jurnal jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 1 radius, diameter, multiplisitas sikel, dan dimensi metrik graf komuting dari grup dihedral abdussakir uin maulana malik ibrahim malang, sakir@mat.uin-malang.ac.id abstrak graf komuting c(g) dari suatu grup non abelian g adalah graf yang memuat semua anggota grup g sebagai himpunan titik dan dua titik akan terhubung langsung di c(g) jika keduanya komutatif di grup g. dalam artikel ini dibahas mengenai graf komuting dari grup dihedral d2n. pembahasan dibatasi pada topik radius, diameter, multiplisitas sikel, dan dimensi metrik. beberapa teorema disajikan terkait topik tersebut disertai bukti kebenarannya. kata kunci: radius, diameter, multiplisitas sikel, dimensi metrik, graf komuting, grup dihedral. abstract commuting graph c(g) of a non abelian group g is graph that contains all element of g as its vertex set and two distinct vertices in c(g) will be adjacent if they are commute in g. in this paper we discuss about commuting graph of dihedral group d2n. we show radius, diameter, cycle multiplicity, and metric dimension of this commuting graph in several theorems with their proof. keywords: radius, diameter, cylce multiplicity, metric dimension, commuting graph, dihedral group. 1. pendahuluan graf 𝐺 memuat 𝑉(𝐺) sebagai himpunan tidak kosong dan berhingga dari objek-objek yang disebut titik dan 𝐸(𝐺) sebagai himpunan pasangan tak berurutan dari titik-titik berbeda di 𝑉(𝐺) yang disebut sisi. e(g) dapat berupa himpunan kosong [1]. jika 𝑒 = (𝑢, 𝑣) adalah sisi di graf 𝐺, maka u dan v disebut terhubung langsung. kardinalitas v(g) disebut order dari g dan kardinalitas dari e(g) disebut ukuran dari g [2]. perkembangan terbaru teori graf juga membahas graf yang dibangun dari grup. misal 𝐺 grup berhingga tak komutatif dan 𝑋 adalah subset dari 𝐺. graf komuting 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 [3]. beberapa penelitian mengenai graf komuting telah dilakukan. vahidi & talebi [3] meneliti bilangan bebas, bilangan clique, dan bilangan cover minimum graf komuting grup d2n dan qn. chelvam, dkk [4] meneliti beberapa sifat pada graf komuting dari grup dihedral dan menemukan bahwa bilangan kromatik pewarnaan titik sama dengan bilangan clique pada graf komuting yang diperoleh dari grup dihedral. rahayuningtyas, dkk [5] meneliti bilangan kromatik perwarnaan titik dan mailto:sakir@mat.uin-malang.ac.id jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 2 pewarnaan sisi pada graf komuting dan non komuting dari grup dihedral. pada penelitian ini dikaji beberapa sifat pada graf komuting dari grup dihedral untuk topik yang belum dibahas pada penelitianpenelitian sebelumnya. kajian difokuskan pada topik radius, diameter, multiplisitas sikel, dan dimensi metrik. 2. kajian teori 2.1 radius dan diameter graf misalkan g graf terhubung. jarak antara titik u dan v di g, dinotasikan dengan d(u, v), adalah panjang lintasan terpendek yang menghubungkan u dan v di g. eksentrisitas titik v di g, dinotasikan dengan e(v), adalah jarak terjauh dari titik v ke semua titik di g. jadi dapat dituliskan e(v) = max{d(u, v)u  v(g)}. titik v dikatakan titik eksentrik dari u jika jarak dari u ke v sama dengan eksentrisitas dari u atau d(u, v) = e(u). radius dari g, dinotasikan dengan rad(g), adalah eksentrisitas minimum dari semua titik di g. jadi, dapat dituliskan rad(g) = min{e(v) v v}. sedangkan diameter dari g, dinotasikan dengan diam(g), adalah eksentrisitas maksimum dari semua titik di g. jadi, dapat dituliskan diam(g) = max{e(v) v v} [6][2]. 2.2 multiplisitas sikel dua sikel atau lebih di dalam suatu graf disebut saling lepas sisi jika sikelsikel tersebut tidak memuat sisi yang sama. multiplisitas sikel dari graf g, dinotasikan dengan cm(g), didefinisikan sebagai bilangan terbesar yang menyatakan banyaknya sikel yang saling lepas sisi yang terdapat dalam graf g [7]. 2.3 dimensi metrik misalkan g graf, s = {x1, x2, ..., xn} himpunan bagian dari v(g), dan v adalah titik di g. representasi dari v terhadap s adalah tuple-n berurutan, r(u|s) = (d(v, x1), d(v, x2), …, d(v, xn)), dengan d(v, xi) adalah jarak antara titik v dan titik xi. himpunan s merupakan himpunan pemisah pada graf g jika untuk setiap titik pada graf g mempunyai representasi jarak yang berbeda terhadap s. himpunan pemisah dengan banyak anggota minimum dinamakan basis dari graf g. dimensi metrik pada graf g adalah kardinalitas minimum pada himpunan pemisah, dan dilambangkan dengan dim(g). dengan kata lain, dim(g) adalah kardinalitas basis dari graf g [8]. 2.4 grup dihedral grup dihedral adalah grup dari himpunan simetri-simetri dari segi-n beraturan, dinotasikan dengan d2n, untuk setiap n bilangan bulat positif dan n ≥ 3. grup dihedral d2n dapat dinyatakan sebagai d2n = {1, r, r 2, ..., rn-1, s, sr, sr2, ..., srn-1}. 2.5 graf komuting misal g adalah grup berhingga tak komutatif dan x adalah subset dari g . graf komuting 𝐶(𝐺, 𝑋) adalah graf dengan x sebagai himpunan titik dan dua elemen berbeda di 𝐶(𝐺, 𝑋) akan terhubung langsung jika keduanya adalah elemen yang saling komutatif di g [3]. dalam hal x = g, maka c(g, x) akan ditulis c(g) 3. hasil penelitian teorema 1. misalkan d2n = {1, r, r 2, ..., rn-1, s, sr, sr2, ..., srn-1} adalah grup dihedral order n dengan n bilangan bulat dan n ≥ 3. a. jika x = {1, r, r2, ..., rn-1}  d2n, maka graf komuting c(d, x) akan membentuk graf komplit order n, yaitu kn. b. jika x = {1, s, sr, sr2, ..., srn-1}  d2n, untuk n ganjil, maka graf komuting c(d, x) akan membentuk graf bintang order (n + 1), yaitu sn. c. jika x = {1, 𝑟 𝑛 2 , s, sr, sr2, ..., srn-1}  d2n, untuk n genap, maka maka graf komuting c(d, x) akan membentuk graf tripartisi komplit order (n + 2), yaitu k(1, 1, n). bukti a. karena masing-masing unsur di x saling komutatif, yaitu rirj = rjri, untuk i, j = 0, 1, 2, ..., n-1, maka ri dan rj akan   jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 3 saling terhubung langsung di c(d2n, x). dengan demikian, maka c(d2n, x) akan membentuk graf komplit order n. b. karena n ganjil, maka unsur sri hanya komutatif dengan 1 di d2n, untuk i = 0, 1, 2, …, n-1. akibatnya sri hanya terhubung langsung dengan 1 di c(d2n, x). jadi, 1 akan menjadi titik pusat dengan derajat n dan sri, i = 0, 1, 2, …, n-1, menjadi titik ujung. dengan demikian, maka c(d2n, x) akan membentuk graf bintang order (n + 1). c. karena n genap, maka center d2n adalah z(d2n) = {1, 𝑟 𝑛 2 }. artinya semua unsur di z(d2n) komutatif dengan semua unsur di d2n. sementara sr i tidak komutatif dengan srj untuk i ≠ j. dengan demikian, maka unsur di c(d2n, x) dapat dipartisi menjadi 3 partisi yaitu {1}, {𝑟 𝑛 2 }, dan {s, sr, sr2, ..., srn-1} sehingga masing-masing unsur di dalam satu partisi tidak saling terhubung langsung tetapi unsur di partisi yang berbeda saling terhubung langsung. jadi, c(d2n, x) membentuk graf tripartisi komplit order (n + 2). teorema 2. misalkan c(d2n) adalah graf komuting dari grup dihedral d2n (n ≥ 3). maka radius dan diameter c(d2n) masing-masing adalah rad(c(d2n)) = 1 dan diam(c(d2n)) = 2. bukti diketahui bahwa 1 ∘ x = x ∘ 1, untuk semua x  d2n. dengan demikian titik 1 akan terhubung langsung dengan semua titik yang lain di c(d2n). dengan demikian, maka e(1) = 1. karena radius c(d2n) adalah eksentrisitas terkecil di c(d2n) maka diperoleh rad(c(d2n)) = 1. karena semua titik terhubung langsung dengan 1, maka jarak sebarang dua titik berbeda di c(d2n) hanya memuat dengan kemungkinan, yaitu 1 atau 2. dua titik berbeda akan berjarak 1 jika saling terhubung langsung dan berjarak 2 jika tidak saling terhubung langsung. karena s dan r tidak komutatif di d2n, maka d(s, r) = 2. dengan demikian, maka e(s) = e(r) = 2. karena diameter adalah eksentrisitas terbesar maka diperoleh diam(c(d2n)) = 2. teorema 3. misalkan c(d2n) adalah graf komuting dari grup dihedral d2n (n ≥ 3). maka multiplisitas sikel dari c(d2n) adalah [ 𝑛2−2𝑛 6 ] untuk n ganjil dan [ 𝑛2 −2𝑛 6 ] + 𝑛 2 untuk n genap. bukti untuk n ganjil, sesuai teorema 1, maka subgraf komplit terbesar di c(d2n) adalah kn. dengan demikian maka multiplisitas sikel graf c(d2n) sama dengan multiplisitas sikel pada kn yaitu [ 𝑛2−2𝑛 6 ], untuk n ganjil. untuk n genap, sesuai teorema 1, maka subgraf komplit terbesar di c(d2n) adalah kn. dengan demikian maka multiplisitas sikel graf c(d2n) sama dengan multiplisitas sikel di kn yaitu [ 𝑛2−2𝑛 6 ] + 𝑛 2 . walaupun 𝑟 𝑛 2 komutatif dengan sri untuk i = 0, 1, 2, …, n-1 dan membentuk sikel-3 dengan titik 1, dalam sikelsikel ini tidak berpengaruh karena sisi (1, 𝑟 𝑛 2 ) sudah terhitung saat menghitung multiplisitas sikel di kn. jadi, multiplisitas sikel di c(d2n) tetap sama dengan multiplisitas sikel di kn yaitu [ 𝑛2−2𝑛 6 ] + 𝑛 2 , untuk n genap. teorema 4. misalkan c(d2n) adalah graf komuting dari grup dihedral d2n (n ≥ 3). maka dimensi metrik dari c(d2n) adalah 2n – 3 untuk n ganjil dan 3𝑛−4 2 untuk n genap. bukti untuk n ganjil, diketahui bahwa rirj = rjri, untuk i, j = 0, 1, 2, 3, …, n-1 di d2n. jadi, r i dan rj saling terhubung langsung di c(d2n). di lain pihak, sri hanya komutatif dengan 1 di d2n, untuk i = 0, 1, 2, …, n-1. jadi, sri tidak terhubung langsung dengan rj, untuk i, j = 0, 1, 2, 3, …, n-1 di c(d2n). ambil s = {s, sr, sr2, …, srn-2, r, r2, …, rn-2}. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 4 jadi, s memuat sebanyak 2n – 3 anggota. akan diperoleh bahwa representasi jarak semua titik di c(d2n) terhadap s adalah berbeda. jadi diperoleh dimensi metrik graf c(d2n) adalah dim(c(d2n))  2n – 3 ambil r himpunan bagian dari v(c(d2n)) dengan r=s1 < s. berarti ada minimal satu sri, i = 0, 2, 3, …, n–2 atau rj , j = 1, 2, 3, …, n–2 yang tidak masuk di r, sebut srp atau rq. akibatnya, srp dan srn-1 akan mempunyai representasi jarak yang sama atau rq dengan rn-1 akan mempunyai representasi jarak yang sama. jadi diperoleh dimensi metrik graf c(d2n) adalah dim(c(d2n)) > 2n – 4 atau dim(c(d2n))  2n – 3 terbukti, dim(c(d2n)) = 2n – 3, untuk n ganjil. untuk n genap, diketahui bahwa rirj = rjri, untuk i, j = 0, 1, 2, 3, …, n-1 di d2n. jadi, r i dan rj saling terhubung langsung di c(d2n). walaupun 𝑟 𝑛 2 komutatif dengan sri, i = 0, 1, 2, …, n-1 tetapi sri tidak komutatif dengan rj untuk j selain 𝑛 2 . ambil s = {s, sr, sr2, …, 𝑠𝑟 𝑛−2 2 , r, r2, …, rn-2}. jadi, s memuat sebanyak 3𝑛−4 2 anggota. akan diperoleh bahwa representasi jarak semua titik di c(d2n) terhadap s adalah berbeda. jadi diperoleh dimensi metrik graf c(d2n) adalah dim(c(d2n))  3𝑛−4 2 ambil r himpunan bagian dari v(c(d2n)) dengan r=s1 < s. berarti ada minimal satu sri, i = 0, 2, 3, …, n-2 atau rj , j = 1, 2, 3, …, n-2 yang tidak masuk di r, sebut srp atau rq. akibatnya, srp dan srn-1 akan mempunyai representasi jarak yang sama atau rq dengan rn-1 akan mempunyai representasi jarak yang sama. jadi diperoleh dimensi metrik graf c(d2n) adalah dim(c(d2n)) > 3𝑛−4 2 1 atau dim(c(d2n))  3𝑛−4 2 terbukti, dim(c(d2n)) = 𝟑𝒏−𝟒 𝟐 , untuk n genap. 4. penutup berdasarkan pembahasan dapat disimpulkan bahwa a. radius dan diameter graf komuting dari grup dihedral masing-masing adalah rad(c(d2n)) = 1 dan diam(c(d2n)) = 2. b. multiplisitas sikel graf komuting dari grup dihedral adalah [ 𝑛2−2𝑛 6 ] untuk n ganjil dan [ 𝑛2−2𝑛 6 ] + 𝑛 2 untuk n genap. c. dimensi metrik graf komuting dari grup dihedral d2n adalah 2n – 3 untuk n ganjil dan 3𝑛−4 2 untuk n genap. penelitian selanjutnya dapat dilakukan pada graf komuting dari grup simetri atau pada graf non komuting dari grup dihedral dan grup tak komutatif lainnya.. referensi [1] g. chartrand, l. lesniak, and p. zhang, graphs and digraphs, 6th ed. florida: chapman and hall, 2015. [2] abdussakir, n. n. azizah, and f. f. novandika, teori graf. malang: uin malang press, 2009. [3] 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. [4] 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. [5] h. rahayuningtyas, a. abdussakir, and a. nashichuddin, “bilangan kromatik graf commuting dan non commuting grup dihedral,” cauchy, vol. 4, no. 1, pp. 16–21, 2015. [6] j. a. bondy and u. s. r. murty, graph theory with applications. new york: elsevier science publishing co., inc, 1976. [7] m. m. a. ali and s. panayappan, “cycle multiplicity of total graph of cn, pn, and k1,n,” int. j. eng. sci. techlogy, vol. 2, no. 2, pp. 54–58, 2010. [8] c. hernando, m. mora, i. m. pelayo, c. seara, and d. r. wood, “extremal graph theory for metric dimension and diameter,” electron. j. comb., vol. 17, no. 1, pp. 1–28, 2010.. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: muhammad fajar, mfajar3600@gmail.com statistics indonesia, indonesia. the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.63-67 a functional form of the zenga curve based on rohde’s version of the lorenz curve muhammad fajar1,2*, setiawan2, nur iriawan2 eko fajariyanto1 1badan pusat statistik, surabaya, indonesia 2institut teknologi sepuluh nopember, surabaya, indonesia article history: received nov 21, 2021 revised may 27, 2022 accepted may 31, 2022 kata kunci: kurva lorenz. kurva zenga, rohde abstrak. kurva zenga adalah alat untuk mengukur ketidakmerataan pendapatan yang merepresentasikan rasio pendapatan antara kelompok pendapatan bawah dan kelompok pendapatan atas. kurva zenga yang tepat adalah kurva zenga yang dapat mendeteksi variasi pada rasio tersebut. dalam paper ini, penulis menurunkan bentuk fungsional kurva zenga yang berasal dari model kurva lorenz versi rohde. hasil penelitian ini menyimpulkan bahwa bentuk fungsional kurva zenga dari model kurva lorenz versi rohde adalah konstanta sehingga dia tidak dapat merepresentasikan fenomena ketidakmerataan yang sesungguhnya terjadi. keywords: lorenz curve, zenga curve, rohde. abstract. the zenga curve is a tool to measure income inequality that represents the income ratio between the bottom income group and the top income group. a proper zenga curve is a zenga curve that can detect variations in the ratio. in this paper, we derive the functional form of the zenga curve from rohde's lorenz curve model. the result of this paper is that the functional form of the zenga curve from rohde's version of the lorenz curve model is a constant. it cannot represent the truly happening phenomenon of inequality. how to cite: m. fajar, setiawan, n. iriawan, and e. fajariyanto, “functional form of the zenga curve based on rohde’s version of the lorenz curve”. j. mat mantik, vol. 8, no. 1, pp. 63-67, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 63-67 issn: 2527-3159 (print) 2527-3167 (online) mailto:mfajar3600@gmail.com http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 1, june 2022, pp.63-67 64 1. introduction the lorenz curve is a popular analytical tool used in research on income inequality. the lorenz curve is a visual representation that can reflect income distribution inequality. this curve build by lorenz [1] to represent the unequal distribution of wealth. furthermore, the lorenz curve can be expressed in the functional form (parametric model) that is not derived directly from inverse of the cumulative distribution function [2 7], but gastwirth [8] formulated a lorenz curve model formula that can be derived from inverse of the cumulative distribution function. the gastwirth formula [8] has a weakness: the researcher must determine the statistical distribution fit for the data before fitting the lorenz curve. on the other hand, several functional forms of lorenz curves are known without first knowing the exact statistical distribution of the data, such as lorenz curve-rohde [2], lorenz curve-raasche [4], lorenz curve-ortega [5], lorenz curve-chotikapanich [9], etc. rohde [2] suggested that his version of the lorenz curve is more fit than the lorenz curve model with one other parameter such as lorenz curve based pareto distribution [2], lorenz curve-gupta [10], lorenz curve-chotikapanich [9], lorenz curve-kakwani [11]. in addition to the lorenz curve, there is also a curve that represents the ratio of the average income of the lowest income group to the average income of the top income group called the zenga curve. the zenga curve can be derived from the lorenz curve [12] [13]. the zenga curve is more sensitive to detect changes in the income structure of the community than the lorenz curve when there is an additional income effect on the community [14]. the authors argue that the zenga curve may not detect the variation in ratio of the average income of the lowest income group to the average income of the top income group. it is presumably due to the lorenz curve specification used in the income distribution modeling. therefore, this study focuses on investigating the functional form of the zenga curve of rohde's lorenz curve. if the functional form of the zenga curve is constant, the arguments previously described are proven. rohde's lorenz curve was chosen based on the explanation at the end of the first paragraph. 2. method 2.1. rohde's version of the lorenz curve suppose 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) represents the cumulative distribution function (cdf) of the random variable x, assuming the value of x is a positive (non-negative) value. then 𝐹−1(𝑝) = inf{𝑥: 𝐹(𝑥) ≥ 𝑝}, it means that the inverse of cdf is a quantile. the lorenz curve, 𝐿(𝑝) formulated as follows: 𝐿(𝑝) = 𝑞 = 1 𝜇𝑋 ∫ 𝐹−1(𝑠)𝑑𝑠 𝑝 0 where 𝜇𝑋 is the mean of x. in this paper, x represents the expenditure/income of the population/households, 𝑞 is the cumulative proportion of expenditure/income (0 ≤ 𝑞 ≤ 1) received by the cumulative proportion of the population or households (𝑝: 0 ≤ 𝑝 ≤ 1). the lorenz curve must have the following characteristics: 𝑑𝐿 𝑑𝑝 > 0, 𝑑2𝐿 𝑑𝑝2 > 0, 𝐿(𝑝) = 0, 𝐿(0) = 0, 𝐿(1) = 1 rohde [2] proposed the lorenz curve model, 𝐿𝑅 (𝑝; 𝛽), as follows: 𝐿𝑅 (𝑝; 𝛽) = (𝛽 − 1)𝑝 𝛽 − 𝑝 , 𝛽 > 1, 0 ≤ 𝑝 ≤ 1. (1) meanwhile, to estimate the parameter in equation (1), the author uses the method proposed by castillo, et al. [15] based on the point (𝑝𝑖 , 𝑞𝑖 ), 𝑖 = 1, . . , 𝑛 on the empirical lorenz curve: m. fajar, setiawan, n. iriawan, and e. fajariyanto functional form of the zenga curve based on rohde’s version of the lorenz curve 65 �̂�𝑖 = 𝑝𝑖 (1 − 𝑞𝑖 ) 𝑝𝑖 − 𝑞𝑖 (2) it happens that 𝑝𝑖 = 𝑞𝑖 = 0 that causing the value of �̂�𝑖 cannot be defined so that it must be deleted which implies that the number of �̂�𝑖 is reduced by one (𝑛 − 1) in equations (4), (5), dan (6). estimation of 𝛽 [15] in equation (1) is: �̂�𝑀 = 1 𝑛 − 1 ∑ �̂�𝑖 𝑛−1 𝑖=1 (3) �̂�𝑀𝑒𝑑 = median (�̂�1, �̂�2, … , �̂�𝑛−1) (4) �̂�𝐿𝑆 = ∑ 𝑝𝑖 (1 − 𝑞𝑖 )(𝑝𝑖 − 𝑞𝑖 ) 𝑛−1 𝑖=1 ∑ (𝑝𝑖 − 𝑞𝑖 ) 2𝑛−1 𝑖=1 (5) 2.2. zenga curve the zenga curve is formulated as follows [11]: 𝑍(𝑝) = 𝑝 − 𝐿(𝑝) 𝑝(1 − 𝐿(𝑝)) . (6) the zenga curve 𝑍(𝑝) measures the inequality between the bottom 100𝑝% of the population against the top 100(1𝑝)% by comparing the mean expenditures of the two groups. equation (6) shows that the zenga curve can be derived from the lorenz curve. then based on equation (6), the zenga index can also be formulated as follows: 𝜁 = ∫ 𝑍(𝑝)𝑑𝑝 1 0 . (7) 3. results and application by combining equation (1) into equation (6), it will get: 𝑍𝑅 (𝑝) = 𝑝 − 𝐿𝑅 (𝑝) 𝑝(1 − 𝐿𝑅 (𝑝)) = 𝑝 − (𝛽 − 1)𝑝 𝛽 − 𝑝 𝑝 (1 − (𝛽 − 1)𝑝 𝛽 − 𝑝 ) = 𝑝(𝛽 − 𝑝) − (𝛽 − 1)𝑝 𝛽 − 𝑝 𝑝 ( 𝛽 − 𝑝 − (𝛽 − 1)𝑝 𝛽 − 𝑝 ) = 𝑝(𝛽 − 𝑝) − 𝑝(𝛽 − 1) 𝑝(𝛽 − 𝑝 − (𝛽 − 1)𝑝) = 𝑝((𝛽 − 𝑝) − (𝛽 − 1)) 𝑝(𝛽 − 𝑝 − (𝛽 − 1)𝑝) = ((𝛽 − 𝑝) − (𝛽 − 1)) (𝛽 − 𝑝 − (𝛽 − 1)𝑝) = −𝑝 + 1 𝛽 − 𝑝 − 𝛽𝑝 + 𝑝 = 1 − 𝑝 𝛽 − 𝛽𝑝 = 1 − 𝑝 𝛽(1 − 𝑝) (8) so, equation (8) can be simplified to: 𝑍𝑅 (𝑝) = 1 𝛽 (9) equation (9) is a function in the form of a constant, meaning that regardless of the value of 𝑝, then 𝑍𝑅 (𝑝) always has a value of 1 𝛽⁄ . the functional form of equation (9) is also known jurnal matematika mantik vol. 8, no. 1, june 2022, pp.63-67 66 as the uniform function or constant. however, it should be noted that in equation (8) the condition 𝛽 ≠ 𝑝 must be fulfilled. then based on equation (7), the zenga index 𝜁𝑅 (10) is obtained whose formulation is the same as equation (9). it's happening because equation (9) is a constant. 𝜁𝑅 = ∫ 𝑍𝑅 (𝑝)𝑑𝑝 1 0 = ∫ 1 𝛽 𝑑𝑝 1 0 = 1 𝛽 ∫ 𝑑𝑝 1 0 = 1 𝛽 (𝑝|0 1 ) = 1 𝛽 (10) the application of the zenga curve formulation and the zenga index using the research results of ref. [16]: 𝐿𝑅 (𝑝; 𝛽) = 0.485𝑝 1.485 − 𝑝 (11) equation (11) uses �̂�𝑀𝑒𝑑 in equation (4) because the estimation results produce a minimum mean squared error (mse) compared to �̂�𝑀 (3) and �̂�𝐿𝑆 (5) in the case of this study. based on equation (10) the zenga curve and zenga index can be derived from equations (9) and (10) are: 𝜁𝑅 = 𝑍𝑅 (𝑝) = 1 1.485 = 0.673. (12) the interpretation of the zenga curve (12) derived from rohde's version of the lorenz curve is that the average household income at each level p of the lowest income group is 67.3% lower than the average income of all levels of the top income group of the population. the zenga curve derived from rohde's lorenz curve has a weakness, namely that it assumes that the ratio of income between the lowest and the top group for all levels is constant (uniform). this does not reflect the reality of what happened. supposedly, the ratio varies at each level of p. because the zenga curve depends on the lorenz curve, the correct functional form of the lorenz curve is the key so that the zenga curve represents the phenomenon. 4. conclusions based on the material, it said that the functional form of the zenga curve from rohde's lorenz curve model is a constant with the condition that 𝛽 ≠ 𝑝, but it has not reflected the reality. it implies that the zenga index has the same formulation as its functional model. the correct functional form of the lorenz curve is the key so that the zenga curve represents the real phenomenon inequality. references [1] m. o. lorenz, “methods of measuring the concentration of wealth,” publications of the american statistical association, vol. 9, no. 70, pp. 209-219, 1905. [2] n. rohde, “an alternative functional form for estimating the lorenz curve,” economics letters 105, pp. 61-63, 2009. [3] n. rohde, “lorenz curve interpolation and the gini coefficient,” journal of income distribution, vol. 19, no. 2, pp. 111 – 123, 2010. [4] r. rasche, j. gaffney, a. y. c. koo, and n. obst, “functional forms for estimating the lorenz curve,” econometrica 48, pp. 1061–1062, 1980. [5] p. ortega, g. martin, a. fernández, m. ladoux, and a. garcía, “a new functional form for estimating lorenz curves,” review of income and wealth, series 37, number 4, 1991. m. fajar, setiawan, n. iriawan, and e. fajariyanto functional form of the zenga curve based on rohde’s version of the lorenz curve 67 [6] t. ogwang and u.l.g. rao. chotikapanich, “a new functional form for approximating the lorenz curve”, economics letters, vol. 52, no. 1, pp. 21 – 29, 1996. [7] s. paul and s. shankar, “an alternative single parameter functional form for lorenz curve,” empirical economics, vol. 59, pp. 1393 – 1402, 2020. [8] j. l. gastwirth and m. glauberman, “the interpolation of the lorenz curve and gini index from grouped data,” econometrica, vol. 44, no. 3, pp. 479-483, 1976. [9] d. chotikapanich, “a comparison of alternative functional forms for the lorenz curve,” economics letters, vol. 41, pp. 129-138, 1993. [10] m. r. gupta, “functional form for estimating the lorenz curve”, econometrica, vol. 52, issue. 5, pp. 1313-1314 [11] n. c. kakwani and n. podder, “on the estimation of lorenz curves from grouped observations,” international economic review, vol. 14, no. 2, pp. 278-292, 1973. [12] m. langel and y. tillé, “inference by linearization for zenga’s new inequality index: a comparison with the gini index, “metrika international journal for theoretical and applied statistics 75, no. 8, pp. 1093–1110, 2012. [13] m. zenga, “inequality curve and inequality index based on the ratios between lower and upper arithmetic means,” statistica e applicazioni 4:3–27, 2007. [14] w. maffenini and m. polisicchio, “how potential is the i(p) inequality curve in the analysis of empirical distributions, technical report,” universita degli studi di milano-bicocca, 2010. [15] e. castillo, a. s. hadi, and j. m. sarabia, ”a method for estimating lorenz curves,” communications in statistics, theory and methods, vol. 27, pp. 2037-2063, 1998. [16] m. fajar, “pemodelan kurva lorenz versi rohde pada pengeluaran rumah tangga pertanian di provinsi papua,” euclid, vol. 8, no.1, pp. 1-5, 2021. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: mohammad nafie jauhari nafie.jauhari@uin-malang.ac.id department of mathematics, uin maulana malik ibrahim, malang, east java 65144 the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.99-104 on the relation of the total graph of a ring and a product of graphs mohammad nafie jauhari universitas islam negeri maulana malik ibrahim, malang, indonesia, article history: received aug 31, 2022 revised, dec 25, 2022 accepted, dec 31, 2022 kata kunci: grup, total graf, isomorfisma, perkalian kartesius abstrak. graf total atas suatu ring 𝑅, dinotasikan dengan 𝑇(γ(𝑅)), didefinisikan sebagai suatu graf dengan himpunan titik 𝑉 (𝑇(γ(𝑅))) = 𝑅 dan dua titik berbeda 𝑢,𝑣 ∈ 𝑉(𝑇(γ(𝑅))) bertetangga jika dan hanya jika 𝑢 + 𝑣 ∈ 𝑍(𝑅), di mana 𝑍(𝑅) merupakan pembagi nol dari 𝑅. perkalian kartesius dari dua graf 𝐺 dan 𝐻 merupakan suatu graf yang dinotasikan dengan 𝐺 × 𝐻 di mana himpunan titiknya adalah 𝑉(𝐺 × 𝐻) = 𝑉(𝐺) × 𝑉(𝐻) dan dua titik berbeda (𝑢1,𝑣1) dan (𝑢2,𝑣2) di 𝑉(𝐺 × 𝐻) bertetangga jika dan hanya jika: 1) 𝑢1 = 𝑢2 dan 𝑣1𝑣2 ∈ 𝐻; atau 2) 𝑣1 = 𝑣2 dan 𝑢1𝑢2 ∈ 𝐸(𝐺). isomorfisma dari graf 𝐺 dan 𝐻 adalah suatu fungsi bijektif 𝜙:𝑉(𝐺) → 𝑉(𝐻) sedemikian sehingga 𝑢,𝑣 ∈ 𝑉(𝐺) bertetangga jika dan hanya jika 𝑓(𝑢),𝑓(𝑣) ∈ 𝑉(𝐻) bertetangga. akan dibuktikan bahwa graf 𝑇(γ(ℤ2𝑝)) isomorf dengan graf 𝑃2 × 𝐾𝑝 untuk setiap bilangan prima 𝑝. keywords: group, total graph, isomorphism, cartesian product abstract. the total graph of a ring 𝑅, denoted as 𝑇(γ(𝑅)), is defined to be a graph with vertex set 𝑉 (𝑇(γ(𝑅))) = 𝑅 and two distinct vertices 𝑢,𝑣 ∈ 𝑉 (𝑇(γ(𝑅))) are adjacent if and only if 𝑢 + 𝑣 ∈ 𝑍(𝑅), where 𝑍(𝑅) is the zero divisor of 𝑅. the cartesian product of two graphs 𝐺 and 𝐻 is a graph with the vertex set 𝑉(𝐺 × 𝐻) = 𝑉(𝐺) × 𝑉(𝐻) and two distinct vertices (𝑢1,𝑣1) and (𝑢2,𝑣2) are adjacent if and only if: 1) 𝑢1 = 𝑢2 and 𝑣1𝑣2 ∈ 𝐻; or 2) 𝑣1 = 𝑣2 and 𝑢1𝑢2 ∈ 𝐸(𝐺). an isomorphism of graphs 𝐺 dan 𝐻 is a bijection 𝜙:𝑉(𝐺) → 𝑉(𝐻) such that 𝑢,𝑣 ∈ 𝑉(𝐺) are adjacent if and only if 𝑓(𝑢),𝑓(𝑣) ∈ 𝑉(𝐻) are adjacent. this paper proved that 𝑇(γ(ℤ2𝑝)) and 𝑃2 × 𝐾𝑝 are isomorphic for every odd prime 𝑝. how to cite: m. n. jauhari, “on the relation of total graph of a ring and a product of graphs”, j. mat. mantik, vol. 8, no. 2, pp. 99-104, december 2022. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 99-104 issn: 2527-3159 (print) 2527-3167 (online) mailto:nafie.jauhari@uin-malang.ac.id https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 99-104 100 1. introduction investigating group or ring properties and its structures from their graph representation become a new trend in graph theoretic research. many authors proved that there are tight bonds between the rings and graphs. aalipour in [1] investigated the chromatic number and clique number of a commutative ring. [2] gave a novel application of a central-vertex complete graph to a commutive ring. in 2008, [3] investigated the commutive graph of rings generated from matrices over a finite filed. three years after the graph of a ring was introduced in [4], [5] proposed useful applications of semirings in mathematics and theoretical computer science. one interest in applying graph invariant on a group also showed in [6] in the properties of zero-divisor graphs. another useful graph generated from group or ring structure is cayley graphs which has many useful applications in solving and understanding a variety problem in several scientific interests [7]. the graph isomorphism itself has many applications in real life and many scientific fields [8]. [9] stated briefly about its application in the atomic structures and [10] showed how it can be applied in biochemical data. to prove the isomorphism of two graphs is an np-problem in which there is no specific algorithm or certain way that works for all graphs in consideration [11]. in 1996, [12] proposed a good graph isomorphism algorithm but still troublesome for a large graphs. considering those applications of ring generated graphs, the applications of the graph isomorphisms, and the isomorphism-related algorithm complexity, finding an isomorphism of ring-structured graphs and the graph obtained from certain operation is a challenging task and a potential new interest in graph theory research. this paper considers the relation between the total graph of ℤ𝑝 and 𝑃2 × 𝐾𝑝 for all odd prime 𝑝. 2. preliminaries a graph 𝐺 is a pair 𝐺 = (𝑉,𝐸) for non-empty set 𝑉 and 𝐸 ⊆ [𝑉]2 (the elements of 𝐸 are 2-element subsets of 𝑉). for terminologies and notations concerning to a graph and its invariants, please consider [13]. this preliminary covers the definitions related to ring and the total graph of a ring. it also provides some definitions related to graph isomorphism and a graph operation. definition 1. ring [14] a ring 𝑅 is a set with two binary operations, addition and multiplication, such that for all 𝑎,𝑏,𝑐 ∈ 𝑅: 1. 𝑎 + 𝑏 = 𝑏 + 𝑎, 2. (𝑎 + 𝑏)+ 𝑐 = 𝑎 + (𝑏 + 𝑐), 3. there is an additive identity 0, 4. there is an element −𝑎 ∈ 𝑅 such that 𝑎 + (−𝑎) = 0, 5. 𝑎(𝑏𝑐) = (𝑎𝑏)𝑐, and 6. 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 and (𝑏 + 𝑐)𝑎 = 𝑏𝑎 + 𝑐𝑎. with this definition, ℤ2𝑝, an integer modulo 2𝑝 set, equipped with addition and multiplication modulo 2𝑝 operation is a ring. definition 2. zero divisor [14] a zero-divisor is a nonzero element 𝑎 of a commutative ring 𝑅 such that there is a nonzero element 𝑏 ∈ 𝑅 with 𝑎𝑏 = 0. mohammad nafie jauhari on the relation of total graph of a ring and a product of graphs 101 definition 3. total graph of a ring [15] let 𝑅 be a ring and 𝑍(𝑅) denotes the zero divisor of 𝑅. the total graph of 𝑅, denoted by 𝑇(γ(𝑅)) is an undirected graph with elements of 𝑅 as its vertices, and for distinct 𝑥,𝑦 ∈ 𝑅, the vertices 𝑥 and 𝑦 are adjacent if and only if 𝑥 + 𝑦 ∈ 𝑍(𝑅). from those definitions of zero divisor and total graph, we will construct a total graph of ℤ2𝑝 for an odd prime 𝑝. definition 4. graph homomorphism and isomorphism [13] let 𝐺 = (𝑉𝐺,𝐸𝐺) and 𝐻 = (𝑉𝐻,𝐸𝐻) be graphs. a map 𝜑:𝑉𝐺 → 𝑉𝐻 is a homomorphism from 𝐺 to 𝐻 if it preserves the adjacency of the vertices. in another word, {𝑥,𝑦} ∈ 𝐸𝐺 ⇒ {𝜑(𝑥),𝜑(𝑦)} ∈ 𝐸𝐻. if 𝜑 is bijective and 𝜑 −1 is also a homomorphism, then 𝜑 is an isomorphism and 𝐺 is said to be isomorphic to 𝐻. definition 5. cartesian product [13] the cartesian product of two graphs 𝐺 and 𝐻 is a graph with the vertex set 𝑉(𝐺 × 𝐻) = 𝑉(𝐺) × 𝑉(𝐻) and two distinct vertices (𝑢1,𝑣1) and (𝑢2,𝑣2) are adjacent if and only if: 1) 𝑢1 = 𝑢2 and 𝑣1𝑣2 ∈ 𝐻; or 2) 𝑣1 = 𝑣2 and 𝑢1𝑢2 ∈ 𝐸(𝐺). an isomorphism of graphs 𝐺 dan 𝐻 is a bijection 𝜙:𝑉(𝐺) → 𝑉(𝐻) such that 𝑢,𝑣 ∈ 𝑉(𝐺) are adjacent if and only if 𝑓(𝑢),𝑓(𝑣) ∈ 𝑉(𝐻) are adjacent. 3. main results in this section we will prove the isomorphism of the total graph of ℤ2𝑝 and 𝑃2 × 𝐾𝑝. we will investigate several properties of 𝑇(γ(ℤ2𝑝)) before we proof the isomorphism. those investigations will be provided as lemmas and theorems equipped with their proofs. to characterize 𝑇(γ(ℤ2𝑝)), we consider its vertex set, the degree of each vertex, and the clique it has as subgraphs, since 𝑃2 × 𝐾𝑝 can easily be considered and seen from those properties. lemma 1. the zero divisor of ℤ2𝑝 is 𝑍(ℤ2𝑝) = {𝑝} ∪ {2𝑛:𝑛 = 1,2,…,𝑛 − 1} for every odd prime 𝑝. proof. for each 𝑥 ∈ ℤ2𝑝, the exactly one of the following holds: 𝑥 = 𝑝, 𝑥 is even, and 𝑥 ≠ 𝑝 is odd. case 1, 𝑥 = 𝑝 since 2𝑝 = 0 and 2 ∈ ℤ2𝑝, we conclude that 𝑝 ∈ 𝑍(ℤ2𝑝). case 2, 𝑥 is even let 𝑥 = 2𝑚 for some 𝑚 ∈ ℤ. since 𝑥𝑝 = 2𝑚𝑝 = 𝑚 ⋅ 2𝑝 = 𝑚 ⋅ 0 = 0 and 𝑝 ∈ ℤ2𝑝, we conclude that 𝑥 ∈ 𝑍(ℤ2𝑝) for all even 𝑥 ∈ ℤ2𝑝. case 3, 𝑥 ≠ 𝑝 is odd if 𝑥 = 1, then 𝑥𝑦 ≠ 0 for all 0 ≠ 𝑦 ∈ ℤ2𝑝. we will show that 1 ≠ 𝑥 ∉ 𝑍(ℤ2𝑝) by using a contradiction. suppose on the contrary, that 𝑥 ∈ 𝑍(ℤ2𝑝). consequently, there exists 0 ≠ 𝑦 ∈ ℤ2𝑝 such that 𝑥𝑦 = 0. it follows that gcd(𝑥,2𝑝) > 1. since the factor of 2𝑝 is 2 and 𝑝, we obtain that 𝑥 divides 𝑝. it is a contradiction since 𝑝 is a prime number. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 99-104 102 lemma 2. let 𝑝 be an odd prime. let 𝐴 ⊆ ℤ2𝑝 be the set of all odd elements of ℤ2𝑝 and 𝐵 ⊆ ℤ2𝑝 be the set of all even elements of ℤ2𝑝. {𝑢,𝑣} ∈ 𝐸(𝑇(γ(ℤ2𝑝))) for all 𝑢,𝑣 ∈ 𝐴 and {𝑥,𝑦} ∈ 𝐸(𝑇(γ(ℤ2𝑝))) for all 𝑥,𝑦 ∈ 𝐵. in another word, the vertices in 𝐴 dan 𝐵 form cliques in 𝑇(γ(ℤ2𝑝)). proof. let 𝑢,𝑣 ∈ 𝐴 and let 𝑢 = 2𝑠 + 1 and 𝑣 = 2𝑡 + 1 for some 𝑠,𝑡 ∈ ℤ. we obtain 𝑢 + 𝑣 = 2𝑠 + 1 + 2𝑡 + 1 = 2(𝑠 + 𝑡 + 1) ∈ 𝑍(ℤ2𝑝). therefore {𝑢,𝑣} ∈ 𝐸(𝑇(γ(ℤ2𝑝))) for all 𝑢,𝑣 ∈ 𝐴. let 𝑥,𝑦 ∈ 𝐴 and let 𝑥 = 2𝑠 and 𝑦 = 2𝑡 for some 𝑠,𝑡 ∈ ℤ. we obtain 𝑥 + 𝑦 = 2𝑠 + 2𝑡 = 2(𝑠 + 𝑡) ∈ 𝑍(ℤ2𝑝). therefore {𝑥,𝑦} ∈ 𝐸(𝑇(γ(ℤ2𝑝))) for all 𝑥,𝑦 ∈ 𝐵. it proves that 𝐴 and 𝐵 form cliques in 𝑇(γ(ℤ2𝑝)). lemma 3. let 𝐴 and 𝐵 be sets defined in lemma 2 and 𝑝 be an odd prime number. for each 𝑣 ∈ 𝐴 there is a unique 𝑥 ∈ 𝐵 such that {𝑣,𝑥} ∈ 𝐸(𝑇(γ(ℤ2𝑝))). proof. for each 𝑣 ∈ 𝐴, choose 𝑥 = 𝑝 − 𝑣. it can be easily verified that 𝑥 ∈ 𝐵 since 𝑝 and 𝑣 are both odd numbers. on the other hand, let 𝑥 ∈ 𝐵 and 𝑥 ≠ 𝑝 − 𝑣. suppose that {𝑣,𝑥} ∈ 𝐸(𝑇(γ(ℤ2𝑝))), that is 𝑣 + 𝑥 ∈ 𝑍(ℤ2𝑝). since 𝑣 is odd and 𝑥 is even, it follows that 𝑣 + 𝑥 is an odd number and 𝑣 + 𝑥 = 𝑝 ⟺ 𝑥 = 𝑝 − 𝑣, a contradiction. this proves that {𝑣,𝑝 − 𝑣} ∈ 𝐸(𝑇(γ(ℤ2𝑝))),∀𝑥 ∈ 𝐴. analogous to this proof, we can easily prove that for each 𝑥 ∈ 𝐵 there is a unique 𝑣 ∈ 𝐴 such that {𝑣,𝑥} ∈ 𝐸(𝑇(γ(ℤ2𝑝))). before we discuss the main problem, consider figure 1 that represents the graph 𝑇(γ(ℤ2𝑝)) for several 𝑝. figure 1. 𝑇(γ(ℤ2𝑝)) for 𝑝 ∈ {3,5,7}. mohammad nafie jauhari on the relation of total graph of a ring and a product of graphs 103 theorem 1. for any odd prime 𝑝, 𝑇(γ(ℤ2𝑝)) is isomorph to 𝑃2 × 𝐾𝑝. proof. let 𝑉(𝑃2) and 𝑉(𝐾𝑝) be labeled as {𝑝1,𝑝2} and {𝑘0,𝑘1,…,𝑘𝑝−1} respectively. the vertices of the resulting graph obtained from the cartesian product, 𝑃2 × 𝐾𝑝, is therefore labeled {(𝑝1,𝑘1),(𝑝1,𝑘2),…,(𝑝1,𝑘𝑝),(𝑝2,𝑘1),(𝑝2,𝑘2),…,(𝑝2,𝑘𝑝)} in which {(𝑝𝑠,𝑘𝑖),(𝑝𝑠,𝑘𝑗)} ∈ 𝐸(𝑃2 × 𝐾𝑝),∀𝑖,𝑗 ∈ {1,2,…,𝑝} and 𝑖 ≠ 𝑗, for 𝑠 ∈ {1,2}. other edges to consider is {(𝑝1,𝑘𝑖),(𝑝2,𝑘(𝑝−𝑖 mod 𝑝)+1)} ∈ 𝐸(𝑃2 × 𝐾𝑝),∀𝑖 ∈ {1,2,…,𝑝}. here, the “mod” in “𝑝 − 𝑖 mod 𝑝” is a modulus operator, not a modulus relation. consider the function 𝜑:𝑉(𝑇(γ(ℤ2𝑝))) → 𝑉(𝑃2 × 𝐾𝑝) defined as follows: 𝜑(𝑥) = { (𝑃1,( 𝑝 − 𝑥 2 mod 𝑝) + 1), if 𝑥 is odd (𝑃2, 𝑥 2 + 1), if 𝑥 is even. since 𝜑 is a bijective function that preserves adjacency of the vertices of 𝑉(𝑇(γ(ℤ2𝑝))) and 𝑉(𝑃2 × 𝐾𝑝), we conclude that 𝑇(γ(ℤ2𝑝)) and 𝑃2 × 𝐾𝑝 are isomorphic. figure 1 and figure 2 show some examples of the mapping result of 𝜑. figure 2. the mapping result of 𝜑 from 𝑇(γ(ℤ2⋅5)) to 𝑃2 × 𝐾5 figure 3. the mapping result of 𝜑 from 𝑇(γ(ℤ2⋅7)) to 𝑃2 × 𝐾7 4. conclusion from the discussion, we conclude that 𝑇(γ(ℤ2𝑝)) and 𝑃2 × 𝐾𝑝 are isomorphic. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 99-104 104 references [1] g. aalipour and s. akbari, “application of some combinatorial arrays in coloring of total graph of a commutative ring,” may 2013. 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[9] m. grohe and p. schweitzer, “the graph isomorphism problem,” commun. acm, vol. 63, no. 11, pp. 128–134, 2020. [10] v. bonnici, r. giugno, a. pulvirenti, d. shasha, and a. ferro, “a subgraph isomorphism algorithm and its application to biochemical data,” bmc bioinformatics, vol. 14, no. suppl7, pp. 1–13, apr. 2013. [11] c. s. calude, m. j. dinneen, and r. hua, “qubo formulations for the graph isomorphism problem and related problems,” theor. comput. sci., vol. 701, pp. 54– 69, nov. 2017. [12] x. y. jiang and h. bunke, “including geometry in graph representations: a quadratic-time graph isomorphism algorithm and its applications,” lect. notes comput. sci. (including subser. lect. notes artif. intell. lect. notes bioinformatics), vol. 1121, pp. 110–119, 1996. [13] r. diestel, “graph theory (5th edition),” springer, 2017. [14] j. gallian, contemporary abstract algebra. 