 substitusikan elemen dan 𝛽𝑙(𝑢𝑖, 𝑣𝑖) ke dalam model mgwr: 𝑦 − 𝑋𝑙𝛽𝑙(𝑢𝑖, 𝑣𝑖) = 𝑋𝑔𝛽𝑔 + 𝑦 − 𝑆𝑙�̂� = 𝑋𝑔𝛽𝑔 + 𝑦 − 𝑆𝑙(𝑦 − 𝑋𝑔𝛽𝑔) = 𝑋𝑔𝛽𝑔 + 𝑦 − 𝑆𝑙𝑦 + 𝑆𝑙𝑋𝑔𝛽𝑔 = 𝑋𝑔𝛽𝑔 + 𝑦 − 𝑆𝑙𝑦 = 𝑋𝑔𝛽𝑔 − 𝑆𝑙𝑋𝑔𝛽𝑔 + (1 − 𝑆𝑙)𝑦 = (1 − 𝑆𝑙)𝑋𝑔𝛽𝑔 + (1 − 𝑆𝑙)𝑦 − (1 − 𝑆𝑙)𝑋𝑔𝛽𝑔 = maka: �̂�𝑔 = (�̂�𝑞+1, �̂�𝑞+2, … , �̂�𝑝) 𝑇 = [𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)𝑋𝑔] −1 𝑋𝑔 𝑇(𝐼 − 𝑆𝑙) 𝑇(𝐼 − 𝑆𝑙)𝑦 5. statistik uji f model mgwr uji f dilakukan pertama kali untuk uji kesesuaian model regresi global dan model mgwr, maka bentuk hipotesisnya adalah (yasin, 2013:532): h0 : = 𝛽𝑘(𝑢𝑖, 𝑣𝑖) = 𝛽𝑘 h1 : minimal ada satu 𝛽𝑘(𝑢𝑖, 𝑣𝑖) = 𝛽𝑘 𝐹 = 𝑦𝑇[(𝑰 − 𝑯) − (𝑰 − 𝑺)𝑡(𝑰 − 𝑺)]𝑦 𝑣1 𝑦𝑇(𝑰 − 𝑺)𝑡(𝑰 − 𝑺)𝑦 𝑢1 mahmuda 177 volume 3 no. 3 november 2014 6. pengujian hipotesis pengujian hipotesis adalah salah satu cara dalam statistika untuk menguji “parameter” populasi berdasarkan statistik sampelnya, untuk dapat diterima atau ditolak pada tingkat signifikansi tertentu. pada prinsipnya pengujian hipotesis ini adalah membuat kesimpulan sementara untuk melakukan penyanggahan atau pembenaran dari permasalahan yang akan ditelaah. sebagai wahana untuk menetapkan kesimpulan sementara tersebut kemudian ditetapkan hipotesis nol dan hipotesis alternatif. jumlah kematian bayi angka kematian bayi (akb) merupakan salah satu indikator keberhasilan pembangunan kesehatan yang telah dicanangkan dalam sistem kesehatan nasional dan bahkan dipakai sebagai indikator sentral keberhasilan pembangunan kesehatan di indonesia. dari sisi penyebabnya, kematian bayi ada dua macam yaitu endogen dan eksogen. kematian bayi endogen atau kematian neonatal disebabkan oleh faktor-faktor yang dibawa anak sejak lahir, yang diperoleh dari orang tuanya pada saat konsepsi. variabel terikat yaitu jumlah kematian bayi di jawa timur tahun 2012, dan variabel bebasadalah jumlah puskesmas, jumlah tenaga medis, jumlah posyandu, pemberian asi eksklusif, pemberian vitamin, kesehatan ibu, kesehatan bayi. metodologi penelitian tahap penelitian a. statistik uji parsial pada model mgwr tahapan penelitian pada statistik uji parsial pada model mgwr yaitu: 1. menetapkan model mgwr 2. melakukan estimasi parameter regresi dan parameter variansi menggunakan metode weighted least square (wls). 3. memeriksa asumsi dari gwr dengan pendekatan model regresi global yang akan digunakan untuk mendeteksi multikolinier dan autokorelasi. 4. menggunakan pengujian uji kesesuaian model untuk menguji model mgwr dengan gwr. 5. mencari statistik uji parsial dan menetapkan hipotesis. b. statistik uji parsial pada data jumlah kematian bayi di jawa timur tahun 2012 tahapan penelitian pada statistik uji parsial pada model mgwr dengan data jumlah kematian bayi yaitu: 1. mendeskripsikan data. 2. mencari nilai vif untuk menguji multikolinier. 3. muncari nilai durbin watson untuk menguji autokorelasi. 4. memasukkan data pada program gwr4.04. 5. menganalisis data dengan cara a. menguji kesesuaian model gwr b. mencari parameter dari model gwr yang signifikan c. mencari parameter yang signifikan dengan statistik uji f d. memasukkan data pada program gwr4.08 sesuai klasifikasi parameter yang telah ditentukan. e. menentukan hasil statistik uji parsial dari model mgwr dengan program gwr4.08. f. membentuk digital peta menggunakan software arcview 3.3. 6. pembahasan hasil analisis. 7. membuat kesimpulan. pembahasan 1. statistik uji parsial pada model mgwr [2] menyatakan bahwa statistik uji parsial pada model mgwr bertujuan untuk menentukan parameter yang berpengaruh secara signifikan terhadap variabel respon. dengan mengikuti hipotesis berikut: h0 : = 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖) = 0 h1 : = 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖) ≠ 0 untuk k=1,2,…,p , h=1,2,…,q dan i=1,2,…,n statistik uji parsial pada model mgwr digunakan untuk mendapatkan estimasi parameter 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖) dan matriks varian kovarian (𝑋𝑇𝑊𝑋)−1𝛿𝑘 2 dengan menggunakan standart eror (se) 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖): 𝑆𝐸(𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖)) = √𝑔𝑘𝑘 gkk adalah elemen diagonal k+1 dari matrik (𝑋𝑇𝑊𝑋)−1𝛿𝑘 2. 𝑆𝐸(𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖)) digunakan untuk menguji tingkat signifikansi masing-masing lokasi dengan menggunakan statistik uji 𝑡 dibawah h0 untuk 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖) = 0, maka: 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = 𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖) 𝑆𝐸(𝛽𝑘ℎ(𝑢𝑖, 𝑣𝑖)) statistik uji parsial pada model mixed geographically weighted regression cauchy – issn: 2086-0382 178 statistik uji parsial global pada model mgwr pada model mgwr prediktor dapat berpengaruh secara global dan sebagian secara lokal. uji 𝑡 untuk parameter global 𝑥𝑘 (𝑞 + 1 ≤ 𝑘 ≤ 𝑝) menggunakan hipotesis: h0 : = 𝛽𝑘 = 0 (variabel global 𝑥𝑘 tidak berpengaruh secara signifikan) h0 : = 𝛽𝑘 ≠ 0 (variabel global 𝑥𝑘 berpengaruh secara signifikan) estimator parameter 𝛽𝑘 akan mengikuti distribusi normal multivariat dengan rata-rata 𝛽𝑘 dan matriks varian kovarian 𝐺𝑖𝐺𝑖 𝑇𝜎2 dengan:         1 t tt t i g l l g g l l      g x i s i s x x i s i s dengan k  adalah rata-rata pada regresi spasial, kk g adalah elemen diagonal ke-k dari matriks 2t i i g g , maka statistik uji yang digunakan adalah: 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = �̂�𝑔 ( �̂� √𝑔𝑘𝑘 ) statistik uji parsial lokal pada model mgwr statistik uji t untuk mengetahui pengaruh signifikan variabel prediktor lokal (1 ) k x k q  terhadap variabel respon model mgwr menggunakan hipotesis sebagai berikut: h0 : 𝛽𝑘(𝑢𝑖, 𝑣𝑖) = 0 (variabel lokal k x pada lokasi ke-i tidak berpengaruh secara signifikan) h1 : 𝛽𝑘(𝑢𝑖, 𝑣𝑖) ≠ 0 (variabel lokal k x pada lokasi ke-i berpengaruh secara signifikan) estimator parameter 𝛽𝑘(𝑢𝑖, 𝑣𝑖) akan mengikuti distribusi normal multivariat dengan rata-rata 𝛽𝑘(𝑢𝑖, 𝑣𝑖) dan matriks varian kovarian 𝑀𝑖𝑀𝑖 𝑇𝜎2. dengan𝛽𝑘(𝑢𝑖, 𝑣𝑖) adalah rata-rata pada regresi spasial, mkk adalah elemen diagonal ke-k dari matriks 𝑀𝑖𝑀𝑖 𝑇𝜎2, maka statistik uji yang digunakan adalah: 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = �̂�𝑘 (𝑢𝑖, 𝑣𝑖) ( �̂� √𝑚𝑘𝑘 ) 2. aplikasi data model mgwr model gwr pada penelitian ini model mgwr diterapkan pada kasus jumlah kematian bayi di propinsi jawa timur tahun 2012. variabel yang diteliti yaitu persentase jumlah kematian bayi (jiwa) sebagai variabel respon (y) dan variabel jumlah puskesmas (x1), jumlah tenaga medis (x2), jumlah posyandu (x3), jumlah pemberian asi eksklusif (x4), jumlah pemberian vitamin (x5), kesehatan ibu (x6), kesehatan bayi (x7) sebagai variabel prediktornya. pada penelitian ini analisis data menggunakan software gwr4 dan digital peta menggunakan software arcview 3.3. langkah pertama adalah menentukan lokasi pengamatan dalam hal ini adalah letak geografis tiap kabupaten/kotamadya di jawa timur. kemudian mencari bandwith optimum berdasarkan koordinat lokasi pengamatan dengan prosedur cross validation (cv). setelah mendapatkan nilai bandwidth optimum, maka langkah selanjutnya adalah menentukan matriks pembobot terbaik untuk mendapatkan model gwr. dari hasil analisis dengan program gwr4 didapatkan model terbaik untuk penentuan jumlah kematian bayi di jawa timur adalah menggunakan jenis pembobot adaptive bisquare , yaitu: tabel 1. hasil uji model gwr dengan pembobot fungsi adaptif bisquare variabel f p-value diff of criterion intercept 1.507602 0.289519 0.750355 x1 3.437276 0.001926* -0.179920 x2 3.978672 0.000572* 0.263310 x3 5.92783 1.4e-05* -1.101479 x4 1.352884 0.434997 -0.135576 x5 5412.07229 7.03e-44* -0.042191 x6 1145829.503 2.83e-75* 0.108179 x7 914566.6336 5.93e-74* -0.004400 dari tabel di atas dapat diketahui variabel yang berpengaruh secara signifikan adalah variabel jumlah rumah sakit (x1), jumlah tenaga medis (x2), jumlah posyandu (x3), pemberian vitamin (x5), kesehatan ibu (x6), dan variabel kesehatan bayi (x7). sehingga model gwr untuk kasus jumlah kematian bayi di jawa timur pada tahun 2012 adalah: 𝑦 = 33,930030 + 3,587866𝑋1 + 1429114𝑋2 + 1429114𝑋3 + 1,611371𝑋5 + 465,611379𝑋6 − 469,063181𝑋7 sedangkan untuk gwr lokal diketahui dari hasil program gwr4 dari tiap lokasi, dalam hal ini peneliti mengambil contoh dari kota malang, diantaranya: tabel 2. hasil estimasi parameter model gwr lokal variabel estimate standard error t (est/se) intercept 34.06744 0.757745 44.95896 mahmuda 179 volume 3 no. 3 november 2014 x1 2.208323 2.047297 1.078653 x2 0.084554 0.994597 0.085013 x3 -0.02908 1.244517 -0.02336 x4 -0.62515 0.648891 -0.96341 x5 0.414951 7.526751 0.05513 x6 475.5051 37.40756 12.71147 x7 -473.876 39.22458 -12.0811 dari tabel di atas dapat diketahui variabel yang berpengaruh secara signifikan adalah variabel kesehatan ibu (x6), dan variabel kesehatan bayi (x7). sehingga model gwr lokal untuk kasus jumlah kematian bayi di kota malang pada tahun 2012 adalah: 𝑦32 = 34,06744 + 475,5051 𝑋6 − 473,876 𝑋7 sedangkan pemetaan model gwr pada kasus kematian bayi di jawa timur pada tahun 2012 adalah sebagai berikut: model mgwr tabel 3. pengujian kesesuain model mgwr sumber keragaman jk db kt f p-value residual global 1097.6 39 30.00 0 gwr improvement 73.068 2.667 27.392 gwr residual 1024.5 71 27.33 3 37.485 7.307 32 1.64e06 berdasarkan tabel di atas maka didapatkan nilai 𝐹ℎ𝑖𝑡𝑢𝑛𝑔 sebesar 7.89544 dan 𝑝 − 𝑣𝑎𝑙𝑢𝑒 sebesar 6.19577e-08. dengan melihat tabel f maka didapatkan nilai 𝐹𝑡𝑎𝑏𝑒𝑙 sebesar 2.33. jika dibandingkan menjadi 𝐹ℎ𝑖𝑡𝑢𝑛𝑔 > 𝐹𝑡𝑎𝑏𝑒𝑙, dan dengan tigkat signifikansi (α) sebesar 5% maka didapatkan 𝑝 − 𝑣𝑎𝑙𝑢𝑒 < α. maka dapat disimpulkan bahwa model gwr memiliki perbedaan yang siginifikan dengan model gwr. setelah dilakukan pengujian kesesuaian model, langkah selanjutnya adalah menguji parameter model mgwr. tabel 4. hasil estimasi model mgwr variable estimate standard error t(estimate/s e) intercept 33.930030 0.982829 34.522839 x1 3.587866 2.374693 1.510876 x2 1.429114 1.482826 0.963777 x3 1.324760 1.462288 0.905950 x5 1.611371 7.436231 0.216692 x6 465.611379 45.789913 10.168427 x7 -469.063181 48.792622 -9.613404 x4 -1.034796 1.012299 -1.022224 berdasarkan hasil dari tabel di atas, dengan membandingkan 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 dan 𝑡𝑡𝑎𝑏𝑒𝑙, maka dapat diketahui variabel apa saja yang berpengaruh secara signifikan terhadap jumlah kematian bayi di jawa timur. dengan 𝛼 = 5% maka dapat diperoleh variabel yang signifikan yaitu: jumlah kesehatan ibu (x6) dan jumlah kesehatan bayi (x7). model mgwr yang didapatkan pada kasus jumlah kematian bayi di jawa timur pada tahun 2012 adalah sebagai berikut: 𝑦 = 33,930030 + 465,611379𝑋6 − 469,063181𝑋7 sedangkan pemetaan model mgwr pada kasus jumlah kematian bayi di jawa timur pada tahun 2012 adalah sebagai berikut: pemetaan untuk pemetaan variabel yang signifikan dengan menggunakan statistik uji pada model mgwr adalah: statistik uji parsial pada model mixed geographically weighted regression cauchy – issn: 2086-0382 180 penutup 1. kesimpulan berdasarkan hasil pada pembahasan, maka dapat diambil kesimpulan sebagai berikut: 1. hasil statistik uji parsial pada model mgwr adalah: a. untuk statistik uji parsial global adalah: 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = �̂�𝑔 ( �̂� √𝑔𝑘𝑘 ) b. untuk statistik uji parsial lokal adalah: 𝑡ℎ𝑖𝑡𝑢𝑛𝑔 = �̂�𝑘 (𝑢𝑖,𝑣𝑖) ( �̂� √𝑚𝑘𝑘 ) 2. pada pemetaan statistik uji pada model mgwr terdapat empat kelompok warna untuk variabel yang signifikan yaitu warna putih, kuning, biru dan merah. untuk warna putih yaitu kabupaten lamongan dengan variabel yang signifikan 𝑋2, 𝑋3, 𝑋6, 𝑋7 (jumlah tenaga medis, jumlah posyandu, jumlah kesehatan ibu dan jumlah kesehatan bayi). sedangkan yang warna biru adalah kabupaten malang dengan variabel yang signifikan: 𝑋1, 𝑋2, 𝑋5, 𝑋6, 𝑋7 (jumlah puskesmas, jumlah tenaga medis, pemberian vitamin, jumlah kesehatan ibu, jumlah kesehatan bayi). untuk yang warna kuning adalah kota batu dengan variabel signifikan: 𝑋1, 𝑋2, 𝑋3, 𝑋5, 𝑋6, 𝑋7 (jumlah puskesmas, jumlah tenaga medis, jumlah posyandu, jumlah pemberian vitamin, jumlah kesehatan ibu dan jumlah kesehatan bayi). sedangkan untuk yang warna adalah daerah yang mempunyai variabel signifikan 𝑋6, 𝑋7 (jumlah kesehatan ibu dan jumlah kesehatan bayi) diantaranya kab. pacitan, kab. ponorogo, kab. trenggalek, kab. tulungagung, kab. blitar, kab. magetan, kab. ngawi, kab. bojonegoro, kab. tuban, kab. tuban, kab. kediri, kab. lumajang, kab. jember, kab. banyuwangi, kab. bondowoso, kab. situbondo, kab. probolinggo, kab. pasuruan, kab. sidoarjo, kab. mojokerto, kab. jombang, kab. nganjuk, kab. madiun, kab. gresik, kab. bangkalan, kab. sampang, kab. pamekasan, kab. sumenep, kota kediri, kota malang, kota probolinggo, kota pasuruan, kota mojokerto, kota madiun, dan kota surabaya. 2. saran dari penelitian ini terdapat beberapa saran yang dapat dilakukan untuk penelitianpenelitian selanjutnya, yaitu: 1. untuk penelitian selanjutnya, perlu ditambahkan variabel-variabel lain yang lebih signifikan berpengaruh secara global agar mendapatkan hasil yang lebih sempurna. 2. untuk peneliti lain, bisa menggunakan metode penaksiran dan fungsi pembobot yang lain . daftar pustaka [1] hasbi yasin, "uji hipotesis model mixed geographically weighted regression dengan metode bootstrap," in universitas diponegoro, jogjakarta, 2013, pp. 528-532. [2] sri harini, purhadi, and mashuri, "statistical test for multivariate geographically weighted regression model using the method of maximum likelihood ratio test.," applied mathematics, pp. 24-30, 2012. [3] hasbi yasin, "pemilihan variabel pada model geographically weighted regression," media statistika, pp. 66-70, 2011. [4] luluk azizah, "pengujian signifikansi pada model geographically weighted regression," malang, 2013. pendahuluan kajian teori 1. geographically weighted regression (gwr) penaksir parameter model gwr 2. statistik uji model gwr 3. mixed geographically weighted regression (mgwr) 4. penaksir parameter model mgwr 5. statistik uji f model mgwr 6. pengujian hipotesis jumlah kematian bayi metodologi penelitian pembahasan 1. statistik uji parsial pada model mgwr statistik uji parsial global pada model mgwr statistik uji parsial lokal pada model mgwr 2. aplikasi data model mgwr model gwr model mgwr penutup 1. kesimpulan 2. saran daftar pustaka analisis grafik kendali np yang distandarisasi untuk pengendalian kualitas dalam proses pendek yayuk nurkotimah dan fachrur rozi jurusan matematika uin maulana malik ibrahim malang e-mail: ocy_cute09@yahoo.com abstrak salah satu grafik kendali yang sering digunakan untuk pengendalian data atribut adalah grafik kendali p. grafik kendali p merupakan grafik kendali proporsi ketidaksesuaian. dalam membentuk grafik kendali p dibutuhkan 20 sampai 30 subgrup. untuk memudahkan interpretasi, maka grafik kendali perlu distandarisasi. standarisasi dilakukan agar data berada di antara � � 3� dan � � 3�. pada praktiknya, terkadang dalam suatu proses produksi banyaknya subgrup yang diperoleh sedikit, sehingga untuk memenuhi banyaknya subgrup yang diinginkan, dibutuhkan lebih dari satu proses produksi, hal ini sering terjadi dalam kasus proses pendek. dalam proses pendek, karena pengamatan yang diperoleh kurang dari standar untuk membentuk grafik kendali, maka perlu dilakukan penyesuaian dengan menambahkan faktor koreksi. faktor koreksi bertujuan agar peluang standarisasi untuk proses pendek ���� �� � dan ���� � � � dengan �� � 3 dan � � �3 dapat memenuhi standar internasional yaitu mendekati 0,00135. dalam penelitian ini, dikembangkan grafik kendali np yang distandarisasi untuk proses pendek, di mana langkah pertama penulis akan mengkaji grafik kendali np, kemudian grafik tersebut distandarisasi berdasarkan distribusi normal standar, setelah diperoleh standarisasi maka data terebut distandarisasi untuk proses pendek. terakhir menganalisis hasil dari standarisasi untuk proses pendek. dengan membandingkan ���� �� � dan ���� � � � serta ���� �� � dan ���� � � �, maka grafik kendali yang berkualitas ditunjukkan oleh grafik kendali np yang distandarisasi untuk proses pendek. kata kunci: faktor koreksi, grafik kendali np, proses pendek, standarisasi. pendahuluan pengendalian kualitas statistik merupakan teknik penyelesaian masalah yang digunakan untuk memonitor, mengendalikan, menganalisis, mengelola, dan memperbaiki produk dan proses menggunakan metode-metode statistik. pengendalian kualitas statistik (pks) berhubungan dengan grafik kendali, di mana grafik kendali digunakan untuk mengendalikan proses sampel yang diperoleh. grafik kendali yang membahas tentang sifat suatu produk disebut grafik kendali data atribut. grafik kendali data atribut menunjukkan karakteristik kualitas yang sesuai dengan spesifikasi atau tidak sesuai dengan spesifikasi. untuk membuat grafik kendali p dan np yang bagus, dibutuhkan 20 sampai 30 subgrup. jika jumlah tersebut tidak terpenuhi maka grafik kendali yang dihasilkan tidak akurat. pada penerapan grafik kendali p dan np sampel yang diinginkan sulit untuk dikumpulkan dalam sekali proses produksi. keadaan seperti ini biasanya terjadi pada proses pendek. proses produksi pendek adalah proses produksi yang memproduksi produk dalam jangka waktu yang pendek, sehingga tidak dapat mengambil sampel sesuai dengan kebutuhan. oleh karena itu, grafik kendali p dan np mengambil beberapa data dari proses produksi yang berbeda yang memiliki proporsi produk cacat (p) berbeda. dari nilai p yang berbeda menghasilkan batas kendali dan garis tengah yang berbeda-beda, jadi grafik kendali p dan np ini harus distandarisasi. proses standarisasi di sini digunakan untuk mempermudah dalam menginterpretasikan grafik kendali p dan np. untuk melakukan standarisai maka harus menghitung simpangan baku dari tiap-tiap sampel. dari penjelasan di atas maka penulis ingin mengembangkan grafik kendali np untuk distandarisasi dengan menggunakan proses pendek agar mendapatkan produk yang berkualitas. kajian teori definisi 1. ekspektasi misalkan x suatu peubah acak dengan distribusi peluang ����. nilai harapan atau rata-rata x adalah � � ���� � ∑ ������ bila x diskrit, dan � � ���� � � �������∞�∞ bila x kontinu. (walpole dan myers, 1995) yayuk nurkotimah dan fachrur rozi 116 volume 2 no. 2 mei 2012 sifat 1. ekspektasi bila � dan � konstanta, maka ���� � �� � ����� � � (walpole dan myers , 1995) sifat 2. ekspektasi misalkan � dan � dua peubah acak yang bebas. maka ����� � �������� (walpole dan myers, 1995) definisi 2. variansi misalkan � peubah acak dengan distribusi peluang ���� dan rata-rata �. variansi � adalah �2 � ���� � ��2� � ∑ �� � ��2����� bila � diskrit dan �2 � ���� � ��2� � � �� � ��2����∞�∞ bila � kontinu. (walpole dan myers, 1995) dalil limit pusat misalkan � ,�" ,… ,�$ adalah sampel acak yang berasal dari distribusi yang memiliki ratarata � dan variansi �". maka � � ∑ %&�$'(&)*+√$ � √$�%-�'� + konvergen dalam distribusi ke suatu peubah acak yang berdistribusi normal dengan rata-rata 0 dan variansi 1 �� . ~012 �0,1�� (dudewicz dan mishra, 1995) grafik kendali proporsi ketidaksesuaian grafik kendali p didefinisikan sebagai perbandingan banyak benda yang ketidaksesuaian dalam suatu populasi dengan banyak benda keseluruhan dalam populasi itu. andaikan p merupakan proporsi ketidaksesuaian dalam proses produksi diketahui. maka garis tengah dan batas kendalian grafik kendali dengan proporsi ketidaksesuaian adalah 567 � 8 � 9 :8�;�8�< (2. 1) 67 � 8 (2. 2)767 � 8 � 9 :8�;�8�< (2. 3) apabila proporsi ketidaksesuaian p tidak dikatahui, maka p harus ditaksir dari data observasi. prosedur yang biasa adalah memilih m pengamatan pendahuluan, masing-masing berukuran n. sebagai aturan umum, m biasanya dipilih antara 20 sampai 25. maka jika ada 2� unit sampel ketidaksesuaian dalam pengamatan ke-k, kita hitung proporsi ketidaksesuaian dalam pengamatan ke-k itu sebagai 8= � >=< ,= � ;,?,9… @ (2. 4) dan rata-rata proporsi ketidaksesuaian dari seluruh pengamatan tersebut adalah 8a � ∑ >b@b);@< � ∑ 8ab@b); @ , b � ;,?,9,… ,@ (2. 5) statistik cd menaksir proporsi ketidaksesuaian p tidak diketahui. garis tengah dan batas kendali grafik kendali untuk proporsi ketidaksesuaian dihitung sebagai berikut: 567 � 8a � 9 :8a�;�8a�< (2. 6) 67 � 8a (2. 7) 767 � 8a � 9 :8a�;�8a�< (2. 8) distribusi normal distribusi normal adalah salah satu distribusi teoritis dari variabel random kontinu. distribusi normal sering disebut distribusi gauss, sesuai nama pengembangnya, yaitu karl gauss pada abad ke-18, seorang ahli matematika dan astronomi. fungsi padat peubah acak normal x dengan rata-rata � dan variansi �2 adalah ���� � 1�√2e f �1 2 ghijk l 2 , � ∞ m � m ∞ karena persamaan kurva normal tersebut di atas tergantung pada nilai � dan �, maka kita mempunyai bermacam-macam bentuk kurva tergantung dengan � dan � tersebut. untuk menyederhanakan dibuat kurva normal standar. kurva normal standar adalah kurva normal yang sudah diubah menjadi distribusi z, di mana distribusi tersebut akan mempunyai � � 0 dan �2 � 1. ini dapat dilakukan melalui transformasi � � %� '+ . distribusi binomial definisi 3. suatu peubah acak � yang mempunyai distribusi bernoulli jika (untuk suatu p, 0 m c m 1). ��� � �� � �%��� � nc ��1 � c� ��,�op� � � 0,1 0,qrsqt u�rv p�orru� w (dudewicz dan mishra, 1995) sifat 3. bila � berdistribusi bernoulli maka ���� � c,x�y ��� � c�1 � c� (dudewicz dan mishra, 1995) sifat 4. (walpole dan myers, 1995) distribusi binomial z��;r,c� mempunyai ratarata dan variansi � � rc dan �2 � rc�1 � c�. pendekatan distribusi normal terhadap distribusi binomial bila x adalah suatu peubah acak binomial dengan rata-rata µ = np dan variansi σ2= np�1 � c�, maka bentuk pelimitan bagi sebaran adalah \ � ]�<8^<8�;�8� untuk r . ∞, adalah sebaran normal baku sebaran normal memberikan hampiran sangat baik pada sebaran binomial bila n besar dan p dekat dengan ½. bahkan bila n kecil dan p analisis grafik kendali np yang distandarisasi untuk pengendalian kualitas dalam proses pendek jurnal cauchy – issn: 2086-0382 117 tidak terlalu dekat pada nol dan satu, hampiran itu masih tetap baik (walpole,1995: 197). seperti diketahui, distribusi binomial bervariabel diskrit sedangkan distribusi normal (kurva normal) bervariabel kontinu. karena itu, penggunaan distribusi normal (kurva normal) untuk menyelesaikan kasus distribusi binomial dapat dilakukan dengan menggunakan aturan (penyesuaian), yaitu menggunakan faktor koreksi. caranya ialah menambahkan atau mengurangi variabel x-nya dengan 0.5 sebagai berikut: 1. untuk batas bawah (kiri), variabel x dikurangi 0.5 2. untuk batas atas (kanan), variabel x ditambah 0.5 dengan demikian, rumus z-nya menjadi: � � �%_0.5��$a^$a�1�a� proses pendek proses pendek adalah proses produksi yang memproduksi produk dalam jangka waktu yang pendek atau singkat, sehingga tidak cukup waktu untuk mengambil sampel dalam jumlah yang dibutuhkan. beberapa bursa kerja ditandai dalam produksi proses pendek, dan bursa kerja ini memproduksi beberapa bagian dari proses produksi yang kurang dari 50 unit. pada situasi ini, dapat membuat penggunaan rutin grafik kendali terlihat seperti sebuah tantangan. untuk menentukan batas control, suatu unit tidak dapat diproduksi dalam satu waktu. masalah ini dapat diselesaikan dengan mudah, karena metode statistik proses kontrol paling sering diterapkan pada karakteristik dari sebuah produk, dengan memperpanjang statistik proses kontrol pada lingkungan bursa kerja dan fokus pada proses karakteristik dari setiap unit produk (montgomery, 1990). pembahasan grafik kendali np grafik kendali np merupakan grafik kendali banyaknya unit ketidaksesuaian. grafik kendali ini dapat dibentuk dari sebuah proses produksi. misal ambil suatu produksi sebanyak m pengamatan, di mana setiap pengamatan ke-k memiliki ukuran sampel pengamatan sebanyak n. misalkan dalam suatu produksi proporsi ketidaksesuaian (p) diketahui dan tiap pengamatan mempunyai ukuran sampel (n) sama maka berdasarkan grafik kendali shewhart dapat diperoleh suatu model baru yaitu: 567 � <8 � 9^<8�; � 8� (3. 1) 67 � <8 (3. 2) 767 � <8 � 9^<8�; � 8� (3. 3) sedangkan jika proporsi ketidaksesuaian (p) tidak diketahui dan tiap pengamatan mempunyai ukuran sampel (n) sama maka p perlu ditaksir, berdasarkan persamaan (2.4) dapat diperoleh: >= � <8= (3. 4) sehingga berdasarkan persamaan (2.5), p dapat ditaksir oleh: 8a � ∑ >b @b); @< � ∑ 8b@b); @ , b � ;,?,… ,@ (3. 5) maka dapat diperoleh grafik kendali baru sebagai berikut: 567 � <8a � 9^<8a�; � 8a� (3. 6) 67 � <8a (3. 7) 767 � <8a � 9^<8a�; � 8a� (3. 8) proses standarisasi grafik kendali np misalkan � ,�" … �$ adalah sampel acak berukuran n yang berdistribusi bernoulli dengan � � c dan �" � c�1 � c� maka berdasarkan dalil limit pusat dapat diperoleh: �� � ∑ �� � r�c�d1 �√r � >=�<8^<8�;�8� ~ef>�g,;� (3. 9) sehingga saat proporsi ketidaksesuaian (p) diketahui dan tiap pengamatan mempunyai ukuran sampel (n) sama maka berdasarkan persamaan (3.4), persamaan (3.9) menjadi: �� � $ah�$a^$a� �a� (3.10) proses standarisasi grafik kendali np untuk proses pendek misal ambil c/n sebagai faktor koreksi, dengan c adalah konstanta dan n adalah ukuran sampel. untuk proporsi ketidaksesuaian (p) diketahui dan ukuran sampel (n) sama maka dari persamaan lai k.chan diperoleh: \= � <8=�<8�i^<8�;�8� (3.11) dalam mencari perumusan proporsi ketidaksesuaian (p) tidak diketahui dan ukuran sampel (n) sama maka p akan ditaksir oleh cd, dari persamaan lai k.chan dapat diperoleh persamaan sebagai berikut: �� � $ah�$ad^$adh� �adh� dengan aturan penyesuaian persamaan �� dimodifikasi menjadi �� dengan menambahkan konstanta : c�c�1� agar �jh � 0 dan �jh2 � 1 sehingga didapatkan: \=k � : @�@�;� <8=�<8a ^<8a�;�8a� (3.12) yayuk nurkotimah dan fachrur rozi 118 volume 2 no. 2 mei 2012 sehingga berdasarkan persamaan lai k.chan dan persamaan (3.12) diperoleh persamaan standarisasi untuk proses pendek adalah \= � : @�@�;� <8=�<8a�i ^<8a�;�8a� (3.13) menganalisis grafik kendali np yang distandarisasi untuk proses pendek dalam penelitian ini penulis meneliti 15 sampel size yang berbeda-beda mulai 50 hingga 750. selain itu, penulis juga meneliti beberapa nilai p yang berbeda-beda yaitu 0,01; 0,05; dan 0,1. sedangkan untuk faktor koreksi penulis menggunakan 0,9; 1,1; 1,3; dan 1,5. misal pilih p = 0,05 dan n = 750 serta faktor koreksi 1,5 maka dapat diperoleh analisisnya sebagai berikut: 1. setelah diplot, ���� �� � dan ���� � � � jauh dari nilai 0,00135 sedangkan ���� �� � dan ���� � � � berada disekitar 0,00135. 2. suatu konstanta faktor koreksi yang baik adalah yang menghasilkan nilai ���� �� � dan ���� � � � yang mendekati 0,00135. semakin besar konstanta faktor koreksi belum tentu nilai ���� �� � dan ���� � � � semakin mendekati 0,00135, begitupun semakin kecil faktor koreksi belum tentu nilai ���� �� � dan ���� � � � mendekati 0,00135. kemudian peneliti juga mengaplikasikan persamaan yang telah diperoleh pada sebuah data. data di ambil pada proses produksi yang berbeda yakni produksi pertama 10 sampel dan produksi kedua 10 sampel. sampel tersebut di buat dalam satu penelitian dengan tujuan untuk memenuhi standar dalam membentuk grafik kendali yakni 20 sampai 30 subgrup. berikut ini datanya: tabel 1. data produksi dari dua run-proses pengamatan r� 2� produksi 1 1 100 20 2 100 25 3 100 35 4 100 10 5 100 30 6 100 5 7 100 45 8 100 20 9 100 10 10 100 10 produksi 2 1 100 5 2 100 2 3 100 3 4 100 8 5 100 4 6 100 1 7 100 2 8 100 6 9 100 3 10 100 4 (sumber: montgomery, 1990) a. grafik kendali np misal ambil pengamatan ke-2 maka dari persamaan (3.6), (3.7) dan (3.8) di peroleh: gambar 1. grafik kendali np pada grafik kendali ini, dapat dilihat bahwa masing-masing proses produksi memiliki batas atas, batas pusat dan batas bawah sendirisendiri. untuk produksi pertama nilai yang keluar dari batas kendali ada 3 sampel sehingga yang terkendali hanya 7 sampel, namun pada produksi yang kedua semua nilai data terkendali. hal tersebut akan sulit untuk dianalisis secara umum, oleh karena itu perlu dilakukan standarisasi. b. grafik kendali np yang distandarisasi misal ambil pengamatan ke-2 maka dari persamaan (3.16) didapatkan: gambar 2. grafik kendali np yang distandarisasi sedangkan untuk grafik ini, dapat diamati bahwa data terkontrol dalam grafik kendali. namun pada data 3 dan 7 berada diluar garis kendali batas atas dan untuk data ke 6 berada diluar batas kendali bawah sehingga dari 20 data yang terkontrol hanya 17 data. c. grafik kendali np yang distandarisasi untuk proses pendek misal ambil pengamatan ke-2 dan ambil faktor koreksi sebesar 0.5 sesuai persamaan (3.36) diperoleh: 0 20 40 60 1 4 7 10 3 6 9 a x is t it le grafik kendali np data np ucl lcl cl -5 0 5 10 1 4 7 10 3 6 9 a x is t it le grafik kendali z data z ucl lcl analisis grafik kendali np yang distandarisasi untuk pengendalian kualitas dalam proses pendek jurnal cauchy – issn: 2086-0382 119 gambar 3. grafik kendali np yang distandarisasi untuk proses pendek dan untuk grafik ini dari 20 data, 16 data berada pada grafik kendali dan untuk data ke 3 dan ke 7 berada di luar batas atas kendali dan data ke 4 dan ke 6 berada di luar batas bawah kendali. jika dibandingkan dengan grafik-grafik sebelumnya maka grafik ini merupakan grafik yang dapat mengkontrol data dengan baik.sehingga dapat disimpulkan bahwa grafik kendali dengan faktor koreksi merupakan grafik kendali yang terbaik dari pada yang lain. penutup berdasarkan dari penjelasan pada bab-bab sebelumnya maka dapat disimpulkan bahwa grafik kendali np yang distandarisasi untuk proses pendek merupakan modifikasi dari grafik kendali np yang distandarisasi klasik yaitu a. ketika p diketahui dan n sama maka persamaan grafik kendali yang distandarisasi klasik ditambah faktor koreksi. b. ketika p tidak diketahui dan n sama maka persamaan grafik kendali yang distandarisasi klasik ditambah dengan faktor koreksi dan konstanta : c�c�1�. penyesuaian tersebut dilakukan agar peluang statistik standarisasi untuk proses pendek mendekati 0,00135. peluang statistik standarisasi lebih dari batas atas atau kurang dari batas bawah grafik kendali np untuk proses pendek dipengaruhi oleh besarnya proporsi ketidaksesuaian (p) yang ditentukan, ukuran sampel (n) dan faktor koreksi yang berbedabeda. terakhir, berdasarkan aplikasi dari sebuah data dari proses yang sama, maka grafik kendali np untuk proses pendek dapat mendeteksi data dengan baik di bandingkan dengan grafik kendali yang lain. penelitian ini, masih dapat dikembangkan keilmuannya sehingga disarankan untuk penelitian selanjutnya agar menggunakan ukuran sampel yang berbeda-beda yang diterapkan dalam grafik kendali u atau grafik kendali c. daftar pustaka [1] lai k. chan., 1996. standardized p control charts for short runs. international journal of quality and reliability management. vol. 13. no. 6. 88-95 [2] dudewicz, edward j dan mishra, satya n. 1995. statistik matematika modern. bandung: itb [3] montgomery, douglas c. 1991. introduction to statistical quality control. singapore: jonh willy &sons,inc [4] tanti octavia.,dkk. 2000, studi tentang peta kendali p yang distandarisasi untuk proses pendek kualitas. jurnal teknik industri. vol. 2. no. 1. 53 – 64 [5] walpole, ronal dan myers, raymond h. 1995. ilmu peluang dan statistika untuk insinyur dan ilmuwan edisi keempat. bandung: institut teknologi bandung -10 0 10 1 4 7 10 3 6 9a x is t it le grafik kendali z* ucl lcl data z* lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf suma’inna dan gugun gumilar kekonvergenan barisan di dalam ruang fungsi kontinu 𝑪[𝒂,𝒃] firdaus ubaidillah1, soeparna darmawijaya, ch. rini indrati 1jurusan matematika fmipa universitas gadjah mada yogyakarta e-mail: firdaus_u@yahoo.com abstrak diberikan c[a,b] merupakan koleksi semua fungsi kontinu bernilai real pada selang tertutup [a,b]. c[a,b] merupakan ruang linear atas lapangan real. dalam tulisan ini dibahas pengertian-pengertian norma, barisan konvergen, terbatas dan monoton, infimum dan supremum dan lain-lain yang semuanya disajikan dalam bahasa fungsi kontinu. selain itu akan ditunjukkan bahwa barisan yang terbatas dan monoton di dalam ruang fungsi kontinu c[a,b] belum tentu konvergen. satu sifat yang menjamin sebuah barisan memiliki supremum atau infimum akan dibahas. kata kunci: barisan konvergen, fungsi kontinu bernilai real, ruang fungsi kontinu abstract let c[a,b] be a collection of all real-valued continuous functions on a closed interval [a, b]. the c[a,b] is a linear space over the real field. in this paper, we discussed some notions of norm, monotone, bounded and convergent sequence, infimum and supremum and others of which are presented in the language of continuous functions. they will also be shown that bounded and monotone sequence in the continuous functions space c[a,b] is not necessarily convergent. a property that ensures a sequence has a supremum or infimum will be discussed. keywords: convergent sequence, continuous real-valued function, continuous function space pendahuluan telah banyak dibahas sifat-sifat fungsi kontinu bernilai real pada selang tertutup [𝑎,𝑏] oleh bartle dan sherbert (2000), diantaranya sifat terbatas, mencapai nilai maksimum dan minimum, dapat didekati dengan fungsi tangga, merupakan fungsi kontinu seragam, terintegral riemann dan lain sebagainya. dalam tulisan ini, 𝐶[𝑎,𝑏] menyatakan koleksi semua fungsi kontinu bernilai real pada selang tertutup [𝑎,𝑏] ⊂ ℝ, yakni 𝑓:[𝑎,𝑏] → ℝ kontinu. pembahasan beberapa sifat dasar 𝐶[𝑎,𝑏] banyak dijumpai dalam ruang banach klasik diantaranya oleh lindenstrauss dan tsafriri (1977), diestel (1984), meyer-nieberg (1991), albiac dan kalton (2006), dan lain-lain. jika didefinisikan penjumlahan dan perkalian skalar di 𝐶[𝑎,𝑏] berturut-turut (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) untuk setiap 𝑓, 𝑔 ∈ 𝐶[𝑎, 𝑏], 𝑥 ∈ [𝑎, 𝑏], dan (𝛼𝑓)(𝑥) = 𝛼𝑓(𝑥) untuk setiap 𝑓 ∈ 𝐶[𝑎,𝑏],𝛼 ∈ ℝ,𝑥 ∈ [𝑎,𝑏], maka 𝛼𝑓,(𝑓 + 𝑔) ∈ 𝐶[𝑎,𝑏]. oleh karena itu 𝐶[𝑎,𝑏] merupakan ruang linear atas lapangan ℝ (yeh, 2006). lebih jauh, jika diberikan norma pada 𝐶[𝑎,𝑏] yang didefinisikan ‖𝑓‖∞ = sup 𝑎≤𝑥≤𝑏 |𝑓(𝑥)| untuk setiap 𝑓 ∈ 𝐶[𝑎,𝑏], maka (𝐶[𝑎,𝑏],‖∙‖∞) merupakan ruang banach (dales, 2003). dua anggota 𝑓 dan 𝑔 di dalam 𝐶[𝑎,𝑏] dikatakan i. 𝑓 = 𝑔 jika 𝑓(𝑥) = 𝑔(𝑥) untuk setiap 𝑥 ∈ [𝑎,𝑏]; ii. 𝑓 < 𝑔 jika 𝑓(𝑥) < 𝑔(𝑥) untuk setiap 𝑥 ∈ [𝑎,𝑏]; iii. 𝑓 ≤ 𝑔 jika 𝑓(𝑥) ≤ 𝑔(𝑥) untuk setiap 𝑥 ∈ [𝑎,𝑏]. berdasarkan pengertian di atas, mudah dipahami bahwa 𝐶[𝑎,𝑏] merupakan himpunan terurut parsial (partially ordered set) terhadap relasi “≤”. dua anggota 𝑓 dan 𝑔 di dalam 𝐶[𝑎,𝑏] dikatakan dapat dibandingkan (comparable) jika 𝑓 ≤ 𝑔 atau 𝑔 ≤ 𝑓 dan dikatakan tidak dapat dibandingkan (incomparable) jika tidak berlaku 𝑓 ≤ 𝑔 dan 𝑔 ≤ 𝑓. selanjutnya jika dianggap penting, penulisan 𝑓 < 𝑔 dan 𝑓 ≤ 𝑔 berturutturut dapat digantikan dengan 𝑔 > 𝑓 dan 𝑔 ≥ 𝑓. kekonvergenan barisan di dalam ruang fungsi kontinu c[a,b] jurnal cauchy – issn: 2086-0382 185 selanjutnya didefinisikan fungsi nol (null function) 𝜃 dan fungsi satuan (unit function) 𝑒 di dalam 𝐶[𝑎,𝑏] berturut-turut dengan 𝜃(𝑥) = 0 dan 𝑒(𝑥) = 1 untuk setiap 𝑥 ∈ [𝑎,𝑏]. karena setiap 𝑓,𝑔,ℎ ∈ 𝐶[𝑎,𝑏] dan 𝑓 ≤ 𝑔 berlaku i. 𝑓 + ℎ ≤ 𝑔 + ℎ ii. 𝛾𝑓 < 𝛾𝑔 untuk setiap bilangan 𝛾 > 0, maka disimpulkan bahwa 𝐶[𝑎,𝑏] merupakan ruang riesz. lebih jauh, jika didefinisikan perkalian di dalam 𝐶[𝑎,𝑏], yakni (𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥) untuk setiap 𝑓,𝑔 ∈ 𝐶[𝑎,𝑏] dan setiap 𝑥 ∈ [𝑎,𝑏] maka 𝑓𝑔 ∈ 𝐶[𝑎,𝑏]. oleh karena itu 𝐶[𝑎,𝑏] merupakan aljabar banach dengan unsur satuan 𝑒 (dales, 2003 ). dalam tulisan ini, akan diperkenalkan pengertian norma dalam “bahasa” fungsi kontinu. untuk membedakan pengertian norma pada umumnya dengan norma dalam bahasa fungsi kontinu, untuk itu digunakan notasi norma*. definisi 1. (darmawijaya, 2012) sebuah fungsi ‖∙‖∗:𝐶[𝑎,𝑏] → 𝐶[𝑎,𝑏] dikatakan norma* (norm*) pada 𝐶[𝑎,𝑏] jika ‖∙‖∗ memenuhi sifat-sifat i. ‖𝑓‖∗ ≥ 𝜃, untuk setiap 𝑓 ∈ 𝐶[𝑎,𝑏]; ‖𝑓‖∗ = 𝜃 jika dan hanya jika 𝑓 = 𝜃; ii. ‖𝛼𝑓‖∗ = |𝛼|‖𝑓‖∗, untuk setiap 𝑓 ∈ 𝐶[𝑎,𝑏] 𝛼 ∈ ℝ; iii. ‖𝑓 + 𝑔‖∗ ≤ ‖𝑓‖∗ + ‖𝑔‖∗, untuk setiap 𝑓,𝑔 ∈ 𝐶[𝑎,𝑏]. sekarang didefinisikan nilai mutlak |∙| di dalam 𝐶[𝑎,𝑏] dengan |𝑓|(𝑥) = |𝑓(𝑥)|, untuk setiap 𝑥 ∈ [𝑎,𝑏]. karena |∙| memenuhi semua sifat dalam definisi 1, maka |∙| merupakan sebuah norma* pada 𝐶[𝑎,𝑏]. berkaitan dengan hal-hal tersebut di atas, tujuan dalam tulisan ini adalah membahas kekonvergenan barisan di dalam 𝐶[𝑎,𝑏] dengan norma* |∙| yang dijelaskan di atas dalam “bahasa” fungsi kontinu, yang mempunyai arti cukup bekerja di dalam 𝐶[𝑎,𝑏]. namun begitu, untuk memudahkan pembuktian teorema dan perhitungan, dalam beberapa kasus akan dibawa ke fungsi real. hasil dan pembahasan dalam bagian ini akan dibahas beberapa definisi dan teorema yang akan digunakan untuk membahas kekonvergenan barisan di dalam 𝐶[𝑎,𝑏]. di akhir bagian ini dibahas syarat suatu barisan memiliki supremum atau infimum. untuk setiap 𝑓,𝑔 ∈ 𝐶[𝑎,𝑏] dan 𝛼 ∈ ℝ, yang dimaksud dengan fungsi i. 𝑓 𝑔 adalah fungsi yang didefinisikan 𝑓 𝑔 (𝑥) = 𝑓(𝑥) 𝑔(𝑥) asalkan 𝑔(𝑥) ≠ 0, ii. 𝑓 ∨ 𝑔 adalah fungsi yang didefinisikan dengan (𝑓 ∨ 𝑔)(𝑥) = sup⁡{𝑓(𝑥),𝑔(𝑥)}, iii. 𝑓 ∧ 𝑔 adalah fungsi yang didefinisikan dengan (𝑓 ∧ 𝑔)(𝑥) = inf⁡{𝑓(𝑥),𝑔(𝑥)}, iv. √𝑓 adalah fungsi yang didefinisikan dengan √𝑓(𝑥) = √𝑓(𝑥) dengan 𝑓(𝑥) ≥ 0, v. 𝑓−1 = 𝑒 𝑓 jika 𝑓 > 𝜃 atau 𝑓 < 𝜃, untuk setiap 𝑥 ∈ [𝑎,𝑏]. telah ditunjukkan oleh bartle dan sherbert (2000), bahwa jika 𝑓,𝑔 ∈ 𝐶[𝑎,𝑏], maka 𝑓 ∨ 𝑔, ⁡⁡𝑓 ∧ 𝑔,√𝑓⁡(𝑓(𝑥) ≥ 0) dan 𝑓 𝑔 ⁡(𝑔(𝑥) ≠ 0) di dalam 𝐶[𝑎,𝑏]. oleh karena itu 𝐶[𝑎,𝑏] merupakan sebuah aljabar (algebra), lebih jauh 𝐶[𝑎,𝑏] sebuah aljabar banach (banach algebra) (dales, 2003). berikut diberikan pengertian himpunan terbatas ke atas, himpunan terbatas ke bawah dan himpunan terbatas di dalam 𝐶[𝑎,𝑏]. definisi 2. sebuah himpunan 𝐴 ⊂ 𝐶[𝑎,𝑏] tidak kosong dikatakan i. terbatas ke atas (bounded above) jika terdapat 𝑞 ∈ 𝐶[𝑎,𝑏] sehingga ℎ ≤ 𝑞 untuk setiap ℎ ∈ 𝐴; selanjutnya 𝑞 disebut batas atas (upper bound) himpunan 𝐴; ii. terbatas ke bawah (bounded below) jika terdapat 𝑝 ∈ 𝐶[𝑎,𝑏] sehingga 𝑝 ≤ ℎ untuk setiap ℎ ∈ 𝐴; selanjutnya 𝑝 disebut batas bawah (lower bound) himpunan 𝐴; iii. terbatas (bounded) jika 𝐴 terbatas ke atas dan terbatas ke bawah atau himpunan 𝐴 dikatakan terbatas jika terdapat 𝑡 ∈ 𝐶[𝑎,𝑏] dan 𝑡 > 𝜃 sehingga |ℎ| ≤ 𝑡 untuk setiap ℎ ∈ 𝐴. selanjutnya diperkenalkan pengertian batas atas terkecil dan batas bawah terbesar dari suatu himpunan. firdaus ubaidillah, soeparna darmawijaya, ch. rini indrati 186 volume 2 no. 4 mei 2013 definisi 3. diberikan himpunan 𝐴 ⊂ 𝐶[𝑎,𝑏] tidak kosong. i. himpunan 𝐴 terbatas ke atas. titik 𝑠 ∈ 𝐶[𝑎,𝑏] disebut batas atas terkecil (least upper bound) atau supremum 𝐴 jika 𝑠 adalah batas atas 𝐴 dan untuk setiap 𝑣 batas atas 𝐴 berlaku 𝑠 ≤ 𝑣. dalam kasus ini ditulis 𝑠 = 𝑠𝑢𝑝(𝐴). ii. himpunan 𝐴 terbatas ke bawah. titik 𝑟 ∈ 𝐶[𝑎,𝑏] disebut batas bawah terbesar (greates lower bound) atau infimum 𝐴 jika 𝑟 adalah batas bawah 𝐴 dan untuk setiap 𝑢 batas bawah 𝐴 berlaku 𝑟 ≤ 𝑢. dalam kasus ini ditulis 𝑟 = 𝑖𝑛𝑓(𝐴). dari pengertian ini, belum tentu benar bahwa setiap himpunan terbatas ke atas (bawah) memiliki supremum (infimum), lihat contoh 10 dan contoh 11. dalam kasus himpunan 𝐴 ⊂ 𝐶[𝑎,𝑏] hingga, maka 𝐴 memiliki supremum dan infimum, dan sup(𝐴) = ⋁ ℎ∈𝐴 ℎ dan inf(𝐴) = ⋀ ℎ∈𝐴 ℎ. dalam definisi 4 berikut ini diberikan pengertian dari barisan konvergen dan barisan cauchy di dalam 𝐶[𝑎,𝑏] dan dilanjutkan dengan sebuah contoh barisan konvergen (contoh 5). definisi 4. sebuah barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan i. konvergen (converges) jika ada 𝑓 ∈ 𝐶[𝑎,𝑏] sehingga untuk setiap bilangan > 0 terdapat bilangan asli 𝐾 sehingga untuk setiap 𝑛 ≥ 𝐾 berlaku |𝑓𝑛 − 𝑓| < 𝑒. jika demikian halnya, barisan {𝑓𝑛} dikatakan konvergen (convergent) ke 𝑓 atau barisan {𝑓𝑛} mempunyai limit 𝑓 untuk 𝑛 → ∞ dan dituliskan dengan 𝑙𝑖𝑚 𝑛→∞ 𝑓𝑛 = 𝑓. ii. barisan cauchy (cauchy sequence) jika setiap bilangan > 0 terdapat bilangan asli 𝐾 sehingga untuk setiap 𝑚,𝑛 ≥ 𝐾 berlaku |𝑓𝑚 − 𝑓𝑛| < 𝑒. contoh 5. diberikan barisan {𝑓𝑛} ⊂ 𝐶[0,1] yang didefinisikan 𝑓𝑛(𝑥) = 𝑥 𝑛 , untuk setiap 𝑥 ∈ [0,1]. barisan {𝑓𝑛} merupakan barisan konvergen ke 𝜃, sebab jika diberikan bilangan > 0 sebarang maka dipilih bilangan asli 𝐾 > 1/ , sehingga jika untuk setiap 𝑛 ≥ 𝐾⁡diperoleh |𝑓𝑛(𝑥) − 𝜃(𝑥)| = | 𝑥 𝑛 − 0| = 𝑥 𝑛 ≤ 1 𝑛 ≤ 1 𝐾 < . jadi lim 𝑛→∞ 𝑓𝑛 = 𝜃. selanjutnya dibahas beberapa sifat dasar barisan konvergen diantaranya ketunggalan limit barisan dan keterbatasan barisan yang konvergen berturut-turut disajikan dalam teorema 6 dan teorema 7. teorema 6. jika barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] konvergen, maka limit barisan {𝑓𝑛} tunggal. bukti: anggap 𝑓,𝑔 ∈ 𝐶[𝑎,𝑏] sehingga barisan {𝑓𝑛} konvergen ke 𝑓 dan 𝑔. jadi untuk setiap bilangan > 0 terdapat dua bilangan asli 𝑛0 dan 𝑚0 sehingga |𝑓𝑛 − 𝑓| < 𝜀 2 𝑒 jika 𝑛 ≥ 𝑛0 dan |𝑓𝑛 − 𝑔| < 𝜀 2 𝑒 jika 𝑛 ≥ 𝑚0. oleh karena itu untuk setiap bilangan asli 𝑛 ≥ 𝑚𝑎𝑘𝑠{𝑛0,𝑚0} diperoleh |𝑓 − 𝑔| ≤ |𝑓𝑛 − 𝑓| + |𝑓𝑛 − 𝑔| < 2 𝑒 + 2 𝑒 = 𝑒. jadi 𝑓 = 𝑔. ∎ teorema 7. jika barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] konvergen, maka {𝑓𝑛} terbatas. bukti: diberikan barisan {𝑓𝑛} konvergen ke 𝑓 ∈ 𝐶[𝑎,𝑏] dan sebarang bilangan > 0. terdapat bilangan asli 𝐾 sehingga untuk setiap 𝑛 ≥ 𝐾 dipunyai |𝑓𝑛 − 𝑓| < 𝑒. untuk setiap 𝑛 ≥ 𝐾 diperoleh |𝑓𝑛| = |𝑓𝑛 − 𝑓 + 𝑓| ≤ |𝑓𝑛 − 𝑓| + |𝑓| < 𝑒 + |𝑓|. selanjutnya diambil 𝑝 = (⋁|𝑓𝑛| 𝐾−1 𝑛=1 )⋁( 𝑒 + |𝑓|). jadi diperoleh |𝑓𝑛| ≤ 𝑝 untuk setiap bilangan asli 𝑛, atau dengan kata lain {𝑓𝑛} terbatas. ∎ berikutnya dibahas pengembangan lebih lanjut sifat-sifat kekonvergenan barisan yang disajikan dalam teorema 8 dan teorema 9. teorema 8. diberikan sebarang bilangan 𝛾 ∈ ℝ dan barisan-barisan {𝑓𝑛},{𝑔𝑛} ⊂ 𝐶[𝑎,𝑏] konvergen berturut-turut ke 𝑓 dan 𝑔. maka barisan {𝛾𝑓𝑛},{𝑓𝑛 + 𝑔𝑛} dan {𝑓𝑛𝑔𝑛} konvergen berturutturut ke 𝛾𝑓,𝑓 + 𝑔 dan 𝑓𝑔. lebih jauh, barisan { 𝑓𝑛 𝑔𝑛 :𝑔𝑛 > 𝜃⁡𝑎𝑡𝑎𝑢⁡𝑔𝑛 < 𝜃⁡𝑢𝑛𝑡𝑢𝑘⁡𝑠𝑒𝑡𝑖𝑎𝑝⁡𝑛} konvergen ke 𝑓 𝑔 bilamana 𝑔 > 𝜃 atau 𝑔 < 𝜃. kekonvergenan barisan di dalam ruang fungsi kontinu c[a,b] jurnal cauchy – issn: 2086-0382 187 bukti: di sini hanya akan dibuktikan untuk barisan {𝑓𝑛𝑔𝑛} konvergen ke 𝑓𝑔 saja. diberikan barisan {𝑓𝑛},{𝑔𝑛} ⊂ 𝐶[𝑎,𝑏] konvergen berturut-turut ke 𝑓 dan 𝑔. berdasarkan teorema 7, terdapat 𝑝 > 𝜃 sehingga |𝑓𝑛| ≤ 𝑝 untuk setiap 𝑛 ∈ ℕ. didefinisikan 𝑡 = 𝑝 ∨ |𝑔|. diberikan sebarang bilangan > 0. karena {𝑓𝑛} konvergen ke 𝑓 dan {𝑔𝑛} konvergen ke 𝑔, maka terdapat bilangan 𝐾1,𝐾2 ∈ ℕ sehingga jika 𝑛 ≥ 𝐾1 berlaku |𝑓𝑛 − 𝑓| < 𝑒 2𝑡 dan jika 𝑛 ≥ 𝐾2 berlaku |𝑔𝑛 − 𝑔| < 𝑒 2𝑡 . dipilih 𝐾 = sup⁡{𝐾1,𝐾2}. oleh karena itu jika 𝑛 ≥ 𝐾 maka diperoleh |𝑓𝑛𝑔𝑛 − 𝑓𝑔| ≤ |𝑓𝑛𝑔𝑛 − 𝑓𝑛𝑔| + |𝑓𝑛𝑔 − 𝑓𝑔| = |𝑓𝑛||𝑔𝑛 − 𝑔| + |𝑓𝑛 − 𝑓||𝑔| < 𝑡 𝑒 2𝑡 + 𝑒 2𝑡 𝑡 = 𝑒. jadi {𝑓𝑛𝑔𝑛} konvergen ke 𝑓𝑔. ∎ teorema 9. barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] konvergen jika dan hanya jika {𝑓𝑛} barisan cauchy. bukti: syarat perlu. diketahui {𝑓𝑛} konvergen, maka terdapat 𝑓 ∈ 𝐶[𝑎,𝑏] sehingga untuk setiap bilangan > 0 terdapat bilangan asli 𝐾 sehingga jika 𝑛 ≥ 𝐾 berlaku |𝑓𝑛 − 𝑓| < 2 𝑒. jadi untuk setiap bilangan asli 𝑛,𝑚 ≥ 𝐾 diperoleh |𝑓𝑛 − 𝑓𝑚| ≤ |𝑓𝑛 − 𝑓| + |𝑓𝑚 − 𝑓| < 2 𝑒 + 2 𝑒 = 𝑒. syarat cukup: diketahui {𝑓𝑛} barisan cauchy. artinya, untuk setiap bilangan > 0 terdapat bilangan asli 𝑁 sehingga jika 𝑚,𝑛 ≥ 𝑁 benar bahwa |𝑓𝑛 − 𝑓𝑚| < 𝑒. jadi untuk setiap 𝑚,𝑛 ≥ 𝑁 berlaku |𝑓𝑛(𝑥) − 𝑓𝑚(𝑥)| < untuk setiap 𝑥 ∈ [𝑎,𝑏]. ini berarti untuk setiap 𝑥 ∈ [𝑎,𝑏], {𝑓𝑛(𝑥)} merupakan barisan cauchy di ℝ. oleh karena itu setiap 𝑥 ∈ [𝑎,𝑏], barisan {𝑓𝑛(𝑥)} konvergen di ℝ. selanjutnya didefinisikan 𝑓(𝑥) = lim 𝑛→∞ 𝑓𝑛(𝑥) untuk setiap 𝑥 ∈ [𝑎,𝑏]. jadi untuk setiap bilangan > 0 terdapat bilangan asli 𝑁 sehingga jika 𝑛 ≥ 𝑁 berlaku |𝑓𝑛(𝑥) − 𝑓(𝑥)| < untuk setiap 𝑥 ∈ [𝑎,𝑏]. (1) karena 𝑓𝑁 ∈ 𝐶[𝑎,𝑏], maka 𝑓𝑁 kontinu seragam pada [𝑎,𝑏]. artinya, untuk setiap bilangan > 0 terdapat bilangan 𝛿 > 0 sehingga untuk setiap 𝑥,𝑦 ∈ [𝑎,𝑏] dengan |𝑥 − 𝑦| < 𝛿 maka berlaku |𝑓𝑁(𝑥) − 𝑓𝑁(𝑦)| < 𝜀 3 . (2) berdasarkan ketaksamaan (1) dan (2), untuk setiap 𝑥,𝑦 ∈ [𝑎,𝑏] dengan |𝑥 − 𝑦| < 𝛿, diperoleh |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑓(𝑥) − 𝑓𝑁(𝑥)| +|𝑓𝑁(𝑥) − 𝑓𝑁(𝑦)| +|𝑓𝑁(𝑥) − 𝑓(𝑦)| < 𝜀 3 + 𝜀 3 + 𝜀 3 = . jadi terbukti 𝑓 ∈ 𝐶[𝑎,𝑏]. selanjutnya, berdasarkan ketaksamaan (1) karena untuk setiap 𝑛 ≥ 𝑁 dan untuk setiap 𝑥 ∈ [𝑎,𝑏], berlaku |𝑓𝑛(𝑥) − 𝑓(𝑥)| ≤ /3 maka untuk setiap 𝑛 ≥ 𝑁 diperoleh |𝑓𝑛 − 𝑓| ≤ 3 𝑒 < 𝑒. jadi {𝑓𝑛} barisan konvergen. ∎ barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan naik monoton (nondecreasing) jika setiap bilangan asli 𝑛 dipunyai 𝑓𝑛 ≤ 𝑓𝑛+1. barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan turun monoton (nonincreasing) jika setiap bilangan asli 𝑛 dipunyai 𝑓𝑛+1 ≤ 𝑓𝑛. sebuah barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan monoton (monotone) jika {𝑓𝑛} naik monoton atau turun monoton. sebuah barisan yang turun (naik) monoton dan terbatas ke bawah (ke atas) belum tentu mempunyai infimum (supremum), seperti diberikan dalam dua contoh berikut. contoh 10. diberikan barisan {𝑓𝑛} ⊂ 𝐶[0,1] yang didefinisikan 𝑓𝑛(𝑥) = 𝑥 𝑛 untuk setiap 𝑛 ∈ ℕ dan setiap 𝑥 ∈ [0,1]. akan ditunjukkan bahwa barisan {𝑓𝑛} turun (naik) monoton dan terbatas tetapi tidak mempunyai infimum. cukup jelas bahwa barisan {𝑓𝑛} terbatas sebab 𝜃 ≤ 𝑓𝑛 ≤ 𝑒 dan turun monoton sebab 𝑓𝑛+1 ≤ 𝑓𝑛 untuk setiap 𝑛. barisan {𝑓𝑛(𝑥)} konvergen titik demi titik ke 𝑓(𝑥) = 0 untuk 𝑥 ∈ [0,1) dan 𝑓(1) = 1, tetapi 𝑓 ∉ 𝐶[0,1]. contoh 11. diberikan barisan {𝑔𝑛} ⊂ 𝐶[0,1] yang didefinisikan 𝑔𝑛(𝑥) = 𝑛𝑥 untuk 0 ≤ 𝑥 < 1 𝑛 dan 𝑔𝑛(𝑥) = 1 untuk 1 𝑛 ≤ 𝑥 ≤ 1 untuk setiap 𝑛 ∈ ℕ dan setiap 𝑥 ∈ [0,1]. akan ditunjukkan bahwa barisan {𝑔𝑛} naik monoton dan terbatas tetapi tidak mempunyai supremum. cukup jelas bahwa barisan {𝑔𝑛} terbatas sebab 𝜃 ≤ 𝑔𝑛 ≤ 𝑒 dan naik monoton sebab 𝑔𝑛 ≤ 𝑔𝑛+1 untuk setiap 𝑛. barisan {𝑔𝑛(𝑥)} konvergen titik demi titik ke 𝑔(𝑥) = 1 untuk 𝑥 ∈ (0,1] dan 𝑔(0) = 0, tetapi 𝑔 ∉ 𝐶[0,1]. teorema 12. jika barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] naik (turun) monoton dan mempunyai supremum (infimum) maka barisan {𝑓𝑛} konvergen ke supremumnya (infimumnya). firdaus ubaidillah, soeparna darmawijaya, ch. rini indrati 188 volume 2 no. 4 mei 2013 bukti: diberikan 𝑠 = sup⁡{𝑓𝑛:𝑛 ∈ ℕ} ∈ 𝐶[𝑎,𝑏]. untuk setiap bilangan > 0, terdapat bilangan asli 𝐾 sehingga 𝑠 − 𝑒 < 𝑓𝐾. nyatanya bahwa barisan {𝑓𝑛} naik monoton, hal ini mengakibatkan 𝑓𝐾 ≤ 𝑓𝑛 untuk setiap 𝑛 ≥ 𝐾, sehingga diperoleh 𝑠 − 𝑒 < 𝑓𝐾 ≤ 𝑓𝑛 ≤ 𝑠 < 𝑠 + 𝑒 untuk setiap 𝑛 ≥ 𝐾. oleh karena itu diperoleh |𝑓𝑛 − 𝑠| < 𝑒 untuk setiap 𝑛 ≥ 𝐾. jadi lim 𝑛→∞ 𝑓𝑛 = 𝑠. bukti untuk infimum serupa. ∎ barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan naik seragam (uniformly nondecreasing) jika setiap bilangan > 0 terdapat bilangan asli 𝑁 sehingga untuk setiap 𝑛 ≥ 𝑁 dipunyai 𝜃 ≤ 𝑓𝑛+1 − 𝑓𝑛 < 𝑒. barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] dikatakan turun seragam (uniformly nonincreasing) jika setiap bilangan > 0 terdapat bilangan asli 𝐾 sehingga untuk setiap 𝑛 ≥ 𝐾 dipunyai 𝜃 ≤ 𝑓𝑛 − 𝑓𝑛+1 < 𝑒. di akhir bagian ini diberikan teorema 13 yang menjamin suatu barisan memiliki supremum atau infimum. teorema 13. diberikan barisan {𝑓𝑛} ⊂ 𝐶[𝑎,𝑏] terbatas. (i). jika {𝑓𝑛} naik seragam maka {𝑓𝑛} memiliki supremum. lebih jauh, barisan {𝑓𝑛} konvergen ke supremumnya. (ii). jika {𝑓𝑛} turun seragam maka {𝑓𝑛} memiliki infimum. lebih jauh, barisan {𝑓𝑛} konvergen ke infimumnya. bukti: (i). diberikan barisan {𝑓𝑛} terbatas dan naik seragam. maka, untuk setiap bilangan > 0 terdapat bilangan asli 𝑁1 sehingga untuk setiap 𝑛 ≥ 𝑁1 dipunyai 𝜃 ≤ 𝑓𝑛+1 − 𝑓𝑛 < 𝑒⁡ ⇔ ⁡⁡0 ≤ 𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥) < untuk setiap 𝑥 ∈ [𝑎,𝑏]. karena {𝑓𝑛} terbatas maka untuk setiap 𝑥 ∈ [𝑎,𝑏] barisan {𝑓𝑛(𝑥)} terbatas di ℝ. oleh karena itu, terdapat bilangan 𝑓(𝑥) = sup⁡{𝑓𝑛(𝑥)} untuk setiap 𝑥 ∈ [𝑎,𝑏]. untuk sebarang bilangan > 0 terdapat bilangan asli 𝑁2 sehingga jika 𝑛 ≥ 𝑁2 maka berlaku 𝑓(𝑥) − 3 < 𝑓𝑛(𝑥) ≤ 𝑓(𝑥) < 𝑓(𝑥) + 3 atau |𝑓𝑛(𝑥) − 𝑓(𝑥)| < 𝜀 3 . (3) jelas bahwa 𝑓 fungsi bernilai real yang terdefinisi pada [𝑎,𝑏]. selanjutnya diambil bilangan asli 𝑁 = sup⁡{𝑁1,𝑁2}. karena 𝑓𝑁 ∈ 𝐶[𝑎,𝑏] maka terdapat bilangan 𝛿 > 0 sehingga untuk setiap 𝑥,𝑦 ∈ [𝑎,𝑏] dengan |𝑥 − 𝑦| < 𝛿 dipunyai |𝑓𝑁(𝑥) − 𝑓𝑁(𝑦)| < 𝜀 3 . (4) akibatnya, berdasarkan ketaksamaan (3) dan (4) disimpulkan bahwa jika setiap 𝑥,𝑦 ∈ [𝑎,𝑏] dengan |𝑥 − 𝑦| < 𝛿 diperoleh |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑓(𝑥) − 𝑓𝑁(𝑥)| + |𝑓𝑁(𝑥) − 𝑓𝑁(𝑦)| +|𝑓𝑁(𝑦) − 𝑓(𝑦)| < 3 + 3 + 3 = . jadi 𝑓 ∈ 𝐶[𝑎,𝑏].⁡ bukti untuk infimum serupa. ∎ referensi [1] albiac, f., dan kalton, nj., (2006), topics in banach space theory, springer-verlag, new york. [2] bartle, r.g. dan sherbert, d.r., (2000), introduction to real analysis, 3rd edition, johnwiley, new york. [3] dales, h.g., (2003), introduction banach algebras, operators, and harmonic analysis, cambridge university press, cambridge. [4] darmawijaya, s., (2012), calculus on the family of continuous functions, seminar nasional knm xvi, universitas padjadjaran sumedang. [5] diestel, j., (1984), sequences and series in banach spaces, springer-verlag, new york. [6] lindenstrauss, j. dan tsafriri, l., (1977), classical banach spaces ii, springer-verlag, berlin. [7] meyer-nieberg, p., (1991), banach lattices, springer-verlag, berlin. [8] yeh, j., (2006), real analysis: theory of measure and integration, 2nd edition, world scientific publishing, singapore pendahuluan hasil dan pembahasan referensi penyelesaian persamaan diferensial parsial fokker-planck dengan metode garis siti muyassaroh mahasiswa jurusan matematika fakultas sains dan teknologi uin maulana malik ibrahim malang e-mail: muy.sms@gmail.com abstrak persamaan fokker-planck merupakan persamaan diferensial parsial yang menggambarkan fungsi distribusi partikel dalam suatu sistem yang berisi banyak partikel yang saling bertumbukan. digunakan metode garis untuk menyelesaikan solusi numerik pada persamaan fokker-planck. metode ini merepresentasikan bentuk persamaan diferensial parsial ke dalam bentuk sistem persamaan diferensial biasa yang ekuivalen pada bentuk persamaan diferensial parsialnya. langkah pertama yang dilakukan untuk menyelesaikan persamaan fokker-planck dengan metode garis yaitu mengganti turunan ruang dengan metode beda hingga pusat, sehingga diperoleh bentuk sistem persamaan diferensial biasa. langkah kedua yaitu menyelesaikan sistem persamaan diferensial biasa yang telah diperoleh pada langkah pertama dengan metode penyelesaian yang berlaku pada persamaan diferensial biasa yaitu metode runga-kutta. hasil solusi numerik dengan metode garis kemudian dibandingkan dengan solusi eksak menghasilkan galat yang sangat kecil atau mendekati nol. sehingga dapat disimpulkan bahwa metode garis merupakan metode yang baik untuk menyelesaikan persamaan fokker-planck. kata kunci: persamaan fokker-planck, metode garis, metode runga-kutta. abstract fokker-planck equation is partially differential equation that describe distribution function of particles on system contain many particles that collide each other. the method of lines is used to solve numerical solution of fokker-planck equation. this method represents form of partially differential equation into the form of ordinary differential equation that equivalent to the form of its partially differential equation. the first step to solve fokker-planck equation using line method is replacing spatial derivative with center finite difference, in order to obtain system of ordinary differential equation. the second step is solving the system of ordinary differential equation that have been obtained in the first step using the method of lines using the solving method that used at ordinary differential equation, that is runga-kutta method of fourth order. then numerical solution obtained by using the method of lines is compared to the exact solution and produce error that very small or tend to zero. therefore, it can be concluded that the method of lines is good method for solving fokker-planck equation. keywords: fokker-planck equation, line method, runge-kutta method. pendahuluan persamaan fokker-planck merupakan persamaan yang menggambarkan fungsi distribusi partikel dalam suatu sistem yang berisi banyak partikel yang saling bertumbukan. persamaan ini berisi komponen difusi partikel dan interaksi antar partikel (palupi, 2010). bentuk umum persamaan fokker-planck adalah: 𝑣𝑡 (𝑥, 𝑡) = −𝐴𝑣𝑥 (𝑥, 𝑡) + 1 2 𝐵𝑣𝑥𝑥 (𝑥, 𝑡) (1) dengan 𝑣 merupakan fungsi distribusi partikel, 𝐴 disebut sebagai koefisien apung (drift coefficient) dan 𝐵 disebut sebagai koefisien diffusi (zauderer, 2006). persamaan fokker-planck termasuk persamaan diferensial parsial karena mengandung turunan parsial, yaitu turunan dengan dua variabel bebas 𝑥 dan 𝑡. salah satu metode untuk menyelesaikan persamaan diferensial parsial adalah dengan metode garis (method of lines). metode garis merupakan metode beda hingga khusus yang menghasilkan solusi numerik yang mendekati solusi sebenarnya. ide dasar metode ini yaitu mengubah bentuk persamaan diferensial parsial ke dalam bentuk persamaan diferensial biasa. metode garis telah banyak diterapkan pada beberapa permasalahan persamaan diferensial parsial. beberapa penelitian terdahulu yang membahas metode garis antara lain yaitu membahas solusi numerik persamaan kortewegde vries dengan metode garis (ozdes & aksan, 2006). penelitian lain membahas penerapan mailto:muy.sms@gmail.com penyelesaian persamaan diferensial parsial fokker-planck dengan metode garis cauchy – issn: 2086-0382 170 metode garis pada persamaan laplace (sadiku & obiozor, 1997). metode garis memiliki beberapa keunggulan antara lain metode ini sangat efisien dalam perhitungan karena menghasilkan solusi yang akurat dengan sedikit waktu yang ditempuh. selain itu metode ini juga mudah dalam menentukan kestabilannya dengan memisahkan antara variabel ruang dan waktu (sadiku & obiozor, 1997). kajian teori persamaan diferensial parsial fokker-planck persamaan fokker-planck merupakan persamaan yang menggambarkan fungsi distribusi partikel dalam suatu sistem yang berisi banyak partikel yang saling bertumbukan (palupi, 2010). persamaan ini pertama kali dikenalkan oleh fokker dan planck. beberapa penerapan persamaan fokker-planck antara lain pada gerakan tidak menentu partikel kecil yang direndam dalam suatu cairan, fluktuasi intensitas sinar laser, dan distribusi kecepatan partikel cairan dalam aliran turbulen. secara umum persamaan fokker-planck dapat diaplikasikan pada sistem keseimbangan maupun ketakseimbangan (frank, 2004). awal terbentuknya persamaan nonlinier fokker-planck merupakan akibat terjadinya tumbukan antara partikel, sehingga mengalami perubahan arah gerak secara acak (brownian motion). partikel yang disebut sebagai partikel brownian tersebut mengalami proses diffusi. gerakan partikel bersifat acak dan gerakan partikel tidak dipengaruhi oleh gerakan partikel sebelumnya (palupi, 2010). persamaan fokker-planck termasuk persamaan diferensial parsial (pdp) karena persamaan ini menggambarkan laju perubahan terhadap dua variabel bebas yaitu waktu dan jarak (ruang). jika dilihat dari persamaan (1), maka persamaan fokker-planck merupakan pdp orde satu terhadap variabel bebas 𝑡 dan orde dua terhadap variabel bebas 𝑥. persamaan fokkerplanck merupakan pdp tipe parabolik. metode beda hingga persamaan fokkerplanck untuk persamaan fokker-planck yang mengandung variabel x dan t, perkiraan beda hingga dilakukan dengan membuat jaringan titik hitungan pada bidang x-t yang dibagi dalam sejumlah pias dengan interval ruang dan waktu adalah δx dan δt. turunan parsial pada setiap titik grid didekati dari nilai-nilai tetangga dengan menggunakan deret taylor. berikut merupakan turunan pertama dan kedua variabel 𝑥 dengan metode beda hingga pusat: 𝑣𝑥 (𝑥𝑖 , 𝑡𝑛) ≈ 𝑣𝑖+1 𝑛 − 𝑣𝑖 𝑛 ∆𝑥 (2) 𝑣𝑥𝑥 (𝑥𝑖 , 𝑡𝑛) ≈ 𝑣𝑖+1 𝑛 − 2𝑣𝑖 𝑛 + 𝑣𝑖−1 𝑛 ∆𝑥 2 (3) sedangkan untuk turunan pada variabel 𝑡 dilakukan dengan cara yang sama seperti di atas. metode garis metode garis merupakan salah satu dari metode numerik yang paling efisien untuk menyelesaikan persamaan diferensial parsial. metode ini banyak diaplikasikan pada beberapa masalah di bidang fisika teori. metode garis pertama kali dikenalkan oleh matematikawan asal jerman bernama erich rothe pada tahun 1930 (pregla, 2008). ide dasar metode garis adalah mengganti turunan ruang (nilai batas) pada persamaan diferensial parsial dengan pendekatan aljabar. setelah ini dilakukan, turunan ruang tidak lagi dinyatakan secara eksplisit dalam variabel bebas ruang. dengan demikian, hanya ada variabel nilai awal saja artinya dengan adanya satu variabel bebas yang tersisa maka diperoleh sistem persamaan diferensial biasa (pdb) yang mendekati persamaan diferensial parsial yang asli. kemudian selanjutnya adalah merumuskan pendekatan sistem persamaan diferensial biasa. setelah ini dilakukan maka bisa diterapkan beberapa pendekatan untuk nilai awal persamaan diferensial biasa guna menghitung solusi numerik dari persamaan diferensial parsial (hamdi, schiesser, & griffiths, 2009). metode runga-kutta penyelesaian pdb dengan metode deret taylor tidak praktis, karena metode tersebut membutuhkan perhitungan turunan 𝑓(𝑥, 𝑦). disamping itu, tidak semua fungsi mudah dihitung turunannya, terutama bagi fungsi yang bentuknya rumit. oleh karena itu metode rungakutta merupakan alternatif dari metode deret taylor yang memberikan ketelitian hasil yang lebih besar dan tidak memerlukan turunan fungsi (triatmodjo, 2002). bentuk umum metode runga-kutta: 𝑣𝑟+1 = 𝑣𝑟 + ℎ𝜃(𝑥𝑟 , 𝑣𝑟 , ℎ) (4) dengan 𝜃(𝑥𝑟 , 𝑦𝑟 , ℎ) adalah fungsi pertambahan yang menggambarkan kemiringan pada interval. ada beberapa tipe metode runga-kutta yaitu metode runga-kutta orde satu, dua, tiga dan empat. metode runga-kutta orde empat banyak digunakan karena mempunyai ketelitian siti muyassaroh 171 volume 3 no. 3 november 2014 yang lebih tinggi (chapra & canale, 2002). berikut merupakan bentuk metode runga-kutta orde empat: 𝑣𝑟+1 = 𝑣𝑟 + 1 6 (𝑘1+ 2𝑘2 + 2𝑘3 + 𝑘4)ℎ (5) dengan: 𝑘1 = 𝑓(𝑥𝑟 , 𝑣𝑟 ) 𝑘2 = 𝑓 (𝑥𝑟 + 1 2 ℎ, 𝑣𝑟 + 1 2 𝑘1) 𝑘3 = 𝑓 (𝑥𝑟 + 1 2 ℎ, 𝑣𝑟 + 1 2 𝑘2) 𝑘4 = 𝑓(𝑥𝑟 + ℎ, 𝑣𝑟 + 𝑘3) pembahasan 1. penerapan metode garis pada penyelesaian persamaan fokker-planck ide dasar metode garis terdiri dari dua langkah, pertama mengganti turunan ruang dengan menggunakan metode beda hingga sehingga diperoleh sistem persamaan diferensial biasa. kemudian menyelesaikan sistem persamaan diferensial biasa yang sudah diperoleh dengan menggunakan metode penyelesaian pada persamaan diferensial biasa, seperti metode euler, metode runga-kutta dan lain-lain. metode ini dinamakan metode garis karena solusi ditentukan pada setiap garis 𝑥 = 𝑥𝑖 dimana daerah solusi dibagi menjadi beberapa garis lurus yang sejajar dengan sumbu-𝑦 pada batas tertentu (sadiku & obiozor, 1997). berikut merupakan model persamaan yang akan diselesaikan dengan metode garis: 𝑣𝑡 − 𝑥𝑒 3𝑡 𝑣𝑥𝑥 − 𝑣𝑥 = 2𝑥𝑒 2𝑡 − 𝑒 2𝑡 (6) dengan nilai awal yang diberikan yaitu 𝑣(𝑥, 0) = 𝑥, untuk 𝑥 ∈ (0,1) dan nilai batas 𝑣(0, 𝑡) = 0, 𝑣(1, 𝑡) = 𝑒 2𝑡. daerah solusi dibatasi pada 0 ≤ 𝑥 ≤ 1 dan 0 ≤ 𝑡 ≤ 1 (hussain & alwan, 2013). langkah pertama yang harus dilakukan pada metode garis adalah mengganti turunan ruang pada persamaan diferensial parsial dengan menggunakan metode beda hingga pusat. transformasi beda pusat untuk turunan pertama dan kedua variabel ruang: 𝑣𝑥 (𝑥𝑖 , 𝑡𝑛) ≈ 𝑣𝑖+1 𝑛 − 𝑣𝑖 𝑛 ∆𝑥 (7) 𝑣𝑥𝑥 (𝑥𝑖 , 𝑡𝑛) ≈ 𝑣𝑖+1 𝑛 − 2𝑣𝑖 𝑛 + 𝑣𝑖−1 𝑛 ∆𝑥 2 (8) subtitusi pada persamaan (6) sehingga akan diperoleh bentuk berikut: 𝜕𝑣 𝜕𝑡 (𝑥𝑖 , 𝑡𝑛) = 𝑥𝑖 𝑒 3𝑡𝑛 ( 𝑣𝑖+1 𝑛 −2𝑣𝑖 𝑛 +𝑣𝑖−1 𝑛 ∆𝑥2 ) + 𝑣𝑖+1 𝑛 −𝑣𝑖−1 𝑛 2∆𝑥 + 2𝑥𝑖 𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 (9) ketika turunan variabel ruang sudah diganti dengan beda hingga, maka turunan ruang tersebut tidak lagi dinyatakan secara eksplisit dalam variabel bebas ruang. sehingga tersisa variabel nilai awal saja yaitu variabel 𝑡. dengan demikian karena tersisa satu variabel bebas saja maka diperoleh sistem persamaan diferensial biasa (pdb) yang mendekati pdp aslinya (hamdi, schiesser, & griffiths, 2009). maka persamaan diferensial parsial di atas berubah menjadi bentuk persamaan diferensial biasa berikut: 𝑑𝑣 𝑑𝑡 (𝑥𝑖 , 𝑡𝑛) = 𝑥𝑖 𝑒 3𝑡𝑛 ( 𝑣𝑖+1 𝑛 −2𝑣𝑖 𝑛 +𝑣𝑖−1 𝑛 ∆𝑥2 ) + 𝑣𝑖+1 𝑛 −𝑣𝑖−1 𝑛 2∆𝑥 + 2𝑥𝑖 𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 (10) jika dipilih ∆𝑥 = 0,25 maka daerah solusi terdiri dari 𝑥𝑖 , dengan 𝑖 = 0,1,2,3,4 dan ∆𝑡 = 0,01 atau 𝑛 = 0,1,2, ⋯ ,101. sehingga akan diperoleh sistem persamaan diferensial biasa sebagai berikut: 𝑑𝑣0 𝑛 𝑑𝑡 = 𝑥0𝑒 3𝑡𝑛 ( 𝑣1 𝑛−2𝑣0 𝑛+𝑣−1 𝑛 ∆𝑥2 ) + 𝑣1 𝑛 −𝑣−1 𝑛 2∆𝑥 + 2𝑥0𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 𝑑𝑣1 𝑛 𝑑𝑡 = 𝑥1𝑒 3𝑡𝑛 ( 𝑣2 𝑛 −2𝑣1 𝑛+𝑣0 𝑛 ∆𝑥2 ) + 𝑣2 𝑛−𝑣0 𝑛 2∆𝑥 + 2𝑥1𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 𝑑𝑣2 𝑛 𝑑𝑡 = 𝑥2𝑒 3𝑡𝑛 ( 𝑣3 𝑛−2𝑣2 𝑛+𝑣1 𝑛 ∆𝑥2 ) + 𝑣3 𝑛−𝑣1 𝑛 2∆𝑥 + 2𝑥2𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 𝑑𝑣3 𝑛 𝑑𝑡 = 𝑥3𝑒 3𝑡𝑛 ( 𝑣4 𝑛−2𝑣3 𝑛+𝑣2 𝑛 ∆𝑥2 ) + 𝑣4 𝑛−𝑣2 𝑛 2∆𝑥 + 2𝑥3𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 𝑑𝑣4 𝑛 𝑑𝑡 = 𝑥4𝑒 3𝑡𝑛 ( 𝑣5 𝑛−2𝑣4 𝑛+𝑣3 𝑛 ∆𝑥2 ) + 𝑣5 𝑛−𝑣3 𝑛 2∆𝑥 + 2𝑥4𝑒 2𝑡𝑛 − 𝑒 2𝑡𝑛 langkah kedua setelah diperoleh sistem pdb adalah menyelesaikan pdb tersebut dengan metode runga-kutta orde empat. bentuk umum metode runga-kutta orde empat yaitu sebagaimana pada persamaan (5). hasil perhitungan solusi metode garis digambarkan dalam tabel berikut. tabel 1. solusi metode garis persamaan fokker-planck (6) iterasi 𝒕 𝒙𝟎 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 1 0 0 0,25 0,5 0,75 1 2 0,01 0 0,2549 0,5100 0,7651 1,0202 3 0,02 0 0,2600 0,5202 0,7804 1,0408 4 0,03 0 0,2652 0,5306 0,7961 1,0618 5 0,04 0 0,2704 0,5412 0,8121 1,0833 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 101 1 0 1,8091 3,9021 5,1894 7,3891 penyelesaian persamaan diferensial parsial fokker-planck dengan metode garis cauchy – issn: 2086-0382 172 2. perbandingan solusi analitik dan solusi numerik metode garis pada persamaan fokker-planck solusi eksak (analitik) persamaan fokker-planck (6) adalah sebagai berikut (hussain & alwan, 2013): 𝑣(𝑥, 𝑡) = 𝑥𝑒 2𝑡 bertujuan untuk menunjukkan bahwa solusi dengan metode garis adalah mendekati solusi analitik maka keduanya digambarkan dalam bentuk plot. solusi analitik persamaan fokker-planck (6) sebagai berikut: gambar 1. solusi analitik persamaan fokker-planck (6) sedangkan plot solusi numerik dengan metode garis sebagai berikut: gambar 2. solusi numerik persamaan fokker-planck (6) penyelesaian secara numerik hanya menghasilkan nilai yang mendekati pada solusi analitiknya. bertujuan untuk mengetahui besarnya error metode garis terhadap solusi eksaknya, dapat dilakukan dengan menghitung selisih antara nilai eksak dan nilai pendekatan dengan metode garis. besarnya error digambarkan pada tabel berikut. tabel 2. perbandingan solusi numerik dan analitik 𝒙 𝒕 solusi numerik solusi eksak nilai error 𝒗(𝒙, 𝒕) 𝒗(𝒙, 𝒕) 0,25 0,01 0,2549 0,2551 0,0002 0,25 0,02 0,2600 0,2602 0,0002 0,25 0,03 0,2652 0,2655 0,0003 0,25 0,04 0,2704 0,2708 0,0004 0,25 0,05 0,2758 0,2763 0,0005 ⋮ ⋮ ⋮ ⋮ ⋮ 0,75 1 5,1894 5,5418 0,3524 sedangkan untuk mengetahui error pemotongan yang dihasilkan oleh persamaan fokker-planck, dilakukan diskritisasi dengan metode beda hingga pusat, sehingga terbentuk skema berikut: 𝑣𝑖 𝑛+1−𝑣𝑖 𝑛−1 2∆𝑡 + 𝐴 ( 𝑣𝑖+1 𝑛 −2𝑣𝑖 𝑛+𝑣𝑖−1 𝑛 ∆𝑥2 ) − 1 2 𝐵 ( 𝑣𝑖+1 𝑛 −𝑣𝑖−1 𝑛 2∆𝑥 ) = 𝑓(𝑥, 𝑡) (11) konsistensi dapat dicari dengan menggunakan ekspansi deret taylor pada masing-masing sukunya. kemudian suku-suku sejenis dikelompokkan, sehingga diperoleh persamaan berikut: ( 𝑣𝑡 + 𝐴 𝑣𝑥 − 𝐵 2 𝑣𝑥𝑥) | 𝑛 𝑖 + 1 6 ∆𝑡 2 𝑣𝑡𝑡𝑡| 𝑛 𝑖 + ∆𝑥2 ( 𝐴 6 𝑣𝑥𝑥𝑥 − 𝐵 24 𝑣𝑥𝑥𝑥𝑥)| 𝑛 𝑖 + 1 120 ∆𝑡 4 𝑣𝑡𝑡𝑡𝑡𝑡 | 𝑛 𝑖 + ∆𝑥4 ( 𝐴 120 𝑣𝑥𝑥𝑥𝑥𝑥 − ⋯ )| 𝑛 𝑖 = 𝑓(𝑥, 𝑡) (12) dari persamaan (12) dapat diketahui bahwa error pemotongan yang dihasilkan mempunyai orde dua yaitu 𝒪(∆𝑡 2, ∆𝑥 2). persamaan (12) dikatakan konsisten jika lim (∆𝑡,∆𝑥)→0 1 6 ∆𝑡 2 𝑣𝑡𝑡𝑡| 𝑛 𝑖 + ∆𝑥 2 ( 𝐴 6 𝑣𝑥𝑥𝑥 − 𝐵 24 𝑣𝑥𝑥𝑥𝑥) | 𝑛 𝑖 = 0 jika ∆𝑡 dan ∆𝑥 sangat kecil maka jumlah dari limit tersebut akan semakin kecil, karena berapapun nilai 𝑣𝑡𝑡𝑡 , 𝑣𝑥𝑥𝑥 , 𝑣𝑥𝑥𝑥𝑥 jika dikalikan dengan nilai dari ∆𝑡 dan ∆𝑥 akan ikut mengecil. error pemotongan yang dihasilkan akan menuju nol untuk ∆𝑡 → 0 dan ∆𝑥 → 0. 3. interpretasi hasil metode garis merupakan metode numerik yang lebih akurat dibandingkan metode beda hingga biasa karena menghasilkan solusi yang akurat dengan waktu perhitungan yang dibutuhkan lebih efisien. penyelesaian dengan metode garis terdiri dari dua tahap, yaitu mengganti turunan variabel ruang dengan metode beda hingga sehingga diperoleh sistem pdb yang mendekati bentuk pdp asli, kemudian langkah kedua yaitu menyelesaikan sistem pdb 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 2 4 6 8 x solusi analitik persamaan fokker-planck t v (x ,t ) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 2 4 6 8 x solusi numerik metode garis t v (x ,t ) siti muyassaroh 173 volume 3 no. 3 november 2014 dengan metode penyelesaian pdb. bentuk metode beda hingga yang digunakan yaitu beda hingga pusat karena titik-titik yang dihitung dipengaruhi oleh titik-titik disekitarnya sehingga beda hingga pusat dianggap lebih baik daripada beda hingga maju dan mundur. kemudian penyelesaian pdb menggunakan metode rungakutta orde empat karena metode ini mempunyai ketelitian yang lebih tinggi bertujuan untuk mengetahui perbandingan solusi analitik dan solusi metode garis, maka keduanya digambarkan dalam bentuk plot di atas. hasil plot menunjukkan bahwa solusi analitik dan solusi metode garis pada persamaan (6) hampir sama atau dapat dikatakan bahwa solusi metode garis mendekati solusi analitiknya. hasil perhitungan menghasilkan nilai error yang sangat kecil menunjukkan bahwa solusi metode garis hampir mendekati solusi analitik. error pemotongan pada persamaan fokkerplanck dilihat dengan cara ekspansi deret taylor pada tiap suku-sukunya sehingga dihasilkan error pemotongan yang mempunyai orde dua yaitu 𝒪(∆𝑡 2, ∆𝑥 2). penutup berdasarkan pembahasan di atas, diperoleh kesimpulan bahwa langkah pertama untuk menyelesaikan persamaan fokker-planck dengan metode garis adalah mengganti turunan variabel ruang dengan metode beda hingga, sehingga diperoleh bentuk sistem persamaan diferensial biasa. langkah kedua yaitu menyelesaikan sistem pdb yang terbentuk dengan metode runga-kutta orde empat. solusi yang dihasilkan dengan metode garis mendekati nilai eksaknya. jadi dapat disimpulkan bahwa metode garis merupakan metode yang baik untuk menyelesaikan permasalahan solusi numerik pada persamaan fokker-planck. daftar pustaka [1] chapra, s.c. dan canale, r.p. (2002). numerical methods for engineers with software and programing applications. fourth edition. new york: the mc graw-hill companies, inc. [2] frank, t. (2004). nonlinear fokker-planck equation. munster: springer-berlin. [3] hamdi s., william e. schiesser, graham w. griffiths. (2009). method of lines. scholarpedia, 5. [4] hussain, e. d. (2013). the finite volume method for solving systems of non-linear initial-boundary value problems for pde's. applied mathematical science, 1737-1755. [5] ozdes a. dan aksan e.n. (2006). the method of lines solution of the korteweg-de vries equation for small times. int. j. contemp. math. science, 639-650. [6] palupi, d. (2010). persamaan fokker-planck dan aplikasinya dalam astrofisika. jurnal berkala fisika, a1-a6. [7] pregla, r. (2008). analysis of elegtromagnetic fields and waves: the method of lines. british: john wiley & sons, ltd. [8] sadiku, m.n.o dan obiozor, c.n. (1997). a simple introduction to the method of lines . international journal of electrical engineering education, 282-296. [9] triatmodjo, b. (2002). metode numerik dilengkapi dengan program komputer. yogyakarta: beta offset. [10] zauderer, e. (2006). partial differential equations of applied mathematich. new york: wiley-interscience. pendahuluan kajian teori persamaan diferensial parsial fokker-planck metode beda hingga persamaan fokker-planck metode garis metode runga-kutta pembahasan 1. penerapan metode garis pada penyelesaian persamaan fokker-planck 2. perbandingan solusi analitik dan solusi numerik metode garis pada persamaan fokker-planck 3. interpretasi hasil penutup daftar pustaka 7. vivi aida analisis sistem persamaan diferensial model predator-prey dengan perlambatan vivi aida fitria dosen stmik stie asia malang e-mail: v2_dz@yahoo.com abstrak model predator-prey dengan perlambatan merupakan model interaksi dua spesies antara mangsa dan pemangsa yang berbentuk sistem persamaan diferensial tak liner. adanya waktu perlambatan sangat mempengaruhi kestabilan titik ekuilibrium sistem persamaan diferensial model predator-prey. penelitian ini bertujuan untuk menganalisis pengaruh waktu perlambatan terhadap kestabilan titik ekuilibrium sistem persamaan diferensial model predator-prey. namun sebelum itu, agar dapat diketahui asal mula pembentukan model predator-prey dengan perlambatan akan dianalisis proses terbentuknya model predator-prey dengan perlambatan. penelitian ini menggunakan penelitian kepustakaan, yaitu dengan menampilkan argumentasi penalaran keilmuan yang memaparkan hasil kajian literatur dan hasil olah pikir peneliti mengenai suatu permasalahan atau topik kajian. hasil penelitian ini menunjukkan bahwa ada beberapa nilai perlambatan yang menyebabkan titik ekuilibrium sistem persamaan diferensial model predator-prey stabil, dan ada beberapa nilai perlambatan yang menyebabkan titik ekuilibrium sistem persamaan diferensial model predator-prey tidak stabil. kata kunci: sistem persamaan diferensial, titik ekuilibrium, kestabilan, perlambatan. abstract predator-prey model with the deceleration is a model of two species interactions between prey and predator in the form of differential equations system is not linear. the existence of deceleration time greatly affects the stability of the equilibrium point differential equations system of predator-prey model. this study aims to analyze the effect of time slowing down of the stability of the equilibrium point differential equations system of predator-prey model. but before that, in order to know the origin of the formation of predator-prey models with the slowdown will be analyzed the process of formation of predator-prey model with deceleration. this study uses the research literature, namely by presenting scientific arguments reasoning that presents the results of literature review and the results if the researcher thinks about an issue or topic of study. the results of this study indicate that there are some values that cause the equilibrium point deceleration system of differential equations model of predatorprey stable, and there are some values that cause the equilibrium point deceleration system of differential equations predator-prey model is unstable. keywords: differential equation system, equilibrium points, stability, deceleration. pendahuluan peningkatan jumlah populasi tanpa batas waktu tertentu tidak akan mungkin terjadi baik di laboratorium maupun di alam. misalnya yang terjadi pada sebuah bakteri. bakteri dapat bereproduksi dengan cara pembelahan setiap 20 menit dengan kondisi laboratorium yang ideal. setelah 20 menit, akan terdapat dua bakteri, empat bakteri setelah 40 menit, dan demikian seterusnya. jika keadaan ini berlangsung terus selama satu setengah hari –hanya 36 jam saja akan terdapat bakteri yang cukup untuk membentuk suatu lapisan setebal satu kaki. darwin menghitung bahwa hanya memerlukan 750 tahun bagi sepasang gajah untuk menghasilkan populasi 19 juta gajah (campbell, 2004 :344). pada tulisan ini akan dibahas tentang analisis kestabilan model predator-prey atau analisis kestabilan model mangsa-pemangsa. dari model predator-prey yang stabil akan terciptalah lingkungan yang seimbang. model ini digambarkan dalam suatu persamaan matematika. persamaan ini merupakan pendekatan terhadap suatu fenomena fisik. persamaan yang digunakan adalah persamaan diferensial. persamaan diferensial adalah persamaan yang di dalamnya terdapat turunanturunan. (frank ayres, 1992 : 1) dengan model dapat digambarkan suatu fenomena sehingga menjadi lebih jelas dalam memahaminya. dan dengan adanya model vivi aida fitria 42 volume 2 no. 1 november 2011 predator-prey ini, memudahkan para ahli untuk dapat memproyeksikan populasi/spesies pada suatu waktu tertentu atau menekan laju populasi agar tetap seimbang. dari waktu ke waktu bentuk model predator-prey dimodifikasi sehingga dapat menggambarkan dengan dengan teliti keadaan sebenarnya. begitupula model predator-prey, berawal dari model yang sederhana yang diperkenalkan oleh lotka-voltera, sampai pada model predator-prey dengan perlambatan. di dalam model predator-prey dengan perlambatan dipertimbangkan waktu tunda dari prey pada saat memasuki masa sebelum melahirkan. dengan adanya waktu perlambatan inilah menyebabkan titik ekuilibrium model tidak stabil. berdasarkan permasalahan tersebut dapat dibahas atau dikaji lebih jauh tentang model predator-prey dengan perlambatan. adapun fokus dalam tulisan ini adalah 1) untuk menganalisis pembentukan model predator-prey dengan perlambatan, dan 2) untuk mengetahui pengaruh waktu perlambatan terhadap kestabilan titik ekuilibrium sistem persamaan diferensial model predator-prey dengan perlambatan. kajian teori 1. sistem persamaan diferensial definisi 1 : sistem persamaan diferensial adalah suatu sistem yang memuat n buah persamaan diferensial, dengan n buah fungsi yang tidak diketahui, dimana n merupakan bilangan bulat positif lebih besar sama dengan 2 (finizio dan ladas, 1982:132). antara persamaan diferensial yang satu dengan yang lain saling keterkaitan dan konsisten. bentuk umum dari suatu sistem n persamaan orde pertama mempunyai bentuk sebagai berikut : = = = ⋮ 1 1 1 2 2 2 1 2 1 2 ( , , ,..., ) ( , , ,..., ) ( , , ,..., ) n n n n n dx g t x x x dt dx g t x x x dt dx g t x x x dt (2.1) dengan 1 2 , ,... n x x x adalah variabel bebas dan t adalah variabel terikat, sehingga = = = 1 1 2 2 ( ), ( ),... ( ) n n x x t x x t x x t , dimana n dx dt merupakan derivatif fungsi xn terhadap t, dan ig adalah fungsi yang tergantung pada variabel 1 2 , ,... n x x x dan t (claudia, 2004:702). 2. sistem otonomus definisi 2 : sistem otonomus adalah suatu sistem persamaan diferensial yang berbentuk =ɺ ( , )x f x y =ɺ ( , )y g x y (2.2) dimana fungsi-fungsi f dan g bebas dari waktu (finizio dan ladas, 1982:287). definisi 3: jika ŷ memenuhi =ˆ( ) 0g y maka ŷ adalah sebuah titik ekuilibrium dari = ( ) dy g y dx (claudia, 2004 : 494) contoh : = −ɺ ,x y =ɺy x (2.3) titik ekuilibrium persamaan (2.3) ditentukan oleh dua persamaan − = =0, 0y x . jadi (0,0) merupakan satu-satunya titik ekuilibriumdari persamaan (2.3). jika sistem otonomus (2.2) linier dengan koefisien konstan, maka sistem otonomus tersebut berbentuk : = +ɺx ax by = +ɺy cx dy (2.4) dengan a,b,c, dan d adalah konstanta. jika dimisalkan ad-bc ≠ 0 maka titik (0,0) adalah satu-satunya titik kritis persamaan (2.4) dan persamaan karakteristiknya berbentuk : λ λ− − + − =2 ( ) ( ) 0a d ad bc (2.5) dengan λ 1 dan λ 2 adalah akar-akar persamaannya. sehingga terdapat teorema berikut : teorema 1 a. titik kritis (0,0) dari sistem (2.4) stabil, jika dan hanya jika, kedua akar dari persamaan (2.5) adalah riil dan negatif atau mempunyai bagian riil takposistif. b. titik kritis (0,0) dari sistem (2.4) stabil asimtotis, jika dan hanya jika, kedua akar dari persamaan (2.5) adalah riil dan negatif atau mempunyai bagian riil negatif. c. titik kritis (0,0) dari sistem (2.4) tak stabil, jika salah satu (atau kedua akar) akar dari persamaan (2.5) adalah riil dan positif atau jika paling sedikit satu akar mempunyai bagian riil posistif (finizio dan ladas, 1982:293). sistem persamaan diferensial tak linier seringkali muncul dalam penerapan, misalnya dalam model predator-prey. tetapi hanya beberapa tipe persamaan diferensial tak linier yang dapat diselesaikan secara eksplisit. sedangkan persamaan diferensial tak linier yang analisis sistem persamaan diferensial model predator-prey dengan perlambatan jurnal cauchy – issn: 2086-0382 43 tidak dapat diselesaikan secara eksplisit, dapat diselesaikan dengan melinierkan terlebih dahulu. sistem (2.2) dari dua persamaan diferensial tak linier dengan dua fungsi yang tak diketahui berbentuk : = + +   = + +  ɺ ɺ ( , ) ( , ) x ax by p x y y cx dy q x y (2.6) dimana a, b, c, d, p, q memenuhi syarat : a. a, b, c dan d konstanta real dan ≠ 0 a c d b b. p(x,y) dan q(x,y) mempunyai derivatif parsial kontinu untuk semua (x,y) dan memenuhi : → → = + + = 2 2 2 2( , ) (0 ,0) ( , ) (0 ,0) ( , ) ( , ) lim lim 0 x y x y p x y q x y x y x y sehingga sistem linearnya berbentuk : = + = + ɺ ɺ x ax by y cx dy (2.7) dari syarat di atas maka berlaku : teorema 2 a. titik kritis (0,0) dari sistem tak linier (2.6) adalah stabil asimtotis jika titik kritis (0,0) dari sistem yang dilinierkan (2.7) adalah stabil asimtotis. b. titik kritis (0,0) dari sistem taklinier (2.6) adalah takstabil jika titik kritis (0,0) dari sistem (2.7) adalah takstabil. teorema ini tidak memberikan kesimpulan mengenai sistem (2.6) bila (0,0) hanya merupakan titik stabil dari sistem (2.7). (finizio dan ladas, 1982:294). 3. model matematika pemecahan masalah dalam dunia nyata dengan matematika dilakukan dengan mengubah masalah tersebut menjadi bahasa matematika, proses tersebut disebut pemodelan secara matematik atau model matematika. (baiduri, 2002:1). jadi pemodelan matematika dapat dipandang sebagai terjemahan dari fenomena atau masalah menjadi permasalahan matematika. informasi matematika yang diperoleh dengan melakukan kajian matematika atas model tersebut dilakukan sepenuhnya dengan menggunakan kaedah-kaedah matematika. syarat utama model yang baik adalah sebagai berikut : a. representatif: model mewakili dengan benar sesuatu yang diwakili, makin mewakili, model makin kompleks. b. dapat dipahami/dimanfaatkan: model yang dibuat harus dapat dimanfaatkan (dapat diselesaikan secara matematis), makin sederhana makin mudah diselesaikan. langkah-langkah dalam pemodelan masalah digambarkan dalam diagram berikut : keterangan : 1. identifikasi masalah, yaitu mampu memahami masalah yang akan dirumuskan sehingga dapat ditranslasi ke dalam bahasa matematika 2. membuat asumsi, yaitu dengan cara menyederhanakan banyaknya faktor yang berpengaruh terhadap kejadian yang sedang diamati dengan mengasumsi hubungan sederhana antara variabel. asumsi tersebut dibagi dalam dua kategori utama : a. klasifikasi variabel pemodel mengidentifikasi variabel terhadap hal-hal yang mempengaruhi tingkah laku pengamatan b. menentukan interelasi antara variabel yang terseleksi untuk dipelajari pemodel membuat sub model sesuai asumsi yang telah dibuat pada model utama, kemidian mempelajari secara terpisah pada satu atau lebih variabel bebas. 3. menyelesaikan atau menginterpretasikan model setelah model diperoleh kemudian diselesaikan secara matematis, dalam hal ini model yang digunakan dan penyelesaiannya menggunakan persamaan diferensial. apabila pemodel mengalami kesulitan untuk menyelesaikan model dan interpretasi model, maka kelangkah 2 dan membuat asumsi sederhana tambahan atau kembali kelangkah 1 untuk membuat definisi ulang dari permasalahan. penyederhanaan atau definisi ulang sebuah model merupakan bagian yang penting dalam matematika model. 4. verifikasi model sebelum menyimpulkan kejadian dunia nyata dari hasil model, terlebih dahulu model tersebut harus diuji. beberapa pertanyaan yang diajukan sebelum melakukan uji dan mengumpulkan data, yaitu : 1) apakah model menjawab masalah yag telah diidentifikasi? 2) apakah model membuat pemikiran yang sehat? 3) apakah data (sebaiknya menggunakan data aktual yang diperoleh vivi aida fitria 44 volume 2 no. 1 november 2011 dari observasi empirik) dapat dikumpulkan untuk menguji dan mengoperasikan model dan apakah memenuhi syarat apabila diuji (baiduri, 2002:15-17). 4. model logistik model logistik atau model verhulst atau kurva pertumbuhan logistik adalah sebuah model pertumbuhan populasi. model logistik termasuk model yang memiliki waktu kontinu. model tersebut dideskripsikan sebagai berikut:  = −    1 dx x rx dt k (2.8) konstanta r, diasumsikan positif. konstanta r adalah laju pertumbuhan intrinsik karena perbandingan laju pertumbuhan untuk x diperkirakan sama dengan r. konstanta positif k biasanya mengarah kepada daya kapasitas kesehatan lingkungan yaitu kemampuan menahan populasi agar tetap maksimum. solusi dari model logistik tersebut adalah : − = + − 0 0 0 ( ) ( ) rt x k x t x k x e (2.9) model logistik mempunyai dua titik ekuilibrium, yaitu x = 0 dan x = k. titik ekulibrium pertama tidak stabil sementara titik ekuilibrium kedua adalah stabil global. beberapa kurva dari solusi model logistik dengan titik awal yang berbeda dapat dilihat pada grafik 2.1. gambar 2.1. grafik model logistik dari persamaan (2.9) dengan k= 100 ,r=1 dan lima kondisi awal masing-masing x(0)= 10, x(0)=30, x(0)=80, x(0) = 120 dan x(0) = 150 5. model logistik dengan perlambatan model logistik tunggal dengan perlambatan adalah ( ) ( ) ( ) τ − = −     1 dx t x t rx t dt k , (2.10) dimana τ adalah sebuah waktu perlambatan dan dianggap positif. suatu titik ekuilibrium positif dari model ini adalah k. hal ini diusulkan oleh hutchinson di gopalsamy, model (2.10) tersebut bisa digunakan pada model pertumbuhan populasi jenis dinamik tunggal terhadap ketahanan level k, dengan sebuah konstanta laju pertumbuhan intrinsik r. bentuk ( )τ − −     1 x t k pada model (2.10) merupakan sebuah kepadatan tergantung pada mekanisme pengaruh arus balik yang mengambil τ satuan waktu untuk menanggapi perubahan pada kepadatan populasi diwakili pada model (2.10) oleh x. model logistik dengan perlambatan (2.10) dikenal sebagai persamaan perlambatan verhulst atau persamaan hutchinson. persamaan hutchinson telah dipelajari di beberapa jurnal dan buku. selanjutnya akan dianalisis stabilitas lokal dari titik ekuilibrium. untuk menganalisis, digunakan sebuah metode standar yaitu metode linierisasi disekitar titik ekuilibrium. misalkan ( ) ( )= −u t x t k , maka ( ) ( )= du t dx t dt dt . mensubtitusi ( ) ( )= +x t u t k ke dalam persamaan (2.10) untuk memperoleh ( ) ( )( ) ( )τ − += + −     1 du t u t k r u t k dt k (2.11) ( ) ( ) ( ) ( )τ τ−= − − − du t r u t u t ru t dt k . karena x(t) tertutup untuk k, ( ) ( )τ−u t u t dapat dihilangkan. selanjutnya didapatkan suatu model linier ( ) ( )τ= − − du t ru t dt . (2.12) untuk memahami stabilitas titik ekuilibrium nol dari model (2.12), dipertimbangkan persamaan karakteristik pada model (2.12). pensubtitusian pada fungsi tes ( ) λτ=x t e ke dalam model (2.12) menghasilkan persamaan karakteristik ( )λ τλτλ −= − te re karena λτ ≠ 0e , maka λτλ −+ = 0re . (2.13) lemma 1 misalkan > 0r dan τ > 0 jika τ ≤ 1 re maka persamaan (2.13) memiliki akar-akar persamaan karakteristik negatif bukti misalkan ( ) λτλ λ −= +f re . dengan catatan bahwa λ bukan bilangan riil nonnegatif. akan dibuktikan bahwa akar-akar dari ( )λf adalah bukan bilangan komplek. karena ( ) λτλ λ −= +f re analisis sistem persamaan diferensial model predator-prey dengan perlambatan jurnal cauchy – issn: 2086-0382 45 maka ( ) λτλ τ −= −' 1f r e dan ( )λ τ τ∗ = 1 ln r adalah titik kritik dari ( )λf oleh karena itu, ( ) λτλ τ −= 2"f r e yang positif. ini berarti bahwa nilai dari titik kritik memberikan nilai minimum untuk ( )λf . selanjutnya karena ( )( )λ τ τ∗ = + 1 ( ) ln 1f r yang sama dengan nol jika τ = 1 r e atau τ =    1 re , dan kurang dari nol jika τ < 1 re maka persamaan (2.13) hanya memiliki satu akar, yaitu τ = 1 re , dan jika ( )λ τ τ = 1 ln r persamaan (2.13) memiliki dua akar riil negatif. jika ( )λ• > 0f , yaitu τ > 1 r e , ini mengakibatkan bahwa tidak ada akar riil dari persamaan karakteristik (2.13). kondisi persamaan karakteristik ini mempunyai akar komplek konjugat. jika dimisalkan λ ρ ω= + i , ρ ∈ r , [ )ω ∈ ∞0, , sebagai sebuah akar dari (2.13), maka ( ) ( ) ( )( ) ρ ω τ ρτ ρ ω ωτ ωτ − + − + = − = − −cos sin i i re re i , maka didapatkan dua persamaan dengan bagian riil dan bagian imajinernya: ( )ρτρ ωτ−= − cosre , (2.14.a) ( )ρτω ωτ−= sinre . (2.14.b) (syamsuddin, 2006:3.7). lemma 2 misalkan > 0r dan τ > 0 . jika π τ< < 1 2re r maka akar dari persamaan karakteristik (2.13) adalah komplek konjugat dengan bagian riil negatif. bukti misalkan ( ) λτλ λ −= +f re . dari persamaan (2.13) bahwa λ bukan bilangan riil nonnegatif. maka ( ) λτλ τ −= −' 1f r e dan ( )λ τ τ• = 1 ln r adalah sebuah titik ekuilibrium untuk ( )λf . selanjutnya ( ) λτλ τ −= 2"f r e adalah positif. ini berarti bahwa nilai dari titik ekuilibrium memberi nilai minimum untuk ( )λf . fungsi ( )λf tidak mempunyai akar riil dimana ( ) ( )( )λ τ τ• = + > 1 ln 1 0f r dan ini terjadi ketika τ< 1 re . sekarang akan ditunjukkan bahwa akar dari ( )λf adalah sebuah bilangan komplek dengan bagian riil negatif. misalkan persamaan (2.13) tersebut mempunyai akar λ ρ ω= + i dengan ρ ≥ 0 . karena λ = 0 adalah bukan akar dari persamaan karakteristik (2.13) dan diasumsikan ω > 0 hal ini menunjukkan (dari persamaan (2.14b)) bahwa ρτ πωτ τ ωτ−< = <0 sin 2 r e hal ini menunjukkan bahwa sisi kiri dari persamaan (2.14a) adalah nonnegatif. karena kontradiksi hal ini membuktikan bahwa ρ < 0 . perhatikan konjugat dari λ membuktikan persamaan karakteristik (2.13) (syamsuddin, 2006: 3.9). berikut adalah kurva dari solusi model logistik dengan beberapa nilai perlambatan berbeda : gambar 2.2. grafik model logistik dari persamaan (2.10) dengan k= 100, r=1 dan tiga nilai perlambatan yaitu τ τ= =1.5, 1.935 dan τ = 2.5 6. model populasi predator-prey dalam subbab ini, dibahas tentang model sederhana dari predator-prey, yang didefinisikan sebagai konsumsi predator terhadap prey. model predator-prey yang paling sederhana didasarkan pada model lotka-volterra (lotka, 1932 ; volterra, 1926) dalam claudia (2004 :760) , yang dideskripsikan dalam kata-kata volterra, sebagai berikut : ”kasus pertama yang saya pertimbangkan adalah bahwa ada dua jenis hubungan. yang pertama menemukan makanan yang cukup di lingkungannya dan akan berkembang terus meskipun hidup sendirian, dan yang kedua mati karena kekurangan makanan jika dibiarkan hidup sendiri. tetapi makanan yang kedua untuk vivi aida fitria 46 volume 2 no. 1 november 2011 makanan yang pertama. sehingga dua jenis ini dapat hidup berdampingan. angka perbandingan dari kenaikan jenis makanan dikurangi jumlah individu dari pertumbuhan jenis makanan, saat tambahan jenis makanan berkembang seiring dengan berkembangnya jumlah individu dari jenis makanan. ” model lotka-voltera tersusun dari pasangan persamaan diferensial yang mendeskripsikan predator-prey dalam kasus yang paling sedehana. model ini membuat beberapa asumsi : 1. populasi prey akan tumbuh secara eksponen ketika tidak adanya predator 2. populasi predator akan mati kelaparan ketika tidak adanya populasi prey 3. predator dapat mengkonsumsi prey dengan jumlah yang tak terhingga 4. tidak adanya lingkungan yang lengkap (dengan kata lain, kedua populasi berpindah secara acak melalui sebuah lingkungan yang homogen) selanjutnya bentuk verbal ini diterjemahkan ke dalam sebuah sistem persamaan diferensial. diasumsikan bahwa populasi prey berkurang ketika predator membunuhnya dan bertahan hidup (tidak mengurangi populasi prey) ketika predator hanya menyerangnya. model dengan laju perubahan dari populasi prey (x) dan populasi predator (y) adalah : α β  = − −    = − + 1 dx x rx xy dt k dy cy xy dt (2.15) parameter model di atas yaitu : k = daya kapasitas r = laju pertumbuhan intrinsik prey c = laju kematian jika predator tanpa prey α = laju perpindahan dari prey ke predator β = laju pepindahan dari predator ke prey model di atas dibentuk dengan analisis sebagai berikut : dimulai dengan memperhatikan apa yang terjadi pada populasi predator ketika tidak adanya prey, tanpa sumber makanan, bilangannya diharapkan berkurang secara eksponensial, dideskripikan oleh persamaan di bawah ini : = − dy cy dt persamaan ini menggunakan hasil kali dari bilangan predator (y) dan kelajuan kematian predator (c). untuk mendeskripsikan penurunan kelajuan (karena tanda negatif pada bagian kanan persamaan) dari populasi predator dengan pengaruh waktu. dengan adanya prey bagaimanapun juga pengurangan ini dilawan oleh laju kelahiran predator, yang ditentukan oleh laju konsumsi ( β xy ). dimana laju penyerangan ( β ) dikalikan dengan bilangan y dan bilangan x. bilangan predator dan prey naik ketika pertemuan predator dan prey lebih sering, tetapi laju aktual dari konsumsi akan tergantung pada laju penyerangan ( β ). persamaan populasi predator menjadi β= − + dy cy xy dt perkalian β y adalah tanggapan predator secara numerik atau peningkatan perkapita dari fungsi prey yang melimpah. dan untuk perkalian β xy menunjukkan bahwa kenaikan populasi predator sebanding dengan perkalian dan prey yang melimpah. beralih pada populasi prey, kita berharap tanpa serangan predator, bilangan prey akan naik secara eksponensial. persamaan di bawah ini mendeskripsikan laju kenaikan populasi prey dengan pengaruh waktu, dimana r adalah laju pertumbuhan intrinsik prey dan x adalah jumlah dari populasi prey. = dx rx dt di hadapan predator, bagaimanapun juga populasi prey dicegah dari peningkatan eksponensial secara terus-menerus. karena model predator prey memiliki waktu yang kontinu dan mengisyaratkan tentang model pertumbuhan populasi maka termasuk dalam model logistik. jadi persamaan di atas menjadi: = −(1 ) dx x rx dt k dengan adanya predator bagaimanapun juga kenaikan ini dilawan oleh laju kematian prey karena adanya penyerangan dari predator, yang ditentukan oleh laju konsumsi ( α xy ).di mana laju penyerangan ( α ) dikalikan dengan bilangan y dan bilangan x. bilangan predator dan prey turun ketika pertemuan predator dan prey lebih sering, tetapi laju aktual dari konsumsi akan tergantung pada laju penyerangan ( α ). persamaan populasi prey menjadi: α= − −(1 ) dx x rx xy dt k adapun analisis numerik model predator prey adalah : 1. titik ekuilibrium model (2.15) adalah e0 = (0,0), e1 = (k,0) dan e* =(x*,y*) = β β αβ  −     ( ) , c r k c k . agar mendapatkan sebuah analisis sistem persamaan diferensial model predator-prey dengan perlambatan jurnal cauchy – issn: 2086-0382 47 titik ekuilibrium yang positif diasumsikan bahwa β −k c > 0. matriks jacobian dari model (2.15) yaitu α α β β  − − − =   − +  2rx r y x j k y c x 2. matriks jacobian pada titik ekuilibrium e* adalah α β β β α − −     =  −     0 rc c k j r k rc k 3. persamaan karakteristik dari matriks jacobian di atas adalah λ λ λ β β β = + + −2( ) ( ) rc c f r k rc k k 4. misalkan β = rc p k dan β β = −( ) c q r k rc k maka nilai eigen dari matrik jacobian di atas adalah λ − ± − = 2 1,2 4 2 p p q karena p dan q adalah bilangan positif, maka nilai eigen dari p dan q memiliki bagian yang riil negatif. hal ini berarti bahwa titik ekuilibrium e* adalah asimtot lokal stabil. karena β −k c > 0 maka titik ekuilibrium e* juga asimtot global stabil. 7. model populasi predator prey dengan perlambatan waktu perlambatan (perlambatan) sangat penting untuk diperhitungkan di dunia permodelan karena keputusan seringkali dibuat berdasarkan pada keterangan realita. merupakan hal yang penting untuk mempertimbangkan model populasi dimana laju pertumbuhan populasi tidak hanya tergantung pada ukuran populasi pada satu waktu tertentu tetapi juga tergantung pada ukuran populasi pada ( )τ−t , dimana τ adalah waktu perlambatan. berikut adalah model populasi predatorprey dengan perlambatan yang diperkenalkan olah may pada tahun 1974: τ α β − = − −    = − + ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) dx t x t rx t x t y t dt k dy t cy t x t y t dt (2.16) dimana τ α, , , ,r k c dan β adalah konstanta positif. parameter model (2.16) yaitu : k = daya kapasitas r = laju pertumbuhan intrinsik prey c = laju kematian jika predator tanpa prey α = laju perpindahan dari prey ke predator β = laju perpindahan dari predator ke prey τ = waktu perlambatan untuk menganalisis kestabilan titik ekuilibrium dari model dengan perlambatan, harus mengelinierisasi model di sekitar titik ekuilibrium, kemudian memeriksa nilai eigen pada persamaan karakteristik. titik ekuilibrium asimtot stabil jika dan hanya jika akar-akar dari persamaan karakteristik mempunyai bagian real negatif . langkah selanjutnya dalam menganalisis titik ekuilibrium dari model predator-prey dengan perlambatan dibutuhkan teorema berikut: teorema 3 misalkan β − > 0k c dan τ ± k didefinisikan pada persamaan (2.16) berbentuk : π π τ ω ω + + + = + / 2 2 k k . dan π π τ ω ω − − − = + 3 / 2 2 , k k maka terdapat sebuah bilangan positif m sedemikian hingga m merubah dari stabil ke tidak stabil dan ke stabil. dengan kata lain, ketika ( ) ( )τ τ τ τ τ τ+ − + − +−∈ ∪ ∪ ∪ 0 0 1 10, ) , ... ,m m , titik ekuilibrium ∗ e pada model (3.2) stabil, dan ketika ( ) ( )τ τ τ τ τ τ τ+ − + − + −− −∈ ∪ ∪ ∪ 0 0 1 1 1 1, ) , ... ,m m , titik ekuilibrium ∗ e tidak stabil. oleh karena itu ada bifurkasi untuk τ τ ±= k , k=0,1,2,… bukti diketahui bahwa titik ekuilibrium ∗ e stabil untuk τ = 0 . maka untuk membuktikan teorema 3 hanya dibutuhkan kondisi secara transversal. τ τ λ τ += > (re ) 0 k d d dan τ τ λ τ −= < (re ) 0 k d d . persamaan λτλ λ −+ + =2 0pe q dideferensialkan menjadi λτ λτ λ λ λ τ τ λ λ τ λ τ − − +  + − − =    2 0, d d pe d d d pe d ( )( )λτ λτλλ λτ λ τ − −+ − = 22 1 . d pe pe d agar lebih mudah dipahami, maka λ τ       d d diubah menjadi λ τ −       1 d d . maka didapatkan : λτ λτ λ λ λτ τ λ λ τ λλ − + −  =    + = − 1 2 2 2 (1 ) 2 d e p d p e p p vivi aida fitria 48 volume 2 no. 1 november 2011 dari persamaan karakteristik λτλ λ −+ + =2 0pe q diketahui bahwa λτ λ λ − = +2 p e q λτ =e maka didapatkan λ λ τ τ λλ λ − − +  = −  +  1 2 2 2 ( ) d q d q oleh karena itu ( ) λ ω λ ω λ ω λ ω λ λ τ τ λ λ λ ω ω ω ω ω ω ω − = = = =     =             − =    +    +   +    − = + − + −   − =   −  = − 1 2 4 2 2 4 2 4 2 2 2 2 4 2 (re ) re 1 re re 1 ( ) i i i i d d sign sign d d sign q q q q sign q q q sign q sign q karena ω ω ω− = − +4 2 4 2 22 ( 2 )q p q maka didapatkan λ ω λ ω ω τ ω =   = − +    = − + 4 2 2 2 2 (re ) (2 ( 2 ) ) (2 ( 2 )) i d sign sign p q d sign p q jadi dapat dibuktikan bahwa kondisi transversal telah terpenuhi. (syamsuddin, 2006:4.10) pembahasan 3.1 analisis pembentukan model predator prey dengan perlambatan berikut adalah model populasi predatorprey dengan perlambatan yang diperkenalkan olah may pada tahun 1974: τ α β − = − −    = − + ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) dx t x t rx t x t y t dt k dy t cy t x t y t dt (3.1) dimana τ α, , , ,r k c dan β adalah konstanta positif. parameter model (3.1) yaitu : k = daya kapasitas r = laju pertumbuhan intrinsik prey c = laju kematian jika predator tanpa prey α = laju perpindahan dari prey ke predator β = laju perpindahan dari predator ke prey τ = waktu perlambatan dimulai dengan menganalisis pembentukan model predator. pertama, dengan memperhatikan apa yang terjadi pada populasi predator ketika tidak adanya prey, tanpa sumber makanan, bilangannya diharapkan berkurang secara eksponensial, dideskripikan oleh persamaan di bawah ini : = − dy cy dt persamaan ini menggunakan hasil kali dari bilangan predator (y) dan kelajuan kematian predator (c). tanda negatif pada bagian kanan persamaan untuk mendeskripsikan penurunan kelajuan dari populasi predator terhadap pengaruh waktu. dengan adanya prey (sebagai konsumsi predator) pengurangan ini dilawan oleh laju kelahiran predator, yang ditentukan oleh laju konsumsi ( β xy ). dimana laju penyerangan ( β ) dikalikan dengan bilangan y dan bilangan x. bilangan predator dan prey naik ketika pertemuan predator dan prey lebih sering, tetapi laju aktual dari konsumsi akan tergantung pada laju penyerangan ( β ). persamaan populasi predator menjadi β= − + dy cy xy dt perkalian β y adalah tanggapan predator secara numerik atau peningkatan perkapita dari fungsi prey yang melimpah. dan untuk perkalian β xy menunjukkan bahwa kenaikan populasi predator sebanding dengan perkalian dan prey yang melimpah. selanjutnya dianalisis populasi prey. diharapkan tanpa serangan predator, bilangan prey akan naik secara eksponensial. persamaan di bawah ini mendeskripsikan laju kenaikan populasi prey dengan pengaruh waktu, dimana r adalah laju pertumbuhan intrinsik prey dan x adalah jumlah dari populasi prey. = dx rx dt di hadapan predator, bagaimanapun juga populasi prey mencegah agar tidak terjadi peningkatan eksponensial secara terus-menerus. karena model predator-prey memiliki waktu yang kontinu dan mengisyaratkan tentang model pertumbuhan populasi maka termasuk dalam model logistik. jadi persamaan di atas menjadi: = −(1 ) dx x rx dt k dengan adanya predator, kenaikan ini dilawan oleh laju kematian prey karena adanya penyerangan dari predator, yang ditentukan oleh laju konsumsi ( α xy ). dimana laju penyerangan ( α ) dikalikan dengan bilangan y dan bilangan x. bilangan predator dan prey turun ketika pertemuan predator dan prey lebih sering, tetapi laju aktual dari konsumsi akan tergantung pada analisis sistem persamaan diferensial model predator-prey dengan perlambatan jurnal cauchy – issn: 2086-0382 49 laju penyerangan ( α ). persamaan populasi prey menjadi: α= − −(1 ) dx x rx xy dt k karena dalam kondisi tertentu pada populasi prey tedapat keterlambatan (waktu τ ) pada laju kelahiran maka persamaan populasi menjadi : τ α − = − − ( ) (1 ) dx x rx xy dt k jadi sistem persamaan diferensial model predator prey dengan perlambatan adalah : τ α β − = − −    = − + ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) dx t x t rx t x t y t dt k dy t cy t x t y t dt 3.2 analisis model predator prey dengan perlambatan berikut ini adalah pembahasan tentang analisis model predator-prey dengan mempertimbangkan waktu perlambatan, may (1974) telah menunjukkan model sistem persamaan diferensial di bawah ini : τ α β − = − −    = − + ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) dx t x t rx t x t y t dt k dy t cy t x t y t dt (3.2) dimana τ α, , , ,r k c dan β adalah konstanta positif. model (3.2) memuat sebuah single diskrit perlambatan. jika masa perlambatan dari prey adalah τ , maka fungsi laju pertumbuhan perkapita akan membawa sebuah waktu perlambatan τ . di dalam tulisan ini akan dianalisis pengaruh waktu perlambatan terhadap kestabilan titik ekuilibrium dari sistem. titik equilibrium model (3.2) ada 2 yaitu : 1. titik (0,0) 2. titik kedua yang memenuhi = 0 dy dt dan = 0 dx dt yaitu : β β β = − + = = = 0 0 dy dt cy xy cy x y c x dan α α α β α β αβ = − − = − = − = − = − = 2 2 0 0 ( ) dx dt rx rx xy rx rx y x rx r y rc r k y r k c y k jadi titik ekuilibrium yang kedua yaitu β β αβ  −     ( ) , c r k c k . dalam tulisan ini akan difokuskan pada analisis kestabilan dari titik equilibrium e*, karena titik equilibrium tersebut berada di kuadran positif dan asimtot stabil ketika tidak ada waktu perlambatan. untuk memahami kestabilan lokal dari titik equilibrium e* pada model (3.2), akan dianalisis model sistem persamaaan diferensial nonlinear setelah model tersebut dilinearisasi. misalkan = − *( ) ( )u t x t x dan ∗= −( ) ( )v t y t y . maka setelah disubtitusi ke dalam model (3.2) didapatkan : ( )( ) ( )( )( ) ( )( ) ( )( )( ) τ α β ∗ ∗ ∗ ∗ ∗ ∗ ∗  − + = + −    − + + = − + + + + ɺ ɺ ( ) ( ) 1 ( ) ( ) ( ) u t x u t r u t x k u t x v t y v t c v t y u t x v t y atau τ τ α α α α ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = + − − − − − − − − − − ɺ 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r u t ru t rx u t u t k r r r x u t x u t x k k k u t v t x v t y u t x y β β β β ∗ ∗ ∗ ∗ ∗ = − − + + + + ɺ( ) ( ) ( ) ( ) ( ) ( ) v t cv t cy u t v t y u t x v t x y setelah menyederhanakan dan mengabaikan hubungan hasil kali, didapat sebuah model linear: τ α∗ ∗= − − −ɺ( ) ( ) ( ) r u t x u t x v t k β ∗=ɺ( ) ( ).v t y u t menganalisis kestabilan lokal dari titik ekulibrium titik ∗ e pada model (3.2) ekuivalen dengan menganalisis kestabilan dari titik ekulibrium nol pada model linear. dari model yang telah dilinearisasi didapatkan matrik jacobian sebagai berikut : vivi aida fitria 50 volume 2 no. 1 november 2011 ∂ ∂   ∂ ∂  = ∂ ∂    ∂ ∂  1 1 2 2 f f u v j f f u v misalkan = ɺ 1 ( )f u t dan = ɺ 1 ( )f v t maka didapatkan matriks jacobian sebagai berikut: λτ α β ∗ − ∗ ∗  − − =     0 r x e x j k y polinom karakteristik dari j adalah : λτ λτ λτ λ α λ λ β λ α β λ λ λ αβ ∗ − ∗ ∗ ∗ − ∗ ∗ ∗ − ∗ ∗   − −   − = −           + =   −  = + +2 0 det( ) det 0 0 det r x e x i j k y r x e x k y r x e x y k jadi persamaan karakteristik dari j adalah sebagai berikut : λτλ λ −+ + =2 0pe q (3.3) dimana ∗= r p x k dan αβ ∗ ∗= .q x y untuk τ = 0 persamaan karakteristik (3.3) menjadi : λ λ+ + =2 0p q (3.4) yang memiliki akar-akar λ − ± − = 2 1,2 4 . 2 p p q (3.5) karena p dan q keduanya adalah bilangan positif , nilai eigen dari persamaan karakteristik (3.4) memiliki bagian riil negatif. sedangkan untuk τ ≠ 0, jika λ ω ω= >, 0,i adalah sebuah akar dari persamaan karakteristik (3.4) maka didapatkan : ωτω ω −− + + =2 0,iip e q ω ω ωτ ω ωτ− + + + =2 cos( ) sin( ) 0.ip p q dengan memisahkan bagian riil dan imajinernya, didapatkan : ω ω ωτ− + + =2 sin( ) 0,p q (3.6) ω ωτ =cos( ) 0.p kedua persamaan diatas dikuadratkan menjadi: ω ωτ ω ω= − +2 2 2 4 2 2sin ( ) 2p q q ω ωτ =2 2 2cos ( ) 0.p kedua persamaan diatas dijumlahkan, menjadi: ω ωτ ω ωτ ω ω ω ω ω + = − + = − + 2 2 2 2 2 2 4 2 2 2 2 4 2 2 sin ( ) cos ( ) 2 2 p p q q p q q kemudian dikelompokkan didapatkan polinomial pangkat empat: ω ω− + + =4 2 2 2( 2 ) 0,p q q (3.7) akar-akar dari persamaan diatas yaitu: { }ω± = + ± + −2 2 2 2 21 ( 2 ) ( 2 ) 4 , 2 p q p q q atau { }ω± = + ± +2 2 4 21 ( 2 ) 4 . 2 p q p p q (3.8) dari persamaan (3.7) dapat diketahui bahwa ada solusi positif pada ω± 2 . selanjutnya dapat ditemukan nilai τ ± k dari subtitusi ω± 2 ke dalam persamaan (3.6) dan penyelesaian untuk τ , yaitu: ωτ =cos 0 ωτ ωτ π ωτ π − + − = = + = + 1 cos 0 90 2 270 2 k k didapatkan π π τ ω ω + + + = + / 2 2 , k k π π τ ω ω − − − = + 3 / 2 2 , k k k=0,1,2,… (3.9) sebagai contoh model (3.2) dimasukkan parameter r =1, k = 200, α = 0.15 , c=1 dan β =0.1. maka didapatkan beberapa grafik sebagai berikut: gambar 3.1 grafik persamaan diferensial dari prey (x(t)) dengan x(0)=9 dan τ = 0.5 analisis sistem persamaan diferensial jurnal cauchy – issn: 2086-0382 gambar 3.2 grafik persamaan diferensial dari predator (y(t)) dengan y(0)=9 dan dan berikut ini adalah jumlah populasi predator dan dan prey pada saat nilai perlambatan τ = 0.5 : hari ke prey (x(t)) predator(y(t)) 1 9,00000 2 11,16000 3 12,91882 . . . . . . 400 9,99372 dari grafik (3.1) dan (3.2) di atas dapat diamati bahwa dengan nilai perlambatan ekuilibrium (10;6.33) sistem persamaan diferensial model predator-prey stabil. gambar 3.3 grafik jumlah populasi preda dan prey dengan τ dari grafik (3.3) dapat diamati bahwa dengan nilai perlambatan τ = 2.7 titik ekuilibrium (10;6.33) sistem persamaan diferensial model predator-prey tidak stabil. analisis sistem persamaan diferensial model predator-prey dengan perlambatan 0382 gambar 3.2 grafik persamaan diferensial dari dengan y(0)=9 dan τ = 0.5 dan berikut ini adalah jumlah populasi dan dan prey pada saat nilai predator(y(t)) 5,00000 5,58000 7,20870 . . . 6,65449 dari grafik (3.1) dan (3.2) di atas dapat diamati bahwa dengan nilai perlambatan τ = 0.5 titik ekuilibrium (10;6.33) sistem persamaan prey stabil. gambar 3.3 grafik jumlah populasi predator τ = 2.7 dari grafik (3.3) dapat diamati bahwa dengan titik ekuilibrium (10;6.33) sistem persamaan diferensial model dari persamaan (3.9) yang telah diubah ke dalam bentuk radian didapatkan nilai sebagai berikut : τ τ τ τ τ τ + + + + + + = = = = = = 0 1 2 3 4 5 1.57080 7.85398 14.13717 20.42035 26.70354 32.98672 τ τ τ τ τ τ − − − − − − 0 1 2 3 4 5 berdasarkan teorema 3 dengan nilai perlambatan (τ ∈ ∪ ∪ ∪ ∪ ∪ (0,1.57080) 5.23599,7.85398 (12.21730,1413717) (19.19862,20.42035) (26.17994,26.70354) (33.16126,32.98672) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator stabil. sedangkan dengan nilai perlambatan τ ∈ ∪ ∪ ∪ ∪ ∪ ∪ (1.57080,5.23599) (7.85398,12.21730) (14.13717,19.19882) (20.42035 26.17994) (26.70354,3316126) (32.98672, 40.14257) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator stabil. untuk mengetahui pengaruh waktu perlambatan terhadap titik ekuilibrium sistem, dapat diamati dengan membandingkan antara model dengan waktu perlambatan dan model tanpa waktu perlambatan. gambar 3.4 grafik persamaan diferensial dari prey (x(t)) tanpa waktu perlambatan prey dengan perlambatan 51 dari persamaan (3.9) yang telah diubah ke dalam bentuk radian didapatkan nilai-nilai perlambatan τ τ τ τ τ τ − − − − − − = = = = = = 0 1 2 3 4 5 5.23599 12.21730 19.19862 26.17994 33.16126 40.14257 berdasarkan teorema 3 dengan nilai perlambatan )∈ ∪ ∪ ∪ ∪ (0,1.57080) 5.23599,7.85398 (12.21730,1413717) (19.19862,20.42035) (26.17994,26.70354) (33.16126,32.98672) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator-prey stabil. sedangkan dengan nilai perlambatan ∈ ∪ ∪ ∪ ∪ ∪ (1.57080,5.23599) (7.85398,12.21730) (14.13717,19.19882) (20.42035 26.17994) (26.70354,3316126) (32.98672, 40.14257) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator-prey tidak untuk mengetahui pengaruh waktu perlambatan terhadap titik ekuilibrium sistem, membandingkan grafik antara model dengan waktu perlambatan dan model tanpa waktu perlambatan. gambar 3.4 grafik persamaan diferensial dari tanpa waktu perlambatan vivi aida fitria 52 gambar 3.5 grafik persamaan diferensial dari prey (x(t)) dengan waktu perlambatan adapun nilai numerik solusi persamaan diferensial dengan waktu perlambatan dan tanpa waktu perlambatan, sebagai berikut : hari ke dengan waktu perlambatan tanpa waktu perlambatan 1 9,00000 2 11,16000 3 12,91882 . . . . . . 280 10,04294 dari nilai numerik pada tabel di atas dapat diamati bahwa titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator prey tanpa waktu perlambatan stabil, terjadi pada hari ke-235. sedangkan titik ekuilibrium (10;6.33) sistem persamaan diferensial dengan waktu perlambatan stabil, terjadi pada hari ke 251, lebih lambat jika dibandingkan dengan kestabilan titik ekuilibrium sistem persamaan diferensial tanpa waktu perlambatan penutup berdasarkan hasil penelitian kesimpulan tentang analisis pembentukan model matematika, yaitu : 1. analisis pembentukan model predator dengan perlambatan a. terjadi penurunan jumlah populasi predator karena tidak ada prey sebagai sumber makanan. oleh karena persamaannya bertanda negatif. b. terjadi kenaikan jumlah populasi predator karena adanya laju kelahiran predator yang ditentukan oleh laju konsumsi predator. volume 2 gambar 3.5 grafik persamaan diferensial dari dengan waktu perlambatan τ = 0.5 adapun nilai numerik solusi persamaan diferensial dengan waktu perlambatan dan tanpa waktu perlambatan, sebagai berikut : tanpa waktu perlambatan 9,00 10,85 12,28 . . . 10,00 dari nilai numerik pada tabel di atas dapat diamati bahwa titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predatorprey tanpa waktu perlambatan stabil, terjadi titik ekuilibrium (10;6.33) sistem persamaan diferensial dengan waktu perlambatan stabil, terjadi pada hari ke251, lebih lambat jika dibandingkan dengan kestabilan titik ekuilibrium sistem persamaan diferensial tanpa waktu perlambatan. penelitian dapat ditarik is pembentukan model analisis pembentukan model predator dengan terjadi penurunan jumlah populasi predator karena tidak ada prey sebagai sumber makanan. oleh karena itu persamaannya bertanda negatif. terjadi kenaikan jumlah populasi predator karena adanya laju kelahiran predator yang ditentukan oleh laju konsumsi predator. 2. analisis pembentukan model prey a. terjadi kenaikan jumlah populasi prey karena tidak adanya pemangsa. oleh karena itu persamaannya bertanda positif. b. model prey termasuk dalam model logistik. c. terjadi penurunan jumlah populasi prey karena adanya laju kematian prey dengan adanya penyerangan dari predator. selanjutnya untuk mengetahu waktu perlambatan terhadap kestabilan titik ekuilibrium pada sistem persamaan diferensial model predator dengan menganalisis titik ekuilibrium β β αβ  −     ( ) , c r k c k . karena titik equilibrium tersebut berada di kuadran positif dan asimtot stabil ketika tidak ada waktu perlambatan. dari hasil analisis pada pembahasan didapatkan nilai perlambatan (τ ∈ ∪ ∪ ∪ ∪ ∪ (0,1.57080) 5.23599,7.85398 (12.21730,1413717) (19.19862,20.42035) (26.17994,26.70354) (33.16126,32.98672) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator stabil. sedangkan dengan nilai p τ ∈ ∪ ∪ ∪ ∪ ∪ ∪ (1.57080,5.23599) (7.85398,12.21730) (14.13717,19.19882) (20.42035 26.17994) (26.70354,3316126) (32.98672, 40.14257) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator tidak stabil. daftar pustaka [1] abdurrahman, bin muhammad. 2008. tafsir al-usyr al-akhir, http://www.tafseer.info. diakses tanggal 7 juli 2009 [2] al katsir, abul fida’. 1993. beirut : darul fikri. [3] al mahalli. jalaluddin. 1990. toha putra : semarang. [4] al maraghi, musthofa ahmad. 1971. maraghi. beirut : darul fikri [5] anonim. 2009. proses pemodelan matematika,http://www.sipoel.unimed.in/ file.php/44/course/bab_ii/bab_2.doc. diakses tanggal 19 mei 2008. [6] ayres, frank. 1992. persamaan diferensial dalam satuan si metric. [7] baiduri. 2002. persamaan diferensial dan matematika model. malang: umm press. no. 1 november 2011 analisis pembentukan model prey terjadi kenaikan jumlah populasi prey karena tidak adanya predator sebagai pemangsa. oleh karena itu persamaannya model prey termasuk dalam model logistik. terjadi penurunan jumlah populasi prey karena adanya laju kematian prey dengan adanya penyerangan dari predator. selanjutnya untuk mengetahui pengaruh waktu perlambatan terhadap kestabilan titik ekuilibrium pada sistem persamaan diferensial model predator-prey yaitu dengan menganalisis titik ekuilibrium . karena titik equilibrium tersebut berada di kuadran positif dan asimtot stabil ketika tidak ada waktu perlambatan. dari hasil analisis pada pembahasan didapatkan nilai perlambatan )∈ ∪ ∪ ∪ ∪ ∪ (0,1.57080) 5.23599,7.85398 (12.21730,1413717) (19.19862,20.42035) (26.17994,26.70354) (33.16126,32.98672) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator-prey stabil. sedangkan dengan nilai perlambatan ∈ ∪ ∪ ∪ ∪ ∪ ∪ (1.57080,5.23599) (7.85398,12.21730) (14.13717,19.19882) (20.42035 26.17994) (26.70354,3316126) (32.98672, 40.14257) titik ekuilibrium (10;6.33) pada sistem persamaan diferensial model predator-prey , bin muhammad. 2008. tafsir http://www.tafseer.info. diakses tanggal 7 juli 2009 , abul fida’. 1993. tafsir ibnu katsir. . jalaluddin. 1990. tafsir jalalain. toha putra : semarang. , musthofa ahmad. 1971. tafsir al beirut : darul fikri . proses pemodelan http://www.sipoel.unimed.in/ file.php/44/course/bab_ii/bab_2.doc. diakses tanggal 19 mei 2008. persamaan diferensial dalam satuan si metric. jakarta : erlangga persamaan diferensial dan malang: umm press. analisis sistem persamaan diferensial model predator-prey dengan perlambatan jurnal cauchy – issn: 2086-0382 53 [8] beals. 1999. predator prey dynamics: lotka voltera, http://www.google.com/htm. diakses tanggal 7 juli 2009 [9] baiduri. 2002. persamaan diferensial dan matematika model. malang: umm press [10] finizio dan ladas. 1998. penerapan diferensial biasa dengan penerapan modern, edisi kedua. terjemahan widiarti santoso. jakarta : erlangga. [11] kasanah, srinur. 2007. analisis model matematika pada interaksi leukimia mielogenous kronik (cml) dengan sel t. skripsi tidak dipublikasikan. malang: uin malang. [12] murray, jd. 2002. mathematical biology i. an introduction third edition. new york: springer [13] neuhauser, claudia. 2004. calculus for biology and medicine. new jersey: pearson education [14] reece, campbell. 2004. biologi. jakarta: erlangga. [15] shihab, m. quraish. 2002. tafsir al-misbah. jakarta: lentera hati. [16] sirin, khaeron. 2008. membangun fiqh bumi, http://www.ptiq.ac.id/index.php? option=com_content&task=view&id=38&item id=34. diakses tanggal 14 juli 2009. [17] toaha, syamsuddin. 2006. stability analysis of sum population model with time delay and harvesting. makasar. department of mathematics hasanuddin university [18] yahya, harun. 2009. menyingkap rahasia alam semesta, http://www.harunyahya.com /indo/buku/menyingkap010.htm. diakses tanggal 7 juli 2009. analisis perilaku model multi agen dengan gangguan sentot achmadi1 dan miswanto2 1prodi teknik informatika itn malang e-mail: sentot_achmadi@yahoo.co.id 2prodi matematika fak. sains dan teknologi unair e-mail: miswanto@fst.unair.ac.id abstrak makalah ini membahas model multi agent pada ruang berdimensi n dengan fungsi penarik dan penolak. pada model ini diberikan fungsi gangguan yang merupakan fungsi terbatas. pada makalah ini juga dibahas tentang kestasioneran dari setiap agen dan analisis kestabilan model dengan lyapunov. dari hasil analitik diperoleh bahwa pusat dari multi agen bersifat stasioner. juga uji kestabilan dengan metode kestabilan lyapunov diperoleh bahwa model yang diajukan merupakan model yang stabil. hasil simulasi numerik menunjukkan bahwa setiap agen konvergen ke suatu daerah yang mempunyai jarak tertentu terhadap pusat dari model multi agen. kata kunci: kestabilan lyapunov, multi agen, stasioner. abstract this paper discuss multi-agent model of the n-dimensional space with the function of attraction and repulsion. in this model is given the disturbance function which is a bounded function. in this paper also discuss about stationary of each agency and stability of the models use stability of lyapunov. from the analytical results obtained center of the multi-agent is stationary. also test the stability with lyapunov method is obtained that the proposed model is a stable model. numerical simulation results show that each agent will converges to a region that has a certain distance to the center of the multi-agent model. keywords: multi-agent, stability of lyapunov, stationary. pendahuluhan beberapa fenomena alam sangat menarik untuk dikaji secara fisika dan dimodelkan secara matematika. salah satu fenomena alam yang menarik untuk dimodelkan secara matematika dan dikaji secara fisika adalah fenomena gerak bergerombol yang dinamakan dengan swarm. fenomena alam ini banyak ditemukan pada beberapa organisme mulai dari yang sederhana (bakteri) sampai pada mamalia, sebagai contoh sekumpulan bakteri, sekawanan burung angsa, sekelompok singa dan sekelompok ikan. dengan melakukan gerak bergerombol, mereka memperoleh beberapa keuntungan seperti terhindar dari pemangsa dan dapat meningkatkan efektifitas dalam pencarian makanan. kelompok binatang (mangsa) memanfaatkan fenomena swarm untuk melindungi anggotanya dari serangan binatang lain (pemangsa). di lain pihak pemangsa memanfaatkan fenomena swarm untuk efektifitas penyerangan terhadap mangsanya. sekawanan burung angsa dalam melakukan terbang selalu membentuk formasi v terbalik. dengan terbang dalam formasi v terbalik, sekawanan burung angsa memperoleh beberapa keuntungan, antara lain dapat meningkatkan jarak terbangnya lebih dari 71 % dan daya terbangnya 24 % lebih cepat dari pada terbang sendirian. fenomena swarm banyak digunakan dalam ilmu teknik seperti optimalisasi gerak robot, gerak satelit, dan sebagainya. dari kajian literatur yang dilakukan, penelitian tentang swarm sudah cukup banyak diantaranya gazi dan passino (2002, 2003 dan 2004), chu et all (2005), tetapi tanpa menyertakan unsur gangguan serta masih terbatas pada perilaku anggota swarm disekitar pusat swarm. fenomena swarm pertama-tama dimodelkan secara matematika oleh breder (1954) yang membahas tentang gaya tarik (attractor force) dan gaya tolak (repellent force) antara dua individu. sedangkan, gazi dan passino (2002, 2003 dan 2004) memodelkan swarm berdasarkan pengaruh perpaduan dari keluarga fungsi attractor dan repellent. model gazi dan passino digeneralisasi oleh wang et all (2004) dengan memasukkan faktor keterkaitan antara anggota swarm (coupling matrix) dan parameter positif. mailto:sentot_achmadi@yahoo.co.id mailto:miswanto@fst.unair.ac.id sentot achmadi dan miswanto 190 volume 2 no. 4 mei 2013 artikel ini mempelajari tentang model swarm yang merupakan modifikasi dari model gazi dan passino (2003), tetapi dengan menyertakan suatu fungsi gangguan yang terbatas. untuk lebih jelasnya ditunjukkan dalam sub bab 2 di bawah ini. model multi agen diberikan model multi agent dengan m anggota pada ruang euclidean berdimensi n sebagai berikut: �̇�𝑖 = ∑ 𝑤𝑖𝑗 𝑓(𝑥𝑖 − 𝑥𝑗 ) + 𝛾𝑖 (𝑡), 𝑀 𝑗=1 (1) dengan �̇�𝑖 = 𝑑𝑥𝑖 𝑑𝑡 , 𝑥𝑖 ∈ 𝑅 𝑁 menyatakan posisi dari agent ke i, 𝐺 = [𝑤𝑖𝑗 ] ∈ 𝑅 𝑁x𝑁 merupakan coupling matrik dengan 𝑤𝑖𝑗 adalah bilangan bulat nonnegative dan matrik g diasumsikan simetri (yaitu 𝑤𝑖𝑗 = 𝑤𝑗𝑖 ). bentuk 𝑓( . ) menyatakan fungsi penarik dan penolak yang didefinisikan sebagai 𝑓(𝑦) = −𝑦 (𝑎 − 𝑟 𝑏+𝑐‖𝑦‖2 ), (2) dengan 𝑎, 𝑏, 𝑐,dan 𝑟 adalah konstanta positif dengan 𝑎 << 𝑟 dan ‖𝑦‖ = √𝑦𝑇 𝑦. sedangkan γ𝑖 (𝑡) menyatakan fungsi gangguan yang didefinisikan sebagai γ𝑖 (𝑡) = 𝐾 exp(−β1 𝑡)sin(β2𝑡), (3) dengan 𝐾, β1, β2 adalah konstanta positif. analisis dari model multi agen pada sub-bab ini dibahas tentang kestasioneran dan kestabilan dari model (1) dengan fungsi penarik dan penolaknya model (2) dan fungsi gangguan (3) yaitu: �̇�𝑖 = − ∑ 𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) 𝑀 𝑗=1 ξij + γ𝑖 (𝑡) (4) dengan 𝜉𝑖𝑗 = 𝑎 − 𝑟 𝑏+𝑐‖𝑥𝑖−𝑥𝑗‖ 2. didefinisikan pusat dari model multi agen (4) sebagai berikut: �̅� = 1 𝑀 ∑ 𝑥𝑖 𝑀 𝑖=1 . (5) teorema 1. pusat �̅�dari model multi agen (4) yang didefinisikan pada persamaan (5) bersifat stasioner. bukti: dengan menurunkan persamaan (5) terhadap t diperoleh: �̇̅� = 1 𝑀 ∑ �̇�𝑖 𝑀 𝑖=1 = 1 𝑀 ∑ (− ∑ 𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) (𝑎 − 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗 ‖ 2 ) 𝑀 𝑗=1 + 𝛾𝑖 (𝑡)) 𝑀 𝑖=1 = − 1 𝑀 ∑ ∑ 𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) (𝑎 − 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗 ‖ 2 ) + 1 𝑀 ∑ 𝛾𝑖 (𝑡) 𝑀 𝑖=1 𝑀 𝑗=1 𝑀 𝑖=1 karena g bersifat simetri dan fungsi (.)f simetri terhadap pusat koordinat maka: �̇̅� = 1 𝑀 ∑ 𝛾𝑖 (𝑡) 𝑀 𝑖=1 = 𝐾 exp(−𝛽1𝑡) sin(𝛽2𝑡) diketahui 𝛾𝑖 (𝑡) merupakan fungsi terbatas. hal ini berarti untuk 𝑡 → ∞ berakibat �̇̅� menuju ke 0. ■ selanjutnya dibahas tentang kestabilan model (4). adapun untuk menganalisis kestabilan model digunakan kestabilan lyapunov. teorema 2: perhatikan model multi agen (4) dengan fungsi gangguan (3). untuk t menuju tak hingga, setiap agen i konvergen pada suatu daerah terbatas yang didefinisikan sebagai berikut: ω = {𝑥: ∑ ‖𝑥𝑖 − �̅�‖ ≤ 𝜌 2𝑀 𝑖=1 } (6) dengan 𝜌 = 𝐴𝐵+2𝐾 √2𝑎𝑤𝑀 , 𝐴 = ∑ 𝑤𝑖𝑗 𝑀 𝑗=1 , 𝐵 = 1 2 𝑟√ 1 𝑏𝑐 , dan 𝑤 elemen terkecil dari 𝑤𝑖𝑗 ,𝑤 ≠ 0. bukti: misal 𝑒𝑖 = 𝑥𝑖 − �̅�. definisikan fungsi lyapunov untuk model (4), yaitu 𝑉 = ∑ 𝑉𝑖 𝑀 𝑖=1 dengan 𝑉 = 1 2 𝑒𝑖 𝑇 𝑒𝑖 sehingga 𝑉 = ∑ 𝑉𝑖 = 𝑀 𝑖=1 1 2 ∑ 𝑒𝑖 𝑇 𝑒𝑖 𝑀 𝑖=1 . dengan menurunkan 𝑉𝑖 terhadap t diperoleh: �̇� = ∑ �̇�𝑖 𝑀 𝑖=1 𝑉 ̇ = ∑ 𝑒𝑖 𝑇 {− ∑ 𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) (𝑎 − 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗‖ 2 ) 𝑀 𝑖=1 + 𝛾𝑖 (𝑡) − 1 𝑀 ∑ 𝛾𝑖 (𝑡) 𝑀 𝑗=1 } 𝑀 𝑖=1 = − ∑ ∑ 𝑒𝑖 𝑇𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) (𝑎 − 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗‖ 2 ) 𝑀 𝑗=1 𝑀 𝑖=1 + ∑ 𝑒𝑖 𝑇 (𝛾𝑖 (𝑡) − 1 𝑚 ∑ 𝛾𝑗 (𝑡) 𝑀 𝑗=1 ) 𝑀 𝑖=1 = −𝑎 ∑ ∑ 𝑒𝑖 𝑇𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) 𝑀 𝑗=1 𝑀 𝑖=1 + ∑ ∑ 𝑒𝑖 𝑇𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) ( 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗‖ 2) 𝑀 𝑗=1 𝑀 𝑖=1 +     ∑ 𝑒𝑖 𝑇 (𝛾𝑖 (𝑡) − 1 𝑀 ∑ 𝛾𝐽 (𝑡) 𝑀 𝑗=1 ) 𝑀 𝑖=1 ambil 𝑤 elemen terkecil dari 𝑤𝑖𝑗 dengan 𝑤 ≠ 0, maka: analisis model multi agen dengan gangguan jurnal cauchy – issn: 2086-0382 191 �̇� ≤ −2𝑎𝑤𝑀𝑉 + ∑ ∑ 𝑒𝑖 𝑇 𝑤𝑖𝑗 (𝑥𝑖 − 𝑥𝑗 ) 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗 ‖ 2 𝑀 𝑗=1 𝑀 𝑖=1 + ∑ 𝑒𝑖 𝑇 𝑀 𝑖=1 (γ𝑖 (𝑡) − 1 𝑀 ∑ γ𝑗 (𝑡) 𝑀 𝑗=1 ) ≤ −2𝑎𝑤𝑀𝑉 + ∑ ∑ 𝑤𝑖𝑗 ‖𝑒𝑖 𝑇 ‖‖𝑥𝑖 − 𝑥𝑗 ‖ 𝑟 𝑏 + 𝑐‖𝑥𝑖 − 𝑥𝑗 ‖ 2 𝑀 𝑗=1 𝑀 𝑖=1 + ∑ 𝑒𝑖 𝑇 𝑀 𝑖=1 (γ𝑖 (𝑡) − 1 𝑀 ∑ γ𝑗 (𝑡) 𝑀 𝑗=1 ) . karena ‖𝑥𝑖 − 𝑥𝑗‖ 𝑟 𝑏+𝑐‖𝑥𝑖−𝑥𝑗‖ 2 adalah fungsi terbatas dan memiliki nilai maksimum pada ‖𝑥𝑖 − 𝑥𝑗 ‖ = √ 𝑏 𝑐 maka, �̇� ≤ −2𝑎𝑤𝑀𝑉 + 1 2 𝑟√ 1 𝑏𝑐 ∑ (‖𝑒𝑖 𝑇 ‖ ∑ 𝑤𝑖𝑗 𝑀 𝑗=1 ) 𝑀 𝑖=1 + ∑ 𝑒𝑖 𝑇 𝑀 𝑖=1 (γ𝑖 (𝑡) − 1 𝑀 ∑ γ𝑗 (𝑡) 𝑀 𝑗=1 ) ≤ −2𝑎𝑤𝑀𝑉 + 1 2 𝑟√ 1 𝑏𝑐 ∑ (‖𝑒𝑖 𝑇 ‖ ∑ 𝑤𝑖𝑗 𝑀 𝑗=1 ) 𝑀 𝑖=1 + ∑‖𝑒𝑖 𝑇 ‖ (‖γ𝑖 (𝑡)‖ + 1 𝑀 ‖∑ γ𝑗 (𝑡) 𝑀 𝑗=1 ‖) . 𝑀 𝑖=1 karena 𝛾𝑖 (𝑡) terbatas maka ‖𝛾𝑖 (𝑡)‖ ≤ 𝐾, sehingga �̇� ≤ −√𝑉 (𝑎𝑤𝑀2√𝑉 − √2(𝐴𝐵 + 2𝐾)) dengan 𝐴 = ∑ 𝑤𝑖𝑗 𝑀 𝑗=1 dan = 1 2 𝑟√ 1 𝑏𝑐 . agar syarat kestabilan lyapunov dipenuhi, maka haruslah 𝑎𝑤𝑀2√𝑉 − √2(𝐴𝐵 + 2𝐾) > 0, atau diperoleh: 𝑉 > ( 𝐴𝐵+2𝐾 √2𝑎𝑤𝑀 ) 2 . hal ini berarti bahwa setiap agen i akan konvergen pada suatu daerah yang terbatas (6) ■ simulasi numerik pada bagian ini diperlihatkan beberapa hasil simulasi numerik yang merupakan ilustrasi dari model (1) pada ruang dimensi 2 dengan fungsi penarik/penolak (2) dan fungsi gangguan (3). pada simulasi ini disimulasikan terhadap 10 agen, dengan posisi agen dapat dilihat pada tabel 1. adapun, nilai konstanta pada fungsi penarik/penolak adalah ,1a ,1b ,2.0c dan 20r , konstanta ini menggunakan konstanta gazi dan passino (2003). sedangkan nilai-nilai konstanta pada fungsi gangguan (3) adalah ,1k 8 1  dan 5.0 2  simulasi numerik di bawah ini menunjukkan perilaku gerak pusat dari multi agen dan anggota dari model multi agen tersebut pada ruang dimensi dua. tabel 1. koordinat/posisi agen agen 1 agen 2 agen 3 agen 4 agen 5 (3,4) (4,3) (6,6) (6,5) (8,7) agen 6 agen 7 agen 8 agen 9 agen 10 (4,6) (9,9) (9,8.5) (8,9) (8,3) pertama-tama disimulasikan gerak pusat dari model multi agen tanpa gangguan (3), kemudian juga disimulasikan pusat dari model multi agen dengan fungsi gangguan (3). hal ini digunakan untuk melihat kestationeran model dengan pengaruh gangguan. hasil simulasi numerik ditunjukkan pada gambar berikut ini. gambar 1. pergerakan pusat multi agen tanpa gangguan hasil simulasi pada gambar 1, nampak bahwa pusat dari model multi agen bersifat stasioner. sedangkan gambar 2 merupakan hasil simulasi pergerakan pusat dari model multi agen dengan gangguan. dari gambar 2, nampak bahwa pusat dari model multi agen juga stasioner. hal ini memperlihatkan bahwa model (4) merupakan model multi agen yang stabil. selanjutnya dilakukan hal yang sama untuk gerak anggota dari model multi agen. analisis kestabilan yang dibuktikan pada teorema 2, secara simulasi numerik ditunjukkan pada gambar 3 dan gambar 4. hasil simulasi pada gambar 3, nampak bahwa gerak dari model multi agen konvergen pada daerah di sekitar pusat dari model multi agen. sedangkan gambar 4 merupakan hasil 4 6 8 10 12 14 16 -6 -4 -2 0 2 4 6 8 sentot achmadi dan miswanto 192 volume 2 no. 4 mei 2013 simulasi pergerakan dari model multi agen dengan gangguan. dari gambar 4, nampak bahwa model multi agen juga konvergen pada daerah di sekitar pusat dari model multi agen. hal ini memperlihatkan bahwa model (4) merupakan model multi agen yang stabil. gambar 2. pergerakan pusat multi agen dengan gangguan gambar 3. pergerakan multi agen tanpa gangguan gambar 4. pergerakan multi agen dengan gangguan penutup dari analisis secara analitik, menunjukkan bahwa model multi agen (4) adalah suatu model yang stasioner dan stabil. hal ini ditunjukkan dengan analisis pada teorema 1 dan teorema 2. hasil ini juga didukung dengan simulasi numerik pada gambar 1, gambar 2, gambar 2, dan gambar 4. referensi [1] breder, c. m., 1954, equation descriptive of fish schools and other animal aggregation, ecology, vol. 35, no.3, pp. 361-370. [2] chu, t., wang, l. dan chen, t., 2005, selforganized motion in a class of anisotropic swarm: convergence vs oscillation, proceedings american control conference, portland. [3] gazi, v. dan passino, k.m., 2002, stability analysis of swarms in an environment with an attractant/repellent profile, proceedings of the american control conference, anchorage. [4] gazi, v. dan passino, k.m., 2003, stability analysis of swarms, ieee transaction on automatic control, vol. 48, no.4. [5] gazi, veysel dan passino, k.m., 2004, stability analysis of social foraging swarms, ieee transaction on system, man, and cyberneticspart b: cybernetics, vol. 34. [6] wang, l., shi, h., chu, t., zhang, w. dan zhang, l., 2004, aggregation of foraging swarms, in advance in artificial intelligence, lecture notes in artificial intelligence, vo. 3339, springer-verlag. [7] shi, h., wang, l. dan chu, t., 2004, swarming behavior of multi-agent system, journal of control and applications, vol. 4, pp. 313 – 318. 4 6 8 10 12 14 16 -6 -4 -2 0 2 4 6 8 0 5 10 15 20 25 -10 -5 0 5 10 15 agen 1 agen 2 agen 3 agen 4 agen 5 agen 6 agen 7 agen 8 agen 9 agen10 pusat agen 0 5 10 15 20 25 -10 -5 0 5 10 15 agen 1 agen 2 agen 3 agen 4 agen 5 agen 6 agen 7 agen 8 agen 9 agen10 pusat agen pendahuluhan model multi agen analisis dari model multi agen simulasi numerik penutup referensi diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel khusnul khamidiyah1, usman pagalay2 1mahasiswa jurusan matematika, fakultas sains dan teknologi uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi uin maulana malik ibrahim malang e-mail : diyah_af@yahoo.co.id, usmanpagalay@yahoo.co.id abstrak persamaan meinhardt merupakan sebuah model matematika yang menggambarkan pola pembentukan sel pada hydra. hans meinhardt menggunakan jenis persamaan difusi untuk menggambarkan bagaimana variabel-variabel berkembang biak, mati, bergerak dan berinteraksi. bentuk model yang dirumuskan oleh meinhardt tersebut merupakan model kontinu, sehingga salah satu studi yang dapat diterapkan pada model meinhardt adalah dilakukannya diskritrisasi. diskritisasi merupakan proses kuantisasi sifat-sifat kontinu. salah satu metode yang dapat memperkirakan bentuk diferensial kontinu menjadi bentuk diskrit ialah metode beda hingga. sehingga dalam penelitian ini akan dilakukan proses diskritisasi pada model pembentukan sel. metode yang digunakan adalah beda hingga skema crank-nicolson yang merupakan pengembangan dari skema eksplisit dan implisit. kelebihan dari skema crank-nicolson adalah nilai error yang lebih kecil dari pada skema eksplisit dan implisit. dalam penelitian ini digunakan beda hingga maju untuk turunan 𝑡 dan beda hingga pusat untuk turunan 𝑥 pada persamaan activator 𝑎(𝑥,𝑡) dan inhibitor 𝑏(𝑥,𝑡). langkah-langkah yang dilakukan adalah dimulai dengan menganalisis persamaan meinhardt dan dilanjutkan dengan diskritisasi. kata kunci: metode beda hingga, persamaan meinhardt, , skema crank-nicolson abstract meinhardt equation is a mathematical model which describes the pattern of cell formation in hydra. hans meinhardt uses the diffusion equation type to describe how the variables are breed, die, move, and interact. the model formulated by meinhardt is a continuous model. thus, one of the studies which can be applied to meinhardt equation is discretization. discretization is a quantization process of continuous properties. one method which can predict continuous differential form into discrete form is by applying finite difference method. thus, the proses of discretization will be conducted in this research on the pattern formation of cell. the method used in this study is finite difference method implementing cranknicolson scheme which is the result of explicit and implicit scheme development. the advantage of the crank nicolson scheme is the smaller error values than the explicit and implicit scheme. this method used finite forward difference for derivatives of 𝑡 and finite centre difference for derivatives of 𝑥 at the activator 𝑎(𝑥,𝑡) and inhibitor 𝑏(𝑥,𝑡). the steps conducted by analyzing meinhardt equation and continued with discretization. keywords: crank-nicolson scheme, finite difference method, meinhardt equation pendahuluan sel merupakan kumpulan materi paling sederhana yang dapat hidup dan termasuk unit penyusun tubuh (building block). seluruh makhluk hidup tersusun atas sel. sel adalah unit dasar kehidupan. dalam kingdom monera dan protista, keseluruhan organisme tersusun atas sel tunggal. pada kebanyakan fungi dan dalam kingdom hewan dan tumbuhan, organisme adalah susunan yang luar biasa kompleks dari sel-sel yang bisa triliunan banyaknya (fried & hademenos, 2005). hydra merupakan metazoan atau hewan bersel banyak yang hidup di kolam atau di sungai yang airnya mengalir. tubuh hydra berbentuk polip yang hidup soliter dalam arti tidak berkoloni, dapat berpindah tempat tetapi biasanya melekat pada obyek, misalnya batubatuan, batang kayu, tanaman air dan lain-lain. tubuhnya berbentuk silindris yang dapat dijulurkan serta dipendekkan. kemampuan untuk dapat menjulur dan memendek ini karena mailto:diyah_af@yahoo.co.id http://id.wikipedia.org/wiki/materi http://id.wikipedia.org/wiki/kehidupan diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel cauchy – issn: 2086-0382 132 memang tubuh hydra memiliki fibril-fibril khusus pada beberapa sel. panjang tubuh hydra mulai dari 2 sampai 20 mm, dengan diameter tubuhnya tidak lebih dari 1 mm (kastawi, 2005). pada organisme yang lebih tinggi, pentingnya proses pembentukan pola dasar pada subdivisi dari sebuah organisme hanya terjadi sekali dan dalam waktu yang relatif singkat. dalam berbagai kasus, embrio lebih sulit dijangkau dalam manipulasi sebuah penelitian. berbeda dengan potongan-potongan kecil pada hydra yang hidup di air tawar dapat meregenerasi tubuhnya setiap saat. oleh karena itu, model organisme yang tepat untuk dipelajari dan diajukan sebagai model serta sistem biologi yang nyata (gierer, 1977). hans meinhardt merumuskan pola pembentukan sel hydra dalam sebuah sistem persamaan diferensial parsial. model yang dirumuskan oleh meinhardt merupakan model kontinu, sehingga salah satu studi yang dapat diterapkan pada model tersebut adalah dilakukannya diskritrisasi. diskritisasi merupakan proses kuantisasi sifat-sifat kontinu. kuantisasi diartikan sebagai proses pengelompokan sifat-sifat kontinu pada selangselang tertentu (step size). kegunaan diskritisasi adalah untuk mereduksi dan menyederhanakan data, sehingga didapatkan data diskrit yang lebih mudah dipahami, digunakan, dan dijelaskan. oleh karena itu, hasil pembelajaran dengan bentuk diskrit dipandang dougherty (1995) sebagai hasil yang cepat dan akurat dibandingkan hasil dari bentuk kontinu (liu, hussain, tan, & dash, 2012). salah satu metode yang dapat memperkirakan bentuk diferensial kontinu menjadi bentuk diskrit ialah dengan metode beda hingga. metode beda hingga yang digunakan oleh penulis dalam penelitian ini adalah skema crank-nicolson yang merupakan pengembangan dari skema eksplisit dan implisit. kelebihan dari skema crank-nicolson adalah nilai error yang lebih kecil. berdasarkan paparan tersebut di atas, penulis tertarik untuk membahas dan mengkaji model meinhardt dalam tulisan ini dengan judul diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel. kajian pustaka 1. persamaan diferensial parsial persamaan diferensial parsial adalah persamaan-persamaan yang mengandung satu atau lebih turunan-turunan parsial. persamaan itu haruslah melibatkan paling sedikit dua variabel bebas. tingkat persamaan diferensial parsial adalah tingkat turunan tertinggi pada persamaan itu (ayres, 1992). 2. orde persamaan diferensial parsial orde dari persamaan diferensial parsial ditentukan berdasarkan orde dari turunan tertinggi pada persamaan diferensial parsial tersebut. sebagai contoh, persamaan diferensial parsial pola pembentukan sel yang kemudian disebut sebagai persamaan meinhardt berikut merupakan contoh persamaan orde satu dan dua: pde orde satu: 𝜕𝑎 𝜕𝑡 = 𝑠𝑏𝑎2 − 𝑟𝑎𝑎 pde orde dua: 𝜕𝑎 𝜕𝑡 = 𝑠𝑏𝑎2 − 𝑟𝑎𝑎 + 𝐷𝑎 𝜕2𝑎 𝜕𝑥2 (sasongko, 2010). 3. linieritas persamaan diferensial parsial persamaan diferensial parsial berikut merupakan bentuk persamaan diferensial orde dua: 𝛼 𝜕2𝑢 𝜕𝑦2 + 𝛽 𝜕2𝑢 𝜕𝑥𝜕𝑦 + 𝛾 𝜕2𝑢 𝜕𝑥2 + 𝛿 = 0 (1) persamaan diferensial parsial juga digolongkan menjadi persamaan linier, kuasilinier dan nonlinier, dengan penjelasan berikut: a. apabila koefisien pada persamaan (1) adalah konstan atau fungsi yang hanya terdiri dari variabel bebas saja [𝛼 = 𝛽 = 𝛾 = 𝛿 = (𝑥,𝑦)], maka persamaan itu disebut persamaan linier; b. apabila koefisien pada persamaan (1) adalah fungsi dari variabel tak bebas (dependent variable) dan/atau merupakan turunan dengan pangkat yang lebih rendah daripada persamaan diferensialnya [𝛼 = 𝛽 = 𝛾 = 𝛿 = (𝑥,𝑦,𝑢, 𝜕𝑢 𝜕𝑥 , 𝜕𝑢 𝜕𝑦 )], maka persamaan itu disebut persamaan kuasilinier; c. apabila koefisiennya merupakan fungsi dengan turunan sama dengan pangkatnya, [𝛼 = 𝛽 = 𝛾 = 𝛿 = (𝑥,𝑦,𝑢, 𝜕2𝑢 𝜕𝑥2 , 𝜕2𝑢 𝜕𝑦2 , 𝜕2𝑢 𝜕𝑥𝜕𝑦 )], maka persamaan itu disebut persamaan nonlinier. (sasongko, 2010). 4. bentuk kanonik persamaan diferensial parsial berikut ini diberikan beberapa bentuk persamaan diferensial parsial: a. persamaan ellips, jika 𝑏2 − 4𝑎𝑐 < 0 b. persamaan parabola, jika 𝑏2 − 4𝑎𝑐 = 0 c. persamaan hiperbola, jika 𝑏2 − 4𝑎𝑐 > 0 (triatmodjo, 2002). khusnul khamidiyah 133 volume 3 no. 3 november 2014 5. deret taylor deret taylor merupakan dasar untuk menyelesaikam masalah dalam metode numerik, terutama penyelesaian persamaan diferensial. jika suatu fungsi 𝑓(𝑥) diketahui di titik 𝑥𝑖 dan semua turunan dari 𝑓 terhadap 𝑥 diketahui pada titik tersebut, maka dengan deret taylor dapat dinyatakan nilai 𝑓 pada titik 𝑥𝑖+1 yang terletak pada jarak ∆𝑥 dari titik 𝑥𝑖 (triatmodjo, 2002). andaikan 𝑓 adalah suatu fungsi dengan turunan ke (𝑛 + 1), 𝑓(𝑛+1)(𝑥), ada untuk setiap 𝑥 pada suatu selang terbuka 𝐼 yang mengandung 𝑎. maka untuk setiap 𝑥 di 𝐼, 𝑓(𝑥) = 𝑓(𝑎) + 𝑓′(𝑎)(𝑥 − 𝑎) + 𝑓′′(𝑎) 2! (𝑥 − 𝑎)2 + ⋯+ 𝑓(𝑛)(𝑎) 𝑛! (𝑥 − 𝑎)𝑛 + 𝑅𝑛(𝑥) dimana sisa (atau kesalahan) 𝑅𝑛(𝑥) diberikan oleh rumus: 𝑅𝑛(𝑥) = 𝑓(𝑛+1)(𝑐) (𝑛 + 1)! (𝑥 − 𝑎)𝑛+1 dengan 𝑐 suatu titik antara 𝑥 dan 𝑎 (purcell & varberg, 1999). persamaan di atas merupakan penjumlahan dari suku-suku (term), yang disebut deret. perhatikanlah bahwa deret taylor ini panjangnya tidak terhingga, untuk memudahkan penulisan suku-suku selanjutnya kita menggunakan tanda ellipsis (⋯). jika dimisalkan 𝑥 − 𝑎 = ℎ, maka 𝑓(𝑥) dapat juga ditulis sebagai: 𝑓(𝑥) = 𝑓(𝑎) + 𝑓′(𝑎)ℎ + 𝑓′′(𝑎) 2! ℎ2 + ⋯ + 𝑓(𝑛)(𝑎) 𝑛! ℎ𝑛 + 𝑅𝑛(𝑥) deret pangkat dari (𝑥 − 𝑎) yang menggambarkan sebuah fungsi dinamakan deret taylor. apabila 𝑎 = 0, deret yang bersangkutan disebut deret maclaurin (purcell & varberg, 1999). 6. diferensial numerik diferensial numerik digunakan untuk memperkirakan bentuk diferensial kontinu menjadi bentuk diskret. diferensial numerik ini banyak digunakan untuk menyelesaikan persamaan diferensial. bentuk tersebut dapat diturunkan berdasar pada deret taylor. persamaan di bawah ini: 𝑦′(𝑥𝑖+1) = 𝑦(𝑥𝑖+1) − 𝑦(𝑥𝑖) (𝑥𝑖+1 − 𝑥𝑖) dikenal dengan metode numerik sebagai beda terbagi berhingga. secara umum dapat dinyatakan sebagai: 𝑓′(𝑥𝑖+1) = ∆𝑓𝑖 ℎ + ℎ dengan ∆𝑓𝑖 disebut beda ke depan pertama dan ℎ disebut ukuran langkah, yaitu panjang selang hampiran. istilah ke depan digunakan karena ada hampiran turunan di titik 𝑖 dipergunakan data pada titik yang bersangkutan dan titik yang di depannya. suku ∆𝑓𝑖 ℎ disebut sebagai beda terbagi berhingga pertama (djojodihardjo, 2000). beda terbagi ke depan ini merupakan salah satu dari beberapa cara yang dapat dikembangkan dari deret taylor guna memperoleh turunan secara numerik. 7. metode beda hingga skema cranknicolson skema crank-nicolson merupakan pengembangan dari skema eksplisit dan implisit. dalam skema eksplisit, ruas kanan ditulis pada waktu ke 𝑛. dalam skema implisit, ruas kanan ditulis untuk waktu 𝑛 + 1. dalam kedua skema tersebut diferensial terhadap waktu ditulis dalam bentuk: 𝜕𝑇 𝜕𝑡 ≈ (𝑇𝑖 𝑛+1 − 𝑇𝑖 𝑛) ∆𝑡 yang berarti diferensial terpusat terhadap waktu 𝑛 + 1 2 . skema crank-nicolson menulis ruas kanan pada waktu 𝑛 + 1 2 yang merupakan nilai rerata dari skema eksplisit dan implisit. skema jaringan titik hitungan diberikan oleh gambar 1 berikut. gambar 1. skema crank-nicolson skema ini menggunakan teknik pembobotan untuk diskritisasi waktu sekarang 𝑡𝑛 dan diskritisasi waktu yang akan datang 𝑡𝑛+1 dengan cara yang lebih fleksibel yaitu dengan menggunakan faktor pemberat waktu. beda hingga terhadap ruang (𝑥): 𝜕2𝑇 𝜕𝑥2 = 𝜃 ( 𝑇𝑖+1 𝑛+1 − 2𝑇𝑖 𝑛+1 + 𝑇𝑖−1 𝑛+1 (∆𝑥)2 ) + (1 − 𝜃) ( 𝑇𝑖+1 𝑛 − 2𝑇𝑖 𝑛 + 𝑇𝑖−1 𝑛 (∆𝑥)2 ) (2) dengan 0 ≤ 𝜃 ≤ 1 adalah faktor pemberat waktu (luknanto, 2003). sedangkan untuk beda hingga terhadap waktu: 𝜕𝑇 𝜕𝑡 = 𝑇𝑖 𝑛+1 − 𝑇𝑖 𝑛 ∆𝑡 dengan 𝜃 adalah koefisien pembobot dengan nilai: diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel cauchy – issn: 2086-0382 134 𝜃 = 0, jika skema adalah eksplisit 𝜃 = 1, jika skema adalah implisit 𝜃 = 1 2 , jika skema adalah crank-nicolson (lapidus & pinder, 1981). sehingga persamaan (2) dapat ditulis sebagai: 𝜕2𝑇 𝜕𝑥2 = 1 2 ( 𝑇𝑖+1 𝑛+1 −2𝑇𝑖 𝑛+1 +𝑇𝑖−1 𝑛+1 (∆𝑥)2 ) + 1 2 ( 𝑇𝑖+1 𝑛 −2𝑇𝑖 𝑛 +𝑇𝑖−1 𝑛 (∆𝑥)2 ) crank dan nicolson (1947) mengajukan dan juga menggunakan sebuah metode yang mereduksi perhitungan pada volume total, dan itu dianggap sah (konvergen dan stabil) untuk semua nilai 𝑟. mereka mempertimbangkan pdp terpenuhi di titik (𝑖ℎ,(𝑛 + 1 2 ) 𝑘) dan mengganti dengan 𝜕2𝑇 𝜕𝑥2 dengan rata-rata perkiraan perbedaan pada saat 𝑛 dan 𝑛 + 1. dengan kata lain mereka mengganti persamaan 𝜕𝑇 𝜕𝑡 = 𝜕2𝑇 𝜕𝑥2 dengan persamaan 𝑇𝑖 𝑛+1 −𝑇𝑖 𝑛 𝑘 = 1 2 ( 𝑇𝑖+1 𝑛+1 −2𝑇𝑖 𝑛+1 +𝑇𝑖−1 𝑛+1 ℎ2 + 𝑇𝑖+1 𝑛 −2𝑇𝑖 𝑛 +𝑇𝑖−1 𝑛 ℎ2 ) dan diberikan, −𝑟𝑇𝑖−1 𝑛+1 + (2 + 2𝑟)𝑇𝑖 𝑛+1 − 𝑟𝑇𝑖+1 𝑛+1 = 𝑟𝑇𝑖−1 𝑛 + (2 − 2𝑟)𝑇𝑖 𝑛 + 𝑟𝑇𝑖+1 𝑛 dimana 𝑟 = 𝑘 ℎ2 (smith, 1985). 8. model meinhardt (meinhardt, 2012) menyajikan sebuah model matematika yang menggambarkan pola pembentukan sel pada hydra, yang berbentuk sistem persamaan diferensial parsial. persamaan meinhardt terdiri dari dua persamaan yang berjenis persamaan difusi. dengan mendefinisikan 𝑎(𝑥,𝑡) sebagai activator (pengaktif) dan 𝑏(𝑥,𝑡) sebagai inhibitor (penghambat) pembentukan sel. persamaan meinhardt ditulis dalam bentuk sebagai berikut: 𝜕𝑎 𝜕𝑡 = 𝑠𝑏𝑎2 − 𝑟𝑎𝑎 + 𝐷𝑎 𝜕2𝑎 𝜕𝑥2 𝜕𝑏 𝜕𝑡 = 𝑏𝑏 − 𝑠𝑏𝑎 2 − 𝑟𝑏𝑏 + 𝐷𝑏 𝜕2𝑏 𝜕𝑥2 9. pembentukan sel hydra seluruh makhluk hidup tersusun atas sel. sel adalah unit dasar kehidupan. dalam kingdom monera dan protista, keseluruhan organisme tersusun atas sel tunggal. pada kebanyakan fungi dan dalam kingdom hewan dan tumbuhan, organisme adalah susunan yang luar biasa kompleks dari sel-sel yang bisa triliunan banyaknya (fried & hademenos, 2005). hydra, satu diantara segelintir cnidaria yang ditemukan di perairan tawar, merupakan hydrozoa yang tak lazim karena hanya terdapat dalam bentuk polip. ketika kondisi lingkungan menguntungkan, hydra bereproduksi secara aseksual. ketika kondisi memburuk, hydra dapat bereproduksi secara seksual, membentuk zigot resisten yang tetap dorman hingga keadaan membaik (campbell, et al., 2012). pembentukan sel yang dipengaruhi oleh zat pengaktif dan zat penghambat terjadi melalui proses difusi. dimana difusi sendiri menurut (villee, 1984) adalah gerakan molekul dari suatu daerah dengan konsentrasi tinggi ke daerah lain dengan konsentrasi lebih rendah yang disebabkan oleh energi kinetik molekul-molekul tersebut. tiap molekul cenderung bergerak lurus sampai terbentur pada molekul lain kemudian terpental dan bergerak ke arah lain. setelah tersebar secara merata, molekul tersebut tetap bergerak, tetapi jika ada molekul yang bergerak dengan cepat dari kiri ke kanan, secepat itu pula ada molekul lain yang bergerak dari kanan ke kiri sehingga keseimbangan dapat dipertahankan. pembahasan variabel-variabel yang digunakan dalam model pengaruh pengaktif dan penghambat terhadap pembentukan sel pada hydra diambil dari jurnal yang dirumuskan oleh hans (meinhardt, 2012) dalam karya tulisnya yang berjudul models of biological pattern formation sebagai berikut: pengaktif (activator) 𝑎(𝑥,𝑡) 𝜕𝑎(𝑥,𝑡) 𝜕𝑡 = 𝑠𝑏(𝑥,𝑡)𝑎2(𝑥,𝑡) − 𝑟𝑎𝑎(𝑥,𝑡) + 𝐷𝑎 𝜕2𝑎(𝑥,𝑡) 𝜕𝑥2(𝑥,𝑡) penghambat (inhibitor) 𝑏(𝑥,𝑡) 𝜕𝑏(𝑥,𝑡) 𝜕𝑡 = 𝑏𝑏 − 𝑠𝑏(𝑥,𝑡)𝑎 2(𝑥,𝑡) − 𝑟𝑏𝑏(𝑥,𝑡) + 𝐷𝑏 𝜕2𝑏(𝑥,𝑡) 𝜕𝑥2(𝑥,𝑡) perubahan konsentrasi pengaktif persatuan waktu sebanding dengan kepadatan sel s yang menggambarkan kemampuan sel tersebut untuk melakukan reaksi dan sebanding dengan berkurangnya kerusakan pengaktif melalui pertukaran dengan sel tetangganya akibat difusi. sedangkan perubahan konsentrasi penghambat persatuan waktu sebanding dengan laju produksi penghambat dan sebanding dengan berkurangnya kerusakan penghambat (meinhardt, 2012). agar pembentukan pola dapat terjadi, penghambat harus berdifusi jauh lebih cepat dari pengaktif (𝐷𝑏 ≥ 𝐷𝑎). pola akan stabil jika tingkat kerusakan penghambat lebih tinggi dibandingkan dengan pengaktif (𝑟𝑏 > 𝑟𝑎), dan jika keadaan sebaliknya terjadi (𝑟𝑎 > 𝑟𝑏) maka osilasi akan terjadi. 𝑏𝑎 menggambarkan laju khusnul khamidiyah 135 volume 3 no. 3 november 2014 produksi activator-independen kecil dari pengaktif yang diperlukan untuk memulai produksi activator autocatalytic pada tingkat rendah pada 𝑎, misalnya selama regenerasi berlangsung. dalam kebanyakan simulasi, kepadatan sel 𝑠 diasumsikan merata kecuali beberapa fluktuasi acak kecil yang memicu pembentukan pola dan yang tetap konstan selama simulasi (meinhardt, 2012). 1. diskritisasi persamaan meinhardt dengan metode beda hingga skema crank-nicolson pada activator berikut merupakan proses diskritisasi model pengaktif, persamaan yang digunakan yaitu persamaan: 𝜕𝑎(𝑥,𝑡) 𝜕𝑡 = 𝑠𝑏(𝑥,𝑡)𝑎2(𝑥,𝑡) − 𝑟𝑎𝑎(𝑥,𝑡) + 𝐷𝑎 𝜕2𝑎(𝑥,𝑡) 𝜕𝑥2(𝑥,𝑡) (3) dinotasikan, 𝑎(𝑥𝑖, 𝑡𝑛) = 1 2 (𝑎𝑖 𝑛 + 𝑎𝑖 𝑛+1) 𝑏(𝑥𝑖, 𝑡𝑛) = 1 2 (𝑏𝑖 𝑛 + 𝑏𝑖 𝑛+1) berdasarkan pernyataan luknanto sebagaimana tercantum dalam kajian pustaka di atas, maka transformasi beda hingga maju untuk turunan 𝑡 dan beda hingga pusat untuk turunan 𝑥 adalah sebagai berikut: 𝜕𝑎(𝑥,𝑡) 𝜕𝑡 = 𝑎𝑖 𝑛+1 − 𝑎𝑖 𝑛 ∆𝑡 𝜕2𝑎 𝜕𝑥2 = 1 2 ( 𝑎𝑖−1 𝑛+1 −2𝑎𝑖 𝑛+1 +𝑎𝑖+1 𝑛+1 ∆𝑥2 + 𝑎𝑖−1 𝑛 −2𝑎𝑖 𝑛 +𝑎𝑖+1 𝑛 ∆𝑥2 ) bentuk beda hingga disubtitusikan ke persamaan (3), sehingga diperoleh bentuk persamaan diskrit sebagai berikut: 𝑎𝑖 𝑛+1 − 𝑎𝑖 𝑛 ∆𝑡 = 1 2 𝑠[𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 + 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2] − 1 2 𝑟𝑎[𝑎𝑖 𝑛+1 + 𝑎𝑖 𝑛] + 1 2 𝐷𝑎 ( 𝑎𝑖−1 𝑛+1 − 2𝑎𝑖 𝑛+1 + 𝑎𝑖+1 𝑛+1 ∆𝑥2 + 𝑎𝑖−1 𝑛 − 2𝑎𝑖 𝑛 + 𝑎𝑖+1 𝑛 ∆𝑥2 ) (4) persamaan (4) juga dapat ditulis dalam bentuk: 𝑎𝑖 𝑛+1 ∆𝑡 = 𝑎𝑖 𝑛 ∆𝑡 + 𝑠 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 + 𝑠 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 − 𝑟𝑎 2 𝑎𝑖 𝑛+1 − 𝑟𝑎 2 𝑎𝑖 𝑛 + 𝐷𝑎 2∆𝑥2 𝑎𝑖−1 𝑛+1 − 𝐷𝑎 ∆𝑥2 𝑎𝑖 𝑛+1 + 𝐷𝑎 2∆𝑥2 𝑎𝑖+1 𝑛+1 + 𝐷𝑎 2∆𝑥2 𝑎𝑖−1 𝑛 − 𝐷𝑎 ∆𝑥2 𝑎𝑖 𝑛 + 𝐷𝑎 2∆𝑥2 𝑎𝑖+1 𝑛 (5) persamaan (5) dikalikan dengan pada kedua ruasnya, sehingga diperoleh: 𝑎𝑖 𝑛+1 = 𝑎𝑖 𝑛 + 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 + 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 − 𝑟𝑎∆𝑡 2 𝑎𝑖 𝑛+1 − 𝑟𝑎∆𝑡 2 𝑎𝑖 𝑛 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛+1 − 𝐷𝑎∆𝑡 ∆𝑥2 𝑎𝑖 𝑛+1 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛+1 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛 − 𝐷𝑎∆𝑡 ∆𝑥2 𝑎𝑖 𝑛 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛 (6) dalam bentuk yang lebih sederhana dan untuk mengelompokkan antara diskritisasi waktu sekarang (𝑛) dan diskritisasi waktu yang akan datang (𝑛 + 1), maka persamaan (6) dapat ditulis dalam bentuk sebagai berikut: − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛+1 + 𝑎𝑖 𝑛+1 + 𝑟𝑎∆𝑡 2 𝑎𝑖 𝑛+1 + 𝐷𝑎∆𝑡 ∆𝑥2 𝑎𝑖 𝑛+1 − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛+1 − 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛 + 𝑎𝑖 𝑛 − 𝑟𝑎∆𝑡 2 𝑎𝑖 𝑛 − 𝐷𝑎∆𝑡 ∆𝑥2 𝑎𝑖 𝑛 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛 + 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (7) persamaan (7) disederhanakan dalam bentuk sebagai berikut: − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛+1 + (1 + 𝑟𝑎∆𝑡 2 + 𝐷𝑎∆𝑡 ∆𝑥2 )𝑎𝑖 𝑛+1 − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛+1 − 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛 + (1 − 𝑟𝑎∆𝑡 2 − 𝐷𝑎∆𝑡 ∆𝑥2 )𝑎𝑖 𝑛 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛 + 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (8) didefinisikan: 𝑃 = 𝐷𝑎∆𝑡 2∆𝑥2 𝑄 = 𝑟𝑎∆𝑡 2 𝑅 = 𝐷𝑎∆𝑡 ∆𝑥2 𝑆 = 𝑠∆𝑡 2 kemudian dengan mensubtitusikan 𝑃,𝑄,𝑅 dan 𝑆 pada persamaan di atas, maka diperoleh: −𝑃𝑎𝑖−1 𝑛+1 + (1 + 𝑄 + 𝑅)𝑎𝑖 𝑛+1 − 𝑃𝑎𝑖+1 𝑛+1 − 𝑆𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑃𝑎𝑖−1 𝑛 + (1 − 𝑄 − 𝑅)𝑎𝑖 𝑛 + 𝑃𝑎𝑖+1 𝑛 + 𝑆𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (9) 2. diskritisasi persamaan meinhardt dengan metode beda hingga skema crank-nicolson pada inhibitor persamaan penghambat dalam pembentukan sel pada hydra sebagaimana persamaan: 𝜕𝑏 𝜕𝑡 = 𝑏𝑏 − 𝑠𝑏𝑎 2 − 𝑟𝑏𝑏 + 𝐷𝑏 𝜕2𝑏 𝜕𝑥2 (10) t diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel cauchy – issn: 2086-0382 136 transformasi beda hingga maju untuk turunan t dan beda hingga pusat untuk turunan x adalah sebagai berikut: 𝜕𝑏(𝑥,𝑡) 𝜕𝑡 = 𝑏𝑖 𝑛+1 − 𝑏𝑖 𝑛 ∆𝑡 𝜕2𝑏 𝜕𝑥2 = 1 2 ( 𝑏𝑖−1 𝑛+1 −2𝑏𝑖 𝑛+1 +𝑏𝑖+1 𝑛+1 ∆𝑥2 + 𝑏𝑖−1 𝑛 −2𝑏𝑖 𝑛 +𝑏𝑖+1 𝑛 ∆𝑥2 ) bentuk beda hingga disubtitusikan ke persamaan (10), sehingga diperoleh bentuk persamaan diskrit sebagai berikut: 𝑏𝑖 𝑛+1 − 𝑏𝑖 𝑛 ∆𝑡 = 𝑏𝑏 − 1 2 𝑠[𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 + 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2] − 1 2 𝑟𝑏[𝑏𝑖 𝑛+1 + 𝑏𝑖 𝑛] + 1 2 𝐷𝑏 ( 𝑏𝑖−1 𝑛+1 − 2𝑏𝑖 𝑛+1 + 𝑏𝑖+1 𝑛+1 ∆𝑥2 + 𝑏𝑖−1 𝑛 − 2𝑏𝑖 𝑛 + 𝑏𝑖+1 𝑛 ∆𝑥2 ) (11) persamaan (11) dapat disederhanakan menjadi: 𝑏𝑖 𝑛+1 ∆𝑡 = 𝑏𝑖 𝑛 ∆𝑡 + 𝑏𝑏 − 𝑠 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 − 𝑠 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 − 𝑟𝑏 2 𝑏𝑖 𝑛+1 − 𝑟𝑏 2 𝑏𝑖 𝑛 + 𝐷𝑏 2∆𝑥2 𝑏𝑖−1 𝑛+1 − 𝐷𝑏 ∆𝑥2 𝑏𝑖 𝑛+1 + 𝐷𝑏 2∆𝑥2 𝑏𝑖+1 𝑛+1 + 𝐷𝑏 2∆𝑥2 𝑏𝑖−1 𝑛 − 𝐷𝑏 ∆𝑥2 𝑏𝑖 𝑛 + 𝐷𝑏 2∆𝑥2 𝑏𝑖+1 𝑛 (12) persamaan (12) dikalikan dengan t pada kedua ruasnya, sehingga diperoleh: 𝑏𝑖 𝑛+1 = 𝑏𝑖 𝑛 + 𝑏𝑏∆𝑡 − 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 − 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 − 𝑟𝑏∆𝑡 2 𝑏𝑖 𝑛+1 − 𝑟𝑏∆𝑡 2 𝑏𝑖 𝑛 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛+1 − 𝐷𝑏∆𝑡 ∆𝑥2 𝑏𝑖 𝑛+1 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛+1 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛 − 𝐷𝑏∆𝑡 ∆𝑥2 𝑏𝑖 𝑛 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛 (13) dalam bentuk yang lebih sederhana dan untuk mengelompokkan antara diskritisasi waktu sekarang (𝑛) dan diskritisasi waktu yang akan datang (𝑛 + 1), maka persamaan (13) dapat ditulis dalam bentuk sebagai berikut: − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛+1 + 𝑏𝑖 𝑛+1 + 𝑟𝑏∆𝑡 2 𝑏𝑖 𝑛+1 + 𝐷𝑏∆𝑡 ∆𝑥2 𝑏𝑖 𝑛+1 − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛+1 + 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑏𝑏∆𝑡 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛 + 𝑏𝑖 𝑛 − 𝑟𝑏∆𝑡 2 𝑏𝑖 𝑛 − 𝐷𝑏∆𝑡 ∆𝑥2 𝑏𝑖 𝑛 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛 − 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (14) dalam bentuk yang lebih sederhana, persamaan (14) dapat ditulis dalam bentuk sebagai berikut: − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛+1 + (1 + 𝑟𝑏∆𝑡 2 + 𝐷𝑏∆𝑡 ∆𝑥2 )𝑏𝑖 𝑛+1 − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛+1 + 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑏𝑏∆𝑡 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛 + (1 − 𝑟𝑏∆𝑡 2 − 𝐷𝑏∆𝑡 ∆𝑥2 )𝑏𝑖 𝑛 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛 − 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (15) didefinisikan: 𝑇 = 𝐷𝑏∆𝑡 2∆𝑥2 𝑈 = 𝑟𝑏∆𝑡 2 𝑉 = 𝐷𝑏∆𝑡 ∆𝑥2 𝑊 = 𝑠∆𝑡 2 kemudian dengan mensubtitusikan 𝑇,𝑈,𝑉 dan 𝑊 pada persamaan (15), maka diperoleh: −𝑇𝑏𝑖−1 𝑛+1 + (1 + 𝑈 + 𝑉)𝑏𝑖 𝑛+1 − 𝑇𝑏𝑖+1 𝑛+1 + 𝑊𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑏𝑏∆𝑡 + 𝑇𝑏𝑖−1 𝑛 + (1 − 𝑈 − 𝑉)𝑏𝑖 𝑛 + 𝑇𝑏𝑖+1 𝑛 − 𝑊𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 (16) dari proses diskritisasi di atas, maka diperoleh sistem persamaan meinhardt diskrit sebagai berikut: −𝑃𝑎𝑖−1 𝑛+1 + (1 + 𝑄 + 𝑅)𝑎𝑖 𝑛+1 − 𝑃𝑎𝑖+1 𝑛+1 − 𝑆𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑃𝑎𝑖−1 𝑛 + (1 − 𝑄 − 𝑅)𝑎𝑖 𝑛 + 𝑃𝑎𝑖+1 𝑛 + 𝑆𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 −𝑇𝑏𝑖−1 𝑛+1 + (1 + 𝑈 + 𝑉)𝑏𝑖 𝑛+1 − 𝑇𝑏𝑖+1 𝑛+1 + 𝑊𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑏𝑏∆𝑡 + 𝑇𝑏𝑖−1 𝑛 + (1 − 𝑈 − 𝑉)𝑏𝑖 𝑛 + 𝑇𝑏𝑖+1 𝑛 − 𝑊𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 penutup berdasarkan penjelasan di atas, maka dapat disimpulkan bahwa bentuk diskrit dari persamaan meinhardt adalah sebagai berikut: 1. activator − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛+1 + (1 + 𝑟𝑎∆𝑡 2 + 𝐷𝑎∆𝑡 ∆𝑥2 )𝑎𝑖 𝑛+1 − 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛+1 − 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖−1 𝑛 + (1 − 𝑟𝑎∆𝑡 2 − 𝐷𝑎∆𝑡 ∆𝑥2 )𝑎𝑖 𝑛 + 𝐷𝑎∆𝑡 2∆𝑥2 𝑎𝑖+1 𝑛 + 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 2. inhibitor khusnul khamidiyah 137 volume 3 no. 3 november 2014 − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛+1 + (1 + 𝑟𝑏∆𝑡 2 + 𝐷𝑏∆𝑡 ∆𝑥2 )𝑏𝑖 𝑛+1 − 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛+1 + 𝑠∆𝑡 2 𝑏𝑖 𝑛+1(𝑎𝑖 𝑛+1)2 = 𝑏𝑏∆𝑡 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖−1 𝑛 + (1 − 𝑟𝑏∆𝑡 2 − 𝐷𝑏∆𝑡 ∆𝑥2 )𝑏𝑖 𝑛 + 𝐷𝑏∆𝑡 2∆𝑥2 𝑏𝑖+1 𝑛 − 𝑠∆𝑡 2 𝑏𝑖 𝑛(𝑎𝑖 𝑛)2 daftar pustaka [1] ayres, f. 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(1984). general zoologi sixth edition. jakarta: erlangga. pendahuluan kajian pustaka 1. persamaan diferensial parsial 2. orde persamaan diferensial parsial 3. linieritas persamaan diferensial parsial 4. bentuk kanonik persamaan diferensial parsial 5. deret taylor 6. diferensial numerik 7. metode beda hingga skema crank-nicolson gambar 1. skema crank-nicolson 8. model meinhardt 9. pembentukan sel hydra pembahasan 1. diskritisasi persamaan meinhardt dengan metode beda hingga skema crank-nicolson pada activator 2. diskritisasi persamaan meinhardt dengan metode beda hingga skema crank-nicolson pada inhibitor penutup daftar pustaka solusi numerik persamaan poisson menggunakan jaringan fungsi radial basis pada koordinat polar fatma mufidah, mohammad jamhuri jurusan matematika uin maulana malik ibrahim malang e-mail: fatma.mufida@gmail.com, m.jamhuri@live.com abstrak persamaan poisson dalam koordinat polar atau lingkaran merupakan persamaan diferensial parsial linier orde dua tipe eliptis. persamaan ini merupakan bentuk non homogen dari persamaan laplace. persamaan poisson pada koordinat polar disini menggambarkan distribusi panas dalam ruang, yang dalam hal ini berbentuk lingkaran. solusi numerik persamaan poisson diperoleh dengan metode jaringan fungsi radial basis. dengan metode ini, setiap fungsi dan turunannya dapat didekati secara langsung dengan sebuah fungsi basis. fungsi basis yang digunakan adalah fungsi basis jenis multiquadrics. solusi numerik menggunakan metode jaringan fungsi radial basis khususnya metode langsung yang diperoleh dari penelitian ini menunjukkan keakuratan yang t inggi dengan diperolehnya galat yang relatif kecil. dengan galat mutlak maksimum terkecil yaitu 0,00088, dengan pemilihan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 . ini menunjukkan bahwa metode jaringan fungsi radial basis cukup efektif dalam mengaproksimasi persamaan poisson dengan domain lingkaran. kata kunci: solusi numerik, persamaan poisson, jaringan fungsi radial basis, koordinat polar. abstract poisson equation on the polar coordinate is one of the second order partial differential equations of elliptic type. this equation is also a special case of laplace equation and inhomogeneous version of laplace equation. poisson equation here describes heat conduction on the polar region. here, we solve poisson equation using radial basis function networks with multiquadrics as a basis function. numerical experiments show that radial basis function network achieves great accuracy. the smallest maximum absolute errors is 0,00088 with ∆𝑟 = 0,1 and ∆𝜃 = 𝜋 45 . from this result, it can be concluded that radial basis function networks method is effective to approximate poisson equation on the polar coordinate. keywords: numerical solution, poisson equation, radial basis function networks, polar coordinate. pendahuluan persamaan poisson merupakan persamaan diferensial parsial orde dua tipe eliptis yang implementasinya banyak digunakan dalam teknik elektro, teknik mesin, dan fisika teori. dinamai dari nama belakang seorang matematikawan perancis yang juga seorang ahli fisika dan geometri siméon denis poisson. dalam fisika teori, persamaan poisson menggambarkan distribusi panas dalam ruang. salah satu metode numerik yang digunakan dalam menyelesaikan masalah persamaan diferensial adalah jaringan fungsi radial basis (radial basis function networks). dengan metode ini, setiap fungsi dan turunannya dapat didekati secara langsung dengan sebuah fungsi basis. jaringan fungsi radial basis telah lama dikenal dalam bidang rekayasa dan teknik sebagai jaringan syaraf tiruan yang menggunakan fungsi basis sebagai fungsi pengaktif. gagasan jaringan fungsi radial basis diperoleh dari teori aproksimasi fungsi. mengaproksimasi suatu fungsi adalah menghampiri fungsi tersebut dengan fungsi lainnya. jaringan fungsi radial basis merupakan pemetaan dari suatu vektor input dengan p-dimensi ke vektor output yang hanya 1 dimensi. secara aljabar disimbolkan dengan f:rp→r1. fungsi f terdiri dari himpunan bobot {𝑤𝑗}𝑗=1 𝑚 dan himpunan fungsi basis {ϕ(𝑥𝑖, 𝑐𝑗)}𝑗=1 𝑚 , dengan {𝑥𝑖}𝑖=1 𝑛 . secara umum, fungsi basis dapat ditulis dengan ϕ(𝑥𝑖, 𝑐𝑗) = √(𝑥𝑖 − 𝑐𝑗) 2 + α2, α = var(𝑥). 𝑥𝑖 adalah vektor input dan 𝑐𝑗 adalah titik center dari 𝑥 ke-𝑖 [3]. koordinat cartesius bukan merupakan satusatunya jalan untuk menunjukkan kedudukan suatu titik pada bidang. karena bentuk geometris di alam tidak selalu berupa kotak-kotak atau persegi panjang, namun adakalanya berbentuk lingkaran. sehingga sistem koordinat cartesius menjadi terbatas penggunaannya. cara lain ialah menggunakan sistem koordinat polar. mailto:fatma.mufida@gmail.com fatma mufidah, mohammad jamhuri 46 volume 3 no. 4 mei 2015 kajian pustaka 1. persamaan poisson strauss [6] menyatakan bahwa jika sebuah persamaan difusi atau persamaan gelombang tidak bergantung pada waktu (𝑡), yaitu 𝑢𝑡 = 0 dan 𝑢𝑡𝑡 = 0, maka persamaan difusi dan persamaan gelombang tersebut menghasilkan persamaan laplace. persamaan poisson merupakan bentuk non homogen dari persamaan laplace. ∇2𝑢 = 𝑓 (1) dengan 𝑓 adalah sebuah fungsi yang diberikan, maka disebut persamaan poisson. bentuk umum persamaan poisson dalam dua dimensi dalam koordinat cartesius yaitu: 𝝏𝟐𝒖(𝒙,𝒚) 𝝏𝒙𝟐 + 𝝏𝟐𝒖(𝒙,𝒚) 𝝏𝒚𝟐 = 𝒇(𝒙, 𝒚) (2) pada domain 𝒙𝒂 < 𝒙 < 𝒙𝒃, 𝒚𝒂 < 𝒚 < 𝒚𝒃. boyce dan diprima [1] menyatakan bahwa masalah kondisi batas untuk persamaan poisson 2 dimensi (𝒙, 𝒚) dalam sistem koordinat cartesius adalah: 𝒖(𝒙𝒂, 𝒚) = 𝒇𝟏(𝒚) 𝒖(𝒙𝒃, 𝒚) = 𝒇𝟐(𝒚) 𝒖(𝒙, 𝒚𝒂) = 𝒇𝟑(𝒙) 𝒖(𝒙, 𝒚𝒃) = 𝒇𝟒(𝒙) (3) dengan 𝒇𝟏(𝒚), 𝒇𝟐(𝒚), 𝒇𝟑(𝒙), 𝒇𝟒(𝒙) adalah fungsifungsi yang menyatakan kondisi pada batas-batas tersebut. persamaan poisson pada sistem koordinat polar (𝒓, 𝜽), dengan 𝒖(𝒓, 𝜽) adalah: 𝝏𝟐𝒖(𝒓,𝜽) 𝝏𝒓𝟐 + 𝟏 𝒓 𝝏𝒖(𝒓,𝜽) 𝝏𝒓 + 𝟏 𝒓𝟐 𝝏𝟐𝒖(𝒓,𝜽) 𝝏𝜽𝟐 = 𝒇(𝒓, 𝜽) (4) dengan domain 𝟎 < 𝒓 < 𝒂 dan 𝟎 ≤ 𝜽 ≤ 𝒃. masalah kondisi batas untuk persamaan poisson yang domainnya berupa lingkaran (𝒓, 𝜽), kondisi batas harus berupa kondisi pada tepi dan pusat lingkaran yaitu: 𝒖(𝒂, 𝜽) = 𝒇(𝜽) 𝒖(𝟎, 𝜽) = 𝒇(𝜽) (5) dengan 𝑎 adalah jari-jari lingkaran dan 𝑓(𝜃) adalah fungsi yang menyatakan kondisi pada batas tersebut. gambar 1 kondisi batas pada koordinat polar 2. sistem koordinat polar pada sistem koordinat polar, sepasang koordinat polar suatu titik ditulis dengan (𝑟, 𝜃). sebagai ilustrasi sistem koordinat polar, kita mulai dengan menggambar sebuah setengah garis tetap yang dinamakan sumbu polar yang berpangkal pada sebuah titik pusat o. titik ini disebut polar atau titik asal. biasanya sumbu polar ini kita gambar mendatar dan mengarah ke kanan dan oleh sebab itu sumbu ini dapat disamakan dengan sumbu x positif pada sistem koordinat cartesius. setiap titik p adalah perpotongan antara sebuah lingkaran tunggal yang berpusat di o dan sebuah sinar tunggal yang memancar dari o. jika r adalah jarijari ingkaran dan θ adalah salah satu sudut antara sinar dan sumbu polar. maka yang dinamakan sepasang koordinat polar dari titik p dan ditulis dengan p (r, θ) adalah (𝑟, 𝜃) [7]. gambar 2 sistem koordinat polar purcell dan varberg [7] menyatakan bahwa andaikan sumbu polar berimpit dengan sumbu x positif pada sistem koordinat cartesius. maka koordinat polar (r, θ) sebuah titik p dan koordinat cartesius (𝒙, 𝒚) titik itu dihubungkan oleh persamaan: 𝒙 = 𝒓 𝐜𝐨𝐬 𝜽 (6) 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 (7 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 (8) 𝐭𝐚𝐧 𝜽 = 𝒚 𝒙 𝜽 = 𝒂𝒓𝒄 𝐭𝐚𝐧 𝒚 𝒙 (9) 3. jaringan fungsi radial basis jaringan fungsi radial basis mulai dikenal pertama kali sejak d. s. broomhead dan david lowe menyampaikan makalahnya yang berjudul radial basis functions, multi-variable functional interpolation and adaptive networks pada tahun 1988 [2]. jaringan fungsi radial basis merepresentasikan sebuah pemetaan dari vektor input dengan p-dimensi ke vektor output yang hanya 1-dimensi. secara matematis jaringan fungsi radial basis ditulis dengan f:rp→r1 [5]. fungsi f pada f:rp→r1 terdiri dari himpunan bobot {𝑤𝑗}𝑗=1 𝑚 dan himpunan fungsi basis solusi numerik persamaan poisson menggunakan jaringan fungsi radial basis pada koordinat polar cauchy – issn: 2086-0382 47 {ϕ(𝑟𝑖, 𝜃𝑖)}𝑖=1 𝑛 , dengan {𝑟𝑖}𝑖=1 𝑛 dan {𝜃𝑖}𝑖=1 𝑛 . secara umum, fungsi basis dapat ditulis dengan ϕ(𝑟𝑖, 𝑐𝑗) = √(𝑟𝑖 − 𝑐𝑗) 2 + α, α = var(𝑟𝑖). 𝑟𝑖 adalah vektor input dan𝑐𝑗 adalah titik center dari 𝑟 ke-𝑖 [3]. fungsi basis untuk 2 variabel (𝑟, 𝜃): 𝛟(𝒓𝒊, 𝜽𝒊, 𝒄𝒋, 𝒅𝒋) = √(𝒓𝒊 − 𝒄𝒋) 𝟐 + (𝜽𝒊 − 𝒅𝒋) 𝟐 + 𝛃 (10) β = var(𝑟𝑖) + var(𝜃𝑖) 2 𝑟𝑖 = vektor input ke-𝑖 𝜃𝑖 = vektor input ke-𝑖 𝑐𝑗 = titik center dari 𝑟 ke-𝑖 𝑑𝑗 = titik center dari 𝜃 ke-𝑖 α = nilai parameter width untuk 1 variabel β = nilai parameter width untuk 2 variabel metode penelitian metode yang digunakan dalam menyelesaikan secara numerik persamaan poisson pada koordinat polar adalah menggunakan pendekatan studi literatur atau library research. adapun langkah-langkah dalam penelitian ini adalah sebagai berikut: a) mentransformasi persamaan poisson beserta kondisi batasnya dari koordinat cartesius ke koordinat polar. b) menurunkan fungsi basis multiquadrics terhadap variabel-variabel bebasnya (𝒓, 𝜽) hingga turunan kedua. c) mendiskritisasi persamaan poisson pada koordinat polar menggunakan jaringan fungsi radial basis serta kondisi batasnya. d) mensubstitukan nilai-nilai input (𝑟𝑖, 𝜃𝑖) dan 𝑓(𝑟𝑖, 𝜃𝑖) ke dalam persamaan poisson dalam bentuk persamaan jaringan fungsi radial basis yang telah diperoleh. e) menghitung nilai bobot 𝑤𝑗. f) menghitung solusi persamaan poisson pada koordinat polar dengan mengalikan nilai bobot 𝑤𝑗 yang telah diperoleh dan fungsi basis multiquadrics yang tanpa diturunkan. g) melakukan simulasi, menggambarkan grafik, serta menganalisis galat. h) memberikan kesimpulan atas hasil penelitian yang telah diperoleh serta saran untuk penelitian selanjutnya. pembahasan 1. transformasi persamaan poisson transformasi persamaan poisson dari koordinat cartesius ke koordinat polar, dengan kaidah rantai (chain rule) diperoleh turunan pertama 𝑢(𝑥, 𝑦) persamaan (2) terhadap 𝑥 dan 𝑦. 𝝏𝒖 𝝏𝒙 = 𝝏𝒖 𝝏𝒓 𝝏𝒓 𝝏𝒙 + 𝝏𝒖 𝝏𝜽 𝝏𝜽 𝝏𝒙 (11) 𝝏𝒖 𝝏𝒚 = 𝝏𝒖 𝝏𝒓 𝝏𝒓 𝝏𝒚 + 𝝏𝒖 𝝏𝜽 𝝏𝜽 𝝏𝒚 (12) kemudian turunan kedua 𝑢(𝑥, 𝑦) persamaan (2) terhadap 𝑥 dan 𝑦 diperoleh: 𝝏𝟐𝒖 𝝏𝒙𝟐 = 𝒙𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 − 𝟐 𝒙𝒚 𝒓𝟑 𝝏𝟐𝒖 𝝏𝜽𝝏𝒓 + 𝒚𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒚𝟒 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 + 𝟐𝒙𝒚 𝒓𝟒 𝝏𝒖 𝝏𝜽 (13) 𝝏𝟐𝒖 𝝏𝒚𝟐 = 𝒚𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 + 𝟐 𝒙𝒚 𝒓𝟑 𝝏𝟐𝒖 𝝏𝜽𝝏𝒓 + 𝒙𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒙𝟒 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 − 𝟐𝒙𝒚 𝒓𝟒 𝝏𝒖 𝝏𝜽 (14) substitusi persamaan (13) dan (14) ke dalam persamaan (2): 𝝏𝟐𝒖(𝒙,𝒚) 𝝏𝒙𝟐 + 𝝏𝟐𝒖(𝒙,𝒚) 𝝏𝒚𝟐 = 𝑓(𝑥, 𝑦) = 𝒇(𝒓, 𝜽) 𝒚𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 +𝟐 𝒙𝒚 𝒓𝟑 𝝏𝟐𝒖 𝝏𝜽𝝏𝒓 + 𝒙𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒙𝟐 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 − 𝟐𝒙𝒚 𝒓𝟒 𝝏𝒖 𝝏𝜽 𝒙𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 −𝟐 𝒙𝒚 𝒓𝟑 𝝏𝟐𝒖 𝝏𝜽𝝏𝒓 + 𝒚𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒚𝟐 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 + 𝟐𝒙𝒚 𝒓𝟒 𝝏𝒖 𝝏𝜽 + 𝒙𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 + 𝒚𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒚𝟐 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 + 𝒚𝟐 𝒓𝟐 𝝏𝟐𝒖 𝝏𝒓𝟐 + 𝒙𝟐 𝒓𝟑 𝝏𝒖 𝝏𝒓 + 𝒙𝟐 𝒓𝟒 𝝏𝟐𝒖 𝝏𝜽𝟐 = 𝒇(𝒓, 𝜽) ( 𝑥2 𝑟2 + 𝑦2 𝑟2 ) 𝜕2𝑢 𝜕𝑟2 + ( 𝑦2 𝑟3 + 𝑥2 𝑟3 ) 𝜕𝑢 𝜕𝑟 + ( 𝑥2 𝑟4 + 𝑦2 𝑟4 ) 𝜕2𝑢 𝜕𝜃2 = 𝑓(𝑟, 𝜃) ( 𝑥2+𝑦2 𝑟2 ) 𝜕2𝑢 𝜕𝑟2 + ( 𝑥2+𝑦2 𝑟3 ) 𝜕𝑢 𝜕𝑟 + ( 𝑥2+𝑦2 𝑟4 ) 𝜕2𝑢 𝜕𝜃2 = 𝑓(𝑟, 𝜃) ( 𝑟2 𝑟2 ) 𝜕2𝑢 𝜕𝑟2 + ( 𝑟2 𝑟3 ) 𝜕𝑢 𝜕𝑟 + ( 𝑟2 𝑟4 ) 𝜕2𝑢 𝜕𝜃2 = 𝑓(𝑟, 𝜃) 𝜕2𝑢 𝜕𝑟2 + ( 1 𝑟 ) 𝜕𝑢 𝜕𝑟 + ( 1 𝑟2 ) 𝜕2𝑢 𝜕𝜃2 = 𝑓(𝑟, 𝜃) sehingga, persamaan poisson dalam koordinat polar diperoleh persamaan (4). dengan domain 0 < 𝑟 < 𝑎 dan 0 ≤ 𝜃 ≤ 𝑏. dan kondisi batas persamaan (5). 2. solusi numerik persaaan poisson langkah pertama untuk menghitung solusi numerik persamaan poisson (4) menggunakan jaringan fungsi radial basis adalah mendiskritisasi persamaan poisson (4). karena persamaan poisson (4) melibatkan turunan kedua 𝒖(𝒓, 𝜽) terhadap 𝒓 dan 𝜽, maka untuk menghampiri fungsi 𝒖(𝒓, 𝜽) pada persamaan poisson (4), fungsi basisnya diturunkan dua kali terhadap 𝒓 dan 𝜽. karena menggunakan metode langsung, sehingga fungsi basisnya diturunkan terhadap variabel bebasnya. 𝑢𝑟(𝑟, 𝜃), 𝑢𝑟𝑟(𝑟, 𝜃), dan 𝑢𝜃𝜃(𝑟, 𝜃) digunakan untuk menghampiri fungsi 𝑢(𝑟, 𝜃) pada persamaan poisson (4). fungsi basis yang digunakan adalah jenis multiquadrics. fatma mufidah, mohammad jamhuri 48 volume 3 no. 4 mei 2015 𝑢(𝑟, 𝜃) = ∑ 𝑤𝑗 𝑚 𝑗=1 ϕ(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) (15) 𝒖𝒓(𝒓, 𝜽) = ∑ 𝒘𝒋 𝒎 𝒋=𝟏 𝛟𝒓(𝒓, 𝜽, 𝒄𝒋, 𝒅𝒋) (16) 𝒖𝒓𝒓(𝒓, 𝜽) = ∑ 𝒘𝒋 𝒎 𝒋=𝟏 𝛟𝒓𝒓(𝒓, 𝜽, 𝒄𝒋, 𝒅𝒋) (17) 𝒖𝜽(𝒓, 𝜽) = ∑ 𝒘𝒋 𝒎 𝒋=𝟏 𝛟𝜽(𝒓, 𝜽, 𝒄𝒋, 𝒅𝒋) (18) 𝒖𝜽𝜽(𝒓, 𝜽) = ∑ 𝒘𝒋 𝒎 𝒋=𝟏 𝛟𝜽𝜽(𝒓, 𝜽, 𝒄𝒋, 𝒅𝒋) (19) dengan ϕ(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) diketahui pada persamaan (10), maka ϕ𝑟 (𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗), ϕ𝑟𝑟(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗), ϕ𝜃(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) dan ϕ𝜃𝜃(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) diperoleh: ϕ𝑟(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) = 𝑟−𝑐𝑗 √(𝑟−𝑐𝑗) 2 +(𝜃−𝑑𝑗) 2 +𝛽 (20) ϕ𝑟𝑟(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) = (𝜃−𝑑𝑗) 2 +𝛽 ((𝑟−𝑐𝑗) 2 +(𝜃−𝑑𝑗) 2 +𝛽) 3 2 (21) ϕ𝜃(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) = 𝜃−𝑑𝑗 √(𝑟−𝑐𝑗) 2 +(𝜃−𝑑𝑗) 2 +𝛽 (22) ϕ𝜃𝜃(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) = (𝑟−𝑐𝑗) 2 +𝛽 ((𝑟−𝑐𝑗) 2 +(𝜃−𝑑𝑗) 2 +𝛽) 3 2 (23) untuk mengaproksimasi persamaan poisson dalam koordinat polar menggunakan jaringan fungsi radial basis digunakan persamaan poisson dalam bentuk persamaan jaringan fungsi radial basis. dengan substitusi persamaan (16), (17), dan (18) ke dalam persamaan (4), maka: 𝜕2𝑢(𝑟,𝜃) 𝜕𝑟2 + 1 𝑟 𝜕𝑢(𝑟,𝜃) 𝜕𝑟 + 1 𝑟2 𝜕2𝑢(𝑟,𝜃) 𝜕𝜃2 = 𝑓(𝑟, 𝜃) ∑ 𝑤𝑗 𝑚 𝑗=1 ϕ𝑟𝑟(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) + 1 𝑟 ∑ 𝑤𝑗 𝑚 𝑗=1 ϕ𝑟(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) + 1 𝑟2 ∑ 𝑤𝑗 𝑚 𝑗=1 ϕ𝜃𝜃(𝑟, 𝜃, 𝑐𝑗, 𝑑𝑗) = 𝑓(𝑟, 𝜃) ∑ 𝑤𝑗 𝑚 𝑗=1 [ϕ𝑟𝑟(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) + 1 𝑟𝑖 ϕ𝑟 (𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) + 1 𝑟𝑖 2 ϕ𝜃𝜃(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗)] = 𝑓(𝑟𝑖, 𝜃𝑖) dengan mensubstitusi persamaan (20), (21), (23) maka diperoleh persamaan poisson dalam bentuk persamaan jaringan fungsi radial basis yaitu persamaan (24). ∑ 𝑤𝑗 𝑚 𝑗=1 [ (𝜃𝑖 − 𝑑𝑗) 2 + 𝛽 ((𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽) 3 2 + 1 𝑟𝑖 𝑟𝑖 − 𝑐𝑗 √(𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽 + (24) 1 𝑟𝑖 2 (𝑟𝑖 − 𝑐𝑗) 2 + 𝛽 ((𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽) 3 2 ] = 𝑓(𝑟𝑖, 𝜃𝑖) pada domain 0 < 𝑟 < 𝑎 dan 0 ≤ 𝜃 ≤ 𝑏. dengan kondisi batas pada persamaan (5). dimana 𝑖 = 1,2, … , 𝑛 dan 𝛽 = 𝑣𝑎𝑟(𝑟𝑖)+𝑣𝑎𝑟(𝜃𝑖) 2 . langkah selanjutnya yaitu menghitung niai bobot 𝑤𝑗 . persamaan (24) akan digunakan untuk menghitung nilai bobot 𝑤𝑗 . misal 𝑨(𝒓𝒊, 𝜽𝒊, 𝒄𝒋, 𝒅𝒋) = (𝜽𝒊−𝒅𝒋) 𝟐 +𝜷 ((𝒓𝒊−𝒄𝒋) 𝟐 +(𝜽𝒊−𝒅𝒋) 𝟐 +𝜷) 𝟑 𝟐 + 𝟏 𝒓𝒊 𝒓𝒊−𝒄𝒋 √(𝒓𝒊−𝒄𝒋) 𝟐 +(𝜽𝒊−𝒅𝒋) 𝟐 +𝜷 + 𝟏 𝒓𝒊 𝟐 (𝒓𝒊−𝒄𝒋) 𝟐 +𝜷 ((𝒓𝒊−𝒄𝒋) 𝟐 +(𝜽𝒊−𝒅𝒋) 𝟐 +𝜷) 𝟑 𝟐 maka persamaan (24) menjadi: ∑ 𝑤𝑗 𝑚 𝑗=1 𝐴(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) = 𝑓(𝑟𝑖, 𝜃𝑖) sehingga, nilai bobot 𝑤𝑗 dapat diperoleh dengan perhitungan: 𝑤𝑗 = 𝐴(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) −1 𝑓(𝑟𝑖, 𝜃𝑖) (25) dengan mensubstitusi nilai-nilai input (𝑟𝑖, 𝜃𝑖) ke persamaan (25) dan 𝑓(𝑟𝑖, 𝜃𝑖) yang didefinisikan dari persamaan poisson pada koordinat polar yang diaproksimasi, dapat dihitung nilai bobot 𝑤𝑗 . langkah selanjutnya yaitu menghitung solusi numerik persamaan poisson dengan cara mengalikan nilai bobot 𝑤𝑗 dengan fungsi basis multiquadrics 2 variabel yang tanpa diturunkan. fungsi basis yang digunakan adalah fungsi basis pada persamaan (10). sehingga, solusi numerik persamaan poisson (4) adalah: �̂�(𝑟𝑖, 𝜃𝑖) = ∑ 𝑤𝑗 𝑚 𝑗=1 ϕ(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) (26) perhitungan galat 𝜀𝑖 diperoleh dengan: [ 𝜀1 𝜀2 ⋮ 𝜀𝑛 ] = | 𝑢(𝑟1, 𝜃1) 𝑢(𝑟2, 𝜃2) ⋮ 𝑢(𝑟𝑛, 𝜃𝑛) − �̂�(𝑟1, 𝜃1) �̂�(𝑟2, 𝜃2) ⋮ �̂�(𝑟𝑛, 𝜃𝑛) | (27) besarnya galat menunjukkan seberapa dekat solusi eksak dengan solusi numeriknya. solusi numerik persamaan poisson menggunakan jaringan fungsi radial basis pada koordinat polar cauchy – issn: 2086-0382 49 3. simulasi untuk lebih memahami cara kerja metode jaringan fungsi radial basis, akan ditunjukkan simulasi solusi numerik persamaan poisson pada koordinat polar. persamaan poisson pada koordinat polar yang akan diselesaikan adalah: 𝝏𝟐𝒖(𝒓,𝜽) 𝝏𝒓𝟐 + 𝟏 𝒓 𝝏𝒖(𝒓,𝜽) 𝝏𝒓 + 𝟏 𝒓𝟐 𝝏𝟐𝒖(𝒓,𝜽) 𝝏𝜽𝟐 = −𝟑 𝐜𝐨𝐬 𝜽, 𝟎 < 𝒓 < 𝟏 dan 𝟎 ≤ 𝜽 ≤ 𝟐𝝅 (28) dengan kondisi batas 𝑢(1, 𝜃) = 0 𝑢(0, 𝜃) = 0 (29) domain pada persamaan (28) kemudian dipartisi menjadi beberapa data diskrit, dengan ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 12 . langkah pertama untuk menghitung solusi numerik persamaan poisson (28) dengan kondisi batas (29) menggunakan jaringan fungsi radial basis adalah diskritisasi domain. domain persamaan poisson kemudian dipartisi dengan ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 12 . 𝜃𝑖 tersebut masih dalam satuan derajat (°), sehingga dirubah terlebih dahulu ke dalam satuan radian supaya dapat dioperasikan dengan 𝑟𝑖. setelah memperoleh 𝑟𝑖 dan 𝜃𝑖 yang telah sama penyebutnya, maka domain pada persamaan poisson dapat digambarkan dengan gambar 3 berikut: gambar 3 diskritisasi domain persamaan poisson setelah membagi domain menjadi beberapa titik diskrit, kemudian persamaan poisson (28) akan dirubah menjadi persamaan jaringan fungsi radial basis, yaitu pada persamaan (30): ∑ 𝑤𝑗 525 𝑗=1 [ (𝜃𝑖 − 𝑑𝑗) 2 + 𝛽 ((𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽) 3 2 + 1 𝑟𝑖 𝑟𝑖 − 𝑐𝑗 √(𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽 + 1 𝑟𝑖 2 (𝑟𝑖 − 𝑐𝑗) 2 + 𝛽 ((𝑟𝑖 − 𝑐𝑗) 2 +(𝜃𝑖 − 𝑑𝑗) 2 + 𝛽) 3 2 ] = −3 cos 𝜃𝑖 (30) dengan kondisi batas pada persamaan (29) yaitu 𝑢(1, 𝜃) = 0 dan 𝑢(0, 𝜃) = 0. artinya, ketika 𝑟𝑖 = 0,00 dan 𝑟𝑖 = 1,00 maka ruas kanan pada persamaan (30) sama dengan 0 atau 𝑓(0, 𝜃𝑖) = 0 dan 𝑓(1, 𝜃𝑖) = 0. langkah selanjutnya, yaitu menghitung nilai bobot 𝑤𝑗. nilai bobot 𝑤𝑗 dihitung dengan persamaan (25): 𝑤𝑗 = 𝐴(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) −1 𝑓(𝑟𝑖, 𝜃𝑖) dengan 𝑖 = 1,2, ⋯ , 525 dan 𝑗 = 1,2, ⋯ , 525. himpunan fungsi basis 𝐴(𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗) yang berbentuk matriks 525×525 dan himpunan 𝑓(𝑟𝑖, 𝜃𝑖) yang berbentuk matriks 525×1 diperoleh dengan mensubstitusikan nilai-nilai input 𝑟𝑖, 𝜃𝑖, 𝑐𝑗, 𝑑𝑗 ke dalam persamaan (25). kemudian diperoleh nilainilai bobot 𝑤𝑗 yang berbentuk matriks 525×1. solusi numerik persamaan poisson (28) dapat dihitung dengan persamaan (31) berikut: �̂�(𝒓𝒊, 𝜽𝒊) = ∑ 𝒘𝒋 𝟓𝟐𝟓 𝒋=𝟏 𝛟(𝒓𝒊, 𝜽𝒊, 𝒄𝒋, 𝒅𝒋) (31) hasil simulasi solusi numerik persamaan (28) dengan kondisi batas (29) kemudian diperoleh �̂�(𝑟𝑖, 𝜃𝑖) seperti dalam gambar 4 berikut: gambar 4 solusi numerik persamaan poisson dengan dengan ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 12 simulasi solusi numerik persamaan poisson tersebut dilakukan dengan program matlab 2008r. persamaan poisson pada koordinat polar (𝑟, 𝜃), domainnya berbentuk lingkaran. persamaan ini menggambarkan distribusi panas atau penyebaran panas dalam suatu ruang dengan keadaan steady state atau tetap. panas dalam suatu ruang yang berbentuk lingkaran menyebar tanpa ada pengaruh waktu, atau dapat dikatakan waktu=konstan. gambar 4 mendeskripsikan bahwa panas menyebar pada seluruh ruang/ domain yang berbentuk lingkaran dengan jari-jari 1. kondisi pada batas persamaan ini adalah untuk 𝑢(1, 𝜃) = 0 dan untuk 𝑢(0, 𝜃) = 0. artinya, pada saat 𝑟𝑖 = 1 dan 𝑟𝑖 = 0, suhu/ panas ruang adalah sebesar 1 satuan panas. gambar 4 juga menjelaskan bahwa panas menyebar dari 𝑟 = 0 ke 𝑟 = 1. dan pada saat tertentu, panas dalam keadaan meningkat/ suhu naik yaitu pada saat 𝑟 = 0,3 sampai 𝑟 = 0,7. dari gambar tersebut terlihat bahwa suhu tertinggi ketika grafik berwarna coklat tua yaitu pada 𝑟 = 0,3. suhu terendah terlihat ketika grafik berwarna biru tua yaitu 𝑟 = −0,3 sampai 𝑟 = −0,7. dengan kondisi batas yang diketahui adalah 𝑢(1, 𝜃) = 0 dan 𝑢(0, 𝜃) = 0, fatma mufidah, mohammad jamhuri 50 volume 3 no. 4 mei 2015 artinya suhu di pusat lingkaran/ pusat ruang sama dengan suhu di batas/ tepi ruang yang pada kasus ini berbentuk lingkaran. perhitungan galat dilakukan untuk mengetahui seberapa dekat solusi analitik atau solusi eksak dengan solusi numeriknya. galat diperoleh dari selisih antara solusi eksak dan solusi numeriknya. solusi eksak dari persamaan poisson pada koordinat polar 𝜕2𝑢(𝑟,𝜃) 𝜕𝑟2 + 1 𝑟 𝜕𝑢(𝑟,𝜃) 𝜕𝑟 + 1 𝑟2 𝜕2𝑢(𝑟,𝜃) 𝜕𝜃2 = −3 cos 𝜃 dengan domain 0 < 𝑟 < 1 dan 0 ≤ 𝜃 ≤ 2𝜋 pada kondisi batas 𝑢(1, 𝜃) = 0 dan 𝑢(0, 𝜃) = 0 tersebut diketahui dalam artikel yang ditulis oleh r. c. mittal dan s. gahlaut [4]: 𝑢(𝑟, 𝜃) = 𝑟(1 − 𝑟) cos 𝜃 = (𝑟 − 𝑟2) ∙ cos 𝜃 = (𝑟 cos 𝜃 − 𝑟2 cos 𝜃) 𝑢(𝑟, 𝜃) = 𝑟 cos 𝜃 − 𝑟2 cos 𝜃 (32) solusi eksak persamaan poisson (28) dengan kondisi batas (29) dapat dilihat pada gambar 5 berikut: gambar 5 solusi eksak persamaan poisson persamaan poisson dengan dengan ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 12 galat solusi numerik persamaan poisson (3.37) menggunakan metode jaringan fungsi radial basis dihitung dari persamaan (3.43) berikut: [ 𝜺𝟏 𝜺𝟐 ⋮ 𝜺𝟓𝟐𝟓 ] = | 𝒖(𝒓𝟏, 𝜽𝟏) 𝒖(𝒓𝟐, 𝜽𝟐) ⋮ 𝒖(𝒓𝟓𝟐𝟓, 𝜽𝟓𝟐𝟓) − �̂�(𝒓𝟏, 𝜽𝟏) �̂�(𝒓𝟐, 𝜽𝟐) ⋮ �̂�(𝒓𝟓𝟐𝟓, 𝜽𝟓𝟐𝟓) | (3.43) dan hasil perhitungannya dapat dilihat pada gambar 6 berikut: gambar 6 galat jaringan fungsi radial basis ketika ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 12 galat mutlak maksimum diketahui dari program matlab yaitu 0,0012. menganalisis galat metode jaringan fungsi radial basis pada solusi numerik persamaan poisson pada domain lingkaran (28) dilakukan beberapa simulasi lagi dengan memperbesar dan memperkecil ∆𝑟 serta ∆𝜃. simulasi kedua dilakukan dengan memperbesar ∆𝑟 dan ∆𝜃. solusi numerik persamaan poisson (28) dengan kondisi batas (29) untuk ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 6 dilakukan dengan program matlab. gambar solusi numerik, solusi eksak, dan galatnya dapat dilihat pada gambar 7, 8, dan 9 berikut: gambar 7 solusi numerik persamaan poisson persamaan poisson dengan dengan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 6 gambar 8 solusi eksak persamaan poisson persamaan poisson dengan dengan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 6 gambar 9 galat jaringan fungsi radial basis ketika ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 6 galat mutlak maksimum diketahui dari program matlab yaitu 0,0219. simulasi ketiga dilakukan dengan memperkecil ∆𝜃. solusi numerik persamaan poisson solusi numerik persamaan poisson menggunakan jaringan fungsi radial basis pada koordinat polar cauchy – issn: 2086-0382 51 (28) dengan kondisi batas (29) untuk ∆𝑟 = 0,05 dan ∆𝜃 = 𝜋 45 dilakukan dengan program matlab. gambar solusi numerik, solusi eksak, dan galatnya dapat dilihat pada gambar 10, 11, dan12 berikut: gambar 10 solusi numerik persamaan poisson dengan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 gambar 11 solusi eksak persamaan poisson dengan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 gambar 12 galat metode jaringan fungsi radial basis ketika ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 gambar 13 trial error untuk ∆𝑟 = 0,1 galat mutlak maksimum diketahui dari program matlab yaitu 0,00088. kemudian dilakukan beberapa simulasi lagi untuk mengetahui hasil galat yang diperoleh. dengan pemilihan ∆𝑟 = 0,1 dan ∆𝜃 yang berbeda-beda, diperoleh hasil trial error dari beberapa simulasi yang telah dilakukan pada gambar 13 diatas. hasil-hasil trial error tersebut, menunjukkan bahwa besarnya galat mutlak maksimum belum tentu dipengaruhi oleh banyaknya iterasi yang dilakukan. semakin banyak iterasi yang disebabkan oleh dengan ∆𝑟 dan ∆𝜃 yang kecil, belum tentu menghasilkan galat mutlak maksimum yang kecil. begitu pula sebaliknya. namun dari simulasi yang telah dilakukan, dapat disimpulkan bahwa metode jaringan fungsi radial basis cukup efektif dalam mengaproksimasi persamaan poisson pada koordinat polar (28) dengan kondisi batas (29) dengan galat mutlak maksimum terkecil yang diperoleh yaitu 0,00088 dari pemilihan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 . kesimpulan berdasarkan hasil pembahasan, dapat diperoleh kesimpulan sebagai berikut: solusi numerik persamaan poisson pada koordinat polar yang diperoleh dalam penelitian ini menunjukkan hasil yang cukup dekat dengan solusi eksaknya galat mutlak maksimum terkecil yang diperoleh yaitu 0,00088 dari pemilihan ∆𝑟 = 0,1 dan ∆𝜃 = 𝜋 45 . besarnya galat mutlak maksimum belum tentu dipengaruhi oleh banyaknya iterasi yang dilakukan. semakin banyak iterasi yang disebabkan oleh dengan ∆𝑟 dan ∆𝜃 yang kecil, belum tentu menghasilkan galat mutlak maksimum yang kecil. begitu pula sebaliknya. referensi [1] w.e. boyce, r.c. diprima, c.w. haines, elementary differential equations and boundary value problems, wiley new york, 1992. [2] d.s. broomhead, d. lowe, radial basis functions, multi-variable functional interpolation and adaptive networks, dtic document, 1988. [3] n. mai-duy, t. tran-cong, approximation of function and its derivatives using radial basis function networks, appl. math. model. 27 (2003) 197–220. fatma mufidah, mohammad jamhuri 52 volume 3 no. 4 mei 2015 [4] r.c. mittal, s. gahlaut, a boundary integral formulation for poisson’s equation in polar coordinates, indian j. pure appi. math. 18 (1987) 965–972. [5] v. olej, p. hajek, municipal creditworthiness modelling by radial basis function neural networks and sensitive analysis of their input parameters, in: artif. neural networks– icann 2009, springer, 2009: pp. 505– 514. [6] w.a. strauss, partial differential equations: an introduction, new york. (1992). [7] d. varberg, e.j. purcell, kalkulus dan geometri analitis jilid 2, jakarta erlangga..(1999). kalkulus dan geom. anal. jilid. 1 (1994). pendahuluan kajian pustaka 1. persamaan poisson 2. sistem koordinat polar 3. jaringan fungsi radial basis metode penelitian pembahasan 1. transformasi persamaan poisson 2. solusi numerik persaaan poisson 3. simulasi kesimpulan referensi suma’inna dan gugun gumilar pemecahan sandi kriptografi dengan menggabungkan metode hill cipher dan metode caesar cipher indria eka wardani jurusan matematika, universitas islam darul ulum lamongan e-mail: arinds.0810@gmail.com abstrak pesan merupakan suatu pernyataan rahasia yang dibuat oleh pembuat pesan dan ditujukan kepada penerima pesan. perkembangan teknologi yang modern mengakibatkan tingkat kerahasiaan suatu pesan tidak terjamin keamanannya. apalagi akhir-akhir ini ulah haeker (pembobol keamanan) yang semakin mengkhawatirkan para pembuat pesan. berdasarkan hal tersebut maka penulis akan mengkaji sandi kriptografi untuk menyampaikan pesan rahasia agar aman sampai pada penerima pesan. dalam makalah ini pesan yang berbentuk sandi kriptografi digunakan untuk menggabungkan metode hill cipher dan caesar cipher. pada pengoperasian enkripsi dan dekripsi sandi kriptografi, langkah-langkahnya yang digunakan adalah metode caesar cipher, sedangkan dalam langkah-langkah tersebut menggunakan kunci dengan memakai metode hill cipher. dalam pembuatan maupun penguraian sandi tersebut hanya pembuat pesan dan penerima pesan yang mengetahui bagaimana teknik pengoperasiannya. dengan demikian diharapkan penyampaian pesan rahasia terjamin keamanannya. kata kunci: sandi kriptografi, metode hill cipher, metode caesar cipher, penggabungan metode hill cipher dan metode caesar cipher . abstract message is a privacy statement which is made by a person and addressed to another person with the consent of people who have made the message. but with the development of modern technology resulted in a high level of privacy for their safety message. especially lately act haeker (security breaker) are increasingly rampant and alarming message makers. to address this, the authors will examine and discuss the cryptographic cipher to convey a secret message privacy. message in the form of a password using cryptography in this paper combining of hill cipher method and caesar cipher methods.in the operation process of encryption and decryption passwords cryptography, the steps using the caesar ciphermethods, whereas in steps using keys by using hill cipher method. in the making and the first decoding the me ssage makers and recipients know how the operation technique. thus the expected delivery of message privacy secured. keywords: password cryptography, methods of hill cipher, caesar cipher method,combining of hill cipher method and caesar cipher method. pendahuluan pesan merupakan pernyataan rahasia yang di buat oleh seseorang dan ditujukan kepada orang lain dengan seijin orang yang telah membuat pesan itu. tetapi dengan perkembangan teknologi yang modern mengakibatkan tingkat kerahasiaan suatu pesan tidak terjamin keamanannya. apalagi akhir-akhir ini ulah haeker (pembobol keamanan) yang semakin merajalela dan mengkhawatirkan para pembuat pesan. maka diperlukan untuk mempelajari ilmu kriptografi yaitu ilmu penyandian terutama dikalangan militer maupun agen-agen rahasia dan yang lainnya mereka sudah tidak asing lagi dalam mempelajari ilmu kriptografi bahkan mereka sudah menggunakan alat-alat penyandian yang modern, sehingga mengkibatkan algoritma klasik menjadi terkesampingkan. sehubungan dengan itu maka masyarakat umum yang kurang mengetahui tentang ilmu kriptografi supaya dapat mempelajarinya juga, karena dalam setiap individu mempunyai hak dalam keamanan suatu pesan.dan dengan algoritma klasik yang tidak terlalu sulit dalam mempelajarinya dapat dijadikan alternatif untuk keamanan suatu pesan. maka penulis ingin memaduhkan bagian dari algoritma klasik yaitu metode hill cipher dan metode caesar cipher. kriptografi kriptografi (chrytography) berasal dari bahasa yunani: “chrytos” artinya “secret” (rahasia), mailto:arinds.0810@gmail.com pemecahan sandi kriptografi dengan menggabungkan metode hill cipher dan metode caesar chipher jurnal cauchy – issn: 2086-0382 233 sedangkan “graphein” artinya “writing” (tulisan). jadi, kriptografi adalah ilmu dan seni untuk menjaga keamanan pesan (munir,2006:2) menezes, van oorschot dan vanstone (1997) menyatakan bahwa kriptografi adalah suatu studi teknik matematika yang berhubungan dengan aspek keamanan informasi seperti kerahasiaan, integritas data, otentikasi entitas dan otentikasi keaslian data. kriptografi tidak hanya berarti penyandian keamanan informasi, melainkan sebuah himpunan teknik-teknik. jadi, secara umum dapat diartikan sebagai seni menulis atau memecahkan cipher (talbot dan welsh,2006). didalam mempelajari ilmu kriptografi terdapat beberapa istilah atau termologi antara lain, pesan (message) adalah data atau informasi yang dapat dibaca dan dimengerti maknanya. keamanan pesan diperoleh dengan menyandikannya menjadi pesan yang tidak mempunyai makna. pesan yang dirahasiakan dinamakan plainteks (plainteks, artinya teks jelas yang dapat dimengerti), sedangkan pesan hasil penyandian disebut cipherteks (ciherteks, artinya teks tersandi). pesan yang telah disandikan dapat dikembalikan lagi kepesan aslinya hanya oleh orang yang berhak (orang yang berhak adalah orang yang mengetahui metode penyandian atau memiliki kunci penyandian). proses menyandikan plainteks menjadi cipherteks disebut enkripsi (encryption) dan proses membalikkan cipherteks menjadi plainteksnya disebut dekripsi (decryption) (rinaldi munir,2005:203). dalam ilmu kriptografi terdapat pula algoritma-algoritma yang bisa dipelajari. algoritma kriptografi atau sering juga disebut dengan cipher adalah suatu fungsi matematis yang digunakan untuk melakukan enkripsi dan dekripsi (schneier, 1996). salah satunya yaitu algoritma simetri (menggunakan satu kunci untuk enkripsi dan dekripsinya) dan disebut juga algoritma klasik. contoh dari algoritma simetri yaitu caesar cipher dan hill cipher. hill cipher hill cipher termasuk dalam salah satu kriptosistem polialfabetik, artinya setiap karakter alphabet bisa dipetakkan ke lebih dari satu macam karakter alphabet. cipher ini ditemukan pada tahun 1929 oleh laster s. hill. misalkan m adalah bilangan bulat positif, dan p = c = (z26)m (dony ariyus, 2008:59). adapun menurut widyanarko,2007. hill cipher merupakan algoritma kunci simetri yang menggunakan kunci yang sama pada proses enkripsi dan dekripsinya hill cipher merupakan penerapan aritmatika modulo pada kriptografi. teknik kriptografi ini menggunakan sebuah matriks persegi sebagai kunci yang digunakan untuk melakukan enkripsi dan dekripsi serta menggunakan m buah persamaan linier. dalam penerapannya, hill cipher menggunakan teknik perkalian matriks dan teknik invers terhadap matriks. ide dari algoritma hill cipher adalah untuk membuat m kombinasi linier dari m karakter alfabetik didalam suatu elemen plainteks, sehingga dihasilkan m karakter alfabetik sebagai elemen dari cipherteks. secara umum jika a adalah matriks mxm atas z26 dan 𝑥 = (𝑥1 … . 𝑥𝑛 ) 𝑇 𝜖 𝑃 sehingga dihitung 𝑦 = 𝑒𝐴(𝑥) = (𝑦1 … . 𝑦𝑚 ) 𝑇 𝜖 𝐶 sebagai berikut: [ 𝑦1 … 𝑦𝑛 ] = [ 𝑎11 ⋯ 𝑎1𝑚 ⋯ ⋯ ⋯ 𝑎𝑚1 ⋯ 𝑎𝑚𝑚 ] [ 𝑥1 ⋯ 𝑥𝑚 ] sehingga dapat ditulis y = ax. fungsi dekripsinya diturunkan dari formula di atas, karena y = ax jika a-1 ada maka x = a-1y. caesar cipher subtitusi cipher yang pertama dalam dunia penyandian pada waktu pemerintahan yulius caesar yang dikenal dengan caesar cipher, dengan mengganti posisi huruf awal dari alphabet. (donny arius,2006:17). pada buku edisi ke dua mengatakan dengan mengganti posisi huruf awal dari alphabet atau disebut juga dengan algoritma rot3. caesar cipher (rot3) plain text encoded text abc def hello khoor attack dwwdfn yang artinya mengganti (menggeser) tiap-tiap huruf dalam alphabet sebanyak tiga huruf kedepan sehingga a, b, c menjadi d, e, f dan 0, 1, 2 menjadi 3, 4, 5. pada algoritma caesar cipher untuk plainteksnya diberikan simbol “p”, cipherteksnya simbolnya “c” dan kunci diberikan symbol “k”. rumus yang digunakan adalah: c = f(p) = (p + k) mod (26) untuk proses enkripsinya sedangkan untuk proses dekripsinya menggunakan rumus: p = f(c) = (c k) mod (26) metode caesar cipher dapat dipecahkan dengan carabrute force attact, suatu bentuk serangan yang dilakukan dengan mencoba-coba berbagai kemungkinan untuk menemukan kunci. karena jumlah kunci sangat sedikit (hanya ada 26 kunci) meskipun membutuhkan waktu yang cukup lama. indria eka wardani 234 volume 2 no. 4 mei 2013 penggabungan metode hill cipher dan metoda caesar cipher. merujuk pada metode caesar cipher dan hill cipher maka penulis ingin membuat penerapan dari kedua metode tersebut. sesuai metode caesar cipher dan hill cipher yang menggunakan z26 maka disini penulis ingin memberikan z30 dengan modulo 30. yaitu dengan menambahkan titik (.), koma (,), tanda tanya (?), dan tanda seru (!). karena dalam penulisan suatu pesan mengandung banyak maksud dan tujuan sehingga supaya pesan yang dikirimkan mengandung maksud serta tujuan yang jelas maka dipergunakan tanda-tanda tersebut. pada proses enkripsi dan dekripsi penulis menggabungkan kedua metode caesar cipher dan hill cipher yaitu untuk proses pengoperasian enkripsi menggunkan metode caesar cipher terlebih dahulu yaitu dengan menggunakan persamaan sebagai berikut. c =f(p) = (p+k) mod (30) setelah itu dilanjut dengan menggunakan metode hill cipher yaitu dengan menggunakan matriks k sebagai kuncinya, rumusnya adalah c = k.p sedangkan untuk proses dekripsinya sama seperti proses enkripsi tetapi terlebih dahulu menggunakan metode hill cipher dimana matriks k harus di inverskan selanjutnya 𝐾. 𝐾 −1 = 𝐼 yaitu menghasilkan matriks identitas, rumus hill cipher pada proses dekripsi adalah 𝑷 = 𝑲−𝟏. 𝑪 selanjutnya tahap berikutnya dengan menggunakan metode caesar cipher yaitu. p =f(c) = ( c-k ) mod (30) dalam persamaan tersebut terdapat p yaitu plainteks (pesan asli), c yaitu cipherteks (pesan yang tersandi) dan k yaitu kuncinya. disini matriks yang digunakan yaitu matriks persegi berordo n x n dan dengan entri-entrinya bilangan bulat. untuk lebih jelasnya maka berikut ini merupakan langkah-langkah proses enkripsi dan dekripsi. sebelum menginjak pada proses enkripsi dan dekripsi hal yang harus dilakukan yaitu membuat tabel korespondensi satu-satu dengan mengkonversikan setiap huruf alphabet ke dalam bilangan bulat. tabel 1.korespondensi satu-satu a b c d e f g h i j k l m n o 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 p q r s t u v w x y z . , ? ! 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 proses enkripsi pada proses enkripsi ini langkah-langkah yang ditempuh hampir sama dengan langkahlangkah proses enkripsi pada metode caesar cipher dan hill cipher. disini penulis menggeser sebanyak tiga huruf kedepan dan menggunakan matriks berordo 2 x 2 dan 3 x 3 sebagai kuncinya. berikut ini merupakan langkah-langkah proses enkripsi. 1. konversikan tiap-tiap huruf alphabet kedalam sebuah bilangan 0 sampai 30 karakter. kemudian geser tiga huruf kedepan. sehingga persamaannya menjadi c =f(p) = (p+3) mod (30) 2. membuat teks aslinya (plainteks) kemudian konversikan kedalam bilangan bulat dengan melihat tabel 1 koresponden satu-satu. selanjutnya masukkan ke persamaannya. 3. setelah diperoleh cipherteks dari metode caesar cipher selanjutnya dengan metode hill cipher dimana persamaannya yaitu c = k.p. 4. memilih matriks 2 x 2 atau 3 x 3 dengan elemen-elemennya bilangan bulat untuk melakukan penyandian dengan syarat k punya invers. karena matriks kunci k berukuran 2 x 2, maka plainteks dibagi menjadi blok yang msingmasing bloknya berukuran k2 karakter. kemudian dienkripsi dengan kunci k dengan persamaan c = k . p mod 30 .jika perkalian tersebut menghasilkan lebih dari angka 29 maka dilakukan modulo 30 pada hasil yang lebih dari 29. 𝐾 = [ 𝑎11 𝑎12 𝑎21 𝑎22 ] 5. mengkonversikan masing-masing pesan yang sudah diurutkan p1, p2 ke vektor kolom 𝑃 = [ 𝑝1 𝑝2 ] dan bentuk perkalian kp, dengan p sebagai vektor teks biasa dan kp disebut vektor teks yang sudah tersandikan 6. mengganti sisa modulo 30 pada masingmasing elemen matriks dengan huruf alphabet sesuai dengan tabel korespondensi satu-satu. dari ilustrasi tersebut maka penulis akan memberikan contoh yaitu: contoh: pesan yang akan disampaikan yaitu: belajar yang rajin langkah-langkah enkripsinya adalah sebagai berikut: 1. konversikan tiap-tiap huruf alphabet ke dalam sebuah bilangan 0 sampai 30 karakter. dengan melihat tabel korespondensi satusatu. pemecahan sandi kriptografi dengan menggabungkan metode hill cipher dan metode caesar chipher jurnal cauchy – issn: 2086-0382 235 belajar yang rajin hasil konversiannya adalah: 1 4 11 0 9 0 17 24 0 13 6 17 0 9 8 13 2. selanjutnya masukkan ke persamaan c =f(p) = (p+3) mod (30) sehingga menghasilkan: 4 7 14 312 3 20 27 3 169 203 12 11 16 menggunakan proses hill cipher 3. disini penulis memilih matriks 2 x 2 sebagai kuncinya, dengan elemen-elemennya bilangan bulat dan mempunyai invers. yaitu matriks 𝐾 = [ 2 3 3 5 ] 4. kelompokkan masing-masing bilangan secara berpasangan 4 7 14 3 12 3 20 27 3 16 9 17 3 12 11 16 5 kemudian pesan teks tersebut dimasukkan kedalam vektor kolom. 𝑃1 = [ 4 7 ], 𝑃2 = [ 14 3 ], 𝑃3 = [ 12 3 ], 𝑃4 = [ 20 27 ], 𝑃5 = [ 3 16 ], 𝑃6 = [ 9 20 ], 𝑃7 = [ 3 12 ], 𝑃8 = [ 11 16 ], selanjutnya kalikan matriks k dengan vektor kolom (p). [ 2 3 3 5 ] [ 4 7 ] = [ 8 + 21 12 + 35 ] = [ 29 47 ] = [ 29 17 ] 𝑚𝑜𝑑 30 = [ ! 𝑅 ] [ 2 3 3 5 ] [ 14 3 ] = [ 28 + 9 42 + 15 ] = [ 37 57 ] = [ 7 27 ] 𝑚𝑜𝑑 30 = [ 𝐻 , ] [ 2 3 3 5 ] [ 12 3 ] = [ 24 + 9 36 + 15 ] = [ 33 51 ] = [ 3 21 ] 𝑚𝑜𝑑 30 = [ 𝐷 𝑉 ] [ 2 3 3 5 ] [ 20 27 ] = [ 40 + 81 60 + 135 ] = [ 121 195 ] = [ 1 15 ] 𝑚𝑜𝑑 30 = [ 𝐵 𝑃 ] [ 2 3 3 5 ] [ 3 16 ] = [ 6 + 48 9 + 80 ] = [ 54 89 ] = [ 24 29 ] 𝑚𝑜𝑑 30 = [ 𝑌 ! ] [ 2 3 3 5 ] [ 9 20 ] = [ 18 + 60 27 + 100 ] = [ 78 127 ] = [ 18 7 ] 𝑚𝑜𝑑 30 = [ 𝑆 𝐻 ] [ 2 3 3 5 ] [ 3 12 ] = [ 6 + 36 9 + 60 ] = [ 42 69 ] = [ 12 9 ] 𝑚𝑜𝑑 30 = [ 𝑀 𝐽 ] [ 2 3 3 5 ] [ 11 16 ] = [ 22 + 48 33 + 80 ] = [ 70 113 ] = [ 10 23 ] 𝑚𝑜𝑑 30 = [ 𝐾 𝑋 ] 6 setelah itu konversikan tiap-tiap hasil penyandian terhadap tabel koresponden satu-satu dan diperoleh: !r h, dv bp y ! sh mj kx proses dekripsi pada proses dekripsi yaitu dengan menggunakan metode hill cipher dan dilanjut dengan metode caesar cipher dimana pada pengoperasian hill cipher menggunakan invers dari matriks k, dengan persamaan sebagai berikut p = k-1.c sdangkan untuk caesar cipher pengorerasiannya menggunakan persamaan: p= f(c) = ( c-k ) mod (30) disini penulis akan menjabarkan beberapa langkah-langkah proses dekripsi 1. mencari invers matriks dari matriks k yang sudah ditetapkan yaitu menggunakan matriks berordo 2 x 2 atau 3 x 3. untuk matriks a = (ai,j) berukuran 2 x 2, nilai determinannya adalah: det a = a1,1 a2,2 – a1,2 a2,1 dan matriks invers dari a adalah: 𝐴−1 = (det 𝐴)−1 [ 𝑎2,2 −𝑎1,2 −𝑎2,1 𝑎1,1 ] (dony ariyus,2006:30) 2. pesan yang sudah tersandi kemudian dikelompokkan secara berurutan kedalam pasangan-pasangan dan menggantinya dengan bilangan bulat. 3. mengkonversikan masing-masing pesan tersandi yang sudah diurutkan c1, c2 ke vektor kolom 𝐶 = [ 𝑐1 𝑐2 ] dan bentuk perkalian k-1 c=p, dengan c sebagai vektor teks yang sudah tersandi dan k -1cdisebut vektor teks aslinya. 4. selanjutnya hasil dari perkalian tersebut masukkan dalam persamaan caesar cipher yaitu p= f(c) = ( c-k ) mod (30) 5. mengganti sisa modulo 30 pada masingmasing hasil persamaan caesar cipher dengan huruf alphabet sesuai dengan tabel korespondensi satu-satu selanjutnya penulis akan memberikan contoh sebagai berikut: contoh: pesan berikut merupakan pesan yang sudah tersandi !r h, dv bp y ! sh mj kx langkah-langkah enkripsinya adalah sebagai berikut: 1. disini penulis menggunakan matriks berordo 2 x 2, yaitu matriks 𝐾 = [ 2 3 3 5 ] sedangkan k-1 adalah 𝐾 −1 = [ 5 −3 −3 2 ] indria eka wardani 236 volume 2 no. 4 mei 2013 2. kelompokkan huruf alphabet secara berpasangan. !r h, dv bp y ! jw mj kx kemudian konversikan kedalam bilangan bulat sesuai dengan tabel korespondensi satusatu. 29 17 7 27 3 21 1 15 2429187 1291023 3. kemudian pesan teks tersebut dimasukkan ke dalam vektor kolom. 𝑃1 = [ 29 17 ], 𝑃2 = [ 7 27 ], 𝑃3 = [ 3 21 ], 𝑃4 = [ 1 15 ], 𝑃5 = [ 24 29 ], 𝑃6 = [ 9 22 ], 𝑃7 = [ 12 9 ], 𝑃8 = [ 10 23 ] selanjutnya kalikan matriks k dengan vektor kolom (p). [ 5 −3 −3 2 ] [ 29 17 ] = [ 145 − 51 −87 + 34 ] = [ 94 −53 ] = [ 4 7 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 7 27 ] = [ 35 − 81 −21 + 54 ] = [ −46 33 ] = [ 14 3 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 3 21 ] = [ 15 − 63 −9 + 42 ] = [ −48 33 ] = [ 12 3 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 1 15 ] = [ 5 − 45 −3 + 30 ] = [ −40 27 ] = [ 20 27 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 24 29 ] = [ 120 − 87 −72 + 58 ] = [ 33 −14 ] = [ 3 16 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 18 7 ] = [ 90 − 21 −54 + 14 ] = [ 69 −40 ] = [ 9 20 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 12 9 ] = [ 60 − 27 −36 + 18 ] = [ 33 −18 ] = [ 3 12 ] 𝑚𝑜𝑑 30 [ 5 −3 −3 2 ] [ 10 23 ] = [ 50 − 69 −30 + 46 ] = [ −19 16 ] = [ 11 16 ] 𝑚𝑜𝑑 30 4. hasilnya adalah: 47 14 3 12 3 20 27 3 16 9 20 3 12 11 16 selanjutnya hasil dari perkalian tersebut masukkan dalam persamaan p= f(c) = ( c-3) mod (30) sehingga menjadi 1 4 11 0 90 17 24 0 136 17 0 98 13 5. mengganti sisa modulo 30 pada masingmasing hasil persamaan caesar cipher dengan huruf alphabet sesuai dengan tabel korespondensi satu-satu be la ja ry an gr aj in sehingga diperoleh pesan asli (plainteks) adalah belajar yang rajin daftar pustaka [1] dony, ariyus. 2006 . kriptografi keamanan data komunikasi. edisi pertama. yogyakarta: graha ilmu [2] dony,ariyus. 2008 . pengantar ilmu kriptografi. edisi dua. yogyakarta: c.v andi offset [3] rinaldi, munir. 2005. matematika diskrit. bandung: informatika pendahuluan kriptografi hill cipher caesar cipher penggabungan metode hill cipher dan metoda caesar cipher. proses enkripsi proses dekripsi daftar pustaka estimasi parameter model arellano dan bond pada regresi data panel dinamis lailatul urusyiyah mahasiswa jurusan matematika uin maulana malik ibrahim malang e-mail : lailatulurusyiyah@ymail.com abstrak data panel merupakan gabungan dari cross section dan time series. terdapat dua model data panel yaitu data panel statis dan dinamis. karena melihat keunggulan model data panel dinamis yang sanggup mengatasi masalah endogenitas terkait penggunaan lag variabel dependen dimana pada model data panel statis penggunaan lag variabel dependen menyebabkan hasil estimasi menjadi bias dan tidak konsisten sehingga penulis meneliti tentang model regresi data panel dinamis. sebagai langkah awal untuk mengestimasi parameter yang tidak diketahui pada model regresi data panel dinamis yaitu dengan hanya memanfaatkan kondisi ortogonalitas yang ada di antara nilai-nilai lag dan error-nya maka model regresi data panel dinamis tersebut menjadi model arellano dan bond. untuk mengestimasi model arellano dan bond maka dilakukan beda pertama pada model tersebut, kemudian mencari matriks varians-kovarians dan mendefinisikan matriks instrumen dari model tersebut. setelah itu, estimasi parameter model arellano dan bond menggunakan metode generalized least square (gls). berdasarkan pembahasan diperoleh rumus estimasi parameter model arellano dan bond adalah sebagai berikut: �̂�𝑔𝑙𝑠 = (∆𝑦−1𝑊(𝑊(𝐼𝑁 ⊗ 𝐺)𝑊) −1𝑊δ𝑦−1) −1δ𝑦−1𝑊(𝑊(𝐼𝑁 ⊗ 𝐺)𝑊) −1δ𝑦 kata kunci: regresi data panel dinamis, model arellano dan bond, estimasi parameter, metode generalized least square (gls). abstract panel data is a combination of cross section and time series. there are two models of panel data namely static panel data and dynamic panel data. because of seeing the advantage of dynamic panel data model is able to handle the problem of endogeneity related to the using of the dependent variable lag when static panel data used the dependent variable for causing the estimation result be biased and inconsistent. it makes the writer research the model of dynamic panel data regression. for the first step to estimate unknown parameters of the regression dynamic panel data model by using orthogonality conditions that existed among lag and an error values, so the regression dynamic panel data becomes arellano and bond model. for estimating arellano and bond model, we differ that model for the first time, then we seek a matrix variance-covarriance and define matrix instruments of the model. then, estimating of the arellano and bond model parameters using generalized least square methods ( gls ). according to the mater we get the estimation formula parameters arellano and bond model as follow: �̂�𝑔𝑙𝑠 = (∆𝑦−1𝑊(𝑊(𝐼𝑁 ⊗ 𝐺)𝑊) −1𝑊δ𝑦−1) −1δ𝑦−1𝑊(𝑊(𝐼𝑁 ⊗ 𝐺)𝑊) −1δ𝑦 keywords: the dynamic panel data regression, arellano and bond model, an estimation of the parameters, a method of generalized least square ( gls ). pendahuluan pada penelitian ini penulis tertarik untuk mengkaji tentang data panel dinamis, dimana data panel merupakan gabungan antara data time series dan cross section. terdapat beberapa model regresi data panel dinamis, yaitu model arellano dan bond, arellano dan bover, kondisi momen ahn dan schmidt, sistem gmm blundel dan bond, serta keane dan runkle. dari beberapa model tersebut, bentuk yang paling sederhana adalah model arellano dan bond, sehingga penulis mengambil judul “estimasi parameter model arellano dan bond pada regresi data panel dinamis”. teori dasar a. pendekatan matriks untuk model regresi linier dengan menggeneralisasikan model regresi linier dua atau tiga variabel, maka model mailto:lailatulurusyiyah@ymail.com estimasi parameter model arellano dan bond pada regresi data panel dinamis jurnal cauchy – issn: 2086-0382 11 regresi populasi k-variabel yang melibatkan variabel tak bebas y dan sebanyak 1k  variabel bebas 𝑥2,𝑥3,… ,𝑥𝐾 dapat ditulis sebagai 𝑦𝑖 = 𝛽1 +𝛽2𝑥2𝑖 + 𝛽3𝑥3𝑖 + ⋯+ 𝛽𝐾𝑥𝐾𝑖 + 𝑖 dimana 𝑖 = 1,2,…,𝑁 dan 𝛽1 adalah intersep, 𝛽2 sampai 𝛽𝐾 adalah koefisien kemiringan parsial, unsur error stokastik, dan 𝑖 adalah observasi ke-𝑖, 𝑁 merupakan besarnya populasi (gujarati, 2004). untuk 𝑖 = 1 hingga 𝑁, persamaan di atas dapat ditulis sebagai: 1 1 2 21 3 31 1 1 2 1 2 22 3 32 2 2 1 2 2 3 3 k k k k n n n k kn n y x x x y x x x y x x x                                  persamaan tersebut dapat ditulis dengan cara lain yang lebih menjelaskan sebagai berikut: 1 21 31 1 10 2 22 32 2 21 2 3 1 1 1 1 1 1 k k n n n kn nk n k nn k y x x x y x x x y x x x                                                         persamaan matriks tersebut dapat ditulis sebagai: y x    b. sifat-sifat transpose menurut anton dan rorres (2004), sifatsifat transpose antara lain: (a)    t t a a (b)   t t t a b a b   dan   t t t a b a b   (c)   t t ka ka , dengan k adalah skalar sebarang (d)   t t t ab b a c. matriks ortogonal anton dan rorres (2004) menyatakan bahwa matriks bujursangkar a yang memiliki sifat 1 t a a   disebut sebagai matriks ortogonal. dari definisi ini diketahui bahwa matriks bujursangkar a ortogonal jika dan hanya jika t t aa a a i  d. metode ordinary least square (ols) dalam penggunaan regresi, terdapat beberapa asumsi dasar yang dapat menghasilkan estimator linier tidak bias yang terbaik dari model regresi yang diperoleh dari metode kuadrat terkecil atau biasa dikenal dengan regresi ols agar estimasi koefisien regresi itu bersifat blue (best linear unbiased estimator). untuk tujuan ini maka perlu memilih parameter  sehingga nilai fungsi, ( ) ( ) t t s y x y x       sekecil mungkin (minimal). persamaan tersebut adalah skalar, sehingga komponen-komponennya juga skalar. akibatnya, transpose skalar tidak mengubah nilai skalar tersebut. sehingga s dapat ditulis sebagai      ( ) ( ) 2 t t t t t t t t t t t t t t t t t t t t t t t t t t t t t s y x y x x y x y y y x x y x x y y x x x x y x y x y x x y y yx x y y x y y                                         untuk meminimumkannya dapat di peroleh dengan melakukan turunan parsial pertama s terhadap  ,  0 2 2 2 2 t t t t t t t t t t ds x y x x x x d x y x x x x x y x x                  dan menyamakannya dengan nol diperoleh t t x x x y  yang dinamakan sebagai persamaan normal, dan 1 ( )ˆ t t ols x x x y   e. metode generalized least square (gls) menurut greene (1997), penanggulangan kasus heteroskedastisitas dapat dilakukan dengan estimasi melalui pembobotan (weighted) yang dapat pula dikatakan sebagai kuadrat terkecil yang diberlakukan secara umum atau disebut generalized least squares (gls). model persamaan linier umum adalah 𝑦 = 𝑋𝛽 + dengan ∈ 𝑁(0,φ), dimana lailatul urusyiyah 12 volume 3 no. 1 november 2013 2 1 2 2 2 2 0 0 0 0 0 0 n                     matriks simetri dan positive definite. karena  matriks simetri dan positive definite maka ada matriks 𝐶 yang ortogonal (𝐶𝑇𝐶 = 𝐶𝐶𝑇 = 𝐼) sedemikian hingga 𝐶𝑇φ𝐶 = 𝐷 adalah matriks diagonal yang elemen-elemennya merupakan nilai-nilai eigen dari  . misalkan 1 2 0 0 0 0 0 0 n d                 dan tulis 1 2 1 0 0 1 0 0 1 0 0 n p                           maka diperoleh 𝑃𝑇𝐷𝑃 = 𝐼. karena 𝐶𝑇φ𝐶 = 𝐷 maka 𝑃𝑇𝐶𝑇φ𝐶𝑃 = 𝑃𝑇𝐷𝑃 = 𝐼. misalkan 𝑊 = 𝑃𝐶 maka 𝐼 = 𝑃𝑇𝐶𝑇φ𝐶𝑃 = 𝑊𝑇φ𝑊 akibatnya diperoleh φ = (𝑊𝑇)−1𝑊−1 = (𝑊𝑇𝑊)−1 atau φ−1 = 𝑊𝑇𝑊. dari persamaan model statistik linier diperoleh transformasi model menjadi 𝑊𝑦 = 𝑊(𝑋𝛽 + ) = 𝑊𝑋𝛽 + 𝑊 atau 𝑦∗ = 𝑋∗𝛽 + ∗ disebut sebagai generalized least squares estimator (glse), yaitu                        1 * * * * 1 1 1 1 1 1 1 1 2 2 1 2 1 1 2 1 1 1 ˆ φ φ 1 gls t t t t t t t t t t t t t t t t x x x y wx wx wx wy x w wx x w wy x x x y x x x y x x x y x x x y                                            f. pengertian data panel menurut rosadi (2006) data panel adalah tipe data yang dikumpulkan menurut urutan waktu dalam suatu rentang waktu tertentu pada sejumlah individu atau kategori. menurut winarno (2007) data panel merupakan gabungan antara data silang (cross section) dengan data runtut waktu (time series). menurut setiawan dan kusrini (2010) ada banyak sebutan untuk data panel ini, misalnya data terkelompok (pooled data), kombinasi berkala (kumpulan data berkala dan tampang lintang), data mikropanel (micropanel data), data bujur (longitudinal data atau studi sekian waktu pada sekelompok objek penelitian), analisis riwayat peristiwa (event history analysis) atau studi sepanjang waktu dari sekumpulan objek sampai mencapai keberhasilan atau kondisi tertentu. menurut gujarati (2003) data panel dapat dibedakan menjadi dua, balanced panel dan unbalanced panel. balanced panel terjadi jika panjangnya waktu untuk setiap unit cross section sama. sedangkan unbalanced terjadi jika panjangnya waktu tidak sama untuk setiap unit cross section. g. model regresi data panel menurut firdaus (2004) analisis regresi adalah teknik analisis yang mencoba menjelaskan bentuk hubungan antara variabel yang mendukung sebab akibat. regresi menggunakan data panel disebut model regresi data panel. bentuk umum model regresi data panel adalah sebagai berikut: 𝑦𝑖𝑡 = 𝑥𝑖𝑡 𝑇𝛽 + 𝑖𝑡 dimana: it y = variabel terikat untuk unit individu kei dan waktu ket ' it x = matriks dengan ordo 1 k  = parameter yang tidak diketahui dengan matriks 1k  it  = error untuk unit individu kei dan waktu ket i = 1, 2, , n untuk unit individu t = 1, 2, , t untuk waktu h. model regresi data panel dinamis analisis data panel dapat digunakan pada model yang bersifat dinamis karena data panel cocok untuk analisis dynamic of adjusment. estimasi parameter model arellano dan bond pada regresi data panel dinamis jurnal cauchy – issn: 2086-0382 13 sejalan dengan adanya model cross section atau time series, hubungan dinamis yang dicirikan oleh data panel dengan memasukkan lag dari peubah atau variabel dependen sebagai regresor dalam regresi. akibatnya muncul masalah endogenitas, sehingga bila model diestimasi dengan pendekatan fixed-effect maupun random-effect akan menghasilkan penduga yang bias dan tidak konsisten. untuk itu maka muncul pendekatan gmm (generalized method of moments). sebagai ilustrasi, dapat diketahui dengan model data panel dinamis yang telah dikemukakan oleh baltagi (2005) sebagai berikut: , 1 i i t i i t t t t y y x u      untuk i = 1, 2, 3, …, n dan t = 1, 2, 3, …, t. dimana  menyatakan besaran skalar, ' it x menyatakan matriks berukuran 1 k dan  berukuran 1k  . dalam hal ini it  diasumsikan mengikuti model one way error component sebagai berikut: it i it u    dimana 2 ~ iid(0, ) i    menyatakan pengaruh individu dan 2~ iid(0, ) it    menyatakan error yang saling bebas satu sama lain. i. kronecker product menurut graham (1981), misalkan matriks   ij a a dengan ordo m n dan makriks   ij b b dengan ordo r s . kronecker product dari kedua matriks tersebut, disimbolkan dengan a b adalah matriks partisi 11 12 1 21 22 2 1 2 n n m m mn a b a b a b a b a b a b a b a b a b a b               a b dipandang matriks berordo mr ns . pembahasan a. model regresi data panel dinamis bentuk umum model data panel dinamis adalah sebagai berikut: , 1 t it i t it it y y x u      (1) dengan it u diasumsikan mengikuti one way error component sebagai berikut: it i it u    (2) dimana 2 ~ iid(0, ) i    dan 2 ~ iid(0, ). it   dengan menggabungkan persamaan (1) dan (2), maka diperoleh: , 1it i t it i it t y xy          (3) dimana: 𝑦𝑖𝑡 = variabel terikat untuk unit individu ke-i dan waktu ke-t 𝑦𝑖,𝑡−1 = variabel bebas untuk unit individu kei dan waktu ket 𝛿 = koefisien lag variabeldependen 𝑥𝑖𝑡 𝑇 = matriks berukuran (1 × 𝐾) yang berisikan variabel bebas untuk unit individu ke-i dan waktu ke-t  = matriks berukuran (1 × 𝐾) yang berisikan parameter variabel bebas it u = error untuk unit individu ke-𝑖 dan waktu ke-𝑡 i  = error untuk cross-section it  = error atau gangguan yang saling bebas satu sama lain i = 1,2,…,𝑁 t = 1,2,3,…,𝑇 karena t it x adalah matriks yang berukuran  1 k dan  adalah matriks yang berukuran  1k  , maka persamaan (3) menjadi:   1 2 , 1 2 , 1 1 2 2 , 1 1 2 1 it i t it kit k i t it kit k i t kit k i it i it k i it k y x x y x x y y x                                                  untuk 𝑡 = 2 hingga 𝑡 = 𝑇 persamaan tersebut menjadi: 1 2 2 3 2 1 2 2 3 1 2 1 2 1 3 , i ki k i ki k k i i i k k i i i k k it i ii t kit k t k y x y x x y y y y                                    atau dapat ditulis dalam bentuk persamaan matriks sebagai berikut: [ 𝑦𝑖2 ⋮ 𝑦𝑖𝑇 ] ⏟ 𝑁(𝑇−1)×1 = 𝛿[ 𝑦𝑖1 ⋮ 𝑦𝑖(𝑇−1) ] ⏟ 𝑁(𝑇−1)×1 +[ 1 𝑥2𝑖2 ⋯ 𝑥𝐾𝑖2 ⋮ ⋮ ⋱ ⋮ 1 𝑥2𝑖𝑇 ⋯ 𝑥𝐾𝑖𝑇 ] ⏟ 𝑁(𝑇−1)×𝐾 [ 𝛽1 ⋮ 𝛽𝐾 ] ⏟ 𝐾×1 + [ 𝜛𝑖 ⋮ 𝜛𝑖 ] ⏟ 𝑁(𝑇−1)×1 + [ 𝜋𝑖2 ⋮ 𝜋𝑖𝑇 ] ⏟ 𝑁(𝑇−1)×1 lailatul urusyiyah 14 volume 3 no. 1 november 2013 dalam penelitian kali ini, peneliti hanya akan mengestimasi parameter  yang merupakan langkah awal (pre-estimation) model regresi data panel dinamis. karena dengan diketahuinya parameter  , maka untuk langkah selanjutnya yaitu mengestimasi parameter  dapat dilakukan dengan mudah menggunakan estimasi ordinary least square (ols) atau maximum likelihood (ml). b. model arellano dan bond pada data panel dinamis arellano dan bond berpendapat bahwa tambahan instrumen dapat diperoleh dalam model data panel dinamis jika memanfaatkan kondisi ortogonalitas yang ada di antara nilai-nilai lag dari it y dan gangguan it v . sehingga tanpa regressor, persamaan (1) dapat ditulis sebagai berikut: , 1it i t it y y     (4) untuk 1, 2, ..., ; dan 2, 3, ...,i n t t  it  pada persamaan (4) diasumsikan mengikuti model one way error component sebagai berikut: it i it v   (5) dimana 𝜇𝑖~𝐼𝐼𝐷(0,𝜎𝜇 2) menyatakan pengaruh individu dan 𝑣𝑖𝑡~𝐼𝐼𝐷(0,𝜎𝜇 2) menyatakan gangguan yang saling bebas satu sama lain. dengan menstubtitusikan persamaan (5) pada persamaan (4), maka diperoleh , 1it i t i it y y v      (6) untuk i = 1, 2, 3, …, n t = 1, 2, 3, …, t untuk 1, 2, ...,i n dan 2, 3, ...,t t adalah sebagai berikut: 12 11 1 12 22 21 2 22 2 1 2 13 12 1 13 23 22 2 23 3 2 3 1 1 , 1 1 1 2 2 , 1 2 2 , 1 n n n n n n n n t t t t t t nt n t n nt y y v y y v y y v y y v y y v y y v y y v y y v y y v                                                 atau dapat ditulis dalam bentuk persamaan matriks sebagai berikut: 2 2 1112 1 2122 2 12 1213 1 2223 2 23 1 , 11 1 2. 12 2 , 1 1 1 nn n nn n tt tt n tnt n n n yy yy yy yy yy yy yy yy yy                                                                                              2 2 12 22 2 13 23 3 1 2 1 1 n n t t nt n n v v v v v v v v v                                                                    c. estimasi parameter model arellano dan bond untuk mendapatkan estimasi yang konsisten dari  dimana n   dengan t tertentu, maka dilakukan beda pertama pada persamaan (6) yaitu   , 1 , 1 , 2 , 1 , 1 , 2 , 1 , 1 , 2 , 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) it i t i t i t i it i i t i t i t i it i i t i t i t it i t y y y y v v y y v v y y v v                                   misalkan , 1it i t y y y     dan , 1it i t v v v     , maka persamaan tersebut menjadi: 1 y y v       dimana , 1 ( ) it i t v v   adalah ma (1) dengan unit root.  untuk 𝑡 = 3 maka 3 2 2 1 3 2 ( ) ( ) i i i i i i y y y y v v     1i y pada persamaan tersebut merupakan variabel instrumental yang valid, karena 1i y berkorelasi dengan 2 1 ( ) i i y y dan tidak berkorelasi dengan error-nya yaitu 3 2 ( ) i i v v selama it v tidak berkorelasi serial.  untuk 𝑡 = 4 maka 4 3 3 2 4 3 ( ) ( ) i i i i i i y y y y v v     dengan variabel instrumental yang valid adalah 2i y .  untuk 𝑡 = 5, maka estimasi parameter model arellano dan bond pada regresi data panel dinamis jurnal cauchy – issn: 2086-0382 15 5 4 4 3 5 4 ( ) ( ) i i i i i i y y y y v v     dengan variabel instrumental yang valid adalah 3i y . begitu seterusnya hingga t t dengan variabel instrumental yang valid adalah , 2i t y  , jadi himpunan variabel instrumental yang valid adalah   1 2 3 , 2 , , , ..., i i i i t y y y y  . jika ditulis dalam bentuk matriks untuk 3t  hingga t t , maka diperoleh hasil sebagai berikut:       3 2 2 1 3 2 4 3 3 2 4 3 5 4 4 3 5 4 , 1 , 1 , 2 , 1 2 1 2 1 2 1 i i i i i i i i i i i i i i i i i i it i t i t i t it i t n t n t n t y y y y v v y y y y v v y y y y v v y y y y v v                                                                        sehingga, matriks varians-kovarians dari error ( it v ) pada persamaan (4) dengan 3t  hingga t t dapat ditulis :       3 2 4 3 3 2 4 3 , 1 1 2 , 1 2 1 3 2 3 2 3 2 , 1 , 1 3 2 , 1 ( ') ( )( ) ( )( ) ( )( ) ( )( i i i i i i i i it i t n t it i t n t i i i i i i it i t it i t i i it i t it v v v v e v v e v v v v v v v v v v v v v v v v e v v v v v v v v                                                      , 1 3 2 3 2 , 1 , 1 3 2 , 1 2 2 3 2 3 2 3 2 3 , 1 2 , 1 3 , 1 3 2 , 1 2 ) var( ) ( ), ( ) ( ), ( ) var( ) 2 i t i i i i it i t it i t i i it i t i i i i i it i it i i t i i t iti i t i iti i t i it v v cov v v v v cov v v v v v v                                                              2 2 , 1 , 1 2 i t iti t                karena 𝑣𝑖𝑡~𝐼𝐼𝐷(0,𝜎𝑣 2) maka 2 2 , dan it v t i    begitu juga kovariansnya adalah 0, sehingga matriks varians-kovarians di atas dapat ditulis sebagai       2 1 0 1 2 0' 2 0 0 2 2 2 e v v v n t n t                     agar terdapat identitas pada persamaan tersebut, maka misalkan 2 1 0 1 2 0 0 0 2 g               yang berordo    2 2t t   , sehingga menurut definisi kronecker product persamaan tersebut dapat ditulis sebagai berikut:        1 0 0 2 1 0 0 1 0 1 2 0' 2 0 0 1 0 0 2 2 2 2 e v v v n n t t i g v n                                                    didefinisikan i w adalah matriks yang elemen diagonal utamanya merupakan matriks dari variabel instrumental yang valid dengan ordo       2 12 2 t t t     yaitu 1 1 2 1 , 2 [ ] 0 0 0 [ , ] 0 0 0 [ , ..., ] i i i i i i t y y y w y y               maka matriks instrumennya adalah   1 2 t t t t n w w w w dengan ordo       2 12 2 t t n t     dan persamaan momennya adalah ( ) 0 t i i e w v  . dengan perkalian kiri t w , didapatkan 1 ( ) t t t w y w y w v      (7) persamaan merupakan persamaan semi linier, sehingga  dapat diestimasi secara gls sebagai berikut, misalkan: * * 1 1 * t t t y w y y w y v w v            maka persamaan (7) menjadi * * * 1 y y v       (8) dimana, 𝐸(δ𝑣∗) = 𝐸(𝑊𝑇δ𝑣) = 𝑊𝑇𝐸(δ𝑣) = 0 dan             * * 2 t t t t t t t t t n e v v e w v w v e w v v w w e v v w w i g w              dengan ordo           2 1 2 1 2 2 t t t t     . lailatul urusyiyah 16 volume 3 no. 1 november 2013 error pada persamaan (8) diasumsikan independent identically distribution (iid) yaitu ∆𝑣∗~𝐼𝐼𝐷(0,φ), sehingga:     2 2 2t t v v n v n w i g w w i g w         matriks simetri dan positive definite. karena  matriks simetri dan positive definite maka ada sebarang matriks c yang orthogonal  t tcc c c i  sedemikian hingga t c c d  adalah matriks yang elemen diagonal utamanya merupakan nilainilai eigen dari  , maka 1 2 0 0 0 0 0 0 n d                 dan misalkan p adalah matriks yang elemen diagonal utamanya adalah 1 √𝜆𝑖 . sehingga dapat ditulis 1 0 0 1 1 0 0 2 1 0 0 p n                             maka diperoleh t p dp i . karena t c c d  , maka t t t p c cp p dp i   . misalkan w pc maka t t t i p c cp w w    , akibatnya diperoleh φ = (𝑊𝑇)−1𝑊−1 = (𝑊𝑇𝑊)−1 φ−1 = 𝑊𝑇𝑊 sehingga bentuk estimator model arellano dan bond adalah sebagai berikut:                           1 * * * * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 wφ wφ ˆ t t t t t t t t t t t gls t t t t t t t t t t t t v v t t t t t n t y y y y w y w y w y w y y ww y y ww y y ww ww y y ww ww y y y y y y w y y w y y w w i g w w w w ww y                                                                          1 1 1 t t n y w w i g w y      yang dikatakan sebagai bentuk akhir estimator parameter  untuk model arellano dan bond. penutup berdasarkan pembahasan diperoleh kesimpulan bahwa model arellano dan bond dapat diperoleh dengan hanya memanfaatkan kondisi ortogonalitas yang ada di antara nilai-nilai lag dan error pada regresi data panel dinamis, sehingga diperoleh: 𝑦𝑖𝑡 = 𝛿𝑦𝑖,𝑡−1 + 𝑖𝑡 it  pada persamaan tersebut diasumsikan mengikuti model one way error component sebagai it i it v   , dimana 𝜇𝑖~𝐼𝐼𝐷(0,𝜎𝜇 2) menyatakan pengaruh individu dan 𝑣𝑖𝑡~𝐼𝐼𝐷(0,𝜎𝑣 2) menyatakan gangguan yang saling bebas satu sama lain. untuk mendapatkan estimasi yang konsisten dari  dimana n   dengan t tertentu, maka dilakukan beda pertama pada model tersebut, kemudian mencari matriks varians-kovarians dari error dan mendefinisikan matriks instrumen dari model tersebut. setelah itu, estimasi parameter model arellano dan bond menggunakan metode generalized least square (gls). berdasarkan pembahasan diperoleh rumus estimasi parameter model arellano dan bond dengan metode gls adalah sebagai berikut: �̂�𝑔𝑙𝑠 = (∆𝑦−1𝑊(𝑊(𝐼𝑁⨂𝐺)𝑊) −1)𝑊∆𝑦−1𝑊(𝑊(𝐼𝑁⨂𝐺)) pada penelitian selanjutnya dapat meneruskan untuk mencari parameter regresi data panel dinamis pada variabel eksogennya dan dapat mengaplikasikan model tersebut pada permasalahan-permasalahan yang terkait dengan ekonometri. daftar pustaka [1] anton, h. dan rorres, c.. 2004. aljabar linier elementer versi aplikasi; alih bahasa, refina indriasari, irzam harmein; editor, amalia safitri edisi delapan jilid 1. jakarta: erlangga. [2] aziz, a.. 2010. ekonometrika teori dan praktek eksperimen dengan matlab. malang: uin-maliki press. [3] baltagi, b.h.. 2005. econometric analysis of panel data, third edition. england: john wiley & son, ltd. [4] dudewicz, e.j. dan mishra, s.h.. 1995. statistika matematika modern; terjemahan rk sembiring. bandung: itb. [5] firdaus, m.. 2004. ekonometrika suatu pendekatan aplikatif. jakarta: bumi aksara. estimasi parameter model arellano dan bond pada regresi data panel dinamis jurnal cauchy – issn: 2086-0382 17 [6] graham, a.. 1981. kronecker products and matrix calculus: with application. england: ieb. [7] greene, w.h.. 1997. econometric analysis. new york: prentice hall international, inc. [8] gujarati, d.n.. 2003. basic econometrics. new york: mcgraw-hill. [9] gujarati, d.n. 2004. basic econometrics, fourth edition. new york: mcgraw-hill. [10] rosadi, d. 2006. diktat kuliah pengantar analisis runtun waktu. yogyakarta: ugm. [11] setiawan & kusrini,d.e. 2010. ekonometrika yogyakarta: c.v andi offset. [12] winarno,w.w..2007. analisis ekonometrika dan statistika eviews. yogyakarta: upp stim ykpn. 5.4 saropah akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo _148-153_ akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo saropah mahasiswa jurusan matematika uin maulana malik ibrahim malang e-mail: haforas@rocketmail.com abstrak salah satu kegunaan yang terpenting dari teori ring dan lapangan adalah perluasan dari suatu lapangan yang lebih luas sehingga suatu polinomial dapat diketahui mempunyai akar. dalam penelitian ini peneliti mengambil modulo prima sebagai koefisien yang mengikuti peubahnya yang akan dicari akarakar penyelesaiannya sehingga dapat diketahui perluasan normalnya. suatu lapangan � yang dikenakan suatu polinomial membentuk himpunan polinomial ����, di mana ���� ini merupakan lapangan yang koefisien suku-sukunya merupakan bilangan modulo prima. dari himpunan polinomial tersebut ada polinomial ���� yang tidak tereduksi, maka perlu adanya perluasan lapangan untuk mengetahui akar-akar penyelesaiannya. misal perluasan lapangan dari � adalah lapangan �. lapangan k disebut perluasan lapangan atas lapangan �, jika lapangan � merupakan sublapangan dari lapangan � dan ���� adalah polinomial tidak tereduksi dalam � maka ���� dapat difaktorkan sebagai hasil kali dari faktor linier dalam lapangan pemisahnya. jika polinomial ���� mempunyai akar yang berlainan dalam lapangan pemisahnya maka polinomial tersebut disebut polinomial separable. pada penelitian ini polinomial yang separable adalah polinomial yang berpangkat ganjil di mana koefisien suku-suku dari polinomial ini terdapat dalam perluasan lapangannya. polinomial ganjil ini dinamakan polinomial separable karena mempunyai akar yang berlainan dalam faktor-faktornya dan salah satu faktornya terdapat dalam polinomial dalam lapangannya. lapangan pemisah yang memuat semua himpunan polinomial separable ini dinamakan perluasan normal. kata kunci: perluasan lapangan, lapangan pemisah, perluasan normal abstract one of the most important uses of the ring and field theory is an extension of a broader field so that a polynomial can be found to have roots. in this study researchers took modulo a prima as follows indeterminate coeffcients to search for his roots extension the solutions of that it can seen normal. a field � is subject to a polynomial form a set of polynomials ����, where ���� is a coefficient field its terms modulo a prime number. of the set of polynomial exists a polynomial ���� is irreducible, it is necessary to extension the field to know the roots of the solution. suppose to extension of the field � is a field �. field � is called extension the field over a field �, if the field � is subfield of the field � and ���� is irreducible polynomial in � then ���� can be factored as a product of linear factors in the splitting field. if the polynomial ���� has different roots in the splitting field the polynomial is called polynomial separable. in this study polynomial separable is contained of odd degree in which the coefficients of the tribes polynomial is contained in the extension field. polynomial is called a polynomial separable odd because it has different roots in the factors and there is one factor in a polynomial in the field. splitting field that contains all the set of polynomials separable is called normal extension. key words: extension field, splitting field, normal extension pendahuluan matematika merupakan ilmu pengetahuan dasar yang dibutuhkan semua manusia dalam kehidupan sehari-hari baik secara langsung maupun tidak langsung. matematika merupakan ilmu yang tidak terlepas dari alam dan agama, semua itu kebenarannya bisa kita lihat dalam al-qur’an (rahman, 2007: 1). salah satu sifat matematika yaitu matematika bersifat abstrak, yang berarti bahwa objek-objek matematika diperoleh melalui abstraksi dari fakta-fakta atau fenomena dunia nyata. karena objek matematika merupakan hasil abstraksi dunia nyata, maka matematika dapat ditelusuri kembali berdasarkan proses abstraksinya. hal inilah yang mendasari bagaimana cara mempelajari matematika (abdussakir, 2007: 15). ilmu aljabar abstrak merupakan bagian dari ilmu matematika. salah satu bahasan dalam aljabar abstrak adalah ring. ring adalah himpunan tak kosong r dengan dua operasi biner akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo jurnal cauchy – issn: 2086-0382 149 yaitu + sebagai operasi pertama dan * sebagai operasi kedua, yang kedua-duanya didefinisikan pada yang memenuhi aksioma-aksioma yang telah ditentukan. sedangkan ring komutatif dengan elemen satuan dan semua unsur di mempunyai invers terhadap operasi kedua kecuali elemen nol (identitas pada operasi pertama) disebut lapangan (raisinghania dan aggarwal, 1980: 313-314). salah satu kegunaan yang terpenting dari teori ring dan lapangan adalah perluasan dari suatu lapangan yang lebih besar atau lebih luas sehingga suatu polinomial (suku banyak) dapat diketahui mempunyai akar. misalkan suatu lapangan � yang dikenakan polinomial ring yang kemudian membentuk himpunan-himpunan polinomial yang koefisien suku-sukunya merupakan elemen dari field �. polinomial-polinomial ini kemudian dicari akar-akar penyelesaiannya, dan ternyata dalam mencari akar-akarnya terdapat polinomial-polinomial yang tidak dapat dicari akar-akarnya, dengan kata lain polinomial tersebut tidak dapat difaktorkan (irreducible). maka dibentuklah perluasan lapangan � untuk memperoleh akar-akar penyelesaian dari polinomial-polinomial tak tereduksi tersebut. akar-akar dari polinomial-polinomial tak tereduksi ini harus berada dalam perluasan lapangannya. berdasarkan jurnal yang di tulis oleh sulastri daruni, bayu surarso, dan bambang irawanto (2004), jika dalam lapangan � terdapat ���� polinomial tak tereduksi maka ���� dapat dibentuk sebagai hasil kali faktor linier dalam ���� di mana ���� adalah himpunan polinomialpolinomial dengan koefisien-koefisien di dalam �. sehingga dapat dikatakan perluasan lapangan � atas lapangan � adalah lapangan pemisah atas lapangan � terhadap polinomial ����. selanjutnya polinomial ���� dapat disajikan sebagai ���� ��� � �� … �� � �� dengan � � 0 � �, �, �, … � � � dan � �� �, �, … �� kemudian jika akar-akar dari polinomial ���� tersebut dalam lapangan pemisahnya mempunyai faktor yang akarakarnya berlainan maka polinomial tidak tereduksi ���� dalam lapangan � adalah separable dalam �. selanjutnya untuk lapangan pemisah yang memuat semua akar-akar yang berlainan dari lapangan tidak tereduksi ���� di mana salah satu akar dari akar-akar berlainan tersebut merupakan faktor dari polinomial pada sublapangannya maka lapangan pemisah ini disebut perluasan normal. pada polinomial ring, jika tidak ada penjelasan mengenai koefisien-koefisie yang menyertai peubahnya masing-masing, maka dianggap sebagai bilangan real. tetapi apabila ada penjelasan lebih lanjut, maka koefisien sesuai dengan ring yang ditunjuk. dari penjelasan di atas maka penulis ingin mengembangkan akarakar polinomial yang koefisien-koefisiennya merupakan elemen dari lapangan yaitu pada ring modulo. oleh karena itu, penulis tertarik untuk mengkaji tentang akar-akar polinomial separable pada ring modulo dengan judul “akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo.” kajian teori 1. lapangan lapangan adalah ring komutatif dengan elemen satuan dan semua unsur di mempunyai invers terhadap operasi kedua kecuali elemen nol (identitas pada operasi pertama) (raisinghania dan aggarwal, 1980: 314). dengan kata lain, untuk setiap elemen bukan nol � � ada ��� � sedemikian hingga � · ��� 1 (wahyudin, 1989: 155). 2. polinomial polinomial ���� berderajat �, didefinisikan sebagai suatu fungsi berbentuk: ���� �0 � ��� � ��� � � � � ��� � dengan �� adalah konstanta riil, 0,1,2, … , � dan �� � 0 di mana: � : merupakan peubah. �0, ��, ��, … , �� : merupakan nilai koefisien persamaan �. � : merupakan orde atau derajat persamaan. 3. polinomial atas lapangan misalkan � adalah lapangan. jika �!, �!��, … , ��, �0 � �, maka sebarang bentuk dari �!� ! � �!��� !�� � � � ��� � �0 disebut polinomial atas � dengan peubah � koefisien �!, �!��, … , ��, �0. semua himpunan polinomial dengan koefisien di � dinotasikan dengan ����. jika � adalah bilangan bulat non negatif paling besar sedemikian hingga �� � 0, maka dapat dikatakan bahwa polinomial ���� ���� � � � �0 mempunyai derajat �, ditulis dengan "����# �, dan �� disebut koefisien pertama dari ����. jika koefisien pertama adalah 1, maka ���� dikatakan polinomial monik (beachy dan blair, 1990: 165). 4. perluasan lapangan suatu sublapangan (lapangan bagian) dari lapangan � adalah suatu subring (ring bagian) � dan juga merupakan lapangan. dalam hal ini, lapangan � disebut perluasan dari lapangan �. misalnya, $ adalah sublapangan dari %, oleh karena itu % adalah suatu perluasan dari lapangan $ (wahyudin, 1989: 234). saropah 150 volume 2 no. 3 november 2012 sebuah lapangan � adalah perluasan lapangan dari �, jika � adalah sublapangan dari � berdasarkan pengertian ini dibuktikan teorema kronecker sebagai berikut (fraleigh, 1994: 394): teorema: teorema kronecker (fraleigh, 1994: 394-395) misalkan f adalah lapangan dan ���� polinomial tidak konstan di dalam ����. maka terdapat perluasan lapangan k dari f dan � � sedemikian sehingga �� � 0. bukti: misalkan ���� � ���� dapat difaktorkan secara tunggal sebagai ���� ���������� … �����, dengan ����� � 1,2, … �� adalah polinomial prima yang tak tereduksi dengan ����/(���� ) adalah suatu lapangan di mana (���� ) himpunan polinomial tidak tereduksi dalam lapangan ����. didefinisikan suatu pemetaan ψ : � → ����/(���� ) dengan ψ�a� a � (����) ψ adalah pemetaan satu-satu, sebab +�, , � � jika ψ(a) = ψ(b) maka � � (����) , � (����) a � b 0 atau a b ⇔ � � , � (���� ) jadi � � , suatu kelipatan ���� yang berderajat 0 1. ψ homomorfisma ring. sehingga ψ(f) = {a + (����) | a ∈ f } ⊆ ����/(���� ) merupakan sublapangan dari ����/(���� ). jadi f ≅ {a + (����)| a ∈ f}. misal � ����/(���� ) maka � merupakan perluasan lapangan dari �. akan dibuktikan f(α) = 0, ambil α∈e dengan α = x + (����). jika ���� �0 � ��� � � � ��� � di mana �� � � maka �� � �0 � ���x � (����)� � � � ���x � (����)� � 0 dalam � ����/(���� ). �� � 0 karena ���� ���������� … … �����, maka �� � 0. 2 5. lapangan pemisah perluasan lapangan � atas lapangan � dikatakan sebagai lapangan pemisah dari polinomial ���� � ���� jika faktor dari ���� merupakan faktor linier di ���� dan ���� bukan faktor linier atas setiap proper sublapangan terhadap � di � (dummit dan foote, 1999: 516). 6. perluasan normal jika � adalah perluasan aljabar � di mana ini merupakan lapangan pemisah atas � untuk koleksi polinomial ���� � ���� maka � disebut perluasan normal � (dummit, 1999: 517). pembahasan adapun untuk memperoleh perluasan normal pada ring modulo dilakukan langkahlangkah sebagai berikut: 1. lapangan dikenakan suatu polinomial 2. dari himpunan polinomial yang terbentuk, terbagi menjadi: a. polinomial konstan b. polinomial tidak konstan yaitu: 1) polinomial tereduksi 2) polinomial tidak tereduksi 3. membentuk perluasan lapangan untuk memperoleh akar-akar dari polinomial tidak tereduksi. 4. perluasan lapangan yang memuat polinomial tidak tereduksi di mana polinomial tidak tereduksi tersebut salah satu faktornya terdapat dalam sublapangan, maka perluasan lapangan ini dinamakan lapangan pemisah. 5. jika dalam lapangan pemisah polinomialnya mempunyai akar-akar yang berlainan, maka polinomial ini dinamakan polinomial separable dalam lapangan. 6. lapangan pemisah yang memuat semua akar-akar berlainan dinamakan perluasan normal. pada penelitian ini penulis memberi contoh dari 3����, 34��� dan 35���. untuk 3���� tidak dapat dicari perluasan normalnya karena tidak ada polinomial yang tidak tereduksi yang menjadi syarat dalam pembentukan lapangan pemisah yaitu polinomial tidak tereduksi dalam lapangan pemisah harus bisa difaktorkan dan faktorisasi tersebut harus berada dalam polinomial lapangannya. pada 34��� dan 35��� telah diketahui bahwa masing-masing terdapat lapangan pemisah. pada 34��� polinomial tidak tereduksi yang merupakan polinomial yang mempunyai akar-akar berlainan dalam lapangan pemisahnya yaitu ��6��� dan untuk 35��� polinomialpolinomial tidak tereduksi yang merupakan polinomial yang tidak mempunyai akar-akar berlainan dalam lapangan pemisahnya yaitu ������, �47���, dan �64���. polinomial-polinomial tidak tereduksi inilah yang termasuk dalam perluasan normal di mana polinomial-polinomial tidak tereduksi ini berderajat lebih dari satu yaitu ���� 8 1. berdasarkan contoh penentuan perluasan normal dari 3�, 34, dan 35 dapat diperoleh bentuk umum perluasan normal sebagai berikut. misalkan suatu ring �3�, �,9� adalah lapangan dengan � prima yang memenuhi aksioma-aksioma di bawah ini: akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo jurnal cauchy – issn: 2086-0382 151 a. �3�, �� gup abelian i. operasi � tertutup di 3� jika �, , � 3� dengan definisi penjumlahan modulo � maka, � � 3�, , � 3� : � � , � 3� +�, , � 3� ii. bersifat asosiatif. untuk semua �, ,, � � 3�, berlaku � � �, � �� �� � ,� � � iii. mempunyai identitas pada operasi � bilangan bulat 0 � 3� adalah elemen identitas di 3� untuk penjumlahan modulo �, untuk setiap bilangan bulat � � 3�, berlaku 0 � � � � 0 � iv. mempunyai invers untuk semua elemen � � 3� kecuali 0 di 3� maka � � 3�, � � 0 : 0 ; � ; � : 0 ; � � � ; � : � � � � 0 dan � � � juga, � � � � � � � � 0 jadi masing-masing bilangan bulat bukan nol � mempunyai invers di . v. bersifat komutatif untuk semua , , � , berlaku � � jadi operasi penjumlahan modulo adalah komutatif di . b. operasi 9 tertutup di 3� misalkan , � berlaku 1 < < � 1 dan 1 < < � 1 dengan definisi dari perkalian modulo maka 9 di mana < � 1 hasilnya dapat dibagi oleh maka 1 < < � 1 dan hal ini menunjukkan bahwa operasi 9 tertutup di . c. bersifat asosiatif untuk semua , , � , berlaku 9 � 9 � � 9 � 9 d. mempunyai elemen identitas bilangan bulat 1 � adalah elemen identitas untuk perkalian biangan modulo , jika menjadi elemen yang berubah-ubah di maka 1 9 sama dengan 9 1 berlaku 1 9 9 1 , + � e. mempunyai elemen invers misalkan � , mengingat himpunan = >1 9 , 2 9 , 3 9 , … , � � 1� 9 ? maka adalah tertutup pada operasi perkalian modulo , mengikuti bahwa anggota dari = adalah anggota dari . semua unsur di = berbeda, jika dan adalah dua bilangan bulat yang berbeda di sedemikian hingga 8 , maka 1 < < � 1, 1 < < � 1 dan 8 . misal 9 9 : · � · dapat dibagi : � � � · dapat dibagi tetapi ini tidak mungkin, ketika 1 < � ; � 1 dan 1 < < � 1 di mana prima, jadi � � � · tidak pernah bisa dibagi dengan . maka dari itu 9 � 9 . dengan demikian himpunan = terdiri dari � � 1� elemen yang berbeda di 3� dan oleh sebab itu bertepatan dengan 3�. jadi satu elemen di 3� = harus 1. misalkan �= 9 � 1, +�= � 3� berakibat �= 9 � � 9 �= 1. hal ini menunjukkan bahwa setiap elemen � di 3� mempunyai invers �= di 3�. f. operasi 9 bersifat komutatif jika �, , � 3� maka berlaku � 9 , , 9 �, +�, , � 3� (raishinghania dan aggarwal, 1980: 49-53). setelah 3� terbukti merupakan lapangan maka dibuktikan bahwa 3� adalah ideal. sesuai dengan teorema 1 berikut: teorema 1. (raishinghania dan aggarwal, 1980: 366) jika 3� adalah ring komutatif dan � � 3�, maka 3� 9 � >@ 9 �: @ � 3�? adalah ideal di 3�. bukti misalkan �3�, �,9� adalah ring komutatif dan � � 3�. maka akan ditunjukkan bahwa 3� 9 � adalah ideal dari 3�. a) 3� 9 � adalah subring misalkan @� 9 � dan @� 9 � adalah dua elemen di 3� 9 � maka �@� 9 �� � �@� 9 �� �@� � @�� 9 � � 3� 9 �. �karena @� � 3�, @� � 3� : @� � @� � 3��. dan untuk �@� 9 �� 9 �@� 9 �� � 9 �@� 9 �� di mana � �@� 9 �� � 3� sebagai @� � 3�, � � 3� : @� 9 � � 3� �� 9 @�� 9 � � 3� 9 � �jadi � � 3�, @� � 3� : � 9 @� � 3�� dapat diketahui bahwa 3� 9 � adalah subring dari 3�. b) misalkan @ adalah sebarang elemen di 3� dan @� 9 � adalah elemen di 3� 9 �, maka @ 9 �@� 9 �� �@ 9 @�� 9 � � 3� 9 � ideal kanan dan �@� 9 �� 9 @ �@� 9 @� 9 � � 3� 9 � ideal kiri. terbukti bahwa 3� 9 � ideal di 3� teorema 2. (raishinghania dan aggarwal, 1980: 366-367) suatu ring komutatif dengan elemen satuan disebut lapangan jika lapangan tidak mempunyai proper ideal. bukti: misalkan 3� ring komutatif dengan elemen satuan yang tidak mempunyai proper ideal yaitu idealnya hanya identitas >0? dan pada dirinya sendiri. kemudian menunjukkan bahwa 3� adalah lapangan dengan memperlihatkan saropah 152 volume 2 no. 3 november 2012 bahwa setiap elemen bukan nol di 3� mempunyai invers pada operasi perkalian di 3�. misalkan � elemen bukan nol di 3�, i >@ 9 �: @ � 3�? di mana i 3� 9 � maka i adalah ideal di 3� ..................... [teorema 1] sekarang 1 � 3� : 1 9 � � � i di mana � adalah suatu elemen bukan nol di 3�. oleh karena itu i adalah ideal bukan nol dalam 3�. ketika 3� tidak mempunyai proper ideal dan i adalah ideal bukan nol dalam 3�. sehingga satusatunya yang mungkin adalah i 3�. sekarang, ketika 1 � 3� dan i 3�, maka harus ada beberapa elemen , � 3� sedemikian hingga , 9 � 1 dengan cara yang sama � 9 , 1. karenanya ��� , � 3� yaitu setiap elemen di 3� mempunyai invers pada operasi perkalian atau operasi kedua di 3�. maka 3� adalah lapangan. 2 pada teorema 1 dan 2 telah terbukti 3� merupakan lapangan dan ideal maka 3� dikenakan pada suatu polinomial ����, misalkan ���� �j� j � ��� � ��� � � � � ��� �, �j, ��, ��, … , �� � 3� di mana koefisien sukusukunya merupakan anggota bilangan dalam 3�. akar-akar dari polinomial ���� dalam lapangan 3� ini ternyata terdapat polinomial yang tidak tereduksi maka dari itu dibentuklah perluasan lapangan yang akar-akarnya terdapat di dalamnya. hal ini akan ditunjukkan sebagai berikut. misalkan polinomial ���� ��� � � �6� 6 � �k� k � �7� 7 � ��j� �j � � � ���� �� tidak tereduksi dalam 3�, polinomial ���� tidak tereduksi dalam 3� karena polinomial ini tidak mempunyai suatu elemen dalam 3� yang menyebabkan ���� 0. diberikan polinomial ���� ��� � � �6� 6 � �k� k � �7� 7 � ��j� �j � � � ���� �� dan di cari akar-akarnya dengan metode horner sebagai berikut: � @ ��� ����� … �7 �k �6 �� l��@ l�j@ l7@ lk@ l6@ l�� l���� l7 lk l6 l� keterangan: l�� ��� l���� l��@ � ����� l���6 l����@ � ����6 m l7 l�j@ � l7 lk l7@ � �k l6 lk@ � �6 l� l6@ � �� untuk @ � 3� maka polinomial ���� tidak tereduksi karena hasil dari pembagiannya bukan nol sehingga sebarang @ � 3� tidak memenuhi ���� 0. oleh sebab itu dibentuklah perluasan lapangan atas 3� yaitu lapangan yang memuat akar-akar dari polinomial ���� . setelah terbentuk perluasan lapangan maka dibentuk lapangan pemisah dengan cara mencari faktorisasi dari polinomial ����. misal diambil polinomial genap ���� ��� � � �6� 6 � �k� k � �7� 7 ����� � �6� � � �k� 6 � �7� k� polinomial ���� dapat dibentuk sebagai hasil kali dari salah satu faktor dalam lapangan 3� maka dari itu polinomial ���� termasuk polinomial separable. tetapi salah faktornya tersebut tidak linier yaitu ��, maka polinomial ���� tidak termasuk dalam perluasan normal. selanjutnya misal ambil lagi polinomial n��� di mana polinomial ini merupakan polinomial ganjil dalam perluasan lapangannya maka n��� ��� � �4� 4 � �5� 5 � �o� o ���� � �4� � � �5� 6 � �o� k� karena salah satu faktor dari n��� adalah linier dan faktor berlainan dengan faktor lainnya maka polinomial n��� merupakan polinomial separable dalam lapangan pemisahnya yang termasuk dalam perluasan normal. penutup hasil penelitian yang dilakukan penulis dengan mengenakan suatu polinomial pada lapangan 3� di mana 3� adalah lapangan yang koefisien suku-sukunya berada dalam modulo prima dengan peubah �, bahwa 3� 9 � ideal dari 3� di mana 3� 9 � adalah sublapangan dari 3�. hal ini menunjukkan bahwa 3� menjadi suatu perluasan dari 3� 9 �. terbentuknya perluasan lapangan dikarenakan terdapat polinomial ���� tidak tereduksi dalam lapangannya sehingga dapat diketahui akar-akar penyelesaiaan pada perluasan lapangan. pada penelitian ini, semua polinomial tidak tereduksi dalam lapangannya termasuk polinomial separable dalam lapangan pemisahnya. namun, ada polinomial separable yang tidak termuat dalam perluasan normal yaitu polinomial separable yang berderajat genap. hal ini dikarenakan adanya faktorisasi nonlinier dalam polinomial tersebut. sementara untuk polinomial separable yang berderajat ganjil �2p � 1� di mana p adalah bilangan bulat positif, maka polinomial separable dalam lapangan pemisahnya ini termasuk dalam perluasan normal. hal ini ditunjukkan dengan adanya polinomial separable ini yang salah satu faktor liniernya termuat dalam polinomial di lapangannya. maka semua polinomial separable yang berderajad ganjil ini termasuk dalam perluasan normal. akar-akar polinomial separable sebagai pembentuk perluasan normal pada ring modulo jurnal cauchy – issn: 2086-0382 153 daftar pustaka [1] abdussakir. 2007. ketika kyai mengajar matematika. malang: uin malang press [2] beachy, j. a. & blair, w. d. 1990. abstract algebra with a concrete introduction. prentice hall, englewood, new jesey 07632 [3] dummit, s. d. & foote, r. m. 1999. abstract algebra, second edition. new york: john wiley & sons, inc. [4] fraleigh, j. b. 1994. a first course in abstract algebra, fifth edition. new york: addition and wisley publishing company, usa. [5] herstein. 1975. topics in algebra 2nd edition. new york: john wiley & sons [6] irawanto, b., dkk. 2004. akar-akar polinomial separabel sebagai pembentuk perluasan normal. yogyakarta: fmipa ugm [7] munir, r. 2008. metode numerik. bandung: informatika [8] rahman, h. 2007. indahnya matematika dalam al-qur’an. malang: uin malang press [9] raisinghania, m. d & aggarwal r. s. 1980. modern algebra. new delhi: s. chand & company ltd. [10] wahyudin. 1989. aljabar modern. bandung: tarsito model regresi logistik multinomial untuk menentukan pilihan sekolah lanjutan tingkat atas pada siswa smp puji subekti mahasiswa program magister matematika universitas brawijaya malang telp : 085649603425; email : ukhti_ajieb88@yahoo.com abstract in general, secondary schools in indonesia contained three institutions namely sma (senior high school), smk (vocational high school) and ma (madrasah aliyah). this study to determine the variables that significantly influence, so we can know the probability of students in choosing junior high school after study. the analysis tool used is the multinomial logistic regression analysis. at the 95% confidence level is obtained that out of 8 (eight) independent variables are used, there are 5 (five) variables that significantly influence. they are the father's education, mother's education, mother's income, number of siblings, and father's occupation. keywords: multinomial logistic regression analyzes, sma, ma, smk, probability selection school. abstrak secara umum, sekolah menengah di indonesia terdapat tiga lembaga yakni sma (sma), smk (sekolah menengah kejuruan) dan ma (madrasah aliyah). penelitian ini untuk mengetahui variabel yang mempengaruhi secara signifikan, sehingga kita dapat mengetahui kemungkinan siswa dalam memilih smp setelah alat analisis study.the digunakan adalah analisis regresi logistik multinomial. pada tingkat kepercayaan 95% diperoleh bahwa dari 8 (delapan) variabel independen yang digunakan, ada 5 (lima) variabel yang mempengaruhi secara signifikan.salah satu diantaranya adalah pendidikan ayah, pendidikan ibu, pendapatan ibu, jumlah saudara kandung, dan pekerjaan ayah. keywords: multinomial logistic regression analyzes, sma, ma, smk, probability selection school. pendahuluan pada umumnya, sekolah menengah di indonesia diwadahi tiga lembaga yakni sma, smk dan ma. sma bertujuan untuk menyediakan dan menyiapkan siswa/i yang hendak melanjutkan studi ke jenjang yang lebih tinggi, akademi atau perguruan tinggi, sedangkan smk lebih ditujukan untuk menyediakan tenaga kerja tingkat menengah. ma, sebagaimana sma bertujuan untuk mengantarkan siswa memasuki perguruan tinggi umum maupun perguruan tinggi islam. kenyataannya tidak semua lulusan sma berkesempatan melanjutkan studi ke jenjang yang lebih tinggi karena berbagai alasan, begitu pula dengan lulusan smk dan ma [1]. beberapa faktor yang diduga mempengaruhi siswa dalam memilih sekolah lanjutan adalah umur siswa, jenis kelamin siswa, asal sekolah siswa, nilai rata-rata mapel siswa pada semester 1, jumlah tanggungan anak dalam keluarga yang masih bersekolah, pendidikan orangtua siswa, pekerjaan orangtua siswa, penghasilan orangtua siswa, minat, dan dukungan keluarga. faktor-faktor tersebut akan diduga mempengaruhi siswa dalam memilih sekolah lanjutan sehingga disebut sebagai variabel independen. sedangkan sekolah lanjutan yaitu: sma, smk, dan ma disebut sebagai variabel terikat. sehubungan dengan hal tersebut untuk menganalisis hubungan antara variabel terikat yang mempunyai kategori lebih dari dua, dengan beberapa variabel independen yang bersifat kontinu, kategorik, atau keduanya adalah dengan menggunakan analisis regresi logistik multinomial. kajian teori model model adalah pola (contoh, acuan, ragam) dari sesuatu yang akan dibuat atau dihasilkan. definisi lain dari model adalah abstraksi dari sistem sebenarnya, dalam mailto:ukhti_ajieb88@yahoo.com puji subekti 92 volume 3 no. 2 mei 2014 gambaran yang lebih sederhana serta mempunyai tingkat prosentase yang bersifat menyeluruh, atau model adalah abstraksi dari realitas dengan hanya memusatkan perhatian pada beberapa sifat dari kehidupan sebenarnya. model yang akan disusun dalam penelitian ini termasuk model simbolis, yaitu model yang menggambarkan sistem yang ditinjau dengan simbol-simbol biasanya dengan simbol-simbol matematik. dalam hal ini sistem diwakili oleh variabel-variabel dari karakteristik sistem yang ditinjau. regresi logistik regresi logistik merupakan salah satu metode yang dapat digunakan untuk mencari hubungan varia-bel respon yang bersifat dichotomous (berskala nominal atau ordinal dengan dua kategori) atau polychotomous (mempunyai skala nominal atau ordinal dengan lebih dari dua kategori) dengan satu atau lebih variabel prediktor dan variabel respon bersifat kontinyu atau kategorik [2]. regresi logistik multinomial regresi logistik multinomial merupakan regresi logistik yang digunakan saat variabel dependen mempunyai skala yang bersifat polichotomous atau multinomial. metode yang digunakan dalam penelitian ini adalah regresi logistik dengan variabel respon berskala nominal dengan tiga kategori. model yang digunakan pada regresi logistik multinomial adalah 𝐿𝑜𝑔𝑖𝑡 𝑃(𝑌 = 1) = ∝ +𝛽1𝑋1 + 𝛽2𝑋2 + … + 𝛽𝑛 𝑋𝑛 dengan menggunakan transformasi logit didapatkan fungsi logit, 𝑃1(𝑥) = 𝑙𝑛 [ 𝑃(𝑌 = 1)1|𝑥 𝑃(𝑌 = 1)0|𝑥 ] = 𝛽10 + 𝛽11𝑋1 + 𝛽12𝑋2 + … + 𝛽1𝑛 𝑋𝑛 = 𝑥′𝛽1 𝑃2(𝑥) = 𝑙𝑛 [ 𝑃(𝑌 = 1)2|𝑥 𝑃(𝑌 = 1)0|𝑥 ] = 𝛽20 + 𝛽21𝑋1 + 𝛽22𝑋2 + … + 𝛽2𝑛 𝑋𝑛 = 𝑥′𝛽2 berdasarkan kedua fungsi logit tersebut maka didapatkan model regresi logistik trichotomous sebagai berikut : 𝜋0(𝑥) = 1 1 + 𝑒𝑥𝑝 𝑃1(𝑥) + 𝑒𝑥𝑝 𝑃2(𝑥) 𝜋1(𝑥) = 𝑒𝑥𝑝 𝑃1(𝑥) 1 + 𝑒𝑥𝑝 𝑃1(𝑥) + 𝑒𝑥𝑝 𝑃2(𝑥) 𝜋2(𝑥) = 𝑒𝑥𝑝 𝑃2(𝑥) 1 + 𝑒𝑥𝑝 𝑃1(𝑥) + 𝑒𝑥𝑝 𝑃2(𝑥) untuk menguji signifikansi β dari model yang telah diperoleh, maka dilakukan uji parsial dan uji serentak. a. uji parsial pengujian signifikansi parameter menggunakan uji wald [3] dengan hipothesis di bawah ini : 𝐻_0: 𝛽_𝑘 = 0, dengan ℎ = 1,2,3, … 𝑘 𝐻_1: 𝛽_𝑘 ≠ 0, dengan ℎ = 1,2,3, … 𝑘 perhitungan statistik uji wald adalah sebagai berikut : 𝑤𝑎𝑙𝑑 = �̂�𝑘 𝑆𝐸(�̂�𝑘 )̂ b. uji serentak hipotesis untuk pengujian adalah sebagai berikut: h0 : 𝛽1 = 𝛽2 = ⋯ = 𝛽𝑝 = 0 h1 : paling tidak ada satu 𝛽𝑖 yang tidak sama dengan 0 di mana 𝑖 = 1,2, … , 𝑛 (n adalah banyaknya lokasi pengamatan) dan 𝑘 = 1,2, … , 𝑝 (p adalah banyaknya variabel prediktor). statistik uji 𝐺 2 atau likelihood ratio test yang dinyatakan sebagai berikut: 𝐺 2 = −2 𝑙𝑛 ( 𝐿(�̂�) 𝐿(�̂�) ) menurut hosmer dan lemeshow (1989), statistik uji g2 mengikuti distribusi chi-square, sehingga untuk memperoleh keputusan dilakukan perbandingan dengan titik kritis 𝜒2 (𝛼,𝑑𝑏) di mana derajat bebasnya adalah p atau banyaknya variabel prediktor. kriteria penolakan (tolak h0) jika nilai 𝐺 2 > 𝜒2 (𝛼,𝑑𝑏) . c. uji kesesuaian model uji kesesuaian model dengan menggunakan statistik uji chi-square adalah sebagai berikut 𝑋2 = ∑ (𝑂𝑘 − 𝑛′𝑘 �̅�𝑘 ) 2 𝑛′ 𝑘�̅�𝑘 (1 − �̅�𝑘 ) 𝑔 𝑘−1 model regrei logistik multinomial untuk menentukan pilihan sekolah lanjutan tingkat atas… cauchy – issn: 2086-0382 93 statistik uji di atas untuk menguji hipotesis sebagai berikut : 𝐻0 : model sesuai (tidak ada perbedaan yang nyata antara hasil observasi dengan kemungkinan hasil prediksi model) 𝐻1 : model tidak sesuai (ada perbedaan yang nyata antara hasil observasi dengan kemungkinan hasil prediksi model). pengambilan keputusan didasarkan pada tolak 𝐻0 jika 𝑋 2 ℎ𝑖𝑡𝑢𝑛𝑔 ≥ 𝑋 2 (𝑑𝑏,∝) dengan db=g-2. pendidikan sma, ma dan smk menurut salim (2008) sistem pendidikan di indonesia secara umum masih dititikberatkan pada kecerdasan kognitif. hal ini dapat dilihat dari orientasi sekolah sekolah yang ada masih disibukkan dengan ujian, mulai dari ujian mid, ujian akhir hingga ujian nasional. ditambah latihan-latihan soal harian dan pekerjaan rumah untuk memecahkan pertanyaan di buku pelajaran yang biasanya tidak relevan dengan kehidupan sehari hari para siswa. setelah jenjang pendidikan menengah pertama, ada beberapa jenis tingkat sekolah diantaranya sma, ma, smk, smea dan lainlain. masing-masing diantara sekolah tersebut berbeda dalam penyusunan kurikulumnya menurut visi dan misi masingmasing. sebagian diantara mereka ada yang lebih menekankan teori dan sebagian lagi lebih mengedepankan praktik. namun tidak perlu mempermasalahkan tentang hal itu, semakin beragam pendidikan di negara kita akan semakin mudah untuk membentuk sumberdaya alam yang sangat bermutu. diantara ciri-ciri kurikulum sma maupun ma adalah sebagai berikut: 1. materi pembelajaran lebih mengarah kepada teori dari pada praktik. 2. tamatannya tidak siap kerja. 3. tempat belajar lebih tertuju di sekolah. ciri-ciri kurikulum smk, diantaranya: 1. materi pembelajaran lebih mengarah kepada praktek dari pada teori, dapat dikatan 60% paraktek dan 40% teori. 2. tamatannya siap kerja 3. tempat belajar di sekolah dan di dunia kerja. metode penelitian rancangan penelitian yang pertama dilakukan pada siswa secara langsung dengan pengisian kuisioner mengenai faktorfaktor internal yang mempengaruhi siswa dalam pemilihan kelanjutan pendidikan setelah lulus dari smp. adapun variabel – variabel dependen dan variabel – variabel independen yang dipergunakan dalam penelitian ini diantaranya adalah : 1. y sebagai variabel dependen yang terdiri atas tiga kategori yaitu sekolah sma (dengan kode 0), ma (dengan kode 1), dan smk (dengan kode 2). 2. x sebagai variabel independen yang terdiri atas 8 variabel yaitu pendidikan ibu (x1), pendidikan ayah(x2) ,penghasilan ibu(x3), penghasilan ayah(x4), jumlah saudara(x5), jumlah tanggungan keluarga (x6) pekerjaan ayah(x7), nilai rata-rata matapelajaran yang diikutkan dalam uan(x8). dalam penelitian ini dipilih siswasiswi smp negeri 2 malang kelas viii sebagai sampel penelitian sebab sebagaimana khalayak umum ketehui bahwa smp negeri 2 malang merupakan salah satu smp yang dianggap oleh masyarakat sebagai smp yang dapat memberikan keluaran siswa-siswi berprestasi khususnya dibidang akademik, langkah-langkah analisis data dalam penelitian ini adalah sebagai berikut : 1. melakukan uji validitas terhadap itemitem soal. 2. melakukan uji reliabilitas terhadap itemitem soal. jika item-item soal telah memenuhi uji validitas dan reabilitas maka dapat dilakukan pada pengujian selanjutnya. 3. melakukan analisis regresi logistik terhadap data: 4. menaksir parameter yang diperoleh. 5. menginterpretasikan model regresi logistik. 6. melakukan uji serentak, parsial dan kesesuaian model 7. evaluasi model puji subekti 94 volume 3 no. 2 mei 2014 hasil dan pembahasan karakteristik siswa smp negeri 2 malang siswa smp negeri 2 malang yang dijadikan sebagai obyek penilitian seperti dijelaskan sebelumnya berjumlah 108 siswa. ditinjau dari segi pendidikan terakhir ibu 18,5% berpendidikan sd, 23,1% berpendidikan smp, 44,4% berpendidikan sma, dan 13,9% berpendidikan sarjana. dari segi pendidikan terakhir ayah juga terdapat 18,5% berpendidikan sd, 20,4% berpendidikan smp, 47,2% berpendidikan sma, dan 13,9% berpendidikan sarjana. jika ditinjau dari penghasilan ibu yang kurang dari 700.000 sebanyak 72%,sebanyak 19,4% antara 700.00 sampai 2.000.000 dan 8,3% diatas 2.000.000. ditinjau dari penghasilan ayah yang kurang dari 700.000 sebanyak 46,3%,sebanyak 29,6% antara 700.00 sampai 2.000.000 dan 824,1% diatas 2.000.000. keluarga siswa yang tidak mempunyai saudara sebanyak 10,2%. sebanyak 19,4% mempunyai 1 saudara, 52,8% mempunyai 2 saudara, 7,4% mempunyai 3 saudara dan 10,2% mempunyai lebih dari 3 saudara. keluarga siswa yang tidak mempunyai tanggungan sebanyak 1,9%. sebanyak 32,4% mempunyai 1 tanggungan, 48,1% mempunyai 2 tanggungan, 13% mempunyai 3 tanggungan dan 4,6% mempunyai lebih dari 3 tanggungan keluarga. ditinjau dari jenis pekerjaan ayah 13,9% seorang pns, 12,0% wiraswasta, 59,3% karyawan swasta, dan 14,8% lain-lain. sedangkan siswa yang mempunyai nilai diatas 75 sebanyak 65,7% dan 33,3% dibawah 75. regresi logistik multinomial secara individu untuk mengetahui pengaruh dari faktor-faktor yang diduga mempengaruhi pemilihan sekolah lanjutan tingkat atas pada siswa smp secara individu, dilakukan uji wald dan p-value disajikan dalam tabel berikut ini : tabel 1 regresi logistik multinomial secara individu logit variabel b wald p-value exp (b) pendidikan ibu sma / ma konstanta 1,099 2,716 0,099 0 sd smp sma sarjana -3,871 -2,351 -2,135 0 9,945 7,219 8,028 0 0,002* 0,007* 0,005* 0 0,021 0,095 0,118 0 smk konstanta sd smp sma sarjana -0,405 -2,367 -0.847 -2.335 0 0,197 2,956 0,622 3,994 0 0,657 0,086 0,430 0,046* 0 0 0,094 0,429 0,097 0 pendidikan ayah sma/ma konstanta pend.ayah sd smp sma sarjana 1,504 -4,212 -3,701 -2,306 0 3,702 10.575 11,743 7,363 0 0,054 0,001* 0,001* 0,017* 0 0 0,015 0,025 0,100 0 smk konstanta pend.ayah sd smp sma sarjana -1,386 -0,754 -0,811 -0,256 0 1,537 0,314 0,364 0,045 0 0,215 0,575 0,546 0,832 0 0 0,471 0,444 0,774 0 penghasilan ayah sma konstanta <700 700<x<2.000 >2.000 1,041 -4,037 -2,140 0 4,810 21,718 10,230 0 0,028 0,000* 0,001* 0 0 0,018 0,118 0 smk konstanta <700 700<x<2.000 >2.000 -1,792 -0,799 0,511 0 2,752 0,418 0,183 0 0,097 0,518 0,668 0 0 0 0,450 1,667 model regrei logistik multinomial untuk menentukan pilihan sekolah lanjutan tingkat atas… cauchy – issn: 2086-0382 95 penghasilan ibu sma konstanta <700 700<x<2.000 >2.000 0,693 -2,266 -0,693 0 0,961 8,417 0,641 0 0,327 0,004* 0,423 0 0 0,104 0,500 0 smk konstanta <700 700<x<2.000 >2.000 -19,868 17,844 18,482 0 631,604 404,726 0 0 0,000* 0,000* 0,000* 0,000* 0 5,618e7 1,063e8 0 jumlah saudara sma konstanta =0 =1 =2 =3 >3 -0,981 0,288 -0,272 0,288 -19,419 0 2,099 0.086 0,095 0,148 0 0 0,147 0,769 0,758 0,700 0 0 0 1,333 0,762 1,333 3,686e-9 0 smk konstanta =0 =1 =2 =3 >3 -21,127 19,335 19,586 19,112 19,335 0 382,576 160,217 244,123 251,905 0 0 0,000* 0,000* 0,000* 0,000* 0,000* 0,000* 0 1,250e7 2,749e7 1,884e7 2,495e8 0 jumlah tanggungan sma konstanta =0 =1 =2 =3 >3 -1.386 1,386 0,197 0,584 0,539 0 1,537 0,591 0,027 0,250 0,168 0 0,215 0,442 0,870 0,617 0,682 0 0 4,000 1,217 1,793 1,714 0 smk konstanta =0 =1 =2 =3 >3 -19,239 0,324 16,797 17,481 17,986 0 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,998 0,000* 0,998 0,998 0,998 0 0 1,382 1,971e7 3,909e7 6,478e7 0 pekerjaan ayah sma konstanta lain-lain swasta wiraswasta pns 1,609 -4,094 -2,886 -2,862 0 4,317 9,959 11,785 6,591 0 0,038 0,002* 0,001* 0,010* 0,000* 0 0,017 0,056 0,057 0 smk konstanta lain-lain swasta wiraswasta pns 0,000 -1,386 -2,663 -1,946 0 0,000 1,357 5,226 1,767 0 1,000 0,244 0,002* 0,184 0 0 0,250 0,070 0,143 0 nilai konstanta <75 >75 -1,962 -0,523 0,219 22,925 0,386 0 0,000 0,534 0,000* 0 0,593 1,244 berdasarkan pada tabel 1 dapat diketahui bahwa variabel pendidikan ibu baik sd, smp, sma dan sarjana memberikan pengaruh yang signifikan terhadap pemilihan sekolah lanjutan tingkat atas yaitu sma maupun ma. berbeda dengan pendidikan terakhir ibu setingkat sd dan smp tidak berpengaruh signifikan terhadap pemilihan sekolah lanjutan smk, namun siswa yang pendidikan terakhir ibunya setingkat sma, dan sarjana yang berpengaruh terhadap pemilihan smk. kesignifikanan ini ditunjukkan dengan adanya nilai p-value < 0,05. begitu juga dengan variabel lain seperti pendidikan ayah yang meliputi sd, smp, sma dan sarjana juga memberikan pengaruh yang signifikan terhadap pemilihan sekolah lanjutan tingkat atas yaitu sma maupun ma puji subekti 96 volume 3 no. 2 mei 2014 dan tidak memberikan pengaruh yang signifikan terhadap pilihan smk. penghasilan ayah yang kurang dari 700.000,00 per bulan juga memberikan pengaruh yang signifikan terhadap pemilihan sma maupun ma namun tidak berpengaruh secara signifikan terhadap pemilihan smk. demikian juga penghasilan ibu yang kurang dari 700.000,00 dan lebih dari 2.000.000,00 juga mempunyai pengaruh signifikan terhadap pemilihan sma maupun ma dibandingkan dengan yang lain , namun disisi lain ketiganya baik pendapatan ibu yang kurang dari 700.000,00 , antara 700.000,00 sampai dengan 2.000.000,00 mapun lebih dari 2.000.000,00 memberikan pengaruh yang signifikan terhadap pemiliah smk. jumlah saudara yang dimiliki siswa memberikan pengaruh yang signifikan terhadap pemilihan smk namun tidak pada pemilihan sma maupun ma. siswa yang memiliki saudara sejumlah 3 orang maupun lebih mempunyai pengaruh yang signifikan terhadap pemilihan smk. namun tidak terjadi pada variabel jumlah tanggungan pada keluarga yang tidak berpengaruh signifikan terhadap pemilihan sma, ma maupun smk, kecuali siswa yang mempunyai tanggungan lebih dari 3 orang. selain itu pekerjaaan ayah juga memberikan pengaruh yang signifikan terhadap pemilihan sekolah lanjutan tingkat atas sma maupun ma namun tidak pada pemilihan smk. hanya siswa yang pekerjaan ayahnya swasta dan pns yang mempunyai pengaruh signifikan terhadap pemilihan smk. nilai rata-rata matapelajaran yang diikutkan dalam uan mulai semester 1 sampai dengan 3 siswa yang mendapatkan diatas kisaran angka 75 lebih memberikan pengaruh yang signifikan terhadap pemilihan sma maupun ma sehingga dapat ditentukan fungsi logit bagi masing-masing variabel yang berpengaruh signifikan diantaranya yaitu sebagai berikut: fungsi logit untuk siswa yang pendidikan terakhir ayahnya adalah sd atau sekolah dasar adalah : 𝑝1(𝑥) = 1,504 − 4,212𝑆𝐷 𝑝2(𝑥) = −1,386 − 0,754𝑆𝐷 fungsi logit untuk siswa yang pendidikan terakhir ayahnya smp adalah : 𝑝1(𝑥) = 1,504 − 3,701𝑠𝑚𝑝 𝑝2(𝑥) = −1,386 − 0,811𝑠𝑚𝑝 fungsi logit untuk siswa yang pendidikan terakhir ayahnya sma adalah : 𝑝1(𝑥) = 1,504 − 2,306𝑠𝑚𝑎 𝑝2(𝑥) = −1,386 − 0,256𝑠𝑚𝑎 fungsi logit untuk siswa yang pendidikan terakhir ayahnya sarjana adalah : 𝑝1(𝑥) = 1,504 𝑝2(𝑥) = −1,386 dari fungsi logit yang didapatkan seperti tertera di atas sehingga didapatkan model regresi logistik trichotomous untuk mengetahui seberapa besar peluang dari masing masing siswa untuk memilih sma, ma maupun smk ditinjau dari masingmasing variabel yang signifikan sebagai berikut : peluang bagi siswa yang pendidikan terakhir ayah sd memilih sma adalah 𝜋0(𝑥) = 1 1 + 𝑒𝑥𝑝 𝑃1(𝑥) + 𝑒𝑥𝑝 𝑃2(𝑥) = 1 1 + 𝑒𝑥𝑝(1,504 − 4,212𝑆𝐷) + 𝑒𝑥𝑝(−1,386 − 0,754𝑆𝐷) = 0,85 peluang bagi siswa yang pendidikan terakhir ayah smp adalah 𝜋0(𝑥) = 1 1 + 𝑒𝑥 𝑝 𝑃1(𝑥) + 𝑒𝑥 𝑝 𝑃2(𝑥) = 1 1 + 𝑒𝑥𝑝 (1,504 − 3,701𝑠𝑚𝑝) +𝑒𝑥 𝑝(−1,386 − 0,811𝑠𝑚𝑝) = 0.82 peluang bagi siswa yang pendidikan terakhir ayah sma adalah 𝜋0(𝑥) = 1 1 + 𝑒𝑥 𝑝 𝑃1(𝑥) + 𝑒𝑥 𝑝 𝑃2(𝑥) = 1 1 + 𝑒𝑥𝑝(1,504 − 2,306𝑠𝑚𝑎) +𝑒𝑥 𝑝(−1,386 − 0,256𝑠𝑚𝑎) = 0,61 peluang bagi siswa yang pendidikan terakhir ayah sarjana adalah 𝜋0(𝑥) = 1 1 + 𝑒𝑥𝑝 𝑃1(𝑥) + 𝑒𝑥𝑝 𝑃2(𝑥) = 1 1 + 𝑒𝑥𝑝(1,504) + 𝑒𝑥𝑝(−1,386) = 0,17 model regrei logistik multinomial untuk menentukan pilihan sekolah lanjutan tingkat atas… cauchy – issn: 2086-0382 97 pengujian individu untuk variabel pendidikan ibu, penghasilan ibu, jumlah saudara dan pekerjaan ayah juga menunjukkan bahwa variabel-variabel tersebut mempunyai pengaruh yang signifikan terhadap pemilihan sekolah lanjutan tingkat atas. dengan menggunakan metode perhitungan yang sama maka didapatkan nilai peluang untuk masingmasing variabel. perhitungan peluang untuk masing-masing varaibel yang tergolong memberikan pengaruh signifikan diberikan pada tabel 2 berikut ini : tabel 2 peluang pemilihan sekolah ditinjau dari variabel yang signifikan variabel dan kategori sma ma smk pendidika n ayah sd smp sma sarjana 0,85 0,82 0,61 0,17 0,062 0,09 0,27 0,78 0,09 0,09 0,1176 0,0043 pendidika n ibu sd smp sma sarjana 0,89 0,636 0,704 0,21 0,056 0,18 0,25 0,642 0,056 0,181 0,045 0,0143 penghasila n ibu <700.000 700.000<x <2.000.00 0 >2.000.00 0 0,745 0,44 0,33 0,156 0,44 0,67 0,098 0,11 0,00000007 6 jumlah saudara jumlah=0 jumlah=1 jumlah=2 jumlah=3 jumlah>3 0,61 0,66 0,71 0,0000 00009 1 0,72 0,306 0,193 0,198 0,000 00001 2 0,27 0,079 0,142 0,09 0,00000000 0000000002 5 0,00000000 48 pekerjaan ayah lain swasta wiraswast a pns 0,75 0,74 0,699 0,143 0,062 0,207 0,200 0,714 0,187 0,051 0,10 0,14 regresi logistik multinomial secara serentak pada model regresi logistik secara serentak ini, variabel-variabel yang tidak signifikan pada regresi logistik multinomial yang individu tidak diikutsertakan dalam model. variabel yang diikutkan dalam model serentak ini adalah pendidikan ayah, pendidikan ibu, penghasilan ibu, jumlah saudara, dan pekerjaan ayah. hasil regresi logistik multinomial secara serentak. dari model regresi logistik multinomial secara serentak didapatkan fungsi logit sebagai berikut : 𝑝1(𝑥) = 5,310 − 1,713𝑋11 − 1,073𝑋12 − 0,322𝑋13 − 2,989𝑋21 − 1,671𝑋22 − 2,005𝑋23 − 2,418𝑋31 − 0,381𝑋32 + 0,695𝑋41 + 0,137𝑋42 + 0,388𝑋43 − 18,386𝑋44 − 3,253𝑋51 − 2,684 𝑋52 − 4,420𝑋53 𝑝2(𝑥) = −34,726 − 1,294𝑋11 − 1,650𝑋12 − 0,016𝑋13 − 0,925𝑋21 − 0,076𝑋22 − 1,817𝑋23 + 18,594𝑋31 + 19,930𝑋32 + 19,657𝑋41 + 18,811𝑋42 + 17,797𝑋43 + 17,149𝑋44 − 1,168𝑋51 − 3,095𝑋52 − 1,850𝑋53 keterangan : x11 = pendidikan ayah sd x12 = pendidikan ayah smp x13 = pendidikan ayah sma x21 = pendidikan ibu sd x22 = pendidikan ibu smp x23 = pendidikan ibu sma 𝑋31 = 𝑝𝑒𝑛𝑔ℎ𝑎𝑠𝑖𝑙𝑎𝑛 𝑖𝑏𝑢 < 700rb 𝑋32 = 𝑝𝑒𝑛𝑔ℎ𝑎𝑠𝑖𝑙𝑎𝑛 𝑖𝑏𝑢 700rb < 𝑥 < 2.000rb x41 = jumlah saudara 0 x42 = jumlah saudara 1 x43 = jumlah saudara 2 x44 = jumlah saudara 3 x51 = pekerjaan ayah lain − lain x52 = pekerjaan ayah swasta x53 = pekerjaan ayah wiraswasta puji subekti 98 volume 3 no. 2 mei 2014 uji kesesuaian model berdasarkan pada hasil perhitungan untuk melihat apakah data empiris cocok atau tidak, diharapkan tidak ada perbedaan antara data empiris dengan model. berdasar pada pustaka bahwa model akan dinyatakan sesuai jika signifikansi di atas 0,05 atau nilai 2 log likelihood kurang dari chi square tabel. keterangan selanjutnya dapat dilihat dari tabel 3 tabel 3 uji kesesuaian model model -2 log likelihood ( 𝑋(𝑑𝑏,∝) 2 ) chi-square (𝑋2 ℎ𝑖𝑡𝑢𝑛𝑔 ) intercept only 160,278 final 11,091 149,187 tampak bahwa x2hitung ≥ x 2 (v,∝) dengan 𝑣 = 𝑔 − 2. pengambilan keputusan didasarkan pada tolak 𝐻0 berarti model adalah fit dan model dinyatakan sesuai dan boleh diinterpretasikan. kesimpulan berdasarkan hasil penelitian yang telah dilakukan dapat diambil kesimpulan sebagai berikut : dari hasil dan pembahasan 8 variabel bebas yang mempengaruhi dalam pemilihan sekolah lanjutan tingkat atas menggunakan regresi logistik multinomial didapatkan 5 variabel bebas yang berpengaruh secara signifikan yaitu pendidikan ayah , pendidikan ibu, penghasilan ibu, jumlah saudara, dan pekerjaan ayah. 1. hasil taksiran peluang pemilihan sekolah lanjutan ditinjau dari variabel yang signifikan siswa smp 2, semakin rendah pendidikan terakhir yang ditempuh ayah maka semakin besar peluang siswa memilih sma yaitu sebesar 70%-80%. demikan juga dengan pendidikan terakhir ibu. sedangkan siswa dengan penghasilan ibu yang kurang dari rp 700.000 lebih besar peluangnya memilih sma dibandingkan ma dan smk. siswa dengan jumlah saudara lebih dari 3 dan jenis pekerjaan ayah yang tidak menentu penghasilanya juga memiliki peluang paling besar untuk memilih sma dibandingkan ma dan smk. selanjutnya saran yang dapat diberikan sehubungan dengan penelitian di atas perlu banyak dilakukan penilitian lagi terhadap variabel-variabel lain yang berpengaruh terhadap pemilihan sekolah lanjutan tingkat atas sehingga dapat dioleh hasil yang lebih baik.para siswa diharapkan mampu mengambil keputusan terbaik dalam memilih sekolah yang dituju dengan mempertimbangkan faktor-faktor yang berpengaruh. pemerintah dianjurkan untuk menambah kuantitas dan kualitas sekolah baik sma, ma maupun ma untuk melayani pendidikan lanjutan bagi siswa sesuai dengan pilihanya masing-masing. bibliography [1] awliya, “minat pilihan melanjutkan sekolah menengah tingkat atas,” kompas, jakarta, 2007. [2] a. agresti, categorical data analysis. john wiley & sons, 2014. [3] d. w. hosmer jr and s. lemeshow, applied logistic regression. john wiley & sons, 2004. pendahuluan kajian teori model regresi logistik regresi logistik multinomial pendidikan sma, ma dan smk metode penelitian hasil dan pembahasan karakteristik siswa smp negeri 2 malang regresi logistik multinomial secara individu regresi logistik multinomial secara serentak uji kesesuaian model kesimpulan bibliography multigroup moderation test in generalized structured component analysis cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 81-85 submitted: 20 february 2016 reviewed: 16 march 2016 accepted: 3 may 2016 multigroup moderation test in generalized structured component analysis angga dwi mulyanto1, solimun1, ni wayan surya wardhani 1, suharno2 1brawijaya university, indonesia 2mercu buana university, indonesia email: angga.dwi.m@gmail.com abstract generalized structured component analysis (gsca) is an alternative method in structural modeling using alternating least squares. gsca can be used for the complex analysis including multigroup. gsca can be run with a free software called gesca, but in gesca there is no multigroup moderation test to compare the effect between groups. in this research we propose to use the t test in pls for testing moderation multigroup on gsca. t test only requires sample size, estimate path coefficient, and standard error of each group that are already available on the output of gesca and the formula is simple so the user does not need a long time for analysis. keywords: gsca, gesca, multigroup moderation test introduction in 2004, hwang and takane introduced an alternative method for path analysis called generalized structured component analysis (gsca). gsca can be used for data reflective and formative, can be used for recursive and non-recursive relationship, doesn’t require normality assumption, and can be used for small sample size [1]. differences and similarities gsca with other method can see in table 1. table 1. sem, pls and gsca criteria sem pls gsca software amos (purchase) smart-pls (free and purchase) gesca (free) outer model reflective reflective and formative reflective and formative inner model recursive and nonrecursive recursive recursive and nonrecursive a few years later, hwang and takane created software for gsca called gesca [2]. gesca already available for multigroup features but have not reached the multigroup moderation test. multigroup moderation test used to compare the effect between groups [3]. in 2000, chin introduced multigroup moderation test using t test on pls [4]. in this research we want to try t test for multigroup moderation on gsca. multigroup moderation test in generalized structured component analysis angga dwi mulyanto 82 theoritical review the formula for multigroup moderation test using t test if standard deviations are equal can be write as [3] 𝑡 = |𝑏1 − 𝑏2| √ (𝑛1−1) 2 𝑛1+𝑛2−2 𝑆𝐸1 2 + (𝑛2−1) 2 𝑛1+𝑛2−2 𝑆𝐸2 2 × √ 1 𝑛1 + 1 𝑛2 ~𝑡𝑛1+𝑛2−2 (1) if standard deviations are unequal, the formula can be write as [3] 𝑡 = |𝑏1 − 𝑏2| √ 𝑛1−1 𝑛1 𝑆𝐸1 2 + 𝑛2−1 𝑛2 𝑆𝐸2 2 (2) degree of freedom for standard deviations are unequal can be write as [3] 𝑑𝑓 = ‖ ( 𝑛1−1 𝑛1 𝑆𝐸1 2 + 𝑛2−1 𝑛2 𝑆𝐸2 2) 2 𝑛1−1 𝑛1 2 𝑆𝐸1 4 + 𝑛2−1 𝑛2 2 𝑆𝐸2 4 − 2‖ (3) with b1 : estimate for path coefficient group 1 b2 : estimate for path coefficient group 2 n1 : sample size group 1 n2 : sample size group 2 se1 : standard error group 1 se2 : standard error group 2 to know that standard deviations are equal or unequal, we can use bartlett test. formula bartlett test for moderation multigroup can be write as [5] 𝜒2 = (𝑛 − 𝑘)ln𝑠𝑝 2 − ∑ (𝑛𝑖 − 1)ln𝑠𝑖 2𝑘 𝑖=1 1 + ( 1 3(𝑘−1) )((∑ 1 𝑛𝑖−1 𝑘 𝑖=1 ) − 1 𝑛−𝑘 ) ~𝜒2 𝑘−1 (4) k is the number of group and n is the total of sample size. because t test can only use for 2 group, equation 4 can rewrite as 𝜒2 = (𝑛 − 2)ln𝑠𝑝 2 − ((𝑛1 − 1)ln𝑠1 2 + (𝑛2 − 1)ln𝑠2 2) 1 + ( 1 3 )( 1 𝑛1−1 + 1 𝑛2−1 − 1 𝑛−2 ) ~𝜒2 1 (5) with 𝑠1 2 = (𝑛1 − 1)𝑆𝐸1 2 (6) 𝑠2 2 = (𝑛2 − 1)𝑆𝐸2 2 (7) 𝑠𝑝 2 = (𝑛1 − 1)𝑠1 2 + (𝑛2 − 1)𝑠2 2 𝑛1 + 𝑛2 − 2 (8) research method the steps of this research are as follows: 1. collecting data a primary data was collected through questionnaires for 4 latent variables to measure the effect of performance expectancy, effort expectancy and social influence on behavioral intention using e-learning with gender as a moderator variable. sample size in this research is 52 males and 48 female students from mercubuana university. multigroup moderation test in generalized structured component analysis angga dwi mulyanto 83 2. gsca the second step is structural modeling with gsca using gesca software. the user can access www.sem-gesca.org to run gsca. 3. bartlett test after obtained the results of gsca, the third step is compare standard deviation group 1 and group 2 using equation 5. if standard deviation group 1 equal with standard deviation group 2 (p-value ≥ 0.05) we can use t test in equation 1, and if standard deviation group 1 unequal with standard deviation group 2 (p-value < 0.05) we can use t test in equation 2. 4. multigroup moderation test the last step is multigroup moderation test using t test according the results of bartlett test. if p-value ≥ 0.05 then the effect of group 1 similar with the effect of group 2, else if p-value < 0.05 then the effect of group 1 difference with the effect of group 2. collecting d ata gs ca us ing gesca bartlett test (equati on 5) equa l standard deviation unequal sta ndard deviation t test (equati on 1) t test (equati on 2) start end figure 1. steps of research model of this research is the part of unified theory of acceptance and use of technology (utaut) can see in figure 2 [6]. http://www.sem-gesca.org/ multigroup moderation test in generalized structured component analysis angga dwi mulyanto 84 performance expectancy effort expectancy social influence behavioral intention gender figure 2. research model results and discussion the first step is generalized structured component analysis with gesca software. the result of gsca can see in table 2. table 2. result of gsca path group 1 group 2 estimate se cr estimate se cr performance expectancy > behavioral intention 0.205 0.183 1.12 0.038 0.22 0.17 effort expectancy > behavioral intention 0.19 0.161 1.18 0.122 0.188 0.65 social influence > behavioral intention 0.379 0.173 2.19* 0.539 0.176 3.06* cr* = significant at 0.05 level as we can see from table 2, performance expectancy and effort expectancy has no effect on behavioral intention both male and female. only social influence has effect on behavioral intention both male and female. after get the result of gsca (estimate path coefficient and standard error of two groups), we can proceed to second step. the second step is bartlett test (equation 5) to know standard deviations are equal or unequal. table 3. result of bartlett test path bartlett test 𝜒2 p-value information performance expectancy > behavioral intention 0.994875 0.318554 standard deviation equal multigroup moderation test in generalized structured component analysis angga dwi mulyanto 85 effort expectancy > behavioral intention 0.632093 0.426589 standard deviation equal social influence > behavioral intention 0.027056 0.869347 standard deviation equal as we can see from table 3 that both groups have the same standard deviation in all paths, so for multigroup moderation test we can use equation 1. table 4. result of t test (multigroup moderation test) path t test t p-value information performance expectancy > behavioral intention 0.592964 0.554571 both group have the same effect effort expectancy > behavioral intention 0.278817 0.780973 both group have the same effect social influence > behavioral intention 0.654298 0.514452 both group have the same effect based on table 4, we can say that there is no difference between male and female in all paths or in other word we can say that gender is not a moderator variable that moderate the effect of performance expectancy, effort expectancy and social influence on behavioral intention using e-learning. conclusion t test only requires estimate path coefficient, sample size and standard error of each group, so it can be used as an alternative test to compare the effect between group in generalized structured component analysis. limitations of the t test can only be used to compare the 2 groups. the results of this research indicate difference with utaut, male and female have similar effect and only social influences which have an effect on behavioral intention on e-learning. to increases behavioral intention using e-learning in male or female student, mercu buana university need more social approach to students. sosial approach can through persuasion from lecturer to student or student to student. reference [1] h. hwang and y. takane, "generalized structured component analysis," psychometrika, vol. 69, no. 1, pp. 81-99, 2004. [2] h. hwang, gesca user’s manual, online user manual, 2011. [3] m. sarstedt, j. henseler and m. christian, "multigroup analysis in partial least squares (pls) path modeling: alternative methods and empirical results," measurement and research methods in international marketing, advances in international marketing, vol. 22, pp. 195-218, 2011. [4] w. w. chin, frequently asked questions – partial least squares and pls-graph, online user manual, 2000. [5] nist/sematech, e-handbook of statistical methods, online user manual, 2012. [6] v. venkatesh, m. g. morris, g. b. davis and f. d. davis, "user acceptance of information technology: toward a unified view," mis quarterly, pp. 425-478, 2003. abstract introduction theoritical review research method results and discussion conclusion reference suma’inna dan gugun gumilar analisis dinamik sudut defleksi pada model vibrasi dawai 1imam mufid, 2ari kusumastuti, 3fachrur rozi 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang 3jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: imam.mufid@outlook.com abstrak model mckenna menggambarkan gerak vertikal dawai dan gerak torsi pada balok yang digantungnya. sudut defleksi merupakan sudut yang terbentuk pada gerak torsi. dengan mengasumsikan bahwa dawai tidak pernah kehilangan ketegangan, maka diperoleh sistem tak berpasangan yang dapat dianalisis secara terpisah. pada artikel ini ditunjukkan pengaruh faktor eksternal terhadap kestabilan dan perilaku dari model sudut defleksi. untuk mengetahui kestabilan dan perilaku dari sudut defleksi digunakan analisis sistem dinamik. berdasarkan hasil analisis, dengan menambahkan massa dari balok yang digantung akan diperoleh tiga solusi yang berbeda berdasarkan nilai eigennya. faktor eksternal memiliki pengaruh terhadap kestabilan dan perubahan besarnya sudut yang terbentuk pada model sudut defleksi, hal ini disebabkan setelah ditambahkan faktor ekstern medan vektor dari model bergerak secara tidak beraturan atau bersifat chaotic. kata kunci: sistem dinamik, model mckenna, sistem tak homogen, sudut defleksi abstract the mckenna is model describes vertical motion on string and torque motion on the beam that is hunged by string. deflection is an angle formed at the torque motion. assuming that string never lost tension, so an unpaired system that can be analyzed separately is obtained. in this article we show the influence of external factor on stability and behaviour of deflection angle of the models. to know the stability and behaviour of the deflection angle model, we used dynamic system analysis. based on the analysis, by adding mass of the beam which is suspended will be obtained three different solutions based on it’s eigenvalues. external factors have an influence on the stability and evolution of angle formed on the deflection angle models, this is due to external factors after adding the vector field of the model moves irregularly or is chaotic. keywords: dynamical system, mckenna model, nonhomogeneous system, deflection angle pendahuluan banyak fenomena alam yang terjadi yang dapat dianalisis dengan menggunakan model matematika, seperti getaran dan gelombang. vibrasi merupakan fenomena alam yang sering terjadi. sehingga akan memberikan manfaat yang besar jika memperlajari fenomena vibrasi ini. salah satu model matematiika yang menggambarkan vibrasi adalah model mckenna yang merepresentasikan sistem gerak yang terdiri dari gerak vertikal dan gerak torsi. berikut adalah model mckenna yang menggambarkan dinamika pergerakan torsi vertikal yang dinyatakan dalam: �̈� = 3𝐾 cos(𝜃) 𝑚𝑙 [(𝑦 − 𝑙 sin(𝜃))+ − (𝑦 + 𝑙 sin(𝜃))+] − 𝛿1�̇� + 𝑓(𝑡) dan �̈� = −𝐾 𝑚 [(𝑦 − 𝑙 sin(𝜃))+ + (𝑦 + 𝑙 sin(𝜃))+] − 𝛿2�̇� + 𝑔, dengan 𝜃(𝑡) merupakan sudut dari horizontal pada dawai pada waktu tertentu yang disebut dengan sudut defleksi. sedangkan 𝑦(𝑡) menyatakan dinamika pergerakan vertikal dawai. sudut defleksi pada skripsi ini menggambarkan gerak torsi dari dawai yang dibebani balok. model dari sudut defleksi adalah salah satu model dinamik karena modelnya bergantung waktu dan dapat merepresentasikan skenario yang dapat berubah sepanjang waktu. tujuan dari analisis dinamik adalah untuk menilai perilaku struktural dalam berbagai beban setiap waktu. berdasarkan teorema hartman-grobman, perilaku dari persamaan tak linier ekuivalen dengan persamaan hasil linierisasinya di sekitar titik kesetimbangan jika titik tetap tersebut hyperbolic. titik kesetimbangan dikatakan hyperbolic jika nilai eigen dari hasil linierisasi di sekitar titik tersebut memiliki bagian real tak nol [1]. karena model mckenna adalah persamaan diferensial tak linier, maka untuk dapat menganalisis model tersebut dilakukan pendekatan dengan melinierisasi model tersebut disekitar titik kesetimbangannya. mailto:imam.mufid@outlook.com analisis dinamik sudut defleksi pada model vibrasi dawai cauchy – issn: 2086-0382 9 artikel ini merupakan upaya ilmiah untuk menganalisis perilaku dari perubahan sudut defleksi pada vibrasi dawai dalam kerangka penelitian pengembangan. skripsi ini dilakukan analisis sistem dinamik terhadap model tersebut, sehingga dapat diketahui respon yang terjadi dengan berbagai parameter yang diberikan dan juga dapat diketahui seberapa besar pengaruh dari faktor eksternal. kajian teori 1. linierisasi fungsi 𝑓(𝑥1,𝑥2) dapat diekspansi dengan menggunakan deret taylor di sekitar titik kesetimbangan (𝑥10,𝑥20), sebagai berikut: 𝑓(𝑥1,𝑥2) = 𝑓(𝑥10,𝑥20)+ 𝜕𝑓 𝜕𝑥1 | 𝑥1=𝑥10 𝑥2=𝑥20 ∙ (𝑥1 − 𝑥10) + 𝜕𝑓 𝜕𝑥2 | 𝑥1=𝑥10 𝑥2=𝑥20 ∙ (𝑥2 − 𝑥20) + 1 2! ( 𝜕2𝑓 𝜕𝑥1 2|𝑥1=𝑥10 𝑥2=𝑥20 ∙ (𝑥1 − 𝑥10)+ 𝜕2𝑓 𝜕𝑥2 2|𝑥1=𝑥10 𝑥2=𝑥20 ∙ (𝑥2 − 𝑥20) 2 + 𝜕2𝑓 𝜕𝑥1𝜕𝑥2 | 𝑥1=𝑥10 𝑥2=𝑥20 ∙ (𝑥1 − 𝑥10)(𝑥2 − 𝑥20)) + ⋯. [1] 2. sistem linier pada dasarnya setiap persamaan diferensial orde 𝑛 dapat diubah menjadi sistem dengan 𝑛 buah persamaan. adapun contoh sistem linier adalah sebagai berikut. diberikan persamaan diferensial linier orde dua: �̈� = −𝑘𝑥 − 𝑏�̇� (1) misal 𝑥1 = 𝑥 dan 𝑥2 = �̇�, sehingga persamaan (1) dapat ditulis sebagai sistem persamaan diferensial linier orde satu. �̇�1 = 𝑥2 �̇�2 = −𝑘𝑥1 − 𝑏𝑥2 (2) [2]. diberikan sistem sebagi berikut: �̇� = 𝑨𝒙 (3) misal matriks 𝑨 dari sistem (3) merupakan matriks berukuran 2 × 2 dan jika sistem (3) mempunyai nilai-nilai eigen yang berbeda, maka solusi umum sistem (3) adalah: 𝒙(𝑡) = 𝐶1𝒗𝑒 𝜆1𝑡 + 𝐶2𝒖𝑒 𝜆2𝑡 (4) dengan 𝒗 dan 𝒖 adalah vektor-vektor eigen yang bersesuaian dengan nilai eigen 𝜆1 dan 𝜆2 [3]. jika persamaan karakteristik (3) mempunyai akar kembar 𝜆1,2 = 𝜆 , dan diperoleh: (a) dua vektor eigen, maka solusinya adalah: 𝒙(𝑡) = 𝐶1𝒗𝑒 𝜆1𝑡 + 𝐶2𝒖𝑒 𝜆1𝑡 (5) (b) satu vektor eigen, maka solusinya adalah: 𝒙(𝑡) = 𝐶1𝒗𝑒 𝜆𝑡 + 𝐶2(𝒗𝑡 − 𝒘)𝑒 𝜆𝑡 (6) dengan (𝑨 − 𝑰)𝒘 = 𝒗 [3]. jika 𝜆1,2 = 𝑎 ± 𝑖𝑏 merupakan akar kompleks dari (3) dan 𝒗1,2 = 𝒖 ± 𝑖𝒘 adalah vektor eigen yang bersesuaian, maka: 𝒙(𝑡) = 𝑒𝑎𝑡𝐶1(cos(𝑏𝑡)𝒖 − sin(𝑏𝑡)𝒘) + 𝑒𝑎𝑡𝐶2(sin(𝑏𝑡)𝒖 + cos(𝑏𝑡)𝒘) (7) adalah masing-masing solusi dari �̇� = 𝑨𝒙 [2]. 3. sistem tak homogen sistem persamaan diferensial linier tak homogen secara umum dapat dituliskan sebagai: �̇� = 𝑨(𝑡)𝒙+ 𝒈(𝑡) (8) solusi 𝒙(𝑡) dari persamaan diferensial linier tak homogen (8) dengan kondisi awal 𝒙(0) = 𝒙0, dapat ditulis sebagai: 𝒙(𝑡) = 𝑒𝑨𝑡 (𝒙0 + ∫ 𝑒 𝑨𝑡𝒈(𝑠) 𝒕 𝟎 𝑑𝑠) [2]. 4. model mckenna penurunan persamaan vibrasi merambat pada dawai mengikuti energi potensial dari dawai dengan konstanta spring 𝐾 dan merentang sejauh 𝑥 dari titik kesetimbangan. sehingga diperoleh: 𝐸𝑃𝑑𝑎𝑤𝑎𝑖 = ∫𝐾𝑥𝑑𝑥 = 1 2 𝐾𝑥2 dengan demikian energi potensial total adalah: 𝐸𝑃𝑑𝑎𝑤𝑎𝑖 𝑡𝑜𝑡𝑎𝑙 = 1 2 𝐾(((𝑦 − 𝑙sin(𝜃))+)2 − ((𝑦 + 𝑙sin(𝜃))+)2) imam mufid, ari kusumastuti 10 volume 3 no. 4 mei 2015 energi potensial 𝐸𝑃𝑏𝑎𝑙𝑜𝑘 karena beban dari balok dengan massa 𝑚 yang mengalami perubahan posisi ke bawah dari titik kesetimbangan dengan jarak 𝑦, diberikan sebagai berikut: 𝐸𝑃𝑏𝑎𝑙𝑜𝑘 = −𝑚𝑦𝑔 dimana 𝑔 adalah gaya gravitasi. sehingga diperoleh energi potensial model dari dawai dan balok yaitu: 𝐸𝑃𝑚 = 𝐸𝑃𝑑𝑎𝑤𝑎𝑖 𝑡𝑜𝑡𝑎𝑙 + 𝐸𝑃𝑏𝑎𝑙𝑜𝑘 𝐸𝑃𝑚 = 𝐾 2 ([(𝑦 − 𝑙sin(𝜃))+]2 − [(𝑦 + 𝑙sin(𝜃))+]2) − 𝑚𝑦𝑔 kemudian dilanjutkan untuk menemukan energi kinetik total, untuk pergerakan vertikal energi kinetik dari pusat massa balok adalah: 𝐸𝐾𝑣𝑒𝑟𝑡𝑖𝑘𝑎𝑙 = 1 2 𝑚�̇�2 dimana �̇� adalah kecepatan dari berat balok, dan persamaan untuk energi kinetik dari gerak torsi yaitu: 𝐸𝐾𝑡𝑜𝑟𝑠𝑖 = 1 6 𝑚𝑙2�̇� dimana �̇� adalah kecepatan dari perubahan sudut. dengan demikian, energi kinetik total diberikan sebagai berikut: 𝐸𝐾𝑚 = 𝐸𝐾𝑣𝑒𝑟𝑡𝑖𝑘𝑎𝑙 + 𝐸𝐾𝑡𝑜𝑟𝑠𝑖 𝐸𝐾𝑚 = 1 2 𝑚�̇�2 + 1 6 𝑚𝑙2�̇�2. sekarang diperoleh lagrangian sebagai berikut: 𝐿 = 𝐸𝐾𝑚 − 𝐸𝑃𝑚 𝐿 = 1 2 𝑚�̇�2 + 1 6 𝑚𝑙2�̇�2 − 𝐾 2 ([(𝑦 − 𝑙sin(𝜃))+]2 + [(𝑦 + 𝑙 sin(𝜃))+]2) + 𝑚𝑦𝑔. berdasarkan pada asas least action, gerakan balok memenuhi persamaan euler-lagrange. 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕�̇� ) − 𝜕𝐿 𝜕𝜃 = 0 dan 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕�̇� ) − 𝜕𝐿 𝜕𝑦 = 0 hasil diperoleh dengan mengevaluasi turunan yang diperlukan pada persamaan euler-lagrange, menambahankan redaman 𝛿1�̇� dan dan 𝑓(𝑡), serta 𝛿2�̇�. sehingga diperoleh: { �̈� = 3𝐾 𝑚𝑙 cos(𝜃)[(𝑦 − 𝑙sin(𝜃))+ − (𝑦 + 𝑙sin(𝜃))+] −𝛿1�̇� + 𝑓(𝑡) �̈� = − 𝐾 𝑚 [(𝑦 − 𝑙sin(𝜃))+ + (𝑦 + 𝑙sin(𝜃))+] − 𝛿2�̇� +𝑔 (9) sistem persamaan di atas merupakan model vibrasi dawai yang diusulkan oleh mckenna [4] pembahasan model sudut defleksi pada skripsi ini adalah model yang menggambarkan gerak torsi dari balok yang digantung oleh dua buah dawai. model yang menggambarkan sistem gerak ini adalah model mckenna. sistem gerak yang digambarkan pada model mckenna adalah gerak vertikal pada dawai dan gerak torsi pada balok. ilustrai sistem gerak dari model mckenna akan ditunjukkan pada gambar 1. gambar 1. ilustrasi model mckenna dengan mengasumsikan bahwa dawai tidak pernah kehilangan ketegangan, maka dimiliki 𝑦 ± 𝑙sin(𝜃) ≥ 0 dan (𝑦 ± 𝑙sin(𝜃))+ = 𝑦 ± 𝑙sin(𝜃). sehingga dengan mensubstitusikan (𝑦 ± 𝑙sin(𝜃))+ = 𝑦 ± 𝑙sin(𝜃) pada sistem persamaan (9) dan setelah disederhanakan, maka diperoleh sistem persamaan diferensial sebagai berikut: { �̈� = − 3𝐾 𝑚 sin(2𝜃) − 𝛿1�̇� + 𝑓(𝑡) �̈� = − 2𝐾 𝑚 𝑦 − 𝛿2�̇� + 𝑔 (10) dengan 𝑓(𝑡) = 𝛽sin(𝜇𝑡) berikut adalah persamaan sudut defleksi yang menggambarkan gerak torsi. �̈� = − 3𝐾 𝑚 sin(2𝜃) − 𝛿�̇� + 𝛽 sin(𝜇𝑡) (11) 𝑦(𝑡) titik kesetimbanga n 𝑙 𝑦 −𝑙 sin(𝜃) 𝑦 + 𝑙sin(𝜃) 𝜃(𝑡) analisis dinamik sudut defleksi pada model vibrasi dawai cauchy – issn: 2086-0382 11 misal : 𝜃1 = 𝜃 dan 𝜃2 = �̇�, dengan menurunkan 𝜃1 dan 𝜃2 terhadap 𝑡 maka akan diperoleh 𝜃1̇ = �̇� dan 𝜃2̇ = �̈�. akibatnya persamaan (11) berubah menjadi sistem sebagai berikut: { �̇�1 = 𝜃2 �̇�2 = − 3𝐾 𝑚 sin(2𝜃1) − 𝛿𝜃2 + 𝛽sin(𝜇𝑡) (12) dari hasil perhitungan di atas diperoleh titik kesetimbangan (𝜃1 ∗ ,𝜃2 ∗) = (𝑛𝜋,0) dengan 𝑛 = 0,1,2,3,…. untuk melakukan ekspansi taylor cukup dibutuhkan satu titik kesetimbangan saja, maka dipilih (𝜃1 ∗ ,𝜃2 ∗) = (0,0). sehingga persamaan (12) menjadi: { 𝜃1̇ = 𝜃2 𝜃2̇ = − 6𝐾 𝑚 𝜃1 − 𝛿𝜃2 + 𝛽sin(𝜇𝑡) (13) untuk menentukan solusi dari sistem (13), terlebih dahulu diabaikan bentuk tak homogennya yaitu 𝛽sin(𝜇𝑡). untuk menentukan solusi homogennya terlebih dahulu harus ditentukan nilai eigennya, yang diperoleh: 𝜆1,2 = −𝛿 ± √𝛿2 − 24𝐾 𝑚 2 karena nilai 𝛿2 − 24𝐾 𝑚 dapat bernilai nol, positif, maupun negatif berakibat solusi homogen memiliki tiga kemungkinan berdasarkan nilai eigennya. setelah diperoleh solusi homogennya dengan menggunakan metode variasi parameter maka diperoleh solusi sistem (13) sebagai berikut: (a) nilai eigen real berbeda, 𝛿2 − 24𝐾 𝑚 > 0 𝜃1(𝑡) = 𝐶1𝑒 −𝛿+𝑃 2 𝑡 + 𝐶2𝑒 −𝛿−𝑃 2 𝑡 + 4𝛽 ( (𝛿2 − 𝑃2 − 4𝜇2)sin(𝜇𝑡) − 4𝜇𝛿cos(𝜇𝑡) (4𝜇2 + (𝛿 − 𝑃)2)(4𝜇2 + (𝑃 + 𝛿)2) ) 𝜃2(𝑡) = 𝐶1 𝑃 − 𝛿 2 𝑒 𝑃−𝛿 2 𝑡 + 𝐶2 −𝛿 − 𝑃 2𝑒 𝛿+𝑃 2 𝑡 + 4𝛽 ( (𝛿2 − 𝑃2 − 4𝜇2)𝜇cos(𝜇𝑡) + 4𝛿𝜇2 sin(𝜇𝑡) (4𝜇2 + (𝛿 − 𝑃)2)(4𝜇2 + (𝑃 + 𝛿)2) ) (14) dengan 𝑃 = √𝛿2 − 24𝐾 𝑚 , dan 𝐶1 = (𝑃 + 𝛿)𝜃1(0) + 2𝜃2(0) 2𝑃 − 4𝛽𝜇 ( 24𝐾 𝑚 − 4𝜇2) + 8𝛽𝜇𝛿(𝑃 − 𝛿) 𝑃(4𝜇2 + (𝛿 − 𝑃)2)(4𝜇2 + (𝑃 + 𝛿)2) 𝐶2 = − 2𝜃2(0) − (𝑃 − 𝛿)𝜃1(0) 2𝑃 + 4𝛽𝜇 ( 24𝐾 𝑚 − 4𝜇2) + 8𝛽𝜇𝛿(𝑃 − 𝛿) 𝑃(4𝜇2 + (𝛿 − 𝑃)2)(4𝜇2 + (𝑃 + 𝛿)2) (b) nilai eigen real kembar 𝛿2 − 24𝐾 𝑚 = 0 𝜃1(𝑡) = 𝑒 − 𝛿 2 𝑡(𝐶1 + 𝐶2) + 𝛽𝑡 𝜇2 + 𝛿2 4 ( 𝛿 2 sin(𝜇𝑡) − 𝜇cos(𝜇𝑡)) 𝛽( (4𝜇𝑡(𝛿2 + 4𝜇2) − 16𝜇𝛿)cos(𝜇𝑡) (𝛿2 + 4𝜇2)2 − (8𝜇2(2 + 𝛿𝑡) + 2𝛿3(𝑡 − 2))sin(𝜇𝑡) (𝛿2 + 4𝜇2)2 ) 𝜃2(𝑡) = 𝑒 − 𝛿 2 𝑡 ( −𝛿 2 𝐶1 + 2 − 𝛿𝑡 2 𝐶2) + 𝛽 2 − 𝛿𝑡 2(𝜇2 + 𝛿2 4 ) ( 𝛿 2 sin(𝜇𝑡)− 𝜇cos(𝜇𝑡)) − 𝛿𝛽 2 ( (4𝜇𝑡(𝛿2 + 4𝜇2) − 16𝜇𝛿) (𝛿2 + 4𝜇2)2 cos(𝜇𝑡)− (8𝜇2(2+ 𝛿𝑡) + 2𝛿3(𝑡 − 2))sin(𝜇𝑡) (𝛿2 + 4𝜇2)2 ) (15) dengan: 𝐶1 = 𝜃1(0)+ ( 16𝜇𝛿 (𝛿2 + 4𝜇2)2 )𝛽 𝐶2 = 𝜃2(0)+ 𝛿 2 𝜃1(0) + 𝛽( 4𝜇 𝛿2 + 4𝜇2 + 8𝜇𝛿2 (𝛿2 + 4𝜇2)2 ) (c) nilai eigen kompleks 𝛿2 − 24𝐾 𝑚 < 0 𝜃1(𝑡) = 𝑒 − 𝛿 2 𝑡 (𝐶1 cos( 𝑄 2 𝑡) + 𝐶2 sin( 𝑄 2 𝑡)) + 𝛽cos( 𝑄 2 𝑡) 𝑄 ( 𝛿 2 cos(( 𝑄 2 + 𝜇)𝑡) + ( 𝑄 2 + 𝜇)sin(( 𝑄 2 + 𝜇)𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 + 𝜇) 2 − 𝛿 2 cos(( 𝑄 2 − 𝜇)𝑡) + ( 𝑄 2 − 𝜇)sin(( 𝑄 2 − 𝜇)𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 − 𝜇) 2 ) + 𝛽 𝑄 sin( 𝑄𝑡 2 )( 𝛿 2 sin(( 𝑄 2 + 𝜇)𝑡) − ( 𝑄 2 + 𝜇)cos(( 𝑄 2 + 𝜇)𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 + 𝜇) 2 imam mufid, ari kusumastuti 12 volume 3 no. 4 mei 2015 − 𝛿 2 sin(( 𝑄 2 − 𝜇)𝑡) − ( 𝑄 2 − 𝜇)cos(( 𝑄 2 − 𝜇)𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 − 𝜇) 2 ) 𝜃1(𝑡) = 𝑒 − 𝛿 2 𝑡 (𝐶1 ( −𝛿 2 cos( 𝑄 2 𝑡) − 𝑄 2 sin( 𝑄 2 𝑡)) + 𝐶2 ( −𝛿 2 sin( 𝑄 2 𝑡) + 𝑄 2 cos( 𝑄 2 𝑡)) + 𝛽 𝑄 ( −𝛿 2 cos( 𝑄 2 𝑡) − 𝑄 2 sin( 𝑄𝑡 2 ))( 𝛿 2 cos( 𝑄+2𝜇 2 𝑡) + 𝑄+2𝜇 2 sin( 𝑄+2𝜇 2 𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 + 𝜇) 2 𝛿 2 cos( 𝑄−2𝜇 2 𝑡) + 𝑄−2𝜇 2 sin( 𝑄−2𝜇 2 𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 − 𝜇) 2 ) + 𝛽 𝑄 ( −𝛿 2 sin( 𝑄 2 𝑡) + 𝑄 2 cos( 𝑄 2 𝑡))( 𝛿 2 sin( 𝑄+2𝜇 2 𝑡) − 𝑄+2𝜇 2 cos( 𝑄+2𝜇 2 𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 + 𝜇) 2 − 𝛿 2 sin(( 𝑄 2 − 𝜇)𝑡) − ( 𝑄 2 − 𝜇)cos(( 𝑄 2 − 𝜇)𝑡) ( 𝛿 2 ) 2 + ( 𝑄 2 − 𝜇) 2 ) (16) dengan 𝑄 = √ 24𝐾 𝑚 − 𝛿2 , 𝐶1 = 𝜃1(0) − 𝛽 𝑄 ( 2𝛿 𝛿2 + (𝑄 + 2𝜇)2 − 2𝛿 𝛿2 + (𝑄 − 2𝜇)2 ) 𝐶2 = 2𝜃2(0) + 𝛿𝜃1(0) 𝑄 − 𝛽 𝑄2 ( 2𝑄2 + 4𝜇𝑄 𝛿2 + (𝑄 + 2𝜇)2 − 2𝑄2 − 4𝜇𝑄 𝛿2 + (𝑄 − 2𝜇)2 ) untuk melihat perilaku dari model sudut defleksi diberikan parameter-parameter sebagai berikut: tabel 1. daftar parameter simbol keterangan 𝑚 massa per satuan panjang balok (kgs2/m) 𝑚 = 657,3 𝐾 konstanta spring (kg/m), 𝐾 = 3,75 𝑙 panjang balok 𝑙 = 60 𝛿 viscous dumping bernilai 0,01 𝜇 konstanta antara 1,2 sampai 1,6 𝛽 amplitudo berkisar antara 0,02 sampai 0,06 sumber: data fiktif, hanya untuk ilustrasi dari data yang diberikan dapat diketahui solusi mana yang akan digunakan dengan cara, subtitusikan parameter ke dalam persamaan berikut 𝛿2 − 24𝐾 𝑚 , diperoleh 𝛿2 − 24𝐾 𝑚 < 0. sehingga solusi yang digunakan adalah persamaan (16). dengan mensubstitusikan parameterparameter dari tabel 1 ke persamaan (16) maka diperoleh grafik sebagai berikut: (a) (b) gambar 2. (a) grafik solusi 𝜃1(𝑡) (b) grafik solusi 𝜃2(𝑡) untuk melihat pengaruh faktor eksternal 𝑓(𝑡) berikut ditunjukkan grafik solusi bentuk homogennya tanpa faktor eksternal. gambar 3. grafik solusi 𝜃1 adapun perilaku dari model dapat dilihat pada medan vektor dan trayekori sebagai berikut: (a) (b) gambar 4. (a) medan vektor model sudut defleksi (b)trayektori solusi 𝜃1(𝑡) dan 𝜃2(𝑡) dari gambar 5 dan gambar 6 menunjukkan bahwa model sudut defleksi tidak stabil, diketahui model sebelum ditambah faktor eksternal memiliki kestabilan jenis spiral sink artinya stabil asimtotik, dan setelah ditambah faktor eksternal menjadikan model sudut defleksi menjadi tidak stabil. analisis dinamik sudut defleksi pada model vibrasi dawai cauchy – issn: 2086-0382 13 gambar 5. potret fase model sudut defleksi sebelum ditambah kesimpulan faktor eksternal 𝑓(𝑡) memiliki dampak yang besar terhadap kestabilan model sudut defleksi. model sudut defleksi tanpa faktor eksternal selalu stabil, tetapi, jika ditambah dengan faktor eksternal grafik solusi menjadi sangat fluktuatif di sekitar titik tetap dan potret fase menunjukkan bentuk yang sangat tidak beraturan atau bersifat chaotic. untuk mendapatkan interpretasi yang lebih baik perlu dilakukan analisis kembali dengan memperhatikan gaya yang terjadi pada dawai yaitu 𝐾(𝑦 ± 𝑙sin(𝜃))+. references [1] g. vries, t. hillen, m. lewis, j. müller and m. schönfisch, a course in mathematical biology: quantitative modeling with mathematical and computational methods, alberta: siam, 2006. [2] a. g. parlos, "linearization of nonlinear dynamics," 2014. [online]. available: http://parlos.tamu.edu/meen651/lineariza tion.pdf. [accessed 17 agustus 2014]. [3] r. c. robinson, an introduction dynamical system continuous and discrete, new jersey: pearson education, 2004. [4] w. e. boyce and r. c. diprima, elementary differential equations and boundary value problems, new york: john wiley & sons, inc, 2009. [5] r. o. kwofie, "a mathematical model of a suspension bridge – case study: adomi bridge, atimpoku, ghana," global advanced research journal of engineering, technology, and innovation, vol. 1(3), pp. 047-062, 2012. pendahuluan kajian teori 1. linierisasi 2. sistem linier 3. sistem tak homogen 4. model mckenna pembahasan kesimpulan references suma’inna dan gugun gumilar analisis teoritis dan empiris uji craps dari diehard battery of randomness test untuk pengujian pembangkit bilangan acaksemu sari agustini hafman1 dan arif fachru rozi2 1,2 lembaga sandi negara e-mail: sari.hafman@lemsaneg.go.id1, arif.fachru@lemsaneg.go.id2 abstrak menurut kerchoffs (1883), keamanan sistem kriptografi harus hanya bergantung pada kunci yang digunakan dalam sistem tersebut. umumnya, kunci dihasilkan oleh pseudo random number generator (prng) atau random number generator (rng). terdapat tiga tipe keacakan yang dihasilkan oleh prng dan rng yaitu pseudorandom sequence (barisan acaksemu), cryptographically secure pseudorandom sequences (barisan acaksemu yang aman secara kriptografi) dan real random sequences (barisan yang acak nyata). untuk memeriksa tipe keacakan yang dihasilkan oleh suatu prng atau rng digunakan berbagai uji statistik, diantaranya diehard battery of test of randomness. mengingat tujuan, dasar pengambilan parameter pengujian serta proses pembentukan statistik uji berhubungan dengan valid atau tidaknya kesimpulan yang dihasilkan dari suatu uji statistik maka dilakukan kajian terhadap salah satu uji yang terdapat dalam diehard battery of randomness test yaitu uji craps. uji yang terinspirasi dari permainan craps ini bertujuan untuk memeriksa apakah suatu prng menghasilkan barisan acaksemu yang berdistribusi identik dan independen (iid). untuk menunjukkan proses pembentukan serta penerapan permainan craps pada statistik uji craps, dilakukan analisis teoritis dengan menerapkan berbagai teori statistik terhadap uji tersebut. selain itu, dilakukan observasi secara empiris dengan menerapkan uji craps pada beberapa prng dengan tujuan untuk memeriksa keefektifan uji tersebut dalam mendeteksi bentuk distribusi dan independensi barisan yang dihasilkan suatu prng. kata kunci: identik dan independen (iid), permainan craps, pseudo random number generator (prng), uji craps abstract according to kerchoffs (1883), the security system should only rely on cryptographic keys which is used in that system. generally, the key sequences are generated by a pseudo random number generator (prng) or random number generator (rng). there are three types of randomness sequences that generated by the rng and prng i.e. pseudorandom sequence, cryptographically secure pseudorandom sequences, and real random sequences. several statistical tests, including diehard battery of tests of randomness, is used t o check the type of randomness sequences that generated by prng or rng. due to its purpose, the principle on taking the testing parameters and the test statistic are associated with the validity of the conclusion produced by a statistical test, then the theoretical analysis is performed by applying a variety of statistical theory to evaluate craps test, one of the test included in the diehard battery of randomness tests. craps test, inspired by craps game, aims to examine whether a prng produces an independent and identically distributed (iid) pseudorandom sequences. to demonstrate the process to produce a test statistics equation and to show how craps games applied on that test, will be carried out theoretical analysis by applying a variety of statistical theory. furthermore, empirical observations will be done by applying craps test on a prng in order to check the test effectiveness in detecting the distribution and independency of sequences which produced by prng. keywords: craps games, craps test, independent and identically distributed (iid), pseudo random number generator (prng) pendahuluan menurut kerchoffs (1883), keamanan sistem kriptografi harus hanya bergantung pada kunci yang digunakan dalam sistem tersebut. umumnya, kunci dihasilkan oleh pseudo random number generator (prng) atau random number generator (rng). terdapat tiga tipe keacakan yang dihasilkan oleh prng dan rng yaitu pseudorandom sequence (barisan acaksemu), cryptographically secure pseudorandom sequences (barisan acaksemu yang aman secara kriptografi) dan real random sequences (barisan yang acak nyata). barisan dikatakan acaksemu jika secara analisis teoritis dan empiris uji craps dari diehard battery of randomness test untuk pengujian … jurnal cauchy – issn: 2086-0382 217 statistik terlihat acak (berdistribusi seragam dan saling bebas). barisan dikatakan aman secara kriptografis bila barisan tersebut secara statistik terlihat acak serta unpredictable (ketidakterdugaan). barisan dikatakan acak nyata bila memenuhi tiga syarat yaitu barisan tersebut secara statistik terlihat acak, ketidakterdugaan dan barisan yang sama tidak dapat dihasilkan kembali (schneier, 1996). untuk memeriksa tipe keacakan yang dihasilkan oleh suatu prng atau rng pendekatan yang umum digunakan adalah membangkitkan barisan kunci dalam jumlah besar dang mengaplikasikan berbagai uji statistik pada barisan tersebut. uji statistik yang banyak digunakan diantaranya diehard battery of test of randomness. informasi yang diperoleh dari hasil pengujian keacakan hanya untuk membedakan barisan kunci tersebut dari barisan kunci yang acak nyata. uji tersebut dianggap sebagai uji yang menggunakan pendekatan black box karena tidak memperhitungkan struktur dari prng/rng yang digunakan untuk menghasilkan barisan tersebut. mengingat tujuan, dasar pengambilan parameter pengujian serta proses pembentukan statistik uji berhubungan dengan valid atau tidaknya kesimpulan yang dihasilkan dari suatu uji statistik maka dilakukan kajian terhadap salah satu uji yang banyak digunakan dalam pengujian keacakan barisan kunci yaitu uji craps yang terdapat dalam diehard battery of randomness. uji yang terinspirasi dari permainan craps ini bertujuan untuk memeriksa apakah suatu prng menghasilkan barisan acaksemu yang berdistribusi identik dan independen (iid) atau tidak. untuk menunjukkan proses pembentukan serta penerapan permainan craps pada statistik uji craps, dilakukan analisis teoritis dengan menerapkan berbagai teori statistik terhadap uji tersebut. selain itu, dilakukan observasi secara empiris dengan menerapkan uji craps pada beberapa prng dengan tujuan untuk memeriksa keefektifan uji tersebut dalam mendeteksi bentuk distribusi dan independensi barisan yang dihasilkan suatu prng. teori dasar distribusi normal distribusi normal adalah model distribusi kontinu yang penting dalam teori probabilitas. distribusi normal memiliki kurva berbentuk lonceng yang simetris. dua parameter yang menentukan distribusi normal adalah mean (𝜇) dan variansi (𝜎 2). fungsi kerapatan probabilitas dari distribusi normal adalah: 𝑓(𝑥) = 1 𝜎√2𝜋 𝑒 − (𝑥−𝜇)2 2𝜎2 𝜇 adalah rata-rata, 𝜎 2 adalah variansi dan 𝜋 = 3,14159 ⋯ teorema 1 : jika x adalah suatu peubah acak binom dengan mean 𝜇 = 𝑛𝑝 dan variansi 𝜎 2 = 𝑛𝑝𝑞 maka bentuk limit dari distibusi 𝑍 = 𝑋 − 𝑛𝑝 √𝑛𝑝𝑞 dengan 𝑛 → ∞ adalah berdistribusi 𝑁(0,1). distribusi chi-square variabel acak kontinu x mempunyai distribusi chi square dengan derajat bebas v jika fungsi kerapatannya adalah 𝑓(𝑥; 𝑣) = { 1 2 𝑣 2γ(𝑣 2 ) 𝑥 𝑣 2 −1𝑒 − 𝑥 2 0, lainnya , 𝑥 > 0 dengan v adalah integer positif. distribusi multinomial definisi 1. (soejati,2005) distribusi multinomial adalah distribusi peluang bersama frekuensi-frekuensi sel 𝑛1, ⋯ , 𝑛𝑘 dalam n trial multinomial dengan parameter 𝑝1, ⋯ , 𝑝𝑘 yang masing-masing merupakan peluang sel. fungsi peluang distribusi multinomial adalah 𝑓(𝑛1, ⋯ , 𝑛𝑘 ) = 𝑛! 𝑛1! ⋯ 𝑛𝑘 ! 𝑝1 𝑛1 ⋯ 𝑝𝑘 𝑛𝑘 untuk ∑ 𝑛𝑖 = 𝑛 𝑘 𝑖=1 . parameter-parameter itu memenuhi ∑ 𝑝𝑖 = 1 𝑘 𝑖=1 nilai ekspektasi dan variansi dari distribusi multinomial adalah 𝐸(𝑛𝑖 ) = 𝑛𝑝𝑖 dan 𝑉𝑎𝑟(𝑛𝑖 ) = 𝑛𝑝𝑖 (1 − 𝑝𝑖 ) dimana 𝑖 = 1,2, ⋯ , 𝑘. teorema 2. misalkan 𝑦1, ⋯ , 𝑦𝑘 berdistribusi multinomial dengan probabilitas 𝑝1, ⋯ , 𝑝𝑘 maka untuk n besar, variabel acak tidak negatif 𝜒2 = ∑ (𝑦𝑖−𝑛𝑝𝑖) 2 𝑛𝑝𝑖 𝑘 𝑖=1 dimana 𝑖 = 1,2, ⋯ , 𝑘 (1) mendekati distribusi chi-square dengan derajat bebas = 𝑘 − 1 dengan harga mean 𝜒2 adalah 𝜇 = 𝑘 − 1. sari agustini hafman dan arif fachru rozi 218 volume 2 no. 4 mei 2013 persamaan 1 pertama kali diperkenalkan dan dipelajari oleh karl pearson pada tahun 1900 sehingga dikenal dengan nama ”pearson’s chi square statistic”. harga mean 𝜒2 hanya tergantung pada banyak sel atau kelas k (banyak kemungkinan yang dapat terjadi pada eksperimen multinomial) dan tidak tergantung pada harga 𝑝𝑖 , 𝑖 = 1,2, ⋯ , 𝑘. bukti : 𝑚𝑒𝑎𝑛(𝜒2) = 𝐸(𝜒2) = ∑ 𝐸(𝑦𝑖 − 𝑛𝑝𝑖 ) 2 𝑛𝑝𝑖 𝑘 𝑖=1 = ∑ 𝑣𝑎𝑟(𝑦𝑖 ) 𝑛𝑝𝑖 𝑘 𝑖=1 = ∑ 𝑛𝑝𝑖 (1 − 𝑝𝑖 ) 𝑛𝑝𝑖 𝑘 𝑖=1 ∑ 1 − 𝑘 𝑖=1 𝑝𝑖 = ∑ 1 𝑘 𝑖=1 − ∑ 𝑝𝑖 𝑘 𝑖=1 = 𝑘 − 1 rumus transformasi 𝜒2 sering ditulis dengan persamaan 𝜒2 = ∑ (𝑦𝑖 − 𝑛𝑝𝑖 ) 2 𝑛𝑝𝑖 𝑘 𝑖=1 = ∑ (𝑂𝑖 − 𝐸𝑖 ) 2 𝐸𝑖 𝑘 𝑖=1 dengan 𝑂𝑖 = 𝑦𝑖 adalah frekuensi sel i yang diobservasi dalam sampel berukuran n, sedangkan 𝐸𝑖 = 𝑛𝑝𝑖 = 𝑚𝑒𝑎𝑛(𝑦𝑖 ) adalah mean atau frekuensi sel i yang diharapkan (nilai ekspektasi). uji chi-square goodness of fit teorema 3. uji chi-square goodness of fit antara frekuensi yang diobservasi dengan frekuensi yang diharapkan, berdasarkan pada ukuran 𝜒2 = ∑ (𝑂𝑖−𝐸𝑖) 2 𝐸𝑖 𝑘 𝑖=1 (2) dengan 𝜒2 adalah nilai variabel acak yang distribusi samplingnya hampir mendekati distribusi chi-square dengan derajat bebas 𝑣 = 𝑘 − 1, 𝑂𝑖 adalah frekuensi yang diobservasi dan𝐸𝑖 adalah frekuensi yang diharapkan untuk setiap sel ke-i. prosedur uji chi-square goodness of fit berdasarkan atas distribusi pendekatan maka prosedur ini sebaiknya tidak digunakan jika frekuensi harapan sangat kecil atau 𝑒𝑖 < 5. jika dalam proses perhitungan terdapat frekuensi harapan yang lebih kecil dari lima maka frekuensi tersebut dapat digabungkan dengan frekuensi yang lain supaya prosedur diatas terpenuhi (soejati, 1985). uji craps ide uji craps berasal dari permainan craps. craps merupakan suatu permainan yang melibatkan dua buah dadu yang dilakukan oleh seorang pemain atau lebih. craps dimainkan dalam babak (round) yang diatur dalam aturan permainan craps. berikut adalah aturan permainan craps. a. jika jumlah mata dadu 7 atau 11 pada lemparan pertama maka pelempar dinyatakan menang. b. jika jumlahnya 2,3, atau 12 maka pelempar kalah. c. jika jumlahnya 4,5,6,8,9 atau 10 maka pelempar dapat melanjutkan lemparannya sampai ia mendapatkan angka yang sama seperti lemparan pertama (dinyatakan menang) atau 7 (dinyatakan kalah). berdasarkan aturan permainan craps tersebut, marsaglia mengajukan uji craps yang terdiri dari dua buah uji statistik yaitu : a. uji untuk menganalisis jumlah kemenangan (uji 1) pada uji ini diasumsikan permainan dilakukan sebanyak n kali, minimal 200.000 kali. dari n kali permainan tersebut akan dihitung jumlah kemenangannya. banyaknya kemenangan harus mendekati distribusi normal dengan rataan 200.000p dan variansi 200.000p(1-p) dengan p = 244/495. b. uji untuk menganalisis banyaknya lemparan sampai permainan selesai (uji 2) pada uji ini akan dihitung banyaknya melakukan lemparan dadu yang dilakukan seorang pemain sampai permainannya selesai. banyaknya lemparan yang dilakukan oleh setiap pemain dapat bervariasi mulai dari 1 sampai tak hingga. tetapi pada uji ini meskipun pemain dapat melakukan lemparan lebih dari 21 kali, lemparan tetap dianggap sebanyak 21 kali sehingga jumlah lemparan hanya akan dikelompokkan kedalam 21 kelas. banyaknya lemparan harus berdistribusi chisquare dengan derajat bebas 20. pseudorandom number generator (prng) definisi 1 (schneier, 1996): prng adalah pembangkit barisan bilangan acaksemu, yang membutuhkan seed (input) dengan proses pembangkitan tiap elemen tergantung dari formulasi matematis yang digunakan pada prng tersebut. analisis teoritis dan empiris uji craps dari diehard battery of randomness test untuk pengujian … jurnal cauchy – issn: 2086-0382 219 proses pembangkitan tiap elemen dari prng memiliki hubungan linier sesuai fungsi matematis yang digunakan, sehingga untuk meminimalisir kelinierannya, digunakan fungsi non-linier dan pengaturan parameter inputnya. untuk memenuhi sifat unpredictable, pada umumnya prng menggunakan input berupa barisan bit acak yang berasal dari suatu rng. terdapat tiga tipe prg berdasarkan prinsip kerjanya yaitu : a. tipe linear tipe ini berdasarkan pada hubungan linear yang berulang yang digunakan untuk menghitung nilai selanjutnya dari nilai sebelumnya. salah satu contoh prng bertipe linear adalah multiply with carry generator (mwc). b. tipe shift register tipe ini mengambil beberapa nilai yang berurutan dari suatu multiple recursive generator (mrg) untuk mengkonstruksi output selanjutnya. contoh prng bertipe shift register adalah shift register generator 31 bit dan shift register generator 32 bit. c. tipe nonlinear tipe prng yang tidak menghasilkan struktur berpola dan menghasilkan output yang berlaku seperti barisan yang acak nyata hampir di seluruh periode. salah satu contoh prng bertipe nonlinear adalah inverse congruential generator (icg). metode penelitian penelitian ini terdiri atas dua tahap yaitu penelitian secara teoritis dan secara empiris. berikut penjelasan dari kedua metode tersebut. penelitian secara teoritis penelitian secara teoritis dilakukan terhadap statistik uji craps baik uji 1 maupun uji 2 dengan menerapkan berbagai teori statistik terhadap uji-uji tersebut penelitian secara empiris data yang digunakan pada penelitian ini merupakan data simulasi yang berasal dari 10 prng yang berasal dari tiga tipe prng. kesepuluh prng beserta parameternya seperti yang diperlihatkan pada tabel 1. jumlah data yang dibangkitkan oleh kesepuluh prng tersebut sebesar 11 mbyte (marsaglia,1985). tabel1. sepuluh prng dan parameternya tipe nama prng parameter linear multiply with carry (mwc) mwc1 a=1791398085 x = 191, c = 17 mwc2 a=1447497129 x=191, c=17 mwc3 a=2083801278 x=191, c=17 shift register shift register generator (srg) 31 bit srg311 l=13, r=18 srg312 l=18, r=3 srg313 l=24, r=7 shift register generator (srg) 32 bit srg321 shift register 17 dan 15 srg322 shift register 15 dan 17 srg323 shift register 13, 17 dan 5 nonlinear inverse congruential generator (icg) a = 9 b = 13 seed = 247 karena uji craps bertujuan untuk menguji sifat keacakan dari suatu prng maka sebelum menerapkan uji craps terlebih dahulu dilakukan pembangkitan barisan kunci sebesar 11 megabyte dari ketiga prng tersebut. kedua statistik uji craps digunakan untuk menghitung p-value. pvalue ini selanjutnya akan dibandingkan dengan tingkat kepercayaan (𝛼). tingkat kepercayaan yang digunakan dalam penelitian ini adalah 0,001. jika 𝑝 − 𝑣𝑎𝑙𝑢𝑒 ≥ 𝛼 maka hipotesis nol diterima atau barisan dikatakan acak. jika 𝑝 − 𝑣𝑎𝑙𝑢𝑒 < 𝛼 maka hipotesis nol ditolak atau barisan dikatakan tidak acak. pembahasan analisis teoritis pada tahap ini dilakukan analisis teoritis dengan menerapkan berbagai teori statistik terhadap proses pembentukan statistik uji pada kedua statistik uji yang terdapat dalam uji craps. hasil analisisnya adalah sebagai berikut. a. proses pembentukan statistik uji pada uji 1 misalkan : g adalah jumlah permainan, n adalah banyaknya lemparan, 𝑋𝑖 , 𝑌𝑖 adalah outcome pada lemparan ke-i, 𝑍𝑖 = 𝑋𝑖 + 𝑌𝑖 adalah jumlah score pada lemparan ke-i, i adalah indikator ketika menang yang dinyatakan dengan : 𝐼 = { 1, jika menang 0, jika kalah a adalah kejadian mendapat mata dadu berjumlah 7 atau 11 sari agustini hafman dan arif fachru rozi 220 volume 2 no. 4 mei 2013 b adalah kejadian mendapat mata dadu berjumlah 4 pada lemparan ke-1 dan ke-2 c adalah kejadian mendapat mata dadu berjumlah 5 pada lemparan ke-1 dan ke-2 d adalah kejadian mendapat mata dadu berjumlah 6 pada lemparan ke-1 dan ke-2 e adalah kejadian mendapat mata dadu berjumlah 8 pada lemparan ke-1 dan ke-2 f adalah kejadian mendapat mata dadu berjumlah 9 pada lemparan ke-1 dan ke-2 g adalah kejadian mendapat mata dadu berjumlah 10 pada lemparan ke-1 dan ke-2 probabilitas memperoleh mata dadu berjumlah 2,3,4,..., 12 pada lemparan pertama diperoleh sebagai berikut : 𝑃(𝑍1 = 2) = 𝑃(𝑍1 = 12) = 1 36 𝑃(𝑍1 = 3) = 𝑃(𝑍1 = 11) = 2 36 𝑃(𝑍1 = 4) = 𝑃(𝑍1 = 10) = 3 36 𝑃(𝑍1 = 5) = 𝑃(𝑍1 = 9) = 4 36 𝑃(𝑍1 = 6) = 𝑃(𝑍1 = 8) = 5 36 𝑃(𝑍1 = 7) = 6 36 (3) probabilitas jumlah kemenangan pada permainan craps diperoleh dari : 1) probabilitas memperoleh mata dadu bernilai 7 atau 11 pada lemparan pertama dari dua dadu 𝑃(𝐴) = 𝑃(𝑍1 = 7) + 𝑃(𝑍1 = 11) = 2 9 2) probabilitas memperoleh mata dadu bernilai sama (4,5,6,8,9 atau 10) baik pada lemparan pertama atau kedua 𝑃(𝐵) = 𝑃(𝑍1 = 4) ∙ 𝑃(𝐼 = 1|𝑍1 = 4) = 𝑃(𝑍1 = 4) ∙ 𝑃(𝑍1 = 4) 𝑃(𝑍1 = 4) + 𝑃(𝑍1 = 7) = 3 36 ∙ 3 36 3 36 + 6 36 = 1 36 𝑃(𝐶) = 𝑃(𝑍1 = 5) ∙ 𝑃(𝐼 = 1|𝑍1 = 5) = 4 36 ∙ 4 10 = 2 45 𝑃(𝐷) = 𝑃(𝑍1 = 6) ∙ 𝑃(𝐼 = 1|𝑍1 = 6) = 5 36 ∙ 5 11 = 25 396 𝑃(𝐸) = 𝑃(𝑍1 = 8) ∙ 𝑃(𝐼 = 1|𝑍1 = 8) = 5 36 ∙ 5 8 = 25 396 𝑃(𝐹) = 𝑃(𝑍1 = 9) ∙ 𝑃(𝐼 = 1|𝑍1 = 9) = 4 36 ∙ 4 10 = 2 45 𝑃(𝐺) = 𝑃(𝑍1 = 10) ∙ 𝑃(𝐼 = 1|𝑍1 = 10) = 3 36 ∙ 3 9 = 1 36 𝑃(𝐼 = 1) = 2 9 + 2 ∙ 1 36 + 2 ∙ 2 45 + 2 ∙ 25 396 = 244 495 sehingga 𝑃(𝐼 = 0) = 251 495 . karena jumlah kemenangan saat permainan craps diulang sebanyak 200.000 atau g = 200000 merupakan peubah acak binom dengan 𝑝 = 244 495 maka diperoleh 𝜇 = 200000 ∙ 244 495 = 98585,86 𝜎 2 = 200000 ∙ 244 495 ∙ 251 495 . dengan menggunakan teorema 1 diperoleh statistik uji pada uji 1 yaitu 𝑧𝑠𝑐𝑜𝑟𝑒 = 𝑋−𝑛𝑝 √𝑛𝑝𝑞 = 𝑋−98585,86 √200000∙ 244 495 ∙ 251 495 . berdasarkan central limit theorem, untuk 𝑛 → ∞ distribusi dari 𝑧𝑠𝑐𝑜𝑟𝑒 = 𝑋−98585,86 √200000∙ 244 495 ∙ 251 495 mendekati 𝑁(0,1) b. proses pembentukan statistik uji pada uji 2 pada proses pembentukan statistik uji pada uji 2, terlebih dahulu harus dihitung probabilitas dari jumlah lemparan yang mungkin dilakukan seorang pemain. seperti yang telah dijelaskan sebelumnya jumlah lemparan dikelompokkan kedalam 21 kelas mulai dari kelas 1 yang merepresentasikan pemain hanya dapat melakukan lemparan sebanyak satu kali atau dengan kata lain kalah pada lemparan pertama atau menang pada lemparan pertama. dikatakan kalah pada lemparan pertama yaitu ketika jumlah mata dadu pada lemparan pertama bernilai 2, 3 atau 12, sedangkan dikatakan menang pada lemparan pertama adalah ketika mata dadu bernilai 7 atau 11. kelas yang lain yaitu kelas 2 sampai 21 terjadi ketika jumlah mata dadu bernilai sama (4,5,6,8,9 atau 10) pada lemparan pertama, kedua dan seterusnya. berikut adalah proses pembentukan statistik ujinya. 1) untuk kelas 1 (𝑁 = 1) karena probabilitas melempar hanya 1 kali adalah 𝑃(𝑁 = 1|𝑍 = 𝑧) = 1. maka probabilitas kelas 1 adalah 𝑃(𝑁 = 1) = 𝑃(𝑍 = 2) ∙ 𝑃(𝑁 = 1|𝑍 = 2) +𝑃(𝑍 = 3) ∙ 𝑃(𝑁 = 1|𝑍 = 3) +𝑃(𝑍 = 12) ∙ 𝑃(𝑁 = 1|𝑍 = 12) +𝑃(𝑍 = 7) ∙ 𝑃(𝑁 = 1|𝑍 = 7) +𝑃(𝑍 = 11) ∙ 𝑃(𝑁 = 1|𝑍 = 11) = 1 36⁄ ∙ 1 + 2 36⁄ ∙ 1 + 1 36⁄ ∙ 1 analisis teoritis dan empiris uji craps dari diehard battery of randomness test untuk pengujian … jurnal cauchy – issn: 2086-0382 221 + 6 36⁄ ∙ 1 + 2 36⁄ ∙ 1 = 12 36⁄ setelah probabilitas dari kelas ke-1 diperoleh maka dapat dihitung nilai harapannya ketika g = 200000 yaitu 𝐸(𝑁 = 1) = 200000 ∙ 𝑃(𝑁 = 1) = 200000 ∙ 12 36⁄ = 66666,7 2) untuk kelas yang lain (𝑁 = 𝑛) dengan 𝑛 = 2,3, ⋯ ,20 probabilitas melempar sebanyak n kali merupakan distrisbusi geometri sehingga 𝑃(𝑁 = 𝑛|𝑍 = 𝑧) = 𝑃𝑍=𝑧 ∙ (1 − 𝑃𝑍=𝑧 ) (𝑛−1)−1 = 𝑃𝑍=𝑧 ∙ (1 − 𝑃𝑍=𝑧 ) 𝑛−2 untuk 𝑛 = 2,3, ⋯ ,20 dengan 𝑃𝑍=𝑧 adalah probabilitas permainan berakhir saat muncul jumlah mata dadu 7(𝑍 = 7) yang pertama. sehingga 𝑃𝑍=𝑧 = 𝑃(𝑍 = 𝑧) + 𝑃(𝑍 = 7). maka untuk 𝑧 = 4,5,6,8,9,10 diperoleh 𝑃𝑍=4 = 𝑃(𝑍 = 4) + 𝑃(𝑍 = 7) = 3 36⁄ + 6 36⁄ = 9 36⁄ 𝑃𝑍=5 = 𝑃(𝑍 = 5) + 𝑃(𝑍 = 7) = 4 36⁄ + 6 36⁄ = 10 36⁄ 𝑃𝑍=6 = 𝑃(𝑍 = 6) + 𝑃(𝑍 = 7) = 5 36⁄ + 6 36⁄ = 11 36⁄ 𝑃𝑍=8 = 𝑃(𝑍 = 8) + 𝑃(𝑍 = 7) = 5 36⁄ + 6 36⁄ = 11 36⁄ 𝑃𝑍=9 = 𝑃(𝑍 = 9) + 𝑃(𝑍 = 7) = 4 36⁄ + 6 36⁄ = 10 36⁄ 𝑃𝑍=10 = 𝑃(𝑍 = 10) + 𝑃(𝑍 = 7) = 3 36⁄ + 6 36⁄ = 9 36⁄ probabilitas kelas ke-n adalah 𝑃(𝑁 = 𝑛) = 𝑃(𝑍 = 4) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 2) +𝑃(𝑍 = 5) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 5) +𝑃(𝑍 = 6) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 6) +𝑃(𝑍 = 8) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 8) +𝑃(𝑍 = 9) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 9) +𝑃(𝑍 = 10) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 10) karena 𝑃(𝑍 = 4) = 𝑃(𝑍 = 10)dan 𝑃(𝑁 = 𝑛|𝑍 = 4) = 𝑃(𝑁 = 𝑛|𝑍 = 10) serta 𝑃(𝑍 = 5) = 𝑃(𝑍 = 9)dan 𝑃(𝑁 = 𝑛|𝑍 = 5) = 𝑃(𝑁 = 𝑛|𝑍 = 9), maka 𝑃(𝑁 = 𝑛) = 2 ∙ 𝑃(𝑍 = 4) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 4) + 2 ∙ 𝑃(𝑍 = 5) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 5) + 2 ∙ 𝑃(𝑍 = 6) ∙ 𝑃(𝑁 = 𝑛|𝑍 = 6) = 2 ∙ 𝑃(𝑍 = 4) ∙ 𝑃𝑍=4(1 − 𝑃𝑍=4) 𝑛−2 +2 ∙ 𝑃(𝑍 = 5) ∙ 𝑃𝑍=5(1 − 𝑃𝑍=4) 𝑛−2 +2 ∙ 𝑃(𝑍 = 6) ∙ 𝑃𝑍=6(1 − 𝑃𝑍=4) 𝑛−2 = 2 ∙ 3 36⁄ ∙ 9 36⁄ ∙ (1 − 9 36⁄ ) 𝑛−2 +2 ∙ 4 36⁄ ∙ 10 36⁄ ∙ (1 − 10 36⁄ ) 𝑛−2 +2 ∙ 5 36⁄ ∙ 11 36⁄ ∙ (1 − 11 36⁄ ) 𝑛−2 = 1 24⁄ ( 3 4⁄ ) 𝑛−2 + 5 81⁄ ( 13 18⁄ ) 𝑛−2 + 55 648⁄ ( 25 36⁄ ) 𝑛−2 untuk 𝑛 = 2,3,4, ⋯ ,20 tabel 2 probabilitas dan nilai harapan dari kelas ke-2 s.d. kelas ke-20 kelas keprobabilitas nilai harapan n p (n=n) e (n=n) 2 0,188271605 37654,3 3 0,134773663 26954,7 4 0,096567311 19313,5 5 0,0692571 13851,4 6 0,049717715 9943,5 7 0,035725128 7145,0 8 0,025695361 5139,1 9 0,018499325 3699,9 10 0,013331487 2666,3 11 0,009616645 1923,3 12 0,006943702 1388,7 13 0,005018575 1003,7 14 0,003630703 726,1 15 0,002629179 525,8 16 0,001905753 381,2 17 0,001382697 276,5 18 0,001004149 200,8 19 0,000729922 146,0 20 0,000531076 106,2 seperti pada kelas ke-1 maka setelah diperoleh probabilitas dari masing-masing kelas maka dapat dihitung nilai harapannya ketika g = 200000 yaitu 𝐸(𝑁 = 𝑛) = 200000 ∙ 𝑃(𝑁 = 𝑛) untuk 𝑛 = 2,3,4, ⋯ ,20. probabilitas dan nilai harapan dari kelas ke-2 s.d. ke-20 ditampilkan pada tabel 2. 3) untuk kelas ke-21 (n = 21) pada uji 2 diasumsikan lemparan di atas 21 kali memiliki kemungkinan yang kecil untuk terjadi (seperti yang ditunjukkan pada tabel 3) maka meskipun pemain dapat melakukan lemparan lebih dari 21 kali, lemparan tersebut dimasukkan dalam kelas ke-21. akibatnya probabilitas kelas ke-21 merupakan gabungan dari probabilitas ketika lemparan mencapai 21 ditambah probabilitas ketika lemparan diatas 21 kali atau dapat dihitung sebagai berikut : 𝑃(𝑁 = 21) = 1 − [𝑃(𝑁 = 1) + 𝑃(𝑁 = 2) + ⋯ + +𝑃(𝑁 = 20)] = 0,001436 sari agustini hafman dan arif fachru rozi 222 volume 2 no. 4 mei 2013 sehingga nilai harapan kelas ke-21 ketika g = 200000 yaitu 𝐸(𝑁 = 21) = 200000 ∙ 0,001436 = 287,1 tabel 3 probabilitas kelas ke-21 s.d. kelas ke-35 kelas keprobabilitas n p (n=n) 21 0,000387 22 0,000282 23 0,000206 24 0,00015 25 0,00011 26 8,03e-05 27 5,88e-05 28 4,31e-05 29 3,16e-05 30 2,32e-05 31 1,7e-05 32 1,25e-05 33 9,19e-06 34 6,77e-06 35 4,98e-06 karena banyaknya lemparan merupakan data berskala nominal dan tujuan dari uji adalah untuk mengetahui apakah banyaknya lemparan membentuk distribusi seperti yang diharapkan untuk suatu barisan acak maka staistik uji yang digunakan adalah chi-square goodness of fit yang dinyatakan dengan persamaan : 𝑐ℎ𝑖𝑠𝑞 = ∑ (𝑂𝑛 − 𝐸(𝑁 = 𝑛)) 2 𝐸(𝑁 = 𝑛) 21 𝑛=1 dengan derajat bebas 20. berdasarkan prosedur uji chi-square goodness of fit yaitu nilai frekuensi harapan tiap kelas 𝑒𝑖 ≥ 5 maka pada uji 2 jumlah permainan yang harus dilakukan minimal harus lebih dari atau sama dengan 9415 kali atau 𝐺 ≥ 9415. karena probabilitas kelas bervariasi maka nilai yang diambil sebagai rujukan adalah probabilitas terkecil. dari kelas ke-1 s.d. kelas ke-21 probabilitas terkecil dimiliki oleh kelas ke-20 yaitu 0,000531076. bukti : 𝑒𝑖 = 𝐺 ∙ 𝑃(𝑁 = 𝑛) dengan minimal 𝑃(𝑁 = 21) = 0,000531076 maka 5 = 𝐺 ∙ 0,000531076 sehingga minimal 𝐺 = 5 0,000531076 = 9414,8485 = 9415. p adalah notasi untuk proporsi. basic step : 𝑝(9415) benar karena 9415 ∙ 0,000531076 = 5,00008054 ≥ 5 inductive step : misalkan 𝑝(𝐺) benar sehingga 𝐺 ∙ 0,000531076 ≥ 5 maka akan ditunjukkan bahwa saat 𝑝(𝐺 + 1) juga benar. 𝑝(𝐺 + 1): (𝐺 + 1) ∙ 0,000531076 ≥ 5 ⇒ (𝐺 + 1) ∙ 0,000531076 = (𝐺 ∙ 0,000531076) +(0,000531076) ≥ 5 menurut hipotesis induksi 𝐺 ∙ 0,000531076 ≥ 5 sedangkan untuk 𝐺 > 9415, nilai 0,000531076 lebih besar dari 0 sehingga 0,000531076 akan memperbesar nilai di ruas kanan persamaan. akibatnya (𝐺 ∙ 0,000531076) + (0,000531076) ≥ 5 jelas benar. jadi g pada uji 2 adalah 𝐺 ≥ 9415. pada uji 2 ini, marsaglia merekomendasikan menggunakan g yang lebih besar dari 9415 yaitu 200000 (marsaglia and tsang, 2002). analisis empiris hasil pengujian dengan menggunakan dua uji dari uji craps pada sepuluh prng ditampilkan baik dalam bentuk tabel maupun gambar. tabel 4 memperlihatkan hasil pengujian dengan menggunakan uji craps ke-1 pada sepuluh prng. tabel 4. hasil pengujian dengan menggunakan uji craps ke-1 generator uji 1 z-score p-value ket. mwc-1 -3,269 0,00054 tdk acak mwc-2 -0,053 0,47885 acak mwc-3 -0,63 0,26435 acak srg31-1 -0,831 0,20291 acak srg31-2 2,152 0,98430 acak srg31-3 -0,178 0,42925 acak srg32-1 0,269 0,60603 acak srg32-1 -0,286 0,38759 acak srg32-3 0,269 0,60603 acak icg 0,73 0,76720 acak pada tabel 4 terlihat bahwa hanya prng mwc-1 yang tidak lulus uji craps ke-1 sedangkan kesembilan prng yang lain lulus uji tersebut. hal ini karena jumlah kemenangan sebenarnya (hasil observasi) mwc-1 memiliki nilai yang jauh lebih kecil dari jumlah kemenangan harapan dengan selisih sebesar 730,86. berbeda dengan kesembilan prng lain yang memiliki selisih tidak terlalu jauh dari nilai harapan yaitu -185,86 s.d. 481,14. informasi mengenai jumlah kemenangan observasi dan jumlah harapan kesepuluh prng tersebut ditampilkan pada tabel 5. berdasarkan informasi yang diperoleh dari tabel 4 dan tabel 5 terlihat bahwa uji craps ke-1 cukup efektif untuk analisis teoritis dan empiris uji craps dari diehard battery of randomness test untuk pengujian … jurnal cauchy – issn: 2086-0382 223 mendeteksi bentuk distribusi dan independensi barisan yang dihasilkan suatu prng. tabel 5 jumlah kemenangan vs jumlah harapan dari sepuluh prng nama prng jumlah menang selisih observasi harapan mwc-1 97855 98585,86 -730,86 mwc-2 98574 98585,86 -11,86 mwc-3 98445 98585,86 -140,86 srg31-1 98400 98585,86 -185,86 srg31-2 99067 98585,86 481,14 srg31-3 98546 98585,86 -39,86 srg32-1 98646 98585,86 60,14 srg32-2 98522 98585,86 -63,86 srg32-3 98646 98585,86 60,14 hasil pengujian pada kesepuluh prng dengan menggunakan uji craps ke-2 diperlihatkan pada tabel 6. pada tabel 6 terlihat bahwa seluruh prng lulus uji ke-2. hal ini karena nilai observasi pada tiap kelas tidak berbeda jauh dengan nilai harapannya. sebagai contoh ditampilkan hasil pengujian dengan menggunakan uji craps ke-2 pada prng mwc-1 secara lengkap pada tabel 7. tabel 6 hasil pengujian dengan menggunakan uji craps ke-2 generator uji 2 chisq p-value ket. mwc-1 26,69 0,855698 acak mwc-2 22,55 0,688596 acak mwc-3 19,82 0,530605 acak srg31-1 15,48 0,251595 acak srg31-2 18,4 0,438732 acak srg31-3 26,39 0,846570 acak srg32-1 16,33 0,303937 acak srg32-1 16 0,283332 acak srg32-3 16,33 0,303937 acak icg 19 0,478151 acak pada tabel 7 terlihat bahwa nilai observasi pada tiap kelas tidak berbeda jauh dengan nilai yang diharapkan yaitu antara -205,1 s.d. 259,7. hal tersebut diperkuat dengan gambar 1. pada gambar 1 terlihat bahwa banyaknya lemparan sebenarnya (hasil observasi) dengan nilai harapan tiap kelas tidak memiliki selisih yang terlalu jauh. tabel 7 hasil pengujian dengan menggunakan uji craps ke-2 pada mwc generator-1 kelas observed expected chisq sum 1 66490 66666.7 .468 .468 2 37914 37654.3 1.791 2.259 3 26889 26954.7 .160 2.419 4 19320 19313.5 .002 2.421 5 14088 13851.4 4.041 6.462 6 10026 9943.5 .684 7.146 7 7105 7145.0 .224 7.370 8 4934 5139.1 8.183 15.554 9 3713 3699.9 .047 15.600 10 2646 2666.3 .155 15.755 11 1904 1923.3 .194 15.949 12 1426 1388.7 1.000 16.949 13 978 1003.7 .659 17.607 14 670 726.1 4.340 21.948 15 512 525.8 .364 22.312 16 383 381.2 .009 22.321 17 268 276.5 .264 22.585 18 195 200.8 .169 22.754 19 160 146.0 1.346 24.099 20 115 106.2 .727 24.826 21 264 287.1 1.861 26.687 gambar 1 grafik banyaknya lemparan vs nilai harapan tiap kelas pada mwc-1 berdasarkan informasi dari tabel 6 dan tabel 7 serta gambar 1, terlihat bahwa uji craps ke-2 cukup efektif untuk mendeteksi bentuk distribusi dan independensi barisan yang dihasilkan suatu prng. penutup berdasarkan hasil penelitian yang telah dilakukan maka dapat disimpulkan bahwa : 1. pada uji craps ke-1, jumlah kemenangan saat permainan craps merupakan peubah acak binom sehingga dengan menggunakan central limit theorem diperoleh bahwa statistik uji ke1 mendekati distribusi normal baku. 2. statistik uji yang digunakan pada uji craps ke2 adalah uji chi-square goodness of fit dengan distribusi hipotesis adalah distribusi sari agustini hafman dan arif fachru rozi 224 volume 2 no. 4 mei 2013 multinomial karena banyaknya lemparan yang dihitung pada uji craps ke-2 merupakan data berskala nominal yang terdiri dari 21 kelas dan tujuan dari uji ke-2 adalah untuk mengetahui apakah banyaknya lemparan membentuk distribusi seperti yang diharapkan untuk suatu barisan acak. 3. sesuai dengan prosedur uji chi-square goodness of fit maka jumlah permainan yang harus dilakukan pada uji craps ke-2 minimal harus lebih atau sama dengan 9415 kali. 4. hasil pengujian terhadap sepuluh prng yang berasal dari tiga tipe prng yang berbeda menunjukkan uji craps baik uji 1 maupun uji2 cukup efektif untuk mendeteksi bentuk distribusi dan independensi barisan yang dihasilkan suatu prng. referensi [1] kerckhoffs a., (1883), la cryptographic militaire. journal des sciences militaires ix. 538. [2] marsaglia g., (1985), a current view of random number generator, keynote addres, proc.statistics and computer science : 16th symposium on the interface, atlanta. [3] marsaglia g. & tsang w.w., (2002), some dificult-to-pass of randomness, journal of statistical software. 7, issue 3. [4] schneier b., (1996), applied cryptography : protocols, algorithms and source code in c 2nd edition, john wiley & sons, canada. [5] soejati z., (1985), metode statistika 2 edisi 1, universitas terbuka, jakarta. pendahuluan teori dasar distribusi normal distribusi chi-square distribusi multinomial uji chi-square goodness of fit uji craps pseudorandom number generator (prng) metode penelitian penelitian secara teoritis penelitian secara empiris pembahasan analisis teoritis analisis empiris penutup referensi suma’inna dan gugun gumilar aplikasi pengambilan keputusan dengan metode tsukamoto pada penentuan tingkat kepuasan pelanggan (studi kasus di toko kencana kediri) 1venny riana agustin, 2wahyu h. irawan 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: agustinvenny@gmail.com abstrak metode tsukamoto merupakan salah satu metode fuzzy inference system pada logika kabur untuk pengambilan keputusan. tahapan pengambilan keputusan pada metode ini yaitu pengaburan (proses mengubah masukan tegas menjadi masukan kabur), pembentukan aturan kabur, analisis logika kabur, penegasan, serta penarikan kesimpulan dan interpretasi hasil. hasil yang diperoleh, metode tsukamoto bekerja melalui tahapan pengaburan (proses mengubah masukan tegas menjadi masukan kabur), pembentukan aturan kabur dengan rumus umum if 𝑎 adalah 𝐴 then 𝑏 adalah 𝐵, analisis logika kabur untuk mendapatkan nilai 𝛼 tiap aturan, penegasan dengan rata-rata terbobot, serta penarikan kesimpulan dan interpretasi hasil. pada contoh kasus pelanggan dengan nilai 16 kualitas pelayanan, nilai 17 kualitas barang, dan nilai 16 pada harga, diperoleh hasil dengan nilai kepuasan 45,29063 dan tingkat kepuasannya adalah rendah. kata kunci: logika kabur, metode tsukamoto, kepuasan pelanggan abstract tsukamoto method is one method of fuzzy inference system on fuzzy logic for decision making. steps of the decision making in this method, namely fuzzyfication (process changing the input into kabur), the establishment of fuzzy rules, fuzzy logic analysis, defuzzyfication (affirmation), as well as the conclusion and interpretation of the results. the results from this research are steps of the decision making in tsukamoto method, namely fuzzyfication (process changing the input into kabur), the establishment of fuzzy rules by the general form if 𝑎 is 𝐴 then 𝑏 is 𝐵, fuzzy logic analysis to get 𝛼 in every rule, defuzzyfication (affirmation) by weighted average method, as well as the conclusion and interpretation of the results. on customers at the case, in value of 16 the quality of services, the value of 17 the quality of goods, and value of 16 a price, a value of the results is 45,29063 and the level is low satisfaction. keyword: fuzzy logic, tsukamoto method, customer satisfaction pendahuluan ilmu matematika semakin berkembang seiring dengan kesadaran kemudahan menggunakannya dalam berbagai hal. berbagai bidang ilmu matematika seperti aljabar, terapan, dan komputer banyak dijumpai untuk menyelesaikan suatu permasalahan. salah satu ilmu matematika dalam bidang aljabar yang dapat diterapkan langsung dalam kehidupan yakni logika kabur. logika kabur merupakan cabang yang menginterpretasikan pernyataan yang samar menjadi sebuah pengertian yang logis. dalam kehidupan sehari-hari, tidak semua perkara bernilai dua hal, benar dan salah. ada pernyataan yang bernilai tidak benar, benar, dan benar sekali. dalam menyelesaikan pernyataan di atas, secara umum logika kabur mempunyai 3 metode yaitu metode mamdani, metode sugeno, dan metode tsukamoto. pada metode tsukamoto, implikasi setiap aturan berbentuk implikasi “sebab-akibat” atau implikasi “input-output” dimana antara sebab (anteseden) dan akibat (konsekuen) harus ada hubungannya. setiap aturan direpresentasikan menggunakan himpunan kabur, dengan fungsi keanggotaan yang monoton. kemudian untuk menentukan hasil tegas digunakan rumus penegasan (defuzzyfikasi). artikel ini merupakan upaya untuk mengaplikasikan logika kabur pada pemasaran dalam rangka pengembangan. penelitian ini dilakukan untuk mengetahui langkah-langkah aplikasi pengambilan keputusan dengan metode tsukamoto dalam menentukan tingkat kepuasan pelanggan toko kencana kediri. mailto:agustinvenny@gmail.com venny riana agustin 11 volume 4 no.1 november 2015 kajian teori 1. himpunan kabur himpunan kabur adalah suatu himpunan yang berisi unsur-unsur yang memiliki tingkat keanggotaan yang bervariasi dari himpunan. hal ini bertolak belakang dengan himpunan klasik, atau tegas, karena anggota himpunan tegas tidak akan menjadi anggota kecuali keanggotaannya penuh atau lengkap. anggota dalam himpunan kabur karena keanggotaannya tidak perlu lengkap maka anggota kabur yang lain juga dapat menjadi anggota di semesta yang sama [1]. 2. fungsi keanggotaan fungsi keanggotaan (membership function) adalah suatu kurva yang menunjukkan pemetaan titik-titik input data ke dalam nilai keanggotaannya (sering juga disebut derajat keanggotaan) yang memiliki interval antara 0 sampai 1. salah satu cara yang dapat digunakan untuk mendapatkan nilai keanggotaan adalah dengan melalui pendekatan fungsi. ada beberapa fungsi yang bisa digunakan [2]. a. fungi keanggotaan linier turun representasi linier turun merupakan kebalikan dari linier naik. garis lurus dimulai dari nilai domain dengan derajat keanggotaan tertinggi pada sisi kiri, kemudian bergerak menurun ke nilai domain yang memiliki derajat keanggotaan lebih rendah. 𝜇(𝑥) = { 0 𝑥 ≥ 𝑏 𝑏 − 𝑥 𝑏 − 𝑎 𝑎 ≤ 𝑥 ≤ 𝑏 1 𝑥 ≤ 𝑎 b. fungsi keanggotaan linier naik garis lurus dimulai pada nilai domain yang memiliki derajat keanggotaan nol bergerak ke kanan menuju ke nilai domain yang memiliki derajat keanggotaan lebih tinggi. 𝜇(𝑥) = { 0 𝑥 ≤ 𝑎 𝑥 − 𝑎 𝑏 − 𝑎 𝑎 ≤ 𝑥 ≤ 𝑏 1 𝑥 ≥ 𝑏 3. aturan kabur jika-maka aturan umum kabur jika-maka mempunyai bentuk “jika 𝑎1 adalah 𝐴1 dan … dan 𝑎𝑛 adalah 𝐴𝑛 maka 𝑏 adalah 𝐵”. dengan 𝐴 dan 𝐵 adalah nilai linguistik yang dinyatakan dengan himpunan kabur dalam semesta pembicaraan. “𝑎1 adalah 𝐴1” dan “𝑎𝑛 adalah 𝐴𝑛” disebut sebagai anteseden, sedangkan “𝑏 adalah 𝐵” disebut konsekuen [3]. 4. sistem kendali kabur sistem kendali kabur merupakan sistem yang berfungsi untuk mengendalikan proses tertentu dengan menggunakan aturan inferensi berdasarkan logika kabur. [4] menyebutkan bahwa pada dasarnya sistem kendali semacam itu terdiri dari 4 unit, yaitu: 1. pengaburan (fuzzyfikasi) karena sistem kendali logika kabur bekerja dengan kaidah dan masukan kabur, maka langkah pertama adalah mengubah masukan yang tegas yang diterima menjadi masukan kabur. untuk masingmasing variabel masukan ditentukan suatu fungsi fuzzyfikasi yang akan mengubah nilai variabel masukan yang tegas (biasanya dinyatakan dalam bilangan real) menjadi nilai pendekatan kabur. jadi, fungsi fuzzyfikasi adalah pemetaan 𝑓:ℝ → 𝐾, dimana 𝐾 adalah suatu kelas himpunan kabur dalam semesta ℝ. 2. basis pengetahuan basis data (data base), memuat fungsi-fungsi keanggotaan dari himpunan-himpunan kabur yang terkait dengan nilai dari variabel linguistik yang dipakai. basis kaidah (rule base), memuat kaidahkaidah berupa implikasi kabur. 3. penalaran logika kabur masukan kabur hasil pengolahan fuzzyfikasi diterima oleh penalaran untuk disimpulkan berdasarkan kaidah-kaidah yang tersedia dalam basis pengetahuan. penarikan kesimpulan itu dilaksanakan berdasarkan aturan modus ponen. 4. penegasan (defuzzyfikasi) kesimpulan/keluaran dari sistem kendali kabur adalah suatu himpunan kabur. karena sistem tersebut hanya dapat mengeksekusikan aplikasi pengambilan keputusan dengan metode tsukamoto pada penentuan tingkat kepuasan pelanggan (studi kasus di toko kencana kediri) cauchy – issn: 2086-0382/e-issn: 2477-3344 12 nilai yang tegas, maka diperlukan suatu mekanisme untuk mengubah nilai kabur keluaran itu menjadi nilai yang tegas. fungsi penegasan adalah suatu pemetaan 𝑡:𝐾 → ℝ, dimana 𝐾 adalah suatu kelas himpunan kabur, yang memetakan suatu himpunan kabur ke suatu bilangan real yang tegas. 5. teknik pengambilan keputusan metode tsukamoto pada metode tsukamoto, sebagai akibat dari aturan masing-masing kabur yang direpresentasikan oleh himpunan kabur dengan fungsi keanggotaan monoton, terkadang disebut fungsi shoulder. output dari masing-masing aturan bernilai tegas yang terimbas dari nilai keanggotaan. keseluruhan output dihitung dengan rata-rata terbobot dari output masing-masing aturan. karena setiap aturan mengambil kesimpulan outputnya bernilai tegas maka keseluruhan output model tsukamoto juga menghindari proses lama dari penegasan. karena fungsi keanggotaan output khusus diperlukan dari metode ini, maka hal ini tidak berguna sebagai pendekatan umum dan harus dikerjakan pada kondisi spesifik [1]. pembahasan pada penentuan tingkat kepuasan pelanggan, variabel input yang digunakan adalah kualitas pelayanan, kualitas barang, dan harga. serta variabel output yang digunakan adalah kepuasan pelanggan. langkah awal yang dilakukan adalah membentuk himpunan kabur kemudian membentuk aturan jika-maka. data input yang diperoleh dikelompokkan berdasarkan himpunan kabur. kemudian memetakan data input ke fungsi keanggotaan untuk diperoleh derajat keanggotaan dari masing-masing data input. derajat keanggotaan digunakan untuk mencari nilai 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 dari masing-masing aturan. penegasan dengan rata-rata terbobot merupakan langkah terakhir yang kemudian dapat ditarik kesimpulan dan interpretasi hasil. berikut ini tahap-tahap dari metode tsukamoto: pengaburan merupakan langkah awal pada analasis logika kabur. karena analisis bersifat kabur, maka data input yang digunakan harus bersifat kabur. oleh karena itu, diperlukan proses mengubah data input tegas menjadi kabur. pembentukan himpunan kabur digunakan untuk mendefinisikan nilai-nilai input tegas. setiap himpunan kabur mempunyai domain yang nilainya terdefinisi di semesta pembicaraan. pada penelitian ini digunakan beberapa varibel dalam menentukan tingkat kepuasan pelanggan. variabel kualitas pelayanan, kualitas barang, dan harga sebagai variabel input. serta variabel kepuasan pelanggan sebagai variabel output. data diperoleh dari hasil kuesioner dan dinyatakan dengan himpunan kabur yang tercantum pada tabel sebagai berikut: tabel 1 himpunan kabur fungsi variabel himpunan kabur semesta pembicaraan domain input kualitas pelayanan tidak ramah ramah sangat ramah [13,21] [13,17] [15,19] [17,21] kualitas barang tidak bagus bagus sangat bagus [12,21] [12,16,5] [14,5,18,5] [16,5,21] harga tidak murah murah sangat murah [12,18] [12,15] [13,5,16,5] [15,18] output kepuasan pelanggan rendah tinggi [0,100] [0,75] [25,100] himpunan kabur diperlukan untuk merepresentasikan variabel kabur dengan membentuk fungsi keanggotaan. fungsi keanggotaan mendefinisikan titik-titik himpunan kabur ke dalam derajat keanggotaan dengan selang tertutup [0,1] pada suatu variabel kabur tertentu. sehingga diperoleh fungsi keanggotaan untuk kualitas pelayanan tidak ramah (ptr) sebagai berikut: 𝜇𝑃𝑇𝑅(𝑥) = { 1 𝑥 ≤ 13 17 − 𝑥 17 − 13 13 ≤ 𝑥 ≤ 17 0 𝑥 ≥ 17 (1) fungsi keanggotaan untuk kualitas pelayanan ramah (pr) sebagai berikut: 𝜇𝑃𝑅(𝑥) = { 1 𝑥 = 17 𝑥 − 15 17 − 15 15 ≤ 𝑥 ≤ 17 19 − 𝑥 19 − 17 17 ≤ 𝑥 ≤ 19 0 𝑥 ≥ 19 ∨ 𝑥 ≤ 15 (2) fungsi keanggotaan untuk kualitas pelayanan sangat ramah (psr) sebagai berikut: 𝜇𝑃𝑆𝑅(𝑥) = { 0 𝑥 ≤ 17 𝑥 − 17 21 − 17 17 ≤ 𝑥 ≤ 21 1 𝑥 ≥ 21 (3) gambar 1 representasi variabel kualitas pelayanan venny riana agustin 13 volume 4 no.1 november 2015 fungsi keanggotaan untuk kualitas barang tidak bagus (btb) sebagai berikut: 𝜇𝐵𝑇𝐵(𝑦) = { 1 𝑦 ≤ 12 16,5 − 𝑦 16,5 − 12 12 ≤ 𝑦 ≤ 16,5 0 𝑦 ≥ 16,5 (4) fungsi keanggotaan untuk kualitas barang bagus (bb) sebagai berikut: 𝜇𝐵𝐵(𝑦) = { 1 𝑦 = 16,5 𝑦 − 14,5 16,5 − 14,5 14,5 ≤ 𝑦 ≤ 16,5 18,5 − 𝑦 18,5 − 16,5 16,5 ≤ 𝑦 ≤ 18,5 0 𝑦 ≥ 18,5 ∨ 𝑦 ≤ 14,5 (5) fungsi keanggotaan untuk kualitas barang sangat bagus (bsb) sebagai berikut: 𝜇𝐵𝑆𝐵(𝑦) = { 0 𝑦 ≤ 16,5 𝑦 − 16,5 21 − 16,5 16,5 ≤ 𝑦 ≤ 21 1 𝑦 ≥ 21 (6) gambar 2 representasi variabel kualitas barang fungsi keanggotaan untuk harga tidak murah (htm) sebagai berikut: 𝜇𝐻𝑇𝑀(𝑧) = { 1 𝑧 ≤ 12 15 − 𝑧 15 − 12 12 ≤ 𝑧 ≤ 15 0 𝑧 ≥ 15 (7) fungsi keanggotaan untuk harga murah (hm) sebagai berikut: 𝜇𝐻𝑀(𝑧) = { 1 𝑧 = 15 𝑧 − 13,5 15 − 13,5 13,5 ≤ 𝑧 ≤ 15 16,5 − 𝑧 16,5 − 15 15 ≤ 𝑧 ≤ 16,5 0 𝑥 ≥ 16,5 ∨ 𝑥 ≤ 13,5 (8) fungsi keanggotaan untuk harga sangat murah (hsm) sebagai berikut: 𝜇𝐻𝑆𝑀(𝑧) = { 0 𝑧 ≤ 15 𝑧 − 15 18 − 15 15 ≤ 𝑧 ≤ 18 1 𝑧 ≥ 18 (9) gambar 3 representasi variabel harga fungsi keanggotaan untuk kepuasan pelanggan rendah (kpr) sebagai berikut: 𝜇𝐾𝑃𝑅(𝑘) = { 1 𝑘 ≤ 0 75 − 𝑘 75 0 ≤ 𝑘 ≤ 75 0 𝑘 ≥ 75 (10) fungsi keanggotaan untuk kepuasan pelanggan tinggi (kpt) sebagai berikut: 𝜇𝐾𝑃𝑇(𝑘) = { 0 𝑘 ≤ 25 𝑘 − 25 100 − 25 25 ≤ 𝑘 ≤ 100 1 𝑘 ≥ 100 (11) gambar 4 representasi variabel kepuasan pelanggan logika kabur bekerja berdasarkan aturanaturan yang dibentuk dengan aturan jika-maka. aturan ini dibentuk untuk menyatakan hubungan inputoutput. dari 3 variabel kabur dan 3 himpunan kabur di atas, diperoleh 27 aturan. setiap aturan mempunyai 3 antesenden yang menggunakan operator dan dan 1 konsekuen. berdasarkan hasil penelitian tentang faktor yang paling dominan pada kepuasan pelanggan toko, diketahui setiap daerah memiliki hasil yang berbeda. sehingga, pada pembentukan aturan kabur digunakan aturan umum bahwa faktor kepuasan pelanggan memiliki pengaruh yang sama besar. tabel 2 aturan-aturan kabur pelayanan barang harga kepuasan [r1] jika tidak ramah sangat bagus sangat murah maka tinggi [r2] jika tidak ramah sangat bagus murah maka tinggi [r3] jika tidak ramah bagus sangat murah maka tinggi [r4] jika ramah sangat bagus sangat murah maka tinggi [r5] jika ramah sangat bagus murah maka tinggi [r6] jika ramah sangat bagus tidak murah maka tinggi [r7] jika ramah bagus sangat murah maka tinggi [r8] jika ramah bagus murah maka tinggi [r9] jika ramah tidak bagus sangat murah maka tinggi [r10] jika sangat ramah sangat bagus sangat murah maka tinggi [r11] jika sangat ramah sangat bagus murah maka tinggi [r12] jika sangat ramah sangat bagus tidak murah maka tinggi aplikasi pengambilan keputusan dengan metode tsukamoto pada penentuan tingkat kepuasan pelanggan (studi kasus di toko kencana kediri) cauchy – issn: 2086-0382/e-issn: 2477-3344 14 [r13] jika sangat ramah bagus sangat murah maka tinggi [r14] jika sangat ramah bagus murah maka tinggi [r15] jika sangat ramah bagus tidak murah maka tinggi [r16] jika sangat ramah tidak bagus sangat murah maka tinggi [r17] jika sangat ramah tidak bagus murah maka tinggi [r18] jika tidak ramah sangat bagus tidak murah maka rendah [r19] jika tidak ramah bagus murah maka rendah [r20] jika tidak ramah bagus tidak murah maka rendah [r21] jika tidak ramah tidak bagus sangat murah maka rendah [r22] jika tidak ramah tidak bagus murah maka rendah [r23] jika tidak ramah tidak bagus tidak murah maka rendah [r24] jika ramah bagus tidak murah maka rendah [r25] jika ramah tidak bagus murah maka rendah [r26] jika ramah tidak bagus tidak murah maka rendah [r27] jika sangat ramah tidak bagus tidak murah maka rendah aturan-aturan kabur di atas merupakan suatu pernyataan implikasi yang antar variabel inputnya dihubungkan dengan operator dan. sehingga untuk mencari nilai 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 tiaptiap aturan digunakan fungsi implikasi min (minimal), yaitu mengambil derajat keanggotaan terkecil dari variabel kualitas pelayanan, kualitas barang, dan harga. 𝛼𝑖 = 𝜇𝑃∩𝐵∩𝐻 = min (𝜇𝑃𝑖(𝑥),𝜇𝐵𝑖(𝑦),𝜇𝐻𝑖(𝑧)) dimana 𝛼 adalah 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 pada aturan ke-i 𝜇𝑃(𝑥) adalah derajat keanggotaan kualitas pelayanan pada aturan ke-i 𝜇𝐵(𝑦) adalah derajat keanggotaan kualitas barang pada aturan ke-i 𝜇𝐻(𝑧) adalah derajat keanggotaan harga pada aturan ke-i. pengambilan keputusan menggunakan metode tsukamoto pada setiap aturan untuk mendapat nilai output (𝑘) yang tegas berupa tingkat kepuasan pelanggan. hasil akhir berupa output nilai 𝑘 diperoleh dengan menggunakan metode rata-rata terbobot: 𝑘 = ∑𝑘𝑖 𝛼𝑖 ∑𝛼𝑖 dimana 𝑘 adalah nilai tingkat kepuasan pelanggan 𝑘𝑖 adalah nilai tingkat kepuasan pelanggan masing-masing aturan 𝛼𝑖 adalah nilai 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 masing-masing aturan. hasil dari penegasan di atas diambil kesimpulan dan diinterpretasikan pada himpunan kabur yang telah dibentuk. hasil output berupa nilai sehingga dapat diketahui tingkatan kepuasan pelanggan toko dari tiap-tiap pelanggan. contoh kasus pelanggan toko memberikan nilai 16 pada kualitas pelayanan, nilai 17 pada kualitas barang, dan nilai 16 pada harga. pemilik toko ingin mengetahui seberapa besar tingkat kepuasan pelanggan tersebut. seperti pada gambar 4.1 dengan fungsi keanggotaan pada persamaan (4.1), (4.2), dan (4.3), diperoleh: 𝜇𝑃𝑇𝑅(16) = 17 − 16 17 − 13 = 0,25 𝜇𝑃𝑅(16) = 16 − 15 17 − 15 = 0,5 𝜇𝑃𝑆𝑅(16) = 0 seperti pada gambar 4.2 dengan fungsi keanggotaan pada persamaan (4.4), (4.5), dan (4.6), diperoleh: 𝜇𝐵𝑇𝐵(17) = 0 𝜇𝐵𝐵(17) = 18,5 − 17 18,5 − 16,5 = 0,75 𝜇𝐵𝑆𝐵(17) = 17 − 16,5 21 − 16,5 = 0,11 seperti pada gambar 4.3 dengan fungsi keanggotaan pada persamaan (4.7), (4.8), dan (4.9), diperoleh: 𝜇𝐻𝑇𝑀(16) = 0 𝜇𝐻𝑀(16) = 16,5 − 16 16,5 − 15 = 0,33 𝜇𝐻𝑆𝑀(16) = 𝑧 − 15 18 − 15 = 0,33 dengan nilai derajat keanggotaan yang diperoleh sehingga menghasilkan 𝛼𝑖 sekaligus 𝑘𝑖 tiap aturan sebagai berikut: [r1] jika pelayanan tidak ramah dan barang sangat bagus dan harga sangat murah maka kepuasan tinggi 𝛼1 = min(0,25, 0,11, 0,33) = 0,11 𝜇𝐾𝑃𝑇(𝑘) = 𝑘1 − 25 100− 25 ⟺ 𝑘1 = 33,25 [r2] jika pelayanan tidak ramah dan barang sangat bagus dan harga murah maka kepuasan tinggi 𝛼2 = min(0,25, 0,11, 0,33) = 0,11 𝜇𝐾𝑃𝑇(𝑘) = 𝑘2 − 25 100− 25 ⟺ 𝑘2 = 33,25 [r3] jika pelayanan tidak ramah dan barang bagus dan harga sangat murah maka kepuasan tinggi 𝛼3 = min(0,25, 0,75, 0,33) = 0,25 𝜇𝐾𝑃𝑇(𝑘) = 𝑘3 − 25 100− 25 ⟺ 𝑘3 = 43,75 venny riana agustin 15 volume 4 no.1 november 2015 [r4] jika pelayanan ramah dan barang sangat bagus dan harga sangat murah maka kepuasan tinggi 𝛼4 = min(0.5, 0,11, 0.33) = 0,11 𝜇𝐾𝑃𝑇(𝑘) = 𝑘4 − 25 100 − 25 ⟺ 𝑘4 = 33,25 [r5] jika pelayanan ramah dan barang sangat bagus dan harga murah maka kepuasan tinggi 𝛼5 = min(0,5, 0,11, 0,33) = 0,11 𝜇𝐾𝑃𝑇(𝑘) = 𝑘5 − 25 100 − 25 ⟺ 𝑘5 = 33,25 [r7] jika pelayanan ramah dan barang bagus dan harga sangat murah maka kepuasan tinggi 𝛼7 = min(0,5, 0,75, 0,33) = 0,33 𝜇𝐾𝑃𝑇(𝑘) = 𝑘7 − 25 100 − 25 ⟺ 𝑘7 = 49,75 [r8] jika pelayanan ramah dan barang bagus dan harga murah maka kepuasan tinggi 𝛼8 = min(0,5, 0,75, 0,33) = 0,33 𝜇𝐾𝑃𝑇(𝑘) = 𝑘8 − 25 100 − 25 ⟺ 𝑘8 = 49,75 [r19] jika pelayanan tidak ramah dan barang bagus dan harga murah maka kepuasan rendah 𝛼19 = min(0,25, 0,75, 0,33) = 0,25 𝜇𝐾𝑃𝑇(𝑘) = 75 − 𝑘19 75 ⟺ 𝑘19 = 56,25 dari proses dan hasil di atas, maka diperoleh nilai tingkat kepuasan pelanggan (𝑘) sebagai berikut: 𝑘 = ∑𝑘𝑖 𝛼𝑖 ∑𝛼𝑖 = 45,29063 jadi, pelanggan toko yang memberikan nilai 16 pada kualitas pelayanan, nilai 17 pada kualitas barang, dan nilai 16 pada harga mempunyai nilai kepuasan sebesar 45,29063 dan tingkat kepuasannya adalah rendah. kesimpulan berdasarkan pembahasan, maka diperoleh kesimpulan bahwa aplikasi metode tsukamoto pada tingkat kepuasan pelanggan bekerja melalui 5 tahap yaitu: a. pengaburan (fuzzyfikasi), mengubah nilai input tegas menjadi nilai input kabur dengan variabel input kualitas pelayanan, kualitas barang, harga, dan variabel output kepuasan pelanggan. b. pembentukan aturan kabur dengan operator antar variabel input adalah operator dan. c. analisis logika kabur yang digunakan adalah fungsi implikasi min karena operator yang digunakan pada aturan adalah operator dan. d. penegasan (defuzzyfikasi) menggunakan metode rata-rata terbobot. e. penarikan kesimpulan dan interpretasi hasil. pada contoh kasus pelanggan yang memberikan nilai 16 pada kualitas pelayanan, nilai 17 pada kualitas barang, dan nilai 16 pada harga mempunyai nilai sebesar 45,29063 dan tingkat kepuasannya adalah rendah. daftar pustaka [1] t. j. ross, fuzzy logic with engineering applications third edition, mexico: john willey & sons, ltd, 2010. [2] s. kusumadewi and h. purnomo, aplikasi logika fuzzy untuk pendukung keputusan, yogyakarta: graha ilmu, 2004. [3] g. chen and t. t. pham, introductions to fuzzy sets, fuzzy logic, and fuzzy control systems, new york: crc press, 2001. [4] f. susilo, himpunan dan logika kabur serta aplikasinya, yogyakarta: graha ilmu, 2006. pendahuluan kajian teori pembahasan kesimpulan daftar pustaka 5.7 siti tabi'atul hasanah pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood di pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld) siti tabi’atul hasanah mahasiswa jurusan matematika uin maulana malik ibrahim malang e-mail: chib_y@yahoo.com abstrak outlier merupakan pengamatan yang jauh berbeda (ekstrim) dari data pengamatan lainnya, atau dapat diartikan data yang tidak mengikuti pola umum model. adakalanya outlier memberikan informasi yang tidak dapat diberikan oleh data yang lainnya. karena itulah outlier tidak boleh begitu saja dihilangkan. outlier dapat juga merupakan pengamatan berpengaruh. banyak sekali metode yang dapat digunakan untuk mendeteki adanya outlier. pada penelitian-penelitian sebelumnya pendeteksian outlier dilakukan pada regresi linier. selanjutnya akan dikembangkan pendeteksian outlier pada regresi nonlinier. regresi nonlinier disini dikhususkan pada regresi nonlinier multiplikatif. untuk mendeteksi yaitu menggunakan metode statistik likelihood displacement. metode statistik likelihood displacement disingkat (ld) adalah suatu metode untuk mendeteksi adanya outlier dengan cara menghilangkan data yang diduga outlier. untuk mengestimasi parameternya maka digunakan metode maximum likelihood, sehingga didapatkan hasil etimasi yang maksimal. dengan metode ld diperoleh ����, yaitu likelihood displacement yang diduga mengandung outlier. selanjutnya keakuratan metode ld dalam mendeteksi adanya outlier ditunjukkan dengan cara membandingkan mse dari ld dengan mse dari regresi pada umumnya. statistik uji yang digunakan adalah . hipotesis awal ditolak ketika ���� �� ������� , sehingga terbukti ���� adalah suatu outlier. kata kunci: likelihood displacement, maximum likelihood estimation, outlier, regresi nonlinier multiplikatif. abstract outlier is an observation that much different (extreme) from the other observational data, or data can be interpreted that do not follow the general pattern of the model. sometimes outliers provide information that can not be provided by other data. that's why outliers should not just be eliminated. outliers can also be an influential observation. there are many methods that can be used to detect of outliers. in previous studies done on outlier detection of linear regression. next will be developed detection of outliers in nonlinear regression. nonlinear regression here is devoted to multiplicative nonlinear regression. to detect is use of statistical method likelihood displacement. statistical methods abbreviated likelihood displacement (ld) is a method to detect outliers by removing the suspected outlier data. to estimate the parameters are used to the maximum likelihood method, so we get the estimate of the maximum. by using ld method is obtained ���� i.e likelihood displacement is thought to contain outliers. further accuracy of ld method in detecting the outliers are shown by comparing the mse of ld with the mse from the regression in general. statistic test used is λ. initial hypothesis was rejected when ��� �� ������� , proved so ���� is an outlier. keywords: likelihood displacement, maximum likelihood estimation, multiplicative nonlinear regression, outlier pendahuluan outlier adalah pengamatan yang jauh berbeda (ekstrim) dari data pengamatan lainnya. salah satu penyebab terjadinya outlier adalah kesalahan pada pengambilan data sehingga menyebabkan data tersebut menjadi ekstrim. adakalanya outlier ini tidak boleh begitu saja dihilangkan, namun dalam hal ini harus hati-hati karena terkadang outlier itu memberikan informasi yang tidak dapat diberikan oleh titik pengamatan lain, misalnya karena adanya kombinasi keadaan yang tidak biasa dan perlu diadakan penyelidikan lebih jauh. suatu outlier dapat dibuang setelah ditelusuri ternyata pengamatan tersebut merupakan akibat dari kesalahan pengukuran atau kesalahan dalam menyiapkan pengukuran. outlier dapat juga merupakan pengamatan berpengaruh. outlier yang bukan pengamatan berpengaruh, tidak memiliki pengaruh yang kuat pada model kecuali outlier tersebut sangat besar. tetapi jika outlier siti tabi’atul hasanah 178 volume 2 no. 3 november 2012 merupakan data berpengaruh, maka akan memberikan dampak pada model (drapper dan smith, 1992:146). misalkan saja pada suatu penelitian tentang sapi penghasil susu. dari suatu data ternyata diperoleh ada beberapa sapi yang menghasilkan hasil susu yang lebih banyak dari biasanya atau dari sapi normalnya. sapi penghasil susu yang tidak sesuai dengan normalnya merupakan suatu outlier, namun jika mengahapus begitu saja data ini berarti telah menghilangkan bibit sapi unggul yang mampu menghasilkan banyak susu sapi. oleh sebab itulah penting untuk mengidentifikasi adanya outlier agar tidak kehilangan suatu data yang memiliki kualitas yang bagus. jika dengan adanya outlier itu kurang baikmaka perlu diidentifikasi dan kemudian dihilangkan data yang mengandung outlier. banyak sekali metode yang dapat digunakan untuk mendeteksi outlier, salah satunya yaitu pendektesian outlier pada model linier univariat telah dikemukaan oleh cook dengan memperkenalkan jarak cook (cook’s distance) sebagai ukuran untuk mendeteksi pengamatan berpengaruh dalam model linier univariat. ukuran jarak cook ini dirumuskan sebagai kombinasi dari studential residual, variansi residual, dan variansi nilai prediksi. selain metode yang dikemukakan oleh cook, masih banyak lagi metode yang digunakan untuk pendeteksian outlier pada model linier (makkulau, 2010:95) xu, abraham dan steiner (2005) mengembangkan jarak cook univariat untuk mendeteksi outlier pada model linier multivariat (model regresi linier multivariat) dengan menggunakan metode statistik likelihood displacement yang disingkat ld. metode ld adalah suatu metode untuk mendeteksi adanya outlier dengan cara menghilangkan pengamatan yang diduga outlier (makkulau, 2010:95). tujuan dari penelitian ini adalahuntuk mengetahui cara mendeteksi outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld). mafaat dari penelitian ini adalah untuk mengembangkan metode yang dapat digunakan untuk mendeteksi adanya outlier. kajian teori 1. outlier secara umum outlier dapat diartikan data yang tidak mengikuti pola umum pada model atau data yang keluar dari model dan tidak berada dalam daerah selang kepercayaan (sembiring, 1995:62). menurut draper dan smith (1992:146) sisaan yang merupakan outlier adalah yang nilai mutlaknya jauh lebih besar dari pada sisaan lainnya dan terletak tiga atau empat kali simpangan baku atau lebih jauh lagi dari rata-rata sisaannya. outlier merupakan suatu keganjilan dan menandakan suatu titik data yang sama sekali tidak tipikal dibandingkan data lainnya. 2. estimasi parameter menurut yitnosumarto (1990:211) penduga (estimator) adalah anggota peubah acak statistik yang mungkin untuk sebuah parameter (anggota peubah yang diturunkkan). parameter adalah nilai yang mengikuti acuan keterangan atau informasi yang dapat menjelaskan batasbatas atau bagian-bagian tertentu dari suatu sistem persamaan. murray dan larry (1999:166) menyatakan terdapat dua jenis estimasi parameter, yaitu: estimasi titik dan estimasi interval. estimasi titik adalah estimasi dari sebuah parameter populasi yang dinyatakan oleh bilangan tunggal disebut sebagai estimasi titik dari parameter tersebut. sebuah nilai yang diperoleh dari sampel dan digunakan sebagai estimasi dari parameter yang nilainya tidak diketahui. misalkan �1,�2,…,�� merupakan sampel acak berukuran n dari x, maka statistik yang berkaitan dengan θ dinamakan estimasi dari θ. setelah sampel diambil, nilai-nilai yang dihitung dari sampel itu digunakan sebagai taksiran titik bagi θ. estimasi dari parameter populasi yang dinyatakan dengan dua bilangan. di antara posisi parameternya diperkirakan berbeda, sehinggga disebut estimasi interval. estimasi interval mengindikasikan adanya tingkat kepresisian atau akurasi dari sebuah estimasi sehingga estimasi interval akan dianggap semakin baik jika mendekati estimasi titik adapun sifat-sifat estimasi titik adalah sebagai berikut: 1. tak bias yusuf wibisono (2005:362) dalam bukunya menyatakan bahwa estimator tak bias bagi parameter θ, jika ����� � 2. konsisten damodar n. gujarati (2007:98) menerangkan estimator parameter �� dikatakan konsisten bila nilai-nilainya mendekati nilai parameter yang sebenarnya meskipun ukuran sampelnya semakin besar. 3. efisien jika distribusi sampling dari dua statistik memiliki mean atau ekspektasi yang sama, maka statistik dengan variansi yang lebih kecil pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld) jurnal cauchy – issn: 2086-0382 179 disebut sebagai estimator efisien dari mean, sementara statistik yang lain disebut estimator tak efisien. adapun nilai-nilai yang berkorespondensi dengan statistik-statistik ini masing-masing disebut sebagai estimasi efisien dan estimasi tak efisien. 4. distribusi suatu peubah acak � berdistribusi normal !"#,$�% bila untuk suatu $� 0 dan * ∞ + # + ∞ (turmudi dan harini, 2008:204). mempunyai fungsi densitas pada � , dengan persamaan: -",% 1$√2/012�34156 7 8 (2.1) distribusi lain yang digunakan yaitu distribusi chi-square. distribusi chi-square merupakan distribusi dengan variabel acak kontinue. simbol untuk chi-square adalah ��. distribusi chi-square sebenarnya merupakan jumlah kuadrat dari variabel-variabel acak yang bebas dan menyebar normal dengan mean 0 dan ragam 1�9~!;�"0,1%�. distribusi ini dapat dinyatakan dengan �� 92� = 9�� = >= 9?� �� @9��� @a b� * #�$� c� � merupakan variabel acak yang tersebar menurut distribusi chi-square dengan derajat bebas sebesar d dan dapat dituliskan �� e �?� dimana �?� yaitu distribusi chi-square dengan derajat bebas d. suatu variabel acak � berdistribusi chisquare dengan derajat bebas d, dinyatakan dengan �?�"0% bila untuk suatu bilangan bulat d 0. (turmudi dan harini, 2008: 210) distribusi ini mempunyai fungsi kepekatan peluang sebagai berikut: -4",% f 1 2?�γ 3d27, 3?�712014�, , g 0 0, selainnya m nilai tengah (mean) dan ragam untuk distribusi �� adalah # d dan n� 2d. distribusi chi-square bergantung pada banyaknya simpangan baku yang bebas antara satu dengan yang lain atau dengan kata lain bergantung pada derajat bebasnya. jika � dan b variabel acak, maka peluang terjadinya � dan b secara serentak dinyatakan sebagai -",,o% disebut distribusi peluang gabungan untuk setiap pasangan ",,o% (herrhyanto, 2009:5). 5. regresi nonlinier analisis regresi merupakan analisis yang menyangkut studi tentang hubungan antara satu variabel yang disebut variabel terikat atau variabel yang dijelaskan dan satu atau lebih variabel yang lain yang disebut variabel bebas atau variabel penjelas (gujarati, 2007:115). regresi yang variabel-variabelnya berbentuk tidak biasa. bentuk grafik regresi nonlinier adalah berupa lekungan (hasan, 2002:297). model regresi nonlinier dapat digolongkan menjadi dua yaitu model linier intrinsik dan model nonlinier intrinsik. jika suatu model dikatakan model linier intrinsik, maka model model ini dapat dinyatakan dalam bentuk linier baku dengan mentransformasikan secara tepat terhadap peubahnya. jika suatu model nonlinier tidak dapat dinyatakan dalam bentuk baku, berarti model ini secara intsinsik adalah nonlinier. berikut ini adalah beberapa model yang dapat dinyatakan dalam linier baku (draper dan smith, 1992:213). 6. regresi multiplikatif regresi multiplikatif adalah salah satu bentuk dari regresi linier intrinsik. bentuk umum dari regresi multiplikatif adalah sebagai berikut: b p �2q��r�srt (2.2) dimana $, u, dan v adalah parameter yang tidak diketahui, dan t adalah galat acak yang bersifat multiplikatif. dengan mengalgoritmakan basis e pada pada persamaan di atas, maka model persamaan di atas menjadi w� b w�x =uw��2 = yw��� = vw��s = w�t. model persamaan tersebut menjadi bentuk linier sehingga dapat ditangani dengan prosedur regresi nonlinier. model tersebut merupakan model linier dalam bentuk w�t. t tidak berdistribusi normal, sebab yang berdistribusi normal adalah w�t (draper dan smith, 1992:213). 7. regresi dalam pendekatan matriks model regresi yang paling sederhana adalah model regresi linier. model regresi linier sederhana terdiri dari satu variabel. model tersebut dapat digeneralisasikan menjadi lebih dari satu atau dalam k variabel. persamaan model regresi linier dengan d peubah adalah sebagai berikut: o uz = u2,2 = u�,� = >= u?,? = t (2.3) pengamatan mengenai o,,2,,�,…,,? dinyatakan masing-masing dengan o�,,�2,,��,…,,�? dan galatnya t�, maka persamaan (2.3) dapat dituliskan sebagai: o� uz = u2,�2 = u�,�� = >= u?,? = t� untuk,^ 1,2, . . ,�. dinotasikan dalam bentuk matriks, sehingga menjadi: _o2o�̀o a _ 1 ,22 … ,2?1 ,�2 … ,�? 1̀ `, 2 `… , ?a_ uzu2̀ u a= _t2t�̀t a siti tabi’atul hasanah 180 volume 2 no. 3 november 2012 misalkan b _o2o�̀o a � _ 1 ,22 … ,2?1 ,�2 … ,�? 1̀ `, 2 `… , ?a u _uzu2̀u a t _t2t�̀t a persamaan (2.11) dapat dinyatakan sebagai: b �u =t (2.4) dengan: b : vektor respon � x 1 � :matriks peubah bebas berukuran � x "d = 1% u : vektor parameter berukuran "d = 1% x 1 t : vektor galat ukuran � x 1 (sembiring,1995:134-135) 8. maximum likelihood statistik inferensia dapat dibagi dalam dua bagian besar, estimasi dan pengujian hipotesis. kedua inferensi tersebut masing-masing bertujuan untuk membuat pendugaan dan pengujian suatu parameter populasi dan informasi sampel yang diambil dari populasi tersebut. gujarati n. damodar (2010:131) menjelaskan bahwa metode dari estimasi titik (point estimation) dengan sifat-sifat teoritis yang lebih kuat dari pada metode ols adalah metode maximum likelihood (ml). fungsi likelihood dari � peubah acak ,2,,�,…,, didefinisikan sebagai fungsi kepadatan bersama dari n peubah acak. fungsi kepadatan bersama -4c,…4d ",2,…,, ;�%, yang mempertimbangkan fungsi dari �. jika ,2,…,, adalah sampel acak dari fungsi kepadatan -",,�%, maka fungsi likelihoodnya adalah -",2;�%-",�;�%…-", ;�% (mood, graybill and boes, 1986:278). maximum likelihood dapat diperoleh dengan menentukan turunan dari l terhadap parameternya dan menyatakannya sama dengan nol. dalam hal ini, akan lebih mudah untuk terlebih dahulu menghitung logaritma kemudian menentukan turunannya. dengan cara ini diperoleh: 1-"�2,�%f-",2,�%f� = >= 1-"� ,�%f-", ,�%f� 0 penyelesaian dari persamaan ini, untuk � dalam bentuk ,?, dikenal sebagai estimator maximum likelihood dari �. 9. metode statistik likelihood displacement (ld) metode ld adalah suatu metode yang dikembangkan dengan cara menghilangkan pengamatan yang diduga outlier. misalkan d adalah pengamatan dikumpulkan pada pengamatan tertentu, dengan d diduga sebagai outlier. indeks g? adalah kumpulan dari d yang diduga outlier. ld dari pengamatan yang mengandung outlier untuk uhi dengan variansi σj� adalah: ���� �uhik$l�� 2mw���uhi,$l��m * mw��3uhin��o,$l�"uhin��o%7p "2.5% dimana σj�"uhin��o adalah mle dari σj� ketika uhi diestimasi oleh uhin��o (makkulau, dkk, 2010:97). pembahasan 1. regresi nonlinier multiplikatif bentuk umum dari regresi nonlinier multiplikatif adalah dinyatakan sebagai berikut: o� uz,�2qc,��q8,�sqr …, ?q� …t� (3.1) persamaan (3.1) dapat dilinierkan dengan melogaritmanaturalkan persamaannya, sehingga modelnya menjadi: w�o� w�uz = u2 w�,2� = u� w�,�� = >= u? w�, ? = >= w�t� (3.2) dengan ^ 1,2,…,� dan d 1,2,…,� dalam penelitian ini diasumsikan bahwa variabel terikat "w�o% berdistribusi normal dengan mean # dan variansi $�. sehingga dalam persamaan (3.1) t berdistribusi log normal, karena yang berdistribusi normal adalah lnt. dengan menggunakan pendekatan matriks, diperoleh: b s 2i � s"?t2 %i u"?t2 %s 2i = t s 2 i (3.3) 2. estimasi parameter regresi noninier multiplikatif dari persamaan (3.3) diketahui bahwa bi "w�o2, w�o�,… , w�o %u adalah variabel random, karena diasumsikan berdistribusi normal, maka bi~!"�iui, ;$�% dengan �i "w�,z� , w�,2� ,…, w�,2?% dan ui "w�uz ,u2,…,u %u dimana ^ 1,2,…,� dan ; adalah matriks identitas. sehingga fungsi distribusi peluang gabungannya adalah -"bi|ui,$�% a 2w�x68c 01 c8y8"zi1{iqi%|"zi1{iqi% (3.4) sehinggga fungsi likelihoodnya adalah: �"ui,$�|bi% a 2w�x68c 01 c8y8"zi1{iqi%|"zi1{iqi% (3.5) dengan menggunakan metode maximum likelihood, estimasi parameter ui dan $� dari persamaan (3.5) adalah sebagai berikut: pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld) jurnal cauchy – issn: 2086-0382 181 uhi "�iu�i%12�iubi (3.6) dan $l� 1�"bi * �iui%u"bi * �iui% (3.7) estimator uhi mempunyai sifat-sifat: uhi mempunyai sifat unbias. bukti: uhi "�iu�i%12�iubi ��uhi� �""�iu�i%12�iubi% "�iu�i%12�iu�"bi% "�iu�i%12�iu�iui ;ui ui selanjutnya akan dibuktikan bahwa estimator uhi adalah estimator efisien. dikatakan estimator efisien apabila mempunyai nilai variansi yang terkecil. sehingga }~� �uhi� "�iu�i%12$� harus sekecil mungkin agar estimator uhi efisien. kemudian sifat estimator yang ketiga yaitu konsisten. dikatakan estimator yang konsisten jika w^� �∞ �"k�� *�k + t 1 sehingga: w^� �∞ �3�� * �����7� w^� �∞"�iu�i%12$� 0. sehingga dapat dikatakan bahwa uhimerupakan estimator yang konsisten selanjutnya menentukan fungsi likelihood dari estimator uhi dan $l� adalah sebagai berikut: ��uhi,$l�� 1"2/% �"$�% � 01 2�68"zi1{iqi%|"zi1{iqi% (3.8) fungsi likelihood ini kemudian dilogaritmakan. sehingga diperoleh: *�2ln"2/%* �2ln"$�%* �2 (3.9) 3. pendeteksian outlier pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld) dilakukan dengan cara menghilangkan pengamatan yang diduga mengandung outlier pada model. misalkan ada d pengamatan yang dikumpulkan dalam suatu himpunan tertentu, dengan d adalah pengamatan yang diduga mengandung outlier. dimana d + �. dan misalkan indeks g? adalah kumpulan dari d pengamatan yang diduga outlier dengan g? ^2, ^�,… , ^?, dan misalkan indeks ^? � �1,2,…,��. dengan mempertimbangkan pengamatan d dalam estimasi parameter, maka likelihood displacement untuk �� uhi,$l�dan ��n��o uhn��oi ,$ln��o� adalah: ���?"u i|$�% 2�ln��uhi,$l��* ln��uhn��oi ,$ln��o� �� (3.10) dimana uhiadalah maximum likelihood estimation dari u i dan $l� adalah maximum likelihood estimation pada keseluruhan pengamatan dan uhn��oi dan $ln��o� adalah mle dari u i dan $� ketika pengamatan dengan indeks g? dihilangkan. pada kasus khusus yaitu �2 "u2i,$2�% subset dari � "ui,$�%, maka fungsi likelihood displacement dapat dimodifikasi menjadi ����""u2i,$2�%|"u�i,$��%% 2mw� ��uhi,$l��m* mln��"uh2i,$l2�%n��o , �uh�i,$l���"uh2i,$l2�%n��o�� dengan: �"�uh�i,$l���"uh2i,$l2�%n��o% �~,"q8i,688%�3"uh2i,$l2�%n��o , �uh�i,$l���7 adalah memaksimumkan fungsi log likelihood pada parameter "u�i,$��% dengan "u2i,$2�% "uh2i,$l2�%n��o maka u2i uh2n��oi dan $2� $l2n��o� adalah maximum likelihood estimation dari "u2i,$2�% ketika pengamatan d dihilangkan. selanjutnya untuk keseluruhan data ketika d pengamatan pada himpunan g? dihilangkan maka modelnya menjadi: bn��oi �n��oi un��oi = tn��o (3.11) dengan bn��oi ~!"0,;$�% estimasi parameter un��oi dan $n��o� dari persamaan (3.11) dengan maximum likelihood diperoleh: uhn��oi uhi * "�iu�i%12���iu�; * ����12t�̂�i estimator uhn��oi adalah estimator tak bias. dan $ln��o� �� * d$� = 1� * dt�̂�u ����; *����12t�̂� dengan: ��� ���i "�iu�i%12���iu t�̂�i b��i * ���i uhi pada kasus khusus seperti yang telah dijelaskan maka estimasi dari $l� 3uhin��o7 dimana $l� 3uhin��o7 adalah maximum likelihood estimation dari $l� ketika ui diestimasi dengan uhin��o. dengan mensubtitusikan uhin��o untuk ui pada $l�, sehingga diperoleh: $l� 3uhin��o7 $l� = 1�t�̂�iu�; *����12���x �; * ����12t�̂�i selanjutnya menentukan fungsi likelihood dari uhin��o,$l� 3uhin��o7 diperoleh: siti tabi’atul hasanah 182 volume 2 no. 3 november 2012 �auhin��o,$l� 3uhin��o7c 132/ �7�$l� 3uhin��o7 �� 01 2�6j83q�i����7azi1{����i q����i c |azi1{����i q����i c fungsi likelihood ini kemudian dilogaritmakan. sehingga diperoleh: *�2w�2/ * �2w�$l� 3uhin��o7* �2 (3.12) 4. metode statistik likelihood displacemen (ld) likelihood displacement dari ui dan $� yang diberikan pada persamaan (2.5) adalah: ����"ui|$�% 2�ln��uhi,$l��* ln�auhin��o,$l� 3uhin��o7cp subtitusikan persamaan (3.9) dan (3.12) ke persamaan (2.5) maka: ����"ui|$�% 2�3*�2w�"2/% * �2w�"$�%7m * m3*�2w�2/ * �2w�$l� 3uhin��o77p 2�*�2w�"2/% * �2w�"$�%= �2w�2/ =m m�2 w�$l� 3uhin��o7p 2�*�2w�"$�%=�2w�$l� 3uhin��o7p *�w�$� = � w�$l� 3uhin��o7 ��*w�$� = w�$l� 3uhin��o7p ��w�$l� 3uhin��o7* w�$�p �w��$l� 3uhin��o7$� � �w��$l� = 1�t�̂�iu�; * ����12����; * ����12t�̂�i$� � misal ��� �; * ����12����; * ����12, maka: ���� �w��$l� = 1�t�̂�iu���t�̂�i$� � �w��$l�$l� = 1�t�̂�iu���t�̂�i$l� � �w��1 = 1�$l� t�̂�iu���t�̂�i � sehingga likelihood displacement yang diduga mengandung outlier adalah sebagai berikut: ���� �w��1 = 1�$l� t�̂�iu���t�̂�i � untuk menunjukkan keakuratan dari hasil metode ld dalam mendeteksi adanya outlier, maka digunakan uji statistik. uji statistik disini dilakukan dengan cara membandingkan mse dari metode ld dengan mse dari regresi pada umumnya (regresi tanpa outlier). statistik uji yang digunakan adalah λ ������������� ~���� dimana  �, ^ 1, 2,…,d, adalah nilai eigen dari ���. ketika nilai ������� lebih besar dari pada ������ maka nilai akan semakin besar. dari hasil uji statistik yang telah dijelaskan, maka diberikan uji hipotesis sebagai berikut: ¡z:g? adalah bukan outlier ¡2:g? adalah outlier ¡z ditolak jika ���� �� ������� dan ¡z diterima jika ���� �� + ������� . penutup berdasarkan pembahasan yang dipaparkan, dapat disimpulkan bahwa metode statistik likelihod displacement (ld) mampu mendeteksi adanya outlier pada regresi nonlinier multiplikatif. sebelum menerapkan metode ld terlebih dahulu harus melinierkan model dengan asumsi bahwa error berdisrtibusi normal kemudian mengestimasi parameter regresi nonlinier multiplikatif dengan metode maximum likelihood estimation. kemudian menerapkan metode statistik likelihood displacement, sehingga diperoleh hasil perumusan ���� likelihood displacement untuk pengamatan yang diduga mengandung outlier. keakuratan metode ld dalam mendeteksi adanya outlier ditunjukkan dengan uji statistik. yaitu dengan membandingkan mse dari ld dengan mse dari regresi pada umumnya. statistik uji yang digunakan adalah . hipotesis awal ditolak ketika ���� �� ������� , sehingga terbukti ���� adalah suatu outlier.. daftar pustaka [1] abdusysyakir. 2007. ketika kyai mengajar matematika. malang : uin-malang press. [2] al-asqolani, i. h. & al-imam, a. 2007. fathul baari penjelas kitab shahih al-bukhari (12). penj. amiruddin. jakarta: pustaka azzam. [3] al-mahally, i. j. & as-suyuthi, i. j. 1990. terjemah tafsir jalalain berikut asbaabun nuzul. bandung: sinar baru. pendeteksian outlier pada regresi nonlinier dengan metode statistik likelihood displacement (ld) jurnal cauchy – issn: 2086-0382 183 [4] al-maraghi, a. m. 1989. tafsir al-maraghi. semarang: cv. thoha putra. [5] amrullah, a. a. 1981. tafsir al-azhar. surabaya: yayasan latimojong [6] draper, n. & harry, s. 1992. analsis regresi terapan (edisi kedua). jakarta: pt. gramedia pustaka utama. [7] ghoffur, a. dkk. 2007. tafsir ibnu katsir (8). bogor: pustaka imam syafi’i. [8] gujarati, d. n. 2007. dasar-dasar ekonometri jilid 1 edisi ke-3. jakarta: penerbit erlangga. [9] hasan, m. i. 2002. pokok-pokok materi metodologi penelitian dan aplikasinya. jakarta:ghalia indonesia. [10] herrhyanto, n. 2007. http: // www. herryanto. blog/ statistika. matematika. i. html (diunduh pada tanggal 26 januari 2012). [11] makkulau, s. l. & purhadi, m. m. 2010. pendeteksian outlier dan penentuan faktorfaktor yang mempengaruhi produksi gula dan tetes tebu dengan metode likelihood displacement statistic-lagrange. jurnal teknik industri, volume 12. no. 2 desember 2010, 95-100. [12] mood, m alexander dkk.1986. introduction to the theory of statistics. mcgrawhill book company.sembiring, rk. 1995. analisis regresi. bandung: itb. [13] murray & larry. 2007. statistik edisi ke-3. jakarta: erlangga. [14] shihab, m. q. 2003. tafsir al-mishbah volume 14. jakarta: lentera hati. [15] sudjana. 2005. metoda statistika. bandung: transito. [16] turmudi & harini, s. 2008. metode statistika pendekatan teoritis dan aplikatif. malang: uin-press. [17] wibisono, y. 2005. metode statistik. yogyakarta: gadjah mada university press. [18] xu, a. & steiner. 1998. outlier detection methods in multivariate regression models. journal of multivariate analysis, 65, 1998, pp. 195-208. [19] yitnosumarto, s. 1990. dasar-dasar statistika. jakarta: rajawali. estimasi nonlinear least trimmed squares (nlts) pada model regresi nonlinier yang dikenai outlier nur laili arofah1, sri harini2 1mahasiswa jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang e-mail: arofah2206@gmail.com1, sriharini21@yahoo.co.id2 abstrak fungsi produksi constant elasticity of substitution (ces) adalah model regresi nonlinier intrinsik yang sering digunakan untuk mengestimasi suatu data dalam sebuah industri. model regresi nonlinier intrinsik merupakan suatu bentuk regresi nonlinier yang tidak dapat dilinierkan, sehingga untuk mengestimasi parameter  model statistik nonlinier yang digunakan adalah nonlinear least square (nls) dengan pendekatan deret taylor orde satu yang digunakan dalam iterasi gauss newton. salah satu masalah yang sering ditemui dalam analisis data adalah outlier, keberadaan outlier dalam analisis data sangat mempengaruhi hasil analisis sehingga kurang tepat dan sifat estimasi menjadi bias. salah satu metode regresi yang kebal terhadap outlier adalah metode nonlinear least trimmed squares. penelitian ini bertujuan untuk mengetahui bagaimana sifat-sifat parameter fungsi produksi ces yang dikenai outlier. hasil penelitian menunjukkan bahwa parameter fungsi produksi ces yang dikenai outlier bersifat bias, tidak konsisten. sehingga fungsi produksi ces yang tidak dikenai outlier lebih baik dari pada yang dikenai outlier. kata kunci: model statistik nonlinier, estimasi parameter, fungsi produksi constant elasticity of substitution (ces), outlier, metode nonlinear least trimmed squares (nlts), iterasi gauss newton. abstract constant elasticity of substitution (ces) production function is the intrinsic nonlinear regression models that are often used to estimate the data in an industry. intrinsic nonlinear regression model is a kind of nonlinear regression that can not be linearized, so as to estimate the  parameters nonlinear statistical model used was nonlinear least squares (nls) using a first order taylor series approach used in the gauss newton iteration. one of the problems often encountered in the analysis of data is an outlier, the presence of outliers in the data analysis greatly influence the results of the analysis so it becomes less valid and the estimation become biased. one method that is resistant to outliers regression is a method of nonlinear least trimmed squares. this research aims to determine the characteristics of parameter ces production function which contains outlier. the result shows that parameter of the production function ces which contains outliers are bias, inconsistent. so the ces production function which does not contain outliers better than the are contains outliers. keywords: nonlinear statistical model, parameter estimation, constant elasticity of substitution (ces) production function, outliers, nonlinear least trimmed square (nlts) method, gauss newton iteration. pendahuluan fungsi produksi constant elasticity of substitution (ces) adalah salah satu bentuk model regresi nonlinier intrinsik dan tidak dapat dilinierkan, fungsi produksi ces banyak diaplikasikan dalam bidang industri untuk menganalis data dan mengambil kesimpulan. namun dalam menganalis data sering ditemukan data pencilan (outlier), outlier mailto:arofah2206@gmail.com1 mailto:sriharini21@yahoo.co.id2 nur laili arofah 23 volume 4 no.1 november 2015 merupakan data yang memiliki nilai residu lebih besar disbanding dengan data yang lainnya. keberadaan data outlier ini sangat mempengaruhi terhadap hasil kesimpulan. jika keberadaan outlier pada suatu data tidak ditangani dengan benar maka dimungkinkan hasil kesimpulannya kurang tepat. oleh karena itu, diperlukan suatu metode yang dapat mengatasi serta mendeteksi terhadap model yang dikenai outlier. metode tersebut harus mampu memberikan hasil yang baik sehingga outlier tersebut tidak lagi berpengaruh buruk terhadap pendugaan parameter dan diharapkan mampu menghasilkan model yang cocok untuk menganalisis data. pada penelitian terdahulu diketahui metode robust yang digunakan untuk mendeteksi outlier adalah metode least trimmed squares, metode least trimmed squares merupakan metode yang mampu mengatasi outlier yang timbul baik dari variabel terikat (dependent) maupun dari variabel bebas (independent) dengan baik pada model regresi linier. namun pada penelitian ini, model regresi yang diteliti adalah model regresi nonlinier yang dikenai outlier, sehingga metode least trimmed squares tidak dapat diterapkan, sehingga pada penelitian ini penulis menggunakan metode nonlinear least trimmed squares. sifat dari metode ini hampir sama dengan metode least trimmed squares yaitu mengatasi outlier dengan cara meminimumkan jumlah kuadrat residu dari h pengamatan, penduga yang dihasilkan oleh metode ini tidak terpengaruh oleh keberadaan outlier. analisis sifat penduga nonlinear least trimmed squares (nlts) dilakukan dengan melihat nilai bias, ragam, dan mean square error (mse). tinjauan pustaka regresi nonlinier model nonlinier merupakan bentuk hubungan antara peubah respon dengan peubah penjelas yang tidak linier dalam parameter. menurut [1], model umum regresi nonlinier adalah: 𝑌𝑖 = 𝑓(𝑋𝑖𝑗 , 𝛽𝑖𝑗 ) + 𝑒𝑖 dengan: 𝑌𝑖 : nilai respon ke-i=1,2,…,n 𝑋𝑖𝑗 : nilai prediktor ke-j pengamatan ke-i 𝛽𝑖𝑗 : nilai parameter ke-u; u=1,2,…,k (k=p+1) 𝑒𝑖 : nilai sisaan ke-i; n : ukuran contoh k : banyaknya parameter p : banyaknya peubah prediktor 𝑓(𝑋𝑖𝑗 , 𝛽𝑖𝑗 ) : fungsi regresi nonlinier menurut [1], model regresi nonlinier diklasifikasikan menjadi linier intrinsik dan nonlinier intrinsik. model linier intrinsik dapat ditransformasikan menjadi bentuk linier, sedangkan model nonlinier intrinsik tidak dapat ditransformasikan menjadi bentuk linier. fungsi produksi constant elasticity of substitution (ces) fungsi produksi adalah suatu hubungan matematis yang menggambarkan suatu cara dengan jumlah dari hasil produksi tertentu tergantung dari jumlah input tertentu yang digunakan [2]. menurut joesron dan fathorrozi dalam [3], terdapat beberapa bentuk fungsi produksi antara lain fungsi produksi cobb-douglas, fungsi produksi constant elasticity of substitution (ces) dan fungsi produksi leontief. fungsi constant elasticity of substitution (ces) dikembangkan oleh arrow, chenery, minhan, dan solow pada tahun 1961. bentuk fungsi produksi constant elasticity of subtitution (ces): 𝑄𝑡 = 𝛽1[𝛽2𝐿𝑡 𝛽3 + (1 − 𝛽2)𝐾𝑡 𝛽3 ] 𝛽4 𝛽2 + 𝑒𝑡 dimana 𝑄𝑡 adalah hasil keluaran (output) produksi pada saat t sebagai nilai respon, 𝐿𝑡 adalah tenaga kerja (labor) pada saat t sebagai variabel pertama, 𝐾𝑡 adalah modal (capital) pada saat t sebagai variabel kedua. parameter-parameter 𝛽 berhubungan secara tak linier dengan variabelnya [4]. sifat-sifat estimasi parameter [5] menyatakan suatu penaksiran dinilai baik apabila memenuhi kriteria unbias, efisien dan konsisten. estimator tak bias (unbiased estimator) estimator dikatakan tidak bias jika mean distribusi sampel adalah sama dengan mean parameter 𝛽 yang tidak diketahui. 𝐸[�̂�] = 𝛽 estimasi nonlinear least trimmed squares (nlts) pada model regresi nonlinier yang dikenai outlier cauchy – issn: 2086-0382/e-issn: 2477-3344 24 dimana 𝛽 merupakan parameter yang diestimasikan, dan �̂� sebagai estimator titik. efisien jika �̂�adalah estimator 𝛽 tak bias, dan tidak ada estimator tak bias lainnya yang memilki varian yang lebih kecil, maka �̂� dikatakan paling efisien atau minimum variance unbiased estimator 𝛽. �̂�1 dan �̂�2 adalah 2 estimator tak bias, maka �̂�1 dikatakan lebih efisien dari �̂�2 jika 𝑉𝑎𝑟 (�̂�1 ) < 𝑉𝑎𝑟 (�̂�2 ). konsisten suatu penduga �̂� dikatakan konsisten apabila semakin mendekati parameter yang diduga. secara matematis dapat dituliskan �̂� penduga yang konsisten karena �̂� → 𝛽 dengan 𝑛 → ∞ atau dengan pernyataan peluang, lim 𝑛→∞ 𝑃(|�̂� − 𝛽| < 𝜀) = 1. iterasi gauss newton menurut [6], iterasi gauss newton merupakan proses iterasi yang akan menghasilkan solusi untuk permasalahan regresi nonlinier. iterasi gauss newton menggunakan deret taylor untuk mendekati model regresi nonlinier dengan persamaan linier dan pada semua proses iterasi dibutuhkan penduga awal. myers dalam [7] menjelaskan penduga awal dapat ditentukan lewat informasi. kegagalan nilai awalan dalam menaksir dapat dikarenakan penduga awal terlalu dekat, terlalu kecil. penduga awal yang baik dapat membuat penduga konvergen jauh lebih cepat [1] outlier secara umum outlier dapat diartikan data yang tidak mengikuti pola umum pada model atau data yang keluar dari model dan tidak berada dalam daerah selang kepercayaan [8]. keberadaan outlier akan menimbulkan beberapa masalah, diantaranya outlier akan mengubah atau mengaburkan kesimpulan yang dibuat oleh peneliti karena nilai penduga parameternya bersifat bias [1]. menurut [9] metode yang digunakan untuk mengidentifikasi adanya outlier yang berpengaruh dalam koefisien regresi antara lain identifikasi dengan grafis, dengan metode laverage value dan the difference in fit statistics. nonlinear least trimmes squares (nlts) rousseuw (1997) dalam [10] menyatakan salah satu metode pendugaan parameter model regresi terhadap data yang mengandung outlier (pencilan) adalah metode penduga least trimmed square (lts). metode ini merupakan salah satu metode pendugaan parameter pada regresi robust yang kekar terhadap keberadaan outlier (pencilan). metode least trimmed square (lts) mempunyai prinsip pendugaan parameter yang sama dengan metode kuadrat terkecil, yaitu meminimumkan jumlah kuadrat galat. hanya saja pada metode least trimmed square (lts), jumlah kuadrat galat yang diminimumkan adalah jumlah kuadrat galat dari h pengamatan yang dianggap bukan outlier (pencilan). menurut [11] metode penduga least trimmed square (lts) memiliki beberapa kekurangan, yaitu ketika metode ini diterapkan pada regresi nonlinier, maka hasil estimasinya kurang maksimal. sehingga ditemukan metode penduga parameter regresi robust untuk regresi nonlinier terhadap data yang mengandung outlier yaitu metode nonlinear least trimmed squares. cizek dalam [12] menjelaskan bahwa penduga parameter nonlinear least trimmed squares dapat diselesaikan dengan metode nonlinier least squares untuk h pengamatan yang terletak dalam interval 𝑛 2 ≤ ℎ ≤ 𝑛. penduga nonlinear least trimmed squares didefinisikan sebagai berikut: �̂�(𝑁𝐿𝑇𝑆,ℎ) = arg 𝑚𝑖𝑛 𝛽 ∈ 𝑅𝑘 ∑ 𝑒[𝑖] 2 ℎ 𝑖=1 = arg 𝑚𝑖𝑛 𝛽 ∈ 𝑅𝑘 ∑{𝑌𝑖 − 𝑓(𝑋𝑖𝑗 , 𝛽𝑖1)} 2 ℎ 𝑖=1 dimana: arg : argument min : minimum 𝑅𝑘 1 : ruang euclidis berdimensi k (k = p+1) h : konstanta pemotongan (trimming constant) 𝑒[𝑖] 2 : statistik peringkat ke-i untuk 𝑒𝑖 2(𝑒[1] 2 ≤ 𝑒[2] 2 ≤ ⋯ 𝑒[𝑛] 2 ) nilai h yang digunakan ketika jumlah data lebih besar atau sama dengan 30 maka untuk mendapatkan nilai h adalah sebagai berikut: nur laili arofah 25 volume 4 no.1 november 2015 ℎ = [ 𝑛 2 ] + [ 𝑝 + 1 2 ] 𝑎𝑡𝑎𝑢 ℎ = [ 𝑛 2 ] + [ 𝑝 + 2 2 ] sejumlah h pengamatan yang memiliki kuadrat sisaan 𝑒[1] 2 ≤ 𝑒[2] 2 ≤ ⋯ 𝑒[𝑛] 2 digunakan untuk menduga parameter nonlinear least trimmed squares. metodologi penelitian data penelitian data dalam penelitian ini berupa data sekunder yang bersumber dari buku ekonometrika teori dan praktik eksperimen dengan matlab [4]. variabel yang diteliti yaitu variabel y adalah output dan labor ( 1 x ), capital ( 2 x ). tahap penelitian 1. estimasi parameter fungsi produksi constant elasticity of substitution (ces) dengan metode nonlinear least trimmed squares (nlts) a. membentuk model fungsi produksi ces b. estimasi parameter model fungsi produksi ces dengan menggunakan nls c. model fungsi produksi ces dikenai outlier d. estimasi parameter model fungsi produksi ces dikenai outlier dengan menggunakan nls. e. menentukan sifat-sifat estimasi parameter (tak bias, efisien, konsisten) 2. aplikasi fungsi produksi constant elasticity of substitution (ces) pada data pengaruh labor dan capital terhadap output produksi 1. estimasi parameter regresi robust menggunakan nonlinear least trimmed squares (nlts), dengan: a. deteksi outlier a. boxplot b. leverage value c. the difference in fit statistics (dfits) b. menghitung nilai h c. membuat urutan data ke-i untuk 𝑒𝑖 2 d. urutkan hasil kuadrat residu dari yang terkecil ke yang terbesar e. mengurutkan ℎ(𝑋𝑖,𝑗 , 𝑌𝑖) berdasarkan 𝑒[𝑖] 2 f. melakukan estimasi parameter pada model fungsi produksi ces dengan menggunakan iterasi gauss newton a. estimasi parameter pada data sebelum data outlier dibuang b. estimasi parameter pada data setelah data outlier dibuang g. pembahasan hasil analisis. h. kesimpulan. pembahasan estimasi parameter fungsi produksi constant elasticity of substitution (ces) dengan metode nonlinear least trimmed squares (nlts) bentuk fungsi produksi constant elasticity of substitution (ces) adalah sebagai berikut: 𝑄𝑡 = 𝛽1[𝛽2𝐿𝑡 𝛽3 + (1 − 𝛽2)𝐾𝑡 𝛽3 ] 𝛽4 𝛽2 + 𝑒𝑡 (1) dengan [𝐿, 𝐾] = 𝑋 maka persamaan (1) dapat ditulis menjadi: 𝑄 = 𝑓(𝑋, 𝛽) (2) persamaan (1) dapat ditulis menjadi: 𝑦 = 𝑄 + 𝑒 atau 𝑦 = 𝑓(𝑋, 𝛽) + 𝑒 (3) dimana persamaan (3) merupakan bentuk fungsi produksi ces yang tidak dikenai outlier. untuk menentukan penduga parameter dari fungsi produksi ces yang tidak dikenai outlier menggunakan bentuk regresi nonlinier dari fungsi produksi ces dengan menggunakan persamaan (3). menurut [13]metode nonlinear least trimmed squares mengestimasi parameter dengan meminimumkan nilai residual. setelah diminimumkan, hasil nilai residual untuk penaksiran parameter 𝛽 pada regresi nonlinier intrinsik ini dilakukan dengan proses iterasi. sehingga diperoleh bentuk umum taksiran modelnya sebagai berikut: 𝛽𝑛+1 = 𝛽𝑛 + (𝑍(𝛽𝑛 )𝑇𝑍(𝛽𝑛 )) −1 𝑍(𝛽𝑛 )𝑇 (𝑦 − 𝑓(𝑋, 𝛽𝑛 )) untuk fungsi produksi ces yang dikenai outlier, bentuk persamaannya sebagai berikut: 𝑦 = 𝑓(𝑋𝜑, 𝛽𝜑) + 𝑒 dimana 𝜑 merupakan outlier. bentuk umum taksiran modelnya sebagai berikut: (𝛽𝜑)𝑛+1 = (𝑍((𝛽𝜑)𝑛 )𝑇𝑍((𝛽𝜑)𝑛 )−1)−1𝑍((𝛽𝜑)𝑛 )𝑇 (𝑦 − 𝑓(𝑋𝜑, (𝛽𝜑)𝑛 )) + 𝑍((𝛽𝜑)𝑛)(𝛽𝜑)𝑛 estimasi nonlinear least trimmed squares (nlts) pada model regresi nonlinier yang dikenai outlier cauchy – issn: 2086-0382/e-issn: 2477-3344 26 menentukan sifat-sifat estimasi parameter suatu parameter dikatakan parameter yang baik jika memenuhi sifat tak bias, efisien, dan konsisten. a. sifat tak bias estimator dikatakan tidak bias jika mean distribusi sampel adalah sama dengan mean parameter 𝛽 yang tidak diketahui. estimasi parameter fungsi produksi ces yang tidak dikenai outlier bersifat tak bias jika 𝐸[�̂�𝑛+1] = 𝛽, dengan melakukan ekspeksi terhadap 𝛽 yang tidak dikenai outlier didapatkan: 𝐸[�̂�𝑛+1] = 𝐸 [𝛽𝑛 + (𝑍(𝛽𝑛 )𝑇 𝑍(𝛽𝑛 )) −1 𝑍(𝛽𝑛 )𝑇 ] + 𝐸[𝑒] sehingga �̂�𝑛+1merupakan penduga yang tak bias bagi 𝛽 yang tidak dikenai outlier. namun pada 𝛽 yang dikenai outlier bersifat bias karena 𝐸[(𝛽𝜑)𝑛+1] ≠ 𝛽𝜑 b. efisien penduga yang efisien adalah suatu penduga yang memiliki nilai variansi minimum. akan dibandingkan nilai varian 𝛽 yang tidak dikenai outlier dengan nilai varian 𝛽 yang dikenai outlier. estimasi parameter fungsi produksi ces yang tidak dikenai outlier memiliki nilai varian 𝑉𝑎𝑟(�̂�𝑛+1) = 𝝈2𝑫𝑫𝑻, dengan 𝑫 = (𝑍(𝛽𝑛 )𝑇 𝑍(𝛽𝑛)) −1 𝑍(𝛽𝑛 )𝑇. sedangkan estimasi parameter fungsi produksi ces yang dikenai outlier memiliki nilai varian 𝑉𝑎𝑟 ((�̂�𝜑) 𝑛+1 ) = 𝝈2𝑪𝑪𝑻, dengan 𝑪 = (𝑍((𝛽𝜑)𝑛 )𝑇𝑍((𝛽𝜑)𝑛 )) −1 𝑍((𝛽𝜑)𝑛 )𝑇 + 𝑍(𝛽𝜑)𝑛 sehingga diperoleh 𝑉𝑎𝑟(�̂�𝑛+1) lebih evisien disbanding 𝑉𝑎𝑟 ((�̂�𝜑) 𝑛+1 ), karena 𝑉𝑎𝑟(�̂�𝑛+1) < 𝑉𝑎𝑟 ((�̂�𝜑) 𝑛+1 ). c. konsisten suatu estimasi disebut konsisten jika 𝐸[(�̂� − 𝐸[�̂�])] 2 → 0 jika 𝑛 → ∞. estimasi parameter fungsi produksi ces yang tidak dikenai outlier adalah: 𝐸[(�̂�𝑛+1 − 𝐸[�̂�𝑛+1])(�̂�𝑛+1 − 𝐸[�̂�𝑛+1])] = 0 sedangkan estimasi parameter fungsi produksi ces yang dikenai outlier adalah: 𝐸 [((�̂�𝜑) 𝑛+1 − 𝐸 [(�̂�𝜑) 𝑛+1 ]) ((�̂�𝜑) 𝑛+1 − 𝐸 [(�̂�𝜑) 𝑛+1 ])] = 𝐸[(𝛽𝜑)𝑛 (𝑍((𝛽𝜑)𝑛 )(𝛽𝜑)𝑛 − 𝑍((𝛽𝜑)𝑛 )(𝛽𝜑)𝑛)𝑇] sehingga dapat disimpulkan bahwa �̂�𝑛+1 lebih konsisten dibandingkan dengan (�̂�𝜑) 𝑛+1 . aplikasi fungsi produksi constant elasticity of substitution (ces) pada data. a. estimasi parameter pada model fungsi produksi constant elasticity of substitution (ces) dengan menggunakan iterasi gauss newton sebelum melakukan estimasi parameter, terlebih dahulu dideteksi keberadaan outlier, metode yang lebih akurat untuk deteksi outlier adalah laverage value dan didapatkan data yang merupakan outlie adalah data ke12, ke-17, ke-18, ke-21, ke-22, ke-25 dan ke30. langkah selanjutnya yaitu memotoh sebesar 13 data yang memiliki nilai residu besar. estimasi parameter dilakukan menggunakan data sebanyak 17, dimana data tersebut merupakan data setelah outlier dibuang. hasil estimasi parameter sebelum data outlier dibuang sebagai berikut: 𝑄 = 𝐴 + [𝐵𝐿𝑐 + (1 − 𝐵)𝐾 𝑐 ] 𝐷 𝐵 dengan: 𝐴 = 1.372418099933 𝐵 = 0.389438365935 𝐶 = 0.309978135216 𝐷 = 0.986867727926 sedangkan hasil estimasi parameter setelah outlier dibuang adalah: 𝑆 = 𝑀 + [𝑁𝐿𝑐 + (1 − 𝑁)𝐾 𝑃] 𝑅 𝑁 𝑀 = 1.372418099933 𝑁 = 0.389438365935 𝑃 = 0.309978135216 𝑅 = 0.986867727926 dari ke-2 hasil estimasi diatas, estimasi pada data setelah outlier dikeluarkan lebih baik dibanding sebelum outlier dikeluarkan. penutup kesimpulan 1. untuk mendeteksi keberadaan outlier pada fungsi produksi constant elasticity of substitution (ces) adalah dengan melihat nilai leverage value, the difference in fit statistics (dfits) atau dengan boxplot. kemudian dilakukan estimasi parameter dengan menggunakan metode nonlinear least trimmed squares, yang menghasilkan estimasi parameter sebagai berikut: nur laili arofah 27 volume 4 no.1 november 2015 a. estimasi parameter pada model constant elasticity of substitution (ces) yang tidak dikenai outlier 𝛽𝑛+1 = 𝛽𝑛 + (𝑍(𝛽𝑛 )𝑇𝑍(𝛽𝑛 )) −1 𝑍(𝛽𝑛 )𝑇 (𝑦 − 𝑓(𝑋, 𝛽𝑛 )) b. estimasi parameter pada model constant elasticity of subsitution yang dikenai outlier (𝛽𝜑)𝑛+1 = (𝑍((𝛽𝜑)𝑛 )𝑇 𝑍((𝛽𝜑)𝑛 )−1)−1 𝑍((𝛽𝜑)𝑛)𝑇 (𝑦 − 𝑓(𝑋𝜑, (𝛽𝜑)𝑛 )) + 𝑍((𝛽𝜑)𝑛 )(𝛽𝜑)𝑛 2. sifat-sifat estimasi parameter pada model constant elasticity of subsitution (ces) yang tidak dikenai outlier lebih baik dibandingkan dengan sifat-sifat estimasi parameter yang dikenai outlier, hal ini terbukti bahwa pada model yang tidak memut outlier sifa-sifat estimasi yaitu unbias, konsisten dan efisien terpenuhi, sedangkan pada model yang dikenai outlier sifat-sifat estimasi tidak terpenuhi. saran pada penelitian ini, model yang digunakan adalah fungsi produksi constant elasticity of substitution (ces) yang dikenai outlier dan yang tidak dikenai outlier dan untuk mengestimasi parameter pada fungsi produksi ini digunakan metode nonlinear least trimmed squares (nlts). bagi pembaca yang ingin melakukan penelitian serupa, maka disarankan menggunkan model nonlinier intrinsik yang lainnya seperti fungsi produksi leontief. daftar pustaka [1] n. draper dan h. smith, analsis regresi terapan, jakarta: pt gramedia pustaka utama, 1992. [2] f. purniawati, aplikasi fungsi produksi cobbdouglas dalam menestimasi pendapatan pajak hotel kota surakarta berdasarkan jumlah tenaga kerja dan pengunjung hotel, surakarta: skripsi tidak dipublikasi, 2009. [3] h. wibisono, analisis efisiensi usaha tani kubis, diponegoro: skripsi tidak dipublikasi, 2011. [4] a. aziz, ekonometrika teori & praktik eksperimen dengan matlab, malang: uin maliki press, 2010. [5] s. adiningsih, statistik edisi pertama, yogyakarta: bpfe-yogyakarta, 2009. [6] j. n. n. kutner c.h, applier linier regression models, new york: mc graw hill, 2004. [7] d. permatasari, pemodelan kurva imbal hasil obligasi korporasi rating aa dan a dengan nelson siegel svensson dan cubic spline smoothing, surakarta: skripsi tidak dipublikasikan, 2009. [8] r. sembiring, analisis regresi, bandung: itb, 1995. [9] soemartini, pencilan (outlier), bandung: universitas padjajaran, 2007. [10] a. azizah, analisis sifat penduga least trimmed squares (lts) pada regresi linier berganda yang mengandung pencilan dengan berbagai ukuran contoh, malang: skripsi tidak dipublikasikan, 2013. [11] p. cizek, “nonlinear least trimmed,” jcmf 2002, pp. 78-86, 2002. [12] r. sundyni, sifat penduga nonlinear least square (nls) dan nonlinear least trimmed square (nlts) akibat pertambahan banyaknya pencilan dan ukuran contoh, malang: skripsi tidak dipublikasi, 2014. [13] c. chen, “robust regression and outlier detection with robustreg procedur,” proceedings, pp. 27-65, 2002. pendahuluan tinjauan pustaka regresi nonlinier fungsi produksi constant elasticity of substitution (ces) sifat-sifat estimasi parameter estimator tak bias (unbiased estimator) efisien konsisten iterasi gauss newton outlier nonlinear least trimmes squares (nlts) metodologi penelitian data penelitian tahap penelitian pembahasan menentukan sifat-sifat estimasi parameter aplikasi fungsi produksi constant elasticity of substitution (ces) pada data. penutup kesimpulan saran daftar pustaka analisis metode binomial dipercepat pada perhitungan harga opsi eropa istiqomah, abdul aziz jurusan matematika universitas islam negeri maulana malik ibrahim malang e-mail: thifa_31@yahoo.com abstrak model umum yang digunakan dalam perhitungan harga opsi eropa adalah model black scholes. kemudian ditemukan suatu metode baru yang merupakan aproksimasi dari model black scholes yaitu metode binomial. akan tetapi, perhitungan harga opsi eropa menggunakan metode binomial membutuhkan partisi waktu yang banyak untuk bisa mendekati model kontinu black scholes. untuk mempercepat kekonvergenan aproksimasi harga opsi eropa maka digunakan pengembangan dari model binomial yaitu binomial dipercepat. langkah yang dilakukan dalam metode binomial dipercepat adalah melakukan pemulusan kurva harga opsi yang disebut dengan middle of tree (mot). sebelum melakukan pemulusan kurva tersebut yang dilakukan terlebih dahulu adalah memisahkan partisi waktu yang digunakan yaitu partisi waktu ganjil dan genap. asumsi yang digunakan pada mot adalah dengan meletakkan harga ketentuan di tengah pohon binomial pada saat jatuh tempo. dari asumsi tersebut didapatkan parameter u dan p yang akan digunakan dalam pemulusan kurva mot. dengan menggunakan parameter mot tersebut diperoleh hasil dari harga opsi eropa yang bisa mendekati harga kekonvergenan black scholes dengan partisi yang lebih sedikit dibandingkan dengan menggunakan metode binomial. kata kunci: opsi eropa, binomial,binomial dipercepat, middle of tree (mot) abstract a common model which is used the calculation of the price of european option is black scholes model. afterwards, there is a new model which in approximation of black scholes model. this model is called as binomial model. but then, the calculation of the price of european option with binomial model requires many iterations to approach the continue model of black scholes. the development of binomial model, accelerated binomial, is used accelerate the convergence of the approximation of european option. on of steps in the accelerated binomial method is the middle of tree (mot). mot is the smoothing of option price’s curve. before doing the smoothing of curve, the first step that must be done is separating the used time; odd and even time. the assumption that is used in mot is placing the price provision among the binomial tree on the maturity time. the result the assumption u and p that will be applied in the smoothing of curve. the using of mot parameter produces a result from the price of european option that convergences to black scholes faster than using binomial model. keywords: european option, binomial, accelerated binomial, middle of tree (mot) pendahuluan pada penelitian sebelumnya tentang perhitungan harga opsi eropa menggunakan metode binomial telah dijelaskan beberapa bentuk yang digunakan untuk menghitung harga opsi eropa. salah satu bentuk yang digunakan dalam perhitungan harga opsi eropa adalah metode binomial bentuk crr. metode binomial bentuk crr ini untuk pertama kali dikembangkan secara simultan oleh cox, ross dan rubinstein [1] serta rendlemen dan bartter [2] dengan mengasumsikan bahwa dalam suatu interval waktu, harga saham akan naik sebesar faktor u (up) dan akan turun sebesar faktor d (down) karena dipengaruhi oleh faktor suku bunga. selanjutnya crr mempertimbangkan bahwa pergerakan harga saham juga dipengaruhi faktor volatilitas. semakin banyak partisi waktu yang digunakan pada metode binomial crr, maka harga opsi yang didapat akan semakin mendekati model kontinyu black scholes [3]. akan tetapi, untuk mendapatkan harga opsi yang mendekati model kontinyu black scholes diperlukan waktu yang cukup lama karena dengan partisi waktu yang semakin banyak maka proses perhitungan harga opsi juga akan semakin banyak. setiap partisi waktu yang berubah pada metode binomial crr mengakibatkan keadaan strike price selalu berubah terhadap node pada waktu jatuh tempo. ini menyebabkan terjadinya osilasi (naik turun) harga opsi terhadap partisi analisis metode binomial dipercepat pada perhitungan harga opsi eropa cauchy – issn: 2086-0382 109 waktu, sehingga kekonvergenan terhadap black scholes sangat lambat [4]. keadaan ini telah dibuktikan pada penelitian sebelumnya tentang perhitungan opsi eropa dengan metode binomial crr. jika menggunakan partisi waktu kecil maka harga saham pada waktu jatuh tempo yang diperoleh sangat kecil sehingga akan menyebabkan keadaan dimana antara opsi call dan opsi put tidak saling imbang. salah satu metode yang digunakan untuk mempercepat kekonvergenan harga opsi metode binomial crr terhadap harga opsi metode black scholes adalah dengan melakukan pemulusan kurva terhadap pohon binomial crr atau biasa disebut dengan middle of tree (mot). dengan melakukan pemulusan kurva binomial crr maka peneliti akan mendapat harga saham pada waktu jatuh tempo yang lebih besar dari metode binomial crr. dengan nilai harga saham pada waktu jatuh tempo yang lebih besar maka dengan partisi waktu berapapun akan didapatkan nilai k yang selalu ada di tengah pohon binomial pada saat jatuh tempo. hal ini menyebabkan nilai opsi call ataupun put selalu tidak nol sehingga kedua opsi tersebut dapat memiliki peluang untuk mendapatkan untung. tinjauan teori 1. binomial definisi 2.1.1 eksperimen binomial: suatu eksperimen dinamakan eksperimen binomial bila dan hanya bila eksperimen yang bersangkutan terdiri dari percobaan-percobaan binomial. anggaplah p adalah probabilitas bahwa suatu kejadian akan terjadi dalam suatu percobaan tunggal sembarang (disebut peluang keberhasilan), maka q =1–p adalah probabilitas bahwa suatu kejadian akan gagal dalam setiap satu percobaan (disebut peluang kegagalan). probabilitas bahwa suatu kejadian akan terjadi tepat x kali dalam n percobaan (artinya, keberhasilan-keberhasilan (sukses) dan n x kegagalan akan terjadi) ditentukan oleh fungsi probabilitas       ! ! ! x n x x n x n n f x p x x p q p q x x n x           dimana variabel acak x melambangkan jumlah keberhasilan dalam n percobaan dan x = 0, 1,…, n. 2. opsi opsi (option) adalah sebuah hak atau suatu kontrak antara writer (penjual) dan holder (pembeli) pemegang opsi yang memberikan hak, kepada holder (pemegang option) untuk membeli atau menjual suatu aset pokok (underlying asset) pada suatu tanggal tertentu untuk suatu harga tertentu. ada dua tipe dasar opsi yaitu call dan put. opsi call adalah hak untuk membeli sejumlah tertentu suatu underlying asset dengan harga sebesar strike price, pada waktu jatuh tempo (maturity date). sedangkan opsi put adalah hak untuk menjual sejumlah tertentu suatu underlying asset dengan harga sebesar strike price, pada waktu jatuh tempo (maturity date) [5]. kontrak opsi adalah suatu perjanjian yang memberikan hak kepada holder untuk membeli suatu underlying asset. pada tingkat harga tertentu (striking price) pada saat tanggal tertentu (maturity date). seorang holder suatu opsi harus membuat suatu keputusan apa yang akan ia lakukan terhadap tanggungan kontrak hak opsi ini. keputusannya akan ditentukan pada situasi pasar, dan tipe opsi ini. misalkan pada call opsi eropa, dia dapat mengabaikan opsi ini bila harga saham (stock price) di pasar pada waktu jatuh tempo (maturity date) lebih rendah dari pada harga call opsi (strike price), karena tidak dapat memberikan keuntungan. ia lebih baik membeli saham serupa di pasar dengan harga yang lebih rendah dari pada membelinya pada writer dengan harga strike price. menurut seydel [3] jika st adalah harga saham di pasar pada waktu t dan k adalah strike price maka keuntungan atau nilai payoff untuk kedua jenis opsi di atas diberikan oleh: untuk opsi call    max , 0t tc s k s k      (1) sedangkan untuk opsi put diberikan oleh    max , 0t tp k s k s      (2) writer memperoleh keuntungan dari biaya atau harga opsi dari holder, baik holder melaksanakan haknya maupun tidak melaksanakan haknya. holder (pemegang opsi) akan mendapatkan untung, jika menggunakan hak opsinya, dari nilai opsi yang diperoleh dari selisih harga saham pada pasar bebas dengan harga saham pada opsi, yang diistilahkan dengan payoff, v, setelah dikurangi dengan harga opsi (option price), yang diistilahkan dengan profit atau keuntungan. keuntungan atau kerugian yang diperoleh holder atau pemegang opsi call pada waktu t: profit = payoff – harga option = max( , 0) c t v c k s c    sebaliknya, bagi pemegang opsi put akan mendapatkan keuntungan atau kerugian: profit = payoff harga option istiqomah, abdul aziz 110 volume 3 no.2 mei 2014 = max( , 0) p t v p k s p    artinya, jika profit bernilai positif maka pemegang opsi mendapatkan keuntungan, dan sebaliknya jika negatif merupakan kerugian yang maksimal sebesar harga opsi. berikut ini adalah gambar kurva fungsi payoff dan profit untuk opsi call dan opsi put. profit diperoleh dari pengurangan harga transaksi pada saat membeli opsi terhadap nilai payoff yang diperoleh [6]. 3. metode binomial eropa pada kenyataannya harga saham di pasar bebas akan selalu berubah naik atau turun dengan perubahan waktu. kemungkinan dua arah perubahan inilah yang digunakan sebagai dasar model binomial. misalkan harga saham saat pembuatan opsi, adalah s0 naik dengan peluang p menjadi su atau akan turun dengan peluang q menjadi sd. sehingga nilai opsi pada saat pembuatan opsi, adalah v0 dan pada saat naik menjadi u atau akan turun menjadi d. permodelan matematika diharapkan dapat membantu untuk memahami keadaan sekarang dan prediksinya pada waktu yang akan datang. oleh karena itu, agar model binomial ini dapat berhasil dengan lebih baik maka harus sesuai dengan keadaan dunia nyata. masalah yang dihadapi sekarang adalah bagaimana kita memilih p, u, dan d sedemikian hingga model binomial ini mendekati pada keadaan dunia nyata [6]. memulai dengan diskritisasi, yaitu menjadikan waktu kontinu t menjadi diskrit dengan menggantikan t oleh waktu yang sama lamanya katakanlah ti. bidang (s,t) diwakili oleh garis-garis lurus paralel dengan jarak ∆t. mengganti nilai-nilai kontinu si sepanjang paralel t = ti dengan nilai-nilai diskrit sji, untuk semua i dan j yang sesuai. untuk lebih memahami lihat gambar 1. gambar ini menunjukkan sebuah hubungan grid, katakanlah perubahan dari t ke t+∆t, atau dari ti ke ti+1. gambar 1 prinsip metode binomial misalkan digunakan notasi sebagai berikut: m = banyaknya selang waktu i = indeks waktu, i t = waktu ke i j = indeks kemungkinan harga saham dengan: t t m  : . , i t i t 0,1, 2, 3,...,i m 0,1, 2, 3,...,j i  :i is s t asumsi-asumsi yang digunakan dalam permodelan ini adalah: 1. harga s, sebagai harga awal, selama setiap periode waktu ∆t hanya dapat berubah dalam dua kemungkinan yaitu naik menjadi su atau turun menjadi sd dengan 0 < d < u. di sini u dan d masing-masing merupakan faktor perubahan naik dan turun yang konstan untuk setiap ∆t. 2. peluang perubahan naik adalah p, p(naik) = p. sehingga p(turun) = q=(1-p). 3. ekspektasi harga saham secara acak kontinu, dengan suku bunga bebas resiko (r), dari harga saham ke i pada waktu ke i menjadi harga saham ke i+1 pada waktu ke i+1 adalah: 1 ( ) . r t i i e s s e    (1) 4. tidak ada pembayaran dividen   selama periode waktu tersebut. skema (tree) untuk fluktuasi harga saham secara diskrit dapat dibangun menggunakan model binomial. sehingga secara umum harga saham pada saat t = ti terdapat i+1 kemungkinan dengan rumus umum: 0 j i j ji s s u d   , i = 0,1,…,m j = 0,1,…i. dan i j (4) pembahasan berdasarkan konstruksi tentang metode binomial untuk perhitungan harga opsi eropa, yang didasarkan pada perhitungan harga saham. harga saham pada waktu tertentu sangat diperlukan oleh holder (pembeli) dan writer (penjual) untuk memprediksi nilai opsi. menghitung harga opsi eropa dan opsi asia eropa terlebih dulu mencari payoff atau nilai opsi. mencari nilai opsi eropa dan opsi asia eropa, pada kontrak opsi yang dilakukan holder dan writer, saat pertama kali perjanjian holder (pembeli) harus menentukan opsi call (hak untuk membeli) atau opsi put (hak untuk menjual), pada kontrak opsi eropa dan opsi asia eropa diketahui harga saham awal  0s dan menentukan harga saham perjanjian (strike s su = u . s sd = d . s p q si+1 si ti+1= t + ∆t ti = t s t analisis metode binomial dipercepat pada perhitungan harga opsi eropa cauchy – issn: 2086-0382 111 price, k ) dalam waktu jatuh tempo (maturity date,t ). menentukan harga saham pada waktu jatuh tempo membutuhkan harga saham awal dengan metode binomial. harga saham di pasar bebas pada waktu tertentu yang akan datang, tidak dapat dipastikan. harga saham pasar bebas kenyataannya selalu mengalami perubahan naik atau turun setiap detiknya atau dengan perubahan waktu. kemungkinan dua arah perubahan inilah yang digunakan sebagai dasar metode binomial. pada metode binomial, diketahui harga saham awal, untuk mengetahui harga saham sampai dengan jatuh tempo menggunakan persamaan (4). gambar 2. skema fluktuasi harga saham secara binomial dari ekspektasi di atas didapatkan rumus diskrit harga saham pada waktu t = m sebagai berikut:     0 m m e s pu qd s  (5) persamaan (4) adalah tidak rekursif, artinya perhitungan yang memerlukan waktu relatif lama, sehingga perlu adanya bentuk rekursif yang diperoleh sebagai berikut, dengan bantuan persamaan   0 r t m e s s e   (6) untuk menghitung harga saham awal dari harga saham jatuh tempo menggunakan diskon r t e   dari persamaan (3) dengan rumus:     0 1 1, 1 , 1 r t r t ji j i j i s e e s s e ps qs           (7) dan untuk mengetahui harga saham sebelumnya menggunakan persamaan (7). dari langkahlangkah ini, dapat menemukan dari harga opsi awal sampai jatuh tempo, terdapat model untuk mencari harga opsi yaitu model opsi eropa yang di bandingkan dengan model black-scholes. 1. perhitungan harga opsi eropa model yang pertama menggunakan opsi eropa. setelah menemukan harga saham waktu jatuh tempo, dari itu dapat memperoleh nilai opsi eropa. sebelum menghitung nilai opsi, menentukan jenis opsi terlebih dahulu, yaitu opsi call atau opsi put. menghitung nilai opsi call (call payoff) yaitu  max , 0jm jmv s k  (8) terdapat beberapa nilai j = 0,1,2,...,i dan i =0,1,2,...,m. gambar 3. skema perubahan payoff secara binomial dari beberapa nilai opsi ini diperoleh secara binomial mundur harga opsi awal,  0 1, 1 , 1 r t j i j i v e pv qv        (9) dan untuk menghitung nilai opsi put (put payoff) yaitu  max , 0jm jmv k s  (10) terdapat beberapa nilai j = 0,1,2,...,i dan i =0,1,2,...,m. dari beberapa nilai opsi ini diperoleh secara binomial mundur harga opsi awal,  0 1, 1 , 1 r t j i j i v e pv qv        (11) supaya model binomial ini mendekati pada keadaan dunia nyata, maka harus mencari nilai parameter u, d dan p. 2. parameter-parameter u,d, dan p metode dengan asumsi u.d=1. untuk menentukan tiga parameter yang belum diketahui, u, d, dan p, diperlukan tiga persamaan, yaitu: 1) menyamakan ekspektasi harga saham model diskrit dengan model kontinu 00 ji v v 11 v 01 v 22 v 12 v 02 v 23v 13 v 03 v i mm v 0m v i t su sd su2 sud sd2 su3 su2d sud2 sd3 1i t  t sum s sdm 33 v j istiqomah, abdul aziz 112 volume 3 no.2 mei 2014 2) menyamakan variansi model diskrit dengan model kontinu 3) menyamakan u . d = 1 persamaan pertama menyamakan ekspektasi harga saham model diskrit dengan model kontinu yaitu menggunakan persamaan (5) dan persamaan (6) pada waktu jatuh tempo m t . sehingga diperoleh nilai untuk u, d dan p yaitu: 2 4 2 u     , 1 d u  , du de p tr     dengan  2r tr t e e        (12) 3. metode binomial dipercepat pada metode binomial dipercepat ini konsep yang digunakan adalah konsep pemulusan kurva harga opsi eropa yang disebut dengan middle of tree (mot). middle of tree (mot) adalah pemulusan kurva dengan meletakkan harga k di tengah pohon binomial pada saat waktu jatuh tempo sehingga harga k selalu tetap terhadap node di setiap nilai m yang berubah. untuk menghilangkan osilasi pada harga opsi eropa menggunakan metode binomial crr, partisi waktu dipisahkan menjadi m ganjil dan m genap [4]. harga k selalu terletak di tengah pohon binomial pada saat waktu jatuh tempo dengan cara merubah parameter u dan d. oleh karena harga k harus terletak di tengah pohon binomial pada saat jatuh tempo, maka 2 2 3 3 0 0 0 0 j i j s ud s u d s u d s u d k       dengan 1, 2, , ; 0,1, 2, ,i m j i  pada saat jatuh tempo, harga k harus terletak di tengah noktah-noktah pohon binomial, akibatnya 2 0 ( ) m s ud k (13) pada metode binomial crr, harga saham akan naik atau turun bergantung pada parameter u dan d sesuai dengan rumus yaitu dan t t u e d e       sedangkan parameter u dan d pada metode binomial dipercepat diubah dengan menambahkan variable c1 dan c2 pada eksponensial parameter u dan d. penambahan variabel c1 dan c2 membuat harga node saham akan bergerak sehingga keadaan k selalu berada di tengah-tengah jumlah node pohon binomial pada saat waktu jatuh tempo sehingga didapatkan parameter u dan d sebagai berikut: 1 2, t c t c u e d e         (14) dengan mensubtitusikan persamaan (13) dan (14) dan dengan melakukan pemilihan nilai c1 dan c2 didapatkan 0 1 2 ln( ) k s c c m   (15) dan dengan mensubtitusikan persamaan (13) ke persamaan (14) didapatkan parameter u dan d sebagai berikut: 0 0 ln( ) ln( ) , k k s s t t m mu e d e         (16) 4. simulasi grafik ilustrasi yang dgunakan untuk menentukan harga opsi call dan harga opsi put dari model opsi eropa dengan perbandingan model black scholes dan model opsi eropa akan diberikan tiga contoh simulasi sebagai berikut. warna hijau menunjukkan harga opsi genap, warna merah menunjukkan harga opsi ganjil warna biru menunjukkan harga black scholes, dan warna biru menunjukkan opsi asia eropa. 4.1 harga saham lebih besar dari harga ketentuan untuk grafik call opsi 0 s = 50, k = 43, r = 0.15,  = 0.24, m = 146 gambar 4. grafik hasil simulasi harga opsi call dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi call menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 13.5056 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. untuk hasil dari mot adalah sebagai berikut: 0 50 100 150 13.35 13.4 13.45 13.5 13.55 13.6 13.65 13.7 13.75 13.8 waktu h ar ga o ps i pergerakan harga opsi call eropa bentuk crr bs crr genap crr ganjil analisis metode binomial dipercepat pada perhitungan harga opsi eropa cauchy – issn: 2086-0382 113 gambar 5. grafik hasil simulasi harga opsi call dengan mot dari grafik hasil pemulusan dapat dilihat bahwa harga opsi eropa menggunakan binomial dipercepat sudah dapat mendekati harga black scholes dengan partisi 101. kekonvergenan ini lebih cepat dibandingkan dengan kekonvergenan menggunakan metode binomial crr di mana harga opsi eropa yang didapat dari metode binomial dipercepat pada partisi waktu 101 baru dapat didapat oleh metode binomial crr pada partisi waktu 146. untuk grafik put opsi 0 s = 50, k = 43, r = 0.15,  = 0.24, m = 146 gambar 6. grafik hasil simulasi harga opsi put dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi put menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 0.5160 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. untuk hasil dari mot adalah sebagai berikut: gambar 7. grafik hasil simulasi harga opsi put dengan mot dari grafik hasil pemulusan dapat dilihat bahwa harga opsi eropa menggunakan binomial dipercepat sudah dapat mendekati harga black scholes dengan partisi 101. kekonvergenan ini lebih cepat dibandingkan dengan kekonvergenan menggunakan metode binomial crr di mana harga opsi eropa yang didapat dari metode binomial dipercepat pada partisi waktu 101 baru dapat didapat oleh metode binomial crr pada partisi waktu 146. 4.2 harga saham sama dengan harga ketentuan untuk grafik call opsi 0 s = 50, k = 50, r = 0.15,  = 0.24, m = 146 gambar 8. grafik hasil simulasi harga opsi call dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi put menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 8.7602 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. untuk hasil dari mot adalah sebagai berikut: gambar 9. grafik hasil simulasi harga opsi call dengan mot grafik yang didapatkan dari pemulusan kurva ini adalah sama dengan grafik dari binomial crr sebelumnya dengan pemisahan partisi waktu. harga opsi awal yang didapat adalah 9.0378 dan pada waktu ke-146 harga opsi call eropa yang didapatkan adalah 8.7515 0 50 100 150 11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4 13.6 waktu h ar ga o ps i pergerakan harga opsi call eropa dengan middle of tree bs mot genap mot ganjil 0 50 100 150 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 waktu h ar ga o ps i pergerakan harga opsi put eropa bentuk crr bs crr genap crr ganjil 0 50 100 150 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 waktu h ar ga o ps i pergerakan harga opsi put eropa dengan middle of tree bs mot genap mot ganjil 0 50 100 150 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9 9.1 waktu h ar ga o ps i pergerakan harga opsi call eropa bentuk crr bs crr genap crr ganjil 0 50 100 150 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9 9.1 waktu h ar ga o ps i pergerakan harga opsi call eropa bentuk crr bs crr genap crr ganjil istiqomah, abdul aziz 114 volume 3 no.2 mei 2014 dan harga black scholes adalah 8.7602. sehingga, hasil yang diperoleh sama dengan hasil yang diperoleh pada binomial crr dengan pemisahan partisi waktu untuk grafik put opsi 0 s = 50, k = 50, r = 0.15,  = 0.24, m = 146 gambar 10. grafik hasil simulasi harga opsi put dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi put menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 1.7956 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. pergerakan harga opsi put eropa dengan menggunakan metode mot sama dengan pergerakan harga opsi call eropa dengan menggunakan metode mot yaitu grafiknya sama dengan hasil perhitungan binomial crr dengan pemisahan partisi waktu. sehingga, untuk mengetahui pergerakan harga opsi put eropa dengan mot dapat dilihat pada pergerakan harga opsi put eropa menggunakan binomial crr dengan pemisahan partisi waktu. 4.3 harga saham kurang dari harga ketentuan untuk grafik call opsi 0 s = 50, k = 57, r = 0.15,  = 0.24, m = 146 gambar 11. grafik hasil simulasi harga opsi call dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi put menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 5.2155 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. untuk hasil dari mot adalah sebagai berikut: gambar 12. grafik hasil simulasi harga opsi call dengan mot dari grafik hasil pemulusan dapat dilihat bahwa harga opsi eropa menggunakan binomial dipercepat sudah dapat mendekati harga black scholes dengan partisi 101. kekonvergenan ini lebih cepat dibandingkan dengan kekonvergenan menggunakan metode binomial crr di mana harga opsi eropa yang didapat dari metode binomial dipercepat pada partisi waktu 101 baru dapat didapat oleh metode binomial crr pada partisi waktu 146. untuk grafik put opsi 0 s = 50, k = 57, r = 0.15,  = 0.24, m = 146 gambar 13. grafik hasil simulasi harga opsi put dengan pemisahan partisi waktu m dari pemisahan partisi ini didapatkan hasil yang yang sama dengan hasil sebelumnya hanya dibedakan partisi m ganjil dan m genap. dari hasil simulasi opsi put menggunakan metode binomial crr didapatkan bahwa holder (pemegang opsi) harus membayar sebesar 4.2758 kepada writer (penulis opsi) dengan acuan perhitungan menggunakan black scholes. untuk hasil dari mot adalah sebagai berikut: 0 50 100 150 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 waktu h ar ga o ps i pergerakan harga opsi put eropa bentuk crr bs crr genap crr ganjil 0 50 100 150 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 waktu h ar ga o ps i pergerakan harga opsi call eropa bentuk crr bs crr genap crr ganjil 0 50 100 150 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 waktu h a rg a o p s i pergerakan harga opsi call eropa dengan middle of tree bs mot genap mot ganjil 0 50 100 150 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 waktu h ar ga o ps i pergerakan harga opsi put eropa bentuk crr bs crr genap crr ganjil analisis metode binomial dipercepat pada perhitungan harga opsi eropa cauchy – issn: 2086-0382 115 gambar 14. grafik hasil simulasi harga opsi put dengan mot dari grafik hasil pemulusan dapat dilihat bahwa harga opsi eropa menggunakan binomial dipercepat sudah dapat mendekati harga black scholes dengan partisi 101. kekonvergenan ini lebih cepat dibandingkan dengan kekonvergenan menggunakan metode binomial crr di mana harga opsi eropa yang didapat dari metode binomial dipercepat pada partisi waktu 101 baru dapat didapat oleh metode binomial crr pada partisi waktu 146. penutup semakin banyak partisi waktu yang digunakan maka aproksimasi harga opsi akan semakin lambat untuk menuju kekonvergenan terhadap black scholes. untuk mempercepat kekonvergenan aproksimasi harga opsi eropa maka digunakan pengembangan dari model binomial yaitu binomial dipercepat. langkah yang dilakukan dalam metode binomial dipercepat adalah melakukan pemulusan kurva harga opsi yang disebut dengan middle of tree (mot). akan tetapi, sebelum melakukan pemulusan kurva tersebut yang dilakukan terlebih dahulu adalah memisahkan partisi waktu yang digunakan yaitu partisi waktu ganjil dan genap. dengan menggunakan parameter mot tersebut diperoleh hasil dari harga opsi eropa yang bisa mendekati harga kekonvergenan black scholes dengan partisi yang lebih sedikit dibandingkan dengan menggunakan metode binomial. saran bagi penelitian selanjutnya, disarankan untuk menggunakan metode binomial dipercepat pada perhitungan harga opsi asia eropa, opsi amerika ataupun opsi asia amerika. bisa juga dikembangkan metode lain untuk mempercepat perhitungan harga opsi selain menggunakan pemulusan kurva mot. bibliography [1] s. m. ross, an introduction to mathematical finance: options and other topics, vol. 36. cambridge university press cambridge, 1999. [2] r. j. rendleman and b. j. bartter, “two state option pricing,” j. finance, vol. 34, no. 5, pp. 1093–1110, 1979. [3] r. seydel and r. seydel, tools for computational finance, vol. 4. springer, 2002. [4] t. r. klassen, “simple, fast, and flexible pricing of asian options,” j. comput. financ., vol. 4, no. 3, pp. 89–124, 2001. [5] z. bodie, essentials of investments. . [6] a. aziz, “empat bentuk nilai parameterparameter u, d, dan p dalam binomial harga saham,” 2004. [online]. available: http://blog.uinmalang.ac.id/abdulaziz/files/2010/08/a bdul-aziz-empat-bentuk-solusi-modelbinomial-fulltext_1.pdf. [accessed: 06nov-2014]. 0 50 100 150 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 waktu h ar ga o ps i pergerakan harga opsi put eropa dengan middle of tree bs mot genap mot ganjil pendahuluan tinjauan teori pembahasan penutup bibliography modified chebyshev (vieta–lucas polynomial) collocation method for solving differential equations 1m. ziaul arif, 2ahmad kamsyakawuni, 3ikhsanul halikin 1 department of mathematics, faculty of sciences, university of jember email: ziaul.fmipa@unej.ac.id abstract this paper presents derivation of alternative numerical scheme for solving differential equations, which is modified chebyshev (vieta-lucas polynomial) collocation differentiation matrices. the scheme of modified chebyshev (vieta-lucas polynomial) collocation method is applied to both ordinary differential equations (odes) and partial differential equations (pdes) cases. finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. it is shown that modified chebyshev collocation method more effective and accurate than fdm for some example given. keywords: vieta-lucas polynomial, collocation method, ordinary differential equations, partial differential equations, finite difference methods introduction chebyshev polynomials have become alternatively numerical solution for differential problems, which has developed after digital computerizing growth rapidly. it was discovered by chebyshev and developed by some mathematicians; clenshaw [1], nidekker, fox and parker [2], etc. after computer was discovered. it is known that there are four kinds of chebyshev polynomial as developed that commonly applied in numerical analysis [3]. for the first through four kinds, it is symbolized by 𝑇𝑁(𝑥),𝑈𝑁(𝑥),𝑉𝑁(𝑥), 𝑎𝑛𝑑 𝑊𝑁(𝑥) respectively. first and second kind of chebyshev polynomial had practiced to solving linear differential equation [4, 5]. furthermore, the other kinds were used to find solution of some problems numerically [6-9]. however, for some reasons of the advantages, modified chebyshev and its statistical properties was discovered by witula and damian and symbolized by ω𝑁(𝑥) [10]. the main objective of the present article is to develop algorithms based on modified chebyshev polynomial (vieta-lucas polynomial) [10] for solving differential equations. and in this paper, it is practiced for some des example and compared with finite difference method. the contents of this article are organized as follows. in section 2, we discuss about a literature review of methods, fdm and modified chebyshev collocations method. in addition, we discuss discovering of differentiation matrices of modified chebyshev (vieta-lucas polynomial) collocation and the result. finally, some concluding remarks are presented in section 3. a literature review finite difference method finite difference method is a superior method to approach the derivative of a function based on the taylor series. the accuracy of finite difference method can be adapted to taking into first part account of the taylor series. in this article, the first and second derivatives are approximated using central difference (centered difference approximation) second order as follows [11]; 𝑓′(𝑥) ≈ 𝑓(𝑥+ℎ)−𝑓(𝑥−ℎ) 2ℎ + 𝛰(ℎ2) (1) 𝑓′′(𝑥) ≈ 𝑓(𝑥+ℎ)−2𝑓(𝑥)+𝑓(𝑥−ℎ) ℎ2 +𝛰(ℎ2) (2) 𝑓(3)(𝑥) ≈ 𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+2𝑓(𝑥−ℎ)−𝑓(𝑥−2ℎ) 2ℎ3 + 𝛰(ℎ2) (3) mailto:ziaul.fmipa@unej.ac.id modified chebyshev collocation method for solving differential equations cauchy – issn: 2086-0382 29 by this scheme, some of differential equations examples shall be solve and compared with the exact solution. chebyshev polynomial collocation method chebyshev collocation is one of spectral methods based on trial functions chebyshev polynomial. many practical problems are performed by mathematician using chebyshev collocation [1, 4, 5, 7-9]. this is because some advantages such as accurate and memory minimizing. there are several kinds of chebyshev polynomial describe below, first kind, 𝑇𝑁(𝑥) = cos (𝑁𝜃). (4) second kind, 𝑈𝑁(𝑥) = 𝑠𝑖𝑛((𝑁+1)𝜃) sin (𝜃) . (5) third kind, 𝑉𝑁(𝑥) = cos((𝑁+ 1 2 )𝜃) cos ( 1 2 𝜃) . (6) fourth kinds, 𝑊𝑁(𝑥) = sin((𝑁+ 1 2 )𝜃) sin ( 1 2 𝜃) . (7) when 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑥) [3] and 𝑥 is collocation points. let we have boundary value problem on range [-1,1]: 𝑑2 𝑑𝑥2 𝑢(𝑥) = 𝑓(𝑥), 𝑢(−1) = 𝑎, 𝑢(+1) = 𝑏, (8) where 𝑓(𝑥) is a function and boundary value is given above. suppose that we approximate 𝑢(𝑥) by, 𝑢𝑁(𝑥) = ∑ 𝑐𝑘 𝑁 𝑘=0 𝜑𝑘(𝑥) (9) it is involving 𝑛 +1 unknown coefficients {𝑐𝑘}, and 𝜑𝑘(𝑥) is a basis function; in this case, we perform chebyshev polynomial as a basis function. then we may select 𝑛 − 1 points {𝑥1,…,𝑥𝑛−1} in the range of integration called collocation points and require 𝑢𝑁(𝑥) satisfying the differential equation (8). this requires us to solve just the system of 𝑛+ 1 linear equations by differentiation matrix of the method. it is well-known as chebyshev collocation method. modified chebyshev collocation method furthermore, in present paper, we perform first kind of modified chebyshev polynomial as a basis function, ω𝑁(𝑥) = 2𝑇𝑁( 𝑥 2 ) (10) where 𝑇𝑁(𝑥) is the first chebyshev polynomial. some properties of modified chebyshev polynomial presented by witula, et all [3]. it is also called n-th vieta-lucas polynomial. results and discussion finite difference scheme to approximate first, second and third derivative, we employ central difference second order as seen in equation (1)-(3). in this part, we can perform its differentiation matrices scheme of fdm equations (1)-(3) respectively as follows, 𝐷𝑁,1 = 1 2ℎ ( 0 1 0 … 0 −1 0 ⋮ 0 −1 1 … 0 ⋱ ⋮ ⋱ 1 0 … −1 0) (11.a) 𝐷𝑁,2 = 1 ℎ2 ( −2 1 0 … 0 1 0 ⋮ −2 1 1 … 0 ⋱ ⋮ ⋱ 1 0 … 1 −2) (11.b) 𝐷𝑁,3 = 1 2ℎ3 ( 0 −2 −1 … 0 2 1 ⋮ 0 2 −2 … ⋮ ⋱ ⋱ −1 ⋱ ⋱ −2 0 … 1 2 0 ) (11.c) however, for solving pdes, the system of ode, which is generated by fdm, can be advanced in time by forward euler scheme or a third order runge-kutta scheme. m. ziaul arif, ahmad kamsyakawuni, ikhsanul halikin 30 volume 3 no. 4 mei 2015 modified chebyshev (vieta-lucas polynomial) differentiation matrices basically, solving des cannot be denied involving system of linear equations. hence, we need differentiation matrices derived by collocation method. in this section, we shall explain how to derive differentiation matrices of modified chebyshev (vieta-lucas polynomial) collocation method. firstly, we define vieta-lucas polynomial as a basis function of collocation approximations. we have first kind of modified chebyshev (vieta-lucas polynomial) collocation method defined by witula [3] as equation (10), consequently, ω𝑁(𝑥) = 2.𝑐𝑜𝑠(𝑁.𝑎𝑟𝑐𝑐𝑜𝑠( 𝑥 2 )) [-2,2] (12) furthermore, discover zero and extrema points of first kind of modified chebyshev (vieta-lucas polynomial) used to construct differentiation matrices. we show that zero points of modified chebyshev (vieta-lucas polynomial) is 𝑥𝑘 = 2.𝑐𝑜𝑠( 2𝑘−1 2𝑁 )𝜋 𝑘 ∈ ℤ (13) by first derivative, 𝑑ω𝑁(𝑥) 𝑑𝑥 = 0 , we discover the extrema points of modified chebyshev as follows, 𝑥𝑗 = 2.𝑐𝑜𝑠( 𝜋𝑗 𝑁 ), 0 ≤ 𝑗 ≤ 𝑁 (14) it is well-known as first kind modified chebyshev (vieta-lucas polynomial) collocation points. considering the chebyshev points above, we construct differentiation matrices of modified chebyshev collocation method as follows. first, we need the following results: ω𝑁(𝑥𝑗) = 2.𝑐𝑜𝑠(𝑁.𝑎𝑟𝑐𝑐𝑜𝑠( 𝑥𝑗 2 )) [-2,2] ω′𝑁(𝑥𝑗) = 𝑁.sin (𝑁𝜃) sin (𝜃) = 0, 0 ≤ 𝑗 ≤ 𝑁 − 1 (15.a) ω′′𝑁(𝑥𝑗) = (−1)𝑗+1.2.𝑁2 (4−𝑥𝑗 2) , 0 ≤ 𝑗 ≤ 𝑁 −1 (15.b) ω′′′𝑁(𝑥𝑗) = (−1)𝑗+1.6.𝑁2𝑥𝑗 (4−𝑥𝑗 2)2 , 0 ≤ 𝑗 ≤ 𝑁 −1 (15.c) ω′𝑁(±2) = (±1) 𝑁+1𝑁2 and ω′′𝑁(±2) = (±1)𝑁𝑁2(𝑁2−1) 6 (15.d) since modified chebyshev ω′𝑁(𝑥) is polynomial and we know all zeros points of it (called as vieta-lucas collocation points) we may write, ω′𝑁(𝑥) = 𝛼ω𝑁 ∏ (𝑥 − 𝑥𝑗) 𝑁−1 𝑗=1 (16) for 𝑥0=−2 and 𝑥𝑁=2, we can write, 𝛾𝑁(𝑥) = ∏ (𝑥 −𝑥𝑗) = 𝛽𝑁(𝑥 2 −4)ω′𝑁(𝑥) 𝑁 𝑗=0 (17) where 𝛽𝑁 is a positive a constant such that coefficient 𝑥𝑁+1 equals to 1. from equation (15.a) and (15.d) and if 𝑥 = 𝑥𝑗 and 0 ≤ 𝑗 ≤ 𝑁, it follows that 𝛾′𝑁(𝑥) = 𝛽𝑁2𝑥ω′𝑁(𝑥)+𝛽𝑁(𝑥 2 −4)ω′′𝑁(𝑥) (18) however, for 0 ≤ 𝑗 ≤ 𝑁 −1, ω′𝑁(𝑥𝑗) = 0 and when 𝑗 = 0 and 𝑗 = 𝑁 as seen in equation (15.d), in consequence, we have 𝛾′𝑁(𝑥𝑗) = (−1) 𝑗𝑐�̃�𝑁 2𝛽𝑁 (19) where 𝑐0̃ = �̃�𝑁 = 4 and 𝑐�̃� = 2 for 1 ≤ 𝑗 ≤ 𝑁 − 1. for an arbitrary function 𝑓(𝑥) and nodes 𝑥 = 𝑥𝑗, we can estimate the function using lagrange’s interpolation polynomial below, 𝑝(𝑥) = ∑ 𝐹𝑗(𝑥).𝑓(𝑥) 𝑁 𝑗=0 (20) where 𝐹𝑗(𝑥) = 𝛾𝑁(𝑥) 𝛾′𝑁(𝑥𝑗)(𝑥−𝑥𝑗) ,0 ≤ 𝑗 ≤ 𝑁 (21) since 𝛾𝑁(𝑥) and 𝛾 ′ 𝑁 (𝑥𝑗) are described by (17) and (19) respectively. by definition of differentiation matrix, hence we obtain differentiation matrix below, 𝐷𝑘𝑗 1 = 𝐹′𝑗(𝑥𝑘) the element of the matrices can be evolved from first derivative of 𝐹(𝑥𝑗) with some point’s defined equation (14). scheme below is determining the element of differentiation matrices; 1. for 1 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁 −1 for equation (21), together with (15.a) and (15.d), leads to 𝐹′𝑗(𝑥𝑘) = (−1)𝑗+𝑘 𝑐�̃�(𝑥𝑘−𝑥𝑗) (22.a) modified chebyshev collocation method for solving differential equations cauchy – issn: 2086-0382 31 2. for 𝑗 ≠ 𝑘 and 𝑘 = 0, 1 ≤ 𝑗 ≤ 𝑁 −1 𝐹′𝑗(𝑥0) = (−1)𝑗.4 𝑐̃𝑗(𝑥0−𝑥𝑗) (22.b) 3. for 𝑗 ≠ 𝑘 and 𝑗 = 𝑁, 1 ≤ 𝑘 ≤ 𝑁 −1 𝐹′𝑁(𝑥𝑘) = (−1)𝑁+𝑘.2 𝑐�̃�(𝑥𝑘−𝑥𝑁) (22.c) 4. for 𝑗 ≠ 𝑘 and 𝑘 = 𝑁, 1 ≤ 𝑗 ≤ 𝑁 −1 𝐹′𝑗(𝑥𝑁) = (−1)𝑗+𝑁.4 𝑐̃𝑗(𝑥𝑁−𝑥𝑗) (22.d) 5. for 𝑗 ≠ 𝑘 and 𝑗 = 0, 1 ≤ 𝑘 ≤ 𝑁 −1 𝐹′0(𝑥𝑘) = (−1)𝑘.2 𝑐0̃(𝑥𝑘−𝑥0) (22.e) 6. for 𝑗 ≠ 𝑘, and 𝑘 = 0,𝑗 = 𝑁, 𝐹′𝑁(𝑥0) = 1 𝑐�̃� (22.f) 7. for 𝑗 ≠ 𝑘, and 𝑗 = 0,𝑘 = 𝑁, 𝐹′0(𝑥𝑁) = − (−1)𝑁 𝑐0̃ (22.g) from equation (22.a) to (22.g), in general, for 0 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁, we have 𝐹′𝑗(𝑥𝑘) = (−1)𝑗+𝑘�̃�𝑘 𝑐�̃�(𝑥𝑘−𝑥𝑗) (23) where 𝑐0̃ = 𝑐�̃� = 4 and 𝑐�̃� = 2. furthermore, by applying l’hospital for 𝑗 = 𝑘 = 0 we have, 𝐹′0(𝑥0) = ( 1+2𝑁2 6 ) (24) and for 𝑗 = 𝑘 = 𝑁 𝐹′𝑁(𝑥𝑁) = −( 1+2𝑁2 6 ) (25) again using l’hospital’s rule for 1 ≤ 𝑗 = 𝑘 ≤ 𝑁 −1 shows that 𝐹′𝑗(𝑥𝑗) = −𝑥𝑗 𝑐̃𝑗(4−𝑥𝑗 2) (26) vieta-lucas differentiation matrices for each 𝑁 ≥ 1, modified chebyshev (vietalucas polynomial) collocation differentiation matrix is defined below (𝐷𝑁 1)00 = ( 1+2𝑁2 6 ) (27.a) (𝐷𝑁 1)𝑁𝑁 = −( 1+2𝑁2 6 ) (27.b) (𝐷𝑁 1)𝑘𝑘 = −𝑥𝑘 2(4−𝑥𝑘 2) , 1 ≤ 𝑘 ≤ 𝑁 −1 (27.c) (𝐷𝑁 1)𝑘𝑗 = (−1)𝑗+𝑘𝑐̃𝑘 𝑐̃𝑗(𝑥𝑘−𝑥𝑗) , 0 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁 (27.d) where 𝑐0̃ = �̃�𝑁 = 4 and 𝑐�̃� = 2. furthermore, for high order differential problem, modified chebyshev (vieta-lucas polynomial) differentiation matrices can be applied by define high order modified chebyshev (vieta-lucas polynomial) differentiation as power of matrix below, 𝐷𝑁 𝑚 = (𝐷𝑁 1)𝑚 (28) examples and solutions for comparison the performance of vieta-lucas collocation scheme and fdm, some of example are given below. table 1. example of boundary and initial value of differential equations problems. no example 1 𝑦(2) + 𝑦 = 𝑥2 + 𝑥 𝑦(−2) = 𝑦(2) = 0,𝑦′(−2) = 2.285 2 𝑦(2) −𝑦(−1) = sin (𝑥) 𝑦(−2) = 𝑦(2) = 0,𝑦′(−2) = 0.67 3 𝑢𝑡 + 𝑢𝑥 = 0, 𝑥 ∈ [−1,1] 𝑢(𝑥,0) = sin (𝜋𝑥) 𝑢(−1,𝑡) = sin (𝜋(−1− 𝑡)) numerical experiment numerical result below is comparison belong to the vieta-lucas collocation, fdm and its exact solution. example of ode no 1 can be solved by vietalucas collocation scheme as equation 𝐷𝑁 2𝑢𝑁 + 𝑢𝑁 = 𝐹. where 𝐹 = (𝑋0,𝑋1,…,𝑋𝑁), x is right hand side of the example of ode no 1 and n=64. while for fdm, we perform the second order of fdm and n=400 to solve the example. the result of both scheme is compared with the exact solution as table and the picture below, m. ziaul arif, ahmad kamsyakawuni, ikhsanul halikin 32 volume 3 no. 4 mei 2015 figure 1. numerical solution of example 1 table 2. error of the example 1 scheme error vieta-lucas 2.3 x 10−6 fdm 1.5 x 10−2 furthermore, for ode example no 2, vieta-lucas scheme can be write as 𝐷𝑁 2𝑢𝑁 +𝐷𝑁𝑢𝑁 = 𝐹 and n=64. whereas fdm is performed with n=400. figure 2. numerical solution of example 2 table 3. error of the example 2 scheme error vieta-lucas 3.9 x 10−5 fdm 3.5 x 10−2 in addition, the boundary and initial problem of linear pde is solved by vieta-lucas collocation for spatial part with n=64, then it is evaluate by forward euler scheme for time differencing as equation u𝑁(𝑡 𝑖+1) = 𝐮𝑁(𝑡 𝑖)+∆𝑡 𝐷𝑁𝐮𝑁(𝑡 𝑖), 𝐮(𝑡0) = 𝑠𝑖𝑛 (𝜋𝑋𝑁). moreover, second order of fdm with n=400 is applied to solve it with time differencing as vieta-lucas collocation. figure 3. numerical solution of example 3 conclusion the problem of solving differential equations occurs frequently in mathematical work. in this paper, we have introduced modified chebyshev collocation scheme called vieta-lucas collocation to for solving differential equation. numerical experiment shows that vieta-lucas collocation scheme s more effective and accurate than fdm even with small number of n. meanwhile, the scheme introduced in this paper can be used to more class of nonlinear differential equations references [1] c. clenshaw, "the numerical solution of linear differential equations in chebyshev series," in mathematical proceedings of the cambridge philosophical society, 1957, pp. 134-149. [2] i. nidekker, "chebyshev polynomials in numerical analysis: l. fox and ib parker. oxford university press, oxford, london, 1968, ix+ 205 pp," zhurnal vychislitel'noi matematiki i matematicheskoi fiziki, vol. 9, pp. 491-491, 1969. [3] j. c. mason and d. c. handscomb, chebyshev polynomials: crc press, 2010. [4] a. akyüz and m. sezer, "chebyshev polynomial solutions of systems of highorder linear differential equations with variable coefficients," applied mathematics and computation, vol. 144, pp. 237-247, 2003. [5] a. akyüz and m. sezer, "a chebyshev collocation method for the solution of linear integro-differential equations," 1.4 1.5 1.6 1.7 1.8 1.9 2 -0.2 -0.15 -0.1 -0.05 0 0.05 x y (x ) example of ode no 1 fdm exact solution vieta-lucas coll 0.6 0.8 1 1.2 1.4 1.6 1.8 -0.15 -0.1 -0.05 0 0.05 x y (x ) example of ode no 2 fdm exact solution vieta-lucas coll -0.528 -0.526 -0.524 -0.522 -0.52 -0.518 -1 -0.995 -0.99 x u ( x ,t ) numeric vs exact 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 t e r r o r absolute error fdm exact solution vieta-lucas fdm vieta-lucas modified chebyshev collocation method for solving differential equations cauchy – issn: 2086-0382 33 international journal of computer mathematics, vol. 72, pp. 491-507, 1999. [6] m. y. hussaini, c. l. streett, and t. a. zang, "spectral methods for partial differential equations," ii. s. army research office, p. 883, 1984. [7] j. mason, "chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms," journal of computational and applied mathematics, vol. 49, pp. 169-178, 1993. [8] w. abd-elhameed, e. doha, and y. youssri, "new wavelets collocation method for solving second-order multipoint boundary value problems using chebyshev polynomials of third and fourth kinds," in abstract and applied analysis, 2013. [9] e. doha, w. abd-elhameed, and m. alsuyuti, "on using third and fourth kinds chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems," journal of the egyptian mathematical society, 2014. [10] r. wituła and d. słota, "on modified chebyshev polynomials," journal of mathematical analysis and applications, vol. 324, pp. 321-343, 2006. [11] s. c. chapra and r. p. canale, numerical methods for engineers: mcgraw-hill higher education new york,2010. introduction a literature review finite difference method chebyshev polynomial collocation method modified chebyshev collocation method results and discussion finite difference scheme modified chebyshev (vieta-lucas polynomial) differentiation matrices vieta-lucas differentiation matrices examples and solutions numerical experiment conclusion references representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 95-99 submitted: 11 january 2016 reviewed: 2 march 2016 accepted: 12 may 2016 representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo muhammad hajarul aswad a1, wahyuni husain2 1department of mathematics education iain palopo 2department of islamic broadcasting communication iain palopo email: as_wad82@yahoo.co.id , yuni_h98@yahoo.com abstract the application of graph theory concept in communication network analysis is interesting to observe. this research was carried out to learn how communication network structure was formed and who had necessary role in the network. it was explorative research and conducted at female students’ dormitory of state islamic institute of palopo (asrama putri iain palopo). the results were interpreted by using microsoft nodexl version 1.0.1.113. it was found that the communication network structure of female students’ who stayed at the dormitory decentralized. it shows that each student had same opportunity to communicate one another directly or indirectly, which 4 to 9 path distance. it was also identified that from 110 people, suarni was the student who had significant influence in the communication network. keywords: communication network analysis, microsoft nodexl, graph theory. introduction graph theory is an applied science which is very suitable to apply in network analysis case in term of communication network. furthermore, it is said that communication network analysis is the application of social network analysis, yet cna focuses more on communication such as identifying communication structure in a system and how actors (person, organization, or institution) in the organization structure. theoritical review looking at the communication network analysis, there are usually some basic terms of graph theory used. the terms are briefly explained as follow: table 1. the basic concept of graph theory in communication network analysis terms graph theory concept the concept of communication network analysis vertex point, dot, node members of network, actors, people, companies, and others depending on network objects edge side, line link, relations degree the number of edges and vertex side by side the number of links/relations occurred at an actor with others isolated vertex vertex with degree 0 actors have no relation to others mailto:yuni_h98@yahoo.com representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo muhammad hajarul aswad a 96 hubs vertex has much more degree compared to other vertices actors have most links in a network the relation of actors (female students) who stay at female students’ dormitory of state islamic institute of palopo (asrama putri iain palopo) was based on emotion among them. this interaction happened not to obtain anything with the exception of comfort and fulfillment of emotional need. such kind of relation was then then called as sentimental relation, where the relation had no direction (undirected). based on the interaction, form of graph/network would be undirected graph. to observe the structure and character in the network, there several things need noticing: 1. density: ratio of link number in a network and the number of links that might appear. the density demonstrates intensity of each actors’ network in communicating. the network with high density is the network which has actors interacting one another. on the contrary, the network with low density is indicated with the less interaction among actors. density in a network can be calculated with the pattern as follows: 𝐷 = 1 𝑁(𝑁 − 1) (1) d= density, i= link number, and n= network size or actor number in a network. the grade of density is about between 0 up to 1, where approaching 1 means that density of a network is getting higher. 2. centralization: an indication how focus a network is used by certain actors. whether relation in a network spreads to many people or only centralize on several users. a network is called centralized when the link points to some certain actors, while a network connects to many users is decentralized. centralization has no connection with density. a network with high density could have centralized and decentralized networks. the same condition happened to the network with a low density. the equation which is used to decide centrality of a network is as follows: 𝐶𝐷 = ∑(𝑀𝑎𝑥(𝐶𝐷𝑖) − 𝐶𝐷𝑖) 𝑁2 − 3𝑁 + 2 (2) cd is centralization, cdi level centrality scores from each actor max cdi maximum score from cdi, and n is the number of actors . 3. distance and diameter: distance is average path required by all actors to communicate. meanwhile, diameter is the farthest distance between two actors in a network. 4. betweennes centrality is the each of actor’s scores that indicate how the actors relate in a network. an actor with the biggest betweenness centrality score illustrates the actors’ existence, which then show others’ existence in a network. if actor a is disconnected in a network, which then causes other actors separated from the network, actor a has high betweenness centrality score. the lowest score of betweenness centrality is 0 and the highest one is 1. closeness centrality is the measurement how important/close an actor to other actors in a network. the lower the closeness centrality score of an actor, the more important/closer the actor with others is. methodology network analysis in this research was complete network analysis where all actors in the network had same attention to observe and analyze. if we look at the analysis, the resea rch used system analysis level where the researcher did not pay attention on actors or certain groups, but more on network structure comprehensively. it was explorative research which would illustrate the students who were mostly chosen for communicating at female students’ dormitory of state islamic institute of palopo (asrama putri iain palopo). furthermore, network structure was identified through network measurements covering network density, centrality, distance, and diameter, betweenness centrality, and closeness centrality. representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo muhammad hajarul aswad a 97 the research was conducted at female students’ dormitory of state islamic institute of palopo (asrama putri iain palopo) at jln. agatis, balandai, palopo city, south sulawesi. the dormitory was utilized by female students that had fulfilled requirement and provision assessed by boarding house committee at iain palopo. the number of undergraduate students who stayed at the dormitory till 10 january 2016 was 110. however, in the further observation process (4 – 10 january 2016) there were only 71 people to take their data sucessfully. the method of collecting data was through questionnaire controlled by a surveyor. the questionnaire content consisted of two parts which were the first one was related to sosiogram data and the second one was connected to the closeness of relationship among female students who stayed at female students’ dormitory of iain palopo. the restricted interview was also administered to several students related to the questions in the questionnaire. the collected data was then analyzed with method of communication network analysis by using microsoft nodexl version 1.0.1.113. results and discussion the data of observation results was firstly tabulated, which then became input in microsoft nodexl. the sosiogram output was as follows: figure 1. communication network sosiogram at female students’ dormitory of iain palopo basic information of the communication network of students at female students’ dormitory of iain palopo on the picture 1 could be showed with the following tables (table 1 and 2). tabel 1. the summary of graph metrics nodexl toward communication network at female students’ dormitory of iain palopo. metrics value graph type undirected vertices 110 unique edges (link) 236 edges with duplicates 0 total edges 236 self-loops 0 connected components 1 single-vertex connected components 0 maximum vertices in a connected component 110 maximum edges in a connected component 236 maximum geodesic distance 9 average geodesic distance 3,89 representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo muhammad hajarul aswad a 98 graph density 0,04 nodexl version 1.0.1.113 looking at table1, it can be seen that graph/network formed a(pict. 1) was undirected graph. the number of vertex/actor/ n=110, and edge number / link / i = 236. the formed graph was connecting graph with the smallest degree 1 and the biggest one 15. based on the equation (1), the density of network was 0.019. it means that interaction among the students at female students’ dormitory of iain palopo was still in the low level. many students were not connected directly or did not know each other. the circumstances happened because the students who stay at the dormitory were heterogeneous in ethnic group, department, and semester. the relation also demonstrates that decentralization did not just focus on several people. it is seen that centralization proportion only accounted for 10.007%. the large amount of network diameter was 9. it illustrates that the communication network formed at female students’ dormitory of iain palopo was relatively large. the distance required by each students at the dormitory in interacting was average 3,89 ≈4. the number describes that the students had to get in touch with 3 people at first before interacting or communicating with the intended student. according to the data at table 2, actor who had the highest betweenness centrality score was actor number 2, suarni. it describes that suarni was the actor that had the significant influence to the students at female students’ dormitory of iain palopo. moreover, suarni also had the lowest closeness centrality score accounting for 2.633. it shows that suarni played important role female students’ dormitory of iain palopo. it is asserted by sosiogram on picture 2 as follows: picture 2. the forms of communication network based on the number of minimum degree 1, 7, 10. conclusion based on research findings and discussions, it can be concluded that communication network structure of students at female students’ dormitory of iain palopo decentralized. it shows that the communication tended to spread evenly, it did not focus only for several people. every female student could still communicate in direct and indirect ways with communication distance average 4 to 9 paths. suarni, actor number 2, was the student who had adequately significant influence in the communication network. reference [1] m. tsvetovat and a. kouznetsov, social network analysis for startups: finding connections on the social web, " o'reilly media, inc.", 2011. [2] p. mathis, "graphs and networks: multilevel modeling," 2007. representation of graph theory in students’ communication network at female students’ dormitory of state islamic institute of palopo muhammad hajarul aswad a 99 [3] d. hansen, b. shneiderman and m. smith, "analyzing social media networks: learning by doing with nodexl," computing, vol. 28, no. 4, pp. 1-47, 2009. [4] k. faust, "centrality in affiliation networks," social networks, vol. 19, no. 2, pp. 157-191, 1997. [5] eriyanto, analisis jaringan komunikasi, prenadamedia group, 2014. abstract introduction theoritical review methodology results and discussion conclusion reference titik dan sisi penutup minimal pada graf bintang ���� ��� dan graf roda ���� �� nurul hijriyah1) dan wahyu h. irawan2) 1) mahasiswa pascasarjana jurusan matematika universitas brawijaya malang 2)jurusan matematika uin maulana malik ibrahim malang e-mail: 1)n.hijriyah@gmail.com abstrak suatu titik dan sisi dikatakan saling menutup pada graf g jika titik dan sisi tersebut berinsiden di g. titik penutup di g merupakan himpunan dari titik-titik yang menutup semua sisi di g dan sisi penutup pada graf g merupakan himpunan sisi-sisi yang menutup semua titik di g. himpunan titik dan sisi penutup di katakan minimal karena banyaknya anggota paling sedikit atau himpunan yang kardinalnya terkecil. titik penutup minimal dilambangkan dengan ��� dan sisi penutup minimal dilambangkan dengan 1���. skripsi ini bertujuan untuk mengetahui rumusan umum titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �����. hasil dari penelitian ini adalah titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �����. kemudian dirumuskan menjadi suatu lemma dan dibuktikan kebenarannya secara umum. 1. graf bintang � �� ��� dengan � � �, � � 3. maka rumusan titik dan sisi penutup minimal masingmasing adalah ���� � 1 dan ����� � �, �� �� ���� � dan ��� �� ���� � � �, �� �� ���� � � 12 �� " 1# ; %�&%� � '(�)* 12 ��� " 1� " 1# ; %�&%� � ')�+,. ��� �� ���� � /01 02 3� 12 � " 1#4 ; %�&%� � '(�)* 12 ��� " 1� " 1# ; %�&%� � ')�+,. 2. graf roda � ����� dengan � � �, � � 3. maka rumusan titik dan sisi penutup minimal masing-masing adalah ���� � 5 �6 � " 1 ; %�&%� � '(�)*�6 �� " 1� " 1 ; %�&%� � ')�+,-. dan ����� � 5 �6 � " 1 ; %�&%� � '(�)*�6 �� " 1� ; %�&%� � ')�+,-. �� ������ � 7 8 �6 � " 19 ; %�&%� � '(�)* 8�6 �� " 1� " 19 ; %�&%� � ')�+,-. dan ��� ������ � 7 8�6 � " 19 ; %�&%� � '(�)* �6 �� " 1�# ; %�&%� � ')�+,-. �� ������ :)� ��� �� ���� ;� /01 02 312 �2�� " � " 2�4 ; � '(�)*, � '(�)* 312 �2�� " � " 3�4 ; � ')�+,-, � '(�)* . �� ������ � < ��� " 1�. :)� ��� �� ���� � = >��� " 1�?; � ')�+,:)� � � �, � � 3. kata kunci: titik penutup, sisi penutup, minimal, graf bintang, graf roda pendahuluan teori graf merupakan salah satu cabang dari ilmu matematika yang masih sangat menarik untuk dibahas karena teori-teorinya masih aplikatif sampai saat ini dan dapat diterapkan untuk memecahkan masalah dalam kehidupan sehari-hari. dengan mengkaji dan menganalisis model atau rumusan, teori graf dapat diperlihatkan peranan dan kegunaannya dalam memecahkan berbagai masalah. permasalahan yang dirumuskan dengan teori graf dibuat sederhana, yaitu diambil aspek-aspek yang diperlukan dan dibuang aspek-aspek lainnya (purwanto, 1998:1). graf telah dikembangkan sejak tahun 1960an yang didefinisikan sebagai himpunan titik (vertex) yang tidak kosong dan himpunan garis atau sisi (edge) yang mungkin kosong. graf itu menghubungkan pasangan dari suatu himpunan, dalam alquran bisa kita hubungkan dengan hablum min an-nas dan hablum min allah. hubungan antara allah dengan manusia dapat dijelaskan bahwa secara umum seluruh alam ini ta’at dan tunduk kepada allah swt, sehingga berfungsi maksimal dan saling memberikan manfaat kepada bagian alam lainnya titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ��� jurnal cauchy – issn: 2086-0382 97 m v1 m v3 m v4m v5 m v6 m v7 m v8 mv9 m vn m v2 1 v1 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1v9 1 vn 1 v2 2 v1 2 v3 2 v42 v5 2 v6 2 v7 2 v8 2v9 2 vn 2 v2 1 v7 1 v8 1 v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v1 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1v9 1 vn 1 v2 1 v7 1 v8 1 v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 2 v2 2 v3 2 v42 v5 2 v6 2 v7 2 v8 2 v9 2 vn 2 v1 m v4 m v3 m v2 m v1 m vn m v5 m v6 m v7 m v8 m v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 m v1 m vn m v2 m v3 m v4m v5 m v6 m v7 m v8 v9 m 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 2 vn 2 v1 2 v2 2 v3 2 v42 v5 2 v6 2 v7 2 v8 v9 2 bahkan tahu cara bertasbih dan sholatnya. secara khusus ternyata seluruh alam semesta ini juga berhijab atau memakai penutup atau pelindung agar berjalan sesuai fungsinya dan selamat dari hal-hal yang membahayakan. contoh kecil seperti pena tanpa penutup maka tintanya akan menjadi kering, seperti rumah juga perlu adanya penutup yakni atap dan dinding. suatu titik dan sisi dikatakan saling menutup pada graf g jika titik dan sisi tersebut terkait langsung di g. titik penutup di g merupakan himpunan dari titik-titik yang menutup semua sisi di g dan sisi penutup pada graf g (tanpa titik terisolasi) merupakan himpunan sisi-sisi yang menutup semua titik di g (chartrand dan lesniak, 1986: 243). berdasarkan latar belakang diatas, maka penelitian ini dapat dirumuskan bagaimana rumus umum titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ���. tujuan dari penelitian ini adalah untuk mengetahui rumus umum titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ���. dalam penelitian ini penulis mendefinisikan untuk beberapa istilah yang digunakan agar tidak terjadi penafsiran ganda terhadap istilah-istilah tersebut yaitu, �� adalah graf bintang dengan � titik, selanjutnya �� ditulis sebagai ���, � �� ��� diperoleh dari ��� sebanyak kali dengan @ab @abc� sisi di � �� ��� , ��� adalah setiap anting graf bintang �� yang diberi � titik, sehingga � �� ��� diperoleh dari ��� sebanyak kali dengan @ab @abc� sisi di � �� ���. d��1�� ���� � <@a�, @��, … , @��f d��2�� ���� � <@a�, @��, … , @��f g <@a6, @�6, … , @�6f d�� h 1������ g <@ai, @�i, … , @�if �� : graf bintang dengan � titik. gambar 1a. graf bintang �� selanjutnya �� ditulis sebagai ��� sehingga, � �� ���: graf bintang ��� sebanyak kali dan @ab @abc� sisi di� �� ���. gambar 1b. graf bintang � �� ��� ���: setiap anting graf bintang �� yang diberi � titik. gambar 1. graf bintang ��� � �� ���: graf bintang �� sebanyak kali yang setiap anting graf bintang �� yang memiliki � titik dan @ab @abc� sisi di� �� ���. gambar 2. graf bintang � �� ��� selanjutnya �� adalah graf roda dengan � titik, selanjutnya �� ditulis sebagai ���, � �� ��� diperoleh dari ��� sebanyak kali dengan @ab @abc� sisi di � �� ��� ,��� adalah setiap sisi graf roda �� yang diberi � titik, sehingga � �� ��� diperoleh dari ��� sebanyak kali dengan @ab @abc� sisi di � �� ���. d��1�� ����� � <@a�, @��, … , @��f d��2�� ���� � <@a�, @��, … , @��f g <@a6, @�6, … , @�6f d�� h 1������ g <@ai, @�i, … , @�if �� ; graf roda dengan � titik gambar 3. graf roda �� selanjutnya �� ditulis sebagai ���: � �� ���: graf roda ��� sebanyak kali dan @ab @abc� sisi di � �� ���. gambar 4. graf roda � �� ��� ��� : setiap sisi yang ada pada graf roda �� diberi � titik. nurul hijriyah dan wahyu h. irawan 98 volume 2 no. 2 mei 2012 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 2 vn 2 v1 2 v2 2 v3 2 v42 v5 2 v6 2 v7 2 v8 2 v9 m vn m v1 m v2 m v3 m v4m v5 m v6 m v7 m v8 m v9 gambar 5. graf roda ��� � �� ���: graf roda ��� sebanyak kali dan @ab @abc� sisi di� �� ���. gambar 6. graf roda � �� ��� kajian teori covering di dalam al-qur’an dalam al-quran kata “penutup” itu diartikan sebagai “hijab dan himar” yang memiliki arti sebagai penutup (aurat) baik laki-laki maupun perempuan dan penutup kepala. memakai hijab yang benar akan mendatangkan kebaikan. dalam kajian matematika khususnya dalam teori graf ada juga kata penutup yaitu penutup (covering) yang di dalam alquran allah swt berfirman dalam surat al-ahzab/33 ayat 53: #sœ î)uρ £ èδθßϑçgø9r'y™ $yè≈ tftβ �∅èδθ è= t↔ ó¡sù  ïβ ï!#u‘ uρ 5>$pg éo 4 öν à6 ï9≡sœ ã� yγ ôû r& öν ä3 î/θ è= à) ï9 £ îγ î/θ è= è% uρ 4 $tβ uρ šχ% x. öν à6 s9 β r& (#ρ èœ ÷σ è? š^θ ß™ u‘ «! $# iω uρ β r& (# þθ ßs å3ζs? … çµy_≡uρ ø— r& . ïβ ÿíν ï‰ ÷è t/ # ´‰ t/ r& 4 ¨βî) öν ä3 ï9≡sœ tβ% ÿ2 y‰ζïã «! $# $̧ϑš ïà tã ∩∈⊂∪ artinya: “apabila kamu meminta sesuatu (keperluan) kepada mereka (isteriisteri nabi), maka mintalah dari belakang tabir. cara yang demikian itu lebih suci bagi hatimu dan hati mereka. dan tidak boleh kamu menyakiti (hati) rasulullah dan tidak (pula) mengawini isteri isterinya selama-lamanya sesudah ia wafat. sesungguhnya perbuatan itu adalah amat besar (dosanya) di sisi allah”. (qs. al-ahzab/33: 53) ini adalah ayat hijab yang di dalamnya mengandung beberapa hukum dan beberapa adat syar’i, di mana sebab turunnya adalah menyetujui perkataan ‘umar ra. dan aku berkata: ‘ya rasulullah, sesungguhnya orang yang baik dan orang yang buruk, terkadang masuk kepada isteri-isterimu, maka kiranya engkau memberikan mereka hijab, lalu allah menurunkan ayat hijab. dalam surat diatas jika di pandang dalam segi matematika yang dimaksud sebagai hijab/khimar adalah suatu covering. covering pada suatu graf menjadi bukti bahwa dengan mengamati petunjuk allah, yang berupa penutup (hijab) tersebut dapat diperoleh suatu formula yang luar biasa yang bermanfaat bagi manusia yaitu dalam bentuk penutup. teori dasar definisi 1. graf j graf g adalah pasangan himpunan (v,e) dengan v adalah himpunan tidak kosong dan berhingga dari objek-objek yang disebut sebagai titik dan e adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di v yang disebut sebagai sisi (chartrand dan lesniak, 1986:4). definisi 2. adjacent dan incident sisi ( � <%, @f dikatakan menghubungkan titik % dan @. jika ( � <%, @f adalah sisi pada graf g, maka % dan @ adalah titik yang terhubung langsung (adjacent), sementara itu % dan ( sama halnya dengan @ dan ( disebut terkait langsung (incident). lebih jauh, jika (� dan (6 berbeda pada � terkait langsung (incident) dengan sebuah titik bersama, maka (� dan (6 disebut sisi adjacent (chartrand dan lesniak, 1986:4). definisi 3. jalan misalkan % dan @ (yang tidak harus berbeda) adalah titik pada graf �. jalan %-@ pada graf � adalah barisan berhingga yang berselang-seling, % � %a, (�, %�, (6, … , %�k�, (�, %� � @ antara titik dan sisi, yang dimulai dari titik % dan diakhiri di titik @, dengan (b � %bk�%b untuk , � 1, 2, … , � (chartrand dan lesniak, 1986:26). definisi 4. trail jalan %-@ yang tidak mengulang sisi atau semua sisinya berbeda disebut trail %-@ (chartrand dan lesniak, 1986:26). definisi 5. lintasan jalan %-@ yang semua titiknya berbeda disebut path (lintasan) %-@. dengan demikian, semua lintasan adalah trail. contoh pada gambar 2.4 yaitu jalan @l, (6, @6, (�, @�, (m, @n, (o, @o, (p, @p adalah lintasan (chartrand dan lesniak, 1986: 26). definisi 6. sirkuit trail tertutup (closed trail) dan tak trivial pada graf � disebut sirkuit �. contoh pada gambar 2.4 yaitu jalan @o, (o, @n, (m, @�, (�, @6, (6, @l, (l, @o adalah sirkuit (chartrand dan lesniak, 1986:28). definisi 7. sikel sirkuit @�, @6, . . , @�, @��� � 3� memiliki � titik dengan @b adalah titik-titik berbeda untuk titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ��� jurnal cauchy – issn: 2086-0382 99 1 r , r � disebut sikel (cycle). contoh pada gambar 2.4 yaitu jalan @o, (o, @n, (n, @6, (6, @l, (l, @o adalah contoh sikel (chartrand dan lesniak, 1986:28). definisi 8. keterhubungan pasangan titik % dan @ dapat dikatakan terhubung (connected), jika terdapat lintasan %-@ di �. suatu graf � dapat dikatakan terhubung (connected) jika untuk setiap titik % dan @ di � terhubung (chartrand dan lesniak, 1986:28). definisi 9. gabungan (union) gabungan (union) dari �� dan �6, ditulis � � �� g �6, adalah graf dengan d��� � d���� gd��6� dan s��� � s���� g s��6�. jika graf � merupakan gabungan dari sebanyak � graf t, � � 2, maka ditulis � � �t (abdussakir, 2009:33). definisi 10. penjumlahan (join) penjumlahan (join) dari �� dan �6, ditulis � � �� " �6, adalah graf dengan d��� � d���� gd��6� dan s��� � s���� g s��6� g <%@|% �d���� .dan @ � d��6�f (abdussakir, 2009:33). definisi 11. graf bintang graf bipartisi komplit v�,� disebut graf bintang (star) dan dinotasikan dengan ��. jadi, �� mempunyai order �� " 1� dan ukuran � (abdussakir, 2009:21-22). definisi 12. graf roda graf roda ���� adalah graf yang memuat satu sikel yang setiap titik pada sikel terhubung langsung dengan titik pusat. graf roda �� diperoleh dengan operasi penjumlahan graf sikel w� dengan graf komplit v�. jadi, �� � w� "v�, � x 2 (chartrand dan lesniak, 1996:8). penutup pada graf definisi 10. titik penutup titik penutup dari graf � adalah y z d��� sedemikian sehingga semua titik di y menutup semua sisi di �, artinya setiap sisi dari � adalah terhubung langsung untuk setidaknya salah satu di titik y (bondy dan murty, 2008:420). titik penutup minimal dari graf � dinotasikan ��� adalah bilangan kardinal terkecil dari himpunan titik penutup yang paling sedikit. definisi 11. sisi penutup sisi penutup dari graf � adalah y z d��� sedemikian hingga semua sisi di y menutup semua titik di �, artinya adalah setiap titik di � berinsiden dengan setidaknya satu sisi di y (bondy dan murty, 2008:420). sisi penutup minimal dari graf � dinotasikan ���� adalah bilangan kardinal terkecil dari himpunan sisi penutup yang paling sedikit. titik dan sisi penutup minimal pada graf lintasan titik penutup minimal dari graf lintasan [� �� � 2) adalah: �[�� � � 12 �; %�&%� � '(�)*12 �� h 1�; %�&%� � ')�+,. sisi penutup minimal dari graf lintasan [� �� � 2) adalah: ��[�� � � 12 �; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. titik dan sisi penutup minimal pada graf sikel titik penutup minimal dari graf sikel w� �� � 3) adalah: �[�� � � 12 �; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. sisi penutup minimal dari graf sikel w� �� � 3) adalah: ��[�� � � 12 �; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. pembahasan suatu titik dan sisi dikatakan saling penutup pada graf g jika titik dan sisi tersebut inciden di g. titik penutup di g merupakan himpunan dari titik-titik yang menutup semua sisi di g dan sisi penutup pada graf g (tanpa titik terisolasi) merupakan himpunan sisi-sisi yang menutup semua titik di g. di katakan minimal karena banyaknya anggota paling sedikit atau himpunan penutup yang kardinalnya terkecil. titik dan sisi penutup pada graf bintang �� dan ������\ perhatikan tabel di bawah ini: tabel 1. titik dan sisi penutup minimal pada graf bintang � �� ��� simpul ]���� ]\���� ]����� ��\� ]\����� ��\� s3 1 3 1 � 3 � s4 1 4 1 � 4 � s5 1 5 1 � 5 � s6 1 6 1 � 6 � s7 1 7 1 � 7 � s8 1 8 1 � 8 � s9 1 9 1 � 9 � s10 .. sn 1 . . 1 10 . . � 1 � . . 10 � . . � � nurul hijriyah dan wahyu h. irawan 100 volume 2 no. 2 mei 2012 berdasarkan tabel 1 maka diperoleh lemma berikut: lemma 1: titik penutup minimal pada graf bintang �� adalah 1. bukti: misal titik pusat adalah @a. karena semua sisi (b insidensi dengan @a sehingga @a menutup semua sisi di ��, atau @a berinsiden dengan (b �, � 1, 2, … , �� maka @a adalah satu-satunya titik yang menutup semua sisi. jadi titik penutup minimal graf bintang ���� adalah 1. lemma 2: sisi penutup minimal pada graf bintang �� adalah �. bukti: misal d���� � <@a, @�, @6, @l, … , @�f. titik @�, @6, @l, … , @� tidak terhubung langsung tetapi titik @�, @6, @l, … , @� terhubung langsung dengan @a sehingga sisi �@b, @a� menutup titik @b :)� @a �, � 1, 2, … , ��. maka sisi penutup minimal pada graf bintang �� terbukti sebanyak �. lemma 3; titik penutup minimal pada graf bintang � �� ��� adalah . bukti: pada graf � �� ��� masing-masing titik pusat yang berurutan adalah terhubung langsung, yaitu �@ab , @abc��. dari lemma 1 titik penutup minimal ��� adalah 1. sehingga diperoleh titik penutup minimal pada graf bintang � �� ��� adalah . lemma 4 sisi penutup minimal pada graf bintang � �� ��� adalah � �. bukti: pada graf � �� ��� masing-masing titik pusat yang berurutan adalah terhubung langsung, yaitu �@ab , @abc��. berdasarkan lemma 2, sisi penutup minimal pada graf bintang �� adalah �. sehingga diperoleh sisi penutup minimal pada graf bintang � �� ��� adalah � �. titik dan sisi penutup pada graf bintang ���� ��� lemma 5 titik penutup minimal pada graf bintang � �� ��� adalah: �� ������ ;� 7 8�6 �� " 19 ; %�&%� � '(�)* 8�6 ��� " 1� " 19 ; %�&%� � ')�+,-. berlaku untuk � � �, � � 3. bukti: 1. untuk � genap ambil @a sebagai titik penutup. maka anting graf ��� berupa lintasan [�c�, dengan �� " 1� bilangan ganjil. jadi �[�c�� � �6 �. karena terdapat sebanyak � anting maka diperoleh ����� � 1 " �6 ���� � �6 �� " 1. 2. untuk � ganjil ambil @a sebagai titik penutup. maka anting graf ��� berupa lintasan [�c�, dengan �� " 1� bilangan genap. jadi �[�c�� � �6 �� " 1�. karena terdapat sebanyak � anting maka diperoleh ����� � �6 ��� " 1� " 1. dari 1 dan 2, karena � �� ��� terdiri dari ��� yang titik pusat berurutan dihubungkan langsung maka titik penutup minimal pada graf bintang � �� ��� ditunjukkan pada gambar 4 sehingga: �� �� ���� ;� � 12 �� " 1# ; %�&%� � '(�)* 12 ��� " 1� " 1# ; %�&%� � ')�+,. berlaku untuk � � �, � � 3. lemma 6 sisi penutup minimal pada graf bintang � �� ��� adalah: ��� �� ���� ;� 7 � 8�6 � " 19# ; %�&%� � '(�)* 8�6 ��� " 1� " 19 ; %�&%� � ')�+, . berlaku untuk � � �, � � 3. bukti: 1. untuk � genap pada graf ��� pandang lintasan 1 anting berikut: @a 1 2 � @6 gambar 9. lintasan [�c6 graf ini berupa [�c6 dengan �� " 2� adalah genap, sehingga ��[�c6� � �6 �� " 2� � �6 � " 1. dengan mengambil �@a, ��� sebagai sisi penutup. anting lainnya berupa [�c� dengan �� " 1� adalah ganjil maka ��[�c�� � �6 >�� " 1� " 1? ��6 � " 1. karena lintasan [�c� sebanyak �� h 1� lintasan maka sisi penutup minimalnya 8�6 � " 19 �� h 1�. sehingga diperoleh ������ � 8�6 � " 19 " 8�6 � " 19 �� h 1� � 8�6 � " 19 �. 2. untuk � ganjil pada graf ��� pandang lintasan 1 anting berikut: @a 1 2 3 � @6 gambar 10. lintasan [�c6 graf ini berupa [�c6 dengan �� " 1� adalah ganjil, maka ��[�c�� � �6 >�2 " 1� " 1? � �6 �� " 3� ��6 �� " 1� " 1. dengan mengambil �@a, ��� sebagai titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ��� jurnal cauchy – issn: 2086-0382 101 sisi penutup. anting lainnya berupa [�c� dengan �� " 1� genap maka ��[�� � �6 �� " 1�. karena lintasan [� sebanyak �� h 1� lintasan maka sisi penutup minimalnya 8�6 �� " 1�9 �� h 1�. sehingga diperoleh ������ � �6 �� " 1� " 1 "8�6 � " 19 �� h 1� � �6 ��� " 1� " 1. dari 1 dan 2, karena � �� ��� terdiri dari ��� yang titik pusat berurutan dihubungkan langsung maka titik penutup minimal pada graf bintang � �� ��� ditunjukkan pada gambar 4 sehingga: ��� �� ���� ;� /01 02 3� 12 � " 1#4 ; %�&%� � '(�)* 12 ��� " 1� " 1# ; %�&%� � ')�+, . berlaku untuk � � �, � � 3. titik dan sisi penutup pada graf roda � dan ���� �\ perhatikan tabel di bawah ini: tabel 2. titik dan sisi penutup minimal pada graf roda � �� ��� berdasarkan tabel 2 maka diperoleh lemma berikut: lemma 9 titik penutup minimal pada graf roda �� adalah ���� � � 12 � " 1 ; %�&%� � '(�)*12 �� " 1� " 1; %�&%� � ')�+,. bukti: misal @a titik pusat pada �� dan �@�, @6, @l, … , @�� adalah titik pada sikel luar. karena @a akan menutup semua sisi di selain sikel luar dan �w�� � � 12 � ; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. maka diperoleh: ���� � � 12 � " 1 ; %�&%� � '(�)*12 �� " 1� " 1; %�&%� � ')�+,. lemma 10: sisi penutup minimal pada graf roda �� adalah ����� � � 12 � " 1 ; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. bukti: misal @a titik pusat pada �� dan �@�, @6, @l, … , @�� adalah titik pada sikel luar. ambil �@a, @�� sebagai sisi penutup, maka pada w�: @�, @6, @l, … , @�, @� titik @� sudah tertutup oleh �@a, @��. ��w�� � �6 �, untuk � genap dan tidak terpengaruh oleh tertutupnya @�. ��w�� � �6 �� " 1�, untuk � ganjil dan terpengaruh oleh tertutupnya @�, sehingga harus dikurangi 1. maka diperoleh: ����� � � 12 � " 1 ; %�&%� � '(�)*12 �� " 1�; %�&%� � ')�+,. lemma 11: titik penutup minimal pada graf roda � �� ��� adalah: �� �� ���� � � 12 � " 1 # ; %�&%� � '(�)* 12 �� " 1� " 1# ; %�&%� � ')�+,. bukti: pada graf � �� ��� masing-masing titik pusat yang berurutan adalah terhubung langsung, yaitu �@ab , @abc��. berdasarkan lemma 9, titik penutup minimal pada graf roda �� adalah adalah �6 � " 1; untuk � genap dan �6 �� " 1� " 1; untuk � ganjil. sehingga diperoleh titik penutup minimal pada graf roda� �� ��� adalah: �� �� ���� � � 12 � " 1 # ; %�&%� � '(�)* 12 �� " 1� " 1# ; %�&%� � ')�+,. lemma 12: sisi penutup minimal pada graf roda � ����� adalah: ��� �� ���� � /01 02 12 � " 1 # ; %�&%� � '(�)* 312 �� " 1�4 ; %�&%� � ')�+,. bukti: pada graf � �� ��� masing-masing titik pusat yang berurutan adalah terhubung langsung, yaitu �@ab , @abc��. berdasarkan lemma 10, sisi penutup minimal pada graf roda �� adalah adalah �6 � " 1; untuk � genap dan �6 �� " 1�; untuk � ganjil. sehingga diperoleh sisi penutup minimal pada graf roda� �� ��� adalah: ��� �� ���� � /01 02 12 � " 1 # ; %�&%� � '(�)* 312 �� " 1�4 ; %�&%� � ')�+,. simpul ]� �� ]\� �� ]����� �\� ]\����� �\� w3 3 2 3 � 2 � w4 3 3 3 � 3 � w5 4 3 4 � 3 � w6 4 4 4 � 4 � w7 5 4 5 � 4 � w8 5 5 5 � 5 � w9 6 5 6 � 5 � w10 6 6 6 � 6 � nurul hijriyah dan wahyu h. irawan 102 volume 2 no. 2 mei 2012 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 titik dan sisi penutup pada graf roda ���� �� lemma 13: titik penutup minimal pada graf roda � �� ��� adalah: �� �� ���� ;� /0 1 02 312 �2�� " � " 2�4 ; � '(�)* :)� � '(�)* 312 �2�� " � " 3�4 ; � ')�+,:)� � '(�)* ��� " 1��; %�&%� � ')�+,-, � � �, � � 3 . bukti: 1. untuk � genap pada graf ��� pandang sikel luarnya: gambar 11. graf sikel w���c�� graf tersebut berupa w���c��. jika � ganjil maka ��� " 1� adalah ganjil, dan jika �genap maka ��� " 1� adalah genap. sehingga >w���c��? ��6 ���� " 1� " 1� untuk � ganjil dan >w���c��? ��6 >��� " 1�? untuk � genap. selanjutnya pandang graf bintang tanpa titik terluarnya: gambar 12. graf bintang ���k� graf ��� tanpa titik terluar ini sama dengan graf ���k�. karena � genap maka � h 1 ganjil. sesuai bukti lemma 5, diperoleh: ����k��: � �6 �>�� h 1� " 1? " 1 � �6 �� " 1 jadi, �����: � /0 1 02 �6 >��� " 1�? " �6 �� " 1� �6 �2�� " � " 2�; untuk � genap�6 ���� " 1� " 1� " �6 �� " 1� �6 �2�� " � " 3�; untuk � ganjil . 2. untuk � ganjil pada graf ��� pandang sikel luarnya: gambar 13. graf sikel cr�sc�� graf tersebut berupa w���c��. karena � ganjil maka �� " 1� genap, sehingga ��� " 1� selalu genap. jadi >w���c��? � �6 >��� " 1�?. selanjutnya pandang graf bintang tanpa titik terluarnya: gambar 14. graf bintang ���k� graf ��� tanpa titik terluar ini sama dengan graf ���k�. karena � ganjil maka � h 1 genap. sesuai bukti lemma 5, maka: ����k��: � �6 ��� h 1� " 1 jadi diperoleh, ����� � 12 >��� " 1�? " 12 ��� h 1� " 1 � �6 �2�� " 2� � �� " 1 dari 1 dan 2, karena � �� ��� terdiri dari ��� yang titik pusat berurutan dihubungkan langsung maka titik penutup minimal pada graf roda � �� ��� ditunjukkan pada gambar 8 sehingga: �� �� ���� ;� /0 1 02 312 �2�� " � " 2�4 ; � '(�)* :)� � '(�)* 312 �2�� " � " 3�4 ; � ')�+,:)� � '(�)* ��� " 1��; %�&%� � ')�+,-, � � �, � � 3 . lemma 14: sisi penutup minimal pada graf roda � �� ��� adalah: ��� �� ���� ;� /0 1 02 312 �2�� " � " 2�4 ; � '(�)* :)� � '(�)* 312 �2�� " � " 3�4 ; � ')�+,:)� � '(�)* ���� " 1��; %�&%� � ')�+,-, � � �, � � 3 . bukti: 1. untuk � genap pada graf ��� pandang sikel luarnya: gambar 15. graf sikel w���c�� graf tersebut berupa w���c��. jika � ganjil maka ��� " 1� adalah ganjil, dan jika �genap maka ��� " 1� adalah genap. sehingga 8w�>�"1?9 �12 >�>� " 1? " 1? untuk � ganjil dan 8w�>�"1?9 �12 8�>� " 1?9 untuk � genap. selanjutnya pandang graf bintang tanpa titik terluarnya: titik dan sisi penutup minimal pada graf bintang � �� ��� dan graf roda � �� ��� jurnal cauchy – issn: 2086-0382 103 1 vn 1 v1 1 v2 1 v3 1 v41 v5 1 v6 1 v7 1 v8 1 v9 gambar 16. graf bintang ���k� graf ��� tanpa titik terluar ini sama dengan graf ���k�. karena � genap maka � h 1 ganjil. sesuai bukti lemma 6, diperoleh: �����k��: � �6 �>�� h 1� " 1? " 1 � �6 �� " 1 jadi, ������: � /0 1 02 �6 >��� " 1�? " �6 �� " 1� �6 �2�� " � " 2�; untuk � genap�6 ���� " 1� " 1� " �6 �� " 1� �6 �2�� " � " 3�; untuk � ganjil . 2. untuk � ganjil pada graf ��� pandang sikel luarnya: gambar 17. graf sikel w���c�� graf tersebut berupa w���c��. karena � ganjil maka �� " 1� genap, sehingga ��� " 1� selalu genap. jadiá>w���c��? � �6 >��� " 1�?. selanjutnya pandang graf bintang tanpa titik terluarnya: gambar 18. graf bintang ���k� graf ��� tanpa titik terluar ini sama dengan graf ���k�. karena � ganjil maka � h 1 genap. sesuai bukti lemma 6, maka: �����k��: � 12 ���� h 1� " 1� " 1 jadi diperoleh, ������ � 12 >��� " 1�? " 12 ���� h 1� " 1� " 1 � ��� " 1�; � � �, � � 3 dari 1 dan 2, karena � �� ��� terdiri dari ��� yang titik pusat berurutan dihubungkan langsung maka sisi penutup minimal pada graf roda � �� ��� ditunjukkan pada gambar 8 sehingga: ]\>���� ��? ;� /0 1 02� 3\u �u�� " � " u�4 ; � vw�xy zx� � vw�xy � 3\u �u�� " � " {�4 ; � vx�|}~ zx� � vw�xy����� " \��; � vx�|}~, � � �, � � { . penutup berdasarkan penelitian yang telah dilaksanakan, dapat disimpulkan bahwa: 1. graf bintang � �� ��� dengan � � �, � � 3. maka rumusan titik dan sisi penutup minimal masing-masing adalah: a. ���� � 1 dan á����� � �. b. �� �� ���� � dan ��� �� ���� � � �. c. �� �� ���� � 7 8�6 �� " 19 ; %�&%� � '(�)* 8�6 ��� " 1� " 19 ; %�&%� � ')�+,-. ��� �� ���� � /01 02 3� 12 � " 1#4 ; %�&%� � '(�)* 12 ��� " 1� " 1# ; %�&%� � ')�+,. 2. graf roda � ����� dengan � � �, � � 3. maka rumusan titik dan sisi penutup minimal masing-masing adalah: a. ���� � 5 �6 � " 1 ; %�&%� � '(�)*�6 �� " 1� " 1 ; %�&%� � ')�+,-. ����� � 5 �6 � " 1 ; %�&%� � '(�)*�6 �� " 1� ; %�&%� � ')�+,-. b. �� ������ � 7 8 �6 � " 19 ; %�&%� � '(�)* 8�6 �� " 1� " 19 ; %�&%� � ')�+,-. ��� ������ � /01 02 12 � " 1# ; %�&%� � '(�)* 312 �� " 1�4 ; %�&%� � ')�+,. c. �� ������: � � 12 �2�� " � " 2�# ; � '(�)*, � '(�)* 12 �2�� " � " 3�# ; � ')�+,-, � '(�)* . ��� ������: � /01 02 312 �2�� " � " 2�4 ; � '(�)*, � '(�)* 312 �2�� " � " 3�4 ; � ')�+,-, � '(�)* . �� ������ � < ��� " 1�. dan ��� �� ���� �= >��� " 1�?; �. ganjil dan � � �, � � 3 daftar pustaka [1] abdullah bin muhammad.2007. tafsir ibnu katsir jilid 6. bogor: pustaka imam asy-syafii [2] abdussakir, dkk. 2009. teori graf. malang: uin press [3] al-maragi, ahmad musthafa.1992. tafsir almaraghi juz 18 dan 22 . semarang:toha putra [4] al-qurthubi, syaikh imam. 2009. tafsir alqurthubi. penj. fathurrahman abdul hamid dkk. jakarta: pustaka azzam nurul hijriyah dan wahyu h. irawan 104 volume 2 no. 2 mei 2012 [5] balakrishnan.v.k. 1995. schaum’s outline of theory and problems of combinatorics. new york: mc graw hill. inc [6] chatrand, gery and lesniak, linda. 1986. graphs and digraphs second edition. california: a division of wadsworth.inc. [7] gallian, j. a. 2007. "a dynamic survey of graphlabeling.(online:http//www.combinat orics.org/surveys/dr6.pdf). diakses 11 oktober 2011 [8] purwanto. 1998. teori graph. malang: ikip malang [9] rosen, kenneth h. 2003. discrete mathemathics ang its application: fifth edition. singapore: mc. graw-hill [10] wilson, robin j dan watkins. 1990. graph and introductory approach. singapore: open universitycourse. lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf 4. daftar isi.pdf 5.1. elva ravita sari studi tentang persamaan fuzzy _54-65.pdf winarni penjadwalan jalur bus dalam kota _15_ penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus (studi kasus busway transjakarta) winarni jurusan matematika fkip universitas adhi buana surabaya email: winarni_its@yahoo.com abstrak jaringan jalur bus dalam kota merupakan salah satu fasilitas transportasi umum yang memegang peranan penting dalam kehidupan masyarakat kota yang mempunyai karakteristik mobilitasnya cukup tinggi. jaringan bus transjakarta (busway) di jakarta merupakan salah satu contohnya. jaringan bus tersebut dibangun antara lain sebagai solusi permasalahan di sektor angkutan umum dan memberikan pilihan solusi untuk mengatasi kemacetan lalu lintas di jakarta. salah satu masalah penting dalam sistem transportasi tersebut yang menjadi keluhan masyarakat adalah mengenai ketidakpastian waktu tunggu kedatangan bus di tiap-tiap halte, hal ini dimungkinkan antara lain karena belum adanya penjadwalan yang baik pada sistem tersebut. dalam penelitian ini akan dilakukan penjadwalan keberangkatan bus menggunakan pendekatan aljabar max-plus dengan terlebih dahulu mengkontruksi model sistem dengan petrinet. studi kasus dalam penelitian ini adalah jaringan bus transjakarta. dari penelitian ini diharapkan memperoleh desain jadwal keberangkatan bus di tiap halte pada masing-masing koridor. kata kunci: jalur bus dalam kota, penjadwalan, aljabar max-plus, petrinet, busway. pendahuluan karakteristik daerah perkotaan yaitu antara lain padat penduduknya dan masyarakatnya mempunyai mobilitas yang cukup tinggi tersebut menjadikan masyarakat kota sangat bergantung pada kebutuhan transportasi. tentunya hal ini berdampak besar pada arus lalu lintas di jalan raya, sehingga kepadatan bahkan kemacetan arus lalu lintas pun hampir tidak bisa dihindarkan masyarakat setiap hari. terlebih lagi dengan semakin meningkatnya taraf perekonomian masyarakat dan semakin mudahnya kredit kendaraan bermotor yang tidak diiringi dengan adanya upaya peningkatan dan perbaikan mutu fasilitas dan regulasi sistem transportasi umum secara optimal mengakibatkan pengguna kendaraan pribadi semakin meningkat. pemandangan kemacetan arus lalu lintas dan dampaknya antara lain polusi udara, stress, pemborosan bahan bakar minyak (bbm) dan lain-lain adalah hal menyedihkan yang harus dihadapi dan dialami masyarakat hampir setiap hari. peningkatan dan perbaikan mutu fasilitas dan regulasi dalam pelayanan transportasi umum secara optimal yang mampu memberikan kepuasan kepada masyarakat, setidaknya dapat memberikan alternatif pada masyarakat untuk lebih memilih menggunakan jasa transportasi umum daripada menggunakan kendaraan pribadi sendiri-sendiri. jika masyarakat sudah lebih tertarik menggunakan jasa transportasi umum, hal ini berarti penggunaan kendaraan pribadi berkurang sehingga mengurangi kemacetan arus lalu lintas di jalan raya dan secara tidak langsung ikut memberikan kontribusi pada penghematan bbm. saat ini di jakarta khususnya sedang terus dikembangkan jaringan transportasi transjakarta dalam upaya untuk mengatasi kemacetan, salah satunya adalah busway. jika dilihat dari perencanaannya dan terlepas dari kesadaran masyarakat mengenai jalur khusus busway yang relatif masih kurang, sistem tersebut mempunyai peluang besar untuk berhasil, terlebih lagi ketika harga bbm semakin mahal. mengutip suatu artikel mengenai busway, dikatakan bahwa peranan busway semakin penting ketika harga bbm naik (http://bataviabusway.blogspot.com). namun, sejauh ini masih banyak keluhan dari pengguna antara lain mengenai ketidakpastian kedatangan bus di tiap-tiap halte, terkadang cepat terkadang cukup lama bahkan ketika bus sudah datang di halte namun bus sudah penuh penumpang. dengan sistem yang ada saat ini, para pengguna jasa bus transjakarta seringkali harus menunggu bus dengan ketidakpastian. meskipun kondisi halte dalam keadaan kosong, tidak menjamin penumpang bisa langsung naik bus yang datang berikutnya, terutama pada jam-jam sibuk (http://transjakartainfo.com). hal ini dimungkinkan karena belum adanya penjadwalan pada sistem tersebut yang dapat mengoptimalkan alokasi jumlah armada sehingga kebutuhan penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 193 pengguna jasa bus transjakarta dapat terpenuhi selain kendala operasional lainnya. kajian yang mengarah pada tujuan memperbaikan sistem transportasi umum perlu terus dikembangkan. terkait dengan masalah ini, mulai tahun 90an hingga saat ini kajian teori aljabar max-plus untuk pemodelan, analisis dan kontrol antara lain dalam jaringan transportasi, bidang manufaktur, jaringan komunikasi dan sistem komputer terus berkembang. berdasarkan uraian di atas, dalam penelitian ini dikaji model petrinet dan model aljabar max-plus untuk mendesaian penjadwalan suatu jaringan transportasi umum dalam hal ini jalur bus dalam kota dengan studi kasus jaringan busway transjakarta. sistem jaringan busway transjakarta jakarta adalah salah satu kota termacet di indonesia. saat ini, di jakarta sedang terus dikembangkan jaringan transportasi dalam upaya untuk mengatasi kemacetan, antara lain jaringan busway transjakarta. adapun denah jaringan busway transjakarta baik yang sudah beroperasi atau sedang dalam proses dapat dilihat pada gambar 1. (http://bataviabusway.blogspot.com) dalam makalah ini, yang akan dikaji adalah jaringan bus transjakarta (busway) di jakarta dengan jumlah koridor yang sudah aktif beroperasi sampai dengan bulan november 2008 yaitu sebanyak 7 koridor seperti dalam gambar 1. ada sedikit perubahan pada koridor 6, yaitu rute ragunan–halimun menjadi ragunan–dukuh atas 2. sedangkan jumlah armada maksimum yang dialokasikan untuk tiap-tiap koridor berdasarkan data yang diperoleh dari kantor badan layanan umum transjakarta dapat dilihat pada tabel 1. (untuk hari kerja dari jam 05.0008.00) gambar 1. denah busway koridor 1 s/d 15 tabel 1. alokasi bus tiap koridor busway koridor 1 (blok m -kota) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) blok m/m.agung kota/h.i 05:00 06:00 4,5 20 12 8 06:00 07:00 1,8 50 17 13 07:00 08:00 1,1 80 18 14 koridor 2 (pulo gadung harmoni) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) pulo gadung/asmi harmoni/pedongkelan 05:00 06:00 5 20 12 8 06:00 07:00 2 43 13 10 07:00 08:00 2 43 0 0 koridor 3 (kalideres harmoni pasar baru) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) kalideres harmoni 05:00 06:00 6 20 8 12 06:00 07:00 2,4 46 16 10 07:00 08:00 2,4 46 0 0 koridor 4 (pulo gadung dukuh atas 2) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) pulogadung dukuh atas 2 05:00 06:00 5 20 12 8 06:00 07:00 3 30 6 4 07:00 08:00 3 30 0 0 koridor 5 (ancol kampung melayu) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) ancol kp. melayu 05:00 06:00 5,6 18 6 12 06:00 07:00 4,3 22 2 3 07:00 08:00 3,7 27 2 2 koridor 6 (ragunan dukuh atas 2) pemberangkatan bus dari pool ke periode headway (menit) alokasi bus (unit) ragunan dukuh atas 2 05:00 06:00 5 20 12 8 06:00 07:00 3 31 7 4 07:00 08:00 3 31 0 0 winarni 194 koridor 7 (kampung melayu periode 05:00 06:00 07:00 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus sebelumnya. adapun data waktu tempuh antar halte di tiap koridor yang diperoleh dari obsevasi di lapangan selama beberapa hari yaitu tanggal 25 29 agustus, 21 2008. namun, karena terbatasnya jumlah hala man makalah ini, data tersebut tidak ditampilkan lengkap dalam makalah ini dan model petrinet petrinet dikembangkan pertama kali oleh c.a. petri pada awal 1960 salah satu alat untuk memodelkan diskrit. pada petrinet, event berkaitan dengan transisi dan keadaan ( place. dalam sistem event diskrit, perubahan keadaan terjadi karena adanya perubahan event. agar suatu event dapat terjadi, beberapa keadaan harus dip berfungsi sebagai input atau output suatu transisi. place sebagai input menyatakan keadaan yang harus dipenuhi agar transisi dapat terjadi. setelah transisi terjadi maka keadaan akan berubah. place yang menyatakan keadaan tersebut adalah output dari transisi. berikut ini adalah definisi petrinet: definisi 1. petrinet adalah 4 p : himpunan berhingga place, winarni koridor 7 (kampung melayu periode headway (menit) 06:00 07:00 08:00 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus sebelumnya. adapun data waktu tempuh antar halte di tiap koridor yang diperoleh dari obsevasi di lapangan selama beberapa hari yaitu tanggal 25 gustus, 21–23 o 2008. namun, karena terbatasnya jumlah hala man makalah ini, data tersebut tidak ditampilkan lengkap dalam makalah ini dan model petrinet petrinet dikembangkan pertama kali oleh c.a. petri pada awal 1960 salah satu alat untuk memodelkan diskrit. pada petrinet, event berkaitan dengan transisi dan keadaan ( place. dalam sistem event diskrit, perubahan keadaan terjadi karena adanya perubahan event. agar suatu event dapat terjadi, beberapa keadaan harus dipenuhi terlebih dahulu. place dapat berfungsi sebagai input atau output suatu transisi. place sebagai input menyatakan keadaan yang harus dipenuhi agar transisi dapat terjadi. setelah transisi terjadi maka keadaan akan berubah. place yang menyatakan keadaan tersebut adalah output dari transisi. berikut ini adalah definisi petrinet: definisi 1. (cassandras, 1993). petrinet adalah 4-tuple : himpunan berhingga place, koridor 7 (kampung melayu headway (menit) 5 3,3 3,3 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus adapun data waktu tempuh antar halte di tiap koridor yang diperoleh dari obsevasi di lapangan selama beberapa hari yaitu tanggal 25 oktober, dan 12 2008. namun, karena terbatasnya jumlah hala man makalah ini, data tersebut tidak ditampilkan lengkap dalam makalah ini dan gambar 2. petrinet dikembangkan pertama kali oleh c.a. petri pada awal 1960-an. ini merupakan salah satu alat untuk memodelkan diskrit. pada petrinet, event berkaitan dengan transisi dan keadaan (state) berkaitan dengan place. dalam sistem event diskrit, perubahan keadaan terjadi karena adanya perubahan event. agar suatu event dapat terjadi, beberapa keadaan enuhi terlebih dahulu. place dapat berfungsi sebagai input atau output suatu transisi. place sebagai input menyatakan keadaan yang harus dipenuhi agar transisi dapat terjadi. setelah transisi terjadi maka keadaan akan berubah. place yang menyatakan keadaan tersebut adalah output dari transisi. berikut ini adalah definisi petrinet: (cassandras, 1993). tuple (p, t, a, w) : himpunan berhingga place, p = { tabel 1 kampung rambutan) headway (menit) alokasi bus (unit) ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus adapun data waktu tempuh antar halte di tiap koridor yang diperoleh dari obsevasi di lapangan selama beberapa hari yaitu tanggal 25 ktober, dan 12 november 2008. namun, karena terbatasnya jumlah hala man makalah ini, data tersebut tidak dapat ditampilkan lengkap dalam makalah ini dan gambar 2. graf jaringan busway petrinet dikembangkan pertama kali oleh an. ini merupakan salah satu alat untuk memodelkan sistem event diskrit. pada petrinet, event berkaitan dengan ) berkaitan dengan place. dalam sistem event diskrit, perubahan keadaan terjadi karena adanya perubahan event. agar suatu event dapat terjadi, beberapa keadaan enuhi terlebih dahulu. place dapat berfungsi sebagai input atau output suatu transisi. place sebagai input menyatakan keadaan yang harus dipenuhi agar transisi dapat terjadi. setelah transisi terjadi maka keadaan akan berubah. place yang menyatakan keadaan tersebut adalah output dari transisi. berikut ini dengan {p1, p2, . . . , pn tabel 1. (lanjutan…) kampung rambutan) alokasi bus (unit) 20 30 30 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus adapun data waktu tempuh antar halte di tiap koridor yang diperoleh dari obsevasi di lapangan selama beberapa hari yaitu tanggal 25– ovember 2008. namun, karena terbatasnya jumlah haladapat ditampilkan lengkap dalam makalah ini dan hanya dirangkum dalam tabel 2. data tersebut selanjutnya diasumsikan tetap. susun gra berikut: jaringan busway petrinet dikembangkan pertama kali oleh an. ini merupakan sistem event diskrit. pada petrinet, event berkaitan dengan ) berkaitan dengan place. dalam sistem event diskrit, perubahan keadaan terjadi karena adanya perubahan event. agar suatu event dapat terjadi, beberapa keadaan enuhi terlebih dahulu. place dapat berfungsi sebagai input atau output suatu transisi. place sebagai input menyatakan keadaan yang harus dipenuhi agar transisi dapat terjadi. setelah transisi terjadi maka keadaan akan berubah. place yang menyatakan keadaan tersebut adalah output dari transisi. berikut ini n}, t : himpunan berhingga transisi, a : himpunan arc, w : fungsi bobot, berarah. node dari gra diambil dari himpunan place diambil dari himpunan transisi petrinet arc untuk menghubungkan dua node atau ekival yang menyatakan jumlah arc. struktur ini d dengan struktur multigraf representasi petrinet secara grafik akan digunakan notasi masing menyatakan himpunan place input ke transisi matematis definisi tersebut dapat ditulis menjadi persamaan berikut (cassandras, 1993) (lanjutan…) pemberangkatan bus dari pool ke kp. melayu 8 0 0 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus hanya dirangkum dalam tabel 2. data tersebut selanjutnya diasumsikan tetap. berdasarkan data susun graf dari jaringan busway adalah sebagai berikut: jaringan busway : himpunan berhingga transisi, : himpunan arc, : fungsi bobot, petrinet berarah. node dari gra diambil dari himpunan place diambil dari himpunan transisi petrinet diperbolehkan menggunakan beberapa arc untuk menghubungkan dua node atau ekivalen dengan memberikan bobot ke setiap arc yang menyatakan jumlah arc. struktur ini d dengan struktur multigraf representasi petrinet secara grafik akan digunakan notasi masing menyatakan himpunan place input ke transisi tj dan output dari transisi matematis definisi tersebut dapat ditulis menjadi persamaan berikut (cassandras, 1993) i( o( volume 1 no. pemberangkatan bus dari pool ke kp. melayu ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus hanya dirangkum dalam tabel 2. data tersebut selanjutnya diasumsikan tetap. berdasarkan data-data di atas dapat di dari jaringan busway adalah sebagai : himpunan berhingga transisi, : himpunan arc, a ⊆ (p × t : fungsi bobot, w : a → {1, 2 dapat digambarkan sebagai graf berarah. node dari graf diambil dari himpunan place diambil dari himpunan transisi diperbolehkan menggunakan beberapa arc untuk menghubungkan dua node atau en dengan memberikan bobot ke setiap arc yang menyatakan jumlah arc. struktur ini d dengan struktur multigraf representasi petrinet secara grafik akan digunakan notasi i(tj) dan masing menyatakan himpunan place input ke dan output dari transisi matematis definisi tersebut dapat ditulis menjadi persamaan berikut (cassandras, 1993) (tj) = {pi : (pi, (tj) = {pi : (tj , keterangan: warna arc bm kt : koridor 1 pg har : koridor 2 kd pb : koridor 3 pg da2 : koridor 4 anc km : koridor 5 rg – da2 : koridor 6 km kr : koridor 7 node kt : kota har : harmoni da1 : dukuh atas 1 da2 : dukuh atas 2 bm : blok m sn : senen sns : senen sentral pg : pulo gadung kd : kalideres pb : pasar baru mn1 : mantraman 1 mn2 : mantraman 2 hl : halimun anc : ancol km : kampung melayu rg : ragunan kr : kampung rambutan : skywalk volume 1 no. 4 pemberangkatan bus dari pool ke kp. rambutan 12 10 0 ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus hanya dirangkum dalam tabel 2. data tersebut selanjutnya diasumsikan tetap. data di atas dapat di dari jaringan busway adalah sebagai : himpunan berhingga transisi, t = {t1, t2 t) ∪ (t × p), 2, 3, . . . }. dapat digambarkan sebagai graf berupa place yan diambil dari himpunan place p atau transisi yang diambil dari himpunan transisi t. pada graf diperbolehkan menggunakan beberapa arc untuk menghubungkan dua node atau en dengan memberikan bobot ke setiap arc yang menyatakan jumlah arc. struktur ini d dengan struktur multigraf. dalam membahas representasi petrinet secara grafik akan ) dan o(tj) yang masing masing menyatakan himpunan place input ke dan output dari transisi tj matematis definisi tersebut dapat ditulis menjadi persamaan berikut (cassandras, 1993) , tj) ∈ a} , pi) ∈ a} koridor 1 koridor 2 koridor 3 koridor 4 koridor 5 koridor 6 koridor 7 kota harmoni dukuh atas 1 dukuh atas 2 blok m senen senen sentral pulo gadung kalideres pasar baru mantraman 1 mantraman 2 halimun ancol kampung melayu ragunan kampung rambutan skywalk mei 2011 pemberangkatan bus dari pool ke kp. rambutan ket: headway adalah interval waktu di ujung koridor antara keberangkatan bus dengan keberangkatan bus hanya dirangkum dalam tabel 2. data tersebut data di atas dapat didari jaringan busway adalah sebagai , t2, . . . , tm}, □ dapat digambarkan sebagai graf berupa place yan1g atau transisi yang . pada graf diperbolehkan menggunakan beberapa arc untuk menghubungkan dua node atau en dengan memberikan bobot ke setiap arc yang menyatakan jumlah arc. struktur ini dikenal . dalam membahas representasi petrinet secara grafik akan ) yang masingmasing menyatakan himpunan place input ke tj. secara matematis definisi tersebut dapat ditulis menjadi penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 195 dengan istilah lain, i(tj) = {pi : (pi, tj) ∈ a} menunjukkan i(tj) adalah himpunan upstream place untuk transisi tj dan o(tj) = {pi : (tj , pi) ∈ a} menunjukkan o(tj) himpunan downstream place untuk transisi tj. notasi yang sama dapat digunakan untuk mendeskripsikan input dan output transisi untuk place pi, yaitu i(pi) = {tj : (tj , pi) ∈ a} o(pi) = {tj : (pi, tj) ∈ a} i(pi) = {tj : (tj , pi) ∈ a}menunjukkan i(pi) adalah himpunan upstream transisi untuk place pi dan o(pi) = {tj : (pi, tj) ∈ a} menunjukkan o(pi) himpunan downstream transisi untuk place pi . grafik petrinet terdiri dari dua macam node yaitu lingkaran dan garis/persegipanjang kecil. lingkaran menyatakan place sedangkan garis/persegipanjang kecil menyatakan transisi. arc disimbolkan dengan panah yang menghubungkan place dan transisi. pada petrinet tidak diperkenankan adanya arc antara place dengan place atau antara transisi dengan transisi. arc yang menghubungkan place pi ke transisi tj berarti pi ∈ i(tj). jika bobot arc dari place pi ke transisi tj adalah k ditulis w(pi, tj) = k , maka terdapat k arc dari place pi ke transisi tj atau sebuah arc dengan bobot k. transisi pada petrinet menyatakan event pada sistem event diskrit dan place merepresentasikan kondisi agar event dapat terjadi. token adalah sesuatu yang diletakkan di place yang menyatakan terpenuhi tidaknya suatu kondisi. secara grafik token digambarkan dengan dot dan diletakkan di dalam place. jika jumlah token banyak maka dituliskan dengan angka. definisi 2. (cassandras, 1993). penanda (marking) x pada petrinet adalah fungsi x : p → {0, 1, 2, . . . }. □ penanda dinyatakan dengan vektor yang berisi bilangan bulat nonnegatif yang menyatakan jumlah token yaitu x = [x(p1), x(p2), . . . , x(pn)]t. jumlah elemen x sama dengan banyak place di petrinet. elemen ke-i pada vektor x merupakan jumlah token pada place pi, x(pi) ∈ {0, 1, 2, . . .}. definisi 3. (cassandras, 1993). petrinet bertanda (marked) adalah 5-tuple (p, t, a, w, x0) dimana (p, t, a, w) adalah petrinet dan x0 adalah penanda awal. □ selanjutnya petrinet bertanda cukup disebut petrinet dan istilah tanda tersebut disebut token. seperti pemodelan sistem pada umumnya, maka harus didefinisikan keadaan (state) pada petrinet. keadaan pada petrinet adalah token petrinet. definisi 4. (cassandras, 1993). keadaan (state) petrinet bertanda adalah x = [x(p1), x(p2), . . . , x(pn)]t. □ jumlah token pada place adalah sebarang bilangan bulat nonnegatif, tidak harus terbatas (bounded). ruang keadaan (state space) x pada petrinet bertanda dengan n place didefinisikan oleh semua vektor berdimensi n dengan elemenelemennya adalah bilangan bulat nonnegatif, sehingga x(pi) î {0, 1, 2, 3, ... }. tetapi, dalam menyusun petrinet perlu dihindari terjadinya ledakan (blow up) token pada satu atau lebih token, karena hal ini menunjukkan petrinet tersebut tidak stabil. definisi 5. (cassandras, 1993). transisi tj ∈ t pada petrinet bertanda dikatakan enabled jika x(pi) ≥ w(pi, tj), ∀ pi ∈ i(tj) □ definisi 6. (cassandras, 1993). fungsi perubahan keadaan, f : {0, 1, 2, . . . }n ×t → {0, 1, 2, . . . }n pada petrinet bertanda (p, t, a, w, x0) terdefinisi untuk transisi tj ∈ t jika dan hanya jika x(pi) ≥ w(pi, tj), ∀ pi ∈ i(tj) jika f(x, tj) terdefinisi maka ditulis x′ = f(x, tj), dimana x′(pi) = x(pi) − w(pi, tj) + w(tj , pi), i = 1, 2, . . . , n, j = 1, 2, . . ., m □ keadaan di mana tidak ada transisi yang enabled disebut keadaan terminal dan petrinet mengalami deadlock. sistem dikatakan tidak stabil jika terjadi ledakan (blow up) pada nilai variabel keadaannya. variabel keadaan dari petrinet adalah jumlah token pada setiap place. jika jumlah token pada satu atau lebih place bertambah menuju ke tak hingga maka dikatakan petrinet tidak stabil. penyusunan petrinet yang baik diharapkan menghindari terjadinya deadlock dan tidak stabil untuk menyusun model petrinet dan model aljabar max-plus yang dibahas dalam penelitian ini, diperlukan spesifikasi fisik sebagai berikut: a) jalur-jalur dalam jaringan busway (koridor), seperti yang sudah dipaparkan di atas. b) jumlah dan distribusi bus di tiap-tiap koridor dapat dilihat pada tabel 2. c) aturan sinkronisasi antar keberangkatan bus berdasarkan data waktu tempuh rata-rata antar halte dan data alokasi bus tiap koridor serta mempertimbangkan kondisi halte ujung dan halte transit, maka diperoleh distribusi bus di tiap lintasan (dengan waktu referensi 07.30) seperti pada tabel 2 di bawah ini. data tersebut selanjutnya diasumsikan tetap. tabel 2. waktu tempuh, distribusi bus, dan pendefinisian variabel winarni 196 volume 1 no. 4 mei 2011 koridor variabel keberangkatan dari menuju ke halte waktu tempuh (menit) jumlah bus yang beroperasi 1 x1 blok m dukuh atas 1 19,71 18 1 x2 dukuh atas 1 harmoni 15,23 12 1 x3 harmoni kota 12,83 7 1 x4 kota harmoni 10,98 9 1 x5 harmoni dukuh atas 1 20,21 17 1 x6 dukuh atas 1 blok m 14,21 5 2 x7 pulo gadung senen 19,5 10 2 x8 senen harmoni 17 9 2 x9 harmoni senen 15,86 8 2 x10 senen pulo gadung 16,75 8 3 x11 kalideres harmoni 24,67 10 3 x12 harmoni pasar baru 8 4 3 x13 pasar baru harmoni 6,24 2 3 x14 harmoni kalideres 26,5 11 4 x15 pulo gadung mantraman 2 24,38 9 4 x16 mantraman 2 halimun 12,61 4 4 x17 halimun dukuh atas 2 5,07 2 4 x18 dukuh atas 2 halimun 2,64 1 4 x19 halimun mantraman 2 10,1 3 4 x20 mantraman 2 pulo gadung 25 8 5 x21 ancol sentral senen 25,47 8 5 x22 sentral senen mantraman 1 15,3 4 5 x23 mantraman 1 kp. melayu 10,31 2 5 x24 kp. melayu mantraman 1 8,63 2 5 x25 mantraman 1 sentral senen 11,47 3 5 x26 sentral senen ancol 17,39 6 6 x27 ragunan halimun 37,69 13 6 x28 halimun dukuh atas 2 5,07 2 6 x29 dukuh atas 2 ragunan 34,32 11 7 x30 kp. melayu kp. rambutan 52,81 16 7 x31 kp. rambutan kp. melayu 43,14 13 berdasarkan observasi di lapangan mengenai kondisi sistem jaringan busway dan berdasarkan hasil perhitungan jumlah bus yang beroperasi pada waktu referensi pukul 07.30 seperti terlihat pada tabel 2 di atas, maka dapat disusun aturan sinkronisasi pada jaringan busway dengan 7 koridor tersebut sebagai berikut i) koridor 1 (blok m kota): bm – har – da1 – kt � keberangkatan bus yang ke-(k+1) dari da1 menuju har (ke arah kt) dan keberangkatan bus yang ke-(k+1) dari da1 menuju bm keduanya menunggu kedatangan bus yang berangkat ke-(k-1) dari hl menuju da2 (yang berasal dari mn2) dan menunggu kedatangan bus yang berangkat ke-(k-1) dari hl menuju da2 (yang berasal dari rg) menuju da2 dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari da2 ke da1. � keberangkatan bus yang ke-(k+1) dari har menuju kt dan keberangkatan bus yang ke-(k+1) dari har menuju da1 (ke arah bm) keduanya menunggu kedatangan bus yang berangkat ke-(k-9) dari kd menuju har (ke arah pb), menunggu kedatangan bus yang berangkat ke-(k-1) dari pb menuju har (ke arah kd), dan menunggu kedatangan bus yang berangkat ke-(k-8) dari sn menuju har (yang berasal dari pg). ii) koridor 2 (pulo gadung harmoni): pg – sn har � keberangkatan bus yang ke-(k+1) dari pg menuju sn (ke arah har) menunggu kedatangan bus yang berangkat ke-(k-7) dari mn2 menuju pg (yang berasal dari da2). � keberangkatan bus yang ke-(k+1) dari sn menuju har menunggu kedatangan bus yang berangkat ke-(k-7) dari anc menuju sns (ke arah km) dan menunggu kedatangan bus yang berangkat ke-(k-2) dari mn1 menuju sns (ke arah anc) dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari sns ke sn. dan sedikit berbeda dengan lintasan lain, halte-halte yang disinggahi dari sn ke har dan dari har ke sn berbeda, sehingga keberangkatan bus yang ke-(k+1) dari sn menuju har juga penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 197 menunggu kedatangan bus yang berangkat ke-(k-8) dari har menuju sn. � keberangkatan bus yang ke-(k+1) dari har menuju sn menunggu kedatangan bus yang berangkat ke-(k-9) dari kd menuju har (ke arah pb), menunggu kedatangan bus yang berangkat ke-(k-1) dari pb menuju har (ke arah kd), menunggu kedatangan bus yang berangkat ke-(k-8) dari kt menuju har (ke arah bm) dan menunggu kedatangan bus yang berangkat ke-(k-11) dari da1 menuju har (ke arah kt). � keberangkatan bus yang ke-(k+1) dari sn menuju pg menunggu kedatangan bus yang berangkat ke-(k-7) dari anc menuju sns (ke arah km) dan menunggu kedatangan bus yang berangkat ke-(k-2) dari mn1 menuju sns (ke arah anc) dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari sns ke sn. iii) koridor 3 (kalideres – pasar baru): kd – har – pb � keberangkatan bus yang ke-(k+1) dari har menuju pb dan keberangkatan bus yang ke-(k+1) dari har menuju kd keduanya menunggu kedatangan bus yang berangkat ke-(k-8) dari kt menuju har (ke arah bm), menunggu kedatangan bus yang berangkat ke-(k-11) dari da1 menuju har (ke arah kt), dan menunggu kedatangan bus yang berangkat ke-(k-8) dari sn menuju har. iv) koridor 4 (pulo gadung – dukuh atas 2) : pg – mn2 – hl – da2 � keberangkatan bus yang ke-(k+1) dari pg menuju mn2 (ke arah da2) menunggu kedatangan bus yang berangkat ke-(k-7) dari sn menuju pg (yang berasal dari har). � keberangkatan bus yang ke-(k+1) dari mn2 menuju hl dan keberangkatan bus yang ke-(k+1) dari mn2 menuju pg keduanya menunggu kedatangan bus yang berangkat ke-(k-3) dari sns menuju mn1 (ke arah km) dan menunggu kedatangan bus yang berangkat ke-(k-1) dari km menuju mn1 (ke arah anc) dan masingmasing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari sns ke sn. � keberangkatan bus yang ke-(k+1) dari da2 menuju hl (ke arah pg) menunggu kedatangan bus yang berangkat ke-(k-17) dari bm menuju da1 (ke arah kt) dan menunggu menunggu kedatangan bus yang berangkat ke-(k-16) dari har menuju da1 (ke arah bm) dan masingmasing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari da1 ke da2. � keberangkatan bus yang ke-(k+1) dari hl menuju mn2 (ke arah pg) menunggu kedatangan bus yang berangkat ke-(k-12) dari rg menuju hl (ke arah da2). v) koridor 5 (ancol – kampung melayu): anc – sns – mn1 – km � keberangkatan bus yang ke-(k+1) dari sns menuju mn1 (ke arah km) dan keberangkatan bus yang ke-(k+1) dari sns menuju anc keduanya menunggu kedatangan bus yang berangkat ke-(k-7) dari har menuju sn (ke arah pg) dan menunggu kedatangan bus yang berangkat ke-(k-9) dari pg menuju sn (ke arah har) dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari sn ke sns. � keberangkatan bus yang ke-(k+1) dari mn1 menuju km dan keberangkatan bus yang ke-(k+1) dari mn1 menuju sns (ke arah anc) keduanya menunggu kedatangan bus yang berangkat ke-(k-8) dari pg menuju mn2 (ke arah da2) dan menunggu kedatangan bus yang berangkat ke-(k-2) dari hl menuju mn2 (ke arah pg) dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari mn2 ke mn1. � keberangkatan bus yang ke-(k+1) dari km menuju mn1(ke arah anc) menunggu kedatangan bus yang berangkat ke-(k-12) dari kr menuju km. vi) koridor 6 (ragunan – dukuh atas 2): rg – hl – da2 / da2 – rg � keberangkatan bus yang ke-(k+1) dari da2 menuju rg menunggu kedatangan bus yang berangkat ke-(k-1) dari hl menuju da2 (yang berasal dari mn2). selain itu juga menunggu kedatangan bus yang berangkat ke-(k-17) dari bm menuju da1 (ke arah kt) dan menunggu kedatangan bus yang berangkat ke-(k-16) dari har menuju da1 (ke arah bm) dan masing-masing ditambah waktu tempuh rata-rata untuk berjalan lewat skywalk dari da1 ke da2. vii) koridor 7 (kampung melayu – kampung rambutan): km – kr � keberangkatan bus yang ke-(k+1) dari km menuju kr menunggu kedatangan bus yang berangkat ke-(k-1) dari mn1 menuju km. berdasarkan spesifikasi fisik di atas, maka dapat disusun model petrinet untuk penjadwalan keberangkatan bus. selanjutnya dari model winarni 198 volume 1 no. 4 mei 2011 petrinet berikut, dapat diterjemahkan menjadi model aljabar max-plus (dan sebaliknya). pertama, disusun petrinet untuk tiap koridor, kemudian disinkronisasi berdasarkan aturan sinkronisasi di atas. petrinet yang disusun berikut ini dimaksukan untuk menggambarkan sinkronisasi antar keberangkatan bus berdasarkan aturan sinkronisasi yang telah diberikan di atas, namun tidak dimaksudkan untuk menggambarkan pergerakan jaringan bus secara simultan. dengan pendefinisian petrinet sebagaimana telah dijelaskan di atas, maka petrinet yang disusun untuk permasalahan ini adalah petrinet dengan waktu yang dikarakterisasi oleh p, t, a, w , x0, dan t , yaitu sebagai berikut: p : himpunan berhingga place, p = {p1, p2, ..., p43}, dengan jumlah token pada place p1, p2, ..., p31 menunjukkan jumlah distribusi bus pada masingmasing rute yang bersesuaian, sedangkan token pada p32, p33, ..., p43 tidak menunjukkan jumlah bus, namun dikondisikan untuk menyusun petrinet yang sesuai untuk permasalahan ini. t : himpunan berhingga transisi, t = {t1, t2, ..., t31}, dalam hal ini akan digunakan notasi untuk masing-masing transisi adalah x yaitu t = {x01, x02, ..., x31}. transisi-transisi tersebut merepresentasikan event keberangkatan bus di halte-halte ujung koridor dan halte transit. hal ini bertujuan agar sesuai dengan notasi yang digunakan dalam penyusunan model aljabar max-plus. a : himpunan arc, a ⊆ (p × t) ∪ (t × p), yaitu a = {(x01, p1), (p1, x02), (x02, p2), (p2, x03), (x03, p3), (p3, x04), (x04, p4), (p4, x05), (x05, p5), (p05, x06), (x06, p6), (p6, x01), (p2, x14), (x14, p2), (p2, x12), (x12, p2), (p2, x09), (x09, p2), (x03, p8), (p8, x03), (x03, p11), (p11, x03), (x03, p13), (p13, x03), (x05, p11), (p11, x05), (x05, p13), (p13, x05), (x05, p8), (p8, x05), (x07, p7), (p7, x08), (x08, p8), (p8, x09), (x09, p9), (p9, x10), (x10, p10), (p10, x07), (x07, p20), (p20, x07), (x08, p9), (p9, x08), (x09, p11), (p11, x9), (x09, p13), (p13, x09), (x09, p4), (p4, x09), (x11, p11), (p11, x12), (x12, p12), (p12, x13), (x13, p13), (p13, x14), (x14, p14), (p14, x11), (x12, p4), (p4, x12), (x14, p4), (p4, x14), (x12, p9), (p9, x12), (x14, p9), (p9, x14), (x15, p15), (p15, x16), (x16, p16), (p16, x17), (x17, p17), (p17, x18), (x18, p18), (p18, x19), (x19, p19), (p19, x20), (x20, p20), (p20, x15), (x15, p10), (p10, x15), (x19, p27), (p27, x19), (x21, p21), (p21, x22), (x22, p22), (p22, x23), (x23, p23), (p23, x24), (x24, p24), (p24, x25), (x25, p25), (p25, x26), (x26, p26), (p26, x21), (x24, p31), (p31, x24), (x27, p27), (p27, x28), (x28, p28), (p28, x29), (x29, p29), (p29, x27), (x29, p17), (p17, x29), (x30, p30), (p30, x31), (x31, p31), (p31, x30), (p32, x02), (p32, x06), (p33, x18), (p33, x29), (p34, x08), (p34, x10), (p35, x22), (p35, x26), (p36, x16), (p36, x20), (p37, x23), (p37, x25), (p38, x02), (p38, x06), (p39, x18), (p39, x29), (p40, x8), (p40, x10), (p41, x22), (p41, x26), (p42, x16), (p42, x20), (p43, x23), (p43, x25), (x17, p32), (x28, p38), (x01, p33), (x05, p39), (x21, p34), (x25, p40), (x07, p35), (x09, p41), (x22, p36), (x24, p42), (x15, p37), (x9, p43)} fungsi bobot, w : a → {1, 2, 3, . . . }, yaitu semua arc dalam himpunan a tersebut bobotnya adalah 1, kecuali w(x17, p32) = w(x28, p38) = w(x01, p33) = w(x05, p39) = w(x21, p34) = w(x25, p40) = w(x07, p35) = w(x09, p41) = w(x22, p36) = w(x24, p42) = w(x15, p37) = w(x9, p43) = 2. x0 = [18 12 7 9 17 5 10 9 8 8 10 4 2 11 9 4 2 1 3 8 7 5 2 2 3 6 13 2 11 16 13 2 2 2 2 2 2 2 2 2 2 2 2], masing-masing menunjukkan jumlah token awal pada p1, p2, ..., p43. t adalah vektor yang elemen-elemennya merepresentasikan waktu tempuh perjalanan tiap-tiap rute, durasi waktu tersebut disertakan pada setiap place yang, dimana τi disertakan pada place pi, untuk p1 s/d p31 masing-masing disertakan τ1 s/d τ31 yang menunjukkan waktu tempuh perjalanan dengan keberangkatan x01 s/d x31, sedangkan untuk p32 s/d p43 masing-masing disertakan τ32 s/d τ43 yang menunjukkan waktu tempuh perjalanan yang bersesuaian ditambah waktu tempuh untuk berjalan di skywalk, karena p32 s/d p43 berkaitan dari kedatangan dan keberangkatan bus dihalte-halte yang ada dihubungkan dengan skywalk. hal ini dapat dilihat lebih lengkap pada gambar 3. inisialisasi x0 untuk place p1, p2, ..., p31 disesuaikan dengan distribusi bus yang beroperasi pada tiap-tiap rute berdasarkan tabel 2, sedangkan jumlah token pada p32, p33, . . . , p43 masing-masing diinisialisasi dengan token sebanyak 2 token. hal ini dimaksudkan agar downstrem transisi dari place p32, p33, ..., p43 enable. selain itu, semua arc dalam petrinet ini diberikan bobot 1, kecuali arc (x17, p32), (x28, p38), (x01, p33), (x05, p39), (x21, p34), (x25, p40), (x07, p35), (x09, p41), (x22, p36), (x24, p42), (x15, p37), (x9, p43) diberi bobot 2. secara fisik, nilai bobot sama dengan 2 dan 2 token pada p32, p33, ..., p43 tersebut maksudnya bahwa kedatangan x17, x28, x01, x05, x21, x25, x07, x09, x22, x24, x15, x19, masing-masing ditunggu oleh 2 keberangkatan bus lainnya, sebagai contoh adalah: kedatangan penumpang dari keberangkatan x01 (dari blok m ke dukuh atas 1) ditunggu oleh keberangkatan x18 (dari dukuh atas 2 ke halimun) dan x29 (dari dukuh atas 2 ke ragunan). hal ini direpresentasikan oleh arc (x01, p33), (p33, x18), dan (p33, x29), perlu diperhatikan pula bahwa durasi waktu yang tertera pada place adalah p33 adalah 19,71 + 5, ini menunjukkan 19,71 menit adalah waktu yang diperlukan untuk perjalanan dengan keberangkatan x01 dan 5 menit adalah waktu yang diperlukan untuk berjalan melalui skywalk dari dukuh atas 1 ke dukuh atas 2. hal ini dimaksudkan untuk menjamin sinkronisasi koridor 4 dengan koridor 1, bahwa penumpang jurnal cauchy dari koridor 1 dapat transit ke koridor 4 melalui skywalk dukuh atas. demikian halnya representasi arc ( dimaksudkan untuk menjamin sinkronisasi koridor 6 dengan koridor 1, bahwa penumpang dari koridor 1 dapat transit ke koridor 6 melalui skywalk dukuh atas. sehingga, model al sistem transportasi dapat dikatakan sebagai sistem dinamik event diskrit (sded) sama halnya sebagai sistem manufakturing, kedinamikan sebagai evolusi perlakuan sistem selama diberikan waktu periodenya. sistem transportasi dinamik diatur dengan sinkronisasi, paralelisasi dan kejadian yang serentak/concurrency (nait sidi-moh, a., dkk., 2008). penggunaan pen dalam sistem even diskrit dinamik adalah karena aljabar max mudah proses sinkronisasi (braker, 1990). pendekatan dengan aljabar max dengan kemampuannya untuk diadaptasikan pada masalah yang event-graph (nait aljabar max dalam sistem manufaktur dan pada masalah masalah yang berkaitan dengan jaringan transportasi yang sering berasal dari kejadian seperti sinkro yang terjadi ketika distribusi resources diperlukan (nait sebelum, menyusun model aljabar max plus, berikut diberikan konsep dasar mengenai struktur aljabar sebagai berikut: rmax r ∪ { =ε jurnal cauchy – issn: 2086 dari koridor 1 dapat transit ke koridor 4 melalui skywalk dukuh atas. demikian halnya representasi arc (x05, dimaksudkan untuk menjamin sinkronisasi koridor 6 dengan koridor 1, bahwa penumpang dari koridor 1 dapat transit ke koridor 6 melalui skywalk dukuh atas. sehingga, model aljabar max sistem transportasi dapat dikatakan sebagai sistem dinamik event diskrit (sded) sama halnya sebagai sistem manufakturing, kedinamikan dari sistem tersebut digambarkan sebagai evolusi perlakuan sistem selama diberikan waktu periodenya. sistem transportasi dinamik diatur dengan sinkronisasi, paralelisasi dan kejadian yang serentak/concurrency (nait moh, a., dkk., 2008). penggunaan pen dalam sistem even diskrit dinamik adalah karena aljabar max-plus dapat menangani dengan mudah proses sinkronisasi (braker, 1990). pendekatan dengan aljabar max dengan kemampuannya untuk diadaptasikan pada masalah yang graph (nait-sidi aljabar max-plus dapat dilihat penerapannya dalam sistem manufaktur dan pada masalah masalah yang berkaitan dengan jaringan transportasi yang sering berasal dari kejadian seperti sinkronisasi antara resources dan konflik yang terjadi ketika distribusi resources diperlukan (nait-sidisebelum, menyusun model aljabar max plus, berikut diberikan konsep dasar mengenai struktur aljabar rmax sebagai berikut: max menotasikan himpunan bilangan real −∞= } dengan dua operasi biner max dan penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max issn: 2086-0382 dari koridor 1 dapat transit ke koridor 4 melalui skywalk dukuh atas. demikian halnya , p39), (p39, x18 dimaksudkan untuk menjamin sinkronisasi koridor 6 dengan koridor 1, bahwa penumpang dari koridor 1 dapat transit ke koridor 6 melalui skywalk dukuh atas. sehingga, x gambar 3 jabar max-plus sistem transportasi dapat dikatakan sebagai sistem dinamik event diskrit (sded) sama halnya sebagai sistem manufakturing, dari sistem tersebut digambarkan sebagai evolusi perlakuan sistem selama diberikan waktu periodenya. sistem transportasi dinamik diatur dengan sinkronisasi, paralelisasi dan kejadian yang serentak/concurrency (nait moh, a., dkk., 2008). penggunaan pendekatan aljabar max dalam sistem even diskrit dinamik adalah karena plus dapat menangani dengan mudah proses sinkronisasi (braker, 1990). pendekatan dengan aljabar max dengan kemampuannya untuk diadaptasikan dapat dimodelkan dengan sidi-moh, a., dkk., 2008). plus dapat dilihat penerapannya dalam sistem manufaktur dan pada masalah masalah yang berkaitan dengan jaringan transportasi yang sering berasal dari kejadian nisasi antara resources dan konflik yang terjadi ketika distribusi resources -moh, a., dkk., 2008). sebelum, menyusun model aljabar max plus, berikut diberikan konsep dasar mengenai max oleh baccelli, dkk menotasikan himpunan bilangan real } dengan dua operasi biner max dan penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max 0382 dari koridor 1 dapat transit ke koridor 4 melalui skywalk dukuh atas. demikian halnya 18), dan (p39, x dimaksudkan untuk menjamin sinkronisasi koridor 6 dengan koridor 1, bahwa penumpang dari koridor 1 dapat transit ke koridor 6 melalui x18 (dari dukuh gambar 3. petrinet sistem transportasi dapat dikatakan sebagai sistem dinamik event diskrit (sded) sama halnya sebagai sistem manufakturing, dari sistem tersebut digambarkan sebagai evolusi perlakuan sistem selama diberikan waktu periodenya. sistem transportasi dinamik diatur dengan sinkronisasi, paralelisasi dan kejadian yang serentak/concurrency (nait dekatan aljabar max-plus dalam sistem even diskrit dinamik adalah karena plus dapat menangani dengan mudah proses sinkronisasi (braker, 1990). pendekatan dengan aljabar max-plus terkenal dengan kemampuannya untuk diadaptasikan dapat dimodelkan dengan moh, a., dkk., 2008). plus dapat dilihat penerapannya dalam sistem manufaktur dan pada masalah masalah yang berkaitan dengan jaringan transportasi yang sering berasal dari kejadian nisasi antara resources dan konflik yang terjadi ketika distribusi resources moh, a., dkk., 2008). sebelum, menyusun model aljabar max plus, berikut diberikan konsep dasar mengenai ccelli, dkk (1992) menotasikan himpunan bilangan real } dengan dua operasi biner max dan penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max dari koridor 1 dapat transit ke koridor 4 melalui skywalk dukuh atas. demikian halnya x29) dimaksudkan untuk menjamin sinkronisasi koridor 6 dengan koridor 1, bahwa penumpang dari koridor 1 dapat transit ke koridor 6 melalui (dari dukuh atas 2 ke halimun) dan ragunan) keduanya harus menunggu kedatangan bus menjamin adanya sinkronisasi antara koridor 1 dan 6 sebagaimana aturan sinkronisasi yang telah diberikan di atas. berikut ini petrinet untuk keberangkatan petrinet untuk jaringan busway sistem transportasi dapat dikatakan sebagai sistem dinamik event diskrit (sded) sama halnya sebagai sistem manufakturing, dari sistem tersebut digambarkan sebagai evolusi perlakuan sistem selama diberikan waktu periodenya. sistem transportasi dinamik diatur dengan sinkronisasi, paralelisasi dan kejadian yang serentak/concurrency (naitplus dalam sistem even diskrit dinamik adalah karena plus dapat menangani dengan mudah proses sinkronisasi (braker, 1990). plus terkenal dengan kemampuannya untuk diadaptasikan dapat dimodelkan dengan moh, a., dkk., 2008). plus dapat dilihat penerapannya dalam sistem manufaktur dan pada masalahmasalah yang berkaitan dengan jaringan transportasi yang sering berasal dari kejadian nisasi antara resources dan konflik yang terjadi ketika distribusi resources sebelum, menyusun model aljabar maxplus, berikut diberikan konsep dasar mengenai 1992) menotasikan himpunan bilangan real } dengan dua operasi biner max dan plus yang masing dan operasi elemen netral untuk operasi dan elemen netral untuk operasi operasi times disebut aljabar max sebagai plus adalah sebagai berikut: suatu barisan untuk 0(x barisan untuk semua dalam hal ini, bus yang ke dan berdasarkan model koridor sebelum sinkronisasi disusun sinkronisasi sebagai berikut: penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max atas 2 ke halimun) dan ragunan) keduanya harus menunggu kedatangan bus x01 dan x05. menjamin adanya sinkronisasi antara koridor 1 dan 6 sebagaimana aturan sinkronisasi yang telah diberikan di atas. berikut ini petrinet untuk keberangkatan jaringan busway plus yang masing dan ⊗ . untuk setiap operasi ⊕ dan maksba def =⊕ elemen netral untuk operasi dan elemen netral untuk operasi operasi ⊕ dibaca times. himpunan r disebut aljabar max sebagai maxℜ = { r adapun bentuk umum model aljabar max plus adalah sebagai berikut: suatu barisan ( (kx untuk 0≥k , di mana 0)0 x= adalah barisan ( )x k dapat ditulis (kx untuk semua k dalam hal ini, bus yang ke-k di suatu halte. berdasarkan data di tabel 1 dan berdasarkan model koridor sebelum sinkronisasi disusun model aljabar max sinkronisasi sebagai berikut: penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max atas 2 ke halimun) dan x29 (dari dukuh a ragunan) keduanya harus menunggu kedatangan . hal tersebut dimaksudkan untuk menjamin adanya sinkronisasi antara koridor 1 dan 6 sebagaimana aturan sinkronisasi yang telah diberikan di atas. berikut ini petrinet untuk keberangkatan bus pada jaringan busway. jaringan busway plus yang masing-masing dinotasikan dengan untuk setiap ba, dan ⊗ dengan ),(maks ba dan elemen netral untuk operasi dan elemen netral untuk operasi dibaca o-plus dan operasi . himpunan rmax dengan operasi disebut aljabar max-plus dan didefinisikan = { rmax, ⊕ , ⊗ adapun bentuk umum model aljabar max plus adalah sebagai berikut: ):)(( nkkx ∈ ()1 kxak ⊗=+ , di mana a h kondisi awal. ( ) dapat ditulis 0) xa k ⊗= ⊗ 0≥ . (heidergott, b., dkk, 2006) )(kx adalah waktu keberangkatan di suatu halte. berdasarkan data di tabel 1 dan berdasarkan model petrinet koridor sebelum sinkronisasi model aljabar max sinkronisasi sebagai berikut: penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max (dari dukuh a ragunan) keduanya harus menunggu kedatangan hal tersebut dimaksudkan untuk menjamin adanya sinkronisasi antara koridor 1 dan 6 sebagaimana aturan sinkronisasi yang telah diberikan di atas. berikut ini petrinet untuk bus pada jaringan busway. masing dinotasikan dengan ∈ rmax, didefinisikan dan aba def +=⊗ elemen netral untuk operasi ⊕ adalah dan elemen netral untuk operasi ⊗ adalah dan operasi ⊗ dengan operasi ⊕ plus dan didefinisikan ⊗ , ε , e }. adapun bentuk umum model aljabar max plus adalah sebagai berikut: dapat dibangun )k ∈a r xn×max , ∈x kondisi awal. secara ekivalen . (heidergott, b., dkk, 2006) adalah waktu keberangkatan di suatu halte. berdasarkan data di tabel 1 dan petrinet masing koridor sebelum sinkronisasi, maka dapat model aljabar max-plus sebelum sinkronisasi sebagai berikut: penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus 199 (dari dukuh atas 2 ke ragunan) keduanya harus menunggu kedatangan hal tersebut dimaksudkan untuk menjamin adanya sinkronisasi antara koridor 1 dan 6 sebagaimana aturan sinkronisasi yang telah diberikan di atas. berikut ini petrinet untuk bus pada jaringan busway. masing dinotasikan dengan ⊕ , didefinisikan b+ adalah ∞−= def ε adalah 0 def e = . dibaca o⊕ dan ⊗ plus dan didefinisikan adapun bentuk umum model aljabar maxbangun oleh (1) ∈ rn dan secara ekivalen (2) . (heidergott, b., dkk, 2006) adalah waktu keberangkatan dan tabel 2, masing-masing , maka dapat plus sebelum winarni 200 volume 1 no. 4 mei 2011 )8(38,24)1( )7(25)1( )1(24,6)1( )3(8)1( )9(67,24)1( )10(5,26)1( )7(86,15)1( )8(17)1( )9(5,19)1( )7(75,16)1( )16(21,20)1( )8(98,10)1( )6(83,12)1( )11(23,15)1( )17(71,19)1( )4(21,14)1( 1516 2015 1314 1213 1112 1411 910 89 78 107 56 45 34 23 12 61 −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx )15(81,52)1( )12(14,43)1( )1(07,5)1( )12(69,37)1( )10(32,34)1( )2(47,11)1( )1(63,8)1( )1(31,10)1( )3(3,15)1( )7(47,25)1( )5(39,17)1( )2(1,10)1( )(64,2)1( )1(01,5)1( )3(61,12)1( 3031 3130 2829 2728 2927 2526 2425 2324 2223 2122 2621 1920 1819 1718 1617 −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ −⊗=+ ⊗=+ −⊗=+ −⊗=+ kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx kxkx (3) dengan aturan sinkronisasi yang sudah dijelaskan pada bagian sebelumnya, maka model aljabar max-plus (3) menjadi sebagai berikut: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ))1(55)1(55)16(21,20)1( )8(17)1(24,6)9(67,24)8(98,10)1( )6(83,12)1( )8(17)1(24,6)9(67,24)11(23,15)1( )1(55)1(55)17(71,19)1( )4(21,14)1( 281756 8131145 34 8131123 281712 61 −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊕−⊗⊕−⊗⊕−⊗=+ −⊗=+ −⊗⊕−⊗⊕−⊗⊕−⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗=+ kxkxkxkx kxkxkxkxkx kxkx kxkxkxkxkx kxkxkxkx kxkx ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ))2(47,113)7(47,253)7(86,15)1( )8(98,10)1(24,6)9(67,24)11(23,15)8(17)1( )2(47,113)7(47,253)7(86,15)9(5,19)1( )7(25)7(75,16)1( 2521910 41311289 2521978 20107 −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊕−⊗⊕−⊗⊕−⊗⊕−⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗⊕−⊗=+ −⊗⊕−⊗=+ kxkxkxkx kxkxkxkxkxkx kxkxkxkxkx kxkxkx ( ) ( ) ( ) ( ) ( ) ( ) ( ))8(17)8(98,10)11(23,15)1(24,6)1( )3(8)1( )8(17)8(98,10)11(23,15)9(67,24)1( )10(5,26)1( 8421314 1213 8421112 1411 −⊗⊕−⊗⊕−⊗⊕−⊗=+ −⊗=+ −⊗⊕−⊗⊕−⊗⊕−⊗=+ −⊗=+ kxkxkxkxkx kxkx kxkxkxkxkx kxkx ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ))1(63,83)3(3,153)2(1,10)1( )12(69,37)(64,2)1( )16(71,195)17(71,195)1(01,5)1( )3(61,12)1( )1(63,83)3(3,153)8(38,24)1( )7(75,16)7(25)1( 24221920 271819 511718 1617 24221516 102015 −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊕⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊕−⊗=+ kxkxkxkx kxkxkx kxkxkxkx kxkx kxkxkxkx kxkxkx (4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ))7(86,153)9(5,193)2(47,11)1( )2(1,103)8(38,243)1(63,8)1( )12(14,43)1(31,10)1( )2(1,103)8(38,243)3(3,15)1( )7(86,153)9(5,193)7(47,25)1( )5(39,17)1( 972526 19152425 312324 19152223 972122 2621 −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊕−⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗⊗⊕−⊗⊗⊕−⊗=+ −⊗=+ kxkxkxkx kxkxkxkx kxkxkx kxkxkxkx kxkxkxkx kxkx ( ) ( ) ( ) ( ))1(07,5)16(21,205)17(71,195)1(07,5)1( )12(69,37)1( )10(32,34)1( 17512829 2728 2927 −⊗⊕−⊗⊗⊗−⊗⊗⊕−⊗=+ −⊗=+ −⊗=+ kxkxkxkxkx kxkx kxkx ( )( ) )16(81,52)1( )1(31,10)12(14,43)1( 3031 233130 −⊗=+ −⊗⊕−⊗=+ kxkx kxkxkx selanjutnya, model (4) dapat dinyatakan dalam bentuk umum model aljabar max-plus pada persamaan (1), yaitu sebagai berikut: ( ))1()1( 1 pkxakx p m p −+⊗=+ ⊕ = (5) dengan ap adalah matriks berukuran n x n dan n adalah jumlah variabel. matriks ap adalah matriks yang berkaitan dengan )1( pkx −+ . dan m dalam hal ini adalah jumlah bus maksimum di antara semua lintasan dalam grap pada gambar 2. berdasarkan tabel 2 di atas, maka n adalah 31 penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 201 dan m adalah 18 yaitu pada koridor 1 lintasan blok m menuju dukuh atas 1. sehingga, nantinya ada 18 buah matriks yaitu a1 sampai dengan a18. dalam hal ini, model pada (4) dapat dinyatakan dalam bentuk umum model max-plus pada persamaan (1) menjadi )()1( kxakx ⊗=+ (6) dengan )(kx vektor berdimensi nm), yang didefinisikan sebagai 1 1 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) t n n n x k x k x k x k x k x k m x k m      = − − + − + −       ⋯ ⋯ ⋯ ⋯ ⋯ (7) di mana notasi t pada persamaan di atas menunjukkan transpose dan a matriks berukuran (n.m) x (nm) yaitu                 = − εεε εεε εεε max max max max 121 i i i i aaaa a mm ⋯ ⋮⋮⋮⋮ ⋯ ⋯ ⋯ (8) dimana maxi adalah matriks n x n dengan elemen diagonalnya adalah e dan elemen lainnya adalah ε , dan e dalam matriks a di atas adalah matriks n x n dengan semua elemennya adalah ε . dan )(kx seperti pada persamaan (7). dalam permasalahan ini, vektor )(kx berdimensi 558 dan matriks a berukuran 558 x 558) dengan scilab dan max-plus toolbox diperoleh bahwa nilai karakteristik matriks a tersebut adalah 3.9542857 (λ = 3.9542857) dan eigen vektor dari matriks a yaitu vx. nilai karakteristik ini menunjukkan performa dari sistem penjadwalan keberangkatan bus, maksudnya bahwa setiap 3.9542857 menit sekali terjadi pemberangkatan bus di tiap-tiap halte atau dengan kata lain periode keberangkatan bus di tiap-tiap halte adalah 3.9542857 menit. jika jadwal keberangkatan bus adalah periodik dan diberikan waktu keberangkatan awal )0(x dengan jumlah bus yang beroperasi sama seperti pada tabel 2, maka dapat disusun jadwal keberangkatan bus untuk keberangkatankeberangkatan selanjutnya, dengan evolusi )0()1( )1( xkx k ⊗=+ +⊗λ (8) nilai dalam vektor vx adalah mewakili menit, karena keterbatasan halaman vector vx tidak dituliskan dalam makalah ini, hal ini dapat dilihat selengkapnya pada (winarni, 2009). misalkan jaringan beroperasi pada pagi hari dimulai pukul 05.00, berarti yang berangkat pertama kali pada jaringan tersebut adalah x3 yaitu pukul 05:00, keberangkatan x1(0) adalah pukul 05:30,001429, x2(0) pukul 05:34,932857, dan seterusnya. periode antar keberangkatan di masing-masing halte adalah sebesar eigenvalue yaitu 3,9542857. diperoleh pula bahwa critical circuit pada jaringan tersebut adalah halte dukuh atas 2 – halimun – mantraman 2 – mantraman 1 – kp melayu – mantraman 1 – mantraman 2 – halimun – dukuh atas 2 karena bobot rata-rata circuit tersebut adalah sama dengan 3,9542857, yaitu maksimum dari bobot rata-rata semua circuit dalam jaringan (eigenvalue). total waktu tempuh circuit tersebut adalah 55.36 menit dengan jumlah bus yang beroperasi pada circuit tersebut adalah 14 bus. semua pembahasan di atas adalah dengan mempertimbangkan halte-halte ujung koridor dan halte transit, belum memperhitungkan haltehalte ’kecil’ yaitu halte-halte di sepanjang koridor selain halte ujung koridor dan halte transit. untuk keberangkatan di halte-halte tersebut cukup ditambahkan waktu tempuh antara halte sebelumnya menuju halte tersebut. sehingga desain jadwal keberangkatan bus untuk seluruh halte pada jaringan busway dengan 7 koridor dapat dilihat pada tabel 3 di bagian lampiran. penutup kesimpulan 1. dalam penelitian ini telah disusun model petrinet dan model aljabar max-plus untuk mendesain jadwal keberangkatan jaringan transjakarta busway. petrinet yang dimaksudkan dapat dilihat pada gambar 3 dan model aljabar max-plus dapat dilihat model (4). dari model petrinet dapat diterjemahkan menjadi model aljabar maxplus dan juga sebaliknya. 2. model (4) dapat dinyatakan dalam bentuk umum model aljabar max-plus yaitu )()1( kxakx ⊗=+ di mana matriks a berukuran 558 x 558. dengan scilab dan maxplus algebra toolbox (subiono, 2008) diperoleh eigenvalue dari mariks a sama dengan )( aλ = 3,9542857. eigenvalue ini menunjukkan bahwa setiap 3,9542857 menit sekali terjadi pemberangkatan bus di tiap-tiap halte atau dengan kata lain periode keberangkatan bus di tiap-tiap halte adalah 3,9542857 menit. dengan mengambil eigenvektor matriks a sebagai x(0) maka akan dapat ditentukan waktu keberangkatan bus ditiap-tiap halte ujung koridor dan halte transit yang ke-(k+1), untuk k = 0, 1, 2, 3, .... dengan )0()1( )1( xkx k ⊗=+ +⊗λ winarni 202 volume 1 no. 4 mei 2011 3. desain jadwal keseluruhan halte pada jaringan tersebut dapat dilihat pada tabel 3 . 4. diperoleh bahwa critical circuit pada jaringan tersebut adalah halte dukuh atas 2 – halimun – mantraman 2 – mantraman 1 – kp melayu – mantraman 1 – mantraman 2 – halimun – dukuh atas 2 karena bobot rata-rata circuit tersebut adalah sama dengan 3,9542857, yaitu maksimum dari bobot rata-rata semua circuit dalam jaringan (eigenvalue). total waktu tempuh circuit tersebut adalah 55.36 menit dengan jumlah bus yang beroperasi pada circuit tersebut adalah 14 bus. saran 1. observasi lapangan yang lebih lama dan pengumpulan data waktu tempuh yang lebih lengkap. 2. melakukan reduksi matriks untuk efisiensi secara komputasi. 3. dilakukan analisa lebih detail lagi pada petrinet yang telah disusun tersebut. 4. melakukan analisa jadwal jika terjadi keterlambatan. 5. melakukan re-alokasi jumlah bus yang beroperasi pada jaringan, sehingga lebih optimal, misalnya dengan mengurangi alokasi bus pada circuit yang minimum dan menambahkannya pada critical circuit dan dengan mengoptimalkan alokasi bus yang disediakan pengelola. 6. jika 8 koridor berikutnya (koridor 9 sampai dengan koridor 15) sudah terealisasi, penelitian ini dapat dikembangkan untuk seluruh koridor transjakarta busway. daftar pustaka [1] baccelli, f., cohen, g., olsder, g. j., dan quadrat, j. p., (1992), synchronisation and linearity, algebra for discrete event systems. john wiley and sons, inc., new york. [2] batavia busway, (16 juni 2008, tanggal akses: 7 agustus 2008), diskusi busway di veteran, http://www.bataviabusway.blogspot.com. [3] batavia busway, (tanggal akses: 24 juli 2008), denah-busway-v080116.jpg, http://www.bataviabusway.blogspot.com. [4] blu transjakarta busway, (2007), company profile transjakarta busway, edisi3. [5] braker, j.-g., (1991), “max-algebra modelling and analysis of time-table dependent transportation networks”. proceedings of the first european control conference (ecc’91), grenoble, france, hal. 1831–1836. [6] cassandras, c.g., (1993), discrete event systems: modelling and performance analysis, richard d. irwin, inc, and aksen associates inc., amherst. [7] dieky, a., (2008), petrinet toolbox, jurusan matematika fmipa institut teknologi sepuluh nopember, surabaya. [8] heidergott, b., olsder g. j., dan woude, j. v. d., (2006), max plus at work modeling and analysis of synchronisation systems: a course on max-plus algebra and its application, princeton university press, new jersey. [9] nait-sidi-moh, a., manier, m. a., el moudni, a., (2008), “spectral analysis for performance evaluation in a bus network”, european journal of operational research, http://www.sciencedirect.com. [10] suaratransjakarta, (20 juni 2008, tanggal akses: 1 agustus 2008), ketidakpastian menunggu bus, http://www.transjakartainfo.com. [11] subiono, (2000), on classes of min-max-plus systems and their application, thesis ph.d., technische universiteit delft, delft. [12] subiono, dieky, a., (2008), max-plus algebra toolbox, jurusan matematika fmipa institut teknologi sepuluh nopember, surabaya. [13] winarni, (2009), penjadwalan jalur bus dalam kota dengan aljabar max-plus, jurusan matematika fmipa institut teknologi sepuluh nopember, surabaya. lampiran tabel 3. desain jadwal periodik keberangkatan busway no. variabel halte keberangkatan halte tujuan kendala 1 x1 blok-m masjid agung 14.21 + x6 atau 4.73 + x6,7 2 x1,1 masjid agung bundaran senayan 8.75 + x1 3 x1,2 bundaran senayan gelora bung karno 3.10 + x1,1 4 x1,3 gelora bung karno polda metro 1.02 + x1,2 5 x1,4 polda metro bendungan hilir 1.17 + x1,3 6 x1,5 bendungan hilir karet 1.74 + x1,4 7 x1,6 karet setia budi 1.37 + x1,5 penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 203 8 x1,7 setia budi dukuh atas 1 1.07 + x1,6 9 x2 dukuh atas 1 tosari 19,71+ x1, 10,07 + x17, 10,07 + x28 atau 1.5 + x1,7, 10,07 + x17, 10,07 + x28 10 x2,1 tosari bunderan hi 1.47 + x2 11 x2,2 bunderan hi sarinah 1.67 + x2,1 12 x2,3 sarinah bank indonesia 1.41 + x2,2 13 x2,4 bank indonesia monumen nasional 3.07 + x2,3 14 x2,5 monumen nasional harmoni 1.59 + x2,4 15 x3 harmoni sawah besar 15.23 + x2, 17 + x8, 24.67 + x11, 6.24 + x13 atau 6.02 + x2,5, 8 + x8,7, 1.5 + x11,11, 3.58 + x13,2 16 x3,1 sawah besar mangga besar 4.76 + x3 17 x3,2 mangga besar olimo 2.57 + x3,1 18 x3,3 olimo glodok 1.05 + x3,2 19 x3,4 glodok kota 1.35 + x3,3 20 x4 kota glodok 12.83 + x3 atau 3.11 + x3,4 21 x4,1 glodok olimo 3.30 + x4 22 x4,2 olimo mangga besar 1.34 + x4,1 23 x4,3 mangga besar sawah besar 1.26 + x4,2 24 x4,4 sawah besar harmoni 2.53 + x4,3 25 x5 harmoni monumen nasional 10.98 + x4, 17 + x8, 24.67 + x11, 6.24 + x13 atau 2.54 + x4,4, 8 + x8,7, 1.5 + x11,11, 3.58 + x13,2 26 x5,1 monumen nasional bank indonesia 5.77 + x5 27 x5,2 bank indonesia sarinah 4.82 + x5,1 28 x5,3 sarinah bunderan hi 2.33 + x5,2 29 x5,4 bunderan hi tosari 1.29 + x5,3 30 x5,5 tosari dukuh atas 1 3.08 + x5,4 31 x6 dukuh atas 1 setia budi 20.21 + x5, 10.07 + x17, 10.07 + x28 atau 2.92 + x5,5, 10.07 + x17, 10.07 + x28 32 x6,1 setia budi karet 0.94 + x6 33 x6,2 karet bendungan hilir 1.23 + x6,1 34 x6,3 bendungan hilir polda metro jaya 1.04 + x6,2 35 x6,4 polda metro jaya gelora bung karno 1.46 + x6,3 36 x6,5 gelora bung karno bundaran senayan 1.08 + x6,4 37 x6,6 bundaran senayan masjid agung 1.39 + x6,5 38 x6,7 masjid agung blok m 2.33 + x6,6 39 x7 pulo gadung bermis 16.75 + x10, 25 + x20 atau 3.15 + x10,10, 11 + x20,9 40 x7,1 bermis pulo mas 3 + x7 41 x7,2 pulo mas asmi 2.5 + x7,1 42 x7,3 asmi pedongkelan 1 + x7,2 43 x7,4 pedongkelan cempaka timur 3 + x7,3 44 x7,5 cempaka timur rs islam 3 + x7,4 45 x7,6 rs islam cempaka tengah 1 + x7,5 46 x7,7 cempaka tengah ps. cempaka putih 1 + x7,6 47 x7,8 ps. cempaka putih rawa selatan 0.75 + x7,7 48 x7,9 rawa selatan galur 0.75 + x7,8 49 x7,10 galur senen 1 + x7,9 50 x8 senen atrium 19.5 + x7, 15.86 + x9, 28.47 + x21, 14.47 + x25 atau 3,5 + x7,10, 3.53 + x9,3,12.82 + x21,4, 9.58 + x25,3 51 x8,1 atrium rspad 1 + x8 52 x8,2 rspad deplu 1 + x8,1 53 x8,3 deplu gambir 1 1 + x8,2 54 x8,4 gambir 1 istiqlal 2 + x8,3 55 x8,5 istiqlal juanda 2 + x8,4 56 x8,6 juanda pecenongan 1 + x8,5 57 x8,7 pecenongan harmoni 1 + x8,6 58 x9 harmoni balai kota 17 + x8, 15.23 + x2, 10.98 + x4, 24.67 + x11 , 6.24 + x13 atau 8 + x8,7, 6.02 + x2,5, 2.54 + x4,4, 1.5 + x11,11, 3.58 + x13,2 59 x9,1 balai kota gambir 2 7.57 + x9 60 x9,2 gambir 2 kwitang 2.10 + x9,1 61 x9,3 kwitang senen 2.66 + x9,2 62 x10 senen galur 15.86 + x9, 28.47 + x21, 14.47 + x25 atau 3.53 + x9,3, 12.82 + x21,4, 9.58 + x25,3 63 x10,1 galur rawa selatan 2.11 + x10 64 x10,2 rawa selatan ps. cempaka putih 1 + x10,1 65 x10,3 ps. cempaka putih cempaka tengah 1.17 + x10,2 66 x10,4 cempaka tengah rs islam 0.95 + x10,3 67 x10,5 rs islam cempaka timur 1.01 + x10,4 68 x10,6 cempaka timur pedongkelan 1.15 + x10,5 69 x10,7 pedongkelan asmi 1.81 + x10,6 winarni 204 volume 1 no. 4 mei 2011 70 x10,8 asmi pulo mas 1.27 + x10,7 71 x10,9 pulo mas bermis 1.03 + x10,8 72 x10,10 bermis pulo gadung 2.10 + x10,9 73 x11 kalideres pesakih 26.5 + x14 atau 4.05 + x14,11 74 x11,1 pesakih sumur bor 2.64 + x11 75 x11,2 sumur bor rawa buaya 0.84 + x11,1 76 x11,3 rawa buaya jembatan baru 1.64 + x11,2 77 x11,4 jembatan baru dispenda 1.36 + x11,3 78 x11,5 dispenda jembatan gantung 1.71 + x11,4 79 x11,6 jembatan gantung taman kota 1.6 + x11,5 80 x11,7 taman kota indosiar 1.71 + x11,6 81 x11,8 indosiar jelambar 3.3 + x11,7 82 x11,9 jelambar grogol trisakti 4.32 + x11,8 83 x11,10 grogol trisakti rs.sumber waras 2.55 + x11,9 84 x11,11 rs.sumber waras harmoni 1.50 + x11,10 85 x12 harmoni pecenongan 24.67 + x11, 15.23 + x2, 10.98 + x4, 17 + x8 atau 1.5 + x11,11, 6.02 + x2,5, 2.54 + x4,4, 8 + x8,7 86 x12,1 pecenongan juanda 3.68 + x12 87 x12,2 juanda pasar baru 2.93 + x12,1 88 x13 pasar baru juanda 8 + x12 atau 1.38 + x12,2 89 x13,1 juanda pecenongan 1.68 + x13 90 x13,2 pecenongan harmoni 0.98 + x13,1 91 x14 harmoni rs.sumber waras 6.24 + x13, 15.23 + x2, 10.98 + x4, 17 + x8 atau 3.58 + x13,2, 6.02 + x2,5, 2.54 + x4,4, 8 + x8,7 92 x14,1 rs.sumber waras grogol trisakti 4 + x14 93 x14,2 grogol trisakti jelambar 1.5 + x14,1 94 x14,3 jelambar indosiar 2 + x14,2 95 x14,4 indosiar taman kota 2 + x14,3 96 x14,5 taman kota jembatan gantung 3.47 + x14,4 97 x14,6 jembatan gantung dispenda 1.03 + x14,5 98 x14,7 dispenda jembatan baru 2.5 + x14,6 99 x14,8 jembatan baru rawa buaya 1.5 + x14,7 100 x14,9 rawa buaya sumur bor 2 + x14,8 101 x14,10 sumur bor pesakih 1 + x14,9 102 x14,11 pesakih kalideres 1.45 + x14,10 103 x15 pulo gadung pasar pulo gadung 25 + x20, 16.75 + x10 atau 11 + x20,9, 3.15 + x10,10 104 x15,1 pasar pulo gadung tu gas 8.17 + x15 105 x15,2 tu gas layur 3.18 + x15,1 106 x15,3 layur velodrome 1.32 + x15,2 107 x15,4 velodrome sunan giri 2.92 + x15,3 108 x15,5 sunan giri unj 1.43 + x15,4 109 x15,6 unj pramuka lia 1 + x15,5 110 x15,7 pramuka lia utan kayu 3.24 + x15,6 111 x15,8 utan kayu pasar genjing 0.99 + x15,7 112 x15,9 pasar genjing mantraman 2 0.88 + x15,8 113 x16 mantraman 2 manggarai 24.38 + x15, 18.3 + x22, 11.63 + x24 atau 1.25 + x15, 8.28 + x22,3, 5.18 + x24,3 114 x16,1 manggarai pasar rumput 6.69 + x16 115 x16,2 pasar rumput halimun 4.12+ x16,1 116 x17 halimun dukuh atas 2 12.61 + x16 atau 1.8 + x16,2 117 x18 dukuh atas 2 halimun 5.07 + x17, 24.71 + x1, 25.21 + x5 atau 5.07 + x17, 6.5 + x1,7, 7.92 + x5,5 118 x19 halimun pasar rumput 2.64 + x18, 37.69 + x27 atau 2.64 + x18, 4.83 + x27,17 119 x19,1 pasar rumput manggarai 2.43 + x19 120 x19,2 manggarai mantraman 2 1.73 + x19,1 121 x20 mantraman 2 pasar genjing 10.1 + x19, 18.3 + x22, 11.63 + x24 atau 5.93 + x19,2, 8.28 + x22,3, 5.18 + x24,3 122 x20,1 pasar genjing utan kayu 2 + x20 123 x20,2 utan kayu pramuka lia 1 + x20,1 124 x20,3 pramuka lia unj 1 + x20,2 125 x20,4 unj sunan giri 3 + x20,3 126 x20,5 sunan giri velodrome 1 + x20,4 127 x20,6 velodrome layur 1 + x20,5 128 x20,7 layur tu gas 3 + x20,6 129 x20,8 tu gas pasar pulo gadung 1 + x20,7 130 x20,9 pasar pulo gadung pulo gadung 1 + x20,8 131 x21 ancol pademangan 17.39 + x26 atau 3 + x26,4 132 x21,1 pademangan jembatan merah 4 + x21 133 x21,2 jembatan merah pasar baru timur 4.96 + x21,1 134 x21,3 pasar baru timur budi utomo 5.05 + x21,2 penjadwalan jalur bus dalam kota dengan model petrinet dan aljabar max-plus jurnal cauchy – issn: 2086-0382 205 135 x21,4 budi utomo senen sentral 1.65 + x21,3 136 x22 senen sentral pal putih 25.47 + x21, 22.5 + x7, 18.86 + x9 atau 9.82 + x21,4, 6.5 + x7,10, 6.53 + x9,3 137 x22,1 pal putih kramat sentiong 5.42 + x22 138 x22,2 kramat sentiong salemba ui 1.58 + x22,1 139 x22,3 salemba ui mantraman 1 3.02 + x22,2 140 x23 mantraman 1 tegalan 15.3 + x22, 27.38 + x15, 13.1 + x19 atau 5.28 + x22,3, 4.25 + x15,9, 8.93 + x19,2 141 x23,1 tegalan slamet riyadi 1.3 + x23 142 x23,2 slamet riyadi kebon pala 1.08 + x23,1 143 x23,3 kebon pala pasar jatinegara 1.66 + x23,2 144 x23,4 pasar jatinegara kampung melayu 2.48 + x23,3 145 x24 kampung melayu kebon pala 10.31 + x23, 43.14 + x31 atau 3.8 + x23,4, 4,29+ x31,12 146 x24,1 kebon pala slamet riyadi 3.78 + x24 147 x24,2 slamet riyadi tegalan 1.61 + x24,1 148 x24,3 tegalan mantraman 1 1.05 + x24,2 149 x25 mantraman 1 salemba ui 8.63 + x24, 27.38 + x15, 13.1 + x19 atau 2.18 + x24,3, 4.25 + x15,9, 8.93 + x19,2 150 x25,1 salemba ui kramat sentiong 2+ x25 151 x25,2 kramat sentiong pal putih 1.58 + x25,1 152 x25,3 pal putih senen sentral 1.30 + x25,2 153 x26 senen sentral budi utomo 11.47 + x25, 22.5 + x7, 18.86 + x9 atau 6.58+ x25,3, 6.5 + x7,10, 6.53 + x9,3 154 x26,1 budi utomo pasar baru timur 3.28 + x26 155 x26,2 pasar baru timur jembatan merah 1.83 + x26,1 156 x26,3 jembatan merah pademangan 3.6 + x26,2 157 x26,4 pademangan ancol 5.69 + x26,3 158 x27 ragunan dep. pertanian 34.32 + x29 atau 3.13 + x29,16 159 x27,1 dep. pertanian smk 57 2.86 + x27 160 x27,2 smk 57 jati padang 2.32 + x27,1 161 x27,3 jati padang pejaten 1.48 + x27,2 162 x27,4 pejaten buncit indah 2.27 + x27,3 163 x27,5 buncit indah warung jati 1.16 + x27,4 164 x27,6 warung jati imigrasi 2.34 + x27,5 165 x27,7 imigrasi duren tiga 2.38 + x27,6 166 x27,8 duren tiga mampang prapatan 1.72 + x27,7 167 x27,9 mampang prapatan kuningan timur 2.91 + x27,8 168 x27,10 kuningan timur patra kuningan 3.12 + x27,9 169 x27,11 patra kuningan depkes 1.23 + x27,10 170 x27,12 depkes gor sumantri 1.37 + x27,11 171 x27,13 gor sumantri karet kuningan 1.87 + x27,12 172 x27,14 karet kuningan kuningan madya a 1.29 + x27,13 173 x27,15 kuningan madya ai setia budi utara 0.96 + x27,14 174 x27,16 setia budi utara latuharhari 1.09 + x27,15 175 x27,17 latuharhari halimun 2.48 + x27,16 176 x28 halimun dukuh atas 2 37.69 + x27 atau 4.83 + x27,16 177 x29 dukuh atas 2 setia budi utara 5.07 + x28, 24.71 + x1, 25.21 + x5, 5.07 + x17 atau 5.07 + x28, 6.5 + x1,7, 7.92 + x5,5, 5.07 + x17 178 x29,1 setia budi utara kuningan madya a 3.59 + x29 179 x29,2 kuningan madya a karet kuningan 1.27 + x29,1 180 x29,3 karet kuningan gor sumantri 1.08 + x29,2 181 x29,4 gor sumantri depkes 1.27 + x29,3 182 x29,5 depkes patra kuningan 2.07 + x29,4 183 x29,6 patra kuningan kuningan timur 1.36 + x29,5 184 x29,7 kuningan timur mampang prapatan 1.34 + x29,6 185 x29,8 mampang prapatan duren tiga 4.03 + x29,7 186 x29,9 duren tiga imigrasi 2.29 + x29,8 187 x29,10 imigrasi warung jati 1.93 + x29,9 188 x29,11 warung jati buncit indah 1.33 + x29,10 189 x29,12 buncit indah pejaten 2.89 + x29,11 190 x29,13 pejaten jati padang 1.02 + x29,12 191 x29,14, jati padang smk 57 2 + x29,13 192 x29,15 smk 57 dep pertanian 2.11 + x29,14 193 x29,16 dep pertanian ragunan 1.62 + x29,15 194 x30 kampung melayu bidara cina 43.14 + x31, 10.31 + x23 atau 4.29 + x31,12, 3.8 + x23,4 195 x30,1 bidara cina gelanggang rmj 2.65 + x30 196 x30,2 gelanggang rmj cawang otista 3.07 + x30,1 197 x30,3 cawang otista bnn 2.87 + x30,2 198 x30,4 bnn cawang uki 1.4 + x30,3 199 x30,5 cawang uki bkn 3.11 + x30,4 winarni 206 volume 1 no. 4 mei 2011 200 x30,6 bkn pgc (cililitan) 2.46 + x30,5 201 x30,7 pgc (cililitan) pasar kramat jati 2.4 + x30,6 202 x30,8 pasar kramat jati pasar induk 6.43 + x30,7 203 x30,9 pasar induk rs. har. bunda 11.03 + x30,8 204 x30,10 rs. har. bunda fly over ry bogor 2.7 + x30,9 205 x30,11 fly over ry bogor kamp.rambutan 2.65 + x30,10 206 x31 kamp. rambutan tanah merdeka 52.81 + x30 atau 12.06 + x30,11 207 x31,1 tanah merdeka fly over ry bogor 3.5 + x31 208 x31,2 fly over ry bogor rs. har.bunda 2.25 + x31,1 209 x31,3 rs. har. bunda pasar induk 3.08 + x31,2 210 x31,4 pasar induk pasar kramat jati 2.71 + x31,3 211 x31,5 pasar kramat jati pgc (cililitan) 12.02 + x31,4 212 x31,6 pgc (cililitan) bkn 3.44 + x31,5 213 x31,7 bkn cawang uki 1.92 + x31,6 214 x31,8 cawang uki bnn 2.21 + x31,7 215 x31,9 bnn cawang otista 2.05 + x31,8 216 x31,10 cawang otista gelanggang rmj 2.02 + x31,9 217 x31,11 gelanggang rmj bidara cina 2.48 + x31,10 218 x31,12 bidara cina kampung melayu 1.18 + x31,11 solusi eksak gelombang soliton: persamaan schrodinger nonlinier nonlokal cauchy – issn: 2086-0382 1 solusi eksak gelombang soliton: persamaan schrodinger nonlinear nonlokal (nnls) riski nur istiqomah dinnullah jurusan pendidikan matematika universitas kanjuruhan malang email: ky2_zahra@yahoo.com abstrak pada penelitian ini dimodelkan perambatan gelombang optik pada medium nonlinear kerr nonlokal dengan menggunakan persamaan schrodinger nonlinear nonlokal (nnls). persamaan nnls tersebut diperoleh dari metode penurunan persamaan helmholtz nonlinear dengan penambahan suatu parameter nonlokal. dengan cara ini ditentukan solusi eksak dari persamaan nnls yang berupa gelombang soliton. solusi eksak yang diperoleh merupakan fungsi implisit. selanjutnya solusi eksak tersebut diplot dengan menggunakan aplikasi matlab untuk melihat perambatan gelombang soliton. hasil plot gambar menunjukkan bahwa lebar soliton dipengaruhi oleh nilai maksimum pada pusat soliton dan parameter nonlokal. jika nilai maksimum pada pusat atau nilai parameter nonlokalnya diperbesar, maka soliton yang terjadi akan semakin lebar. kata kunci: soliton, medium nonlinear kerr nonlokal, persamaan schrodinger nonlinear nonlokal, persamaan helmholtz nonlinear. pendahuluan di era global saat ini, teknologi serat optik telah banyak dipelajari dalam berbagai bidang ilmu pengetahuan terutama dalam bidang komunikasi optik. dengan menggunakan laser berintensitas tinggi, serat optik telah menjadi lahan penelitian yang menarik dan memiliki potensi besar untuk dimanfaatkan sebagai media transmisi. pada tahun 1988, linn f. mollenauer mempelopori penggunaan soliton untuk sistem komunikasi optik. soliton adalah sebuah gelombang soliter atau gelombang tunggal yang memiliki sifat terlokalisasi dan merambat tanpa perubahan bentuk maupun kecepatan. pada sistem komunikasi optik, informasi (data) dapat ditransfer dalam kapasitas yang besar pada jarak yang sangat jauh dan memiliki kecepatan transmisi yang sangat tinggi. sistem komunikasi optik yang berbasiskan soliton ini menjadi salah satu teknologi penting dalam pengembangan serat optik yang mampu memberikan peluang besar untuk merevolusi dunia telekomunikasi dan teknologi komputer. beberapa desain soliton sangat potensial untuk diaplikasikan sebagai alloptical switching, all-optical logic gates, all-optical processing, dan lain sebagainya [1]. studi perambatan soliton banyak sekali dilakukan pada medium nonlinear kerr. pada medium ini, efek nonlinear kerr dari medium dapat muncul ketika cahaya yang berintensitas tinggi mengenai medium sehingga menyebabkan adanya perubahan indeks bias. pengaruh nonlinearitas ini bersifat lokal. namun pada eksperimen yang meneliti kristal fotorefraktif sebagai medium nonlinear telah dibuktikan bahwa respon nonlinear dari suatu medium dapat bersifat nonlokal [2]. pada medium nonlinear nonlokal, gelombang yang terlokalisasi pada daerah yang sempit dapat memberikan respon yang sangat lebar. hal ini terjadi karena respon nonlokal dari suatu medium tidak hanya dipengaruhi oleh perubahan intensitas gelombang pada suatu daerah tertentu, tetapi juga dipengaruhi oleh intensitas gelombang di sekitar daerah tersebut [3]. perambatan gelombang optik pada medium nonlinear kerr nonlokal dimodelkan oleh persamaan schrodinger nonlinear nonlokal (nnls). persamaan nnls merupakan persamaan diferensial parsial nonlinear yang solusi eksaknya secara umum sulit untuk diperoleh sehingga diperlukan metode pendekatan, misalnya pendekatan numerik. namun, pada beberapa fenomena gelombang nonlinear solusi eksaknya dapat dicari secara analitik dalam bentuk fungsi implisit. pada penelitian ini ditentukan solusi eksak gelombang soliton pada medium nonlinear kerr nonlokal dari persamaan nnls. solusi eksak soliton dapat digunakan untuk mengetahui sifatsifat perambatan soliton pada medium nonlinear kerr nonlokal [4]. perubahan respon nonlokal pada suatu medium dapat dipengaruhi oleh sifatsifat perambatan soliton. dengan melihat sifatsifat perambatan soliton pada medium nonlinear mailto:ky2_zahra@yahoo.com riski nur istiqomah dinnullah 40 volume 3 no. 4 mei 2015 nonlokal, memungkinkan soliton dapat diaplikasikan untuk membuat desain peralatan komunikasi optik seperti self-optical switching ataupun self-optical routing [5]. oleh karena itu, penelitian ini penting dilakukan untuk memahami respon nonlinear nonlokal pada gelombang soliton. tinjauan pustaka 1. persamaan maxwell teori propagasi cahaya pada medium dielektrik yang melibatkan medan elektrik dan medan magnetik dikembangkan pertama kali oleh maxwell pada tahun 1860. dalam notasi vektor, persamaan maxwell dapat ditulis sebagai ∇⃗⃗ × �⃗� = − 𝜕�⃗� 𝜕𝑡 ∇⃗⃗ × �⃗⃗� = 𝐽 + 𝜕�⃗⃗� 𝜕𝑡 (1) ∇⃗⃗ . �⃗⃗� = 𝜌 ∇⃗⃗ . �⃗� = 0 dengan �⃗� , 𝐻⃗⃗ ⃗ , �⃗⃗� , �⃗� , 𝐽 dan 𝜌 berturut-turut adalah medan elektrik, medan magnetik, rapat fluks elektrik, rapat fluks magnetik, rapat arus, dan rapat muatan elektrik. pembahasan dibatasi untuk material nonmagnetik, tanpa muatan dan arus elektrik, yaitu: j = 0 dan 𝜌 = 0 d⃗⃗ = 𝜀0�⃗� + �⃗� (2) b⃗⃗ = 𝜇0�⃗⃗� dengan 𝜀0, 𝜇0 dan �⃗� berturut-turut adalah permitivitas ruang hampa, permeabilitas ruang hampa, dan polarisasi elektrik. persamaan maxwell dapat digunakan untuk memperoleh persamaan schrodinger yang menggambarkan perambatan cahaya dalam medium nonmagnetik [6]. 2. persamaan helmholtz nonlinear dalam pembahasan ini, diasumsikan bahwa medium perambatan gelombang adalah material nonlinear kerr, sehingga rapat fluks elektrik akan berbentuk d⃗⃗ = 𝜀0 (1 + 𝜒 (1) + 𝜒(3)|�⃗� | 2 ) �⃗� (3) dengan 𝜒 adalah konstanta. dengan menggunakan persamaan (2) dan (3), persamaan maxwell (1) dapat direduksi menjadi ∇⃗⃗ (∇⃗⃗ . e⃗⃗ ) − ∇⃗⃗ 2e⃗⃗ + 𝜇0𝜀0 𝜕2 𝜕𝑡2 (1 + 𝜒(1) + 𝜒(3)|�⃗� | 2 ) �⃗� =0. (4) menurut suryanto (2004), persamaan (4) merupakan persamaan gelombang elektromagnetik dimana persamaan tersebut melibatkan vektor kompleks dimensi tiga. permasalahan tersebut dapat disederhanakan dengan mengasumsikan bahwa medium berbentuk lempengan (slab waveguide), sedemikian sehingga ketergantungan medan elektrik dan magnetik terhadap salah satu dimensi dapat diabaikan. persamaan gelombang elektromagnetik tersebut dapat disederhanakan dengan mempertimbangkan dua tipe solusi yaitu gelombang transverse electric mode (te) dan gelombang transverse magnetic mode (tm). pada kasus ini pembahasan hanya dibatasi pada gelombang te yaitu vektor medan listrik tegak lurus dengan arah perambatan. arah perambatan dipilih searah dengan sumbu z, sehingga vektor medan listrik dan magnetik berturut-turut adalah �⃗� = [0, 𝐸𝑦(𝑥, 𝑧, 𝑡), 0] (5) �⃗⃗� = [𝐻𝑥(𝑥, 𝑧, 𝑡), 0, 𝐻𝑧(𝑥, 𝑧, 𝑡)]. (6) oleh karena itu, ∇⃗⃗ . �⃗� = 𝜕 𝜕𝑦 𝐸𝑦(𝑥, 𝑧, 𝑡) = 0 (7) dengan demikian persamaan (4) dapat ditulis sebagai persamaan skalar yaitu − ( 𝜕2 𝜕𝑥2 𝐸𝑦 + 𝜕2 𝜕𝑧2 𝐸𝑦) + 1 𝑐2 𝜕2 𝜕𝑡2 (1 + 𝜒(1) + 𝜒(3)|�⃗� 𝑦| 2 ) �⃗� 𝑦 = 0 (8) dengan 𝑐 = 1 √𝜀0𝜇0 adalah kecepatan cahaya di ruang hampa. berdasarkan asumsi bahwa gelombang yang dipancarkan dalam material kerr mempunyai frekuensi tunggal atau gelombang monokromatik, yaitu 𝐸𝑦(𝑥, 𝑧, 𝑡) = 𝐸(𝑥, 𝑧)𝑒 𝑖𝜔𝑡, (9) sedemikian sehingga persamaan (9) menjadi persamaan helmholtz nonlinear ∇⃗⃗ 2𝐸 + 𝜔2 𝑐2 𝑛2𝐸 = 0 , (10) dimana n adalah indeks bias nonlinear kerr. berdasarkan asumsi bahwa 𝑛2 ≪ 1 maka 𝑛 2 dapat diaproksimasi menjadi 𝑛2 ≈ 𝑛0 2 + 2𝑛0𝑛2|�⃗� | 2 (11) dengan 𝑛0 = √1 + 𝜒 (1) adalah indeks bias linear dan 𝑛2 = 𝜒(3) 2𝑛0 adalah koefisien indeks bias nonlinear kerr. dengan demikian perubahan indeks bias ∆𝑛 akibat pengaruh nonlinearitas adalah solusi eksak gelombang soliton: persamaan schrodinger nonlinier nonlokal cauchy – issn: 2086-0382 41 -1 , 𝑛2 < 0 1 , 𝑛2 ≥ 0 ∆𝑛 = 𝑛2|�⃗� | 2 (12) [6]. 3. persamaan schrodinger nonlinear (nls) persamaan helmholtz nonlinear (10) merupakan persamaan diferensial parsial yang secara umum sangat sulit untuk diselesaikan. berdasarkan persamaan tersebut akan diturunkan persamaan schrodinger nonlinear. sebelumnya, akan dicari terlebih dahulu solusi e dari persamaan (10) yang berbentuk 𝐸(𝑥, 𝑧) = 𝐴(𝑥, 𝑧)𝑒𝑖𝑘0𝑧, (13) dengan 𝐴(𝑥, 𝑧) adalah selubung gelombang yang diasumsikan berubah secara lambat sepanjang arah perambatan dan 𝑘0 adalah bilangan gelombang yang memenuhi hubungan 𝑘0 = 𝜔𝑛0 𝑐 . berdasarkan persamaan (13), setelah dibagi dengan 2𝑘0 2 𝑒𝑖𝑘0𝑧 maka persamaan helmholtz nonlinear dapat dituliskan sebagai 1 𝑘0 𝜕𝐴 𝜕𝑧 + 1 2𝑘0 2 𝜕2𝐴 𝜕𝑥2 + 1 2𝑘0 2 𝜕2𝐴 𝜕𝑧2 + 𝑛2 𝑛0 |𝐴|2𝐴 = 0. (14) tergantung pada material kerr, koefisien indeks bias 𝑛2 akan bernilai positif pada material selffocusing dan bernilai negatif pada material selfdefocusing. oleh karena itu persamaan (2.36) ditulis menjadi 1 𝑘0 𝜕𝐴 𝜕𝑧 + 1 2𝑘0 2 𝜕2𝐴 𝜕𝑥2 + 1 2𝑘0 2 𝜕2𝐴 𝜕𝑧2 + 𝑠𝑖𝑔𝑛(𝑛2) �̃�2 𝑛0 |𝐴|2𝐴 = 0 , (15) dengan 𝑠𝑖𝑔𝑛(𝑛2) = untuk menormalkan bentuk persamaan maka didefinisikan suatu parameter kecil 𝜅, 0 < 𝜅 ≤ 1. kemudian diasumsikan bahwa selubung gelombang berubah secara lambat sepanjang arah perambatan z. untuk melihat perubahan selubung gelombang pada jarak perambatan yang cukup jauh tetapi masih berhingga jaraknya maka didefinisikan variabel lambat dan sangat lambat, yaitu 𝑋 = 𝜅𝑘0𝑥 (16) 𝑍 = 𝜅2𝑘0𝑧, (17) dan menskala selubung gelombang sebagai 𝐴(𝑋, 𝑍) = 𝜅√ 𝑛0 �̃�2 𝐵(𝑋, 𝑍). (18) selanjutnya dengan mensubstitusikan persamaan (14), maka diperoleh persamaan schrodinger nonlinear (nls) sebagai berikut 𝑖 𝜕𝐵 𝜕𝑧 + 1 2 𝜕2𝐵 𝜕𝑥2 + 𝑠𝑖𝑔𝑛(𝑛2)|𝐵| 2𝐵 = 0 (19) [6]. 4. gelombang soliton gelombang soliton adalah gelombang soliter (gelombang tunggal) yang memiliki sifat-sifat terlokalisasi dan merambat tanpa perubahan bentuk dan kecepatan. gelombang ini memiliki sifat seperti partikel, yaitu jika gelombang soliton bertumbukan dengan gelombang soliton yang lain, profil dan kecepatannya tidak mengalami perubahan setelah tumbukan. fenomena soliton tidak hanya diprediksi secara teori tetapi juga telah dibuktikan secara eksperimen. pengamatan pertama dilakukan pada tahun 1844 oleh ilmuwan skotlandia, john scott-russel (1808-1882). pada tahun 1834, russel mengamati fenomena gelombang air di kanal eidenberg-glaslow. russel menyebut ini sebagai gelombang besar translasi. gelombang air tersebut menjalar dengan bentuk yang tidak berubah sepanjang kanal dalam rentang waktu yang relatif lama [2]. metodologi penelitian pada penelitian ini, persamaan schrodinger nonlinear nonlokal (nnls) diturunkan dari persamaan helmholtz nonlinear dengan penambahan suatu parameter nonlokal �̃�. selanjutnya, melalui persamaan nnls tersebut akan ditentukan solusi eksak dari gelombang soliton yang berupa fungsi implisit. alur kerangka pemikiran dari penelitian ini diberikan dalam gambar 1, yaitu sebagai berikut: riski nur istiqomah dinnullah 42 volume 3 no. 4 mei 2015 hasil dan pembahasan 1. metode penurunan persamaan schrodinger nonlinier nonlokal (nnls) persamaan helmholtz nonlinear diasumsikan bahwa medium perambatan gelombang adalah material nonlinear bertipe kerr nonlokal, sehingga rapat fluks elektrik akan berbentuk d⃗⃗ = 𝜀0 (1 + 𝜒 (1) + 𝜒(3)|�⃗� | 2 + 𝛾 |�⃗� | 2 𝜕𝑥2 ) �⃗� (20) dengan menggunakan penurunan persamaan helmholtz nonlinear yang sama, maka diperoleh ∇⃗⃗ 2𝐸 + 𝜔2 𝑐2 𝑛2𝐸 = 0. (21) indeks bias nonlinear kerr nonlokal dapat ditulis sebagai 𝑛 = 𝑛0 + 𝑛2|�⃗� | 2 + 𝛾 2𝑛0 |�⃗� | 2 𝜕𝑥2 (22) dengan 𝑛2 = 𝜒(3) 2𝑛0 adalah koefisien indeks bias nonlinear kerr. jika (22) dikuadratkan maka diperoleh 𝑛2 = (𝑛0 + 𝑛2|�⃗� | 2 + 𝛾 2𝑛0 |�⃗� | 2 𝜕𝑥2 ) 2 (23) berdasarkan asumsi bahwa 𝑛2 ≪ 1 dan 𝛾 ≪ 1 maka 𝑛2 dapat diaproksimasi menjadi 𝑛2 ≈ 𝑛0 2 + 2𝑛0𝑛2|�⃗� | 2 + 𝛾 𝜕2|�⃗� | 2 𝜕𝑥2 . (24) dengan demikian perubahan indeks bias ∆𝑛 akibat pengaruh nonlinearitas adalah ∆𝑛 = 𝑛2|�⃗� | 2 + 𝛾 𝜕2|�⃗� | 2 𝜕𝑥2 . (25) persamaan schrodinger nonlinear nonlokal (nnls) berdasarkan persamaan helmholtz nonlinear (21) akan diturunkan persamaan schrodinger nonlinear nonlokal. dalam hal ini dicari terlebih dahulu solusi e dari persamaan yang berbentuk 𝐸 (𝑥, 𝑧) = 𝐴(𝑥, 𝑧)𝑒𝑖𝑘0𝑧 , (26) pada pembahasan ini diasumsikan bahwa 𝑛2 bernilai positif. dengan menggunakan langkah yang sama untuk menemukan persamaan schrodinger nonlinear. selanjutnya diasumsikan juga bahwa  bernilai positif, sehingga diperoleh 𝑖 𝜕𝐵 𝜕𝑍 + 1 2 𝜕2𝐵 𝜕𝑋2 + 𝜅2 𝜕2𝐵 𝜕𝑍2 + |𝐵|2𝐵 + �̃� 𝜕2|𝐵|2 𝜕𝑋2 𝐵 =0. (27) karena nilai 𝜅2 sangat kecil maka nilai 𝜅2 dan suku yang memuat turunan kedua terhadap z dapat diabaikan, sehingga persamaan (27) dapat ditulis menjadi 𝑖 𝜕𝐵 𝜕𝑍 + 1 2 𝜕2𝐵 𝜕𝑋2 + |𝐵|2𝐵 + �̃� 𝜕2|𝐵|2 𝜕𝑋2 𝐵 = 0. (28) persamaan (28) merupakan persamaan schrodinger nonlinear nonlokal (nnls). latar belakang: soliton dengan medium perambatan gelombang adalah material nonlinear bertipe kerr nonlokal persamaan helmholtz nonlinear dengan penambahan parameter �̃� rapat fluks pada media nonlinear bertipe kerr nonlokal: d⃗⃗ = 𝜀0 (1 + 𝜒 (1) + 𝜒(3)|�⃗� | 2 + 𝛾 |�⃗� | 2 𝜕𝑥2 ) �⃗� persamaan schrodinger nonlinear nonlokal (nnls) solusi soliton pada medium nonlinear kerr nonlokal aplikasi matlab untuk melihat perambatan gelombang soliton analisis hasil dan pembahasan kesimpulan gambar 1. alur kerangka pemikiran solusi eksak gelombang soliton: persamaan schrodinger nonlinier nonlokal cauchy – issn: 2086-0382 43 2. solusi soliton pada medium nonlinear kerr nonlokal asumsikan bahwa selubung gelombang berbentuk 𝐵(𝑋, 𝑍) = 𝑈(𝑋)𝑒𝑖γ𝑧, (29) jika persamaan (29) di substitusikan ke persamaan (28), maka diperoleh persamaan −2𝑈γ + 𝜕2𝑈 𝜕𝑋2 + 2𝑈3 + 2�̃� 𝜕2|𝑈|2 𝜕𝑋2 𝑈 = 0. (30) jika persamaan (30) dikali dengan 𝜕𝑈 𝜕𝑋 dan diintegralkan terhadap x, maka diperoleh (1 + 4�̃�𝑈2) ( 𝜕𝑈 𝜕𝑋 ) 2 + (𝑈2 − 2γ)𝑈2 = 𝐶, (31) dengan c adalah konstanta integrasi. kemudian diasumsikan bahwa u dan 𝜕𝑈 𝜕𝑋 bernilai 0 pada 𝑋 → ±∞ sehingga diperoleh nilai konstanta 𝐶 = 0 dan persamaan (31) dapat dituliskan sebagai berikut (1 + 4�̃�𝑈2) ( 𝜕𝑈 𝜕𝑋 ) 2 + (𝑈2 − 2γ)𝑈2 = 0. (32) selain itu, diasumsikan juga bahwa soliton berpusat di 𝑋 = 0 sehingga 𝜕𝑈 𝜕𝑋 = 0 pada saat 𝑋 = 0. dengan menggunakan asumsi tersebut dapat diketahui bahwa hubungan antara konstanta propagasi γ dan amplitudo 𝑈0 adalah sebagai berikut 𝑈0 2 = 2γ, (33) sehingga persamaan (32) dapat disederhanakan menjadi ( 𝜕𝑈 𝜕𝑋 ) 2 = (𝑈0 2−𝑈2)𝑈2 1+4�̃�𝑈2 . (34) untuk kasus �̃� < 0, solusi (35) ada jika dan hanya jika intensitas puncak 𝑈0 2 = 𝜌0 memiliki nilai lebih kecil dari nilai kritis (𝑈𝑐𝑟) 2 = 1 |4�̃�| = 𝜌𝑐𝑟. nilai kritis tersebut diperoleh dari 1 + 4�̃�𝑈 2 yang nilainya selalu positif, namun karena �̃� < 0 maka 1 + 4�̃�𝑈2 > 0 𝜌𝑐𝑟 = 1 |4�̃�| dengan 𝑈2 = 𝜌 adalah intensitas soliton. selanjutnya persamaan (34) diintegralkan terhadap x, sehingga diperoleh ±𝑋 = 1 𝑈0 𝑡𝑎𝑛ℎ−1 ( 𝜎 𝑈0 ) + √4�̃�𝑡𝑎𝑛−1√4�̃�, (35) dengan 𝜎 = √ (𝑈0 2−𝑈2) 1+4�̃�𝑈2 . fungsi implisit (35) merupakan solusi dari soliton dalam medium kerr nonlokal. jika �̃� = 0 maka diperoleh profil 𝑈(𝑋) = 𝑈0sech (𝑈0𝑋) yang merupakan solusi perambatan soliton pada medium lokal. pusat soliton memiliki intensitas yang tinggi, sedangkan di sekitar pusat tersebut intensitasnya sangat rendah (hampir nol). plot soliton untuk parameter nonlokal �̃� yang berbeda ditunjukkan pada gambar 2. dari gambar 2 dapat dilihat dengan jelas bahwa perubahan lebar dari gelombang soliton tergantung pada parameter nonlokal. selanjutnya, soliton dengan parameter nonlokal �̃� = 0 , �̃� = 0.25 , dan �̃� = 0.5 pada 𝑈0 = 1 dan 𝑈0 = 2 diplot pada gambar 3. gambar 2. plot gelombang soliton untuk 𝑼𝟎 = 𝟏 dengan �̃� = 𝟎, �̃� = 𝟎. 𝟐𝟓 dan �̃� = 𝟎. 𝟓 riski nur istiqomah dinnullah 44 volume 3 no. 4 mei 2015 gambar 3 menunjukkan bahwa gelombang soliton ini memiliki nilai maksimum 𝑈0 pada titik pusat 𝑋 = 0. perubahan dari nilai 𝑈0 mempengaruhi perubahan lebar dari soliton. kesimpulan berdasarkan hasil pembahasan yang telah diuraikan pada hasil dan pembahasan, maka dapat diambil kesimpulan bahwa dengan menyelesaikan persamaan schrodinger nonlinear nonlokal (nnls) diperoleh solusi eksak gelombang soliton pada medium nonlinear kerr nonlokal. solusi tersebut berbentuk fungsi implisit yaitu ±𝑋 = 1 𝑈0 𝑡𝑎𝑛ℎ−1 ( 𝜎 𝑈0 ) + √4�̃�𝑡𝑎𝑛−1√4�̃� degnan 𝜎 = √ (𝑈0 2−𝑈2) 1+4�̃�𝑈2 dan �̃� merupakan parameter nonlokal. perubahan parameter nonlokal  ~ dapat mempengaruhi lebar soliton. semakin besar nilai parameter nonlokal �̃� , soliton akan semakin lebar. sementara itu, gelombang soliton memiliki nilai maksimum 𝑈0 pada titik pusat 𝑋 = 0 . perubahan dari nilai 𝑈0 juga mempengaruhi perubahan lebar soliton. soliton akan semakin lebar seiring dengan semakin besarnya nilai 𝑈0. bibliography [1] g. p. agrawal, application of nonlinear fiber optic, san diego: academic pres inc., 2001. [2] y. s. kivshar and g. p. agrawal, optical solution: from fibers to photonic crystal, amsterdam: academic press inc., 2003. [3] m. a. karpierz, a. d. boardman and g. i. stegeman, "nonlocal soliton," in proc. of spie, 2005. [4] w. krolikowski and o. bang, "soliton in nonlocal nonlinear media: exact result," phys. rev. e., p. 63, 2000. [5] f. garzia, c. sabilia and m. bertolotti, "new phase modulation technique based on spatial soliton switching," ieee journal of lightwave technology, pp. 337-338, 2001. [6] a. suryanto, pemodelan matematika, malang: fmipa universitas brawijaya, 2004. gambar 3. plot soliton untuk (a) �̃� = 𝟎, (b) �̃� = 𝟎. 𝟐𝟓, dan (c) �̃� = 𝟎. 𝟓 dengan 𝑼𝟎 = 𝟏 dan 𝑼𝟎 = 𝟐 pendahuluan tinjauan pustaka 1. persamaan maxwell 2. persamaan helmholtz nonlinear 3. persamaan schrodinger nonlinear (nls) 4. gelombang soliton metodologi penelitian hasil dan pembahasan 1. metode penurunan persamaan schrodinger nonlinier nonlokal (nnls) 2. solusi soliton pada medium nonlinear kerr nonlokal kesimpulan bibliography penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana 1alfi fadliana, 2fachrur rozi jurusan matematika universitas islam negeri maulana malik ibrahim malang email: 1alfifadliana@gmail.com, 2fachrurkibar@yahoo.com abstrak metode agglomerative hierarchical clustering merupakan metode analisis cluster yang bertujuan untuk mengelompokkan objek-objek berdasarkan karakteristik yang dimilikinya, yang dimulai dengan objek-objek individual sampai objek-objek tersebut bergabung menjadi satu cluster tunggal. metode agglomerative hierarchical clustering terbagi menjadi beberapa algoritma, di antaranya metode single linkage, complete linkage, average linkage, dan ward. penelitian ini membandingkan keempat metode dalam agglomerative hierarchical clustering dengan tujuan untuk mendapatkan solusi cluster terbaik dalam kasus pengklasifikasian kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana (kb). hasil penelitian menunjukkan bahwa berdasarkan hasil uji validitas cluster, diketahui bahwa metode average linkage memberikan solusi cluster yang lebih baik bila dibandingkan dengan metode agglomerative hierarchical clustering lainnya (single linkage, complete linkage, dan ward). solusi cluster pada metode average linkage menghasilkan 4 cluster dengan karakteristik yang berbeda. cluster 1 memiliki karakteristik tingkat kualifikasi klinik kb dan tingkat kompetensi tenaga pelayanan kb “sangat rendah”. cluster 2 memiliki karakteristik tingkat kualifikasi klinik kb “cukup baik”, dan tingkat kompetensi tenaga pelayanan kb “rendah”. cluster 3 memiliki karakteristik tingkat kualifikasi klinik kb “rendah” dan tingkat kompetensi tenaga pelayanan kb “sedang”. cluster 4 terdiri dari empat kabupaten dengan karakteristik tingkat kualifikasi klinik kb “sedang” dan tingkat kompetensi tenaga pelayanan kb “cukup baik”. kata kunci: analisis cluster, agglomerative hierarchical clustering, uji validitas cluster, keluarga berencana (kb) abstract agglomerative hierarchical clustering methods is cluster analysis method whose primary purpose is to group objects based on its characteristics, it begins with the individual objects until the objects are fused into a single cluster. agglomerative hierarchical clustering methods are divided into single linkage, complete linkage, average linkage, and ward. this research compared the four agglomerative hierarchical clustering methods in order to get the best cluster solution in the case of the classification of regencies/cities in east java province based on the quality of “keluarga berencana” (kb) services. the results of this research showed that based on calculation of cophenetic correlation coefficient, the best cluster solution is produced by average linkage method. this method obtained four clusters w ith the different characteristics. cluster 1 has an “extremely bad condition” on the qualification of kb clinics and the competence of kb service personnel. cluster 2 has a “good condition” on the qualification of kb clinics and “bad condition” on the competence of kb service personnel. cluster 3 has a “bad condition” on the qualification of kb clinics and “medium condition” on the competence of kb service personnel. cluster 4 have a “medium condition” on the qualification of kb clinics and a “good condition” on the competence of kb service personnel. keywords: cluster analysis, agglomerative hierarchical clustering, cophenetic correlation coefficient, keluarga berencana (kb) pendahuluan analisis cluster merupakan salah satu teknik dalam analisis statistik multivariat yang mempunyai tujuan utama mengelompokkan objekobjek pengamatan menjadi beberapa kelompok berdasarkan karakteristik yang dimilikinya [1]. secara umum analisis cluster dibagi menjadi dua, yaitu hierarchical clustering methods (metode clustering hirarki) dan nonhierarchical clustering mailto:alfifadliana@gmail.com mailto:2fachrurkibar@yahoo.com alfi fadliana 36 volume 4 no.1 november 2015 methods (metode clustering nonhirarki). metode clustering hirarki terdiri atas dua bagian, yaitu metode agglomerative (penyatuan) dan divisive (penyebaran). dalam metode agglomerative dikenal beberapa metode untuk membentuk cluster, di antaranya yaitu metode single linkage, complete linkage, average linkage, dan ward. pada penelitian ini penulis akan membandingkan keempat metode agglomerative hierarchical clustering yang telah disebutkan sebelumnya, dengan tujuan untuk mengetahui solusi cluster terbaik yang dihasilkan. dalam penelitian ini, penulis mencoba menerapkan metode agglomerative hierarchical clustering dalam kasus klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana (kb). penelitian ini diharapkan dapat memberikan solusi cluster/pengelompokan terbaik untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb sehingga diperoleh gambaran tentang kondisi dan potensi klinik kb di wilayah jawa timur, yang dapat dijadikan sebagai acuan dalam merumuskan dan menetapkan kebijakan yang sesuai dengan kebutuhan dari masing-masing wilayah di provinsi jawa timur. sehingga ke depannya, program kependudukan dan keluarga berencana nasional dapat mencapai sasaran yang telah ditetapkan. kajian teori 2.1 korelasi pada analisis cluster terdapat asumsi yang harus dipenuhi, yaitu asumsi nonmultikolinieritas (tidak terdapat korelasi antar variabel). untuk mengetahui apakah asumsi tersebut dipenuhi, dilakukan pengujian asumsi korelasi dengan menghitung koefisien korelasi pearson 𝒓 = 𝒏 ∑ 𝑿𝒊𝟏𝑿𝒊𝟐 − (∑ 𝑿𝒊𝟏 𝒏 𝒊=𝟏 )(∑ 𝑿𝒊𝟐 𝒏 𝒊=𝟏 ) 𝒏 𝒊=𝟏 √[𝒏 ∑ 𝑿𝒊𝟏 𝟐 − (∑ 𝑿𝒊𝟏 𝒏 𝒊=𝟏 ) 𝟐𝒏 𝒊=𝟏 ][𝒏 ∑ 𝑿𝒊𝟐 𝟐 − (∑ 𝑿𝒊𝟐 𝒏 𝒊=𝟏 ) 𝟐𝒏 𝒊=𝟏 ] [2]. 2.2 analisis komponen utama analisis komponen utama merupakan teknik penanggulangan masalah multikolinieritas, yang dilakukan dengan cara menghilangkan korelasi di antara variabel bebas melalui transformasi variabel bebas asal ke variabel baru yang tidak berkorelasi sama sekali. variabel baru yang terbentuk ini merupakan kombinasi linier dari variabel asal. prosedur analisis komponen utama adalah sebagai berikut: 1. menentukan matriks inputan: a. matriks varian-kovarian (𝚺), apabila variabel penelitian memiliki unit satuan yang sama. 𝚺 = [ 𝒔𝟏 𝟐 𝒔𝟏𝟐 … 𝒔𝟏𝒑 𝒔𝟐𝟏 𝒔𝟐 𝟐 … 𝒔𝟐𝒑 ⋮ 𝒔𝒑𝟏 ⋮ 𝒔𝒑𝟐 ⋮ … ⋮ 𝒔𝒑 𝟐 ] b. matrik korelasi (𝐙), apabila variabel penelitian tidak memiliki unit satuan yang sama. 𝐙 = [ 𝑍11 𝑍12 … 𝑍1𝑝 𝑍21 𝑍22 … 𝑍2𝑝 ⋮ 𝑍𝑝1 ⋮ 𝑍𝑝2 ⋮ … ⋮ 𝑍𝑝𝑝] dimana z𝑖 adalah nilai baku variabel penelitian. 2. menentukan nilai eigen, 𝜆1, 𝜆2, … , 𝜆𝑝 dari matriks inputan. 3. menentukan vektor eigen ke-𝑗 untuk nilai eigen ke-𝑗 (𝑗 = 1, 2, 3, … , 𝑝), 𝒂𝒋 = (𝑎1𝑗, 𝑎2𝑗, … , 𝑎𝑝𝑗). 4. menghitung skor komponen utama 𝑌𝑝 dengan persamaan komponen yang diperoleh: a. matriks varian-kovarian 𝒀𝟏 = 𝒂′𝟏𝑿 = 𝒂𝟏𝟏𝑿𝟏 + 𝒂𝟏𝟐𝑿𝟐 + ⋯ + 𝒂𝒋𝟏𝑿𝒑 𝒀𝟐 = 𝒂′𝟐𝑿 = 𝒂𝟐𝟏𝑿𝟏 + 𝒂𝟐𝟐𝑿𝟐 + ⋯ + 𝒂𝒋𝟐𝑿𝒑 ⋮ 𝑌𝑝 = 𝒂′𝒑𝑿 = 𝑎1𝑝𝑋1 + 𝑎2𝑝𝑋2 + ⋯ + 𝑎𝑗𝑝𝑋𝑝 b. matriks korelasi 𝒀𝟏 = 𝒂′𝟏𝒁 = 𝒂𝟏𝟏𝒁𝟏 + 𝒂𝟏𝟐𝒁𝟐 + ⋯ + 𝒂𝒋𝟏𝒁𝒑 𝒀𝟐 = 𝒂′𝟐𝒁 = 𝒂𝟐𝟏𝒁𝟏 + 𝒂𝟐𝟐𝒁𝟐 + ⋯ + 𝒂𝒋𝟐𝒁𝒑 ⋮ 𝑌𝑝 = 𝒂′𝒑𝒁 = 𝑎1𝑝𝑍1 + 𝑎2𝑝𝑍2 + ⋯ + 𝑎𝑗𝑝𝑍𝑝 skor komponen utama yang diperoleh digunakan sebagai input dalam analisis selanjutnya, sebagai pengganti dari nilai data variabel awal. 2.3 ukuran kedekatan ukuran kedekatan yang digunakan adalah ukuran ketidakmiripan/ukuran jarak euclidean yang dihitung dengan menggunakan persaman berikut 𝑑𝑖𝑘 = √∑(𝑋𝑖𝑗 − 𝑋𝑘𝑗) 2 𝑝 𝑗=1 dimana 𝑿𝒊𝒋 = objek ke-𝒊 pada variabel ke-𝒋 penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana cauchy – issn: 2086-0382/e-issn: 2477-3344 37 𝑿𝒌𝒋 = objek ke-𝒌 pada variabel ke-𝒋 𝒊, 𝒌 = 𝟏, 𝟐, . . . , 𝒏; 𝒊 ≠ 𝒌 𝒋 = 𝟏, 𝟐, . . . , 𝒑 𝑝 = banyaknya variabel [3]. 2.4 agglomerative hierarchical clustering metode agglomerative, proses pengelompokan dimulai dengan objek-objek yang individual. jadi, banyaknya cluster sama dengan banyaknya objek. objek-objek yang paling mirip pertama kali bergabung membentuk cluster, demikian seterusnya sampai membentuk satu cluster. 1. single linkage merupakan prosedur pengelompokan agglomerative yang didasarkan pada jarak minimum/jarak terdekat antar objek. prosedur pengelompokan single linkage pada awalnya dipilih jarak terkecil dalam 𝑫 = {𝑑𝑖𝑗} dan menggabungkan objek-objek yang bersesuaian misalnya 𝑈 dan 𝑉 untuk mendapatkan cluster (𝑈𝑉). langkah berikutnya, jarak di antara (𝑈𝑉) dan cluster lainnya, misalnya 𝑊. 𝑑(𝑈𝑉)𝑊 = 𝑚𝑖𝑛(𝑑𝑈𝑊, 𝑑𝑉𝑊) dimana, 𝑑𝑈𝑊 = jarak antara tetangga terdekat dari cluster 𝑈 dan 𝑊, dan 𝑑𝑉𝑊 = jarak antara tetangga terdekat dari cluster 𝑉 dan 𝑊 [3]. 2. complete linkage didasarkan pada jarak maksimum/jarak terjauh antar objek. algoritma metode complete linkage dimulai dengan menemukan elemen minimum dalam 𝑫 = {𝒅𝒊𝒋}, selanjutnya menggabungkan objek-objek yang bersesuaian misalnya 𝑼 dan 𝑽 untuk mendapatkan cluster (𝑼𝑽). tahap berikutnya, jarak di antara (𝑼𝑽) dan cluster lainnya, misalnya 𝑾. 𝑑(𝑈𝑉)𝑊 = 𝑚𝑎𝑥(𝑑𝑈𝑊, 𝑑𝑉𝑊) dimana, 𝒅𝑼𝑾 = jarak antara tetangga terjauh dari cluster 𝑼 dan 𝑾, dan 𝒅𝑽𝑾 = jarak antara tetangga terjauh dari cluster 𝑽 dan 𝑾 [3]. 3. average linkage memperlakukan jarak antar dua cluster sebagai jarak rata-rata antara semua pasangan objek data dalam satu cluster dengan seluruh objek pada cluster lain. prosedur average linkage dimulai dengan mendefinisikan matrik 𝑫 = {𝒅(𝒊𝒋)} untuk memperoleh objek-objek paling dekat, sebagai contoh 𝑼 dan 𝑽. kemudian objek ini digabung ke dalam bentuk cluster (𝑼𝑽). selanjutnya, jarak antara (𝑼𝑽) dan cluster lainnya, 𝑾. 𝒅(𝑼𝑽)𝑾 = ∑ ∑ 𝒅𝒊𝒋𝒋𝒊 𝒏(𝑼𝑽)𝒏𝒘 dimana, 𝒅𝒊𝒋 = jarak antara objek 𝒊 pada cluster (𝑼𝑽) dan objek 𝒋 pada cluster 𝑾, 𝒏(𝑼𝑽) = banyaknya anggota dalam cluster 𝑼𝑽, 𝒏𝑾 = banyaknya anggota pada cluster 𝑾 [3]. 4. ward didasarkan pada sum square of error (sse) 𝑆𝑆𝐸𝑢𝑣 = ∑(𝑋𝑖 − �̅�𝑢𝑣)′(𝑋𝑖 − �̅�𝑢𝑣) 𝑛𝑢𝑣 𝑖=1 dimana: 𝑿𝒊 = nilai objek ke-𝒊 �̅� = rata-rata nilai objek dalam cluster 𝒊 = 𝟏, 𝟐, 𝟑, … , 𝒏 𝒏 = banyaknya objek 𝒏𝑼 = banyaknya titik pada cluster 𝑼 𝒏𝑽 = banyaknya titik pada cluster 𝑼 dan 𝑽 𝒏𝑼𝑽 = banyaknya titik pada cluster 𝑼𝑽 �̅�𝒖𝒗 = 𝒏𝑼�̅�𝒖 + 𝒏𝑽�̅�𝒗 𝒏𝑼 + 𝒏𝑽 [4]. 