2021. [15] d. f. anderson and a. badawi, “the total graph of a commutative ring,” j. algebr., vol. 320, no. 7, pp. 2706–2719, oct. 2008. contact: sizar mohammed, sizar@uod.go college basic education, university of duhok, iraq the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.140-146 a comparative study between adm and mdm for a system of volterra integral equation sizar mohammed1, hela shawkat ahmed2 1,2college basic education, university of duhok, iraq article history: received, jul 25, 2021 revised, oct 19, 2021 accepted, oct 31, 2021 kata kunci: sistem persamaan integral volterra, adomain decomposition method, modified decomposition method abstrak. dalam penelitian ini, studi perbandingan antara adomain decomposition method (adm) dan modified decomposition method (mdm) untuk sistem persamaan integral volterra. dari contoh ilustrasi terlihat bahwa solusi eksak lebih kecil di kedua metode, metode dekomposisi yang dimodifikasi lebih bagus daripada cara tradisional, metode ini tidak terlalu rumit, membutuhkan lebih sedikit waktu untuk mendapatkan solusi dan yang terpenting solusi eksak dicapai dalam dua iterasi. keywords: system of volterra integral equations, adomain decomposition method, modified decomposition method abstract. in this paper, a comparative study between adomain decomposition method (adm) and modified decomposition method (mdm) for a system of volterra integral equation. from the illustrate examples it is observed that the exact solution is smaller in both method, the modified decomposition method is more proficient than its traditional ones it is less complicated, needs less time to get to the solution and most importantly the exact solution is achieved in two iterations. how to cite: s. mohammed and h. s. ahmed, “a comparative study between adm and mdm for a system of volterra integral equation, j. mat. mantik, vol. 7, no. 2, pp. 140-146, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 140-146 issn: 2527-3159 (print) 2527-3167 (online) mailto:sizar@uod.go http://u.lipi.go.id/1458103791 s. mohammed and h. s. ahmed a comparative study between adm and mdm for a system of volterra integral equation 141 1. introduction the subject of integral equations is one of the most useful mathematical tools in both pure and applied mathematics, and also they have enormous applications in many physical problems, in engineering, chemistry and biological problems. many initial and boundary value problems associated with the ordinary and partial differential equations can be transformed in to the integral equations. an integral equation is the equation in which the unknown function 𝑢(𝑥) appears inside an integral sign. in this paper we make a comparative study between two methods. the most standard type of integral equation in 𝑢(𝑥) is of the form: 𝑢(𝑥) = 𝑓(𝑥) + 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢(𝑡)𝑑𝑡, ℎ(𝑥) 𝑔(𝑥) where 𝑔(𝑥) and ℎ(𝑥) are the limits of integration, 𝜆 is a constant parameter, and 𝑘(𝑥, 𝑡) is a known function of two variables 𝑥 and 𝑡, called the kernel. the unknown function 𝑢(𝑥) that will be determined appears inside the integral sign. in many other cases, the unknown function 𝑢(𝑥) appears inside and outside the integral sign [3]. it is to be noted that the limits of integrations 𝑔(𝑥) and ℎ(𝑥) may be both variables, constants or mixed. in the past decades, many researchers have studied some numerical and analytical methods for solving different types of integral equations, for example, for example, the author in [1] investigated a numerical solution for volterra-fredholm integral equations via least squares method. in [2] the authors obtained the approximate solutions for the linear part of volterra integral equations of second kind by using two accurate quadrature rules and also in general wazwaz in [3] explained different types of integral equations. the systems of linear and nonlinear integral equations have been solved using the adomian decomposition method in [4]. the authors in [5] are developed a runge-kutta method theory for integrated volterra equations of the second kind. in [6] the author concerning the singular cauchy kernel were used to find integral equation formulations for the laplace equation. in [7] the authors extended smooth methods through the use of partitioned quadrature based on the collocation methods, to allow the efficient numerical solution of linear, scalar volterra integral equations of the second kind with. the numerical solution of the volterra integral equations with delay was obtained using block methods in [8]. in [9] the author introduced a new approach which is the galerkin method with hermite polynomials for estimating the numerical solutions of volterra's integral equations. in [10] the authors applied the galerkin residual weighted method to solve volterra integral equations of the first and second type with normal and single kernel. the author in [11] obtained the numerical solution by aggregation method that was formulated and justified for fredholm equations of the second type. in reference [12], the authors apply both the aggregation method and chebyshev polynomials to obtain numerical solutions to the volterra integral equations. the authors in [13] used a modified trapezoid quadrature method to solve linear integral equations of the second kind. tahmasbi in [14] is introduced a new approach, namely the power series method for solving volterra integral equation of the second kind. the galerkin weighted residual approximation method was applied to obtain a numerical approach to volterra's integral equations in [15] and in [16] wazwaz focused on recent developments in approximate methods for solving linear and nonlinear integral equations with applications. the aim of this study is to solve a system of volterra integral equation of the second kind by two accurate methods, which are the adomian decomposition method and the modified decomposition method. 2. adomain decomposition method the adomain decomposition method (adm) was introduced and developed by george adomain [17-18]. it consists of decomposing the unknown function 𝑢(𝑥) of any jurnal matematika mantik vol 7, no 2, october 2021, pp. 140-146 142 equation in to a sum of an infinite number of components defined by the decomposition series 𝑢(𝑥) = ∑ 𝑢𝑛(𝑥) , ∞ 𝑛=0 (1) or equivalently 𝑢(𝑥) = 𝑢0(𝑥) + 𝑢1(𝑥) + 𝑢2(𝑥) + ⋯, (2) the decomposition method is concerned with finding the components 𝑢0, 𝑢1, 𝑢2, …. individually. to establish the recurrence relation, we substitute (1) into equation [3] to get ∑ 𝑢𝑛 (𝑥) = 𝑓(𝑥) + 𝜆 ∫ 𝑘(𝑥, 𝑡)( 𝑥 0 ∑ 𝑢𝑛 (𝑥))𝑑𝑡 ∞ 𝑛=0 ∞ 𝑛=0 (3) or equivalently 𝑢0(𝑥) + 𝑢1(𝑥) + 𝑢2(𝑥) + ⋯ = 𝑓(𝑥) + 𝜆 ∫ 𝑘(𝑥, 𝑡) 𝑥 0 [𝑢0(𝑡) + 𝑢1(𝑡) + 𝑢2(𝑡) + ⋯ ]𝑑𝑡. (4) the components 𝑢𝑗 (𝑥), 𝑗 ≥ 1 of the unknown function 𝑢(𝑥) are completely determined by setting the recurrence relation: 𝑢0(𝑥) = 𝑓(𝑥), 𝑢𝑛+1(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢𝑛 (𝑡)𝑑𝑡, 𝑛 ≥ 0, 𝑥 0 (5) or equivalently 𝑢0(𝑥) = 𝑓(𝑥), 𝑢1(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢0 𝑥 0 (𝑡)𝑑𝑡, 𝑢2(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢1(𝑡)𝑑𝑡, 𝑥 0 (6) 𝑢3(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢2(𝑡)𝑑𝑡, 𝑥 0 and so on for other components. as a result the components 𝑢1(𝑥), 𝑢2(𝑥), 𝑢3(𝑥), … are completely determined, and then the solution 𝑢(𝑥) of the volterra integral equation (6) is readily obtained in a series from by using the series assumption in (1). the decomposition method converts the integral equation into an elegant determination of components. if an exact solution exists for the problem, then the obtained series converges very rapidly to that exact solution. however, for concrete problems, where a closed from solution is not obtainable. the more components we use the higher accuracy we obtain [16]. example 1. consider the volterra integral equation of the second kind 𝑢(𝑥) = 6𝑥 − 3𝑥2 + ∫ 𝑢(𝑡)𝑑𝑡, 𝑥 0 (7) using the adomain decomposition method, we notice that 𝑓(𝑥) = 6𝑥 − 3𝑥2, 𝜆 = 1, 𝑘(𝑥, 𝑡) = 1. recall that the solution 𝑢(𝑥) is assumed to have series from given in (1). substituting the decomposition series (1) into both sides of (7) gives ∑ 𝑢𝑛 (𝑥) = 6𝑥 − 3𝑥 2 + ∫ ∑ 𝑢𝑛 (𝑡)𝑑𝑡, ∞ 𝑛=0 𝑥 0 ∞ 𝑛=0 or equivalently s. mohammed and h. s. ahmed a comparative study between adm and mdm for a system of volterra integral equation 143 𝑢0(𝑥) + 𝑢1(𝑥) + 𝑢2(𝑥) + ⋯ = 6𝑥 − 3𝑥 2 + ∫[𝑢0(𝑡) + 𝑢1(𝑡) + 𝑢2(𝑡) + ⋯ ]𝑑𝑡. 𝑥 0 we identify the zeroth component by all terms that are not included under the integral sign. therefore, we obtain the following recurrence relation: 𝑢0(𝑥) = 6𝑥 − 3𝑥 2, 𝑢𝑛+1(𝑥) = ∫ 𝑢𝑛 (𝑡)𝑑𝑡, 𝑥 0 𝑛 ≥ 0. so that 𝑢0(𝑥) = 6𝑥 − 3𝑥 2, 𝑢1(𝑥) = ∫ 𝑢0(𝑡)𝑑𝑡 = ∫ 6𝑡 − 3𝑡 2𝑑𝑡 = 3𝑥2 𝑥 0 𝑥 0 − 𝑥3, 𝑢2(𝑥) = ∫ 𝑢1(𝑡)𝑑𝑡 = ∫( 𝑥 0 𝑥 0 3𝑡2 − 𝑡3)𝑑𝑡 = 𝑥3 − 𝑥4 4 , 𝑢3(𝑥) = ∫ 𝑢2(𝑡)𝑑𝑡 = ∫(𝑡 3 𝑥 0 𝑥 0 − 𝑡4 4 )𝑑𝑡 = 𝑥4 4 − 𝑥5 20 , 𝑢4(𝑥) = ∫ 𝑢3(𝑡)𝑑𝑡 = ∫( 𝑥 0 𝑥 0 𝑡4 4 − 𝑡5 20 )𝑑𝑡 = 𝑡5 20 − 𝑡6 120 , the solution in a series from is given by 𝑢(𝑥) = 6𝑥 − 3𝑥2 + 3𝑥2 − 𝑥3 + 𝑥3 − 𝑥4 4 + 𝑥4 4 − 𝑥5 20 + 𝑥5 20 − 𝑥6 120 + ⋯ . we can easily notice the appearance of identical of terms with opposite signs this phenomenon of such terms is called noise term phenomenon canceling the identical terms with opposite terms gives the exact solution: 𝑢(𝑥) = 6𝑥. 3. modified decomposition method as shown before, the adomain decomposition method provides the solution in an infinite series of components. the components 𝑢𝑗 , 𝑗 ≥ 0 are easily computed if the inhomogeneous term 𝑓(𝑥) in the volterra integral equation: 𝑢(𝑥) = 𝑓(𝑥) + 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢(𝑡)𝑑𝑡, 𝑥 0 (8) consists of a polynomial. however, if the function 𝑓(𝑥) consists of a combination of two or more of polynomials, trigonometric functions, hyperbolic functions, and others, the evaluation of the components 𝑢𝑗 , 𝑗 ≥ 0 require more work. a reliable modification of the adomain decomposition method was developed by wazwaz [4]. the modified decomposition method will facilitate the computational process and further accelerate the convergence of the series solution. this will be applied whenever it is appropriate to all integral equations and differential equations of any order. it is important to note that the modified decomposition method relies mainly on splitting the function 𝑓(𝑥) into two parts; therefore it can not be used if the function 𝑓(𝑥) consists of only one term. to explain this technique, we recall that the standard adomain decomposition method admits the use of the recurrence relation: 𝑢0(𝑥) = 𝑓(𝑥), jurnal matematika mantik vol 7, no 2, october 2021, pp. 140-146 144 𝑢𝑛+1(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢𝑛 𝑥 0 (𝑡)𝑑𝑡, 𝑛 ≥ 0, (9) where the solution 𝑢(𝑥) is expressed by an infinite sum of components defined by 𝑢(𝑥) = ∑ 𝑢𝑛 ∞ 𝑛=0 (𝑥). (10) in virtue of (9), the components 𝑢𝑛 , 𝑛 ≥ 0 can easily be evaluated. the modified decomposition method introduces slight variation to the recurrence relation (9). that will lead to be determination of the components of 𝑢(𝑥) in an easier and faster manner. for many cases, the function 𝑓(𝑥) can be set as the sum of two partial functions, namely 𝑓2(𝑥). in other words, we can set 𝑓(𝑥) = 𝑓1(𝑥) + 𝑓2(𝑥) (11) in virtue of (11), we introduce a qualitative change in the formation of the recurrence relation (9). to reduce the calculations, we will introduce of the modified decomposition method into recurrence relation: 𝑢0(𝑥) = 𝑓1(𝑥) 𝑢1(𝑥) = 𝑓2(𝑥) + 𝜆 ∫ 𝑘(𝑥, 𝑡) 𝑥 0 𝑢0(𝑡)𝑑𝑡, (12) 𝑢𝑛+1(𝑥) = 𝜆 ∫ 𝑘(𝑥, 𝑡)𝑢𝑛 (𝑡)𝑑𝑡, 𝑛 ≥ 1 𝑥 0 . this shows that the formation of the first two components 𝑢0(𝑥)𝑎𝑛𝑑 𝑢1(𝑥) is only the difference between the standard recurrence relation (9) and the modified recurrence relation (12). the others components 𝑢𝑗 , 𝑗 ≥ 2 remain the same in the two recurrence relations. this variation in the formation of 𝑢0(𝑥)𝑎𝑛𝑑 𝑢1(𝑥) is important to accelerate the convergence of the solution and in minimizing the size of computational work [3]. example 2. consider the volterra integral equations of the second kind 𝑢(𝑥) = 6𝑥 − 3𝑥2 − ∫ 𝑢(𝑡)𝑑𝑡. 𝑥 0 using the modified decomposition method, we first split 𝑓(𝑥) 𝑓(𝑥) = 6𝑥 − 3𝑥2, into two parts, namely 𝑓1(𝑥) = 6𝑥, 𝑓2(𝑥) = −3𝑥 2. next, use the modified recurrence formula (2.12) to obtain 𝑢0(𝑥) = 𝑓1(𝑥) = 6𝑥, 𝑢1(𝑥) = 6𝑥 − 3𝑥 2 − ∫ 𝑢0 𝑥 0 (𝑡)𝑑𝑡 = 0, 𝑢𝑛+1(𝑥) = − ∫ 𝑘(𝑥, 𝑡)𝑢𝑛 𝑥 0 (𝑡)𝑑𝑡 = 0, 𝑛 ≥ 1. it is obvious that each component of 𝑢𝑗 , 𝑗 ≥ 1 is zero. this in turn gives the exact solution by 𝑢(𝑥) = 6𝑥. s. mohammed and h. s. ahmed a comparative study between adm and mdm for a system of volterra integral equation 145 4. conclusion it is clearly seen that the decomposition method converted the integral equation into an elegant determination of computable components. it was formally shown that if an exact solution exists for such problems, then the obtained series converges very rapidly to that exact solution. we here emphasize on the two important remarks, first, by proper selection of the functions 𝑓1(𝑥) and 𝑓2(𝑥), the exact solution solution 𝑢(𝑥) may be obtained by using very few iterations, and sometimes by evaluating only two components. the success of this modification depends only on the proper choice of 𝑓1(𝑥) and 𝑓2(𝑥), and this can be made through trials only. second, if 𝑓(𝑥) consist of one term only, the modified decomposition method cannot be used in this case. this confirms our belief that the adomian decomposition method and the modified decomposition method introduce the solution of volterra integral equation in the form of a rapidly convergent power series with elegantly computable term. however, if 𝑓(𝑥) consist of more than one term, the modified decomposition method minimizes the volume of the computational work. the obtained result showed that the modified decomposition method is more accurate and effective than adomian decomposition method, needs less time to get to the solution and most importantly the exact solution is achieved in two iterations. the essential condition for that to succeed is that the zeroth component should include the exact solution. references [1] s. s. ahmed, numerical solution for volterra-fredholm integral equation of the second kind by using least squares technique. iraqi journal of science 52, no. 4 (2011): 504-512. [2] m. u. s. a. aigo, on the numerical approximation of volterra integral equations of the second kind using quadrature rules. international journal of advanced scientific and technical research 1, no. 3 (2013): 558-564. [3] a. m. wazwaz, a first course in integral equations, world scientific publishing, london,(2015). [4] j. biazar and m. pourabd, a maple program for solving systems of linear and nonlinear integral equations by adomian decomposition method. int. j. contemp. math. sciences 2, no. 29 (2007): 1425-1432. [5] h. brunner, e. hairer and s. p. njersett, runge-kutta theory for volterra integral equations of the second kind. mathematics of computation 39, no. 159 (1982): 147-163. [6] w. hackbusch, integral equations: theory and numerical treatment, birkhauser verlag, basel, (1995). [7] a. isaacson and m. kirby, numerical solution of linear volterra integral equations of the second kind with sharp gradients. journal of computational and applied mathematics 235, no. 14 (2011): 4283-4301. [8] m. mustafa, numerical solution of volterra integral equations with delay using block methods. al-fatih journal 36 (2008). [9] m. m. rahman, numerical solution s of volterra integral equations using calerkin method with hernite polynomials, pure and appl. math.,(2013). [10] md a. rahman, md s. islam, and m. m. alam, numerical solutions of volterra integral equations using laguerre polynomials. journal of scientific research 4, no. 2 (2012): 357-357 [11] a. g. ramm, a collocation method for solving integral equations. international journal of computing science and mathematics 2, no. 3 (2009): 222-228. [12] j. rashidinia, e. najafi and a. arzhang, an iterative scheme for numerical solution of volterra integral equations using collocation method and chebyshev polynomials, rashidinia et al. mathematical sciences 2012. jurnal matematika mantik vol 7, no 2, october 2021, pp. 140-146 146 [13] j. saeri-nadja and m. heidari, solving linear integral equations of the second kind with repeated modified trapezoid quadrature method. appl . math. comput., 189 (2007), 980-985. [14] a. tahmasbi, a new approach to the numerical solution of linear volterra integral equations of the second kind. int. j. contemp. math. sciences 3, no. 32 (2008): 16071610. [15] m. s. islam and md. a. rahman, solutions of linear and nonlinear volterra integral equations using hermite and chebyshev polynomials. international journal of computers & technology 11 (2013): 2910-2920. [16] a. m. wazwaz, linear and nonlinear integral equations: methods and applications (2011). higher education, springer, beijing, berlin. [17] g. adomian, nonlinear stochastic systems and applications to physics, kluwer,(1989). [18] g. adomian, g.e. adomian, a global method for solution of complex systems, math. model, 5 (1984) 521-568. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: siti hadijah hasanah, sitihadijah@ecampus.ut.ac.id department of statistics, universitas terbuka, jl. pd. cabe raya, pd cabe udik pamulang, tangerang selatan, 15418, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.68-77 application of machine learning for heart disease classification using naive bayes siti hadijah hasanah department of statistics, faculty of science and technology, universitas terbuka , article history: received nov 21, 2021 revised may 27, 2022 accepted may 31, 2022 kata kunci: penyakit jantung, klasifikasi, machine learning, naive bayes abstrak. naive bayes classifier menggunakan pendekatan dari suatu teorema bayes dengan penggabungan pengetahuan sebelumnya dengan yang baru. tujuan penelitian ini adalah untuk mengembangkan machine learning dengan menggunakan teknik klasifikasi naive bayes dan sebagai sistem keputusan dalam menghasilkan ketepatan klasifikasi yang cepat dan akurat dalam mendiagnosis penyakit kardiovaskuler seperti penyakit jantung. penyakit kardiovaskular merupakan penyebab kematian utama, 32% dari seluruh kematian global dimana 85% di antaranya disebabkan oleh stroke dan serangan jantung. berdasarkan hasil analisis didapatkan bahwa akurasi ketepatan klasifikasi di data training pada data pasien diklasifikasikan tepat memiliki dan tidak memiliki penyakit jantung masing-masing sebesar 83,21% dan 83,81%. pada data testing persentase data pasien diklasifikasi tepat memiliki dan tidak memiliki penyakit jantung masing-masing sebesar 83,78% dan 87,50%. berdasarkan nilai auc di data training dan data testing masing-masing sebesar 83,15% dan 85,24%. maka dari hasil tersebut dapat disimpulkan bahwa metode naive bayes baik digunakan pada klafisikasi data pasien penyakit jantung. keywords: heart disease, classification, machine learning, naive bayes abstract. the naive bayes classifier uses an approximation of a bayes theorem by combining previous knowledge with new ones. the purpose of this research is to develop machine learning using naive bayes classification techniques and as a decision system in producing fast and accurate classification accuracy in diagnosing cardiovascular diseases such as heart disease. cardiovascular disease is the leading cause of death, 32% of all global deaths, of which 85% are caused by stroke and heart disease. based on the results of the analysis, it was found that the accuracy of classification accuracy in the training data on patient data was classified as having and not having heart disease, respectively 83,21% and 83,1%. in data testing, the percentage of patient data classified as having and not having heart disease was 83,78% and 87,50%, respectively. based on the auc values in the training data and testing data, they are 83,15% and 85,24%, respectively. so, from these results, it can be concluded that the naive bayes method is good for classifying heart disease patient data. how to cite: s. h. hasanah, “application of machine learning for heart disease classification using naive bayes”. j. mat mantik, vol. 8, no. 1, pp. 68-77, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 67-77 issn: 2527-3159 (print) 2527-3167 (online) mailto:sitihadijah@ecampus.ut.ac.id http://u.lipi.go.id/1458103791 siti hadijah hasanah application of machine learning for heart disease classification using naive bayes 69 1. introduction according to who globally cardiovascular disease (cvds) is the leading cause of death, in 2019 an estimated 17.9 million died from cdv, and this number represents 32% of all global deaths of which 85% are due to stroke and heart attack [1]. currently, globally, developing lowand middle-income countries have the highest cardiovascular risk (cvd) and are common in adults aged 40 years and over [2]. several factors that cause cardiovascular disease include heredity, uncontrolled high blood pressure, increased prevalence of diabetes, and obesity [3]. the rapid development of science in all fields automatically produces a very large amount of data, data mining has a role in handling large amounts of data. the basic functions of data mining include classification, clustering, and association [4]. classification is an important data mining technique with a wide range of applications in classifying various types of data [5]. the application of classification can be predicted quickly and accurately by using several machine learning algorithms such as support vector learning (svm) [6], [7], random forest, and cart [8], k-nearest neighbor, genetic algorithm [4], and artificial neural network (ann) [9]. the purpose of this research is to develop machine learning using naive bayes classification techniques and as a decision system in producing fast and accurate classification accuracy in diagnosing cardiovascular diseases such as heart disease. naive bayes is a data mining algorithm that is quite popular in classification [10]. a naive bayes classifier uses an approximation of a bayes theorem by combining previous knowledge with new ones [11]. the advantage of this method is the use of a simple algorithm [12] and has high accuracy [11]. several studies that apply the naive bayes classification method include in the field of education such as regarding the interest of students in determining majors in high school [13], in the economic field, namely the classification of data for underprivileged citizens so that the aid funds provided by the government to them are right on target [13], [14], classifying customers who are right on target in receiving credit and as a way to avoid the risk of default in the future [15], in the social sector regarding the analysis of public sentiment towards the development of e-sports education [16]. this article will present the application of classification using naive bayes whose implementation is assisted by the python program, in the hope that it can continue our research. in the future, the author will apply several machine learning methods and at the same time use optimization in machine learning. 2. methods the data for this article was obtained from uci machine learning in the form of a heart disease data set [17], with a total of 330 data records. the data is divided into 2, namely training data (80%) and testing data (20%) [18], and consists of 14 variables including 13 predictor variables, 1 response variable which is seen in table 1. the response variable describes the target in the classification with 2 categories, namely if < 50 % diameter narrowing means that they do not have heart disease (category 0) and if > 50% diameter is narrowing then it has heart disease (category 1) and flowchart of naive bayes classification can be seen in figure 1. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.68-77 70 figure 1. flowchart of naive bayes classification the method used in this classification is naive bayes which aims to classify patients who have the potential to have or do not have heart disease. table 1. characteristics of the heart disease data set variable description scale category x1 age x2 sex nominal 0 = male 1 = female x3 chest pain type (cp) nominal 1 = typical angina 2 = atypical angina 3 = non-anginal pain 4 = asymptomatic x4 resting blood pressure (trestbps) x5 serum cholesterol (chol) x6 fasting blood sugar > 120 nominal 0 = false siti hadijah hasanah application of machine learning for heart disease classification using naive bayes 71 variable description scale category mg/dl (fbs) 1 = true x7 resting 0 = normal electrocardiographic results nominal 1 = having st-t wave abnormality (restecg) 2 = showing probable x8 maximum heart rate achieved (thalach) x9 exercise induced angina nominal 0 = no (exang) 1 = yes x10 st depression induced by exercise relative to rest (oldpeak) x11 the slope of the peak nominal 1 = upsloping exercise st segment (slope) 2 = flat 3 = downsloping x12 number of major vessels (0–3) colored by fluoroscopy (ca) x13 thal 3 = normal 6 = fixed defect 7 = reversable defect y target/diagnosis of heart disease nominal 0 = < 50% diameter narrowing 1 = > 50% diameter narrowing 3. results and discussion 3.1 naive bayes naive bayes is one of the algorithm methods used in classification techniques in the field of statistics [19]. this classification can predict the probability of a class which can be calculated based on the following bayes theorem: 𝑃(𝐻|𝑋) = 𝑃(𝑋|𝐻) 𝑃(𝐻) 𝑃(𝑋) (1) where: 𝑋 : unknown data 𝐻 : hypothesis from class data 𝑃(𝐻|𝑋) : probability of the value hypothesis based on the condition of the value of 𝑋 𝑃(𝐻) : hypothesis probability 𝐻 value 𝑃(𝑋|𝐻) : probability of the value of 𝑋 based on 𝐻 value hypothesis 𝑃(𝑋) : probability of value 𝑋 3.2. multicollinearity and normalization test if there is a linear relationship between each variable, it may indicate the existence of multicollinearity, and there are several ways to overcome multicollinearity, one of which is eliminating one of the variables that have a high correlation. the high correlation can be jurnal matematika mantik vol. 8, no. 1, june 2022, pp.68-77 72 seen from the value of the correlation coefficient which is getting closer to the value 1. the normalization process in data is very important in producing classification outcomes, the use of normalization in classification produces better output results [20]. 3.3. receiver operating characteristics (roc) and area under curve (auc) roc is a graph that describes the performance of a binary classification system between sensitivity on the y-axis and 1-specificity on the x-axis. sensitivity is a measure of the classification accuracy of an expected event, while specificity is a measure of the classification accuracy of an unexpected event [21]. the overall indication of the diagnostic accuracy of the roc curve is the area under curve (auc) value. auc values can be divided into several groups as follows [22]: table 2. classification of data based on the auc value auc classification auc>0.9 outstanding classification 0.8≥auc>0.7 excellent classification 0.7≥auc>0.6 acceptable classification 0.6≥auc>0.5 poor classification 0,50 no classification 3.4. descriptive statistics based on [23] data on patients who have heart disease can be categorized based on age and gender, so from the exploration results obtained descriptive statistical results are as follows: figure 2. heart disease data by age based on figure 2, a person's vulnerable age is at risk of having heart disease, namely when they are over 40 years old. however, based on these data, the age under 40 years does not rule out the risk of heart disease even though it has a smaller percentage when compared to those over 40 years old. 1,82% 6,06% 23,64% 19,39% 23,03% 13,33% 9,70% 3,03% 0,00% 5,00% 10,00% 15,00% 20,00% 25,00% 29-34 35-40 41-46 47-52 53-58 59-64 65-70 71-76 siti hadijah hasanah application of machine learning for heart disease classification using naive bayes 73 figure 3. heart disease data by gender based on the results of the percentage of patients who have heart disease in figure 3, it is found that women patients have a greater percentage of 12.72% than men patients. figure 4. correlation results between variables figure 4 shows that the old peak variable with slope has a negative correlation with the medium category, which is 0,8. so to overcome multicollinearity, the authors delete one of the two variables, namely by eliminating the slope variable in the next analysis. 56,36% 43,64% 0,00% 10,00% 20,00% 30,00% 40,00% 50,00% 60,00% female male jurnal matematika mantik vol. 8, no. 1, june 2022, pp.68-77 74 table 3. training data class naive bayes result (%) 0 1 0 83,81% 16,79% 1 16,19% 83,21% based on table 3, it was found that the percentage of data for patients classified as having no heart disease was 83,81% and patients classified as having heart disease was 83,21%. so it can be concluded that the naive bayes method is good for classification in training data. table 4. testing data class naive bayes result (%) 0 1 0 87,50% 16,22% 1 12,50% 83,78% based on table 4, it was found that the percentage of patients classified as having heart disease was 87,50% and the patients were classified as having heart disease at 83,78%. so, it can be concluded that the naive bayes method is good for classification in testing data. figure 5. auc of training data figure 5 shows that the auc value is 83,15%, so it can be concluded that the diagnostic accuracy of the classification of patients having or not having heart disease in the training heart disease data can be classified well at 83,15%. siti hadijah hasanah application of machine learning for heart disease classification using naive bayes 75 figure 6. auc of testing data figure 6 shows that the auc value is 85,24%, so it can be concluded that the diagnostic accuracy of the classification of patients having or not having heart disease in the training heart diseases data can be classified as good at 85,24%. the weakness of this research is that it has not explored the classification of heart diseases using several machine learning methods. researchers will continue this research by comparing naive bayes with several machine learning methods such as random forest (rf), support vector machine (svm), artificial neural network (ann) and at the same time applying optimization in machine learning. 4. conclusions the results of the accuracy of the classification machine learning with naive bayes are good classification, with the results of the accuracy of the classification in the training data on patient data that is classified correctly as not having heart disease by 83,81% and the right patient being classified as having heart disease by 83,21%. in data testing, the percentage of patients classified as having heart disease was 87.50% and the patients were classified as having heart disease at 83,78%. likewise, the auc values in the training data and testing data are 83,15% and 85,24%, respectively. references [1] who, “cardiovascular diseases (cvds),” 2021. https://www.who.int/newsroom/fact-sheets/detail/cardiovascular-diseases-(cvds) (accessed jan. 21, 2022). 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[21] k. hajian-tilaki, “receiver operating characteristic (roc) curve analysis for medical diagnostic test evaluation,” casp. j. intern. med., vol. 4, no. 2, pp. 627– 635, 2013. siti hadijah hasanah application of machine learning for heart disease classification using naive bayes 77 [22] s. yang and g. berdine, “the receiver operating characteristic (roc) curve,” southwest respir. crit. care chronicles, vol. 5, no. 19, p. 34, 2017, doi: 10.12746/swrccc.v5i19.391. [23] f. babic, j. olejar, z. vantova, and j. paralic, “predictive and descriptive analysis for heart disease diagnosis,” proc. 2017 fed. conf. comput. sci. inf. syst. fedcsis 2017, no. october, pp. 155–163, 2017, doi: 10.15439/2017f219. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: joko riyono jokoriyono@trisakti.ac.id faculty of industrial technology, universitas trisakti,indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.113-123 simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia joko riyono1, christina eni pujiastuti1 1universitas trisakti, jakarta, indonesia article history: received sep 1, 2022 revised nov 30, 2022 accepted dec 31, 2022 kata kunci: defisit anggaran, clustering, cluster optimal, kedalaman kemiskinan, keparahan kemiskinan abstrak. salah satu cara agar progam dan bantuan pemerintah ke tiap provinsi bisa tepat sasaran adalah dengan membuat model pengelompokan atau clustering provinsi di indonesia didasarkan pada tingkat kemiskinan. algoritma k means merupakan salah satu metode clustering di data mining untuk membagi n pengamatan menjadi k kelompok sedemikian hingga tiap pengamatan berada dalam kelompok dengan rata-rata terdekat. dalam penelitian ini akan dibuat clustering tingkat kemiskinan provinsi di indonesia didasarkan pada tiga indikator tingkat kemiskinan yaitu prosentase penduduk miskin (p0), kedalaman kemiskinan (p1) dan keparahan kemiskinan (p2) dengan algoritma k-means menggunakan metode elbow berbantukan progam phyton. diperoleh hasil 5 cluster optimal tingkat kemiskinan provinsi di indonesia. keywords: budget deficit, clustering, optimal cluster, poverty depth, poverty severity abstract. one way to ensure that government programs and assistance for each province are right on target is to create a model of grouping or clustering provinces in indonesia based on poverty levels. algorithm k means is one of the clustering methods in data mining to divide n observations into k groups so that each observation is in the group with the closest mean. in this study, provincial poverty level clustering in indonesia will be made based on three poverty level indicators, namely the percentage of poor population (p0), poverty depth (p1), and poverty severity (p2) with the k-means algorithm using the elbow method assisted by the python program. the results obtained are 5 optimal clusters of provincial poverty rates in indonesia. how to cite: j. riyono and c. e. pujiastuti, “simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia”, j. mat. mantik, vol. 8, no. 2, pp. 113-123, december 2022 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 113-123 issn: 2527-3159 (print) 2527-3167 (online) https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 113-123 114 1. introduction the covid-19 pandemic that occurred in indonesia is part of a global pandemic which has a major impact on the country's economy. this pandemic has resulted in many companies going out of business, so many companies have laid off their employees. the large number of companies that have terminated employment has made unemployment more common in society, resulting in a decrease in people's purchasing power for goods and services, all of which have had an effect on increasing the level of poverty in society and slowing economic growth. according to indonesian economic data from the central bureau of statistics, the indonesian economy will experience a decline of up to 2.41% in 2021. the government has implemented various strategies in an effort to help the people's economic growth. the wide area coverage and the large number of people with different economic problems require different strategies to overcome them. there are many factors that serve as reference material for the actions that should be taken so that the assistance provided can be appropriate as needed. one way to ensure that government programs and assistance for each province are right on target is to create a model of grouping or clustering provinces in indonesia based on poverty levels. the results of poverty mapping in the form of clustering are expected to provide benefits for the government in making policies and programs to overcome poverty problems and allocate budgets effectively and have a positive impact on poverty alleviation. in this study, provincial poverty level clustering in indonesia will be made based on three poverty level indicators, namely the percentage of poor population (p0), poverty depth (p1) and poverty severity (p2) with the k-means algorithm using the elbow method assisted by the python program. clustering analysis is a multivariate technique to group similar observations into a number of clusters based on the observed values of several variables for each individual. the purpose of the clustering process is to minimize the occurrence of the objective function that is set in the clustering process, which is generally used to minimize variations within a cluster and maximize variations between clusters[1]. one of the most frequently used algorithms in statistics and machine learning is k-means clustering. k-mean clustering is one of the unsupervised learning algorithms which is included in the nonhierarchical cluster analysis which is used to group data based on variables or features. the purpose of k-means clustering, like other cluster methods, is to obtain clusters of data by maximizing the similarity of characteristics within the cluster and maximizing the differences between clusters. the k-means clustering algorithm groups data based on the distance between the data and the cluster centroid point obtained through an iterative process. the performance of the k-means clustering algorithm depends on the highly efficient cluster it forms. determining the optimal number of clusters is an important step in creating clusters with k-means clustering. there are several different ways to find the optimal number of clusters, one of the most popular methods for finding the number of clusters is the elbow method. the elbow method is a method used to generate information in determining the best number of clusters by looking at the percentage of the comparison between the number of clusters that will form an angle at a point. this method uses the within cluster sum of squares (wcss) value concept, which defines the total variation within a cluster. this method provides ideas or ideas by selecting the cluster value and then adding the cluster value to be used as a data model in determining the best cluster. and besides that the percentage of the resulting calculation becomes a comparison between the number of clusters added. the results of different percentages of each cluster value can be shown by using a graph as a source of information. clustering is one method of data mining, data mining can be defined as the extraction of useful information or drawing patterns of knowledge from data stored in large quantities [2]. data mining analysis runs on the best data and techniques to get the most feasible conclusions [3]. data mining also has several jokoriyono, christina eni pujiastuti simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia 115 names such as knowledge discovery in database (kdd), knowledge extraction (knowledge extraction), business intelligence (business intelligence), and others. one of the clustering algorithms is the k-means clustering algorithm. k-means clustering is a data analysis method or data mining method that performs a modeling process without supervision where k is a constant for the number of clusters and means is the average value of a data group defined as a cluster [4].k-means clustering is one of the no hierarchical cluster analysis methods that group objects based on their characteristics so that they are in the same cluster and objects that have different characteristics are grouped together with similar objects in other clusters. the k-means algorithm process can be described in a flowchart diagram or flow chart as follows [5]: figure 1. k-means algorithm flowchart based on figure 1, the initial stage of applying the k-means clustering algorithm is the selection of the value of k randomly based on the number of clusters needed, determining the centroid or center point of each cluster, calculating the distance of the object matrix to the center point or centroid, grouping each object into clusters based on distance. minimum or shortest, and iterate over each process starting from determining the centroid based on the final result of the temporary cluster. if there is no change in the group members of each cluster, the iteration can be stopped and the final results of clustering will be seen using the k-means method. an explanation of the stages of k-means clustering can be formulated as follows [6]: a. determine the value of k randomly determination of the value of k randomly is done based on the needs or desires of researchers. the value of k is used as the number of clusters to be formed. determination of the value of k can be done through several considerations, both conceptual and theoretical [7]. b. determine the center point or k centroid determination of the center point or initial centroid can be done randomly from the available objects. the number of central points determined must be based on the jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 113-123 116 number of k values because the center point in question is the center point of each cluster. in the iteration process, the determination of the i-th cluster centroid can be done by the formula [8]: v= σ𝑖−1 χi 𝑛 𝑛 ;=1,2,3, ... 𝑛..................................................................................(1) information: v : centroid on cluster xi : i-th object n : the number of objects/ number of objects that are members of the cluster. c. calculate the distance of each object to each centroid calculation of the distance between each object and the centroid can be done by calculating the euclidian distance with the following formula: (𝑥,𝑦)=‖𝑥−𝑦‖=√∑ (𝑥𝑖 −𝑦𝑖 ) 2𝑛 𝑖−1 ,i=1,2,3,...𝑛 ................................................(2) information: xi : ith x object yi : y-th power n : number of objects d. group each object into clusters the grouping of each object into clusters is based on the minimum distance or the shortest distance that each object has to each center point or centroid. in the iteration process, grouping objects into each cluster can be done with hard k-means or fuzzy c-means [9]. e. doing iteration the iteration process is carried out from determining the centroid based on the final results of the temporary cluster and the next process is carried out as previously done. if in the new iteration there is no change in the cluster group, then the iteration can be stopped and the final results of the cluster can be seen. if the results are the same then the k-means cluster analysis algorithm has converged, but if it is different, then it has not converged so it is necessary to do the next iteration [10] 2. method the method used in this study is a quantitative method using literacy studies, where data is collected using measuring instruments, then analyzed statistically and quantitatively. in accordance with the intent and purpose of the study, the data taken as a population is data on the percentage of poor population, poverty depth and poverty severity in 34 provinces in indonesia while the data taken as a sample is data on percentage of poor population, poverty depth and poverty severity in 34 provinces in semester i of 2021, semester ii of 2021 and semester i of 2022. data was taken through a literature study on the official website of the central statistics agency with the website address https://bps.go.en/ [8]. through this sample data will be analyzed to obtain clusters of population poverty levels in this case the provinces in indonesia using the k-means clustering algorithm based on the elbow method with python software. the selection of quantitative methods is based on existing research results, including: quantitative methods emphasize objective measurements and statistical, mathematical or numerical analysis of the data obtained [11]. research with quantitative methods has several advantages, including [12]: a. based on accurate data and measurements b. supported by advanced statistical techniques and software c. predictability validity that can be extended into the future jokoriyono, christina eni pujiastuti simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia 117 d. can control at any time the validity of the relationship between the dependent and independent variables figure 2. poverty percentage data, poverty depth index data, and poverty severity index data in 34 provinces in indonesia in the first semester of 2021, second semester of 2021 and first semester of 2022 caption: column a: the percentage of poor population (p0) semester i 2021 column b: the percentage of poor population (p0) semester ii 2021 column c: the value of the percentage of poor people (p0) in the first semester of 2022 column d: poverty depth score (p1) semester i 2021 column e: poverty depth score (p1) in semester ii 2021 column f: poverty depth score (p1) semester i 2022 column g: poverty severity score (p2) semester i 2021 column h: poverty severity score (p2) semester ii 2021 column i: poverty severity score (p2) semester i 2022 indicator column p(0) : average value of the percentage of poor population (p0) 20212022 indicator column p(1) : average value of poverty depth (p1) 2021-2022 indicator column p(2) : average score of poverty severity (p2) 2021-2022 standardization is one of the processing processes by equating the units of each attribute. in cases where the data has a much different range of values it can cause the calculation to be ineffective [14].so standardization is done so that the data has the same range of values so it is hoped that data with large or small values will not affect the final analysis results. one way to standardize data is to use z-score, basically standardizing using z-score is to change the original data value into z form (normally distributed data) with the formula [15]: 𝑧 = 𝑥𝑖−�̅� 𝑆 with �̅� = ∑ 𝑥𝑖 𝑛 𝑖=1 𝑛 and s=√ 1 𝑛−1 ∑ (𝑥𝑖 − �̅�) 2𝑛 𝑖=1 ..................................................(3) jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 113-123 118 3. results and discussion the research process with the k-means clustering algorithm used in this study is to group 34 provinces in indonesia based on poverty level indicators from semester i 2021, semester ii 2021 and semester i 2022 with the help of python software. the following are the stages in the research process: figure 3. research flowchart a. input the data set percentage of poor population (p0), poverty depth index p(1), and poverty severity index p(2) found in 34 provinces in indonesia in semester i 2021, semester ii 2021 and semester i 2022 using microsoft excel. b. pre-processing data with z-score normalization method jokoriyono, christina eni pujiastuti simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia 119 figure 4. results of pre-processing data c. determine the value of k clusters or the number of clusters through the elbow method. elbow method is a heuristic used in determining the number of clusters in a data set by plotting the variation described as a function of the number of clusters and selecting the angle of the curve as the number of clusters to be used. figure 5. wss data value jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 113-123 120 figure 6. wss vs cluster from figure 5 and 6 with the elbow method, the elbow point is found in cluster 5, therefore it is concluded that the number of clusters is 5. d. processing the results of pre-processing data in step 2 using the k-means algorithm in python software by inputting the value of k=5 or the number of clusters obtained from step 3. figure 7. results of k-means clustering with python software jokoriyono, christina eni pujiastuti simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia 121 figure 8. image of the clustering map of the poverty level of the indonesian province with k means clustering using the python software from figures 7 and 8, the results show that there are 5 optimal clusters of provincial poverty levels in indonesia obtained by the elbow method, the five clusters are: the first cluster contains the provinces of papua and west papua with an average poverty rate indicator of 10.644. the second cluster contains the provinces of maluku and east nusa tenggara with an average poverty rate indicator of 7.79. the third cluster contains the provinces of aceh, bengkulu, west nusa tenggara, gorontalo with an average poverty rate indicator of 4.5635. the fourth cluster contains the provinces of south sumatra, lampung, central java, yogyakarta, east java, central sulawesi, southeast sulawesi, west sulawesi with an average poverty rate indicator of 4.77 and the fifth cluster contains the provinces of north sumatra, west sumatra, riau, jambi, bangka islands. belitung, riau islands, jakarta, west java, banten, bali, west kalimantan, south kalimantan, central kalimantan, east kalimantan, north kalimantan, north sulawesi, south sulawesi, north maluku with an average poverty rate indicator of 2.58. based on the average population poverty rate, it can be seen that the provinces of papua and west papua are two provinces that need attention, even though in terms of natural resources the two provinces have very abundant wealth, this can be used as a topic for further research on why conditions like this can occur. 4. conclusion the conclusion that can be drawn from the grouping of provincial poverty levels in indonesia using the elbow method is that there are 5 poverty level clusters. then using the k-means clustering algorithm, clusters with very high poverty rates are found in the provinces of papua and west papua. high poverty rates are found in maluku and east nusa tenggara. poverty levels are moderate in the provinces of aceh, bengkulu, west nusa tenggara, gorontalo. low poverty rates are found in south sumatra, lampung, central java, yogyakarta, east java, central sulawesi, southeast sulawesi, west sulawesi and very low poverty rates are found in the provinces of north sumatra, west sumatra, riau, jambi, bangka 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[14] e. s. y. pandie, “implementation of naive bayes data mining algorithms in cooperatives,” j-icon, vol. 6, no. 1, pp. 15–20, 2018. jokoriyono, christina eni pujiastuti simulation of the k-means clustering algorithm with the elbow method in making clusters of provincial poverty levels in indonesia 123 [15] t. alfina, t. alfina, b. santosa, and a. r. barakbah, "a comparative analysis of hierarchical clustering methods, k-means and the combined both in cluster data (case study: its industrial engineering practice work problems)," its engineering journal, vol. 1, no. 1, pp. a521–a525, sept. 2012, doi: 10.12962/j23373539.v1i1.1794. contact: siti hadijah hasanah sitihadijah@ecampus.ut.ac.id statistics department, faculty of science and technology, universitas terbuka, tangerang selatan 15437, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.51-58 multivariate adaptive regression splines (mars) for modeling the student status at universitas terbuka siti hadijah hasanah department of statistics, universitas terbuka, indonesia article history: received dec 23, 2020 revised may 10, 2021 accepted may 30, 2021 kata kunci: fungsi basis, gcv, multivariate, recursive, splines abstrak. multivariate adaptive regression splines (mars) digunakan untuk memodelkan status mahasiswa aktif program studi statistika universitas terbuka serta untuk mengetahui faktor-faktor yang mempengaruhi variabel respon. penelitian ini terdiri dari 9 variabel yaitu jenis kelamin, umur, pendidikan, status pernikahan, pekerjaan, tahun registrasi awal, jumlah registrasi, sks, dan ipk, tetapi setelah dilakukan pemodelan menggunakan metode mars maka variabel penjelas yang dapat mempengaruhi variabel respon adalah tahun registrasi awal, jumlah registrasi, ipk, dan sks. berdasarkan hasil output r dan menggunakan selang kepercayaan 95%, maka masing-masing fungsi basis 1 sampai 10 adalah signifikan secara parsial dengan nilai-p dari fungsi basis 1-10 lebih kecil dari 0,05 dan secara simultan dengan nilai nilai-p lebih kecil dari 0,05, sehingga model di atas memiliki pengaruh yang signifikan secara parsial maupun simultan terhadap variabel respon. dari hasil tersebut maka disimpulkan bahwa model mars layak digunakan untuk menentukan faktor-faktor yang mempengaruhi status aktif siswa. keywords: basis function, gcv, multivariate, recursive, splines abstract. multivariate adaptive regression splines (mars) used to model the active student’s status in the department of statistics at universitas terbuka and determine the factors that influence the response variable. this study consists of 9 variables, namely gender, age, education, marital status, job, initial registration year, number of registrations, credits, and gpa, but after modeling using the mars method, the explanatory variable can affect the response variable is the initial registration year. several registrations, gpa, and credits. based on the results of the r output and using a 95% confidence interval, each base 1 to 10 function is partially significant with the pvalue of the base 1-10 function being smaller than 0.05 and simultaneously with a smaller p-value. of 0.05, so that the above model has a significant effect partially or simultaneously on the response variable. from these results, it is concluded that the mars model is suitable for determining the factors that affect the active status of students. how to cite: s. h. hasanah, “multivariate adaptive regression splines (mars) for modeling the student status at universitas terbuka”, j. mat. mantik, vol. 7, no. 1, pp. 51-58, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 51-58 issn: 2527-3159 (print) 2527-3167 (online) mailto:sitihadijah@ecampus.ut.ac.id https://doi.org/10.15642/mantik.2021.7.1.51-58 https://doi.org/10.15642/mantik.2021.7.1.51-58 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 51-58 52 1. introduction universitas terbuka has 39 service offices spread from sabang to merauke. the advantages of universitas terbuka are that there are no restrictions on the period of completion of the study, no implementation of the dropout system, no restrictions on the year of graduation or age, and registration time throughout the year [1]. based on the advantages and ease of studying at universitas terbuka there is a problem that must be faced one of them is the active status of ut students, students who have been registered at universitas terbuka are given the facility to register at any time so many students with inactive status in a semester and do not know when status as an active student again. this study was conducted by analyzing the factors that affect the status of student active, especially in the department of statistics based on several indicators, namely age, gender, education, marital status, employment status, year of initial registration, number of registrations, credits, and gpa. several statistical methods used to determine the effect of explanatory variables on categorical response variables include the binary logistic regression method and the multivariate adaptive regression spline (mars) [2]. both methods are included in regression analysis, which is a statistical method that studies the mathematical relationship pattern between one or more explanatory variables and the response variable [3]. logistic regression has the advantage that it does not need to fulfill assumptions [4], can be used on non-linear data, and is easy to interpret. however, logistic regression has a weakness, namely that there is multicollinearity between the explanatory variables which is one of the problems that make the parameter estimation results unstable [5]. the mars method is able to accommodate interactions between variables even for high-dimensional data [6], [7]. mars is a relatively flexible classification method used to determine the pattern of relationships between explanatory variables and response variables without using initial assumptions about the form of functional relationships. this method is a complex combination of spline and recursive partitioning and involves a high data dimension, namely the number of observations and a large number of variables [8]. in addition, mars can effectively explore the hidden non-linear relationship between response variables and predictor variables as well as the interaction effects on complex data structures [9]. mars can solve the problem of high and non-continuous dimensions in nodes. this method can analyze 50-1000 amounts of data with 3-20 explanatory variables [10]. dynamic non-linear patterns and interactions can be explained by this method [11]. mars can also be applied in credit assessment [12], [13], transportation [9], and software engineering [14]. so the method we use to determine the active status of universitas terbuka students in this study is multivariate adaptive regression splines (mars). this research is an alternative study program to determine the best step in overcoming the many inactive student statuses, especially in the statistics study program. 2. multivariate adaptive regression splines (mars) suppose 𝑦 a single response variable depends on 𝑛 predictor variables 𝑥, where 𝑥′ = (𝑥1,𝑥2,… ,𝑥𝑛) then the regression model is : 𝑦 = 𝑓(𝑥1,𝑥2,…,𝑥𝑛)+ 𝜀 (1) assume 𝑓 is a linear combination of the base functions of 𝐵𝑚(𝑥), with the base of each subregion 𝑚 = 1,2,…,𝑀. 𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚𝐵𝑚(𝑥) 𝑀 𝑚=1 ( (2) siti hadijah hasanah multivariate adaptive regression splines (mars) for modeling the student status at universitas terbuka 53 𝑎0,𝑎1,𝑎2,… ,𝑎𝑀 = regression coefficient 𝐵𝑚(𝑥) = m base function 𝑀 = maximum base function each base function is a truncated power splines function. univariate truncated power basis can be described as an indicator function. the functions of the base of univariate splines base from right and left are as follows : 𝑏𝑞 +(𝑥,𝑐) = [+(𝑥 −𝑐)]+ 𝑞 , 𝑏𝑞 −(𝑥,𝑐) = [−(𝑥 − 𝑐)]+ 𝑞 a single base function can be specified: 𝐵𝑚(𝑥) = ∏ [𝑠ℎ𝑚(𝑥𝑖(ℎ,𝑚) − 𝑐ℎ𝑚)] 𝐻𝑚 ℎ=1 + 𝑞 (3) hm = the number of interactions on the m based function shm = value ±1, if the knot is located to the right or left of the subregion xi(h,m) = predictor variables (1,2, …, n), h-interactions (1,2, …, m) and base of each m subregion (1,2,…, m) chm = knot point position q = order of splines the mars model can be stated as follows : 𝑓(𝑥) = 𝑎0 +∑ 𝑎𝑚 𝑀 𝑚=1 ∏ [𝑠ℎ𝑚(𝑥𝑖(ℎ,𝑚) − 𝑐ℎ𝑚)] 𝐻𝑚 ℎ=1 (4) location selection and the number of knots on mars using forward stepwise and backward stepwise steps: 1. forward stepwise this stage is to determine the location of the knot and the maximum base function based on the data by minimizing the average sum of square residual (asr) [15]. the addition of the base function is continued until it reaches the maximum base function. 2. backward stepwise this stage is to determine the size of the appropriate base function. at this stage, the removal of the base function that contributes to the estimated value of the small response until a balance between bias and variety and a suitable model is obtained, that is, by minimizing the value of generalized cross-validation (gcv) [16]. the minimum gcv functions are as follows : 𝐺𝐶𝑉(𝑀) = 𝐴𝑆𝑅 [1− 𝐶(�̂�) 𝑛 ] 2 = 1 𝑛 ∑ [𝑦𝑖 − 𝑓𝑀(𝑥𝑖)] 2𝑛 𝑖=1 [1 − 𝐶(�̂�) 𝑛 ] 2 (5) yi = response variable 𝑓𝑀(𝑥𝑖) = the value of the response variable estimates on the m base function n = many observations 𝐶(�̂�) = 𝐶(𝑀)+𝑑𝑀 𝐶(𝑀) = trace [𝐵(𝐵𝑀 𝑇𝐵𝑀) −1𝐵𝑀 𝑇 ] +1 d = value when each base function achieves optimization ( 2 ≤ 𝑑 ≤ 4 ) 2.1 decomposition anova interpretation of the mars model through decomposition anova [17] : jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 51-58 54 𝑓(𝑥) = 𝑎0 + ∑ 𝑎𝑚 𝑀 𝑚=1 [𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)] + ∑ 𝑎𝑚 𝑀 𝑚=1 [𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)][𝑠2𝑚(𝑥𝑖(2,𝑚) − 𝑐2𝑚)] + ∑ 𝑎𝑚 𝑀 𝑚=1 [𝑠1𝑚(𝑥𝑖(1,𝑚) − 𝑐1𝑚)][𝑠2𝑚(𝑥𝑖(2,𝑚) − 𝑐2𝑚)][𝑠3𝑚(𝑥𝑖(3,𝑚) − 𝑐3𝑚)]+ ⋯ (6) can be written as follows: in general, it can be written as follows: 𝑓(𝑥) = 𝑎0 + ∑ 𝑓𝑖(𝑥𝑖) 𝐻𝑚=1 +∑ 𝑓 𝑖𝑗 (𝑥 𝑖 ,𝑥𝑗 𝐻𝑚=2 ) +∑ 𝑓 𝑖𝑗𝑘 (𝑥 𝑖 ,𝑥𝑗,𝑥𝑘 𝐻𝑚=3 ) + ⋯ (7) 2.2 estimation of mars model parameters suppose the base function 𝐵𝑚(𝑥), 𝑚 = 0,1,2,…,𝑀, to estimate the regression coefficient 𝑎𝑚 using the smallest square method. �̂�𝑀𝐾𝑇 = (𝐵 𝑇𝐵)−1𝐵𝑇 𝑌, (8) 𝑎 = (𝑎0,𝑎1,… ,𝑎𝑀) 𝑇 (9) 𝑌 = (𝑦1,𝑦2,… ,𝑦𝑛) 𝑇 (10) 𝐵 = ( 1 ∏ 𝑠1𝑚(𝑥1(1,𝑚) − 𝑐1𝑚)… 𝐻1 ℎ=1 ∏ 𝑠𝑀𝑚(𝑥1(𝑀,𝑚) − 𝑐𝑀𝑚) 𝐻𝑀 ℎ=1 1 ∏ 𝑠1𝑚(𝑥2(1,𝑚) − 𝑐1𝑚)… 𝐻1 ℎ=1 ∏ 𝑠𝑀𝑚(𝑥2(𝑀,𝑚) − 𝑐𝑀𝑚) 𝐻𝑀 ℎ=1 … … 1 ∏ 𝑠1𝑚(𝑥𝑛(1,𝑚) − 𝑐1𝑚)… 𝐻1 ℎ=1 ∏ 𝑠𝑀𝑚(𝑥𝑛(𝑀,𝑚) − 𝑐𝑀𝑚) 𝐻𝑀 ℎ=1 ) (11) 2.3 mars model significance test the significance test of the mars model is on the base function which includes simultaneous and partial tests, simultaneous tests with the following hypotheses: 𝐻0: 𝑎1 = 𝑎2 = ⋯ = 𝑎𝑚 = 0 𝐻1: there is at least one 𝑎𝑚 ≠ 0;𝑚 = 1,2,…,𝑀 f test : 𝐹 = 𝑆𝑆𝑅 𝑆𝑆𝐸 ~𝐹(𝑀,𝑁−𝑀−1) (12) partial test with the following hypotheses: 𝐻0:𝑎𝑚 = 0 𝐻1:𝑎𝑚 ≠ 0 ;𝑚 = 1,2,…,𝑀 t test : 𝑡 = �̂�𝑚 𝑠𝑒(�̂�𝑚) ~𝑡 ( 𝑎 2 ,𝑁−𝑀−1) (13) siti hadijah hasanah multivariate adaptive regression splines (mars) for modeling the student status at universitas terbuka 55 3. data and methods the data used in this study is secondary data, which is the data of the student characteristics of the department of statistics, universitas terbuka from year 2009.1 to 2019.2. the data is divided into two parts, namely training data 1046 and testing data 447 which consists of 9 explanatory variables and one response variable : table 1. characteristics students at department of statistics, universitas terbuka variable information scale category x1 gender nominal 1 = man 2 = woman x2 age interval x3 education ordinal 1 = senior high school 2 = associate degree 3 = bachelor degree 4 = master degree 5 = doctoral degree x4 marital status nominal 1 = single 2 = married x5 job nominal 1 = unemployment 2 = private employees 3 = entrepreneur 4 = civil servants 5 = army/police x6 initial registration year interval x7 number of registrations interval x8 credits interval x9 gpa interval y student status nominal 0 = not active 1 = active the steps for applying the mars method are as follows: 1. determining the number of base functions base function limit between 2 4 times the number of explanatory variables. 2. determination of maximum interaction (mi) the mi used are 1, 2, and 3. 3. determination of minimum observations in each knot (mo) 4. perform a trial and error process combine base, mi, and mo functions to get minimum gcv value. 5. test the significance of the regression coefficient partial test statistics with t and f test statistics to test the significance of the regression coefficient simultaneously. 4. result by using the r software and simulating the number of basis functions, mi, mo, and gcv, the mars model is obtained as follows: 𝑌 = −0,003𝐵𝐹1 − 0,082𝐵𝐹2 + 0,029𝐵𝐹3 −0,027𝐵𝐹4 − 0,233𝐵𝐹5 −0,272𝐵𝐹6 +0,254𝐵𝐹7 −0,004𝐵𝐹8 − 0,008𝐵𝐹9 +0,006𝐵𝐹10 (14) jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 51-58 56 𝐵𝐹1 = ℎ(20152 −𝑌𝑒𝑎𝑟_ 𝐸𝑎𝑟𝑙𝑦_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛) 𝐵𝐹2 = ℎ(10 −𝑇𝑜𝑡𝑎𝑙_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛) 𝐵𝐹3 = ℎ(𝑌𝑒𝑎𝑟_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛_𝐸𝑎𝑟𝑙𝑦 − 20121) 𝐵𝐹4 = ℎ(𝑌𝑒𝑎𝑟_𝐸𝑎𝑟𝑙𝑦_𝑅𝑒𝑔𝑖𝑠𝑡𝑟𝑎𝑡𝑖𝑜𝑛 −20142) 𝐵𝐹5 = ℎ(𝐺𝑃𝐴−0,33) 𝐵𝐹6 = ℎ(0,33 − 𝐺𝑃𝐴) 𝐵𝐹7 = ℎ(𝐺𝑃𝐴−1,63) 𝐵𝐹8 = ℎ(𝐶𝑟𝑒𝑑𝑖𝑡𝑠 − 21) 𝐵𝐹9 = ℎ(21 −𝐶𝑟𝑒𝑑𝑖𝑡𝑠) 𝐵𝐹10 = ℎ(𝐶𝑟𝑒𝑑𝑖𝑡𝑠 −70) table 2. mars model interpretation basis function (bf) interpretation 1 base 1 function contributes to the model of -0.003, when students register early in 2015 semester 2 then the student status is active. 2 base 2 function contributes to the model by -0.082, if the student register 10 times then the student status is active. 3 base 3 function contributes to the model of 0.029, when the student registers early in 2012 semester 1 then the student status is active. 4 base 4 function contributes to the model of -0.027, if the student registers early in 2014 semester 2 then the student status is active. 5 base 5 function contributes to the model by -0.233, if the student has a gpa of 0.33 then the student status is active. 6 base 6 function contributes to the model by -0.272, if the student has a gpa of 0.33 then the student status is active. 7 base 7 function contributes to the 0,254 models. if the student has a gpa of 1.63 then the student status is active. 8 the base 8 functions contribute to the model of -0.004, when a student takes 21 credits then the student status is active. 9 the base 9 functions contribute to the model of -0.008, when a student takes 21 credits then the student status is active. 10 the base 10 function contributes to the model of 0.006, when a student takes 70 credits then the student status is active. the significance test is then performed on each of the above basis functions as follows: table 3. significance test basis function t p-value f p-value 𝐵𝐹1 -2,252 0,024537 297,7 < 2,2e-16 𝐵𝐹2 -16,277 < 2e-16 𝐵𝐹3 12,425 < 2e-16 𝐵𝐹4 -10,58 < 2e-16 𝐵𝐹5 -7,577 7,86e-14 𝐵𝐹6 -2,327 0,02014 𝐵𝐹7 4,945 8,90e-07 𝐵𝐹8 -4,684 3,18e-06 𝐵𝐹9 -3,873 0,000114 𝐵𝐹10 4,976 7,60e-07 previously, based on research data, there were 9 explanatory variables used to determine the effect of student activeness status, but after modeling using the mars siti hadijah hasanah multivariate adaptive regression splines (mars) for modeling the student status at universitas terbuka 57 method. these explanatory variables could affect the active status of students majoring in statistics were initial registration year, number of registrations, gpa, and credits. based on the output of the r software in table 3 and using a 95% confidence interval, each of the base 1 to 10 functions is partially significant (t-statistic) with the p-value of the base function 1-10 being smaller than 0,05 and automatically simultaneous (f-statistic) with pvalue less than 0,05 so that the above model has a partially and simultaneously significant influence on the response variable. from these results, it is concluded that the mars model is suitable for determining the factors. 5. conclusions there are nine explanatory variables used to determine the effect of student active status, including gender, age, education, marital status, job, initial registration year, number of registrations, credits, gpa, but after modeling using the mars method, the explanatory variables can affect statistics student active status is the initial registration year, number of registrations, gpa, and credits. based on the results of the r application output and using a 95% confidence interval, each of the base 1 to 10 functions is partially significant (t-statistic) with the p-value of the base function 1-10 less than 0,05 and simultaneously (f-statistic) with a p-value less than 0,05, so that the above model has a significant effect partially or simultaneously on the response variable. from these results, it is concluded that the mars model is suitable for determining the factors that affect student active status. the results of this study are as a way to find solutions for study programs in overcoming the many inactive student statuses, especially in the statistics study program, namely by paying attention to these four factors, the study program will provide treatment in 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[14] y. zhou and h. leung, “predicting object-oriented software maintainability using multivariate adaptive regression splines,” j. syst. softw., vol. 80, no. 8, pp. 1349– 1361, 2007, doi: 10.1016/j.jss.2006.10.049. [15] r. biswas, b. rai, p. samui, and s. s. roy, “estimating concrete compressive strength using mars, lssvm and gp,” eng. j., vol. 24, no. 2, pp. 41–52, 2020, doi: 10.4186/ej.2020.24.2.41. [16] s. sekulic and b. r. kowalski, “mars : a tutorial,” vol. 6, no. april, pp. 199– 216, 1992. [17] e. kartal koc and h. bozdogan, “model selection in multivariate adaptive regression splines (mars) using information complexity as the fitness function,” mach. learn., vol. 101, no. 1–3, pp. 35–58, 2015, doi: 10.1007/s10994-014-5440-5. contact: m. fajar, mfajar@bps.go.id. statistics indonesia, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.132-139 the testing of existence wagner's law in papua province muhammad fajar1,2*, yuyun guna winarti1 1badan pusat statistik, indonesia 2department of mathematics, institut teknologi sepuluh nopember, surabaya, indonesia article history: received, jun 6, 2021 revised, oct 10, 2021 accepted, oct 31, 2021 kata kunci: hukum wagner, papua, kointegrasi, ekonomi abstrak. hukum wagner diduga tidak bisa diterapkan secara universal. masih ada perdebatan mengenai kebenaran hukum ini. oleh karena itu, tujuan dari penelitian ini adalah untuk mengkaji implementasi hukum wagner dalam perekonomian provinsi papua. data yang digunakan adalah pdrb dan pengeluran pemerintah dalam bentuk logaritma natural (tahun dasar 2000) dari seluruh kota/kabupaten di provinsi papua dari tahun 2000 sampai 2013. sumber data berasal dari badan pusat statistik. metode yang digunakan dalam penelitian ini adalah uji kointegrasi kao dan uji kausalitas granger. dua metode ini mampu mengindikasikan kemungkinan adanya pengaruh jangka pendek dan jangka panjang antar variabel ekonomi. hasil penelitian ini menyimpulkan bahwa ada hubungan yang signifikan antara pengeluaran pemerintah dan pdb. hasil ini membawa konsekuensi bahwa hukum wagner tidak terbukti dalam perekonomian papua. hal ini didukung oleh uji kausalitas yang menunjukkan bahwa pengeluaran pemerintah menyebabkan (granger cause) terjadinya perubahan gdp riil, tetapi tidak sebaliknya. meskipun demikian, kedua variabel tersebut kointegrasi keywords: wagner’s law, papua, cointegration, economy abstract. the law of wagner is supposedly not universally applicable. there are debates about the truth of this law. based on this phenomenon, the purpose of this study is to examine the existence of wagner's law in the economy of papua province. data used in this research are grdp (gross regional domestic product) at the constant price (real grdp) and government expenditure at the constant price (real gov) in the form of natural logarithms (the base year 2000) from all municipalities in papua province from 2000 to 2013. data source comes from badan pusat statistik-statistics indonesia. the method used in this research is the kao cointegration test and the granger causality test. these methods indicate the possibility of short-term and long-term effects across economic variables. the result of this study concludes that there is a significant relationship between government expenditure and gross regional domestic product (grdp). this result means that the wagner’s law is not proven in the papuan economy. this is supported by a causality test which shows government expenditure is granger cause of the real gdp, but not vice versa. however, the two variables are cointegrating. how to cite: m. fajar and y. g. winarti, “the testing of validity wagner’s law in papua province, j. mat. mantik, vol. 7, no. 2, pp. 132-139, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 132-139 issn: 2527-3159 (print) 2527-3167 (online) mailto:mfajar@bps.go.id http://u.lipi.go.id/1458103791 muhammad fajar, yuyun guna winarti the testing of validity wagner’s law in papua province 133 1. introduction the relationship between government expenditure and gross domestic product is one of the most topics interesting to observe. government expenditure affects the gdp in a country. the government plays an important role in the economy. government expenditure is one of the components of aggregate demand that can increase domestic products. that’s why government expenditure affects economic activities. adolph wagner is the first researcher who demonstrates a positive relationship between economic growth and public sector expenditures. in “grundlegung der politischen ökonomie”, published in 1893, wagner investigated the law of increasing public expenditures and state activities. the result of this investigation is that the direction of long-term causality between state activities and economic growth has been found from national income to public expenditures by wagner’s law. wagner’s law explains that the direction of causality in a relationship over three reference points. the first one is the function of the government that takes place of the private sector within the industrialization process. the second one for the direction of causality is the change in income elasticity induced by economic growth increase demand for public goods which have socio-cultural character. the last one is an inability to make requiring projects of the public sector aimed to improve the welfare of industrialized societies and the obligation of carrying out these projects with public finance point out the direction of causality among public expenditures and economic growth. responsibilities and duties are improved in connection with economic growth and economic development in developed countries. wagner’s law was used by some previous researchers. lall [1] used cross-section data for the period 1962-1964 from 46 developing countries. his result is that wagner's law did not occur in the economies of those 46 countries [1]. landau [2] showed a negative relationship between government expenditure and the growth of gdp per capita. singh and sahni [3] used the granger causality test to determine the direction of causality between national income and public expenditure in india. their study results showed that there was no causality between the two variables. kormendi and meguire [4], in their study, concluded that there was no relationship between government expenditure and gdp growth rate. barro showed that large government expenditure reduces the growth in per capita production [5]. ansari et al. in their study to determine the direction of causality between government expenditure and national income in three countries, namely ghana. kenya and south africa with the granger causality test. the results of his study indicate that there was no long-term balance between government expenditure and national income in these three countries. wagner's law only applies to ghana on a short-term basis [6]. islam shows that there is cointegration between the relative size of government expenditure and the real gni per capita [7]. rafiee and zibaii used the ardl model to explore the relationship between the amount of government expenditure and iran's economic growth, which concluded that the amount of government expenditure has a significant positive effect on the growth of agriculture sector [8]. based on the findings of previous studies and the author's considerations, the authors use the peacock-wiseman specification to test wagner's law in the papua province economy. this model uses main variables and not derivative variables, such as the share of government expenditure on gdp, gdp per capita, and government expenditure per capita. on the other hand, the province of papua was chosen with the consideration that much social assistance was disbursed from both the central and regional governments to improve the welfare of the community. the gross domestic product (gdp) per capita of papua from the period 1996 to 2002 had a downward trend. meanwhile, from the period 2003 to 2013, gdp per capita of papua began to increase in growth although moving slowly. changes in gdp trends indicate that there are determinant factors behind them. it must be jurnal matematika mantik vol 7, no 2, october 2021, pp. 132-139 134 tested with wagner’s law. this paper aims to examine the existence of wagner's law in the economy of papua province. 