2.5 uji validitas cluster uji validitas cluster diperlukan untuk melihat kebaikan (goodness) atau kualitas (quality) hasil analisis cluster. ukuran yang digunakan untuk menguji validitas hasil clustering pada penelitian ini adalah koefisien korelasi cophenetic. koefisien korelasi cophenetic merupakan koefisien korelasi antara elemen-elemen asli matriks ketidakmiripan (matriks jarak euclidean) dan elemen-elemen yang dihasilkan oleh dendrogram (matriks cophenetic) (silva & dias, 2013). formula yang digunakan untuk menghitung koefisien korelasi cophenetic sebagai berikut 𝑟𝐶𝑜𝑝ℎ = ∑ (𝑑𝑖𝑘 − 𝑑̅)(𝑑𝐶 𝑖𝑘 − 𝑑 ̅ 𝐶)𝑖<𝑘 √[∑ (𝑑𝑖𝑘 − 𝑑̅) 2 𝑖<𝑘 ] [∑ (𝑑𝐶 𝑖𝑘 − 𝑑 ̅ 𝐶) 2 𝑖<𝑘 ] dimana: 𝒓𝑪𝒐𝒑𝒉 = koefisien korelasi cophenetic 𝒅𝒊𝒌 = jarak asli (jarak euclidean) antara objek 𝒊 dan 𝒌 �̅� = rata-rata 𝒅𝒊𝒌 𝒅𝑪𝒊𝒌 = jarak cophenetic objek 𝒊 dan 𝒌 �̅�𝑪 = rata-rata 𝒅𝑪𝒊𝒌 [5]. nilai 𝑟𝐶𝑜𝑝ℎ berkisar antara −1 dan 1; nilai 𝑟𝐶𝑜𝑝ℎ mendekati 1 berarti solusi yang dihasilkan dari proses clustering cukup baik. metode penelitian data yang digunakan dalam penelitian ini adalah data sekunder yang berkaitan dengan alfi fadliana 38 volume 4 no.1 november 2015 kualitas pelayanan kb di provinsi jawa timur, yaitu data mengenai kualifikasi klinik kb dan kompetensi tenaga pelayanan kb di provinsi jawa timur tahun 2014 yang diperoleh dari laporan pelayanan kontrasepsi (pelkon) tahun 2014 provinsi jawa timur dalam situs resmi badan kependudukan dan keluarga berencana nasional (bkkbn). variabel penelitian yang digunakan dalam penelitian ini adalah: 1. kualifikasi klinik kb (𝑋1), yang terdiri dari empat sub-variabel: a. jumlah klinik kb sederhana (𝑋1,1) b. jumlah klinik kb lengkap (𝑋1,2) c. jumlah klinik kb sempurna (𝑋1,3) d. jumlah klinik kb paripurna (𝑋1,4) 2. kompetensi tenaga pelayanan kb (𝑋2), yang terdiri dari 11 sub-variabel: a. jumlah dokter yang telah mendapat pelatihan iud (𝑋2,1) b. jumlah dokter yang telah dilatih mow (𝑋2,2) c. jumlah dokter yang telah mendapat pelatihan mop (𝑋2,3) d. jumlah dokter yang telah dilatih implan (𝑋2,4) e. jumlah dokter yang telah mendapat pelatihan kip (𝑋2,5) f. jumlah bidan yang telah dilatih iud (𝑋2,6) g. jumlah bidan yang telah dilatih implan (𝑋2,7) h. jumlah bidan yang telah mendapat pelatihan kip (𝑋2,8) i. jumlah bidan yang telah dilatih r/r (𝑋2,9) j. jumlah perawat yang telah mendapat pelatihan kip (𝑋2,10) k. jumlah perawat yang telah dilatih r/r (𝑋2,11) langkah-langkah analisis dalam penelitian ini adalah sebagai berikut: 1. melakukan analisis statistika deskriptif untuk mengetahui karakteristik umum kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb. 2. melakukan analisis statistika inferensi untuk mengetahui penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb. a. melakukan uji asumsi korelasi untuk memeriksa apakah terdapat korelasi antar variabel atau tidak. apabila terdapat korelasi, maka perlu dilakukan penanggulangan dengan menggunakan analisis komponen utama, namun bila tidak terdapat korelasi maka langsung melanjutkan ke langkah b. b. menghitung jarak euclidean sebagai ukuran kedekatan antar objek. c. melakukan analisis cluster pada data kualitas pelayanan kb provinsi jawa timur menggunakan metode agglomerative hierarchical clustering. d. menginterpretasikan hasil/solusi clustering yang dihasilkan dari langkah c. e. melakukan uji validitas cluster. 3. menarik kesimpulan. pembahasan 3.1 karakteristik umum kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana berdasarkan hasil analisis statistika deskriptif diketahui bahwa lebih dari 90% klinik kb di provinsi jawa timur merupakan klinik kb sederhana. sedangkan klinik kb lengkap, sempurna dan paripurna menunjukkan persentase yang cukup kecil. sedangkan pada variabel kompetensi tenaga pelayanan kb, diketahui bahwa dari 2.792 orang dokter yang tercatat memberikan pelayanan kb di provinsi jawa timur tahun 2014, sekitar 64,72% belum mendapat pelatihan iud, 86,32% belum mendapat pelatihan mow, 89,22% belum mendapat pelatihan mop, 67,87% belum mendapat pelatihan implan dan 73,78% belum mendapat pelatihan kip; sekitar 44,09% bidan di provinsi jawa timur belum mendapat pelatihan iud, 45,61% belum mendapat pelatihan implan, 60,66% belum mendapat pelatihan kip, dan 71,78% belum mendapat pelatihan r/r; dan lebih dari 90% tenaga perawat kesehatan belum mendapat pelatihan kip dan r/r. 3.2 penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana hasil uji asumsi korelasi menunjukkan bahwa data penelitian mengindikasikan adanya masalah multikolinieritas. sehingga harus dilakukan penanggulangan multikolinieritas dengan menggunakan analisis komponen utama. persamaan komponen utama yang terbentuk adalah sebagai berikut penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana cauchy – issn: 2086-0382/e-issn: 2477-3344 39 𝑲𝑼𝟏 = −𝟎, 𝟐𝟔𝟗𝑿𝟏,𝟏 − 𝟎, 𝟎𝟎𝟑𝑿𝟏,𝟐 − 𝟎, 𝟎𝟎𝟏𝑿𝟏,𝟑 − 𝟎, 𝟎𝟖𝟎𝑿𝟐,𝟏 − 𝟎, 𝟎𝟐𝟗𝑿𝟐,𝟐 − 𝟎, 𝟎𝟐𝟓𝑿𝟐,𝟑 + 𝟎, 𝟎𝟕𝟐𝑿𝟐,𝟒 − 𝟎, 𝟎𝟔𝟑𝑿𝟐,𝟓 − 𝟎, 𝟓𝟕𝟔𝑿𝟐,𝟔 − 𝟎, 𝟓𝟒𝟑𝑿𝟐,𝟕 − 𝟎, 𝟒𝟑𝟐𝑿𝟐,𝟖 − 𝟎, 𝟑𝟎𝟐𝑿𝟐,𝟗 − 𝟎, 𝟎𝟓𝟔𝑿𝟐,𝟏𝟎 − 𝟎, 𝟎𝟓𝟎𝑿𝟐,𝟏𝟏𝑲𝑼𝟐 = 𝟎, 𝟖𝟖𝟓𝑿𝟏,𝟏 + 𝟎, 𝟎𝟑𝟗𝑿𝟏,𝟐 + 𝟎, 𝟎𝟎𝟑𝑿𝟏,𝟑 + 𝟎, 𝟎𝟎𝟓𝑿𝟏,𝟒 − 𝟎, 𝟏𝟏𝟑𝑿𝟐,𝟏 − 𝟎, 𝟎𝟓𝟒𝑿𝟐,𝟐 − 𝟎, 𝟎𝟐𝟔𝑿𝟐,𝟑 + 𝟎, 𝟎𝟖𝟑𝑿𝟐,𝟒 − 𝟎, 𝟎𝟕𝟗𝑿𝟐,𝟓 − 𝟎, 𝟐𝟔𝟎𝑿𝟐,𝟔 − 𝟎, 𝟐𝟐𝟕𝑿𝟐,𝟕 − 𝟎, 𝟎𝟐𝟖𝑿𝟐,𝟖 + 𝟎, 𝟐𝟒𝟔𝑿𝟐,𝟗 − 𝟎, 𝟎𝟔𝟓𝑿𝟐,𝟏𝟎 − 𝟎, 𝟎𝟑𝟑𝑿𝟐,𝟏𝟏 skor komponen utama yang digunakan sebagai inputan, pengganti variabel asal, dihitung berdasarkan persamaan komponen utama yang terbentuk. klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb tahun 2014 menggunakan metode single linkage menghasilkan solusi cluster sebanyak 4, complete linkage sebanyak 2, average linkage sebanyak 4, dan ward sebanyak 2. berdasarkan hasil uji validitas cluster dengan koefisien korelasi cophenetic pada tabel 1 di atas, diketahui bahwa metode average linkage clustering memberikan solusi cluster yang lebih baik bila dibandingkan dengan ketiga metode yang lain (single linkage, complete linkage, dan ward). tabel 1 koefisien korelasi cophenetic solusi cluster metode koefisien korelasi cophenetic single linkage 0,6853 complete linkage 0,6978 average linkage 0,7169 ward 0,6530 karakterisasi hasil klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb tahun 2014 menggunakan metode clustering terbaik (average linkage) adalah sebagai berikut: tabel 2 karakterisasi hasil klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb tahun 2014 menggunakan metode average linkage cluster anggota cluster karakterisasi 1 kab. pacitan, kab. lumajang, kab. situbondo, kab. pasuruan, kab. madiun, kab. magetan, kab. bangkalan, kab. sampang, kab. pamekasan, kab. sumenep, kota kediri, kota tingkat kualifikasi klinik kb dan tingkat kompetensi tenaga pelayanan kb yang “sangat rendah” blitar, kota malang, kota probolinggo, kota pasuruan, kota mojokerto, kota madiun, dan kota batu 2 kab. ponorogo tingkat kualifikasi klinik kb “cukup baik”, dan tingkat kompetensi tenaga pelayanan kb “rendah” 3 kab. trenggalek, kab. tulungagung, kab. blitar, kab. kediri, kab. jember, kab. bondowoso, kab. probolinggo, kab. mojokerto, kab. jombang, kab. nganjuk, kab. bojonegoro, kab. tuban, kab. lamongan, kab. gresik, dan kota surabaya tingkat kualifikasi klinik kb “rendah” dan tingkat kompetensi tenaga pelayanan kb “sedang” 4 kab. malang, kab. banyuwangi, kab. sidoarjo, dan kab. ngawi tingkat kualifikasi klinik kb “sedang” dan tingkat kompetensi tenaga pelayanan kb “cukup baik” penutup berdasarkan uji validitas cluster dengan koefisien korelasi cophenetic diketahui bahwa metode average linkage clustering memberikan solusi cluster yang lebih baik bila dibandingkan dengan metode agglomerative hierarchical clustering yang lain dalam penerapannya untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan kb tahun 2014. saran untuk penelitian selanjutnya dapat menggunakan metode clustering, ukuran kedekatan, dan/atau uji validitas cluster yang berbeda dengan yang telah digunakan oleh penulis; atau menerapkannya pada data/kasuskasus lain. daftar pustaka [1] j. hair, w. black, b. babin and r. anderson, multivariate data analysis, 7th edition, new jersey: pearson prentice hall, 2010. [2] a. bluman, elementary statistics: a step by step approach, 5th edition, new york: mc alfi fadliana 40 volume 4 no.1 november 2015 graw-hill, 2004. [3] r. johnson and d. wichern, applied multivariate statistical analysis, 6th edition, upper saddle river, new jersey 07458: prentice education, inc, 2007. [4] a. rencher, methods of multivariate analysis, 2nd edition, new york: john wiley & sons, inc, 2002. [5] s. saraçli, n. doğan and i. doğan, "comparison of hierarchical cluster analysis methods by cophenetic correlation," journal of inequalities and applications 2013, p. 203, 2013. pendahuluan kajian teori 2.1 korelasi 2.2 analisis komponen utama 2.3 ukuran kedekatan 2.4 agglomerative hierarchical clustering 2.5 uji validitas cluster metode penelitian pembahasan 3.1 karakteristik umum kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana 3.2 penerapan metode agglomerative hierarchical clustering untuk klasifikasi kabupaten/kota di provinsi jawa timur berdasarkan kualitas pelayanan keluarga berencana penutup daftar pustaka application of pontryagin’s minimum principle in optimum time of missile manoeuvring cauchy journal of pure and applied mathematics volume 4(3) (2016), pages 107-111 submitted: june, 28 2016 reviewed: october, 21 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3534 application of pontryagin’s minimum principle in optimum time of missile manoeuvring sari cahyaningtias1 ,subchan2 1adi buana university 2kalimantan institute of technology email: scahyaningtias@gmail.com, s.subchan@gmail.com abstract missile is a guided weapon and designed to protect outermost island from a thread of other country. it, commonly, is used as self defense. this research presented surface-to-surface missile in final dive manoeuvre for fixed target. furthermore, it was proposed manoeuvring based on unmanned aerial vehicle (uav), autopilot system, which needs accuration and minimum both time and thrust of missile while attacking object. this paper introduced pontryagin’s minimum principle, which is useable to solve the problem. the numerical solution showed that trajectory of the missile is split it up in three sub-intervals; flight, climbing, and diving. the numerical simulation showed that the missile must climb in order to satisfy the final dive condition and the optimum time of a missile depend on initial condition of the altitude and the terminal velocity. keywords: missile, optimal control, pontryagin’s minimum principle, thrust, angle of attack. introduction missile is a military rocket which has automatic control system to target an object and suit the direction. missile is one of the unmanned aeral vehicle (uav) which uses both in military and civilian needed such as controlling forest from illegal logging and sea from illegal fishing. it clearly is known that uav can be handled using autopilot system to follow the references trajectory which could minimized both the risk and cost [1] trajectory tracking of surface-to-surface missile is divided in three sub-interval, namely: minimum altitude flight, climbing, and diving [2] [3]. restrictiveness of fuel supply in maoeuvring can be handled by minimizing time manoeuvring to attack the target. mathematics modelling of mass centre missile follow several assumption: the earth is spherical and non-rotation and ignoring lateral movement of the missile. some boundary problems also were considered in this research such as massa of missile remains constant 𝑑𝑚 𝑑𝑡 ≅ 0; there was no any obstacle while manoeuvring; the thrust and angle of attack are the controller of this problem which should be attacked fixed target. trajectory problem of missile has been studied by many researches both using classical control theory and modified optimal control. in 2013, wang and dong using hybrid algorithm to intercept trajectory optimization for multi-stage air defense missile which the missile always flies in the plane formed by the earth centre, launch point, and the predicted intercept point [4]. maopeng at. al. in 2014, integrated missile guidance and control law using adaptive fuzzy sliding mode control. they also conducted stability of close loop system [5]. http://dx.doi.org/10.18860/ca.v4i3.3534 mailto:scahyaningtias@gmail.com mailto:s.subchan@gmail.com application of pontryagin’s minimum principle in optimum time of missile manoeuvring sari cahyaningtias 108 this paper presents computational result of the optimal trajectory of missile which minimized the time manoeuvring along the optimal trajectory. furthermore, vertical dive manoeuvre was chosen as final maneuvers by considering effect of its attacking. it would have shot to pieces the target permanently. optimal control is applied by forcing the thrust and angle of attack to get optimum time using pontryagin’s minimum principle. methods purpose of this research is to attack motionless target precisely and to minimize time manoeuvring. here was given the definition of missile axes and angles. figure 1 definition of missile axes and angles based on figure 1, the dynamic equations of a point mass missile, which may be written as [6]: 𝑑𝑉 𝑑𝑡 = 1 𝑚 [𝑇 cos 𝛼 − 𝐷] − 𝑔 sin 𝛾 (1) [7] 𝑑𝛾 𝑑𝑡 = 1 𝑚𝑉 [𝐿 + 𝑇 sin 𝛼] − 𝑔 𝑉 cos 𝛾 (2) 𝑑𝑥 𝑑𝑡 = 𝑉 cos 𝛾 (3) 𝑑ℎ 𝑑𝑡 = 𝑉 sin 𝛾 (4) there are four state variables, namely: speed 𝑉, 𝛾 flight path angle, horizontal position 𝑥, and altitude ℎ. the control variables are thrust 𝑇 and angle of attack 𝛼. axial aerodynamics forces and normal aerodynamics forces are function of altitude, speed, and angle of attack, which are defined as: 𝐷(ℎ, 𝑉, 𝛼) = 1 2 𝐶𝐷𝜌𝑉 2𝑆 𝐶𝐷 = 𝐴1𝛼 2 + 𝐴2𝛼 + 𝐴3 𝐿(ℎ, 𝑉, 𝛼) = 1 2 𝐶𝐿 𝜌𝑉 2𝑆 𝐶𝐿 = 𝐵1𝛼 + 𝐵2 where 𝜌, air density, is given by 𝜌 = 𝐶1ℎ 2 + 𝐶2ℎ + 𝐶3 whereas s reference area of missile, 𝑚 massa and 𝑔 gravity. table 1. physical modelling parameter parameter nilai unit m 1005 kg g 9.81 m/s² s 0.3376 m² a1 -1.9431 a2 -0.1499 a3 0.2359 b1 21.9 b2 0 c1 3.312 10-9 kg m-5 c2 -1.142 10-4 kg m-4 c3 1.224 kg m-3 table 2. boundary condition and constraints parameter nilai unit v(0) 210 m/s2 𝛾(0) 0 rad 𝑥(0) 0 m ℎ(0) 30, 100, 200 m 𝑉(𝑡𝑓 ) 290, 310 m/s 2 application of pontryagin’s minimum principle in optimum time of missile manoeuvring sari cahyaningtias 2 𝛾(𝑡𝑓 ) − 𝜋 2 rad 𝑥(𝑡𝑓 ) 10000 meter 𝑇𝑚𝑎𝑘𝑠 6000 n 𝑇𝑚𝑖𝑛 1000 n boundary of control 𝑇𝑚𝑖𝑛 ≤ 𝑇 ≤ 𝑇𝑚𝑎𝑘𝑠 the problem of this research is how to get the minimum time of missile to attack the fixed target in vertical dive manoeuvre. performance index of this problem is function of time which is given by: j = ∫ dt tf t0 (5) where 𝑡 time of manoeuvring, 𝑡0 ≤ 𝑡 ≤ 𝑡𝑓 in which 𝑡0 initial time and 𝑡𝑓 final time. results and discussion mathematical analysis showed that some parts of those solution couldn’t be gained using analytic method. that is why, computational analysis was applied in that case. this paper used optimal control toolbox, namely miser3 which is developed by jennings [8] the trajectory of surface-to-surface missile with vertical dive manoeuvre is split of three subinterval: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 2000 2500 3000 waktu (normalisasi) k e t in g g ia n ( m e te r) v(tf)=310 v(tf)=290 figure 4 performance of angle of attack missile toward time figure 5 performance of thrust of missile toward time figure 2 performance of missile’s altitude toward normalization time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 waktu (normalisasi) g a y a d o r o n g ( n e w to n ) v(tf)=310 v(tf)=290 figure 3 performance of missile’s velocity toward nirmalization time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 waktu (normalisasi) s u d u t s e r a n g ( ra d ia n ) v(tf)=310 v(tf)=290 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 200 220 240 260 280 300 320 waktu (normalisasi) k e c e p a t a n ( m /d e ti k ) v(tf)=310 v(tf)=290 109 application of pontryagin’s minimum principle in optimum time of missile manoeuvring sari cahyaningtias 110 1. minimum altitude flight the missile launches at current altitude and therefore the altitude constraints are active directly at the start of the manoeuvre. in this case the speed of the missile remains constant on the minimum value hmin (see figures 2) until the missile must start climbing while the thrust is on the maximum value. the flight time of first arc depends to the final-speed. figure 5 shows that the thurst of missile remains at maximum, 6000 n, while the speed slightly higher until the next sub-interval in figure 3. 2. climbing the speed of missile after flight in minimum altitude decreased gradually until reached the optimum altitude to start manoeuvre which is vertical dive manoeuvre. the thrust of missile remains maximum until time-0.6 (normalization time), furthermore there suddenly switched into lowest boundary, 1000 n. it showed that missile was turning down and starting dive manoeuvre. 3. diving the missile must gain the power to reach the target therefore the speed increase rapidly since the initial diving speed is lower than the final speed. the normal acceleration is saturated on the minimum value for this arc. whereas the thrust remains constant around 1000 n. the angle of attack convergenly is moving at –pi/40 radian. here presents computational solution of minimizing time problem depends on initial altitude and final speed. table 3. optimum time by computation initial altitude (meter) final speed (m/s) objective function (s) 30 290 48.0781385 310 44.8815557 100 290 47.9329181 310 44.8199641 200 290 47.7656314 310 44.6765635 conclusion in this paper, analysis of surface-to-surface missile which considered objective and dynamical system showed that trajectory tracking of this missile was split up in three subintervals, namely flying, climbing, and diving in which had different performances of speed. those paths were result by controlling thrust and angle of attack. first path, the missile launched at h=30m and remained stable in this altitude. second part, velocity of missile decreased dramatically while climbing to reach optimum altitude before manoeuvring. furthermore the last path, dive manoeuvring, the speed of missile got faster rapidly to final speed. it would be happened while missile was at 30 meters above the target with angle of attack’s -90 deg. travel time of missile was influenced by starting position while it was shoot and final speed of missile itself. the higher launch site and the bigger final speed of missile affected the shorter its time travel. another variable that should be considered in missile manoeuvring is energy. it can be minimize in objective function. this problem can be conducted in future reserach. application of pontryagin’s minimum principle in optimum time of missile manoeuvring sari cahyaningtias 111 references [1] s. subchan, "pythagorean hodograph path planning for tracking airborne contaminant using sensor swarn," in international instrumentation and measurement technology conference victoria, canada, 2008. [2] s. subchan, "trajectory happing of surface-to-surface missile with terminal impact angle constraint," makara teknologi indonesia university, pp. 65-70, 2007. [3] s. subchan and r. zbikowski, computational optimal control: tools and practice, uk: john wiley and sons ltd, 2009. [4] f. b. wang and c. h. dong, "fast intercept trajectory optimization for multi-stage air defense missile using hybrid algorthm," elsivier procedia engineering, pp. 447-456, 2013. [5] e. a. maoping, "backstepping design of missile guidance and control based on adaptive fuzzy sliding mode control," chinese journal of aeronautics, pp. 634-642, 2014. [6] g. siouris, missile guidance and control systems, usa: springer, 2003. [7] d. s. naidu, optimal control systems, usa: crc press llc, 2002. [8] l. s. jennings, m. e. fisher, k. l. teo and c. j. goh, miser3 optimal control software, australia: the university of western australia, 2002. suma’inna dan gugun gumilar sistem pendukung keputusan metode sugeno dalam menentukan tingkat kepribadian siswa berdasarkan pendidikan (studi kasus di mi miftahul ulum gondanglegi malang) 1wildan hakim, 2turmudi , 3wahyu h. irawan 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang 3jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: supriseforu@gmail.com metode sugeno merupakan salah satu metode fuzzy inference system pada logika kabur untuk pengambilan keputusan. dalam metode sugeno orde-nol dalam logika kabur terdiri dari empat tahap, yaitu 1. pengaburan (fuzzifikasi). 2. aplikasi fungsi implikasi, fungsi implikasi yang digunakan adalah fungsi min (minimum). 3. komposisi aturan menggunakan fungsi max (maksimum). 4. penegasan (defuzzifikasi) menggunakan metode perhitungan rata-rata terbobot (weight average). berdasarkan kasus 1, masing-masing siswa dengan pendidikan nonformal sebesar 12 dan pendidikan informal sebesar 19 mempunyai nilai kepribadian 20,7 dan variabel linguistiknya adalah sedang. saran untuk penelitian selanjutnya dapat menggunakan parameter lain dalam menentukan tingkat kepribadian siswa dengan logika kabur. kata kunci: sistem pendukung keputusan metode sugeno, tingkat kepribadian siswa abstract sugeno method is one method of fuzzy inference system on fuzzy logic for decision making. in-zero order sugeno method in fuzzy logic consists of four stages: 1. fuzzification. 2. the application functionality implications, the implication function used is function min (minimum). 3. the composition rule using the function max (maximum). 4. defuzzification weight average. based on case 1, each student with non-formal education and informal education by 12 by 19 has a value of 20.7 and a variable linguistic personality is medium. suggestions for further research can use other parameters in determining the level of personality of students with fuzzy logic. keyword: decision support systems sugeno method, the level of student personality pendahuluan logika adalah ilmu yang mempelajari secara sistematis kaidah-kaidah penalaran yang absah (valid). dewasa ini terdapat 2 konsep logika, yaitu logika tegas dan logika kabur. logika tegas hanya mengenal dua keadaan yaitu: ya atau tidak, on atau off, high atau low, 1 atau 0. logika semacam ini disebut dengan logika himpunan tegas. sedangkan logika kabur adalah logika yang menggunakan konsep sifat kesamaran. sehingga logika kabur adalah logika dengan tak hingga banyak nilai kebenaran yang dinyatakan dalam bilangan real dalam selang [0,1] [1]. secara teori sudah ada cara untuk menentukan tingkat kepribadian siswa, namun perhitungan tingkat kepribadian tersebut menggunakan himpunan crisp (tegas). pada himpunan tegas, suatu nilai mempunyai tingkat keanggotaan satu jika nilai tersebut merupakan anggota dalam himpunan dan nol jika nilai tersebut tidak menjadi anggota himpunan. hal ini sangat kaku, karena dalam tingkat kepribadian siswa tidak hanya “tinggi” atau “rendah” saja melainkan ada yang mengandung kekaburan (ketidakpastian) seperti “sedang”. himpunan kabur digunakan untuk mengantisipasi hal tersebut, karena dapat memberikan toleransi terhadap nilai sehingga dengan adanya perubahan sedikit pada nilai tidak akan memberikan perbedaan yang signifikan. metode yang dapat digunakan dalam sistem pendukung keputusan adalah metode mamdani, metode tsukamoto, dan metode sugeno. sistem pendukung keputusan dalam menetukan tingkat kepribadian siswa berdasarkan pendidikan ini menggunakan metode sugeno karena metode ini output (konsekuen) sistem tidak berupa himpunan kabur, melainkan berupa konstanta atau persamaan linear sehingga tingkat keakurasiannya lebih tinggi dibandingkan dengan metode-metode yang lain. mailto:supriseforu@gmail.com wildan hakim 49 volume 4 no.1 november 2015 sistem pendukung keputusan dalam menentukan tingkat kepribadian siswa berdasarkan pendidikan ini dapat memberikan kemudahan dalam mendapatkan informasi pada siswa, guru dan orang tua sehingga dapat dilakukan penanganan dengan keputusan yang tepat. kajian teori 1. konsep himpunan kabur himpunan adalah suatu kumpulan atau koleksi objek-objek yang mempunyai kesamaan sifat tertentu [1]. himpunan kabur merupakan suatu pengembangan lebih lanjut tentang konsep himpunan dalam matematika. himpunan kabur adalah rentang nilai-nilai, masing-masing nilai mempunyai derajat keanggotaan antara 0 sampai dengan 1. suatu himpunan kabur ã dalam semesta pembicaraan u dinyatakan dengan fungsi keanggotaan μã , yang nilainya berada dalam interval [0,1]. menurut kusumadewi [2] himpunan kabur memiliki dua atribut, yaitu: a. linguistik, yaitu penamaan suatu kelompok yang mewakili suatu keadaan atau kondisi tertentu dengan menggunakan bahasa alami, seperti: lambat, sedang, cepat. b. numeris, yaitu suatu nilai (angka) yang menunjukkan ukuran dari suatu variabel, seperti: 40, 50, 60 dan sebagainya. hal-hal yang perlu diketahui dalam memahami sistem kabur, yaitu: a. variabel kabur variabel kabur merupakan variabel yang akan dibahas dalam suatu sistem kabur, seperti: umur, berat badan, tinggi badan, dan sebagainya. b. himpunan kabur himpunan kabur merupakan suatu kelompok yang mewakili suatu keadaan tertentu dalam suatu variabel kabur. c. semesta pembicaraan semesta pembicaraan adalah keseluruhan nilai yang diperbolehkan untuk dioperasikan dalam suatu variabel kabur. nilai semesta pembicaraan dapat berupa bilangan positif maupun negatif. sebagai contoh, semesta pembicaraan untuk variabel laju kendaraan adalah [0,160]. d. domain domain himpunan kabur adalah keseluruhan nilai yang diperbolehkan untuk dioperasikan dalam suatu himpunan kabur. nilai domain dapat berupa bilangan positif maupun negatif. 2. fungsi keanggotaan menurut klir [3]setiap himpunan kabur ã di dalam himpunan universal x, x ∈ x dipetakan ke dalam interval [0,1]. pemetaan dari x ∈ x pada interval [0,1] disebut fungsi keanggotaan. fungsi keanggotaan dari himpunan kabur ã di dalam semesta x dapat ditulis ã:x → [0,1] a. fungi keanggotaan linier pada representasi linear, pemetaan input ke derajat keanggotannya digambarkan sebagai suatu garis lurus. ada dua keadaan himpunan kabur linear, yaitu linear naik dan linear turun. representasi himpunan kabur linear naik seperti yang ditunjukkan pada gambar 2.5. x rendah b 1 0 (x) a fungsi keanggotaan: μ[x] = { 0;x ≤ a (x − a) (b − a) ;a ≤ x ≤ b 1;x ≥ b representasi himpunan kabur linear turun seperti yang ditunjukkan pada gambar 2.6. x rendah b 1 0 (x) a fungsi keanggotaan: μ[x] = { (b − x) (b − a) ;a ≤ x ≤ b 0;x ≥ b b. representasi kurva segitiga kurva segitiga pada dasarnya merupakan gabungan antara dua garis (linear) seperti terlihat pada gambar 2.7. x rendah c 1 0 (x) a b fungsi keanggotaan: sistem pendukung keputusan metode sugeno dalam menentukan tingkat kepribadian siswa berdasarkan pendidikan (studi kasus di mi miftahul ulum gondanglegi malang) cauchy – issn: 2086-0382/e-issn: 2477-3344 50 μ[x] = { 0;x ≤ a atau x ≥ c (x − a) (b − a) ;a ≤ x ≤ b (c − x) (c − a) ;b ≤ x ≤ c c. representasi kurva bahu himpunan kabur bahu digunakan untuk mengakhiri variabel suatu daerah kabur. bentuk kurva bahu berbeda dengan kurva segitiga, yaitu salah satu sisi pada variabel tersebut mengalami perubahan turun atau naik, sedangkan sisi yang lain tidak mengalami perubahan atau tetap. bahu kiri bergerak dari benar ke salah, demikian juga bahu kanan bergerak dari salah ke benar. gambar 2.9 menunjukkan variabel temperatur dengan daerah bahunya. hangat1 0 panas 40 (x) normal1 0 (x) 1 0 (x) x 1 0 (x) sejukdingin 28 3. operasi himpunan kabur menurut wang [4] ada tiga operasi dasar dalam himpunan kabur yaitu komplemen, irisan (intersection), dan gabungan (union). a. komplemen operasi komplemen pada himpunan kabur adalah sebagai hasil operasi dengan operator not diperoleh dengan mengurangkan nilai keanggotaan elemen pada himpunan yang bersangkutan dari 1. ã ̅(x) = 1 − ã(x) b. irisan (intersection) operasi irisan (intersection) pada himpunan kabur adalah sebagai hasil operasi dengan operator and diperoleh dengan mengambil nilai keanggotaan terkecil antar elemen pada himpunan-himpunan yang bersangkutan. (ã ∩ b̃)(x) = min[ã(x),b̃(x)] c. gabungan operasi gabungan (union) pada himpunan kabur adalah sebagai hasil operasi dengan operator or diperoleh dengan mengambil nilai keanggotaan terbesar antar elemen pada himpunan-himpunan yang bersangkutan. (ã ∪ b̃)(x) = max[ã(x),b̃(x)] 4. fungsi implikasi kabur menurut wulandari [5] tiap-tiap aturan (proposisi) pada basis pengetahuan kabur akan berhubungan dengan suatu relasi kabur. bentuk umum dari aturan yang digunakan dalam fungsi implikasi adalah: jika x adalah a maka y adalah b dengan x dan y adalah skalar, dan a dan b adalah himpunan kabur. proposisi yang mengikuti jika disebut sebagai anteseden, sedangkan proporsi yang mengikuti maka disebut sebagai konsekuen. secara umum, ada dua fungsi implikasi yang dapat digunakan, yaitu: a. min (minimum) pengambilan keputusan dengan fungsi min, yaitu dengan cara mencari nilai minimum berdasarkan aturan ke-i dan dapat dinyatakan dengan: 𝛼𝑖 = 𝜇𝐴𝑖(𝑥)∩ 𝜇𝐵𝑖(𝑥) = 𝑚𝑖𝑛{𝜇𝐴𝑖(𝑥),𝜇𝐵𝑖(𝑥)} b. dot (product) pengambilan keputusan dengan fungsi dot yang didasarkan pada aturan ke-i dinyatakan dengan: αi . μci(z) 5. logika kabur dalam pengambilan keputusan salah satu aplikasi logika kabur yang telah berkembang amat luas dewasa ini adalah sistem inferensi kabur (fuzzy inference system/fis), yaitu kerangka komputasi yang didasarkan pada teori himpunan kabur, aturan kabur berbentuk jika-maka, dan penalaran kabur. misalnya dalam penentuan status gizi, produksi barang, sistem pendukung keputusan, penentuan kebutuhan kalori harian, dan sebagainya. ada tiga metode dalam sistem inferensi kabur yang sering digunakan, yaitu metode tsukamoto, metode mamdani, dan metode takagi sugeno. penalaran dengan metode sugeno hampir sama dengan penalaran mamdani, hanya saja output (konsekuen) sistem tidak berupa himpunan kabur, melainkan berupa konstanta atau persamaan linear. metode ini diperkenalkan oleh takagi sugeno kang pada tahun 1985, sehingga metode ini sering juga dinamakan dengan metode tsk. metode tsk terdiri dari 2 jenis, yaitu : a. model kabur sugeno orde-nol secara umum bentuk model kabur sugeno ordenol adalah : if(x1 is a1) ∩ (x2 is a2) ∩ (x3 is a3) ∩ … ∩ (xn is an)then z = k dengan an adalah himpunan kabur ke-n sebagai anteseden, dan k adalah suatu konstanta (tegas) sebagai konsekuen. wildan hakim 51 volume 4 no.1 november 2015 b. model kabur sugeno orde-satu secara umum bentuk model kabur sugeno orde-satu adalah if(x1 is a1) ∩ …∩ (xn is an)then z = p1∗x1 + ⋯+ pn∗xn + q dengan an adalah himpunan kabur ke-n sebagai anteseden, dan pn adalah suatu konstanta (tegas) sebagai konsekuen. dan q juga merupakan suatu konstanta (tegas) sebagai konsekuen. dalam metode sugeno untuk mendapatkan output, diperlukan 4 tahapan : a. pembentukan himpunan kabur pada metode sugeno, baik variabel input maupun variabel output dibagi menjadi satu atau lebih himpunan kabur. b. aplikasi fungsi implikasi pada metode sugeno, fungsi implikasi yang digunakan adalah min (minimum). c. komposisi aturan apabila sistem terdiri dari beberapa aturan, maka inferensi diperoleh dari gabungan antar aturan. ada tiga metode yang digunakan dalam melakukan inferensi sistem kabur, yaitu: max, additive dan probabilistik or (probor). d. penegasan (defuzzifikasi) input dari proses penegasan adalah suatu himpunan kabur yang diperoleh dari suatu komposisi aturan-aturan kabur, sedangkan output yang dihasilkan merupakan suatu bilangan pada himpunan kabur tersebut. sehingga jika diberikan suatu himpunan kabur dalam range tertentu, maka harus dapat diambil suatu nilai crisp tertentu sebagai output. dalam metode sugeno, penegasan dilakukan dengan cara mencari rata-rata terbobot (weight average) wa = ∑ αizi n i=1 ∑ αi n i=1 pembahasan dalam penentuan tingkat kepribadian berdasarkan pendidikan, sistem pendukung keputusan kabur digunakan untuk mengubah input yang berupa pendidikan nonformal (nonformal) dan pendidikan informal (informal) sehingga mendapatkan output berupa tingkat kepribadian. dalam penentuan tingkat kepribadian digunakan metode sugeno orde-nol, pada tahap pengujian input, pembentukan kombinasi aturan kabur harus disesuaikan dengan data output dengan menyertakan semua variabel dimana pada metode ini anteseden dipresentasikan dengan aturan dalam himpunan kabur, sedangkan konsekuen dipresentasikan dengan sebuah konstanta. selanjutnya dilakukan dengan mencari derajat keanggotaan nilai tiap variabel dari salah satu data. kemudian mencari α − predikat untuk setiap aturan kombinasi aturan kabur. dengan menggunakan rata-rata terbobot (weighted average), hasil α − predikat yang tidak nol digunakan untuk mencari nilai rata-rata yang juga merupakan penegasan (defuzzifikasi). berikut adalah beberapa tahap, yaitu: 1. pengaburan (fuzzifikasi) pengaburan yaitu proses dimana data inputan nilai yang bersifat tegas (crips input) kedalam input kabur. pada penelitian ini digunakan beberapa variabel yang digunakan dalam penentuan tingkat kepribadian dengan parameter pendidikan nonformal (nonformal) dan pendidikan informal (informal). pada metode sugeno, baik variabel input maupun variabel output dibagi menjadi satu atau lebih himpunan kabur. dalam penentuan tingkat kepribadian dengan parameter pendidikan nonformal (nonformal) dan pendidikan informal (informal), variabel input dibagi menjadi dua yaitu variabel pendidikan nonformal dan pendidikan informal. serta satu variabel output, yaitu variabel tingkat kepribadian. variabel ini dibentuk berdasarkan klasifikasi pendidikan nonformal (nonformal) dan pendidikan informal (informal). penentuan variabel yang digunakan dalam penelitian ini, terlihat pada tabel 4.1 tabel 4.3 semesta pembicaraan untuk setiap variabel kabur. dari variabel yang telah dimunculkan, kemudian disusun domain himpunan kabur. berdasarkan domain tersebut, selanjutnya ditentukan fungsi keanggotaan dari masingfungsi nama variabel semesta pembicaraan input pendidikan nonformal (nonformal) [8,19] pendidikan informal (informal) [11,22] output tingkat kepribadian [13,26] sistem pendukung keputusan metode sugeno dalam menentukan tingkat kepribadian siswa berdasarkan pendidikan (studi kasus di mi miftahul ulum gondanglegi malang) cauchy – issn: 2086-0382/e-issn: 2477-3344 52 masing variabel seperti terlihat pada tabel 4.2 berikut adalah perancangan himpunan kabur pada penentuan tingkat kepribadian dengan parameter pendidikan nonformal dan pendidikan informal: tabel 4.4 himpunan kabur himpunan kabur beserta fungsi keanggotaan dari variabel pendidikan nonformal dan pendidikan informal direpresentasikan sebagai berikut: a. himpunan kabur variabel pendidikan nonformal (nonformal) pada variabel nonformal didefinisikan tiga himpunan kabur yaitu rendah, sedang, dan tinggi. untuk merepresentasikan variabel nonformal digunakan bentuk kurva bahu kiri untuk himpunan kabur rendah, bentuk kurva segitiga untuk himpunan kabur sedang dan bentuk kurva bahu kanan untuk himpunan kabur tinggi. gambar himpunan kabur untuk variabel nonformal ditunjukkan pada gambar 4.1. rendah sedang x 17,513,59,5 1 0 tinggi (x) gambar 4.1 variabel nonformal dimana sumbu vertikal merupakan nilai input dari variabel nonformal, sedangkan sumbu horizontal merupakan tingkat keanggotaan dari nilai input. μrendah[x] = { 1;x ≤ 9,5 13,5 − x 4 ;9,5 ≤ x ≤ 13,5 0;x ≥ 13,5 μsedang[x] = { 0;x ≤ 9,5,x > 17,5 x − 9,5 4 ;9,5 ≤ x ≤ 13,5 17,5 − x 4 ;13,5 ≤ x ≤ 17,5 μtinggi[x] = { 1;x ≥ 17,5 x − 13,5 2 ;13,5 ≤ x ≤ 17,5 0;x ≤ 13,5 b. himpunan kabur variabel pendidikan informal (informal) pada variabel informal didefinisikan tiga himpunan kabur yaitu rendah, sedang, dan tinggi. untuk merepresentasikan variabel informal digunakan bentuk kurva bahu kiri untuk himpunan kabur rendah, bentuk kurva segitiga untuk himpunan kabur sedang, dan bentuk kurva bahu kanan untuk himpunan kabur tinggi. gambar himpunan kabur untuk variabel informal ditunjukkan pada gambar 4.2. rendah sedang x 20,516,512,5 1 0 tinggi (x) gambar 4.2 variabel informal dimana sumbu horizontal merupakan nilai input dari variabel informal, sedangkan sumbu vertikal merupakan tingkat keanggotaan dari nilai input. fungsi keanggotaan diperoleh dengan cara yang sama sebagaimana dalam variabel informal, sehingga menjadi sebagai berikut: μrendah[x] = { 1;x ≤ 12,5 16,5 − x 4 ;12,5 ≤ x ≤ 16,5 0;x ≥ 16,5 μsedang[x] = { 0;x ≤ 12,5x > 20,5 x − 12,5 4 ;12,5 ≤ x ≤ 16,5 20,5 − x 4 ;16,5 ≤ x ≤ 20,5 μtinggi[x] = { 1;x ≥ 20,5 x − 16,5 4 ;16,5 ≤ x ≤ 20,5 0;x ≤ 16,5 2. aplikasi fungsi implikasi berdasarkan kategori dalam penentuan himpunan kabur di atas diperoleh 9 aturan implikasi sebagai berikut: tabel 4.3 aturan implikasi dalam penentuan tingkat kepribadian ini menggunakan metode sugeno orde-nol, yaitu: if(x1 is a1) ∩ (x2 is a2)∩ …∩ (xn is an)then z = k keterangan xn = variabel input an = kategori variabel himpunan domain fungsi keanggotaan pendidikan nonformal (nonformal) rendah [0 − 13,5] bahu kiri sedang [9,5 − 17,5] segitiga tinggi [13,5 − 19] bahu kanan pendidikan informal (informal) rendah [0 − 16,5] bahu kiri sedang [12,5 − 20,5] segitiga tinggi [16,5 − 22] bahu kanan wildan hakim 53 volume 4 no.1 november 2015 z = { rendah → k1 = 15 sedang → k2 = 20 tinggi → k3 = 25 3. komposisi aturan komposisi aturan menggunakan fungsi max, yaitu solusi himpunan kabur diperoleh dengan cara mengambil nilai maksimum aturan. jika semua proposisi telah dievaluasi, maka output akan berisi suatu himpunan kabur yang merefleksikan konstribusi dari tiap-tiap proposisi. secara umum dapat dituliskan: usf[xi] = max(usf[xi],ukf[xi]) 4. penegasan (defuzzifikasi) penegasan yaitu mengkonversi himpunan kabur keluaran ke bentuk bilangan tegas (crips). dalam metode sugeno menggunakan metode perhitungan rata-rata terbobot (weight average). wa = ∑ αizi n i=1 ∑ αi n i=1 6. kasus penerapan sistem pendukung keputusan metode sugeno dalam menentukan tingkat kepribadian siswa masing-masing siswa dengan nilai pendidikan nonformal sebesar 12 dan nilai pendidikan informal sebesar 19 ingin mengetahui tingkat kepribadiannya langkah 1. menentukan himpunan kabur untuk variabel nonformal didefinisikan pada tiga himpuanan kabur yaitu: rendah, sedang, dan tinggi. setiap himpuanan kabur memiliki interval keanggotaan, yakni seperti terlihat pada g ambar berikut adalah gambar tingkat keanggotaan pada variabel nonformal sebesar 12: 0000 rendah sedang x 17,513,59,5 1 0 tinggi (x) gambar 4.3 variabel nonformal pendidikan nonformal sebesar 12 termasuk ke dalam himpunan kabur rendah dan sedang dengan tingkat keanggotaan sesuai fungsi, sehingga diperoleh sebagai berikut: 𝜇𝑅𝐸𝑁𝐷𝐴𝐻[𝑥] = { 1;𝑥 ≤ 9,5 13,5 − 𝑥 4 ;9,5 ≤ 𝑥 ≤ 13,5 0;𝑥 ≥ 13,5 𝜇𝑆𝐸𝐷𝐴𝑁𝐺[𝑥] = { 0;𝑥 ≤ 9,5,𝑥 > 17,5 𝑥 − 9,5 4 ;9,5 ≤ 𝑥 ≤ 13,5 17,5 − 𝑥 4 ;13,5 ≤ 𝑥 ≤ 17,5 sehingga diperoleh 𝜇𝑅𝐸𝑁𝐷𝐴𝐻[12] = 13,5 − 12 4 = 0,375 𝜇𝑆𝐸𝐷𝐴𝑁𝐺[12] = 12 − 9,5 4 = 0.625 𝜇𝑇𝐼𝑁𝐺𝐺𝐼[12] = 0 untuk informal didefinisikan pada tiga himpuanan kabur yaitu: rendah, sedang, dan tinggi. setiap himpuanan kabur memiliki interval keanggotaan, yakni seperti terlihat pada gambar berikut adalah gambar tingkat keanggotaan pada variabel informal sebesar 19: rendah sedang x 20,516,512,5 1 0 tinggi (x) gambar 4.4 variabel informal pendidikan informal sebesar 19 termasuk ke dalam himpunan kabur sedang dan tinggi dengan tingkat keanggotaan sesuai fungsi keanggotaan, sehingga diperoleh sebagai berikut: no pendidikan nonformal pendidikan informal tingkat kepribadian 1 jika rendah dan rendah maka rendah 2 jika rendah dan sedang maka sedang 3 jika rendah dan tinggi maka rendah 4 jika sedang dan rendah maka sedang 5 jika sedang dan sedang maka tinggi 6 jika sedang dan tinggi maka sedang 7 jika tinggi dan rendah maka rendah 8 jika tinggi dan sedang maka sedang 9 jika tinggi dan tinggi maka tinggi sistem pendukung keputusan metode sugeno dalam menentukan tingkat kepribadian siswa berdasarkan pendidikan (studi kasus di mi miftahul ulum gondanglegi malang) cauchy – issn: 2086-0382/e-issn: 2477-3344 54 𝜇𝑆𝐸𝐷𝐴𝑁𝐺[𝑥] = { 0;𝑥 ≤ 12,5𝑥 > 20,5 𝑥 − 12,5 4 ;12,5 ≤ 𝑥 ≤ 16,5 20,5 − 𝑥 4 ;16,5 ≤ 𝑥 ≤ 20,5 𝜇𝑇𝐼𝑁𝐺𝐺𝐼[𝑥] = { 1;𝑥 ≥ 20,5 𝑥 − 16,5 4 ;16,5 ≤ 𝑥 ≤ 20,5 0;𝑥 ≤ 16,5 sehingga diperoleh 𝜇𝑅𝐸𝑁𝐷𝐴𝐻[19] = 0 𝜇𝑆𝐸𝐷𝐴𝑁𝐺[19] = 20,5 − 19 4 = 0,375 𝜇𝑇𝐼𝑁𝐺𝐺𝐼[19] = 19 − 16,5 4 = 0,625 langkah 2. aplikasi fungsi implikasi fungsi implikasi yang digunakan dalam proses ini adalah fungsi min (minimum), yaitu dengan mengambil tingkat keanggotaan yang minimum dari variabel input sebagai outputnya. dari dua data kabur input tersebut variabel nonformal: rendah (0,375) dan sedang (0,625) dan variabel informal: sedang (0,375) dan tinggi (0,625). berdasarkan 9 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 yang disebutkan pada tabel fungsi implikasi, maka diperoleh hanya 2 𝛼 − 𝑝𝑟𝑒𝑑𝑖𝑘𝑎𝑡 yang tidak nol, yaitu: [r2] jika nonformal adalah rendah dan informal adalah sedang, maka tingkat kepribadiannya adalah sedang. 𝛼2 = 𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑅𝐸𝑁𝐷𝐴𝐻 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺 = 𝑚𝑖𝑛(𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙 _𝑅𝐸𝑁𝐷𝐴𝐻 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺) = 𝑚𝑖𝑛 (0,375, 0,375) = 0,375 [r3] jika nonformal adalah rendah dan informal adalah tinggi, maka tingkat kepribadiannya adalah rendah. 𝛼3 = 𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑅𝐸𝑁𝐷𝐴𝐻 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑇𝐼𝑁𝐺𝐺𝐼 = 𝑚𝑖𝑛(𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙 _𝑅𝐸𝑁𝐷𝐴𝐻 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑇𝐼𝑁𝐺𝐺𝐼) = 𝑚𝑖𝑛 (0,375, 0,625) = 0,375 [r5] jika nonformal adalah sedang dan informal adalah sedang, maka tingkat kepribadiannya adalah tinggi. 𝛼5 = 𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺 = 𝑚𝑖𝑛(𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙 _𝑆𝐸𝐷𝐴𝑁𝐺 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺) = 𝑚𝑖𝑛 (0,625, 0,375) = 0,625 [r6] jika nonformal adalah sedang dan informal adalah tinggi, maka tingkat kepribadiannya adalah sedang. 𝛼6 = 𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑆𝐸𝐷𝐴𝑁𝐺 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑇𝐼𝑁𝐺𝐺𝐼 = 𝑚𝑖𝑛(𝜇𝑛𝑜𝑛𝑓𝑜𝑟𝑚𝑎𝑙 _𝑆𝐸𝐷𝐴𝑁𝐺 ∩ 𝜇𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑙_𝑇𝐼𝑁𝐺𝐺𝐼) = 𝑚𝑖𝑛 (0,625, 0,625) = 0,625 langkah 3. komposisi aturan komposisi aturan menggunakan fungsi max, komposisi aturan merupakan kesimpulan secara keseluruhan dengan mengambil tingkat keanggotaan maksimum dari tiap konsekuen aplikasi fungsi implikasi dan menggabungkan dari semua kesimpulan masing-masing aturan, sehingga diperoleh daerah solusi kabur sebagai berikut: 𝑈𝑠𝑓[𝑥𝑖] = 𝑚𝑎𝑥(𝑈𝑠𝑓[𝑥𝑖],𝑈𝑘𝑓[𝑥𝑖]) keterangan: 𝑈𝑠𝑓[𝑥𝑖] = nilai keanggotan solusi kabur sampai aturan ke-i 𝑈𝑘𝑓[𝑥𝑖] = nilai keanggotan solusi kabur sampai aturan ke-i sehingga diperoleh tingkat kepribadian: 1. rendah = 𝑚𝑎𝑥 (0,375) = 0,375 2. sedang = 𝑚𝑎𝑥 (0,375, 0,625) = 0,625 3. tinggi = 𝑚𝑎𝑥 (0,625) = 0,625 langkah 4. penegasan (defuzzifikasi) pada metode sugeno penegasan menggunakan perhitungan rata-rata terbobot (weight-average) : 𝑊𝐴 = ∑ 𝛼𝑖𝑧𝑖 𝑁 𝑖=1 ∑ 𝛼𝑖 𝑁 𝑖=1 = 0,375(15)+ 0,625(20) + 0,625(25) 0,375 + 0,625+ 0,625 = 20,7 jadi dengan menggunakan metode sugeno, m asing m asing siswa dengan pendidikan no nf o rm al s e b e s ar 12 da n p end idik a n i nfo rm al s e b e s ar 19 mempunyai nilai kepribadian 20,7 dengan variabel linguistiknya adalah sedang. kesimpulan berdasarkan pembahasan penentuan tingkat kepribadian siswa berdasarkan pendidikan dengan menggunakan metode sugeno dalam logika kabur, dapat diambil kesimpulan bahwa : 1. dalam perhitungan tingkat kepribadian siswa ini menggunakan metode sugeno orde-nol dalam logika kabur terdiri dari empat tahap, a. pengaburan (fuzzifikasi) yaitu pembentukan himpunan kabur variabel pendidikan nonformal dan pendidikan informal masingmasing dibagi menjadi tiga himpunan kabur yaitu variabel rendah, variabel sedang, dan varibel tinggi. fungsi keanggotaan yang digunakan adalah representasi segitiga dan repsentasi kurva bahu. b. aplikasi fungsi implikasi, fungsi implikasi yang digunakan adalah fungsi min yaitu masing-masing anteseden (proposisi yang mengikuti if) dicari nilai minimum berdasarkan aturan-aturan kabur. berdasarkan variabel linguistik dalam wildan hakim 55 volume 4 no.1 november 2015 penentuan himpunan kabur diperoleh 9 aturan. c. komposisi aturan, metode yang diunakan yaitu metode max yaitu solusi himpunan kabur diperoleh dengan cara mengambil nilai maksimum aturan, kemudian menggunakannya untuk memodifikasi daerah kabur, dan mengaplikasikannya ke output dengan menggunakan operator or (union). jika semua proposisi telah dievaluasi, maka output akan berisi suatu himpunan kabur yang merefleksikan konstribusi dari tiap-tiap proposisi. d. penegasan (defuzzifikasi) merupakan suatu himpunan kabur yang diperoleh dari suatu komposisi aturan-aturan kabur, sedangkan output yang dihasilkan merupakan suatu bilangan pada himpunan kabur tersebut. sehingga jika diberikan suatu himpunan kabur dalam range tertentu, maka harus dapat diambil suatu nilai crisp tertentu sebagai output. defuzzifikasi dilakukan dengan cara mencari rata-rata terbobot (weight average). 2. berdasarkan kasus, masing-masing siswa dengan pendidikan nonformal sebesar 12 dan pendidikan nonformal sebesar sebesar 19 mempunyai nilai kepribadian 20,7 dengan variabel linguistik sedang. daftar pustaka [1] f. susilo, himpunan dan logika kabur serta aplikasinya, yogyakarta: graha ilmu, 2006. [2] s. kusumadewi and h. purnomo, aplikasi logika fuzzy untuk pendukung keputusan, yogyakarta: graha ilmu, 2004. [3] g. j. k. a. b. yuan, fuzzy set theory foundations and aplications, new jersey: prentice hall international, inc, 1997. [4] l. x. wang, a course in fuzzy system and control, new jersey: prentice hall international, inc, 1997. [5] y. wulandari, aplikasi mamdani dalam penentuan status gizi dengan indeks massa (imt) menggunakan logika fuzzy, yogyakarta: tidak diterbitkan, 2011. pendahuluan kajian teori pembahasan kesimpulan daftar pustaka keterkaitan antara modul bebas dengan modul dilihat dari sifat-sifat homomorfisme modul khusnul afifa1, abdussakir2 1mahasiswa jurusan matematika uin maulana malik ibrahim malang 2dosen jurusan matematika uin maulana malik ibrahim malang e-mail: apheecute@gmail.com, abdussakir@gmail.com abstrak dalam artikel ini akan dibahas tentang cara untuk mengetahui suatu 𝑅-modul adalah modul bebas atau bukan dengan memanfaatkan suatu modul bebas sebagai 𝑅-modul melalui media homomorfisma modul. penelitian ini menggunakan metode kajian kepustakaan (library research), yaitu melakukan penelitian untuk memperoleh data-data dan informasi serta objek yang digunakan dalam pembahasan masalah tersebut. berdasarkan pembahasan dapat diperoleh bahwa suatu 𝑅-modul merupakan modul bebas jika 𝑅-modul tersebut isomorfik dengan suatu modul bebas sebagai 𝑅-modul. artinya, suatu 𝑅modul merupakan modul bebas jika terdapat suatu isomorfisma dari 𝑅-modul tersebut ke suatu modul bebas yang juga merupakan suatu 𝑅-modul. lebih jauh lagi, jika suatu 𝑅-modul adalah modul bebas, maka 𝑅-modul tersebut isomorfik dengan 𝑅𝑛 , dimana 𝑛 adalah kardinalitas dari basis bagi 𝑅-modul tersebut. kata kunci : basis, homomorfisme modul, modul, modul bebas, ring abstract in this paper, the way to know whether 𝑅-module is free module or not will be discussed by using free module as 𝑅-module through module homomorphism media. in this research, the author used library research, which is conducting the research to obtain data and information about object that used in the discussion. based on the discussion, it was obtained that 𝑅-module was free module if the 𝑅-module was isomorphic to free module as 𝑅-module. it means that 𝑅-module was free module if there is isomorphism from the 𝑅-module to the free module which is an 𝑅-module. furthermore, if an 𝑅-module was free module, then the 𝑅-module would be isomorphic with 𝑅𝑛 , where 𝑛 is cardinality from basis for the 𝑅module. keywords: basis, free module, module, module homomorphism, ring. pendahuluan struktur aljabar yang dikembangkan dalam dua himpunan yang tidak kosong dengan dua operasi biner dan memenuhi syarat tertentu, yaitu distributif kanan, distributif kiri, assosiatif, dan mempunyai elemen identitas disebut dengan modul [1]. modul sendiri juga dapat dikembangkan menjadi beberapa sub pembahasan di antaranya adalah homomorfisme. seperti halnya ring di dalamnya dibahas mengenai homomorfisme ring, maka di dalam modul juga dibahas mengenai homomorfisme modul. homomorfisme modul merupakan suatu pemetaan dari suatu modul 𝑀 ke modul 𝑁 yang mengawetkan kedua operasi yang ada dalam modul. homomorfisme modul dibedakan menjadi 3, yaitu homomorfisme yang merupakan pemetaan satu-satu (one to one/ injektif) disebut monomorfisme modul, homomorfisme yang merupakan pemetaan pada (onto/surjektif) disebut epimorfisme modul, dan homomorfisme yang mempunyai sifat keduaduanya (injektif dan surjektif) atau yang dikenal dengan istilah bijektif disebut isomorfisme modul [2]. suatu modul yang memiliki basis atau himpunan pembangun disebut modul bebas. jika 𝑀 adalah 𝑅-modul dan terdapat 𝑋 ⊆ 𝑀 dengan 𝑋 merupakan basis untuk 𝑀, maka 𝑀 disebut modul bebas [3]. penelitian ini dilakukan untuk mengetahui suatu 𝑅-modul adalah modul bebas atau bukan dengan memanfaatkan suatu modul bebas sebagai 𝑅-modul melalui media homomorfisme modul. mailto:apheecute@gmail.com mailto:apheecute@gmail.com khusnul afifa 153 volume 3 no. 3 november 2014 teori dasar 1. fungsi suatu fungsi dari himpunan s ke t adalah aturan yang mengaitkan setiap anggota s dengan tepat satu anggota t. anggota s disebut domain dari fungsi, dan himpunan t disebut kodomain [4]. 2. operasi biner operasi + pada suatu himpunan tidak kosong 𝐺 adalah biner jika dan hanya jika 𝑎 ∈ 𝐺, 𝑏 ∈ 𝐺 maka 𝑎 + 𝑏 ∈ 𝐺, ∀𝑎, 𝑏 ∈ 𝐺. sifat tersebut dari operasi di 𝐺 dikatakan tertutup dan jika sifat ini memenuhi operasi + di 𝐺 [5]. 3. grup misalkan 𝐺 adalah suatu himpunan tak kosong dan pada 𝐺 didefinisikan operasi biner +. sistem aljabar (𝐺, +) disebut grup jika memenuhi aksioma-aksioma: a. operasi + bersifat assosiatif di 𝐺 (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐), ∀𝑎, 𝑏, 𝑐 ∈ 𝐺 b. 𝐺 mempunyai unsur identitas terhadap operasi + misalkan 𝑒 unsur di 𝐺 sedemikian hingga 𝑎 + 𝑒 = 𝑒 + 𝑎 , ∀𝑎 ∈ 𝐺 maka 𝑒 disebut unsur identitas. c. setiap unsur di 𝐺 mempunyai invers terhadap operasi + untuk setiap 𝑎 ∈ 𝐺 ada a−1∈ 𝐺 yang disebut sebagai invers dari a, sehingga 𝑎−1 + 𝑎 = 𝑎 + 𝑎−1 = 𝑒, dimana e adalah unsur identitas di 𝐺 [5]. grup (𝐺, +) dikatakan grup komutatif jika untuk setiap unsur 𝑎 dan 𝑏 di 𝐺 berlaku 𝑎 + 𝑏 = 𝑏 + 𝑎 [6]. 4. ring 𝑅 adalah himpunan tak kosong dengan dua operasi biner yang dilambangkan dengan + dan × (penjumlahan pada operasi pertama dan perkalian pada operasi kedua) disebut ring jika memenuhi syarat-syarat sebagai berikut: i. (𝑅, +) adalah grup komutatif ii. operasi × bersifat asssosiatif (𝑎 × 𝑏) × 𝑐 = 𝑎 × (𝑏 × 𝑐), ∀𝑎, 𝑏, 𝑐 ∈ 𝑅 iii. operasi × bersifat distributif terhadap + di 𝑅, ∀𝑎, 𝑏, 𝑐 ∈ 𝑅 (𝑎 + 𝑏) × 𝑐 = (𝑎 × 𝑐) + (𝑏 × 𝑐) (distributif kanan) 𝑎(𝑏 + 𝑐) = (𝑎 × 𝑏) + (𝑎 × 𝑐) (distributif kiri) [7]. misalkan 𝑅 dan 𝑆 adalah ring. homomorfisme ring adalah pemetaan 𝜑: 𝑅 → 𝑆 jika memenuhi syarat-syarat berikut: i. 𝜑(𝑎 + 𝑏) = 𝜑(𝑎) + 𝜑(𝑏), ∀𝑎, 𝑏 ∈ 𝑅 ii. 𝜑(𝑎𝑏) = 𝜑(𝑎)𝜑(𝑏), ∀𝑎, 𝑏 ∈ 𝑅 [7]. 5. modul misalkan (𝑅, +,×) adalah ring. 𝑅-modul di 𝑅 adalah himpunan 𝑀 yang memenuhi: i. (𝑀, +) adalah grup komutatif ii. diberikan pemetaan 𝑅 × 𝑀 → 𝑀, dimana 𝑟𝑚, ∀𝑟 ∈ 𝑅, 𝑚 ∈ 𝑀 yang memenuhi: a. distributif kanan (𝑟 + 𝑠)𝑚 = 𝑟𝑚 + 𝑠𝑚, ∀𝑟, 𝑠 ∈ 𝑅, 𝑚 ∈ 𝑀 b. distributif kiri 𝑟(𝑚 + 𝑛) = 𝑟𝑚 + 𝑟𝑛, ∀𝑟 ∈ 𝑅, 𝑚, 𝑛 ∈ 𝑀 c. assosiatif (𝑟𝑠)𝑚 = 𝑟(𝑠𝑚), ∀𝑟, 𝑠 ∈ 𝑅, 𝑚 ∈ 𝑀 jika r mempunyai unsur identitas 1 maka d. 1𝑚 = 𝑚, ∀𝑚 ∈ 𝑀 [7]. misal 𝑅 adalah ring dan 𝑀 adalah 𝑅modul. 𝑅-submodul di 𝑅 adalah 𝑁 subgrup dari 𝑀 yang bersifat tertutup terhadap elemenelemen ring, yaitu 𝑟𝑛 ∈ 𝑁, ∀𝑟 ∈ 𝑅, 𝑛 ∈ 𝑁 [7]. teorema 1 misalkan 𝑅 adalah ring dan 𝑀 adalah 𝑅modul. subset 𝑁 di 𝑀 adalah submodul di 𝑀 jika dan hanya jika: a. 𝑁 ≠ ∅ b. 𝑥 + 𝛼𝑦 ∈ 𝑁, ∀𝛼 ∈ 𝑅, ∀ 𝑥, 𝑦 ∈ 𝑁 [7]. bukti a. jika 𝑁 adalah submodul di 𝑀 maka 0 ∈ 𝑁 jadi 𝑁 ≠ ∅. b. 𝑁 bersifat tertutup terhadap operasi penjumlahan misal 𝛼 = −1, maka 𝑥 + (−1)𝑦 = 𝑥 + (−𝑦) = 𝑥 − 𝑦 ∈ 𝑁 maka 𝑥 + 𝛼𝑦 ∈ 𝑁, ∀𝛼 ∈ 𝑅, ∀𝑥, 𝑦 ∈ 𝑁. misalkan 𝑅 adalah ring dan misalkan 𝑀 dan 𝑁 adalah 𝑅-modul. pemetaan 𝜑: 𝑀 → 𝑁 disebut homomorfisme modul jika pemetaan itu memenuhi syarat sebagai berikut: a) 𝜑(𝑥 + 𝑦) = 𝜑(𝑥) + 𝜑(𝑦), ∀𝑥, 𝑦 ∈ 𝑀 b) 𝜑(𝛼𝑥) = 𝛼𝜑(𝑥), ∀𝛼 ∈ 𝑅, 𝑥 ∈ 𝑀 [7]. 6. modul bebas diketahui 𝑀 adalah 𝑅-modul. jika terdapat 𝑋 ⊆ 𝑀 dengan 𝑋 merupakan basis untuk 𝑀, maka 𝑀 disebut modul bebas [3]. misalkan 𝑀 suatu r-modul dan 𝑋 ⊆ 𝑀. unsur 𝑦 ∈ 𝑀 dikatakan kombinasi linier dari 𝑋 jika untuk semua 𝑥 ∈ 𝑋 dapat diungkapkan dalam bentuk 𝑦 = 𝛼1𝑥1 + 𝛼2𝑥2 + ⋯ + 𝛼𝑛𝑥𝑛 dimana 𝛼1, 𝛼2, … , 𝛼𝑛 adalah skalar [8]. misalkan 𝑀 suatu r-modul dan 𝑋 ⊆ 𝑀. jika untuk semua 𝑥 ∈ 𝑋 dapat dinyatakan keterkaitan antara modul bebas dengan modul dilihat dari sifat-sifat homomorfisme modul cauchy – issn: 2086-0382 154 sebagai kombinasi linier maka dikatakan bahwa 𝑋 merentang 𝑀 [8]. misalkan 𝑀 suatu r-modul dan 𝑋 ⊆ 𝑀. maka persamaan 𝛼1𝑥1 + 𝛼2𝑥2 + ⋯ + 𝛼𝑛𝑥𝑛 = 0 mempunyai paling sedikit satu pemecahan, yakni 𝛼1 = 0, 𝛼2 = 0, … , 𝛼𝑛 = 0 jika ini adalah satu-satunya pemecahan, maka 𝑋 dinamakan himpunan bebas linier. jika ada pemecahan lain, maka 𝑆 dinamakan himpunan tak bebas linier [8]. diketahui 𝑀 adalah 𝑅-modul dan 𝑋 ⊆ 𝑀. himpunan 𝑋 dikatakan basis untuk 𝑀 jika dan hanya jika: a. 𝑋 merentang 𝑀 b. 𝑋 bebas linier [8]. metode penelitian metode yang digunakan dalam penelitian ini adalah metode penelitian kepustakaan (library research). adapun langkah-langkah yang akan digunakan oleh peneliti dalam membahas penelitian ini adalah: 1. mendefinisikan kembali tentang modul, modul bebas, dan homomorfisme modul serta membuat contohnya dan contoh yang salah 2. mendefinisikan monomorfisme modul, epimorfisme modul, dan isomorfisme modul 3. membuat contoh dan contoh yang salah dari monomorfisme modul, epimorfisme modul, dan isomorfisme modul dengan menggunakan domain modul bebas dan kodomainnya modul 4. dari poin 3 didapatkan dua teorema baru dan membuktikan teorema tersebut serta memberikan contohnya. pembahasan homomorfisme modul merupakan pemetaan dari suatu modul ke modul yang lain yang mengawetkan kedua operasi yang ada dalam modul tersebut. misalkan 𝑅 adalah ring dan misalkan 𝑀 dan 𝑁 adalah 𝑅-modul. pemetaan 𝜑: 𝑀 → 𝑁 disebut homomorfisme modul jika pemetaan itu memenuhi syarat sebagai berikut: a) 𝜑(𝑥 + 𝑦) = 𝜑(𝑥) + 𝜑(𝑦), ∀𝑥, 𝑦 ∈ 𝑀 b) 𝜑(𝛼𝑥) = 𝛼𝜑(𝑥), ∀𝛼 ∈ 𝑅, 𝑥 ∈ 𝑀 [7]. contoh: diberikan 𝑅 dan 𝑅2 sebagai 𝑅-modul. melalui pemetaan 𝜑: 𝑅2 → 𝑅 yang didefinisikan dengan 𝜑(𝑥) = 0𝑅 . akan ditunjukkan 𝜑 adalah homomorfisme modul. berdasarkan definisi homomorfisme modul, 𝜑 dikatakan homomorfisme modul jika memenuhi sifat-sifat berikut: i. 𝜑(𝑥 + 𝑦) = 𝜑(𝑥) + 𝜑(𝑦) 𝜑(𝑥 + 𝑦) = 0𝑅 ( berdasarkan definisi 𝜑, karena 𝑥 + 𝑦 ∈ 𝑅2 ) dilain pihak, 𝜑(𝑥) + 𝜑(𝑦) = 0𝑅 + 0𝑅 = 0𝑅 jadi 𝜑(𝑥 + 𝑦) = 𝜑(𝑥) + 𝜑(𝑦) ii. 𝜑(𝛼𝑥) = 𝛼𝜑(𝑥) 𝜑(𝛼𝑦) = 0𝑅 ( berdasarkan definisi 𝜑, karena 𝛼𝑦 ∈ 𝑅2 ) dilain pihak, 𝛼𝜑(𝑦) = 𝛼0𝑅 = 0𝑅 𝜑(𝛼𝑥) = 𝛼𝜑(𝑥) jadi 𝜑 terbukti homomorfisme modul. secara garis besar, homomorfisme modul dibedakan menjadi tiga, yaitu monomorfisme, epimorfisme, dan isomorfisme. misalkan 𝑅 adalah ring, 𝑀 dan 𝑁 adalah 𝑅 -modul, jika homomorfisme modul dari 𝑀 ke 𝑁 bersifat injektif (satu-satu) maka disebut monomorfisme modul [7]. contoh: diberikan 𝑍 dan 𝑍 × 𝑍2 sebagai 𝑍-modul. pemetaan 𝜑 ∶ 𝑍 → 𝑍 × 𝑍2 didefinisikan sebagai 𝜑(𝑥) = (𝑥, 0). pemetaan 𝜑 ini adalah pemetaan monomorfisme. ambil sebarang 𝑥, 𝑦 ∈ 𝑍 dan 𝛼 ∈ 𝑍, maka 1. 𝜑(𝑥 + 𝑦) = (𝑥 + 𝑦, 0) = (𝑥, 0) + (𝑦, 0) = 𝜑(𝑥) + 𝜑(𝑦) 2. 𝜑(𝛼𝑥) = (𝛼𝑥, 0) = (𝛼𝑥, 𝛼0) = 𝛼(𝑥, 0) = 𝛼𝜑(𝑥) dari 1 dan 2, dapat disimpulkan bahwa 𝜑 adalah suatu homomorfisme modul. kemudian, untuk sebarang 𝜑(𝑥), 𝜑(𝑦) ∈ 𝑅𝜑 dengan 𝜑(𝑥) = 𝜑(𝑦), berlaku 𝟎 = 𝜑(𝑥) − 𝜑(𝑦) = (𝑥, 0) − (𝑦, 0) = (𝑥 − 𝑦, 0) oleh karena itu, 𝑥 − 𝑦 = 0. dengan kata lain 𝑥 = 𝑦. jadi 𝜑 adalah pemetaan 1-1. dilain pihak, terdapat (1,1) ∈ 𝑍𝑥𝑍2 sehingga (1,1) ≠ 𝜑(𝑥) untuk setiap 𝑥 ∈ 𝑍. jadi 𝜑 bukan pemetaan pada. jadi dapat disimpulkan bahwa 𝜑 adalah monomorfisme modul. misal 𝑅 adalah ring, 𝑀 dan 𝑁 adalah 𝑅modul, jika homomorfisme modul dari 𝑀 ke 𝑁 bersifat surjektif (pada/onto), maka disebut epimorfisme modul [7]. contoh: diberikan 𝑍 dan 𝑍2 sebagai 𝑍-modul. pemetaan 𝜑 ∶ 𝑍 → 𝑍2 didefinisikan sebagai 𝜑(𝑥) = 𝑥 (𝑚𝑜𝑑2). pemetaan 𝜑 ini adalah pemetaan epimorfisme. ambil sebarang 𝑥, 𝑦 ∈ 𝑍 dan 𝛼 ∈ 𝑍, maka 1. 𝜑(𝑥 + 𝑦) = 𝑥 + 𝑦 (𝑚𝑜𝑑2) = 𝑥 (𝑚𝑜𝑑2) + 𝑦 (𝑚𝑜𝑑2) = 𝜑(𝑥) + 𝜑(𝑦) khusnul afifa 155 volume 3 no. 3 november 2014 2. 𝜑(𝛼𝑥) = 𝛼𝑥 (𝑚𝑜𝑑2) = 𝛼(𝑥 (𝑚𝑜𝑑2)) = 𝛼𝜑(𝑥) dari 1 dan 2, dapat disimpulkan bahwa 𝜑 adalah suatu homomorfisme modul. kemudian, untuk sebarang 𝑧 ∈ 𝑍2, maka i) untuk 𝑧 = 0, maka terdapat 0 ∈ 𝑍 sehingga 𝜑(0) = 0 ii) untuk 𝑧 = 1, maka terdapat 3 ∈ 𝑍 sehingga 𝜑(3) = 1 jadi 𝜑 adalah pemetaan pada. dilain pihak, terdapat 1,3 ∈ 𝑍 dengan 1 ≠ 3, tetapi 𝜑(3) = 1 = 𝜑(1). jadi 𝜑 bukan pemetaan 1-1. jadi dapat disimpulkan bahwa 𝜑 adalah epimorfisme modul. misalkan 𝑅 adalah ring, 𝑀 dan 𝑁 adalah 𝑅-modul, jika homomorfisme modul dari 𝑀 ke 𝑁 bersifat bijektif (satu-satu) dan surjektif (pada), dengan kata lain homomorfisme modul dari 𝑀 ke 𝑁 bersifat bijektif, maka disebut isomorfisme modul. jika terdapat suatu isomorfisme dari 𝑀 ke 𝑁, maka 𝑀 isomorfik dengan 𝑁 atau 𝑁 isomorfik dengan 𝑀 [7]. contoh: diberikan 𝑍4 dan 𝑀 = {( 𝑎11 𝑎12 𝑎21 𝑎22 ) |𝑎11, 𝑎12, 𝑎21, 𝑎22 ∈ 𝑍} adalah 𝑍modul. pemetaan 𝜑 ∶ 𝑀 → 𝑍4 didefinisikan sebagai 𝜑(𝑎) = (𝑎11, 𝑎12, 𝑎21, 𝑎22) dimana 𝑎 = ( 𝑎11 𝑎12 𝑎21 𝑎22 ). pemetaan 𝜑 ini adalah pemetaan isomorfisme. ambil sebarang 𝑥, 𝑦 ∈ 𝑀 dan 𝛼 ∈ ℤ, maka i) 𝜑(𝑥 + 𝑦) = ( 𝑥11 + 𝑦11 𝑥12 + 𝑦12 𝑥21 + 𝑦21 𝑥22 + 𝑦22 ) = (𝑥11 + 𝑦11 , 𝑥12 + 𝑦12, 𝑥21 + 𝑦21, 𝑥22 + 𝑦22) = (𝑥11, 𝑥12, 𝑥21, 𝑥22) + (𝑦11, 𝑦12, 𝑦21 , 𝑦22) = 𝜑(𝑥) + 𝜑(𝑦) ii) 𝜑(𝛼𝑥) = 𝜑 (( 𝛼𝑥11 𝛼𝑥12 𝛼𝑥21 𝛼𝑥22 )) = (𝛼𝑥11, 𝛼𝑥12, 𝛼𝑥21, 𝛼𝑥22) = 𝛼(𝑥11, 𝑥12, 𝑥21 , 𝑥22) = 𝛼𝜑(𝑥) dari (i) dan (ii), dapat disimpulkan bahwa 𝜑 adalah suatu homomorfisme modul. kemudian untuk sebarang (𝑚11, 𝑚12, 𝑚21, 𝑚22) ∈ 𝑍, maka terdapat 𝑚 = ( 𝑚11 𝑚12 𝑚21 𝑚22 ) anggota 𝑀 sehingga 𝜑(𝑚) = (𝑚11, 𝑚12, 𝑚21, 𝑚22). jadi 𝜑 adalah pemetaan pada. di lain pihak, untuk sebarang 𝜑(𝑥), 𝜑(𝑦) ∈ 𝑅𝜑 dengan 𝜑(𝑥) = 𝜑(𝑦), berlaku 𝟎 = 𝜑(𝑥) − 𝜑(𝑦) = (𝑥11, 𝑥12, 𝑥21, 𝑥22) − (𝑦11, 𝑦12, 𝑦21, 𝑦22) = (𝑥11 − 𝑦11, 𝑥12 − 𝑦12, 𝑥21 − 𝑦21, 𝑥22 − 𝑦22) oleh karena itu, 𝑥𝑖𝑗 − 𝑦𝑖𝑗 = 0 untuk setiap 𝑖, 𝑗 = 1,2. dengan kata lain 𝑥𝑖𝑗 = 𝑦𝑖𝑗 untuk setiap 𝑖, 𝑗 = 1,2. sehingga diperoleh 𝑥 = 𝑦. jadi 𝜑 adalah pemetaan 1-1. jadi dapat disimpulkan bahwa 𝜑 adalah isomorfisma modul. pada contoh monomorfisme dan epimorfisme, domainnya adalah 𝑍 sebagai 𝑍modul. telah diketahui bahwa 𝑍 sebagai 𝑍modul adalah modul bebas dengan basis {1}. pada contoh epimorfisme, mudah diketahui bahwa 𝑍2 sebagai 𝑍-modul bukan modul bebas disebabkan satu-satunya pembangun di 𝑍2, yaitu {1} tak bebas linear, karena terdapat 2 ∈ 𝑍 dimana 2 ≠ 0, berlaku 2 ∙ 1 = 0 di 𝑍2. jadi epimorfisme bukan jaminan untuk kodomain merupakan modul bebas saat domain modul bebas. selanjutnya, pada contoh monomorfisme, 𝑍 × 𝑍2 juga bukan modul bebas ( hal ini akan dibuktikan dengan teorema terakhir pada bab ini ). jadi monomorfisme bukan jaminan untuk kodomain merupakan modul bebas saat domain modul bebas. terakhir, pada contoh ketiga, yaitu isomorfisme, 𝑍4 sebagai 𝑍-modul adalah modul bebas dengan basis {(1,1,1,1)}. begitu pula dengan 𝑀 sebagai 𝑍-modul juga merupakan modul bebas dengan basis {( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 1 0 ) , ( 0 0 0 1 )}. dari contoh isomorfisme ini, ada kemungkinan bahwa isomorfisme bisa jadi jaminan untuk kodomain modul bebas saat domain adalah modul bebas. hal ini dijawab oleh teorema berikut. teorema 2 misalkan 𝑀 dan 𝐹 adalah 𝑅-modul. jika 𝑀 adalah modul bebas dan 𝑀 isomorfik dengan 𝐹, maka 𝐹 modul bebas. bukti misalkan 𝑋 = {𝑥1 , 𝑥2, … , 𝑥𝑛 } adalah basis untuk 𝑀 dan 𝜑 adalah isomorfisme dari 𝑀 ke 𝐹. selanjutnya akan ditunjukkan bahwa 𝑉 = {𝜑(𝑥1), 𝜑(𝑥1), … , 𝜑(𝑥𝑛 )} adalah basis bagi 𝐹. i) 𝑉 membangun 𝐹 ambil 𝑦 ∈ 𝐹. karena 𝜑 pemetaan pada, maka terdapat 𝑚 ∈ 𝑀 sehingga 𝜑(𝑚) = 𝑦. karena 𝑀 modul bebas, maka terdapat 𝑟1, 𝑟2, … , 𝑟𝑛 ∈ 𝑅 sehingga 𝑚 = ∑ 𝑟𝑖 𝑥𝑖 = 𝑟1𝑥1 + 𝑟2𝑥2 + 𝑟3𝑥3 + ⋯ + 𝑟𝑛 𝑥𝑛 𝑛 𝑖=1 selain itu, karena 𝜑 suatu homomorfisme, maka 𝜑(𝑚) = 𝜑(∑ 𝑟𝑖 𝑥𝑖 𝑛 𝑖=1 ) = 𝜑(𝑟1𝑥1 + 𝑟2𝑥2 + ⋯ + 𝑟𝑛 𝑥𝑛 ) = 𝜑(𝑟1𝑥1) + 𝜑(𝑟2𝑥2) + ⋯ + 𝜑(𝑟𝑛 𝑥𝑛 ) = 𝑟1𝜑(𝑥1) + 𝑟2𝜑(𝑥2) + ⋯ + 𝑟𝑛 𝜑(𝑥𝑛 ) = ∑ 𝑟𝑖 𝜑(𝑥𝑖 ) 𝑛 𝑖=1 = 𝑦 keterkaitan antara modul bebas dengan modul dilihat dari sifat-sifat homomorfisme modul cauchy – issn: 2086-0382 156 jadi 𝑉 membangun 𝐹. ii) 𝑉 bebas linear berikutnya, persamaan ∑ 𝑠𝑖 𝑛 𝑖=1 𝜑(𝑥𝑖 ) = 0𝐹 dimana 𝑠1, 𝑠2, … , 𝑠𝑛 ∈ 𝑅, dapat dituliskan menjadi 𝜑(∑ 𝑠𝑖 𝑥𝑖 𝑛 𝑖=1 ) = 0𝐹 . karena 𝜑 pemetaan 1-1, maka prapeta dari 0𝐹 adalah 0𝑀 . dengan kata lain ∑ 𝑠𝑖 𝑥𝑖 𝑛 𝑖=1 = 0𝑀 . karena 𝑋 adalah basis bagi 𝑀, maka persamaan ∑ 𝑠𝑖 𝑥𝑖 𝑛 𝑖=1 = 0𝑀 hanya dipenuhi oleh 𝑠1 = 𝑠2 = ⋯ = 𝑠𝑛 = 0. oleh karena itu persamaan ∑ 𝑠𝑖 𝑛 𝑖=1 𝜑(𝑥𝑖 ) = 0𝐹 hanya dipenuhi oleh skalar 𝑠1 = 𝑠2 = ⋯ = 𝑠𝑛 = 0. jadi 𝑉 bebas linear. oleh karena itu, 𝑉 basis bagi 𝐹. jadi 𝐹 adalah modul bebas. teorema di atas adalah jaminan mengetahui suatu modul adalah modul bebas dengan memanfaatkan modul bebas yang lain melalui isomorfisme. namun, masih diperlukan cara untuk menentukan modul pada domain (atau kodomain ) tersebut modul bebas atau bukan, tentu saja tidak dengan memanfaatkan kebebasan dari modul pada kodomain ( atau domain ) karena tentu saja hal ini seperti berputar ditempat yang sama. teorema berikut dapat dijadikan sebagai prosedur untuk mengetahui apakah suatu 𝑅-modul adalah modul bebas atau bukan. teorema ini menjadi teorema penutup pada bab pembahasan ini. teorema 3 misalkan 𝑀 adalah 𝑅-modul. jika 𝑀 adalah modul bebas maka 𝑀 isomorfik dengan 𝑅𝑛 , dimana 𝑛 adalah kardinalitas dari basis 𝑀. bukti misalkan 𝑋 = {𝑥1, 𝑥2, … , 𝑥𝑛 } adalah basis untuk 𝑀. maka setiap 𝑚 di 𝑀 dapat dituliskan secara tunggal sebagai 𝑚 = ∑ 𝑚𝑖𝑥𝑖 𝑛 𝑖=1 untuk suatu 𝑚1, 𝑚2, … , 𝑚𝑛 ∈ 𝑅. sekarang, misalkan pemetaan 𝜑: 𝑀 → 𝑅𝑛 didefinisikan sebagai 𝜑(𝑚) = (𝑚1, 𝑚2, … , 𝑚𝑛) dimana 𝑚 = ∑ 𝑚𝑖 𝑥𝑖 𝑛 𝑖=1 untuk 𝑚1, 𝑚2, … , 𝑚𝑛 ∈ 𝑅. ambil sebarang 𝑧, 𝑦 ∈ 𝑀 dan 𝛼 ∈ 𝑍, maka i) 𝜑(𝑧 + 𝑦) = (∑ (𝑧𝑖 + 𝑦𝑖 )𝑥𝑖 𝑛 𝑖=1 ) = (𝑧1 + 𝑦1, 𝑧2 + 𝑦2, 𝑧3 + 𝑦3, … , 𝑧𝑛 + 𝑦𝑛) = (𝑧1, 𝑧, … , 𝑧𝑛) + (𝑦1, 𝑦2, … , 𝑦𝑛 ) = 𝜑(𝑧) + 𝜑(𝑦) ii) 𝜑(𝛼𝑦) = 𝜑(∑ (𝛼𝑦𝑖 )𝑥𝑖 𝑛 𝑖=1 ) = (𝛼𝑦1 , 𝛼𝑦2, 𝛼𝑦3 , … , 𝛼𝑦𝑛 ) = 𝛼(𝑦1 , 𝑦2, 𝑦3, … , 𝑦𝑛) = 𝛼𝜑(𝑦) dari (i) dan (ii), dapat disimpulkan bahwa 𝜑 adalah suatu homomorfisme modul. kemudian, untuk sebarang (𝑚1, 𝑚2, … , 𝑚𝑛 ) ∈ 𝑅 𝑛 , maka terdapat 𝑚 = ∑ 𝑚𝑖𝑥𝑖 𝑛 𝑖=1 anggota 𝑀 sehingga 𝜑(𝑚) = (𝑚1, 𝑚2, … , 𝑚𝑛 ). jadi 𝜑 adalah pemetaan pada. di lain pihak, untuk sebarang 𝜑(𝑧), 𝜑(𝑦) ∈ 𝑅𝜑 dengan 𝜑(𝑧) = 𝜑(𝑦), berlaku 𝟎 = 𝜑(𝑧) − 𝜑(𝑦) = (𝑧1, 𝑧2, … , 𝑧𝑛 ) − (𝑦1 , 𝑦2, … , 𝑦𝑛 ) = (𝑧1 − 𝑦1, 𝑧2 − 𝑦2, 𝑧3 − 𝑦3, … , 𝑧𝑛 − 𝑦𝑛 ) oleh karena itu, 𝑥𝑖 − 𝑦𝑖 = 0 untuk setiap 𝑖 = 1,2, … , 𝑛, dengan kata lain 𝑥𝑖 = 𝑦𝑖 untuk setiap 𝑖 = 1,2, … , 𝑛. sehingga diperoleh 𝑧 = 𝑦, jadi 𝜑 adalah pemetaan 1-1. jadi dapat disimpulkan bahwa 𝜑 adalah isomorfisme modul. oleh karena itu, 𝑀 isomorfik dengan 𝑅𝑛 . akibat dari teorema 2 jika 𝑀 isomorfik dengan 𝑅𝑛 dan 𝑀 adalah modul bebas, maka 𝑅𝑛 modul bebas. penutup 1. kesimpulan dari pembahasan pada bab 3, dapat disimpulkan bahwa suatu 𝑅-modul merupakan modul bebas jika 𝑅-modul tersebut isomorfik dengan suatu modul bebas sebagai 𝑅-modul. artinya, suatu 𝑅-modul merupakan modul bebas jika terdapat suatu isomorfisme dari 𝑅-modul tersebut ke suatu modul bebas yang juga merupakan suatu 𝑅-modul. lebih jauh lagi, jika suatu 𝑅-modul adalah modul bebas, maka 𝑅modul tersebut isomorfik dengan 𝑅𝑛 , dimana 𝑛 adalah kardinalitas dari basis bagi 𝑅-modul tersebut. 2. saran dalam studi modul, dikenal pula modul notherian. untuk penelitian selanjutnya, dapat mengkaji tentang bagaimana mengetahui suatu modul adalah modul notherian atau bukan, metodenya mungkin melalui media homomorfisma modul juga, atau mungkin menggunakan media yang lain. daftar pustaka [1] yunita wildaniati, "penjumlahan langsung pada modul," malang, 2009. [2] khusniyah, "kajian homomorfisme modul atas ring komutatif," malang, 2007. [3] wijna. (2009, 5 februari) http://wijna.web.ugm.ac.id. [4] john r. durbin, modern algebra an introduction third edition. new york: john willey & sons, inc, 1992. [5] m.d raisinghania and r.s aggarwal, modern algebra. new delhi: ram nagar, 1980. [6] achmad arifin, aljabar. bandung: itb khusnul afifa 157 volume 3 no. 3 november 2014 bandung, 2000. [7] david s dummit and richard m. foote, abstract algebra. new york: prentice-hall international, inc, 1991. [8] howard anton, aljabar linear elementer edisi kelima. jakarta: erlangga, 1987. [9] m.d raisinghania and r.s aggarwal, modern algebra. new delhi: ram nagar, 1980. [10] anton howard, aljabar linear elementer edisi kelima. jakarta: erlangga, 1987. . pendahuluan teori dasar 1. fungsi 2. operasi biner 3. grup 4. ring 5. modul 6. modul bebas metode penelitian pembahasan penutup 1. kesimpulan 2. saran daftar pustaka suma’inna dan gugun gumilar model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa feni ira puspita1, vita ratnasari2, purhadi3 jurusan statistika, fmipa, institut teknologi sepuluh nopember e-mail: 1irapuspitafeni@gmail.com, 2vita_ratna@statistika.its.ac.id, 3purhadi@gmail abstrak indeks pembagunan manusia (ipm) merupakan suatu indeks komposit yang mencakup tiga dimensi pokok pembangunan manusia yang dinilai mencerminkan status kemampuan dasar penduduk yaitu kesehatan, pencapaian pendidikan, dan daya beli masyarakat. data ipm dikategorikan/ diklasifikasikan menjadi empat, yaitu rendah, menengah ke bawah, menengah ke atas, dan tinggi. oleh karena itu, penentuan data ipm dapat dilakukan dengan pendekatan regresi probit. secara data didapati bahwa terdapat kemiripan nilai ipm antar wilayah yang berdekatan secara geografis yang mengakibatkan klasifikasi ipm wilayah yang berdekatan tersebut sama. hal ini diduga karena adanya depen-densi/keterkaitan hubungan antar wilayah. fenomena ini diduga karena adanya dependensi spasial yang dapat dijabarkan melalui metode spasial. dari penjabaran di atas, maka data ipm dalam penelitian ini didasarkan pada metode regresi probit spasial yang akan dibandingkan dengan metode probit. penelitian ini bertujuan mengkaji penaksir parameter dan pengujian penaksir parameter yang diaplikasikan pada data klasifikasi ipm di pulau jawa. mcmc digunakan sebagai metode pada pengkajian penaksir parameter.sedangkan pengkajian pengujian parameter yang digunakan ialah 25% sampai 75% dari estimasi parameter. kata kunci: ipm, mcmc, probit,probit spasial abstract human pembagunan index (hdi) is a composite index that includes three basic dimensions of human development is considered to reflect the status of the population's basic abilities of health, educational attainment, and purchasing power. data ipm classified into four, namely low, lower middle, upper middle, and high. therefore, determining the hdi can be done with the data probit regression approach. in the data found that there were similarities between the hdi value of geographically adjacent regions which resulted in the classification of the adjacent hdi same region. this is presumably due to the inter-regional dependency. this phenomenon is suspected because of the spatial dependencies can be described through spatial methods. from the explanation above, the hdi of data in this study are based on the spatial probit regression method are compared with the probit method. this study aims to assess the predictors for estimating parameters a nd testing parameters were applied to the data classification ipm in java. mcmc is used as a method of estimating the parameters in the assessment. while the assessment test parameters used are 25% to 75% of the estimated parameters. key words: ipm, mcmc, probit, spatial probit pendahuluan hakekat pembangunan dalam suatu wilayah adalah proses multidimensional yang mencakup perubahan yang mendasar meliputi struktur-struktur sosial, sikap-sikap masyarakat dan institusi-institusi nasional dengan tetap mengejar akselerasi pertumbuhan ekonomi, penanganan ketimpangan pendapatan serta pengentasan kemiskinan. keberhasilan pembangunan khususnya pembangunan manusia menempatkan manusia sebagai tujuan akhir dari pembangunan bukan alat dari pembangunan. keberhasilan pembangunan manusia dapat dilihat dari seberapa besar permasalahan mendasar di masyarakat dapat teratasi. permasalahan-permasalahan tersebut meliputi kemiskinan, pengangguran, gizi buruk, dan buta huruf (bps, 2009). salah satu alat ukur yang dianggap dapat merefleksikan status pembangunan manusia adalah indeks pembangunan manusia (ipm) atau human development index (hdi). undp sejak tahun 1990 menggunakan ipm untuk mengukur perkembangan pembangunan manusia. ipm merupakan suatu indeks komposit yang mencakup tiga dimensi pokok pembangunan manusia yang dinilai mencerminkan status kemampuan dasar penduduk yaitu kesehatan, pencapaian model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa jurnal cauchy – issn: 2086-0382 199 pendidikan, dan daya beli masyarakat. di samping mengukur kinerja pembangunan ma-nusia, perlu disadari bahwa ipm yang sebagai indeks komposit hanya dapat memperlihatkan perbandingan antar daerah provinsi atau kabupaten/kotamadya dan perkembangan antar waktu. oleh karena itu, diperlukan juga untuk melihat komponenkomponen yang membentuk ipm tersebut sehingga diketahui pencapaian dari masingmasing komponennya. menurut laporan bps (2011), kondisi pembangunan manusia indonesia mengalami kemajuan yang pesat. selama tahun tahun 19962010, angka ipm telah meningkat 4,57point menjadi 72,27. hal ini juga konsisten dengan peningkatan komponen pembentuknya, kecuali komponen daya beli yang sempat merosot akibat guncangan krisis ekonomi 1998. penelitian mengenai faktor-faktor yang mempengaruhi nilai ipm pernah dilakukan oleh yurviany (2007) dan salam (2008) yang masingmasing menggunakan metode analisis regresi logistik dan pemodelan regresi logistik ordinal sedangkan diana (2009)menggunakan regresi multivariat. sementara arianti (2009) dengan menggunakan cluster analysis, dan faidah (2010) dalam penelitiannya menggunakan regresi probit ordinal dan mengklasifikasikan kabupaten/kota berdasarkan model yang telah diperoleh. data ipm diklasifikasikan menjadi empat, yaitu rendah, menengah ke bawah, menengah ke atas, dan tinggi (bps, 2008). oleh karena itu, pengklasifikasian ipmdapat dilakukan dengan pendekatan regresi probit. dalam penelitian sosial, analisis statistika yang sering digunakan mengarah pada hubungan antara variabel respon dan variabel prediktor dengan data yang sering diamati merupakan data kategorik termasuk data biner. adapun model yang sering digunakan untuk menganalisis variabel respon berskala biner adalah model probit (gujarati, 2003, dan green, 2008). adanya perbedaan nilai ipm di suatu wilayah tertentu dengan wilayah yang lain disebabkan oleh adanya perbedaan tingkat kualitas masyarakat/penduduk di wilayah tersebut. secara data didapati bahwa terdapat kemiripan nilai ipm antar wilayah yang berdekatan secara geografis yang mengakibatkan klasifikasi ipm wilayah yang berdekatan tersebut sama. hal ini diduga karena adanya dependensi/keterkaitan hubungan antar wilayah. penelitian lesage dan smith (2002) menjabarkan bahwa model probit sering digunakan untuk menjelaskan variasi dalam pilihan individu yang dapat menjelaskan pengaruh interaksi spasial berdasarkan variasi lokasi spasial dari para pembuat keputusan. oleh karena itu individu-individu yang berlokasi pada titik yang berdekatan akan cenderung menunjukkan suatu perilaku yang mirip dalam membuat keputusan. sementara penggunaan data spasial pada regresi linier klasik dapat menyebabkan hasil yang kurang tepat, karena asumsi error yang saling bebas dan homogen tidak terpenuhi. pada data respon yang berskala biner yang melibatkan aspek keterkaitan antara wilayah satu dengan wilayah yang lain diperlukan metode khusus yang menggabungkan metode regresi probit dengan aspek spasial. dari penjabaran di atas, maka data ipm dalam penelitian ini dapat didekati dengan metode regresi probit spasial. penelitian mengenai regresi probit spasial pernah dilakukan oleh wang, iglesias, dan wooldridge (2009) yang menggunakan data cross section pada variabel respon yang berskala biner dengan metode penaksiran maximum likelihood estimation (mle). berdasarkan penelitian tersebut nilai fungsi ln likelihood untuk model probit spasial lebih kecil jika dibandingkan dengan model probit sehingga dapat disimpulkan bahwa model probit spasial lebih baik jika dibandingkan dengan model probit. wilayah pulau jawa terdiri dari 6 provinsi besar di indonesia, dan mempunyai kabupaten/kota terbanyak di indonesia yaitu 118 kabupaten/kota. berdasarkan peringkat ipm tahun 2010 , provinsi-provinsi di pulau jawa masuk dalam 20 besar dibandingkan dengan 33 provinsi di indonesia, kecuali provinsi banten menduduki peringkat 23. peringkat pertama ipm nasional diduduki oleh provinsi dki jakarta, kemudian provinsi diy dengan peringkat 4, disusul provinsi jawa tengah menduduki peringkat 14, sedangkan provinsi jawa barat dan jawa timur masing-masing menduduki peringkat 15 dan 18 (bps,2011). pada penelitian probit spasial sebelumnya belum dikaji secara rinci tentang penaksiran parameter dan pengujian parameter model, maka kajian utama pada penelitian ini adalah penaksiran dan pengujian parameter model regresi probit spasial dengan mengaplikasikan pada kasus klasifikasi ipm di pulau jawa. pada penelitian ini diharapkan penggunaan model regresi probit spasial mampu menghasilkan model klasifikasi ipm yang spesifik di setiap daerah sehingga semakin informatif dan aplikatif sehingga dapat diketahui faktor-faktor yang berpengaruh terhadap klasifikasi ipm di pulau jawa. permasalahan yang diangkat dalam penelitian ini adalah bagaimana mendapatkan penaksiran dan pengujian parameter model regresi probit spasial dengan mengkaji secara rinci dan mengaplikasikan pada kasus klasifikasi ipm di pulau jawa. feni ira puspita, vita ratnasari, purhadi 200 volume 2 no. 4 mei 2013 jika ditinjau dari segi kelimuan, manfaat penelitian ini adalah untuk meningkatkan wawasan pengetahuan dan keilmuan mengenai metode pemodelan untuk data kualitatif; khususnya model regresi probit spasial. selain itu, diharapkan penelitian ini dapat memberikan manfaat bagi pemerintah khususnya pemerintah propinsi pulau jawa yaitu dengan mengetahui faktor-faktor yang mempengaruhi klasifikasi ipm untuk dijadikan acuan dalam mengambil keputusan kebijakan dalam meningkatkan kualitas indeks pembangunan manusia di pulau jawa. teori dasar 1. model regresi probit menurut greenberg (1980) analisis probit pertama kali dikemukakan oleh chester ittner bliss pada tahun 1934 yang dalam penelitiannya mengenai pestisida untuk mengendalikan serangga yang hidup pada daun dan buah anggur. dalam suatu artikel di jurnal science, bliss (1934) mengemukakan bahwa istilah probit dalam model regresi probit berasal dari kata probability unit, dengan kata lain model regresi probit merupakan suatu model regresi yang berkatian dengan unitunit probabilitas. model regresi probit adalah salah satu model regresi yang dapat digunakan untuk mengetahui pengaruh variabel prediktor terhadap variabel respon yang bersifat biner. menurut o’donnel dan connor (1996), serta greene (2008) pemodelan regresi probit diawali dengan melihat model sebagai berikut : 𝑌∗ = 𝛽𝑇 𝑥 + 𝜀 (1) dimana 𝑌∗ adalah variabel respon yang merupakan variabel kontinu, 𝛽 adalah vektor parameter koefisien dengan 𝛽 = [𝛽0, 𝛽1, … , 𝛽𝑝], 𝑥 adalah vektor variabel prediktor, dengan 𝑥 = [1, 𝑥1, … , 𝑥𝑝 ] 𝑇 dan 𝜀 adalah error yang diasumsikan berdistribusi 𝑁(0, 𝜎 2). dengan demikian 𝐸(𝑌∗) = 𝛽𝑇 𝑥, 𝑉𝑎𝑟(𝑌∗) = 𝜎 2 dan 𝑓(𝑌∗) = 1 √2𝜋𝜎 𝑒𝑥𝑝 [− 1 2𝜎2 (𝑌∗ − 𝛽𝑇 𝑥)2] yang dapat dilihat pada gambar berikut. dari gambar 1, terlihat bahwa untuk setiap luasan mempunyai probabilitas sebagai berikut : 𝑃(𝑌∗ > 𝑘) = 1 − 𝑃(𝑌∗ ≤ 𝛾𝑘−1) = 1 − φ(𝛾𝑘−1) = ∫ 𝑓(𝑌∗) ∞ 𝛾𝑘−1 𝑑𝑌∗ (2) kemudian dari persamaan (2) dilakukan transformasi ke dalam bentuk 𝑧 = 𝑌∗−𝛽𝑇𝑥 𝜎 dimana 𝑍~𝑁(0,1), sehingga (3) menjadi persamaan (4) adalah sebagai berikut: 𝑃(𝑌∗ > 𝑘) = 1 − 𝑃(𝑌∗ ≤ 𝛾𝑘−1) = 1 − 𝑃 (𝑍 ≤ 𝛾𝑘−1 − 𝛽 𝑇 𝑥 𝜎 ) (3) 𝑃(𝑌∗ > 𝑘) = 1 − φ ( 𝛾𝑘−1 − 𝛽 𝑇 𝑥 𝜎 ) (4) dimana 𝛾1, 𝛾2, … , γk adalah batasan (threshold) dan φ(⋅) adalah seperti pada (2). kemudian dilakukan pengkategorian terhadap 𝑌∗ yaitu untuk 𝑌∗ ≤ 𝛾1 dikategorikan dengan 𝑌 = 1, untuk 𝛾1 < 𝑌 ∗ ≤ 𝛾2 dikategorikan dengan 𝑌 = 2, ... , untuk 𝛾𝑘−2 < 𝑌 ∗ ≤ 𝛾𝑘−1 dikategorikan dengan 𝑌 = 𝑘 − 1, ..., untuk 𝑌∗ > 𝛾𝑘−1 dikategorikan dengan 𝑌 = 𝑘, sehingga diperoleh model regresi probit pada persamaan (4). gambar 1. grafik fungsi distribusi probabilitas dari 𝑌∗ menginterpretasikan model probit pada persamaan (5) digunakan efek marginal (green, 2008). efek marginal diperoleh dari turunan probabilitas tiap kategorinya. efek marginal variabel xpada model probit adalah sebagai-mana persamaan (4) berikut : 𝜕𝑃(𝑌 = 𝑘) 𝜕𝑋𝑖 = 𝜙 ( 𝛾𝑘−1 − 𝛽 𝑇 𝑥 𝜎 ) 𝛽𝑖 𝜎 (5) persamaan (5) menunjukkan besarnya pengaruh variabel 𝑋𝑖 terhadap 𝑃(𝑌 = 𝑘). sehingga untuk mengetahui distribusi model regresi probit pada persamaan (2) dipandang 𝑌 = [𝑌1 𝑌2 𝑌3 … 𝑌𝑘 ] 𝑇 dimana 𝑌𝑖 = 0,1 untuk 𝑖 = 1, … , 𝑘; 𝑌𝑘 = 1 − (𝑌1 + ⋯ + 𝑌𝑘−1) dan probabilitas untuk 𝑌 = 𝑘 adalah 𝑃(𝑌 = 𝑘) = 1 − [𝑃(𝑌 = 1) + ⋯ + 𝑃(𝑌 = 𝑘 − 1)], dengan demikian variabel 𝑌 mengikuti distribusi multinomial yang dapat dituliskan seperti berikut. 𝑌~𝑀[1; 𝑃(𝑌 = 1|𝑥), 𝑃(𝑌 = 2|𝑥), 𝑃, 𝐿 𝑃(𝑌 = 𝑘 − 1|𝑥)] (6) dan model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa jurnal cauchy – issn: 2086-0382 201 𝑃[𝑌1 = 𝑦1, … , 𝑌𝑘−1 = 𝑦𝑘−1] = {(φ [ 𝛾1 − 𝛽 𝑇 𝑥 𝜎 ]) 𝑦1 (φ [ 𝛾2 − 𝛽 𝑇 𝑥 𝜎 ] − φ [ 𝛾1 − 𝛽 𝑇 𝑥 𝜎 ]) 𝑦2 ⋯ (φ [ 𝛾𝑘−1 − 𝛽 𝑇 𝑥 𝜎 ] − φ [ 𝛾𝑘−2 − 𝛽 𝑇 𝑥 𝜎 ]) 𝑦𝑘−2 (1 − φ [ 𝛾𝑘−1 − 𝛽 𝑇 𝑥 𝜎 ]) 𝑦𝑘−1 } (7) 2. analisis spasial penelitian yang berkaitan dengan region atau kewilayahan sering disebut dengan spasial. menurut tobler (1979) dalam anselin (1988) mengatakan bahwa segala sesuatu yang saling berpengaruh satu sama lain, yang artinya wilayah yang lebih dekat cenderung akan memberikan pengaruh yang lebih besar dari pada wilayah yang lebih jauh jaraknya. pada data spasial, seringkali pengamatan di suatu wilayah bergantung pada pengamatan di wilayah lain yang berdekatan (neighboring). sifat data spasial ada dua macam yaitu spatial dependence yang terjadi akibat adanya dependensi dalam data cross-section dan spatial heterogeneity terjadi akibat adanya perbedaan antara satu wilayah dengan wilayah lainnya (anselin, 1995). anselin (1988) telah mengembangkan beberapa metode spasial dengan menggunakan data cross section. bentuk umum model regresi spasial adalah sebagai berikut : 𝐲 = ρ𝐖𝟏𝐲 + 𝐗𝛃 + 𝐮 𝐮 = 𝜆𝐖𝟐𝐮 + 𝛆 (9) dimana 𝛆~𝑁(0, 𝜎 2𝐼𝑛 ), 𝐲 merupakan vektor variabel respon yang berukuran (n x 1) dan 𝐗 adalah matriks variabel prediktor yang berukuran 𝑛 × (𝑘 + 1). 𝛃 adalah parameter koefisien regresi yang berukuran (𝑘 + 1) × 1, ρ adalah parameter koefisien spasial lag dari variabel respon, dan 𝜆 adalah parameter koefisien spasial lag pada error. adapun 𝐖𝟏dan 𝐖𝟐 adalah matriks pembobot yang berukuran(𝑛 × 𝑛) yang elemen diagonalnya bernilai nol. 𝐮 adalah vektor regresi yang saling berkorelasi dengan ukuran (n x 1). ada beberapa turunan model yang bisa diperoleh dari model umum pada persamaan (9), yaitu : apabila nilai 𝜌 = 0 dan 𝜆 = 0, maka persamaannya menjadi : 𝒚 = 𝑿𝜷 + 𝜺 (10) persamaan (10) disebut model regresi klasik atau model regresi ordinary least square (ols) yaitu model regresi yang tidak mempunyai efek spasial. apabila nilai 𝜌 ≠ 0 dan 𝜆 = 0, maka persamaannya menjadi : 𝐲 = 𝜌𝐖𝟏𝐲 + 𝐗𝛃 + 𝛆𝐲 = 𝜌𝐖𝟏𝐲 + 𝐗𝛃 + 𝛆 (11) persamaan (11) disebut sebagai regresi spatial lag model (slm). menurut lesage (1999) istilah lain model ini adalah model spatial autoregressive (sar). apabila nilai 𝜆 ≠ 0 dan 𝜌 = 0, maka persamaannya menjadi : 𝐲 = 𝐗𝛃 + 𝐮 𝐮 = 𝜆𝐖𝟐𝐮 + 𝛆 (12) persamaan (2.16) disebut sebagai regresi spatial error model (sem). apabila nilai 𝜌 ≠ 0 dan 𝜆 ≠ 0, maka persamaannya menjadi : 𝐲 = 𝜌𝐖𝟏𝐲 + 𝐗𝛃 + 𝐮 𝐮 = 𝜆𝐖𝟐𝐮 + 𝛆 (13) persamaan (13) disebut sebagai general spatial model, atau anselin (1988) menyebutnya model spatial autoregressive moving average (sarma). 3. uji dependensi spasial anselin (1988) menyatakan bahwa untuk mengetahui adanya dependensi spasial digunakan dua metode yaitu metode moran’s i dan lagrange multiplier (lm). berikut ini akan dijabarkan mengenai kedua metode tersebut : moran’s i indeks moran’s i adalah ukuran korelasi antara pengamatan wilayah yang saling berdekatan. moran’s i (selanjutnya dinotasikan i) membandingkan nilai pengamatan di suatu wilayah tertentu dengan nilai pengamatan di wilayah yang lainnya. menurut lee dan wong (2011) indeks moran’s i dapat dihitung dengan persamaan (2.18) dibawah ini : 𝐼 = 𝑛 ∑ ∑ 𝑤𝑖𝑗 (𝑦𝑖 − �̅�) (𝑦𝑗 − �̅�) 𝑛 𝑗=1 𝑛 𝑖=1 (∑ ∑ 𝑤𝑖𝑗 𝑛 𝑖=1 𝑛 𝑖=1 ) ∑ (𝑦𝑖 − �̅�) 2𝑛 𝑖=1 (14) keterangan : 𝑛 : banyaknya jumlah pengamatan 𝑤𝑖𝑗 : elemen matriks pembobot spasial 𝑥𝑖 : nilai pengamatan pada lokasi ke-i 𝑥𝑗 : nilai pengamatan pada lokasi ke-j �̅� : nilai rata-rata pengamatan identifikasi pola persebaran antar wilayah dapat menggunakan kriteria nilai indeks i (lee, dan wong, 2001). jika nilai 𝐈 > 𝐈𝟎 maka mempunyai pola mengelompok, dan jika nilai 𝐈 < 𝐈𝟎 maka mempunyai pola menyebar (tidak ada autokorelasi), sedangkan nilai 𝐈 < 𝐈𝟎 dikatakan memiliki pola menyebar jika 𝐈𝟎 = −1/(𝑛 − 1). feni ira puspita, vita ratnasari, purhadi 202 volume 2 no. 4 mei 2013 langrange multiplier test (lm test) langrange multiplier test diperoleh berdasarkan asumsi model di bawah 𝐻0. ada tiga hipotesis yang diajukan, yaitu : 𝐻0 𝜌 = 0 ; 𝐻1 ∶ 𝜌 ≠ 0 (untuk model spatial lag model) 𝐻0 ∶ 𝜆 = 0 ; 𝐻1 ∶ 𝜆 ≠ 0 (untuk model spatial error model) 𝐻0 ∶ 𝜌, 𝜆 = 0 ; 𝐻1 𝜌, 𝜆 ≠ 0 (untuk model sarma) digunakan statistik lm test yang mempunyai bentuk : 𝐿𝑀 = 𝐸−1 {(𝑅𝑦) 2 𝑇22 − 2𝑅𝑦𝑅𝑒 𝑇12 + (𝑅𝑒) 2(𝐷 + 𝑇11)}~𝜒𝑚 2 (15) dengan m adalah jumlah parameter spasial, untuk slm = 1, sem = 1, dan sarma = 2 𝑅𝑦 = 𝐞𝑇 𝐖𝟏𝐲 𝜎 2 𝑅𝑒 = 𝐞𝑇 𝐖𝟐𝐞 𝜎 2 𝐌 = 𝐈 − 𝐗(𝐗𝑇 𝐗)−1 𝐗𝑇 𝑇𝑖𝑗 = 𝑡𝑟{𝐖𝐢𝐖𝐣 +𝐖𝐢 𝑇 𝐖𝐣) 𝐷 = 𝜎 −2(𝐖𝟏𝐗𝛃) 𝑇 𝐌(𝐖𝟏𝐗𝛃) 𝐸 = (𝐷 + 𝑇11)𝑇22 − (𝑇12) 2 jika matriks pembobot spasialnya sama (𝐖𝟏 = 𝐖𝟐 = 𝐖), maka: 𝑇11 = 𝑇12 = 𝑇22 = 𝑇 = 𝑡𝑟{(𝐖 𝑇 + 𝐖)𝐖} (16) tolak 𝐻0 jika nila lm > 𝜒(𝑚) 2 . 4. model regresi probit spasial suatu model probit dengan dependensi spasial pertama kali diteliti oleh mc. millen (1992) yaitu suatu algoritma em diperkenalkan untuk menghasilkan suatu dugaan yang konsisten (maksimum likelihood) untuk model tersebut. akan tetapi model tersebut sangat bergantung pada asymtotic dan membutuhkan sampel yang besar. alternatif lainnya, diperkenalkan suatu model probit nonspasial dengan pendekatan hirarki bayessian oleh albert dan chib (1993) yang dapat diaplikasikan untuk data dengan sampel kecil. lesage (2000) kemudian memperkenalkan suatu model yang dikembangkan dari model albert dan chib dengan mengikutsertakan suatu dependensi spasial antar data. model regresi probit spasial merupakan gabungan model regresi probit dan model spasial. menurut anselin (1988) dan dikembangkan lagi oleh anselin (1996), anselin dan bera (1998), anselin (2001a) dan anselin (2001b) bentuk model regresi probit spasial adalah sebagai berikut : 𝑌∗ = 𝜌𝑊𝑦∗ + 𝑋𝛽 + 𝜀 (17) dimana 𝐲∗ adalah vektor variabel laten berukuran (n x 1), 𝐗 adalah matriks kovariat yang berukuran (n x k), 𝐖 yaitu matriks pembobot berukuran (n x n), dan ρ adalah parameter spasial. persamaan (2.21) dapat ditulis dalam bentuk direduksi adalah sebagai berikut : 𝑦∗ = (𝐼 − 𝜌𝑊)−1𝑋𝛽 + 𝑢 𝑢 = (𝐼 − 𝜌𝑊)−1𝜀 (18) link variabel laten 𝐲∗ untuk pengamatan biner, y, melalui persamaan berikut 𝑦 = { 0, 𝑗𝑖𝑘𝑎 𝑦∗ ≤ 𝛾1 1, 𝑗𝑖𝑘𝑎 𝛾1 < 𝑦 ∗ ≤ 𝛾2 probabilitas untuk pengamatan ke-i adalah salah satunya dapat dihitung sebagai berikut : 𝑃(𝑌 = 𝑘) = 𝑃(𝑌∗ > 𝛾𝑘−1) = 1 − 𝑃(𝑌∗ ≤ 𝛾𝑘−1) = 1 − 𝑃 (𝑍 ≤ 𝛾𝑘−1 − 𝜌𝑊𝑦 ∗ − 𝑋𝛽 𝜎 ) = 1 − φ ( 𝛾𝑘−1 − 𝜌𝑊𝑦 ∗ − 𝑋𝛽 𝜎 ) 5. indeks pembangunan manusia (ipm) salah satu alat ukur yang dianggap dapat merefleksikan status pembangunan manusia adalah indeks pembangunan manusia (ipm) atau human development index (hdi). undp sejak tahun 1990 menggunakan ipm untuk mengukur perkembangan pembangunan manusia. ipm merupakan suatu indeks komposit yang mencakup tiga dimensi pokok pembangunan manusia yang dinilai mencerminkan status kemampuan dasar penduduk yaitu kesehatan, pencapaian pendidikan, dan standar hidup layak. ketiga dimensi tersebut memiliki pengertian sangat luas karena terkait banyak faktor. dimensi kesehatan yang diukur dari angka harapan hidup, dimensi pencapaian pendidikan yang diukur dari rata-rata lama sekolah dan angka melek huruf, sedangkan dimensi standar hidup layak yang diindikasikan dengan logaritma normal dari produk domestik bruto perkapita penduduk dalam paritas daya beli (bps, 2008). ipm dapat mengetahui kondisi pembangunan di daerah dengan alasan : a) ipm menjadi indikator penting untuk mengukur keberhasilan dalam pembangunan kualitas manusia. model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa jurnal cauchy – issn: 2086-0382 203 b) ipm menjelaskan tentang bagaimana manusia mempunyai kesempatan untuk mengakses hasil dari proses pembangunan, sebagai bagian dari haknya seperti dalam memperoleh pendapatan, kesehatan, pendidikan, dan kesejahteraan. c) ipm digunakan sebagai salah satu ukuran kinerja daerah, khususnya dalam hal evaluasi terhadap pembangunan kualitas hidup masyarakat/penduduk. d) ipm belum tentu mencerminkan kondisi sesungguhnya namun untuk saat ini ipm merupakan satu-satunya indikator yang dapat digunakan untuk mengukur pembangunan kualitas hidup manusia. salah satu alat ukur yang dianggap dapat merefleksikan status pembangunan manusia adalah human development index (hdi) atau ipm. ipm merupakan sauatu indeks komposit yang mencakup tiga bidang pembangunan manusia yang dianggap sangat mendasar yaitu usia angka harapan hidup, tingkat pendidikan, dan standar hidup layak (bps, 2009). angka harapan hidup pembangunan manusia harus lebih mengupayakan agar penduduk dapat mencapai usia hidup yang panjang dan sehat. kemampuan untuk bertahan hidup lebih lama yang akan dijalani oleh bayi yang baru lahir pada suatu tahun tertentu yang diukur dengan angka harapan hidup pada saat lahir (life expectacy at birth). angka harapan hidup dapat digunakan untuk mengevaluasi kinerja pemerintah dalam meningkatkan kesejahteraan penduduk khususnya meningkatkan derajat kesehatan penduduk. semakin tinggi angka harapan hidup suatu wilayah, mengindikasikan bahwa semakin tinggi kualitas fisik penduduk pada wilayah tersebut (bps, 2008). tingkat pendidikan komponen pendidikan diukur dengan angka melek huruf dan rata-rata lama sekolah. sebagai catatan, undp dalam publikasi tahunan hdr sejak 1995 menggunakan indikator partisipasi sekolah dasar, menengah, dan tinggi sebagai pengganti rata-rata lama sekolah karena sulitnya memperoleh data rata-rata lama sekolah secara global. indikator angka melek huruf diperoleh dari variabel membaca dan menulis, sedangkan indikator rata-rata lama sekolah dihitung dengan menggunakan dua variabel secara simultan, yaitu tingkat/kelas yang sedang/pernah dijalani dan jenjang pendidikan tertinggi yang ditamatkan (bps, 2008). standar hidup layak dimensi ketiga ipm adalah standar hidup layak (decent living standard) yang dapat mengindikasikan tingkat kesejahteraan masyarakat atau dikenal dengan istilah kemampuan daya beli daya masyarakat/ penduduk. sebagai catatan, undp menggunakan indikator pdrb per kapita riil yang telah disesuaikan sebagai ukuran komponen tersebut hal ini dikarenakan tidak tersedia indikator lain yang lebih baik untuk keperluan perbandingan antar negara. sementara bps dalam menghitung standar hidup layak menggunakan rata-rata pengeluaran per kapita riil yang disesuaikan dengan formula atkinson (bps, 2008). indeks pembangunan manusia (ipm) menjadi indikator penting untuk mengukur keberhasilan dalam upaya membangun kualitas hidup manusia yang dapat menjelaskan bagaimana penduduk dapat mengakses hasil pembangunan dalam memperoleh penadapatan, kesehatan, dan pendidikan. kualitas sumber daya manusia suatu daerah sangat tergantung dari tingkat pendidikan penduduknya, karena pendidikan yang berkualitas dapat menghasilkan output pendidikan yang berkualitas. apabila tingkat pendidikan masyarakat rendah, kapabilitasnya juga rendah. hal ini akan berpengaruh terhadap rendahnya kemampuan untuk menangkap peluang, sehingga akan berakibat pada kemiskinan (naja, 2006). kajian mengenai indikator ipm telah banyak dilakukan diantaranya penelitian yang dilakukan oleh wachid (2000) tentang faktorfaktor yang mempengaruhi kualitas hidup manusia dengan analisis regresi berganda. hasil penelitian tersebutadalah faktor-faktor yang mempengaruhi kualitas hidup manusia adalah rata-rata lama sekolah, angka buta huruf, angka kematian bayi, angka kematian ibu, tingkat pengangguran terbuka dan persentase penduduk miskin. dalam penelitian setiawan (2004) mengungkapkan bahwa faktor lokasi tempat tinggal (perkotaan atau pedesaan) cenderung berpengaruh terhadap tingkat pendidikan dan menunjukkan hubungan yang tinggi dengan tingkat pendapatan dan pengeluaran. sementara penelitian indikator ipm juga dilakukan oleh suanita (2006), hasil penelitiannya menyimpulkan bahwa faktor-faktor yang mempengaruhi rata-rata pengeluaran konsumsi per kapita adalah pdrb perkapita, rasio ketergantungan penduduk dan peranan sektor pertanian dalam pdrb. secara umum pengeluaran konsumsi setiap anggota rumah tangga dibagi menjadi dua kelompok yaitu pengeluaran konsumsi untuk makanan dan bukan makanan. menurut bps (2008) rasio ketergantungan feni ira puspita, vita ratnasari, purhadi 204 volume 2 no. 4 mei 2013 merupakan gabungan indikator rasio ketergantungan anak (rka) dan rasio ketergantungan usia lanjut (rkl) yang menunjukkan total rasio ketergantungan penduduk usia tidak produktif pada usia produktif. kegunaan rka adalah menunjukkan besarnya beban tanggungan anak bagi penduduk usia di suatu daerah pada suatu waktu tertentu sedangkan kegunaan rkl menggambarkan besarnya beban tanggungan penduduk usia lanjut bagi penduduk usia produktif. secara umum dapat menggambarkan beban tanggungan ekonomi kelompok usia produktif (penduduk usia 15-65 tahun) terhadap kelompok usia tidak produktif (penduduk usia 0-14 tahun dan usia si atas 65 tahun). sehingga semakin besar rasio ketergantungan akan menyebabkan pengeluaran konsumsi per kapita semakin kecil, karena meningkatnya jumlah tanggungan menyebabkan jumlah pendapatan yang diperoleh seseorang akan dibagi kepada lebih banyak orang yang juga berdampak pada tingkat pendidikan dan kesehatan (suanita, 2006). menurut zairin (2006) dalam penelitiannya yang menggunakan rumus regresi sederhana ada korelasi positif antara alokasi belanja dalam apbd dengan pencapaian ipm. semakin besar alokasi belanja dalam apbd yang relevan dengan komponen pembentuk ipm, maka semakin tinggi tingkat pencapaian ipm. itu, salah satu upaya untuk meningkatkan pencapaian ipm yaitu meningkatkan besaran alokasi belanja dalam apbd untuk komponen yang terkait secara langsung maupun secara tidak langsung dengan aspek pendidikan, kesehatan dan kemampuan daya beli masyarakat/penduduk. besarnya pengeluaran sektor industri terhadap pdrb akan berpengaruh terhadap pengeluaran konsumsi penduduk, mengingat daerah yang maju sektor industrinya akan meningkatkan tingkat pendidikan dan kesejahteraan masyarakatnya (suanita, 2006). sementara semakin tinggi angka keluhan kesehatan maka akan mengurangi angka harapan hidup sehingga mengakibatkan penurunan ipm suatu daerah (bps, 2009). tingkat pengangguran terbuka adalah salah satu indikator kemampuan daya beli masyarakat. sedikitnya jumlah pengangguran mengindikasikan bahwa meningkatkan nilai daya beli masyarakat (bps, 2010). rata-rata umur kawin pertama wanita mempunyai pengaruh yang positif terhadap nilai ipm. semakin tinggi rata-rata umur kawin pertama wanita di suatu provinsi menyebabkan nilai ipm di provinsi tersebut semakin tinggi (bps, 2008). metode penelitian data yang digunakan dalam penelitian ini adalah data sekunder yang yang diperoleh dari badan pusat statistik (bps) yaitu data yang diambil dari publikasi maupun data dari hasil survei. data publikasi yang diambil adalah data publikasi provinsi dalam angka tahun 2010 dan publikasi laporan pembangunan manusia indonesia tahun 2010 sedangkan data survei yang diambil adalah data survei sosial ekonomi nasional (susenas) tahun 2010 untuk pulau jawa. unit pengamatan pada penelitian ini adalah 118 kabupaten/kota di pulau jawa. adapun variabel yang diamati dalam penelitian ini adalah terdiri dari satu variabel respon dan sembilan variabel prediktor. variabel yang berperan sebagai veriabel respon (y) adalah indeks pembangunan manusia (ipm). variabel respon ini bersifat kategorik. menurut bps (2008) ipm dibagi menjadi empat kategori yaitu sebagai berikut : 0 = rendah, jika ipm ≤ 50 1 = menengah rendah jika 50 < ipm ≤ 66 2 = menengah tinggi jika 66< ipm ≤ 80 3 = tinggi jika ipm > 80 pada penelitian ini jika ipm dibagi sesuai dengan ketentuan bps yaitu empat kategori maka terdapat beberapa kategori yang kosong yaitu pada kategori rendah dan tinggi. oleh karena itu pada penelitian ini menggunakan dua kategori yaitu ketegori menengah rendah dan menengah tinggi sesuai dengan kategori yang ada di bps, pembagian kategori tersebut adalah sebagai berikut : 0 = menengah rendah, jika ipm ≤ 66 1 = menengah tinggi jika 66< ipm ≤ 80 berikut merupakan beberapa variabel prediktor yang akan dilibatkan dalam penelitian ini adalah sebagai berikut: persentase penduduk yang tinggal di perkotaan (x1). persentase penduduk yang tinggal di perkotaan adalah jumlah penduduk yang tinggal di daerah perkotaan dalam jangka waktu tertentu. persentase penduduk yang berpendidikan di atas sltp (x2) penduduk yang berpendidikan diatas sltp adalah penduduk yang telah menamatkan pendidikan setingkat sltp atau jenjang pendidikan yang lebih tinggi. model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa jurnal cauchy – issn: 2086-0382 205 rata-rata pendapatan per kapita (x3) pendapatan per kapita (per capita income) adalah pendapatan rata-rata penduduk suatu negara pada suatu periode tertentu, yang biasanya satu tahun. rasio ketergantungan penduduk (x4) rasio ketergantungan menunjukkan beban tanggungan penduduk usia produktif (15-64 tahun) terhadap penduduk usia muda (kurang dari 15 tahun) dan penduduk usia tua (65 tahun atau lebih) (bps, 2008). peranan sektor industri dalam pdrb (x5) pdrb merupakan jumlah nilai tambah atas barang dan jasa yang dihasilkan oleh berbagai unit produksi di suatu wilayah dalam jangka waktu tertentu. persentase penduduk miskin (x6) penduduk miskin adalah penduduk yang tidak mempunyai kemampuan dalam memenuhi kebutuhan dasar untuk kehidupan yang layak, baik kebutuhan dasar makanan maupun kebutuhan dasar bukan makanan (urip, 2007). persentase penduduk yang mengalami keluhan kesehatan (x7) semakin tinggi angka keluhan kesehatan maka akan mengurangi angka harapan hidup sehingga mengakibatkan penurunan ipm suatu daerah (bps, 2009). tingkat pengangguran terbuka (x8) tingkat pengangguran terbuka adalah salah satu indikator kemampuan daya beli masyarakat. sedikitnya jumlah pengangguran mengindikasikan bahwa meningkatkan nilai daya beli masyarakat (bps, 2010). rata-rata umur kawin pertama wanita (x9) rata-rata umur kawin pertama wanita mempunyai pengaruh yang positif terhadap nilai ipm. semakin tinggi rata-rata umur kawin pertama wanita di suatu provinsi menyebabkan nilai ipm di provinsi tersebut semakin tinggi (bps, 2008). pulau jawa terdiri dari 84 kabupaten dan 34 kota yang menyebar di enam provinsi (lihat tabel 3.3). secara geografis, letak dari 118 kabupaten/kota di pulau jawa tersebut masingmasing tidak saling simetris. peta digital pulau jawa digunakan sebagai dasar pembentukan penimbang spasial menggunakan metode queen contiguity, wilayah-wilayah tersebut saling berbatasan sisi atau sudut yang dianggap saling mempengaruhi. tabel 1. jumlah kabupaten/kota di pulau jawa menurut provinsi provinsi kabupaten kota jumlah dki jakarta 1 5 6 jawa barat 17 9 26 jawa tengah 29 6 35 daerah istimewa yogyakarta 4 1 5 jawa timur 29 9 38 banten 4 4 8 jumlah 84 34 118 sumber : badan pusat statistik, 2011 alasan penggunaan metode queen contiguityadalah adanya dua wilayah yang saling bersinggungan antara kabupaten sleman provinsi daerah istimewa yogyakarta dengan kabupaten boyolali provinsi jawa tengah. metode dan tahapan analisis yang akan digunakan untuk mencapai tujuan penelitian ini adalah sebagai berikut : 1. menentukan bentuk umum model regresi probit spasial. 2. menyusun langkah-langkah estimasi parameter metode mcmc sebagai berikut: a) menentukan teknik pembangkitan angka random yang berdistribusi normal terpotong untuk melengkapi data yang tersensor. b) menentukan bentuk persamaan likelihood dari model regresi spasial c) menentukan probabilita prior dari masing-masing parameter yang saling independen berdasarkan hasil studi literatur d) menentukan distribusi posterior bersama dari model regresi spasial e) menentukan distribusi posterior bersyarat dari masing-masing parameter dengan syarat parameter lainnya diketahui. untuk menyusun algoritma pemrograman dalam rangka pembentukan model regresi probit spasial, dilakukan langkah-langkah sebagai berikut: 1. menyusun algoritma dan pemrograman untuk pengujian efek korelasi spasial dan pengujian heteroskedastisitas dari model regresi probit spasial yang akan dibentuk. 2. menyusun algoritma gibbs sampler untuk estimasi parameter model regresi probit spasial dari distribusi posterior bersyarat feni ira puspita, vita ratnasari, purhadi 206 volume 2 no. 4 mei 2013 yang telah terbentuk pada langkah sebelumnya. 3. menyusun algoritma metropolis within gibbs untuk estimasi parameter model regresi probit spasial ( atau ) dari distribusi posterior bersyarat yang telah terbentuk pada langkah sebelumnya. 4. menyusun algoritma slice sampling untuk membangkitkan angka random yang berdistribusi normal terpotong dalam rangka melengkapi data yang tersensor. diskusi / pembahasan persebaran ipm di pulau jawa menurut kabupaten/kota disajikan pada gambar 1.1. kabupaten/kota yang memiliki nilai klasifikasi ipm terendah sebesar 59.70 yaitu terdapat di kabupaten sampang dan tertinggi sebesar 79.52 terdapat di kota yogyakarta. degradasi warna yang terdapat pada gambar menunjukkan klasifikasi ipm. gambar 1.1. sebaran ipm di pulau jawa tahun 2010 berdasarkan tabel 2 dapat diketahui bahwa 6.78% kabupaten/kota di pulau jawa tergolong dalam wilayah yang memiliki nilaiklasifikasi ipm yang menengah rendah. sebagian besar kabupaten/kota di pulau jawa tergolong dalam wilayah yang memiliki nilai klasifikasi ipm menengah tinggi yaitu mencapai 93.22%, tabel 2. persentase kelompok kabupaten/kota berdasarkan klasifikasi ipm di pulau jawa eksplorasi data statistika deskriptif di gunakan untuk mengetahui perbedaan karakteristik antara kelompok wilayah yang memiliki nilai klasifikasi ipm rendah, menengah dan tinggi. karakteristik variabel prediktor yang diduga berpengaruh terhadap klasifikasi ipm disajikan pada tabel 3. tabel 3. statistik deskriptif variabel prediktor setelah mengetahui karakteristik dari vaeriabel prediktor yang diduga berpengaruh terhadap klasifikasi ipm di pulau jawa maka langkah selanjutnya adlah menguji ada atau tidaknya multikolinearitas diantara variabelvariabel prediktor. kriteria yang digunakan untuk memeriksa multikolinearitas antar variabel adalah dengan menggunakan nilai variance inflation factor (vif) pada variabel-variabel prediktor. nilai vif yang lebih besar dari 10 menunjukkan adanya kolinearitas antar variabel prediktor. tabel 4. nilai vif variabel prediktor variabel vif keterangan x1 4,844 tidak ada multikolinearitas x2 14,05 ada multikolinearitas x3 1,318 tidak ada multikolinearitas x4 2,965 tidak ada multikolinearitas x5 1,316 tidak ada multikolinearitas x6 2,649 tidak ada multikolinearitas x7 2,303 tidak ada multikolinearitas x8 5,415 tidak ada multikolinearitas berdasarkan tabel 4 terlihat bahwa ada satu variabel prediktor yang mempunyai nilai vif lebih dari 10 yaitu variabel persentase penduduk yang berpendidikan di atas sltp (x2). karena terdapat multikolinearitas antar variabel prediktor, maka untuk memodelkan data klasifikasi ipm di pulau jawa diperlukan metode backward elimination untuk mengatasi adanya multikolinearitas. identifikasi pola hubungan antara variabel prediktor dan variabel respon langkah awal dalam proses penyusunan model regresi probit spasial dilakukan identifikasi jumlah persentase menengah rendah 8 6.78 menengah tinggi 110 93.22 total 118 100.00 klasifikasi ipm pulau jawa rata-rata std dev min maks menengah rendah 28.79 11.73 12.92 44.32 menengah tinggi 60.22 30.67 9.28 100.00 menengah rendah 22.64 4.77 14.01 27.52 menengah tinggi 42.38 13.73 20.14 70.18 menengah rendah 10.33 2.78 6.04 15.33 menengah tinggi 8.87 2.34 4.38 15.76 menengah rendah 47.89 5.14 42.01 56.56 menengah tinggi 48.45 5.68 35.69 63.23 menengah rendah 8.14 6.72 1.03 18.83 menengah tinggi 22.08 18.22 0.31 76.94 menengah rendah 22.53 6.42 13.27 32.47 menengah tinggi 12.64 5.27 1.67 24.58 menengah rendah 2.80 1.39 1.60 5.79 menengah tinggi 7.76 3.87 0.87 19.84 menengah rendah 17.00 0.70 16.00 17.81 menengah tinggi 19.00 1.31 17.00 22.15 deskriptif klasifikasi ipmvariabel x5 x4 x3 x2 x1 x8 x7 x6 model probit spasial pada faktor-faktor yang mempengaruhi klasifikasi ipm di pulau jawa jurnal cauchy – issn: 2086-0382 207 pola hubungan antara variabel prediktor dan variabel respon melalui analisis korelasi dan diagram pencar. berdasarkan tabel 4.3 dapat diketahui bahwa hasil pengujian dengan menggunakan 𝛼 = 5% diperoleh kedelapan variabel prediktor, yaitu persentase penduduk yang tinggal di perkotaan (x1), persentase penduduk yang berpendidikan di atas sltp (x2), rasio ketergantungan (x3), peranan sektor industri dalam pdrb (x4), persentase penduduk miskin (x5), persentase penduduk yang mengalami keluhan kesehatan (x6), tingkat pengangguran terbuka (x7), dan rata-rata umur kawin pertama wanita (x8) memiliki hubungan yang nyata terhadap ipm (y). tabel 4.3 korelasi antara variabel prediktor dan variabel respon variabel prediktor koefisien korelasi pvalue keterangan x1 0.736 0.000 ada korelasi x2 0.873 0.000 ada korelasi x3 -0.288 0.002 ada korelasi x4 -0.506 0.000 ada korelasi x5 0.237 0.010 ada korelasi x6 -0.666 0.000 ada korelasi x7 0.380 0.000 ada korelasi x8 0.830 0.000 ada korelasi terdapat lima variabel prediktor yang berkorelasi positif terhadap ipm yaitu x1, x2, x5, x7, dan x8. korelasi positif ini berarti bahwa jika terjadi peningkatan pada variabel x1, x2, x5, x7, dan x8 maka akan mengakibatkan semakin tingginya nilai ipm begitu juga sebaliknya. sementara itu variabel x3, x4 dan x6 berkorelasi negatif yang berarti bahwa jika terjadi penurunan pada variabel tersebut maka akan berakibat pada peningkatan nilai ipm. nilai moran’s i langkah berikutnya untuk megidentifikasi adanya dependensi spasial pada ipm digunakan nilai moran’s i. berdasarkan hasil yang telah diperoleh dengan menggunakan software matlab menunjukkan nilai moran’s i sebesar 0.3110. apabila dibandingkan dengan i0 yang bernilai 0.0288 maka moran’s i pada ipm memiliki nilai yang lebih tinggi. hal ini mengindikasikan bahwa ipm di pulau jawa menunjukkan pola mengelompok dan memiliki karakteristik pada wilayah yang berdekatan. pengujian model spasial pengujian model spasial pada penelitian ini menggunakan lagrange multiplier dan robust lagrange multiplier. hasil pengujian yang diperoleh digunakan untuk mengetahui model spasial yang sesuai untuk memodelkan ipm di pulau jawa. berdasarkan hasil yang telah diperoleh dari software matlab untuk masingmasing pengujian adalah sebagai berikut : lagrange multiplier lag lagrange multiplier lag digunakan untuk mengetahui dependensi spasial lag pada ipm di pulau jawa. hipotesis yang digunakan adalah sebagai berikut : h0 : 𝜌 = 0 (tidak adanya dependensi spasial autoregressive dalam model) h1 : 𝜌 ≠ 0 (ada dependesi spasial autoregressive dalam model) berdasarkan hasil perhitungan diperoleh nilai lm-lag adalah 45.9881 yang lebih besar dari nilai chi square dengan derajat bebas satu yaitu 3.841. hal ini diperkuat dengan nilai p-value sebesar 0 yang lebih kecil dari 5%  sehingga tolak h0. dengan demikian dapat disimpulkan bahwa terdapat dependensi spasial lag dalam memodelkan ipm. lagrange multiplier error lagrange multiplier error digunakan untuk mengetahui adanya dependensi error pada pemodelan ipm di pulau jawa. hipotesis yang digunakan adalah sebagai berikut : h0 : 𝜆 = 0 (tidak ada dependensi error spasial) h1 : 𝜆 ≠ 0 (ada dependensi error spasial) nilai lm error yang dihasilkan dari software matlab adalah sebesar 20.9876 yang lebih besar jika dibandingkan dengan nilai chi square dengan derajat bebas satu yaitu 3.841. hal ini diperkuat dengan nilai p-value sebesar 0 yang lebih kecil dari 𝛼 = 5% sehingga tolak h0. dengan demikian dapat disimpulkan bahwa terdapat dependensi spasial error. berdasarkan hasil pengujian lagrange multiplier, baik koefisien spasial lag maupun koefisien spasial error signifikan pada 𝛼 = 5%. oleh karena itu, untuk mengetahui model spasial apa yang sesuai untuk memodelakn ipm di pulau jawa maka digunakan pengujian robust lagrange multiplier. robust lagrange multiplier lag pengujian robust lagrange multiplier lag digunakan untuk mengetahui adanya dependensi spasial lag dengan asumsi bahwa koefisien spasial error tidak bernilai nol. feni ira puspita, vita ratnasari, purhadi 208 volume 2 no. 4 mei 2013 hipotesis yang digunakan adalah sebagai berikut : h0 : 𝜌 = 0 dan 𝜆 = 0 h1 : 𝜌 ≠ 0 dan 𝜆 ≠ 0 berdasarkan hasil perhitungan diperoleh nilai lm-lag adalah 25.974 yang lebih besar jika dibandingkan dengan nilai chi square dengan derajat bebas satu yaitu 3.841. hal ini diperkuat dengan nilai p-value sebesar 0 yang lebih kecil dari 𝛼 = 5% sehingga tolak h0. dengan demikian dapat disimpulkan bahwa terdapat dependensi spasial lag pada saat koefisien spasial error tidak sama dengan nol. oleh karena itu, untuk memodelkan ipm di pulau jawa mengikuti model spatial autoregressive (sar). roburt lagrange multiplier error pengujian robust lagrange multiplier error digunakan untuk mengetahui adanya dependensi spasial error dengan asumsi bahwa koefisien spasial lag tidak bernilai nol. hipotesis yang digunakan adalah sebagai berikut : h0 : 𝜌 ≠ 0 dan 𝜆 = 0 h1 : 𝜌 ≠ 0 dan 𝜆 ≠ 0 berdasarkan hasil perhitungan diperoleh nilai robust lagrange multiplier error adalah 0.8669 yang lebih kecil jika dibandingkan dengan nilai chi square dengan derajat bebas satu yaitu 3.841. hal ini diperkuat dengan nilai p-value sebesar 0 yang lebih kecil dari 𝛼 = 5% sehingga gagal tolak h0. dengan demikian dapat disimpulkan bahwa tidak terdapat dependensi spasial error pada saat koefisien spasial lag tidak sama dengan nol. regresi probit spasial tabel 5 hasil estimasi mcmc parameter model regresi probit spasial lag variabel estimasi pr(>|z|) konstan -19.73480 0.01062 x6 -0.157550 0.01007 x8 1.3104200 0.00298 rho 0.3953000 0.00666 berdasarkan hasil tabel 5, variabel yang signifikan terhadap model regresi probit spasial adalah x6 dan x8. sehingga variabel prediktor yang berpengaruh pada klasifikasi ipm di pulau jawa adalah persentase penduduk yang mengalami keluhan kesehatan (x6), dan rata-rata umur kawin pertama wanita (x8). penutup metode estimasi parameter yang digunakan pada probit spasial adalah teknik markov chain monte carlo (mcmc) dengan algoritma gibbs sampler pendekatan inferensia bayesia, atau disingkat mcmc gibss sampler. variabel-variable yang mempengaruhi klasifikasi ipm di pulau jawa adalah persentase penduduk yang mengalami keluhan kesehatan (x6), dan rata-rata umur kawin pertama wanita (x8). berdasarkan hasil penelitian yang telah diperoleh, pengembangan lebih lanjut dapat dilakukan dengan menggunakan highest posterior density (hpd) dan bayes faktor sebagai metode pengujian parameter dan model. penelitian ini masih menggunakan matriks pembobot queen contiguity, sehingga pada penelitian selanjutnya dapat dikembangkan menggunakan matriks pembobot lain, misalnya pembobot jarak. lebih lanjut, metode mcmc gibbs sampler untuk pemodelan regresi probit spasial ini dapat digunakan untuk data dan kasus lain yang lebih aplikatif. ucapan terimakasih pada penelitian ini 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[35] yurviany., (2007). analisis regresi logistik pada data indeks pembangunan manusia di propinsi jawa timur. skripsi jurusan statistika fmipa its. surabaya. pendahuluan teori dasar 1. model regresi probit 2. analisis spasial 3. uji dependensi spasial moran’s i langrange multiplier test (lm test) 4. model regresi probit spasial 5. indeks pembangunan manusia (ipm) angka harapan hidup tingkat pendidikan standar hidup layak metode penelitian diskusi / pembahasan nilai moran’s i pengujian model spasial lagrange multiplier lag lagrange multiplier error robust lagrange multiplier lag roburt lagrange multiplier error regresi probit spasial penutup ucapan terimakasih referensi correlation test application of supplier’s ranking using topsis and ahp-topsis method cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 65-73 submitted: 8 february 2016 reviewed: 4 march 2016 accepted: 1may 2016 correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati postgraduate statistics program, brawijaya university, malang email: ikayuniwati@yahoo.com abstract the supplier selection process can be done using multi-criteria decision making (mcdm) methods in firms. there are many mcdm methods, but firms must choose the method suitable with the firm condition. company a has analyzed supplier’s ranking using topsis method. topsis method has a marjor weakness in its subjective weighting. this flaw is overcome using ahp method weighting having undergone a consistency test. in this study, the comparison of supplier’s ranking using topsis and ahp-topsis method used correlation test. the aim of this paper is to determine different result from two methods. data in suppliers’ ranking is ordinal data, so this process used spearman’s rank and kendall’s tau b correlation. if most of the ranked scored are same, kendall’s tau b correlation should be used. the other way, spearman rank should be used. the result of this study is that most of the ranked scored are different, so the process used spearman rank p-value of spearman’s rank correlation of 0.505. it is greater than 0.05, means there is no statistically significant correlation between two methods. furthermore, increment or decrement of supplier’s ranking in one method is not significantly related to the increment or decrement of supplier’s ranking in the second method. keywords: supplier’s ranking, topsis method, ahp-topsis method, and correlation test. introduction in the competitive business environmet, the companies have to follow strategies to achieve reduced costs and higher quality. one of them, they must select the right suppliers. the supplier selection process requires evaluating various criteria in suppliers performance. supplier ranking methods are required to know the right future supplier. company a has implemented entry of technique for order preference by similiarity to ideal solution (topsis) matrix with loss of performance of suppliers and subjective weighting for topsis method. the main advantages of using this method is its simplicity. it takes into account all types of criteria (negative or positive criteria). it is also rational and understandable method. company a wants to know the ranking of suppliers with ahp-topsis method. in topsis method, the entry of matrix topsis is the company's losses in 50 times procurement of supplier performance in each criteria. in this study, ahp-topsis method used entry matrix topsis with oppinion of six experts. then, the subjective weightings were replaced by analytic hierarchy process (ahp) method weighting. ahp method weighting having undergone a consistency test. data of supplier’s ranking on both methods are ordinal scale. because of the data characteristic, we used spearman’s rank or kendals tau b correlation. the spearman correlation is less sensitive than the pearson correlation to strong outliers that are in the tails of both samples. mailto:ikayuniwati@yahoo.com correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 66 that is because spearman's rho limits the outlier to the value of its rank. while kendall rank correlation coefficient, commonly referred to as kendall's tau b coefficient is a statistic used to measure the ordinal association between two measured quantities. it is a measure of rank correlation, the similarity of the orderings of the data when ranked by each of the quantities. the result of correlation test was used to answer problem of company a. the purpose of this study is to determine the relationship of the decrease or increase of supplier ranking in topsis and ahp-topsis method. then company a can make decision in supplier selection problem, it will choose topsis or ahp-topsis method to identify criteria for evaluation of suppliers. there are six criterias of supplier selection in company a. they are quality material, price, delivery time, purchase time, supply capability, and response to claim. theoritical review supplier selection problem have been discussed since 1960s. the study conducted by dickson [1] identified 23 criteria to evaluate 170 buyers, namely: quality; delivery; performance history; warranties and claim policies, production facilities and capacity; price; technical capability; financial position; procedural compliance; communication system; reputation and position in industry; desire for business; management and organitation; operating controls; repair services; attitude; impression; packaging ability; labor relation records; geographical location; amount of pass bussiness; training aids; and reciprocal arrangement. ellram [2] studied three principal criteria, namely: financial statement of supplier; organizational culture of supplier; and technological state of supplier. wirdianto and unbersa [3] developed criteria of supplier selection using analytic hierarchy process in company x. there are six criterias, those are condition of company, completeness of company document, price, quality, delivery, and service. various multi-criteria decision making (mcdm) methods have been suggested for supplier selection problem. jannah, et al [4] used ahp to reassure raw material of good quality. the researchers used quality, cost, delivery, flexibility, responsiveness (qcdfr) model. there are several suppliers which come from four different areas, namely madura, bondowoso, tulungagung, and malang. moreover, supplier’s ranking according to area were madura, tulungagung, bondowoso, and malang. wangchen and phipon [5] applied ahp and topsis for supplier selection problem. based on ahp, they have calculated weights for each criterion and inputted those weights to topsis method to rank suppliers. then shahroudi and roydel [6] used anp-topsis to evaluate suppliers in iran’s auto industry. the purpose of researchers integrated approach of anp-topsis choosed the best supplier and defined the optimum quantities order among selected suppliers by using multi-objective linier programming (molp). lavanpriya, et al [7] integrated taguchi loss function and topsis method to select optimal supplier in manufacturing industry. they used two characteristics of taguchi loss function, namely higher is better and lower is better. loss of each criterion were inputted in matrix topsis to rank suppliers. afterwards, onder and dag [8] combined ahp and topsis approaches to select cable company supplier. the company detected eight criteria for procurement of electrolytic copper cathode. these are origin, quality, availability, cost, delivery requerements, cost of conveyance, reliability of supplier, and quality certificates. ginting, et al [9] evaluated raw material suppliers using ahp and loss function. they calculated ahp weighting and loss function weighting. then they multiplyed ahp weighting and loss function weighting to determine total loss. bas ed on total loss, they selected the best suppliers performance. moreover, bhatt [10] integrated ahp and topsis approach to select the best supplier in automotive industry. using topsis method to rank the suppliers technique for others preference by similarity to ideal solution (topsis) method was first introduced by hwang and yoon [11], the main idea topsis method based on the chosen alternative should have the shortest geometric distance from the positive ideal solution (the optimal solution) and should have the longest geometric distance from the negative ideal solution correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 67 (non-optimal solutions). according wangchen and phipon [5]; tzeng and huang [12] general measures of topsis method consists of seven steps. 1. establish a decision matrix for ranking. when given a set of alternatives, 𝐴 = {𝐴𝑘 |𝑘 = 1, … , 𝑛 }, and a group of criteria 𝐶 = {𝑋𝑗 |𝑗 = 1, … , 𝑚 }, where 𝐶 = {𝑋𝑗 |𝑗 = 1, … , 𝑚 } indicates a set of performance ratings and 𝑊 = {𝑤𝑗 |𝑗 = 1, … , 𝑚 } set of weighting. the structure of the matrix can be expressed in the equation (1). 𝐷 = ( 𝑥11 𝑥12 ⋯ 𝑥1𝑚 𝑥21 𝑥22 ⋯ 𝑥2𝑚 ⋮ 𝑥𝑛1 ⋮ 𝑥𝑛2 ⋮ ⋯ ⋮ 𝑥𝑛𝑚 ) (1) 2. normalize the decision matrix 𝐷 by using the following formula (2) 𝑟𝑘𝑗 = 𝑥𝑘𝑗 √∑ 𝑥𝑘𝑗 2𝑛 𝑖=1 𝑘 = 1, … 𝑛; 𝑗 = 1, … , 𝑚. (2) 3. construct the weighted normalized decision matrix by multiplying the normalized decision matrix by its associated weights. the weighted normalized value (3) is calculated as 𝑣𝑘𝑗 = 𝑤𝑗 𝑟𝑘𝑗 , 𝑘 = 1, … 𝑛; 𝑗 = 1, … , 𝑚 (3) 4. determine the positive ideal solution and negative ideal solution the ideal solution positive is denoted a+ by and negative ideal solution is denoted by a-, they can be expressed in equation (4) and (5) 𝐴+ = {(max 𝑣𝑘𝑗 |𝑗 ∈ 𝐽)(min 𝑣𝑘𝑗 |𝑗 ∈ 𝐽 ′), 𝑘 = 1,2,3, … 𝑛} = {𝑣1 +, 𝑣2 +, … 𝑣𝑗 +, … , 𝑣𝑚 + } (4) 𝐴− = {(𝑚𝑖𝑛 𝑣𝑘𝑗 |𝑗 ∈ 𝐽)(max 𝑣𝑘𝑗 |𝑗 ∈ 𝐽 ′), 𝑘 = 1,2,3, … 𝑛} = {𝑣1 −, 𝑣2 −, … , 𝑣𝑗 −, … , 𝑣𝑚 − } (5) where 𝑣𝑘𝑗 : element of matrix 𝑉 in kth row and jth column 𝐽 ∶ {𝑗 = 1, 2, 3, … , 𝑛 and 𝑗 is associated with benefit criteria} 𝐽′: {𝑗 = 1, 2, 3, … , 𝑛 and 𝑗 is associated with cost criteria} 5. calculate the separation measure separation measure is a measurement of the geometric distance of an alternative to the positive ideal solution and negative ideal solution. the separation measure of each alternative from the positive ideal one (6) is given by : 𝑆𝑘 + = √∑ (𝑣𝑘𝑗 − 𝑣𝑗 +) 2 𝑛 𝑗=1 , where k = 1, 2, 3, … n (6) similarly, the separation of each alternative from negative ideal one (7) is given by : correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 68 𝑆𝑘 − = √∑ (𝑣𝑘𝑗 − 𝑣𝑗 −) 2 𝑛 𝑗=1 , where k = 1, 2, 3, … n (7) 6. calculate the relative closeness to the ideal solution the relative closeness of with ai respect to a+ is defined as equation (8) 𝐶𝑘 = 𝑆𝑘 − 𝑆𝑘 − + 𝑆𝑘 + , where 0 < 𝐶𝑘 < 1 and ∀𝑘 = 1,2,3, … 𝑚 (8) 7. ank the preference order the larger the 𝐶𝑘 value, the better performance of the suppliers using ahp-topsis method to rank the suppliers ahp-topsis method is mcdm method which combined ahp and topsis method. based on ahp, we have calculated weights for each criteria and inputted those weights to topsis method to rank suppliers. analytic hierarchy process (ahp) developed in the 1970s by thomas saaty [12]. it has beeen used for analyzing complex decision. there are several steps in ahp method. the first step covers define the problem, determine criteria, structure the decision hierarchy. afterwards, we collected data from experts using gradient scale of saaty in table 1. tabel 1. gradient scale of pairwise comparison in ahp intencity of importance definition explanation 1 equal importance two activities have equal contribute to the objective 3 moderate importance experience and judgment slightly favor one activity over another 5 strong importance experience and judgment strongly favor one activity over another 7 very strong on demonstrated importance an activity is favored very strongly over another 9 extreme importance the evidence favoring one activity over another is of the highest possible order of affirmation 2, 4, 6, 8 for compromise between the above values (fuzzy input) sometimes one needs to interpolate a compromise judgment numerically reciprocal 𝑎𝑗𝑖 = 1/𝑎𝑖𝑗 sources: bushan and rai [13], saaty [14] weighting the criteria by multiple experts avoid the bias decision making. saaty [14] said geometric mean to obtain a sigle assesment of variety different opinions on the same criteria. geometric mean calculated by the following formula (9) �̅�𝑔 = √∏ 𝑋𝑖 𝑛 𝑖=1 𝑛 (9) where �̅�𝑔 : geometric mean 𝑛 : number of expert 𝑋𝑖 : expert scored correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 69 afterwards, the result of pairwise comparison in each criterion inputted in a square matrix. it can be represented by matrix a in equation (10) 𝐴 = [𝑎𝑖𝑗 ] = [ 1 𝑎12 ⋯ 𝑎1𝑛 𝑎21 1 ⋯ 𝑎2𝑛 ⋯ 𝑎𝑛1 ⋯ 𝑎𝑛2 ⋯ ⋯ ⋯ 1 ] , where 𝑖, 𝑗 = 1,2,3, … 𝑛 (10) the next step of ahp, normalize matrix by using formula (11) 𝑎𝑖𝑗 ′ = 𝑎𝑖𝑗 ∑ 𝑎𝑖𝑗 𝑛 𝑖=1 (11) after normalization matrix process, it will be determined weighting in each criterion by formula (12) 𝑤𝑖 = ∑ 𝑎𝑖𝑗 ′𝑛 𝑗=1 𝑗 (12) saaty [15] said to avoid inconsistency in ahp, we must use maximum eigen value (𝜆𝑚𝑎𝑥 ). it purposed for calculate effectiveness of decision. mukherjee [16] calculated maximum eigen value (13) by 𝜆𝑚𝑎𝑥 = ∑ 𝑎𝑖𝑗 𝑤𝑖 𝑛 𝑖=1 𝑛 , 𝑖 = 1,2,3, … 𝑛 (13) for measure consistency index (ci) adopt the value (14) 𝐶𝐼 = 𝜆𝑚𝑎𝑥 − 𝑛 𝑛 − 1 (14) accept the estimate of w if the consistency ratio (cr) of ci that random matrix is significant small. the cr is obtained comparing the ci with an average random consistency index (ri). saaty [15] showed that the matrix is consistent, then the cr value ≤0,05 for n = 3, cr value ≤0,08 for n = 4, and cr value ≤0,1 for n≥5. ri value can be seen in table 2. tabel 2. list of random consistency index (ri) value n 3 4 5 6 7 8 9 10 11 12 13 14 15 ri 0,58 0,90 1,12 1,24 1,32 1,41 1,45 1,49 1,51 1,48 1,56 1,57 1,59 sources: tzeng and huang [12] in ahp-topsis method, ahp method weighting in formula (12) used in formula (3). based on the process we got supplier’s ranking from formula (8) in the second method. correlation test according to karami [17] comparison of two methods of mcdm can be done using spearman rank correlation test. spearman rank correlation coefficient (rho) shows the strength of the relationship of two variables or two mcdm methods were compared. the formula to get the value of rho [18] as follows: 𝑟𝑠 = 1 − 6 ∑ 𝑑𝑖 2𝑛 𝑖=1 𝑛2 − 𝑛 (15) where 𝑑𝑖 : the difference between ranking correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 70 n: the sample size (supplier) hypothesis of spearman rank correlation test as follows h0 : 𝑟𝑠 = 0 h1 : 𝑟𝑠 ≠ 0 significance test p-value is known through the test t with 𝑡 = 𝑟𝑠 √ 𝑛−2 1−𝑟𝑠 2. if the p-value less than or equal to 0.05, then reject h0, the conclusion is there is no statistically significant correlation between two variable, so increase and decrease in one variable is not significantly related to increase or decrease in second variable. karami [17] also suggests a comparison of two methods of mcdm can use kendall's tau b correlation to determine the strength of the relationship between two variables or two methods of mcdm. the test is used when a lot of ranked score are same. the formula used to calculate the correlation kendall's tau b [18]: 𝑇 = 2𝑆 𝑛(𝑛 − 1) (16) where s = total score of all (grand total) = sum of (concordant discordant) n = sample size hypothesis of correlation kendall's tau b as follows h0 : 𝑇 = 0 h1 : 𝑇 ≠ 0 significance test p-value is known through the t test, if the p-value less or equal to 0.05, then reject h0, thus conclusion there is not relationship between these two variables. so increase and decrease in one variable is not significantly related to increment or decrement in the second variable. results and discussion in this study, we used case study in company a which have six criterias of supplier performance. then we analyzed their performance in topsis and ahp-topsis method. by using topsis method, the ranking of alternative suppliers are calculated. table 3 shows the evaluation result and final ranking of alternative suppliers. tabel 3. topsis method result alternatives 𝑆𝑘 + 𝑆𝑘 − 𝐶𝑘 supplier a 0.026 0.330 0.926 supplier b 0.098 0.293 0.749 supplier c 0.082 0.184 0.691 supplier d 0.294 0.147 0.333 supplier e 0.169 0.168 0.498 sources: data calculated by researcher depends of the 𝐶𝑘 value, the supplier’s ranking in topsis method from top to bottom order are supplier a, supplier b, supplier c, supplier e, and supplier d. afterwards, we analyzed supplier performance in ahp-topsis method. the weights of criteria to be used in evaluation process are calculated by using ahp method. in this phase, the expert team are given the task of forming individual pairwise comparison by using gradient scale. then, we calculated geometric mean in square matrix which expressed in table 4. correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 71 tabel 4. the pairwise comparison matrix for criteria c1 c2 c3 c4 c5 c6 c1 1.000 1.308 1.201 4.155 2.840 2.720 c2 0.765 1.000 1.000 2.646 1.661 3.271 c3 0.833 1.000 1.000 4.583 3.460 4.524 c4 0.241 0.378 0.218 1.000 1.886 2.498 c5 0.352 0.602 0.289 0.530 1.000 3.979 c6 0.368 0.306 0.221 0.400 0.251 1.000 sources: data calculated by researcher where c1 is quality material, c2 is price, c3 is delivery time, c4 is purchase time, c5 is supply capability, c6 is response to claim. then the next step of ahp method determined eigen value. tabel 5. eigen value in each criteria criteria of supplier selection eigen value quality material 0,2651 price 0,2029 delivery time 0,2689 purchase time 0,0982 supply capability 0,1091 response to claim 0,0557 sources: data calculated by researcher table 5 shows that the most important criteria in supplier selection in company a is delivery time. based on eigen value, the importance of criteria can be rated from top to bottom order are delivery time criteria, quality material criteria, price criteria, supply capability criteria, purchase time criteria, and response to claim criteria. moreover, consistency test will be done by determined 𝜆𝑚𝑎𝑥 , ci, and cr. the result of 𝜆𝑚𝑎𝑥 and ci are 0.6543 and 0.091. then ci value were compared with ir in ordo 6 x 6. the result of cr is 0.073, so eigen value can be used in topsis method weighting. it expressed in figure 1. figure 1. topsis weighting in ahp-topsis method after consistency test of ahp method, the process continued in topsis method. by using ahp weighting method, the result of ahp-topsis method can be showed in table tabel 6. ahp-topsis method result alternatives 𝑆𝑘 + 𝑆𝑘 − 𝐶𝑘 supplier a 0.037 0.226 0.860 supplier b 0.170 0.136 0.443 supplier c 0.158 0.110 0.441 supplier d 0.137 0.164 0.543 supplier e 0.199 0.072 0.267 correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 72 sources: data calculated by researcher depends of the 𝐶𝑘 value in table 6, the supplier’s ranking in topsis method from top to bottom order are supplier a, supplier d, supplier b, supplier c, and supplier e. correlation test indicated relationship between two methods, so each supplier should be given scored rank. it can be showed in table 7. afterwards they will be observed to determine number of the same scored. tabel 7. supplier’s ranking in two methods alternatives topsis method ahp-topsis method supplier a 1 1 supplier b 2 3 supplier c 3 4 supplier d 5 2 supplier e 4 5 sources: data calculated by researcher most of scored rank in two methods are different, so we used speraman rank correlation. the result of spearman rho can be expressed in table 8. sig. (2-tailed) in spearman’s rho is 0.505. it is less than 0.05, so the conclusion is there is no statistically significant correlation between two variable, so increase and decrease in one variable is not significantly related to increase or decrease in second variable. conclusion multi-criteria decision making becomes important strategic decision in complex and competitive business life. the company should choose the right method of mcdm, in order to they can select the best supplier in the future. correlation test (speraman’s rho and kendall’s tau b correlation) can be used to determine supplier’s ranking between two methods. if suppliers’s ranking of two method is different, the process can be continued with sensitivity and accuracy method. it purposed to determine the best mcdm. reference [1] g. w. dickson, "an analysis of vendor selection systems and decisions," pp. --, 1996. [2] l. m. ellram, g. a. zsidisin, s. p. siferd and m. j. stanly, "the impact of purchasing and supply management activities on corporate success," journal of supply chain management, vol. 38, no. 4, pp. 4-17, 2002. [3] e. wirdianto and e. unbersa, "aplikasi metode analytical hierarchy process dalam menentukan kriteria penilaian supplier," jurnal teknik industri, vol. 2, no. 29, pp. --, 2008. tabel 8. correlations test result metode_topsis metode_ahptopsis spearman's rho metode_topsis correlation coefficient 1.000 .400 sig. (2-tailed) . .505 n 5 5 metode_ahptopsis correlation coefficient .400 1.000 sig. (2-tailed) .505 . n 5 5 correlation test application of supplier’s ranking using topsis and ahp-topsis method ika yuniwati 73 [4] m. jannah, m. fakhry and rahmawati, "pengambilan keputusan untuk pemilihan supplier bahan baku dengan pendekatan analytic hierarchy process di pt pahala sidoarjo," jurnal agrointek, vol. 5, pp. 88-97, 2011. [5] p. w. bhutia and r. phipon, "appication of ahp and topsis method for supplier selection problem," iosr journal of engineering (iosrjen) volume, vol. 2, pp. 43-50, 2012. [6] k. shahroudi and h. rouydel, "using a multi-criteria decision making approach (anp-topsis) to evaluate suppliers in iran†™s auto industry," international journal of applied operational researchan open access journal, vol. 2, no. 2, pp. 0-0, 2012. [7] c. lavanpriya, v. m. mauralidaran and c. laksmanpriya, "a case study of integrating the taguchi loss function and topsis method to select an optimal supplier in a manufacturing industry," journal ifbm, vol. 1, pp. 18-22, 2013. [8] e. onder and s. dag, "combining analytical hierarchy process and topsis approaches for supplier selection in a cable company," journal of business, economics and finance (jbef), vol. 2, pp. 56-74, 2013. [9] e. s. ginting, "evaluasi supplier bahan baku pada pt xyz dengan menggunakan metode ahp dan loss function," pp. --, 2015. [10] n. bhatt, "an integrated ahp-topsis approach in supplier selection: an automotive industry as a case study," the international journal of business \& management, vol. 3, no. 8, pp. 160--, 2015. [11] k. p. yoon and c.-l. hwang, multiple attribute decision making: an introduction, vol. 104, sage publications, 1995, pp. --. [12] g.-h. tzeng and j.-j. huang, multiple attribute decision making: methods and applications, crc press, 2011, pp. --. [13] n. bhushan and k. rai, strategic decision making: applying the analytic hierarchy process, springer science \& business media, 2007, pp. --. [14] t. l. saaty, "pengambilan keputusan bagi para pemimpin," terjemahan). pt pustaka binaman pressindo, jakarta, pp. --, 1993. [15] t. l. saaty, "the analytic hierarchy process: planning, priority setting, resources allocation," new york: mcgraw, pp. --, 1980. [16] k. mukherjee, "analytic hierarchy process and technique for order preference by similarity to ideal solution: a bibliometric analysis †˜ from†™past, present and future of ahp and topsis," international journal of intelligent engineering informatics, vol. 2, no. 2-3, pp. 96-117, 2014. [17] a. karami, "utilization and comparison of multi attribute decision making techniques to rank bayesian network options," pp. --, 2011. [18] s. siegel, statistic nonparametric for behavioral sciences, mcgraw-hill book company, inc, 1956. [19] t. saaty, "fundamental decision making and priority thinking with the ahp," pp. --, 1994. abstract introduction theoritical review using topsis method to rank the suppliers using ahp-topsis method to rank the suppliers correlation test results and discussion conclusion reference suma’inna dan gugun gumilar aproksimasi numerik belah dua dan newton-raphson pada estimasi parameter distribusi weibull indira p. kinasih jurusan pendidikan matematika ikip mataram e-mail : indiraputeri@gmail.com abstrak distribusi masa aktif produk bergaransi, misalkan kendaraan bermotor dan mesin fotokopi merupakan hal yang menarik untuk dipelajari. faktanya, distribusi kerusakan atau masa hidup produk bergaransi seringkali memiliki plot fungsi kepadatan peluang yang condong ke kanan. terkait hal ini, distribusi weibull termasuk salah satu ragam distribusi non-negatif yang biasanya sesuai untuk memodelkan distribusi data. karena fungsi distribusi kerusakan merupakan fungsi dari parameter, maka estimasi parameter selalu menjadi kebutuhan utama pada masalah pemodelan distribusi kerusakan. ketiadaan solusi analitis pada maksimasi fungsi persekitaran distribusi weibull mengakibatkan pendekatan numerik menjadi solusi pilihan. dua macam metode numerik, yaitu newton-raphson dan belah dua disajikan pada makalah ini untuk diamati perbandingan kecepatan dan keakuratan keduanya dalam pencarian solusi. data yang digunakan adalah data kerusakan mesin fotokopi berdasarkan usia dan banyak salinan yang terekam pada saat kerusakan selama 4,5 tahun, bersumber dari penelitian bulmer dan eccleston [1]. kata kunci : distribusi weibull, newton-raphson, belah dua, parameter , fungsi persekitaran abstract life time distribution of warranted product, for example like automobile and photocopier is an interesting studies. based on the fact that mostly, life distribution of warranted product is somewhat skewed to the right, this suggests that usual failure distribution such as weibull distribution may provide a reasonable fit to the data. related to these studies, parameter estimation is directly applicable since life distribution function is a function of parameters. there are no closed solution for weibull likelihood function maximation. therefore, numerical approximation could be an alternative solution. in this paper, newton -raphson and bisection method were employed to estimate the scale and shape parameter of weibull distribution for photocopier failure data for since 4,5 years. performance of those method were compared based on their approximation result and how fast they get into the solution. data were taken from bulmer and eccleston (2003) research about photocopier realibility model. keywords : weibull distribution, newton-raphson, bisection, parameter, likelihood function pendahuluan berdasarkan ross (1976), distribusi weibull merupakan distribusi yang banyak digunakan dalam praktek mekanika atau permesinan. hal ini disebabkan kesesuaian sifatnya terhadap berbagai permasalahan bidang tersebut (versatility). secara khusus, distribusi weibull terkait erat dengan pemodelan kekuatan bahan, waktu rusak komponen, peralatan, serta sistem elektronik dan mekanik. stone dan heeswijk (1977) memodelkan distribusi kerusakan komponen elektronika berdasarkan distribusi weibull. bulmer dan eccleston [1] juga telah memodelkan keandalan mesin fotokopi menggunakan beragam model distribusi weibull. pemodelan distribusi masa hidup tentu melibatkan fungsi padat peluang, fungsi distribusi, fungsi ketahanan, fungsi laju kerusakan dan fungsi karakteristik lainnya. fungsi-fungsi tersebut tak lain merupakan fungsi parameter. oleh karena itu, pengetahuan mengenai estimasi parameter distribusi selalu menjadi bagian penting pada permasalahan pemodelan distribusi masa hidup suatu objek. adapun metode estimasi yang sering digunakan antara lain adalah estimasi persekitaran maksimum (mle), metode estimasi kuadrat terkecil (least square), dan metode estimasi bayesian. pada kasus distribusi weibull univariat, stone dan heeswijk (1977) menguraikan estimasi parameter weibull 3 parameter dengan menggunakan metode grafik dan mle untuk data tersensor. estimasi ini digunakan untuk model distribusi kerusakan alat-alat elektronik. sedangkan bhattacharya dan bhattacharjee [2] , mailto:indiraputeri@gmail.com indira p. kinasih 194 volume 2 no. 4 mei 2013 telah mempelajari estimasi parameter weibull 2 parameter pada model kecepatan angin yang digunakan untuk tenaga pembangkit listrik. metode yang digunakan adalah mle dan least square. pada banyak pembahasan mengenai estimasi parameter distribusi weibull dengan metode persekitaran maksimum, aproksimasi numerik newton-raphson seringkali digunakan untuk mendapatkan solusi dari fungsi persekitaran. hal ini wajar, dikarenakan oleh ketiadaan solusi analitis atau tertutup dari fungsi persekitaran distribusi weibull. makalah ini meninjau kembali permasalahan estimasi persekitaran maksimum untuk parameter weibull dua parameter dengan mengujicobakan metode aproksimasi numerik lainnya, yaitu belah dua (biseksi). selain itu, metode newton-raphson tetap disajikan untuk kemudian dibandingkan akurasi dan kecepatannya dalam menemukan solusi. distribusi weibull dua parameter dan estimasi persekitaran maksimumnya suatu variabel acak kontinu non-negatif 𝑋, dikatakan memiliki distribusi weibull dua parameter ditulis sebagai 𝑋 ∼ 𝑊𝑒𝑖𝑏𝑢𝑙𝑙(𝛼, 𝛽) dengan 𝛼 > 0 dan 𝛽 > 0, apabila memiliki fungsi padat peluang yang memenuhi persamaan berikut 𝑓(𝑥; 𝛼, 𝛽) = 𝛽 𝛼 𝛽 𝑥 𝛽−1 ⋅ 𝑒 1 𝛼𝛽 𝑥𝛽 (1) 𝛼 merupakan parameter skala yang menunjukkan karakteristik hidup distribusi weibull. sedangkan 𝛽 merupakan parameter bentuk yang menunjukkan ukuran penyebaran data waktu hidup. semakin besar nilai 𝛽, grafik fungsi padat peluang akan melandai, demikian pula sebaliknya. gambar 1. plot fungsi distribusi kumulatif weibull dua parameter misalkan 𝑋 dipandang sebagai peubah acak yang merepresentasikan masa hidup atau waktu antar kerusakan suatu objek, fungsi padat peluang dapat diartikan sebagai peluang objek tersebut bertahan selama 𝑥. artinya, jika dituliskan 𝑓(25), fungsi ini menunjukan nilai peluang suatu objek dapat hidup selama 25 satuan waktu (detik, menit, jam, hari, bulan, tahun, dst). dalam hal ini, nilai 𝛼 menunjukkan bahwa 63,212% masa hidup objek berada pada nilai 𝑥 = 𝑎. gambar 2. plot fungsi padat peluang distribusi weibull dua parameter. kesimpulan ini dapat diperoleh apabila dilakukan pengamatan terhadap fungsi distribusi kumulatif weibull, yang tak lain merupakan integrasi persamaan (1), yaitu 𝐹(𝑥) = 1 − 𝑒 − 1 𝛼𝛽 𝑥𝛽 (2) pada nilai 𝑥 = 𝛼 = 1, 𝐹(𝛼) akan selalu mencapai nilai 1 − 𝑒 −1 = 0.63212. visualisasi peranan parameter α dan β dapat diamati masingmasing pada plot fungsi distribusi kumulatif dan fungsi padat peluang di gambar 1 dan gambar 2. estimasi persekitaran maksimum diawali dengan mengkonstruksi fungsi persekitaran weibull. fungsi inilah yang nantinya akan dimaksimasi, dengan mencari nilai maksimum dari �̂� dan �̂�. konstruksi fungsi persekitaran didasarkan pada fungsi padat peluang di persamaan 1. misalkan 𝑋1, … , 𝑋𝑛 merupakan peubah acak yang berdistribusi identik dan independen dengan fungsi kepadatan peluang 𝑓(𝑥; 𝛼, 𝛽). fungsi persekitaran weibull diberikan oleh 𝐿(𝛼, 𝛽; 𝑥𝑖 ) = ∏ 𝑓(𝛼, 𝛽; 𝑥𝑖 ) 𝑛 𝑖=1 = ∏ 𝛽 𝛼 ( 𝑥𝑖 𝛼 ) 𝛽−1 ⋅ 𝑒 −( 𝑥𝑖 𝛼 ) 𝛽 𝑛 𝑖=1 (3) persamaan 3 perlu ditransformasi supaya dapat lebih representatif untuk proses selanjutnya. perhatikan bahwa terdapat komponen eksponensial pada persamaan 3. aproksimasi numerik belah dua dan newton-raphson pada estimasi parameter distribusi weibull jurnal cauchy – issn: 2086-0382 195 dengan demikian, transformasi yang tepat adalah transformasi logaritma, sehingga diperoleh fungsi logaritma persekitaran berikut : 𝑙(𝛼, 𝛽) = ln 𝐿(𝛼, 𝛽; 𝑥𝑖 ) = 𝑛 ln 𝛽 − 𝑛𝛽 ln 𝛼 + (𝛽 − 1) ∑ ln(𝑥𝑖 ) 𝑛 𝑖=1 − 1 𝛼 𝛽 ∑(𝑥𝑖 ) 𝛽 𝑛 𝑖=1 (4) nilai maksimum dari ̂ dan ̂ diperoleh dari turunan pertama fungsi logaritma persekitaran, masing-masing terhadap  dan  . tentunya, sebelumnya telah dijamin eksistensi dan ketunggalan solusi dari persamaan 4 sebagaimana pada kinasih [3]. berikut merupakan turunan pertama fungsi logaritma persekitaran weibull terhadap masing-masing parameternya : 𝜕𝑙(𝛼, 𝛽) 𝜕𝛼 = −𝑛𝛽(𝛼 𝛽+1) + 𝛼𝛽 ∑ 𝑥𝑖 𝛽 𝑛 𝑖=1 (5𝑎) 𝜕𝑙(𝛼, 𝛽) 𝜕𝛼 = 𝑛 𝛽 − 𝑛 ln 𝛼 + ∑ 𝑥𝑖 𝛽 𝑛 𝑖=1 + ln 𝛼 ∑ 𝑥𝑖 𝛽𝑛 𝑖=1 − ∑ 𝑥𝑖 𝛽𝑛 𝑖=1 ln 𝑥𝑖 𝛼 𝛽 (5𝑏) perhatikan, nilai kritis untuk persamaan 5a dan 5b tidak dapat diperoleh secara eksplisit. fakta inilah yang mengakibatkan digunakannya aproksimasi numerik untuk mendapatkan solusi yang diinginkan. metode numerik belah dua (biseksi) dan newton-raphson (n-r) pada estimasi parameter distribusi weibull, aproksimasi numerik sebenarnya digunakan hanya untuk mendapatkan akar atau nilai kritis persamaan 5b. persamaan 5a dapat dinyatakan sebagai fungsi dari parameter beta, seperti pada persamaan 6. sehingga ketika telah diperoleh nilai ̂ , maka nilai ̂ dapat diperoleh dengan mudah melalui substitusi. 𝛼 = ( 1 𝑛 ∑ 𝑥𝑖 𝛽 𝑛 𝑖=1 ) 1 𝛽 (6) sebelum menerapkan metode aproksimasi numerik belah dua ataupun newton-raphson, persamaan 6 terlebih dahulu disubstitusikan pada persamaan 5b, sehingga didapat fungsi beta seperti pada persamaan 7. 𝑓(𝛽) = 1 𝛽 + ∑ ln 𝑥𝑖 𝑛 𝑖=1 𝑛 − ∑ 𝑥𝑖 𝛽 ln 𝑥𝑖 𝑛 𝑖=1 ∑ 𝑥 𝑖 𝛽𝑛 𝑖=1 (7) metode belah dua (biseksi) metode numerik biseksi dapat digunakan untuk mendapatkan akar persamaan 7, dimana 𝛽 ∈ ℜ. f adalah fungsi kontinu yang terdefinisi di selang [𝛽1, 𝛽2] dengan 𝑓(𝛽1) > 0 dan 𝑓(𝛽2) < 0, atau sebaliknya. ide metode biseksi adalah menggunakan interval [𝛽1, 𝛽2] untuk mengurung akar 𝑓(𝛽). sesuai dengan teorema nilai tengah, f pasti memiliki paling tidak satu nilai akar pada interval [𝛽1, 𝛽2]. prinsip kerja metode biseksi adalah sebagai berikut : 1. mendapatkan titik tengah interval [𝛽1, 𝛽2], yaitu 𝛽3 = (𝛽1+𝛽2) 2 ; 2. jika 𝑓(𝛽3) > 0 maka gantikan 𝛽1 dengan 𝛽3 sehingga interval menjadi [𝛽3, 𝛽2], lakukan biseksi selanjutnya; 3. jika 𝑓(𝛽3) < 0 maka gantikan 𝛽2 dengan 𝛽3 sehingga interval menjadi [𝛽1, 𝛽3], lakukan biseksi selanjutnya; 4. jika 𝑓(𝛽3) = 0, iterasi dihentikan. artinya 𝛽3 adalah solusi dari persamaan 𝑓(𝛽) = 0 gambar 3. ilustrasi aproksimasi numerik belah dua (biseksi) metode newton-raphson berbeda dengan metode belah dua, newton-raphson menggunakan konsep garis singgung fungsi untuk mendapatkan akar suatu persamaan. idenya adalah memilih satu titik awal untuk dicari nilai fungsinya, kemudian pada nilai fungsi titik awal tersebut dibuatkan suatu garis singgung yang juga memotong sumbu horisontal. titik potong itulah yang menjadi titik baru, dugaan nilai akar fungsi tersebut. demikian seterusnya hingga tercapai nilai mutlak selisih indira p. kinasih 196 volume 2 no. 4 mei 2013 antar titik yang kurang dari suatu nilai galat yang ditentukan di awal. gambar 4. ilustrasi aproksimasi numerik newton-raphson karena itu, diperlukan turunan pertama dari persamaan 7 yaitu : 𝑓 ′(𝛽) = − 1 𝛽2 − ∑ 𝑥𝑖 𝛽 (ln 𝑥𝑖 ) 2𝑛 𝑖=1 ∑ 𝑥 𝑖 𝛽𝑛 𝑖=1 + [∑ 𝑥𝑖 𝛽 (ln 𝑥𝑖 ) 𝑛 𝑖=1 ] 2 [∑ 𝑥 𝑖 𝛽𝑛 𝑖=1 ] (8) untuk dapat menjalankan rumus iteratif newton-raphson berikut : �̂�𝑛+1 = �̂�𝑛 − 𝑓(�̂�𝑛 ) 𝑓 ′(�̂�𝑛) (9) adapun tahapan metode newton-raphson dapat disajikan sebagai berikut : 1. tentukan nilai awal �̂�𝑛 . nilai ini dapat berupa sebarang nilai tebakan, misalkan simpangan baku data; 2. jalankan iterasi sesuai persamaan 9; 3. periksa nilai galat yaitu nilai mutlak dari selisih �̂�𝑛+1 dan �̂�𝑛 untuk dibandingkan dengan toleransi galat yang ditentukan, misalkan 𝑒 = 10−5. 4. iterasi dihentikan jika nilai galat telah kurang dari 𝑒. artinya, nilai �̂�𝑛+1 itulah yang dijadikan nilai aproksimasi untuk �̂�. hasil dan pembahasan selain diujicobakan terhadap data kerusakan mesin fotokopi, metode belah dua dan newton-raphson terlebih dahulu disimulasikan pada data acak hasil bangkitan beragam ukuran. hasil simulasi dapat dilihat pada tabel 1. selanjutnya, data kerusakan mesin fotokopi berdasarkan waktu kerusakan dan banyak salinan saat kerusakan yang terdistribusi weibull [1] juga diestimasi menggunakan dua macam aproksimasi numerik ini. hasilnya dapat dilihat pada tabel 2. tabel 1. hasil simulasi aproksimasi numerik belah dua dan newton-raphson tabel 2. hasil estimasi parameter data waktu dan banyak salinan antar kerusakan mesin fotokopi [1] penutup ketiadaan solusi analitis bagi metode estimasi persekitaran distribusi weibull berakibat pada penggunaan aproksimasi numerik. makalah ini menggunakan metode numerik belah dua dan newton-raphson untuk menemukan estimator parameter bentuk �̂�, untuk selanjutnya mendapatkan estimator parameter skala �̂� melalui substitusi. dapat diperhatikan bahwa metode belah dua relatif lebih praktis karena algoritma metode ini hanya membutuhkan turunan pertama fungsi persekitaran terhadap beta seperti pada persamaan 7. sedangkan metode newtonraphson masih memerlukan turunan pertama dari persamaan 7 untuk dapat menjalankan algoritmanya. hasil simulasi menunjukkan bahwa metode belah dua juga terpantau memiliki waktu eksekusi lebih singkat dan hasil aproksimasi yang relatif lebih mendekati nilai parameter sesungguhnya. sedangkan, hasil uji coba terhadap data kerusakan fotokopi baik berdasarkan waktu kerusakan dan banyak salinan saat kerusakan menunjukkan hasil aproksimasi numerik kedua metode mendekati aproksimasi numerik belah dua dan newton-raphson pada estimasi parameter distribusi weibull jurnal cauchy – issn: 2086-0382 197 hasil estimasi bulmer dan eccleston [1]. namun kali ini, metode numerik newton-raphson cenderung memiliki hasil yang paling mendekati estimator di penelitian mereka. referensi [1] m. bulmer and j. eccleston, photocopier reliability modeling using evolutianary algorithm., john wiley & sons, 2003. [2] p. bhattacharya and r. bhattacharjee, "a study on weibull distribution for estimating the parameters," journal of applied quantitative methods, vol. 2, p. 5, 2010. [3] i. p. kinasih, "penaksiran parameter distribusi weibull bivariat menggunakan algoritma genetika," tesis, 2012. pendahuluan distribusi weibull dua parameter dan estimasi persekitaran maksimumnya metode numerik belah dua (biseksi) dan newton-raphson (n-r) metode belah dua (biseksi) metode newton-raphson hasil dan pembahasan penutup referensi 5.1 aisyah pengembangan model dasar eoq dengan integrasi produksi œ distribusi untuk produk deteriorasi _120 pengembangan model dasar eoq dengan integrasi produksi – distribusi untuk produk deteriorasi dengan kebijakan backorder (studi kasus pada ud. bagus agrista mandiri, batu) siti aisyah1, sobri abusini2, marsudi3 1,2,3fakultas mipa, jurusan matematika universitas brawijaya jl. veteran 2 malang 65145 email : aisyah_rasyid84@yahoo.com, sobri@ub.ac.id, marsudi61@ub.ac.id abstrak model persediaan digunakan untuk menentukan kebijakan mengawasi tingkat persediaan. oleh sebab itu keberadaan persediaan perlu dikelola dengan baik sehingga diperoleh kinerja yang optimal. penelitian ini bertujuan untuk menghasilkan model integrasi produksi – distribusi untuk produk deteriorasi dengan kebijakan backorder. model persediaan economic order quantity (eoq) single item digunakan sebagai dasar pengembangan model. algoritma pencarian solusi model dibuat untuk mendapatkan solusi dari model. selain itu pada bagian akhir diberikan studi kasus implementasi model di ud. bagus agrista mandiri, batu. kata kunci : model persediaan eoq, backorder, deteriorasi pendahuluan dalam melakukan pengelolaan persediaan, jenis barang atau produk juga harus menjadi perhatian dalam menentukan kebijakan yang optimal. hal ini disebabkan tidak semua jenis produk atau barang dapat tahan lama selama dalam persediaan sehingga dapat menimbulkan kerugian bagi pemasok atau pembeli. barang yang mengalami deteriorasi tersebut antara lain produk makanan, gasoline, obat-obatan, bank darah, dan bahan makanan. bentuk deteriorasi yang terjadi bermacam-macam, seperti damage, spoilage atau drynase. penelitian pengendalian persediaan dengan mempertimbangkan deteriorasi juga telah banyak dilakukan. su dan lin (2001) mengembangkan model persediaan produksi dengan mempertimbangkan laju produksi tergantung demand dan tingkat persediaan. wee dan law (1999) mengembangkan model persediaan produksi dengan mempertimbangkan nilai waktu dan uang. wu (2001) mengembangkan model persediaan dengan laju permintaan tipe ramp dan mengijinkan partial backorder. penelitian model jels untuk produk yang mengalami deteriorasi masih sedikit dan tidak mengijinkan shortage , yang merupakan konsekuensi ekonomi dari pesanan yang tidak terpenuhi dan akan dipenuhi kemudian. shortage pada pembeli akan mengakibatkan dampak munculnya biaya backorder dan resiko ketidakpastian konsumen yang dapat berakibat pada kerugian jangka panjang. namun, dalam situasi di mana biaya backorder memang ada dan dapat ditentukan, sebuah manfaat ekonomi dapat dilakukan dengan mengizinkan stockouts terjadi (tersine, 1994). dengan memungkinkan stockouts, kelebihan permintaan akan dilakukan dengan backorder dan dipenuhi dengan pengiriman berikutnya. oleh karena itu penulis ingin mengembangkan model integrasi produksidistribusi untuk produk yang mengalami deteriorasi dengan kebijakan backorder yang dipenuhi dengan pengiriman berikutnya sehingga produk di persediaan menjadi lebih sedikit dan, dan menghasilkan biaya persediaan yang lebih rendah. pertanyaan penelitian yang aka dijawab dalam penelitian ini adalah: 1) bagaimana mengembangakan model dasar eoq untuk produk yang mengalami deteriorasi (penurunan nilai setelah waktu tertentu) dengan kebijakan backorder?, dan 2) bagaimana implementasi dan analisis sensitivitasnya dari model yang dihasilkan? metode penelitian a. formulasi model 1. notasi notasi–notasi yang digunakan dalam formulasi model matematis ini adalah sebagai berikut: a. notasi untuk rantai pasok n = jumlah pengiriman persiklus produksi q = ukuran produksi per batch (unit) t = total waktu siklus (waktu) q = ukuran pengiriman (unit) � = laju deteriorasi ��= biaya deteriorasi per unit (rp) pengembangan model dasar eoq dengan integrasi produksi – distribusi untuk produk deteriorasi… jurnal cauchy – issn: 2086-0382 121 b. notasi untuk pembeli � = laju permintaan (unit/waktu) a = biaya pesan (rp/pesan) h� = biaya simpan (rp/unit/waktu) f = biaya tetap transportasi (rp) v = biaya variabel transportasi per unit (rp) s���= luasan dibawah kurva persediaan pembeli j = ukuran stockout (unit) k = biaya backorder (rp/unit/waktu) c. notasi untuk pemasok p = laju produksi (unit/waktu) c = biaya setup per siklus batch produksi (rp/setup) hs = biaya simpan persediaan (rp/unit/waktu) ssup= luasan dibawah kurva persediaan pemasok 2. asumsi – asumsi a. situasi deterministik tetap dan diketahui oleh pembeli dan pemasok. b. laju item deteriorasi adalah konstan c. � � �. d. biaya transportasi dan biaya handling ditanggung oleh pembeli. e. biaya item deteriorasi adalah konstan. f. biaya backorder adalah tetap g. tidak terjadi lost sale, permintaan yang tidak terpenuhi akan dilakukan backorder. h. semua parameter biaya yang terkait dengan inventori dan produksi adalah diketahui dan tetap. i. pemenuhan untuk backorder dilakukan seketika. pembahasan a. model matematik total biaya sistem �������� merupakan penjumlahan dari biaya – biaya yang dikeluarkan oleh pembeli ����� dan biaya – biaya yang dikeluarkan oleh pemasok����� . 1. model biaya yang dikeluarkan oleh pembeli biaya – biaya yang dikeluarkan pembeli: a. biaya pesan pembeli per periode = a / t b. biaya simpan pembeli per periode = ������� c. biaya transportasi = ! " #$%� d. biaya kekurangna pembeli = &' e. biaya deteriorasi pembeli = ()*����� di mana : � + ,$%,-.*% (1) ' + /0$,1 + /0,% (2) jadi total biaya yang dikeluarkan pembeli adalah: ��� + 2� " 34546�� " ! " 7 8� "���546�� ����8, ,:� + ; -$% " *,$<�2 " ! "7 8�";%, = : " /0,%<�34 "���� " & /0,% 2. model biaya yang dikeluarkan pemasok biaya – biaya yang dikeluarkan pemasok: a. biaya setup pemasok per periode = c/t b. biaya simpan pemasok per periode = 3�5�6>/� c. biaya deteriorasi pemasok = ()*�@�a� di mana: 5�6> + �* + 8�;-b = c, " $, = -$,b< (3) jadi total biaya pemasok adalah: ��� + (� " �@�@�a� " ()*�@�a� ����8, � + "�3� " ����8;-b = c, " $, = -$,b< 3. model integrasi pemasok pembeli �������8, ,:� + ��� " ��� �������8, ,:� + c� �2 " ! " 7 8 " �� "d4�34 " ���� "&' " �3� " ����8;-b = c, " $, =-$ ,b< (4) b. solusi model hitung nilai optimal dari 8, dan : dengan menurunkan �������8, ,:� terhadap 8, dan : sama dengan nol, sehingga didapat: 8e + f ,-�g.(.$h�$i���.()*�.��@.()*�j�0kl�mm .$nco.*#n �p�qr)s�0�p�qr)sqt�u (5) :e + ���.()*�%���.()*.v� (6) e + w �,-.*%��g.(�b��@.(@*�.%0�bn-� (7) untuk + 1 (batas bawah), 8 akan menjadi batas atas sehingga 8e y w ,-b �g.(.$h�b���.()*�.-��@.()*�.*#bnb�0���.()*.v� ketika meningkat, maka nilai 8 menurun sehingga dari persamaan (5) didapatkan 8e [ w ,-b �g.(.$h�\���.()*�.*#]\,-.$�bn-�]$ ketika � 1 hal ini membuat persamaan (7) menjadi 1 y y 6 +���.()*�.*#��@.�)*� g.(h "w,-\���.()*�*#].*�g.(�b��@.()*��bn-� (8) c. implementasi model sebagai implementasi model, digunakan data yang diambil dari ud. bagus agrista mandiri siti aisyah, sobri abusini, dan marsudi 122 volume 2 no. 3 november 2012 batu dimana perhitungan dilakukan dengan bantuan software matlab dengan � + 0 = 0.2. tabel 1. nilai parameter keterangan nilai biaya deteriorasi ���� rp 1.500.000 laju produksi ��� 486 unit biaya setup ��� 972.000 / batch biaya simpan pemasok �3�� rp 800 / unit biaya kekurangan �&� rp 1.500.000 jumlah permintaan ��� 443 unit biaya pesan pembeli �2� rp 15.000/pesan biaya simpan pembeli �34� rp 900 / unit biaya transportasi �!� rp 1.500.000 biaya variabel �7� rp 100 / unit hasil perhitungan pada tabel 2 integrasi pemasok pembeli dengan backorder untuk � + 0.1 diperoleh: a. ukuran lot pengiriman �8e� = 82 b. jumlah pengiriman � e� = 3 c. ukuran stock out �:e� = 41 d. ukuran batch produksi �a� = 248 e. waktu siklus pemesanan ��� = 201 hari f. total biaya sistem �������� =rp19.831.284 perbandingan hasil perhitungan model integrasi dengan dan tanpa backorder dari tabel (3) dan (4) di bawah ini diperoleh bahwa ketika laju deteriorasi meningkat, ukuran produksi optimal �a� dan waktu siklus ��� menurun. hal ini tentu tidak diharapkan karena ketika laju deteriorasi meningkat akan mengakibatkan turunnya ukuran produksi yang berhubungan dengan waktu siklus dapat mengurangi keuntungan dalam sistem suplaychain tabel 2. perhitungan integrasi pemasok pembeli � e 8e :e � a ������ 0.1 1 102 51 83 103 21.817.356 2 88 44 144 178 20.124.975 3 82 41 201 248 19.831.284 4 78 39 255 315 19.908.353 5 75 38 306 379 20.132.754 6 73 37 358 443 20.427.088 tabel 3. hasil perhitungan model integrasi dengan backorder untuk � berbeda-beda � 0 0.025 0.05 0.075 0.1 0.125 0.150 0.175 0.2 ukuran batch produksi �a� 3672 575 332 281 248 228 210 198 189 waktu siklus ���) 3025 469 270 228 201 183 169 159 151 jum.pengiriman � � 4 4 3 3 3 3 3 3 3 b. pengiriman �8� 918 143 110 93 82 75 69 65 62 ukuran stockout �:� 5 29 37 40 41 42 41 41 41 (������) 1730063 10886607 14765958 17563065 19831284 21780169 23514598 25091298 26549758 tabel 4. hasil perhitungan model integrasi tanpa backorder untuk � berbeda � 0 0.025 0.05 0.075 0.1 0.125 0.150 0.175 0.2 ukuran produksi �a� 3668 546 390 318 279 246 226 210 199 total waktu siklus ���) 3022 447 318 259 226 199 183 170 160 jumlah pengiriman � � 4 4 4 4 4 4 4 4 4 ukuran pengiriman �8� 917 136 97 79 69 61 56 52 49 (������) 1732507 11432717 16073868 19652046 22676243 25345276 27760565 29984614 32057737 d. analisis sensitivitas analisis sensitivitas dilakukan dengan merubah nilai parameter pada model. besar perubahan parameter adalah sebesar 10%, 25%, dan 50% dari nilai parameter pada implementasi model, sedang laju deteriorasi yang digunakan adalah � + 0.1. 1. perubahan biaya deteriorasi dari hasil perhitungan pada tabel (5) di bawah ini dapat disimpulkan bahwa perubahan pengembangan model dasar eoq dengan integrasi produksi – distribusi untuk produk deteriorasi… jurnal cauchy – issn: 2086-0382 123 biaya deteriorasi pada sistem supplay chain mengakibatkan perubahan yang signifikan pada ukuran lot dan biaya total sistem. tabel 5. perubahan biaya deteriorasi pemasok % nilai �� a � 8 : ������ %������ -50 750000 334 269 3 110 37 14811958 -25.3101 -25 1125000 282 227 3 93 40 17586043 -11.3217 -10 1350000 261 211 3 86 41 18979685 -4.2942 0 1500000 248 201 3 82 41 19831284 0 10 1650000 239 194 3 79 41 20633069 4.0430 25 1875000 227 184 3 75 42 21757213 9.7116 50 2250000 209 169 3 69 41 23468701 18.3418 2. perubahan kekurangan dari hasil perhitungan yang disajikan pada tabel (6) di bawah ini terlihat perubahan biaya kekurangan memberi pengaruh cukup besar terhadap ukuran lot pesan maupun terhadap total biaya. perubahan biaya kekurangan sebesar -50% dari biaya semula mengakibatkan turunnya biaya sebesar 5.4%. tabel 6. perubahan biaya kekurangan �&� pada pembeli % & nilai & a � 8 : ������ %������ -50 75000 264 213 3 87 58 18768913 -5.3570 -25 112500 255 206 3 84 48 19382891 -2.2610 -10 135000 252 203 3 83 44 19667188 -0.8275 0 150000 248 201 3 82 41 19831284 0 10 165000 248 201 3 82 39 19978639 0.7430 25 187500 245 198 3 81 36 20173221 1.7242 50 225000 242 196 3 80 32 20442893 3.0841 3. perubahan biaya transportasi dari hasil perhitungan tabel (7) di bawah ini dapat dilihat bahwa perubahan biaya transportasi memberi pengaruh cukup besar terhadap total biaya. perubahan biaya sebesar 50 % dari biaya transportasi awal menyebabkan perubahan total biaya sebesar 18.8% tabel 7. perubahan biaya transportasi (f) pengiriman % ! nilai ! a � 8 : ������ %������ -50 750000 238 193 4 59 30 15033855 -24.1912 -25 1125000 279 226 4 69 35 17639072 -11.0543 -10 1350000 239 194 3 79 40 18999060 -4.1965 0 1500000 248 201 3 82 41 19831284 0 10 1650000 261 211 3 86 43 20630629 4.0307 25 1875000 273 220 3 90 45 21775060 9.8016 50 2250000 298 240 3 98 49 23560875 18.8066 penutup hasil penelitian menunjukan ketika laju deteriorasi meningkat maka ukuran produksi menurun sebesar 15.72 % dari produksi semula, tetapi menyebabkan kenaikan total biaya sistem sebesar 18.3%. sedang untuk peningkatan biaya transportasi perpengiriman mengakibatkan kenaikan total biaya sistem sebesar 18.8% dan menyebabkan ukuran produksi meningkat sebesar 20.16%. begitu pula perubahan pada biaya setup memberikan pengaruh yang signifikan terhadap ukuran lot produksi yaitu meningkat sebesar 32% tetapi pada biaya sistem hanya meningkat sebesar 3.8%. untuk penelitian lanjutan model dapat dikembangkan untuk multi produk, multi pembeli, atau multi pemasok. dan dapat juga dikembangkan dalam manajemen rantai pasok produksi dan distribusi, aspek persediaan dan distribusi. siti aisyah, sobri abusini, dan marsudi 124 volume 2 no. 3 november 2012 daftar pustaka [1] bhunia, a.k. & maiti, m. 1998. deterministic inventory model for deteriorating items with finite rate of replenishment dependent on inventory level. computers & operations research. 25 (11), 997. [2] gaither, n. 1992. production and operations management, sixth edition. new york: fort worth philadelphia sandiego. [3] goyal, s.k & giri, b.c. 2001. recent trends in modelling of deteriorating inventory. european journal of operational research. 134, 1-16. [4] mulyono, s. 1991. operation reseach. jakarta: lembaga penerbit fe ui [5] rangkuti, f. 2004. manajemen persediaan (aplikasi di bidang bisnis). jakarta: pt. raja grafindo persada. [6] simchi-levi, d., kaminsky, p., & simchilevi, e. 2000. designing and managing the supply chain. mcgraw-hill int. ed. [7] siswanto. 2007. operation research. jilid 1. jakarta: erlangga. [8] sukmana, a., & i. lokman. model matematika sistem persediaan dengan pengadaan darurat. integral. vol 10: no 1. [9] tersine, r. j. 1994. principle of inventory and materials management. new jersey: prentice hall. [10] yan, c., banerjee, a., & yang l. 2010. an integrated production – distribution model for a deteriorating inventory item. international journal of production economics. article in press. suma’inna dan gugun gumilar algoritma ford-fulkerson untuk memaksimumkan flow dalam pendistribusian barang 1fahrun nisa’, 2wahyu henky irawan 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: fahrunnisa33@gmail.com abstrak algoritma ford fulkerson digunakan untuk mencari flow maksimum pada jaringan yang mempunyai satu titik sumber dan satu titik tujuan. dengan mendefinisikan jalur pendistribusian barang sebagai jaringan, maka jaringan tersebut memiliki beberapa titik sumber dan beberapa titik tujuan. untuk menyelesaikan permasalahan flow maksimum pada jaringan ini maka digunakan modifikasi algoritma ford-fulkerson. pada artikel ini akan ditunjukkan langkah-langkah mencari flow maksimum serta hasil yang diperoleh dengan menggunakan modifikasi algoritma ford-fulkerson pada data simulasi tentang pendistribusian barang dengan lima titik sumber dan lima titik tujuan. hasil dari penelitian ini merupakan bentuk modifikasi algoritma ford-fulkerson, yaitu membentuk jaringan baru dengan menambahkan satu titik sumber utama dan satu titik tujuan utama pada jaringan baru, membentuk kapasitas di busur dari titik sumber utama ke beberapa titik sumber serta membentuk kapasitas di busur dari beberapa titik tujuan ke titik tujuan utama dengan nilai kapasitas maksimum dan memberi nilai flow awal sebesar nol. selanjutnya memaksimumkan flow dari titik sumber utama ke titik tujuan utama menggunakan algoritma fordfulkerson, yaitu dengan melakukan pelabelan titik, menggunakan prosedur balik, dan mencari lintasan peningkatan sampai semua titik yang terlabel telah teramati dan titik tujuan utama tidak terlabel sehingga iterasi dihentikan. berdasarkan hasil perhitungan didapatkan flow maksimum pada jaringan yang tidak dipartisi, pada jaringan yang dipartisi serta dengan membuat program di matlab r2010a dengan nilai 45. kata kunci: algoritma ford-fulkerson, flow maksimum, jaringan, jaringan bipartisi abstract ford-fulkerson algorithms used to find the maximum flow in a network that has a single point source and a sink point . by defining the goods distribution channels as the network , then the network has several source point and several sink point. to solve the problems of the maximum flow in the network is then used a modified ford-fulkerson algorithm. this article can be shown the steps looking for maximum flow and obtained results by using a modified ford-fulkerson algorithm on simulated data on the distribution of goods with a five source point and five sink.point. the results of this study is a modified form of the ford -fulkerson algorithm , which formed a new network by adding a main source point and a sink point on a major new network ,and formed capacity in an arc from primary sources point to a few source point and established capacity in the bow of some sink point to point the main goal with the value of the maximum capacity and provide initial flow value of zero . furthermore, to maximize a flow of the main source point to a major sink point by using the ford fulkerson algorithm , namely by labeling a point , using the reverse procedure and to seek increase trajectory till all the labeled points have been observed and a major sink point has not labeled so the iteration is stopped . based on the results of the calculation, the maximum flow on a unpartitioned network on a partitioned network and by creating a program in matla b r2010a with a value of 45. keywords: ford-fulkerson algorithm, flow maximum, network, bipartite network pendahuluan 1. latar belakang teori graf merupakan salah satu cabang dari ilmu matematika, dimana teori-teori yang digunakan tergolong aplikatif dan dapat digunakan untuk memecahkan masalah alam kehidupan sehari-hari. salah satu pembahasan dalam teori graf adalah jaringan (network). permasalahan dalam suatu jaringan dapat dicari penyelesaiannya yaitu salah satunya dengan mencari flow maksimum. artikel mailto:fahrunnisa33@gmail.com fahrun nisa’, wahyu henky irawan 182 volume 3 no. 4 mei 2015 ini membahas tentang pendistribusian barang yang didefinisikan sebagai jaringan. dalam konsep pendistribusian barang, semakin banyak barang yang dijual, maka kemungkinan barang yang laku terjual semakin banya, sehingga keuntungan yang didapatkan oleh perusahaan akan semakin banyak. berdasarkan prinsip ekonomi “dengan modal sekecil-kecilnya memperoleh keuntungan sebesar-besarnya”, maka permasalahan matematika yang sesuai dalam kasus ini yaitu pencarian flow maksimum, sehingga flow maksimum digunakan dalam jaringan ini. sebuah penelitian yang dilakukan oleh [1] yaitu tentang pencarian flow maksimum pada jaringan yang memiliki satu titik sumber dan satu titik tujuan menggunakan algoritma ford-fulkerson. menurut farizal, algoritma ini selalu memberikan hasil yang benar dalam mencari lintasan peningkatan (augmenting-path) sehingga algoritma ford-fulkerson sesuai dengan permasalahan dalam artikel ini. dalam penelitian ini akan dicari flow maksimum pada jaringan pendistribusian barang, dimana jaringan tersebut memiliki beberapa titik sumber dan beberapa titik tujuan, sehingga akan dilakukan modifikasi algoritma ford-fulkerson. harapan dari adanya penelitian ini yaitu pendistribusian barang dapat dilakukan secara maksimum sehingga keuntungan yang diperoleh suatu perusahaan akan lebih banyak. 2. metode penelitian penelitian ini menggunakan jenis penelitian kualitatif dengan langkah-langkah penelitiannya: a. menyediakan data simulasi yaitu tentang pendistribusian barang. b. data simulasi yang disediakan memiliki lima titik sumber dan lima titik tujuan, sehingga dilakukan modifikasi algoritma fordfulkerson agar algoritma ini dapat digunakan untuk mencari flow maksimum pada jaringan. c. menyusun flow maksimum pada jaringan. d. membuat program di matlab r2010a. kajian teori 1. graf suatu graf 𝐺 adalah suatu himpunan tidak kosong dan berhingga dari objek-objek yang disebut sebagai vertex (titik) dan himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda pada 𝐺 yang disebut edge (sisi). contoh: gambar 1. graf 𝐺 graf 𝐺 pada gambar 1 dapat dinyatakan dengan 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) dengan 𝑉(𝐺) = {𝑎, 𝑏, 𝑐, 𝑑} dan 𝐸(𝐺) = {𝑎𝑏, 𝑎𝑑, 𝑎𝑐, 𝑏𝑐, 𝑐𝑑} [2]. sebuah sisi graf yang menghubungkan sebuah titik dengan dirinya sendiri disebut gelung (loop). jika terdapat lebih dari satu sisi yang menghubungkan dua titik pada suatu graf, maka sisi-sisi tersebut disebut sisi rangkap atau sisi ganda. graf yang tidak mempunyai sisi rangkap dan tidak memiliki gelung disebut graf sederhana [3]. 2. digraf graf berarah atau digraf 𝐷 adalah struktur yang terdiri dari himpunan tak kosong dan berhingga, yang unsurnya disebut titik, beserta himpunan pasangan berurutan dari dua titik berbeda yang disebut busur. contoh: gambar 2. digraf 𝐺 algoritma ford-fulkerson untuk memaksimumkan flow dalam pendistribusian barang cauchy – issn: 2086-0382 183 pada gambar 2 titik 𝑢 terhubung langsung ke 𝑣. di lain pihak, titik 𝑣 tidak terhubung langsung ke 𝑢 [2]. 3. derajat titik pada digraf misalkan 𝑣 adalah titik pada digraf 𝐷. banyaknya titik yang terhubung langsung ke 𝑣 disebut derajat masuk (indegree) dari 𝑣, dan dinotasikan dengan 𝑖𝑑(𝑣). banyaknya titik yang terhubung langsung dari 𝑣 disebut derajat keluar (outdegree) dari 𝑣, dan dinotasikan dengan 𝑜𝑑(𝑣) [2]. contoh: gambar 3. derajat titik pada graf 𝐻 4. digraf terhubung misalkan 𝐷 adalah suatu graf berarah. graf berarah 𝐷 dikatakan terhubung kuat jika untuk setiap dua titik di 𝐷 ada lintasan berarah yang menghubungkan dua titik tersebut. sedangkan graf 𝐷 dikatakan terhubung lemah jika 𝐷 tidak terhubung kuat tetapi graf tak berarah yang bersesuaian dengan graf 𝐷 terhubung [2]. contoh: gambar 4. graf 𝐷 terhubung lemah; graf 𝐻 terhubung kuat 5. jaringan (network) jaringan 𝑁 = (𝑉(𝑁), γ(𝑁)) merupakan sebuah graf berarah sederhana terhubung lemah yang setiap unsurnya dikaitkan dengan bilangan real non negatif. bilangan real non negatif yang dikaitkan pada busur (𝑉𝑖, 𝑉𝑗 ) di jaringan 𝑁 disebut kapasitas busur dan dilambangkan 𝑐(𝑉𝑖 , 𝑉𝑗 ). sebuah titik 𝑠 di jaringan 𝑁 disebut titik sumber jika 𝑖𝑑(𝑠) = 0 dan sebuah titik 𝑡 di jaringan 𝑁 disebut titik tujuan jika 𝑜𝑑(𝑡) = 0, sedangkan titik yang lain di jaringan 𝑁 disebut titik-titik antara [3]. contoh: gambar 5. jaringan 𝑁 dengan titik sumber 𝑣𝑠 dan titik tujuan 𝑣𝑡 6. pemutus pada jaringan misalkan 𝑁 sebuah jaringan dengan titik sumber 𝑠 dan titik tujuan 𝑡. misalkan himpunan 𝑋 adalah himpunan bagian tak kosong dari 𝑉(𝑁) dan 𝑋1 = 𝑉(𝑁) − 𝑋. jika 𝑠 ∈ 𝑋, dan 𝑡 ∈ 𝑋1, maka himpunan busur 𝐵(𝑋, 𝑋1) disebut sebuah pemutus−(𝑠, 𝑡) dari jaringan 𝑁, karena penghapusan semua busur 𝐵(𝑋, 𝑋1) dari 𝑁, memutus semua lintasan berarah dari titik 𝑠 ke titik 𝑡 pada jaringan 𝑁 [3]. 7. konsep flow pada jaringan sebuah flow di jaringan 𝑁 dari titik sumber 𝑠 dan titik tujuan 𝑡 adalah sebuah fungsi yang memetakan setiap busur (𝑖, 𝑗) di 𝑁 dengan sebuah bilangan non negatif yang memenuhi syarat-syarat berikut: 1. 0 ≤ 𝑓(𝑖, 𝑗) ≤ 𝑐(𝑖, 𝑗) ∀(𝑖, 𝑗) ∈ γ(𝑁) (disebut kapasita pembatas) 2. ∑ 𝑓(𝑖, 𝑗)(𝑖,𝑗)∈𝑂(𝑠) = ∑ 𝑓(𝑖, 𝑗)(𝑖,𝑗)∈𝐼(𝑡) (disebut nilai flow 𝑓) 3. ∑ 𝑓(𝑖, 𝑗)(𝑖,𝑗)∈𝑂(𝑠) = ∑ 𝑓(𝑖, 𝑗)(𝑖,𝑗)∈𝐼(𝑡) ∀𝑥 ∈ 𝑉(𝑁) − {𝑠, 𝑡} (disebut sebagai “konservasi flow”) [3]. 8. flow maksimum pada jaringan flow 𝑓 bernilai 𝑓𝑠,𝑡 dari titik 𝑠 ke titik tujuan 𝑡 pada jaringan 𝑁 dikatakan flow maksimum jika 𝑓𝑠,𝑡 = min {𝑐(𝑋, 𝑋1)|𝐵(𝑋, 𝑋1) suatu pemutus−(𝑠, 𝑡) pada jaringan 𝑁} [3]. 9. algoritma ford-fulkerson secara sistematis, langkah-langkah pada algoritma ford-fulkerson yaitu: fahrun nisa’, wahyu henky irawan 184 volume 3 no. 4 mei 2015 input : jaringan 𝑁 = (𝑉, 𝐺) dengan titik sumber 𝑠 dan titik tujuan 𝑡. langkah 1 : misalkan 𝑓 sebuah flow dari 𝑠 ke 𝑡 pada 𝑁. (boleh dimulai dengan flow 𝑓 bernilai nol, yaitu 𝑓(𝑖, 𝑗) = 0, ∀ (𝑖, 𝑗) ∈ γ). dilanjutkan ke rutinpelabelan. langkah 2 : rutin-pelabelan. a. label 𝑣𝑠 = (𝑠, +, 𝜀(𝑠) = ~). titik 𝑣𝑠 telah terlabel dan belum “teramati”. (catatan: semua titik 𝑣 dikatakan telah teramati jika semua titik yang dapat dilabel dari titik 𝑣 sudah terlabel). b. pilih sebarang titik yang terlabel tetapi belum teramati, misalkan titik tersebut 𝑣𝑥 . untuk ∀ 𝑣𝑦 ∃ (𝑦, 𝑥) ∈ γ, 𝑣𝑦 belum terlabel dan 𝑓(𝑦, 𝑥) > 0, maka label 𝑣𝑦 = (𝑥, −, 𝜀(𝑦)) dengan 𝜀(𝑦) = min {𝜀(𝑥), 𝑓(𝑦, 𝑥)}. sekarang titik 𝑣𝑦 telah terlabel, tetapi belum teramati. untuk ∀ 𝑣𝑦 ∃ (𝑦, 𝑥) ∈ γ, 𝑣𝑦 belum terlabel dan 𝑐(𝑥, 𝑦) > 𝑓(𝑥, 𝑦), maka label 𝑣𝑦 = (𝑥, +, 𝜀(𝑦)) dengan 𝜀(𝑦) = min {𝜀(𝑥), 𝑐(𝑥, 𝑦) − 𝑓(𝑥, 𝑦)}. sekarang titik 𝑣𝑦 terlabel, tetapi belum teramati. diubah label 𝑣𝑥 dengan cara melingkari tanda + atau -. sekarang titik 𝑣𝑥 terlabel dan teramati. c. ulangi langkah a. sampai: 1) titik 𝑣𝑡 terlabel atau semua titik terlabel telah teramati tetapi titik 𝑣𝑡 tak terlabel. 2) jika titik 𝑣𝑡 terlabel, lanjut ke langkah 3 atau jika semua titik yang terlabel telah teramati tetapi titik 𝑣𝑡 tak terlabel, maka iterasi dihentikan dan flow 𝑓 adalah flow maksimum dana jaringan 𝑁. langkah 3 : dengan prosedur “balik”, temukan lintasan peningkatan 𝑃 dengan 𝑖(𝑃) adalah label 𝑣𝑡 . langkah 4 : tingkatkan nilai flow 𝑓 sebesar label 𝑣𝑡 , berdasarkan lintasan peningkatan 𝑃 dengan menggunakan “rutin peningkatan”. rutin peningkatan: a. misalkan 𝑍 = 𝑡 lanjutkan ke langkah b. b. jika label 𝑣𝑧 = (𝑞, +, 𝜀(𝑡)) tingkatkan nilai 𝑓(𝑞, 𝑧) dengan 𝜀(𝑡) = 𝑖(𝑃). c. jika 𝑞 = 𝑠, hapus semua label. pada tahap ini diperoleh flow 𝑓 baru dengan nilai = 𝑖(𝑃) + nilai flow 𝑓 lama. ganti flow 𝑓 dengan flow 𝑓 yang baru, dan kembali ke langkah 1. [3]. 10. jaringan bipartisi sebuah jaringan 𝐺 = (𝑉, 𝐸) dikatakan bipartisi jika himpunan titik 𝑉 dapat dipartisi menjadi dua subhimpunan 𝑉1 dan 𝑉2 sehingga semua sisi dari 𝐺 mempunyai ujung di 𝑉1 dan ujung lainnya di 𝑉2, sebagaimana contoh berikut [4]: gambar 6. jaringan bipartisi pembahasan data simulasi yang disediakan merupakan data pendistribusian barang yang memiliki lima titik sumber dan lima titik tujuan dimana untuk mencari flow maksimum pada data tersebut, maka dengan memodifikasi algoritma ford-fulkerson. bentuk modifikasi algoritma ini adalah: input : jaringan 𝐴 = (𝑉, 𝐺) dengan beberapa titik sumber (𝑠𝑖 ) dan beberapa titik tujuan 𝑡(𝑡𝑗 ). langkah 1 : dibentuk sebuah jaringan baru dari 𝐴 yang dimisalkan 𝐴∗, dengan algoritma ford-fulkerson untuk memaksimumkan flow dalam pendistribusian barang cauchy – issn: 2086-0382 185 menambahkan satu titik sumber 𝑠 dan satu titik tujuan 𝑡 sedemikian hingga 𝑠 merupakan satu-satunya titik sumber di jaringan 𝐴∗ dan 𝑡 merupakan satu-satunya titik tujuan di 𝐴∗. langkah 2 : dibuat kapasitas busur (𝑠, 𝑠𝑖 ) dan (𝑡𝑖 , 𝑡) dengan nilai kapasitas tak hingga atau dilambangkan ∞, dan nilai flow dapat dimulai dari 0. langkah selanjutnya yaitu memaksimumkan flow 𝑠 − 𝑡 dengan menggunakan algoritma flow maksimum, yaitu: langkah 3 : lakukan rutin pelabelan. langkah 4 : gunakan prosedur balik. langkah 5 : lakukan rutin peningkatan. langkah 6 : apabila titik-titik di 𝐴∗ yang terlabel telah teramati semua, dan titik 𝑡 tidak terlabel maka iterasi dihentikan. selanjutnya data dibentuk menjadi sebuah jaringan yang dimisalkan jaringan 𝑀, sebagaimana dapat dicontohkan pada gambar berikut: gambar 7. jaringan 𝑀 jaringan 𝑀 pada gambar 7 akan dicari flow maksimumnya menggunakan modifikasi algoritma ford-fulkerson dengan langkahlangkah sebagai berikut: langkah 1 : dibentuk sebuah jaringan baru 𝑀∗ dari 𝑀 dengan menambahkan satu titik sumber baru dan satu titik tujuan baru sedemikian hingga merupakan titik sumber utama dan titik tujuan utama di 𝑀∗ dimana titik sumber utama dan titik tujuan utama pada jaringan 𝑀∗ diberi warna biru. langkah 2 : dibuat kapasitas busur (𝑠, 𝑠1), (𝑠, 𝑠2), (𝑠, 𝑠3), (𝑠, 𝑠4) dan (𝑠, 𝑠5) serta (𝑡1, 𝑡), (𝑡2, 𝑡), (𝑡3, 𝑡), (𝑡4, 𝑡) dan (𝑡5, 𝑡) yang juga diberi warna biru dengan nilai kapasitas utama yaitu stok persediaan barang dengan nilai 45 dan nilai flow awal 0. gambar 8. jaringan 𝑀∗ dengan nilai flow awal 𝑓 = 0 jaringan 𝑀∗ pada gambar 8 mempunyai kapasitas di titik sumber utama dengan nilai 45, sedangkan total kapasitas di beberapa titik sumber dengan nilai 50. sebagaimana prosedur algoritma fordfulkerson pada bab sebelumnya, agar nilai flow segera terpenuhi maka strategi flow maksimum yang dilakukan yaitu dipilih lintasan dari titik sumber utama ke titik yang mempunyai busur terhubungnya paling banyak serta titik yang mempunyai kapasitas terbesar, sehingga urutan iterasi untuk lintasan peningkatan yaitu di titik 𝑠2, 𝑠4, 𝑠1, 𝑠3, 𝑠5. langkah selanjutnya dalam pencarian flow maksimum menggunakan modifikasi algoritma ford-fulkerson untuk iterasi pertama yaitu sebagai berikut: 1) dari titik 𝑠, diberi label titik 𝑠1, 𝑠2, 𝑠3, 𝑠4 dan 𝑠5 masing-masing 𝑠1 = (𝑠, +,45), 𝑠2 = (𝑠, +,45), 𝑠3 = (𝑠, +,45), 𝑠4 = (𝑠, +,45) dan 𝑠5 = (𝑠, +,45). sekarang titik 𝑠 telah terlabel dan teramati dengan label 𝑠 = (𝑠,⊕ ,45). 2) dari titik 𝑠2, diberi label untuk titik 𝑡1, 𝑡2, 𝑡3 masing-masing 𝑡1 = (𝑠2, +,14), 𝑡2 = (𝑠2, +,2), 𝑡3 = (𝑠2, +,7). titik 𝑠2 telah terlabel dan teramati dengan label 𝑠2 = (𝑠, +,45). fahrun nisa’, wahyu henky irawan 186 volume 3 no. 4 mei 2015 3) dari titik 𝑡1, diberi label untuk titik 𝑡 dengan label 𝑡 = (𝑡1, +, 45). titik t1 telah terlabel dan teramati dengan label 𝑡1 = (𝑠2, +,14). langkah 4 : gunakan prosedur balik. titik 𝑡 dilabel dari titik 𝑡1, titik 𝑡1 dilabel dari 𝑠2 dan titik 𝑠2 dilabel dari titik 𝑠. dengan demikian diperoleh lintasan peningkatan 𝑃 = (𝑠, 𝑠2, 𝑡1, 𝑡) dan 𝑖(𝑃) = min (45,14,45) = 14. langkah 5 : rutin peningkatan. titik 𝑡 = (𝑡1, +, 45) maka nilai flow pada busur (𝑡1, 𝑡) ditambah 14. titik 𝑡1 =(𝑠2, +, 14) maka nilai flow pada busur (𝑠2, 𝑡1) ditambah 14. titik 𝑠2 = (𝑠, +,45) maka nilai flow pada busur (𝑠, 𝑠2) ditambah 14. setelah dilakukan penghapusan terhadap semua label, diperoleh nilai flow baru 𝑓1 = 14. lintasan peningkatan pada jaringan 𝑀∗ diberi warna orange dan ditunjukkan sebagaimana pada gambar berikut: gambar 9. jaringan 𝑀∗ dengan nilai flow baru 𝑓1 = 14 iterasi selanjutnya diulangi pencarian flow maksimum menggunakan algoritma fordfulkerson dari titik sumber ke titik tujuan di 𝑀∗ dengan kembali ke langkah 2 sampai iterasi ke 8 yaitu untuk lintasan peningkatan di titik 𝑠5 dan diperoleh nilai flow maksimum sebesar 45. berdasarkan jaringan yang telah terbentuk pada gambar 7, jaringan 𝑀 merupakan bentuk jaringan bipartisi dimana himpunan titik-titik di jaringan 𝑀 dapat dipartisi menjadi dua sub-himpunan, sehingga jaringan 𝑀 akan dipartisi berdasarkan titik sumber dan dicari flow maksimumnya untuk menunjukkan bahwasanya nilai flow maksimumnya sama antara jaringan yang tidak dipartisi dengan jaringan yang dipartisi. jaringan 𝑀 dipartisi menjadi lima karena memiliki lima titik sumber, karena mempunyai lima titik tujuan maka dengan memanfaatkan modifikasi algoritma ford-fulkerson, akan ditambahkan satu titik tujuan baru dengan kapasitasnya adalah stok persediaan barang sebesar 45. selanjutnya dilakukan iterasi dengan mencari lintasan peningkatan dari masing-masing titik sumber ke titik tujuan utama sampai tidak ditemukan lagi lintasan peningkatan, yaitu pada saat semua titik sumber telah diselidiki. sehingga nilai flow maksimum pada jaringan diperoleh dari total flow maksimum untuk semua titik sumber di jaringan, dan didapatkan flow maksimum dengan iterasi sebanyak lima nilainya 45. pencarian flow maksimum pada jaringan 𝑀 juga dapat dilakukan dengan membuat program menggunakan matlab r2010a dengan langkah iterasi sebagaimana prosedur yang dimiliki algoritma ford-fulkerson. tujuan membuat program ini yaitu agar pencarian flow maksimum dapat diketahui dengan lebih mudah, dimana hasilnya dengan iterasi sebanyak 8 nilai flow maksimum sebesar 45. berdasarkan hasil contoh pendistribusian barang dengan pencarian flow maksimum pada jaringan 𝑀 di atas didapatkan flow maksimum dengan nilai 𝑓8 = 45, pada jaringan yang dipartisi didapatkan flow maksimum dengan nilai 𝑓5 = 45 serta dengan program di matlab r2010a dengan iterasi sebanyak 8 didapatkan flow maksimum sebesar 45. kesimpulan bentuk modifikasi algoritma ford-fulkerson untuk menentukan flow maksimum pada jaringan yaitu: a. membuat jaringan baru dari jaringan distribusi barang dengan menambahkan satu titik sumber utama dan satu titik tujuan utama di jaringan baru tersebut. b. membuat kapasitas busur dari titik sumber utama ke beberapa titik sumber dan membuat kapasitas busur dari beberapa titik tujuan ke titik tujuan utama dengan kapasitas maksimum dan memberikan nilai flow awal pada semua busur di jaringan baru dengan nilai 0. algoritma ford-fulkerson untuk memaksimumkan flow dalam pendistribusian barang cauchy – issn: 2086-0382 187 setelah itu memaksimumkan flow dari titik sumber baru ke titik tujuan baru dengan: c. melakukan pelabelan titik. d. menggunakan prosedur balik. e. mencari lintasan peningkatan sampai titik-titik di jaringan yang terlabel telah teramati semua dan titik tujuan baru tidak terlabel maka iterasi dihentikan. dari bentuk modifikasi di atas, didapatkan flow maksimum pada jaringan tersebut dengan nilai 𝑓8 = 45, pada jaringan yang dipartisi didapatkan flow maksimum dengan nilai 𝑓5 = 45 serta dengan membuat program di matlab r2010a didapatkan flow maksimum dengan iterasi sebanyak 8 nilainya 45. daftar pustaka [1] t. farizal, "pencarian flow maksimum dengan algoritma ford-fulkerson," universitas negeri semarang, semarang, 2013. [2] g. charland and l. lesniak, graf and digraf, london: chapman & hall/crc, 1996. [3] i. budayasa, teori graph dan aplikasinya, surabaya: unesa university press, 2007. [4] r. k. ahuja, j. b. orlin, c. stein and r. e. tarjan, "improved algorithms for bipartite network flow," siamj comput, pp. 906-933, 1994. pendahuluan 1. latar belakang 2. metode penelitian kajian teori 1. graf 2. digraf 3. derajat titik pada digraf 4. digraf terhubung 5. jaringan (network) 6. pemutus pada jaringan 7. konsep flow pada jaringan 8. flow maksimum pada jaringan 9. algoritma ford-fulkerson 10. jaringan bipartisi pembahasan kesimpulan daftar pustaka some properties from construction of finite projective planes of order 2 and 3 cauchy journal of pure and applied mathematics volume 4(3) (2016), pages 131-137 submitted: august, 14 2016 reviewed: october, 21 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3633 some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati1, corina karim2 department of mathematics fmipa,brawijaya university, malang email : virahari@ub.ac.id, co_mathub@ub.a.c.id abstract in combinatorial mathematics, a steiner system is a type of block design. specifically, a steiner system s(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. it is well-known that a finite projective plane is one examples of steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. in this paper, we observe some properties from construction of finite projective planes of order 2 and 3. also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group. keywords: automorphism group, finite projective planes, steiner system, symmetric group. introduction a finite projective plane is a geometric construction that extends the concept of a plane. it consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. meanwhile in combinatorial mathematics, a steiner system is a type of block design. specifically, a steiner system s(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. it is well-known that a finite projective plane is one examples of steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. or we can state that a finite projective plane of order q, denoted by pg(2, q), with the lines as blocks, is an steiner system s(2, q+1, q2+q+1). arithmetically, when a steiner system exists, there can be no more than v-tck-t mutually disjoint of the steiner systems, which is called large set. in other words, a large set is a partition of the set of all k-subsets of v-set into steiner systems s(t, k, v). for finite projective planes, according to [1], there is only known a large set of projective plane of order 3 or steiner system s(2, 4, 13) [2], while a large set of projective plane of order 2 or steiner system s(2, 3, 7) do not exist and the existence of large set of projective plane of order 4 or steiner system s(2, 5, 21) is still unsettled problem. due to the fact, we observe some properties from construction of finite projective planes of order 2 and order 3 which is as a comparison. also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group. http://dx.doi.org/10.18860/ca.v4i3.3633 mailto:virahari@ub.ac.id mailto:co_mathub@ub.a.c.id some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 132 to observe and check some conditions, we use gap software (groups, algorithms, and programming). results and discussion based on definition 1 and theorem 1, we analyse some properties from construction of finite projective planes of order 2 and 3 by observing the characteristics of points and lines. 1. finite projective plane of order 2 finite projective plane of order 2 is a projective plane whose the smallest order, or it is called steiner system s(2, 3, 7). based on definition 1 and theorem 1, it consists of 7 points and 7 lines which there are 3 points in each line and dually every point lies on 3 lines. also, pg(2, 2) satisfies that every 2 points of p is contained in one line and dually every 2 lines of incident with one point. construction of finite projective plane of order 2 let be a finite projective plane of order 2 or denoted by pg(2, 2), the set of points is p , and l is the set of lines of . by observing these conditions, we get some characteristics which can be used to construct a finite projective plane of order 2. some properties of the constructions are contained in some following propositions. proposition 1. there are 3 important lines in l to generate a finite projective plane of order 2 such that other lines can be found uniquely. the 3 generator lines consist of : 2 lines from l , and 1 line from l . proposition 2. there are 2 possibilities conditions to choose 3 lines in l. there exist 1 point which is contained in 3 lines. then, the number possibilities of such 3 lines in l equal to 7. 3 lines in the (i) property are generator lines which satisfy proposition 1. then, the number possibilities of 3 such generator lines in l equal to 28. proposition 3. there are 6 distinct finite projective plane of order 2 which contain one same line. and, the total number of distinct finite projective plane of order 2 is 30. it is well-known that the automorphism group of is isomorphic to projective special linear group, psl(3,2). by using gap, we can find that psl(3,2) consists of some following elements which is described in table 1. table 1. all element of psl(3, 2) based on the cycle order cycle type the total number 1 1 2 21 3 56 4 42 7 48 t o t a l 168    7,,2,1  ipi  7,,2,1  ili   l lpi   'l 'lpi   7 1 23 2.1 21 3.1 111 4.2.1 1 7 some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 133 corollary 1. there is unique up to isomorphic pg(2, 2), which is cyclic. proof. let g be the automorphism group of . since and by deduction of lagrange’s theorem, g contain a single cycle of length 7, thus is cyclic. and since the index of subgroup g in the symmetric group , , there is unique up to isomorphic . ■ according to proposition 1, to construct finite projective plane of order 2 there are 3 important lines in l such that other lines can be found uniquely. let x be the set of such 3 important lines. based on proposition 2, the number possibilities of x equal to 28. by using gap, we can determine the stabilizer of this type x, . 1.2 the existence of intersection between two finite projective planes of order 2 now, we analyse the number of common lines between two finite projective planes of order 2. let and be two distinct pg(2, 2). also, let l be the set of lines of and l be the set of lines of . if │l l │= s, it means that l and l intersect in s lines and if │l l │= 0, then l and l are mutually disjoint. proposition 4. there are no and such that │l l │= 2. proof. supposse there is and such that │l l │= 2. let l l = . since in pg(2,2), every 2 lines incident with 1 point, let and every point lies on 3 lines, then we can determine which contain point p uniquely. thus, │l l │= 3 ≠ 2. it is a contradiction. hence, there is no , such that │l l │= 2. ■ proposition 5. there are no and such that │l l │= 4, 5, or 6. proof. let i = l l and . we divide the proof into two cases : case 1. if there are 3 generator lines in i which satisfy proposition 1, then . case 2. assume i do not contain generator lines based on proposition 1. according to the construction of pg(2, 2) and due to , there exist one point which incident with 3 lines in i. thus, every 2 lines of such 3 lines satisfy proposition 1 part (i). since it still has at least one line in i and based on the properties of pg(2, 2), the line satisfy proposition 1 part (ii), so . hence, it contradicts with the asumption. therefore, there is no , such that │l l │= 4, 5, or 6. ■ based on proposition 4 and 5, the possibilities of intersection between two pg(2, 2) is 0, 1, 3, or 7 lines. to examine the exsistence of this intersection, we fix pg(2, 2), and then determine the orbit of under some elements of symmetric group . proposition 6. for all , x with cycle type then │l (l )x│= 3. proposition 7. for all , x with cycle type then │l (l )x│= 1. proposition 8. for all , x with cycle type such that there is one line in l which contains 3 points in x then │l (l )x│= 0. also, according to proposition 4, proposition 5, and the existence of common lines in proposition 6 until proposition 8, we can state the following theorem. theorem 2. │l l │= 0, 1, 3, or 7, for any and are finite projective plane of order 2. moreover based on corollary 1, there is unique up to isomorphism . thus by using the orbit of fixed under all elements in the right transversal of g, we can calculate how many numbers of such that │l l │= s, where .   168732 3 g  7s   )}2,2(:{#30 168 !7 : 7 7 pg g s gs      3sxstabg   '   ' '  '  '  '  '  '  '  '  '  '  21, ll  pll  21 3l  '  '  '  '  '  ' 64  i '  4i '   '  '   7s 7sx  15 2.1   7sx  14 3.1   7sx  13 4.1     '  '   '  '  7,3,1,0s some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 134 remark 1. let be a fix pg(2, 2). the number of finite projective plane of order 2 whose have the such possibilities of size common lines with l , as stated in theorem 2 is showed in the following table 2. we can get it by using gap. table 2. the number of pg(2,2) based on the size of common lines │ l l │= s #{ : pg(2, 2) | │ l l │= s } 0 8 1 14 3 7 7 1 t o t a l 30 2. finite projective plane of order 3 based on definition 1 and theorem 1, finite projective plane of order 3 or steiner system s(2, 4, 13) consists of 13 points and 13 lines which there are 4 points in each line and dually every point lies on 4 lines. also, pg(2, 3) satisfies that every 2 points of p is contained in one line and dually every 2 lines of incident with one point. 2.1. construction of finite projective plane of order 3 let be a finite projective plane of order 3 or denoted by pg(2, 3), the set of points is p , and l is the set of lines of . by observing these conditions, we get some characteristics which can be used to construct a finite projective plane of order 3. some properties of the constructions are contained in some following propositions. proposition 9. there are 6 important lines in l to generate a finite projective plane of order 3 such that other lines can be found uniquely. let be any points of p and , then the 6 generator lines consist of : 3 lines from l , 2 lines from l , and 1 line from l , where . proposition 10. there are 4 possibilities conditions to choose 6 lines in l. there exists one point which is contained in 4 lines and another point lies on one of such 4 lines and other 2 lines. then the number possibilities of such 6 lines in l equal to 468. there exist 2 points which appear in 3 lines. then the number possibilities of such 6 lines in l equal to 78. there exist 3 points which appear in 3 lines. then the number possibilities of such 6 lines in l equal to 936. there exist 4 points which is contained in 3 lines. then the number possibilities of such 6 lines in l equal to 234. thus, the number of possibilities of 6 generator lines in l based on proposition 9 are part (iii) dan (iv), i.e., 936 + 234 = 1.170. proposition 11. there are 20.160 distinct projective plane of order 3 that contain one same line. and, the total number of distinct finite projective plane of order 3 is 1.108.800. it is well-known that the automorphism group of is isomorphic to projective special linear group, psl(3, 3). by using gap, we can find that psl(3, 3) consists of some following elements which is described in table 3. table 3. all element of psl(3, 3) based on the cycle order cycle type the total number 1 1 2 117 3 104 624  '   ' '  '    13,,2,1  ipi  13,,2,1  ili  21 , pp 121, lpp   l lp 1  'l ',' 12 lplp   ''l '',,'' 21 lpplpi   211 ,\ pplpi   13 1 45 2.1 34 3.1 41 3.1 some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 135 4 702 6 936 8 1404 13 1728 t o t a l 5616 corollary 2. there is unique up to isomorphic pg(2, 3), which is cyclic. proof. let g be the automorphism group of . since and by deduction of lagrange’s theorem, g contain a single cycle of length 13, thus is cyclic. and since the index of subgroup g in the symmetric group , , there is unique up to isomorphic . ■ according to proposition 9, to construct finite projective plane of order 3 there are 6 important lines in l such that other lines can be found uniquely. let x be the set of such 6 important lines. based on proposition 10, we can state that there are 2 types of x : there are 3 points that incident with 3 lines in x. the number possibilities of such x equal to 936. by using gap, we can determine the stabilizer of this type x, . there are 4 points that incident with 3 lines in x. the number possibilities of such x equal to 234. by using gap, we can determine the stabilizer of this type x, . 2.2 the existence of intersection between two finite projective planes of order 3 in this section, we analyse the number of common lines between two finite projective planes of order 3. let and be two distinct pg(2, 3). also, let l be the set of lines of and l be the set of lines of . if │l l │= s, it means that l and l intersect in s lines and if │l l │= 0, then l and l are mutually disjoint. proposition 12. there are no and such that │l l │= 6. proposition 13. there are no and such that │l l │= 8, 9, 10, 11, or 12. based on proposition 12 and 13, the possibilities of intersection between two pg(2, 3) is 0, 1, 2, 3, 4, 5, 7, or 13 lines. to observe the exsistence of this intersection, we fix pg(2, 3), and then determine the orbit of under some elements of symmetric group . proposition 14. for all , x with cycle type then │l (l )x│= 7. proof. from the cycle type of x, x fixes 11 points and let . since in every 2 points appear in the one line, so there is unique l such that then (l )x. and, since every point lies on 4 lines, l l , then such 6 lines satisfy l because , so do the line . also, the other 6 lines are fixed by x, since they just consist of fix points. thus, │l (l )x│= 7. ■ proposition 15. for all , x with cycle type then │l (l )x│= 1, 3, or 5. proposition 16. for all , x with cycle type then │l (l )x│= 4. proposition 17. for all , x with cycle type then │l (l )x│= 1, 2, or 3. proposition 18. for all , x with cycle type then │l (l )x│= 0, 1, or 2. according to proposition 12 and proposition 13, also the existence of common lines in proposition 14 until proposition 18, we can state the following theorem. 221 4.2.1 1112 6.3.2.1 111 8.4.1 1 13   616.51332 34 g  13s   )}3,2(:{#800.108.1 616.5 !13 : 13 13 pg g s gs      3sxstabg    4sxstabg   '   ' '  '  '  '  '  '  '  '  '   13 s 13sx  111 2.1    21 , ppx   l   lpp 21,    ll x   '# l  ',' 21 lplp    ''# l   3'','' 21  lplp   xl'    3''  x ll ''l   13sx  29 2.1   13sx  110 3.1   13sx  19 4.1   13sx  18 5.1   some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 136 theorem 3. │l l │= 0, 1, 2, 3, 4, 5, 7 or 13, for all and are finite projective plane of order 3. based on corollary 2, there is unique up to isomorphism . thus by using the orbit of fixed under all elements in the right transversal of g, we can calculate how many numbers of such that │l l │= s, where . remark 2. let be a fix pg(2, 3). the number of finite projective plane of order 3 whose have the such possibilities of size common lines with l , as stated in theorem 3 is showed in the following table 4. we can get it by using gap. table 4. the number of pg(2, 3) based on the size of common lines │ l l │= s #{ : pg(2,3) | │ l l │= s } 0 891.972 1 179.712 2 30.888 3 4.914 4 884 5 351 7 78 13 1 t o t a l 1.108.800 conclusion a finite projective plane is a geometric construction that extends the concept of a plane. it consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. in this paper we observe some properties from construction of finite projective planes of order 2 and order 3 which is as a comparison. also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group. generally, there are some analysis results of finite projective planes of order 2, pg(2, 2) and order 3, pg(2, 3). the construction of pg(2, 2) depends on 3 generator lines satisfying specific conditions such that 4 other lines can be determined uniquely, which are stated in proposition 1. while the construction of pg(2, 3) depends on 6 generator lines satisfying certain conditions such that 7 other lines can be found uniquely, which are stated in the proposition 9. furthermore, the existence of intersection between two pg(2, 2) is 0, 1, 3, or 7 common lines as listed in the theorem 2 and the number of distinct pg(2, 2) which have same intersection described in table 2. meanwhile, the existence of intersection between two pg(2, 3) is 0, 1, 2, 3, 4, 5 or 13 common lines as listed in the theorem 3 and the number of distinct pg(2, 3) which have same intersection described in table 4. based on the results, there are significant differencs between the characteristics of finite projective plane of order 2 and 3. therefore, we need to analyse further for observing the existence of the large set.  '  '   '  '  13,7,5,4,3,2,1,0s  '   ' '  ' some properties from construction of finite projective planes of order 2 and 3 vira hari krisnawati 137 references [1] s. s. magliveras, "large sets of t-designs from groups," math. slovaca 59, pp. 19-38, 2009. [2] l. g. chouinard ii, "partitions of the 4 subsets of a 13-set into disjoint projective planes, discrete mathematics," 1983. studi tentang persamaan fuzzy elva ravita sari dan evawati alisah jurusan matematika fakultas sains dan teknologi universitas islam negeri maulana malik ibrahim malang e-mail: mbemvie@gmail.com abstrak bilangan fuzzy merupakan konsep perluasan dari bilangan tegas. misalkan a adalah himpunan fuzzy dalam semesta himpunan semua bilangan riil r, maka a disebut bilangan fuzzy jika memenuhi empat sifat diantaranya yaitu: himpunan fuzzy normal, mempunyai support s�a� yang terbatas, semua a� merupakan interval tertutup untuk semua α � 0,1 dalam r, dan konveks. operasi aritmetika pada bilangan fuzzy dilakukan dengan memanfaatkan α-cut yang berbentuk interval tertutup dengan menggunakan fungsi keanggotaan bentuk segitiga, karena bentuk ini sederhana dan sudah memenuhi syarat keanggotaan bilangan fuzzy, dan sudah mewakili dari representasi fungsi keanggotaan bentuk yang lainnya. persamaan fuzzy adalah kombinasi dari bilangan fuzzy dan operasi aritmetika fuzzy. misalkan a dan b adalah bilangan fuzzy pada semesta r dengan fungsi keanggotaan masing-masing µ� dan µ�. � adalah empat operasi aritmetika dasar pada r, maka operasi a� b yaitu �a � b�� � a� � b�. selama �a � b�� adalah interval tertutup untuk setiap α � 0,1 , dan a dan b adalah bilangan fuzzy, maka a� b juga bilangan fuzzy. prosedur penyelesaian persamaan fuzzy yaitu dengan merepresentasikan bilangan fuzzy dalam bentuk α cut menggunakan fungsi keanggotaan segitiga. kemudian mengoperasikannya menggunakan operasi aritmetika pada bilangan fuzzy. penentuan hasil operasi aritmetika pada persamaan fuzzy dilakukan dengan merepresentasikan ulang bilangan fuzzy tersebut dengan α-cut, sehingga didapatkan bilangan fuzzy baru sebagai hasil penyelesaian dari persamaan fuzzy. pada skripsi ini hanya memfokuskan pokok bahasan pada persamaan fuzzy. maka dari itu, disarankan kepada pembaca untuk mengkaji lebih lanjut tentang bentuk aljabar klasik yang lainnya yaitu pertidaksamaan linier yang dikembangkan menjadi pertaksamaan fuzzy. dapat juga mengkaji lebih lanjut tentang sistem persamaan fuzzy. kata kunci: bilangan fuzzy, operasi aritmetika, dan persamaan fuzzy. pendahuluan aljabar merupakan cabang dari matematika yang tergolong klasik, tetapi dalam perkembangannya memiliki beberapa keunggulan terutama dalam aplikasinya dengan ilmu-ilmu eksak lainnya. aljabar dasar mencatat sifat-sifat operasi bilangan riil, menggunakan simbol sebagai pengganti untuk menandakan konstanta dan variabel, dan mempelajari aturan tentang ungkapan dan persamaan matematis yang melibatkan simbol-simbol. persamaan matematis tersebut seperti: (�) 2� �6 � 4 �3� , ( � ) �� 3� � 4 , dan ( ! ) 2� 3� � 4�� 1 . suatu persamaan disebut persamaan linier dalam suatu variabel jika pangkat tertinggi dari variabel dalam persamaan tersebut adalah 1 . suatu persamaan disebut persamaan kuadrat dalam variabel jika pangkat tertinggi dari variabel tersebut adalah 2 . persamaan (�) adalah persamaan linier dalam satu variabel, persamaan (�) adalah persamaan kuadrat dalam satu variabel, dan persamaan (!) adalah persamaan linier dalam dua variabel. pada persamaan linear ini berlaku hukum: (1) ruas kiri dan ruas kanan dapat ditambahkan atau dikurangi bilangan yang sama, dan (2) ruas kiri dan ruas kanan dapat dikalikan atau dibagi dengan bilangan yang sama (ayres dan schmidt, 2004:17). aljabar ini selanjutnya dinamakan aljabar klasik. dari sini muncul keinginan untuk mengembangkan persamaan aljabar fuzzy. kajian teori persamaan linier persamaan linier adalah sebuah persamaan aljabar yang setiap sukunya merupakan konstanta atau perkalian konstanta dengan variabel tunggal. secara umum persamaan linier dengan # variabel �$,��,… ,�& didefinisikan dengan '()( '*)* + ',), � (2.1) dengan �$,��,…,�& dan � adalah konstanta-konstanta riil. sebuah persamaan linier tidak melibatkan sesuatu hasil kali atau akar variabel. semua veriabel hanya terdapat sampai dengan derajat pertama dan tidak muncul sebagai argumen untuk fungsi trigonometri, fungsi logaritma, atau fungsi eksponensial (anton, 2000:17-20). studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 55 logika fuzzy (fuzzy logic) logika fuzzy merupakan peningkatan dari logika boolean yang diperkenalkan oleh dr. lotfi zadeh dari universitas california, berkeley pada tahun 1965. dalam logika boolean menyatakan bahwa segala sesuatu hanya dapat diekspresikan dalam dua nilai, yaitu 0 dan 1, hitam dan putih, atau ya dan tidak. dalam logika fuzzy memungkinkan nilai keanggotaan antara 0 dan 1, sehingga dalam logika fuzzy mengenal istilah “hitam, keabuan dan putih”, atau “sedikit, lumayan dan sangat”. himpunan fuzzy (fuzzy set) pada himpunan klasik, keberadaan suatu elemen � dalam suatu himpunan . , hanya memiliki dua kemungkinan keanggotaan, yaitu � menjadi anggota . atau � tidak menjadi anggota .. suatu nilai yang menunjukkan seberapa besar tingkat keanggotaan suatu elemen � dalam suatu himpunan . biasa disebut dengan nilai keanggotaan, yang biasa ditulis dengan /0���. pada himpunan klasik, nilai keanggotaan hanya memasangkan nilai 0 atau 1 untuk unsur-unsur pada semesta pembicaraan, yang menyatakan anggota atau bukan anggota. nilai keanggotaan untuk himpunan . adalah fungsi /0:3 4 50,16 dengan /0��� � 71, 89:� � � .0, 89:� � ; . �<�= � � .>? (klir dan yuan, 1995:6). fungsi ini, pada himpunan fuzzy diperluas sehingga nilai yang dipasangkan pada unsurunsur dalam semesta pembicaraan tidak hanya 0 dan 1 saja, tetapi keseluruhan nilai dalam interval 0,1 yang menyatakan derajat keanggotaan suatu unsur pada himpunan yang dibicarakan. fungsi ini disebut fungsi keanggotaan, dan himpunan yang didefinisikan dengan fungsi keanggotaan ini disebut himpunan fuzzy. fungsi keanggotaan himpunan fuzzy . pada himpunan semesta 3, dinotasikan dengan /0, yaitu: /0:3 4 0,1 operasi dasar himpunan fuzzy operasi “dan” (intersection) a “dan” b merupakan himpunan fuzzy dari 3, ditunjukkan sebagai derajat keanggotaan dari . bc adalah hasil yang diperoleh dengan mengambil nilai keanggotaan terkecil antarelemen pada himpunan-himpunan yang bersangkutan /0bd � min /0���,/d��� , � � 3 operasi “atau” (union) a “atau” b merupakan himpunan fuzzy dari 3, ditunjukkan sebagai derajat keanggotaan dari . hc adalah hasil yang diperoleh dengan mengambil nilai keanggotaan terbesar antarelemen pada himpunan-himpunan yang bersangkutan /0hd � max /0���,/d��� , � � 3 operasi “tidak” (complement) operasi “tidak” a merupakan himpunan fuzzy dari 3 , ditunjukkan sebagai derajat keanggotaan dari .’ (. komplemen) adalah hasil yang diperoleh dengan mengurangkan nilai keanggotaan elemen pada himpunan yang bersangkutan dari 1 /0k��� � 1 � /0��� (kusumadewi dan purnomo, 2004:24-25). fungsi keanggotaan fungsi keanggotaan (membership function) adalah suatu kurva yang menunjukkan pemetaan titik-titik input data ke dalam nilai keanggotaannya yang memiliki interval antara 0 sampai 1. salah satu cara yang dapat digunakan untuk mendapatkan nilai keanggotaan adalah melalui pendekatan fungsi. nilai keanggotaan didapatkan dari beberapa fungsi keanggotaan, dalam buku yang ditulis oleh sri kusumadewi dan hari purnomo (2004:8) dijelaskan bahwa ada beberapa fungsi yang dapat digunakan untuk memperoleh nilai keanggotaan, yaitu: a. representasi linier (linier naik dan turun) b. representasi kurva segitiga c. representasi kurva trapesium d. representasi kurva-s (pertumbuhan dan penyusutan) e. representasi kurva bentuk lonceng (bell curve) i. kelas kurva phi (m) ii. kelas kurva beta (n) iii. kurva gauss (o) p-qrs definisi 1: t-!=< adalah himpunan tegas dari himpunan fuzzy . yang mempunyai derajat keanggotaan lebih dari atau sama dengan derajat keanggotaan yang ditentukan yang dapat didefinisikan dengan .u � 5� � .,/0��� v t6. selain itu juga terdapat strong t-!=<, yakni himpunan dari himpunan fuzzy . yang mempunyai derajat keanggotaan lebih dari derajat keanggotaan yang ditentukan atau dengan kata lain .>u � 5� � .,/0��� w t6 (dubbois dan prade, 1980:19). definisi 2: misalkan . adalah himpunan fuzzy pada 3. core dari . adalah himpunan tegas yang memuat semua anggota . yang mempunyai derajat keanggotaan 1 (sivanandam, sumathi dan deepa, 2007:73). jadi, core dari . adalah x� � .µ0��� � 1y. definisi 3: elva ravita sari dan evawati alisah 56 volume 2 no. 2 mei 2012 misalkan . adalah himpunan fuzzy pada 3 . support dari . adalah himpunan tegas yang memuat semua anggota . yang mempunyai derajat keanggotaan tidak nol (klir dan yuan, 1995:21). berdasarkan definisi support, secara matematis dapat ditulis sebagai berikut: z�.� � x� � .µ0��� w 0y dalam konteks 3 � [, maka support dari ., atau z�.� , dikatakan terbatas di atas (�\=#]^] ��\_^) jika terdapat ` � [ sehingga � a `, untuk setiap � � z�.�. support dari ., atau z�.� , dikatakan terbatas di bawah (�\=#]^] �^b\c ) jika terdapat ` � [ sehingga ` a �, untuk setiap � � z�.�. selanjutnya, z�.� dikatakan terbatas (�\=#]^]) jika terbatas di atas dan terbatas di bawah (zhang dan liu, 2006:6). selanjutnya, misalkan . himpunan fuzzy dengan fungsi keanggotaan µ0��� . dibentuk himpunan fuzzy baru .u yang didefinisikan dengan .u � d� � . /0e��� � t f χ0e���g maka akan diperoleh bahwa . � h .uu� i,$ hal ini mengakibatkan bahwa himpunan fuzzy . telah didekomposisi ke dalam gabungan dari .u, untuk semua t � 0,1 . pada himpunan fuzzy .u yang didefinisikan di atas diperoleh /0e��� � t ∧ χ0e��� � 7t, � � .u0, � ; .u ? terdapat korespondensi 1 � 1 antara µ0��� dengan.u , untuk t � 0,1 . dengan demikian, himpunan fuzzy dapat dinyatakan hanya dalam bentuk t !=< tanpa menyatakan fungsi keanggotaan (susilo, 2006:74). klasifikasi berdasarkan fungsi keanggotaan berdasarkan grafik fungsi keanggotaan, himpunan fuzzy dapat diklasifikasikan ke dalam beberapa klasifikasi yaitu normal, subnormal, konvek dan takkonvek. sebelumnya akan diberikan definisi titik crossover dan tinggi suatu himpunan fuzzy. definisi 4: misalkan . adalah himpunan fuzzy pada 3. titik crossover pada . adalah titik yang mempunyai derajat keanggotaan 0.5 (sivanandam, sumathi dan deepa, 2007:74). dengan demikian, � � . disebut titik crossover jika /0��� � 0.5. definisi 5: misalkan . adalah himpunan fuzzy pada 3. tinggi dari ., dinotasikan dengan k�.�, adalah nilai maksimum dari fungsi keanggotaan himpunan . (sivanandam, sumathi dan deepa, 2007:74). dengan redaksi yang berbeda george j. klir dan bo yuan (1995:21) mendefinisikan k�.� sebagai derajat keanggotaan terbesar yang dicapai oleh sebarang unsur di . . secara simbolik, dapat dinyatakan bahwa k�.� � l��xµ0���y atau k�.� � m=nxµ0���y definisi 6: misalkan . adalah himpunan fuzzy pada 3 . . sebagai himpunan fuzzy normal jika k�.� � 1 dan subnormal jika k�.� o 1 (klir dan yuan, 1995:21). definisi 7: misalkan . adalah himpunan fuzzy pada 3 . himpunan fuzzy . disebut konvek jika fungsi keanggotaannya monoton naik, atau menoton turun, atau monoton naik dan monoton turun pada saat nilai unsur pada himpunan semesta semakin naik. himpunan fuzzy . disebut tak konvek jika fungsi keanggotaannya tidak monoton naik, atau tidak menoton turun, atau monoton naik dan monoton turun pada saat nilai unsur pada himpunan semesta semakin naik (sivanandam, sumathi dan deepa, 2007:75). sebagai ilustrasi, perhatikan gambar berikut: gambar 1. himpunan fuzzy normal dan subnormal gambar 2. himpunan fuzzy konvek dan himpunan fuzzy tak konvek operasi aritmetika fuzzy operasi aritmetika dasar pada bilangan fuzzy merupakan konsep perluasan dari operasi aritmetika dasar pada umumnya, yaitu dengan mengikut sertakan derajat keanggotaannya. operasi bilangan fuzzy dilakukan dengan memanfaatkan α !=< yang berbentuk interval tertutup. 0 1 µ0��� 0 1 µ0��� 0 1 µ0��� 0 1 µ0��� studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 57 misalkan � adalah operasi aritmetika pada interval tertutup (meliputi operasi penjumlahan, pengurangan, perkalian, dan pembagian), maka �,� � !,] � xp � q� a p a �,! a q a ]y merupakan aturan umum untuk operasi aritmetika pada interval tertutup, kecuali bahwa �,� / !,] tidak didefinisikan jika 0 � !,] . hasil operasi aritmetika pada interval tertutup juga merupakan interval tertutup. operasi aritmetika pada interval tertutup �,� dan !,] , dengan � a � dan ! a ] didefinisikan sebagai berikut. a. penjumlahan �,� !,] � min5� !,� ],� !,� ]6,? ?max5� !,� ],� !,� ]6 � � !,� ] b. pengurangan �,� – !,] � min5� � !,� � ],� � !,� � ]6,? ?max5� � !,� �],� �!,� �]6 � � � ],� � ! c. perkalian �,� ⋅ !,] � min5�!,�],�!,�]6 ,max5�!,�],�!,�]6 d. pembagian �,� / !,] � �,� ⋅ t1! ,1]u � tmin7�! ,�] ,�! ,�]v ,max7�! ,�] ,�! ,�]vu dengan syarat 0 ; !,] (klir dan yuan, 1995:103). definisi 8: misalkan . dan c adalah bilangan fuzzy pada semesta [ dengan fungsi keanggotaan masingmasing µ0��� dan µd��� . misalkan � adalah empat operasi aritmetika dasar pada [ . didefinisikan operasi . � c dengan menggunakan definisi t !=< , �. � c�u sebagai persamaan berikut: �. � c�u � .u � cu untuk setiap t � 0,1 (klir dan yuan, 1995:105). sehingga . � c dapat ditulis sebagai . � c � h �. � c�uu��i,$ dengan �. �c�u � t f χ�0�d�e��� � 7t,0, � � �. � c�u� ; �. � c�u ? dan χ�0�d�e��� � 71,0, � � �. � c�u� ; �. �c�u ? selama �. � c�u adalah interval tertutup untuk setiap t � 0,1 , . dan c adalah bilangan fuzzy, maka . � c juga bilangan fuzzy. sehingga secara khusus, diperoleh penjumlahan : �. c�u � .u cu pengurangan : �. � c�u � .u � cu perkalian : �. ·c�u � .u xcu pembagian : �./c�u � .u cu⁄ , dengan syarat 0 ; cu. bilangan fuzzy definisi 9: misalkan . adalah himpunan fuzzy pada [ . . disebut bilangan fuzzy jika memenuhi syaratsyarat berikut: 1. . merupakan himpunan fuzzy normal, 2. .u merupakan interval tertutup untuk semua t � �0,1 , dan 3. z�.�, atau .iz, merupakan himpunan terbatas (klir dan yuan, 1995:97). syarat bahwa .u merupakan interval tertutup untuk semua t � �0,1 sama dengan syarat bahwa . merupakan himpunan konvek. bilangan fuzzy sebagai himpunan fuzzy normal dan konvek, dan setiap α-!=< merupakan interval tertutup. jadi, bilangan fuzzy adalah himpunan konvek, normal, dan merupakan interval tertutup (chen dan pham, 2001:42). pembahasan secara umum bilangan fuzzy didefinisikan sebagai himpunan fuzzy dalam semesta himpunan semua bilangan riil [ yang memenuhi empat sifat diantaranya yaitu: normal, mempunyai support yang terbatas, semua t-!=< adalah selang tertutup dalam [, dan konveks. suatu bilangan fuzzy bersifat normal jika mempunyai nilai fungsi keanggotaannya sama dengan 1 dan sifat lainnya digunakan untuk dapat mendefinisikan operasi-operasi aritmetika (penjumlahan, pengurangan, perkalian, dan pembagian) pada bilangan fuzzy. operasi bilangan fuzzy dilakukan dengan memanfaatkan α-!=< yang berbentuk interval tertutup. misalkan .{ dan c| adalah bilangan fuzzy pada semesta [ dengan fungsi keanggotaan masing-masing µ0��� dan µd���, dan � adalah empat operasi aritmetika dasar pada [ . maka �. �c�u � .u � cu untuk setiap t � 0,1 . gambar 3. fungsi keanggotaan segitiga ��;�,�,!� fungsi keanggotaan segitiga diperoleh dari persamaan garis dari titik ke titik, yaitu dengan 0 1 � µ��� � � ! elva ravita sari dan evawati alisah 58 volume 2 no. 2 mei 2012 titik ���,0�, ���,1�, dan !�!,0�. yang selanjutnya fungsi keanggotaan dari bilangan fuzzy segitiga .{ dapat dinyatakan dengan /0��� � ~�� ��� � �� � � , � a � a �! � �! � �, � o � a !0, � o � atau � w ! ? kemudian akan ditentukan nilai t !=< dari bilangan fuzzy dengan fungsi keanggotaan segitiga. dari bilangan .{, dengan menyatakan t � � � �� � � diperoleh � � �� � ��t � dan untuk t � ! � �! � � diperoleh � � ! � �! � ��t jadi bilangan fuzzy .{pada semesta [ dapat ditulis ulang menjadi: .u � �� ���α �,! � �! � ��α selanjutnya untuk mendapatkan generalisasi dari masing-masing operasi aritmetika, diberikan contoh fungsi keanggotaan bilangan fuzzy dengan interval 1 satuan, interval 2 satuan, interval 3 satuan, dan interval # satuan. contoh pertama operasi aritmetika penjumlahan �� �| � ��, dimana �� dan �| adalah bilangan fuzzy dalam [. 1. operasi aritmetika penjumlahan dengan interval 1 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �1,�,� 1� dan �| � m^q9<9q���;� �1,�,� 1� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � t � � 1,� 1� t dan �u � t � � 1,� 1 �t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u �u � 2t � � � 2,� � 2� 2t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� � �2,� �,� � 2�� � �� 2. operasi aritmetika penjumlahan dengan interval 2 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �2,�,� 2� dan �| � m^q9<9q���;� �2,�,� 2� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 2t � � 2,� 2� 2t dan �u � 2t � � 2,� 2 �2t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u �u � 4t � � � 4,� � 4� 4t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� � �4,� �,� � 4�� � �� 3. operasi aritmetika penjumlahan dengan interval 3 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �3,�,� 3� dan �| � m^q9<9q���;� �3,�,� 3� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 3t � � 3,� 3� 3t dan �u 3t � �3,� 3� 3t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u �u � 6t � � � 6,� � 6� 6t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� � �6,� �,� � 6�� � �� 4. operasi aritmetika penjumlahan dengan interval # satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� � #,�,� #� dan �| � m^q9<9q���;� � #,�,� #� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � #t � � #,� # � #t dan �u � #t � � #,� # �#t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u �u � 2#t � � � 2#,� � 2# �2#t studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 59 untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� � � 2#,� �,� � 2#�� � �� dari contoh operasi aritmetika penjumlahan di atas, dapat digeneralisasikan sebagai berikut: misalkan bilangan fuzzy .{ dan c| di [ berturutturut yaitu .{ � m^q9<9q���;�,�,!� dan c| � m^q9<9q���;],^,p� maka .{ c| � m^q9<9q���;� ],� ^,! p� contoh kedua untuk operasi aritmetika pengurangan �� � �| � �� , dimana �� dan �| adalah bilangan fuzzy dalam [ 1. operasi aritmetika pengurangan dengan interval 1 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �1,�,� 1� dan �| � m^q9<9q���;� �1,�,� 1� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � t � � 1,� 1� t dan �u t � �1,� 1� t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u � �u � 2t � � � � 2,� � � 2� 2t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� � �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� �� �2,� ��,� � � 2�� � � �� 2. operasi aritmetika pengurangan dengan interval 2 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �2,�,� 2� dan �| � m^q9<9q���;� �2,�,� 2� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 2t � � 2,� 2� 2t dan �u � 2t � � 2,� 2� 2t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u � �u � 4t � � � � 4,� � � 4� 4t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� � �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� �� �4,� ��,� � � 4�� � � �� 3. operasi aritmetika pengurangan dengan interval 3 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �3,�,� 3� dan �| � m^q9<9q���;� �3,�,� 3� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 3t � � 3,� 3� 3t dan �u � 3t � � 3,� 3 �3t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u � �u � 6t � � � � 6,� � � 6� 6t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� � �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� �� �6,� ��,� � � 6�� � � �� 4. operasi aritmetika pengurangan dengan interval # satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� � #,�,� #� dan �| � m^q9<9q���;� � #,�,� #� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � #t � � #,� # � #t dan �u � #t � � #,� # �#t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u � �u � 2#t � � � � 2#,� � � 2# �2#t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �� � �| � �� adalah: �� � h ��uu� i,$ � m^q9<9q���;� � � � 2#,� ��,� � � 2#�� � � �� elva ravita sari dan evawati alisah 60 volume 2 no. 2 mei 2012 dari contoh operasi aritmetika pengurangan di atas, dapat digeneralisasikan sebagai berikut: misalkan bilangan fuzzy .{ dan c| di [ berturutturut yaitu .{ � m^q9<9q���;�,�,!� dan c| � m^q9<9q���;],^,p� maka .{ � c| � m^q9<9q���;� � p,� � ^,! �]� contoh ketiga untuk operasi aritmetika perkalian �� ·�| � �� , dimana �� dan �| adalah bilangan fuzzy dalam [ 1. operasi aritmetika perkalian dengan interval 1 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �1,�,� 1� dan �| � m^q9<9q���;� �1,�,� 1� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � t � � 1,� 1� t dan �u � t � � 1,� 1 �t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u · �u � �min��t� �� � � 2�t �� � � �� 1�,��t� ��� � � � 2�t �� � �� �1�,��t� �� �� 2�t �� �� �� 1�,�t� � �� � 2�t �� � � 1��,max��t� �� � �2�t �� �� � � 1�,��t� ��� � � � 2�t �� � �� �1�,��t� �� �� 2�t �� �� �� 1�,�t� � �� � 2�t �� � � 1��� untuk setiap t � 0,1 . 2. operasi aritmetika perkalian dengan interval 2 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �2,�,� 2� dan �| � m^q9<9q���;� �2,�,� 2� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 2t � � 2,� 2� 2t dan �u � 2t � � 2,� 2 �2t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u · �u � �min��4t� �2� 2� � 8�t �� � 2�� 2� 4�,��4t�� �2� � 2� � 8�t �� 2�� 2� �4�,��4t� �2� � 2� 8�t �� � 2� 2� �4�,�4t�� �2� 2� 8�t �� 2� 2� 4��,max��4t� �2� 2� � 8�t �� � 2�� 2� 4�,��4t�� �2� � 2� � 8�t �� 2�� 2� �4�,��4t� �2� � 2� 8�t �� � 2� 2� �4�,�4t�� �2� 2� 8�t �� 2� 2� 4��� untuk setiap t � 0,1 . 3. operasi aritmetika perkalian dengan interval 3 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �3,�,� 3� dan �| � m^q9<9q���;� �3,�,� 3� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 3t � � 3,� 3� 3t dan �u � 3t � � 3,� 3 �3t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u · �u � �min��9t� �3� 3� � 18�t �� � 3�� 3� 9�,��9t�� �3� � 3� � 18�t �� 3� � 3� � 9�,��9t� �3� � 3� 18�t ��� 3� 3� � 9�,�9t�� �3� 3� 18�t �� 3� 3� 9��,max��9t� �3� 3� � 18�t ��� 3� � 3� 9�,��9t�� �3� � 3� � 18�t �� 3� � 3� � 9�,��9t� �3� � 3� 18�t ��� 3� 3� � 9�,�9t�� �3� 3� 18�t �� 3� 3� 9��� untuk setiap t � 0,1 . 4. operasi aritmetika perkalian dengan interval # satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� � #,�,� #� dan �| � m^q9<9q���;� � #,�,� #� studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 61 dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � #t � � #,� # � #t dan �u � #t � � #,� # �#t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u · �u � �min��#�t� �#� #� � 2#��t �� � #�� #� #��,��#�t�� �#� � #� � 2#��t �� #� � #� � #��,��#�t� �#� � #� 2#��t ��� #� #� � #��,�#�t�� �#� #� 2#��t �� #� #� #��� ,max��#�t� �#� #� � 2#��t ��� #� � #� #��,��#�t�� �#� � #� � 2#��t �� #� � #� � #��,��#�t� �#� � #� 2#��t ��� #� #� � #��,�#�t�� �#� #� 2#��t �� #� #� #���� untuk setiap t � 0,1 . contoh keempat untuk operasi aritmetika pembagian ���| � �� , dimana �� dan �| adalah bilangan fuzzy dalam [ 1. operasi aritmetika pembagian dengan interval 1 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �1,�,� 1� dan �| � m^q9<9q���;� �1,�,� 1� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � t � � 1,� 1� t dan �u � t � � 1,� 1 �t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u�u� tmin�t � � 1t � � 1,t � �1� 1� t,� 1 � tt � � 1,?? ??� 1 � t� 1 � t�,max�t � � 1t � �1,t � � 1� 1 � t,?? ??� 1 �tt � � 1,� 1� t� 1� t�u untuk setiap t � 0,1 . 2. operasi aritmetika pembagian dengan interval 2 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �2,�,� 2� dan �| � m^q9<9q���;� �2,�,� 2� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 2t � � 2,� 2� 2t dan �u � 2t � � 2,� 2 �2t dengan menggunakan definisi 9 operasi aritmetika pada interval, diperoleh �u � �u�u� tmin�2t � � 22t � � 2,2t � � 2� 2 �2t ,� 2� 2t2t � � 2,?? ?? � 2 � 2t� 2 � 2t�,max�2t � � 22t � � 2,2t � � 2� 2 � 2t ,?? ??� 2 �2t2t � � 2, � 2 �2t� 2 �2t�u untuk setiap t � 0,1 . 3. operasi aritmetika pembagian dengan interval 3 satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� �3,�,� 3� dan �| � m^q9<9q���;� �3,�,� 3� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � 3t � � 3,� 3� 3t dan �u � 3t � � 3,� 3 �3t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u�u� tmin�3t � � 33t � � 3,3t � � 3� 3 �3t ,� 3� 3t3t � � 3,?? ??� 3 �3t� 3 � 3t�,max�3t � � 33t � �3,3t � � 3� 3 � 3t ,?? ??� 3 �3t3t � � 3,� 3� 3t� 3 �3t�u untuk setiap t � 0,1 . 4. operasi aritmetika pembagian dengan interval # satuan bilangan fuzzy �� dan �| mempunyai fungsi keanggotaan �� � m^q9<9q���;� � #,�,� #� dan �| � m^q9<9q���;� � #,�,� #� dari bilangan fuzzy �� dan �| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu �u � #t � � #,� # � #t dan �u � #t � � #,� # �#t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � �u�u� tmin�#t � � ##t � � #,#t � � #� # � #t ,� # � #t#t � � #,?? elva ravita sari dan evawati alisah 62 volume 2 no. 2 mei 2012 ??� # � #t� # � #t�,max�#t � � ##t � � #,#t � � #� # � #t ,?? ???� # � #t#t � � #,� # � #t� # � #t�?u untuk setiap t � 0,1 . contoh 1: cari penyelesaian persamaan fuzzy 7| 5| � ��. bilangan fuzzy 7| dan 5| mempunyai fungsi keanggotaan 7| � m^q9<9q���;4,7,10� dan 5| � m^q9<9q���;3,5,7� dari bilangan 7| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 7u � 3t 4,10 � 3t dan dari bilangan 5|dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 5u � 2t 3,7 � 2t dengan menggunakan definisi 2.9 operasi aritmetika pada interval, diperoleh �u � 5t 7,17 �5t , t � 0,1 sehingga selesaian dari persamaan fuzzy 7| 5| � �� adalah �� � ~�� �� 0, � a 7 dan � v 17� � 75 , 7 a � a 1217� �5 , 12 a � a 17 ? atau dapat ditulis sebagai berikut �� � h ��u � m^q9<9q���;7,12,17� � 12�u� i,$ contoh 2: cari penyelesaian persamaan fuzzy 15� � 5| � ��. bilangan fuzzy 5| dan 15� mempunyai fungsi keanggotaan 15� � m^q9<9q���;10,15,20� dan 5| � m^q9<9q���;3,5,7� dari bilangan 15� dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 15u � 5t 10,20� 5t dan dari bilangan 5| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 5u � 2t 3,7 � 2t dengan menggunakan definisi 2.9 operasi aritmetika pada interval, diperoleh �u � 7t 3,17 �7t , t � 0,1 sehingga selesaian dari persamaan fuzzy 15� � 5| � �� adalah �� � ~�� �� 0, � a 3 dan � v 17� � 37 , 3 a � a 1017� �7 , 10 a � a 17 ? atau dapat ditulis sebagai berikut �� � h ��u � m^q9<9q���;3,10,17� � 10�u� i,$ contoh 3: cari penyelesaian persamaan fuzzy 3| ·2| � ��. bilangan fuzzy 3| dan 2| mempunyai fungsi keanggotaan 3| � m^q9<9q���;2,3,4� dan 2| � m^q9<9q���;�1,2,5� dari bilangan 3| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 3u � t 2,4� t dan dari bilangan 2| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 2u � 3t �1,5 � 3t dengan menggunakan definisi operasi aritmetika pada interval, diperoleh �u � 7 �3t� 13t � 4,3t� � 17t 20 , t � 0,0.5 3t� 5t � 2,3t� � 17t 20 , t � 0.5,1 ? sehingga selesaian dari persamaan fuzzy 3| ·2| � �� adalah �� � h ��uu� i,$ � ~� �� �� � 0, � a �4 dan � v 20�13 √121� 12��6 , �4 a � a 0�5 √49 12�6 , 0 a � a 617 � √49 12�6 , 6 a � a 20 ? contoh 4: cari penyelesaian persamaan fuzzy $��� � ��. bilangan fuzzy 1| dan 3| mempunyai fungsi keanggotaan 1| � m^q9<9q���;�1,1,3� dan 3| � m^q9<9q���;2,3,4� dan dari bilangan 1| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 1u � 2t �1,3� 2t dari bilangan 3| dapat dinyatakan dalam t-!=< untuk t � 0,1 , yaitu 3u � t 2,4 � t dengan menggunakan definisi 2.9 operasi aritmetika pada interval, diperoleh �u � �t 2t �1t 2 ,3 � 2tt 2 u, t � 0,0.5 t2t �14� t ,3 � 2tt 2 u, t � 0.5,1 ? sehingga selesaian dari persamaan fuzzy $��� � �� adalah studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 63 �� � h ��uu� i,$ � ~� �� �� � 0, � a �12 dan � v 322� 12 � � , �12 a � a 04� 1� 2 , 0 a � a 133 � 2�� 2 , 13 a � a 32 ? persamaan fuzzy terdiri dari bilangan fuzzy dan operasi aritmetika pada bilangan fuzzy. berikut ini adalah karakterisasi sifat dari persamaan fuzzy yang sangat sederhana yaitu �� �� � �| dan �� ·�� � �| , dimana �� dan �| adalah bilangan fuzzy yang diketahui, dan �� adalah bilangan fuzzy yang merupakan selesaian dari persamaan fuzzy tersebut (klir dan yuan, 1995:114). misalkan diketahui bilangan fuzzy �� dan �|, dan akan dicari bilangan fuzzy ��. t-!=< dari ��, �|, dan �� berturut-turut adalah �u � �u�,�uz , �u � �u�,�uz , dan �u � �u�,�uz . dari persamaan fuzzy �� �� � �| , dengan menggunakan t !=< diperoleh �u� �u�,�uz �uz � �u�,�uz untuk t � 0,1 . jadi �u� �u� � �u� dan �uz �uz � �uz , sehingga �u� � �u� � �u� dan �uz � �uz � �uz . agar t !=< dari bilangan fuzzy �� adalah �u � �u�,�uz � �u� ��u�,�uz � �uz , maka harus dipenuhi syarat berikut. a. �u� � �u� a �uz � �uz untuk setiap t � 0,1 , dan b. jika t a n , maka �u� � �u� a ��� � ��� a ��z ���z a �uz � �uz. jika kedua syarat tersebut dipenuhi, maka selesaian persamaan fuzzy �� �� � �| yaitu �� � h ��uu� i,$ dan untuk persamaan fuzzy �� ·�� � �| dimana �� dan �| adalah bilangan fuzzy pada [z, maka solusi dari persamaan fuzzy harus memenuhi syarat berikut. a. �u�/�u� a �uz/�uz untuk t � 0,1 , dan b. jika t a n, maka �u�/�u� a ���/��� a ��z/��z a�uz/�uz. jika kedua syarat tersebut terpenuhi, maka selesaian dari persamaan fuzzy �� ·�� � �| yaitu �� � h ��uu� i,$ (klir dan yuan, 1995:115-116). contoh 5: misalkan diberikan persamaan fuzzy 4| ·�� 8|5| �1| � 3| dengan bilangan fuzzy 1| � m^q9<9q��0,1,2� 3| � m^q9<9q���1,3,7� 4| � m^q9<9q��3,4,5� 5| � m^q9<9q��4,5,6� 8| � m^q9<9q��6,8,10� representasi 1|, 3|, 4|, 5|, dan 8| yaitu 1u � t,2 � t , 3u � 4t � 1,7 � 4t , 4u � t 3,5� t , 5u � t 4,6 �t , dan 8u � 2t 6,10 � 2t . dari persamaan fuzzy 4| ·�� 8|5| �1| � 3| misal: 4| ·�� 8|5| � `̃ 4| ·�� 8| � m̃ 4| ·�� � <̃ maka, `̃ � 1| � 3| �3.1� m̃5| � `̃ �3.2� <̃ 8| � m̃ �3.3� 4| · �� � <̃ �3.4� untuk persamaan 3.1, dengan menggunakan t-!=< diperoleh ù� � �2 � t�, ùz � t � 4t � 1,7 � 4t untuk t � 0,1 . jadi ù� ��2 � t� � 4t � 1 dan ùz � t � 7� 4t. sehingga ù� � 3t 1 ùz � 7� 3t kemudian diuji bahwa ù� a ùz dan jika 0 a t a n a 1 maka 3t 1 a 3n 1 a 7 �3n a 7 �3t , agar t-!=< dari bilangan fuzzy `̃ adalah ù � ù�, ùz . sehingga ù � 3t 1,7 � 3t untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy `̃ � 1| � 3| adalah `̃ � h `̃uu� i,$ � ~�� �� 0, � a 1 dan � v 7� � 13 , 1 a � a 47� �3 , 4 a � a 7 ? karena `̃ � 4|, maka persamaan 3.2 menjadi �̃�� �4|. dari persamaan �̃�� � 4| dengan menggunakan t-!=< diperoleh � �e���u , �e�uz�� � 3t 1,7 � 3t untuk t � 0,1 . jadi �e���u � 3t 1 dan �e�uz� � 7� 3t . sehingga mu� � �3t� 17t 6 muz � �3t� �5t 28 kemudian diuji bahwa mu� a muz dan jika 0 a t a n a 1 maka �3t� 17t 6 a �3n� 17n 6 a �3n� �5n 28 a �3t� � 5t 28 , agar t !=< dari bilangan fuzzy m̃ adalah mu � mu�, muz . sehingga mu � �3t� 17t 6,�3t� � 5t 28 untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy �̃�� � 4| adalah elva ravita sari dan evawati alisah 64 volume 2 no. 2 mei 2012 m̃ � h m̃uu� i,$ � ~�� �� 0, � a 6 dan � v 28�17 √361 �12��6 , 6 a � a 205 � √361� 12��6 , 20 a � a 28 ? karena m̃ � 20� , maka persamaan 3.3 menjadi <̃ 8| � 20� . dari persamaan <̃ 8| � 20� dengan menggunakan t !=< diperoleh <u� �2t 6�,<uz �10 � 2t� � �3t� 17t 6,�3t� �5t 28 untuk t � 0,1 . jadi <u� �2t 6� ��3t� 17t 6 dan <uz �10 � 2t� � �3t� �5t 28. sehingga <u� � �3t� 15t <uz � �3t� � 3t 18 kemudian diuji bahwa <u� a <uz dan jika 0 a t a n a 1 maka �3t� 15t a �3n� 15n a �3n� � 3n 18 a �3t� �3t 18, agar t-!=< dari bilangan fuzzy <̃ adalah <u � <u�, <uz . sehingga <u � �3t� 15t,�3t� � 3t 18 untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy <̃ 8| � 20� adalah <̃ � h <̃uu� i,$ � ~�� �� 0, � a 0 dan � v 18�15 √225 �12��6 , 0 a � a 123 � √225� 12��6 , 12 a � a 18 ? karena <̃ � 12� , maka persamaan 3.4 menjadi 4| ·�� � 12� . dari persamaan 4| ·�� � 12� dengan menggunakan t !=< diperoleh �t 3� ·�u�, �5 � t� ·�uz � �3t� 15t,�3t� � 3t 18 untuk t � 0,1 . jadi �t 3� ·�u� � �3t� 15t dan �5 � t� ·�uz � �3t� � 3t 18 . sehingga �u� � �3t� 15tt 3 �uz � �3t� � 3t 185 � t kemudian diuji bahwa �u� a �uz dan jika 0 a t a n a 1 maka ��u�z$�uuz� a ����z$���z� a�������z$���� a ��u���uz$���u , agar t-!=< dari bilangan fuzzy �� adalah �u � �u�,�uz . sehingga �u � ��3t� 15tt 3 ,�3t� �3t 185 �t   untuk setiap t � 0,1 . sehingga selesaian dari persamaan fuzzy ��·¡�z���� � 1| � 3| adalah �� � h ��uu� i,$ � ~�� � ��� 0, � a 0 ]�# � v 185� � 15 √�� �66� 225�6 , 0 a � a 33� � � √�� � 66� 225�6 , 3 a � a 185 ? penutup kesimpulan persamaan fuzzy adalah kombinasi dari bilangan fuzzy dan operasi aritmetika pada bilangan fuzzy. bilangan fuzzy merupakan konsep perluasan dari bilangan tegas yang memenuhi empat sifat diantaranya yaitu: himpunan fuzzy normal, mempunyai support yang terbatas, semua t-!=< merupakan interval tertutup untuk semua t � 0,1 , dan konvek. dan operasi aritmetika dasar pada bilangan fuzzy merupakan konsep perluasan dari operasi aritmetika dasar pada umumnya, yaitu dengan mengikutsertakan derajat keanggotaannya. prosedur penyelesaian persamaan fuzzy yaitu dengan merepresentasikan bilangan fuzzy dalam bentuk t !=< menggunakan fungsi keanggotaan segitiga. kemudian mengoperasikannya menggunakan operasi aritmetika dasar pada bilangan fuzzy. penentuan hasil operasi aritmetika pada persamaan fuzzy dilakukan dengan merepresentasikan ulang bilangan fuzzy tersebut dengan t-!=<, sehingga didapatkan bilangan fuzzy baru sebagai hasil penyelesaian dari persamaan fuzzy. saran pada skripsi ini, penulis hanya membahas pada persamaan fuzzy. maka dari itu, penulis menyarankan kepada pembaca untuk mengkaji bentuk aljabar klasik yang lainnya, yaitu tentang pertidaksamaan linier yang dikembangkan menjadi pertaksamaan fuzzy. dapat juga mengkaji lebih lanjut tentang sistem persamaan fuzzy. daftar pustaka [1] anton, howard. 2000. dasar-dasar aljabar linier. batam: interaksara. [2] ayres, frank dan schmidt, philip a. 2004. schaum’s outline of theory and problem’s of college mathematics. terjemahan alit bondan. jakarta: erlangga. [3] chen, guanrong dan pham, trung tat. 2000. introduction to fuzzy sets, fuzzy logic, and fuzzy control systems. london: crc press. studi tentang persamaan fuzzy jurnal cauchy – issn: 2086-0382 65 [4] dubbois, deider dan prade, henri. 1980. fuzzy sets and systems, theory and applications. new york: academic press. [5] klir, george j. dan yuan, bo. 1995. fuzzy set and fuzzy logic: theory and applications. new jersey: prentice hall international, inc. [6] kusumadewi, sri. 2002. analisis dan desain system fuzzy menggunakan toolbox matlab. yogyakarta: graha ilmu. [7] kusumadewi, sri dan purnomo, hari. 2004. aplikasi logika fuzzy untuk pendukung keputusan. yogyakarta: graha ilmu. [8] sivanandam, sumathi dan deepa. 2006. introduction to fuzzy logic using matlab. india: springer. [9] susilo, frans. 2006. himpunan dan logika kabur serta aplikasinya. yogyakarta: graha ilmu. [10] utomo, tri. 2012. operasi aritmetika dasar pada bilangan fuzzy dan sifat-sifatnya. skripsi. universitas islam negeri maulana malik ibrahim malang. [11] yudha d, made. no. 36 pebruari 1997. sistem fuzzy: sebuah kecenderungan. science. hal: 8-9. [12] zadeh, lotfi a. 2000. fuzzy sets and fuzzy information-granulation theory. beijing: beijing normal university press. [13] zhang, huaguang dan liu, derong. 2006. interval type-2 fuzzy hidden markov models. hungary: proc. of ieee fuzz conference, budapest lembar kosong (5).pdf 2.dewan editor.pdf 0 lembar kosong (1).pdf 3. pengantar redaksi.pdf lembar kosong (6).pdf 4. daftar isi.pdf 5.1. elva ravita sari studi tentang persamaan fuzzy _54-65.pdf diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m volume 4, issue 3, november 2016 jurnal matematika murni dan aplikasi malang november 2016 cauchy no. 3 pages: vol. 4 issn 2086-0382 e-issn 2477-3344 issn 2086-0382 e-issn 2477-3344 cauchy  editor in chief : fachrur rozi, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. dr. usman pagalay, m.si, maulana malik ibrahim state islamic university of malang, indonesia. dr. abdussakir, m.pd, maulana malik ibrahim state islamic university of malang, indonesia. ari kusumastuti, m.si, maulana malik ibrahim state islamic university of malang, indonesia. assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id jurnal matematika murni dan aplikasi cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy editorial board jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we welcome authors for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 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(2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy table of contents application of pontryagin’s minimum principle in optimum time of missile manoeuvring ....................................................................................................... 107 – 111 power of test path analysis and partial least square analysis ........................ 112 – 114 estimation parameters and modelling zero inflated negative binomial ........... 115 – 119 leontief input-output method for the fresh milk distribution linkage analysis ............................................................................................................................. 120 – 124 on the local metric dimension of line graph of special graph ...................... 125 – 130 some properties from construction of finite projective planes of order 2 and 3 ............................................................................................................................ 131 137 leontief input-output method for the fresh milk distribution linkage analysis cauchy – jurnal matematika murni dan aplikasi volume 4(3) (2016), pages 120-124 submitted: october, 14 2016 reviewed: october, 21 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3718 leontief input-output method for the fresh milk distribution linkage analysis riski nur istiqomah dinnullah1, trija fayeldi 2 1,2mathematics education department, kanjuruhan university malang email: ky2_zahra@unikama.ac.id, trija_fayeldi@unikama.ac.id abstract this research discusses about linkage analysis and identifies the key sector in the fresh milk distribution using leontief input-output method. this method is one of the application of mathematics in economy. the current fresh milk distribution system includes dairy farmers→collectors→fresh milk processing industries→processed milk distributors→consumers. then, the distribution is merged between the collectors’ axctivity and the fresh milk processing industry. the data used are primary and secondary data taken in june 2016 in kecamatan jabung kabupaten malang. the collected data are then analyzed using leontief input-output matrix and maple software. the result is that the merging of the collectors’ and the fresh milk processing industry’s activities shows high indices of forward linkages and backward linkages. it is shown that merging of the two activities is the key sector which has an important role in developing the whole activities in the fresh milk distribution. keywords: fresh milk distribution, leontief input-output method, forward linkages, backward linkages, key sector introduction mathematics is a subject which underlies and serves many other subjects that are needed in the development of modern science and technology [1]. mathematics functions as a tool to help finding the right solution to solve problems. using mathematics, every problem that occurs in many kinds of fields can be solved through mathematical approach and a mathematical modeling is formed. one of these mathematical applications is the leontief inputoutput method. leontief input-output method is a mathematical modeling which is used to determine the value of the production output that needs to be expensed by the industries in the same economic system in order to preserve other industrial processes. this is the basic method to illustrate the economic activities as a linkage system between goods and services, and also to analyze the intersectoral or industrial linkages in economy [2]. leontief input-output method has different kinds of applications in economy. in the article from [3], and [4], the leontief input-output method has been applied in many modern science and technology development. some of the method’s application on the pollution control problems have been discussed in [5], [6], and [7]. basically, the leontief input-output method uses inverse matrix or linear system of equation to construct an economic model. this method shows how the output from one industry can be the input for other industries and determines the linkage between each of the industries or sectors in economy [8]. this method is used to calculate the index of backward linkage that provides how much the sector demands from the other sectors of the economy, whereas the index of forward linkage provides the quantity of http://dx.doi.org/10.18860/ca.v4i3.3718 leontief input-output method for the fresh milk distribution linkage analysis riski nur istiqomah dinnullah 121 products demanded from the related sectors to the sector [9]. this linkage analysis was originated by [10], which was then used as a mean to identify the key sector by [11]. when one sector has high index values of backward and forward linkage, the sector has overall above average linkages of economy sector; thus, it is assumed that the sector is the key sector in the economy [9]. the aim of this article is to discuss about the leontief input-output method which is used to analyze the linkages and to identify the key sector in the fresh milk distribution. the fresh milk distribution system found in this article is: dairy farmers→collectors→fresh milk processing industries→processed milk distributors→consumers. however, in this research the collectors’ and the fresh milk processing activities are merged. the data used are the primary and secondary data obtained through interview, questionnaire and observation in june 2016. these data were taken from the dairy farmers, koperasi agro niaga jabung as the collector and the fresh milk processing industry, and the distributors within the area of kecamatan jabung kabupaten malang. next, the data were analysed using the leontief input-output matrix and maple software. in the economic activity, fresh milk is a dairy product which might potentially improve the national economy in the future. beside consumed freshly, milk is also the raw material for the fresh milk processing industry. several products produced from fresh milk are pasteurized milk, butter, cheese, yoghurt, skimmed milk or non-fat milk and some other food products. this fact indicates that the growth in dairy farming industries will affect other industries. furthermore, the growth in the fresh milk processing industry has strong linkages with the growth in the dairy farming sector. most of this fresh milk processing industries are located in the area whose majority of the citizens are dairy farmers. through this linkage analysis on the fresh milk distribution, the fresh milk processing industry has both backward linkages towards the raw material related to the dairy farmers, and forward linkages towards the food and beverage industry. methods this research is an industrial mathematic research focusing on the problems within the fresh milk distribution system in kecamatan jabung on june 2016. the objects of this research are several groups of dairy farmers in kecamatan jabung, koperasi agro niaga jabung as the collector as well as the fresh milk processing industry, and the processed milk distributors in kecamatan jabung. the explanation level of this research is up to description level. thus, this research will describe the phenomenon of the research objects. this research uses the qualitative and quantitative analysis techniques. the qualitative analysis is conducted through direct observation of the condition of the fresh milk distribution system in kecamatan jabung. the data is obtained through interview with several groups of dairy farmers, the koperasi as the fresh milk collector as well as the fresh milk processing industry, and the processed milk distributor in kecamatan jabung. the quantitative analysis includes income and added value analysis. the data obtained are then analyzed. before analyzing the data, the first step is to process it using raw data tabulation. to fill in the table 1, we conduct income and added value analysis for each of the activity. after that, we analyze backward and forward linkage using input-output method. income analysis the amount of income from each sector within 1 month can be formulated as follows: i𝑖 = q𝑖 x p𝑖 (8) where: i𝑖 = income of sector-i for 1 month q𝑖 =quantity of product sold by sector-i within 1 month p𝑖 = price per product sold by sector-i within 1 month. leontief input-output method for the fresh milk distribution linkage analysis riski nur istiqomah dinnullah 122 added value analysis the amount of added value is obtained from the margin of operational cost and income of each of the sector within 1 month. these operational costs include production cost, trade system cost, maintenance cost, shipping cost, or other costs which are adjusted to the expense of each sector. the added value analysis can be formulated as follows: va𝑖 = i𝑖 − oc𝑖 (9) where: va𝑖 = added value of sector-i within 1 month oc𝑖 = operational cost of sector-i within 1 month. result and discussion in this research, we merge the collectors’ and the fresh milk processing industries’ activities in the fresh milk distribution system model as follows: figure 1. fresh milk distribution system model by merging the collectors’ and the fresh milk processing industries’ activities using equation (8) and (9), the data is processed and the result is put into the inputoutput table of fresh milk distribution as follows: table 1. input-output table of fresh milk distribution by merging the collectors’ and the fresh milk processing industries’ activities (in million rupiah) output input farmers collectors and fresh milk processing industries distributors final demand total output farmers 0 7.856,859 0 0 7.856,859 collectors and fresh milk processing industries 2.526,24 0 357,312 0 2.883,552 distributors 0 0 0 488,927 488,927 added value 3.510 2.124,631 478,727 total input 6.036,24 9.981,49 836,039 (source: data processing result) table 2 shows that the output or the dairy farming product, that is the fresh milk, is used as the input or raw material for the collectors and the fresh milk processing industries, in this case is the koperasi agro niaga (kan) jabung, with the total value of rp7,856.859 million. the collectors distribute the fresh milk to the fresh milk processing industries located outside of processed milk distributors dairy farmers collectors & fresh milk processing industries consumers leontief input-output method for the fresh milk distribution linkage analysis riski nur istiqomah dinnullah 123 desa jabung, but these collectors also function as the small scale fresh milk processing industries. besides selling the fresh milk to the processing industries, the collectors also give an output in the form of cattle food that is used as the input for the dairy farmers with the total value of rp2,526.24 million. then, the output or the product from the fresh milk processing industries by the collectors is distributed to the processed milk distributors with the total value of rp357.312 million. next, the output from the distributors is sold to the consumers with the total value of rp488.927 million. the added value or the income of each of the activities are rp3,510 million, rp2,124.631 million, and rp478.727 million. from table 2, we can generate a matrix as follows 𝐴 = [ 0 7.856,859 0 2.526,24 0 357,312 0 0 0 ]. (10) substitusing matrix a from (10) into equation (4) and we obtain leontief inverse matrix as follow 𝛼 = [ 1.4913 1.1738 0.5017 0.6241 1.4913 0.6373 0 0 1 ]. (11) furthermore, substitute 𝛼 from (11) into equation (6) and (7). using maple software, we obtain indices of backward linkages and forward linkages which can be seen on table 3. table 2. linkage indices of merging the collectors’ and the fresh milk processing industries’ activities in the fresh milk distribution activities forward linkages activities backward linkages farmers 1.3730 farmers 0.9171 collectors & fresh milk processing industries 1.1935 collectors & fresh milk processing industries 1.1555 distributors 0.4336 distributors 0.9274 (source: data processing result) on table 3, we can see that the farmers’ activity has low index of backward linkages, that is 0.9171 (bl<1), while the forward linkages have high index, that is 1.3730 (fl>1). these numbers show that the farmers’ activity can only shove the activities in front of them, but it cannot attract the growth of the activities behind. whereas for the distributors’ activity, the indices of forward linkages and backward linkages are low, those are 0.4336 (fl<1) and 0.9274 (bl<1). these numbers show that the distributors’ activity does not influence the growth of others’ activities. on the other hand, the merging of the collectors’ and the fresh milk processing industries’ activities has high indices of both forward linkages and backward linkages, those are 1.1935 (fl>1) and 1.1555 (bl>1). these numbers show that the merging of the two activities has high forward and backward linkages so that it can be concluded that this might be the key sector, which can improve the growth of all activities in the fresh milk distribution. thus, we know that the merging of the collectors’ and the fresh milk processing industries’ activities can be the key sector within the fresh milk distribution. therefore, developments in the merging of the two activities will have great influences to the growth of the whole activities within the fresh milk distribution. conclusion leontief input-output method can illustrate the intersectoral linkages and identify the key sector in the fresh milk distribution. it is clearly shown that the merging of the collectors’ and the fresh milk processing industries’ activities offers high indices of both backward linkages and forward linkages, so that it can be the key sector in the fresh milk distribution. the key sector has strong attraction and thrust towards the growth of upstream as well as downstream industries. therefore, the merging of the two activities has great influences and needs more leontief input-output method for the fresh milk distribution linkage analysis riski nur istiqomah dinnullah 124 attention in terms of its relation with the developments of the whole activities in the fresh milk distribution. references [1] y. rahayu and b. nurhadiyono, "implementasi matriks pada matematika bisnis dan ekonomi," techno.com, pp. 74-81, 2012. [2] . f. j. m. tanaka, "applications of leontief’s input-output analysis in our economy," faculty of economics journal university of nagasaki, vol. 45, no. 1, pp. 29-96, 2011. [3] a. a. ebiefung and k. m., "the generalized leontief input-output model and its application to the choice of new technology," annals of operations research, vol. 44, pp. 161-172, 1993. [4] a. a. ebiefung , "choice of technology, industrial pollution, and the vertical linear complementarity problem," global journal of mathematical sciences, vol. 9, no. 2, pp. 113-120, 2010. [5] a. a. ebiefung, "a generalization of the input-output pollution control model and product selection," applied mathematics, vol. 4, pp. 360-362, 2013. [6] a. a. ebiefung and i. isaac, "an input-output pollution control model and product selection," journal of mathematical research, vol. 4, no. 5, pp. 1-7, 2012. [7] w. leontief , input-output economics, new york: oxford university press, 1986. [8] c. obikwere and a. a. ebiefung, "the leontief input-ooutput production model and its application to inventory control," asian journal of mathematics and applications, pp. 1-7, 2014. [9] t. s. silveira, t. smaniotto, d. r. fabris, a. n. neto, c. a. g. jùnior, b. f. cardoso and p. f. a. shikida, "input-output analysis for agricultural and livestock sector in the brazilian economy," rivista di economia agraria, vol. lxx, no. 1, pp. 33-54, 2015. [10] p. n. rasmussen , studies inintersectoral relations, amsterdam: north-holland, 1956. [11] a. o. hirschman , the strategy of economic development, new haven: yale university press, 1958. diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it 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reviewers in this issue was very appreciated abdussakir (maulana malik ibrahim state islamic university of malang) ari kusumastuti (maulana malik ibrahim state islamic university of malang) fachrur rozi (maulana malik ibrahim state islamic university of malang) sri harini (maulana malik ibrahim state islamic university of malang) usman pagalay (maulana malik ibrahim state islamic university of malang) thoufina kurniyati deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana thoufina kurniyati mahasiswa jurusan matematika fakultas sains dan teknologi uin maulana malik ibrahim malang e-mail: thoufinakurniyati@yahoo.co.id abstrak sistem bandul ganda sederhana merupakan pengembangan sistem bandul sederhana. penurunan model bandul ganda sederhana berasal dari persamaan euler-lagrange. sistem bandul ganda sederhana didapatkan dengan asumsi besar sudut perpindahan benda pertama maupun benda kedua sangat kecil. penelitian terdahulu [1] membahas mengenai kestabilan dan solusi eksak dari sistem bandul ganda sederhana, selanjutnya pada penelitian ini difokuskan untuk mendeskripsikan kestabilan perilaku pada sistem dengan parameter yang berbeda. hasil penelitian ini menunjukkan sistem memiliki titik tetap trivial, nilai eigen imajiner murni yang berarti sistem berayun di sekitar titik tetap, dan solusi sistem berupa solusi periodik untuk perubahan besar sudut benda pertama dan kedua (𝜃1,𝜃2) serta solusi quasiperiodic untuk laju kecepatan benda satu dan benda dua (�̈�1, �̈�2). perubahan parameter tidak mempengaruhi kestabilan sistem bandul ganda sederhana. kata kunci: sistem bandul ganda sederhana, analisis perilaku, titik tetap, nilai eigen, vektor eigen, solusi periodik, solusi quasiperiodic abstract simple double pendulum system is the development of a simple pendulum system. the decline in the simple model of a double pendulum is derived from the euler-lagrange equation. simple double pendulum system obtained by assuming a large angle displacement of the object first and second objects is very small. previous research by [1] discuss the stability and the exact solution of a simple double pendulum system, further research is focused to describe the stability of the behavior of the system with different parameters. the results of this study indicate the system has a nontrivial fixed point, purely imaginary eigenvalues which means the system swinging around a fixed point, and system solutions in the form of periodic solutions for a major change angle first and second objects (𝜃1,𝜃2) and quasi-periodic solutions to the rate of velocity one and two object (�̈�1, �̈�2). parameter changes do not affect the stability of a simple double pendulum system keywords : simple double pendulum system, behavioral analysis, fixed point, eigen value, eigen vector, periodic solutions, quasiperiodic solutions pendahuluan peranan teori dan peranan penerapan matematika tidak dapat dipisahkan. banyak konsep abstrak matematika yang dikembangkan karena kebutuhan untuk menjawab permasalahan dari dunia nyata dan bidang ilmu lain [2]. pemodelan matematika dapat diterapkan ke berbagai disiplin ilmu lain, salah satunya adalah sistem bandul ganda sederhana. sistem bandul ganda sederhana merupakan pengembangan bandul sederhana, pengertian bandul sederhana yaitu benda ideal yang terdiri dari sebuah titik massa, yang digantungkan pada tali ringan yang tidak dapat mulur [3]. penelitian sebelumnya [1] membahas mengenai solusi eksak dan kestabilan sistem bandul ganda sederhana, selanjutnya pada penelitian ini difokuskan mendeskripsikan kestabilan perilaku dengan parameter yang berbeda, parameter tersebut adalah massa benda pertama 𝑚1. salah satu permasalahan yang menggunakan sistem bandul ganda adalah pada sistem kerja tim sar (search and rescue) dan suplai makanan atau amunisi ke barak analisis dengan menggunakan helikopter. berbagai bencana alam yang ada di indonesia akhir-akhir ini, menyebabkan kerja tim sar semakin tinggi. keefektifan kerja tim sar sangat diperlukan mailto:thoufinakurniyati@yahoo.co.id thoufina kurniyati 125 volume 3 no. 3 november 2014 agar bantuan makanan dan, pakaian ataupun obat-obatan dapat tersebar merata. sesuai dengan latar belakang yang dikemukakan untuk melanjutkan penelitian sebelumnya, maka dalam penelitian ini penulis mengambil tema “deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana”. tinjauan pustaka 1. bandul ganda sederhana sistem bandul ganda sederhana adalah sistem yang terdiri dari dua benda 𝐵1 dan 𝐵2 dengan massa masing-masing benda adalah 𝑚1 dan 𝑚2. selain itu benda tersebut masingmasing dihubungkan dengan dua helai kawat yang kuat tapi ringan 𝐿1 dan 𝐿2 dengan panjang masing-masing kawat adalah 𝑙1 dan 𝑙2, benda 𝐵1 terpasang pada ujung kawat 𝐿1 (ujung kawat 𝐿1 lainnya terpasang mantap pada sebuah bidang). sementara itu benda 𝐵2 terpasang pada ujung kawat 𝐿2 di bawah pengaruh grafitasi (ujung kawat 𝐿2 lainnya mantap terpasang pada benda pertama 𝐵1). sistem bandul ganda memiliki 4 (empat) parameter yakni 𝑙1, 𝑙2,𝑚1 dan 𝑚2 dengan dipengaruhi oleh grafitasi, bandul ganda berosilasi pada bidang vertikal dengan sudut perpindahan untuk suatu waktu adalah 𝜃1(𝑡) dan 𝜃2(𝑡) [1]. sistem bandul sederhana adalah sebagai berikut { (𝑚1 + 𝑚2)𝑙1 2 �̈�1 + 𝑚2𝑙1𝑙2�̈�2 + (𝑚1 + 𝑚2)𝑙1𝑔𝜃1 = 0 𝑚2𝑙1𝑙2�̈�1 + 𝑚2𝑙2 2 �̈�2 + 𝑚2𝑙2𝑔𝜃2 = 0 variabel dan parameter yang digunakan adalah 𝑚1 : massa benda pertama dalam satuan slug 𝑚2 : massa benda kedua dalam satuan slug 𝑙1 : panjang kawat/tali pertama dalam satuan kaki 𝑙2 : panjang kawat/tali kedua dalam satuan kaki 𝑔 : gravitasi bumi dalam satuan kaki/s2 𝜃1(𝑡) : sudut perpindahan benda pertama pada waktu 𝑡 dalam satuan radian 𝜃2(𝑡) : sudut perpindahan benda kedua pada waktu 𝑡 dalam satuan radian �̈�1(𝑡) : laju kecepatan benda pertama terhadap waktu dalam satuan kaki/s2 �̈�2(𝑡) : laju kecepatan benda kedua terhadap waktu dalam satuan kaki/s2 2. analisis model bandul ganda sederhana pada bagian ini akan dianalisis model bandul ganda sederhana yang dibangun ke dalam bentuk sistem persamaan diferensial. analisis dimulai dengan penurunan dari persamaan euler-lagrange. dalam mekanika lagrangian, evolusi sistem dijelaskan dalam hal koordinat umum dan kecepatan umum. dalam kasus ini, sudut defleksi bandul 𝜃1, 𝜃2 dan kecepatan angular dapat diambil sebagai variabel umum. dengan menggunakan variabel-variabel tersebut, dibangun persamaan lagrangian untuk bandul ganda kemudian persamaan diferensial eulerlagrange. koordinat bandul pertama didefinisikan 𝑥1 = 𝑙1 sin𝜃1 , 𝑦1 = − 𝑙1 cos𝜃1. energi kinetik (𝑇) secara umum dinyatakan 𝑇 = 1 2 𝑚𝑣2. energi kinetik benda pertama (𝑇1) dinyatakan 𝑇1 = 𝑚1 2 (𝑙1 2 �̇�1 2 ), (1) dan energi kinetik benda kedua (𝑇2) dinyatakan 𝑇2 = 𝑚2 2 [𝑙1 2 �̇�1 2 + 𝑙2 2 �̇�2 2 + 2𝑙1𝑙2�̇�1�̇�2cos (𝜃1 − 𝜃2)]. (2) energi potensial (𝑉) secara umum dinyatakan 𝑉 = 𝑚𝑔ℎ. energi potensial benda pertama (𝑉1) dinyatakan 𝑉1 = −𝑚1𝑔 𝑙1𝑐𝑜𝑠 𝜃1, (3) dan energi potensial benda kedua dinyatakan 𝑉2 = −𝑚2𝑔 (𝑙1𝑐𝑜𝑠 𝜃1 + 𝑙2𝑐𝑜𝑠 𝜃2). (4) persamaan lagrangian adalah selisih antara energi kinetik dengan energi potensial. 𝐿 = 𝑇 − 𝑉 = 𝑇1 + 𝑇2 − (𝑉1 + 𝑉2) (5) substitusi (1)-(4) ke persamaan (5) maka didapatkan 𝐿 = ( 𝑚1 2 + 𝑚2 2 )𝑙1 2 �̇�1 2 + 𝑚2 2 𝑙2 2 �̇�2 2 + 𝑚2𝑙1𝑙2�̇�1�̇�2cos (𝜃1 − 𝜃2) + (𝑚1 + 𝑚2)𝑔𝑙1cos 𝜃1 + 𝑚2𝑔𝑙2cos 𝜃2 . (6) fungsi trigonometri cos 𝜃1, dan cos 𝜃2 dapat digantikan oleh ekspresi perkiraan berikut cos 𝜃1 ≈ 1 − �̇�1 2 2 , cos 𝜃2 ≈ 1 − �̇�2 2 2 . deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana cauchy – issn: 2086-0382 126 diasumsikan bahwa sudut 𝜃1 dan 𝜃2 sangatlah kecil maka nilai cos (𝜃1 − 𝜃2) dapat dinyatakan sebagai berikut cos (𝜃1 − 𝜃2) ≈ cos𝜃1 cos𝜃2 + sin𝜃1 sin𝜃2 ≈ 1 maka didapatkan 𝐿 = ( 𝑚1 2 + 𝑚2 2 )𝑙1 2 �̇�1 2 + 𝑚2 2 𝑙2 2 �̇�2 2 + 𝑚2𝑙1𝑙2�̇�1�̇�2 −( 𝑚1 2 + 𝑚2 2 )𝑔𝑙1�̇�1 2 + 𝑚2 2 𝑔𝑙2�̇�1 2 . persamaan euler lagrange dinyatakan sebagai berikut 𝑑 𝑑𝑡 𝜕𝐿 𝜕�̇�𝑖 − 𝜕𝐿 𝜕𝜃𝑖 = 0, 𝑖 = 1,2 sehingga diperoleh sistem bandul ganda dengan asumsi sudut osilasi yang kecil { 𝑑 𝑑𝑡 [(𝑚1 + 𝑚2)𝑙1 2 �̇�1 + 𝑚2𝑙1𝑙2�̇�2]+ (𝑚1 + 𝑚2)𝑔𝑙1𝜃1 = 0 𝑑 𝑑𝑡 [𝑚2𝑙2 2 �̇�2 + 𝑚2𝑙1𝑙2�̇�1]− 𝑚2𝑔𝑙2𝜃2 = 0 atau { (𝑚1 + 𝑚2)𝑙1 2 �̈�2 + 𝑚2𝑙1𝑙2�̈�2 + (𝑚1 + 𝑚2)𝑔𝑙1𝜃1 = 0 𝑚2𝑙2 2 �̈�2 + 𝑚2𝑙1𝑙2�̈�1 − 𝑚2𝑔𝑙2𝜃2 = 0 . (2.24) 3. nilai eigen dan vektor eigen misalkan matriks 𝐴 adalah matriks 𝑛 × 𝑛, maka vektor tak nol x di dalam ℛ𝑛 dinamakan vektor eigen (eigenvector) dari 𝐴 adalah kelipatan skalar dari x; yakni, 𝐴x = 𝜆x untuk suatu skalar 𝜆. skalar 𝜆 dinamakan nilai eigen dari 𝐴 dan x dikatakan vektor eigen yang bersesuaian dengan λ [4]. 4. solusi dan potret fase dengan nilai eigen kompleks persamaan diferensial ẋ = 𝑎x memiliki solusi x(𝑡) = x0(𝑡)𝑒 𝑎𝑡 dengan x(𝑡) merupakan variabel yang bergantung pada waktu 𝑡, 𝑎 merupakan parameter dan kondisi awal x0 yang berbeda [5]. selanjutnya untuk solusi dari nilai eigen kompleks, diperlukan untuk memahami eksponensial dengan eksponen kompleks. dengan membandingkan ekspansi deret kuasa, dapat dilihat bahwa 𝑒𝑖 𝛽𝑡 = cos(𝛽𝑡) + 𝑖 sin(𝛽𝑡), 𝑒(𝛼+𝑖𝛽)𝑡 = 𝑒𝛼𝑡𝑒𝑖 𝛽𝑡 = 𝑒𝛼𝑡(cos(𝛽𝑡) + 𝑖 sin(𝛽𝑡)) [5]. demikian, jika 𝜆 = 𝛼 + 𝑖𝛽 adalah nilai eigen kompleks dengan vektor kompleks v = u + 𝑖 w (dengan 𝛼 dan 𝛽 adalah bilangan real serta u dan w adalah vektor real), maka 𝑒(𝛼+𝑖𝛽)𝑡(u + 𝑖 w) = 𝑒𝛼𝑡(cos(𝛽𝑡) + 𝑖 sin(𝛽𝑡))(u + 𝑖 w) berikutnya adalah menggambar potret fase untuk sepasang nilai eigen kompleks. diasumsikan sebuah nilai eigen adalah 𝜆 = 𝛼 ± 𝑖𝛽 dengan 𝛽 ≠ 0 [5] 1. jika 𝛼 = 0, maka titik tetap adalah elliptic center, dengan solusi periodik. arah gerakan yaitu searah jarum jam atau berlawanan dengan jarum jam. 2. jika 𝛼 < 0, maka titik tetap adalah stable focus. arah gerakan yaitu searah jarum jam atau berlawanan dengan jarum jam. 3. jika 𝛼 > 0, maka solusi spiral keluar dan titik tetap adalah unstable focus, dengan arah gerak spiral yaitu searah jarum jam atau berlawanan dengan jarum jam. 4. dalam tiga kasus di atas, arah solusi menuju ke sekitar titik tetap dapat ditentukan dengan mengecek apakah �̇�1 adalah positif atau negatif ketika 𝑥1 = 0. jika �̇�1 positif maka solusi searah dengan jarum jam dan jika �̇�1 negatif maka solusi berlawanan dengan jarum jam. pembahasan 1. analisis perilaku sistem bandul ganda sederhana riset pendahuluan bandul ganda sederhana [1] telah dilakukan penondimensionalan terhadap sistem bandul ganda sederhana, dimisalkan 𝑎 = (𝑚1 + 𝑚2)𝑙1𝑔, 𝑏 = (𝑚1 + 𝑚2)𝑙1 2 , ℎ = 𝑚2𝑙2𝑔, 𝑗 = 𝑚2𝑙1𝑙2, 𝑘 = 𝑚2𝑙2 (7) dengan 𝑎,𝑏,𝑗,ℎ, dan 𝑘 > 0. oleh karena itu sistem persamaan bandul ganda dapat ditulis sebagai berikut { 𝑏�̈�1 + 𝑗�̈�2 = −𝑎 𝜃1 𝑗�̈�1 + 𝑘�̈�2 = −ℎ 𝜃2 . (8) misalkan 𝑥1 = 𝜃1, 𝑥2 = �̇�1, �̇�2 = �̈�1, 𝑥3 = 𝜃2, 𝑥4 = �̇�4 dan �̇�4 = �̈�2 maka diperoleh { �̇�1 = �̇�1 �̇�2 = �̈�1 �̇�3 = �̇�2 �̇�4 = �̈�2 (9) atau dapat ditulis { �̇�1 = 𝑥2 �̇�2 = −𝑎𝑘 −𝑗2+𝑏𝑘 𝑥1 + 𝑗ℎ −𝑗2+𝑏𝑘 𝑥3 �̇�3 = 𝑥4 �̇�4 = −𝑎𝑗 𝑗2−𝑏𝑘 𝑥1 + 𝑏ℎ 𝑗2−𝑏𝑘 𝑥3 (10) sistem (10) menunjukkan bahwa laju kecepatan benda pertama dipengaruhi oleh besar −𝑎𝑘 −𝑗2+𝑏𝑘 dari perpindahan sudut bandul pertama ditambah besar 𝑗ℎ −𝑗2+𝑏𝑘 dari thoufina kurniyati 127 volume 3 no. 3 november 2014 perpindahan sudut benda kedua. laju kecepatan bandul kedua dipengaruhi oleh besar −𝑎𝑗 𝑗2−𝑏𝑘 dari perpindahan sudut bandul pertama dan besar 𝑏ℎ 𝑗2−𝑏𝑘 dari perpindahan sudut bandul kedua. analisis titik tetap analisis titik tetap dikerjakan dengan mengasumsikan bahwa �̇�1 = 𝑑𝑥1 𝑑𝑡 = 0, �̇�2 = 𝑑𝑥2 𝑑𝑡 = 0, �̇�3 = 𝑑𝑥3 𝑑𝑡 = 0, �̇�4 = 𝑑𝑥4 𝑑𝑡 = 0. maka, didapatkan sistem linier untuk model bandul ganda sederhana sebagai berikut: 0 = 𝑥2 0 = −𝑎𝑘 −𝑗2+𝑏𝑘 𝑥1 + 𝑗ℎ −𝑗2+𝑏𝑘 𝑥3 0 = 𝑥2 0 = −𝑎𝑗 𝑗2−𝑏𝑘 𝑥1 + 𝑏ℎ 𝑗2−𝑏𝑘 𝑥3 (11) oleh karena itu, sistem (11) mempunyai pemecahan trivial sehingga didapatkan titik tetap yaitu 𝐹 = (𝑥1,𝑥2,𝑥3,𝑥4) = (0,0,0,0) sistem bandul ganda sederhana hanya memiliki satu titik tetap yaitu 𝐹 = (𝑥1,𝑥2,𝑥3,𝑥4) = (0,0,0,0) yang berarti bahwa posisi seimbang pada bandul ganda sederhana ketika sudut perpindahan benda pertama suatu waktu, laju kecepatan benda pertama terhadap waktu, sudut perpindahan benda kedua suatu waktu dan laju kecepatan benda kedua terhadap waktu sama dengan nol (𝜃1(𝑡) = �̈�1(𝑡) = 𝜃2(𝑡) = �̈�2(𝑡) = 0). posisi seimbang merupakan posisi dimana energi potensial mencapai harga minimum dan merupakan posisi untuk keseimbangan stabil. contoh kasus contoh analisis perilaku sistem bandul ganda sederhana diberikan untuk memahami lebih dalam mengenai perilaku pada sistem bandul ganda sederhana. contoh kasus dengan parameter 𝑚1 = 937,5 slug, 𝑚2 = 312,5 slug, 𝑙1 = 16 kaki, 𝑙2 = 16 kaki, dan 𝑔 = 32 kaki/s 2 dengan kondisi awal yang berbeda 𝑥1(0) = −0,5 rad, 𝑥2(0) = −1 rad/s, 𝑥3(0) = 1 rad, dan 𝑥4(0) = 2 rad/s. pandang sistem linier (10) menjadi sebuah matriks ẋ = 𝐴x dengan ẋ = [ ẋ1 ẋ2 ẋ3 ẋ4 ], 𝐴 = [ 0 1 −𝑎𝑘 −𝑗2 + 𝑏𝑘 0 0 0 𝑗ℎ −𝑗2 + 𝑏𝑘 0 0 0 −𝑎𝑗 𝑗2 − 𝑏𝑘 0 0 1 𝑏ℎ 𝑗2 − 𝑏𝑘 0 ] , x = [ x1 x2 x3 x4 ]. substitusi parameter ke persamaan (7) kemudian ke persamaan (10) sehingga didapatkan [ ẋ1 ẋ2 ẋ3 ẋ4 ] = [ 0 1 −512 192 0 0 0 128 192 0 0 0 512 192 0 0 1 −512 192 0 ] [ x1 x2 x3 x4 ], (12) kondisi awal yang diberikan adalah [ x1(0) = −0,5 x2(0) = −1 x3(0) = 1 x4(0) = 2 ]. sistem (12) memiliki solusi sebagai berikut 𝐱 = 𝐱𝟎𝑒 𝐴𝑡 atau dapat ditulis [ 𝑥1(𝑡) 𝑥2(𝑡) 𝑥3(𝑡) 𝑥4(𝑡) ] = [ 𝑒𝐴1,1𝑡 𝑒𝐴1,2𝑡 𝑒𝐴2,1𝑡 𝑒𝐴2,2𝑡 𝑒𝐴1,3𝑡 𝑒𝐴1,4𝑡 𝑒𝐴2,3𝑡 𝑒𝐴2,4𝑡 𝑒𝐴3,1𝑡 𝑒𝐴3,2𝑡 𝑒𝐴4,1𝑡 𝑒𝐴4,2𝑡 𝑒𝐴3,3𝑡 𝑒𝐴3,4𝑡 𝑒𝐴4,3𝑡 𝑒𝐴4,4𝑡 ][ −0,5 −1 1 2 ]. menentukan perilaku dari 𝑒𝐴𝑡 saat 𝑡 → ∞ dengan memenuhi 𝑒𝐴𝑡 = 𝐯𝑒𝜆𝑡𝐯−𝟏 dimana 𝜆 merupakan nilai eigen dan 𝐯 merupakan vektor eigen yang bersesuaian dengan 𝜆 serta 𝐯−𝟏 merupakan invers dari vektor eigen. selanjutnya, menentukan nilai eigen dari persamaan karakteristik yang memenuhi det(𝐴 − λi) = 0. persamaan karakteristik 𝐴 adalah 𝜆4 + 1( 128 192 )1( 512 192 ) − 1( −512 192 )1( −512 192 ) = 0. sehingga dengan menggunakan bantuan maple 12 nilai-nilai eigen dari matriks 𝐴 adalah 𝜆1 = 2 3 √3 𝑖 , 𝜆2 = − 2 3 √3 𝑖 , 𝜆3 = 2 𝑖 , 𝜆4 = −2 𝑖. (13) vektor eigen yang bersesuaian dengan nilai eigen adalah deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana cauchy – issn: 2086-0382 128 𝐯 = [ − 1 4 √3𝑖 1 4 √3𝑖 1 2 1 2 1 4 𝑖 − 1 4 𝑖 − 1 2 − 1 2 − 1 2 √3𝑖 1 2 √3𝑖 1 1 − 1 2 𝑖 1 2 𝑖 1 1 ] . selanjutnya menentukan masing-masing solusi umum dari x1(𝑡), x2(𝑡), x3(𝑡) dan x4(𝑡). x1(𝑡) = 𝐶1 (− 1 4 sin(2𝑡)) + 𝐶2 ( 1 4 cos(−2𝑡)) +𝐶3 ( 1 4 √3sin( 2 3 √3𝑡)) +𝐶4 ( 1 4 √3 cos(− 2 3 √3𝑡)) (3.38) x2(𝑡) = 𝐶1 (− 1 2 cos(2𝑡)) + 𝐶2 ( 1 2 sin(−2𝑡)) + 𝐶3 ( 1 2 cos( 2 3 √3𝑡)) +𝐶4 ( 1 2 sin(− 2 3 √3𝑡)) (3.39) x3(𝑡) = 𝐶1 ( 1 2 sin(2𝑡)) + 𝐶2 (− 1 2 cos(−2𝑡)) +𝐶3 ( 1 2 √3sin( 2 3 √3𝑡)) +𝐶4 (− 1 2 √3 cos(− 2 3 √3𝑡)) (3.40) x4(𝑡) = 𝐶1cos(2𝑡) + 𝐶2 sin(−2𝑡) +𝐶3 cos( 2 3 √3𝑡) +𝐶4sin(− 2 3 √3𝑡) (3.41) solusi khusus dengan 𝐶1 = 1,𝐶2 = 1, 𝐶3 = 1, dan 𝐶4 = − 3 √3 adalah x1(𝑡) = − 1 4 sin(2𝑡) + 1 4 cos(−2𝑡) + 1 4 √3sin( 2 3 √3𝑡) − 3 4 cos(− 2 3 √3𝑡) (3.48) x2(𝑡) = − 1 2 cos(2𝑡) + 1 2 sin(−2𝑡) + 1 2 cos( 2 3 √3𝑡) − 3 2√3 sin(− 2 3 √3𝑡) (3.49) x3(𝑡) = 1 2 sin(2𝑡) − 1 2 cos(−2𝑡) + 1 2 √3sin( 2 3 √3𝑡) + 3 2 cos(− 2 3 √3𝑡) (3.50) x4(𝑡) = cos(2𝑡) + sin(−2𝑡) +cos( 2 3 √3𝑡) − 3 √3 sin(− 2 3 √3𝑡) (3.51) berikut adalah grafik perilaku dari sistem bandul ganda dengan 𝑡 = 0…50 detik. gambar 1 grafik solusi x1(𝑡) terhadap 𝑡 gambar 1 merupakan grafik perubahan sudut perpindahan benda pertama terhadap waktu dengan rentang waktu 50 detik. benda pertama mulai berayun pada sudut -0,5 rad (benda pertama berada di sebelah kiri posisi setimbang) dan bergerak mendekati posisi seimbang (0,0) akan tetapi setelah benda pertama berada di posisi seimbang perlahan benda pertama menjauhi posisi seimbang, hal tersebut terjadi secara periodik. pada gambar tersebut menunjukkan bahwa pergerakan benda pertama berubah secara periodik baik besar maupun arahnya di sekitar titik keseimbangan dengan periode 4,9 detik, simpangan maksimum yaitu 1,21 rad pada saat benda pertama berayun selama 5,49 detik dan selama rentang waktu 50 detik benda pertama tetap berosilasi di sekitar posisi seimbang. gambar 2 grafik x2(𝑡) terhadap 𝑡 gambar 2 merupakan grafik laju kecepatan benda pertama terhadap waktu selama rentang waktu 50 detik. pada gambar tersebut menunjukkan bahwa laju kecepatan awal benda pertama terhadap waktu sebesar 1 rad/s2 (benda pertama berada di sebelah kanan posisi seimbang) dan kecepatannya berubah secara quasiperiodic baik besar maupun arahnya. benda kedua memiliki kecepatan maksimal saat 8,9 detik. thoufina kurniyati 129 volume 3 no. 3 november 2014 gambar 3 grafik x3(𝑡) terhadap 𝑡 gambar 3 merupakan grafik perubahan sudut perpindahan benda kedua terhadap waktu selama kurung waktu 50 detik. pada gambar tersebut menunjukkan bahwa pergerakan benda kedua berubah secara periodik baik besar maupun arahnya di sekitar titik keseimbangan dengan periode 4,91 detik dan simpangan maksimum yaitu 16,76 rad saat 𝑡 = 2,43 detik. gambar 4 grafik x4(𝑡) terhadap 𝑡 gambar 4 merupakan grafik laju kecepatan bandul kedua terhadap waktu dengan 𝑡 = 0…50 detik. pada gambar tersebut menunjukkan bahwa laju kecepatan awal sebesar -0.5 rad, bandul kedua terhadap waktu berubah secara quasiperiodic baik besar maupun arahnya. pada saat 𝑡 = 19,94 detik bandul kedua memiliki laju kecepatan maksimal. 2. deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana parameter massa benda pertama lebih besar daripada massa benda kedua (𝒎𝟏 > 𝒎𝟐) simulasi pertama dengan massa benda pertama sebesar 412.5 slug dan massa benda kedua sama dengan 312.5 slug. bandul pertama mulai bergerak saat sudut perpindahan sebesar -0,5 rad (benda pertama berada di sebelah kiri posisi seimbang sebesar -0,5 rad) dan laju kecepatannya 1 rad/s2 (benda pertama berada di sebelah kanan posisi seimbang dengan kecepatan 1 rad/s2). gambar 5 trayektori dengan 𝑚1 > 𝑚2 gambar 5 memperlihatkan bahwa trayektori bergerak searah jarum jam untuk mendekati titik tetap sampai akhirnya mulai menjauhi titik tetap saat 𝑡 > 0 detik artinya kestabilan dari sistem adalah tidak stabil. periode dari sistem bandul ganda sederhana berangsur-angsur membesar sampai 𝑡 → ∞. parameter massa benda pertama sama dengan massa benda kedua (𝒎𝟏 = 𝒎𝟐) simulasi kedua dengan massa benda pertama sebesar 312,5 slug dan massa benda kedua sama dengan 312,5 slug. benda pertama mulai bergerak saat sudut perpindahan sebesar -0,5 rad dan laju kecepatannya 1 rad/s2. gambar 6 trayektori dengan 𝑚1 = 𝑚2 gambar 6 memperlihatkan bahwa trayektori bergerak melawan arah jarum jam untuk mendekati titik tetap sampai akhirnya mulai menjauhi titik tetap saat 𝑡 > 0 detik artinya kestabilan dari sistem adalah tidak stabil. periode dari sistem bandul ganda sederhana berangsur-angsur membesar sampai 𝑡 → ∞. deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana cauchy – issn: 2086-0382 130 parameter massa benda pertama lebih besar daripada massa benda kedua 𝒎𝟏 < 𝒎𝟐 simulasi pertama dengan massa benda pertama sebesar 212.5 slug dan massa benda kedua sama dengan 312.5 slug. bandul pertama mulai bergerak saat sudut perpindahan sebesar -0,5 rad dan laju kecepatannya 1 rad/s2. gambar 7 trayektori dengan 𝑚1 < 𝑚2 gambar 7 memperlihatkan bahwa trayektori bergerak melawan arah jarum jam untuk mendekati titik tetap sampai akhirnya mulai menjauhi titik tetap saat 𝑡 > 0 detik artinya kestabilan dari sistem adalah tidak stabil. periode dari sistem bandul ganda sederhana berangsur-angsur membesar sampai 𝑡 → ∞. oleh karena itu, perubahan parameter tidak mempengaruhi kestabilan sistem melainkan hanya arah trayektorinya saja. penutup 1. kesimpulan berdasarkan pembahasan di atas maka dapat diperoleh kesimpulan sebagai berikut: 1. model bandul ganda sederhana tersebut memiliki titik tetap saat 𝐹 = (𝑥1,𝑥2,𝑥3,𝑥4) = (0,0,0,0) artinya sistem memiliki posisi seimbang ketika sudut perpidahan benda pertama dan kedua suatu waktu serta laju kecepatan benda pertama dan kedua terhadap waktu sama dengan nol. nilai eigen dari sistem bandul ganda dengan parameter tertentu adalah imajiner murni berarti kestabilan dari sistem bandul ganda yaitu tak stabil dan tipe kestabilan sistem bandul ganda sederhana adalah elliptic center. grafik perpindahan bandul pertama dan kedua terhadap waktu adalah periodik dan grafik laju kecepatan bandul pertama dan kedua adalah quasiperiodic. hal tersebut menunjukkan bahwa sistem bandul ganda bergerak bolak-balik melalui lintasan yang sama. perubahan besar parameter benda pertama tidak mempengaruhi kestabilan sistem. 2. saran penelitian ini difokuskan pada masalah analisis perilaku sistem bandul ganda sederhana. pada penelitian selanjutnya penulis menyarankan untuk analisis bifurkasi pada sistem bandul ganda sederhana karena telah diketahui bahwa sistem bandul ganda sederhana memiliki nilai eigen murni yang merupakan syarat cukup untuk analisis bifurkasi. daftar pustaka [1] amanto and la zakaria, "solusi eksak dan kestabilan sistem bandul ganda sederhana," jurnal sains mipa, pp. 23-32, 2008. [2] asep k. supriatna, "matematika dan contoh penerapan matematika dalam disiplin ilmu lain," matematika integratif, pp. 1-7, 2002. [3] robert resnick and david halliday, physics, 3'th edition. jakarta: erlangga, 1978. [4] howard anton, elementary linier algebra, jilid i. bandung: erlangga, 1997. [5] r c robinson, an introduction to dynamical system continous and discrete. western: american mathematical society , 2012. pendahuluan tinjauan pustaka 1. bandul ganda sederhana 2. analisis model bandul ganda sederhana 3. nilai eigen dan vektor eigen 4. solusi dan potret fase dengan nilai eigen kompleks pembahasan 1. analisis perilaku sistem bandul ganda sederhana analisis titik tetap contoh kasus 2. deskripsi pengaruh parameter terhadap kestabilan perilaku sistem bandul ganda sederhana parameter massa benda pertama lebih besar daripada massa benda kedua (,𝒎-𝟏.>,𝒎-𝟐.) parameter massa benda pertama sama dengan massa benda kedua ,(𝒎-𝟏.=,𝒎-𝟐.) parameter massa benda pertama lebih besar daripada massa benda kedua ,𝒎-𝟏.