2. method data used in this research are grdp (gross regional domestic product) at the constant price (real grdp) and government expenditure at the constant price (real gov) in the form of natural logarithms (the base year 2000) from all municipalities in papua province from 2000 to 2013 (panel data). data source comes from badan pusat statistik-statistics indonesia. the stages of the procedure in this study are: a. determination of the optimum lag applied to unit root, cointegration, and causality tests with the schwarz information criterion. b. stationarity check of real grdp and real gov by detecting the presence of unit root as a prerequisite for cointegration test. if the two data series are integrated in the same order (non-zero order integration), further analysis can be applied (cointegration test and causality test can be applied). however, if the two data series are not integrated in the same order, further analysis cannot be applied (cointegration test and causality test cannot be applied) and the analysis only uses ordinary regression (the implication is that it does not conclude that wagner's law exists or does not exist). c. cointegration test between real grdp and real gov. if there is cointegration between real grdp and real gov, then the two variables have a long-term relationship so that this can be continued to the causality testing stage. however, if there is no cointegration, there is no long-term relationship and cannot proceed to the causality test (the implication is that there is no conclusion that wagner's law exists or does not exist). d. conducting a causality test with the toda-yamamoto procedure. based on the results of causality testing, if real grdp causes real gov then wagner's law exists. however, if real gov causes real grdp then wagner's law doesn't exist. e. draw conclusions based on points (2), (3), (4), and (5). 2.1. lag criteria determination of the lag time for unit root testing, cointegration, and causality in the study using the schwarz information criterion (sic). the formulation of the sic is as follows: 𝑆𝐼𝐶 = −2(𝑙 𝑂⁄ ) + 𝑘 𝑙𝑜𝑔(𝑂) 𝑂⁄ (1) where: l is the log-likelihood function of a model, 𝑘 is the number of estimated parameters, and 𝑂 is the number of data observations (for balanced panel data, 𝑂 = 𝑁𝑇 where t: time index and n: number of cross-section). 2.2. unit root test the hadri test for unit root testing on panel data is similar to the kpss test [9]. it tests the null hypothesis that there is no unit root on panel data. the hadri test is based on the residuals of the individual ols regressions of 𝑦𝑖𝑡 (lgrdp_real and lgov_real) over a constant or on constants and trends. for example, if we enter constants and trends, we can derive the estimate from: 𝑦𝑖𝑡 = 𝛿𝑖 + 𝜂𝑖 𝑡 + 𝑖𝑡 , 𝑖 = 1, … , 𝑁, 𝑡 = 1, … , 𝑇 (2) muhammad fajar, yuyun guna winarti the testing of validity wagner’s law in papua province 135 where 𝑦𝑖𝑡 is the interested variable, 𝛿𝑖 is intercept i-th, 𝜂𝑖 is coefficient i-th. estimating the residual ( ̂) from individual regression, we find the lm statistic, namely: 𝐿𝑀1 = 1 𝑁 (∑ (∑ 𝑆𝑖 (𝑡) 2/𝑇 2 𝑇 ) 𝑁 𝑖=1 /𝑓0) (3) where 𝑆𝑖 (𝑡) is the cumulative summation of the residuals. 𝑆𝑖 (𝑡) = ∑ �̂�𝑡 𝑡 𝑆=1 (4) 𝑓0 is the average of the individual estimators of the residual spectrum at zero frequency: 𝑓0 = ∑ 𝑓𝑖0 𝑁 𝑖=1 /𝑁 (5) an alternative form of lm statistic accommodating for heteroscedastic: 𝐿𝑀2 = 1 𝑁 (∑ (∑ 𝑆𝑖 (𝑡) 2/𝑇 2 𝑇 ) 𝑁 𝑖=1 /𝑓𝑖0) (6) hadri pointed out that [9]: 𝑧 = √𝑁(𝐿𝑀2 − 𝜉) 𝜉 → 𝑁(0,1) (7) 2.3. kao's cointegration test the kao test [10] follows the same basic approach as the pedroni test [11], but the cross-section specifics at the intercept and the coefficient are homogeneous for the first stage regressor. for example for two variables 𝑦𝑖𝑡 = 𝛼𝑖𝑡 + 𝛽𝑥𝑖𝑡 + 𝑒𝑖𝑡 (8) where: 𝑦𝑖𝑡 = 𝑦𝑖𝑡−1 + 𝑢𝑖𝑡 (9) 𝑥𝑖𝑡 = 𝑥𝑖𝑡−1 + 𝜖𝑖𝑡 (10) for t = 1,…, t; i = 1,…., n. 𝛼𝑖𝑡 is the heterogeneous intercept, β is the homogeneous coefficient for the cross-section, 𝑒𝑖𝑡 , 𝜖𝑖𝑡 , 𝑢𝑖𝑡 are error terms, and all trend coefficients are assumed to be absent. kao then runs a pooled regression, namely: 𝑒𝑖𝑡 = 𝜌𝑒𝑖𝑡−1 + 𝑣𝑖𝑡 or another version, that is 𝑒𝑖𝑡 = �̅�𝑒𝑖𝑡−1 + ∑ 𝜓δ𝑒𝑖𝑡−𝑗 𝑝 𝑗=1 + 𝑣𝑖𝑡 (11) under the null hypothesis: no cointegration occurs, kao shows the following statistics [10]: 𝐷𝐹𝜌 = 𝑇√𝑁(�̂� − 1) + 3√𝑁 √10.2 (12) jurnal matematika mantik vol 7, no 2, october 2021, pp. 132-139 136 𝐷𝐹𝑡 = √10.2𝑡𝜌 + √1.875𝑁 (13) 𝐷𝐹𝜌 ∗ = 𝑇√𝑁(�̂� − 1) + 3 �̂�𝑣 2 �̂�0𝑣 2⁄ √𝑁 √3 + 36 �̂�𝑣 4 (5�̂�0𝑣 4 )⁄ (14) 𝐷𝐹𝑡 ∗ = 𝑡𝜌 + √6𝑁/2�̂�0𝑣 √ �̂�0𝑣 2 2�̂�𝑣 2 + 3�̂�𝑣 2 10�̂�0𝑣 2 (15) for 𝑝 > 0, then 𝐴𝐷𝐹 = 𝑡𝜌 + √6𝑁/2�̂�0𝑣 √ �̂�0𝑣 2 2�̂�𝑣 2 + 3�̂�𝑣 2 10�̂�0𝑣 2 (16) converts asymptotically towards 𝑁(0,1) , where the estimated variance is �̂�𝑣 2 = �̂�𝑢 2 − �̂�𝑢𝜖 2 �̂�𝜖 2, and the long run variance estimate is �̂�0𝑣 2 = �̂�0𝑢 2 − �̂�0𝑢𝜖 2 �̂�0𝜖 2 covariance of 𝑤𝑖𝑡 = [ 𝑢𝑖𝑡 𝜖𝑖𝑡 ] estimated as follows: σ̂ = [ �̂�𝑢 2 �̂�𝑢𝜖 �̂�𝑢𝜖 �̂�𝜖 2 ] = 1 𝑁𝑇 ∑ ∑ �̂�𝑖𝑡 �̂�𝑖𝑡 ′ 𝑇 𝑡=1 𝑁 𝑖=1 (17) the long run covariance is estimated with the kernel estimator, namely: ω̂ = [ �̂�0𝑢 2 �̂�0𝑢𝜖 �̂�0𝑢𝜖 �̂�0𝜖 2 ] = 1 𝑁 ∑ [ 1 𝑇 ∑ �̂�𝑖𝑡 �̂�𝑖𝑡 ′ 𝑇 𝑡=1 + 1 𝑇 ∑ 𝜅(𝜏 𝑏⁄ ) ∑ �̂�𝑖𝑡 �̂�𝑖𝑡−𝜏 ′ + �̂�𝑖𝑡−𝜏�̂�𝑖𝑡 ′ 𝑇 𝑡=𝜏+1 ∞ 𝜏=1 ] 𝑁 𝑖=1 (18) where 𝜅(. )is the kernel function and 𝑏 is the bandwidth. 2.4. granger causality test with toda-yamamoto procedure the var (𝑝) of the panel data: 𝑌𝑖,𝑡 = 𝑎0 + 𝑎1,𝑖 𝑌𝑖,𝑡−1 + 𝑎2,𝑖 𝑌𝑖,𝑡−2 + ⋯ + 𝑎𝑝,𝑖 𝑌𝑖,𝑡−𝑝 + 𝑏1,𝑖 𝑋𝑖,𝑡−1 + 𝑏2,𝑖 𝑋𝑖,𝑡−2 + ⋯ + 𝑏𝑝,𝑖 𝑋𝑖,𝑡−𝑝 + 𝑢𝑖,𝑡 (19) 𝑋𝑖,𝑡 = 𝑐0 + 𝑐1,𝑖 𝑌𝑖,𝑡−1 + 𝑐2,𝑖 𝑌𝑖,𝑡−2 + ⋯ + 𝑐𝑝,𝑖 𝑌𝑖,𝑡−𝑝 + 𝑑1,𝑖 𝑋𝑖,𝑡−1 + 𝑑2,𝑖 𝑋𝑖,𝑡−2 + ⋯ + 𝑑𝑝,𝑖 𝑋𝑖,𝑡−𝑝 + 𝑣𝑖,𝑡 (20) then we test it with: 𝐻0: 𝑏1 = 𝑏2 = ⋯ = 𝑏𝑝 = 0 versus 𝐻1: 𝑛𝑜𝑡 𝑎𝑙𝑙 𝑏𝑖 = 0 and muhammad fajar, yuyun guna winarti the testing of validity wagner’s law in papua province 137 𝐻0: 𝑑1 = 𝑑2 = ⋯ = 𝑑𝑝 = 0 versus 𝐻1: 𝑛𝑜𝑡 𝑎𝑙𝑙 𝑑𝑖 = 0 toda-yamamoto procedures [12]: 1. checking the root unit on the time series data. 2. find the maximum integration order (𝑚) in the time series data group. if there are two data series, one series integrates into i(1) and the other into i(2). then, the maximum order of integration is 𝑚 = 2, and so on. 3. create a var(𝑝) model based on level data. 4. determine the optimum lag on var, namely p, with the information criterion with sic. 5. make a var (𝑝 + 𝑚) model based on the level data. perform the granger test as usual by producing the wald test statistics which is asymptotically distributed as chi-squared with degrees of freedom p. 3. result and discusions 3.1. unit root table 1. unit root hadri test results (constants and trends) on panel data method lgrdp_real lgov_real level 1st difference level 1st difference hadri z-stat 10.256 8.926 2.558 6.992 p-value: 0.000 p-value: 0.000 p-value: 0.000 p-value: 0.000 heteroscedastic consistent z-stat 10.312 13.845 2.558 6.992 p-value: 0.000 p-value: 0.000 p-value: 0.0000 p-value: 0.000 note: lgrdp_real is the natural logarithm of real grdp lgov_real is the natural logarithm of municipality/regency government real expenditure based on the results of hadri's test on the unit root on panel data with a significance level of five percent, it shows that the lpdrb_real and lgov_real series are stationary integrated in the same order, namely i(1). it means that these two variables can be analyzed for cointegration and causality testing 3.2 cointegration test based on the results of the kao cointegration test, the adf value was -4.480 with a p-value of 0.00. so, it can be concluded that there is cointegration between real grdp and real expenditure of the municipality government at the five percent significance level. table 2. kao cointegration test results adf residual variance hac variance t-statistic: -4.480 p-value : 0.000 0.012 0.0189 3.3 causality test before the causality test, sic is used to determine the lag so that the optimum lag for the test is 8, then the order of integration between lgrdp_real and lgov_real (i(1)) is 1, so the lag used in granger causality testing with the toda yamamoto procedure is 9. based on wagner's law, the direction of causality must be from regional output to local jurnal matematika mantik vol 7, no 2, october 2021, pp. 132-139 138 government expenditure. but if we look at the results of the granger causality test in table 3 with the toda yamamoto procedure, it turns out that at the five percent significance level, it shows that real government expenditure causes real grdp. based on these results, wagner's law is not proven in the case of the economy of papua province. it means that wagner's law does not apply universally to economic activity for all regions/countries. table 3. granger causality test results hypothesis null chi-squared statistic degree of freedom p-value lgrdp_real does not granger cause lgov_real 8.821 9 0.454 lgov_real does not granger cause lgrdp_real 22.017 9 0.009 this results are consistence with lall [1], landau [2], and kormendi and meguire [4]. the cause of the absence of wagner's law through the specification of the model used is that the contribution of household consumption expenditure is more than 50% to grdp, compared to government expenditure, which has a little share. so that, grdp and regional economic growth are still dominantly supported by household consumption. 4. conclusion based on the results of the discussion in the previous chapter, wagner's law does not exist in the economic mechanism of papua province. it is indicated from government expenditure granger causes grdp, and not vice versa. even though there is cointegration between government expenditure and grdp, it can be explained because wagner's law is not universal for all countries/regions. the cause of the absence of wagner's law through the specification of the model used is that the contribution of household consumption expenditure is more than 50% to grdp, compared to government expenditure, which has a little share. so that, grdp and regional economic growth are still dominantly supported by household consumption. for further research, model specifications beside peacock-wiseman can be used to test wagner's law and in terms of analytical tools, you can use non-parametric cointegration tests, so that a broader spectrum of conclusions can be obtained. for stakeholders in papua, government expenditure can increase economic growth and people's welfare. if government expenditure is used on a basic economic sector, that is under regional potential and more equitable development of community infrastructure, as well as tighter supervision of budget, use to prevent corruption. references [1] s. lall, “a note on government expenditures in developing countries,” the economic journal 79, pp. 413-417, 1969. [2] d. landau, “government expenditure and economic growth: a cross-country study,” southern economic journal, vol. 49, no. 3, pp. 783-792, 1983. [3] b. singh, b.s. sahni, “causality between public expenditure and national income,” the review of economics and statistics, vol. 66, pp. 630-644, 1984. [4] r. kormendi, p. meguire, "macroeconomics determinants of growth: crosscountry evidence," journal of monetary economics, 16, 4, 141-163, 1985. [5] r. j. barro, “government spending in a simple model of endogenous growth," journal of political economy, 98 (5), pp. s103-5126. 1990. [6] m.i., ansari, d.v. gordon, c. akuamoach, “keynes versus wagner: public expenditure and national income for three african countries,” applied economics, muhammad fajar, yuyun guna winarti the testing of validity wagner’s law in papua province 139 29, pp. 543-550, 1997. [7] a.m. islam, “wagner’s law revisited: cointegration and exogeneity tests for the usa,” applied economics letters, 8, pp. 509-515, 2001. [8] h. rafiee, m. zibaii, “government size, economic growth and labor productivity in agricultural sector seasonal,” journal of agriculture economic development. 43-44, pp. 75-88, 2003. [9] k. hadri, “testing for stationarity in heterogeneous panels,” the econometrics journal 3, pp. 148-161, 2000. [10] c. d. kao, “spurious regression and residual-based tests for cointegration in panel data,” journal of econometrics 90, pp. 1–44, 1999. [11] p. pedroni, “panel cointegration; asymptotic and finite sample properties of pooled time series tests with an application to the ppp hypothesis,” econometric theory, 20, pp. 597–625, 2004. [12] h.y. toda, t. yamamoto,” statistical inferences in vector autoregressions with possibly integrated processes”, journal of econometrics, 66, pp. 225-250, 1995. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: fitriani, fitriani.1984@fmipa.unila.ac.id department of mathematics, universitas lampung, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.45-52 the implementation of rough set on a group structure ananto adi nugraha1, fitriani1*, muslim ansori1, ahmad faisol1 1department of mathematics, universitas lampung, indonesia article history: received jul 6, 2021 revised, may 9, 2022 accepted, may 31, 2022 kata kunci: aproksimasi bawah, aproksimasi atas, himpunan rough, grup rough, sentralizer abstrak. diberikan himpunan tak kosong u dan relasi ekuivalensi r pada u. pasangan berurut (𝑈, 𝑅) disebut ruang aproksimasi. relasi ekuivalensi pada u membentuk kelas-kelas ekuivalensi yang saling asing. jika 𝑋 ⊆ 𝑈, maka dapat dibentuk aproksimasi bawah dan aproksimasi atas dari 𝑋. pada penelitian ini dikonstruksi grup rough, subgrup rough pada ruang aproksimasi (𝑈, 𝑅) terhadap operasi biner yang bersifat komutatif maupun non-komutatif. keywords: lower approximation, upper approximation, rough set, rough group, centralizer. abstract. let 𝑈 be a non-empty set and 𝑅 an equivalence relation on 𝑈. then, (𝑈, 𝑅) is an approximation space. the equivalence relation on 𝑈 forms disjoint equivalence classes. if 𝑋 ⊆ 𝑈, we can form a lower approximation and an upper approximation of 𝑋. if x⊆u, then we can form a lower approximation and an upper approximation of x. in this research, rough group and rough subgroups are constructed in the approximation space (𝑈, 𝑅) for commutative and non-commutative binary operations. how to cite: a. a. nugraha, fitriani, m. ansori, and a. faisol, “the implementation of rough set on a group structure”, j. mat. mantik, vol. 8, no. 1, pp. 19-26, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.45-52 issn: 2527-3159 (print) 2527-3167 (online) mailto:fitriani.1984@fmipa.unila.ac.id https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 1, june 2022, pp.45-52 46 1. introduction zdzislaw pawlak [1] first introduced the rough set theory in 1982 as a mathematical technique to deal with vagueness and uncertainty problems. various studies have discussed this theory and the possibility of its applications, for example, in data mining [2] and some algebraic structures. in [3], biswaz and nanda introduce the rough group and rough ring. furthermore, miao et al. [4] improve definitions of a rough group and rough subgroup and prove their new properties. in [5], jesmalar investigates the homomorphism and isomorphism of the rough group. furthermore, in [6], bagirmaz and ozcan give the concept of rough semigroups on approximation spaces. then, kuroki in [7] gives some results about the rough ideal of semigroups. in [8], davvaz investigates roughness in the ring, and in [9], davvaz and mahdavipour give a roughness in modules. in [10], isaac and neelima introduce the concept of the rough ideal. moreover, in [11], zhang et al. give some properties of rough modules. davvaz and malekzadeh give roughness in modules [12]. they use the notion of reference points. furthermore, ozturk and eren give the multiplicative rough modules [13]. then, sinha and prakash introduce the rough exact sequence of rough modules [14]. they also give the injective module based on rough set theory [15]. in [16], kazancı and davvaz give the rough prime in a ring. jun in [17] investigate the roughness of ideals in bck-algebras. moreover, dubois and prade [18] define the rough fuzzy sets. this research focuses on the algebraic aspects by applying a rough set theory to construct a rough group and its subgroups on an approximation space. moreover, in this research, we discuss the centralizer and the center of a rough group. 2. preliminaries in this section, there will be several definitions and theorems that can be helpful for this article. those definitions are written as follows: definition 1 [19] define 𝐶𝐺 (𝐴) = {𝑔 ∈ 𝐺 | 𝑔𝑎𝑔 −1 = 𝑎 for all 𝑎 ∈ 𝐴}. this subset of 𝐺 is called the centralizer of 𝐴 in 𝐺. since 𝑔𝑎𝑔−1 = 𝑎 if and only if 𝑔𝑎 = 𝑎𝑔, 𝐶𝐺 (𝐴) is the set of elements of 𝐺 which commute with every element of 𝐴. definition 2 [19] define 𝑍(𝐺) = {𝑔 ∈ 𝐺 | 𝑔𝑥 = 𝑥𝑔 for all 𝑥 ∈ 𝐺}, the set of elements commuting with all the elements of 𝐺. this subset of 𝐺 is called the center of 𝐺. definition 3 [20] let 𝑅 be an equivalence relation on 𝐴 and 𝑎 ∈ 𝐴. then the equivalence class of 𝑎 under 𝑅 is [𝑎]𝑅 = {𝑥 ∶ 𝑥 ∈ 𝐴 and 𝑎𝑅𝑥}. in other words, the equivalence class of 𝑎 under 𝑅 contains all the elements in 𝐴 to which 𝑎 is related by 𝑅. definition 4 [3] let (𝑈, 𝑅) be an approximation space and 𝑋 be a subset of 𝑈, the sets, 𝑋 = {𝑥 | [𝑥]𝑅 ∩ 𝑋 ≠ ∅} (1) 𝑋 = {𝑥 | [𝑥]𝑅 ⊆ 𝑋} (2) are called upper approximation and lower approximation of 𝑋. definition 5 [1] let 𝑅 be an equivalence relation on universe set 𝑈, a pair (𝑈, 𝑅) is called an approximation space. a subset 𝑋 ⊆ 𝑈 can be defined if 𝑋 = 𝑋, in the opposite case, if 𝑋 − 𝑋 ≠ ∅ then 𝑋 is called a rough set. a. a. nugraha, fitriani, m. ansori, and a. faisol the implementation of rough set on a group structure 47 definition 6 [3] let 𝐾 = (𝑈, 𝑅) be an approximation space and ∗ be a binary operation defined on 𝑈. a subset 𝐺 of universe 𝑈 is called a rough group if the following properties are satisfied: i. ∀𝑥, 𝑦 ∈ 𝐺, 𝑥 ∗ 𝑦 ∈ 𝐺; ii. association property holds in 𝐺; iii. ∃𝑒 ∈ 𝐺 such that ∀𝑥 ∈ 𝐺, 𝑥 ∗ 𝑒 = 𝑒 ∗ 𝑥 = 𝑥; 𝑒 is called the rough identity element of 𝐺; iv. ∀𝑥 ∈ 𝐺, ∃𝑦 ∈ 𝐺 such that 𝑥 ∗ 𝑦 = 𝑦 ∗ 𝑥 = 𝑒; 𝑦 is called the rough inverse element of 𝑥 in 𝐺. we will give the example of rough group in section 3. the following theorem gives the characteristics of a rough group. theorem 1. [3] a necessary and sufficient condition for a subset 𝐻 of rough group 𝐺 to be a rough subgroup is that: (i) ∀𝑥, 𝑦 ∈ 𝐻, 𝑥 ∗ 𝑦 ∈ 𝐻; (ii) ∀𝑥 ∈ 𝐻, 𝑥−1 ∈ 𝐻. several steps will be taken to achieve the objectives of this research. those steps are written as follows: 1. determine a set 𝑈, where 𝑈 ≠ ∅. 2. define a relation 𝑅 on 𝑈. 3. shows that a relation 𝑅 is the equivalence relation on 𝑈. 4. determine equivalence classes on 𝑈. 5. determine a set 𝐺, where 𝐺 ⊆ 𝑈 and 𝐺 ≠ ∅. 6. determine the approximation space, lower approximation on 𝐺 (𝐺), and upper approximation on 𝐺 (𝐺). 7. determine a rough set 𝐴𝑝𝑟(𝐺) = (𝐺, 𝐺). 8. determine a binary operation ∗ on the set 𝐺. 9. shows that 〈𝐺,∗〉 is a rough group in the approximation space that has been constructed. 10. determine a rough subgroup 〈𝐻,∗〉 from a rough group 〈𝐺,∗〉. 3. rough group construction 3.1 commutative rough group construction in this section, we will give the construction of commutative rough group. example 3.1. given a non-empty set 𝑈 = {0,1,2,3, … ,99}. we define a relation 𝑅 on the set 𝑈, that is, for every 𝑎, 𝑏 ∈ 𝑈 apply 𝑎𝑅𝑏 if and only if 𝑎 − 𝑏 = 7𝑘 where 𝑘 ∈ ℤ. furthermore, it can be shown that relation 𝑅 is reflexive, symmetrical, and transitive. so, relation 𝑅 is an equivalence relation on 𝑈. as a result, relation 𝑅 produces some disjoint partitions called equivalence classes. the equivalence classes are written as follows: 𝐸1 = [1] = {1,8,15,22,29,36,43,50,57,64,71,78,85,92,99}; 𝐸2 = [2] = {2,9,16,23,30,37,44,51,58,65,72,79,86,93}; 𝐸3 = [3] = {3,10,17,24,31,38,45,52,59,66,73,80,87,94}; 𝐸4 = [4] = {4,11,18,25,32,39,46,53,60,67,74,81,88,95}; 𝐸5 = [5] = {5,12,19,26,33,40,47,54,61,68,75,82,89,96}; jurnal matematika mantik vol. 8, no. 1, june 2022, pp.45-52 48 𝐸6 = [6] = {6,13,20,27,34,41,48,55,62,69,76,83,90,97}; 𝐸7 = [0] = {0,7,14,21,28,35,42,49,56,63,70,77,84,91,98}. given a non-empty subset 𝑋 ⊆ 𝑈 that is 𝑋 = {10,20,30,40,50,60,70,80,90}. because the set 𝑈 ≠ ∅ and 𝑅 is an equivalence relation on 𝑈, a pair (𝑈, 𝑅) is the approximation space. furthermore, it can be obtained the lower approximation and upper approximation of 𝑋, that is: 𝑋 = ∅. 𝑋 = 𝐸1 ∪ 𝐸2 ∪ 𝐸3 ∪ 𝐸4 ∪ 𝐸5 ∪ 𝐸6 ∪ 𝐸7 = 𝑈. after determining the lower approximation and upper approximation of 𝑋, then given a binary operation +100 on 𝑋. here is given table cayley of 𝑋 with the operation +100. table 1. table cayley of 𝑋 with the operation +100 +𝟏𝟎𝟎 𝟏𝟎 𝟐𝟎 𝟑𝟎 𝟒𝟎 𝟓𝟎 𝟔𝟎 𝟕𝟎 𝟖𝟎 𝟗𝟎 𝟏𝟎 20 30 40 50 60 70 80 90 0 𝟐𝟎 30 40 50 60 70 80 90 0 10 𝟑𝟎 40 50 60 70 80 90 0 10 20 𝟒𝟎 50 60 70 80 90 0 10 20 30 𝟓𝟎 60 70 80 90 0 10 20 30 40 𝟔𝟎 70 80 90 0 10 20 30 40 50 𝟕𝟎 80 90 0 10 20 30 40 50 60 𝟖𝟎 90 0 10 20 30 40 50 60 70 𝟗𝟎 0 10 20 30 40 50 60 70 80 i. based on table 1, it is proved that for each 𝑥, 𝑦 ∈ 𝑋, apply 𝑥(+100)𝑦 ∈ 𝑋. ii. for each 𝑥, 𝑦, 𝑧 ∈ 𝑋, the associative property that is (𝑥(+100)𝑦)(+100)𝑧 = 𝑥(+100)(𝑦(+100)𝑧) holds in 𝑋. the operation +100 is associative in 𝑋. iii. there is a rough identity element 𝑒 ∈ 𝑋 that is 0 ∈ 𝑋 such that for each 𝑥 ∈ 𝑋, 𝑥(+100)𝑒 = 𝑒(+100)𝑥 = 𝑥. table 2. table of element inverse of the set 𝑋 𝒙 10 20 30 40 50 60 70 80 90 𝒙−𝟏 90 80 70 60 50 40 30 20 10 iv. for each 𝑥 ∈ 𝑋, there is a rough inverse element of 𝑥 that is 𝑥−1 ∈ 𝑋 such that 𝑥(+100)𝑥 −1 = 𝑥−1(+100)𝑥 = 𝑒. based on table 2, it can be seen that each element 𝑥 in the set 𝑋, then the inverse element 𝑥−1 is also in 𝑋. since those four conditions have been satisfied, then 〈𝑋, +100〉 is a rough group. 3.2 non-commutative rough group construction in this section, we will give the construction of non-commutative rough group. example 3.2. given a permutation group 𝑆3 to the operation of permutation multiplication " ∘. " for example, take a subgroup 𝐺 = {(1), (12)} of the group 𝑆3. for 𝑥, 𝑦 ∈ 𝑆3, define a relation 𝑅 that is 𝑥𝑅𝑦 if and only if 𝑥 ∘ 𝑦 −1 ∈ 𝐺. furthermore, it can be shown that relation 𝑅 is reflexive, symmetrical, and transitive. so, relation 𝑅 is an equivalence relation on 𝑆3. as a result, relation 𝑅 produces some disjoint partitions called equivalence classes. suppose 𝑎 is the element in 𝑆3, the equivalence class containing 𝑎 defined as follows: [𝑎]𝑅 = {𝑥 ∈ 𝑆3 | 𝑥𝑅𝑎} = {𝑥 ∈ 𝑆3 | 𝑥 ∘ 𝑎 −1 ∈ 𝐺} a. a. nugraha, fitriani, m. ansori, and a. faisol the implementation of rough set on a group structure 49 = {𝑥 ∈ 𝑆3 | 𝑥 ∘ 𝑎 −1 = 𝑔, 𝑔 ∈ 𝐺} = {𝑥 ∈ 𝑆3 | 𝑥 = 𝑔 ∘ 𝑎, 𝑔 ∈ 𝐺} = {𝑔 ∘ 𝑎 | 𝑔 ∈ 𝐺} (3) based on the equation (3), this is corresponding to the definition of the right coset of 𝐺 in 𝑆3 that is 𝐺𝑎 = {𝑔 ∘ 𝑎 | 𝑔 ∈ 𝐺}. thus, the right cosets of 𝐺 in 𝑆3 as follows: 𝐺 ∘ (1) = 𝐺 ∘ (12) = {(1), (1 2)}; 𝐺 ∘ (1 3) = 𝐺 ∘ (1 2 3) = {(1 3), (1 2 3)}; 𝐺 ∘ (2 3) = 𝐺 ∘ (1 3 2) = {(2 3), (1 3 2)}. given a non-empty subset 𝑌 ⊆ 𝑆3 that is 𝑌 = {(1), (1 2), (1 2 3), (1 3 2)}. furthermore, it can be obtained the lower approximation and upper approximation of 𝑌, that is: 𝑌 = {(1), (1 2)}. 𝑌 = {(1), (1 2)} ∪ {(1 3), (1 2 3)} ∪ {(2 3), (1 3 2)} = 𝑆3. after determining the lower approximation and upper approximation of 𝑌, then we give a permutation multiplication " ∘ " on 𝑌. we give a table cayley of 𝑌 with the operation of permutation multiplication as follows. table 3. table cayley of 𝑌 with the operation of permutation multiplication ∘ (1) (1 2) (1 2 3) (1 3 2) (1) (1) (1 2) (1 2 3) (1 3 2) (1 2) (1 2) (1) (2 3) (1 3) (1 2 3) (1 2 3) (1 3) (1 3 2) (1) (1 3 2) (1 3 2) (2 3) (1) (1 2 3) i. based on table 3, it is proved that for each 𝑥, 𝑦 ∈ 𝑌, apply 𝑥 ∘ 𝑦 ∈ 𝑌. ii. for each 𝑥, 𝑦, 𝑧 ∈ 𝑌, the associative property that is (𝑥 ∘ 𝑦) ∘ 𝑧 = 𝑥 ∘ (𝑦 ∘ 𝑧) holds in 𝑌. the operation ∘ is associative in 𝑌. iii. there is a rough identity element 𝑒 ∈ 𝑌 that is (1) ∈ 𝑌 such that for each 𝑦 ∈ 𝑌, 𝑦 ∘ 𝑒 = 𝑒 ∘ 𝑦 = 𝑦. table 4. table of inverse element of 𝑌 𝒚 (1) (1 2) (1 2 3) (1 3 2) 𝒚−𝟏 (1) (1 2) (1 3 2) (1 2 3) iv. for each 𝑦 ∈ 𝑌, there is a rough inverse element of 𝑦 that is 𝑦−1 ∈ 𝑌 such that 𝑦 ∘ 𝑦−1 = 𝑦−1 ∘ 𝑦 = 𝑒. based on table 4, it can be seen that each element 𝑦 in the set 𝑌, then the inverse element 𝑦−1 is also in the set 𝑌. since those four conditions have been satisfied, then 〈𝑌,∘〉 is a rough group. 4. subgroup construction of the rough group after constructing a commutative rough group and a non-commutative rough group, we will construct subgroups of each of the previously constructed rough groups. 4.1 subgroup construction of commutative rough group before it has been obtained, a commutative rough group 𝑋 with the operation " +100 ". furthermore, we will construct several subgroups that can be formed from the rough group 𝑋. based on theorem 1, we can obtain several subgroups from the rough group 𝑋 that written as follows: jurnal matematika mantik vol. 8, no. 1, june 2022, pp.45-52 50 1. 〈{20,30,40,50,60,70,80}, +100〉; 2. 〈𝑋, +100〉. after determining several subgroups from the rough group 𝑋 that is commutative, then we will determine the centralizer and the center of subgroups in rough group 𝑋. suppose all subgroups of rough group 𝑋 above are denoted by 𝐴. based on definition 1, the centralizer 𝐴 in 𝑋 is the set where is the element of 𝑋 is commutative with each element of 𝐴. here is given the table that shows the centralizer of subgroups 𝐴 in rough group 𝑋. table 5. table of the centralizer of subgroups 𝐴 in rough group 𝑋 𝑨 𝑪𝑿 (𝑨) {20,30,40,50,60,70,80} 𝑋 𝑋 = {10,20,30,40,50,60,70,80,90} 𝑋 since the operation +100 of rough group 𝑋 is commutative, the centralizer of subgroups in rough group 𝑋 is 𝑋 itself. based on definition 2, the center of 𝑋 is the set of elements that is commutative with all elements of 𝑋. because rough group 𝑋 using commutative operation, the center of rough group 𝑋 is 𝑋 itself, or it can be written as 𝑍(𝑋) = 𝑋. using theorem 1, we will show that the center of rough group 𝑋 that is 𝑍(𝑋) = 𝑋 is a rough subgroup of rough group 𝑋. i. based on table 1, it is proved that for each 𝑥, 𝑦 ∈ 𝑍(𝑋) = 𝑋, apply 𝑥(+100)𝑦 ∈ 𝑍(𝑋) = 𝑋 = 𝑈. ii. for each 𝑥 ∈ 𝑍(𝑋) = 𝑋, there is an inverse element of 𝑥 that is 𝑥−1 ∈ 𝑍(𝑋) = 𝑋. based on table 2, it can be seen that if each element 𝑥 in the set 𝑋 then the inverse element of 𝑥 also in the set 𝑋. two conditions on theorem 1 have been satisfied, so it is proved that the center of rough group 𝑋 that is 𝑍(𝑋) = 𝑋 is a rough subgroup of rough group 𝑋. 4.2 subgroup construction of non-commutative rough group before it has been obtained a non-commutative rough group 𝑌 with the operation of permutation multiplication " ∘. " furthermore, we will construct several subgroups that can be formed from the rough group 𝑌. based on theorem 1, we can obtain several subgroups from the rough group 𝑌 that written as follows: 1. 〈{(1)},∘〉; 2. 〈{(1), (1 2)},∘〉; 3. 〈{(1), (1 2 3), (1 3 2)},∘〉; 4. 〈{(1 2), (1 2 3), (1 3 2)},∘〉; 5. 〈𝑌,∘〉. after determining several subgroups from the rough set 𝑌 that are non-commutative, then we will determine the centralizer and the center of subgroups in rough group 𝑌. suppose all subgroups of rough group 𝑌 above are denoted by 𝐵. based on definition 1, the centralizer 𝐵 in 𝑌 is the set where is the element of 𝑌 is commutative with each element of 𝐵. here is given the table that shows the centralizer of subgroups 𝐵 in rough group 𝑌. table 6. table of the centralizer of subgroups 𝐵 in rough group 𝑌 a. a. nugraha, fitriani, m. ansori, and a. faisol the implementation of rough set on a group structure 51 𝑩 𝑪𝒀(𝑩) {(1)} 𝑌 {(1), (1 2)} {(1), (12)} {(1), (1 2 3), (1 3 2)} {(1)} {(1 2), (1 2 3), (1 3 2)} {(1)} 𝑌 = {(1), (1 2), (1 2 3), (1 3 2)} {(1)} based on definition 2, the center of 𝑌 is the set of elements that is commutative with all elements of 𝑌. from the definition 2, the center of rough group 𝑌 is an identity element, or it can be written as 𝑍(𝑌) = {(1)}. using theorem 1, we will show that the center of rough group 𝑌 that is 𝑍(𝑌) = {(1)} is a rough subgroup of rough group 𝑌. previously, determine the upper approximation of 𝑍(𝑌) that is 𝑍(𝑌) = {(1), (1 2)}. i. for (1) ∈ 𝑍(𝑌), apply (1) ∘ (1) = (1) ∈ 𝑍(𝑌). ii. for (1) ∈ 𝑍(𝑌), there is an inverse element of (1) that is (1) ∈ 𝑍(𝑌). based on theorem 1, because the two conditions have been satisfied, it is proved that the center of rough group 𝑌 that is 𝑍(𝑌) = {(1)} is a rough subgroup of rough group 𝑌. 5 conclusions based on the results, we construct a rough group, a rough subgroup in the case of the commutative and non-commutative binary operation. furthermore, the centralizer of a commutative rough subgroup is also a rough group. in comparison, the centralizer of the subgroup of a non-commutative rough group must contain the identity element and the center. the center of each rough group, both commutative and non-commutative, are subgroups of each rough group. references [1] z. pawlak, “rough sets,” int. j. comput. inf. sci., vol. 11, no. 5, pp. 341–356, 1982, doi: https://doi.org/10.1007/bf01001956. 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[20] w. barnier and n. feldman, introduction to advanced mathematics. new jersey: prentice hall, inc., 1990. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 59-66 contact: nihaya alivia coraima dewi, alivianihaya@gmail.com department of mathematics, universitas billfath, lamongan, jawa timur 62261, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.59-66 optimization of balanced menu for pregnant women in grobogan-central java using simplex method nihaya alivia coraima dewi1*, fitroh resmi2, pukky t. bantining ngastiti3 1,2,3department of mathematics, universitas billfath, lamongan, indonesia article history: received nov 9, 2020 revised may 27, 2021 accepted may 30, 2021 kata kunci: ibu hamil, metode simpleks, stunting abstrak. penelitian ini bertujuan untuk menentukan optimasi komposisi makanan seimbang bagi ibu hamil. penentuan optimasi makanan seimbang dilakukan dengan membentuk model linear beserta kondisi batas dan fungsi tujuan, serta menginputkan data usia ibu hamil, usia kandungan, dan kebutuhan nutrisi ibu. kemudian dilakukan perhitungan dengan metode simpleks sehingga diperoleh berat bahan makanan yang harus dikonsumsi untuk mendapatkan gizi yang seimbang yaitu dengan 75 kombinasi yang telah dianalisis terhadap kelompok ibu hamil usia 19-29 tahun dan usia 30-49 tahun pada tiga trimester, meliputi jenis makanan pokok, sayur (bayam, sawi hijau, kembang kol, kangkung, wortel), buah, lauk pauk, kacang-kacangan, gula dan susu dengan angka kecukupan gizi yang dianjurkan untuk kandungan data angka kecukupan air, energi, protein, lemak, karbohidrat (kh), serat, vitamin a, b1, b2, b3 dan vitamin c. terhadap kelompok ibu hamil usia 19-29 tahun dan usia 30-49 tahun pada tiga trimester, diperoleh bahwa kombinasi 55 adalah kombinasi yang optimal dengan bahan makanan beras, kangkung, semangka, dan tahu. keywords: pregnant women, simplex method, stunting abstract. this study aims to determine the optimization of balanced dietary composition for pregnant women. determination of the optimization of balanced food is carried out by forming a linear model along with boundary conditions and objective functions, as well as inputting data on the age of pregnant women, age of pregnancy and maternal nutritional needs, then the calculation is carried out using the simplex method in order to obtain the weight of food ingredients that must be consumed to get a balanced nutrition, namely with 75 combinations that have been analyzed on groups of pregnant women aged 19-29 years and 30-49 years in three trimesters, including staple foods, vegetables (spinach, green mustard, cauliflower, kale, carrots), fruit, side dishes vegetables, nuts, sugar and milk with the recommended nutritional adequacy rate for the data content of water, energy, protein, fat, carbohydrate (kh), fiber, vitamin a, b1, b2, b3 and vitamin c. in the group of pregnant women aged 19-29 years and women aged 30-49 years in the three trimesters, it was found that the combination of 55 was the optimal combination with rice, kale, watermelon, and tofu. how to cite: n. a. c. dewi, f. resmi, and p. t. b. ngastiti, “optimization of balanced menu for pregnant women in grobogan-central java using simplex method”, j. mat. mantik, vol. 7, no. 1, pp. 5966, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 59-66 issn: 2527-3159 (print) 2527-3167 (online) mailto:alivianihaya@gmail.com https://doi.org/10.15642/mantik.2021.7.1.59-66 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp.59-66 60 1. introduction stunting is one of the nutritional problems experienced by toddlers in the world today. indonesia is included in the third country with the highest prevalence, in the southeast asia region. the average prevalence of toddler stunting in indonesia from 2005 to 2017 was 36.4%. the nutritional condition of the mother before and during pregnancy and after childbirth affects fetal growth and the risk of stunting. other factors in the mother's influence are the mother's posture (short), the distance of pregnancy that is too close, the mother who is still a teenager, as well as the lack of nutrient intake at the time of pregnancy. the government designated 1,000 villages a priority for stunting interventions located in 100 districts/cities and 34 provinces. one of the 100 districts/cities that the government prioritizes in stunting is grobogan regency, central java province [2]. the nutritional needs of pregnant women are very important for the growth and development of the fetus, therefore the pregnant woman must be sufficient to intake her nutrients so that the fetus can develop normally, malnutrition in pregnant women will result in chronic energy deficiency (kek) of the case itself due to a lack of food intake that is not in accordance with the needs of the food intake of pregnant women this results in the development of the fetus is hampered. data collection is done with field surveys to get data on the market price of food that pregnant women can consume. determination of balanced foods by forming linear models along with the conditions of the limits and functions of the destination, as well as the nutritional needs of pregnant women, then the data of food ingredients, nutritional content, and the price of ingredients, then done calculations with simplex method so that obtained the weight of food ingredients that must be consumed to get balanced nutrition. research on the optimization of nutrition for pregnant women in each trimester has been carried out in ampana tete district, tojo una-una district, central sulawesi province, and in this study will be investigated more deeply regarding the optimization of nutrition in pregnant women in each trimester according to the age category located in grobogan regency. this research aims to determine the optimization of the composition of balanced foods for pregnant women using simplex method by considering the price of groceries so that there will be a minimum cost in grobogan regency. course composition that fulfills nutrition for pregnant women in the 1st, 2nd, and 3rd trimesters based on age categories with minimum costs can be solved using a linear programming model. the simplex method can be used in linear programming which functions to find the optimum solution. 2. research methods the research began by conducting preliminary studies to observe stunting problems and conclude mathematical methods used for problem solving. furthermore, literature studies are conducted to study the theories used to achieve research objectives. identification of data needs required during research, then data collection is done with field surveys to obtain data on the market price of food that pregnant women can consume. to determine the optimization of balanced foods by input data on the nutritional adequacy of pregnant women according to the age group and trimester of pregnant women, then food data in the form of nutritional content and the price of ingredients, then done calculations with simplex method so that it obtained the weight of food ingredients that should be consumed to get balanced nutrition. based on the results of determining health status and calculation of food composition can be used as a discussion to establish conclusions. nihaya alivia coraima dewi, fitroh resmi, pukky tetralian bantining ngastiti optimization of balanced menu for pregnant women in grobogan-central java using simplex method 61 𝑍𝑚𝑖𝑛 = 𝑐1𝑥1 + 𝑐2𝑥2 + 𝑐3𝑥3 + 𝑐4𝑥4 + 𝑐5𝑥5 + 𝑐6𝑥6 + 𝑐7𝑥7 (1) constraints: 𝑎11𝑥1 + 𝑎12𝑥2 + 𝑎13𝑥3 + 𝑎14𝑥4 + 𝑎15𝑥5 + 𝑎16𝑥6 + 𝑎17𝑥7 ≥ 𝑏1 (2) 𝑎21𝑥1 + 𝑎22𝑥2 + 𝑎23𝑥3 + 𝑎24𝑥4 + 𝑎25𝑥5 + 𝑎26𝑥6 + 𝑎27𝑥7 ≥ 𝑏2 (3) 𝑎31𝑥1 + 𝑎32𝑥2 + 𝑎33𝑥3 + 𝑎34𝑥4 + 𝑎35𝑥5 + 𝑎36𝑥6 + 𝑎37𝑥7 ≥ 𝑏3 (4) 𝑎41𝑥1 + 𝑎42𝑥2 + 𝑎43𝑥3 + 𝑎44𝑥4 + 𝑎45𝑥5 + 𝑎46𝑥6 + 𝑎47𝑥7 ≥ 𝑏4 (5) 𝑎51𝑥1 + 𝑎52𝑥2 + 𝑎53𝑥3 + 𝑎54𝑥4 + 𝑎55𝑥5 + 𝑎56𝑥6 + 𝑎57𝑥7 ≥ 𝑏5 (6) 𝑎61𝑥1 + 𝑎62𝑥2 + 𝑎63𝑥3 + 𝑎64𝑥4 + 𝑎65𝑥5 + 𝑎66𝑥6 + 𝑎67𝑥7 ≥ 𝑏6 (7) 𝑎71𝑥1 + 𝑎72𝑥2 + 𝑎73𝑥3 + 𝑎74𝑥4 + 𝑎75𝑥5 + 𝑎76𝑥6 + 𝑎77𝑥7 ≥ 𝑏7 (8) 𝑎81𝑥1 + 𝑎82𝑥2 + 𝑎83𝑥3 + 𝑎84𝑥4 + 𝑎85𝑥5 + 𝑎86𝑥6 + 𝑎87𝑥7 ≥ 𝑏8 (9) 𝑎91𝑥1 + 𝑎92𝑥2 + 𝑎93𝑥3 + 𝑎94𝑥4 + 𝑎95𝑥5 + 𝑎96𝑥6 + 𝑎97𝑥7 ≥ 𝑏9 (10) 𝑎101𝑥1 + 𝑎102𝑥2 + 𝑎103𝑥3 + 𝑎104𝑥4 + 𝑎105𝑥5 + 𝑎106𝑥6 + 𝑎107𝑥7 ≥ 𝑏10 (11) 𝑎111𝑥1 + 𝑎112𝑥2 + 𝑎113𝑥3 + 𝑎114𝑥4 + 𝑎115𝑥5 + 𝑎116𝑥6 + 𝑎117𝑥7 ≥ 𝑏11 (12) where 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥7 ≥ 0 (13) details: 𝑎1𝑖 is the amount of water in the food. 𝑎2𝑖 is the amount of energy in the food. 𝑎3𝑖 is the amount of protein in the food. 𝑎4𝑖 is the amount of fat in the food. 𝑎5𝑖 is the amount of carbohydrate in the food. 𝑎6𝑖 is the amount of fiber in the food. 𝑎7𝑖 is the amount of vitamin a in the food. 𝑎8𝑖 is the amount of vitamin b1 in the food. 𝑎9𝑖 is the amount of vitamin b2 in the food. 𝑎10𝑖 is the amount of vitamin b3 in the food. 𝑎11𝑖 is the amount of vitamin c in the food. where 𝑖 = 1,2,3,4,5,6,7 is sequence of food ingredient. 2.1 data collection data collection is done in tanggungharjo market, grobogan regency of central java to get the price data of groceries to be researched. the food ingredients surveyed included staple foods, vegetables, fruit, side dishes, nuts, sugar and milk. with the following data acquisition: tabel 1. list of prices of groceries on the market no. food price price no. food price price (rp/kg) (rp/g) (rp/kg) (rp/g) 1 rice 9.000 9 10 green beans 21.000 21 2 sugar 12.000 12 11 peanut 12.000 12 3 tofu 8.000 8 12 watermelon 6.000 6 4 tempe 16.000 16 13 banana 20.000 20 5 spinach 12.000 12 14 red dragon fruit 17.000 17 6 green mustard 9.000 9 15 starfruit 14.000 14 7 cabbage 16.000 16 16 apple 25.000 25 8 kale 7.500 7,5 17 milkfish 25.000 25 9 carrot 7.000 7 18 milk powder 100.000 100 based on the indonesian food composition table in 2017 published by the directorate general of public health, directorate of public nutrition, ministry of health of the republic of indonesia obtained food nutritional data for amount of water, energy, fat, carbohydrate (kh), fiber, vitamin a, vitamin b1, vitamin b2, vitamin b3 and vitamin jurnal matematika mantik vol. 7, no. 1, may 2021, pp.59-66 62 c as follows: table 2. indonesian food composition table (per 1 gram) no. food water (g) energy (kcal) protein (g) fat (g) kh (g) fiber (g) vit. a (mcg) vit. b1 (mg) vit. b2 (mg) vit. b3 (mg) vit. c (mg) 1 rice 0,120 3,57 0,084 0,017 0,771 0,002 0 0,0020 0,0008 0,026 0 2 sugar 0,054 3,94 0,0 0,0 0,94 0,0 0 0,00 0,00 0,0 0 3 tofu 0,822 0,80 0,109 0,047 0,008 0,001 1,18 0,0001 0,0008 0,001 4 tempe 0,683 1,50 0,14 0,077 0,091 0,014 0 0,0017 0,0044 0,036 5 spinach 0,945 0,16 0,09 0,004 0,029 0,007 22,93 0,0004 0,0010 0,001 0,41 6 green mustard 0,922 0,28 0,023 0,003 0,04 0,025 64,60 0,0009 0,0023 0,007 0,10 7 cabbage 0,917 0,25 0,024 0,002 0,049 0,016 0,9 0,0011 0,0009 0,006 0,69 8 kale 0,910 0,28 0,034 0,007 0,039 0,02 55,42 0,0007 0,0036 0,002 0,17 9 carrot 0,899 0,36 0,01 0,006 0,079 0,01 71,25 0,0004 0,0004 0,01 0,18 10 green bean 0,155 3,23 0,229 0,015 0,568 0,075 2,23 0,0046 0,0015 0,015 0,10 11 peanut 0,096 5,25 0,279 0,427 0,174 0,024 0,3 0,0044 0,0027 0,014 12 watermelon 0,921 0,28 0,005 0,002 0,069 0,004 5,90 0,0005 0,0005 0,003 0,06 13 banana 0,658 1,2 0,012 0,002 0,318 0,053 9,50 0,0006 0,0014 0,012 0,1 14 red dragon fruit 0,857 0,71 0,017 0,031 0,091 0,032 0 0,0050 0,0030 0,005 0,01 15 starfruit 0,90 0,36 0,004 0,004 0,088 0,032 1,70 0,0003 0,35 16 apple 0,841 0,58 0,003 0,004 0,149 0,026 0,9 0,0004 0,0003 0,001 0,05 17 milkfish 0,740 1,23 0,20 0,048 0,0 0,0 0,45 0,0005 0,0010 0,06 0 18 milk powder 0,035 5,13 0,246 0,30 0,362 0,0 4,76 0,0029 0,0139 0,016 0,06 the recommended nutritional adequacy figures for indonesians are contained in the minister of health regulation of the republic of indonesia no. 28 of 2019 on the recommended nutritional adequacy figures for indonesians, the data required are the age group 19-29 years and 30-49 years for the gender of women and pregnant with the division of trimester 1, trimester 2 and trimester 3. data on water adequacy, energy, protein, fat, carbohydrate (kh), fiber, vitamin a, vitamin b1, vitamin b2, vitamin b3, and vitamin c. table 3. recommended nutritional adequacy rate (per person per day) age group weight height water energy protein fat kh fiber vit a vit b1 vit b2 vit b3 vit c (kg) (cm) (ml) (kcal) (g) total (g) (g) (mcg) (mg) (mg) (mg) (mg) woman (y.o.) 19 29 55 159 2.350 2.250 60 65 360 32 600 1,1 1,1 14 75 30 49 56 158 2.350 2.150 60 60 340 30 600 1,1 1,1 14 75 pregnant trimester 1 +300 +180 +1 +2.3 +25 +3 +300 +0.3 +0.3 +4 +10 trimester 2 +300 +300 +10 +2.3 +40 +4 +300 +0.3 +0.3 +4 +10 trimester 3 +300 +300 +30 +2.3 +40 +4 +300 +0.3 +0.3 +4 +10 nihaya alivia coraima dewi, fitroh resmi, pukky tetralian bantining ngastiti optimization of balanced menu for pregnant women in grobogan-central java using simplex method 63 3. result and discussion the decision variables used include the types of staple foods, vegetables, fruit, side dishes, nuts, sugar, and milk with the following descriptions: 𝑥1 is a staple type of food in the combination of food (rice) 𝑥2 is a type of vegetable in a combination of foods (spinach, green mustard, cabbage, kale, carrots) 𝑥3 is a type of fruit in a combination of foods (watermelon, banana, red dragon fruit, starfruit, apple) 𝑥4 is a type of side dish in a combination of foods (tofu, tempe, fish) 𝑥5 is a type of nuts in a combination of foods (green beans, peanuts) 𝑥6 is sugar in a combination of foods (white sugar) 𝑥7 is milk in a combination of foods (milk powder) based on the food ingredients used are rice, spinach, green mustard, cabbage, kale, carrots, watermelon, banana, red dragon fruit, starfruit, apple, tofu, tempe, fish, green beans, peanuts, white sugar, and milk obtained probably 75 combinations of food ingredients that some will analyze. the simplex method used to solve the problem of balanced nutrition optimization in pregnant women with the age group 19-29 years and 30-49 years for the gender of women and pregnant with the division of trimester 1, trimester 2 and trimester 3 with minimum cost is shaped ≥ by variable amount 7 and constraints 11 namely water, energy, protein, fat, carbohydrate (kh), fiber, vitamin a, vitamin b1, vitamin b2, vitamin b3 and vitamin c in nutrient adequacy. to facilitate the above calculation can be completed using the help of winqsb program with the basic algorithm used in problem solving is simplex method with the following results: table 4. the calculation result of food combination of pregnant women 19-29 y.o. 1st trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 379,1835 1.595,97 319,6961 1.043,46 0 0 0 25.648,29 61 389,9379 1.958,05 0 999,2493 0 0 0 26.188,80 19 357,9672 1.238,59 522,1503 1.194,65 0 0 0 27.221,13 based on the calculation results on 75 combinations obtained, the 55th combination is the optimal combination for pregnant women aged 19-29 years old of the 1st trimester with 𝑥1 is rice as much as 379,1835 g, 𝑥2 is kale as much as 1.595,97 g, 𝑥3 is watermelon as much as 319,6961 g, 𝑥4 is tofu as much as 1.043,46 g and the total prices is rp.25.648,29 or can be rounded up rp.25.700 per day. table 5. the calculation result of food combination of pregnant women 19-29 y.o. 2nd trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 400,095 1.654,08 272,225 1.029,26 0 0 0 25.873,93 61 409,252 1.962,40 0 991,616 0 0 0 26.334,18 19 396,762 1.283,69 482,051 1.185,96 0 0 0 27.504,03 jurnal matematika mantik vol. 7, no. 1, may 2021, pp.59-66 64 based on the calculation results on 75 combinations obtained, the 55th combination is the optimal combination for pregnant women aged 19-29 years old of the 2nd trimester with 𝑥1 is rice as much as 400,095 g, 𝑥2 is kale as much as 1.654,08 g, 𝑥3 is watermelon as much as 272,225 g, 𝑥4 is tofu as much as 1.029,26 g, and the total price is rp.25.873,93 or can be rounded up rp.25.900 per day. table 6. the calculation result of food combination of pregnant women 19-29 y.o. 3rd trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 400,0948 1.654,08 272,2251 1.029,26 0 0 0 25.873,93 61 409,2523 1.962,40 0 991,6156 0 0 0 26.334,18 19 396,7615 1.283,69 482,0511 1.185,96 0 0 0 27.504,03 based on the calculation results on 75 combinations obtained, the 55th combination is the optimal combination for pregnant women aged 19-29 years old of the 3rd trimester with 𝑥1 is rice as much as 400,0948 g, 𝑥2 is kale as much as 1.654,08 g, 𝑥3 is watermelon as much as 272,2251 g, 𝑥4 is tofu as much as 1.029,26 g and the total price rp.25.873,93 or can be rounded up rp. 25.900 per day. table 7. the calculation result of food combination of pregnant women 30-49 y.o. 1st trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 342,0348 1.461,65 530,3547 961,5567 0 0 0 24.915,25 61 359,8766 2.062,31 0 888,2111 0 0 0 25.811,92 19 339,0892 1.134,35 715,7697 1.100,02 0 0 0 26.355,71 based on the calculation results on 75 combinations obtained, the 55th combination is the optimal combination for pregnant women aged 30-49 years old of the 1st trimester with 𝑥1 is rice as much as 342,0348 g, 𝑥2 is kale as much as 1.461,65 g, 𝑥3 is watermelon as much as 530,3547 g, 𝑥4 is tofu as much as 961,5567 g and the total prices is rp.24.915,25 or can be rounded up rp. 25.000 per day. table 8. the calculation result of food combination of pregnant women 30-49 y.o. 2nd trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 362,9461 1.519,76 482,8837 947,3579 0 0 0 25.140,89 61 379,1901 2.066,66 0 880,5773 0 0 0 25.957,29 19 359,8835 1.179,45 675,6705 1.091,33 0 0 0 26.638,62 based on the calculation results on 75 combinations obtained, the 55th combination is the optimal combination for pregnant women aged 30-49 years old of the 2nd trimester with 𝑥1 is rice as much as 342,0348 g, 𝑥2 is kale as much as 1.461,65 g, 𝑥3 is watermelon as much as 530,3547 g, 𝑥4 is tofu as much as 961,5567 g and the total prices is rp.24.915,25 or can be rounded up rp. 25.000 per day. nihaya alivia coraima dewi, fitroh resmi, pukky tetralian bantining ngastiti optimization of balanced menu for pregnant women in grobogan-central java using simplex method 65 table 9. the calculation result of food combination of pregnant women 30-49 y.o. 3rd trimester comb x1 x2 x3 x4 x5 x6 x7 prices (rp) 55 362,9461 1.519,76 482,8837 947,3579 0 0 0 25.140,89 61 379,1901 2.066,66 0 880,5773 0 0 0 25.957,29 19 359,8835 1.179,45 675,6705 1.091,33 0 0 0 26.638,62 based on the calculation results on 75 combinations obtained that the 55th combination is the optimal combination for pregnant women aged 30-49 years old of the 3rd trimester with 𝑥1 is rice as much as 362,9461 g, 𝑥2 is kale as much as 1.519,76 g, 𝑥3 is watermelon as much as 482,8837 g, 𝑥4 is tofu as much as 947,3579 g and the total prices is rp.25.140,89 or can be rounded up rp. 25.200 per day. 4. conclusions based on the 75 combinations that have been analyzed, based on food ingredients including staple foods, vegetables (spinach, green mustard, cauliflower, kale, carrots), fruit, side dishes, nuts, sugar and milk with the recommended nutritional adequacy rate for the data content of the adequacy of water, energy, protein, fat, carbohydrate (kh), fiber, vitamin a, vitamin b1, vitamin b2, vitamin b3 and vitamin c, for pregnant women aged 19-29 years and 30-49 years old. three trimesters, the results show that the combination of 55 is the optimal combination with food ingredients such as rice, kale, watermelon, and tofu. references [1] hardinsyah, kecukupan energi, lemak, protein dan karbohidrat. skripsi. bogor: ipb, 2012. 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[8] subagyo, p., asri, m., handoko, t. h., dasar-dasar operations research, mataram: bpfe, 2009. jurnal matematika mantik vol. 7, no. 1, may 2021, pp.59-66 66 [9] suryanto, nugroho, e. s., putra, r. a. k, analisis optimasi keuntungan dalam produksi keripik daun singkong dengan linier programming melalui metode simpleks, jurnal manajemen vol. 11(2), 226-236, 2019. [10] united nations children’s fund, world health organization, world bank group, levels and trends in child malnutrition: key findings of the 2018 edition of the joint child malnutrition estimates, 2018. [11] veronica, yunita, program bantu penetuan bahan makanan untuk menu diet bagi penderita komplikasi dengan metode simpleks. universitas kristen duta wacana : yogyakarta, 2004. [12] wibowo, irwan, penerapan metode simpleks untuk menyusun komposisi pakan unggas. universitas kristen duta wacana : yogyakarta, 2000. [13] winston, wayne, l., “operations research; applications and algorithms”, thomson learning, inc., 4th ed, 2004. [14] supranto, j., linear programming. fakultas ekonomi universitas indonesia jakarta, 1980. [15] zulhajrah, jaya, a. i., sahari, a., optimalisasi kebutuhan gizi pada menu makanan ibu hamil anemia menggunakan metode brach and bound, jurnal ilmiah matematika dan terapan vol 15: 209-219, 2018. aswar anas and marsidi michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity contact: marsidi, marsidiarin@gmail.com departement of mathematics education universitas pgri argopuro jember, jember, jawa timur 68121, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.107-114 michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity aswar anas1, marsidi2* 1,2department of mathematics education, universitas pgri argopuro, jember, indonesia article history: received jun 16, 2021 revised, jul 14, 2021 accepted, oct 31, 2021 kata kunci: model rantai makanan, model michaelis-menten, model mangsa dan pemangsa abstrak. pemodelan rantai makanan saat ini sedang berkembang pesat. ekosistem terlindungi dari rantai proses makan dan memakan. semua makhluk hidup saling membutuhkan, namun jika proses memakannya tidak seimbang, maka kepunahan makhluk hidup akan terjadi. salah satunya adalah model mangsa dan pemangsa yang berfungsi sebagai penyeimbang dalam sistem rantai makanan. model michaelis-menten merupakan model mangsapemangsa yang pada intinya menjaga kepunahan mangsa. permasalahannya adalah bagaimana menjaga mangsa tidak punah namun dengan pemanenan maksimal di suatu tempat dan jumlah minimum mangsa dengan waktu yang tepat. metode yang digunakan untuk mengatasi masalah tersebut adalah menambah dua variabel baru pada model michaelis-menten, yaitu jumlah minimum mangsa dan kapasitas tempat yang akan ditempati. terlihat bahwa sistem akan berada dalam keseimbangan jika tingkat kematian pemangsa besar, sehingga mangsa terjaga dari kepunahan sampai pemanenan. selain itu waktu yang tepat untuk perkembangbiakan yang baik juga dapat ditentukan. dari model ini didapatkan waktu yang tepat untuk pemanenan agar tidak terjadi kepunahan mangsa adalah ℎ = 𝑛( 𝐾2−𝑚2 4 𝐾 ). keywords: food chain model, michaelis-menten model, prey-predator model abstract. food chain modeling is currently developing rapidly. the ecosystem is protected from the chain of eating and eating processes. all living things need each other, but if the process of eating them is not balanced, then the extinction of living things will occur. one of them is the prey and predator model that serves as a balancer in the food chain system. the michaelis-menten model is a preypredator model that essentially prevents prey extinction. the problem is how to keep the prey from becoming extinct but with maximum harvesting in one place and the minimum amount of prey at the right time. the method used to overcome this problem is to add two new variables to the michaelis-menten model, namely the minimum number of prey and the capacity of the place to be occupied. it is seen that the system will be in equilibrium if the predator mortality rate is large so that the prey is kept from extinction until harvesting. in addition, the right time for good breeding can also be determined. from this model, it is found that the right time for harvesting so that prey extinction does not occur is ℎ = 𝑛( 𝐾2−𝑚2 4 𝐾 ). how to cite: a. anas and marsidi, “michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity”, j. mat. mantik, vol. 7, no. 2, pp. 107-114, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 107-114 issn: 2527-3159 (print) 2527-3167 (online) mailto:marsidiarin@gmail.com http://u.lipi.go.id/1458103791 jurnal matematika mantik vol 7, no 2, october 2021, pp. 107-114 108 1. introduction the food chain is a system created in nature. the dependence of living things on other living things is a normal interaction on this earth. this interaction occurs because of the needs of one party or two parties. the relationship between prey and predators is a onesided relationship that harms one party, this is very closely related because predators can only survive if there is prey. as a result, the chances of predators experiencing extinction are small. in addition, the predator also functions as a controller of the growth rate of the prey [1]. dongmei et al. stated that population extinction occurred because the initial population was too low [2]. if this happens, the prey population will be decrease and extinction may occur. as a result, predator populations are increasingly threatened indirectly. with the extinction of the prey, the predator also experiences extinction. the prey in this model is harvested while the predator population is not harvested because it has no commercial value. they are states that the commercial value is determined by the maximum value of h because if the prey harvested exceeds the value of h, the prey will become extinct. the cause of the extinction of the prey is also influenced by over-harvesting. therefore, it is necessary to limit the number of prey so that the number of prey remains under control and does not exceed the existing capacity [3] . the lotka–volterra model is frequently used to describe the dynamics of ecological systems in which two species interact, one a predator and one its prey. the model is simplified with the following assumptions: (1) only two species exist: fox and rabbit; (2) rabbits are born and then die through predation or inherent death; (3) foxes are born and their birth rate is positively affected by the rate of predation, and they die naturally [4]. the michaelis-menten model is a predator-prey model which is a generalization of the lotkavolterra model [5]. this model is used for events if the interaction system between individuals in a population has limited capacity. however, in this case, harvesting is also applied. previously, researchers researched the lotka-volterra proportional predation and prey model on the von bertallanfy [6] logistics modifications model . in this model, the researcher concludes that the number of predators in a place is highly dependent on the initial number of prey, the birth rate of the predator, and the birth rate of the prey. if the initial number of prey is large and the birth rate of prey is also large, the growth rate of predators increases. and if the birth constant of predators is small and the number of predators is smaller than the prey, there will be a decrease in the growth rate of prey. from previous research, researchers are interested in researching the michaelis-menten model, because this model prevents the extinction of both prey and predators. however, the researchers added the variables of space capacity and the minimum amount of prey that must be given. 2. research method this research method is a literature study of the addition of the [7] variable with the capacity of the place and the minimum number of prey, the equation comes from: according to [7], the equation for population growth and space capacity is formed by the formula: 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 ) (1) from [8] with the addition of the assumption that the minimum number of population also affects the rate of population growth, then the form is: 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) ) (2) aswar anas and marsidi michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity 109 where n is the coefficient of the growth rate of the prey population, p(t) is the number of prey populations at t, k is the capacity of the place, and m is the minimum number of populations. equation (2) is a logistic model that is influenced by the capacity of the place and the minimum number population. if the growth rate is also affected by the number of harvests of h, then equation (2) changes to: 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) )−ℎ (3) where 0 ≤ ℎ ≤ ℎ𝑚𝑎𝑥, h is the prey harvest rate constant and is the maximum harvested prey. xiao, et al describe the general formula for the persistence model in prey as follows [9]: 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )− 𝑐𝑃(𝑡)𝑦(𝑡) 𝑟𝑦(𝑡)+𝑃(𝑡) −ℎ 𝑑𝑦(𝑡) 𝑑𝑡 = 𝑦(𝑡)(−𝐷 + 𝑓𝑃(𝑡) 𝑟𝑦(𝑡)+𝑃(𝑡) ) (4) where 𝑟 is the satisfaction level of the predator, 𝑐 the number of prey captured, f the cconversion factor between the number of predators born for each prey captured, 𝐷 is the mortality rate of the predator. by adding the factor of the minimum number of prey population to reproduce, it is obtained 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) )− 𝑐𝑃(𝑡)𝑦(𝑡) 𝑟𝑦(𝑡)+𝑃(𝑡) −ℎ 𝑑𝑦(𝑡) 𝑑𝑡 = 𝑦(𝑡)(−𝐷 + 𝑓𝑃(𝑡) 𝑟𝑦(𝑡)+𝑃(𝑡) ) (5) by [10] equation (5) is sought a solution so that the prey does not experience extinction with: a. look for the maximum harvest value if predator is not present. b. search for fixed points. c. analysis of system stability at points 𝑇1 and 𝑇2. 3. discussion the following will discuss certain limits of harvesting a population to prevent extinction, then look for a fixed point from the michaelis-menten model to analyze the stability of the system at each of these fixed points and also perform simulations with different parameters. 3.1. finding of harvesting maximum value (𝒉𝒎𝒂𝒙) without predator condition in equation (2) we have maximum points if: 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) ) = 0 (6) by [11] with a condition half of the carrying capacity and the minimum number of prey population 𝑃(𝑡). in this case, fixed point occurs in 𝑃(𝑡) = 𝑚 and 𝑃(𝑡) = 𝐾. if 𝑃(𝑡) = 𝑚 then this point is not stable. it is caused by the population 𝑃(𝑡) > 𝑚. it means that the population will grow rapidly and avoid the 𝑃(𝑡) = 𝑚 and will attain 𝑃(𝑡) = 𝐾. let 𝑚 ≤ 𝑃0 < ( 𝐾+𝑚 2 ), point 𝑃0 moves rapidly towards the maximum point when 𝑃0 = ( 𝐾+𝑚 2 ). jurnal matematika mantik vol 7, no 2, october 2021, pp. 107-114 110 whereas if ( 𝐾+𝑚 2 ) < 𝑃0 ≤ 𝐾, then this point moves slowly towards the stable point by k. if 0 ≤ 𝑃0 < 𝑚, then the population will be moving down so that the population experiences extinction. while if 𝐾 < 𝑃0 , then this point will be moving slowly down to stable point k. since harvesting results must be maximum, then the predator must be eliminated or equal to zero and 𝑃(𝑡) = 𝐾+𝑚 2 , such that we have 𝑑𝑃(𝑡) 𝑑𝑡 = 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) )−ℎ = 0 ⟹ 𝑛𝑃(𝑡)(1− 𝑃(𝑡) 𝐾 )(1− 𝑚 𝑃(𝑡) ) = ℎ ⟹ 𝑛(𝐾−𝑃(𝑡))(𝑃(𝑡)−𝑚) 𝐾 = ℎ ; 𝑃(𝑡) = 𝐾+𝑚 2 so that: 𝑛( 𝐾+𝑚 2 )(1− 𝐾+𝑚 2𝐾 )(1− 2𝑚 (𝐾+𝑚) ) = ℎ ⟹ 𝑛( 𝐾+𝑚 2 )( 2𝐾−𝐾+𝑚 2𝐾 )( 𝐾+𝑚−2𝑚 𝐾+𝑚 ) = ℎ ⟹ 1 2 𝑛( 𝐾+𝑚 2𝐾 )(𝐾 −𝑚) = ℎ ⟹ 𝑛( 𝐾2−𝑚2 4𝐾 ) = ℎ therefore, the maximum harvesting is bounded by ℎ = 𝑛( 𝐾2−𝑚2 4𝐾 ). 3.2. finding fixed point let 𝑓1(𝑃,𝑦) = 𝑛𝑝(1− 𝑃 𝐾 )(1− 𝑚 𝑃 )− 𝑐𝑃𝑦 𝑟𝑦+𝑃 −ℎ 𝑓2(𝑃,𝑦) = 𝑦(−𝐷 + 𝑓𝑝 𝑟𝑦+𝑃 ) (7) suppose 𝑓1(𝑃,𝑦) = 0; 𝑓2(𝑃,𝑦) = 0 such that we have 𝑦 = 0 or (−𝐷 + 𝑓𝑝 𝑟𝑦+𝑃 ) = 0. (−𝐷 + 𝑓𝑝 𝑟𝑦+𝑃 ) = 0 ⟺ 𝑓𝑝 𝑟𝑦+𝑃 = 𝐷 ⟺ 𝑓𝑃 = 𝐷𝑟𝑦 +𝑃𝐷 ⟺ 𝑓𝑃−𝑃𝐷 𝐷𝑟 = 𝑦 then we have 𝑦1 = 0; 𝑦2 = 𝑃(𝑓−𝐷) 𝐷𝑟 . furthermore, finding the points 𝑃1 and 𝑃2 by substituting 𝑦1 = 0 to 𝑓1(𝑃,𝑦1) = 0. 𝑓1(𝑃,𝑦) = 𝑛𝑝(1− 𝑃 𝐾 )(1− 𝑚 𝑃 )− 𝑐𝑃0 𝑟0+𝑃 −ℎ = 0 ⟺ 𝑛𝑝(1− 𝑃 𝐾 )(1− 𝑚 𝑃 )−ℎ = 0 aswar anas and marsidi michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity 111 ⟺ 𝑛𝑝( 𝐾−𝑃 𝐾 )( 𝑝−𝑚 𝑃 )−ℎ = 0 ⟺ 𝑛 𝐾 (𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)−ℎ = 0 ⟺ 𝑛𝑃 −𝑛𝑚 + 𝑃 𝐾 𝑚𝑛 − 𝑛 𝐾 𝑃2 −ℎ = 0 ⇔ − 𝑛 𝐾 𝑃2 +𝑃(𝑛 + 𝑚𝑛 𝐾 )−(ℎ +𝑛𝑚) = 0 ⇔ 𝑛 𝐾 𝑃2 −(𝑛 + 𝑚𝑛 𝐾 )𝑃 +(ℎ +𝑛𝑚) = 0 such that 𝑃1,2 = (𝑛+ 𝑚𝑛 𝐾 )±√(−(𝑛+ 𝑚𝑛 𝐾 )) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) 𝑃1,2 = (𝑛+ 𝑚𝑛 𝐾 )±√(𝑛+ 𝑚𝑛 𝐾 ) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) for 𝑦2 = 𝑃(𝑓−𝐷) 𝐷𝑟 we have 𝑛𝑝(1− 𝑃 𝐾 )(1− 𝑚 𝑃 )−( 𝑐𝑃 𝑃(𝑓−𝐷) 𝐷𝑟 𝑟 𝑃(𝑓−𝐷) 𝐷𝑟 +𝑃 )−ℎ = 0 ⇔ 𝑛𝑝(1 − 𝑃 𝐾 )(1− 𝑚 𝑃 )−( 𝑐𝑃 𝑃(𝑓−𝐷) 𝐷𝑟 𝑟 𝑃(𝑓−𝐷)+𝑃𝐷𝑟 𝐷𝑟 )−ℎ = 0 ⇔ 𝑛𝑝(1 − 𝑃 𝐾 )(1− 𝑚 𝑃 )− (𝑐𝑃2𝑓−𝑐𝑃2𝐷) 𝑟𝑃𝑓−𝑃𝐷𝑟+𝑃𝐷𝑟 −ℎ = 0 ⇔ 𝑛𝑝(1 − 𝑃 𝐾 )(1− 𝑚 𝑃 )− (𝑐𝑃2𝑓−𝑐𝑃2𝐷) 𝑟𝑃𝑓 −ℎ = 0 ⇔ 𝑛 𝐾 (𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)− (𝑐𝑃2𝑓−𝑐𝑃2𝐷) 𝑟𝑃𝑓 −ℎ = 0 ⇔ 𝑟𝑃𝑓𝑛 𝐾 (𝐾𝑃 −𝐾𝑚 +𝑃𝑚 −𝑃2)−(𝑐𝑃2𝑓 −𝑐𝑃2𝐷)−ℎ𝑟𝑃𝑓 = 0 ⇔ 𝑃( 𝑟𝑓𝑛 𝐾 (𝐾𝑃 −𝐾𝑚 +𝑃𝑚−𝑃2)−(𝑐𝑃𝑓 −𝑐𝑃𝐷)−ℎ𝑟𝑓) = 0 ⇔ 𝑃((𝑟𝑓𝑛𝑃 −𝑟𝑓𝑛𝑚 + 𝑟𝑓𝑛𝑃𝑚 𝐾 − 𝑟𝑓𝑛𝑃2 𝐾 )−𝑐𝑃𝑓 +𝑐𝑃𝐷 = ℎ𝑟𝑓) = 0 ⇔ 𝑃(− 𝑟𝑓𝑛 𝐾 𝑃2 +𝑃(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓 +𝑐𝐷)−(ℎ𝑟𝑓 +𝑟𝑛𝑓𝑚)) = 0 ⇔ 𝑃( 𝑟𝑓𝑛 𝐾 𝑃2 −(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓 +𝑐𝐷)𝑃 +(ℎ𝑟𝑓 +𝑟𝑛𝑓𝑚)) = 0 thus 𝑃 = 0 or 𝑃1,2 = (𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓+𝑐𝐷)±√(−(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓+𝑐𝐷)) 2 −4( 𝑟𝑓𝑛 𝐾 )(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚) 2( 𝑟𝑓𝑛 𝐾 ) 𝑃1,2 = (𝑟𝑓𝑛 (𝐾+𝑚) 𝐾 −𝑐𝑓+𝑐𝐷)±√(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓+𝑐𝐷) 2 −4( 𝑟𝑓𝑛 𝐾 )(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚) 2( 𝑟𝑓𝑛 𝐾 ) it is impossible for 𝑃 = 0. since this model must have a minimum amount so that the prey can grow. from the results above, we have a fixed points as follows: jurnal matematika mantik vol 7, no 2, october 2021, pp. 107-114 112 𝑇1:(𝑃1,𝑦1) = ( (𝑛+ 𝑚𝑛 𝐾 )+√((𝑛+ 𝑚𝑛 𝐾 )) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) ,0 ) 𝑇2:(𝑃2,𝑦1) = ( (𝑛+ 𝑚𝑛 𝐾 )−√(𝑛+ 𝑚𝑛 𝐾 ) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) ,0) 𝑇3:(𝑃1 ∗,𝑦2 ∗) = ( (𝑟𝑓𝑛 (𝐾+𝑚) 𝐾 −𝑐𝑓+𝑐𝐷)+√(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓+𝑐𝐷) 2 −4( 𝑟𝑓𝑛 𝐾 )(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚) 2( 𝑟𝑓𝑛 𝐾 ) , 𝑃1 ∗(𝑓−𝐷) 𝐷𝑟 ) 𝑇4:(𝑃2 ∗,𝑦2 ∗) = ( (𝑟𝑓𝑛 (𝐾+𝑚) 𝐾 −𝑐𝑓+𝑐𝐷)−√(𝑟𝑓𝑛+ 𝑟𝑓𝑛𝑚 𝐾 −𝑐𝑓+𝑐𝐷) 2 −4( 𝑟𝑓𝑛 𝐾 )(ℎ𝑟𝑓+𝑟𝑛𝑓𝑚) 2( 𝑟𝑓𝑛 𝐾 ) , 𝑃2 ∗(𝑓−𝐷) 𝐷𝑟 ) equilibrium analysis is carried out to find out the points that cause the system to be in equilibrium and not, to analyze it, look for real eigenvalues at each equilibrium point [12]. to find the stability of the equilibrium point of a model, you can use the jacobian matrix with the order 2×2. this matrix is discussed in the next sub-chapter. 3.3. jacobian matrix in [13] let the equation system equation (5) written by: 𝑑𝑃 𝑑𝑡 = 𝑓1(𝑃,𝑦) 𝑑𝑌 𝑑𝑡 = 𝑓2(𝑃,𝑦) such that, the jacobian matrix can be formed by: 𝐽 = [ 𝜕𝑓1 𝜕𝑃 𝜕𝑓1 𝜕𝑦 𝜕𝑓2 𝑃 𝜕𝑓2 𝜕𝑦 ] 𝐽 = [ − 2𝑛𝑝 𝐾 + 𝑛(𝐾+𝑚) 𝐾 − 𝑟𝑐𝑦2+𝑐𝑦𝑃+𝑟𝑐𝑃𝑦2 (𝑟𝑦+𝑃)2 − 𝑐𝑃2 (𝑟𝑦+𝑃)2 𝑓𝑟𝑦2 (𝑟𝑦+𝑃)2 −𝐷 + 𝑓𝑟𝑦2+𝑓𝑦𝑃−𝑓𝑃𝑦𝑟 (𝑟𝑦+𝑃)2 ] the stability of the system of equations (5) will be known by analyzing the eigen values of the jacobian matrix. 3.4. fixed point stable analysis the following discussion discusses the stability analysis of fixed points using the jacobi matrix of order 2×2 and only discussed in point 𝑇1(𝑃1,𝑦1), 𝑇2(𝑃2,𝑦1). 3.4.1 stable system at fixed point 𝑻𝟏 by [14] we take the fixed point aswar anas and marsidi michaelis-menten models with constant harvesting of restricted prey populations minimum place and amount capacity 113 𝑇1:(𝑃1,𝑦1) = ( (𝑛+ 𝑚𝑛 𝐾 )+√(𝑛+ 𝑚𝑛 𝐾 ) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) ,0). we substitute the point 𝑇1 to jacobian matrix as follows. 𝐽1 = [ − 2𝑛𝑝 𝐾 + 𝑛(𝐾+𝑚) 𝐾 − 𝑟𝑐𝑦2+𝑐𝑦𝑃+𝑟𝑐𝑃𝑦2 (𝑟𝑦+𝑃)2 − 𝑐𝑃2 (𝑟𝑦+𝑃)2 𝑓𝑟𝑦2 (𝑟𝑦+𝑃)2 −𝐷 + 𝑓𝑟𝑦2+𝑓𝑦𝑃−𝑓𝑃𝑦𝑟 (𝑟𝑦+𝑃)2 ] then we have: 𝐽1 = [ − 2𝑛𝑝 𝐾 + 𝑛(𝐾+𝑚) 𝐾 − 𝑟𝑐0+𝑐0𝑃+𝑟𝑐𝑃0 (𝑟0+𝑃)2 − 𝑐𝑃2 (𝑟0+𝑃)2 0 −𝐷 ] 𝐽1 = [ − 2𝑛𝑝 𝐾 + 𝑛(𝐾+𝑚) 𝐾 −𝑐 0 −𝐷 ] 𝐽1 = [ −𝑛 − 𝑚𝑛 𝐾 −√(𝑛 + 𝑚𝑛 𝐾 )− 4𝑛(𝑚𝑛+ℎ) 𝐾 + 𝑛(𝐾+𝑚) 𝑚 −𝑐 0 −𝐷 ] 𝐽1 = [ −√(𝑛 + 𝑚𝑛 𝐾 ) 2 − 4𝑛(𝑚𝑛+ℎ) 𝐾 −𝑐 0 −𝐷 ] the eigen value can be obtained by 𝑑𝑒𝑡(𝐴−𝜆1𝐼) = 0. (−√(𝑛+ 𝑚𝑛 𝐾 ) 2 − 4𝑛(𝑚𝑛+ℎ) 𝐾 −𝜆)(−𝐷 −𝜆) = 0 ⇔ 𝜆1 = −√(𝑛 + 𝑚𝑛 𝐾 ) 2 − 4𝑛(𝑚𝑛+ℎ) 𝐾 or 𝜆2 = −𝐷 with ℎ = 𝑛( 𝐾2−𝑚2 4𝐾 ) we have 𝜆1 = −√2√− 𝑛2𝑚(𝐾−𝑚) 𝐾2 or 𝜆2 = −𝐷. from the equation above we know that 𝜆1 is not real eigen value because 𝑛,𝑚,𝐾 is a positive parameter with 𝐾 > 𝑚 and 𝜆2 ≤ 0 if 𝐷 is a positive integer. it show that 𝑇1 is stable. 3.4.2 stable system at fixed point 𝑻𝟐 let 𝑇2:(𝑃2,𝑦1) = ( (𝑛+ 𝑚𝑛 𝐾 )−√(𝑛+ 𝑚𝑛 𝐾 ) 2 −4( 𝑛 𝐾 )(ℎ+𝑛𝑚) 2( 𝑛 𝐾 ) ,0). with the same way we obtain that 𝜆1 = √(𝑛 + 𝑚𝑛 𝐾 ) 2 − 4𝑛(𝑚𝑛+ℎ) 𝐾 or 𝜆2 = −𝐷. wit ℎ = 𝑛( 𝐾2−𝑚2 4𝐾 ) we get 𝜆1 = √2√− 𝑛2𝑚(𝐾−𝑚) 𝐾2 or 𝜆2 = −𝐷, because 𝜆1 is not real eigen value and 𝜆2 ≤ 0 if 𝐷 is a positive integer. it show that 𝑇2 is stable [15]. jurnal matematika mantik vol 7, no 2, october 2021, pp. 107-114 114 4. conclusion from the calculation results obtained 4 fixed points. in here we are only discussing points that are 𝑇1and 𝑇2. the analysis carried out at two fixed points states that the equilibrium and stability of the population is influenced by the level of capacity, the minimum number of population, and the death rate of the predator. from the discussion section we know that maximum harvesting can be done if ℎ = 𝑛( 𝐾2−𝑚2 4𝐾 ), it’s aim is to ensure that prey and predator populations do not become extinct. 5. acknowledgments we gratefully acknowledge the support from universitas pgri argopuro jember 2021. references [1] s. o. lehtinen, “ecological and evolutionary consequences of predator-prey role reversal: allee effect and catastrophic predator extinction,” j. theor. biol., vol. 510, p. 110542, 2021, doi: 10.1016/j.jtbi.2020.110542. 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[15] t. pnv, dynamical system, an introduction with application in economics and biology. springerverlag. heidelberg, germany, 1994. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 5 aplikasi metode singular spectral analysis (ssa) dalam peramalan pertumbuhan ekonomi indonesia tahun 2017 rina sri kalsum siregar1, dina prariesa2, gumgum darmawan3 magister statistika terapan, universitas padjajaran, jl. dipati ukur no.35, bandung 40132 email: rienhaaa@gmail.com1, dinaprariesa@gmail.com2, gumstat@gmail.com3 abstrak tujuan dari penelitian ini adalah untuk melihat pola musiman pada data produk domestik bruto (pdb) riil triwulanan tahun 2000-2016 dan melihat penerapan dari singular spectral analysis (ssa) pada data pdb tersebut untuk meramal data pdb pada tahun 2017. adapun metode ssa yang digunakan adalah metode recurrent forecasting dengan bootstrap confidence interval untuk melihat selang kepercayaannya. sumber data berasal dari data pdb tahun 2000-2016 tahun dasar 2000 yang dikumpulkan oleh badan pusat statistik (bps). hasilnya menunjukkan bahwa metode ssa dapat dijadikan metode yang handal dan dapat dikatakan valid dilihat dari nilai ukuran mape 0.82 dan ukuran tracking signal sebesar -4.00. kata kunci: produk domestik bruto, metode singular spectral analysis, peramalan abstract the purpose of this study was to look at seasonal patterns in the data of gross domestic product (gdp) quarterly in the year 2000-2016 and the implementation of singular spectral analysis (ssa) in the data of gdp to predict the data of gdp in 2017. the ssa method used is the method of recurrent forecasting with bootstrap confidence interval to look at its beliefs of interval. the source of data derived from the data of gdp in 2000-2016 with the base year in 2000 compiled by the central statistics agency (csa). the results showed that the ssa method can be used as a reliable method and can be valid that view from the value of mape size is 0.82 and the size of the tracking signal at -4.00. keywords: gross domestic product, singular spectral analysis method, forecasting 1. pendahuluan pertumbuhan ekonomi merupakan perubahan aktifitas perekonomian dalam menghasilkan tambahan pendapatan masyarakat suatu negara secara berkesinambungan selama periode tertentu. selain itu, pertumbuhan ekonomi juga dapat dikatakan salah satu indikator keberhasilan perekonomian suatu negara [1]. salah satu indikator penting untuk mengetahui kondisi ekonomi di suatu negara dalam suatu periode tertentu adalah data produk domestik bruto (pdb), baik atas dasar harga berlaku maupun atas dasar harga konstan [2]. pada dasarnya, pdb merupakan jumlah nilai tambah yang dihasilkan oleh seluruh unit usaha dalam suatu negara tertentu, atau merupakan jumlah nilai barang dan jasa akhir yang dihasilkan oleh seluruh unit ekonomi. pdb atas dasar harga berlaku menggambarkan nilai tambah barang dan jasa yang dihitung menggunakan harga yang berlaku pada setiap tahun, sedangkan pdb atas dasar harga konstan menunjukkan nilai tambah barang dan jasa tersebut yang dihitung menggunakan harga yang berlaku pada satu tahun tertentu sebagai dasar. pdb nominal (pdb atas dasar harga berlaku) merujuk kepada nilai pdb tanpa memperhatikan pengaruh harga, sedangkan pdb riil (pdb atas dasar harga konstan) mengoreksi pdb nominal dengan memasukkan pengaruh dari harga. pdb atas dasar harga berlaku dapat digunakan untuk melihat pergeseran dari struktur ekonomi, sedangkan pdb atas dasar harga konstan digunakan untuk jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 6 mengetahui pertumbuhan perekonomian dari tahun ke tahun. badan pusat statistik [2] dalam berita resmi statistik menyatakan bahwa pertumbuhan ekonomi indonesia pada tahun 2016 triwulan ke-2 adalah sebesar 5,18%. peramalan (forecasting) dalam pdb rill (pertumbuhan ekonomi) menjadi suatu hal yang sangat penting untuk melihat bagaimana perkembangan pdb atau pertumbuhan ekonomi suatu negara, sehingga pemerintah dapat mengambil kebijakan yang perlu berdasarkan hasil yang didapat. kegiatan peramalan merupakan bagian dari ilmu statistik dengan berbagai model yang ditawarkan, salah satunya model singular spectrum analysis (ssa) [3]. beberapa hasil penelitian menunjukkan bahwa model ssa memiliki beberapa keuntungan dibandingkan model time series yang lain, seperti arima [4], [5], [6], [7], [8], [10], dan [13]. keunggulan dari model ssa terlihat lebih fleksibel dan mampu memodelkan musiman dengan waktu multi periode dan musiman yang kompleks [6], [7], dan [8]. selain itu, model ssa terhindar dari banyaknya syarat seperti independensi dan normalitas residual, sebagaimana pada model arima [10] dan [13]. suatu metode peramalan yang paling baik dapat di lihat dari nilai ukuran mean absolute percentage error (mape) yang kecil dan ukuran tracking signal pada batas yang dapat diterima [6], [7], [8], dan [11]. berdasarkan penjelasan di atas, peneliti ingin melihat pola musiman atau tren pada data produk domestik bruto (pdb) riil triwulanan tahun 20002016 dan menerapkan model ssa pada data tersebut untuk meramal data pdb riil pada tahun 2017. selain itu, penelitian ini juga ingin melihat sejauh sejauh mana keunggulan model ini berdasarkan ukuran mape dan tracking signal-nya. hasil peramalan data pdb 2017 menggunakan metode recurrent forecasting dibandingkan metode bootstrap forecasting dengan melihat selang kepercayaan. 2. metode penelitian metode penelitian yang digunakan dalam penelitian ini adalah analisis data statistik menggunakan program r. jenis penelitian ini merupakan jenis penelitian eksperimen dengan membandingkan 2 jenis metode analisis data statistik, yaitu metode recurrent forecasting dibandingkan metode bootstrap forecasting dengan melihat selang kepercayaan [14]. data yang digunakan adalah data produk domestik bruto atas dasar harga konstan (pdb adhk/pdb riil) deret berkala triwulanan dari tahun 2000-2016 dengan tahun dasar 2000, yang bersumber dari badan pusat statistik [8]. analisis data pdb adhk menggunakan metode ssa dan metode bootstrap forecasting. 3. hasil dan pembahasan 3.1. identifikasi data pdb adhk triwulanan data pdb triwulanan dianggap sangat baik untuk menggambarkan perkembangan perekonomian suatu negara atau wilayah. analisis time series data pdb triwulan sangat penting peranannya untuk mengetahui perkembangan perekonomian suatu negara atau daerah dari waktu ke waktu dalam periode waktu yang relatif pendek, yakni 3 (tiga) bulan. hasil analisis ini sangat efektif dalam mendukung proses penentuan arah kebijakan ekonomi suatu negara atau daerah tersebut dengan cepat. pdb merupakan salah satu indikator untuk mengetahui kondisi perekonomian suatu negara. jadi, kondisi perekonomian indonesia dapat digambarkan melalui besarnya nilai pdb indonesia pada suatu waktu. secara umum, pdb indonesia selalu mengikuti pola trend naik. artinya, besarnya pdb indonesia selalu mengalami pertumbuhan dari waktu ke waktu. selain itu, juga terlihat adanya pola fluktuasi beraturan yang hampir selalu terjadi di setiap tahun. hal ini merupakan indikator bahwa pdb indonesia dipengaruhi oleh suatu pola musiman. efek musiman terbesar rata-rata terjadi di setiap triwulan iii. jadi, setiap triwulan iii terdapat fenomena yang menyebabkan angka pdb indonesia lebih besar dibandingkan triwulan lainnya. pola musiman yang terjadi di setiap tahun terlihat merupakan efek musiman aditif. efek musiman aditif ini ditunjukkan dengan fluktuasi musiman dari tahun ke tahun yang besarnya hampir sama. kondisi pdb atas dasar harga konstan dari tahun 2000 sampai tahun 2016 dengan tahun dasar 2000 dapat di lihat pada gambar 1. gambar 1. pdb atas dasar harga konstan tahun 2000-2016 tahun dasar 2000 pdb triwulan indonesia waktu (triwulan) p d b t ri w u la n 0 10 20 30 40 50 60 70 4 e + 0 5 5 e + 0 5 6 e + 0 5 7 e + 0 5 8 e + 0 5 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 7 perekonomian indonesia tahun 2000-2016 ketika digambarkan melalui pdb triwulanan terlihat adanya fluktuasi-fluktuasi nilai pdb yang menunjukkan bahwa setiap triwulan terdapat perbedaan kondisi ekonomi yang cukup signifikan. hal ini dikarenakan di setiap triwulan terdapat fenomena yang berbeda-beda yang mempengaruhi besarnaya nilai pdb setiap triwulan. perbedaan fenomena yang berdampak pada nilai pdb inilah yang disebut dengan efek musiman. dengan kata lain, fenomena musiman yang terjadi di setiap triwulan berbeda-beda. untuk memastikan apakah pdb terpengaruh faktor musiman, dapat dilihat melalui hasil pengujian terhadap adanya efek musiman [3]. pengujian adanya efek musiman ini menghasilkan fungsi ‘data mengikuti pola musiman dengan periode 3.94’. artinya data mempunyai pola musiman berulang dengan periode ulangan setiap 4 periode. karena data pdb yang dihitung menggunakan data pdb triwulanan maka bisa diartikan perulangan pola musiman ini terjadi setiap triwulannya. 3.2. metode ssa pada peramalan pdb adhk triwulanan peramalan diperlukan untuk memperkirakan kejadian di masa yang akan datang [10]. peramalan sangat membantu dalam kegiatan perencanaan dan pengambilan keputusan akan suatu kebijakan. apalagi jika ada tuntutan untuk dapat mengambil kebijakan secara cepat yang menyangkut masa depan perekonomian suatu negara atau wilayah. selain itu, peramalan dapat digunakan sebagai sarana antisipasi terhadap kejadian yang tidak diinginkan di periode mendatang. dalam penelitian ini digunakan metode ssa untuk meramalkan pdb adhk triwulanan pada tahun mendatang dengan 2 tahapan besar, yaitu dekomposisi dan reconstruction. 