<,𝒎-𝟐. penutup 1. kesimpulan 2. saran daftar pustaka application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in way kanan district, lampung cauchy – jurnal matematika murni dan aplikasi volume 4 (4) (2017), pages 155-160 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 16 january 2017 reviewed: 31 january 2017 accepted: 17 may 2017 doi: http://dx.doi.org/10.18860/ca.v4i4.3979 application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in way kanan district, lampung i kadek yama rinaldi1, maria bernadetha theresia1, achmad efendi1, henny pramoedyo1 1department of statistics, brawijaya university, malang email: kadekyamarinaldi112@gmail.com, dethamitakda@yahoo.com abstract modification of spatial k'luster analysis by tree edge removal method (skater) is one of the regionalization method for clustering based on the location by spatial autocorrelation and spatial patterns. modification of skater is a development of ordinary skater that modifying the calculation of weights and the partition process to accommodate the homogeneity within the cluster and heterogeneity between clusters. covariance normalized distance (cnd) is an edge weight for the formation of minimum spanning tree (mst) using prim's algorithm which connects two location. after mst partition has been formed, then we partitioning mst sequentially called between cluster dissimilarity or cluster stretch which created a group with individually connected to the mst. the results from skater grouping are based on the sum of squared deviations (ssd), the smaller value, the better ssd partitions are used. the purpose of this research is to identificate the autocorrelation and spatial distribution patterns also applying the modification of skater method on rate of crime data in way kanan district, lampung. this research using secondary data obtained from 14 sub districts in way kanan police resort. the results show that 14 districts in way kanan autocorrelated and has a pattern of spatial distribution and four level of cluster (𝑘) has been obtained with average; 𝑘1 (bumi agung, 12.73%) ≥ 𝑘2 (blambangan umpu, 10.9%) ≥ 𝑘3 (gunung labuan, baradatu, kasui and rebang tangkas, 7.83%) ≥ 𝑘4 (way tuba, 7.81%). keywords: spatial, clustering, minimum spanning tree, skater, crime introduction in analysis of cluster often found that there is a spatial effect which affects the characteristics of the cluster [1]. spatial data is data that are geographically oriented and has a coordinate system [2]. if the spatial analysis carried out without involving spatial element would result in conclusions that are less precise due to the assuming the error is independent [3]. to find out whether they are met or not then the assumption needs to be tested with a spatial autocorrelation moran's test. analysis of cluster which involves spatial element in its analysis called spatial clustering [4]. one of the specific methods of regionalization spatial clustering is spatial k'luster analysis by tree edge removal (skater) method introduced by assuncao, et al. (2006) [5]. skater is a clustering method based on the spatial relationship between dependence of locations. this method uses an algorithm to partition adjacent location and does not have the same characteristics. skater uses graph theory approach with the minimum spanning tree technique [6]. http://dx.doi.org/10.18860/ca.v4i4.3979 mailto:kadekyamarinaldi112@gmail.com mailto:dethamitakda@yahoo.com application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in district of way kanan, lampung i kadek yama rinaldi 156 modification of skater aims to partition locations that are not homogeneous into homogeneous location, it also to get a homogenous group of internal and external heterogeneous cluster. the modifications are out on four ordinary skater method; three changes in the weight calculation edges and one-partitioning process. the first modification is called orthogonal normalized distance. the second step is the weight calculation of edges, where 𝑑 (𝑖, 𝑗) connecting the vertices 𝑖 and 𝑗. the third step, these modifications using weights edge called covariance normalized distance (cnd). the last modified is the third step change from ordinary skater is the partitioning process called dissimilarity between cluster (cluster stretch). several methods of skater have been developed and implemented by bekti [7], assuncao et al. [5] and lage et al. [8]. the implementation and execution research on modification of skater are still rare, so this study will apply the modification of skater in crime data comprise of spoliation, robbery and gambling cases in way kanan district, lampung. methods the location of this research is the police department resort way kanan district office which consists of 14 districts (figure 1). this study uses data which obtained from the crime rate resort district police way kanan, lampung in 2014, include: the number of spoliation cases, robbery and gambling. clustering method used is a modification of spatial k'luster edge analysis by tree removal (skater). modification of skater clustering method is performed on rstudio software with the package; rgdal, spdep, rcolorbrewer, classint and maptools. analysis procedure is: the first procedure is identification of dispersion pattern and spatial autocorrelation test to the crime rate with moran's i test [9]. the second procedure, calculating the edges weight with covariance normalized distance, first step is calculating the average of attributes and variance-covariance matrix. the next is counting input variance-covariance matrix in the euclidean distance. covariance normalized distance was used as edges weights for connecting nodes and edges at each locations. the next procedure is, establishment of mst by prim's algorithm based on graph connectivity by removing the higher edges weight [10]. the form of mst describe the closeness between locations based on variables. last procedure, is mst partition by eliminating edge into the connections between locations in a sequence to produce the subtrees which representing every region at the study site. partitioning results will generate clusters based on specific characteristics of three variables, namely the number of spoliation, robbery and gambling. figure 1. map of way kanan district (14 districts) application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in district of way kanan, lampung i kadek yama rinaldi 157 description of districts name in figure 1: 1. pakuan ratu 2. rebang tangkas 3. kasui 4. banjit 5. baradatu 6. gunung labuan 7. way tuba 8. negeri agung 9. blambangan umpu 10. buay bahuga 11. bumi agung 12. bahuga 13. negeri besar 14. negara batin results and discussion the test results of moran's i spatial autocorrelation in the crime rate in way kanan district is presented in table 1. table 1. testing results of moran's i spatial autocorrelation table 1 shows that there are autocorrelation between sites, there are linkages crime rate in one location with the crime rate in other locations. the next pattern conducted to determine the common characteristics using spatial distribution of crime rate in 2014 cases between district in way kanan, the detail is presented in figure 1. (a) (b) research variables p-value test statistics z decision spoliation 0.0036** 2.9142 reject 𝐻0 robbery 0.0168** 2.3985 reject 𝐻0 gambling 0.0377* 2.0836 reject 𝐻0 application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in district of way kanan, lampung i kadek yama rinaldi 158 (c) figure 2. (a) the spread pattern of spoliation case (b) the spread pattern of robbery case, and (c) the spread pattern of gambling case the spread pattern of the spoliations, robberies and gambling cases show their spatial clustering. spatial clustering indicated by districts that have the highest robbery case (red) surrounded by other districts with high case anyway. thus the similarity of characteristics between adjacent locations on the spatial distribution patterns in accordance with the identification of moran's i (spatial autocorrelation). results of generating minimum spanning tree are presented in figure 2 in the form of connectivity lines among 14 districts. the first connection connects location 13 and 14, then 14 and 1 to the last connection is location 7 and 11 are the result of generating mst. figure 3. results of generating minimum spanning tree mst partition results in figure 3. mst partition edge process performed to obtain homogeneity between in the cluster and to generate subtrees that represent each district. mst partition results for the spoliation, robbery and gambling case generate 4 clusters (k) in 14 districts are indicated by lines with different colors. application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in district of way kanan, lampung i kadek yama rinaldi 159 figure 4. results of partition minimum spanning tree in detail about clustering modification of skater presented in table 2 and figure 4. table 2. clustering description of skater modification in crime case figure 5. the results of cluster from skater modification on crime cases in way kanan district. table 2 shows that the highest mean of spoliation (14.28%) and gambling cases (23.08%) are in cluster 1 and the lowest are in cluster 3 and 2 (5.25%) and (0%). cluster 4 has the lowest mean of robbery case (3.30%) while the highest (18.23%) is in cluster 2. the variance of spoliation cases (29.52%) and robbery (6.47%) in cluster 4 and gambling cases (99.06%) in cluster 3 show that the crime rate is more fluctuate (there is a variation among districts) than the other cluster. cluster 1 and 2 do not have variance because they only consist of one single district. cluster spoliation (%) robbery (%) gambling (%) 𝒌𝟏 mean 14.82 15.12 23.08 𝒌𝟐 mean 12.04 18.23 0.00 𝒌𝟑 mean 5.25 6.94 9.40 variance 15.66 5.93 99.06 𝒌𝟒 mean 6.11 3.30 3.42 variance 29.52 6.47 38.57 application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in district of way kanan, lampung i kadek yama rinaldi 160 conclusion the spread pattern of the spoliation, robberies and gambling cases in way kanan district showed their spatial clustering. modification of skater clustering method produce the 4-rise cluster, it is because the mean level of crime is 𝑘1 ≥ 𝑘2 ≥ 𝑘3 ≥ 𝑘4. references [1] j. aldstadt and a. getis, "using amoeba to create a spatial weights matrix and identify spatial clusters," geogr anal, vol. 38 (4), pp. 327-343, 2010. [2] e. prahasta, sistem informasi geografi, bandung: informatika, 2001. [3] l. anselin, spatial econometrics : methods and models, dordrecht: kluwer, 1988. [4] j. lee and d. w. s. wong, statistic for spatial data, new york: john wiley & sons, 2001. [5] r. m. assuncao, m. c. neves, g. camaras and f. costa, "efficient regionalization techniques for socioeconomic geograph-icalunits using minimum spanning trees," international journal of geographical information science, vol. 20:7, pp. 797-811, 2006. [6] ia. reis, g. camaras, r. assuncao and amv. monteiro, "data-aware clustering for geosensor networks data collection," anais xiii simposiolobrasileiro de sensoriamento remoto, florianopolis, brasil, pp. 6059-6066, 2007. [7] r. d. bekti, "metode spasial skater untuk pengelompokan lokasi berdasarkan fasilitas air bersih dan sanitasi," jurnal teknologi, vol. 8, pp. 53-58, 2015. [8] j. p. lage, r. m. assuncao, and ia. reis, "a minimal spanning tree algorithm applied to spatial cluster analysis," electronic notes in discrete mathematics, vol. 7, pp. 1-23, 2002. [9] m. m. fischer and a. getis, "handbook of applied spatial analysis," in springer, berlin, heidelberg and new york, 2010, pp. 255-278. [10] r. munir, matematika diskrit, bandung: informatika, 2005. fixed point theorems in complex-valued b-metric spaces cauchy – jurnal matematika murni dan aplikasi volume 4 (4) (2017), pages 138-145 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 14 nopember 2016 reviewed: 29 nopember 2016 accepted: 17 may 2017 doi: http://dx.doi.org/10.18860/ca.v4i4.3669 fixed point theorems in complex-valued b-metric spaces dahliatul hasanah mathematics department, universitas negeri malang jl. semarang no. 5 malang jawa timur indonesia 65145 email: dahliatul.hasanah.fmipa@um.ac.id abstract fixed point theorems in complex-valued metric spaces have been extensively studied following the extension of the definition of a metric to a complex-valued metric. the definition of b-metric, which is considered as an extension of a metric, is also extended to complex-valued b-metric. this extension causes the study of existence and uniqueness of fixed point in complex-valued b-metric spaces. fixed point theorems that have been studied in b-metric spaces are now of interest whether the theorems are still satisfied in complex-valued b-metric spaces. in this paper, we study the existence and uniqueness of fixed point in complex valued b-metric spaces. we consider some fixed point theorems in b-metric spaces to be extended to fixed point theorems in complex valued b-metric spaces. by using the definition of complex-valued b-metric spaces and the properties of partial order on complex numbers, we have obtained some fixed point theorems in complex-valued b-metric spaces which are derived from the corresponding theorems in b-metric spaces. keywords: b-metric spaces, complex-valued b-metric spaces, fixed point theorems introduction fixed point theorems have been the main topic of many researches in various spaces including metric spaces. fixed point theorems in metric spaces have been studied extensively by many researchers as in [1], [2], and [3]. after bakhtin [4] introduced the notion of b-metric spaces in 1989, researchers started to develop fixed point theorems in the spaces, for example in [5], [6], and [7]. in 2011, azzam et. al [8] introduced a new metric space called complex-valued metric space as an extension of the classical metric space in terms of the defined metric. fixed point theorems are still of interest to study in the new space. this is shown by many articles talked about fixed point theorems in complex valued-metric spaces as in [9], [10], dan [11]. the extension of the notion of metric into complex-valued metric causes the extension of b-metric spaces into complex-valued b-metric spaces. in this paper, we observe fixed point theorems which have been established in b-metric space whether the theorems are still applicable in the spaces. methods first, we recall some basic definitions and properties of b-metric spaces given in [5] and of complex-valued b-metric spaces given in [12]. http://dx.doi.org/10.18860/ca.v4i4.3669 mailto:dahliatul.hasanah.fmipa@ fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 139 b-metric spaces a 𝑏-metric space is considered as a generalization of a metric space regarding the definition of the metric as defined as follows. definition 1 (see [5]). let 𝑋 be a nonempty set and let 𝑠 ≥ 1 be a given real number. a function 𝑑: 𝑋 × 𝑋 → [0, ∞) is called a b-metric space if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 the following conditions are satisfied. 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦; 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥); 𝑑(𝑥, 𝑦) ≤ 𝑠[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)]. the pair (𝑋, 𝑑) is called a b-metric space. the number 𝑠 ≥ 1 is called the coefficient of (𝑋, 𝑑). example 2. let (𝑋, 𝑑) be a metric space. the function 𝜌(𝑥, 𝑦) is defined by 𝜌(𝑥, 𝑦) = (𝑑(𝑥, 𝑦)) 2 . then (𝑋, 𝜌) is a 𝑏-metric space with coefficient 𝑠 = 2. this can be seen from the nonnegativity property and triangle inequality of metric to prove the property (iii). complex-valued b-metric spaces unlike in real numbers which has completeness property, order is not well-defined in complex numbers. before giving the definition of complex-valued b-metric spaces, we define partial order in complex numbers (see [12]). let ℂ be the set of complex numbers and 𝑧1, 𝑧2 ∈ ℂ. define partial order ≼ on ℂ as follows. we have 𝑧1 ≼ 𝑧2 if and only if 𝑅𝑒(𝑧1) ≤ 𝑅𝑒(𝑧2) and 𝐼𝑚(𝑧1) ≤ 𝐼𝑚(𝑧2). this means that we would have 𝑧1 ≼ 𝑧2 if and only if one of the following conditions holds: (i) re(𝑧1) = re(𝑧2) and im(𝑧1) < im(𝑧2), (ii) re(𝑧1) < re(𝑧2) and im(𝑧1) = im(𝑧2), (iii) re(𝑧1) < re(𝑧2) and im(𝑧1) < im(𝑧2), (iv) re(𝑧1) = re(𝑧2) and im(𝑧1) = im(𝑧2). if one of the conditions (i), (ii), and (iii) holds, then we will write 𝑧1 ⋨ 𝑧2. particularly, we have 𝑧1 ≺ 𝑧2 if the condition (iv) is satisfied. from the definition of partial order on complex numbers, partial order has the following properties: (i) if 0 ≼ 𝑧1 ⋨ 𝑧2, then |𝑧1| < |𝑧2|; (ii) 𝑧1 ≼ 𝑧2 and 𝑧2 ≺ 𝑧3 imply 𝑧1 ≺ 𝑧3; (iii) if 𝑧 ∈ ℂ, 𝑎, 𝑏 ∈ ℝ and 𝑎 ≤ 𝑏, then 𝑎𝑧 ≼ 𝑏𝑧. a b-metric on a b-metric space is a function having real value. based on the definition of partial order on complex numbers, real-valued b-metric can be generalized into complex-valued b-metric as follows. definition 3 (see [13]). let 𝑋 be a nonempty set and let 𝑠 ≥ 1 be a given real number. a function 𝑑: 𝑋 × 𝑋 → ℂ is called a complex-valued b-metric on 𝑋 if, for all 𝑥, 𝑦, 𝑧 ∈ ℂ, the following conditions are satisfied: (i) 0 ≼ d(x, y) and 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦, (ii) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), (iii) 𝑑(𝑥, 𝑦) ≼ s[d(x, z) + d(z, y)]. the pair (𝑋, 𝑑) is called a complex valued b-metric space. example 4. let 𝑋 = ℂ. define the mapping 𝑑: ℂ × ℂ → ℂ by 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + 𝑖|𝑥 − 𝑦|2 for all 𝑥, 𝑦 ∈ 𝑋. then (ℂ, 𝑑) is complex-valued b-metric space with 𝑠 = 2. definition 5 (see [13]). let (𝑋, 𝑑) be a complex-valued b-metric space. (i) a point 𝑥 ∈ 𝑋 is called interior point of set 𝐴 ⊆ 𝑋 if there exists 0 ≺ 𝑟 ∈ ℂ such that 𝐵(𝑥, 𝑟) = {𝑦 ∈ 𝑌: 𝑑(𝑥, 𝑦) ≺ 𝑟} ⊆ 𝐴. fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 140 (ii) a point 𝑥 ∈ 𝑋 is called limit point of a set 𝐴 if for every 0 ≺ 𝑟 ∈ ℂ, 𝐵(𝑥, 𝑟) ∩ (𝐴 − {𝑥}) ≠ ∅. (iii) a subset 𝐴 ⊆ 𝑋 is open if each element of 𝐴 is an interior point of 𝐴. (iv) a subset 𝐴 ⊆ 𝑋 is closed if each limit point of 𝐴 is contained in 𝐴. definition 6 (see [13]). let (𝑋, 𝑑) be a complex valued b-metric space, (𝑥𝑛 ) be a sequence in 𝑋 and 𝑥 ∈ 𝑋. (i) (𝑥𝑛 ) is convergent to 𝑥 ∈ 𝑋 if for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that for all 𝑛 ≥ 𝑁, 𝑑(𝑥𝑛 , 𝑥) ≺ 𝑟. thus 𝑥 is the limit of (𝑥𝑛 ) and we write lim(𝑥𝑛 ) = 𝑥 or (𝑥𝑛 ) → 𝑥 as 𝑛 → ∞. (ii) (𝑥𝑛 ) is said to be cauchy sequence if for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that for all 𝑛 ≥ 𝑁, 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 ) ≺ 𝑟, where 𝑚 ∈ ℕ. (iii) if for every cauchy sequence in 𝑋 is convergent, then (𝑋, 𝑑) is said to be a complete complex valued b-metric space. lemma 7 (see [13]). let (𝑋, 𝑑) be a complex valued b-metric space ad let (𝑥𝑛 ) be a sequence in 𝑋. then (𝑥𝑛 ) converges to 𝑥 if and only if |𝑑(𝑥, 𝑦)| → 0 as 𝑛 → ∞. lemma 8 (see [13]). let let (𝑋, 𝑑) be a complex valued b-metric space ad let (𝑥𝑛 ) be a sequence in 𝑋. then (𝑥𝑛 ) is a cauchy sequence if and only if |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| → 0 as 𝑛 → ∞, where 𝑚 ∈ ℕ. results and discussions the following fixed point theorem is studied by mishra,et.al [5] in b-metric spaces: theorem 9. let (𝑋, 𝜌) be a complete b-metric space and let 𝑇: 𝑋 → 𝑋 be a function with the following 𝜌(𝑇𝑥, 𝑇𝑦) ≤ 𝑎𝜌(𝑥, 𝑇𝑥) + 𝑏𝜌(𝑦, 𝑇𝑦) + 𝑐𝜌(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 where 𝑎, 𝑏, and 𝑐 are non-negative real numbers and satisfy 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝑇 has a uniqe fixed point. this theorem is extended to a fixed point theorem in complex valued b-metric spaces as follows. theorem 10. let (𝑋, 𝑑) be a complete complex valued b-metric space and let 𝑇: 𝑋 → 𝑋 be a function with the following 𝑑(𝑇𝑥, 𝑇𝑦) ≼ 𝑎𝑑(𝑥, 𝑇𝑥) + 𝑏𝑑(𝑦, 𝑇𝑦) + 𝑐𝑑(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 where 𝑎, 𝑏, and 𝑐 are non-negative real numbers and satisfy 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝑇 has a uniqe fixed point. proof. let 𝑥0 ∈ 𝑋 and (𝑥𝑛 ) be a sequence in x such that 𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇 𝑛𝑥0. note that for all 𝑛 ∈ ℕ, we have 𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) ≼ 𝑎𝑑(𝑥𝑛+1, 𝑇𝑥𝑛+1) + 𝑏𝑑(𝑥𝑛 , 𝑇𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛+1, 𝑥𝑛 ) ≼ 𝑎𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑏𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑐𝑑(𝑥𝑛+1, 𝑥𝑛 ) ≼ 𝑎𝑑(𝑥𝑛+2, 𝑥𝑛+1) + (𝑏 + 𝑐)𝑑(𝑥𝑛+1, 𝑥𝑛 ) (1 − 𝑎)𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ (𝑏 + 𝑐)𝑑(𝑥𝑛+1, 𝑥𝑛 ) 𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ ( 𝑏 + 𝑐 1 − 𝑎 ) 𝑑(𝑥𝑛+1, 𝑥𝑛 ). if we take 𝛾 = 𝑏+𝑐 1−𝑎 and continuing this process, then we have 𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ 𝛾 𝑛+1𝑑(𝑥1, 𝑥0). for 𝑚 ∈ ℕ, 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠 𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1) + 𝑠 2𝛾𝑛+1𝑑(𝑥0, 𝑥1) + 𝑠 3𝛾𝑛+2𝑑(𝑥0, 𝑥1) + ⋯ + 𝑠 𝑚 𝛾𝑛+𝑚−1𝑑(𝑥0, 𝑥1) ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾) 2 + ⋯ + (𝑠𝛾)𝑚−1] fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 141 now, |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| ≤ |𝑠𝛾 𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾) 2 + ⋯ + (𝑠𝛾)𝑚−1]| = 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾) 2 + ⋯ + (𝑠𝛾)𝑚−1] since 𝑎 + 𝑠(𝑏 + 𝑐) < 1 for 𝑠 ≥ 1 then 𝛾 < 1 and 𝑠𝛾 < 1. taking limit 𝑛 → ∞, we have 𝛾𝑛 → 0. this implies |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 )| → 0 as 𝑛 → ∞. by lemma 8, the sequence (𝑥𝑛 ) is cauchy sequence in x. since 𝑋 is a complete complex valued b-metric space then (𝑥𝑛 ) is a convergent sequence. let 𝑥∗ be the limit of (𝑥𝑛 ). we show that 𝑥 ∗ is a fixed point of 𝑇. we have 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥 ∗)] = 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑇𝑥𝑛−1, 𝑇𝑥 ∗)] ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑏𝑑(𝑥 ∗, 𝑇𝑥∗) + 𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)] (1 − 𝑏𝑠)𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)] 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠 1 − 𝑏𝑠 [𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)] 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠 1 − 𝑏𝑠 [𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)] taking the absolute value of both sides, we get |𝑑(𝑥∗, 𝑇𝑥∗)| ≤ 𝑠 1 − 𝑏𝑠 |𝑑(𝑥∗, 𝑥𝑛 ) + 𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)| ≤ 𝑠 1 − 𝑏𝑠 (|𝑑(𝑥∗, 𝑥𝑛 )| + |𝑎𝑑(𝑥𝑛−1, 𝑥𝑛 )| + |𝑐𝑑(𝑥𝑛−1, 𝑥 ∗)|) ≤ 𝑠 1 − 𝑏𝑠 (|𝑑(𝑥∗, 𝑥𝑛 )| + 𝑎𝛾 𝑛 |𝑑(𝑥1, 𝑥0)| + 𝑐|𝑑(𝑥𝑛−1, 𝑥 ∗)|) since (𝑥𝑛 ) converges to 𝑥 ∗, taking 𝑛 → ∞ implies |𝑑(𝑥∗, 𝑥𝑛 )| → 0 and |𝑑(𝑥𝑛−1, 𝑥 ∗)| → 0. thus for 𝑛 → ∞, we have lim|𝑑(𝑥∗, 𝑇𝑥∗)| = 0. this yields 𝑥∗ = 𝑇𝑥∗. hence 𝑥∗ is a fixed point of 𝑇. now we show the uniqueness of the fixed point of 𝑇. we assume that there are two fixed points of 𝑇 which are 𝑥 = 𝑇𝑥 and 𝑦 = 𝑇𝑦. thus, 𝑑(𝑥, 𝑦) = 𝑑(𝑇𝑥, 𝑇𝑦) ≼ 𝑎𝑑(𝑥, 𝑇𝑥) + 𝑏𝑑(𝑦, 𝑇𝑦) + 𝑐𝑑(𝑥, 𝑦) ≼ 𝑎𝑑(𝑥, 𝑥) + 𝑏𝑑(𝑦, 𝑦) + 𝑐𝑑(𝑥, 𝑦) ≼ 𝑐𝑑(𝑥, 𝑦). taking the absolute value of both sides, we have |𝑑(𝑥, 𝑦)| ≤ |𝑐𝑑(𝑥, 𝑦)|. this implies 𝑑(𝑥, 𝑦) = 0. based on the properties of complex valued b-metric, we get 𝑥 = 𝑦. this completes the proof. corollary 11. let (𝑋, 𝑑) be a complete complex valued b-metric space with the coefficient 𝑠 ≥ 1. let 𝑇: 𝑋 → 𝑋 be a mapping (for some fixed 𝑛) satisfying 𝑑(𝑇 𝑛𝑥, 𝑇 𝑛 𝑦) ≼ 𝑎𝑑(𝑥, 𝑇 𝑛 𝑥) + 𝑏𝑑(𝑦, 𝑇 𝑛 𝑦) + 𝑐𝑑(𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋 where 𝑎, 𝑏, 𝑐 are non-negative real numbers with 𝑎 + 𝑠(𝑏 + 𝑐) < 1. then 𝑇 has a unique fixed point in 𝑋. proof. suppose 𝑆 = 𝑇 𝑛, then by the theorem 10, there is a fixed point 𝑢 such that 𝑆(𝑢) = 𝑢. thus 𝑇𝑛 (𝑢) = 𝑢. we have 𝑑(𝑇(𝑢), 𝑢) = 𝑑(𝑇(𝑇 𝑛 (𝑢)), 𝑢) = 𝑑 (𝑇 𝑛 (𝑇(𝑢)), 𝑇 𝑛(𝑢)) ≼ 𝑎𝑑 (𝑇(𝑢), 𝑇 𝑛 (𝑇(𝑢))) + 𝑏𝑑(𝑢, 𝑇 𝑛 (𝑢)) + 𝑐𝑑(𝑇(𝑢), 𝑢) = 𝑎𝑑 (𝑇(𝑢), 𝑇(𝑇 𝑛 (𝑢))) + 𝑏𝑑(𝑢, 𝑢) + 𝑐𝑑(𝑇(𝑢), 𝑢) fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 142 = 𝑎𝑑(𝑇(𝑢), 𝑇(𝑢)) + 𝑏𝑑(𝑢, 𝑢) + 𝑐𝑑(𝑇(𝑢), 𝑢) = 𝑐𝑑(𝑇(𝑢), 𝑢). taking the absolute value of both sides, we have |𝑑(𝑇(𝑢), 𝑢)| ≤ 𝑐|𝑑(𝑇(𝑢), 𝑢)|. hence (1 − 𝑐)|𝑑(𝑇(𝑢), 𝑢)| ≤ 0. this must be |𝑑(𝑇(𝑢), 𝑢)| = 0, which is 𝑇(𝑢) = 𝑢 = 𝑇 𝑛(𝑢). so 𝑇 has a fixed point. to prove its uniqueness, suppose 𝑣 is also a fixed point of 𝑇. we need to prove that 𝑢 = 𝑣. we see that 𝑑(𝑢, 𝑣) = 𝑑(𝑇 𝑛 (𝑢), 𝑇 𝑛 (𝑣)) ≼ 𝑎𝑑(𝑢, 𝑇 𝑛 (𝑢)) + 𝑏𝑑(𝑣, 𝑇 𝑛(𝑣)) + 𝑐𝑑(𝑢, 𝑣) = 𝑎𝑑(𝑢, 𝑢) + 𝑏𝑑(𝑣, 𝑣) + 𝑐𝑑(𝑢, 𝑣) = 𝑐𝑑(𝑢, 𝑣). taking the absolute value of both sides we get (1 − 𝑐)𝑑(𝑢, 𝑣) ≤ 0. since 𝑐 < 1, this yields 𝑑(𝑢, 𝑣) = 0. thus 𝑢 = 𝑣. this completes the uniqueness of the fixed point. theorem 12. let (𝑋, 𝑑) be a complete complex-valued b-metric space with constant 𝑠 ≥ 1 and let 𝑇: 𝑋 → 𝑋 be a mapping such that 𝑑(𝑇𝑥, 𝑇𝑦) ≼ 𝛼𝑑(𝑥, 𝑇𝑦) + 𝛽𝑑(𝑦, 𝑇𝑥) for every 𝑥, 𝑦 ∈ 𝑋, where 𝛼, 𝛽 ≥ 0 with 𝛼𝑠 < 1 1+𝑠 or 𝛽𝑠 < 1 1+𝑠 . then 𝑇 has a fixed point in 𝑋. moreover, if 𝛼 + 𝛽 < 1, then 𝑇 has a unique fixed point in 𝑋. proof. let 𝑥0 ∈ 𝑋 and (𝑥𝑛 ) be a sequence in x such that 𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇 𝑛𝑥0. now, for every 𝑛 ∈ ℕ we have 𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) ≼ 𝛼𝑑(𝑥𝑛+1, 𝑇𝑥𝑛 ) + 𝛽𝑑(𝑥𝑛 , 𝑇𝑥𝑛+1) = 𝛼𝑑(𝑥𝑛+1, 𝑥𝑛+1) + 𝛽𝑑(𝑥𝑛 , 𝑥𝑛+2) = 𝛽𝑑(𝑥𝑛 , 𝑥𝑛+2) ≼ 𝑠𝛽𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠𝛽𝑑(𝑥𝑛+1, 𝑥𝑛+2) (1 − 𝛽𝑠)𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼ 𝑠𝛽𝑑(𝑥𝑛 , 𝑥𝑛+1) 𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼ 𝑠𝛽 1 − 𝛽𝑠 𝑑(𝑥𝑛 , 𝑥𝑛+1). taking 𝛾 = 𝑠𝛽 1−𝛽𝑠 and continuing the process yields 𝑑(𝑥𝑛+1, 𝑥𝑛+2) ≼ 𝛾𝑑(𝑥𝑛 , 𝑥𝑛+1) ≼ 𝛾2𝑑(𝑥𝑛−1, 𝑥𝑛 ) ⋮ ≼ 𝛾𝑛+1𝑑(𝑥0, 𝑥1). on the other hand, 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠 𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1) + 𝑠 2𝛾𝑛+1𝑑(𝑥0, 𝑥1) + 𝑠 3𝛾𝑛+2𝑑(𝑥0, 𝑥1) + ⋯ + 𝑠 𝑚 𝛾𝑛+𝑚−1𝑑(𝑥0, 𝑥1) ≼ 𝑠𝛾𝑛 𝑑(𝑥0, 𝑥1)[1 + 𝑠𝛾 + (𝑠𝛾) 2 + ⋯ + (𝑠𝛾)𝑚−1]. fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 143 since 𝛽𝑠 < 1 1+𝑠 then 𝛾 ∈ (0, 1 𝑠 ) and 𝑠𝛾 < 1. taking limit 𝑛 → ∞, we have 𝛾𝑛 → 0. this implies |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚)| → 0 as 𝑛 → ∞. by lemma 8, the sequence (𝑥𝑛 ) is cauchy sequence in x. since 𝑋 is a complete complex valued b-metric space then (𝑥𝑛 ) is a convergent sequence. suppose that (𝑥𝑛 ) converges to 𝑥 ∗ ∈ 𝑋. we show that 𝑥∗ is a fixed point of 𝑇. 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥 ∗)] = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝑑(𝑇𝑥𝑛−1, 𝑇𝑥 ∗) = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝛼𝑑(𝑥𝑛−1, 𝑇𝑥 ∗) + 𝑠𝛽𝑑(𝑇𝑥𝑛−1, 𝑥 ∗) = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝛼𝑑(𝑥𝑛−1, 𝑇𝑥 ∗) + 𝑠𝛽𝑑(𝑥𝑛 , 𝑥 ∗) = (𝑠 + 𝑠𝛽)𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠 2𝛼𝑑(𝑥𝑛−1, 𝑥 ∗) + 𝑠2𝛼𝑑(𝑥∗, 𝑇𝑥∗) (1 − 𝛼𝑠2)𝑑(𝑥∗, 𝑇𝑥∗) ≼ (𝑠 + 𝑠𝛽)𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠 2𝛼𝑑(𝑥𝑛−1, 𝑥 ∗). since (𝑥𝑛 ) converges to 𝑥 ∗ that is |𝑑(𝑥∗, 𝑥𝑛 )| → 0 as 𝑛 → ∞. and taking the absolute value of both sides, we have |𝑑(𝑥∗, 𝑇𝑥∗)| ≤ 0. this means that |𝑑(𝑥∗, 𝑇𝑥∗)| = 0, i.e. 𝑥∗ = 𝑇𝑥∗. so the limit of the sequence is a fixed point of 𝑇. now we suppose that 𝛼 + 𝛽 < 1. assume that there are 𝑢, 𝑣 ∈ 𝑋 such that 𝑢 = 𝑇𝑢 and 𝑣 = 𝑇𝑣. 𝑑(𝑢, 𝑣) = 𝑑(𝑇𝑢, 𝑇𝑣) ≼ 𝛼𝑑(𝑢, 𝑇𝑣) + 𝛽𝑑(𝑣, 𝑇𝑢) = 𝛼𝑑(𝑢, 𝑣) + 𝛽𝑑(𝑣, 𝑢) = (𝛼 + 𝛽)𝑑(𝑢, 𝑣). this means that |𝑑(𝑢, 𝑣)| ≤ (𝛼 + 𝛽)|𝑑(𝑢, 𝑣)|. since 𝛼 + 𝛽 < 1, then we must have |𝑑(𝑢, 𝑣)| = 0, i.e. 𝑢 = 𝑣. the proof of the uniqueness of the fixed point of 𝑇 when 𝛼 + 𝛽 < 1 is complete. mir [14] studied the following fixed point theorem in b-metric spaces and we generalize it into complex-valued b-metric spaces. theorem 13. let (𝑋, 𝑑) be a complete b-metric space with constant 𝑠 ≥ 1 and define the sequence (𝑥𝑛 ) by the recursion 𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇 𝑛𝑥0, 𝑛 = 1,2, … let 𝑇: 𝑋 → 𝑋 be a mapping for which there exists 𝜇 ∈ [𝑜, 1 2 ) such that 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜇[𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)] for all 𝑥, 𝑦 ∈ 𝑋. then there exists 𝑥∗ ∈ 𝑋 such that 𝑥𝑛 → 𝑥 ∗ and 𝑥∗ is unique fixed point of 𝑇. theorem 14. let (𝑋, 𝑑) be a complete complex-valued b-metric space with constant 𝑠 ≥ 1. let 𝑇: 𝑋 → 𝑋 be a mapping for which there exists 𝜇 ∈ [𝑜, 1 2 ) such that 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜇[𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)] for all 𝑥, 𝑦 ∈ 𝑋. then there exists unique fixed point of 𝑇. proof. we construct a sequence (𝑥𝑛 ) in 𝑋 by 𝑥𝑛 = 𝑇𝑥𝑛−1 = 𝑇 𝑛𝑥0, 𝑛 = 1,2, … for any 𝑥0 ∈ 𝑋. we show that this sequence (𝑥𝑛 ) is cauchy sequence in 𝑋 and hence is convergent sequence. for every 𝑛 ∈ ℕ we have 𝑑(𝑥𝑛+2, 𝑥𝑛+1) = 𝑑(𝑇𝑥𝑛+1, 𝑇𝑥𝑛 ) ≼ 𝜇(𝑑(𝑥𝑛+1, 𝑇𝑥𝑛+1) + 𝑑(𝑥𝑛 , 𝑇𝑥𝑛 )) = 𝜇(𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑑(𝑥𝑛 , 𝑇𝑥𝑛 )) (1 − 𝜇)𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ 𝜇𝑑(𝑥𝑛 , 𝑇𝑥𝑛 ) 𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ 𝜇 1 − 𝜇 𝑑(𝑥𝑛+1, 𝑥𝑛 ) if this process continues, for all 𝑛 ∈ ℕ we have 𝑑(𝑥𝑛+2, 𝑥𝑛+1) ≼ ( 𝜇 1 − 𝜇 ) 𝑛+1 𝑑(𝑥1, 𝑥0). therefore, 𝑑(𝑥𝑛 , 𝑥𝑛+𝑚) ≼ 𝑠[𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑑(𝑥𝑛+1, 𝑥𝑛+𝑚 )] ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 2𝑑(𝑥𝑛+2, 𝑥𝑛+𝑚 ) fixed point theorems in complex-valued b-metric spaces dahliatul hasanah 144 ≼ 𝑠𝑑(𝑥𝑛 , 𝑥𝑛+1) + 𝑠 2𝑑(𝑥𝑛+1, 𝑥𝑛+2) + 𝑠 3𝑑(𝑥𝑛+2, 𝑥𝑛+3) + ⋯ + 𝑠 𝑚 𝑑(𝑥𝑛+𝑚−1, 𝑥𝑛+𝑚) ≼ 𝑠 ( 𝜇 1 − 𝜇 ) 𝑛 𝑑(𝑥0, 𝑥1) + 𝑠 2 ( 𝜇 1 − 𝜇 ) 𝑛+1 𝑑(𝑥0, 𝑥1) + 𝑠 3 ( 𝜇 1 − 𝜇 ) 𝑛+2 𝑑(𝑥0, 𝑥1) + ⋯ + 𝑠𝑚 ( 𝜇 1 − 𝜇 ) 𝑛+𝑚−1 𝑑(𝑥0, 𝑥1). ≼ 𝑠 ( 𝜇 1 − 𝜇 ) 𝑛 𝑑(𝑥0, 𝑥1) [1 + 𝑠 ( 𝜇 1 − 𝜇 ) + (𝑠 ( 𝜇 1 − 𝜇 )) 2 + ⋯ + (𝑠 ( 𝜇 1 − 𝜇 )) 𝑚−1 ]. note that 𝜇 ∈ [𝑜, 1 2 ) implies ( 𝜇 1−𝜇 ) ∈ [0,1). taking the absolute value of both sides gives |𝑑(𝑥𝑛 , 𝑥𝑛+𝑚 )| → 0 as 𝑛 → ∞. thus (𝑥𝑛 ) is cauchy sequence in 𝑋 and is therefore a convergent sequence. suppose 𝑥 ∗ ∈ 𝑋 is the limit of sequence (𝑥𝑛 ). we show that 𝑥 ∗ is a fixed point of 𝑇. using the properties of b-metric and the mapping 𝑇, we get 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠[𝑑(𝑥∗, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑇𝑥 ∗)] = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝑑(𝑇𝑥𝑛−1, 𝑇𝑥 ∗) = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑇𝑥𝑛−1) + 𝑠𝜇𝑑(𝑥 ∗, 𝑇𝑥∗) = 𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥 ∗, 𝑇𝑥∗) (1 − 𝑠𝜇)𝑑(𝑥∗, 𝑇𝑥∗) ≼ 𝑠𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑠𝜇𝑑(𝑥𝑛−1, 𝑥𝑛 ) = 𝑠𝑑(𝑥 ∗, 𝑥𝑛 ) + 𝑠𝜇 ( 𝜇 1 − 𝜇 ) 𝑛−1 𝑑(𝑥1, 𝑥0) 𝑑(𝑥∗, 𝑇𝑥∗) ≼ 1 (1 − 𝑠𝜇) (𝑠𝑑(𝑥∗, 𝑥𝑛 ) + 𝑠𝜇 ( 𝜇 1 − 𝜇 ) 𝑛−1 𝑑(𝑥1, 𝑥0)). taking the absolute value of both sides and taking 𝑛 → ∞, we have lim 𝑑(𝑥∗, 𝑇𝑥∗) = 0. this implies 𝑥∗ is a fixed point of 𝑇, i.e. 𝑥∗ = 𝑇𝑥∗. now we assume that there exists another fixed point 𝑣 of 𝑇, that is 𝑣 = 𝑇𝑣. we have 𝑑(𝑥∗, 𝑣) = 𝑑(𝑇𝑥∗, 𝑇𝑣) ≼ 𝜇(𝑑(𝑥∗, 𝑇𝑥∗) + 𝑑(𝑣, 𝑇𝑣)) = 𝜇(𝑑(𝑥∗, 𝑥∗) + 𝑑(𝑣, 𝑣)) = 0. taking the absolute value of both sides we obtain |𝑑(𝑥∗, 𝑣)| ≤ 0 which is a contradiction to the distinct fixed points. hence 𝑥∗ is the unique fixed point of 𝑇. conclusion complex valued b-metric spaces are considered as a generalisation of b-metric spaces by changing the definition of real valued metric into complex valued metric. this change is expected to bring wider applications of fixed point theorems. in this paper, we extend fixed point theorems in b-metric spaces into fixed point theorems in complex valued b-metric spaces. by using the properties of partial order on complex numbers, a mapping 𝑇 on a real valued b-metric space which has unique fixed point still has unique fixed point in a complex valued b-metric space. references [1] 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[14] m. kir and h. kiziltunc, "on some well known fixed point b-metric spaces," turkish journal of analysis and number theory, vol. 1, no. 1, pp. 13-16, 2013. diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi volume 4, issue 4, may 2017 issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behavior of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development 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decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. jurnal matematika murni dan aplikasi volume 4, issue 4, may 2017 issn 2086-0382 e-issn 2477-3344 cauchy preface cauchy is a national journal published by mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang. this is the second issue of this year. it contains 6 (six) article from all over the country, not only from local area of east java. those articles covered graph theory, numerical analysis, applied mathematics, statistics, economic mathematics, and algebra. this issue was authored by 13 authors and co-authors. in the first article, the author discussed the existence and uniqueness of fixed point in complex valued b-metric spaces. she considers some fixed-point theorems in bmetric spaces to be extended to fixed point theorems in complex valued b-metric spaces. some fixed-point theorems in complex-valued b-metric spaces which are derived from the corresponding theorems in b-metric spaces are obtained in this research. the second article focuses on graph theory and its applications. the paper showed that super (𝑎, 𝑑) − 𝐵𝑚-antimagic total labeling of 𝐴𝑚𝑎𝑙(𝐹𝑛, 𝑃𝑛, 𝑚) and 𝑠𝐴𝑚𝑎𝑙(𝐹𝑛,𝑃𝑛,𝑚) for some feasible 𝑑. the author also suggests to do some other research concerning to this field since there are a lot of things to determine yet the obtained results are still very limited. the third article, entitled “application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in way kanan district, lampung” elaborated the modification of spatial k'luster analysis by tree edge removal. the result shows that the spread pattern of the spoliation, robberies and gambling cases in way kanan district showed their spatial clustering. it also explained that modification of skater clustering method produce the 4-rise cluster, it is because the mean level of crime is 𝑘1 ≥ 𝑘2 ≥ 𝑘3 ≥ 𝑘4. the fourth article emphasized the discussion of graph theory especially on harmonious labeling on pleated of the dutch windmill graphs. it showed that the pleated of the dutch windmill graphs and the union pleated of the dutch windmill graphs admitted odd harmonious labeling. the authors suggest an open problem for further research including the question if there exists odd harmonious labeling on another kind of pleated of the dutch windmill graph. jurnal matematika murni dan aplikasi volume 4, issue 4, may 2017 issn 2086-0382 e-issn 2477-3344 cauchy preface the fifth article covered the discussion of almost surjective epsilon-isometry in the reflexive banach spaces. in this paper, the author discussed some applications of almost surjective 𝜀-isometry mapping, one of them is in lorentz space (𝐿𝑝,𝑞 − space). the last article entitled “on the spectra of commuting and non-commuting graph on dihedral group”. this paper integrates the algebraic field of study with the application of graph theory. in this paper, the author investigates adjacency spectrum, laplacian spectrum, signless laplacian spectrum, and detour spectrum of commuting and non-commuting graph of dihedral group 𝐷2𝑛. 72 analisis perilaku dinamik pada sel t cd𝟒+ dan sel t cd𝟖+ terhadap infeksi mikobakterium tuberkulosis alfi nur rochmatin, usman pagalay jurusan matematika universitas islam negeri maulana malik ibrahim malang e-mail: alfinurrochmatin@yahoo.co.id abstrak model matematika pada infeksi mikobakterium tuberkulosis yang berbentuk sistem persamaan diferensial nonlinear orde satu. penelitian ini telah mengkonstruksi model matematika pada interaksi makrofag, sel t cd4+ dan sel t cd8+ dengan pengaruh usia. solusi numerik pada model matematika ini dengan menggunakan ode 45 berbantuan matlab. analisis kestabilan diamati melalui titik tetap dengan mencari matriks jacobian dan nilai eigen dari titik tetap tersebut, maka dapat diperoleh bahwa semua titik tetap tersebut tidak stabil. berdasarkan analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+pada usia muda dan usia tua maka akan diperoleh bahwa sel t cd4+dan sel t cd8+ lebih banyak mempengaruhi populasi bakteri mikobakterium tuberkulosis dari pada saat usia muda. kata kunci: perilaku dinamik, analisis kestabilan, mikobakterium tuberkulosis, sel t 𝐂𝐃𝟒+ dan sel t 𝐂𝐃𝟖+ abstract mathematics model to mycobacterium tuberculosis infections which in from sistem of non linear differential equation first order. in this research a mathematical model of interactions of macrophages, 𝐶𝐷4+ t cells and 𝐶𝐷8+ t cells with influence of age has been to construct. numerical solution of this mathematical model is using ode 45 assisted matlab. stability analysis refer to fixed point by finding the jacobian matrix and the eigenvalues of the fixed point, then it can be obtained that all the fixed points are unstable. based on the analysis of dynamic behavior of the 𝐶𝐷4+ t cells and 𝐶𝐷8+ t cells in old age we obtain that 𝐶𝐷4+ t cells and 𝐶𝐷8+ t cells affect population of micobacterium tuberculosis bacteria move than it does at young age. keywords: dynamic behavior, stabillity analysis, mycobacterium tuberculosis, t cells 𝐶𝐷4+ and t cells 𝐶𝐷8+ pendahuluan model matematika pada infeksi mikobakterium tuberkulosis ini, menggunakan 7 persamaan yaitu berupa persamaan diferensial biasa linier dan non linier. dan menganalisis perilaku model matematika pada infeksi mikobakterium tuberkulosis, penting dilakukan untuk mengetahui bagaimana model matematika yang dituangkan dalam suatu sistem persamaan matematika tersebut, menggambarkan interaksi pada semua variabel. dalam pembahasan ini penulis mengkhususkan pembahasan pada perilaku dinamik terhadap infeksi mikobakterium tuberkulosis dengan pengaruh usia yang melibatkan populasi sel t cd4+ dan sel t cd8+. dalam model ini konsentrasi sitokin diabaikan karena tidak berpengaruh nyata dan juga sitokin bisa diproduksi oleh makrofag dan limfosit t. namun kendala utamanya adalah kemungkinan terjadinya ketidakstabilan. teori dasar 1. titik tetap misal diberikan sistem persamaan diferensial 𝑑𝑇4 𝑑𝑡 = 𝑇4̇ = 𝑓(𝑇4) (1) titik tetap merupakan titik gerak dari vektor keadaan konstan. atau dengan kata lain, titik tetap merupakan solusi yang tetap konstan walaupun waktu berganti. maka titik tetap dari persamaan (1) didapat jika 𝑑𝑇4 𝑑𝑡 = 0. adapun istilah lain dari titik tetap adalah titik equilibrium, titik stasioner, fixed point, atau singularity. untuk lebih jelasnya diberikan contoh, yaitu: misal 𝑓(𝑇4) = 𝑇4 2 − 𝑇4 − 6, maka untuk mencari titik tetapnya yaitu dengan cara 𝑓(𝑇4) = 0 atau menyamadengankan nol pada turunan pertamanya, sehingga diperoleh: 𝑓(𝑇4) = 𝑇4 2 − 𝑇4 − 6 = 0 mailto:alfinurrochmatin@yahoo.co.id analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 73 (𝑇4 − 3)(𝑥 + 2) = 0 sehingga diperoleh titik tetap yaitu: 𝑇4 = 3 atau 𝑇4 = −2 [1]. 2. kestabilan titik tetap penentuan kestabilan titik tetap dapat diperoleh dengan melihat nilai-nilai eigennya, yaitu 𝜆𝑖, 𝑖 = 1,2,3, … , 𝑛 yang diperoleh dari persamaan karakteristik dari 𝐴, yaitu (𝐴 − 𝜆𝐼) = 0. secara umum kestabilan titik tetap mempunyai tiga perilaku sebagai berikut: 1. stabil yaitu suatu titik kestabilan 𝑥∗ stabil jika setiap nilai eigen real adalah negatif (𝜆, < 0, 𝑖 = 1,2, … , 𝑛). 2. tidak stabil yaitu suatu titik kestabilan 𝑥∗ tidak stabil jika setiap nilai eigen real adalah positif (𝜆𝑖 > 0, untuk setiap 𝑖). 3. pelana (saddle) yaitu suatu titik kestabilan 𝑥∗ adalah pelana jika perkalian dua nilai eigen real adalah negatif (𝜆𝑖𝜆𝑗 < 0, untuk setiap 𝑖 dan 𝑗 sembarang) [2]. 3. sel t 𝐂𝐃𝟒+ dan sel t 𝐂𝐃𝟖+ sel t cd4+ memainkan 2 peran utama di dalam infeksi mikobakterium tuberkulosis. pertama adalah dalam produksi sitokin dalam memerintahkan respon yang diperantarai oleh sel, kedua adalah mengeliminasi makrofag yang sudah terinfeksi melalui apoptosis (pagalay, 2009:48). berdasarkan fungsinya sel t cd4+ dibedakan menjadi 2 sub populasi yaitu sel th1 dan th2. baik th1 dan th2 berpengaruh terhadap manifestasi infeksi oleh bi [3]. sel t cd8+ dapat juga menghancurkan sel yang terinfeksi bakteri intraselular. sel t cd8+mengenal komples antigen mhc-i yang dipresentasikan apc. molekul mhc-i ditemukan pada semua sel tubuh yang bernukleus. fungsi utama sel cd8+ yaitu dapat menyingkirkan sel terinfeksi virus, menghancurkan sel ganas dan sel histoin kompatibel yang menimbulkan penolakan pada transplantasi [4]. 4. mikobakterium tuberkulosis mikobakterium tuberkulosis merupakan bakteri yang dapat menyebabkan tb. bakteri mikobakterium tuberkulosis memiliki panjang sekitar 1 − 4 mikron dan lebar sekitar 0,2 − 0,8 mikron dengan bentuk batang tipis, lurus atau agak bengkok, bergranular atau tidak mempunyai selubung, tetapi mempunyai lapisan luar tebal yang terdiri dari lipoid. mikobakterium tuberkulosis adalah bakteri yang berbentuk batang dan bersifat tahan asam. bakteri ini pertama kali ditemukan pada tanggal 24 maret1882 oleh robert koch. bakteri ini juga disebut baksil koch [3]. setiap tahunnya tb menyebabkan kematian 3 juta penduduk dunia dan diperkirakan sepertiga penduduk dunia telah terinfeksi kuman tb, yang dapat berkembang menjadi penyakit tb di masa mendatang. selain itu jumlah kematian dan infeksi tb yang sangat besar, pertambahan kasus baru tb amat signifikan, mencapat sembilan juta kasus baru setiap tahun. penyebaran penyakit tb adalah melalui udara terkontaminasi mycobacterium tuberculosis yang terhirup kemudian masuk ke dalam paru-paru, menyerang dinding saluran pernafasan dengan membentuk rongga yang berisi nanah dan bakteri tb. apabila penderita tb batuk atau bersin akan ikut mengeluarkan bakteri tb ke udara. apabila terhirup oleh orang yang rentan penyakit tb, orang tersebut akan dapat terinfeksi bakteri tb [5]. 5. pembentukan model identifikasi dimulai dengan menganalisis pembentukan model pada populasi bakteri pada makrofag yang terinfeksi. diasumsikan bahwa bakteri intraseluler tumbuh pada laju maksimal 𝛼𝐼. bakteri ini tumbuh dengan berkurangnya persamaan hill, yaitu pada koefisien hill dan 𝑁𝑀𝐼. 𝑁𝑀𝐼 merupakan jumlah bakteri pada bakteri intraseluler yang sudah mencapai kapasitas maksimum 𝑁 dalam makrofag yang terinfeksi. makrofag inilah yang akan meledak dan melepaskan bakteri. sehingga diperoleh pertumbuhannya yaitu: 𝛼𝐼𝐵𝐼 (1 − 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 ) (2) ketika bakteri ekstraseluler masuk dan makrofag gagal untuk membunuh bakteri, maka makrofag resting akan menjadi terinfeksi oleh bakteri ekstraseluler. jumlah bakteri pada makrofag yang terinfeksi akan tergantung pada populasi bakteri ekstraseluler yang menginfeksi makrofag resting. sehingga diperoleh perkembangannya yaitu: 𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 (3) dinamika bakteri pada makrofag yang terinfeksi dipengaruhi oleh pembebasan bakteri akibat lisis (pecah) dari makrofag terinfeksi. makrofag inilah yang meledak dan mengakibatkan kerugian pada bakteri pada makrofag yang terinfeksi. dimana bakteri ini akan dilepaskan http://id.wikipedia.org/wiki/24_maret http://id.wikipedia.org/wiki/24_maret http://id.wikipedia.org/wiki/1882 http://id.wikipedia.org/wiki/robert_koch http://id.wikipedia.org/wiki/robert_koch alfi nur rohmatin, usman pagalay 74 volume 3 no. 2 mei 2014 pada bakteri ekstraseluler. sehingga diperoleh perkembangan meledaknya makrofag terinfeksi yaitu: 𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 (4) bakteri intraseluler selain dipengaruhi oleh pembebasan bakteri akibat lisis (pecah) dari makrofag terinfeksi, pembebasan bakteri intraseluler dengan laju 𝑘3, perkembangannya yaitu: 𝑛1𝑘3𝐵𝐼 (5) bakteri intraseluler juga dipengaruhi oleh pembebasan bakteri intraseluler dengan laju 𝑘4, perkembangannya yaitu: 𝑛2𝑘4𝐵𝐴 (6) dari persamaan (2) − (6) maka persamaan model untuk dinamika populasi bakteri intraseluler yaitu sebagai berikut: 𝑑𝐵𝐼 𝑑𝑡 = 𝛼𝐼𝐵𝐼 (1 − 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 ) +𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 − 𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 −𝑛1𝑘3𝐵𝐼 + 𝑛2𝑘4𝐵𝐴 (7) bakteri di dalam makrofag teraktivasi tumbuh pada laju maksimal 𝛼𝐴. sehingga diperoleh pertumbuhannya yaitu: 𝛼𝐴𝐵𝐴 (8) bakteri ini akan berkurang karena mengalami deaktivasi (penurunan kemampuan untuk aktif kembali) makrofag teraktifasi dengan laju 𝑘4, perkembangannya yaitu: 𝑛2𝑘4𝐵𝐴 (9) pertumbuhan bakteri di dalam makrofag teraktivasi juga dipengaruhi adanya pertambahan bakteri pada makrofag yang terinfeksi dengan laju 𝑘3, perkembangannya yaitu: 𝑛1𝑘3𝐵𝐼 (10) akan tetapi, bakteri ini akan hilang karena kematian makrofag aktif secara alami. 𝑛2𝜇𝑀𝐴𝐵𝐴 (11) dari persamaan (8) − (11) maka persamaan model untuk dinamika populasi bakteri di dalam makrofag teraktivasi yaitu sebagai berikut: 𝑑𝐵𝐴 𝑑𝑡 = 𝛼𝐴𝐵𝐴 − 𝑛2𝑘4𝐵𝐴 + 𝑛1𝑘3𝐵𝐼 − 𝑛2𝜇𝑀𝐴𝐵𝐴 (12) bakteri intraseluler tumbuh pada laju maksimal 𝛼𝐸. sehinggal diperoleh pertumbuhannya yaitu: 𝛼𝐸𝐵𝐸 (13) berkurangnya makrofag resting juga mempengaruhi pertumbuhan dari bakteri ekstraseluler. makrofag ini akan menjadi terinfeksi oleh bakteri ketika bakteri ekstraseluler masuk dan makrofag gagal untuk membunuhnya. sehingga diperoleh perkembangannya yaitu: 𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 (14) kebalikan dari bakteri intraseluler, dinamika bakteri ekstraseluler dipengaruhi oleh bakteri yang pecah dari makrofag yang terinfeksi. dimana makrofag akan meledak yang mengakibatkan bertambahnya bakteri ekstraseluler. diperoleh perkembangan meledaknya makrofag yang terinfeksi yaitu: 𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 (15) pengambilan bakteri ekstraseluler oleh makrofag teraktivasi menyebabkan berkurangnya bakteri dengan laju 𝑘5, perkembangannya yaitu: 𝑘5𝑀𝐴𝐵𝐸 (16 selain itu, bakteri ekstraseluler juga tumbuh disebabkan oleh kematian dari bakteri aktif dengan pertumbuhannya adalah 𝑛2𝜇𝑀𝐴𝐵𝐴 (17) dari persamaan (13) − (17) maka persamaan model untuk dinamika populasi bakteri ekstraseluler yaitu sebagai berikut: 𝑑𝐵𝐸 𝑑𝑡 = 𝛼𝐸𝐵𝐸 − 𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 + 𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 −𝑘5𝑀𝐴𝐵𝐸 + 𝑛2𝜇𝑀𝐴𝐵𝐴 (18) populasi makrofag terinfeksi berasal dari makrofag resting yang terinfeksi oleh bakteri ekstraseluler, bakteri ini akan masuk ke dalam tubuh dan berkembangbiak. sehingga diperoleh pertumbuhannya yaitu: 𝑘1𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 (19) bakteri yang masuk akan terus menerus berkembangbiak di dalam makrofag, ketika jumlah bakteri mencapai kapasitas maksimal 𝑁, analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 75 makrofag yang terinfeksi ini akan meledak karena adanya peningkatan jumlah bakteri dengan perkembangannya yaitu: 𝑘2𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 (20) makrofag yang terinfeksi akan menjadi makrofag aktif yang terinfeksi dengan laju 𝑘3, perkembangannya yaitu: 𝑘3𝑀𝐼 (21) makrofag ini juga akan mengalami deaktivasi makrofag aktif dengan laju 𝑘4, perkembangannya yaitu: 𝑘4𝑀𝐴 (22) dan makrofag akan mengalami kematian secara alami pada laju 𝜇𝑀𝐼, 𝜇𝑀𝐼𝑀𝐼 (23) dari persamaan (19) − (23) maka persamaan model untuk dinamika populasi makrofag terinfeksi yaitu sebagai berikut: 𝑑𝑀𝐼 𝑑𝑡 = 𝑘1𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 −𝑘2𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 𝑘3𝑀𝐼 +𝑘4𝑀𝐴 − 𝜇𝑀𝐼𝑀𝐴 (24) untuk populasi makrofag teraktivasi, kegagalan deaktivasi makrofag aktif dengan laju 𝑘4, perkembangannya yaitu: 𝑘4𝑀𝐴 (25) makrofag teraktivasi juga berasal dari makrofag yang terinfeksi dengan laju 𝑘3, perkembangannya yaitu: 𝑘3𝑀𝐼 (26) dan makrofag akan mengalami kematian secara alami pada laju 𝜇𝑀𝐴, perkembangannya yaitu: 𝜇𝑀𝐴𝑀𝐴 (27) selain itu, pada populasi makrofag teraktivasi dapat diperoleh dari makrofag resting yang terinfeksi oleh bakteri ekstraseluler yang datang dengan laju 𝑘6, perkembangannya yaitu: 𝑘6𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐5 (28) dari persamaan (25) − (28) maka persamaan model untuk dinamika populasi makrofag teraktivasi yaitu sebagai berikut: 𝑑𝑀𝐴 𝑑𝑡 = −𝑘4𝑀𝐴 + 𝑘3𝑀𝐼 − 𝜇𝑀𝐴𝑀𝐴 +𝑘6𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐5 (29) dinamika populasi sel t cd4+ tergantung pada aktivasi makrofag pada mhc-ii dengan fungsi 𝜆𝑧 yang bergantung pada waktu, perkembangannya yaitu 𝜆𝑧𝑀𝐴 (30) selain itu, dinamika sel t cd4+ juga tergantung pada poliferasi sel t dengan laju 𝑘13, perkembangannya yaitu 𝑘13𝑇4 (31) dan dihambat oleh kematian sel t cd4+ sendiri secara alami pada laju 𝜇𝑇4, yaitu: 𝜇𝑇4𝑇4 (32) dari persamaan (30) − (32) maka persamaan model untuk dinamika populasi sel t cd4+ yaitu sebagai berikut: 𝑑𝑇4 𝑑𝑡 = 𝜆𝑧𝑀𝐴 + 𝑘13𝑇4 − 𝜇𝑇4𝑇4 (33) dan yang terakhir adalah dinamika populasi sel t cd8+ tergantung dari makrofag aktif dan makroag yang terinfeksi pada mhc-i dengan fungsi 𝜆𝑧 yang bergantung pada waktu, perkembangannya yaitu: 𝜆𝑥(𝑀𝐴 + 𝑀𝐼) (34) selain itu, dinamika sel t cd8+ juga tergantung pada poliferasi sel t dengan laju 𝑘14, perkembangannya yaitu 𝑘14𝑇8 (35) dan dihambat oleh kematian sel t cd8+ sendiri secara alami pada laju 𝜇𝑇4, yaitu: 𝜇𝑇8𝑇8 (36) dari persamaan (34) − (36) maka persamaan model untuk dinamika populasi sel t cd8+ yaitu sebagai berikut (friedman, 2008:3-5): 𝑑𝑇8 𝑑𝑡 = 𝜆𝑥(𝑀𝐴 + 𝑀𝐼) + 𝑘14𝑇8 − 𝜇𝑇8𝑇8 (37) pembahasan 1. identifikasi model matematika berikut ini merupakan gambar skema perubahan dan interaksi setiap populasi sel pada model matematika, yaitu: alfi nur rohmatin, usman pagalay 76 volume 3 no. 2 mei 2014 berikut ini merupakan gambaran singkat tentang bakteri, makrofag,dan populasi sel t yang telah disajikan pada gambar (1) makrofag didefinisikan menjadi tiga subpopulasi yaitu: makrofag resting (𝑀𝑅), makrofag terinfeksi (𝑀𝐼) dan makrofag teraktivasi (𝑀𝐴). dan juga populasi sel t hanya meliputi populasi sel t cd4+ (𝑇4) dan sel t cd8+ (𝑇8). gambar (1). skema perubahan dan interaksi setiap populasi pada model sebuah makrofag resting (𝑀𝑅) menjadi teraktivasi oleh sejumlah bakteri kecil dengan laju 𝑘6. makrofag teraktivasi (𝑀𝐴) mampu mengendalikan pertumbuhan mikobakteri dan penyajian antigen ke sel t cd8+ (𝑇8) melalui mhc-i dengan laju aktivasi 𝜆𝑥(𝑡). sedangkan penyajian antigen ke sel t cd4+ (𝑇4) melalui mhcii dengan laju aktivasi 𝜆𝑧(𝑡). sebuah makrofag resting (𝑀𝑅) menjadi makrofag terinfeksi (𝑀𝐼) apabila terinfeksi akibat sejumlah bakteri kecil dengan laju 𝑘1. dan sebuah makrofag terinfeksi (𝑀𝐼) gagal mengontrol pertumbuhan mikobakteri dan dapat meledak jika melebihi kapasitas n maksimal, sedangkan untuk penyajian antigen ke sel t cd8+ (𝑇8) melalui mhc-i dengan laju aktivasi 𝜆𝑥(𝑡). makrofag terinfeksi (𝑀𝐼) dan makrofag teraktivasi (𝑀𝐴) juga akan mengalami kematian secara alami pada laju masing-masing 𝜇𝑀𝐼 dan 𝜇𝑀𝐴. jumlah makrofag resting (𝑀𝑅) tetap tidak berubah selama perkembangan penyakit, yaitu ketika beberapa makrofag resting (𝑀𝑅) menjadi makrofag teraktivasi (𝑀𝐴) dan makrofag terinfeksi (𝑀𝐼). populasi bakteri dibagi menurut tempat tinggal mereka yaitu: bakteri intraseluler (𝐵𝐼), bakteri di dalam makrofag teraktivasi (𝐵𝐴) dan bakteri ekstraselular (𝐵𝐸). bakteri di dalam makrofag teraktivasi (𝐵𝐴) berada di dalam makrofag yang teraktivasi (𝑀𝐴) dan tumbuh dengan laju 𝛼𝐴. bakteri intraseluler (𝐵𝐼) berada di dalam makrofag yang terinfeksi (𝑀𝐼) dan tumbuh dengan laju 𝛼𝐼. sedangkan bakteri ekstraselular (𝐵𝐸) berada diluar makrofag dengan laju tumbuh 𝛼𝐸 . 2. interpretasi model matematika berikut ini merupakan interpretasi pada persamaan model interaksi bakteri intraseluler (𝐵𝐼), bakteri di dalam makrofag teraktivasi (𝐵𝐴), bakteri ekstraselular (𝐵𝐸), makrofag terinfeksi (𝑀𝐼), makrofag teraktivasi (𝑀𝐴), sel t cd4 + (𝑇4) dan sel t cd8+ (𝑇8) ditulis sebagai berikut: 𝑑𝐵𝐼 𝑑𝑡 = 𝛼𝐼𝐵𝐼 (1 − 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 ) +𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 −𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 −𝑛1𝑘3𝐵𝐼 + 𝑛2𝑘4𝐵𝐴 (36) perubahan populasi bakteri intraseluler yang bergantung waktu dipengaruhi oleh beberapa faktor, antara lain: pertumbuhan bakteri pada laju maksimal 𝛼𝐼 kemudian dikurangi dengan persamaan hill yang bergantung pada daya ledakan bakteri 𝑁 dalam makrofag terinfeksi, bertambahnya makrofag resting yang berubah menjadi terinfeksi dengan laju 𝑘1, pembebasan bakteri intraseluler akibat lisis (pecah) dari makrofag yang terinfeksi dengan laju 𝑘2 yang bergantung pada daya ledak bakteri 𝑁 dan pembebasan bakteri intraseluler dengan laju 𝑘3 serta deaktivasi makrofag aktif dengan laju 𝑘4. 𝑑𝐵𝐴 𝑑𝑡 = 𝛼𝐴𝐵𝐴 − 𝑛2𝑘4𝐵𝐴 + 𝑛1𝑘3𝐵𝐼 −𝑛2𝜇𝑀𝐴𝐵𝐴 (37) perubahan populasi bakteri di dalam makrofag teraktivasi yang bergantung pada waktu itu dipengaruhi oleh beberapa faktor, antara lain: pertumbuhan bakteri pada laju maksimal 𝛼𝐴 dikurangi dengan deaktivasi makrofag teraktifasi dengan laju 𝑘4, bertambahnya bakteri pada makrofag yang terinfeksi dengan laju 𝑘3. 𝑑𝐵𝐸 𝑑𝑡 = 𝛼𝐸𝐵𝐸 − 𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 (38) analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 77 +𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 𝑘5𝑀𝐴𝐵𝐸 +𝑛2𝜇𝑀𝐴𝐵𝐴 perubahan bakteri ekstraseluler yang bergantung pada waktu dipengaruhi oleh beberapa faktor antara lain: pertumbuhan bakteri pada laju maksimal 𝛼𝐸 yang diambil oleh makrofag resting yang berubah menjadi terinfeksi dengan laju 𝑘1, bertambahnya bakteri pada makrofag yang terinfeksi yang pecah dari makrofag yang terinfeksi dengan laju 𝑘2 yang bergantung pada daya ledak 𝑁, pengambilan bakteri oleh makrofag teraktivasi pada laju 𝑘5. 𝑑𝑀𝐼 𝑑𝑡 = 𝑘1𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 −𝑘2𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 𝑘3𝑀𝐼 +𝑘4𝑀𝐴 − 𝜇𝑀𝐼𝑀𝐴 (39) perubahan populasi makrofag terinfeksi yang bergantung pada waktu dipengaruhi oleh beberapa faktor antara lain: makrofag resting yang terinfeksi dengan laju 𝑘1 dikurangi dengan ledakan makrofag yang terinfeksi dengan laju 𝑘2 dan aktivasi makrofag yang terinfeksi dengan laju 𝑘3, deaktivasi makrofag aktif dengan laju 𝑘4, dan kematian makrofag yang terinfeksi dengan laju 𝜇𝑀𝐼. 𝑑𝑀𝐴 𝑑𝑡 = −𝑘4𝑀𝐴 + 𝑘3𝑀𝐼 − 𝜇𝑀𝐴𝑀𝐴 +𝑘6𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐5 (40) pertumbuhan populasi makrofag teraktivasi yang bergantung waktu itu dipengaruhi oleh beberapa faktor, antara lain: kegagalan deaktivasi makrofag aktif dengan laju 𝑘4, bertambahnya aktivasi makrofag yang terinfeksi dengan laju 𝑘3, kematian makrofar teraktivasi pada laju 𝜇𝑀𝐴, dan penambahan aktivasi makrofag resting oleh bakteri ekstraseluler yang datang dengan laju 𝑘6. 𝑑𝑇4 𝑑𝑡 = 𝜆𝑧𝑀𝐴 + 𝑘13𝑇4 − 𝜇𝑇4𝑇4 (41) perubahan populasi sel t cd4+ yang bergantung pada waktu dipengaruhi oleh makrofag teraktivasi pada mhc-ii dengan laju 𝜆𝑧 dan penambahan poliferasi sel t dengan laju 𝑘13 dan sel t cd4+ rusak pada rata-rata 𝜇𝑇4. 𝑑𝑇8 𝑑𝑡 = 𝜆𝑥(𝑀𝐴 + 𝑀𝐼) + 𝑘14𝑇8 − 𝜇𝑇8𝑇8 (42) perubahan populasi sel t cd8+ yang bergantung pada waktu dipengaruhi oleh makrofag teraktivasi dan makrofag terinfeksi pada mhc-i dengan laju 𝜆𝑥 dan penambahan poliferasi sel t dengan laju 𝑘14 dan sel t cd8 + rusak pada rata-rata 𝜇𝑇8. 3. nilai parameter model tabel 1. nilai awal variabel nilai muda tua 𝐵𝐼 36.000 4000 𝐵𝐴 1000 9000 𝐵𝐸 1000 1000 𝑀𝐼 1800 200 𝑀𝐴 200 1800 𝑇4 200.000 100.000 𝑇8 80.000 80.000 tabel 2. nilai parameter nam a nilai muda tua 𝛼𝐼 0.5 0.5 𝛼𝐸 0 0 𝛼𝐴 0 0 𝑘1 0.4 0.4 𝑘2 0.81139 0.81139 𝑘3 0.023415 0.025440 𝑘4 0.28876 0.61707 𝑘5 0.000081301 0.000081301 𝑘6 0.077068 0.13539 𝑘13 0.1638 0.14789 𝑘14 0.01638 0.01413 𝑐1 1000.000 1000.000 𝑐5 100.000 100.000 𝑀𝑅 500.000 500.000 𝑁 25 25 𝑛1 20 20 𝑛2 5 5 𝑛3 10 10 𝜇𝑀𝐴 0.015 0.015 𝜇𝑀𝐼 0.2 0.2 𝜇𝑇4 0.33 0.33 𝜇𝑇8 0.33 0.33 𝜆𝑧 0.010532 0.010532 𝜆𝑥 0.005266 0.0022854 4. titik tetap dari model interaksi antara infeksi mikobakterium tuberkulosis dengan sel imun ditunjukkan dalam model yang berbentuk sistem persamaan diferensial berikut: 𝑑𝐵𝐼 𝑑𝑡 = 𝛼𝐼𝐵𝐼 (1 − 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 ) +𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 (43) alfi nur rohmatin, usman pagalay 78 volume 3 no. 2 mei 2014 −𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 𝑛1𝑘3𝐵𝐼 +𝑛2𝑘4𝐵𝐴 𝑑𝐵𝐴 𝑑𝑡 = 𝛼𝐴𝐵𝐴 − 𝑛2𝑘4𝐵𝐴 + 𝑛1𝑘3𝐵𝐼 −𝑛2𝜇𝑀𝐴𝐵𝐴 𝑑𝐵𝐸 𝑑𝑡 = 𝛼𝐸𝐵𝐸 − 𝑘1𝑛3𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 +𝑘2𝑁𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 −𝑘5𝑀𝐴𝐵𝐸 + 𝑛2𝜇𝑀𝐴𝐵𝐴 𝑑𝑀𝐼 𝑑𝑡 = 𝑘1𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐1 − 𝑘2𝑀𝐼 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 −𝑘3𝑀𝐼 + 𝑘4𝑀𝐴 − 𝜇𝑀𝐼𝑀𝐴 𝑑𝑀𝐴 𝑑𝑡 = −𝑘4𝑀𝐴 + 𝑘3𝑀𝐼 − 𝜇𝑀𝐴𝑀𝐴 +𝑘6𝑀𝑅 𝐵𝐸 𝐵𝐸 + 𝑐5 𝑑𝑇4 𝑑𝑡 = 𝜆𝑧𝑀𝐴 + 𝑘13𝑇4 − 𝜇𝑇4𝑇4 𝑑𝑇8 𝑑𝑡 = 𝜆𝑥(𝑀𝐴 + 𝑀𝐼) + 𝑘14𝑇8 − 𝜇𝑇8𝑇8 terdapat 2 macam titik tetap, yaitu titik tetap bebas penyakit dan titik tetap dengan terinfeksi penyakit. 5. titik tetap pertama (titik tetap bebas penyakit) pada kasus ini, merupakan kasus titik tetap bebas penyakit yang menyatakan bahwa dalam keadaan seimbang pada saat belum ada infeksi. dengan kata lain tidak ada bakteri yang disajikan. akibatnya tidak ada bakteri intraseluler, bakteri di dalam makrofag teraktivasi, bakteri ekstraseluler, makrofag terinfeksi, maupun makrofag teraktivasi. secara analitik untuk mencari titik tetap yang pertama, yaitu karena 𝐵𝐼 ∗ = 0, 𝐵𝐴 ∗ = 0, 𝐵𝐸 ∗ = 0, 𝑀𝐼 ∗ = 0, 𝑀𝐴 ∗ = 0 maka 𝜆𝑧𝑀𝐴 + 𝑘13𝑇4 − 𝜇𝑇4𝑇4 = 0 𝜆𝑧(0) + 𝑘13𝑇4 − 𝜇𝑇4𝑇4 = 0 𝑘13𝑇4 − 𝜇𝑇4𝑇4 = 0 𝑇4(𝑘13 − 𝜇𝑇4) = 0 𝑇4 ∗ = 0 𝜆𝑥(𝑀𝐴 + 𝑀𝐼) + 𝑘14𝑇8 − 𝜇𝑇8𝑇8 = 0 𝜆𝑥(0 + 0) + 𝑘14𝑇8 − 𝜇𝑇8𝑇8 = 0 𝑘14𝑇8 − 𝜇𝑇8𝑇8 = 0 𝑇8(𝑘14 − 𝜇𝑇8) = 0 𝑇8 ∗ = 0 pada titik tetap bebas penyakit (titik tetap pertama), populasi dari semua spesies yang tercakup pada interaksi sistem imun diperoleh titik tetap pertama dari sistem persamaan terhadap usia muda tersebut menjadi: 𝐸1(𝑚𝑢𝑑𝑎) = (𝐵𝐼 ∗ , 𝐵𝐴 ∗ , 𝐵𝐸 ∗ , 𝑀𝐼 ∗ , 𝑀𝐴 ∗ , 𝑇4 ∗ , 𝑇8 ∗ ) = (0,0,0,0,0, 0,0) dan titik tetap pertama dari sistem persamaan terhadap usia tua tersebut menjadi: 𝐸1(𝑡𝑢𝑎) = (𝐵𝐼 ∗ , 𝐵𝐴 ∗ , 𝐵𝐸 ∗ , 𝑀𝐼 ∗ , 𝑀𝐴 ∗ , 𝑇4 ∗ , 𝑇8 ∗ ) = (0,0,0,0,0, 0,0) 6. titik tetap kedua (titik tetap dengan terinfeksi penyakit) pada titik tetap kedua, makrofag menjadi terinfeksi secara kronik, dan makrofag resting berubah menjadi teraktivasi. titik tetap ini mewakili dua kemungkinan dari penyakit tersebut, yakni penyakit laten dan penyakit primer. terjadinya infeksi secara laten, bergantung pada parameter-parameternya dan ketika parameter-parameternya bervariasi maka penyakit primerpun terjadi. makrofag yang terinfeksi secara kronik, meledak melepaskan bakteri intraseluler ke lingkungan ekstraseluler, sehingga terjadi pengerahan sel t cd4+ dan sel t cd8+ ke tempat yang terjadi infeksi. dengan menggunakan maple maka akan diperoleh nilai titik tetap kedua dari sistem persamaan terhadap usia muda yaitu: 𝐸2(𝑚𝑢𝑑𝑎) = 𝐵𝐼 ∗ = 2,914438433 × 107; 𝐵𝐴 ∗ = 8,986249133 × 106; 𝐵𝐸 ∗ = 1,327579621 × 105; 𝑀𝐼 ∗ = 3,250903282 × 105; 𝑀𝐴 ∗ = 97414,38980; 𝑇4 ∗ = 6173,094785; 𝑇8 ∗ = 7094,285585 sedangkan diperoleh nilai titik tetap kedua dari sistem persamaan terhadap usia tua, yaitu: 𝐸2(𝑡𝑢𝑎) = 𝐵𝐼 ∗ = 9,246406350 × 107; 𝐵𝐴 ∗ = 1,488623586 × 107; 𝐵𝐸 ∗ = 2,591041477 × 105; 𝑀𝐼 ∗ = 8,567107395 × 105; 𝑀𝐴 ∗ = 1,117576282 × 105; 𝑇4 ∗ = 6463,298775; 𝑇8 ∗ = 7007,115609 7. kestabilan titik tetap untuk melihat kestabilan dari sistem (43) dapat dilihat dari akar-akar persamaan karakteristik (nilai eigen 𝜆 matriks jacobian). akan ditinjau dua kasus yaitu kestabilan pada titik tetap bebas penyakit dan kestabilan pada titik tetap dengan terinfeksi penyakit. matrik jacobian analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 79 untuk 𝐵𝐼, 𝐵𝐴, 𝐵𝐸, 𝑀𝐼, 𝑀𝐴, 𝑇4, dan 𝑇8 yaitu sebagai berikut: 1) matriks jacobian pada baris pertama 𝜕𝑓1 𝜕𝐵𝐼 = 𝛼𝐼 (1 − 𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 ) + 𝛼𝐼𝐵𝐼 (− 2𝐵𝐼 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 + 2𝐵𝐼 3 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 ) − 2𝑘2𝑁𝑀𝐼𝐵𝐼 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 + 2𝑘2𝑁𝑀𝐼𝐵𝐼 3 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 − 𝑛𝐼𝑘3 = 𝛼1 𝜕𝑓1 𝜕𝐵𝐴 = 𝑛2𝑘4 = 𝛼2; 𝜕𝑓1 𝜕𝐵𝐸 = 𝑘1𝑛3𝑀𝑅 𝐵𝐸 + 𝑐1 − 𝑘1𝑛3𝑀𝑅𝐵𝐸 (𝐵𝐸 + 𝑐1) 2 = 𝛼3 𝜕𝑓1 𝜕𝑀𝐼 = 2𝑎𝐼𝐵𝐼 3 𝑁2𝑀𝐼 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 − 𝑘2𝑁𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 + 2𝑘2𝑁 3𝑀𝐼 2 𝐵𝐼 2 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 = 𝛼4 𝜕𝑓1 𝜕𝑀𝐴 = 0 𝜕𝑓1 𝜕𝑇4 = 0 𝜕𝑓1 𝜕𝑇8 = 0 1. matriks jacobian pada baris kedua 𝜕𝑓2 𝜕𝐵𝐼 = 𝑛1𝑘3 = 𝛼5 𝜕𝑓2 𝜕𝐵𝐴 = 𝛼𝐴 − 𝑛2𝑘4 − 𝑛2𝜇𝑀𝐴 = 𝛼6 𝜕𝑓2 𝜕𝐵𝐸 = 0 𝜕𝑓2 𝜕𝑀𝐼 = 0 𝜕𝑓2 𝜕𝑀𝐴 = 0 𝜕𝑓2 𝜕𝑇4 = 0 𝜕𝑓2 𝜕𝑇8 = 0 2. matriks jacobian pada baris ketiga 𝜕𝑓3 𝜕𝐵𝐼 = 2𝑘2𝑀𝐼𝐵𝐼 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 2𝑘2𝑁𝑀𝐼𝐵𝐼 3 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 = 𝛼7 𝜕𝑓3 𝜕𝐵𝐴 = 𝑛2ꩤ𝑀𝐴 = 𝛼8 𝜕𝑓3 𝜕𝐵𝐸 = 𝛼𝐸 − 𝑘1𝑛3𝑀𝑅 𝐵𝐸 + 𝑐1 + 𝑘1𝑛3𝑀𝑅𝐵𝐸 (𝐵𝐸 + 𝑐1) 2 − 𝑘5𝑀𝐴 = = 𝛼9 𝜕𝑓3 𝜕𝑀𝐼 = 𝑘2𝑁𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 − 2𝑘2𝑁 3𝑀𝐼 2 𝐵𝐼 2 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 = 𝛼10 𝜕𝑓3 𝜕𝑀𝐴 = −𝑘5𝐵𝐸 = 𝛼11 𝜕𝑓3 𝜕𝑇4 = 0 𝜕𝑓3 𝜕𝑇8 = 0 3. matriks jacobian pada baris keempat 𝜕𝑓4 𝜕𝐵𝐼 = − 2𝑘2𝑀𝐼𝐵𝐼 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 + 2𝑘2𝑀𝐼𝐵𝐼 3 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 = 𝛼12 𝜕𝑓4 𝜕𝐵𝐴 = 0 𝜕𝑓4 𝜕𝐵𝐸 = 𝑘1𝑀𝑅 𝐵𝐸 + 𝑐1 − 𝑘1𝑀𝑅𝐵𝐸 (𝐵𝐸 + 𝑐1) 2 = 𝛼13 𝜕𝑓4 𝜕𝑀𝐼 = − 𝑘2𝐵𝐼 2 𝐵𝐼 2 + (𝑁𝑀𝐼) 2 + 2𝑘2𝑀𝐼 2 𝐵𝐼 2 𝑁2 (𝐵𝐼 2 + 𝑁2𝑀𝐼 2) 2 − 𝑘3 = 𝛼14 𝜕𝑓4 𝜕𝑀𝐴 = 𝑘4 − 𝜇𝑀𝐼 = 𝛼15 𝜕𝑓4 𝜕𝑇4 = 0 𝜕𝑓4 𝜕 8 = 0 4. matriks jacobian pada baris kelima 𝜕𝑓5 𝜕𝐵𝐼 = 0 𝜕𝑓5 𝜕𝐵𝐴 = 0 𝜕𝑓5 𝜕𝐵𝐸 = 𝑘6𝑀𝑅 𝐵𝐸 + 𝑐5 − 𝑘6𝑀𝑅𝐵𝐸 (𝐵𝐸 + 𝑐5) 2 = 𝛼16 𝜕𝑓5 𝜕𝑀𝐼 = 𝑘3 = 𝛼17 𝜕𝑓5 𝜕𝑀𝐴 = −𝑘4 − 𝜇𝑀𝐴 = 𝛼18 𝜕𝑓5 𝜕𝑇4 = 0 𝜕𝑓5 𝜕𝑇8 = 0 5. matriks jacobian pada baris keenam                                                                                                                                            8 7 8 6 5 8 4 4 7 4 6 4 5 4 4 7 6 5 4 7 6 5 4 7 6 5 4 7 6 5 4 7 6 5 4 8 3 4 333333 8 2 4 222222 8 1 4 111111 t f t f bi f t f t f t f t f t f m f m f m f m f m f m f m f m f b f b f b f b f b f b f b f b f b f b f b f b f t f t f m f m f b f b f b f t f t f m f m f b f b f b f t f t f m f m f b f b f b f j a a a a i i i i e e e e e e e e i i i i aieei aieei aieei alfi nur rohmatin, usman pagalay 80 volume 3 no. 2 mei 2014 𝜕𝑓6 𝜕𝐵𝐼 = 0 𝜕𝑓6 𝜕𝐵𝐴 = 0 𝜕𝑓6 𝜕𝐵𝐸 = 0 𝜕𝑓6 𝜕𝑀𝐼 = 0 𝜕𝑓6 𝜕𝑀𝐴 = 𝜆𝑧 = 𝛼19 𝜕𝑓6 𝜕𝑇4 = 𝑘13 − 𝜇𝑇4 = 𝛼20 𝜕𝑓6 𝜕𝑇8 = 0 6. matriks jacobian pada baris ketujuh 𝜕𝑓6 𝜕𝐵𝐼 = 0 𝜕𝑓6 𝜕𝐵𝐴 = 0 𝜕𝑓6 𝜕𝐵𝐸 = 0 𝜕𝑓6 𝜕𝑀𝐼 = 𝜆𝑥 = 𝛼21 𝜕𝑓6 𝜕𝑀𝐴 = 𝜆𝑥 = 𝛼22 𝜕𝑓6 𝜕𝑇4 = 0 𝜕𝑓6 𝜕𝑇8 = 𝑘14 − 𝜇𝑇8 = 𝛼23 sehingga matriks jacobian dari persmaan (3.8) dapat ditulis: 𝐽 = [ 𝛼1 𝛼2 𝛼3 𝛼4 0 0 0 𝛼5 𝛼6 0 0 0 0 0 𝛼7 𝛼8 𝛼9 𝛼10 𝛼11 0 0 𝛼12 0 𝛼13 𝛼14 𝛼15 0 0 0 0 𝛼16 𝛼17 𝛼18 0 0 0 0 0 0 𝛼19 𝛼20 0 0 0 0 𝛼21 𝛼22 0 𝛼23] 8. kestabilan pada titik tetap bebas penyakit matriks jacobian dari titik tetap pertama pada usia muda 𝐸1(𝑚𝑢𝑑𝑎) = (𝐵𝐼 ∗ , 𝐵𝐴 ∗ , 𝐵𝐸 ∗ , 𝑀𝐼 ∗ , 𝑀𝐴 ∗ , 𝑇4 ∗ , 𝑇8 ∗ ) = (0,0,0,0,0, 0,0) maka diperoleh matriks jacobian dari titik tetap 𝐸1(𝑚𝑢𝑑𝑎) adalah: 𝐽1(𝑚𝑢𝑑𝑎) = [ 𝛼1 𝛼2 𝛼3 0 0 0 0 𝛼5 𝛼6 0 0 0 0 0 0 𝛼8 𝛼9 0 0 0 0 0 0 𝛼13 𝛼14 𝛼15 0 0 0 0 𝛼16 𝛼17 𝛼18 0 0 0 0 0 0 𝛼19 𝛼20 0 0 0 0 𝛼21 𝛼22 0 𝛼23] jika nilai parameter pada tabel (2) di substitusikan pada matriks jacobian di atas diperoleh: 𝐽1(𝑚𝑢𝑑𝑎) = [ 0,03 1,44 2,00 0 0 0 0 0,47 −1,52 0 0 0 0 0 0 0,07 −2,00 0 0 0 0 0 0 −0,20 0,02 0,09 0 0 0 0 0,38 0,02 −0,30 0 0 0 0 0 0 0,01 −0,17 0 0 0 0 0,005 0,005 0 −0,31] maka perhitungan nilai eigen untuk titik tetap pertama pada usia muda adalah sebagai berikut: det(𝜆𝐼 − 𝐽1(𝑚𝑢𝑑𝑎)) = 0, det [ 𝜆 − 0,03 1,4438 2,00 0 0 0 0 0,47 𝜆 + 1,52 0 0 0 0 0 0 0,07 𝜆 + 2,00 0 0 0 0 0 0 −0,20 𝜆 − 0,02 0,09 0 0 0 0 0,38 0,02 𝜆 + 0,30 0 0 0 0 0 0 0,01 𝜆 + 0,17 0 0 0 0 0,005 0,005 0 𝜆 + 0,31] = 0 untuk mencari determinan dari matriks tersebut, penulis menggunakan bantuan progam maple, maka diperoleh nilai eigen yaitu sebagai berikut: (𝜆 + 2,115885255)(𝜆 + 1,744601249) (𝜆 − 0,3733865037)(𝜆 + 0,3099935398) (𝜆 − 0,02964853979)(𝜆 − 0,1662) (𝜆 + 0,31362) = 0 sehingga diperoleh nilai eigen sebagai berikut: 𝜆1 = −2,115885255, 𝜆2 = −1,744601249, 𝜆3 = 0,3733865037, 𝜆4 = −0,3099935398, 𝜆5 = 0,02964853979, 𝜆6 = 0,1662, 𝜆7 = 0,31362 karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu pada 𝜆3, 𝜆5, 𝜆6 dan 𝜆7 maka dapat dikatakan bahwa titik tetap yang pertama terhadap usia muda tidak stabil. matriks jacobian dari titik tetap pertama pada usia tua. jika nilai parameter pada tabel (2) di substitusikan pada matriks jacobian di atas diperoleh: 𝐽1(𝑡𝑢𝑎) = [ −0,008 3,08 2,00 0 0 0 0 0,509 −3,16 0 0 0 0 0 0 0,07 −2,00 0 0 0 0 0 0 0,20 0,02 0,41 0 0 0 0 0,6769 0,0254 −0,6320 0 0 0 0 0 0 0,0105 −0,1821 0 0 0 0 0,002285 0,0023 0 −0,3159] maka perhitungan nilai eigen untuk titik tetap pertama pada usia tua adalah sebagai berikut: det(𝜆𝐼 − 𝐽1(𝑡𝑢𝑎)) = 0, analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 81 det [ 𝜆 + 0,008 3,08 2,00 0 0 0 0 0,509 𝜆 + 3,16 0 0 0 0 0 0 0,07 𝜆 + 2,00 0 0 0 0 0 0 0,20000 𝜆 − 0,02 0,42 0 0 0 0 0,6769 0,0254 𝜆 + 0,632 0 0 0 0 0 0 0,0105 𝜆 + 0,18 0 0 0 0 0,0023 0,0023 0 𝜆 + 0,31] = 0 untuk mencari determinan dari matriks tersebut, penulis menggunakan bantuan progam maple, maka diperoleh nilai eigen yaitu sebagai berikut: (𝜆 + 3,609497480)(𝜆 + 1,980411714) (𝜆 − 0,4207592944)(𝜆 + 0,6478293115) (𝜆 − 0,0419931150)(𝜆 + 0,41707) (𝜆 − 0,63207) = 0 sehingga 𝜆1 = −3,609497480, 𝜆2 = −1,980411714, 𝜆3 = 0,4207592944, 𝜆4 = −0,6478293115, 𝜆5 = 0,0419931150, 𝜆6 = 0,41707, 𝜆7 = −0,63207 karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu pada 𝜆3, 𝜆5, dan 𝜆6 maka dapat dikatakan bahwa titik tetap yang pertama terhadap usia tua tidak stabil. 9. kestabilan pada titik tetap dengan terinfeksi penyakit matriks jacobian dari titik tetap kedua pada usia muda 𝐸2(𝑚𝑢𝑑𝑎) = 𝐵𝐼 ∗ = 2,914438433 × 107; 𝐵𝐴 ∗ = 8,986249133 × 106; 𝐵𝐸 ∗ = 1,327579621 × 105; 𝑀𝐼 ∗ = 3,250903282 × 105; 𝑀𝐴 ∗ = 97414,38980; 𝑇4 ∗ = 6173,094785; 𝑇8 ∗ = 7094,285585 maka diperoleh matriks jacobian dari titik tetap 𝐸2(𝑚𝑢𝑑𝑎) adalah: 𝐽2(𝑚𝑢𝑑𝑎) = [ 𝛼1 𝛼2 𝛼3 𝛼4 0 0 0 𝛼5 𝛼6 0 0 0 0 0 𝛼7 𝛼8 𝛼9 𝛼10 𝛼11 0 0 𝛼12 0 𝛼13 𝛼14 𝛼15 0 0 0 0 𝛼16 𝛼17 𝛼18 0 0 0 0 0 0 𝛼19 𝛼20 0 0 0 0 𝛼21 𝛼22 0 𝛼23] jika nilai parameter pada tabel (2) di substitusikan pada matriks jacobian di atas diperoleh: 𝐽2(𝑚𝑢𝑑𝑎) = [ −1,36 1,44 1,56 6996754,7 0 0 0 0,47 −1,52 0 0 0 0 0 0,003 0,07 −9,48 1,84 −10,8 0 0 −0,0001 0 0,15 −0,88 0,09 0 0 0 0 0,07 0,02 −0,30 0 0 0 0 0 0 0,01 −0,17 0 0 0 0 0,005 0,005 0 −0,31] maka perhitungan nilai eigen untuk titik tetap kedua pada usia muda adalah sebagai berikut: det(𝜆𝐼 − 𝐽2(𝑚𝑢𝑑𝑎)) = 0, det [ 𝜆 + 1,36 1,44 1,56 6996754,7 0 0 0 0,47 𝜆 + 1,52 0 0 0 0 0 0,003 0,07 𝜆 + 9,48 1,84 −10,8 0 0 −0,0001 0 0,15 𝜆 + 0,88 0,09 0 0 0 0 0,07 0,02 𝜆 + 0,30 0 0 0 0 0 0 0,01 𝜆 + 0,17 0 0 0 0 0,005 0,005 0 𝜆 + 0,31] = 0 untuk mencari determinan dari matriks tersebut, penulis menggunakan bantuan progam maple, maka diperoleh nilai eigen yaitu sebagai berikut: (𝜆 + 15,21869890)(𝜆 + 0,2097069375) (𝜆 − 1,387438442) (𝜆2 − 1,288908970𝜆 + 918,2471335) = 0 sehingga 𝜆1 = −15,21869890, 𝜆2 = −0,2097069375, dan 𝜆2 − 1,288908970𝜆 + 918,2471335 karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu salah satunya adalah 𝜆3 maka dapat dikatakan bahwa titik tetap yang kedua terhadap usia muda tidak stabil. 2) matriks jacobian dari titik tetap kedua pada usia tua 𝐸2(𝑡𝑢𝑎) = 𝐵𝐼 ∗ = 9,246406350 × 107; 𝐵𝐴 ∗ = 1,488623586 × 107; 𝐵𝐸 ∗ = 2,591041477 × 105; 𝑀𝐼 ∗ = 8,567107395 × 105; 𝑀𝐴 ∗ = 1,117576282 × 105; 𝑇4 ∗ = 6463,298775; 𝑇8 ∗ = 7007,115609 maka diperoleh matriks jacobian dari titik tetap 𝐸2(𝑡𝑢𝑎) adalah: 𝐽2(𝑚𝑢𝑑𝑎) = [ 𝛼1 𝛼2 𝛼3 𝛼4 0 0 0 𝛼5 𝛼6 0 0 0 0 0 𝛼7 𝛼8 𝛼9 𝛼10 𝛼11 0 0 𝛼12 0 𝛼13 𝛼14 𝛼15 0 0 0 0 𝛼16 𝛼17 𝛼18 0 0 0 0 0 0 𝛼19 𝛼20 0 0 0 0 𝛼21 𝛼22 0 𝛼23] jika nilai parameter pada tabel (2) di substitusikan pada matriks jacobian di atas diperoeh: 𝐽2(𝑡𝑢𝑎) = [ −1,43 3,08 1,26 19292037,7 0 0 0 0,508 −3,16 0 0 0 0 0 0,001 0,07 −10,3 1,89 −21,1 0 0 −0,007 0 0,13 −0,09 0,42 0 0 0 0 0,05 0,02 −0,63 0 0 0 0 0 0 0,01 −0,18 0 0 0 0 0,002 0,002 0 −0,31] maka perhitungan nilai eigen untuk titik tetap kedua pada usia tua adalah sebagai berikut: det(𝜆𝐼 − 𝐽2(𝑡𝑢𝑎)) = 0, det [ 𝜆 + 1,43 3,08 1,26 19292037,7 0 0 0 0,508 𝜆 + 3,16 0 0 0 0 0 0,002 0,07 𝜆 + 10,3 1,89 −21,1 0 0 −0,007 0 0,13 𝜆 + 0,09 0,42 0 0 0 0 0,05 0,02 𝜆 + 0,63 0 0 0 0 0 0 0,01 𝜆 + 0,18 0 0 0 0 0,002 0,002 0 𝜆 + 0,31] = 0 alfi nur rohmatin, usman pagalay 82 volume 3 no. 2 mei 2014 untuk mencari determinan dari matriks tersebut, penulis menggunakan bantuan progam maple, maka diperoleh nilai eigen yaitu sebagai berikut: maka akan diperoleh: (𝜆 + 17,03285449)(𝜆 + 0,1137376750) (𝜆 − 0,7987421765) (𝜆2 − 0,6790282736𝜆 + 1486,999279) = 0 sehingga 𝜆1 = −17,03285449, 𝜆2 = −0,1137376750, 𝜆3 = 0,7987421765, dan 𝜆2 − 0,6790282736𝜆 + 1486,999279 karena terdapat nilai eigen yang akarakarnya bernilai positif, yaitu salah satunya adalah 𝜆3 maka dapat dikatakan bahwa titik tetap yang kedua terhadap usia tua tidak stabil. 10. simulasi numerik model matematika pada bagian ini akan dibahas mengenai perilaku dinamik pada sel t cd4+ dan sel t cd8+ dengan menaikkan dan menurunkan nilai parameter 𝑘13 dan 𝑘14 . selanjutnya hasil ini akan dibandingkan dengan grafik pada saat belum mengalami perubahan parameter. penelitian ini dilakukan selama 60 hari dengan menggunakan bantuan program matlab. (a) (b) gambar 2. grafik simulasi populasi sel t cd4+ gambar 2 menujukkkan perubahan populasi sel t cd4+ dengan nilai parameter k13 yang berbeda. pada grafik (a) menunjukkan grafik sel t cd4+ pada usia muda, sedangkan pada grafik (b) menunjukkan grafik sel t cd4+ pada usia tua. untuk grafik (a), ketika laju poliferasi sel t cd4+ meningkat (k13 = 0,1638 menjadi k13 = 0,20) secara otomatis populasi sel t cd4+ juga meningkat mencapai 4.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. sedangkan ketika laju poliferasi sel t cd4+ menurun (k13 = 0,1638 menjadi k13 = 0,020) secara otomatis populasi sel t cd4+ juga menurun mencapai 2.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. sedangkan untuk grafik (b), ketika laju poliferasi sel t cd4+ meningkat (k13 = 0,14789 menjadi k13 = 0,20) secara otomatis populasi sel t cd4+ juga meningkat mencapai 7.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. sedangkan ketika laju poliferasi sel t cd4+ menurun (k13 = 0,14789 menjadi k13 = 0,020) secara otomatis populasi sel t cd4+ juga menurun mencapai 3.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. gambar 3 menujukkkan perubahan populasi sel t cd8+ dengan nilai parameter k14 yang berbeda. pada grafik (a) menunjukkan grafik sel t cd8+ pada usia muda, sedangkan pada grafik (b) menunjukkan grafik sel t cd8+ pada usia tua. (a) untuk grafik (a), ketika laju poliferasi sel t cd8+ meningkat (k14 = 0,01638 menjadi k14 = 0,020) secara otomatis populasi sel t cd8+ juga meningkat mencapai 3.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. sedangkan ketika laju poliferasi sel t cd8+ menurun (k14 = 0,01638 menjadi k13 = 0,0020) secara otomatis populasi sel t cd8+ juga menurun mencapai 2.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 simulasi sel t cd4+ muda waktu (hari) p e ru b a h a n s e l t c d 4 + m u d a ( s e l/ m il il it e r) k13=0.1638 k13=0.20 k13=0.020 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 9 10 x 10 4 simulasi sel t cd4+ tua waktu (hari) p er ub ah an s el t c d 4+ t ua ( se l/m ili lit er ) k13=0.14789 k13=0.20 k13=0.020 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 x 10 4 simulasi sel t cd8+ muda waktu (hari) p e ru b a h a n s e l t c d 8 + m u d a ( s e l/ m il il it e r) k14=0.01638 k14=0.020 k14=0.0020 analisis perilaku dinamik pada sel t cd4+ dan sel t cd8+ terhadap infeksi… cauchy-issn: 2086-0382 83 (b) gambar 3. grafik simulasi populasi sel t cd8+ untuk grafik (b), ketika laju poliferasi sel t cd8+ meningkat (k14 = 0,01638 menjadi k14 = 0,020) secara otomatis populasi sel t cd8+ juga meningkat mencapai 3.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. sedangkan ketika laju poliferasi sel t cd8+ menurun (k14 = 0,01638 menjadi k14 = 0,0020) secara otomatis populasi sel t cd8+ juga menurun mencapai 2.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. penutup berdasarkan hasil pembahasan, maka dapat diambil kesimpulan bahwa analisis kestabilannya yaitu untuk titik tetap yang pertama terhadap usia muda dikatakan tidak stabil, karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu pada 𝜆3, 𝜆5, 𝜆6 dan 𝜆7. sedangkan untuk titik tetap yang pertama terhadap usia tua dikatakan tidak stabil juga, karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu pada 𝜆3, 𝜆5, dan 𝜆6. kemudian, untuk titik tetap yang kedua terhadap usia muda dikatakan tidak stabil juga, karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu salah satunya adalah 𝜆3. dan untuk titik tetap yang kedua terhadap usia tua dikatakan tidak stabil juga, karena terdapat nilai eigen yang akar-akarnya bernilai positif, yaitu salah satunya adalah 𝜆3. sedangkan kesimpulan dari perubahan populasi sel t cd4+ dan sel t cd8+ pada usia tua akan lebih cepat terinfeksi mikobakterium tuberkulosis dari pada saat usianya masih muda. begitu juga dengan manusia saat usianya menginjak tua akan lebih rentan terhadap suatu penyakit, seperti halnya penyakit tuberkulosis. dari analisis perilaku dinamik pada sel t cd4+ didapatkan bahwa ketika laju poliferasi sel t cd4+ meningkat secara otomatis populasi sel t cd4+ juga meningkat mencapai 4.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. sedangkan ketika laju poliferasi sel t cd4+ menurun (secara otomatis populasi sel t cd4+ juga menurun mencapai 2.000 sel/mililiter. dan grafik sel t cd4+ stabil pada hari ke 30. sedangkan dari analisis perilaku dinamik pada sel t cd8+ didapatkan bahwa ketika laju poliferasi sel t cd8+ meningkat secara otomatis populasi sel t cd8+ juga meningkat mencapai 3.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. sedangkan ketika laju poliferasi sel t cd8+ menurun secara otomatis populasi sel t cd8+ juga menurun mencapai 2.000 sel/mililiter. dan grafik sel t cd8+ stabil pada hari ke 18. pembaca dapat menganalisis perilaku dinamik lain yang kestabilan pada titik tetap adalah stabil untuk menyelesaikan sistem persamaan diferensial biasa non linier dengan penyakit lain, misalnya penyakit kanker, tumor, dbd, dan lain sebagainya. bibliography [1] s. wiggins, s. wiggins, and m. golubitsky, introduction to applied nonlinear dynamical systems and chaos, vol. 2. springer, 1990. [2] n. finizio and g. ladas, “persamaan diferensial biasa dengan penerapan modern,” jakarta: erlangga, 1988. [3] a. subagyo, t. y. aditama, d. k. sutoyo, and l. g. partakusuma, “pemeriksaan interferon-gamma dalam darah untuk deteksi infeksi tuberkulosis,” j. tuberkulosis indones., vol. 3, no. 2, pp. 6– 19, 2006. [4] k. g. baratawidjaja and i. rengganis, “imunologi dasar,” edisi, vol. 7, pp. 76–77, 2006. [5] l. prihutami and s. sutimin, “analisis kestabilan model penyebaran penyakit tuberculosis.” diponegoro university, 2009. 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 x 10 4 simulasi sel t cd8+ tua waktu (hari) p e ru b a h a n s e l t c d 8 + t u a ( s e l/ m il il it e r) k14=0.01413 k14=0.020 k14=0.0020 pendahuluan teori dasar 1. titik tetap 2. kestabilan titik tetap 3. sel t ,𝐂𝐃𝟒-+. dan sel t ,𝐂𝐃𝟖-+. 4. mikobakterium tuberkulosis 5. pembentukan model pembahasan 1. identifikasi model matematika 2. interpretasi model matematika 3. nilai parameter model 4. titik tetap dari model 5. titik tetap pertama (titik tetap bebas penyakit) 6. titik tetap kedua (titik tetap dengan terinfeksi penyakit) 7. kestabilan titik tetap 8. kestabilan pada titik tetap bebas penyakit 9. kestabilan pada titik tetap dengan terinfeksi penyakit 10. simulasi numerik model matematika penutup bibliography sifat aljabar banach komutatif dan elemen identitas pada kelas 𝑫(𝑲) malahayati program studi matematika fakultas sains dan teknologi uin sunan kalijaga yogyakarta e-mail: malahayati_01@yahoo.co.id abstrak himpunan semua fungsi baire kelas satu yang terbatas pada k ditulis 𝐵1(𝐾), dengan k sembarang ruang metrik separabel. salah satu kelas bagian terpenting dari 𝐵1(𝐾) adalah 𝐷(𝐾), yang menotasikan kelas semua fungsi pada 𝐾 yang merupakan selisih fungsi-fungsi semikontinu terbatas pada k. pada paper ini dibuktikan bahwa sifat aljabar banach komutatif dan elemen identitas berlaku di kelas 𝐷(𝐾). kata kunci: ruang metrik separabel, aljabar banach, baire kelas satu, fungsi semikontinu, kelas 𝑫(𝑲). abstract the first baire class of bounded functions on separable metric spaces k denoted by 𝐵1(𝐾). one of the most important subclass of 𝐵1(𝐾) is d(k), by d(k) is denoted the class of all functions on k which are differences of bounded semicontinuous functions. in this paper we proved that d(k) is abelian banach algebra and identity element. keywords: separable metric spaces, banach algebra, the first baire class, semicontinuous functions, the class d(k) pendahuluan himpunan semua fungsi baire kelas satu yang terbatas pada k ditulis 𝐵1(𝐾), dengan k sembarang ruang metrik separabel. salah satu kelas bagian terpenting dari 𝐵1(𝐾) adalah 𝐷(𝐾), yang menotasikan kelas semua fungsi pada 𝐾 yang merupakan selisih fungsi-fungsi semikontinu terbatas pada k. kelas 𝐷(𝐾) pertama kali dikenalkan oleh a.s kechris dan louveau pada tahun 1990. banyak peneliti yang telah membahas tentang kelas fungsi 𝐷(𝐾). sejalan dengan kemajuan sains dan teknologi, kajian tentang 𝐷(𝐾) juga mengalami perkembangan sehingga muncul beberapa pengertian tentang 𝐷(𝐾) dan norma pada 𝐷(𝐾), seperti yang ditulis oleh haydon, odell, rosenthal (1991) dan rosenthal (1994) serta farmaki (1996). kelas fungsi 𝐷(𝐾) memiliki peranan penting dalam cabang matematika diantaranya analisis fungsional, khususnya dalam pengaplikasian teori ruang banach. oleh karena itu penulis tertarik untuk membuktikan sifat aljabar banach komutatif dengan elemen identitas pada kelas 𝐷(𝐾). teori dasar pada bagian ini akan diberikan beberapa pengertian dasar dan sifat yang merupakan konsep awal untuk dipahami agar mudah mengikuti pembahasan selanjutnya. pengertianpengertian dan sifat-sifat yang disajikan diadopsi dari beberapa literatur yang disebutkan pada daftar pustaka. definisi 2.1 (aljabar banach) aljabar adalah ruang linear 𝐴 yang di dalamnya didefinisikan operasi multiplikasi sehingga untuk setiap 𝑥, 𝑦, 𝑧 ∈ 𝐴 dan skalar 𝛼, berlaku 1) 𝑥𝑦 ∈ 𝐴 2) 𝑥(𝑦𝑧) = (𝑥𝑦)𝑧 3) 𝑥(𝑦 + 𝑧) = 𝑥𝑦 + 𝑥𝑧 4) (𝑥 + 𝑦)𝑧 = 𝑥𝑧+yz 5) 𝛼(𝑥𝑦) = (𝛼𝑥)𝑦 = 𝑥(𝛼𝑦). selanjutnya 𝐴 dikatakan komutatif (abelian) jika untuk setiap 𝑥, 𝑦 ∈ 𝐴 berlaku 𝑥𝑦 = 𝑦𝑥. aljabar 𝐴 dikatakan mempunyai elemen identitas jika terdapat dengan tunggal elemen 𝑒 ≠ 𝟎 sehingga 𝑒𝑥 = 𝑥𝑒 = 𝑥 untuk setiap 𝑥 ∈ 𝐴 . elemen 𝑒 ini disebut elemen identitas. aljabar bernorma 𝐴 ialah suatu aljabar yang dilengkapi dengan norma ‖. ‖ sehingga ‖𝑥𝑦‖ ≤ ‖𝑥‖‖𝑦‖ untuk setiap 𝑥, 𝑦 ∈ 𝐴. sedangkan aljabar bernorma yang lengkap disebut aljabar banach. selanjutnya diberikan definisi fungsi semikontinu, yang akan digunakan dalam mendefinisikan kelas fungsi 𝐷(𝐾). fungsi – fungsi yang dibicarakan bernilai real dan didefinisikan pada 𝐸, dengan e himpunan bagian dari ruang metrik x. sebelumnya disepakati terlebih dahulu bahwa setiap pengambilan infimum dan supremum dari suatu himpunan pada bagian ini, sifat aljabar banach komutatif dan elemen identitas pada kelas d(k) jurnal cauchy – issn: 2086-0382 27 himpunan yang dimaksud merupakan himpunan bagian dari �̅�, dengan �̅� = 𝑹 ∪ {−∞, ∞}. dalam mendefinisikan fungsi semikontinu diperlukan konsep limit atas dan limit bawah, oleh karena itu akan diberikan definisi limit atas dan limit bawah terlebih dahulu. definisi 2.2 (limit atas dan limit bawah) diberikan fungsi f yang di definisikan pada 𝐸 dan 𝑥0 ∈ 𝐸. 1) limit atas (upper limit) fungsi f ketika x mendekati 𝑥0 ditulis dengan lim 𝑥→𝑥0 𝑓(𝑥) dan didefinisikan lim 𝑥→𝑥0 𝑓(𝑥) = inf {𝑀𝜀 (𝑓, 𝑥0): 𝜀 > 0}, dengan 𝑀𝜀 (𝑓, 𝑥0) = sup{𝑓(𝑥): 𝑥 ∈ 𝑁𝜀 (𝑥0) ∩ 𝐸}. 2) limit bawah (lower limit) fungsi f ketika x mendekati 𝑥0 ditulis dengan lim 𝑥→𝑥0 𝑓(𝑥) dan didefinisikan lim 𝑥→𝑥0 𝑓(𝑥) = sup {𝑚𝜀 (𝑓, 𝑥0): 𝜀 > 0} , dengan 𝑚𝜀 (𝑓, 𝑥0) = inf {𝑓(𝑥): 𝑥 ∈ 𝑁𝜀 (𝑥0) ∩ 𝐸}. pada definisi 2 diatas, nilai limitnya selalu ada dan dapat bernilai berhingga, +∞, atau −∞. definisi 2.3 (fungsi semikontinu) diberikan fungsi f yang didefinisikan pada 𝐸 dan 𝑥0 ∈ 𝐸. 1) fungsi 𝑓 dikatakan semikontinu atas (upper semicontinuous) di 𝑥0 apabila 𝑓(𝑥0 ) = lim 𝑥→𝑥0 𝑓(𝑥). selanjutnya, fungsi 𝑓 dikatakan semikontinu atas pada e apabila fungsi f semikontinu atas disetiap 𝑥0 ∈ 𝐸. 2) fungsi 𝑓 dikatakan semikontinu bawah (lower semicontinuous) di 𝑥0 apabila 𝑓(𝑥0 ) = lim 𝑥→𝑥0 𝑓(𝑥). selanjutnya, fungsi 𝑓 dikatakan semikontinu bawah pada e apabila fungsi f semikontinu bawah disetiap 𝑥0 ∈ 𝐸. 3) fungsi yang semikontinu atas atau semikontinu bawah dinamakan fungsi semikontinu. pada pembuktian sifat-sifat kelas 𝐷(𝐾), pemakaian definisi kelas 𝐷(𝐾) secara langsung cukup menyulitkan. oleh karena itu, diperlukan suatu hasil yang lebih memudahkan dalam pembahasan yang dimaksud. fungsi–fungsi yang dibicarakan bernilai real dan didefinisikan pada 𝐾, dengan k sebarang ruang metrik separabel. selain itu, himpunan semua fungsi–fungsi kontinu pada k dinotasikan dengan 𝐶(𝐾). lemma 2.4 fungsi 𝑓 ∈ 𝐷(𝐾) jika dan hanya jika terdapat fungsi-fungsi semikontinu bawah terbatas u dan v pada k, sehingga 𝑓 = 𝑢 − 𝑣. bukti : (syarat cukup). diketahui fungsi-fungsi semikontinu bawah terbatas u dan v pada k, sehingga 𝑓 = 𝑢 − 𝑣. menurut definisi 𝐷(𝐾), jelas 𝑓 ∈ 𝐷(𝐾). (syarat perlu). diketahui 𝑓 ∈ 𝐷(𝐾), berarti terdapat fungsi–fungsi semikontinu terbatas u dan v pada k, sehingga 𝑓 = 𝑢 − 𝑣. dalam hal ini ada beberapa kemungkinan, yaitu: kemungkinan pertama: jika 𝑢 dan 𝑣 fungsi-fungsi semikontinu atas terbatas pada k, maka diperoleh 𝑓 = 𝑢 − 𝑣 = (−𝑣) − (−𝑢). karena 𝑢 dan 𝑣 fungsifungsi semikontinu atas, maka – 𝑢 dan – 𝑣 fungsifungsi semikontinu bawah. oleh karena itu, apabila 𝑢′ = −𝑣 dan 𝑣′ = −𝑢 maka diperoleh 𝑢′, 𝑣′ fungsi-fungsi semikontinu bawah terbatas pada k dan 𝑓 = 𝑢′ − 𝑣′ . kemungkinan kedua: jika 𝑢 fungsi semikontinu bawah terbatas pada k dan 𝑣 fungsi semikontinu atas terbatas pada k, maka 𝑓 = 𝑢 − 𝑣 = (𝑢 − 𝑣) − 0. karena 𝑣 fungsi semikontinu atas, maka – 𝑣 fungsi semikontinu bawah sehingga 𝑢 − 𝑣 fungsi semikontinu bawah. oleh karena itu, apabila 𝑢′ = 𝑢 − 𝑣 dan 𝑣′ = 0 maka diperoleh 𝑢′, 𝑣′ fungsifungsi semikontinu bawah terbatas pada k dan 𝑓 = 𝑢′ − 𝑣′ . kemungkinan ketiga : jika 𝑢 fungsi semikontinu atas terbatas pada k dan 𝑣 fungsi semikontinu bawah terbatas pada k, maka 𝑓 = 𝑢 − 𝑣 = 0 − (𝑣 − 𝑢). karena 𝑢 fungsi semikontinu atas, maka – 𝑢 fungsi semikontinu bawah sehingga 𝑣 − 𝑢 fungsi semikontinu bawah. oleh karena itu, jika 𝑢′ = 0 , dan 𝑣′ = 𝑣 − 𝑢 maka diperoleh 𝑢′, 𝑣′ fungsi-fungsi semikontinu bawah terbatas pada k dan 𝑓 = 𝑢′ − 𝑣′.∎ untuk selanjutnya, apabila 𝑓 sebarang fungsi yang didefinisikan pada k, notasi 𝑓 ≥ 0 dimaksudkan 𝑓(𝑥) ≥ 0 untuk semua 𝑥 ∈ 𝐾. lemma 2.5 fungsi 𝑓 ∈ 𝐷(𝐾) jika dan hanya jika terdapat fungsi–fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k, sehingga 𝑓 = 𝑢 − 𝑣. bukti : (syarat cukup). diketahui fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k sehingga 𝑓 = 𝑢 − 𝑣. oleh karena itu, menurut lemma 4 di atas, jelas 𝑓 ∈ 𝐷(𝐾). (syarat perlu). diketahui 𝑓 ∈ 𝐷(𝐾), maka menurut lemma 4 terdapat fungsi-fungsi semikontinu bawah terbatas 𝑔 dan ℎ pada k sehingga 𝑓 = 𝑔 − ℎ. karena 𝑔 fungsi semikontinu bawah terbatas pada k, maka terdapat barisan {𝜑𝑛 } di 𝐶(𝐾) sehingga 𝜑0 ≤ 𝜑1 ≤ 𝜑2 ≤ 𝜑3 ≤ ⋯ ≤ 𝜑𝑛 ≤ 𝜑𝑛+1 ≤ ⋯ dengan 𝜑0 = 𝟎 dan {𝜑𝑛} konvergen titik demi titik ke 𝑔. oleh karena itu, diperoleh malahayati 28 volume 3 no. 1 november 2013 𝑔(𝑥) = lim 𝑛→∞ 𝜑𝑛 (𝑥) = lim 𝑛→∞ ∑ (𝜑𝑗 − 𝜑𝑗−1) 𝑛 𝑗=1 (𝑥) = ∑ (𝜑𝑗 − 𝜑𝑗−1)(𝑥) ∞ 𝑗=1 , untuk setiap 𝑥 ∈ 𝐾. selanjutnya, karena ℎ juga fungsi semikontinu bawah terbatas pada k, maka terdapat barisan {𝜓𝑛} ⊆ 𝐶(𝐾) sehingga 𝜓0 ≤ 𝜓1 ≤ 𝜓2 ≤ 𝜓3 ≤ ⋯ dengan 𝜓0 = 𝟎 dan {𝜓𝑛 } konvergen titik demi titik ke ℎ. oleh karena itu, diperoleh ℎ(𝑥) = lim 𝑛→∞ 𝜓𝑛 (𝑥) = lim 𝑛→∞ ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) 𝑛 𝑗=1 = ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) ∞ 𝑗=1 , untuk setiap 𝑥 ∈ 𝐾. akibatnya, untuk sebarang 𝑥 ∈ 𝐾 diperoleh 𝑓(𝑥) = 𝑔(𝑥) − ℎ(𝑥) = ∑ (𝜑𝑗 − 𝜑𝑗−1)(𝑥) ∞ 𝑗=1 − ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) ∞ 𝑗=1 . selanjutnya, namakan 𝑢 = ∑ (𝜑𝑗 − 𝜑𝑗−1) ∞ 𝑗=1 dan 𝑣 = ∑ (𝜓𝑗 − 𝜓𝑗−1) ∞ 𝑗=1 . karena 𝜑𝑗 − 𝜑𝑗−1 ≥ 0 dan 𝜓𝑗 − 𝜓𝑗−1 ≥ 0 untuk setiap 𝑗 = 1,2, ⋯ , maka diperoleh 𝑢, 𝑣 ≥ 0. jadi terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k sehingga 𝑓 = 𝑢 − 𝑣. pembahasan pada bagian ini akan dibuktikan sifat aljabar banach komutatif dengan elemen identitas pada kelas 𝐷(𝐾). berdasarkan definisi 1.1 di atas, beberapa langkah harus dibuktikan terlebih dahulu. berikut ini akan ditunjukkan bahwa 𝐷(𝐾) merupakan ruang linear. lemma 3.1 diberikan ruang metrik separabel 𝐾, 𝐷(𝐾) merupakan ruang linear. bukti : diambil sembarang 𝑓, 𝑔 ∈ 𝐷(𝐾) dan skalar 𝛼, maka terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑠, 𝑡 ≥ 0 pada k, dengan sifat 𝑓 = 𝑢 − 𝑣 dan 𝑔 = 𝑠 − 𝑡, sehingga diperoleh 1) 𝑓 + 𝑔 = (𝑢 − 𝑣) + (𝑠 − 𝑡) = (𝑢 + 𝑠) − (𝑣 + 𝑡). karena 𝑢, 𝑠, 𝑡 dan 𝑣 fungsi –fungsi semikontinu bawah, maka 𝑢 + 𝑠 dan 𝑣 + 𝑡 juga fungsi – fungsi semikontinu bawah. oleh karena itu, apabila 𝑢′ = 𝑢 + 𝑠 dan 𝑣′ = 𝑣 + 𝑡 maka diperoleh fungsi-fungsi semikontinu bawah terbatas 𝑢′, 𝑣′ ≥ 0 pada k, sehingga 𝑓 + 𝑔 = 𝑢′ − 𝑣′ . akibatnya 𝑓 + 𝑔 ∈ 𝐷(𝐾). 2) 𝛼𝑓 = 𝛼(𝑢 − 𝑣) = 𝛼𝑢 − 𝛼𝑣. apabila 𝛼 ≥ 0, maka 𝛼𝑢, 𝛼𝑣 fungsi-fungsi semikontinu bawah. oleh karena itu, apabila 𝑢′ = 𝛼𝑢 dan 𝑣′ = 𝛼𝑣, maka diperoleh fungsi-fungsi semikontinu bawah terbatas 𝑢′, 𝑣′ ≥ 0 pada k, sehingga 𝛼𝑓 = 𝑢′ − 𝑣′. dilain pihak, apabila 𝛼 < 0, maka – 𝛼𝑢 dan −𝛼𝑣 fungsi – fungsi semikontinu bawah. karena itu, apabila 𝑢" = −𝛼𝑣 dan 𝑣 " = −𝛼𝑢, maka diperoleh fungsi – fungsi semikontinu bawah terbatas 𝑢", 𝑣" ≥ 0 pada k, sehingga 𝛼𝑓 = 𝑢" − 𝑣". akibatnya, 𝛼𝑓 ∈ 𝐷(𝐾). jadi, terbukti 𝐷(𝐾) ruang linear.∎ berikut diberikan definisi fungsi yang sangat berperan dalam pembahasan pada bagian ini. definisi 3.2 diberikan ruang metrik separabel k, didefinisikan fungsi ‖. ‖𝐷 : 𝐷(𝐾) → 𝑹, dengan ‖𝑓‖𝐷 = 𝑖𝑛𝑓{‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, 𝑑𝑒𝑛𝑔𝑎𝑛 𝑢, 𝑣 ≥ 0 𝑓𝑢𝑛𝑔𝑠𝑖 − 𝑓𝑢𝑛𝑔𝑠𝑖 𝑠𝑒𝑚𝑖𝑘𝑜𝑛𝑡𝑖𝑛𝑢 𝑏𝑎𝑤𝑎ℎ 𝑡𝑒𝑟𝑏𝑎𝑡𝑎𝑠 𝑝𝑎𝑑𝑎 𝐾}, untuk setiap 𝑓 ∈ 𝐷(𝐾). selanjutnya akan dibuktikan bahwa 𝐷(𝐾) merupakan ruang bernorma terhadap fungsi ‖. ‖𝐷 . terlebih dahulu dibuktikan beberapa lemma yang akan digunakan dalam pembuktian. lemma 3.3 jika 𝑓 ∈ 𝐷(𝐾) maka ‖𝑓‖∞ ≤ ‖𝑓‖𝐷 . bukti : diambil sembarang 𝑓 ∈ 𝐷(𝐾) dan fungsifungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k, dengan sifat 𝑓 = 𝑢 − 𝑣. menurut definisi norma supremum, diperoleh ‖𝑓‖∞ = sup 𝑥∈𝐾 |𝑓(𝑥)| = sup 𝑥∈𝐾 |𝑢(𝑥) − 𝑣(𝑥)| ≤ sup 𝑥∈𝐾 |𝑢(𝑥) + 𝑣(𝑥)| = ‖𝑢 + 𝑣‖∞. dengan kata lain, ‖𝑓‖∞ merupakan batas bawah dari himpunan {‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi fungsi semikontinu bawah terbatas pada 𝐾} oleh karena itu, diperoleh ‖𝑓‖∞ ≤ inf{‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣 dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbataspada 𝐾}. jadi, terbukti bahwa ‖𝑓‖∞ ≤ ‖𝑓‖𝐷. lemma 3.4 jika 𝑓 ∈ 𝐷(𝐾) dan 𝛼 ∈ 𝑹 maka berlaku {‖𝛼(𝑢 + 𝑣)‖∞: 𝛼𝑓 = 𝛼𝑢 − 𝛼𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾} = {‖ℎ + 𝑔‖∞ ∶ 𝛼𝑓 = ℎ − 𝑔, dengan ℎ, 𝑔 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾}. bukti : diambil sembarang 𝑓 ∈ 𝐷(𝐾) dan 𝛼 ∈ 𝑹. untuk kemudahan dalam pembuktian, namakan 𝐴 = {‖𝛼(𝑢 + 𝑣)‖∞: 𝛼𝑓 = 𝛼𝑢 − 𝛼𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾}, dan 𝐵 = {‖ℎ + 𝑔‖∞ ∶ 𝛼𝑓 = ℎ − 𝑔, dengan ℎ, 𝑔 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾}. sifat aljabar banach komutatif dan elemen identitas pada kelas d(k) jurnal cauchy – issn: 2086-0382 29 akan dibuktikan 𝐴 = 𝐵. diambil sembarang 𝑎 ∈ 𝐴, maka terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k, dan 𝛼𝑓 = 𝛼𝑢 − 𝛼𝑣, sehingga diperoleh 𝑎 = ‖𝛼(𝑢 + 𝑣)‖∞. dalam hal ini ada dua kemungkinan, yaitu 𝛼 ≥ 0 atau 𝛼 < 0. jika 𝛼 ≥ 0 maka 𝛼𝑢 , 𝛼𝑣 ≥ 0. berarti terdapat fungsi – fungsi semikontinu bawah terbatas ℎ = 𝛼𝑢 dan 𝑔 = 𝛼𝑣 pada k, sehingga 𝛼𝑓 = ℎ − 𝑔 dan diperoleh 𝑎 = ‖ℎ + 𝑔‖∞ , dengan kata lain 𝑎 ∈ 𝐵. jika 𝛼 < 0 maka dapat dipilih fungsi-fungsi semikontinu bawah terbatas ℎ = −𝛼𝑣 dan 𝑔 = −𝛼𝑢, sehingga 𝛼𝑓 = ℎ − 𝑔 dan diperoleh 𝑎 = ‖ℎ + 𝑔‖∞, dengan kata lain 𝑎 ∈ 𝐵. akibatnya, diperoleh 𝐴 ⊆ 𝐵. sebaliknya, diambil sembarang 𝑏 ∈ 𝐵, maka terdapat fungsi-fungsi semikontinu bawah terbatas ℎ, 𝑔 ≥ 0 pada k dengan 𝛼𝑓 = ℎ − 𝑔, sehingga diperoleh 𝑏 = ‖ℎ + 𝑔‖∞. apabila 𝛼 = 0 maka jelas terbukti 𝐵 ⊆ 𝐴. selanjutnya, apabila 𝛼 ≠ 0 maka dapat dipilih 𝑢 = ℎ 𝛼 dan 𝑣 = 𝑔 𝛼 . dalam hal ini ada dua kemungkinan, yaitu 𝛼 > 0 atau 𝛼 < 0. jika 𝛼 > 0 maka diperoleh fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k dan 𝛼𝑓 = 𝛼𝑢 − 𝛼𝑣, sehingga 𝑏 = ‖𝛼(𝑢 + 𝑣)‖∞. dengan kata lain, 𝑏 ∈ 𝐴. jika 𝛼 < 0 maka diperoleh 𝛼𝑓 = ℎ − 𝑔 = 𝛼𝑢 − 𝛼𝑣 = (−𝛼𝑣) − (−𝛼𝑢) = 𝛼(−𝑣) − 𝛼(−𝑢). oleh karena itu, apabila 𝑢′ = (−𝑣) dan 𝑣′ = (−𝑢), maka diperoleh fungsi – fungsi semikontinu bawah terbatas 𝑢′, 𝑣′ ≥ 0 pada k dan 𝛼𝑓 = 𝛼𝑢′ − 𝛼𝑣′, sehingga 𝑏 = ‖𝛼(𝑢′ + 𝑣′)‖∞. dengan kata lain, 𝑏 ∈ 𝐴. akibatnya diperoleh 𝐵 ⊆ 𝐴. jadi, terbukti 𝐴 = 𝐵. seperti yang telah disebutkan sebelumnya, dengan menggunakan lemma 3.3 dan lemma 3.4, dapat dibuktikan bahwa fungsi ‖. ‖𝐷 adalah norma pada 𝐷(𝐾). teorema 3.5 fungsi ‖. ‖𝐷 adalah norma pada kelas 𝐷(𝐾). bukti : (1). akan dibuktikan bahwa ‖𝑓‖𝐷 ≥ 0, untuk setiap 𝑓 ∈ 𝐷(𝐾), dan ‖𝑓‖𝐷 = 0 jika dan hanya jika 𝑓 = 𝟎. diambil sembarang 𝑓 ∈ 𝐷(𝐾). karena ‖𝑓‖𝐷 = inf{‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾}, maka diperoleh ‖𝑓‖𝐷 ≥ 0. selanjutnya, jika ‖𝑓‖𝐷 = 0 maka menurut lemma 3.3 diperoleh ‖𝑓‖∞ = 0. oleh karena itu, 𝑓(𝑥) = 0 untuk setiap 𝑥 ∈ 𝐾, dengan kata lain 𝑓 = 𝟎. sebaliknya, jika 𝑓 = 𝟎 maka terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢 = 𝟎 dan 𝑣 = 𝟎, sehingga 𝑓 = 𝑢 − 𝑣 = 𝟎, akibatnya diperoleh ‖𝑓‖𝐷 = inf{‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾} = 0 (2). akan dibuktikan ‖𝛼𝑓‖𝐷 = |𝛼|. ‖𝑓‖𝐷 , untuk setiap 𝑓 ∈ 𝐷(𝐾) dan skalar 𝛼. diambil sembarang 𝑓 ∈ 𝐷(𝐾) dan 𝛼 ∈ 𝑹. berdasarkan lemma 3.4, diperoleh |𝛼|. ‖𝑓‖𝐷 = |𝛼| inf{‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾} = inf{|𝛼|‖𝑢 + 𝑣‖∞ ∶ 𝑓 = 𝑢 − 𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾} = inf{‖𝛼(𝑢 + 𝑣)‖∞ ∶ 𝛼𝑓 = 𝛼𝑢 − 𝛼𝑣, dengan 𝑢, 𝑣 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾 } = inf{‖ℎ + 𝑔‖∞ ∶ 𝛼𝑓 = ℎ − 𝑔 dengan ℎ, 𝑔 ≥ 0 fungsi − fungsi semikontinu bawah terbatas pada 𝐾} = ‖𝛼𝑓‖𝐷. (3). akan dibuktikan ‖𝑓 + 𝑔‖𝐷 ≤ ‖𝑓‖𝐷 + ‖𝑔‖𝐷 , untuk setiap 𝑓, 𝑔 ∈ 𝐷(𝐾). diambil 𝑓, 𝑔 ∈ 𝐷(𝐾) dan 𝜀 > 0 sembarang, maka terdapat fungsi – fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑠, 𝑡 ≥ 0 dengan 𝑓 = 𝑢 − 𝑣 dan 𝑔 = 𝑠 − 𝑡, sehingga berlaku ‖𝑢 + 𝑣‖∞ < ‖𝑓‖𝐷 + 𝜀 2 dan ‖𝑠 + 𝑡‖∞ < ‖𝑔‖𝐷 + 𝜀 2 . oleh karena itu, diperoleh ‖𝑓‖𝐷 + ‖𝑔‖𝐷 + 𝜀 > ‖𝑢 + 𝑣‖∞ + ‖𝑠 + 𝑡‖∞ ≥ ‖(𝑢 + 𝑣) + (𝑠 + 𝑡)‖∞ = ‖(𝑢 + 𝑠) + (𝑣 + 𝑡)‖∞ ≥ ‖𝑓 + 𝑔‖𝐷 . karena berlaku untuk 𝜀 > 0 sembarang, maka diperoleh ‖𝑓 + 𝑔‖𝐷 ≤ ‖𝑓‖𝐷 + ‖𝑔‖𝐷 . berdasarkan (1), (2), dan (3), maka terbukti bahwa fungsi ‖. ‖𝐷 adalah norma pada 𝐷(𝐾). ∎ pengertian norma pada 𝐷(𝐾) dapat juga disajikan lain, yang tertuang dalam teorema berikut ini. teorema 3.6 fungsi 𝑓 ∈ 𝐷(𝐾) jika dan hanya jika terdapat barisan {𝑓𝑛} di 𝐶(𝐾) dan 𝑠𝑢𝑝 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| <𝑛 ∞ sehingga ∑ 𝑓𝑛𝑛 = 𝑓 titik demi titik. lebih lanjut, ‖𝑓‖𝐷 = 𝑖𝑛𝑓{‖∑ |𝑓𝑛| ∞ 𝑛=0 ‖∞ ∶ {𝑓𝑛}𝑛=0 ∞ ⊂ 𝐶(𝐾) 𝑑𝑎𝑛 malahayati 30 volume 3 no. 1 november 2013 𝑠𝑢𝑝 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=0 < ∞ sehingga ∑ 𝑓𝑛 ∞ 𝑛=0 = 𝑓 𝑡𝑖𝑡𝑖𝑘 𝑑𝑒𝑚𝑖 𝑡𝑖𝑡𝑖𝑘}. bukti : (syarat perlu). diketahui 𝑓 ∈ 𝐷(𝐾), maka terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada 𝐾, sehingga 𝑓 = 𝑢 − 𝑣. karena 𝑢 fungsi semikontinu bawah terbatas, maka terdapat barisan {𝜑𝑛 } ⊆ 𝐶(𝐾) sehingga 𝜑0 ≤ 𝜑1 ≤ 𝜑2 ≤ 𝜑3 ≤ ⋯ dengan 𝜑0 = 𝟎 dan {𝜑𝑛 } konvergen titik demi titik ke 𝑢. oleh karena itu, diperoleh 𝑢(𝑥) = lim 𝑛→∞ 𝜑𝑛 (𝑥) = lim 𝑛→∞ ∑ (𝜑𝑗 − 𝜑𝑗−1) 𝑛 𝑗=1 (𝑥) = ∑ (𝜑𝑗 − ∞ 𝑗=1 𝜑𝑗−1)(𝑥), untuk setiap 𝑥 ∈ 𝐾. selanjutnya, karena 𝑣 juga fungsi semikontinu bawah terbatas, maka terdapat barisan {𝜓𝑛} ⊆ 𝐶(𝐾) sehingga 𝜓0 ≤ 𝜓1 ≤ 𝜓2 ≤ ⋯ dengan 𝜓0 = 𝟎 dan {𝜓𝑛 } konvergen titik demi titik ke 𝑣. oleh karena itu, diperoleh 𝑣(𝑥) = lim 𝑛→∞ 𝜓𝑛 (𝑥) = lim 𝑛→∞ ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) 𝑛 𝑗=1 = ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) ∞ 𝑗=1 , untuk setiap 𝑥 ∈ 𝐾. sehingga untuk sebarang 𝑥 ∈ 𝐾 diperoleh 𝑓(𝑥) = 𝑢(𝑥) − 𝑣(𝑥) = ∑ (𝜑𝑗 − 𝜑𝑗−1)(𝑥) ∞ 𝑗=1 − ∑ (𝜓𝑗 − 𝜓𝑗−1)(𝑥) ∞ 𝑗=1 = ∑ (𝜑𝑗 − 𝜑𝑗−1 − 𝜓𝑗 + 𝜓𝑗−1)(𝑥) ∞ 𝑗=1 . selanjutnya, namakan 𝑓𝑗 = 𝜑𝑗 − 𝜑𝑗−1 − 𝜓𝑗 + 𝜓𝑗−1, untuk setiap 𝑗 ∈ 𝑵 dengan 𝑓0 = 𝟎, sehingga diperoleh barisan {𝑓𝑛 } ⊆ c(𝐾). oleh karena itu, 𝑓(𝑥) = ∑ 𝑓𝑛(𝑥) ∞ 𝑛=1 , untuk setiap 𝑥 ∈ 𝐾. dilain pihak, karena 𝑢 dan 𝑣 terbatas, maka terdapat 𝑀1, 𝑀2 > 0 sehingga |𝑢(𝑥)| ≤ 𝑀1 dan |𝑣(𝑥)| ≤ 𝑀2, untuk setiap 𝑥 ∈ 𝐾. akibatnya, untuk sembarang 𝑥 ∈ 𝐾 berlaku ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=1 = ∑ |(𝜑𝑛 − 𝜑𝑛−1 − 𝜓𝑛 + 𝜓𝑛−1)(𝑥)| ∞ 𝑛=1 ≤ 𝑀1+𝑀2 karena berlaku untuk sebarang 𝑥 ∈ 𝐾, maka diperoleh sup 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=1 < ∞. dengan kata lain, ada barisan {𝑓𝑛} ⊆ 𝐶(𝐾) dan sup 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| < ∞𝑛 sehingga ∑ 𝑓𝑛𝑛 = 𝑓 titik demi titik. oleh karena itu, diperoleh ‖𝑓‖𝐷 ≥ inf{‖∑ |𝑓𝑛| ∞ 𝑛=0 ‖∞ ∶ {𝑓𝑛}𝑛=0 ∞ ⊂ 𝐶(𝐾) dan sup 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=0 < ∞ sehingga ∑ 𝑓𝑛 ∞ 𝑛=0 = 𝑓 titik demi titik}. (syarat cukup). diketahui barisan {𝑓𝑛} ⊆ 𝐶(𝐾) dan sup 𝑥∈𝐾 ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=0 < ∞ sehingga ∑ 𝑓𝑛 ∞ 𝑛=0 = 𝑓 titik demi titik. untuk sembarang 𝑥 ∈ 𝐾, berlaku 𝑓(𝑥) = ∑ 𝑓𝑛(𝑥) ∞ 𝑛=1 = lim 𝑘→∞ ∑ 𝑓𝑛(𝑥) 𝑘 𝑛=1 = lim 𝑘→∞ ∑ (𝑓𝑛 + − 𝑓𝑛 − )(𝑥)𝑘𝑛=1 = lim 𝑘→∞ ∑ (𝑓𝑛 +(𝑥) − 𝑓𝑛 −(𝑥))𝑘𝑛=1 = lim 𝑘→∞ ∑ (𝑓𝑛 + )(𝑥) − lim 𝑘→∞ ∑ (𝑓𝑛 −)(𝑥)𝑘𝑛=1 𝑘 𝑛=1 = ∑ (𝑓𝑛 + )(𝑥) − ∑ (𝑓𝑛 −)(𝑥)∞𝑛=1 . ∞ 𝑛=1 selanjutnya, namakan 𝑢 = ∑ (𝑓𝑛 + )∞𝑛=1 dan 𝑣 = ∑ (𝑓𝑛 −)∞𝑛=1 . karena 𝑓𝑛 + , 𝑓𝑛 − ≥ 0 maka diperoleh 𝑢, 𝑣 ≥ 0. diperhatikan bahwa ∑ (𝑓𝑛) +(𝑥)𝑘𝑛=1 ≤ ∑ (𝑓𝑛) +(𝑥)𝑘+1𝑛=1 untuk setiap 𝑥 ∈ 𝐾, dan lim 𝑘→∞ ∑ (𝑓𝑛) +(𝑥) =𝑘𝑛=1 ∑ (𝑓𝑛) +(𝑥)∞𝑛=1 . akibatnya, u merupakan fungsi semikontinu bawah terbatas pada k. dengan cara yang sama, diperoleh v merupakan fungsi semikontinu bawah terbatas pada k. oleh karena itu, terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada 𝐾, sehingga 𝑓 = 𝑢 − 𝑣. dengan kata lain, 𝑓 ∈ 𝐷(𝐾). akibatnya, diperoleh ‖𝑓‖𝐷 ≤ inf{‖∑ |𝑓𝑛| ∞ 𝑛=0 ‖∞ ∶ {𝑓𝑛}𝑛=0 ∞ ⊂ 𝐶(𝐾) dan , sup x∈k ∑ |𝑓𝑛(𝑥)| ∞ 𝑛=0 < ∞ sehingga ∑ 𝑓𝑛 ∞ 𝑛=0 = 𝑓 titik demi titik}. menggunakan teorema 3.6, pengertian norma pada 𝐷(𝐾) dapat disajikan dalam bentuk lain, yang tertuang dalam teorema berikut. teorema 3.7 fungsi 𝑓 ∈ 𝐷(𝐾) jika dan hanya jika terdapat barisan {𝑓𝑛} di 𝐶(𝐾) dan c < ∞, sehingga {𝑓𝑛} konvergen titik demi titik ke 𝑓 dengan 𝑓0 = 𝟎 dan ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ≤ c ∞ 𝑛=0 untuk setiap 𝑥 ∈ 𝐾. lebih lanjut, ‖𝑓‖𝐷 = 𝑖𝑛𝑓{c ∶ {𝑓𝑛} ⊆ 𝐶(𝐾), 𝑓0 = 𝟎 𝑑𝑎𝑛 ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ≤ 𝐶, ∀ 𝑥 ∈ ∞ 𝑛=0 𝐾, 𝑠𝑒ℎ𝑖𝑛𝑔𝑔𝑎 {𝑓𝑛} 𝑘𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛 𝑡𝑖𝑡𝑖𝑘 𝑑𝑒𝑚𝑖 𝑡𝑖𝑡𝑖𝑘 𝑘𝑒 𝑓}. bukti : (syarat perlu). diketahui 𝑓 ∈ 𝐷(𝐾) maka menurut teorema 3.6, terdapat barisan {𝑔𝑛 } ⊆ 𝐶(𝐾), sehingga ∑ 𝑔𝑛𝑛 = 𝑓 titik demi titik dan sup 𝑥∈𝐾 ∑ |𝑔𝑛 (𝑥)| < ∞𝑛 . namakan c = sup 𝑥∈𝐾 ∑ |𝑔𝑛 (𝑥)|𝑛 , dan untuk setiap 𝑛 ∈ 𝑵 dibentuk 𝑓𝑛 = ∑ 𝑔𝑖 𝑛 𝑖=1 dengan 𝑓0 = 𝟎, maka diperoleh barisan {𝑓𝑛} di 𝐶(𝐾) dan {𝑓𝑛} konvergen titik demi titik ke 𝑓. untuk sembarang 𝑥 ∈ 𝐾 diperoleh ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ∞ 𝑛=0 = ∑ |∑ 𝑔𝑖 𝑛+1 𝑖=1 (𝑥) − ∑ 𝑔𝑖 (𝑥) 𝑛 𝑖=1 | ∞ 𝑛=0 = ∑ |𝑔𝑛+1(𝑥)| ∞ 𝑛=0 ≤ c. dengan kata lain, terdapat barisan {𝑓𝑛} ⊆ 𝐶(𝐾) dan c < ∞, sehingga {𝑓𝑛} konvergen titik demi titik ke 𝑓 dengan 𝑓0 = 𝟎 dan ∑ |𝑓𝑛+1(𝑥) − ∞ 𝑛=0 𝑓𝑛(𝑥)| ≤ c untuk setiap 𝑥 ∈ 𝐾. oleh karena itu, diperoleh ‖𝑓‖𝐷 ≥ inf{c ∶ {𝑓𝑛} ⊆ 𝐶(𝐾), 𝑓0 = 𝟎 dan ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ≤ 𝐶, ∞ 𝑛=0 ∀ 𝑥 ∈ 𝐾 sehingga {𝑓𝑛} konvergen titik demi titik ke 𝑓}. (syarat cukup). untuk setiap 𝑛 ∈ 𝑵, dibentuk 𝑔𝑛 = 𝑓𝑛+1 − 𝑓𝑛 dengan 𝑔0 = 𝟎 . akibatnya, sifat aljabar banach komutatif dan elemen identitas pada kelas d(k) jurnal cauchy – issn: 2086-0382 31 diperoleh barisan {𝑔𝑛 } ⊆ 𝐶(𝐾), dan ∑ 𝑔𝑛 ∞ 𝑛=0 = 𝑓 titik demi titik. selanjutnya, untuk sebarang 𝑥 ∈ 𝐾, diperoleh ∑ |𝑔𝑛(𝑥)| ∞ 𝑛=0 = ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ∞ 𝑛=0 ≤ c < ∞. karena berlaku untuk sembarang 𝑥 ∈ 𝐾, maka diperoleh sup 𝑥∈𝐾 ∑ |𝑔𝑛(𝑥)| < ∞𝑛 . dengan kata lain, terbukti 𝑓 ∈ 𝐷(𝐾). oleh karena itu, diperoleh ‖𝑓‖𝐷 ≤ inf{c ∶ {𝑓𝑛} ⊆ 𝐶(𝐾), 𝑓0 = 𝟎 dan ∑ |𝑓𝑛+1(𝑥) − 𝑓𝑛(𝑥)| ≤ 𝐶, ∞ 𝑛=0 ∀ 𝑥 ∈ 𝐾 sehingga {𝑓𝑛} konvergen titik demi titik ke 𝑓} menggunakan teorema 3.7, akan dibuktikan bahwa 𝐷(𝐾) merupakan ruang banach. teorema 3.8 diberikan ruang metrik separabel k, 𝐷(𝐾) merupakan ruang banach. bukti : berdasarkan teorema 3.5, (𝐷(𝐾), ‖. ‖𝐷) merupakan ruang bernorma, selanjutnya akan dibuktikan 𝐷(𝐾) lengkap. diambil sebarang barisan cauchy {𝑓𝑛} ⊆ 𝐷(𝐾). oleh karena itu, dapat diasumsikan ‖𝑓𝑛+1 − 𝑓𝑛‖𝐷 < 1 2𝑛 , untuk setiap 𝑛 ∈ 𝑵. karena 𝑓𝑛+1 − 𝑓𝑛 ∈ 𝐷(𝐾), maka untuk setiap 𝑛 ∈ 𝑵 terdapat barisan {𝜑𝑚 𝑛 }𝑚=1 ∞ di 𝐶(𝐾), sehingga {𝜑𝑚 𝑛 }𝑚=1 ∞ konvergen titik demi titik ke 𝑓𝑛+1 − 𝑓𝑛 dan memenuhi ∑ |𝜑𝑚+1 𝑛 (𝑥) − 𝜑𝑚 𝑛 (𝑥)| ≤ ∞𝑚=0 1 2 𝑛, untuk setiap 𝑥 ∈ 𝐾. karena {𝑓𝑛} barisan cauchy di 𝐷(𝐾), maka diperoleh ‖𝑓𝑚 − 𝑓𝑛‖∞ → 0, untuk 𝑛, 𝑚 → ∞. oleh karena itu, untuk setiap 𝑥 ∈ 𝐾 diperoleh {𝑓𝑛(𝑥)} barisan cauchy di r. karena r lengkap, maka untuk setiap 𝑥 ∈ 𝐾, terdapat 𝑓(𝑥) ∈ 𝑹 sehingga {𝑓𝑛(𝑥)} konvergen ke 𝑓(𝑥). akibatnya diperoleh ‖𝑓𝑛 − 𝑓‖∞ → 0. diambil sembarang 𝑛0 ∈ 𝑵. dibentuk 𝑔𝑛 = 𝑓𝑛+1 − 𝑓𝑛, untuk setiap 𝑛 ∈ 𝑵 dan 𝜓𝑛 = (𝜑𝑛 𝑛0 + ⋯ + 𝜑𝑛 𝑙 ) + (𝜑𝑛+1 𝑙+1 − 𝜑𝑛 𝑙+1) + ⋯ + (𝜑𝑛+1 𝑛 − 𝜑𝑛 𝑛), untuk setiap 𝑙, 𝑛 ∈ 𝑵 dengan 𝑛0 ≤ 𝑙 < 𝑛. oleh karena itu didapat barisan {𝜓𝑛 } ⊆ 𝐶(𝐾) dan berlaku ∑ 𝑔𝑛 ∞ 𝑛=𝑛0 = lim 𝑘→∞ ∑ 𝑔𝑛 𝑘 𝑛=𝑛0 = lim 𝑘→∞ ∑ (𝑓𝑛+1 − 𝑓𝑛) 𝑘 𝑛=𝑛0 = lim 𝑘→∞ (𝑓𝑘+1 − 𝑓𝑛0) = 𝑓 − 𝑓𝑛0. selanjutnya, akan dibuktikan 𝑓 − 𝑓𝑛0 ∈ 𝐷(𝐾). untuk setiap 𝑙, 𝑛 ∈ 𝑵 dengan 𝑛0 ≤ 𝑙 < 𝑛, diperoleh ‖𝜓𝑛 − (𝜑𝑛 𝑛0 + ⋯ + 𝜑𝑛 𝑙 )‖ ∞ = ‖(𝜑𝑛+1 𝑙+1 − 𝜑𝑛 𝑙+1) + ⋯ + (𝜑𝑛+1 𝑛 − 𝜑𝑛 𝑛 )‖ ∞ ≤ ∑ 1 2𝑖 𝑛 𝑖=𝑙+1 ≤ 1 2𝑙 . oleh karena itu, apabila 𝑛 → ∞ maka untuk setiap 𝑥 ∈ 𝐾 dan 𝑙 ≥ 𝑛0, diperoleh lim 𝑛→∞ |𝜓𝑛(𝑥) − (𝜑𝑛 𝑛0 + ⋯ + 𝜑𝑛 𝑙 )(𝑥)| ≤ 1 2𝑙 . akibatnya, diperoleh 𝑔𝑛0(𝑥) + ⋯ + 𝑔𝑙 (𝑥) − 1 2𝑙 ≤ lim 𝑛→∞ 𝜓𝑛 (𝑥) ≤ 𝑔𝑛0(𝑥) + ⋯ + 𝑔𝑙 (𝑥) + 1 2𝑙 . selanjutnya, apabila 𝑙 → ∞, maka diperoleh {𝜓𝑛 } konvergen titik demi titik ke 𝑓 − 𝑓𝑛0. disisi lain, diperoleh ∑ |𝜓𝑛+1(𝑥) − 𝜓𝑛 (𝑥)| ≤ ∞ 𝑛=0 ∑ |𝜑𝑛+1 𝑛0 (𝑥) −∞𝑛=0 𝜑𝑛 𝑛0(𝑥)| + ⋯ + ∑ |𝜑𝑛+1 𝑙 (𝑥) − 𝜑𝑛 𝑙 (𝑥)|∞𝑛=0 + ∑ |𝜑𝑛+1 𝑙+1 (𝑥) −∞𝑛=0 𝜑𝑛 𝑙+1(𝑥)| + ⋯ + ∑ |𝜑𝑛+1 𝑛 (𝑥) − 𝜑𝑛 𝑛(𝑥)|∞𝑛=0 + ∑ |𝜑𝑛+2 𝑙+1 (𝑥) −∞𝑛=0 𝜑𝑛+1 𝑙+1 (𝑥)| + ⋯ + ∑ |𝜑𝑛+2 𝑛+1(𝑥) − 𝜑𝑛+1 𝑛+1(𝑥)|∞𝑛=0 ≤ ∑ 1 2𝑖 𝑛+1 𝑖=𝑛0 ≤ 1 2 , untuk setiap 𝑥 ∈ 𝐾. dengan demikian terdapat barisan {𝜓𝑛 } ⊆ 𝐶(𝐾) yang konvergen titik demi titik ke 𝑓 − 𝑓𝑛0, dan ∑ |𝜓𝑛+1(𝑥) − 𝜓𝑛(𝑥)| ≤ ∞ 𝑛=0 1 2 . dengan kata lain benar bahwa 𝑓 − 𝑓𝑛0 ∈ 𝐷(𝐾). karena 𝐷(𝐾) ruang linear, maka diperoleh 𝑓 ∈ 𝐷(𝐾). selanjutnya, berdasarkan asumsi diawal pembuktian, maka diperoleh ‖𝑓 − 𝑓𝑛0‖𝐷 = ‖∑ 𝑔𝑛 ∞ 𝑛=𝑛0 ‖ 𝐷 = ‖∑ 𝑓𝑛+1 − 𝑓𝑛 ∞ 𝑛=𝑛0 ‖ 𝐷 ≤ ∑ ‖𝑓𝑛+1 − 𝑓𝑛‖𝐷 ∞ 𝑛=𝑛0 ≤ ∑ 1 2𝑛 ∞ 𝑛=𝑛0 ≤ 1 2𝑛0−1 , untuk setiap 𝑛0 ∈ 𝑵. karena berlaku untuk sembarang 𝑛0 ∈ 𝑵, maka diperoleh barisan {𝑓𝑛} konvergen ke 𝑓. jadi, terbukti 𝐷(𝐾) ruang banach. berdasarkan teorema 3.8, akan ditunjukkan bahwa 𝐷(𝐾) merupakan aljabar banach komutatif dan mempunyai elemen identitas. teorema 3.9 diberikan ruang metrik separabel k, 𝐷(𝐾) merupakan aljabar banach komutatif dan mempunyai elemen identitas. bukti: berdasarkan teorema 3.8, 𝐷(𝐾) merupakan ruang banach, selanjutnya akan dibuktikan 𝐷(𝐾) aljabar yang komutatif dan mempunyai elemen identitas. 1) diambil sembarang 𝑓, 𝑔, ℎ ∈ 𝐷(𝐾), maka terdapat fungsi – fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑤, 𝑥, 𝑦, 𝑧 ≥ 0 pada k, sehingg 𝑓 = 𝑢 − 𝑣, 𝑔 = 𝑤 − 𝑥, dan ℎ = 𝑦 − 𝑧. oleh karena itu, diperoleh 𝑓(𝑔 + ℎ) = (𝑢 − 𝑣) ((𝑤 − 𝑥) + (𝑦 − 𝑧)) = (𝑢 − 𝑣) (𝑤 − 𝑥) + (𝑢 − 𝑣)(𝑦 − 𝑧) = 𝑓𝑔 + 𝑓ℎ. dengan kata lain, 𝑓(𝑔 + ℎ) = 𝑓𝑔 + 𝑓ℎ, untuk setiap 𝑓, 𝑔, ℎ ∈ 𝐷(𝐾). 2) diambil sembarang 𝑓, 𝑔, ℎ ∈ 𝐷(𝐾), maka terdapat fungsi – fungsi semikontinu bawah malahayati 32 volume 3 no. 1 november 2013 terbatas 𝑢, 𝑣, 𝑤, 𝑥, 𝑦, 𝑧 ≥ 0 pada k, sehingga 𝑓 = 𝑢 − 𝑣, 𝑔 = 𝑤 − 𝑥, dan ℎ = 𝑦 − 𝑧. oleh karena itu, diperoleh (𝑔 + ℎ) 𝑓 = ((𝑤 − 𝑥) + (𝑦 − 𝑧))(𝑢 − 𝑣) = (𝑤 − 𝑥)(𝑢 − 𝑣) + (𝑦 − 𝑧)(𝑢 − 𝑣) = 𝑔𝑓 + ℎ𝑓. dengan kata lain, (𝑔 + ℎ) 𝑓 = 𝑔𝑓 + ℎ𝑓, untuk setiap 𝑓, 𝑔, ℎ ∈ 𝐷(𝐾). 3) diambil sembarang 𝑓, 𝑔 ∈ 𝐷(𝐾) dan skalar 𝛼, maka terdapat fungsi–fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑤, 𝑥 ≥ 0 pada k, sehingga 𝑓 = 𝑢 − 𝑣, dan 𝑔 = 𝑤 − 𝑥. oleh karena itu, diperoleh 𝛼(𝑓𝑔) = 𝛼 ((𝑢 − 𝑣) (𝑤 − 𝑥)) = (𝛼(𝑢 − 𝑣))(𝑤 − 𝑥) = (𝛼𝑓)𝑔. dengan kata lain, 𝛼(𝑓𝑔) = (𝛼𝑓)𝑔, untuk setiap 𝑓, 𝑔 ∈ 𝐷(𝐾) dan skalar 𝛼. 4) diambil 𝑓, 𝑔 ∈ 𝐷(𝐾) dan 𝜀 > 0 sembarang, maka terdapat fungsi-fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑤, 𝑥 ≥ 0 pada k, dengan 𝑓 = 𝑢 − 𝑣 dan 𝑔 = 𝑤 − 𝑥, sehingga berlaku ‖𝑢 + 𝑣‖∞ < ‖𝑓‖𝐷 + 𝜀 (‖𝑔‖𝐷+1) dan ‖𝑤 + 𝑥‖∞ < ‖𝑔‖𝐷 + 𝜀 (‖𝑓‖𝐷+1) . oleh karena itu, diperoleh ‖𝑓𝑔‖𝐷 ≤ ‖(𝑣𝑥 + 𝑢𝑤) + (𝑢𝑥 + 𝑣𝑤)‖∞ = ‖(𝑢 + 𝑣)(𝑤 + 𝑥)‖∞ ≤ ‖𝑢 + 𝑣‖∞‖𝑤 + 𝑥‖∞ < ‖𝑓‖𝐷 ‖𝑔‖𝐷 + 2𝜀 + 𝜀 2. karena berlaku untuk 𝜀 > 0 sembarang maka diperoleh ‖𝑓𝑔‖𝐷 ≤ ‖𝑓‖𝐷‖𝑔‖𝐷. dengan kata lain, ‖𝑓𝑔‖𝐷 ≤ ‖𝑓‖𝐷 ‖𝑔‖𝐷 , untuk setiap 𝑓, 𝑔 ∈ 𝐷(𝐾). 5) diambil sembarang 𝑓, 𝑔 ∈ 𝐷(𝐾), maka terdapat fungsi – fungsi semikontinu bawah terbatas 𝑢, 𝑣, 𝑤, 𝑥 ≥ 0 pada k, sehingga 𝑓 = 𝑢 − 𝑣 dan 𝑔 = 𝑤 − 𝑥. oleh karena itu, diperoleh 𝑓𝑔 = (𝑢 − 𝑣)(𝑤 − 𝑥) = (𝑤 − 𝑥)(𝑢 − 𝑣) = 𝑔𝑓. dengan kata lain, 𝑓𝑔 = 𝑔𝑓, untuk setiap 𝑓, 𝑔 ∈ 𝐷(𝐾). dengan demikian, terbukti bahwa 𝐷(𝐾) merupakan aljabar banach komutatif. selanjutnya akan dibuktikan d(k) mempunyai elemen identitas. diambil sembarang 𝑓 ∈ 𝐷(𝐾), maka terdapat fungsi–fungsi semikontinu bawah terbatas 𝑢, 𝑣 ≥ 0 pada k, sehingga 𝑓 = 𝑢 − 𝑣 . untuk sebarang 𝑥 ∈ 𝐾 didefinisikan 𝑒(𝑥) = 1. akibatnya, diperoleh 𝑒 ∈ 𝐷(𝐾). oleh karena itu, diperoleh 𝑒𝑓 = 𝑒 (𝑢 − 𝑣) = (𝑢 − 𝑣)𝑒 = 𝑢 − 𝑣 = 𝑓. dengan kata lain, e merupakan elemen identitas pada 𝐷(𝐾). jadi, terbukti bahwa sifat aljabar banach komutatif dan mempunyai elemen identitas berlaku pada kelas 𝐷(𝐾). penutup berdasarkan hasil pembahasan dapat disimpulkan bahwa kelas 𝐷(𝐾) mempunyai sifat aljabar banach komutataif dan mempunyai elemen identitas. sifat aljabar banach komutatif dapat pula diselidiki pada kelas bagian 𝐵1(𝐾) lainnya, diantaranya kelas 𝐵1 4⁄ (𝐾). daftar pustaka [1] ash, r.b., 2007, real variables with basic metric space topology, department of mathematics university of illionis at urbana-champaign. [2] dugundji, j., 1966, topology, allyn and bacon, inc., boston. [3] farmaki, v., 1996, on baire-1 4⁄ functions, trans. amer. math. soc, 348, 10. [4] gordon, r.a., 1994, the integral of lebesgue, denjoy, perron and henstock, american mathematical society, usa. [5] haydon, r., odell, e. dan rosenthal, h.p., 1991, on certain classes of baire-1 functions with applications to banach space theory, lecture notes in math., 1470, springer, new york. [6] kreyszig, e., 1978, introductory functional analysis with applications, john wiley and sons, inc., canada. [7] mcshane, e.j., 1944, integration, princeton university press, princeton. [8] rosenthal, h.p., 1994, a characterization of banach spaces containing c0, j. amer. math. soc, 7, 3, 707-748. [9] rosenthal, h.p., 1994, differences of bounded semi-continuous functions i, http://www.arxiv.org/abs/math/9406217, 20 juni 1994, diakses pada tanggal 27 agustus 2009. [10] royden, h.l., 1989, real analysis, macmillan publishing company, new york. http://www.arxiv.org/abs/math/9406217 power of test path analysis and partial least square analysis cauchy – jurnal matematika murni dan aplikasi volume 4(3) (2016), pages 112-114 submitted: august, 09 2016 reviewed: october, 31 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3593 power of test path analysis and partial least square analysis arif kurniawan1, loekito2, solimun3 faculty of mathematics and natural sciences1, 2, 3 brawijaya university malang email: arkur111@gmail.com, arkur112@gmail.com, arkur113@gmail.com abstract path analysis and partial least square (pls) was used to analyze many variables. both methods use the least squares method (ols) that can be compared between the two to determine the best method in a study to get an assessment of the behavior of civil servants in the government of kediri. the purpose of this study is comparing path analysis with partial least square (pls) on the power of the test and the value r2. path method is able to provide the value of r2 higher than analysis of partial least square (pls) also the value of the test power analysis path is higher than using analysis of partial least square (pls). usage analysis methods path analysis and partial least square (pls) produces behavioral assessment of civil servants in the government of kediri is nearly equal results and discussion. based on the analysis to prove that the behavior of civil servants in the government of kediri not meet eligibility based on the grade levels and echelons of the civil service. keywords: partial least square, least square, path method. introduction problems that exist in the world of work are many and complex, most of it is not simple and only includes a direct causal relationship. regression analysis is a statistical analysis that has been widely known and easier understanding, use and application. regression analysis can be used to determine the relationship of more than 2 variables but in the regression analysis has met some of the assumptions in advance that multicollinearity, heteroscedasticity, autocorrelation and normality. path analysis basically have a broader scope than the regression analysis which can both be used to analyze more than two points (variables), which has a form of certain relationship. path analysis can determine the influence of the independent variables relationships dependent variable either directly or indirectly directly. partial least square (pls) analysis was used to estimate the more complex models involving latent variables (variables measured from the other variables are indicators). the existence of variables that influence and are influenced other variables show a causal relationship between more than two variables. path analysis and partial least square can create a diagram that connects between several independent variables and some dependent variable. therefore, the use of path analysis and partial least square is appropriate to analyze a problem that has many variables [1]. the problems in this study is whether there is a difference between the power of the test and r2 path analysis with partial least square restrictions problem in this research is the use of a formative model with variables observed. the objectives of this research are comparing authorization test between path analysis with partial least square, is there a difference between path r2 analysis with partial least square, how do the results of behavioral assessment of civil servants working in the city government of kediri. http://dx.doi.org/10.18860/ca.v4i3.3593 mailto:arkur111@gmail.com mailto:arkur112@gmail.com mailto:arkur113@gmail.com power of test path analysis and partial least square analysis arif kurniawan 113 the benefits that can be achieved through this research are powerful test shows difference between path analysis with partial least square, shows the difference r2 path analysis with partial least square, shows the working behavior of civil servants in the government of kediri. work behavior in the organization cannot be separated from four things: cooperation, integrity, commitment and discipline. commitment good work will improve work habits, as well as the level of cooperation, discipline and integrity held by workers within an organization. so, the value of work behavior obtained from the overall value of collaboration, integrity, commitment and discipline [2]. methods data of the study include primary data about the behavior of employment and length of employment of civil servants, secondary data on the length of service, basic salary per grade and salary allowances echelon primary pns. data obtained by collecting data directly at the sites. data were collected through interviews to civil servants in kediri city government as well as through direct observation, so that the research data that will be obtained is accurate and up to date. in addition to the primary data used, this study also used secondary data is data compiled from existing data. implementation of research conducted in the area of research time kediri city government in february to march 2016 sampling for civil servants structural in city government kediri done randomly as many as 97 people, using the formula 𝑛 = 𝑁 1+𝑁𝛼2 , where 𝑁 is the number of population is the number structural civil servants in kediri city government as much as 3,234 people with 𝛼 = 0.1. results and discussion significant relationships shown in endogenous variables basic salary per class and work behavior, while the effective working time is not significant. it is clear that the higher the grade and salary of civil servants in the town of kediri government does not guarantee increasing the effective working time of civil servants in carrying out its duties. it can be said that civil servants in the town of kediri based on the career paths and salaries amounted to only 3.5% of civil servants who work in a professional manner, namely by increasing the effective working time. the strong influence on civil servants to work behavior of kediri only given by the effective working time with a value of 0.797 while the period of employment, and salary below 0.5, meaning that the higher salary and years of service does not guarantee provides work behavior is increasing. this applies to civil servants in the government of kediri. it can be seen that all the behavioral indicators of civil servants working in kediri city government have a significant influence. this suggests that the scoring system is better done by colleagues in other words co-workers could be involved in addition to official appraiser or supervisor in work behavior assessment system of civil servants. in summary pls results indicate that there are four significant influence with a confidence level of 0.05, namely : the influence of effective working time on work behaviors with p value = 0.000, the effect of tenure on the behavior of working with p value = 0.043 , the effect of salary subject to the behavior and working with p value = 0.017 and the effect of tenure on the basic salary bracket with a p value of 0.000. the results of analysis by using path analysis and partial least square ( pls ) analysis provides a discussion of analysis results are almost the same but the value is not the same calculation. to determine the value of better calculation between the two can be seen by comparing the value of r and the power of the test as follows : based on the research that the above comparison shows that r2 on path analysis produces a value greater than the pls analysis. this proves that in this study the use of path analysis is better than the use of pls analysis. power of test path analysis and partial least square analysis arif kurniawan 114 table 1. compare r2 path analysis and pls analysis no. variable path analysis pls analysis 1. the period of employment, basic salary, salary allowances echelon, the effective working time to work behavior 0,699 0,599 2. the period of employment, basic salary, salary allowances echelon to the effective working time 0,035 0,005 3. the period of employment to basic salary 0,778 0,599 power comparison test on the response variable indicates that the power test on path analysis produces a value greater than the pls analysis. this proves that in this study the use of path analysis is better than the use of pls analysis of the power of the test. table 2. compare power of test path analysis and pls analysis no. variable path analysis pls analysis 1. the period of employment, basic salary, salary allowances echelon, the effective working time to work behavior 0,999778 0,999655 2. the period of employment, basic salary, salary allowances echelon to the effective working time 0,985788 0,897423 3. the period of employment to basic salary 0,999853 0,999655 conclusion the conclusion of this study is the use path analysis provides better value than pls at r2 value and in terms of the power of the test. behavioral conditions of work of civil servants in kediri city government did not follow the rules of the science of human resources in terms of election officials echelon structural or civil servants. advice that can be given is a benchmarking test r2 and the power to use a sample model of a broader research eg civil servants in the province in order to obtain a very significant research model with population models . references [1] akter, a., d’ambra, j., ray, p., an evaluation of pls based complex models: the roles of power analysis, predictive relevance and gof index. 2011. [2] leka, s., griffiths, a., cox, t., work organisation & stress. university of nottingham. 1999. 5.2 fuad adi saputra bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf fuad adi saputra mahasiswa jurusan matematika uin maulana malik ibrahim malang e-mail: tee_fu@yahoo.com abstrak graf � dengan pewarnaan sisi disebut pelangi sisi terhubung, jika setiap titik pada graf � dihubungkan oleh lintasan yang memiliki sisi-sisi dengan warna yang berbeda. rainbow connection pada graf � yang terhubung, disimbolkan oleh ����� yaitu bilangan terkecil dari warna yang dibutuhkan untuk membuat graf � menjadi pelangi sisi terhubung. sedangkan graf � dengan pewarnaan titik adalah pelangi titik terhubung, jika setiap titik pada graf � dihubungkan oleh lintasan yang memiliki titik-titik interior dengan warna yang berbeda. rainbow vertex-connection pada graf � yang terhubung disimbolkan oleh ������ yaitu bilangan terkecil dari warna yang dibutuhkan untuk membuat graf � menjadi pelangi titik terhubung. penelitian ini menganalisis besarnya bilangan ����� dan ������ dari graf hasil penjumlahan dan perkalian kartesius dua sebarang graf. penjumlahan dua graf ��dan �� yang dinotasikan � �� �� mempunyai himpunan titik ���� ����� � ����� dan himpunan sisi ��� ���� � ���� � ���|� � �������� � � ������. bilangan rainbow connection dari graf � adalah: 1) ����� 1, �� dan �� adalah graf komplit, dan 2) ����� � 2, ��atau �� adalah bukan graf komplit sedangkan bilangan rainbow vertex-connection dari graf � adalah: ������ �0, ��dan �� adalah graf komplit1, ��atau �� adalah bukan graf komplit � graf � hasil kali kartesius adalah graf yang dinotasikan � �� �� dan mempunyai titik ���� ����� �����, dan dua titik ���,��� dan ���,��� dari graf � terhubung langsung jika dan hanya jika �� �� dan ���� � ���� atau �� �� dan ���� � ����. bilangan rainbow connection dari graf � adalah: �"�# ���� �"�# ���� $ ����� $ ������ ������ sedangkan bilangan rainbow vertexconnection dari graf � adalah: �"�# ���� �"�# ���� % 1 $ ������ $ ������� ������� 1 kata kunci: graf penjumlahan, graf perkalian kartesius, rainbow connection, rainbow vertex-connection abstract an edge-colored graph � is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. the rainbow connection of a connected graph �, denoted by �����, is the smallest number of colors that are needed in order to make � rainbow edge-connected. a vertexcolored graph � is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. the rainbow vertex-connection of a connected graph �, denoted by ������, is the smallest number of colors that are needed in order to make � rainbow vertex-connected. this research was analysis about number of ����� and ������ from the join and cartesian product of two graphs. the join � �� �� has ���� ����� � ����� and ��� ���� � ���� � ���|� � ����� ��� � � ������. the number of rainbow connection from graph � is: 1) ����� 1, ��and �� are complete graph, and 2) ����� � 2, ��or �� are non-complete graph and then the number of rainbow vertexconnection from graph � is: ������ �0, ��and �� are complete graph1, ��or �� ��' non % complete graph � the cartesian product � �� �� has ���� ����� �����, and two vertices ���,��� and ���,��� of � adjecent if only if either �� �� and ���� � ���� or �� �� and ���� � ����. the number of rainbow connection from graph � is: �"�# ���� �"�# ���� $ ����� $ ������ ������ and then the number of rainbow vertex-connection from graph � is: �"�# ���� �"�# ���� % 1 $������ $ ������� ������� 1 keywords: cartesian product, join graph, rainbow connection, rainbow vertex-connection fuad adi saputra 126 volume 2 no. 3 november 2012 pendahuluan matematika merupakan raja dan pelayan bagi disiplin ilmu lain atau pun dalam lini kehidupan. teori graf merupakan salah satu cabang matematika yang penting dan banyak manfaatnya karena teori-teorinya dapat diterapkan untuk memecahkan masalah dalam kehidupan sehari-hari. (purwanto, 1998:1). graf g adalah pasangan himpunan (v, e) dengan v adalah himpunan tidak kosong dari obyek-obyek yang disebut sebagai titik dan e adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di v yang disebut sebagai sisi. himpunan titik di g dinotasikan dengan v(g) dan himpunan sisi dinotasikan dengan e(g) (chartrand dan lesniak, 1986: 4). pewarnaan pada graf � adalah pemetaan warna-warna ke titik atau sisi dari � sedemikian hingga titik atau sisi yang terhubung langsung mempunyai warna-warna yang berbeda. dalam teori graf konsep pewarnaan terus mengalami perkembangan, salah satunya adalah tentang rainbow connection. rainbow connection dibagi menjadi 2 jenis, yang pertama adalah pelangi sisi terhubung (rainbow edge-connected) yang didefinisikan sebagai pewarnaan sisi pada graf � jika setiap titik pada graf � dihubungkan oleh lintasan yang memiliki sisi-sisi dengan warna yang berbeda, sedangkan yang kedua adalah pelangi titik terhubung (rainbow vertexconnected) yang didefinisikan sebagai pewarnaan titik pada graf � jika setiap titik pada graf � dihubungkan oleh lintasan memiliki titik-titik interior dengan warna yang berbeda. bilangan rainbow connection pada graf terhubung disimbolkan oleh ����� yaitu bilangan warna terkecil pada sisi yang dibutuhkan untuk membuat � menjadi pelangi sisi yang terhubung. bilangan rainbow vertex-connection pada graf terhubung disimbolkan oleh ������ yaitu bilangan warna terkecil pada titik yang dibutuhkan untuk membuat � menjadi pelangi titik terhubung (krivelevich dan yuster, 2010:1) jurnal yang ditulis oleh michael krivelevich dan raphael yuster (2010) menjelaskan mengenai bilangan rainbow connection yang dibangun oleh derajat terkecil dari suatu graf umum. mereka mengembangkan dari kajian yang ditulis oleh y. caro, a. lev, y. roditty, z. tuza, dan r. yuster (2008) dalam jurnalnya yang berjudul on rainbow connection. dalam jurnal on rainbow connection hasil dari bilangan rainbow connection ����� dan ������ masih dibatasi oleh suatu variabel yang belum jelas, misalkan ����� $ て�/*, dimana + adalah variabel. kemudian oleh krivelevich dan yuster dikembangkan lagi dan berhasil menentukan nilai variabelnya menjadi ����� $ 20�/* sehingga batas nilai + menjadi lebih jelas. selain itu juga dijelaskan bahwa bilangan ����� ��"�#��� penelitian yang dilakukan oleh krivelevich dan yuster menggunakan objek graf yang umum. sedangkan untuk graf dari hasil operasi belum diteliti, khususnya pada operasi penjumlahan dan perkalian kartesius, sehingga perlu dilakukan penelitian lagi untuk objek graf tersebut. kajian teori 1. graf graf g adalah pasangan himpunan (v,e) dengan v adalah himpunan tidak kosong dari obyek-obyek yang disebut sebagai titik, dan e adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di v yang disebut sebagai sisi. himpunan titik di g dinotasikan dengan v(g) dan himpunan sisi dinotasikan dengan e(g). 2. adjacent dan incident sisi e = (u, v) dikatakan menghubungkan titik u dan v. jika e = (u, v) adalah sisi di graf g, maka u dan v disebut terhubung langsung (adjacent), u dan e serta v dan e disebut terkait langsung (incident). untuk selanjutnya, sisi e = (u,v) akan ditulis e = uv. derajat titik v di graf g, ditulis degg (v), adalah banyaknya sisi di g yang terkait langsung dengan v (chartrand dan lesniak, 1986:4). contoh: gambar 2.1 graf g gambar 1. contoh adjacent dan incident dari gambar 1 titik v4 dan sisi e2, e3 dan e4 adalah terkait langsung. sedangkan titik v3 dan v4 adalah terhubung langsung tetapi v3 dan v2 tidak. 3. graf terhubung suatu jalan (walk) u-v pada graf g adalah barisan berhingga (tak kosong) w : u = u0, e1, u1, e2, . . ., un-1, en, un = v yang berselang seling antara titik dan sisi, yang dimulai dari titik u dan diakhiri dengan titik v, dengan iii uue 1−= untuk i = 1, 2, . . ., n adalah sisi di g. u0 disebut titik awal, un disebut titik akhir, u1, u2, ..., un-1 disebut titik internal, dan n menyatakan panjang dari w (chartrand dan lesniak, 1986:26). jalan u-v yang semua sisinya berbeda disebut trail u-v. jalan u-v yang semua sisi dan titiknya berbeda disebut lintasan (path) u-v. p : u 4v 3v 1v 2v e3 e2 e4 e1 bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 127 = u0, e1, u1, e2, . . ., un-1, en, un = v, u0 disebut titik awal, un disebut titik akhir. sedangkan u1, u2, ..., un1 disebut titik internal, dan n menyatakan panjang dari p. dengan demikian, semua lintasan adalah trail. graf lintasan dengan � titik dinotasikan dengan ,(chartrand dan lesniak, 1986:26). contoh: g: gambar 2. graph terhubung dari graf di atas v1, e1, v2, e5, v5, e6, v4, e4, v2, e2, v3 adalah trail, sedangkan v1, e1, v2, e5, v5, e6, v4 adalah lintasan. misalkan u dan v titik berbeda pada graf g. maka titik u dan v dapat dikatakan terhubung (connected), jika terdapat lintasan u–v di g. sedangkan suatu graf g dapat dikatakan terhubung (connected), jika untuk setiap titik u dan v di g terhubung (chartrand dan lesniak, 1986:28). misalkan u dan v titik berbeda pada graf g, maka jarak antara dua titik di � adalah panjang lintasan terpendek antara kedua titik tersebut yang dinotasikan dengan �"./0��,��. sedangkan eksentrisitas titik � � ���� adalah '����� max ��"./0��,��:� � ����. radius dari � adalah ���� min �'�����:� � ����� dan diameter dari � adalah �"�#��� max �'�����:� � ����� (chartrand dan lesniak, 1986:29). 4. operasi pada graf penjumlahan dua graf ��dan �� yang dinotasikan � �� �� mempunyai himpunan titik ���� ����� � ����� dan himpunan sisi ��� ���� � ���� � ���|� � �������� � �������(chartrand dan lesniak, 1986: 11). perhatikan contoh di bawah ini. g1: g2: g: gambar 3. penjumlahan graf � �� �� hasil kali kartesius adalah graf yang dinotasikan � �� �� dan mempunyai titik ���� ����� �����, dan dua titik ���,��� dan ���,��� dari graf � terhubung langsung jika dan hanya jika �� �� dan ���� � ���� atau �� �� dan ���� � ����(chartrand dan lesniak, 1986: 11). perhatikan contoh berikut, g1: g2: g1 x g2: gambar 4. graf hasil kali kartesius 5. jenis graf a. graf komplit (complete graph) adalah graf dengan setiap pasang titik yang berbeda dihubungkan langsung oleh satu sisi. graf komplit dengan � titik dinyatakan dengan kn (purwanto, 1998:21). b. graf bipartisi komplit (complete bipartite graph) adalah graf bipartisi dengan himpunan partisi x dan y sehingga masing-masing titik di x dihubungkan dengan masing-masing titik di y oleh tepat satu sisi. jika |x| = m dan |y| = n, maka graf bipartisi tersebut dinyatakan dengan km,n. (purwanto, 1998:22). c. graf sikel cn adalah graf terhubung n titik yang setiap titiknya berderajat 2. misal graf sikel cn mempunyai himpunan titik v(cn) = , maka graf tersebut mempunyai himpunan sisi e(cn) = dimana untuk setiap i=1,2,…,n. d. graf roda adalah graf yang dibentuk dari operasi penjumlahan antara graf sikel (+-) dan graf komplit dengan satu titik (5-�. graf roda dinotasikan dengan 6dan � � 3 (harary, 1969:46) e. graf kipas dibentuk dari penjumlahan graf komplit (5�� dan graf lintasan (,-� yaitu 8 5� ,-. dengan demikian graf kipas mempunyai (� 1� titik dan (2� % 1) sisi ( gallian, 2007: 16) f. graf kipas ganda dibentuk dari penjumlahan antara gabungan dua graf komplit (5�� dan graf lintasan (,-� yaitu 〰8 25� ,-. dengan demikian graf kipas mempunyai (� 2� titik dan (3� % 1) sisi. g. graf tangga yang dinotasikan sebagai 9 adalah suatu graf yang dibentuk dari operasi hasil kali kartesius antara graf lintasan dengan dua titik dan graf lintasan dengan n titik yaitu 9 ,�: ,(galian, 2007:12) 6. pewarnaan graf pewarnaan titik dari graf g adalah suatu proses pemberian warna pada titik-titik suatu graf sehingga tidak ada dua titik yang terhubung langsung pada graf tersebut berwarna sama. graf },...,,{ 21 nvvv },...,,{ 21 neee 1+= iii vve v1 v2 v3 e1 e2 e5 v5 v4 e6 e3 e4 u u v v v v v v u u (u2,v2) (u1,v2) (u1,v3) (u2,v3) (u1,v1) (u2,v1) vv vu u fuad adi saputra 128 volume 2 no. 3 november 2012 g berwarna n jika terdapat pewarnaan dari g yang menggunakan n warna (chartrand dan lesniak, 1986:271). suatu pewarnaan sisi-k untuk graf g adalah suatu penggunaan k warna untuk mewarnai semua sisi di g sehingga setiap pasang sisi yang mempunyai titik persekutuan diberi warna yang berbeda. jika g mempunyai pewarnaan sisi-n, maka dikatakan sisi-sisi di g diwarnai dengan n warna. indeks kromatik g dinotasikan dengan adalah bilangan n terkecil sehingga sisi di g dapat diwarna dengan n warna (purwanto, 1998:80). 7. rainbow connection pewarnaan sisi pada graf � disebut pelangi sisi terhubung (rainbow edge-connected) jika setiap titik pada graf � dihubungkan oleh lintasan yang memiliki sisi-sisi dengan warna yang berbeda. rainbow connection pada graf � yang terhubung (connected graph) disimbolkan oleh ����� yaitu bilangan terkecil dari warna yang dibutuhkan untuk membuat graf g menjadi pelangi sisi terhubung (rainbow edge-connected) ( krivelevich dan yuster, 2010:1). lintasan pelangi (rainbow path) adalah lintasan antara dua titik sehingga tidak ada dua sisi pada lintasan tersebut yang memiliki warna yang sama. jika lintasan pelangi tersebut ada di setiap antara dua titik maka pewarnaan tersebut dinamakan pewarnaan pelangi (rainbow colouring). sedangkan bilangan minimum pada warna yang diinginkan dinamakan bilangan pelangi yang terhubung (rainbow connection number rc(g)). ( l. sunil chandran,2011:3). pewarnaan titik pada graf � adalah pelangi titik terhubung (rainbow vertex-connected) jika setiap titik pada graf � dihubungkan oleh lintasan memiliki titik-titik interior dengan warna yang berbeda. rainbow vertex-connection pada graf � yang terhubung (connected graph) disimbolkan oleh ������ yaitu bilangan terkecil dari warna yang dibutuhkan untuk membuat graf g menjadi pelangi titik terhubung (rainbow vertexconnected) ( krivelevich dan yuster, 2010: 2). misalkan graf � memiliki � titik, maka ����� $ � % 1. kemudian jika graf sikel + dengan � � 3 maka ����� $ ;-�<. sedangkan jika dihubungkan dengan �"�# ���, maka ����� � �"�# ��� dan ������ � �"�# ��� % 1 ( krivelevich dan yuster, 2010:1-2). 1 1 1 2 3 3 3 4 5 5 5 ? ? ? @ @ @ a a a gambar 5. graf pelangi sisi terhubung dan pelangi titik terhubung gambar 5 merupakan contoh dari graf pelangi sisi terhubung dan pelangi titik terhubung dengan 5 warna sisi dan 3 warna titik. pada gambar di atas mempunyai bilangan ����� 5 dan ������ 3. pembahasan dalam pembahasan ini, sebelum mengkaji inti permasalahan yaitu mengkaji bilangan rainbow connection dan bilangan rainbow vertex-connection pada graf hasil operasi penjumlahan dan perkalian kartesius dua graf dengan menggunakan sebarang graf, maka akan dibahas terlebih dahulu �����dan ������ dari jenis graf yang dihasilkan oleh operasi penjumlahan dan perkalian kartesius. 1. bilangan rainbow connection pada jenis graf hasil penjumlahan pada graf khusus hasil penjumlahan akan diberikan 5 contoh graf yaitu graf komplit, graf bipartisi komplit, graf roda, graf kipas, dan graf kipas ganda. a. graf komplit tabel 1. pola bilangan ���5-� dan ����5-� no jenis graf bc�de� bfc�de� 1. 5� 0 0 2. 5� 1 0 3. 5g 1 0 4. 5h 1 0 5. 5i 1 0 6. 5j 1 0 �. 51 0 dari pola yang ditunjukan oleh bilangan rainbow connection dan bilangan rainbow vertexconnection di atas dapat diperoleh teorema sebagai berikut: teorema 1 pada graf komplit 5banyak titik � � 2, bilangan rainbow connection ���5-� 1, dan bilangan rainbow vertex-connection ����5-� 0. b. graf bipartisi komplit secara umum graf 5-,k dapat dibentuk menjadi pelangi sisi terhubung hanya dengan menggunakan 4 warna. hal ini dapat dijelaskan dengan model lintasan antara 2 titik sebagai berikut: �" �" 1 �" 2 �" �" 1 1 3 4 2 �" 2 �" 1 1 3 4 2 dan �" 1 �" 3 �" 2 gambar 6. model lintasan graf bipartisi 5k, )(' gχ bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 129 teorema 2 graf bipartisi komplit 5-,k dengan � $ # dan �,# � l, maka bilangan rainbow connection pada graf 5-,k adalah: ��m5-,kn o #, jika � 12, jika 2� #3, jika 2r # $ 3-4, jika lainnya � bilangan rainbow vertex-connection pada graf 5-,k adalah: ����5-,k� 1 bukti: graf bipartisi komplit 5-,k terdiri dari 2 partisi, yaitu : dan u, dengan |:| # dan |u| �.misalkan �v � ��:�, " 1,2, . . . ,# dan �v � ��u�, " 1,2,…,�. maka setiap dua titik �v dan setiap dua titik �v tidak terhubung langsung, akan tetapi setiap titik �v dengan �v akan dihubungkan dengan satu sisi. terdapat 4 kasus, yaitu: (i) jika � 1, maka ��m5�,kn #. andaikan ��m5�,kn r #, maka ada dua sisi di 5�,k yang berwarna sama. misal��v,��� dan m�x,��n dengan " y z. akibatnya lintasan �v % �x bukan lintasan pelangi karena hanya ada 1 warna. jadi 5�,k bukan pelangi sisi yang terhubung, jadi ��m5�,kn � #. karena pewarnaan sisi dengan # warna menghasilkan 5�,k graf pelangi sisi, maka ��m5�,kn $ #. disimpulkan ��m5�,kn #. (ii) ��m5-,kn 2 jika 2� #. akan dibuktikan jika 2� #, maka ��m5-,kn 2, deg��v� � dan deg��v� # jika ��m5-,kn 2 maka setiap dua titik, maksimal ada lintasan dengan 2 warna sisi yang berbeda. untuk itu jika setiap lintasan �v % �x dan setiap lintasan �v % �x terbentuk lintasan pelangi maka susunan warna yang dikenakan pada sisi yang terkait langsung pada �v berbeda dengan �x, begitu juga pada �v berbeda dengan �x. selanjutnya karena deg��v� � $ deg��v� #, maka jika setiap susunan warna yang dikenakan pada sisi yang terkait langsung dengan �v berbeda, maka susunan warna yang dikenakan pada sisi yang terkait langsung dengan setiap �v juga berbeda. sehingga cukup dianalisis susunan warna pada sisi yang terkait langsung dengan titik �v � ��:�, " 1,2,…,#. diketahui ��m5-,kn 2 dan deg��v� , jadi dari 2 warna tersebut akan disusun ke-� tempat dengan perulangan, sehingga diperoleh banyaknya susunan 2� 2� 2g … 2 2-.kemudian susunan tersebut diberikan pada setiap titik �v dan �x dengan " y z. jika banyaknya susunan sebanyak 2lebih banyak dari banyaknya titik �v yang sebanyak #, maka setiap �v mempunyai susunan warna sisi yang terkait langsung berbeda dengan �x. sehingga lintasan �v % �x akan membentuk lintasan pelangi. jadi terbukti jika 2� # maka ��m5-,kn 2. (iii) jika ��m5-,kn 3 jika 2$ # $ 3 akan dibuktikan jika 2$ # maka ��m5-,kn � 3 dan jika # $ 3maka ��m5-,kn 3. ambil ��m5-,kn 2, jika 2$ # maka akan dibuktikan ada dua titik yang semua lintasannya mempunyai warna sisi yang sama. banyak susunan warna 2-, artinya ��,��,… . ,��[ susunan warna sisi yang terkait langsung berbeda. karena 2$ # maka ada titik ��[\] yang mempunyai susunan warna yang sama dengan �^, dengan 1 $ � $ �# % 2-� dan 1 $ _ $2-.misalkan ada fungsi �: m5-,kn ` �1,2�, maka ����[\],�v� ���^,�v�, sehingga lintasan �^ % ��[\] tidak membentuk lintasan pelangi, karena semua lintasannya pasti mempunyai warna sisi yang sama. sehingga terbukti jika 2$ # maka ��m5-,kn � 3. sekarang ��m5-,kn 3 dan deg��v� �, jadi dari 3 warna tersebut akan disusun ke-� tempat dengan perulangan, sehingga diperoleh banyaknya susunan 3� 3� 3g … 3 3-. kemudian susunan tersebut diberikan pada setiap titik �v dan �x dengan " y z. jika banyaknya susunan sebanyak 3lebih banyak dari banyaknya titik �v yang sebanyak #, maka setiap �v mempunyai susunan warna sisi yang terkait langsung berbeda dengan �x. sehingga lintasan �v % �x akan membentuk lintasan pelangi. jadi terbukti jika # $ 3maka ��m5-,kn 3. (iv) ��m5-,kn 4, jika lainnya jika sudah tidak memenuhi semua ketentuan di atas, untuk membentuk graf 5k,menjadi pelangi sisi terhubung, maka graf 5k,dapat diwarnai minimal mengunakan 4 warna. misalkan ada fungsi pewarnaan sisi �: m5-,kn ` �1,2,3,4� maka: �m�v,�xn 1, " y z �'�a�� " a��z"b, z a��z"b, �m�v,�xn 3, " y z �'�a�� " a'��c,z a��z"b, fuad adi saputra 130 volume 2 no. 3 november 2012 �m�v,�xn 2, " y z �'�a�� " a��z"b, z a'��c, �m�v,�xn 4, " y z �'�a�� " a'��c,z a'��c, dengan pewarnaan di atas maka setiap dua titik pada graf 5k,terdapat lintasan dengan warna sisi yang berbeda, sehingga graf 5k, akan membentuk graf pelangi sisi terhubung, jadi dengan demikian terbukti ��m5k,n 4. (v) ���m5k,n 1 akan dibuktikan ���m5k,n � 1 dan ���m5k,n $ 1. diketahui �"�#m5k,-n 2, sehingga ���m5k,n � 2 % 1 1. untuk membuktikan ���m5k,n $ 1, maka akan dibuktikan bahwa dengan 1 warna titik, dapat membentuk pelangi titik yang terhubung, artinya setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. ambil ���m5k,n 1,misalkan ada fungsi pewarnaan �:�m5-,kn ` �1� maka: ���v� 1, dan ���v� 1, sehingga setiap lintasan �v % �x akan melewati titik interior �v dimana ���v� 1, sedangkan setiap lintasan �v % �x akan melewati titik interior �v dimana ���v� 1, dengan demikian setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. jadi terbukti ���m5k,n $ 1. karena ���m5k,n � 1 dan ���m5k,n $ 1, maka ���m5k,n 1. c. graf roda secara umum graf 6dapat dibentuk menjadi graf pewarnaan sisi yang terhubung, dengan warna sisi minimal 3, dan juga graf 6 dapat dibentuk menjadi graf pewarnaan titik yang terhubung, dengan warna titik minimal 1, yang ditampilkan dalam model pewarnaan berikut: �1 �1 ��%1 �5 �4 �3 �2 1 2 1 1 2 2 3 1 ? ? ? ? ? ? �� ? 3 3 3 3 3 1 de: gambar 7. graf roda 6 teorema 3 graf � adalah graf roda 6dengan � � l, maka bilangan rainbow connection pada graf � adalah: ���6-� e1, jika � 32, jika 4 $ � $ 63, jika � � 7� bilangan rainbow vertex-connection pada graf 6 adalah: ����6-� 1, jika � � 4 bukti: graf roda 6dengan � � 3adalah graf yang terbentuk dari operasi penjumlahan antara graf sikel (+-) dan graf komplit dengan satu titik (5��. misalkan �v � ��+-�, " 1,2,…,�, maka �-\1 ��,dan � � ��5��, serta untuk semua �v terhubung langsung dengan �. (i) jika � 3, akan dibuktikan 6g adalah graf 5h. untuk semua ��,��,�g � �+g� terhubung langsung dengan �,sehingga diperoleh deg���� deg���� deg��g� deg��� 3. karena graf 6g dengan 4 titik beraturan%3 maka 6g adalah graf komplit, jadi terbukti dengan � 3 maka ���6g� 1. (ii) jika 4 $ � $ 6 maka ���6-� 2 akan dibuktikan ���6-� � 2 dan ���6-� $ 2 graf 6dengan 4 $ � $ 6 bukan merupakan graf komplit, karena deg��v� 3 sedangkan jumlah titiknya 5 $ |6-| $ 7, sehingga ����� � 2. kemudian untuk membuktikan ����� $ 2 maka akan dibuktikan bahwa dengan 2 warna dapat membentuk 6menjadi pelangi sisi yang terhubung, sehingga setiap dua titik terdapat lintasan dengan warna sisi yang berbeda. fungsi pewarnaan sisi �: �6-� `�1,2� yang didefinisikan oleh ���v,�� 1 jika " ganjil, ���v,�� 2 jika " genap, ���v,�v\1� 1 jika " ganjil, ���v,�v\1� 2 jika " genap. setiap lintasan �v % � atau � % �v hanya terdapat satu warna sisi. kemudian lintasan �v % �x, jika " genap dan �v dan z ganjil pasti berbentuk lintasan pelangi 2 warna. sedangkan untuk lintasan �v % �x dengan " dan zsama-sama genap atau "dan z samasama ganjil. jika �v % �v\2 maka membentuk lintasan pelangi yang sisi-sinya +-.tetapi jika lintasan �v % �v\h, dengan 4 $ � $ � % 2 maka � 4, karena � 6 diperoleh 4 $ � $ 6 % 2.untuk " 1, dengan lintasan �� % �i, maka dapat dibentuk lintasan pelangi melewati titik �j, sedangkan untuk " 2 dengan lintasan �� % �j, maka dapat dibentuk lintasan pelangi melewati titik ��, terbukti setiap dua titik terdapat lintasan dengan warna sisi yang berbeda, sehingga diperoleh ����� $ 2. jadi terbukti untuk 4 $ � $ 6 maka ����� 2. bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 131 (iii) jika � � 7 maka ����� 3 akan dibuktikan ����� � 3 dan ����� $ 3. dibuktikan ����� $ 3 maka akan dibuktikan bahwa dengan 3 warna dapat membentuk 6menjadi pelangi sisi yang terhubung, sehingga setiap dua titik terdapat lintasan dengan warna sisi yang berbeda. jika fungsi pewarnaan sisi �: �6-� ` �1,2,3� yang didefinisikan oleh ���v,�� 1 jika " ganjil, ���v,�� 2 jika " genap, dan ��i� 3,ji � �+-� maka akan membentuk pelangi sisi terhubung, sehingga ���6-� $ 3. selanjutnya dibuktikan ����� � 3, dari hasil di atas didapat 6bukan graf komplit sehingga ����� � 2. ambil ����� 2 misalkan fungsi pewarnaan sisi �: �6k� `�1,2� didefinisikan �� �,�� 1,maka ���h,�� 2 dan ���i,�� 2,karena tidak mungkin menggunakan lintasan sisi +yang panjangnya 3. kemudian jika ���v,�v\1� 1 dengan " ganjil, ���v,�v\1� 2 dengan " genap, maka ���-l1,�� 1 membentuk lintasan pelangi, karena panjang lintasan dengan sisi +sama dengan 2 dan warnanya berbeda. tetapi jika ���g,�� 1,lintasan �g % �-l1 warnanya akan sama dan kalau lintasannya menggunakan sisi +k juga tidak mungkin karena panjangnya � 3, jadi haruslah ���g,�� 2. selanjutnya ����,�� harus sama dengan 1, karena ���i,�� 2, akan tetapi pewarnaan demikian akan membuat antara titik �� dan �-l1 semua lintasannya akan mempunyai warna yang sama, sehingga haruslah ����,�� 3, hal tersebut berlaku jika � � 7 karena lintasan �� % �-l1 minimal mempunyai panjang 3, sehingga ���6-� � 3. karena ���6-� � 3 dan ���6-� $ 3 maka terbukti ���6-� 3, dengan � � 7. (iv) jjika � � 4 maka ����6-� 1 akan dibuktikan ����6-� � 1dan ����6-� $ 1. diketahui �"�#�6-� 2, sehingga ����6-� � 2 % 1 1. untuk membuktikan ����6-� $ 1, maka akan dibuktikan bahwa dengan 1 warna titik, dapat membentuk pelangi titik yang terhubung, artinya setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. ambil ����6-� 1, misalkan ada fungsi pewarnaan �:�m5-,kn ` �1� maka: ���� 1, sehingga setiap lintasan �v % �x akan melewati titik interior � dimana ���� 1, sedangkan setiap lintasan �v % � tidak ada titik interior karena terhubung langsung, dengan demikian setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. jadi terbukti ����6-� $ 1. karena ����6-� � 1dan ����6-� $ 1, maka ����6-� 1. d. graf kipas secara umum graf 8dapat dibentuk menjadi graf pewarnaan sisi yang terhubung, dengan warna sisi minimal 3, dan juga graf 8 dapat dibentuk menjadi graf pewarnaan titik yang terhubung, dengan warna titik minimal 1, yang ditampilkan dalam model pewarnaan berikut: �1 1 2 �1 �2 �3 �4 ��%1 1 2 2 3 ? ? ? ? ? ? 3 3 3 ? �� 1 me: gambar 8. graf kipas 8 teorema 4 graf kipas 8dengan � � l, maka bilangan rainbow connection pada graf 8adalah: ���8-� e1, z"n� � 22, z"n� 3 $ � $ 63, z"n� � � 7� bilangan rainbow vertex-connection pada graf 8 adalah: ����8-� 1, z"n� � � 2 bukti: graf kipas 8dibentuk dari penjumlahan graf komplit (5�� dan graf lintasan (,-� yaitu 8 5� ,-. dengan demikian graf kipas mempunyai (� 1� titik dan (2� % 1) sisi. misalkan �v � ��,-�, " 1,2,… ,�, dan � � ��5��, serta untuk semua �v terhubung langsung dengan �. (i) jika � 2, akan dibuktikan 8� adalah graf 5g. untuk semua ��,�� � �,�� terhubung langsung dengan �,sehingga diperoleh deg���� deg���� deg��� 2. karena graf 8� dengan 3 titik beraturan%2 maka 8� adalah graf komplit, jadi terbukti dengan � 2 maka ����� 1. (ii) jika 3 $ � $ 6 maka ����� 2 akan dibuktikan ����� � 2 dan ����� $ 2. graf 8dengan 4 $ � $ 6 bukan merupakan graf komplit, karena deg��v� $ 3 sedangkan jumlah titiknya 4 $ |8-| $ 7, sehingga ����� � 2. kemudian untuk membuktikan ����� $ 2 maka akan dibuktikan bahwa dengan 2 warna dapat membentuk 8 menjadi pelangi sisi yang terhubung, sehingga setiap dua titik terdapat lintasan dengan warna sisi yang berbeda. fungsi fuad adi saputra 132 volume 2 no. 3 november 2012 pewarnaan sisi �: �8-� ` �1,2� yang didefinisikan oleh ���v,�� 1 jika 1 $ " $o-�p, ���v,�� 2 jika o-�p $ " $ �, ���v,�v\1� 1 jika " ganjil, ���v,�v\1� 2 jika " genap. setiap lintasan �v % � atau � % �v hanya terdapat satu warna sisi. kemudian lintasan �v % �x, dengan 1 $ " $ ;-�< dan ;-�< $ z $ � pasti berbentuk lintasan pelangi 2 warna. sedangkan untuk lintasan �v % �x dengan " dan z sama-sama 1 $ " $ ;-�< atau dengan " dan z sama-sama ;-�< $ " $ �. lintasan terpanjang adalah �� % �h dengan � ;-�< , karena � $ 6 maka lintasan terpanjangnya �� % �g dan �h % �j maka membentuk lintasan pelangi 2 warna yang sisi-sinya anggota ,-, sehingga terbukti setiap dua titik terdapat lintasan dengan warna sisi yang berbeda, sehingga diperoleh ����� $ 2. jadi terbukti untuk 3 $ � $ 6 maka ����� 2. (iii) jika � � 7 maka ����� 3 akan dibuktikan ����� � 3 dan ����� $ 3. dibuktikan ����� $ 3 maka akan dibuktikan bahwa dengan 3 warna dapat membentuk 8menjadi pelangi sisi yang terhubung, sehingga setiap dua titik terdapat lintasan dengan warna sisi yang berbeda. jika fungsi pewarnaan sisi �: �8-� ` �1,2,3� yang didefinisikan oleh ���v,�� 1 jika " ganjil, ���v,�� 2 jika " genap, dan ��i� 3,ji � �,-� maka akan membentuk pelangi sisi terhubung, sehingga ���8-� $ 3. selanjutnya dibuktikan ���8-� � 3, dari hasil di atas didapat 8bukan graf komplit sehingga ���8-� � 2.ambil ���8-� 2 misalkan fungsi pewarnaan sisi �: �8-� `�1,2� didefinisikan ����,�� 1, maka ����\h,�� 2,dengan 3 $ � $ � % 1 karena tidak mungkin menggunakan lintasan sisi ,yang panjangnya � 3. sedangkan jika ���v,�v\1� 1 dengan " ganjil, ���v,�v\1� 2 dengan " genap, maka ����,�� ���g,�� 1 membentuk lintasan pelangi, karena panjang lintasan dengan sisi ,sama dengan 2 dan warnanya berbeda. pewarnaan demikian akan membuat antara titik ��\h dengan � 3 dan ��\h dengan � � % 1 semua lintasannya akan mempunyai warna yang sama, karena ���h,�� ���-,�� 2 dan lintasan dengan sisi ,-dengan � � 7 panjangnya lebih dari sama dengan 3, maka terbukti ���8-� � 3. karena ���8-� � 3 dan ���8-� $ 3 maka terbukti ���8-� 3, dengan � � 7. (iv) jika � � 2 maka ����8-� 1 akan dibuktikan ����8-� � 1 dan ����8-� $ 1. diketahui �"�#�8-� 2, sehingga ����8-� � 2 % 1 1. untuk membuktikan ����8-� $ 1, maka akan dibuktikan bahwa dengan 1 warna titik, dapat membentuk pelangi titik yang terhubung, artinya setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. ambil ����8-� 1,misalkan ada fungsi pewarnaan �:��8-� ` �1� maka: ���� 1, sehingga setiap lintasan �v % �x akan melewati titik interior � dimana ���� 1, sedangkan setiap lintasan �v % � tidak ada titik interior karena terhubung langsung, dengan demikian setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. jadi terbukti ����8-� $ 1. karena ����8-� � 1 dan ����8-� $ 1, maka ����8-� 1. e. graf kipas ganda secara umum graf �8dapat dibentuk menjadi graf pewarnaan sisi yang terhubung, dengan warna sisi minimal 3, dan juga graf �8 dapat dibentuk menjadi graf pewarnaan titik yang terhubung, dengan warna titik minimal 1, yang ditampilkan dalam model pewarnaan berikut: �1 �2 �1 �3 �4 �6 �5 1 �2 1 1 2 3 3 3 2 2 2 3 3 3 qme: 2 2 3 3 3 3 1 1 1 2 2 2 2 2 �7 �8 �9 �10 ��%1 �� 2 2 2 2 2 2 gambar 9. graf kipas ganda �8 teorema 5 graf kipas ganda �8dengan jumlah titik � � 2, maka bilangan rainbow connection pada graf �8 adalah: ����8-� t2, � $ 123, � � 13� bilangan rainbow vertex-connection pada graf �8 adalah: �����8-� 1 bukti: graf kipas ganda dibentuk dari penjumlahan antara gabungan dua graf komplit (25�� dan graf lintasan (,-� yaitu �8 25� ,-. dengan demikian graf kipas mempunyai (� 2� titik dan (3� % 1) sisi. bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 133 (i) ����8-� 2 jika 2 $ � $ 12. akan dibuktikan ����8-� � 2, dan ����8-� $ 2. diketahui �"�# ��8-� 2,karena ����� � �"�#��� maka diperoleh ����8-� � 2. misalkan �v � ��,-� dengan 2 $ � $ 12, " 1,2,… ,� dan �v � ��25�� dengan " 1,2, terdapat fungsi pewarnaan sisi �: ��8-� `�1,2� yang didefinisikan oleh ���v,��� 1 jika 1 $ " $ ;-�< , ���v,��� 2 jika o-�p r " $12, ���v,�v\1� 1 jika " ganjil, ���v,�v\1� 2 jika " genap, serta ���v,��� 1 jika 1 $ " $ ;-h< dan ;-�< r " $ ;gu-h <, kemudian ���v,��� 2 jika o-hp r " $ ;-�< dan ;gu-h < r " $ �, maka akan membentuk graf �8menjadi graf pelangi sisi yang terhubung dimana setiap dua titik terdapat lintasan dengan warna sisi yang berbeda. hal ini membuktikan bahwa ����8-� $ 2, jadi diperoleh dengan 2 $ � $ 12 maka ����8-� 2. (ii) ����8-� 3 jika � � 13. akan dibuktikan ����8-� � 3, dan ����8-� $ 3. pertama dibuktikan ����8-� � 3, ambil ����8-� 2, maka dibuktikan dengan menggunakan 2 warna sisi akan ada 2 titik yang semua lintasannya memiliki warna yang sama. misalkan �v � ��,-� dengan � � 13, " 1,2,…,� dan �v � ��25�� dengan " 1,2, terdapat fungsi pewarnaan sisi �: ��8-� ` �1,2�, akan dibentuk setiap dua titik dihubungkan oleh lintasan pelangi. misalkan lintasan �v % �x,karena ���v,�v\1� 1 jika " ganjil, ���v,�v\1� 2 jika " genap, maka z � " 3. jika ���v,��� 1 agar lintasan �v % �x terbentuk lintasan pelangi maka 〳m�x,��n 2. kemudian lintasan �v % �x\1, �v % �x\2, �v % �x\3 maka haruslah �m�x\1,��n �m�x\�,��n �m�x\3,��n 2,kemudian untuk lintasan �x % �x\3, karena �m�x,��n �m�x\3,��n 2 dan panjang lintasan dengan sisi ,sama dengan 3 pasti lintasan tersebut memiliki warna yang sama, sehingga �m�x,��n 1 dan �m�x\3,��n 2. selanjutnya lintasan �x % �x\4, �x % �x\5, �x % �x\6 maka haruslah �m�x\4,��n �m�x\5,��n �m�x\6,��n 2. pewarnaan demikian akan membuat lintasan �x\3 % �x\6 memiliki warna sisi yang sama, sehingga dengan 2 warna sisi saja maksimal cukup untuk titik �x sampai �x\5. begitu juga untuk �v sampai �v\5. sehingga dengan 2 warna sisi berlaku untuk 12 titik, sedangkan untuk � � 13 tidak bisa membuat �8menjadi graf pelangi sisi yang terhubung. jadi terbukti ����8-� � 3. selanjutnya jika fungsi pewarnaan sisi �: ��8-� ` �1,2,3� yang didefinisikan oleh ���v,��� 1 jika " ganjil, ���v,��� 2 jika " genap, ���v,��� 2 dan ��i� 3,ji � �,-� maka akan membentuk pelangi sisi terhubung, sehingga ����8-� $ 3.terbukti dengan � � 13 maka ����8-� 3. (iii) jika � � 2 maka �����8-� 1 akan dibuktikan �����8-� � 1 dan �����8-� $ 1. diketahui �"�#��8-� 2, sehingga �����8-� � 2 % 1 1. untuk membuktikan �����8-� $ 1, maka akan dibuktikan bahwa dengan 1 warna titik, dapat membentuk pelangi titik yang terhubung, artinya setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. ambil �����8-� 1, misalkan ada fungsi pewarnaan �:���8-� ` �1� maka: ����� 1, sehingga setiap lintasan �v % �x akan melewati titik interior �� dimana ����� 1, untuk lintasan �� % �� jika ���v� 1, maka akan melewati titik interior �v dimana ���v� 1, sedangkan setiap lintasan �v % �� dan �v % �� tidak ada titik interior karena terhubung langsung, dengan demikian setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. jadi terbukti �����8-� $ 1.karena �����8-� � 1 dan �����8-� $ 1, maka �����8-� 1. 2. bilangan rainbow connection pada jenis graf hasil perkalian kartesius pada graf khusus hasil perkalian kartesius akan ditampilkan 1 contoh graf yaitu graf tangga. graf tangga dibentuk dari perkalian kartesius graf lintasan dengan 2 titik (,�) dengan graf lintasan dengan n titik (,-�. tabel 2. pola bilangan ���9-� dan ����9-� no jenis graf bc�ve� bfc�ve� 1. 9� 2 1 2. 9g 3 2 3. 9h 4 3 4. 9i 5 4 5. 9j 6 5 �. 9� � % 1 teorema 6 pada graf tangga 9dengan banyak titik � �2,bilangan rainbow connection ���9-� �,dan fuad adi saputra 134 volume 2 no. 3 november 2012 bilangan rainbow vertex-connection ����9-� � % 1. bukti: graf tangga yang dinotasikan sebagai 9adalah suatu graf yang dibentuk dari operasi hasil kali kartesius antara graf lintasan dengan dua titik dan graf lintasan dengan n titik yaitu 9 ,�: ,-, dengan �v � ��,-�, " 1,2,…,� dan �v � ��,��, " 1,2. (i) akan dibuktikan ���9-� � �, dan ����9-� � � % 1. diketahui graf tangga 9memiliki panjang diameter �"�#�9-� �. karena ����� � �"�#��� dan ������ � �"�#��� %1, maka terbukti ����� � � dan ������ �� % 1. (ii) akan dibuktikan ���9-� $ �, dan ����9-� $ � % 1. misalkan graf 9dibagi 2 himpunan titik : dan u. di mana ��v,��� � ��:� dan ��v,��� � ��u�, terdapat fungsi pewarnaan sisi �: �9g� ` �1,2,…,��, maka: �m��v,���,��v\�,���n ", dengan " �1,2,…,� % 1�, �m��v,���,��v,���n � �m��v,���,��v\�,���n ", dengan " �1,2,…,� % 1�. pewarnaan sisi demikian akan membuat graf 9menjadi pelangi sisi yang terhubung, sehingga ���9-� $ �. selanjutnya misal fungsi pewarnaan titik �w:��9j� ` �1,2,…,� % 1�, maka: �w��v,��� ", dengan " �1,2,…,� % 1�, �w��v,��� ", dengan " �1,2,…,� % 1�, �w��-,��� �w��-,��� 1 pewarnaan titik demikian akan membuat graf 9menjadi pelangi titik yang terhubung, sehingga ����9-� $ � % 1. dari (i) dan (ii) terbukti ���9-� � dan ����9-� � % 1. 3. bilangan rainbow connection pada sebarang graf pada pembahasan ini, telah didapat model atau teorema dari bilangan rainbow connection dan bilangan rainbow vertexconnection pada jenis graf dari hasil penjumlahan dan perkalian kartesius dua graf. dengan berpikir induktif, maka dari pola-pola tersebut dapat disimpulkan ����� dan ������ pada sebarang graf. graf disini merupakan sebarang graf berhingga dan jika dioperasikan akan menghasilkan graf yang terhubung. kemudian dengan berpikir deduktif maka pola-pola bilangan rainbow connection dan bilangan rainbow vertex-connection pada sebarang graf tersebut dapat dibuktikan kebenarannya. a. bilangan rainbow connection pada graf hasil penjumlahan untuk menentukan bilangan rainbow connection pada graf hasil penjumlahan dengan obyek sebarang graf yaitu dengan cara menganalisis bilangan rainbow connection dari jenis graf hasil penjumlahan dua graf yang telah ditampilkan di atas. terdapat 5 contoh graf, yaitu graf komplit, graf bipartisi komplit, graf roda, graf kipas, dan graf kipas ganda. dari kelima graf tersebut dibagi menjadi dua bagian. bagian pertama adalah graf hasil dari penjumlahan dua graf komplit, sedangkan yang kedua adalah graf hasil dari penjumlahan dua bukan graf komplit. untuk bagian pertama yaitu graf hasil dari penjumlahan dua graf komplit dicontohkan oleh graf komplit 5-, sedangkan bagian kedua yaitu graf hasil dari penjumlahan dua bukan graf komplit dicontohkan oleh graf bipartisi komplit 5k,, graf roda 6-, graf kipas 8dan graf kipas ganda �8-. graf komplit 5bilangan ���5-� 1, sedangkan ����5-� 0. pada graf bipartisi 5k, komplit bilangan ��m5k,-n � 2 dan ���m5k,-n 1. untuk graf roda 6bilangan ���6-� � 2 dan bilangan ����6-� 1. graf kipas 8bilangan ���8-� � 2 dan ����8-� 1. sedangkan pada graf kipas ganda �8bilangan ����8-� � 2 dan �����8-� 1. melihat hasil di atas diperoleh suatu teorema bilangan rainbow connection dan bilangan rainbow vertex-connection pada graf hasil dari penjumlahan dua sebarang graf sebagai berikut: teorema 7 misalkan graf � adalah graf hasil penjumlahan sebarang graf �� dan sebarang graf ��, maka bilangan rainbow connection dari graf � adalah: ����� 1, �� dan �� adalah graf komplit ����� � 2, ��atau �� adalah bukan graf komplit sedangkan bilangan rainbow vertex-connection dari graf � adalah: ������ e0, ��dan �� adalah graf komplit1, ��atau �� adalah bukan graf komplit � bukti: penjumlahan dua graf ��dan �� yang dinotasikan � �� �� mempunyai himpunan titik ���� ����� � ����� dan himpunan sisi ��� ���� � ���� � ���|� � �������� � �������. [1] jika �� dan �� adalah graf komplit. misalkan �� 5k dan �� 5maka banyak titik dari graf � adalah # �. anggap � � �����dan � � ������, sebelum dioperasikan deg��� # % 1, sedangkan deg��� � % 1. kemudian dioperasikan penjumlahan antara �� dan ��, diperoleh bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 135 setiap titik pada graf ��terhubung langsung dengan setiap titik pada ��, ����|� � �������� � � �������. sehingga derajat setiap titik � bertambah sebanyak � menjadi deg��� # � % 1 dan derajat setiap titik � bertambah sebanyak # menjadi deg��� # � % 1. artinya setiap titik � ,� � ���� beraturan-�# � % 1� sehingga graf � adalah graf 5k\-. terbukti bahwa graf dari penjumlahan dua graf komplit mempunyai bilangan rainbow connection dan bilangan rainbow vertex-connection berturut-turut sebesar 1 dan 0. [2] jika �� atau �� adalah graf bukan komplit dan � �� ��. misalkan �v,�x � �����, dengan ", z 1,2,…,�, serta ��,�������, dengan �,. 1,2,…,# maka �,�,�,� � ����. akan dibuktikan �"�#��� 2, ada 3 kemungkinan: 1) �� graf komplit dan �� bukan graf komplit. jika �� graf komplit maka �"�#���� 1. �� bukan graf komplit maka �"�#���� n � 0. misalkan lintasan �� % �� adalah lintasan dengan panjang kurang dari sama dengan n. karena �� terhubung langsung dengan �v dan �� juga terhubung langsung dengan �v. maka lintasan �� % �� dapat dibuat melewati �v, sehingga panjang lintasannya menjadi 2. dengan demikian �"�#��� 2. 2) �� bukan graf komplit dan �� graf komplit. jika �� graf komplit maka �"�#���� 1. �� bukan graf komplit maka �"�#���� b � 0. misalkan lintasan �v % �x adalah lintasan dengan panjang kurang dari sama dengan b. karena �v terhubung langsung dengan �� dan �x juga terhubung langsung dengan ��. maka lintasan �v % �x dapat dibuat melewati ��, sehingga panjang lintasannya menjadi 2. dengan demikian �"�#��� 2. 3) �� bukan graf komplit dan �� bukan graf komplit. �� bukan graf komplit maka �"�#���� b �0. misalkan lintasan �v % �x adalah lintasan dengan panjang kurang dari sama dengan b. karena �v terhubung langsung dengan �� dan �x juga terhubung langsung dengan ��. maka lintasan �v % �x dapat dibuat melewati ��, sehingga panjang lintasannya menjadi 2. �� bukan graf komplit maka �"�#���� n �0. misalkan lintasan �� % �� adalah lintasan dengan panjang kurang dari sama dengan n. karena �� terhubung langsung dengan �v dan �� juga terhubung langsung dengan �v. maka lintasan �� % �� dapat dibuat melewati �v, sehingga panjang lintasannya menjadi 2. dengan demikian �"�#��� 2. dari 3 kasus di atas disimpulkan jika � �� �� kemudian ��atau �� adalah bukan graf komplit maka �"�# ��� 2. karena ����� � �"�# ��� maka terbukti ����� � 2. selanjutnya dibuktikan ������ � 1. akan dibuktikan �����8-� � 1 dan �����8-� $ 1. diketahui �"�#��� 2, sehingga ������ � 2 % 1 1. untuk membuktikan ������ $ 1, maka akan dibuktikan bahwa dengan 1 warna titik, dapat membentuk pelangi titik yang terhubung, artinya setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. ambil ������ 1, misalkan ada fungsi pewarnaan �:���� ` �1� maka: ���v� 1, sehingga setiap lintasan �� % �� akan melewati titik interior �v dimana ���� 1, untuk lintasan �v % �x jika ����� 1, maka akan melewati titik interior �v dimana ����� 1, sedangkan setiap lintasan �� % �v tidak ada titik interior karena terhubung langsung, dengan demikian setiap dua titik terdapat lintasan dengan warna titik interior yang berbeda. jadi terbukti ������ $ 1. karena ������ � 1dan ������ $ 1, maka ������ 1. b. bilangan rainbow connection pada graf hasil perkalian kartesius untuk menentukan bilangan rainbow connection graf hasil perkalian kartesius dengan sebarang graf yaitu dengan cara menganalisis bilangan rainbow connection dari graf khusus hasil perkalian kartesius dua graf yang dengan contoh graf tangga 9-. dari hasil pembahasan mengenai bilangan rainbow connection dan bilangan rainbow vertex-connection pada graf tangga 9-, diperoleh bilangan ����� � dan bilangan ������ � % 1. kedua pola tersebut dipengaruhi oleh bilangan ����� dan ������ dari graf sebelum dioperasikan oleh perkalian kartesius, yaitu ,� dengan ,-. untuk ����� �, didapat dari penjumlahan antara ����� pada ,� dengan ����� pada ,yang masing besarnya 1 dan � % 1, dimana penjumlahan kedua akan menghasilkan ����� 1 �� % 1� �. sedangkan untuk ������ � % 1, didapat dari penjumlahan antara ������ pada ,� dengan ������ pada ,yang masing besarnya 0 dan � % 2 yang kemudian dijumlah lagi dengan 1, sehingga akan dihasilkan ������ 0 �� % 2� 1 � % 1. fuad adi saputra 136 volume 2 no. 3 november 2012 dengan menggunakan contoh sebarang graf lainnya, misalkan graf sikel +h dikalikan kartesius dengan graf bipartisi 5�,�. x graf +4 ����� = 2 ������ = 1 graf 51,2 ����� = 2 ������ = 1 graf +4 x 51,2 ����� = 2 + 2 = 4 ������ = 1 + 1 + 1 = 3 x gambar 10. perkalian kartesius antara graf +h dengan graf 5�,� dari hasil di atas diperoleh bahwa �฀��� pada graf +h 5�,� adalah 4, yang didapat dari penjumlahan ����� pada graf +h yang bernilai 2 dengan ����� pada graf 5�,� yang juga bernilai 2. sedangkan ������ pada graf +h 5�,� adalah 3, yang didapat dari penjumlahan ������ pada graf +h yang bernilai 1 dengan ������ pada graf 5�,� yang juga bernilai 1 dan ditambah dengan 1. hasil ini bukanlah suatu kebetulan yang muncul dari contoh-contoh yang telah ditampilkan. akan tetapi ini adalah suatu pola yang terdapat pada setiap graf dari hasil perkalian kartesius dua graf. setiap sebarang graf �� jika dikalikan dengan sebarang graf ��, maka hasilnya akan isomorfik dengan graf �� yang titik-titiknya adalah graf ��. begitu juga sebaliknya hasilnya akan isomorfik dengan graf �� yang titik-titiknya adalah graf ��. dari contoh di atas, misalnya graf �� adalah graf sikel +h akan dikalikan kartesius dengan �� yaitu graf bipartisi 5�,�. maka menghasilkan graf dengan graf bipartisi 5�,� akan menggantikan titik-titik pada graf sikel +h. graf �2 graf �2 graf �2 graf �2 gambar 11. graf +h�5�,� dengan memperhatikan hasil-hasil di atas diperoleh sebagai berikut: teorema 8 graf � adalah graf hasil perkalian kartesius sebarang graf terhubung �� dan sebarang graf terhubung ��. bilangan rainbow connection dari graf � adalah: �"�# ���� �"�# ���� $ �����$ ������ ������ sedangkan bilangan rainbow vertex-connection dari graf � adalah: �"�# ���� �"�# ���� % 1 $ ������$ ������� ������� 1 bukti: graf � graf hasil kali kartesius adalah graf yang dinotasikan � ��� �� dan mempunyai titik ���� ������ �����, dan dua titik ���,��� dan ���,��� dari graf � terhubung langsung jika dan hanya jika �� �� dan ���� � ���� atau �� �� dan ���� � ����. �i� akan dibuktikan jika � ��� �� maka ����� � �"�# ���� �"�# ���� dan ����� $ ������ ������ diketahui jika � ��� �� maka �"�# ��� �"�# ���� �"�# ����,karena ����� � �"�# ��� maka ����� ��"�# ���� �"�# ����. selanjutnya misalkan ������ n, artinya dengan bilangan warna sisi minimum sebanyak n akan membuat graf �� menjadi graf pelangi sisi yang terhubung, dan ������ b, artinya dengan bilangan warna sisi minimum sebanyak b akan membuat graf �� menjadi graf pelangi sisi yang terhubung. kemudian �v,�x � �����, dengan ", z 1,2,…,�,serta ��,�� � �����, dengan �,. 1,2,…,# maka ��v,���,m�x,��n � ����. jika lintasan �v % �x membentuk lintasan pelangi dengan � warna sisi maka � $ n, dan jika �� %て� membentuk lintasan pelangi dengan _ warna sisi maka _ $ b. akan dibuktikan lintasan ��v,��� % ��x,��� membentuk lintasan pelangi dengan warna sisi $ n b. ada 4 kemungkinan, yaitu: 1) lintasan ��v,��� % ��x,��� dengan " z ��� � y .. himpunan dari titik-titik ��v,���,��x,��� dengan " z ��� � y . akan membentuk graf yang isomorfik dengan graf ��. dua titik tersebut bisa terhubung langsung dan juga bisa tidak terhubung langsung akan tetapi keduanya pasti dihubungkan oleh lintasan. karena setiap �� % �� membentuk lintasan pelangi dengan _ warna sisi di mana _ $ b. maka lintasan ��v,��� % ��v,��� juga akan membentuk lintasan pelangi dengan _ warna sisi di mana _ $ b. 2) lintasan ��v,��� % ��x,��� dengan " yz dan � . himpunan dari titik-titik ��v,���,��x,��� dengan " y z dan � . akan membentuk bilangan rainbow connection dari hasil operasi penjumlahan dan perkalian kartesius dua graf jurnal cauchy – issn: 2086-0382 137 graf yang isomorfik dengan graf ��. dua titik tersebut bisa terhubung langsung dan juga bisa tidak terhubung langsung akan tetapi keduanya pasti dihubungkan oleh lintasan. karena setiap �v % �x membentuk lintasan pelangi dengan � warna sisi di mana � $ n. maka lintasan ��v,��� % ��v,��� juga akan membentuk lintasan pelangi dengan � warna sisi di mana � $ n. 3) lintasan ��v,��� % ��x,��� dengan " z dan � . jika terdapat dua titik ��v,��� dan ��x,��� dengan " z dan � ., maka kedua titik tersebut sama, sehingga panjang lintasan sama dengan 0. 4) lintasan ��v,��� % ��x,��� dengan " yz dan � y . jika terdapat dua titik ��v,��� dan ��x,��� dengan " y z dan � y ., maka kedua titik tersebut tidak mungkin terhubung langsung. sehingga kemungkinan lintasan dengan panjang sisi terbesar adalah antara kedua titik ini. lintasan antara kedua titik ��v,��� dan ��x,��� akan membentuk lintasan ��v,��� %m�x,��n % m�x,��n, sehingga terdapat dua langkah. pertama lintasan ��v,��� %��x,���, lintasan ini akan membentuk lintasan pelangi dengan � warna sisi di mana � $ n. kemudian ditambah dengan lintasan m�x,��n% m�x,��n, yang membentuk lintasan pelangi dengan _ warna sisi di mana _ $ b. sehingga lintasan ��v,��� % ��x,��� akan membentuk lintasan pelangi dengan � _ warna sisi di mana � $ n dan _ $ b. jadi terbukti ����� � _ $ n b ������ ������. (ii) akan dibuktikan jika � ��� �� maka ������ � �"�# ���� �"�# ���� % 1 dan ������ $ ������� ������� 1 diketahui jika � ��� �� maka �"�# ��� �"�# ���� �"�# ����, karena ������ � �"�# ��� % 1 maka ������ ��"�# ���� �"�# ���� % 1. selanjutnya ������� � �"�# ���� % 1 dan ������� � �"�# ���� % 1, ini berakibat �"�# ���� $ ������� 1 dan �"�# ���� $������� 1. karena ������ � �"�# ���� �"�# ���� % 1 maka ������ $ ������� 1 ������� 1 % 1 sehingga diperoleh: ������ $ ������� ������� 1. jadi terbukti, jika � .� � �� maka : �"�# ���� �"�# ���� % 1 $ ������ $������� ������� 1. penutup berdasarkan hasil pembahasan, maka dapat diambil kesimpulan sebagai berikut : 1. penjumlahan dua graf ��dan �� yang dinotasikan � �� ��, diperoleh bilangan rainbow connection dari graf � adalah: ����� 1, �� dan �� graf komplit ����� � 2, ��atau �� bukan graf komplit sedangkan bilangan rainbow vertexconnection dari graf � adalah: ������ �0, ��dan �� graf komplit1, ��atau �� bukan graf komplit� 2. graf � graf hasil kali kartesius adalah graf yang dinotasikan � ��� ��, diperoleh bilangan rainbow connection dari graf � adalah: �"�# ���� �"�# ���� $ �����$ ������ ������ sedangkan bilangan rainbow vertexconnection dari graf � adalah: �"�# ���� �"�# ���� % 1 $ ������$ ������� ������� 1 daftar pustaka [1] abdullah bin muhammad. 2005. tafsir ibnu katsir jilid 8. bogor: pustaka imam as-syafi’i [2] abdussakir. 2007. ketika kiai mengajar matematika. malang: uin malang press [3] al-hifnawi, m. i. 2009. tafsir al-qurthubi. jakarta: pustaka azam. [4] al-maraghi, a. m. 1989. tafsir al-maraghi. semarang: cv. toha putra. [5] al-qarni , ‘a. 2008. tafsir muyassar, jilid 3. jakarta: qitshi press. [6] alisah, e & dharmawan, e. p. filsafat dunia matematika. jakarta: prestasi pustaka publisher. [7] bondy, j. a & murty, u.s.r. 1976. graph theory with applications. london: macmillan press [8] chandran, l. sunil. rainbow coloring of graph. (online: www.tcs.tifr.res.in/.../ nitk.../combinatore.pdf diakses pada tanggal 31 april 2012) [9] chartrand, g. & lesniak, l. 1986. graphs and digraphs second edition. california: a division of wadsworth, inc. [10] harary, f. 1969. graph theory. amerika: addison-wesley publishing company, inc. fuad adi saputra 138 volume 2 no. 3 november 2012 [11] j. a. gallian. 2007. a dynamic survey of graph labeling. electronic journal combinatorics. dynamic survey d#56 [12] krivelevich, m. & yuster, r. 2010. the rainbow connection of a graph is (at most) reciprocal to its minimum degree, school of mathematics, tel aviv university. [13] purwanto. 1998. matematika diskrit. malang: ikip malang. [14] quthb, s. 2004. tafsir fi zhilalil qur’an. jakarta: gema insani [15] y. caro, a. lev, y. roditty, z. tuza, and r. yuster, on rainbow connection, electronic journal of combinatorics 15 (2008), #r57 nova nevisa a.f faktorisasi graf baru yang dihasilkan dari pemetaan titik graf sikel pada bilangan bulat positif nova nevisa auliatul faizah1, h. wahyu h. irawan2 1mahasiswa jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang e-mail: novanevisa@yahoo.com, henky_lily@yahoo.com abstrak faktor merupakan subgraf merentang dari suatu graf. subgraf merentang terdiri dari himpunan pasangan titik yang tidak saling terhubung dan selalu berbentuk graf beraturan satu, ini dapat disebut sebagai graf yang memiliki 1-faktor. ketika himpunan titik dari graf sikel 𝐶𝑛 dipetakan pada bilangan bulat positif yang dibatasi oleh derajatnya maka akan menghasilkan graf baru 𝐶𝑛 ∗yang memiliki 1-faktor dengan ciri-ciri fungsi tertentu. tujuan penelitian ini adalah untuk mengetahui ciri-ciri fungsi yang menghasilkan graf baru 𝐶𝑛 ∗ yang dihasilkan dari graf 𝐶𝑛 akan memiliki 1-faktor. adapun langkah-langkah untuk memperoleh hasil dari penelitian ini adalah: (1) menggambar graf sikel 𝐶𝑛, (2) menentukan kemungkinan-kemungkinan dari fungsi 𝑓(𝐶𝑛) → {1,2}, (3) menentukan 𝐷(𝑥), (4) menentukan 𝑠(𝑥) dan 𝑆(𝑥), (5) menentukan graf baru 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗), (6) faktorisasi graf baru 𝐶𝑛 ∗ dengan menunjukkan himpunan pasangannya. hasil dari penelitian ini adalah ciri-ciri fungsi yang menghasilkan graf baru 𝐶𝑛 ∗ yang memiliki 1faktor dengan membedakan untuk banyak titik ganjil dan banyak titik genap sebagaimana berikut: 1. fungsi dengan banyak 𝑛 atau satu titik dipetakan ke 2 untuk 𝑛 ganjil 2. fungsi dengan banyak 𝑛 titik dipetakan ke 2 atau 1 untuk 𝑛 genap bagi penelitian selanjutnya diharapkan dapat mengembangkan penelitian ini untuk graf lainnya. kata kunci: faktorisasi, 1-faktor, f-faktor, graf sikel (𝐶𝑛 ) abstract factor is a spanning subgraph of a graph. spanning subgraph consist of a set of couples of vertices that disconnected each other and always organized as regular graph one, called as a graph that has 1factor. when a set of points of cycle graph (𝐶𝑛) mapped on a positive integerbordered by its degree, it will produce a new graph of 𝐶𝑛 ∗ that has 1-factor with the particular function’s characteristics. the aim of this study is to determine the function’s characteristics that produc a new graph of 𝐶𝑛 ∗ obtained from 𝐶𝑛 of having 1-factor. therefore, the steps to provide the result of the research are: (1) drawing the cycle graph 𝐶𝑛, (2) determining the possibility function of 𝑓(𝐶𝑛 ) → {1, 2}, (3) determining 𝐷(𝑥), (4) determining 𝑠(𝑥) dan 𝑆(𝑥), (5) determining a new graph 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗), (6) factorizing the new graph 𝐶𝑛 ∗ with showing a set matching. the results of this research are the function’s characteristics that producing a new graph of 𝐶𝑛 ∗ of having 1-factor with differentiate for many odd points and even points as follow: 1. a function of 𝑛 or one vertex has mapped to 2 for 𝑛 odd 2. a function of 𝑛 vertices is mapped to 2 or 1 for 𝑛 even. for the further research is expected to another graph. keyword: factoritation, 1-factor, f-factor, graph cycle (𝐶𝑛) pendahuluan teori graf merupakan salah satu cabang ilmu matematika yang sebenarnya sudah ada sejak dua ratus tahun yang lalu. graf digunakan untuk mempresentasikan objek-objek diskrit dan hubungan antara objek-objek tersebut. representasi visual dari graf adalah dengan menyatakan objek sebagai noktah, bulatan, atau titik, sedangkan hubungan antara objek dinyatakan dengan garis (munir, 2012). secara sistematis, menurut (chartrand & lesniak, 1986) mailto:novanevisa@yahoo.com mailto:henky_lily@yahoo.com nova nevisa a.f 139 volume 3 no. 3 november 2014 graf didefinisikan sebagai himpunan titik 𝑉(𝐺) yang tak kosong dan himpunan sisi 𝐸(𝐺) yang mungkin kosong yang mana keduanya saling berhubungan. dalam perkembangannya, jurnal pertama yang ditulis tentang teori graf muncul pada tahun 1736 oleh matematikawan terkenal dari swiss bernama euler (budayasa, 2007). perkembangan teori ini tidak hanya secara teoritis, tetapi juga secara aplikatif. dalam perkembangan teoritis, konsep-konsep yang ada dapat dibuktikan secara deduksi dan induksi. sedangkan perkembangan secara aplikatif merupakan terapan dari hasil teoritis sebagai penyelesaian dari suatu masalah nyata. salah satu topik yang menarik adalah membahas tentang faktorisasi. menurut (chartrand & lesniak, 1986), faktor adalah subgraf merentang dari suatu graf dan faktorisasi adalah penjumlahan sisi-sisi yang disjoint dari faktor-faktor suatu graf. selanjutnya r-reguler faktor dapat dinyatakan sebagai r-faktor. karena faktorisasi dari suatu graf akan selalu menghasilkan faktor-faktor graf yang beraturan satu, maka graf tersebut dapat dinyatakan memiliki 1-faktor. sedangkan pasangan sempurna adalah himpunan pasangan titik yang membentuk sisi pada suatu graf yang tidak saling terhubung langsung. melihat definisi pasangan sempurna ini dapat mewakili definisi dari faktor maka dapat dinyatakan bahwa suatu graf yang mengandung pasangan sempurna akan memiliki 1-faktor. sejauh ini masalah yang sudah dikaji tentang faktorisasi adalah faktorisasi graf komplit oleh vera mandailina tahun 2009 dan faktorisasi graf beraturan-r dengan order genap oleh asna bariroh tahun 2010, kedua masalah ini hanya membahas tentang faktorisasi pada graf yang hanya menghasilkan berapa banyak faktor pada suatu graf. jika permasalahan faktorisasi ini dikaitkan dengan sebuah pemetaan dari himpunan titik suatu graf pada bilangan bulat positif yang dibatasi oleh derajatnya maka akan berakibat terbentuknya graf baru yang mana graf baru tersebut tidak selalu memiliki 1-faktor. oleh karena itu peneliti tertarik untuk mengembangkannya lebih lanjut. metode penelitian penelitian ini bertujuan untuk mengembangkan permasalahan tentang faktorisasi, sehingga metode penelitian ini merupakan metode kepustakaan, pendekatan yang digunakan ialah pendekatan kualitatif yang pada umumnya tidak terjun ke lapangan dalam pencarian dan pengumpulan sumber datanya. sedangkan pola pembahasannya dari induktif ke deduktif, hal ini berarti pembahasannya dimulai dari hal-hal khusus menuju pada hal-hal umum (generalisasi). secara garis besar langkah penelitian ini sebagai berikut: 1) menggambar graf sikel (𝐶𝑛) dengan 𝑛 bilangan ganjil. 2) menentukan fungsi 𝑓: 𝑉(𝐶𝑛) → 𝑍 + dengan ketentuan 𝑓(𝑥) ≤ 𝑑(𝑥) ∀𝑥 ∈ 𝑉(𝐶𝑛), karena derajat pada graf sikel (𝐶𝑛) selalu dua maka himpunan 𝑍+ = {1,2} sehingga fungsinya dapat ditulis 𝑓: 𝑉(𝐶𝑛) → {1,2}. kemudian menuliskan semua kemungkinan pemetaan yang terjadi. 3) menentukan 𝐷(𝑥) = {𝑥𝛼 |𝛼 ∈ 𝐸(𝐶𝑛), 𝛼 adalah sisi yang terkait langsung dengan 𝑥; ∀𝑥 ∈ 𝑉(𝐶𝑛)}. 4) menentukan 𝑠(𝑥) = 𝑑(𝑥) − 𝑓(𝑥) ∀𝑥 ∈ 𝑉(𝐶𝑛) yang didefinisikan sebagai bilangan yang dihasilkan dari selisih antara derajat titik di graf sikel (𝐶𝑛) dan bilangan bulat positif dari fungsi 𝑓: 𝑉(𝐶𝑛) → 𝑍 +, kemudian menentukan 𝑆(𝑥) = {𝑥(𝑖)|1 ≤ 𝑖 ≤ 𝑠(𝑥)} yang berupa himpunan titik dari 𝑠(𝑥). 5) menentukan graf baru 𝐶𝑛 ⋆ = (𝑉 ⋆, 𝐸⋆), yang masing-masing dari 𝑉 ⋆ dan 𝐸⋆ didefinisikan sebagai berikut: a. 𝑉 ⋆ = 𝑉 ⋆1⋃𝑉 ⋆ 2 ; 𝑉 ⋆ 1 = ⋃ 𝐷(𝑥)𝑥∈𝑉 (𝐶𝑛) ; 𝑉 ⋆2 = ⋃ 𝑆(𝑥)𝑥∈𝑉 (𝐶𝑛) b. 𝐸⋆ = 𝐸⋆1⋃𝐸 ⋆ 2 ; 𝐸 ⋆ 1 = {𝑥𝛼 𝑦𝛼 |𝛼 = 𝑥𝑦 ∈ 𝐸(𝐶𝑛) ; 𝐸⋆ 2 = {𝑢𝑣|𝑢 ∈ 𝐷(𝑥) 𝑑𝑎𝑛 𝑣 ∈ 𝑆(𝑥), 𝑥 ∈ 𝑉 (𝐶𝑛)}. 6) faktorisasi graf baru 𝐶𝑛 ⋆. 7) membuat suatu konjektur berdasarkan ciriciri yang didapatkan. 8) merumuskan konjektur sebagai suatu teorema, kemudian dibuktikan kebenarannya sehingga dapat dinyatakan benar secara umum. 9) menuliskan laporan penelitian. teori dasar 1. definisi graf sikel sikel dengan 3 atau lebih titik adalah suatu graf sederhana yang titiknya tersusun mengikuti putaran seperti suatu jalan dengan dua titik yang terhubung langsung jika keduanya berturutan dan tidak terhubung langsung dengan lainnya. sebuah sikel dengan satu titik disebut loop, dan sikel dengan dua titik yang dihubungkan oleh dua sisi disebut sisi rangkap (bondy & murty, 2008). contoh 2.2: graf sikel 𝐶𝑛 dengan 𝑛 = 3,4,5,6 gambar 1 graf 𝐶3, 𝐶4, 𝐶5, 𝐶6 faktorisasi graf baru yang dihasilkan dari pemetaan titik graf sikel pada bilangan bulat positif cauchy – issn: 2086-0382 140 2. definisi subgraf merentang subgraf h dari graf g yang memiliki himpunan titik yang sama pada g atau jika subgraf h dengan 𝑉(𝐻) = 𝑉(𝐺), maka h disebut spanning subgraf (subgraf merentang) dari g (chartrand & lesniak, 1986). dari definisi sub graf merentang, peneliti memberikan contoh sebagai berikut: g g1 gambar 2 subgraf merentang graf g1 dari graf g 3. definisi matching matching (pasangan) pada graf g adalah himpunan pasangan titik yang membentuk sisi pada graf g yang tidak saling terhubung langsung (chartrand & lesniak, 1986). 4. definisi perfect matching m disebut sebagai perfect matching (pasangan sempurna) pada graf g, jika m merupakan suatu pasangan pada graf g, dan semua sisi di m menutup semua titik di g (bondy & murty, 2008). 5. definisi perfect matching m disebut sebagai perfect matching (pasangan sempurna) pada graf g, jika m merupakan suatu pasangan pada graf g, dan semua sisi di m menutup semua titik di g (bondy & murty, 2008). contoh diberikan graf g dan graf h sebagai berikut: 𝐺 𝐻 gambar 3 graf 𝐺 dan graf 𝐻 dari gambar 3 graf 𝐺 dengan 𝑀 = {𝑒1, 𝑒4} merupakan pasangan tidak sempurna dan graf 𝐻 dengan 𝑀 = {𝑒1, 𝑒3, 𝑒5} adalah pasangan sempurna. 6. definisi faktor faktor dari graf g adalah suatu subgraf merentang dari graf g (chartrand & lesniak, 1986). jika 𝐺1, 𝐺2, … , 𝐺𝑛 adalah faktor yang disjoint sisi pada graf g sedemikian hingga ⋃𝑖=1 𝑛 𝐸(𝐺𝑖 ) = 𝐸(𝐺) dimana 𝐺 = 𝐺1 ⊕ 𝐺2 ⊕ … ⊕ 𝐺𝑛 disebut sebagai penjumlahan sisi dari faktor-faktor 𝐺1, 𝐺2, … , 𝐺𝑛 . 7. definisi faktorisasi faktorisasi dari graf g adalah penjumlahan sisi dari faktor-faktor graf (chartrand & lesniak, 1986). suatu r-reguler faktor dari graf g dapat dinyatakan sebagai r-faktor dari g. oleh karena itu, suatu graf memiliki 1-faktor jika dan hanya jika mengandung suatu perfect matching. contoh faktorisasi dari graf g sebagai berikut: g g1 g2 g3 gambar 2.9 garf g dan faktor-faktornya berdasarkan buku extremal graph theory peneliti menyimpulkan bahwa untuk membangun graf baru 𝐺 ∗ dari graf 𝐺 yang memiliki 1-faktor adalah dengan langkah-langkah sebagai berikut: 1) menentukan fungsi 𝑓: 𝑉(𝐺) → 𝑍+ dengan ketentuan 𝑓(𝑥) ≤ 𝑑(𝑥) ∀𝑥 ∈ 𝑉(𝐺). 2) menentukan 𝐷(𝑥) = {𝑥𝛼 |𝛼 ∈ 𝐸(𝐺), 𝛼 adalah sisi yang terkait langsung dengan 𝑥; ∀𝑥 ∈ 𝑉(𝐺)}. 3) menentukan 𝑠(𝑥) = 𝑑(𝑥) − 𝑓(𝑥) ∀𝑥 ∈ 𝑉(𝐺)), kemudian menentukan 𝑆(𝑥) = {𝑥(𝑖)|1 ≤ 𝑖 ≤ 𝑠(𝑥)}. 4) menentukan graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) dengan 𝑉 ⋆ dan 𝐸⋆ didefinisikan sebagai berikut: a. 𝑉⋆ = {𝑉⋆1⋃𝑉 ⋆ 2|𝑉 ⋆ 1 = ⋃ 𝐷(𝑥)𝑥∈𝑉(𝐺) 𝑑𝑎𝑛 𝑉 ⋆ 2 = ⋃ 𝑆(𝑥)}𝑥∈𝑉(𝐺) b. 𝐸⋆ = {𝐸⋆1⋃𝐸 ⋆ 2|𝐸 ⋆ 1 = {𝑥𝛼 𝑦𝛼 |𝛼 = 𝑥𝑦 ∈ 𝐸(𝐺)} 𝑑𝑎𝑛 𝐸⋆2 = {𝑢𝑣|𝑢 ∈ 𝐷(𝑥), 𝑣 ∈ 𝑆(𝑥), 𝑥 ∈ 𝑉 (𝐺)}}. 5. faktorisasi graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) dengan menunjukkan adanya himpunan pasangan sempurna. dari graf baru 𝐺 ∗ yang dibangun tersebut maka terbentuklah lemma sebagai berikut: lemma 1 𝐺 memiliki f-faktor jika dan hanya jika 𝐺 ∗ memiliki 1-faktor (bollobas, 1978). bukti: misalkan 𝐺 memiliki f-faktor dengan himpunan sisi 𝐹 ⊂ 𝐸. maka 𝜓(𝐹) memuat sisi yang bebas dan dalam setiap himpunan 𝐷(𝑥) pasti titik 𝑠(𝑥) tidak tertutup oleh 𝜓(𝐹). untuk setiap 𝑥 kita menambahkan 𝑠(𝑥) yang bebas dengan sisi 𝐷(𝑥) − 𝑆(𝑥) yang menutup titik-titik yang lain dari 𝐷(𝑥). dengan cara ini kita memperoleh 𝐺 ∗ memiliki 1-faktor. sebaliknya jika 𝐺 ∗ mempunyai 1-faktor dengan himpunan sisi 𝐹∗ ⊂ 𝐸∗ kemudian 𝜓−1(𝐹∗ ∩ 𝐸∗) adalah himpunan sisi dari f-faktor di 𝐺. contoh diberikan graf komplit 𝐾4 nova nevisa a.f 141 volume 3 no. 3 november 2014 gambar 4 garf 𝐾4 1. menentukan fungsi 𝑓: 𝑉(𝐾4) → 𝑍 + dengan ketentuan 𝑓(𝑥) ≤ 𝑑(𝑥) ∀𝑥 ∈ 𝑉(𝐾4). gambar 5 salah satu kemungkinan fungsi 𝑓: 𝑉(𝐾4) → 𝑍 + 2. menentukan 𝐷(𝑥) = {𝑥𝛼 |𝛼 ∈ 𝐸(𝐾4), 𝛼 adalah sisi yang terkait langsung dengan 𝑥; ∀𝑥 ∈ 𝑉(𝐾4)}. 𝐷(𝑎) = {𝑎𝛼 |𝛼 ∈ 𝐸(𝐾4) dengan 𝛼 = 1, 𝛼 = 4 dan 𝛼 = 5} maka diperoleh 𝐷(𝑎) = {𝑎1, 𝑎4, 𝑎5} 𝐷(𝑏) = {𝑏𝛼 |𝛼 ∈ 𝐸(𝐾4) dengan 𝛼 = 1, 𝛼 = 2 dan 𝛼 = 6} maka diperoleh 𝐷(𝑏) = {𝑏1, 𝑏2, 𝑏6} 𝐷(𝑐) = {𝑐𝛼 |𝛼 ∈ 𝐸(𝐾4) dengan 𝛼 = 2, 𝛼 = 3 dan 𝛼 = 5} maka diperoleh 𝐷(𝑐) = {𝑐2, 𝑐3, 𝑐5} 𝐷(𝑑) = {𝑑𝛼 |𝛼 ∈ 𝐸(𝐾4) dengan 𝛼 = 3, 𝛼 = 4 dan 𝛼 = 6} maka diperoleh 𝐷(𝑑) = {𝑑3, 𝑑4, 𝑑6} 3. menentukan 𝑠(𝑥) = 𝑑(𝑥) − 𝑓(𝑥) ∀𝑥 ∈ 𝑉(𝐾4)), kemudian menentukan 𝑆(𝑥) = {𝑥(𝑖)|1 ≤ 𝑖 ≤ 𝑠(𝑥)}. 𝑠(𝑎) = 𝑑(𝑎) − 𝑓(𝑎) = 3 − 2 = 1 maka diperoleh 𝑆(𝑎) = {𝑎(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑎(1)} 𝑠(𝑏) = 𝑑(𝑏) − 𝑓(𝑏) = 3 − 2 = 1 maka diperoleh 𝑆(𝑏) = {𝑏(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑏(1) } 𝑠(𝑐) = 𝑑(𝑐) − 𝑓(𝑐) = 3 − 2 = 1 maka diperoleh 𝑆(𝑐) = {𝑐(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑐(1)} 𝑠(𝑑) = 𝑑(𝑑) − 𝑓(𝑑) = 3 − 2 = 1maka diperoleh 𝑆(𝑑) = {𝑑(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑑(1)} 4. menentukan graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) a. 𝑉⋆ = 𝑉⋆1 ⋃ 𝑉 ⋆ 2 𝑉⋆1 = ⋃ 𝐷(𝑥)𝑥∈𝑉{𝐾4) = {𝐷(𝑎) ∪ 𝐷(𝑏) ∪ 𝐷(𝑐) ∪ 𝐷(𝑑)} = {𝑎1, 𝑎4, 𝑎5, 𝑏1, 𝑏2, 𝑏6, 𝑐2, 𝑐3, 𝑐5, 𝑑3, 𝑑4, 𝑑6} 𝑉⋆ 2 = ⋃ 𝑆(𝑥)𝑥∈𝑉(𝐾4) = {𝑆(𝑎) ∪ 𝑆(𝑏) ∪ 𝑆(𝑐)} = {𝑎(1), 𝑏(1), 𝑐(1), 𝑑(1)} sehingga untuk 𝑉 ⋆ = 𝑉 ⋆1 ⋃ 𝑉 ⋆ 2 diperoleh 𝑉⋆ = {𝑎1, 𝑎4, 𝑎5, 𝑏1, 𝑏2, 𝑏6, 𝑐2, 𝑐3, 𝑐5, 𝑑3, 𝑑4, 𝑑6, 𝑎(1), 𝑏(1), 𝑐(1), 𝑑(1)} b. 𝐸⋆ = 𝐸⋆1 ⋃ 𝐸 ⋆ 2 𝐸⋆1 = {𝑥𝛼 𝑦𝛼 |𝛼 = 𝑥𝑦 ∈ 𝐸(𝐾4)} 𝛼 = 1 maka diperoleh 𝑎1𝑏1 𝛼 = 2 maka diperoleh 𝑏2𝑐2 𝛼 = 3 maka diperoleh 𝑐3𝑑3 𝛼 = 4 maka diperoleh 𝑑4𝑎4 𝛼 = 5 maka diperoleh 𝑎5𝑐5 𝛼 = 6 maka diperoleh 𝑏6𝑑6 jadi, 𝐸⋆1 = {𝑎1𝑏1, 𝑏2𝑐2, 𝑐3𝑑3, 𝑑4𝑎4, 𝑎5𝑐5, 𝑏6𝑑6} 𝐸⋆2 = {𝑢𝑣| 𝑢 ∈ 𝐷(𝑥), 𝑣 ∈ 𝑆(𝑥), 𝑥 ∈ 𝑉(𝐾4)} = {𝑎1𝑎(1), 𝑎4𝑎(1), 𝑎5𝑎(1), 𝑏1𝑏(1), 𝑏2𝑏(1), 𝑏6𝑏(1), 𝑐2𝑐(1), 𝑐3𝑐(1), 𝑐5𝑐(1), 𝑑3𝑑(1), 𝑑4𝑑(1), 𝑑6𝑑(1)} sehingga untuk 𝐸⋆ = 𝐸⋆1 ⋃ 𝐸 ⋆ 2 diperoleh 𝐸⋆ = {𝑎1𝑏1, 𝑏2𝑐2, 𝑐3𝑑3, 𝑑4𝑎4, 𝑎5𝑐5, 𝑏6𝑑6, 𝑎1𝑎(1), 𝑎4𝑎(1), 𝑎5𝑎(1), 𝑏1𝑏(1), 𝑏2𝑏(1), 𝑏6𝑏(1), 𝑐2𝑐(1), 𝑐3𝑐(1), 𝑐5𝑐(1), 𝑑3𝑑(1), 𝑑4𝑑(1), 𝑑6𝑑(1)} jadi, graf baru 𝐾4 ⋆ = (𝑉 ⋆, 𝐸⋆) adalah gambar 6 graf baru 𝐾4 ⋆ 5. faktorisasi graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) dengan menunjukkan adanya himpunan pasangan sempurna, yaitu 𝑀 = {𝑎1𝑏1, 𝑎5𝑐5, 𝑐3𝑑3, 𝑏6𝑑6, 𝑏2𝑏(1), 𝑐2𝑐(1), 𝑑4𝑑(1)}. karena graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) memuat pasangan sempurna maka graf baru 𝐺 ⋆ = (𝑉 ⋆, 𝐸⋆) memiliki 1-faktor. pembahasan 1. faktorisasi graf baru 𝑪𝟑 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟑) → {𝟏, 𝟐} pertama menggambar graf sikel 𝐶3 dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, dan 𝑐𝑎 = 3 gambar 7 graf sikel 𝐶3 selanjutnya menentukan semua kemungkinan fungsi 𝑓: 𝑉(𝐶3) → {1, 2} adalah sebagai berikut: gambar 8 semua kemungkinan fungsi 𝑓: 𝑉(𝐶3) → {1, 2} kemudian untuk langkah selanjutnya akan dikerjakan pada salah satu kemungkinan dari gambar 8 sebagai sampel kemungkinan yang dijadikan acuan untuk mengerjakan kemungkinan-kemungkinan fungsi yang lain. salah satu kemungkinan tersebut adalah: faktorisasi graf baru yang dihasilkan dari pemetaan titik graf sikel pada bilangan bulat positif cauchy – issn: 2086-0382 142 gambar 9 salah satu kemungkinan fungsi 𝑓: 𝑉(𝐶3) → {1, 2} selanjutnya menentukan 𝐷(𝑥) = {𝑥𝛼 |𝛼 = 𝐸(𝐶3), 𝛼 terkait langsung dengan 𝑥 ∈ 𝑉(𝐶3)} 𝐷(𝑎) = {𝑎𝛼 |𝛼 ∈ 𝐸(𝐶3) dengan 𝛼 = 1 dan 𝛼 = 3} maka diperoleh 𝐷(𝑎) = {𝑎1, 𝑎3} 𝐷(𝑏) = {𝑏𝛼 |𝛼 ∈ 𝐸(𝐶3) dengan 𝛼 = 1 dan 𝛼 = 2} maka diperoleh 𝐷(𝑏) = {𝑏1, 𝑏2} 𝐷(𝑐) = {𝑐𝛼 |𝛼 ∈ 𝐸(𝐶3) dengan 𝛼 = 2 dan 𝛼 = 3} maka diperoleh 𝐷(𝑐) = {𝑐2, 𝑐3} selanjutnya menentukan 𝑠(𝑥) = 𝑑(𝑥) − 𝑓(𝑥) ∀𝑥 ∈ 𝑉(𝐶3), kemudian menentukan 𝑆(𝑥) = {𝑥(𝑖)|1 ≤ 𝑖 ≤ 𝑠(𝑥)} 𝑠(𝑎) = 𝑑(𝑎) − 𝑓(𝑎) = 2 − 1 = 1 maka diperoleh 𝑆(𝑎) = {𝑎(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑎(1)} 𝑠(𝑏) = 𝑑(𝑏) − 𝑓(𝑏) = 2 − 2 = 0 maka diperoleh 𝑆(𝑏) = {𝑏(𝑖)|1 ≤ 𝑖 ≤ 0} = { } 𝑠(𝑐) = 𝑑(𝑐) − 𝑓(𝑐) = 2 − 1 = 1 maka diperoleh 𝑆(𝑐) = {𝑐(𝑖)|1 ≤ 𝑖 ≤ 1} = {𝑐(1)} selanjutnya menentukan graf baru 𝐶3 ⋆ = (𝑉 ⋆, 𝐸⋆) a. 𝑉 ⋆ = 𝑉 ⋆1 ⋃ 𝑉 ⋆ 2 𝑉 ⋆1 = ⋃ 𝐷(𝑥)𝑥∈𝑉{𝐶3) = {𝐷(𝑎) ∪ 𝐷(𝑏) ∪ 𝐷(𝑐)} = {𝑎1, 𝑎3, 𝑏1, 𝑏 2, 𝑐2, 𝑐3} 𝑉 ⋆ 2 = ⋃ 𝑆(𝑥)𝑥∈𝑉(𝐶3) = {𝑆(𝑎) ∪ 𝑆(𝑏) ∪ 𝑆(𝑐)} = {𝑎(1), 𝑐(1)} sehingga untuk 𝑉 ⋆ = 𝑉 ⋆1 ⋃ 𝑉 ⋆ 2 diperoleh 𝑉 ⋆ = {𝑎1, 𝑎3, 𝑏1, 𝑏2, 𝑐2, 𝑐3, 𝑎(1), 𝑐(1)} b. 𝐸⋆ = 𝐸⋆1 ⋃ 𝐸 ⋆ 2 𝐸⋆1 = {𝑥𝛼 𝑦𝛼 |𝛼 = 𝑥𝑦 ∈ 𝐸(𝐶3)} 𝛼 = 1 maka diperoleh 𝑎1𝑏1 𝛼 = 2 maka diperoleh 𝑏2𝑐2 𝛼 = 3 maka diperoleh 𝑐3𝑎3 jadi, 𝐸⋆1 = {𝑎1𝑏1, 𝑏2𝑐2, 𝑐3𝑎3} 𝐸⋆2 = {𝑢𝑣| 𝑢 ∈ 𝐷(𝑥), 𝑣 ∈ 𝑆(𝑥), 𝑥 ∈ 𝑉(𝐶3)} = {𝑎1𝑎(1), 𝑎3𝑎(1), 𝑐2𝑐(1), 𝑐3𝑐(1)} sehingga untuk 𝐸⋆ = 𝐸⋆1 ⋃ 𝐸 ⋆ 2 diperoleh 𝐸⋆ = {𝑎1𝑏1, 𝑏2𝑐2, 𝑐3𝑎3, 𝑎1𝑎(1), 𝑎3𝑎(1), 𝑐2𝑐(1), 𝑐3𝑐(1)} jadi, graf baru 𝐶3 ⋆ = (𝑉 ⋆, 𝐸⋆) dari kemungkinan fungsi 𝑓(𝑎) = 1, 𝑓(𝑏) = 2, dan 𝑓(𝑐) = 1 adalah: gambar 10 graf baru 𝐶3 ⋆ dari sampel kemungkinan fungsi 𝑓: 𝑉(𝐶3) → {1, 2} selanjutnya faktorisasi graf baru 𝐶3 ⋆. dari gambar 10 didapatkan pasangan 𝑀 = {𝑏2𝑐2, 𝑐(1)𝑐3, 𝑎3𝑎(1), 𝑎1𝑏1 } sebagai himpunan pasangan titik yang membentuk sisi yang tidak saling terhubung langsung. karena titik-titik pada gambar 10 bisa berpasang-pasangan dan merupakan pasangan sempurna, maka berakibat graf baru 𝐶3 ⋆ dari fungsi 𝑓(𝑎) = 1, 𝑓(𝑏) = 2, dan 𝑓(𝑐) = 1 memiliki 1-faktor. selanjutnya untuk pembahasan kemungkinankemungkinan dari fungsi 𝑓: 𝑉(𝐶3) → {1, 2} yang lain dapat dapat dilakukan dengan cara yang sama dan disimpulkan bahwa kemungkinankemungkinan fungsi yang menghasilkan graf baru 𝐶3 ⋆ yang memiliki 1-faktor adalah sebagai berikut: gambar 11 kemungkinan fungsi 𝑓: 𝑉(𝐶3) → {1, 2} yang memiliki 1-faktor maka dapat dibuat dugaan mengenai ciri-ciri fungsi yang mengakibatkan graf baru 𝐶3 ⋆ yang dihasilkan dari fungsi 𝑓: 𝑉(𝐶3) → {1, 2} akan memiliki 1-faktor dengan melihat banyaknya titik yang dipetakan yaitu fungsi dengan banyak 𝑛 titik atau hanya satu titik dipetakan ke 2. untuk graf 𝑪𝒏 selanjutnya akan dilakukan dengan cara yang sama dan hanya akan ditunjukkan fungsi yang memiliki 1-faktor. 2. faktorisasi graf baru 𝑪𝟒 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟒) → {𝟏, 𝟐} gambar graf sikel (𝐶4) dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, 𝑐𝑑 = 3, 𝑑𝑒 = 4 gambar 12 graf sikel 𝐶4 kemungkinan-kemungkinan fungsi yang menghasilkan graf baru 𝐶4 ⋆ yang memiliki 1faktor adalah sebagai berikut: gambar 13 kemungkinan fungsi 𝑓: 𝑉(𝐶4) → {1, 2} yang memiliki 1-faktor 3. faktorisasi graf baru 𝑪𝟓 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟓) → {𝟏, 𝟐} gambar graf sikel 𝐶5 dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, 𝑐𝑑 = 3, 𝑑𝑒 = 4, 𝑒𝑎 = 5 gambar 14 graf sikel 𝐶5 kemungkinan-kemungkinan fungsi yang menghasilkan graf baru 𝐶5 ⋆ yang memiliki 1faktor adalah sebagai berikut: nova nevisa a.f 143 volume 3 no. 3 november 2014 gambar 15 kemungkinan fungsi 𝑓: 𝑉(𝐶5) → {1, 2} yang memiliki 1-faktor 4. faktorisasi graf baru 𝑪𝟔 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟔) → {𝟏, 𝟐} gambar graf sikel (𝐶6) dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, 𝑐𝑑 = 3, 𝑑𝑒 = 4, 𝑒𝑓 = 5, 𝑓𝑎 = 6 gambar 3.10 graf sikel 𝐶6 kemungkinan-kemungkinan fungsi yang menghasilkan graf baru 𝐶6 ⋆ yang memiliki 1faktor adalah sebagai berikut: gambar 15 kemungkinan fungsi 𝑓: 𝑉(𝐶6) → {1, 2} yang memiliki 1-faktor 5. faktorisasi graf baru 𝑪𝟕 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟕) → {𝟏, 𝟐} gambar graf sikel (𝐶7) dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, 𝑐𝑑 = 3, 𝑑𝑒 = 4, 𝑒𝑓 = 5, 𝑓𝑔 = 6, 𝑔𝑎 = 7 gambar 16 graf sikel 𝐶7 kemungkinan-kemungkinan fungsi yang menghasilkan graf baru 𝐶7 ⋆ yang memiliki 1faktor adalah sebagai berikut: gambar 17 kemungkinan fungsi 𝑓: 𝑉(𝐶7) → {1, 2} yang memiliki 1-faktor 6. faktorisasi graf baru 𝑪𝟖 ⋆ yang dihasilkan dari fungsi 𝒇: 𝑽(𝑪𝟖) → {𝟏, 𝟐} pertama menggambar graf sikel (𝐶8) dengan 𝑎𝑏 = 1, 𝑏𝑐 = 2, 𝑐𝑑 = 3, 𝑑𝑒 = 4, 𝑒𝑓 = 5, 𝑓𝑔 = 6, 𝑔ℎ = 7, ℎ𝑎 = 8 gambar 18 graf sikel 𝐶8 kemungkinan-kemungkinan fungsi yang menghasilkan graf baru 𝐶8 ⋆ yang memiliki 1faktor adalah sebagai berikut: gambar 19 kemungkinan fungsi 𝑓: 𝑉(𝐶8) → {1, 2} yang memiliki 1-faktor 7. dugaan ciri-ciri fungsi 𝒇: 𝑽(𝑪𝒏) → {𝟏, 𝟐} yang mengakibatkan graf baru 𝑪𝒏 ⋆ memiliki 1-faktor ciri-ciri fungsi yang mengakibatkan graf baru 𝐶𝑛 ⋆ yang dihasilkan dari fungsi 𝑓: 𝑉(𝐶𝑛) → {1, 2} akan memiliki 1-faktor adalah: untuk 𝑛 ganjil a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 b. fungsi dengan sebanyak hanya satu titik dipetakan ke 2. untuk 𝑛 genap a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 b. fungsi dengan sebanyak 𝑛 titik dipetakan ke 1. teorema 1: fungsi yang mengakibatkan graf baru 𝐶𝑛 ⋆ yang dihasilkan dari kemungkinankemungkinan fungsi 𝑓: 𝑉(𝐶𝑛 ) → {1, 2} dapat memiliki 1-faktor untuk 𝑛 ganjil (𝑛 ≥ 3) adalah fungsi dengan sebanyak 𝑛 titik atau hanya satu titik dipetakan ke 2. bukti: misal 𝐶𝑛 adalah graf sikel dengan 𝑛 ganjil (𝑛 ≥ 3) dengan 𝑉(𝐶𝑛) = {𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑛 } dan 𝐸(𝐶𝑛) = {𝑒1, 𝑒2, 𝑒3, … , 𝑒𝑛} dengan 𝑒𝑖 = 𝑣𝑖 𝑣𝑖+1 untuk 𝑖 = 1,2, … , 𝑛 − 1 dan 𝑒𝑛 = 𝑣𝑛 𝑣1. misalkan 𝐶𝑛 ∗ adalah graf baru yang dihasilkan dari kemungkinan 𝑓: 𝑉(𝐶𝑛) → {1, 2}. akan ditunjukkan bahwa 𝐶𝑛 ∗ memiliki 1-faktor jika fungsinya memetakan sebanyak 𝑛 titik atau hanya satu titik dipetakan ke 2. misal gambar graf 𝐶𝑛 adalah: gambar 20 graf sikel 𝐶𝑛 untuk 𝑛 ganjil selanjutnya menentukan fungsi berdasarkan ciri-ciri yang telah ditentukan, yaitu: a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 misalkan 𝑓 fungsi dari 𝑉(𝐶𝑛) ke {1, 2} dengan 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛. maka diperoleh: 𝐷(𝑥) = {𝑣1 𝑒𝑛 , 𝑣1 𝑒1 , 𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 …, 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝑆(𝑥) = { } 𝑉 ⋆ = {𝑣1𝑒𝑛 , 𝑣1𝑒1 , 𝑣2 𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 …, 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝐸⋆ = {𝑣1 𝑒1 𝑣2𝑒1 , 𝑣2 𝑒2 𝑣3 𝑒2 , … , 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 } faktorisasi graf baru yang dihasilkan dari pemetaan titik graf sikel pada bilangan bulat positif cauchy – issn: 2086-0382 144 jadi graf baru 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗) dengan fungsi 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛 adalah: gambar 21 graf baru 𝐶𝑛 ∗ untuk 𝑛 ganjil dari fungsi 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛 selanjutnya dari gambar 21 dilakukan faktorisasi dengan menunjukkan adanya pasangan yaitu dengan melihat pengembangan titik yang terjadi dari setiap titik di graf sikel 𝐶𝑛 berdasarkan masingmasing pemetaannya sebagaimana berikut: untuk 𝑣1 dipetakan ke 2, maka diperoleh 𝐷(𝑣1) = {𝑣1 𝑒𝑛 , 𝑣1𝑒 1 } dan 𝑆(𝑣1) = { }. jadi titik 𝑣1 berkembang menjadi {𝑣1 𝑒𝑛 , 𝑣1𝑒 1 } untuk 𝑣2 dipetakan ke 2, maka diperoleh 𝐷(𝑣2) = {𝑣2 𝑒1 , 𝑣2 𝑒2 } dan 𝑆(𝑣2) = { }. jadi titik 𝑣2 berkembang menjadi {𝑣2𝑒1 , 𝑣2 𝑒2 } ⋮ untuk 𝑣𝑛 dipetakan ke 2, maka diperoleh 𝐷(𝑣𝑛 ) = { 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } dan 𝑆(𝑣𝑛 ) = { }. jadi titik 𝑣𝑛 berkembang menjadi {𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 }. dari pengembangan titik ini dapat dilihat bahwa untuk setiap titik yang dipetakan ke 2 selalu berkembang menjadi 2 titik, karena sebanyak 𝑛 titik yang dipetakan ke 2 maka banyak pengembangannya adalah 2𝑛 titik, dan karena 𝑛 = 2𝑘 + 1 maka 2𝑛 = 2(2𝑘 + 1) bernilai genap. kemudian dari simulasi gambar 3.17 terlihat bahwa sisi-sisi yang terbentuk pada graf baru 𝐶𝑛 ∗ berupa sisisisi yang berselang-seling dengan 𝑀 = {𝑣1𝑒1 𝑣2 𝑒1 , 𝑣2 𝑒2 𝑣3 𝑒2 , … , 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 }, maka dapat dipastikan graf baru 𝐶𝑛 ∗ dengan fungsi sebanyak 𝑛 titik dipetakakan ke 2 akan selalu memiliki 1-faktor. b. fungsi dengan sebanyak hanya satu titik dipetakan ke 2 misalkan 𝑓 fungsi dari 𝑉(𝐶𝑛) ke {1, 2} dengan 𝑓(𝑣𝑖 ) = 1 dan 𝑓(𝑣1) = 2 untuk 𝑖 = 2, 3, … , 𝑛. maka diperoleh: 𝐷(𝑥) = {𝑣1 𝑒𝑛 , 𝑣1 𝑒1 , 𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 …, 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝑆(𝑥) = {𝑣2(1), 𝑣3(1), … , 𝑣𝑛(1)} 𝑉 ⋆ = {𝑣1𝑒𝑛 , 𝑣1𝑒1 , 𝑣2 𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 … , 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 , 𝑣2(1), 𝑣3(1), … , 𝑣𝑛 (1)} 𝐸⋆ = {𝑣1 𝑒1 𝑣2 𝑒1 , 𝑣2𝑒2 𝑣3𝑒2 , … , 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 , 𝑣2 𝑒1 𝑣2(1), 𝑣2 𝑒2 𝑣2(1), 𝑣3 𝑒2 𝑣3 (1), 𝑣3 𝑒3 𝑣3(1), …, 𝑣𝑛 𝑒𝑛−1 𝑣𝑛 (1), 𝑣𝑛 𝑒𝑛 𝑣𝑛 (1)} jadi graf baru 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗) dengan fungsi 𝑓(𝑣𝑖 ) = 1 dan 𝑓(𝑣1) = 2 untuk 𝑖 = 2, 3, … , 𝑛 adalah: gambar 22 graf baru 𝐶𝑛 ∗ untuk 𝑛 ganjil dari fungsi 𝑓(𝑣𝑖 ) = 1 dan 𝑓(𝑣1) = 2 untuk 𝑖 = 2, 3, … , 𝑛 selanjutnya dari gambar 22 dilakukan faktorisasi dengan menunjukkan adanya pasangan yaitu dengan melihat pengembangan titik yang terjadi dari setiap titik di graf sikel 𝐶𝑛 berdasarkan masingmasing pemetaannya sebagaimana berikut: untuk 𝑣1 dipetakan ke 2, maka diperoleh 𝐷(𝑣1) = {𝑣1 𝑒1 , 𝑣1 𝑒𝑛 } dan 𝑆(𝑣1) = { }. jadi titik 𝑣1 berkembang menjadi {𝑣1 𝑒1 , 𝑣1 𝑒𝑛 } untuk 𝑣2 dipetakan ke 1, maka diperoleh 𝐷(𝑣2) = {𝑣2 𝑒1 , 𝑣2 𝑒2 } dan 𝑆(𝑣2) = { 𝑣2(1)}. jadi titik 𝑣2 berkembang menjadi {𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣2(1)} ⋮ untuk 𝑣𝑛 dipetakan ke 1, maka diperoleh 𝐷(𝑣𝑛 ) = {𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } dan 𝑆(𝑣𝑛 ) = {𝑣𝑛 (1) }. jadi titik 𝑣𝑛 berkembang menjadi {𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 , 𝑣𝑛(1)}. dari pengembangan titik ini dapat dilihat bahwa untuk setiap titik yang dipetakan ke 2 selalu berkembang menjadi 2 titik, karena sebanyak satu titik yang dipetakan ke 2 maka banyak pengembangannya adalah 2 titik, sedangkan untuk setiap titik yang dipetakan ke 1 selalu berkembang menjadi 3 titik, karena sebanyak 𝑛 − 1 titik yang dipetakan ke 1 maka banyak pengembangannya adalah 3(𝑛 − 1 ). jadi, secara keseluruhan perkembangan titiknya adalah sebesar 2 + 3(𝑛 − 1 ) titik. karena 𝑛 = 2𝑘 + 1 maka 2 + 3(𝑛 − 1 ) = 6𝑘 + 2 bernilai genap. kemudian dari simulasi gambar 3.18 terlihat bahwa sisisisi yang terbentuk pada graf baru 𝐶𝑛 ∗ akan selalu berbentuk lintasan, sehingga dapat ditunjukkan himpunan pasangannya adalah 𝑀 = {𝑣1 𝑒1 𝑣2𝑒1 , 𝑣2 𝑒2 𝑣2(1), 𝑣3𝑒2 𝑣3(1) … , 𝑣𝑛 𝑒𝑛−1 𝑣𝑛 (1), 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 }, maka dapat dipastikan graf baru 𝐶𝑛 ∗ dengan fungsi sebanyak hanya satu titik dipetakakan ke 2 akan selalu memiliki 1-faktor. teorema 2: fungsi yang mengakibatkan graf baru 𝐶𝑛 ⋆ yang dihasilkan dari kemungkinannova nevisa a.f 145 volume 3 no. 3 november 2014 kemungkinan fungsi 𝑓: 𝑉(𝐶𝑛 ) → {1, 2} dapat memiliki 1-faktor untuk 𝑛 genap (𝑛 ≥ 4) adalah fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 atau ke 1. bukti misal 𝐶𝑛 adalah graf sikel dengan 𝑛 genap (𝑛 ≥ 4) dengan 𝑉(𝐶𝑛) = {𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑛 } dan 𝐸(𝐶𝑛) = {𝑒1, 𝑒2, 𝑒3, … , 𝑒𝑛} dengan 𝑒𝑖 = 𝑣𝑖 𝑣𝑖+1 untuk 𝑖 = 1,2, … , 𝑛 − 1 dan 𝑒𝑛 = 𝑣𝑛 𝑣1. misalkan 𝐶𝑛 ∗ adalah graf baru yang dihasilkan dari kemungkinan 𝑓: 𝑉(𝐶𝑛) → {1, 2}. akan ditunjukkan bahwa 𝐶𝑛 ∗ memiliki 1-faktor jika fungsinya memetakan sebanyak 𝑛 titik dipetakan ke 2 atau ke 1. misal gambar graf 𝐶𝑛 adalah: gambar 23 graf sikel 𝐶𝑛 untuk 𝑛 genap selanjutnya menentukan fungsi berdasarkan ciri-ciri yang telah ditentukan, yaitu: a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 misalkan 𝑓 fungsi dari 𝑉(𝐶𝑛) ke {1, 2} dengan 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛. maka diperoleh: 𝐷(𝑥) = {𝑣1 𝑒𝑛 , 𝑣1 𝑒1 , 𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 …, 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝑆(𝑥) = { } 𝑉 ⋆ = {𝑣1𝑒𝑛 , 𝑣1𝑒1 , 𝑣2 𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 … , 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝐸⋆ = {𝑣1 𝑒1 𝑣2𝑒1 , 𝑣2 𝑒2 𝑣3 𝑒2 , … , 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 } jadi graf baru 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗) dengan fungsi 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛 adalah: gambar 24 graf baru 𝐶𝑛 ∗ untuk 𝑛 genap dari fungsi 𝑓(𝑣𝑖 ) = 2 untuk 𝑖 = 1, 2, … , 𝑛 selanjutnya dari gambar 24 dilakukan faktorisasi dengan menunjukkan adanya pasangan yaitu dengan melihat pengembangan titik yang terjadi dari setiap titik di graf sikel 𝐶𝑛 berdasarkan masingmasing pemetaannya sebagaimana berikut: untuk 𝑣1 dipetakan ke 2, maka diperoleh 𝐷(𝑣1) = {𝑣1 𝑒𝑛 , 𝑣1𝑒 1 } dan 𝑆(𝑣1) = { }. jadi titik 𝑣1 berkembang menjadi {𝑣1 𝑒𝑛 , 𝑣1𝑒 1 } untuk 𝑣2 dipetakan ke 2, maka diperoleh 𝐷(𝑣2) = {𝑣2 𝑒1 , 𝑣2 𝑒2 } dan 𝑆(𝑣2) = { }. jadi titik 𝑣2 berkembang menjadi {𝑣2𝑒1 , 𝑣2 𝑒2 } ⋮ untuk 𝑣𝑛 dipetakan ke 2, maka diperoleh 𝐷(𝑣𝑛 ) = { 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } dan 𝑆(𝑣𝑛 ) = { }. jadi titik 𝑣𝑛 berkembang menjadi {𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 }. dari pengembangan titik ini dapat dilihat bahwa untuk setiap titik yang dipetakan ke 2 selalu berkembang menjadi 2 titik, karena sebanyak 𝑛 titik yang dipetakan ke 2 maka banyak pengembangannya adalah 2𝑛 titik, dan karena 𝑛 = 2𝑘 + 2 maka 2𝑛 = 2(2𝑘 + 2) bernilai genap. kemudian dari simulasi gambar 3.20 terlihat bahwa sisi-sisi yang terbentuk pada graf baru 𝐶𝑛 ∗ berupa sisisisi yang berselang-seling dengan 𝑀 = {𝑣1𝑒1 𝑣2 𝑒1 , 𝑣2 𝑒2 𝑣3 𝑒2 , … , 𝑣𝑛+1𝑒𝑛+1 𝑣𝑛+2𝑒𝑛+1 , 𝑣𝑛+2𝑒𝑛+2 𝑣1 𝑒𝑛+2 }, maka dapat dipastikan graf baru 𝐶𝑛 ∗ dengan fungsi sebanyak 𝑛 titik dipetakakan ke 2 akan selalu memiliki 1faktor. b. fungsi dengan banyak 𝑛 titik dipetakan ke 1 misalkan 𝑓 fungsi dari 𝑉(𝐶𝑛) ke {1, 2} dengan 𝑓(𝑣𝑖 ) = 1 untuk 𝑖 = 1, 2, … , 𝑛. maka diperoleh: 𝐷(𝑥) = {𝑣1 𝑒𝑛 , 𝑣1 𝑒1 , 𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 …, 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } 𝑆(𝑥) = { 𝑣1(1), 𝑣2(1), 𝑣3(1), … , 𝑣𝑛 (1)} 𝑉 ⋆ = {𝑣1𝑒𝑛 , 𝑣1𝑒1 , 𝑣2 𝑒1 , 𝑣2 𝑒2 , 𝑣3 𝑒2 , 𝑣3 𝑒3 … , 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 , 𝑣1(1), 𝑣2(1), 𝑣3(1), … , 𝑣𝑛 (1)} 𝐸⋆ = {𝑣1 𝑒1 𝑣2 𝑒1 , 𝑣2𝑒2 𝑣3𝑒2 , … , 𝑣𝑛 𝑒𝑛 𝑣1 𝑒𝑛 , 𝑣1 𝑒𝑛 𝑣1(1), 𝑣1 𝑒1 𝑣1(1), 𝑣2 𝑒1 𝑣2(1), 𝑣2 𝑒2 𝑣2(1), 𝑣3 𝑒2 𝑣3(1), 𝑣3 𝑒3 𝑣3(1), … , 𝑣𝑛 𝑒𝑛−1 𝑣𝑛 (1), 𝑣𝑛 𝑒𝑛 𝑣𝑛 (1)} jadi graf baru 𝐶𝑛 ∗ = (𝑉 ∗, 𝐸∗) dengan fungsi 𝑓(𝑣𝑖 ) = 1 untuk 𝑖 = 1, 2, … , 𝑛 adalah: gambar 25 graf baru 𝐶𝑛 ∗ untuk 𝑛 genap dari fungsi 𝑓(𝑣𝑖 ) = 1 untuk 𝑖 = 1, 2, … , 𝑛 selanjutnya dari gambar 25 dilakukan faktorisasi dengan menunjukkan adanya pasangan yaitu dengan melihat pengembangan titik yang terjadi dari setiap titik di graf sikel 𝐶𝑛 berdasarkan masingmasing pemetaannya sebagaimana berikut: untuk 𝑣1 dipetakan ke 1, maka diperoleh 𝐷(𝑣1) = {𝑣1 𝑒𝑛 , 𝑣1𝑒 1 } dan 𝑆(𝑣1) = { 𝑣1(1)}. jadi titik 𝑣1 berkembang menjadi {𝑣1𝑒𝑛 , 𝑣1𝑒 1 , 𝑣1(1)} untuk 𝑣2 dipetakan ke 1, maka diperoleh 𝐷(𝑣2) = {𝑣2 𝑒1 , 𝑣2 𝑒2 } dan 𝑆(𝑣2) = {𝑣2(1) }. jadi faktorisasi graf baru yang dihasilkan dari pemetaan titik graf sikel pada bilangan bulat positif cauchy – issn: 2086-0382 146 titik 𝑣2 berkembang menjadi {𝑣2𝑒1 , 𝑣2 𝑒2 , 𝑣2(1) } ⋮ untuk 𝑣𝑛 dipetakan ke 1, maka diperoleh 𝐷(𝑣𝑛 ) = { 𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 } dan 𝑆(𝑣𝑛 ) = {𝑣𝑛 (1) }. jadi titik 𝑣𝑛 berkembang menjadi {𝑣𝑛 𝑒𝑛−1 , 𝑣𝑛 𝑒𝑛 , 𝑣𝑛(1)}. dari pengembangan titik ini dapat dilihat bahwa untuk setiap titik yang dipetakan ke 1 selalu berkembang menjadi 3 titik, karena sebanyak 𝑛 titik yang dipetakan ke 1 maka banyak pengembangannya adalah 3𝑛 titik, dan karena 𝑛 = 2𝑘 + 2 maka 3𝑛 = 3(2𝑘 + 2) bernilai genap. kemudian dari simulasi gambar 3.21 terlihat bahwa sisi-sisi yang terbentuk pada graf baru 𝐶𝑛 ∗ berupa lintasan, sehingga dapat ditunjukkan himpunan pasangannya adalah 𝑀 = {𝑣3𝑒3 𝑣3(1), 𝑣3𝑒3 𝑣2𝑒2 , 𝑣2 𝑒1 𝑣2(1), 𝑣1𝑒1 𝑣1(1), 𝑣1 𝑒𝑛 𝑣𝑛 𝑒𝑛 , 𝑣𝑛 𝑒𝑛−1 𝑣𝑛 (1)}, maka dapat dipastikan graf baru 𝐶𝑛+2 ∗ dengan fungsi sebanyak 𝑛 titik dipetakakan ke 2 akan selalu memiliki 1faktor. penutup 1. kesimpulan berdasarkan pembahasan, maka dapat diperoleh kesimpulan bahwa ciri-ciri fungsi yang mengakibatkan graf baru 𝐶𝑛 ⋆ yang dihasilkan dari kemungkinan-kemungkinan fungsi 𝑓: 𝑉(𝐶𝑛) → {1, 2} dapat memiliki 1faktor adalah: untuk 𝑛 ganjil a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 b. fungsi dengan sebanyak hanya satu titik dipetakan ke 2 untuk 𝑛 genap a. fungsi dengan sebanyak 𝑛 titik dipetakan ke 2 b. fungsi dengan sebanyak 𝑛 titik dipetakan ke 1 2. saran bagi penelitian selanjutnya disarankan untuk melanjutkan penelitian pada graf lain. daftar pustaka [1] bollobas, b. (1978). extremal graph theory. san francisco: academic press. [2] bondy, j., & murty, u. (2008). graph theory. usa: springer. [3] budayasa. (2007). teory graph dan aplikasinya. surabaya: unesa university press. [4] chartrand, g., & lesniak, l. (1986). graph and digraphs. washington: chapman & hall/crc. [5] munir, r. (2012). matematika diskrit. bandung: informatika bandung. abstrak abstract pendahuluan metode penelitian teori dasar 1. definisi graf sikel 2. definisi subgraf merentang 3. definisi matching 4. definisi perfect matching 5. definisi perfect matching 6. definisi faktor 7. definisi faktorisasi pembahasan 1. faktorisasi graf baru ,,𝑪-𝟑. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟑.)→{𝟏, 𝟐} 2. faktorisasi graf baru ,,𝑪-𝟒. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟒.)→{𝟏, 𝟐} 3. faktorisasi graf baru ,,𝑪-𝟓. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟓.)→{𝟏, 𝟐} 4. faktorisasi graf baru ,,𝑪-𝟔. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟔.)→{𝟏, 𝟐} 5. faktorisasi graf baru ,,𝑪-𝟕. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟕.)→{𝟏, 𝟐} 6. faktorisasi graf baru ,,𝑪-𝟖. -⋆. yang dihasilkan dari fungsi 𝒇:𝑽(,𝑪-𝟖.)→{𝟏, 𝟐} 7. dugaan ciri-ciri fungsi 𝒇:𝑽(,𝑪-𝒏.)→{𝟏, 𝟐} yang mengakibatkan graf baru ,,𝑪-𝒏. -⋆. memiliki 1-faktor penutup 1. kesimpulan 2. saran daftar pustaka binti tsamrotul fitria penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan binti tsamrotul fitria1 , mohammad jamhuri2 1mahasiswa jurusan matematika, fakultas sains dan teknologi, uin maulana malik ibrahim malang 2dosen jurusan matematika, fakultas sains dan teknologi, uin maulana maulana ibrahim malang e-mail: binti_tsamrotulfitria@yahoo.com, m.jamhuri@live.com abstrak penelitian ini membahas tentang penurunan model makroskopis masalah traffic flow berdasarkan hukum-hukum kesetimbangan, yaitu hukum kesetimbanganmassa dan hukum kesetimbangan momentum.asumsi yang digunakan adalah bahwa sepanjang interval jalan tidak ditemukan persimpangan yang menyebabkan perubahan jumlah kendaraan.langkah-langkah dalam penurunan model persamaan tersebut adalah: (1)menurunkan persamaan kontinuitas dan persamaan momentum sebagai persamaan pengatur, (2) menentukan variabel-variabel yang mempengaruhi traffic flow yaitu kepadatan, kecepatan dan fluks kendaraan, (3) menurunkan model berdasarkan hukum-hukum kesetimbangan tersebut. model yang dihasilkan dalam skripsi ini dikenal sebagai persamaan transport, dimana persamaan tersebut menyatakan kepadatan kendaraan per satuan luas jalan yang dipengaruhi oleh kecepatan. untuk kecepatan kendaraan yang konstan, maka model tersebut menjadi model linier. sedangkan bila kecepatan kendaraan bergantung pada kepadatan kendaraan maka persamaan tersebut menjadi non linier. bentuk non linier dari persamaan traffic flow ini dikenal sebagai persamaan burger.solusi dari model yang dihasilkan didapat dengan menggunakan metode finite differenceskema ftbs untuk bentuk yang linier dan menggunakan metode lax wendroffskema ftcs untuk bentuk yang non linier. kata kunci: traffic flow, model makroskopis, hukum kesetimbangan, metode lax wendroff abstract this study discusses the derivation of macroscopic model of traffic flow problems based on the laws of conservation, those are conservation law of mass and conservation law of momentum. the asumtionused is that in the whole of intervals there is no junction which causes the number of vehiclesto change. the steps in the derivation of the equation model are: 1. derivingcontinuity equation and momentum equation as the regulator equation, 2. determining variables which influencetraffic flow, namely density, velocity and flux of vehicle, 3. deriving model based on the laws of conservation. the resulting modelin this thesisis known as the transport equation, speed of vehicle where, the equation states the vehicle density per unit area which is affected by the speed road. for a constant vehicle speed, the model becomes linear. while, when the speed of the vehicle depends on the density of the vehicle, then the equation becomes nonlinear. nonlinear form of the traffic flow equation is known as the burger equation. the solution of the resulting model is obtained by using the method of finite difference implementing ftbs scheme for linear form and using the method of lax wendroff implementing ftcs scheme for non-linear form. key words: traffic flow, macroscopic model, the conservation law, lax wendroff method. pendahuluan arus lalu lintas kendaraan masih menjadi masalah yang cukup serius. kepadatan kendaraan yang terus bertambah membuat kemacetan yang terjadi di kota besar semakin parah, terutama pada ruas jalan yang sempit, bercabang, dan naik turun. perencanaan dan desain pembangunan jalan sangat penting peranannya dalam pengaturan lalu lintas. agar terciptanya jalur lalu lintas yang teratur, lancar, dan bebas hambatan sehingga membuat nyaman bagi pengendara maupun penumpang lainnya. untuk membangun jalan raya perlu memperhatikan luas jalan yang harus dibangun dan peletakan rambu-rambu lalu lintas atau traffic light. untuk mengatur tata letak kota tersebut perlu didukung oleh teori traffic flow. nagel [1] menjelaskan teori traffic flow adalah suatu teori yang membahas masalah transportasi, yang menghubungkan antara tiga variabel fundamental yaitu kecepatan, kepadatan, dan mailto:binti_tsamrotulfitria@yahoo.com mailto:m.jamhuri@live.com binti tsamrotul fitria 159 volume 3 no. 3 november 2014 flow dari kendaraan itu sendiri. solusi dari hubungan tersebut, dengan kondisi awal dan masalah batasnya mungkin bisa menjadi informasi yang berguna pada perencanaan dan optimalisasi masalah traffic flow. lebih dari setengah abad yang lalu, para ahli matematika dan teknik telah menggabungkan teori dinamika fluida dengan masalah transportasi. dimulai pada tahun 1950an ketika lighthill dan witham mengenalkan onedimensional method mengenai traffic flow yang menyatakan bahwa masalah transportasi bisa dipelajari dan dimodelkan dengan menggunakan metode dinamika fluida [2]. sejak saat itu pembahasan mengenai traffic flow menjadi topik yang menarik untuk diteliti sehingga banyak para ilmuwan mengembangkan teori tersebut seperti daganzo [3] dan rascle [4] yang mengembangkan model makroskopis dari traffic flow yang telah ditemukan oleh lighthill dan whitam [5]. model dari arus lalu lintas, terbagi menjadi mikroskopis dan makroskopis. sedangkan immers dan logghe [6] menjelaskan bahwa makroskopis adalah pendekatan yang mengamati kendaraan secara keseluruhan dan sangat bergantung pada kepadatan di suatu ruas jalan, sedangkan mikroskopis adalah pendekatan yang mengamati kendaraan secara terpisah, sehingga lebih menekankan pada jarak dan hubungan antar dua kendaraan yang saling berdekatan. tinjauan pustaka 1. persamaan pengatur persamaan pengatur adalah persamaan yang diturunkan melalui hukum-hukum kesetimbangan. adapun persamaan pengatur dalam penelitian ini adalah: persamaan kontinuitas sebuah sistem didefinisikan sebagai kumpulan dari isi yang tidak berubah. prinsip kekekalan massa berbunyi, laju perubahan massa terhadap waktu sama dengan nol. olson [7] mengatakan bahwa pada persamaan kontinuitas mensyaratkan bahwa massa fluida harus bersifat kekal, yakni tidak dapat diciptakan atau dimusnahkan. bentuk dari persamaan kontinuitas yang diturunkan melalaui hukum kekekalan massa sebagai berikut 𝜕𝜌 𝜕𝑡 = − 𝜕(𝜌𝑢) 𝜕𝑥 − 𝜕(𝜌𝑣) 𝜕𝑦 − 𝜕(𝜌𝑤) 𝜕𝑧 sedangkan [ 𝜕 𝜕𝑥 , 𝜕 𝜕𝑦 , 𝜕 𝜕𝑧 ] = ∇. sehingga persamaan (2.5) di atas dapat ditulis 𝜕𝜌 𝜕𝑡 = 𝜌(�̅� ∇) a. persamaan momentum bentuk dari persamaan momentum yang diturunkan berdasarkan hukum kesetimbangan adalah sebagai berikut 𝜕𝑞 𝜕𝑡 + 𝑞𝛻𝑞 = − 1 𝜌 𝛻𝑝 + 𝑔𝛻𝑧 dimana �̅� = (𝑢, 𝑣, 𝑤) [ 1 𝜌 𝜕𝑝 𝜕𝑥 , 1 𝜌 𝜕𝑝 𝜕𝑦 , 1 𝜌 𝜕𝑝 𝜕𝑧 ] = 1 𝜌 ( 𝜕𝑝 𝜕𝑥 , 𝜕𝑝 𝜕𝑦 , 𝜕𝑝 𝜕𝑧 ) = 1 𝜌 ∇p (𝑢, 𝑣, 𝑤) [ 𝜕𝑢 𝜕𝑥 , 𝜕𝑣 𝜕𝑦 , 𝜕𝑤 𝜕𝑧 ] = 𝑞∇𝑞 2. variabel makroskopis a. ukuran interval interval s didefinisikan sebagai daerah yang terletak pada ruang 𝑡 − 𝑥 seperti yang dijelaskan pada gambar berikut: gambar 1. ukuran interval 𝑆1 dan 𝑆2 𝑆1 merupakan daerah interval tertutup yang berbentuk segiempat ini mewakili ruas jalan sepanjang ∆𝑥 dengan waktu sekecil-kecilnya yaitu 𝑑𝑡. 𝑆2 merupakan daerah segiempat ini mewakili daerah dimana jarak yang sekecilkecilnya dalam waktu yang berbeda. 𝑆3 adalah ukuran interval yang berubah-ubah sesuai ruang dan waktu. seperti yang dijelaskan melalui gambar berikut gambar 2. ukuran interval 𝑆3 b. kepadatan kendaraan kepadatan yang dinotasikan dengan (𝜌) menyatakan jumlah kendaraan per kilometernya di jalan (immers dan logghe, 2002). dalam interval satuan waktu tertentu, misalnya daerah penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan cauchy – issn: 2086-0382 160 𝑆1, 𝜌 dapat dicari dengan menghitung banyaknya kendaraan per partisi panjang jalan sebesar ∆𝑥, secara matematis dapat ditulis 𝜌 = 𝑛 ∆𝑥 dengan 𝑛 merupakan jumlah kendaraan yang melintas sepanjang interval jalan ∆𝑥. c. laju alir kendaraan (fluks) laju alir merupakan jumlah kendaraan yang melintas suatu ruas jalan pada interval waktu tertentu (immers dan logghe, 2002). misalnya pada interval waktu ∆𝑇 dan pada lokasi 𝑥2 maka laju alirnya bisa dicari dengan 𝑞 = 𝑚 ∆𝑇 indeks 𝑚 merupakan jumlah kendaraan yang melintas pada lokasi 𝑥2. d. kecepatan kecepatan 𝑣 adalah hasil bagi antara laju alir dengan kepadatan. dengan kata lain, kecepatan adalah fungsi atas lokasi, waktu dan ukuran intervalnya [6]. secara matematis dapat ditulis 𝑣 = 𝑞 𝜌 = 𝑡𝑜𝑡𝑎𝑙 𝑗𝑎𝑟𝑎𝑘 𝑘𝑒𝑛𝑑𝑎𝑟𝑎𝑎𝑛 𝑑𝑖 𝑆2 𝑡𝑜𝑡𝑎𝑙 𝑤𝑎𝑘𝑡𝑢 e. hubungan antara ketiga variabel makroskopis untuk memberikan gambaran tersebut akan diberikan beberapa kasus di antaranya adalah jika dipunyai pergerakan kendaraan dengan kecepatan konstan 𝑣 dengan kepadatan (𝜌). karena setiap kendaraan melaju dengan kecepatan yang sama, maka jarak antara masingmasing kendaraan akan menyisakan konstan. sehingga kepadatan lalu lintas tidak berubah. untuk mengukur aliran traffic pada saat 𝑡 jam, kita ingat kembali rumus 𝑗𝑎𝑟𝑎𝑘 = 𝑘𝑒𝑐𝑒𝑝𝑎𝑡𝑎𝑛 × 𝑤𝑎𝑘𝑡𝑢. sehingga pada saat 𝑡 jam setiap kendaraan akan menempuh jarak 𝑣𝑡, sehingga jumlah kendaraan yang lewat dan diamati sebanyak 𝑡 jam merupakan sejumlah kendaraan pada jarak 𝑣𝑡. karena 𝜌 adalah sejumlah kendaraan perkilometer dan jarak sebenarnya berupa 𝑣𝑡 kilometer, maka 𝜌𝑣𝑡 adalah sejumlah kendaraan yang lewat dan teramati sepanjang 𝑡 jam. sehingga sejumlah kendaraan per jam yang disebut aliran lalu lintas 𝑞 adalah 𝑞 = 𝜌𝑣 karena variabel lalu lintas bergantung pada jarak dan waktu, maka dapat ditulis 𝑞(𝑥, 𝑡) = 𝜌𝑣 pembahasan 1. penurunan model traffic flow pada bagian ini akan dijelaskan penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan. hukum kekekalan massa mensyaratkan bahwa perubahan massa per satuan waktu yaitu 𝑝𝑒𝑟𝑢𝑏𝑎ℎ𝑎𝑛 𝑚𝑎𝑠𝑠𝑎 𝑡𝑒𝑟ℎ𝑎𝑑𝑎𝑝 𝑤𝑎𝑘𝑡𝑢 = 𝑚𝑎𝑠𝑠𝑎 𝑦𝑎𝑛𝑔 𝑚𝑎𝑠𝑢𝑘 − 𝑚𝑎𝑠𝑠𝑎 𝑦𝑎𝑛𝑔 𝑘𝑒𝑙𝑢𝑎𝑟 jika 𝑚 menunjukkan jumlah kendaraan yang melintasi suatu ruas jalan, 𝜌 menunjukkan kepadatan dan 𝑞 adalah fluks kendaraan, dimana 𝑞|𝑥 menunjukkan fluks kendaraan yang memasuki suatu ruas jalan sedangkan 𝑞|𝑥+∆𝑥 adalah fluks kendaraan yang keluar dari ruas jalan maka dapat dinyatakan sebagai 𝜕𝑚 𝜕𝑡 = 𝑞|𝑥 − 𝑞|𝑥+∆𝑥 (1) berdasarkan rumus dari massa jenis (𝜌 ) 𝜌 = 𝑚 𝑉 untuk 𝑚 = massa dan 𝑉 = volume sehingga 𝑚 = 𝜌𝑉 oleh karena itu persamaan (1) dapat ditulis 𝜕𝜌𝑉 𝜕𝑡 = 𝑞|𝑥 − 𝑞|𝑥+∆𝑥 (2) karena objek pembahasannya adalah jalan raya dan hanya berdimensi satu, maka volume yang dimaksud adalah panjang interval jalan sebesar ∆𝑥. sehingga persamaan (2) menjadi 𝜕𝜌 ∆𝑥 𝜕𝑡 = 𝑞|𝑥 − 𝑞|𝑥+∆𝑥 dengan membagi kedua ruas dengan ∆𝑥 maka 𝜕𝜌 𝜕𝑡 = 𝑞|𝑥 − 𝑞|𝑥+∆𝑥 ∆𝑥 bila ∆𝑥 → 0 lim ∆𝑥→0 𝑞|𝑥 − 𝑞|𝑥+∆𝑥 ∆𝑥 = − 𝜕𝑞 𝜕𝑥 sehingga 𝜕𝜌 𝜕𝑡 = − 𝜕𝑞 𝜕𝑥 karena 𝑞 = 𝜌𝑣 , dimana 𝑣 adalah kecepatan, maka 𝜕𝜌 𝜕𝑡 = − 𝜕(𝜌𝑣) 𝜕𝑥 atau 𝜕𝜌 𝜕𝑡 + 𝜕(𝜌𝑣) 𝜕𝑥 = 0 (3) 2. hubungan kepadatan dengan kecepatan pada penelitian ini diasumsikan bahwa kepadatan adalah faktor yang paling mempengaruhi kecepatan. sedangkan untuk faktor lain diabaikan. sehingga dapat ditulis 𝑣 = 𝑣(𝜌) binti tsamrotul fitria 161 volume 3 no. 3 november 2014 jika tidak terdapat kendaraan lain atau tidak ada mobil lain yang melewati interval (𝑎, 𝑏) artinya 𝜌 = 0 maka kendaraan akan berjalan dengan kecepatan maksimum, yaitu 𝑣𝑚𝑎𝑥 , akan tetapi jika terdapat peningkatan kepadatan maka laju kecepatannya menjadi pelan, sehingga dapat ditulis 𝑣(𝜌) ≤ 0 sedangkan bila kepadatan kendaraan menjadi maksimum (𝜌maks ), maka kondisi jalan menjadi bumper to bumper, dalam keadaan ini kendaraan tidak dapat berjalan atau berhenti. bila dipaksakan maka kendaraan akan bertabrakan, karena sudah tidak ada ruang lagi untuk bergerak. sehingga dapat ditulis 𝑣(𝜌maks ) = 0 sehingga 𝜌maks = 1 𝐿 dengan l adalah panjang kendaraan. oleh karena itu hubungan antara kepadatan dan kecepatan dapat digambarkan sebagai berikut gambar 3. hubungan kepadatan dengan kecepatan berdasarkan gambar 3 di atas didapatkan 2 buah titik yaitu (0, 𝑣𝑚𝑎𝑘𝑠 ) dan (𝜌𝑚𝑎𝑘𝑠 , 0) sehingga bentuk persamaan garisnya dapat dituliskan 𝑣 − 𝑣𝑚𝑎𝑘𝑠 −𝑣𝑚𝑎𝑘𝑠 = 𝜌 − 𝜌𝑚𝑎𝑘𝑠 𝜌𝑚𝑎𝑘𝑠 𝑣 = −𝜌 𝑣𝑚𝑎𝑘𝑠 𝜌𝑚𝑎𝑘𝑠 + 𝑣𝑚𝑎𝑘𝑠 𝑣 = 𝑣maks (1 − 𝜌 𝜌maks ) ansgar jungel [8] menyatakan model kecepatan yang bergantung kepadatan dalam bentuk persamaan 𝑣(𝜌) = 𝑣maks (1 − 𝜌 𝜌maks ), 0 ≤ 𝜌 ≤ 𝜌maks (4) kemudian persamaan (4) disubstitusikan pada model traffic flow pada persamaan (4.3) menjadi 𝜌𝑡 + [𝑣maks 𝜌 (1 − 𝜌 𝜌maks )] 𝑥 = 0 𝜌𝑡 + [𝑣maks 𝜌 − 𝑣maks 𝜌maks 𝜌2] 𝑥 = 0 (5) dengan mensubstitusikan data-data yang sudah didapatkan bisa ditentukan suatu persamaan garis lurus yang menggambarkan hubungan antar kecepatan dengan kepadatan lalu lintasnya. observasi yang dilakukan menghasilkan data 𝑣𝑚𝑎𝑥 = 27. 8 𝑚/𝑑𝑡 pada saat kepadatan mendekati 0, sedangkan 𝑣 ≈ 0 pada saat 𝜌𝑚𝑎𝑥 = 0.67. sehingga didapatkan 2 titik (0, 𝑣) dan (𝜌, 0) yaitu (0,27. 89) dan (0.67,0). gambar 4. persamaan garis dengan mensubstitusikan data di atas ke persamaan (4) maka didapatkan 𝑣(𝜌) = 𝑣maks (1 − 𝜌 𝜌maks ) 𝑣(𝜌) = 27.89 (1 − 𝜌 0.67 ) 3. penskalaan skala adalah perbandingan ukuran pada model dengan ukuran sebenarnya. baik mengubah ke ukuran yang lebih kecil, maupun ke ukuran yang lebih besar dengan tanpa menghilangkan karakteristiknya. selanjutnya kita kenalkan variabel lain dengan mengikuti penelitian sebelumnya [8] yaitu 𝜃 dan 𝜏 dimana 𝜃 menyatakan jarak dan 𝜏 waktu, bila 𝜃 𝜏 = 𝑣maks maka akan ada 𝑥𝑠 = 𝑥 𝜃 , 𝑡𝑠 = 𝑡 𝜏 dan 𝑢 = 1 − 2𝜌 𝜌maks sehingga 𝜌𝑡 = 𝜕𝜌 𝜕𝑡𝑠 ∙ 𝜕𝑡𝑠 𝜕𝑡 karena 𝑢 = 1 − 2𝜌 𝜌maks maka 𝜌 = 𝜌maks 2 (1 − 𝑢) sehingga 𝜌𝑡 = [ 𝜌maks 2 (1 − 𝑢)] 𝑡𝑠 1 𝜏 𝜌𝑡 = 1 𝜏 [ 𝜌maks 2 (1 − 𝑢)] 𝑡𝑠 𝜌𝑡 = − 𝜌maks 2𝜏 𝑢𝑡𝑠 (6) sedangkan untuk penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan cauchy – issn: 2086-0382 162 𝜌𝑥 = 𝜕𝜌 𝜕𝑥𝑠 ∙ 𝜕𝑥𝑠 𝜕𝑥 𝜌𝑥 = [ 𝜌maks 2 (1 − 𝑢)] 𝑥𝑠 1 𝜃 𝜌𝑥 = 1 𝜃 [ 𝜌maks 2 (1 − 𝑢)] 𝑥𝑠 𝜌𝑥 = − 𝜌maks 2𝜃 𝑢𝑥𝑠 (7) selanjutnya dengan mensubstitusikan persamaan (6) dan (7) ke persamaan (5) maka 𝜌𝑡 + [𝑣maks 𝜌 − 𝑣maks 𝜌maks 𝜌2] 𝑥 = 0 𝜌𝑡 + 𝑣maks 𝜌x − 𝑣maks 𝜌maks (2𝜌𝜌x) = 0 − 𝜌maks 2𝜏 𝑢𝑡𝑠 + 𝑣maks (− 𝜌maks 2𝜃 𝑢𝑥𝑠 ) − 𝑣maks 𝜌maks (2 ( 𝜌maks 2 (1 − 𝑢)) (− 𝜌maks 2𝜃 𝑢𝑥𝑠 )) = 0 − 𝜌maks 2𝜏 𝑢𝑡𝑠 + 𝑣maks (− 𝜌maks 2𝜃 𝑢𝑥𝑠 ) + 𝑣maks 𝜌maks 2𝜃 ((1 − 𝑢) 𝑢𝑥𝑠 ) = 0 karena 𝜃 𝜏 = 𝑣maks maka − 𝜌maks 2𝜏 𝑢𝑡𝑠 − 𝜃 𝜏 𝜌maks 2𝜃 ( 𝑢𝑥𝑠 − 𝑢𝑥𝑠 + 𝑢 𝑢𝑥𝑠 ) = 0 − 𝜌maks 2𝜏 𝑢𝑡𝑠 − 𝜌maks 2𝜏 (𝑢 𝑢𝑥𝑠 ) = 0 bila kedua ruas dibagi dengan − 𝜌maks 2𝜏 maka menjadi 𝑢𝑡𝑠 + 𝑢 𝑢𝑥𝑠 = 0 model tersebut dikenal sebagai model traffic flow. untuk selanjutnya 𝑡𝑠 akan ditulis sebagai 𝑡 dan 𝑥𝑠 akan ditulis sebagai 𝑥 . sehingga modelnya menjadi 𝑢𝑡 + 𝑢 𝑢𝑥 = 0 (8) 4. solusi dan simulasi a. model linier dari persamaan traffic flow pada persamaan (3) bila 𝑣(𝜌) merupakan suatu konstanta maka persamaan tersebut menjadi persamaan transport linier. sehingga persamaan (3) dapat dituliskan 𝜕𝜌 𝜕𝑡 + 𝑣 𝜕𝜌 𝜕𝑥 = 0 (9) dengan 𝜌(0, 𝑡) = 3 𝜌(1680, 𝑡) = 2 𝜌(𝑥, 0) = −𝑥 1680 + 3 kondisi awal 𝜌 diperoleh dengan mengasumsikan jarak 𝑥 berada pada selang interval (0,1680) yang akan didapatkan suatu persamaan garis lurus yang menghubungkan 𝜌 di batas kiri dengan 𝜌 di batas kanan. melalui rumus persamaan garis lurus akan didapatkan 𝜌 − 3 2 − 3 = 𝑥 1680 (𝜌 − 3) = 𝑥 1680 (−1) 𝜌 = −𝑥 1680 + 3 dengan menggunakan metode beda hingga skema ftbs (forward time backward space) yaitu aproksimasi dengan menggunakan skema beda maju untuk waktu dan skema beda mundur untuk ruang, maka akan diperoleh bentuk diskrit seperti berikut 𝜌𝑡 = 𝜌𝑗 𝑛+1 − 𝜌𝑗 𝑛 ∆𝑡 𝜌𝑥 = 𝜌𝑗 𝑛 − 𝜌𝑗−1 𝑛 ∆𝑥 sehingga persamaan (9) menjadi 𝜌𝑗 𝑛+1 − 𝜌𝑗 𝑛 ∆𝑡 + 𝑣 𝜌𝑗 𝑛 − 𝜌𝑗−1 𝑛 ∆𝑥 = 0 𝜌𝑗 𝑛+1 = 𝜌𝑗 𝑛 − 𝑣 ∆𝑡 ∆𝑥 [𝜌𝑗 𝑛 − 𝜌𝑗−1 𝑛 ] (10) selanjutnya akan dilakukan simulasi dari persamaan (10), dengan menggunakan program matlab. dengan mengambil ∆𝑥 = 0.99 m dan ∆𝑡 = 0.099 satuan waktu, dan mengambil 𝑣 = 3 𝑚 per satuan waktu. sehingga perubahan kepadatan kendaraan dapat dilihat sebagai berikut gambar 5. simulasi kasus 1 dari gambar 5 di atas dapat dilihat bahwa sepanjang waktu 𝑡 pada batas kiri kepadatannya sebesar 3 kendaraan per satuan luas dan pada batas kanan kepadatannya 2 kendaraan per binti tsamrotul fitria 163 volume 3 no. 3 november 2014 satuan luas. hal ini menunjukkan bahwa pada posisi 𝑥 = 0 selalu terdapat 3 kendaraan, dan pada posisi 𝑥 = 1680 selalu terdapat 2 kendaraan. sedangkan untuk mengetahui kondisi kepadatan di setiap posisi 𝑥 pada saat 𝑡 tertentu maka digambarkan pada grafik berikut gambar 6. simulasi kasus 1 dengan waktu berbeda berdasarkan gambar 6 tersebut dapat diketahui bahwa pada saat 𝑡 = 0 kepadatan kendaraan sebesar 2 kendaraan per satuan luas, setelah 𝑡 = 49.95 kendaraan mulai bertambah. karena jumlah kendaraan di batas kiri lebih banyak daripada batas kanan, sehingga kendaraan menumpuk di titik-titik awal. dan untuk 𝑡 menuju tak hingga kepadatan di sepanjang jalan besarnya sama, yaitu 3 kendaraan per satuan luasnya, tetapi di ujung jalan tetap ada 2 kendaraan per satuan luasnya. simulasi pada kasus kedua, bila diasumsikan bahwa kepadatan kendaraan di daerah batas berkebalikan dengan kasus pertama. jika terdapat 2 kendaraan di batas kiri dan terdapat 3 kendaraan maka dengan melakukan hal yang sama akan didapat grafik kepadatannya sebagai berikut gambar 7. simulasi kasus 2 pada kasus kedua ini, kepadatan kendaraan di batas kiri sebanyak 2 kendaraan per satuan luas, sedangkan kepadatan di batas kanan sebanyak 3 kendaraan per satuan luas, sehingga dapat diketahui bahwa penumpukan kendaraan terjadi di ujung interval yang akan berdampak pada kemacetan. tetapi setelah 𝑡 = 1000 kondisi jalan telah mencapai kepadatan yang seimbang, artinya, sepanjang jalan kepadatannya sama; yaitu 2 kendaraan per satuan luas jalan. seperti yang ditunjukkan pada gambar berikut gambar 8. simulasi kasus 2 dengan waktu berbeda selanjutnya dengan menggunakan metode karakteristik bisa didapatkan solusi analitik dari model linier 𝜕𝜌 𝜕𝑡 + 𝑣 𝜕𝜌 𝜕𝑥 = 0 𝜌(𝑥, 0) = −𝑥 1680 + 3 persamaan 𝜕𝜌 𝜕𝑡 + 𝑣 𝜕𝜌 𝜕𝑥 = 0 adalah turunan berarah dari 𝜌 dalam suatu vektor dengan arah 𝕍 = [1, 𝑣] = 𝑖 + 𝑣𝑗 dimana kurva dari persamaan tersebut memiliki gradien 𝑑𝑥 𝑑𝑡 = 𝑣 1 = 𝑣 dalam hal ini selalu bernilai 0 atau 𝜌(𝑥, 𝑡) = 𝑐 dalam arah 𝕍. vektor [𝑣, −1] adalah orthogonal terhadap 𝕍. sedangkan garis yang sejajar dengan 𝕍 adalah −𝑥 = 𝑐 , dan persamaan ini disebut persamaan karakteristik. solusi pdp di atas selalu konstan dalam masing-masing karakteristik ini, sehingga tergantung hanya pada 𝑣𝑡 − 𝑥. dengan demikian solusinya adalah 𝜌(𝑥, 𝑡) = 𝑓(𝑣𝑡 − 𝑥) atau bisa dengan menggunakan metode koordinat. dalam sistem koordinat 𝑡, 𝑥 dapat kita transportasikan ke dalam sistem gambar 9. transportasi sistem koordinat misal ditetapkan 𝑡 ′ = 𝑡 + 𝑣𝑥 dan 𝑥 ′ = 𝑣𝑡 − 𝑥 dengan aturan turunan rantai maka turunan 𝜌(𝑥 ′, 𝑦′) terhadap 𝑥 dan 𝑦 adalah 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( p ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( p ) penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan cauchy – issn: 2086-0382 164 𝜌𝑡 = 𝜕𝜌 𝜕𝑡 = 𝜕𝜌 𝜕𝑡′ ∙ 𝜕𝑡′ 𝜕𝑡 + 𝜕𝜌 𝜕𝑥′ ∙ 𝜕𝑥′ 𝜕𝑡 = 𝜌𝑡 ′ + 𝑣𝜌𝑥′ 𝜌𝑥 = 𝜕𝜌 𝜕𝑥 = 𝜕𝜌 𝜕𝑥′ ∙ 𝜕𝑥′ 𝜕𝑥 + 𝜕𝜌 𝜕𝑡′ ∙ 𝜕𝑡′ 𝜕𝑥 = −𝜌𝑥′ + 𝑣𝜌𝑡 ′ = 𝑣𝜌𝑡 ′ −𝜌𝑥′ selanjutnya substitusikan ke dalam persamaan 𝜕𝜌 𝜕𝑡 + 𝑣 𝜕𝜌 𝜕𝑥 = 0 sehingga 𝜌𝑡 ′ + 𝑣𝜌𝑥′ + 𝑣(𝑣𝜌𝑡 ′ −𝜌𝑥′ ) = 0 𝜌𝑡 ′ + 𝑣𝜌𝑥′ + 𝑣 2𝜌𝑡 ′ − 𝑣𝜌𝑥′ = 0 𝜌𝑡 ′ + 𝑣 2𝜌𝑡 ′ = 0 (1 + 𝑣2)𝜌𝑡 ′ = 0 untuk (1 + 𝑣2) ≠ 0 maka 𝜌𝑡 ′ = 0 ∫ 𝜌𝑡 ′ 𝑑𝑡 ′ = ∫ 0 𝑑𝑡 ′ 𝜌(𝑥, 𝑡) = 𝑓(𝑥 ′) sehingga 𝜌(𝑥, 𝑡) = 𝑓(𝑣𝑡 − 𝑥) merupakan solusi analitiknya. dengan mensubstitusikan nilai awal ke model (9) sedangkan untuk 𝑣 = 3 maka 𝜌(𝑥, 0) = 𝑓(3.0 − 𝑥) 𝜌(𝑥, 0) = 𝑓(−𝑥) 𝑓(−𝑥) = −𝑥 1680 + 3 sehingga 𝑓(3𝑡 − 𝑥) = (3𝑡 − 𝑥) 1680 + 3 maka solusi analitiknya 𝜌(𝑥, 𝑡) = (3𝑡 − 𝑥) 1680 + 3 𝜌(𝑥, 𝑡) = (3𝑡 − 𝑥) 1680 + 3 solusi analitik digunakan untuk membandingkan keakuratan dari metode numerik yang dipakai dalam skripsi ini. untuk mengetahui galat pada solusi numerik dari model linier tersebut akan dilakukan simulasi dengan mengambil domain 0 < 𝑥 < 1 dan 0 < 𝑡 < 1 dengan ∆𝑥 = ∆𝑡 = 025. sedangkan untuk solusi numeriknya yaitu persamaan (10) dengan kondisi batasnya 𝜌𝑗 𝑛+1 = 𝜌𝑗 𝑛 − 𝑣 ∆𝑡 ∆𝑥 [𝜌𝑗 𝑛 − 𝜌𝑗−1 𝑛 ] 𝜌(0, 𝑡) = 3𝑡 1680 + 3 𝜌(1, 𝑡) = 3𝑡 − 1 1680 + 3 maka didapatkan solusi sebagai berikut: tabel 1. perbandingan solusi analitik dan solusi numerik x t analitik numerik galat 0 0 3.0000 3.0000 0 0 0.25 3.0004 3.0004 0 0 0.5 3.0009 3.0009 0 0 0.75 3.0013 3.0013 0 0 1 3.0018 3.0018 0 0.25 0 2.9999 2.9999 0 0.25 0.25 3.0003 3.0003 0 0.25 0.5 3.0007 3.0007 0.0444 x 10-14 0.25 0.75 3.0012 3.0012 -0.0888 x 10-14 0.25 1 3.0016 3.0016 0.1776 x 10-14 0.5 0 2.9997 2.9997 0 0.5 0.25 3.0001 3.0001 0 0.5 05 3.0006 3.0006 0 0.5 0.75 3.0010 3.0010 0.0444 x 10-14 0.5 1 3.0015 3.0015 -0.3553 x 10-14 0.75 0 2.9996 2.9996 0 0.75 0.25 3.0000 3.0000 0 0.75 0.5 3.0004 3.0004 0 0.75 0.75 3.0009 3.0009 0.0444 x 10-14 0.75 1 3.0013 3.0013 0.0444 x 10-14 1 0 2.9994 2.9994 0 1 0.25 2.9999 2.9999 0 1 0.5 3.0003 3.0003 0 1 0.75 3.0007 3.0007 0 1 1 3.0012 3.0012 0 dari hasil solusi di atas dapat diketahui bahwa galat maksimumnya sebesar 0.1776 x 1014 pada saat 𝑥 = 0.25 dan 𝑡 = 1. sedangkan untuk yang lain galatnya sangat kecil bahkan hampir keseluruhan galatnya 0. hal ini bisa dikatakan bahwa metode ftbs ini sudah cukup baik untuk mengaproksimasi model tersebut. adapun untuk grafik solusi bisa dilihat di bawah ini table 2. grafik solusi untuk solusi analitik dan numerik solusi analitik solusi numerik b. model non linier persamaan traffic flow pada persamaan (3) di atas jika kecepatannya bergantung pada kepadatan kendaraan yang berbeda di setiap posisi, maka persamaan tersebut menjadi persamaan non linier. persamaan (10) adalah persamaan (3) yang telah diskalakan sehingga menjadi persamaan burger. misal diasumsikan pada batas interval (0,1680) adalah 𝑥 = 0 batasnya 3 kendaraan dan 𝑥 = 1680 batasnya 2 kendaraan. artinya pada posisi 𝑥 = 0 selalu terdapat 3 kendaraan per satuan luas. dan pada posisi 𝑥 = 1680 selalu terdapat 2 kendaraan per satuan luas. persamaan (10) beserta kondisi batasnya dapat dituliskan sebagai berikut 𝑢𝑡 + 𝑢𝑢𝑥 = 0 , 𝑡 > 0 dan 0 < 𝑥 < 1680 dengan 𝑢(0, 𝑡) = 3 𝑢(1680, 𝑡) = 2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 2.999 3 3.001 3.002 3.003 posisi grafik solusi analitik waktu k e p a d a ta n 0 0.2 0.4 0.6 0.8 1 0 0.5 1 2.999 3 3.001 3.002 3.003 posisi (x) solusi numerik waktu (t) k e p a d a ta n k e n d a ra a n ( u ) binti tsamrotul fitria 165 volume 3 no. 3 november 2014 𝑢(𝑥, 0) = −𝑥 1680 + 3 selanjutnya akan dilakukan pendiskritan pada persamaan (10) tersebut dengan menggunakan metode lax wendroff skema ftcs (forward time center space) dengan mensubstitusikan 𝑢𝑡 ke bentuk 𝑢𝑥 . sehingga bentuk diskritnya sebagai berikut 𝑢𝑡 + 𝑢𝑢𝑥 = 0 𝑢𝑡 = −𝑢𝑢𝑥 𝑢𝑡𝑡 = −(𝑢𝑢𝑥 )𝑡 𝑢𝑡𝑡 = − 𝑢𝑡 𝑢𝑥 − 𝑢𝑢𝑥𝑡 berdasarkan ekspansi taylor 𝑢𝑗 𝑛+1 = 𝑈𝑗 𝑛 + ∆𝑡 𝑢𝑡 |𝑗 𝑛 + 1 2 ∆𝑡 2𝑢𝑡𝑡 |𝑗 𝑛 + ⋯ 𝑢𝑗 𝑛+1 = 𝑢𝑗 𝑛 + ∆𝑡 (−𝑢 𝑢𝑥 )|𝑗 𝑛 + 1 2 ∆𝑡 2(− 𝑢𝑡 𝑢𝑥 − 𝑢 𝑢𝑥𝑡 )|𝑗 𝑛 (11) karena pada persamaan (11) ruas kanan masih terdapat unsur 𝑢𝑡 maka (− 𝑢𝑡 𝑢𝑥 − 𝑢𝑢𝑥𝑡 ) = (−(−𝑢𝑢𝑥 )𝑢𝑥 ) − 𝑢𝑢𝑥𝑡 = (𝑢𝑢𝑥 )𝑢𝑥 − 𝑢(𝑢𝑡 )𝑥 = 𝑢(𝑢𝑥 ) 2 − 𝑢(−𝑢𝑢𝑥 )𝑥 = 𝑢(𝑢𝑥 ) 2 + 𝑢(𝑢𝑥 𝑢𝑥 + 𝑢𝑢𝑥𝑥 ) = 𝑢(𝑢𝑥 ) 2 + 𝑢(𝑢𝑥 ) 2 + 𝑢2𝑢𝑥𝑥 = 2𝑢(𝑢𝑥 ) 2 + 𝑢2𝑢𝑥𝑥 (12) selanjutnya substitusi persamaan (11) ke persamaan (12) 𝑢𝑗 𝑛+1 = 𝑢 + ∆𝑡 (−𝑢𝑢𝑥 )|𝑗 𝑛 + 1 2 ∆𝑡 2(2. 𝑢(𝑢𝑥 ) 2 + 𝑢2𝑢𝑥𝑥 )|𝑗 𝑛 𝑢𝑗 𝑛+1 = 𝑢𝑗 𝑛 − ∆𝑡 (𝑢𝑢𝑥 )|𝑗 𝑛 + ∆𝑡 2(𝑢(𝑢𝑥 ) 2 + 1 2 𝑢2 𝑢𝑥𝑥)|𝑗 𝑛 𝑢𝑗 𝑛+1 = 𝑢𝑗 𝑛 − ∆𝑡 (𝑢𝑗 𝑛 [ 𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 2∆𝑥 ]) + ∆𝑡 2 (𝑢𝑗 𝑛 [ 𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 2∆𝑥 ] 2 + 1 2 𝑢2 [ 𝑢𝑗+1 𝑛 − 2𝑢𝑗 𝑛 + 𝑢𝑗−1 𝑛 ∆𝑥2 ]) 𝑢𝑗 𝑛+1 = 𝑢𝑗 𝑛 − ∆𝑡 2∆𝑥 (𝑢𝑗 𝑛[𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 ]) + ∆𝑡 2 4∆𝑥2 (𝑢𝑗 𝑛[𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 ] 2 ) + ∆𝑡 2 2∆𝑥2 ((𝑢𝑗 𝑛)2[𝑢𝑗+1 𝑛 − 2𝑢𝑗 𝑛 + 𝑢𝑗−1 𝑛 ]) (13) selanjutnya pada kasus ketiga ini akan dilakukan simulasi untuk model non linier dengan kondisi batas dirichlet. dengan mengambil ∆𝑥 dan ∆𝑡 sama seperti kasus sebelumnya maka kepadatan kendaraan pada kasus ini dapat dilihat sebagai berikut gambar 10. simulasi kasus 3 dari gambar 10 dapat dilihat bahwa pada batas kiri kepadatannya sebesar 3 dan pada batas kanan kepadatannya 2. gambar tersebut menunjukkan hubungan antara kepadatan berdasarkan ruang dan waktu. sedangkan untuk mengamati perubahan kepadatan di tiap waktu tertentu maka disajikan dalam bentuk plot berikut gambar 11. simulasi kasus 3 dengan waktu yang berbeda dari gambar 11 tersebut bisa diamati perubahan kepadatan di beberapa waktu. pada 𝑡 awal kepadatan kendaraan terjadi di daerah batas kiri, tetapi lama-lama terjadi penumpukan kendaraan sampai ujung interval. hal ini dikarenakan jumlah kendaraan yang bisa melewati batas kanan hanya 2 kendaraan, sehingga terjadi antrian di sepanjang interval jalan. pada kasus ketiga ini model yang digunakan adalah model non linier, sehingga kepadatan sepanjang ruas jalan pun akan mengalami fluktuasi pada posisi tertentu. seperti pada gambar di bawah ini gambar 12. fluktuasi kasus 3 pada x =10 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( u ) 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t = 99.999 posisi k e p a d a ta n penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan cauchy – issn: 2086-0382 166 pada gambar 12 tersebut dapat diketahui bahwa pada saat 0 < 𝑥 < 30 kepadatan kendaraan sudah mengalami fluktuasi dari awal, hal ini karena model yang digunakan adalah model non linier yang bergantung pada kecepatan dan kepadatan kendaraan. simulasi keempat dilakukan pada model yang sama dengan kasus ketiga di atas, yaitu dengan model non linier. akan tetapi kondisi batas yang digunakan berbalik, yaitu posisi 𝑥 = 0 terdapat 2 kendaraan dan posisi 𝑥 = 1680 terdapat 3 kendaraan. dengan langkah-langkah yang sama dengan kasus ketiga maka didapatkan hubungan kepadatan berdasarkan ruang dan waktunya sebagai berikut gambar 13. simulasi kasus 4 pada kasus keempat ini kepadatan lebih fluktuatif. dapat dilihat dari gambar tersebut bahwa fluktuasi kepadatan sudah terjadi saat 𝑥 = 3. tetapi kepadatan kendaraan terjadi ujung jalan karena adanya penumpukan kendaraan di ujung interval jalan. lebih jelasnya bisa dilihat pada gambar di bawah ini gambar 14. simulasi kasus 4 dengan waktu berbeda selanjutnya bila diasumsikan pada posisi 𝑥 = 0 terdapat 3 kendaraan yang masuk pada interval, dan pada 𝑥 = 1680 terdapat 2 kendaraan yang keluar pada interval, maka persamaan (4.10) beserta kondisi batasnya dapat dituliskan sebagai berikut 𝑢𝑡 + 𝑢𝑢𝑥 = 0 dengan kondisi sedangkan bila kita mendiskritkan kondisi batas (14) dan (15) dengan menggunakan skema beda pusat menjadi: 𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 2∆𝑥 = 3 untuk 𝑗 = 1 𝑢2 𝑛 − 𝑢0 𝑛 2∆𝑥 = 3 𝑢0 𝑛 = 𝑢2 𝑛 − 6∆𝑥 (16) dan untuk batas kanan 𝑢𝑗+1 𝑛 − 𝑢𝑗−1 𝑛 2∆𝑥 = 2 untuk 𝑗 = 𝑙 𝑢𝑙+1 𝑛 − 𝑢𝑙−1 𝑛 2∆𝑥 = 2 𝑢𝑙+1 𝑛 = 4∆𝑥+𝑢𝑙−1 𝑛 (17) sehingga bila kita substitusikan (16) dan (17) ke skema (13) akan didapatkan kondisi batas kiri sebagai berikut 𝑢1 𝑛+1 = 𝑢1 𝑛 − ∆𝑡 2∆𝑥 (𝑢1 𝑛 [𝑢2 𝑛 − 𝑢0 𝑛 ]) + ∆𝑡 2 4∆𝑥2 (𝑢1 𝑛[𝑢2 𝑛 − 𝑢0 𝑛]2) + ∆𝑡 2 2∆𝑥2 ((𝑢1 𝑛 )2[𝑢2 𝑛 − 2𝑢1 𝑛 + 𝑢0 𝑛]) 𝑢1 𝑛+1 = 𝑢1 𝑛 − ∆𝑡 2∆𝑥 (𝑢1 𝑛 [𝑢2 𝑛−𝑢2 𝑛 + 6∆𝑥]) + ∆𝑡 2 4∆𝑥2 (𝑢1 𝑛[𝑢2 𝑛 − 𝑢2 𝑛 + 6∆𝑥]2) + ∆𝑡 2 2∆𝑥2 ((𝑢1 𝑛)2[𝑢2 𝑛 − 2𝑢1 𝑛 + 𝑢2 𝑛 + 6∆𝑥]) 𝑢1 𝑛+1 = 𝑢1 𝑛 − ∆𝑡 2∆𝑥 (𝑢1 𝑛 [6∆𝑥]) + ∆𝑡 2 4∆𝑥2 (𝑢1 𝑛[6∆𝑥]2) + ∆𝑡 2 2∆𝑥2 ((𝑢1 𝑛)2[2𝑢2 𝑛 − 2𝑢1 𝑛 + 6∆𝑥]) dan kondisi batas kanan sebagai berikut 𝑢𝑙 𝑛+1 = 𝑢𝑙 𝑛 − ∆𝑡 2∆𝑥 (𝑢𝑙 𝑛 [𝑢𝑙+1 𝑛 − 𝑢𝑙−1 𝑛 ]) + ∆𝑡 2 4∆𝑥2 (𝑢𝑙 𝑛[𝑢𝑙+1 𝑛 − 𝑢𝑙−1 𝑛 ]2) + ∆𝑡 2 2∆𝑥2 ((𝑢𝑙 𝑛)2[𝑢𝑙+1 𝑛 − 2𝑢𝑙 𝑛 + 𝑢𝑙−1 𝑛 ]) 𝑢𝑙 𝑛+1 = 𝑢𝑙 𝑛 − ∆𝑡 2∆𝑥 (𝑢𝑙 𝑛[4∆𝑥+𝑢𝑙−1 𝑛 − 𝑢𝑙−1 𝑛 ]) + ∆𝑡 2 4∆𝑥2 (𝑢𝑙 𝑛[4∆𝑥+𝑢𝑙−1 𝑛 − 𝑢𝑙−1 𝑛 ]2) + ∆𝑡 2 2∆𝑥2 ((𝑢𝑙 𝑛)2[4∆𝑥+𝑢𝑙−1 𝑛 − 2𝑢𝑙 𝑛 + 𝑢𝑙−1 𝑛 ]) 𝑢𝑙 𝑛+1 = 𝑢𝑙 𝑛 − ∆𝑡 2∆𝑥 (𝑢𝑙 𝑛[4∆𝑥]) + ∆𝑡 2 4∆𝑥2 (𝑢𝑙 𝑛[4∆𝑥]2) + ∆𝑡 2 2∆𝑥2 ((𝑢𝑙 𝑛)2[4∆𝑥+2𝑢𝑙−1 𝑛 − 2𝑢𝑙 𝑛 ]) 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( u ) 𝑢𝑥 (0, 𝑡) = 3 (14) 𝑢𝑥 (1680, 𝑡) = 2 (15) 𝑢(𝑥, 0) = −𝑥 1680 + 3 binti tsamrotul fitria 167 volume 3 no. 3 november 2014 simulasi untuk kasus kelima ini dilakukan untuk model non linier dengan batas neuman. dengan cara yang sama dengan kasus sebelumnya maka didapatkan bentuk grafik kepadatan berdasarkan ruang dan waktu adalah sebagai berikut gambar 15. simulasi kasus 5 dari gambar 15 di atas dapat dilihat bahwa penumpukan kendaraan terjadi di waktu pertama. setelah 𝑡 waktu kemudian kepadatan menurun. artinya berdasarkan model ini jalan bisa dikatakan sepi. penumpukan jumlah kendaraan terjadi ujung interval yang menyebabkan kepadatan di ujung mencapai 4 kendaraan persatuan luas. hal ini menunjukkan di ujung jalan terjadi kemacetan yang parah. berikut disajikan grafik pada beberapa waktu pengamatan yang berbeda gambar 16. simulasi kasus 5 dengan waktu berbeda gambar 16 di atas menunjukkan kondisi jalan di beberapa waktu yang berbeda. dari beberapa posisi yang diamati, dapat diketahui bahwa pada waktu mula-mula kepadatan kendaraan sangatlah tinggi, akan tetapi, lamalama kepadatannya menurun meski terjadi fluktuasi sepanjang interval jalan. simulasi keenam adalah simulasi yang dilakukan dengan model yang sama dengan simulasi kelima. yaitu dengan menggunakan persamaan burger dan kondisi batas neumann. akan tetapi kondisi batas yang dipakai berkebalikan dengan simulasi sebelumnya. persamaan beserta kondisi batasnya adalah sebagai berikut 𝑢𝑡 + 𝑢𝑢𝑥 = 0 dengan kondisi 𝑢𝑥 (0, 𝑡) = 2 𝑢𝑥 (1680, 𝑡) = 3 𝑢(𝑥, 0) = 𝑥 1680 + 2 dengan cara yang sama dengan kasus kelima, maka didapatkan kondisi batas kiri 𝑢1 𝑛+1 = 𝑢1 𝑛 − ∆𝑡 2∆𝑥 (𝑢1 𝑛 [4∆𝑥]) + ∆𝑡 2 4∆𝑥2 (𝑢1 𝑛[4∆𝑥]2) + ∆𝑡 2 2∆𝑥2 ((𝑢1 𝑛)2[2𝑢2 𝑛 − 2𝑢1 𝑛 + 4∆𝑥]) dan kondisi batas kanan sebagai berikut 𝑢𝑙 𝑛+1 = 𝑢𝑙 𝑛 − ∆𝑡 2∆𝑥 (𝑢𝑙 𝑛[6∆𝑥]) + ∆𝑡 2 4∆𝑥2 (𝑢𝑙 𝑛[6∆𝑥]2) + ∆𝑡 2 2∆𝑥2 ((𝑢𝑙 𝑛)2[6∆𝑥+2𝑢𝑙−1 𝑛 − 2𝑢𝑙 𝑛 ]) sehingga hasil simulasi dengan menggunakan matlab sebagai berikut gambar 17. simulasi kasus 6 grafik yang dihasilkan tidak jauh berbeda dengan kasus kelima. keduanya memiliki kepadatan di waktu yang pertama dan kemudian kepadatannya menurun setelah 𝑡 waktu. sehingga bisa dikatakan untuk model non linier dengan batas neumann kepadatan kendaraan di sepanjang ruas jalan rendah (tidak ada kemacetan), sehingga memungkinkan para pengemudi untuk memaksimalkan kecepatan kendaraannya. untuk perubahan kepadatan per satuan waktu bisa dilihat dari gambar berikut 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( u ) penurunan model traffic flow berdasarkan hukum-hukum kesetimbangan cauchy – issn: 2086-0382 168 gambar 18. simulasi kasus 6 dengan waktu berbeda penutup berdasarkan pembahasan di atas dapat disimpulkan bahwa model traffic flow yang diturunkan dari hukum-hukum kesetimbangan adalah sebagai berikut 𝜕𝜌 𝜕𝑡 + 𝜕(𝜌𝑣) 𝜕𝑥 = 0 dimana 𝜌 menyatakan kepadatan kendaraan, 𝑣 merupakan kecepatan kendaraan. bila kecepatan 𝑣 dalam model tersebut adalah konstan, maka model traffic flow tersebut menjadi persamaan transport (linier) 𝜌𝑡 + 𝑣𝜌𝑥 = 0 sedangkan bila kecepatan 𝑣 merupakan suatu fungsi yang bergantung pada kepadatan kendaraan di setiap ∆𝒙 nya maka model tersebut menjadi persamaan burger (tansport non linier), dan dapat dituliskan sebagai 𝑢 + 𝑢𝑢𝑥 = 0 sedangkan untuk mencari solusi dari kedua model yang telah dihasilkan, dilakukan dengan menggunakan metode beda hingga skema ftbs untuk persamaan transport, dan metode lax wendroff skema ftcs untuk persamaan burger. berdasarkan hasil solusi numeriknya dapat diketahui bahwa model linier tersebut stabil dengan syarat 0 ≤ 𝑣 ∆𝑡 ∆𝑥 ≤ 1, sedangkan model non liniernya stabil bila dilihat dari grafiknya, akan tetapi perlu adanya analisis konvergensi untuk mengetahui kestabilan dan kekonsistenan dari model non linier tersebut. daftar pustaka [1] k. nagel, “particle hopping vs. fluiddynamical models for traffic flow,” la ur, p. 4018, 1995. [2] d. s. & j. lv, “in-depth analysis of traffic congestion using computational fluid dynamic (cfd) modelling method,” j. mod. transp., vol. 19, no. 1, pp. 58–67, 2011. [3] c. f. daganzo, “requiem for second-order fluid approximations of traffic flow,” transp. res. part b methodol., vol. 29, pp. 277–286, 1995. [4] a. aw and m. rascle, “resurrection of ‘second order’ models of traffic flow,” siam journal on applied mathematics, vol. 60. pp. 916–938, 2000. [5] g. whitham, linier and non linier waves. canada: a wilev-imterscience series, 1974. [6] immers dan logghe, traffic flow theory, no. may. heverlee belgium: a wilevimterscience series, 2002. [7] r. m. olson, dasar dasar mekanika fluida teknik. 1993. [8] a. jungel, “modeling and numerical approximation of traffic flow problems,” 2002. 0 500 1000 1500 0 1 2 3 4 5 t = 49.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 249.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 399.95 posisi (x) k e p a d a ta n ( u ) 0 500 1000 1500 0 1 2 3 4 5 t = 999.95 posisi (x) k e p a d a ta n ( u ) pendahuluan tinjauan pustaka 1. persamaan pengatur a. persamaan momentum 2. variabel makroskopis a. ukuran interval b. kepadatan kendaraan c. laju alir kendaraan (fluks) d. kecepatan e. hubungan antara ketiga variabel makroskopis pembahasan 1. penurunan model traffic flow 2. hubungan kepadatan dengan kecepatan 3. penskalaan 4. solusi dan simulasi a. model linier dari persamaan traffic flow b. model non linier persamaan traffic flow penutup daftar pustaka the application of quadratic bezier curve on rotational and symmetrical lampshade cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 100-106 submitted: 12 february 2016 reviewed: 17 march 2016 accepted: 1 may 2016 the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih, imam sujarwo mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang email: erny.octafia1710@gmail.com, imamsoejarwo@gmail.com abstract the procedure of constructing lampshade is through parameter merger and selection of bezier surfaces shapeshifters; thus it produces perfect and varied sitting-lamp shades. constructing sitting-lamp shades requires the study of the physical (lighting) and geometry aspects. in terms of geometry, the existed sittinglamp shade creation models is generally monotonous and constructed from objects pieces. in line with these problems, this research is divided into four stages: first, preparing the data to build a sitting-lamp shade. second, technical studying to construct the symmetricity of the sitting-lamp shade shape. third, constructing the parts of the sitting-lamp shade, namely the base, the main part, the roof. fourth, constructing the complete sitting-lamp shade. the results of this research to obtain the procedures of constructing the sitting-lamp shade namely: first, dividing the major axis into three non-homogeneous sub segments. second, constructing parts of the sitting-lamp shade namely the base, the main part, and the roof by combining the sitting-lamp shade components from the geometrical objects deformation. third, filling each sub segment of non-homogeneous parts with parts of the lampshade and creating the boundary curves producing a varied innovative symmetrical models of sitting-lamp shade. keywords: sitting-lamp shade, hermit curve, bezier curve. introduction the fabrication process of lampshade requires several treatments. first, building a model of the object to be constructed. second, converting raw materials into semi-finished object. third, forming semi-finished object to a subtler form. and the last, smoothing the surface of objects. the problem is that it turns out the existed design and fabrication techniques, often causes industry losses. first, using mall design technique that widely used by the craftsmen, the obtained results are often fails to meet the export contract orders, it is because the size of the object become incompatible with the buyer’s order. second, trial and error technique fabrication requires more expensive cost, since it produces many risks and errors on fabrications. it also requires extra time and effort for the production process. in addition, the lack of awareness of the manufactured object needs, many raw materials are wasted. constructing sitting-lamp shades requires the study of the physical (lighting) and geometry aspects. in terms of geometry, the existed sitting-lamp shade creation models is generally monotonous and constructed from objects pieces. it can be seen that the industrial sitting-lamp shade products are still simple and the used design techniques are still conventional. the technique takes a very long time so the customer orders are often not completed on time. mailto:erny.octafia1710@gmail.com1 mailto:imamsoejarwo@gmail.com the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 101 besides, the resulted products are generally constants, no improvements in terms of art s and innovations needed by customers. the diversity is in the form of the level of symmetricity, harmony, and model variations as well as from the types and sizes of the offered products. this article is a scientific effort to obtain innovative and varied sitting-lamp shade models. this article attempts to construct a sitting-lamp shade, so that the procedure of its construction can be determined. theoritical review quadratic hermitian curve the hermitian curve is in the form of 𝑃(𝑢) = 𝑃(0)𝐾1(𝑢) + 𝑃(1)𝐾2(𝑢) + 𝑃 ′(1)𝐾3(𝑢) (1) with 𝐾1(𝑢) = (1 − 2𝑢 + 𝑢 2) 𝐾2(𝑢) = (2𝑢 − 𝑢 2) 𝐾3(𝑢) = (−𝑢 + 𝑢 2) 𝑃(0) = initial point of the curve in the form of (𝑥0, 𝑦0, 𝑧0) 𝑃(1) = end point of the curve in the form of (𝑥1, 𝑦1, 𝑧1) 𝑃′(1) = control point of curvature of the curve with 0 ≤ 𝑢 ≤ 1 [1] second order bezier curve the second order bezier curve is represented in parametric form: 𝑉(𝑢) = 𝐾0(1 − 2𝑢 + 𝑢 2) + 𝐾1(2𝑢 − 2𝑢 2) + 𝐾2(𝑢 2) (2) with 0 ≤ 𝑢 ≤ 1 [2]. rotation rotation is the change of an object coordinates into the new position by moving the whole point coordinates defined in the initial form with an angle on an axis of rotation. the coordinate system 𝑅3 has three rotary axes, the rotation of each axis can be written as follows, with 𝜃 denotes the rotary angle. a. rotation on 𝑥-axis (𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞) = (𝑥𝑝 , 𝑦𝑝 ⋅ cos 𝜃 − 𝑧𝑝 ⋅ sin 𝜃 , 𝑦𝑝 ⋅ sin 𝜃 + 𝑧𝑝 ⋅ cos 𝜃) (3) b. rotation on 𝑦-axis (𝑥𝑞 , 𝑦𝑞, 𝑧𝑞) = (𝑥𝑝 ⋅ cos 𝜃 + 𝑧𝑝 ⋅ sin 𝜃 , 𝑦𝑝 𝑥𝑝 ⋅ (−sin 𝜃) + 𝑧𝑝 ⋅ cos 𝜃) (4) c. rotation on 𝑧-axis (𝑥𝑞 , 𝑦𝑞, 𝑧𝑞) = (𝑥𝑝 ⋅ cos 𝜃 + 𝑦𝑝 ⋅ sin 𝜃 , 𝑥𝑝 ⋅ (− sin 𝜃) + 𝑦𝑝 ⋅ cos 𝜃 , 𝑧𝑝) (5) [3] displacement if 𝑃(𝑥𝑝, 𝑦𝑝, 𝑧𝑝) is the initial position, 𝑄(𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞) is the position after point is displaced, 𝐼 is the identity matrix, and (𝑡𝑟𝑥 , 𝑡𝑟𝑦 , 𝑡𝑟𝑧 ) is a constant value that indicates the amount of displacement in each coordinate axes, then the displacement can be expressed as[3] (𝑥𝑞 , 𝑦𝑞, 𝑧𝑞) = (𝑥𝑝 + 𝑡𝑟𝑥 , 𝑦𝑝 + 𝑡𝑟𝑦 , 𝑥𝑝 + 𝑡𝑟𝑥 ) (6) the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 102 interpolation of line segments and curves in space let 𝐴𝐵̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ be line segments defined by: 𝐴(𝑥1, 𝑦1, 𝑧1), 𝐵(𝑥2, 𝑦2, 𝑧3), 𝐶(𝑥3, 𝑦3, 𝑧3) and 𝐷(𝑥4, 𝑦4, 𝑧4) in parametrical form 𝐼1(𝑢) and 𝐼2(𝑢). parametrical surface obtained from linear interpolation of to line segments is formulated as follows 𝑆(𝑢, 𝑣) = (1 − 𝑣)𝐼1(𝑢) + 𝑣𝐼2(𝑢) (7) with 0 ≤ 𝑢 ≤ 1 and 0 ≤ 𝑣 ≤ 1. there are special cases of linear interpolations form of these two line segments. if 𝐴 = 𝐵, then the interpolation result of (7) produces a triangle. if 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ , then it is generally produces a quadrangle plane. if the plane is obtained from interpolation of two crossed lines, then it produces a non-flat surface (can be curved of twisted) in some regions of the surface.[4] curved surfaces can be constructed from space curve interpolations using the following equation 𝑆(𝑢, 𝑣) = (1 − 𝑣)𝐶1(𝑢) + 𝑣𝐶2(𝑢) (8) where 𝐶1(𝑢) and 𝐶2(𝑢) are boundary curves (figure 2)[4] figure 1. special case of linear interpolation of two line segments figure 2. a linear interpolation on curves point dilatation on ℝ𝟑 dilatations that maps point 𝑃(𝑥, 𝑦, 𝑧) to 𝑃′(𝑥′, 𝑦 ′, 𝑧 ′) is formulated to: [ 𝑥′ 𝑦 ′ 𝑧 ′ ] = [ 𝑘1 0 0 0 𝑘2 0 0 0 𝑘3 ] [ 𝑥 𝑦 𝑧 ] (9) in this case 𝑘1, 𝑘2, 𝑘3 represent the scale factors on 𝑋-axis, 𝑌-axis, and 𝑍-axis respectively. if 𝑘1 = 𝑘2 = 𝑘3, then the resulting objects are congruent with the original objects (enlarged, reduced, or remain unchanged).[1] for instance, a triangle 𝑃𝑄𝑅 with angle points 𝑃(𝑥1, 𝑦1, 𝑧1), 𝑄(𝑥2, 𝑦2, 𝑧2) and 𝑅(𝑥3, 𝑦3, 𝑧3) is dilated with multiplication factor 𝑘 > 1, then we obtain triangular map 𝑃′𝑄′𝑅′ with angle points 𝑃′(𝑘𝑥1, 𝑘𝑦1, 𝑘𝑧1), 𝑄 ′(𝑘𝑥2 , 𝑘𝑦2, 𝑘𝑧2) and 𝑅 ′(𝑘𝑥3, ᡐ𝑦3, 𝑘𝑧3) showed in figure 3. the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 103 figure 3. second order bezier curve the representation of geometry space objects a. representation of a tube let a tube is known with the base center 𝑃1(𝑥1, 𝑦1, 𝑧1), radius 𝑟, and height 𝑡, it can be drawn using the following equations: if the central axis is parallel to 𝑍-axis 𝑇(𝜃, 𝑧) = (𝑥1 + 𝑟 cos 𝜃 , 𝑦1 + 𝑟 sin 𝜃 , 𝑧) (10) with 0 ≤ 𝜃 ≤ 2𝜋 and r ∈ ℝ. if the central axis is parallel to 𝑋-axis 𝑇(𝜃, 𝑧) = (𝑥, 𝑦1 + 𝑟 sin 𝜃 , 𝑧1 + 𝑟 cos 𝜃) (11) with 0 ≤ 𝜃 ≤ 2𝜋 and 𝑥1 < 𝑥 ≤ 𝑥1 + 𝑡. if the central axis is parallel to 𝑌-axis 𝑇(𝜃, 𝑧) = (𝑥1 + 𝑟 cos 𝜃 , 𝑦, 𝑧1 + 𝑟 sin 𝜃) (12) with 0 ≤ 𝜃 ≤ 2𝜋 and 𝑦1 ≤ 𝑦 ≤ 𝑦1 + 𝑡. [5] figure 4. tubes with different central axis b. representation of a ball a ball with center 𝑄(𝑎, 𝑏, 𝑐) and radius 𝑟 is formulated to: 𝐵(𝜃, 𝛽) = (𝑟 sin 𝜃 cos 𝛽 + 𝑎, 𝑟 cos 𝜃 + 𝑐, 𝑟 sin 𝜃 sin 𝛽 + 𝑏) (13) with parameter 0 ≤ 𝜃 ≤ 2𝜋 and 0 ≤ 𝛽 ≤ 2𝜋 [5]. results and discussion this article describes the procedures of constructing the sitting-lamp shade. the first procedure is to construct the used components of siting-lamp shade. the next step is to construct parts of the sitting-lamp shade namely the base, and the main part of the roof. the latter procedure is to construct the full sitting-hood. a. constructing sitting-lamp shade components 1. tube deformation suppose given a tube of radius 𝑟, the minimum radius of the tube is 10 cm, while the maximum limit of the tube radius is 20 cm, the minimum height tube is 10 cm while the maximum height of the tube is 30 cm, and the base is centered in 𝑃(𝑥0, 𝑦0, 𝑧0). thus, the sizes of the tube is 10𝑐𝑚 ≤ 𝑡 ≤ 30𝑐𝑚 and 10𝑐𝑚 ≤ 𝑟 ≤ 20𝑐𝑚. the selection of the value of 𝑟 and 𝑡 in the interval aims to differences in size of shape components of sittinglamp shade. based on these data designed various forms of constituent components of the sitting-lamp shade using surface curve modification techniques and surface arch dilatation technique. the algorithms of tube deformation with the modification of the surface curve are as follows: the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 104 a. determined a center point on the tube base circle (𝑥1, 𝑦1, 𝑧1) = (0, 0, 0), build the tube base circle using the circle equation and set the value 𝜃 = 0 and obtain the point 𝑃(0). 𝑃 (0) = (𝑥1 + 𝑟1𝑐𝑜 s 𝜃 , 𝑦1 + 𝑟1𝑠𝑖 n 𝜃 , 𝑧1) = (𝑟1, 0,0 ). b. determined center point on the tube roof circle namely (𝑥1, 𝑦1, 𝑧1) = (0, 0, 𝑡), build tube roof circle using the circle equation and set the value 𝜃 = 0 and obtain one point, namely 𝑃(1). 𝑃 (1) = (𝑥1 + 𝑟2 cos 𝜃 , 𝑦1 + 𝑟2 sin 𝜃 , 𝑧1) = (𝑟2, 0, 𝑡). c. determined control point 𝑃′(1) to control the curvature of the hermit curve so that 𝑃′(1) = (𝑥, 0, 𝑧) with −2𝑟 ≤ 𝑥, 𝑧 ≤ 2𝑡 and 𝑥, 𝑧 ∈ 𝑅. d. hermit curve constructed by substituting value 𝑃 (0), 𝑃 (1) and 𝑃′(1) to the equation (1) thus obtained 𝑃 (𝑢) = (𝑟𝐻1(𝑢) + 𝑟𝐻2(𝑢) + 𝑥𝐻3(𝑢), 0𝐻1(𝑢) + 0𝐻2(𝑢) + 0𝐻3(𝑢), 0𝐻1(𝑢) + 𝑡𝐻2(𝑢) + 𝑧𝐻3(𝑢)) with 𝐻_1 (𝑢) = (1 − 𝑢 − 𝑢 ^ 2) 𝐻_2 (𝑢) = (2𝑢 − 𝑢 ^ 2) 𝐻_3 (𝑢) = (− 𝑢 + 𝑢 ^ 2) 0 ≤ 𝑢 ≤ 1. e. hermit curve is rotated about the 𝑍 axis using equation (5) and 0 ≤ 𝜃 ≤ 360° so that 𝑃(𝑢) = ((𝑟𝐻1(𝑢) + 𝑟𝐻2(𝑢) + 𝑥𝐻3(𝑢)) cos 𝜃 , (𝑟𝐻1(𝑢) + 𝑟𝐻2(𝑢) + 𝑥𝐻3(𝑢)) sin 𝜃 , 0𝐻1(𝑢) + 𝑡𝐻2(𝑢) + 𝑡𝐻3(𝑢)) figure 5. different results of tube deformations 2. hexagonal prism deformation suppose given a regular hexagonal prism with coordinate pairs [𝐾𝑖 (𝑥𝑖 , 𝑦𝑖, 𝑧𝑖), 𝐾𝑖 ′(𝑥𝑖 , 𝑦𝑖, 𝑧𝑖 + 𝑡)] with 𝑖 = 1,2, 3, . . . , 6 and 𝑡 5𝑐𝑚 ≤ 𝑡 ≤ 15𝑐𝑚. all of its closure has center of gravity on 𝐾(𝑥0, 𝑦0, 𝑧0) and 𝐾 ′(𝑥0, 𝑦0, 𝑧0 + 𝑡). the distance of 𝐾 to 𝐾𝐼 and 𝐾′ to 𝐾𝑖 ′ is 6𝑐𝑚 ≤ 𝑟 ≤ 10𝑐𝑚. in this case, 𝐾𝐾′̅̅ ̅̅ ̅̅ is taken as the axis of symmetry of hexagonal prism deformation. steps of straight side deformation of the prism into curved-concaved namely: a. determined 𝐾𝑖 and 𝐾𝑖 ′ with 𝑖 = 0, 1, 2, 3, 4, 5 as a control point for multiple bezier curves using the linear equation of the circle by setting 𝜃 = 𝑖𝜋 3 and (𝑥1, 𝑦1, 𝑧1) = (0, 0, 0). 𝐾𝑖 (𝜃) = (𝑟 cos 𝑖𝜋 3 , 𝑟 sin 𝑖𝜋 3 , 0) 𝐾𝑖 ′(𝜃) = (𝑟 cos 𝑖𝜋 3 , 𝑟 sin 𝑖𝜋 3 , 𝑡). b. defined control points 𝑄 to control the curvature of the quadratic bezier curve the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 105 𝑄 = (𝑥0, 𝑦0, 𝑧) with 𝑧 ∈ [𝑧0, 𝑡]. c. bezier curve of degree two for each pair of control points (𝐾𝑖 , 𝑄, 𝐾𝑖 ′) is built using the equation (2) 𝑉𝑖(𝑢) = (1 − 𝑢) 2𝐾𝑖 + 2 (1 − 𝑢)(𝑢)𝑄 + 𝑢 2𝐾𝑖 ′ with 0 ≤ 𝑢 ≤ 1. d. bezier curves are consecutively linearly interpolated in pairs using equation (8) counter clockwise s(u, v) = ((1 − 𝑣)(1 − 𝑢)2 𝑟 cos 𝑖𝜋 3 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑥0 + (1 − 𝑣)𝑢 2𝑟 cos 𝑖𝜋 3 + 𝑣(1 − 𝑢)2𝑟 cos (𝑖+1)𝜋 3 + 𝑣(2𝑢 − 2𝑢2)𝑥0 + 𝑣𝑢 2 𝑟 cos (𝑖+1)𝜋 3 , (1 − 𝑣)(1 − 𝑢)2𝑟 sin 𝑖𝜋 3 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑦0 + (1 − 𝑣)𝑢 2𝑟 sin 𝑖𝜋 3 + 𝑣(1 − 𝑢)2 𝑟 sin (𝑖+1)𝜋 3 + 𝑣(2𝑢 − 2𝑢2)𝑦0 + 𝑣𝑢2 𝑟 sin (𝑖+1)𝜋 3 , (1 − 𝑣)(1 − 𝑢)20 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑧 + (1 − 𝑣)𝑢20 + 𝑣 0(1 − 𝑢)2 + 𝑣(2𝑢 − 2𝑢2)𝑧 + 𝑣0𝑢2) with 0 ≤ 𝑣 ≤ 1 and 0 ≤ 𝑢 ≤ 1. figure 6. variety of tube deformation b. constructing full lampshade furthermore, to obtain an intact form of sitting-lamp shade combined continuously in this section the concatenation of several basic components of sitting-lamp shade by combining the components parts of the sitting-lamp shade is conducted. steps of constructing full sitting-lamp shade: 1. the part of the line segment 𝑆3𝑆4̅̅ ̅̅ ̅̅ is filled with the base shade component of sitting-lamp. 2. the part of the line segment 𝑆2𝑆3̅̅ ̅̅ ̅̅ is filled with the main part of the sitting-lamp shade and adjust its height. 3. the part of the line segment 𝑆2𝑆1̅̅ ̅̅ ̅̅ is filled with the roof parts and adjust the sitting-lamp shade height. 4. some parts sitting-lamp shade is combined by establishing the boundary between two adjacent components with the following procedures. a. defined circle or a regular hexagon as a cap of the sitting-lamp shade base component with radius 𝑟1 as boundary curve 𝐶1(𝑢). b. defined circle or a regular hexagon as an upper cap of the sitting-lamp shade of the main components with radius 𝑟2 as boundary curve 𝐶2(𝑢). c. the boundary between 𝐶1(𝑢) and 𝐶2(𝑢) is constructed using linear interpolation of equation (8). d. do steps (a) to (c) to establish the boundary between the main body of the sitting-lamp shade with roofing components. the application of quadratic bezier curve on rotational and symmetrical lampshade erny octafiatiningsih 106 figure 7. another variety of sitting-lamp shade with different control points reference [1] kusno, geometri rancang bangun studi tentang desain dan pemodelan benda dengan kurva dan permukaan berbantu komputer, jember: jember university press, 2010. [2] kusno, a. cahaya and m. darsin, "modelisasi benda onyx dan marmer melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan putar bezier," jurnal ilmu dasar, pp. 175-185, 2014. [3] a. cristiyyanto, "perancangan objek tiga dimensi dengan teknik flat shading dan gouraund menggunakan bahasa turbo pascal 7.0," skripsi tidak dipublikasikan, 2003. [4] m. roifah, "modelisasi knop melalui penggabungan benda dasar hasl deformasi tabung, prisma segienam beraturan dan permukaan putar," skripsi tidak dipublikasikan, 2013. [5] a. bastian, "desain kap lampu duduk melalui penggabungan benda-benda geometi ruang," skripsi tidak dipublikasikan, 2010. abstract introduction theoritical review quadratic hermitian curve second order bezier curve rotation displacement interpolation of line segments and curves in space point dilatation on ,ℝ-𝟑. the representation of geometry space objects results and discussion reference 5.5 shofwan ali fauji analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk gelombang spektra babi dan sapi shofwan ali fauji mahasiswa pps institut teknologi sepuluh nopember surabaya e-mail: shofwan.ali.fauji@gmail.com abstrak jaringan syaraf tiruan (jst) sedikit demi sedikit mulai mampu menggantikan tugas seorang pakar, bahkan dengan jst mampu dibuat alat untuk menggantikan seorang dokter. salah satu jenis jst adalah jaringan backpropagation, jaringan ini dapat digunakan untuk melakukan pelatihan agar program mampu mengenali apakah itu gelombang spektra babi atau sapi. untuk menentukan keluaran dalam pelatihan backpropagation dibutuhkan fungsi aktivasi yang cocok. karena itu pada penelitian ini akan dibandingkan beberapa fungsi aktivasi yang dapat digunakan dalam pelatihan. fungsi aktivasi-fungsi aktivasi itu diuji coba dengan tes rasio untuk mengetahui interval kekonvergenannya. setelah diuji coba dengan tes rasio didapatkan bahwa fungsi aktivasi tanh� adalah fungsi aktivasi yang paling bagus untuk digunakan pelatihan jaringan backpropagation, karena memiliki rentang bobot yang dapat memenuhi metode-metode yang digunakan dalam penentuan bobot. setelah diuji coba dengan data, fungsi aktivasi tanh� mampu mengenali data uji coba dengan tepat semua. diharapkan pada penelitian selanjutnya untuk meneliti interval bobot yang membuat pelatihan mencapai konvergensi dengan cepat dan errornya sedikit. kata kunci:fungsi sigmoid biner, gelombang spektra sapi dan bab, jaringan backpropagation abstract artificial neural network (ann) is beginning little by little to replace the task of an expert, even with the ann can be a tool to replace a doctor. one of kind of ann is backpropagation networks, this network can be used to training programs in order to be able to recognize whether it is pig or cow wave spectra. to determine the output in backpropagation training required suitable activation functions. therefore, in this research will be compared to some of the activation function that can be used in training. activation functions will be tested with the ratio test to determine the interval convergence. after tested with the ratio test it was found that the activation function tanh z was the best activation function to use the backpropagation network training, because it has a weight range that can meet the methods used in the determination of weights. when tested with the data, the activation function tanh z is able to recognize correctly all trial datas. expected in future research to examine the weight that makes the interval training to achieve fast convergence and the error bit. keywords:backpropagation networks, binary sigmoid function, wave spectra in the case of cows and pigs pendahuluan babi adalah hewan yang haram dimakan. berbagai penelitian ilmiah dan medis membuktikan bahwa dibandingkan hewanhewan yang lain, babi tergolong paling berpotensi membawa virus dan bakteri yang membahayakan tubuh manusia (ahmad, 2008). semua jenis makanan yang diharamkan allah swt selalu bersifat kotor dan semua yang kotor mencakup semua obyek yang dapat merusak kehidupan, kesehatan, harta dan moral manusia. haramnya babi dapat dilihat pada surat al-an’am ayat 145. sebagai mahluk hidup babi juga memiliki protein, dari protein itulah dapat diketahui gelombang spektra babi. dan untuk membandingkannya maka digunakan gelombang spektra makanan yang halal dan sering digunakan sebagai bahan makanan masyarakat luas, yaitu sapi. berikut gambar gelombang spektra sapi dan babi yang digambar dengan alat fourier transform (ftir). gambar 1. turunan kedua spektra infra merah sapi dan babi dengan alat ftir analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk gelombang … jurnal cauchy – issn: 2086-0382 155 selanjutnya dalam penelitian ini akan dikaji kaitan jaringan syaraf tiruan (jst) dalam pendeteksian makanan yang mengandung babi atau tidak. jst dipilih karena jst sistem pemroses informasi yang memiliki karakteristik mirip dengan jaringan syaraf biologi (siang, 2009). seperti halnya otak manusia, jst juga terdiri beberapa neuron dan ada hubungan antara neuron-neuron tersebut. neuron-neuron tersebut akan mentransformasikan informasi yang diterima melalui sambungan keluarnya menuju ke neuron-neuron yang lain. pada jst informasi tersebut disimpan pada suatu nilai tertentu yang disebut bobot. jst dibentuk sebagai generalisasi model matematika dari jaringan syaraf biologi, dengan asumsi bahwa: • pemrosesan informasi terjadi pada banyak elemen sederhana (neuron). • sinyal dikirimkan diantara neuron-neuron melalui penghubung-penghubung. • penghubung antar neuron memiliki bobot yang akan memperkuat dan memperlemah sinyal. • untuk menentukan keluaran, setiap neuron menggunakan fungsi aktivasi (biasanya non linier) yang dikenakan pada jumlahan masukan yang diterima. besarnya keluaran ini selanjutnya dibandingkan dengan suatu batas ambang. kajian teori 1. matriks matriks adalah jajaran elemen berbentuk empat persegi panjang (purwanto, et al., 2005). bentuk (ukuran) matriks ditentukan oleh banyaknya baris dan kolom. untuk selanjutnya matriks matriks yang mempunyai � baris dan kolom dituliskan sebagai matriks berukuran � (misalnya matriks a berukuran 3 4), dan ukuran ini biasa disebut dengan ordo matriks. berikut ini adalah beberapa contoh matriks. � �3 21 4�� � � 6 7� matriks-matriks di atas ber-ordo, � 2 2, � � 2 1. matriks yang hanya mempunyai satu kolom disebut matriks kolom seperti matriks �, atau sering juga disebut dengan vektor. seperti halnya bilangan riil, matriksmatriks juga bisa ditambahkan, dikurangkan dan dikalikan dengan cara yang berguna (anton, 2000). definisi. dua matriks didefinisikan sama jika keduanya mempunyai ukuran yang sama dan unsur-unsurnya berpadanan sama. dalam notasi matriks, � ����� dan � � ����� mempunyai ukuran yang sama, maka � � jika dan hanya jika ��� � ��� untuk semua � dan �. contohnya adalah: � �3 46 5�� � � 3 4 6 5� dari contoh itu terlihat bahwa matriks dan � sama secara ukuran dan unsur-unsurnya. definisi. jika dan � adalah matriksmatriks berukuran sama, maka jumlah � adalah matriks yang diperoleh dengan menambahkan unsur-unsur matriks dengan unsur-unsur � yang berpadanan. !� adalah matriks yang diperoleh dengan mengurangkan unsur-unsur dengan unsur-unsur � yang berpadanan. matriks-matriks berukuran berbeda tidak bisa ditambahkan atau dikurangkan. definisi. jika adalah sebarang matriks dan " adalah sebarang skalar, maka hasil kali " adalah matriks yang diperoleh dengan mengalikan setiap unsur dan ". definisi. jika adalah sebuah matriks � # dan � adalah sebuah matriks # , maka hasil kali � adalah matriks � yang unsurunsurnya didefinisikan sebagai berikut. untuk mencari unsur dalam baris � dan kolom � dari �, pilih baris � dari matriks dan kolom � dari matriks �. kalikan unsur-unsur yang berpadanan dari baris dan kolom secara bersama-sama dan kemudian jumlahkan hasil kalinya. definisi. perkalian matriks mensyaratkan bahwa jumlah kolom matriks sama dengan jumlah baris matriks � untuk membentuk hasil kali �. jika syarat ini tidak terpenuhi, hasil kalinya tidak terdefinisi. 2. citra citra adalah suatu representasi (gambaran), kemiripan, atau imitasi dari suatu objek (sutoyo, et al., 2009). citra sebagai keluaran suatu sistem perekaman data dapat bersifat optik berupa foto, bersifat analog berupa sinyal-sinyal video seperti gambar pada monitor televisi atau bersifat digital yang dapat langsung disimpan pada suatu media penyimpanan. citra dapat dibagi menjadi dua, yaitu citra analog dan citra digital. citra analog adalah citra yang bersifat kontinu, seperti gambar pada monitor televisi, foto sinar x, foto yang tercetak di kertas foto, lukisan, pemandangan alam, hasil ct scan dan lain sebagainya. citra analog tidak dapat dipresentasikan dalam komputer sehingga tidak bisa diproses di komputer secara langsung. oleh sebab itu, agar citra ini dapat diproses di komputer, proses konversi analog ke digital harus dilakukan terlebih dahulu. citra analog dihasilkan dari alat-alat analog di antaranya adalah video kamera analog, kamera foto analog dan ct scan. citra digital adalah citra yang dapat diolah oleh komputer. shofwan ali fauji 156 volume 2 no. 3 november 2012 3. akuisisi citra akuisisi citra adalah tahap awal untuk mendapatkan citra digital (sutoyo, et al., 2009). tujuan akuisisi citra adalah untuk menentukan data yang diperlukan dan memilih metode perekaman citra digital. tahap ini dimulai dari objek yang akan diambil gambarnya, persiapan alat-alat, sampai pada pencitraan. pencitraan adalah proses untuk mentranformasi citra analog menjadi citra digital. beberapa alat yang dapat digunakan untuk pencitraan di antaranya adalah video kamera, kamera digital, scanner dan sinar infra merah. hasil dari akuisisi citra ini ditentukan oleh kemampuan sensor untuk mendigitalisasi sinyal yang terkumpul pada sensor tersebut. 4. jaringan syaraf biologi jaringan syaraf biologi terdiri atas banyak elemen pemroses sederhana yang disebut neuron, sel, unit atau simpul (putra, 2010). sebagai bahan perbandingan, otak seekor cacing diperkirakan memiliki 1000 neuron dan otak manusia memiliki sekitar 100 miliar. setiap sel syaraf berhubungan dengan sel syaraf lainnya memakai saluran komunikasi yang teratur dengan suatu bobot penghubung. neuron bekerja berdasarkan impuls atau sinyal yang diberikan pada neuron (siang, 2009). neuron meneruskannya pada neuron lain. diperkirakan manusia memiliki 10%& neuron dan 6,10%' sinapsis. dengan jumlah yang begitu banyak, otak mampu mengenali pola, melakukan perhitungan dan mengontrol organ-organ tubuh dengan kecepatan yang lebih tinggi dibandingkan komputer digital. sebagai perbandingan, pengenalan wajah seseorang yang sedikit berubah (misal memakai topi, memiliki jenggot, dan lain-lain) akan lebih cepat dilakukan manusia dibandingkan komputer. pada waktu lahir, otak mempunyai struktur yang menakjubkan karena kemampuannya membentuk sendiri aturanaturan atau pola berdasarkan pengalaman yang diterima. jumlah dan kemampuan neuron berkembang seiring dengan pertumbuhan fisik manusia, terutama pada umur 0-2 tahun. pada 2 tahun pertama umur manusia terbentuk 1 juta sinapsis per-detiknya. pada dasarnya neuron memiliki 4 daerah utama, yaitu: sel tubuh atau soma sel tubuh atau soma merupakan jantungnya sel yang memiliki inti (nucleus). soma bertugas memproses nilai masukan dari semua dendrit yang terhubung dengannya menjadi suatu keluaran. soma memiliki dua cabang yaitu dendrit dan axon. dendrit dendrit merupakan suatu perluasan dari soma yang menyerupai rambut dan bertindak sebagai saluran untuk menerima masukan dari sel syaraf lainnya melaui sinapsis. axon neuron biasanya hanya memiliki satu axon yang tumbuh dari bagian soma dan disebut dengan axon hillock. axon menyalurkan sinyal elektrik yang dihasilkan pada bagian bawah dari axon hillock. sinyal elektrik digunakan oleh neuron untuk menyampaikan informasi atau sinyal ke otak dengan semua sinyal sama. oleh karena itu, otak menentukan jenis informasi yang diterima berdasarkan jalur yang membawa sinyal. otak kemudian menganalisis dan menafsirkan jenis informasi yang diterima. myelin adalah materi lemak yang melindungi syaraf. fungsinya seperti lapisan pelindung pada kabel listrik dan memudahkan syaraf untuk mengirim impulsnya dengan cepat. tidak semua bagian axon terbungkus dengan myelin. bagian yang tidak terbungkus ini disebut nodus ranvier. pada nodus ini, sinyal yang mengalir dan mengalami penurunan akan diperkuat lagi. hal ini akan memastikan bahwa perjalanan sinyal pada axon mengalir cepat dan tetap konsisten. sinapsis sinapsis merupakan bagian kontak (tempat) terjadinya pertukaran sinyal antar dua neuron. neuron sebenarnya secara fisik tidak berhubungan. mereka dipisahkan oleh synaptic cleft. neuron yang mengirim sinyal disebut dengan sel presynaptic, sedangkan neuron yang menerima sinyal disebut dengan sel postsynaptic (putra, 2010). neuron biologi merupakan sistem yang “fault tolerant” dalam 2 hal (siang, 2009). pertama, manusia dapat mengenali sinyal masukan yang agak berbeda dari yang pernah kita terima sebelumnya. sebagai contoh, manusia sering dapat mengenali seseorang yang wajahnya pernah dilihat dari foto, atau dapat mengenali seseorang yang wajahnya agak berbeda karena sudah lama tidak ditemuinya. kedua, otak manusia tetap mampu bekerja meskipun beberapa neuronnya tidak mampu bekerja dengan baik. jika sebuah neuron rusak, neuron lain kadang-kadang dapat dilatih untuk menggantikan fungsi sel yang rusak tersebut. 5. jaringan syaraf tiruan jaringan syaraf tiruan merupakan model jaringan syaraf yang meniru prinsip kerja dari neuron otak manusia (putra, 2010). jst pertama analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk gelombang … jurnal cauchy – issn: 2086-0382 157 kali muncul setelah model sederhana dari neuron buatan diperkenalkan oleh mcculloch dan pitts pada tahun 1943. model sederhana tersebut dibuat berdasarkan fungsi neuron biologis yang merupakan dasar unit pensinyalan dari sistem syaraf. jst memiliki beberapa kemampuan seperti yang dimiliki otak manusia, yaitu: a. kemampuan untuk belajar dari pengalaman. b. kemampuan melakukan perumpamaan terhadap masukan baru dari pengalaman yang dimilikinya. c. kemampuan memisahkan karakteristik penting dari masukan yang mengandung data yang tidak penting. jst ditentukan oleh 3 hal (siang, 2009): pola hubungan antar neuron (disebut arsitektur jaringan) a. metode untuk menentukan bobot penghubung (disebut metode training/learning/algoritma) b. fungsi aktivasi sebagai contoh, perhatikan neuron ( pada gambar 2 gambar 2. jaringan syaraf tiruan sederhana jst dapat diaplikasikan kedalam pengenalan pola. secara umum pengenalan pola adalah suatu ilmu untuk mengklasifikasikan atau menggambarkan sesuatu berdasarkan pengukuran kuantitatif fitur (ciri) atau sifat utama dari suatu objek (putra, 2010). pola sendiri adalah suatu entitas yang terdefinisi dan dapat diidentifikasikan serta diberi nama. sidik jari adalah suatu contoh pola. pola bisa merupakan kumpulan hasil pengukuran atau pemantauan dan bisa dinyatakan dalam notasi matriks atau vektor. pengenalan pola telah dikembangkan adalah pengenalan secara otomatis karakter tulisan tangan (angka atau huruf), dengan variasi yang sangat banyak di ukuran, posisi dan gaya tulisan (fausett, 1994). seperti jaringan backpropagation yang telah digunakan untuk mengenali tulisan tangan kode pos. jst dapat dibagi lagi menurut jenis pelatihannya. dalam jst ada dua macam pelatihan yang dikenal, yaitu pelatihan terbimbing dan tak terbimbing (siang, 2009). dalam pelatihan terbimbing, terdapat sejumlah pasangan data (masukan dan keluaran) yang digunakan untuk melatih jaringan hingga diperoleh bobot yang diinginkan. pasangan data tersebut berfungsi sebagai “guru” untuk melatih jaringan hingga diperoleh bentuk yang terbaik. “guru” akan memberikan informasi yang jelas tentang bagaimana sistem harus mengubah dirinya untuk meningkatkan kerjanya. pada setiap kali pelatihan, suatu input diberikan ke jaringan. jaringan akan memproses dan mengeluarkan keluaran. selisih antara keluaran jaringan dengan target (keluaran yang diinginkan) merupakan error. jaringan akan memodifikasi bobot sesuai dengan error tersebut. sebaliknya, dalam pelatihan tak terbimbing tidak ada guru yang akan mengarahkan proses pelatihannya, perubahan bobot jaringan dilakukan berdasarkan parameter tertentu dan jaringan dimodifikasi menurut ukuran parameter tersebut. contohnya adalah jaringan lapisan kompetitif. selain jenis pelatihan yang tidak kalah penting dalam jst adalah fungsi aktivasi. kegunaan fungsi aktivasi sudah dijelaskan di atas, yaitu untuk menentukan keluaran. banyak fungsi aktivasi yang dapat digunakan di antaranya fungsi-fungsi goniometri dan hiperboliknya, fungsi unit step, impuls, dan sigmoid (purnomo, et al., 2006). tetapi yang lazim digunakan adalah fungsi sigmoid, karena dianggap lebih mendekati kinerja sinyal pada otak manusia. pada umumnya fungsi aktivasi membangkitkan sinyal-sinyal unipolar atau bipolar. fungsi sigmoid, dapat juga dibuat untuk jenis unipolar maupun bipolar. 6. jaringan backpropagation backpropagation merupakan algoritma pembelajaran yang terbimbing(kusumadewi, 2004). algoritma backpropagation menggunakan error keluaran untuk mengubah nilai bobotbobotnya dalam arah mundur. untuk mendapatkan error ini, tahap perambatan maju harus dilakukan terlebih dahulu. arsitektur yang digunakan dalam jaringan backpropagation merupakan arsitektur lapisan jamak (siang, 2009). karena merupakan jaringan lapisan jamak, maka backpropagation memiliki lapisan tersembunyi. dan dalam jaringan backpropagation, fungsi aktivasi yang digunakan harus memenuhi beberapa syarat, yaitu: kontinu, terdifferensial dengan mudah dan merupakan fungsi yang tidak turun. salah satu fungsi yang memenuhi ketiga syarat tersebut sehingga sering digunakan adalah fungsi sigmoid biner. dan fungsi lain yang sering digunakan adalah fungsi sigmoid bipolar yang memiliki range (-1,1). shofwan ali fauji 158 volume 2 no. 3 november 2012 7. konvergensi definisi. sebuah barisan ) � * +, di dikatakan konvergen untuk setiap . -, atau dikatakan limit * +,, jika untuk setiap / 0 0 ada bilangan asli 1*/, sedemikian sehingga untuk setiap 2 1*/,, + memenuhi | + ! | 4 /(bartle dan sherbert, 2000). jika sebuah barisan memiliki limit, disebut konvergen. tapi jika tidak memiliki limit maka disebut divergen. berikut ini adalah beberapa konsep dasar tentang definisi yang berkaitan dengan limits yang digunakan sebagai dasar untuk mengevaluasi konvergensi atau sifat asimtotis dari suatu estimator (suhartono, 2007). definisi.misalkan �+ adalah suatu barisan bilangan riil. jika ada suatu bilangan riil � dan jika untuk setiap bilangan riil 5 0 0 ada suatu bilangan bulat 6*5,sedemikan hingga untuk semua 2 6*5,, |�+ ! �| 4 5 maka � merupakan limit dari barisan 7�+8. definisi. (i) suatu barisan 7�+8 dikatakan sebanyak-banyaknya pada orde 9, dinotasikan �+ � :* ;,, jika untuk beberapa bilangan riil terbatas ∆0 0, ada suatu bilangan bulat terbatas 6 sedemikan hingga untuk semua 2 6,| =%�+| 4 ∆. (ii) suatu barisan 7�+8adalah pada orde lebih kecil dari 9,dinotasikan �+ � >* 9,, jika untuk setiap bilangan riil 5 0 0 ada suatu bilangan bulat terbatas 6*5, sedemikian hingga untuk semua 2 6*5,, | =%�+| 4 5, yaitu =%�+ ? 0. 8. tes rasio dan deret taylor tes rasio sangat berguna menentukan apakah deret yang diberikan konvergen (stewart, 2010). a. jika lim+?∞ defghef d � i 4 1, maka deret ∑ �+ ∞+k% adalah konvergen. b. jika lim+?∞ defghef d � i 0 1 atau lim+?∞ defghef d � ∞ maka deret ∑ �+ ∞+k% adalah divergen. c. jika lim+?∞ defghef d � 1, maka tes rasio tidak dapat ditentukan. tidak ada hasil yang bisa ditentukan apakah ∑ �+∞+k% konvergen atau divergen. deret taylor teorema taylor. andaikan l sebuah fungsi yang memiliki turunan dari semua tingkatan dalam suatu selang �� ! #,� #�. syarat perlu dan cukup agar menjadi deret taylor adalah l*�, l;*�,* ! �, l ;;*�,* !�,& 2! n l* ,* !�,+ ! (1) apabila � � 0, maka deret tersebut disebut deret maclaurin. 9. spektroskopi infra merah spektroskopi infra merah merupakan salah satu alat yang banyak digunakan untuk mengidentifikasi senyawa baik alami maupun buatan (habsari, 2010). bila sinar infra merah dilewatkan melalui cuplikan senyawa organik, maka sejumlah frekuensi akan diserap sedang frekuensi yang lain diteruskan atau ditransmisikan tanpa diserap. gambaran antara persen absorbansi atau persen transmitansi lawan frekuensi akan menghasilkan suatu spektrum infra merah. transisi yang terjadi didalam serapan infra merah berkaitan dengan perubahan-perubahan vibrasi dalam molekul. daerah radiasi spektroskopi infra merah berkisar pada bilangan gelombang 1280-10 cm-1 atau pada panjang gelombang 0,78-1000 μm. dilihat dari segi aplikasi dan instrumentasi spektroskopi infra merah dibagi ke dalam tiga jenis radiasi yaitu infra merah dekat, infra merah pertengahan, dan infra merah jauh. ada 2 jenis instrmentasi untuk absorbsi infra merah yaitu, instrumentasi dispersi (konvensional) yang hanya digunakan untuk analisis kualitatif dan instrumentasi yang menggunakan fourier transform (ftir) dapat digunakan untk analisis kuantitatif dan kualitatif . spektroskopi ftir (fourier transform infrared) merupakan salah satu teknik analitik yang sangat baik dalam proses identifikasi struktur molekul suatu senyawa. komponen utama spektroskopi ftir adalah interferometer michelson yang mempunyai fungsi menguraikan (mendispersi) radiasi infra merah menjadi komponen-komponen frekuensi. penggunaan interferometer michelson tersebut memberikan keunggulan metode ftir dibandingkan metode spektroskopi infra merah konvensional maupun metode spektroskopi yang lain. di antaranya adalah informasi struktur molekul dapat diperoleh secara tepat dan akurat (memiliki resolusi yang tinggi). keuntungan yang lain dari metode ini adalah dapat digunakan untuk mengidentifikasi sampel dalam berbagai fase (gas, padat atau cair). kesulitan-kesulitan yang ditemukan dalam identifikasi dengan spektroskopi ftir dapat ditunjang dengan data yang diperoleh dengan menggunakan metode spektroskopi yang lain. uji kandungan lemak babi dengan metode ftir telah ditemukan adanya kekhasan vibrasi ulur c-h pada sampel lemak babi yang berbeda dengan lemak hewani lainnya. hasil yang diperoleh menunjukkan bahwa mesin ftir sangat berpotensi digunakan sebagai alat untuk mendeteksi lemak babi secara cepat dengan hasil yang konsisten. analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk gelombang … jurnal cauchy – issn: 2086-0382 159 pembahasan sebelum melakukan pelatihan data, maka yang harus disiapkan adalah data pelatihan, dan data pelatihan diambil dari data gelombang spektra sapi dan babi, disini + adalah bilangan gelombang dan o+ adalah absorbansi (sinar-x yang diserap oleh benda dan diteruskan untuk mengetahui vibrasi molekulnya). dan kemudian akan diinterpolasi lagrange untuk mengetahui apropksimasi fungsi dari data berpasangan. berikut adalah data bilangan gelombang dan absorbansi untuk data babi. tabel 1. bilangan gelombang dan absorbansi pada data babi bilangan gelombang absorbansi pq � 970,192166 or � -0,000009 ps � 972,120977 o% � 0,000002 pt � 974,049789 o& � 0,000003 pu � 975,9786 ov � -0,000002 pw � 977,907411 ox � -0,000006 py � 979,836223 oz � -0,000003 p[ � 981,765034 o\ � -0,000004 p] � 983,693846 o^ � -0,000015 p_ � 985,622657 o' � -0,000007 data bilangan gelombang dan absorbansi pada tabel di atas diinnterpolasi lagrange, sehingga menghasilkan persamaan 2 berikut ini `* , � !1,035741567042484 .10=%% ' (2) 8,104872585865455.10=' ^ ! 2,774703359707413.10=x \ 0,542808208062731 z ! 6,636724626885896.10& x 5,193243594766944.10z v ! 2,539809476507496.10' & 7,097788748923267.10%r ! 8,678018906807824.10%& sehingga didapatkan gambar 3 sebagai grafik data pelatihan, dan gambarnya adalah sebagai berikut gambar 3. gelombang spektra babi untuk data sapi dengan alat ftir, didapatkan bilangan gelombang dan data absorbansi pada tabel 2 berikut. tabel 2. bilangan gelombang dan absorbansi pada data sapi bilangan gelombang absorbansi pq � 970,192166 or � 0.00005 ps � 972,120977 o% � 0.000034 pt � 974,049789 o& � -0.000006 pu � 975,9786 ov � -0.000043 pw � 977,907411 ox � -0.000061 py � 979,836223 oz � -0.00006 p[ � 981,765034 o\ � -0.000041 p] � 983,693846 o^ � -0.000011 p_ � 985,622657 o' � 0.000013 setelah itu data bilangan gelombang dan absorbansi diinnterpolasi lagrange, sehingga didapatkan aproksimasi fungsi data berpasangannya adalah, `* , � 7,897573065875410.10=%& ' (3) !6,180374054490586.10=' ^ 2,115981828265967.10=x \ !0.413970186280669 z 5,061791113327844.10& x !3,961114096958705.10z v 1,937351871176862.10' & !5,414515674767383.10%r 6,62043892617714810%& sehingga didapatkan gambar 4 berikut ini sebagai grafiknya untuk dijadikan data pelatihan. gambar 4. gelombang spektra sapi fungsi aktivasi yang akan dibanding adalah fungsi a. tan�, b. sin�, c. tanh�, d. sinh�, e. arctan�, f. arcsin�, g. arctanh�, h. arcsinh� delapan fungsi aktivasi itu dipilih karena sebagai fungsi non linier yang cukup memadai untuk digunakan sebagai fungsi aktivasi (kim, 2001). bentuk kompleks dari fungsi aktivasi yang akan dibandingkan adalah. -0.00002 -0.00001 0 0.00001 960 970 980 990 bilangan gelombang babi 100% shofwan ali fauji 160 volume 2 no. 3 november 2012 tan� � f �g ! f=�g � *f�g f=�g, sin� � f �g ! f=�g 2 � tanh� � f g ! f=g fg f=g sinh� � f g ! f=g 2 arctan� � 12� lnh �� 1 1! ��i arcsin� � 1� lnj�� k1! � &l arctanh� � 12lnh � 1 1 ! �i arcsinh� � ln j� k�& 1l kedelapan fungsi aktivasi diata diuji dengan tes rasio untuk mengetahui interval kekonvergenannya. dan berikut ini hasill dari tes rasio dalam mendeteksi interval kekonvergenannya. tabel 3. perbandingan fungsi aktivasi setelah dicari interval kekonvergenan dengan tes rasio fungsi aktivasi interval kekonvergenan mnop !3 4 � 4 3 qrop !1 4 � 4 1 mnosp !3 4 � 4 3 qrosp !1 4 � 4 1 ntumnop !1 4 � 4 1 ntuqrop !1 4 � 4 1 ntumnosp !1 4 4 1 !1 4 4 1 untuk interval !1 4 4 1 dianggap terlalu terbatas karena metode penentuan bobot yang biasa digunakan adalah metode nguyen widrow yang mampu menghasilkan bobot lebih dari satu. sehingga fungsi aktivasi ini terlalu riskan untuk digunakan sebagai fungsi aktivasi. untuk tan , memang memiliki interval yang bagus sebagai fungsi aktivasi, tapi setelah dimasukkan nilai ternyata malah menghasilkan nilai kompleks, padahal algoritma yang digunakan dalam pengenalan pola kali ini adalah algoritma bilangan riil. dan algoritma bilangan riil tidak bisa digunakan untuk bilangan kompleks. berbeda dengan tanh , fungsi aktivasi ini memiliki interval kekonvergenan yang bagus dan bila dimasukkan nilai menghasilkan nilai bilangan riil. sehingga fungsi aktivasi tanh sangat bagus untuk digunakan sebagai fungsi aktivasi agar pelatihan bisa mengenali perbedaan karakteristik bentuk gelombang spektra babi dan sapi. untuk simulasi program. berarti akan dibuat program pelatihan backpropagation untuk mengenali gelombang spektra adalah. clc,clear fa=zeros(120*100,8); fb=zeros(120*100,8); for i=1:8 a=imread(['sapi' num2str(i) '.jpg']); [m,n,o]=size(a); a2=rgb2gray(a); a3=im2bw(a2,graythresh(a2)); a4=imresize(a3,[120 100]); a5=reshape(a4,[120*100 1]); fa(:,i)=a5; b=imread(['babi' num2str(i) '.jpg']); [m,n,o]=size(b); b2=rgb2gray(b); b3=im2bw(b2,graythresh(b2)); b4=imresize(b3,[120 100]); b5=reshape(b4,[120*100 1]); fb(:,i)=b5; end p=[fa fb]; t=[ones(1,8) zeros(1,8); zeros(1,8) ones(1,8)]; p1=[p p p p p p p p p p]; t1=[t t t t t t t t t t]; clear a a2 a3 a4 b b2 b3 b4 b5 i m n o a5 fa fb p t net=newff(minmax(p1),[50,2],{'logsig','pur elin'},'traingd') net.trainparam.goal=0.00001 net.iw{1,1}; net.b{1}; net.lw{2,1}; net.b{2}; [y,pf,af,e,perf]=sim(net,p1,[],[],t1) net=train(net,p1,t1) gambar 5 dibawah ini menunjukkan saat koding program ini dimasukkan dalam matlab. gambar 5 koding program dimasukkan ke dalam matlab berikutnya akan dianalisis mengenai tingkat tingginya mse (mean square error) bila dilihat dari banyaknya jumlah unit tersembunyi dalam satu lapisan tersembunyi dengan laju analisis fungsi aktivasi jaringan syaraf tiruan untuk mendeteksi karakteristik bentuk gelombang … jurnal cauchy – issn: 2086-0382 161 pelatihan � 0,0001. berikut tabel jumlah error pelatihan backpropagation tabel 4. mse berdasarkan banyaknya unit dalam satu lapisan tersembunyi banyak unit mse 3 0,06082 4 0,020942 5 0,020208 30 0,0053801 40 0,0019251 dari kelima uji coba pada tabel di atas dapat disimpulkan bahwa semakin banyak jaringan dalam satu lapisan tersembunyi, maka semakin kecil errornya sehingga semakin bagus hasil pelatihannya. setelah melakukan pelatihan, maka hasil dari pelatihan disimpan dalam “net”. kemudian penulis akan menguji coba hasil pelatihan dengan data uji coba, apakah hasil pelatihan mampu mengenali data uji coba dengan benar atau tidak. pertama-tama disediakan gambar gelombang spektra babi dan sapi. berikut gambar gelombang spektra babi dan sapi. gambar 6. gelombang spektra babi yang pertama kemudian berikut gambar gelombang spektra sapi. gambar 7. gelombang spektra sapi yang pertama dua gambar di atas adalah perwakilan dari tiga gambar gelombang spektra babi dan sapi yang digunakan untuk menguji coba hasil pelatihan backpropagation. gambar yang dipanggil pertama adalah gambar gelombang spektra babi. dan setelah program dijalankan, maka hasilnya adalah gambar 8. program menebak bahwa gambar gelombang spektra itu adalah gelombang spektra babi cara membaca gambar tebakan di atas dapat dijelaskan dengan gambar di bawah ini. gambar 9. tata cara membaca tebakan program dan saat gambar gelombang spektra yang dipanggil adalah gelombang spektra sapi, maka hasilnya adalah gambar 10. program menebak bahwa gambar gelombang spektra itu adalah gelombang spektra sapi dari 6 tebakan program dapat menebak jenis ke-6 gambar gelombang spektra dengan benar. berarti hasil pelatihan dengan backpropagation yang menggunakan hasil analisis matematik tadi sudah bagus. shofwan ali fauji 162 volume 2 no. 3 november 2012 penutup berdasarkan hasil pembahasan, dapat disimpulkan bahwa fungsi aktivasi tangen hiperbolik adalah fungsi yang terbaik untuk menbedakan gelombang spektra sapi dan babi. karena menghasilkan bilangan riil dan meiliki interval konvergensi yang cukup memadai untuk beberapa metode seperti nguyen widrow ataupun bilangan acak -0,5 sampai 0,5 maupun bilangan acak -1 sampai 1. setelah diuji coba pun fungsi aktivasi ini mampu melakukan pelatihan dengan baik, sehingga hasil pelatihan mampu mengenali gambar uji coba dengan sempurna. fungsi aktivasi dan interval bobot adalah hal yang sangat esensial dalam pelatihan jaringan backpropagation. karena bila salah menggunakan fungsi aktivasi ataupun bobotnya tidak sesuai sehingga bisa divergen maka pelatihan akan gagal. dengan fungsi aktivasi tangen hiperbolik dan interval kekonvergenannya telah mampu membedakan gelombang spektra sapi dan babi. bila menginginkan pelatihan yang lebih baik, dapat menambah jumlah unit jaringan tersembunyi dalama satu lapisan tersembunyi karena telah diuji coba sampai 40 unit jaringan, bahwa semakin banyak unit jaringan semakin kecil mse. dalam tulisan masih dibahas mengenai fungsi aktivasi dan interval kekonvergenan. tapi masi belum dibahas mengenai interval bobot yang memiliki konvergensi cepat dan menghasilkan pelatihan yang bagus. sehingga disarankan untuk meneliti tentang interval bobot yang bagus sehingga menghasilkan pelatihan yang bagus dan konvergensinya yang cepat dengan error yang lebih kecil. ucapan terima kasih penulis ucapkan kepada semua pihak yang telah membantu penulis baik berupa bantuan maupun dukungan. sekian trima kasih. daftar pustaka [1] ‘abdullah bin muhammad bin ‘abdurrahman bin ishaq alu syaikh. 2007. tafsir ibnu katsir jilid 3. bogor: pustaka imam asy-syafi’i [2] ahmad, y. a. 2008. al-qur’an kitab kedokteran. yogyakarta: sajadah press [3] aizenberg, i. 2011. complex valued neural networks with multi-valued neurons. berlin: springer [4] al-ghazali, i. 2007. rahasia halal haram. bandung: mizania [5] anita s. g, p.k. kalra, & d.s. chauhan. (2009). inversion of complex backpropagation algorithm [6] anton, h. 2000. dasar-dasar aljabar linier. batam: interaksara [7] as sayyid, a. a. m. 2006. pola makan rasulullah. jakarta: almahira [8] bartle, r. g., & sherbert, d. r. 2000. introduction to real analysis. usa: john willey & sons [9] fausett, l. 1994. fundamentals of neural networks. new jersey: prentice hall [10] habsari, l. y. d. 2010. identifikasi pola khas spektra infra merah protein daging sapi dan babi rebus menggunakan metode second derivative (2d). skripsi tidak diterbitkan. malang: uin maliki malang [11] kim, t. & adali, t. 2001. complex backpropagation neural network using elementary transcendental activation functions. [12] kusumadewi, s. 2004. membangun jaringan syaraf tiruan (menggunakan matlab dan excel link). jakarta: graha ilmu [13] munir, r. 2008. metode numerik. bandung: informatika [14] purnomo, m. h. & kurniawan, a. 2006. supervised neural networks dan aplikasinya. yogyakarta: graha ilmu [15] purwanto, h. dkk. 2005. aljabar linier. jakarta: pt. ercontara rajawali [16] putra, d. 2010. pengolahan citra. yogyakarta: andi [17] siang, j. j. 2009. jaringan syaraf tiruan dan pemrogamannya menggunakan matlab. yogyakarta: andi [18] stewart, j. 2010. calculus concepts and contexts. belmont: cengage learning [19] suhartono. 2007. feedforward neural networks untuk pemodelan runtun waktu. disertasi tidak diterbitkan. yogyakarta: pps universitas gadjah mada [20] sutoyo, t. dkk, 2009. teori pengolahan citra. yogyakarta: andi [21] wunsh, a. d. 2005, complex varibles with application, usa: pearson education [22] yaqub, a. m. 2009. kriteria halal haram. jakarta: pt pustaka firdaus on the spectra of commuting and non commuting graph on dihedral group cauchy –jurnal matematika murni dan aplikasi volume 4 (4) (2017), pages 176-182 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 17 may 2017 reviewed: 20 may 2017 accepted: 30 may 2017 doi: http://dx.doi.org/10.18860/ca.v4i4.4211 on the spectra of commuting and non commuting graph on dihedral group abdussakir1, rivatul ridho elvierayani2, muflihatun nafisah3 1,2,3) mathematics department, universitas islam negeri maulana malik ibrahim malang email: sakir@mat.uin-malang.ac.id abstract study about spectra of graph has became interesting work as well as study about commuting and non commuting graph of a group or a ring. in this paper, we investigate adjacency spectrum, laplacian spectrum, signless laplacian spectrum, and detour spectrum of commuting and non commuting graph of dihedral group d2n. keywords: graph, spectrum, commuting graph, non commuting graph, dihedral group. introduction let g be a graph of order n (n ≥ 1) and let its vertex set is v(g) = {v1, v2, …, vn}. the adjacency matrix a(g) = [aij] of g is nn matrix where aij = 1 if vivj  e(g) and aij = 0 if vivj  e(g)) [1]. so, the adjacency matrix of a graph is real and symmetric that contains 0 in all of main diagonal entries [2]. the degree matrix d(g) = [dij] of g is a diagonal matrix where dii = degg(vi) and dij = 0, for i ≠ j. the laplacian matrix l(g) of g is defined by l(g) = d(g) – a(g) [3] and the signless laplacian matrix q(g) of g is defined by q(g) = d(g) + a(g) [4]. the detour matrix dd(g) = [eij] of g is nn matrix where eij is equal to the length of the longest path between vi and vj [5]. let 0, 1, …, 𝜆𝑘−1 where 0 > 1 > … > 𝑘−1 are distinct eigenvalues from a matrix of a graph, and let m(0), m(1), …, m(𝑘−1) are multiplicities of i, for i = 0, 1, 2, ..., k – 1. the multiplicity of i as roots of characteristic equation is equal to the dimension of the space of eigenvectors corresponding to i [6]. matrix of order 2k that contains 0, 1, …, 𝑘−1 for the first row and m(0), m(1), …, m(𝑘−1) for the second row is called spectrum of graph g and denoted by spec(g) [6]. according to yin [7], we can write spectrum of graph g by spec(g) = [ 𝜆0 𝜆1 ⋯ 𝜆𝑘−1 𝑚(𝜆0) 𝑚(𝜆1) ⋯ 𝑚(𝜆𝑘−1) ]. spectrum that obtained from the adjacency matrix a(g), the laplacian matrix l(g), the signless laplacian matrix q(g), and the detour matrix dd(g) are called the adjacency spectrum denoted by speca(g), the laplace spectrum denoted by specl(g), the signless laplacian spectrum denoted by specq(g), and the detour spectrum denoted by specdd(g) respectively. let g be a non abelian group with center z(g). the commuting graph c(g, x), where x is a subset of g, is the graph with vertex set x and distinct vertices being joined by an edge whenever xy = yx in g [8]. in the case x = g, we denote the commuting graph of g by c(g) http://dx.doi.org/10.18860/ca.v4i4.4211 mailto:sakir@mat.uin-malang.ac.id on the spectra of commuting and non commuting graph on dihedral group abdussakir 177 instead of c(g, x). the non commuting graph γ𝐺 of g, is the graph whose vertices are the non central elements g\z(g) and whose edges join those vertices x, y  g\z(g) for which xy  yx in g [9]. the non commuting graph γ𝐺 of g is always connected [10]. the study of spectrum of graph was begun by work of norman bigg [6] in his famous book algebraic graph theory. some investigations about spectrum of graphs had been done before, for example yin [7], brouwer and haemers [4], ayyaswamy & balachandran [5], and jog & kotambari [11]. researches on commuting and non commuting graph of a group and ring had also been done before, for example akbari et al. [12], abdollahi et al. [10], darafsheh [13], abdollahi et al. [9], vahidi & talebi [14], rahayuningtyas et al. [15], chelvam et al. [16], woodcock [17], and nawawi & rowley [8]. in this study, we determine the spectra of commuting and non commuting graph of dihedral group, including adjacency spectrum, laplacian spectrum, signless laplacian spectrum, and detour spectrum. result and discussion here we present several results from our studies. for adjacency spectrum, we observe and get result just for non commuting graph of dihedral group. lemma 1: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n > 3. the characteristic polynomial for adjacency matrix 𝐴(γ𝐷2𝑛) of γ𝐷2𝑛 is 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))). proof: for dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is odd positive integer, n > 3, we have z(𝐷2𝑛) = {1}. so, the non commuting graph γ𝐷2𝑛 of d2n has { r, r2, …, rn-1, s, sr, sr2, …, srn-1} as its vertex set. because n is odd, we have 𝑠𝑟𝑖 ≠ 𝑟𝑖𝑠, 𝑖 = 1, 2, … , 𝑛 − 1, so s and 𝑟𝑖 are adjacent in γ𝐷2𝑛. and also, 𝑠𝑟 𝑖𝑠𝑟𝑗 ≠ 𝑠𝑟𝑗𝑠𝑟𝑖, for i = 1, 2,..., n – 1 and 𝑗 = 1, 2, … , 𝑛 − 1, where 𝑖 ≠ 𝑗, so 𝑠𝑟𝑖 and 𝑠𝑟𝑗 are adjacent in γ𝐷2𝑛 . then, we have the adjacency matrix for γ𝐷2𝑛 as the following 𝑟 𝑟2 ⋯ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋯ 𝑠𝑟𝑛−1 𝐴(γd2𝑛) = 𝑟 𝑟2 ⋮ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋮ 𝑠𝑟𝑛−1 [ 0 0 … 0 1 1 1 ⋯ 1 0 0 … 0 1 1 1 ⋯ 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 1 1 1 ⋯ 1 1 1 ⋯ 1 0 1 1 ⋯ 1 1 1 ⋯ 1 1 0 1 ⋯ 1 1 1 ⋯ 1 1 1 0 ⋯ 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1 1 ⋯ 1 1 1 1 ⋯ 0] we know that the characteristic polynomial of 𝐴(γd2n) is de𝑡(𝐴(γd14) − 𝐼). using the gaussian elimination procedure we obtain this upper triangular matrix on the spectra of commuting and non commuting graph on dihedral group abdussakir 178 𝑟 𝑟2 ⋯ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋯ 𝑠𝑟𝑛−1 𝑟 𝑟2 ⋮ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋮ 𝑠𝑟𝑛−1 [ 1 1 … 1 −𝜆 1 1 ⋯ 1 0 𝜆 … 𝜆 1 − 𝜆2 1 + 𝜆 1 + 𝜆 ⋯ 1 + 𝜆 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 𝜆 (𝑛 − 2) − 𝜆2 (𝑛 − 2) + 𝜆 (𝑛 − 2) + 𝜆 ⋯ (𝑛 − 2) + 𝜆 0 0 ⋯ 0 1 + 𝜆 −𝜆 − 1 0 ⋯ 0 0 0 ⋯ 0 0 1 + 𝜆 −𝜆 − 1 ⋯ 0 0 0 ⋯ 0 0 0 1 + 𝜆 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 0 ⋯ −𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))] because the determinant for this upper triangular matrix is the multiplication of all entries in its main diagonal, so we obtain the characteristic polynomial of 𝐴(γd2n) as follows 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))).  theorem 1: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n > 3. the adjacency spectrum of γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐴(𝛤𝐷2𝑛) = [ 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 0 −1 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 1 𝑛 − 2 𝑛 − 1 1 ]. proof: from lemma 1, the characteristic polynomial of 𝐴(γd2n) is 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))). by solving the equation 𝑝(𝜆) = 0, we have 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , 𝜆2 = 0, 𝜆3 = −1, 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 as eigenvalues for 𝐴(γd2n). now, we will determine the multiplicity for each eigenvalue. because the multiplicity is equal to number of basis for space of eigenvectors corresponding to i, i = 1, 2, 3, and 4, we substitute i to 𝐴(γd2n) − 𝑖𝐼 and apply gaussjordan elimination to this matrix to obtain row-reduced eselon matrix and then we can see the number of zero rows of it. for 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , by eliminating 𝐴(γd2n) − 1𝐼, we obtained rowreduced eselon matrix with one zero row. so, for 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , we have its algebraic multiplicity is 1. for 𝜆2 = 0, by eliminating 𝐴(γd2n) − 2𝐼, we obtain row-reduced eselon matrix with n – 2 zero rows. so, for 𝜆2 = 0, we have its algebraic multiplicity is n – 2. for 𝜆3 = −1, by eliminating 𝐴(γd2n) − 3𝐼, we obtain row-reduced eselon matrix with n – 1 zero rows. so, for 𝜆2 = −1, we have its algebraic multiplicity is n – 1. on the spectra of commuting and non commuting graph on dihedral group abdussakir 179 for 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , by eliminating 𝐴(γd2n) − 4𝐼, we obtain rowreduced eselon matrix with one zero rows. so, for 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , we have its algebraic multiplicity is 1. we conclude that speca(𝛤𝐷2𝑛) = [ 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 0 −1 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 1 𝑛 − 2 𝑛 − 1 1 ].  theorem 2: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  6. the adjacency spectrum of γ𝐷2𝑛 is speca(𝛤𝐷2𝑛 ) = [ 𝑛−2 2 + √ 5𝑛2−12𝑛+4 4 0 −2 𝑛−2 2 − √ 5𝑛2−12𝑛+4 4 1 𝑛−2 2 3𝑛−6 2 1 ]. the next results are laplacian spectrum of commuting and non commuting graph of dihedral group. lemma 2: let 𝐶(𝐷2𝑛) be a commuting graph of dihedrral group 𝐷2𝑛 where n is positive integer and n  3. the characteristic polynomial for laplacian matrix 𝐿(𝐶(𝐷2𝑛)) of 𝐶(𝐷2𝑛) is 𝑝(𝜆) = −𝜆(𝜆 − 𝑛)𝑛−2(𝜆 − 1)𝑛(𝜆 − 2𝑛), for odd n and 𝑝(𝜆) = −𝜆(𝜆 − 𝑛)𝑛−3((𝜆 − 4)(𝜆 − 2)) 𝑛 2(𝜆 − 2𝑛)2, for even n. theorem 3: let 𝐶(𝐷2𝑛) be a commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. the laplacian spectrum of 𝐶(𝐷2𝑛) is 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷2𝑛)) = [ 2𝑛 𝑛 1 0 1 𝑛 − 2 𝑛 1 ]. proof: from lemma 2, for odd n, we have 𝜆1 = 2𝑛, 𝜆2 = 𝑛, 𝜆3 = 1, and 𝜆4 = 0 as eigenvalues for the laplacian matrix of 𝐶(𝐷2𝑛). and the multiplicity for each eigenvalue is 𝑚(𝜆1) = 1, 𝑚(𝜆2) = 𝑛 − 2, 𝑚(𝜆3) = 𝑛, and 𝑚(𝜆4) = 1.  theorem 4: let 𝐶(𝐷2𝑛) be a commuting graph of dihedrral group 𝐷2𝑛 where n is even positive integer and n  6. the laplacian spectrum of 𝐶(𝐷2𝑛) is 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷2𝑛)) =[ 2𝑛 𝑛 4 2 0 2 𝑛 − 3 𝑛 2 𝑛 2 1]. proof: from lemma 2, for even n and n ≥ 6, we have 𝜆1 = 2𝑛, 𝜆2 = 𝑛, 𝜆3 = 4, 𝜆4 = 2, and 𝜆5 = 0 as eigenvalues for the laplacian matrix of 𝐶(𝐷2𝑛). and the multiplicity for each eigenvalue is 𝑚(𝜆1) = 2, 𝑚(𝜆2) = 𝑛 − 3, 𝑚(𝜆3) = 𝑛 2 , 𝑚(𝜆4) = 𝑛 2 , and 𝑚(𝜆5) = 1. from our investigation, the laplacian spectrum of 𝐶(𝐷8) is on the spectra of commuting and non commuting graph on dihedral group abdussakir 180 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) = [ 8 4 2 0 2 3 2 1 ]. if we apply theorem 4 for n = 4 then we have 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) =[ 8 4 4 2 0 2 1 2 2 1 ] according to spectrum definition, we must write the laplacian spectrum for 𝐶(𝐷8) as 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) =[ 8 4 2 0 2 3 2 1 ] so, by this reason, theorem 4 also holds for n = 4. lemma 3: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. the characteristic polynomial for the laplacian matrix 𝐿(γ𝐷2𝑛) of γ𝐷2𝑛 is 𝑝(𝜆) = −𝜆(𝑛 − 𝜆)𝑛−2((−2𝑛 + 1) + 𝜆) 𝑛 . theorem 5: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. the laplacian spectrum of γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐿 (𝛤𝐷2𝑛 ) = [ 2𝑛 − 1 𝑛 0 𝑛 𝑛 − 2 1 ]. proof: from lemma 3, for odd n and n ≥ 3, we have 𝜆1 = 2𝑛 − 1, 𝜆2 = 𝑛, and 𝜆3 = 0. as eigenvalues for the laplacian matrix of γ𝐷2𝑛 . and the multiplicity for each eigenvalue is 𝑚(𝜆1) = 𝑛, 𝑚(𝜆2) = 𝑛 − 2, and 𝑚(𝜆3) = 1. theorem 6: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  6. the laplacian spectrum of γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐿 (𝛤𝐷2𝑛 ) = [ 2(𝑛 − 1) 2(𝑛 − 2) 𝑛 0 𝑛 2 𝑛 2 𝑛 − 3 1 ]. proof: the characteristic polynomial for the laplacian matrix 𝐿(γ𝐷2𝑛) of γ𝐷2𝑛 where n is even positive integer and n  6 is 𝑝(𝜆) = −𝜆(𝑛 − 𝜆)𝑛−3(2(𝑛 − 1) + 𝜆) 𝑛 2(2(𝑛 − 2) + 𝜆) 𝑛 2 so we have 𝜆1 = 2(𝑛 − 1), 𝜆2 = 2(𝑛 − 2), 𝜆3 = 𝑛, and 𝜆4 = 0 as eigenvalues for the laplacian matrix of γ𝐷2𝑛. and the multiplicity for each eigenvalue is 𝑚(𝜆1) = 𝑛 2 , 𝑚(𝜆2) = 𝑛 2 , 𝑚(𝜆3) = 𝑛 − 3, and 𝑚(𝜆4) = 1.  as for the adjacency spectrum, we just have the signless laplacian spectrum for non commuting graph of dihedral group. we present the result without proof. theorem 7: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  8. the signless laplacian spectrum of γ𝐷2𝑛 is 𝑠𝑝𝑒𝑐𝑄(𝛤𝐷2𝑛 ) = [ (2𝑛 − 3) + √2𝑛2 − 8𝑛 + 9 2(𝑛 − 2) 2(𝑛 − 3) 𝑛 (2𝑛 − 3) − √2𝑛2 − 8𝑛 + 9 1 𝑛 2 𝑛−2 2 𝑛 − 3 1 ] on the spectra of commuting and non commuting graph on dihedral group abdussakir 181 the last two theorems are results for detour spectrum of non commuting graph on dihedral group. we start by this lemma. lemma 4: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is positive integer and n  3. the longest path between two distinct vertices x, y  v(γ𝐷2𝑛 ) is 2n – 2 for odd n and 2n – 3 for even n. proof: for dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is odd positive integer, n  3, we have z(𝐷2𝑛) = {1}. so, the non commuting graph γ𝐷2𝑛 of d2n has {r, r2, …, rn-1, s, sr, sr2, …, srn-1} as its vertex set. hence the order of γ𝐷2𝑛 is 2n – 1. because n is odd, we have 𝑠𝑟𝑖 ≠ 𝑟𝑖𝑠, 𝑖 = 1, 2, … , 𝑛 − 1, so s and 𝑟𝑖 are adjacent in γ𝐷2𝑛 . and also, 𝑠𝑟𝑖𝑠𝑟𝑗 ≠ 𝑠𝑟𝑗𝑠𝑟𝑖, for i = 1, 2,..., n – 1 and 𝑗 = 1, 2, … , 𝑛 − 1, where 𝑖 ≠ 𝑗, so 𝑠𝑟𝑖 and 𝑠𝑟𝑗 are adjacent in γ𝐷2𝑛. on the other hand r i and rj are not adjacent in γ𝐷2𝑛 because r irj = rjri in d2n. we have that the sum of degree of any two non adjacent vertices in γ𝐷2𝑛 is 2n > 2𝑛−1 2 and γ𝐷2𝑛 is hamiltonian graph (see [10]). so, the length of the longest path between two distinct vertices is 2n – 2 for odd n. for dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is even positive integer, n > 3, we have 𝑍(𝐷2𝑛) = {1, 𝑟 𝑛 2}. so, the non commuting graph γ𝐷2𝑛 of d2n has {r, r2, …, 𝑟 𝑛 2 −1 , 𝑟 𝑛 2 +1 , ..., rn-1, s, sr, sr2, …, srn-1} as its vertex set. hence the order of γ𝐷2𝑛 is 2n – 2. we also have that γ𝐷2𝑛 is hamiltonian graph. so, the length of the longest path between two distinct vertices is 2n – 3 for even n.  theorem 8: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. the detour spectrum of γ𝐷2𝑛 is specdd(γ𝐷2𝑛) = [ (2𝑛 − 2)2 −(2𝑛 − 2) 1 2𝑛 − 2 ]. proof: by lemma 4, we have dd(γ𝐷2𝑛) is (2n – 1)(2n – 1) matrix where all entries in its main diagonal is 0 and 2n – 2 elsewhere, for odd n. then, we have the characteristic polinomial is p(𝜆) = (𝜆 + (2𝑛 − 2)) 2𝑛−2 (𝜆 − (2𝑛 − 2)2). from the characteristics polinomial, we have 𝜆1 = (2𝑛 − 2) 2 and 𝜆2 = 2𝑛 − 2 as eigenvalues of dd(γ𝐷2𝑛) with their multiplicity are 𝑚(𝜆1) = 1 and 𝑚(𝜆2) = 2𝑛 − 2. so, we get the desired result.  theorem 9: let γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. the detour spectrum of γ𝐷2𝑛 is specdd(γ𝐷2𝑛) = [ (2𝑛 − 3)2 −(2𝑛 − 3) 1 2𝑛 − 3 ]. proof: by lemma 4, we have dd(γ𝐷2𝑛) is (2n – 2)(2n – 2) matrix where all entries in its main diagonal is 0 and 2n – 3 elsewhere, for even n. then, we have the characteristic polinomial is p(𝜆) = (𝜆 + (2𝑛 − 2)) 2𝑛−2 (𝜆 − (2𝑛 − 2)2). from the characteristic polinomial, we have 𝜆1 = (2𝑛 − 3) 2 and 𝜆2 = 2𝑛 − 3 as eigenvalues of dd(γ𝐷2𝑛) with their multiplicity are 𝑚(𝜆1) = 1 and 𝑚(𝜆2) = 2𝑛 − 3. this lead us to the detour spectrum.  on the spectra of commuting and non commuting graph on dihedral group abdussakir 182 conclusion from our results, several spectrums for several cases of n are not determined yet. so, the remaining cases can be done in further research. investigation for spectra of commuting and non commuting graph of symmetric group are also interesting to be done. references [1] abdussakir, n. n. azizah, and f. f. novandika, teori graf. malang: uin malang press, 2009. [2] g. chartrand, l. lesniak, and p. zhang, graphs and digraphs, 6th edition. florida: chapman and hall, 2015. [3] b. mohar, “laplace eigenvalues of graphs-a survey,” discrete math., vol. 109, no. 1–3, pp. 171–183, 1992. [4] a. e. brouwer and w. h. haemers, spectra of graphs: monograph. new york: springer, 2011. [5] s. k. ayyaswamy and s. balachandran, “on detour spectra of some graphs,” int. j. math. comput. phys. electr. comput. eng., vol. 4, no. 7, pp. 1038–1040, 2010. [6] n. biggs, algebraic graph theory, 2nd edition. new york: cambridge university press, 1993. [7] s. yin, “investigation on spectrum of the adjacency matrix and laplacian matrix of graph gl,” wseas trans. syst., vol. 7, no. 4, pp. 362–372, 2008. [8] a. nawawi and p. rowley, “on commuting graphs for elements of order 3 in symmetric groups,” electron. j. comb., vol. 22, no. 1, pp. 1–21, 2015. [9] a. abdollahi, a. azad, a. m. hassanabadi, and m. zarrin, “on the clique numbers of non-commuting graphs of certain groups,” algebr. colloq., vol. 17, no. 4, pp. 611–620, 2010. [10] a. abdollahi, s. akbari, and h. r. maimani, “non-commuting graph of a group,” j. algebr., vol. 298, no. 2, pp. 468–492, 2006. [11] s. r. jog and r. kotambari, “on the adjacency, laplacian, and signless laplacian spectrum of coalescence of complete graphs,” j. math., vol. 2016, pp. 1–11, 2016. [12] s. akbari, m. ghandehari, m. hadian, and a. mohammadian, “on commuting graphs of semisimple rings,” linear algebra appl., vol. 390, no. 1–3, pp. 345–355, 2004. [13] m. r. darafsheh, “groups with the same non-commuting graph,” discret. appl. math., vol. 157, no. 4, pp. 833–837, 2009. [14] j. vahidi and a. a. talebi, “the commuting graphs on groups d2n and qn,” j. math. comput. sci., vol. 1, no. 2, pp. 123–127, 2010. [15] h. rahayuningtyas, abdussakir, and a. nashichuddin, “bilangan kromatik graf commuting dan non commuting grup dihedral,” cauchy, vol. 4, no. 1, pp. 16–21, 2015. [16] t. t. chelvam, k. selvakumar, and s. raja, “commuting graphs on dihedral group main results,” j. math. comput. sci., vol. 2, no. 2, pp. 402–406, 2011. [17] t. woodcock, “the commuting graph of the symmetric group sn,” int. j. contemp. math. sci., vol. 10, no. 6, pp. 287–309, 2015. diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and 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editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy preface cauchy is a national journal published by mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang. this is the second issue of this year. it contains 6 (six) article from all over the country, not only from local area of east java. those articles covered graph theory, numerical analysis, applied mathematics, statistics, economic mathematics, and algebra. this issue was authored by 13 authors and co-authors. in the first article, the author discussed the application of pontryagin’s minimum principle in optimum time of missile manoeuvring. analysis of surface-to-surface missile which considered objective and dynamical system showed that trajectory tracking of this missile was split up in three sub-intervals, namely flying, climbing, and diving in which had different performances of speed. the second article discussed about power of test path analysis and partial least square analysis. the result showed that the usage of path analysis provides better value than 𝑃𝐿𝑆 at 𝑅2 value in terms of the power of the test. behavioral conditions of work of civil servants in kediri city government did not follow the rules of the science of human resources in terms of election officials echelon structural or civil servants. the third article, entitled “estimation parameters and modelling zero inflated negative binomial” elaborated that the estimation parameter of zero inflated negative binomial (zinb) model was conducted using maximum likelihood estimation (mle) and to maximize the likelihood function used the em (expectation maximization) algorithm. for parameter test predictor variable that has significant effect on the number of cases of tetanus neonatorum are pregnant mothers visit k4 (x1) and maternal mothers assisted by health workers (x3) for the negative binomial state models, while zero inflation state model predictor variable that has significant effect on the number of cases of tetanus neonatorum include the percentage of neonates visit (x4). the fourth article emphasized the discussion to leontief input-output method for the fresh milk distribution linkage analysis. leontief input-output method can illustrate the intersectoral linkages and identify the key sector in the fresh milk distribution. it is clearly shown that the merging of the collectors’ and the fresh milk processing industries’ activities offers high indices of both backward linkages and forward linkages, so that it can be the key sector in the fresh milk distribution. the fifth article covered the local metric dimension of line graph of special graphs. the local metric dimension number of line graph of special graphs, namely line graph of path, jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy preface the last article entitled “some properties from construction of finite projective planes of order 2 and 3”. it showed that a finite projective plane is a geometric construction that extends the concept of a plane. it consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. in this paper some properties from construction of finite projective planes of order 2 and order 3 which is as a comparison are observed. jurnal matematika murni dan aplikasi volume 4, issue 3, november 2016 issn 2086-0382 e-issn 2477-3344 cauchy preface parameter estimation of structural equation modeling using bayesian approach cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 86-94 submitted: 27 february 2016 reviewed: 10 march 2016 accepted: 1 may 2016 parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari, ni wayan surya wardhani, suci astutik department of statistics, faculty of mathematics and natural sciences brawijaya university, malang email: dewi.kurnia.14@gmail.com abstract leadership is a process of influencing, directing or giving an example of employees in order to achieve the objectives of the organization and is a key element in the effectiveness of the organization. in addition to the style of leadership, the success of an organization or company in achieving its objectives can also be influenced by the commitment of the organization. where organizational commitment is a commitment created by each individual for the betterment of the organization. the purpose of this research is to obtain a model of leadership style and organizational commitment to job satisfaction and employee performance, and determine the factors that influence job satisfaction and employee performance using sem with bayesian approach. this research was conducted at statistics fni employees in malang, with 15 people. the result of this study showed that the measurement model, all significant indicators measure each latent variable. meanwhile in the structural model, it was concluded there are a significant difference between the variables of leadership style and organizational commitment toward job satisfaction directly as well as a significant difference between job satisfaction on employee performance. as for the influence of leadership style and variable organizational commitment on employee performance directly declared insignificant. keywords: sem, bayesian, leadership style, organizational commitment, job satisfaction, employee performance. introduction leadership is a process of influencing, directing and giving examples of his subordinates in order to achieve organizational goals. every leader must have style, personality traits, habits, character and personality are different. leadership is a key element in an organization's effectiveness [1]. based on this, it can be said that the role of a leader is very important for achieving the vision, mission and goals of an organization. moreover, the success of an organization or company in achieving its objectives can be influenced also by the commitment of the organization. where organizational commitment is a commitment made by each individual to the organization's progress. the commitment is visible when individuals can exercise their rights and obligations in accordance with the duties and functions of each organization. therefore, an organization must give your full attention and make employees believe the organization, so it will obtain high organizational commitment. many ways to measure the impact of leadership style and organizational commitment to employee performance, one of which is the structural equation modeling is a multivariate analysis parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 87 technique that combines the measurement model as in the confirmatory factor analysis with structural model on regression analysis or analysis of lines. one of the assumptions that must be met in the sem is the sample size should be large enough. other assumptions that must be met is an indicator to follow a multivariate normal distribution and the indicator variables with latent variables and between latent variables have a linear relationship. the use of small sample size in a sem with a classical approach can generate sample covariance matrix is singular. if the requirements analysis structural equation modeling (sem) is not fulfilled such a small sample size, it would require alternative methods to resolve the issue is through a bayesian approach. some of the advantages include: (1) bayesian methods more emphasis on the use of individual data rather than sample covariance matrix, (2) latent variable can be predicted directly (3) using prior information. other virtues are not taking into account the size of the sample that will affect the operating costs of research. this study was conducted to determine the effect of leadership style and organizational commitment to job satisfaction and employee performance in the fni statistics using structural equation modeling (sem) through a bayesian approach. theoritical review basic concepts sem analysis with bayesian approach analysis of the relationship between latent variables need to be developed so that the resulting model to be more precise, especially for complex problems. one method of analysis that is often used to test whether there is influence between the latent variable is the analysis of structural equation modeling (sem). one of the assumptions that must be fulfilled in such method is the sample size should be large enough. according to [2] suggested sample size to use maximum likelihood estimation (mle) is approximately 100-200 or be greater when using asymptotically approaches distribution free (adf) to handle the data distribution is not normal. the classical method in the analysis of sem depends on the matrix variance covariance sample s and not on the response variable vector. the structure of matrix σ(θ) is the variance covariance matrix containing the parameter θ with large dimensions. estimation of θ by minimizing some objective function to calculate σ(θ) estimation methods commonly used in standard sem is the maximum likelihood estimation (mle) or generalized least squares (gls). analysis of variance covariance based heavily dependent on the asymptotic normality of the matrix s. using a small sample in sem with a classical approach can produce a negative variance [3] as well as the sample covariance matrix is singular [4]. in addition, the use of mle when the small sample size could lead to biased results of estimation parameters [3]. based on this, we need an alternative method to solve the problem of small sample size, namely through a bayesian approach. development of bayesian sem is intended to allow estimation of the model can be done even if some assumptions are not met. sem testing method with the bayesian approach is not based on the variance covariance matrix but based on the number of individual data (observations). in sem with maximum likelihood approach, θ is not considered as a random variable where θ is φ, ψ, λ, γ, β. while the bayesian approach sem, θ is considered as a random variable that has a distribution. furthermore, the distribution referred to as the prior distribution. if m is a sem equation with vector parameter θ is unknown, the density function chances of θ is 𝑝(θ|𝑀). if the bayesian inference is based on data observations y then the joint distribution of y and θ on m is 𝑝(y, θ|𝑀). with 𝑝(y, θ|𝑀) = 𝑝(𝑌|𝜃, 𝑀)𝑝(θ) = 𝑝(θ|y, 𝑀)𝑝(y|𝑀) (2.1) and that 𝑝(y|𝑀) does not depend on θ and with regard y has been determined and constant, then: 𝐿𝑜𝑔𝑝(θ|y, 𝑀) ∝ log 𝑝(y|θ, 𝑀) + log 𝑝(θ) (2.2) where, m : any form of sem with a vector parameter θ is unknown y : data observations of size n 𝑝(θ|𝑀) : prior distribution of θ in m model 𝑝(y, θ|𝑀) : probability join distribution from y and θ with m model is known, parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 88 𝑝(θ|y, 𝑀) : probability distribution from posterior 𝑝(y|θ, 𝑀) : likelihood function in equation (2.2) 𝑝(y|θ, 𝑀) is influenced by the number of instances while p (θ) is not. for a large number of examples of the log 𝑝(y|θ, 𝑀) will dominate log 𝑝(θ), so it has little prior and posterior density function will be approaching the log-likelihood function of 𝑝(y|θ, 𝑀). on the contrary, for a small sample, prior distribution of θ has a significant role. prior distribution prior was the initial information parameter values in the model. basically there are two types of prior distribution is non-informative prior distribution and the distribution of informative priors. for example, 𝑌 = (𝑦1, … . . , 𝑦𝑛) is the matrix of the data observation and ω = (𝜔1, … . , 𝜔𝑛 ) is the matrix of factor score of latent variable and θ is the structure of vector parameter which cover unknown element of λ, ψ , π, γ, φ, ψ𝛿 . in posterior analysis, data observations y was added with matrix of latent variable ω and the joint posterior distribution [𝜃, ω|y]. to derive the formula p[θ|ω, y], is required prior to the distribution of the components θ and lee (2007) using a type of conjugate prior distribution. λ𝑘 𝑇 is row of-k λ so that conjugate prior distribution from (λk, ψek) are as follows: 𝜓𝑒𝑘 𝐷 ⇒ 𝐼𝑛𝑣𝑒𝑟𝑡𝑒𝑑𝐺𝑎𝑚𝑚𝑎 (𝛼∗0𝑒𝑘 , 𝛽 ∗ 0𝑒𝑘 ) or equivalent with 𝜓𝑒𝑘 −1 𝐷 ⇒ 𝐺𝑎𝑚𝑚𝑎 (𝛼∗0𝑒𝑘 , 𝛽 ∗ 0𝑒𝑘 ) and [λ𝑘 |𝜓𝑒𝑘 ] 𝐷 ⇒ 𝑁[λ0𝑘 , 𝜓𝑒𝑘hoyk] (2.3) where 𝛼0𝑒𝑘, 𝛽0𝑒𝑘 , 𝛼 ∗ 0𝑒𝑘 , 𝛽∗ 0𝑒𝑘 and the element of λ0𝑘 and hoyk is the hyperparameters and hoyk is the matrix definite positive. conjugate prior distribution from φ −1 is follow wishart distribution with q dimension and be define φ−1 ~𝑊𝑞 [𝑅0, 𝜌0] or equivalent with φ ~𝐼𝑊𝑞 [𝑅 ∗ 0, 𝜌0] (2.4) where 𝑊𝑞[𝑅0, 𝜌0] is wishart distribution q dimension with 𝜌0 and matrix definite positive 𝑅0 as hyperparameters while 𝐼𝑊𝑞 [𝑅 ∗ 0, 𝜌0] is inverse from wishart distribution with 𝜌0 and matrix definite positive 𝑅0 as hyperparameters. mcmc application with gibbs sampling in this research, mcmc application with gibbs sampling is done to get the estimation of the posterior distribution on each of the unknown parameters including latent variables. to get the required characteristics of the posterior distribution of observation sufficient. for that, raised a number of observations such that the resulting empirical distribution approaches the actual distribution. an example for a sem model with y = (y1 … . yn) is the matrix of data observation, ω = (ω1 … . ωn) is the matrix of latent variable, and θ is the vector matrix which consist of unknown parameter are λ, ψ, φ. the stages of gibbs sampling in raising the posterior distribution is as follows: 1. take an initiation ω(𝑗), λ(j), ψε (𝑗) , φ(𝑗) 2. generating value ω(𝑗+1) dari 𝑝(ω| ψε (𝑗) , λ(𝑗), φ(j), y) 3. generating value ψε (𝑗+1) dari 𝑝(ψε| ω (𝑗+1), λ(𝑗), φ(j), y) 4. generating value λ(𝑗+1) dari 𝑝(λ| ω(𝑗+1), ψε (𝑗+1) , φ(j), y) 5. generating value φ(𝑗+1) dari 𝑝(φ| ω(𝑗+1), ψε (𝑗+1) , λ(𝑗+1), y) parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 89 posterior distribution concept of probability distribution with gibbs sampling in bayesian sem applying the parameters are θ = (θ1, … , θa) is (λ, φ, ψε). as for the set of latent variables ω where ω = (ω1, … , ω𝑏) a set of latent variables(ω1, ω2), with ω1 is a set of endogenous latent variable and ω2 is a set of exogenous latent variable. as for the members of each set ω1 and ω2 can be written by ωi. iterations are performed in the algorithm gibbs sampler used by probability distribution 𝑝(ω|ψε, λ, φ, y) with 𝑝(ω|ψε, λ, φ, y) = 𝑝(ω|θ, y) and the equation is based on the definition of the conditional distribution of random vectors yi and ωi, provided that 𝑖 = 1, … , 𝑛 and ωi is free (mutually independent) and yi is free of (ωi, θ), will be obtained equation (lee,2007): 𝒑(ω|θ, y) = ∏ 𝑝( n i=1 ωi|yi, θ) ∝ ∏ 𝑝( n i=1 ωi|θ)𝒑(yi|ωi, θ) (𝟐. 𝟓) probability distribution ωi is known with θ and yi conditional (ωi, θ) follow normal distribution 𝑁(0, φ) and 𝑁(λωi, ψ ). thus, distribution ωi conditional (yi, θ) is: (ωi| yi, θ)~n [(φ −1 + λtψε −1 λ) −1 λtψε −1 yi, (𝛷 −1 + 𝛬𝑇 𝛹 −1 𝛬) −1 ] (2.6) thus, the conditional distribution ω with (y, θ) is known can be compute based on the equation (2.5) and (2.6). posterior analysis of conditional distribution where θ with the provision of (y, ω) proportional to𝑝(θ) 𝑝(y, ω|θ). model yi = λωi + εi is known with ω is the result of regression model of yi and ω only depend of λ and ψε while φ only involving ωi distribution. so it is known that the prior distribution (λ, ψε) and φ is independent and can be written by: 𝑝(θ) = 𝑝(λ, φ, ψε) = 𝑝(λ, ψε) 𝑝(φ) (2.7) conditional distribution y which is given ω only depend on λ and ψε, while ω distribution is depend on φ. based on these case, can be conclude that: 𝑝(λ, ψε, φ|y, ω) = 𝑝(θ|y, ω) ∝ 𝑝(y, ω |θ)𝑝(θ) = 𝑝(y|θ, ω) 𝑝(ω|θ) 𝑝(θ) = 𝑝(y|θ, ω) 𝑝(ω|θ) 𝑝(λ, ψε) 𝑝(φ) = [𝑝(y|λ, ψε, ω) 𝑝(λ, ψε)][𝑝(ω|φ) 𝑝(φ)] (2.8) prior join distribution is used in prior distribution of (λ, ψε) and φ can be explain with the illustration if element 𝜓 𝑘 is a diagonal element from ψε and λk t is the k-row from λ. so it can be written: 𝜓 𝑘 −1 ~ 𝐺𝑎𝑚𝑚𝑎 (𝛼0𝑒𝑘 , 𝛽0𝑒𝑘 ) (2.9) (λ𝑘 | 𝜓 𝑘) ~ 𝑁(λ0𝑒𝑘 , 𝜓 𝑘 h𝑜𝑦𝑘 ) (2.10) φ−1 ~ 𝑊𝑞 (r0, 𝜌0) (2.11) to get 𝑝(φ|y, ω) based on the equation (2.8) with ωi is free, then obtained a conditional distribution 𝑝(φ|y, ω) as follows: 𝑝(φ|y, ω) ∝ 𝑝 (φ) ∏ 𝑝(ωi|φ) n i=1 (2.12) based on prior distribution of φ−1 in the equation (2.11) can be conclude that φ is follow invers wishart distribution with q dimension: φ~𝐼𝑊𝑞 (r0 −1 , 𝜌0). based on ωi distribution with conditional φ is 𝑁(0, φ) then the resulting: 𝑝( φ|y, ω) ∝ |φ| −(n+ρ0 +q+1) 2 ⁄ exp (− 1 2 tr[φ−1 (ωωt + r0 −1 )]) so that, (φ|y, ω) ~ iwq[(ωω t + r0 −1 , n + ρ0)] (2.13) likelihood function is known from y is (lee, 2007): parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 90 𝑝(y|λ, ψ𝛿 , ω) ∝ |ψ𝛿 | −𝑛 2⁄ exp [− 1 2 ∑(𝑦𝑖 − λωi) 𝑇 ψ𝛿 −1(𝑦𝑖 − 𝑛 𝑖=1 λωi)] (2.14) based on likelihood and join prior from λωk and 𝑣𝛿𝑘 then posterior distribution of (λωk, 𝑣δk) with given y, ω is (lee, 2007): (𝑣δk|y, ω) ~ 𝐺𝑎𝑚𝑚𝑎 ( 𝑛 2 + 𝛼0𝛿𝑘 , 𝛽𝛿𝑘 ) (2.15) (λ𝜔𝑘 | 𝑌, ω, 𝑣δk) ~ 𝑁(a𝜔𝑘 , 𝜓𝛿𝑘 −1 a𝜔𝑘 ) (2.16) parameter significance test significance testing parameters on bayesian method is done by using credible interval which generated by the predictive posterior distribution of bayesian mcmc approach (ntzoufras 2009). credible interval is the interval region or area of the posterior distribution opportunities. the parameter value on bayesian methods indicated by the mean value obtained from the average value of the resurrection of the posterior distribution. 95% credible interval indicated by the lower limit percentiles of 2.5% and the upper limit percentiles 97.5% of the value of the resurrection of the posterior distribution. here is a hypothesis that is used to test the significance of the parameters in the bayesian sem methods: 1. 𝐻0: 𝛾𝑖𝑗 = 0 ; there is no influence of exogenous variables on the endogenous variables 𝐻1: 𝛾𝑖𝑗 ≠ 0 ; there is the influence of exogenous variables on the endogenous variables 2. 𝐻0: 𝛽𝑖𝑗 = 0 ; there is no influence of endogenous variables on the endogenous variables 𝐻1: 𝛽𝑖𝑗 ≠ 0 ; there is the influence of endogenous variables on the endogenous variables the acceptance or rejection of based on the presence or absence of zero value in a credible interval on each parameter. parameter is said to be significant (reject 𝐻0)) if credible interval does not contain the zero value and means that there are significant predictor variables on the response variable. conversely, a parameter is said to be not significant (accept 𝐻0)) if credible interval includes zero value means there is no influence on the response variable predictor variables. research method data that used in this study are primary data obtained from interviews with employees fni statistics in malang which totaled 15 people on leadership styles, job satisfaction and organizational commitment well as interviews with corporate leaders regarding employee performance. analysis of structural equation modeling (sem) using a bayesian approach with the following steps: parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 91 determining diagram research js x1.4 λx1.4 x1.5 y1.1 λx1.5 λy1.1 y1.2 y1.3 λy1.3 λy1.2 ls ep oc x1.3 x1.1 x1.2 λx1.3 λx1.2 λx1.1 x2.2 x2.3 y2.3 y2.7y2.4 y2.5 x2.1 x2.4 λy2.4 λx2.4 λx2.3 λx2.2 λx2.1 λy2.3 γ11 γ21 β21 y2.2 λy2.2 γ12 γ22 λy2.7 λy2.5 δ6 δ7 δ8 δ9 ζ2 ε8 ε7ε6ε5ε4 ζ1 δ1 δ2 δ3 δ4 δ5 ε1 ε2 ε3 a. determine parameter model x : vector indicator for exogenous latent variable y : vector indicator for endogenous latent variable ω : matrix of exogenous and endogenous latent variable furthermore, there will be estimating the parameters in the model are unknown, namely θ consisting of: φ : covariance matrix of 𝜉(exogenous latent variable 𝜉1: leadership style, 𝜉2: organizational commitment) ψ : covariance matrix of 𝜁(measurement error of endogenous latent variable job satisfaction (𝜁1) and employee performance (𝜁2)) λ : matrix of factor loading from exogenous and endogenous indicator γ : matrix coefficient of relationship between exogenous to endogenous latent variables β : matrix coefficient of relationship between endogenous to endogenous latent variables b. prior choice prior used in this study refers to lee (2007) is a conjugate prior distribution c. calculations posterior bayesian sem d. mcmc application with gibbs sampling to get the estimation of the posterior distribution. e. parameter significance test significance testing parameters on bayesian method is done by using credible interval. f. parameter accuracy test the accuracy test of the parameters is done by seeing mc error value where the smaller mc error or getting close to zero mean that parameter estimation value is better. results and discussion prior distribution hyperparameters value on each prior determined using a type of conjugate prior distribution, and pseudo informative priors. determination of the proper prior analysis will produce a more accurate and precise prediction. hiperparameter any prior structure is presented in table 4.1: table 4.1. hyperparameters and distribution prior used no parameters distribution 1 θ𝛿 ~ invers gamma (9,8) parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 92 no parameters distribution 2 θ ~ invers gamma (9,8) 3 θ𝜂 ~ invers gamma (9,8) 5 [λ𝑥 |𝜃𝛿 ] ~ normal (0.393, 𝜃𝛿 ) 6 [λ𝑦|𝜃 ] ~ normal (0.393, 𝜃 ) 7 [𝛾1.1| 𝜓] ~ normal (0.326, 4𝜓) 8 [𝛾1.2| 𝜓] ~ normal (0.380, 4𝜓) 9 [𝛾2.1| 𝜓] ~ normal (0.138, 4𝜓) 10 [𝛾2.2| 𝜓] ~ normal (0.348, 4𝜓) 11 [𝛽2.1| 𝜓] ~ normal (0.679, 10𝜓) 12 𝜓 ~ invers gamma (9,8) significance testing parameters and building model in structural equation modeling (sem), significance testing carried out on the model parameter measurement model and structural model. in the measurement model, significance testing was conducted to determine whether each of the significant indicators measure the latent variable or not. while the structural model, the parameter significance testing was conducted to determine whether there is influence between exogenous variables on endogenous variables or between variables endogen itself. parameter estimation using sem method with the bayesian approach is done with the help of winbugs14 program. summary predictive value of each parameter models and their significance testing parameters are presented in table 4.2 and is graphically shown in figure 4.1. table 4.2. summary of parameter estimation values and significance parameter mean sd percentile 2.5% percentile 97.5% summary 𝜆𝑥1.1 1.000 significant 𝜆𝑥1.2 0.623 0.146 0.346 0.923 significant 𝜆𝑥1.3 0.890 0.194 0.538 1.308 significant 𝜆𝑥1.4 0.688 0.152 0.408 0.999 significant 𝜆𝑥1.5 0.721 0.158 0.426 1.047 significant 𝜆𝑥2.1 1.000 significant 𝜆𝑥2.2 0.734 0.163 0.431 1.074 significant 𝜆𝑥2.3 0.803 0.176 0.475 1.173 significant 𝜆𝑥2.4 0.723 0.163 0.418 1.060 significant 𝜆𝑦1.1 1.000 significant 𝜆𝑦1.2 0.632 0.158 0.344 0.964 significant 𝜆𝑦1.3 0.826 0.188 0.487 1.214 significant 𝜆𝑦2.2 1.000 significant 𝜆𝑦2.3 0.743 0.163 0.4505 1.094 significant 𝜆𝑦2.4 0.633 0.146 0.3657 0.9428 significant 𝜆𝑦2.5 0.739 0.156 0.4559 1.069 significant parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 93 parameter mean sd percentile 2.5% percentile 97.5% summary 𝜆𝑦2.7 0.634 0.144 0.3727 0.9391 significant 𝛾1.1 0.440 0.216 0.02218 0.8674 significant 𝛾1.2 0.510 0.225 0.07161 0.9622 significant 𝛾2.1 0.185 0.239 -0.2801 0.6511 not significant 𝛾2.2 0.384 0.252 -0.1086 0.8775 not significant 𝛽2.1 0.676 0.171 0.337 1.012 significant based on the results presented in table 4.2 and figure 4.1 is known that the measurement model, all significant indicators to measure each of the latent variables. while the structural model, it is known that of the five parameters tested, there are three significant parameters that 𝛾1.1, 𝛾1.2 and 𝛽2.1. it can be concluded that there are significant differences between the variables of leadership style and organizational commitment toward job satisfaction directly as well as a significant difference between job satisfaction on employee performance. as for the influence of leadership style and variable organizational commitment on employee performance directly declared insignificant. job satisfaction x1.4 x1.5 y1.1 1.000 y1.2 y1.3 0.8256 0.6319 leadership style employee performance organizational commitment x1.3 x1.1 x1.2 0.8898 1.000 x2.2 x2.3 y2.3 y2.7 y2.4 y2.5 x2.1 x2.4 0.6327 0.7229 0.8029 0.7236 1.000 0.7431 0.4398 0.1850 0.6757 y2.2 1.000 0.5102 0.3844 0.6339 0.7393 0.6228 0.6875 0.7213 -0.1247 0.0835 0.0595 -0.1193 0.1606 0.1726 0.1645 0.1362 0.1923 0.1235 0.1311 -0.1595 0.1155 0.2078 -0.2114 0.1266 0.1159 0.1077 0.1161 note : not significant significant gambar 4.1. full model structural equation modeling parameter estimation test one way to check the accuracy of parameter estimation in bayesian sem is to look at the mc error is generated, where the smaller the value mc error or getting close to zero then the better parameter estimation results. error mc value for each parameter can be seen in table 4.3: table 4.3. mc error value of each parameter parameter mc error parameter mc error 𝜆𝑥1.2 0,002327 𝜆𝑦2.3 0,003308 𝜆𝑥1.3 0,003417 𝜆𝑦2.4 0,002648 𝜆𝑥1.4 0.002139 𝜆𝑦2.5 0.002527 parameter estimation of structural equation modeling using bayesian approach dewi kurnia sari 94 parameter mc error parameter mc error 𝜆𝑥1.5 0.002136 𝜆𝑦2.7 0.002352 𝜆𝑥2.2 0.002271 𝛾1.1 0.004573 𝜆𝑥2.3 0.002907 𝛾1.2 0.00446 𝜆𝑥2.4 0.002294 𝛾2.1 0.00494 𝜆𝑦1.2 0.002345 𝛾2.2 0.005634 𝜆𝑦1.3 0.00339 𝛽2.1 0.002884 it is seen that the mc error of all parameters on the structural equation model is very small or approaching closer 0 (zero) value, so the results of estimation parameters generated a good result parameter estimation. conclusion based on the analysis and discussion, it can be concluded as follows: 1. model of leadership style and organizational commitment to job satisfaction and employee performance using the estimation method parameters structural equation modeling (sem) through a bayesian approach mathematically is job satisfaction = 0.4398 leadership style + 0.5102 organizational commitment + 0.0835 and employee performance = 0.1850 leadership style + 0.3844 commitments organizations + job satisfaction 0.6757 + 0.0595. 2. in the measurement model, all significant indicators to measure latent variables respectively. while the structural model, it was concluded that the five parameters tested, there are three significant parameters that 𝛾1.1, 𝛾1.2 and 𝛽2.1. that there are a significant difference between the variables of leadership style and organizational commitment toward job satisfaction directly as well as a significant difference between job satisfaction on employee performance. as for the influence of leadership style and variable organizational commitment on employee performance directly declared insignificant. reference [1] i. ntzoufras, bayesian modeling in winbugs. hoboken, nj, usa: john wiley & sons, inc, 2009, pp. --. [2] s.-y. lee, structural equation modeling: a bayesian approach, vol. 711, john wiley & sons, 2007, pp. --. [3] l. kogan, "small-sample inference and bootstrap," small, vol. 1, pp. 2--, 2010. [4] j. f. hair, w. c. black, b. j. babin, r. e. anderson and r. l. tatham, multivariate data analysis, vol. 6, pearson prentice hall upper saddle river, nj, 2006, pp. --. [5] i. ghozali, "model persamaan struktural: konsep dan aplikasi dengan program amos ver. 16.0," semarang, indonesia: badan penerbit universitas diponegoro, pp. --, 2004. [6] arrizal, "pemimpin gaya kepemimpinan transaksional mencapai sukses melalui manajemen sumber daya manusia," jurnal kajian bisnis sekolah tinggi ilmu ekonomi widya wiwaha, no. 22, 2001. abstract introduction theoritical review prior distribution mcmc application with gibbs sampling posterior distribution parameter significance test research method results and discussion prior distribution significance testing parameters and building model parameter estimation test conclusion reference bilangan kromatik graf commuting dan noncommuting grup dihedral 1handrini rahayuningtyas, 2abdussakir, 3achmad nashichuddin 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang 3jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: handrienie05@gmail.com abstrak graf commuting adalah graf yang memiliki himpunan titik x dan dua titik berbeda akan terhubung langsung jika saling komutatif di 𝐺. misal 𝐺 grup non abelian dan 𝑍(𝐺) adalah center dari 𝐺. graf noncommuting adalah suatu graf yang mana titik-titiknya merupakan himpunan dari 𝐺\𝑍(𝐺) dan dua titik x dan y terhubung langsung jika dan hanya jika 𝑥𝑦 ≠ 𝑦𝑥. pewarnaan titik pada graf 𝐺 adalah pemberian sebanyak k warna pada titik sehingga dua titik yang terhubung langsung tidak diberi warna yang sama. pewarnaan sisi pada graf 𝐺 adalah dua sisi yang berasal dari titik yang sama diberi warna yang berbeda. bilangan terkecil k sehingga suatu graf dapat diberi k warna pada titik dan sisi inilah yang dinamakan bilangan kromatik. pada artikel ini didapatkan rumus umum bilangan kromatik dari graf commuting dan noncommuting yang dibangun dari suatu grup yaitu grup dihedral. kata kunci: bilangan kromatik, pewarnaan titik, pewarnaan sisi, graf commuting dan noncommuting, grup dihedral. abstract commuting graph is a graph that has a set of points x and two different vertices to be connected directly if each commutative in 𝐺. let 𝐺 non abelian group and 𝑍(𝐺) is a center of 𝐺. noncommuting graph is a graph which the the vertex is a set of 𝐺\𝑍(𝐺) and two vertices x and y are adjacent if and only if 𝑥𝑦 ≠ 𝑦𝑥. the vertex colouring of 𝐺 is giving k colour at the vertex, two vertices that are adjacent not given the same colour. edge colouring of g is two edges that have common vertex are coloured with different colour. the smallest number k so that a graph can be coloured by assigning 𝑘 colours to the vertex and edge called chromatic number. in this article, it is available the general formula of chromatic number of commuting and noncommuting graph of dihedral group. keywords: chromatic number, vertex colouring, edge colouring, commuting and noncommuting graph, dihedral group. pendahuluan graf g adalah pasangan (𝑉(𝐺), 𝐸(𝐺)) dengan 𝑉(𝐺) adalah himpunan tidak kosong dan berhingga dari objek-objek yang disebut titik, dan 𝐸(𝐺) adalah himpunan (mungkin kosong) pasangan tak berurutan dari titik-titik berbeda di 𝑉(𝐺) yang disebut sisi. banyaknya unsur di 𝑉(𝐺) disebut order dari g dan dilambangkan dengan 𝑝(𝐺), dan banyaknya unsur di 𝐸(𝐺) disebut ukuran dari g dan dilambangkan dengan 𝑞(𝐺). jika graf yang dibicarakan hanya graf 𝐺, maka order dan ukuran dari 𝐺 masing-masing cukup ditulis 𝑝 dan 𝑞. graf dengan order 𝑝 dan ukuran 𝑞 dapat disebut graf (𝑝, 𝑞) [1]. sisi e = (u,v) dikatakan menghubungkan titik u dan v. jika e = (u,v) adalah sisi di graf g, maka u dan v disebut terhubung langsung (adjacent), v dan e serta u dan e disebut terkait langsung (incident), dan titik u dan v disebut ujung dari e. dua sisi berbeda e1 dan e2 disebut terhubung langsung (adjacent), jika terkait langsung pada satu titik yang sama. untuk selanjutnya, sisi e = (u, v) akan ditulis e = uv [2]. perkembangan terbaru teori graf yaitu membahas graf yang dibangun oleh suatu grup. misal 𝐺 grup berhingga dan 𝑋 adalah subset dari 𝐺. graf commuting 𝐶(𝐺, 𝑋) adalah graf yang memiliki himpunan titik 𝑋 dan dua titik berbeda akan terhubung langsung jika saling komutatif di 𝐺. jadi, titik x dan y akan terhubung langsung di 𝐶(𝐺, 𝑋) jika dan hanya jika 𝑥𝑦 = 𝑦𝑥 di 𝐺 (vahidi & talebi, 2010:123). sebaliknya, misal 𝐺 grup non abelian dan 𝑍(𝐺) adalah center dari 𝐺. graf noncommuting γ𝐺 adalah suatu graf yang mana titik-titiknya merupakan himpunan dari 𝐺\𝑍(𝐺) dan dua titik 𝑥 dan 𝑦 terhubung langsung jika dan hanya jika 𝑥𝑦 ≠ 𝑦𝑥 [3]. perkembangan berikutnya muncul bilangan kromatik pewarnaan titik dan pewarnaan sisi pada graf. pewarnaan titik pada graf 𝐺 adalah pemberian sebanyak 𝑛 warna pada titik sehingga mailto:handrienie05@gmail.com handrini rahayuningtyas 17 volume 4 no.1 november 2015 dua titik yang saling terhubung langsung tidak diberi warna yang sama. pewarnaan sisi pada graf 𝐺 adalah pemberian sebanyak 𝑛 warna pada sisi sehingga dua sisi yang saling terkait langsung tidak diberi warna yang sama. bilangan 𝑛 terkecil sehingga graf 𝐺 dapat diwarnai dengan cara tersebut dinamakan bilangan kromatik. bilangan kromatik titik ditulis 𝜒(𝐺) dan bilangan kromatik sisi ditulis 𝜒′(𝐺) [1]. kajian teori 1. graf 𝑮 graf g adalah pasangan (v(g), e(g)) dengan v(g) adalah himpunan tidak kosong dan berhingga dari objek-objek yang disebut titik, dan e(g) adalah himpunan (mungkin kosong) pasangan takberurutan dari titik-titik berbeda di v(g) yang disebut sisi [1]. sehingga jika 𝐺 = (𝑉(𝐺), 𝐸(𝐺)), maka 𝑉(𝐺) = {𝑣1, 𝑣2, … , 𝑣𝑛 } dan 𝐸(𝐺) = {𝑒1, 𝑒2, … , 𝑒𝑛 }, dimana 𝑣𝑖 ∈ 𝑉(𝐺), 𝑖 = 1,2, … , 𝑛 disebut titik (vertex) dan 𝑒𝑗 = 1,2, … , 𝑚 disebut sisi (edge). 2. derajat titik jika v adalah titik pada graf 𝐺, maka himpunan semua titik di 𝐺 yang terhubung langsung dengan 𝑣 disebut lingkungan dari v dan ditulis 𝑁𝐺(𝑣). derajat dari titik 𝒗 di graf 𝐺, ditulis deg𝐺 (𝑣), adalah banyaknya sisi di 𝐺 yang terkait langsung dengan v. derajat total g adalah jumlah derajat semua titik dalam g. dalam konteks pembicaraan hanya terdapat satu graf g, maka tulisan deg𝐺 (𝑣) disingkat menjadi deg(v) dan ng(v) disingkat menjadi n(v). jika dikaitkan dengan konsep lingkungan, derajat titik v di graf g adalah banyaknya anggota dalam n(v) [2]. gambar 1. graf 𝑮 dengan himpunan titik 𝑽(𝑮) berdasarkan gambar, diperoleh bahwa: 𝑁(𝑎) = {𝑏, 𝑐, 𝑑} 𝑁(𝑏) = {𝑎, 𝑐} 𝑁(𝑐) = {𝑎, 𝑏, 𝑑} 𝑁(𝑑) = {𝑎, 𝑐}. dengan demikian, maka deg(𝑎) = 3 deg(𝑏) = 2 deg(𝑐) = 3 deg(𝑑) = 2 3. graf terhubung suatu graf g dikatakan terhubung jika untuk setiap titik u dan v di g terdapat lintasan u-v di g. sebaliknya, jika ada dua titik u dan v di g, tetapi tidak ada lintasan u-v di g, maka g dikatakan tak terhubung (disconnected) [2]. 4. bilangan kromatik pewarnaan titik pada graf g adalah pemberian sebanyak n warna pada titik sehingga dua titik yang saling terhubung langsung tidak diberi warna yang sama. pewarnaan sisi pada graf g adalah pemberian sebanyak n warna pada sisi sehingga dua sisi yang saling terkait langsung tidak diberi warna yang sama. bilangan n terkecil sehingga graf g dapat diwarnai dengan cara tersebut dinamakan bilangan kromatik. bilangan kromatik titik ditulis 𝜒(𝐺) dan bilangan kromatik sisi ditulis 𝜒′(𝐺) [1]. 5. grup dihedral grup adalah suatu struktur aljabar yang dinyatakan sebagai (𝐺,∗) dengan 𝐺 adalah himpunan tak kosong dan ∗ adalah operasi biner di 𝐺 yang memenuhi sifat-sifat berikut: 1. (𝑎 ∗ 𝑏) ∗= 𝑎 ∗ (𝑏 ∗ 𝑐), untuk semua 𝑎, 𝑏, 𝑐 ∈ 𝐺 (yaitu assosiatif ). 2. ada suatu elemen 𝑒 di 𝐺 sehingga 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎, untuk semua 𝑎 ∈ 𝐺 (𝑒 disebut identitas di 𝐺). 3. untuk setiap 𝑎 ∈ 𝐺 ada suatu element 𝑎−1 di 𝐺 sehingga 𝑎 ∗ 𝑎−1 = 𝑎−1 ∗ 𝑎 = 𝑒 (𝑎−1 disebut invers dari 𝑎) adapun grup (𝐺,∗) disebut abelian (grup komutatif) jika 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 untuk semua 𝑎, 𝑏 ∈ 𝐺 [4]. grup dihedral adalah grup dari himpunan simetri-simetri dari segi-n beraturan, dinotasikan𝐷2𝑛, untuk setiap 𝑛 bilangan bulat positif dan 𝑛 ≥ 3. dalam buku lain ada yang menuliskan grup dihedral dengan 𝐷𝑛 [5]. adapun himpunan anggota grup dihedral 𝐷2𝑛 yaitu 𝐷2𝑛 = {1, 𝑟, 𝑟 2, … , 𝑠, 𝑠𝑟, 𝑠𝑟2, … , 𝑠𝑟𝑛−1}. 6. graf commuting dan noncommuting misal 𝐺 adalah grup berhingga dan 𝑋 adalah subset dari 𝐺, graf commuting 𝐶(𝐺, 𝑋) adalah graf dengan 𝑋 sebagai himpunan titik dan dua elemen berbeda di 𝐶(𝐺, 𝑋) terhubung langsung jika bilangan kromatik graf commuting dan noncommuting grup dihedral cauchy – issn: 2086-0382/e-issn: 2477-3344 18 keduanya adalah elemen yang saling komutatif di 𝐺 [6]. misal 𝐺 grup non abelian dan 𝑍(𝐺) adalah center dari 𝐺. graf non commuting γ𝐺 adalah suatu graf yang mana titik-titiknya merupakan himpunan dari 𝐺\𝑍(𝐺) dan dua titik 𝑥 dan 𝑦 terhubung langsung jika dan hanya jika 𝑥𝑦 ≠ 𝑦𝑥 [3]. pembahasan pewarnaan titik pada graf g adalah pemberian sebanyak n warna pada titik sehingga dua titik yang saling terhubung langsung tidak diberi warna yang sama. pewarnaan sisi pada graf g adalah pemberian sebanyak n warna pada sisi sehingga dua sisi yang saling terkait langsung tidak diberi warna yang sama. dari pewarnaan titik dan sisi inilah dapat diketahui bilangan kromatiknya, baik graf commuting maupun graf noncommuting. bilangan kromatik pewarnaan titik dan sisi graf commuting grup dihedral perhatikan tabel di bawah ini: tabel 1. bilangan kromatik pewarnaan titik dan sisi graf commuting grup dihedral 𝐶(𝐷2𝑛 ) 𝜒(𝐶(𝐷2𝑛 )) 𝜒 ′(𝐶(𝐷2𝑛 )) 𝐷6 3 5 𝐷8 4 7 𝐷10 . . . 5 . . . 9 . . . 𝐷2𝑛 𝑛 2𝑛 − 1 sumber: penulis berdasarkan tabel 1, didapatkan teorema berikut: teorema 1 misal 𝐶(𝐷2𝑛 ) adalah graf commuting dari grup dihedral-2n (𝐷2𝑛 ). maka bilangan kromatik pewarnaan titik graf commuting dari grup dihedral-2n (𝐷2𝑛 ) adalah 𝜒(𝐶(𝐷2𝑛 )) = 𝑛. bukti: untuk 𝑛 ganjil dan genap, misal diketahui 𝑣 = {1, 𝑟, 𝑟2, … , 𝑟𝑛−1} di 𝐷2𝑛, untuk 𝑖 ≠ 𝑗. kemudian 𝑟𝑖 ∘ 𝑟𝑗 = 𝑟𝑗 ∘ 𝑟𝑖 untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 di 𝐷2𝑛, maka 𝑟𝑖 dan 𝑟𝑗 saling terhubung langsung di 𝐶(𝐷2𝑛 ). karena 𝑟 𝑖 dan 𝑟𝑗 saling komutatif, maka terdapat (𝑟𝑖 , 𝑟𝑗 ) ∈ 𝐶(𝐷2𝑛 ) yang membentuk subgraf komplit-𝑛. sehingga dibutuhkan sebanyak 𝑛 warna, atau dengan kata lain bilangan kromatik pewarnaan titik 𝑟𝑖 dan 𝑟𝑗 yaitu 𝑛. misal 𝑤 = {𝑠, 𝑠𝑟, 𝑠𝑟1, … , 𝑠𝑟𝑛−1} dimana 𝑤 hanya komutatif dengan 1 di 𝐷2𝑛. artinya 𝑠𝑟 𝑖 dan 𝑠𝑟𝑗 , 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 tidak saling komutatif. karena tidak saling komutatif, maka dapat diberi warna yang sama. pada 𝑛 ganjil, 𝑠𝑟𝑖 tidak komutatif dengan 𝑟𝑗 , 𝑗 = 1,2, … , 𝑛 − 1. 𝑠𝑟 ∘ 𝑟 = 𝑠𝑟1+1 = 𝑠𝑟2 𝑟 ∘ 𝑠𝑟 = 𝑠𝑟1−1 = 𝑠 karena 𝑠𝑟𝑖 tidak komutatif dengan 𝑟𝑗, 𝑗 = 1,2, … , 𝑛 − 1, maka warna titik 𝑠𝑟𝑖 berlaku sama dengan titik 𝑟𝑗 , 𝑗 = 1,2, … , 𝑛 − 1. pada 𝑛 genap, terdapat 𝑠𝑟 𝑛 2 ∘ 𝑟 𝑛 2 = 𝑟 𝑛 2 ∘ 𝑠𝑟 𝑛 2 , sehingga warna titik 𝑠𝑟 𝑛 2 tidak boleh sama dengan 𝑟 𝑛 2 . selain itu, terdapat 𝑠𝑟 𝑛 2 ∘ 𝑟𝑖 = 𝑟𝑖 ∘ 𝑠𝑟 𝑛 2 , sehingga warna titik 𝑠𝑟 𝑛 2 boleh sama dengan warna titik 𝑟𝑖 , atau dengan kata lain 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 dapat diberi warna yaitu memilih dari 𝑛 2 warna. jadi, diperoleh 𝜒(𝐶(𝐷2𝑛 )) = 𝑛, untuk 𝑛 ganjil maupun genap. teorema 2 misal 𝐶(𝐷2𝑛 ) adalah graf commuting dari grup dihedral-2n (𝐷2𝑛 ). maka bilangan kromatik pewarnaan sisi graf commuting dari grup dihedral2n (𝐷2𝑛 ) adalah 𝜒′(𝐶(𝐷2𝑛 )) = 2𝑛 − 1 bukti: untuk 𝑛 ganjil, diketahui bahwa 𝑟𝑖 ∘ 𝑟𝑗 = 𝑟𝑗 ∘ 𝑟𝑖 , untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 di 𝐷2𝑛, untuk 𝑖 ≠ 𝑗. jadi, 𝑟𝑖 dan 𝑟𝑗 saling terhubung langsung di 𝐺. di samping itu, 1 komutatif dengan semua elemen 𝑟𝑖 dan 𝑠𝑟𝑖 , sehingga 1 terhubung langsung dengan semua elemen 𝑟𝑖 dan 𝑠𝑟𝑖, 𝑖 = 0,1,2, … , 𝑛 − 1 di 𝐺. karena 1 terhubung langsung dengan 2𝑛 − 1 elemen di 𝐺, maka minimal warna yang digunakan pewarnaan sisinya yaitu sebanyak 2𝑛 − 1 warna. berdasarkan aturan pewarnaannya, setiap sisi yang terkait dengan 1 titik yang sama diberi warna yang berbeda. adapun 𝑟𝑖 dan 𝑟𝑗 , 𝑖 = 0,1,2, … ,2𝑛 − 1 yang saling terhubung langsung dapat diberi warna dari 2𝑛 − 1 warna yang telah digunakan sebelumnya. sehingga didapatkan bilangan kromatik sisinya yaitu 𝜒′(𝐶(𝐷2𝑛 )) = 2𝑛 − 1, untuk 𝑛 ganjil. untuk 𝑛 genap, diketahui bahwa untuk 𝑛 ganjil, diketahui bahwa 𝑟𝑖 ∘ 𝑟𝑗 = 𝑟𝑗 ∘ 𝑟𝑖 , untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 di 𝐷2𝑛, untuk 𝑖 ≠ 𝑗. jadi, 𝑟 𝑖 dan 𝑟𝑗 saling terhubung langsung di 𝐺. walaupun 𝑟 𝑛 2 handrini rahayuningtyas 19 volume 4 no.1 november 2015 komutatif dengan 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1, tetapi 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 tidak komutatif dengan 𝑟𝑗 untuk 𝑗 selain 𝑛 2 . elemen 1 komutatif dengan 𝑟𝑖 dan 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 yaitu sebanyak 2𝑛 − 1 elemen. karena 1 komutatif dengan semua elemen 𝑟𝑖 dan 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 yaitu sebanyak 2𝑛 − 1 elemen, maka 1 memiliki derajat tertinggi di 𝐺. artinya, minimal warna yang dibutuhkan adalah sebesar derajat tertinggi di 𝐺 yaitu 1. 1 terhubung langsung dengan 2𝑛 − 1 elemen, maka minimal warna yang digunakan dalam pewarnaan sisi di 𝐺 sebanyak 2𝑛 − 1 warna. jadi, diperoleh bilangan kromatik sisinya yaitu 𝜒′(𝐶(𝐷2𝑛 )) = 2𝑛 − 1, untuk 𝑛 genap. bilangan kromatik pewarnaan titik dan sisi graf noncommuting grup dihedral perhatikan tabel berikut: tabel 2. bilangan kromatik graf noncommuting graf noncommuting grup dihedral γ(𝐷2𝑛 ) 𝜒(γ(𝐷2𝑛 )) 𝜒 ′(γ(𝐷2𝑛 )) γ(𝐷6) 4 5 γ(𝐷8) 3 5 γ(𝐷10) 6 9 γ(𝐷12) 4 9 γ(𝐷14) 8 13 γ(𝐷16) . . . 5 . . . 13 . . . γ(𝐷2𝑛 ) 𝑛 + 1, 𝑛 ganjil 𝑛 2 + 1, 𝑛 genap 2𝑛 − 1, n ganjil 2𝑛 − 3, n genap sumber: penulis berdasarkan tabel 2, diperoleh teorema sebagai berikut: teorema 3 misal γ(𝐷2𝑛 ) adalah graf noncommuting dari grup dihedral-2n (𝐷2𝑛 ), maka bilangan kromatik dari pewarnaan titik graf noncommuting pada grup dihedral-2n (𝐷2𝑛 ) adalah 𝜒(γ(𝐷2𝑛 )) = 𝑛 + 1 untuk 𝑛 ganjil dan 𝜒(γ(𝐷2𝑛 )) = 𝑛 2 + 1 untuk 𝑛 genap. bukti: untuk 𝑛 ganjil, diperoleh himpunan 𝑆 = {𝑟, 𝑠, 𝑠𝑟, … , 𝑠𝑟𝑛−1} saling tidak komutatif di 𝐷2𝑛, untuk 𝑖 ≠ 𝑗. dapat dikatakan bahwa 𝑟𝑖 dan 𝑠𝑟𝑗 , untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 saling terhubung langsung. dengan demikian, 𝑆 = {𝑟, 𝑠, 𝑠𝑟, … , 𝑠𝑟𝑛−1} akan membentuk subgraf komplit terbesar di 𝐺. karena 𝑆 = {𝑟, 𝑠, 𝑠𝑟, … , 𝑠𝑟𝑛−1} membentuk subgraf terbesar di 𝐺, maka bilangan clique atau order subgraf komplit terbesar graf 𝐺 adalah 𝑛 + 1, yaitu kardinalitas himpunan 𝑆. karena order dari subgraf komplit terbesarnya adalah 𝑛 + 1, maka pewarnaan titik pada graf 𝐺 membutuhkan minimal warna sebanyak 𝑛 + 1 warna. dengan demikian didapatkan bilangan kromatik titik pada graf noncommuting grup dihedral yaitu 𝜒(γ(𝐷2𝑛 )) = 𝑛 + 1, untuk 𝑛 ganjil. untuk 𝑛 genap, diketahui bahwa 𝑍(𝐺) = {1, 𝑟 𝑛 2 }. karena 𝑟𝑖 dan 𝑠𝑟𝑗 , untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 tidak saling komutatif, maka 𝑟𝑖 dan 𝑠𝑟𝑗 , untuk 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 terhubung langsung di 𝐺, untuk 𝑖 ≠ 𝑗. karena 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 2 saling komutatif dengan 𝑠𝑟𝑗 , 𝑗 = 𝑛 2 , 𝑛 2 + 1, 𝑛 2 + 2, … , 𝑛 − 1, maka 𝑠𝑟𝑖 tidak terhubung langsung dengan 𝑠𝑟𝑗 . namun demikian, 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 2 tidak komutatif satu sama lain. maka 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 2 akan membentuk subgraf komplit. karena 𝑠𝑟𝑖, 𝑖 = 0,1,2, … , 𝑛 2 terhubung langsung dengan 𝑟, maka diperoleh subgraf komplit terbesar yang memuat 𝑛 2 + 1 titik. dengan kata lain, bilangan clique atau order subgraf komplit terbesar graf 𝐺 adalah 𝑛 2 + 1. karena order dari subgraf komplit terbesar 𝐺 adalah 𝑛 2 + 1, maka pewarnaan titik pada graf 𝐺 membutuhkan minimal warna sebanyak 𝑛 2 + 1 warna. dengan demikian, dapat disimpulkan bahwa bilangan kromatik titik graf noncommuting grup dihedral yaitu 𝜒(γ(𝐷2𝑛 )) = 𝑛 2 + 1, untuk 𝑛 genap. teorema 4 misal γ(𝐷2𝑛 ) adalah graf noncommuting dari grup dihedral-2n (𝐷2𝑛 ), maka bilangan kromatik dari pewarnaan sisi graf noncommuting pada grup dihedral-2𝑛 (𝐷2𝑛 ) adalah 𝜒 ′(γ(𝐷2𝑛 )) = 2𝑛 − 1 untuk 𝑛 ganjil dan 𝜒′(γ(𝐷2𝑛 )) = 2𝑛 − 3 untuk 𝑛 genap. bukti: untuk 𝑛 ganjil, diketahui 𝑟𝑖 dan 𝑠𝑟𝑗 , 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 tidak komutatif, artinya 𝑟𝑖 dan 𝑠𝑟𝑗 , 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 saling terhubung langsung di 𝐷2𝑛, untuk 𝑖 ≠ 𝑗. karena 𝑟 𝑖 dan 𝑠𝑟𝑗 saling terhubung langsung, maka membentuk subgraf komplit di γ(𝐷2𝑛). misal 𝑣 merupakan titik di γ(𝐷2𝑛). diketahui bahwa banyaknya titik berderajat ganjil pada sebuah graf adalah genap, dapat ditulis ∑ 𝑑𝑒𝑔(𝑣) = 2𝑛𝑣∈γ(𝐷2𝑛) . pada graf noncommuting, pada 𝑛 ganjil banyaknya bilangan kromatik graf commuting dan noncommuting grup dihedral cauchy – issn: 2086-0382/e-issn: 2477-3344 20 ∑(𝑍(𝐷2𝑛 )) adalah 1. karena pada graf noncommuting center grup tidak dimunculkan, maka dapat ditulis 𝐷(γ(𝐷2𝑛 )) = ∑ deg(𝑣) 𝑣∈γ(𝐷2𝑛) − ∑(𝑍(𝐷2𝑛 )) = 2𝑛 − 1. dengan demikian, minimum warna yang digunakan yaitu 2𝑛 − 1. sehingga didapatkan (γ(𝐷2𝑛 )) = 2𝑛 − 1 untuk 𝑛 ganjil. untuk 𝑛 genap, diketahui 𝑟𝑖 dan 𝑠𝑟𝑗, 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 tidak saling komutatif, maka 𝑟𝑖 dan 𝑠𝑟𝑗 , 𝑖, 𝑗 = 0,1,2, … , 𝑛 − 1 terhubung langsung di γ(𝐷2𝑛 ), 𝑖 ≠ 𝑗. diketahui bahwa banyaknya derajat titik pada sebuah graf adalah dua kali banyak sisi. misal 𝑣 merupakan titik di γ(𝐷2𝑛 ), maka dapat ditulis ∑ 𝑑𝑒𝑔(𝑣) = 2𝑛𝑣∈γ(𝐷2𝑛) . diketahui pada graf noncommuting 𝑍(𝐷2𝑛 ) = {1, 𝑟 𝑛 2 }, maka ∑(𝑍(𝐷2𝑛 )) = 2. dari banyaknya titik dan center grup di γ(𝐷2𝑛 ), dapat dikatakan 𝐷(γ(𝐷2𝑛 )) = 2𝑛 − 2. karena 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 saling komutatif dengan 𝑠𝑟𝑗 , 𝑗 = 𝑛 2 , 𝑛 2 + 1, 𝑛 2 + 2, … , 𝑛 − 1, maka 𝑠𝑟𝑖 , 𝑖 = 0,1,2, … , 𝑛 − 1 tidak terhubung langsung dengan 𝑠𝑟𝑗, 𝑗 = 𝑛 2 , 𝑛 2 + 1, 𝑛 2 + 2, … , 𝑛 − 1, sehingga 𝐷(γ(𝐷2𝑛 )) = 2𝑛 − 3. karena 𝐷(γ(𝐷2𝑛 )) = |𝑁(𝑣 ∈ 𝐷2𝑛 )| yaitu 2𝑛 − 3, maka dapat dikatakan minimal warna yang digunakan sebanyak 2𝑛 − 3 warna. dengan demikian (γ(𝐷2𝑛 )) = 2𝑛 − 3 untuk 𝑛 genap. kesimpulan berdasarkan pembahasan pada penelitian ini, maka dapat diambil kesimpulan mengenai bilangan kromatik graf commuting dan noncommuting dari grup dihedral yaitu sebagai berikut: 1. bilangan kromatik dari pewarnaan titik graf commuting grup dihedral yaitu 𝜒(𝐶(𝐷2𝑛 )) = 𝑛, untuk 𝑛 ganjil dan genap. 2. bilangan kromatik dari pewarnaan sisi graf commuting grup dihedral ialah 𝜒′(c(𝐷2𝑛 )) = 2𝑛 − 1, untuk 𝑛 ganjil dan genap. 3. bilangan kromatik dari pewarnaan titik graf noncommuting grup dihedral ialah: 𝜒(γ(𝐷2𝑛 )) = { 𝑛 + 1, 𝑛 𝑔𝑎𝑛𝑗𝑖𝑙 𝑛 2 + 1, 𝑛 𝑔𝑒𝑛𝑎𝑝 4. bilangan kromatik dari pewarnaan sisi graf noncommuting grup dihedral ialah: 𝜒′(γ(𝐷2𝑛 )) = { 2𝑛 − 1, 𝑛 𝑔𝑎𝑛𝑗𝑖𝑙 2𝑛 − 3, 𝑛 𝑔𝑒𝑛𝑎𝑝 daftar pustaka [1] g. chartrand and l. lesniak, graph and digraph 2nd edition, california: wadsworth, inc, 1986. 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[40] j. kampf, ocean modelling for beginners, new york: springer, 2009. pendahuluan kajian teori 1. graf 𝑮 2. derajat titik 3. graf terhubung 4. bilangan kromatik 5. grup dihedral 6. graf commuting dan noncommuting pembahasan bilangan kromatik pewarnaan titik dan sisi graf commuting grup dihedral teorema 1 teorema 2 bilangan kromatik pewarnaan titik dan sisi graf noncommuting grup dihedral teorema 3 teorema 4 kesimpulan daftar pustaka seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 57-64 submitted: 17 february 2016 reviewed: 1 march 2016 accepted: 1 may 2016 seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts siti choirun nisak department of statistics, faculty of mathematics and natural sciences, brawijaya university, malang, indonesia email: nisak.statistika@gmail.com abstract time series forecasting models can be used to predict phenomena that occur in nature. generalized space time autoregressive (gstar) is one of time series model used to forecast the data consisting the elements of time and space. this model is limited to the stationary and non-seasonal data. generalized space time autoregressive integrated moving average (gstarima) is gstar development model that accommodates the non-stationary and seasonal data. ordinary least squares (ols) is method used to estimate parameter of gstarima model. estimation parameter of gstarima model using ols will not produce efficiently estimator if there is an error correlation between spaces. ordinary least square (ols) assumes the variance-covariance matrix has a constant error 𝜀𝑖𝑗 ~𝑁𝐼𝐷(𝟎, 𝝈 𝟐) but in fact, the observatory spaces are correlated so that variance-covariance matrix of the error is not constant. therefore, seemingly unrelated regression (sur) approach is used to accommodate the weakness of the ols. sur assumption is 𝜀𝑖𝑗 ~𝑁𝐼𝐷(𝟎, 𝚺) for estimating parameters gstarima model. the method to estimate parameter of sur is generalized least square (gls). applications gstarima-sur models for rainfall data in the region malang obtained gstarima models ((1)(1,12,36),(0),(1))-sur with determination coefficient generated with the average of 57.726%. keywords: space time, gstarima, ols, sur, gls, rainfall. introduction time series data is data based on the sequence of time within a certain time span and modeled by time series models either univariate or multivariate [1]. if the data consists of the elements of time and space is modeled using multivariate models of space-time. one of the multivariate models are most often used for data modeling space-time is a generalized spacetime autoregressive (gstar) model introduced by [2]. gstar has limitations that can only be used for data space-time that are stationary and non-seasonal. this condition tends to not be met at the data that is not stationary and containing a seasonal pattern. gstarima first implemented by sun, et al. (2010) for forecasting traffic flow data in beijing. gstarima is the development of gstar model for data non-stationary and seasonal. gstarima more flexible and practical for every space observatory has its own real-time parameter and not influenced by other changes in the observation location. a method that can be used in parameter estimation of gstarima include ordinary least square method (ols) and approach to the system of equations seemingly unrelated regression (sur). ols assumes the variance-covariance matrix has a constant error. thus, the equations system of seemingly unrelated regression (sur) is used to overcome the weakness of the ols. mailto:nisak.statistika@gmail.com seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts siti choirun nisak 58 sur equations system is used because it can accommodate correlations between space of the rainfall observatories. parameter estimation of sur can use the generalized least square (gls) method. discussion and implementation of the model gstarima still a bit to do so in this research will be modeling gstarima approach sur forecasting rainfall data in the region malang with an observation location jabung, tumpang, turen, tumpuk renteng, tangkilsari, wajak, blambangan, bululawang, tajinan and poncokusumo. literature review gstar (generalized space time autoregressive) model gstar with order autoregressive (p) and spatial order (𝜆𝑝), or represented by gstar (𝑝, 𝜆𝑝) can be written in the equation: 𝒛(𝒕) = ∑ [𝜱𝒌𝟎𝑾 (𝟎)𝒛(𝒕−𝒌) + ∑ 𝜱𝒌𝒔 𝜆𝑝 𝑠=1 𝑾(𝒔)𝒛(𝒕−𝒌)] + 𝒆(𝒕) 𝑝 𝑘=1 (1) where : 𝒛(𝒕) : vector of location 𝑚 at time t 𝜆𝑝 : spatial order from form of autoregressive p 𝑾(𝑠) : weighted matrix with size of 𝑚 × 𝑚 𝜱𝒌𝒔 : autoregresive parameter at time lag 𝑘 and spatial lag 𝑠 𝒆(𝒕) : error vector with white noise and normal multivariate distribution gstarima (generalized space time integrated autoregressive and moving average) model min, et al. (2010), defines the model gstarima as follows: ∆𝒅𝒛(𝒕) = ∑ ∑ 𝚽𝒌𝒍 𝑾 (𝒍) 𝝀𝒌 𝒍=𝟎 𝒑 𝒌=𝟏 ∆𝑑 𝑧(𝑡 − 𝑘) − ∑ ∑ 𝚯𝒌𝒍𝑾 (𝒍)𝜖 𝒎𝒌 𝒍=𝟎 𝒒 𝒌=𝟏 (𝑡 − 𝑘) + 𝝐𝒕 (2) with: 𝚽𝒌𝒍 = 𝑑𝑖𝑎𝑔(𝜙𝑘𝑙 1 , 𝜙𝑘𝑙 2 , … 𝜙𝑘𝑙 𝑁 ) = [ 𝜙𝑘𝑙 1 ⋮ 0 … ⋱ … 0 ⋮ 𝜙𝑘𝑙 𝑁 ] 𝚯𝒌𝒍 = 𝑑𝑖𝑎𝑔(𝜃𝑘𝑙 1 , 𝜃𝑘𝑙 2 , … 𝜃𝑘𝑙 𝑁 ) = [ 𝜃𝑘𝑙 1 ⋮ 0 … ⋱ … 0 ⋮ 𝜃𝑘𝑙 𝑁 ] where, p and q is the order of the autoregressive and moving average, 𝜆𝑘 and 𝑚𝑘 is the spatial order k to autoregressive and moving average, and ∆𝒅𝒛(𝒕) is the observation vector 𝒛(𝒕) in order differencing to d. 𝝓𝒌𝒍 and 𝜽𝒌𝒍 are autoregressive and moving average parameters on the lag time to spatial lag k and l. 𝑾(𝒍) is the weighted matrix of size n x n and 𝝐(𝒕) is the vector of the residual that is random and normal with size n x 1. seemingly unrelated regression (sur) seemingly unrelated regression (sur) is an equation parameter estimation using general least square (gls). sur model with m equations stated, siti choirun nisak 59 ( 𝒚𝟏 𝒚𝟐 ⋮ 𝒚𝒎 ) = ( 𝑿𝟏 𝟎 ⋮ 𝟎 𝟎 𝑿𝟐 ⋮ 𝟎 … … ⋱ … 𝟎 𝟎 ⋮ 𝑿𝒎 ) ( 𝜷𝟏 𝜷𝟐 ⋮ 𝜷𝒎 ) + ( 𝜺𝟏 𝜺𝟐 ⋮ 𝜺𝒎 ) = 𝑿𝜷 + 𝜺 (3) the assumption of the model is a residual 𝜀𝑖𝑟 is independent at all times, but between equality / contemporary correlated location. 𝐸[𝜀𝑖𝑟 𝜀𝑗𝑠|𝑋] = 0 when r≠s and 𝐸[𝜀𝑖𝑟 𝜀𝑗𝑟|𝑋] = σij. variance-covariance matrix is denoted by ω. 𝛀 = [ σ11𝐈𝑅 σ12𝐈𝑅 … σ1𝑚𝐈𝑅 σ21𝐈𝑅 σ22𝐈𝑅 … σ2𝑚 𝐈𝑅 ⋮ σ𝑚1 𝐈𝑅 ⋮ σ𝑚2 𝐈𝑅 ⋱ … ⋮ σ𝑚𝑚 𝐈𝑅 ] = 𝚺 ⊗ 𝐈𝑅 (4) sur estimation by adding variance covariance matrix residual is stated as follows, �̂� = (𝑿′(�̂�−𝟏 ⊗ 𝐈𝑹)𝑿) −𝟏𝑿′(�̂�−𝟏 ⊗ 𝐈𝑹)𝐘 (5) with 𝑟(�̂�) = 𝐸 [(�̂� − 𝜷) 𝟐 ] = 𝐸[(�̂� − 𝜷)(�̂� − 𝜷)′] = 𝐸[{(𝑿′𝑿)−𝟏𝑿′𝜺}{(𝑿′𝑿)−𝟏𝑿′𝜺}′] = 𝐸 [(𝑿′𝑿)−𝟏𝑿′𝜺𝜺′𝑿(𝑿 ′𝑿) −𝟏 ] = (𝑿′𝑿)−𝟏𝑿′𝑬[𝜺𝜺′]𝑿(𝑿′𝑿)−𝟏 = (𝑿′𝑿)−𝟏𝑿′𝛀𝑿(𝑿′𝑿)−𝟏 = (𝑿′𝛀−𝟏𝑿) −𝟏 . assuming 𝜺~𝑵(𝟎, 𝚺). sur estimators use the information system more efficient than ols because of the diversity in each equation is smaller, [3]. the precision model with mse and r2 criteria for the good of the model can be determined based on the residual is the mean square error (mse). 𝑀𝑒𝑎𝑛 𝑆𝑞𝑢𝑎𝑟𝑒 𝐸𝑟𝑟𝑜𝑟 (mse) = 1 n ∑ et 2 n t=1 (6) where et = zn+1 − �̂�𝑛(𝑙) and n is account of data. the coefficient of determination stating how great the diversity of the dependent variable (y) can be explained by the independent variable (x). according makridakis, et al. (1999), r2 is obtained from, 𝑅2 = 1 − ∑ (∑ (𝑍𝑖(𝑡) − �̂�𝑖(𝑡)) 2𝑇 𝑡=1 ) 𝑁 𝑖=1 ∑ (∑ (𝑍𝑖(𝑡) − �̅�𝑖(𝑡)) 2𝑇 𝑡=1 ) 𝑁 𝑖=1 (7) where : 𝑍𝑖(𝑡) : dependen variable i �̅�𝑖(𝑡) : mean of 𝑍𝑖(𝑡) seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts siti choirun nisak 60 �̂�𝑖(𝑡) : predicted value of 𝑍𝑖(𝑡) methodology location of the study is the rainfall observatory stations in the in malang southern region districts. the stations are tumpang, wajak, tajinan, jabung, poncokusumo, turen, tumpuk renteng, tangkilsari, blambangan, and bululawang. data used is 10 days period of rainfall (dasarian) since the beginning of the dry season is determined based on the amount of rainfall in a single dasarian (10 days) of less than 50 mm and is followed by several subsequent dasarian. meanwhile, the beginning of the rainy season is determined based on the amount of rainfall in a single dasarian (10 days) is equal or more than 50 mm and is followed by several subsequent dasarian [4]. rainfall data used to build gstarima-sur model is a sample data for forecasting (insample). the insample data is dasarian rainfall data for the period may 2000 to april 2015. the other data is called outsample data that is used to validate the gstarima-sur models of the data in the period may-june 2015. steps taken to form gstarima-sur model is started by exploration of rainfall data in the ten rainfall observatory stations, identification of univariate model (acf and pacf) and multivariate model (mpacf and mccf) to determine the order gstarima, determination parameters of the model, model validation, and the last is forecasting rainfall data in the malang southern region districts and the surrounding region. results and discussion exploration data is used to easy for viewing the information on the data. exploration data can be presented in the form of graphs and descriptive data. graph data movement precipitation 10 locations this year may 2000 to april 2015 is shown in the form of a time series plot as in figure 4.1 below. figure 4.1. time series plot of rainfall data based on figure 4.1, known that movement pattern of rainfall data in 10 locations tend to be similar. high and low altitude change of rainfall data recorded in each period indicates the same pattern every year. identification of the model is used to find the order of autoregressive and moving average for gstarima model. the order of autoregressive lag is obtained from the identification of the real mpacf and the order of moving average lag is obtained from the identification of the real macf, then from some real lag is chosen the best use of aic. lag which has the smallest aic value will be used as the order of autoregressive and moving average for gstarima model. moreover to identification seasonal pattern use univariate identification by seeing acf and pacf plot. results of identification rainfall in malang southern region districts is gstarima((1)(1,12,36),(0),(1))-sur. siti choirun nisak 61 estimation of parameters is done by inserting a weighted into the equation to describe the spatial relationship between the location of the post of rain. the weight of the locations used in this study is the inverse distance weighting location. table 4.1 inverse distance weighted value 10 pos rain gstarima-sur models for rainfall in the malang southern region districts can be expressed as follows: 𝒁(𝒕) = 𝚽𝟎𝟏 (𝟏)𝒁(𝒕 − 𝟏) + 𝚽𝟏𝟏 (𝟏)𝑾𝒁(𝒕 − 𝟏) + 𝚽𝟎𝟏 (𝟐)𝒁(𝒕 − 𝟏𝟐) + 𝚽𝟏𝟏 (𝟐)𝑾𝒁(𝒕 − 𝟏𝟐) + 𝚽𝟎𝟏 (𝟑)𝒁(𝒕 − 𝟑𝟔) + 𝚽𝟏𝟏 (𝟑)𝑾𝒁(𝒕 − 𝟑𝟔) + 𝒆(𝒕) − 𝛉𝟎𝟏 (𝟏) 𝒆(𝒕 − 𝟏) + 𝛉𝟏𝟏 (𝟏) 𝑾𝒆(𝒕 − 𝟏) based on the results of parameter estimation gstarima ((1)(1,12,36),(0),(1))-sur to tumpang rain post is as follows: �̂�1(𝑡) = 0.236𝑧1(𝑡−1) + 0.102𝑧2(𝑡−1) + 0.038𝑧3(𝑡−1) + 0.024𝑧4(𝑡−1) + 0.015𝑧5(𝑡−1) + 0.007𝑧6(𝑡−1) + 0.05𝑧7(𝑡−1) + 0.026𝑧8(𝑡−1) + 0.037𝑧9(𝑡−1) + 0.041𝑧10(𝑡−1) + 0.026𝑧1(𝑡−12) + 0.009𝑧2(𝑡−12) − 0.017𝑧3(𝑡−12) + 0.003𝑧4(𝑡−12) + 0.005𝑧5(𝑡−12) − 0.004𝑧6(𝑡−12) + 0.005𝑧7(𝑡−12) − 0.002𝑧8(𝑡−12) − 0.004𝑧9(𝑡−12) + 0.001𝑧10(𝑡−12) + 0.147𝑧1(𝑡−36) + 0.121𝑧2(𝑡−36) + 0.023𝑧3(𝑡−36) + 0.007𝑧4(𝑡−36) + 0.006𝑧5(𝑡−36) + 0.02𝑧6(𝑡−36) + 0.029𝑧7(𝑡−36) + 0.014𝑧8(𝑡−36) + 0.022𝑧9(𝑡−36) + 0.019𝑧10(𝑡−36) + 𝑒1(𝑡) + 0.081𝑒1(𝑡−1) + 0.009𝑒2(𝑡−1) + 0.019𝑒3(𝑡−1) + 0.002𝑒4(𝑡−1) + 0.004𝑒5(𝑡−1) − 0.005𝑒6(𝑡−1) + 0.011𝑒7(𝑡−1) + 0.012𝑒8(𝑡−1) + 0.003𝑒9(𝑡−1) + 0.015𝑒10(𝑡−1) rainfall in the tumpang rain post is influenced by the rain heading more in ten days and twenty days earlier and was influenced by weighting the location. in addition there is seasonality in the rainfall in the period a quarter of the year and annually. based on the above model prediction results can be obtained in the data sample for the tumpang rain post as follows: figure 4.2 forecasting using gstarima((1)(1,12,36),(0),(1))-sur in tumpang rain post pos hujan turen tumpuk renteng wajak tumpang jabung poncokus umo bululawang tajinan tangkilsari blambangan turen 0 0.286 0.140 0.057 0.044 0.068 0.103 0.087 0.092 0.122 tumpuk renteng 0.226 0 0.179 0.055 0.041 0.067 0.111 0.098 0.101 0.121 wajak 0.123 0.200 0 0.082 0.055 0.118 0.097 0.130 0.103 0.090 tumpang 0.059 0.073 0.097 0 0.191 0.223 0.075 0.128 0.090 0.064 jabung 0.063 0.074 0.089 0.260 0 0.141 0.083 0.121 0.097 0.072 poncokusumo 0.072 0.090 0.141 0.227 0.105 0 0.078 0.129 0.090 0.069 bululawang 0.068 0.093 0.073 0.048 0.039 0.049 0 0.115 0.280 0.235 tajinan 0.069 0.099 0.117 0.098 0.068 0.097 0.138 0 0.218 0.097 tangkilsari 0.062 0.087 0.080 0.059 0.046 0.058 0.288 0.187 0 0.133 blambangan 0.099 0.125 0.084 0.050 0.041 0.053 0.290 0.100 0.159 0 seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts siti choirun nisak 62 inspection accuracy of the model or the model validation is done by looking at the value of rmse and r2 of the model. table 4.2 accuracy test model of gstarima ((1),(1,12,36)(0)(1))-sur model in ten rain post location rmse r2 prediction tumpang 5.668 0.5052 tajinan 5.51 0.5716 poncokusumo 5.916 0.5146 blambangan 5.697 0.5935 bululawang 5.175 0.5805 wajak 5.899 0.5136 tumpuk renteng 6.007 0.6476 jabung 5.337 0.5671 turen 6.119 0.6308 tangkilsari 5.263 0.6481 r2 prediction for 10 observation locations is more than 50%. the greater the value of r2 prediction obtained the greater, model can explain the rainfall distribution. the biggest prediction of r2 is tangkilsari, that is equal to 0.6481. it can meaned that about 64.81% in tangkilsari rainfall distribution can be explained by the model gstarima ((1), (1,12,36) (0) (1))-sur. the lowest r2 prediction is in tumpang area rainfall, but the prediction obtained r2 can still be said to be good for rainfall prediction for prediction obtained r2 values> 50%. if viewed from the rmse and r2 prediction can be said that the rainfall distribution in ten observation locations have the same result by using modeling gstarima ((1) (1,12,36) (0) (1))-sur. result forecast rainfall using gstarima ((1),(1,12,36)(0)(1))-sur model is: table 4.3 forecasting rainfall period of may-june 2015 month may june locations 1 2 3 1 2 3 tumpang 7.157 8.254 6.642 5.503 3.241 1.438 tajinan 3.318 15.397 10.754 1.504 1.842 1.495 poncokusumo 5.702 3.069 6.390 6.610 3.309 1.179 blambangan 1.180 1.515 1.100 1.382 1.259 0.397 bululawang 9.392 2.592 6.521 4.620 3.943 0.490 wajak 1.509 1.143 1.343 1.401 1.468 0.031 tumpuk renteng 5.360 3.440 3.025 4.704 2.654 1.073 jabung 5.981 1.204 1.291 0.851 1.090 0.962 turen 8.348 7.754 3.159 5.744 3.885 1.703 tangkilsari 7.406 1.766 2.010 1.669 1.947 1.160 siti choirun nisak 63 figure 4.3 forecast for period may-june 2015 in malang southern region districts model validation is done by comparing the actual data of rainfall dasarian the period from may to june 2015 on the results of data using models forecasting gstarima ((1),(1,12,36)(0)(1))sur. if seen from figure 4.3 can be said that result of forecasting data can approach the actual data. to better know the data equation then done two paired samples t test. based on the results of two sample paired t test obtained by value t amounted to 1,958 with significant value 0095. t table with db = 59 values obtained 2,301, for t <t table (1.9585 <2,301) and the p-value is more than 0.05 (0095> 0.05) it was decided to accept h0. the conclusion of this validation test is that the rainfall dasarian data forecasting results do not differ significantly from the actual data, so the model gstarima-sur formed from in sample data can be used for forecasting rainfall dasarian next period. conclusion our analysis found that the gstarima ((1)(1,12,36)-sur model can be used to forecast the dasarian seasonal rainfall in the malang southern region districts. by using seasonal patterns in lag 1,12,36 more accurate forecasting result is obtained. to validate the model, mse and r2 prediction can be used. in this research, the largest r2 prediction is 57.726%. reference [1] j. rosadi, pengantar analisis runtun waktu, diktat kuliah, program studi statistika, fakultas matematika dan ilmu pengetahuan alam, universitas gajah mada, yogyakarta, universitas gajah mada, 2006. [2] s. borovkova, h. lopuhaa and b. ruchjana, "generalized star model with experimental weights," in proceedings of the 17th international workshop on statistical modelling, 2002. [3] h. r. moon and b. perron, "seemingly unrelated regressions," the new palgrave dictionary of economics, pp. 1-9, 2006. [4] bmkg, modul diklat badan meteorologi klimatologi dan geofisika karangploso malang, bmkg, 2000. [5] a. zellner, "an efficient method of estimating seemingly unrelated regressions and tests for aggregation bias," journal of the american statistical association, vol. 57, no. 298, pp. 348-368, 1962. [6] p. e. pfeifer and s. jay deutsch, "stationarity and invertibility regions for low order starma models: stationarity and invertibility regions," communications in statistics-simulation and computation, vol. 9, no. 5, pp. 551-562, 1980. [7] x. min, j. hu and z. zhang, "urban traffic network modeling and short-term traffic flow forecasting based on gstarima model," in intelligent transportation systems (itsc), 2010 13th international ieee conference on, 2010. seemingly unrelated regression approach for gstarima model to forecast rain fall data in malang southern region districts siti choirun nisak 64 [8] a. iriany, suharningsih, b. n. ruchjana and setiawan, "prediction of precipitation data at batu town using the gstar (1,p)-sur model," journal of basic and application scientific research, vol. 3, no. 6, pp. 860-865, 2013. abstract introduction literature review gstar (generalized space time autoregressive) model gstarima (generalized space time integrated autoregressive and moving average) model seemingly unrelated regression (sur) the precision model with mse and r2 methodology results and discussion conclusion reference suma’inna dan gugun gumilar implementasi transformasi wavelet daubechies pada kompresi citra digital suma’inna1 dan gugun gumilar2 1,2 program studi matematika fakultas sains dan teknologi uin syarif hidayatulallah jakarta abstrak proses kompresi merupakan proses mereduksi ukuran suatu data untuk menghasilkan representasi digital yang padat atau mampat, namun tetap dapat mewakili kuantitas informasi yang terkandung pada data tersebut. pada citra, video, dan suara, kompresi mengarah pada minimalisasi jumlah bit untuk representasi citra digital. dalam penelitian ini, proses kompresi citra menggunakan dua wavelet yang terkenal, yaitu wavelet haar dan daubechies db4. citra yang digunakan terbatas hanya citra grayscale berformat .jpg. tujuan dari penelitian ini adalah menentukan wavelet manakah yang baik dalam kompresi citra. dari hasil yang didapat, wavelet haar dan daubechies db4 berturut-turut mempunyai nilai rmse sebesar 6,54 dan 6,5. sedangkan rasio kompresi untuk wavelet haar adalah 25%, lebih besar dibanding wavelet daubchies db4, yaitu sebesar 22%. kata kunci : kompresi, wavelet haar, wavelet daubchies db4, rmse pendahuluan teknologi kompresi data berkembang seiring dengan kemajuan teknologi informasi. melalui teknologi kompresi ini, penyebaran data menjadi lebih cepat karena ukuran data yang lebih kecil dari ukuran aslinya, sehingga mempermudah proses pengiriman data atau dapat mengurangi kebutuhan terhadap kapasitas media penyimpanan. salah satu teknik kompresi citra ialah dengan transformasi wavelet. transformasi adalah proses pengubahan data atau sinyal ke dalam bentuk lain agar lebih mudah dianalisis. transformasi wavelet memiliki kemampuan untuk menganalisis suatu data dalam domain waktu dan domain frekuensi secara simultan. kompresi berbasis wavelet didasari pada prinsip bahwa koefisien-koefisien hasil proses transformasi wavelet yang melakukan dekorelasi pixel pada citra , dapat dikodekan lebih efisien dibandingkan dengan citra aslinya [1]. aplikasi dari wavelet semakin meluas untuk berbagai macam bidang seperti teknik penyelesaian persamaan differensial [2], pemrosesan sinyal dan analisis high dimentional data statistic [3-7]. dalam proses perhitungan matematis, wavelet dapat menganalisa frekuensi dan waktu secara simultan. sedangkan dalam sudut pandang fisika wavelet adalah teknik matematika yang digunakan untuk membagi atau memisah sinyal ke dalam komponen-komponen frekuensi yang berbeda dengan menggunakan sebuah penyaring (filterisasi) dengan menggunakan metode discrete wavelet transform (dwt), sehingga setiap komponen dapat diperiksa atau dipelajari pencocokan resolusinya. dwt yang cukup populer dalam pemerosesan citra adalah dwt haar (db1) dan daubechies (db). transformasi wavelet daubechies ini telah muncul sebagai salah satu ujung tombak teknologi, dalam bidang sinyal & analisis citra. daubechies (1988, 1992) mengembangkan teori dan kontruksi wavelet wavelet ortonormal yang mempunyai compact support. fungsi-fungsi yang mempunyai compact support mempunyai sifatsifat yang menarik, seperti dilasi dan penskalaan. pada penelitian ini, mencoba membandingkan dua tipe wavelet yang terkenal, yaitu daubechies (db2) dan haar (db1). dari kedua wavelet tersebut, manakah yang mampu memberikan hasil yang optimum dalam proses kompresi citra digital. hasil penelitian ini dapat menjadi masukan bagi peneliti lain dalam menentukan wavelet manakah yang optimum dalam memampatkan data selain data citra. landasan teori a. transformasi wavelet transformasi wavelet memberikan representasi frekuensi-waktu pada sebuah sinyal. pada awalnya, transformasi wavelet digunakan untuk menganalisa sinyal non-stasioner. pada transformasi wavelet, sinyal non-stasioner dianalisis menggunakan analisis multi-resolusi. umumnya, teknik analisis multi-resolusi adalah teknik yang digunakan untuk menganalisis frekuensi yang berbeda menggunakan resolusi yang berbeda [8]. analisis multiresolusi berisi keluarga subruang tertutup {𝑉𝑚 , 𝑚 ∈ ℤ} dari 𝐿 2(𝑅) yang memenuhi: suma’inna dan gugun gumilar 212 volume 2 no. 4 mei 2013 1. 𝑉𝑚 ⊂ 𝑉𝑚+1, untuk setiap 𝑚 ∈ ℤ. 2. 𝑓(𝑥) ∈ 𝑉𝑚 jika dan hanya jika 𝑓(2𝑥) ∈ 𝑉𝑚+1. 3. ⋂ 𝑉𝑚 = {0}𝑚∈ℤ 4. ∪𝑚∈𝑧 𝑉𝑚 = 𝐿 2(𝑅) 5. terdapat fungsi skala 𝜙(𝑥) sedemikian sehingga {𝜙0,𝑛 = 𝜙(𝑥 − 𝑛), 𝑛 ∈ ℤ} merupakan basis ortonormal untuk 𝑉0. untuk 𝑚 → ∞, 𝑉∞ = 𝐿 2 dan untuk 𝑚 → −∞ = {0} [9]. b. wavelet haar definisi fungsi scala wavelet haar (father wavelet) adalah [8]: 𝜙(𝑥) = { 1, jika 0 ≤ 𝑥 ≤ 1 0, untuk 𝑥 yang lain misal 𝑉𝑚 adalah ruang fungsi di 𝐿 2(𝑅) yang konstan pada interval [2−𝑚 𝑛, 2−𝑚 (𝑛 + 1)], 𝑘 ∈ 𝑍, maka {𝑉𝑚 : 𝑚 ∈ 𝑍} adalah sebuah analisis multiresolusi yang bersesuaian dengan wavelet haar. wavelet haar yang dikontruksi dari analisis multi-resolusinya diperoleh dari mengeneralisir (mendilasi atau menggeser) fungsi skala (father wavelet) 𝜙(𝑥) = 𝜒[ 0,1)(𝑥) yang bersesuaian dengan mra. wavelet haar memenuhi persamaan dilasi (2.6), dimana 𝐶𝑛 = √2 ∫ 𝜙(𝑥)𝜙(2𝑥 − 𝑛) ∞ −∞ 𝑑𝑥. dengan demikian diperoleh 𝐶0 = 𝐶1 = 1 √2 dan 𝐶𝑛 = 0, 𝑛 ≠ 0, 1. untuk wavelet haar atau dikenal juga daubechies 1 (db1) dengan koefisien 𝐶0 = 𝐶1 = 1 √2 merupakan high pass filter [8]. c. wavelet daubechies wavelet daubechies adalah pengembangan dari wavelet haar. daubechies 1 (db1) dengan panjang filter 2 merupakan wavelet haar. daubechies 2 disingkat (db2) adalah wavelet daubechies dengan banyak filter 4 yang db3 adalah wavelet daubechies dengan banyak filter 6 dan seterusnya. selengkapnya koefisien filter sampai db5 dapat dilihat pada tabel 4.1. d. transformasi wavelet diskrit transformasi wavelet diskrit (twd) merupakan pentransformasian sinyal diskrit menjadi koefisien-koefisien wavelet yang diperoleh dengan cara menapis sinyal menggunakan dua buah tapis yang berlawanan low pass filter dan high pass filter. secara garis besar proses dalam teknik ini adalah dengan melewatkan sinyal yang akan dianalisis pada filter dengan frekuensi dan skala yang berbeda. analisis frekuensi yang berbeda dengan menggunakan resolusi yang berbeda inilah yang disebut dengan multi-resolution analysis. pembagian sinyal menjadi frekuensi tinggi dan frekuensi rendah dalam proses filterisasi highpass filter dan lowpass filter disebut sebagai dekomposisi [10]. tabel 1. filter wavelet daubechies proses dekomposisi citra yang dimulai dengan melakukan dekomposisi melalui terhadap baris dari data citra, kemudian dilanjutkan dengan dekomposisi terhadap kolom (down sampling), seperti ditunjukkan pada gambar 1. gambar 1. transformasi wavelet citra level 1 komponen ll disebut komponen aproksimasi sementara komponen lh, hl, dan hh disebut komponen detil. dekomposisi pada transformasi wavelet untuk level 2, 3, dan seterusnya dilakukan dengan cara yang sama, hanya saja dilakukan pada bagian ll. kompresi citra menggunakan transformasi wavelet proses kompresi merupakan proses mereduksi ukuran suatu data untuk menghasilkan representasi digital yang padat atau mampat, namun tetap dapat mewakili kuantitas informasi yang terkandung pada data tersebut. pada citra, video, dan suara, kompresi mengarah pada minimalisasi jumlah bit untuk representasi citra digital. transformasi wavelet pada penelitian ini digunakan untuk mengompres matriks gambar dengan melewatkamya pada low pass filter dan high pass filter untuk mendapatkan aproksimasinya. kompresi gambar menggunakan dwt dikaitkan dengan dekomposisi gambar. dekomposisi gambar seperti yang ditunjukkan pada gambar 2 menghasilkan rentang informasi implementasi transformasi wavelet daubechies pada kompresi citra digital jurnal cauchy – issn: 2086-0382 213 frekuensi yang berbeda yaitu ll, frekuensi rendah rendah, lh, frekuensi rendah tinggi, hl, frekuensi tinggi rendah, dan hh, frekuensi tinggi tinggi [11]. gambar 2. two levels pyramid decomposition tujuan utama kompresi citra adalah untuk mengurangi data berlebihan (redundancy data), sehingga ukuran data menjadi lebih kecil dan lebih ringan dalam proses transmisi data. setelah image ditansformasi wavelet, tahapan selanjutnya dalam compresi citra adalah kuantisasi. kuantisasi adalah suatu metode yang bekerja dengan cara mengurangi derajat keabuan, sehingga jumlah bit yang dibutuhkan untuk merepresentasikan citra berkurang. prosesnya adalah dengan melakukan pengalian dengan koefisien komponen detil dari citra yang sudah ditransformasi dengan wavelet. nilai koefisiennya dikalikan dengan suatu konstanta pengali yang berkisar antara 0-1 dengan menggunakan rumus berikut [12]. 𝑘𝑜𝑒𝑓 2 = 𝑘𝑜𝑒𝑓 1 × 𝑘𝑜𝑛𝑠𝑡. 𝑝𝑒𝑛𝑔𝑎𝑙𝑖 × 1 2𝑗 dimana :  koef 2 adalah nilai koefisien komponen detil setelah dikuantisasi  koef 1 adalah nilai koefisien komponen detil sebelum dikuantisasi  j adalah level dekomposisi pengalian dengan 0 (nol) akan menghasilkan rasio kompresi yang paling besar. tetapi akibatnya akan menghasilkan gambar kompresi yang buruk, karena semua informasi dari komponen detil dihilangkan. sedangkan pengalian dengan 1 (satu) akan mengahasilkan gambar kompresi yang sangat baik. tetapi dalam hal ini bukan merupakan sistem pengkompresian, karena tidak ada proses kuantisasi. oleh karena itu koefisien komponen detil dikalikan dengan konstanta pengali dari antara 0 dan 1 [12]. tahapam-tahapan yang dilakukan dapat dilihat pada flowchart di bawah ini: gambar 3. flowchart tahapan kompresi dan dekompresi a. kualitas kompresi citra (fidelity) kualitas citra hasil kompresi dapat diukur secara kuantitatif dengan melihat nilai root mean square error (rmse) 𝑅𝑀𝑆𝐸 = √ 1 𝑀 × 𝑁 ∑ ∑ (𝑓(𝑥, 𝑦) − 𝑓(𝑥, 𝑦)) 2 𝑁 𝑦=1 𝑀 𝑥=1 dan peak signal to noise ratio (psnr) 𝑃𝑆𝑁𝑅 = 20 × 𝑙𝑜𝑔10 ( 𝑚𝑎𝑘𝑠𝑓(𝑥, 𝑦) 𝑅𝑀𝑆𝐸 ) dalam hal ini 𝑓(𝑥, 𝑦) dan 𝑓(𝑥, 𝑦) berturutturut merupakan fungsi dari citra hasil kompresi dan fungsi dari citra asli, sedangkan maks adalah nilai intensitas terbesar. semakin besar nilai psnr, maka citra hasil pemampatan semakin mendekati citra aslinya, dengan kata lain semakin bagus kualitas citra hasil kompresi tersebut [13]. untuk lossy compression, nilai psnr yang baik berkisar antara 30 dan 50 db. b. rasio kompresi citra rasio kompresi citra adalah ukuran persentase citra yang telah berhasil dimampatkan. secara matematis, rasio pemampatan citra dituliskan sebagai berikut [13]: 𝑅𝑎𝑠𝑖𝑜 = 100% − [ �̂� 𝑎 𝑥100%] keterangan : �̂� = hasil kompresi (bit) 𝑎 = citra asli (bit) c. proses dekompresi proses dekompresi ini adalah proses untuk mengembalikan citra pada ukuran semula melalui kuantisasi balik dekuantisasi dan transformasi balik (invers) wavelet. untuk melakukan proses kuantisasi balik (dekuantisasi) untuk mengembalikan data yang sebelumnya di kuantisasi saat proses kompresi. suma’inna dan gugun gumilar 214 volume 2 no. 4 mei 2013 proses ini akan memperoleh kembali komponen-komponen detil, yaitu horizontal, vertikal, dan diagonal dengan menggunakan rumus. 𝑘𝑜𝑒𝑓 1 = 𝑘𝑜𝑒𝑓 2 × 2𝑗 𝑘𝑜𝑛𝑠𝑡. 𝑝𝑒𝑛𝑔𝑎𝑙𝑖 sebelum melakukan transformasi wavelet balik, terlebih dahulu menggabungkan seluruh komponen hasil dekuantisasi dengan komponen aproksimasi kemudian dilakukan invers transformasi. pembahasan citra yang digunakan dalam penelitian ini adalah tiga buah citra yang diperoleh dari kamera handphone dengan resolusi 3.2 mp. citra tersebut diberi nama citra ukiran, patung, dan pura gambar 4. masing-masing berukuran 256 x 256 piksel. gambar 4. data citra a. transformasi wavelet maju pada tahap ini, citra pada gambar 4. ditransformasi dengan menggunakan transformasi wavelet haar dan daubechies db4 satu level. pada gambar 5a proses transformasi dilakukan 1 kali (1 level) dekomposisi. b. kuantisasi komponen detil hasil dari transformasi wavelet haar dan daubechies db4 1 level adalah 4 buah komponen matrik citra, yaitu komponen aproksimasi (ll), komponen horizontal (lh), komponen diagonal (hh), dan komponen vertikal (hl). komponen aproksimasi (ll) disimpan ke dalam media penyimpanan agar komponen ini tidak berubah. sedangkan untuk komponen komponen detil dilakukan kuantisasi, pemilihan konstanta pengali dilakukan berdasarkan variansi dari ketiga komponen detil tersebut. komponen yang mempunyai variansi yang paling besar dikalikan dengan konstanta yang paling besar pula dan sebaliknya. dalam penelitian ini digunakan konstanta pengali sebesar 0.5, 0.6, dan 0.4. ukuran komponen detil setelah proses kuantisasi dapat dilihat pada tabel 2. (a) (b) gambar 5. (a) transformasi wavelet haar 1 level (b) transformasi wavelet daubechies db4 1 level tabel 2 ukuran komponen detil sebelum dan setelah dikuantisasi dari tabel 2 dapat diketahui bahwa rasio kompresi citra dengan wavelet haar lebih tinggi dibanding wavelet daubechies db4. c. dekompresi (rekonstruksi) citra untuk mengembalikan citra terkompresi ke citra semula adalah dengan melakukan dekantisasi dan invers tyransformasi wavelet. hasil dari transformasi wavelet balik dapat dilihat pada gambar 5b. komponen detil hasil proses dekuantisasi akan berbeda dengan komponen detil sebelum kuantisasi. hal ini terjadi karena pada saat proses kuantisasi terjadi pembulatan ke bilangan terdekat. sehingga ketika komponen detil didekuantisasi, hasilnya tidak sama persis dengan komponen detil sebelum kuantisasi. walaupun demikian, citra hasil transformasi wavelet balik pada gambar 6a apabila dibandingkan dengan citra asli pada gambar 4, maka terlihat tidak ada perbedaan yang signifikan antara citra asli dengan citra rekonstruksi. implementasi transformasi wavelet daubechies pada kompresi citra digital jurnal cauchy – issn: 2086-0382 215 gambar 6 transformasi wavelet balik (a) wavelet haar (b) wavelet daubechies db4 referensi [1] sonja grgic, mislav grgic, and branka zovkocihlar performance analysis of image compression using wavelets ieee transactions on industrial electronics, vol. 48, no. 3, june 2001, 682 – 695 [2] dahmen, wolfgang c.s., “multiscale wavelet method for partial differensial equations”, academic pres, 1997. [3] mallat, stephanie, “a wavelet tour of signal processing, 2nd edition, academic press, 1999. [4] rafiee, j c.s., “wavelet basis functions in biomedical signal processing”, j. rafiee cs. 2011, expert system with applications 38, 2011, 6190–6201. [5] zhang, xiaoqun c.s., “wavelet inpainting by nonlocal total variation”, inverse problems and imaging, volume 4, no. 1, 2010, 191–210. [6] varletti, martin c.s., “fourier and wavelet signal processing”, copyright (c) 2011 by copyright under the attributionnoncommercial-noderivs 3.0 unported license from creative commons. [7] ergun gumus dkk, “evaluation of face recognition techniques using pca, wavelet and svm “, journal of expert systrem with application 37, 2010, (6404 – 6408), elsivier. [8] e. hernandez and g. weis, a first course on wavelets, crc press, new york, 1996. [9] lokenath debnath, wavelet transform and their applications, birkhäuser, boston-baselberlin, 2002. [10]terzija, nataša, robust digital image watermarking algorithms for copyright protection. universität duisburgessen, 2006 [11] graps, amara, an introduction to wavelet, institute of electrical electronics engineers, 1995. [12] gonzalez rc et al, digital image processing using matlab. new jersey: pearson prentice hall, 2002. [13]sutoyo, t, dkk. 2009, “teori pengolahan citra digital”, penerbit andi. yogyakarta hal 9 27. pendahuluan landasan teori a. transformasi wavelet b. wavelet haar c. wavelet daubechies d. transformasi wavelet diskrit kompresi citra menggunakan transformasi wavelet a. kualitas kompresi citra (fidelity) b. rasio kompresi citra c. proses dekompresi pembahasan a. transformasi wavelet maju b. kuantisasi komponen detil c. dekompresi (rekonstruksi) citra referensi suma’inna dan gugun gumilar penerapan kurva bezier karakter simetrik dan putar pada model kap lampu duduk menggunakan maple 1juhari, 2erny octafiatiningsih 1,2jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: jo_alkanderi57@yahoo.co.id, erny.octafia1710@gmail.com abstrak penelitian ini bertujuan untuk memperoleh prosedur mengkonstruksi bentuk kap lampu duduk melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan bezier. sehingga, menghasilkan kap lampu duduk secara utuh yang simetri dan bervariasi. pada pembuatan kap lampu duduk memerlukan studi tentang aspek fisis (pencahayaan) maupun geometri. dari segi geometri, model pembuatan kap lampu duduk yang telah ada pada umumnya monoton dan terbangun dari suatu model potongan benda. sehubungan dengan permasalahan tersebut maka penelitian ini dibagi menjadi empat tahap yaitu: pertama, menyiapkan data untuk membangun kap lampu duduk. kedua, studi teknik untuk membangun kesimetrian bentuk kap lampu duduk. ketiga, mengkonstruksi bagianbagian kap lampu duduk (bagian alas, bagian utama, bagian atap), dan keempat, mengkonstruksi kap lampu duduk secara utuh. hasil penelitian ini mendapatkan prosedur mengkonstruksi kap lampu duduk yaitu: pertama, membagi sumbu utama menjadi tiga sumbu sub segmen non homogen. kedua, membangun bagian-bagian dari kap lampu duduk (bagian alas, bagian utama, bagian atap) dengan cara menggabungkan komponen-komponen kap lampu duduk hasil deformasi benda-benda geometri. ketiga, mengisi setiap bagian sub segmen non homogen dengan bagian-bagian dari kap lampu dan membangun kurva batas sehingga menghasilkan model kap lampu duduk yang bervaiasi, inovasi dan simetri. kata kunci: kap lampu duduk, kura hermit, kurva bezier abstract this research aimed to obtain construction procedures lampshade form through incorporation and election of parameters shape shifter bezier surface. thus, it product a sholid lampshade that both symmetrical and varied. in contruction lampshade it requires learning about the physical (expose) and geometrical aspects. in terms of geometry model-making of lampshade sitting which has existed in general still monotone and built of object cut model. dealing with the problem, so this research is divided into four stages: firstly, prepare the data of building sitting lampshade. secondly, study about technique of building symmetrical sitting lampshade. thirdly, construct overall lampshade. the results of this research is procedures by contruction of sitting lampshade: first, the main axis split into three sub segments axis non-homogeneous. second, build parts of the sitting lampshade (the base, the main part, the roof) by combining the components lampshade deformation results geometry objects. third, fill each sub-segment of non-homogeneous parts with parts of the lampshade and build a boundary curve resulting lampshade varied models, innovation, and symmetry. keywords: bezier curve, hermit curve, standing lamp shading pendahuluan geometri merupakan cabang matematika yang mempelajari tentang garis, sudut, bidang, benda-benda ruang dan sifat-sifat serta hubungnnya dengan yang lain. geometri mempunyai banyak kegunaan dalam kehidupan sehari-hari. benda-benda yang ada di alam raya ini mempunyai bentuk geometri berbentuk bidang maupun ruang. walaupun benda-benda yang dijumpai tidak sempurna, namun dapat digambarkan atau ditunjukkan kemiripannya terhadap bangun geometri tertentu. pada perkembangannya geometri dapat digolongkan berdasarkan ruang atau bidang kajian, yaitu geometri bidang (dua-dimensi), geometri ruang (tiga-dimensi), dan geometri dimensi 𝑛. geometri bidang dan ruang dapat digunakan sebagai sarana untuk mendesain model kerajinan, seperti kap lampu, vas bunga, knop, guci, dan lain-lain. kap lampu duduk merupakan salah satu aksesoris didalam desain interior ruangan. selain berfungsi sebagai penerangan, lampu kini mengalami perkembangan dengan banyak inovasi. pada dasarnya kap lampu duduk dapat ditempatkan di setiap sudut ruangan. akan tetapi, mailto:jo_alkanderi57@yahoo.co.id mailto:erny.octafia1710@gmail.com1 erny octafiatiningsih 29 volume 4 no. 1 november 2015 tidak dapat sebarang memilih kap lampu duduk yang akan dipakai didalam ruangan. ragam model dan ukuran kap lampu duduk yang bervariasi dapat disesuaikan dengan kebutuhan ruangan. bentuk dan model yang selalu up to date dan cahayanya dapat membuat ruangan terlihat lebih indah. pembuatan kap lampu duduk memerlukan studi tentang aspek fisis (pencahayaan) maupun geometris. dari segi geometris, model pembuatan kap lampu duduk yang telah ada pada umumnya masih monoton dan terbangun dari satu model potongan benda. hal ini dapat dilihat dari produk industri kap lampu duduk yang masih sederhana dan teknik desain yang digunakan masih menggunakan cara konvensional. teknik tersebut membutuhkan waktu yang sangat lama sehingga pesanan pelanggan sering tidak selesai pada waktunya. selain itu produk yang dihasilkan pengrajin yang menggunakan teknik desain konvensional pada umumnya model yang dihasilkan tidak berubah (tetap), tidak diimbangi oleh peningkatan seni dan inovasi yang dibutuhkan oleh pelanggan yang sangat beragam ditinjau dari aspek tingkat kesimetrian, keserasian, dan variasi model maupun dari aspek ragam jenis dan ukuran barang yang ditawarkan. artikel ini merupakan upaya ilmiah untuk memperoleh model-model kap lampu duduk yang inovatif dan bervariasi. artikel ini dilakukan upaya mengknstruksi kap lampu duuk, sehingga dapat diketahui prosedur mengkonstruksi kap lampu duduk. kajian teori 1. kurva hermit kuadratik bentuk kurva hermit adalah 𝑃(𝑢) = 𝑃(0)𝐾1(𝑢) + 𝑃(1)𝐾2(𝑢) + 𝑃 ′(1)𝐾3(𝑢) (1) dengan: 𝐾1(𝑢) = (1 − 2𝑢 + 𝑢 2) 𝐾2(𝑢) = (2𝑢 − 𝑢 2) 𝐾3(𝑢) = (−𝑢 + 𝑢 2) 𝑃(0) = titik awal kurva berbentuk (𝑥0, 𝑦0, 𝑧0) 𝑃(1) = titik akhir kurva berbentuk (𝑥1, 𝑦1, 𝑧1) 𝑃′(1) = titik kontrol kelengkungan kurva dengan 0 ≤ 𝑢 ≤ 1 [1] 2. kurva bezier berderajat dua kurva bezier berderajat-dua dinyatakan dalam bentuk parametrik yaitu: 𝑉(𝑢) = 𝐾0(1 − 2𝑢 + 𝑢 2) + 𝐾1(2𝑢 − 2𝑢 2) + 𝐾2(𝑢 2) ( 2) dengan 0 ≤ 𝑢 ≤ 1 [2] contoh: diketahui 𝐾0 = (5,0,0); 𝐾1 = (10,0,3) dan 𝐾2 = (7,0,5), maka kurva bezier berderajat dua dapat dinyatakan sebagai berikut: 𝑉(𝑢) = (5,0,0)(1 − 2𝑢 + 𝑢2) + (10,0,5)(2𝑢 − 2𝑢2) + (7,0,5)(𝑢2) = (5(1 − 2𝑢 + 𝑢2), 0(1 − 2𝑢 + 𝑢2), 0(1 − 2𝑢 + 𝑢2)) + (10(2𝑢 − 2𝑢2), 0(2𝑢 − 2𝑢2), 5(2𝑢 − 2𝑢2)) + (7(𝑢2), 0𝑢2, 5(𝑢2)) = (5 − 10𝑢 + 5𝑢2, 0,0) + (20𝑢 − 20𝑢2, 0,10𝑢 − 10𝑢2) + (7𝑢2, 0,5𝑢2) = (5 + 10𝑢 − 8𝑢2, 0 ,6𝑢 − 𝑢2) dengan 0 ≤ 𝑢 ≤ 1 sehingga diperoleh kurva seperti pada (gambar 1). gambar 1. kurva bezier berderajat dua 3. perputaran (rotasi) rotasi adalah perubahan dari suatu koordinat objek ke dalam kedudukan baru dengan menggerakkan seluruh titik koordinat yang didefinisikan pada bentuk awal dengan suatu besaran sudut pada suatu sumbu putar. jika 𝑄(𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) adalah posisi setelah rotasi pada sumbu putar, 𝑃(𝑥𝑝, 𝑦𝑝, 𝑧𝑝 ) adalah posisi awal sebelum dilakukan rotasi, dan 𝑅 adalah matriks rotasi pada suatu sumbu putar. sistem koordinat 𝑅3 mempunyai tiga sumbu putar, maka rotasi setiap sumbu dapat ditulis sebagai berikut, dengan 𝜃 menunjukkan sudut putar. a. rotasi terhadap sumbu 𝑋 (𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) = (𝑥𝑝, 𝑦𝑝 ⋅ cos 𝜃 − 𝑧𝑝 ⋅ sin 𝜃 , 𝑦𝑝 ⋅ sin 𝜃 + 𝑧𝑝 ⋅ cos 𝜃) ( 3) b. rotasi terhadap sumbu 𝑌 (𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) = (𝑥𝑝 ⋅ cos 𝜃 + 𝑧𝑝 ⋅ sin 𝜃 , 𝑦𝑝 𝑥𝑝 ⋅ (−sin 𝜃) + 𝑧𝑝 ⋅ cos 𝜃) ( 4) c. rotasi terhadap sumbu 𝑍 (𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) = (𝑥𝑝 ⋅ cos 𝜃 + 𝑦𝑝 ⋅ sin 𝜃, 𝑥𝑝 ⋅ (− sin 𝜃) + 𝑦𝑝 ⋅ cos 𝜃 , 𝑧𝑝) ( 5) [3] 1 k 0 k 2 k penerapan kurva bezier karakter simetrik dan putar pada model kap lapu duduk menggunakan maple cauchy – issn: 2086-0382/e-issn: 2477-3344 30 4. pergeseran jika 𝑃(𝑥𝑝, 𝑦𝑝, 𝑧𝑝) adalah posisi titik asal, 𝑄(𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) adalah posisi setelah titik digeser, 𝐼 adalah matriks identitas, dan (𝑡𝑟𝑥 , 𝑡𝑟𝑦 , 𝑡𝑟𝑧 ) merupakan nilai konstanta yang menunjukkan besarnya penggeseran pada setiap sumbu koordinat, maka hasil penggeseran dapat dinyatakan sebagai berikut [3]. (𝑥𝑞 , 𝑦𝑞 , 𝑧𝑞 ) = (𝑥𝑝 + 𝑡𝑟𝑥 , 𝑦𝑝 + 𝑡𝑟𝑦 , 𝑥𝑝 + 𝑡𝑟𝑥 ) ( 6) 5. interpolasi di antara segmen garis dan kurva di ruang misalkan terdapat dua segmen garis 𝐴𝐵̅̅ ̅̅ dan 𝐶𝐷̅̅ ̅̅ didefinisikan masing-masing oleh 𝐴(𝑥1, 𝑦1, 𝑧1), 𝐵(𝑥2, 𝑦2, 𝑧3 ), 𝐶(𝑥3, 𝑦3, 𝑧3) dan 𝐷(𝑥4, 𝑦4, 𝑧4) dalam bentuk parametrik 𝐼1(𝑢) dan 𝐼2(𝑢), maka permukaan parametrik hasil interpolasi linier kedua segmen garis tersebut diformulasikan, 𝑆(𝑢, 𝑣) = (1 − 𝑣)𝐼1(𝑢) + 𝑣𝐼2(𝑢) ( 7) dengan 0 ≤ 𝑢 ≤ 1 dan 0 ≤ 𝑣 ≤ 1 terdapat beberapa kasus khusus bentuk interpolasi linier kedua garis tersebut. jika 𝐴 = 𝐵 maka hasil interpolasi persamaan (7) akan menghasilkan bidang segitiga. jika 𝐴𝐵̅̅ ̅̅ ∕∕ 𝐶𝐷̅̅ ̅̅ maka secara umum akan membentuk bidang segiempat. jika bidang tersebut dibentuk dari interpolasi dua garis yang bersilang maka menghasilkan permukaan yang tidak datar (dapat berbentuk lengkung maupun puntiran) di sebagian permukaan tersebut [4] dapat dibangun permukaan lengkung hasil interpolasi kurva ruang melalui persamaan berikut: 𝑆(𝑢, 𝑣) = (1 − 𝑣)𝐶1(𝑢) + 𝑣𝐶2(𝑢) ( 8) dengan 𝐶1(𝑢) dan 𝐶2(𝑢) merupakan kurva batas (gambar 3) [4] gambar 2. contoh kasus khusus interpolasi linier dua segmen garis gambar 3. interpolasi linier pada kurva 6. dilatasi titik pada 𝑹𝟑 transformasi dilatasi yang memetakkan titik 𝑃(𝑥, 𝑦, 𝑧) ke 𝑃′(𝑥 ′, 𝑦 ′, 𝑧′) didefinisikan dengan bentuk formula berikut: [ 𝑥 ′ 𝑦 ′ 𝑧′ ] = [ 𝑘1 0 0 0 𝑘2 0 0 0 𝑘3 ] [ 𝑥 𝑦 𝑧 ] ( 9) dalam hal ini pemilihan nilai 𝑘1 menyajikan sekala ke arah sumbu 𝑋, 𝑘2 ke arah sumbu 𝑌 dan 𝑘3 menyajikan skala ke arah sumbu 𝑍, jika 𝑘1 = 𝑘2 = 𝑘3, maka peta objek yang didapat sebangun dengan objek aslinya (diperbesar, diperkecil; atau tetap) [1]. misalkan segitiga 𝑃𝑄𝑅 dengan titik-titik sudut 𝑃(𝑥1, 𝑦1, 𝑧1), 𝑄(𝑥2, 𝑦2, 𝑧2) dan 𝑅(𝑥3, 𝑦3, 𝑧3) didilatasikan dengan faktor pengali 𝑘 > 1, sehingga didapatkan segitiga bayangan 𝑃′𝑄′𝑅′ dengan titik-titik sudut 𝑃′(𝑘𝑥1, 𝑘𝑦1, 𝑘𝑧1), 𝑄′(𝑘𝑥2, 𝑘𝑦2 , 𝑘𝑧2) dan 𝑅 ′(𝑘𝑥3, ᡐ𝑦3, 𝑘𝑧3) seperti terlihat pada (gambar 4) gambar 4. kurva bezier berderajat dua 7. penyajian benda-benda geometri ruang a. penyajian tabung jika diketahui tabung dengan pusat alas 𝑃1(𝑥1, 𝑦1, 𝑧1) dengan jari-jari 𝑟 dan tinggi 𝑡 dapat di gambar dengan menggunakan persamaan sebagai berikut: jika sumbu pusat tabung sejajar sumbu 𝑍 𝑇(𝜃, 𝑧) = (𝑥1 + 𝑟 cos 𝜃 , 𝑦1 + 𝑟 sin 𝜃 , 𝑧) ( 10) dengan 0 ≤ 𝜃 ≤ 2𝜋 dan r ∈ ℝ jika sumbu pusat tabung sejajar smbu 𝑋 𝑇(𝜃, 𝑧) = (𝑥, 𝑦1 + 𝑟 sin 𝜃 , 𝑧1 + 𝑟 cos 𝜃) ( 11) dengan 0 ≤ 𝜃 ≤ 2𝜋 dan 𝑥1 < 𝑥 ≤ 𝑥1 + 𝑡 jika sumbu pusta tabung sejajar sumbu 𝑌 𝑇(𝜃, 𝑧) = (𝑥1 + 𝑟 cos 𝜃 , 𝑦, 𝑧1 + 𝑟 sin 𝜃) ( 12) dengan 0 ≤ 𝜃 ≤ 2𝜋 dan 𝑦1 ≤ 𝑦 ≤ 𝑦1 + 𝑡 [5]. gambar 5. tabung dengan beragam sumbu pusat b. penyajian bola p z x y r q 'p 'r 'q erny octafiatiningsih 31 volume 4 no. 1 november 2015 menyatakan persamaan parametrik bola dengan pusat 𝑄(𝑎, 𝑏, 𝑐) dan jari-jari 𝑟, yaitu: 𝐵(𝜃, 𝛽) = (𝑟 sin 𝜃 cos 𝛽 + 𝑎, 𝑟 cos 𝜃 + 𝑐, 𝑟 sin 𝜃 sin 𝛽 + 𝑏) ( 13) dengan paramer 0 ≤ 𝜃 ≤ 2𝜋 dan 0 ≤ 𝛽 ≤ 2𝜋 [5]. 8. konstruksi objek pada program maple disajikan contoh bahasa pemrograman menggunakan softwere maple 15 untuk mengkonstruksi objek geometri. a. mengkonstruksi segmen garis untuk membangun segmen garis 𝐴𝐵̅̅ ̅̅ dengan titik 𝐴(3,2,4) dan titik 𝐵(9,8,12) pada maple 15 (gambar 2.23), dengan menggunakan persamaan sebagai berikut: 𝑃(𝑢) = 𝑎 + 𝑏𝑢 𝑃(0) = 𝑎 𝑃(1) = 𝑎 + 𝑏 sehingga 𝑃(𝑢) = 𝑃(0) + 𝑃(1)𝑢 dimana 𝑃(0) = titik awal kurva 𝑃(1) = titik akhir kurva (𝑥, 𝑦, 𝑧) = (𝑥𝐴 + 𝑢 ⋅ (𝑥𝐵 − 𝑥𝐴), 𝑦𝐴 + 𝑢 ⋅ (𝑦𝐵 − 𝑦𝐴 ), 𝑧𝐴 + 𝑢 ⋅ (𝑧𝐵 − 𝑧𝐴)) maka dapat ditulis dengan script program yaitu, >plot3d([3+u*(9-3),2+u*(8-2),4+u*(124)],u=0..1); gambar 6. segmen garis pada maple 15 b. megkonstruksi tabung tabung adalah sebuah bidang yang dibentuk oleh lingkaran yang berjari-jari 𝑟 dan bergerak secara paralel pada sumbu pusat sepanjang 𝑡. contoh penulisan script pada maple 15 untuk mengkonstruksi tabung dengan menggunakan persamaan (11) seperti pada (gambar 7) adalah: >plot3d([4*cos(u)+4,4*sin(u)+4,4*v],u=0..2 *pi,v=0..4); tabung terbentuk dari bidang lingkaran berpusat di 𝑥 = 4 dan 𝑦 = 4 dan 𝑟 = 4 dengan ketinggian 𝑧 = 4𝑣 dengan 𝑣 interval dari 0 sampai 4. gambar 7. tabung pada maple 15 c. mengkonstruksi bola untuk mengkonstruksi bola dengan jari-jari 3 dan berpuata di titik (1,4,3) dengan menggunakan persamaan (13) seperti pada (gambar 8) pada maple 15 adalah: >plot3d([3*sin(v)*cos(u)+1, 3*sin(v)*sin(u)+4, 3*cos(v)+3], u = 0 .. 2*pi, v = 0 .. 2*pi); gambar 8. bola pada maple 15 pembahasan di dalam artikel ini dijelaskan bagaimana prosedur untuk mengkonstruksi kap lampu duduk. prosedur yang pertama yaitu mengkonstruksi komponen-komponen kap lampu duduk duduk yang akan digunakan. langkah selanjutnya yaitu mengkonstruksi bagian-bagian kap lampu duduk yaitu bagian alas, bagian utama dan bagian atap. prosedur yang terakhir yaitu mengkonstruksi kap duduk secarah utuh. a. mengkonstruksi kompnen-komponen kap lampu duduk 1. mendeformasi tabung misalkan diberikan tabung yang berjari-jari 𝑟, batas minimum jari-jari tabung yaitu 10 cm sedangkan batas maximum jari-jari tabung yaitu 20 cm, tinggi minimum tabung yaitu 10 cm sedangkan tinggi maximum tabung yaitu 30 cm, dan alas berpusat di 𝑃(𝑥0, 𝑦0, 𝑧0). sehingga, selang ukuran tabung yaitu 10 cm ≤ 𝑡 ≤ 30 cm dan 10 cm ≤ 𝑟 ≤ 20 cm. pemilihan nilai 𝑟 dan 𝑡 dalam selang tersebut bertujuan untuk perbedaan ukuran bentuk komponen penyusun kap lampu duduk. berdasarkan data tersebut didesain beragam bentuk komponen penyusun kap lampu duduk menggunakan teknik modifikasi kurva selimut dan teknik dilatasi lengkung selimut. algoritma untuk mendoformasi tabung dengan modifikasi pada kurva selimut adalah sebagai berikut: a. ditentukan titik pusat pada lingkaran alas tabung yaitu (𝑥1, 𝑦1, 𝑧1) = (0,0,0), bangun lingkaran alas tabung dengan menggunakan persamaan lingkaran dan menetapkan nilai 𝜃 = 0 sehingga didapat satu titik yaitu 𝑃(0). 𝑃(0) = (𝑥1 + 𝑟1 cos 𝜃 , 𝑦1 + 𝑟1 sin 𝜃 , 𝑧1) = (𝑟1, 0,0) b. ditentukan titik pusat pada lingkaran atap tabung yaitu (𝑥1, 𝑦1, 𝑧1 ) = (0,0, 𝑡), bangun lingkaran atap tabung dengan dengan menggunakan persamaan lingkaran dan menetapkan nilai 𝜃 = 0 sehingga didapatkan satu titik yaitu 𝑃(1). 𝑃(1) = (𝑥1 + 𝑟2 cos 𝜃 , 𝑦1 + 𝑟2 sin 𝜃 , 𝑧1) = (𝑟2, 0, 𝑡) penerapan kurva bezier karakter simetrik dan putar pada model kap lapu duduk menggunakan maple cauchy – issn: 2086-0382/e-issn: 2477-3344 32 c. ditentukan titik kontrol 𝑃′(1) untuk mengontrol kelengkungan kurva hermit sehingga 𝑃′(1) = (𝑥, 0, 𝑧) dengan −2𝑟 ≤ 𝑥, 𝑧 ≤ 2𝑡 dan 𝑥, 𝑧 ∈ ℝ. d. kurva hermit dibangun dengan mensubtitusikan nilai 𝑃(0), 𝑃(1) dan 𝑃′(1) ke persamaan (1) sehingga didapat 𝑃(𝑢) = (𝑟𝐻1 (𝑢) + 𝑟𝐻2 (𝑢) + 𝑥𝐻3 (𝑢), 0𝐻1 (𝑢) + 0𝐻2 (𝑢) + 0𝐻3 (𝑢), 0𝐻1 (𝑢) + 𝑡𝐻2 (𝑢) + 𝑧𝐻3 (𝑢) ) dengan 𝐻1(𝑢) = (1 − 𝑢 − 𝑢 2) 𝐻2(𝑢) = (2𝑢 − 𝑢 2) 𝐻3(𝑢) = (−𝑢 + 𝑢 2) 0 ≤ 𝑢 ≤ 1 e. kurva hermit diputar terhadap sumbu 𝑍 dengan menggunakan persamaan (5) dan 0 ≤ 𝜃 ≤ 3600 sehingga 𝑃(𝑢) = ((𝑟𝐻1(𝑢) + 𝑟𝐻2(𝑢) + 𝑥𝐻3(𝑢)) cos 𝜃, (𝑟𝐻1(𝑢) + 𝑟𝐻2(𝑢) + 𝑥𝐻3(𝑢)) sin 𝜃, 0𝐻1(𝑢) + 𝑡𝐻2(𝑢) + 𝑡𝐻3(𝑢)) gambar 9. variasi hasil deformasi tabung 2. mendeformas prisma segi enam misalkan diberikan prisma segienam beraturan dengan koordinat pasangan titik ujung rusuk [𝐾𝑖(𝑥𝑖, 𝑦𝑖 , 𝑧𝑖 ), 𝐾𝑖 ′ (𝑥𝑖, 𝑦𝑖 , 𝑧𝑖 + 𝑡)] dengan 𝑖 = 1,2, 3, … ,6 dan tinggi 𝑡 dengan 5 cm ≤ 𝑡 ≤ 15 cm. masing-masing tutupnya bertitik berat dititik 𝐾(𝑥0, 𝑦0, 𝑧0) dan 𝐾 ′(𝑥0, 𝑦0, 𝑧0 + 𝑡). jarak titik k ke 𝐾𝐼 dan 𝐾′ ke 𝐾𝑖 ′ adalah 6 cm ≤ 𝑟 ≤ 10 cm. dalam hal ini, 𝐾𝐾′̅̅ ̅̅ ̅ diambil sebagai sumbu simetri deformasi prisma segienam. langkah-langkah deformasi sisi tegak prisma menjadi lengkung cekung yaitu: a. ditentukan 𝐾𝑖 dan 𝐾𝑖 ′ dengan 𝑖 = 0,1,2,3,4,5 sebagai titik kontrol untuk beberapa kurva bezier linier dengan menggunakan persamaan lingkaran dengan menetapkan 𝜃 = 𝑖𝜋 3 dan (𝑥1, 𝑦1, 𝑧1) = (0,0,0). 𝐾𝑖(𝜃) = (𝑟 cos 𝑖𝜋 3 , 𝑟 sin 𝑖𝜋 3 , 0) 𝐾𝑖 ′(𝜃) = (𝑟 cos 𝑖𝜋 3 , 𝑟 sin 𝑖𝜋 3 , 𝑡) b. ditetapkan titik kontrol 𝑄 untuk mengontrol kelengkungan kurva bezier kuadratik 𝑄 = (𝑥0, 𝑦0, 𝑧) dengan 𝑧 ∈ [𝑧0, 𝑡]. c. kurva bezier berderajat dua untuk setiap pasang titik kontrol (𝐾𝑖 , 𝑄, 𝐾𝑖 ′) dibangun dengan menggunakan persamaan (2) 𝑉𝑖 (𝑢) = (1 − 𝑢) 2𝐾𝑖 + 2(1 − 𝑢)(𝑢)𝑄 + 𝑢 2𝐾𝑖 ′ dengan 0 ≤ 𝑢 ≤ 1. d. diinterpolasikan secara linier masing-masing kurva bezier melalui persamaan (8) secara berpasangan dan berurutan berlawanan arah jarum jam s(u, v) = ((1 − 𝑣)(1 − 𝑢)2 𝑟 cos 𝑖𝜋 3 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑥0 + (1 − 𝑣)𝑢2𝑟 cos 𝑖𝜋 3 + 𝑣(1 − 𝑢)2𝑟 cos (𝑖+1)𝜋 3 + 𝑣(2𝑢 − 2𝑢2)𝑥0 + 𝑣𝑢2 𝑟 cos (𝑖+1)𝜋 3 , (1 − 𝑣)(1 − 𝑢)2𝑟 sin 𝑖𝜋 3 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑦0 + (1 − 𝑣)𝑢 2𝑟 sin 𝑖𝜋 3 + 𝑣(1 − 𝑢)2 𝑟 sin (𝑖+1)𝜋 3 + 𝑣(2𝑢 − 2𝑢2)𝑦0 + 𝑣𝑢 2 𝑟 sin (𝑖+1)𝜋 3 , (1 − 𝑣)(1 − 𝑢)20 + (1 − 𝑣)(2𝑢 − 2𝑢2)𝑧 + (1 − 𝑣)𝑢20 + 𝑣 0(1 − 𝑢)2 + 𝑣(2𝑢 − 2𝑢2)𝑧 + 𝑣0𝑢2) dengan 0 ≤ 𝑣 ≤ 1 dan 0 ≤ 𝑢 ≤ 1 gambar 10. variasi hasil deformasi tabung b. mengkonstruksi bagian-bagian kap lampu dudu sebelum dilakukan perangkaian bagianbagian kap lampu duduk yang dilakukan terlebih dahulu yaitu menetapkan sumbu sepanjang tinggi kap lampu duduk kemudian membagi segmen tersebut menjadi tiga sub segmen non-homogen yaitu 𝑆3𝑆4̅̅ ̅̅ ̅̅ bagian alas, 𝑆2𝑆3̅̅ ̅̅ ̅̅ bagian utama, dan 𝑆1𝑆2̅̅ ̅̅ ̅̅ bagian atap. 1. merangkai alas kap lampu duduk langkah-langkah penyusunan alas kap lampu duduk: a. bagian 𝑆3𝑆4̅̅ ̅̅ ̅̅ diisi dengan komponenkomponen kap lampu duduk. b. dipilih parameter-parameter pengubah bentuk permukaan seperti ukuran dan titik kontrol kelengkungan sehingga didapat bentuk kap lampu duduk bagian alas yang diinginkan. c. dibangun bidang tutup bawah dengan prosedur sebagai berikut. 1) ditetapkan lingkaran tutup bawah bagian alas dengan jari-jari 𝑟1; erny octafiatiningsih 33 volume 4 no. 1 november 2015 2) dibangun bola dengan jari-jari 𝑟1 dengan 𝑧 = 0 sebagai tutup bawah dengan persamaan (13). gambar 11. beberapa variasi alas kap lampu duduk 2. mengkonstruksi bagian utama kap lampu duduk langkah-langkah penyusunan bagian utama kap lampu duduk yang terbentuk dari beberapa benda dasar sebagi berikut: a. segmen garis 𝑆2𝑆3̅̅ ̅̅ ̅̅ dibagi menjadi beberapa sub segmen. b. setiap sub segmen diisi dengan komponenkomponen kap lampu duduk. 1) sub segmen bagian bawah diisi dengan komponen kap lampu duduk. 2) dipilih parameter pengubah bentuk permukaan komponen kap lampu duduk seperti ukuran dan titik kontrol kelengkungan sehingga didapat komponen kap lampu duduk sesuai keinginan. 3) dilakukan langkah 1) dan 2) untuk mengisi sub segmen selanjutnya kemudian translasikan komponen tersebut searah sumbu 𝑍 sejauh panjang sub segmen sebelumnya. 4) dilakukan langkah 1) sampai 3) untuk mengisi sub segmen selanjutnya hingga bagian sub segmen dari segmen garis 𝑆2 𝑆3̅̅ ̅̅ ̅̅ terisi semua. c. beberapa bangun komponen bagian utama kap lampu duduk digabung dengan cara membangun bidang batas antara dua komponen berdekatan dengan prosedur sebagai berikut. 1) ditetapkan lingkaran tutup atas bagian komponen kap lampu duduk yang pertama dengan jari-jari 𝑟1 sebagai kurva batas 𝐶1(𝑢). 2) ditetapkan lingkaran tutup bawa bagian komponen kap lampu duduk yang kedua dengan jari-jari 𝑟2 sebagai kurva batas 𝐶2(𝑢). 3) dibangun bidang batas antara 𝐶1(𝑢) dan 𝐶2(𝑢) dengan interpolasi linier menggunaan persamaan (8). 4) dilakukan langkah (a) sampai (c) untuk membangun bidang batas antara bagian komponen-komponen utama kap lampu duduk. gambar 12. variasi bagian utama 3. mengkonstruksi bagian atap kap lampu duduk langkah-langkah penyusunan atap kap lampu duduk: a. bagian 𝑆1𝑆2̅̅ ̅̅ ̅̅ diiisi dengan komponenkomponen kap lampu duduk. b. dibangun bidang tutup atas dengan prosedur sebagai berikut. jika permukaan atas berbentuk lingkaran. 1) ditetapkan lingkaran tutup atas bagian alas dengan jari-jari 𝑟1. 2) dibangun bola dengan jari-jari 𝑟1 dengan 𝑧 = 0 tutup bawah dengan persamaan (13). jika permukaan atas berbentuk segienam. 1) ditetapkan titik kontrol 𝐾𝑖 dengan 𝑖 = 0, 1, 2, 3, 4, dan 5 pada poligon segienam bagian atas atap kap lampu duduk dengan menggunakan persamaan (10), menetaapkan 𝜃 = 𝑖𝜋 3 dan 𝑟 merupakan jarak antara titik 𝑆1 dengan 𝐾𝑖 sehingga 𝐾𝑖 = (𝑟 cos 𝑖𝜋 3 , 𝑟 sin 𝑖𝜋 3 , 𝑡) 2) diinterpolasikan secara linier masingmasing pasangan titik kontrol dengan menggunakan persamaan (8) sehingga 𝑆(𝑢, 𝑣) = ((1 − 𝑣) 𝑟 cos 𝑖𝜋 3 + (1 − 𝑣) 𝑢 𝑟 cos (𝑖+1)𝜋 3 + (1 − 𝑣) 𝑢 𝑟 cos 𝑖𝜋 3 + 0𝑣, (1 − 𝑣)𝑟 sin 𝑖𝜋 3 + (1 − 𝑣) 𝑢 𝑟 sin (𝑖+1)𝜋 3 + (1 − 𝑣) 𝑢 𝑟 sin 𝑖𝜋 3 + 0𝑣, (1 − 𝑣)𝑡 + (1 − 𝑣) 𝑢 𝑡 + (1 − 𝑣) 𝑢 𝑡 + 𝑡𝑣) penerapan kurva bezier karakter simetrik dan putar pada model kap lapu duduk menggunakan maple cauchy – issn: 2086-0382/e-issn: 2477-3344 34 gambar 13. variasi bagian utama c. mengkonstruksi kap lampu secarah utuh selanjutnya untuk mendapatkan bentuk utuh kap lampu duduk yang tergabung secara kontinyu pada bagian ini dilakukan perangkaian beberapa benda-benda dasar komponen kap lampu duduk dengan cara menggabungkan komponen-komponen bagian dari kap lampu duduk. langkah-langkah perangkaian kap lampu duduk secara utuh: 1. bagian segmen garis 𝑆3𝑆4̅̅ ̅̅ ̅̅ diisi dengan komponen bagian alas kap lampu duduk. 2. bagian segmen garis 𝑆2𝑆3̅̅ ̅̅ ̅̅ diisi dengan komponen bagian utama kap lampu duduk dan sesuaikan tingginya. 3. bagian segmen garis 𝑆2𝑆1̅̅ ̅̅ ̅̅ diisi dengan komponen bagian atap kap lampu duduk dan sesuaikan tingginya. 4. beberapa komponen bagian kap lampu duduk digabung dengan cara membangun bidang batas antara dua komponen berdekatan dengan prosedur sebagai berikut. a. ditetapkan lingkaran atau poligon segienam beraturan tutup atas bagian alas komponen kap lampu duduk dengan jarijari 𝑟1 sebagai kurva batas 𝐶1(𝑢). b. ditetapkan lingkaran atau poligon segienam beraturan tutup bawa bagian utama komponen kap lampu duduk dengan jari-jari 𝑟2 sebagai kurva batas 𝐶2(𝑢). c. bidang batas antara 𝐶1(𝑢) dan 𝐶2(𝑢) dibangun dengan interpolasi linier menggunaan persamaan (8). d. dilakukan langkah (a) sampai (c) untuk membangun bidang batas antara komponen utama kap lampu duduk dengan komponen atap kap lampu duduk. gambar 14. variasi bagian utama kesimpulan berdasarkan hasil penelitian, diperoleh bahwa: 1. merangkai komponen kap lampu duduk secarah utuh yaitu mengkonstruksi komponen-komponen kap lampu duduk, mengkonstruksi bagian-bagian dari kap lampu duduk (bagian alas, bagian utama dan bagian atap), langkah yang terakhir yaitu merangkai komponen kap lampu duduk secarah utuh. 2. pemilihan parameter-parameter dan titik kontrol kelengkungan kurva akan menghasilkan bentuk-bentuk kelengkungan kurva yang berbeda sehingga bentuk kap lampu yang diperoleh bervariatif. daftar pustaka [1] kusno, geometri rancang bangun studi tentang desain dan pemodelan benda dengan kurva dan permukaan berbantu komputer, jember: jember university press, 2010. [2] kusno, a. cahaya and m. darsin, "modelisasi benda onyx dan marmer melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan putar bezier," jurnal ilmu dasar, pp. 175-185, 2014. [3] a. cristiyyanto, "perancangan objek tiga dimensi dengan teknik flat shading dan gouraund menggunakan bahasa turbo pascal 7.0," skripsi tidak dipublikasikan, 2003. [4] m. roifah, "modelisasi knop melalui penggabungan benda dasar hasl deformasi tabung, prisma segienam beraturan dan permukaan putar," skripsi tidak dipublikasikan, 2013. [5] a. bastian, "desain kap lampu duduk melalui penggabungan benda-benda geometi ruang," skripsi tidak dipublikasikan, 2010. pendahuluan kajian teori pembahasan kesimpulan daftar pustaka diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m volume 5, issue 1, november 2017 jurnal matematika murni dan aplikasi malang november 2017 cauchy no. 1 pages: 1-41 vol. 5 issn 2086-0382 e-issn 2477-3344 issn 2086-0382 e-issn 2477-3344 cauchy  editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : dr mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno kusno, university of jember, indonesia slamin slamin, university of jember, indonesia dr riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia elly susanti, universitas islam negeri 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by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we welcome authors for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics jurnal matematika murni dan aplikasi volume 5, issue 1, november 2017 issn 2086-0382 e-issn 2477-3344 cauchy indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. sherpa/romeo (2016-,)-(http://www.sherpa.ac.uk/romeo/search.php) 3. infobase index (2016-,)-(http://infobaseindex.com/journal-accepted.php) 4. index copernicus (2015-,)-(http://journals.indexcopernicus.com/form/login.php) 5. 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service (2014-,)-(http://www.sindexs.org/journallist.aspx?id=2193) 28. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 29. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 30. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsnf6omofbk7q0o2q-9xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 31. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) jurnal matematika murni dan aplikasi volume 5, issue 1, november 2017 issn 2086-0382 e-issn 2477-3344 cauchy table of contents applied hierarchical cluster analysis with average linkage algoritm .................................................................................. 1-7 estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency .................................................................................................................................................................................. 8-14 geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data ............................................................................................................................................ 15-19 local stability analysis of an svir epidemic model ................................................................................................................... 20-28 modelling of multi input transfer function for rainfall forecasting in batu city ...................................................... 29-35 the simulation study to test the performance of quantile regression method with heteroscedastic error variance............................................................................................................................................................................................................ 36-41 cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 29-35 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 11 july 2017 reviewed: 12 july 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.4288 modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama, ni wayan surya wardhani, rahma fitriani department of statistics, brawijaya university, malang email: priska.arindya@gmail.com, wswardhani@yahoo.com, rahmafitriani@ub.ac.id abstract the aim of this research is to obtain the appropriate transfer function to model rainfall data in batu city and to figure out how accurate the results of rainfall forecasting in batu city using the transfer fu nction model based on air temperature, humidity, wind speed and cloud. transfer function model is a multivariate time series model which consists of an output series (yt) sequence expected to be affected by an input series (xt) and other inputs in a group called a noise series (nt). air temperature, humidity, wind speed and cloud are used as an input series (xt) and rainfall as an output series (yt). multi input transfer function model obtained is (𝑏1, 𝑠1, 𝑟1) (𝑏2, 𝑠2, 𝑟2) (𝑏3, 𝑠3, 𝑟3) (𝑏4, 𝑠4, 𝑟4) (𝑝𝑛 ,𝑞𝑛 ) = (0,0,0) (23,0,0) (1,2,0) (0,0,0) ([5,8],2) and shows that rainfall on a certain day is affected by air temperature and cloud on that day, air humidity in the previous 23 days and wind speed in the previous day. the results of rainfall forecasting in batu city using multi input with single output transfer function model is accurate based on the result of model validation using 𝑡 test and mape statistic that less than 20%. keywords: multi input transfer function model, rainfall, forecasting introduction time series analysis is a data analysis considering the effect of time. time series or time series data is a sequence of observations taken sequentially based on time with the same interval, ie daily, weekly, monthly, yearly or other time periods [1]. the modeling in time series analysis is classified into two, ie univariate and multivariate. one univariate time series model is autoregressive integrated moving average (arima), whereas one multivariate time series model is a transfer function model. the transfer function model consists of an output series (yt) that is influenced by the input series (xt) and other inputs combined in a group called the noise series (nt) [2]. the transfer function model combines several characteristics of the univariate arima model and multiple regression analysis or combines time series analysis with a causal approach. the purpose of modeling the transfer function is to establish a simple model that can connect the output series (yt) with the input series (xt) and the noise series (nt). some forecasting research using transfer function model is done by tankersley, et. al. [3] about groundwater fluctuations in florida showed that the transfer function model is more accurate than arima. while edlurd and karlsson [4] who studied the unemployment rate in sweden showed that the forecasting results of the transfer function model is better when compared with other classical forecasting models such as vector autoregressive (var) and mailto:priska.arindya@gmail.com mailto:wswardhani@yahoo.com mailto:rahmafitriani@ub.ac.id modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 30 arima. in addition thomakos and geurard [5] who examined unemployment forecasting in st. louis united states showed that the transfer function model is better when compared to the naive model and arima. rainfall data information becomes an important thing that affects the agricultural output to determine the beginning of the growing season. one of the areas in east java province that has potential in the agricultural sector is batu city, so it is important to link rainfall with the agricultural sector in batu city. the agricultural sector is a leading sector which is expected to synergize with other sectors such as tourism, trade and industry [6] . some factors that can affect rainfall are air temperature, humidity, wind speed and cloud [7]. some previous research on rainfall forecasting is done by faulina [8] about comparison of accuracy of ensemble arima in batu city and tresnawati, et. al. [9] examined the kalman filter method with sst nino 3.4 predictors in purbalingga. research of rainfall forecasting on preceding still has a weakness that only involves one variable. while research that has been done by badan meteorologi, klimatologi dan geofisika (bmkg) in rainfall forecasting uses ensemble method (combination of individual method using concept of merging some forecasting model) and bma (bayesian model averaging) modelling but the forecasting rarely gives the accurate results. therefore, in this research will discuss rainfall forecasting involving more than one variable. air temperature, humidity, wind speed and clouds can be used to model and forecast rainfall using the transfer function model. in this study, rainfall is a response variable (output series), while air temperature, air humidity, wind speed and cloud are predictor variables (input series). the transfer function model used in this research is called the multi-input transfer function model, because the predictor variable or input series is used more than one variable (input). the existence of factors that affect the rainfall as a predictor variable (input series) is expected to provide accurate forecasting results. fundamental theories the steps of formation of the transfer function model consists of five steps, namely prewhitening process of input series and output series, identification of impulse response function and transfer function, identification of noise model, prediction and testing of parameter significance of transfer function model, and diagnostic test of transfer function model [10]. prior to the prewhitening process, the first step is to prepare the input series and the output series. in this preparation step, the identification of time series plots, stationarity test and arima modeling of input series and output series is conducted. stationarity test consists of two parts, namely stationary to the variance and the mean. stationarity test against variance can be seen from the box-cox plot and the value of λ. the data is said to be stationary to the range if the value of λ is equal to or near 1. the non-stationary time series data can be overcome by doing the box-cox transformation. while the stationarity test against the mean can be seen from the autocorrelation plot (acf). the stationary data on the mean on the acf plot will not be significant after the second or third lag. time series data that is not stationary to the mean can be overcome by differencing to become a stationary series. after obtaining a stationary series then the next step is to identify the arima model in the input sequence. the identification of arima model is performed on time series data which has been stationary to the variance and the mean by looking at acf and pacf plots. the arima model will be used in the prewhitening process of the input series and output series. the first step of identification of the impulse response function and transfer function is the calculation phase of cross correlation and the autocorrelation of the input series and the prewhitened output series. the cross correlation function (ccf) of the input series xt and the output series yt between the k-lags is expressed by 𝜌𝑥𝑦 (𝑘) = γxy(k) σx σy [10]. the second step is the direct prediction of the impulse response weight used to establish the order of the model of the defined transfer function vk̂ = cαβ(k) sα2 = ρ̂αβ(k) sβ sα [2]. the last step is the assignment of values (b, r, s) of the transfer function model, b, r and s are three keys of parameter in the transfer function model. modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 31 determining the value of b, r, s can be done by identifying the cross-correlation values of the sample or by directly estimating the weight of the impulse response. the phase identification model of noise series consists of two steps, the first is the initial estimate of the noise series (nt) declared �̂�𝑡 = 𝑦𝑡 − �̂�(𝐵) �̂�(𝐵) 𝐵𝑏 𝑥𝑡 [10] and the second is the arima modeling (pn,0,qn) for the noise series (nt). the estimation of transfer function model parameters used maximum likelihood method and significance test of transfer function model parameters using t test statistic. the last formation step of the transfer function model is the diagnostic test of the transfer function model which is done by testing the remaining autocorrelation and cross-correlation test (ccf) between the input and the batch input series using the ljung-box (q) test statistic. multi-input transfer function model with single output is formulated as follows: 𝑦𝑡 = ∑ 𝜈𝑗 (𝐵)𝑥𝑗𝑡 𝑘 𝑗=1 + 𝑛𝑡 where 𝑦𝑡 is output series that has been stationary, 𝑥𝑗𝑡 is a j-inputs series that have been stationary, 𝑛𝑡 is a noise series and 𝜈𝑗 (𝐵) is the transfer function for the j-input sequence (xjt). results and discussion this research used daily secondary data of rainfall (𝑌𝑡 ), air temperature (𝑋1𝑡 ), humidity (𝑋2𝑡 ), wind speed (𝑋3𝑡 ) and cloud (𝑋4𝑡 ) in batu city obtained from the web www.worldweatheronline.com. daily data for the period of january 2016-december 2016 was used as training data to form a multi-input transfer function model. while the daily data for january 2017 period was used as data testing for model validation. the time series plot of each input series and output series is shown in figure 1. (a) (b) (c) (d) (e) 36032428825221618014410872361 50 40 30 20 10 0 time r a in fa ll ( m m ) time series plot of rainfall 36032428825221618014410872361 32 31 30 29 28 time a ir t e m p e ra tu re ( °c ) time series plot of air temperature 36032428825221618014410872361 85 80 75 70 65 60 55 50 time h u m id it y ( % ) time series plot of humidity 36032428825221618014410872361 14 12 10 8 6 4 2 time w in d s p e e d ( m p h ) time series plot of wind speed modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 32 figure 1. (a) time series plot of rainfall (b) time series plot of air temperature (c) time series plot of humidity (d) time series plot of wind speed (e) time series plot of cloud based on the time series plot each input series and the output series indicated that the rainfall, air temperature, humidity, wind speed and cloud daily throughout the year 2016 has a pattern up and down and does not show the trend pattern. the stationarity test results to the variance indicated that each input series and output series are not stationary, which is shown by the box-cox plot and the λ values of each input and output series are not close to 1. then, the boxcox transformation is done for each the input series and the output series. while the results of stationary test against the mean indicated by acf and pacf plots also showed that each input series and output series has not been stationary to the mean. therefore, differencing is done once for each of input and output series so it becomes stationary. furthermore, the results of arima model identification for the input series obtained arima (1,1,2) for the air temperature input series (𝑋1𝑡 ), arima ([3,5,7],1,2) for air humidity input series (𝑋2𝑡 ), arima (1,1,1) for wind speed input series (𝑋3𝑡 ) and arima (6,1,1) for cloud input series (𝑋4𝑡 ). based on the arima model obtained from each input series, it can be determined prewhitening model on each input series and output series, such as: 𝛼1𝑡 = 𝑥1𝑡 − 0.26995𝑥1𝑡−1 + 0.63921𝛼1𝑡−1 + 0.21176𝑎1𝑡−2 (1) 𝛼2𝑡 = 𝑥2𝑡 + 0.39196𝑥2𝑡−1 − 0.24251𝑥2𝑡−2 + 0.19483𝑥2𝑡−3 + 0.18366𝑥2𝑡−5 + 0.36474𝑥2𝑡−7 + 0.31646𝛼2𝑡−1 + 0.72181𝛼2𝑡−2 (2) 𝛼3𝑡 = 𝑥3𝑡 − 0.51882𝑥3𝑡−1 + 0.94334𝛼3𝑡−1 (3) 𝛼4𝑡 = 𝑥4𝑡 − 0.58649𝑥4𝑡−1 + 0.14606𝑥4𝑡−2 + 0.42116𝑥4𝑡−3 + 0.42977𝑥4𝑡−4 + 0.16236𝑥4𝑡−5 + 0.34483𝑥4𝑡−6 + 0.90410𝑎4𝑡−1 (4) equations (1), (2), (3), and (4) are prewhitening models for the input air temperature series, humidity, wind speed and cloud. the prewhitening model for the output series is the same as the input series, only 𝛼𝑗𝑡 and 𝑎𝑗𝑡 are replaced by 𝛽𝑗𝑡 , 𝑥𝑗𝑡 is replaced by 𝑦𝑗𝑡 where j is the index of the input series. the results of the cross-correlation and the autocorrelation of the prewhitened input and output series are shown in the cross-correlation plot of each input air temperature, air humidity, wind speed, cloud series with prewhitened rainfall output are presented in figure 2. 36032428825221618014410872361 70 60 50 40 30 20 10 0 time c lo u d ( % ) time series plot of cloud modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 33 figure 2. crosscorrelation plot of prewhitened input series and output series based on the results of cross-correlation and the weight of the impulse response for each input series with prewhitened output series, it can be identified the values of b, s, r for each input series. the value of b, s, r for the air temperature input series and the rainfall output series are (b = 0, s = 0, r = 0), for the humidity input series and the rainfall output series are (b = 23, s = 0, r = 0), while for the wind speed input series and the rainfall output series are (b = 1, s = 2, r = 0) and for the cloud input series and the rainfall output series are (b = 0, s = 0, r = 0). the noise series model (𝑛𝑡 ) obtained based on the value of the impulse response weight is formulated as the following: 𝑛𝑡 = 𝑦𝑡 − 𝑦�̂� = 𝑦𝑡 − ∑ 𝑣�̂� (𝐵)𝑥𝑗𝑡 𝑘 𝑗=1 𝑛𝑡 = 𝑦𝑡 − {(−0,1166)𝑥1𝑡 + ⋯ + (−0,8663)𝑥1𝑡−23} − {(0,1391)𝑥2𝑡 + ⋯ + (0,3001)𝑥2𝑡−23} − {(−0,2078)𝑥3𝑡 + ⋯ + (−0,4434)𝑥3𝑡−23} − {(0,1681)𝑥4𝑡 + ⋯ + (0,0439)𝑥4𝑡−23} based on the equation we can get the value 𝑛24, 𝑛25, … , 𝑛365 where the value 𝑛1 … 𝑛23 is set equal to zero. arima model (𝑝𝑛 , 0, 𝑞𝑛 ) for noise series (𝑛𝑡 ) is arima ([5,8],0,2) and can be formulated 𝑛𝑡 = (1 + 0.0378𝐵 − 0.9109𝐵2) (1 + 0.72949𝐵 − 0.23092𝐵2 − 0.06788𝐵3 − 0.01765𝐵4 + 0.1179𝐵5 + 0.10773𝐵8) 𝑎𝑡 identify the values of b, s, r for the multi-input transfer function model based on the incorporation of b, s, r values from the single input transfer function model in each input series to the output series. the possible multi-input transfer function model is (𝑏1, 𝑠1, 𝑟1) (𝑏2, 𝑠2, 𝑟2) (𝑏3, 𝑠3, 𝑟3) (𝑏4, 𝑠4, 𝑟4) (𝑝𝑛 ,𝑞𝑛) = (0,0,0) (23,0,0) (1,2,0) (0,0,0) ([5,8],2). the prediction and significance test of the parameter of multi-input transfer function model is shown in table 1. modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 34 table 1. significance test results parameter multi input transfer function model input series orde (b,s,r) parameter estimation t p-value conclusion 𝑿𝟏𝒕 b=0,s=0,r=0 𝜔01 0.1042 3.84 0.0089 significant 𝑿𝟐𝒕 b=23,s=0,r=0 𝜔02 0.2459 2.03 0.0423 significant 𝑿𝟑𝒕 b=1,s=2,r=0 𝜔03 0.4318 3.53 0.0180 significant 𝑿𝟒𝒕 b=0,s=0,r=0 𝜔04 0.1992 4.12 <.0001 significant table 1 shows that all parameters in the multi-input transfer function model are significant, this is based on the p-value of each parameter smaller than the value of α. the diagnostic results of the multi-input transfer function model performed by testing the residual autocorrelation using the ljung-box test are shown below: table 2. diagnostic test result of multi input transfer function model multi input transfer function model lag chi-square p-value conclusion (𝒃𝟏, 𝒔𝟏, 𝒓𝟏) (𝒃𝟐, 𝒔𝟐, 𝒓𝟐) (𝒃𝟑, 𝒔𝟑, 𝒓𝟑) (𝒃𝟒, 𝒔𝟒, 𝒓𝟒) (𝒑𝒏,𝒒𝒏) = (0,0,0) (23,0,0) (1,2,0) (0,0,0) ([5,8],2) 12 2.80 0.5921 fit 18 7.12 0.7142 24 9.66 0.8840 based on the ljung-box test, a model is said to be feasible if the whole lag of a model yields a pvalue greater than α = 0.05. the result of model test using ljung-box test stated that the multi input transfer function model with single output is feasible to use. so the model of multi input transfer function with single output for rainfall forecasting in batu city stated 𝑦𝑡 = 0.10418𝑥1𝑡 + 0.24597𝑥2𝑡−23 + 0.43178𝑥3𝑡−1 + 0.19916𝑥4𝑡 + 𝑛𝑡 the model shows that rainfall on a certain day is affected by air temperature and cloud on that day, air humidity in the previous 23 days and wind speed in the previous day. the precision of forecasting was measured by calculating the mape (mean absolute percentage error) statistic. mape statistic for testing data was 16.35%. the results of model validation using t paired test showed that the value of forecasting with actual value was not significantly different. conclusion the results of analysis showed that based on the multi input with single output transfer function model, the rainfall in batu city on certain days is affected by air temperature and cloud on that day, humidity in the previous 23 days, and wind speed in the previous day. the results of rainfall forecasting in batu city using multi input with single output transfer function model is accurate based on the result of model validation using 𝑡 test and mape statistic that less than 20%. modelling multi input transfer function for rainfall forecasting in batu city priska arindya purnama 35 references [1] g. box, g. jenkins dan j. reisel, time 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[10] w. wei, time series analysis: univariate and multivariate methods, 2nd penyunt., addison-wesley publishing co, 2006. abstract introduction fundamental theories results and discussion conclusion references suma’inna dan gugun gumilar solusi persamaan keseimbangan massa reaktor menggunakan metode pemisahan variabel 1moh. syaiful arif, 2mohammad jamhuri 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: arif.ipoel@gmail.com abstrak persamaan keseimbangan massa reaktor menyatakan perubahan konsentrasi massa zat yang masuk dan keluar pada sistem tertutup. persamaan ini mempunyai kondisi batas tak homogen, yaitu kondisi pada saat zat masuk pada reaktor (𝐶𝑖𝑛) dan kondisi pada saat zat keluar dari reaktor (𝐶𝑜𝑢𝑡). pada penelitian ini, konsentrasi massa zat yang dihasilkan setelah proses reaksi di dalam reaktor adalah nol, atau 𝐶𝑜𝑢𝑡 = 0. pada kondisi batas tak homogen, dengan menggunakan metode pemisahan variabel, terdapat kendala untuk menyelesaikan persamaan tersebut. sehingga perlu dilakukan transformasi terlebih dahulu. transformasi dilakukan dengan tujuan untuk mengubah kondisi batas yang awalnya tak homogen menjadi kondisi batas yang homogen, sehingga metode pemisahan variabel dapat digunakan untuk menyelesaikan persamaan diferensial parsial yang mempunyai kondisi batas homogen. hasil analisa diperoleh, semakin cepat zat yang menyebar pada reaktor, maka semakin sedikit jumlah konsentrasi massa zat yang mengalami perubahan, semakin besar koefisien massa zat yang bereaksi dalam reaktor, maka semakin banyak jumlah konsentrasi massa zat yang mengalami perubahan kata kunci: kondisi batas homogen, kondisi batas tak homogen, metode pemisahan variabel. abstract mass balance of reactor equation express the change of mass concentration of substances in and out of the closed system. this equation has inhomogeneous boundary conditions, that is the conditions at the time of its entry to the reactor (𝐶𝑖𝑛) and the conditions under which the substance out of the reactor (𝐶𝑜𝑢𝑡). in this study, the mass concentration of substances produced after the reaction in the reactor is zero, or 𝐶𝑜𝑢𝑡 = 0. in the inhomogeneous boundary conditions, using the method of separation of variables, there are obstacles to complete the equation. so we need to first transformation. transformation is done with the aim to change the conditions which originally inhomogeneous boundary into a homogeneous boundary condition, so the method of separation of variables can be used to solve partial differential equations that have a homogeneous boundary conditions. the results obtained by the analysis, the faster a substance that spreads to the reactor, the less amount of mass concentration of substances that undergo a change; the greater the mass coefficient of substances that react in the reactor, the more the number of mass concentration of substances that are subject to change keywords: homogeneous boundary condition, inhomogeneous boundary condition, method of separation of variabel. pendahuluan hukum kekekalan massa adalah suatu hukum yang menyatakan bahwa massa pada suatu sistem tertutup tidak akan berubah meskipun terjadi berbagai macam reaksi didalam sistem tersebut. tempat atau alat yang digunakan selama proses reaksi disebut reaktor [1]. hukum kekekalan massa umumnya digunakan pada bidang kimia industri. hukum kekekalan massa dapat dinyatakan sebagai keseimbangan suatu massa zat yang masuk dan keluar dari sistem tertutup (reaktor). keseimbangan massa pada reaktor terjadi apabila akumulasi massa zat di dalam reaktor sama dengan selisih antara massa zat yang masuk dengan massa zat yang keluar serta massa yang dihasilkan di dalam sistem tertutup (caldwell, 2004). kondisi tersebut, dalam ilmu matematika diformulasikan menjadi bentuk persamaan diferensial parsial linier yang disebut dengan persamaan keseimbangan massa reaktor. sehingga untuk menganalisa permasalahan tersebut, cukup dengan cara mencari solusi mailto:arif.ipoel@gmail.com moh. syaiful arif 42 volume 4 no.1 november 2015 penyelesaian dari persamaan keseimbangan massa reaktor. salah satu metode yang digunakan untuk memperoleh solusi analitik persamaan diferensial parsial adalah metode pemisahan variabel. metode ini dilakukan dengan cara memisahkan masingmasing variabel independent dengan suatu fungsi dan konstanta pemisahan, atau disebut dengan fungsi eigen dan nilai eigen [2]. kemudian dengan menggunakan kondisi batas homogen, masingmasing fungsi eigen dan nilai eigen dapat ditentukan. akan tetapi, pada kondisi batas tak homogen terdapat kendala untuk menentukan fungsi eigen dan nilai eigen, dengan demikian perlu dilakukan transformasi yang bertujuan untuk mengubah kondisi batas tak homogen menjadi kondisi batas homogen. sehingga fungsi yang sudah dipisahkan tersebut dapat diselesaikan secara terpisah. oleh karena itu, penelitian ini bermaksud menentukan solusi analitik untuk model keseimbangan massa reaktor yang dipengaruhi oleh perubahan waktu dengan kondisi batas tak homogen menggunakan metode pemisahan variabel. kajian teori 1. persamaan keseimbangan massa reaktor persamaan keseimbangan massa reaktor menyatakan perubahan konsentrasi massa zat yang masuk dan keluar pada sistem tertutup [1]. [3] menyebutkan bahwa perubahan konsentrasi massa zat di dalam sistem tertutup atau reaktor, dinyatakan dalam persamaan berikut: 𝜕𝐶(𝑥,𝑡) 𝜕𝑡 = 𝐷 𝜕2𝐶(𝑥,𝑡) 𝜕𝑥2 − 𝑈 𝜕𝐶(𝑥,𝑡) 𝜕𝑥 −𝛾𝐶(𝑥,𝑡) (1) dengan kondisi awal dan kondisi batas yang digunakan yaitu: 𝐶(𝑥,0) = 0, 0 < 𝑥 < 𝑙 𝐶(0,𝑡) = 𝐶𝑖𝑛, 𝐶(𝑙, 𝑡) = 0, 𝑡 > 0 (2) pada persamaan (1) dan (2), 𝐶𝑖𝑛 adalah menyatakan massa zat yang masuk pada suatu reaktor dengan satuan 𝑚𝑜𝑙 𝑚3⁄ , 𝐷 menyatakan koefisien penyebaran zat dengan satuan 𝑚2 𝑑𝑒𝑡⁄ , 𝑈 menyatakan kecepatan fluida yang mengalir dalam reaktor dengan satuan 𝑚 𝑑𝑒𝑡⁄ , sedangkan 𝛾 adalah koefisien reaksi dengan satuan 𝑚𝑜𝑙 𝑑𝑒𝑡⁄ . 2. persamaan diferensial biasa persamaan diferensial biasa merupakan suatu persamaan yang menyatakan relasi fungsi, yang mempunyai satu peubah dengan turunan fungsi terhadap peubah tersebut. dengan demikian turunan fungsi dinotasikan dengan suatu operator 𝑑 [4]. 3. persamaan diferensial parsial persamaan diferensial parsial merupakan suatu persamaan yang menyatakan relasi antara suatu fungsi dependent dengan variabel independent yang lebih dari satu variabel beserta turunan parsialnya [5]. [4] menggunakan notasi 𝜕 sebagai operator persamaan diferensial parsial. untuk menyederhanakan penulisan, turunan parsial bisa dinotasikan 𝜕𝐶(𝑥,𝑡) 𝜕𝑥 = 𝐶𝑥(𝑥,𝑡), 𝜕2𝐶(𝑥,𝑡) (𝜕𝑥)2 = 𝐶𝑥𝑥(𝑥,𝑡), dan seterusnya [2]. suatu persamaan diferensial parsial disebut linier apabila terdapat suatu operator diferensial ℒ yang diterapkan pada fungsi c, sehingga berlaku ℒ𝐶 = 𝑔 (3) pada persamaan (3), apabila 𝑔 = 0 maka disebut sebagai persamaan diferensial parsial linier homogen. sedangkan apabila 𝑔 ≠ 0, maka disebut sebagai persamaan diferensial parsial linier tak homogen [5]. dengan demikian, persamaan keseimbangan massa reaktor (1) merupakan persamaan diferensial parsial linier homogen. 4. kondisi batas homogen dan kondisi batas tak homogen misalkan 𝐶(𝑥,𝑡) adalah solusi dari suatu persamaan diferensial parsial pada batas 0 < 𝑥 < 𝑙, kondisi batas pada saat 𝑥 = 0 dan 𝑥 = 𝑙 dituliskan seperti persamaan berikut: 𝐶(0,𝑡) = 𝑔(𝑡) (4) 𝐶(𝑙,𝑡) = ℎ(𝑡) (5) persamaan (4) dan (5) merupakan contoh kondisi batas dirichlet. apabila 𝑔(𝑡) = 0 dan ℎ(𝑡) = 0 maka disebut sebagai kondisi batas homogen. sedangkan apabila 𝑔(𝑡) ≠ 0 atau ℎ(𝑡) ≠ 0, maka disebut sebagai kondisi batas tak homogen [5]. 5. metode pemisahan variabel salah satu metode yang digunakan untuk memperoleh solusi persamaan diferensial parsial linier adalah metode pemisahan variabel. metode ini dilakukan dengan cara memisahkan masingmasing variabel independent dengan suatu fungsi yang mengandung variabel independent dan konstanta pemisahan. masing-masing fungsi dan konstanta pemisahan disebut sebagai fungsi eigen dan nilai eigen [2]. misalkan c(𝑥,𝑡) = 𝑋(𝑥)𝑇(𝑡) adalah solusi dari suatu persamaan diferensial parsial, maka 𝑋(𝑥) merupakan fungsi eigen yang memuat variabel 𝑥, sedangkan 𝑇(𝑡) adalah fungsi eigen yang memuat variabel 𝑡. sehingga masing-masing solusi persamaan keseimbangan massa reaktor menggunakan metode pemisahan variabel cauchy – issn: 2086-0382/e-issn: 2477-3344 43 fungsi eigen dapat diselesaikan secara terpisah [6]. pada kondisi batas homogen, untuk menentukan nilai eigen dapat dilakukan dengan cara mensubstitusikan kondisi batas pada masingmasing fungsi yang sudah diselesaikan secara terpisah [2]. sedangkan pada kondisi batas tak homogen terdapat kendala untuk menentukan nilai eigen. sehingga penyelesaian persamaan diferensial parsial yang mempunyai kondisi batas tak homogen dapat dilakukan dengan cara mentransformasikan kondisi batas yang awalnya tak homogen diubah menjadi homogen [5]. 6. deret fourier [2] menyebutkan bahwa, misalkan terdapat domain 𝑥 pada suatu interval 0 < 𝑥 < 𝑙, sedemikian sehingga terdapat suatu deret sinus yang konvergen menuju suatu fungsi 𝜙(𝑥) yang didefinisikan seperti persamaan berikut: 𝜙(𝑥) = ∑𝐴𝑛 sin( 𝑛𝜋𝑥 𝑙 ) ∞ 𝑛=1 (6) pada persamaan (6), 𝐴𝑛 merupakan koefisien deret sinus, sedangkan 𝜙(𝑥) adalah suatu fungsi tertentu. dengan demikian nilai 𝐴𝑛 dapat ditentukan seperti berikut: 𝐴𝑛 = { 0, 𝑛 ≠ 𝑚 2 𝑙 ∫ 𝜙(𝑥)sin( 𝑛𝜋𝑥 𝑙 ) 𝑙 0 𝑑𝑥, 𝑛 = 𝑚 (7) pembahasan pada bab sebelumnya telah dijelaskan, terdapat kendala dalam menyelesaikan persamaan keseimbangan massa reaktor (1) yang mempunyai kondisi batas tak homogen (2). sehingga perlu dilakukan transformasi kondisi batas, yang awalnya tak homogen diubah menjadi kondisi batas homogen. sehingga solusi persamaan keseimbangan massa reaktor ditrasformasikan seperti berikut: 𝐶(𝑥,𝑡) = 𝑉(𝑥,𝑡) + 𝐶(𝑥) (8) dari persamaan (2), dapat diketahui bahwa kondisi batas yang digunakan 𝐶(0,𝑡) dan 𝐶(𝑙, 𝑡) adalah kondisi batas tak homogen, sehingga 𝑉 mempunyai kondisi batas homogen apabila kondisi batas yang digunakan 𝐶(0) dan 𝐶(𝑙) sama dengan kondisi batas yang digunakan 𝐶(0,𝑡) dan 𝐶(𝑙,𝑡), atau ditulis 𝐶(0) = 𝐶𝑖𝑛 dan 𝐶(𝑙) = 0. fungsi 𝐶(𝑥) merupakan suatu fungsi yang hanya bergantung pada variabel 𝑥 saja. hal ini dikarenakan apabila fungsi 𝐶(𝑥) hanya bergantung pada variabel 𝑥 dan 𝑡, mengakibatkan kondisi batas 𝑉 tidak homogen. dengan demikian, transformasi (8) dapat menjamin 𝐶(𝑥,𝑡) merupakan solusi persamaan keseimbangan massa reaktor yang mempunyai kondisi batas tak homogen. dari penjelasan di atas, 𝐶(𝑥) merupakan solusi persamaan keseimbangan massa reaktor pada kondisi laju perubahan konsentrasi massa tidak dipengaruhi oleh perubahan waktu (steadystate). sehingga untuk memperoleh solusi pada persamaan (8), perlu diselesaikan persamaan keseimbangan massa reaktor pada kondisi steadystate seperti berikut: 𝐷 𝑑2𝐶(𝑥) 𝑑𝑥2 − 𝑈 𝑑𝐶(𝑥) 𝑑𝑥 − 𝛾𝐶(𝑥) = 0 (9) persamaan (9) merupakan persamaan diferensial biasa homogen orde dua. sehingga apabila 𝑈2 + 4𝐷𝛾 > 0 maka solusi 𝐶(𝑥) mempunyai akar-akar real berbeda, apabila 𝑈2 + 4𝐷𝛾 = 0 maka solusi 𝐶(𝑥) mempunyai akar-akar kembar, sedangkan apabila 𝑈2 + 4𝐷𝛾 < 0 maka solusi 𝐶(𝑥) mempunyai akar-akar komploks. pada kondisi 𝑈2 + 4𝐷𝛾 > 0 solusinya diperoleh 𝐶(𝑥) = 𝑘1𝑒 ( 𝑈+√𝑈2+4𝐷𝛾 2𝐷 )𝑥 + 𝑘2𝑒 ( 𝑈−√𝑈2+4𝐷𝛾 2𝐷 )𝑥 (10) dengan menggunakan kondisi batas 𝐶(0) = 𝐶𝑖𝑛 dan 𝐶(𝑙) = 0, maka nilai 𝑘1 dan 𝑘2 dapat diperoleh: 𝑘1 = 𝐶𝑖𝑛𝑒 ( 𝑈−√𝑈2+4𝐷𝛾 2𝐷 )𝑙 𝑒 ( 𝑈−√𝑈2+4𝐷𝛾 2𝐷 )𝑙 − 𝑒 ( 𝑈+√𝑈2+4𝐷𝛾 2𝐷 )𝑙 dan 𝑘2 = − 𝐶𝑖𝑛𝑒( 𝑈+√𝑈2+4𝐷𝛾 2𝐷 ) 𝑙 𝑒( 𝑈−√𝑈2+4𝐷𝛾 2𝐷 ) 𝑙 −𝑒( 𝑈+√𝑈2+4𝐷𝛾 2𝐷 ) 𝑙 pada kondisi 𝑈2 + 4𝐷𝛾 = 0, dengan cara yang sama, solusinya diperoleh 𝐶(𝑥) = 𝐶𝑖𝑛𝑒 𝑈 2𝐷 𝑥 − 𝐶𝑖𝑛 𝑙 𝑥𝑒 𝑈 2𝐷 𝑥 (11) sedangkan pada kondisi 𝑈2 + 4𝐷𝛾 < 0, dengan cara yang sama, solusinya diperoleh 𝐶(𝑥) = 𝐶𝑖𝑛𝑒 𝑈 2𝐷 𝑥 𝑐𝑜𝑠( √|𝑈2 + 4𝐷𝛾| 2𝐷 𝑥) (12) +𝑘4𝑒 𝑈 2𝐷 𝑥 𝑠𝑖𝑛( √|𝑈2 + 4𝐷𝛾| 2𝐷 𝑥) moh. syaiful arif 44 volume 4 no.1 november 2015 dengan 𝑘4 = −𝐶𝑖𝑛𝑐𝑜𝑡( √|𝑈2+4𝐷𝛾| 2𝐷 𝑙). selanjutnya, untuk menjamian solusi 𝑉(𝑥,𝑡) memenuhi persamaan keseimbangan massa reaktor, maka persamaan (8) disubstiusikan pada persamaan (1), sehingga dapat diperoleh persamaan baru seperti berikut: 𝑉𝑡(𝑥,𝑡) = 𝐷𝑉𝑥𝑥(𝑥,𝑡) − 𝑈𝑉𝑥(𝑥,𝑡) − 𝛾𝑉(𝑥,𝑡) (13) dengan mensubstitusikan kondisi batas 𝐶(0,𝑡), 𝐶(𝑙,𝑡) dan 𝐶(0), 𝐶(𝑙) pada persamaan (8), maka kondisi batas 𝑉 pada saat 𝑥 = 0 dan 𝑥 = 𝑙 adalah 𝑉(0,𝑡) = 0, 𝑉(𝑙, 𝑡) = 0 (14) sedangkan kondisi awal 𝑉 pada saat 𝑡 = 0 adalah 𝑉(𝑥,0) = −𝐶(𝑥) (15) langkah selanjutnya, 𝑉(𝑥,𝑡) dapat diperoleh dengan cara menyelesaikan persamaan (13). sehingga dimisalkan 𝑉(𝑥,𝑡) = 𝑇(𝑡)𝑋(𝑥) (16) sebagai solusi dari persamaan (13). kemudian dengan mensubstitusikan persamaan (16) pada (13), maka dapat diperoleh 𝑇′(𝑡) 𝑇(𝑡) = 𝐷 𝑋′′(𝑥) 𝑋(𝑥) − 𝑈 𝑋′(𝑥) 𝑋(𝑥) − 𝛾 𝑇′(𝑡) 𝑇(𝑡) + 𝛾 = 𝐷 𝑋′′(𝑥) 𝑋(𝑥) − 𝑈 𝑋′(𝑥) 𝑋(𝑥) (17) pada persamaan (17), ruas kiri hanya bergantung pada variabel 𝑡, sedangkan ruas kanan hanya bergantung pada variabel 𝑥. sehingga kondisi tersebut hanya dapat dipenuhi apabila kedua ruas bernilai konstanta yang tidak bergantung pada variabel 𝑡 dan 𝑥. misalkan konstanta yang dimaksud adalah 𝜆, maka persamaan (17) dapat dipisahkan menjadi seperti berikut: 𝐷 𝑋′′(𝑥) 𝑋(𝑥) − 𝑈 𝑋′(𝑥) 𝑋(𝑥) = 𝜆 𝐷𝑋′′(𝑥)− 𝑈𝑋′(𝑥)− 𝜆𝑋(𝑥) = 0 (18) dan 𝑇′(𝑡) 𝑇(𝑡) + 𝛾 = 𝜆 𝑇′(𝑡)− (𝜆 − 𝛾)𝑇(𝑡) = 0 (19) pada persamaan (18) dan (19), 𝜆 adalah suatu konstanta, sehingga 𝜆 dapat bernilai nol, positif, atau negatif, secara terurut dinotasikan dengan 𝜆 = 0, 𝜆 > 0, atau 𝜆 < 0. dengan demikian, untuk menentukan solusi dari persamaan (18) dan (19), ditentukan terlebih dahulu nilai 𝜆 yang memenuhi. untuk 𝜆 = 0, maka solusi persamaan (18) adalah 𝑋(𝑥) = 𝑚3 + 𝑚4𝑒 𝑈 𝐷 𝑥 dengan menggunakan kondisi batas (14), maka dapat diperoleh 𝑚3 = 0 dan 𝑚4 = 0. dengan demikian, untuk 𝜆 = 0, solusi 𝑋(𝑥) adalah nol (trivial solution). untuk 𝜆 > 0, dimisalkan 𝜆 = 𝛽2 maka solusi persamaan (18) adalah 𝑋(𝑥) = 𝑚5𝑒 𝑈+√𝑈2+4𝐷𝛽2 2𝐷 𝑥 + 𝑚6𝑒 𝑈−√𝑈2+4𝐷𝛽2 2𝐷 𝑥 dengan menggunakan kondisi batas (14), maka dapat diperoleh 𝑚5 = 0 dan 𝑚4 = 0. dengan demikian, untuk 𝜆 > 0, solusi 𝑋(𝑥) adalah nol (trivial solution). sedangkan untuk 𝜆 < 0, dimisalkan 𝜆 = −𝛽2 maka solusi persamaan (18) adalah 𝑋(𝑥) = 𝑚7𝑒 𝑈+√𝑈2−4𝐷𝛽2 2𝐷 𝑥 + 𝑚8𝑒 𝑈−√𝑈2−4𝐷𝛽2 2𝐷 𝑥 dari persamaan tersebut dapat diketahui bahwa, apabila 𝑈2 − 4𝐷𝛽2 > 0, maka seperti pada kasus 𝜆 > 0 solusinya adalah nol (trivial solution). dengan demikian, nilai dipilih 𝑈2 − 4𝐷𝛽2 < 0. artinya 𝑈2 − 4𝐷𝛽2 harus bernilai negatif. dengan demikian, 𝑈2 − 4𝐷𝛽2 = −(4𝐷𝛽2 − 𝑈2) dapat menjamin nilainya selalu negatif. sehingga solusinya diperoleh 𝑋(𝑥) = 𝑚7𝑒 𝑈+√−(4𝐷𝛽2−𝑈2) 2𝐷 𝑥 + 𝑚8𝑒 𝑈−√−(4𝐷𝛽2−𝑈2) 2𝐷 𝑥 = 𝑚7𝑒 𝑈+𝑖√4𝐷𝛽2−𝑈2 2𝐷 𝑥 + 𝑚8𝑒 𝑈−𝑖√4𝐷𝛽2−𝑈2 2𝐷 𝑥 persamaan tersebut dapat diubah menjadi seperti berikut: 𝑋(𝑥) = 𝑒 𝑈 2𝐷 𝑥 (𝑚9 cos( √4𝐷𝛽2 − 𝑈2 2𝐷 𝑥)) +𝑒 𝑈 2𝐷 𝑥 (𝑚10 sin( √4𝐷𝛽2 − 𝑈2 2𝐷 𝑥)) dengan menggunakan kondisi batas (14), maka dapat diperoleh 𝑚9 = 0, 𝑚10 ≠ 0, dan 𝛽 = √ (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 untuk 𝑛 menuju tak hingga (𝑛 = 1,2,…). sehingga solusi 𝑋(𝑥) dapat disederhanakan menjadi 𝑋(𝑥) = 𝑚10𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 ) (20) dikarenakan nilai 𝜆 yang memenuhi adalah 𝜆 = −𝛽2, maka solusi 𝑇(𝑡) dari persamaan (20) adalah 𝑇(𝑡) = 𝑚12𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 (21) langkah selanjutnya, dengan mensubstitusikan persamaan (20) dan (21) pada solusi persamaan keseimbangan massa reaktor menggunakan metode pemisahan variabel cauchy – issn: 2086-0382/e-issn: 2477-3344 45 persamaan (16), maka solusi 𝑉(𝑥,𝑡) dapat diperoleh seperti berikut 𝑉(𝑥,𝑡) = ∑𝐸𝑛𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 ) ∞ 𝑛=0 𝐸𝑛 merupakan suatu nilai yang dapat diperoleh dengan mensubstitusikan kondisi awal. dari persamaan tersebut dapat diketahui pada saat 𝑛 = 0, mengakibatkan nilai dari sin(0) = 0, sehingga solusi 𝑉(𝑥,𝑡) = 0. dengan demikian pada saat 𝑛 = 0, nilai 𝐸0 tidak mempengaruhi hasil dari penjumlahanya, jadi solusinya yaitu: 𝑉(𝑥,𝑡) = ∑𝐸𝑛𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 ) ∞ 𝑛=1 kemudian dengan menggunakan kondisi awal (15), maka pada saat 𝑡 = 0 diperoleh −𝐶(𝑥) = ∑𝐸𝑛 sin( 𝑛𝜋𝑥 𝑙 ) ∞ 𝑛=1 dengan menggunakan ekspansi deret fourier seperti persamaan (7), maka nilai 𝐸𝑛 dapat diperoleh seperti berikut: en = − 2 𝑙 ∫ 𝐶(𝑥) 𝑙 0 sin( 𝑛𝜋𝑥 𝑙 )𝑑𝑥 kemudian dengan mengintegralkan persamaan tersebut, apabila 𝑈2 + 4𝐷𝛾 > 0 diperoleh en = 2𝑛𝜋(𝑘1(𝑒 𝐴1𝑙cos(𝑛𝜋)−1)+𝑘2(𝑒 −𝐴1𝑙cos(𝑛𝜋)−1)) (𝑛𝜋)2+(𝑙𝐴1) 2 , dengan 𝐴1 = √|𝑈2+4𝐷𝛾| 2𝐷 . apabila 𝑈2 + 4𝐷𝛾 = 0, maka 𝐸𝑛 = − 2𝐶𝑖𝑛 𝑛𝜋 . sedangkan apabila 𝑈2 + 4𝐷𝛾 < 0, maka dapat diperoleh 𝐸𝑛 = 𝐶𝑖𝑛((cos(𝐴1𝑙+𝑛𝜋)−1))+𝑘3(sin(𝐴1𝑙+𝑛𝜋)) (𝐴1𝑙+𝑛𝜋) − 𝐶𝑖𝑛((cos(𝐴1𝑙−𝑛𝜋)−1))+𝑘3(sin(𝐴1𝑙−𝑛𝜋)) (𝐴1𝑙−𝑛𝜋) . substitusikan masing-masing solusi 𝐶(𝑥) dan 𝑉(𝑥,𝑡) pada persamaan (14), maka solusi persamaan keseimbangan massa reaktor yang mempunyai kondisi batas tak homogen yaitu: 1. jika 𝑈2 + 4𝐷𝛾 > 0, maka diperoleh 𝐶(𝑥,𝑡) = ∑ 𝐸𝑛𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 )∞𝑛=1 +𝑘1𝑒 ( 𝑈+√𝑈2+4𝐷𝛾 2𝐷 )𝑥 + 𝑘2𝑒 ( 𝑈−√𝑈2+4𝐷𝛾 2𝐷 )𝑥 (22) 2. jika 𝑈2 + 4𝐷𝛾 = 0, maka diperoleh 𝐶(𝑥,𝑡) = ∑𝐸𝑛𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 ) ∞ 𝑛=1 𝐶𝑖𝑛𝑒 𝑈 2𝐷 𝑥 − 𝐶𝑖𝑛 𝑙 𝑥𝑒 𝑈 2𝐷 𝑥 (23) 3. jika 𝑈2 + 4𝐷𝛾 < 0, maka diperoleh 𝐶(𝑥,𝑡) = ∑ 𝐸𝑛𝑒 −( (2𝑛𝜋𝐷)2+𝑈2𝑙2 4𝐷𝑙2 +𝛾)𝑡 𝑒 𝑈 2𝐷 𝑥 sin( 𝑛𝜋𝑥 𝑙 )∞𝑛=1 +𝐶𝑖𝑛𝑒 𝑈 2𝐷 𝑥 𝑐𝑜𝑠( √|𝑈2+4𝐷𝛾| 2𝐷 𝑥) +𝑘4𝑒 𝑈 2𝐷 𝑥 𝑠𝑖𝑛( √|𝑈2+4𝐷𝛾| 2𝐷 𝑥) (24) setelah solusi persamaan keseimbangan massa reaktor diperoleh, maka untuk mengetahui perubahan konsentrasi massa zat di sepanjang reaktor, ditentukan parameternya terlebih dahulu. misalkan masing-masing parameternya ditentukan seperti berikut: 𝐶𝑖𝑛 = 100 𝑚𝑜𝑙/𝑚 3, 𝑈 = 0.008 𝑚 𝑑𝑒𝑡⁄ , 𝛾 = 0.018𝑚𝑜𝑙 𝑑𝑒𝑡⁄ , kemudian penyebaran zat yang awalnya 𝐷 = 0.083 𝑚2 𝑑𝑒𝑡⁄ dipercepat menjadi 𝐷 = 0.8 𝑚2 𝑑𝑒𝑡⁄ , maka perubahan jumlah massa zat pada reaktor sepanjang 𝑥 = [0,10] dapat dilihat seperti gambar di bawah ini: (a) (b) gambar 1. (a) perubahan konsentrasi zat dengan 𝐷 = 0.083 𝑚2 𝑑𝑒𝑡⁄ (b) perubahan konsentrasi zat dengan 𝐷 = 0.8 𝑚2 𝑑𝑒𝑡⁄ moh. syaiful arif 46 volume 4 no.1 november 2015 gambar 1.a menunjukkan grafik perubahan konsentrasi massa zat dengan memperlambat penyebaran zat di dalam reaktor. sedangkan gambar 1.b menyatakan perubahan konsentrasi massa zat dengan mempercepat penyebaran zat di dalam reaktor. dari kedua gambar di atas, dapat diketahui bahwa semakin cepat zat yang menyebar pada reaktor maka semakin sedikit konsentrasi zat yang mengalami perubahan. begitupun sebaliknya, semakin lambat zat yang menyebar pada reaktor, maka semakin banyak jumlah konsentrasi zat yang mengalami perubahan. selanjutnya, dengan parameter 𝐶𝑖𝑛 = 100 𝑚𝑜𝑙/𝑚3, 𝐷 = 0.083 𝑚2 𝑑𝑒𝑡⁄ , dan 𝑈 = 0.008 𝑚 𝑑𝑒𝑡⁄ , maka perubahan konsentrasi massa zat dengan memperbesar dan memperkecil koefesien reaksi zat (𝛾), dapat dilihat seperti gambar di bawah ini: (a) (b) gambar 2. (a) perubahan konsentrasi zat dengan 𝛾 = 0.2𝑚𝑜𝑙 𝑑𝑒𝑡⁄ (b) perubahan konsentrasi zat dengan 0.02𝑚𝑜𝑙 𝑑𝑒𝑡⁄ gambar 2.a menunjukkan grafik perubahan konsentrasi massa zat dengan memperkecil koefisien reaksi zat di dalam reaktor. sedangkan gambar 2.b menyatakan perubahan konsentrasi massa zat dengan memperbesar koefisien reaksi zat di dalam reaktor. dari gambar 2.a dan 2.b, dapat diketahui bahwa semakin besar koefisien reaksi zat pada reaktor maka semakin banyak konsentrasi zat yang mengalami perubahan di sepanjang reaktor. begitupun sebaliknya, semakin kecil koefisien reaksi zat pada reaktor, maka semakin sedikit jumlah konsentrasi zat yang mengalami perubahan di sepanjang reaktor. pada gambar (1) sampai dengan (2), parameter yang digunakan memenuhi kondisi 𝑈2 + 4𝐷𝛾 > 0. parameter yang memenuhi kondisi 𝑈2 + 4𝐷𝛾 = 0 dan 𝑈2 + 4𝐷𝛾 < 0 dapat dilihat seperti gambar (3) di bawah ini: (a) (b) gambar 3. (a) perubahan konsentrasi massa zat dengan 𝑈2 + 4𝐷𝛾 = 0 (b) perubahan konsentrasi massa zat dengan 𝑈2 + 4𝐷𝛾 < 0 dari gambar 3.a dan 3.b, dapat diketahui pemilihan parameter yang memenuhi 𝑈2 + 4𝐷𝛾 = 0 membutuhkan waktu lebih lama dari pada kondisi 𝑈2 + 4𝐷𝛾 > 0 untuk mencapai kondisi steady-state, sedangkan pemilihan parameter yang memenuhi 𝑈2 + 4𝐷𝛾 < 0 membutuhkan waktu paling lama untuk mencapai kondisi steady-state. perubahan massa zat dengan parameter yang sama seperti pada gambar 2.b, selama kurun waktu 15 detik dapat dilihat melalui gambar di bawah ini: solusi persamaan keseimbangan massa reaktor menggunakan metode pemisahan variabel cauchy – issn: 2086-0382/e-issn: 2477-3344 47 gambar 3. perubahan konsentrasi massa zat selama kurun waktu 15 detik kesimpulan faktor pemilihan parameter yang sedemikian sehingga 𝑈2 + 4𝐷𝛾 > 0, paling efesien digunakan untuk menyelesaikan permasalahan terkait perubahan konsentrasi massa zat pada suatu raktor. dengan demikian, semakin cepat zat yang menyebar pada reaktor, maka semakin sedikit jumlah konsentrasi massa zat yang mengalami perubahan; semakin besar koefisien reaksi zat dalam reaktor, maka semakin banyak jumlah konsentrasi massa zat yang mengalami perubahan di sepanjang reaktor. daftar pustaka [1] b. barnes and g. r. fulford, mathematical modelling with case studies a differential equation using maple and matlab second edition, london: crc press, 2009. [2] w. a. strauss, partial differential equation an introduction, singapore: john wiley & sons.inc, 1992. [3] j. caldwell, mathematical modelling case studies and projects, new york: kluwer academic publishers, 2004. [4] w. e. boyce and r. c. diprima, elementarry differential equation and boundary value proplem ninth edition, usa: john wiley & sons, inc, 2009. [5] m. a. pinsky, partial differential equation and boundary-value problem with aplication third edition, rhode island: waveland press, 2003. [6] r. p. agarwal and d. o'regan, ordinary and partial differential equation with special function, fourier series and boundary value problems, new york: spinger scince + business media, 2009. pendahuluan kajian teori pembahasan kesimpulan daftar pustaka selection of ideal contraceptive tool using hybrid entropy-topsis method cauchy –jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 67-72 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 20 november 2017 reviewed: 27 february 2018 accepted: 16 may 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.4510 selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari1, kwardiniya andawaningtyas2, corina karim3 1,2,3 department of mathematics, universitas brawijaya email: ewahyu-math @ub.ac.id, dina_math@ub.ac.id, co_mathub@ub.ac.id abstract the hybrid entropy-topsis method is a combination of two methods, entropy and topsis. the combination of the two methods is used in the decision-making model to improve the quality of a decision. in this research, the hybrid entropy-topsis method was applied to determine the ideal contraceptive tool based on the acceptor criteria. entropy is a method of weighted criteria, while topsis is a method of decision making through the alternative ranking process based on weighted criteria. the criteria are factors which influence to keluarga berencana acceptors for selecting contraceptives. the used criteria in this research were the age, blood pressure, menstrual cycle, the use of contraceptives, and the cost. while the alternative is contraceptive tool itself. the selected alternatives here were pill, injection, condoms, implant, iud and mop / mow. based on the results of the questionnaire data and the simulation, it was obtained that pill and implants has the highest ranking and they indicated that pill and implants as an alternative selection of the ideal contraceptives. keywords: contraceptive tool, entropy, topsis introduction the increasing number of population is the problem for developing country like indonesia. the population of indonesia in 2010 as many as 237.6 million people. according to badan pusat statistik [1], the population growth tend to increase, so if it is not controlled, it is predicted that in 2035 the population of indonesia is about 305.6 million people. the increasing number of population in indonesian is the impact of the increasing number of birth. thus, we need to a commitment from the government and society to suppress the birth rate. keluarga berencana is one of methods to suppress the population growth rates by planning the sum and the distance between the children. some of which are family planning and the contraceptive selection. keluarga berencana is one of the basic health services and the most important for women. keluarga berencana acceptors has to determine the appropriate contraceptive tool, which is safe to use, but it is not easy. before keluarga berencana acceptor chooses the contraceptive tool, the acceptor has a right to get the information about contraceptive tool. all of contraceptive methods have side effects that must be known by the user. there are various contraceptive tools, so women who use contraceptives have to make their own choice and considered appropriate. the research on the classification of the contraceptive selection of an opportunity-based using the naive bayes classifier method has been done by sobri abusini [2], which was implemented in the form of an application program. on the other hand, endang wahyu [3] use backpropagation neural network algorithm to classify the effectiveness of contraceptive. in her study, keluarga berencana acceptors will be helpful to determine the effective contraceptive tool based on their criteria, such as age, menstrual cycle, and blood pressure. further research on keluarga berecana acceptors is to examine the patterns of changes in effective use of contraceptive based on the primary data of them. that research has been done by endang wahyu [4]. acceptors who become the object of that research was acceptors who make a change to the use of contraceptives more than one. the method for searching the pattern is the k-nearest neighbors (knn) algorithm to determine the shortest mailto:%20ewahyu-math%20@ mailto:dina_math@ub.ac.id mailto:co_mathub@ub.ac.id selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari 68 distance from several determinant factors of contraceptive and generalized sequential pattern algorithm which is one of the algorithms in data mining. different from the previous research, in this research, the used decision-making method is based on multi attribute decision making method (madm). technique for order of preference by similarity to ideal solution (topsis) which is one of the multicriteria decision-making methods where it was introduced in 1981. the topsis method is one of the effective methods for madm with qualitative and quantitative features. first, we find the entropy by using the weighted criteria of the topsis method. there are some articles that are related to the hybrid entropy – topsis, that is, evaluation model and empirical study of human all-round development based on entropy weight and topsis written by li gang et al. [5]. in 2014, yancang li et al. has writen an article entitled comprehensive assessment on sustainable development of highway transportation capacity based on entropy weight and topsis [6]. in this research we will use hybrid from two methods, entropy-topsis, to select the ideal type of contraceptive. methods the source of data this research was conducted at the hospital of malang muhammadiyah university and family planning clinic in dau district, malang regency and also the respondents were the tocologists and keluarga berencana officer in that location. methodology and data analysis the variables were the questionnaire data in the form of information about the detemined criteria and the weighted criteria to be achieved against the alternative. the used criteria and alternatives in this research was variable criteria, including, the age, blood pressure (systole and diastole), menstrual cycle, the cost, the experience of contraceptive methods used by the acceptor. for alternatives, it was including pills, injections, condoms, implants, iud, and mop / mow. the research model to make a decission is a madm by combining the weighted on entropy and the ranking on topsis to select the ideal contraceptive. after obtaining the necessary informations and weighted, it were processed using the entropy-topsis method. this method was used to determine decision preferences on the data that has different units, qualitative and quantitative in each criteria. the topsis was used to evaluate the obtained alternative weighted in the entropy mehod. the steps are the following: 1. identifications of the problem and formulations of the problem. determining the criteria variable, which is the contraceptive selection factor, and alternative, which is the used contraceptive tool. 2. conducting literature studies to find relevant theory references with the formulation of the problem. 3. conducting surveys, interviews and questionnaires distribution related to the criteria and alternatives (questionnaire draft attached). the purposive sampling was used as a sampling technique. the number of keluarga berencana officers as respondents was determined by the highest and lowest percentage of keluarga berencana acceptors in the subdistrict. 4. data processing of questionnaire result from criteria to alternative was using entropy method. the output of the process is the final weighted criteria (final entropy). 5. the final weighted criteria were used as the topsis method input. the output of the topsis method was alternative ranking (the used contraceptive) which is the ideal decision solution. selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari 69 results and discussion acceptor data retrieval based on the results to a number of respondents in malang, the respondents were the general hospital malang muhammadiyah university and the tocologist clinics in dau district, malang regency with a total of 17 respondents. the result of questionnaire is combination of six alternatives and six criteria which are shown in table 1 below: table 1. alternative data to criteria age diastole pressur e sistole pressu re co st menstrual cycle experience of contraceptive tool pill 26 30 90 – 99 100 109 low normal fair injections 31 35 100 – 109 100 109 fair normal fair condomss 31 35 100 – 109 100 109 fair normal fair implant 26 30 90 – 99 120 129 high normal good iud 31 35 100 – 109 140 149 high normal fair mow 41 45 100 – 109 140 149 high less normal good conversion table to get the value of each alternative criterion are shown as follow: table 2. convertion in each criterion criterion conversi on age diastole pressure sistole pressure cost menstrual cycle experience of contraceptive tool 5 26 – 30 100 – 109 100 109 very low normal very good 4 31 – 35 90 – 99 110 – 119 low quite normal good 3 36 – 40 80 – 89 120 – 129 fair less normal fair 2 41 – 45 70 – 79 130 -139 high abnormal bad 1 > 45 < 70 > 139 very high very abnormal very bad the entropy method according to wen-yon-zhou [7], the entropy method is used to determine a weighted on a criterion with variations of each criterion where the highest weighted is in the highest value of variation. calculation of the final weighted entropy let  025.00334.01333.0775.00083.0025.0 . conversion for each desired criteria by keluarga berencana worker (tocologist) adjusted to the conditions of the keluarga berencana acceptor, it will get the final weighted entropy. selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari 70   654321 * ;;;;; wwwwwww   as a simulation if the decision maker uses the following weighting. table 3. weighted simulation criterion acceptor data value of conversions age 36 – 40 3 diastole pressure 80 – 89 3 sistole pressure 130 – 139 2 cost low 4 menstrual cycle less normal 3 experience of previous contraceptive tool bad 2 the weighted value for the final entropy is the following 6,5,4,3,2,1 6 1 *    j w w j jj jj j    and the valued of  2;3;4;2;3;3*  w get the final weighted entropy, i.e:  0,0214320,0429490,2285470,6643810,0106730,032147*  hybrid method entropy-topsis the topsis is one of a method that can help the optimally of decision-making process to solve the decision problem in a practical way. according to smaradance and ali [8], the topsis is one of the multicriteria decision-making methods which is introduced by yoon and hwang in 1981. the topsis consider the distance to the positive ideal solution and the distance to the negative ideal solution simultaneously. according to sandeep et al [9], the topsis is a method used to deal with the alternative ranking issues from the best to the worse. the topsis is a method that used to solve an alternative rank from the best to the worst. here, we use topsis method for alternate ranking of contraceptives using the final weighted entropy  * . the topsis procedures are as follows. 1. build a decision matrix. here is the x decision matrix refers to the m alternative and will be evaluated based on n criteria                      432152 352154 452345 353554 353554 354545 x 2. make a normalized decision matrix. each element in the x matrix is normalized to obtain the normalized r matrix. using the equation (2.10), it was obtained the normalized decision matrix from the r work rating matrix as follows selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari 71 table 4. the result of normalisastion (r) 0,495074 0,371391 0,539164 0,589768 0,431934 0,3638034 0,396059 0,464238 0,539164 0,442326 0,431934 0,3638034 0,396059 0,464238 0,539164 0,442326 0,431934 0,3638034 0,495074 0,371391 0,323498 0,294884 0,431934 0,4850713 0,396059 0,464238 0,107833 0,294884 0,431934 0,3638034 0,19803 0,464238 0,107833 0,294884 0,259161 0,4850713 3. the weighted on a normalized matrix using equations 𝑣𝑖𝑗 = 𝑤𝑗 𝑟𝑖𝑗 , 4. determine the matrix of positive ideal solutions, negatives ideal solutions, separation and close to relative to the positive ideal solutions. 5. in this research, the advantages of criteria are the age, sistole pressure, diastolic pressure, menstrual cycle, and contraceptive experience. meanwhile, the cost criterion is the cost incurred in accordance with the contraceptive used. the positive ideal solution matrix element, the negative ideal solution matrix element, and the separation were obtained based to table 5 below. in the topsis method, the optimal alternative has the nearest positive ideal solution and the furthest negative ideal solution. after an alternative distance was obtained, then the positive and negative ideal solution were calculated on the close to relative of each alternative to the two solutions, so it was obtained as follows. table 5. the positive ideal solution, the negative ideal solution, the separation and the close to relative to  i v i  i v  i v  i s  i s  i c 1 0.015915 0.006366 0 0.09613 1 2 0.005284 0.002114 0.004873 0.04673 0.905559 3 0.328918 0.263134 0.004873 0.04673 0.905559 4 0.045259 0.113148 0 0.09613 1 5 0.021263 0.008505 0.004873 0.04673 0.905559 6 0.1061 0.004244 0.043861 0.131567 0.749978 6. sorting or ranking the alternatives. the alternative was sorted from the largest to the smallest  i c value. the alternative with the largest  i c value is the best alternative. the alternative ranking results based on the simulation are shown as follows. table 6. the ranking of entropy-topsis method ranking entropy-topsis preference  i c value alternative 1. 1 pill and implants 2. 0,905559 injections, condoms, and iud 3. 0,749978 mow the analysis of the calculation results using topsis method of 6 criteria and 6 alternatives was obtained that pill and implant has the highest preference value of 1. meanwhile, injections, condoms, and iud were in the second ranked which has a preference value of 0.905559. furthermore, mow was in the last ranked which has a preference value of 0.749978. selection of ideal contraceptive tool using hybrid entropy-topsis method endang wahyu handamari 72 conclusions 1. based on the results of questionnaire data processing, it was obtained work rating matrix. furthermore, by using entropy method, it was obtained the eight entropy vector, i.e.:  025.00334.01333.0775.00083.0025.0 . by using a weighted simulation  2;3;4;2;3;3*  w , it was obtained the final weighted entropy value, that is  0,0214320,0429490,2285470,6643810,0106730,032147*  . 2. the final determintaion entropy value was used to rank alternatives using the hybrid entropy-topsis method. the ranking results indicate the pills and implants ae the alternative selection of the ideal contraceptives. references [1] badan pusat statistik. proyeksi penduduk indonesia. https://www.bappenas.go.id/files/.../proyeksi_penduduk_indonesia_2010-2035.pdf. retrieved on august 10th, 2017 [2] abusini, s. 2013. klasifikasi pemakaian alat kontrasepsi yang potensional dengan metode naïve bayes classifier. research report boptn. [3] wahyu, e. 2014. klasifikasi alat kontrasepsi menggunakan metode backpropagation. research report dpp/spp. [4] wahyu, e. 2016. eksplorasi pola pemakaian alat kontrasepsi yang efektif. research report dpp/spp. [5] gang l.,tai c.g., qiu c.y. 2011. evaluation model and empirical study of human all-round development based on entropy weight and topsis. journal of system engineering 03. [6] li y., zhao l. suo j. 2014. comprehensive assessment on sustainable development of highway transportation capacity based on entropy weight and topsis. journal sustainability 6(7) [7] wen-yon-zhou. 2013. multiple criteria decision analysis state of the art surveys. amerika: springer science. retrieved on april 12th, 2017. [8] smaradanche, f dan ali, m. 2013. neutrosophic sets and systems. vol.1. mexico:university of mexico. diakses pada 12 april 2017. [9] sandeep, p.c tewari, prakash, u., dan khanduja, k. 2015. ranking of sintered material for high loaded automobile application by applying entropy-topsis method. elsevier. india. https://www.bappenas.go.id/files/.../proyeksi_penduduk_indonesia_2010-2035.pdf suma’inna dan gugun gumilar ring stabil berhingga samsul arifin program studi pendidikan matematika, stkip surya, tangerang email: samsul.arifin@stkipsurya.ac.id abstract dalam tulisan ini akan dibahas mengenai karakteristik ring dengan sifat stabil berhingga, motivasi dan sifat-sifatnya. akan dibahas juga mengenai sifat stabil berhingga kuat kanan yang akan membantu dalam memahami sifat stabil berhingga, beserta sifat-sifatnya, dan tidak lupa akan dikaji juga kaitan antara keduanya. kata kunci: stably finite, right strong stably finite, dedekind finite abstract in this paper, we will discuss about the characterization of stably finite rings and their properties. we will also discuss about right strong stably finite rings and their relationship, and also its characterization. keywords: stably finite, right strong stably finite, dedekind finite pendahuluan perlu diasumsikan sebelumnya di sini bahwa setiap ring bersifat asosiatif dengan 1 ≠ 0, dan setiap modulnya adalah modul unital. berdasarkan [6], r disebut ring stabil berhingga jika 𝑀𝑛(𝑅) adalah ring dedekind finite. di lain pihak, r disebut ring dedekind finite jika untuk setiap 𝑥, 𝑦 ∈ 𝑅 berlaku 𝑥𝑦 = 1 ⇒ 𝑦𝑥 = 1. dalam tulisan ini akan dibahas kaitan ring stabil berhingga dengan ring stabil berhingga kuat (kanan), modul co-hopfian lemah dan ring dengan kondisi rank kuat kanan. modul kanan 𝑀𝑅 disebut modul co-hopfian jika sebarang endomorfisma injektif dari m adalah suatu isomorfisma, dan 𝑀𝑅 disebut modul co-hopfian lemah jika sebarang endomorfisma injektif f dari m adalah essensial, yaitu berlaku 𝑓(𝑀) ⊆𝑒 𝑀. ring stabil berhingga sebelum dijelaskan mengenai ring stabil berhingga, akan dijelaskan terlebih dahulu ring dedekind finite karena terkait satu sama lain. 1.1. definisi (ring dedekind finite): ring r dikatakan ring yang dedekind (directly atau von neumann) finite jika untuk setiap 𝑥, 𝑦 ∈ 𝑅 berlaku 𝑥𝑦 = 1 ⇒ 𝑦𝑥 = 1. 1.2. definisi (stably finite): ring r dikatakan ring yang stably-finite jika 𝑀𝑛(𝑅) adalah ring dedekind finite ∀𝑛 ≥ 1. 1.3. akibat: ring r bersifat stably finite jika dan hanya jika ring r bersifat dedekind finite. sifat-sifat dari ring stabil berhingga tertuang dalam proposisi di bawah ini, dimana suatu ring stabil berhingga r dapat dilihat dari sisi jumlahan langsung r-modul bebasnya dengan rmodul lain, epimorfisma r-modul bebas ke dirinya sendiri dan sifat dedekind finite. 1.4. proposisi: pernyataan-pernyataan di bawah ini ekuivalen: (i) ring r adalah ring stabil berhingga. (ii) untuk setiap 𝑛 ∈ ℤ+, jika 𝑅𝑛 ≅ 𝑅𝑛 ⊕ 𝐾 sebagai r-modul kanan maka 𝐾 = 0. (iii) untuk setiap 𝑛 ∈ ℤ+, setiap epimorfisma 𝑅𝑛 → 𝑅𝑛 adalah isomorfisma. (iv) end(𝑅𝑅 ) adalah ring dedekind finite. bukti: (𝑖) ⇒ (𝑖𝑖) diketahui ring r memenuhi sifat stabil berhingga. diambil sebarang 𝑛 ∈ ℤ+ dengan 𝑅𝑛 ≅ 𝑅𝑛 ⊕ 𝐾 sebagai r-modul kanan. karena 𝑅𝑛 ≅ 𝑅𝑛 ⊕ 𝐾 maka ada isomorfisma modul 𝑓: 𝑅𝑛 → 𝑅𝑛 ⊕ 𝐾 dengan matriks representasi [𝑓] ∈ 𝑀𝑚×𝑛 (𝑅) dan 𝑚 = 𝑛 + 𝑥 untuk setiap m dimensi dari 𝑅𝑛 ⊕ 𝐾 dan x dimensi dari k. selanjutnya ada 𝑔: 𝑅𝑛 ⊕ 𝐾 → 𝑅𝑛 dengan matriks representasi [𝑔] ∈ 𝑀𝑛×𝑚 (𝑅) sedemikian hingga 𝑓 ∘ 𝑔 = 1𝑅𝑚 dan 𝑔 ∘ 𝑓 = 1𝑅𝑛 . akibatnya diperoleh [𝑓][𝑔] = 𝐼𝑚 dan [𝑔][𝑓] = 𝐼𝑚 ∈ 𝑀𝑛(𝑅). berdasarkan (𝑖), 𝑀𝑛(𝑅) adalah ring dedekind finite sehingga karena [𝑔][𝑓] = 𝐼𝑚 ∈ 𝑀𝑛(𝑅) maka berlaku [𝑓][𝑔] = 𝐼𝑛 ∈ 𝑀𝑛 (𝑅). di lain pihak [𝑓][𝑔] = 𝐼𝑚 ∈ 𝑀𝑚 (𝑅), 𝑚 = 𝑛 + 𝑥 atau 𝐼𝑚 = [𝑓][𝑔] = 𝐼𝑛 , sehingga akibatnya diperoleh 𝑥 = 0 atau 𝐾 = 0. mailto:samsul.arifin@stkipsurya.ac.id samsul arifin 226 volume 2 no. 4 mei 2013 (𝑖𝑖) → (𝑖𝑖𝑖) diambil sebarang 𝑛 ∈ ℤ+ dan epimorfisma 𝑓: 𝑅𝑛 → 𝑅𝑛 .. karena 𝑅𝑛 adalah r-modul bebas, maka f split, akibatnya dapat dibentuk barisan 0 → ker(𝑓) ⟶ 𝑖 𝑅𝑛 ⟶ 𝑓 𝑅𝑛 → 0 dengan 𝑅𝑛 = 𝑅𝑛 ⊕ ker (𝑓), sehingga berdasarkan (𝑖𝑖) diperoleh ker(𝑓) = 0 yang artinya f injektif. karena f surjektif dan injektif sekaligus, maka f adalah isomorfisma. (𝑖𝑖𝑖) ⇒ (𝑖) diambil sebarang 𝑛 ∈ ℤ+ dan 𝑋, 𝑌 ∈ 𝑀𝑛 (𝑅) dengan 𝑋𝑌 = 𝐼𝑛 . akibatnya diperoleh 𝑅𝑛 ⟶ 𝑔 𝑅𝑛 ⟶ 𝑓 𝑒𝑝𝑖 𝑅𝑛 dengan matriks-matriks representasi 𝑋 = [𝑓] ∈ 𝑀𝑛(𝑅), 𝑌 = [𝑔] ∈ 𝑀𝑛(𝑅). karena f adalah epimorfisma, maka berdasarkan (𝑖𝑖𝑖) f adalah isomorfisma, yang artinya ada 𝑍 ∈ 𝑀𝑛(𝑅) dengan 𝑍𝑋 = 𝐼𝑛 . perhatikan bahwa dengan mengalikan z dari kiri pada persamaan 𝑋𝑌 = 𝐼𝑛 diperoleh: 𝑍(𝑋𝑌) = 𝑍𝐼𝑛 ⇔ (𝑍𝑋)𝑌 = 𝑍 ⇔ 𝐼𝑛 𝑌 = 𝑍 ⇔ 𝑌 = 𝑍 dengan demikian diperoleh 𝑋𝑌 = 𝐼𝑛 yang artinya 𝑀𝑛(𝑅) ring dedekind finite atau ring r stabil berhingga. (𝑖𝑖𝑖) ⇒ (𝑖𝑣) diketahui untuk setiap 𝑛 ∈ ℤ+, setiap epimorfisma 𝑅𝑛 → 𝑅𝑛 adalah isomorfisma. diambil sebarang 𝑎, 𝑏 ∈ end(𝑅𝑅 ) dengan 𝑏 = 1end(𝑅𝑅). akibatnya diperoleh bahwa a surjektif, dan berdasarkan (𝑖𝑖𝑖) diperoleh bahwa a adalah isomorfisma, sehingga terdapat 𝑐 ∈ end(𝑅𝑅 ) sedemikian hingga 𝑎 = 1end(𝑅𝑅). perhatikan bahwa: 𝑐(𝑎𝑏) = 𝑐1 ⇔ (𝑐𝑎)𝑏 = 𝑐 ⇔ 1𝑏 = 𝑐 ⇔ 𝑏 = 𝑐, sehingga dapat diperoleh 𝑏𝑎 = 1, dan terbukti bahwa end(𝑅𝑅 ) adalah ring dedekind finite. (𝑖𝑣) ⇒ (𝑖𝑖𝑖) diketahui end(𝑅𝑅 ) adalah ring dedekind finite. diambil sebarang 𝑛 ∈ ℤ+ dan epimorfisma 𝑓: 𝑅𝑛 → 𝑅𝑛 . karena 𝑅𝑛 adalah r-modul bebas maka f split. jadi, terdapat 𝑔: 𝑅𝑛 → 𝑅𝑛 dengan 𝑓 ∘ 𝑔 = 1𝑅𝑛 . dari sini sudah bisa dikatakan bahwa 𝑓, 𝑔 ∈ 𝐸, dan berdasarkan (𝑖𝑣), dapat diperoleh 𝑔 ∘ 𝑓 = 1𝑅𝑛 . dengan demikian terbukti bahwa f adalah suatu isomorfosma. 1.5. contoh 1. ring komutatif memiliki sifat stabil berhingga, karena setiap ring komutatif merupakan ring dedekind finte. berdasarkan akibat 2.3 di atas, diperoleh bahwa setiap ring komutatif adalah ring stabil berhingga. 2. dalam [1], dijelaskan bahwa ring noetherian adalah ring stabil berhingga, dengan skema pembuktian sebagai berikut: 𝑅 noetherian ⇓ (𝑅𝑛)𝑅 noetherian ⇓ setiap epimorfisma 𝑓: 𝑅𝑛 → 𝑅𝑛 adalah isomorfisma ⇕ 𝑅 stabil berhingga perhatikan kembali bahwa jika 𝑔: 𝑅 → 𝑆 adalah suatu homomorfisma ring dengan ring s bersifat stabil berhingga, maka belum tentu ring r juga merupakan ring stabil berhingga. berikut merupakan syarat pemetaan di atas agar berlaku untuk sifat stabil berhingga, yang dijelaskan dalam teorema di bawah ini. 1.6. teorema: diberikan r adalah subring s. jika ring s adalah bersifat stabil berhingga maka r juga bersifat stabil berhingga. bukti: karena r adalah subring s maka ada pemetaan embedding 𝑔: 𝑅 → 𝑆. selanjutnya, dengan 𝑅 = 𝑆 = 𝑔(𝑅), maka elemen identitas e di ring r adalah elemen idempoten di ring s, dengan elemen idempoten komplemen 𝑓 = 1 − 𝑒 yang memenuhi 𝑅𝑓 = 𝑓𝑅 = 0. kemudian diambil sebarang 𝐴, 𝐵 ∈ 𝑀𝑛(𝑅) dengan 𝐴𝐵 = 𝑒𝐼𝑛. sebelumnya perhatikan bahwa ∀(𝐴 + 𝐹𝐼𝑛 ), (𝐵 + 𝑓𝐼𝑛 ) ∈ 𝑀𝑛 (𝑆) berlaku: (𝐴 + 𝑓𝐼𝑛 )(𝐵 + 𝑓𝐼𝑛 ) = 𝐴𝐵 + 𝑓 2𝐼𝑛 = 𝑒𝐼𝑛 + 𝑓𝐼𝑛 = (𝑒 + 𝑓)𝐼𝑛 = 𝐼𝑛 karena diketahui bahwa ring s adalah ring stabil berhingga, maka berlaku juga: 𝐼𝑛 = (𝐵 + 𝑓𝐼𝑛 )(𝐴 + 𝑓𝐼𝑛 ) = 𝐵𝐴 + 𝑓𝐼𝑛 sehingga diperoleh 𝐵𝐴 = 𝐼𝑛 − 𝑓𝐼𝑛 = (1 − 𝑒)𝐼𝑛 = 𝑒𝐼𝑛 . terbukti bahwa 𝑀𝑛 (𝑅) adalah ring dedekind-finite atau dengan kata lain r adalah ring stabil berhingga. dengan demikian terbukti bahwa jika r adalah subring s dengan s adalah ring stabil berhingga, maka ring r juga merupakan ring stabil berhingga. selanjutnya, untuk suatu ideal i di ring r, jika ring r adalah ring stabil berhingga maka belum tentu ring 𝑅/𝐼 adalah ring stabil berhingga. ring stabil berhingga jurnal cauchy – issn: 2086-0382 227 berikut adalah syarat bagi idealnya agar hal tersebut berlaku, yang dijelaskan dalam teorema di bawah ini. 1.7. teorema: diberikan ideal 𝐼 ⊆ rad (𝑅). ring r adalah ring stabil behingga jika dan hanya jika ring 𝑅/𝐼 adalah ring stabil berhingga. bukti: karena ring r adalah ring stabil berhingga, maka 𝑀𝑛(𝑅) adalah ring dedekind finite. berdasarkan [6], diperoleh bahwa s adalah ring dedekind finite jika dan hanya jika 𝑆̅ = 𝑆/𝐽 ring dedekind finite. untuk ring r yang diberikan dan sebarang 𝑛 ≥ 1, jika dimisalkan 𝑆 = 𝑀𝑛(𝑅) maka diperoleh: 𝐽 = 𝑀𝑛(𝐼) ⊆ 𝑀𝑛 (rad(𝑅)) = rad(𝑀𝑛 (𝑅)) = rad (𝑆), dan 𝑆/𝐽 = 𝑀𝑛(𝑅)/𝑀𝑛(𝐼) ≅ 𝑀𝑛 (𝑅/𝐼) dari sini akan diperoleh bahwa ring 𝑆 = 𝑀𝑛(𝑅) adalah ring dedekind finite jika dan hanya jika 𝑆/𝐽 ≅ 𝑀𝑛(𝑅/𝐼) ring dedekind finite. karena ini berlaku untuk setiap 𝑛 ∈ ℤ+, maka 𝑅/𝐼 adalah ring stabil berhingga. dengan demikian terbukti bahwa untuk suatu ideal 𝐼 ∈ rad (𝑅), berlaku r adalah ring stabil berhingga jika dan hanya jika 𝑅/𝐼 ring stabil berhingga. 1.8. akibat: diberikan ∏ 𝑖∈𝐼 𝑅𝑖 adalah hasil kali langsung dari 𝑅𝑖 . ∏ 𝑖∈𝐼 𝑅𝑖 adalah ring stabil berhingga jika dan hanya jika 𝑅𝑖 untuk setiap 𝑖 ∈ 𝐼 adalah ring stabil berhingga. bukti: ⇒ diketahui ∏ 𝑖∈𝐼 𝑅𝑖 adalah ring stabil berhingga. karena ada pemetaan embedding 𝜀: 𝑅𝑖 → ∏ 𝑖∈𝐼 𝑅𝑖 maka berlaku 𝑅𝑖 juga merupakan ring stabil berhingga. dengan demikian terbukti bahwa jika ∏ 𝑖∈𝐼 𝑅𝑖 adalah ring stabil berhingga, maka 𝑅𝑖 juga merupakan ring stabil berhingga. ⇐ diketahui 𝑅𝑖 adalah ring stabil berhingga untuk setiap 𝑖 ∈ ℤ+. berdasarkan [6], diperoleh bahwa untuk setiap 𝑛 ∈ ℤ+, 𝑀𝑛(𝑅𝑖 ) adalah ring stabil berhingga untuk setiap 𝑖 ∈ ℤ+. akibatnya diperoleh bahwa 𝑀𝑛(𝑅𝑖 ) adalah ring dedekind finite. selanjutnya perhatikan bahwa untuk sebarang (( 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏11 ⋯ 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) , ⋯ ) , (( 𝑎′11 ⋯ 𝑎′1𝑛 ⋮ ⋱ ⋮ 𝑎′𝑛1 ⋯ 𝑎′𝑛𝑛 ) , ( 𝑏′11 ⋯ 𝑏′1𝑛 ⋮ ⋱ ⋮ 𝑏′𝑛1 ⋯ 𝑏′𝑛𝑛 ) , ⋯ ) ∈ ∏ 𝑀𝑛(𝑅𝑖 ) 𝑖∈𝐼 dengan (( 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏11 ⋯ 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) , ⋯ ) (( 𝑎′11 ⋯ 𝑎′1𝑛 ⋮ ⋱ ⋮ 𝑎′𝑛1 ⋯ 𝑎′𝑛𝑛 ) , ( 𝑏′11 ⋯ 𝑏′1𝑛 ⋮ ⋱ ⋮ 𝑏′𝑛1 ⋯ 𝑏′𝑛𝑛 ) , ⋯ ) ∈ ∏ 𝐿𝑖 𝑖 berlaku: ( 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) ( 𝑏11 ⋯ 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) = 𝐼𝑖 = ( 𝑏11 ⋯ 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) ( 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) untuk setiap ( 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) ( 𝑏11 ⋯ 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) ∈ 𝑀𝑛 (𝑅𝑖 ), sehingga pasti berlaku (( 𝑎′11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎′𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏′11 … 𝑏′1𝑛 ⋮ ⋱ ⋮ 𝑏′𝑛1 ⋯ 𝑏′𝑛𝑛 ) , . . . ) (( 𝑎11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏11 … 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) , . . . ) = (( 𝑎′11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎′𝑛1 ⋯ 𝑎𝑛𝑛 ) ( 𝑎11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏′11 … 𝑏′1𝑛 ⋮ ⋱ ⋮ 𝑏′𝑛1 ⋯ 𝑏′𝑛𝑛 ) ( 𝑏11 … 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) , . . . ) = (( 𝑎11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ) ( 𝑎′11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎′𝑛1 ⋯ 𝑎𝑛𝑛 ) , ( 𝑏11 … 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 ⋯ 𝑏𝑛𝑛 ) ( 𝑏′11 … 𝑏′1𝑛 ⋮ ⋱ ⋮ 𝑏′𝑛1 ⋯ 𝑏′𝑛𝑛 ) , . . . ) = ∏ 𝐼𝑖 𝑖 , yang artinya bahwa ∏ 𝑀𝑛(𝑅𝑖)𝑖∈𝐼 adalah ring dedekind finite. dari sini diperoleh bahwa 𝑀𝑛(∏ 𝑅𝑖𝑖∈𝐼 ) juga merupakan ring dedekind finite, karena 𝑀𝑛(∏ 𝑅𝑖𝑖∈𝐼 ) ≅ ∏ 𝑀𝑛(𝑅𝑖 )𝑖∈𝐼 dengan ( (𝑎11, 𝑏11, … ) … (𝑎1𝑛 , 𝑏1𝑛, … ) ⋮ ⋱ ⋮ (𝑎𝑛1, 𝑏𝑛1, … ) … (𝑎𝑛𝑛 , 𝑏𝑛𝑛, … ) ) ↦ (( 𝑎11 … 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 … 𝑎𝑛𝑛 ) , ( 𝑏11 … 𝑏1𝑛 ⋮ ⋱ ⋮ 𝑏𝑛1 … 𝑏𝑛𝑛 ) , … ) untuk setiap 𝑎𝑖 ∈ 𝐴𝑖, yang artinya bahwa 𝑀𝑛(∏ 𝑅𝑖𝑖∈𝐼 ) juga merupakan ring stabil berhingga. oleh karena itu diperoleh bahwa ∏ 𝑅𝑖𝑖∈𝐼 adalah ring stabil berhingga. dengan demikian terbukti bahwa jika 𝑅𝑖 adalah ring stabil berhingga maka ∏ 𝑅𝑖𝑖∈𝐼 juga merupakan ring stabil berhingga. sebelumnya akan dijelaskan proses konstruksi dari ring matriks triangular formal. dari ring a, b, (𝐴, 𝐵)-bimodul 𝑀 dapat dibentuk ring: 𝑇 = ( 𝐴 0 𝑀 𝐵 ) = {( 𝑎 0 𝑚 𝑏 ) |𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵, 𝑚 ∈ 𝑀} yang disebut ring matriks triangular formal. operasi penjumlahan dan pergandaan pada ring t adalah sebagai berikut: samsul arifin 228 volume 2 no. 4 mei 2013 a) ( 𝑎 0 𝑚 𝑏 ) + ( 𝑎′ 0 𝑚′ 𝑏′ ) = ( 𝑎 + 𝑎′ 0 𝑚 + 𝑚′ 𝑏 + 𝑏′ ) b) ( 𝑎 0 𝑚 𝑏 ) ( 𝑎′ 0 𝑚′ 𝑏′ ) = ( 𝑎𝑎′ 0 𝑚𝑎′ + 𝑏𝑚′ 𝑏𝑏′ ) untuk setiap ( 𝑎 0 𝑚 𝑏 ) , ( 𝑎′ 0 𝑚′ 𝑏′ ) ∈ 𝑇. kemudian jika diberikan ring-ring a dan b, suatu grup abelian m disebut (𝐴, 𝐵)–bimodul, dinotasikan 𝑀𝐴 𝑏 , jika m adalah a-modul kiri dan b-modul kanan serta berlaku (𝑎𝑚)𝑏 = 𝑎(𝑚𝑏) untuk setiap 𝑎 ∈ 𝐴, 𝑚 ∈ 𝑀, dan 𝑏 ∈ 𝐵. berikut adalah kaitan antara ring stabil berhingga dan ring matriks triangular formal. 1.9. proposisi diberikan ring a, b, dan 𝑀𝐵 𝐴 adalah (𝐵, 𝐴)bimodul. ring 𝑇 = ( 𝐴 0 𝑀 𝐵 ) adalah ring stabil berhingga jika dan hanya jika ring a dan b adalah ring stabil berhingga. bukti: ⇒ diketahui ring 𝑇 = ( 𝐴 0 𝑀 𝐵 ) adalah ring stabil berhingga. perhatikan bahwa ada pemetaan embedding 𝜀1: 𝐴 → ( 𝐴 0 𝑀 𝐵 ) dengan ( 𝑎 0 0 0 ) ↦ ( 𝑎 0 𝑚 𝑏 ) untuk setiap ( 𝑎 0 0 0 ) ∈ ( 𝐴 0 0 0 ) ≅ 𝐴, dan juga ada embedding 𝜀2: 𝐵 → ( 𝐴 0 𝑀 𝐵 ) dengan ( 0 0 0 𝑏 ) ↦ ( 𝑎 0 𝑚 𝑏 ) untuk setiap ( 0 0 0 𝑏 ) ∈ ( 0 0 0 𝐵 ) ≅ 𝐵. akibatnya, karena ring 𝑇 = ( 𝐴 0 𝑀 𝐵 ) adalah ring stabil berhingga maka berakibat masing-masing ring a dan b juga merupakan ring stabil berhingga. dengan demikian terbukti bahwa jika ring 𝑇 = ( 𝐴 0 𝑀 𝐵 ) adalah ring stabil berhingga maka ring a dan b juga merupakan ring stabil berhingga. ⇐ diketahui ring a dan b adalah ring stabil berhingga. perhatikan kembali bahwa ideal 𝐼 = ( 0 0 𝑀 0 ) ⊆ 𝑇 adalah ideal nilpotent dan 𝑇/𝐼 ≅ 𝐴 × 𝐵. selanjutnya, jika diberikan 𝑛 ∈ ℤ+ sebarang dan dinotasikan ring 𝑋𝑛 = 𝑀𝑛(𝑋), maka akan diperoleh: 𝑇𝑛/𝐼𝑛 = 𝑀𝑛(𝑇)/𝑀𝑛(𝐼) = 𝑀𝑛 (( 𝐴 0 𝑀 𝐵 )) 𝑀𝑛 (( 0 0 𝑀 0 ))⁄ ≅ 𝑀𝑛 (( 𝐴 0 𝑀 𝐵 ) ( 0 0 𝑀 0 )⁄ ) = 𝑀𝑛(𝑇/𝐼) = (𝑇/𝐼 )𝑛 ≅ (𝐴 × 𝐵)𝑛 perhatikan bahwa ring a dan b adalah ring stabil berhingga jika dan hanya jika ring 𝐴 × 𝐵 adalah ring stabil berhingga, sehingga ring 𝑀𝑛 (𝐴 × 𝐵) = (𝐴 × 𝐵)𝑛 ≅ 𝑇𝑛 /𝐼𝑛 = 𝑀𝑛 (( 𝐴 0 𝑀 𝐵 )) 𝑀𝑛 (( 0 0 𝑀 0 ))⁄ adalah ring dedekind finte. karena 𝐼𝑛 = 𝑀𝑛 (𝐼) adalah ideal nilpotent maka diperoleh bahwa 𝑇𝑛 = 𝑀𝑛 ( 𝐴 0 𝑀 𝐵 ) adalah ring dedekind finte, sehingga diperoleh bahwa ring 𝑇 = 𝑀𝑛 ( 𝐴 0 𝑀 𝐵 ) adalah ring stabil berhingga. dengan demikian terbukti bahwa jika a dan b adalah ring stabil berhingga maka ring 𝑇 = ( 𝐴 0 𝑀 𝐵 ) juga merupakan ring stabil berhingga. dalam [1], dijelaskan juga karakteristik lain dari ring stabil berhingga, yang termuat dalam skema berikut: 𝑀𝑛(𝑅) 𝑖𝑠 "𝑆𝐵" ⇔ 𝑅 𝑖𝑠 "𝑆𝐵" ⇔ 𝑅[𝑥] 𝑖𝑠 "𝑆𝐵" ⇕ 𝑅[[𝑥]] 𝑖𝑠 "𝑆𝐵" dengan sb = stabil berhingga, serta untuk sebarang ring tak nol berlaku: stabil berhingga ⇒ kondisi rank ⇒ ibn. modul co-hopfian lemah dan ring stabil berhingga kuat kanan sebelum dijelaskan mengenai ring stabil berhingga kuat kanan, akan dijelaskan terlebih dahulu mengenai modul co-hopfian (lemah) karena saling berkaitan satu sama lain. ring stabil berhingga jurnal cauchy – issn: 2086-0382 229 1.10. definisi (modul co-hopfian (lemah)): diberikan ring r dan 𝑀𝑅 adalah modul kanan. m disebut modul co-hopfian jika sebarang endomorfisma injektif dari m adalah suatu isomorfisma, dan m disebut modul co-hopfian lemah jika sebarang endomorfisma injektif f dari m adalah essensial, yaitu berlaku 𝑓(𝑀) ⊆𝑒 𝑀. dalam [4] dijelaskan, ada teorema yang menyatakan bahwa definisi modul co-hopfian lemah bisa dinyatakan dalam enam kalimat, yaitu sebagai berikut: 1.11. teorema pernyataan-pernyataan berikut ekuivalen untuk suatu r-modul kanan m: (1) 𝑀𝑅 adalah modul co-hopfian lemah. (2) untuk sebarang r-modul kanan n, jika ada rmonomorfisma 𝑀 ⊕ 𝑁 → 𝑀 maka 𝑁 = 0. (2’) untuk sebarang r-modul kanan n, jika 𝑀 ⊕ 𝑁 → 𝑀 adalah monomorfisma essensial maka 𝑁 = 0. (3) m adalah modul dedekind finite dan image dari sebarang endomorfisma injektif dari m adalah essensial atau direct summand sejati. (4) terdapat suatu submodul essensial invariant penuh yang bersifat co-hopfian lemah. (5) endomorfisma injektif dari 𝑀𝑅 memetakan submodul-submodul essesnsial ke submodulsubmodul essensial. (6) image inverse dari sebarang submodul tak nol dari sebarang endomorfisma injektif dari m adalah tak nol. berikut adalah sifat-sifat dari modul co-hopfian lemah: 1.12. akibat: suatu direct summand dari modul co-hopfian lemah juga merupakan modul co-hopfian lemah. proposisi berikut menjelaskan kaitan antara modul co-hopfian (lemah) dan modul dedekind finite, yaitu sebagai berikut. 1.13. proposisi: pernyataan-pernyataan berikut berlaku: (1) untuk suatu r-modul m berlaku: m co-hopfian ⇒ m co-hopfian lemah ⇒ m dedekind finite. (2) jika m adalah modul quasi-injektif, maka berlaku: a) m co-hopfian ⇔ m co-hopfian lemah ⇔ m dedekind finite. b) jika m adalah modul co-hopfian lemah maka sebarang submodul dan sebarang direct sum berhingga yang memuat m adalah modul co-hopfian lemah. c) jika n adalah submodul essensial invariant penuh, maka berlaku: n co-hopfian ⇔ m co-hopfian ⇔ 𝐸(𝑀) co-hopfian berikut adalah sifat-sifat modul co-hopfian: 1.14. proposisi: pernyataan-pernyataan berikut berlaku: a) jika 𝑀𝑅 adalah modul kanan dengan sebarang submodul co-hopfian lemah, maka m adalah modul co-hopfian lemah. b) diberikan n adalah submodul invariant penuh yang co-hopfian di modul kanan 𝑀𝑅. jika 𝑀 𝑁⁄ adalah modul co-hopfian (lemah) maka m modul co-hopfian (lemah). c) jika submodul-submodul nonessensial di 𝑀𝑅 adalah submodul noetherian, maka m adalah modul co-hopfian lemah. berikut adalah akibat dari proposisi di atas: 1.15. akibat: pernyataan-pernyataan berikut berlaku: a) suatu modul semisederhana m adalah modul co-hopfian lemah jika dan hanya jika sebarang komponen homogen di m dibangun secara berhingga. b) jika m adalah modul hopfian dan bersifat kondisi rantai turun, maka submodulsubmodulnya adalah co-hopfian. c) jika m memiliki sifat kondisi rantai turun pada submodul-submodul non-co-hopfiannya, maka m adalah modul co-hopfian. setelah dijelaskan mengenai modul cohopfian (lemah) dan sifat-sifatnya, selanjutnya akan dibahas mengenai ring stabil berhingga kuat. 1.16. definisi (right strong stably finite): suatu ring r disebut ring stabil berhingga kuat kanan jika 𝑅𝑅 𝑛 adalah modul co-hopfian lemah untuk setiap 𝑛 ≥ 1. berikut adalah karakteristik ring stabil berhingga kuat kanan. dalam [4] dijelaskan bahwa samsul arifin 230 volume 2 no. 4 mei 2013 ring stabil berhingga kuat kanan dapat dinyatakan dalam tiga kalimat yaitu tertuang dalam teorema berikut: 1.17. teorema: pernyataan-pernyataan berikut ekuivalen: (1) r adalah ring stabil berhingga kuat kanan. (2) untuk sebarang 𝑛 ≥ 1, jika 𝑢1, . . . , 𝑢𝑛 adalah elemen-elemen yang r-bebas linear di dalam modul bebas 𝑅𝑅 𝑛 maka 𝑢1𝑅+. . . +𝑢𝑛 𝑅 adalah submodul essensial. (3) untuk sebarang 𝑛 ≥ 1, 𝑀𝑛(𝑅) adalah ring cohopfian kanan. 1.18. definisi: ring r dikatakan ore kanan jika untuk setiap 𝑎, 𝑏 ∈ 𝑅 dengan b adalah elemen regular, terdapat 𝑐, 𝑑 ∈ 𝑅 dengan d adalah elemen regular, sehingga berlaku 𝑎𝑑 = 𝑏𝑐. dalam [7] dijelaskan kaitan antara ring ore kanan dan ring stabil berhingga kuat kanan, yaitu sebagai berikut: 1.19. teorema: jika r adalah ring dengan 𝑅[𝑥] adalah ring ore kanan, maka r adalah ring stabil berhingga kuat kanan. kaitan antara ring stabil berhingga dengan ring stabil berhingga kuat kanan berikut adalah kaitan antara ring stabil berhingga dan ring stabil berhingga kuat kanan. dalam [7] dijelaskan kaitan antara ring stabil berhingga, ring stabil berhingga kuat kanan, ring dengan kondisi rank kuat kanan dan ring komutatif, yaitu sebagai berikut: 𝑆𝐵 ⇑ 𝑢. dim𝑅𝑅 < ∞ ⇒ 𝑆𝐵𝐾𝐾 ⇒ 𝐾𝑅𝐾𝐾 ⇑ 𝐾𝑜𝑚𝑢𝑡𝑎𝑡𝑖𝑓 dengan sb = stabil berhingga, sbkk = stabil berhingga kuat kanan dan krkk = kondisi rank kuat kanan. 1.20. proposisi: ring komutatif ⇒ ring stabil berhingga kuat kanan ⇒ ring stabil berhingga. proposisi berikut menjelaskan kaitan antara ring stabil berhingga dan ring stabil berhingga kuat kanan. 1.21. proposisi: jika ring r adalah ring stabil berhingga kuat kanan, maka ring r adalah ring stabil berhingga. lebih lanjut, untuk suatu modul injektif 𝑅𝑅 , ring r adalah ring stabil berhingga kuat kanan jika dan hanya jika ring r adalah ring stabil berhingga. bukti: misalkan r adalah ring stabil berhingga kuat kanan dan 𝑛 ∈ ℤ+ dengan 𝑅𝑛 ⊕ 𝐾 ≅ 𝑅𝑛 untuk suatu 𝐾 ∈ 𝑀𝑜𝑑 − 𝑅. berdasarkan [4, 1.1], dapat diperoleh bahwa 𝑅𝑛 ⊕ 𝐾 dapat diembeddingkan ke dalam 𝑅𝑛 sehingga berakibat 𝐾 = 0. dengan menggunakan [6, 1.7], maka dapat dibuktikan bahwa r adalah ring stabil berhingga. sebaliknya, misalkan 𝑅𝑅 adalah modul injektif dan r adalah ring stabil berhingga. dengan menggunakan [4, 1.4], maka diperoleh bahwa 𝑅𝑅 adalah modul co-hopfian, sehingga berdasarkan [4, 1.5(i)], diperoleh bahwa 𝑅𝑅 𝑛 adalah modul cohopfian, (∀𝑛 ∈ ℤ+). oleh karena itu terbukti bahwa r adalah ring stabil berhingga kuat kanan. proposisi berikut menjelaskan kaitan antara ring stabil berhingga kuat kanan dan ring dengan kondisi rank kuat kanan. 1.22. proposisi: jika r adalah ring stabil berhingga kuat kanan, maka ring r bersifat kondisi rank kuat kanan. lebih lanjut, untuk suatu daerah integral r, r adalah ring stabil berhingga kuat kanan jika dan hanya jika ring r bersifat kondisi rank kuat kanan. bukti: diasumsikan bahwa 0 → 𝑅𝑚 → 𝑅𝑛 adalah barisan eksak di dalam 𝑀𝑜𝑑 − 𝑅 dengan 𝑚 > 𝑛. dari sini dapat diperoleh bahwa 𝑅𝑛 ⊕ 𝑅𝑚−𝑛 ≅ 𝑅𝑚 dapat diembeddingkan ke dalam 𝑅𝑛 . karena 𝑅𝑛 diasumsikan adalah modul co-hopfian lemah, maka diperoleh kontradiksi dengan 𝑅𝑚−𝑛 = 0. oleh karena itu terbukti bahwa ring r bersifat kondisi rank kuat kanan. sebaliknya, dari [6, 1.32] diperoleh bahwa untuk suatu daerah integral, sifat kondisi rank kuat kanan ekuivalen dengan sifat ore kanan, sehingga menjadi seragam. ada cara lain untuk untuk membuktikan hal ini yaitu: misalkan r adalah daerah integral dengan sifat kondisi rank kuat kanan dan 𝑁 ⊆ 𝑅𝑘 sedemikian hingga 𝑁 ≅ 𝑅𝑅 𝑘 . jika n bukan submodul esensial, maka terdapat suatu submodul tak nol l di modul 𝑅𝑘 dengan 𝑁 ∩ 𝐿 = 0. untuk sebarang elemen tak ring stabil berhingga jurnal cauchy – issn: 2086-0382 231 nol 𝑙 ∈ 𝐿 berlaku 𝑙𝑅 ≅ 𝑅 karena r adalah daerah integral, sehingga berlaku 𝑅𝑘 ⊕ 𝑅 ≅ 𝑁 ⊕ 𝑙𝑅 ⊆ 𝑅𝑘 . hal ini kontradiksi dengan r yang bersifat kondisi rank kuat kanan. oleh karena itu, pastilah n adalah submodul esensial sehingga r adalah ring stabil berhingga kuat kanan. selanjutnya akan dibahas mengenai sifatsifat ring stabil berhingga kuat kanan. proposisi berikut menjelaskan kaitan ring stabil berhingga kuat kanan r dan vektor-vektor yang r-bebas linear pada modul bebas 𝑅𝑅 𝑛 . 1.23. proposisi: r adalah ring stabil berhingga kuat kanan jika dan hanya jika ∀𝑛 ≥ 1, jika 𝑢1 , 𝑢2 , . . . , 𝑢𝑛 vektor-vektor yang r-bebas linear di modul 𝑅𝑅 𝑛 , maka 𝑢1 𝑅 + 𝑢2𝑅+. . . +𝑢𝑛 𝑅 adalah suatu submodul esensial di modul 𝑅𝑅 𝑛 . bukti: misalkan r adalah ring stabil berhingga kuat kanan dan 𝑢1, 𝑢2, . . . , 𝑢𝑛 adalah vektor-vektor yang r-bebas linear di modul n r r . pemetaan 𝑓: 𝑅𝑛 → 𝑅𝑛 dengan 𝑓(𝑟1, 𝑟2, . . . , 𝑟𝑛 ) = 𝑢1𝑟1 + 𝑢2𝑟2+. . . 𝑢𝑛𝑟𝑛 adalah suatu r-monomorfisma kanan dengan im(𝑓) = 𝑢1𝑅 + 𝑢2𝑅+. . . 𝑢𝑛𝑅. karena 𝑅 𝑛 adalah modul co-hopfian maka diperoleh im(𝑓) ⊆𝑒 𝑅𝑅 𝑛 . sebaliknya, misalkan g adalah endomorfisma injektif dari 𝑅𝑅 𝑛 dengan 𝑔(𝑒𝑖 ) = 𝑢𝑖 di mana 𝑒𝑖 = (0, . . . ,1,0, . . . ,0). karena g injektif maka jika 𝑢1 𝑟1 + 𝑢2𝑟2 +. . . 𝑢𝑛 𝑟𝑛 = 0 akan berakibat 𝑟1 = 𝑟2 =. . . = 𝑟𝑛 = 0. oleh karena itu diperoleh bahwa 𝑢1 , 𝑢2 , . . . , 𝑢𝑛 adalah vektor-vektor yang r-bebas linear sehingga berdasarkan asumsi yang digunakan, im(𝑔) = 𝑢1𝑅 + 𝑢2𝑅+. . . +𝑢𝑛𝑅 adalah submodul esensial. 1.24. teorema: setiap ring komutatif adalah ring stabil berhingga kuat. bukti: kita gunakan proposisi 4.4 di atas pada kasus suatu ring komutatif r. misalkan 𝑢1, 𝑢2, . . . , 𝑢𝑛 adalah vektor-vektor yang r-bebas linear di modul 𝑅𝑅 𝑛 dan himpunan i = 𝑢1𝑅 + 𝑢2𝑅+. . . 𝑢𝑛𝑅. akan ditunjukkan bahwa untuk sebarang vector tak nol 𝑢𝑛+1 ∈ 𝑅 𝑛 berlaku 𝐼 ∩ 𝑢𝑛+1𝑅 ≠ 0. tulis 𝑢𝑗 = ∑ 𝑒𝑖 𝑎𝑖𝑗𝑖 (𝑗 = 1, . . . , 𝑛 + 1 dan 𝑖 = 1, . . . , 𝑛) dengan 𝑎𝑖𝑗 ∈ 𝑅, kemudian notasikan s sebagai subring dari r yang dibangun oleh elemen-elemen 𝑎𝑖𝑗 ∈ 𝑅 dan 1𝑅. berdasarakan teorema basis hilbert, s adalah ring noetherian, sehingga s memiliki dimensi seragam yang berhingga. akibatnya s adalah ring stabil berhingga kuat. perhatikan bahwa 𝑢1, 𝑢2, . . . , 𝑢𝑛+1 ∈ 𝑆 𝑛 dan 𝑢1, 𝑢2, . . . , 𝑢𝑛 adalah sbebas linear. oleh karena itu berlaku (𝑢1𝑆 + 𝑢2𝑆+. . . 𝑢𝑛𝑆) ∩ 𝑢𝑛+1𝑆 ≠ 0 dan berdasarkan proposisi di atas maka berlaku 𝐼 ∩ 𝑢𝑛+1𝑅 ≠ 0 atau 𝐼 ⊆𝑒 𝑅 𝑛 . terdapat kaitan yang sangat erat antara ring stabil berhingga kuat kanan, modul cohopfian lemah, dan ring yang memenuhi kondisi rank kuat kanan. hal tersebut dijelaskan dalam [4], yaitu dalam proposisi berikut: 1.25. proposisi: diberikan r adalah daerah integral. pernyataanpernyataan berikut ekuivalen: (1) rr adalah modul co-hopfian lemah (2) r adalah ring stabil berhingga kuat kanan (3) ring r memenuhi kondisi rank kuat kanan (4) sebarang dua elemen di ring r tidak r-bebas linear kanan referensi: [1] arifin, s., 2011, characteristic of ibn, rank condition and stably finite rings, proc. of the seams-gmu conference, 6, 223-232. [2] breaz, s., calugareanu, g., and schultz, p., 1991, modules with dedekind finite endomorphism rings, babes bolyai university, 1-13. [3] haghany, a., varadarajan, k., 2002, ibn and related properties for rings, acta mathematica hungarica, 94, 251-261. [4] haghany, a., vedadi, m.r., 2001, modules whose injective endomorphisms are essential, journal of algebra, 243, 765-779 [5] lam, t. y., exercises in modules and rings, 2007, springer verlag, new york. [6] lam, t. y., lectures on modules and rings, 1999, springer verlag, new york. [7] vedadi, m.r., 2009, strong stably finite rings and some extensions, acta math. univ. comenianae, 1, 137-144 pendahuluan ring stabil berhingga 1.1. definisi (ring dedekind finite): 1.2. definisi (stably finite): 1.3. akibat: 1.4. proposisi: 1.5. contoh 1.6. teorema: 1.7. teorema: 1.8. akibat: 1.9. proposisi modul co-hopfian lemah dan ring stabil berhingga kuat kanan 1.10. definisi (modul co-hopfian (lemah)): 1.11. teorema 1.12. akibat: 1.13. proposisi: pernyataan-pernyataan berikut berlaku: 1.14. proposisi: 1.15. akibat: pernyataan-pernyataan berikut berlaku: 1.16. definisi (right strong stably finite): 1.17. teorema: 1.18. definisi: 1.19. teorema: kaitan antara ring stabil berhingga dengan ring stabil berhingga kuat kanan 1.20. proposisi: 1.21. proposisi: 1.22. proposisi: 1.23. proposisi: 1.24. teorema: 1.25. proposisi: referensi: almost surjective epsilon-isometry in the reflexive banach spaces cauchy – jurnal matematika murni dan aplikasi volume 4 (4) (2017), pages 167-175 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 13 march 2017 reviewed: 26 may 2017 accepted: 30 may 2017 doi: http://dx.doi.org/10.18860/ca.v4i4.4100 almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman department of islamic studies, faculty of tarbiyah, stai ma’had aly al-hikam, malang, indonesia email: miminanira@gmail.com abstract in this paper, we will discuss some applications of almost surjective -isometry mapping, one of them is in lorentz space (𝐿𝑝,𝑞 -space). furthermore, using some classical theorems of 𝑤 ∗-topology and concept of closed subspace 𝛼-complemented, for every almost surjective -isometry mapping f : x → y, where y is a reflexive banach space, then there exists a bounded linear operator t : y → x with ‖𝑇‖ ≤ 𝛼 such that ‖𝑇𝑓(𝑥) − 𝑥‖ ≤ 4 for every 𝑥 ∈ 𝑋. keywords: almost surjectivity; -isometry; lorentz space; reflexive space. introduction suppose 𝑋 and 𝑌 be real banach spaces. a mapping 𝑈: 𝑋 → 𝑌 is called an isometry if ‖𝑈(𝑥) − 𝑈(𝑦)‖𝑌 = ‖𝑥 − 𝑦‖𝑋 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. mazur-ulam showed that if 𝑈 is a surjective isometry, then 𝑈 is affine. in other word, a surjective isometry mapping can be translated. this result lead to the following definition. definition 1.1. let 𝑋 and 𝑌 be real banach spaces. a mapping 𝑓: 𝑋 → 𝑌 is called an isometry if there exists ≥ 0 such that |‖𝑓(𝑥) − 𝑓(𝑦)‖𝑌 − ‖𝑥 − 𝑦‖𝑋 | ≤ , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. if = 0, then 0-isometry is just an isometry. there are other names of -isometry, some of them are nearisometry [13], approximate isometry ([1], [7]), and perturbed metric preserved [3]. to simplify, the norm fuction is just written as ‖∙‖ without mentioning the vector space. definition 1.1 above begs a question, “for every -isometry f, does always exist an isometry mapping u and 𝛾 > 0 such that ‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 𝛾 (1.1) for all 𝑥 ∈ 𝑋?”. this problem was first investigated by hyers-ulam in 1945 [7] (therefore, -isometry problems are also called as hyers-ulam problems) and they found that for every surjective isometry mapping f in the euclidean spaces, there always exists a surjective isometry mapping u http://dx.doi.org/10.18860/ca.v4i4.4100 mailto:miminanira@gmail.com almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 168 satisfying (1.1) with 𝛾 = 10. some years later, d. g bourgin studied -isometry mapping in the lebesgue space with 𝛾 = 12. in 1983, gevirtz delivered 𝛾 = 5 which was hold for any banach space x and y [6]. this result was sharpened by omladić and šemrl [10]. theorem 1.2. let x and y be banach spaces and f : x → y is a surjective -isometry, there exists a linear surjective isometry mapping u: x → y such that ‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 2 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. there are many applications of -isometry mapping, such as dai-dong who use -isometry mapping to determine the smoothness and rotundity of banach spaces. šemrl and väisälä proposed a definition of almost surjective mapping as follows [13]. definition 1.3. let f : x → y is a mapping, 𝑌1 is a closed subspace of y, and 𝛿 ≥ 0. a mapping f is called almost surjective onto y if for every 𝑦 ∈ 𝑌1, there exists 𝑥 ∈ 𝑋 with ‖𝑓(𝑥) − 𝑦‖ ≤ 𝛿 and for every 𝑢 ∈ 𝑋, there exists 𝑣 ∈ 𝑌1 with ‖𝑓(𝑢) − 𝑣‖ ≤ 𝛿. using definition 1.3, šemrl and väisälä weakened the surjective condition become almost surjective [13] (see also [14]). theorem 1.4. let e and f be hilbert spaces and f : e → f is an almost surjective -isometry with 𝑓(0) = 0 that satisfies sup ‖𝑦‖=1 lim |𝑡|→∞ inf ‖𝑡𝑦 − 𝑓(𝐸) 𝑡 ‖ < 1 , (1.2) then there exists a linear surjective isometry 𝑈: 𝐸 → 𝐹 such that ‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 2 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝐸. šemrl and väisälä showed that this result is true for e and f are lebesgue spaces [13]. vestfrid decrease the value 1 become ½ in (1.2) and is valid for any banach space [15]. on the other hand, figiel proved that for any isometry mapping u : x → y with u(0) = 0, there exists an operator 𝜙 ∶ 𝑠𝑝𝑎𝑛̅̅ ̅̅ ̅̅ ̅ 𝑈(𝑥) → 𝑋 with ‖𝜙‖ = 1 such that 𝜙 ° 𝑈 = 𝐼, the identity on x [5]. using figiel’s theorem, cheng et. al. give the following lemma [2] (see also qian [11]). lemma 1.5. if f : ℝ → 𝑌 is a surjective -isometry with f(0) = 0, then there exists 𝜙 ∈ 𝑌∗ with ‖𝜙‖ = 1 such that |〈𝜙, 𝑓(𝑡)〉 − 𝑡| ≤ 3 , 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑡 ∈ ℝ. using vestfrid’s theorem and lemma 1.5, minan et. al. showed that the following lemma is true [12]. theorem 1.6. let x and y be real banach spaces, f : x → y is an -isometry with f(0) = 0. if the mapping f satisfies almost surjective condition, i. e. sup 𝑦∈𝑆𝑌 lim |𝑡|→∞ inf ‖𝑡𝑦 − 𝑓(𝑋) 𝑡 ‖ < 1 2 , then for every 𝑥∗ ∈ 𝑋∗, there exists a linear functional 𝜙 ∈ 𝑌∗ with ‖𝜙‖ = ‖𝑥∗‖ = 𝑟 such that |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 𝑟, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 169 results and discussion we will discuss some applications of theorem 1.6. therefore, this paper will be organized as follows. firstly, we will discuss two applications of theorem 1.6, one of them is to determine the stability of almost surjective -isometry in 𝐿𝑝,𝑞 -spaces (also called as lorentz spaces). next, we will discuss the stability of almost surjective -isometry in reflexive banach spaces. the used notations and terminology are standard. 𝑤 ∗ (pronounced “weak star”) topology of the dual of normed space x is the smallest topology of 𝑋∗ such that for every 𝑥 ∈ 𝑋, the linear functional 𝑥∗ → 𝑥∗𝑥 on 𝑋∗ is continuous. the author is greatly indebted to anonymous referee for a number of important suggestions that help the author to simplify the proof. 1. applications of theorem 1.6 let x and y be banach spaces and 𝑌 ∈ 𝔅(𝑋, 𝑌). for bounded set 𝐶 ⊂ x, is defined ‖𝑇‖𝐶 = sup{‖𝑇𝑥‖ ∶ 𝑥 ∈ 𝐶}. if there is 𝑐 ∈ 𝐶 such that ‖𝑇𝑐‖ = ‖𝑇‖𝐶 , the t attains its supremum over c. operator t is called as norm-attaining operator. theorem 2.1. let 𝑋 and 𝑌 be real banach spaces, f : x → y is an almost surjective -isometry with 𝑓(0) = 0 satisfies sup 𝑦∈𝑆𝑌 lim |𝑡|→∞ inf ‖𝑡𝑦 − 𝑓(𝑋) 𝑡 ‖ < 1 2 , (2.1) then ‖∑ 𝜆𝑖 𝑓(𝑥𝑖 )‖ + 4 ≥ ‖∑ 𝜆𝑖 𝑥𝑖 ‖ for every 𝑥1, … , 𝑥𝑛 ∈ 𝑋 and every 𝜆1, … , 𝜆𝑛 ∈ ℝ , where ∑ 𝜆𝑖 𝑛 𝑖=1 = 1. proof. let 𝑥1, … , 𝑥𝑛 ∈ 𝑋 and 𝑋𝑛 = 𝑠𝑝𝑎𝑛 (𝑥1, … , 𝑥𝑛 ). banach spaces x and y that satisfy (2.1) are strictly convex [12]. in addition, 𝑋∗ and 𝑌∗ are 𝑤 ∗-compact convex, so that for every 𝑥∗ ∈ 𝑆𝑋𝑛∗ , there exists a linear functional 𝜑𝑥∗ ∈ 𝑆𝑌∗ such that |〈𝜑𝑥∗ , 𝑓(𝑥)〉 − 〈𝑥 ∗, 𝑥〉| ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋𝑛 . since f is an almost surjective -isometry, then for every 𝑦 ∈ 𝑆𝑌, there exists 𝑥0 ∈ 𝑋 such that f(𝑥0) = 𝑦. therefore, ‖𝑦 − 𝑓(𝑋) 𝑡 ‖ = ‖𝑓(𝑥0) − 𝑓(𝑡𝑥) 𝑡 ‖ ≥ |‖𝜑𝑥∗ ‖‖𝑓(𝑥0)‖ − ‖𝜑𝑥∗ ‖ ‖ 𝑓(𝑡𝑥) 𝑡 ‖| − ‖𝑥∗‖‖𝑥0 − 𝑥‖ + ‖𝑥 ∗‖‖𝑥0 − 𝑥‖ ≥ |‖𝜑𝑥∗ ‖‖𝑓(𝑥0)‖ − ‖𝜑𝑥∗ ‖‖𝑓(𝑥)‖| − ‖𝑥 ∗‖‖𝑥0 − 𝑥‖ + ‖𝑥 ∗‖‖𝑥0 − 𝑥‖ ≥ sup 𝑥∗∈𝑆𝑋𝑛 ∗ |〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥 ∗, 𝑥0 − 𝑥〉| + |〈𝜑𝑥∗ , 𝑓(𝑥0)〉 − 〈𝜑𝑥∗ , 𝑓(𝑥)〉| = sup 𝑥∗∈𝑆𝑋𝑛 ∗ |〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥 ∗, 𝑥0 − 𝑥〉| + |〈𝜑𝑥∗ , 𝑓(𝑥0) − 𝑓(𝑥)〉| ≥ sup 𝑥∗∈𝑆𝑋𝑛 ∗ |〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥 ∗, 𝑥0 − 𝑥〉 − 〈𝜑𝑥∗ , 𝑓(𝑥0) − 𝑓(𝑥)〉| almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 170 recall that 𝑋 and 𝑌 are strictly convex, so 𝑓(𝑥0) − 𝑓(𝑥) = ∑ |𝜆𝑖 |𝑓(𝑥𝑖 ) 𝑛 𝑖=1 and 𝑥0 − 𝑥 = ∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 . since 𝑥 ∗ ∈ 𝑆𝑋𝑛∗ , i. e. ‖𝑥 ∗‖ = 1, then 𝑥∗ is norm attaining functional. hence, sup 𝑥∗∈𝑆𝑋𝑛 ∗ |〈𝑥∗, ∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 〉| = ‖𝑥 ∗(∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 )‖ = ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖. therefore, ‖∑ |𝜆𝑖 |𝑓(𝑥𝑖 ) 𝑛 𝑖=1 ‖ ≥ ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖ − |〈𝑥∗, ∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 〉 − 〈𝜑𝑥∗ , ∑ |𝜆𝑖 |𝑓(𝑥𝑖 ) 𝑛 𝑖=1 〉| = ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖ − ∑ |𝜆𝑖 | 𝑛 𝑖=1 |〈𝑥∗, ∑ 𝑥𝑖 𝑛 𝑖=1 〉 − 〈𝜑𝑥∗ , ∑ 𝑓(𝑥𝑖 ) 𝑛 𝑖=1 〉| = ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖ − |〈𝑥∗, ∑ 𝑥𝑖 𝑛 𝑖=1 〉 − 〈𝜑𝑥∗ , ∑ 𝑓(𝑥𝑖 ) 𝑛 𝑖=1 〉| ≥ ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖ − 4 or ‖∑ |𝜆𝑖 |𝑓(𝑥𝑖 ) 𝑛 𝑖=1 ‖ + 4 ≥ ‖∑ |𝜆𝑖 |𝑥𝑖 𝑛 𝑖=1 ‖ . ∎ next, we will discuss an application of theorem 1.6 in 𝐿𝑝,𝑞 -spaces. let 𝐿(ω) is an algebra of equivalence classes of real valued measured function over (ω, ∑, 𝜇). the distribution 𝛿𝑓 (𝑠) for 𝑓 ∈ 𝐿(ω) is defined as 𝛿𝑓 (𝑠) ∶= 𝜇(|𝑓| > 𝑠) 𝐿𝑝,𝑞-spaces are collections of all measured function over ω such that ‖𝑓‖𝑝,𝑞 ∶= ( 𝑞 𝑝 ∫ 𝑓 ∗(𝑡)𝑞 𝑡 ( 𝑞 𝑝 ) − 1 𝑑𝑡 ∞ 0 ) 1 𝑞 < ∞ where 𝑓 ∗(𝑡) ∶= inf{𝑠 > 0 ∶ 𝛿𝑓 (𝑠) < 𝑡} , 𝑡 > 0. if p = q, then 𝐿𝑝,𝑝(ω) is just lebesgue space 𝐿𝑝(ω). the following theorem shows that there exists an isomorphism operator [8] in 𝐿𝑝,𝑞 -spaces. theorem 2.2. suppose that 1 ≤ 𝑝,𝑞 ≤ ∞, 𝑝 ≠ 𝑞, 𝑝 ≠ 1, 𝑝 ≠ 2. then there exists no weakly sequence {𝑓𝑘 }𝑘=1 ∞ in 𝐿𝑝,𝑞 (0,1) such that for some 𝐶 > 0 and for any subsequence {𝑓𝑘 ′}𝑘=1 ∞ of {𝑓𝑘 }𝑘=1 ∞ , the estimate 𝐶−1𝑁1/𝑝 ≤ ‖∑ 𝑓𝑘 ′ 𝑁 𝑘=1 ‖ 𝑝,𝑞 ≤ 𝐶𝑁1/𝑝 holds. now, using theorem 2.2, the author will show that the following theorem is true. theorem 2.3. let x =𝐿𝑝,𝑞 and y be banach space. if f : x → y is an almost surjective -isometry with f(0) = 0, then there exists an operator s : y → x with ‖𝑆‖ = 1 such that ‖𝑆𝑓(𝑥) − 𝑥‖ ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. proof. from theorem 2.2, then 𝐵𝑋∗ is a 𝑤 ∗-closed convex hull of 𝑤 ∗-exposed point in 𝐿𝑝,𝑞 . let 𝑡 ∈ (0,1) and 𝑥∗ ∈ 𝑆𝑋∗ . theorem 1.6 implies that there exists 𝜙𝑡 ∈ 𝑆𝑌∗ such that |〈𝜙𝑡 , 𝑓(𝑥)〉 − 〈𝑥𝑡 ∗, 𝑥〉| ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 171 let s : y → x is defined by s(y) = 〈𝜙𝑡 , 𝑓(𝑥)〉𝑒𝑡 = 𝑥0. clearly ‖𝑆‖ = 1 and ‖𝑆𝑓(𝑥) − 𝑥‖𝑝,𝑞 = ( 𝑞 𝑝 ∫(𝑥0(𝑡) 𝑞 − 𝑥(𝑡)𝑞 )𝑡 ( 𝑞 𝑝 ) − 1 𝑑𝑡 1 0 ) 1 𝑞 ≤ sup 𝑡∈(0,1) |〈𝜙𝑡 , 𝑓(𝑥)〉 − 〈𝑥𝑡 ∗, 𝑥〉| ≤ 4 . ∎ 2. almost surjective 𝜺-isometry in the reflexive banach spaces a normed space is reflexive if the natural mapping 𝜋(𝑥) from 𝑋 into 𝑋∗∗ which is defined by 𝜋(𝑥) : f → f(x) where 𝑓 ∈ 𝑋∗, is onto 𝑋∗∗, or 𝑋 ≈ 𝑋∗∗. therefore, every reflexive normed space is banach space. in addition, every closed subspace of reflexive banach space is reflexive [4]. proposition 3.1 (megginson [9], proposition 1.11.11). let x be reflexive normed space. every functional 𝑥∗ ∈ 𝑋∗ is norm-attaining functional. the following definitions are classic (for more detail see [4]). let x be banach space and 𝑀 ⊂ 𝑋. the polar set of m is 𝑀∘ ∶= { 𝑥∗ ∈ 𝑋∗ ∶ 〈𝑥∗, 𝑥〉 ≤ 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑀}. conversely, 𝑀° ∶= { 𝑥 ∈ 𝑋 ∶ 〈𝑥∗, 𝑥〉 ≤ 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥∗ ∈ 𝑀°}° . the annihilator of m is 𝑀⊥ ∶= { 𝑥∗ ∈ 𝑋∗ ∶ 〈𝑥∗, 𝑥〉 = 0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑀}, and preannihilator 𝑀⊥ is 𝑀⊥ ∶= { 𝑥 ∈ 𝑋: 〈𝑥∗, 𝑥〉 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 ⊥ 𝑥 ∗ ∈ 𝑀⊥}. for an almost surjective -isometry mapping f : x → y with f(0) = 0 and > 0, 𝐶(𝑓) is a closed convex hull of f(x), e is an annihilator of subspace f⊂ 𝑌∗ where f is a set of all bounded functionals over 𝐶(𝑓), and 𝑀𝜀 ∶= {𝜙 ∈ 𝑌 ∗: 𝜙 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝛽 𝑜𝑣𝑒𝑟 𝐶(𝑓), 𝑓𝑜𝑟 𝛽 > 0}. obviously, 𝐶(𝑓) is symmetric, then 𝑀𝜀 is linear subspace of 𝑌 ∗ with 𝑀𝜀 = ⋃ 𝑛𝐶(𝑓) °∞ 𝑛=1 . therefore, 𝐸 = ⋂{𝑘𝑒𝑟𝜙 ∶ 𝜙 ∈ 𝑀}. the following lemma is easy to be proved (see [2]). lemma 3.2. let y be banach space, the following statements are equicalence. (1) 𝐶(𝑓) ⊂ 𝐸 + 𝐹 for a bounded 𝐵 ⊂ 𝑌; (2) 𝑀𝜀 is 𝑤 ∗-closed; (3) 𝑀𝜀 is closed. for every almost surjective -isometry f, is defined a linear mapping ℓ : 𝑋∗ → 2𝑌 ∗ by ℓ𝑥∗ ∶= {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝛽 > 0 𝑎𝑛𝑑 𝑎𝑙𝑙 𝑥 ∈ 𝑋}. since ℓ is a linear mapping, the definition of 𝑀𝜀 implies ℓ0 = {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉 − 〈0, 𝑥〉| ≤ 𝛽 } = {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉| ≤ 𝛽 } = {𝜙 ∈ 𝑌∗: 𝜙 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝛽 𝑜𝑣𝑒𝑟 𝐶(𝑓)} = 𝑀𝜀 and ℓ𝑥 ∗ = ℓ(𝑥∗ + 0) = ℓ𝑥∗ + ℓ0 ⊃ 𝜙 + ℓ0 = 𝜙 + 𝑀𝜀. lemma 3.3. if 𝑥∗ ≠ 𝑦∗, then ℓ𝑥∗ ∩ ℓ𝑦∗ = ∅. proof. suppose that ℓ𝑥∗ ∩ ℓ𝑦∗ is nonempty, then there exists 𝜑 ∈ ℓ𝑥∗ ∩ ℓ𝑦∗. assume 𝜑𝑥∗ = {𝜑 ∈ 𝑌 ∗: |〈𝜑, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋) and almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 172 𝜑𝑦∗ = {𝜑 ∈ 𝑌 ∗: |〈𝜑, 𝑓(𝑥)〉 − 〈𝑦∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋). for 𝑥∗ ≠ 𝑦∗, there exists 𝜑 ∈ ℓ𝑥∗ ∩ ℓ𝑦∗ such that 𝜑𝑥∗ = 𝜑𝑦∗ . this is impossible because 𝜑 is a linear functional and the proof is complete. ∎ let 𝑈 ∈ 𝔅(𝑋, 𝑌). then 𝑈∗ ∈ 𝔅(𝑌∗, 𝑋∗) is called an adjoint operator such that 〈𝑈𝑥, 𝑓〉 = 〈𝑥, 𝑈∗𝑓〉 for all 𝑥 ∈ 𝑋. two results below are classic. proposition 3.4 (fabian et. al. [4], proposition 2.6). let m is a closed subspace of banach space x. then (𝑋/𝑌)∗ is isometric with 𝑀⊥ and 𝑀∗ is isometric with 𝑋∗/𝑀⊥. theorem 3.5 (megginson [9], proposition 3.1.11). let x and y be normed spaces. if 𝑈 ∈ 𝔅(𝑋, 𝑌), then 𝑈∗ is 𝑤 ∗-to-𝑤 ∗ continuous. conversely, if 𝑄 is 𝑤 ∗-to-𝑤 ∗ linear continuous operator from 𝑌∗ to 𝑋∗, then there exists 𝑈 ∈ 𝔅(𝑋, 𝑌) such that 𝑈∗ = 𝑄. now, using proposition 3.4 and theorem 3.5, we will show that the following theorem is true. theorem 3.6. let x and y be banach spaces, f : x → y is an almost surjective -isometry with f(0) = 0, and 𝑀 = ℓ̅0. then (1) 𝑄: 𝑋∗ → 𝑌∗/𝑀, which is defined by 𝑄𝑥∗ = ℓ𝑥∗ + 𝑀, is linear isometry. (2) if m is 𝑤 ∗-closed, then 𝑄 is an adjoint operator of 𝑈: 𝐸 → 𝑋 with ‖𝑈‖ = 1. proof. (1) from lemma 3.3 and definition of ℓ, it is clear that 𝑄 is linear. lemma 3.2 implies 𝑀 = ℓ̅0 = 𝑀𝜀̅̅ ̅̅ . for every 𝑥 ∗ ∈ 𝑋∗, ‖𝑄𝑥∗‖ = inf{‖𝜙 − 𝑚‖ : 𝜙 ∈ ℓ𝑥∗, 𝑚 ∈ 𝑀} = inf{‖𝜙 − 𝑚‖ : 𝜙 ∈ ℓ𝑥∗, 𝑚 ∈ 𝑀𝜀 }. from definition of ℓ𝑥∗ and 𝑀𝜀, we have ‖𝑄𝑥∗‖ = inf{‖𝜙‖ : 𝜙 ∈ ℓ𝑥∗}. theorem 1.6 gives inf{‖𝜙‖ : 𝜙 ∈ ℓ𝑥∗} ≤ ‖𝑥∗‖. (3.1) to show the conversely, from definition of ℓ, there exists 𝛽 > 0 such that |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. for any 𝛿 > 0, we can choose 𝑥0 ∈ 𝑆𝑋 such that 〈𝑥∗, 𝑥0〉 > ‖𝑥 ∗‖ − 𝛿. for all 𝑛 ∈ ℕ, we get |〈𝜙, 𝑓(𝑛𝑥0) 𝑛 〉 − 〈𝑥∗, 𝑥0〉| ≤ 𝛽 𝑛 as 𝑛 → ∞, ‖𝜙‖ ≥ lim sup 𝑛 |〈𝜙, 𝑓(𝑛𝑥0) 𝑛 〉| = 〈𝑥∗, 𝑥0〉 > ‖𝑥 ∗‖ − 𝛿. almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 173 since 𝛿 is arbitrary, then ‖𝜙‖ ≥ ‖𝑥∗‖. (3.2) from (3.1) and (3.2), then ‖𝜙‖ = ‖𝑥∗‖ which shows that 𝑄 is an isometry. (2) since 𝑀 is 𝑤 ∗-closed, then 𝑀 = { 𝑀⊥ }⊥ = { 𝑀𝜀 ⊥ }⊥ = 𝐸⊥. therefore, 𝑌∗/𝑀 = 𝑌∗/𝐸⊥. from definitions of 𝐸 and 𝑀𝜀, proposition 3.4 implies 𝑌 ∗/𝐸⊥ = 𝐸∗. we will prove that 𝜙 is 𝑤 ∗-to-𝑤 ∗ continuous over the unit ball 𝐵𝑋∗ . let 𝐼 is an index set and 𝛼 ∈ 𝐼. suppose there exists net (𝑥∗) ⊂ 𝐵𝑋∗ which is 𝑤 ∗-converging to 𝑥∗ ∈ 𝑋∗. using theorem 1.6, there exists net (𝜙𝛼 ) ⊂ 𝑌 ∗ with ‖𝜙𝛼 ‖ = ‖𝑥𝛼 ∗ ‖ = 𝑟𝛼 ≤ 1 such that |〈𝜙𝛼 , 𝑓(𝑥)〉 − 〈𝑥𝛼 ∗ , 𝑥〉| ≤ 4 𝑟𝛼 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. since (𝜙𝛼 ) is 𝑤 ∗-compact, then there exists 𝑤 ∗-limit point ∈ 𝑌∗ such that |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 𝑟, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋 therefore, 𝑄𝑥𝛼 ∗ = ℓ𝑥𝛼 ∗ + 𝑀 = 𝜙𝛼 + 𝑀 is 𝑤 ∗-convergence to 𝜙 + 𝑀 = 𝑄𝑥∗. this result shows that 𝑄: 𝐵𝑋∗ → 𝐸 ∗ is 𝑤 ∗-to-𝑤 ∗ continuous. from theorem 3.5, there exists 𝑈: 𝐸 → 𝑋 such that 𝑈∗ = 𝑄. definition of 𝐸 shows that 𝑈 is a surjective mapping. since 𝑈∗ = 𝑄 is a linear isometry, then ‖𝑈‖ = 1. ∎ before discussing the stability of an almost surjective -isometry in reflexive banach space, we need some resuls below. proposition 3.7 (fabian et. al. [4], proposition 3.114). let y be reflexive banach space. if 𝑀 is a closed subspace of y, then 𝑀 is banach space. corollary 3.8. if y be reflexive banach space, then the mapping 𝑄 in theorem 3.6 is an adjoint operator. proof. since y is reflexive, from proposition 3.7, 𝑀 is 𝑤 ∗-closed. the proof can be completed by theorem 3.6. definition 3.9. let x be banach space and 0 ≤ α < ∞. a closed subspace 𝑀 ⊂ 𝑋 is called 𝛼complemented in x if there is a closed subspace 𝑁 ⊂ 𝑋 with 𝑀 ∩ 𝑁 = {0} and projection 𝑃: 𝑋 → 𝑀 𝑎𝑙𝑜𝑛𝑔 𝑁 such that 𝑋 = 𝑀 + 𝑁 and ‖𝑃‖ ≤ 𝛼. theorem 3.10. let x and y be banach spaces where y is reflexive, and f : x → y is an almost surjective -isometry, then there exists a bounded linear operator t : y → x and ‖𝑇‖ ≤ 𝛼 such that ‖𝑇𝑓(𝑥) − 𝑥‖ ≤ 4 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. proof. since y is reflexive, from corollary 3.8 and theorem 3.6, there exists a surjective operator 𝑈: 𝐸 → 𝑋 with ‖𝑈‖ = 1 such that 𝑄 = 𝑈∗. since 𝐸 is 𝛼-complemented in y, there is a closed subspace 𝐹 ⊂ 𝑌 and 𝐸 ∩ 𝐹 = {0} such that 𝐸 + 𝐹 = 𝑌 and projection 𝑃: 𝑌 → 𝐸 𝑎𝑙𝑜𝑛𝑔 𝐹 satisfies ‖𝑃‖ ≤ 𝛼. let 𝑇 = 𝑈𝑃, then ‖𝑇‖ = ‖𝑈𝑃‖ ≤ ‖𝑈‖‖𝑃‖ = ‖𝑃‖ ≤ 𝛼. furthermore, 〈𝑄𝑥∗, 𝑃𝑦〉 = 〈𝑄𝑥∗, 𝑦〉 for all 𝑥∗ ∈ 𝑋∗ and 𝑦 ∈ 𝑌. therefore, almost surjective epsilon-isometry in the reflexive banach spaces minanur rohman 174 |〈𝑄𝑥∗, 𝑃𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| = |〈𝑄𝑥∗, 𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 ‖𝑥∗‖, (3.3) for all 𝑥 ∈ 𝑋 and 𝑥∗ ∈ 𝑋∗. definition of adjoint operator gives 〈𝑄𝑥∗, 𝑃𝑓(𝑋)〉 = 〈𝑥∗, 𝑄∗𝑃𝑓(𝑋)〉 = 〈𝑥∗, 𝑈𝑃𝑓(𝑋)〉 = 〈𝑥∗, 𝑇𝑓(𝑋)〉. (3.4) substitute (3.4) into (3.3), then |〈𝑥∗, 𝑇𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| = |〈𝑥∗, 𝑇𝑓(𝑋) − 𝑥〉| ≤ 4 ‖𝑥∗‖ for all 𝑥 ∈ 𝑋 and 𝑥∗ ∈ 𝑋∗. in addition, |〈𝑥∗, 𝑇𝑓(𝑋) − 𝑥〉| ≤ ‖𝑥∗‖‖𝑇𝑓(𝑋) − 𝑥‖ ≤ 4 ‖𝑥∗‖ or ‖𝑇𝑓(𝑋) − 𝑥‖ ≤ 4 for all 𝑥 ∈ 𝑋. ∎ conclusion in this paper, the author has shown simple applications of theorem 1.6. in addition, almost surjective -isometries are remain stable in the reflexive banach spaces. open problem 1. this paper has shown that almost surjective -isometries are stable in lorentz spaces when the conditions in theorem 2.2 are satisfied. is it possible to generalize these restrictions? open problem 2. according to the results of this paper, is it possible to determine the stability of almost surjective -isometries in the higher dual of banach spaces? references [1] d. g. bourgin, "approximate isometries", bull. amer. math. soc., vol. 52, no. 8, pp. 704-713, 1954. 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[15] i. a. vestfrid, "stability of almost surjective ε-isometries of banach spaces", j. funct. anal., vol. 269, no. 7, pp. 2165-2170, 2015. hybrid model gstar-sur-nn for precipitation data cauchy journal of pure and applied mathematics volume 4(2) (2016), pages 74-80 submitted: 19 january 2016 reviewed: 1 march 2016 accepted: 2 may 2016 hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono, waego hadi nugroho, rahma fitriani, atiek iriany department of statistics, faculty of mathematics and natural sciences brawijaya university, malang email: agus.dwi.sulistyono@gmail.com, atiekiriany@yahoo.com abstract spatio-temporal model that have been developed such as space-time autoregressive (star) model, generalized space-time autoregressive (gstar), gstar-ols and gstar-sur. besides spatio-temporal phenomena, in daily life, we often find nonlinear phenomena, uncommon patterns and unidentified characteristics of the data. one of current developed nonlinear model is a neural network. this study is conducted to form a hybrid model gstar-sur-nn to develop spatio-temporal model that has better prediction. this research is conducted on ten-daily rainfall data at 2005 2015 for blimbing, singosari, karangploso, dau, and wagir region. based on the results of this research, indicated that the accuracy of gstar ((1), 1,2,3,12,36)-sur model used cross-covariance weight has relatively similar to gstar ((1), 1,2,3 , 12.36)-sur-nn (25-14-5) for blimbing and singosari region with 5% error level. while karangploso, dau, and wagir, gstar ((1), 1,2,3,12,36)-sur-nn (25-14-5) model has better accuracy in predicting the precipitation at three locations with the value of r2prediction for each location is 0.992, 0.580, and 0.474. keywords: spatio temporal, precipitation, hybrid model, neural network introduction one of the spatio-temporal model that has been developed is space-time autoregressive (star) which introduced by pfeifer and deutsch[1]. star model did not fit for the data which had heterogeneous characteristics of locations. it was the star model's weaknesses and it can be addressed by the generalized space-time autoregressive (gstar) model and gstar-ols that developed by ruchjana[2], [3]. the latest development of spatio-temporal models is gstar-sur developed by iriany [4] to address for non-stationary and seasonal pattern data. the use of locations weights on the formation of spatio temporal models also contribute to the accuracy of the model. the location weights that often used are uniform weight, inverse distance, and normalized cross correlation weight [5], [6]. the location weight consider the neighborhood between locations. for data that has a high variability, it is necessary to consider the location weight with variability aspects of observational data, cross covariance weights. the use of cross covariance weights have been studied and applied by apanasovich and genton [7] to predict pollution in california and efromovich and smirnova [8] to process the fmri imaging with wavelate approach. in addition to the phenomenon of spatio-temporal, in daily life we often find nonlinear phenomena. time series models that have been explained are a time series model with a linear approach. there is many limitations in modeling with linear approach, especially is the fulfillment of the assumptions underlying the linear model. linear time series modeling is not appropriate mailto:agus.dwi.sulistyono@gmail.com mailto:atiekiriany@yahoo.com hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 75 and difficult to do on the data with nonlinear pattern. some nonlinear time series models have been developed and applied by tong [9], priestley [10], lee et al. [11], as well as granger and terasvirta [12]. nonlinear time series model that most developed and applied is artificial neural network. therefore, this study was conducted to form a hybrid model gstar-sur-nn to develop spatio-temporal models that have better forecasts. theoritical review location weight the problems that arise in the modeling of space-time is the use of location weight. there are several methods to determine the weight location in space-time model of the application [6], such as : a. uniform weight uniform weights can be calculated using the formula 𝑤𝑖𝑗 = 1 𝑛𝑖 with ni is the number of neighborhood locations with ith locations. weight of the location is only used for homogeneous characteristics or have the same distance between locations b. inverse distance weight inverse distance weight is calculated based on the actual distance between locations. the closer the distance between locations, the greater weight is. thus, a location adjacent have greater weight. c. normalized cross-correlation weight this weight is the result of the normalization of cross-correlation between the location of the corresponding time lag [5]. the normalization of cross-correlation weight was first introduced by suhartono and atok [6] and more applied research by [5]. d. normalized cross-covariance weight the use of cross covariance weight had been studied and applied by apanasovich and genton [7] to predict pollution in california, as well as efromovich and smirnova [8]to process the fmri imaging with wavelate approach. gstar model ruchjana [3] suggested that gstar was a generalization and extension of space time autoregresssive models (star) by pfeifer [1]. the main difference are on spatial dependent and weight matrix. gstar more realistic because it is in fact more prevalent models with different parameters for different locations [13]. gstar with p order and 𝜆1, 𝜆2, … 𝜆𝑝 spatial order, gstar (𝑝𝜆1, 𝜆2, … 𝜆𝑝 ) formulated as follows [14]: 𝑍(𝑡) = 𝝁(𝑡) + [𝚽01 + 𝚽01𝑊]𝑍(𝑡 − 1) + 𝜀(𝑡) (6) or ( 𝑍1(𝑡) 𝑍2(𝑡) ⋮ 𝑍𝑛 (𝑡) ) = ( 𝜙01 0 … 0 0 𝜙02 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 𝜙0𝑛 ) ( 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) ⋮ 𝑍𝑛 (𝑡 − 1) ) + ( 𝜙11 0 … 0 0 𝜙12 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 𝜙1𝑛 ) ( 0 𝑤12 … 𝑤1𝑛 𝑤21 0 … 𝑤2𝑛 ⋮ ⋮ ⋱ ⋮ 𝑤𝑛1 𝑤𝑛2 … 𝑤𝑛𝑛 ) ( 𝑍1(𝑡 − 1) 𝑍2(𝑡 − 1) ⋮ 𝑍𝑛(𝑡 − 1) ) + ( 𝑒1(𝑡) 𝑒2(𝑡) ⋮ 𝑒𝑛 (𝑡) ) (7) seemingly unrelated regression (sur) seemingly unrelated regression (sur) is an equation that parameter estimation use general least square (gls). iriany (2013) explains that gls is the regression coefficient estimator hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 76 that count the relationship between the equation error. the error value is obtained from the estimated ordinary least squares (ols) that will be used in the calculation to estimate the regression coefficients in the sur system equation. sur models with m equation expressed by: 𝒚𝒊 = 𝑿𝒊 𝜷𝒊 + 𝜺𝒊, 𝑖 = 1, … , 𝑁 (8) where 𝒚𝒊 is vector with size r×1, 𝑋𝑖 ’s size is r×𝑘𝑖 dan 𝛽𝑖 vector with size 𝑘𝑖 × 1. according to greene [15], equation (8) is sur model with the assumption 𝐸[𝜀|𝑿𝟏,𝑿𝟐, … , 𝑿𝑵] = 0 and 𝐸[𝜀𝜀 ′|𝑿𝟏,𝑿𝟐, … , 𝑿𝑵] = 𝛀 with 𝛀 is variance-covariance matrices. gls method use the error variance : cov(𝜀) = 𝐸(𝜀𝜀′) = σ2𝚺 = 𝛀. matrix 𝛀 describe the error correlation with : 𝛀 = e(𝜀𝜀 ′) = [ e(e1e1 ′ ) e(e1e2 ′ ) … e(e1e𝑁 ′ ) e(e2e1 ′ ) e(e2e2 ′ ) … e(e2e𝑁 ′ ) ⋮ e(e𝑁e1 ′ ) ⋮ e(e𝑁e2 ′ ) ⋱ … ⋮ e(e𝑁e𝑁 ′ ) ] since 𝐸(𝜀𝑖𝜀𝑗 ) = 𝜎𝑖𝑗 𝐼𝑇, 𝛀 = [ σ11𝐈𝑁𝑇 σ12𝐈𝑁𝑇 … σ1𝑁𝐈𝑁𝑇 σ21𝐈𝑁𝑇 σ22𝐈𝑁𝑇 … σ2𝑁 𝐈𝑁𝑇 ⋮ σ𝑁1𝐈𝑁𝑇 ⋮ σ𝑁2𝐈𝑁𝑇 ⋱ … ⋮ σ𝑁𝑁 𝐈𝑁𝑇 ] = 𝚺 ⊗ 𝐈𝑁𝑇 𝚺 = [ σ11 σ12 … σ1𝑚 σ21 σ22 … σ2𝑚 ⋮ σ𝑚1 ⋮ σ𝑚2 ⋱ … ⋮ σ𝑚𝑚 ] (9) with 𝐈𝑁𝑇 is identity matrix sized (𝑁𝑇 × 𝑁𝑇) and 𝚺 is matrix sized (𝑁 × 𝑁) with 𝜎𝑖𝑗 error variance from each equation for i= j and error covariance between equation for i≠ j. the parameter model estimation is obtained by estimating parameter 𝜷 in equation (8). 𝜺∗ ′ 𝜺∗ = 𝜺 ′𝛀−𝟏𝛆 = 𝒀′𝛀−𝟏𝐘 − 𝟐𝐘′𝛀−𝟏𝐗𝜷 + 𝜷′𝑿′𝛀 −𝟏 𝐗 𝜷 (10) equation (10) is differenced by 𝜷 and equal to zero. so : (𝑿′𝛀 −𝟏 𝑿)�̂� = 𝑿′𝛀 −𝟏 𝒀 (𝑿′𝛀−𝟏𝑿)−𝟏(𝑿′𝛀−𝟏𝑿)�̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝒀 𝑰�̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀 −𝟏 𝒀 �̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀 −𝟏 𝒀 hybrid model gstar-sur-nn hybrid model by integrating the neural network with conventional forecasting model was proved capable of producing more accurate forecasts. some hybrid models using neural network are feed forward neural network for time series data [16], auto regressive integrated moving average with exogenous factor-neural network (arimax-nn) for data inflation in indonesia [17], and neural network multiscale autoregressive (nn-mar) for forecasting the number of tourists [18]. general space-time autoregressive with seemingly unrelated regression neural network (gstar-sur-nn) is an integration/fusion (hybrid) between gstar model with neural network. gstar-sur is used to obtain the most suitable nn architecture, so it can obtain the best forecasting performance. architecture in the meaning is the amount of input variables that will be used in modeling nn. hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 77 results and discussion based on the result of descriptive statistics analysis, we can describe the descriptive statistics of precipitation data for each location : table 1. description of precipitation data in five location location n mean (mm) standard deviation (mm) minimum (mm) maximum (mm) blimbing 360 5.682 6.909 0 33.5 singosari 360 3.93 5.575 0 41.75 karangploso 360 4.302 5.71 0 25.36 dau 360 4.564 5.825 0 36.38 wagir 360 7.08 8.187 0 43.63 table 1 shows that precipitation of each location has a high variability. it can be seen that the standard deviation of precipitation of all location greater than the average. high variability indicates that there is fluctuation to the extreme point of precipitation, especially during the rainy season. here is the result of homogeneity test of variance the precipitation data: figure 1. the result of homogeneity test of variance figure 2 describe the results of homogeneity test of variance precipitation data using bartlett and levene test that was obtained p-value of 0.000 (p < 0.05), indicating that the precipitation data at blimbing, karangploso, singosari, wagir, and dau region have different variance. model identification is done by comparing the value of aic at some time lag. here the value of aic with varmax procedure in sas: table 2. the value of aic with varmax procedure time lag aic value 1 14.67906 2 14.52868 3 14.52445 hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 78 based on table 2 above, indicated that at the time lag of 1-3, the lowest aic is in 3rd time lag. thus, the order gstar model is gstar ((1), 1,2,3)-sur. based on acf of precipitation data at each location there is seasonal pattern in the time lag of 12 and 36. therefore, the appropriate model is gstar ((1) (1,2,3,12 , 36)) sur. the architecture of gstar-sur-nn model is as follows: figure 2. the architecture of gstar((1),1,2,3,12,36)-sur-nn (25-14-5) model the neural network architecture consist of 25 input variables, 14 hidden layers, dan 5 output layers. therefore, hybrid model of gstar-sur-nn is gstar((1),1,2,3,12,36)-sur-nn (2514-5). follow is the result of validation test of gstar ((1)(1,2,3,12,36))-sur model and gstar((1),1,2,3,12,36)-sur-nn (25-14-5) model : table 3. the result of gstar-sur and gstar-sur-nn model validation test model t-statistics p-value gstar((1),1,2,3,12,36)-sur 2.619 0.010 gstar((1),1,2,3,12,36)-sur-nn (25-14-5) 0.332 0.741 based on the results of validation test of gstar ((1), 1,2,3,12,36)-sur using cross covariance weight in table 3, showed that, at α = 5%, gstar ((1),1,2,3,12,36)-sur model has pvalue less than 0.05 which implies that there is a significant difference between average of actual precipitation data with average of predicted results. or in other words, gstar ((1), 1,2,3,12,36)sur using cross-covariance weights still have low accuracy. while, gstar ((1), 1,2,3,12,36) -surnn (25-14-5) was obtained p-value of 0.741. p-value is more than 0.05 implies that there is no significant difference between the average of actual precipitation with the average precipitation predicted results. or in other words, gstar ((1),1,2,3,12,36)-sur-nn (25-14-5) model have high accuracy. from this comparison, it has been proven that gstar ((1), 1,2,3,12,36)-sur-nn (25-145) model has better accuracy rate in predicting the precipitation data. table 4. the comparison gstar-sur and gstar-sur-nn model performance location gstar-sur model gstar-sur-nn model data testing rmse r2prediction data testing rmse r2prediction blimbing 10.132 0.539 10.672 0.640 singosari 0.451 0.458 hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 79 karangploso 0.623 0.992 dau 0.527 0.580 wagir 0.365 0.474 table 4 explains the performance comparison between gstar ((1),1,2,3,12,36)-sur with the hybrid model gstar ((1), 1,2,3,12,36)-sur-nn (25-14 -5). from these comparison, indicated that gstar-sur model has rmse value lower than the hybrid model gstar-sur-nn. judging from the value of rmse, gstar-sur model have better performance because it produces lower rmse value. but, if it judge from the r2prediction value, indicated that the r2prediction value on the hybrid model gstar-sur-nn is higher than gstar-sur model. in fact, the r2prediction value of karangploso region reached 0.992. based on this comparison, it is proved that the hybrid model gstar ((1),1,2,3,12,36)-sur-nn (25-14-5) has better accuracy rate in predicting precipitation in malang especially blimbing, singosari, karangploso, dau, and wagir region. conclusion based on the results of this study concluded that the hybrid model gstar ((1), 1,2,3,12,36)-sur-nn (25-14-5) produces precipitation forecast better than gstar ((1),1,2,3,12,36)-sur model. reference [1] p. e. pfeifer and s. j. deutsch, “identification and interpretation of first order space-time arma models,” technometrics, vol. 22, no. 3, pp. 397–408, 1980. [2] b. n. ruchjana, “study on the weight matrix in the space-time autoregressive model,” in proceeding of the 10th international symposium on applied stochastic models and data analysis (asmda), france, 2001, vol. 2, pp. 789–794. [3] b. n. ruchjana, “a generalized space time autoregressive model and its application to oil production data,” institut teknologi bandung, bandung, 2002. [4] a. iriany, suhariningsih, b. n. ruchjana, and setiawan, “prediction of precipitation data at batu town using the gstar (1,p)-sur model,” j. basic appl. sci. res., vol. 3, no. 6, pp. 860–865, 2013. [5] s. suhartono and s. subanar, “the optimal determination of space weight in gstar model by using cross-correlation inference,” quant. methods, vol. 2, no. 2, pp. 45–53, dec. 2006. [6] suhartono and r. m. atok, “pemilihan bobot lokasi yang optimal pada model gstar,” in prosiding konferensi nasional matematika xiii, semarang, 2006. [7] t. v. apanasovich and m. g. genton, “cross-covariance functions for multivariate random fields based on latent dimensions,” biometrika, vol. 97, no. 1, pp. 15–30, mar. 2010. [8] e. smirnova, “statistical analysis of large cross-covariance and cross-correlation matrices produced by fmri images,” j. biom. biostat., vol. 5, no. 2, 2013. [9] h. tong, non-linear time series: a dynamical system approach. oxford university press, 1993. [10] m. b. priestley, non-linear and non-stationary time series analysis, 2nd edition, 2nd ed. london: academic press, 1991. [11] t.-h. lee, h. white, and c. w. j. granger, “testing for neglected nonlinearity in time series models: a comparison of neural network methods and alternative tests,” j. econom., vol. 56, no. 3, pp. 269–290, apr. 1993. [12] c. w. . granger and t. terasvirta, modeling nonlinear economic relationships. united kingdom: oxford university press. [13] w. dhoriva urwatul and s. suhartono, “peramalan deret waktu multivariat seasonal pada data pariwisata dengan model var-gstar,” in seminar nasional matematika dan pendidikan matematika 2009, 2009. [14] s. a. borovkova, h. p. lopuha, and b. n. ruchjana, “generalized s-tar with random weights,” in the 17th international workshop on statistical modeling, chania-greece. [15] w. h. greene, econometric analysis, 7th ed. boston: prentice hall, 2012. [16] suhartono, “feedforward neural networks untuk pemodelan runtun waktu,” universitas gajah mada, yogyakarta, 2007. hybrid model gstar-sur-nn for precipitation data agus dwi sulistyono 80 [17] h. suryono, “auto regressive integrated moving average with exogeneous factor-neural network (arimax-nn) pada data inflasi di indonesia,” institut teknologi sepuluh november, surabaya, 2009. [18] b. sutijo, s. suhartono, and a. j endharta, “forecasting tourism data using neural networksmultiscale autoregressive model,” j. mat. sains, vol. 16, no. 1, pp. 35–42, 2011. abstract introduction theoritical review location weight gstar model seemingly unrelated regression (sur) hybrid model gstar-sur-nn results and discussion conclusion reference fundamental solution of elliptic equation with positive definite matrix coefficient cauchy –jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 64-66 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 17 january 2018 reviewed: 09 april 2018 accepted: 16 may 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.4717 fundamental solution of elliptic equation with positive definite matrix coefficient khoirunisa1, corina karim2 1,2department of mathematics, universitas brawijaya email: khoir.n97@gmail.com, co_mathub@ub.ac.id abstract we study the fundamental solution of elliptic equations with real constant coefficients ∑∑𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢) 𝑛 𝑗=1 𝑛 𝑖=1 = 0, where 𝑎𝑖𝑗 is a positive definite matrix. we obtained by searching the radial solution such that we solved the equation into ordinary differential equations. keywords: fundamental solution, elliptic equation, positive definite matrix introduction we consider the linear differential operator 𝐿𝑢 = ∑ ∑ 𝑎𝑖𝑗(𝑥) 𝜕2𝑢 𝜕𝑥𝑖𝜕𝑥𝑗 + ∑ 𝑏𝑖(𝑥) 𝜕𝑢 𝜕𝑥𝑖 + 𝑐(𝑥)𝑢𝑛𝑖=1 𝑛 𝑗=1 𝑛 𝑖=1 , (1) with coefficient defined in an 𝑛 −dimensional domain 𝐷, where 𝑏𝑖(𝑥) and 𝑐(𝑥) are any functions that depend on 𝑥. 𝐿 is said to be of elliptic type (or elliptic) at a point 𝑥0 if the matrix 𝑎𝑖𝑗(𝑥 0) is positive definite, i.e., if for any real vector 𝜉 ≠ 0, ∑ ∑ 𝑎𝑖𝑗𝜉𝑖𝜉𝑗 > 0 𝑛 𝑗=1 𝑛 𝑖=1 [3]. laplace's equation ∆𝑢 = 0 is the simplest and most basic example of elliptic equation, and the laplacian of 𝑢 is ∆𝑢 = ∑ 𝑢𝑥𝑖𝑥𝑖 𝑛 𝑖=1 [1]. while, a symmetric matrix 𝐴 is called a positive definite matrix if a quadratic form 𝑥𝑇 𝐴𝑥 > 0 for all 𝑥 ≠ 0,𝑥 ∈ ℝ [5]. a fundamental solution of the differential operator 𝐿 in ω is a function 𝐾(𝑥,𝑧) defined for 𝑥𝜖ω,𝑧𝜖ω,𝑥 ≠ 𝑧 and satisfying the following property: for any bounded domain ℝ with smooth boundary 𝜕ℝ and for any 𝑧𝜖ℝ, 𝑢(𝑧) = ∫ 𝐾(𝑥,𝑧)𝐿∗𝑢(𝑥)̅̅ ̅̅ ̅̅̅̅ ̅̅ ̅̅ ̅̅ ̅𝑑𝑥, ℝ for every 𝑢 ∈ 𝐶0 𝑚(ℝ), and 𝐿∗ is formal adjoint of 𝐿 [4]. here, we define the following equation as elliptic equation with positive definite matrix ∑ ∑ 𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢) = 0, 𝑛 𝑗=1 𝑛 𝑖=1 (2) where 𝑎𝑖𝑗 is element of positive definite matrix, 𝑢 = 𝑢(𝑥) and 𝑥 ∈ 𝑈, 𝑈 ⊂ ℝ 𝑛. in 2016, fitri studied the holder regularity of weak solutions to linear elliptic partial differential equations with continuous coefficients, campanato type estimates are obtained for the validity of regularity of solutions (see [2] ). in this paper, we study the fundamental solution of elliptic equation with positive definite matrix coefficient. due to the symmetry of elliptic equation, radial solutions are natural to look for since the given partial differential equation can be reduced to an ordinary differential equation which is easier to solve. in this way, we can reduce the higher dimensional problems to one dimensional problems. mailto:khoir.n97@gmail.com mailto:co_mathub@ fundamental solution of elliptic equation with positive definite matrix coefficient khoirunisa 65 results and discussion let the elliptic equation with positive definite matrix as in (2) then to find a solutions 𝑢 of elliptic equations , it consequently seems advisable to search for radial solutions, that is functions of 𝑟. theorem 3.1 let 𝑟 = √∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗 𝑛 𝑗=1 𝑛 𝑖=1 , where 𝑏𝑖𝑗 is element of coffactor matrix 𝑎𝑖𝑗. then partial derivative of r with respect to xj and partial derivative of r with respect to xixj are defined by 𝜕𝑟 𝜕𝑥𝑗 = 1 𝑟 ∑∑𝑏𝑖𝑗𝑥𝑖 𝑛 𝑗=1 𝑛 𝑖=1 , and 𝜕2𝑢 𝜕𝑥𝑖𝜕𝑥𝑗 = 1 𝑟 ∑∑𝑏𝑖𝑗 − 1 𝑟3 𝑛 𝑗=1 𝑛 𝑖=1 ∑∑(∑ 𝑏𝑖𝑘𝑥𝑘 𝑛 𝑘=1 ∑ 𝑏𝑗𝑘𝑥𝑘 𝑛 𝑘=1 ) 𝑛 𝑗=1 𝑛 𝑖=1 . theorem 3.2 suppose 𝑢 (𝑥) = 𝑣 (𝑟) , then the second derivative of 𝑢 with respect to 𝑥𝑖𝑥𝑗 denoted 𝐷𝑖(𝐷𝑗𝑢) = 𝜕2𝑢 𝜕𝑥𝑖𝜕𝑥𝑗 is the total derivative of 𝑣 (𝑟) respect to 𝑥𝑖𝑥𝑗. corollary 3.3 if the second derivative of (4) is substituted into the equation (2) then we get 𝑣′′(𝑟) 𝑟2 ∑∑𝑎𝑖𝑗 (∑ 𝑏𝑖𝑘𝑥𝑘 𝑛 𝑘=1 ∑ 𝑏𝑗𝑘𝑥𝑘 𝑛 𝑘=1 ) + 𝑣′(𝑟) 𝑟 𝑛 𝑗=1 𝑛 𝑖=1 ∑∑𝑎𝑖𝑗𝑏𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 − 𝑣′(𝑟) 𝑟3 ∑∑𝑎𝑖𝑗 (∑ 𝑏𝑖𝑘𝑥𝑘 𝑛 𝑘=1 ∑ 𝑏𝑗𝑘𝑥𝑘 𝑛 𝑘=1 ) 𝑛 𝑗=1 𝑛 𝑖=1 = 0. corollary 3.4 if 𝐴 is positive definite matrix with 𝑎𝑖𝑗 element, and 𝐵 is cofactor matrix of 𝐴 with 𝑏𝑖𝑗 element then 𝐼(det𝐴) = 𝐴𝐵. corollary 3.5 if 𝐴 matrix and 𝐵 matrix are symmetric, then ∑ ∑ 𝑎𝑖𝑗𝑏𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 can be simplified to be ∑∑𝑎𝑖𝑗𝑏𝑖𝑗 = 𝑛det𝐴. 𝑛 𝑗=1 𝑛 𝑖=1 next, to simplify the sum of bracket in the first and the last terms in (5), we denoted 𝑌𝑖 for 𝑖 = 1,2,…,𝑛 and 𝑗 = 1,2,…,𝑛, so that we have 𝑌𝑖 = (det𝐴)∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗 𝑛 𝑗=1 𝑛 𝑖=1 = (det𝐴)𝑟2, thus, the equation (5) can be rewrite as 𝑣′′ 𝑣′ = 1 − 𝑛 𝑟 , then we have (3) (5) (4) fundamental solution of elliptic equation with positive definite matrix coefficient khoirunisa 66 𝑣′ = 𝑎𝑟1−𝑛. since, the domain 𝑥 is at 𝑛-dimension, 𝑛 ≥ 2 then 𝑣(𝑟) has the following general form: 𝑣(𝑟) = { 𝑎log𝑟 + 𝑏, 𝑛 = 2 𝑐 𝑟𝑛−2 + 𝑏, 𝑛 ≥ 3. 𝑎,𝑏,𝑐 ∈ ℝ. by rescaling (6) to 𝑢(𝑥) = 𝑣(𝑟), where 𝑟 = (∑ ∑ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗 𝑛 𝑗=1 𝑛 𝑖=1 ) 1 2, then we get the fundamental solution of (2) is 𝑣(𝑟) = { 𝑎log(∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗 𝑛 𝑗=1 𝑛 𝑖=1 ) 1 2 + 𝑏, 𝑛 = 2 𝑐 (∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗 𝑛 𝑗=1 𝑛 𝑖=1 ) 2−𝑛 2 + 𝑏, 𝑛 ≥ 3 where 𝑎,𝑏,𝑐 ∈ ℝ. remarks: we suppose the identity matrix 𝐼𝑛×𝑛 as a coefficient of (2). 𝐼 is a positive definite matrix, then the elliptic equation can be formed as follows ∑∑𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢) = 0, 𝑛 𝑗=1 𝑛 𝑖=1 where 𝑎𝑖𝑗 = { 1, 𝑖 = 𝑗 0, 𝑖 ≠ 𝑗. then coffactor matrix 𝑏𝑖𝑗 = { 1, 𝑖 = 𝑗 0, 𝑖 ≠ 𝑗. according to 𝑎𝑖𝑗 and 𝑏𝑖𝑗 as above, equation (7) can be written as ∑𝐷𝑖(𝐷𝑖𝑢) = 0, 𝑛 𝑖=1 where (8) is equivalent to the laplace’s equation ∑ 𝑢𝑥𝑖𝑥𝑖 𝑛 𝑖=1 . references [1] evans, l. c. (1998). partial differential equation. second edition. american mathematical society: usa [2] fitri, s. (2016). campanato type estimates for solutions of an elliptic systems class. international interdisciplinary studies seminar proceeding. 12-17. malang: universitas brawijaya. [3] friedman, a. (1964). partial differential equation of parbolic type. dover publication,inc: new york. [4] friedman, a. (1969). partial differential equation. dover publication,inc: new york. [5] leon, s. j. (2001). aljabar linear dan aplikasinya. erlangga: jakarta. (6) (7) (8) suma’inna dan gugun gumilar analisis klaster k-means dari data luas grup sunspot dan data grup sunspot klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan 𝐇𝛂 1siti jumaroh, 2nanang widodo, 3wahyu h.irawan 1jurusan matematika, universitas islam negeri maulana malik ibrahim malang 2lembaga penerbangan dan antariksa nasional 3jurusan matematika, universitas islam negeri maulana malik ibrahim malang email: sjumaroh09@gmail.com abstrak analisis klaster merupakan teknik interpendensi yang mengelompokkan suatu objek berdasarkan kemiripan dan kedekatan jarak antar objek. pengelompokan objek dengan jumlah banyak membutuhkan waktu yang lama. salah satu analisis klaster yang dapat digunakan dalam situasi ini adalah analisis klaster non hierarki, yaitu k-means. pada artikel ini mengelompokkan data luas grup sunspot dan data grup sunspot klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan hα. untuk mengetahui luas grup sunspot dan grup sunspot klasifikasi mc.intosh yang berpeluang membangkitkan flare soft x-ray dan hα dengan intensitas ledakan yang tinggi dan rendah. berdasarkan hasil analisis, diperoleh dua klaster yaitu klaster pertama yang tergolong mampu membangkitkan flare soft x-ray dan hα dengan intensitas yang tinggi. sedangkan klaster kedua yang tergolong mampu membangkitkan flare soft x-ray dan hα dengan intensitas yang rendah. kata kunci: analisis klaster, flare, luas grup sunspot dan grup sunspot klasifikasi mc.intosh abstract cluster analysis is an interdependence technique which classify an object based on the similarity and proximity distance between objects. objects clasifying by a lot of number takes long time. one of the cluster analysis can be used in this situation is the analysis of non-hierarchy cluster, it is kmeans. this article classifies extensive data group sunspot and data group sunspot classification mc.intosh that evokes a soft x-ray flares and hα. to know the broad group sunspot sunspot group classification and mc. intosh who could evoke the soft x-ray flares and hα intensity blast of high and low. based on the results of the analysis, retrieved two clusters i.ethe first cluster belongs is capable of arousing flares of soft x-ray and hα intensity high. while the second included cluster capable of arousing flares of soft x-ray and hα intensity is low. keywords: cluster analysis, flare, extensive sunspot group and group sunspot classification mc.intosh pendahuluan matahari merupakan salah satu bintang yang dapat memancarkan cahaya. beberapa energi di dalamnya bermanfaat bagi keberlangsungan makhluk hidup di muka bumi, namun sebagaian energi yang dipancarkan matahari berdampak negatif dapat menyebabkan kanker, contohnya sinar x lemah (soft x-ray (sxr)). peristiwa tersebut disebabkan oleh aktivitas matahari yaitu fenomena flare. grup sunspot memiliki tingkat kompleksitas yang berbeda-beda yang ditunjukkan oleh formasi spot-spot dan panumbra-panumbra pada bagian preceeding dan following. tingkat kompleksitas ini memiliki hubungan dengan fenomena flare, bahwa semakin besar tingkat kompleksitas suatu sunspot maka peluang terjadinya fenomena flare semakin besar. perubahan luas grup sunspot tersebut juga mempengaruhi fenomena flare, karena semakin besar tingkat kompleksitas suatu sunspot maka kemungkinana luas grup sunspot semakin besar, sehingga fenomena flare kemungkinan besar terjadi. fenomena flare melontarkan berbagai energi yang dapat diteliti pada panjang gelombang yaitu sxr, hα, radio, dan sinar 𝛾. pada penelitian ini objek-objek yang digunakan adalah klasifikasi mc,intosh dan luas grup sunspot yang membangkitkan flare. sedangkan variabel-variabel yang digunakan antara lain flare sxr (𝑊𝑎𝑡𝑡 ∕ 𝑚2), termasuk di dalamnya terdapat kelas c, m dan x dan flare hα kelas sub flare, 1, 2 dan 3 dengan satuan siti jumaroh 2 volume 4 no.1 november 2015 (perbandingan luas flare terhadap luas permukaan matahari × 106disk matahari. dalam statistik, jika variabel yang digunakan lebih dari dua maka metode yang dapat digunakan yaitu analisis multivariat. salah satunya yaitu analisis klaster. analisis klaster terdiri dari dua jenis yaitu analisis klaster hierarki dan non hierarki. sedangkan analisis klaster non hierarki yaitu k-means. pengklasteran non hierarki lebih cepat daripada metode hierarki dan lebih menguntungkan kalau jumlah objek atau kasus besar sekali [1]. hal ini yang melatar belakangi dilakukannya penelitian yang berjudul ”analisis klaster k-means dari data luas grup sunspot dan data grup sunspot klasifikasi mc.intosh yangmembangkitkan flare soft x-ray dan hα”. kajian teori 1. analisis multivariat analisis multivariat adalah analisis statistik yang menggunakan banyak variabel secara simultan [2]. teknik analisis multivariat diklasifikasikan menjadi dua yaitu metode ketergantungan (dependence method) dan metode saling ketergantungan (interdependence method). 2. analisis klaster analisis klaster merupakan suatu teknik analisis statistik yang ditujukan untuk membuat klasifikasi objek-objek ke dalam kelompokkelompok lebih kecil yang berbeda satu dengan yang lain [3] .terdapat beberapa langkah dalam analisis klaster yaitu: [4] . 1. pengukuran jarak sebagai ukuran kemiripan jarak euclid merupakan jarak langsung dan lurus dari satu titik ke titik lainnya [5]. ukuran jarak antara objek ke-i dengan ke-j dapat diperoleh melalui jarak euclid sebagai berikut: 𝑑𝑖𝑗 = ∑(𝑥𝑖𝑘 − 𝑥𝑗𝑘 ) 2 𝑝 𝑘=1 (1) 2. eliminasi data pencilan (outlier) data outlier adalah data yang secara nyata berbeda dengan data-data yang lain. deteksi terhadap outlier dapat dilakukan dengan menentukan nilai batas yang akan dikategorikan sebagai data outlier yaitu dengan cara mengkonversi nilai data ke dalam skor standardized atau yang biasa disebut z-score [6]. 3. membentuk klaster ada dua pilihan dalam membentuk klaster yaitu: a. metode hierarki metode ini memulai pengelompokan dengan dua atau lebih objek yang mempunyai objek paling dekat. kemudian proses diteruskan dengan meneruskan ke objek lain yang memiliki kedekatan kedua. demikian seterusnya. metode yang ada pada analisis ini adalah metode agglomerative terdiri dari single linkage, complete linkage, average linkage serta metode ward dan metode devisif. b. metode non hierarki (metode k-means) metode k-means pada dasarnya adalah metode partisi yang digunakan untuk menganalisis data dan memperlakukan pengamatan data sebagai objek berdasarkan lokasi dan jarak antara tiap data [7]. proses pengelompokan dengan metode k-means adalah menentukan besarnya nilai k, yaitu banyaknya klaster, menentukan centroid (pusat) di setiap klaster, menghitung jarak tiap objek dengan setiap centroid, menghitung kembali rataan untuk klaster yang baru terbentuk dan mengulang langkah kedua sampai tidak ada lagi pemindahan objek antar klaster [1] 3. analisis varian teknik analisis varian (anova) digunakan untuk menguji perbedaan rata-rata hitungan jika kelompok sampel yang diuji lebih dari dua buah populasi yang berbeda. berdasarkan banyaknya klasifikasi anova dibagi menjadi dua, yaitu anova satu jalan dan anova dua jalan [8]. 4. matahari matahari merupakan suatu bola gas yang berukuran sangat besar dengan ukuran diameter mencapai 1,4 juta kilometer. pada lapisan matahari terdiri dari beberapa fenomena di antaranya: 1. sunspot sunspot adalah suatu daerah di fotosfer yang memiliki medan magnetik yang kuat. sunspot mengalami perubahan jumlah, letak, luas dan medan magnet dari polaritas sederhana menjadi kompleks. 2. klasifikasi mc.intosh klasifikasi grup sunspot mc.intosh merupakan perubahan dan penyempurnaan dari klasifikasi grup sunspot zurich. klasifikasi grup mc.intosh dinyatakan dengan penulisan tiga huruf. huruf pertama menunjukkan modifikasi klasifikasi zurich, huruf kedua menunjukkan bentuk penumbra pada spot terbesar di dalam grup, dan huruf ketiga menunjukkan distribusi spot yang membentuk grup. contohnya kelas dao, eao, ekc, fai, analisis klaster k-means dari data luas grup sunspot dan data klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan h𝜶 cauchy – issn: 2086-0382/e-issn: 2477-3344 3 3. flare flare adalah ledakan kuat yang terjadi di kromosfer matahari di atas sunspot. flare dapat diklasifikasikan di antaranya yaitu a. klasifikasi flare sxr menurut [9] ledakan yang terjadi di matahari diklasifikasikan dalam beberapa kelas berdasarkan kecerlangannya pada panjang gelombang sinar sxr antara 1-8 angstroms antara lain flare kelas-x, flare kelas-m dan flare kelas-c b. klasifikasi flare hα menurut [10] klasifikasi flare h dibagi menjadi 5 kelas yaitu sub flare, kelas1, 2, 3 dan 4. sedangkan berdasarkan tingkat kecerahan dibedakan dalam tiga kriteria yaitu: faint (f), normal (n) dan bright (b). metode penelitian data yang digunakan dalam melakukan penelitian ini berupa data sekunder yang diambil secara online berupa data luas grup sunspot dan data grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dan hα dari tanggal 2 januari 2000 sampai dengan 18 januari 2005. kemudiam diklasifikasikan menjadi 4 jenis data, yaitu luas grup sunspot yang membangkitkan flare sxr, luas grup sunspot yang membangkitkan flare hα, grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα. kegiatan penelitian ini dilakukan di balai pengamatan dirgantara lembaga penerbangan dan antariksa nasional (lapan) watukosek, pasuruan, jawa timur. adapun langkah-langkah dalam menganalisis data adalah: 1. mendeskripsikan data 2. melakukan analisis klaster i. menghitung standarisasi variabel pada masing-masing data. ii. menghitung jarak euclid, untuk mengukur kemiripan suatu objek. 3. melakukan proses clustering dengan metode kmeans i. menentukan rata-rata di tiap kelas ii. menentukan banyaknya klaster (k) iii. menghitung jarak tiap objek dengan setiap rata-rata iv. menghitung kembali rataan untuk klaster yang baru terbentuk v. mengulangi langkah (iii) sampai tidak ada lagi pemindahan objek antar klaster. vi. melakukan validasi klaster dengan anova 4. interprestasi hasil klaster yang diperoleh pembahasan 1. deskripsi data data yang telah diperoleh harus dideskripsikan terlebih dahulu. deskripsi data dapat dilakukan dengan membuat plot sebaran data dari semua variabel dan objek. hasil sampling data intensitas flare sxr dan luas grup sunspot sebagai berikut: luas grup sunspot (x10^-6) in t e n s it a s f la r e s x r ( x 1 0 ^ 6 ) 25002000150010005000 600 500 400 300 200 100 0 gambar 1. grafik sebaran data luas grup sunspot dengan rata-rata flare sxr gambar 1. menunjukkan sebaran data luas grup sunspot dengan flare sxr. dimana sumbu x menyatakan luas grup sunspot dan sumbu y menyatakan intensitas flare sxr. flare sxr kelas c banyak terjadi pada luas grup sunspot yang memiliki luas 10 sampai 1500 × 10−6 disk matahari. flare sxr kelas m tersebar merata. pada kisaran luas grup sunspot 10 sampai 2000 × 10−6disk matahari. flare sxr kelas x yang mempunyai intensitas besar hanya mampu dibangkitkan dari grup sunspot yang mempunyai luas antara 250 sampai 2200 × 10−6disk matahari. sedangkan hasil sampling data intensitas flare hα dan luas grup sunspot sebagai berikut: luas grup sunspot (x 10^-6) l u a s f la r e h a lf a ( x 1 0 ^ 6 d is k m a t a h a r i) 25002000150010005000 800 700 600 500 400 300 200 100 0 gambar 2. grafik sebaran data luas grup sunspot dengan rata-rata flare hα pada gambar 2. menunjukkan sebaran data luas grup sunspot dengan nilai flare hα. flare hα kelas sf dibangkitkan oleh semua luas grup sunspot dengan luas 10 sampai 2200 × 10−6disk matahari. flare hα kelas 1 mayoritas terjadi pada luas grup siti jumaroh 4 volume 4 no.1 november 2015 sunspot 10 sampai 1700 dengan luas penampang 100 sampai 250 × 10−6 disk matahari. namun ada 1 flare hα kelas 1 yang muncul pada luas penampang 299 × 10−6disk matahari. hal ini diduga kemungkinan luas grup sunspot memiliki energi kecil untuk membangkitkan flare hα kelas 2. flare hα kelas 2 mayoritas terjadi pada luas grup sunspot dengan luas penampang 250 sampai 600 × 10−6disk matahari. terdapat lima peristiwa flare hα kelas 3. empat peristiwa diantaranya dibangkitkan dari grup sunspot dengan luas penampang lebih dari 500 × 10−6 disk matahari. berikut sampel data grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr ditampilkan pada gambar 3 dibawah ini: nilai kelas mc.intosh in te n s it a s f la re s x r ( x 1 0 ^ -6 ) 6050403020100 600 500 400 300 200 100 0 gambar 3. grafik sebaran data nilai kelas mc.intosh dengan flare sxr pada gambar 3 menunjukkan bahwa sebanyak 38 nilai kelas mc.intosh dapat menimbulkan flare sxr dengan tingkatan intensitas yang berbeda-beda. flare kelas c tersebar merata pada kelas mc.intosh yang bernilai 5 sampai 60. flare sxr kelas m mayoritas terjadi pada kisaran kelas mc.intosh yang bernilai 20 sampai 60 dengan nilai flare sxr 60 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. hanya 7 flare sxr kelas x yang muncul pada nilai kelas 30 sampai 60. karena memiliki intensitas besar, sehingga diperlukan potensi energi yang sangat besar dari grup sunspot yang besar pula. hasil sampling data flare hα yang dibangkitkan dari grup sunspot dengan nilai 5 sampai dengan 60 klasifikasi mc.intosh ditampilkan dalam gambar 4 berikut. nilai kelas mc.intosh l u a s f la r e h a lf a ( x 1 0 ^ 6 d is k m a t a h a r i) 6050403020100 800 700 600 500 400 300 200 100 0 gambar 4 grafik sebaran data nilai kelas mc.intosh dengan flare h𝛼 pada gambar 4 menunjukkan bahwa flare h𝛼 kelas sf banyak terjadi pada rentang nilai mc.intosh 20 sampai 60. flare hα kelas 1 dibangkitkan dari kelas mc intosh di atas 20. flare hα kelas 2 sering terjadi pada nilai kelas mc.intosh 20 sampai 60. namun, flare hα kelas 2 jarang ditimbulkan dari nilai kelas mc.intosh 5 sampai 20. hal ini diduga bahwa kelas mc.intosh rendah (cro, cso, dll) ini mendapat tambahan energi dari lapisan dibawahnya atau dari grup sunspot besar yang ada disekitarnya sehingga dapat membangkitkan flare hα kelas 2. hanya terdapat 3 peristiwa flare yaitu pada grup sunspot dso, fki dan fhc. flare hα kelas 3 ditimbulkan oleh kelas mc.intosh pada nilai 25 ke atas. 2. analisis klaster masing-masing data akan dilakukan proses analisis klaster dengan menggunakan metode non hierarki yaitu metode k-means. a. standarisasi variabel perhitungan jarak euclid sangat rentan terhadap perbedaan skala pengukuran, yang biasanya ditunjukkan oleh perbedaan variansi antar variabel. sehingga perlu dilakukan standarisasi variabel. penulis menggunakan minitab 14. b. menghitung jarak euclid data standar tersebut kemudian digunakan untuk mengukur jarak euclid. misalnya hasil dari jarak euclid setiap objek luas grup sunspot pada data flare sxr kelas c, m dan x dengan hasil sebagai berikut: tabel 1. hasil jarak euclid data luas grup sunspot yang membangkitkan flare sxr objek 10 30 … 2200 10 0 0,9458 … 1,1537 20 1,6291 1,9418 … 2,2252 22 0,0209 0,9660 … 1,1677 ⁞ ⁞ ⁞ ⁞ ⁞ 2200 1,1537 0,7742 … 0 berdasarkan tabel 1. menunjukkan bahwa luas ke-10 dengan 2200 memiliki jarak lebih pendek daripada luas ke-10 dengan 36. perbedaan jarak tersebut menunjukkan bahwa luas ke-10 dengan 2200 memiliki karakteristik yang lebih mirip daripada luas ke-10 dengan 36. demikian seterusnya penafsiran untuk jarak tiap luas dengan luas lainnya. hal ini juga berlaku pada data luas grup sunspot yang membangkitkan flare hα, data grup sunspot klasifikasi mc.intosh yang analisis klaster k-means dari data luas grup sunspot dan data klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan h𝜶 cauchy – issn: 2086-0382/e-issn: 2477-3344 5 membangkitkan flare sxr dan data grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα. 3. proses clustering dengan metode k-means setiap perhitungan dengan metode ini dilakukan dengan menggunakan microsoft excel 2010. i. menentukan rata-rata di tiap klaster langkah pertama dalam melakukan pengklasteran yaitu menentukan rata-rata di tiap klaster. misalnya rata-rata tiap klaster pada luas grup sunspot yang membangkitkan flare sxr adalah tabel 2. hasil rata-rata tiap luas grup sunspot yang membangkitkan flare sxr no luas grup sunspot c m x ratarata 1 10 1,84 0 0 1,84 2 20 2,41 79 0 40,70 3 22 1,8 0 0 1,8 ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ 132 2200 0 0 120 120 pada tabel 2 menunjukkan bahwa nilai 0 tidak terjadi fenomena flare dan rata-rata terbesar terjadi pada luas 1160 sebesar 540 dan terkecil adalah 710 sebesar 1,2. sedangkan pada luas grup sunspot yang membangkitkan flare hα, rata-rata terkecil adalah luas 1280 sebesar 12 dan terbesar adalah 1525 sebesar 744. pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr rata-rata terbesar adalah kelas ekc sebesar 119,42 dan terkecil adalah dsc sebesar 1,3. dan pada data grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα rata-rata terbesar adalah kelas fki sebesar 333,56 dan terkecil adalah dsc sebesar 10. ii. menentukan banyak klaster (k) jumlah klaster (k) pada penelitian ini ditetapkan sebanyak 2, dengan berdasarkan pada ukuran jarak rata-rata terbesar dan terkecil. pada luas grup sunspot yang membangkit flare sxr, rata-rata tekecil pada luas 710 (c; m; x) masing-masing dengan ratarata intensitas (1,2; 0; 0). sedangkan rata-rata terbesar pada luas 1160 (c; m; x) masingmasing dengan rata-rata intensitas (0; 0; 540). sedangkan pada luas grup sunspot yang membangkit flare hα, rata-rata tekecil pada luas 1280 (sf; 1; 2; 3) masing-masing dengan rata-rata luas flare (12; 0; 0; 0) dan rata-rata terbesar pada 1525 (sf; 1; 2; 3) masing-masing dengan rata-rata luas flare (0; 0; 0; 744). adapun pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr, ratarata terkecil pada kelas dso (c; m; x) masingmasing dengan rata-rata intensitas (1,3; 0; 0). sedangkan rata-rata terbesar pada kelas ekc (c; m; x) masing-masing dengan rata-rata intensitas (3,90; 39,36; 315). pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα, rata-rata terkecil pada dsc (sf; 1; 2; 3) masing-masing dengan rata-rata luas flare (10; 0; 0; 0). sedangkan rata-rata terbesar pada fki (sf; 1; 2; 3) masing-masing dengan rata-rata luas flare (41,40; 132,50; 411,33; 749). iii. menghitung jarak tiap objek ke tiap centroid (rata-rata) setelah menentukan banyaknya klaster, maka langkah selanjutnya yaitu menghitung jarak setiap objek dari rata-rata terbesar (𝑐1) dan rata-rata terkecil (𝑐2) yang dihitung dengan menggunakan jarak euclid pada masing-masing data. misalnya hasil jarak tiap objek ke tiap centroid flare sxr pada luas grup sunspot sebagai berikut: tabel 3 hasil jarak tiap objek ke tiap centroid flare sxr pada luas grup sunspot dari hasil perhitungan jarak tiap objek ke tiap rata-rata terbesar (𝑐1) dan rata-rata terkecil (𝑐2) lalu dibandingkan. jika jarak suatu kelas memiliki jarak dekat dengan 𝑐1 maka akan masuk klaster 1. begitu sebaliknya. hal ini juga berlaku pada keempat jenis data. iv. menentukan rata-rata (centroid) baru langkah selanjutnya yaitu menghitung rata-rata baru. rata-rata baru diperoleh dari nilai rataan dari ketiga variabel (c; m; x) pada tiap klaster baru untuk luas grup sunspot dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr. sedangkan ratarata baru diperoleh dari nilai rataan dari keempat variabel (sf; 1; 2; 3) pada tiap klaster baru untuk luas grup sunspot dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα. misalnya pada luas grup sunspot yang membangkitkan flare sxr. hasil rata-rata terbesar baru pada luas grup luas grup sunspot 𝑐1 𝑐2 10 540,00 0,64 20 545,75 79,01 22 540,00 0,60 30 540,64 26,11 ⁞ ⁞ ⁞ 2200 420,00 120,01 siti jumaroh 6 volume 4 no.1 november 2015 sunspot yang membangkitkan flare sxr adalah sebagai berikut: tabel 4. nilai rata-rata terbesar baru luas grup sunspot yang membangkitkan flare sxr pada masing-masing objek sedangkan tabel rata-rata terkecil baru pada luas grup sunspot yang membangkitkan flare sxr: tabel 5. nilai rata-rata terkecil baru luas grup sunspot yang membangkitkan flare sxr pada masing-masing objek no luas grup sunspot c m x 1 10 1,84 0,00 0,00 2 20 2,41 79,00 0,00 3 22 1,80 0,00 0,00 ⁞ ⁞ ⁞ ⁞ ⁞ 127 2200 0,00 0,00 120,00 rata-rata 4,20 32,83 154,41 berdasarkan tabel 4 dan 5 menunjukkan nilai rata-rata baru terbesar dan terkecil pada luas grup sunspot yang membangkitkan flare sxr (c; m; x) dari tiap klaster adalah 𝑅1𝑎 (ratarata terbesar baru luas grup sunspot yang membangkitkan flare sxr)= (4,23; 33,66; 450) dan 𝑅2𝑎 (rata-rata terkecil baru luas grup sunspot yang membangkitkan flare sxr) = (4,20; 32,83; 154,41). sedangkan pada luas grup sunspot yang membangkitkan flare hα (sf; 1; 2; 3), nilai rata-rata baru terkecil (𝑅1𝑏 ) adalah (29,10; 180; 0,00; 703,80) dan nilai rata-rata baru terkecil adalah (𝑅2𝑏 ) adalah (36,01; 152,88; 377,20; 0). sedangkan pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr (c; m; x), nilai rata-rata baru terkecil (𝑅1𝑐 ) adalah (4,22; 29,48; 221,94) dan nilai rata-rata baru terkecil adalah (𝑅2𝑐 ) adalah (3,39; 24,95; 110). dan pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα (sf; 1; 2; 3), nilai rata-rata baru terkecil (𝑅1𝑑 ) adalah (46,12; 155,97; 410,83; 689,17) dan nilai rata-rata baru terkecil adalah (𝑅2𝑑 ) adalah (38,68; 155,39; 365,10; 0). v. menghitung jarak tiap objek ke tiap centroid (rata-rata) baru. setelah mendapatkan rata-rata baru maka langkah selanjutnya yaitu menghitung jarak tiap objek ke tiap rata-rata baru dengan menggunakan jarak euclid. seperti halnya langkah (iii), misalnya pada luas grup sunspot yang membangkitkan flare sxr, sehingga diperoleh hasil sebagai berikut: tabel 6 hasil jarak tiap objek ke tiap centroid baru flare sxr pada luas grup sunspot luas grup sunspot r1a r2a 10 463,17 157,88 20 465,68 161,18 22 463,17 157,88 30 462,50 154,56 ⁞ ⁞ ⁞ 2200 331,74 160,39 dari hasil perhitungan jarak tiap objek ke tiap rata-rata terbesar dan rata-rata terkecil lalu dibandingkan. jika jarak suatu kelas memiliki jarak dekat dengan rata-rata terbesar maka akan masuk klaster 1. begitu sebaliknya. misalnya pada tabel 6, perbandingan jarak grup sunspot luas 20 terhadap r1a dan r2a yang memberikan nilai r2a < r1a menyatakan bahwa grup sunspot luas 20 tergolong klaster 2. sehingga diperoleh luas grup sunspot yang tergolong klaster 1 adalah 400, 600, 730, 1160, dan 1630. sedangkan yang tergolong klaster 2 adalah luas 10, 20, 22, 30, 36, 40, 48, 50, 60, 70, dll.pada tabel 6 ini menunjukkan bahwa hasil dari pengklasteran jarak tiap objek ke tiap ratarata baru ini memiliki anggota yang sama dengan jarak tiap objek ke tiap rata-rata pada tabel 3, sehingga proses pengklasteran berhenti. dari hasil pengklasteran luas grup sunspot yang membangkitkan flare sxr dapat ditunjukkan pada gambar 5 di bawah ini: gambar 5. hasil dari pengklasteran luas grup sunspot yang membangkitkan flare sxr no luas grup sunspot c m x 1 400 4,13 36,5 360 2 600 3,9 51,5 400 3 730 4,65 13 570 4 1160 0 0 540 5 1630 0 0 380 rata-rata 4,23 33,66 450 analisis klaster k-means dari data luas grup sunspot dan data klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan h𝜶 cauchy – issn: 2086-0382/e-issn: 2477-3344 7 pada gambar 5 mengilustrasikan adanya pemisahan klaster 1 dan klaster 2 sesuai dengan objek-objek yang saling berdekatan jaraknya. klaster 1 terdiri dari objek 400, 600, 730, 1160 dan 1630 mempunyai kesamaan atau kemiripan karakter dalam membangkitkan flare sxr. hal ini juga berlaku pada kedua jenis data, yaitu luas grup sunspot yang membangkitkan flare hα, dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα. namun, pada grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr terjadi sebanyak tiga kali iterasi, karena pada dua kali iterasi ada pemindahan klaster yaitu eac yang semula masuk klaster 1 ke klaster 2. vi. validasi klaster dengan uji anova setelah melakukan proses clustering, langkah selanjutnya yaitu mengecek apakah variabel-variabel yang telah membentuk klaster tersebut merupakan variabel pembeda atau bukan. misalnya hasil anova pada luas grup sunspot yang membangkitkan flare sxr adalah sebagai berikut: tabel . hasil anova pada luas grup sunspot yang membangkitkan flare sxr vari abel rata rata jumlah kuadrat df kesalaha n ratarata jumlah kuadrat df fhitung ftabel c 352872 83 185527 48 1,9 3,8 m 325098 63 260478 68 1,3 3,8 x 181472 16 306871 115 0,6 3,8 adapun langkah pertama dalam pengujian anova yaitu menetapkan hipotesis. hipotesis yang akan di uji dalam penelitian ini adalah h0 = variabel c, m dan x bukan pembeda dalam pengklasteran h1 = variabel c, m dan x pembeda dalam pengklasteran kriteria uji tolak h0 jika fhitung > ftabel (f∝,k−1,n−k). tingkat signifikansi yang dipakai dalam penelitian ini adalah 5%. pada luas grup sunspot yang membangkitkan flare sxr , nilai n = 132 dan k = 2. sehingga ftabel = f∝,k−1,n−k = f0,05,1,130 = 3,8. dari tabel 7 menunjukkan bahwa semua ketiga variabel c, m dan x menerima h0, karena nilai jika fhitung < ftabel. dengan demikian ketiga variabel tersebut merupak variabel bukan pembeda dalam pengklasteran dalam luas grup sunspot yang membangkitkan flare sxr. hal ini juga berlaku pada luas grup sunspot yang membangkitkan flare hα, dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα dan grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr. bahwa semua variabel pada ketiga jenis data tersebut bukan pembeda dalam pengklasteran. 4. interprestasi klaster interprestasi klaster merupakan proses terakhir dari pengklasteran, yang bertujuan untuk memberi ciri spesifik atau menggambarkan isi klaster yang terbentuk. berdasarkan penjelasan subbab (v) di atas luas grup sunspot yang membangkitkan flare sxr dapat dikelompokkan menjadi dua yaitu 1. klaster pertama, terdiri dari 5 objek yaitu luas grup sunspot 400, 600, 730, 1160, dan 1630. luas grup sunspot ini mampu membangkitkan flare sxr kelas x dengan intensitas sebesar 450 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. sehingga objek-objek dalam klaster ini berpotensi membangkitkan flare sxr tinggi. 2. klaster kedua, terdiri dari 127 objek, misalnya luas grup sunspot 10, 20, 22, 2170, 2180 dll. luas grup sunspot yang tergolong klaster kedua ini mayoritas membangkitkan flare sxr kelas c dan m. namun ada luas grup sunspot yang mampu membangkitkan flare sxr kelas x dengan intensitas kurang dari 250 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. sehingga klaster kedua digolongkan menjadi kelompok luas grup sunspot yang membangkitkan flare sxr rendah sedangkan luas grup sunspot yang membangkitkan flare hα dapat dikelompokkan sebagai berikut: 1) klaster pertama, terdiri dari 5 objek yaitu 90, 630, 800, 1525 dan 2200. luas flare hα yang dihasilkan dari objek-objek ini mempunyai luas rata-rata di atas 600 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2disk matahari. terdapat perkecualian, luas grup sunspot 90 ini mampu membangkitkan flare hα kelas 3, karena hal ini diduga adanya suplai energi dari grup sunspot didekatnya. sehingga klaster pertama digolongkan menjadi kelompok luas grup sunspot yang membangkitkan flare hα tinggi. 2) klaster kedua, terdiri dari 127 objek misalnya 10, 20, 200, 210, 1230, dll. klaster ini memiliki rata-rata luas grup sunspot yang membangkitkan flare hα relatif lebih rendah dari klaster pertama. sehingga klaster kedua dapat digolongkan menjadi kelompok luas grup sunspot yang membangkitkan flare hα rendah. grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dapat dikelompokkan sebagai berikut: 1) klaster pertama, terdiri dari 5 objek yaitu eko, eki, fki, ekc dan fkc. grup sunspot klasifikasi mc.intosh ini mampu membangkitkan flare sxr kelas x dengan intensitas lebih dari 150 × siti jumaroh 8 volume 4 no.1 november 2015 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. grup sunspot klasifikasi mc.intosh yang masuk ke klaster 1 ini merupakan grup sunspot yang mempunyai dua kutub dengan ukuran besar dan luas daerah aktif lebih dari 100 bujur. sehingga klaster pertama dapat digolongkan menjadi kelompok grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dengan intensitas yang besar. 2) klaster kedua, terdiri dari 33 objek misalnya cro, cao, cso, dll. objek-objek dalam klaster ini rata-rata hanya mampu membangkitkan flare sxr kelas c dan m. namun, ada grup sunspot klasifikasi mc.intosh yang membangkitkan flare kelas x dengan intensitas kurang dari 150 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. sehingga klaster kedua dapat digolongkan menjadi kelompok kelas grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dengan intensitas rendah. sedangkan grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα dapat dikelompokkan sebagai berikut: 1. klaster pertama, terdiri dari 3 objek yaitu dso, fki, dan fkc. grup sunspot klasifikasi mc.intosh ini mampu membangkitkan flare hα kelas 3 dengan luas penampang lebih dari 600 × 10−6disk matahari. sehingga klaster pertama dapat digolongkan menjadi kelompok kelas mc.intosh yang membangkitkan flare hα tinggi. 2. klaster kedua, terdiri dari 35 objek misalnya kelas cro, cao, cso dll. grup sunspot klasifikasi mc.intosh yang masuk ke klaster ini mayoritas membangkitkan flare hα kelas sf, 1 dan 2. sehingga klaster kedua dapat digolongkan menjadi kelompok kelas mc.intosh yang membangkitkan flare hα rendah. kesimpulan berdasarkan hasil analisis data luas grup sunspot dan data grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr dan hα menggunakan metode k-means. dapat disimpulkan tergolong menjadi 2 klaster yaitu berdasarkan rata-rata terbesar dan rata-rata terkecil. berikut hasil klaster yang diperoleh: 1. klaster luas grup sunspot yang membangkitkan flare sxr adalah: a. klaster pertama terdiri dari 5 objek tergolong grup sunspot yang membangkitkan flare sxr kelas x, dengan intensitas rata-rata 280,245 × 10−6𝑊𝑎𝑡𝑡 ∕ 𝑚2. b. klaster kedua terdiri dari 127 objek dan digolongkan menjadi klaster yang mampu membangkitkan flare sxr kelas m, dengan intensitas rata-rata 27,87 × 10−6𝑊𝑎𝑡𝑡 𝑚2⁄ . 2. klaster luas grup sunspot yang membangkitkan flare hα adalah: a. klaster pertama tergolong luas grup sunspot yang mampu membangkitkan flare hα kelas 2, dengan luas rata-rata 491,45 × 10−6 disk matahari. b. klaster kedua tergolong klaster yang mampu membangkitkan flare hα kelas 1, dengan luas rata-rata 116,84 × 10−6 disk matahari. 3. klaster grup sunspot klasifikasi mc.intosh yang membangkitkan flare sxr adalah: a. klaster pertama tergolong klaster yang mampu membangkitkan flare sxr kelas m dengan intensitas rata-rata 85,21 × 10−6𝑊𝑎𝑡𝑡 𝑚2⁄ . b. klaster kedua tergolong klaster yang mampu membangkitkan flare sxr kelas m dengan intensitas rata-rata 14,59 × 10−6𝑊𝑎𝑡𝑡 𝑚2⁄ . 4. klaster grup sunspot klasifikasi mc.intosh yang membangkitkan flare hα adalah: a. klaster pertama tergolong klaster yang mampu membangkitkan flare hα kelas 2, dengan luas rata-rata 314,92 × 10−6disk matahari. b. klaster kedua tergolong klaster yang dapat membangkitkan flare hα kelas 2 dengan luas rata-rata 130,37 × 10−6 disk matahari. daftar pustaka [1] r. sitepu, irmehyana and b. gulton, "analisis cluster terhadap tingkat pencemaran udara pada sektor industri di sumatera selatan," jurnal penelitian sains, pp. 11-17, 2011. [2] widarjono, analisis statistika multivariat terapan, yogjakarta: unit penerbit dan penerbit sekolah tinggi ilmu manajemen ykpn, 2010. [3] narimawati, teknik-teknik analisis multivariat riset ekonomi, yogyakarta: graha ilmu, 2008. [4] a. abdillah, analisis klaster pada grup sunspot klasifikasi mc. intosh yang berpotensi membangkitkan flare (data noaa, studi kasus di bpd lapan watukosek)., malang: uin maulana malik ibrahim malang., 2014. [5] gundono, analisis data multivariat, yogjakarta: bpfe-yogjakarta, 2011. [6] s. yulianto and k. h. hidayatullah, "analisis klaster untuk pengelompokan kabupaten/kota di provinsi jawa tengah berdasarkan indikator kesejahteraan rakyat," jurnal statistik, pp. 56-63, 2014. [7] s. ghosh and s. k. dubey, "comperative analisis klaster k-means dari data luas grup sunspot dan data klasifikasi mc.intosh yang membangkitkan flare soft x-ray dan h𝜶 cauchy – issn: 2086-0382/e-issn: 2477-3344 9 analysis of k-means and fuzzy c-means alogarithms," internasional journal of advancea computer science and applications, pp. 35-39, 2013. [8] b. nurgiyantoro, statistik terapan, yogjakarta: gadja mada university press, 2009. [9] a. yamami, "klasifikasi flare matahari," 8 agustus 2010. [online]. available: http://langitselatan.com/2010/08/08/klasifi kasi-flare-matahari/. [accessed 15 februari 2015]. [10] m. nathanael, "flare matahari dan pengamatannya," 31 agustus 2010. [online]. available: http://langitselatan.com/2010/08/31/flarematahari-danpengamatannya/. [accessed 15 februari 2015]. pendahuluan kajian teori 1. analisis multivariat 2. analisis klaster 3. analisis varian 4. matahari metode penelitian pembahasan 1. deskripsi data 2. analisis klaster 3. proses clustering dengan metode k-means 4. interprestasi klaster kesimpulan daftar pustaka estimation parameters and modelling zero inflated negative binomial cauchy – jurnal matematika murni dan aplikasi volume 4(3) (2016), pages 115-119 submitted: august, 20 2016 reviewed: october, 31 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3656 estimation parameters and modelling zero inflated negative binomial cindy cahyaning astuti1, angga dwi mulyanto2 1 muhammadiyah university of sidoarjo, sidoarjo, indonesia 2 alpha research, malang, indonesia email: cindy.cahyaning@umsida.ac.id, angga.dwi.m@gmail.com abstract regression model between predictor variables and the poisson distributed response variable is called poisson regression model. since, poisson regression requires an equality between mean and variance, it is not appropriate to apply this model on overdispersion. poisson regression can be used to analyze count data but it has not been able to solve problem of excess zero value on the response variable. an alternative model which is more suitable for overdispersion data and can solve the problem of excess zero value on the response variable is zero inflated negative binomial (zinb). in this research, zinb is applied on the case of tetanus neonatorum in east java. the aim of this research is to examine the likelihood function and to form an algorithm to estimate the parameter of zinb and also applying zinb model in the case of tetanus neonatorum in east java. maximum likelihood estimation (mle) method is used to estimate the parameter on zinb and the likelihood function is maximized using expectation maximization (em) algorithm. test results of zinb regression model showed that the predictor variable have a partial significant effect at negative binomial model is the percentage of pregnant women visits and the percentage of maternal health personnel assisted, while the predictor variables that have a partial significant effect at zero inflation model is the percentage of neonatus visits. keywords: overdispersion, tetanus neonatorum, zero inflation, zero inflated negative binomial (zinb) introduction regression analysis is used to determine relationship between one or several response variable (y) with one or several predictor variables (x). in the classical linear model assumptions are response variables follow a normal distribution, but in fact often found the response variable did not follow the normal distribution. to overcome this there is development in the classical linear model, namely the generalized linear model (glm) [1]. glm assuming the response variable follows the exponential family distribution, which has a more general characteristic. in some research, there are often data with response variable that follows a poisson distribution, regression analysis is used to this kind of data is the poisson regression analysis. poisson regression model is commonly used to analyze the data count (data count). poisson regression there is an assumption on y ~ poisson (μ). a key assumption in the poisson regression analysis is the variance should be equal to the average, the condition is called equidispersion. on the type of count data often encountered zero value is more than 50 percent on the response variable (zero inflation) [2]. data proportion that has exaggeration zero value can lead to the accuracy of inference. poisson regression can be used to analyze the data count but still cannot resolve the problem of excessive zero value. in modelling count data if there ar many zero observations on response variable it can be overcome by using zero inflated poisson regression (zip) model [3]. however, if there are many http://dx.doi.org/10.18860/ca.v4i3.3656 mailto:cindy.cahyaning@umsida.ac.id mailto:angga.dwi.m@gmail.com estimation parameters and modelling zero inflated negative binomial cindy cahyaning astuti 116 zero observations and occurs overdispersion then zero inflated poisson regression (zip) inappropriately used. overdispersion can be defined as a condition in which the poisson distribution variance is greater than average. if in modelling count data (data count) there are many zero observations on response variable (zero inflation) and occurs overdispersion then the regression model can be used is zero inflated generalized poisson [2]. in progress there are other alternatives to modelling many zero observations and occurs overdispersion besides using zero inflated generalized poisson (zigp), the regression model is zero inflated negative binomial (zinb). zero inflated negative binomial (zinb) model is formed of poisson gamma mixture distribution [4]. zero inflated negative binomial (zinb) can be used as an alternative to modelling many zero observations and occurs overdispersion because this model does not require the variance should be equal with average, in addition zero inflated negative binomial (zinb) model also has a dispersion parameter that useful to describe the variation of the data, which is commonly denoted by κ (kappa). the purpose of this research is examine the likelihood form, estimation parameters of zero inflated negative binomial (zinb) model and modelling zero inflated negative binomial (zinb) on neonatorum tetanus cases. methods in this research used secondary data sourced from east java health profile 2012 [5]. unit of observation in this research was 38 districts/cities in east java province which covers 29 districts and 9 cities. the response variable (y) used in this research is number of cases of tetanus neonatorum in each district/city in east java province, while the predictor variable (x) is used as much as 4 variables. operational definition of each variable response and predictor variables will be described as a. the response variable (y): number of cases of tetanus neonatorum b. predictor variable (x) 1. the percentage of pregnant mothers visit k4 (x1) 2. percentage of immunization tetanus toxoid (tt) in pregnant women (x2) 3. percentage of maternal mothers assisted by health workers (x3) 4. the percentage of neonates visits (x4) the method of analysis in this study is. a. knowing the probability function of zero inflated negative binomial (zinb) model. b. determining the likelihood function of zero inflated negative binomial (zinb) model based on probability function that are already known. c. develop algorithms for estimation parameter process based on the likelihood function that is already known. parameter estimation of zero inflated negative binomial (zinb) model. was performed using mle method and solved using em algorithm. d. modelling zero inflated negative binomial (zinb) model on neonatorum tetanus cases in east java province e. significance test of parameters model carried out simultan and partial test. statistical tests are used for simultan test is the test statistic g and to partial test used test statistics t. results and discussion estimation parameter zero inflated negative binomial (zinb) was conducted using maximum likelihood estimation (mle) and to maximize the function is used em (expectation maximization) algorithm. probability function of zero inflated negative binomial (zinb) model can be defined as : 𝑃(𝑌𝑖 = 𝑦𝑖) = { 𝜋𝑖 + (1 − 𝜋𝑖)( 1 1+𝜿𝜇𝑖 ) 1 𝜿 ,𝑓𝑜𝑟 𝑦 𝑖 = 0 (1 − 𝜋𝑖)  (𝑦𝑖+ 1 𝜿 )  (1 𝜿 )𝑦𝑖! ( 1 1+𝜿𝜇𝑖 ) 1 𝜿 ( 𝜿𝜇𝑖 1+𝜿𝜇𝑖 ) 𝑦𝑖 ,𝑓𝑜𝑟 𝑦 𝑖 > 0 estimation parameters and modelling zero inflated negative binomial cindy cahyaning astuti 117 em algorithm consists of two stage, expectation and maximization stage. expectation stage is expectation calculation of ln likelihood the function, the next stage maximization is calculation to look for estimation parameter which maximizes the likelihood function. probability function of zinb model consist of two conditions, yi = 0 and yi > 0. response variable is also composed of two conditions, namely zero state and negative binomial state. to describe in detail the condition yi, then it will be redefined variables yi with latent variable zi. 