3.3. dekomposisi pada dekomposisi terdapat dua tahapan yaitu embedding dan singular value decomposition (svd). pada tahapan dekomposisi data dibagi menjadi data insample dan outsample dengan periode musiman 4 sesuai hasil pengujian sebelumnya. parameter yang memiliki peran penting dalam dekomposisi adalah window length (l). window length (l) merupakan salah satu parameter utama dari ssa. pada tahapan ini diperlukan penentuan parameter window length (l) dengan ketentuan 2 < l < 𝑁 2 . namun untuk penentuan nilai window length tidak ada aturan yang baku sehingga pengulangan analisis ssa dengan berbagai nilai window length yang berbeda perlu dilakukan untuk mendapatkan model yang cocok. pada penelitian ini window length yang diuji sebesar 34, 24, 14, 11, 9, 4, dan 2 yang menghasilkan nilai mape terkecil pada window length 9. sehingga pada tahapan selanjutnya window length (l) yang digunakan adalah l = 9. 1. embedding langkah pertama dalam ssa adalah embedding dimana f ditransformasi ke dalam matriks lintasan berukuran l x k. matriks lintasan ini merupakan matriks dimana semua elemen pada anti diagonalnya bernilai sama. matriks lintasan tersebut dinamakan matriks hankel dengan dimensi l x k dari data insample dengan l = 9 dan k = n1 (jumlah sampel insample) – l + 1 = 64 – 9 + 1 = 56. berikut ini merupakan matriks lintasan yang terbentuk. 𝐗 = (𝑥𝑖𝑗 )𝑖,𝑗=1 9,56 dengan 𝑋𝑖𝑗 = [ 342.852,4 340.865,2 … 699.526,3 340.865,2 355.289,5 … 705.934,3 ⋮ ⋮ ⋱ ⋮ 368.650,4 375.720,9 … 775.447,1 ] 2. singular value decomposition (svd) pada tahapan ini kita membuat singular value decomposition dari matrik lintasan x. matrik x terbentuk dari eigen vector 𝑈𝑖, singular value √𝜆𝑖 , dan principal component 𝑉𝑖 𝑇 . ketiga elemen pembentuk svd ini disebut dengan eigentriple. svd dari matrik lintasan dapat didenotasikan sebagai berikut: x = 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑑 = 𝑈1√𝜆1𝑉1 𝑇 + 𝑈2√𝜆2𝑉2 𝑇 + ⋯ + 𝑈𝑑 √𝜆𝑑 𝑉𝑑 𝑇 = ∑ 𝑈𝑖 √𝜆𝑖 𝑉𝑖 𝑇𝑑 𝑖=1 plot yang terbentuk dari eigen value dan eigen vectors pada window length 9 dapat dilihat pada gambar 2. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 8 gambar 2. eigenvalue dan eigenvectors (l = 9) 3.4. reconstruction pada rekontruksi terdapat dua tahapan yaitu grouping dan diagonal averaging. parameter yang memiliki peranan penting dalam rekontruksi adalah grouping effect (r). tujuan dari tahapan grouping ini adalah memisahkan komponen aditif dari data deret waktu. pada tahapan ini diputuskan pengelompokan yang sesuai dan bagaimana merekontruksinya. dengan kata lain, diperlukan identifikasi eigentriples yang sesuai dengan data deret waktu yang diteliti. terdapat dua cara dalam menentukan pengelompokan yang pertama yaitu dengan mengobservasi data deret waktu secara keseluruhan dan cara kedua yaitu mengeksplorasi periodogram deret waktu. dari pengelompokan eigenvalue dan eigenvector pada gambar 2 dengan window length l = 9, secara subjektif pada bagian pertama dapat dilihat terdapat 3 patahan kasar, pada bagian kedua juga terbentuk 3 macam bentuk eigenvector yang berbeda. sehingga apabila dilakukan observasi data deret waktunya secara keseluruhan (cara 1) kelompok yang kita bentuk ada 3 macam. kelompok pertama merupakan pengelompokan berdasarkan trend, sedangkan kelompok kedua dan ketiga merupakan pengelompokan berdasarkan musiman dengan periodogram yang berbeda. cara kedua yaitu mengeksplorasi periodegram deret waktu dan pengelompokan berdasarkan periodegram terdekat. sebelum itu kita melakukan analisis komponen utama pada data eigenvalue untuk menentukan banyaknya komponen utama dan didapatkan output sebanyak 5 komponen utama (pada baris ke 5 diambil karena mampu menjelaskan sebanyak 99.99%), seperti tampak pada gambar 3. gambar 3. data eigen value kemudian kita melakukan pengujian periodegram pada matriks lintasan diambil 5 kolom pertama. dari pengujian ini kita dapatkan 2 macam periode musiman sehingga data dapat kita kelompokkan menjadi 3 kelompok data trend, season 1, dan season 2. 1. grouping berdasarkan penjelasan di atas kita membagi pengelompokan menjadi 3 macam yaitu trend, season 1, dan season 2. dengan kelompok trend terdiri dari komponen utama 1, kelompok season 2 terdiri dari komponen utama 2 dan 3 dan kelompok season 3 terdiri dari komponen utama 4 dan 5 dengan plot korelasi antar kelompok yang terbentuk dapat dilihat pada gambar 4. kita dapat melihat adanya strong separability saat masing masing kelompok tidak lagi berkorelasi, seperti tampak pada gambar 4. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 9 gambar 4. matriks korelasi w 2. diagonal averaging pada tahapan ini dilakukan transformasi dari hasil pengelompokan matriks xii dari penguraian kelompok x = xi1 + xi2 + ⋯ + xim menjadi series baru dengan panjang n. hasil dari tahapan ini merupakan matriks komponen dengan dimensi k x n dari data insample yaitu 3 x 64. kemudian melalui perhitungan didapatkan rataan diagonal averaging yang ditunjukkan pada hasil output, seperti tampak pada tabel 1. deret rekontruksi yang dihasilkan dari elementary grouping ini dinamakan deret rekontruksi elementary. tabel 1. hasil rataan diagonal averaging tahapan selanjutnya adalah melakukan peramalan dari serangkaian model yang dibentuk melalui pemecahan data deret waktu di atas dengan metode ssa. peramalan ini dimaksudkan untuk meramalkan keadaan perekonomian pada tahun 2017 yang digambarkan melalui nilai pdb atas dasar harga konstan (pdb riil) selama triwulan i-iv tahun 2017. hasil output peramalan pdb dapat dilihat pada gambar 5. gambar 5. hasil output peramalan pdb dari peramalan pdb triwulanan atas dasar harga konstan (pdb riil) melalui metode singular spectrum analysis (ssa) didapatkan ramalan nilai pdb riil untuk triwulan i (januari – maret) 2017 sebesar rp. 825.939,2 milyar dan naik menjadi rp. 852.082,9 milyar pada triwulan ii (april – juni). pdb riil triwulan iii (juli – september) naik lagi menjadi 873.569,3 milyar kemudian mengalami sedikit penurunan pada triwulan iv (oktober – desember) ke rp. 864.650,9 milyar. secara total nilai pdb atas dasar harga konstan 2017 sebesar rp. 3.416.242,3 milyar dimana apabila dibandingkan dengan nilai pdb atas dasar harga konstan 2016 sebesar rp. 3.218.694,3 milyar terjadi kenaikan pertumbuhan prekonomian sebesar 6.13%. dari model dan hasil peramalan untuk mengukur ketepatan peramalan pdb riil melalui metode ssa di atas kita menggunakan nilai mean squared error (mse) dan nilai mean absolute percentage error (mape) melalui syntax pada lampiran 9 bagian akurasi peramalan. pada metode peramalan ssa dengan window length l = 9 dan gruping sebanyak 3 komponen didapatkan nilai berikut (setelah melakukan metode ssa dengan window length yang berbeda dan pengelompokan alternatif nilai yang tertera pada output merupakan pembanding dengan nilai terkecil, seperti tampak pada gambar 6). gambar 6. hasil output pembanding jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 10 perhitungan menggunakan mad dan mse akan menghasilkan nilai yang besar bahkan sampai ribuan, untuk menghindari hal ini maka dapat digunakan alternative nilai mape atau mean absolute percent error, dimana kesalahan dihitung berdasarkan persen kesalahannya absolutenya. mape menghitung deviasi antara data aktual dengan nilai peramalan kemudian dihitung persen rataratanya. adapun nilai mape yang dihasilkan cukup kecil yaitu sebesar 0.82, setelah melakukan beberapa percobaan dengan window length yang berbeda beda dan pengelompokan alternatif maka nilai mape ini merupakan nilai yang minimum sehingga diharapkan model peramalan yang terbentuk sudah cukup baik untuk meramalkan nilai pdb atas dasar harga konstan (pdb riil) triwulanan tahun 2017. setelah mengukur ketepatan model peramalan maka analisis dilanjutkan untuk mengukur validitas peramalan yang diukur menggunakan tracking signal. apabila nilai-nilai tracking signal berada di luar batas yang dapat diterima yaitu ± 5 [12], maka model peramalan harus ditinjau kembali dan akan dipertimbangkan model baru. hasil output tracking signal dapat dilihat pada gambar 7. gambar 7. hasil output tracking signal nilai tracking signal yang terbentuk dari 4 periode peramalan adalah sekitar -4 dimana masih berada dalam batas keandalan peramalan yang dapat diterima, seperti tampak pada gambar 7. hal ini dapat diartikan bahwa model peramalan pdb riil triwulanan tahun 2017 menggunakan metode ssa merupakan metode yang handal dan dapat dikatakan valid. 3.5. bootstrap ssa forecasting confidence interval atau selang kepercayaan merupakan informasi tambahan tentang keakurasian dan kestabilan peramalan [5, 8]. dalam menentukan selang kepercayaan dapat digunakan metode monte carlo maupun metode bootstrap. kedua metode ini sangat baik digunakan untuk membandingkan model terbaik dengan metode ssa recurrent forecasting atau vector forecasting. pada penelitian ini kita melihat bagaimana hasil peramalan pdb triwulanan atas dasar harga konstan (pdb riil) tahun 2017 dan confidence interval dengan menggunakan bootstrap reccurent forecasting. dengan taraf signifikan 95%, dibangun model menggunakan syntax bootstrap, sehingga dihasilkan output seperti tampak pada tabel 2. tabel 2. hasil output bootstrap forecasting hasil forecast 2.5% 97.5% 827.961,8 822.748,6 833.739,8 854.062,4 848.600,5 859.943,3 874.484,7 868.779,0 880.453,8 866.158,2 860.217,2 872.250,1 hasil output pada tabel 2, menunjukkan bahwa nilai peramalan pdb riil melalui syntax bootstrap didapatkan ramalan nilai pdb riil untuk triwulan i (januari-maret) 2017 sebesar rp. 827.961,8 milyar dengan selang kepercayaan antara 822.748,6 milyar sampai 833.739,8 milyar dan triwulan berikutnya (april-juni) naik menjadi rp. 854.062,4 milyar dengan selang kepercayaan antara 848.600,5 milyar sampai 859.943,3 milyar. pdb riil triwulan iii (juli-september) mengalami kenaikan menjadi 874.484,7 milyar dengan selang kepercayaan antara 868.779,0 milyar sampai 880.453,8 milyar kemudian mengalami penurunan pada triwulan iv (oktober-desember) ke rp. 866.158,2 milyar dengan selang kepercayaan antara 860.217,2 milyar sampai 872.250,1 milyar. secara total nilai pdb atas dasar harga konstan 2017 sebesar rp. 3.422.667,1 milyar dimana apabila dibandingkan dengan nilai pdb atas dasar harga konstan 2016 sebesar rp. 3.218.694,3 milyar terjadi kenaikan pertumbuhan prekonomian sebesar 6.33%. 3.6. perbandingan hasil peramalan setelah melakukan seluruh tahapan estimasi, perbandingan antar metode dapat dilihat pada tabel 3. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 11 tabel 3. perbandingan hasil peramalan berdasarkan hasil perbandingan pada tabel 3, dapat ditunjukkan bahwa nilai peramalan pdb riil baik menggunakan basic recurrent maupun boostrap recurrent masih berada dalam batas nilai selang kepercayaan dalam tiap triwulan. nilai pertumbuhan ekonomi yang dihasilkan dengan peramalan bootstrap lebih tinggi dari nilai peramalan basic recurrent, tapi selisihnya sangat kecil. untuk perbandingan plot data nya bisa dilihat pada gambar 8. gambar 8. plot data peramalan pada gambar 8 terlihat jelas plot nilai peramalan pdb riil dengan menggunakan basic recurrent (berwarna merah) maupun dengan menggunakan boostrap recurrent (berwarna biru) dalam tiap triwulan memiliki fluktuasi yang hampir sama yaitu nilai peramalan pdb naik pada triwulan pertama sampai ketiga kemudian turun pada triwulan keempat. 4. kesimpulan berdasarkan hasil dan pembahasan di atas, dapat disimpulkan bahwa pdb indonesia selama tahun 2000-2006 menunjukkan adanya tren yang meningkat dan dipengaruhi oleh suatu pola musiman. efek musiman terbesar rata-rata terjadi di setiap triwulan iii. selain itu, berdasarkan peramalan pdb triwulanan atas dasar harga konstan (pdb riil) melalui metode singular spectrum analysis (ssa) didapatkan ramalan nilai pdb riil untuk triwulan i (januari-maret) 2017 sebesar rp. 825.939,2 milyar dan naik menjadi rp. 852.082,9 milyar pada triwulan ii (apriljuni). pdb riil triwulan iii (juli-september) naik lagi menjadi 873.569,3 milyar kemudian mengalami sedikit penurunan pada triwulan iv (oktober-desember) ke rp. 864.650,9 milyar. secara total nilai pdb atas dasar harga konstan 2017 sebesar rp. 3.416.242,3 milyar dimana apabila dibandingkan dengan nilai pdb atas dasar harga konstan 2016 sebesar rp. 3.218.694,3 milyar terjadi kenaikan pertumbuhan perekonomian sebesar 6.13% sedangkan dengan menggunakan bootstrap recurrent didapatkan ramalan kenaikan pertumbuhan prekonomian sebesar 6.33%. terakhir, model peramalan pdb riil triwulanan tahun 2017 menggunakan metode ssa merupakan metode yang handal dan dapat dikatakan valid dilihat dari nilai ukuran mape yang kecil dan ukuran tracking signal pada batas yang dapat diterima. referensi [1] prahmana, r.c.i. (2008). penentuan harga opsi untuk model black-scholes menggunakan metode beda hingga crank-nicolson. (skripsi). yogyakarta: universitas gadjah mada. [2] badan pusat statistik. (2010). seasonal adjustment dan peramalan pdb triwulanan. jakarta: badan pusat statistik. [3] darmawan, g. (2016). identifikasi pola data curah hujan pada proses grouping dalam metode singular spectrum analysis. prosiding sempoa: seminar nasional, pameran alat peraga, dan olimpiade matematika (pp. 415-424). surakarta: universitas muhammadiyah surakarta. [4] darmawan, g., hendrawati, t., & arisanti, r. (2015). model auto singular spectrum untuk meramalkan kejadian banjir di bandung dan sekitarnya. prosiding seminar nasional matematika dan pendidikan matematika uny 2015 (pp. 457-462). yogyakarta: universitas negeri yogyakarta. [5] efron, b., & thibshirani, r. (1986). bootstrap method for standar errors, confidence intervals, and other measures of statistical accuracy. statistical science, 1(1), 54-75. basic recurrent (milyar) bootstrap recurrent (milyar) i rp. 825.939,2 rp. 827.961,8 rp. 822.748,6 s.d rp. 833.739,8 ii rp. 852.082,9 rp. 854.062,4 rp. 848.600.5 s.d rp. 859.943,3 iii rp. 873569,3 rp. 874.484,7 rp. 868.779.0 s.d rp. 880.453,8 iv rp. 864.650,9 rp. 866.158,2 rp. 860.217.2 s.d rp. 872.250,1 pertumbuhan ekonomi 6,13% 6,33% nilai peramalan pdb rill triwulan (2017) selang kepercayaan 95 % (milyar) jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 12 [6] golyandina, n., & korobeynikov, a. (2012). basic singular spectrum analysis and forecasting with r. rusia: faculty of mathematics and mechanic, st. petersburgh state university. [7] golyandina, n., & zhigljavsky, a.a. (2013). singular spectrum analysis for time series. new york: springer. [8] golyandina, n., nekrutkin, v., & zhigljavsky, a.a. (2001). analysis of time series structure: ssa and related techniques. new york: chapman & hall/crc. [9] https://bps.go.id/subjek/view/id/11#subjekviewta b3|accordion-daftar-subjek1. [10] leeuw, j.d., & crutcher, p. (2009). singular spectrum analysis in r. los angeles: department of statistics, university of california. [11] sungkawa, i., & megasari, r.t. (2011). penerapan ukuran ketepatan nilai ramalan data deret waktu dalam selesi model peramalan volume penjualan pt satria mandiri citra mulia. jakarta: departemen matematika dan statistik, universitas binus. [12] abraham, b., & ledolter, j. (1983). statistical methods for forecasting. new york: john wiley. [13] darmawan, g., handoko, b., & suparman, y. (2016). seasonal test for non-stationary time series data by means of periodogram analysis. prosiding the 2nd international conference on applied statistics (pp. 74-81). bandung: fakultas matematika dan ilmu pengetahuan, universitas padjajaran. [14] prahmana, r.c.i. (2017). design research (teori dan implementasinya: suatu pengantar). jakarta: rajawali pers. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: ahmad khairul umam akumam@billfath.ac.id department of mathematics, universitas billfath, lamongan, east java 62261 the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.105-112 existence and uniqueness of fixed points in cone metric spaces for 𝝎-distance ahmad khairul umam1, nihaya alivia c. dewi1, pukky tetralian b. ngastiti1 1universitas billfath, lamongan, indonesia article history: received nov 19, 2021 revised may 14, 2022 accepted dec 31, 2022 kata kunci: titik tetap, metrik cone, jarak-𝜔 abstrak. di dalam penelitian ini, fungsi jarak yang digunakan adalah fungsi jarak-𝜔. beberapa teorema, pembuktian teorema, dan contoh tentang ruang metrik cone dibahas pada penelitian ini. jika diberikan ruang metrik cone lengkap dengan fungsi jarak-𝜔, cone normal 𝑃 di 𝑋, fungsi kontraksi 𝑓: 𝑋 → 𝑋 dengan bentuk 𝜔(𝑓(𝑥), 𝑓(𝑦)) ≼ 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑓(𝑥) + 𝜔(𝑦, 𝑓(𝑦)] + 𝛾[𝜔(𝑥, 𝑓(𝑦) + 𝜔(𝑦, 𝑓(𝑥)] untuk setiap 𝑥, 𝑦 ∈ 𝑋, dan 𝛼, 𝛽, 𝛾 adalah bilangan real tak negatif dimana 𝛼 + 2𝛽 + 2𝛾 < 1, maka fungsi 𝑓 memiliki titik tetap tunggal di 𝑋. keywords: fixed point, cone metric, 𝜔-distance abstract. in this study, the distance function used is the distance function-𝜔. several theorems, proof of theorem, and example of cone metric space are discussed in this study. if given a complete cone metric space with distance function, the cone is normal at contraction function 𝑓: 𝑋 → 𝑋 with 𝜔(𝑓(𝑥), 𝑓(𝑦)) ≼ 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑓(𝑥) + 𝜔(𝑦, 𝑓(𝑦)] + 𝛾[𝜔(𝑥, 𝑓(𝑦) + 𝜔(𝑦, 𝑓(𝑥)] for every 𝑥, 𝑦 ∈ 𝑋, and 𝛼, 𝛽, 𝛾 is a non-negative real number where 𝛼 + 2𝛽 + 2𝛾 < 1, then function 𝑓 has unique fixed point at 𝑋. how to cite: a. k. umam, n. h. a. dewi, and p. t. b. ngastiti, “existence and uniqueness of fixed points in cone metric spaces for 𝝎-distance”, j. mat. mantik, vol. 8, no. 2, pp. 105-112, december 2022. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 105-112 issn: 2527-3159 (print) 2527-3167 (online) mailto:akumam@billfath.ac.id http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 105-112 106 1. introduction fixed point has many useful for solving linear equation, ordinary differential equation, partial differential equation, intergral equation. the famous fixed point theorem is banach fixed point theorem. according to [1], the banach fixed point guarantee the existence and uniqueness of fixed point for function in complete space and contractive function. in this paper we discuss some fixed point theorems in cone metric space with 𝜔distance. according to [2], cone metric space is generalization of metric space. range of cone metric space is banach space. we use real banach space and 𝜔-distance for mertic. according to [3], a 𝜔-distance is a function in metric spaces with three condition: symmetry, lower semicontinuous function, and relationship 𝜔-distance with metric itself. 2. preliminaries definition 1. [4] let 𝑉 is non-empty set with addition and scalar (real number) multiplication operation. addition operation: �̅�, �̅� ∈ 𝑉, �̅� + �̅� ∈ 𝑉; and scalar multiplication operation: 𝑘 ∈ ℝ, �̅� ∈ 𝑉, 𝑘�̅� ∈ 𝑉. 𝑉 is called vector space if satisfy v1. ∀�̅�, �̅� ∈ 𝑉, �̅� + �̅� ∈ 𝑉; v2. ∀�̅�, �̅�, �̅� ∈ 𝑉, (�̅� + �̅�) + �̅� = �̅� + (�̅� + �̅�); v3. ∃0̅ ∈ 𝑉, ∀�̅� ∈ 𝑉, 0̅ + �̅� = �̅� + 0̅ = �̅�; v4. ∀�̅� ∈ 𝑉, ∃ − �̅� ∈ 𝑉, �̅� + (−�̅�) = −�̅� + �̅� = 0̅; v5. ∀�̅�, �̅� ∈ 𝑉, �̅� + �̅� = �̅� + �̅�; v6. ∀𝑘 ∈ ℝ, ∀�̅� ∈ 𝑉, 𝑘�̅� ∈ 𝑉; v7. ∀𝑘 ∈ ℝ, ∀�̅�, �̅� ∈ 𝑉, 𝑘(�̅� + �̅�) = 𝑘�̅� + 𝑘�̅�; v8. ∀𝑘, ℎ ∈ ℝ, ∀�̅� ∈ 𝑉, (𝑘 + ℎ)�̅� = 𝑘�̅� + ℎ�̅�; v9. ∀𝑘, ℎ ∈ ℝ, ∀�̅� ∈ 𝑉, 𝑘(ℎ�̅�) = 𝑘ℎ(�̅�); v10. ∃1 ∈ ℝ, ∀�̅� ∈ 𝑉, 1. �̅� = �̅�. definition 2. [5] let 𝑉 is vector space over the field 𝔽. function ‖∙‖: 𝑉 → ℝ is called norm of 𝑉 if satisfy n1. ‖𝑥‖ ≥ 0 for all 𝑥 ∈ 𝑉; n2. if 𝑥 ∈ 𝑉 dan ‖𝑥‖ = 0 then 𝑥 = 0; n3. ‖𝛼𝑥‖ = |𝛼|‖𝑥‖ for all 𝑥 ∈ 𝑉 and 𝛼 ∈ 𝔽; n4. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖ for all 𝑥, 𝑦 ∈ 𝑉. a normed space is a vector space 𝑉 together with a norm ‖∙‖. definition 3. [6] all complete normed vector space is called banach space. definition 4. [7] a metric on a set 𝑋 is a function 𝑑: 𝑋 × 𝑋 → ℝ that satisfies the following properties: m1. 𝑑(𝑥, 𝑦) ≥ 0 for all 𝑥, 𝑦 ∈ 𝑋. (positivity) m2. 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦. (definiteness) m3. 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋. (symmetry) m4. 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. (triangle inequality) a metric space (𝑋, 𝑑) is a set 𝑋 together with a metric 𝑑 on 𝑋. definition 5. [8] let 𝔼 is real banach space and 𝑃 a subset of 𝔼. a set 𝑃 is called a cone if only if: i 𝑃 is closed, nonempty, and 𝑃 ≠ 0; ii 𝑎𝑥 + 𝑏𝑦 ∈ 𝑃 for all 𝑥, 𝑦 ∈ 𝑃 dan 𝑎, 𝑏 ∈ ℝ+ ∪ {0}; ahmad khairul umam, nihaya alivia c. dewi, and pukky tetralian b. n. existence and uniqueness of fixed points in cone metric spaces for 𝝎-distance 107 iii 𝑃 ∩ (−𝑃) = 0. further, if 𝑃 ⊆ 𝔼 is cone, then we define partial ordering “≼” with respect to 𝑃 by 𝑥 ≼ 𝑦 if only if 𝑦 − 𝑥 ∈ 𝑃. and then 𝑥 < 𝑦 is interpreted 𝑥 ≼ 𝑦 and 𝑥 ≠ 𝑦. whereas 𝑥 < 𝑦 is interpreted 𝑦 − 𝑥 ∈ 𝑖𝑛𝑡 𝑃 (interior of 𝑃). definition 6. [9] the cone 𝑃 is called normal if there is a number 𝑀 > 0 such that for all 𝑥, 𝑦 ∈ 𝔼, 0 ≤ 𝑥 ≤ 𝑦 implies ‖𝑥‖ ≤ 𝑀‖𝑦‖. (1) the least positive 𝑀 satisfying (1) is called the normal constant of 𝑃. definition 7. [2] a cone metric on non-empty set 𝑋 is a function 𝑑: 𝑋 × 𝑋 → 𝔼 that satisfies the following properties: c1. 0 < 𝑑(𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋 and 𝑑(𝑥, 𝑦) = 0 if only if 𝑥 = 𝑦; c2. 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋; c3. 𝑑(𝑥, 𝑦) ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦) for all 𝑥, 𝑦 dan 𝑧 ∈ 𝑋. a cone metric spaces (𝑋, 𝑑) is a set 𝑋 together with a metric 𝑑 on 𝑋. definition 8. [10] let (𝑋, 𝑑) be a cone metric space, 〈𝑥𝑛 〉 be a sequence in 𝑋. if for any 𝑐 ∈ 𝔼 with 0 < 𝑐, there is 𝑁 such that for all 𝑛 > 𝑁, 𝑑(𝑥, 𝑥𝑛 ) < 𝑐, then 〈𝑥𝑛 〉 is called a convergent sequence to a point 𝑥 ∈ 𝑋. definition 9. [2] let (𝑋, 𝑑) be a cone metric space, 〈𝑥𝑛 〉 be a sequence in 𝑋. if for any 𝑐 ∈ 𝔼 with 0 < 𝑐, there is 𝑁 such that for all 𝑚, 𝑛 > 𝑁, 𝑑(𝑥𝑛 , 𝑥𝑚 ) < 𝑐, then 〈𝑥𝑛 〉 is called a cauchy sequence in 𝑋. definition 10. [11] let (𝑋, 𝑑) be a cone metric space, 〈𝑥𝑛 〉 be a sequence in 𝑋. (𝑋, 𝑑) is a complete cone metric space if every cauchy sequence is convergent. definition 11. [12] a function 𝜓 from a metric space (𝑋, 𝑑) to ℝ is called lower semicontinuous if for every 𝑦 ∈ 𝑋, 𝑙𝑖𝑚 𝑥→𝑦 𝑖𝑛𝑓 𝜓(𝑥) ≥ 𝜓(𝑦). example 1. let a function 𝑓(𝑥) = { 2, 𝑥 > 2 1, 𝑥 ≤ 2 . the function 𝑓(𝑥) is lower semicontinuous at 𝑥 = 2. 𝑦 2 1 -1 0 1 2 3 4 𝑥 figure 1. example of lower semicontinuous function definition 12. [13] let (𝑋, 𝑑) be a metric space and a function 𝑓: 𝑋 → 𝑋. a point 𝑥 ∈ 𝑋 is called a fixed point of function 𝑓 if 𝑥 = 𝑓(𝑥). definition 13. [14] let metric space (𝑋, 𝑑). function 𝑓: 𝑋 → 𝑋 is said contraction function if there is a real number 𝑐 where 0 ≤ 𝑐 < 1 such that: 𝑑(𝑓(𝑥), 𝑓(𝑦)) ≤ 𝑐𝑑(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋. theorem 1. [15] (banach’s fixed point). let (𝑋, 𝑑) is complete metric space. if function 𝑓: 𝑋 → 𝑋 is contraction function in 𝑋, then function 𝑓 has unique fixed point. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 105-112 108 defintion 14. [16] a function 𝜔: 𝑋 × 𝑋 → [0, ∞) is a 𝜔-distance on 𝑋 if it satisfies the following conditions for any 𝑥, 𝑦, 𝑧 ∈ 𝑋: ω1. 𝜔(𝑥, 𝑧) ≤ 𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑧) ω2. the function 𝜔(𝑥,⋅): 𝑋 → [0, ∞) is lower semicontinuous ω3. for any > 0, there exists 𝛿 > 0 such that 𝜔(𝑧, 𝑥) ≤ 𝛿 and 𝜔(𝑧, 𝑦) ≤ 𝛿 imply 𝑑(𝑥, 𝑦) ≤ . of course, the metric 𝑑 is a 𝜔-distance on 𝑋. 3. result and discussion theorem 2. let complete cone metric space (𝑋, 𝑑) with 𝜔-distance. let normal cone 𝑃 in 𝑋 and function 𝑓: 𝑋 → 𝑋 that satisfy 𝜔(𝑓(𝑥), 𝑓(𝑦)) ≼ 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑓(𝑥)) + 𝜔(𝑦, 𝑓(𝑦))] +𝛾[𝜔(𝑥, 𝑓(𝑦)) + 𝜔(𝑦, 𝑓(𝑥))] (2) for all 𝑥, 𝑦 ∈ 𝑋 where 𝛼, 𝛽, 𝛾 are non-negative real numbers such that 𝛼 + 2𝛽 + 2𝛾 < 1, then 𝑓 has unique fixed point in 𝑋. proof: let a sequence 〈𝑥𝑛 〉 in cone metric space (𝑋, 𝑑). a sequence 〈𝑥𝑛 〉 satisfies this property: 𝑥𝑛 = 𝑓(𝑥𝑛−1) = 𝑓(𝑓(𝑥𝑛−2)) = 𝑓 2(𝑥𝑛−2) = ⋯ = 𝑓 𝑛(𝑥0) where 𝑛 ∈ ℕ. according (2), we get 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽[𝜔(𝑥0, 𝑓(𝑥0)) + 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0))] +𝛾[𝜔(𝑥0, 𝑓 2(𝑥0)) + 𝜔(𝑓(𝑥0), 𝑓(𝑥0))] = 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾𝜔(𝑥0, 𝑓 2(𝑥0)) + 𝛾𝜔(𝑓(𝑥0), 𝑓(𝑥0)) = 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾𝜔(𝑥0, 𝑓 2(𝑥0)) + 0 = 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾𝜔(𝑥0, 𝑓 2(𝑥0)). (3) we use triangle inequality and we get 𝜔(𝑥0, 𝑓 2(𝑥0)) ≼ 𝜔(𝑥0, 𝑓(𝑥0)) + 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) so that (3) become 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾𝜔(𝑥0, 𝑓 2(𝑥0)) ≼ 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾[𝜔(𝑥0, 𝑓(𝑥0)) + 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0))] = 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) +𝛾𝜔(𝑥0, 𝑓(𝑥0)) + 𝛾𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) = 𝛼𝜔(𝑥0, 𝑓(𝑥0)) + 𝛽𝜔(𝑥0, 𝑓(𝑥0)) + 𝛾𝜔(𝑥0, 𝑓(𝑥0)) +𝛽𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) + 𝛾𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) = (𝛼 + 𝛽 + 𝛾)𝜔(𝑥0, 𝑓(𝑥0)) + (𝛽 + 𝛾)𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) or we can write 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ (𝛼 + 𝛽 + 𝛾)𝜔(𝑥0, 𝑓(𝑥0)) +(𝛽 + 𝛾)𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) − (𝛽 + 𝛾)𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ (𝛼 + 𝛽 + 𝛾)𝜔(𝑥0, 𝑓(𝑥0)) [1 − (𝛽 + 𝛾)]𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ (𝛼 + 𝛽 + 𝛾)𝜔(𝑥0, 𝑓(𝑥0)) ahmad khairul umam, nihaya alivia c. dewi, and pukky tetralian b. n. existence and uniqueness of fixed points in cone metric spaces for 𝝎-distance 109 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ 𝛼 + 𝛽 + 𝛾 1 − (𝛽 + 𝛾) 𝜔(𝑥0, 𝑓(𝑥0)). suppose 𝐾 = 𝛼+𝛽+𝛾 1−(𝛽+𝛾) according to definition 8 about contraction function, then 𝜔(𝑓(𝑥0), 𝑓 2(𝑥0)) ≼ 𝐾𝜔(𝑥0, 𝑓(𝑥0)) (4) where 0 ≤ 𝐾 = 𝛼+𝛽+𝛾 1−(𝛽+𝛾) < 1. according (4), we get 𝜔(𝑓 𝑛(𝑥0), 𝑓 𝑛+1(𝑥0)) ≼ 𝐾𝜔(𝑓 𝑛−1(𝑥0), 𝑓 𝑛(𝑥0)) ≼ ⋯ ≼ 𝐾 𝑛 𝜔(𝑥0, 𝑓(𝑥0)) where 0 ≤ 𝐾 < 1. let 𝑚 > 𝑛 ≥ 𝑁 ∈ ℕ, according to triangle inequality then 𝜔(𝑓 𝑛(𝑥0), 𝑓 𝑚 (𝑥0)) ≼ 𝜔(𝑓 𝑛(𝑥0), 𝑓 𝑛+1(𝑥0)) + 𝜔(𝑓 𝑛+1(𝑥0), 𝑓 𝑛+2(𝑥0)) + ⋯ +𝜔(𝑓 𝑚−2(𝑥0), 𝑓 𝑚−1(𝑥0)) + 𝜔(𝑓 𝑚−1(𝑥0), 𝑓 𝑚(𝑥0)) ≼ 𝐾𝑛 𝜔(𝑥0, 𝑓(𝑥0)) + 𝐾 𝑛+1𝜔(𝑥0, 𝑓(𝑥0)) + ⋯ +𝐾𝑚−2𝜔(𝑥0, 𝑓(𝑥0)) + 𝐾 𝑚−1𝜔(𝑥0, 𝑓(𝑥0)) = (𝐾𝑛 + 𝐾𝑛+1 + ⋯ + 𝐾𝑚−2 + 𝐾𝑚−1)𝜔(𝑥0, 𝑓(𝑥0)) = 𝐾𝑛 (1 + 𝐾 + 𝐾2 + ⋯ 𝐾𝑚−𝑛−1)𝜔(𝑥0, 𝑓(𝑥0)) = 𝐾𝑛 ( ∑ 𝐾𝑖 𝑚−𝑛−1 𝑖=0 ) 𝜔(𝑥0, 𝑓(𝑥0)) ≼ 𝐾𝑛 (∑ 𝐾𝑖 ∞ 𝑖=0 ) 𝜔(𝑥0, 𝑓(𝑥0)) = 𝐾𝑛 ( 1 1 − 𝐾 ) 𝜔(𝑥0, 𝑓(𝑥0)) = ( 𝐾𝑛 1 − 𝐾 ) 𝜔(𝑥0, 𝑓(𝑥0)). suppose 𝜔(𝑥0, 𝑓(𝑥0)) = 𝑐, then 𝜔(𝑓 𝑛 (𝑥0), 𝑓 𝑚(𝑥0)) ≼ ( 𝐾𝑛 1 − 𝐾 ) 𝜔(𝑥0, 𝑓(𝑥0)) = 𝑐𝐾𝑛 1 − 𝐾 . choose 𝑁 ∈ ℕ with 𝑁 < 𝐾 log (1 − 𝐾) 𝑐 such that for all 𝑚, 𝑛 ≥ 𝑁, we get 𝜔(𝑓 𝑛(𝑥0), 𝑓 𝑚(𝑥0)) ≼ 𝑐𝐾𝑛 1 − 𝐾 ≼ 𝑐𝐾𝑁 1 − 𝐾 < 𝑐 1 − 𝐾 . (1 − 𝐾) 𝑐 = . so, sequence 〈𝑥𝑛 〉 is cauchy sequence. because space (𝑋, 𝑑) is complete cone metric space, then sequence 〈𝑥𝑛 〉 is convergent to point 𝑥 ∈ 𝑋 or we can write 〈𝑥𝑛 〉 → 𝑥. and then, we will proof that point 𝑥 is fixed point of function 𝑓. according to triangle inequality and (2), then 𝜔(𝑓(𝑥), 𝑥) ≼ 𝜔(𝑓(𝑥), 𝑓(𝑥𝑛 )) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽[𝜔(𝑥, 𝑓(𝑥)) + 𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 ))] +𝛾[𝜔(𝑥, 𝑓(𝑥𝑛 )) + 𝜔(𝑥𝑛 , 𝑓(𝑥))] + 𝜔(𝑓(𝑥𝑛 ), 𝑥) = 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥, 𝑓(𝑥)) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) +𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) or we can write 𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥, 𝑓(𝑥)) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) +𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) 𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑓(𝑥), 𝑥) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) +𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 105-112 110 𝜔(𝑓(𝑥), 𝑥) − 𝛽𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) +𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) (1 − 𝛽)𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) +𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) (1 − 𝛽)𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) +𝛾𝜔(𝑥, 𝑓(𝑥𝑛 )) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) (1 − 𝛽)𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) +𝛾𝜔(𝑓(𝑥𝑛 ), 𝑥) + 𝜔(𝑓(𝑥𝑛 ), 𝑥) (1 − 𝛽)𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼𝜔(𝑥, 𝑥𝑛 ) + 𝛽𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾𝜔(𝑥𝑛 , 𝑓(𝑥)) +(𝛾 + 1)𝜔(𝑓(𝑥𝑛 ), 𝑥) 𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼 (1 − 𝛽) 𝜔(𝑥, 𝑥𝑛 ) + 𝛽 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑓(𝑥𝑛 )) + 𝛾 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑓(𝑥)) + (𝛾 + 1) (1 − 𝛽) 𝜔(𝑓(𝑥𝑛 ), 𝑥) 𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼 (1 − 𝛽) 𝜔(𝑥, 𝑥𝑛 ) + 𝛽 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑥𝑛+1) + 𝛾 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑓(𝑥)) + (𝛾 + 1) (1 − 𝛽) 𝜔(𝑥𝑛+1, 𝑥) because sequence 〈𝑥𝑛 〉 is convergent for 𝑛 → ∞, then: 𝜔(𝑓(𝑥), 𝑥) ≼ 𝛼 (1 − 𝛽) 𝜔(𝑥, 𝑥𝑛 ) + 𝛽 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑥𝑛+1) + 𝛾 (1 − 𝛽) 𝜔(𝑥𝑛 , 𝑓(𝑥)) + (𝛾 + 1) (1 − 𝛽) 𝜔(𝑥𝑛+1, 𝑥) = 𝛼 (1 − 𝛽) (0) + 𝛽 (1 − 𝛽) (0) + 𝛾 (1 − 𝛽) (0) + (𝛾 + 1) (1 − 𝛽) (0) = 0. (5) because 0 ≼ 𝜔(𝑓(𝑥), 𝑥), according (5) then 𝜔(𝑓(𝑥), 𝑥) = 0, or we can write 𝑥 = 𝑓(𝑥). so, point 𝑥 is fixed point of function 𝑓. furthermore, we will proof uniqueness of a fixed point. suppose 𝑥 and 𝑦 are fixed points of function 𝑓, such that 𝑥 = 𝑓(𝑥) and 𝑦 = 𝑓(𝑦) then 𝜔(𝑥, 𝑦) = 𝜔(𝑓(𝑥), 𝑓(𝑦)) ≼ 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑓(𝑥)) + 𝜔(𝑦, 𝑓(𝑦))] + 𝛾[𝜔(𝑥, 𝑓(𝑦)) + 𝜔(𝑦, 𝑓(𝑥))] = 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑥) + 𝜔(𝑦, 𝑦)] + 𝛾[𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑥)] = 𝛼𝜔(𝑥, 𝑦) + 𝛽[0 + 0] + 𝛾[𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑥)] = 𝛼𝜔(𝑥, 𝑦) + 𝛽[0] + 𝛾[𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑥)] = 𝛼𝜔(𝑥, 𝑦) + 0 + 𝛾[𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑥)] = 𝛼𝜔(𝑥, 𝑦) + 𝛾[𝜔(𝑥, 𝑦) + 𝜔(𝑦, 𝑥)] = 𝛼𝜔(𝑥, 𝑦) + 𝛾𝜔(𝑥, 𝑦) + 𝛾𝜔(𝑦, 𝑥) = 𝛼𝜔(𝑥, 𝑦) + 𝛾𝜔(𝑥, 𝑦) + 𝛾𝜔(𝑥, 𝑦) = 𝛼𝜔(𝑥, 𝑦) + 2𝛾𝜔(𝑥, 𝑦) = (𝛼 + 2𝛾)𝜔(𝑥, 𝑦) because 𝛼 + 2𝛽 + 2𝛾 < 1 and 𝛼 + 2𝛾 < 1, so that 𝜔(𝑥, 𝑦) ≼ (𝛼 + 2𝛾)𝜔(𝑥, 𝑦) is contradiction. so, point 𝑥 is unique fixed point of function 𝑓 in 𝑋. ahmad khairul umam, nihaya alivia c. dewi, and pukky tetralian b. n. existence and uniqueness of fixed points in cone metric spaces for 𝝎-distance 111 example 2. let set 𝔼 = ℝ2, 𝑃 = {(𝑥, 𝑦) ∈ 𝔼: 𝑥, 𝑦 ≥ 0} ⊂ ℝ2, 𝑋 = ℝ, and 𝑑: 𝑋 × 𝑋 → 𝔼 such that 𝑑(𝑥, 𝑦) = (|𝑥 − 𝑦|, 𝛼|𝑥 − 𝑦|) where 𝛼 ≥ 0 is a constant, then (𝑋, 𝑑) is cone metric space. example 3. let metric space (𝑋, 𝑑) and function 𝜔: 𝑋 × 𝑋 → [0, ∞). function 𝜔(𝑥, 𝑦) = 𝑐 for every 𝑥, 𝑦 ∈ 𝑋 is 𝜔-distance on 𝑋 (𝑐 is positive real number). function 𝜔 is not metric space because 𝜔(𝑥, 𝑥) = 𝑐 ≠ 0 for any 𝑥 ∈ 𝑋. 4. conclusion cone metric space is generalization of metric space. range of cone metric in cone metric space is banach space. in this research, we use 𝜔-distance for metrik. a 𝜔-distance is a function in metric spaces with three condition: symmetry, lower semicontinuous function, and relationship 𝜔-distance with metric itself. fixed point has many useful for solving linear equation, ordinary differential equation, partial differential equation, integral equation. the famous fixed-point theorem is banach fixed point theorem. the banach fixed point guarantee the existence and uniqueness of fixed point for function in complete space and contractive function. let complete cone metric space with 𝜔-distance. let normal cone 𝑃 in 𝑋 and function 𝑓: 𝑋 → 𝑋 that satisfy 𝜔(𝑓(𝑥), 𝑓(𝑦)) ≼ 𝛼𝜔(𝑥, 𝑦) + 𝛽[𝜔(𝑥, 𝑓(𝑥)) + 𝜔(𝑦, 𝑓(𝑦))] + 𝛾[𝜔(𝑥, 𝑓(𝑦)) + 𝜔(𝑦, 𝑓(𝑥))] for all 𝑥, 𝑦 ∈ 𝑋 where 𝛼, 𝛽, 𝛾 are non negative real numbers such that 𝛼 + 2𝛽 + 2𝛾 < 1, then 𝑓 has unique fixed point in 𝑋. references [1] s. oltra, and o. valero, " banach’s fixed point theorem for partial metric spaces," rend. istit. mat. univ. trieste, vol. 36, pp. 17-26, 2004. 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[14] a. k. umam, a. isro'il, and p. t. b. ngastiti, "beberapa teorema titik tetap untuk pemetaan kontinu di ruang metrik kompak," jms: jurnal matematika & sains, vol. 1, no. 2, pp. 81-86, 2021. [15] e. kreyszig, introductory functional analysis with aplication, john wiley and sons, inc, 1978. [16] a. latif, “generalized caristi’s fixed-point theorems,” fixed point theory and applications, vol. 2009, no. 170140, pp. 1-7, 2009. jurnal matematika mantik vol 7, no 2, october 2021, pp. 20-29 contact: e. susanti, eka_susanti@mipa.unsri.ac.id departement of mathematics, universitas sriwijaya palembang, sumatera selatan 30662, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.124-131 optimization of inventory level using fuzzy probabilistic exponential two parameters model eka susanti1, indrawati2, robinson sitepu3, karita ondhiana4, widya dwi wulandari5 1,2,3,4,5 department of mathematics, universitas sriwijaya, palembang, indonesia article history: received nov 30, 2020 revised, may 30, 2021 accepted, oct 31, 2021 kata kunci: inventori fuzzy probabilistik, distribusi eksponensial, distribusi pareto abstrak. pengendalian persediaan adalah faktor penting dalam kegiatan perdagangan. pengendalian persediaan bertujuan untuk menjamin ketersediaan produk. terdapat beberapa faktor yang mempengaruhi tingkat persediaan diantaranya faktor tingkat permintaan, persediaan maksimal produk dan tingkat kerusakan. jika faktor yang mempengaruhi tidak dapat didefinisikan dengan pasti dan mengikuti distribusi statisik tertentu maka pendekatan fuzzy probabilistik dapat diterapkan. pada penelitian ini dibahas masalah optimasi persediaan cabai merah pada tingkat pengecer. tingkat kerusakan diasumsikan mengikuti distribusi eksponensial dan permintaan mengikuti distribusi pareto. parameter statistik diestimasi dengan metode maksimum likelihood dan parameter biaya dinyatakan dengan bilangan fuzzy segitiga. berdasarkan hasil perhitungan untuk beberapa nilai beta diperoleh total biaya tertinggi sebesar rp 405143.6 dengan tingkat persediaan maksimum sebanyak 15 kg dan waktu siklus pemesanan selama 0.923 hari. keywords: fuzzy probabilistic inventory, exponential distribution, pareto distribution abstract. inventory control is an important factor in trading activities. inventory control aims to ensure product availability. several factors affect the level of inventory including the level of demand factor, maximum inventory, and the level of deterioration. if the influencing factors cannot be defined with certainty and follow a certain statistic distribution then the fuzzy probabilistic approach can be applied. this research discusses the problem of optimizing the inventory of red chillies at the retail level. the level of deterioration is assumed to follow an exponential distribution and demand follows a pareto distribution. statistical parameters are estimated using the maximum likelihood method and cost parameters are expressed by triangular fuzzy numbers. based on the calculation results for several beta values, the highest total cost is 405143.6 rupiah, a maximum inventory level of 15 kg, and an order cycle time of 0.923 days. how to cite: e. susanti et al, “optimization of inventory level using fuzzy probabilistic exponential two parameters model, j. mat. mantik, vol. 7, no. 2, pp. 124-131, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 124-131 issn: 2527-3159 (print) 2527-3167 (online) mailto:eka_susanti@mipa.unsri.ac.id http://u.lipi.go.id/1458103791 jurnal matematika mantik vol 7, no 2, october 2021, pp. 124-131 125 1. introduction inventory planning is an important stage in production, distribution and trading activities. planning is carried out to ensure product availability in fulfilling consumer demand activities. the inventory model can be applied to inventory planning activities. the inventory system is given operational policies related to product storage control, such as how much is the maximum inventory, when to order and stock out time to minimize the total cost of ordering. research related to inventory methods has been developed and applied in various fields. inventory and supply chain problems to determine the optimal location, optimal route for distribution activities are discussed by [1]. the problem of inventory optimization with various payment systems is discussed by [2]. eoq model with nonlinear constraints in inventory problem discussed by [3]. the solution method used by [4] for the inventory model is the algebraic method. the fruit fly algorithm modification method was introduced by [5] to solve the problem of inventory and allocation optimization. the problem of inventory for perishable products is discussed by [6]. the concept of inventory to the health sector introduced by [7]. the research that has been mentioned is an implementation of the deterministic inventory concept with demand and the parameters that affect it can be stated with certainty. in some cases, the parameters cannot be stated with certainty, for example, data obtained from the forecasting process, unequal and unpredictable demand data in the next period. fuzzy probabilistic and stochastic approaches can be used to solve inventory problems with uncertainty. the following is research related to inventory problems with uncertainty. stochastic inventory model to the problem of perishable product inventory introduced by [8]. the queuing theory is applied by [9] to solve the stochastic inventory problems. a tri-level optimization model and a stochastic model with demand and lead time in uncertainty were introduced by [10]. in [11] discussed the problem of stochastic inventory with demand depending on price. if the parameters are considered following a certain statistical probability distribution and contain an element of uncertainty, then the probabilistic inventory model is appropriate to use, either the deterministic approach or the fuzzy approach. the probabilistic inventory model was introduced by [12]. a review of some literature related to inventory problems with a fuzzy approach is given by [13]. in [14] discussed probabilistic inventory with time-dependent storage costs and considering the extent of the damage. a two-parameter probabilistic inventory model was introduced by [15] and applied to chemical inventory. in [16] discussed a probabilistic inventory model for waiting times. fuzzy concepts can be applied to inventory problems with uncertainty. the following is research related to the fuzzy concept. the fuzzy concept to the forecasted data is discussed by [17]. in [18] studied the relationship between fuzzy and probabilistic concepts. the method for solving nonlinear problems with fuzzy intervals was introduced by [19]. the concept of zero set in fuzzy optimization problems is used by [20]. the fuzzy concept is also discussed by [21] and applied to fuzzy transportation problems. in this study, a two-parameter probabilistic inventory model introduced by [15] was implemented in the red chilli inventory planning problem where the cost and time parameters are represented by triangular fuzzy numbers. the tolerance value to the left and right of the triangular fuzzy number is taken from the deviation value of the data. the defuzzification stage uses the concept of α-cut. statistical parameters were estimated using the maximum likelihood. this research discusses optimizing the inventory level of red chillies and the optimal ordering cycle at one of the traders located in the modern market of palembang city. the merchant places orders every day. unsold red chillies are not stored in a special room. the level of damage to red chillies is assumed to follow an exponential distribution. the demand for unsold red chillies sold every day is limited, in this study the level of demand is assumed to follow the pareto distribution. eka susanti et al. optimization of inventory level using fuzzy probabilistic exponential two parameters model 126 2. research method the data used in this research is primary data. data were collected by interviewing one of the traders. the data collection period is 30 days. interviews were conducted by telephone. primary data consist of purchase data, the number of red chillies sold every day. following is the procedure for solving the problem of optimization of inventory levels and the cycle time for ordering red chillies. a. defining parameters and variables there are six cost parameters (𝐶𝑝, 𝐶𝑜, 𝐶𝑏 , 𝐶ℎ , 𝐶𝑑 , 𝐾𝑑 ), namely the cost of purchasing unit items for one order cycle, ordering costs, backlogged costs, storage costs, damage costs, and goals related to the estimated cost of damage. based on 30 days records, the mean and deviation for each cost were obtained. cost and deviation data are given in table 1. statistical parameters of the two-parameter exponential distribution and the pareto distribution were estimated using the maximum likelihood method. estimated data are data on the level of damage and the level of demand. b. fuzzification stage at this stage, the parameter values are defined in the form of triangular fuzzy numbers. the definition of values is based on the data obtained. this parameter is a cost parameter. following are given the values of fuzzy parameters and statistical parameters. �̃�𝑝, �̃�𝑜, �̃�𝑏 , �̃�ℎ , �̃�𝑑, �̃�𝑑; 𝜎, 𝜂 c. defuzzification stage at the defuzzification stage, the fuzzy parameters are transformed into a deterministic form using the α-cut concept. d. formulation of a two-parameter fuzzy probabilistic model the following gives a fuzzy probabilistic model with a pareto distribution of demand and the level of damage with an exponential distribution introduced by [15]. minimize 𝐸(𝑇𝐶(𝑄𝑚 , 𝑛𝑑 , 𝑛1, 𝑁, )) (1) subject to 𝐸(𝐷𝐶(𝑄𝑚 , 𝑛𝑑 , 𝑛1, 𝑁, )) ≤ 𝑘�̃� (2) where 𝐸(𝑇𝐶) : expected annual total cost 𝐸(𝐷𝐶) : expected varying deteriorating cost 𝐸(𝑉𝐶) : expected the salvage cost 𝐸(𝐵𝐶) : expected backorder cost 𝑄𝑚 : maximum inventory level (kg) 𝑛𝑑 : time of deterioration (days) 𝑛1 : time of stock out (days) 𝑁 : time of review cycle (days) 𝑘�̃� : the fuzzy goal associated to expected deterioration cost (rupiah) �̃�𝑝 : the fuzzy purchase cost unit item at a cycle ordering (rupiah) �̃�𝑜 : the fuzzy ordering cost unit item at a cycle ordering (rupiah) �̃�𝑏 : the fuzzy backlogged cost unit item at a cycle ordering (rupiah) �̃�ℎ : the fuzzy storage cost unit item at a cycle ordering (rupiah) �̃�𝑑 : the fuzzy damage cost unit item at a cycle ordering (rupiah) e. the nonlinear model solution obtained in step (4) uses lingo software model formulations and completion procedures are given in the results and discussion sections. jurnal matematika mantik vol 7, no 2, october 2021, pp. 124-131 127 3. result and discussions in this study, it was determined the optimal supply and ordering time on the optimization problem of supply of red chilli from one of the traders. cost parameters were measured for 30 days of recording. the following are given the average and deviation of each cost parameter. table 1. costs parameter data cost parameter average (rupiah) deviation (rupiah) 𝐶𝑝 36800 7500 𝐶𝑜 6000 2200 𝐶𝑏 38400 500 𝐶ℎ 17500 6900 𝐶𝑑 1900 2000 𝐾𝑑 7000 1000 in this study, a fuzzy probabilistic model with two-parameter exponential distribution and a pareto distribution was implemented. the cost parameter values are expressed in the triangular fuzzy numbers. based on the data in table 1, the triangular fuzzy number can be determined as introduced by [15]. the following parameter values are given. 𝐶�̃� = (𝐶𝑝 − 𝜔1, 𝐶𝑝 , 𝐶𝑝 + 𝜔2) = (29300; 36800 ; 44400) 𝐶�̃� = (𝐶𝑜 − 𝜔3, 𝐶𝑜 , 𝐶𝑜 + 𝜔4) = (3800; 6000; 8400) 𝐶ℎ̃ = (𝐶ℎ − 𝜔5, 𝐶ℎ , 𝐶ℎ + 𝜔6) = (17000; 17500; 19000) 𝐶�̃� = (𝐶𝑏 − 𝜔7, 𝐶𝑏 , 𝐶𝑏 + 𝜔8) = (31500; 38400; 45400) 𝐶�̃� = (𝐶𝑑 − 𝜔9, 𝐶𝑑 , 𝐶𝑑 + 𝜔10) = (4000; 6000; 19000) 𝐾�̃� = (𝐾𝑑 − 𝜔11, 𝐾𝑑 , 𝐾𝑑 + 𝜔12) = (6000; 7000 ; 9000) where 𝜔𝑖 , 𝑖 = 1,2, … ,12 are arbitrary positive numbers that satisfy the following constraints. 0 ≤ 𝜔1 ≤ 𝐶𝑝, 𝜔2 ≥ 0 0 ≤ 𝜔3 ≤ 𝐶𝑜, 𝜔4 ≥ 0 0 ≤ 𝜔5 ≤ 𝐶ℎ , 𝜔6 ≥ 0 0 ≤ 𝜔7 ≤ 𝐶𝑏 , 𝜔8 ≥ 0 0 ≤ 𝜔9 ≤ 𝐶𝑑 , 𝜔10 ≥ 0 0 ≤ 𝜔11 ≤ 𝐾𝑑 , 𝜔12 ≥ 0 using the concept of α-cut introduced by [15] at the defuzzification stage obtained the following parameter values. • �̃�𝑝 = 𝐶𝑝 + 1 4 (𝜔2 − 𝜔1) �̃�𝑝 = 36800 + 1 4 (7600 − 7500) �̃�𝑝 = 36800 + 1 4 (100) �̃�𝑝 = 36825 • �̃�ℎ = 𝐶ℎ + 1 4 (𝜔6 − 𝜔5) �̃�ℎ = 17500 + 1 4 (1500 − 500) �̃�ℎ = 17500 + 1 4 (1000) �̃�ℎ = 17750 eka susanti et al. optimization of inventory level using fuzzy probabilistic exponential two parameters model 128 • �̃�𝑏 = 𝐶𝑏 + 1 4 (𝜔8 − 𝜔7) �̃�𝑏 = 38400 + 1 4 (7000 − 6900) �̃�𝑏 = 38400 + 1 4 (1100) �̃�𝑏 = 38425 • �̃�𝑑 = 𝐶𝑑 + 1 4 (𝜔10 − 𝜔9) �̃�𝑑 = 6000 + 1 4 (13000 − 5600) �̃�𝑑 = 6000 + 1 4 (7400) �̃�𝑑 = 7850 • �̃�𝑑 = 𝐾𝑑 + 1 4 (𝜔12 − 𝜔11) �̃�𝑑 = 7000 + 1 4 (2000 − 1000) �̃�𝑑 = 7000 + 1 4 (1000) �̃�𝑑 = 7250 the parameter estimation for the level of deterioration has an exponential distribution of two parameters and the level of demand has a pareto distribution. parameter estimation using maximum likelihood. 𝐿(𝜎, 𝜇) = ∏ 1 𝜎 𝑒 − (𝑡−𝜇) 𝜎 𝑛 𝑖=1 = ( 1 𝜎 ) 𝑛 exp ( − ∑(𝑥𝑖−𝜇) 𝜎 ) , ∀𝑥𝑖 ≥ 𝜇 𝑑 ln 𝐿(𝜎,𝜇) 𝑑𝜎 = − 𝑛 𝜎 + ∑(𝑥𝑖−𝜇) 𝜎2 = 0, �̂� = ∑(𝑥𝑖−�̂�) 𝑛 𝐿(𝜂, 𝛿) = ∏ 𝜂𝛿𝜂 (𝑥+𝛿)𝜂+1 𝑛 𝑖=1 = 𝜂𝑛 𝛿 𝑛𝜂 ∏ 1 (𝑥+𝛿)𝜂+1 𝑛 𝑖=1 = ln 𝜂𝑛 + ln 𝛿 𝑛𝜂 + (−𝜂 − 1) ∑ ln 𝑥𝑖 𝑛 𝑖=1 + 𝛿 = 𝜕 𝜕𝜂 [ln 𝜂𝑛 + ln 𝛿 𝑛𝜂 + (−𝜂 − 1) ∑ ln 𝑥𝑖 𝑛 𝑖=1 + 𝛿] = 𝑛 𝜂 + 𝑛 ln 𝛿 − ∑ ln 𝑥𝑖 𝑛 𝑖=1 + 𝛿 = 0 𝑛 𝜂 = ∑ ln 𝑥𝑖 𝑛 𝑖=1 + 𝛿 − ln 𝛿 �̂�𝐿 = 𝑛 ∑ ln 𝑥𝑖+𝛿 𝛿 𝑛 𝑖=0 , 𝛿 = min 𝑥𝑖 the estimated value are 𝛿 = 1,47 𝜂 = 0,8618 , 𝜎 = 0,08 . the formulation of a twoparameter fuzzy probabilistic model on the problem of determining the level of inventory red chilis is as follows. min 𝐸(𝑇𝐶) = 𝐸(𝑂𝐶) + 𝐸(𝑃𝐶) + 𝐸(𝐷𝐶) + 𝐸(𝑉𝐶) + 𝐸(𝐵𝐶) (3) subject to jurnal matematika mantik vol 7, no 2, october 2021, pp. 124-131 129 𝐸(𝐷𝐶) ≤ �̃�𝑑 (4) 𝑁 ≥ 0 (5) where 𝐸(𝑂𝐶) = �̃�0 = 6050 𝐸(𝑃𝐶) = �̃�𝑝 �̃� ∫ 𝑥 𝑓(𝑥) 𝑑𝑥 ∞ 𝑥=0 𝐸 (𝑃𝐶) = 36.825 �̃� ∫ 𝑥 0,8618 × 1,470,8618 (𝑥 + 1,47)0,8618 𝑑𝑥 ∞ 𝑥=0 𝐸 (𝐷𝐶) = �̃�𝑑 𝑁 𝛽 (�̃�𝑚 − ∫ 𝑥 𝑓(𝑥)𝑑𝑥 �̃�𝑚 𝑥=0 ) = 7850 𝑁𝛽 (�̃�𝑚 − ∫ 𝑥 0,8618 × 1,470,8618 (𝑥 + 1,47)0,8618 𝑑𝑥 �̃�𝑚 𝑥=0 ) 𝐸 (𝑉𝐶) = �̃�𝑑 𝛾 𝑁 𝛽 (𝑄𝑚 − ∫ 𝑥 𝑓(𝑥) 𝑑𝑥 𝑄𝑚 𝑥=0 ) = 7850 (0,02) 𝑁𝛽 (𝑄𝑚 − ∫ 𝑥 0,8618 × 1,470.8618 (𝑥 + 1,47)0,8618 𝑑𝑥 𝑄𝑚 𝑥=0 ) 𝐸 (𝐵𝐶) = �̃�𝑏 [∫ 𝑥 −�̃�𝑚 ∈ (1 − 1 ∈(�̃�−�̃�1) ln[1+ ∈ (�̃� − �̃�1)]) 𝑓(𝑥)𝑑𝑥 ∞ �̃�𝑚 ] 𝐸 (𝐵𝐶) = 38425 [ ∫ 𝑥 − �̃�𝑚 0,5 (1 ∞ �̃�𝑚 − 1 0,5(�̃� − �̃�1) ln[1 + 0,5(�̃� − �̃�1)]) 0,8618 × 1,470,8618 (𝑥 + 1,47)0,8618 𝑑𝑥] the optimal solution is determined for some 𝛽 value. 𝛽 is an arbitrary positive number with 0 ≤ 𝛽 ≤ 1. for the 𝛽 = 0,1; 0,3; 0,5; 0,9; 1. nonlinear models (3), (4), (5) are solved using the lingo software, the following solution is obtained. table 2. solution nonlinear model 𝛽 𝑁 𝑄𝑚 𝐸(𝑇𝐶) 𝐸(𝑇𝐶)/𝑄𝑚 0,1 0 10 397748,6 39774,86 0,3 0 10 397748,6 39774,86 0,5 0,128 15 405022 27001,466 0,9 0,247 15 405240,2 27001,48 1 0,923 10 405143,6 40514,36 the minimum total cost is influenced by the variable value 𝛽, 𝑁 and 𝑄𝑚. based on table 2, it can be seen that the more the value approaches 1, the ordering cycle is getting closer to 1 and the total costs incurred are increasing. the maximum inventory is 15 kg and the largest total cost is rp 405143,6. the largest total cost occurs when the value of 𝑁 increases. the length of the time of review cycle is affected by the remaining inventory and the remaining product incurs storage costs. base on the results obtained, the optimal eka susanti et al. optimization of inventory level using fuzzy probabilistic exponential two parameters model 130 amount of red chillies that can be ordered so that the minimum cost is 10 kg with the length of the time of review cycle is 0. a value of 𝑁 equal to zero means that the red chillies ordered were sold out immediately or the seller can order as much as 15 kg with a review cycle duration of 0,128 days or approximately 3 hours. 4. conclusion based on the results and discussion, it can be concluded that the fuzzy approach can be applied to inventory optimization problems with uncertainty. the choice of 𝛽 variable value influences the solution obtained. the ordering cycle is getting closer to one day and the total costs incurred are increasing for the 𝛽 value close to 1. 𝛽 value affects the value of 𝑁. the greater the 𝛽 value, the greater the value of 𝑁. consequently, the total costs are increasing. the maximum inventory level of red chillies is 15 kg. acknowledgements this research is supported by universitas sriwijaya through sains teknologi dan seni (sateks) research scheme with the number of the research assignment contract number 0163.175/un9/sb3.lppm.pt/2020. references [1] x. zheng, m. yin, and y. zhang, “integrated optimization of location, inventory and routing in supply chain network design,” transp. res. part b, vol. 121, pp. 1–20, 2019. 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[20] h. wu, “applying the concept of null set to solve the fuzzy optimization problems,” fuzzy optim. decis. mak., vol. 18, no. 3, pp. 279–314, 2019. [21] p. t. ngastiti, b. surarso, and s. sutimin, “comparison between zero point and zero suffix methods in fuzzy transportation problems,” j. mat. “mantik,” vol. 6, no. 1, pp. 38–46, 2020, doi: 10.15642/mantik.2020.6.1.38-46. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: adi damanhuri, adidamanhuri@uinsby.ac.id department of islamic astronomy uin sunan ampel surabaya, east java, 60237 the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.28-35 ideal data to determine accurate fajr time adi damanhuri1, akh mukarrom1 1department of islamic astronomy, uin sunan ampel, surabaya, indonesia article history: received dec 4, 2021 revised, jun 2, 2022 accepted, jun 28, 2022 kata kunci: cahaya fajar, waktu subuh, sky quality meter abstrak. penelitian awal waktu fajar banyak dilakukan oleh berbagai pihak dengan beragam teknik, salah satunya menggunakan alat fotometri sky quality meter (sqm). data pengamatan dari berbagai daerah yang memiliki level malam bervariasi menghasilkan awal waktu fajar yang bervariasi juga. dengan memperhatikan pengaruh kualitas langit yang direpresentasikan level malam pada lokasi pengamatan, penelitian ini ingin menjawab apakah batas level malam 20mpsas sebagai data ideal dengan melihat koefisien korelasi antara level malam dengan solusi titik belok. dari 1068 data dengan level malam bervariasi menunjukkan koefisien korelasi (𝑅2) antara level malam dengan solusi titik beloknya adalah 0,42 yang berarti ada pengaruh, sedangkan untuk data-data dengan level malam minimal 20mpsas koefisien korelasinya sebesar 0,07 yang berarti tidak ada pengaruh. berdasarkan hasil analisis, level malam 20 mpsas bisa menjadi batas minimal untuk melakukan penelitian awal waktu fajar yang ideal. dari 241 data pengamatan yang ideal dari 6 stasiun pengamatan lapan, awal waktu fajar hadir saat sudut elevasi matahari -16,51°. awal waktu fajar juga merupakan awal waktu subuh, dengan standar waktu subuh yang digunakan di indonesia yaitu -20° atau berbeda 3,49° dengan hasil analisis, jika dikonversi ada perbedaan 13 menit 57 detik. keywords: dawn light, fajr time, sky quality meter abstract. early fajr time research was carried out by various parties with various techniques, one of which was using a sky quality meter (sqm) photometric tool. observational data from various regions that have varying night levels result in a varying early fajr time as well. by paying attention to the effect of sky quality represented by the night level at the observation location, this research wants to answer whether the 20 mpsas night level limit is ideal data by looking at the correlation coefficient between night level and the turning point solution. from 1068 data with varying night levels, the correlation coefficient (𝑅2) between the night level and the turning point solution is 0,42 which means there is an effect, while for data with a minimum night level of 20mpsas the correlation coefficient is 0,07 which means there is no influence. based on the results of the analysis, the night level of 20mpsas can be the minimum limit for conducting an ideal early fajr time research. from 241 ideal observation data from 6 lapan observation stations, early fajr time presents when the sun's elevation angle is -16,51°. early fajr time is also the beginning of subuh prayer time, with its standard used in indonesia, which is 20° or 3,49° different from the analysis results, if it converted there is a difference of 13 minutes 57 seconds. how to cite: a. damanhuri and a. mukarrom, “ideal data to determine accurate fajar time”, j. mat. mantik, vol. 8, no. 1, pp. 28-35, jun. 2022 jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 28-35 issn: 2527-3159 (print) 2527-3167 (online) mailto:adidamanhuri@uinsby.ac.id https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 a. damanhuri and akh. mukarrom ideal data to determine accurate fajar time 29 1. introduction the beginning of the presence of the dawn light on the western horizon in addition to marking the end of the night, it is also for muslims as a sign of the entry of the subuh prayer time. when the sun is below the horizon with a certain altitude, sunlight interacts with the upper atmosphere so that the earth's surface will appear slightly reddish [1]⁠. in practice, the presence of dawn is represented by the elevation angle of the sun or the position of the sun below the horizon. the sun's elevation angle below the horizon, which indicates the presence of dawn, also indicates the beginning of subuh prayer time. in indonesia, the elevation angle of the sun used for the beginning of subuh prayer time is -20° or zenith=110° [2], lately this standard has been questioned because it is assumed that when the sun is in that position, the dawn light on the western horizon has not yet appeared. dawn light research has been carried out with various approaches, some using the naked eye [3]⁠⁠, using a digital camera [4][5]⁠, and some using simple photometric tools, namely sky quality meter (sqm) [2], [6]–[8]. sqm is a simple photometric tool that can quantify the quality of sky darkness in units of magnitude per arc second square (mpsas) [9]–[12]. research of the beginning of subuh prayer time using sqm was carried out by looking for turning points from sqm data which indicate the presence of dawn due to changes in sky conditions from dark to bright due to the presence of dawn light. then the obtained brightness level also indicates the level of light pollution, the higher the brightness in the mpsas, the lower the level of light pollution [13]–[15] ⁠. light pollution has an impact on the emergence of dawn light in the form of pseudo-night which is caused by the absorption between sunlight and pollutant particles in the atmosphere, to overcome this, it is necessary to select observational data from the night which has a night level of at least 20 mpsas which corresponds to grade 4 on the bortle scale [16], [17] ⁠or bright category [18]⁠. there are several techniques used to determine inflection points, including using polynomial functions as has been done by [8]⁠, with this method, the determination of the inflection point uses a degree of polynomial that varies from polynomial of degree 3, polynomial of degree 4, polynomial degree 5, to polynomial degree 6, depending on the sqm data itself, even several times have to reduce unnecessary data. in addition, in the research conducted by saksono, et al. used less ideal data because the data used came from a location have sky quality below 20mpsas or fall into category 5 and above on the bortle scale [16], [17]⁠ . in this study, determining the turning point of sqm data uses the solver method, which is a method that uses the exponential function but to determine the variables used as parameters for its function using the solver menu [19]–[21]⁠, besides the data used for determining the elevation angle of the sun that indicates the presence of dawn light is only data that has been selected with the condition that it comes from data that has a minimum night level of 20mpsas or category 4 on the bortle scale in order to reduce the effects of air pollution that can interfere with sqm signal capture. by considering light pollution which can cause interference during observation [15],[16] the data selected with the main criteria must come from data that has a night level of at least 20 mpsas or class 4 on the bortle scale [16], [17]⁠ or bright class [18]⁠. by connecting the night level and the turning point solution using a regression function to obtain a correlation coefficient (𝑅2), this research tries to answer whether the minimum 20mpsas night level limit show no effect between the night level and the turning point solution or still shows a significant effect. the correlation level will be consulted with the guilford empirical rules correlation table [22]. if the night level of at least 20mpsas shows no effect between the night level and the turning point solution, then the ideal data to determine the beginning of fajr time is data from observations with a night level of at least 20mpsas, and what the elevation angle of the sun as a marker of the beginning of fajr time based on the ideal data. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.28-35 30 2. material and method 2.1 sky quality meter sky quality meter (sqm) is a simple and very easy to use photometric tool for measuring sky brightness levels. sqm is produced by unihedron, and has been widely used for sky quality mapping research [23], [24] determine astronomical site [25], [26]⁠, light pollution research [25], [27]–[29] research in the health sector [28]⁠, behavioural research birds [30], [31]⁠, to research on determining the time prayer [2], [8], [13], [15] ⁠. sqm uses a light frequency sensor tsl237, sqm in this study is set to quantify in mpsas every 1 minute with the direction of the sensor to the zenith [32]. observations were made throughout the day in 2019. the data generated from sqm is in the form of ansi data which contains information on sky brightness levels, time and date data in universal time (ut) as well as local time, solar elevation angle data, temperature data and count data. sky brightness level is expressed in magnitude per square arc second (mpsas), where sqm has a precision level of 0.1 mpsas [14]. this will be used for this study only data on sky brightness, time and date data, as well as data on the elevation angle of the sun. 2.2 location and data selection sqm data is taken from the observation data from 6 lapan observation stations, namely in agam, garut, kupang, pasuruan, pontianak and sumedang. all data that can be analyzed are 1,068 data. even though observations are made throughout the year, not all data can be processed for several reasons, including because of interference at critical moments such as from the moonlight, the data is disturbed due to the presence of clouds, to disturbances due to rain. as shown in table 1, the data to be used in determining the early dawn of time must fulfill the requirements, namely that the night level is at least 20 mpsas. the night level is obtained by averaging the mpsas from 00.00 am until the height of the sun reaches -20° (z=110°). to determine the dawn time in this study only selected data were used. table 1. location of observation location latitude longitude number of data agam, west sumatera -0.204430 100.320057 118 garut, west java -7.650062 107.692214 65 kupang, east nusa tenggar -10.142009 123.731231 224 pasuruan, east java -7.567506 112.673702 239 pontianak, west borneo -0.007800 109.365000 221 sumedang, west java -6.913079 107.837213 201 2.3 method of analysis to determine the turning point on the graph, two data are used, namely the solar elevation angle data and the mpsas value, then the mpsas data is approached with the exponential function obtained using formula (1) of normal distribution [33], [34]. to simplify the calculation, the solver menu is used, so that the prediction function gets an optimal minimum difference from the original data. 𝑓(𝑥) = 𝐶 − 𝑁 × 1 𝜎√2𝜋 𝑒 −( 𝑥−𝜇 2𝜎 ) 2 (1) where c is a constant level, n is normalized, µ is mean, and σ is the standard deviation. the values for c, n, µ, and σ will be obtained using the menu solver, respectively, with the variables c, n, µ, and σ being used as boundary variables, or chisquare being selected as the objective variable. meanwhile, to determine the inflection point, it is obtained from the formula (2): a. damanhuri and akh. mukarrom ideal data to determine accurate fajar time 31 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = 𝜇 − 3𝜎 (2) the result of the calculation using turning point formula is the final result that shows the elevation angle of the sun at dawn. for example, for analysis on 10 january 2019 for the pontianak location, the model is obtained as shown in figure 1. figure 1. plot of solver analysis after the turning point of each data is known, the next step is to look at the correlation or effect between the solution or the turning point which represents the presence of dawn light with the night level which represents the quality of the sky. correlation is represented by the correlation coefficient or r2, the smaller the value of r2 shows that the correlation is getting smaller. to obtain the correlation coefficient (r2) with the formula: 𝑅2 = 𝑆𝑆𝑅 𝑆𝑆𝑅+𝑆𝑆𝐸 (3) where r 2 is coefficient correlation, ssr is a total regression sum of squares, sse is an error sum of squares. plotting and calculating correlation coefficients using the gnu plot [35]. furthermore, to determine the level of correlation, r 2 is compared with the guilford empirical rules correlation table as shown in table 2 below. table 2. table guilford empirical rules 𝑹𝟐 interpretation 0,00 < 0,20 0,20 < 0,40 0,40 < 0,70 0,70 < 0,90 0,90 ≤ 1,00 very weak relationship (ignored, presumed non-existent) low relationship medium or moderate relationship strong or high relationship very strong or very high relationship 3. result and discussion from 1,068 data with various night level conditions, it shows r 2 = 0,42 as shown in figure 2(a) while for a minimum night level of 20mpsas, 241 data is obtained and shows r 2 = 0,07 as shown in figure 2(b). jurnal matematika mantik vol. 8, no. 1, june 2022, pp.28-35 32 (a) (b) figure 2. (a) plotting all data, (b) plotting for minimum night level 20 mpsas based on observations of sky brightness using the sky quality meter at 6 lapan observation station as shown in table 1, obtained 1.068 daily data with the lowest night level of 14,64mpsas and the highest 23,86 mpsas and an average of 18,49 mpsas. from the 1.068 data, the solution for the lowest turning point is obtained with the sun's elevation angle of -21,71°, the highest sun's elevation angle of -8,16°, and an average of -14,86°. the correlation coefficient (r2) between the turning point solution and the night level is 0,42 which indicates that there is still a significant effect of the night level on the turning point solution. as for the data with a minimum night level of 20mpsas, there are only 241 data, with the lowest turning point solution with a sun’s elevation angle of -21,71° and the highest turning point solution with a sun’s elevation angle of -8,91° with an average of -16,51 ° with a correlation coefficient of 0,07. with a night level of at least 20mpsas or at grade 4 on the bortle scale, it shows the agreement with r2 between the turning point solution and the night level is small, which is 0,07. this shows that the effect of night level on the turning point solution is very small or can be ignored. 4. conclusions data with a night level of at least 20 mpsas or grade 4 on the bortle scale shows a correlation coefficient (r2) of 0,07 which is based on the guilford empirical rules table showing the correlation between night levels which represent night quality and the solution of the turning point or elevation angle of the sun which indicates the fajr time is very weak and can be ignored or considered non-existent. and from 241 data with a night level of at least 20mpsas, it shows that on average the fajr time is present when the sun's elevation is at -16,51° or 3,49° different from official standard and if it is converted into a time of about 13 minutes 57 seconds. if rounded, -16,51° becomes -17° and differs by about 3° or if converted to about 12 minutes later than the beginning of the official early subuh prayer time. 5. acknowledgement thanks to the national aviation and space agency of republic indonesia or lembaga penerbangan dan antariksa nasional (lapan) for providing sky observation data with sqm to the public. and 2021 research funding from uin sunan ampel surabaya with cluster interdisciplinary research. a. damanhuri and akh. mukarrom ideal data to determine accurate fajar time 33 references [1] j. b. kaler, the ever-changing sky: a guide to the celestial sphere, 1st ed. cambridge: cambridge university press, 2002, p. 495. 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[35] t. williams et al., “gnuplot 5.2 an interactive plotting program,” vol. 8, no. december, 2019, [online]. available: http://www.gnuplot.info/docs_5.2/gnuplot_5.2.pdf jurnal matematika mantik vol 7, no 2, october 2021, pp. 20-29 contact: abdul karim, abdulkarim@walisongo.ac.id universitas islam negeri walisongo, semarang, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.147-154 regional economic growth: a spatial durbin model approach abdul karim universitas islam negeri walisongo, semarang, indonesia article history: received, mar 24, 2021 revised, aug 19, 2021 accepted, oct 31, 2021 kata kunci: spatial durbin model, spatial autoregressive model, model spatial, spatial autocorrelation, pertumbuhan ekonomi. abstrak. tujuan penelitian ini adalah untuk mengetahui pengaruh ketergantungan spasial (spatial dependence) terhadap produk domestik regional bruto (pdrb) di provinsi jawa tengah. spatial durbin model (sdm) merupakan model regresi yang terdiri dari struktur data spasial yang merupakan pengembangan dari spatial autoregressive model (sar). terdapat tambahan efek spasial pada komponen variabel independen yang tidak terdapat dalam model sar atau biasa disebut efek tidak langsung pada variabel independen. hal ini mengindikasikan sdm memiliki kelebihan dibandingkan dengan sar karena terdapat efek spasial pada variabel dependen dan independent, matriks pembobot spasial yang digunakan dalam penelitian adalah rownormalized binary contiguity. data yang digunakan dalam penelitian ini bersumber dari badan pusat statistik (bps) jawa tengah tahun 2019 untuk 35 kabupaten dan kota, dimana terdiri dari pdrb sebagai variabel dependen dan tenaga kerja, sumber daya manusia, dan infrastruktur jalan sebagai variabel independen. berdasarkan hasil analisis, nilai aic menunjukkan bahwa sdm secara signifikan lebih baik daripada model ordinary least square (ols) dan sar. hasil sdm menunjukkan bahwa sumber daya manusia memiliki tanda postifi dan pengaruh langsung (direct effect) sebesar 88.5 persen serta memiliki pengaruh tidak langsung (indirect effect) sebesar 13.1 persen. selain itu, variabel tenaga kerja memiliki pengaruh tidak langsung (indirect effect) terhadap pdrb sebesar 22.2 persen. keywords: spatial durbin model, spatial autoregressive model, spatial modeling, spatial autocorrelation, economy growth. abstract. the purpose of this study is to determine the effect of spatial dependence on gross regional domestic product (grdp) in central java province. spatial durbin model (sdm) is a regression model consisting of a spatial data structure which is the development of the spatial autoregressive model (sar). there is an additional spatial effect on the component of the independent variable that is not included in the sar model or commonly referred to as an indirect effect on the independent variable. this indicates that sdm has advantages compared to sar because there are spatial effects on the dependent and independent variables, the spatial weighted matrix used in this study is row-normalized binary contiguity. the data used in this study is sourced from the central java statistics agency (bps) in 2019 for 35 districts and cities, which grdp as the dependent variable, labor, human resources, and road infrastructure as independent variables. based on the results of the analysis, the aic value shows that sdm is significantly better than the ordinary least square (ols) and sar models. sdm results show that human resources have a positive sign and a direct effect of 88.5 percent and an indirect effect of 13.1 percent. in addition, the labor variable has an indirect effect on grdp of 22.2 percent. how to cite: abdul karim, “regional economic growth: a spatial durbin model approach, j. mat. mantik, vol. 7, no. 2, pp. 147-154, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 147-154 issn: 2527-3159 (print) 2527-3167 (online) mailto:abdulkarim@walisongo.ac.id http://u.lipi.go.id/1458103791 jurnal matematika mantik vol 7, no 2, october 2021, pp. 147-154 148 1. introduction gross regional domestic product (grdp) is one of important indicators to determine the economic conditions of a region within a particular period. grdp is defined as the amount of added value produced by all business units in an area or the total value of final goods and services produced by all economic units. it is not only an important indicator in determining the success of regional economic growth achieved, but grdp can also be used as a basis for determining the direction of development in the future, [1] . grdp is a record of total value in rupiah from the final goods and services produced by an economy in a region for one year, [2]. in other words, an area experiences economic growth when there is an increase in real grdp with the production added value created by the increase of all economic activity sectors (business field) in a region. it can be used as a success indication of economic development. central java is a province with a quite potential contribution in national economic growth. as the province with the third largest population after west java and east java, the economy of central java is relatively stable. based on the data from the indonesian central statistics agency of 2012 and 2013, indonesia's economic growth was at the level of 5.34 percent and 5.14 percent respectively, while there was an upward trend in 2014 of 5.42 percent and 5.44 percent in 2015. different from the indonesia's declining economic growth in the last three years, central java's economic growth has experienced an upward trend higher than the level of national economic growth. the other form of spatial data-based regression analysis in regression model is spatial regression, and one of the most popular models is spatial autoregressive model (sar). this model defines the effect of lag spatial on response variable [3]. the development of this model is spatial durbin model (sdm). sdm is a special case of spatial autoregressive, which is the addition of lag spatial effect on dependent and independent variables [4]. sdm involves lag spatial from dependent and independent variables, resulting in different estimates for the β parameters of general regression. this model is able to define the indirect impacts arising from the changes in dependent and independent variables. according to [5], many studies in spatial model for economic growth had been conducted, for example, the regional spillovers from transportation infrastructure by applying spatial durbin model for the period of 1978-2009 and three sub-periods of 19781990, 1991-2000, and 2001-2009. the results show that the spillovers are positively at national level in each period due to the characteristics of transportation infrastructure connectivity. at regional level, the spillover effect of transportation infrastructure varied greatly throughout the year in four major regions of china. sdm was utilized by [6] to define transportation infrastructure to grdp in 47 cities in spain. the result is that the variable of road infrastructure has a positive significant effect on grdp. similar conclusions are also derived from the analysis of [7] that transport infrastructure plays a major role in the new silk road economic belt (nsreb) regional economic growth, and economic growth strengthens development in the surrounding region. in addition, the mode of road transportation influences regional economic growth is higher than the mode of railway transportation. the aim of this study was to determine the direct and indirect effects of transportation infrastructure through spatial dependency test to the dependent and independent variables of the data of the gross regional domestic product (pdrb) in central java province using spatial durbin model. abdul karim regional economic growth: a spatial durbin model approach 149 2. theoretical framework 2.1. spatial dependence regression models for spatial data require diagnostic testing for spatial dependence in errors using moran’s i test. spatial dependence test is needed to measure the autocorrelation between regions or observations.[8] the following is moran’s i test when using an unstandardized weighted matrix w*, 0 n ' * m i s = e w e e'e (1) 0 1 1 with * n n ij i j s w = = =  and when the standardized weighted matrix w is used, equation 1 is simplified to ' m i = e we e'e (2) because of s0 = n, im interpretable as the coefficient of an ols regression of w*e on e or we on e, respectively. e represent n x 1 vector of ols residuals, w is a weighted matrix, with elements wij = 1 when two cities share a common border, and 0 otherwise. 2.2. spatial durbin model spatial auto regressive (sar) model includes a lagged-response regressor and is specified in equation 3,[9] 1 . n i i ij j i i i j y w y x    = = + + + (3) when written in the form of matrix: y = wy + xβ +ε (4) where y is an n by 1 vector of the dependent variable, ρ is spatial lag coefficient, w is a row-normalized binary contiguity weighted matrix, with the element of wij = 1 when two cities share a common border, and 0 otherwise. x is an n by k matrix of the independent variables, β is a k by 1 vector of parameters and ε is an n by 1 vector errors. the development of the spatial autoregressive (sar) model is a spatial mixed regressive-autoregressive model, which is also called the spatial durbin model (sdm). there is an addition of spatial lag to the independent variables. this model was developed because the spatial dependencies do not only occur in the dependent variables. [10] this model has the following equations, ( )++++++++=  = i1ki1k2i121i110 n 1j jiji ........ ll xxxxywy  n n n n 21 ij 1j 22 ij 2j 2k ij kj 2 ij j i j 1 j 1 j 1 j 1 .... .... . l l w x w x w x w x ε    = = = =   + + + + + +        (5) n n i ij j 0 1k ki 2k ij kj i j 1 k 1 k 1 j 1 . l l y w y x w x ε    = = = = = + + + +    (6) when written in the form of matrix: ,= + +y wy zβ ε (7) with jurnal matematika mantik vol 7, no 2, october 2021, pp. 147-154 150  ,=z i x wx   , t = 0 1 2 β β β β the method of maximum likelihood estimation (mle) was used to estimate the parameters. from equation 7 above, the likelihood function is formed, and the formation of the likelihood function is done through error ε. ,− = + +y wy zβ ε ( ) ,− = +i w y zβ ε ( ) ( ) 1 1 ,  − − = − + −y i w zβ i w ε ( ) .