𝑍𝑖 = { 1,𝑖𝑓 𝑦𝑖 𝑓𝑟𝑜𝑚 𝑧𝑒𝑟𝑜 𝑠𝑡𝑎𝑡𝑒 0,𝑖𝑓 𝑦𝑖 > 0𝑓𝑟𝑜𝑚 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 zero inflated negative binomial regression (zinb) model can be defined as two models that are : model for negative binomial �̂�𝑖 𝑙𝑛�̂�𝑖 = �̂�0 + ∑�̂�0𝑥𝑖𝑗 𝑝 𝑗=1 , 𝑖 = 1,2,…,𝑛 𝑎𝑛𝑑𝑗 = 1,2, . . ,𝑝 model for zero inflation �̂�𝑖 𝑙𝑜𝑔𝑖𝑡�̂�𝑖 = 𝛾0 + ∑𝛾0𝑥𝑖𝑗 𝑝 𝑗=1 , 𝑖 = 1,2,…,𝑛 𝑎𝑛𝑑𝑗 = 1,2, . . ,𝑝 em algorithm is alternative methods to maximize likelihood function on the data containing latent variables defining new variables such as variable zi. em algorithm consists of two stage: the expectation stage and maximization stage. expectation stage is calculation of the ln likelihood function, the next stage is maximization calculation stage to look for parameter estimation which maximizes the likelihood function ln results from stage earlier expectations. estimation parameter and parameter test of zero inflated negative binomial (zinb) on neonatorum tetanus cases in east java province using sas software, the result can see at table 1. table 1. estimation parameter and parameter test of zinb parameter estimation se t value (pr >| t|) �̂� 0 -5,847 3,602 -1,623 0,105 �̂� 1 -0,145 0,055 -2,644 0,008* �̂� 2 -0,006 0,010 -0,599 0,549 �̂� 3 0,233 0,101 2,295 0,022* �̂� 4 -0,023 0,067 -0,339 0,735 �̂� 0 11,325 13,409 0,845 0,398 �̂� 1 0,223 0,169 1,316 0,188 �̂� 2 -0,296 0,179 -1,653 0,098 �̂� 3 0,835 0,503 1,660 0,096 �̂� 4 -1,078 0,539 -2,000 0,045* test statistic g = 581,24 results of simultan parameter test based on test statistic g. g test is 581.24. value of g test is greater than 2 (0,05;8) 15, 507  . this shows that simultaneously the predictor variables x1, x2, x3 and x4 have significant effect on response variable. while results of partial parameter test based on test statistical t. according to table 1, there are two predictor variables in negative binomial state model and one predictor variables in zero inflation state model that has t value greater than or equal to t (α / 2; 37 = 2.00) and has p-value less than α (0.05). this indicates that the predictor variables were partial significant effect in negative binomial state model are pregnant mothers visit k4 (x1) and maternal mothers assisted by health workers (x3), while the predictor estimation parameters and modelling zero inflated negative binomial cindy cahyaning astuti 118 variables were partial significant effect in zero inflation state model is the percentage of neonates visits (x4). so that zero inflated negative binomial (zinb) model can be defined as : a. negative binomial state model for �̂� �̂� = 𝑒𝑥𝑝(−5,847 − 0,145 𝑋1 − 0,006 𝑋2 + 0,233 𝑋3 − 0,023 𝑋4) b. zero inflation state model for �̂� �̂� = exp (11,325 + 0,223 𝑋1 − 0,296 𝑋2 + 0,835 𝑋3 − 1,078 𝑋4) 1 + exp (11,325 + 0,223 𝑋1 − 0,296 𝑋2 + 0,835 𝑋3 − 1,078 𝑋4) all coefficient parameter which aren’t significant still is exist in negative binomial state zero inflation state model because it is intended to determine the contribution of each predictor variable on the response variable can be defined as : zero inflation model for ˆ  1. each additional 1 percent of pregnant mothers visit k4 (x1) it will increase the chances of the number of tetanus neonatorum by exp (0.223) = 1.249 times the number of cases of tetanus neonatorum original, if the other variables constant value. 2. each additional 1 percent immunization tetanus toxoid (tt) in pregnant women (x2) will decrease the chances of the number of tetanus neonatorum by exp (0.296) = 1.344 times the number of cases of tetanus neonatorum original, if the other variables constant value. 3. each additional 1 percent of maternal mothers assisted by health workers (x3) then it will increase the chances of the number of tetanus neonatorum by exp (0.835) = 2.305 times the number of cases of tetanus neonatorum original, if the other variables constant value. 4. each additional 1 percent of neonates visits (x4) will decrease the chances of the number of tetanus neonatorum by exp (1.078) = 2.939 times the number of cases of tetanus neonatorum original, if the other variables constant value. let’s discussion, based on the negative binomial state model and zero inflation state model, there are signs of regression coefficient as opposed to the theory are percentage of maternal mothers assisted by health workers (x3) to model negative binomial state model and the percentage of pregnant mothers visit k4 (x1) and the percentage of maternal mothers assisted by health workers (x3) for zero inflation state model . the existence of the regression coefficient has a sign contrary to the theory of probability caused by the effect of the multikolinieritas. moreover sign contrary to the theory also caused by the shape of the data pattern of the predictor variables that have a positive correlation with the response variable. in a subsequent study if there are multikolinieritas the predictor variables can be addressed using principal component analysis (pca). conclusion based on the results, estimation parameter of zero inflated negative binomial (zinb) model was conducted using maximum likelihood estimation (mle) and to maximize the likelihood function used the em (expectation maximization) algorithm. for parameter test predictor variable that has significant effect on the number of cases of tetanus neonatorum are are pregnant mothers visit k4 (x1) and maternal mothers assisted by health workers (x3) for the negative binomial state models, while zero inflation state model predictor variable that has significant effect on the number of cases of tetanus neonatorum include the percentage of neonates visit (x4). estimation parameters and modelling zero inflated negative binomial cindy cahyaning astuti 119 references [1] a. agresti, categorical data analysis, new york: john wiley and sons, inc., 2002. [2] f. famoye dan k. p. singh, “zero inflated poisson regression model with an applications domestic violence to accident data,” journal of data science, pp. 117-130, 2006. [3] d. lambert, “zero inflated poisson regression, with an application to defect in manufacturing,” technometric, vol. 34, no. 1, 1992. [4] j. m. hilbe, negative binomial regression, new york: cambridge university press, 2011. [5] dinas kesehatan provinsi jawa timur, profil kesehatan provinsi jawa timur tahun 2012, surabaya: dinas kesehatan provinsi jawa timur, 2013. cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 8-14 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 26 july 2016 reviewed: 18 october 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.4305 estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto, henny pramoedyo, suci astutik department of statistics, faculty of mathematics and natural sciences brawijaya university, malang email: danangariyanto75@gmail.com, pramoedyohp@yahoo.com, suci_sp@yahoo.com abstract regression is a method connected independent variable and dependent variable with estimation parameter as an output. principal problem in this method is its application in spatial data. geographically weighted regression (gwr) method used to solve the problem. gwr is a regression technique that extends the traditional regression framework by allowing the estimation of local rather than global parameters. in other words, gwr runs a regression for each location, instead of a sole regression for the entire study area. the purpose of this research is to analyze the factors influencing wet land paddy productivities in tulungagung regency. the methods used in this research is gwr using cross validation bandwidth and weighted by adaptive gaussian kernel function. this research using four variables which are presumed affecting the wet land paddy productivities such as: the rate of rainfall(x1), the average cost of fertilizer per hectare(x2), the average cost of pesticides per hectare(x3) and allocation of subsidized npk fertilizer of food crops subsector(x4). based on the result, x1, x2, x3 and x4t has a different effect on each district. so, to improve the productivity of wet land paddy in tulungagung regency required a special policy based on the gwr model in each district. keywords: spatial data, geographically weighted regression, gwr, cross validation, adaptive gaussian kernel introduction the conventional spatial analysis techniques use a single equation to assess the overall relationships between the dependent and independent variables across space, known as a global analytic approach. one important assumption underlying this approach is that the relationships of interest are stationary or homogeneous spatially. while the global perspective is effective in handling spatial dependence and generating less unbiased estimates (than the non-spatial modeling), it is not capable of exploring spatial non-stationarity (or heterogeneity) or identifying place-specific associations [1]. geographically weighted regression (gwr) is a local spatial statistical technique used to analyze spatial non-stationarity, defined as when the measurement of relationships among variables differs from location to location. unlike conventional regression, which produces a single regression equation to summarize global relationships among the explanatory and dependent variables, gwr generates spatial data that express the spatial variation in the mailto:danangariyanto75@gmail.com mailto:pramoedyohp@yahoo.com mailto:suci_sp@yahoo.com estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 9 relationships among variables [2]. this approach includes locational information and smoothing techniques into regression models. in contrast to the global approach, gwr has proved to be a useful local spatial analysis tool that helps researchers to generate nuanced insights into existing literature [1]. in this research, the parameter estimation on the gwr method requires a weights matrix that calculated by adaptive gaussian kernel function. the data weighting is according to the proximity of the 𝑖-th observation location. cross validation is used for estimating the kernel bandwidth. the case study investigated the productivity of wet land paddy in tulungagung regency, there are 18 districts in tulungagung regency and 1 district has no wet land paddy productivity. this research using four variables which are presumed affecting the wet land paddy productivities such as: the rate of rainfall(x1), the average cost of fertilizer per hectare(x2), the average cost of pesticides per hectare(x3) and allocation of subsidized npk fertilizer of food crops sub-sector(x4). the final result of this gwr method will be obtained productivity model of wet land paddy in tulungagung regency. in addition, the mapping of paddy productivity per district is expected to be useful and add information especially in agriculture to increase wet land paddy productivity in tulungagung regency. methods geographically weighted regression is an extension of the traditional multiple linear regression toward a local regression, in which regression coefficients are specific to a location rather than being global estimates. the specification of a basic gwr model is: 𝑦𝑖 = 𝛽0(𝑢𝑖, 𝑣𝑖 ) + ∑ 𝛽𝑘 (𝑢𝑖 , 𝑣𝑖 )𝑥𝑖𝑘 + 𝜀𝑖 ; 𝑖 = 1,2, … , 𝑛 (1) 𝑝 𝑘=1 where yi is the dependent variable at location i, 𝑥𝑖𝑘 is the value of the kth explanatory variable at location i, the 𝛽𝑘 (𝑢𝑖 , 𝑣𝑖 )𝑥𝑖𝑘 is the local regression coefficient for the kth explanatory variable at location i, 𝛽0(𝑢𝑖 , 𝑣𝑖 ) is the intercept parameter at location i, and 𝜀𝑖 is the random disturbance at location i, which may follow an independent normal distribution with zero mean and homogeneous variance [3]. to facilitate the exposition, it is convenient to express the gwr model in matrix notation: 𝒀𝒘 = 𝑿𝒘𝜷𝒘 + 𝜺𝒘 (2) weighted least square (wls) is used for geographically weighted regression parameter estimation, so: 𝐿 = 𝜺𝒘 𝑻𝜺𝒘 = (𝒀𝒘 − 𝑿𝒘𝜷𝒘) 𝑻(𝒀𝒘 − 𝑿𝒘𝜷𝒘) = 𝒀𝒘 𝑻𝒀𝒘 − 𝜷𝒘 𝑻 𝑿𝒘 𝑻𝒀𝒘 − 𝒀𝒘 𝑻𝑿𝒘 𝜷𝒘 + 𝜷𝒘 𝑻 𝑿𝒘 𝑻𝑿𝒘𝜷𝒘 = 𝒀𝒘 𝑻𝒀𝒘 − 𝟐𝜷𝒘 𝑻 𝑿𝒘 𝑻𝒀𝒘 + 𝜷𝒘 𝑻 𝑿𝒘 𝑻𝑿𝒘𝜷𝒘 (3) as we learned in calculus, a univariate optimization involves taking the derivative and setting equal to 0. this gives us, 𝜕𝐿 𝜕𝜷𝒘 = 𝜕(𝒀𝒘 𝑻𝒀𝒘 − 𝟐𝜷𝒘 𝑻 𝑿𝒘 𝑻𝒀𝒘 + 𝜷𝒘 𝑻 𝑿𝒘 𝑻𝑿𝒘 𝜷𝒘) 𝜕𝜷𝒘 = 0 𝜕𝐿 𝜕𝜷𝒘 = −𝟐𝑿𝒘 𝑻𝒀𝒘 + 𝟐𝑿𝒘 𝑻𝑿𝒘�̂�𝒘 = 0 𝜕𝐿 𝜕𝜷𝒘 = −𝑿𝒘 𝑻𝒀𝒘 + 𝑿𝒘 𝑻𝑿𝒘�̂�𝒘 = 0 estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 10 𝑿𝒘 𝑻𝑿𝒘�̂�𝒘 = 𝑿𝒘 𝑻𝒀𝒘 �̂�𝒘 = (𝑿𝒘 𝑻𝑿𝒘) −𝟏𝑿𝒘 𝑻𝒀𝒘 (4) if 𝑾 1 2𝒀 = 𝑾 1 2𝑿𝜷 + 𝑾 1 2𝜀 is equal to 𝒀𝒘 = 𝑿𝒘 𝜷𝒘 + 𝜺𝒘 so, = [(𝑾 1 2𝑿)𝑻𝑾 1 2𝑿] −𝟏 (𝑾 1 2𝑿)𝑻𝑾 1 2𝒀 = (𝑿𝑻 𝑾 1 2𝑾 1 2𝑿)−𝟏𝑿𝑻𝑾 1 2𝑾 1 2𝒀 �̂�𝒘 = (𝑿 𝑻 𝑾𝑿)−𝟏𝑿𝑻𝑾𝒀 (5) for each i-th point, the geographically weighted regression model parameter estimation is performed by matrix operation: �̂�(𝑢𝑖 , 𝑣𝑖 ) = (𝑿 𝑻 𝑾(𝑢𝑖 , 𝑣𝑖 ) 𝑿) −𝟏𝑿𝑻 𝑾(𝑢𝑖 , 𝑣𝑖 ) 𝒀 𝑖 = 1,2, … , 𝑛 (7) with, 𝑾𝒊𝒋(𝑢𝑖 , 𝑣𝑖 ) = [ 𝒘𝑖1 0 ⋯ 0 0 𝒘𝑖2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 𝒘𝑖𝑛 ] the first step to estimate parameters in gwr, it is important to decide the spatial weighting matrix, which can be calculated by different methods. one method is to specify wij as a continuous and monotonic decreasing function of distance dij between points i and j. for adaptive kernel size, the weight of each point can be calculated by applying the gaussian function [3]: 𝑤𝑖𝑗 (𝑢𝑖, 𝑣𝑖 ) = 𝑒𝑥𝑝(− ( 𝑑𝑖𝑗 ℎ𝑖(𝑞) ) 2 ) (8) where 𝑤𝑖𝑗 (𝑢𝑖, 𝑣𝑖 ) is the weight of location j in the space at which data are observed for estimating the dependent variable at location i, andℎ𝑖(𝑞) is referred as a bandwidth. 𝑑𝑖𝑗 is eucledian distance between points i and j, 𝑑𝑖𝑗 = √(𝑢𝑖 − 𝑢𝑗 ) 2 + (𝑣𝑖 − 𝑣𝑗 ) 2 . bandwidth is used to specifies how the extent of the kernel should be determined. it controls the degree of smoothing in the model. crossvalidation (cv) is an iterative process that searches for the kernel bandwidth that minimizes the prediction error of all the y(s) using a subset of the data for prediction [1]. 𝐶𝑉(ℎ) = ∑ (𝑦𝑖 − �̂�≠𝑖 (ℎ)) 2𝑛 𝑖=1 (9) where �̂�≠𝑖 (ℎ) is the predicted value of observation 𝑖 with calibration location 𝑖 left out of the estimation dataset. the second step is testing for spatial heterogeneity. spatial heterogeneity indicates the variation between location. so, each location has different relationship structures and parameters. spatial data heterogeneity can be tested using the breach-pagan (bp) test [4]: h0: 1 1 2 2 2 2 2 2 ( , ) ( , ) ( , )n nu v u v u v        , 2 2 ( , )i iu v   (there is no spatial heterogeneity) h1: at least one 2 2 ( , )i iu v   (there is spatial heterogeneity) 𝐵𝑃 = ( 1 2 ) 𝒇 𝑇 𝒁 (𝒁𝑇 𝒁)−1𝒁𝑇 𝒇 + ( 𝟏 𝑇 ) [ 𝑒𝑇𝑾𝑒 𝝈𝟐 ] 𝟐 ~ 𝜒2(𝑘+1) (10) estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 11 where 𝒇𝒊 = 𝒆𝒊 𝝈𝟐 − 𝟏, 𝒆𝒊 is a vector of ols residuals, 𝝈 𝟐 is the variance based on ols residual, t= trace [ wtw + w2] and 𝒁 is an n by (k+1) matrix of normal standard score (z). the third step is testing parameters of gwr model partially using t test with hypothesis as follows [5]: h0: 𝛽𝑗 (𝑢𝑖 , 𝑣𝑖 ) = 0 h1: 𝛽𝑗 (𝑢𝑖 , 𝑣𝑖 ) ≠ 0 ; 𝑗 = 1,2, … , 𝑘 t test statistic can be written as follows: �̂�𝑘(𝑢𝑖,𝑣𝑖) �̂�√𝑐𝑗𝑗 ∼ 𝑡(𝑛−𝑘−1) (11) where 𝑐𝑗𝑗 is a diagonal element of the cct matrix, with 𝑪 = (𝑿 𝑻 𝑾(𝒖𝒊, 𝒗𝒊)𝑿) −𝟏𝑿𝑻𝑾(𝒖𝒊, 𝒗𝒊). the fourth step is testing parameter of gwr model simultaneously, the hypothesis is: h0: ( , ) j i i j u v  , where 𝑗 = 1, 2, … 𝑘. h1: at least one ( , ) j i i u v has a relation with location ( , )i iu v the statistic test is: 2 1 2 2 1 1 2 * ( , 1) 0 ( ) / ( ) / 1 n k sse h f sse h n k               (12) where: 0 ( )sse h  t t y (i l) (i l)y 1 ( ) sse h  t y (i s)y   1 trace  t i l (i l)    2 2 trace  t i l (i l) 1 1 1 1 1 1 1 2 2 2 2 2 ( ) 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) t t t t t t n n t t t n n n n n x x w u v x x w u v x x w u v x x w u v x x w u v x x w u v                            l  t -1 t s x(x x) x i = identity matrix ordo n results and discussion the results of breusch-pagan (bp) test in wet land paddy productivities in tulungagung regency is: 𝐵𝑃 = ( 1 2 ) 𝒇 𝑇 𝒁 (𝒁𝑇𝒁)−1𝒁𝑇 𝒇 + ( 𝟏 𝑇 ) [ 𝑒𝑇 𝑾𝑒 𝝈𝟐 ] 𝟐 = 7438,636 estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 12 because bp test > 𝜒2(0,05; 5) = 11,070, the decision is reject 𝐻0. there is spatial heterogeneity in wet land paddy productivities in tulungagung regency, so we can use geographically weighted regression to estimate parameter model wet land paddy productivities in tulungagung regency. the next step is to determine the parameter estimation of the gwr model based on the equation 7. table 1 show the parameters estimation gwr model. table 1. parameters estimation gwr model no district 0  1  2  3  4  1 besuki 26,553 -0,02175 0,0000165 0,0000363 -0,0015 2 bandung 40,055 0,01395 0,0000014 0,0000374 -0,00065 3 pakel 27,372 -0,02315 0,0000179 0,0000311 -0,0016 4 campurdarat 35,107 -0,01342 0,0000077 0,0000396 -0,00154 5 kalidawir 42,804 0,01075 -0,0000009 0,0000370 -0,00036 6 pucanglaban 59,237 0,12682 -0,0000260 0,0000264 0,007461 7 rejotangan 55,276 0,08255 -0,0000189 0,0000297 0,005735 8 ngunut 40,515 0,03892 -0,0000185 0,0000672 0,010068 9 sumbergempol 41,269 0,05628 -0,0000239 0,0000751 0,011815 10 boyolangu 40,946 0,06148 -0,0000224 0,0000706 0,0117 11 tulungagung 41,456 0,05585 -0,0000207 0,0000664 0,011093 12 kedungwaru 41,122 0,05185 -0,0000186 0,0000621 0,01075 13 ngantru 42,624 0,03504 -0,0000128 0,0000508 0,007392 14 karangrejo 45,889 0,04003 -0,0000111 0,0000406 0,005076 15 kauman 59,895 0,10582 -0,0000251 0,0000278 0,0076 16 gondang 59,386 0,17687 -0,0000308 0,0000234 0,008816 17 pagerwojo 60,541 0,16031 -0,0000305 0,0000244 0,009065 18 sendang 41,348 0,02341 -0,0000006 0,0000365 -0,000099 the gwr model for district besuki based on table 1 can be written as follows: �̂�1 = 26,553 0,0217x1 + 0,0000165x2 + 0,0000363x3 0,00150x4 (13) other gwr models for the other districts has the same way as in equation 8 based on table 1. testing parameters of gwr model simultaneously conducted to determine the effect of weighting in the process of parameter estimation on the case of paddy productivity in tulungagung regency. results of simultan parameter test based on equation 12. f test is 8,904. value of f test is greater than f (0,05;10,3) = 8,785. this shows that simultaneously the predictor variables x1, x2, x3 and x4 have significant effect spatially on response variable. the gwr model of each district is formed on the basis of influential parameters, so partial parameter testing is performed by table 2 using t test. estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 13 table 2. t test statistics for parameter of gwr model no district t test statistic 0  t test statistic 1  t test statistic 2  t test statistic 3  t test statistic 4  1 besuki 5,429 -1,23768 2,974011 7,616609 -1,81654 2 bandung 12,447 0,907966 0,456556 11,73849 -0,89729 3 pakel 5,814 -1,27889 2,910577 4,486399 -1,91835 4 campurdarat 10,604 -0,82625 2,479856 11,72245 -1,91886 5 kalidawir 18,321 0,886951 -0,41056 13,04989 -0,58983 6 pucanglaban 13,264 5,188448 -5,04069 7,326105 4,716729 7 rejotangan 14,778 3,647647 -4,34952 8,891485 3,844718 8 ngunut 19,811 2,362104 -8,07605 15,72467 11,62082 9 sumbergempol 14,223 3,046957 -5,58308 10,28998 10,56913 10 boyolangu 14,431 3,073174 -6,91394 12,4301 11,11775 11 tulungagung 19,018 3,071231 -7,73936 13,62258 11,04025 12 kedungwaru 19,701 2,941217 -8,31189 14,96989 11,01064 13 ngantru 18,828 2,111468 -7,95199 18,32336 9,755659 14 karangrejo 17,363 2,105942 -6,30701 13,46622 5,196659 15 kauman 13,516 3,402 -4,47746 6,19037 3,55978 16 gondang 5,404 1,296335 -4,49924 2,7131 3,188156 17 pagerwojo 9,567 2,655057 -4,6735 5,152779 3,895126 18 sendang 12,836 1,525182 -0,21392 11,48837 -0,13862 significant if t test > t(0,05;13) = 2,16 based on table 2, the farmers need to add the average cost of pesticides per hectare(x3) to each district because the effect is positive significant to the wet land paddy productivity. in the districts of ngantru, karangrejo and gondang need to add the average cost of fertilizer per hectare(x2) because it has a significant positive effect on paddy productivity. the addition of x2 and x3 is not continuously. there are 12 districts that need to be added by the allocation of subsidized npk fertilizer of food crops sub-sector(x4) because it proved to have a significant positive effect on wet land paddy productivity. the district is grouping according to the variables that significantly affect the gwr model, the result is listed at tabel 3. table 3. the grouping according to the significant variables of gwr model district group significant variables pucanglaban, rejotangan, ngunut, sumbergempol, boyolangu, tulungagung, kedungwaru, kauman, pagerwojo yellow x1, x2, x3 and x4 ngantru, karangrejo, gondang red x2, x3 and x4 besuki, pakel, campurdarat purple x2 and x3 bandung, kalidawir, sendang green x3 based on table 3, a spread pattern of variables that has a significant effect on paddy productivity by districts in tulungagung regency presented in figure 1. estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 14 figure 1. the spread pattern of variables that significantly affected by district on wet land paddy productivities in tulungagung regency the yellow group is the group where the four variables, x1, x2, x3 dan x4 is significant to the productivity of wet land paddy. the yellow group consisted of 9 districts. only variable x2, x3 and x4 had a significant effect on the red group, there were 3 districts in the red group. purple group is a group with 2 variables that have significant effect that is x2 and x3. there are 3 districts that enter the purple group. the last group is green group where only x3 has significant effect. green group consists of 3 districts. the district of tanggunggunung has no wet land paddy productivity so the colour is white. conclusion based on the result, x1, x2, x3 and x4 has a different effect on each district. so, to improve the productivity of wet land paddy in tulungagung regency required a special policy based on the gwr model in each district. references [1] a. s. fotheringham, c. brundson and m. charlthon, geographically weighted regression : the analysis of spatially varying relationships, united kingdom: john wiley and sons, ltd, 2002. [2] c. l. mei, geographycally weighted regression technique for spatial data analysis, china: jiaotong university, 2005. [3] m. m. f. a. a. getis, handbook of applied spatial analysis, berlin: heidelberg and new york, 2010, pp. 255-278. [4] l. anselin, spatial econometrics : methods and models, netherlands: kluwer academic publishers, 1988. [5] y. leung, c. l. mei and w. x. zhang, "statistical test for spatial non-stationary based on the geographically weighted regression model," environment and planning a, vol. 32, pp. 9-32, 2000. estimation of geographically weighted regression case study on wet land paddy productivities in tulungagung regency danang ariyanto 15 abstract introduction methods where ,𝒇-𝒊.=,,𝒆-𝒊.-,𝝈-𝟐..−𝟏, ,𝒆-𝒊. is a vector of ols residuals, ,𝝈-𝟐. is the variance based on ols residual, t= trace [ wtw + w2] and 𝒁 is an n by (k+1) matrix of normal standard score (z). results and discussion conclusion references diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi volume 4, issue 4, may 2017 issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an 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javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5449') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5448') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5443') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5437') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5047') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5055') diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi volume 5, issue 1, november 2017 issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behavior of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the journal editor, the peer reviewer, the publisher and the society. as publisher of pure and applied mathematics journal, we take our duties to back up over all stages of publishing seriously and we recognize our ethical and other responsibilities. we are committed to ensuring that advertising, reprint or other commercial revenue has no impact or influence on editorial decisions. publication decisions the editor of cauchy is responsible for deciding which of the articles submitted to the journal should be published. the validation of the work in question and its importance to researchers and readers must always drive such decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. jurnal matematika murni dan aplikasi volume 5, issue 1, november 2017 issn 2086-0382 e-issn 2477-3344 cauchy preface cauchy is a national journal published by mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang. this is the second issue of this year. it contains 6 (six) article from all over the country, not only from local area of east java. those articles covered graph theory, numerical analysis, applied mathematics, statistics, economic mathematics, and algebra. this issue was authored by 13 authors and co-authors. in the first article, the author discussed the application of hierarchical cluster analysis. there are four clusters were formed based on the result of the study. in general, cluster 1 is a sub-district with good education condition, cluster 2 is a sub-district with good agricultural condition, in cluster 3 it is necessary to increase the education condition, utilizing dominant land area as well, and cluster 4 is a sub-district with good industrial condition. the second article focuses on estimation using wet land paddy productivities in tulungagung. based on the result, each parameter has different effects on each district. accordingly, to improve the productivity of wet land paddy in tulungagung regency a special policy based on the gwr model in each district is required. the third article, entitled “geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data” elaborated the gwr method with the weighted fixed gaussian kernel. the study concluded that from this model a r2 greater than the weighted queen contiguity. this result indicates that the weighted fixed gaussian kernel use for dengue fever case data in this study. the fourth article emphasized the discussion of disease modelling especially on svir epidemic model. from the results of endemic equilibrium point analysis, the author defined the basic reproduction number parameter. furthermore, 𝑅0 is a necessary condition for the existence of the two points of equilibrium as well as its local stability. results of local stability analysis indicates if 𝑅0 ≤ 1, then the diseasefree equilibrium point 𝐸0 is locally asymptotically stable. jurnal matematika murni dan aplikasi volume 5, issue 1, november 2017 issn 2086-0382 e-issn 2477-3344 cauchy preface the fifth article covered the discussion of modelling multi input transfer function. the results of analysis showed that based on the multi input with single output transfer function model, the rainfall in batu city on certain days is affected by air temperature and cloud on that day, humidity in the previous 23 days, and wind speed in the previous day. the last article entitled “the simulation study to test the performance of quantile regression method with heteroscedastic error variance” discussed modeling the data containing non-uniform variance problem (heteroscedasticity). this study resulted that all parameter estimated are close to the initial values. the value of pseudo r2 for all proposed model at any selected quantile points are quite large, more than 80%. it also stated that based on simulation study, the value of parameter estimated are within 95% confidence intervals indicating that parameter estimated could be accepted. this study also result that quantile regression method is able to produce small value of mse. therefore, it could be concluded here that quantile regression methods is unbiased estimator method and could result the acceptable model although in the due of heteroscedasticity problem of error variance. local stability analysis of an svir epidemic model cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 20-28 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 26 august 2017 reviewed: 10 october 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.4388 local stability analysis of an svir epidemic model joko harianto1, titik suparwati2 1, 2 mathematics department, cenderawasih university email: joharijpr88@gmail.com abstract in this paper, we present an svir epidemic model with deadly deseases. initially the basic formulation of the model is presented. two equilibrium point exists for the system; disease free and endemic equilibrium point. the local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. if the value of 𝑅0 less than or equal one then the desease free equilibrium point is locally asymptotically stable, and if its exceeds, the endemic equilibrium point is locally asymptotically stable. the numerical results are presented for illustration. keywords: svir model, local stability, basic reproduction number introduction health problems are closely associated with a disease. it is motivating humans to study health as well as illness. one of the goals of a nation to achieve prosperity in the future is the creation of a healthy living environment. one of the indicators of a healthy environment involves a high level of quality health services. quality health care issues need to be addressed systematically, effectively and efficiently. analysis of the spread of a disease is one step to address the problem of quality health services. the spread of disease due to virus or bacteria that enter into the body will lead to human health disorders and will affect the socio-economic development of the community. efforts to prevent a spreading disease can work optimally if several stages such as research, development of various diagnostic tools, drugs, and new vaccines, has achieved. mathematical models are one of a useful tool for reviewing or analyse the patterns of a disease spread. according to [2] and [4] the spread of infectious diseases can be described mathematically through a model, such as sir and sirs. the most basic procedure in modelling the spread of disease is by using the compartment model. in this case, the population is divided into three different classes namely, susceptible, infected, and recovered which then shortened to sir. vaccination can be considered as the addition of a class naturally into the epidemic base model for some types of diseases. along with the development of knowledge in the field of health, control of an epidemic disease can be done by vaccination action. then the sir model develops with the addition of a vaccinated population class. some researchers have proposed models of the dynamics of vaccination. among others is svi type, which has been developed by [5], [6] and [7]. several studies have stated that the threshold mailto:joharijpr88@gmail.com local stability analysis of an svir epidemic model joko hariantoianto 21 to determine the occurrence of endemic disease seen from the value of basic reproduction number (𝑅0). the disease distribution model will be locally stable if 𝑅0 ≤ 1 and the disease will be present if 𝑅0 > 1. paper [8] discusses the svir epidemic model for non-lethal diseases that does not involve the individual's mortality rate due to the disease. from these studies, we can build some further assumptions related to the svir epidemic model that developed in public life. in this paper, we analyse the svir epidemic model for a deadly disease. we hope the model can be useful to analyse the disease outbreaks so that we can take optimal precautions. methods this research used literature study with literature sources from some reputable journal. the simulation of model svir used mathematical software, i.e. maple 2016. results and discussion 2.1. formulation model the following are assumptions used in modelling: 1. closed population (no migration) 2. births occur in every sub populations and enter in vulnerable subpopulations. 3. natural death occurs on each sub populations at the same rate. 4. illness can cause death. 5. individuals who have cured cannot return to susceptive class (permanent cure). 6. short incubation period. 7. vaccination has given to susceptible subpopulations. 8. we do not distinguish the rate of recovery for a child has been infected and for the adult. 9. vaccination will reach the level of immunity over time and finally enters the sub populations that heals. 10. individuals who have vaccinated become infected if they lose immunity. 11. contact between infected and susceptive individuals can lead to disease transmission, and so are contact between infected and those have vaccinated. we investigate the basic model formulation by dividing the total population into four compartments, such as s(t) for susceptible, v(t) for vaccinated, i(t) for infected and r(t) for recovered individuals. let 𝛽 is the rate of contact that is sufficient to transmit the disease. we also assume a constant recovery rate 𝛾. the rate at which the susceptible population is vaccinated is 𝛼. we assume that there can be disease related death and define 𝜔 to be the rate of disease related death, while 𝜇 is the rate of natural death that is not related to the disease. we assume that all newborns enter the susceptible class at the constant rate of 𝜇. let 𝛾1 be the average rate for susceptible individuals to obtain immunity and moved into recovered population. we do not distinguish the natural immunity and vaccine-induced immunity here because vaccine-induced immunity can also last for a long term. we assume that before obtaining immunity the vaccinees still have possibility of infection with a disease transmission rate 𝛽1 while contacting with infected individuals. 𝛽1 may be assume to be less than 𝛽 because the vaccinating individuals may have some partial immunity during the process or they may recognize the transmission characters of the disease and hence decrease the effective contacts with infected individuals. figure 1 shows the details of the population transfer diagram of the svir model. local stability analysis of an svir epidemic model joko hariantoianto 22 𝜔𝑅 figure.1 transfer diagram of the svir model the following form presents the system model of differential equations. 𝑑𝑆 𝑑𝑡 = 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 (1) 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 𝑑𝑅 𝑑𝑡 = 𝛾1𝑉 +𝛾𝐼 −𝜇𝑅 where 𝑆(0) > 0,𝑉(0) > 0,𝐼(0) > 0,𝑅(0) = 0, 𝑆 +𝑉 +𝐼 +𝑅 = 1,∀ 𝑡 ≥ 0 and all parameters are positive. the parameters with their description are presented in table 1. table 1. parameter description parameter description 𝜇 natural death rate (and equally, the birth rate) 𝛽 the transmission rate of disease when susceptible individuals contact with infected individuals 𝛼 the rate of migration of susceptible individuals to the vaccination process 𝛽1 the transmission rate for vaccination individuals to be infected before gaining immunity 𝛾1 the average rate for vaccination individuals to obtain immunity and move into recovered population 𝜔 the death rate due to disease 𝛾 recovery rate 2.2 . equilibrium point next, we will study system dynamics in equation (1). since the last equation does not depend on other equations, we simply study the following system. 𝑑𝑆 𝑑𝑡 = 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 (2) 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 local stability analysis of an svir epidemic model joko hariantoianto 23 furthermore, we performed a svir model stability analysis around the equilibrium point of the system (2). as a first step, we define system equilibrium points (disease free points and endemic points). in general, according to [1] and [3] there exist two equilibrium points of epidemic model, i.e. the disease free equilibrium point and the endemic equilibrium point. the two equilibrium points are reviewed based on the existence of the disease in a population with a continuous time. the disease-free equilibrium point is the point at which the disease is unlikely to spread in an area because the infected population is equal to zero (i=0) for 𝑡 → ∞. while the endemic equilibrium point is the point at which the disease must spread (i>0) for 𝑡 → ∞, in a defined closed area. theorem 1 there are two equilibrium points in the system (2), namely: a) the disease-free equilibrium point, that is, 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 𝜇 𝛼 +𝜇 , 𝛼𝜇 (𝛼 +𝜇)(𝛾1 +𝜇) ,0) b) the endemic equilibrium point, that is, 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) = ( 𝜇 (𝛼 +𝜇 +𝛽𝐼∗) , 𝛼𝜇 (𝛼 +𝜇 +𝛽𝐼∗)(𝜇 +𝛾1 +𝛽1𝐼 ∗) , 𝐼∗), where 𝐼∗ is the positive root of 𝐴1𝐼 2 +𝐴2𝐼 +𝐴3(1−𝐶) = 0, for c > 1, and 𝐴1 = (𝛾 +𝜇 +𝜔)𝛽𝛽1 > 0 𝐴2 = ( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇)𝛽1 + (𝛾1 + 𝜇)𝛽))−𝛽1𝛽𝜇 𝐴3=(𝛾 +𝜇 +𝜔)( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 proof: the equilibrium point of system (2) exists when 𝑑𝑆 𝑑𝑡 = 0, 𝑑𝑉 𝑑𝑡 = 0, 𝑑𝐼 𝑑𝑡 = 0, so that: 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 (3) 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 = 0 (4) 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇𝐼 −𝜔𝐼 = 0 (5) from equation (5), we have 𝐼(𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇 −𝜔) = 0, and the solutions are 𝐼 = 0 or 𝛽𝑆 + 𝛽1𝑉 −𝛾 −𝜇 −𝜔 = 0, consequently, we have to situation, namely: a) situation at 𝑰 = 𝟎 it is the necessary condition for the disease-free equilibrium point. note that,  from equation (3), 𝜇 −𝜇𝑆 −𝛼𝑆 = 0 ⟺ 𝑺 = 𝝁 𝜶+𝝁  from (4), 𝛼𝑆 −𝛾1𝑉 −𝜇𝑉 = 0 ⟺ 𝑽 = 𝛼𝝁 (𝜶+𝝁)(𝜸𝟏+𝝁) so we get the equilibrium point of disease-free, that is, 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 𝜇 𝛼 +𝜇 , 𝛼𝜇 (𝛼 +𝜇)(𝛾1 +𝜇) ,0) b) situation at 𝑰 ≠ 𝟎, or 𝑰 > 0, this situation is a necessary condition for an endemic equilibrium point. note that:  from equation (3), 𝜇 −𝜇𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 ⟺ 𝑺∗ = 𝝁 (𝜶+𝝁+𝜷𝐼∗)  from (4): 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇𝑉 = 0 ⟺ 𝑽 ∗ = 𝛼𝝁 (𝜶+𝝁+𝜷𝐼∗)(𝜸𝟏+𝝁+𝜷𝟏𝐼 ∗) from (5), if 𝐼 ≠ 0, then 𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇 −𝜔 = 0 , consequently, with substitute 𝑺 ∗ and 𝑽∗ we have: 𝛼𝜇𝛽1 (𝛼 +𝜇 +𝛽𝐼∗)(𝛾1 +𝜇 +𝛽1𝐼 ∗) = 𝜇 +𝛾 +𝜔 − 𝛽𝜇 (𝛼 +𝜇 +𝛽𝐼∗) ⟺ 𝛽1𝛽(𝜇 +𝛾 +𝜔) 𝐼 ∗2 +((𝜇 +𝛾 +𝜔)(( 𝛼 +𝜇)𝛽1 +(𝜇 +𝛾1)𝛽)−𝛽1𝛽𝜇) 𝐼 ∗ + local stability analysis of an svir epidemic model joko hariantoianto 24 (𝜇 + 𝛾 + 𝜔)(𝜇 + 𝛾1)(𝛼 + 𝜇)(1− ( 𝛽𝜇 (𝛼 +𝜇)(𝜇 + 𝛾 + 𝜔) + 𝛼𝛽1𝜇 (𝛼 + 𝜇)(𝜇 + 𝛾1)(𝜇 + 𝛾 + 𝜔) )) = 0 by applying: 𝐴1 = (𝛾 +𝜇 +𝜔)𝛽1𝛽 > 0 𝐴2 =( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇) 𝛽1 + (𝛾1 + 𝜇)𝛽)− 𝛽1𝛽𝜇 𝐴3=(𝛾 +𝜇 +𝜔)( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 𝐶 = 𝛽𝜇 (𝛼 +𝜇)(𝜇 +𝛾 +𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔) , the above equation becomes: 𝐴1𝐼 ∗2 +𝐴2𝐼 ∗ +𝐴3(1−𝐶) = 0 (6) with the roots of equation (6), 𝐼1,2 ∗ = −𝐴2 ±√𝐴2 2 −4𝐴1𝐴3(1−𝐶) 2𝐴1 where 𝐼∗ > 0 and 𝐶 > 1. thus, we obtain an endemic equilibrium point, that is, 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) = ( 𝜇 (𝛼 +𝜇 +𝛽𝐼∗) , 𝛼𝜇 (𝛼 +𝜇 +𝛽𝐼)(𝜇 +𝛾1 +𝛽1𝐼 ∗) , 𝐼∗). 2.3. basic reproduction number (𝑹𝟎) the equilibrium point stability analysis on the svir model depends on the value of basic reproduction number 𝑅0 (the number of susceptible individuals who are then infected if they interact with patients in an entirely vulnerable population). a selection of this number is by observing the state of the endemic equilibrium point. consider eq. (6), an endemic equilibrium point only applicable to positive roots (𝐼∗ > 0) when c > 1. in conclusion, the value of a positive endemic equilibrium point lies in the value of c, so that it can be defined the value basic reproduction number, that is, 𝑅0 = 𝛽𝜇 (𝛼 +𝜇)(𝜇 +𝛾 +𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔) 2.4. local stability of equilibrium point we will determine the stability of the equilibrium point of the svir model around the disease-free equilibrium and the endemic equilibrium point. equations (3), (4), and (5) are nonlinear equations, so by using taylor series at each equilibrium point, there will be a system of linear differential equations. theorem 2 defined: 𝑅0 = 𝛽𝜇 (𝛼 +𝜇)(𝜇 +𝛾 +𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔) i. if 𝑅0 ≤ 1, then there is exist one (unique) equilibrium of system (1), that is, the diseasefree equilibrium point 𝐸0 . furthermore, if 𝑅0 > 1, then there are two equilibrium points of system (1), that is, the disease-free equilibrium points 𝐸0 dan 𝐸 ∗. ii. if 𝑅0 ≤ 1, then the equilibrium point 𝐸0 is locally asymptotically stable and if 𝑅0 > 1, then the equilibrium point 𝐸0 is unstable. iii. if 𝑅0 > 1, then the endemic equilibrium 𝐸 ∗ is locally asymptotically stable. proof: (i) we have discussed the existence of the equilibrium point of disease-free in theorem 1. in the discussion, we obtain a point of equilibrium when i = 0 and is singular if 𝑅0 ≤ 1. the equilibrium point is the disease-free equilibrium point. next, let's look at the equation 𝐴1𝐼 2 +𝐴2𝐼 +𝐴3(1−𝑅0) = 0, if 𝑅0 > 1, then 𝐴3(1−𝑅0) < 0 and 𝐴1 > 0, consequently, the equation has two real roots (positive and negative). therefore, a necessary condition that ensures the existence and unique equilibrium 𝐸∗ has been fulfilled. furthermore, when 𝑅0 ≤ 1, we will show that the above quadratic equation has no positive roots. note that, local stability analysis of an svir epidemic model joko hariantoianto 25 𝑅0 ≤ 1 ⟹ 𝛽𝜇 ≤ (𝛼 +𝜇)(𝛾 +𝜇 +𝜔) ⟹ 𝐴2 ≥ ( 𝛾 +𝜇 +𝜔)(( 𝛼 +𝜇) 𝛽1 + (𝛾1 + 𝜇)𝛽)− 𝛽1(𝛼 +𝜇)(𝛾 +𝜇 +𝜔) > ( 𝛾 +𝜇 +𝜔)(𝛾1 + 𝜇)𝛽 > 0 so if 𝐴1 > 0, 𝐴3(1−𝑅0) > 0 then the function of 𝐴1𝐼 2 +𝐴2𝐼 +𝐴3(1−𝑅0) will rise and 𝐴1𝐼 2 + 𝐴2𝐼 +𝐴3(1−𝑅0) > 𝐴3(1−𝑅0) ≥ 0, for 𝐼 > 0, which means the quadratic function has no positive roots. (ii) the jacobian matrix of system (2) is 𝐽(𝑆,𝑉,𝐼) = ( −𝜇 −𝛼 −𝛽𝐼 0 −𝛽𝑆 𝛼 −𝜇 −𝛾1 −𝛽1𝐼 −𝛽1𝑉 𝛽𝐼 𝛽1𝐼 𝛽𝑆 +𝛽1𝑉 −𝜇 −𝛾 −𝜔 ) so the jacobian matrix around the point 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 𝜇 𝛼+𝜇 , 𝛼𝜇 (𝛼+𝜇)(𝛾1+𝜇) ,0) is 𝐽(𝐸0) = ( −𝜇 −𝛼 0 −𝛽𝑆0 𝛼 −𝜇 −𝛾1 −𝛽1𝑉0 0 0 𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔 ) the characteristic equation is ⟺ [−(𝜇 +𝛼)−𝜆][−(𝜇 +𝛾1)−𝜆][ 𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔 −𝜆] = 0 where 𝜆1 = −(𝜇 +𝛼) < 0 𝜆2 = −(𝜇 +𝛾1)< 0 𝜆3 = 𝛽𝑆0 +𝛽1𝑉0 −𝜇 −𝛾 −𝜔 = (𝛾 +𝜇 +𝜔)(𝑅0 −1) it is clear if 𝑅0 < 1 then all the eigenvalues of 𝐽(𝐸0) are negative, hence the point 𝐸0 is locally asymptotically stable. meanwhile, if 𝑅0 > 1, then there is a positive eigenvalue, hence the point 𝐸0 is unstable. (iii) it is clear that the endemic point 𝐸∗ exist when 𝑅0 > 1, so the jacobian matrix around the point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is 𝐽(𝐸∗) = ( −𝜇 −𝛼 −𝛽𝐼∗ 0 −𝛽𝑆 𝛼 −𝜇 −𝛾1 −𝛽1𝐼 ∗ −𝛽1𝑉 𝛽𝐼∗ 𝛽1𝐼 ∗ 𝛽𝑆∗ +𝛽1𝑉 ∗ −𝜇 −𝛾 −𝜔 ) we can modify the element 𝑎11,𝑎22, 𝑎33, from matrix above into −𝜇 −𝛼 −𝛽𝐼∗ = − 𝜇 𝑆∗ −𝜇 −𝛾1 −𝛽1𝐼 ∗ = − 𝛼𝑆∗ 𝑉∗ based on the evaluation results around the endemic equilibrium point we are getting 𝛽𝑆∗ +𝛽1𝑉 ∗ −𝜇 −𝛾 −𝜔 = 0, and the matrix become 𝐽(𝐸∗) = ( − 𝜇 𝑆∗ 0 −𝛽𝑆 𝛼 − 𝛼𝑆∗ 𝑉∗ −𝛽1𝑉 𝛽𝐼∗ 𝛽1𝐼 ∗ 0 ) the characteristic equation of matrix 𝐽(𝐸∗) is 𝜆3 +𝑎1𝜆 2 +𝑎2𝜆 +𝑎3 = 0 where 𝑎1 = 𝜇 𝑆∗ + 𝛼𝑆∗ 𝑉∗ > 0 𝑎2 = 𝛼𝑆∗ 𝑉∗ +𝛽1 2 𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗ > 0 𝑎3 = 𝛼𝛽1𝛽𝑆 ∗𝐼∗ + 𝛼𝛽2𝑆∗ 2 𝐼∗ 𝑉∗ + 𝜇𝛽1 2 𝑉∗𝐼∗ 𝑆∗ > 0 hence, 𝑎1𝑎2 −𝑎3 = 𝛼𝜇2 𝑆∗𝑉∗ +(𝜇 +𝛽𝐼∗)𝛽2𝑆∗𝐼∗ + 𝛼2𝜇𝑆∗ 𝑉∗2 +𝛼𝐼∗𝑆∗(𝛽 −𝛽1) 2 +𝛼𝛽1𝛽𝑆 ∗𝐼∗ > 0 local stability analysis of an svir epidemic model joko hariantoianto 26 has been fulfilled. according to the routh-hurwitz criterion, all the eigenvalues of the above characteristic equations have a negative real part. so, the equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is locally asymptotically stable. based on the results, we will interpret the results biologically as follows,  if 𝑅0 ≤ 1, then for 𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸0 = (𝑆0,𝑉0, 𝐼0), the solution of system (1) will move to 𝐸0 = (𝑆0,𝑉0, 𝐼0). it means that if 𝑅0 ≤ 1, then for the number of susceptible, vaccinated and infected individuals close to 𝐸0 = (𝑆0,𝑉0, 𝐼0), the disease will not plague and tends to disappear indefinitely time. this condition is then called asymptotically stable around the equilibrium point 𝐸0 = (𝑆0,𝑉0, 𝐼0), and we interpret as a disease-free equilibrium point.  if 𝑅0 > 1, then for 𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸 ∗ = (𝑆∗,𝑉∗, 𝐼∗), the solution of system (1) will move to 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗). this means if 𝑅0 > 1, then for the number of susceptible, vaccinated and infected individuals close to 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), the disease will plague but does not reach extinction in an infinite time. this condition is then called asymptotically stable around the equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), and we interpret as an endemic equilibrium point. 2.5. numerical results in this section, we investigate the numerical solution of the system (1) by using the rungekutta order four scheme. the state variables are chosen with same initials conditions. the numerical results are shown in figure 1 and 2. figure 1, illustrate the fact, when the basic reproduction number less than unity, when the value of basic reproduction number exceeds than unity, figure 2 illustrate this fact. figure 1. the dynamical behavior of system (1), for same initial conditions and different parameters: when 𝑅0 = 0,22 ≤ 1,the desease free equilibrium point is locally asymptotically stable where 𝜇 = 1, 𝛽 = 10,𝛽1 = 2,𝛾 = 4,𝛾1 = 8,𝛼 = 10,𝜔 = 0,01. local stability analysis of an svir epidemic model joko hariantoianto 27 figure 2. the dynamical behavior of system (1), for same initial conditions and different parameters: when 𝑅0 = 3,17 > 1, endemic equilibrium is locally asymptotically stable where 𝜇 = 1, 𝛽 = 20,𝛽1 = 15,𝛾 = 1,𝛾1 = 2,𝛼 = 10,𝜔 = 0,01. conclusion from the results of endemic equilibrium point analysis, we define the basic reproduction number parameter, that is, 𝑅0 = 𝛽𝜇 (𝛼 +𝜇)(𝜇 +𝛾 +𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇)(𝜇 +𝛾1)(𝜇 +𝛾 +𝜔) furthermore, 𝑅0 is a necessary condition for the existence of the two points of equilibrium as well as its local stability. when 𝑅0 ≤ 1, there is only one (unique) equilibrium point of disease-free, that is, 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 𝜇 𝛼+𝜇 , 𝛼𝜇 (𝛼+𝜇)(𝛾1+𝜇) ,0). conversely, if 𝑅0 > 1, then there are two equilibrium points, that is, 𝐸0 and an endemic equilibrium point 𝐸 ∗ = (𝑆∗,𝑉∗, 𝐼∗) = ( 𝜇 (𝛼+𝜇+𝛽𝐼∗) , 𝛼𝜇 (𝛼+𝜇+𝛽𝐼)(𝜇+𝛾1+𝛽1𝐼 ∗) , 𝐼∗) where 𝐼∗ is the positive root of the equation 𝐴1𝐼 2 +𝐴2𝐼 + 𝐴3(1−𝑅0) = 0. results of local stability analysis indicates if 𝑅0 ≤ 1, then the disease free equilibrium point 𝐸0 is locally asymptotically stable. it means if the terms 𝑅0 ≤ 1 fulfilled, then in a long time, there would have no spread of disease in susceptible and vaccinated subpopulations, or in other words, the epidemic will stop. conversely, if 𝑅0 > 1 then the endemic equilibrium point e  is locally asymptotically stable. it means, in a long time, the disease will always exist in the population with the proportion of each subpopulation is equal to 𝑆∗,𝑉∗,dan 𝐼∗. references [1] diekmann,o., and heesterbeek,j.a.p., mathematical epidemiology of infectious diseases: model building, analysis, and interpretation, john wiley and sons, chichester, 2000. local stability analysis of an svir epidemic model joko hariantoianto 28 [2] driessche, p.v.d., watmough, j., "reproduction number and sub-threshod endemic equilibria for compartmental models of disease transmission," mathematical biosciences 180, pp. 29-48, 2002. http://dx.doi.org/10.1016/s0025-5564(02)00108-6 [3] h.w. hethcote, j.w.v. ark, epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, math. biosci. 84 (1987) 85 [4] kermack, w. o. and mckendrick, a. g., "a contribution to the mathematical theory of epidemics,"proc. roy. soc. lond., a 115, pp. 700-721, 1927. http://dx.doi.org/10.1098/rspa.1927.0118 [5] khan, muhammad altaf, saeed islam and others, "stability analysis of an svir epidemic model with non-linear saturated incidence rate," applied mathematical sciences, vol. 9, no. 23, .1145-1158, 2015. http://dx.doi.org/10.12988/ams.2015.41164 [6] kribs-zaleta, c., velasco-hernandez, j., "a simple vaccination model with multiple endemic state," mathematical biosciences, vol. 164, pp. 183-201, 2000. [7] liu, xianing, and others, "svir epidemic models with vaccination strategies," journal of theoretical biology 253, 1-11, 2007. http://dx.doi.org/10.1016/j.jtbi.2007.10.014 [8] zaman, g., and others, "stability analysis and optimal vaccination of an sir epidemic model," bio systems 93, pp. 240-249, 2008. http://dx.doi.org/10.1016/j. biosystems.2008.05.004 issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 cauchy vol. 5 no. 2 pages: 42 – 79 malang may 2018 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 editorial board elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 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http://www.cosmosimpactfactor.com/page/journals_details/943.html http://globalimpactfactor.com/cauchy/ http://www.sindexs.org/journallist.aspx http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 table of contents a discrete numerical scheme of modified leslie gower with harvesting model . 42 – 47 comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region ................................................... 48 – 54 fitting the first birth interval in indonesia using weibull proportional hazards model ............................................................................................................................................. 55 – 63 fundamental solution of elliptic equation with positive definite matrix coefficient .................................................................................................................................... 64 – 66 selection of ideal contraceptive tool using hybrid entropy-topsis method ........ 67 – 72 the construction of patient loyalty model using bayesian structural equationmodeling approach ............................................................................................... 73 – 79 cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. 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murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated abdussakir (maulana malik ibrahim state islamic university of malang) elly susanti (maulana malik ibrahim state islamic university of malang) eridani (airlangga university) sri harini (maulana malik ibrahim state islamic university of malang) usman pagalay (maulana malik ibrahim state islamic university of malang) cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 1-7 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 21 december 2016 reviewed: 31 january 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.3862 applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti1, rahmania sri untari1 1muhammadiyah university of sidoarjo, sidoarjo, indonesia email: cindy.cahyaning@umsida.ac.id, rahmania.sriuntari@umsida.ac.id abstract this research was conducted in sidoarjo district where source of data used from secondary data contained in the book "kabupaten sidoarjo dalam angka 2016" .in this research the authors chose 12 variables that can represent sub-district characteristics in sidoarjo. the variable that represents the characteristics of the subdistrict consists of four sectors namely geography, education, agriculture and industry. to determine the equitable geographical conditions, education, agriculture and industry each district, it would require an analysis to classify sub-districts based on the sub-district characteristics. hierarchical cluster analysis is the analytical techniques used to classify or categorize the object of each case into a relatively homogeneous group expressed as a cluster. the results are expected to provide information about dominant sub-district characteristics and non-dominant sub-district characteristics in four sectors based on the results of the cluster is formed. keywords: hierarchical cluster analysis, sub-district introduction sub-district characteristics is general overview of sub-district that need to be developed optimally, thus providing a positive impact on the sub-district progress. sub-district characteristics divided into several sectors, geography, education, government, social, agriculture, industry, commerce, communications, finance and prices and regional income. sidoarjo district is divided into 18 sub-districts that are buduran sub-district, candi sub-district, porong sub-district, krembung sub-district, tulangan sub-district, tanggulangin sub-district, jabon sub-district, krian sub-district, balongbendo sub-district, wonoayu sub-district, tarik sub-district, prambon sub-district, taman sub-district, waru sub-district, gedangan sub-district, sedati sub-district and sukodono subdistrict. potential sidoarjo district evenly spread over the 18 sub-districts and is reflected on the sub-district characteristics. in order equitable development to improve people's welfare, goverment of sidoarjo ditrict collaboration with bps sidoarjo district published "sidoarjo dalam angka 2016" which contains sub-district characteristics in sidoarjo. this book is expected to provide benefits for the implementation of development as well as helping to evaluate and supervise development outcomes of sidoarjo district. data of sub-district characteristics in "sidoarjo dalam angka 2016" has been analyzed only use descriptive analysis, therefore the authors consider that these data have a lot of information if further analysis. the focus of research using hierarchical cluster analysis are mailto:cindy.cahyaning@umsida.ac.id mailto:rahmania.sriuntari@umsida.ac.id applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 2 the four sectors that represent sub-district characteristics, namely geography, education, agriculture and industry. the results are expected to provide information about dominant sub-district characteristics and non-dominant sub-district characteristics in four sectors based on the results of the cluster is formed. methods cluster analysis is a technique used to classify objects into relatively homogeneous groups, called clusters. objects in each group tend to resemble each other and differ greatly with objects from other clusters. cluster analysis using the principal components analysis can use interval and ratioscaled data. cluster analysis is also called classification analysis or taxonomy numerical analysis because it deals with clustering procedure where each object is only fit into one cluster only, to avoid overlapping [1]. there are several terms used dapam cluster analysis. the terms include the following [2]: • aglomeration schedule, is to schedule that provides information about the object or the case will be merged or entered in clusters on each stage, in a process of hierarchical cluster analysis. • cluster centroid, is the average value of all variable objects or cases in a particular cluster. • cluster centers, is the starting point of the start of the grouping in non-hierarchical cluster analysis. • cluster membership, membership is showing the clusters, where each object or a case of being members. • dendogram, is a graphical tool to present the results of cluster analysis, or upright vertical lines represent the merged cluster together. line position on the scale indicates the distance which were merged cluster. dendogram should be read from left to right. terms of normality, linearity, and homoscedasticity highly considered in the multivariate analysis, but not in the cluster analysis. in cluster analysis, researchers should be more concerned with how large a sample representative in population and the presence or absence of multicollinearity. the first step in formulating cluster analysis of the problem of defining the variables used for basic grouping. then measure the exact distance should be selected. the distance measure determine similarity or dissimilarity of the object to be grouped. to determine the number of clusters requires subjective judgment of the researchers, in addition based on the calculation results objectively. cluster obtained should be interpreted and expressed in the variables used for the basic formation of clusters. the equation commonly used for calculating the distance between the item x to item y is a euclidean distance. the equation used to calculate the euclidean distance is as follows [3]: 𝑑(𝑥, 𝑦) = √(𝑥1 − 𝑦1) 2 + (𝑥2 − 𝑦2) 2 + ⋯ + (𝑥𝑝 − 𝑦𝑝) 2 = √(𝑋 − 𝑌)′(𝑋 − 𝑌) (1) there are two types of cluster analysis is hierarchical cluster analysis and non-hierarchical cluster analysis. in the method of hierarchical cluster there are two basic types namely agglomerative (concentration) and divisive (the spread). in agglomerative method, any object or observation is considered as a separate cluster. in the next stage, the two clusters which has some similarities are combined into a new cluster and so on. instead, the divisive methods, from a large cluster consisting of all objects or observation. furthermore, the object or observation that the highest value does not resemble separated and so on [4]. there are five kinds of algorithms to form a group with a hierarchical method, namely [5]: • single-linkage applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 3 single linkage method defines the similarity between clusters based on the shortest distance from any object in one cluster to any other object. if there is a third object which has the closest distance to one of the objects in the group that has been formed, then the object can be merged into the group. this process continues to form a single group. this method is the most flexible method aglomeratif. • complete linkage this method is basically the same as the single linkage method. it's just the distance used is the maximum distance. reasons have the maximum distance is that objects that have little in common can be connected. • average-linkage average linkage method also has similarities with two single linkage method. only the distance used is the average distance of all objects in a group with other objects outside the group. grouping objects with one another based on the average minimum. because using the average, then this method is considered more stable, and no bias. • centroid method the distance used in this method is the distance between the center point of the two groups. where is the center point of the group is the middle value of each variable object in one group. in this method each time a new group is formed, then the center point changes. the advantage of this method is the small effect of outliers in the formation of the group. • ward's method in the ward method, distance calculations based on the sum of the squares between the two groups for all variables. this method can be used if the number of observations is not too large. in general, the distance used is a euclidean distance squared. the opposite of hierarchy cluster analysis method is non-hierarchy cluster analysis. in this method does not include the "treelike construction" but through the process by placing objects into the cluster at once, forming a particular cluster. the first step in the method is to choose a cluster nonhirarki as initial cluster centers, and all objects within a certain distance placed on cluster formation. then select the next cluster and the placement of objects continued until all are placed. the objects can be placed again if the distance was closer to the other cluster than the cluster of origin.non-hierarchy cluster analysis methods associated with the k-means cluster, and there are three approaches used to place each observation on a single cluster. such approaches include the following [6]: • sequential threshold, threshold sequential method start by selecting one cluster and placing all the objects that are at a certain distance into it. if all objects that are at a certain distance has been entered, then the second cluster selected and put all objects within a certain thereto. then the third cluster is selected and the process continues as before. • parallel threshold, threshold parallel method is the opposite of the first approach by selecting a number of clusters simultaneously and placing objects into clusters that have the distance between the nearest face. in the process, the distance between the face can be specified to include some objects into clusters -cluster. also some variation on this method, the rest of the objects are not grouped if it is outside a certain distance of a cluster. • optimization, the third method is similar to the previous method except that this method makes it possible to put objects back into the cluster closer. in this research used secondary data sourced from “kabupaten sidoarjo dalam angka 2016”. unit of observation in this research was 18 sub-district in sidoarjo district. in this research the authors chose 12 variables that can represent sub-district characteristics. operational definition of each variable will be described as 𝑋1: surface area (km2) 𝑋2: total population 𝑋3: the number of national and private elementary school 𝑋4: the number of national and private elementary school students applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 4 𝑋5: the number of national and private junior high school 𝑋6: the number of national and private junior high school student 𝑋7: the number of national and private high school 𝑋8: the number of national and private high school students 𝑋9: harvest land area (ha) 𝑋10: paddy production (kw) 𝑋11: the number of large and small industries 𝑋12: the number of workers in large and small industries the method of analysis in this research is. 1. standardization of data that have variability research unit 2. correlation analysis and principal component analysis on research variables 2. classify sub-district in sidoarjo with hierarchical cluster analysis with average linkage algorithm processing of data by hierarchical cluster analysis with average linkage algorithm performed with spss 20, before the cluster analysis, principal component analysis is done to overcome the correlation between variables, as well as the standardization of variables in order to obtain a variable with the same unit, making it eligible cluster analysis. the results of the cluster analysis then be concluded and interpreted. results and discussion sidoarjo district consists of 18 sub-districts formed four clusters, details of four clusters with members that include within each cluster can be seen in table 1. grouping four clusters based on the data that provide a general overview of the sub-district characteristics in sidoarjo, represented by the four sectors, geography, education, agriculture and industry. table 1. clusters membership case 4 clusters 1:sidoarjo 1 2:buduran 2 3:candi 2 4:porong 2 5:krembung 2 6:tulangan 2 7:tanggulangin 2 8:jabon 3 9:krian 2 10:balongbendo 2 11:wonoayu 2 12:tarik 2 13:prambon 2 14:taman 2 15:sukodono 2 16:gedangan 2 17:waru 4 18:sedati 3 cluster 1 consists of one sub-district, cluster 2 consists of 14 sub-district, cluster consists of two sub-districts and cluster 4 consists of one sub-district. to identify the characteristics of each cluster conducted a descriptive analysis. members of the group can be described as follows: applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 5 1. members of cluster 1: sidoarjo sub-district 2. members of the cluster 2: buduran sub-district, candi sub-district, porong sub-district, krembung sub-district, tulangan sub-district, tanggulangin sub-district, krian sub-district, balongbendo subdistrict, wonoayu sub-district, tarik sub-district, prambon sub-district, taman sub-district, gedangan sub-district and sukodono sub-district. 3. members of the cluster 3: jabon sub-district and sedati sub-district 4. members of cluster 4: waru sub-district once known the number of clusters formed and the members of each cluster then performed a descriptive analysis of each cluster. to search for characteristics which are most dominant in each cluster, then look for the highest average of the variables for each cluster. summary of average value in each cluster can be seen in table 2. table 2. summary of average value in each custer variabel cluster 1 cluster 2 cluster 3 cluster 4 𝑋1 62,56 32,9257 80,215 30,32 𝑋2 194051 98389,00 71228,5 231298 𝑋3 52 29,71 20 38 𝑋4 22379 7812,43 5427,5 18476 𝑋5 21 7,64 6 20 𝑋6 10813 3755,36 2934,5 8475 𝑋7 13 2,86 2 6 𝑋8 7691 1441,29 504 1876 𝑋9 633 1871,29 1663 109 𝑋10 42825 141781,79 124964,5 7445 𝑋11 42 29,29 12,5 151 𝑋12 5268 5820,64 1892,5 35770 : dominant variable : non-dominant variable based on table 3 can known dominant characteristics and non-dominant characteristics of the four clusters are formed with the following description : 1. cluster 1 consist of sidoarjo sub-district has dominant variables are 𝑋2, 𝑋3, 𝑋4, 𝑋5, 𝑋6, 𝑋7 and 𝑋8, this indicates that the average number of national and private elementary school sector, the average number of national and private elementary school students, the average number of national and private junior high school, the average number of national and private junior high school students, the average number of national and private high school and the average number of national and private high school students most are in sidoarjo sub-district. sixth dominant variables are 𝑋2, 𝑋3, 𝑋4, 𝑋5, 𝑋6, 𝑋7 and 𝑋8 represents of education condition, shows that education condition in the sidoarjo sub-district has the most good progress compared to 17 other sub-districts. applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 6 2. cluster 2 consist of buduran sub-district, candi sub-district, porong sub-district, krembung sub-district, tulangan sub-district, tanggulangin sub-district, krian sub-district, balongbendo sub-district, wonoayu sub-district, tarik sub-district, prambon sub-district, taman subdistrict, gedangan sub-district and sukodono sub-district has dominant variable is 𝑋9 and 𝑋10, this indicates that 14 sub-district in cluster 2 have an the average number of harvest land area and the average number of paddy production more than four other sub-districts. both the dominant variables are 𝑋9 and 𝑋10 represent agricultural condition, it indicates that the agricultural condition in 14 sub-districts incorporated in cluster 2 has the most good progress compared to four other sub-districts in sidoarjo. 3. cluster 3 consist of jabon sub-district and sedati sub-district has dominant variable is 𝑋1, this indicates jabon sub-district and sedati sub-district had an the number of average surface area (km2) which wider than the 16 other sub-districts. while non-dominant variables are 𝑋2, 𝑋3, 𝑋4, 𝑋5, 𝑋6, 𝑋7, 𝑋8, 𝑋11 and 𝑋12. six variables are non-dominant are 𝑋2, 𝑋3, 𝑋4, 𝑋5, 𝑋6, 𝑋7 and 𝑋8 represents education condition while the two others variables that are non-dominant are 𝑋11 and 𝑋12 represents industry condition, this shows that education and industry condition in the two sub-districts has less well development compared to 16 other sub-districts. 4. cluster 4 consist of waru sub-district has dominant variable are 𝑋2, 𝑋11 and 𝑋12, this indicates that the average number of total population, the average number of large and small industries the average number of workers in large and small industries most are in waru sub-district. both the dominant variables are 𝑋11 and 𝑋12 represent industry condition, it indicates that industry condition in waru sub-district has the most good progress compared to 17 other sub-districts in sidoarjo. while non-dominant variables are 𝑋1, 𝑋9 and 𝑋10. both the non-dominant variables are 𝑋9 and 𝑋10 represent agricultural condition this shows that agricultural condition in waru sub-districts has less well development compared to 17 other sub-districts. conclusion based on the results of hierarchical cluster analysis with average linkage algorithm is known that there are four clusters were formed. in general, cluster 1 is a sub-district with good education condition, in cluster 2 is a sub-district with good agricultural condition are very good, in cluster 3 is necessary to the increase the education condition, utilizing dominant land area as well as in cluster 4 is a sub-district with good industrial condition. based on the results of the analysis, the government may retain the dominant characteristics and increase non-dominant characteristics in each cluster to perform construction and development in their respective sub-districts. besides equitable development must also consider the potential possessed by each sub-district, so that the education condition, the agricultural condition and the industrial condition has good development. references [1] o. yim dan k. t. ramdeen, “hierarchical cluster analysis: comparison of three linkage measures and application to psychological data,” journal of tutorials in quantitative methods for psychology, vol. 11, no. 1, pp. 1-4, 2015. [2] j. supranto, analisis multivariat : arti & interpretasi, jakarta: rineka cipta, 2010. [3] r. a. johnson dan d. w. winchern, applied multivariate statistical analysis, new jersey: prentice hall international inc., 2007. [4] s. saracli, n. dogan dan i. dogan, “comparison of hierarchical cluster analysis methods by cophenetic correlation,” journal of inequalities and applications 2013, vol. 1, pp. 203-210, 2013. [5] j. f. j. hair, r. e. anderson, r. l. thatham dan w. c. black, multivariate data analysis, new jersey: prentice hall international inc., 2009. applied hierarchical cluster analysis with average linkage algoritm cindy cahyaning astuti 7 [6] p. trebuna dan j. halcinova, “mathematical tools of cluster analysis,” journal of applied mathematics, vol. 4, pp. 814-816, 2013. abstract introduction methods 1. standardization of data that have variability research unit 2. correlation analysis and principal component analysis on research variables 2. classify sub-district in sidoarjo with hierarchical cluster analysis with average linkage algorithm processing of data by hierarchical cluster analysis with average linkage algorithm performed with spss 20, before the cluster analysis, principal component analysis is done to overcome the correlation between variables, as well as the standardization ... results and discussion conclusion references on the local metric dimension of line graph of special graph cauchy – jurnal matematika murni dan aplikasi volume 4(3) (2016), pages 125-130 submitted: october, 04 2016 reviewed: october, 21 2016 accepted: november, 29 2016 doi: http://dx.doi.org/10.18860/ca.v4i3.3694 on the local metric dimension of line graph of special graph marsidi1, dafik2 ,ika hesti agustin3, ridho alfarisi4 1mathematics edu. depart. ikip jember indonesia 2mathematics edu. depart. university of jember indonesia 3mathematics depart. university of jember indonesia 4mathematics depart. its surabaya indonesia email: marsidiarin@gmail.com, d.dafik@gmail.com, ikahestiagustin@gmail.com, alfarisi38@gmail.com. abstract let g be a simple, nontrivial, and connected graph. 𝑊 = {𝑤1,𝑤2,𝑤3 ,… ,𝑤𝑘} is a representation of an ordered set of k distinct vertices in a nontrivial connected graph g. the metric code of a vertex v, where 𝑣 ∈ g, the ordered 𝑟(𝑣|𝑊) = (𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2), . . . ,𝑑(𝑣,𝑤𝑘)) of k-vector is representations of v with respect to w, where 𝑑(𝑣,𝑤𝑖) is the distance between the vertices v and wi for 1≤ i ≤k. furthermore, the set w is called a local resolving set of g if 𝑟(𝑢|𝑊) ≠ 𝑟(𝑣|𝑊) for every pair u,v of adjacent vertices of g. the local metric dimension ldim(g) is minimum cardinality of w. the local metric dimension exists for every nontrivial connected graph g. in this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. the line graph l(g) of a graph g has a vertex for each edge of g, and two vertices in l(g) are adjacent if and only if the corresponding edges in g have a vertex in common. keywords: metric dimension, local metric dimension number, line graph, resolving set. introduction all graph in this paper are simple, nontrivial, and undirected, for more detail basic definition of graph, see [1]. in [2] define the distance 𝑑(𝑢,𝑣) between two vertices u and v in a connected graph g is the length of a shortest path between these two vertices. suppose that 𝑊 = {𝑤1,𝑤2,𝑤3 ,… ,𝑤𝑘} is an ordered set of vertices of a nontrivial connected graph g. the metric representation of v with respect to w is the k-vector 𝑟(𝑣|𝑊) = (𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2), . . . ,𝑑(𝑣,𝑤𝑘)). distance in graphs has also been used to distinguish all of the vertices of a graph. the set w is called a resolving set for g if distinct vertices of g have distinct representations with respect to w. the metric dimension dim(g) of g is the minimum cardinality of resolving set for g [3]. furthermore, we consider those ordered sets w of vertices of g for which any two vertices of g having the same code with respect to w are not adjacent in g. if 𝑟(𝑢|𝑊) ≠ 𝑟(𝑣|𝑊) for every pair u, v of adjacent vertices of g, then w is called a local metric set of g. the minimum k for which g has a local metric k-set is the local metric dimension of g, which is denoted by ldim(g) [4]. in this paper, we study the local metric dimension number of line graph of special graphs, namely path, cycle, generalized star, and wheel graph. line graphs are a special case of intersection graphs. the line graph l(g) of a graph g has a vertex for each edge of g, and two vertices in l(g) are adjacent if and only if the corresponding edges in g have a vertex in common. thus, the line graph l(g) is the intersection graph corresponding to the endpoint sets of the edges http://dx.doi.org/10.18860/ca.v4i3.3694 mailto:marsidiarin@gmail.com mailto:d.dafik@gmail.com mailto:ikahestiagustin@gmail.com mailto:alfarisi38@gmail.com on the local metric dimension of line graph of special graph marsidi 126 of g. as for an example, figure 1 shows a graph g and its line graph l(g) [5]. the results show that distinct vertices of each l(g), with g is a special graph, namely path, cycle, generalized star, and wheel graph have distinct representations with respect to w, and their local metric dimension attain a minimum number. furthermore, [6] showed that on the metric dimension, the upper dimension and the resolving number of graphs, [7] showed the metric dimension of amalgamation of cycles, [8] studied the constant metric dimension of regular graphs, [2] obtained resolvability in graphs and the metric dimension of a graph. the last, [9] showed the fault-tolerant metric and partition dimension of graph. figure 1. a graph and its line graph results and discussions we have some observation to show the result of line graph of path, cycle, generalized star, and wheel graph. thus, we have four main theorem about local metric dimension to discuss as follows. observation 1. the order and the size of 𝐿(𝑃𝑛) are |𝑉(𝐿(𝑃𝑛))| = 𝑛−1 and |𝐸(𝐿(𝑃𝑛))| = 𝑛−2, respectively. proof. line graph of path 𝐿(𝑃𝑛) is connected, simple, and undirected graph with vertex set 𝑉(𝐿(𝑃𝑛)) = {𝑣𝑖;1 ≤ 𝑖 ≤ 𝑛−1} and edge set 𝐸(𝐿(𝑃𝑛)) = {𝑣𝑖𝑣𝑖+1;1 ≤ 𝑖 ≤ 𝑛−2}. thus, |𝑉(𝐿(𝑃𝑛))| = 𝑛−1 and |𝐸(𝐿(𝑃𝑛))| = 𝑛−2. see figure 2 (a) and 2 (b) for illustration. ∎ theorem 1. for 𝑛 ≥ 2, the local metric dimension of line graph of path is 𝑙𝑑𝑖𝑚(𝐿(𝑃𝑛)) = 1. proof. by observation 1, line graph of path 𝐿(𝑃𝑛) is isomorphic to 𝑃𝑛−1. thus, the vertex set and the edge set of 𝐿(𝑃𝑛) are 𝑉(𝐿(𝑃𝑛)) = {𝑣𝑖;1 ≤ 𝑖 ≤ 𝑛−1} and 𝐸(𝐿(𝑃𝑛)) = {𝑣𝑖𝑣𝑖+1;1 ≤ 𝑗 ≤ 𝑛−2}, respectively. the number of vertices |𝑉(𝐿(𝑃𝑛))| = 𝑛−1 and the size |𝐸(𝐿(𝑃𝑛))| = 𝑛−2. by theorem 3, 𝑙𝑑𝑖𝑚(𝑃𝑛) = 1, thus 𝑙𝑑𝑖𝑚(𝐿(𝑃𝑛)) = 1. it concludes the proof. see figure 2 (c) for illustration . ∎ figure 2. (a) 𝑃8 , (b) 𝐿{𝑃8}, (c) construction of local resolving set 𝑊 = {𝑣1} observation 2. the order and the size of 𝐿(𝐶𝑛) are |𝑉(𝐿(𝐶𝑛)) | = 𝑛 and |𝐸(𝐿(𝑃𝑛))| = 𝑛, respectively. on the local metric dimension of line graph of special graph marsidi 127 proof. line graph of cycle 𝐿(𝐶𝑛) is connected, simple, and undirected graph with vertex set 𝑉(𝐿(𝐶𝑛)) = {𝑣𝑖;1 ≤ 𝑖 ≤ 𝑛} and edge set 𝐸(𝐿(𝐶𝑛)) = {𝑣𝑖𝑣𝑖+1;1 ≤ 𝑗 ≤ 𝑛−1}∪{𝑣𝑛𝑣1;}. thus, |𝑉(𝐿(𝐶𝑛))| = 𝑛 and |𝐸(𝐿(𝐶𝑛))| = 𝑛. see figure 3 (a) and 3 (b) for illustration. ∎ theorem 2. for 𝑛 ≥ 3, the local metric dimension of line graph of cycle is 𝑙𝑑𝑖𝑚(𝐿(𝐶𝑛)) = { 1, 𝑓𝑜𝑟 𝑛 𝑒𝑣𝑒𝑛 2, 𝑓𝑜𝑟 𝑛 𝑜𝑑𝑑 proof. by observation 2, line graph of cycle 𝐿(𝐶𝑛) is isomorphic to 𝐶𝑛. thus, the vertex set and the edge set of 𝐿(𝐶𝑛) are 𝑉(𝐿(𝐶𝑛)) = {𝑣𝑖;1 ≤ 𝑖 ≤ 𝑛} and 𝐸(𝐿(𝐶𝑛)) = {𝑒𝑗;1 ≤ 𝑗 ≤ 𝑛}, respectively. the order of vertices |𝑉(𝐿(𝐶𝑛))| = 𝑛 and the size |𝐸(𝐿(𝐶𝑛))| = 𝑛. by theorem 3, 𝑙𝑑𝑖𝑚(𝐶𝑛) = { 1, for 𝑛 even 2, for 𝑛 odd thus 𝑙𝑑𝑖𝑚(𝐿(𝐶𝑛)) = { 1, for 𝑛 even 2, for 𝑛 odd it concludes the proof. see figure 3 (c) for illustration. ∎ figure 3. (a) 𝐶8 , (b) 𝐿{𝐶8}, (c) construction of local resolving set 𝑊 = {𝑣1} observation 3. the order and the size of 𝐿(𝑆𝑛,𝑚) are |𝑉(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛 and |𝐸(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛+ 𝑛(𝑛−3) 2 , respectively. proof. line graph of generalized star 𝐿(𝑆𝑛,𝑚) is connected, simple, and undirected graph with vertex set 𝑉(𝐿(𝑆𝑛,𝑚)) = {𝑥𝑖,𝑗;1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑚} and edge set 𝐸(𝐿(𝑆𝑛,𝑚)) = {𝑥𝑖,𝑗𝑥𝑖,𝑗+1;1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑚−1}∪{𝑥𝑖,1𝑥𝑡,1; 𝑖 ≠ 𝑡,1 ≤ 𝑖,𝑡 ≤ 𝑛}. thus, |𝑉(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛 and |𝐸(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛+ 𝑛(𝑛−3) 2 . see figure 4 (a) and 4 (b) for illustration. ∎ theorem 3. for 𝑛 ≥ 3, the local metric dimension of line graph of generalized star is 𝑑𝑖𝑚(𝐿(𝑆𝑛,𝑚)) = 𝑛−1. proof. by observation 3, the order and the size of 𝐿(𝑆𝑛,𝑚) are |𝑉(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛 and |𝐸(𝐿(𝑆𝑛,𝑚))| = 𝑚𝑛+ 𝑛(𝑛−3) 2 , respectively. suppose the lower bound is 𝑙𝑑𝑖𝑚(𝐿(𝑆𝑛,𝑚)) ≥ 𝑛−2 with the resolving set 𝑊′ = {𝑥𝑖,1;1 ≤ 𝑖 ≤ 𝑛−2}. thus two vertices of 𝑉(𝐿(𝑆𝑛,𝑚))−𝑊 ′ with respect to 𝑊′ are 𝑟(𝑥𝑛−1,1|𝑊 ′) = 𝑟(𝑥𝑛,1|𝑊 ′) = ( 1,…,1⏟ 𝑛−2 𝑡𝑖𝑚𝑒𝑠 ) it is easy to see that the line graph of generalized star has the same vertex representation respecting to 𝑊′. thus, the lower bound is 𝑙𝑑𝑖𝑚(𝐿(𝑆𝑛,𝑚)) ≥ 𝑛−1. now, we will show that 𝑙𝑑𝑖𝑚(𝐿(𝑆𝑛,𝑚)) ≤ 𝑛−1 by determining the resolving set 𝑊 = {𝑥𝑖,1;1 ≤ 𝑖 ≤ 𝑛−1} and the vertex representation of 𝑉(𝐿(𝑆𝑛,𝑚))−𝑊 respect to 𝑊, as follows on the local metric dimension of line graph of special graph marsidi 128 𝑟(𝑥𝑛,𝑗|𝑊) = ( 𝑗,…,𝑗⏟ 𝑛−1 𝑡𝑖𝑚𝑒𝑠 );1 ≤ 𝑗 ≤ 𝑚 𝑟(𝑥𝑖,𝑗|𝑊) = ( 𝑗,…,𝑗⏟ 𝑖−1 𝑡𝑖𝑚𝑒𝑠 , 𝑗−1, 𝑗,…,𝑗⏟ 𝑛−𝑖−1 𝑡𝑖𝑚𝑒𝑠 );1 ≤ 𝑖 ≤ 𝑛−1,2 ≤ 𝑗 ≤ 𝑚 it easy to see that ∀𝑢,𝑣 ∈ 𝑉(𝐿(𝑆𝑛,𝑚))−𝑊 have a different representation respect to w for every pair 𝑢,𝑣 of adjacent vertices in 𝐿(𝑆𝑛,𝑚). the cardinality of resolving set 𝐿(𝑆𝑛,𝑚) is 𝑛−1, thus 𝑙𝑑𝑖𝑚(𝐿(𝑆𝑛)) ≤ 𝑛−1. it concludes the proof. see figure 4 (c). ∎ figure 4. (a) 𝑆6,3, (b) 𝐿(𝑆6,3), (c) construction of local resolving set 𝑊 = {𝑥1,1,𝑥2,1,𝑥3,1,𝑥4,1,𝑥5,1} observation 4. the order and the size of 𝐿(𝑊𝑛) are |𝑉(𝐿(𝑊𝑛))| = 2𝑛 and |𝐸(𝐿(𝑊𝑛))| = 𝑛(𝑛+5) 2 , respectively. proof. line graph of wheel 𝐿{𝑊𝑛} is connected, simple, and undirected graph with vertex set 𝑉(𝐿(𝑊𝑛)) = {𝑥𝑖,𝑗;1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 2} and edge set 𝐸(𝐿(𝑆𝑛,𝑚)) = {𝑥𝑖,1𝑥𝑖,2;1 ≤ 𝑖 ≤ 𝑛}∪ {𝑥𝑖,1𝑥𝑡,1; 𝑖 ≠ 𝑡,1 ≤ 𝑖,𝑡 ≤ 𝑛}∪{𝑥𝑖,2𝑥𝑖+1,2;1 ≤ 𝑖 ≤ 𝑛−1}∪{𝑥𝑖,2𝑥𝑖+1,1;1 ≤ 𝑖 ≤ 𝑛−1}∪{𝑥1,2𝑥𝑛,2} ∪{𝑥1,1𝑥𝑛,2}. thus, |𝑉(𝐿(𝑊𝑛))| = 2𝑛 and |𝐸(𝐿(𝑊𝑛))| = 𝑛(𝑛+5) 2 . see figure 5 (a) and 5 (b) for illustration. ∎ theorem 4. for 𝑛 ≥ 3, the local metric dimension of line graph of wheel is 𝐿𝐷𝑖𝑚(𝐿(𝑊𝑛)) = 𝑛−1. proof. by observation 4, the order and the size of 𝐿(𝑊𝑛) are |𝑉(𝐿(𝑊𝑛))| = 2𝑛 and |𝐸(𝐿(𝑊𝑛))| = 𝑛(𝑛+5) 2 , respectively. suppose the lower bound of 𝐿(𝑊𝑛) is 𝑙𝑑𝑖𝑚(𝐿(𝑊𝑛)) ≥ 𝑛−2 with the resolving set 𝑊′ = {𝑥𝑖,1;1 ≤ 𝑖 ≤ 𝑛−2}. thus two vertices of 𝑉(𝐿(𝑊𝑛))−𝑊 ′ with respect to 𝑊′ are 𝑟(𝑥𝑛−1,1|𝑊 ′) = 𝑟(𝑥𝑛,1|𝑊 ′) = ( 1,…,1⏟ 𝑛−2 𝑡𝑖𝑚𝑒𝑠 ) it is easy to see that the line graph of wheel possess the same vertex representation respecting to 𝑊′. thus the lower bound is 𝑙𝑑𝑖𝑚(𝐿{𝑊𝑛}) ≥ 𝑛−1. now, we will show that 𝑙𝑑𝑖𝑚(𝐿{𝑊𝑛}) ≤ 𝑛−1 on the local metric dimension of line graph of special graph marsidi 129 by determining the resolving set 𝑊 = {𝑥𝑖,1;1 ≤ 𝑖 ≤ 𝑛−1} and the vertex representation of 𝑉(𝐿{𝑊𝑛})−𝑊 respect to 𝑊, as follows. 𝑊 = {𝑥1,1,𝑥2,1,𝑥3,1,𝑥4,1,𝑥5,1,𝑥6,1,𝑥7,1} 𝑟(𝑥𝑛,1|𝑊) = ( 1,…,1⏟ 𝑛−1 𝑡𝑖𝑚𝑒𝑠 ) 𝑟(𝑥𝑖,2|𝑊) = ( 2,…,2⏟ 𝑖−1 𝑡𝑖𝑚𝑒𝑠 ,1, 2,…,2⏟ 𝑛−𝑖−1 𝑡𝑖𝑚𝑒𝑠 );2 ≤ 𝑖 ≤ 𝑛−1 𝑟(𝑥1,2|𝑊) = (1,1, 2…,2⏟ 𝑛−3 𝑡𝑖𝑚𝑒𝑠 ) 𝑟(𝑥𝑛,2|𝑊) = (1, 2,…,2⏟ 𝑛−2 𝑡𝑖𝑚𝑒𝑠 ) it easy to see that ∀𝑢,𝑣 ∈ 𝑉(𝐿(𝑊𝑛))−𝑊 have a different representation respect to 𝑊 for every pair 𝑢,𝑣 of adjacent vertices in 𝐿(𝑊𝑛). the cardinality of resolving set 𝐿(𝑊𝑛) is 𝑛−1, thus 𝑙𝑑𝑖𝑚(𝐿(𝑊𝑛)) ≤ 𝑛−1. it concludes the proof. see figure 5 (c) for illustration. ∎ figure 5. (a) 𝑊8, (b) 𝐿{𝑊8}, (c) construction of local resolving set conclusion we have shown the local metric dimension number of line graph of special graphs, namely line graph of path, cycle, star, and wheel. the results show that the local metric dimension numbers attain the best lower bound. however we have not found the sharpest lower bound for any connected graph, therefore we proposed the following open problem. open problem 1. let 𝐺 be a connected graph, obtain the best lower bound of the local metric dimension of any graph 𝐺. on the local metric dimension of line graph of special graph marsidi 130 references [1] g. chartrand, e. salehi, and p. zhang, “the partition dimension of a graph,” aequationes math., vol. 59, pp. 45–54, 2000. [2] g. chartrand, l. eroh, and m. a. johnson, “resolvability in graphs and the metric dimension of a graph,” discrate appl. math., vol. 105, pp. 99–113, 2000. 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[9] m. a. chaudhry, i. javaid, and m. salman, “fault-tolerant metric and partition dimension of graphs,” util. math., 2010. diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m volume 4, issue 4, may 2017 jurnal matematika murni dan aplikasi malang may 2017 cauchy no. 4 pages: 138-182 vol. 4 issn 2086-0382 e-issn 2477-3344 issn 2086-0382 e-issn 2477-3344 cauchy  editor in chief : ari kusumastuti, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. dr. usman pagalay, m.si, maulana malik ibrahim state islamic university of malang, indonesia. dr. abdussakir, m.pd, maulana malik ibrahim state 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.............................. 138 – 145 a super (a,d)-𝐵𝑚-antimagic total covering of ageneralized amalgamation of fan graphs ................................................................................................................. 146 – 154 application of modified spatial k'luster analysis by tree edge removal (skater) method on the level of crime in way kanan district, lampung ........................ 155 – 160 odd harmonious labeling on pleated of the dutch windmill graphs .............. 161 – 166 almost surjective epsilon-isometry in the reflexive banach spaces ............. 167 – 175 on the spectra of commuting and non commuting graph on dihedral group ................................................................................................................ ……….176 – 182 simulation study of the using of bayesian quantile regression in non-normal error cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 121-126 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 04 october 2018 reviewed: 08 october 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5633 simulation study of the using of bayesian quantile regression in nonnormal error catrin muharisa1, ferra yanuar2*, dodi devianto3 1,2,3 department of mathematics, faculty of mathematics and natural science, andalas university, kampus limau manis, 25163, padang indonesia email: catrinmuharisa@gmail.com, ferrayanuar@sci.unand.ac.id, ddevianto@sci.unand.ac.id *corresponding author email: ferrayanuar@sci.unand.ac.id abstract the purposes of this paper are to introduce the ability of the bayesian quantile regression method in overcoming the problem of the non-normal errors. in this research we do simulation study to apply the proposed method. we generate data and assume error by asymmetric laplace distribution. in this research, we solve the non-normal problem using quantile regression method and bayesian quantile regression method and then we compare. the quantile regression approach we divide the data into any quantiles, then estimated the conditional quantile function and minimize absolute error that is asymmetrical. bayesian regression method used the asymmetric laplace distribution in likelihood function. markov chain monte carlo method using gibbs sampling algorithm is applied then to estimate the parameter in bayesian regression method. convergence and confidence interval of parameter estimated are also checked. bayesian quantile regression method results has more significance parameter and smaller confidence interval than quantile regression method. the best regression equation model is bayesian quantile method used quantile value 0.50. this study proves that bayesian quantile regression method can produce acceptable parameter estimate for non-normal error. keywords: quantile regression; asymmetric laplace distribution; gibbs sampling; markov chain monte carlo; bayesian quantile regression introduction regression analysis is a tool in experienced statistics development and used in many areas of life. the analysis has a purpose to estimate the relationship between dependent variable with independent variable [1]. there are several methods used to estimate parameters in the regression equation, one of the most commonly used is the ordinary least square (ols) method. the used of ols method is based on several assumptions, one of which is the assumption of normality. furthermore, developed quantile regression method which is generally used in case of econometrics. in this quantile regression approach, we divide the data into any quantiles, then estimated the conditional quantile function and minimize absolute error that is asymmetrical. in the method of quantile regression usually requires large data size. bayes introduced a method to estimate parameters by utilizing initial information called prior distribution. this method is known as the bayesian method. prior distribution can be derived from the prior research data or based on the researcher's intuition[2]. prior information from the distribution of parameters is then combined with information from data obtained from sampling or so-called likelihood function so that posterior mailto:catrinmuharisa@gmail.com mailto:ferrayanuar@sci.unand.ac.id mailto:ddevianto@sci.unand.ac.id mailto:ferrayanuar@sci.unand.ac.id simulation study of the using of bayesian quantile regression in non-normal error catrin muharisa 122 distribution is obtained. the mean and variance of this posterior distribution or the posterior mean and posterior variance then can be estimated. bayesian method uses mcmc algorithm (markov chain monte carlo). mcmc can be easily used to obtain posterior distribution even in that situation complex [3]. in this research will be estimated model parameters with combining the quantile regression method and the bayesian regression method called the bayesian quantile regression method. methods in this section, we will introduce two models, quantile regression and bayesian quantile regression. here, we denote y as the dependent variable, x is the independent variable. 1. quantile regression quantile function denoted by 𝑄𝜃 where 𝜃, 0 ≤ 𝜃 ≤ 1. given 𝑌 be a random variable with a cumulative distribution function 𝐹𝑌 = 𝑃(𝑌 ≤ 𝑦). the quantile regression 𝜃𝑡ℎ of 𝑌 can be simply written as follows [4]: 𝑄𝜃(𝑌) ≔ 𝐹𝑌 −1(𝜃) = inf⁡{𝑦:𝐹𝑌(𝑦) ≥ 𝜃} (1) given 𝑌 is dependent variable, and 𝑋 is independent variable have 𝑝 dimension. let 𝐹𝑌(𝑦|𝑋 = 𝑥) = 𝑃(𝑌 ≤ 𝑦|𝑋 = 𝑥) conditional cumulative function notation of 𝑌 given 𝑋 = 𝑥. the conditional quantile 𝜃𝑡ℎ of 𝑌 defined as bellows [4]: 𝑄𝜃(𝑌|𝑋 = 𝑥) = inf⁡{𝑦:𝐹𝑌(𝑦|𝑥) ≥ 𝜃}. (2) based on the median concept of estimate for 𝛽 from the quantile regression 𝜃𝑡ℎ obtained by minimizing the absolute number of errors by weighting 𝜃 for positive error and weighting 1 − 𝜃 for negative error [5] : 𝑎𝑟𝑔𝑚𝑖𝑛 𝑥𝑖 𝑇𝛽0∈𝑅 𝜃 ∑ 𝜌𝜃 𝑛 𝑖=1 ⁡(𝑦𝑖 − 𝑄𝜃(𝑌|𝑋))⁡ (3) where 𝜽 is quantile indeks ∈ (𝟎,𝟏) and 𝝆𝜽 is asymmetrice loss function for 𝑸𝜽(𝒀|𝑿) = 𝑿𝑻𝜷. 2. bayesian quantile regression the bayesian method uses markov monte carlo chain (mcmc) to estimate the posterior distribution. given 𝒚 = (𝑦1,𝑦2,…,𝑦𝑛), where the prior distribution of 𝛽 is 𝑝(𝛽). the prior distribution taken in this research is prior informative those originating from previous research [6]. determination of prior distribution parameters are very subjective, depending on the researcher's intuition. a variable 𝑌 is said to follows asymmetric laplace distribution with the density function of the probability as follows [2]: 𝑓𝑝(𝑦) = 𝑝(1 − 𝑝)exp⁡{𝜌𝜃(𝑦𝑖 − 𝜇)} (4) and likelihood function as follows : 𝐿(𝑦|𝛽) = 𝑝𝑛(1 − 𝑝)𝑛exp⁡{−∑ 𝜌𝜃𝑖 (𝑦𝑖 − 𝑥𝑖 𝑇𝛽)} . (5) simulation study of the using of bayesian quantile regression in non-normal error catrin muharisa 123 then the posterior distribution of 𝛽, 𝑓(𝛽|𝑦) is given by 𝑓(𝛽|𝑦) ∝ 𝐿(𝑦|𝛽)⁡𝑝(𝛽) ∝ 𝑝𝑛(1 − 𝑝)𝑛exp⁡{−∑ 𝜌𝜃𝑖 (𝑦𝑖 − 𝑥𝑖 𝑇𝛽)}⁡𝑝(𝛽) (6) results and discussion in this study data generated by software r that consists of two independent variables (𝑋1,𝑋2) and one response variable (𝑌), each independent variable (𝑋1,𝑋2) spreads according to the normal distribution (𝑋1⁡~⁡𝑁(0,1)) and (𝑋2⁡~⁡𝑁(0,1)). while the response variable (𝑌), is set value 𝑦 = 0,7⁡𝑥1 +⁡𝑥2 + 𝜀, where 𝜀~𝐴𝐿𝐷(0,1,0.75). each variable measured 150 sample data. estimation results of each model parameter for each quantile using the quantile regression method can be seen in the following table 1 table 1. estimated parameter model of quantile regression method quantile (𝜽𝒕𝒉) 𝜷𝟏 se 𝜷𝟐 se 0.05 0.25 0.50 0.75 0.95 0,97237 (0,30104) 0,7057 (0,11758) 1,00287 (0,00031**) 1,20465 (0,0000**) 1,12467 (0,1943) 0,9369 0,44829 0,27177 0,19633 0,86254 4,43896 (0,000055**) 0,92898 (0,06935) 1,25697 (0,00007**) 1,22291 (0,0000**) 1,27023 (0,1956) 1,06125 0,50779 0,30784 0,30784 0,97702 **significant on level 𝛼=0.05 se=standart error based on table 1, the parameter of 𝛽1 significant in quantities to 0.50 and 0.75. while the parameter 𝛽2 is significant in quantiles to 0.05, 0.50 and 0.75. hence, the coefficient parameter of each quantile satisfied when used quantile 0.50. estimation results for each quantile using the bayesian quantile regression method can be seen in the table 2 table 2. estimated parameter model of bayesian quantile regression method quantile (𝜽𝒕𝒉) 𝜷𝟏 𝜷𝟐 0.05 0,527 3,499 0.25 0,749 1,037 0.50 1,04 1,22 simulation study of the using of bayesian quantile regression in non-normal error catrin muharisa 124 0.75 1,077 1,195 0.95 1,08 1,48 based on the coefficient parameter of each quantile satisfied when used quantile 0.25. based on the point estimated value of the bayesian quantile regression closer to the beta value than the quantile regression. estimation of model parameters with quantile regression method and bayesian quantile regression method as previously obtained. next will be compared by using the confidence interval. the comparison results are shows in the following table 3 table 3. confidence interval of quantile regression and bayesian quantile regression quantile (𝜽𝒕𝒉) 𝜷𝟏 𝜷𝟐 qr bqr qr bqr 0.05 0.25 0.50 0.75 0.95 lower limit upper limit difference lower limit upper limit difference lower limit upper limit difference lower limit upper limit difference lower limit upper limit difference -1,84973 2,54512 4,39485 -0,07067 1,51635 1,58702 0,7346 1,53139 0,79679 0,49631 1,46396 0,96765 0,6571 1,64607 0,98897 -0,819 1,17 1,989 0,189 1,33 1,141 0,706 1,41 0,704 0,601 1,5283 0,9273 0,369 1,72 1,351 -2,8179 4,63321 7,45111 0,34168 1,84061 1,49893 0,69714 1,54094 0.8438 0,87919 1,4644 0,58521 0,58451 2,69848 2,11397 1,663 4,77 3,107 0,462 1,63 1,168 0,801 1,61 0,809 0,795 1,5898 0,7948 0,337 2,7 2,363 qr=quantile regression bqr=bayesian quantile regression based on table 3, it can be seen in the quantile 0.05 by using the quantile regression method on 𝛽1 parameter the result is not significant, where bayesian quantile regression method is significant. on the parameter of 𝛽2 to the quantile 0.05 using the quantile regression method and bayesian quantile regression method not significant. in quantiles 0.25, 0.50, 0.75, and 0.95 it can be seen that using the bayesian quantile and quantile regression methods has been significant. from the analysis, the best regression equation model is obtained when using bayesian quantile regression analysis with 0.50 quantile value. the next step in bayesian quantile regression approach is convergence test of convergence of model parameters that have been estimated parameter. test is using history trace plot and density plot [3]. figure 1 and figure 2 presents a trace plot and density plot for some selected parameters. simulation study of the using of bayesian quantile regression in non-normal error catrin muharisa 125 figure 1. trace plot of parameter 𝛽1 for 𝜃 = 0.50 figure 2. trace plot of parameter 𝛽2 for 𝜃 = 0.50 figure 1 and figure 2 it can be concluded that the assumption of convergence is related. data distribution has been stable as it is between two parallel horizontal lines. figure 3. density plot of parameter 𝛽1 for 𝜃 = 0.50 figure 4. density plot of parameter 𝛽2 for 𝜃 = 0.50 in figure 3 and figure 4, the density plot for some parameters show the normal distributed curve. this result informs us that the selected parameters model is convergent. based on the convergence examination of the trace plot and density plot, it can be concluded that the alleged model has satisfied the criterion of convergence. conclusions in this research, we use the analysis quantile and bayesian quantile regression to analyze simulation study in non-normal error. obtained the best regression equation model is bayesian quantile method used quantile value 0.50. bayesian quantile regression method using gibbs algorithm sampling better estimator than quantile regression method. because the bayesian method has more parameter significance and confidence interval values smaller. simulation study of the using of bayesian quantile regression in non-normal error catrin muharisa 126 references [1] walpole, r.e and myers, r. h. 1995. ilmu peluang dan statistika untuk insinyur dan ilmuwan edisi ke-4. itb : bandung. [2] yu, k. and moyeed, r. 2001. bayesian quantile regression. statistics & probability letters, 54(4), 437-447. [3] ntzoufras, i. 2009. bayesian modeling using winbugs. john wiley sons, inc: new jersey. [4] davino, c., furno, m. and vistocco, d. 2014. quantile regression theory and applications. john wiley and sons, ltd. [5] koenkar,r and basset,g.jr. 1978. regression quantiles. econometrica,46: 33-50 [6] benoit, d.f and van den poel, d. 2017. bayesqr: a bayesian approach to quantile regression.journal of statistical software, 76(7), 1-32. restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 80-87 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 18 nopember 2016 reviewed: 15 march 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.3777 restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah, loekito adi soehono, suci astutik department of statistics, faculty of mathematics and natural sciences, brawijaya university, malang, indonesia email: dwi.masrokhah@gmail.com abstract students are part of the community who have an income. the income of student is pocket money, scholarships, part-time jobs and so forth. they are trying to become trendsetter for their dress style. the consumption patterns are very influential in the behavior of saving. if the savings increases, not only the public funds will increase but also the investment. if the investment increases, the economic growth will also increase. the purpose of this research is to estimate multiple regression parameters using reml methods in modeling the student’s saving in faculty of mathematics and natural science, brawijaya university. the variables used were: the student’s age, the amount of income of student’s parent, the amount of student’s pocket money, the amount of student’s additional income, the amount of student’s consumption and the amount of student’s saving. reml method can overcome heteroscedasticity of error variance and provide unbiased estimator. the model of student’s saving using reml method is as follows: �̂�𝑖 = −1609 + 112 𝑋1 + 0.0088𝑋2 + 0.0504 𝑋3 + 0.4706 𝑋4 − 0.636𝑋5 student’s saving is affected significantly by: student’s age (𝑋1), the amount of student’s additional income (𝑋4), and the amount of student’s consumption (𝑋5). keywords: student’s saving, reml, regression, assumptions, heteroscedasticity introduction the regression analysis is used to create a functional model of the data to explain or predict a natural phenomenon based on the other phenomena. regression analysis was introduced by sir francis galton in 1822-1911. the purpose of regression analysis is for prediction based on the relationship between the predictor variables and the response variables [1]. based on the shape of the relationship, the regression analysis can be divided into linear regression and non-linear regression. linear regression is an approach for modeling the relationship between a dependent variable y and one or more explanatory variables (or independent variables) denoted by x. parameter estimation methods which are often used in multiple linear regression is ordinary least squares (ols). the ols method minimizes the sum squared of residuals (error). the ols method require some classical assumptions in order to achieve estimator which is best linear unbiased estimator (blue). the assumptions related to errors that is generated from that model. the assumptions that must be met, namely the normality of error, nonautocorrelation, homoscedasticity, and non-multicollinearity. homoscedasticity is one of the important assumptions in the regression analysis, where the variance of the error term is constant otherwise heteroscedasticity. the effect of heteroscedasticity will give much weight to a small subset of data (namely the subset where the error variance is largest) when estimating regression parameters. restricted maximum likelihood (reml) is known as an unbiased parameter estimation method. reml method can be applied to models that have a normal experimental error, mailto:dwi.masrokhah@gmail.com https://en.wikipedia.org/wiki/dependent_variable https://en.wikipedia.org/wiki/explanatory_variable restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 81 interrelated and different variance. the used of reml variance component estimation can be done even if the data did not meet the assumptions of analysis of variance [2]. economics is one of social field that is often use regression analysis to make decisions. one of the developed economic theory is consumption theory. consumption theory states that any individual who has income, is assumed to set aside their part of revenue after being deducted by consumption [3]. the consumption pattern is significantly affecting the saving’s behavior. indonesian society is known as a consumer society, it could lead to the low motivation of savings. the benefits of savings are degrading consumerist patterns, practicing thrift and as a reserve fund. if the savings increases, not only the public funds will increase but also the investment [4]. students are part of the community who have an income. the purpose of this research to estimate multiple regression parameters using reml methods in modeling student’s saving at the faculty of mathematics and natural science, brawijaya university. methods the parameters used in this study consisted of a response variable and five predictor variables. the response variable is student’s saving (y). five predictor variables that affect student savings and used in the research are: 𝑋1 = the student’s age (years), 𝑋2 = the amount of income of student’s parent (thousand rupiah), 𝑋3 = the amount of student’s pocket money (thousand rupiah), 𝑋4 = the amount of student’s additional income (thousand rupiah) and 𝑋5 = the amount of student’s consumption (thousand rupiah). linear regression analysis is a statistical method that is useful to model the relationship between the response variable and predictor variables. the relationships model derived from regression analysis can be used as a description of the phenomenon of data. the regression model can also be used for predicting the values of the response variable. the concept of predicting in the regression analysis can only be done in the data range of the predictor variables used to establish the regression model [5]. the response variable is also called dependent variable and denoted by y. predictor variables are called independent variables and denoted by x. multiple linear regression model is a model where one response variable is determined as a function of more than one predictor variable (p): 𝑌𝑖 = 𝛽0 + 𝛽1𝑋1𝑖 + 𝛽2𝑋2𝑖 + ⋯ + 𝛽𝑝𝑋𝑝𝑖 + 𝜀𝑖 (1) where: 𝑖 = 1,2, ⋯ , 𝑛 𝑌𝑖 = response variable 𝑋1𝑖 , 𝑋2𝑖 , ⋯ , 𝑋𝑝𝑖 = predictor variables 𝛽0, 𝛽1, ⋯ , 𝛽𝑝 = regression coefficients 𝜀𝑖 = error 𝑝 = number of predictor variables. equation (1) has (𝑝 + 1) unknown parameters, with {𝑥1𝑖 , . . . , 𝑥𝑝𝑖 , 𝑖 = 1, ⋯ , 𝑛} is assumed fix and {𝜀𝑖 } assumed variables are independent, normal distribution with average 0 and variance 𝜎 2: restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 82 using a matrix of the equation (1) can be denoted: [ 𝑦1 𝑦2 ⋮ 𝑦𝑛 ] = [ 1 𝑥11 … 𝑥𝑝𝑖 1 𝑥21 … 𝑥2𝑘 1 ⋮ ⋱ ⋮ 1 𝑥𝑛1 … 𝑥𝑛𝑘 ] [ 𝛽0 𝛽1 ⋮ 𝛽𝑛 ] + [ 𝜀1 𝜀2 ⋮ 𝜀𝑛 ] or 𝐘 = 𝐗𝛃 + 𝛆 (2) where 𝐘 = respons vector of size (𝑛 × 1) 𝐗 = predictor matrix size (𝑛 × (𝑝 + 1)) 𝛃 = regression coefficient measuring ((p + 1) × 1) 𝛆 = error vector size (n × 1) the steps of data analysis are as follows: 1. estimate the parameters by using the ols. ordinary least squares method is one of the parameter estimations in regression analysis by minimizing the sum of squared errors. by using ols, the obtained estimators for the parameter β is �̂�. based on the model (2), it is obtained: 𝛆 = 𝐘 − 𝐗𝛃 (3) so that 𝑆(𝛽) = ∑ 𝜀𝑖 2𝑛 𝑖=1 = 𝜺 𝑻𝜺 = (𝐘 − 𝐗𝛃)𝐓(𝐘 − 𝐗𝛃) = 𝐘𝐓𝐘 − 𝛃𝐓𝐗𝐓𝐘 − 𝐘𝐓𝐗𝛃 + 𝛃𝐓𝐗𝐓𝐗𝛃 = 𝐘𝐓𝐘 − 𝟐𝛃𝐓𝐗𝐓𝐘 + 𝛃𝐓𝐗𝐓𝐗𝛃 by using the properties of the inverse matrix, 𝛃𝐓𝐗𝐓𝐘 = 𝐘𝐓𝐗𝛃 is a scalar, then the least squares estimators must meet: 𝜕𝑆 𝜕𝛽 |�̂� = −𝟐𝐗𝐓𝐘 + 𝟐𝐗𝐓𝐗�̂� = 0 be simplified, 𝐗𝐓𝐗�̂� = 𝐗𝐓𝐘 (4) multiply the final form of the matrix equation (4) both sides with (𝐗𝐓𝐗)−𝟏, produces the least squares estimator for β is: (𝐗𝐓𝐗)−𝟏𝐗𝐓𝐗�̂� = (𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘 𝐈�̂� = (𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘 �̂� = (𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘 (5) 2. test the classical assumption of multiple linear regression analysis. the model derived from multiple regression analysis must meet the assumptions of the classical regression analysis. the assumptions include: error normally distributed error, homoscedasticity of error variance, non-autocorrelation and non-multicollinearity. the normality assumption of error is an error value (𝜀𝑖 ) obtained from the regression model should follow the normal distribution. one of the methods to detect normality of error is shapiro-wilk [6] . the hypotheses tested: 𝐻0 ∶ 𝜀𝑖𝑗 is normally distributed 𝐻1 ∶ 𝜀𝑖𝑗 is not normally distributed if 𝐻0 is true, shapiro-wilk test statistics: 𝐺 = 𝑏𝑛 + 𝑐𝑛 ln [ 𝑇3−𝑑𝑛 1−𝑇3 ] ~𝑍(0,1) (6) where 𝑇3 = 1 𝐷 [∑ 𝑎𝑖 𝑛 𝑖=1 (𝑋(𝑛−𝑖+1) − 𝑋𝑖 )] 2 restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 83 𝐷 = ∑ (𝜀𝑖 − 𝜀)̅ 2𝑛 𝑖=1 g value can be approximated by the normal distribution as the z value is the value of the coefficient counting. the value of 𝑎𝑖 is shapiro-wilk’s value with certain n. value 𝑏𝑛, 𝑐𝑛, and 𝑑𝑛is the conversion value shapiro-wilk statistical approaches a normal distribution for n (many observations), if g value less than the critical value of z distribution, then it can be decided to accept h0, which means that the experimental error is normally distributed [7]. one of the assumptions of classical regression model is homoscedasticity [8]. if the variance is not constant, is expressed as heteroscedasticity. one of the methods to detect the presence of heteroscedasticity is by using glejser test. after getting 𝑒𝑖 from regression with ols method, glejser suggest regressing the absolute 𝑒𝑖 as a response to the predictor variables based on the hypotheses: 𝐻0 : σ1 2 = σ2 2 = ⋯ = σj 2 = σ2 ; σ𝑗 2 = 𝜎2 𝐻1 : at least one j where σ𝑗 2 ≠ 𝜎2 if 𝐻0true, the test statistic 𝑀𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑀𝑆𝑒𝑟𝑟𝑜𝑟 ~𝐹(𝑝,(𝑛−(𝑝+1))) (7) where: 𝑀𝑆𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 = 𝛃𝐗′𝐘 − 𝑛�̅�2 𝑝⁄ 𝑀𝑆𝑒𝑟𝑟𝑜𝑟 = 𝐘 ′𝐘 − 𝛃𝐗′𝐘 (𝑛 − (𝑝 + 1)) ⁄ if the test statistic is less than the critical point 𝐹(𝑝,(𝑛−𝑝−1)), then it is decided to accept h0, which means that the error variance is homogeneous [8] autocorrelation is the correlation between members of a series of observations which are sorted by time (time series) or space (data cross-section). to detect the presence of autocorrelation, the durbin watson’s test was used based on: 𝐻0: 𝜌 = 0 (error are independent) 𝐻1:: 𝜌 ≠ 0 (error are not independent) statistical test: 𝑑 = ∑ (𝑒𝑖−𝑒𝑖−1) 2𝑛 𝑖=2 ∑ 𝑒𝑖 2𝑛 𝑖=1 (8) where: 𝑑 : durbin watson statistic 𝑒𝑖 : the 𝑖 − 𝑡ℎ error value 𝑒𝑖−1 : the (𝑖 − 1) error value 𝐻0 rejected if 𝑑 < 𝑑𝐿 𝑜𝑟 𝑑 > 4 − 𝑑𝐿 𝐻0 acceptable if 𝑑𝑈 < 𝑑 < 4 − 𝑑𝑈 no decision if dl <d <du or 4-du <d <4-dl one of the assumptions that must be met in the establishment of regression model with multiple variables predictor is non-multicollinearity. multicollinearity is the presence of high linear relationship between the predictor variables. multicollinearity could be detected by using variance inflation factor (vif) [9], based on hypotheses: 𝐻0: no multicollinearity between variables 𝐻1 : there multicollinearity between variables 𝑉𝐼𝐹 = 1 1−𝑅𝑗 2 (11) 𝑅𝑗 2 = 𝐽𝐾𝑟𝑒𝑔𝑟𝑒𝑠𝑖 𝐽𝐾𝑡𝑜𝑡𝑎𝑙 = 𝛃𝐗′𝐘−𝑛�̅�2 𝒀′𝐘−𝑛�̅�2 restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 84 where 𝑅𝑗 2 is coefficient of determination of auxiliary regression, regressing between 𝑋𝑗 and (𝑝 − 1) the other predictor variable. auxiliary regression equation is 𝑋𝑗𝑖 = 𝛾0 + ∑ 𝛾𝑗 𝑋𝑘𝑖 𝑝−1 𝑘=1|𝑘≠𝑗 + 𝑢𝑖 ; 𝑘 = 𝑗 = 1,2, ⋯ , 𝑝 (9) if vif is less than 10 then h0 accepted, so the assumption of non-multicollinearity is met. 3. if the assumptions are not met homoscedasticity, then if is followed by suspected regression parameter using reml. restricted maximum likelihood is an alternative variance estimation parameter derived from the maximum likelihood method (mlm)[2]. the parameters obtain from reml estimators is divided into two parts, namely fixed effects parameter by parameter 𝛽 and 𝜎2. as an example of a random sample that has a normal distribution, then: 𝑌 − 𝜇 = (𝑌 − �̅�) + (�̅� − 𝜇) with the μ=xβ so: 𝐘 − 𝐗𝛃 = (𝐘 − 𝐗�̂�) + (𝐗�̂� − 𝐗𝛃) = (𝐘 − 𝐗�̂�) + 𝐗(�̂� − 𝛃) then the likelihood function is: (𝐘 − 𝐗𝛃)𝐓(𝐘 − 𝐗𝛃) = [(𝐘 − 𝐗�̂�) + 𝐗(�̂� − 𝛃)] t [(𝐘 − 𝐗�̂�) + 𝐗(�̂� − 𝛃)] = (𝐘 − 𝐗�̂�) t (𝐘 − 𝐗�̂�) − 2(𝐘 − 𝐗�̂�) t 𝐗(�̂� − 𝛃) + (�̂� − 𝛃) t 𝐗t𝐗(�̂� − 𝛃) based on these equations obtained 2(𝐘 − 𝐗�̂�) 𝐓 𝐗(�̂� − 𝛃) = 2(𝐗𝐓𝐘 − 𝐗𝐓𝐗�̂�) 𝐓 (�̂� − 𝛃) by completing 𝐗t𝐘 − 𝐗t𝐗�̂� = 𝟎 to (p + 1) parameters, resulted (𝐘 − 𝐗𝛃)t(𝐘 − 𝐗𝛃) = (𝐘 − 𝐗𝐛)t(𝐘 − 𝐗�̂�) + (�̂� − 𝛃) t 𝐗t𝐗(�̂� − 𝛃) (10) equation (6) is a function of (p + 1) parameters (β). for this function free of the parameter β can be written: (𝐘 − 𝐗�̂�) 𝐓 (𝐘 − 𝐗�̂�) = (𝐘 − 𝐗(𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘)𝐓(𝐘 − 𝐗(𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘) = 𝐘𝐓(𝐈 − 𝐗(𝐗𝐓𝐗)−𝟏𝐗𝐓)𝐓(𝐈 − 𝐗(𝐗𝐓𝐗)−𝟏𝐗𝐓)𝐘 since, 𝐈 − 𝐗(𝐗t𝐗)−1𝐗t is symmetric and idempotent, then: (𝐘 − 𝐗�̂�) t (𝐘 − 𝐗�̂�) = 𝐘t(𝐈 − 𝐗(𝐗t𝐗)−1𝐗t)𝐘 the log likelihood of multiple linear regression is: 𝑙𝑜𝑔𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝐘 = −p+1 2 ln(2π) − p+1 2 ln(σ2) − 1 2σ2 (�̂� − 𝛃) t 𝐗t𝐗(�̂� − 𝛃) − n−1−p 2 ln(2π) − n−1−p 2 ln(σ2) − 1 2σ2 𝐘t(𝐈 − 𝐗(𝐗t𝐗)−1𝐗t)𝐘 with minimum variance the obtained results estimator 𝜎2 𝜎2 = (𝐘−𝐗�̂�) t (𝐘−𝐗�̂�) n−1−p = 𝐘t(𝐈−𝐗(𝐗t𝐗) −1 𝐗t)𝐘 n−1−p (11) results and discusion estimation of regression parameter using ols linear regression analysis is a statistical method that is useful to model the relationship between response variable and predictor variables[5]. the relationships derived from regression analysis that can be used as a descript of the phenomenon of data. in this study, the model obtained using ordinary least squares (ols) method is: �̂�𝑖 = −1855 + 121,5𝑋1 + 0,0098𝑋2 + 0.0524 𝑋3 + 0.559 𝑋4 − 0.585𝑋5 restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 85 testing of the classical assumption of multiple linear regression analysis testing the assumption of normality of errors using the shapiro-wilk test based on the hypothesis: 𝐻0 ∶ an error is normally distributed 𝐻1 ∶ an error is not normally distributed the p-value of 0.294 for the test and the shapiro wilk test statistic g of 0.9784. based on testing criteria, because the value of p > 𝛼 and statistic’s test g < critical point z (1,96) , h0 accepted and concluded that the error distributes normally with a confidence level of 95%. detection of autocorrelation using durbin watson test[10]. hypotheses were tested: 𝐻0 ∶ 𝜌 = 0 (error is independent) 𝐻1 ∶ 𝜌 ≠ 0 (error is not independent) d test statistic of 1.928. according to the durbin watson table then obtained a value of 1.464 dl and du value of 1.768. the test of the statistic is between the value d and 4-du du then h0 accepted, can be conclude non-autocorrelation assumptions are met. the variance inflation factor (vif) is one of the values used to detect the presence of multicollinearity [9]. hypotheses were tested: 𝐻0 ∶ non multicollinierity 𝐻1 ∶ multicollinierity table 1. vif value of each predictor variable predictor value of vif information 𝑋1 1,04 𝐻0 accepted 𝑋2 1,13 𝐻0 accepted 𝑋3 1,05 𝐻0 accepted 𝑋4 1,15 𝐻0 accepted 𝑋5 1,06 𝐻0 accepted table 1 showed vif value of each predictor variable <10 so h0 is accepted, nonmulticollinearity assumptions are met. an error variance assumption test homoscedasticity using glejser test [8]. after getting 𝑒𝑖 from ols method, glejser suggest regressing the absolute error and predictor variables. hypothesis were tested: 𝐻0 : σ1 2 = σ2 2 = ⋯ = σj 2 = σ2 ; σ𝑗 2 = 𝜎2 𝐻1 : at least one 𝑗 where σ𝑗 2 ≠ 𝜎2 p value of glejser test 0,006, then 𝐻0 rejected and concluded that error variance not homogen. estimation of regression parameter using reml restricted maximum likelihood (reml) is an alternative variance estimation parameter derived from the maximum likelihood method (mlm) [2]. the outlinesof the parameters reml estimators into two parts, namely fixed effects parameter by parameter 𝛽 and 𝜎2. the model obtained using restricted maximum likelihood (reml) method is: �̂�𝑖 = −1609 + 112 𝑋1 + 0.0088𝑋2 + 0.0504 𝑋3 + 0.4706 𝑋4 − 0.636𝑋5 the results of testing each parameter using wald test is shown in table 4.2 restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 86 table 2. the result of partial test use reml method parameter value information x1 0,001 𝐻0 rejected x2 0,615 𝐻0 accepeted x3 0,218 𝐻0 accepted x4 0,017 𝐻0 rejected x5 0,017 𝐻0 rejected based on table 2, variables that affect student’s saving are student’s age (years), the amount of student’s additional income (thousand rupiah), and the amount of student’s consumption (thousand rupiah). model validation to find out whether the model obtained in the study is in accordance with the actual conditions in the field, the model validation is performed. then, the comparison between the predicted values from reml method with the actual value of observations is tested using paired t test. the summary result of paired t test is presented in table 4.3. table 3. paired t test predicted value from reml method and actual value data n average standard deviation y observation 29 408,62 364,76 y prediction 29 399,19 184,22 difference 29 9,43 182,10 p-value = 0,904 the result of paired t test provided in table 3 decided to accept h0 because p value is larger than 0.05, lead to conclusion that the model of reml method can be used to predict student’s saving. conclusion the model of student’s saving using ols method as follows: �̂�𝑖 = −1855 + 121,5𝑋1 + 0,0098𝑋2 + 0.0524 𝑋3 + 0.559 𝑋4 − 0.585𝑋5 student’s saving is affected significantly by: student’s age (𝑋1), the amount of student’s additional income (𝑋4), and the amount of student’s consumption (𝑋5). reml method can overcome heteroscedasticity error variance and producing unbiased estimator. the model of student saving using reml method as follows: �̂�𝑖 = −1609 + 112 𝑋1 + 0.0088𝑋2 + 0.0504 𝑋3 + 0.4706 𝑋4 − 0.636𝑋5 student’s saving is affected significantly by: student’s age (𝑋1), the amount of student’s additional income (𝑋4), and the amount of student’s consumption (𝑋5). based on the results and the discussion it can be concluded that: reml method can overcome heteroscedasticity error variance and unbiased estimator. references [1] chatterjee, s., & hadi, a. s. (2006). regression analysis by example fourth edition. new york: john wiley and sons, inc. [2] o' neill, m. (2010). anova & reml: a guide to linier mixed models in an experimental design context. statsitical advsory training service pty ltd. [3] browning, m., & lusardi, a. (1996). household saving: micro theories and micro facts. journal of ecomomic literature. restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression dwi masrokhah 87 [4] dupas, p., & robinson, s. (2009). savings constraints and microenterprise development: evidence from a field experiment in kenya. national bureau research working paper. [5] draper, n. r., & smith, n. r. (1998). applied regression analysis. third edition. new york: john wiley and sons. [6] shapiro, s. s., & wilk, m. b. (1965). an anlytics of variance test for normality. biometrika. [7] razali, n., & wah, y. p. (2011). power comparisons of shapiro wilk, kolmogorov smirnov, lilliefors and anderson darling. journal of statistical modeling and analytics. [8] gujarati, d. (2004). basic econometrics fourth edition. new york: the mcgraw-hill. [9] alaudin, m., & nghiem, h. s. (2010). do instructional attributes pose multicollinierity problems? am empirical exploration. economic analysis and policy. [10] gujarati, d., & porter, d. c. (2012). dasar-dasar ekonometrika (jilid 2) terjemahan raden carlos mangunsong. jakarta: salemba empat. mathematical modelling on transportation method apllication for rice distribution cost optimization cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 195-202 p-issn: 2086-0382; e-issn: 2477-3344 submitted: march 14, 2018 reviewed: march 15, 2018 accepted: may 31, 2019 doi: http://dx.doi.org/10.18860/ca.v5i4.4893 mathematical modelling on transportation method apllication for rice distribution cost optimization nuri lutvi azizah1 1muhammadiyah university of sidoarjo, sidoarjo, indonesia email: nurillutviazizah@umsida.ac.id abstract raskin/rastra is a rice distribution program for the poor family with the aim of improving food security starting from the household scale. the aim of this study is to determine whether the mathematics modelling can use in the transportation methods to provide an efficient solution of costs on the rice distribution in the region of sidoarjo. the method used in this research is the nwc (northwest corner) method used to analyze the initial fiscal solution, and refined by modi (modified distribution) to analyze the most optimal cost. from the calculation by transportation method, the optimum cost is lower than the company's calculation of rp 85.186.040 while the cost of the company's calculation is rp 87.209.690,750. thus the use of transportation methods can save rastra's distribution cost of rp 2.023.650,750. keywords: mathematics modelling, transportation methods, optimum cost introduction raskin/rastra is a staple food subsidy in the form of rice intended for the poor as the government's efforts to increase food security and provide protection to poor families. bulog has two activities, namely service activities and commercial business activities. but in its implementation almost 90% of the activities of the company is a service activity that is the assignment of the government [1]. “perum bulog sub divre sidoarjo” as the implementer of rastra program for some areas such as balongbendo, buduran, candi, gedangan, jabon, krembung, krian, porong, prambon, sedati and sidoarjo spent considerable funds for distribution activities. to minimize the cost of this distribution it is necessary to plan for the distribution of rastra so that the cost of distribution issued is as optimal as possible. one method that can be used to optimize distribution costs is by transportation method. the transportation methods require a valid mathematical model before the transporatation’s calculation. mathematical model is a description of a system using mathematical concepts and language, and the process of developing a mathematical model is termed mathematical modelling. the method of transportation is a method or means used to solve the problem of distribution from sources that provide the same product, to the place where the need optimally so that the cost of distribution issued is minimal [2]. the goal is to minimize the cost of distributing rastra from one location to another in the sidoarjo area, so that the needs of each region are met according to capacity with minimal distribution cost. mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 196 methods a. transportation method the mathematical model is used to determine the mathematical equation related to the problems encountered in the problem of distribution cost optimization. the method of transportation is related to the distribution of a single product from multiple sources with limited supply, to some destinations with a particular request to obtain a minimum distribution charge. because only one kind of product then a destination can meet the demand of one or more sources. transportation issues have several characteristics that are [3]: a) there are a number of sources and a number of specific goals b) the quantity of goods distributed from each source and requested by any destination is certain c) the quantity or quantity of goods sent from a source of destination in accordance with the request or capacity of the source d) the cost of transportation from a source to a destination is certain mathematically, transportation problems can be modeled as follows: objective function 𝑀𝑖𝑛𝑖𝑚𝑢𝑚𝑘𝑎𝑛 𝑍 = ∑ ∑ 𝐶𝑖𝑗 𝑋𝑖𝑗 𝑛 𝑗=1 𝑚 𝑖=1 (1) constrain function ∑ 𝐶𝑖𝑗 𝑚 𝑖=1 𝑋𝑖𝑗 = 𝑎𝑖 ; 𝑖 = 1, 2, … . . , 𝑚 (2) ∑ 𝐶𝑖𝑗 𝑛 𝑗=1 𝑋𝑖𝑗 = 𝑏𝑗 ; 𝑗 = 1, 2, … . . , 𝑛 (3) operator definition of each variable will be described as cij = transportation cost per unit of goods from source i to destination j xij = number of goods distributed from source i to destination j ai = quantity of goods offered or capacity from source i bj = quantity of goods requested or ordered by destination j m = number of sources n = number of goals b. transportation models a transport problem is said to be balanced (balance program) if the amount of supply at source i is equal to the amount of demand on the destination [4].can be written: ∑ 𝑎𝑖 = ∑ 𝑏𝑗 𝑛 𝑗=1 𝑚 𝑖=1 (4) mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 197 the first step to solve the problem of transportation is to determine the initial feasible solution. there are three methods for determining the initial feasible solution [5] : 1. north west corner (nwc) method 2. the least cost method 3. vogel approximation method (vam) after obtaining the initial feasible solution then sought the optimal solution. there are two methods to determine the optimal solution [5]: 1. the stepping stone method (stepping stone) 2. modified method of distribution (modi) c. step of nwc methods the method discussed in this research is northwest corner method for initial solution and modi for optimal solution. the step of the nwc methods are [6]: 1. step 1: allocate the maximum amount available to the selected cell and adjust the associated supply and demand quantities by subtracting the allocated quantity. 2. step 2: exit the row or the column when the supply or demand reaches zero and cross it out, to show that it cannot make any more allocations that row or column. if a row or column simultaneously reach zero, only cross out one (the row or the column) and leave a zero supply (demand) in the row (column) that is not crosses out. 3. step 3: if exactly one row or column is left that is not crosses out, stop. otherwise, advance to the cell to the right if a column has just been crossed out, or to the cell below if a row was crossed out. continue with step 1. d. step of modi methods while the modi method steps are [6]: 1. step 1: determine the 𝑈𝑖 values for each row and 𝑉𝑗 values for each column by using the relation 𝐶𝑖𝑗 = 𝑈𝑖 + 𝑉𝑗 for all base variables and set that the value of 𝑈𝑖 is zero. 2. step 2 : calculate the cost charges for each non based variable using 𝑋𝑖𝑗 = 𝐶𝑖𝑗 − 𝑈𝑖 − 𝑉𝑗 3. step 3: if there is a negative 𝑋𝑖𝑗 value, then the solution is not optimal. select the 𝑋𝑖𝑗 variable with the largest negative value as the entering variable. 4. step 4 : allocated goods entering variable 𝑋𝑖𝑗 according stepping stone or nwc process 5. step 5: repeat step 1 through step 4 until all 𝑋𝑖𝑗 values are zero or positive. e. data collection technique the main problem faced in this research is not optimal the fund spent for rice distribution cost in “perum bulog sub divre sidoarjo”, so after mathematical modelling used in this research, the transportation method will be applied to get the most optimal cost solution. here is a data collection technique used are: a. study of literature b. secondary data retrieval this study uses secondary data obtained from “perum bulog sub divre sidoarjo”, and can be done using assumptions to be used for similar problems. c. observation this activity is done to synchronize the secondary data obtained by doing direct observation of the spaciousness to ensure the data obtained is valid. d. system planning using mathematics model e. implementation of transportation methods mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 198 results and discussion a. rice distribution data for raskin/rastra the rastra distribution data consist of the amount of rastra rice distributed per month in kilograms (kg), for each area sub-div described in several districts/cities. in each sub div of the districts, there are number of distribution points (td), and at the village level with the number of target household (rts) receiving rastra [7]. in this research, rastra distribution in point emphasizes on the distribution from warehouse to the subdistrict level to find out how much the cost from the warehouse to the distribution place in the area to the sub district where rice distribution area. for the value to be measured for completion in this research is the amount of distribution or distribution of rastra rice per month in kilogram (kg) then be converted in tons/year as needed. the amount of rastra rice distribution per month in north surabaya region can be seen in table 1. table 1. rastra rice distribution per month no subdivre distribution per month kab/kota rts-pm kg i surabaya utara 221.845 3.327.675 1. kota surabaya 65.991 989.865 2. kab. sidoarjo 78.103 1.171.545 3. kab. gresik 77.751 1.166.265 sidoarjo district consist of 18 sub-district formed five clusters for distribution of rice rastra, details for five clusters with members called distribution point (td) that include within each cluster can be seen in table 2. table 2. capacity data for each region in rastra distribution no warehouse location capacity (per kg) 1 warehouse 1 gedangan 265045 2 warehouse 2 gedangan 257500 3 warehouse 3 gedangan 230000 4 warehouse 4 porong 201000 5 warehouse 5 porong 218000 total 1171545 b. description of distance data distance data is retrieved through google maps data source to know the distance of warehouse with the sub-district or rice distribution/distribution area for rastra [7]. the verification of the distance data that has been collected can be seen in table 3, it contains the relative distance data collection from the warehouse to the sub-districts where the distribution to the rts. mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 199 table 3. relative distance data from warehouse to the sub-district no. location warehouse/sub district distance(km) 1 balongbendo 28,02 2 buduran 0,5 3 candi 9 4 gedangan 3,5 5 jabon 19,3 6 krembung 24,9 7 krian 18,3 8 porong 17,6 9 prambon 27,7 10 sedati 7,8 11 sidoarjo 6,1 12 sukodono 7,5 13 taman 12,2 14 tanggulangin 12,5 15 tarik 32,8 16 tulangan 18,8 17 waru 10,8 18 wonoayu 15,1 the following data on the amount of rastra rice required depend on the distance data by google maps application and can be seen on table 4. table 4. raskin rice required amount in each region in kg of rice no warehouse sub district rice capacity (kg)/demand total 1 warehouse 1 buduran 69000 280425 sedati 71035 gedangan 71985 sidoarjo 68405 2 warehouse 2 candi 67600 246160 tanggulangin 53500 wonoayu 60035 tulangan 65025 3 warehouse 3 sukodono 50025 234635 taman 52000 krian 75545 waru 57065 4 warehouse 4 balongbendo 68050 198080 tarik 61030 prambon 69000 5 warehouse 5 krembung 67000 212245 porong 77245 mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 200 jabon 68000 total 1171545 1171545 c. mathematical modelling preparation of data is done by translating the problems and descriptions into the form of mathematical models. formation of the mathematical model of the data obtained need to pay attention to things that affect the calculation process, such as: 1. decision variable 2. objective function 3. constrains the steps to be taken in the transport model are the determination of decision variables, objective functions, and constraints. 1. decision variable here are the variables for delivery allocation to each warehouse alocation distribusion from gbb banjar kemantren 1 (warehouse 1) a. allocation from warehouse 1 to several point distribution 𝑿𝟏𝟏 = allocation from warehouse 1 to point distribution 1, 𝑿𝟏𝟐 = allocation from warehouse 1 to point distribution 2, 𝑿𝟏𝟑 = allocation from warehouse 1 to point distribution 3, 𝑿𝟏𝟒 = allocation from warehouse 1 to point distribution 4, 𝑿𝟏𝟓 = allocation from warehouse 1 to point distribution 5 b. allocation from warehouse 2 to several point distribution 𝑿𝟐𝟏 = allocation from warehouse 2 to point distribution 1 𝑿𝟐𝟐 = allocation from warehouse 2 to point distribution 2 𝑿𝟐𝟑 = allocation from warehouse 2 to point distribution 3 𝑿𝟐𝟒 = allocation from warehouse 2 to point distribution 4 𝑿𝟐𝟓 = allocation from warehouse 2 to point distribution 5 c. allocation from warehouse 3 to several point distribution 𝑿𝟑𝟏 = allocation from warehouse 3 to point distribution 1 𝑿𝟑𝟐 = allocation from warehouse 3 to point distribution 2 𝑿𝟑𝟑 = allocation from warehouse 3 to point distribution 3 𝑿𝟑𝟒 = allocation from warehouse 3 to point distribution 4 𝑿𝟑𝟓 = allocation from warehouse 3 to point distribution 5 d. allocation from warehouse 4 to several point distribution 𝑿𝟒𝟏 = allocation from warehouse 4 to point distribution 1 𝑿𝟒𝟐 = allocation from warehouse 4 to point distribution 2 𝑿𝟒𝟑 = allocation from warehouse 4 to point distribution 3 𝑿𝟒𝟒 = allocation from warehouse 4 to point distribution 4 𝑿𝟒𝟓 = allocation from warehouse 4 to point distribution 5 e. allocation from warehouse 5 to several point distribution 𝑿𝟓𝟏 = allocation from warehouse 5 to point distribution 1 𝑿𝟓𝟐 = allocation from warehouse 5 to point distribution 2 𝑿𝟓𝟑 = allocation from warehouse 5 to point distribution 3 𝑿𝟓𝟒 = allocation from warehouse 5 to point distribution 4 𝑿𝟓𝟓 = allocation from warehouse 5 to point distribution 5 mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 201 2. objective function optimization the function 𝑍 𝑍 = ∑ ∑ 𝐶𝑖𝑗 𝑋𝑖𝑗 𝑛 𝑗=1 𝑚 𝑖=1 (5) so that the formulation based on the data became a linear programming formulation and can be formulated by matrix calculation. 𝑍 = 71,25𝑋11 + 75𝑋12 + 80𝑋13 + 110,25𝑋14 + 102,675𝑋15 + 73𝑋21 + 77,5𝑋22 + 76,15𝑋23 + 117,15𝑋24 + 81,765𝑋25 + 70,5𝑋31 + 78,25𝑋32 + 82,75𝑋33 + 105,25𝑋34 + 88,85𝑋35 + 80𝑋41 + 85,75𝑋42 + 76,75𝑋43 + 71,05𝑋44 + 71,75𝑋45 + 101,75𝑋51 + 90,25𝑋52 + 80,75𝑋53 + 71,85𝑋54 + 70,25𝑋55. (6) 3. constrains constrains in this research can be divided into two categories. first constrain is for demanding problem and the second is for bargaining problem. here is the same about demanding constrain and bargaining constrains. 𝑋11 + 𝑋21 + 𝑋31 + 𝑋41 + 𝑋51 = 280425 (7) 𝑋12 + 𝑋22 + 𝑋32 + 𝑋42 + 𝑋52 = 246160 (8) 𝑋13 + 𝑋23 + 𝑋33 + 𝑋43 + 𝑋53 = 234635 (9) 𝑋14 + 𝑋24 + 𝑋34 + 𝑋44 + 𝑋54 = 198080 (10) 𝑋15 + 𝑋25 + 𝑋35 + 𝑋45 + 𝑋55 = 21224 (11) d. program formulation the compilation from the formulation and equations is using the computational program as a qm for windows script. the problem solving for rastra rice distribution cost optimization can be seen on the table 5 by northwest corner methods to determine initial fiscal point. table 5. solution report for northwest corner (nwc) methods solution value district 1 district 2 district 3 district 4 district 5 $87440020 warehouse 1 265045 (-0.75) (0.25) (35.7) (-29.725) warehouse 2 15380 242120 (-5.85) (40.85) (-7.065) warehouse 3 (-3.25) 4040 225969 (28.2) (13.4) warehouse 4 (12.25) (13.5) 8675 192325 (2.3) warehouse 5 (33.2) (17.2) (3.2) 5755 212245 after the initial value or point obtained from nwc method, then the calculation and the program compilation is continued with modi method to get the optimal solution, it can be seen on the table 6. table 6. solution report for modi by using initials nwc method solution value district 1 district 2 district 3 district 4 district 5 $85186040 warehouse 1 18885 246160 (5.6) (41.55) (35.575) warehouse 2 31540 (0.75) 225960 (46.7) (12.915) warehouse 3 230000 (4) (9.1) (37.3) (22.5) mathematical modelling on transportation method apllication for rice distribution cost optimization nuril lutvi azizah 202 warehouse 4 (6.4) (8.4) 8675 192325 (2.3) warehouse 5 (27.35) (12.1) (3.2) 5755 212245 the solution report from qm for windows compilation obtained that the nwc method has the solution value about $87440020. the modi method by using the initial fiscal point nwc method has the solution value about $85186040. conclusion mathematical modelling on the transportation methods will produce a mathematical equations useful for optimization problem solving if the model used is a valid model. based on the result of qm for windows compilations is known that transportation with the initial nwc method to determine the initial value has the solution value about rp 87.440.020. the modi method is used to determine the optimal solution by using initial fiscal point in nwc method has the solution value about rp 85.186.040. the modi method obtain the optimal solution and it can save cost about rp 2.253.980 from the nwc method. the secondary data that given from the calculation “perum bulog sub-divre sidoarjo” has minimal cost about rp 87.209.690,750. so that, with the modi method can save rastra rice distribution cost about rp 2.023.650,750. references [1] h, a. taha. (1997). operation research an introduction. university of arkansas, fayetteville: prentice hall, vol. 6, pp. 202-230. [2] clark, s. d. (2002). handbook of transportation science. journal of the operational research society (vol. 53, pp. 470–471). https://doi.org/10.1017/cbo9781107415324.004 [3] t, d, a, dimyati. operation research. (2003). handbook of operation research. bandung : sinar baru algesindo, vol. 3, pp. 304-325. [4] shay, e., combs, t. s., findley, d., kolosna, c., madeley, m., & salvesen, d. (2016). identifying transportation disadvantage: mixed-methods analysis combining gis mapping with qualitative data. transport policy, 48, 129–138. https://doi.org/10.1016/j.tranpol.2016.03.002 [5] simbolon, l. d. situmorang., n. napitupulu. aplikasi metode transportasi dalam optimasi biaya distribusi beras miskin (raskin) pada perum bulog sub divre medan. jurnal saintia matematika, vol.02, 2337-9197. [6] siswanto. (2007).operation research. handbook of operation research. jakarta : penerbit erlangga, vol. 01, pp. 175-215. [7] riskiet, al. (2007).implementasi dan penyaluran raskin perum bulog divre di indonesia. jurnal aplikasi matematika, vol. 01, pp 01-07. https://doi.org/10.1017/cbo9781107415324.004 https://doi.org/10.1016/j.tranpol.2016.03.002 the rainbow vertex-connection number of star fan graphs cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 112-116 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 28 august 2018 reviewed: 08 october 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5516 the rainbow vertex-connection number of star fan graphs ariestha widyastuty bustan1, a.n.m. salman2 1mathematics department, faculty of mathematics and natural sciences, universitas pasifik morotai 2combinatorial mathematics research group, faculty of mathematics and natural sciences, institut teknologi bandung email: ariesthawidyastutybustan@gmail.com , msalman@math.itb.ac.id abstract a vertex-colored graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) is said to be rainbow vertex-connected, if for every two vertices 𝑢 and 𝑣 in 𝑉(𝐺), there exists a 𝑢 − 𝑣 path with all internal vertices have distinct colors. the rainbow vertexconnection number of 𝐺, denoted by 𝑟𝑣𝑐(𝐺), is the smallest number of colors needed to make 𝐺 rainbow vertex-connected. in this paper, we determine the rainbow vertex-connection number of star fan graphs. keywords: rainbow vertex-coloring; rainbow vertex-connection number; star fan graph; fan introduction all graph considered in this paper are finite, simple, and undirected. we follow the notation and terminology of diestel [1]. a vertex-colored graph 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) is said to be rainbow vertex-connected, if for every two vertices 𝑢 and 𝑣 in 𝑉(𝐺), there exists a 𝑢 − 𝑣 path with all internal vertices have distinct colors. the rainbow vertex-connection number of 𝐺, denoted by 𝑟𝑣𝑐(𝐺), is the smallest number of colors needed to make 𝐺 rainbow vertex-connected. it was introduced by krivelevich and yuster [2]. let 𝐺 be a connected graph, 𝑛 be the size of 𝐺, and diameter of 𝐺 denoted by 𝑑𝑖𝑎𝑚(𝐺), then they stated that 𝑑𝑖𝑎𝑚(𝐺) − 1 ≤ 𝑟𝑣𝑐(𝐺) ≤ 𝑛 − 2 (1) besides that, if 𝐺 has 𝑐 cut vertices, then 𝑟𝑣𝑐(𝐺) ≥ 𝑐 (2) in fact, by coloring the cut vertices with distinct colors, we obtain 𝑟𝑣𝑐(𝐺) ≥ 𝑐. it is defined that 𝑟𝑣𝑐(𝐺) = 0 if 𝐺 is a complete graph there are many interesting results about rainbow vertex-connection numbers. some of them were stated by li and liu[3] and simamora and salman[4] and bustan [5]. li and liu determined the rainbow vertex-connection number of a cycle 𝐶𝑛 of order 𝑛 ≥ 3. based on it, they proved that for a connected graph 𝐺 with a block decomposition 𝐵1, 𝐵2, … , 𝐵𝑘 and 𝑐 cut vertices, 𝑟𝑣𝑐(𝐵1) + 𝑟𝑣𝑐(𝐵2) + ⋯ + 𝑟𝑣𝑐(𝐵𝑘 ) + 𝑡. in 2015 simamora and salman determined the rainbow vertex-connection number of pencil graph. in 2016 bustan determined the rainbow vertex-connection number of star cycle graph. in this paper, we introduce a new class of graph that we called star fan graphs and we determine the rainbow vertex-connection number of them. star fan graphs are divided into two classes based on the selection of a vertex of the fan graph ie a vertex with 𝑛 degree and vertex with 3 degree. mailto:ariesthawidyastutybustan@gmail.com mailto:msalman@math.itb.ac.id the rainbow vertex-connection number of star fan graphs ariestha widyastuty bustan 113 figure 1. 𝑆(6, 𝐹6, 𝑣𝑖,1) figure 2. 𝑆(6, 𝐹6, 𝑣𝑖,7) results and discussion definition 1. let 𝑚 and 𝑛 be two integers at least 3, 𝑆𝑚 be a star with 𝑚 + 1 vertices. 𝐹𝑛 be a fan with 𝑛 + 1 vertices, 𝑣 ∈ 𝑉(𝐹𝑛 ) and 𝑣 is a vertex with 𝑛 degree. a star fan graph is a graph obtained by embedding a copy of 𝐹𝑛 to each pendant of 𝑆𝑚, denoted by 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1) 𝑖 ∈ [1, 𝑚], such that the vertex set and the edge set, respectively, as follows. 𝑉 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) = {𝑣𝑖,𝑗 |𝑖 ∈ [1, 𝑚], 𝑗 ∈ [1, 𝑚 + 1]} ∪ {𝑣𝑚+1}, 𝐸 (𝑆(𝑚, 𝐹𝑛, 𝑣𝑖,1)) = {𝑣𝑚+1𝑣𝑖,1|𝑖 ∈ [1, 𝑚]} ∪ {𝑣𝑖,1𝑣𝑖,𝑗 | 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚 + 1]} ∪ {𝑣𝑖,𝑗 𝑣𝑖,𝑗+1| 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚]} the rainbow vertex-connection number of star fan graphs ariestha widyastuty bustan 114 theorem 1. let 𝑚 and 𝑛 be two integers at least 3 and 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1) be a star fan graph, then 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1))=𝑚 + 1 proof. based on equation (2), we have 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) ≥ 𝑐 = 𝑚 + 1 (3) in order to proof 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) ≤ 𝑚 + 1, define a vertex-coloring 𝛼: 𝑉(𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖1)) → [1, 𝑚 + 1] as follows. 𝛼(𝑣𝑖,𝑗 ) = { 𝒊, 𝑓𝑜𝑟 𝑗 = 1 𝒎 + 𝟏, 𝑜𝑡ℎ𝑒𝑟𝑠 we are able to find a rainbow path for every pair vertices 𝑢 and 𝑣 in 𝑉 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) as shown in table 1 tabel 1. the rainbow vertex 𝑢 − 𝑣 path for graph 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1) 𝒖 𝒗 condition rainbow-vertex path 𝑢 is adjacent to v trivial 𝒗𝒊,𝒋 𝒗𝒌,𝒍 𝒊, 𝒌 ∈ [𝟏, 𝒎] 𝒋, 𝒍 ∈ [𝟏, 𝒎 + 𝟏] 𝒗𝒊,𝒋, 𝒗𝒊,𝟏, 𝒗𝒎+𝟏, 𝒗𝒌,𝟏,𝒗𝒌,𝒍 so we conclude that 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) ≤ 𝑚 + 1 (4) from equation (3) and (4), we have 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,1)) = 𝑚 + 1 definition 2. let 𝑚 and 𝑛 be two integers at least 3, 𝑆𝑚 be a star with 𝑚 + 1 vertices. 𝐹𝑛 be a fan with 𝑛 + 1 vertices, 𝑣 ∈ 𝑉(𝐹𝑛 ) and 𝑣 is a vertex with 3 degree. a star fan graph is a graph obtained by embedding a copy of 𝐹𝑛 to each pendant of 𝑆𝑚, denoted by 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ) 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚] such that the vertex set and the edge set, respectively, as follows. 𝑉 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) = {𝑣𝑖,𝑗 |𝑖 ∈ [1, 𝑚], 𝑗 ∈ [, 𝑚 + 1]} ∪ {𝑣𝑚+1}, 𝐸 (𝑆(𝑚, 𝐹𝑛, 𝑣𝑖,𝑗 )) = {𝑣𝑚+1𝑣𝑖,𝑗 |𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚 + 1]} ∪ {𝑣𝑖,1𝑣𝑖,𝑗 | 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚 + 1]} ∪ {𝑣𝑖,𝑗 𝑣𝑖,𝑗+1| 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [2, 𝑚]} theorem 2. let 𝑚 and 𝑛 be two integers at least 3 and 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ) be a star fan graph, then 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ))=𝑚 + 2 proof.  case 1. 𝒎 = 𝟑 based on equation (1), we have 𝑟𝑣𝑐 (𝑆(3, 𝐹3, 𝑣𝑖,4)) ≥ 𝑑𝑖𝑎𝑚 − 1 = 6 − 1 = 5. we may define a rainbow vertex 5-coloring on 𝑆(𝑚, 𝐹, 𝑣𝑖,7) as shown in figure 2. the rainbow vertex-connection number of star fan graphs ariestha widyastuty bustan 115 figure 3. 𝑆(3, 𝐹3, 𝑣𝑖,4)  case 2. 𝒎 ≥ 𝟒 based on equation (2), we have 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) ≥ 𝑚 + 1. suppose that there is a rainbow vertex 𝑚 + 1-coloring on 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ). without loss of generality, color the vertices as follows: 𝛽′(𝑣𝑚+1) = 𝑚 + 1 𝛽′(𝑣𝑖,𝑚+1) = 𝑖, 𝑖 ∈ [1, 𝑚] look at the vertex 𝑣1,2 and 𝑣2,2 who can not use the same color. to obtain rainbow vertex path between them, should be passed the path of 𝑣1,2, 𝑣1,1, 𝑣1,𝑚+1, 𝑣𝑚+1, 𝑣2,𝑚+1, 𝑣2,1, 𝑣2,2. certainly 𝑣1,1 should be colored by the color which used at the cut vertices, beside color 1, 𝑚 + 1 and 2. suppose that 𝑣1,1 being color with 𝑘. it’s impacted there is no rainbow vertex path between vertex 𝑣𝑘,2 and 𝑣𝑖,2 for 𝑖 ≠ 𝑘. so that, graph 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ) cannot be colored with 𝑚 + 1 colors, so we obtain 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) ≥ 𝑚 + 2 (5) in order to proof 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) ≤ 𝑚 + 2, define a vertex-coloring 𝛽: 𝑉 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) → [1, 𝑚 + 2] as follows. 𝛽(𝑣𝑚+1) = 𝑚 + 2 𝛽(𝑣𝑖,𝑗 ) = (𝑖 + 𝑗)𝑚𝑜𝑑(𝑚 + 1), 𝑖 ∈ [1, 𝑚], 𝑗 ∈ [1, 𝑚 + 1] we are able to find a rainbow path for every pair vertices 𝑢 and 𝑣 in 𝑉 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) as shown in table 2. tabel 2. the rainbow vertex 𝑢 − 𝑣 path for graph 𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 ) 𝒖 𝒗 condition rainbow-vertex path 𝑢 is adjacent to v trivial 𝒗𝒊,𝒋 𝒗𝒌,𝒍 𝒊, 𝒌 ∈ [𝟏, 𝒎] 𝒋, 𝒍 ∈ [𝟏, 𝒎 + 𝟏] 𝒌 ≠ 𝒊 + 𝒍, 𝒌 ≠ 𝒊 − 𝒍, 𝒌 ∈ [𝟐, 𝒎 − 𝟏] if 𝒊 = 𝟏, 𝒌 ≠ 𝒎 others 𝒖, 𝒗𝒊,𝒋+𝟏, 𝒗𝒊,𝒋+𝟐, … , 𝒗𝒊,𝒎+𝟏, 𝑣𝒎+𝟏,𝒗𝒌,𝒎+𝟏, 𝒗𝒌,𝟏, 𝒗 𝒖, 𝒗𝒊,𝟏 , 𝒗𝒊,𝒎+𝟏, 𝒗𝒎+𝟏,𝒗𝒌,𝒎+𝟏, 𝒗𝒌,𝟏, 𝒗 so we conclude that the rainbow vertex-connection number of star fan graphs ariestha widyastuty bustan 116 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) ≤ 𝑚 + 2 (6) from equation (5) and (6), we have 𝑟𝑣𝑐 (𝑆(𝑚, 𝐹𝑛 , 𝑣𝑖,𝑗 )) = 𝑚 + 2 figure 4. 𝑆(6, 𝐹6, 𝑣𝑖,7) references [1] r. diestel, graph theory 4th edition, springer, 2010. [2] m. krivelevich and r. yuster, "the rainbow connection of a graph is (at reciprocal to its minimum degree," journal of graph theory, vol. 63, no. 2, pp 185-191, 2010. [3] x. li and s. liu, "rainbow vertex-connection number of 2-connected graphs," arxiv preprint arxiv: 1110.5770, 2011. [4] d. n. simamora and a. n. m. salman, "the rainbow (vertex) connection number of pencil graphs," procedia computer science, vol. 74, pp 138-142, 2015. [5] a. w. bustan, " bilangan terhubung titik pelangi untuk graf lingkaran bintang (smcn)," barekeng: jurnal ilmu matematika dan terapan, vol. 10, no. 2, pp. 77-81, 2016. cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 preface cauchy is a national journal published by mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang. this is the second issue of this year. it contains 6 (six) article from all over the country, not only from local area of east java. those articles covered numerical analysis, statistics, mathematical analysis, and decision making. this issue was authored by 15 authors and co-authors. in the first article, the author elaborated an alternative parameter estimation in heteroscedastic regression. it models the consumption patter of students in faculty of mathematics and natural science, brawijaya university. it also implemented the reml methods in modeling the students’ saving. the second article discussed about the metric dimension of some operation graphs. the paper showed that the local metric dimension of some graph operation such as joint graph, amalgamation of parachute, amalgamation of a fan and shack. the third article, entitled “simulation study of the implementation of quantile bootstrap method on autocorrelated error” proved that the estimated value with quantile regression was within the bootstrap percentile confidence interval. it also proved that 10 times replication resulted in a better estimation value compared to the other replication measures. the fourth article emphasized the discussion to the epidemic model with non-constant population. the results of local stability analysis indicated that if 𝑅0 < 1, then the disease-free equilibrium point 𝐸0 is locally asymptotically stable. it means if the terms fulfilled, then in a long time, there would have no spread of disease in susceptible and vaccinated subpopulations, or in other words, the epidemic will stop the fifth article discussed on the rainbow vertex-connection number of star fan graphs. it proved several theorems concerning to the rainbow connection number. the interesting part of this article that it generates general theorems in terms of the related graphs. in the sixth article, the authors take advantage of danp-topsis method to improve the guidance learning (lbb) satisfaction in malang city. the marketing strategy that needs to be improved on lbb avicenna education malang is cluster d4, which is classroom criteria, additional consultation, and activities outside kbm. by understanding the marketing strategy to be improved, it is expected to improve the quality of lbb customer satisfaction. the seventh article generated the simulation study of bayesian quantile regression use in nonnormal error. it introduced the ability of the bayesian quantile regression method in solving the problem of the non-normal errors. it also jumped to conclusion that the best regression equation model is bayesian quantile method used quantile value of 0.50. the eighth article, discussing structural equation modeling based on variance density index. it is implemented on larvae of the rainy season in the city of banjarbaru. the results of the study showed with sem based variance approach that this research model meets the required criteria. cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 preface the ninth article discussed geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan. it concluded that the prevalence of hypertension spread spatially because there are heterogenitas between the location of the observation that means that observations of a location depends on the observations in another location that the distance is near so do spatial regression modeling with adaptive gaussian kernel function, to result five groups. the tenth article elaborated the vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district. it concluded that the pattern of the number of the prevalence of malaria tends to be uniform from the four tribes, the number of the prevalence of malaria javanese there is a relationship with the other tribes and the number of the prevalence of malaria tribe bugis with other tribes. the last article entitled “the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan”, focused the discussion to health problem of arti. it concluded that the number of the prevalence of acute (rti) in south kalimantan province follow poisson distribution, so used poisson regression modeling. simulation study of the implementation of quantile bootstrap method on autocorrelated error cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 95-101 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 26 july 2018 reviewed: 10 october 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5349 simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri1, ferra yanuar2*, dodi devianto3 1,2,3department of mathematics, faculty of mathematics and natural science, andalas university, kampus limau manis, 25163, padang indonesia email: ovidelviyanti@gmail.com, ferrayanuar@sci.unand.ac.id, ddevianto@sci.unand.ac.id *corresponding author e-mail: ferrayanuar@sci.unand.ac.id abstract quantile regression is a regression method with the approach of separating or dividing data into certain quantiles by minimizing the number of absolute values from asymmetrical errors to overcome unfulfilled assumptions, including the presence of autocorrelation. the resulting model parameters are tested for accuracy using the bootstrap method. the bootstrap method is a parameter estimation method by resampling from the original sample as much as r replication. the bootstrap trust interval was then used as a algorithm consistency test constructed on the estimator by the quantile regression method. and test the uncommon quantile regression method with bootstrap method. the data obtained in this test were 10 times replication data. the biasness was calculated from the difference between the quantile estimate and bootstrap estimation. quantile estimation methods were said to be unbiased if the standard deviation bias was less than the standard bootstrap deviation. this study proved that the estimated value with quantile regression was within the bootstrap percentile confidence interval and proved that 10 times replication produces a better estimation value compared to other replication measures. quantile regression method in this study was also able to produce unbiased parameter estimation values. keywords: quantile regression; bootstrap method; autocorrelation error introduction regression analysis is a statistical analysis that is often used in all fields of science. this analysis aims to model the relationship between two types of variables, the dependent variable (y) with one or more independent variables (x) in a system. the relationship between these variables is expressed in a regression model which is generally stated as exfy  )( , by e stating the error component. the model connects independent and dependent variables through a parameter named as a regression parameter, denoted by  . regression model can be obtained by estimating the model parameters. to estimate the value of this regression parameter is usually used the least squares method (ols). this ols method is applied if some assumptions are met, such as there is no autocorrelation error, normal distribution error, there is no multicollinearity between the independent variables and homogeneous error [1]. all assumptions must be fulfilled so that the blue parameter estimators can be obtained (best linear unbiased estimator). but this method becomes no longer blue and is inefficient if there are unfulfilled assumptions. in many cases, not all assumptions are often met. the problem then is how to overcome this violated assumption. in this study the focus was on how to overcome the problem of data whose model errors were autocorrelated. autocorrelation is if the mailto:ovidelviyanti@gmail.com mailto:ferrayanuar@sci.unand.ac.id mailto:ddevianto@sci.unand.ac.id mailto:ferrayanuar@sci.unand.ac.id simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 96 remainder i  , 𝑖 = 1,2, . . . , 𝑛, correlated with each other. time-dependent data such as inflation, rupiah exchange rate, monthly exports, daily rainfall habits often have autocorrelation [2]. quantile regression method then appears to overcome the weaknesses in mkt. the quantile regression method was first introduced by koenker and bassett [3]. this method uses the parameter estimation approach by separating or dividing the data into quantiles by assuming the conditional quantile function on a distribution of the data and minimizes the absolute error of an asymmetrical weight. quantile regression analysis is used to overcome unfulfilled assumptions, including autocorrelation, normal assumptions, no multicollinearity and homogeneity of variance [4]. furthermore, the parameter values of the models produced using the quantile regression method must be tested for accuracy, which is to ensure that the values obtained have produced true values. as for previous researchers such as feng [5] studied the wild bootstrap in the quantile regression method on errors containing heteroskedacity. oktafia [6] examined the quantile regression parameter estimator simulation with the bootstrap method on errors containing heteroscedacity. this bootstrap method was a method of parameter estimation by re-sampling from the original sample [7]. the bootstrap method could generate statistical values to create a bootstrap trust interval [8]. this bootstrap confidence interval was then used as a consistency test statistic algorithm constructed on the estimator using the quantile regression method. thus, in this study bootstrap method was used in quantile regression to overcome errors that contained autocorrelation problems and applied on simulation data. methods 1. data simulation study was done by generating a set of data using r software. the data used in this study consisted of two independent variables ( 1 x and 2 x ) and one dependent variable ( y ). the data for the independent variables ( 1x dan 2x ) were each distributed according to the normal distribution ( 1x ∼  1,0n and 1x ∼  1,0n ), whereas the dependent variable ( y ) was set to the value  ttt xxy 21 25.0 where in tt zseq  )1.0,3.18,1.0(sin(  with t z ∼  1.0,0n , 183,...,3,2,1t . each variable was 183 pieces of data. 2. bootstrap quantile regression quantile regression was an approach in regression analysis introduced by koenker and basset [10]. this approach assumed the various quantile functions of a y distribution as a function of x. the use of this regression method was done by dividing or splitting data into several groups which having different estimations on quantile results. quantile function was denoted by 𝑄𝜃 with value 𝜃, 0 ≤ 𝜃 ≤ 1. for example 𝑌 was dependent variable, dan 𝑋 was independent variable dimension p. for example 𝐹𝑌(𝑦|𝑋 = 𝑥) = 𝑃(𝑌 ≤ 𝑦|𝑋 = 𝑥) was cumulative distribution function from 𝑌 given 𝑋 = 𝑥. the quantile function was defined as: 𝑄𝜃 (𝑌|𝑋 = 𝑥) = 𝑖𝑛𝑓{𝑦: 𝐹𝑌(𝑦|𝑥) ≥ 𝜃} (1) the estimation by the 𝜃-th quantile regression method was obtained by minimizing the weighted absolute sum of deviations. that was weighted sum of absolute simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 97 deviations, where a(1 − 𝜃)weight was assigned to the negative deviations and a 𝜃 weight was used for the positive deviations. the previous minimization problem became: 𝑄𝜃 = 𝑎𝑟𝑔𝑚𝑖𝑛𝛽𝜖ℝ𝑝 ∑ 𝜌𝜃 (𝑦𝑖 − 𝑄𝜃 (𝑌|𝑋)) 𝑛 𝑖=1 (2) can be written in the following equation: 𝑄𝜃 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑥𝑖 ′𝛽𝜃∈ℝ {∑ 𝜃|𝑦𝑖 − 𝑥𝑖 ′𝛽|𝑖𝜖𝑖|𝑦𝑖≥𝑥𝑖 ′𝛽𝜃 + ∑ (1 − 𝜃)|𝑦𝑖 − 𝑥𝑖 ′𝛽|𝑖𝜖𝑖|𝑦𝑖≥𝑥𝑖 ′𝛽𝜃 } (3) where 𝜌𝜃 denoted an asymmetric loss function from quantile regression and 𝜃 𝑤𝑎𝑠 quantile indeks∈ (0,1). quantile regression estimation analysis was performed by bootstrap method. the bootstrap method was a method used to estimate an unknown population distribution where the empirical distribution was obtained from the repeating process as much as r replication [6]. bootstrap method was a nonparametric approach to estimating various statistical quantiles such as mean, standard deviation, and estimation bias or to form confidence intervals by following certain algorithms. based on [5] the mean values of bootstrap parameters were:     q r b bq r     1 ˆ1ˆ (4) where   qb ̂ with rb ,....,2,1 . while the confidence interval used in this research was the percentile confidence interval. percentile method was based on the lower limit    lowqb ̂ percentile -th and based on the upper limit    upqb ̂ percentile  1 the cumulative distribution function of the estimated bootstrap vector parameters were:            1ˆ,ˆˆ,ˆ ff uplow (5) the indicators for the goodness of fit for parameter estimation were based on the smallest of confidence intervals and biasness. results and discussion simulation study was done by generating a set of data using r software. the data used in this study consisted of two independent variables ( 1x and 2x ) and one dependent variable ( y ). the data for the independent variables ( 1x dan 2x ) were each distributed according to the normal distribution ( 1x ∼  1,0n and 1x ∼  1,0n ), whereas the dependent variable ( y ) was set to the value  ttt xxy 21 25.0 for autocorrelated errors, the error setting becomes tt zseq  )1.0,3.18,1.0(sin(  with t z ∼  1.0,0n , 183,...,3,2,1t . each variable was 183 pieces of data. after the data is generated, the model parameters estimation with ols will be carried out on the simulation data and test the significance of each result parameter estimate. the following are the results of estimating the model parameters with ols. the resulting model parameters are tested for accuracy using the bootstrap method. the bootstrap method is a parameter estimation method by resampling from the original sample as much as r replication. the bootstrap trust interval was then used as a algorithm consistency test constructed on the estimator by the simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 98 quantile regression method. and test the uncommon quantile regression method with bootstrap method in the initial stage, the estimation of the model parameters with ols will be carried out on the simulation data and testing the significance of each result parameter estimate. the following are the results of estimating the model parameters with ols. table 1. estimation of model parameters with ols parameter coefficient standar deviation t-value p-value 0  0.0083 0.0528 0.1580 0.8750 1  0.5335 0.0505 10.5730 0.0000 2  1.9608 0.0527 37.2130 0.0000 the f test value on this simulation data is 732.5 with a value 05.0p . therefore, the p value is less than the real level 05.0 , so there is at least one significant independent variable in the model. the analysis continued with the t test where the value is presented in table 1. based on table 1 it can be seen that the estimated regression coefficients are for 1  and 2  significant at the level 05.0 , because the p-value is smaller than 05.0 . next estimation of the parameters by quantile regression based on equation (3) can be seen in table 2. table 2. quantile regression parameter estimation results quantile quantile regression parameter estimation 1  p-value 2  p-value 0.1 0.5513 0.0000 2.0149 0.0000 0.0366* 0.0382* 0.25 0.5714 0.0000 1.9681 0.0000 0.0733* 0.0766* 0.5 0.5883 0.0000 1.8656 0.0000 0.1083* 0.1131* 0.75 0.4868 0.0000 2.0523 0.0000 0.0581* 0.0606* 0.9 0.4797 0.0000 1.9829 0.0000 0.0201* 0.0210* *sd (standar deviation value) in table 2. it was found that the estimation of the regression parameters for 1 and 2  was significant at the level 05.0 for all quantiles. the value of the regression coefficient for 1x and 2x all was positive on each selected quantity means that these two independent variables significantly influenced y . simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 99 the analysis continued with the durbin watson test, where the dw value in the 0.1th quantile in this simulation data was 2.0355 with a value 05.0p . in the dw table with n = 183 and k = 2 and 7249.1 l d with 7915.1 u d . based on the durbin watson test if the dw value is located between u d and u d4 stated that uu dd  4 the model error does not contain autocorrelation. likewise, with the quantile 0.25, 0.5, 0.75 and 0.9. then the accuracy of the quantile regression method was tested in the quantiles 0.1, 0.25, 0.5, 0.75 and 0.9 using the bootstrap method for cases of autocorrelation errors. estimation of the value of the quantile regression parameter for each data by using bootstrap method, where the number of replications used was 10 times, 25 times, 50 times and 100 times. then do the bootstrap percentile confidence interval for each replication was according to equation (5). table 3. the result of estimation of quantile regression parameter with bootstrap method on quantile 0.1 replication parameter quantile regression estimation value mean of bootstrap confidence interval of bootstrap percentile lower limit upper limit 10 1  0.5513 0.4741 0.3657 0.5906 2  2.0149 1.9582 1.7967 2.1501 25 1  0.5513 0.4653 0.1746 0.6177 2  2.0149 1.9189 1.6560 2.1652 50 1  0.5513 0.5184 0.2128 0.7587 2  2.0149 1.8993 1.6544 2.1432 100 1  0.5513 0.5143 0.2650 0.8283 2  2.0149 1.9257 1.6997 2.2117 based on table 3. it was found that the value of the quantile regression estimation for all model parameters was within the confidence interval of the bootstrap percentile for each replication. furthermore, the estimation value of quantile regression parameter 1  and 2  for 10 times replication in a row that was 0.5513 and 2.0149 was close enough to the initial population model value that was set to 0.5 for the parameter value 1 and 2 for the parameter value 2 , as well as 25 times, 50 times and 100 times . similar results were also obtained on the 0.25, 0.5, 0.75 and 0.9 quantiles. then the difference between the two intervals from the bootstrap percentile confidence interval was calculated for each replication to select the best replication. according to [8] the best replication is the replication that has the smallest confidence interval difference and close to the replication value that has the smallest confidence interval gap and closer to the actual parameter estimation value. difference in bootstrap percentile confidence interval on autocorrelated error data for each quantile of the calculation results was presented in the following table table 4. difference between bootstrap percentile confidence intervals in quantile 0.1 simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 100 replication parameters difference between bootstrap percentile confidence intervals 10 1  0.2249 2  0.3533 25 1  0.4430 2  0.5092 50 1  0.5460 2  0.4888 100 1  0.5633 2  0.5120 based on table 4. it was found that the smallest bootstrap percentile confidence interval was on 10 times replication was 0.2249. thus, the best replication of the 0.1quantile for data with an autocorrelated error was 10 times replication. the same thing was done for each quantile, so it was obtained that for quantile 0.25, the best replication was 10 times replication, for quantile 0.5, the best replication was 10 times replication, for quantile 0.75, the best replication was 10 times replication, and for quantile 0.9, the best replication was 10 times replication. it can be concluded that the confidence interval of the smallest bootstrap percentile census tended to be 10 times replication. furthermore, test for the unbiased quantile regression method was done with bootstrap method. as for the data obtained in this test was the 10 times replication data, because the replication tended to produce the best estimation value. bias was calculated from the difference between quantile estimation and bootstrap estimation. quantile estimation method was said to be unbiased if the standard deviation bias was less than the standard bootstrap deviation. table 5. estimation of quantile regression parameters by determining unbiased quantiles sd* bootstrap of parameters bias of parameters sd (bias) 1  2  1  2 0.1 0.0756 0.1133 0.0772 0.0567 0.0145 0.25 0.0882 0.1236 0.0529 0.0327 0.0143 0.5 0.0538 0.0657 0.0186 0.0072 0.0057 0.75 0.0720 0.0527 0.0157 0.1727 0.1110 0.9 0.1442 0.1748 0.0822 0.0956 0.0095 based on table 5 it can be informed that the standard deviation bias on the 0.1 quantile was 0.0145. this value was less than the value of the bootstrap standard deviation value for 1 (0.0756) and also 2 (0.1133). this meant that the quintile estimation and bootstrap estimation resulted in an unbiased value in the 0.1 quintile. the simulation study of the implementation of quantile bootstrap method on autocorrelated error ovi delviyanti saputri 101 same way can be seen for the 0.25, 0.5, 0.75 and 0.9 quantiles. from the results of the analysis, it was found that the quantile estimation method produced an unbiased value. thus, it can be said that in this case, the quantile estimation method was able to produce unbiased and accurate values. conclusions quantile regression method used to model simulation data was accurate based on the accuracy test with the bootstrap method. this study proved that the estimated value with quantile regression was within the bootstrap percentile confidence interval. of several replication measures selected, 10, 25, 50 and 100 replications. this study proved that 10 times replication resulted in a better estimation value compared to the other replication measures. quantile regression method in this study was also able to produce unbiased parameter estimation values. based on the unbiased test, it was found that the standard deviation estimation result of the quantile regression method was smaller than the standard deviation of the bootstrap simulation result. references [1] bain, l. j., and engelhardt, m. 1992.introduction to probability and mathematical statistics second edition. duxbury press, california. [2] gujarati, d. n. 2003. basic econometrics. 4th edition. the mcgraw-hill companies, new jersey. [3] koenker, r., and basset, g. j. 1978. regression quantiles. econometrica, 46, 33-50. [4] koenker, r. 2005. quantile regression. cambridge university press, new york. [5] feng, x. 2011. wild bootstrap for quantile regression. journal biometrika 98, 4, 995-999. [6] oktafia, m., yanuar, f., and maiyastri. 2015. simulasi penduga parameter regresi kuantil dengan metode bootstrap. jurnal matematika unand, 5(1), 125-130. [7] efron, b., and tibshirani, j. r. 1993. an introduction to the bootstrap. new york: champman and hall, inc. [8] mansyur, a., efendi, s., firmansyah, and togi. 2010. estimator parameter model regresi linear dengan metode bootstrap. jurnal bulletin of mathematics, 03, 189202. fitting the first birth interval in indonesia using weibull proportional hazards model cauchy – jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 55-63 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 20 pebruary 2017 reviewed: 31 december 2017 accepted: 29 march 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.4060 fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk1, endro setyo cahyono2, ning eliyati3 1,2,3department of mathematics, faculty of mathematics and natural science, sriwijaya university email: alfensifaruk@unsri.ac.id, endrosetyo_c@yahoo.co.id, ningeliyati@gmail.com abstract the first birth interval (fbi) is defined as the duration of time spent by married couples to have their first child since the day of marriage. it is one of the indicators of women’s fertility rate. the long fbi indicates the low fertility rate. therefore, many governments of populous countries, such as indonesia, attempted to prolong the fbi in order to prevent their country from overpopulation. in order to control the factors related to the fbi, it is important to determine the most significant factors and to quantify the effects of the significant factors on the fbi data. these tasks can be conducted by using survival analysis techniques. in particular, this research proposes weibull proportional hazards (ph) model to analyze several socioeconomic and demographic factors, which may affect the fbi data in indonesia. the data were obtained from 2012 indonesian demographic and health survey (idhs) and consisted of 27488 ever married women aged 15-49 at the time of interview. in addition to the weibull ph model, the data were also analyzed by using the other approaches, namely kaplan-meier (km) method, logrank test, and ph model. the results indicated that the weibull ph model performs very well compared to ph model. moreover, the fitting of the weibull ph model showed that age at the first birth, place of residence, women’s educational level, husband’s educational level, contraceptive knowledge, wealth index, and employment status had the significant effects on the fbi data in indonesia. keywords: first birth interval, fertility rate, weibull ph model introduction indonesia is the most densely populated country in south east asia. according to statistics indonesia [1], there are about 259 million people of indonesia in 2016. this has made indonesia in the fourth rank as the world’s most populous country behind china, india, and the united states. however, the huge amount of the indonesian population is not along with the national economic condition which is still categorized as one of the third world countries. these facts show that the population density is still the main problem in indonesia and need to be resolved. fertility is one of the factors that influence the fluctuation of the number of population. one of the indicators of fertility rate is the total fertility rate (tfr), which can be defined as the average number of children that would be born to a woman over her reproductive age. according to islam [2], tfr can be reduced by increasing the age at marriage. however, this strategy is difficult to apply in the developed countries such as indonesia due to the influence of social and cultural factors. another alternative strategy is by controlling the fbi, which is defined as the time interval of a married woman to give birth her first child since the time of first marriage. if the fbi can be controlled, the next birth time is also automatically controlled [2]. many works employed survival analysis to evaluate the factors that affect the fbi. it due to the fact that the fbi data commonly contain censored observations, that is the subjects who did not give birth until the end of the observation time. consequently, the conventional statistical methods, such as multiple linear regression or logistic regression, are no longer appropriate to analyze the data set. otherwise, survival analysis provides various statistical tools to handle any types of censored data [3]. one of the examples is the work by hidayat et al. [4] which showed that the area of residence, education, and age were factors affecting the fbi in three provinces in indonesia. they applied the cox model and cox extended model in analyzing the fbi dataset. by using parametric models, shayan et al. [5] showed that age at marriage, level of women's mailto:alfensifaruk@unsri.ac.id mailto:endrosetyo_c@yahoo.co.id mailto:ningeliyati@gmail.com fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 56 education, and menstrual status had highly significant effects on the fbi in pakistan. in most recent work, faruk [6] concluded that age, women’s educational level, and wealth index significantly affect the fbi in south sumatra province, indonesia. in this study, the weibull ph model and ph model are considered to evaluate the socioeconomic and demographic factors that could significantly affect the fbi. comparisons were made to find the best multivariate model for the fbi data in indonesia. the weibull ph model was used since the weibull distribution is commonly fitted to survival data [7]. meanwhile, the ph model was presented because this model is the most applied model in the time-to-event study [8]. in this research, the dataset contains the sample from all 33 provinces in indonesia. the data, then, were also analyzed by using km method and logrank test in addition to weibull ph model and cox model. the data analysis in this research was supported by rstudio version 1.1.383. the remainder of this paper is organized as follows: the next section will outline the materials and methods of this research. in section 3, both univariate and multivariate survival analysis techniques such as the km method, the logrank test, the ph model, and the weibull ph model are implemented to analyze the fbi data in indonesia. finally, the last section presents the main conclusions of this work. materials and methods the fbi data were obtained from 2012 idhs. statistics indonesia and united states agency for international development (usaid) conducted the survey from may to june 2012. they used stratified two-stage cluster sampling technique to determine the sample of the study. in particular, the cluster blocks, which are based on the indonesia population census in 2010, are the first stage unit sampling. meanwhile, the households in each cluster block are the second stage unit sampling. the surveyors used questionnaires to gain the information needed from the respondents. in addition, only the data from the complete questionnaires are included in this research. the fbi (in months) is the survival time which was measured from the day of marriage up to the time of first give birth. moreover, the time was considered as the variable of interest. in the analysis, eight covariates which consisted of one continuous covariate and seven categorical covariates were used in this work. these covariates were socioeconomic and demographic characteristics of the respondents based on the 2012 idhs (table 1). in order to achieve the research objectives, this work consists of several steps. firstly, the km method was applied to the fbi data in order to estimate the survival and hazards functions. after that, the km estimation results were plotted based on the groups in each categorical covariate. then, the logrank test was applied to evaluate the differences between two or more survival functions in every categorical covariate. next, the ph model and the weibull ph model was fitted to the fbi data. finally, the performance of the ph model and the weibull ph was evaluated by using akaike information criterion (aic). results and discussion 3.1. description of the dataset a total of 27448 ever married women aged 15-49 at the time of interview were included in this research. of total of respondents, there were 25564 women who gave birth and 1884 women who did not give birth until the end of observation time. the latter were categorized as the right censored survival data. the summary of the frequency based on the characteristics and the status of the survival time can be seen in table 1. table 1. descriptive statistics of covariates and their survival time status covariate categories frequency status #censored #event age at the first birth 27448 1884 25564 fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 57 place of residence rural 14522 944 13578 urban 12926 940 11986 women's educational level no education 1121 93 1028 primary 10238 519 9719 secondary & above 16089 1272 14817 husband's educational level no education 752 71 681 primary 9733 520 9213 secondary & above 16963 1293 15670 contraceptive knowledge no 435 114 321 yes 27013 1770 25243 access to mass media no 2630 203 2427 yes 24818 1681 23137 wealth index poor 12388 893 11495 middle & above 15060 991 14069 employment status no 10984 730 10254 yes 16464 1154 15310 3.2. the estimation of survival function by using km method let 𝑇 be a continuous random variable of survival times and 𝑡 ≥ 0 is the realization of 𝑇, then the probability density function of 𝑇 is 𝑓(𝑡) = 𝑙𝑖𝑚 ∆𝑡→0 pr(𝑡<𝑇<𝑡+∆𝑡) ∆𝑡 , (1) where 𝑓(𝑡) is a nonnegative function and also known as the unconditional failure rate [9]. the survival function, denoted by 𝑆 (𝑡), is defined as the probability of the occurrence of an event at more than time 𝑡 and can be formulated by 𝑆(𝑡) = pr(𝑇 > 𝑡) = ∫ 𝑓(𝑥)𝑑𝑥 = 1 − 𝐹(𝑡) ∞ 𝑡 , (2) where 𝐹(𝑡) is a cumulative distribution function at time 𝑡. 𝑆(𝑡) is a nondecreasing function and has the properties 𝑆(𝑡) = 1 at 𝑡 = 0 and 𝑆(𝑡) = 0 at 𝑡 = ∞. to estimate survival function 𝑆(𝑡) by using km method, the 𝑛 survival times (include censored time) were firstly arranged in order of increasing magnitude such that 𝑡1 ≤ 𝑡2 ≤ ⋯ ≤ 𝑡𝑛. then, the survival function can be estimated using the formula �̂�(𝑡) = ∏ 𝑛−𝑟 𝑛−𝑟+1𝑡𝑟≤𝑡 , (3) where 𝑡𝑟 is uncensored time for 𝑟 ∈ {1,2, … , 𝑛}, and 𝑛 − 𝑟 is the number of individuals in the sample surviving longer than time 𝑡𝑟 . in this work, the km estimate of the survival function, �̂�(𝑡), was estimated by using equation (3) and conducted by using rstudio. the estimation results for each group in categorical covariates of the fbi dataset in indonesia were plotted in figure 1. fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 58 figure 1. km plots for survival of fbi in indonesia according to the plots in figure 1, the obvious difference between survival curves can be seen in the survival curves which based on women’s educational level, husband’s educational level, contraceptive knowledge, access to mass media, and wealth index. meanwhile, the difference of the survival curves which based on the other covariates such as place of residence and employment status is not obvious due to the estimate curves are very close each other. therefore, a statistical test is needed to obtain more objective conclusion about the difference among the survival curves. in this work, the logrank test is applied to test the difference between two or more survival curves. 3.3. comparing survival functions by using the logrank test in addition to km plot, a statistical test is needed to obtain more convincing conclusion about the difference between the survival curves in each categorical covariates. for comparing two survival groups, the null hypothesis is h0 : 𝑆1(𝑡) = 𝑆2(𝑡) (no difference between two survival curves) with the test statistic (𝑂2−𝐸2) 2 𝑉𝑎𝑟 (𝑂2−𝐸2) ~𝜒𝑑𝑓;𝛼 2 , (4) fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 59 where 𝑂2 is the total number of failures in group 2, 𝐸2 is the expected number of failures in group 2, 𝑑𝑓 is degree of freedom which equals to 1, and 𝛼 is level of significance. the null hypothesis (h0) will be rejected if the 𝜒 2 > 𝜒1;𝛼 2 or p-value < 𝛼. to compare the survival curves for more than two groups, the null hypothesis is h0 : 𝑆1(𝑡) = 𝑆2(𝑡) = ⋯ = 𝑆𝑛 (𝑡) (survival curves are similar) and the test statistic is [10] ∑ (𝑂𝑖−𝐸𝑖) 2 𝐸𝑖 𝑛 𝑖=1 ~𝜒𝑑𝑓;𝛼 2 , (5) where 𝑂𝑖 is the total number of failures in group 𝑖, 𝐸𝑖 is the expected number of failures in the group 𝑖, 𝑛 is the total number of groups, and 𝑑𝑓 is degree of freedom which equal to 𝑛 − 1. if the 𝜒2 > 𝜒𝑛−1;𝛼 2 or p-value < 𝛼, h0 will be rejected. by using rstudio with 5% level of significance, the results of the logrank test between two or more survival curves of the fbi dataset in indonesia are summarized in table 2. table 2. results of the logrank tests covariate categories df 𝝌𝟐 𝝌𝒅𝒇;𝟎,𝟎𝟓 𝟐 p-value conclusio n place of residence rural 1 8 0 3.481 0 reject h0 urban women's educational level no education 2 4 8 6 5.591 0 reject h0 primary secondary & above husband's educational level no education 2 3 2 3 5.591 0 reject h0 primary secondary & above contraceptive knowledge no 1 1 8 5 3.481 0 reject h0 yes access to mass media no 1 9 0 . 5 3.481 0 reject h0 yes wealth index poor 1 1 6 2 3.481 0 reject h0 middle & above employment status no 1 4 6 . 6 3.481 8.27e12 reject h0 yes as shown in table 2, the 𝜒2 scores are more than 𝜒𝑑𝑓;0.05 2 and p–values for all logrank tests of each categorical covariate are less than the level of significance (𝛼) = 0.05. therefore, the null hypothesis h0 can be rejected at the 5% level of significance. in other words, the difference of survival curves between categories for every categorical covariate based on the logrank test is statistically significant. the results are quite different with the work by gebeyehu [11], which concluded that the difference between survival curves based on access to mass media was insignificant. 3.4. fitting ph model ph model is represented by the relationship of the hazard function, the baseline hazard function, and one or more covariates in the form ℎ(𝑡) = ℎ0(𝑡) exp(𝛃 t𝐗), (6) where ℎ(𝑡) is the hazard function, ℎ0(𝑡) is the baseline hazard function which is left unspecified, 𝛃 is a column vector of the regression coefficients, and 𝐗 is a column vector of the covariates. the ph model assumes that there is proportionality of the hazard rate between any two individuals in the population. the hazard ratio is defined as the ratio of the hazard functions for two subjects with different values of covariate 𝑋1 and 𝑋2. the formula of the hazard ratio is 𝐻(𝑡) = ℎ0(𝑡) exp(𝛽2𝑋2) ℎ0(𝑡) exp(𝛽1𝑋1) = exp(𝛽2𝑋2) exp(𝛽1𝑋1) = 𝑒 𝛽 ′(𝑋2−𝑋1). (7) fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 60 it can be seen that the hazard ratio in equation (7) is independent of time. in other words, the hazard ratio for two individuals is constant over time. to estimate the regression coefficients of ph model, the partial likelihood method, which uses the partial likelihood function instead of the likelihood function as in the maximum likelihood estimation (mle) method [12], should be used. the results of fitting the ph model to the fbi data in indonesia in case of full model can be seen in table 3. this fitting tests the hypothesis h0 : �̂�𝑖 = 0, ∀𝑖 = 1 … 𝑛 (covariate 𝑋𝑖 is not statistically significant). the hypothesis h0 is rejected if p-value < level of significance (𝛼). table 3. results of fitting the ph model covariate category 𝑬𝒙𝒑 (�̂�) p-value age at the first birth 1.004 0.005 a place of residence rural b urban 0.993 0.604 women's educational level no education b primary 1.079 0.032 a secondary & above 1.267 1.97e-10 a husband's educational level no education b primary 1.151 0.001 a secondary & above 1.224 3.14e-06 a contraceptive knowledge no b yes 1.706 < 2e-16 a access to mass media no b yes 1.017 0.463 wealth index poor b middle & above 1.045 0.002 a employment status no b yes 0.938 5.53e-07 a a: significant at 10% level , b: reference category in the results of fitting ph model, as shown in table 3, place of residence and access to mass media are two covariates which do not significantly affect the survival time at 10% level. meanwhile, the rest of covariates have the significant effects on the fbi data in indonesia at 10% level. 3.5. fitting weibull proportional hazards model in cox ph model, the baseline hazards (ℎ0(𝑡)) is left unspecified. however, if ℎ0(𝑡) is assumed to follow a certain survivor distribution, the model is called as parametric ph model. this model still holds the ph assumption as in the cox ph, but it is a full parametric model instead of semiparametric. to estimate the unknown parameters, maximum likelihood estimation (mle) method can be applied. among the survivor distributions, weibull distribution is the most popular distribution. it because the survival and hazards functions of the weibull distribution have the closed forms [13]. the ph model with a weibull baseline distribution is given by ℎ(𝑡) = 𝑝 𝜆 ( 𝑡 𝜆 ) 𝑝−1 exp(𝛃t𝑿), (8) where ℎ(𝑡) is the hazard function at time 𝑡 > 0, 𝑝 is shape parameter, 𝜆 is scale parameter, 𝛃 is a column vector of the regression coefficients, and 𝐗 is a column vector of the covariates. in rstudio, fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 61 the estimation of equation (8) can be employed by using ‘eha’ package and phreg() function. the results of fitting the equation (8) to the fbi data are summarized in table 4. table 4. results of fitting the weibull ph model covariate category 𝑬𝒙𝒑 (�̂�) p-value age at the first birth 1.002 0.063 a place of residence rural b urban 0.958 0.002 a women's educational level no education b primary 1.145 0.000 a secondary & above 1.385 0.000 a husband's educational level no education b primary 1.259 0.000 a secondary & above 1.358 0.000 a contraceptive knowledge no b yes 2.054 0.000 a access to mass media no b yes 1.023 0.329 wealth index poor b middle & above 1.030 0.039 a employment status no b yes 0.901 0.000 a a:significant at 10% level , b:reference category according to table 4, access to mass media is insignificant at 10% level and along with the results of ph model (table 3). meanwhile, the significance of place of residence is not similar to the results in table 3, which concluded that place of residence is insignificant at 10% level. in addition, the results in table 4 are also not along with the results of the work by [6] although it is consistent with the results of [11]. in order to assess the performance of ph model and weibull ph model, it is common to use the aic value. a regression model is said the better model if it has the lower aic value than the other models. by using ‘suvival’ package with step() function in rstudio, it is obtained that the aic values of ph model and weibull ph model for the last step (best model) of backward stepwise procedure are equal to 477946.9 and 221723.4, respectively. it means that the performance of weibull ph model is better than the ph model in fitting the fbi data in indonesia. the best model of weibull ph model based on the variable selection procedure is given in table 5. table 5. the best model of weibull ph model using backward stepwise procedure covariate category 𝑬𝒙𝒑 (�̂�) p-value age at the first birth 1.002 0.065 a place of residence rural b urban 0.959 0.003 a women's educational level no education b primary 1.149 0.000 a secondary & above 1.392 0.000 a fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 62 husband's educational level no education b primary 1.263 0.000 a secondary & above 1.362 0.000 a contraceptive knowledge no b yes 2.069 0.000 a wealth index poor b middle & above 1.033 0.026 a employment status no b yes 0.901 0.000 a a:significant at 10% level , b:reference category all covariates, that included in the best model in table 5, are already significant at 10% level. the significant covariates based on the backward stepwise are similar with the results in table 4. the clear difference between the models in table 4 and table 5 is that the best model excludes access to mass media from the best model. the exclusion of such covariate decreases the aic value of the model in the previous step of the backward elimination procedure. conclusions although the km method and logrank test are commonly used in the applications of survival analysis, they cannot account the effects of multiple covariates on the survival time simultaneously. therefore, a typical survival regression model is needed to overcome such drawback. in this work, the ph model and weibull ph model were applied to determine and quantify the effects of several socioeconomic and demographic factors on the fbi data in indonesia. the model comparison results showed that the weibull ph model was better than the ph model to fit the data. moreover, the fitting of weibull ph model by using backward stepwise procedure concluded that age at the first birth, place of residence, women’s educational level, husband’s educational level, contraceptive knowledge, wealth index, and employment status had the significant effects on the fbi data in indonesia. in this article, the ph assumption was assumed to the data. however, this assumption was not checked yet. further research should account for this by using a graphical method or goodness of fit tests. if the data do not hold the ph assumption, further research also should consider the alternative models, such as accelerated failure time (aft) model or cox extended model. another extension is also can be made by also accounting the individual heterogeneity in the data by using frailty model. acknowledgments this project was sponsored by sriwijaya university through penelitian sains teknologi dan seni (sateks) 2017. the authors are thankful to united states agency for international development (usaid) for contributing materials for use in the project. references [1] statistics indonesia, statistik indonesia 2016 statistical yearbook of indonesia 2016. statistics indonesia, 2016. [2] m. s. islam, “differential determinants of birth spacing since marriage to first live birth in rural bangladesh,” pertanika journal of social science and humanities, vol. 17, no. 1, pp. 1– 6, 2009. [3] i. gijbels, “censored data,” wiley interdisciplinary reviews: computational statistics, vol. 2, no. 2, pp. 178–188, 2010. [4] r. hidayat, h. sumarno, and e. h. nugrahani, “survival analysis in modeling the birth fitting the first birth interval in indonesia using weibull proportional hazards model alfensi faruk 63 interval of the first child in indonesia,” open journal of statistics, vol.4, pp. 198–206, 2014. [5] z. shayan, s. m. t. ayatollahi, n. zare, and f. moradi, “prognostic factors of first birth interval using the parametric survival models,” iranian journal of reproductive medicine, vol. 12, no. 2, pp. 125–130, 2014. [6] a. faruk, “aplikasi regresi cox pada selang kelahiran anak pertama di provinsi sumatera selatan,” jurnal matematika, vol. 7, no. 1, pp. 19–29, 2017. [7] n. juhan, n. a. razak, y. z. zubairi, n. n. naing, c. h. c. hussin, and m. a. daud, “comparison of stratified weibull model and weibull accelerated failure time (aft) model in the analysis of cervical cancer survival,” jurnal teknologi (sciences & engineering), vol. 78, pp. 21–26, 2016. [8] a. faruk, a. amran, and n. nasir, “aplikasi model proportional hazard cox pada waktu tunggu kerja lulusan jurusan matematika fakultas mipa universitas sriwijaya,” jurnal penelitian sains, vol. 17, no. 1, pp. 5–8, 2014. [9] e. t. lee and j. w. wang, statistical methods for survival data analysis, third edit., john wiley & sons, inc., 2003. [10] d. g. kleinbaum and m. klein, survival analysis a self-learning text, second edi., springer, 2005. [11] a. gebeyehu, survival analysis of time-to-first birth after marriage among women in ethiopia : application of parametric shared frailty model., 2015. [12] a. faruk, “estimasi parameter data tersensor tipe i berdistribusi log-logistik menggunakan maximum likelihood estimate dan iterasi newton-rhapson,” prosiding seminar nasional mipa 2014, pp. 21–25, 2014. [13] g. broström, event history analysis with r., crc press, 2012. issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 cauchy vol. 5 no. 3 pages: 80 – 168 malang november 2018 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 editorial board elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 table of contents restricted maximum likelihood method as an alternative parameter estimation in heteroscedastic regression ...................................................................... 80 – 87 on the metric dimension of some operation graphs ..................................................... 88 – 94 simulation study the implementation of quantile bootstrap method on autocorrelated error .............................................................................................................. 95 – 101 svir epidemic model with non constant population ...................................................... 102 – 111 the rainbow vertex-connection number of star fan graphs ..................................... 112 – 116 improving the guidance learning (lbb) consumer satisfaction in malang using danp topsis method .......................................................................................................... 117 – 120 simulation study the using of bayesian quantile regression in nonnormal error .......................................................................................................................................................... 121 – 126 structural equation modeling based on variance the density index of larvae of the rainy season in the city of banjarbaru ................................................................... 127 – 139 geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan ........................................................... 140 – 149 vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district ............................................................................................................. 150 – 160 the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantans . 161 – 168 http://ejournal.uin-malang.ac.id/index.php/math/article/view/3777 http://ejournal.uin-malang.ac.id/index.php/math/article/view/3777 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5331 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5349 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5349 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5511 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5516 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5541 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5541 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5633 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5877 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5877 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5879 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5879 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5880 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5880 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5881 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5881 cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the journal editor, the peer reviewer, the publisher and the society. as publisher of pure and applied mathematics journal, we take our duties to back up over all stages of publishing seriously and we recognize our ethical and other responsibilities. we are committed to ensuring that advertising, reprint or other commercial revenue has no impact or influence on editorial decisions. publication decisions the editor of cauchy is responsible for deciding which of the articles submitted to the journal should be published. the validation of the work in question and its importance to researchers and readers must always drive such decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics contribution to editorial decisions peer review assists the editor in making editorial decisions and through the editorial communications with the author may also assist the author in improving the paper. promptness any selected referee who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the editor and excuse himself from the review process. standards of objectivity reviews should be conducted objectively. personal criticism of the author is inappropriate. referees should express their views clearly with supporting arguments. acknowledgement of sources reviewers should identify relevant published work that has not been cited by the authors. any statement that an observation, derivation, or argument had been previously reported should be accompanied by the relevant citation. a reviewer should also call to the editor's attention any substantial similarity or overlap between the manuscript under consideration and any other published paper of which they have personal knowledge. disclosure and conflict of interest privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers. reporting standards authors of reports of original research should present an accurate account of the work performed as well as an objective discussion of its significance. underlying data should be represented accurately in the paper. a paper should contain sufficient detail and references to permit others to replicate the work. fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. data access and retention authors are asked to provide the raw data in connection with a paper for editorial review and should be prepared to provide public access to such data (consistent with the alpspstm statement on data and databases), if practicable, and should in any event be prepared to retain such data for a reasonable time after publication. originality and plagiarism the authors should ensure that they have written entirely original works, and if the authors have used the work and/or words of others that this has been appropriately cited or quoted. cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics multiple, redundant or concurrent publication an author should not in general publish manuscripts describing essentially the same research in more than one journal or primary publication. submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. acknowledgement of sources proper acknowledgment of the work of others must always be given. authors should cite publications that have been influential in determining the nature of the reported work. authorship of the paper authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. all those who have made significant contributions should be listed as co-authors. where there are others who have participated in certain substantive aspects of the research project, they should be acknowledged or listed as contributors. the corresponding author should ensure that all appropriate co-authors and no inappropriate co-authors are included on the paper, and that all co-authors have seen and approved the final version of the paper and have agreed to its submission for publication. hazards and human or animal subjects if the work involves chemicals, procedures or equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated abdussakir (universitas islam negeri maulana malik ibrahim malang) fachrur rozi (universitas islam negeri maulana malik ibrahim malang) sri harini (universitas islam negeri maulana malik ibrahim malang) usman pagalay (universitas islam negeri maulana malik ibrahim malang) cauchy jurnal matematika murni dan aplikasi volume 5, issue 2, may 2018 issn : 2086-0382 e-issn : 2477-3344 preface cauchy is a national journal published by mathematics department, science and technology faculty, maulana malik ibrahim state islamic university of malang. this is the second issue of this year. it contains 6 (six) article from all over the country, not only from local area of east java. those articles covered numerical analysis, statistics, mathematical analysis, and decision making. this issue was authored by 15 authors and co-authors. in the first article, the author discussed a discrete numerical scheme of modified leslie-gower with harvesting model. from this paper, a discrete numerical scheme of modified leslie-gower with harvesting model has been investigated. the author used euler-method to discretize the model. it is shown that the discrete model is dynamically consistent with its continuous model only for relatively small step-size. the second article discussed about comparing inverse distance weighted and natural neighbour interpolation method on air temperature in malang. the research concluded that based on the rmse value, the idw interpolation method with the power parameter 2 yields much more accurate prediction values than the natural neighbor interpolation method. areas with low air temperatures are located around kota batu and kecamatan poncokusumo, while areas with high temperatures are in the vicinity of malang and bantur subdistricts. the third article, entitled “fitting the first birth interval in indonesia using weibull proportional hazards model” elaborated that although the km method and logrank test are commonly used in the applications of survival analysis, they cannot account the effects of multiple covariates on the survival time simultaneously. therefore, a typical survival regression model is needed to overcome such drawback. in this article, the ph assumption was assumed to the data. however, this assumption was not checked yet. the author suggested for further research to account for this by using a graphical method or the advantages of fit tests. the author also suggested to consider the alternative models, such as accelerated failure time (aft) model or cox extended model. the fourth article emphasized the discussion to the solution of elliptic equations using positive definite matrix coefficient. the research came to the result that by determining the radial solution, the used method is able to solve the equation into ordinary differential equations. the fifth article discussed on the selection of ideal contraceptive tool using hybrid entropy-topsis method. the research obtained work rating matrix. the final determination entropy value was used to rank alternatives using the hybrid entropy-topsis method. the ranking results indicate the pills and implants ae the alternative selection of the ideal contraceptives. in the last article, the authors take advantage of bayesian structural equation modeling approach for the construction of patient loyalty model. in this research, the author used the analysis technique of bayesian sem approach to analyze the relationship between service quality and patient satisfaction to patient loyalty at puskesmas in padang city. based on the model, it is found that the service quality and patient satisfaction is significantly correlated to the patient loyalty on health services and the quality of service and patient satisfaction is an indicator to measure patient loyalty at puskesmas in padang city. cauchy jurnal matematika murni dan aplikasi volume 6, issue 1, november 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the journal editor, the peer reviewer, the publisher and the society. as publisher of pure and applied mathematics journal, we take our duties to back up over all stages of publishing seriously and we recognize our ethical and other responsibilities. we are committed to ensuring that advertising, reprint or other commercial revenue has no impact or influence on editorial decisions. publication decisions the editor of cauchy is responsible for deciding which of the articles submitted to the journal should be published. the validation of the work in question and its importance to researchers and readers must always drive such decisions. the editors may be guided by the policies of the journal's editorial board and constrained by such legal requirements as shall then be in force regarding libel, copyright infringement and plagiarism. the editors may confer with other editors or reviewers in making this decision. fair play an editor at any time evaluates manuscripts for their intellectual content without regard to race, gender, sexual orientation, religious belief, ethnic origin, citizenship, or political philosophy of the authors. confidentiality the editor and any editorial staff must not disclose any information about a submitted manuscript to anyone other than the corresponding author, reviewers, potential reviewers, other editorial advisers, and the publisher, as appropriate. any manuscripts received for review must be treated as confidential documents. they must not be shown to or discussed with others except as authorized by the editor. disclosure and conflicts of interest unpublished materials disclosed in a submitted manuscript must not be used in an editor's own research without the express written consent of the author. cauchy jurnal matematika murni dan aplikasi volume 6, issue 1, november 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics contribution to editorial decisions peer review assists the editor in making editorial decisions and through the editorial communications with the author may also assist the author in improving the paper. promptness any selected referee who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the editor and excuse himself from the review process. standards of objectivity reviews should be conducted objectively. personal criticism of the author is inappropriate. referees should express their views clearly with supporting arguments. acknowledgement of sources reviewers should identify relevant published work that has not been cited by the authors. any statement that an observation, derivation, or argument had been previously reported should be accompanied by the relevant citation. a reviewer should also call to the editor's attention any substantial similarity or overlap between the manuscript under consideration and any other published paper of which they have personal knowledge. disclosure and conflict of interest privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers. reporting standards authors of reports of original research should present an accurate account of the work performed as well as an objective discussion of its significance. underlying data should be represented accurately in the paper. a paper should contain sufficient detail and references to permit others to replicate the work. fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. data access and retention authors are asked to provide the raw data in connection with a paper for editorial review and should be prepared to provide public access to such data (consistent with the alpspstm statement on data and databases), if practicable, and should in any event be prepared to retain such data for a reasonable time after publication. originality and plagiarism the authors should ensure that they have written entirely original works, and if the authors have used the work and/or words of others that this has been appropriately cited or quoted. cauchy jurnal matematika murni dan aplikasi volume 6, issue 1, november 2019 issn : 2086-0382 e-issn : 2477-3344 publication etics multiple, redundant or concurrent publication an author should not in general publish manuscripts describing essentially the same research in more than one journal or primary publication. submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. acknowledgement of sources proper acknowledgment of the work of others must always be given. authors should cite publications that have been influential in determining the nature of the reported work. authorship of the paper authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. all those who have made significant contributions should be listed as co-authors. where there are others who have participated in certain substantive aspects of the research project, they should be acknowledged or listed as contributors. the corresponding author should ensure that all appropriate co-authors and no inappropriate co-authors are included on the paper, and that all co-authors have seen and approved the final version of the paper and have agreed to its submission for publication. hazards and human or animal subjects if the work involves chemicals, procedures or equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 6, issue 1, november 2019 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated kartick chandra mondal (jadavpur university, salt lake campus, india) abdussakir (universitas islam negeri maulana malik ibrahim malang) eridani (airlangga university) sri harini (universitas islam negeri maulana malik ibrahim malang) usman pagalay (universitas islam negeri maulana malik ibrahim malang) edy tri baskoro (institut teknologi bandung) diterbitkan oleh program studi matematika fsaintek secara berkala terbit 2 kali setahun pada bulan juni dan desember yang m jurnal matematika murni dan aplikasi issn 2086-0382 e-issn 2477-3344 cauchy publication ethics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. peer-reviewed articles support and embody the scientific method. it is therefore important to agree upon standards of expected ethical behavior for all parties involved in the act of publishing: the author, the 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university of malang, indonesia  abdussakir abdussakir, maulana malik ibrahim state islamic university of malang, indonesia volume 5, issue 1, november 2017 javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5449') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5448') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5443') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5437') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5047') javascript:openrtwindow('http://ejournal.uin-malang.ac.id/index.php/math/about/editorialteambio/5055') a discrete numerical scheme of modified leslie-gower with harvesting model cauchy –jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 42-47 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 8 january 2017 reviewed: 14 march 2018 accepted: 10 april 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.4716 a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah1, trija fayeldi2 1,2 department of mathematics education, kanjuruhan university email: ky2_zahra@unikama.ac.id, trija_fayeldi@unikama.ac.id abstract recently, exploitation of biological resources and the harvesting of two populations or more are widely practiced, such as fishery or foresty. the simplest way to describe the interaction of two species is by using predator prey model, that is one species feeds on another. the leslie-gower predator prey model has been studied in many works. in this paper, we use euler method to discretisize the modified leslie-gower with harvesting model. the model consists of two simultanious predator prey equations. we show numerically that this discrete numerical scheme model is dynamically consistent with its continuous model only for relatively small step-size. by using computer simulation software, we show that equlibrium points can be stable, saddles, and unstable. it is shown that the numerical simulations not only illustrate the results, but also show the rich dynamics behaviors of the discrete system. keywords: leslie-gower; euler method; rich dynamics; dynamically consistent introduction for many years, the predator-prey relationship has been the most popular topics in mathematical modelling. one of the model is known as leslie-gower predator-prey model. it was introduced by [1] and written as follow. 𝑑𝑥 𝑑𝑡 = (𝑟1 − 𝑎1𝑥)𝑥 − 𝑝(𝑥)𝑦 𝑑𝑦 𝑑𝑡 = (𝑟2 − 𝑎2𝑦 𝑥 ) 𝑦 (1) where 𝑥(𝑡) represents the density of the prey at time t with intrinsic growth rate 𝑟1; while 𝑦(𝑡) represents the density of the predator at time t with intrinsic growth rate 𝑟2. 𝑎1 and 𝑎2 are the strength of competition among individuals of the prey and predator respectively. the main difference between leslie-gower model and lotka-volterra that is in leslie-gower model carrying capacity of the predator is proportional to the number of prey [2]. since then, the model has been modified by numerous experts, such as [3] and [4]. finite difference has been used to discretize the continous model to find its numerical solution approximation. the common method is euler method [5]. this technique has been applied to various models, for instance [6], [7] , [8] , and [9]. it is found that the obtained discrete model may show rich dynamics. however, divergent approximation may be occur. hence, stepsize plays an important role to avoid inconsistency of the model. mailto:zahra@unikama.ac.id mailto:fayeldi@unikama.ac.id a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah 43 methods numerical discretization and its equilibria the model we use in this paper is written as follow. 𝑑𝐻 𝑑𝑡 = (𝑟1 − 𝑎1𝑃 − 𝑏1𝐻)𝐻 − 𝑐1𝐻 𝑑𝑃 𝑑𝑡 = (𝑟2 − 𝑎2𝑃 1+𝐻 ) 𝑃 − 𝑐2𝑃 (2) where h and p represent the density of the prey and predator in time t respectively. in this model, we also assume that 0 < 𝑐𝑖 < 𝑟𝑖 , 𝑖 = 1,2 to control the density of the prey and predator. we are now derive the discrete version of the (2). applying euler scheme to the first equation of (2) leads us to the following form: 𝐻𝑛+1−𝐻𝑛 ℎ = (𝑟1 − 𝑎1𝑃𝑛 − 𝑏1𝐻𝑛)𝐻𝑛 − 𝑐1𝐻𝑛 𝐻𝑛+1 − 𝐻𝑛 = ℎ[(𝑟1 − 𝑎1𝑃𝑛 − 𝑏1𝐻𝑛)𝐻𝑛 − 𝑐1𝐻𝑛 ] 𝐻𝑛+1 = 𝐻𝑛 + ℎ[(𝑟1 − 𝑎1𝑃𝑛 − 𝑏1𝐻𝑛 )𝐻𝑛 − 𝑐1𝐻𝑛] 𝐻𝑛+1 = 𝐻𝑛 + ℎ𝐻𝑛 [(𝑟1 − 𝑎1𝑃𝑛 − 𝑏1𝐻𝑛)𝐻𝑛 − 𝑐1] (3) applying euler scheme to the second equation of (2) leads us to the following form: 𝑃𝑛+1−𝑃𝑛 ℎ = (𝑟2 − 𝑎2𝑃𝑛 1+𝐻𝑛 ) 𝑃𝑛 − 𝑐2𝑃𝑛 𝑃𝑛+1 − 𝑃𝑛 = ℎ [(𝑟2 − 𝑎2𝑃𝑛 1+𝐻𝑛 ) 𝑃𝑛 − 𝑐2𝑃𝑛 ] 𝑃𝑛+1 = 𝑃𝑛 + ℎ [(𝑟2 − 𝑎2𝑃𝑛 1+𝐻𝑛 ) 𝑃𝑛 − 𝑐2𝑃𝑛] 𝑃𝑛+1 = 𝑃𝑛 + ℎ𝑃𝑛 [(𝑟2 − 𝑎2𝑃𝑛 1+𝐻𝑛 ) − 𝑐2] (4) combining (3) and (4) leads us to the following system: 𝐻𝑛+1 = 𝐻𝑛 + ℎ𝐻𝑛 [(𝑟1 − 𝑎1𝑃𝑛 − 𝑏1𝐻𝑛)𝐻𝑛 − 𝑐1] 𝑃𝑛+1 = 𝑃𝑛 + ℎ𝑃𝑛 [(𝑟2 − 𝑎2𝑃𝑛 1+𝐻𝑛 ) − 𝑐2] (5) having found the discrete version, the next step is find the equilibria of (5). suppose that (𝐻∗, 𝑃∗) is the equilibrium of (5), then we have 𝐻∗ = 𝐻∗ + ℎ𝐻∗((𝑟1 − 𝑎1𝑃 ∗ − 𝑏1𝐻 ∗) − 𝑐1) 0 = ℎ𝐻∗[(𝑟1 − 𝑎1𝑃 ∗ − 𝑏1𝐻 ∗) − 𝑐1] (6) since step-size ℎ > 0 then we have 0 = 𝐻∗((𝑟1 − 𝑎1𝑃 ∗ − 𝑏1𝐻 ∗) − 𝑐1) (7) that is either 𝐻∗ = 0 (8) a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah 44 or 0 = (𝑟1 − 𝑎1𝑃 ∗ − 𝑏1𝐻 ∗) − 𝑐1 𝑏1𝐻 ∗ = 𝑟1 − 𝑎1𝑃 ∗ − 𝑐1 𝐻∗ = 𝑟1−𝑎1𝑃 ∗−𝑐1 𝑏1 (9) on the other hand we have 𝑃∗ = 𝑃∗ + ℎ𝑃∗ [(𝑟2 − 𝑎2𝑃 ∗ 1+𝐻∗ ) − 𝑐2] 0 = ℎ𝑃∗ [(𝑟2 − 𝑎2𝑃 ∗ 1+𝐻∗ ) − 𝑐2] (10) since h > 0 then we have 0 = 𝑃∗ [(𝑟2 − 𝑎2𝑃 ∗ 1+𝐻∗ ) − 𝑐2] (11) that is either 𝑃∗ = 0 (12) or 0 = 𝑟2 − 𝑎2𝑃 ∗ 1+𝐻∗ − 𝑐2 𝑎2𝑃 ∗ 1+𝐻∗ = 𝑟2 − 𝑐2 𝑃∗ = 𝑟2−𝑐2 𝑎2 (1 + 𝐻∗) (13) by looking at (8) and (12), we have the first equilibrium, that is 𝐸1 = (0,0) (14) next, by substituting (8) to (13) we have the second equilibrium, that is 𝐸2 = (0, 𝑟2−𝑐2 𝑎2 ) (15) the third equilibrium is found by substituting (12) to (9), that is 𝐸3 = ( 𝑟1−𝑐1 𝑏1 , 0) (16) the last equilibrium is found by solving (9) and (13), that is 𝐸4 = ( 𝑘1−𝑘2 1+𝑘2 , 𝑏1 𝑎1 (1 + 𝑘1−𝑘2 1+𝑘2 ) 𝑘2) (17) where 𝑘1 = 𝑟1−𝑐1 𝑏1 and 𝑘2 = 𝑟2−𝑐2 𝑎2 . a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah 45 results and discussion numerical simulation in this section, we apply the model along with its numerical simulation. for our purpose, we use numerical simulation software, i.e matlab. for the first simulation, we use the following parameters: 𝑟1 = 0.5; 𝑐1 = 0.3; 𝑟2 = 0.4; 𝑐2 = 0.1; 𝑎1 = 0.1; 𝑎2 = 0.1; and 𝑏1 = 0.05. figure 1. numerical solution approach 𝐸2 in figure 1, we use (0.5; 0.2) as the initial condition and step-size ℎ = 0.1. we found that the numerical solution approach the 𝐸2 = (0,3). from this simulation, we can infer that the equilibrium point 𝐸2 is stable for relatively small step-size. for the second simulation, we use parameters as follow: 𝑟1 = 0.8; 𝑐1 = 0.1; 𝑟2 = 0.6; 𝑐2 = 0.5; 𝑎1 = 0.5; 𝑎2 = 0.8; and 𝑏1 = 0.8. figure 2. numerical solution of second simulation in figure 2, we use (0.1; 0.1) as the initial condition. we found that the numerical solution is close to 𝐸4 but never reach the point. from the simulation, we can conclude that 𝐸4 is an unstable equilibrium. a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah 46 in the third simulation, we use parameters as follow: 𝑟1 = 0.8; 𝑐1 = 0.1; 𝑟2 = 0.4; 𝑐2 = 0.1, ; 𝑎1 = 0.9; 𝑎2 = 0.2; and 𝑏1 = 0.1. figure 3. numerical solution of third simulation in figure 3, we use (0.01; 0.01) as initial condition. we found numerically that 𝐸1 is an unstable equilibrium while 𝐸2 is a stable equilibrium. finally, for the last simulation, we use parameters as follow: 𝑟1 = 0.5; 𝑐1 = 0.2; 𝑟2 = 0.9; 𝑐2 = 0.5, ; 𝑎1 = 1.8; 𝑎2 = 0.3; and 𝑏1 = 0.2. figure 4. numerical solution of the fourth simulation in figure 4, we use (2; 0.5) as the initial condition. from numerical simulation, we found that the solution close to 𝐸3 but at the end of computation reach the 𝐸2 equilibrium. so we can conclude that 𝐸3 is an unstable equilibrium. a discrete numerical scheme of modified leslie-gower with harvesting model riski nur istiqomah dinnullah 47 conclusion from this paper, a discrete numerical scheme of modified leslie-gower with harvesting model has been investigated. we use euler-method to discretisize the model. we found four equilibrum points with their properties. the 𝐸1, 𝐸3, and 𝐸4 are unstable equilibrium, while 𝐸2 is the stable one. it is shown that the discrete model is dynamically consistent with its continuous model only for relatively small step-size h. further works, especially about stability-analysis need to be considered. furthermore, the discrete model may show complex dynamical behaviors, such as bifurcation and chaos phenomenon. references [1] p.h. leslie, "some further notes on the use of matrices in population mathematics," biometrika, vol. 35, pp. 213–245, 1948. [2] q. yue, "dynamics of a modified lesliegower predatorprey model with holling-type ii schemes and a prey refuge," springerplus, vol. 5, pp. 1–12, 2016. [3] e. gonzlez-olivares and c. paulo, "a leslie gower-type predator prey model with sigmoid functional response," international journal of computer mathematics, vol. 92, pp. 1895– 1909, 2015. [4] a. kang, y. xue, and j. fu, "dynamic behaviors of a leslie-gower ecoepidemiological model," discrete dynamics in nature and society, vol. 2015, 2015. [5] t. fayeldi, "skema numerik persamaan leslie gower dengan pemanenan," cauchy, vol. 3, pp. 214–218, 2015. [6] p. das, d. mukherjee, and a. sarkar , "study of an s-i epidemic model with nonlinear incidence rate: discrete and stochastic version," applied mathematics and computation, vol. 218, no. 6, pp. 2509 – 2515, 2011. [7] a. e. a. elsadany, h. a. el-metwally, e. m. elabbasy, and h. n. agiza, "chaos and bifurcation of a nonlinear discrete prey-predator system," computational ecology and software, vol. 2, pp. 169–180, 2012. [8] z. hu, z. teng, and h. jiang, "stability analysis in a class of discrete sir sepidemic models," nonlinear analysis: real world applications, vol. 13, no. 5, pp. 2017 – 2033, 2012. [9] l. li, g.q. sun, and z. jin, "bifurcation and chaos in an epidemic model withnonlinear incidence rates," applied mathematics and computation, vol. 216, no. 4, pp. 1226 – 1234, 2010. cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 15-19 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 10 october 2017 reviewed: 18 october 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.4393 geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data grissila yustisia1, waego hadi nugroho 1, maria bernadetha theresia mitakda 1 1department of statistics, brawijaya university, malang email: y_grissila@yahoo.com abstract regression analysis is a method for determining the effect of the response and predictor variables, yet simple regression does not consider the different properties in each location. methods geographically weighted regression (gwr) is a technique point of approach to a simple regression model be weighted regression model. the purpose of this study is to establish a model using geographically weighted regression (gwr) with a weighted fixed gaussian kernel and queen contiguity in cases of dengue fever patients and to determine the best weighting between the weighted euclidean distance as well as the queen contiguity based on the value of r2. results from the study showed that the modeling geographically weighted regression (gwr) with a weighted fixed gaussian kernel showed that all predictor variables affect the number of dengue fever patients, whereas the weighted queen contiguity, not all predictor variables affect the dengue fever patients. based on the value of r2 is known that a weighted fixed gaussian kernel is better used. keywords: geographically weighted regression, fixed gaussian kernel, queen contiguity introduction spatial data is the measurement data containing location information. methods geographically weighted regression (gwr) is a technique point of approach to a simple regression model be weighted regression model [1]. spatial weighting matrix is used to determine the closeness of the relationship between the regions. gwr weighting role model is very important because it represents a weighted value of research data layout. weighted grouped by distance (distance) and region (contiguity) [2]. in gwr models often use a weighting based on the distance (distance) without considering other weighting associated with the area (contiguity). in gwr models often use a weighting based on the distance (distance) without considering other weighting associated with the area (contiguity). in general, dengue fever clustered in specific locations. things to note about the location other than the one with the distance (distance), which is the state of the location or about the area (contiguity). therefore, in this study will consider the distance (distance) and region (contiguity) as weighting to search for the best model in dengue cases. mailto:y_grissila@yahoo.com geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data grissila yustisia 16 methods 2.1 testing the effect of spatial testing of spatial heterogeneity can be done by using the test statistic breusch-pagan (bp), is based on the hypothesis h0: σ1 2= σ2 2= ⋯= σj 2= σ2 h1: there are at least one j where σj 2 ≠ σ2 if 𝐻0 true test statistic: bp= ( 1 2 )𝒇(1𝑥𝑛) 𝑇 𝒁(𝒏𝒙𝒏)(𝒁(𝑛𝑥𝑛) 𝑇 𝒁(𝒏𝒙𝒏)) -1 𝒁(𝑛𝑥𝑛) 𝑇 𝒇(𝒏𝒙𝟏)+ ( 1 t )[ 𝑒(1𝑥𝑛) 𝑇 𝑾(𝒏𝒙𝒏)𝒆(𝒏𝒙𝟏) σ2 ] 2 ~ χ(p+1) 2 (1) 2.2 fixed gaussian kernel fixed gaussian kernel is the weighting matrix based on the proximity of the location of the observation point i and point to another location. fixed weighted gaussian kernel as follows [1]: 𝑤𝑖𝑗 = exp⁡[−((di𝑗/ℎ)/2) 2] (2) if the location of the i located at coordinates⁡(ui,vi) it will obtain the euclidean distance (dij) between all locations i and j are: di𝑗=√(uiu𝑗) 2 + (viv𝑗) 2 (3) bandwidth is a circle of radius from the center point location. methods to determine the bandwidth is cross validation (cv) [1]): cv= ∑ (y i -ŷ ≠i (ℎ)) 2n i=1 (4) 2.3 queen contiguity observations on the adjacent locations tend to be similar compared to locations far apart, because they relate to weighted location [3]. wij = { 1,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓⁡𝑟𝑒𝑔𝑖𝑜𝑛𝑠⁡𝑖⁡𝑎𝑛𝑑⁡𝑗⁡𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑤ℎ𝑒𝑟𝑒⁡⁡⁡𝑖 ≠ 𝑗 0,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑎𝑛𝑜𝑡ℎ𝑒𝑟⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ wi⏟ wij = [ 𝑤11 𝑤12 … 𝑤1𝑛 𝑤21 𝑤22 ⋯ 𝑤2𝑛 ⋮ 𝑤𝑛1 ⋮ 𝑤𝑛2 ⋱ ⋮ … 𝑤𝑛𝑛 ] 𝑤1 𝑤2 ⋮ 𝑤𝑛 (5) wi = ∑⁡wq(𝑛𝑥𝑛) wij = ⁡wq(𝑛𝑥𝑛) wi where: wi : sum total row wij : the weighting matrix row i column j 2.4 geographically weighted regression spatial data is the measurement data containing location information. methods geographically weighted regression (gwr) is a technique point of approach to a simple regression model be weighted regression model [1]. according to yasin [4] model of gwr is: 𝑦𝑖 = 𝛽0(𝑢𝑖,𝑣𝑖)+∑ 𝛽𝑗(𝑢𝑖,𝑣𝑖)𝑥𝑖𝑗 + 𝑖 𝑝 𝑗=1 (𝑢𝑖,𝑣𝑖)= coordinates (longitude, latitude) point i to a geographical location. 2.5 testing geographically weighted regression model parameters geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data grissila yustisia 17 testing geographically weighted regression model parameters is done simultaneously and partially [5]: 1. simultaneous testing to determine the effect of predictor variables together against the response variable. if 𝐻0 true test statistic: 𝐽𝐾𝑠𝑖𝑠𝑎(𝐻1) 𝛿1 2 𝛿2 ⁄ 𝐽𝐾𝑠𝑖𝑠𝑎(𝐻0) 𝑛−(𝑝+1)⁄ ~𝐹 ( 𝛿1 2 𝛿2 ,𝑛−(𝑝+1⁡) ∗ (6) 𝛿1 = 𝑡𝑟𝑎𝑐𝑒{(𝑰−𝑳) 𝑇(𝑰−𝑳)} 𝛿2 = 𝑡𝑟𝑎𝑐𝑒{(𝑰−𝑳) 𝑇(𝑰−𝑳)}2 where 𝐽𝐾𝑠𝑖𝑠𝑎(𝐻0) = 𝒀(𝟏𝒙𝒏) 𝑻 (𝑰(𝒏𝒙𝒏) −𝑳(𝒏𝒙𝒏)) 𝑻 (𝑰(𝒏𝒙𝒏) −𝑳(𝒏𝒙𝒏))𝒀(𝒏𝒙𝟏) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐽𝐾𝑠𝑖𝑠𝑎(𝐻1) = 𝒀(𝟏𝒙𝒏) 𝑻 (𝑰(𝒏𝒙𝒏) −𝑯(𝒏𝒙𝒏))𝒀(𝒏𝒙𝟏) 𝑯 = 𝑿(𝒏𝒙𝒑)(𝑿(𝒑𝒙𝒏) 𝑻 𝑿(𝒏𝒙𝒑)) −𝟏𝑿(𝒑𝒙𝒏) 𝑻 𝑳(𝑛𝑥𝑛) = ( 𝒙1 𝑇[𝑿𝑇𝑾(u1,v1)𝑿] −1𝑿𝑻𝑾(u1,v1) 𝒙2 𝑇[𝑿𝑇𝑾(u2,v2)𝑿] −1𝑿𝑻𝑾(u2,v2) ⋮ 𝒙𝑛 𝑇[𝑿𝑇𝑾(un,vn)𝑿] −1𝑿𝑻𝑾(un,vn)) i = identity matrix of order n using criteria reject h0 if the test statistic |𝐹| > 𝐹 (𝛼, 𝛿1 2 𝛿2 ,𝑛−(𝑝+1)) ∗ 2. partial testing to determine which predictor variables that influence the response variable for each observation location, using the t test statistic is based on the hypothesis: h0: βj(ui,vi) = 0 h1: βj(ui,vi) ≠ 0, j = 1, 2, ⋯, p t test statistic can be written as follows: β̂j(ui,vi) σ̂√cjj ~t( n-(p+1)) (7) 𝜎2 = 𝐽𝐾𝑠𝑖𝑠𝑎 𝑑𝑏𝑔𝑎𝑙𝑎𝑡 = 𝒆′𝒆 𝑛 −(𝑝 +1) where cjj a diagonal matrix element cct, 𝑪 = (𝑿(𝑝𝑥𝑛) 𝑇 𝑾(𝑢𝑖,𝑣𝑖)(𝑛𝑥𝑛)𝑿(𝑛𝑥𝑝)) −1 (8) reject if the test statistic |t| > t ( α 2 , n-(p+1)) 2.5 testing assessment geographically weighted regression model the coefficient of determination can describe the magnitude of the response variable diversity can be explained by the predictor variables. gwr r2 value obtained by the following mathematical equation [1]: 𝑅2(𝑢𝑖,𝑣𝑖) = 𝐽𝐾𝑅𝑊 𝐽𝐾𝑇𝑊 = ∑ 𝑾𝑖𝑗(𝑦𝑖−�̂�𝑖) 2𝑝 𝑗=1 ∑ 𝑾𝑖𝑗(𝑦𝑖−�̅�𝑖) 2𝑝 𝑗=1 (9) geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data grissila yustisia 18 results and discussion the results of statistical calculations breusch-pagan test with both weighting are presented in table 1. table 1. testing result breusch-pagan table 1 shows that the critical point test χ2 with error level 𝛼 = 0,05 and degrees of freedom (p+1) is 11,707 then reject h0 so the conclusion is that there is spatial heterogeneity in the data dengue cases. further testing gwr model parameters simultaneously, the test results are presented in table 2. table 2. model parameter testing results gwr simultaneous table 2 shows that the predictor variables with gaussian kernel weighting fixed effecting simultaneously the response variable for f⁡> (3,305)𝐹0.05(⁡2,31), where the weighted predictor variables queen contiguity no effect along the response variable for f > (2,922) 𝐹0.05(3,30). the partial test shows throughout predictor variables with fixed gaussian kernel weighting affect the response variable at each location and weighted queen contiguity affect the response variable at each location. comparison of methods performed to determine the best weighting. criteria for selection of the best weighting by using⁡𝑅2 are presented in table 3. best weighting is weighted with the largest value of 𝑅2. table 3. comparison of 𝑅2 at gwr model rated 𝑅2 for a model with gaussian kernel weighting fixed bigger than queen contiguity weighted, so that it can be concluded that the weighting fixed weighting gaussian kernel is better used for data dengue cases in this study. weighting breuschpagan fixed gaussian kernel 13.341 queen contiguity 18.089 weighting f fixed gaussian kernel 4.397 queen contiguity 0.21 weighting 𝑹𝟐 fixed gaussian kernel 0.42 queen contiguity 0.06 geographically weighted regression (gwr) modelling with weighted fixed gaussian kernel and queen contiguity for dengue fever case data grissila yustisia 19 conclusion the gwr method with the weighted fixed gaussian kernel yields a r2 greater than the weighted queen contiguity. this result indicates that the weighted fixed gaussian kernel use for dengue fever case data in this study. references [1] h. yasin, "pemilihan variabel pada model geographically weighted regression," jurnal media statistika, vol. iv, no. 2, pp. 63-72, 2011. [2] m. ilham and i. dwi, "pemodelan data kemiskinan di provinsi sumatera barat dengan metode geographically weighted regression," jurnal media statistika, vol. vi, no. 1, pp. 37-49, 2013. [3] a. s. fotheringham, c. brundson and m. charlthon, geographically weighted regression: the analysis of spatially varying relationships, 2002. [4] l. anselin, i. syabri and k. youngihn, geoda: an introduction to spatial data analysis, urbana: university of illinois, 2004. [5] j. lesage and r. k. pace, introduction to spatial econometrics, boca ration: crc press, 2009. abstract introduction methods results and discussion conclusion references issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 cauchy vol. 6 no. 2 pages: 49 – 99 malang may 2020 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, universitas islam negeri maulana malik ibrahim malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, maulana malik ibrahim state islamic university of malang, indonesia. managing editor : mohammad jamhuri, m.si, maulana malik ibrahim state islamic university of malang, indonesia. juhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial board : mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, maulana malik ibrahim state islamic university of malang, indonesia abdussakir, maulana malik ibrahim state islamic university of malang, indonesia ari kusumastuti, maulana malik ibrahim state islamic university of malang, indonesia fachrur rozi, maulana malik ibrahim state islamic university of malang, indonesia cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 editorial board elly susanti, universitas islam negeri maulana malik ibrahim malang, indonesia assistant editor : mohammad nafie jauhari, m.si, maulana malik ibrahim state islamic university of malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 table of contents cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas ..................................................................................................................... 49 – 57 basis existence of internal n-graph space ........................................................................... 58 – 65 pyramid population prediction using age structure model .......................................... 66 – 76 mathematical model quartic curve bezier of modification cubic curve bezier .... 77 – 83 on square-integrable representations of a lie group of 4-dimensional standard filiform lie algebra ................................................................................................................. 84 – 90 sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission .............................................................................................................................. 91 – 99 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5516 http://ejournal.uin-malang.ac.id/index.php/math/article/view/5516 5.3 muhammad izzat ubaidillah -proyeksi geometri fuzzy pada ruang _139-147_x proyeksi geometri fuzzy pada ruang muhammad izzat ubaidillah mahasiswa jurusan matematika uin maulana malik ibrahim malang e-mail: izzatbja@yahoo.co.id abstrak geometri fuzzy merupakan perkembangan dari geometri tegas, yang mana pada geometri tegas unsur-unsurnya hanya ada dan tidak ada, akan tetapi pada geometri fuzzy unsur-unsur tersebut berkembang dengan ketebalan yang dimiliki oleh masing-masing unsur tersebut. proyeksi geometri tegas merupakan pembentukan bayangan suatu unsur geometri yang diproyeksikan terhadap unsur proyektor, dengan sifat tegak lurus yang diwakili oleh masing-masing unsurnya, pembahasannya difokuskan pada koordinat hasil proyeksi. sedangkan proyeksi geometri fuzzy mempunyai pembahasan yang lebih luas, yang mencakup tentang koordinat hasil proyeksi, keeratan relasi masing-masing unsur dan ketebalan masing-masing unsur tersebut. penelitian ini dilakukan untuk mendeskripsikan dan menganalisis prosedur proyeksi geometri fuzzy pada ruang serta menjelaskan perbedaan antara proyeksi geometri tegas dan proyeksi geometri fuzzy pada ruang. kata kunci: geometr fuzzy, proyeksi geometri fuzzy, relasi fuzzy. abstract fuzzy geometry is an outgrowth of crisp geometry, which in crisp geometry elements are exist and not exist, but also while on fuzzy geometry elements are developed by thickness which is owned by each of these elements. crisp projective geometries is the formation of a shadow of geometries element projected on the projectors element, with perpendicular properties which are represented by their respective elemental, the discussion focused on the results of the projection coordinates. while the fuzzy projective geometries have richer discussion, which includes about coordinates of projection results, the mutual relation of each element and the thickness of each element. this research was conducted to describe and analyzing procedure fuzzy projective geometries on the plane and explain the differences between crisp projective geometries and fuzzy projective geometries on plane. keywords: fuzzy geometry, fuzzy relations, fuzzy projection geometries pendahuluan teori himpunan fuzzy merupakan perluasan dari teori himpunan tegas. pada teori himpunan tegas, keberadaan suatu elemen pada suatu himpunan � hanya akan memiliki dua kemungkinan keanggotaan, yaitu menjadi anggota � atau tidak menjadi anggota �. suatu nilai yang menunjukkan seberapa besar tingkat keanggotaan suatu elemen ��� dalam suatu himpunan ���, sering dikenal dengan nama derajat keanggotaan dinotasikan dengan �����. pada himpunan tegas, hanya ada dua nilai keanggotaan, yaitu ����� � 1 untuk � anggota �, dan ����� � 0 untuk � bukan anggota �. himpunan fuzzy didasarkan pada gagasan untuk memperluas jangkauan fungsi keanggotaan sedemikian hingga fungsi stersebut akan mencakup bilangan real pada interval 0,1�, dengan demikian menunjukkan bahwa derajat keanggotaan suatu elemen ��� dalam suatu himpunan ��� tidak hanya ada 0 dan 1, namun juga nilai yang terletak di dalamnya (kusumadewi, 2002). berdasarkan teori tersebut, titik dan garis yang pada geometri tegas hanya ada dan tidak ada, maka dalam geometri fuzzy akan berkembang, titik dan garis tidak hanya direpresentasikan dengan ada dan tidak ada, tetapi berkembang dengan ketebalan yang berbeda. kajian teori a. proyeksi geometri tegas pembahasan proyeksi tegas pada ruang terdiri dari tiga poin, yaitu proyeksi titik ke garis, titik ke bidang, dan proyeksi garis ke bidang. 1. proyeksi titik ke garis proyeksi titik ke garis merupakan pembentukan bayangan suatu titik terhadap suatu garis proyektor, dengan syarat garis hubung titik dengan titik hasil proyeksinya harus tegak lurus dengan garis proyektor. sedangkan hasil proyeksi yang berupa bayangan titik tersebut dapat ditemukan koordinatnya, dengan diketahui koordinat titik yang diproyeksikan dan persamaan garis proyektornya. muhammad izzat ubaidillah 140 volume 2 no. 3 november 2012 gambar 1. proyeksi titik ke garis 2. proyeksi titik ke bidang proyeksi titik ke bidang merupakan pembentukan bayangan suatu titik terhadap suatu bidang proyektor, dengan syarat garis hubung titik dengan titik hasil proyeksinya harus tegak lurus dengan bidang proyektor. sedangkan hasil proyeksi yang berupa bayangan titik tersebut dapat ditemukan koordinatnya, dengan diketahui koordinat titik yang diproyeksikan dan persamaan bidang proyektornya. gambar 2. proyeksi titik ke bidang 3. proyeksi garis ke bidang proyeksi garis ke bidang merupakan pembentukan bayangan suatu garis yang diproyeksikan terhadap bidang proyektor, dengan sifat tegak lurus yang diwakili oleh masing-masing unsurnya. pada proyeksi garis ke bidang terdapat tiga kemungkinan, yaitu garis yang diproyeksikan tegak lurus dengan bidang proyektor, garis yang diproyeksikan sejajar dengan bidang proyektor, dan garis yang diproyeksikan tidak tegak lurus dan tidak sejajar dengan bidang proyektor. gambar 3. proyeksi garis yang tegak lurus dengan bidang proyektor proyeksi garis ke bidang, dimana garis yang di proyeksikan saling tegak lurus dengan bidang. proyeksinya diwakili titik, dan hasilnya juga berupa titik bukan berupa garis. kemungkinan proyeksi ini terjadi jika garis saling tegak lurus dengan setiap garis pada bidang (krismanto, 2008). gambar 4. proyeksi garis yang sejajar dengan bidang proyektor proyeksi garis ke bidang, dimana garis yang diproyeksikan sejajar dengan bidang proyektor, proyeksi ini terjadi bila garis dan bidang tidak mempunyai titik potong. hasil proyeksi ini didapatkan misalnya titik a dan b terletak pada garis, titik a’ dan b’ merupakan proyeksi titik a dan b pada bidang. ruas garis a’b’ adalah proyeksi garis pada bidang. gambar 5. proyeksi garis yang tidak tegak lurus dan tidak sejajar dengan bidang proyektor proyeksi garis ke bidang, dimana garis yang diproyeksikan tidak saling tegak lurus dan tidak sejajar dengan bidang proyektor. untuk hasil proyeksinya diwakili titik � dan titik dan hasilnya berupa titik �’ dan titik ’ yang apabila dihubungkan menjadi garis �’ ’. kemungkinan ini terjadi jika terdapat garis tidak tegak lurus dengan bidang, serta garis menembus bidang (krismanto, 2008). b. teori himpunan fuzzy menurut kusumadewi dkk (2006), teori himpunan fuzzy merupakan kerangka matematis yang digunakan untuk mempresentasikan ketidakpastian, dan ketidakjelasan karena nilai kebenarannya tidak hanya bernilai benar atau salah. nilai 0 menunjukkan salah, dan nilai 1 menunjukkan benar dan masih ada nilai-nilai yang terletak antara benar dan salah. secara matematis suatu himpunan fuzzy � dalam semesta pembicaraan � dapat dinyatakan sebagai himpunan pasangan terurut � � ���|������|� � �� dimana �� adalah fungsi keanggotaan dari himpunan fuzzy �, yang merupakan suatu proyeksi geometri fuzzy pada ruang jurnal cauchy – issn: 2086-0382 141 pemetaan dari himpunan semesta � ke selang tertutup 0,1�(susilo, 2006). apabila semesta � adalah himpunan yang kontinu, maka himpunan fuzzy � dinyatakan dengan � � � �|����� ��� dimana lambang � di sini bukan lambang integral seperti yang dikenal dalam kalkulus, tetapi melambangkan keseluruhan unsur-unsur � � � bersama dengan derajat keanggotaannya dalam himpunan fuzzy �. apabila semesta � adalah himpunan yang diskrit, maka himpunan fuzzy � dinyatakan dengan � � � �|����� ��� dimana lambang σ di sini tidak melambangkan operasi penjumlahan seperti yang dikenal dalam aritmatika, tetapi melambangkan keseluruhan unsur-unsur � � � bersama dengan derajat keanggotaannya dalam himpunan fuzzy �. terdapat tiga oprasi dasar pada himpunan fuzzy,yaitu operasi irisan (intersection), operasi gabungan (union), dan operasi tidak (complement). operasi irisan berhubungan dengan operasi interseksi pada himpunan tegas. �predikat sebagai hasil operasi dengan operator “dan” diperoleh dengan mengambil nilai keanggotaan terkecil antar element pada himpunan-himpunan yang bersangkutan. ditunjukkan sebagai � � adalah suatu fuzzy subset � dari � sehingga � � � � dan derajat keanggotaannya ���� � min��� ��, �� �� operasi gabungan berhubungan dengan operasi union pada himpunan tegas. �-predikat sebagai hasil operasi dengan operator “atau” diperoleh dengan mengambil nilai keanggotaan terbesar antar element pada himpunanhimpunan yang bersangkutan. ditunjukkan sebagai � ! adalah suatu fuzzy subset " dari � sehingga " � � ! dan derajat keanggotaannya ��!� � maks��� ��, �� �� operasi tidak berhubungan dengan operasi komplemen pada himpunan tegas. �predikat sebagai hasil operasi dengan operator “tidak” diperoleh dengan mengurangkan nilai keanggotaan elemen pada himpunan yang bersangkutan dari 1. ditunjukkan sebagai a’ (a komplemen) dan derajat keanggotaannya �� � 1 # �� c. relasi fuzzy relasi fuzzy $% antara elemen-elemen dalam himpunan � dengan elemen–elemen dalam himpunan & didefinisikan sebagai himpunan bagian fuzzy dari perkalian cartesius� ' &, yaitu himpunan fuzzy $% � ����, �|�(��, ��|��, � � � ' &� relasi fuzzy $% disebut juga relasi fuzzy pada himpunan (semesta) � ' &. jika � � &, maka $% disebut relasi fuzzy pada himpunan �. relasi tegas hanya menyatakan adanya atau tidak adanya hubungan antara elemenelemen dari suatu himpunan dengan elemenelemen dari himpunan lainnya, sedangkan relasi fuzzy lebih luas dari itu, karena juga menyatakan derajat keeratan hubungan relasi tersebut. pembahasan a. geometri fuzzy proyeksi geometri fuzzy merupakan pengembangan dari proyeksi geometri tegas dengan himpunan fuzzy yang didefinisikan sebagai berikut: misal )��*, *, +*� merupakan suatu titik di koordinat ruang �$,�, )% � -)|�*%. himpunan fuzzy di ), jadi )% � -�*, *,+*| �*%. dinamakan titik fuzzy, misal / 0��1� 2 1 2 �1+ � "� ��3� 2 3 2 �3+ � "� 4 merupakan suatu persamaan garis di koordinat ruang �$,�, /5 � -/|�65. himpunan fuzzy di /, jadi /5 � 0��1� 2 1 2 �1+ � "� ��3� 2 3 2 �3+ � "� |�657 dinamakan garis fuzzy, dan misal 8 9 �� 2 2 �+ � " merupakan persamaan bidang dikoordinat ruang �$,�, 8% � -8| �:;. himpunan fuzzy di 8, jadi 8% � -�� 2 2 �+ � " | �:;. dinamakan bidang fuzzy. b. proyeksi geometri fuzzy 1. proyeksi titik fuzzy pada garis fuzzy gambar 6. proyeksi titik fuzzy pada garis fuzzy muhammad izzat ubaidillah 142 volume 2 no. 3 november 2012 misal sebuah titik fuzzy )%��<, <, +<|�<5� diproyeksikan terhadap garis fuzzy /5 9 =0��1� 2 1 2 �1+ 2 " � 0� ��3� 2 3 2 �3+ 2 " � 0� 4 |�65>, untuk mengetahui hasil proyeksinya, dapat dicari dengan langkah-langkah sebagai berikut: pertama, diberikan titik fuzzy )%��<, <, +<?�<5� sebagai unsur yang diproyeksikan dan garis fuzzy /5 9 =0��1� 2 1 2 �1+ 2 " � 0� ��3� 2 3 2 �3+ 2 " � 0� 4 |�65> sebagai unsur proyektor. kedua, dicari koordinat hasil proyeksi tegas titik fuzzy )%��<, <, +<|�*%� terhadap garis fuzzy /5 9 =0��1� 2 1 2 �1+ 2 " � 0� ��3� 2 3 2 �3+ 2 " � 0� 4 |�65>, yang didefinisikan dengan @%< pada garis /5. selanjutnya mencari jarak antara titik )% dengan titik @%<, setelah itu mencari jarak antara titik fuzzy )% dengan titik fuzzy @%a (titik-titik fuzzy pada garis fuzzy /5, @%a � /5�. ketiga, dicari derajat keanggotaan relasi antara titik fuzzy )% dengan garis fuzzy /5 yang didefinisikan dengan ��(%�, dimana $% merupakan relasi dari titik fuzzy )% pada garis fuzzy /5, �$% b )% ' /5�. $% � c��)%, @%a�|�(%�)%, @%a��|�)%, @%a� � )% ' /5d pada proyeksi geometri fuzzy dari titik fuzzy ke garis fuzzy, hasil proyeksi tidak hanya berupa satu titik fuzzy seperti pada proyeksi geometri tegas, akan tetapi semua titik fuzzy pada garis fuzzy proyektor dengan derajat keanggotaan ketebalan yang dipengaruhi oleh derajat keanggotaan global relasi fuzzy. derajat keanggotaan relasi ��(%� bisa diketahui dengan fungsi berikut: �(%�)%, @%a� � ef�ghfi�0,5 dimana j � jarak terdekat titik fuzzy )% pada @%< ka � jarak titik fuzzy )% pada @%a derajat keanggotaan relasi tersebut merupakan representasi dari seberapa besar atau kuat relasi antara titik fuzzy )% dengan garis fuzzy /5, dengan berasumsi jika jarak antara titik fuzzy )% dengan @%a � /5 semakin dekat, maka kekuatan �(% semakin besar. sedangkan jika jarak antara titik fuzzy )% dengan @%a semakin jauh maka berlaku sebaliknya. akan tetapi, kekuatan relasi tersebut masih berada dalam interval 0,1�. relasi tersebut dapat digambar ke dalam tabel berikut. tabel 1. tabel matrik relasi titik fuzzy )% dengan garis fuzzy /5, � �(%�)%, @%a�� $% @%<fl … @%<f1 @%< )% �(%�)%, @%<fl� … �(%�)%, @%<f1� �(%�)%, @%<� $% @%<n1 … @%<nl )% �(%�)%,@%<n1� … �(%�)%, @%<nl� sumber: djauhari, 1990:55 sehingga diperoleh derajat keanggotaan relasi titik )% dengan garis /5 �(%�)%, @%a� sebagai berikut �(%�)%, @%a � � -��)%, @%<fl�|�( ; �)%, @%<fl��, … , ��)%, @%<f1�|�(%�)%, @%<f1��, ��)%, @%<�|�(%�)%, @%<��, ��)%, @%<n1�|�(%�)%, @%<n1��, …, ��)%, @%<nl�|�(%�)%, @%<nl��. keempat, setelah diketahui �(%�)%, @%a�, dicari hasil perkalian derajat keanggotaan ketebalan titik yang diproyeksikan �*% dan derajat keanggotaan relasi �(%�)%, @%a�, yang didefinisikan dengan �o�)%, @%a�. selanjutnya mencari derajat keanggotaan ketebalan masing-masing titik hasil proyeksi � �*pq�)%, @%a��, yang merupakan irisan (intersection) dari �o�)%, @%a� dengan �65. �*r q �)%, @%<fl� � min ��o�)%, @%<fl�, �65� v �*r q �)%, @%<f1� � min ��o�)%, @%<f1�, �65� �*r q �)%, @%<� � min ��o�)%, @%<�, �65� �*r q �)%, @%<n1� � min ��o�)%, @%<n1�, �65� v �*r q �)%, @%<nl� � min ��o�)%, @%<nl�, �65� jadi didapatkan hasil proyeksi yaitu berupa garis fuzzy )w; � 0��1� 2 1 2 �1+ 2 " � 0� ��3� 2 3 2 �3+ 2 " � 0� 4 |�*pq , dengan koordinat yang sama dengan garis proyektor /5 karena )w; � /5, dan dengan �*r q sebagai berikut: �*r q �)%, @%a� � -��)%, @%<fl�|�*r q �)%, @%<fl��, …, x�)%, @%<f1�y�*r q �)%, @%<f1�z, ��)%, @%<�|�*r q �)%, @%<��, ��)%, @%<n1�|�*r q �)%, @%<n1��, …, ��)%, @%<nl�|�*r q �)%, @%<nl��. proyeksi geometri fuzzy pada ruang jurnal cauchy – issn: 2086-0382 143 2. proyeksi titik fuzzy pada bidang fuzzy gambar 7. proyeksi titik fuzzy pada bidang fuzzy misal sebuah titik fuzzy )%��<, <, +<|�<5� diproyeksikan terhadap bidang fuzzy 8% � -�� 2 2 �+ � " | �:;., untuk mengetahui hasil proyeksinya, dapat dicari dengan langkahlangkah sebagai berikut: pertama, diberikan titik fuzzy )%��<, <, +<?�<5� sebagai unsur yang diproyeksikan dan bidang fuzzy 8% � -�� 2 2 �+ � " | �:;., sebagai unsur proyektor. kedua, dicari koordinat hasil proyeksi tegas titik fuzzy )%��<, <, +<|�*%� terhadap bidang fuzzy 8% � -�� 2 2 �+ � " | �:;., yang didefinisikan dengan @%< pada bidang 8% . selanjutnya mencari jarak antara titik )% dengan titik @%<, setelah itu mencari jarak antara titik fuzzy )% dengan titik fuzzy @%a (titik-titik fuzzy pada garis fuzzy 8% , @%a � 8%�. ketiga, dicari derajat keanggotaan relasi antara titik fuzzy )% dengan bidang fuzzy 8% yang didefinisikan dengan ��(%�, dimana $% merupakan relasi dari titik fuzzy )% pada bidang fuzzy 8% , �$% b )% ' 8%�. $% � c��)%, @%a�|�(%�)%, @%a��|�)%, @%a� � )% ' 8%d pada proyeksi geometri fuzzy dari titik fuzzy ke bidang fuzzy, hasil proyeksi tidak hanya berupa satu titik fuzzy seperti pada proyeksi geometri tegas, akan tetapi semua titik fuzzy pada bidang fuzzy proyektor dengan derajat keanggotaan ketebalan yang dipengaruhi oleh derajat keanggotaan global relasi fuzzy. derajat keanggotaan relasi ��(%� bisa diketahui dengan fungsi berikut: �(%�)%, @%a� � ef�ghfi�[,\ dimana j � jarak terdekat titik fuzzy )% pada @%< ka � jarak titik fuzzy )% pada @%a derajat keanggotaan relasi tersebut merupakan representasi dari seberapa besar atau kuat relasi antara titik fuzzy )% dengan bidang fuzzy 8% , dengan berasumsi jika jarak antara titik fuzzy )% dengan @%a � 8% semakin dekat, maka kekuatan �(% semakin besar. sedangkan jika jarak antara titik fuzzy )% dengan @%a semakin jauh maka berlaku sebaliknya. akan tetapi, kekuatan relasi tersebut masih berada dalam interval 0,1�. relasi tersebut dapat digambar ke dalam tabel berikut tabel 2. tabel matrik relasi titik fuzzy )% dengan bidang fuzzy 8% , � �(%�)%, @%a�� $% @%<fl] … @%<f1] @%< )% �(%�)%, @%<fl]� … �(%�)%, @%<f1]� �(%�)%, @%<� $% @%<n1] … @%<nl] )% �(%�)%, @%<n1]� … �(%�)%, @%<nl]� sumber: djauhari, 1990:55 sehingga diperoleh derajat keanggotaan relasi titik )% dengan bidang 8% �(%�)%, @%a� sebagai berikut �(%�)%, @%a � � -��)%, @%<fl]�|�( ; �)%, @%<fl]��, … , ��)%, @%<f1]�|�(%�)%, @%<f1]��, ��)%, @%<�|�(%�)%, @%<��, ��)%, @%<n1]�|�(%�)%, @%<n1]��, …, ��)%, @%<nl]�|�(%�)%, @%<nl]��. keempat, setelah diketahui �(%�)%, @%a�, dicari hasil perkalian derajat keanggotaan ketebalan titik yang diproyeksikan �*% dan derajat keanggotaan relasi �(%�)%, @%a�, yang didefinisikan dengan �o�)%, @%a�. selanjutnya mencari derajat keanggotaan ketebalan masing-masing titik hasil proyeksi � �*pq�)%, @%a��, yang merupakan irisan (intersection) dari �o�)%, @%a� dengan �:;. �*r q �)%, @%<fl]� � min ��o�)%, @%<fl]�, �65� v �*r q �)%, @%<f1]� � min ��o�)%, @%<f1]�, �65� �*r q �)%, @%<� � min ��o�)%, @%<�, �65� �*r q �)%, @%<n1]� � min ��o�)%, @%<n1]�, �65� v �*r q �)%, @%<nl]� � min ��o�)%, @%<nl]�, �65� jadi didapatkan hasil proyeksi berupa bidang fuzzy )p; � -�� 2 2 �+ � " | �*pq., dengan koordinat yang sama dengan bidang proyektor 8% karena )w; � 8% , dan dengan �*r q sebagai berikut: �*r q �)%, @%a� � -��)%, @%<fl]�|�*r q �)%, @%<fl]��, …, x�)%, @%<f1]�y�*r q �)%, @%<f1]�z, muhammad izzat ubaidillah 144 volume 2 no. 3 november 2012 ��)%, @%<�|�*r q �)%, @%<��, ��)%, @%<n1]�|�*r q �)%, @%<n1]��, …, ��)%, @%<nl]�|�*r q �)%, @%<nl]��. 3. proyeksi garis fuzzy pada bidang fuzzy pada proyeksi geometri fuzzy pada garis fuzzy pada bidang fuzzy, sama halnya pada proyeksi geometri tegas yaitu terdapat tiga bentuk proyeksi diantaranya garis fuzzy yang diproyeksikan tegak lurus dengan bidang fuzzy proyektor, garis fuzzy yang diproyeksikan sejajar dengan bidang fuzzy proyektor, dan garis fuzzy yang diproyeksikan tidak sejajar dan tidak saling tegak lurus dengan bidang fuzzy proyektor. a. proyeksi garis fuzzy _̂ pada bidang fuzzy ;̀ jika garis _̂ saling tegak lurus dengan bidang ;̀ gambar 8. proyeksi garis fuzzy /5 pada bidang fuzzy 8% yang saling tegak lurus misal sebuah garis fuzzy /5 diproyeksikan terhadap bidang fuzzy 8% , untuk mengetahui hasil proyeksinya, dapat dicari dengan langkahlangkah sebagai berikut: pertama, diberikan garis fuzzy /5 sebagai unsur yang diproyeksikan dan bidang fuzzy 8% , sebagai unsur proyektor. kedua, dicari koordinat hasil proyeksi tegas garis fuzzy /5 terhadap bidang fuzzy 8% , yang di-definisikan dengan @%< pada bidang 8% . selanjutnya mencari jarak antara titik /5< yang mana titik /5< � /5 dan /5< merupakan titik dengan jarak terdekat dengan titik @%<, setelah itu mencari jarak antara titik /5a � /5 dengan titik fuzzy @%a (titik-titik fuzzy pada bidang fuzzy 8% , @%a � 8%�. ketiga, dicari derajat keanggotaan relasi antara titik /a_ � /5 dengan bidang fuzzy 8% yang didefinisikan dengan ��(%�, dimana $% merupakan relasi dari titik-titik fuzzy pada /5 terhadap bidang fuzzy 8% , �$% b /5 ' 8%�. $% � c��/5a, @%a�|�(%�/5a, @%a��|�/5a, @%a� � /5 ' 8%d pada proyeksi geometri fuzzy dari garis fuzzy pada bidang fuzzy, hasil proyeksi tidak hanya berupa satu titik fuzzy seperti pada proyeksi geometri tegas, akan tetapi semua titik fuzzy pada bidang fuzzy proyektor dengan derajat keanggotaan ketebalan yang dipengaruhi oleh derajat keanggotaan global relasi fuzzy. derajat keanggotaan relasi ��(%� bisa diketahui dengan fungsi berikut: �(%�)%, @%a� � ef�ghfi�[,\ dimana j � jarak terdekat garis /5 pada bidang 8% ka � jarak titik /5 dengan @%a derajat keanggotaan relasi tersebut merupakan representasi dari seberapa besar atau kuat relasi antara titik fuzzy /5a dengan bidang fuzzy 8% , dengan berasumsi jika jarak antara titik fuzzy /5adengan @%a � 8% semakin dekat, maka kekuatan �(% semakin besar. sedangkan jika jarak antara titik fuzzy /5adengan @%a semakin juah, maka berlaku sebaliknya yaitu kekuatan �(% semakin kecil, akan tetapi kekuatan masing-masing relasi tersebut masih berada pada interval [0, 1]. relasi tersebut dapat digambar ke dalam tabel berikut tabel 3. tabel matriks relasi garis fuzzy /5 dengan bidang fuzzy 8% , /5 tegak lurus 8% , ��(%�/5a, @%a�� $% @%<flh … @%<f1 @%< /51 �(%�/51, @%<flh� … �(%�/51, @%<f1� �(%�/51, @%<� /53 �(%�/53, @%<flh� … �(%�/53, @%<f1� �(%�/53, @%<� /5, �(%�/5,, @%<flh� … �(%�/5,, @%<f1� �(%�/5,, @%<� v v v v /5l �(%�/5l, @%<flh� … �(%�/5l, @%<f1� �(%�/5l, @%<� $% @%<n1 … @%<flh /51 �(%�/51, @%<n1� … �(%�/51, @%<flh� /53 �(%�/53, @%<n1� … �(%�/53, @%<flh� /5, �(%�/5,, @%<n1� … �(%�/5,, @<flh� v v v /5l �(%�/5l, @%<n1� … �(%�/l, @%<flh� sumber: djauhari, 1990:55 sehingga diperoleh derajat keanggotaan relasi garis fuzzy /5 pada bidang fuzzy 8% dengan �(%�/5, @%a� sebagai berikut: �(%�/5, @%a� � -��/5, @%<flh�|�( ; �/5, @%<flh��, …, ��/5, @%<f1�|�(%�/5, @%<f1��, ��/5, @%<�|�(%�/5, @%<��, ��/5, @%<n1�|�(%�/5, @%<n1��, …, ��/5, @%<nlh�|�(%�/5, @%<nlh��. proyeksi geometri fuzzy pada ruang jurnal cauchy – issn: 2086-0382 145 keempat, setelah diketahui �(%�/5, @%a�, dicari hasil perkalian derajat keanggotaan ketebalan titik yang diproyeksikan �65 dan derajat keanggotaan relasi �(%�/5, @%a�, yang didefinisikan dengan �o�/5, @%a�. selanjutnya mencari derajat keanggotaan ketebalan masing-masing titik hasil proyeksi � �6pk�/5, @%a��, yang merupakan irisan (intersection) dari �o�/5, @%a� dengan �:;. �6r q �/5, @%<flh� � min��o�/5, @%<flh�, �:;� v �6r q �/5, @%<f1� � min��o�/5, @%<f1�, �:;� �6r q �/5, @%<� � min ��o�/5, @%<�, �:;� �6r q �/5, @%<n1� � min ��o�/5, @%<n1�, �:;� v �6r q �/5, @%<nlh� � min ��o�/5, @%<nlh�, �:;� jadi didapatkan hasil proyeksi berupa bidang /w; � 8% dengan /p; 9 l�� 2 2 �+ 2 " � 0y�6r q m dengan koordinat yang sama dengan bidang proyektor 8% , dan dengan �6r q sebagai berikut: �6r q �/5, @%a� � -��/5, @%<flh�|�6r q �/5, @%<flh��, …, ��/5, @%<f1�|�6r q �/5, @%<f1��, ��/5, @%<�|�6r q �/5, @%<��, ��/5, @%<n1�|�6r q �/5, @%<n1��, …, ��/5, @%<nlh�|�6r q �/5, @%<nlh��. b. proyeksi garis fuzzy _̂ pada bidang fuzzy ;̀ jika garis _̂ saling sejajar dengan bidang ;̀ gambar 9. proyeksi garis fuzzy /5 pada bidang fuzzy 8% yang saling sejajar misal sebuah garis fuzzy /5 diproyeksikan terhadap bidang fuzzy 8% , untuk mengetahui hasil proyeksinya, dapat dicari dengan langkahlangkah sebagai berikut: pertama, diberikan garis fuzzy /5 sebagai unsur yang diproyeksikan dan bidang fuzzy 8% , sebagai unsur proyektor. kedua, dicari jarak terdekat antara garis fuzzy /5 dengan bidang fuzzy 8% , yang dinotasikan j dengan syarat j merupakan panjang garis fuzzy hubung /5 dengan bidang fuzzy 8% , dimana j tegak lurus dengan garis fuzzy /5 dan bidang fuzzy 8% , setelah itu mencari jarak antara titik /5a � /5 dengan titik fuzzy @%a (titik-titik fuzzy pada bidang fuzzy 8% , @%a � 8%�. ketiga, dicari derajat keanggotaan relasi antara titik /a_ � /5 dengan bidang fuzzy 8% yang didefinisikan dengan ��(%�, dimana $% merupakan relasi dari titik-titik fuzzy pada /5 terhadap bidang fuzzy 8% , �$% b /5 ' 8%�. $% � c��/5a, @%a�|�(%�/5a, @%a��|�/5a, @%a� � /5 ' 8%d derajat keanggotaan relasi ��(%� bisa diketahui dengan fungsi berikut: �(%�)%, @%a� � ef�ghfi�[,\ dimana j � jarak terdekat garis /5 pada bidang 8% ka � jarak titik /5 dengan @%a derajat keanggotaan relasi tersebut merupakan representasi dari seberapa besar atau kuat relasi antara titik fuzzy /5a dengan bidang fuzzy 8% , dengan berasumsi jika jarak antara titik fuzzy /5adengan @%a � 8% semakin dekat, maka kekuatan �(% semakin besar. sedangkan jika jarak antara titik fuzzy /5adengan @%a semakin juah, maka berlaku sebaliknya yaitu kekuatan �(% semakin kecil, akan tetapi kekuatan masing-masing relasi tersebut masih berada pada interval [0, 1]. relasi tersebut dapat digambar ke dalam tabel berikut tabel 4. tabel matriks relasi garis fuzzy /5 dengan bidang fuzzy 8% , dimana /5 sejajar 8%, ��(%�/5a, @%a�� $% @%<flh … @%<f1 @%< /51 �(%�/51, @%<flh� … �(%�/51, @%<f1� �(%�/51, @%<� /53 �(%�/53, @%<flh� … �(%�/53, @%<f1� �(%�/53, @%<� /5, �(%�/5,, @%<flh� … �(%�/5,, @%<f1� �(%�/5,, @%<� v v v v /5l �(%�/5l, @%<flh� … �(%�/5l, @%<f1� �(%�/5l, @%<� $% @%<n1 … @%<flh /51 �(%�/51, @%<n1� … �(%�/51, @%<flh� /53 �(%�/53, @%<n1� … �(%�/53, @%<flh� /5, �(%�/5,, @%<n1� … �(%�/5,, @<flh� v v v /5l �(%�/5l, @%<n1� … �(%�/l, @%<flh� sumber: djauhari, 1990:55 sehingga diperoleh derajat keanggotaan relasi garis fuzzy /5 pada bidang fuzzy 8% dengan �(%�/5, @%a� sebagai berikut: muhammad izzat ubaidillah 146 volume 2 no. 3 november 2012 �(%�/5, @%a� � -��/5, @%<flh�|1�, …, ��/5, @%<f1�|1�, ��/5, @%<�|1�,��/5, @%<n1�|1�,…, ��/5, @%<nlh�|1�. dimana �(%�/5a, @%a� � 1 , terjadi ketika relasi antara titik fuzzy /5a dan titik fuzzy @%a bernilai sama dimana ka � j, sehingga harga maksimum dari �(%�/5a, @%a� yang relatif terhadap variabel /5a bernilai 1. keempat, setelah diketahui �(%�/5, @%a�, dicari hasil perkalian derajat keanggotaan ketebalan titik yang diproyeksikan �65 dan derajat keanggotaan relasi �(%�/5, @%a�, yang didefinisikan dengan �o�/5, @%a�. selanjutnya mencari derajat keanggotaan ketebalan masing-masing titik hasil proyeksi � �6pk�/5, @%a��, yang merupakan irisan (intersection) dari �o�/5, @%a� dengan �:;. �6r q �/5, @%<flh� � min��65 , �:;� v �6r q �/5, @%<f1� � min��65 , �:;� �6r q �/5, @%<� � min��65 , �:;� �6r q �/5, @%<n1� � min��65 , �:;� v �6r q �/5, @%<nlh� � min��65 , �:;� jadi didapatkan hasil proyeksi yaitu berupa bidang /w; � 8% dengan /p; 9 l�� 2 2 �+ 2 " � 0y�6r q m dengan koordinat yang sama dengan bidang proyektor 8% , dan dengan �6r q sebagai berikut: �6r q �/5, @%a� � �65 � �:; c. proyeksi garis fuzzy _̂ pada bidang fuzzy ;̀ jika garis _̂ tidak saling tegak lurus dan tidak saling sejajar gambar 10. proyeksi garis fuzzy /5 pada bidang fuzzy 8% yang tidak saling tegak lurus dan tidak saling sejajar misal sebuah garis fuzzy /5 diproyeksikan terhadap bidang fuzzy 8% , untuk mengetahui hasil proyeksinya, dapat dicari dengan langkahlangkah sebagai berikut: pertama, diberikan garis fuzzy /5 sebagai unsur yang diproyeksikan dan bidang fuzzy 8% , sebagai unsur proyektor. kedua, dicari jarak terdekat antara garis fuzzy /5 dengan bidang fuzzy 8% , yang dinotasikan j dengan syarat j merupakan panjang garis fuzzy hubung /5 dengan bidang fuzzy 8% , dimana j tidak berpotongan dengan bidang fuzzy 8% . jika garis fuzzy / berpotongan dengan bidang fuzzy 8% , maka j bernilai 0, setelah itu mencari jarak antara titik /5a � /5 dengan titik fuzzy @%a (titik-titik fuzzy pada bidang fuzzy 8% , @%a � 8%�. ketiga, dicari derajat keanggotaan relasi antara titik /a_ � /5 dengan bidang fuzzy 8% yang didefinisikan dengan ��(%�, dimana $% merupakan relasi dari titik-titik fuzzy pada /5 terhadap bidang fuzzy 8% , �$% b /5 ' 8%�. $% � c��/5a, @%a�|�(%�/5a, @%a��|�/5a, @%a� � /5 ' 8%d derajat keanggotaan relasi ��(%� bisa diketahui dengan fungsi berikut: �(%�)%, @%a� � ef�ghfi�[,\ dimana j � jarak terdekat garis /5 pada bidang 8% ka � jarak antara titik /5 dengan @%a derajat keanggotaan relasi tersebut merupakan representasi dari seberapa besar atau kuat relasi antara titik fuzzy /5a dengan bidang fuzzy 8% , dengan berasumsi jika jarak antara titik fuzzy /5adengan @%a � 8% semakin dekat, maka kekuatan �(% semakin besar. sedangkan jika jarak antara titik fuzzy /5adengan @%a semakin juah, maka berlaku sebaliknya yaitu kekuatan �(% semakin kecil, akan tetapi kekuatan masing-masing relasi tersebut masih berada pada interval [0, 1]. relasi tersebut dapat digambar ke dalam tabel berikut tabel 5. tabel matriks relasi garis fuzzy /5 dan bidang fuzzy 8% , dimana /5 tidak tegak lurus dan tidak sejajar 8% , �(%�/5a, @%a� $% @%<flh … @%<f1 @%< /51 �(%�/51, @%<flh� … �(%�/51, @%<f1� �(%�/51, @%<� /53 �(%�/53, @%<flh� … �(%�/53, @%<f1� �(%�/53, @%<� /5, �(%�/5,, @%<flh� … �(%�/5,, @%<f1� �(%�/5,, @%<� v v v v /5l �(%�/5l, @%<flh� … �(%�/5l, @%<f1� �(%�/5l, @%<� $% @%<n1 … @%<flh /51 �(%�/51, @%<n1� … �(%�/51, @%<flh� /53 �(%�/53, @%<n1� … �(%�/53, @%<flh� /5, �(%�/5,, @%<n1� … �(%�/5,, @<flh� v v v /5l �(%�/5l, @%<n1� … �(%�/l, @%<flh� sumber: djauhari, 1990:55 proyeksi geometri fuzzy pada ruang jurnal cauchy – issn: 2086-0382 147 sehingga diperoleh derajat keanggotaan relasi garis fuzzy /5 pada bidang fuzzy 8% dengan �(%�/5, @%a� sebagai berikut: �(%�/5, @%a� � -��/5, @%<flh�|�( ; �/5, @%<flh��, …, ��/5, @%<f1�|�(%�/5, @%<f1��, ��/5, @%<�|�(%�/5, @%<��, ��/5, @%<n1�|�(%�/5, @%<n1��, …, ��/5, @%<nlh�|�(%�/5, @%<nlh��. keempat, setelah diketahui �(%�/5, @%a�, dicari hasil perkalian derajat keanggotaan ketebalan titik yang diproyeksikan �65 dan derajat keanggotaan relasi �(%�/5, @%a�, yang didefinisikan dengan �o�/5, @%a�. selanjutnya mencari derajat keanggotaan ketebalan masing-masing titik hasil proyeksi � �6pk�/5, @%a��, yang merupakan irisan (intersection) dari �o�/5, @%a� dengan �:;. �6r q �/5, @%<flh� � min��o�/5, @%<flh�, �:;�� v �6r q �/5, @%<f1� � min��o�/5, @%<f1�, �:;� �6r q �/5, @%<� � min��o�/5, @%<�, �:;� �6r q �/5, @%<n1� � min��o�/5, @%<n1�, �:;� v �6r q �/5, @%<nlh� � min��o�/5, @%<nlh�, �:;� jadi didapatkan hasil proyeksi berupa bidang /w; � 8% dengan /p; 9 l�� 2 2 �+ 2 " � 0y�6r q m dengan koordinat yang sama dengan bidang proyektor 8% , dan dengan �6r q sebagai berikut: �6r q �/5, @%a� � -��/5, @%<flh�|�6r q �/5, @%<flh��, …, ��/5, @%<f1�|�6r q �/5, @%<f1��, ��/5, @%<�|�6r q �/5, @%<��, ��/5, @%<n1�|�6r q �/5, @%<n1��, …, ��/5, @%<nlh�|�6r q �/5, @%<nlh��. penutup perbedaan proyeksi geometri tegas dan proyeksi geometri fuzzy yaitu pada proyeksi geometri tegas unsur yang diproyeksikan dan unsur proyektor hanya bersifat bivalue, yaitu ada dan tidak ada. sedangkan pada proyeksi geometri fuzzy unsur geometri bersifat multivalue, dengan ketebalan yang yang direpresentasikan dengan derajat keanggotaan dalam interval [0,1]. pada proyeksi geometri fuzzy cenderung lebih global dibandingkan dengan proyeksi geometri tegas karena pada proyeksi geometri tegas terdapat syarat tegak lurus antara unsur yang diproyeksikan dengan unsur proyektor, sehingga hasil proyeksi terbatas antara ada dan tidak adanya hasil proyeksinya pada syarat tersebut, sedangkan pada geometri fuzzy semua anggota unsur proyektor dianggap sebagai hasil dari unsur yang diproyeksikan tersebut dengan derajat keanggotaan ketebalan tertentu. derajat keanggotaan ketebalan tersebut dipengaruhi oleh derajat keanggotaan kekuatan relasi antara unsur-unsur yang diproyeksikan dan unsurunsur proyektor. daftar pustaka [1] alisah, e. & idris. 2009. buku pintar matematika. jogjakarta: mitra pelajar. [2] casse, r. 2006. projective geometry an introduction. new york: oxford university press inc. [3] djauhari, m. 1990. himpunan fuzzy. jakarta: karunika universitas terbuka. [4] krismanto. 2008. pembelajaran sudut dan jarak dalam ruang dimensi tiga di sma. yogyakarta: pusat pengembangan dan pemberdayaan pendidik dan tenaga kependidikan matematika. [5] kusumadewi, s. 2002. analisis & desain sistem fuzzy menggunakan toolbox matlab. yogyakarta: graha ilmu. [6] kusumadewi, s. dkk. 2006. fuzzy multiattribute decision making. yogyakarta: graha ilmu. [7] kusumadewi, s. & purnomo, h. 2004. aplikasi logika fuzzy untuk pendukung keputusan. yogyakarta: graha ilmu. [8] larson, r. & edwards, b. h. 2010. calculus. belmont: cengage learning. [9] rich, b. 2002. geometri. jakarta: erlangga [10] soebari. 1995. geometri analitik. malang: fmipa ikip malang. [11] spiegel, r. m. 1999. analisis vektor. jakarta: erlangga. [12] stein, k. s. & barchellos, a. 1992. calculus and analytic geometry. us: mc.graw hill. [13] sundawa, d. 2009. teorema pythagoras dan garis-garis pada segitiga. (online: http://www.crayonpedia.org/mw/bse:teor ema pythagoras dan garis-garis pada segitiga 8.1 (bab 5). diakses 7 mei 2012). [14] susilo, f. 2006. himpunan dan logika fuzzy serta aplikasinya. yogyakarta: graha ilmu. application of rectangular matrices: affine cipher using asymmetric keys cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 181-185 p-issn: 2086-0382; e-issn: 2477-3344 submitted: september 26, 2017 reviewed: october 09, 2017 accepted: may 31, 2019 doi: http://dx.doi.org/10.18860/ca.v5i4.4408 application of rectangular matrices: affine cipher using asymmetric keys maxrizal1, baiq desy aniska prayanti2 1stmik atma luhur pangkalpinang 2universitas bangka belitung email: maxrizal@atmaluhur.c.id, baiqdesyaniska@gmail.com abstract in cryptography, we know the symmetric key algorithms and asymmetric key algorithms. we know that the asymmetric key algorithm is more secure than the symmetric key algorithm. affine cipher uses a symmetric key algorithm. in this paper, we introduce the affine cipher using asymmetric keys. asymmetrical keys are formed from rectangular matrices. keywords: cryptography, symmetric key algorithms, asymmetric key algorithms, affine cipher using asymmetric keys, rectangular matrices introduction nowadays, cryptography becomes the main option for securing messaging or data over a communications network. in cryptography, the sender encrypts the message (plaintext) into ciphertext. furthermore, the receiver describes ciphertext into plaintext. encryption and decryption in cryptography require key algorithm. if the encryption and decryption use a similar key, the algorithm is called symmetric key algorithms. if the encryption and decryption use a different key, it is called asymmetric key algorithm. the asymmetric key algorithm is also known as modern cryptographic systems. one example of an asymmetric key algorithm is rsa [1,2,3,4]. affine cipher is one of the cryptographic algorithms that uses a symmetric key algorithm. it including substitution cipher type. in affine cipher [5,6], each plaintext encrypted as  modi ic ap b m  and described as a   1 mod i i p a c b m    , with a and b are integers as private keys. based on the above facts, in this study, we will modify the affine cipher using asymmetric keys. we will generate a private key and a public key on affine cipher. asymmetrical keys are formed from rectangular matrices. methods this research is a study of literature research. some properties of asymmetric keys are obtained from [1,2,3,4]. furthermore, the properties of affine cipher is derived from [5,6]. application of rectangular matrices: affine cipher using asymmetric keys maxrizal 182 results and discussion affine cipher and affine cipher using asymmetric keys in the affine cipher [5,6], applied encryption  modi ic ap b m  and descriptions  1 modi ip a c b m    , where a must be invertible on mod m and b is an arbitrary integer. the sender and receiver must save the private key a and b for encryption and description. in this study, we formed               * 1 1 1 1 1 * 1 1 1 1 1 1 1 * 1 1 1 1 1 1 1 1 * 1 1 1 1 1 1 1 1 1 * 1 1 1 1 1 1 1 1 mod mod mod mod mod (1) n n n n n n n n n n n n n n n n n n n n n n n n n n c y p b m x c x y p b m x c x y p x b m x c x b m x y p p x y x c x b m                                           if we suppose 1 1n n x y a    , * 1 1n n x c c    and 1 1n n x b b    , we have the similar decryption as affine cipher. next, we get encryption  * 1 1 1 1 1 modn n nc y p b m     , with the public key  1 1,n ny b  and description     1 * 1 1 1 1 1 1 1 1 mod n n n n n n p x y x c x b m           , with the private key  1 nx  . furthermore, encryption and description above, we call as affine cipher using asymmetric keys. key generating algorithm on affine cipher using asymmetric keys steps to generate of keys algorithm:  choose 1 n x  .  choose 1n y  .  calculate 1 1n n x y   . if the matrix 1 1n n x y   has no inverse over mod m , we repeat steps 1) and 2). if matrix   1 1 1n n x y    exists on mod m , we have one of a public key 1n y  and private key 1 n x  .  choose an arbitrary matrix 1n b  . so, we get the public key  1 1,n ny b  and a private key  1 nx  . security analysis on affine cipher and affine cipher using asymmetric keys on affine cipher using asymmetric key, apply encryption  * 1 1 1 1 1 modn n nc y p b m     , with the public key  1 1,n ny b  . if the sender sent the message, the attacker (unauthorized recipients) will get a public key  1 1,n ny b  and ciphertext * 1n c  . however, the attacker would be difficult to describe the ciphertext, because the matrix   1 1n y   does not exist.             * 1 1 1 1 1 * 1 1 1 1 1 1 * 1 1 1 1 1 mod mod mod 2 n n n n n n n n n c y p b m c b m y p p y c b m                    it means that the plaintext remains safe. furthermore, we show a comparison affine cipher and affine cipher using asymmetric keys. application of rectangular matrices: affine cipher using asymmetric keys maxrizal 183 affine cipher affine cipher using asymmetric keys  symmetric key.  asymmetric keys.  the key is an integer.  the key is matrix pair.  elements on the plaintext and ciphertext one-to-one correspondence (bijective function).  elements on the plaintext and ciphertext do not one-to-one correspondence ( it is not bijective function). example nisca will send a message to max. max generate the public key and private key. key generating algorithm:  max chooses  1 2 1x  .  he chooses 3 1 4 y           .  he calculates   3 1 2 1 1 9 4 xy            . note that 19 mod 26 3  .  he chooses an arbitrary matrix 2 1 5 b           .  max gets the public key 3 2 1 , 1 4 5                         and private key   1 2 1 . encryption: niska received a public key 3 2 1 , 1 4 5                         from max. she will send “use” to max. she converts “use” to 21-19-5. she encrypts  * 1 1 1 1 1 modn n nc y p b m     . she gets  *1 3 2 13 1 21 1 mod 26 22 4 5 11 c                                      *2 3 2 7 1 19 1 mod 26 20 4 5 3 c                                      *3 3 2 17 1 5 1 mod 26 6 4 5 25 c                                     niska gets 13-22-11-7-20-3-17-6-25. she converts and sends “mvkgtcqfy” to max. application of rectangular matrices: affine cipher using asymmetric keys maxrizal 184 description: max received “mvkgtcqfy” from nisca. he converts 13-22-11-7-20-3-17-6-25. max knows the public key 3 2 1 , 1 4 5                         and saves the private key   1 2 1 . he describes     1 * 1 1 1 1 1 1 1 1 mod n n n n n n p x y x c x b m           . he calculates  1 1 3 1 2 1 1 9 4 n n x y              . he gets   1 1 1 3 n n x y     . next, he gets    1 1 13 2 3 1 2 1 22 1 2 1 1 mod 26 21 11 5 p                               1 1 7 2 3 1 2 1 20 1 2 1 1 mod 26 19 3 5 p                               1 1 17 2 3 1 2 1 6 1 2 1 1 mod 26 5 25 5 p                            max gets 21-19-5. he converts “use”. conclusion from this study, we obtained several conclusions, including:  we can form affine cipher using asymmetric keys.  affine cipher using asymmetric keys more secure than affine cipher. . acknowledgments the research was done well because there is good support from stmik atma luhur pangkalpinang. therefore, the author’s thank you for the support of funds and policies that are in stmik atma luhur pangkalpinang so this research can be done and completed. references [1] khairnar, d.b., and sandeep, k., " secure rsa: pair-wise key distribution using modified rsa algorithm", international journal of advanced research in computer science and software engineering, vol.6, issue. 4, pp. 383-387, 2016. [2] puneeth, m., farha, j. s., sandhya, n., and yamini, m., "rsa and modified rsa algorithm using c programming", international journal of advanced engineering research and science, vol.2, issue. 2, pp. 15-20, 2015. application of rectangular matrices: affine cipher using asymmetric keys maxrizal 185 [3] hussain, a, k., "a modified rsa algorithm for security enhancement and redundant messages elimination using k-nearest neighbor", international journal of innovative science, engineering, and technology, vol.2, issue. 1, pp.159-163, 2015. [4] deshmukh, s., and patil, r., “hybrid cryptography technique using modified diffiehellman and rsa", international journal of computer science and information technology, vol.5, issue. 6, pp. 7302-7304, 2014. [5] mokhtari, m., and naraghi, h., "analysis and design of affine and hill cipher", journal of mathematics research, vol.4, no. 1, pp. 67-77, 2012. [6] hussein, l, a., "internal affine stream cipher", computer engineering and information technology journal, vol.1, issue. 1, pp. 1-5, 2015. the study geometry fractals designed on batik motives cauchy –jurnal matematika murni dan aplikasi volume 6(1) (2019), pages 34-39 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 22, 2019 reviewed: november 25, 2019 accepted: november 30, 2019 doi: http://dx.doi.org/10.18860/ca.v6i1.8081 the study geometry fractals designed on batik motives juhari mathematics department, universitas islam negeri maulana malik ibrahim malang email: juhari@uin-malang.ac.id abstract this research was conducted to gain some patterns of fractals julia set and seirpinski that applied on batik then creating batik that has many varied motives and multifaceted. there are three steps in formulating the patterns of fractals of julia set and seirpinski. first, build the fractals by analyzing the function of fractals julia set and determine the plane’s coordinate which you want to use. in this case, we use square and rectangle which will be created by using fractals patterns seirpinski. second, create a batik motives from fractals pattern julia set and seirpinski by using geometry transformation. the geometry transformation which will be used are translation, dilatation, reflection, and rotation. the last, combine some batik motives which were created by using image processing. it was summation of two images processing. the result is batik motives that has many variated and multifaceted. keywords: fractals; julia set; seirpinski introduction creative industries models are a like a building whom will make indonesia economy strong, with the runway, pillars and roofs as the elements for the building. it will bring the creative industries from origin point to achieve the vision and the mission of creative industries 2013 (destination point). however, by expending the numbers of creative industries, it will cause a selling power and a market value some products in a domestic market or an export market to be weakened. during 2nd quarter 2017, small and medium industrial growth are being flagging, especially the buying power. the central of bureau of statistics wrote that manufacturing small and micro industry had already being slow. the sector was just grown 2,5% in 2nd quarter 2017. during 2nd quarter 2016 and 2nd quarter 2015, imk grew 6,56% and 4,57%. the main problem is products that originally they produce still have same system, they have not been counterbalanced with the increasing of an art quality and innovation from batik motives that was needed by customers whether looking from the level of symmetry, harmony, and variation of modelling or the size of the product. in addition, the purpose of this article is to gain the solution of fractals formula and designing the combination from some fractals formulas becoming new design of batik fractals formula. batik formula fractals has been introduced by [1] about woven batik motives, but [2] and [3] introduced about fractals application on batik motives from their own regions. for the variation of fractals has been discussed by [4]. in this article, it was written for explaining about fractals formula in different degrees choosing varied complex numbers to create fractals motives and multifaceted. after that, we combine the design from the fractals formula that we got before. http://dx.doi.org/10.18860/ca.v6i1.8081 mailto:juhari@uin-malang.ac.id the study geometry fractals designed on batik motives juhari 35 methods the measures that used to generate batik fractals formula design are, first, generate fractals julia set and fractals sierpinski pattern to gain woven batik motives which involving square and rectangle by using mathematics operation. many iteration on fractals are determined to know the shape of fractals which will combine. second, shape up batik motives and woven fractals from the archetype that we have created using geometry transformation. geometry transformation which we use are translation, dilatation, reflection, and rotation [5]. last, combining some batik motives which created using image processing that is two summations of it. results and discussion julia set julia set was made for by having a certain points and multiplying every others points repeatedly, then adding the result to the original points. the iteration processing that belongs to a function is to do a modification on function composition itself. it is built from polynomial function [6]. the function for julia set is 𝑓𝑎(𝑧) = 𝑧 2 + 𝑐. z is a variable where z is part of complex number which is 𝑥 + 𝑦𝑖, where 𝑥, 𝑦 𝜖 𝑅, and c is a parameter constant that can be divided into three, those are: a. it can be only real number. b. it can be only imaginer number. c. it can be mixed real number and imaginer number. in some literatures, making motives differently is only created a change on the iteration. (n). it makes adding iteration to add infinite new details, but when an image is zoomed in, then there is still the resemblance characteristic. however, in this research, it will be exposed new modification in terms of squared and parameter value c. here the research of the program using maple and matlab with squared and different parameters. picture 1. modification of fractals julia set picture (1) was built from julia set function itself. it was from quadratic equation complex number 𝑓(𝑧) = 𝑧2 by setting firstly the parameter of c -0.2 – 0.7. meanwhile the real value and the imaginer value in an open interval with the purpose to show the shape of fractals julia set roundly. doing iteration n times in each points that have been (1) f(z)=z2+c dengan c=-0.2-0.7 (2) f(z)=z5+a*eit*z dengan a=1.0001 dan t=pi/1000 the study geometry fractals designed on batik motives juhari 36 determined, then the result of the iteration will make an orbit from those points and those orbits show the shape of fractals julia set. picture (2) was built from polynomial function complex numbers 𝑓(𝑧) = 𝑧5 by adding linier function𝑓(𝑧) = 𝑎𝑧, where a is complex constant number which not zero, and z is complex number = 𝑥 + 𝑦𝑖 . applying that a = 1.001 and multiplying linier function by 𝑧0 in a polar equation 𝑧0 = 𝑟𝑒 𝑖𝑡 with 𝑡 = 𝑃𝑖 100 . similar with pictures (3) and (4) which have same process to create the pattern. these are some others results modification of fractals julia set to add motives in fractals batik design. julia set is made by choosing certain points and multiplying every others points repeatedly, then the result is added to the original one. the processing of iteration which is made for a function is to do a change on a function composition itself repeatedly (recursive). suppose that f(z)=z5+a*e2pi*i/g*z with a =1.0001 is function z, the second iteration of z is f 2(z) = f (f(z)) = f (z5+a*e2pi*i/g*z) = (z5+a*e2pi*i/g*z)5 + a*e2pi*i/g* (z5+a*e2pi*i/g*z) so on until you are in certain iteration then you get the fractals model julia set like these picture: picture 2. modification of fractals julia set julia set analyzing domain julia set for function 𝑓(𝑧) = 𝑧2 + 𝑐 is a set for every points which have limit orbits [7]. julia set is a limit between filled julia set and set points that have bounded orbits (escape set). look at function 𝑓(𝑧) = 𝑧2, we know that from the quadratic equation complex number, we know that: f(z)=1/(z3+dz), d=-1.7+2.4i f(z)=1/(z3+dz+c), c=0 dan d=-3(1+i) the study geometry fractals designed on batik motives juhari 37  if 1 0 z then orbit will head 0 and it will be bounded (bounded);  if 1 0 z then orbit will always be unit circle and this orbit will be also bounded (bounded);  if 1 0 z then orbit will head a limit and this orbit is unbounded (unbounded); by seeing these orbits, the we can conclude that julia set for equation 𝑓(𝑧) = 𝑧2 + 𝑐 where c = 0 is a unit circle. making fractals julia set, first step has to be determined the parameter value of c, for real number and imaginer number in closed interval [-2,2]. if it is changed the parameter value of c form the julia set domain on the function 𝑓(𝑧) = 𝑧2 + 𝑐 with c = -0.2+0.7i, and fetching six points (x+iy), those are: point 1 = (1.00,0.00), point 2 = (0,50,0.25) point 3 = (0.00,0.88), point 4= (0.000,0.000) point 5 = (0.500-0.250), point 6 = (-0.250,0.50) by doing it n times iteration on every points, then result of iteration will create orbits from the points itself and those orbits will show the fractals motives on julia set. these are the results of the calculating every points. orbit 1: z0 = 1 + 0i, z1 = cz  2 0 = (1 + 0i)2 + (-0.2 + 0.7i) = 0.80 + 0.7i z2 = cz  2 1 = (0.80 + 0.7i)2 + (-0.2 + 0.7i) = -0.05 + 1.82i z3 = cz  2 2 = (-0.05 + 1.82i)2 + (-0.2 + 0.7i) = -3.2099 + 0.182i  infinite orbit 2: z0 = 0.50 + 0.25i, z1 = cz  2 0 = (0.50 + 0.25i) 2 + (-0.2 + 0.7i) = -0.17265625 + 0.793750i z2 = cz  2 1 = (-0.17265625 + 0.793750i) 2 + (-0.2 + 0.7i) = -0,7942 + 0.2733i z3 = cz  2 2 = (-0,7942 + 0.2733i) 2 + (-0.2 + 0.7i) = 0,3561 + 0.2659i bounded to orbit 0 orbit 3: z0 = 0.00 + 0.88i, z1 = cz  2 0 = (0.00 + 0.88i,) 2 + (-0.2 + 0.7i) = 0.9744 + 0.7i z2 = cz  2 1 = (0.9744 + 0.7i) 2 + (-0.2 + 0.7i) = 0.2594 – 0.66416i z3 = cz  2 2 = (0.2594 – 0.66416i) 2 + (-0.2 + 0.7i) = -0.57382 + 0.4446i bounded to orbit 0 it can be concluded that orbit 1 is infinite and this orbit goes to unbounded, while orbit 2 and orbit 3 are bounded to orbit 0. the process of calculating orbit on point 4 until point 6 equal the calculating on orbit 1 until orbit 3. if it is shows in fractals julia set like picture (1). the result of picture 1 can be used to create varied and multiset batik motives formula. these are batik motives fractals, the result by combining some modifications of julia set. the study geometry fractals designed on batik motives juhari 38 picture 3. merging some of julia set fractal modifications conclusions basic pattern of batik motives formula fractals julia set is built by using iterative on a function whether polynomial or polar. iteration process that is done for a function is to do a change on a composition function itself repeatedly (recursion) that raised from the polynomial function and polar. julia set function is 𝑓𝑎(𝑧) = 𝑧 2 + 𝑐. z is a variable where z is one of the part of complex number x + yi, where 𝑥, 𝑦 ∈ 𝑅. the amalgamation of fractals julia set generates 4 models like picture 2. the model that existed can be added with other motive models. it can be created by combining different fractals julia set (picture 2) then resulting many variated of batik models and multifaset. acknowledgments greatest appreciation to lp2m universitas islam negeri maulana malik ibrahim malang for the funding support in this research phase to make this article. references [1] h. situngkir, “cellular-automata and innovation within indonesian traditional weaving crafts,” ssrn, vol. 3, no. 2357389, pp. 1–8, 2013. [2] f. dan and a. lingkaran, “aplikasi pola batik menggunakan metode fraktal dan algoritma lingkaran 8 way simetris,” j. teknol. inf., vol. 10, no. 2, pp. 1–9, 2014. [3] y. romadiastri, “batik fraktal : perkembangan aplikasi geometri,” delta j. ilm. the study geometry fractals designed on batik motives juhari 39 pendidik. mat., vol. 1, no. 2, pp. 158–164, 2013. [4] e. susanti, “variasi motif batik palembang menggunakan sistem fungsi teriterasi dan himpunan julia,” j. mat., vol. 5, no. 1, pp. 36–44, 2015. [5] a. cahya and p. mahros, “modelisasi benda onyx dan marmer melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan putar bezier,” j. ilmu dasar, vol. 8, no. 2, pp. 175–185, 2007. [6] r. kodri and j. titaley, “variasi motif batik minahasa berbasis julia set,” j. mipa unsrat online, vol. 6, no. 2, pp. 81–85, 2017. [7] a. k. jaya and n. aliansa, “orbit fraktal himpunan julia,” j. mat. stat. dan komputasi, vol. 13, no. 2, pp. 162–170, 2017. modification of chaos game with rotational variation on a square cauchy –jurnal matematika murni dan aplikasi volume 6(1) (2019), pages 27-33 p-issn: 2086-0382; e-issn: 2477-3344 submitted: may 10, 2019 reviewed: september 17, 2019 accepted: november 30, 2019 doi: http://dx.doi.org/10.18860/ca.v6i1.6936 modification of chaos game with rotational variation on a square kosala dwidja purnomo1, indy larasati2, ika hesti agustin3, firdaus ubaidillah4 department of mathematics, university of jember, indonesia email: 1) kosala.fmipa@unej.ac.id, 2) indylarasati92@gmail.com, 3) ikahesti.fmipa@unej.ac.id, 4) firdaus_u@yahoo.com abstract chaos game is a game of drawing a number of points in a geometric shape using certain rules that are repeated iteratively. using those rules, a number of points generated and form some pattern. the original chaos game that apply to three vertices yields sierpinski triangle pattern. chaos game can be modified by varying a number of rules, such as compression ratio, vertices location, rotation, and many others. in previous studies, modification of chaos games rules have been made on triangles, pentagons, and n-facets. modifications also made in the rule of random or nonrandom, vertex choosing, and so forth. in this paper we will discuss the chaos game of quadrilateral that are rotated by using an affine transformation with a predetermined compression ratio. affine transformation is a transformation that uses a matrix to calculate the position of a new object. the compression ratio r used here is 2. it means that the distance of the formation point is 1/2 of the fulcrum, that is α = 1/r = 1/2. variations of rotation on a square or a quadrilateral in chaos game are done by using several modifications to random and non-random rules with positive and negative angle variations. finally, results of the formation points in chaos game will be analyzed whether they form a fractal object or not. keywords: chaos game, rotational variation, random, non-random. introduction the original chaos game is a game of drawing a number of points on an equilateral triangle with certain rules that are repeated iteratively. sierpinski triangle is an example of a fractal object formed by chaos game. devaney [1] has modified the sierpinski triangle with rotational variations that form fractal "films". the modification of sierpinski triangle can be done by varying the angle of rotation, compression ratio, or location of the vertices for chaos game problems. based on the research conducted by devaney, this study will discuss the formation of chaos game on a square which are rotated by using an affine transformation with a certain compression ratio. the production rules are by using random and non-random rules, where random rules are built by selecting random points that can be modified with certain conditions, while non-random rules for point selection are carried out regularly and sequentially as many points as desired. the study was conducted to analyze whether the chaos game on a square varied by rotation with certain rules will still make fractals form such as sierpinski triangle. fractals are defined as complex forms without limits because they have infinite details in a limited volume of space. this form will always look identical like an infinite pattern. however, if an object is made from elementary particles, then the details of the number of geometric shapes are limited. this amount is determined by the number of elementary particles and their relative position. the amount may be very large, but it is http://dx.doi.org/10.18860/ca.v6i1.6936 modification of chaos game with rotational variation on a square kosala dwidja purnomo 28 always limited [2]. some examples of fractal forms include cantor's middle third set, von koch curve, sierpinski triangle, and cantor dust [3]. chaos game with three vertices is one method to generate sierpinski triangle. the shape of vertices in chaos games can also be rectangles, pentagons, hexagons, and nfacets. chaos games are generated using the specified compression ratio, namely the boundary between random points and vertices. the compression ratio is written with the symbol r and the result is the new distance point 𝑍𝑛+1 from the right point is 1/𝑟 from the previous distance 𝑍𝑛+1 to the vertex [4]. chaos game is a point game where the production rules can still be developed, for example what if changing the number of vertices or changing the distance between the starting point and the specified vertex [5]. if the starting point in game chaos is located in a triangle, then the next marked point will always be located in the sierpinski triangle. if the starting point is outside the triangle, then in some iteration a marked point will occur located in the triangle. therefore, for some iteration the marked points are guaranteed to always be in the triangle [6]. sierpinski triangle is a linear type of fractal that has identical self-similarity from one iteration to the next iteration. sierpinski triangle is a triangle consisting of other triangles which are in the same shape but with half size. if a small part of the triangle is enlarged, it will look like another larger part. affine transformation can generate sierpinski triangle on chaos game with three vertices [7]. transformation can be interpreted as a method that can be used to manipulate the location of a point. if the transformation is imposed on a set of points, then the object will change. changes in this case are changes from the initial object location to the new location. the most commonly used method in graphical computer is the affine transformation method. an affine transformation is a transformation that uses a matrix to calculate the position of a new object. transformation on two-dimensional objects can be in the form of translation, scale, and rotation. translation can be interpreted by changing locations from the initial location to the new location. the point 𝑃(𝑥, 𝑦) is shifted by a number of 𝑇𝑟𝑥 on the x-axis and shifted by a number of 𝑇𝑟𝑦 on the y-axis. therefore the general formula will be obtained as follows 𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) + 𝑇𝑟 = 𝑃(𝑥 + 𝑇𝑟𝑥 , 𝑦 + 𝑇𝑟𝑦 ). scale can be interpreted by an initial location multiplied by the desired quantity. the original location is multiplied by the amount 𝑆𝑥 on the x-axis and 𝑆𝑦 on the y-axis. so that the general formula will be obtained as follows 𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) x 𝑆(𝑥, 𝑦) = 𝑃(𝑥 x 𝑆𝑥 , 𝑦 x 𝑆𝑦 ). rotation can be interpreted by moving the initial location of an object (object) by rotating at an angle θ to the x-axis or y-axis. it is assuming that the centre point (0,0) with a positive angle is counter-clockwise and a negative angle is in the direction clockwise. the rotation formula is as follows 𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) x [ cos 𝜃 − sin 𝜃 sin 𝜃 cos 𝜃 ]. two-dimensional transformation uses two axes, namely the x-axis and the y-axis. the x-axis represents the length, while the y-axis represents the width [8]. we will modify the original chaos game on a square/quadrilateral by using geometric transformations, namely rotation, as devaney done. the rules of affine transformation with linear algebraic principles for building a "fractal" film are formulated 𝑇 ( 𝑥 𝑦) = ( 1/𝑟 0 0 1/𝑟 ) ( cos 𝜃 − sin 𝜃 sin 𝜃 cos 𝜃 ) ( 𝑥 − 𝑥0 𝑦 − 𝑦0 ) + ( 𝑥0 𝑦0 ). the affine transformation combines the calculations of translation, scales and rotation modification of chaos game with rotational variation on a square kosala dwidja purnomo 29 matrices with a compression ratio written as r, and α is a distance that its value 1/r. the centre of rotation is (𝑥0, 𝑦0) with distance α, and θ is the angle of rotation [9]. results and discussion generation of modified chaos game on a square/quadrilateral will be done using random and non-random rules. the generation using random rules is done with some modifications. the compression ratio used is r=2, which means that the distance from the formation point is 1/2 of the weight point (that is α=1/r). there are four modifications here: 1. the last point that has been chosen cannot be chosen again; 2. the facing point cannot be chosen; 3. the left entry point cannot be chosen; 4. the right side point cannot be chosen. the left and right points are seen from sequential patterns 𝐴1− 𝐴2 − 𝐴3 − 𝐴4, where the left point is the previous point, and the right point is the point after it. the exception is point 𝐴3 where the left point is point 𝐴4, then after choosing point 𝐴3 it is not allowed to choose point 𝐴4 and otherwise. facing points also apply only to rectangles. the rules will be different if applied to pentagon and other aspects. non-random rules are also carried out with several modifications, namely nonrandom 1 with 4 regular and sequential point selection patterns (𝐴1− 𝐴2 − 𝐴3 − 𝐴4), nonrandom 2 with 16 point selection patterns, that is in multiples of 4 repeat from the last chosen point (𝐴1 − 𝐴2 − 𝐴3 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴4), non-random 3 with 20 point selection patterns, that is in multiples of 5 jumping over one point after that and in 𝑡 iterations namely 𝑡 ≡ 4 𝑚𝑜𝑑 (5) the chosen point is the point facing the last point (𝐴1 − 𝐴2 − 𝐴3 − 𝐴1 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴3 − 𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴2 − 𝐴4), non-random 4 with 24 point selection patterns, that is in multiples of 6 and in 𝑡 iterations namely 𝑡 ≡ 5 𝑚𝑜𝑑 (6) repeats once like the previous point, then at 𝑡 ≡ 4 𝑚𝑜𝑑 (6) jumps over one point after its last point (𝐴1 − 𝐴2 − 𝐴3 − 𝐴3 − 𝐴3 − 𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴4− 𝐴4 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴1 − 𝐴 1 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴2 − 𝐴2 − 𝐴4), and non-random 5 with 28 point selection patterns, in 𝑡 iterations namely 𝑡 ≡ 4 𝑚𝑜𝑑 (7) jumps over one point from the last point, in the iteration 𝑡 ≡ 5 𝑚𝑜𝑑 (7) repeats like the last point selected, at the iteration 𝑡 ≡ 6 𝑚𝑜𝑑 (7) jumps over one point from the last point again, and in multiples of 7 repeats one point from the last point chosen (𝐴1 − 𝐴2 − 𝐴3 − 𝐴1 − 𝐴1 − 𝐴3 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴4 − 𝐴2 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴3 − 𝐴3 − 𝐴1 − 𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴2 − 𝐴2 − 𝐴4 − 𝐴4). steps to make a chaos game: a. make a rectangle and give a name to each rectangular vertex, for example 𝐴1, 𝐴2, 𝐴3, and 𝐴4; b. choose a random starting point and is named 𝑇0; c. select one vertex in the square to be associated with the starting point 𝑇0 to make the game chaotic; d. determining the new point named 𝑇1 and the distance (α) is half the distance of the vertex to the starting point according to the predetermined compression ratio, namely 𝑟 = 2; e. repeating steps (c) with the starting point is a new point obtained from the previous iteration and may be repeated by selecting the vertex as a reference point; f. do the desired number of points to iterations; modification of chaos game with rotational variation on a square kosala dwidja purnomo 30 the rotation rule used is by using an affine transformation. suppose the rotation is done with the rotation angle given 30° and the fulcrum is 𝐴4. then the new point formed will shift by 30° towards the chosen fulcrum 𝐴4. the difference in random and nonrandom game chaos formation lies in the selection of random points where points may be chosen randomly according to the desired modification rules, while the selection of points on non-random rules is determined by the desired pattern, in this study 4 patterns were selected, 16 patterns, 20 patterns, 24 patterns and 28 point repetition patterns. the results of the forms obtained by random rules and non-random chaos games with positive and negative angle values will be compared, and the fulcrum of rotation is at the quadrilateral angle. the visualization results in chaos game on a square by random rules and their modification is done by rotation of angles every 10°, 𝛼=1/2 and iterations of 10,000. all objects generated in the rotation of positive angles are random rules and their modifications can be said to be fractal objects because they have similar shapes to each other (self-similarity). the rotation at the negative angle is also a fractal object. the difference is in the direction of rotation, where the rotation of the positive angle rotates counter-clockwise while the rotation of the negative angle rotates clockwise. the visualization for the positive angle are shown in figure 1 to figure 5. figure 1. rotational variation with random rules figure 2. rotational variation with modification 1 figure 3. rotational variation with modification 2 modification of chaos game with rotational variation on a square kosala dwidja purnomo 31 figure 4. rotational variation with modification 3 figure 5. rotational variation with modification 4 the visualization of the rules of non-random chaos games is done with 𝛼 = 1/2, 5,000 iterations, and at positive angles of rotation every 30° it does not produce a clear shape, but only like far apart points. the visualization of non-random rules simulation and their modifications shown in figure 6 to figure 7. figure 6. rotational variation with non-random 1 figure 7. rotational variation with non-random 2 the resulted points of non-random chaos game at a certain coordinate are due to the continuous selection of vertices that are patterned regularly. the points of the formation are spread unevenly and only "convergent" to a certain point as much as the pattern specified, and not converging on the four points of the support point as stated in previous studies. this causes the results of the chaos game with non-random rules not a modification of chaos game with rotational variation on a square kosala dwidja purnomo 32 fractals form because they have no similarity to each other (have no self-similarity). the coordinates of the new points generated from the last few iterations of non-random rules 1 to non-random rules 2 shown in figure 8. figure 8. new coordinate points of non-random 1 (left) and non-random 2 (right) it is different from random rules, where all the modifications form fractal objects because the selection of points is done randomly. according to [3] this is because the main principle of game chaos is to use chaos theory, which is a description of the complex movements and behavior of certain non-linear dynamic systems whose movements depend heavily on the initial state. chaos formation of games with point selection is done non-randomly with a regular pattern that is determined as much as anything but still will not form a fractal due to the selection of regular and recurring points, so that the point will gather at a certain point. previous studies that combined random and non-random rules also did not show fractal objects due to the involvement of non-random patterns, so they did not form fractals, although there were some patterns that were almost fractal, but less identical and still said that the object was not is a fractal. conclusions rotational variation of chaos game on a square/quadrilateral are built by random rules and all their modifications produce fractal objects because they have similar shapes (have self-similarity). whereas the chaos game with non-random rules which the pattern is repeating points every 4 patterns, 16 patterns, 20 patterns, 24 patterns are not fractal objects because the formation points are far apart so they do not have similarity in form to one another points, but they have the characteristic of "convergent" repeating at certain coordinate points as much as the pattern of selection of a specified point. based on the results of the study, we get an open problem that can be done for further research: 1. shows other modifications such as a combination of rotation variations and compression ratio; 2. show analytic studies to prove that a chaos game object is a fractal modification of chaos game with rotational variation on a square kosala dwidja purnomo 33 acknowledgements we gratefully acknowledge the support from keris gerbang mata and lp2m – university of jember of year 2019. references [1] devaney, robert l. 2003. fractal patterns and chaos game. department of mathematics: boston university, boston ma 02215. [2] shenker, o. r. 1994. fractal geometry is not the geometry of nature. jerusalem: the hebrew university of jerusalem. [3] sulistiyantoko, d. 2008. application of sequence row on simple fractal geometry calculation formations. essay. mathematics education program, faculty of science and technology, sunan kalijaga uin yogyakarta. [4] miller, c. 2011. communicating mathematics iii: z-corp 650. http://www.maths.dur.ac.uk. [5] [6] [7] [8] [9] yunaning, f. 2018. study of the rules for non-random chaos games in the triangle. essay. faculty of mipa, university of jember. zohuri, b. 2015. dimensional analysis and self-similarity methods for engineers and scientist. switzerland: springer international publishing. purnomo, kosala d. 2016. study of sierpinski triangle establishment using affine transformation. journal of mathematics: faculty of mipa, universty of udayana, vol. vi no. 2:86-92. sulastri. 2007. build transformation in three-dimensional space using visual basic 6.0. journal of dynamic information technology volume xii, no. 1: 88-100. stikubank university faculty of information technology. sundbye, l. 1997. chaos and fractals on the ti graphing calculator. department of mathematical and computer sciences: metropolitan state college of denver. http://www.maths.dur.ac.uk/ selection of specialization class using support vector machine method in sma negeri 1 ambon cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 162-168 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 26, 2020 reviewed: november 10, 2020 accepted: march 25, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.8882 selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela1, yopi andry lesnussa2, venn yan ishak ilwaru3 1mathematics department, pattimura university email: annytamaela@gmail.com, yopi_a_lesnussa@yahoo.com, vennilwaru007@gmail.com abstract the curriculum is a system or education plan that is made by a government to form the abilities and character of children based on a standard, one of its forms is the division of specialization classes at the senior high school level. the 2013 curriculum emphasizes that all students in indonesia can practice their abilities based on their interests and talents, therefore students no longer choose majors but choose abilities (interests) they specialize in. this research uses the support vector machine (svm) method in the decision making system (dms) of specialization classes at sma negeri 1 ambon. using the factors driving student acceptance and selection as input data, the svm method is processed with matlab software and produces a classification of interest class with an accuracy rate of 100%. keywords: 2013 curriculum; classification; support vector machine introduction the curriculum is a plan about forming the abilities and character of children based on a standard [1], the 2013 curriculum is a curriculum that simplification [2] and thematic-integrative, adding rainy learning hours to encourage students or students, one of the forms is the division of specialization classes at the high school level (sma)[3]. specialization aims to provide opportunities for students to develop their interests in a group of subjects by their scientific interests in tertiary institutions [4] and developing their interest in a particular discipline or skill [5]. decision making system is the process of choosing between two or more alternative actions to achieve goals or objectives. several studies have concluded that there are various decision-making methods can determine reliable results. in this case, the method to be applied is the support vector machine (svm) algorithm, the svm method is a technique for making predictions, both in the case of classification and regression [6][7][8]. svm was introduced by boser, guyon, vanpik, dan first presented in 1992 [9][10] at the annual workshop on computational learning theory. the basic concept of svm is a harmonization of computation theories that have existed tens of years before, such as the hyperplane margin introduced by aronszajn in 1950 [11]. however, until 1992 there had never been an effort to assemble these components. the basic principle of svm is linear classifier [12], and further developed to be able to work on non-linear problems by incorporating the concept of kernel tricks in highhttp://dx.doi.org/10.18860/ca.v6i4.8882 mailto:annytamaela@gmail.com mailto:yopi_a_lesnussa@yahoo.com mailto:vennilwaru007@gmail.com selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 163 dimensional space [13]. svm is a classification method with a relatively fast computative time and several studies have shown that svm is superior in classifying. in addition, there is a research about the classification of acoustic events using svm-based clustering schemes. several classifiers based on support vector machines (svm) are developed using confusion matrix based clustering schemes to deal with the multiclass problem [14][15]. the main purpose of svm is to increase the speed of training and testing, meaning that svm can be used for large data [16]. support vector machine (svm) is used along with continuous wavelet transform (cwt), an advanced signalprocessing tool, to analyze the frame vibrations during start-up [17]. methods this type of research is a case study. namely, the svm method determines the choice of interest in tenth-grade students. the material used in this study is secondary data obtained from the sekolah menengah atas (sma) negeri 1 ambon. secondary data taken from sma negeri 1 ambon is in the form of a value criterion as a measure of interest selection include the initial test scores in sma 1 or a comparison score to determine the choice of interest in tenth-grade students. in detail, the flowchart diagram of the support vector machine (svm) can be seen in the following figure 1: figure 1. flowchart diagram of support vector machine (svm) this research uses the linear svm method with matlab software to analyze data. the ratio of training and testing data used is 7:3 from each grouping that has been determined. start grouping data dividing data for training and for testing classification svm score > 0 science major classification results no finish social major yes selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 164 for this research method the variables used for data are the national exam score (𝑋1), initial test score (𝑋2), psychology score (𝑋3). while the target or output is major/ interest (𝑌). data obtained as follows: table 1. svm method data no national exam score (rec) initial test score psychology score final score major/ interest 1 82 35 106.7 74.57 science 2 80.2 34 86.2 66.8 social 3 80 34 100 71.33 science 4 81.5 34 94.4 69.97 social 5 80 34 93.9 69.30 social 6 85.5 33 111.8 76.77 science 7 84.5 33 111.9 76.47 science 8 80 33 86.1 66.37 social 9 81.5 33 109.6 74.70 science 10 80.8 32 106.3 73.03 science 11 80 32 97 69.67 social 12 83 32 106.7 73.90 science 13 82.5 32 92.6 69.03 social 14 81.9 31 107.7 73.53 science 15 80.5 31 92.6 68.03 social 16 80 31 98.7 69.90 social 17 81.5 31 83.3 65.27 social 18 80 31 91.3 67.43 social 19 85.5 30 90.6 68.70 social 20 84.5 30 101.3 71.93 science 21 80 30 89.7 66.57 social 22 80 30 93.6 67.87 social 23 83 30 88.3 67.10 social 24 82.5 30 111.4 74.63 science 25 81.9 30 97.7 69.87 social 26 80.5 30 109.6 73.37 science 27 80 30 104.5 71.50 science 28 81.5 30 100 70.50 science 29 83 30 91.1 68.03 social 30 82.5 30 103.9 72.13 science 31 81.9 30 100 70.63 science 32 80.5 28 104.2 70.90 science 33 83.9 28 106 72.63 science 34 84 28 92.9 68.30 social 35 80.5 28 87.7 65.40 social 36 80.4 28 86.9 65.10 social 37 81.3 28 111.9 73.73 science selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 165 no national exam score (rec) initial test score psychology score final score major/ interest 38 81.9 28 92.5 67.47 social 39 81 28 86.8 65.27 social 40 80.2 28 105 71.07 science 41 82 27 109.9 72.97 science 42 83.6 27 110.5 73.70 science 43 80 27 93.8 66.93 social 44 82.5 27 94.9 68.13 social 45 81.6 27 91.3 66.63 social 46 85 27 106.6 72.87 science 47 82.9 27 106 71.97 science 48 83.5 27 94.7 68.40 social 49 80 27 93.3 66.77 social 50 80 27 111.1 72.70 science the amount of data in this study amounted to 50 data that will be divided into data testing and training. the target data of the department is divided into 3 groups with the following conditions: science major for y > 80 with the very smart requirement neutral major for 70 ≤ y ≤ 80 with the average requirement social major for y < 70 hint: y = major results and discussion svm linear search for hyperplane with the largest margin, known as maximum marginal hyperplane (mmh). based on the lagrangian formulation mentioned, mmh can be rewritten as a boundary decision: 𝑑(𝑋𝑇) = ∑ 𝑦𝑖𝛼𝑖𝑋𝑖𝑋 𝑇 + 𝑏0 𝑙 𝑖=1 (1) where 𝑦𝑖 is the label of support vector classes 𝑋𝑖. 𝑋 𝑇 is the tuple; 𝛼𝑖 and 𝑏0 are the numerical parameter determined automatically by the svm optimization algorithm, and l is the number of support vector. the results of the program for the grouping of testing data are as follows: table 2. software processing results for testing data no original data svmstruct1 svmstruct2 svmstruct3 software results 1 0 0 0 0 0 2 0 1 0 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 6 0 0 0 0 0 7 0 0 0 0 0 8 0 1 0 0 0 selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 166 no original data svmstruct1 svmstruct2 svmstruct3 software results 9 1 1 1 1 1 10 1 1 1 1 1 11 1 1 1 1 1 12 1 1 1 1 1 13 1 0 1 1 1 14 1 1 1 1 1 15 1 0 1 1 1 16 1 1 1 1 1 from table 2 above it can be seen, the svm method classification shows the output with 99% accuracy for testing data. this can be seen from the comparison between the target data and the output target. with hyperplane obtained from each class as follows: figure 2. hyperplane for svmstruct1 based on figure 2 above it can be explained that the svmstruct1 data shows that the data in the two classes are not completely separate can be seen from several red circles whose distribution is around the green circle area. figure 3. hyperplane for svmstruct2 selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 167 based on figure 3 above it can be explained that the data in svmstruct2 is properly classified in class 0 (which is the science major group) or class 1 (which is the social major group) while those that are support vector are patterns within a circle. figure 4. hyperplane for svmstruct3 based on figure 4 above it can be explained that all svmstruct3 data is properly classified into class 0 (which is the science major) or class 1 (which is the social major), while those that are support vector are patterns within a circle. conclusions from the results of the software output that is matched with the real data, it is seen that 100% of the output data is the same as the real data. this shows for the svm method of class division of interests carried out by the school according to the ability of children. based on the results of research with the svm method there is no difference between the results of the assessment using software and real data. in other words the results of the svm method are said to be perfect. references [1] m. lestari, “implementation of citizenship character formation by the study of civic education on senior high school in the district of bantul,” e-civics, 2016. [2] suyatmini, “implementasi kurikulum 2013 pada pelaksanaan pembelajaran akuntansi di sekolah menengah kejuruan,” j. pendidik. ilmu sos., 2017. [3] subandi, “pengembangan kurikulum 2013,” j. pendidik. dan pembelajaran dasar, 2014. [4] b. l. julien, l. lexis, j. schuijers, t. samiric, and s. mcdonald, “using capstones to develop research skills and graduate capabilities: a case study from physiology,” j. univ. teach. learn. pract., 2012. [5] direktorat pembinaan sma, modul pelatihan implementasi kurikulum 2013 sma tahun 2018. 2018. [6] a. s. nugroho, a. b. witarto, and d. handoko, “application of support vector machine in bioinformatics,” 2003. [7] a. m. puspitasari, d. e. ratnawati, and a. w. widodo, “klasifikasi penyakit gigi dan mulut menggunakan metode support vector machine,” j. pengemb. teknol. inf. dan selection of specialization class using support vector machine method in sma negeri 1 ambon stevanny tamaela 168 ilmu komput., vol. 2, no. 2, pp. 802–810, 2018. [8] p. a. octaviani, yuciana wilandari, and d. ispriyanti, “penerapan metode klasifikasi support vector machine (svm) pada data akreditasi sekolah dasar (sd) di kabupaten magelang,” j. gaussian, vol. 3, no. 8, pp. 811–820, 2014, [online]. available:http://download.portalgaruda.org/article.php?article=286497&val=47 06&title=penerapan metode klasifikasi support vector machine (svm) pada data akreditasi sekolah dasar (sd) di kabupaten magelang. [9] y. lin, h. tseng, and c. fuh, “using support vector machine,” image process., 2003. [10] m. ichwan, i. a. dewi, and z. m. s, “klasifikasi support vector machine (svm) untuk menentukan tingkatkemanisan mangga berdasarkan fitur warna,” mind j., vol. 3, no. 2, pp. 16–23, 2019, doi: 10.26760/mindjournal.v3i2.16-23. [11] y. wang, j. wong, and a. miner, “anomaly intrusion detection using one class svm,” 2004, doi: 10.1109/iaw.2004.1437839. [12] m. a. oskoei and h. hu, “support vector machine-based classification scheme for myoelectric control applied to upper limb,” ieee trans. biomed. eng., 2008, doi: 10.1109/tbme.2008.919734. [13] s. vijayakumar and s. wu, “sequential support vector classifiers and regression,” 1999. [14] s. aprilia, m. t. furqon, and m. a. fauzi, “klasifikasi penyakit skizofrenia dan episode depresi pada gangguan kejiwaan dengan menggunakan metode support vector machine (svm),” j. pengemb. teknol. inf. dan ilmu komput., vol. 2, no. 11, pp. 5611–5618, 2018. [15] a. temko and c. nadeu, “classification of acoustic events using svm-based clustering schemes,” pattern recognit., vol. 39, no. 4, pp. 682–694, 2006, doi: 10.1016/j.patcog.2005.11.005. [16] h. amalia, a. f. lestari, and a. puspita, “penerapan metode svm berbasis pso untuk penentuan kebangkrutan perusahaan,” none, 2017. [17] p. dhanalakshmi, s. palanivel, and v. ramalingam, “classification of audio signals using svm and rbfnn,” expert syst. appl., vol. 36, no. 3 part 2, pp. 6069–6075, 2009, doi: 10.1016/j.eswa.2008.06.126. the simulation study to test the performance of quantile regression method with heteroscedastic error variance cauchy –jurnal matematika murni dan aplikasi volume 5(1)(2017), pages 36-41 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 15 may 2017 reviewed: 24 july 2017 accepted: 2 november 2017 doi: http://dx.doi.org/10.18860/ca.v5i1.4209 the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar1*, laila hasnah2, dodi devianto3 1,2,3jurusan matematika fmipa universitas andalas kampus limau manis 25163 padang * corresponding author. e-mail: ferrayanuar@yahoo.co.id abstract least square estimator has many limitations. this estimator will not be a best linear unbiased estimator (blue) in the condition of the variance error term have heteroscedasticity problem. quantile regression is a robust approach in situations where the limitation addressed above present for least square estimator. the purpose of this study is to describe the performance of quantile regression method in modeling a data set which contain the heteroscedasticity problem. to achieve the goal, a data set is generated and statistical framework quantile method then applied to the data. the consistency of the proposed model is then checked by doing a simulation study. this study proves that the quantile regression method is able to produce acceptable parameter model since the proposed models have large pseudo r2 and small mean square error (mse) for all parameter estimated. it could be conclude here that quantile regression method is an unbiased estimator method and able to result acceptable model althought in the present of heteroscedasticity problem of variance error. keywords: heteroscedasticity, quantile regression, simulation study, pseudo r2, mean square error (mse). introduction in modeling the relationship between covariates and responses, it need estimator method to estimate the parameter model. to estimate the value of parameters, it usually use the ordinary least squares (ols). the principle of this method is to minimize the sum of the squares of the error. this ols is applied if all model assumptions are met (independent observations, linearity of conditional means, normality of response variable and homogeneity of error variance). in all model assumptions are met, the estimator method is called as blue (best linear unbiased estimator). however, if one or more of the assumptions are not met, the results could be misleading [1]. in regression, an error is how far a point deviates from the regression line. ideally, our data should be homoscedastic (i.e. the variance of the errors should be constant). in many real world applications, this situation rarely happens. most data is heteroscedastic by nature. due to these limitatios, an alternative approach to classical linear regression is in demand. quantile regression is a robust approach in situations where the limitations addressed above [2] present for ordinary least square estimator. the quantile method is one of the regression modeling methods by dividing a batch of data into the same parts after the data is sorted from the smallest or the largest [1, 3]. quantile regression is an approach in regression analysis introduced by koenker and basset [4]. quantile mailto:ferrayanuar@yahoo.co.id the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar 37 regression in his theory is able to overcome the violation of normality assumptions, heteroscedasticity, multicollinearity problems and so on. this method uses the parameter estimation approach by separating or dividing the data into quantities, by assuming the conditional quantization function on a distribution of data and minimizing the absolute asymmetry of unsymmetric weighted error and presupposes a conditional quantile function on a distribution of data [5]. in this paper, we adopt the quantile regression approach to modeling groups of data with non homogeneity of error variance. section 2 of the paper, describes the theoretical framework of quantile regression and its indicators to determine the goodness of fit of the proposed model. in section 3, we illustrate the implementation of quantile regression through a simulated case-study. we choose two covariates in our model hypothesis.as the predictors to the response variable. we end with a short discussion in section 4. fundamental theories and related works in stricly linear models, a simple approach to estimating the conditional quantiles is suggested in koenker and basset [2]. based on the classical regression model, we have: 𝑦𝑖 = 𝒙𝒊′𝜷 + 𝜀𝑖 , 𝑖 = 1,2, … , 𝑛 (1) where 𝑦𝑖 are dependent variable for each data , 𝑥𝑖 are independent matrix 𝑛𝑥𝑝, with (𝑦𝑖 ) = 𝒙𝒊′𝜷, 𝜷 is vector of parameter model size 𝑝𝑥1, 𝜀𝑖 are error for each data . the parameter estimate by classical regression, by minimizing the sum of the error squares, is written as 𝑚𝑖𝑛 ∑ (𝑦𝑖 − 𝑦�̂�) 2𝑛 𝑖=1 (2) where are estimated value for each data . 2.1. quantile regression method the prediction based on the median, that is, by minimizing the absolute number of errors can be written using following equation : 𝑚𝑖𝑛 ∑ |𝑦𝑖 − 𝑦�̂�| 2𝑛 𝑖=1 (3) furthermore, the linear equations for -th quantil can be written as follows : 𝑦𝑖 = 𝒙𝒊′𝜷𝝉 + 𝜀𝑖 , 𝑖 = 1,2, . . , 𝑛 (4) noted that 𝑦�̂� = 𝒙𝒊′𝜷𝝉, the parameter estimate for th quantile is to minimize the absolute value of the error by weighting τ for the positive and weighted error (1τ) for the negative error [4]. the 𝜏th (0 < 𝜏 < 1) quantile of 𝜀𝑖 is the value, 𝑄𝜏, for which 𝑃 (𝜀𝑖 < 𝑄𝑌(𝜏)) = 𝜏. the 𝜏th conditional quantile of 𝑦𝑖 given 𝒙𝒊 is then simply [6,7] : 𝑄𝜏(𝑦𝑖 |𝒙𝒊) = 𝒙𝒊′𝜷𝝉, where 𝜷𝝉, is a vector of coefficients dependent of 𝜏. the 𝜏th regression quantile is defined as any solution, �̂�𝝉, to the quantile regression minimasation problem : 𝑚𝑖𝑛𝛽𝜖ℛ ∑ 𝜌𝜏 𝑛 𝑖=1 (𝑦𝑖 − 𝒙𝒊′𝜷𝝉), (5) where the loss function : 𝜌𝜏(𝑢) = 𝑢(𝜏 − 𝐼(𝑢 < 0)). (6) equivalently, we may rewrite (5) as : 𝑚𝑖𝑛𝛽𝜖ℛ {∑ 𝜏|𝑦𝑖 − 𝒙𝒊′𝜷𝝉| + ∑ (1 − 𝜏)|𝑦𝑖 − 𝒙𝒊′𝜷𝝉| 𝑛 𝑖=1 𝑛 𝑖=1 } (7) the meaning which can be added to explain the equation (7), namely that all observations greater than the quantile value, multiplied by the weighting τ and the observations whose value is less than the quantile multiplied by 1 − 𝜏. 2.2 goodness of fit using pseudo r2 simple quantile regression model with n independent variables can be formed as follows: 𝑄𝜏(�̂�|𝒙) = �̂�0(𝜏) + �̂�1(𝜏)𝒙 + ⋯ + �̂�𝑛 (𝜏)𝒙 (8) i i ˆ iy i   the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar 38 the indicator of the goodness of fit for the model can be predicted with pseudo r2 as defined below [2]: pseudo 𝑅𝜏 2 = 1 − 𝑅𝐴𝑊𝑆𝜏 𝑇𝐴𝑆𝑊𝜏 (9) where : 𝑅𝐴𝑊𝑆𝜏 = ∑ 𝜏|𝑦𝑖 − �̂�0(𝜏) − �̂�1(𝜏)𝒙𝒊 − ⋯ − �̂�𝑛 (𝜏)𝒙𝒊| 𝑦𝑖≥�̂�0(𝜏)+�̂�1(𝜏)𝒙𝒊+⋯+�̂�𝑛(𝜏)𝒙 + ∑ (1 − 𝜏)|𝑦𝑖 − �̂�0(𝜏) − �̂�1(𝜏)𝒙𝒊 − �̂�𝑛(𝜏)𝒙𝒊|𝑦𝑖<�̂�0(𝜏)+�̂�1(𝜏)𝒙𝒊+⋯+�̂�𝑛(𝜏)𝒙 (10) and 𝑇𝐴𝑆𝑊𝜏 = ∑ 𝜏|𝑦𝑖 − �̂�| +𝑦𝑖≥𝜏 ∑ (1 − 𝜏)|𝑦𝑖 − �̂�|𝑦𝑖<𝜏 (11) the value of 𝑅𝐴𝑊𝑆𝜏 (residual absolute sum of weighted) is always less than the value of 𝑇𝐴𝑆𝑊𝜏 (total absolute sum of weighted) so that the pseudo 𝑅𝜏 2 will be in the range 0 to 1. the closer the pseudo r2 value to one the model will be better. however, the virtues of pseudo r2 can not be used to test the overall goodness of fit for the model, it can only be used to test the merits of the selected quantile [1,2]. 2.3 mean square error (mse) the parameter estimate obtained is said to be good if it has a small bias and small variance. therefore, to see the goodness of estimating the parameters based on the bias and variance values simultaneously, represented in the value of mean square error (mse) [8, 9, 10], formulated as follows : 𝑀𝑆𝐸(�̂�𝑗 (𝜏𝑞 )) = 𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 )) + 𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 )) 2 (12) where : 𝑀𝑆𝐸(�̂�𝑗 (𝜏𝑞 )) : value of mse for 𝑗 = 1,2, … 𝑝 𝜏𝑞 = 𝑞𝑢𝑎𝑛𝑡𝑖𝑙 𝑞 = 1,2, … , 𝑘 𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 )) : variance for selected quantile 𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 )) = 𝑛 ∑ (�̂�𝑗(𝜏𝑞)) 2𝑛 𝑖=1 −(∑ (�̂�𝑗(𝜏𝑞)) 𝑛 𝑖=1 ) 2 𝑛(𝑛−1) 𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 )) : the value of bias for selected quantile is obtained from the mean of the difference of the expected value and the estimated value, or : 𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 )) = 1 𝑛 ∑ ((�̂�𝑗 (𝜏𝑞 )) − 𝑛 𝑖=1 (𝛽𝑗 (𝜏𝑞 )) result and discussion we describe our approach to quantile regression by conducting the simulation study. in this research, we design two covariates each measuring 100 samples. the response variable, 𝑦𝑖 is generated from the model : 𝑦𝑖 = 𝛽0 + 𝛽1𝑥𝑖1 + 𝛽2𝑥𝑖2 + 𝜀𝑖 , 𝑖 = 1, … ,100 (13) where covariate 𝑥𝑖1 is generated from a standard normal distribution and 𝑥𝑖2 is generated from exponential with one degrees of freedom. the parameter 𝛽0, 𝛽1 and 𝛽2 are set to 1.2, 1, and 1.7 respectively. the data for error is also generated by taking the mean at zero and its variances have heteroscedasticity problems. we consider the heteroscedastic normal, 𝑁(0, √0.01 x (𝑿𝜷)2) for distribution of error term. in this example, we choose 𝜏 = 0.10, 0.25, 0.50, 0.75 and 0.90 as the quantile points for estimated. table 1 below shows the parameter estimated and its corresponding standard error. the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar 39 table 1. quantile regression estimates 𝜏th quantile parameter estimates standard error 0.10 b0 1.0251* 0.0308 b1 0.8004* 0.0220 b2 1.4783* 0.0219 0.25 b0 1.0891* 0.0310 b1 0.8438* 0.0222 b2 1.5931* 0.0220 0.50 b0 1.1840* 0.0417 b1 0.9955* 0.0298 b2 1.7640* 0.0296 0.75 b0 1.2829* 0.0435 b1 1.0871* 0.0312 b2 1.8561* 0.0309 0.90 b0 1.3215* 0.0287 b1 1.1514* 0.0205 b2 2.0225* 0.0204 (* significant at α = 0,05) table 1 informs us that estimated coefficient (�̂�0, �̂�1 and �̂�2) for all quantile points are close to initial value, 𝑏0 = 1.2 𝑏1 = 1 and 𝑏2 = 1.7. for example, consider at 0.50th quantile, proposed parameter estimated are �̂�0 = 1.1840, �̂�1 = 0.9955 and �̂�2= 1.7640. the next analysis in the quantile regression is a consistency test of the proposed model to reveal the performance of the quantile approach and its associated algorithm in recovering the true parameters of the quantile regression analysis. consistency test is done by doing simulation study. simulation study does so by generating a set of new data set by sampling with replacement from the original data set, and fitting the model to each new data set [11, 12]. to compute standard errors for calculating the 95% confidence interval of all parameters in this study, roughly 25 model fits are determined. the goodness of fit of each model are also calculated. table 2 presents the result taken from the simulation study. table 2. simulation results of 25 data sets using quantile regression approach -th quantile parameter paramter estimated (standard error) 95% interval confidence pseudo r2 0.10 b0 1.0317* (0.0439) (1.0233 ; 1.0401) 0.8104 b1 0.8974* (0.0312) (0.8898 ; 0.9049) b2 1.5091* (0.0321) (1.4973 ; 1.5208) 0.25 b0 1.0992* (0.0322) (1.0928 ; 1.0992) 0.8288 b1 0.9334* (0.0229) (0.9270 ; 0.9398) b2 1.1613* (0.0229) (1.6000 ; 1.6226) 0.50 b0 1.1912* (0.0301) (1.1861 ; 1.1972) 0.8477 b1 0.9949* (0.0220) (0.9906 ; 0.9992) b2 1.6973* (0.0218) (1.6897 ; 1.7050) 0.75 b0 1.2837* (0.0341) (1.2747 ; 1.2928) 0.8655 b1 1.0533* (0.0243) (1.0478 ; 1.0589) b2 1.8074* (0.0244) (1.7925; 1.8223) 0.90 b0 1.3693* (0.0414) (1.3550 ; 1.3837) 0.8854 b1 1.1064* (0.0299) (1.0961 ; 1.1168) b2 1.9164* (0.0295) (1.8976 ; 1.9352) (* significant at α = 0,05) table 2 informs us that the estimated value of the model parameter coefficients (�̂�0, �̂�1 and �̂�2) on each quantile are close to the initial value (𝑏0 = 1.2 𝑏1 = 1 and 𝑏2 = 1.7). for example,  the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar 40 consider the 50th quantile, the estimated value for �̂�0, �̂�1 and �̂�2 are 1.1916, 0.9949, and 1.6973 respectively. based on table 2, we also know that all parameter estimated values fall within 95% confidence intervals obtained from the simulation study. it means quantile 95% confidence interval seem to work well here and parameter estimated are acceptable. the goodness of fit for the each quantile regression model is presented by the value of pseudo r2, shown in the last column in table 2. all pseudo r2 values obtained here are more than 80%, indicating that all proposed model are adequate and could be accepted. in this study we also determine the value of of mse (mean square error) to ensure that parameter estimated have small bias and small variance. table 3 below presents the mse value of the quantile regression method for all three parameter estimated at corresponding quantile points. table 3. the value of mse for any quantile points parameter mse (mean square error) 0.10 0.25 0.50 0.75 0.90 b0 0.0011 0.0011 0.0008 0.0022 0.0054 b1 0.0015 0.0012 0.0005 0.0008 0.0028 b2 0.0037 0.0033 0.0015 0.0058 0.0093 table 3 above gives information that all parameter estimated have small mse. these results indicate that quantile regression method is able to produce unbiased parameter model since it has small bias and small variance. based on any results here, we could believe that the power of our quantile regression method result the best fit for the model althought in the existence of heteroscedastic error variance. conclusions this present study purposes to describe the performace of quantile regression method in modeling the data containing non-uniform variance problem (heteroscedasticity). a data set with two covariates is generated, each measuring 100 samples. the distribution for error term is heteroscedastic normal, 𝑁(0, √0.01 x (𝑿𝜷)2) is designed to generate a response variable. each parameter model are also set as initial values. a simulation study is done to check the power of quantile regression algorithm. this study resulted that all parameter estimated are close to the initial values. the value of pseudo r2 for all proposed model at any selected quantile points are quite large, more than 80%. based on simulation study, the value of parameter estimated are within 95% confidence intervals indicating that parameter estimated could be accepted. this study also result that quantile regression method is able to produce small value of mse. therefore, it could be concluded here that quantile regression methods is unbiased estimator method and could result the acceptable model although in the due of heteroscedasticity problem of error variance. references [1] ferra y, hazmira y and izzati r. 2016. penerapan metode regresi kuantil pada kasus pelanggaran asumsi kenormalan sisaan. eksakta, 1 (xvii) : 33 – 37. [2] davino c, furno m, and vistocco d. 2014. quantile regression: theory and applications. john wiley & sons, ltd. [3] arbia, g. 2006. spatial econometrics: statistical foundation application to regional convergence. springer, berlin [4] koenkar.r. and basset,g.jr.1978. quantiles regression. econometrica. 46. 33-50 [5] kozumi h and kobayashi g. 2011. gibbs sampling methods for bayesian quantile regression. journal of statistical computation and simulation, 81 (11) : 1565 – 1578. [6] feng x and zhu l. 2016. estimation and testing of varying coefficients in quantile regression. journal of the american statistical association 111, 266 – 274. the simulation study to test the performance of quantile regression method with heteroscedastic error variance ferra yanuar 41 [7] yanuar f. 2013. quantile regression approach to determine the indicator of health status. scientific research journal , i (iv): 17 – 23. [8] yang y, wang hj and he x. 2015. posterior inference in bayesian quantile regression with asymmetric laplace likelihood. international statistical review, 0 0, 1-18 doi: 10.1111.insr.12114 [9] he x. and zhu lx. 2011. a lack of fit test for quantile regression. journal of the american statistical association, 98 (464) : 1013 1022 [10] feng, x., he, x., and hu, j. 2011. wild boostrap for quantile regression. biometrica, 98, 995–999. [11] yanuar f, ibrahim k and jemain aa. 2013. bayesian structural equation modeling index in the health model. journal of applied statistics, 40 (6) : 1254–1269. [12] yanuar f. 2014. the estimation process in bayesian structural equation modeling approach. journal of physics : conference series, 495, 012047. on the local adjacency metric dimension of generalized petersen graphs cauchy –jurnal matematika murni dan aplikasi volume 6(1) (2019), pages 10-17 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 22, 2019 reviewed: october 02, 2019 accepted: november 30, 2019 doi: http://dx.doi.org/10.18860/ca.v6i1.6487 on the local adjacency metric dimension of generalized petersen graphs marsidi1, dafik2, ika hesti agustin3, ridho alfarisi4 1mathematics edu. depart. ikip pgri jember indonesia 2mathematics edu. depart. university of jember indonesia 3mathematics depart. university of jember indonesia 4elementary school teacher edu. depart. university of jember indonesia email: marsidiarin@gmail.com, d.dafik@gmail.com, ikahesti@fmipa.unej.ac.id. abstract the local adjacency metric dimension is one of graph topic. suppose there are three neighboring vertex 𝑎, 𝑏, 𝑐 in path 𝑎 − 𝑐. path 𝑎 − 𝑐 is called local if 𝑎,𝑏,𝑐 where each has representation: a is not equals 𝑏 and 𝑎 may equals to 𝑐. let’s say, 𝑥,𝑦 ∈ 𝑉(𝐺). for an order set of vertices 𝐻 = {ℎ1,ℎ2, . . . ,ℎ𝑘}, the adjacency representation of 𝑣 with respect to 𝐻 is the ordered 𝑘-tuple 𝑟𝐴(𝑥𝑗|𝐻) = (𝑑𝐴(𝑥,ℎ1),𝑑𝐴(𝑥,ℎ2), . . . ,𝑑𝐴(𝑥,ℎ𝑘)), where 𝑑𝐴(𝑥,ℎ) represents the adjacency distance 𝑥 − ℎ. the distance 𝑑𝐴(𝑥,ℎ) defined by 0 if 𝑥 = ℎ𝑖, 1 if 𝑥 adjacent with ℎ, and 2 if 𝑥 does not adjacent with ℎ. the set 𝐻 is a local adjacency resolving set of 𝐺 if for every two distinct vertices 𝑥, 𝑦 and 𝑥 adjacent with y then 𝑟𝐴(𝑥𝑗|𝐻) =𝑟𝐴(𝑦𝑗|𝐻). a minimum local adjacency resolving set in 𝐺 is called local adjacency metric basis. the cardinality of vertices in the basis is a local adjacency metric dimension of 𝐺, denoted by (𝑑𝑖𝑚𝐴,𝑙(𝐺)). next, we investigate the local adjacency metric dimension of generalized petersen graph. keywords: local resolving set; local (adjacency) metric dimension; adjacency metric dimension; generalized petersen. introduction a graph 𝐺 is defined by set of 𝑉(𝐺) and 𝐸(𝐺), the set of vertices and the set of edges of 𝐺, for more details of the definition in [1,2]. the metric dimension is one of interesting studied graph topics. local means that every adjacent two vertices or two edges has distinct representation. let’s say, there are three neighboring vertex in a path, 𝑎, 𝑏, 𝑐 where each has representation: 𝑎 = 𝑏 and 𝑎 may equals 𝑐. then, the path 𝑎 − 𝑐 is called local. the local adjacency metric dimension is combination of local metric dimension and adjacency metric dimension [3]. let 𝐺 = (𝑉,𝐸) be a connected simple finite graph and 𝑢, 𝑣 in 𝐺. for an ordered set of vertices 𝑋 = {𝑥1,𝑥2, . . . ,𝑥𝑘}, the adjacency representation of 𝑣 with respect to 𝑋 is the ordered 𝑘-tuple 𝑟𝐴(𝑣|𝑋) = (𝑑𝐴(𝑣,𝑥1),𝑑𝐴(𝑣,𝑥2), . . . , 𝑑_𝐴 (𝑣,𝑥_𝑘)), where 𝑑𝐴(𝑢,𝑣) represents the adjacency distance 𝑢 − 𝑣. 𝑑𝐴(𝑢,𝑣) defines by 0 if 𝑢 = 𝑣𝑖, 1 if 𝑢 ∼ 𝑣, and 2 if 𝑢 ≉ 𝑣. we called 𝑋 is a local adjacency resolving set of g if for every two distinct vertices 𝑢, 𝑣 and 𝑢 ∼ 𝑣 then 𝑟𝐴(𝑢|𝑋) =𝑟𝐴(𝑣|𝑋). a local adjacency metric basis of g is a minimum local adjacency resolving set in g. the cardinality of vertices in the basis is a local adjacency metric dimension of 𝐺(𝑑𝑖𝑚𝐴,𝑙(𝐺)). the research is originated by rodriguez, et al. [3] about local adjacency metric dimension of corona graphs. marsidi, et. al. determined the local metric dimension of line graph of special graphs. then in 2017 rinurwati, et al. [5] researched about local adjacency metric dimension of some wheel related graphs with pendant vertices. http://dx.doi.org/10.18860/ca.v6i1.6487 mailto:d.dafik@gmail.com on the local adjacency metric dimension of generalized petersen graphs marsidi 11 recently, darmaji, et al. [4] studied about local adjacency metric dimension of sun graph and stacked book graph this year. results and discussion in this section, we investigate the local adjacency metric dimension of generalized petersen graph 𝐺𝑃(𝑛,𝑘) for 𝑘 = 2 as follows theorem 1. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+2 7 , for 𝑛 ≡ 4 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacensy resolving set 𝑊 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑣𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑣𝑛−2}| = 3𝑛+2 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛+2 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛+2 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛+2 7 , we choose |𝐻| = 3𝑛+2 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) =. {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+2 7 , for 𝑛 ≡ 4 𝑚𝑜𝑑 7. ∎ theorem 2. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+6 7 , for 𝑛 ≡ 5 𝑚𝑜𝑑 7. on the local adjacency metric dimension of generalized petersen graphs marsidi 12 proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛+6 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛+6 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛+6 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛+6 7 , we choose |𝐻| = 3𝑛+6 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2))= 3𝑛+6 7 , for 𝑛 ≡ 5 𝑚𝑜𝑑 7. ∎ theorem 3. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+3 7 , for 𝑛 ≡ 6 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,…,2}⏟ ( 3𝑛+2 7 ) , for 𝑖 ≡ 6 𝑚𝑜𝑑 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} on the local adjacency metric dimension of generalized petersen graphs marsidi 13 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛+3 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛+3 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛+3 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛+3 7 , we choose |𝐻| = 3𝑛+3 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+3 7 , for 𝑛 ≡ 6 𝑚𝑜𝑑 7. ∎ theorem 4. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛 7 , for 𝑛 ≡ 0 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 on the local adjacency metric dimension of generalized petersen graphs marsidi 14 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛+2 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛 7 , we choose |𝐻| = 3𝑛 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2))= 3𝑛 7 , for 𝑛 ≡ 0 𝑚𝑜𝑑 7. ∎ theorem 5. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+4 7 , for 𝑛 ≡ 1 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻 ) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛+4 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛+4 7 . furthermore, we prove that the lower bound of local on the local adjacency metric dimension of generalized petersen graphs marsidi 15 adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛+4 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛+4 7 , we choose |𝐻| = 3𝑛 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2))= 3𝑛+4 7 , for 𝑛 ≡ 1 𝑚𝑜𝑑 7. ∎ theorem 6. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛+1 7 , for 𝑛 ≡ 2 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛+1 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛+1 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛+1 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛+1 7 , we choose |𝐻| = 3𝑛+1 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2))= 3𝑛+1 7 , for 𝑛 ≡ 2 𝑚𝑜𝑑 7. ∎ on the local adjacency metric dimension of generalized petersen graphs marsidi 16 theorem 7. the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) = 3𝑛−2 7 , for 𝑛 ≡ 3 𝑚𝑜𝑑 7. proof. the vertex set of 𝐺𝑃(𝑛,2) is 𝑉(𝐺𝑃(𝑛,2)) = {𝑥𝑖,𝑦𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛}. we choose the local adjacency resolving set 𝐻 = {𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7}⋃{𝑦𝑛 −2} such that we have the vertex representation respect to 𝐻 as follows. 𝑟(𝑦𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 3 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑣𝑖|𝐻) = {2,2,…,2}, for 𝑖 ≡ 6 𝑚𝑜𝑑 7 3𝑛+2 7 𝑟(𝑦𝑛−1|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑛|𝐻) = {2,2,…,2} 3𝑛+2 7 𝑟(𝑦𝑖|𝐻) = {2,2,…,2,1,2,2,…,2}, for 𝑖 ≡ 2 𝑚𝑜𝑑 7 𝑖−2 7 3𝑛−𝑖−3 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 4,5 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 𝑟(𝑦𝑖|𝐻) = {2,2,2,…,2,2,2,2,…,2,1,2,…,2,2,2,2,…,2,2,2}, for 𝑖 ≡ 0 𝑚𝑜𝑑 7 𝑖−3 7 𝑖−2 7 3𝑛−𝑖−3 7 3𝑛−𝑖−4 7 the representation in vertex 𝑥𝑖 ∈ 𝑉(𝐺𝑃(𝑛,2)) follows the representation in vertex 𝑦𝑖 such that we have the cardinality of local adjacency resolving set is |𝐻| = |{𝑦𝑖; 𝑖 ≡ 1 𝑚𝑜𝑑 7} ∪{𝑦𝑛−2}| = 3𝑛−2 7 . thus, the upper bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≤ 3𝑛−2 7 . furthermore, we prove that the lower bound of local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) ≥ 3𝑛−2 7 . assume that 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2)) < 3𝑛−2 7 , we choose |𝐻| = 3𝑛−2 7 − 1 such that we have the same representation for two adjacent vertices in 𝐺𝑃(𝑛,2) namely 𝑟(𝑦𝑛|𝐻) ≠ 𝑟(𝑦𝑛−2|𝐻) = {2,…,2}⏟ ( 3𝑛−5 7 ) . thus, the local adjacency metric dimension of 𝐺𝑃(𝑛,2) is 𝑑𝑖𝑚𝐴,𝑙(𝐺𝑃(𝑛,2))= 3𝑛−2 7 , for 𝑛 ≡ 3 𝑚𝑜𝑑 7. ∎ conclusions we have discussed about the local adjacency metric dimension of generalized petersen graph 𝐺𝑃(𝑛,𝑘) for 𝑘 = 2. accordingly, we have some problem for 𝑘 ≥ 3 as follows. open problem 1. find the local adjacency metric dimension of generalized petersen graph 𝐺𝑃(𝑛,𝑘) for 𝑘 ≥ 3 ?. on the local adjacency metric dimension of generalized petersen graphs marsidi 17 acknowledgments we gratefully acknowledge the support from ikip pgri jember 2019 and cgantuniversity of jember of year 2019. references [1] j. l. gross, j. yellen and p. zhang, handbook of graph theory, second edition crc press taylor and francis group, 2014. [2] g. chartrand and l. lesniak, graphs and digraphs 3rd ed london: chapman and hall, 2000. [3] j. a. rodriguez-velazquez and h. fernau, " on the (adjacency) metric dimension of corona and strong product graphs and their local variants", combinatorial and computational results, arxiv: 1309.2275.v1[math.co]. [4] a. y. badri and darmaji, ”local adjacency metric dimension of sun graph and stacked book graph”, journal of physics: conf. series, vol. 974, no. 012069, pp. 01-05, 2018. [5] [6] rinurwati, h. suprajitno, and slamin, ”on local adjacency metric dimension of some wheel related graphs with pendant points”, aip conference proceedings, vol. 1867, no. 020065, pp. 01-06, 2017. marsidi, m., dafik, d., agustin, i. h., & alfarisi, r. (2016). on the local metric dimension of line graph of special graph. cauchy, 4(3), 125-130. stability of cancerous chemotherapy model with obesity effect cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 186-194 p-issn: 2086-0382; e-issn: 2477-3344 submitted: december 07, 2017 reviewed: march 14, 2018 accepted: may 31, 2019 doi: http://dx.doi.org/10.18860/ca.v5i4.4558 stability of cancerous chemotherapy model with obesity effect indah yanti1, ummu habibah2 1,2mathematics department, brawijaya university email: indah_yanti@ub.ac.id abstract in this paper we present stability of cancerous chemotherapy model with obesity effect. this is a four-population model that includes immune cells, cancer cells, normal cells, and fat cells. the analytical result shows that there are four equilibrium points in case the drugs given and fat cells were not equal to zero, i.e., dead equilibrium, total cancer invasion equilibrium, cancer-free equilibrium, and coexistence equilibrium. some numerical simulation also presented to illustrate the results. keywords: cancer; obesity; chemotherapy introduction in the last few years, many studies have shown a relationship between obesity and cancers such as [1], [2], [5], [6], [7], [8]. not only research in the medical field, but also the relationship between cancer and obesity also studied mathematically. in 2015, kucarrillo, delgadillo, and chen-charpentier construct mathematics model for the effect of the obesity on cancer growth and on the immune system response [3]. and the next year, they do research on the effect of the obesity on optimal control schedules of chemotherapy on a cancerous tumor [4]. this paper orginized as follows. in section 2, we describe methods that we used to do the research. in section 3, we find the equilibrium points and its stability, and numerical simulations. finally, conclusions are presented in section 4. methods this research was conducted by doing few steps as folows. we use the model that originally discussed by ku-carrillo                                                 1 1 1 1 1 2 3 5 2 2 2 4 3 3 3 6 4 2 1 1 1 1 1 1 1 u u u u di it s c it d i a e i dt f t dt r t b t c it c tn c tf a e t dt dn r n b n c tn a e n dt df r f b f c tf a e f dt du v d u dt (1) stability of cancerous chemotherapy model with obesity effect indah yanti 187 where i denotes the number of immune cells, t denotes the tumor cells, n denotes the normal cells, f denotes the fat cells which is stored in adipocytes. the constants i r , i b , i c , i a denote cells growth rate, inverse of the population carrying capacity, competition coefficients, and the kill effectiveness of the drug on population respectively. the function v(t) models the application protocol of chemotherapy. parameters , , , ,s u   denote basal response, the immune response stimulated by the cancer cells, the immune response caused by tumor respectively. we refer to system (1) that will be analyzed then. we will determined of the equilibrium points by solving the nullcline equations of system (1). and then observe the local stability of the equilibrium points by using eigen values of the jacobian matrix of each equilibrium points. results and discussion 3.1. mathematical model of cancer growth with obesity effect on the immune system response in this paper the model that we investigate is originally discussed by ku-carrillo [4], that is                                               1 1 1 1 1 2 3 5 2 2 2 4 3 3 3 6 4 1 1 1 1 1 1 1 u u u u di it s c it d i a e i dt f t dt r t b t c it c tn c tf a e t dt dn r n b n c tn a e n dt df r f b f c tf a e f dt . (2) in this system we ignore the last equation from the original ones because of the aimed of the research is to find the stability of equilibrium points by which the drug dose is not equal to zero. where in the origin drug dose system is given by control procedure. 3.2. equilibrium points and existence equilibrium point of system (2) was hold by solving the following system                                               1 1 1 1 1 2 3 5 2 2 2 4 3 3 3 6 4 1 0 1 1 0 1 1 0 1 1 0 u u u u it s c it d i a e i f t r t b t c it c tn c tf a e t r n b n c tn a e n r f b f c tf a e f . system (2) has eight equilibrium which is can be separated to two groups, i.e., four equilibrium under the zero fat condition and four equilibrium under the certain amount of fat condition. our discussion is focused on the second condition. the second condition is impossible because there is always amount of fat in human body. the equilibrium points are the dead point:    4 0 3 3 31 1 11 , 0, 0, 1 u u a es e b b ra e d             , the total cancer invansion point: stability of cancerous chemotherapy model with obesity effect indah yanti 188       *4 6 1* 1 1* * 3 3 31 1 1 1 1 * 4 6 1 * 1 3 3 3 11 , , 0, 1 11 u u u a e c ts e t b b rt c t d a e a e c t t b b r                                          , where 3 2 3 3 6 3 b p a bc ad q p a c r a       1 1 2 a c x x  6 7 1 2 1 2 4b x x x x c x x    2 2 1 5 4 6 2 4 7 c sc x x x x x x x x     2 3 4 5d sc x x x    1 u x e    5 6 1 1 1 3 3 c c x b r b r   6 2 3 3 1 c x b r   3 4 3 3 3 r a x x b r   3 3 4 5 4 1 2 3 3 c r a c x x r a x b r       5 1 1 3x d a x x    6 1 1 3 x c c x     7 1 1 x d a x  the cancer-free equilibrium point:      3 4 2 2 2 2 3 3 31 1 1 11 1 , 0, , 1 u u u a e a es e b b r b b ra e d                , and the coexist equilibrium point  * * * *3 3 3 3 3, , ,e i t n f , where stability of cancerous chemotherapy model with obesity effect indah yanti 189             * 3 * *3 1 3 1 1* 4 6 3 * 3 3 3 3 3 3 * 2 2 2 23 3 3 * 3 4 3* 3 2 2 2 * 4 6 3* 3 3 3 3 1 11 11 , 11 . u u u u s i t c t d a e a e c t t b b r t q q r p q q r p p a e c t n b b r a e c t f b b r                                          3 2 3 3 6 3 b p a bc ad q p a c r a       5 8 a x x 6 8 4 5 b x x x x  2 6 3 2 4 6 8 c sc c x sc x x x    4 7 2 2 1 3 d x x sc sc x x   1 u x e    1 3 4 x r a x  2 1 1 x d a x  3 3 3 x b r    3 5 14 2 3 1 2 2 2 3 3 c c x x r a x r a x b r b r       5 1 31x c x   6 1 2 1 1 3 6 2 3 x c x c x x c x x      7 1 2 3 2 6 x x x x x c   3 4 5 6 8 1 1 2 2 3 3 c c c c x b r b r b r    3.3. stability of equilibrium points the local stability of equilibrium points of system (1) is investigate by linearizing system (1) around the points. jacobian matrix is obtained from this process which is its eigen values can be used to determine the stability of each points. the jacobian matrix of system (2) is                 1 1 1 1 2 2 1 1 1 1 2 3 5 2 3 5 4 2 2 2 2 4 3 6 3 3 3 3 6 4 (1 ) 0 1 1 0 1 1 0 0 0 1 1 u n u u u t i it it c t d a e c i f t f t f t f t c t r b t b rt c i c n c f a e c t c tj c n r b n b r n c t a e c f r b f b r f c t a e                                                                    stability of cancerous chemotherapy model with obesity effect indah yanti 190 the eigen value of  0j e are  1 4 31 u a e r     ,  2 3 21 u a e r      ,  3 1 11 u a e d      , and          2 1 2 3 3 1 4 5 1 3 1 3 2 3 1 3 1 5 3 4 5 1 3 2 3 3 1 1 3 5 1 3 4 1 1 3 3 1 1 1 u u u a a b r a a c e a b r r a b d r a c r a c d e b c r s b d r r c d r a e d b r                   . thus 0e is stable when  4 31 u a e r    ,  3 21 u a e r    , and       2 1 2 3 3 1 4 5 1 3 1 3 2 3 1 3 1 5 3 4 5 1 3 2 3 3 1 1 3 5 1 3 1 1 0 u u a a b r a a c e a b r r a b d r a c r a c d e b c r s b d r r c d r              . at 2e , the eigen values of jacobian matrix are  1 4 31 u a e r     ,  2 3 21 u a e r     ,  3 1 11 u a e d      , and             2 1 3 3 3 3 1 4 2 5 2 1 2 2 3 2 3 4 1 1 2 2 3 3 1 2 3 1 2 3 1 3 3 2 3 3 3 3 1 3 1 2 5 2 3 4 2 5 1 2 2 2 3 1 2 3 1 1 2 2 3 3 2 3 1 1 2 3 2 2 3 2 3 3 3 1 2 3 2 5 1 1 1 1 1 u u u u a a b c r a a b c r a a b b r r e a e d b r b r a b b r r r a b c r r a b c d r a b c r r a b c d r a b b d r r e a e d b r b r b b d r r r sc b b r r b c d r r b c d r                              2 3 1 1 2 2 3 3 . 1 u r a e d b r b r    the first three eigen values are the same with  0j e eigen values. and this point is stable under the conditions 0 i   for i = 1, 2, 3, 4. the jacobian of the total cancer invasion equilibrium and the coexist equilibrium points are produce very long and complex term eigen values, so here we just do some numerical analysis to proof its stability. 3.4. numerical simulations to illustrate the analytical result, we do some numerical simulations by using the parameters in table 1. table 1 parameters values parameter value simulation i simulation ii simulation iii simulation iv s 0.33 0.125 0.125 0.2  0.25 0.75 0.25 0.5  0.3 0.7 0.3 0.005  0.013 0.03 0.015 0.8 1 c 1 0.001 0.5 0.85 1 d 0.2 0.04 0.3 0.005 1 a 0.2 0.1 0.4 0.025 1 r 0.45 0.45 0.5 0.1 1 b 1 0.2 0.2 0.1 2 c 0.5 0.201 0.6 0.025 3 c 1 0.24 0.5 0.02 5 c 0.13 0.13 0.5 1 2 a 0.5 0.2 0.8 0.6 2 r 0.04 0.04 0.5 0.1 stability of cancerous chemotherapy model with obesity effect indah yanti 191 2 b 1 0.9 0.2 0.1 4 c 1 0.3 0.5 0.025 3 a 0.1 0.02 0.4 0.05 3 r 1 0.07 0.5 0.1 3 b 1 1.5 0.2 0.1 6 c 1 0.72 0.4 0.025 4 a 0.5 0.1 0.1 0.05 u 1 1 1 1 numerical simulation is performed by taking parameter values on the table. this simulation is aimed to show the stability of the dead equilibrium point, the total cancer invasion equilibrium, the cancer-free equilibrium, and the coexistence equilibrium. for each equilibrium point we use initial value i(0) = 0 , t(0) = 0.01 , n(0) = 1, f(0) = 0.8. figure 1 number of total cells every time of the dead equilibrium point the first simulation is aimed to show the stability of the dead equilibrium. by using simulation i parameters we get the eigen values are 1 0.6839   , 2 0.0232   , 3 0.3264   , and 4 0.2826   . and the dead equilibrium point is  0 1.0110, 0, 0, 0.6839e  . from figure 1 we observe that the dead equilibrium point reach after day 200. stability of cancerous chemotherapy model with obesity effect indah yanti 192 figure 2 number of total cells every time of the cancer-free equilibrium point the second simulation produce 1 0.0068   , 2 0.0274   , 3 0.1032   , dan 4 0.0938   and cancer-free equilibrium  1 1.2111, 0, 07599, 0.0646e  . the result of second simulation is presented in figure 2. the result of next simulation are 1 0.9038   , 2 0.0470 0.2695i    , 3 0.0470 0.2695i    , 4 0.2508   with the total cancer invasion equilibrium point is  3 0.1454, 0.9958, 0, 0.3845e  . it is shown in figure 3 that the complex eigen value conduce oscillation part of the number of total cells in the first time the drug is given. but along the given drug, the total cells number converge to equilibrium point. figure 3 number of total cells every time of the total cancer invansion equilibrium point stability of cancerous chemotherapy model with obesity effect indah yanti 193 figure 4 number of total cells every time of the coexist equilibrium point the last simulation is conducted to explore the last equilibrium points behaviour. from the figure 4 we can find that although in the end the drug no longer has an effect on the cells number, but the number of cancer cells is more than the number of normal cells, for parameter that is used in this simulation. but it is possible with different parameter values will be produced different result. conclusion we have shown that cancerous chemotherapy model with obesity effect has four equilibriums, in case fat cells not equal to zero, namely the dead equilibrium point (e0), the cancer-free equilibrium point (e1), the total cancer invansion point (e2), and the coexist equilibrium point (e3). all of them are stable under some condition. acknowledgments this work was supported by faculty of mathematics and natural science, brawijaya university: indonesia: contract number: 24/un10.f09.01/pn/2017 dated august 31, 2017. references [1] ehsanipour, e.a., x. sheng, j.w. behan, x. wang, a. butturini, v.i. avramis, s.d. mittelman, adipocytes cause leukemia cell resistance to l-asparaginase via release of glutamine, cancer research, vol. 73, no. 10, pp. 2998–3006, 2013. [2] hursting, s. d., minireview: the year in obesity and cancer, molecular endocrinology, vol. 26, no. 12, pp. 1961-1966, 2012 [3] ku-carrillo, r. a., sandra e. d., b. m. chen-charpentier, a mathematical model for the effect of obesity on cancer growth and on the immune system response. applied mathematical modelling, vol. 40, pp. 4908-4920, 2015. stability of cancerous chemotherapy model with obesity effect indah yanti 194 [4] ku-carrillo, r. a., sandra e. d., b. m. chen-charpentier, effects of the obesity on optimal control schedules of chemotherapy on a cancerous tumour, journal of computational and applied mathematics, vol. 309, pp. 603-610, 2016 [5] taubes, g. , unravelling the obesity-cancer connection, science, vol. 335, no. 6064, pp. 28-32, 2012. [6] smith, e. (25 august 2017). how exactly does obesity cause cancer. url scienceblog.cancerresearchuk.org. [7] rubin, h., promotion and selection by serum growth factors drive field cancerization, which is anticipated in vivo by type 2 diabetes and obesity, proceedings of the national academy of sciences, vol. 110, no. 34, pp.13927-13931, 2013. [8] suskic, a., suskic, s.h., opric, d. and maksimovic, s., obesity as f significant risk factor for endometrial cancer, international journal of reproduction, contraception, obstetrics and gynecology, vol. 5, no. 9, pp.2949-2951, 2017 comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region cauchy – jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 48-54 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 18 january 2018 reviewed: 14 march 2018 accepted: 18 april 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.4722 comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi1, henny pramoedyo, rahma fitriani2 1,2department of statistics, brawijaya university, malang email: jakapratama193@gmail.com, hennyp@ub.ac.id, rahmafitriani@ub.ac.id abstract the purpose of this study was to compare the results of inverse distance weighted (idw) and natural neighbor interpolation methods for spatial data of air temperature in the malang region. interpolation is one way to determine a point of events from several points around the known value. spatial interpolation can be used to estimate an area that does not have a data record using the value of its known surroundings. 38 points observation air temperature of malang region in 2016 is used as a sample point to interpolate the surrounding air temperature. obtained optimum parameter power value is 2 for idw interpolation method. the rmse comparison results show that idw method is better to be used than the natural neighbor interpolation method with the rmse values of 1,2292 for the idw method and 1,6173 for the nn method. keywords: inverse distance weighted, natural neighbor, interpolation, air temperature, rmse introduction geostatistical has been developed and able to explain the diversity that occurs due to natural phenomena and human-made on earth. geostatistics can be used to predict data at unobserved locations. to expect values in unsampled locations, a statistical method called interpolation can be used. the process of estimating data at a location that can not be observed requires a model, but at different cases do not have to get a specific model. geostatistical have very important role in it, using the estimation method based on the model. one of the geostatistical techniques for estimation is the interpolation method. interpolation is a method for obtaining data based on known data. in cartography, interpolation is the process of estimating values in areas that are not sampled or measured, to form a map or distribution of values across the entire region. one of the interpolation methods that can be used is inverse distance weighted (idw). idw method has a local influence on the distances that give greater weight to the nearest point compared to the farther point [1]. idw has the advantage which that the interpolation characteristics can be controlled by limiting the input points in the interpolation process. any input that have small spatial correlations or even no spatial correlation can be removed from the calculation method. idw interpolation has weakness that it can not predict the value above the maximum value and below the minimum value of the sample points or can be called the bullseye effect [2]. another interpolation method with a weighting method similar to idw is natural neighbor interpolation. the natural neighbor interpolation is known as sibson interpolation or "areastealing" where this method works by looking for points adjacent to the sample point and apply the weights at those points. the basic of this interpolation method is uses only the sample that is around the point to be interpolated and the results obtained are similar to the sample point as the input value of the interpolation process [3]. the logic in the spatial interpolation mentioned in tobler's geography law is that the value of the adjacent observation point will have the same value (close to) as compared to the value at the farther point. spatial interpolation assumes that attributes are continuous in space and spatial dependencies. both assumptions indicate that the data attribute estimate can be done mailto:jakapratama193@gmail.com mailto:hennyp@ub.ac.id mailto:rahmafitriani@ub.ac.id comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 49 based on the locations at the surrounding points [4]. according to watson [5], the idw interpolation technique uses the nearest distance weight, while the natural neighbor interpolation technique uses the average weight of the surrounding area. both of these techniques are local interpolation techniques which of course can be compared. global warming is an increasing average temperature in the atmosphere, sea, and land including one of the earth's big damage. global warming is indicated by rising earth surface temperatures and the process which is called the greenhouse effect. the greenhouse effect is formed because of troposphere gas that exceeds the natural state. causes of global warming include population growth, industry, refrigeration, transportation, forest burning, land use diversion and greenhouse gas in the atmosphere. transportation is a very valuable part of society. the other side of the excessive use of transportation causing a negative effect. air temperature increase is most prevalent in vehicle dense points or congestion-prone. the number of vehicles also contributed in rising air temperature. the vehicle's exhaust gases cause an increase in the greenhouse effect so that solar radiation to the earth can not bounce back into the atmosphere. the number of transportation continues to increase especially in big cities in indonesia, one of the big cities in indonesia is the city of malang. the city of malang is known as a city of tourism and education. the development of malang city can be seen on the increase of transportation and population which is accompanied by government facility and investor. the area of malang raya is located between several mountains so that it has a very wide geographical condition and is widely spread, thus allowing for an unsampled location for air temperature data. this is an interesting phenomenon, but there has been no research done to interpolate the air temperature data in this city, especially regarding the study of idw and natural neighbor interpolation methods. this study uses two interpolation methods by considering the ability of the method to estimate simple but accurate to interpolate air temperature in malang. interpolation results are presented in a prediction map to determine the spread of air temperature in malang. this research is expected to be valuable and useful for various parties, especially for the agricultural sector to provide an overview of the air temperature in the future considering the increase in air temperature will affect the development of plants. methods spatial interpolation steps consist of 6 stages: first, we preparing maps with spatial data attributes, then testing spatial autocorrelation, and we do spatial interpolation of idw and natural neighbor. after that we can draw two contour maps from the results of the two interpolation methods. for validation we do cross-validation, and then testing the validation of results research. first we must fulfilled the assumtion of spatial autocorrelation test. autocorrelation can be defined as a series of inter elements in the series and elements of the same circuit and separated by an interval. one of the spatial autocorrelation tests is using the moran i test. gumprecht [6] describes moran i statistics as the most widely known and used method for the measurement and testing of spatial dominance. the test measures the intensity of spatial autocorrelation. the autocorrelation coefficient of moran is denoted by i as follows: 𝐼 = 𝑛 ∑ ∑ 𝑤𝑖𝑗(𝑥𝑖 − �̅�)(𝑥𝑗 − �̅�) 𝑛 𝑗=1 𝑛 𝑖=1 (∑ ∑ 𝑤𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 )∑ (𝑥𝑖 − �̅�) 𝑛 𝑖=1 where 𝑥𝑖 and 𝑥𝑗 is a observation value at location i and j, �̅� is average of the observed value, 𝑤𝑖𝑗 is the weight value between location i and j, and n is the sample size. moran i autocorrelation test has a value between -1 to 1. the value -1 shows a strong negative autocorrelation, while the value of 1 shows a strong positive correlation. the expected value of i under 𝐻0is that autocorrelation is not expected to be 0 but given by 𝐼0 [7]. in the autocorrelation test should be considered also spatial pattern. in general, spatial patterns can be described as 3 types of clustered, dispersed (chessboard) and random. spatial comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 50 autocorrelation is positive if in an adjacent area has a similar value and if it is poured in the image it forms a hoop pattern. spatial autocorrelation is negative if in an adjacent area has a much different value and if it is depicted to form a chessboard pattern. between the two mentioned patterns there is a random pattern showing no spatial autocorrelation. spatial interpolation is one way of knowing the value of a point location based on the value of some point around its known value. for example, to make a precipitation map of a country, an available weather stations will never be evenly distributed to provide data on the territory of the country. spatial interpolation can estimate the value of an area that does not have a data record using the value of its known surroundings [8]. after the spatial autocorrelation assumption test has been fulfilled, we can continue to the interpolation method, which is idw and natural neighbor method. the inverse distance wighted (idw) method assumes that each input point has a local effect on distance. this method gives higher weight to the closest cell to the data point compared to the further cells. points on a given radius are used in determining the output value for each location. the general weighting function of idw is inverse of the squared distance formulated in the following equation: 𝑍∗ = ∑𝜔𝑖𝑍𝑖 𝑛 𝑖=1 where 𝑍𝑖represents the value of the data to be interpolated by a number of n points and weights (𝜔𝑖) formulated as 𝜔𝑖 = ℎ𝑖 −𝑝 ∑ ℎ 𝑗 −𝑝𝑛 𝑗=0 p is a positive value that can be changed is called the power parameter and ℎ𝑗 is the distance to the point of interpolation which is described as: ℎ𝑖 = √(𝑥 − 𝑥𝑖) 2 + (𝑦 − 𝑦𝑖) 2 (x, y) is the coordinate of the interpolation point and (𝑥𝑖, 𝑦𝑖) is the coordinate of the default data point. the natural neighbor interpolation method was first introduced by sibson [3] as a weighted average weighted interpolation method such as idw. the natural neighbor interpolation does not use the distance as a weight but instead forms the delauney triangulation of the input points and selects the nearest cell that forms the convex hull around the interpolation point and uses the proportional area as weight. this method is used when the sample data points are distributed with uneven density. to interpolate natural neighbor, the first process is to build a polygon for all input points in interpolation, then the new voronoi diagram will be created around the interpolation points. if the point that wanted to be interpolated z is inserted into the data set, the area around the voronoi diagram will decrease. if 𝑝𝑖 and 𝑞𝑖 are the voronoi diagram areas of the sample points 𝑧𝑖 before and after the addition of 𝑧 ∗, the weight for the sample point 𝑧𝑖 [9] : 𝜔𝑖 = (𝑝𝑖 − 𝑞𝑖) 𝑝𝑖 according to armstrong [10], cross-validation is a way to know the accuracy of some interpolation methods. cross validation test using root mean square error (rmse) is described as follows : figure 1. voronoi diagram (thiessen polygon) comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 51 𝑅𝑀𝑆𝐸 = √ ∑ (�̂�𝑖 − 𝑦𝑖) 2𝑛 𝑖=1 𝑛 results and discussion this study uses secondary data of the average monthly air temperature of malang raya area which includes malang city, malang regency, and batu town obtained from upt water resources management of malang city. monthly air temperature data for 1 year in ℃/year in 2016 from 38 points of observation which located spread in malang region. the autocorrelation assumption test results using moran's i is shown in figure 2. figure 2 shows that with 95% confidence level can be said that formed spatial pattern clustered, which means that the temperature data of malang region in 2016 has positive spatial autocorrelation. in addition to using spatial distribution pattern, this assumption test can also be seen from moran i test statistic. the result of moran i test resulted z test statistic (i) of 2.57 with moran i index of 0.22. based on test criteria with 95% confidence level, because z (i)> z (𝛼 2⁄ ) then 𝐻0 is rejected and concluded that air temperature data of malang region in 2016 has positive spatial autocorrelation. interpolation idw uses two methods of determining area and a power parameters. malang raya area is located between several mountains so it has a geographical condition that is very varied and widely spread. based on this situation, we use variable search radius method on idw interpolation. in addition to the area determination, also used some weighting for power parameters. in this study we use the power values 1,2,3,4 and 5. determination of value must be positive and some power values are used only by selecting multiple values to see the resulting difference. the power parameters to be used are the most optimum, ie by looking at the rmse value for each power. the rmse value for each power parameters can be seen in table 1. figure 2. moran i test result from air temperature 2016 in malang region comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 52 table 1. rmse value for each power power rmse 1 1.257122 2 1.229251 3 1.261345 4 1.318822 5 1.376546 table 1 shows that a large rmse value (1.376546) is obtained from a larger power value (power value 5), because the observation points that affected by this interpolation point are less. otherwise, if the value of small power (power value 2), resulting rmse value of 1.229251. from the rmse value for air temperature in malang region, it can be said that it would be better to use idw interpolation method with power value 2. after the most optimum power parameters are obtained, the next step is to interpolate idw. interpolation results are used to create a prediction map at all points in malang region which presented with raster data format in figure 3. figure 3 is a prediction map of idw interpolation result using power value 2 using air temperature data of malang area. based on figure 3 it is observed that the highest interpolation figure 3. idw interpolation result comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 53 value of air temperature is in the malang city and bantur subdistrict, while the lowest interpolation air temperature is in the mountains such as batu city and poncokusumo subdistrict. the natural neighbor interpolation method is local, where only the samples are located around the points to be interpolated, so that the air temperature data to be interpolated will be similar to the value of the air temperature at the sample point. this interpolation produces a smooth surface which is shown in fig. 4. using this interpolation method resulting more smoothed topography surface compared to the idw interpolation method, caused by the theory contained in this method where the thiessen polygon system is used to calculate the point interpolation value based on the area that affecting the interpolation point. cross validation test result is rmse value. the interpolation method with the smallest rmse value is the best method. the results of rmse calculations for each interpolation method are presented in table 2. figure 4. natural neighbor interpolation result comparison of inverse distance weighted and natural neighbor interpolation method at air temperature data in malang region jaka pratama musashi 54 table 2. rmse value of each interpolation method interpolation method rmse natural neighbor 1.617307 idw power = 1 1.257122 power = 2 1.229251 power = 3 1.261345 power = 4 1.318822 power = 5 1.376546 conclusion based on the rmse value, the idw interpolation method with the power parameter 2 yields much more accurate prediction values than the natural neighbor interpolation method. areas with low air temperatures are located around kota batu and kecamatan poncokusumo, while areas with high temperatures are in the vicinity of malang and bantur subdistricts. references [1] d. f. watson dan g. m. phillip, “a refinement of inverse distance weighted interpolation,” geoprocessing, vol. 2, pp. 315-327, 1985. [2] g. h. pramono, “akurasi metode idw dan krigging untuk interpolasi sebaran sedimen tersuspensi,” forum geografi, vol. 22, no. 1, pp. 97-110, 2008. [3] r. sibson, “a brief description of natural neighbor interpolation (chapter 2),” dalam interpreting multivariate data, chichester, john wiley, 1981, pp. 21-36. [4] s. anderson, an evaluation of spatial interpolation methods on air temperature in phoenix, arizona: department of geography, arizona university, 2001. [5] d. f. watson and g. m. phillip, "neigborhood based interpolation," geobyte, vol. 2, pp. 12-16, 1987. [6] d. gumprecht, “treatment of far-off objects in moran's i test,” vienna university of economics and business administration, vienna, 2007. [7] m. j. fortin dan m. r. t. dale, spatial analysis: a guide for ecologist, newyork: cambridge university press, 2005. [8] s. naoum dan i. k. tsanis, “ranking spatial interpolation techniques using a gis-based dss,” global nest journal, vol. 6, no. 1, pp. 1-20, 2004. [9] v. merwade, d. r. maidment dan j. a. goff, “anisotropic considerations while interpolating river channel bathymetry,” journal of hydrology, vol. 331, no. 3, pp. 731-741, 2006. [10] m. amstrong, basic linear geostatistics, newyork: springer, 1998. basis existence of internal n-graph space cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 58-65 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 16, 2019 reviewed: april 03, 2020 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.8002 basis existence of internal n-graph space ahmad lazwardi university of muhammadiyah banjarmasin email: lazwardiahmad@gmail.com abstract graph of real valued continuous function with special addition and multiplication has already proven that is isomorphic to real number system. furthermore, the graph of continous real valued function forms a field. the aim of this research was to generalize such concept to its n-tuple cartesian product and to prove that interchange of basis still able to be executed. the result of this research is n-tuple cartesian product of graph function forms a vector space over ℝ and interchange of basis still able to be executed. keywords: n-graph space; general vector space; internal n-graph space introduction by associating the special operation to a graph of continuous function, such graph can be claimed as a vector space. this follows from the fact that every graph of continuous real valued function has a bijection to its domain i.e. the real number system. furthermore, they are homeomorphic. graph of real valued continuous function has a unique characteristic. it has continuous shape of curve along real line ℝ. more detail will be described as follows [1] definition 1. let real valued function 𝑓: ℝ → ℝ. the graph of f is defined as ℝ𝑓 = {(𝑥, 𝑓(𝑥)): 𝑥 ∈ ℝ} (1) continuity of f indicates that ℝ𝑓 topologically equivalent to ℝ. its described briefly as follows; definition 2 [2]. for each 𝑈 ⊂ ℝ. define the image of 𝑈 over ℝ𝑓 as 𝑈 × 𝑓(𝑈) = {(𝑥, 𝑓(𝑥)): 𝑥 ∈ 𝑈} (2) define the associated topology for ℝ𝑓 as 𝜏𝑓 = {𝑈 × 𝑓(𝑈): 𝑈 𝑜𝑝𝑒𝑛} (3) it can be shown trivially that such topology implies ℝ𝑓 and ℝ are homeomorphics . the fact that ℝ𝑓 and ℝ are homeomorphics describes that even though their graph geometrically has different shapes, but they still have similarity in views of topology[3]. it motivates us to explore more special properties of ℝ𝑓. now let see it further. definition 3. let real valued function 𝑓: ℝ → ℝ. define the addition for ℝ𝑓 as follow http://dx.doi.org/10.18860/ca.v6i2.8002 mailto:lazwardiahmad@gmail.com basis existence in graph of real valued continous functions ahmad lazwardi 59 𝑥𝑓⨁𝑦𝑓 = (𝑥 + 𝑦, 𝑓(𝑥 + 𝑦)) for any 𝑥𝑓, 𝑦𝑓 ∈ ℝ𝑓 . define the scalar multiplication as follow (𝛼)𝑥𝑓 = 𝛵𝑓 (𝛼𝛵 −1 𝑓(𝑥, 𝑓(𝑥))) = (𝛼𝑥, 𝑓(𝛼𝑥)) for any 𝑥𝑓 ∈ ℝ𝑓 and scalar  , and tf is natural bijection generated by real valued function f such as 𝛵𝑓(𝑥) = (𝑥, 𝑓(𝑥)). by associating ℝ𝑓 with those operations, ℝ𝑓 become real vector space. moreover ℝ𝑓 and ℝ are isomorphics [4]. methods the method of this research is done by following method: first was to prove that ℝ𝑓 has dimension 1 and isomorphics to ℝ. next step was to analyize more general space i.e. n tuple cartesian product of ℝ𝑓 denoted by ℝ𝑓 𝑛. in generalization of ℝ𝑓 𝑛 was to prove that such space is a vector space over ℝ and was able to change of bases. finally, was to find out the briefly method to change of bases as well as in real vector spaces. results and discussion it was proven previously that for each continuous function 𝑓: ℝ → ℝ, the space ℝ𝑓 generated by f forms a field. this fact becomes the basic to generalize the idea by constructing new n-tuple cartesian product of ℝ𝑓 which preserves vector space properties [5]. before any further discussion, first define some necessary terms in order to help in generalization. we mean linear combination is 𝑧𝑓 = (∝)𝑥𝑓 + (𝛽)𝑦𝑓 for each 𝑧𝑓 ∈ ℝ𝑓 and some scalars 𝛼, 𝛽 ∈ ℝ [6]. the set of linear combinations of 𝑥𝑓, 𝑦𝑓 is named as span {𝑥𝑓, 𝑦𝑓 } [7]. the set 𝑈 ⊂ ℝ𝑓 is said to be linearly independent if none of its members is able to expressed as linear combination of other members. here is definition of basis: definition 4. the set },...,,{ 21 n fff xxx is called basis of subspace 𝑈 ⊂ ℝ𝑓 if },...,,{ 21 n fff xxx are linearly independent and span },...,,{ 21 n fff xxx =u. recall that, dimension of 𝑈 is defined as base cardinality of 𝑈. next theorem is an important result. theorem 1. ℝ𝑓 has dimension 1. proof: chose 1𝑓 ∈ ℝ𝑓. for each 𝑥𝑓 ∈ ℝ𝑓, it’s obvious that x𝑓 = (𝑥, 𝑓(𝑥)) = 𝛵𝑓(𝑥) = 𝛵𝑓(𝑥. 1) = (𝑥. 1, 𝑓(𝑥. 1)). = (𝑥) × 1𝑓 by last equation, we conclude that 𝑆𝑝𝑎𝑛{1𝑓} = ℝ𝑓. furthermore, based on the above results, vector space theory of ℝ𝑓 can be developed: basis existence in graph of real valued continous functions ahmad lazwardi 60 definition 5. let 𝑆 = {𝑓1, 𝑓2, … , 𝑓𝑛} be a finite collection of real valued continuous functions, define n-graph space as ℝ𝑓𝑝 𝑛 = ∏ ℝ𝑓𝑘 𝑛 𝑘=1 = {�⃗�: {1,2, … , 𝑛} → ⋃ ℝ𝑓𝑘: �⃗�(𝑖) ∈ ℝ𝑓𝑖 𝑛 𝑘=1 }. (4) definition 6. let 𝑆 = {𝑓1, 𝑓2, … , 𝑓𝑛} be a finite collection of real valued continuous functions. define addition on ℝ𝑓𝑝 𝑛 as . �⃗�⨁�⃗⃗⃗� = (�⃗�(1)⨁�⃗⃗⃗�(1), �⃗�(2)⨁�⃗⃗⃗�(2), . . . , �⃗�(𝑛)⨁�⃗⃗⃗�(𝑛)) (5) and scalar multiplication as . (∝)�⃗� = ((∝)�⃗�(1), (∝)�⃗�(2), . . . , (∝)�⃗�(𝑛)) (6) for each 𝛼 ∈ ℝ. theorem 2. let 𝑆 = {𝑓1, 𝑓2, … , 𝑓𝑛} be a finite collection of real valued continuous function. n-graph space ℝ𝑓𝑝 𝑛 is a vector space over ℝ under operation ⨁ and scalar multiplication (∝)�⃗�. in special case which 𝑓1 = 𝑓2 = 𝑓3 = ⋯ = 𝑓𝑛, graph space ℝ𝑓𝑝 𝑛 is called internal n-graph space i.e. cartesian product n-tuple of ℝ𝑓 itself. one can write ℝ𝑓𝑝 𝑛 = ℝ𝑓 𝑛. euclidean space ℝ𝑛 is one of finest example of graph space which f is defined as identity mapping. one of the most important tools to analyze relation between two internal n-graph space is linear transformation[8], here we still able to define linear transformation as well as done on commonly vector spaces. definition 7. let two graph spaces ℝ𝑓 𝑚, ℝ𝑓 𝑛. mapping 𝐿: ℝ𝑓 𝑚 → ℝ𝑓 𝑛 is linear transformation if the following properties hold 𝐿(�⃗�⨁�⃗⃗⃗�) = 𝐿(�⃗�)⨁𝐿(�⃗⃗⃗�), ∀�⃗�, �⃗⃗⃗� ∈ ℝ𝑓 𝑚 𝐿((∝)�⃗�) = (∝)𝐿(�⃗�), ∀∝∈ ℝ the set of all linear transformation from ℝ𝑓 𝑚 to ℝ𝑓 𝑛 is denoted as 𝐿𝑖𝑛(ℝ𝑓 𝑚, ℝ𝑓 𝑛) [7]. one of necessary example of linear transformation is linear mapping from ℝ𝑓 𝑚 to its coordinate i.e. 𝑣𝑖⃗⃗⃗ ⃗: ℝ𝑓 𝑚 → ℝ by specific formula 𝑣𝑖⃗⃗⃗ ⃗(�⃗�) = �⃗�(𝑖), 𝑖 = 1,2,3, … , 𝑚. the term of linear transformation is very useful in constructing theory change bases in graph space[9]. on internal graph space ℝ𝑓 𝑚 change of bases still able to be constructed. definition 8. let �⃗�1, �⃗�2, . . . , �⃗�𝑘 ∈ ℝ𝑓 𝑚 are said to be linearly independent if for each �⃗�𝑖 ⃗⃗⃗ ⃗ can’t be expressed as linear combination of some others. the term of linear combination refers to equation �⃗� = (∝1)𝑣1⃗⃗⃗⃗⃗⨁ . . . ⨁ (∝𝑘)𝑣𝑘⃗⃗⃗⃗⃗ (7) another way to express coordinate transformation is by defining linear transformation which maps a vector to its scalars corresponding to the basis used to. basis existence in graph of real valued continous functions ahmad lazwardi 61 definition 9. let v be a nontrivial vector space over field f. let 𝑆 = {𝑠1⃗⃗⃗⃗ , 𝑠2⃗⃗⃗⃗ , 𝑠3⃗⃗⃗⃗ , … , 𝑠𝑚⃗⃗ ⃗⃗⃗} be a basis of v. define coordinate transformation as ∅𝑆: 𝑉 → 𝐹 𝑚 such that for each �⃗� = 𝑣1𝑠1⃗⃗⃗⃗ + 𝑣2𝑠2⃗⃗⃗⃗ + ⋯ + 𝑣𝑚𝑠𝑚⃗⃗ ⃗⃗⃗, the map ∅𝑆(�⃗�) = [�⃗�]𝑆 = [ 𝑣1 𝑣2 ⋮ 𝑣𝑚 ] . it’s easy to check that coordinate transformation is an isomorphism [10]. the concept of coordinate mapping above is used to construct the concept of change of basis. first, we have to decide standard basis which lies on ℝ𝑓 𝑚. standard basis of ℝ𝑓 𝑚 is 𝑣𝑖⃗⃗⃗ ⃗ = (1𝑓, 0𝑓, 0𝑓, 0𝑓, . . . , 0𝑓) 𝑣2⃗⃗⃗⃗⃗ = (0𝑓, 1𝑓, 0𝑓, 0𝑓, . . . , 0𝑓) 𝑣𝑚⃗⃗⃗⃗⃗⃗ = (0𝑓, 0𝑓, 0𝑓, 0𝑓, . . . , 1𝑓) now let’s see how the change of basis works: for each �⃗� ∈ ℝ𝑓 𝑚. by previous definition, one can write �⃗� = (�⃗�(1), �⃗�(2), . . . , �⃗�(𝑚)) (8) which �⃗�(𝑖) ∈ ℝ𝑓 for each i = 1,2, . . ., m. therefore �⃗� = �⃗�(1)�⃗�1⨁�⃗�(2)�⃗�2⨁ . . . ⨁�⃗�(𝑚)�⃗�𝑚 for corresponding coordinate �⃗�(𝑖) ∈ ℝ𝑓. this is how the above coordinate and corresponding scalars related, let’s see it as follows: �⃗�(1). 1𝑓 = �⃗�(1) = 𝛵𝑓(𝛵𝑓 −1(�⃗�(1))) = 𝛵𝑓(𝛵𝑓 −1(�⃗�(1). 1) = (𝛵𝑓 −1(�⃗�(1). 1), 𝑓(�⃗�(1). 1)) = (𝛵𝑓 −1(�⃗�(1). 1) . 1𝑓 hence �⃗� = (𝛵𝑓 −1(�⃗�(1)) . 𝑣1⃗⃗⃗⃗⃗⨁ (𝛵𝑓 −1(�⃗�(2)) . 𝑣2⃗⃗⃗⃗⃗⨁ . . . ⨁ (𝛵𝑓 −1(�⃗�(𝑚)) . 𝑣𝑚⃗⃗⃗⃗⃗⃗ (9) for some (𝛵𝑓𝑘 −1(�⃗�(1)) ∈ ℝ. the last equation can be expressed in term of more visual vector addition as follows: �⃗� = (𝛵𝑓 −1(�⃗�(1)) . 𝑣1⃗⃗⃗⃗⃗⨁. . . ⨁ (𝛵𝑓 −1(�⃗�(𝑚)) . 𝑣𝑚⃗⃗⃗⃗⃗⃗ = (𝛵𝑓 −1(�⃗�(1)) . [ 1𝑓 0𝑓 ⋮ 0𝑓] ⨁. . . ⨁ (𝛵𝑓 −1(�⃗�(𝑚)) [ 0𝑓 0𝑓 ⋮ 1𝑓] = [ 1𝑓 0𝑓 0𝑓 ⋮ 0𝑓 … … 0𝑓 … 0𝑓 … … … 0𝑓 ⋮ 1𝑓] [ 𝛵𝑓 −1(�⃗�(1)) … … 𝛵𝑓 −1(�⃗�(𝑚)) ] = 𝐼𝑓𝛵𝑓 −1(�⃗�) the above equation gives a consequence that ℝ𝑓 𝑚 is isomorphic to ℝ𝑚. the next theorem will ensure that ℝ𝑓 𝑚 has more than just standard basis. teorema 3. if 𝑆 = {𝒔1, 𝒔2, . . . , 𝒔𝑚} is basis for ℝ 𝑚 then 𝑇𝑓(𝑆) = {𝑇𝑓(𝒔𝟏), 𝑇𝑓(𝒔2), . . . , 𝑇𝑓(𝒔𝑚)} forms a basis for ℝ𝑓 𝑚. bukti: basis existence in graph of real valued continous functions ahmad lazwardi 62 suppose that �⃗� ∈ ℝ𝑓 𝑚. let’s see the following fact, another fine expression for vector in ℝ𝑓 𝑚 is �⃗� = [ �⃗�(1) �⃗�(2) ⋮ �⃗�(𝑚) ] . thus we have 𝛵𝑓 −1(�⃗�) = [ 𝛵𝑓 −1(�⃗�(1)) 𝛵𝑓 −1(�⃗�(2)) ⋮ 𝛵𝑓 −1(�⃗�(𝑚))] is actually lies in ℝ𝑚. therefore 𝛵𝑓 −1(�⃗�) = [ 𝛵𝑓 −1(�⃗�(1)) 𝛵𝑓 −1(�⃗�(2)) ⋮ 𝛵𝑓 −1(�⃗�(𝑚))] = (∝1) [ 𝑠11 𝑠12 ⋮ 𝑠1𝑚 ] + . . . +(∝𝑚) [ 𝑠𝑚1 𝑠𝑚2 ⋮ 𝑠𝑚𝑚 ] , by 𝒔𝑖 = [ 𝑠𝑖1 𝑠𝑖2 ⋮ 𝑠𝑖𝑚 ] for each i. one can have �⃗� = 𝛵𝑓 (𝛵𝑓 −1(�⃗�)) = 𝛵𝑓((𝛼1)𝒔1 + (𝛼2)𝒔2+ . . . +(𝛼𝑚)𝒔𝑚) = ((𝛼1)𝒔1 + (𝛼2)𝒔2+ . . . +(𝛼𝑚)𝒔𝑚, 𝑓((𝛼1)𝒔1 + (𝛼2)𝒔2+ . . . +(𝛼𝑚)𝒔𝑚)) = (𝛼1)𝛵𝑓(𝒔1)⨁(𝛼2)𝛵𝑓(𝒔2)⨁ . . . ⨁(𝛼𝑚)𝛵𝑓(𝒔𝑚). in other words, {𝛵𝑓(𝒔1), 𝛵𝑓(𝒔2), . . . , 𝛵𝑓(𝒔𝑚)} spans ℝ𝑓 𝑚. the rest is to prove that 𝛵𝑓(𝒔) is linearly independent. suppose the statement is not true, then there exist 𝛵𝑓(𝒔) which become linear combination of other members. let’s assume that is 𝛵𝑓(𝒔𝒊) then we have 𝛵−1 (𝛵𝑓(𝒔𝑖)) = 𝒔𝑖 = (𝛼1)𝒔1+ . . . +(𝛼𝑖−1)𝒔𝑖−1 + (𝛼𝑖+1)𝒔𝑖+1+ . . . + (𝛼𝑚)𝒔𝑚 but since {𝒔𝟏, 𝒔𝟐, . . . , 𝒔𝒎} is linearly independent, then it should be a contradiction. the above theorem indirectly explains that the internal graph space ℝ𝑓 𝑚 has infinitely many vectors that can form a basis for ℝ𝑓 𝑚. the next theorems will be discussing about how the basis related each other. let 𝑆 = {𝑠1⃗⃗⃗⃗ , 𝑠2⃗⃗⃗⃗ , . . . , 𝑠𝑚⃗⃗ ⃗⃗⃗} be a basis for ℝ𝑓 𝑚. then for each �⃗� ∈ ℝ𝑓 𝑚, one can write �⃗� = (𝛼1 𝑠)𝑠1⃗⃗⃗⃗ + (𝛼2 𝑠)𝑠2⃗⃗⃗⃗ + . . . +(𝛼𝑚 𝑠 )𝑠𝑚⃗⃗ ⃗⃗⃗ dengan (𝛼𝑖 𝑠) ∈ ℝ. (10) = (𝛼1 𝑠) [ 𝑠1(1) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ 𝑠1(2) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⋮ 𝑠1(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗] + (𝛼2 𝑠) [ 𝑠2(1) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗ 𝑠2(2) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗ ⋮ 𝑠2(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗] +…+(𝛼𝑚 𝑠 ) [ 𝑠𝑚(1) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ 𝑠𝑚(2) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⋮ 𝑠𝑚(𝑚) ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ] = [ 𝑠1(1) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ 𝑠1(2) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⋮ 𝑠1(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑠2(1) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗ 𝑠2(2) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗ ⋮ 𝑠2(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ … 𝑠𝑚(1) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ 𝑠𝑚(2) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⋮ 𝑠𝑚(𝑚) ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ] [ (𝛼1 𝑠) (𝛼2 𝑠) ⋮ (𝛼𝑚 𝑠 ) ] =[𝑀]𝑆,𝑓[𝛼]𝑆 by [𝑀]𝑆,𝑓 is matrics which for each entry lies in ℝ𝑓 𝑚 and [𝛼]𝑆 is real scalar vector. then we have 𝑇𝑓 −1(�⃗�) = [ 𝑇𝑓 −1(𝑠1(1) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗) 𝑇𝑓 −1(𝑠1(2) ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗) ⋮ 𝑇𝑓 −1(𝑠1(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗) 𝑇𝑓 −1(𝑠2(1) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗) 𝑇𝑓 −1(𝑠2(2) ⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗) ⋮ 𝑇𝑓 −1(𝑠2(𝑚) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗) … 𝑇𝑓 −1(𝑠𝑚(1) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) 𝑇𝑓 −1(𝑠𝑚(1) ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) ⋮ 𝑇𝑓 −1(𝑠𝑚(𝑚) ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ )] [ (𝛼1 𝑠) (𝛼2 𝑠) ⋮ (𝛼𝑚 𝑠 ) ] basis existence in graph of real valued continous functions ahmad lazwardi 63 therefore, 𝑇𝑓 −1(�⃗�)=[𝑇𝑓 −1(𝑆)][𝛼]𝑆 (11) the above equation explains how the basis s in internal graph space ℝ𝑓 𝑚 mapped to ℝ𝑚. it will make us easier to change basis from the old to the new one. let’s pay attention to the following discussion; let 𝑆 = {𝑠1⃗⃗⃗⃗ , 𝑠2⃗⃗⃗⃗ , . . . , 𝑠𝑚⃗⃗ ⃗⃗⃗} be a basis of ℝ𝑓 𝑚 and 𝑊 = {𝑤1⃗⃗ ⃗⃗⃗, 𝑤2⃗⃗⃗⃗⃗⃗ , . . . , 𝑤𝑚⃗⃗⃗⃗⃗⃗⃗} another basis. we want to change the old basis s into the new one w. we have �⃗� = [𝑀]𝑆,𝑓[𝛼]𝑆 on the other hand �⃗� = [𝑀]𝑊,𝑓[𝛼]𝑊 therefore 𝑇𝑓 −1(�⃗�) =[𝑇𝑓 −1(𝑆)][𝛼]𝑆 )( 1 )( 1 vt fst f    =[𝛼]𝑆 there exists transition metrics   )(),( 11 wtst ffm  such that the following works )( 1 )( 1 vt fwt f    =    swtst ffm )(),( 11  = [𝛼]𝑊. now we send it back through the map )( 1 )( 1 )( 11 vt fwtwt ff      = [𝑇𝑓 −1(𝑊)][𝛼]𝑊. hence �⃗�= [𝑀]𝑆,𝑓[𝛼]𝑆 = [𝑀]𝑊,𝑓[𝛼]𝑊. that’s how the basis change. for more understanding, let’s see the example let 𝑓(𝑥) = 𝑒𝑥. the corresponding isomorphism is 𝑇𝑓(𝑥) = (𝑥, 𝑒 𝑥) for each 𝑥 ∈ ℝ. the graph is ℝ𝑓 = {𝑎𝑓 = (𝑎, 𝑒 𝑎): 𝑎 ∈ ℝ}. by applying the method to find the corresponding addition, we have 𝑎𝑓⨁𝑏𝑓 = (𝑎 + 𝑏, 𝑒 𝑎+𝑏) , 𝑎, 𝑏 ∈ ℝ (12) and the scalar multiplication (𝛼)(𝑥𝑓) = (𝛼𝑥, 𝑒 𝛼𝑥), 𝑥 ∈ ℝ (13) now the internal graph space dimension 2 has the form ℝ𝑓 2 = {( 𝑥𝑓 𝑦𝑓 ) : 𝑥𝑓, 𝑦𝑓 ∈ ℝ𝑓} (14) now let see how the basis change ℝ𝑓 2. choose a basis 𝑆 = {( (2, 𝑒2) (0,1) ) , ( (0,1) (2, 𝑒2) )} and 𝑊 = {( (1, 𝑒) (0,1) ) , ( (1, 𝑒) (−1, 𝑒−1) )}. suppose �⃗� = ( (5, 𝑒5) (2, 𝑒2) ). it will be shown that �⃗� is linear combination of s. let’s pay attention to this �⃗� = (𝛼) ( (2, 𝑒2) (0,1) ) ⨁(𝛽) ( (0,1) (2, 𝑒2) ) basis existence in graph of real valued continous functions ahmad lazwardi 64 =( (2𝛼, 𝑒2𝛼) (0,1) ) ⨁ ( (0,1) (2𝛽, 𝑒2𝛽) ) = ( (2𝛼, 𝑒2𝛼) (2𝛽, 𝑒2𝛽) ) =( (5, 𝑒5) (2, 𝑒2) ) hence 2𝛼 = 5, 2𝛽 = 2, 𝑒2𝛼 = 𝑒5, 𝑒2𝛽=𝑒2. those imply 𝛼 = 5 2 and 𝛽 = 1. therefore �⃗� = [ 2𝑓 0𝑓 0𝑓 2𝑓 ] [ 5 2 1 ] =[𝑀]𝑆,𝑓[𝛼]𝑆 on the other hand, �⃗� = (𝛼) ( (1, 𝑒) (0,1) ) ⨁(𝛽) ( (1, 𝑒) (−1, 𝑒−1) ) =( (𝛼, 𝑒𝛼) (0,1) ) ⨁ ( (𝛽, 𝑒𝛽) (−𝛽, 𝑒−𝛽) ) = ( (𝛼 + 𝛽, 𝑒𝛼+𝛽) (−𝛽, 𝑒−𝛽) ) =( (5, 𝑒5) (2, 𝑒2) ) we have 𝛼 + 𝛽 = 5, −𝛽 = 2, 𝑒𝛼+𝛽 = 𝑒5, 𝑒−𝛽=𝑒2. those imply 𝛼 = 7 and 𝛽 = −2. therefore �⃗� = [ 1𝑓 1𝑓 0𝑓 −1𝑓 ] [ 7 −2 ] =[𝑀]𝑊,𝑓[𝛼]𝑊. its already shown how the vector �⃗� to be expressed as linear combination of s and w. to change basis from s to w, we must transfer all members of s to the ℝ2 i.e. 𝑇𝑓 −1(𝑆) = {( 2 0 ) , ( 0 2 )} and 𝑇𝑓 −1(𝑊) = {( 1 1 ) , ( 0 −1 )} . by elementary calculation we have ( 2 0 ) = 2 ( 1 1 ) + 2 ( 0 −1 ) ( 0 2 ) = 0 ( 1 1 ) − 2 ( 0 −1 ) thus, the transition metrics is )( 1 )( 1 vt fwt f    =[ 2 0 2 −2 ] [ 5 2 1 ] =[ 7 −2 ]. that’s the new coordinate. conclusions for each real valued function f, the corresponding internal graph space forms a vector space over ℝ. the concept of linear mapping and linear combinations still can be adapted from the graph space and still well defined. basis existence in graph of real valued continous functions ahmad lazwardi 65 references [1] j. doboš, “on the set of points od discontinuity for functions with closed graphs,” vol. 110, no. 1, 1985. [2] a. v arhangel, “relative topological properties and relative topological spaces,” vol. 8641, no. 95, 1996. [3] t. banakh, k. mine, and k. sakai, “classifying homeomorphism groups of infinite graphs,” topol. appl., vol. 156, no. 17, pp. 2845–2869, 2009. [4] a. lazwardi, “topologi grafik fungsi real kontinu,” in prosiding seminar nasional pendidikan matematika, 2017, no. 3185, p. 51. [5] w. a. trybulec, “basis of vector space,” vol. 2, no. 1, pp. 2–4, 2003. [6] b. hou and s. gao, “the structure of some linear transformations,” linear algebra appl., vol. 437, no. 9, pp. 2110–2116, 2012. [7] s. roman, advanced lienar algebra, 3th ed. united states: springer, 2008. [8] y. zhang, h. tam, and f. guo, “invertible linear transformations and the lie algebras,” vol. 13, pp. 682–702, 2008. [9] s. waldron, “frames for vector spaces and affine spaces,” linear algebra appl., vol. 435, no. 1, pp. 77–94, 2011. [10] a. aleman, k. perfekt, s. richter, and c. sundberg, “linear graph transformations on spaces of analytic,” j. funct. anal., vol. 268, no. 9, pp. 2707–2734, 2015. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 127-139 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 03 july 2018 reviewed: 15 august 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5877 structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati1,3, bambang widjanarko otok2, suharto3, arief wibowo3 1health polytechnic banjarmasin, ministry of health & ph.d student faculty of public health, airlanga university, surabaya 2laboratory of environmental and health statistic, institut teknologi sepuluh nopember, surabaya, indonesia 3 department of biostatistics and demography, faculty of public health, airlanga university, surabaya, email: 1isna.husaini1@gmail.com , 2dr.otok.bw@gmail.com abstract climate change causes changes rainfall, temperature, air humidity and wind direction so that affect the reproduction of vectors of diseases such as the mosquito aedes, malaria, etc. that it needs to be monitored the increase in many cases db. free number of larvae (abj) is one of the larva density indicator, although abj has more than 90 percent but morbidity remains high. the condition of the abj not describes the density of larvae jentik, so that the need to study the density jentik indicator that more can describe as the larvae density index with sem based variance approach. the results of the study showed that the structural model nonparametric to larva density is the best model based on the criteria of r2 and q2. the ministry of health and behavior, environment condition and breeding place/site effect on the larva density of 87.7%. the dominant indicator counseling on health services, knowledge on the behavior of the temperature of the water on the conditions in the environment and the material container on the breeding place/sites. while on the larva density each indicator provides value loading, larvae density index (0.864), house index (0.459), container index (0.894), and breateau index (0.925). environmental conditions the dominant factor in affecting larva density decline of 32.4%, with each indicator larvae density index (28%), house index (15%), container index (29%), and breateau index (30%). keywords: sem based variance, larva density, r2, q2 introduction causes of extraordinary events (klb) is a condition of the emergence or increased incidents of pain or death by epidemiologis on a region in a specific period of time. in 2013, patients with dengue fever which spread all over the country by 112511 cases, while in the year 2014, cases dengue fever decline of 100347 cases with death as much as 907 people [1]. in indonesia dengue fever is still the main health problems and until this time was interested in medicine and not yet available when this disease can be transmitted quickly by the mosquito aedes aegypti so that potential to cause a pandemic [2]. the end of the year 2014 was recorded as much as 433 from 508 regency/city (85.2%) has been infected with dengue fever with the total number of cases 100347 (morbidity: 39.83 per 100000 inhabitants) with 907 cases died (death rate: 0.90%), and in 2015 recorded 9 provinces that have increased cases of dengue fever and there are 2 provinces that have mailto:isna.husaini1@gmail.com mailto:dr.otok.bw@gmail.com structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 128 morbidity (inciden rate/ir) dengue fever above national targets 49 per 100000 inhabitants, south kalimantan and east kalimantan [1]. the case of dengue fever in south kalimantan in the last three years there has been increased with ir 2013 (31.31 per 100000 inhabitants), the year 2014 (21.53 per 100000 inhabitants), and there is an increase in the 2015 namely 70.17 per 100000 inhabitants of 49 per 100000 population [3]. from 13 regencies/cities in south kalimantan banjarbaru city in the last three years also has increased, even 2015 ir jumped sharply until 230,73 per 100000 inhabitants of ir in the previous year 89.48 per 100000 inhabitants and on the years down to the numbers 11.45 per 100000 population [3]. various intervention done for pressing the case of dengue fever is still high as fogging, eradication of the nest lice (psn) or 3m plus, but abj until 2015 still fluctuate from 2013 of 68.02% and prevent larva behavior according to [4], 77.4 percent new is still below the target. the year 2014 and 2015 abj of 93.46% [3]. up to this moment this disease can only be controlled with termination chain of transmission vektornya because drugs and vaccines of this disease is still not yet [5] and a research in australia shows that stop the supervision of mosquitoes can improve epidemic management costs compared with the strategy sustainable surveillance and early detection of cases [6]. one of them is the supervision of the larvae. in the case of dengue fever there is some latent variables such as environmental conditions, vectors and remain indifferent. one of analysis techniques that are often used to describe the relationship or the influence of the complex between the variable is the modeling technique statistics structural equation modellng (sem) who have the ability to analyze the pattern of the relationship between the latent konstrak and the indicators, or the latent konstrak with each other, and measurement error directly. sem allows analysis done among some endogenous dependent variables and independent exogenous supply directly [7]. many counting provides data analysis model biased on parametrik method. data analysis approach is also directed as nonparametrik technique and if the available information about the regression curve is limited and difficult to make the assumptions of regression form, then the largest part of the information located on the data pattern, so to suspect the regression curve can be used non parametrik regression that analytical is not based on a specific spread [8]. the diversity of data and there is a parameter or indicators that form the latent variable of the conditions in the analysis of the structural model and some variables which contribute in genesis a disease can be formed from various types of data. nominal, ordinal, interval or ratio data and conditions if that is more flexible to be analyzed with sem nonparametric, including a small number of sample, the existence of lost data (missing values), and multicolinierity. according to [7] perform the estimation of model parameters partial least square (pls) on poverty issues, and [9] shows that in the decision-making in the field of health sem pls more perspective and realistic for empirical analysis. according to [10] using nonparametric model structure only to estimation income and environmental emissions, while [11], and [12] trying to find a suitables parameters in a model of sem is used as a basis for inference and prediction. with the increasing case of dengue fever, it needs to be examined larva density model with the approach of nonparametric sem, so that by knowing the factors that affect the density of larva decline expected can be used as a reference for the prevention and control of a case of dengue fever. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 129 methods the research population is all housewife in twelve kelurahan endemic with 296 household in the city banjarbaru on 2016 and sample research is a housewife who was selected in the determination of the sample size based on the cluster method rapid survey [13]. the size of the samples used in this research is the cluster 30 household in twelve kelurahan endemic in the working area of the clinic and each cluster with 7 samples so that the number of the whole sample is 210 house wife who is location in the city of banjarbaru. the variables examined consists of the endogenous latent variables and exogenous supply along with the variables the indicators. the latent variables were five laten, namely latent variable larva density with four indicator variables larva density index, house index, container index, and breateu index. behavior consists of three indicator variables, namely knowledge, attitudes and actions. the latent variable breeding places/site in this case is container consists of indicator variable, material, type, cover, and the color of the container. the latent variable environmental conditions consist of four indicator variables are the source of water, water temperature and ph of the water in the container and the latent variable health services consists of four indicator variables, namely larvasidasi, fogging, periodic larva examination, and conselling service. the conceptual research presented as follows. figure 1. the framework of the concept of larva density prediction aedes sp in the city banjarbaru [14] the data in this research using the primary data and then done analysis with the modeling of sem pls. the modeling of sem that is done by using partial least square (pls) with the steps as follows [15]: 1. outer model, covers the validity test is seen from the results of loading factor, and reliability tests seen from the value of composite reliability. the indicator is called structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 130 valid if has a value of loading factor > 0.5 and said reliable if the value of composite reliability > 0.6. 2. inner model, test this can be seen as a result of the value of the inner weight to test the formulation of research hypothesis through t test on the bootstrap samples and goodness of fit model. the model can be stated to have the goodness of fit if has a value r-square > 0 and the value of q2 > 0.35 provides high accuracy. results and discussion description of this research includes the mean and standard deviation from each indicator on the research variables. now in detail is presented in the following table. table 1. descriptive statistic of research variables no the characteristic percentage (%) ministry of health 1. larvasidasi : less (< 60%) just (60%-75%) good (>75%) 72.4 8.1 19.5 2. fogging less (< 60%) just (60%-75%) good (>75%) 77,6 9.0 13.3 3. periodic larva examination (pjb) less (< 60%) just (60%-75%) good (>75%) 84,8 2.9 12.4 4. counseling services less (< 60%) just (60%-75%) good (>75%) 81,9 4.3 13.8 behavior 1. knowledge : less enough good 20.8 51.9 26.4 2. attitude: less good 4.8 95.2 3. action: less enough good 50.5 29.5 20.0 structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 131 table 2. descriptive statistic of research variables (continues) no the characteristic percentage (%) environment condition 1. type of water/clean water sources : non piping dug wells, boreholes others) pipelines (pdam) 61.4 38.6 2. the water temperature : support the growth (25oc-27oc) less support growth (<25oc and >27oc) 45.7 54.3 3. water ph : support the growth (ph 5.8-8.6) less support growth (ph <5.8 or ph >8.6) 0 100 breeding place/site 1. container material: the metal contacts/metal cement plastic 5.7 31.0 63.3 2. container type: water reservoirs (tpa) non-water reservoirs (non tpa) experience 71.36 27.17 1.45 3. container cover: there is no cover there is the cover 60,0 40 4. container color: dark (black, gray) the light (red, pink blue, green, white orange) 21.4 78.6 larva density 1 larva density index/ldi 18.21 2 house index /hi (%) 37.62 3 container index/ci (%) 14.32 4 breuteu index /bi (%) 55.71 5 numbers free jentik/abj (%) 62.38 6 density figure /df 5.67 table 2 shows that most of the mother of households with health services from the health officer at 5.2 percent to 19.5 percent stated that the ministry of health by the officers in the prevention of vector diseases dengue fever is good. more intensive health services done in the rainy season so that health officials in the ministry of health to become more active. the behavior of the mother of the household in which related with knowledge, attitudes, customs and pencegahannnya efforts against vector dengue fever by 51.9 percent is good. environmental conditions in this research is related to water reservoirs (tpa) as a place to live jentik mosquitoes aedes aegypti, most of the sources of water used population is not water piping water dug wells, boreholes, river) namely ranges 61.4 percent. physical condition of water a qualify namely odourless, no color, odorless. the temperature of the water in the containers ranging between 250c-310c, according to the ministry [16] in general temperature to the place of the multiplication of mosquitoes aedes ranged between the temperature of 250c-270c. the development of optimally for living water at a temperature 250c until 270c larvae will die at a temperature less than 100c or more than 40oc. water temperature can also be structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 132 influenced by air temperature, normally, water temperature ranged between ± 30c with air temperature [17]. water ph rainy season conditions support, ranging from 4-8 ph level of water affects the level of o2 and co, while o2 will be deposited. ph is normal for the development of mosquitoes from laying eggs to become pupae ranged between 4 9, and ph of the water in the container is good for the reproduction of larva aedes aegypty ranged between 5.8 – 8.6 [18]. massive ph range from 0 (very acid) to 14 (very base/alkalis). the value of ph less than 7 shows an environment of acid while the value above 7 indicates the base environment (alkaline). while ph = 7 is called as neutral [19]. the level of o2 and co in the water also affect the formation of the enzyme oxidation sinokrom aedes aegypti larvae sp. and aedes albopictus sp. the fluctuations in ph water is in specify by alkalinity water. the manner vector diseases dengue fever or jentik larvae mosquitoes aedes, sp called breeding place/ site. at the place that is observed is container, good ingredients, type, there is whether or not the cover, and the color of the container. the container with the rainy season more dominant namely 63.3 percent, the ingredients are the most used by the housewife is a plastic 39.11 percent. close the container that is used is also there that can be filled with water so that allows for a place larva to multiply. the color of the container that used household most or more than 70 percent bright as red, blue, green. larva density can be seen from some of the size of such as house index (hi), container index (ci), breuteu index (bi), density figure (df), and larval density index (ldi). in the rainy season jentik density of each house is relatively homogeneous namely range between the lowest infection rate was from 0.00 to 58,14 jentik or the average range 18.21 around 18-19 jentik each house. the type of jentik or the larvae found on the containers in household on rainy season is 77.8 percent larva aedes aegypty and 22.2 percent aedes albopictus. index larva a house index (hi) is also able to show the size for free number larva (abj), namely if the rainy season with hi between 23.8167.86 percent with the average 37.62 percent means numbers free larva is 62,38 percent. breuteu index (bi) the existence of larva seen each 100 houses, the rain around 56 houses which have a positive container. the rainy season df (5.33), including regions are being or need to be cautious in the transmission of the disease by the larvae, and epidemiology df > 1 including risk in the transmission of dengue fever [1], [20], [21]. data analysis the research using sem by involving the validity test and reliability using through confirmatory factor analysis (cfa) and contruct reliability using the program smartpls. validity test is intended to determine whether the questions in the questionnaire just representative. validity test is done by using confirmatory factor analysis on each of the latent variable. the second measure test is reliable, namely the index shows the extent to which reliable measure or can be trusted. reliability is the size of the internal consistency of the indicators a variables proxies which shows the degree to which each indicator it indicates a variable common adjectives more detailed results are presented in table 3. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 133 table 3. validity and reliability convergence of the indications on laten variables the variables the indicator validity convergence validity discriminant composite reliability loading factor p-value conclusion ministry of health (x) larvasidasi (x1) 0.691 0.686 8.185 0.695 0.799 fogging (x2) 0.714 0.715 6.646 pjb (x3) 0.601 0.575 5.003 counseling service (x4) 0.795 0.772 9.052 behavior (y1) knowledge (y1.1) 0.716 0.723 6.360 0.643 0.669 attitude (y1.2) 0.708 0.678 6.766 action (y1.3) 0.562 0.543 5.522 environment condition (y2) type of water (y2.1) 0.649 0.647 9.832 0.603 0.670 the water temperature (y2.2) 0.665 0.657 10.539 ph water (y2.3) 0.588 0.590 8.075 breeding place/site (y3) container materials (y3.1) 0.963 0.962 124.888 0.898 0.708 the type of containers (y3.2) 0.950 0.950 116.528 the container cover (y3.3) 0.920 0.919 64.784 the color of the container (y3.4) 0.957 0.956 169.037 larva density (y4) larva density index (y4.1) 0.864 0.864 56.534 0.652 0.973 house index (y4.2) 0.459 0.454 11.051 container index (y4.3) 0.894 0.895 57.346 breateau index (y4.4) 0.925 0.922 67.551 after the validity and reliability test on all latent variables which valid results and reliabel, then done test the suitability of the structural model on the larva density. the full model larva density chart form as follows. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 134 figure 2. the relationship of ministry health, behavior, environment condition, braiding place against the larva density on rainy season the results of the complete model testing in smartpls program can be seen from the values r-square that illustrates the goodness of the fit of a model. the value of rsquare that is recommended is greater than zero. the results of this research data processing using smartpls provides value r-square as is shown in table 3. table 4. goodness of fit model larva density on rainy season the variables r-square ministry health (x1)  behavior (y1) 0.204 behavior (y1), breeding place/site (y3)  environment condition (y2) 0.229 environment condition (y2)  breeding place/site (y3) 0.303 ministry health (x1), behavior (y1), environment condition (y2), breeding place/site (y3)  larva density (y4) 0.571 table 4 explains that:  contributions or the proportion of ministry health (x1) variables in explaining the variations around the behavior (y1) of variables of gaining of 0.204.  contributions or the proportion of behavior variables (y1), breeding place/ site (y3) in explaining the variation around environmental conditions variable (y2) of 0.229.  contributions or the proportion of environmental conditions variable (y2) in explaining the variations around the variables breeding place/site (y3) of 0.303. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 135  contributions or the proportion of health services variables (x1), the behavior (y1), the condition of the environment (y2), breeding place/site (y3) in explaining the variations around the larva density (y4) variable of 0.571. the results from all the value r-square showed that all values r-square greater than zero. this means that the model of this research is already meet the goodness of fit is required. the results of the calculation of the value of q square from the table 3., obtained the following result: q2 = 1((1-0.204) x (1-0.229) x (1-0.303) x (1-0.571)) = 0.877 this can be interpreted that the model is able to explain the larva density (y4) of 87.7%, and the rest is explained by other variables outside the model. from the appropriate model, so it can be in interprets each path coefficient. the coefficient of the path coefficient is the hypothesis in this research that can be performed in the following structural equation: y1 = 0.066 x1 , r2 = 0.204 y2 = 0.148 y1 + 0.091 y3 , r2 = 0.229 y3 = 0.059 y1 , r2 = 0.303 y4 = 0.169 x1 – 0.205 y1 0.324 y2 – 0.054 y3 , r2 = 0.571 structural model tests (inner weight) demonstrated through the results of the structural path coefficient. the results of the structural path coefficient (inner weight) and the value of the full significance is shown in table 5. table 5. test the inner weight against the larva density rainy season with the bootstrap the influence coefficient orginal (bootstrap b=500) coefficient |t-statistics| the ministry of health (x1)  behavior (y1) 0.066 0.073 1.988 the behavior (y1)  environment condition (y2) 0.148 0.147 5.257 breeding place/site (y3)  environment condition (y2) 0.091 0.097 3.262 the ministry of health (x1)  larva density (y4) -0.169 -0.170 5.714 the behavior (y1)  larva density (y4) -0.205 -0.212 6.676 the condition of the environment (y2)  larva density (y4) -0.324 -0.329 11.956 breeding place/site (y3)  larva density (y4) -0.054 -0.056 1.993 the behavior (y1)  breeding place/site (y3) 0.059 0.061 2.138 on the test of the bootstrap samples b = 500 give significant results, then continued in the analysis with chart form as follows: structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 136 figure 3. the relationship of ministry health, behavior, environment condition, braiding place against the larva density on rainy season with estimates of the bootstrap based on the table 5 and figure 3, interpretation of each path coefficient is as follows:  the ministry of health (x) influential significant and positive toward the behavior (y1). this can be seen from the path marked by the positive coefficient of 0.066 with t-value of 1.988 statistics greater than t-table =1.96. thus the health services (x) directly impact on the behavior of (y1) of 0.066 which means that every there is increasing health services (x) then will increase the behavior (y1) of 0.066.  the behavior (y1) influential significant and positive impact on the condition of the environment (y2). this can be seen from the path marked by the positive coefficient of 0.148 with t-value of 5.257 statistics greater than t-table =1.96. thus the behavior (y1) directly impact on the condition of the environment (y2) of 0.148 which means that every there is increasing the behavior (y1) then will improve the condition of the environment (y2) of 0.148.  breeding place/site (y3) influential significant and positive impact on the condition of the environment (y2). this can be seen from the path marked by the positive coefficient of 0.091 with t-value of 3.262 statistics greater than t-table =1.96. thus the breeding place/site (y3) directly impact on the condition of the environment (y2) of 0.091 which means that every there is increasing breeding place/ site (y3) then will improve the condition of the environment (y2) of 0.091.  the ministry of health (x) significant and negative effect on the larvae density (y4). this can be seen from the path marked by the negative coefficient of 0.169 with tvalue of 5.714 statistics greater than t-table =1.96. thus the health services (x) directly impact on the larvae density (y4) of -0.169 which means that every there is increasing health services (x) then will reduce the larvae density (y4) of 0.169. structural equation modeling based on variance density index of larvae of the rainy season in the city of banjarbaru isnawati 137  the behavior (y1) significant and negative effect on the larvae density (y4). this can be seen from the path marked by the negative coefficient of 0.205 with t-value of 6.676 statistics greater than t-table =1.96. thus the behavior (y1) directly impact on the larvae density (y4) of -0.205 which means that every there is increasing the behavior (y1) then will reduce the larvae density (y4) of 0.205.  the condition of the environment (y2) significant and negative effect on the larvae density (y4). this can be seen from the path marked by the negative coefficient of 0.324 with t-value of 11.956 statistics greater than t-table =1.96. thus the condition of the environment (y2) directly impact on the larvae density (y4) of -0.324 which means that every there is increasing environmental conditions (y2) then will reduce the larvae density (y4) of 0.324.  breeding place/site (y3) significant and negative effect on the larvae density (y4). this can be seen from the path marked by the negative coefficient of 0.054 with tvalue of 1.993 statistics greater than t-table =1.96. thus the breeding place/site (y3) directly impact on the larvae density (y4) of -0.054 which means that every there is increasing breeding place/site (y3) then will reduce the larvae density (y4) of 0.054.  the behavior (y1) influential significant and positive for the breeding place/site (y3). this can be seen from the path marked by the positive coefficient of 0.059 with the value of the t statistics of 2.138 greater than t-table =1.96. thus the behavior (y1) directly impact on the breeding place/site (y3) of 0.059 which means that every there is increasing the behavior (y1) then will increase the breeding place/site (y3) of 0.059. the relationship is not directly between the exogenous latent variables (health services (x)). with latent variable endogenous mediator/ intervening (behavior (y1). the condition of the environment (y2). breeding place/site(y3)) and endogenous latent variable (larvae density)). the influence of indirect behavior (y1) on health services (x) against larvae density (y4) of -0.014. environment condition (y2) on the behavior (y1) to larvae density (y4) of -0.048. and on breeding place/site (y3) to larvae density (y4) of -0.029. while the influence of indirect breeding place/site (y3) on the behavior (y1) to larvae density (y4) of -0.003. conclusions the results of the study showed with sem based variance approach that this research model meets the goodness of fit is required. larva density model is the best model based on the criteria of r2 and q2, which is capable of described by the ministry of health, the behavior, the condition of the environment, breeding place/site of 87.7%, and the rest is explained by other variables outside the model. the ministry of health, the behavior, the condition of the environment, and breeding place/site effect on the larva density. the ministry of health with the highest indicator counseling, latent variable behavior with the highest indicator knowledge, latent variable environmental conditions with the highest indicator water temperature, latent variable breeding place/site with the highest indicator ingredients container, and latent variable larva density with the highest indicator breateau index. the ministry of health in explaining the variations around the variables behavior model of gaining 0.204, behavior, breeding place/site in explaining the variation around environmental conditions variable of 0.229, environment condition in explaining the variations around the variables breeding place/site of 0.303, and health services, the behavior, the condition of the 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[21] l. c. chan, singapore dengue haemorragic fever control programme a case study on the succesfull, control of aedes aegypty and aedes albopictus using mainly environmental measures as a part of integrated vector control, seamic, tokyo, 1985. matrix approach to the direct computation method for the solution of fredholm integro-differential equations of the second kind with degenerate kernels cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 100-108 p-issn: 2086-0382; e-issn: 2477-3344 submitted: march 14, 2020 reviewed: april 04, 2020 accepted: november 01, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.8960 matrix approach to the direct computation method for the solution of fredholm integro-differential equations of the second kind with degenerate kernels nathaniel kamoh1, geoffrey kumlengand2, joshua sunday3 1,2,3department of mathematics, university of jos, jos, nigeria email: mahwash1477@gmail.com, gkumleng@gmail.com, joshuasunday2000@yahoo.com abstract in this paper, a matrix approach to the direct computation method for solving fredholm integrodifferential equations (fides) of the second kind with degenerate kernels is presented. our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. the proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. the result of this research is the solution of the second type fredholm integro-differential equation (fide) with a numerically accurate kernel degenerate. some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method. keywords: fredholm; matrix; direct solution; integro-differential equation; integral introduction the subject integro-differential equations (ides) is one of the most important mathematical tools in both pure and applied mathematics. in recent year’s mathematical modeling of real-life usually results in functional equations such as differential equations, integral and integro-differential equations (ides) these equations play very important role in modern science and technology application such as the theory of signal processing, neutral networks, heat transfer, diffusion process, neutron diffusion and biological species. these equations can be classified into fredholm equations and volterra equations. the upper bound of the region for integral part of the volterra type is a variable, while it is a fixed number for that of fredholm type. more details and sources where these equations can be found are in the areas of physics, biology, engineering and social sciences and have been extensively studied both at theoretical and practical level. most integro-differential equations are usually very difficult to solve analytically and so accurate, acceptable and efficient numerical method is required to approximate the solution (see [1], [4], [5], [6], [13], [15] and [16]). in recent times, extensive efforts have been devoted to the numerical methods of solution for fredholm integro-differential equations by many researchers. [2] considered non-standard finite difference method for the numerical solution of linear fredholm integro-differential equations. the method and the repeated composite http://dx.doi.org/10.18860/ca.v6i3.8960 mailto:mahwash1477@gmail.com mailto:gkumleng@gmail.com mailto:joshuasunday2000@yahoo.com matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 101 trapezoidal quadrature method were used to transform the fredholm integrodifferential equation into a system of non-linear algebraic equations and experiments on some linear model problems showed the simplicity and efficiency of the proposed method. the wavelet method for the numerical solution of fredholm integro-differential equation was used in [14]. [3] developed a finite difference hybrid method by a combination of power series and the shifted legendre polynomial to solve fredholm integro-differential equation. a new and efficient approach for the numerical solution of fredholm integro-differential equations (fides) of the second kind with an unbounded domain with degenerate kernel based on operational matrices with respect to generalized laguerre polynomials (glps) was introduced in [8].the adomian’s decomposition method which is a well-known method for solving functional equations in recent times was used to solve linear fredholm integro-differential equations by [9]. the result obtained gives more accurate approximation as compared to two other methods. [10], applied the legendre polynomials for the solution of the linear fredholm integro-differential-difference equation of high order and the results obtained by the developed technique were more accurate than the results reported for the taylor and the wavelet galerkin methods the purpose of this paper is to solve fredholm integro-differential equation of the second kind with separable kernels by the approach of matrix method for the direct computation method, this approach is considered simple, accurate and easy to implement. methods consider the standard form of the fredholm integro-differential equation of the second kind given by 𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∫ 𝑘(𝜃, 𝜉)𝜓(𝜉)𝑑𝜉 𝑏 𝑎 (1) 𝜓(𝑘)(0) = 𝑞𝑘 , 0 ≤ 𝑘 ≤ 𝑛 − 1 (2) where 𝜓(𝑛)(𝜃)indicates the 𝑛𝑡ℎ derivative of 𝜓(𝜃) with respect to 𝜃. because (1)combines differential operator and the integral operator, then it is necessary to define initial conditions given in (2) for the determination of the particular solution 𝜓(𝜃) of (1). suppose that we wish to determine the approximate solution of the theoretical solution 𝜓(𝜃) of problem (1) at the domain [𝑎, 𝑏]. let the separable or degenerate kernel𝐾(𝜃, 𝜉)of (1) be approximated by∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) 𝑛 𝑘=1 . thus the integrodifferential equation (1) may be written as 𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∫ ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) 𝑛 𝑘=1 𝜓(𝜉)𝑑𝜉, 𝑏 𝑎 (3) where 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) 𝑛 𝑘=1 is a finite sum of products𝜏𝑘 (𝜃) and 𝜙𝑘 (𝜉), 𝜏𝑘 (𝜃) is a function of 𝜃 only and 𝜙𝑘 (𝜉) is a function of 𝜉 only. equation (3) can be rewritten as 𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃) 𝑛 𝑘=1 ∫ 𝜙𝑘 (𝜉)𝜓(𝜉)𝑑𝜉 𝑏 𝑎 (4) discretizing the integral part of (4) by letting 𝜇𝑘 = ∫ 𝜙𝑘 (𝜉)𝜓(𝜉)𝑑𝜉 𝑏 𝑎 , (4) becomes matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 102 𝜓(𝑛)(𝜃) = 𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃)𝜇𝑘 𝑛 𝑘=1 (5) now, multiplying both sides of (5) by the integral operator𝜒−𝑛 define by (𝜒−𝑛(⋆) = ∫(⋆)𝑑𝜃) with the application of the initial conditions (2), it follows that (5) can be written as 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 𝑛 𝑖=1 𝜓(𝑖−1)(0) = 𝜒−𝑛 [𝜔(𝜃) + 𝜆 ∑ 𝜏𝑘 (𝜃)𝜇𝑘 𝑛 𝑘=1 ] (6) simplifying (6), we obtain 𝜓(𝜃) − ∑ 𝜃 (𝑖−1) (𝑖 − 1)! 𝑛 𝑖=1 𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + 𝜆 ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 𝑛 𝑘=1 (7) multiplying both sides of (7) by𝜙𝑚(𝜃), 𝑚 = 1,2, . . . , 𝑛 and integrating from 𝑎 to 𝑏 over 𝜃 leads to a matrix equation that facilitates the determination of the𝜇𝑘 ’s in (5). thus equation (7) becomes ∫ 𝜓(𝜃)𝜙𝑚(𝜃)𝑑𝜃 𝑏 𝑎 − ∫ ∑ 𝜃(𝑖−1) (𝑖 − 1)! 𝑛 𝑖=1 𝑏 𝑎 𝜓(𝑖−1)(0)𝜙𝑚(𝜃)𝑑𝜃 = ∫ (𝜔⋇(𝜃) + 𝜆 ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 𝑛 𝑘=1 ) 𝑏 𝑎 𝜙𝑚(𝑥)𝑑𝜃 (8) if we define𝜇𝑚,𝜔𝑚,𝑎𝑚𝑘 as∫ 𝜓(𝜃)𝜙𝑚(𝜃)𝑑𝜃 𝑏 𝑎 ,∫ 𝜔⋆(𝜃)𝜙𝑚(𝜃)𝑑𝜃 𝑏 𝑎 and∫ 𝜏 ∗𝑘 (𝜃)𝜙𝑚(𝜃)𝑑𝜃 𝑏 𝑎 respectively,then (8)can be written as 𝜇𝑚 − 𝜆 ∑ 𝜇𝑘 𝑎𝑚𝑘 = 𝜔𝑚, 𝑛 𝑘=1 𝑚 = 1,2, . . . , 𝑛 (9) equation (9) gives a nonhomogeneous system of 𝑛 linear equations in 𝜇1, 𝜇2, 𝜇3, … , 𝜇𝑛 unknown.𝜔𝑚 and 𝜏𝑚𝑘 are known since 𝜙𝑚(𝜃), 𝜔 ⋆(𝜃) and𝜏 ∗𝑘 (𝜃) are all given. putting (9) in matrix equation form, it follows that (𝐼 − 𝜆 ∑ 𝑎𝑚𝑘 𝑛 𝑘=1 ) 𝜇𝑘 = 𝜔𝑚 (10) where 𝜇𝑘 = [ 𝜇1 ⋮ 𝜇𝑛 ] , 𝜔𝑚 = [ 𝜔1 ⋮ 𝜔𝑛 ] , 𝑎𝑚𝑘 = [ 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑛1 ⋯ 𝑎𝑛𝑛 ] equation (10)can be used to determine the unknown 𝜇1, 𝜇2, 𝜇3, … , 𝜇𝑛 which are then substituted into 𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 𝑛 𝑘=1 for the particular solution of (1) results and discussion we now apply the method presented in this paper to solved four model problems to illustrate the above mentioned approach and demonstrate its computational accuracy. this method differs from the direct computation method since the unknown constants are determined at once by the introduced matrix equation matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 103 problem1 consider the linear fredholm integro-differential equation given in [1] 𝑦′′′(𝑥) = 5𝑙𝑛2 − 3 − 𝑥 + 4𝑐𝑜𝑠ℎ𝑥 − ∫ (𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡, 𝑦(0) = 𝑦′′(0) = 0 𝑙𝑛2 0 , 𝑦′(0) = 4, 0 ≤ 𝜃 ≤ 𝜋 the analytical solution to the problem is 𝑦(𝑥) = 4𝑠𝑖𝑛ℎ(𝑥) let 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉) 3 𝑘=1 𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −1, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉 let the required solution be given as 𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 3 𝑘=1 multiplying the given fide through with the integral operator𝜒−3 to obtain 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 3 𝑖=1 𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 3 𝑘=1 or 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 3 𝑖=1 𝜓(𝑖−1)(0) = 2𝑒𝜃 + 5 6 𝜃3𝑙𝑛2 − 1 2 𝜃3 − 1 24 𝜃4 − 2 𝑒𝜃 − ( 1 24 𝜃4𝜇1 − 1 6 𝜃3𝜇2) where 𝜔1 = 1 5 𝑙𝑛25 − 1 8 𝑙𝑛24 + 1, 𝜔2 = 5𝑙𝑛2 − 1 10 𝑙𝑛25 + 23 144 𝑙𝑛26 − 3 𝑎11 = 1 120 𝑙𝑛25, 𝑎12 = − 1 24 𝑙𝑛24, 𝑎21 = 1 120 𝑙𝑛26,𝑎22 = − 1 30 𝑙𝑛25 using (10) to solve for𝜇𝑘, 𝑘 = 1, 2, we have 𝜇1 = 1, 𝜇2 = 5𝑙𝑛2 − 3 substituting in 𝜓(𝜃) = 2𝑒𝜃 + 5 6 𝜃3𝑙𝑛2 − 1 2 𝜃3 − 1 24 𝜃4 − 2 𝑒𝜃 − ( 1 24 𝜃4𝜇1 − 1 6 𝜃3𝜇2) we have 𝜓(𝜃) = 4sinh(𝜃)as the exact solution. table 1 below reveals the performance of the proposed method for problems 1 table 1: the performance results of the proposed method for problem 1 values (𝑥 = 𝜃) exact solution proposed method absolute error 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.000000000 0.400667000 0.805344010 1.218081174 1.643009303 2.084381222 2.546614328 3.034334807 3.552423929 4.106066904 4.700804776 0.000000000 0.400667000 0.805344010 1.218081174 1.643009303 2.084381222 2.546614328 3.034334807 3.552423929 4.106066904 4.700804776 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 0.00e+00 matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 104 problem2 consider the linear fredholm integro-differential equation in [1] given by 𝑦′′′(𝑥) = 𝑒 − 2 − 𝑥 + 𝑒 𝑥 (3 + 𝑥) + ∫ (𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡, 𝑦(0) = 0 1 0 , 𝑦′(0) = 1, 𝑦′′(0) = 2, 0 ≤ 𝜃 ≤ 𝜋 the analytical solution to the problem is 𝑦(𝑥) = 𝑥𝑒 𝑥 let 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉) 3 𝑘=1 𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −1, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉 the required solution is given as 𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 3 𝑘=1 multiplying the given fide with the integral operator𝜒−3, we have 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 3 𝑖=1 𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 3 𝑘=1 or 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 3 𝑖=1 𝜓(𝑖−1)(0) = 1 6 𝑒𝜃3 − 1 3 𝜃3 − 1 24 𝜃4 + 𝜃𝑒𝜃 + ( 1 24 𝜃4𝜇1 − 1 6 𝜃3𝜇2) where 𝜔1 = 109 120 + 1 24 𝑒, 𝜔2 = − 1493 720 + 31 30 𝑒 𝑎11 = 1 120 , 𝑎12 = − 1 24 , 𝑎21 = 1 144 , 𝑎22 = − 1 30 using (10) to solve for 𝜇𝑘, 𝑘 = 1, 2 and substituting into 𝜓(𝜃) = 1 6 𝑒𝜃3 − 1 3 𝜃3 − 1 24 𝜃4 + 𝜃𝑒 𝜃 + ( 1 24 𝜃4𝜇1 − 1 6 𝜃3𝜇2) we obtain𝜓(𝜃) = 𝜃𝑒𝜃 giving the same result as the exact solution. problem 3 consider the linear fredholm integro-differential equation in [1] given as 𝑦′′(𝑥) = 4𝑥 − sin(𝑥) + ∫ (𝑥 − 𝑡)2𝑦(𝑡)𝑑𝑡, 𝑦(0) = 0 𝜋 2 − 𝜋 2 , 𝑦′(0) = 1, − 𝜋 2 ≤ 𝜃 ≤ 𝜋 2 the analytical solution to the problem is 𝑦(𝑥) = 𝑠𝑖𝑛(𝑥) let 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 𝜉) 2 2 𝑘=1 𝜏1(𝜃) = 𝜃 2, 𝜏2(𝜃) = −2𝜃, 𝜏3(𝜃) = 1 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉, 𝜙3(𝜉) = 𝜉 2 matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 105 the required solution is given as 𝜓(𝜃) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 2 𝑘=1 multiplying the given fide with the integral operator𝜒−2, we have 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 2 𝑖=1 𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 3 𝑘=1 or 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 2 𝑖=1 𝜓(𝑖−1)(0) = 𝑠𝑖𝑛𝜃 + 2 3 𝜃³ + ( 1 2 𝜃²𝜇₃ − 1 3 𝜃³𝜇₂ + 1 12 𝜃⁴𝜇₁) where 𝜔1 = 0, 𝜔2 = 1 12 𝜋⁵ + 2, 𝜔2 = 0 𝑎11 = 1 960 𝜋⁵, 𝑎12 = 0, 𝑎13 = 1 24 𝜋³ 𝑎21 = 0, 𝑎22 = − 1 240 𝜋⁵, 𝑎23 = 0 𝑎31 = 1 5376 𝜋⁷, 𝑎32 = 0, 𝑎33 = 1 160 𝜋⁵ using (10) to solve for 𝜇𝑘, 𝑘 = 1, 2, 3 and substituting in 𝜓(𝜃) = 𝑠𝑖𝑛𝜃 + 2 3 𝜃³ + ( 1 2 𝜃²𝜇₃ − 1 3 𝜃³𝜇₂ + 1 12 𝜃⁴𝜇₁) gives 𝜓(𝜃) = 𝑠𝑖𝑛(𝜃) having the same result as the exact solution. problem 4 consider the linear fredholm integro-differential equation given by 𝑦(𝑖𝑣)(𝑥) = 2𝑥 − 𝜋 + sin(𝑥) + 𝑐𝑜𝑠(𝑥) − ∫ (𝑥 − 2𝑡)𝑦(𝑡)𝑑𝑡, 𝜋 2 0 𝑦(0) = 𝑦′(0) = 1, 𝑦′′(0) = 𝑦′′′(0) = −1, − 𝜋 2 ≤ 𝜃 ≤ 𝜋 2 the analytical solution to the problem is 𝑦(𝑥) = 𝑠𝑖𝑛(𝑥) + 𝑐𝑜𝑠(𝑥) let 𝑘(𝜃, 𝜉) = ∑ 𝜏𝑘 (𝜃)𝜙𝑘 (𝜉) = (𝜃 − 2𝜉) 4 𝑘=1 𝜏1(𝜃) = 𝜃, 𝜏2(𝜃) = −2, 𝜏3(𝜃) = 0, 𝜏4(𝜃) = 0, 𝜙1(𝜉) = 1, 𝜙2(𝜉) = 𝜉, 𝜙3(𝜉) = 0, 𝜙4(𝜉) = 0 the required solution is given as 𝜓(𝜃) = 𝜔⋇(𝜃) + 𝛾 ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 4 𝑘=1 multiplying the given fide with the integral operator𝜒−4, we have 𝜓(𝜃) − ∑ 𝜃(𝑖−1) (𝑖 − 1)! 4 𝑖=1 𝜓(𝑖−1)(0) = 𝜔⋇(𝜃) + ∑ 𝜏𝑘 ⋇(𝜃)𝜇𝑘 4 𝑘=1 matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 106 or 𝜓(𝜃) − ∑ 𝜃 (𝑖−1) (𝑖 − 1)! 4 𝑖=1 𝜓(𝑖−1)(0) = 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 − 1 24 𝜋𝜃⁴ + 1 60 𝜃⁵ + ( 1 12 𝜃⁴𝜇₂ − 1 120 𝜃⁵𝜇₁) where 𝜔1 = 2 − 1 4608 𝜋⁶, 𝜔2 = 1 2 𝜋 − 29 322560 𝜋⁷ 𝑎11 = − 1 46080 𝜋⁶, 𝑎12 = 1 1920 𝜋⁵ 𝑎21 = − 1 107520 𝜋⁷, 𝑎22 = 1 4608 𝜋⁶ using (10) to solve for 𝜇𝑘, 𝑘 = 1, 2 and substituting in 𝜓(𝑥) = 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 − 1 24 𝜋𝜃⁴ + 1 60 𝜃⁵ + ( 1 12 𝜃⁴𝜇₂ − 1 120 𝜃⁵𝜇₁) we obtain 𝜓(𝜃) = 𝑐𝑜𝑠(𝜃) + 𝑠𝑖𝑛(𝜃) giving the same result with the exact solution. conclusions this paper deals with the solution of linear fredholm integro-differential equations of the second kind with separable kernels. our approach was based on the matrix approach which reduces the fredholm integrodifferential equation into a set of linear algebraic equations for the determination of the unknown constants. the advantage of this method over the direct computation method is that the constants in this method are obtained at once instead of the successive substitution approach inherent in the direct computation method. the method was tested on some model problems from the literature; the results obtained by the technique developed were the same with the exact solution revealing the effectiveness of the proposed method. acknowledgments the authors express their sincere thanks to the referees for the careful and details reading of their earlier version of the paper and for the very helpful suggestions. references [1] abdul-majid wazwaz, linear and nonlinear integral equations methods and applications. higher education press, beijing, 2011 [2] pandey, p. k., numerical solution of linear fredholm integro-differential 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integro-differential equations, applied mathematics and computation (118): pp.327-342, 2001 [16] hosseini, s.m. and shahmorad, s., numerical solution of a class of integrodifferential equations by the tau method with error estimation. applied mathematics and computation (136):pp. 559–570, 2003 https://doi.org/10.1108/mmms-12-2017-0149 matrix approach to the direct computation method for the solution of fredholm integrodifferential equations of the second kind with degenerate kernels nathaniel kamoh 108 the algorithm for the implementation of the given problems using the maple programme is outline below: problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 169-180 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 18, 2017 reviewed: october 09, 2017 accepted: may 31, 2019 doi: http://dx.doi.org/10.18860/ca.v5i4.4294 problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori1, mohammad imam jauhari 2 1,2 department of mathematics state islamic university sunan kalijaga, indonesia email: borymuch@yahoo.com, jonycrish@gmail.com abstract matching is a part of graph theory that discusses pair. a matching m is called to be maximum if m has the highest number of elements. a blossom which is encountered in non-bipartite graph can cause failure in process of finding the maximum matching in non-bipartite graph. one of the algorithms that can be used to find a maximum matching in non-bipartite graph is edmonds’ cardinality matching algorithm. shrinking process is done in each blossom bi that is encountered to become pseudovertex bi, in a way that each blossom does not interfere the process of finding a maximum matching in non-bipartite graph. in order to accelerate the finding, simple greedy method is used to perform initialization of matching and bfs algorithm is also used in constructing an alternating tree in a non-bipartite graph. the research discussed the finding of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm. in addition, this research gave a sample of its application in the resolution of the battle of britain case. the result obtained is a maximum matching in non-bipartite graph. the maximum matching obtained is a solution to the case of the battle of britain. keywords: edmonds’ cardinality matching, maximum matching, the battle of britain introduction a matching 𝑀 in 𝐺 is a subset of 𝐸(𝐺) where there are no two edges in 𝑀 that meet in the same vertex [3]. matching concept can be used in many things especially for the problem of pairing. in the discussion about matching, there is a term called maximum matching. maximum matching is matching 𝑀 with the highest cardinality [1]. generally, the finding of maximum matching is performed in bipartite graph, but it can also be executed in non-bipartite graph. the process of finding the maximum matching in non-bipartite graph is a bit different from that in bipartite graph where in non-bipartite graph, it should contain at least an odd cycle that issues a blossom. a blossom can cause a failure in some algorithms of finding the maximum matching or it cannot be engaged in finding the maximum matching in non-bipartite graph. one of the algorithms that can be used in finding maximum matching in nonbipartite graph is edmonds’ cardinality matching algorithm. this algorithm can handle a problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 170 blossom found in non-bipartite graph. hence the process of finding maximum matching can be done without any obstacle even though there is a blossom while finding the maximum matching. methods the research was begun with studying the basic concept about graph theory which cover definition of basic graph, types of graph, subgraph and skelter subgraph, operation of graph, and basic concept about tree graph and matching concept in 𝐺 graph. besides, the research also explains about breadth-first search (bfs) algorithm to arrange tree construction of alternating 𝑇. then, it discusses edmonds’ cardinality matching algorithm and its application in finding a maximum matching in a graph and in the last chapter, there is a sample of pairing a pilot in a war of britain in 1940 which was taken from m. gondran and minoux’s book entitled “graphs and algorithms”. results and discussion definition: matching [6] a matching 𝑀 in 𝐺 is a subset of a set 𝐸(𝐺) where there are no two edges in 𝑀 that meet at the same vertex. a 𝐺 graph with matching 𝑀 ⊆ 𝐸(𝐺) can be written in (𝐺, 𝑀). example: a graph 𝐺1 with matching 𝑀 = {𝑣2𝑣3, 𝑣4𝑣5}. 𝐺1: there are two vertex terms in matching graph: 1. saturated vertex is a vertex which is incident with the edges in matching 𝑀. [8] example: in 𝐺1, the saturated vertex is vertex 𝑣2, 𝑣3, 𝑣4, 𝑣5. 2. exposed vertex is a vertex which is not incident with each edge in matching m. [8]. example: in 𝐺1, the exposed vertex is vertex 𝑣1 and 𝑣6. apart from vertex, path in the matching graph is also divided into two kinds: 1. alternating path is a path with the edges that alternate between the edge that is included in 𝑀 and not included in 𝑀. [7] example: in 𝐺1 path 𝑃: 𝑣1 − 𝑣2 − 𝑣3 − 𝑣4 − 𝑣5 is an alternating path 2. augmenting path is a path with the beginning and ending vertex that are exposed vertex. [7] example: in 𝐺1 path 𝑃: 𝑣1 − 𝑣2 − 𝑣3 − 𝑣4 − 𝑣5 − 𝑣6 is an augmenting path problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 171 maximum matching matching 𝑀 in a 𝐺 graph is called maximum if there is no matching 𝑀′in 𝐺 with |𝑀′| > |𝑀|. [1] example: graph with a matching that is not maximum yet graph with a matching that is already maximum matching initialization initialization process is used to shorten the finding of maximum matching in a graph. this process uses a help of simple greedy method as follows: [9] 1. 𝑀 = ∅ 2. if there is no more e edge which can be added into 𝑀, then stop 3. otherwise, select e edge that does not have the same vertex with the edges in 𝑀 3.1 𝑀 = (𝑀 ∪ 𝑒) 4. turn back to 𝑀. blossom a blossom 𝐵 in a matching graph (𝐺, 𝑀) is an odd cycle with 𝑀 ∩ 𝐵 which is the maximum matching in 𝐵 with base which is an exposed vertex or outer vertex for 𝑀 ∩ 𝐵. [3] example: in a case of bipartite graph, it is not possible to find a blossom while constructing alternating tree 𝑇 because bipartite graph does not have two vertices that are adjacent to each other at the same partition. therefore, it is impossible for two vertices at the same layer to join each other [10]. the following below is the illustration: in non-bipartite graph in bipartite graph problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 172 a blossom in non-bipartite graph can raise a possibility of not finding an augmenting path while constructing alternating tree 𝑇 so that it causes a failure in the finding process of maximum matching [10]. shrinking blossom to prevent the problem caused by a blossom 𝐵, the process of shrinking and unshrinking should be done to a blossom 𝐵 that is encountered when the alternating tree 𝑇 is shrinking. a shrinking blossom is a shrinking process to odd cycle in (𝐺, 𝑀) by replacing all of the vertices and edges in blossom 𝐵 with a labeled vertex, for example 𝑏 with 𝑏 = 𝐵/𝐵. vertex 𝑏 is called pseudovertex. when the unshrink process toward 𝑏 is not executed yet (by deleting label 𝑏 then replacing it with 𝐵 as it was), ignore all of the vertices in 𝐵 and all of the edges which join one vertex to the other in 𝐵. besides, the unshrink process toward 𝑏 is a returning process of all vertices and edges included in blossom 𝐵 [3]. example: shrink unshrink below is a lemma which ensures that shrinking and unshrink process can be used in finding a maximum matching. lemma of shrinking cycle [7] supposed 𝐺 is a graph with matching 𝑀, and r is an exposed vertex in (𝐺, 𝑀). supposed that during the process of establishing an alternating tree t with the root 𝑟, it encounters blossom 𝐵. if vertex 𝑤 is a base of blossom 𝐵 and graph 𝐺 ′ = 𝐺/𝐵 is a graph produced from the shrinking of blossom 𝐵 into pseudovertex 𝑏. graph (𝐺, 𝑀) will contain an augmenting path that begins in vertex 𝑟 if and if only graph (𝐺 ′, 𝑀′) with matching 𝑀′ = 𝑀/𝐵 contains an augmenting path that starts in vertex 𝑟. problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 173 alternating tree an alternating tree 𝑇 is a tree whose edges alternate between the edge that is included in 𝑀 and not included in 𝑀 and are rooted in an exposed vertex of which each edge relates an inner vertex with an outer vertex thus every inner vertex in 𝑇 is exactly incident with two edges in 𝑇. inner vertex is a vertex that occupies odd layer (1,3,5, … ), while outer vertex is a vertex that occupies even layer (0,2,4, … ) [3]. the following is the steps to construct an alternating tree 𝑇 using bfs: [7] (root): choose an exposed vertex 𝑟 in graph (𝐺, 𝑀). make the exposed vertex 𝑟 as the root (layer 0) in alternating tree 𝑇. 1. in layer 1, add all the vertices 𝑎1, 𝑎2, 𝑎3, … , 𝑎𝑝 which are adjacent to the root 𝑟. keep in mind that all of the vertices that are added are only saturated ones. 2. add vertex 𝑏𝑖 = 𝑚𝑎𝑡𝑒(𝑎𝑖 ) on layer 2 (even) in 𝑇. 3. for the next layer, add all the vertices 𝑐1, 𝑐2, 𝑐3, … , 𝑐𝑝 where 𝑐𝑖 is adjacent to at least one 𝑏𝑖 , while the edge that relates 𝑐𝑖 with 𝑏𝑖 is not contained in 𝑀. continue step 1, 2, and 3 until finding one of these two possibilities as follows: (a) an augmenting path, (b) an odd cycle. 4. if x is a vertex in layer 2i, and 𝑦 ≠ 𝑚𝑎𝑡𝑒(𝑥) is a vertex that is adjacent to 𝑥. so, there are four possibilities: case 1: y is an exposed vertex (and not yet contained in 𝑇). so, an augmenting path has been found. therefore, the constructing process of alternating tree 𝑇 is discontinued. case 2: 𝑦 is not an exposed vertex; neither 𝑦 nor 𝑚𝑎𝑡𝑒(𝑦) is contained in 𝑇. so, add 𝑦 into layer 2𝑖 + 1 and m 𝑚𝑎𝑡𝑒(𝑦) into layer 2𝑖 + 2. case 3: 𝑦 is already contained in 𝑇 as an inner vertex. so, a cycle with an even length has been found. therefore, ignore if the condition below is found. problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 174 case 4: 𝑦 is already contained in 𝑇 as an outer vertex. blossom has been found. stop the constructing process of alternating tree t and do the shrinking process of the blossom. the followings are theorems that become the basis of edmonds’ cardinality matching algorithm in finding a maximum matching in a graph [7]. theorem: [2] 𝑀1 and 𝑀2 are matching in graph 𝐺. if 𝐻 is a spanning subgraph of 𝐺, then in every component of 𝐻 with (𝐸(𝐻) = (𝑀1 − 𝑀2) ∪ (𝑀2 − 𝑀1) thing that should fulfill the following requirements: 1. an isolated vertex, 2. an even cycle in which the edge of the cycle alternates in 𝑀1 dan 𝑀2, 3. a path in which the edge of the path alternates in 𝑀1 dan 𝑀2 thus every end of the vertex is unsaturated in 𝑀1 or 𝑀2 but not in both. theorem: [7] matching 𝑀 in graph 𝐺 is called maximum if and if only there is no augmenting path in 𝐺 with respect to 𝑀. edmonds’ cardinality matching algorithm the followings are the steps from edmonds’ cardinality matching algorithm: [4] step 1 (initialization) graph 𝐺 is a non-bipartite graph with matching 𝑀0 = ø. do the matching initialization in 𝐺 by using greedy algoritm. step 2 (an examination of exposed vertex) 2.1 if all vertices in (𝐺, 𝑀) are incident with a matching edge, go to step 5. 2.2 if there is only one unexamined exposed vertex in (𝐺, 𝑀), go to step 5. 2.3 otherwise, select any exposed vertex 𝑟 in (𝐺, 𝑀) as a root and construct alternating tree 𝑇: 2.3.1 if the alternating tree 𝑇 ends with an exposed vertex 𝑠 wheres ≠ 𝑟, an augmenting path from 𝑟 to 𝑠 has been found. go to step 3. 2.3.2 if the alternating tree 𝑇 encountered an odd cycle, stop the constructing process and go to step 4. step 3 (augmenting path): 3.1 if the augmenting path 𝑃 obtained from the alternating tree 𝑇 contains pseudovertex 𝑏𝑖 (𝑖 = 1,2,3 … ), then unshrink the pseudovertex 𝑏𝑖 into a blossom 𝐵𝑖 as previously. by using path 𝑃, trace backwards starting from the exposed vertex s to vertex r (root) through an alternating path with an even length in blossom 𝐵𝑖 . do the same steps until augmenting path in (𝐺, 𝑀) is found in which it does not contain any pseudovertex 𝑏𝑖 . augment matching m in (𝐺, 𝑀) with 𝑀′ = 𝑀 ⊕ 𝑃. go to step 3.2. problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 175 3.2 if the augmenting path 𝑃 that has been obtained from the alternating tree 𝑇 does not contain pseudovertex 𝑏𝑖 (𝑖 = 1,2,3 … ), then augment matching m in (𝐺, 𝑀) with 𝑀′ = 𝑀 ⊕ 𝑃. go to step 2. step 4 (odd cycle) this step is reached only if the alternating tree 𝑇 encounters a cycle with an odd length (called blossom). suppose a blossom 𝐵𝑖 (𝑖 = 1,2,3 … ) is encountered in the constructing process of alternating tree 𝑇, then shrink blossom 𝐵𝑖 into a pseudovertex 𝑏𝑖 . continue the construction process of alternating tree 𝑇 with root 𝑟 as step 2. step 5 the matching on graph 𝐺 is maximum. stop the finding process. sample case (the battle of britain) in the battle of britain (1940), the royal air force provided several aircrafts to thwart the german air force attack. the aircrafts provided by the royal air force could be flown only by two people as a pilot and co-pilot. to get the best results in the mission to thwart german air force attack, the royal air force brought some of the best pilots from all regions in britain. however, there was a problem to solve by the royal air force: among all the pilots that were called, some of them could not fly together in one aircraft because there were differences in terms of language, the ability of flying techniques, as well as in terms of flying experience. suppose the royal air force obtained 18 best pilots from all regions in britain that were ready to fly the aircraft. based on the problem, the royal air force wished to pair the 18 pilots where each pair consisted of two pilots in such a way that the number of all the possible pairs is the highest number. in order to solve such problem, the royal air force conducted a series of tests to the 18 pilots including language test, flying technique test, and psychological test to determine the suitability of the 18 pilots to fly together in one aircraft as a pilot and co-pilot [5]. table 1 table of pairing between each aircraft pilot the pilot’s number 𝒊 = (𝟏, 𝟐, … , 𝟏𝟖) the result of pairing between each aircraft pilot 1st pilot pilot: 2nd and 17th 2nd pilot pilot: 1st, 3rd, and 5th 3rd pilot pilot: 2nd, 4th, 5th, and 6th 4th pilot pilot: 3rd, 7th, and 9th 5th pilot pilot: 2nd, 3rd, and 6th 6th pilot pilot: 3rd, 5th, and 15th 7th pilot pilot: 4th, 8th, 9th, 11th, and 13th 8th pilot pilot: 7th, 9th, and 10th 9th pilot pilot: 4th, 7th, 8th, 10th, 13th, and 15th 10th pilot pilot: 8th, 9th, and 16th 11th pilot pilot: 7th, 12th, 13th, and 14th 12th pilot pilot: 11th and 14th 13th pilot pilot: 7th, 9th, 11th, and 14th 14th pilot pilot: 11th, 12th, and 13th 15th pilot pilot: 6th, 9th, 16th, and 18th 16th pilot pilot: 10th, 15th, and 18th 17th pilot pilot 1th 18th pilot pilot: 15th and 16th by transforming the problem into the form of graph, where the pilots are represented as a vertex and the corresponding relationship between each pilot is problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 176 represented as an edge, the following graph is obtained: (the following form of graph is taken from [7]): 𝐺2: by following the steps from edmonds’ cardinality matching algorithm, we obtain maximum matching on graph 𝐺2 as follows: step 1 (initialization): by doing the process of matching initialization to graph 𝐺2, we obtain graph (𝐺2, 𝑀) with 𝑀 = {𝑣1𝑣2, 𝑣3𝑣4, 𝑣5𝑣6, 𝑣7𝑣8, 𝑣9𝑣10, 𝑣11𝑣12, 𝑣13𝑣14, 𝑣15𝑣16} as follows: (𝐺2, 𝑀): since there are still two exposed vertices left that are 𝑣17 and 𝑣18, then go to step 2.3 step 2.3 : vertex 𝑣17 is selected as the root, by using bfs we obtain an alternating tree 𝑇1 that corresponds to graph (𝐺2, 𝑀) as follows: problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 177 tree 𝑇1: since blossom 𝐵 = {𝑣4, 𝑣8, 𝑣9, 𝑣10} has been encountered, then go to step 4. step 4: shrink blossom 𝐵 = {𝑣4, 𝑣8, 𝑣9, 𝑣10} to obtain a graph as follows: (𝐺2 ′ , 𝑀1): by continuing the constructing process of alternating tree that corresponds to graph (𝐺2 ′ , 𝑀1), we obtain: tree 𝑇2: problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 178 since blossom 𝐵′ = {𝑏, 𝑣2, 𝑣3, 𝑣5, 𝑣6, 𝑣15, 𝑣16} has been encountered, then go to step 4: shrink blossom 𝐵′ = {𝑏, 𝑣2, 𝑣3, 𝑣5, 𝑣6, 𝑣15, 𝑣16} to obtain a graph as follows: (𝐺2 ′′, 𝑀2): by continuing the constructing process of alternating tree that corresponds to graph (𝐺2 ′′, 𝑀2), we obtain: tree 𝑇3: since an augmenting path 𝑃′′: 𝑣18 − 𝑏 ′ − 𝑣1 − 𝑣17 that contains pseudovertex 𝑏′ has been obtained, then go to step 3.1 step 3.1: by unshrinking the pseudovertex 𝑏′, we obtain augmenting path 𝑃′: 𝑣18 − 𝑣15 − 𝑣16 − 𝑏 − 𝑣3 − 𝑣2 − 𝑣1 − 𝑣17 that contains pseudovertex 𝑏, then go to step 3.1 step 3.1: by usnhrinking the pseudovertex 𝑏, we obtain augmenting path 𝑃: 𝑣18 − 𝑣15 − 𝑣16 − 𝑣10 − 𝑣9 − 𝑣4 − 𝑣3 − 𝑣2 − 𝑣1 − 𝑣17 which does not contain any pseudovertex , then go to step 3.2 step 3.2: augment the initial matching 𝑀 using augmenting path 𝑃, we obtain a new matching in (𝐺2, 𝑀) as follows: problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad abrori 179 (𝐺2, 𝑀′): since all vertices in (𝐺2, 𝑀′) are incident with a matching edge, then go to step 5. step 5: matching 𝑀′ in graph 𝐺2 is maximum, therefore the finding process is discontinued. from the steps that have been done, we obtain a maximum matching 𝑀′ in 𝐺2 with cardinality of matching 𝑀’ that is |𝑀′| = 9. matching 𝑀′ that has been obtained is the solution of the problem of pairing the 18 pilots as described above with details as follows: 1. pilot number 1 pairs up with pilot number 17 2. pilot number 2 pairs up with pilot number 3 3. pilot number 4 pairs up with pilot number 9 4. pilot number 5 pairs up with pilot number 6 5. pilot number 7 pairs up with pilot number 8 6. pilot number 10 pairs up with pilot number 16 7. pilot number 11 pairs up with pilot number 12 8. pilot number 13 pairs up with pilot number 14 9. pilot number 15 pairs up with pilot number 18 and the maximum number of aircrafts that can be flown is as many as 9 aircrafts. conclusion matching 𝑀 in graph 𝐺 is a subset of 𝐸(𝐺) edge set where there are no two edges in matching 𝑀 that meet each other at the same vertex in graph 𝐺. generally, edmonds’ cardinality matching algorithm is divided into three steps where the first step is the process of matching initialization. the second step is shrinking and unshrinking of blossom 𝐵𝑖 . the third step is to execute augmenting𝑀 if the augmenting 𝑃 path has been found. edmonds’ cardinality matching algorithm can be applied to find for maximum matching especially towards non-bipartite graph. the process of matching initialization is the effort to find the first matching to fasten the finding of maximum matching. alternating 𝑇 tree is a 𝑇 graph concept that is used to find an augmenting path to obtain a new matching that has bigger cardinality than the first matching from the process of matching initialization. problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case muchammad 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[2] chartrand, g., and lesniak, l., 1996, graph dan digraphs (third edition), london: chapman dan hill. [3] edmons, jack, 1965, paths, trees, and flowers, canadian j.math., 17, 449-467. [4] evans, james r., and edward minieka, 1992, optimization algorithms for networks and graphs, new york: marcel dekker, inc. [5] gondran, m., and m. minoux, 1984, graphs and algorithms, new york: jhon wiley & sons. [6] gross, jonathan l., and jay yellen (editors), 1999, handbook of graph theory, usa: crc press. [7] jungnickel, dieter, 2008, graphs, networks, and algorithms (third edition), berlin: springer – verlag. [8] lovasz, l., and m. d. plummer, 1986, matching theory, usa: north – holland. [9] winter, 2005, maximum matching, cs105, www.cs.dartmouth.edu/~ac/teach/cs105.../kavathekar-scribe.pdf [10] zwick, u., 2009, maximum matching in bipartite and non-bipartite graphs, http://www.cs.tau.ac.il/~zwick/grad-algo-0910/match.pdf http://www.cs.dartmouth.edu/~ac/teach/cs105.../kavathekar-scribe.pdf http://www.cs.tau.ac.il/~zwick/grad-algo-0910/match.pdf on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 84-90 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 15, 2020 reviewed: april 17, 2020 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.9094 on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi department of mathematics of fmipa of universitas padjadjaran email: edi.kurniadi@unpad.ac.id abstract in this paper, we study irreducible unitary representations of a real standard filiform lie group with dimension equals 4 with respect to its basis. to find this representation we apply the orbit method introduced by kirillov. the corresponding orbit of this representation is generic orbits of dimension 2. furthermore, we show that obtained representation of this group is squareintegrable. moreover, in such case , we shall consider its duflo-moore operator as multiple of scalar identity operator. in our case that scalar is equal to one. keywords: duflo-moore operator; irreducible unitary representation; square-integrable representation, standard filiform lie algebra. introduction in this paper, all lie algebras are determined over ℝ. the notion of filiform lie algebras can be found in [1]. in this work, we can find the classification of filiform lie algebras of dimension ≤ 8 and the computation of their index. roughly speaking, a lie algebra 𝔤 of dimension 𝑛 is said to be filiform if there exists a smallest positive integer 𝜍 that equals 𝑛 − 1 such that 𝔤𝜍 = {0}. we recall that 𝔤𝑛 is defined as follows : 𝔤1 = 𝔤, 𝔤2 = [𝔤, 𝔤], … , 𝔤𝑛+1 = [𝔤, 𝔤𝑛]. particularly, we shall work on a lie group 𝐺 of a 4-dimensional standard filiform lie algebra 𝔤. we determine the irreducible unitary representations of 𝐺 by applying the orbit method. the orbit method that shall be applied in this paper comes from [2] and [3]. to bring it down to earth of this method, we can enjoy some examples in [4]. we review this method from [2] and [4] as follows. definition 1 [5]. let 𝐺 be a lie group whose lie algebra is 𝔤. let 𝑔 be element of 𝐺. we define the conjugation map given by 𝐶𝑔: 𝐺 ∋ 𝑥 ↦ 𝑔𝑥𝑔 −1 ∈ 𝐺 which is a lie group homomorphism whose differential of 𝐶𝑔 is denoted by ad(𝑔) ≔ (𝐶𝑔)∗ . the adjoint representations of 𝐺 is given as group homomorphism ad ∶ 𝐺 → gl(𝔤). the infinitesimal adjoint representation of ad is denoted by ad. this map is given by ad: 𝔤 → 𝔤𝑙(𝔤) which is defined by ad(𝑥)𝑦 = [𝑥, 𝑦] for each 𝑥, 𝑦 ∈ 𝔤. moreover, the representations ad can be defined as a dual representation on a dual vector space 𝔤∗. namely, we have http://dx.doi.org/10.18860/ca.v6i2.9094 mailto:edi.kurniadi@unpad.ac.id on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 85 〈ad∗(𝑔)𝑓, 𝑋〉 = 〈𝑓, ad(g−1)x〉, (g ∈ 𝐺, 𝑓 ∈ 𝔤∗, 𝑋 ∈ 𝔤) (1) the coadjoint orbit of 𝑓 ∈ 𝔤∗ is defined as a set {ad∗(𝑔)𝑓 ; 𝑔 ∈ 𝐺} ⊂ 𝔤∗. let 𝐺 be a lie group whose lie algebra is 𝔤. let 𝔤∗ be a dual vector space of 𝔤. to obtain the formula of irreducible unitary representation of 𝐺 we should consider some steps as follows. 1. for orbit ω ⊂ 𝔤∗, we choose a point 𝑓 ∈ ω. 2. we consider a polarization υ. υ is said to be a polarization of 𝔤 if υ is a subalgebra of 𝔤 and it has maximal dimension, namely its codimension equals 1 2 dim ω, which is subordinate to 𝑓. this means 〈𝑓, [υ, υ]〉 = 0. 3. let η ≔ exp υ be a subgroup and let 𝜓𝑓 be a one dimensional irreducible unitary representations of η given by 𝜓𝑓 (exp 𝑋) = 𝑒 2𝜋𝑖〈𝑓,𝑋〉, (𝑓 ∈ ω, 𝑋 ∈ 𝔤). (2) 4. identifying the coset space 𝐺/η by m and considering a section 𝑠: m → g. (3) 5. solving the master equation 𝑠(𝑥)𝑔 = ℎ(𝑥, 𝑔)𝑠(𝑥. 𝑔), (𝑥 ∈ m, 𝑔 ∈ 𝐺, ℎ(𝑥, 𝑔) ∈ h), (4) which is called a master equation, and computing the measure on m, we have 𝜋(𝑔)𝑓(𝑥) = 𝜓𝑓 (ℎ(𝑥, 𝑔))𝑓(𝑥. 𝑔). (5) the notion of square-integrable representations can be found in ( [6], [7], [8], and [9]). let (𝜋, 𝜘𝜋) be a irreducible unitary representations locally compact group 𝐺. the irreducible unitary representation (𝜋, 𝜘𝜋) is said to be square-integrable if there exists a non zero vector 𝜈 ∈ 𝜘𝜋 such that ∫ |(𝜈|𝜋(𝑔)𝜈)𝜘𝜋 | 2 𝑑𝑔 < ∞ 𝐺 . (6) such vector is called admissible. furthermore, in the case the representation (𝜋, 𝜘𝜋 ) is square-integrable, duflo-moore in [6] found a densely defined and positive self-adjoint unique operator given by 𝐾𝜋 : 𝜘𝜋 → 𝜘𝜋. (7) which satisfies two following conditions: 1. the neccessary and sufficient conditions for 𝜈 to be admissible are 𝜈 in domain 𝐾𝜋. 2. if vectors 𝑓1, 𝑓3 ∈ 𝜘𝜋 and 𝑓2, 𝑓4 ∈ dom 𝐾𝜋, then we have ∫ (𝑓1|𝜋(𝑥)𝑓2)𝜘𝜋 (𝜋(𝑥)𝑓4|𝑓3)𝜘𝜋 𝑑𝑥𝐺 = (𝑓1|𝑓3)𝜘𝜋 (𝐾𝜋 (𝑓4)|𝐾𝜋 (𝑓2))𝜘𝜋 . (8) let 𝔤 be a 4-dimensional standard filiform lie algebra with basis δ ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥3} whose lie group is 𝐺. the non zero brackets of 𝔤 are given by [𝑥1, 𝑥2] = 𝑥3, [𝑥1, 𝑥3] = 𝑥4 (9) we mention here that in previous work in [2] and [10], we found the representation of 𝐺 is realized on hilbert space l2(ℝ) as follows: on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 86 𝜋𝛿,𝛽 (𝑔)𝑓(𝑥) ≔ 𝑒 2𝜋𝑖(𝛽𝑏+𝛿𝑑+𝛿𝑥𝑐+( 1 2 )𝛿𝑥2𝑏+( 1 2 )𝛿𝑥𝑎𝑏+( 1 2 )𝛿𝑎𝑐+( 1 6 )𝛿𝑎2𝑏) 𝑓(𝑥 + 𝑎), (10) with 𝑔 ≔ 𝑔(𝑎, 𝑏, 𝑐, 𝑑) ∈ 𝐺 and 𝑓 ∈ l2(ℝ). the representation (1) corresponds to 2dimensional generic coadjoint orbits ω𝛿,𝛽 ≔ {𝛿𝑥4 ∗ + 𝑠𝑥3 ∗ + (𝛽 + 𝑠2 2𝛿 ) 𝑥2 ∗ + 𝑢𝑥1 ∗ ; 𝑠, 𝑢 ∈ ℝ, 𝛿 ≠ 0}. (11) contrary to [10] and [2], in this paper we consider the representations of 𝐺 on hilbert space l2(ℝ) with respect to the basis δ ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥3} of 𝔤. we thought that our computations are simpler than previous results and another reason is to attract indonesian young researcher to study representation theory of lie groups. we shall claim two our main proposition as follows: proposition 1. let 𝐺 be a lie group of a 4-dimensional standard filiform lie algebra 𝔤 with basis 𝛥 ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥3} whose its non zero brackets is given in (9). then, the irreducible unitary representation of 𝐺 on the hilbert space l2(ℝ) corresponding to coadjoint orbits ω𝛿,𝛽 in (11) can be written as follows: 𝜋𝜇 (𝑒 𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎), 𝜋𝜇 (𝑒 𝑏𝑥2 )𝑓(𝑥) = 𝑒 2𝜋𝑖(𝛽𝑏+( 1 2 )𝛿𝑥2𝑏) 𝜑(𝑥), (12) 𝜋𝜇 (𝑒 𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 𝜋𝜇 (𝑒 𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥), where 𝜇 ≔ 𝛽𝑥2 ∗ + 𝛿𝑥4 ∗ ∈ ω𝛿,𝛽 ⊂ 𝔤 ∗ and 𝜑 ∈ l2(ℝ). furthermore, we investigate square-integrability of 𝜋𝜇 of 𝐺 and we have. proposition 2. the irreducible unitary representation 𝜋𝜇 of 𝐺 as written in eqs. (12) is square-integrable and its duflo-moore operator is a multiple of identity. methods the method of this research is based on literature survey, specially from [10]. we give another approach in construction of irreducible unitary representation of a lie group 𝐺 whose lie algebra is 4-dimensional standard filiform. we compute it with respect to basis of 𝔤. therefore, the computations are more simpler than previous work. results and discussion in this section, we shall prove our main propositions as already mentioned in introduction proposition 1. let 𝐺 be a lie group of a 4-dimensional standard filiform lie algebra 𝔤 with basis 𝛥 ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥4} whose its non zero brackets is given in (9). then, the irreducible on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 87 unitary representation of 𝐺 on the hilbert space l2(ℝ) corresponding to coadjoint orbits ω𝛿,𝛽 in (12) can be written as follows: 𝜋𝜇 (𝑒 𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎), 𝜋𝜇 (𝑒 𝑏𝑥2 )𝑓(𝑥) = 𝑒 2𝜋𝑖(𝛽𝑏+( 1 2 )𝛿𝑥2𝑏) 𝜑(𝑥), 𝜋𝜇 (𝑒 𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 𝜋𝜇 (𝑒 𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥), where 𝜇 ≔ 𝛽𝑥2 ∗ + 𝛿𝑥4 ∗ ∈ ω𝛿,𝛽 ⊂ 𝔤 ∗ and 𝜑 ∈ l2(ℝ). proof. let 𝛥∗ ≔ {𝑥1 ∗, 𝑥2 ∗, 𝑥3 ∗, 𝑥4 ∗} be a basis for 𝔤∗. the detail computations for coadjoint orbits of 𝐺 can be found in ([10], p. 33—34) or ([2], p. 77—80). let 𝑓 = 𝛼𝑥1 ∗ + 𝛽𝑥2 ∗ + 𝛾𝑥3 ∗ + 𝛿𝑥4 ∗ ∈ 𝔤∗ . we recall the result of 2-dimensional coadjoint orbits as follows: ω𝛿,𝛽 ≔ {𝛿𝑥4 ∗ + 𝑠𝑥3 ∗ + (𝛽 + 𝑠2 2𝛿 ) 𝑥2 ∗ + 𝑢𝑥1 ∗ ; 𝑠, 𝑢 ∈ ℝ, 𝛿 ≠ 0}. to obtain the irreducible unitary representations of 𝐺 corresponding to generic coadjoint orbits ω𝛿,𝛽 , we consider the following steps step 1. we choose a point 𝑓𝛽,𝛿 ≔ 𝛽𝑥2 ∗ + 𝛿𝑥4 ∗ ∈ ω𝛿,𝛽. step 2. we determine a subalgebra υ ≔ span{𝑥2, 𝑥3, 𝑥4} of 𝔤. in this case υ is commutative since [𝑥𝑖 , 𝑥𝑗 ] = 0 for 1 ≤ 𝑖, 𝑗 ≤ 3. furthermore, since υ has maximal dimension i.e. codim υ = 1 2 dim ω𝛿,𝛽 = 1 2 . 2 = 1. on the other hand, υ is subordinate to 𝑓𝛽,𝛿 since 〈𝑓𝛽,𝛿 , [υ, υ]〉 = 0. thus, υ is a polarization of 𝔤. step 3. let η ≔ exp υ be subgroup and let 𝜓𝑓 be a 1-dimensional irreducible unitary representations of η given by 𝜓𝑓𝛽,𝛿 (exp(𝑎𝑥1 + 𝑏𝑥2 + 𝑐𝑥3 + 𝑑𝑥4) = 𝑒 2𝜋𝑖〈𝑓𝛽,𝛿, 𝑋〉 = 𝑒2𝜋𝑖(𝛽𝑏+𝛿𝑑), (𝑓𝛽,𝛿 ∈ ω𝛿,𝛽, 𝑋 = 𝑎𝑥1 + 𝑏𝑥2 + 𝑐𝑥3 + 𝑑𝑥4 ∈ 𝔤). step 4. identifying the coset space 𝐺/η with ℝ by ℝ ∋ 𝑥 ↦ η exp(𝑎𝑥1) ∈ 𝐺/η, we have a section given by 𝑠: 𝐺/η ≅ ℝ ∋ 𝑥 ↦ exp 𝑥𝑥1 ∈ 𝐺. step 5. we now solve the master equation with respect to (w.r.t) basis 𝛥 ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥4}. a. w.r.t 𝑒𝑎𝑥1 exp 𝑥𝑥1 exp 𝑎𝑥1 = exp(𝑥 + 𝑎)𝑥1 . b. w.r.t 𝑒𝑏𝑥2 on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 88 exp 𝑥𝑥1 exp 𝑏𝑥2 = exp(𝑒 ad 𝑥𝑥1 . 𝑎𝑥2) exp 𝑥𝑥1 , = exp(𝑏𝑥2 + 𝑏𝑥𝑥3 + ( 1 2 ) 𝑥2𝑏𝑥4) exp 𝑥𝑥1. c. w.r.t 𝑒𝑐𝑥3 exp 𝑥𝑥1 exp 𝑐𝑥3 = exp(𝑒 ad 𝑥𝑥1 . 𝑐𝑥3) exp 𝑥𝑥1 , = exp(𝑐𝑥3 + 𝑐𝑥𝑥4) exp 𝑥𝑥1. d. w.r.t 𝑒𝑑𝑥4 exp 𝑥𝑥1 exp 𝑑𝑥4 = exp(𝑒 ad 𝑥𝑥1 . 𝑑𝑥4) exp 𝑥𝑥1 , = exp(𝑑𝑥4) exp 𝑥𝑥1. using induced representation on 1-dimensional irreducible unitary representations 𝜓𝑓 of η, we obtain the explicit formulas for ireducible unitary representations of 𝐺 realized on hilbert space l2(ℝ) written as follows. 𝜋𝜇 (𝑒 𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎), 𝜋𝜇 (𝑒 𝑏𝑥2 )𝑓(𝑥) = 𝑒 2𝜋𝑖(𝛽𝑏+( 1 2 )𝛿𝑥2𝑏) 𝜑(𝑥), 𝜋𝜇 (𝑒 𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 𝜋𝜇 (𝑒 𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥), where 𝜇 ≔ 𝛽𝑥2 ∗ + 𝛿𝑥4 ∗ ∈ ω𝛿,𝛽 ⊂ 𝔤 ∗ and 𝜑 ∈ l2(ℝ). ∎ proposition 2. the irreducible unitary representation 𝜋𝜇 of 𝐺 as written in eqs. (12) is square-integrable and its duflo-moore operator is a multiple of identity. proof. let 𝜑1, 𝜑2 be elements of l 2(ℝ). for these functions we shall compute the integral ∫ |(𝜑1|𝜋(𝑔)𝜑2)l2(ℝ)| 2 𝐺 𝑑𝑔 . let us put 𝑔 = 𝑔′𝑒𝑎𝑥1 where 𝑔′ = 𝑒𝑝𝑥4 𝑒𝑐3 𝑒𝑏𝑥2 . we have (𝜑1|𝜋(𝑔)𝜑2)l2(ℝ) = ∫ 𝜑1(𝑥)𝑒 2𝜋𝑖(𝛿𝑝+𝛿𝑐𝑥+𝛽𝑏+( 1 2 )𝛿𝑥2𝑏) 𝜋(𝑒𝑎𝑥1 )𝜑(𝑥) 𝑑𝑥 ℝ by plancherel’s theorem and fubini’s theorem of integral we have on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 89 ∫ |(𝜑1|𝜋(𝑔)𝜑2)l2(ℝ)| 2 ℝ4 𝑑𝑝𝑑𝑐𝑑𝑏𝑑𝑎 = ∫ |𝜑1(𝑥)| 2{∫ |𝜑2(𝑥 + 𝑎)| 2 𝑑𝑎}𝑑𝑥 ℝℝ = ∫ |𝜑1(𝑥)| 2 𝑑𝑥 ∫ |𝜑2(𝑎′)| 2 𝑑𝑎′ ℝℝ (we put 𝑎′ = 𝑥 + 𝑎 ) = ‖𝜑1‖l2(ℝ) 2 ‖𝜑2‖l2(ℝ) 2 . the latter equation gives information that 𝜋𝜇 is a square-integrable representation. therefore, its duflo-moore operator is equal to scalar multiple of identity which scalar equals one. ∎ we mention here some results in [8] regarding unimodular groups as follows. 1. if 𝐺 is unimodular then the duflo-moore operator 𝐾𝜋 in (8) is the scalar multiple of identity. therefore, the equation (8) is of the form ∫ (𝑓1|𝜋(𝑥)𝑓2)𝜘𝜋 (𝜋(𝑥)𝑓4|𝑓3)𝜘𝜋 𝑑𝑥𝐺 = 𝜆 (𝑓1|𝑓3)𝜘𝜋 (𝑓4|𝑓2)𝜘𝜋 . (13) 2. if 𝐺 is unimodular then all vectors in 𝜘𝜋 are admissible. in fact, since tr ∘ ad = 0 then the lie group 𝐺 of 4-dimensional standard filiform lie algebra 𝔤 is unimodular. therefore, the computations is immediately true as desired. namely, the irreducible unitary representation 𝜋𝜇 is square-integrable and its duflomoore operator is equal to the scalar 𝜆 multiple with 𝜆 = 1. conclusion we conclude that irreducible unitary representation 𝜋𝜇 of lie group of 4dimensional filiform lie algebra with respect to its basis is obtained. furthermore, this representation 𝜋𝜇 is square-integrable and we obtain the duflo-moore operator is a scalar multiple of identity and its scalar equals one. references [1] a. hadjer and a. makhlouf, “index of graded filiform and quasi filiform lie algebras,” no. may 2014, 2012. [2] a. a. kirillov, “lectures on the orbit method, graduate studies in mathematics,” am. math. soc., vol. 64, 2004. [3] a. . kirillov, “unitary representations of nilpotent lie groups,” uspekhi mat.nauk, vol. 17, pp. 57--110, 1962. [4] r. berndt, representation of linear groups. an introduction based on examples from physics and number theory. wiesbaden: vieweg, 2007. [5] j. . lee, introduction to smooth manifolds, graduate text in mathematics,. new york: springer-verlag, 2003. on square-integrable representations of a lie group of 4 dimensional standard filiform lie algebra edi kurniadi 90 [6] m. duflo and c. c. moore, “on the regular representation of a nonunimodular locally compact,” j. funct. anal., vol. 21, pp. 209–243, 1976. [7] a. . carey, “square-integrable representation of nonunimodular groups,” bull.austral.math.soc, vol. 15, pp. 1--12, 1976. [8] a. grossmann, j. morlet, and t. paul, “transform associated to square-integrable group of representations i,” j.math.phys, vol. 26, pp. 2473--2479, 1985. [9] a. grossmann, j. morlet, and t. paul, “trsansform associated to square-integrable group representations .ii. examples,” ann.inst.h.poincare phys.theor, vol. 45, pp. 293--309, 1986. [10] l. j. corwin and f. . greenleaf, representations of nilpotent lie groups and their applications. part i. basic theory and examples,. cambridge: cambridge university press, 1990. geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 140-149 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 03 july 2018 reviewed: 15 august 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5879 geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto1,3, bambang widjanarko otok2, suharto3, arief wibowo3 1 health polytechnic banjarmasin, ministry of health & ph.d student faculty of public health, airlanga university, surabaya 2 laboratory of environmental and health statistic, ‘sepuluh nopember’ institute of technology (its), surabaya 3department of biostatistics and demography, faculty of public health, airlanga university, surabaya, indonesia email: 1surotojahrani@yahoo.com, 2dr.otok.bw@gmail.com abstract hypertension is one of the disease is not contagious diseases which is a public health problem. uncontrolled hypertension can trigger a degenerative disease such as congestive heart failure, renal failure and vascular disease. hypertension is called the silent killer because his nature the condition is asymptomatic and can cause a fatal stroke. with the increasing prevalence of cases of degenerative diseases, one only hypertension, then the researchers want to predict the variables very big role as one of the risk factors of genesis hypertension. with clearly know the risk factors that play against genesis hypertension is expected to be used as a reference for the prevention and control so that they can reduce the prevalence of hypertension and prevent deaths from degenerative diseases, especially hypertension. the results of the study showed that the results of the modeling the prevalence of hypertension in south kalimantan province using linier regression there is no factor that affect the genesis of hypertension. the prevalence of hypertension spread spatially because there are heterogenitas between the location of the observation that means that observations of a location depends on the observations in another location that the distance is near so do spatial regression modeling with adaptive gaussian kernel function, the result 5 groups. group i consists of the districts tanah laut and tanah bumbu; group ii, kota baru; group iii consists of banjar, kota banjar baru, kota banjarmasin; group iv on the barito kuala regency and the group v consists of tapin, h s selatan, h s tengah, h s utara, tabalong, balangan. keywords: gwr, kernel function, adaptive gaussian, prevelance hypertension introduction a research influenced by aspects of territorial characteristic (spatial) then need to be considered spatial data on the model. spatial data is data that contains the location information. on spatial data, often observations in a location dependent on observation in other locations near (neighboring). the first law of geography advanced by tobler in 1979, stated that all things are related to each one with the other but something close more had the effect of something far [1]. the law is the basis of the examination of the problems based on the effect of the location or spatial method. in the modeling language, when the classic regression model used as a tool of analysis on spatial data, then can cause the conclusion that less accurate mailto:surotojahrani@yahoo.com mailto:dr.otok.bw@gmail.com geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 141 because of the assumption of the error free to each other and the assumption homogenitas not met [2]. hypertension is one of the disease is not contagious diseases which is a public health problem. uncontrolled hypertension can trigger a degenerative diseases such as congestive heart failure, renal failure and vascular disease. hypertension is called the silent killer because his nature the condition is asymptomatic and can cause a fatal stroke [3]. although not treated, prevention and managements can decrease the occurrence of hypertensi and the accompanying disease. hypertension is the cause of the death of number 3 after stroke and tuberculosis, namely 6.7 % from the population of death in all age in indonesia. the problem of hypertension that occurs in south kalimantan would not escape from the factors causing the hypertension essential where genetic factors, environment, behavior, health services also contribute to cause a high case of hypertension in south kalimantan province [4]. to identify the multitude of causative factors that affect it may need to be an analysis or the development of the model is spatial [5]. spatial effects testing done with test heterogenitas and spatial dependencies. if there is a settlement of securities heterogenity is by using point approach. spatial regression points between the other geographically weighted regression (gwr) with the scale of the measurement of the response variable is the interval and ratio,geographically weighted poisson regression (gwpr) with the variable data is response count, geographically weighted logistic regression (gwlr) with the scale of the measurement of the response variable is the nominal [6]. based on the [7], the prevalence of hypertension in indonesia that acquired through the measurement at the age > 18 years of 25.8%, the highest in bangka belitung securities have totaled 30.9%, followed south kalimantan 30,8 % , east kalimantan 29.6% and west java 29.4%. the prevalence of hypertension in indonesia that is obtained through the questionnaire diagnosed health workers of 9.4%, diagnosed health workers or is drinking drugs of 9.5%. so there are 0.1% that medication itself. respondents who have a normal blood but is drinking hypertension medications 0.7%. so the prevalence of hypertension in indonesia by 26.5%. the problem of hypertension that occurs in south kalimantan would not escape from the factors causing the hypertension essential where genetic factors, environment, behavior, health services also contribute to cause a high case of hypertension in south kalimantan province. to identify the multitude of causative factors that affect it may need to be an analysis or development models. with the increasing prevalence of hypertension in a region, then the researchers want to predict the variables very big role as one of the risk factors of genesis hypertension. with clearly know the risk factors that play against genesis hypertension with gwr approach is expected to be used as a reference for the prevention and control so that they can reduce the prevalence of hypertension. methods the data used is the basic health research data [7]. the data will be analyzed in this research is data genesis hypertension in regency/city of south kalimantan province. based on the results of the analysis of the previous library, hypertension served on the following conceptual framework [8]. geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 142 figure 1. gwr modeling flow in the case of hypertension the response variable y that is used in this research is the percentage of patients with hypertension are diagnosed per sub-districts in south kalimantan province. while the variables predictors xi used there in table 1. table 1. the variables predictors [9] variables description x1 the percentage of the inhabitants of gender male x2 the percentage of population with education completed sd/mi x3 the percentage of population with smoking habit every day x4 the percentage of the population of physical activity x5 percentage of the population who consume the fruits of 7 times in 1 weeks x6 percentage of the population who consume vegetables 7 times in 1 weeks x7 percentage of the population who consume salty food more than 1 times per day x8 the percentage of the population consuming fatty food consumption/ order/ fried more than 1 times per day x9 the percentage of the population with ownership of health insurance the variables x1 and x2 used to describe the spread of environmental aspects according to their geographic region. aspects of the behavior described by the variables x3, x4, x5, x6, x7 and x8. while aspects of the ministry of health is portrayed through the variable x9. result and discussion description of the prevalence of hypertension description of this research includes the mean and standard deviation from each of the research variables. now in detail is presented in the following table. descriptive statistics gwr hypertension environment behavior ministry health geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 143 table 2. descriptive data the prevalence of hypertension in south kalimantan province [9] variables [7] minimum maximum mean stdev persentase patients with hypertension is diagnosed (y) 1.471 29.508 14.978 5.764 the percentage of the inhabitants of gender male (x1) 32.787 55.556 47.522 3.904 the percentage of population with education completed sd/mi (x2) 7.320 69.230 33.660 12.340 the percentage of population with smoking habit every day (x3) 9.375 66.667 23.750 7.507 the percentage of the population physical activity (x4) 54.348 100,000 86.855 9.988 percentage of the population who consume the fruits of 7 times in 1 weeks (x5) 0.000 41.584 20.289 7.070 percentage of the population who consume vegetables 7 times in 1 weeks (x6) 13.390 93.100 55.470 15.200 percentage of the population who consume salty food more than 1 times per day (x7) 22.220 86.360 45.080 13.170 the percentage of the population consuming fatty food consumption/ order/ fried more than 1 times per day (x8) 0.000 79.550 31.670 15.620 the percentage of the population with ownership of health insurance (x9) 0.000 54.023 15.002 11.946 table 2. shows that the percentage of the highest hypertension prevalence on the sub-district pulau laut tanjung selayar of 29.508 percent and lowest on sub-district tatah makmur of 1.471 percent while the average percentage of the prevalence of hypertension by 14.978 percent with standard deviation of 5.764 percent. the percentage of the inhabitants of gender male (x1) highest on the sub-district danau panggang (55.556 percent), and the lowest on the sub-district pulau laut tanjung selayar (32.787). the percentage of population with education completed sd/mi (x2) highest on the sub-district pamukan barat of 69.230 percent and lowest on sub-district tatah makmur of 7.320 per cent while the average percentage of the prevalence of hypertension by 33.660 percent with standard deviation of 12.340 percent. the percentage of population with smoking habit every day (x3) has an average of 23.750 percent with standard deviation 7.507 percent. the percentage of the population physical activity (x4) has an average of 86.855 percent with standard deviation 9.988 percent. percentage of the population who consume the fruits of 7 times in 1 weeks (x5) has an average of 20.289 percent with standard deviation 7.070 percent. percentage of the population who consume vegetables 7 times in 1 weeks (x6) has an average of 55.470 percent with standard deviation 15.2 percent. percentage of the population who consume salty food more than 1 times per day (x7) has an average of 45.08 percent with standard deviation 13.170 per cent. the percentage of the population consuming fatty food consumption/ order/ fried more than 1 times per day (x8) has an average of 31.670 percent with standard deviation 15.62 percent and the percentage of the population with ownership of health insurance (x9) has an average of 15.002 percent with standard deviation 11.946 percent. geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 144 regression modeling the prevalence of hypertension before performing the analysis using spatial regression method, done regression modeling double linier first. regression modeling linear to the prevalence of hypertension and factors suspected to influence using parameter assessment method ordinary least square (ols) [10] which aims to know the variables significant on the prevalence of hypertension globally. the first step is to detect multicolinearity to know whether or not the relationship between the free variable (predictors), then continued with double linier regression modeling (global) covers the review of the significance of the parameters simultaneously or partially, and residual assumptions iidn test [11]. multicolinearity detection one of the conditions in the multiple regression analysis with some variables predictors is no cases multikolinieritas or not there is a variable predictors that have a correlation with other predictors variable. tracing muticolinearity done based on the value of variance inflation factor (vif). the following is the value of vif in each of the variables predictors. table 3. the value of their respective vif predictors variables variables x1 x2 x3 x4 x5 x6 x7 x8 x9 vif 1.287 1.279 1.313 1.093 1.153 1.347 1.194 1.208 1.422 based on the table 3, obtained the information that all variables predictors have a vif value less 10. this detects that there are not cases multikolinieritas or not there is a variable predictors that have a correlation with other predictors variable. the significance of parameters linear regression tests the prevalence of hypertension the following is the significance test good linier regression parameters simultaneously or partial to know the influence of predictors variables used. the hypothesis to test the significance of the parameters simultaneously on the linier regression is as follows [6]. h0 : β1 = β2 = ...= β9 = 0 (the parameters do not affect the significant impact on model) h1 : at least one βk ≠ 0 ; k = 1.2, ... ,9 (at least one parameters that affect the significant impact on model) table 4. anova table the prevalence of hypertension in south kalimantan province source sum of squares df mean of square f p regression 598.32 9 66.48 2.14 0.030 residual 4418.32 142 31.11 total 5016.64 151 geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 145 table 4., produce f-statistic value of 2.14 and p-value of 0.030. based on the level of the significance () of 5 percent and f(0.05;9;142) of 1,.46, obtained the decision reject h0 because the value of f-statistic > f(0.05;9;142) or p-value < 0.05. this can be interpreted that there is at least one parameters that affect the significant impact on the prevalence of hypertension. next to know the variables predictors anywhere that provides significantly influence, then the test is done the significance of parameters partially presented in table 5. the following is the hypothesis test the significance of the parameters spatially against linier regression model (global) [6]. h0 : βk = 0, h1 : βk ≠ 0, k = 1,2,3,...,9 table 5. the results of the test regression model parameters linear partial parameters coefficient se coefficient t-statistic sig. the decision 0 28.9260 6.53000 4.43 0.000 1 -0.1728 0.08935 -1.93 0.055 fail to reject h0 2 0.0181 0.04155 0.44 0.663 fail to reject h0 3 0.0031 0.05605 0.05 0.956 fail to reject h0 4 -0.0447 0.04784 -0.94 0.351 fail to reject h0 5 -0.1015 0.06937 -1.46 0.146 fail to reject h0 6 0.0284 0.03464 0.82 0.414 fail to reject h0 7 -0.0160 0.03780 -0.42 0.673 fail to reject h0 8 -0.0555 0.03205 -1.73 0.086 fail to reject h0 9 0.0252 0.04528 0.56 0.579 fail to reject h0 based on the results of the test in table 5, with significant level () of 5 percent and    2 0.025,142; 1 1.977 n p t t      , obtained the information that all values t count smaller than t table. this shows that there is no predictors variables that affect the prevalence of hypertension. testing the assumptions of a residual iidn after testing the significance of the parameters simultaneously and partially, then the next step is to test the assumptions residual of identical, independent, and normal distribution (iidn). test the assumption of identical residual one of the assumption of the test in the ols regression is a residual must be homoscedasticitys variance (is identical) or in cases of heteroscedasticity. how to identify the existence of the case of heteroskedastisitas is to create a regression model between a residual and predictor variables. when there are variables predictors that affect the model significantly, it can be said that the residual is not identical or in cases of heteroscedasticity. testing the assumption of identical residual provides information that there are not cases heteroscedasticity or identic residual with significant () of 0.05 and     ; 1 0,05;9,142 1.946 n p f f      . this is due to the value of the p-value of 0.002 smaller geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 146 than  and f-statistic greater 3.06 than  0,05;9,142 1.946f  then happened heteroscedasticity. test the assumption of independent residual [11] test the assumption of independent residual used to know whether or not the relationship between the residual periods. the test statistics used is durbin-watson. based on the attachment 4 obtained the value 2.07875d  with the value 1.0201 l d  and 1.9198 u d  . so that the decision can be taken is to fail to reject h0 because 1.9198 (4 ) 2.0802 u u d d d     . it shows that there is no relationship between the residual, so that the assumption of independent residual have been fulfilled. test the assumption normal distribution the assumption of the normal distribution test is done with the following kolmogorov-smirnov test. h0: data normal distribution h1: data not normal distribution 20100-10-20 99.9 99 95 90 80 70 60 50 40 30 20 10 5 1 0.1 rediual p e r c e n t mean -2.07904e-14 stdev 5.409 n 152 ks 0.035 p-value >0.150 probability plot of residual normal figure 2. probability plots of residual of the hypertension prevalence based on the figure 2 obtained the information that the points of red spread near linear line (normal) which means that the data has been normal distribution. in addition, also can be seen from the value of p-value that is more > 0.15. so that the decision can be taken is to fail to reject h0 on equal significant () of 5%, because the value of the pvalue is greater than . this means that the data has been meet the assumption normal distribution. based on the results of the test of assumption, it can be concluded that a residual on the linear regression model (global) meets the independent assumption, and data has been normal distribution, but the assumption is identical not fulfilled. geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 147 spatial regression modeling the prevalence of hypertension the analysis using the gwr method aims to know the variables that affect the prevalence of genesis hypertension on each observation location regency/city in south kalimantan province. the first step is done to get the gwr model is to determine the coordinates of the point latitude and longitude on each location to count the distance euclidean and determine the optimum bandwidth values based on the criteria of cross validation (cv). the next step is to determine the matrix pembobot with kernel functions: fixed gaussian, fixed bi-square, adaptive gaussian, adaptive bi-square and assess the gwr model parameters. the matrix pembobot obtained for each location and then used to form a model, so that obtained the model vary in each location of observation [12]. gwr model hypothesis testing consists of two test, test the suitability of the gwr model and test the significance of the parameters gwr model. the following is the results of the hypothesis testing gwr model [6]. h0 : kiik vu  ),( ; (there is no significant difference between the linear regression model (global) and gwr model) h1 : at least one kiik vu  ),( k = 1, 2, ....,9 (no difference between significant linear regression model (global) and gwr model) table 6. the estimation of gwr on the weight of the kernel function statistic weight fixed gaussian fixed bi-square adaptive gaussian* adaptive bisquare bandhwith 0.879990 1.802952 148.831349 152.000000 mse 28.141 27.681 26.794 28.429 r2 0.265928 0.290850 0.302022 0.252414 aicc 956.170600 956.662511 956.153605 957.642999 note: *) best gwr model table 6 shows the comparison of the model with pembobot gwr estimates that vary. testing the suitability of the gwr model is done by using the difference in the number of residual square gwr model and global regression model. gwr model will vary significantly with global regression model if can reduce the amount of residual square significantly. table 6., shows that the value of the smallest aicc is the gwr model with adaptive gaussian kernel pembobot amounting 956.153. so using the significance level   5 percent so it can be concluded that the gwr model differ significantly with global regression model. this means that the kernel pembobot gwr model with adaptive gaussian more worthy to illustrate the percentage of the prevalence of hypertension in south kalimantan province. next is a test of the significance of the parameters gwr model with adaptive gaussian kernel pembobot partially to know any parameters that affect the prevalence of hypertension in each location of observation. sub division that have common geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 148 variables which affect the significant impact on the prevalence of hypertension is presented in table 7. table 7. grouping of districts based on a significant variable group significant variables variable description tanah laut tanah bumbu x3,x4,x6 persentase patients with hypertension is diagnosed (y) the percentage of the inhabitants of gender male (x1) the percentage of population with education completed sd/mi (x2) the percentage of population with smoking habit every day (x3) the percentage of the population physical activity (x4) percentage of the population who consume the fruits of 7 times in 1 weeks (x5) percentage of the population who consume vegetables 7 times in 1 weeks (x6) percentage of the population who consume salty food more than 1 times per day (x7) the percentage of the population consuming fatty food consumption/ order/ fried more than 1 times per day (x8) the percentage of the population with ownership of health insurance (x9) x3,x4,x6 kota baru x4,x8 banjar kota banjarmasin banjar baru x4,x5,x6,x7 x4,x5,x6,x7 x4,x5,x6,x7 barito kuala x3,x4,x9 tapin hulu sungai selatan hulu sungai tengah hulu sungai utara tabalong balangan x1,x5 x1,x5 x1,x5 x1,x5 x1,x5 x1,x5 clustering based on a significant variable is divided into 5 groups. there are districts that have common variables which significantly influence with the surrounding district, but there is a district that has its own uniqueness because the variables that influence significant is not the same. conclusion the results of the modeling the prevalence of hypertension in south kalimantan province based on using sub-linier regression there is no factor that affect the genesis of hypertension. the prevalence of hypertension spread spatially because there are heterogenitas between the location of the observation that means that observations of a location depends on the observations in another location that the distance is near so do spatial regression modeling with adaptive gaussian kernel function, to result 5 groups. group i consists of the districts of tanah laut and tanah bumbu with the characteristics of the percentage of the population with the smoking habit every day (x3), the percentage of the population physical activity (x4), the percentage of the population who consume vegetables 7 times in 1 weeks (x6). the group ii, kota baru with the characteristics of the percentage of physical activity (x4), the percentage of the population who consume fatty food consumption/ order/ fried more than 1 times per day (x8). group iii consists of banjar, kota banjar baru, kota banjarmasin, with characteristic of the percentage of the population physical activity (x4), the percentage of the population who consume the fruits of 7 times in 1 weeks (x5), the percentage of the population who consume vegetables 7 times in 1 weeks (x6), the percentage of the population who consume salty food more than 1 times per day (x7). the group iv on the barito kuala regency with the characteristics of a percentage of the population with the smoking habit every day (x3), the percentage of the population physical activity (x4), geographically weighted regression to predict the prevalence of hypertension based on the risk factors in south kalimantan suroto 149 the percentage of the population with the ownership of health insurance (x9), and the group v consists of tapin, h s selatan, h s tengah, h s utara, tabalong, balangan with characteristics of the percentage of the population of the sexes men (x1), the percentage of the population who consume the fruits of 7 times in 1 weeks (x5). references [1] l. anselin, spatial econometrics: methods and models. kluwer academic, dordrecht, 1988 [2] h. j. miller, ‘tobler’s first law and spatial analysis’. annals of the association of america geographers, 94(2), hal.284-289, 2004. [3] h. j. tudor, f. tom, tanya jawab seputar tekanan darah tinggi, edisi 2, arcan, jakarta, 2010. [4] l. marliani, 100 question & answers hipertensi. pt alex media komputindo gramedia. jakarta, 2007. [5] c. chasco, i. garcia, and j. vicens, ‘‘modeling spatial variations in household disposible income with geographically weighted regression“, munich personal repec arkhive (mpra) working papper no. 1682, 2007. [6] a. s. fotheringham, c. brunsdon, and m. charlton, geographically weighted regression, jhon wiley & sons, chichester, uk, 2002. [7] kementrian kesehatan ri, riset kesehatan dasar (riskesdas 2013), badan penelitian dan pengembangan kesehatan, jakarta, 2014. [8] a. mansjoer, kapita selekta kedokteran, edisi 3 , media aesculapius, fk ui, jakarta, 2005. [9] dinas kesehatan propinsi kalimantan selatan, profil kesehatan propinsi kalimantan selatan, dinkes prop. kalsel, banjarmasin, 2014. [10] j. p. lesage, a family of geographically weighted regression, departement of economics university of toledo, 2001. [11] y. leung, c. l. may, and w. x. zhang, “testing for spatial autocorrelation among the residuals of the geographically weighted regression", environment and planning a, 32, 871-890, 2000. [12] y. leung, c. l. may, and w. x. zhang, “statistic tests for spatial non-stationarity based on the geographically weighted regression model”, environment and planning a, 32 9-32, 2000. on the metric dimension of some operation graphs cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 88-94 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 23 july 2018 reviewed: 08 october 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5331 on the metric dimension of some operation graphs marsidi1, ika hesti agustin2, dafik3, ridho alfarisi4, hendrik siswono5 1mathematics edu. depart. ikip pgri jember indonesia 2mathematics depart. university of jember indonesia 3mathematics edu. depart. university of jember indonesia 4elementary school teacher edu. depart. university of jember indonesia 5majoring in early childhood edu. depart. ikip pgri jember indonesia email: marsidiarin@gmail.com, ikahesti@fmipa.unej.ac.id, d.dafik@gmail.com abstract let 𝐺 be a simple, finite, and connected graph. an ordered set of vertices of a nontrivial connected graph 𝐺 is 𝑊 = {𝑤1,𝑤2,𝑤3,…,𝑤𝑘} and the 𝑘-vector 𝑟(𝑣|𝑊) = (𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2),…,𝑑(𝑣,𝑤𝑘)) represent vertex 𝑣 that respect to 𝑊, where 𝑣 ∈ 𝐺 and 𝑑(𝑣,𝑤𝑖) is the distance between vertex 𝑣 and 𝑤𝑖 for 1 ≤ 𝑖 ≤ 𝑘. the set 𝑊 called a resolving set for 𝐺 if different vertex of 𝐺 have different representations that respect to 𝑊. the minimum cardinality of resolving set of 𝐺 is the metric dimension of 𝐺, denoted by dim⁡(𝐺). in this paper, we give the local metric dimension of some operation graphs such as joint graph 𝑃𝑛 + 𝐶𝑚, amalgamation of parachute, amalgamation of fan, and 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚). keywords: metric dimension, resolving set, operation graphs. introduction all graphs in this paper are simple, finite and connected, for basic definition of graph we can see in chartrand [1]. chartrand [2] define the length of a shortest path between two vertices 𝑢 and 𝑣 is the distance 𝑑(𝑢,𝑣) between two vertices in a connected graph g. an ordered set of vertices of a nontrivial connected graph 𝐺 is 𝑊 = {𝑤1,𝑤2,𝑤3,…,𝑤𝑘} and the 𝑘-vector 𝑟(𝑣|𝑊) = (𝑑(𝑣,𝑤1),𝑑(𝑣,𝑤2),…,𝑑(𝑣,𝑤𝑘)) represent vertex 𝑣 that respect to 𝑊. the set 𝑊 called a resolving set for 𝐺 if different vertex of 𝐺 have different representations that respect to 𝑊. the minimum of cardinality of resolving set of g is the metric dimension of 𝐺, denoted by dim(𝐺) [3]. there are many articles explained about metric dimension such as [2], [4], [5], [6], and [7]. [8] defined a shackle graphs 𝑠ℎ𝑎𝑐𝑘(𝐺1,𝐺2,…,𝐺𝑘) constructed by nontrivial connected graphs 𝐺1,𝐺2,…,𝐺𝑘 such that 𝐺𝑖 and 𝐺𝑗 have no a common vertex for every 𝑖, 𝑗 ∈ [1,𝑘] with |𝑖 − 𝑗| ≥ 2, and for every 𝑙 ∈ [1,𝑘 − 1], 𝐺𝑙 and 𝐺𝑙+1 share exactly one common vertex (called linkage vertex) and the 𝑘 − 1 linking vertices are all different. [9] defined an amalgamation of graphs constructed from isomorphic connected graphs 𝐻 and the choice of the vertex 𝑣𝑗 as a terminal is irrelevant. for any 𝑘 positive integer, we denote such an amalgamation by 𝑎𝑚𝑎𝑙(𝐻,𝑘), where 𝑘 denotes the number of copies of 𝐻. proposition 1. [2] let 𝐺 be a connected graph or order 𝑛⁡ ≥ ⁡2, then the following hold: a. 𝑑𝑖𝑚(𝐺) = 1⁡if and only if graph 𝐺 is a path graph b. 𝑑𝑖𝑚(𝐺) = 𝑛 − 1 if and only if graph 𝐺 is a complete graph c. for 𝑛 ≥ 3, 𝑑𝑖𝑚(𝐶𝑛) = 2 mailto:marsidiarin@gmail.com,%20ikahesti@fmipa.unej.ac.id mailto:d.dafik@gmail.com on the metric dimension of some operation graphs marsidi 89 d. for 𝑛 ≥ 4, 𝑑𝑖𝑚(𝐺) = 𝑛 − 2 if and only if 𝐺 = 𝐾𝑝,𝑞⁡(𝑝,𝑞 ≥ 1),𝐺 = 𝐾𝑝 + 𝐾𝑞̅̅̅̅ (𝑝 ≥ 1,𝑞 ≥ 2). results and discussion theorem 2.1. for 𝑛 ≥ 2 and 𝑚 ≥ 7, the metric dimension of joint graph 𝑃𝑛 + 𝐶𝑚 is 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) = ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋. proof. the joint of path and cycle graph, denoted by 𝑃𝑛 + 𝐶𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 + 𝐶𝑚) = {𝑥𝑗; ⁡1 ≤ 𝑗 ≤ 𝑛} ∪ {𝑦𝑙; ⁡1 ≤ 𝑙 ≤ 𝑚} and edge set 𝐸(𝑃𝑛 + 𝐶𝑚) = {𝑥𝑗𝑦𝑙; ⁡1 ≤ 𝑗 ≤ 𝑛; ⁡1 ≤ 𝑙 ≤ 𝑚} ∪ {𝑥𝑗𝑥𝑗+1; ⁡1 ≤ 𝑗 ≤ 𝑛 − 1} ∪ {𝑦𝑙𝑦𝑙+1; ⁡1 ≤ 𝑙 ≤ 𝑚 − 1} ∪ {𝑦𝑛𝑦1}. the cardinality of vertex set and edge set, respectively are |𝑉(𝑃𝑛 + 𝐶𝑚)|⁡= ⁡𝑛⁡ + ⁡𝑚 and |𝐸(𝑃𝑛 + 𝐶𝑚)| = ⁡𝑛(𝑚 + 1) + 𝑚. if we show that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) = ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋ for 𝑛⁡ ≥ 2 dan 𝑚 ≥ 7, then we will show the lower bound namely 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) ≥ ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋ − 1. assume that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) < ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋. this can be shown with take resolving set 𝑊 = {𝑥1,𝑦1,𝑦5} so that it obtained the representation of the vertices 𝑥,𝑦 ∈ 𝑉(𝑃2 + 𝐶7) respect to 𝑊. it can be seen that there is at least two vertices in 𝑃𝑛 + 𝐶𝑚 which have the same representation respect to 𝑊, one of them is 𝑟(𝑦4|𝑊)⁡= (1,2,1) and 𝑟(𝑦6|𝑊)⁡= (1,2,1) such that we have the cardinality of resolving set of 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) ≥ ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋. furthermore, we will prove that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚) ≤ ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋ with determine the resolving set 𝑊 = {𝑥𝑗; ⁡1 ≤ 𝑗 ≤ 2⌊ 𝑛 2 ⌋ ; ⁡𝑖 ∈ 𝑜𝑑𝑑} ∪ {𝑦𝑙; ⁡1 ≤ 𝑙 ≤ 2⌊ 𝑚−1 2 ⌋ ; ⁡𝑗 ∈ 𝑜𝑑𝑑}. so, we have the cardinality of resolving set of 𝑃𝑛 + 𝐶𝑚 namely |𝑊| = 2⌈ 𝑛 2 ⌉ 2 ⁡+ 2⌈ 𝑚−1 2 ⌉ 2 = ⌊ 𝑛 2 ⌋ + ⌊ 𝑚−1 2 ⌋. the representation of the vertices 𝑦 ∈ 𝐹𝑛 and 𝑥 ∈ 𝐹𝑛 respect to 𝑊 as follows. 𝑟(𝑥𝑗|𝑊)⁡=⁡{(𝑎𝑖𝑗); ⁡1 ≤ 𝑗 ≤ 𝑛,1 ≤ 𝑖 ≤ (⌈ 𝑛 2 ⌉ + 1) + (⌊ 𝑚−1 2 ⌋)}, where 𝑎𝑖𝑗 = { 0;𝑓𝑜𝑟⁡𝑖 = 𝑗+1 2 ,1 ≤ 𝑗 ≤ 𝑛,𝑗 ∈ 𝑜𝑑𝑑⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1;𝑓𝑜𝑟⁡(⌈ 𝑛 2 ⌉ + 1) ≤ 𝑖 ≤ (⌈ 𝑛 2 ⌉ + 1) + (⌊ 𝑚−1 2 ⌋) ,1 ≤ 𝑗 ≤ 𝑛⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑜𝑟⁡𝑖 = 𝑗 2 ,2 ≤ 𝑗 ≤ 𝑛,𝑗 ∈ 𝑒𝑣𝑒𝑛⁡𝑜𝑟⁡𝑖 = 𝑗 2 + 1,2 ≤ 𝑗 ≤ 𝑛,𝑗 ∈ 𝑒𝑣𝑒𝑛 2;𝑓𝑜𝑟⁡𝑖, 𝑗 = 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑟(𝑦𝑙|𝑊) = {(𝑎𝑖𝑗); ⁡1 ≤ 𝑗 ≤ 𝑚,1 ≤ 𝑖 ≤ (⌈ 𝑛 2 ⌉ + 1) + (⌊𝑚−1 2 ⌋)}, where 𝑎𝑖𝑗 = { 0;𝑓𝑜𝑟⁡𝑖 = (⌈ 𝑛 2 ⌉) + 𝑗+1 2 ,1 ≤ 𝑗 ≤ 𝑚 − 2,𝑗 ∈ 𝑜𝑑𝑑 1;𝑓𝑜𝑟⁡1 ≤ 𝑖 ≤ (⌈ 𝑛 2 ⌉) ,1 ≤ 𝑗 ≤ 𝑚 − 2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑜𝑟⁡𝑖 = (⌈ 𝑛 2 ⌉) + 𝑗 2 ,2 ≤ 𝑗 ≤ 𝑚,𝑗 ∈ 𝑒𝑣𝑒𝑛⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑜𝑟⁡𝑖 = (⌈ 𝑛 2 ⌉) + 𝑗+1 2 + 1,2 ≤ 𝑗 ≤ 𝑚,𝑗 ∈ 𝑒𝑣𝑒𝑛⁡⁡⁡⁡ 2;𝑓𝑜𝑟⁡𝑖, 𝑗 = 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ it can be seen that every vertex in 𝑃𝑛 + 𝐶𝑚 have distinct representation respect to 𝑊, such that the cardinality of resolvng set in 𝑃𝑛 + 𝐶𝑚 is ⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋ or 𝑑𝑖𝑚(𝐹𝑛) ≤ ⌈ 𝑛 2 ⌉⁡+ ⁡⌊ 𝑚−1 2 ⌋. thus, we conclude that 𝑑𝑖𝑚(𝑃𝑛 + 𝐶𝑚)⁡=⁡⌈ 𝑛 2 ⌉⁡+⁡⌊ 𝑚−1 2 ⌋ for 𝑛 ≥ 2 and 𝑚 ≥ 7. ∎ on the metric dimension of some operation graphs marsidi 90 fig 1. the metric dimension of joint graph 𝑃6 + 𝐶4. theorem 2.2. for 𝑛 ≥ 7, the metric dimension of amalgamation of parachute 𝑎𝑚𝑎𝑙⁡(𝑃𝐶7,𝑣,𝑚) is 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣 = 𝐴,𝑚)) = 6𝑚 2 . proof. the amalgamation of parasut graph, denoted by 𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚) is a connected graph with vertex set 𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) = {𝑥𝑖 𝑗 ; ⁡1 ≤ 𝑖 ≤ 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑦𝑖 𝑗 ; ⁡1 ≤ 𝑖 ≤ 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝐴} and edge set 𝐸(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) = {𝐴⁡𝑥𝑖 𝑗 ; ⁡1 ≤ 𝑖 ≤ 7; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥𝑖 𝑗 ⁡𝑥𝑖+1 𝑗 ; ⁡1 ≤ 𝑖 ≤ 6; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑦𝑖 𝑗 ⁡𝑦𝑖+1 𝑗 ; ⁡1 ≤ 𝑖 ≤ 6; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥1 𝑗 ⁡𝑦1 𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥7 𝑗 ⁡𝑦7 𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. the cardinality of vertex set and edge set, respectively are |𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚))|⁡= 14m⁡ + ⁡1 and |𝐸(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚))|⁡= ⁡21𝑚. if we show that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) ≥ 6𝑚 2 r 𝑛⁡ = ⁡7, then we will show the best lower bound namely 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) ≥ 7𝑚 2 − 1. assume that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) < 6𝑚 2 .⁡this can be shown with take resolving set 𝑊 = {𝑥1 1,𝑥4 1,𝑥6 1,𝑥1 2,𝑥4 2,𝑥6 2,𝑥1 3,𝑥4 3,𝑥6 3,𝑥1 4,𝑥4 4,𝑥6 4} so that it obtained the representation of the vertices 𝑥,𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) respect to 𝑊. it can be seen that there is at least two vertices in 𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,4) which have the same representation respect to 𝑊, one of them is 𝑟(𝑥3 1|𝑊)⁡= (2,1,2,2,2,2,2,2,2,2,2) and 𝑟(𝑥5 1|𝑊)⁡= (2,1,2,2,2,2,2,2,2,2,2) such that we have the cardinality of resolving set of (𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) ≥ ⌈ 6𝑚 2 ⌉. furthermore, we will prove that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) ≤ ⌈ 6𝑚 2 ⌉ with determine the resolving set 𝑊 = {𝑥𝑖 𝑗 ; ⁡4 ≤ 𝑖 ≤ 7; ⁡2 ≤ 𝑗 ≤ 𝑚;⁡𝑖 = 𝑜𝑑𝑑} ∪ {𝑥1 𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. so, we have the cardinality of resolving set of 𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚) namely |𝑊|⁡=⁡|{𝑥𝑖 𝑗 ; ⁡4 ≤ 𝑖 ≤ 7; ⁡2 ≤ 𝑗 ≤ 𝑚;𝑖 = 𝑜𝑑𝑑} ∪ {𝑥1 𝑗 ;1 ≤ 𝑗 ≤ 𝑚}| = ( 4 2 𝑚)⁡+ 𝑚 = ( 6𝑚 2 ). the representation of the vertices 𝑦 ∈ (𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚 = 4)) and 𝑥 ∈ (𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚 = 4)) respect to 𝑊 as follows. 𝑟(𝑥𝑖 𝑗 |𝑊) = {(𝑎𝑖𝑘 𝑗 ); ⁡1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑚,1 ≤ 𝑘 ≤ 6𝑚 2 }, where 𝑎𝑖𝑘 = { 0;𝑓𝑜𝑟⁡𝑘 = 1,𝑘 = 2𝑖,2 ≤ 𝑖 ≤ (⌊ 𝑛 2 ⌋) ,1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1;𝑓𝑜𝑟⁡𝑘 = 𝑖+1 2 ,3 ≤ 𝑖 ≤ 𝑛,𝑖 ∈ 𝑜𝑑𝑑⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 𝑖−1 2 ,5 ≤ 𝑖 ≤ 7,𝑖 ∈ 𝑜𝑑𝑑,1 ≤ 𝑗 ≤ 𝑚 2;𝑓𝑜𝑟⁡1 ≤ 𝑗 ≤ 𝑚,𝑘,𝑖 = 𝑜𝑡ℎ𝑒𝑟⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ on the metric dimension of some operation graphs marsidi 91 𝑟(𝑦|𝑊) = {(𝑎𝑖𝑘); ⁡1 ≤ 𝑖 ≤ 7,1 ≤ 𝑘 ≤ 6𝑚+2 2 }, where 𝑎𝑖𝑘 = { 1;𝑓𝑜𝑟⁡𝑘 = 3𝑗 − 2,𝑖 = 1,1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 2;𝑓𝑜𝑟⁡𝑘 = 6𝑚+2 2 , 𝑖 = 1,7,1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2,𝑖 = 2,⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 7,1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 3; ⁡𝑓𝑜𝑟⁡𝑘 = 6𝑚+2 2 , 𝑖 = 2,6,1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2,𝑖 = 3,⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 6,1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑖 = 1,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 = 6𝑚+2 2 ⁡𝑜𝑟⁡𝑖 = 7,𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 = 6𝑚+2 2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡ 4; ⁡𝑓𝑜𝑟⁡𝑘 = 6𝑚+2 2 , 𝑖 = 3,5,1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗 − 2,𝑖 = 4,⁡⁡⁡⁡⁡⁡⁡⁡ 1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑘 = 3𝑗, 𝑖 = 5,1 ≤ 𝑗 ≤ 𝑚⁡𝑜𝑟⁡𝑖 = 2,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 = 6𝑚+2 2 ⁡𝑜𝑟⁡𝑖 = 6,𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 = 6𝑚+2 2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡ 5; ⁡𝑓𝑜𝑟⁡𝑘 = 6𝑚+2 2 , 𝑖 = 3,3𝑗⁡𝑜𝑟⁡𝑘 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑘 ≠ 6𝑚+2 2 , 𝑖 = 3, 𝑜𝑟⁡𝑘 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑘 ≠ 6𝑚+2 2 , 𝑖 = 5⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 6;𝑓𝑜𝑟⁡𝑖 = 4⁡𝑎𝑛𝑑⁡𝑖 ≠ 3𝑗⁡𝑎𝑛𝑑⁡𝑖 ≠ 3𝑗 − 2⁡𝑎𝑛𝑑⁡𝑖 ≠ 6𝑚+2 2 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ it can be seen that every vertex in 𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚) have distinct representation respect to 𝑊, such that the cardinality of resolvng set in 𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚) is 6𝑚 2 or 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚)) ≤ 6𝑚 2 . thus, we conclude that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝑃𝐶7,𝑣,𝑚))⁡=⁡ 6𝑚 2 . ∎ fig 2. the metric dimension of amalgamation of parachute 𝐴𝑚𝑎𝑙⁡(𝑃𝐶7,𝑣,3). theorem 2.3. for 𝑛 ≥ 6, the metric dimension of amalgamation of fan graph 𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚) is: on the metric dimension of some operation graphs marsidi 92 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝐴,𝑚)) = { 𝑛𝑚 2 − 1,𝑓𝑜𝑟⁡𝑛⁡is⁡even 𝑛𝑚 − 𝑚 2 ,𝑓𝑜𝑟⁡𝑛⁡is⁡odd⁡⁡⁡⁡ proof. the amalgamation of fan graph, denoted by 𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚) is a connected graph with vertex set 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) = {𝑥𝑖 𝑗 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 1; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑦𝑗; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑥𝑛 𝑚} and edge set 𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) = {⁡𝑥𝑖 𝑗 ⁡𝑥𝑖+1 𝑗 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 2; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {𝑦𝑗𝑥𝑖 𝑗 ; ⁡1 ≤ 𝑖 ≤ 𝑛 − 1; ⁡1 ≤ 𝑗 ≤ 𝑚} ∪ {⁡𝑥𝑛−1 𝑗 ⁡𝑥1 𝑗+1 ; ⁡1 ≤ 𝑗 ≤ 𝑚 − 1} ∪ {⁡𝑥𝑛−1 𝑚 ⁡𝑥𝑛 𝑚} ∪ {𝑦𝑗𝑥1 𝑗+1 ; ⁡1 ≤ 𝑗 ≤ 𝑚 − 1} ∪ {𝑦𝑚𝑥𝑛 𝑚}. the cardinality of vertex set and edge set, respectively are |𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚))|⁡= nm⁡ + ⁡1 and |𝐸(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚))| = 𝑚(2𝑛 − 1). if we show that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) = 𝑛𝑚 2 − 1 for𝑛 ≥ 7 and 𝑛 is even, then we will show the best lower bound namely 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) ≥ 𝑛𝑚 2 − 1. assume that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) < 𝑛𝑚 2 − 1.⁡this can be shown with take resolving set 𝑊 = {𝑥1 1,𝑥4 1,𝑥1 2,𝑥4 2,𝑥6 2,𝑥1 3,𝑥4 3,𝑥6 3,𝑥1 4,𝑥4 4} so that it obtained the representation of the vertices 𝑦 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐹6,𝑣 = 𝑦,4)) and 𝑥𝑖 𝑗 ∈ 𝑉(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 6,4)) respect to 𝑊. it can be seen that there is at least two vertices in 𝑎𝑚𝑎𝑙(𝐹6,𝑣 = 𝑦,4) which have the same representation respect to 𝑊, one of them is 𝑟(𝑥6 1|𝑊)⁡= (2,2,2,2,2,2,2,2,2,2) and 𝑟(𝑥6 4|𝑊)⁡= (2,2,2,2,2,2,2,2,2,2) such that we have the cardinality of resolving set of 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) ≥ 𝑛𝑚 2 − 1. furthermore, we will prove that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚)) ≤ 𝑛𝑚 2 − 1 with determine the resolving set 𝑊 = {𝑥𝑖 𝑗 ; ⁡4 ≤ 𝑖 ≤ 𝑛; ⁡2 ≤ 𝑗 ≤ 𝑚;⁡𝑖 = 𝑜𝑑𝑑} − {𝑥𝑛 𝑚} ∪ {𝑥1 𝑗 ; ⁡1 ≤ 𝑗 ≤ 𝑚}. so, we have the cardinality of resolving set of 𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣 = 𝑦,𝑚) namely |𝑊| = ⁡|{𝑥𝑖 𝑗 ; ⁡4 ≤ 𝑖 ≤ 𝑛; ⁡1 ≤ 𝑗 ≤ 𝑚;𝑖⁡is⁡even} − {𝑥𝑛 𝑚} ∪ {𝑥1 𝑗 ;1 ≤ 𝑗 ≤ 𝑚}| = ( 𝑛−2 2 )𝑚⁡ + 𝑚 − 1 = ( 𝑛𝑚 2 − 1). the representation of the vertices 𝑦 ∈ 𝐹𝑛 and 𝑥 ∈ 𝐹𝑛 respect to 𝑊′ as follows. 𝑟(𝑥𝑖 𝑗 |𝑊) = {(𝑎𝑖𝑘 𝑗 ); ⁡1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑚,1 ≤ 𝑘 ≤ 𝑛𝑚 2 − 1}, where 𝑎𝑖𝑘 = { 0;𝑓𝑜𝑟⁡𝑘 = 1,𝑘 = 2𝑖,2 ≤ 𝑖 ≤ (⌊ 𝑛 2 ⌋) ,1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 1;𝑓𝑜𝑟⁡𝑘 = 𝑖+1 2 ,3 ≤ 𝑖 ≤ 𝑛,𝑖 ∈ 𝑜𝑑𝑑,1 ≤ 𝑗 ≤ 𝑚⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑘 = 𝑖−1 2 ,5 ≤ 𝑖 ≤ 𝑛,𝑖 ∈ 𝑜𝑑𝑑,1 ≤ 𝑗 ≤ 𝑚⁡𝑎𝑛𝑑⁡(𝑘 ≠ 𝑚 ∩ 𝑖 ≠ 𝑛) 2;𝑓𝑜𝑟⁡1 ≤ 𝑗 ≤ 𝑚,𝑘,𝑖 = 𝑜𝑡ℎ𝑒𝑟𝑠⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ 𝑟(𝑦|𝑊) = {(𝑎𝑖𝑘); ⁡1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑘 ≤ 𝑛−2 2 }, where 𝑎𝑖𝑘 = {1;𝑓𝑜𝑟⁡1 ≤ 𝑘 ≤ 𝑛𝑚−𝑛 2 , 𝑖 = 1 it can be seen that every vertex in 𝑎𝑚𝑎𝑙(𝐹6,𝑣,4) have distinct representation respect to 𝑊, such that the cardinality of resolvng set in 𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣,𝑚) is 𝑛𝑚 2 − 1 or 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣,𝑚)) ≤ 𝑛𝑚 2 − 1. thus, we conclude that 𝑑𝑖𝑚(𝑎𝑚𝑎𝑙(𝐹𝑛,𝑣,𝑚)) =⁡ 𝑛𝑚 2 − 1. ∎ on the metric dimension of some operation graphs marsidi 93 fig 3. the metric dimension of amalgamation of fan graph a𝑚𝑎𝑙(𝐹6,𝑣 = 𝑦,3). theorem 2.4. for 𝑚 ≥ 2, the metric dimension of 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) = 2. proof. the shackle of fan graph, denoted by 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)is a connected graph with vertex set 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) = {𝑥𝑗; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} ∪ {𝑦𝑗; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} and edge set 𝐸(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) = {𝑥𝑗𝑦𝑗; ⁡1 ≤ 𝑗 ≤ 𝑚 + 1} ∪ {𝑥𝑗𝑦𝑗+1; ⁡1 ≤ 𝑗 ≤ 𝑛} ∪ {𝑥𝑗+1𝑦𝑗; ⁡1 ≤ 𝑗 ≤ 𝑚}. the cardinality of vertex set and edge set, respectively are |𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚))| = ⁡2𝑚 + 2 and |𝐸(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚))| = 3𝑚 + 1. the proof that the lower bound of 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) ≥ 2. based on proposition 1, that 𝑑𝑖𝑚(𝐺) = 1 if only if 𝐺 ≅ 𝑃𝑛. the graph 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚) does not isomorphic to path 𝑃𝑛 such that 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) ≥ 2. furthermore, we proof that the upper bound of 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) ≤ 2, we choose the resolving set 𝑊 = {𝑥1,𝑦1}. the representation of the vertices 𝑣 ∈ 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) respect to 𝑊 as follows. 𝑟(𝑥𝑗|𝑊) =⁡(𝑗 − 1,𝑗);𝑗 ∈ 𝑜𝑑𝑑 𝑟(𝑦𝑗|𝑊) =⁡(𝑗, 𝑗 − 1);𝑗 ∈ 𝑜𝑑𝑑 𝑟(𝑥𝑗|𝑊) =⁡(𝑗, 𝑗 − 1);𝑗 ∈ 𝑒𝑣𝑒𝑛 𝑟(𝑦𝑗|𝑊) =⁡(𝑗 − 1,𝑗);𝑗 ∈ 𝑒𝑣𝑒𝑛 vertex 𝑣 ∈ 𝑉(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) are distict. so, we have the cardinality of resolving set 𝑊 is |𝑊| = 2. thus, the upper bound of 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚) is 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) ≤ 2. it concludes that 𝑑𝑖𝑚(𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚)) = 2. ∎ on the metric dimension of some operation graphs marsidi 94 fig 4. the metric dimension of sℎ𝑎𝑐𝑘(𝐻2 2,𝑒,3). conclusions in this paper, the result show that the local metric dimension of some graph operation such as joint graph 𝑃𝑛 + 𝐶𝑚, amalgamation of parachute, amalgamation of fan, and 𝑠ℎ𝑎𝑐𝑘(𝐻2 2,𝑒,𝑚). acknowledgments we gratefully acknowledge the support from drpm kemenristekdikti 2018 indonesia. references [1] g. chartrand, e. salehi, and p. zhang, "the partition dimension of a graph," aequationes math., vol. 59, pp. 45-54, 2000. [2] g. chartrand, l. eroh, and m. a. johnson, " resolvability in graphs and the metric dimension of a graph," discrate appl. math., vol. 105, pp. 99-113, 2000. [3] marsidi, dafik, i. h. agustin, and r. alfarisi, “on the local metric dimension of line graph of special graph,” cauchy, vol. 4, no. 3, pp. 125-130, 2016. [4] i. g. yero, d. kuziak, and j. s. rodríguez-velázquez, “on the metric dimension of corona product graphs,” computers and mathematics with applications, vol. 61, pp. 2793-2798, 2011. [5] h. fernau, p. heggernes, p. hof, d. meister, and r. saei, “computing the metric dimension for chain graphs,” information processing letters, vol. 115, pp. 671-676, 2015. [6] j. cáceres, c. hernando, m. mora, i. m. pelayo, and m. l. puertas, “on the metric dimension of infinite graphs,” discrete applied mathematics, vol. 160, pp. 2618-2626, 2012. [7] m. fehr, s. gosselin, and o. r. oellermann, “the metric dimension of cayley digraphs,” discrete mathematics, vol. 360, pp. 31-41, 2006. [8] t. k. maryati, a. n. m. salman, and e. t. baskoro, “on h-supermagic labelings for certain shackles and amalgamations of a connected graph,” utilitas mathematica, 2010. [9] i. h. agustin, dafik, s. latifah, and r. m. prihandini, “a super (a,d)-bm-antimagic total covering of a generalized amalgamation of fan graphs,” cauchy, vol. 4, no. 4, pp. 146154, 2017. improving the guidance learning (lbb) consumer satisfaction in malang using danp topsis method cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 117-120 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 31 august 2018 reviewed: 05 september 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5541 improving the guidance learning (lbb) consumer satisfaction in malang using danp topsis method kwardiniya andawaningtyas1, endang wahyu handamari 2, corina karim3 mathematics department, universitas brawijaya email: dina_math@ub.ac.id abstract decision analysis of multiple attribute decision making (madm) model is used to assess the performance, not only in a rank but also in a plan of marketing strategy as an effort to increase consumers’ satisfaction by combining dematel-based analytical network process (danp) method and the technique for order of preference by similarity to ideal solution (topsis) method. one of the industrial services in the education nowadays is the services of the guidance learning (lbb). this article has 3 alternatives to 6 criteria. the questionnaire was distributed to 80 lbb’ students and 55 lbb’ mentors. the result of the dominant criteria affecting customer satisfaction of lbb in malang by danp method is the mentor quality. meanwhile, the topsis result showed that the lbb of avicenna education malang is the best alternative to the marketing strategy. keywords: dominant criteria, alternative, lbb, danp, topsis introduction the system of education in indonesia is growing rapidly that affecting the students’ learning patterns. in order to improve the learning ability of students outside school hours, one of the solutions is to engage students at the guidance learning (lbb). lbb services are considered helping parents who have difficulties in teaching and guiding their children's assignments from the school. however, we found some lbb in malang so that it was difficult for parents to select the best lbb for their children [1]. such conditions can increase the competition among the similar educational institutions in terms of offering the best products to meet the consumer needs [1]. the guidance business can be said to be successful if it has many satisfied consumers and has new customers who come in this lbb. the right marketing strategy is the success key and the target of success in running a business in education is the fulfillment of consumers’ satisfaction. some research on the lbb decision-making system was established such as the election decision support system of a private tutor for lbb’ students by ahp method [2], the election decision support system of tutoring institution [3], and the implementation of topsis method for the lbb selection decision support system [4]. in addition, the research on the high school students’ motivation in taking part in lbb also has been done by [5]. in early 2013’s, [6] combines dematel and anp to evaluate the implementation of viable information systems. infact, the research undertaken by [7] combines the decision-making method of dematel-based analytical network process (danp) and visekriterijumsko kompromisno rangiranje (vikor) to improve e-store business services. in this paper, we will be combining the danp and topsis methods to assess the lbb's marketing strategy not only in a rank but also to improve and create marketing strategies to enhance the lbb's customer satisfaction. the criteria which we will use here mailto:dina_math@ub improving the guidance learning (lbb) consumer satisfaction in malang using danp – topsis method kwardiniya andawaningtyas 118 is according to [8] without wi-fi criteria. methods determine the value of dominant criteria by dematel-based anp (danp) method. the analytical network process (anp) method was published in [9]. the first step of anp procedure is comparing the criteria through the systems to form the unweighted super matrix by pairwise comparisons. then the weighted super matrix is obtained by summing up each column to obtain a value of one. each element in the column is divided by the number of clusters, such that the elements in the column are one. on the basic concept of anp, each cluster is assumed to have the same value, this assumption will obtain the weighted super matrix become irrational as there are different degrees [10]. the weakness of the basic concept of anp is improved by adopting the dematel method to determine the degree of the affecting criteria and applying this method to normalize unweighted super matrix in the anp method. the refinement of the anp basic concepts is called dematel-based anp (danp). the input of the danp method is tc matrix in dematel method, that is, 80 lbb students by grouping criteria on several clusters, namely cluster 1 (d1) for the cost criteria (a), cluster 2 (d2) for mentor quality criteria (b), cluster 3 (d3) for location criteria (c) and cluster 4 facility (d4) for classroom (d) criteria, additional consultation (e) and activities outside kbm (f) (see in [11]). determine the analysis of marketing strategy ranking in order to increase the consumer's satisfaction of each alternative by topsis method. topsis is one of the methods to find the optimal decision-making. according to [12], topsis is one of the multi-criteria decision-making methods that first introduced by yoon and hwang in 1981. topsis considers the distance to the positive ideal solution and the distance to the negative ideal solution simultaneously. according to [13], topsis is the method used to deal with the problem of alternative rank from the best to the worst. the input of this method is questionnaire of 55 lbb mentors. this method obtained the highest index value (cj+) as the best alternative. the alternative ranking in the topsis method depends on the selection of the profitable criteria or cost criteria. the cost criteria are obtained from affecting criteria and profitable criteria are obtained from the criteria influenced by the map digraph impact result by dematel method [11]. step of data analysis above can be described as below: figure 1. step of data analysis improving the guidance learning (lbb) consumer satisfaction in malang using danp – topsis method kwardiniya andawaningtyas 119 result and discussion 1. use tc in [11] to determine the value of dominant criteria using danp method by calculating the limiting super matrix which presented in table 1. table 1. value criteria. criteria a b c d e f value 0,243 0,249 0,247 0,085 0,089 0,087 the table 1 showed that the highest criteria are the criteria b (mentor quality) of 24.9%. meanwhile, the lowest criteria are the criterion d (classroom) of 8.5%. the calculation of the value criteria is obtained based on the error value of questionnaire data that is 2.022%, so the result of the questionnaire test is valid because the error value is <5%, with the validity value is 97,978%. 2. determine the analysis of lbb marketing strategy ranking of each alternative by topsis method 2.1. from [11] the selection of cost criteria or profitable criteria is obtained based on the map digraph impact of dematel method. the cost criteria are the price criteria (a), mentor quality (b) and the location (c). meanwhile, the profitable criteria are the classroom (d) criteria, additional consultation (e) and activities outside teaching and learning activities (kbm) (f) [12]. the map digraph impact can be seen in figure 2. figure 2. the map of digraph impact the cost criteria is a criterion affecting the lbb selection, which the consumers expect the price, mentor quality, and the location commensurate with paid cost. otherwise, the profitable criteria is the criteria affected by lbb selection, which is the consumers will obtain the profit with the paid cost by obtaining the classroom criteria, additional consultation, and activities outside the kbm. generally, in this case, the profitable criteria is the facilities cluster (d4) of the danp method. 2.2. calculate the value of proximity relative to the positive ideal solution (cj+). the alternatives are sorted from the largest cj+value to the smallest value. the first alternative rank with the largest cj+ value is the avicenna education (aii) with a value of 0.543. the second alternative rank is bu lastri’s tutoring (aiii) with a value of 0,509. meanwhile, the improving the guidance learning (lbb) consumer satisfaction in malang using danp – topsis method kwardiniya andawaningtyas 120 lbb progressive private center (p2c) (ai) as the third alternative rank with a value of 0,457. the criteria of marketing strategy that must be upgraded to fulfill lbb's consumer satisfaction at avicenna education is the criteria of the activity outside kbm (f), at the lbb progressive private center (p2c) is additional consultation criteria (e), and at the bu lastri’s guidance is the classroom criteria (d). generally, to improve the lbb marketing strategy in malang, all the alternative lbbs have to improve the cluster facility (d4). conclusion the best alternative for this case is obtained from the highest cj+ value in the lbb avicenna education malang. the marketing strategy that needs to be improved on lbb avicenna education malang is cluster d4, which is classroom criteria, additional consultation, and activities outside kbm. by understanding the marketing strategy to be improved, it is expected to improve the quality of lbb customer satisfaction. references [1] firdaus, f., peran lembaga bibingan belajar terhadap peningkatan motivasi belajar anak,http://www.kompasiana.com/fachrifirdaus/peran-lembaga-bimbinganbelajar-terhadap-peningkatan-motivasi-belajar anak_552990c0f17e612f07d623b1, 2013. [2] kristiyanti, l., sugiharto, a., dan wibawa, h.a., sistem pendukung keputusan pemilihan pengajar les privat untuk siswa lembaga bimbingan belajar dengan metode ahp (studi kasus lbb system cerdas), jurnal masyarakat informatika 4.7, 2013, 39-47. [3] sutrisno, j., sistem pendukung keputusan pemilihan lembaga kursus dan pelatihan berbasis web, doctoral dissertation, universitas muhammadiyah surakarta, 2013. [4] mayasari, r.p., penerapan metode topsis untuk sistem pendukung keputusan pemilihan lembaga bimbingan belajar di purwokerto, phd thesis, universitas muhammadiyah purwokerto, 2016. [5] widodo, f., motivasi siswa sma mengikuti lembaga bimbingan belajar di kota surakarta, sosialitas, jurnal ilmiah pend. sos. ant 5.2., 2016. [6] shih, k.h., lin, w.r., wang y.h. dan hung, t.e., applying dematel-anp for assessing organizational information system development decisions, knowledge management & innovation, international conference 2013. [7] chiu, w.y., tzeng, g.h. dan li h.l., a new hybrid mcdm model combining danp with vikor to improve e-stores business, knowledge-based systems, 37, 2013, 4861. [8] putri, e.n.d., perbandingan metode dematel dan fuzzy dematel dalam menentukan kriteria terbaik yang mempengaruhi kepuasan konsumen (studi kasus: lbb airlangga sengkaling), jurnal mahasiswa matematika, vol 5, no 1, 2017. [9] saaty, l., fundamentals of the analytic network process, 1999. [10] ou yang, shieh, h., leu, j., dan tzeng, a novel hybrid mcdm model combined with dematel and anp with applications. international journal of operations research vol. 5, no. 3, 2008, 160-168. [11] andawaningtyas, k., identification factor affecting of learning guidance institution (lbb) selection in malang by dematel method, poster of basic international conference, 2018. modification of the curve and the surface polynomial bezier using de casteljau algorithm cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 211-215 p-issn: 2086-0382; e-issn: 2477-3344 submitted: pebruary 04, 2019 reviewed: may 16, 2019 accepted: may 31, 2019 doi: http://dx.doi.org/10.18860/ca.v5i4.6346 modification of the curve and the surface polynomial bezier using de casteljau algorithm juhari1 1mathematics department, uin maulana malik ibrahim malang e-mail: juhari@uin-malang.ac.id abstract research carried out to obtain a bezier curve of degree six resulting curvature of the curve is more varied and multifaceted. stages in formulating applications bezier surfaces revolution in design, there are three marble objects. first, calculate the parametric representation revolution bezier surface and shape modification in a number of different forms. second, formulate bezier parametric surfaces that are continuously incorporated. lastly, apply the formula to the design objects using computer simulation. results marble obtained are bezier curves of degree six modified version of the bezier curve of degree five and some form of revolution bezier surfaces are varied and multifaceted. keywords: revolution bezier surfaces, modeling, continuity and modification form introduction model development of creative industries is like a building that would strengthen economic development, with the foundation, pillars and roofs as elements of the building. in practice, the fabrication process industry craft objects, such as industrial ceramic or marble handicrafts requires treatment stages in shaping it. first, establish the model to be constructed. second, convert the raw materials into semi-finished goods, thirdly, to form semi-finished goods into goods more refined, last, smoothing the surface of objects, but with the continuously growing number of creative industry such, it resulted in the marketability and price of items manufactured in domestic or export markets is waning. primarily problems are generally the products they produce the pattern still remains, has not been matched by an increase in artistic quality and diversification/innovation shapes required by customers both from the level of symmetry, harmony, and variations of the model as well as from the aspect of different types of goods. design and fabrication of objects of industrial output is still done manually. how to design the shape of the object before it was fabricated, mostly done by duplicating the object already. these issues, this article is intended to get technique count and parametric formula of bezier curves of degree six that will be used as the basic component of the play character crafts industry. formula natural shape already covered by faux [1] and kusno [2], the study continuous merger of two pieces of bezier curves and surfaces have been covered by zheng[3], merging bezier curves with geometric objects developed by juhari [4] by adopting the theories of geometry transformation [5], application development order polynomial bezier curve has been done previously during the bezier curve of order five modified version of the bezier curve cuartik[6], in this paper, further developed rotary defining bezier surfaces which will be equipped with the basic shape of the surface modifier parameters. furthermore, the merger will be calculated continuous parametric bezier surfaces rotate as well as some form of revolution mailto:juhari@uin-malang.ac.id mailto:juhari@uin-malang.ac.id modification of the curve and the surface polynomial bezier using de casteljau algorithm juhari 212 bezier surfaces are varied and multifaceted. methods stages in formulating applications bezier surfaces revolution in design, there are three marble objects. first, calculate the parametric representation revolution bezier surfaces and modified forms in several different forms. second, formulate bezier parametric surfaces that are continuously incorporated. lastly, apply the formula on a marble design objects using computer simulations. results and discussion the general formula bezier curve of degree n defined in terms of 𝐶𝑛(𝑢) = ∑𝑃𝑖𝐵𝑖 𝑛(𝑢) 𝑛 𝑖=0 with and 𝐵𝑖 𝑛(𝑢) = ( 𝑛 𝑖 )(1 − 𝑢)𝑛−𝑖.𝑢𝑖0 ≤ 𝑢 ≤ 1 the formula bezier five order 𝑝(𝑢) = 𝑎 + 𝑏𝑢 + 𝑐𝑢2 + 𝑑𝑢3 + 𝑒𝑢4 + 𝑓𝑢5 interpolation point is the first point and the sixth, while the second point to the fifth point is the point approximation. thus, for the i-th segment that forms the control points p1, p2, p3, p4, p5, p6 is defined as follows: 𝑉(0) = 𝑃0 𝑉(1) = 𝑃5 𝑉,(0) = 5(𝑃1 − 𝑃0) 𝑉,(1) = 5(𝑃5 − 𝑃4) 𝑉,,(0) = 20(𝑃0 − 2𝑃1 + 𝑃2) 𝑉,,(1) = 20(𝑃3 − 2𝑃4 + 𝑃5), 𝑀𝑀𝐻a matrix resulting the hermit curve of degree five. so bezier curve of degree five in parametric form, namely: 𝑉(𝑢) = 𝑃0(1 − 5𝑢 + 10𝑢 2 − 10𝑢3 + 5𝑢4 − 𝑢5) + 𝑃1(5𝑢 − 20𝑢 2 + 30𝑢3 − 20𝑢4 + 5𝑢5) + 𝑃2(10𝑢 2 − 30𝑢3 + 30𝑢4 − 10𝑢5) + 𝑃3(10𝑢 3 − 20𝑢4 + 10𝑢5) + 𝑃4(10𝑢 3 − 15𝑢4 + 5𝑢5) + 𝑃5(𝑢 5) figure 1. bezier curve of degree lima modification of the curve and the surface polynomial bezier using de casteljau algorithm juhari 213 modification a curve bezier order five to six order suppose a bezier curve of degree five is expressed in the form 𝐶5(𝑢) = ∑𝑃𝑖𝐵𝑖 5(𝑢) 5 𝑖=0 with. bezier polygon with a point of view of control is defined as follows,0 ≤ 𝑢 ≤ 1[𝑃0,𝑃1,𝑃2,𝑃3,𝑃4,𝑃5]𝑊51,𝑊52,𝑊53,𝑊54,𝑊55 𝑊51 = 𝜆51𝑃1 + (1 − 𝜆51)𝑃0 𝑊52 = 𝜆52𝑃2 + (1 − 𝜆52)𝑃1 𝑊53 = 𝜆53𝑃3 + (1 − 𝜆53)𝑃2 𝑊54 = 𝜆54𝑃4 + (1 − 𝜆54)𝑃3 𝑊55 = 𝜆55𝑃5 + (1 − 𝜆55)𝑃4 with 0 ≤ 𝜆51,𝜆52,𝜆53,𝜆54,𝜆55 ≤ 1 and 𝜆51,𝜆52,𝜆53,𝜆54,𝜆55 figure 2. modification of the bezier curve bezier curves of degree five become a degree six with the control points the new polygon ω = [𝑃0,𝑊51,𝑊52,𝑊53,𝑊54,𝑊55,𝑃5] on the model of bezier curve of degree five 𝐶5(𝑢) could be modified into a bezier curve of degree six 𝐶6(𝑢) with polygon control points ω in the following, suppose a bezier curve of degree six expressed in the form 𝐶6(𝑢) = ∑𝑃𝑖𝐵𝑖 6(𝑢) 6 𝑖=0 with 𝐵𝑖 6(𝑢) = ( 6 𝑖 )(1 − 𝑢)6−𝑖.𝑢𝑖 and 0 ≤ 𝑢 ≤ 1, then the six-degree bezier curves results in the form of parametric modifications are: 𝑉(𝑢) = 𝑃0(𝑢 6 − 6𝑢5 + 15𝑢4 − 20𝑢3 + 15𝑢2 − 6𝑢 + 1) + 𝑊51(−6𝑢 6 + 30𝑢5 − 60𝑢4 + 60𝑢3 − 30𝑢2 + 6𝑢) + 𝑊52(15𝑢 6 − 60𝑢5 + 90𝑢4 − 60𝑢3 + 15𝑢2) + 𝑃3(−20𝑢 6 + 60𝑢5 − 60𝑢4 + 20𝑢3) + 𝑃4(15𝑢 6 − 30𝑢5 + 15𝑢4) + 𝑊55(−6𝑢 6 + 6𝑢5) + 𝑃5(𝑢 6) p0 p1 p3 p4 p2 p5 w 51 w5 2 w5 3 w5 4 w5 5 modification of the curve and the surface polynomial bezier using de casteljau algorithm juhari 214 with 0 ≤ 𝑢 ≤ 1. the following are examples with kuartik bezier curve with control points 𝑃0 = (1,0,1),𝑃1 = (1,0,3),𝑃2 = (8,0,6),𝑃3 = (8,0,9),𝑃4 = (1,0,12),𝑃5 = (1,0,15). the example 𝜆 = [0.1,0.1,0.1,0.1,0.1] figure 3 bezier curve of degree five modifications become bezier curve of degree six conclusion the introduction of parameters shapeshifters swivel base bezier surface can produce a variety of new forms of rotary bezier surfaces. so the impact on the creation of models of industrial objects easier, more varied, and multifaceted. the forms of the rotary bezier surface of degree six results from the modification of bezier curves berdrajat five with some data selection control points 𝑃0,𝑃1,𝑃2,𝑃3,𝑃4 and 𝑃5 a new control point 𝑊51,𝑊52,𝑊53,𝑊54 and being influenced by the administration of parameter values 𝜆51,𝜆52,𝜆53,𝜆54,𝜆55 present different results. the steps undertaken to construct a marble craft is to determine the height and radius of the object. so it can be used to model the other objects ceramics industry. acknowledgement the author would like to thank the lembaga penelitian dan pengabdian masyarakat universitas islam negeri maulana malik ibrahim malang in 2018 for financial support in conducting research, the results of which were in the form of this paper. references [1] faux, “(mathematics & its applications numbered hardcover) i. d. faux, michael j. pratt-computational geometry for design and manufacture-horwood (1979).pdf.” 1987. [2] kusno, cahya and p. mahros, “modelisasi benda onyx dan marmer melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan putar bezier ( onyx and marmer objects modeling by joining and choosing parametric modifications of bezier revolution surfaces ) a ) b ) c ),” vol. 8, no. 2, pp. 175–185, 2007. [3] zheng, g. wang, and y. liang, “curvature continuity between adjacent rational bezier patches,” comput. aided geom. des., vol. 9, no. 5, pp. 321–335, 1992. [4] juhari and e. octafiatiningsih, “penerapan kurva bezier karakter simetrik dan putar pada model kap lampu duduk menggunakan maple,” cauchy, vol. 4, no. 1, modification of the curve and the surface polynomial bezier using de casteljau algorithm juhari 215 pp. 28–34, 2015. [5] juhari, “pemodelan pertumbuhan zea mays l. menggunakan sthochastic l-system,” cauchy, vol. 3, no. 1, pp. 49–54, 2013. [6] maharani, “aplikasi kurva bezier berderajat lima dari modifikasi kurva kuartik pada desain keramik,” 2016. forecasting financial system stability using vector error correction model approach cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 109-116 p-issn: 2086-0382; e-issn: 2477-3344 submitted: june 29, 2020 reviewed: october 19, 2020 accepted: november 05, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.9811 forecasting financial system stability using vector error correction model approach setiawan1, moch. trianto utomo2, alfira mulya astuti3, m. sjahid akbar4, imam safawi ahmad5 1,2,4,5department of statistics, institut teknologi sepuluh nopember, indonesia 3department of mathematics education, universitas islam negeri mataram, indonesia email: setiawan@statistika.its.ac.id, antoksmada@gmail.com, alfiramulyaastuti@uinmataram.ac.id, m_syahid_a@statistika.ac.id, safawi@statistika.ac.id abstract indonesia is one of the developing countries whose economic system is still very dependent on other developed countries. this reliance often becomes one of the causes of the occurrence of economic turmoil sectors that interfere with financial system stability in indonesia. therefore, to forecast financial system stability indicators, primarily macroeconomic variables, become essential to do to provide an accurate index value. this research aims to forecast indicators that affect indonesia's financial stability using the vector error correction model (vecm) approach. the indicators used are banking stability index, jakarta stock exchange composite index, inflation, and exchange rate. forecasting using the vecm method produces two models i.e., deterministic model using the intercept and deterministic model without intercept. the best model is a deterministic model with intercept and without a trend. the result of forecasting for bank stability index, inflation, and exchange rate can be used to forecast 12 periods ahead. the banking stability index affects exchange rate, the jakarta stock exchange composite index, and inflation. keyword: financial; macroeconomic; forecasting; vecm approach introduction the stability of the financial system is a condition in which the economic mechanisms in pricing, allocation, and risk management function and supports economic growth [1]. the instability of the financial system of a country has a different cause in each state, depending on the components that dominate in the preparation of the financial system. indonesia as a developing country has the elements of an economic system dominated by the banking sector, it can be seen from the results of the study by bank indonesia stated that the banking sector holds 77% of the total assets of the components of the financial system in indonesia [2]. indonesia’s financial stability is related to two interrelated sources, namely external and internal. based on the results of the financial stability review (fsr) no. 30 march 2018 conducted by bank indonesia said that external factors which will affect the stability of the financial system in indonesia that us trade protectionism policy and internal factors are expected to shake the financial stability in indonesia, namely the volatility of inflation and exchange rate rupiah to dollar [3]. research of financial system stability has been conducted by several researchers, including wijaya [4], qoirul [5], akinsola dan ikhide [6], nadri et al. [7], elijah and hamzah [8], and mande et al [9]. based on previous studies, there is no justification regarding indicators that affect financial stability in indonesia. therefore, this research will forecast http://dx.doi.org/10.18860/ca.v6i3.9811 mailto:setiawan@statistika.its.ac.id mailto:setiawan@statistika.its.ac.id mailto:antoksmada@gmail.com mailto:alfiramulyaastuti@uinmataram.ac.id mailto:m_syahid_a@statistika.ac.id mailto:safawi@statistika.ac.id forecasting financial system stability using vector error correction model approach setiawan 110 indicators that affect indonesia's financial stability using the vector error correction model (vecm) approach. the financial stability indicators used in this study are inflation, exchange rate, bank stability index (bsi), and the jakarta stock exchange composite index (jseci). the application of these indicators is based on the results of the financial stability study (ksk) no. 30, march 2018. this method's advantages can understand the existence of a good relationship between economic variables in the form of a structured long-term relationship model [10]. this study is based on research conducted by jenkin [11] who uses the vecm approach because the data used is a time series with first-order or more and with cointegration conditions (long term). the best selection criteria for the model used are root mean square error (rmse) and symmetric mean absolute percentage error (smape). methods vector error correction model vector error correction model (vecm) is an interpreted vector autoregressive model, where the assumptions that must be fulfilled, namely the time series data, must be stationary in order one or more, and there is cointegration between variables [12]. the following is the specification of the model used in this study: v v v v 4 n i 1 j 1 k 1 l 1 n 1                     n,t i 1,t-i j 2,t-j k 3,t-k l 4,t-l n,t-1 tδy μ γ δy γ δy γ δy γ δy α β y u (1) measure of evaluation the measure of evaluation model to know the models created can be used in the prediction of how the period by using root mean square error (rmse) dan symmetric mean absolute percentage error (smape) [13]. the following are the equations for the rmse and smape method:   2 1 n t t t y y rmse n     (2)  1 1 / 2 n t t t t t y y smape n y y     (3) where t y is actual data at t=1, 2, ..., 12, and the forecast results at the same t. if the two indicators have smaller values, the results of the predictions produced will be better. forecast error variance decomposition forecast error variance decomposition (fevd) is a tool used to describe the shock (shock) on a variable to components of other variables used in modeling vecm. the variance decomposition in shock j caused variable i with a lag of n can be written in the equation: forecasting financial system stability using vector error correction model approach setiawan 111     2 0 2 0 1 n ij k n h ij k j k k        (4) where i and j is the number of variables used in this research and is the effect of structural shock on lag-k. result and discussion characteristics of indicators of financial system stability in indonesia characteristics of the indicators of financial system stability in indonesia are shown by the time series plot. the financial system stability indicators are used bsi, inflation, jseci, and exchange rates. in figure 1, time series plots from bsi, inflation, jseci, and exchange rates will be presented: figure 1. time series plot of bsi, jseci, exchange rates, and inflations figure 1 shows that the development of bsis in indonesia from january 2011 to december 2018 is still in the normal position. the bsi value is below the 103 basis point (bps) figure. however, the condition of the bsi at the interval had been on the alert categories, namely september 2013 and early 2014. in figure 1, it can also show that there was a decline in the largest jseci value in 2015, from 2011 to 2017. the value of the jseci opened at the level of rp. 5,450,294 and closed at the level of rp. 4,615,163 in december 2015. it can be seen in the period of january 2015. the highest increase in jseci occurred in 2017. it can be seen in the period of january 2017. the value of the jseci opened at the level of rp. 5,386,691 and closed at the level of rp. 6,605.63 in december 2017, wherein 2017, the jseci increased by 22.6% on an annual basis. year month 20182017201620152014201320122011 janjanjanjanjanjanjanjan 7000 6500 6000 5500 5000 4500 4000 3500 j a k a rt a s to c k e x c h a n g e c o m p o s it e i n d e x year month 20182017201620152014201320122011 janjanjanjanjanjanjanjan 3 2 1 0 -1 in fl a ti o n forecasting financial system stability using vector error correction model approach setiawan 112 based on figure 1, it can be seen that the value of the rupiah exchange rate against the us dollar experienced the weakest point in october 2018, which was rp. 15,679. the condition of the rupiah value was due to the trade war escalation between the united states and china. it caused a depreciation of the rupiah to the highest point. in figure 1, it can also show that inflation is highest at this period interval, namely in july 2013 and december 2014. the surge in inflation in the period of july 2013 was due to the increase in subsidized fuel prices in june 2013 which affected supply disruptions of some food commodities, resulting in the period of 2013 to be the year with the rate of inflation (year on year), the highest during the period january 2011 to december 2018 which amounted to 8.61%. modeling the indicator of financial system stability with the vecm approach the vector error correction model approach is used to forecast the indicators of financial system stability in indonesia. the modeling of time series data using the vecm approach must meet two assumptions are stationary and cointegration. the following are the results of the stationarity test using the adf test. table 1. stationarity test variable p-value adf test bsi 0.2794 -2.01637 jseci 0.9148 -0.33086 inflation 0.0000 -8.21506 exchange rate 0.8217 -0.77115 the stationarity test in table 1 shows that the jseci, bsi, and exchange rate data are not stationary both in the mean and in the variance. it can be seen from table 1, the p-value of jseci, bsi, and exchange rate variables are more than the significant level (0.05). hence, box-cox transformation and differencing will be carried out. after the data is stationary, cointegration testing will be carried out using the johansen test. table 2 is the results of the cointegration test. table 2. cointegration test rank data trend none, no intercept, no trend none, intercept, no trend linear, intercept, no trend linear, intercept, trend quadratic, intercept, trend akaike information criterion 0 21,5616 21,56180 21,6652 21,6652 21,76294 1 21,4969 21,47900 21,5559 21,5460 21,61712 2 21,5246 21,443* 21,4929 21,4754 21,52122 3 21,6219 21,54704 21,5736 21,56524 21,59116 4 21,8325 21,71441 21,7144 21,71323 21,71323 rank data trend none, no intercept, no trend none, intercept, no trend linear, intercept, no trend linear, intercept, trend quadratic, intercept, trend schwarz information criterion 0 25,0225* 25,0225* 25,24960 25,24960 25,47092 1 25,20494 25,21788 25,38750 25,40850 25,57230 2 25,47978 25,45943 25,57165 25,61604 25,72360 3 25,82430 25,84212 25,89957 25,98391 26,04074 4 26,28206 26,28759 26,28759 26,41001 26,41001 forecasting financial system stability using vector error correction model approach setiawan 113 the akaike information criterion (aic) and schwarz information criterion (sic) criteria were used to test trend data from variable group data. based on table 2, the deterministic model is determined from the location of stars (*) for each indicator both sic and aic, where the best selection for both indicators is based on the smallest value of each indicator. from table 2, it can be seen that the selected deterministic models are based on the values of aic and sic, namely models without trends and without intercepts and models without trends and with intercepts. two models were obtained from modeling with the vecm approach, namely the intercept and the model without intercept. furthermore, the best model selection will be based on the value of rmse and smape. the best model will be used to forecast the sample data and predict the next 12 months of data after the testing data period. in table 3, the values of rmse and smape are presented from the results of vecm modeling. table 3. comparison of rmse and smape values from all models data model rmse smape bsi model without intercept 0,10862 107,5686 model with intercept 0,09728 103,4109 inflation model without intercept 0,25487 20,2801 model with intercept 0,24986 21,2447 jseci model without intercept 120,37740 94,7484 model with intercept 120,34530 94,5959 exchange rate model without intercept 9,52625 88,8382 model with intercept 9,32525 88,0502 table 3 shows that the best forecasting model chosen is the deterministic model with intercept and without a trend, for the five indicators of financial system stability in indonesia. it can be seen from the data in the error rate on a smaller sample than a deterministic model with no intercept and no trend. the following is a model with an intercept with bsi as an endogenous variable: ∆𝐵𝑆𝐼𝑡 = −1,33(𝐵𝑆𝐼𝑡−1 − 0,00066𝐽𝑆𝐸𝐶𝐼𝑡−1 − 5,05𝐼𝑁𝐹𝑡−1 + 0,0124𝐸𝑅𝑡−1 + 1,411) + 0,245∆𝐵𝑆𝐼𝑡−1 + 0,181∆𝐵𝑆𝐼𝑡−2 + 0,129∆𝐵𝑆𝐼𝑡−3 + 0,216∆𝐵𝑆𝐼𝑡−4 + 0,12∆𝐵𝑆𝐼𝑡−5 + 0,27∆𝐵𝑆𝐼𝑡−6 + 0,23∆𝐵𝑆𝐼𝑡−7 + 0,00002∆𝐽𝑆𝐸𝐶𝐼𝑡−1 − 0,000188∆𝐽𝑆𝐸𝐶𝐼𝑡−2 − 0,000298∆𝐽𝑆𝐸𝐶𝐼𝑡−3 − 0,000236∆𝐽𝑆𝐸𝐶𝐼𝑡−4 − 0,000078∆𝐽𝑆𝐸𝐶𝐼𝑡−5 − 0,000056∆𝐽𝑆𝐸𝐶𝐼𝑡−6 + 0,000016∆𝐽𝑆𝐸𝐶𝐼𝑡−7 − 0,307∆𝐼𝑁𝐹𝑡−1 − 0,35∆𝐼𝑁𝐹𝑡−2 − 0,282∆𝐼𝑁𝐹𝑡−3 − 0,191∆𝐼𝑁𝐹𝑡−4 − 0,219∆𝐼𝑁𝐹𝑡−5 − 0,157∆𝐼𝑁𝐹𝑡−6 − 0,11∆𝐼𝑁𝐹𝑡−7 + 0,0003∆𝐸𝑅𝑡−1 − 0,00173∆𝐸𝑅𝑡−2 + 0,000124∆𝐸𝑅𝑡−3 − 0,000826∆𝐸𝑅𝑡−4 + 0,00092∆𝐸𝑅𝑡−5 + 0,001238∆𝐸𝑅𝑡−6 + 0,00006∆𝐸𝑅𝑡−7 (5) forecasting the indicators of financial system stability with the vecm approach forecasting on indicators of financial system stability performed on training data, the data testing, and forecast in january 2019 until december 2019 using the best models were selected based on the smallest value of rmse and smape. in figure 2, time series plots from indicators of financial system stability will be presented: forecasting financial system stability using vector error correction model approach setiawan 114 figure 2. time series plot forecasting of bsi, jseci, exchange rates, and inflation figure 2 shows the results of predictions on testing data and the results of the forecast 12 months ahead for inflation, bsi, jseci, and exchange rates variables. in figure 2, it can be seen that the forecasting model using the best model for bsi, jseci, inflation rate, and exchange rate each capable of use for 12 periods, 12 periods, a period, and 12 periods forward. it is shown in the forecast data pattern testing that follows the pattern of actual data of each variable. forecast error variance decomposition variance decomposition (vd) is used to see how much changes or the shock of an indicator contributes to other indicators or to itself. this study focused more on jseci's shock contribution, inflation, and rupiah exchange rate on the endogenous variable of the banking stability index (bsi) in ten periods. results of analysis of variance decomposition of each exogenous variable on endogenous variables (bsi) as presented in table 4. table 4. variance decomposition: banking stability index period s.e. bsi jseci inflation e.r. 1 0,131 100,000 0,000 0,000 0,000 2 0,133 98,180 0,656 0,887 0,276 3 0,139 89,066 7,025 2,560 1,349 4 0,262 76,210 6,819 0,925 16,047 5 0,280 67,588 6,494 7,988 17,931 6 0,284 66,376 8,401 7,779 17,443 7 0,302 60,792 7,924 10,467 20,816 8 0,310 59,010 7,624 10,946 22,420 9 0,320 58,325 7,195 10,939 23,540 10 0,328 58,011 7,757 10,837 23,395 table 4 shows the summary of the variance decomposition analysis results for bsis from shocks given by each variable, including itself. the vd analysis in table 5 shows that in the short term, for the first month, the biggest shock is given by bsi itself, which is equal forecasting financial system stability using vector error correction model approach setiawan 115 to 100%, while for other variables, it still does not give shocks to the bsi. however, in the second period to the tenth shocks of jseci, exchange rate, and inflation against bsis tended to increase, and shocks caused by themselves decreased. in the long-term period, namely the 10th period, it can be seen that the shocks caused by the bsi itself, as well as the shocks caused by jseci, exchange rate, and inflation are 74.07%, 10.087%, 8.71%, respectively. 7.13%. the shock is caused by the shock of each variable against endogenous as shown in figure 3. figure 3. variance decomposition: bank stability index conclusions the best model is a deterministic model with intercept and without a trend. the banking stability index (bsi) affects exchange rate, jakarta stock exchange composite index, and inflation variables. the shocks or changes that occur at the bsi are responded fluctuatively by all of the exogenous variables. the result of forecasting for bank stability index, inflation, and exchange rate can be used to forecast 12 periods ahead. the jakarta stock exchange composite index (jseci) based on the rmse additive value can be used to predict for a period ahead. the results of forecast error variance decomposition show that in the long term, namely the 10th period, contributions itself getting weaker and the contribution of exogenous variables is increasing. references [1] r. siswosoemarto, d. v. hasibuan dan d. d. iskandar, intelijen ekonomi, jakarta: pt gramedia pusaka utama, 2012. [2] bank indonesia, "stabilitas sistem keuangan," 2012. [online]. available: http://www.bi.go.id. [3] bank indonesia, "bank indonesia," march 2018.. [online]. available: http://www.bi.go.id. [4] p. wijaya, "impact of internal and external shocks monetary policy on financial system stability in indonesia," faculty of economics and business, universitas airlangga, surabaya, 2015. [5] r. qoirul, "effect of banking industry competition, loan intensity, bank size, and bi forecasting financial system stability using vector error correction model approach setiawan 116 rate on financial system stability in indonesia," universitas airlangga, surabaya, 2016. [6] f. akinsola and s. ikhide, "can bank capital adequacy changes amplify the business cycle in south africa?," economic research southern africa (ersa), southern africa, 2017. [7] k. nadri, s. ebrahimi and a. fadaie, "assessment of financial stability in the banking," journal of money and economy, vol. 13, no. 4, pp. 501-523, 2018. [8] s. elijah and n. hamza, "the relationship between financial sector development and economic growth in nigeria: cointegration with structural break approach," international journal of engineering and advanced technology (ijeat), vol. 8, no. 5c, pp. 1081-1088, 2019. [9] b. t. mande, a. a. salisu, a. n. jimoh, f. dosumu, and g. h. adamu, "financial stability and income growth in emerging markets," bulletin of monetary economics and banking, vol. 23, no. 2, pp. 201 220, 2020. [10] w. enders, applied econometric time series, fourth edition, new york: john wiley & sons, 2014. [11] g. jenkin, "determinants of gdp: a vecm forecasting and granger causality analysis for eight european countries," maynooth university, national university of ireland, ireland, 2014. [12] d. gujarati and d. c. porter, basic econometrics fifth edition, new york: mcgrawhiii/irwin, 2008. [13] w. w. s. wei, time series analysis univariate and multivariate methods (second edition), usa: pearson education, 2006. the construction of patient loyalty model using bayesian structural equation modeling approach cauchy –jurnal matematika murni dan aplikasi volume 5(2) (2018), pages 73-79 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 04 april 2018 reviewed: 20 april 2018 accepted: 16 may 2018 doi: http://dx.doi.org/10.18860/ca.v5i2.5039 the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita1, ferra yanuar2*, dodi devianto3 1,2,3 department of mathematics, faculty of mathematics and natural science, andalas university, kampus limau manis, 25163, padang – indonesia * corresponding author. email: ferrayanuar@sci.unand.ac.id abstract the information on the health status of an individual is often gathered based on a health survey. patient assessment on the quality of hospital services is important as a reference in improving the service so that it can increase a patient satisfaction and patient loyalty. the concepts of health service are ofte n involve multivariate factors with multidimensional sructure of corresponding factors. one of the methods that can be used to model such these variables is sem (structural equation modeling). structural equation modelling (sem) is a multivariate method that incorporates ideas from regression, path-analysis and factor analysis. a bayesian approach to sem may enable models that reflect hypotheses based on complex theory. bayesian sem is used to construct the model for describing the patient loyalty at puskesmas in padang city. the convergence test with the history of trace plot, density plot and the model consistency test were performed with three different prior types. based on bayesian sem approach, it is found that the quality of service and patient satisfaction significantly related to the patient loyalty. keywords: bayesian methods; patient loyalty; structural equation modeling introduction health is a basic requirement of each individual. not only in developed countries that make health a top priority for communities but also in developing countries like indonesia. health problems are an important consideration because the quality of life of a community become an indicator o measure the welfare of corresponding community. in fulfilling the need for health care for communities, the government has provided health facilities, such as puskesmas (pusat kesehatan masyarakat), rsud or private hospitals. through development in the field of health is expected to further improve the degree of public health and health services can be felt by all levels of community. the information on the health status of an individual is often gathered based on a health survey. it can be used by researchers to allocate resources prudently when planning of activities which aims at improving the overall health status. patient assessment on the quality of hospital services is important as a reference in improving the service so that the creation of a patient satisfaction and create a patient loyalty. model of community loyalty in terms of three factors are service quality, patient satisfaction and patient loyalty. they are latent variables that can not be mesured directly. the measurement of latent variables needs to be calculated by several indicators or so-called indicator variables [1]. the bayesian sem approach will be applied in modeling patient loyalty on health service at puskesmas in padang city. bayesian sem method is used in this study which is an inference method that combines the current data with the data previous research (prior data) [2]. the distinguishing feature of bayesian inference is the specification of the prior distribution for the model parameters. the difficulty arises in how a researcher goes about choosing prior distributions for the model parameters. we can distinguish between two types of priors, noninformative and informative priors. in this study use the informative prior. one type of informative prior is based on the notion of a “conjugate prior” distribution, which is one that, when combined with the likelihood function, mailto:ferrayanuar@sci.unand.ac.id the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 74 yields a posterior distribution that is in the same distributional family as the prior distribution [3]. in this study, bayesian structural equation modeling is used to construct the model for describing the patient loyalty. many researchers such as lee et al [4], lee and shi [3] and ferra y et al [5] [6] proposed the use of bayesian approach in sem to overcome these problems. the bayesian sem approach is attractive since it allows the user to use the prior information for updating the current information regarding the parameters of interest. the interrelationship among the latent variables such as service quality and patient satisfaction with patient loyalty and their respective manifest variables are determined using the data found from the survey that were undertaken at puskesmas in padang city, indonesia methods data this study uses data from the survey conducted by center for local political studies and autonomy andalas university area [6]. the data is the result of a survey on the perception of the community who got the health service at selected puskesmas in padang city. the survey involving 150 respondents was conducted in march up to may 2015. all the data involved is continuous, binary and ordinal. the information gathered in the survey includes information about service quality (𝜉1), patient satisfaction (𝜉2), and patient loyalty (𝜂1), in puskesmas padang. the indicators of service quality factor are reliability (𝑋11), responsiveness (𝑋21), assurance (𝑋31), empathy (𝑋41) and tangible (𝑋51) [7], [8]. the indicators of patient satisfaction factor are the ability of the officer in serving the patient (𝑋62), the feeling of pleasure after receiving treatment (𝑋72), the service as expected (𝑋82) and the overall service satisfactory (𝑋92) [9]. the indicators of patient loyalty factor in this study are recommend the health services to others (𝑌13), often discussing the quality of service to others (𝑌23), telling positive things (𝑌33), and invite others to use the health service (𝑌43) [6], [10]. the respondents are asked about reliability, responsiveness, assurance, empathy and tangible of hospital in serving patients. their responses are measured using likert scale, 1 until 5 from strongly disagree to strongly agree. bayesian structural equation modeling structural equation modeling is composed of two main model components, the measurement model and the structural model. generally, the measurement model can be written as [4], [11] 𝑦𝑖 = λ𝜔𝑖 + 𝑖 , (1) where 𝑦𝑖 is an 𝑝 × 1 vector of indicators. describing the 𝑞 × 1 random vector of latent variables 𝜔𝑖 is distributed as 𝑁[0, φ], let 𝜔𝑖 = (𝜂𝑖 𝑇 , 𝜉𝑖 𝑇 ) be a patition of 𝜔𝑖 into 𝑞1 × 1 dependent latent vector 𝜂𝑖 , and 𝑞2 × 1 independent latent vector 𝜉𝑖 . λ is 𝑝 × 𝑞 matrices of the loading coefficients and 𝑖 is 𝑝 × 1 random vectors of the measurement errors which follow 𝑁[0, ψ ], where 𝜓 is a diagonal covariance matrix. 𝑖 and 𝜔𝑖 are independent. the structural equation for explaining the interrelationship among the latent factors 𝜂𝑖 and 𝜉𝑖 , is given by [4], [12] 𝜂𝑖 = 𝐵𝜂𝑖 + γ𝜉𝑖 + 𝛿𝑖 , (2) where 𝐵 is 𝑞1 × 𝑞1 matrices of structural parameters governing the relationship among the endogenous latent variables which is assumed to have zeros in the diagonal and γ is 𝑞1 × 𝑞2 unknown parameter matrices of regression coefficients for relating the endogenous latent variables and exogenous latent variables. 𝛿𝑖 is structural error 𝑞1 × 1 vector of disturbances which is assumed 𝑁[0, ψ𝛿 ], where 𝜓𝛿 is a diagonal covariance matrix. it is also assumed that 𝛿𝑖 is the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 75 uncorrelated with 𝜉𝑖 . since only one endogenous latent variable is involved in this study, then 𝐵𝜂𝑖 = 0, so equation (2) can be rewritten as [4] 𝜂𝑖 = γ𝜉𝑖 + 𝛿𝑖 (3) in this study, we take prior distributions for all the parameters via the following conjugate type distributions. consider conjugate type prior distributions 𝜓 𝑘 −1 ∼ 𝐺𝑎𝑚𝑚𝑎(𝛼0 𝑘 , 𝛽0 𝑘 ), (4) 𝜓𝛿𝑘 −1 ∼ 𝐺𝑎𝑚𝑚𝑎(𝛼0𝛿𝑘 , 𝛽0𝛿𝑘 ), (5) (λ𝑘 |𝜓 𝑘 −1) ∼ 𝑁(λ0𝑘 , 𝜓 𝑘 𝐻0𝑦𝑘 ), (6) (λ𝜔𝑘 |𝜓𝛿𝑘 −1) ∼ 𝑁(λ0𝜔𝑘 , 𝜓𝛿𝑘 𝐻0𝜔𝑘 ), (7) φ−1 ∼ 𝑊𝑞 [𝑅0, 𝜌0] (8) the bayesian method is used to obtain posterior distribution. the estimation process of each parameter is obtained by calculating the average posterior distribution for each model parameter. researchers can apply the maximum likelihood estimation method to derive a marginal posterior distribution. however, high dimensional integration is required. therefore, it is difficult to apply the ml estimation method in this study. so that, this problem is solved using the markov chain monte carlo (mcmc) approach [1], [13]. the estimation process using the mcmc method is performed by repeated random sampling through a full conditional posterior distribution. in this way, we can know the characteristics for each parameter without calculating or knowing how the marginal function of the parameter is. the mcmc method which is chosen in this analysis gibbs sampler. the conditional distribution are required in rhe implementation of the gibbs sampler. let 𝑌 = (𝑦1, 𝑦2, … , 𝑦𝑛 ) be the observed data matrix. ω = (𝜔1, 𝜔2, … , 𝜔𝑛 ) be the matrix of latent variables and let 𝜃 be the vector of unknown parameters in γ, ψ𝜖 , ψ𝛿 , λ and φ. the observed data 𝑌 are augmented with the latent data ω in the posterior analysis. a sufficiently large sample of (𝜃, ω) from the joint posterior ditribution (θ, ω|y) are generated by the following gibbs sampler algorithm. at the (𝑗 + 1)th iteration with current values of ω(𝑗), ψ𝜖 (𝑗), ψ𝛿 (𝑗), λ(𝑗) and φ(𝑗) [4] 1. generate ω(𝑗+1) from 𝑝(ω|ψ𝜖 (𝑗), ψ𝛿 (𝑗) , λ(𝑗), φ(𝑗), y ) 2. generate ψ𝜖 (𝑗+1) from 𝑝(ψ𝜖 |ω (𝑗+1), ψ𝛿 (𝑗), λ(𝑗), φ(𝑗), y ) 3. generate ψ𝛿 (𝑗+1) from 𝑝(ψ𝛿 |ω (𝑗+1), ψ𝜖 (𝑗+1)λ(𝑗), φ(𝑗), y ) 4. generate λ(𝑗+1) from 𝑝(λ|ω(𝑗+1), ψ𝜖 (𝑗+1), ψ𝛿 (𝑗+1), φ(𝑗), y ) 5. generate φ(𝑗+1) from 𝑝(φ|ω(𝑗+1), ψ𝜖 (𝑗+1), ψ𝛿 (𝑗+1), λ(𝑗+1), y ) based on the hypothesis, the model is described by figure 1. in figure 1, the hypothesized model which involves measurement and structural components are used to illustrate the patient loyalty model. the hypothesis model designed in this study is described by figure 1: the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 76 figure 1. illustration of patient loyalty model results and discussion in our bayesian analysis, we use the conjugate prior distribution for updating the current information on the parameter. the conjugate prior distribution for this analysis are as written in the equation (4), (5), (6), (7) and (8). the analysis bayesian sem model use the help of package program winbugs 1.4. with an iteration of 4000 times, the model parameter estimation process has reached convergent. estimated parameters are shown in the following table 1 table 1. bayesian estimates of the structural and measurement equation parameters estimate parameters 𝛾1 0,5418 𝛾2 0,4902 𝜆11 0,78399 𝜆21 0,60864 𝜆31 0,49585 𝜆41 0,7046 𝜆51 0,5267 𝜆62 0,52165 𝜆72 0,60153 𝜆82 0,67166 𝜆92 0,61533 𝜆13 0,96125 𝜆23 0,61531 𝜆33 0,55416 𝜆43 0,40913 the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 77 the results of the estimate process is presented in table 1. the estimated structural equation that address the relationship between the patient loyalty with service quality and paient satisfaction for the bayesian sem which are given by 𝜂 = 0,5418 𝜉1 + 0,4902 𝜉2 + 𝛿 it informed from the table that the effect of service quality to patient loyalty (𝛾1) is 0,5418. the effect of patient satisfaction to patient loyalty (𝛾2) is 0,4902. these estimated structural equations indicated that service quality (𝜉1) has the greatest effect on the patient loyalty (𝜂). the next step in bayesian sem approach is convergency test of convergence of model parameters that have been estimated in value. test is using history trace plot and density plot. figure 3 and figure 4 presents a trace plot and density plot for some selected parameters, 𝛾1is loading factor latent variable of patient satisfaction to patient loyalty and 𝜆11 is loading factor indicator variable reliability of latent variable service quality. figure 2. trace plot for some parameters based on figure 2 it can be concluded that the assumption of convergence is related. data distribution has been stable as it is between two parallel horizontal lines. figure 3. density plot for some parameters in figure 3, the density plot for some parameters show the normal distributed curve. this inform that the selected parameters model have convergen. thus, based on the convergence examination of the trace plot and density plot, it can be concluded that the alleged model has satisfed the criterion of convergence. the next analysis is test of the sensitivity of the algorithm which is constructed to design the bayesian sem model. in this study, we design three different prior types. the following table 3 presents the result of this sensitivity test. table 3. bayesian sem for three different prior parameters mean of some parameters type prior 1 type prior 2 type prior 3 𝛾1 0,5418 0,4138 0,8508 𝛾2 0,4902 0,4790 0,5859 𝜆11 0,78399 0,84057 0,63421 𝜆21 0,60864 0,46539 0,94161 𝜆62 0,52165 0,55141 0,45123 𝜆72 0,60153 0,48091 0,89735 𝜆23 0,61531 0,98395 0,94336 the parameter estimates obtained under various prior inputs are reasonably close. we could conclude here that the statistics found based on the bayesian sem is not sensitive to these three different prior input. accordingly, for the purpose of discussion or discussion of the results found the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 78 using bayesian sem, we will use the results obtained using type i prior. these results are provided in figure 4. figure 4. the fitted model of bayesian sem conclusions in this research, we use the analysis technique of bayesian sem approah to analyze the relationship between service quality and patient satisfaction to patient loyalty at puskesmas in padang city. quality of service and patient satisfaction are assumed as indicator variables for patient loyalty. based on data processing using software winbugs 1.4, the compiled model has been suitable used in analyzing the relationship between service quality and patient satisfaction to patient loyalty in padang city. furthermore, some types of tests are tested for convergence and consistency test on the resulting algorithm. this test aims to determine whether the estimated value obtained from the bayesian sem technique is the result of estimation obtained is accurate and the algorithm calculation has been properly compiled. the parameters fitted to model if the parameter estimation values have convergence and output of different prior types generate which is not much different. therefore, the bayesian sem model is believed to have been produce the proper and reliable value. based on the model estimation, it is found that the inuence of service quality to the patient loyalty is 0.5418 and the inuence of patient satisfaction on patient loyalty is 0.4902. thus, service quality and patient satisfaction is significantly correlated to the patient loyalty on health services and the quality of service and patient satisfaction is an indicator to measure patient loyalty at puskesmas in padang city. the construction of patient loyalty model using bayesian structural equation modeling approach astari rahmadita 79 references [1] a. ansari, k. jedidi, and s. jagpal, “a hierarchical 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[8] a. parasuraman, v. a. zeithaml, and l. l. berry, “reassessment of expectations as a comparison standard in measuring service quality : implications for further research,” journal of marketing, vol. 58, no. 1. pp. 111–124, 1994. [9] i. m. mattjik, a. a. and sumertajaya, sidik peubah ganda. bogor: institut pertanian bogor, 2011. [10] s. l. bowen, j. t. and chen, “the relationship between customer loyalty and customer satisfaction,” international journal of contemporary hospitality management. mcb up ltd, pp. 213–217, 2001. [11] k. a. bollen, structural equations with latent variables. new york: john wiley & sons, 1989. [12] j. b. ullman, “structural equation modeling: reviewing the basics and moving forward,” journal of personality assessment, vol. 87, no. 1. pp. 35–50, 2006. [13] x. y. lee, s. y. and song, “evaluation of the bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes,” multivariate behav. res., vol. 39, no. 4, pp. 653–686, 2004. gstar-x-sur model with neural network approach on residuals cauchy –jurnal matematika murni dan aplikasi volume 5(4) (2019), pages 203-210 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 10, 2018 reviewed: pebruary 11, 2019 accepted: may 31, 2019 gstar-x-sur model with neural network approach on residuals diana rosyida1, atiek iriany2, nurjannah3 1,2,3department of statistics, faculty of mathematics and natural science, university of brawijaya, malang, indonesia email: dianarosyida94@gmail.com, atiekiriany@yahoo.com, nj_anna@ub.ac.id abstract one of the models that combine time and inter-location elements is generalized space time autoregressive (gstar) model. gstar model involving exogenous variables is gstar-x model. the exogenous variables which are used in gstar-x model can be both metrical and non-metrical data. the case study of this research is the forecasting of precipitation, which is the exogenous variable is non-metrical data of precipitation intensity of a certain location. the approach of parameter estimation method employed by seemingly unrelated regression (sur) model, which can solve the correlation between residual models. now, the phenomenon of precipitation possesses patterns and characteristics is difficult to identify, and can be interpreted as a non-linear model. the non-linear model is much developed by neural network (nn). this research employed gstarx-sur modelling with neural network approach on residuals. the data used in this research were the records of 10-day precipitations in four regions in west java, namely cisondari, lembang, cianjur, and gunung mas, from 2005 to 2015. the gstarx-sur nn modelling resulted in precipitation deviation average of the forecast and the actual data at 4.1385 mm. this means that this model can be used as an alternative in forecasting precipitation. keywords: gstar-x, sur, neural network, precipitation introduction multivariate time series analysis has been implemented in forecasting since there are many problems that cannot be solved with univariate forecasting [1]. one of them is space time autoregressive (star) model that was first introduced by pfeifer and deutsch [2]. that model considers the element of time and location. the star model was developed by ruchjana [3] to be a more general model for all locations, which is known as the generalized space time autoregressive (gstar). in time series model, there is a model that contains exogenous variables, namely arimax and varimax. similarly, space time model has gstarx model. exogenous variable which is used in gstar model can be in a form of metrical data and non-metrical data. some researches on gstar using exogenous variables with metrical data are kurnia, et al [4] and astuti, et al [5], while that of exogenous variable with non-metrical data are ditago, et al [6] and suhartono, et al [7]. one of the methods in estimating the parameter in multivariate analysis is seemingly unrelated regression (sur) [8] [9]. parameter estimation of gstar model with ordinary least square method conducted by ruchjana [3] found residuals which are mostly correlated and thus the estimation value acquired is not efficient [10]. one of the non-linear phenomena is precipitation. currently, precipitation possesses patterns which are difficult to be identified and predicted. neural network mailto:dianarosyida94@gmail.com mailto:atiekiriany@yahoo.com mailto:nj_anna@ub.ac.id gstar-x-sur model with neural network approach on residuals diana rosyida 204 model is a non-linear model which has been commonly developed. some researches which implement neural network in space time model are suhartono [11] and sulistyono, et al [12]. these researches employed gstarx-sur-nn modelling aiming at acquiring accurate forecast. methods the data used in this research include 10-day precipitation data in four locations in west java, namely cisondari, lembang, cianjur, and gunung mas regions. the period of precipitation data used is from 2005 to 2015, resulting in 360 observations from each observed region. the exogenous variable used is dummy variable. there are four dummy variables employed, namely precipitation of more than 20 mm in cisondari, precipitation of more than 20 mm in lembang, precipitation of more than 25 mm in cianjur, and precipitation of more than 30 mm in gunung mas. the code of dummy variables is 1 if the data fill the criteria of four exogenous variable and 0 when the data do not. this ways is analog with the reasearch of ditago [6] uses ramadhan month criteria, in which the data in ramadhan month are given 1 and 0 for the data in other months. location value used in the gsrarx modelling is the normalization value of cross correlation initially introduced by suhartono and atok [13]. the normalization value of cross correlation taking place in the corresponding lag [14] 𝑊𝑖𝑗 = 𝑟𝑖𝑗(𝑠) ∑ |𝑟𝑖𝑘(𝑠)|𝑘≠𝑠 where 𝑖 ≠ 𝑗 and this value meets ∑ 𝑊𝑖𝑗 = 1𝑖≠𝑗 . sur model possesses the assumption 𝐸[𝑒|𝑿𝟏, 𝑿𝟐, … , 𝑿𝒎] = 0 and 𝐸[𝜺′𝜺|𝑿𝟏, 𝑿𝟐, … , 𝑿𝒎] = 𝛀 where 𝛀 is the variance covariance matrix [15]. 𝛀 = [ 𝜎11𝑰𝒎𝑻 𝜎12𝑰𝒎𝑻 … 𝜎1𝑚𝑰𝒎𝑻 𝜎21𝑰𝒎𝑻 𝜎22𝑰𝒎𝑻 … 𝜎2𝑚𝑰𝒎𝑻 ⋮ ⋮ ⋱ ⋮ 𝜎𝑚1𝑰𝒎𝑻 𝜎𝑚2𝑰𝒎𝑻 … 𝜎𝑚𝑚𝑰𝒎𝑻 ] (1) where 𝑰𝒎𝑻 as the identity matrix with (𝑚𝑇 𝑥 𝑚𝑇) size and 𝜎𝑖𝑗 is the error variance from each equation for 𝑖 = 𝑗 and covariance error between equations 𝑖 ≠ 𝑗. therefore, estimation of parameter using sur is shown in the equation below (2) �̂� = (𝑿′𝛀−𝟏𝑿)−𝟏𝑿′𝛀−𝟏𝒀 (2) essential components in the neural network model are the layer input, hidden layer, and output layer. input layer used in this case is the residual of gstar-sur model. this current research merely employs one hidden layer; however, the number of neurons used in the hidden layer is based on the lowest rmse value. the output layer employed involves four variables. resilient algorithm backpropagation is used in estimating the neural network model. this algorithm has been employed in researches by apriliyah, et al [16] and fadil, et al [17] in estimating the sales of electricity load. gstar-x-sur model with neural network approach on residuals diana rosyida 205 results and discussion the illustration of precipitation data plot in cisondari, lembang, cianjur, and gunung mas is presented below. ( a ) ( b ) ( c ) ( d ) figure 1. precipitation data plot figure 1.a is precipitation data plot in cisondari, figure 1.b for lembang, figure 1.c for cianjur, and figure 1.d for gunung mas. in figure 1, it can be seen that the precipitation data patterns in these four different locations are randomly horizontal. gunung mas region has shown higher level of precipitation compared to those in the other three regions, with the precipitation of 50 mm being the highest. the highest precipitation in cisondari is 28.15 mm; while precipitations in lembang and cianjur respectively are at 24.45 mm and 36.18 mm. the increase of precipitation in these four regions has happened in different time period and therefore it is hard to predict. precipitation data plot illustrated in figure 1 shows that the data possess big variance. therefore, box-cox transformation must be conducted in the precipitation data in the four regions. if the lambda in box-cox transformation equals to 1, the data can be classified as stationary against the variance. table 1. box-cox transformation location transformation cisondari (𝑍1𝑡 + 1) −0.16 lembang [ln(𝑍4𝑡 + 1) + 1] 0.25 cianjur (𝑍3𝑡 + 1) 0.21 gunung mas (𝑍4𝑡 + 1) 0.12 precipitation data from the transformation based on table 1 have shown to be variance stationary. it is then tested to find if the data are stationary against the average 0 5 10 15 20 25 30 1 2 7 5 3 7 9 1 0 5 1 3 1 1 5 7 1 8 3 2 0 9 2 3 5 2 6 1 2 8 7 3 1 3 3 3 9 0 10 20 30 40 1 2 5 4 9 7 3 9 7 1 2 1 1 4 5 1 6 9 1 9 3 2 1 7 2 4 1 2 6 5 2 8 9 3 1 3 3 3 7 0 5 10 15 20 25 30 1 2 5 4 9 7 3 9 7 1 2 1 1 4 5 1 6 9 1 9 3 2 1 7 2 4 1 2 6 5 2 8 9 3 1 3 3 3 7 0 10 20 30 40 50 60 1 2 5 4 9 7 3 9 7 1 2 1 1 4 5 1 6 9 1 9 3 2 1 7 2 4 1 2 6 5 2 8 9 3 1 3 3 3 7 gstar-x-sur model with neural network approach on residuals diana rosyida 206 value. dickey fuller test result has shown that the stationary variance data are stationary against the average value. this means that differencing is not necessary. gstar model order is determined by the mpacf and macf schemes; while spatial order used is order 1. ( a ) ( b ) figure 2. mpacf and macf schemes figure 2.a is of a mpacf scheme and figure 2.b is of a macf scheme. figure 2 shows that mpacf scheme cuts off at lag 1 and macf scheme possesses sinus wave pattern, so the model developed is autoregressive with lag 1 [18]. therefore, in this research, the order of gstarx-sur model used is gstarx-sur (11). neural network modelling is to be performed on the residuals from gstarx model (11). there are four inputs of input layers used, namely 𝑎1,𝑡−1, 𝑎2,𝑡−1, 𝑎3,𝑡−1, 𝑎𝑛𝑑 𝑎4,𝑡−1. the inputs are layers used by suhartono and endharta [19] as one of the inputs in neural network modelling. there are four neurons in the output layer of some variables used in gstarx-sur (11). hidden layer is limited to 1 layer, where neuron used is limited to 1 to 10 neurons. the selection of the number of neurons used in hidden layer is drawn from the lowest rmse value. table 2. the rmse value in numbers of neurons in hidden layer the number of neurons in hidden layer rmse value the number of neurons in hidden layer rmse value 1 0.1435 6 0.1346 2 0.1430 7 0.1352 3 0.1421 8 0.1403 4 0.1411 9 0.1327 5 0.1405 10 0.1303 based on table 2, 10 neurons possess the lowest rmse values. therefore, the neural network model developed is nn (4, 10, 4). the best architectural design for neural network based on the residual gstarx-sur model (11) is shown in figure 3. gstar-x-sur model with neural network approach on residuals diana rosyida 207 the mathematical equations developed from the gstarx-sur (11) – nn (4,10,4) from each location are as follows. 1. cisondari �̂�1𝑡 = 𝑍1𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍1𝑡,𝑁𝑁 �̂�1𝑡 = 0.7717 𝑍1,𝑡−1 + 0.0516 𝑍2,𝑡−1 + 0.0468 𝑍3,𝑡−1 + 0.0404 𝑍4,𝑡−1 − 0.1656 𝐷1 − 0.0786 𝐷2 𝑍1𝑡,𝑁𝑁 = −0.615 − 0.0473 𝑓(ℎ1) + 0.6355 𝑓(ℎ2) − 0.2415 𝑓(ℎ3) + 0.8345 𝑓(ℎ4) + 0.2273 𝑓(ℎ5) − 0.5837 𝑓(ℎ6) − 0.4523 𝑓(ℎ7) − 0.2578 𝑓(ℎ8) − 0.6448 𝑓(ℎ9) + 1.1062 𝑓(ℎ10) (3) 2. lembang �̂�2𝑡 = 𝑍2𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍2𝑡,𝑁𝑁 �̂�2𝑡 = 0.4027 𝑍2,𝑡−1 + 0.1928 𝑍1,𝑡−1 + 0.2077 𝑍3,𝑡−1 + 0.1976 𝑍4,𝑡−1 + 0.0994 𝐷1 + 0.0796 𝐷2 𝑍2𝑡,𝑁𝑁 = 1.1521 + 0.0785 𝑓(ℎ1) − 0.7041 𝑓(ℎ2) − 0.1386 𝑓(ℎ3) − 1.1537 𝑓(ℎ4) − 0.4983 𝑓(ℎ5) + 0.9914 𝑓(ℎ6) + 0.0946 𝑓(ℎ7) + 0.5594 𝑓(ℎ8) − 0.4155 𝑓(ℎ9) − 0.5196 𝑓(ℎ10) (4) 3. cianjur �̂�3𝑡 = 𝑍3𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍3𝑡,𝑁𝑁 �̂�3𝑡 = 0.4711 𝑍3,𝑡−1 + 0.2305 𝑍1,𝑡−1 + 0.2713 𝑍2,𝑡−1 + 0.2191 𝑍4,𝑡−1 + 0.1923 𝐷1 + 0.1197 𝐷2 + 0.2225 𝐷3 𝑍3𝑡,𝑁𝑁 = −0.9406 + 0.2818 𝑓(ℎ1) + 0.2951 𝑓(ℎ2) − 1.1130 𝑓(ℎ3) − 2.5219 𝑓(ℎ4) − 1.2281 𝑓(ℎ5) + 1.7891 𝑓(ℎ6) + 1.0893 𝑓(ℎ7) + 1.424 𝑓(ℎ8) + 0.8216 𝑓(ℎ9) + 0.7601 𝑓(ℎ10) (5) 4. gunung mas �̂�4𝑡 = 𝑍4𝑡,𝐺𝑆𝑇𝐴𝑅𝑋 + 𝑍4𝑡,𝑁𝑁 �̂�4𝑡 = 0.2083𝑍4,𝑡−1 + 0.2482 𝑍1,𝑡−1 + 0.3161 𝑍2,𝑡−1 + 0.268 𝑍3,𝑡−1 + 0.1239 𝐷1 + 0.2187 𝐷4 𝑍4𝑡,𝑁𝑁 = 0.1742 + 0.1249 𝑓(ℎ1) − 1.1336 𝑓(ℎ2) − 0.1868 𝑓(ℎ3) − 0.84 𝑓(ℎ4) − 0.777 𝑓(ℎ5) + 0.7616 𝑓(ℎ6) + 2.6908 𝑓(ℎ7) + 0.8405𝑓(ℎ8) − 0.2835 𝑓(ℎ9) − 0.4847 𝑓(ℎ10) (6) 𝑓(ℎ𝑖 ) is the activation function of logistic sigmoid in the hidden unit which is defined as follow 𝑓(ℎ𝑖 ) = 1 1 + 𝑒−(ℎ𝑖) , 𝑖 = 1,2, … ,10 (7) where, ℎ1 = −54.5045 + 179.7641 𝑎1,𝑡−1 + 491.5302 𝑎2,𝑡−1 − 35.5407 𝑎3,𝑡−1 − 316.5188 𝑎4,𝑡−1 ℎ2 = 0.3103 + 7.4173 𝑎1,𝑡−1 − 1.4297 𝑎2,𝑡−1 + 1.7059 𝑎3,𝑡−1 + 0.4799 𝑎4,𝑡−1 ℎ3 = −0.7417 − 21.0108 𝑎1,𝑡−1 − 4.3276 𝑎2,𝑡−1 − 2.0921 𝑎3,𝑡−1 + 4.6632 𝑎4,𝑡−1 ℎ4 = 0.3549 + 14.6467 𝑎1,𝑡−1 − 3.4917 𝑎2,𝑡−1 − 0.5961 𝑎3,𝑡−1 + 0.5499 𝑎4,𝑡−1 ℎ5 = 7.1522 − 29.3058 𝑎1,𝑡−1 + 128.514 𝑎2,𝑡−1 − 183.3922 𝑎3,𝑡−1 + 3.3083 𝑎4,𝑡−1 ℎ6 = 0.1041 + 11.7434 𝑎1,𝑡−1 − 8.4182 𝑎2,𝑡−1 − 0.516 𝑎3,𝑡−1 + 2.8594 𝑎4,𝑡−1 ℎ7 = −0.4036 + 0.481 𝑎1,𝑡−1 − 0.5068 𝑎2,𝑡−1 + 0.1383 𝑎3,𝑡−1 + 0.4666 𝑎4,𝑡−1 ℎ8 = 3.1722 − 10.3635 𝑎1,𝑡−1 + 64.1415 𝑎2,𝑡−1 − 84.2384 𝑎3,𝑡−1 − 2.4503 𝑎4,𝑡−1 ℎ9 = 9.2704 − 5.0167 𝑎1,𝑡−1 + 6.2685 𝑎2,𝑡−1 − 18.4776 𝑎3,𝑡−1 + 15.4932 𝑎4,𝑡−1 gstar-x-sur model with neural network approach on residuals diana rosyida 208 ℎ10 = −0.3992 − 13.9493 𝑎1,𝑡−1 − 0.9574𝑎2,𝑡−1 − 2.3135 𝑎3,𝑡−1 + 1.5081 𝑎4,𝑡−1 figure 3. the best architectural design for gstarx-sur residual model neural network (11) the already developed model is to be forecasted and compared to the actual data in order to find out whether or not the models are able to illustrate the real condition of precipitation in cisondari, lembang, cianjur, and gunung mas. ( a ) ( b ) -6.17883 3.56837 9.0 191 9 28 .1 77 57 55 7. 4 55 66 -3 2 .5 3 8 2 7 -0 .2 2 8 5 5 -0 .4 4 2 2 4 1 .1 2 4 4 1 0 .7 5 1 5 1 a4_1 -2 .8 3 5 9 9 -9 .07 985 30.0238 -21.19769 -267 .027 31 25 .9 56 14 2. 94 5 44 -0 .5 9 5 0 8 -5 .2 1 4 7 8 4 .4 0 7 9 8 a3_1 6 .8 3 2 6 -1 .1 2 1 8 7 -3 4 .3 1 3 2 -2 .59 808 171.26618 1.16981 -2.307 33-3 .21 48 5 1. 04 4 7 -2 2 .0 4 4 3 2 a2_1 -3 .8 7 5 0 1 -2 .8 3 9 2 -3 .1 2 8 9 -5 1 .8 0 5 5 9 -1 7 3 .4 4 6 9 5 88 .59 308 0.75884 0.0353 -2.05644 -2 .46 36 8 a1_1 -1 .22 63 6 -0 .7 7 8 6 4 -0 .1 3 6 2 1 0 .6 0 6 4 9 -0.814 84 -1 .2 5 47 8 -0 .5 4 1 4 7 -0 .1 8 8 9 2 -0.09376 -0 .32 54 9 -0 .0 9 4 9 3 0 .0 9 8 9 7 0.60965 1.696 50 .6 08 88 -0 .3 3 4 3 6 -0 .21 894 -0.3947 -0 .1 64 39 0 .0 6 2 8 0 .5 4 7 5 7 1.59246 0.59 8560 .3 0 90 9 -1 .0 3 8 3 4 -0 .52 779 0.50509 2. 48 74 3-2 .6 9 7 6 5 -6 .7 7 8 8 9 -2.13069 -0.1 512 8 1 .1 8 9 2 3 1 .7 6 6 7 1.047 81 1.38444 0 .2 8 6 6 8 0 .6 3 0 0 8 0 .2 2 6 6 7 -0.16043 a4 a3 a2 a1 3 .5 5 8 2 4 1 .1 3 2 9 9 -0 .7 1 1 0 3 1 0 .8 1 2 6 5 9 8 .0 5 8 2 2 -1 7 .9 9 5 9 1 -1 .7 0 1 4 8 -2 .9 5 7 8 0.747 4 -0.514111 0 .8 0 2 9 -0 .5 0 3 1 5 -0 .6 3 1 5 2 -1 .47 196 1 error: 11.983662 steps: 86824 -5 0 5 10 15 20 25 1 3 5 7 9 11131517192123252729313335 actual gstar nn 0 5 10 15 20 1 3 5 7 9 11131517192123252729313335 actual gstar nn gstar-x-sur model with neural network approach on residuals diana rosyida 209 ( c ) ( d ) figure 4. precipitation forecast plot in 2015 figure 4.a is precipitation forecast result plot of cisondari, figure 4.b, 4.c, and 4.d respectively are precipitation forecast result plots of lembang, cianjur, and gunung mas. the precipitation forecast result using gstarx-sur (11) – nn (4,10,4) possesses similar pattern to the actual data. however, the performance of gstarx-sur (11) – nn (4,10,4) model cannot be seen in detailed from the graph. the performance of gstarx-sur (11) – nn (4,10,4) model can be seen from the rmse and mad values. table 3. gstarx-sur (11) – nn (4,10,4) performance model regions rmse value mad value cisondari 5.4012 3.5767 lembang 4.9883 3.3882 cianjur 5.7111 4.2282 gunung mas 7.8792 5.3608 the result of precipitation forecast in lembang has shown the lowest rmse and mad values. the average deviation of precipitation from the forecast result and the actual data of lembang is only at 3.3882 mm. on the other hand, gunung mas has the highest rmse value and precipitation average deviation from forecast result and actual data at 5.3882 mm. conclusion the model of precipitation forecasting in cisondari, lembang, cianjur, and gunung mas that are formed with gstarx-sur (11) nn (4,10,4). based on the value of forecasting in each location and compared by the actual data, the result of forecasting is good because the average of mad value is 4.1385 mm, so gstax-sur model with a neural network approach on the side can be used as a good alternative to predict precipitation. acknowledgments acknowledgments to bmkg, the head of cisondari, chincona, cianjur and gunung mas villages so that this research can be complete and help people in the villages. 0 10 20 30 40 1 3 5 7 9 11131517192123252729313335 actual gstar nn 0 10 20 30 40 50 60 1 3 5 7 9 11131517192123252729313335 actual gstar nn gstar-x-sur model with neural network approach on residuals diana rosyida 210 references [1] j. g. de gooijer and r. j. hyndman, “25 years of time series forecasting,” int. j. forecast., vol. 22, pp. 443–473, 2006. 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[19] suhartono and a. j. endharta, “double seasonal recurrent neural networks for forecasting short term electricity load demand in indonesia,” recurr. neural networks temporal data process., 2011. the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 140-148 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 14, 2020 reviewed: october 13, 2020 accepted: november 10, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.9939 the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah1, agustin absari wahyu kuntarini 2 1,2department of mathematics, faculty of science and technology uin walisongo semarang email: khasanah.nur@walisongo.ac.id abstract the application of centrosymmetric matrix on engineering takes their part, particularly about determinant rule. this basic rule needs a computational process for determining the appropriate algorithm. therefore, the algorithm of the determinant kind of hessenberg matrix is used for computing the determinant of the centrosymmetric matrix more efficiently. this paper shows the algorithm of lower hessenberg and sparse hessenberg matrix to construct the efficient algorithm of the determinant of a centrosymmetric matrix. using the special structure of a centrosymmetric matrix, the algorithm of this determinant is useful for their characteristics. key words : hessenberg; determinant; centrosymmetric introduction one of the widely used studies in the use of centrosymmetric matrices is how to get determinant from centrosymmetric matrices. besides this special matrix has some applications [1], it also has some properties used for determinant purpose [2]. special characteristic centrosymmetric at this entry is evaluated at [3] resulting in the algorithm of centrosymmetric matrix at determinant. due to sparse structure of this entry, the evaluation of the determinant matrix has simpler operations than full matrix entries. one special sparse matrix having rules on numerical analysis and arise at centrosymmetric the determinant matrix is the hessenberg matrix. the role of hessenberg matrix decomposition is the important role of computing the eigenvalue matrix. in the discussion, the recursive algorithm is explained to compute the n-per-n determinant of the hessenberg matrix [4]. this study is the evaluation of some researches before about the determinant hessenberg matrix [5,6]. the rule of the hessenberg matrix for computing the determinant of general centrosymmetric matrix based on block matrix has been done by [7]. moreover, this study continued by [8] on evaluating the previous work and shows only lower hessenbeg matrix as block centrosymmetric matrix can use this algorithm. then necessary and sufficient condition for this algorithm is constructed also that can be evaluated for the determinant process [9]. a numerical example also shows for more understanding at a special centrosymmetric matrix on its block. furthermore, the determinant of sparse hessenberg matrix is also constructed for resulting in the new algorithm for a pentadiagonal matrix[10]. some studies are also http://dx.doi.org/10.18860/ca.v6i3.9939 the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 141 focussing on the determinant algorithm for computational process [11-17]. based on the previous study, we propose a new form of pentadiagonal matrix with construction entries based on a centrosymmetric matrix called pentadiagonal centrosymmetric matrix. we then continue applying the previous algorithm to compute our new form of pentadiagonal centrosymmetric matrix for determinant purpose. finally, this paper shows two algorithms of centrosymmetric matrix determinant by using the algorithm of the determinant of the hessenberg matrix. preliminaries before we discuss the main result, let introduce some definitions and properties used to explain the determinant of centrosymmetric matrix using hessenberg’s determinant. definition 1 [3] the form of n -byn lower hessenberg matrix is written as the matrix with the entries as follow                    nnnnnn nnnnnn hhhh hhhh hhh hh ,1,2,1, ,11,12,11,1 3,22,221 2,111 0 00 000   h . (1) definition 2 [10] the sparse hessenberg matrix with order n is the matrix with construction as                           nnnn nnnnnn nn hh hhh h hhhh hhhh hhh ,1, ,11,12,1 ,2 5,34,33,32,3 4,23,22,221 3,12,111  s . (2) definition 3 [2] let   nn nnij ra   a be a centrosymmetric matrix, if the entries of the matrix satisfy 1,1   jninij aa for ni 1 , nj 1 or                  1112,1 2122,2 ,22221 ,11211 aaa aaa aaa aaa n n n n      a . (3) the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 142 definition 4 [7] the centrosymmetric matrix has a special structure that can be constructed as aajj nn  , where  11nnn eeej ,,,  and ie is the unit vector with the i-th element 1 and others 0. definition 5 [8] the centrosymmetric matrix with n order where mn 2 is even number order can be partitioned with the form        mm mm bjjc cjjb a . (4) results and discussion this part shows the rule of the determinant of lower hessenberg and sparse hessenberg matrix for determinant concept of the centrosymmetric matrix. further discussion is also given for explaining the result and deeper understanding. determinant lower hessenberg matrix for centrosymmetric matrix’s determinant the algorithm of the determinant of lower hessenberg is presented based on the definition and properties of this matrix. the following is the lemma of the determinant of lower hessenberg, which is constructed as a block matrix. lemma 6 [7] let                                 1 0 0 00 0000 0001 0~ ,1,2,1, ,11,12,11,1 3,22,221 2,111 nnnnnn nnnnnn hhhh hhhh hhh hh      n t 1 eh e h is the partition matrix and                                 1 0 0 00 0000 0001 0~ ,1,2,1, ,11,12,11,1 3,22,221 2,111 nnnnnn nnnnnn hhhh hhhh hhh hh      n t 1 eh e h as the inverse matrix of its matrix, then the determinant of this matrix is written as          1 1 1, 1det n i ii n hhh . this algorithm is used for evaluating the algorithm of the determinant of centrosymmetric matrix with lower hessenberg as a block matrix. the algorithm of the determinant of the centrosymmetric matrix with lower hessenberg as block matrix can be proposed as follow. input : matrix        mm mm bjjc cjjb a output :  adet 1) operate                n m cjb cjb app m mt 0 0 0 0 the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 143 2) construct nm, are hessenberg matrices become          m t 1 em e m 0~ ,          m t 1 en e n 0~ 3) let inverse matrices        t m mm1 β lα m m h ~ ,         t n nn1 β lα n n h ~ 4) compute          nmp n m ph t detdetdet 0 0 detdetdet        5)              1 1 1 1 1,1, 11det m i m i iin m iim m qhghh . or we can write this algorithm by the following theorem for computing the determinant of centrosymmetric matrix with applying the algorithm of the determinant of lower hessenberg matrix. this algorithm is only applied on a specific centrosymmetric matrix based on the special block matrix. theorem 7 [9] let centrosymmetric matrix with its block matrix, lower hessenberg matrix, and its notation, such 11 nmnmph  ~ , ~ ,,,, , then the determinant of the centrosymmetric matrix is written as         1 1 1,1, det m i iiiimn qghhh . (5) proof. from the definition of the centrosymmetric matrix, this matrix can be formed as a block matrix        mm mm bjjc cjjb a . by the orthogonal matrix         mm mm jj ii p 2 2 , this block matrix can be constructed as                n m cjb cjb hpp m mt 0 0 0 0 . based on this matrix, the determinant of centrosymmetric matrix only calculate on block matrices the determinant. these block matrices nm, are hessenberg form, therefore the algorithm of determinant lower hessenberg can be applied for this determinant. this step becomes the main necessary for the next algorithm. by the algorithm of lower hessenberg matrix, let’s form          m t 1 em e m 0~ ,          m t 1 en e n 0~ , and assume         t m mm1 β lα m m h ~ ,          t n nn1 β lα n n h ~ , where nm, α , nm, l , nm, β are matrices with the size of 1n , nn  , 1n respectively and nm h , is scalar. by analytical process at lower hessenberg algorithm, then we find 0 m h and 0nh . finally, the determinant of the centrosymmetric matrix based on lower hessenberg form is          nmp n m ph t detdetdet 0 0 detdetdet        .∎ (6) the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 144 determinant sparse hessenberg matrix for centrosymmetric matrix’s determinant based on the previous study, [10] shows the efficient algorithm general pentadiagonal matrix. then, this paper gives a specific discussion on the determinant of pentadiagonal centrosymmetric matrix by applying a pentadiagonal determinant matrix. there are some basic definitions for further study on the determinant of pentadiagonal centrosymmetric matrix. definition [17,18] let   njiij d   ,1 d is called nn  general pentadiagonal matrix which the entry 0 ij d for 2 ji or it can be constructed as                             nnnnnn nnnnnnnn nn ddd dddd d ddddd dddd ddd ,1,2, ,11,12,13,1 ,2 3534333231 24232221 131211  d . (7) this performance of the general pentadiagonal matrix has been described, and moreover, the algorithm of determinant general pentadiagonal is also evaluated [10]. this paper will give a different point of view from a general structure pentadigonal matrix, which can be constructed as a centrosymmetric matrix. this work combines the general pentadiagonal matrix’s definition, which has a centrosymmetric matrix structure, then we call the pentadiagonal centrosymmetric matrix. therefore, we construct the algorithm of the pentadiagonal centrosymmetric matrix based on the algorithm of the determinant of the general pentadiagonal matrix. the algorithm has the following steps. 1. construct pentadiagonal centrosymmetric matrix based on the definition of pentadiagonal and centrosymmetric matrix, then the pentadiagonal centrosymmetric matrix is written as                             nnnnnn nnnnnnnn nn ddd dddd d ddddd dddd ddd ,1,2, ,11,12,13,1 ,2 3534333231 24232221 131211  d where the entries have a centrosymmetric structure 1,1   jninij aa for ni 1 , nj 1 . 2. transform into sperse hessenberg matrix according to the form of sparse hessenberg matrix, then the construction of sparse hessenberg matrix   njiij d   ,1 d is written as the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 145                             nnnn nnnnnn nn dd ddd d dddd dddd ddd ,1, ,11,12,1 ,2 35343332 24232221 131211  h . (8) the form of (8) matrix is resulted by choosing i1,i λ  from the matrix of                    1 ,1 1 1 1 1 in ii i i i i  λ . (9) for instance, let pentadiagonal centrosymmetric matrix with the size 88 and choose 1,2 1,3 2,3 d d  for transforming matrix as follows.                                                          888786 78777675 6867666564 5756555453 5645444342 3534333231 24232221 131211 1,2 1,3 1 1 1 1 1 1 1 1 ddd dddd ddddd ddddd ddddd ddddd dddd ddd d d dλ 3                            888786 78777675 6867666564 5756555453 5645444342 35343332 24232221 131211 ~ ddd dddd ddddd ddddd ddddd dddd dddd ddd . by the same way, taking 76 86 87 32 42 3,4 ,, d d d d    and the process will have the following hessenberg matrix as follow. the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 146                            8887 787776 68676665 57565554 56454443 35343332 24232221 131211 ~~ ~~~ ~~~ ~~~ ~~~ ~~~ dd ddd dddd dddd dddd dddd dddd ddd dλλλλ 3678  . (10) generally, the formula of the value of ii ,1  is defined as 1, 1,1 ,1 ~     ii ii ii d d  with 2121 ~ dd  . remembering that, 31     nn v is the matrix which formed by λ matrix , having the form of hvd  . consequently,          hvddvd detdetdetdetdet  , where   1det v (11) by the entries of d which integer numbers or zd i j , then it can use the following matrix                     1 1,1,1 1 1 ~ 1 in iiii i i i dd i λ . (12) furthermore, to compute the determinant of the pentadiagonal centrosymmetric matrix is only on computing the determinant of the hessenberg matrix. then, the next step is how to compute the hessenberg matrix. 3. compute the hessenberg matrix by applying the previous algorithm, this determinant is constructed by using the two-term recurrence. this step takes the following explanation. let nn  matrix          n1n t 1n1n n sr qp z , where the size of block matrices is 1n z  has    11  nn , n1n s,q  are the scalar and 1n t r  has  11  n . then the determinant of n z recursively is written as : 1 0 f 1iii fαf   , where 11 dα  , 1i 1 1i t 1iii qzrsα     then, the determinant of i z matrix is   ii fz det , where ni ,,2,1  . now, we apply the previous algorithm for our sparse hessenberg matrix, and we have the following recursive.   1nn1,n2nn2,n1n 1 1n t 1nn ,nn , ededhehdα     n the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 147   1n1n 1 1n t n1,n2n1n 1 1n t n2,n1nn ,nn , ehedehedddα       n , where e is a vector unit. by substituting the equations 2n1,n 1n 3n 2n1n 1 1n t d f f ehe        and 1n 2n 1n1n 1 1n t f f ehe       then become                 n1,n 1n 2n n2,n2n1,n 1n 3n 1nn ,nn , d f f dd f f ddα n . then, by multiply the equation with 1n f we have   n1,n2nn2,n2n1,n3n1nn,nn,1n1nnn dfddfddffαf   . 4. construct the algorithm of determinant pentadiagonal centrosymmetric matrix 111 df  1221221 dddff  2   2311321323323 dfddddff  for ni ,,5,4    iiiiiiiiiiiii dfddfddf ,12,22,131,,1  f end   n fd det . this algorithm shows the rule of the determinant of hessenberg matrix can be contructed for the general determinant of the pentadiagonal centrosymmetric matrix. based on two different algorithms above, the rule of determinant hesseberg matrix can evaluate the determinant of the centrosymmetric matrix. for efficient algorithm, the first algorithm evaluates the determinant of centrosymmetric matrix applying the algorithm of the determinant of the lower hessenberg matrix. it happens caused by the lower hessenberg matrix appear as its block matrix, then the algorithm of centrosymmetric matrix only a half working. on the next algorithm, the algorithm of the sparse hessenberg matrix is applied to construct the algorithm of determinant general pentadiagonal centrosymmetric matrix. this algorithm is used caused by the same structure of the main matrix is a sparse hessenberg matrix, therefore is applicable. conclusions the different cases of the application of determinant hesseberg matrix are applied to a different form of a centrosymmetric matrix . it is necessary to identify the form of the main general centrosymmetric from the beginning. this part determines the best algorithm for the computational process more efficient. this paper shows the different hessenberg appearance on a centrosymmetric matrix results different algorithms of the computing of the determinant. the rule of hessenberg matrix for computing the determinant of centrosymmetric matrices nur khasanah 148 references [1] datta and morgera, "on the reducibility of centrosymmetric matrices-aplication in engineering problems", circuits system signal process., 8 (1) pp. 71-95, 1989. 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[17] r.a. sweet, a recursive relation for the determinant of a pentadiagonal matrix, comm. acm 12 (1969) 330–332. cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an article in a peer-reviewed cauchy is an essential building block in the development of a coherent and respected network of knowledge. it is a direct reflection of the quality of the work of the authors and the institutions that support them. 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equipment that have any unusual hazards inherent in their use, the author must clearly identify these in the manuscript. disclosure and conflicts of interest all authors should disclose in their manuscript any financial or other substantive conflict of interest that might be construed to influence the results or interpretation of their manuscript. all sources of financial support for the project should be disclosed. fundamental errors in published works when an author discovers a significant error or inaccuracy in his/her own published work, it is the author’s obligation to promptly notify the journal editor or publisher and cooperate with the editor to retract or correct the paper. cauchy jurnal matematika murni dan aplikasi volume 5, issue 3, november 2018 issn : 2086-0382 e-issn : 2477-3344 acknowledgment to reviewers in this issue contributions and valuable comments of the following reviewers in this issue was very appreciated abdussakir (universitas islam negeri maulana malik ibrahim malang) fachrur rozi (universitas islam negeri maulana malik ibrahim malang) sri harini (universitas islam negeri maulana malik ibrahim malang) usman pagalay (universitas islam negeri maulana malik ibrahim malang) cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 102-111 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 28 august 2018 reviewed: 08 october 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5511 svir epidemic model with non constant population joko harianto1, titik suparwati2 1,2mathematics department, cenderawasih university email: joharijpr88@gmail.com abstract in this article, we present an svir epidemic model with deadly deseases and non constant population. we only discuss the local stability analysis of the model. initially the basic formulation of the model is presented. two equilibrium point exists for the system; disease free and endemic equilibrium point. the local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. if the value of 𝑅0 less than one then the desease free equilibrium point is locally asymptotically stable, and if its exceeds, the endemic equilibrium point is locally asymptotically stable. the numerical results are presented for illustration. keywords: svir model, basic reproduction number, numerical results introduction health problems are closely associated with a disease. it is motivating humans to study health as well as illness. one of the goals of a nation to achieve prosperity in the future is the creation of a healthy living environment. one of the indicators of a healthy environment involves a high level of quality health services. quality health care issues need to be addressed systematically, effectively and efficiently. analysis of the spread of a disease is one step to address the problem of quality health services. the spread of disease due to virus or bacteria that enter into the body will lead to human health disorders and will affect the socio-economic development of the community. efforts to prevent a spreading disease can work optimally if several stages such as research, development of various diagnostic tools, drugs, and new vaccines, has achieved. mathematical models are one of a useful tool for reviewing or analyse the patterns of a disease spread. according to [2] and [4] the spread of infectious diseases can be described mathematically through a model, such as sir and sirs. the most basic procedure in modelling the spread of disease is by using the compartment model. in this case, the population is divided into three different classes namely, susceptible, infected, and recovered which then shortened to sir. vaccination can be considered as the addition of a class naturally into the epidemic base model for some types of diseases. along with the development of knowledge in the field of health, control of an epidemic disease can be done by vaccination action. then the sir model develops with the addition of a vaccinated population class. some researchers have proposed models of the dynamics of vaccination. among others is svi type, which has been developed by [5], [6] and [8]. several studies have stated that the threshold to determine the occurrence of endemic disease seen from the value of basic reproduction number (𝑅0). the disease distribution model will be locally stable if 𝑅0 ≤ 1 and the disease will be present if 𝑅0 > 1. paper [9] discusses the svir epidemic model for non-lethal diseases that does not involve the individual's mortality mailto:joharijpr88@gmail.com svir epidemic model with non constant population joko harianto 103 rate due to the disease. from these studies, we can build some further assumptions related to the svir epidemic model that developed in public life. in this article we consider local stability analysis of an svir epidemic model with deadly disease and non constant population based on svir epidemic model developed in [8]. we hope the model can be useful to analyse the disease outbreaks so that we can take optimal precautions. methods this research used literature study with literature sources from some reputable journal. the simulation of model svir used mathematical software, i.e. maple 2016. results and discussion the following are assumptions used in modelling: 1. closed population (no migration) 2. births occur in every sub populations and enter in vulnerable subpopulations. 3. natural birth occurs on each sub populations at the same rate 4. natural death occurs on each sub populations at the different rate. 5. illness can cause death. 6. individuals who have cured cannot return to susceptive class (permanent cure). 7. short incubation period. 8. vaccination has given to susceptible subpopulations. 9. we do not distinguish the rate of recovery for a child has been infected and for the adult. 10. vaccination will reach the level of immunity over time and finally enters the sub populations that heals. 11. individuals who have vaccinated become infected if they lose immunity. 12. contact between infected and susceptive individuals can lead to disease transmission, and so are contact between infected and those have vaccinated. we investigate the basic model formulation by dividing the total population into four compartments, such as s(t) for susceptible, v(t) for vaccinated, i(t) for infected and r(t) for recovered individuals. let 𝛽 is the rate of contact that is sufficient to transmit the disease. we also assume a constant recovery rate 𝛾. the rate at which the susceptible population is vaccinated is 𝛼. we assume that there can be disease related death and define 𝜔 to be the rate of disease related death, while 𝜇 is the rate of natural death that is not related to the disease. let 𝜇1 be the natural death rate for susceptible population, 𝜇2 be the natural death rate for vaccination population, 𝜇3 be the natural death rate for infected population, 𝜇4 be the natural death rate for recovery population. we assume that all newborns enter the susceptible population at the constant rate of 𝜇. let 𝛾1 be the average rate for susceptible individuals to obtain immunity and moved into recovered population. we do not distinguish the natural immunity and vaccine-induced immunity here because vaccine-induced immunity can also last for a long term. we assume that before obtaining immunity the vaccinees still have possibility of infection with a disease transmission rate 𝛽1 while contacting with infected individuals. 𝛽1 may be assume to be less than 𝛽 because the vaccinating individuals may have some partial immunity during the process or they may recognize the transmission characters of the disease and hence decrease the effective contacts with infected individuals. figure 1 shows the details of the population transfer diagram of the svir epidemic model. svir epidemic model with non constant population joko harianto 104 figure 1. transfer diagram of the svir epidemic model with deadly deseases and non constant population the following form presents the system model of differential equations. 𝑑𝑆 𝑑𝑡 = 𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 (1) 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 𝑑𝑅 𝑑𝑡 = 𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 where 𝑆(0) > 0,𝑉(0) > 0,𝐼(0) > 0,𝑅(0) = 0, with 𝑆(0)+𝑉(0)+𝐼(0)+𝑅(0) = 1, and all parameters are positive. the parameters with their description are presented in the following table. table 1. parameter description parameter description 𝜇 birth rate 𝛽 the transmission rate of disease when susceptible individuals contact with infected individuals 𝛼 the rate of migration of susceptible individuals to the vaccination process 𝛽1 the transmission rate for vaccination individuals to be infected before gaining immunity 𝛾1 the average rate for vaccination individuals to obtain immunity and move into recovered population 𝜔 the death rate due to disease 𝛾 recovery rate 𝜇1 natural death rate of the susceptible population 𝜇2 natural death rate of the vaccination population 𝜇3 natural death rate of the infected population 𝜇4 natural death rate of the recovery population 𝜇 𝛾𝐼 s 𝜇1𝑆 𝜇3𝐼 𝜇4𝑅 𝜔𝐼 𝛽𝑆𝐼 i r 𝛽1𝑉𝐼 𝛼𝑆 v 𝜇2𝑉 𝛾1𝑉 svir epidemic model with non constant population joko harianto 105 equilibrium point as a first step, we define equilibrium points of the system (disease free equilibrium point and endemic equilibrium point). in general, according to [1] and [3] there exist two equilibrium points of epidemic model, i.e. the disease free equilibrium point and the endemic equilibrium point. the two equilibrium points are reviewed based on the existence of the disease in a population with a continuous time. the disease-free equilibrium point is the point at which the disease is unlikely to spread in an area because the infected population is equal to zero (i=0) for 𝑡 → ∞. while the endemic equilibrium point is the point at which the disease must spread (i>0) for 𝑡 → ∞, in a defined closed area. the equilibrium point of system (1) exists when 𝑑𝑆 𝑑𝑡 = 0 , 𝑑𝑉 𝑑𝑡 = 0 , 𝑑𝐼 𝑑𝑡 = 0 , 𝑑𝑅 𝑑𝑡 = 0 so that: 𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 (2) 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 = 0 (3) 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 = 0 (4) 𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 = 0 (5) from equation (4), we have 𝐼(𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −𝜔) = 0, and the solutions are 𝐼 = 0 or 𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −𝜔 = 0, consequently, we have to situation, namely: a) situation at 𝑰 = 𝟎 it is the necessary condition for the disease-free equilibrium point. note that,  from equation (2), we have 𝜇 −𝜇1𝑆 −𝛼𝑆 = 0 ⟺ 𝑆 = 𝜇 𝛼+𝜇1  from equation (3), we have 𝛼𝑆 −𝛾1𝑉 −𝜇2𝑉 = 0 ⟺ 𝑉 = 𝛼𝑺 (𝛾1+𝜇2) = 𝛼𝝁 (𝜶+𝝁𝟏)(𝜸𝟏+𝝁𝟐)  from equation (5), we have 𝛾1𝑉 +𝛾𝐼 −𝜇4𝑅 = 0 ⟺ 𝑅 = 𝛾1𝑽 𝜇4 = 𝛾1𝛼𝜇 𝜇4(𝛼+𝜇1)(𝛾1+𝜇2) so we get the disease-free equilibrium, that is, 𝐸0 = (𝑆0,𝑉0, 𝐼0,𝑅0) = ( 𝜇 𝛼 +𝜇1 , 𝛼𝜇 (𝛼 +𝜇1)(𝛾1 +𝜇2) ,0 , 𝛾1𝛼𝜇 𝜇4(𝛼 +𝜇1)(𝛾1 +𝜇2) ) b) situation at 𝑰 > 0, this situation is a necessary condition for an endemic equilibrium point. note that:  from equation (2), we have 𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 = 0 ⟺ 𝑆 ∗ = 𝝁 (𝜶+𝝁𝟏+𝜷𝐼 ∗)  from equation (3), we have 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 = 0 ⟺ 𝑉 ∗ = 𝛼𝝁 (𝜶+𝝁𝟏+𝜷𝐼 ∗)(𝜸𝟏+𝝁𝟐+𝜷𝟏𝐼 ∗) from equation (4), if 𝐼 ≠ 0, then 𝛽𝑆 +𝛽1𝑉 −𝛾 −𝜇3 −𝜔 = 0 , consequently, with substitute 𝑆∗ and 𝑉∗ we have: 𝛽𝜇 (𝛼 +𝜇1 +𝛽𝐼 ∗) + 𝛽1𝛼𝜇 (𝛼 +𝜇1 +𝛽𝐼 ∗)(𝛾1 +𝜇2 +𝛽1𝐼 ∗) −(𝜇3 +𝛾 +𝜔) = 0 ⟺ (𝜇3 +𝛾 +𝜔)− 𝛽𝜇 (𝛼 +𝜇1 +𝛽𝐼 ∗) − 𝛼𝜇𝛽1 (𝛼 +𝜇1 +𝛽𝐼 ∗)(𝛾1 +𝜇2 +𝛽1𝐼 ∗) = 0 ⟺ (𝜇3 +𝛾 +𝜔)− 𝛽𝜇 (𝛼 +𝜇1 +𝛽𝐼 ∗) − 𝛽1𝛼𝜇 𝑘 = 0 ⟺ (𝜇3 +𝛾 +𝜔)𝑘 −𝛽𝜇(𝛾1 +𝜇2 +𝛽1𝐼 ∗)−𝛼𝜇𝛽1 = 0 with 𝑘 = (𝛼 +𝜇1 +𝛽𝐼 ∗)(𝛾1 +𝜇2 +𝛽1𝐼 ∗) = 𝛽𝛽1𝐼 ∗ +[(𝛼 +𝜇1)𝛽1 +(𝛾1 +𝜇2)𝛽]𝐼 ∗ −𝛽1𝛼𝜇 and then we have svir epidemic model with non constant population joko harianto 106 𝐴1 𝐼 ∗2 +𝐴2𝐼 ∗ +𝐴3 (1−( 𝛽𝜇 (𝛼 + 𝜇 1 )(𝜇 3 + 𝛾 + 𝜔) + 𝛽1𝛼𝜇 (𝛼 + 𝜇 1 )(𝛾 1 + 𝜇 2 )(𝜇 3 + 𝛾 + 𝜔) )) = 0 where: 𝐴1 = (𝜇3 + 𝛾 + 𝜔)𝛽1𝛽 > 0 𝐴2 = (𝜇3 + 𝛾 + 𝜔)[( 𝛼 +𝜇1) 𝛽1 + (𝛾1 + 𝜇2)𝛽]− 𝛽1𝛽𝜇 𝐴3 = (𝜇3 + 𝛾 + 𝜔) ( 𝛼 +𝜇)( 𝛾1 + 𝜇) > 0 let 𝐶 = 𝛽𝜇 (𝛼 + 𝜇 1 )(𝜇 3 + 𝛾 + 𝜔) + 𝛽1𝛼𝜇 (𝛼 + 𝜇 1 )(𝛾 1 + 𝜇 2 )(𝜇 3 + 𝛾 + 𝜔) , so, the above equation becomes: 𝐴1𝐼 ∗2 +𝐴2𝐼 ∗ +𝐴3(1−𝐶) = 0 (6) with the roots of equation (6), i.e. 𝐼1,2 ∗ = −𝐴2 ±√𝐴2 2 −4𝐴1𝐴3(1−𝐶) 2𝐴1 in this case, the proportion of the infected population must be positive. so, the existence of the endemic equilibrium point of infected population depends on the value of c. from equation (6) we can have positive root (𝐼∗ > 0) if 𝐶 > 1. the value of a positive endemic equilibrium point lies in the value of c, so that it can be defined the value basic reproduction number, that is, 𝑅0 = 𝛽𝜇 (𝛼 +𝜇1)(𝜇3 +𝛾 +𝜔) + 𝛽1𝛼𝜇 (𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +𝛾 +𝜔) in epidemiology, the basic reproduction number shows about the disease spread and control. if 𝑅0 < 1, then the disease free equilibrium is stable, the disease dies out from population. if 𝑅0 > 1 the endemic equilibrium exists, and disease permanently exist in the population. it is observed that 𝐼∗ exist if 𝑅0 > 1. thus, we obtain an endemic equilibrium point if 𝑅0 > 1 , that is, 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗,𝑅∗) = ( 𝜇 (𝛼 +𝜇1 +𝛽𝐼 ∗) , 𝛼𝜇 (𝛼 +𝜇1 +𝛽𝐼)(𝛾1 +𝜇2 +𝛽1𝐼 ∗) , 𝐼∗, 𝛾1𝑉 ∗ +𝛾𝐼∗ 𝜇4 ). the existence of disease free equilibrium point is not depends on 𝑅0 condition. thus we conclude that, if 𝑅0 ≤ 1, then there is exist one (unique) equilibrium of system (1), that is, the disease-free equilibrium point 𝐸0 . furthermore, if 𝑅0 > 1, then there are two equilibrium points of system (1), that is, the disease-free equilibrium point 𝐸0 and the endemic equilibrium point 𝐸∗. local stability analysis of equilibrium point we will determine the stability of the equilibrium point of the svir model around the disease-free equilibrium and the endemic equilibrium point. next, we will study system dynamics in equation (1). since the last equation does not depend on other equations, we simply study the following system. 𝑑𝑆 𝑑𝑡 = 𝜇 −𝜇1𝑆 −𝛽𝑆𝐼 −𝛼𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 −𝛽1𝑉𝐼 −𝛾1𝑉 −𝜇2𝑉 (7) svir epidemic model with non constant population joko harianto 107 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 +𝛽1𝑉𝐼 −𝛾𝐼 −𝜇3𝐼 −𝜔𝐼 furthermore, we performed a svir model stability analysis around the equilibrium point of the system (7). to discuss the local stability of system (7), we compute the jacobian matrix of system (7). the signs of the real parts of the eigenvalues of the jacobian matrix evaluated at a given equilibrium determine its stability. the entries of general jacobian matrix are given by differentiating the right hand side of system (7) with respect to 𝑆,𝑉,𝐼. the disease free stability and the endemic stability are presented in the following results. theorem 1 defined: 𝑅0 = 𝛽𝜇 (𝛼 + 𝜇 1 )(𝜇 3 + 𝛾 + 𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +𝛾 +𝜔) i. if 𝑅0 < 1, then the equilibrium point 𝐸0 is locally asymptotically stable and if 𝑅0 > 1, then the equilibrium point 𝐸0 is unstable. ii. if 𝑅0 > 1, then the endemic equilibrium 𝐸 ∗ is locally asymptotically stable. proof: (i). the jacobian matrix of system (2) is 𝐽(𝑆,𝑉,𝐼) = ( −(𝛼 +𝜇1 +𝛽1𝐼) 0 −𝛽𝑆 𝛼 −(𝛾1 +𝜇2 +𝛽1𝐼) −𝛽1𝑉 𝛽𝐼 𝛽1𝐼 𝛽𝑆 +𝛽1𝑉 −(𝜇3 +𝛾 +𝜔) ) so the jacobian matrix around the point 𝐸0 = (𝑆0,𝑉0, 𝐼0) = ( 𝜇 𝛼+𝜇1 , 𝛼𝜇 (𝛼+𝜇1)(𝛾1+𝜇2) ,0) is 𝐽(𝐸0) = ( −(𝛼 +𝜇1) 0 −𝛽𝑆0 𝛼 −(𝛾1 +𝜇2) −𝛽1𝑉0 0 0 𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔) ) the characteristic equation is ⟺ [𝜆 +(𝛼 +𝜇1)][𝜆 +(𝛾1 +𝜇2)][ 𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔)] = 0 where 𝜆1 = −(𝛼 +𝜇1) < 0 𝜆2 = −(𝛾1 +𝜇2)< 0 𝜆3 = 𝛽𝑆0 +𝛽1𝑉0 −(𝜇3 +𝛾 +𝜔) = (𝜇3 +𝛾 +𝜔)(𝑅0 −1) it is clear if 𝑅0 < 1 then all the eigenvalues of 𝐽(𝐸0) are negative, hence the point 𝐸0 is locally asymptotically stable. meanwhile, if 𝑅0 > 1, then there is a positive eigenvalue, hence the point 𝐸0 is unstable. (ii). it is clear that the endemic point 𝐸∗ exist when 𝑅0 > 1, so the jacobian matrix around the point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is 𝐽(𝐸∗) = ( −(𝛼 +𝜇1 +𝛽𝐼 ∗) 0 −𝛽𝑆 𝛼 −(𝛾1 +𝜇2 +𝛽1𝐼 ∗) −𝛽1𝑉 𝛽𝐼∗ 𝛽1𝐼 ∗ 𝛽𝑆∗ +𝛽1𝑉 ∗ −(𝜇3 +𝛾 +𝜔) ) we can modify the element 𝑎11,𝑎22, 𝑎33, from matrix above into 𝛼 +𝜇1 +𝛽𝐼 ∗ = 𝜇 𝑆∗ 𝛾1 +𝜇2 +𝛽1𝐼 ∗ = 𝛼𝑆∗ 𝑉∗ based on the evaluation results around the endemic equilibrium point we are getting 𝛽𝑆∗ +𝛽1𝑉 ∗ −𝜇3 −𝛾 −𝜔 = 0, and the matrix become svir epidemic model with non constant population joko harianto 108 𝐽(𝐸∗) = ( − 𝜇 𝑆∗ 0 −𝛽𝑆∗ 𝛼 − 𝛼𝑆∗ 𝑉∗ −𝛽1𝑉 ∗ 𝛽𝐼∗ 𝛽1𝐼 ∗ 0 ) the characteristic equation of matrix 𝐽(𝐸∗) is 𝜆3 +𝑎1𝜆 2 +𝑎2𝜆 +𝑎3 = 0 (8) where 𝑎1 = 𝜇 𝑆∗ + 𝛼𝑆∗ 𝑉∗ > 0 𝑎2 = 𝛼𝜇 𝑉∗ +𝛽1 2 𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗ > 0 𝑎3 = 𝛼𝛽1𝛽𝑆 ∗𝐼∗ + 𝛼𝛽2𝑆∗ 2 𝐼∗ 𝑉∗ + 𝜇𝛽1 2 𝑉∗𝐼∗ 𝑆∗ > 0 hence, 𝑎1𝑎2 −𝑎3 = ( 𝜇 𝑆∗ + 𝛼𝑆∗ 𝑉∗ )( 𝛼𝜇 𝑉∗ +𝛽1 2 𝑉∗𝐼∗ + 𝛽2𝑆∗𝐼∗)−(𝛼𝛽1𝛽𝑆 ∗𝐼∗ + 𝛼𝛽2𝑆∗ 2 𝐼∗ 𝑉∗ + 𝜇𝛽1 2 𝑉∗𝐼∗ 𝑆∗ ) = 𝛼𝜇2 𝑆∗𝑉∗ +𝜇𝛽2𝐼∗ + 𝛼2𝜇𝑆∗ 𝑉∗ 2 +𝛼𝛽1 2𝑆∗𝐼∗ −𝛼𝛽1𝛽𝑆 ∗𝐼∗ = 𝛼𝜇2 𝑆∗𝑉∗ +(𝛼 +𝜇1 +𝛽𝐼 ∗)𝛽2𝑆∗𝐼∗ + 𝛼2𝜇𝑆∗ 𝑉∗ 2 +𝛼𝑆∗𝐼∗(𝛽 −𝛽1) 2 +𝛼𝛽1𝛽𝑆 ∗𝐼∗ > 0. it can be seen that all of the coefficients 𝑎𝑖 > 0,𝑖 = 1,2,3 are positive. the routh-hurwitz criteria that are necessary and sufficient for the local asymptotic stability of the endemic equilibrium are that the coefficient are positive and the hurwitz determinants 𝐻𝑖 are all positive [7]. for a third degree polynomial, the hurwitz determinants criteria are 𝐻1 = 𝑎1 > 0, 𝐻2 = 𝑎3 > 0 and 𝐻3 = 𝑎1𝑎2 −𝑎3 > 0. by routh-hurwitz criteria, roots of equation (7) have negative real parts if 𝑎1𝑎2 −𝑎3 > 0. note that all term in 𝑎1𝑎2 −𝑎3 is always positive. thus, the routh-hurwitz criteria satisfied for all parameter values. therefore, by routh-hurwitz criteria the endemic equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗) is locally asymptotically stable. based on the results, we will interpret the results biologically as follows,  if 𝑅0 < 1, then for 𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸0 = (𝑆0,𝑉0, 𝐼0), the solution of system (1) will reach the value 𝐸0 = (𝑆0,𝑉0, 𝐼0). it means that if 𝑅0 < 1, then for the number of susceptible, vaccinated and infected individuals close to 𝐸0 = (𝑆0,𝑉0, 𝐼0), the disease will not plague and tends to disappear indefinitely time. this condition is then called asymptotically stable around the equilibrium point 𝐸0 = (𝑆0,𝑉0, 𝐼0), and we interpret as a disease-free equilibrium point.  if 𝑅0 > 1, then for 𝑡 → ∞ and (𝑆,𝑉,𝐼) that close enough to 𝐸 ∗ = (𝑆∗,𝑉∗, 𝐼∗), the solution of system (1) will reach the value 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗). this means if 𝑅0 > 1, then for the number of susceptible, vaccinated and infected individuals close to 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), the disease will plague but does not reach extinction in an infinite time. this condition is then called asymptotically stable around the equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗), and we interpret as an endemic equilibrium point. numerical results to substantiate the above analitycal findings, the model is studied numerically. we will discuss the results of the model by taking numerical example. we consider the parameters 𝜇 = 0,03, 𝜇1 = 0,04 , 𝜇2 = 0,035 , 𝜇3 = 0,06, 𝜇4 = 0,045 , 𝛽 = 0,5, 𝛽1 = 0,05, 𝛾 = 0,1 ,𝛾1 = 0,1 , 𝛼 = 0,2 , 𝜔 = 0,1 and the basic reproduction number is svir epidemic model with non constant population joko harianto 109 𝑅0 = 0,28 < 1. the diseases free equilibrium point for this set of parameters are (0.125,0.185,0 ,0.41). the numerical results are shown in figure 1 and 2. figure 1, illustrate the fact, when the basic reproduction number less than unity, the infected proportion will reach to zero so that the desease free equilibrium point is locally asymptotically stable. figure 2. basic reproduction number 𝑅0 versus infected proportion 𝐼(𝑡) figure 3. the dynamical behavior of system (1), when 𝑅0 < 1 and the initial conditions are 𝑆(0) = 0.5 ,𝑉(0) = 0.3 , 𝐼(0) = 0.2 ,𝑅(0) = 0. we consider the parameters 𝜇 = 0,03, 𝜇1 = 0,04 , 𝜇2 = 0,035 , 𝜇3 = 0,06, 𝜇4 = 0,045 , 𝛽 = 0,5, 𝛽1 = 0,05, 𝛾 = 0,1 ,𝛾1 = 0,1 , 𝛼 = 0,02 , 𝜔 = 0,1 and the basic reproduction number is 𝑅0 = 1,17 > 1. the endemic equilibrium point for this set of parameters are (0.315,0.046,0.07 ,0.26). the numerical results are shown in figure 3. svir epidemic model with non constant population joko harianto 110 figure 4. the dynamical behavior of system (1), when 𝑅0 > 1 and the initial conditions are 𝑆(0) = 0.5 ,𝑉(0) = 0.3 , 𝐼(0) = 0.2 ,𝑅(0) = 0. conclusions from the results of endemic equilibrium point analysis, we define the basic reproduction number parameter, that is, 𝑅0 = 𝛽𝜇 (𝛼 + 𝜇 1 )(𝜇 3 + 𝛾 + 𝜔) + 𝛼𝛽1𝜇 (𝛼 +𝜇1)(𝛾1 +𝜇2)(𝜇3 +𝛾 +𝜔) furthermore, 𝑅0 is a necessary condition for the existence of the two points of equilibrium as well as its local stability. when 𝑅0 ≤ 1, there is only one (unique) equilibrium point of disease-free, that is, 𝐸0 = (𝑆0,𝑉0, 𝐼0,𝑅0) = ( 𝜇 𝛼+𝜇1 , 𝛼𝜇 (𝛼+𝜇1)(𝛾1+𝜇2) ,0 , 𝛾1𝛼𝜇 𝜇4(𝛼+𝜇1)(𝛾1+𝜇2) ). conversely, if 𝑅0 > 1, then there are two equilibrium points, that is, 𝐸0 and an endemic equilibrium point 𝐸∗ = (𝑆∗,𝑉∗, 𝐼∗,𝑅∗) = ( 𝜇 (𝛼+𝜇1+𝛽𝐼 ∗) , 𝛼𝜇 (𝛼+𝜇1+𝛽𝐼)(𝛾1+𝜇2+𝛽1𝐼 ∗) , 𝐼∗, 𝛾1𝑉 ∗+𝛾𝐼∗ 𝜇4 ) where 𝐼∗ is the positive root of the equation 𝐴1𝐼 2 +𝐴2𝐼 +𝐴3(1−𝑅0) = 0. results of local stability analysis indicates if 𝑅0 < 1, then the disease free equilibrium point 𝐸0 is locally asymptotically stable. it means if the terms 𝑅0 ≤ 1 fulfilled, then in a long time, there would have no spread of disease in susceptible and vaccinated subpopulations, or in other words, the epidemic will stop. conversely, if 𝑅0 > 1 then the endemic equilibrium point e is locally asymptotically stable. it means, in a long time, the disease will always exist in the population with the proportion of each subpopulation is equal to 𝑆∗,𝑉∗, 𝐼∗and 𝑅∗. svir epidemic model with non constant population joko harianto 111 references [1] diekmann,o., and heesterbeek,j.a.p., mathematical epidemiology of infectious diseases: model building, analysis, and interpretation, john wiley and sons, chichester, 2000. [2] driessche, p.v.d., watmough, j., "reproduction number and sub-threshod endemic equilibria for compartmental models of disease transmission," mathematical biosciences 180, pp. 29-48, 2002. http://dx.doi.org/10.1016/s0025-5564(02)00108-6 [3] h.w. hethcote, j.w.v. ark, epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, math. biosci. 84 (1987) 85 [4] kermack, w. o. and mckendrick, a. g., "a contribution to the mathematical theory of epidemics,"proc. roy. soc. lond., a 115, pp. 700-721, 1927. http://dx.doi.org/10.1098/rspa.1927.0118 [5] khan, muhammad altaf, saeed islam and others, "stability analysis of an svir epidemic model with non-linear saturated incidence rate," applied mathematical sciences, vol. 9, no. 23, .1145-1158, 2015. http://dx.doi.org/10.12988/ams.2015.41164 [6] kribs-zaleta, c., velasco-hernandez, j., "a simple vaccination model with multiple endemic state," mathematical biosciences, vol. 164, pp. 183-201, 2000. [7] lancaster, p.,"theory of matrices", academic press, new york, 1969. [8] liu, xianing, and others, "svir epidemic models with vaccination strategies," journal of theoretical biology 253, 1-11, 2007. http://dx.doi.org/10.1016/j.jtbi.2007.10.014 [9] zaman, g., and others, "stability analysis and optimal vaccination of an sir epidemic model," bio systems 93, pp. 240-249, 2008. http://dx.doi.org/10.1016/j. biosystems.2008.05.004 cauchy jurnal matematika murni dan aplikasi volume 6, issue 2, may 2020 issn : 2086-0382 e-issn : 2477-3344 publication etics journal cauchy is a peer-reviewed electronic national journal. this statement clarifies ethical behaviour of all parties involved in the act of publishing an article in this journal, including the author, the chief editor, the editorial board, the peer-reviewer and the publisher (mathematics department of maulana malik ibrahim state islamic university of malang). this statement is based on cope’s best practice guidelines for journal editors. ethical guideline for journal publication the publication of an 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http://www.cosmosimpactfactor.com/page/journals_details/943.html http://globalimpactfactor.com/cauchy/ http://www.sindexs.org/journallist.aspx http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 6, issue 1, november 2019 issn : 2086-0382 e-issn : 2477-3344 table of contents a discrete numerical solution of the sir model with horizontal and vertical transmission ............................................................................................................................... 1 – 9 on the local adjacency metric dimension of generalized petersen graphs .......... 10 – 17 a finitely generated module over a valuation domain .................................................. 18 – 26 modification of chaos game with rotation variation on a square .............................. 27 – 33 the study geometry fractals designed on batik motives .............................................. 34 – 39 on the local edge antimagic coloring of corona product of path and cycle ........ 40 – 48 http://ejournal.uin-malang.ac.id/index.php/math/article/view/6413 http://ejournal.uin-malang.ac.id/index.php/math/article/view/6413 http://ejournal.uin-malang.ac.id/index.php/math/article/view/6487 http://ejournal.uin-malang.ac.id/index.php/math/article/view/6798 http://ejournal.uin-malang.ac.id/index.php/math/article/view/6936 http://ejournal.uin-malang.ac.id/index.php/math/article/view/8081 http://ejournal.uin-malang.ac.id/index.php/math/article/view/8054 issn 2086-0382 e-issn 2477-3344 cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 cauchy vol. 5 no. 4 pages: 169 – 215 malang may 2018 issn 2086-0382 e-issn 2477-3344 𝜒 cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 cauchy is a mathematical journal published twice a year on may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. this journal includes research papers, literature studies, analysis, and problem solving in mathematics (algebra, analysis, statistics, computing and applied mathematics). editorial board editor in chief : dr. sri harini, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia. managing editor : mohammad jamhuri, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia. juhari, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia. editorial board : mario rosario guarracino, computational and data science laboratory high performance computing and networking institute national research council of italy, italy kartick chandra mondal, jadavpur university, salt lake campus, india rowena alma l. betty, university of the philippines diliman, philippines subanar seno, gadjah mada university, indonesia toto nusantara, state university of malang, indonesia edy tri baskoro, institut teknologi bandung, indonesia eridani eridani, airlangga university, indonesia abdul halim abdullah, university of technology malaysia, malaysia kusno, university of jember, indonesia slamin, university of jember, indonesia riswan efendi, uin sultan syarif kasim riau, indonesia arief fatchul huda, uin sunan gunung djati bandung, indonesia usman pagalay, universitas islam negeri maulana malik ibrahim malang, indonesia abdussakir, universitas islam negeri maulana malik ibrahim malang, indonesia ari kusumastuti, universitas islam negeri maulana malik ibrahim malang, indonesia fachrur rozi, universitas islam negeri maulana malik ibrahim malang, indonesia cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 editorial board assistant editor : mohammad nafie jauhari, m.si, universitas islam negeri maulana malik ibrahim malang, indonesia. editorial office mathematics department, maulana malik ibrahim state islamic university of malang gajayana st. 50 malang, east java, indonesia 65144 phone (+62) 81336397956, faximile (+62) 341 558933 e-mail: cauchy@uin-malang.ac.id cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 focus and scope cauchy-jurnal matematika murni dan aplikasi is a mathematical journal published twice a year in may and november by the mathematics department, faculty of science and technology, maulana malik ibrahim state islamic university of malang. we we lc om e a u t h or s for original articles (research), review articles, interesting case reports, special articles illustrations that focus on the mathematics pure and applied. subjects suitable for publication include, but are not limited to the following fields of: 1. actuaria 2. algebra 3. analysis 4. applied 5. computing 6. econometry 7. statistics cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 indexing and abstracting cauchy-jurnal matematika murni dan aplikasi has been covered (indexed and abstracted) by following services: 1. doaj (2016-,)(https://doaj.org/toc/2477-3344) 2. d i m e n s i o n s 3. moraref (2015-,)-(http://moraref.or.id/index.php/browse/index/36) 4. onesearch indonesia (2015-,)-(http://onesearch.id/search/results?filter[]=repoid:ios2732) 5. mendeley (2013-,)-(https://www.mendeley.com/groups/5034091/cauchy/papers/) 6. indonesian scientific journal database (isjd) (2013-,)-(http://isjd.pdii.lipi.go.id/index.php/direktorijurnal.html) 7. google scholar (2009-,)-(https://scholar.google.co.id/citations?hl=en&view_op=list_works&gmla=ajsn f6omofbk7q0o2q-9 xuimca1zi8oz9lp2ehctubhl9dcisxnyh9saieau0g0udt8tym6jk3z666zu46vrsbyz6vjc2a_w&user=dr k-5hkaaaaj) 8. ipi (2009-,)-(http://id.portalgaruda.org/?ref=browse&mod=viewjournal&journal=5272) http://moraref.or.id/index.php/ http://onesearch.id/search/results http://www.mendeley.com/groups/5034091/cauchy/papers/ http://isjd.pdii.lipi.go.id/index.php/ http://id.portalgaruda.org/ cauchy jurnal matematika murni dan aplikasi volume 5, issue 4, may 2019 issn : 2086-0382 e-issn : 2477-3344 table of contents problem of maximum matching in non-bipartite graph using edmonds’ cardinality matching algorithm and its application in the battle of britain case ................................................................................................................................................. 169 – 180 application of rectangular matrices: affine cipher using asymmetric keys ......... 181 – 185 stability of cancerous chemotherapy model with obesity effect ............................... 186 – 194 mathematical modelling on transportation method apllication for rice distribution cost optimization ........................................................................................... 195 – 202 gstar-x-sur model with neural network approach on residuals .......................... 203 – 210 modification of the curve and the surface polynomial bezier using de casteljau algorithm ..................................................................................................................................... 211 – 215 cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 49-57 p-issn: 2086-0382; e-issn: 2477-3344 submitted: august 18, 2019 reviewed: november 12, 2019 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.7544 cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono1, hartawati2, ni wayan suryawardhani3 , atiek iriany3 , aniek iriany2 1 faculty of fisheries and marine science, brawijaya university, indonesia 2 department of agrotechnology, faculty of agriculture and animal husbandry, university of muhammadiyah malang, indonesia 3 department of statistics faculty of mathematics and natural sciences, brawijaya university, indonesia email: agus.dwi.sulistyono@gmail.com abstract the use of location weights on the formation of the spatio-temporal model contributes to the accuracy of the model formed. the location weights that are often used include uniform location weight, inverse distance, and cross-correlation normalization. the weight of the location considers the proximity between locations. for data that has a high level of variability, the use of the location weights mentioned above is less relevant. this research was conducted with the aim of obtaining a weighting method that is more suitable for data with high variability. this research was conducted using secondary data derived from 10 daily rainfall data obtained from bmkg karangploso. the data period used was january 2008 to december 2018. the points of the rain posts studied included the rain post of the blimbing, karangploso, singosari, dau, and wagir regions. based on the results of the research forecasting model obtained is the gstar ((1), 1,2,3,12,36) -sur model. the cross-covariance model produces a better level of accuracy in terms of lower rmse values and higher r2 values, especially for karangploso, dau, and wagir areas. keywords: cross-covariance; gstar model; rainfall; spatio-temporal introduction there are several spatio-temporal models that have been developed. for the first time, space-time autoregressive (star) model was introduced by pfeifer & deutsch [1], [2]. the space-time autoregressive (star) model had the assumption that the variance between locations is the same/homogeneous. however, in fact, it often gets heterogeneity between observation sites. thus the star model is not suitable for data that has heterogeneous location characteristics. this is the weakness of the star model and this weakness can be handled by the generalized space-time autoregressive (gstar) and gstar-ols models developed by borovkova, lopuha, & ruchjana [3] and ruchjana [4], [5]. the gstar model developed is used for data that meets stationary assumptions. the latest development of this spatio-temporal model is the gstar-sur model developed by iriany [6] to overcome data that is not stationary and has a seasonal pattern. furthermore, the use of the gstar-sur-nn hybrid model was also developed for data that has a nonlinear pattern [7]–[9] the use of location weights in the formation of spatiotemporal models also http://dx.doi.org/10.18860/ca.v6i2.7544 mailto:agus.dwi.sulistyono@gmail.com cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 50 contributes to the level of accuracy of the model formed. there are some types of location weights that are used to build models, there are uniform location weight, inverse distance, and cross-correlation normalization [10], [11]. the weight of the location considers the proximity between locations. for data that has a high level of variability, the use of the location weights mentioned above is less relevant. therefore, we need location weights that consider the various aspects of observational data. one of the location weights that have been developed is the weight of the variance ratio which has proven to have a better level of accuracy [12]. the weight of other locations developed is the weight of crosscovariance. the use of the weight of cross-covariance has been researched and applied in the research of apanosovich and genton [13] to predict pollution in california and the research of efromovich & smirnova [14] for fmri imaging processes with a wavelet approach. this research was conducted to determine the accuracy of the gstar model that was built using the weight of cross-covariance and compare the level of accuracy with the gstar model that was built with the weight of cross-correlation. methods the data period used is january 2008 to december 2018, where data for conducting a training (in-sample) is data from january 2008 to december 2017. while data from january 2018 to december 2018 is used as testing data (out-of-sample). the first step taken is testing the stationary data on rainfall. the stationary test on the average is done using the augmented dickey-fuller test. while the stationarity test for the variance was carried out by the box-cox test. the next step is to identify the real macf and mpacf lags to determine the order that will be used as an estimate of the gstar model. next, the cross-correlation normalization weighting is calculated with equation 1 [11]: (1) (1) ij ij ik k i r w r    (1) and the normalization weight of cross-covariance is calculated with equation 2 [13], [14] :     22 1 1 1 ( ) ( ) ( ) ( ) ( ) n n n i i j j ij i i j j t k t t z t z z t k z r k z t z z t k z                             (2) the next process is gstar-ols analysis to get the residual value with equation 3  01 11( ) ( )(t) t (t 1) t   μ φ φz w z  (3) next, calculate the var ( ) ε ω matrix with the equation 𝚺 = [ σ11 σ12 … σ1𝑚 σ21 σ22 … σ2𝑚 ⋮ σ𝑚1 ⋮ σ𝑚2 ⋱ … ⋮ σ𝑚𝑚 ] (4) the next step is estimating the gstar (1, p) -sur parameter using the formula ( ) -1 -1 -1 β x'ω x x'ω y . the best model is chosen based on rmse and r2 prediction values. the research data analysis process was carried out using r and sas software. cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 51 results and discussion this research was carried out by taking daily rainfall data obtained from the rain heading point for the blimbing, karangploso, singosari, dau, and wagir regions. the following is a description of the statistics of rainfall data in the five locations presented in table 1: table 1. description of rainfall data statistics in five research locations location n mean (mm) standard deviation (mm) minimum (mm) maximum (mm) blimbing 360 5.682 6.909 0 33.5 singosari 360 3.93 5.575 0 41.75 karangploso 360 4.302 5.71 0 25.36 dau 360 4.564 5.825 0 36.38 wagir 360 7.08 8.187 0 43.63 based on table 1 above, it is descriptively shown that the average rainfall in wagir district is the highest and singosari district has the lowest average rainfall. in all study locations, the standard deviation value was greater than the average, indicating a high level of rainfall variation in all study locations. in addition, the heterogeneity of the observation location can be measured by calculating the gini index. the higher the index value, the more heterogeneous the location will be. this index calculation for the five locations in this study is: 𝐺𝑛 = 1 + 1 𝑛 − 2 𝑛2𝑦�̅� ∑ 𝑦𝑖 𝑁 𝑖=1 = 1 + 1 360 − 2 36025.111 9200.68 = 0.975 based on the results of the gini index calculation, the gini index value is 0.975, close to 1. from the gini index calculation, it is shown that heterogeneous locations so that modeling using the gstar-sur model can be done. stationary testing of variance was carried out using a box-cox plot. the stationarity of variance is said to be fulfilled if the box-cox plot results in a value of λ = 1. however, if the value of λ ≠ 1, then the data transformation process is carried out. the following are the results of stationary testing of the variance in rainfall data for each location: table 2. stationary testing of the variance of rainfall in each location location λ transformation i transformation λ blimbing -0.27 z-0.27 1.0 singosari -0.68 z-0.68 1.0 karangploso -0.50 z-0.5 1.0 dau -0.50 z-0.5 1.0 wagir -0.19 z-0.19 1.0 based on table 4.3 the initial λ values for all study locations have not been worth 1. this shows that the rainfall data in each location is not yet stationary in variety so that box-cox transformation is needed. the box-cox i transformation results show the value cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 52 of λ = 1, which means that the data has been stationary to the variety and the transformation is stopped. in addition to stationary variety, stationary testing is also carried out on the average. stationary to average testing was carried out using the augmented dickey-fuller (adf) test. stationarity on the average is said to be fulfilled if the results of the adf test obtained the p-value of less than 0.05. if the adf test results obtained the p-value of more than 0.05, it is necessary to do a differencing process. following are the results of the adf test: table 3. stationary testing on the average rainfall of each location location t-statistics p-value blimbing -5.974 0.000 singosari -5.556 0.000 karangploso -7.831 0.000 dau -7.215 0.000 wagir -7.425 0.000 based on the results of the stationary test on the average using the adf test in table 3, at each location p-value was less than 0.05. from this test, it is shown that the stationary data of rainfall on the average has been fulfilled. the gstar model identification process is done by looking at the matrix partial autocorrelation function (mpacf) scheme. table 4. matrix partial autocorrelation scheme (mpacf) based on the mpacf matrix scheme in table 4 it can be seen that there is a real mpacf lag in lag 1 to lag 3. then in lag 4, there is no significant partial autocorrelation. then in 5 lags and so on there are some significant partial autocorrelations. based on the mpacf scheme, it is shown that significant partial autocorrelation is truncated at lag 4. so, the determination of the var order (p) is done by looking at the smallest aic value for real lag. the following is the aic value in lag 1 to lag 3: table 5. aic values for gstar order determination order aic value 1 14.67906 2 14.52868 3 14.52445 based on the aic value in table 5 it can be seen that the lowest aic value is obtained in the 3rd order. thus, the gstar model used has a 3rd order. in addition to determining the order with the aic value, identification of the gstar model is also carried out by univariate acf and pacf plots at each location. based on the acf plot it is shown that rainfall data at each location is indicated by seasonal patterns. this can be seen in the acf cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 53 plot which has a repetitive pattern at a certain time lag. based on the pacf plot in appendix 8 shows that at some time lag there is a pacf that crosses the 5% boundary line. when combined in 5 locations, it was found that the five locations had pacf that passed 12 and 36 of time lags. therefore, the results of the identification of seasonal patterns indicated that the appropriate model was gstar ((1), 1,2,3,12,36) this study uses five locations with 𝑛 𝑖 (1) or the number of locations adjacent to the ith location is 4 locations so that the cross-correlation normalization matrix is as follows: 0 0.2353 0.2294 0.2688 0.2665 0.2664 0 0.2143 0.2675 0.2518 0.2734 0.2100 0 0.2765 0.2401 0.2950 0.2174 0.2327 0 0.2549 0.2694 0.2235 0.2420 0.2650 0 while the magnitude of cross-covariance normalization weighting based on calculations is as follows: the results of the estimation of the parameters of the gstar ((1), 1,2,3,12,36) -sur model with the weight of the cross correlation normalization location for blimbing district are as follows: �̂�1t = 0.119 z1(t-1) + 0.009 z2(t-1) 0.009 z3(t-1) + 0.022 z4(t-1) + 0.017 z5(t-1) + 0.205 z1(t-2) + 0.087 z2(t-2) + 0.084 z3(t-2) + 0.009 z4(t-2) + 0.075 z5(t-2) 0.086 z1(t-3) 0.074 z2(t-3) + 0.037 z3(t-3) + 0.023 z4(t-3) + 0.005 z5(t-3) + 0.059 z1(t-12) 0.056 z2(t-12) 0.01 z3(t-12) + 0.019 z4(t12) 0.025 z5(t-12) + 0.246 z1(t-36) + 0.052 z2(t-36) 0.093 z3(t-36) + 0.034 z4(t-36) + 0.069 z5(t36) while the result of the parameter estimation of the gstar ((1), 1,2,3,12,36) -sur model with cross-covariance normalization location weights is as follows : �̂�1t = 0.116 z1(t-1) + 0.003 z2(t-1) 0.006 z3(t-1) + 0.022 z4(t-1) + 0.015 z5(t-1) + 0.212 z1(t-2) + 0.083 z2(t-2) + 0.081 z3(t-2) + 0.018 z4(t-2) + 0.072 z5(t-2) 0.092 z1(t-3) 0.066 z2(t-3) + 0.031 z3(t-3) + 0.025 z4(t-3) + 0 z5(t-3) + 0.06 z1(t-12) 0.048 z2(t-12) 0.016 z3(t-12) + 0.016 z4(t-12) 0.023 z5(t-12) + 0.256 z1(t-36) + 0.045 z2(t-36) 0.085 z3(t-36) + 0.025 z4(t-36) + 0.068 z5(t-36) the result of parameter estimation of gstar ((1), 1,2,3,12,36) -sur model with the weight of cross-correlation normalization location for singosari subdistrict is as follows: �̂�2t = 0.024 z1(t-1) + 0.171 z2(t-1) 0.009 z3(t-1) + 0.02 z4(t-1) + 0.012 z5(t-1) + 0.017 z1(t-2) 0.255 z2(t-2) + 0.083 z3(t-2) + 0.009 z4(t-2) + 0.052 z5(t-2) + 0.055 z1(t-3) + 0.649 z2(t-3) + 0.036 z3(t3) + 0.021 z4(t-3) + 0.004 z5(t-3) 0.017 z1(t-12) + 0.335 z2(t-12) 0.01 z3(t-12) + 0.018 z4(t-12) 0.018 z5(t-12) + 0.031 z1(t-36) 0.084 z2(t-36) 0.092 z3(t-36) + 0.031 z4(t-36) + 0.048 z5(t-36) while the result of the parameter estimation of the gstar ((1), 1,2,3,12,36) -sur model with cross-covariance normalization location weights is as follows : �̂�2t = 0.024 z1(t-1) + 0.206 z2(t-1) 0.006 z3(t-1) + 0.021 z4(t-1) + 0.01 z5(t-1) + 0.013 z1(t-2) 0.281 z2(t-2) + 0.081 z3(t-2) + 0.016 z4(t-2) + 0.05 z5(t-2) + 0.055 z1(t-3) + 0.639 z2(t-3) + 0.03 z3(t-3) + 0 0.2060 0.2056 0.2458 0.3426 0.2754 0 0.1831 0.2331 0.3084 0.2847 0.1764 0 0.2427 0.2962 0.3058 0.1818 0.1993 0 0.3130 0.3085 0.2065 0.2290 0.2559 0 wij = wij = cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 54 0.023 z4(t-3) + 0 z5(t-3) 0.017 z1(t-12) + 0.301 z2(t-12) 0.015 z3(t-12) + 0.015 z4(t-12) 0.016 z5(t-12) + 0.025 z1(t-36) 0.04 z2(t-36) 0.084 z3(t-36) + 0.023 z4(t-36) + 0.047 z5(t-36) the result of parameter estimation of gstar ((1), 1,2,3,12,36) -sur model with the weight of cross-correlation normalization location for karangploso subdistrict is as follows: �̂�3t = 0.033 z1(t-1) + 0.008 z2(t-1) + 0.411 z3(t-1) + 0.023 z4(t-1) + 0.016 z5(t-1) + 0.023 z1(t-2) + 0.081 z2(t-2) 0.318 z3(t-2) + 0.01 z4(t-2) + 0.072 z5(t-2) + 0.076 z1(t-3) 0.069 z2(t-3) 0.056 z3(t-3) + 0.024 z4(t-3) + 0.005 z5(t-3) 0.023 z1(t-12) 0.052 z2(t-12) + 0.029 z3(t-12) + 0.02 z4(t-12) 0.024 z5(t-12) + 0.042 z1(t-36) + 0.048 z2(t-36) + 0.752 z3(t-36) + 0.035 z4(t-36) + 0.066 z5(t-36) while the result of the parameter estimation of the gstar ((1), 1,2,3,12,36) -sur model with cross-covariance normalization location weights is as follows : �̂�3t = 0.033 z1(t-1) + 0.003 z2(t-1) + 0.407 z3(t-1) + 0.023 z4(t-1) + 0.014 z5(t-1) + 0.018 z1(t-2) + 0.078 z2(t-2) 0.348 z3(t-2) + 0.019 z4(t-2) + 0.069 z5(t-2) + 0.076 z1(t-3) 0.061 z2(t-3) 0.028 z3(t-3) + 0.026 z4(t-3) + 0 z5(t-3) 0.023 z1(t-12) 0.045 z2(t-12) + 0.067 z3(t-12) + 0.017 z4(t-12) 0.022 z5(t-12) + 0.035 z1(t-36) + 0.042 z2(t-36) + 0.744 z3(t-36) + 0.026 z4(t-36) + 0.065 z5(t-36) the result of parameter estimation of gstar ((1), 1,2,3,12,36) -sur model with the weight of cross-correlation normalization location for dau subdistrict is as follows: �̂�4t = 0.033 z1(t-1) + 0.009 z2(t-1) 0.009 z3(t-1) + 0.253 z4(t-1) + 0.016 z5(t-1) + 0.023 z1(t-2) + 0.089 z2(t-2) + 0.086 z3(t-2) + 0.13 z4(t-2) + 0.069 z5(t-2) + 0.076 z1(t-3) 0.075 z2(t-3) + 0.038 z3(t-3) + 0 z4(t-3) + 0.005 z5(t-3) 0.023 z1(t-12) 0.057 z2(t-12) 0.01 z3(t-12) 0.119 z4(t-12) 0.023 z5(t-12) + 0.042 z1(t-36) + 0.053 z2(t-36) 0.096 z3(t-36) + 0.169 z4(t-36) + 0.064 z5(t-36) while the result of the parameter estimation of the gstar ((1), 1,2,3,12,36) -sur model with cross-covariance normalization location weights is as follows : �̂�4t = 0.034 z1(t-1) + 0.003 z2(t-1) 0.006 z3(t-1) + 0.25 z4(t-1) + 0.014 z5(t-1) + 0.018 z1(t-2) + 0.085 z2(t-2) + 0.084 z3(t-2) + 0.083 z4(t-2) + 0.066 z5(t-2) + 0.077 z1(t-3) 0.067 z2(t-3) + 0.032 z3(t3) 0.008 z4(t-3) + 0 z5(t-3) 0.023 z1(t-12) 0.049 z2(t-12) 0.016 z3(t-12) 0.113 z4(t-12) 0.021 z5(t-12) + 0.035 z1(t-36) + 0.046 z2(t-36) 0.087 z3(t-36) + 0.203 z4(t-36) + 0.063 z5(t-36) the result of parameter estimation of gstar ((1), 1,2,3,12,36) -sur model with the weight of cross-correlation normalization location for wagir subdistrict is as follows: �̂�5t = 0.031 z1(t-1) + 0.009 z2(t-1) 0.01 z3(t-1) + 0.023 z4(t-1) + 0.243 z5(t-1) + 0.022 z1(t-2) + 0.092 z2(t-2) + 0.092 z3(t-2) + 0.01 z4(t-2) + 0.07 z5(t-2) + 0.072 z1(t-3) 0.078 z2(t-3) + 0.04 z3(t-3) + 0.024 z4(t-3) + 0.048 z5(t-3) 0.022 z1(t-12) 0.059 z2(t-12) 0.011 z3(t-12) + 0.02 z4(t-12) + 0.047 z5(t-12) + 0.04 z1(t-36) + 0.055 z2(t-36) 0.102 z3(t-36) + 0.035 z4(t-36) + 0.206 z5(t-36) while the result of the parameter estimation of the gstar ((1), 1,2,3,12,36) -sur model with cross-covariance normalization location weights is as follows : �̂�5t = 0.032 z1(t-1) + 0.003 z2(t-1) 0.007 z3(t-1) + 0.023 z4(t-1) + 0.252 z5(t-1) + 0.017 z1(t-2) + 0.088 z2(t-2) + 0.089 z3(t-2) + 0.018 z4(t-2) + 0.071 z5(t-2) + 0.072 z1(t-3) 0.07 z2(t-3) + 0.034 z3(t-3) + 0.026 z4(t-3) + 0.061 z5(t-3) 0.022 z1(t-12) 0.051 z2(t-12) 0.017 z3(t-12) + 0.017 z4(t-12) + 0.042 z5(t-12) + 0.033 z1(t-36) + 0.047 z2(t-36) 0.093 z3(t-36) + 0.026 z4(t-36) + 0.204 z5(t-36) the following is a plot to predict rainfall data in each location: cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 55 figure 1. actual rainfall and prediction of gstar ((1), 1,2,3,12,36) -sur with the weight of cross-correlation and cross-covariance examination of the accuracy of the gstar ((1), 1,2,3,12,36) -sur models was done by calculating the rmse and r2 prediction values in the model with the weighing location of the normalized cross-correlation and cross-covariance. the lower the rmse value and the higher the r2 prediction value, the better the accuracy of the gstar ((1), 1,2,3,12,36) sur model in generating the forecast value. following is the examination of the accuracy of the gstar ((1), 1,2,3,12,36) -sur models presented in table 6: table 6. accuracy examination of gstar ((1), 1,2,3,12,36) -sur location cross-correlation weight model cross-correlation weight model rmse data training rmse data testing r2 prediction rmse data training rmse data testing r2 prediction blimbing 5.796 10.471 0.579 5.779 10.433 0.558 singosari 0.609 0.599 karangploso 0.707 0.720 dau 0.565 0.595 wagir 0.328 0.336 based on the results of the accuracy of the gstar ((1), 1,2,3,12,36) -sur models in table 6, the gstar ((1), 1,2,3,12,36) -sur models that use correlation weights cross has rmse training data value of 5.796 and rmse on testing data is 10.471. whereas in the gstar ((1), 1,2,3,12,36) -sur model which uses the cross-covariance weight, model, the rmse value of training data is 5,779 and rmse testing data is 10,433. if the rmse values cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 56 in the two models are compared, the rmse values of the two models are relatively the same. besides being done by calculating the rmse value, checking the accuracy of the model is also done by calculating the r2 prediction value at each location. as shown in table 6, r2 prediction values on gstar ((1), 1,2,3,12,36) -sur models that use cross-covariance weights, are higher than r2 prediction in models with cross-correlation weights, except in locations blimbing and singosari districts. conclusions the cross-covariance model produces a better level of accuracy in terms of lower rmse values and higher r2 values, especially for karangploso, dau, and wagir areas. acknowledgments we would like to thank the university of muhammadiyah malang and brawijaya university for funding and support of this research. references [1] p. e. pfeifer and s. j. deutsch, “identification and interpretation of first-order spacetime arma models,” technometrics, 1980. [2] p. e. pfeifer and s. j. deutsch, “seasonal space-time arima modeling,” geogr. anal., 1981. [3] borovkova, lopuha, and b. n. ruchjana, “generalized s-tar with random weights,” in proceeding of the 17th international workshop on statistical modeling, 2002. [4] b. n. ruchjana, “study on the weight matrix in the space-time autoregressive model,” in proceeding of the tenth international symposium on applied stochastic models and data analysis (asmda), 2001, pp. 789–794. [5] b. n. ruchjana, “pemodelan kurva produksi minyak bumi menggunakan model generalisasi star,” bogor, 2001. [6] a. iriany, suhariningsih, b. n. ruchjana, and setiawan, “prediction of precipitation data at batu town using the gstar (1,p) -sur model,” j. basic appl. sci. res., vol. 3, no. 6, pp. 860–865, 2013. [7] a. d. sulistyono, w. h. nugroho, r. fitriani, and a. iriany, “hybrid model gstarsur-nn for precipitation data,” cauchy, vol. 4, no. 2, p. 74, 2016. [8] a. d. sulistyono, w. hadi nugroho, and a. iriany, “development of hybrid model gstar-sur-nn and application for rainfall forecasting,” in 1st international conference pure applied resources univ. muhammadiyah malang, 2015, p. 104. [9] a. iriany, w. m. firdaus, w. h. nugroho, and a. d. sulistyono, “rainfall forecasting using gstar-sur-nn approach in west java province,” in international conference on science, engineering, built environment, and social science, 2016, p. 1. [10] suhartono and r. m. atok, “pemilihan bobot lokasi yang optimal pada model gstar,” in prosiding konferensi nasional matematika xiii, 2006. [11] suhartono and subanar, “the optimal determination of space weight in gstar model by using cross-correlation inference,” j. quant. methods, vol. 2, no. 2, pp. 45– 53, 2006. [12] a. d. sulistyono, w. h. nugroho, and a. iriany, “location weight of gstar model for heterogeneity variance of precipitation data,” in international conference on cross-covariance weight of gstar-sur model for rainfall forecasting in agricultural areas agus dwi sulistyono 57 science, engineering, built environment, and social science, 2016, p. 6. [13] t. v. apanasovich and m. g. genton, “cross-covariance functions for multivariate random fields based on latent dimensions,” biometrika, vol. 97, no. 1, pp. 15–30, 2010. [14] s. efromovich and e. smirnova, “statistical analysis of large cross-covariance and cross-correlation matrices produced by fmri images,” j. biom. biostat., vol. 05, no. 02, 2013. pyramid population prediction using age structure model cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 66-76 p-issn: 2086-0382; e-issn: 2477-3344 submitted: february 22, 2020 reviewed: march 09, 2020 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.8859 pyramid population prediction using age structure model heni widayani1, nuning nuraini2, anita triska3 1,2,3department of mathematics, faculty of mathematics and natural sciences, institut teknologi bandung email: heni.w@s.itb.ac.id, nuning@math.itb.ac.id, a.triska@students.itb.ac.id abstract population composition in a country by sex and age-structure often illustrated through the population pyramid. in this study, an age-structure model will be constructed to predict the population pyramid shape in the coming year. it is assumed that changes in population are affected by natality and mortality number in each age group, ignoring migration rates. the proposed age structure model formulated as a first-order partial differential equation with the non-negative initial condition. the boundary condition is given by the number of births which is proportional to the number of women at childbearing age. then, this age structure model implemented utilizing united nations data to predict population pyramids of indonesia, brazil, japan, the usa, and russia. the population pyramid prediction of the five countries shows different characteristics, according to whether it is a developing or developed country. the results of this study indicate that the age structure model can be used to predict the composition of the population in a country in the next few years. indonesia is predicted to be the highest populated country in 2066, compared to the other four countries. this result can be used as a reference for the government to plan policies and strategies according to age groups to control population explosion in the future. keywords: population pyramid; prediction; age-structure introduction the study of population characteristics which we called demography can be complex. population pyramids are a useful tool for understanding the structure and composition of populations because they graphically portray many aspects of a population, such as sex ratios and age structure [1]. structured population models distinguish among individuals according to characteristics such as age to determine the birth and death rates. age is one of the most important characteristics in the modeling of populations [2]. individuals of different ages may have different reproduction and survival capacities. therefore, it is often useful to divide a population into different groups by age, for obtaining a better description of the population dynamics. mathematical models of populations incorporating spatial structure, age structure, or another structuring of individuals with continuously varying properties have been developed by many researchers [3]. nevertheless, these models are mathematically complex and their scientific applicability depends on extensive parametric inputs. lealramirez [4] proposed a discrete age-structured population model within a fuzzy cellular structure. russell [5] shows that a fluctuating of mortality and reproductive rate gives a cumulative extinction probability analytically. abia [6] considers the numerical solution http://dx.doi.org/10.18860/ca.v6i2.8859 mailto:heni.w@s.itb.ac.id mailto:nuning@math.itb.ac.id mailto:a.triska@students.itb.ac.id pyramid population prediction using age structure model heni widayani 67 of age-structured population models using characteristic curves methods and finite difference methods. euler suggested to modeled the growth of the human population geometrically [7]. this idea was developing by malthus [8] who predicted the exponential growth of the human population through time. this model assumed that the population depended on constant fertility and mortality rates. denoted that the constant fertility rate as  , the constant mortality rate as  , and the size of the human population in time as ( )u t then the population dynamics was determined by the malthus law 𝑑𝑈(𝑡) 𝑑𝑡 = (𝛼 − 𝜇)𝑈(𝑡) (1) verhulst [9] modified the malthus model became the famous logistic model by introducing a maximum size 𝐾 for the population size which can be written as 𝑑𝑈(𝑡) 𝑑𝑡 = (𝛼 − 𝜇)(1 − 𝑈(𝑡) 𝐾 )𝑈(𝑡) (2) this logistic model was applied by lotka [10] to fit the growth curve of the united states population in the thirties. these experiments provided growth curves that agreed with the prediction of the logistic equations. all of this model cannot capture the composition age of the population, since it just a function of time. for demography purposes, we also need to know the age composition of the population as an important issue. the importance of population age-structured prediction of a country due to the government can plan the right strategies based on age-group for the people welfare. the specific regulation with targeting a specific age group expected to be more efficient. for young people, the government can focus on improving nutrition and health so that infant or child mortality can be reduced. for adolescents, policies related to education become the primary. socialization about the negative side of promiscuity and drugs also become a major issue. for adulthood, economic policies become the main issues because this age is a productive age in which the individual became an economic milestone in a country. methods age modeled as a continuous variable determined by individual year of birth. these continuous variables are then categorized by the 5 yearly age group as a discretetime interval. the 3-year age group consists of children who have just celebrated their third birthday, children who are 3.5 years old, 3.8 years old, and so on. in this agestructured model, all individuals in the same age-group are assumed to have the same death rate. the age of an individual denoted by 𝑥. since the newborn age is 0 years and the oldest is 105 years, then 𝑥 ∈ [0,105]. individuals over 105 years will be classified into the oldest age-group. let 𝑈(𝑥,𝑡) denote the number of individuals of age 𝑥 living at time 𝑡 in a certain domain. to model the population dynamic we have to consider an aging process, i.e., 𝑈(𝑥,𝑡 + 𝛿𝑡) = 𝑈(𝑥 − 𝛿𝑡,𝑡) (3) besides the aging processes, we also take into account the death process in which the individual dies at any age with age-dependent probability, denoted by 𝜇(𝑥). we neglect pyramid population prediction using age structure model heni widayani 68 the immigration process in the domain since the immigration number insignificant compared to the total population of a country. based on aging and death process, we consider the following discrete model 𝑈(𝑥,𝑡 + 𝛿𝑡) = 𝑈(𝑥 − 𝛿𝑡,𝑡) − 𝜇(𝑥)𝑈(𝑥,𝑡)𝛿𝑡 (4) a taylor expansion for 0 ≪ 𝛿𝑡 ≪ 1 yields { 𝑈(𝑥,𝑡 + 𝛿𝑡) = 𝑈(𝑥,𝑡) + 𝛿𝑡 𝜕𝑈(𝑥,𝑡) 𝜕𝑡 + 1 2 (𝛿𝑡)2 𝜕𝑈(𝑥,𝑡)2 𝜕𝑡2 + ℴ(𝛿𝑡3) 𝑈(𝑥,𝑡 − 𝛿𝑡) = 𝑈(𝑥,𝑡) − 𝛿𝑡 𝜕𝑈(𝑥,𝑡) 𝜕𝑡 + 1 2 (𝛿𝑡)2 𝜕𝑈(𝑥,𝑡)2 𝜕𝑡2 + ℴ(𝛿𝑡3) (5) and hence we obtain in leading order 𝜕𝑈(𝑥,𝑡) 𝜕𝑡 + 𝜕𝑈(𝑥,𝑡) 𝜕𝑥 = −𝜇(𝑥)𝑈(𝑥,𝑡) (6) where (𝑥,𝑡) ∈ ℝ2 + with non-negative initial condition 𝑈(𝑡 = 0,𝑥) = 𝑈0(𝑥) (7) the 𝑈0(𝑥) is the age structure at time 𝑡 = 0. the boundary condition has interpretation as the number of birth in time 𝑡 (population who have zero-age 𝑥 = 0) which is proportional to the number of female people in reproductive age [11]. the number of new-born in year 𝑡𝑛+1 assumed to be proportional to the women at childbearing age in year 𝑡𝑛. a new-born is assumed has equal probabilities to be a male or female baby. the birth rate is a function of age 𝑥 and denoted by 𝛼(𝑥), then we get the boundary condition for (6) is 𝑈(𝑡,𝑥 = 0) = ∫ 𝛼(𝑠)𝑈(𝑡,𝑠) 𝑥1 𝑥0 𝑑𝑠 (8) so, the age-structured population model including birth rate can be written as { 𝜕𝑈(𝑥,𝑡) 𝜕𝑡 + 𝜕𝑈(𝑥,𝑡) 𝜕𝑥 = −𝜇(𝑥)𝑈(𝑥,𝑡) 𝑈(𝑥,0) = 𝑈0(𝑥) 𝑈(0,𝑡) = ∫ 𝛼(𝑥)𝑈(𝑥,𝑡) 𝑥1 𝑥0 𝑑𝑥 (9) introduce discrete time points 𝑡𝑘 = 𝑘ℎ,𝑘 = 0,…,𝑚 where 𝑚ℎ = 𝑇 denotes the final time for our simulation. similarly, we introduce discrete ages 𝑥𝑖 = 𝑖ℎ,𝑖 = 0,…,𝑛 where 𝑛ℎ ≫ 1 is large enough to cover the maximal life span of an individual. for the partial derivatives, we use the finite differences method where { 𝜕 𝜕𝑡 𝑈(𝑡𝑘,𝑥𝑖) ≅ 𝑈(𝑡𝑘+1,𝑥𝑖) − 𝑈(𝑡 𝑘,𝑥𝑖) ℎ 𝜕 𝜕𝑥 𝑈(𝑡𝑘,𝑥𝑖) ≅ 𝑈(𝑡𝑘,𝑥𝑖) − 𝑈(𝑡 𝑘,𝑥𝑖−1) ℎ (10) then we have the finite difference scheme of (6) as pyramid population prediction using age structure model heni widayani 69 𝑈𝑖 𝑘+1 − 𝑈𝑖−1 𝑘 = −ℎ𝜇(𝑥𝑖)𝑈𝑖 𝑘 (11) where 𝑈𝑖 𝑘 = 𝑈(𝑥𝑖, 𝑡 𝑘). the time step ℎ can be chosen as one year. replacing on the righthand side 𝑈𝑖 𝑘 by 𝑈𝑖−1 𝑘 we get 𝑈𝑖 𝑘+1 = 𝑈𝑖−1 𝑘 (1 − ℎ𝜇(𝑥𝑖)) (12) where the factor 1 − ℎ𝜇(𝑥𝑖) can be interpreted as the rate of survival of the age-cohort 𝑖 − 1 from the past year. in the sequel, we will use (12) instead of (11) since it allows for an easier interpretation. the integral in the boundary condition can be discretized using any numerical quadrature, e.g. 𝑈0 𝑘 = ℎ ∑ 𝛼(𝑥𝑖)𝑈𝑖 𝑘 𝑖1 𝑖=𝑖0 (13) where [𝑖0, 𝑖1] denotes the age interval, when individuals reproduce. from (9), we obtain the following numerical scheme for all 𝑖 as { 𝑈𝑖 𝑘+1 = 𝑈𝑖−1 𝑘 (1 − ℎ𝜇(𝑥𝑖)) 𝑈𝑖 0 = 𝑈0(𝑥𝑖) 𝑈0 𝑘+1 = ℎ ∑𝛼(𝑥𝑖)𝑈𝑖 𝑘+1 𝑖1 𝑖=𝑖0 (14) we can apply this mathematical model to the population pyramid of a nation. the age structure model is used in the demography to project population structure based on age categories in the future. the population in a country is recorded periodically. in indonesia, we do 10 years census (sensus populasi, sp) and 5 years census (survey penduduk antar sensus, supas) [12]. population data is categorized into five-yearly age groups namely 0-4 years, 5-9 years, 10-14 years, 15-19 years, 20-24 years, 25-29 years, 30-34 years, 35-39 years, 40-44 years, 45-49 years, 50-54 years, 55-59 years, 60-64 years, 65-69 years, 70-74 years, 75-79 years, 80-84 years, 85-89 years, 90-94 years, 95-99 years, and 100 years and above. this population data is displayed in the population pyramid or age-sex pyramids. we refer to these graphs as pyramids because they are usually shaped like triangles, though as we will see shortly, population pyramids also take other shapes. based on the population pyramid shapes, the policymakers can arrange strategies based on age-group in social, economic, educational, and human resource [13]. (a) expansive (b) stationary (c) constrictive figure 1. population pyramid types there are three types of population pyramids: expansive, constrictive, and stationary. expansive population pyramids depict populations that have a larger pyramid population prediction using age structure model heni widayani 70 percentage of people in younger age groups. populations with this shape usually have high fertility rates with lower life expectancies. many third world countries have expansive population pyramids. constrictive population pyramids are named so because they are constricted at the bottom. there is a lower percentage of younger people. constrictive population pyramids show declining birth rates since each succeeding age group is getting smaller and smaller. the united states has a constrictive population pyramid. stationary population pyramids are those that show a somewhat equal proportion of the population in each age group. there is not a decrease or increase in population; it is stable. austria has a stationary population pyramid. these three types of population pyramid are illustrated in figure 1. the shape of a population pyramid can be used to interpret a population. for example, a pyramid with a very wide base and a narrow top section suggests a population with both high fertility and death rates. whereas, a pyramid with a wider top half and a narrower base would suggest an aging population with low fertility rates. population pyramids can also be used to speculate a population’s future development. an aging population that is not reproducing would eventually run into issues such as having enough offspring to care for the elderly. other theories such as the “youth bulge” state that when there’s a wide bulge around the 16-30 age range, particularly in males, this leads to social unrest, war, and terrorism. population pyramids are ideal for detecting changes or differences in population patterns. multiple population pyramids can be used to compare patterns across nations or selected population groups. this makes population pyramids useful for fields such as ecology, sociology, and economics. results and discussion the above model will be implemented in five countries which have different population pyramid shape, i.e. indonesia, usa, brazil, russia, and japan. data that we used are taken from un data 2015. the death rate categories by the age-sex group can be shown in figure 2, while figure 3 shows the birth rate categories by the age-sex group. figure 2 shows the mortality rate in these five countries has increased with age. the mortality rate of males is higher compared to females. this indicated by a widening up and leaning to the left. the 90-94 age group in brazil has the highest mortality rate compared to all age-group. the chance of death of age above 90 in indonesia is zeros because of the data from the united nation website for indonesia only available up to age 85. japan and the usa have a high level of elderly well-being which is indicated by the highest mortality rate at the age of over 100 years. the highest death rate in newborn age occurs in indonesia while japan gets the lowest death rate for zeros ages. this indicates that the knowledge level about new-born health related to sanitation and nutrition in indonesia is lower than in japan. furthermore, newborns are very susceptible to any bacteria and viruses. some people in indonesia choose not to vaccinate their children for various reasons. this certainly will increase the chance that the baby died due to infectious diseases. pyramid population prediction using age structure model heni widayani 71 figure 2. death rate categories by age-sex group figure 3. birth rate categories by mother’s age group figure 3 shows the birth rate categories by mother’s age group. brazil has the youngest age of mothers at 10-14 years, while indonesia has the oldest age of mothers, i.e. 50-54 years. if the usa and russia have the youngest mother age at 15-19 years, japan women begin to be a mother at 20-24 years old. the range of productive age in japan pyramid population prediction using age structure model heni widayani 72 shifts 5 years older than women in russia or the usa. this is because the majority of japanese women prefer to prioritize career or education before 20 years and only decided to have children after 20 years. moreover, this also becomes the reason why the highest birth rate in japan given by 35-39 years age group. at this age-group, japan women started to become economically established and start thinking about having a family. from the five countries, indonesia has the highest birth rates in each age group. the highest maternal age groups for indonesia, the usa, and russia are in the same age group, i.e., 2529 years. (a) 2016 and 2021 (b) 2016 and 2031 (c) 2016 and 2041 (d) 2016 and 2066 figure 4. pyramid population indonesia (2016) and prediction using the model (9) with the initial condition is the same with age structure data from the un [14], we can make a pyramid population for each country. beginning with the prediction population pyramid of indonesia which shown in figure 4. the population pyramid based on 2016 data is shown by a straight line while the prediction is shown by a dashed line. in 2021, the oldest age in indonesia is 85-89 years. in the following ten years, the oldest age becomes 90-94 years. if group 5-9 year become the largest population in 2016, then in 2041 the largest population will be 20-24 years. from 2016 to 2041, the pyramid has type as a stationary pyramid in which the percentage of adolescents and adults (25-69 years) almost the same. in 2066, the pyramid classified as an expansive population pyramid where the younger age groups have the largest percentage. in this year, indonesia has high fertility rates with lower life expectancies. figure 5 shows the prediction of the population pyramid for brazil. the population pyramid of brazil in 2016 has a contracting type in which it has a lower percentage of the youngest age. most age groups in 2016 are at the age of 30-34 years. this age group becomes the largest age group of the next 15 years where the individual age of this group becomes 45-49 years. in 2041, the largest age group is 55-59 years with a slimmer pyramid population prediction using age structure model heni widayani 73 pyramid shape. in 2066, the percentage of the old population is more than the young age. the shape of the pyramid that is formed is getting slimmer. if this continues, the population of brazil will decrease from year to year. (a) 2016 and 2026 (b) 2016 and 2031 (c) 2016 and 2041 (d) 2016 and 2066 figure 5. pyramid population brazil (2016) and prediction the other four countries before show a balanced composition of men and women in the population pyramid shape, but the russian population pyramid shows an asymmetrical shape. the pyramid shape of the russian population in 2016 is more inclined to the right which means that the percentage of the female population is higher than the male population. the population pyramid tends to contract type where the number of younger age is smaller. in 2066, the population pyramid shape is not as smooth as the shape of the pyramid of the other four countries which are shown in figure 8. pyramid population prediction using age structure model heni widayani 74 (a) 2016 and 2026 (b) 2016 and 2031 (c) 2016 and 2041 (d) 2016 and 2066 figure 6 pyramid population usa (2016) and prediction figure 6 shows the population pyramid prediction for the usa. from 2016 to 2066 the population pyramid has a stationary type where each age group has an equal proportion. the population is neither decrease nor increase. we called this population is stable. this characteristic is owned by developed countries where the government can control population growth well so that population growth still exists but does not explode. the prediction of the japanese population pyramid is shown in figure 7. the pyramid type from 2016 to 2066 is a contracting type in which the number of young population is less than the old age. in 2016, the 40-44 and 65-69 age groups became the largest age group. this resulted in the year 2041, the largest age group was at the age of 65-69 years. the shape of the pyramid is increasingly symmetrical and smoother by 2066. (a) 2016 and 2026 (b) 2016 and 2031 (c) 2016 and 2041 (d) 2016 and 2066 pyramid population prediction using age structure model heni widayani 75 figure 7. pyramid population japan (2016) and prediction (a) 2016 and 2026 (b) 2016 and 2031 (c) 2016 and 2041 (d) 2016 and 2066 figure 8. pyramid population russia (2016) and prediction conclusion in this article, the age structure model for a population in a particular domain has been constructed. the proposed model has form as a linear one-order partial differential equation with non-negative initial conditions. the boundary condition is modeled from the cumulative birth rate of all reproductive age. the proposed model is implemented to predict the population pyramids from 5 countries i.e. indonesia, brazil, the usa, japan, and russia. these five countries were chosen because the population pyramid of the population represents the three types of population pyramid shape. from the simulation results, it can be seen that the age structure model can be used to predict the composition of a population in the future through the population pyramid. to improve the accuracy of the model, birth and death data in each age group must be complete and new for each year. this is because the rate of births and deaths in each year is very likely to change due to factors causing other than natural death. this model can also be developed by including the rate of immigration and emigration from a country. data limitations are one challenge to predict population growth more accurately. acknowledgments this research is funded by itb through the program penelitian, pengabdian kepada masyarakat, dan inovasi (p3mi) 2019. pyramid population prediction using age structure model heni widayani 76 references [1] richmond, m. (2020, february 21). human population pyramids. retrieved february 21, 2020, from http://www/fsl.orst.edu/pnwerc/wrb/atlas_web_compressed/. [2] christina, kuttler. 2011. lecture notes: reaction-diffusion equations with applications. munich: tu munich. [3] auger, pierre, pierre magal, and shigui ruan. 2008. structured population models in biology and epidemiology. vol. 1936. berlin: springer. [4] leal-ramirez, cecilia & castillo, oscar & melin, patricia & rodríguez-díaz, antonio. 2011. simulation of the bird age-structured population growth based on an interval type-2 fuzzy cellular structure. inf. sci.. 181. 519-535. 10.1016/j.ins.2010.10.011. [5] lande, russell, and steven hecht orzack. 1988. extinction dynamics of agestructured populations in a fluctuating environment. proceedings of the national academy of sciences 85.19 (1988): 7418-7421. [6] abia, l. m., o. angulo, and j. c. lópez-marcos. 2005. age-structured population models and their numerical solution. ecological modelling 188.1 (2005): 112-136. [7] bacaër, nicolas. 2011. a short history of mathematical population dynamics. springer science & business media. [8] malthus, thomas robert, donald winch, and patricia james. 1992. malthus:'an essay on the principle of population'. cambridge university press. [9] verhulst, pierre-françois. deuxième mémoire sur la loi d'accroissement de la population. 1847. mémoires de l'académie royale des sciences, des lettres et des beaux-arts de belgique 20 : 1-32. [10] lotka, alfred j. 1931. the structure of a growing population. human biology 3.4: 459493. [11] thomas, gotz. 2018. compact course: numerics for reaction-diffusion equations. depok: university of indonesia. [12] hayes, adrian, and diahhadi setyonaluri. 2015. taking advantage of the demographic dividend in indonesia: a brief introduction to theory and practice. united nations population fund: new york, ny, usa. [13] pitoyo, a. j., et al. 2018. system dynamics modeling of indonesia population projection model. iop conference series: earth and environmental science. vol. 145. no. 1. iop publishing. [14] united nation. 2016. demographic yearbook 2016. retrieved from https://unstats.un.org/unsd/demographic-social/products/dyb/dyb_2016/ http://www/fsl.orst.edu/pnwerc/wrb/atlas_web_compressed/ mathematical model quartic curve bezier of modification cubic curve bezier cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 77-83 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 08, 2020 reviewed: april 14, 2020 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.9054 mathematical model quartic curve bezier of modification cubic curve bezier juhari mathematics department, universitas islam negeri maulana malik ibrahim malang email: juhari@uin-malang.ac.id abstract the creative industries have become the government's attention for contributing to economic accretion. but due to the lack of artistic creativity and appeal, the evolution of the creative industries' craft section is not optimal. so that it was needed a variation of relief items to increase attractiveness. in general, an industrial object's design is still limited to the space geometry objects or a bezier curve of degree two. therefore, bezier curves of degree are selected and modified it into a quartic bezier forms and then applied to the design of industrial objects (glassware). the purpose of this research is to determine the formula of the quartic bezier form of cubic bezier modifications and to determine the rotary surface shape of quartic bezier from cubic bezier modifications. then, from some form of the revolving surface of modified cubic bezier, the glassware designs are generated. the results of this research are, first, the formula of the quartic bezier result of bezier cubic modifications. second, the form of the revolving surface of modified cubic bezier which is influenced by five control points 𝑃0, 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33, 𝑃3 and parameter selection 𝜆31, 𝜆32, 𝜆33. for further research, it is expected to develop a modification of cubic bezier into bezier of degree-𝑛 keywords: cubic bezier modification; quartic bezier; modeling introduction geometry is one of the mathematical sciences that discuss the shape, size, position of an object, both space and field objects. in its development, the geometry consists of space geometry and field geometry. the merger of both geometries can be utilized in object design. according to [1], the bezier curve is a polynomial-based curve that is often utilized to design objects and form the surface of objects. the bezier curve is an ideal standard to represent a more complex polynomial curve. previous studies have covered the bezier quadratic curve, which is utilized to design objects [2]. however, there is an issue of lack of curvature variation of curves and the design of objects confined to the geometry of a particular space. the applicability of the metric bezier character curves and swivel on the lampshade model using the maple 13 performed by juhari [3]. in the study discussed the bezier quadratic curve applied to the components of the lampshade design. the lamp hood is formed intact and joined continuously. however, there is no modification of the curve into the shape of a higher degree curve. so that the lack of variation of curvature curve and design is only limited to the geometry objects of certain space. wahyudi [4] has conducted research on the design of industrial objects with rotating objects that are implemented for the design of vase, glassware, jug, or knop. the drawback is that the swivel surface obtained in the study generally has a single and flat http://dx.doi.org/10.18860/ca.v6i2.9054 mailto:juhari@uin-malang.ac.id mathematical model quartic curve bezier of modification cubic curve bezier juhari 78 curved surface so that variations of the shape obtained are less varied. in addition, roifah [5] has been conducting research on knop or handle modelization using the merger of tube objects, irregular hexagons prisms, and rotary surfaces. the excess is the modification of the tube blanket curve using the quartic curve of hermit and bezier, thus producing a more convenient and versatile knop model creation. the drawback is that a knop fabrication is only done on the geometric objects of a tube and a rectangular prism. in addition, the relief knop offered should be modified again on the rotating surface of the curve. arinda [6] has conducted research on the construction of vases of flowers through the merger of several geometric objects of space. the excess is to obtain a variance form of flower vase through the construction of a space-building such as a prism, pyramid, or a tube. the disadvantage is the manufacture of flower vase simply by constructing the building of the prism chamber, pyramid, or the jar of tubes. in connection with these issues, the development of the bezier quadratic curve is needed to higher degrees with the addition of a curve modification to provide a variation of curvature of the curve that can be utilized for the modelization of industrial objects (glassware). in this study, further discussed equations and bezier quartic swivel surfaces of the bezier cubic curve modified on the design of the glassware. methods the steps used in determining bezier's quartic curve formula are, first to create a hermit quartic matrix, second to determine the new polygon control points ω = [𝑃0, 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33, 𝑃3], third, making bezier's quartic equation from the modified bezier cubic curve, finally making bezier's quartic rotational surface shapes from bezier's cubic modification. in making these surface forms, several steps are required as follows: a. determine point data. b. construct a cubic bezier curve resulting from cubic modification using point data to provide curvature. c. rotate or interpolate bezier curves. d. bezier quartic curve simulation with point data determined using maple 13. results and discussion hermit cubic curve according to [7], the hermit curve was discovered by a french mathematician named charles hermit in 1822-1901. the hermit cubic curve segment is an interpolated curve of the two control points 𝑃0 dan 𝑃1 where this curve requires the well-knew tangent vectors of the u-parameter variables of the control points that are 𝑃0 𝑢 dan 𝑃1 𝑢. the algebraic form of a parametric cubic curve 𝑃(𝑢) is for supposing by following three polynomials: 𝑥(𝑢) = 𝑎𝑥 + 𝑏𝑥𝑢 + 𝑐𝑥𝑢 2 + 𝑑𝑥𝑢 3 𝑦(𝑢) = 𝑎𝑦 + 𝑏𝑦𝑢 + 𝑐𝑥𝑢 2 + 𝑑𝑥𝑢 3 𝑧(𝑢) = 𝑎𝑍 + 𝑏𝑧𝑢 + 𝑐𝑍𝑢 2 + 𝑑𝑥𝑢 3 (1) where 𝑎, 𝑏, 𝑐, and 𝑑 are the coefficients that follow the 𝑥, 𝑦, and 𝑧 equations with 𝑢 parameter that are delimited in intervals 0 ≤ 𝑢 ≤ 1 or 𝑢 ∈ [0,1]. next of the shape curve (1), is written in the parametric form: mathematical model quartic curve bezier of modification cubic curve bezier juhari 79 𝑃(𝑢) = 𝑎 + 𝑏𝑢 + 𝑐𝑢2 + 𝑑𝑢3 (2) then based on equations (2) specified the following conditions: 𝑃(0) = 𝑎 𝑃(1) = 𝑎 + 𝑏 + 𝑐 + 𝑑 𝑃𝑢(0) = 𝑏 𝑃𝑢(1) = 𝑏 + 2𝑐 + 3𝑑 (3) if the system equation (3) is resolved, it is acquired hermit matrix and obtained vectorvectors 𝑎, 𝑏, c, and 𝑑 as follows: 𝑎 = 𝑃0 𝑏 = 𝑃0 𝑢 𝑐 = −3𝑃0 + 3𝑃1 − 2𝑃0 𝑢 − 𝑃1 𝑢 𝑑 = 2𝑃0 − 2𝑃1 + 𝑃0 𝑢 + 𝑃1 𝑢 (4) equations (4) substituted into equations (2) hence the acquired cubic curve equation of hermit. 𝑃(𝑢) = 𝐻1(𝑢)𝑃0 + 𝐻2(𝑢)𝑃1 + 𝐻3(𝑢)𝑃0 ′ + 𝐻4(𝑢)𝑃1 ′ (5) with 𝐻1(𝑢) = (1 − 3𝑢 2 + 2𝑢3) 𝐻2(𝑢) = (3𝑢 2 − 2𝑢3) 𝐻3(𝑢) = (𝑢 − 2𝑢 2 + 𝑢3) 𝐻4(𝑢) = (−2𝑢 2 + 𝑢3) (6) the shape of the equation (5) is called the presentation of curves in the geometric form, and the 𝑃0, 𝑃1, 𝑃0 𝑢, a𝑛𝑑 𝑃1 𝑢 are called geometric by. while the functions of 𝐻1(𝑢), 𝐻2(𝑢), 𝐻3(𝑢) and 𝐻4(𝑢) in equations (6) are called the base of hermit, and its curve is called the hermit curves [8]. bezier cubic curve the bezier curve is a polynomial curve consisting of multiple sets of control points. the interpolated point of the curve is the first and last point, while the second to point 𝑃𝑛−1 is the approximate tangent and magnitude used to control the curvature of the curve. bezier polynomial curves n expressed in the parametric form are: 𝐶(𝑢) = ∑ 𝑃𝑖𝐵𝑖 𝑛(𝑢)𝑛𝑖=0 , 𝑢 ∈ [0,1] (7) where its function base is 𝐵𝑖,𝑛(𝑢) = ( 𝑛 𝑖 ) 𝑢𝑖(1 − 𝑢)𝑛−1 ( 𝑛 𝑖 ) = 𝑛! 𝑖! (𝑛 − 𝑖)! in such equations, the points of the 𝑃𝑖 are called the geometric coefficient or 𝐶(𝑢) curve control point. these points are of real value. for 4 dots and 𝑛 = 3: 𝐵0,3 = (1 − 𝑢) 3 𝐵1,3 = 3𝑢(1 − 𝑢) 2 𝐵2,3 = 3𝑢 2(1 − 𝑢) mathematical model quartic curve bezier of modification cubic curve bezier juhari 80 𝐵3,3 = 𝑢 3 thus obtained a bezier cubic curve with the u parameter as follows: 𝐶3(𝑢) = (1 − 𝑢) 3𝑃0 + 3(1 − 𝑢) 2𝑢𝑃1 + 3(1 − 𝑢)𝑢 2𝑃2 + 𝑢 3𝑃3 (8) hermit quartic matrix suppose taken parametric quartic curve with 𝑃(𝑢) expressed in the form of algebraic: 𝑥(𝑢) = 𝑎𝑥 + 𝑏𝑥𝑢 + 𝑐𝑥𝑢 2 + 𝑑𝑥𝑢 3 + 𝑒𝑥𝑢 4 𝑦(𝑢) = 𝑎𝑦 + 𝑏𝑦𝑢 + 𝑐𝑥𝑢 2 + 𝑑𝑥𝑢 3 + 𝑒𝑦𝑢 4 𝑧(𝑢) = 𝑎𝑍 + 𝑏𝑧𝑢 + 𝑐𝑍𝑢 2 + 𝑑𝑧𝑢 3 + 𝑒𝑧𝑢 4 (9) where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are the coefficients that follow the 𝑥, 𝑦, and 𝑧 equations with the u parameter delimited in intervals of 0 ≤ 𝑢 ≤ 1 or 𝑢 ∈ [0,1]. 𝑢 value restriction is intended to keep the curve segment awake limited and easy to control. based on the equation (1) it can be written into a parametric form so that it becomes: 𝑃(𝑢) = 𝑎 + 𝑏𝑢 + 𝑐𝑢2 + 𝑑𝑢3 + 𝑒𝑢4 (10) in which coefficients 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 in equation p (u) are vectors that have three components, such as 𝑎 = [𝑎𝑥, 𝑎𝑦, 𝑎𝑧]. then in getting the first derivative of 𝑃(𝑢) is: 𝑃𝑢(𝑢) = 𝑏 + 2𝑐𝑢 + 3𝑑𝑢2 + 4𝑒𝑢3 the second derivative of 𝑃(𝑢) is: 𝑃𝑢𝑢(𝑢) = 2𝑐 + 6𝑑𝑢 + 12𝑒𝑢2 then set some of the following conditions: 𝑃(𝑢 = 0) 𝑃(𝑢 = 1) 𝑃𝑢(𝑢 = 0) 𝑃𝑢(𝑢 = 1) 𝑃𝑢𝑢(= 0) = 𝑃0 = 𝑎 = 𝑃1 = 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 = 𝑃0 𝑢 = 𝑏 = 𝑃1 𝑢 = 𝑏 + 2𝑐 + 3𝑑 + 4𝑒 = 𝑃0 𝑢𝑢 = 2𝑐 hence the modification of hermit is derived: mmh = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 −4 4 −3 −1 −1 3 −3 2 1 1 2 ] (11) quartic bezier curve of modification cubic bezier curve suppose given the new bezier polygon control dots ω = [𝑃0, 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33, 𝑃3], then the control point between 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33 is defined as the following: 𝑁𝑃31 = 𝜆31𝑃1 + (1 − 𝜆31)𝑃0 𝑁𝑃32 = 𝜆32𝑃2 + (1 − 𝜆32)𝑃1 𝑁𝑃33 = 𝜆33𝑃3 + (1 − 𝜆33)𝑃2 with 0 ≤ 𝜆31, 𝜆32, 𝜆33 ≤ 1. furthermore, the new control points defined 𝑃0, 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33 and 𝑃3 to create a bezier quartic equation of the bezier cubic modification as follows, 𝐵(0) = 𝑃0 mathematical model quartic curve bezier of modification cubic curve bezier juhari 81 𝐵(1) = 𝑃3 𝐵′(0) = 4(𝑁𝑃31 − 𝑃0) 𝐵′(1) = 4(𝑃3 − 𝑁𝑃33) 𝐵"(0) = 12(𝑃0 − 2𝑁𝑃31 + 𝑁𝑃32) thus modification of the bezier cubic curve into the bezier quartic form, 𝐵4(𝑢) = [(1 − 4𝑢 + 6𝑢2 − 4𝑢3 + 𝑢4)𝑃0 + (4𝑢 + 12𝑢 2 + 12𝑢3 − 4𝑢4)𝑁𝑃31 +(6𝑢2 − 12𝑢3 + 6𝑢4)𝑁𝑃32 + (4𝑢 3 − 4𝑢4)𝑁𝑃33 + 𝑢 4𝑃3] (12) with 0 ≤ 𝑢 ≤ 1... in figure 3.1 is shown a bezier cubic curve difference using equations (2.13) and the bezier quartic curve of a bezier cubic modification using equations (3.4) with point 𝑃0 = (9, 0, 3), 𝑃1 = (4, 0, 5), 𝑃2 = (4, 0, 9), and 𝑃3 = (9, 0, 12) with the parameter value 𝜆31 = 0.2, 𝜆32 = 0.6, and 𝜆33 = 0.9. figure 1. example cubic bezier curve and quartic bezier curve quartic bezier surface from modified cubic bezier bezier quartic shapes modified the bezier cubic curve is dependent on the determination of the 𝑃0, 𝑃1, 𝑃2 and 𝑃3 control points. because of the case of point modification, a new control point or a modification point in 𝑁𝑃31, 𝑁𝑃32, and 𝑁𝑃33 in the equation (3.4). the point determination is used to produce a smoother curve. figure 1 is an example of a bezier quartic rotary surface of a bezier cubic modification with some selection of 𝑃0, 𝑃1, 𝑃2, and 𝑃3 control points and a parameter of 𝜆31, 𝜆32, 𝜆33 on the different modifications of 𝑁𝑃31, 𝑁𝑃32, and 𝑁𝑃33. bezier cubic rotating surface bezier quartic rotary surface modified bezier cubic curve. 𝑃0 = (11,0,0), 𝑃1 = (10,0,3) 𝑃2 = (20, 0,4), dan 𝑃3 = (17.5,0,10) 𝑃0 = (11,0,0), 𝑃1 = (10,0,3) 𝑃2 = (20, 0,4), dan 𝑃3 = (17.5,0,10) 𝜆31 = 0.75, 𝜆32 = 1, 𝜆33 = 0 bezier cubic curve bezier quartic curve result cubic modification 𝑃3 𝑃2 𝑃1 𝑃0 mathematical model quartic curve bezier of modification cubic curve bezier juhari 82 𝑃0 = (3.5,0,0), 𝑃1 = (0,0,2) 𝑃2 = (4, 0,4), dan 𝑃3 = (3.5,0,12) 𝑃0 = (3.5,0,0), 𝑃1 = (0,0,2) 𝑃2 = (4, 0,4), dan 𝑃3 = (3.5,0,12) 𝜆31 = 0.1, 𝜆32 = 0.5, 𝜆33 = 1 𝑃0 = (0,0,0), 𝑃1 = (8,0,1.5) 𝑃2 = (1, 0,2), dan 𝑃3 = (1,0,7) 𝑃0 = (0,0,0), 𝑃1 = (8,0,1.5) 𝑃2 = (1, 0,2), dan 𝑃3 = (1,0,7) 𝜆31 = 0.1, 𝜆32 = 0.05, 𝜆33 = 1 figure 2. cubic bezier surfaces and modifications subsequently, some bezier quartic swivel surfaces from the bezier cubic modification results can be converted to model industrial objects (glassware) such as flower vase, teplik lamps, marble chairs, and other glassware. figure 2 illustrates the design example of glassware using some form of bezier quartic of the bezier cubic curve modification. figure 3. example of design of glassware using a bezier quartic of a bezier cubic modification conclusions the bezier quartic equation results from the bezier cubic modification obtained from the hermit modification matrix and the determination of a new polygon control point. the bezier cubic modified form of bezier modification has 5 control points: 𝑃0, 𝑁𝑃31, 𝑁𝑃32, 𝑁𝑃33, 𝑃3. forms of scartic bezier turndown surface of bezier cubic modification is influenced by the control point values of 𝑃0, 𝑃1, 𝑃2, 𝑃3, and selection of parameters of 𝜆31, 𝜆32, 𝜆33 values in new control point data 𝑁𝑃31, 𝑁𝑃32 and 𝑁𝑃33. jar teplik lamp marble chairs mathematical model quartic curve bezier of modification cubic curve bezier juhari 83 references [1] e. mortenson, mathematics for computer grapichs applications, new york: industrial press, inc, 1999. [2] a. c. p. m. d. kusno, "modelisasi benda onyx dan marmer melalui penggabungan dan pemilihan parameter pengubah bentuk permukaan putar bezier," jurnal ilmu dasar, vol. 8, no. 2, pp. 175-185, 2007. [3] e. o. juhari, "penerapan kurva bezier karakter simetrik dan putar pada model kap lampu duduk menggunakan maple," cauchy: jurnal matematika murni dan aplikasi, vol. 4, no. 1, pp. 28-34, 2015. [4] wahyudi, "perancangan objek-objek industri dengan benda permukaan putar," universitas jember, jember, 2001. [5] roifah, "modelisasi knop melalui penggabungan benda dasar hasil deformasi tabung, prisma segienam beraturan, dan permukaan putar," universitas jember, jember, 2013. [6] arinda, "konstruksi vas bunga melalui penggabungan beberapa benda geometri ruang," universitas jember, jember, 2007. [7] e. mortenson, geometric modelling, new york: willey komputer publishing, 1996. [8] kusno, geometri rancang bangun studi surfas putar transformasi titik dan proyeksi, jember: jurusan matematika fakultas mipa universitas jember, 2003. optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm cauchy –jurnal matematika murni dan aplikasi volume 6(4) (2021), pages 169-180 p-issn: 2086-0382; e-issn: 2477-3344 submitted: march 09, 2020 reviewed: october 26, 2020 accepted:february 09, 2021 doi: http://dx.doi.org/10.18860/ca.v6i4.8933 optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia universitas islam darul ulum lamongan email: dinitarahmalia@gmail.com abstract in weather clustering, there are many variables which can be observed such as air temperature, humidity, sunlight intensity, and so on. in this research, takagi-sugeno fuzzy inference system (fis) will be used for forecasting the sunlight intensity based on temperature and humidity and fuzzy clustering means (fcm) will be used for clustering them based on fuzzy set. from the data consisting of temperature, humidity, and sunlight intensity, we will forecast sunlight intensity and cluster them into two clusters, three clusters, and four clusters by fcm method. in fis method, the membership degree are often generated by trial and error. also, the optimization of the initial of membership degree are required in fcm. because the initial of membership degree are often generated by trial and error, in this research, we use heuristic method like firefly algorithm to optimize the membership degree. from the simulations, firefly algorithm can optimize the membership degree of fis for forecasting the data with minimum mean square error (mse) and the initial of membership degree of fcm with two clusters, three clusters, and four clusters with minimum objective value. keywords: fuzzy clustering means; fuzzy inference system; firefly algorithm; membership degree introduction weather clustering is important work such as determining hot, warm or cold weather. in weather clustering, there are many variables which can be observed such as air temperature, humidity, sunlight intensity, and so on. from previous research [1], the temperature, humidity, and sunlight intensity have good correlation so that in this research, fuzzy clustering means (fcm) will be used for clustering them based on fuzzy set. fuzzy set was introduced by l.a. zadeh in 1965. the difference of the fuzzy set and crisp set is in which fuzzy set has membership function. there are many applications in fuzzy system: optimization, control, estimation. the application of fuzzy set which can be used on clustring problem is fuzzy clustering means (fcm) [2]. in previous researches, some methods have been applied in clustering problem like k-means [3], kohonen network or neural network [4], genetic algorithm [5]. fuzzy inference system (fis) is the method for forecasting the data using membership function in their inputs. there are two types of fis such as mamdani and http://dx.doi.org/10.18860/ca.v6i4.8933 mailto:dinitarahmalia@gmail.com optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 170 takagi sugeno. the difference between mamdani and ts is that ts output membership functions are either linear or constant [12],[13]. fuzzy c means (fcm) is the method for clustering the data where each data belongs to a cluster that is specified by membership degree. this method was founded by bezdek in 1981 [2]. in fcm, we update membership degree so that minimizing the objective function value. from the data consisting of temperature, humidity, and sunlight intensity, we will forecast sunlight intensity and cluster them into two clusters, three clusters, and four clusters by fcm method. in fis method, the membership degree are often generated by trial and error.in fcm method, there is the initialization of membership degree randomly between 0-1. therefore, the optimizations of the initial of membership degree are required. in previous researches, particle swarm optimization (pso) has been applied in optimizing the initial of membership degree [6], [7]. in this research, we use heuristic method like firefly algorithm to optimize the initial of membership degree. the advantages of firefly algorithm are they use brightness as attractiveness. the less bright firefly will approach to the brighter firefly. firefly algorithm has been applied in some optimization problem such as linear programming with constrains [8]. xin-she yang discovered firefly algorithm in 2008. it is inspired on behavior of flashing characteristics of fireflies. theysearch the prey, and find mates by bioluminescence, and communicate. one of characteristics of fireflies is the less bright firefly will approach to the brighter firefly. brighter firefly indicates better objective function as fitness function [9]. from the simulations, firefly algorithm can optimize the initial of membership degree of fcm with two clusters, three clusters, and four clusters with minimum objective value. methods takagi-sugeno fuzzy inference system (fis) general form of rule-based fuzzy models are there are antecedent proposition and cause the consequent proposition. depending on the form of the consequent, the types of rule based fuzzy model are: 1. linguistic fuzzy model both antecedent and the consequent are fuzzy propositions 2. takagi – sugeno (ts) fuzzy model the antecedent is a fuzzy proposition and the consequent is a crisp function. suppose the rule is : if x is i a and y is j b then z is kc , 1, 2,...,i n , 1, 2,...,j m , 1, 2,...,k nm then, the fuzzy inference system can be computed as follows : 1. compute the degree of membership function of two inputs ( ( ), ( )) k ai bj f f x y  (1) 2. derive the output of fuzzy sets ' min( , ) k k k c f c (2) 3. aggregate the output fuzzy sets ' max( , 1, 2,..., ) k b c k nm  (3) optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 171 4. compute the center of gravity ( )cog defuzzification 1 1 d i i i d i i b z cog b      (4) with d is the number of elements of z takagi – sugeno (ts) fuzzy inference is introduced in 1985. it is similar to mamdani method, but the difference between mamdani and ts is that ts output membership functions are either linear or constant [12],[13]. the rule in a ts fuzzy model has the form: if x is i a and y is j b then k k k kf p x q y r   , 1, 2,...,i n , 1, 2,...,j m generally, , 1, 2,..., k f k nm are outputs with nm is the number of rules. the firing strength k w are: ( ( ), ( )) k ai bj w and x y  the inference formula of the ts model is as follows: 1 1 nm k k k nm k k w f cog w      (5) fuzzy clustering means (fcm) clustering is the process to groupthe data into many groups or clusters.the objects in the same cluster have high similarity. the similarities are determinedby the attribute and distance measures [10]. the purpose of clustering is for identifying natural groupings from data set to produce representation of the data [11]. bezdek found fuzzy clustering means (fcm) in 1981 [2]. fcm is the method for clustering the data in which each data belongs to a cluster that is specified by membership degree. fcm is the minimization of the following objective function in equation 6.     d i k j ji m ijm cxj 1 1 2  (6) with the parameters are : m : fuzzy partition matrix exponent to control the degree of fuzzy, with 1m d : the number of data k : the number of clusters ij  : the degree of membership of ix in the j-th cluster i x : the i-th data j c : the center of the j-th cluster for every data i x , 1, 2,...,i d the sum of the membership valuesfor all clusters is one as in equation 9. optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 172 11 12 1 21 22 2 1 2 k k d d dk                       (7) )1,0(~ u ij  , 1, 2,...,i d , 1, 2,...,j k (8) 1 1 k ij j    , 1, 2,...,i d (9) the computation of fcm is as follows [10] : 1. initializing the cluster membership values ij  subject to the constrains in equation 8 and equation 9 randomly. 2. calculating the cluster centers j c in equation 10 with 2m  .     d i m ij d i i m ij j x c 1 1   1, 2,...,j k (10) 3. updating the membership degree ij  based on equation 11.               n k m ki ji ij cx cx 1 1 2 1  di ,...,2,1 1, 2,...,j k (11) 4. calculate the objective function m j in equation 6. 5. repeat step 2 – 4 until convergent. firefly algorithm xin-she yang discovered firefly algorithm (fa) in 2008. it is inspired on behavior of flashing characteristics of fireflies. they search the prey, and find mates by bioluminescence, and communicate. the fa is based on the rules [9]: 1. all fireflies are unisex so they attract one another. 2. attractiveness is the brightness of firefly. the brightness of a firefly is affected by the objective function ( )f y . in the fireflyalgorithm, attractiveness of a firefly is determined by its brightness as objective function. in fis, we need to optimize the membership functions because they are often generated by trial and error. because the membership value depend on parameter of membership function [14],[15], then we use firefly algorithm to optimize the parameter of membership function. suppose that the membership function used is gaussian as in equation (12).                 2 exp)( a cx x (12) optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 173 then the parameter of membership function is ca, . in takagi sugeno fis, the output is k k k k f p x q y r   , 1, 2,...,k nm and parameters kkk rqp ,, are computed by least square estimation (lse) in equation (13) with dk is the number of data in rule k 1 1 1 1 1 1 1 1 1 1 1 , 1, 2,... dk dk dk ik ik ik i i i kdk dk dk dk ik ik ik ik ik k ik ik i i i i kdk dk dk dk ik i ik ik ik ik ik i i i i dk x y z r x x x x y p x z k nm q y y x y y y z                                                                 (13) in fcm, we need to optimize the initial of membership degree )1,0(~ u ij  by firefly algorithm because they are also generated by trial and error. in firefly algorithm, the decision variable as position of fireflies is represented in equation 14. 11 12 1 21 22 21 1 2 k kk d d dk y                      , 1 1, 2,..., maxk pop (14) with d is the number of data and k is the number of clusters. the algorithm for optimizationusing fa are as follows : 1. generating initial population position of fireflies 1, 1 1, 2,...maxky k pop and compute the fitness value 1( ), 1 1, 2,...maxkf y k pop in equation 6 with one cycle. 2. determining the best firefly  min 1 1 1 arg min ( ), 1 1, 2,..., max k k k f y k pop  (15)   min 1 1 1 arg min ( ), 1 1, 2,..., max k k k y y f y k pop  (16) 3. do the iteration as follows : for 1 1: maxk pop for 2 1: maxk pop if 2 1( ( ) ( ))k kf y f y a. computing the distance between firefly 1k and firefly 2k 1 2 1 2 2 1 2 1 ( ) t k k k k k k t t t r y y y y      (17) b. computing the attractiveness of firefly 0 ijre      (18) c. generating 1 1 ( ) 2 k u rand  , with ~ (0,1)rand u d. updating the movement of firefly 1k optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 174 1 1 2 1 (1 ) k k k k y y y u     (19) 4. generating min 1 1 ( ) 2k u rand  , with ~ (0,1)rand u 5. updating the movement of best firefly min min min 1 1 1 k k k y y u  (20) 6. repeat step 3. results and discussion data used are weather data in jakarta consisting of temperature (celcius), humidity (%), and sunlight intensity (watt/m2) during january until july 2012. figure 1 shows the 3-dimention graph consisting of temperature 1 ( )x , humidity 2 ( )x , and sunlight intensity 3 ( )x during january until july 2012 with 200 days. figure 1. the plot of temperature, humidity, and sunlight intensity data from the data consisting of temperature, humidity, and sunlight intensity, we will forecast the sunlight intensity based on temperature and humidity by fis method andcluster them into two clusters, three clusters, and four clusters by fcm method for proving the performance of fcm. in both if fis and fcm method, there are the initialization of membership degree randomly between 0-1. therefore, the optimizations of the membership degree are required. in fis method, before the simulation is applied, the temperature, humidity, and sunlight intensity data are normalized between 0.1 to 0.9. when the forecasting result are obtained, we de-normalize the data. the research, we use firefly algorithm to optimize the initial of membership degree. the parameters of firefly algorithm used are 0 1, 5, 0.1     with the number of fireflies are 20 and maximum iterations are 30. forecasting the data the firefly algorithm optimization process can be seen in figure 2. at the early, the position of fireflies are random so that the fitness function is relative large. in the optimization, the brightness of fireflies is updated to the brighter firefly with minimum fitness function. the optimalparameter of membership degree are : optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 175 , , , , , arg , arg , , , , , arg , arg 0.7081 0.1 0.9021 0.5 0.1845 0.9 0.5244 0.1 0.5680 0.5 0.7482 0.9 tem small tem small tem medium tem medium tem l e tem l e hum small hum small hum medium hum medium hum l e hum l e a c a c a c a c a c a c             with mean square error (mse) as objective function is 816.4335. and the optimal parameters of rules computed by least square estimation (lse) are in table 1, with i p is the i-th rule coefficient of temparature, i q is the i-th rule coefficient of humidity, i r is the constants. table 1. optimal parameters of rules rule-i ip iq ir 1 1,0785 -0,1668 0,1740 2 0,6472 -0,0995 0,2340 3 0,6472 -0,0995 0,2340 4 0,6472 -0,0995 0,2340 5 0,9224 0,0284 0,0420 6 0,8010 -0,2378 0,1353 7 0,6472 -0,0995 0,2340 8 0,6472 -0,0995 0,2340 9 0,3093 -0,6442 0,6249 figure 2. firefly algorithm simulation for optimizing fis in forecasting after the optimalparameter of membership degree are obtained, they will be used in fis simulation.figure 3(a) show the optimal membership function of temperature data with three types of gauss membership function such as small, medium, and large and figure 3(b) show the optimal membership function of humidity data with three types of gauss membership function such as small, medium, and large. figure 4 shows the forecasting result of sunlight intensity based on temperature and humidity. optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 176 figure 3. (a) optimal membership function of temperature data. (b) optimal membership function of temperature data figure 4. forecasting result of sunlight intensity clustering into two clusters the firefly algorithm optimization process can be seen in figure 5. at the early, the position of fireflies are random so that the fitness function is relative large. in the optimization, the brightness of fireflies is updated to the brighter firefly with minimum fitness function. the optimal initial of membership degree ij  are obtained with minimum objective function in one cycle as minimum fitness is 286094. figure 5. firefly algorithm simulation for optimizing fcm into two clusters after the optimal initial of membership degree are obtained, they will be used in fcm simulation until 100 iterations with optimized objective value is 176534. figure 6(a) optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 177 shows the convergence process of fcm into two clusters. figure 6(b) shows the plot of temperature, humidity, and sunlight intensity data after clustering into two clusters with different colors. (a) (b) figure 6. (a) convergence process of fcm into two clusters. (b) the plot of temperature, humidity, and sunlight intensity data after clustering into two clusters clustering into three clusters similar process can be seen in figure 7. the optimal initial of membership degree ij  are obtained with minimum objective function in one cycle as minimum fitness is 281248. figure 7. firefly algorithm simulation for optimizing fcm into three clusters after the optimal initial of membership degree are obtained, they will be used in fcm simulation until 100 iterations with optimized objective value is 125442. figure 8(a) shows the convergence process of fcm into three clusters. figure 8(b) shows the plot of temperature, humidity, and sunlight intensity data after clustering into three clusters with different colors. optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 178 (a) (b) figure 8. (a) convergence process of fcm into three clusters. (b) the plot of temperature, humidity, and sunlight intensity data after clustering into three clusters clustering into four clusters similar process can be seen in figure 9. the optimal initial of membership degree ij  are obtained with minimum objective function in one cycle as minimum fitness is 288235. figure 9. firefly algorithm simulation for optimizing fcm into four clusters after the optimal initial of membership degree are obtained, they will be used in fcm simulation until 100 iterations with optimized objective value is 90865. figure 10(a) shows the convergence process of fcm into four clusters. figure 10(b) shows the plot of temperature, humidity, and sunlight intensity data after clustering into four clusters with different colors. optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 179 (a) (b) figure 10. (a) convergence process of fcm into four clusters. (b) the plot of temperature, humidity, and sunlight intensity data after clustering into four clusters conclusions takagi-sugeno fuzzy inference system (fis) can be used for forecasting the sunlight intensity based on temperature and humidity and fuzzy clustering means (fcm) can be used for clustering them based on fuzzy set. from the data consisting of temperature, humidity, and sunlight intensity, we will forecast sunlight intensity and cluster them into two clusters, three clusters, and four clusters by fcm method. in fis method, the membership degree are often generated by trial and error. also, the optimization of the initial of membership degree are required in fcm. because the initial of membership degree are often generated by trial and error, in this research, we use heuristic method like firefly algorithm to optimize the membership degree. from the simulations, firefly algorithm can optimize the membership degree of fis for forecasting the data with minimum mean square error (mse)and the initial of membership degree of fcm with two clusters, three clusters, and four clusters with minimum objective value.clustering by four clusters results the smallest objective value. the developments of this research are classifying the data with larger dimentions so that resulting maximum accuration or minimum error. references [1] d. rahmalia, t. herlambang, "prediksi cuaca menggunakan algoritma particle swarm optimization-neural network (psonn)," prosiding seminar nasional matematika dan aplikasinya, pp. 41-48, 2017 [2] j.c. bezdec, pattern recognition with fuzzy objective function algorithms, plenum press, 1981 [3] a. rohmatullah, d. rahmalia and m.s.pradana, "klasterisasi data pertanian di kabupaten lamongan menggunakan algoritma k-means dan fuzzy c means," jurnal ilmiah teknosains, vol. 5, no. 2, pp. 86-93, 2019 [4] d. rahmalia and t.herlambang, "application kohonen network and fuzzy c means for clustering airports based on frequency of flight,"kinetik : game technology, information system, computer network, computing, electronics, and control, vol. 3, no. 3, pp. 229-236, 2018 optimizing the membership degree of fuzzy inference system (fis) and fuzzy clustering means (fcm) in weather data using firefly algorithm dinita rahmalia 180 [5] h. maheshwar, k. kaushik, v. arora, " a hybrid data clustering using firefly algorithm based improved genetic algorithm," procedia computer science 18, pp. 249-256, 2015 [6] r. kakar, a. singh, " image segmentation using hybrid pso-fcm,"international journal of innovative science, engineering & technology, vol. 2, no. 7, pp. 415-418, 2015 [7] h. izakian, a. abraham, " fuzzy clustering using hybrid fuzzy c-means and fuzzy particle swam optimization," world congress on nature & biologically inspired computing, pp. 1690-1694, 2009 [8] d. rahmalia and a.m.rohmah, "optimisasi perencanaan produksi pupuk menggunakan firefly algorithm,"jurnal matematika mantik, vol. 4, no. 1, pp. 1-6, 2018 [9] x.s. yang, z. chui, r. xiao, swarm intelligent and bio-inspired computation, elsevier, 2013 [10] j. han,m. kamber,j. pei, data mining concept and techniques, elsevier, 2012 [11] m.j. zaki, w. meira, data mining and analysis. fundamental concepts and algorithms, cambridge university press, 2014 [12] l.a. zadeh, fuzzy set. information and control, 1965 [13] h.j. zimmermann, fuzzy set theory and its applications, springer science+business media, 2001 [14] d. rahmalia and a. rohmatullah, "pengaruhkorelasi data padaperamalankelembabanudaramenggunakan adaptive neuro fuzzy inference system (anfis),"applied technology and computing science journal, vol. 2, no. 1, pp. 10-24, 2019 [15] d. karaboga and e.kaya, " training anfis by using the artificial bee colony algorithm,"turkish journal of electrical engineering and computer science, vol. 25, pp. 1669-1679, 2017 cauchy –jurnal matematika murni dan aplikasi volume 6(2) (2020), pages 91-99 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 27, 2020 reviewed: april 28, 2020 accepted: may 31, 2020 doi: http://dx.doi.org/10.18860/ca.v6i2.9165 sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan1, lailany yahya2 1,2 department of mathematics, universitas negeri gorontalo jl. prof. dr. ing. bj. habibie, tilongkabila, bone bolango, gorontalo, indonesia email: resmawan@ung.ac.id, lailany.math@gmail.com abstract the study was aimed to introduce a new model construction regarding the transmission of coronavirus disease (henceforth, covid-19) in human population. the mathematical model was constructed by taking into consideration several epidemiology parameters that are closely identical with the real condition. the study further conducted an analysis on the model by identifying the endemicity parameters of covid-19, i.e., the presence of disease-free equilibrium (dfe) point and basic reproductive number. the next step was to carry out sensitivity analysis to find out which parameter is the most dominant to affect the disease’s endemicity. the results revealed that the parameters 𝜂, 𝜁𝑠𝑒, 𝛼,, and 𝜎 in sequence showed the most dominant sensitivity index towards the basic reproductive number. moreover, the results indicated positive index in parameters 𝜂 and 𝜁𝑠𝑒 that represented transmission chances during contact as well as contact rate between susceptible individuals and exposed individual. this suggests that an increase in the previous parameter value could potentially enlarge the endemicity of covid-19. on the other hand, parameters 𝛼 and 𝜎, representing movement rate of exposed individuals to the quarantine class and proportion of quarantined exposed individuals, showed negative index. the numbers indicate that an increase in the parameter value could decrease the disease’s endemicity. all in all, the study concludes that treatments for covid-19 should focus on restriction of interaction between individuals and optimization of quarantine. keywords: sensitivity analysis; mathematical model; coronavirus disease; covid-19; basic reproductive number introduction coronavirus disease (henceforth, covid-19) has shocked many thanks to its very rapid spread. firstly, identified to occur in wuhan city of china, the disease has shortly become one of the main talking points as it reached the whole world and took thousands of death toll in a very short period. the disease somewhat instigates all parties to conduct active measures in finding options to the best treatments and anticipatory means to prevent damage on a much wider scale. in mathematical perspective, the concern is closely related to implementation of mathematical model to identify potential solutions. mathematical modelling is one of the key tools in epidemic preparation, including the covid-19 pandemic. the system allows one to comprehend and identify the correlation between covid-19 spread and several epidemiology parameters, conduct preparatory measures for future planning, and implement best practices of pandemic http://dx.doi.org/10.18860/ca.v6i2.9165 mailto:resmawan@ung.ac.id mailto:lailany.math@gmail.com sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 92 treatment. previous studies, albeit little in number, has begun to address this problem and design mathematical model for covid-19 transmission [1][2][3][4]. the model involved accurate and effective public health interventions. on top of that, a study compared between the outbreaks of current covid-19 and previous mers disease that spread in middle eastern countries and korea [5]. other studies designed mathematical model that predict covid-19 cases in different countries [6][7]. several models proposed in previous studies has discussed that the virus started from an unknown source and eventually began to spread to human population. the virus source, further referred to as reservoir variable, is suspected to be the place of first infection-to-human case. the present study introduces a different approach on mathematical modelling to the virus transmission by also involving the epidemiology parameters; a variable that is not discussed in previous studies. previous models have assumed that the virus transmission only occurs in interactions between individuals that have contracted the virus; differing from that, this article takes into consideration transmission cases caused by susceptible individuals and exposed individuals. it views the importance of involving such parameters, considering the number of infections that occurred in the interaction between exposed individuals yet to be detected as infected. moreover, the model lays its emphasis on the pattern of transmission between human after the virus has become epidemic or pandemic, therefore, disregarding the reservoir variable. the model thus overlooks the process of first human infection and focuses on how the virus has spread within human-to-human interactions. in addition to that, the model employs new parameter representing death cases of covid-19, pertaining to the fact that the virus has taken numbers of death toll. the model also takes into account cases of quarantined individuals that were identified to be exposed to the virus. the following section elaborates construction of mathematical model in this study. further, the article presents the research results in the form of model analysis. within this section, the study focuses on the construction of basic reproductive number and sensitivity analysis to identify which parameter is the most sensitive to the change in basic reproductive number value. finally, the last section proposes several conclusions to the research findings and discussion. methods in this model, the total of human population was denoted in 𝑁(𝑡) and divided into six classes: susceptible individuals 𝑆(𝑡), exposed individuals in incubation period 𝐸(𝑡), asymptotically infected individuals 𝐴(𝑡), symptomatic infected individuals 𝐼(𝑡), quarantined individuals 𝑄(𝑡), and individuals that have recovered/remove from covid19 𝑅(𝑡). therefore, the total population was stated in 𝑁(𝑡) = 𝑆(𝑡) + 𝐸(𝑡) + 𝐴(𝑡) + 𝐼(𝑡) + 𝑄(𝑡) + 𝑅(𝑡). recruitment rate of natural human natality and mortality is given the parameter π and 𝜇 sequentially. susceptible individuals (𝑆) will be infected from enough contact with exposed individuals (𝐸), symptomatic infected individuals (𝐼), and asymptotically infected individuals (𝐴) in respective rate of 𝜂𝜁𝑠𝑒𝑆𝐸, 𝜂𝜁𝑠𝑖𝑆𝐼, and 𝜂𝜁𝑠𝑎𝑆𝐴, in which 𝜂 is the infection probability during contact between individuals. 𝜁𝑠𝑒 , 𝜁𝑠𝑖, and𝜁𝑠𝑎 each states the contact rate between susceptible individuals (𝑆) and individual groups of 𝐸, 𝐼, and 𝐴. the parameter 𝜃 and 𝜎 in respective order is the proportion of asymptotically infected individuals and quarantined exposed individuals, while parameter 𝛼 states movement rate of exposed individuals to the quarantined individuals. parameter 𝜔 and 𝜛 each sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 93 represent transmission rate after incubation period and status change to 𝐼 and 𝐴 class. quarantined individuals can be transferred to the class of infected individuals with symptoms at the rate of 𝜚 and proportion of 𝜑. parameter 𝜏, 𝛽, 𝜌 each states the recovery rate of infected individuals without symptoms, quarantined individuals, and infected individuals with symptoms to be transferred to recovered individuals class 𝐼. further, death rate of covid-19 in 𝐼 class is represented in 𝛿. schematically, the transmission pattern of covid-19 is displayed in figure 1 below. figure 1. compartment diagram of covid-19 transmission based on the schematic diagram of virus transmission in figure 6, the article presents model in the form of differential equation system as follows: 𝑑𝑆 𝑑𝑡 = π − 𝜂𝑆(𝜁𝑠𝑒𝐸 + 𝜁𝑠𝑖𝐼 + 𝜁𝑠𝑎𝐴) 𝑁 − 𝜇𝑆 𝑑𝐸 𝑑𝑡 = 𝜂𝑆(𝜁𝑠𝑒𝐸 + 𝜁𝑠𝑖𝐼 + 𝜁𝑠𝑎𝐴) 𝑁 − (𝜃𝜛 + 𝜎𝛼 + 𝜇 + 𝜔 − 𝜃𝜔 − 𝜎𝜔)𝐸 𝑑𝐴 𝑑𝑡 = 𝜃𝜛𝐸 − (𝜏 + 𝜇)𝐴 𝑑𝑄 𝑑𝑡 = 𝜎𝛼𝐸 − (𝜑𝜚 + 𝛽 − 𝛽𝜑 + 𝜇)𝑄 𝑑𝐼 𝑑𝑡 = (1 − 𝜃 − 𝜎)𝜔𝐸 + 𝜑𝜚𝑄 − (𝜌 + 𝜇 + 𝛿)𝐼 𝑑𝑅 𝑑𝑡 = 𝜏𝐴 + (1 − 𝜑)𝛽𝑄 + 𝜌𝐼 − 𝜇𝑅 with early condition: 𝑆(0) = 𝑆0 ≥ 0, 𝐸(0) = 𝐸0 ≥ 0, 𝐴(0) = 𝐴0 ≥ 0, 𝑄(0) = 𝑄0 ≥ 0, 𝐼(0) = 𝐼0 ≥ 0, 𝑅(0) = 𝑅0 ≥ 0. dynamics of population in total is acquired by totaling the five equations in model 𝑆 𝐴 𝐼 𝑄 𝐸 𝑅 π 𝜇𝑆 𝜇𝐴 𝜇𝐼 𝜇𝑅 𝜏𝐴 𝜇𝐸 𝜌𝐼 𝜃𝜛𝐸 𝜂𝑆 (𝜁𝑠𝑒𝐸 + 𝜁𝑠𝑖𝐼 + 𝜁𝑠𝑎𝐴) 𝑁 𝛿𝐼 (1 − 𝜃 − 𝜎)𝜔𝐸 𝜇𝑄 𝜑𝜚𝑄 𝜎𝛼𝐸 (1 − 𝜑)𝛽𝑄 (1) sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 94 (1), resulting in: 𝑑𝑁 𝑑𝑡 = π − 𝜇𝑁 − 𝛿𝐼. positive invariant region that meets the model (1) is given by ω = {(𝑆(𝑡), 𝐸(𝑡), 𝐴(𝑡), 𝑄(𝑡), 𝐼(𝑡), 𝑅(𝑡)) ∈ 𝑅+ 6 : 𝑁(𝑡) ≤ π 𝜇 }. results and discussion the disease-free equilibrium point and the basic reproductive number the equilibrium point is acquired at, 𝑑𝑆 𝑑𝑡 = 𝑑𝐸 𝑑𝑡 = 𝑑𝐴 𝑑𝑡 = 𝑑𝑄 𝑑𝑡 = 𝑑𝐼 𝑑𝑡 = 𝑑𝑅 𝑑𝑡 = 0 therefore, the system of equation (1) indicates that the dfe point denoted by 𝐸𝑑𝑓𝑒, i.e., 𝐸𝑑𝑓𝑒 = (𝑆0, 0,0,0,0,0) = ( π 𝜇 , 0,0,0,0,0) the basic reproductive number denoted by 𝑅0 is the expected value of infection rate per time unit. the infection occurs in a susceptible population, caused by an infected individual. based on the equation system (1), the article generates equation that involves the classes of exposed population, infected population without symptom, and infected population with symptom, as follows: 𝑑𝐸 𝑑𝑡 = 𝜂𝑆(𝜁𝑠𝑒𝐸 + 𝜁𝑠𝑖𝐼 + 𝜁𝑠𝑎𝐴) 𝑁 − (𝜃𝜛 + 𝜎𝛼 + 𝜇 + 𝜔 − 𝜃𝜔 − 𝜎𝜔)𝐸 𝑑𝐴 𝑑𝑡 = 𝜃𝜛𝐸 − (𝜏 + 𝜇)𝐴 𝑑𝑄 𝑑𝑡 = 𝜎𝛼𝐸 − (𝜑𝜚 + 𝛽 − 𝛽𝜑 + 𝜇)𝑄 𝑑𝐼 𝑑𝑡 = (1 − 𝜃 − 𝜎)𝜔𝐸 + 𝜑𝜚𝑄 − (𝜌 + 𝜇 + 𝛿)𝐼 further, the basic reproductive number is calculated by referring to the job [8]. from the equation (6), the study generates matrix ℱ and 𝒱, i.e. ℱ = [ 𝜂𝑆(𝜁𝑠𝑒𝐸 + 𝜁𝑠𝑖𝐼 + 𝜁𝑠𝑎𝐴) 𝑁 0 0 0 ] dan 𝒱 = [ (𝜃𝜛 + 𝜎𝛼 + 𝜇 + 𝜔 − 𝜃𝜔 − 𝜎𝜔)𝐸 (𝜏 + 𝜇)𝐴 − 𝜃𝜛𝐸 (𝜑𝜚 + 𝛽 − 𝛽𝜑 + 𝜇)𝑄 − 𝜎𝛼𝐸 (𝜌 + 𝜇 + 𝛿)𝐼 − (1 − 𝜃 − 𝜎)𝜔𝐸 − 𝜑𝜚𝑄] using the dfe point in equation (5), jacobian matrix 𝐹 and 𝑉 is generated, i.e. (4) (5) (2) (3) (6) (7) sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 95 𝐹 = [ 𝜂𝜁𝑠𝑒𝑆0 𝑁 0 0 0 𝜂𝜁𝑠𝑎𝑆0 𝑁 0 0 0 0 0 0 0 𝜂𝜁𝑠𝑖𝑆0 𝑁 0 0 0 ] = [ 𝜂𝜁𝑠𝑒 0 0 0 𝜂𝜁𝑠𝑎 0 0 0 0 0 0 0 𝜂𝜁𝑠𝑖 0 0 0 ] and 𝑉 = [ 𝜃𝜛 + 𝜎𝛼 + 𝜇 + 𝜔 − 𝜃𝜔 − 𝜎𝜔 −𝜃𝜛 −𝜎𝛼 −(1 − 𝜃 − 𝜎)𝜔 0 𝜏 + 𝜇 0 0 0 0 𝜑𝜚 + 𝛽 − 𝛽𝜑 + 𝜇 −𝜑𝜚 0 0 0 𝜌 + 𝜇 + 𝛿 ] the basic reproductive number 𝑅0 is acquired from the largest positive eigenvalue of next generation matrix, 𝐾 = 𝐹𝑉−1, i.e. 𝑅0 = 𝜌(𝐹𝑉 −1), 𝜌 = dominant eigenvalue. therefore, it is generated that: 𝑅0 = 𝜂𝜁𝑠𝑒(𝜏 + 𝜇) + 𝜂𝜁𝑠𝑎𝜃𝜛 (𝜏 + 𝜇)(𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔) + 𝜂𝜁𝑠𝑖𝜑𝜚𝛼𝜎 + 𝜂𝜁𝑠𝑖(𝜔 − 𝜎𝜔 − 𝜃𝜔)(𝜇 + 𝛽 − 𝛽𝜑 + 𝜑𝜚) (𝜇 + 𝛿 + 𝜌)(𝜇 + 𝛽 − 𝛽𝜑 + 𝜑𝜚)(𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔) further, 𝑅0 can be stated in the form: 𝑅0 = 𝑅1 + 𝑅2 with 𝑅1 = 𝜂𝜁𝑠𝑒(𝜏 + 𝜇) + 𝜂𝜁𝑠𝑎𝜃𝜛 (𝜏 + 𝜇)(𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔) 𝑅2 = 𝜂𝜁𝑠𝑖𝜑𝜚𝛼𝜎 + 𝜂𝜁𝑠𝑖(𝜔 − 𝜎𝜔 − 𝜃𝜔)(𝜇 + 𝛽 − 𝛽𝜑 + 𝜑𝜚) (𝜇 + 𝛿 + 𝜌)(𝜇 + 𝛽 − 𝛽𝜑 + 𝜑𝜚)(𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔) baseline parameter values parameter value implemented in the equation model (1) involves parameter value as a result of fitting process from several valid references based on real cases of covid19 that has been reported and published. the complete parameter values implemented in the present study are displayed in the following table 1. by referring to the parameter values in table 1 in equation (10), the study acquires estimation of basic reproductive number of 𝑅0 ≈ 3.097. this indicates that every infected individual is potential to infect minimum of three new individuals. sensivity analysis to determine the most optimum approach in suppressing the number of infected individuals, one needs to identify several factors that contribute to the transmission of the (8) (9) (10) (11) sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 96 virus and its prevalence. the first case of covid-19 transmission closely relates to 𝑅0 and persons exposed to covid-19, as of 𝐸, 𝐴, 𝑄, and 𝐼 group. table 1. estimation of parameter values in covid-19 cases parameter description value source 𝜇 𝜂 𝜁𝑠𝑒 𝜁𝑠𝑎 𝜁𝑠𝑖 𝜃 𝜎 𝛼 𝜔 𝜛 𝜚 𝜑 𝜏 𝛽 𝜌 𝛿 natural mortality rate transmission probability during contact rate of contact between susceptible individual and exposed individuals rate of contact between susceptible individual and asymptotically infected individuals rate of contact between susceptible individual and symptomatic infected individuals proportion of asymptotically infected individuals proportion of quarantined exposed individuals movement rate of exposed individuals to quarantined individuals transmission rate after incubation period and transferred to symptomatic infected class transmission rate after incubation period and transferred to asymptotically infected class movement rate of quarantined individual to symptomatic infected individuals proportion of quarantined symptomatic infected individuals recovery rate of asymptotically infected individuals and transferred to r class recovery rate of quarantined individuals and transferred to r class recovery rate of symptomatic infected individualsand transferred to r class mortality rate due to covid-19 3.57 × 10−5 0.2 0.09 0.07 0.05 0.12 0.04 0.13266 0.005 0.00048 0.1259 0.05 0.854302 0.11624 0.33029 1.78 × 10−5 estimation estimation estimation estimation [4] estimation estimation [3] [4] [4] [3] estimation [4] [2] [2] [3] in this section, the study calculates the sensitivity index of each parameter model that correlates with the basic reproductive numbers, 𝑅0. this index provides information about the importance of each parameter to the model representing the transmission of covid-19. the index is used to identify the parameter that has the most significant impact on 𝑅0which is later served as the intervention target. the parameter with a high impact in 𝑅0shows that the parameter has a dominant influence on the endemicity of covid-19. the parameter of the sensitivity index towards the basic reproductive number is calculated using the approach similar to the one seen in [9]. based on the explicit form of 𝑅0in equation (10), the study derivate the analytic expression for the sensitivity 𝑅0to the parameter 𝑝 that is 𝐶𝑝 𝑅0 = 𝜕𝑅0 𝜕𝑝 × 𝑝 𝑅0 . (12) sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 97 table 2. model parameter index that is related to basic reproductive number parameter description sensitivity index 𝜂 𝜁𝑠𝑒 𝛼 𝜎 𝜔 𝜛 𝜇 𝜁𝑠𝑖 𝜌 𝜑 𝜚 𝛽 𝜃 𝜁𝑠𝑎 𝜏 𝛿 transmission probability during contact rate of contact between susceptible individual and exposed individuals movement rate of exposed individuals to quarantined individuals proportion of quarantined exposed individuals transmission rate after incubation period and transferred to symptomatic infected class transmission rate after incubation period and transferred to asymptotically infected class natural mortality rate rate of contact between susceptible individual and symptomatic infected individuals recovery rate of symptomatic infected individualsand transferred to r class proportion of quarantined symptomatic infected individuals movement rate of quarantined individual to symptomatic infected individuals recovery rate of quarantined individuals and transferred to r class proportion of asymptotically infected individuals rate of contact between susceptible individual and asymptotically infected individuals recovery rate of asymptotically infected individuals and transferred to r class mortality rate due to covid-19 in class of asymptotically infected individuals +1.000 +0.999 −0.911 −0.908 −0.0715 −0.0098 −0.0061 +0.0012 −0.0011 +0.0005 +0.0005 −0.0005 +0.0004 +0.00005 −0.00005 −6.4 × 10−8 by referring to the formulation of equation (12) and parameter value in table 1, it is generated the sensitivity index of each parameter in the basic reproductive number 𝑅0, presented in table 2. as an example, the sensitivity index of 𝑅0 towards parameter 𝜁𝑠𝑒 and 𝛼 is 𝐶𝜁𝑠𝑒 𝑅0 = 𝜕𝑅0 𝜕𝜁𝑠𝑒 × 𝜁𝑠𝑒 𝑅0 = 𝜂 (𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔) × 𝜁𝑠𝑒 𝑅0 = 𝜂𝜁𝑠𝑒 (𝜇 + 𝜔 + 𝜃𝜛 + 𝛼𝜎 − 𝜃𝜔 − 𝜎𝜔)𝑅0 = 0.999 𝐶𝛼 𝑅0 = 𝜕𝑅0 𝜕𝛼 × 𝜁𝛼 𝑅0 sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 98 = 𝜂𝜎 (−ζse − ζsa𝜃𝜛 𝜇+𝜏 + ζsi(− 𝛼𝜚𝜎𝜑 𝛽+𝜇−𝛽𝜑+𝜚𝜑 +(𝜃+𝜎−1)𝜔) 𝛿+𝜇+𝜌 + ζsi𝜚𝜑(𝜇+𝜃𝜛+𝛼𝜎+𝜔−(𝜃+𝜎)𝜔) (𝛿+𝜇+𝜌)(𝛽+𝜇−𝛽𝜑+𝜚𝜑) ) (𝜇 + 𝜃𝜛 + 𝛼𝜎 + 𝜔 − (𝜃 + 𝜎)𝜔) 2 × 𝜁𝛼 𝑅0 = −0.911 the sensitivity index in table 2 sequentially shows the parameter with the highest sensitivity to the lowest sensitivity. in general, four parameters are significant to the changes in the basic reproductive number. those parameters include the chances of transmission during contact (𝜂), the rate of contact between susceptible individuals and exposed individuals (𝜁𝑠𝑒 ), the movement rate of individuals to the quarantine class (𝛼), and the proportion of quarantined exposed individuals (𝜎). the parameter 𝜂 and 𝜁𝑠𝑒 has a positive sensitivity index, while the parameter 𝛼 and 𝜎 have a negative sensitivity index. parameters with a positive sensitivity index represent the positive significance in the increase of basic reproductive numbers. thereby, increasing (or decreasing) the value of the parameter, while the other parameters’ value remains the same, will contribute to the increases (or decreases) in the basic reproductive numbers. parameters with a negative sensitivity index represent the negative significance to the increase of basic reproductive numbers. in other words, increasing (or decreasing) the value of the parameter, while the other parameters’ value remains the same, will contribute to the decreases (or increases) in the basic reproductive numbers. the sensitivity index shows that contact with exposed individuals and the chance of transmission during the contact is the most dominant parameters contributing to the covid-19 transmission. on the one hand, the proportion of quarantined exposed individuals is the parameter that suppresses the transmission of the disease the most. since 𝐶𝜁𝑠𝑒 𝑅0 = +0.999, increasing (or decreasing) the parameter 𝜁𝑠𝑒 by 10% will result in a increases (or decreases) of the value 𝑅0 by 9.99%. the result that 𝐶𝛼 𝑅0 = −0.911 signifies that increasing (or decreasing) the parameter 𝛼 by 10% will lead to a decreases (or increases) in the value of 𝑅0 by 9.11%. interrupting the rate of transmission with exposed individuals and maximizing quarantine for exposed individuals and suspects are crucial. however, inadequate knowledge and lack of medical kits to detect those who have been contacted with covid19 are now hindering the countermeasures for combating the disease. it can be said that there is no strong evidence to determine those who have been contacted with covid-19 and those who are not carrying the virus. due to the situations, preventive measures, such as social and physical distancing, are still the only option to halt the transmission of covid-19. conclusions the index of the sensitivity parameter shows that the parameter 𝜂 and 𝜁𝑠𝑒 that represents the chance of transmission during the contact and the rate of contact between the susceptible individuals and exposed individuals have the most dominant sensitivity to raise the endemicity of covid-19. on the other hand, the parameter α and σ, which represent the movement rate of exposed individuals to the quarantine and the proportion of quarantined exposed individuals, are the ones that can decrease the endemicity of covid-19 the most. the effectiveness of the measurement against covid-19, therefore, can be assured by suppressing the interaction rate between susceptible individuals and sensitivity analysis of mathematical model of coronavirus disease (covid-19) transmission resmawan 99 exposed individuals and maximizing treatment in the quarantine for those who are infected with the disease. all these attempts can be optimized by adhering to the health protocol, such as by keeping space between oneself and the other, or well-known as social or physical distancing. acknowledgments this study is financially supported by lppm universitas negeri gorontalo, as stated in the decree number b/105/un47.d1/pt.01.03/2020. references [1] b. tang et al., “estimation of the transmission risk of the 2019-ncov and its implication for public health interventions,” journal of clinical medicine, vol. 9, no. 2, p. 462, 2020, doi: 10.3390/jcm9020462. [2] b. tang et al., “the effectiveness of quarantine and isolation determine the trend of the covid-19 epidemics in the final phase of the current outbreak in china,” int j infect dis, 2020, doi: 10.1016/j.ijid.2020.03.018. [3] b. tang, n. l. bragazzi, q. li, s. tang, y. xiao, and j. wu, “an updated estimation of the risk of transmission of the novel coronavirus (2019-ncov),” infectious disease modelling, vol. 5, pp. 248–255, 2020, doi: 10.1016/j.idm.2020.02.001. [4] m. a. khan and a. atangana, “modeling the dynamics of novel coronavirus (2019ncov) with fractional derivative,” alexandria engineering journal, 2020, doi: 10.1016/j.aej.2020.02.033. [5] t. m. chen, j. rui, q. p. wang, z. y. zhao, j. a. cui, and l. yin, “a mathematical model for simulating the phase-based transmissibility of a novel coronavirus,” infectious diseases of poverty, vol. 9, no. 1, pp. 1–8, 2020, doi: 10.1186/s40249-020-00640-3. [6] h. sun, y. qiu, h. yan, y. huang, y. zhu, and s. x. chen, “tracking and predicting covid-19 epidemic in china mainland,” medrxiv, no. february, p. 2020.02.17.20024257, 2020, doi: 10.1101/2020.02.17.20024257. [7] t. kuniya, “prediction of the epidemic peak of coronavirus disease in japan, 2020,” journal of clinical medicine, vol. 9, no. 3, p. 789, 2020, doi: 10.3390/jcm9030789. [8] p. van den driessche and j. watmough, “further notes on the basic reproduction number,” in mathematical epidemiology, victoria: springer, 1945. [9] n. chitnis, j. m. hyman, and j. m. cushing, “determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” bulletin of mathematical biology, vol. 70, no. 5, pp. 1272–1296, jul. 2008, doi: 10.1007/s11538-008-9299-0. abstract introduction methods results and discussion the disease-free equilibrium point and the basic reproductive number baseline parameter values sensivity analysis conclusions acknowledgments references finitely generated module over a discrete valuation domain cauchy –jurnal matematika murni dan aplikasi volume 6(1) (2019), pages 18-26 p-issn: 2086-0382; e-issn: 2477-3344 submitted: april 24, 2019 reviewed: october 18, 2019 accepted: november 30, 2019 doi: http://dx.doi.org/10.18860/ca.v6i1.6798 a finitely generated module over a discrete valuation domain dwi mifta mahanani department of mathematics brawijaya university email: mdwimifta@yahoo.com abstract this article discusses some properties of a finitely generated module over a pid and that over a valuation domain which are equivalent. in general, pid is not a valuation domain and also a valuation domain is not a pid. but there is a domain which is pid as well as a valuation domain. the domain is called a discrete valuation domain (dvr). therefore, the equivalent properties which will be considered are obtained from a finitely generated module over a dvr. those properties are related to a decomposition into direct sum of cyclic submodules and height of an element of the module. keywords: dvr; finitely generated module; height of element; pid; valuation domain introduction in general, the properties of an 𝑅 −module depend on the properties of ring 𝑅. for example, every finitely generated module over pid can be decomposed into a direct sum of cyclic submodules. however, that property does not satisfy in every finitely generated modules over a valuation domains. that is to say, not all of those modules can be decomposed into a direct sum of cyclic submodules. moreover, those which can be decomposed are those which the goldie dimension is equal to the length of their series of 𝑅𝐷 −submodules. the example of finitely generated modules over a valuation domain which can not be decomposed can be seen in [5]. in general, a pid is not a valuation domain. an integral domain ℤ, for instance, is a pid but not a valuation domain because there are two ideals of ℤ, ℤ2 and ℤ5, which are not subset each other. otherwise, a valuation domain is not a pid. note the example below. example let 𝐾 be a field with transcendental 𝑥, 𝐾𝑖 = 𝐾 [𝑥 1 2𝑖 ] , 𝑃𝑖 = 𝐾 [𝑥 1 2𝑖 ] 𝑥 1 2𝑖 where 𝑖 = 0,1,2, ⋯ . it can be seen obviously that 𝐾𝑖 is a pid and 𝑃𝑖 is it’s prime ideal. moreover, there are domains 𝑂𝑖 = 𝐾𝑖 𝑃𝑖 = { 𝑓(𝑥 1 2𝑖 ) 𝑔(𝑥 1 2𝑖 ) |𝑓 (𝑥 1 2𝑖 ) ∈ 𝐾𝑖 , 𝑔 (𝑥 1 2𝑖 ) ∈ 𝐾𝑖 − 𝑃𝑖 }. (1) http://dx.doi.org/10.18860/ca.v6i1.6798 a finitely generated module over a discrete valuation domain dwi mifta mahanani 19 a domain 𝑂 =∪𝑖 𝑂𝑖 is a non-noetherian valuation domain. hence 𝑂 is not a pid. the detail of the example can be seen in [3] and [6]. meanwhile, there is a domain which is a pid as well as a valuation domain. in [2], this domain is called discrete valuation domain (dvr). because a dvr is an intersection of two different domains, then there must be some properties of finitely generated modules over a pid and those over a valuation domain which are equivalent. these equivalent properties will be discussed in the result section. materials and methods the method which is used to write this article is by literature studying on some papers and books. there are some properties of a module over a dvr which have been discussed in the papers and the books. it will be developed some new properties by considering the previous properties. before we discuss the results, we have to know some definitions and properties that we use in the proof of the results. the readers are assumed to know about the definition of a module over a ring and some basic properties of a module. therefore, in this article there are no definition and basic properties of a module over a ring. definition 1 [1, pp. 115] let 𝑀 be a finitely generated 𝑅 −module. the minimal number of generators of 𝑀 is denoted by 𝜇(𝑀). if 𝑀 is not finitely generated as 𝑅 −module, then 𝜇(𝑀) = ∞. unlike in vector spaces, in modules, not all of those have a basis. for those which have a basis are called free modules as which have been explained in [8]. we follow [1] that every basis of a free module over a pid contains 𝜇(𝑀) elements . there is a submodule of modules over a valuation domain which has an important role in those modules. this kind of submodule is an 𝑅𝐷 −submodule. the definition of this submodule is presented below. definition 2 [4, pp. 38] let 𝑁 be an 𝑅 −submodule of a module 𝑀. this submodule is a relatively divisible submodule (𝑅𝐷 −submodule) of 𝑀 if 𝑟𝑁 = 𝑁 ∩ 𝑟𝑀 for all 𝑟 ∈ 𝑅. there are some properties of 𝑅𝐷 −submodule which will be used to prove the equivalence of two theorems in a finitely generated free module over a pid and in that over a valuation domain. these properties will be given without any proof. lemma 3 [4, pp. 39] let 𝑀 be an 𝑅 −module and 𝐿, 𝑁 are 𝑅 − submodules of 𝑀. then these properties are satisfied. 1 direct summand of 𝑀 is an 𝑅𝐷 −submodule of 𝑀. 2 if 𝐿 ≤ 𝑁 ≤ 𝑀 and 𝐿, 𝑁 are 𝑅𝐷 −submodules of 𝑀, then 𝐿 is an 𝑅𝐷 −submodule of 𝑁. using those properties of 𝑅𝐷 −submodule to proof that a submodule is an 𝑅𝐷 −submodule is easier than using the definition. the examples related to a finitely generated module over a discrete valuation domain dwi mifta mahanani 20 𝑅𝐷 −submodule can be seen below. 1. ring ℤ15 can be assumed to be a ℤ −module. note that ℤ15 = ℤ3̅ ⊕ ℤ5̅. then by lemma 3, ℤ3̅ and ℤ5̅ are 𝑅𝐷 −submodules of ℤ15. 2. a domain ℤ can be considered as a module over itself. we know that every submodule of this module is an ideal of ℤ. furthermore, every ideal of ℤ is generated by single element (principal ideal). therefore every two nonzero ideals of ℤ always have a nonzero intersection element. hence, ℤ as a ℤ −module does not have any proper nonzero 𝑅𝐷 −submodule. in a dvr, there is a unique element (up to associates) which has an important role to define a height of an element of a module over this domain. definition 4 [1, pp.98] let 𝑅 be a ring and 𝑟 ∈ 𝑅. an element 𝑟 is a non nilpotent element of 𝑅 if there is no positive integer 𝑛 such that 𝑟𝑛 = 0. because of the uniqueness of a non nilpotent in a dvr, we can write every element in this domain by a multiplication of a unit element and this non nilpotent element. moreover, this can also be written uniquely. note the lemma below. lemma 5 [7, pp. 5009] let 𝑎 be a nonzero element of a dvr 𝑅. then 𝑎 can be written uniquely as 𝑎 = 𝑢𝑝𝑘 where 𝑢 is a unit of 𝑅, 𝑝 is a non nilpotent element of 𝑅 and 𝑘 ∈ ℤ≥0. the theorem below explains about the decomposition of a finitely generated module over a pid into a direct sum of finite cyclic submodules. moreover, the annihilator of generators of these cyclic submodules can be arranged such that they form a non decreasing chain. note this theorem below. theorem 6 [1, pp. 156] let 𝑀 ≠ 0 be a finitely generated module over a pid 𝑅. if 𝜇(𝑀) = 𝑛, then 𝑀 is isomorphic to 𝑅𝑤1 ⊕ 𝑅𝑤2 ⊕ ⋯ ⊕ 𝑅𝑤𝑛 (2) such that 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑤𝑛) ⊆ ⋯ ⊆ 𝐴𝑛𝑛(𝑤2) ⊆ 𝐴𝑛𝑛(𝑤1) ≠ 𝑅. (3) moreover, for 1 ≤ 𝑖 ≤ 𝑛, 𝐴𝑛𝑛(𝑤𝑖 ) = 𝐴𝑛𝑛 ( 𝑀 𝑅𝑤𝑖+1 + ⋯ + 𝑅𝑤𝑛 ). (4) different from a finitely generated module over a pid, in that over a valuation domain, we can find a series of 𝑅𝐷 −submodules. moreover, the chain can be chosen so a finitely generated module over a discrete valuation domain dwi mifta mahanani 21 that the annihilators of the factors forms a non decreasing chain. theorem 7 [5, pp. 1798] let 𝑀 ≠ 0 is a finitely generated module over a valuation domain 𝑅. there exist a series of 𝑅𝐷 −submodule of 𝑀 0 = 𝑀0 < 𝑀1 < 𝑀2 < ⋯ < 𝑀𝑛 = 𝑀 (5) such that 1. every 𝑀𝑖 is an 𝑅𝐷 −submodule of 𝑀, 2. every factor module 𝑀𝑖+1/𝑀𝑖 is cyclic. moreover, it can be chosen that 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑀1) ≤ 𝐴𝑛𝑛 ( 𝑀2 𝑀1 ) ≤ ⋯ ≤ 𝐴𝑛𝑛 ( 𝑀𝑛 𝑀𝑛−1 ). (6) there is a possibility for a finitely generated module over a valuation domain to have more than one 𝑅𝐷 −submodules series. nevertheless, there is another property of this module stating that every two 𝑅𝐷 −submodules series are isomorphic. hence, we can choose any series of 𝑅𝐷 −submodule. now consider a finitely generated free module over a pid. in this module, an 𝑅𝐷 −submodule is the same as a complemented submodule. note the theorem below. theorem 8 [1, pp. 172] a submodule of a finitely generated free module over a pid is an 𝑅𝐷 −submodule if and only if it is complemented. theorem 8 will be used to construct a direct sum of cyclic submodules of a finitely generated free module over a dvr. theorem 9 [5, pp. 1801] let 𝑀 be a finitely generated 𝑅 −module where 𝑅 is a valuation domain. let 0 ≠ 𝑥 ∈ 𝑀 be any element of 𝑀. there exist 𝑟 ∈ 𝑅 such that 𝑥 ∈ 𝑟𝑀 − 𝑟𝑃𝑀 where 𝑃 is a maximal ideal of 𝑅. the theorem above ensures the existence of an element 𝑟 ∈ 𝑅 which has a property that 𝑥 always in 𝑟𝑀 but not in 𝑟𝑃𝑀. if there is another element 𝑠 ∈ 𝑅 which satisfies the property in the theorem 9, then the ideal 𝑅𝑟 will be equal to ideal 𝑅𝑠. theorem 9 and that argument motivate a definition of height of an element in a module over a valuation domain. the definition is presented below. definition 10 [5, pp. 1802] let 𝑀 be a finitely generated 𝑅 −module where 𝑅 is a valuation domain. let 0 ≠ 𝑥 ∈ 𝑀 be any element of 𝑀. height of 𝑥, which is denoted by ℎ𝑀(𝑥), is defined as an ideal 𝑅𝑟 such that 𝑥 ∈ (𝑟𝑀 − 𝑟𝑃𝑀) where 𝑃 is a maximal ideal of 𝑅. height of 0 is defined as a zero ideal. the definition of height of an element of a finitely generated module over a a finitely generated module over a discrete valuation domain dwi mifta mahanani 22 valuation domain and that over a dvr are different. in a module over a valuation domain, the height of an element is in the form of an ideal. however, in a module over a dvr, it is a non negative integer. in fact, dvr is a special case of valuation domain. those two statements actually are equivalent. this equivalence will be proven in the next theorem. but before doing the proof, note the definition of height of an element of a module over a dvr. definition 11 [7, pp. 5033] let 𝑎 ∈ 𝑀 be any nonzero element where 𝑀 is an 𝑅 −module over a dvr. height of 𝑎 is the greatest non negative integer 𝑛 which equation 𝑝𝑛𝑥 = 𝑎 is soluble. if there is no such integer, it means the height is ∞. element 𝑝 in the equation is a non nilpotent element in 𝑅. from the definition above, the height of zero element in 𝑀 is ∞. it is because there is no greatest non negative integer 𝑛 such that 𝑝𝑛𝑥 = 0 is soluble. results and discussion we know that a dvr is a noetherian valuation domain. therefore, every property in a module over a valuation domain always satisfies in that over a dvr. one of those properties is the existence of a series of rd-submodules. note an illustration below. let 𝑀 be a finitely generated 𝑅 −module over a pid and 𝜇(𝑀) = 𝑛. it will be constructed a series of 𝑅𝐷 −submodule of 𝑀. based on theorem 6, module 𝑀 is isomorphic to 𝑅𝑤1 ⊕ 𝑅𝑤2 ⊕ ⋯ ⊕ 𝑅𝑤𝑛. (7) with this direct sum, we can make a series. observe this series below. 0 = 𝑀0 < 𝑅𝑤𝑛 < 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 < ⋯ < 𝑅𝑤𝑛 ⊕ ⋯ ⊕ 𝑅𝑤2 ⊕ 𝑅𝑤1 = 𝑀. (8) based on lemma 3, submodule 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 where 1 ≤ 𝑘 ≤ 𝑛 is an 𝑅𝐷 −submodule of 𝑀. hence, there exist a series of 𝑅𝐷 −submodule of 𝑀. observe that ( 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ) (9) is cyclic with the generator 𝑅(𝑤𝑘−1 + (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 )) for 1 ≤ 𝑙 ≤ 𝑛 − 1. let 𝑟 ∈ 𝐴𝑛𝑛 ( 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ). (10) then 𝑟𝑤𝑘−1 ∈ (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ). therefore, 𝑟𝑤𝑘−1 ∈ 𝑅𝑤𝑘−1 ∩ (𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ). (11) a finitely generated module over a discrete valuation domain dwi mifta mahanani 23 because 𝑅𝑤𝑘−1 + 𝑅𝑤𝑘 + ⋯ + 𝑅𝑤𝑛 is a direct sum, then it causes 𝑟𝑤𝑘−1 = 0. it means that 𝑟 ∈ 𝐴𝑛𝑛(𝑤𝑘−1). based on theorem 6, 𝐴𝑛𝑛(𝑤𝑘−1) = 𝐴𝑛𝑛 ( 𝑀 𝑅𝑤𝑘+⋯+𝑅𝑤𝑛 ). it is obtained that 𝑟𝑤𝑘−2 = 𝑟𝑘 𝑤𝑘 + ⋯ + 𝑟𝑛𝑤𝑛 for some 𝑟𝑖 ∈ 𝑅. note that 𝑟𝑤𝑘−2 + (𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1) = (𝑟𝑘 𝑤𝑘 + ⋯ + 𝑟𝑛𝑤𝑛) + (𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1) = 𝑅𝑤𝑛 + ⋯ + 𝑅𝑤𝑘−1. (12) hence 𝑟 ∈ 𝐴𝑛𝑛 ( 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−2 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1 ). (13) therefore, it is proved that 𝐴𝑛𝑛 ( 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘 ) ≤ 𝐴𝑛𝑛 ( 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−2 𝑅𝑤𝑛 ⊕ 𝑅𝑤𝑛−1 ⊕ ⋯ ⊕ 𝑅𝑤𝑘−1 ). (14) now, let 0 ≠ 𝑀 is a finitely generated free module over a dvr 𝑅. based on theorem 7, there exist a series of 𝑅𝐷 −submodule 0 = 𝑀0 < 𝑀1 < 𝑀2 < ⋯ < 𝑀𝑛 = 𝑀 (15) with every factor module is cyclic. note that 𝑀𝑖 is an 𝑅𝐷 −submodule of 𝑀𝑖+1 because 𝑀𝑖 is an 𝑅𝐷 −submodule of 𝑀 for all 𝑖. based on theorem 8 𝑀𝑖 is complemented in 𝑀𝑖+1. therefore it can be written 𝑀𝑖+1 = 𝑀𝑖 ⊕ 𝑁𝑖+1. (16) note that 𝑀𝑖+1/𝑀𝑖 ≅ 𝑁𝑖+1 and 𝑀𝑖+1/𝑀𝑖 is cyclic for every 𝑖. hence 𝑁𝑖+1 is cyclic. it can be written 𝑁𝑖+1 = 𝑅𝑎𝑖+1. therefore, it is obtained 𝑀 = 𝑀𝑛−1 ⊕ 𝑅𝑎𝑛 = 𝑀𝑛−2 ⊕ 𝑅𝑎𝑛−1 ⊕ 𝑅𝑎𝑛 ⋮ = 𝑅𝑎1 ⊕ 𝑅𝑎2 ⊕ ⋯ ⊕ 𝑅𝑎𝑛. (17) now consider that 𝐴𝑛𝑛 ( 𝑀𝑖+1 𝑀𝑖 ) = 𝐴𝑛𝑛(𝑁𝑖+1) = 𝐴𝑛𝑛(𝑎𝑖+1). if we choose the series of 𝑅𝐷 −submodule so that 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑀1) ≤ 𝐴𝑛𝑛 ( 𝑀2 𝑀1 ) ≤ ⋯ ≤ 𝐴𝑛𝑛 ( 𝑀𝑛 𝑀𝑛−1 ), (18) a finitely generated module over a discrete valuation domain dwi mifta mahanani 24 then we have a chain 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑎1) ≤ 𝐴𝑛𝑛(𝑎2) ≤ ⋯ ≤ 𝐴𝑛𝑛(𝑎𝑛). (19) by renumbering, that is 𝑎𝑛−𝑖 = 𝑤𝑖+1, we have a chain 𝐴𝑛𝑛(𝑀) = 𝐴𝑛𝑛(𝑤𝑛) ≤ 𝐴𝑛𝑛(𝑤𝑛−1) ≤ ⋯ ≤ 𝐴𝑛𝑛(𝑤1). (20) illustration above shows that theorem 6 and theorem 7 are equivalent if 𝑀 is a finitely generated free module over a dvr. theorem 12 in a finitely generated free module over a dvr, theorem 6 and theorem 7 are equivalent. proof. ∎ beside the equivalent statement, there are also some properties of the module which related to the property of the domain. in this case, we will assume the domain is a valuation domain. as we know, the properties will also satisfy if the domain is a dvr. the comparable property of ideals of a valuation domain is inherited to submodules of a cyclic module over a valuation domain. note the lemma below. lemma 13 let 0 ≠ 𝑅𝑚 is a cyclic module over a valuation domain 𝑅. if 𝑁 and 𝑁′ are submodules of 𝑅𝑚, then 𝑁 ⊆ 𝑁′ or 𝑁′ ⊆ 𝑁. proof. let 𝐼 = {𝑎 ∈ 𝑅|𝑎𝑚 ∈ 𝑁} and 𝐽 = {𝑏 ∈ 𝑅|𝑏𝑚 ∈ 𝑁′}. these two sets are ideals of 𝑅. we know that 𝑅 is a valuation domain. hence we have 𝐼 ⊆ 𝐽 or 𝐽 ⊆ 𝐼. let 𝑥 = 𝑎𝑚 ∈ 𝑁 be any element. then 𝑎 ∈ 𝐼. if 𝐼 ⊆ 𝐽, then 𝑎𝑚 ∈ 𝑁′. therefore, 𝑁 ⊆ 𝑁′. we can prove analogously for 𝑁′ ⊆ 𝑁. ∎ by lemma 13, a cyclic module over a valuation domain can not be decomposed into a direct sum of its nonzero submodules. hence a cyclic module over a valuation domain is an indecomposable module. from this fact, we have this corollary below. corollary 14 if 𝑀 is a finitely generated module over a dvr 𝑅, then 𝑀 is isomorphic to direct sum of cyclic indecomposable submodules. proof. use theorem 6 and lemma 13. ∎ a finitely generated module over a discrete valuation domain dwi mifta mahanani 25 lemma 15 non nilpotent element of a dvr 𝑅 is a prime element. proof. let 𝑝 be a non nilpotent element of 𝑅 and 𝑝|(𝑎𝑏) where 𝑎, 𝑏 ∈ 𝑅. to show that 𝑝 is a prime element of 𝑅, we need to show that 𝑝|𝑎 or 𝑝|𝑏. based on lemma 5, let 𝑎 = 𝑢1𝑝 𝑛1 and 𝑏 = 𝑢2𝑝 𝑛2 with 𝑢1, 𝑢2 are unit elements of 𝑅 and 𝑛1 ∈ ℤ>0 or 𝑛2 ∈ ℤ>0. then it is obvious that 𝑝|𝑢1𝑝 𝑛1 or 𝑝|𝑢2𝑝 𝑛2 . hence 𝑝 is a prime element of 𝑅. ∎ if there is another prime element of 𝑅, namely 𝑞, then every element of 𝑅 can also be written in the form of 𝑢𝑞𝑛 uniquely where 𝑢 is a unit of 𝑅 and 𝑛 ∈ ℤ≥0. note this explanation below. if 𝑞 is another prime element of 𝑅, then 𝑅𝑞 is a maximal ideal of 𝑅. the uniqueness of maximal ideal of 𝑅 makes 𝑅𝑝 = 𝑅𝑞. hence we have 𝑝 = 𝑟𝑞 for some 𝑟 ∈ 𝑅. it means 𝑝|𝑟𝑞. because 𝑝 is prime element, then 𝑝|𝑟 or 𝑝|𝑞. if 𝑝|𝑟, then 𝑝𝑠 = 𝑟 for some 𝑠 ∈ 𝑅. we have 𝑝 = 𝑝𝑠𝑞. it means 𝑠𝑞 = 1. therefore 𝑞 is a unit element of 𝑅. it is contradiction because 𝑞 is prime element. therefore, we have 𝑝|𝑞. let 𝑝𝑡 = 𝑞 for some 𝑡 ∈ 𝑅. then we have 𝑝 = 𝑟𝑡𝑝. it means 𝑟𝑡 = 1. hence, 𝑟 is a unit of 𝑅. therefore, 𝑝 and 𝑞 are associate. definition 10 is defined in a finitely generated module over a valuation domain. it means the definition satisfies also for that over a dvr. hence the definition 10 and definition 11 must be equivalent for a dvr. but before proving this equivalence, we need the property below. lemma 16 let 𝑅 be a dvr and 𝑎 is a non nilpotent element of 𝑅. then 𝑅𝑎 is a maximal ideal of 𝑅. proof. by lemma 15, 𝑎 is a prime element of 𝑅. then 𝑅𝑎 is a prime ideal which is a maximal ideal of 𝑅. ∎ theorem 17 definition 10 and definition 11 are equivalent in a finitely generated module over a dvr. proof. let 𝑃 be a maximal ideal of 𝑅 and 𝑎 be any nonzero element of 𝑀. because 𝑅 is a dvr, then 𝑃 = 𝑅𝑝 for 𝑝 is a non nilpotent element of 𝑅. based on definition 10 𝑎 ∈ 𝑟𝑀 − 𝑟𝑃𝑀 for some 𝑟 ∈ 𝑅. we can write 𝑎 = 𝑟𝑚 = 𝑢𝑝𝑛 𝑚 with 𝑢 is a unit of 𝑅, 𝑛 ∈ ℤ≥0, 𝑚 ∈ 𝑀 and 𝑟 = 𝑢𝑝𝑛. because 𝑎 ∉ 𝑟𝑃𝑀 = 𝑟𝑝𝑀, then 𝑎 ∉ 𝑝𝑛+1𝑀. hence 𝑛 is the greatest non negative integer such that 𝑝𝑛𝑥 = 𝑎 have a solution. now let 𝑛 be the greatest non negative integer such that 𝑝𝑛𝑥 = 𝑎 is soluble. it means that 𝑎 ∈ 𝑝𝑛𝑀 − 𝑝𝑛+1𝑀. hence based on definition 10, the height of 𝑎 is 𝑅𝑝𝑛. ∎ a finitely generated module over a discrete valuation domain dwi mifta mahanani 26 conclusions in a finitely generated free module over a dvr, we can make a series of it’s 𝑅𝐷 −submodules by arranging it’s cyclic submodules which are the summand of it’s decomposition. furthermore, from this series, we also get a special series which has a property that it’s sequence of the annihilator of it’s factor is ordered non-decreasingly. otherwise, if we have a series of 𝑅𝐷 −submodule, we can decompose the module into a direct sum of cyclic submodules. moreover, these cyclic submodules are indecomposable submodules. besides, the height of an element of a finitely generated module over a dvr can be considered as an ideal of it’s domain or a non negative integer which is the prime power of a generator of the ideal. references [1] adkins, a.a. and weintraub, s.h., "algebra an approach via module theory", springer-verlag,1992. [2] dummit, d.s. and foote, r.m., "abstract algebra", usa: john wiley & sons inc, 2004. [3] knaf, h. (https://math.stackexchange.com/users/2479/hagen-knaf), "examples of non-noetherian valuation rings", url (version: 2014-09-02): https://math.stackexchange.com/q/363201. [4] laszlo, f. and luigi, s, "modules over non-noetherian domains", rhode island: american mathematical society, 2001. [5] luigi, s. and paolo, z., "finitely generated modules over valuation ring". comm. algebra. 12:15, pp. 1795-1812. 1984. [6] mahanani, d.m., "thesis: karakterisasi dekomposisi modul yang dibangun secara hingga atas daerah valuasi", itb, 2014. [7] piotr k., askar a.t., "modules over discrete valuation domains i", journal of mathematics science, volume 145, no.4, pp. 5008-5048, 2007. [8] roman, s., "advanced linear algebra third edition", springer, 2000. https://math.stackexchange.com/q/363201 the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 161-168 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 03 july 2018 reviewed: 15 august 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5881 the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah1, suharto2, arief wibowo2, bambang widjanarko otok3 1health polytechnic banjarmasin, ministry of health & ph.d student faculty of public health, airlanga university, surabaya 2 department of biostatistics and demography, faculty of public health, airlanga university, surabaya, 3laboratory of environmental and health statistic, institut teknologi sepuluh nopember, surabaya, indonesia email: 1mahpolah@gmail.com, 3dr.otok.bw@gmail.com abstract acute (rti) is still an important health problem because the cause of the death of infants and children under five high enough, 1 from 4 death that happens. the purpose of this research examines the factors that affect the genesis acute (rti) using poisson regression approach with estimates of the maximum likelihood estimator (mle) and generalized method moment (gmm). this research done in the area of health clinic in south kalimantan. the results of the study showed that the estimates of the gmm method on poisson regression model gives better performance in terms of the significance of the parameters than the mle method. the factors that affect an increasing number of the prevalence of acute (rti) a region namely persentase breast feeding non-exclusive (0.0279), the percentage of low birth weight (0.0569), the percentage of shelter density (0.028), the percentage of the existence of smoker family members in the house (0.0308), the percentage of immunization is not complete (0.0193). while the factors that affect a downturn in the number of the prevalence of acute (rti) in a region which is the percentage of the number of infants less than 2 (0.0364), the percentage of normal nutrition status (0.0224), the percentage of mothers education on high school (0.0339), and the percentage of social economy (ump enough to top) (0.0194). keywords: poisson regression, mle, gmm, acute (rti) introduction acute respiratory tract infection (acute (rti)) is a disease of the respiratory tract with special attention on the lung inflammation (phenomonia), and not the disease ear and throat. acute (rti) is acute respiratory tract, this term covers three basic namely, respiratory tract infections and acute. the infection is the entry of germs or microorganisms into the human body and multiply so that cause symptoms of disease. the respiratory tract is an organ of such as the nose until the alveoli and adneksnya organs such as the sinus sinus middle ear cavity and pleora. acute (rti) are anatomically include infection of the respiratory tract infection [1]. each child is expected to experience 3-6 episodes acute (rti) every year, 40 % -60 % from the visit in the clinic is by disease acute (rti) [2]. mailto:mahpolah@gmail.com the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 162 acute (rti) more occurs in developing countries compared to developed countries with the percentage of each by 25%-30% and 10%-15%. the death of infants due to acute (rti) in southeast asia as much as 2.1 million infants in 2004. india, bangladesh, indonesia and myanmar are a country with the case of the death of infants due to acute (rti) majority votes [3]. each year the number of patients with acute (rti) in kalsel shows a significant increase, 2010 is no more than 100 thousand cases occurred in south kalimantan. the case of acute (rti) dyspnea, data public health service south kalimantan until october 2010 find as much as 963 cases spread in 13 districts. then to acute (rti) not breathing difficulty there are 119.350 cases, also spread in 13 regencies/cities [4]. according to the data from the district health office tanah bumbu years 2014 on the moon january-december acquired infants who could anticipate acute (rti) as much as 27.457 [5]. the research that related with acute (rti), [6] and [7], gmm estimation method on logistic regression model gives better performance in terms of the significance and interpretation than mle method. the prevalence of acute (rti) on children 6-12 months more often occurs in children who are not given breast milk, on children with low birth weight (bblr) (< 2500 grams). according [8], which means there is a correlation between the level of zinc hair with genesis acute (rti), and there is a correlation that means between the level of zinc with genesis diarrhea in infants stunting of entities exist and infants under normal. on the condition of the zinc that low in the body more vulnerable to the bacteria that produce the toxin. bad breast feeding in indonesia, limited food supplies in the household levels and limited access infants sick toward quality health services cause 5 million children suffered malnourished [9]. risk factors that can cause the genesis acute (rti) is the internal factors consist of the age of approximately 2 months, bblr, male, setatus nutrient deficiency of vitamin a, membendong children, (blanketed excessive force), providing additional food too early while factor outward signs are exclusive breastfeeding, immunization, air pollution (smoking habit family members normally housed infants live ), density shelter, fentilasi less sufficient and social economy [10]. based on that, this research examines the factors that affect the genesis acute (rti) based on risk factor in the clinic in south kalimantan mnggunakan poisson regression with gmm approach. methods research data is secondary data, namely the number of the prevalence of acute (rti) [11] and risk factors in the region of south kalimantan clinic. the framework of the concept of the genesis acute (rti) as follows. the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 163 figure 1. conceptual framework application research [12][7] based on the figure 1, then the method used is poisson regression with gmm approach [13][14]. poisson regression used to analyze verified response variable discrete graphics, where observabel response (y) be cacahan object that is a function of a number of specific characteristics (x). probability of y "many a genesis" poisson distribution namely [15].  ; , 0,1, 2, ! y e p y y y y        (y = 0,1,2,...) (1) with µ is the average number of genesis that berdistribusi poisson. parameters μ are very dependent on the specific unit or a specific period of time, the distance, wide, volume and etc. poisson distribution is used for are modeling genesis which is relatively rare during a specific time interval. the poisson regression model used the log function is ln (μi) = ηi, so that the function of the relationship is served on the common (2) and similarities (3) [13] 0 1 1 2 2 ln ... i i i k ki x x x         (2) (3) gmm method is an extension of the estimation of the moment method [16]. the moment the condition of the population to represent the information that will be used. gmm method takes the concept of the estimation of the moment method, where if at the moment method the number of moment conditions with the number of parameters that are being estimated, while for gmm method the number of moment conditions is greater or equal to the number of parameters that are being estimated. the moment the condition is a function of the model parameters and data that has been determined so that the value of the hope of the function is zero on the real value of the parameter. the steps the implementation of poisson regression analysis with estimates of the gmm as follows [17]: a. the identification of the data research b. perform a descriptive analysis of the response variable and variables predictors of research data c. use and modify the algorithm gmm on software r based on the package )...exp()exp( 22110 kikiii xxx   βx t i the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 164 d. the interpretation of the parameter coefficient through the value exp( )i  t i x β results and discussion the response variables used in this research is the number of the prevalence of acute (rti) in south kalimantan province 2016. the number of cases of acute (rti) follow poisson distribution, this is demonstrated through the kolmogorov smirnov test with the value asymp. sig. (2 tailed) = 0.329 greater than α=5 percent then failed to reject ho, which means that the data follow poisson distribution. descriptive statistics variables predictors that affect the amount of the prevalence of acute (rti) in south kalimantan are presented in the following table. table 1. research variable descriptive statistics variable min max mean std. deviation the percentage of non-exclusive breastfeeding (x1) 0.00 7.00 4.03 2.34 the percentage of low birth weight (x2) 0.00 51.61 14.57 11.46 the percentage of the number of infants less than 2 (x3) 2.89 28.26 14.58 5.35 the percentage of normal nutrition status (x4) 54.35 100.00 85.27 10.53 the percentage of mothers education on high school (x5) 56.31 99.27 87.49 9.55 the percentage of social economy (ump enough to top) (x6) 0.00 37.09 21.05 6.77 the percentage of shelter density (x7) 22.22 86.36 46.15 14.09 the percentage of the existence of smokers family members in the house (x8) 0.00 33.96 20.47 6.43 the percentage of immunization is not complete (x9) 0.00 79.54 30.63 15.77 table 1 shows that the average percentage of non-exclusive breastfeeding (x1) of 4.03% with standard deviation 2.34%, the average percentage of low birth weight (x2) of 14.57% and standard deviation 11.46%, the average percentage of the number of infants less than 2 (x3) of 14.58% and standard deviation 5.35%, the average percentage of normal nutrition status (x4) of 85.27% with standard deviation 10.53%, the average percentage of mothers education on high school (x5) of 87.49% with standard deviation 9.55%, the average percentage of economic social (ump enough to top) (x6) of 21.05% and standard deviation 6.77%, the average percentage of shelter density (x7) of 46.15% with standard deviation 14.09%, the average percentage of the existence of smokers family members in the house (x8) of 20.47% with standard deviation 6.43%, and the average percentage of immunization is not complete (x9) of 30.63% and standard deviation 15.77%. the results of gmm estimation in the regression model poisson processed through package r, as follows: table 2. the estimation of poisson regression model parameters on genesis acute (rti) variables poisson regression mle gmm estimate std. error sig. i estimate std. error sig. i intercept 1.3078 0.478 0.006 3.6980 3.9395 0.937 0.000 51.395 the percentage of non-exclusive breastfeeding (x1) -0.0003 0.004 0.922 0.9997 0.0279 0.008 0.000 1.0283 the percentage of low birth weight (x2) 0.0200 0.006 0.002 1.0202 0.0569 0.014 0.000 1.0585 the percentage of the number of infants less than 2 (x3) -0.0061 0.003 0.051 0.9939 -0.0364 0.006 0.000 0.9642 the percentage of normal nutrition 0.0033 0.004 0.349 1.0033 -0.0224 0.009 0.009 0.9778 the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 165 status (x4) the percentage of mothers education on high school (x5) -0.0030 0.005 0.567 0.9970 -0.0339 0.010 0.000 0.9666 the percentage of social economy (ump enough to top) (x6) 0.0023 0.002 0.346 1.0023 -0.0194 0.009 0.027 0.9808 the percentage of shelter density (x7) 0.0031 0.005 0.574 1.0031 0.0280 0.010 0.003 1.0284 the percentage of the existence of smokers family members in the house (x8) -0.0029 0.002 0.203 0.9971 0.0308 0.008 0.000 1.0313 the percentage of immunization is not complete (x9) 0.0001 0.003 0.967 1.0001 0.0193 0.007 0.005 1.0194 table 2 shows with mle method that the variables x3, x5, x6 significant because the value of sig smaller than α=5%, while with gmm method, all variables are statistically significant. the model with the value of goodness of fit a small said a good model, gmm method is better than the mle method. the model is obtained from the results of the gmm estimation is as follows: 1 2 3 4 5 6 7 8 9 ˆlog( ) 3.940 0.028 0.057 0.036 0.022 0.034 0.019 0.028 0.031 0.019 i x x x x x x x x x           then, 1 2 3 4 5 6 7 8 9 ˆ exp(3.940 0.028 0.057 0.036 0.022 0.034 0.019 0.028 0.031 0.019 ) i x x x x x x x x x           so that the interpretation of each variable to genesis acute (rti) as follows: the percentage of non-exclusive breastfeeding (x1) each there is increasing the percentage of non-exclusive breastfeeding (x1) then the average number of the prevalence of acute (rti) will increase by 1.0283 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. this is in line with the research of [18] stating that the things that can affect the genesis pneumonia in children is the lack of breast milk as the food naturally can optimize the immune system in the body of the son. the percentage of low birth weight (x2) each there is increasing the percentage of low birth weight (x2) then the average number of the prevalence of acute (rti) will increase by 1.0585 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. this is also in line with the research of [18] stating that the things that can affect the genesis pneumonia in children is bblr. infants with low birth weight (bblr) shows a tendency to more susceptible to infection than infants with birth weight normal (bbln) and it is the cause of the high level of the death of the baby. the percentage of the number of infants less than 2 (x3) each there is increasing the percentage of the number of infants less than 2 (x3) then the average number of the prevalence of acute (rti) will fell 0.9642 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. this is in line with the research of [19], which states that the number of infants who live together and genesis acute (rti) depends on the origin of the disease, namely tetular or not the percentage of normal nutrition status (x4) each there is increasing the percentage of normal nutrition status (x4) then the average number of the prevalence of acute (rti) will fell 0.9778 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 166 the percentage of mothers education on high school (x5) each there is increasing the percentage of maternal education on high school (x5) then the average number of the prevalence of acute (rti) will fell 0.9666 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. the percentage of social economy (ump enough to top) (x6) each there is increasing the percentage of social economy (ump enough to top) (x6) then the average number of the prevalence of acute (rti) will fell 0.9808 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. children with socio-economic status less has the possibility of experiencing acute (rti) higher compared with children with high economic social status. this can be explained that the socioeconomic status influence on education and other factors such as nutrition, lingkuingan and acceptance of health services. parents who come from high economic social level better able to provide a healthy food vitamins and supplements that can help improve the health status of the family. children who come from families with low socio-economic status have a greater risk of experiencing acute (rti). the percentage of shelter density (x7) each there is increasing the percentage of shelter density (x7) then the average number of the prevalence of acute (rti) will increase by 1.0284 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. the percentage of the existence of smoker family members in the house (x8) each there is increasing the percentage of the existence of smoker family members in the house (x8) then the average number of the prevalence of acute (rti) will increase by 1.0313 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. this shows that one of the causes of children experiencing the attack acute (rti) because the family members are smoking so that children were exposed to smoke that cause an attack acute (rti) the percentage of immunization is not complete (x9) each there is increasing the percentage of immunization is not complete (x9) then the average number of the prevalence of acute (rti) will increase by 1.0194 times from an average of arata prevalence number of acute (rti) conditions with all other variables constant. children with incomplete immunization status has the possibility of experiencing higher compared with children with complete immunization status. it is related with measles, pertussis and some other disease can increase the risk acute (rti) and weighed acute (rti) itself, but actually it can be prevented. measles, pertussis and difteri together can cause 15-25 percent of all deaths related to acute (rti). conclusions based on the analysis of the data and the discussion it can be concluded that the number of the prevalence of acute (rti) in south kalimantan province follow poisson distribution, so used poisson regression modeling. the results of the study showed that the estimates of the gmm method on poisson regression model gives better the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 167 performance in terms of the significance than mle estimation method. the factors that affect an increasing number of the prevalence of acute (rti) a region caused namely persentase breast feeding non-exclusive (x1), the percentage of low birth weight (x2), the percentage of shelter density (x7), the percentage of the existence of smoker family members in the house (x8), the percentage of immunization is not complete (x9). while the factors that affect a downturn in the number of the prevalence of acute (rti) in a region because of the variables the percentage of the number of infants less than 2 (x3), the percentage of normal nutrition status (x4), the percentage of mothers education on high school (x5), and the percentage of social economy (ump enough to top) (x6). references [1] j. m. wantania, r. naning, a. wahani, infeksi respiratori akut. buku ajar respirologi anak edisi pertama, ikatan dokter anak indonesia, jakarta, 2008. [2] w. purnomo, hubungan antara pengetahuan dan sikap ibu dengan upaya pencegahan ispa pada balita di puskesmas ngoresan surakarta, fakultas ilmu kesehatan universitas muhammadiyah surakarta, surakarta, 2008. [3] who, penanganan ispa pada anak di rumah sakit kecil negara berkembang, penerbit buku kedokteran egc, jakarta, 2012. [4] dinas kesehatan propinsi kalimantan selatan, profil kesehatan propinsi kalimantan selatan, dinkes prop. kalsel, banjarmasin, 2014. [5] dinas kesehatan kabupaten tanah bumbu, profil kesehatan kabupaten tanah bumbu, dinkes prop. kalsel, banjarmasin, 2014. [6] mahpolah, suharto, w. arief, b. w. otok, ”generalized method moment logistic regression model on the prevalence of acute respiratory tract infections,” j. appl. environ. biol. sci., 7(10)92-96, 2017. [7] s. e. purwanengsi, kuntoro, p. windu, b. w. otok, ”the modeling level of birth weight using a maximum likelihood estimation and generalized method of moment,” j. appl. environ. biol. sci., 6(7)125-128, textroad publication, 2016. [8] d. arini, b. w. otok, e. dwi, ”the modeling of acute respiratory tract infections (rti) on children 6-12 months with multinomial logit approach,” j. basic. appl. sci. res., 6(9) 1-6, 2016. [9] a. arwin dan n. zakiudin, alergi-imunologi anak, edisi 2, jakarta, badan penerbit idai, 2010. [10] depkes ri, rencana pembangunan jangka panjang bidang kesehatan 2005-2025, direktorat jendral bina kesehatan masyarakat, jakarta, 2009. [11] d. pramono, besar sampel dalam penelitian kesehatan yogyakarta: gadjah mada university press. edisi terjemahaan dari: lemeshow, s., hosmer, d.w., klar, j., lwanga. (1990) adequacy of sample size in health studies. who: john willey & sons, 1997. [12] g. lawrence and a. ruth, breastfeeding: a guide for the medical profession, fourth edition, st. louis: masby: 9-40, 1994. [13] a.c. cameron and p.k. trivedi, regression analysis of count data, cambridge: cambridge university press, 1998. [14] p. chausse, ”computing generalized method of moments with r,” journal of statistica, vol.34, 2010. [15] f. famoye, s. bae, j.t. wulu, a.a. bartolucci, k.p. singh, ”on the generalized poisson regression model with an application to accident data,” central the estimation of generalized method moment poisson regression model on the prevalence of acute respiratory tract infection (rti) in south kalimantan mahpolah 168 michigan university, bureau of primary health care unt health science center, journal of data science, 2(2004), 287-295, 2004. [16] a. r. hall, ”generalized method of moments: advanced text in econometrics,” oxford university press inc: new york, 2005. [17] t. r. muhammad, b. w. otok, i. n. latra, ”estimation of generalized method of moment in logistic regression model,” prosiding seminar nasional matematika, universitas jember, 167-174, 2014. [18] c.g. voctora, b. r. kirkwood, b.r. ashworth, r. e. black, s. rogers. “potential interventions for the prevention of the childhood pneumonia in developing coutries: improving nutrition,” am j clinnutr, vol.70, pp. 309 – 320, 1999. [19] r. b. darmawan, buku ajar respirologi anak. edisi i. badan penerbit idai, jakarta, 2010. on the local edge antimagic coloring of corona product of path and cycle cauchy –jurnal matematika murni dan aplikasi volume 6(1) (2019), pages 40-48 p-issn: 2086-0382; e-issn: 2477-3344 submitted: november 19, 2019 reviewed: november 26, 2019 accepted: november 30, 2019 doi: http://dx.doi.org/10.18860/ca.v6i1.8054 on the local edge antimagic coloring of corona product of path and cycle siti aisyah1, ridho alfarisi3, rafiantika m. prihandini2, arika indah kristiana2, ratna dwi c.1 1mathematics department, kaltara university of north kalimantan 2 mathematics education department, jember university of jember 3 primary school department, jember university of jember email: aisyah_rasyid84@yahoo.com, alfarisi.fkip@unej.ac.id, arikakristiana@gmail.com, r_christyanti@yahoo.com abstract let 𝐺(𝑉,𝐸) be a nontrivial and connected graph of vertex set 𝑉 and edge set 𝐸. a bijection 𝑓:𝑉(𝐺) → {1,2,3,…,|𝑉(𝐺)|} is called a local edge antimagic labeling if for any two adjacent edges 𝑒1 and 𝑒2,𝑤(𝑒1) ≠ 𝑤(𝑒2), where 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺),𝑤(𝑒) = 𝑓(𝑢) + 𝑓(𝑣). thus, the local edge antimagic labeling induces a proper edge coloring of 𝐺 if each edge 𝑒 assigned the color 𝑤(𝑒). the color of any edge 𝑒 = 𝑢𝑣 is assigned by 𝑤(𝑒) which is defined by the sum of both vertices labels 𝑓(𝑢) and 𝑓(𝑣). the local edge antimagic chromatic number, denoted by 𝛾𝑙𝑎𝑒(𝐺) is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of 𝐺. in our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, and cycle corona cycle. keywords: local antimagic; edge coloring; corona product; path; cycle. introduction the local antimagic vertex coloring of a graph 𝐺 introduced by arumugam et. al [1]. furthermore, agustin, et. al [2] defined local edge antimagic coloring of the graph. a bijection 𝑓:𝑉(𝐺) → {1,2,3, . . . , |𝑉(𝐺)|}, is called a local edge antimagic labeling if every two adjacent edges 𝑒1 and 𝑒2, 𝑤(𝑒1) ≠ 𝑤(𝑒2), where 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺), and 𝑤(𝑒) = 𝑓(𝑢) + 𝑓(𝑣). thus, the local edge antimagic labeling induces a proper edge coloring of 𝐺 if any edge 𝑒 is assigned the color 𝑤(𝑒). the color of each edge 𝑒 = 𝑢𝑣 are assigned by 𝑤(𝑒) which is defined by the sum of label both and vertices 𝑓(𝑢) and 𝑓(𝑣). the local edge antimagic chromatic number, denoted by 𝛾𝑙𝑎𝑒(𝐺), is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of graph 𝐺. agustin, et. al [2] establish the local edge antimagic chromatic number of path graph and cycle graph. dettlaff, et. al in [3], a corona graph of 𝐺 and 𝐻, denoted by 𝐺 ⊙ 𝐻, is obtained by joining each vertex of hi to the vertex uj of g. ramya in [4], discussed an acyclic coloring of a corona graph and yero in [5] studied coloring, location, and domination of a corona graph. kristiana, et.al in [6] found the lower bound of the r-dynamic chromatic number of a corona product by wheel graphs. many papers presented a corona product topics for example in [7], [2], and [8]. however, the local edge antimagic coloring of corona product still has nothing to discuss. in our paper, we investigate the local edge antimagic coloring of corona product of path and cycle, namely http://dx.doi.org/10.18860/ca.v6i1.8054 mailto:aisyah_rasyid84@yahoo.com mailto:alfarisi.fkip@unej.ac.id mailto:arikakristiana@gmail.com mailto:r_christyanti@yahoo.com on the local edge antimagic coloring of corona product of path and cycle siti aisyah 41 path corona cycle, cycle corona path, path corona path, and cycle corona cycle. the results of local edge antimagic labeling are as follows. conjecture 1.1. every connected graph other than 𝐾2 is local antimagic. observation 1.1. [6]for any graph 𝐺, 𝛾𝑙𝑎𝑒(𝐺) ≥ 𝛾(𝐺) − 1. observation 1.2. [8]for any graph g, χla(g) ≥ χ(g), where χ(g) is a vertex chromatic number of g. observation 1.3. [6] for any graph 𝐺, 𝛾𝑙𝑎𝑒(𝐺) ≥ 𝛾(𝐺), where 𝛾(𝐺) is an edge chromatic number of g. theorem 1.1. [6] if δ(𝐺) is maximum degrees of 𝐺, then we have 𝛾𝑙𝑎𝑒(𝐺) ≥ δ(𝐺). proposition 1.1. [6] let g be a connected graph, we have a) if 𝐺 ≅ 𝑃𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 2. b) if 𝐺 ≅ 𝐶𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 3. c) if 𝐺 ≅ 𝐿𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 3. d) if 𝐺 ≅ 𝐾𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 2𝑛 − 3. e) if 𝐺 ≅ 𝑊𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 𝑛 + 2. f) if 𝐺 ≅ 𝑆𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 𝑛. g) if 𝐺 ≅ 𝐹𝑛 , then 𝛾𝑙𝑎𝑒(𝐺) = 2𝑛 + 1. h) if 𝐺 ⊙ 𝑚𝐾1, then 𝛾𝑙𝑎𝑒(𝐺 ⊙ 𝑚𝐾1) = 𝛾𝑙𝑎𝑒(𝐺) + 𝑚. results and discussion in our paper, we consider the local edge antimagic chromatic number of a corona product of path and cycle, including path corona cycle, cycle corona path, path corona path, cycle corona cycle. furthermore, we determine the exact values of local edge antimagic chromatic number of corona product in the following theorems. theorem 2.1. the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 for 𝑛 odd and 𝑛,𝑚 ≥ 3 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) = 2(𝑛 + 1) + 𝑚. proof. the graph 𝑃𝑛 ⊙ 𝑃𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖:1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑗 𝑖:1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛 } and edge set 𝐸(𝑃𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑗 𝑖𝑥𝑗+1 𝑖 : 1 ≤ 𝑗 ≤ 𝑚 − 1; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗 𝑖: 1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛}. the cardinality of the vertex set is |𝑉(𝑃𝑛 ⊙ 𝑃𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is |𝐸(𝑃𝑛 ⊙ 𝑃𝑚)| = 2𝑚𝑛 − 1. we define a bijection 𝑓:𝑉(𝑃𝑛 ⊙ 𝑃𝑚) → {1,2,3,…, |𝑉(𝑃𝑛 ⊙ 𝑃𝑚)|} for the graph 𝑃𝑛 ⊙ 𝑃𝑚 to be local edge antimagic labeling as follows. 𝑓(𝑥𝑖) = { 𝑖 + 1 2 , if 𝑖 is odd 𝑛 − ( 𝑖 − 2 2 ), if 𝑖 is even on the local edge antimagic coloring of corona product of path and cycle siti aisyah 42 𝑓(𝑥𝑗 𝑖) = { 𝑛 + 1 + ( 𝑖 − 2 2 ) + ( 𝑗 − 2 2 )𝑛, if 𝑖 and 𝑗 are even 2𝑛 + 𝑛⌈ 𝑚 2 ⌉ + 1 + ( 𝑖 − 2 2 ) − 𝑛( 𝑗 − 1 2 ), if 𝑖 is even and 𝑗 is odd 2𝑛 − ( 𝑖 − 1 2 ) + ( 𝑗 − 2 2 )𝑛, if 𝑖 is odd and 𝑗 is even 𝑚𝑛 + 𝑛 − ( 𝑖 − 1 2 ) − 𝑛( 𝑗 − 1 2 ), if 𝑖 and 𝑗 are odd it is clear that 𝑓 is a local antimagic labeling of 𝑃𝑛 ⊙ 𝑃𝑚 and the edge weights are as follows: 𝑤(𝑥𝑖𝑥𝑖+1) = { 𝑛 + 1, if 𝑖 is odd 𝑛 + 2, if 𝑖 is even 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) = { 𝑚𝑛 + 3𝑛 − (𝑖 − 1), if 𝑖 and j are odd 𝑚𝑛 + 2𝑛 − (𝑖 − 1), if 𝑖 is odd and 𝑗 is even 𝑚𝑛 + 𝑛 + 𝑖, if 𝑖 is even and 𝑗 is odd 𝑚𝑛 + 𝑖, if 𝑖 and 𝑗 are even 𝑤(𝑥𝑖𝑥𝑗 𝑖) = { 𝑚𝑛 + 1 + 𝑛( 𝑗 − 3 2 ), if 𝑗 is odd 2𝑛 + 1 + 𝑛( 𝑗 − 2 2 ), if 𝑗 is even hence, we get that the upper bound of the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≤ 2(𝑛 + 1) + 𝑚. furthermore, we prove that lower bound of the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≥ 2(𝑛 + 1) + 𝑚. by contradiction, we assume that 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) < 2(𝑛 + 1) + 𝑚. without lost of generality, we assume that 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗 𝑖). based on proposition 1, 𝛾𝑙𝑎𝑒(𝑃𝑛) = 2 and 𝛾𝑙𝑎𝑒(𝑃𝑚) = 2 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| = 2, |{𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 )}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 2(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1 such that |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| +|{𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 )}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = 2 + 𝑚 + 2(𝑛 − 1) + 1 |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = 𝑚 + 2𝑛 + 1 if |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1, then we obtain at least two edges which have same edge weight, it is a contradiction. thus, we receive that the lower bound of the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≥ 𝑚 + 2𝑛 + 2 = 2(𝑛 + 1) + 𝑚. it concludes that the local antimagic edge chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) = 2(𝑛 + 1) + 𝑚.∎ theorem 2.2 the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 for 𝑛,𝑚 odd and 𝑛,𝑚 ≥ 4 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) = 2 + 3𝑛 + 𝑚. proof. the graph 𝑃𝑛 ⊙ 𝐶𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖:1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑗 𝑖:1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛 } and edge set 𝐸(𝑃𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑗 𝑖𝑥𝑗+1 𝑖 : 1 ≤ 𝑗 ≤ 𝑚 − 1; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑚 𝑖 𝑥1 𝑖 ,1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗 𝑖: 1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛}. the cardinality of the vertex set is |𝑉(𝑃𝑛 ⊙ 𝐶𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is |𝐸(𝑃𝑛 ⊙ 𝐶𝑚)| = 2𝑚𝑛 + 𝑛 − 1. we define a function bijection 𝑓:𝑉(𝑃𝑛 ⊙ 𝐶𝑚) → {1,2,3,…,|𝑉(𝑃𝑛 ⊙ 𝐶𝑚)|} for the graph 𝑃𝑛 ⊙ 𝐶𝑚 to be local edge antimagic labeling as follows. on the local edge antimagic coloring of corona product of path and cycle siti aisyah 43 𝑓(𝑥𝑖) = { 𝑖 + 1 2 , if 𝑖 is odd 𝑛 − 𝑖 − 2 2 , if 𝑖 is even 𝑓(𝑥𝑗 𝑖) = { 𝑚𝑛 + 1 + ( 𝑖 − 2 2 ) − ( 𝑗 − 2 2 )𝑛, if 𝑖 and 𝑗 are even 𝑛 + 1 + ( 𝑖 − 2 2 ) + ( 𝑗 − 1 2 )𝑛, if 𝑖 is even and 𝑗 is odd 𝑚𝑛 + 𝑛 − ( 𝑖 − 1 2 ) − ( 𝑗 − 2 2 )𝑛, if 𝑖 is odd and 𝑗 is even 2𝑛 − ( 𝑖 − 1 2 ) + 𝑛( 𝑗 − 1 2 ), if 𝑖 and 𝑗 are odd 𝑓(𝑥𝑚 𝑖 ) = { 𝑛⌈ 𝑚 2 ⌉ + 𝑛 − ( 𝑖 − 1 2 ), if 𝑖 is odd 𝑛⌈ 𝑚 2 ⌉ + 1 + ( 𝑖 − 2 2 ), if 𝑖 is even it is easy to see that 𝑓 is a local edge antimagic labeling of 𝑃𝑛 ⊙ 𝐶𝑚 and the edge weights are as follows: 𝑤(𝑥𝑖𝑥𝑖+1) = { 𝑛 + 1, if 𝑖 is odd 𝑛 + 2, if 𝑖 is even 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) = { 𝑚𝑛 + 4𝑛 − (𝑖 − 1), if 𝑖 and 𝑗 are odd 𝑚𝑛 + 3𝑛 − (𝑖 − 1), if 𝑖 is odd and 𝑗 is even 𝑚𝑛 + 2(𝑛 + 1) + (𝑖 − 2), if 𝑖 is even and 𝑗 is odd 𝑚𝑛 + 𝑛 + 2 + (𝑖 − 2), if 𝑖 and 𝑗 are even 𝑤(𝑥𝑚 𝑖 𝑥1 𝑖) = { 𝑛⌈ 𝑚 2 ⌉ + 3𝑛 − (𝑖 − 1), if 𝑗 is odd 𝑛⌈ 𝑚 2 ⌉+ 𝑛 + 2 + (𝑖 − 2), if 𝑗 is even 𝑤(𝑥𝑖𝑥𝑗 𝑖) = { 2𝑛 + 1 + 𝑛( 𝑗 − 1 2 ), if 𝑗 is odd 𝑚𝑛 + 𝑛 − 𝑛( 𝑗 − 2 2 ), if 𝑗 is even 𝑤(𝑥𝑖𝑥𝑚 𝑖 ) = { 𝑛⌈ 𝑚 2 ⌉+ 𝑛 + 1 − ( 𝑖 − 1 2 ), if 𝑗 is odd 𝑛⌈ 𝑚 2 ⌉ + 2 + ( 𝑖 − 2 2 ), if 𝑗 is even hence, we get that the upper bound of the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≤ 2 + 3𝑛 + 𝑚. furthermore, we prove that the lower bound of the on the local edge antimagic coloring of corona product of path and cycle siti aisyah 44 local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≥ 2 + 3𝑛 + 𝑚. by contradiction, we assume that 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) < 2 + 3𝑛 + 𝑚 . without of generality, 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) ≠ 𝑤(𝑥1 𝑖𝑥𝑚 𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗 𝑖). based on proposition 1 that 𝛾𝑙𝑎𝑒(𝑃𝑛) = 2 and 𝛾𝑙𝑎𝑒(𝐶𝑚) = 3 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| = 2, |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 3(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2 such that |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}|+|{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| = 2 + 𝑚 + 3(𝑛 − 1) + 2 |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| = 𝑚 + 3𝑛 + 1 if |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2, then we obtain at least two edges which have the same edge weight, which is a contradiction. accordingly, the lower bound of the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≥ 𝑚 + 3𝑛 + 2. it concludes that the local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) = 2 + 3𝑛 + 𝑚. ∎ theorem 2.3. the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 for 𝑛,𝑚 even and 𝑛,𝑚 ≥ 4 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) = 3(𝑛 + 1) + 𝑚. proof. the graph 𝐶𝑛 ⊙ 𝐶𝑚 is a connected graph with vertex set 𝑉(𝐶𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖:1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑗 𝑖:1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛 } and edge set 𝐸(𝐶𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥1𝑥𝑛} ∪ {𝑥𝑗 𝑖𝑥𝑗+1 𝑖 : 1 ≤ 𝑗 ≤ 𝑚 − 1; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗 𝑖: 1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑚 𝑖 𝑥1 𝑖 ,1 ≤ 𝑖 ≤ 𝑛}. the cardinality of the vertex set is |𝑉(𝐶𝑛 ⊙ 𝐶𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is |𝐸(𝐶𝑛 ⊙ 𝐶𝑚)| = 2𝑚𝑛 + 𝑛. we define a function bijection 𝑓:𝑉(𝐶𝑛 ⊙ 𝐶𝑚) → {1,2,3,…,|𝑉𝐶𝑛 ⊙ 𝐶𝑚)|} for the graph 𝐶𝑛 ⊙ 𝐶𝑚 to be local edge antimagic labeling as follows. 𝑓(𝑥𝑖) = { 𝑖 + 1 2 , if 𝑖 is odd 𝑛 − 𝑖 − 2 2 , if 𝑖 is even 𝑓(𝑥𝑗 𝑖) = { 𝑚𝑛 + 1 + ( 𝑖 − 2 2 ) − ( 𝑗 − 1 2 )𝑛, if 𝑖 and 𝑗 are even 𝑛 + 1 + ( 𝑖 − 2 2 ) + ( 𝑗 − 1 2 )𝑛, if 𝑖 is even and 𝑗 is odd 𝑚𝑛 + 𝑛 − ( 𝑖 − 1 2 ) − ( 𝑗 − 2 2 )𝑛, if 𝑖 is odd and 𝑗 is even 2𝑛 − ( 𝑖 − 1 2 ) + 𝑛( 𝑗 − 1 2 ), if 𝑖 and 𝑗 are odd 𝑓(𝑥𝑚 𝑖 ) = { 𝑛⌈ 𝑚 2 ⌉ + 𝑛 − ( 𝑖 − 1 2 ), 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑛⌈ 𝑚 2 ⌉ + 1 + ( 𝑖 − 2 2 ), 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 it is easy to see that 𝑓 is a local edge antimagic labeling of 𝐶𝑛 ⊙ 𝐶𝑚 and the edge weights are as follows: on the local edge antimagic coloring of corona product of path and cycle siti aisyah 45 𝑤(𝑥𝑖𝑥𝑖+1) = { 𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥1𝑥𝑛) = 𝑛 + 2 2 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) = { 𝑚𝑛 + 4𝑛 − (𝑖 − 1), 𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑 𝑚𝑛 + 3𝑛 − (𝑖 − 1), 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑚𝑛 + 2(𝑛 + 1) + (𝑖 − 2), 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑 𝑚𝑛 + 𝑛 + 2 + (𝑖 − 2), 𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑒𝑣𝑒𝑛 𝑤(𝑥𝑚 𝑖 𝑥1 𝑖) = { 𝑛⌈ 𝑚 2 ⌉ + 3𝑛 − (𝑖 − 1),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑 𝑛⌈ 𝑚 2 ⌉ + 𝑛 + 2 + (𝑖 − 2), 𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥𝑖𝑥𝑗 𝑖) = { 2𝑛 + 1 + 𝑛( 𝑗 − 1 2 ), 𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑 𝑚𝑛 + 𝑛 − 𝑛( 𝑗 − 2 2 ),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥𝑖𝑥𝑚 𝑖 ) = { 𝑛⌈ 𝑚 2 ⌉+ 𝑛 + 1 − ( 𝑖 − 1 2 ),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑 𝑛⌈ 𝑚 2 ⌉+ 2 + ( 𝑖 − 2 2 ),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 hence, we get that the upper bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≤ 3(𝑛 + 1) + 𝑚. furthermore, we prove that lower bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≥ 3(𝑛 + 1) + 𝑚. by contradiction, we assume that 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) < 3(𝑛 + 1) + 𝑚 . without lost of generality, we gives that 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥1𝑥𝑛) ≠ 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) ≠ 𝑤(𝑥1 𝑖𝑥𝑚 𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗 𝑖). based on proposition 1 that 𝛾𝑙𝑎𝑒(𝐶𝑛) = 3 and 𝛾𝑙𝑎𝑒(𝐶𝑚) = 3 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| = 3, |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 3(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2 such that |𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝐶𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 3 + 𝑚 + 3(𝑛 − 1) + 2 = 𝑚 + 3𝑛 + 2 if |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2, then we obtain at least two edges which have same edge weight, which is a contradiction. thus, we receive that the lower bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≥ 𝑚 + 3𝑛 + 3. it concludes that the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) = 3(𝑛 + 1) + 𝑚.∎ on the local edge antimagic coloring of corona product of path and cycle siti aisyah 46 theorem 2.4. the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 for 𝑛 odd and 𝑛,𝑚 ≥ 3 is𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) = 3 + 2𝑛 + 𝑚. proof. the graph 𝐶𝑛 ⊙ 𝑃𝑚 is a connected graph with vertex set 𝑉(𝐶𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖:1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑗 𝑖:1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛 } and edge set 𝐸(𝐶𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥1𝑥𝑛} ∪ {𝑥𝑗 𝑖𝑥𝑗+1 𝑖 : 1 ≤ 𝑗 ≤ 𝑚 − 1; 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗 𝑖: 1 ≤ 𝑗 ≤ 𝑚; 1 ≤ 𝑖 ≤ 𝑛}. the cardinality of the vertex set is |𝑉(𝐶𝑛 ⊙ 𝑃𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is |𝐸(𝐶𝑛 ⊙ 𝑃𝑚)| = 2𝑚𝑛 + 𝑛 − 1. we define a function bijection 𝑓:𝑉(𝐶𝑛 ⊙ 𝑃𝑚) → {1,2,3,…,|𝑉(𝐶𝑛 ⊙ 𝑃𝑚)|} for the graph 𝐶𝑛 ⊙ 𝑃𝑚 to be local edge antimagic labeling as follows. 𝑓(𝑥𝑖) = { 𝑖 + 1 2 , 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑛 − 𝑖 − 2 2 , 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑓(𝑥𝑗 𝑖) = { 𝑛 + 1 + ( 𝑖 − 2 2 ) + ( 𝑗 − 2 2 )𝑛, 𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑒𝑣𝑒𝑛 2𝑛 + 𝑛⌈ 𝑚 2 ⌉ + 1 + ( 𝑖 − 2 2 ) − 𝑛( 𝑗 − 1 2 ), 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑 2𝑛 − ( 𝑖 − 1 2 ) + ( 𝑗 − 2 2 )𝑛, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑚𝑛 + 𝑛 − ( 𝑖 − 1 2 ) − 𝑛( 𝑗 − 1 2 ), 𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑 it is easy to see that 𝑓 is a local edge antimagic labeling of 𝑃𝑛 ⊙ 𝐶𝑚 and the edge weights are as follows: 𝑤(𝑥𝑖𝑥𝑖+1) = { 𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥1𝑥𝑛) = 𝑛 + 2 2 (𝑥𝑖𝑥𝑖+1) = { 𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) = { 𝑚𝑛 + 3𝑛 − (𝑖 − 1),𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑 𝑚𝑛 + 2𝑛 − (𝑖 − 1), 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑚𝑛 + 𝑛 + 𝑖, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑 𝑚𝑛 + 2 + 𝑖, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑤(𝑥𝑖𝑥𝑗 𝑖) = { 𝑚𝑛 + 1 + 𝑛( 𝑗 − 3 2 ),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑 2𝑛 + 1 + 𝑛( 𝑗 − 2 2 ),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛 on the local edge antimagic coloring of corona product of path and cycle siti aisyah 47 hence, we get that the upper bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≤ 3 + 2𝑛 + 𝑚. furthermore, we prove that the lower bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≥ 3 + 2𝑛 + 𝑚. by contradiction, we assume that 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) < 3 + 2𝑛 + 𝑚 . without lost of generality, we gives that 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥𝑗 𝑖𝑥𝑗+1 𝑖 ) ≠ 𝑤(𝑥1 𝑖𝑥𝑚 𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗 𝑖). based on proposition 1 that 𝛾𝑙𝑎𝑒(𝐶𝑛) = 3 and 𝛾𝑙𝑎𝑒(𝑃𝑚) = 2 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| = 3, |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 2(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1 such that |𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗 𝑖)}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| |𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = 3 + 𝑚 + 2(𝑛 − 1) + 1 |𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = 𝑚 + 2𝑛 + 2 if |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1, then we obtain at least two edges which have same edge weight, which is a contradiction. thus, we receive that the lower bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≥ 𝑚 + 2𝑛 + 3. it concludes that the local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) = 3 + 2𝑛 + 𝑚.∎ conclusions in this paper we have given the result on the local edge antimagic chromatic number of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle. open problem 1. what is the upper bound of local edge antimagic coloring of corona product of a connected graph? open problem 2. what is the lower bound of local edge antimagic coloring of corona product of a connected graph? acknowledgment we gratefully acknowledge the support from dikti indonesia trough the “penelitian dosen pemula” ristekdikti 2019 research project. references [1] s. arumugam, k. premalatha, m. bača and a. semaničová-feňovčı́ková, "local antimagic vertex coloring of a graph," graphs and combinatorics, vol. 33, pp. 275-285, 2017. [2] r. alfarisi, "dafik, ih agustin, and ai kristiana," in journal of physics: conference series, 2018. [3] m. dettlaff, j. raczek and i. g. yero, "edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs," opuscula mathematica, vol. 36, pp. 575-588, 2016. [4] n. ramya, "on coloring of corona graphs," indian journal of science and technology, vol. 7, p. 9, 2014. [5] i. g. yero, d. kuziak and a. r. aguilar, "coloring, location and domination of corona graphs," aequationes mathematicae, vol. 86, pp. 1-21, 2013. on the local edge antimagic coloring of corona product of path and cycle siti aisyah 48 [6] a. i. kristiana, m. i. utoyo and dafik, "the lower bound of the r-dynamic chromatic number of corona product by wheel graphs," in aip conference proceedings, 2018. 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[9] r. m. prihandini, "local edge antimagic coloring of graphs," 2017. vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district cauchy –jurnal matematika murni dan aplikasi volume 5(3) (2018), pages 150-160 p-issn: 2086-0382; e-issn: 2477-3344 submitted: 03 july 2018 reviewed: 15 august 2018 accepted: 30 november 2018 doi: http://dx.doi.org/10.18860/ca.v5i3.5880 vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair1, sarmanu2, santi martini3, bambang widjanarko otok4 1health polytechnic banjarmasin, ministry of health & ph.d student faculty of public health, airlanga university, surabaya 2 faculty veterinary medicine, airlanga university, surabaya, 3 department of biostatistics and demography, faculty of public health, airlanga university, surabaya, 4laboratory of environmental and health statistic, institut teknologi sepuluh nopember, surabaya, indonesia email: ulunkhair@gmail.com, prof.sarmanu@gmail.com, santi-m@fkm.unair.ac.id, dr.otok.bw@gmail.com abstract the number of malaria in this area always has the tendency of the most compared to the city/district in south kalimantan province. behavior is internalisation factor from the level of knowledge, attitudes and actions of a person who influenced by customs, customs and belief in certain things that has been handed down by his ancestors. the behavior of a community group can be different from the other groups so that they formed a group behavior or can be said tribal behavior.the purpose of this research predicts that the number of the prevalence of malaria in tanah bumbu tribal based with y1t : the tribe of banjar, y2t : javanese, y3t : the tribe of bugis, and y4t : other tribes using vector autoregressive (var) model. the results of the study showed with granger causality approach there is a relationship between the amount of the prevalence of malaria javanese with other tribes, bugis tribe with other tribes. the relationship is strengthened in the var model, which is the number of the prevalence of malaria javanese influenced by the number of the prevalence of malaria javanese at period t-1, and the number of the prevalence of malaria tribe bugis at period t-1. while the number of the prevalence of malaria tribe bugis influenced by the number of the prevalence of malaria other tribes in the period t-2. the model of divination the prevalence of malaria based on ethnicity as follows: 1 1 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.325 0.277 0.198 0.302 0.645 0.955 .732 0.811 2 t t t t t t t t t t y y y y y y y y y y                   2 2 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.085 .347 0.419 0.059 0.844 0.525 .125 0.5010 + 0 t t t t t t t t t t y y y y y y y y y y                   3 3 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.041 .125 0.136 0.181 0.318 0.074 .399 0.6530 + 0 t t t t t t t t t t y y y y y y y y y y                   4 4 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.019 .026 0.101 0.0140 + 0.0960.136 .075 0.0390 t t t t t t t t t t y y y y y y y y y y                   the ability of divination number of the prevalence of malaria based on the tribal areas with the criteria mape provides good performance of 28.5 percent. keywords: malaria, tanah bumbu, var, tribe banjar, javanese, tribe bugis, other tribes introduction genesis malaria of course is associated with the existence of contact with the mosquito anopheles as penular. some of the factors that related with the possibility of contact between people in a place with the mosquito namely characteristics, climate mailto:ulunkhair@gmail.com mailto:prof.sarmanu@gmail.com mailto:santi-m@fkm.unair.ac.id, vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 151 change, behavior, socio-economic conditions and others. malaria is one of the infectious disease caused by the plasmodium. however as other communicable disease, infection of malaria is very influenced by many factors. the things that related with the environment, behavior, and mosquitoes usually most discussed in the transmission of malaria in a region. various researches have been done. most research related to the mosquito anopheles [1]. the types of mosquitoes is indeed the gnats penular known to play a role in the transmission of malaria from one person to another. modeling the relevance of these factors in the genesis of malaria on a certain period used vector autoregressive approach (var) [2]. var model was built in order to catch the phenomenon that is with good. var nonstructural is a model of the relations between the variables, while cointegration and vector error correction models (vecm) is a solution to create a model that has a long and short term are relationships of variables which have the problem of nonstationary [3]. var has several benefits such as (1) this method is simple, researchers do not need to determine where the endogenous variable and which exogenous variable because all the variables in the var is the endogenous variable; (2) estimation method can use ordinary least square (ols) and can be done for each equation separately; (3) the result of divination with this method in many cases better than divination with a complex simultaneous equation method [4]. south kalimantan province, if reviewed number from the diversity of genesis malaria 2011, a province with the number of malaria sixth largest. the number of malaria absolutely 2011 in south kalimantan province as much as 6663 genesis [5]. if compared to the year 2007, then the incidence of malaria 2013 in south kalimantan province. tanah bumbu is endemic areas malaria because in the last 3 years (2011 until 2013) always found cases of malaria. the number of malaria in this area always has the tendency of the most compared to the city/another district in south kalimantan province. the discovery of cases of malaria in tanah bumbu on 2012 totaled 6044 cases from the total 29212 cases in south kalimantan [6]. the results of research related to the genesis of malaria are the temperature and humidity in research [7], [8], [9], and [10]. the height of the location in research [7] and [11]. biological environment in research [12] and [1]. the socio-economic research in the [12], [13], [14] and [1]. the behavior in research [15], [12], [16], [17], [18] and [19]. the status of the population in research [13], [20] and [1]. research and modeling of divination genesis diseases including done by [21], [22] and [23]. however, model of divination that obtained in the three research is considering only the case of previously according to time and location as the variable wizards (predictors). based on the need to be examined the predictions of malaria cases in the district of the land of spices that based on ethnicity and behavior with a model var. methods the unit of analysis in this research is data number of genesis malaria per month according to the sub-districts which are recorded in the system of recording and reporting on the health of tanah bumbu year 2012-2015. the basic concept in epidemiology often termed the trias epidemiology provides the relationship between the three main factors that have a role in a disease and other health problems, namely host, agent, and environment. the three factors is a dynamic unity in the balance (equilibrium) on the healthy. if the balance is disrupted, will cause the status of the sick [24]. based on the research and theoretical studies description, then the framework of vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 152 the concept of the research is presented in figure 1. figure 1. the conceptual framework of genesis malaria based on the tribal areas with vector autoregressive model [24] genesis malaria can had been modeled according to the theory of h.l. 's taxonomy. in the theory of genesis a disease can be influenced by 4 (four) factors, factors, descendants of health services, the behavior and the environment. specific locations have certain environmental factors that can be different from the other locations. so the variables can be used as location) are typical of a genesis pain. incidents or genesis a disease according to the time will be able to form a specific model. the model can be used to predict the future events that will come. community groups in tanah bumbu become heterogeneous community groups that can have a specific tribal behavior. this tribal behavior patterns will be used in the development of the model var in predicting genesis malaria in tanah bumbu. analysis steps on the modeling var as follows [25], [26]: 1. pre-processing data in the tribal areas of tanah bumbu include month and year. 2. test stationary in variance and the average stationary data is data that is not experience growth or decline. stationary data have a horizontal pattern along the time axis, or fluctuations in the data is located around an average value a constant, not depending on time and varians from fluctuations remain constant each time. 3. do the identification of the model to obtain the vector autoregressive order (var) the first model identifkasi determine p order from var(p), and then q order from x. the determination of the prevailing from var(p) used mpacf plots and aic minimum without involving exogenous variable. while the determination of order x involving exogenous variable. the value of the aic as follows: (wei, 2006) 2 2 ( ) ln(| |) p m p aic p n    where: insiden time (t-1) time (t-2) .................... time (t-n) h.l bloom descendants ministry health behavior tribal environmental location vector autoregressive model vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 153 1 1 ( ) ' n p t tn t    u u is based on the model residual kovarian penduga var(p), tu is a residual on the time to the t model var(p) m is the number of endogenous variables in the model. 4. perform the estimation of model parameters var used the smallest square method 5. to test the significance of model parameters var 6. checking the assumption of white noise, normal multivariate on a residual 7. perform data forecasts out sample results and discussion unit will analysis in this research is data number of genesis malaria per month according to the sub-districts which are recorded in the system of recording and reporting on the health of tanah bumbu year 2014-2017. genesis malaria seen according to each of the tribal areas on all months starting january the year 2014 until may 2017 is shown in table 1. genesis plots timeseries malaria based on ethnicity served as follows: figure 2. genesis plots timeseries malaria tribal based figure 2 shows the pattern of the number of the prevalence of malaria from 41 observations on the tribe of banjar, java, bugis and other tribes have the trend is down, although on the other tribes paring losses more entry ramp. this can be in it that the pattern of genesis malaria tends to be uniform from the four tribes. vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 154 in detail is presented in table 1 below. table 1. cross-tabulation of genesis malaria in tanah bumbu according to the years, the month and the tribal areas the year the month tribal areas banjar central java bugis others f % f % f % f % 2014 january 45 51.7 21 24.1 15 17.2 6 6.9 2014 february 42 48.8 27 31.4 13 15.1 4 4.7 2014 march 25 43.9 18 31.6 12 21.1 2 3.5 2014 april 37 45.7 28 34.6 11 13.6 5 6.2 2014 mei 15 38.5 18 46.2 3 7.7 3 7.7 2014 june 17 44.7 12 31.6 7 18.4 2 5.3 2014 july 17 58.6 8 27.6 4 13.8 0 0.0 2014 august 15 75.0 1 5.0 3 15.0 1 5.0 2014 september 13 52.0 8 32.0 3 12.0 1 4.0 2014 october 4 44.4 2 22.2 3 33.3 0 0.0 2014 november 6 66.7 3 33.3 0 0.0 0 0.0 2014 december 6 66.7 3 33.3 0 0.0 0 0.0 2015 january 26 46.4 22 39.3 7 12.5 1 1.8 2015 february 23 46.0 19 38.0 3 6.0 5 10.0 2015 march 13 56.5 8 34.8 1 4.3 1 4.3 2015 april 18 75.0 5 20.8 0 0.0 1 4.2 2015 mei 5 26.3 9 47.4 1 5.3 4 21.1 2015 june 12 41.4 13 44.8 2 6.9 2 6.9 2015 july 3 33.3 2 22.2 3 33.3 1 11.1 2015 august 4 57.1 2 28.6 0 0.0 1 14.3 2015 september 5 50.0 4 40.0 1 10.0 0 0.0 2015 october 12 60.0 3 15.0 4 20.0 1 5.0 2015 november 1 33.3 1 33.3 1 33.3 0 0.0 2015 december 1 11.1 6 66.7 1 11.1 1 11.1 2016 january 3 25.0 6 50.0 2 16.7 1 8.3 2016 february 11 52.4 7 33.3 2 9.5 1 4.8 2016 march 11 73.3 3 20.0 0 0.0 1 6.7 2016 april 13 81.3 2 12.5 1 6.3 0 0.0 2016 mei 35 61.4 18 31.6 3 5.3 1 1.8 2016 june 6 66.7 1 11.1 2 22.2 0 0.0 2016 july 3 60.0 2 40.0 0 0.0 0 0.0 2016 august 5 55.6 4 44.4 0 0.0 0 0.0 2016 september 8 50.0 6 37.5 2 12.5 0 0.0 2016 october 12 60.0 6 30.0 2 10.0 0 0.0 2016 november 5 50.0 5 50.0 0 0.0 0 0.0 2016 december 4 57.1 2 28.6 1 14.3 0 0.0 2017 january 4 57.1 2 28.6 1 14.3 0 0.0 2017 february 10 62.5 4 25.0 1 6.3 1 6.3 2017 march 41 80.4 9 17.6 1 2.0 0 0.0 2017 april 41 82.0 7 14.0 2 4.0 0 0.0 2017 mei 0 0.0 1 50.0 1 50.0 0 0.0 total 577 53.9 328 30.6 119 11.1 47 4.4 table 1 shows that the genesis malaria on the tribe of banjar each month is relatively always higher than genesis malaria on the other tribes. when compared with the other tribes, then genesis malaria on the tribe of banjar the average range between 11.1 percent to 82.0%. now the variables used are: the tribe of banjar (y1t): the number of the prevalence of malaria tribe of banjar against all the tribes in tanah bumbu the tribe of central java (y2t): the number of the prevalence of malaria javanese against all the tribes in tanah bumbu the tribe of bugis origin ((y3t): the number of the prevalence of malaria tribe bugis vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 155 against all the tribes in tanah bumbu other tribes (y4t): the number of the prevalence of malaria tribes besides (banjar, java, bugis) against all the tribes in tanah bumbu. next time series plots, acf and pacf each research variables presented in figure 3. 3632282420161284 20 10 0 -10 -20 -30 index d 1 _ b a n ja r time series plot of d1_banjar 302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 lag a u t o c o r r e la t io n autocorrelation function for d1_banjar (with 5% significance limits for the autocorrelations) 302520151051 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 lag p a r t ia l a u t o c o r r e la t io n partial autocorrelation function for d1_banjar (with 5% significance limits for the partial autocorrelations) a) the tribe of banjar b) javanese c) the tribe of bugis d) other tribes figure 3. acf and pacf plots differencing 1 prevalence of malaria based on ethnicity based on the figure 3 shows that the number of patients with malaria tribes (banjar, java, bugis, others) form the pattern that stationary on timeseries plots, while on the acf and pacf plots there is a lag that out of bounds. it can be said that the data the number of patients with malaria other tribes stationary in the average and variance. the test stationary through augmented dickey-fuller presented in the following table. vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 156 table 2. the value of the augmented dickey-fuller test statistic the tribe of t-statistic prob decision banjar -3.812047 0.0063 reject h0 java -3.251855 0.0252 reject h0 bugis -3.624860 0.0104 reject h0 others -4.238311 0.0021 reject h0 table 2 shows that the prob. on all tribes (banjar, java, bugis, others) smaller than α=0.05, which means giving the decision reject ho. it can be said that the number of the prevalence of malaria on the tribe of banjar, javanese, bugis tribe and other tribes have been stationary in mean and varians. furthermore, done causalitas granger tests, presented in table 3. table 3. granger causality test null hypothesis obs f-statistic prob. java does not granger cause banjar 34 0.13943 0.8704 banjar does not granger cause java 0.83067 0.4459 bugis does not granger cause banjar 34 0.86838 0.4303 banjar does not granger cause bugis 0.28776 0.7521 others does not granger cause banjar 34 0.76900 0.4727 banjar does not granger cause others 1.58712 0.2218 bugis does not granger cause java 34 1.41222 0.2599 java does not granger cause bugis 0.10857 0.8975 others does not granger cause java 34 0.04465 0.9564 java does not granger cause others 3.03627 0.0635 others does not granger cause bugis 34 1.42283 0.2574 bugis does not granger cause others 2.74853 0.0807 table 3. shows that with the level of the significance α=10%, value prob. smaller than ten percent is the number of the prevalence of malaria javanese with other tribes and tribal bugis with other tribes. this shows that there is the influence between the number of the prevalence of malaria javanese with other tribes and the number of the prevalence of malaria tribe bugis with other tribes. next, done modeling var in the tribe of banjar (y1t), javanese (y2t), the tribe of bugis origin (y3t), and other tribes (y4t), which is presented in the following table. vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 157 table 4. the value of the estimator, standard deviation, t-statistics var model number of the prevalence of malaria based on ethnicity the tribe of banjar (y1t) javaness (y2t) the tribe of bugis (y3t) other tribes (y4t) the tribe of banjar (y1t-1) 0.325115 -0.085934 -0.041048 -0.019274 (0.33166) (0.24043) (0.07731) (0.04538) [ 0.98027] [-0.35742] [-0.53094] [-0.42472] the tribe of banjar (y1t-2) 0.277228 0.346611 0.124575 0.026518 (0.31300) (0.22690) (0.07296) (0.04283) [ 0.88572] [ 1.52756] [ 1.70739] [ 0.61919] the tribe of central java (y2t-1) 0.198695 0.418743 0.136616 0.101222 (0.56290) (0.40807) (0.13122) (0.04702) [ 0.35298] [ 1.02616] [ 1.04115] [ 2.15274] the tribe of central java (y2t-2) 0.302650 -0.059674 -0.181024 0.013926 (0.46587) (0.33773) (0.10860) (0.06375) [ 0.64964] [-0.17669] [-1.66691] [ 0.21846] the tribe of bugis (y3t-1) 0.645513 0.843973 0.317533 0.136022 (0.90863) (0.65870) (0.21181) (0.06853) [ 0.71042] [ 1.28127] [ 1.49916] [ 1.98485] the tribe of bugis (y3t-2) -0.955027 -0.525865 0.074744 -0.096987 (0.84783) (0.61463) (0.19764) (0.11601) [-1.12643] [-0.85558] [ 0.37819] [-0.83603] other tribes (y4t-1) -1.731879 -0.124645 -0.399394 -0.074635 (1.85035) (1.34139) (0.43133) (0.25318) [-0.93597] [-0.09292] [-0.92596] [-0.29479] other tribes (y4t-2) 0.811660 -0.501651 0.653682 -0.039508 (1.55322) (1.12599) (0.32207) (0.21253) [ 0.52257] [-0.44552] [ 2.02962] [-0.18590] r-squared 0.230273 0.299488 0.585486 0.361856 table 4 shows that the results of the estimation of var models with lag two monthly provide the determination coefficient value for the equation number of the prevalence of malaria based on ethnicity dimasing of each tribe is the tribe of banjar (0.230), javanese (0.299), the tribe of bugis (0.585), and other tribes (0.362). this indicates that the variables are not yet strong enough to explain the fluctuations in the number of the prevalence of malaria. this is because the fluctuations in the number of variables the prevalence of malaria was influenced by many good variables that originate from external and internal, for example the factors behavior. model var based on ethnicity as follows. 1 1 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.325 0.277 0.198 0.302 0.645 0.955 .732 0.811 2 t t t t t t t t t t y y y y y y y y y y                   2 2 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.085 .347 0.419 0.059 0.844 0.525 .125 0.5010 + 0 t t t t t t t t t t y y y y y y y y y y                   3 3 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.041 .125 0.136 0.181 0.318 0.074 .399 0.6530 + 0 t t t t t t t t t t y y y y y y y y y y                   4 4 1 1 1 1 2 2 1 2 2 3 1 3 2 4 1 4 2 0.019 .026 0.101 0.0140 + 0.0960.136 .075 0.0390 t t t t t t t t t t y y y y y y y y y y                   to view the model stability, used the criteria of the value of the root of modulus less than 1. modulus values from the var model is presented in the following table. vector autoregressive modeling on cases of malaria based on the tribal in tanah bumbu district abdul khair 158 table 5. review of the condition of the stability of the var root modulus 0.831296 0.831296 0.5344060.338320 i 0.632496 + 0.5344060.338320 i 0.632496 0.577297 0.577297 0.378896-0.371162 i 0.530399 + 0.378896-0.371162 i 0.530399 -0.521198 0.521198 0.165044 0.165044 table 5 indicates whether the results of the estimation of the model var stable or stationary. if all the roots of has modulus less than one and the model is stable. all root on the table 5 shows that the roots have modulus less than one so that the var model has been stable. because the model var stable then the results of the forecast for out of sample number of the prevalence of malaria based on ethnicity presented on the following table. table 6. the results of the forecast number of the prevalence of malaria var model tribal based year month banjar javaness bugis others actual forecast actual forecast actual forecast actual forecast 2017 january 4 6 2 6 1 1 0 0 2017 february 10 11 4 15 1 3 1 1 2017 march 41 17 9 9 1 1 0 1 2017 april 41 36 7 4 2 0 0 0 2017 mei 0 2 1 1 1 1 0 0 mape 68% 36% 8% 2% table 6 shows that the forecast of the number of the prevalence of malaria in five next period on the tribe of banjar mape value of 68% (entered in the ability of bad forecasts), on javanese mape value of 36% (entered in the ability of divination enough), in the tribe of the bugis provides value mape is 8% (entered in the ability of divination is very good), and on the other tribes provide a 2 percent mape value (entered in the ability of divination is very good). the ability of divination number of the prevalence of malaria on the whole of 28.5 %, so that entered in the category of good. conclusions based on the results of the analysis that has been done, it can be concluded that the pattern of the number of the prevalence of malaria tends to be uniform from the four tribes, the number of the prevalence of malaria javanese there is a relationship with the other tribes and the number of the prevalence of malaria tribe bugis with other tribes. var model tribal based on the prevalence of malaria is strong enough to explain the fluctuations in the number of the prevalence of malaria among the tribe. the number of the prevalence of malaria javanese influenced by the number of the prevalence of malaria javanese at period t-1, and the number of the prevalence of malaria tribe bugis at period t-1. while the number of the prevalence of malaria tribe bugis influenced by the number of the prevalence of malaria other 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[26] h.p. lopuhaa, space time autoregressive models. delft university of technology. jerman, 2002. cauchy –jurnal matematika murni dan aplikasi volume 6(3) (2020), pages 122-132 p-issn: 2086-0382; e-issn: 2477-3344 submitted: july 08, 2020 reviewed: october 02, 2020 accepted: november 05, 2020 doi: http://dx.doi.org/10.18860/ca.v6i3.9891 local dynamics of an svir epidemic model with logistic growth joko harianto1, inda puspita sari2 1department of mathematics, cenderawasih university, 2faculty of medicine, cenderawasih university, email: joharijpr88@gmail.com, indasarira@yahoo.co.id abstract discussion of local stability analysis of svir models in this article included in the scope of applied mathematics. the purpose of this discussion was to provide results of local stability analysis that had not to discuss in some articles related to the svir model. the svir models discussed in this article involve logistics growth in the vaccinated compartment. the results obtained, i.e., if the basic reproduction number ℜ0 < 1 and 𝑚 > 0, then there is one equilibrium point i.e. 𝐸 0 is locally asymptotically stable. in the field of epidemiology, this means that the disease will disappear from the population. however, if ℜ0 > 1, 𝛽1 > 𝛽 and 𝑚 > 0, then there are two equilibrium points, i.e., disease-free equilibrium point denoted by 𝐸0 and the endemic equilibrium point denoted by 𝐸1 ∗. in this case, the endemic equilibrium point 𝐸1 ∗ locally asymptotically stable. in the field of epidemiology, this means that the disease will remain in the population. the numerical simulation supports these results. keywords: local dynamics; svir model; logistic growth introduction the mathematical modeling of infectious disease spread is one of the areas of research for mathematicians and epidemiologists. the formulated model is often said to be a model epidemic. epidemic models sometimes involve complexity and non-linearity. therefore, not all epidemic models can found solutions with analytical methods. epidemic models usually formulate with a deterministic or stochastic approach. kermack and mckendrick first introduced a deterministic model in the article [1], which was later generalized by capasso and serio in the paper [2]. in the article [2], the population is divided into three parts, i.e., susceptible, infected, and recovered. each population contains some people whose health conditions correspond to the name of the population. the model is known as the sir model. the sir model represents the dynamic epidemic of a person infected to a susceptible person through direct contact with the closed population. the sir model formulate as an initial value condition of the ordinary differential equation system, and it is analyzed mathematically. this sir model is then continuously developed by many researchers for various cases occurring today. based on the sir model developed by capasso and serio, the spread of infectious diseases can formulate by dividing the population into some compartment that depends on health conditions. therefore, the vaccine given to a person can also be considered a specific situation to add to its compartment. people who have given vaccines are included in the v population [3]. the v population is then added to the si model to study the dynamics of the spread of tuberculosis disease. the siv model later generalizes into the svir model in the article [4], which discusses the possibility that http://dx.doi.org/10.18860/ca.v6i3.9891 mailto:joharijpr88@gmail.com mailto:indasarira@yahoo.co.id local dynamics of an svir epidemic model with logistic growth joko harianto 123 individuals can recover from the disease. furthermore, the svir model develops to study the transmission dynamics of influenza disease discussed in articles [5] and [6]. the vaccination strategy discussed in some of the above articles is a continuously given vaccination. some cases found that vaccines sometimes administer for a certain period or seasonal. vaccination strategies for this state are usually called pulse vaccination strategy (pvs). a comparison between the continuous vaccination model and the pvs discuss in the article [7]. later, the model of the continuous vaccination strategy mentioned by liu et al. was developed by khan et al., islam, harianto, and suparwati. in the article [8], the non-linear saturation rate adds to the svir model with a continuous vaccination strategy. the addition of population migration factors in the susceptible population of the svir model with a continuous vaccination strategy discuss in the article [9]. in the paper [10], the svir model with a continuous vaccination strategy was developed by adding a death factor due to disease infection. in other words, the infectious disease is a deadly disease. the discussion also provided numerical simulation to support the analysis results. the death rate of each population in the svir model is assumed to be identical. a year later, harianto and suparwati continued discussing the svir model by adding different death rate assumptions for each population in the svir model. its numerical analysis and simulation discuss in the article [11]. in the paper [1] until [11], the population growth assumes to increase exponentially. the models in these articles are more realistic if their population growth uses logistic growth. in other words, the population assumes to be growing in logistics. the logistics growth factor on the sir model discusses in the articles [12],[13],[14],[15], and [16]. the paper in [17] and [18] also discusses other compartment models that use logistical growth. li et al. discuss the paper [12] as a sir model with a saturation incidence rate. according to the article, these discussions will be modified svir models in [7] by involving the logistic growth in the population studied in [12]. furthermore, it discussed the existence of the equilibrium point and local stability. the last part makes a simulated model that has analysis. methods this study uses the literature review method. the references used are in the form of several reputable international scientific books and journals. the theories used to support the results of this study are the equilibrium point stability theory of ordinary differential equation systems described in a book written by perko [19] and the theory related to the construction of the basic reproduction number of an epidemiological model in article [20]. the flow of thought in this discussion, first adds assumptions to the model that has been studied, then forms the model. next, determine the equilibrium point of the system and analyze its stability. finally, a numerical simulation is given to check the correctness of the analysis results obtained and a sensitivity analysis of the basic reproduction number. results and discussion model the model discussed in this section is a modification of the svir model with a continuous vaccination strategy adopted from the article [7]. the modification of the model is the growth of its population using logistics growth and the death rate caused by disease involved. this model divides the human population into four epidemiological classes, i.e., susceptible (s), infected (i), vaccinated (v), and recovered (r). the dynamic spread of the disease for the case formulate as the following ordinary differential equation system. local dynamics of an svir epidemic model with logistic growth joko harianto 124 𝑑𝑆 𝑑𝑡 = 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝛽𝑆𝐼 − (𝜇 + 𝛼)𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 − 𝛽1𝑉𝐼 − (𝜇 + 𝛾1)𝑉 (1) 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 − (𝛾 + 𝜇 + 𝜔)𝐼 𝑑𝑅 𝑑𝑡 = 𝛾1𝑉 + 𝛾𝐼 − 𝜇𝑅 with the initial conditions 𝑆(0) = 𝑆0 > 0, 𝑉(0) = 𝑉0 > 0, 𝐼(0) = 𝐼0 > 0 dan 𝑅(0) = 𝑅0 ≥ 0. 𝑟 denotes the intrinsic growth rate of susceptible population, 𝐾 is the carrying capacity of the area. 𝛽 is the contact rate between susceptibles and infected, 𝜇 is the natural death rate, 𝛼 is the rate at which an individual leaves the class of the susceptible by becoming the vaccination process. 𝛽1 is the contact rate between vaccinated and infected, 𝛾1 is the recovery rate from the vaccinated class, 𝛾 is the natural recovery rate from the infected class. 𝜔 is the rate of death caused by disease. theorem 1 all solutions of the system (1) with initial conditions (𝑆0, 𝑉0, 𝐼0, 𝑅0) ∈ ℝ+ 4 are positive and bounded for all 𝑡 ≥ 0. proof: the system (1) can be written 𝑑�̃� 𝑑𝑡 = 𝑓(�̃�) with �̃� = (𝑆, 𝑉, 𝐼, 𝑅) ∈ ℝ4 and 𝑓(�̃�) = (𝑓1(�̃�), 𝑓2(�̃�), 𝑓3(�̃�), 𝑓4(�̃�)) ∈ ℝ 4. clearly 𝑓(�̃�) differentiable continuous for all �̃� ∈ ℝ4 such that the system (1) has a unique solution. if we let 𝐺(𝑡) = 𝛽1𝐼(𝑡) + 𝜇 + 𝛾1, then the second equation of the system (1) can be written 𝑑𝑉(𝑡) 𝑑𝑡 + 𝐺(𝑡)𝑉(𝑡) = 𝛼𝑆(𝑡) (2) multiplying both sides of the equation (2) by exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) gives 𝑑𝑉(𝑡) 𝑑𝑡 exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) + 𝐺(𝑡)𝑉(𝑡) exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) = 𝛼𝑆(𝑡) exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) by using multiplication rules on derivatives we have 𝑑 𝑑𝑡 [𝑉(𝑡) exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 )] = 𝛼𝑆(𝑡) exp (∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) (3) by integrating two sides of the equation (3) with 𝑡 gives 𝑉(𝑡) = 𝑉0 exp (− ∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) + [∫ 𝛼𝑆(𝜏) exp (∫ 𝐺(𝑢)𝑑𝑢 𝜏 0 ) 𝑑𝜏 𝑡 0 ] exp (− ∫ 𝐺(𝑢)𝑑𝑢 𝑡 0 ) > 0. similarly, we can prove that 𝑆(𝑡), 𝐼(𝑡) dan 𝑅(𝑡) are nonnegative for all 𝑡 > 0. consequently, from the first equation of the system (1), we have 𝑑𝑆 𝑑𝑡 = 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝛽𝑆𝐼 − (𝜇 + 𝛼)𝑆 < 𝑟𝑆 (1 − 𝑆 𝐾 ) hence, there exists 𝑀 = max{𝑆0, 𝐾} such that lim 𝑡→∞ sup 𝑆(𝑡) < 𝑀 let 𝑁(𝑡) = 𝑆(𝑡) + 𝑉(𝑡) + 𝐼(𝑡) + 𝑅(𝑡), total population change is 𝑑𝑁 𝑑𝑡 = 𝑑𝑆 𝑑𝑡 + 𝑑𝑉 𝑑𝑡 + 𝑑𝐼 𝑑𝑡 + 𝑑𝑅 𝑑𝑡 local dynamics of an svir epidemic model with logistic growth joko harianto 125 = 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝜇𝑁 − 𝜔𝐼 < 𝑟𝑆 − 𝜇𝑁 < 𝑟𝑀 − 𝜇𝑁 then we have 0 < 𝑁(𝑡) < 𝑟𝑀 𝜇 for 𝑡 → ∞. therefore, all solutions of the system (1) are positive and bounded all 𝑡 ≥ 0. equilibria and basic reproduction number we see that 𝑅 does not appear in three equations in the system (1). hence, the fourth equation on the system (1) can be ignored, and the discussion is focused on three equations as follows: 𝑑𝑆 𝑑𝑡 = 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝛽𝑆𝐼 − (𝜇 + 𝛼)𝑆 𝑑𝑉 𝑑𝑡 = 𝛼𝑆 − 𝛽1𝑉𝐼 − (𝜇 + 𝛾1)𝑉 (4) 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 − (𝛾 + 𝜇 + 𝜔)𝐼 the equilibrium point of the system (4) obtained by resolving the following equations: 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝛽𝑆𝐼 − (𝜇 + 𝛼)𝑆 = 0 (5) 𝛼𝑆 − 𝛽1𝑉𝐼 − (𝜇 + 𝛾1)𝑉 = 0 (6) 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 − (𝛾 + 𝜇 + 𝜔)𝐼 = 0 (7) clearly from the equation (3), (4), and (5) that if 𝑚 = 𝑟 − (𝜇 + 𝛼) > 0, then system (4) has a unique disease-free equilibrium point. the disease-free equilibrium point of the system (4) denoted by 𝐸0 = (𝑆0, 𝑉0, 𝐼0) with 𝑆0 = 𝐾𝑚 𝑟 , 𝑉0 = 𝐾𝛼𝑚 𝑟(𝜇+𝛾1) , 𝐼0 = 0. the basic reproduction number in epidemiology used to indicate the number of diseases and control spread. the basic reproduction number denoted by ℜ0. if ℜ0 < 1 then the spread of disease in the population predicted to disappear. however, if ℜ0 > 1 then the virus will remain in the population. the basic reproduction number of the system (4) in this discussion is determined based on the theory learned in the article [21]. let 𝑋 = (𝑆, 𝑉), 𝑋0 = (𝑆0, 𝑉0), 𝑍 = (𝐼), ℎ(𝑋, 𝑍) = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 − 𝑛𝐼 with 𝑛 = 𝛾 + 𝜇 + 𝜔 > 0 and 𝑓(𝑋, 𝑍) = [ 𝑟𝑆 (1 − 𝑆 𝐾 ) − 𝛽𝑆𝐼 − (𝜇 + 𝛼)𝑆 𝛼𝑆 − 𝛽1𝑉𝐼 − (𝜇 + 𝛾1)𝑉 ] derivating ℎ(𝑋0, 𝑍) by 𝑍 gives 𝐷𝑧ℎ(𝑋 0, 𝑍)|𝑧=0 = 𝛽𝑆 0 + 𝛽1𝑉 0 − 𝑛 we have 𝐴 = 𝐷𝑧ℎ(𝑋 0, 𝑍)|𝑧=0 = 𝛽𝑆 0 + 𝛽1𝑉 0 − 𝑛 because 𝐴 = 𝑀 − 𝐷 then gives 𝑀 = 𝛽𝑆0 + 𝛽1𝑉 0 and 𝐷 = 𝑛. hence, we have ℜ0 = 𝜌(𝑀𝐷 −1) = 𝛽𝑆0 + 𝛽1𝑉 0 𝑛 = 𝛽𝐾𝑚(𝜇 + 𝛾1) + 𝛽1𝐾𝛼𝑚 𝑟𝑛(𝜇 + 𝛾1) with 𝑚 = 𝑟 − (𝜇 + 𝛼) and 𝑛 = 𝛾 + 𝜇 + 𝜔. the following discussed the existence of the endemic equilibrium point denoted by 𝐸∗ = (𝑆∗, 𝑉∗, 𝐼∗). assume that 𝑆∗, 𝐼∗, 𝑉∗ ≠ 0. from equation (5), we have 𝑟 (1 − 𝑆 𝐾 ) − 𝛽𝐼 − (𝜇 + 𝛼) = 0 then local dynamics of an svir epidemic model with logistic growth joko harianto 126 𝑆∗ = 𝐾 𝑟 (𝑚 − 𝛽𝐼∗) from equation (6), we have 𝑉∗ = 𝛼𝑆∗ 𝛽1𝐼 ∗ + 𝜇 + 𝛾1 = 𝐾𝛼(𝑚 − 𝛽𝐼∗) 𝑟(𝛽1𝐼 ∗ + 𝜇 + 𝛾1) from equation (7), we have 𝛽𝑆∗ + 𝛽1𝑉 ∗ − 𝑛 = 0 (8) with 𝑛 = 𝛾 + 𝜇 + 𝜔. if 𝑆∗ = 𝐾 𝑟 (𝑚 − 𝛽𝐼∗) and 𝑉∗ = 𝐾𝛼(𝑚−𝛽𝐼∗) 𝑟(𝛽1𝐼 ∗+𝜇+𝛾1) substituted to equation (8), then 𝐴1𝐼 ∗2 + 𝐴2𝐼 + 𝐴3 = 0 (9) with 𝐴1 = 𝐾 𝑟 𝛽2𝛽1 𝐴2 = 𝐾 𝑟 𝛽2(𝜇 + 𝛾1) + 𝐾 𝑟 𝛽𝛽1𝛼 + 𝑛𝛽1 − 𝐾 𝑟 𝛽𝛽1𝑚 𝐴3 = 𝑛(𝜇 + 𝛾1) − 𝐾 𝑟 𝛽𝑚(𝜇 + 𝛾1) − 𝐾 𝑟 𝛽1𝑚𝛼 = 𝑛(𝜇 + 𝛾1)(1 − ℜ0) hence, 𝐼∗ is the root of the equation (9), i.e.: 𝐼1,2 ∗ = −𝐴2 ± √𝐴2 2 − 4𝐴1𝑛(𝜇 + 𝛾1)(1 − ℜ0) 2𝐴1 clearly that if ℜ0 < 1 then 𝑟𝑛 > 𝐾𝛽𝑚, consequently 𝐴2 > 0. thus, the existence of the endemic equilibrium point depends on ℜ0. the existence of disease-free and the endemic equilibrium points associated with basic reproductive numbers summarized in the following theorems. theorem 2 we define ℜ0 = 𝛽𝐾𝑚(𝜇 + 𝛾1) + 𝛽1𝐾𝛼𝑚 𝑟𝑛(𝜇 + 𝛾1) 𝑔(𝐼∗) = 𝐴1𝐼 ∗2 + 𝐴2𝐼 + 𝑛(𝜇 + 𝛾1)(1 − ℜ0) 𝐴1 = 𝐾 𝑟 𝛽2𝛽1 𝐴2 = 𝐾 𝑟 𝛽2(𝜇 + 𝛾1) + 𝐾 𝑟 𝛽𝛽1𝛼 + 𝑛𝛽1 − 𝐾 𝑟 𝛽𝛽1𝑚 with 𝑚 = 𝑟 − (𝜇 + 𝛼) and 𝑛 = 𝛾 + 𝜇 + 𝜔 > 0. i. if ℜ0 < 1 and 𝑚 > 0 then 𝐴2 > 0 such that there exists a unique equilibrium point of the system (4), i.e. the disease-free equilibrium point denoted by 𝐸0 = ( 𝐾𝑚 𝑟 , 𝐾𝛼𝑚 𝑟(𝜇+𝛾1) , 0). ii. if ℜ0 > 1 and 𝑚 > 0, then system (4) has only two equilibrium point, i.e. the disease-free equilibrium point denoted by 𝐸0 = ( 𝐾𝑚 𝑟 , 𝐾𝛼𝑚 𝑟(𝜇+𝛾1) , 0) and the endemic equilibrium point denoted by 𝐸1 ∗ = (𝑆1 ∗, 𝑉1 ∗, 𝐼1 ∗) with 𝑆1 ∗ = 𝐾 𝑟 (𝑚 − 𝛽𝐼∗), 𝑉1 ∗ = 𝐾𝛼(𝑚−𝛽𝐼∗) 𝑟(𝛽1𝐼 ∗+𝜇+𝛾1) and 𝐼1 ∗ is the positive root of 𝑔(𝐼∗). iii. if ℜ0 = 1, 𝑚 > 0 and 𝐴2 < 0, then system (4) has only two equilibrium point, i.e. the disease-free equilibrium point denoted 𝐸0 = ( 𝐾𝑚 𝑟 , 𝐾𝛼𝑚 𝑟(𝜇+𝛾1) , 0) and the endemic local dynamics of an svir epidemic model with logistic growth joko harianto 127 equilibrium point denoted 𝐸2 ∗ = (𝑆2 ∗, 𝑉2 ∗, 𝐼2 ∗) with 𝑆2 ∗ = 𝐾 𝑟 (𝑚 − 𝛽𝐼∗), 𝑉2 ∗ = 𝐾𝛼(𝑚−𝛽𝐼∗) 𝑟(𝛽1𝐼 ∗+𝜇+𝛾1) and 𝐼2 ∗ = − 𝐴2 𝐴1 . local stability analysis this section discusses the local stability analysis of disease-free equilibrium and endemic points of the system (4). the local stability of the equilibrium point of the system (4) is determined using the jacobian matrix (linearization method) studied in [19]. the jacobian matrix of systems (4) written as follows: 𝐽 = [ −2𝑟𝑆 𝐾 − 𝛽𝐼 + 𝑚 0 −𝛽𝑆 𝛼 −𝛽1𝐼 − (𝜇 + 𝛾1) −𝛽1𝑉 𝛽𝐼 𝛽1𝐼 𝛽𝑆 + 𝛽1𝑉 − 𝑛 ] with 𝑚 = 𝑟 − (𝜇 + 𝛼) and 𝑛 = 𝛾 + 𝜇 + 𝜔 > 0. local stability analysis of the disease-free equilibrium point of the system (4) presented in the following theorems. theorem 3 if ℜ0 < 1 and 𝑚 > 0, then the disease-free equilibrium point of the system (4), i.e. 𝐸 0 = ( 𝐾𝑚 𝑟 , 𝐾𝛼𝑚 𝑟(𝜇+𝛾1) , 0) is locally asymptotically stable. proof. the jacobian matrix of the system (4) in 𝐸0, i.e. 𝐽(𝐸0) = [ −𝑚 0 −𝛽𝑆0 𝛼 −(𝜇 + 𝛾1) −𝛽1𝑉 0 0 0 𝑛(ℜ0 − 1) ] hence, the characteristics equation of 𝐽(𝐸0), i.e. (𝜆 − 𝑛(ℜ0 − 1))(𝜆 + 𝑚)(𝜆 + 𝜇 + 𝛾1) = 0 then we have eigenvalue 𝜆1 = 𝑛(ℜ0 − 1), 𝜆2 = −𝑚 dan 𝜆3 = −𝜇 − 𝛾1 < 0. therefore, if ℜ0 < 1 and 𝑚 > 0, then all the eigenvalues of 𝐽(𝐸 0) are negative consequently according to [19], 𝐸0 is locally asymptotically stable. local stability analysis of the endemic equilibrium point of the system (4) presented in the following theorems. theorem 4 if ℜ0 > 1, 𝛽1 > 𝛽 and 𝑚 > 0, then the endemic equilibrium point of the system (4) denoted by 𝐸1 ∗ = (𝑆1 ∗, 𝑉1 ∗, 𝐼1 ∗) is locally asymptotically stable with 𝑆1 ∗ = 𝐾 𝑟 (𝑚 − 𝛽𝐼∗), 𝑉1 ∗ = 𝐾𝛼(𝑚−𝛽𝐼∗) 𝑟(𝛽1𝐼 ∗+𝜇+𝛾1) and 𝐼1 ∗ is the root of 𝑔(𝐼∗) = 𝐴1𝐼 ∗2 + 𝐴2𝐼 + 𝑛(𝜇 + 𝛾1)(1 − ℜ0), 𝐴1 = 𝐾 𝑟 𝛽2𝛽1 , 𝐴2 = 𝐾 𝑟 𝛽2(𝜇 + 𝛾1) + 𝐾 𝑟 𝛽𝛽1𝛼 + 𝑛𝛽1 − 𝐾 𝑟 𝛽𝛽1𝑚. proof: the jacobian matrix of the system (4) in 𝐸1 ∗, i.e. 𝐽(𝐸1 ∗) = [ − 2𝑟 𝐾 𝑆1 ∗ − 𝛽𝐼1 ∗ + 𝑚 0 −𝛽𝑆1 ∗ 𝛼 −𝛽1𝐼1 ∗ − (𝜇 + 𝛾1) −𝛽1𝑉1 ∗ 𝛽𝐼1 ∗ 𝛽1𝐼1 ∗ 𝛽𝑆1 ∗ + 𝛽1𝑉1 ∗ − 𝑛 ] because 𝑆1 ∗ ≠ 0 and according to equation (5) we have 𝑟 − 𝑟 𝐾 𝑆1 ∗ − 𝛽𝐼1 ∗ − (𝜇 + 𝛼) = 0 ⇒ 𝑚 − 2𝑟 𝐾 𝑆1 ∗ − 𝛽𝐼1 ∗ = − 𝑟 𝐾 𝑆1 ∗ local dynamics of an svir epidemic model with logistic growth joko harianto 128 from equation (6) we have 𝛽1𝐼1 ∗ + 𝜇 + 𝛾1 = 𝛼𝑆1 ∗ 𝑉1 ∗ and clearly 𝛽𝑆1 ∗ + 𝛽1𝑉1 ∗ − 𝑛 = 0. hence, 𝐽(𝐸1 ∗) equivalent with 𝐽(𝐸1 ∗) = [ − 𝑟 𝐾 𝑆1 ∗ 0 −𝛽𝑆1 ∗ 𝛼 − 𝛼𝑆1 ∗ 𝑉1 ∗ −𝛽1𝑉1 ∗ 𝛽𝐼1 ∗ 𝛽1𝐼1 ∗ 0 ] thus the characteristics equation of 𝐽(𝐸1 ∗), i.e. 𝑎1𝜆 2 + 𝑎2𝜆 + 𝑎3 = 0 with 𝑎1 = 𝛼𝑆1 ∗ 𝑉1 ∗ + 𝑟 𝐾 𝑆1 ∗ 𝑎2 = 𝑟𝛼𝑆1 ∗2 𝐾𝑉1 ∗ + 𝛽 2𝑆1 ∗𝐼1 ∗ + 𝛽1 2𝑉1 ∗𝐼1 ∗ 𝑎3 = 𝛽𝛽1𝛼𝑆1 ∗𝐼1 ∗ + 𝛽2𝛼 𝑆1 ∗2 𝑉1 ∗ 𝐼1 ∗ + 𝑟 𝐾 𝛽1 2𝑆1 ∗𝑉1 ∗𝐼1 ∗ 𝑎1𝑎2 − 𝑎3 = 𝑟𝛼2𝑆1 ∗3 𝐾𝑉1 ∗2 + 𝑟2𝛼𝑆1 ∗3 𝐾2𝑉1 ∗ + 𝑟𝛽2 𝐾 𝑆1 ∗2𝐼1 ∗ + 𝛽1𝛼𝑆1 ∗𝐼1 ∗(𝛽1 − 𝛽) clearly 𝑎1 > 0, 𝑎2 > 0, 𝑎3 > 0 and if 𝛽1 > 𝛽 then 𝑎1𝑎2 − 𝑎3 > 0. therefore, based on the routh-hurtwiz criteria if 𝛽1 > 𝛽, then all the eigenvalues of 𝐽(𝐸1 ∗) are negative consequently the endemic equilibrium point of the system (4) denoted by 𝐸1 ∗ locally asymptotically stable. numerical simulations the following given numerical simulations to confirm the results obtained analytic related to local stability of disease-free equilibrium points and endemic to the system (1). figure 1 is the local dynamics of the solution on the system (1) with the parameter value 𝐾 = 1, 𝑟 = 0,04 , 𝜇 = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,002, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , 𝜔 = 0,02 , 𝛾1 = 0,004 and initial conditions 𝑆0 = 0,8 , 𝑉0 = 0,1, 𝐼0 = 0,1 , 𝑅0 = 0. from the value of the parameter then we can calculate that ℜ0 = 0,04 < 1, 𝑚 = 0,02 > 0 and 𝐸 0 = (𝑆0, 𝑉0, 𝐼0, 𝑅0) = (0,5 ; 0,36; 0 ; 0,14). according to theorem 3, the disease-free equilibrium point denoted by 𝐸0 is locally asymptotically stable. these results correspond to the numerical simulation given. it means that the disease will disappear from the population. figure 2 is the local dynamics of the solution on the system (1) with the parameter value 𝐾 = 1, 𝑟 = 0,04 , 𝜇 = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,1, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , 𝜔 = 0,02 , 𝛾1 = 0,004 and initial conditions 𝑆0 = 0,8 , 𝑉0 = 0,1, 𝐼0 = 0,1 , 𝑅0 = 0. clearly that 𝛽1 > 𝛽 then we can calculate that ℜ0 = 1,66 > 1, 𝑚 = 0,02 > 0 and the endemic equilibrium point denoted by 𝐸1 ∗ = (𝑆1 ∗, 𝑉1 ∗, 𝐼1 ∗, 𝑅1 ∗), i.e 𝐸1 ∗ = (0,3 ; 0,21; 0,08 ; 0,09). according to theorem 3, the endemic equilibrium point denoted by 𝐸1 ∗ locally asymptotically stable. these results correspond to the numerical simulation given. it means that the disease will remain in the population. local dynamics of an svir epidemic model with logistic growth joko harianto 129 figure 1. dynamics of the solution on the system (1) for the case ℜ0 = 0,04 < 1 figure 2. dynamics of the solution on the system (1) for the case ℜ0 = 1,66 > 1 local dynamics of an svir epidemic model with logistic growth joko harianto 130 sensitivity analysis of the basic reproduction number the sensitivity analysis performed was a local sensitivity ([22], [23], and [24]). in this section, the parameters analyzed are 𝛽, 𝛼, 𝛾1, 𝛾, 𝛽1. the sensitivity index for the parameter 𝛽 is 𝐶 𝛽 ℜ0 = 𝜕ℜ0 𝜕𝛽 × 𝛽 ℜ0 = 𝐾𝛽𝑚(𝜇 + 𝛾1) 𝐾𝛽𝑚(𝜇 + 𝛾1) + 𝛽1𝐾𝛼𝑚 with the same steps, we can obtain a sensitivity index for the other parameters. the following table 1 is given the five parameter sensitivity to the basic reproduction number. the parameter 𝛽, 𝛼, 𝛾1, 𝛾, 𝛽1 values used are 𝐾 = 1, 𝑟 = 0,04 , 𝜇 = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,002, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , 𝜔 = 0,02 , 𝛾1 = 0,004. table 1. sensitivity index of parameters to the basic reproduction number parameter sensitivity index 𝛽 +0,861 𝛽1 +0.005 𝛼 -0,138 𝛾 -0,001 𝛾1 -0.132 sensitivity index of parameter 𝛽 and 𝛽1 has a positive value to ℜ0, it's mean that when the parameter 𝛽 and 𝛽1 increases, then ℜ0 value also increases and conversely. the parameter sensitivity index of parameter 𝛼, 𝛾, 𝛾1has a negative value to ℜ0, it's mean that when the parameter 𝛼, 𝛾, 𝛾1 increases, then ℜ0 value decreases, and conversely. the most influential parameters of the five parameters are transmission rate (𝛽) and vaccine rate (𝛼). conclusions this article is discussed the local stability analysis of the equilibrium point of the svir model adapted from liu et al. the svir model discussed in this article was modified, assuming the growth rate on susceptible classes (v) approached with the growth of logistics. first, we determine the disease-free equilibrium point. then, we find the basic reproduction number by using a disease-free equilibrium point that obtains. next, we determine the existence of the endemic equilibrium point associated with the basic reproduction number (ℜ0). lastly, we determine the stability of the equilibrium point and 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