= − −ε i w y zβ jacobian value is obtained from the result of the equation: j   = = −  ε i w y so that it results in; ( ) ( ) ( ) 2 1 2 2 222, , ; 2 , tnn l j e       − −−  = ε ε β y ( ) ( ) ( ) ( )( ) ( )( ) 2 1 2 2 222, , ; 2 . tnn i l i e          −  −    −    −  = − w y-zβ iw y-zβ β y w the natural logarithm operation (ln likelihood) in the equation above: ( ) ( )2 2 2 1 ln , , ; ln ln 2 2 n l c i     = − + − −β y w ( )( ) ( )( ) .        t i w y zβ i w y -zβ from the equation above, the parameter estimations of β̂ ,  and 2 ̂ are obtained. 2.3. goodness of fit in this paper, we use akaike information criteria (aic) for goodness of fit. according to [11], aic is a tool to measure of information that contains the best measurements in the feasibility test of model estimates. aic is usually used to choose which model is best among the models obtained. selection of the model is based on the least expected error results that form new observation data (error) that are equally distributed from the data used, aic defined: ( )2 l o g 2a i c l p= − + (8) where l represent the likelihood under the fitted model p is the number of parameters in the model. the best model is the model that has the minimum aic value. 3. methods 3.1. description of data and variables the data used in this study were the data obtained from the central java provincial statistics agency (bps) in 2019 for 35 districts and cities. the variables used in this study referred to the research of [12][13]. the followings are the research variables used: abdul karim regional economic growth: a spatial durbin model approach 151 table 1. definition of operational variables no variables indicator analysis unit dependent variable 1) economic growth (grdp) gross regional domestic product (grdp) million rupiah independent variables 2) labor (l) labor value for each city in central java person 3) human capital (hc) human capital is the number of residents with the lowest education of junior high school for each city in central java person 4) transportation infrastructure (inf) length of roads with the conditions of good and medium (km) for each city in central java km 3.2. model specification the estimated model specification in this study was based on log linear cobb-douglas production product [6]. the following is the estimated spatial durbin model: 0 1 2 3 1 2 3 i i i i i i i i i grdp grdp inf l hc inf l hc          = + + + + + + + + w w w w (9) where the variables from the equation in logarithms with ε as the error term and the subscripts states the cities in central java. the dependent variable of economic growth was measured based on the added value of each business field originated from the statistics of central java. then, the independent variable of labor was measured by each city in central java. human capital was measured by total labor with the lowest education for each city in central java, and transportation infrastructure was measured by length of roads with the conditions of good and medium for each city in central java. in addition, the equation above includes the spatial lags (w) in the dependent and independent variables recognized as spillover effect. w is defined as a row-normalized binary contiguity weighted matrix, with the element of wij = 1 when two cities share a common border, and 0 otherwise. 4. result and discussion we used moran’s i test of the ols model residuals as an assumption in spatial modeling. the assumption for sdm is that there is a spatial dependence. to test spatial dependencies, moran’s i test was used. the moran’s i test results show the moran’s i value of 3.982 and p-value 0.000. it can be concluded that there are spatial dependencies in the ols model so that spatial modeling can be performed involving spatial lag on the dependent variable. the moran’s i value is along with its significance presented in the table: table 2. moran’s i value of ols residuals moran’s i expectation p-value 3.982 -0.029 0.000 at the beginning of the discussion section, it needs to be carefully re-understood in terms of the interpretation of the model that had been produced. according to [13], sdm allows for spatial lag effects on the dependent variable and does not allow for spatial lag effects on error terms. spatial durbin model simplifies the interpretation of the direct impacts represented by the parameter of model β and the indirect impact on γ. in other jurnal matematika mantik vol 7, no 2, october 2021, pp. 147-154 152 words, the global and local multipliers that exist in the spatial durbin model facilitated the interpretation of the model estimates. table 3 shows the estimation results of the ordinary least square (ols) and sdm models in which the parameters of the two models appeared to have the effects on the grdp of central java province. based on these results, the aic value indicates that the spatial durbin model is better than the ordinary least square model because the aic value of the spatial durbin model is smaller than that of the ordinary least square model. table 3. the parameter estimation results of the ols and sdm models parameter ols coefficient (p-value) sar coefficient (p-value) sdm coefficient (p-value) 0  -0.000 0,0150 4.429 (1.000) (0,8807) (0.000)* 1  -0.009 -0,0845 -0.095 (0.559) (0,5438) (0.471) 2  -0.198 -0,1911 0.002 (0.405) (0,3738) (0.991) 3  0.978 0,9536 0.885 (0.000)* (0,000)* (0.000)* 1  -0.102 (0.051)** 2  -0.222 (0.016)* 3  0.131 (0.031)*  0,0127 0.0083 (0,53348) (0.7232) aic 74.221 75,8330 72.929 notes: *significant at α = 5 percent; ** significant at α = 10 percent the direct effect of human capital on the grdp of central java province was positive and significant. this result is in line with [14]. moreover, the results are very stable at approximately 0.88. in other hand, the results of the direct effect estimation from transportation infrastructure and labor were not suitable with the model specifications. they are negative which are similar to the estimation results conducted by [15] and [6]. however, these two variables had no significant effect on the grdp of central java province. this result can be interpreted that the grdp of central java province was influenced by human capital globally. in the spatial durbin model, although the indirect effects of all variables had a significant effect on the grdp of central java province, transportation infrastructure and labor were not suitable with the negative model specifications. the indirect effects of the spatial durbin model results can be interpreted that transportation infrastructure and labor for each city have a close relationship with the surrounding cities. the negative mark proves that the uneven development of transportation infrastructure and the uneven distribution of labor are still evident in central java. some cities have good transportation infrastructures and a lot of labor, but the surrounding areas have the opposite condition. in addition, human capital has a tendency for positive dependence between cities. furthermore, the value of rho in the spatial durbin model is not significant indicating that there is no spatial lag on economic growth. in other words, there is no association between the economic growth of a city and other cities. the economic progress in a city does not have an impact on the increase of the economy in the surrounding cities. abdul karim regional economic growth: a spatial durbin model approach 153 thus, the results of this analysis provide the evidence that the presence of unequal transportation infrastructure has an impact on the inequality in economic growth in central java. 5. conclusion today, indonesia is involved in the construction of large-scale transportation infrastructures, while the municipal economy faces quite high disparities. furthermore, we used the spatial durbin model to analyze the impact of road transportation infrastructure and other economic factors on the economic growth in 35 cities in central java province. the conclusions of this paper are as follows: first, road transportation infrastructure and labor do not have direct impact on economic growth, whereas human capital has a positive and significant direct effect on economic growth. second, transportation infrastructure and labor have a negative spatial spillover effect on economic growth, while the spatial spillover effect of human capital is positive. therefore, we recommend that the government should improve the connectivity of transportation infrastructures and optimize transportation layouts. the local governments may improve the coordination and break the barriers with the central government. references [1] n. feriyanto, “determinant of gross regional domestic product (grdp) in yogyakarta special province,” econ. j. emerg. mark., vol. 6, no. 2, pp. 131–140, oct. 2014, doi: 10.20885/ejem.vol6.iss2.art6. [2] m. egerer, e. langmantel, and m. zimmer, “gross domestic product,” in regional assessment of global change impacts, cham: springer international publishing, 2016, pp. 147–152. [3] r. k. pace, j. p. lesage, and s. zhu, “spatial dependence in regressors and its effect on estimator performance,” ssrn electron. j., 2011, doi: 10.2139/ssrn.1801241. [4] a. tientao, d. legros, and m. c. pichery, “technology spillover and tfp growth: a spatial durbin model,” int. econ., apr. 2015, doi: 10.1016/j.inteco.2015.04.004. [5] n. yu, m. de jong, s. storm, and j. mi, “spatial spillover effects of transport infrastructure: evidence from chinese regions,” j. transp. geogr., vol. 28, pp. 56– 66, apr. 2013, doi: 10.1016/j.jtrangeo.2012.10.009. [6] p. arbués, j. f. baños, and m. mayor, “the spatial productivity of transportation infrastructure,” transp. res. part a policy pract., vol. 75, pp. 166–177, may 2015, doi: 10.1016/j.tra.2015.03.010. 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[11] h. de-g. acquah, “on the comparison of akaike information criterion and consistent akaike information criterion in selection of an asymmetric price jurnal matematika mantik vol 7, no 2, october 2021, pp. 147-154 154 relationship: bootstrap simulation results,” agris on-line pap. econ. informatics, vol. 5, no. 1, pp. 3–9, mar. 2013, doi: 1804-1930. [12] a. karim, a. faturohman, s. suhartono, d. d. prastyo, and b. manfaat, “regression models for spatial data: an example from gross domestic regional bruto in province central java,” j. ekon. pembang. kaji. masal. ekon. dan pembang., vol. 18, no. 2, p. 213, dec. 2017, doi: 10.23917/jep.v18i2.4660. [13] a. karim, suhartono, and d. d. prastyo, “spatial spillover effect of transportation infrastructure on regional growth,” econ. reg., vol. 16, no. 3, pp. 911–920, 2020, doi: 10.17059/ekon.reg.2020-3-18. [14] j. p. lesage and r. k. pace, “spatial econometric modeling of origin-destination flows,” j. reg. sci., vol. 48, no. 5, pp. 941–967, 2008, doi: 10.1111/j.14679787.2008.00573.x. [15] m. a. márquez, j. ramajo, and g. j. d. hewings, “a spatio-temporal econometric model of regional growth in spain,” j. geogr. syst., vol. 12, no. 2, pp. 207–226, jun. 2010, doi: 10.1007/s10109-010-0119-3. [16] p. cantos, m. gumbau‐albert, and j. maudos, “transport infrastructures, spillover effects and regional growth: evidence of the spanish case,” transp. rev., vol. 25, no. 1, pp. 25–50, jan. 2005, doi: 10.1080/014416410001676852. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran student group dynamic model based on understanding in mathematics subjects m. ivan ariful fathoni1*, anisa fitri2, hanifahtul husnah3 1,2,3department of mathematics education, universitas nahdlatul ulama sunan giri, bojonegoro, indonesia article history: received nov 21, 2020 revised may 6, 2021 accepted may 30, 2021 kata kunci: sistem dinamik, model matematika, titik ekuilibrium, interaksi peserta didik, proses pembelajaran abstrak. penelitian ini membahas tentang interaksi peserta didik dengan sudut pandang pemodelan matematika. interaksi tersebut melibatkan peserta didik yang memahami dan belum memahami materi mata pelajaran matematika. proses interaksi antar grup peserta didik dimodelkan dalam suatu sistem persamaan diferensial dua dimensi. variabel a adalah persentase peserta didik yang memahami materi, dan variabel b adalah persentase peserta didik yang kurang memahami materi. hasil analisis dinamik diperoleh satu titik ekuilibrium trivial dan tiga titik ekuilibrium non-trivial yang eksis dengan beberapa syarat. berdasarkan analisis kestabilan titik ekuilibrium non-trivial, diperoleh kondisi tanpa adanya peserta didik yang kurang memahami materi pelajaran matematika. kondisi inilah yang menjadi tujuan dari penelitian ini, dimana dengan melibatkan interaksi antar peserta didik, dapat meningkatkan keberhasilan proses pembelajaran. keywords: dynamical system, mathematical model, equilibrium points, student interaction, learning process abstract. this study discusses the interaction of students with a mathematical modeling point of view. this interaction involves students who understand and do not understand mathematics subject matter. the interaction process between groups is modeled in a twodimensional system of differential equations. variable a is the percentage of students who understand the material, and variable b is the percentage of students who do not understand the material. the dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. this condition is the goal of this study, which involves interaction between students; it can increase the learning process's success. how to cite: m. i. a. fathoni, a. fitri, and h. husnah, “student group dynamic model based on understanding in mathematics subjects”, j. mat. mantik, vol. 7, no. 1, pp. 41-50, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 issn: 2527-3159 (print) 2527-3167 (online) contact: m. ivan ariful fathoni, fathoni@unugiri.ac.id department of mathematics education, universitas nahdlatul ulama sunan giri, bojonegoro, jawa timur 62115, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.41-50 mailto:fathoni@unugiri@ac.id https://doi.org/10.15642/mantik.2021.7.1.41-50 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 42 1. introduction education means increased skill or development of knowledge and understanding as a result of practice, study, or experience [1]. the importance of education places it at the highest level of human needs. in the introduction to the book written by klaus dieter bieter, nelson mandela calls education a powerful force that builds every human being. all countries in the world place education as one of the human rights [2]. likewise, in indonesia, education is one area that is the responsibility of the state. law of the republic of indonesia states that: education is a conscious, planned effort to create an atmosphere of learning and the learning process to actively develop their potential to have spiritual strength, self-control, personality, intelligence, noble morals, and skills needed for themselves, society, nation, and state [3] [4]. in general, education aims to increase the nation's intelligence by increasing students' understanding of the subject matter. there are many obstacles for a country to create good quality education, especially in developing countries. one of the problems faced in the world of education is the weak learning process. students are encouraged to develop thinking skills in the learning process, but in reality, students are only directed to accept and memorize information or knowledge provided by the teacher. the teaching method is one of the supporting factors for the achievement of national education goals [5]. as one of the basic sciences, mathematics has an essential role in the effort to master science and technology. mathematics in schools needs to function as a vehicle to develop intelligence, abilities, and skills and shape students' personalities [6]. not a few students think that mathematics is a difficult subject and causes various problems that are difficult to solve, resulting in low student learning outcomes. this results from the lack of interactive mathematics learning methods, where students are only asked to memorize formulas, receive the material, and work on practice questions. therefore, there is a need for an interactive process between students in the mathematics learning process. interaction can be interpreted as communication or a reciprocal relationship between two or more people for a specific purpose [7]. interaction in a lesson is critical and necessary to help develop students' understanding and achieve educational goals. learning interaction is an interactive activity of various components to realize the learning objectives set when planning to learn [8]. learning mathematics as the construction and abstraction of mathematical concepts can be maximized by solving mathematical problems. the problem-solving process will be achieved at a higher level if students work in cooperative groups, especially heterogeneously [9]. in other words, the interaction between student’s needs to be done without differentiating each student's abilities. based on the learning objectives, students with moderate and low abilities should solve problems and be confident and communicate them effectively. a learning strategy is needed to accommodate student interactions with the learning environment to achieve this goal. one of the learning strategies that support this is a learning strategy that uses cooperative groups, or so-called cooperative learning [10]. in this study, the cooperative learning process was examined by grouping students based on their understanding and mastery of mathematics material. according to murdock-stewart [11], understanding consists of three general categories, which include: (1) understanding structural progress, (2) understanding as a form of knowing, (3) understanding as a process. students are said to understand something to connect the ideas in their minds and abstract for the next step [12]. so understanding is an organizational process that involves cognitive activities to solve problems. in this study, the measurement of students' understanding was carried out through test scores in mathematics. students who have a value above minimum learning completeness are m. ivan ariful fathoni, anisa fitri, hanifahtul husnah student group dynamic model based on understanding in mathematics subjects 43 assumed to have understood the material. in contrast, students who have a value below minimum learning completeness are supposed to have not understood the material. the development of science in the field of mathematics has a role in helping to analyze the problems of everyday life. one of these problems is in the field of education. many occurrences in the surroundings can be observed and analyzed in the form of a mathematical model. mathematical models are models that represent a problem in the real world into mathematical equations [13]. in this study, we analyzed the dynamics of students' level of understanding in mathematics based on time through a modeling approach. in this modeling, the mathematical interpretation of the differential equation system model in the student population will be known based on the level of mathematical understanding. several studies on interactions in the learning process have been carried out several times [14]–[16]. in previous research, no one has been able to predict the success of learning in the future. in contrast to previous studies, students' interaction in the learning process was studied from the mathematical modeling perspective. this approach is an alternative method in solving problems in the world of education that have not been widely researched. mathematical modeling can solve a problem by simplifying cases in everyday life into a mathematical problem where the solution can be obtained. the mathematical model's solution obtained can be used to predict the conditions for the success of the learning process. 2. model formulation this study discusses the interaction of students with a mathematical modeling point of view. this interaction involves students who understand and do not yet understand the subject matter of mathematics. the interaction process between groups is modeled in a twodimensional system of differential equations. variable a is the percentage of students who understand the material, and variable b is the percentage of students who do not understand the material. the compartment model in fathoni's research [18] is a reference in this study. the basic model in this research is the predator and prey model and the logistic model. the relationship between predator and prey is the primary basis of studying ecology. one of the important components in this relationship is the speed of predators in preying on their prey. the level of prey per capita of predators or the response function is often called the basis for the predator-prey theory applied in this study. the group that acts as a predator is a group of students who understand mathematics (a). meanwhile, the group that acts as prey is students who do not understand mathematics (b). logistic models are often used to describe the growth rate, including the growth rate of the prey population [19]. the prey population does not always increase continuously, but it depends on the population's size if there is a limit to the environment's carrying capacity. likewise, the student group does not always experience an increase. the capacity of existing classrooms limits the student group. the logistic model in putranto's research [20] is applied in this study, where the increase in students' group follows the growth of logistics. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 44 figure 1. the compartment diagram by the compartment diagram on figure 1, we construct a mathematical model with two-dimensional system of the first order ordinary differential equation (ode). our model is formulated as the following, see system (1). the parameters used in the model are assumed to be positive, with the information in table 1. based on the model in equation (1), a dynamic analysis process is carried out. the discussion of dynamic analysis conducted by ningsih [21] regarding the growth in the number of students is a reference in the discussion of dynamic analysis in this study. the research carried out includes searching for the equilibrium points, analyzing the equilibrium points' existence, and analyzing the equilibrium points' stability. 𝑑𝐴 𝑑𝑡 = 𝛼𝐴 ( 1 − 𝐴 𝐾 ) + 𝜇𝐴𝐵 − 𝛾𝐴 (1) 𝑑𝐵 𝑑𝑡 = 𝛽𝐵 ( 1 − 𝐵 𝐾 ) − 𝜇𝐴𝐵 − 𝛾𝐵 table 1. parameter description parameter description unit 𝛼 student group a growth rate. 1/𝑑𝑎𝑦 𝛽 student group b growth rate. 1/𝑑𝑎𝑦 k carrying capacity of each student group in the class % 𝜇 rate of interaction between student group a and student group b 1/% /𝑑𝑎𝑦 𝛾 student reduction rate 1/𝑑𝑎𝑦 3. existence of equilibrium points the system (1) has four equilibrium points consisting of one trivial equilibrium point and three non-trivial equilibrium points. however, we consider the analysis of the three nontrivial equilibrium points. the existence of those equilibrium points is presented in theorem 1. theorem 1. the existences of equilibrium points 𝐸(𝐴, 𝐵) of system (1) (i) the equilibrium point 𝐸1 = (0, 𝐾 (−𝛾+𝛽) 𝛽 ) exist if 𝛽 > 𝛾 (ii) the equilibrium point 𝐸2 = ( 𝐾(−𝛾+𝛼) 𝛼 , 0) exist if 𝛼 > 𝛾 𝛽𝐵 𝜇𝐴𝐵 𝛼𝐴 a b 𝛼𝐴2 𝐾 𝜕𝐴 𝜕𝐵 𝛽𝐵2 𝐾 m. ivan ariful fathoni, anisa fitri, hanifahtul husnah student group dynamic model based on understanding in mathematics subjects 45 (iii) the equilibrium point 𝐸3 = ( 𝐾 (−𝜇𝛾𝐾+ 𝜇𝛽𝐾−𝛾𝛽+𝛼𝛽) 𝛽𝛼+ 𝜇2𝐾2 , 𝐾 (𝜇𝛾𝐾− 𝜇𝛼𝐾−𝛾𝛼+𝛼𝛽) 𝛽𝛼+ 𝜇2𝐾2 ) exist if −𝜇𝛾𝐾 + 𝜇𝛽𝐾 > 𝛾𝛽 − 𝛼𝛽 and 𝜇𝛾𝐾 − 𝜇𝛼𝐾 > 𝛾𝛼 − 𝛼𝛽 4. stability analysis we apply the linearization method to determine the stability of the equilibrium point of system (1) and we have the jacobian matrix as the following, 𝐽 = [ 𝛼 (1 − 𝐴 𝐾 ) − 𝛼𝐴 𝐾 + 𝜇𝐵 − 𝛾 𝜇𝐴 −𝜇𝐵 𝛽 (1 − 𝐵 𝐾 ) − 𝛽𝐵 𝐾 − 𝜇𝐴 − 𝛾 ] (2) based on the jacobian matrix (2), the eigenvalues can be obtained from the characteristic equation |𝐽 − 𝜆𝐼| = 0 evaluated at each point of equilibrium. 4.1 equilibrium point 𝑬𝟏 the jacobian matrix obtained from the linearization of system (1) near the equilibrium point 𝐸1 is 𝐽(𝐸1) = [ 𝛼 − 𝜇𝐾(𝛾 − 𝛽) 𝛽 − 𝛾 0 𝜇𝐾(𝛾 − 𝛽) 𝛽 𝛽 (1 + 𝛾 − 𝛽 𝛽 ) − 𝛽 ] (3) the characteristic equation formed from matrix (3) is − (−𝐾𝛽𝜇 + 𝐾𝜇𝛾 − 𝛼𝛽 + 𝛽𝛾 + 𝜆𝛽)(𝛾 − 𝛽 − 𝜆) 𝛽 = 0 which has the eigenvalues 𝜆1 = 𝛾 − 𝛽 and 𝜆2 = 𝐾𝛽𝜇−𝐾𝜇𝛾+𝛼𝛽−𝛽𝛾 𝛽 . the stability conditions for the equilibrium point 𝐸1 are obtained when 𝛾 < 𝛽 and 𝐾𝛽𝜇 − 𝐾𝜇𝛾 < −𝛼𝛽 + 𝛽𝛾. 4.2 equilibrium point 𝑬𝟐 the jacobian matrix obtained from the linearization of system (1) near the equilibrium point 𝐸2 is 𝐽(𝐸2) = [ 𝛼 (1 + −𝛼 + 𝛾 𝛼 ) − 𝛼 − 𝜇(−𝛼 + 𝛾)𝐾 𝛼 0 𝛽 + 𝜇(−𝛼 + 𝛾)𝐾 𝛼 − 𝛾 ] (4) the characteristic equation formed from matrix (4) is (−𝐾𝛼𝜇 + 𝐾𝜇𝛾 + 𝛼𝛽 − 𝛼𝛾 − 𝜆𝛼)(𝛾 − 𝛼 − 𝜆) 𝛽 = 0 which has the eigenvalues 𝜆1 = 𝛾 − 𝛼 and 𝜆2 = −𝐾𝛼𝜇+𝐾𝜇𝛾+𝛼𝛽−𝛼𝛾 𝛼 . the stability conditions for the equilibrium point 𝐸2 are obtained when 𝛾 < 𝛼 and −𝐾𝛼𝜇 + 𝐾𝜇𝛾 < −𝛼𝛽 + 𝛼𝛾. 4.3 equilibrium point 𝑬𝟑 linearization of system (1) near the equilibrium 𝐸3 produces the characteristic equation as the following jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 46 1 𝐾2𝜇2 + 𝛼𝛽 (−𝛽(𝐾𝜇 − 𝛽 + 𝛾 − 𝜆)𝛼2 + ((𝐾𝜇 − 𝛾 + 𝜆)𝛽2 + (−𝐾2𝜇2 + (𝛾 − 𝜆)2)𝛽 + 𝐾𝛾𝜇(𝐾𝜇 + 𝛾 − 𝜆))𝛼 + 𝐾(𝛾(𝐾𝜇 − 𝛾 + 𝜆)𝛽 − 𝐾𝜇(𝛾 − 𝜆)(𝛾 + 𝜆))𝜇) = 0 the stability of this equilibrium point was analyzed using the routh hurwitz criteria and the stability requirements were obtained, 0 < −𝐾2𝜇2 − 𝛼𝛽 𝐾2𝜇2 + 𝛼𝛽 0 < −𝛽𝛼2 + (𝐾𝛾𝜇 − 𝛽2 + 2𝛾𝛽)𝛼 − 𝐾𝛽𝛾𝜇 𝐾2𝜇2 + 𝛼𝛽 0 < ((𝐾𝜇 − 𝛽 + 𝛾)𝛼 − 𝐾𝜇𝛾)(𝛼𝛽 + (𝐾𝜇 − 𝛾)𝛽 − 𝐾𝜇𝛾) 𝐾2𝜇2 + 𝛼𝛽 or 0 > −𝐾2𝜇2 − 𝛼𝛽 𝐾2𝜇2 + 𝛼𝛽 0 > −𝛽𝛼2 + (𝐾𝛾𝜇 − 𝛽2 + 2𝛾𝛽)𝛼 − 𝐾𝛽𝛾𝜇 𝐾2𝜇2 + 𝛼𝛽 0 > ((𝐾𝜇 − 𝛽 + 𝛾)𝛼 − 𝐾𝜇𝛾)(𝛼𝛽 + (𝐾𝜇 − 𝛾)𝛽 − 𝐾𝜇𝛾) 𝐾2𝜇2 + 𝛼𝛽 5. numerical simulations the simulation in this study uses assumption data from a condition at school. however, the research subject of real data in a school is still an open problem in this study. simulations are carried out based on the assumption of the percentage of the number of students in each group in one class to the total number of students in one education level. the percentage of students who understand mathematics subject matter (group a) was 20%, while students who did not understand mathematics subject matter (group b) were 15%. these numbers are assumed to be the initial condition after a teacher conducted a math ability test. furthermore, through the cooperative learning process, where learning is carried out that focuses on the interaction between students, it is expected to increase students who understand the material. in the first case, a simulation based on the system's differential equation model (1) is carried out by using the assumed parameter values 𝛼 = 0.6, 𝛽 = 0.7, 𝐾 = 0.4, 𝜇 = 0.01, 𝛾 = 0.2. this case shows a very low interaction process between students, characterized by a very small interaction rate (𝜇). assuming such a parameter value, the only stable equilibrium point is 𝐸3. the simulation results are obtained in figure 2. the number of students who understood the material increased from 0.20 to 0.269. the same thing happened to students who did not understand, which increased from 0.15 to 0.284. the simulation results show that there are still many students who do not understand. it is certainly not a goal in the success of the learning process. m. ivan ariful fathoni, anisa fitri, hanifahtul husnah student group dynamic model based on understanding in mathematics subjects 47 figure 2. the dynamic of changes in the student’s number in each group (case 1) in the second case, the simulation is carried out based on the assumed parameter values 𝛼 = 0.6, 𝛽 = 0.7, 𝐾 = 0.4, 𝜇 = 1.9, 𝛾 = 0.02. there is a significant increase in the interaction rate from the first case. in this second case, students' movement out of class is also increasingly limited by reducing the student reduction rate (𝛾). assuming such a parameter value, the only stable equilibrium point is 𝐸2. the equilibrium point 𝐸3 becomes not exist. the simulation results are obtained in figure 3. the number of students in group a increased from 0.2 to 0.387. the opposite happened in group b, which continued to decrease from 0.15 until no more students did not understand mathematics subject matter. in the second case, the learning process's objective has been achieved but can still be improved. figure 3. the dynamic of changes in the student’s number in each group (case 2) in the third case, the simulation is carried out based on the assumed parameter values 𝛼 = 0.9, 𝛽 = 0.4, 𝐾 = 0.4, 𝜇 = 1.9, 𝛾 = 0.02. we increase the student group a growth rate (𝛼) and decrease the student group b growth rate (𝛽). the simulation results are obtained in figure 4. under these conditions, the stability of the equilibrium point is the same as in the second case. however, there was an addition of students who understood (group a) to 0.391, while group b students continued to decrease to zero. another difference is that the time the solution moves to equilibrium is faster than the second case. in the third case, equilibrium occurs at 16 days. the goal of successful learning happens more quickly. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 48 figure 4. the dynamic of changes in the student’s number in each group (case 3) the simulation results are shown in figures 1, 2, and 3 explain the process of changing the conditions of students' level of understanding in real-time. based on this model, predictions can be made when learning success can be achieved, so that teachers can apply appropriate learning methods before starting the learning process. the implication of this study's results can help teachers make teaching more effective and directed under learning outcomes. 6. concluding remarks mathematical modeling can solve a problem by simplifying cases in everyday life into a mathematical problem where the solution can be obtained. this study departs from the background above issues, namely, students' interaction in the mathematics class. this interaction involves two groups of students who understand and do not understand the subject matter of mathematics. the two groups will interact with each other, creating a positive atmosphere in the learning process. the interaction process between groups is modeled in a system of differential equations. the solution of the system of differential equations is expected to reflect the teaching and learning process's success. the results of the dynamical analysis obtained are strengthened by numerical simulation evidence that supports the analysis. in numerical simulations, it can be concluded that by doing learning that focuses on the interaction between students and limits students' movement out of the classroom, the learning process can run successfully. all students in the class will understand the learning material. better results are obtained if the teacher adds more students who understand and reduces the more students who do not understand each group. to increase and decrease can be done by moving students from or to another class. approaching and teaching specifically for some students who do not understand can also be done not to be included in the observation group. it will help accelerate the success of the learning process. the findings in this study can be used as a new research field in the world of education. references [1] e. humphrey, encyclopedia internasional. new york: grolier, 1975. [2] e. bruce-jones, “beiter, ‘the protection of the right to education by international law,’” menschenrechtsmagazin, vol. 3. universitätsverlag potsdam, pp. 325–326, 2006. m. ivan ariful fathoni, anisa fitri, hanifahtul husnah student group dynamic model based on understanding in mathematics subjects 49 [3] sekretariat negara ri, undang-undang republik indonesia no.20 tahun 2003 tentang sistem pendidikan nasional. 2003, p. 2. [4] sekretariat negara ri, undang-undang republik indonesia no.14 tahun 2005 tentang guru dan dosen. 2005. [5] m. yusuf, pengantar ilmu pendidikan, 1st ed. palopo: lembaga penerbit kampus iain palopo, 2018. [6] a. c. sari, “meningkatkan kemampuan pemecahan masalah matematis siswa smp dengan model pembelajaran think talk write,” j. math. educ. sci., vol. 1, no. april, pp. 7–13, 2018. [7] n. k. roestiyah, masalah pengajaran sebagai suatu sistem. rineka cipta, 1994. 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[17] c. febriyanti and s. seruni, “peran minat dan interaksi siswa dengan guru dalam meningkatkan hasil belajar matematika,” form. j. ilm. pendidik. mipa, vol. 4, no. 3, 2015. [18] m. i. a. fathoni, gunardi, f. a. kusumo, and s. h. hutajulu, “mathematical model analysis of breast cancer stages with side effects on heart in chemotherapy patients,” in aip conference proceedings, 2019, vol. 2192, doi: 10.1063/1.5139153. [19] w. e. boyce, r. c. diprima, and d. b. meade, elementary differential equations. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 41-50 50 john wiley & sons, 2017. [20] y. w. putranto, “analisis titik ekuilibrium dan solusi model interaksi pemangsamangsa menggunakan metode dekomposisi adomian,” universitas sanata dharma, 2017. [21] w. ningsih and r. khusniah, “analisis dinamis model matematika pertumbuhan jumlah mahasiswa program studi pendidikan matematika stkip pgri pasuruan,” j. math. educ. sci., vol. 1, no. october, pp. 61–66, 2018. the rarity of joint probability between interest and inflation rates at the 1998 economic crisis in indonesia using copula contact: mohamad khoirun najib, mkhoirun_najib@apps.ipb.ac.id department of mathematics, ipb university, bogor, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.10-17 rarity of joint probability between interest and inflation rates in the 1998 economic crisis in indonesia and their comparison over three time periods mohamad khoirun najib1*, sri nurdiati1, faiqul fikri1 1department of mathematics, ipb university, bogor, indonesia article history: received sep 29, 2021 revised, may 7, 2022 accepted, may 30, 2022 kata kunci: copula, ketidakpastian, krisis ekonomi, periode ulang bersama, tingkat inflasi, tingkat suku bunga, abstrak. setelah lebih dari dua puluh tahun, tidak ada krisis ekonomi separah tahun 1998 berdasarkan inflasi dan suku bunga. menarik untuk dibandingkan kondisi sebelum dan setelah krisis tahun 1998 serta kondisi ekonomi pada dekade terakhir di indonesia. oleh karena itu, penelitian ini bertujuan menganalisis hubungan antara tingkat inflasi dan suku bunga menggunakan distribusi bersama berbasis copula. dari distribusi bersama tersebut, periode ulang bersama pada krisis ekonomi tahun 1998 diestimasi. hasil penelitian menunjukkan bahwa copula gumbel adalah fungsi copula bivariat yang paling cocok untuk membangun distribusi bersama antara tingkat inflasi dan suku bunga pada tahun 1990-2019 dengan dependensi ekor atas sebesar 0.6224. periode ulang bersama antara tingkat inflasi dan suku bunga yang lebih parah daripada tahun 1998 secara bersamaan adalah 389 tahun dengan interval kepercayaan 95% yaitu [47, ∞] tahun. hasil ini sangat tidak pasti karena banyak faktor yang mempengaruhi tingkat inflasi dan suku bunga. tingkat inflasi mengalami penurunan pada periode setelah krisis 1998. pada dekade terakhir, tingkat inflasi dan suku bunga jauh lebih rendah dibandingkan dua periode sebelumnya. keywords: copula, economic crisis, inflation rate, interest rate, joint return period, uncertainty abstract. after more than twenty years, there has been no economic crisis as severe as 1998 based on inflation and interest rates. it is interesting to compare the conditions before and after the 1998 crisis and the economic conditions in the last decade in indonesia. therefore, this study aims to analyze the relationship between inflation and interest rates using a copula-based joint distribution. the joint return period of the 1998 economic crisis is estimated from this joint distribution. the results showed that the gumbel copula is the most suitable bivariate copula to construct a joint distribution between inflation and interest rates in 1990-2019, with an upper tail dependency of 0.6224. moreover, the joint return period between inflation and interest rates more severe than 1998 is 389 years with a 95% confidence interval of [47, ∞] years. this result is uncertain because many factors affect inflation and interest rates. the inflation rate decreased after the 1998 crisis. meanwhile, in the last decade, the inflation and interest rates were much lower than in the two previous periods. how to cite: m. k. najib, s. nurdiati, and f. fiqri, “rarity of joint probability between interest and inflation rates in the 1998 economic crisis in indonesia and their comparison over three time periods”, j. mat. mantik, vol. 8, no. 1, pp. 10-17, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.10-17 issn: 2527-3159 (print) 2527-3167 (online) mailto:mkhoirun_najib@apps.ipb.ac.id http://u.lipi.go.id/1458103791 m. k. najib, s. nurdiati, and f. fiqri rarity of joint probability between interest and inflation rates in the 1998 economic crisis in indonesia and their comparison over three time periods 11 1. introduction the economic crisis in 1998 played an essential role in global history, including in indonesia [1]. the devaluation of the indonesian currency (rupiah) causes inflation and reduces actual public spending on health. household spending on health also declined, in absolute terms and as a percentage of overall expenditure. self-reported morbidity increased sharply from 1997 to 1998 in indonesia’s rural and urban areas [2]. the crisis led to a substantial reduction in health care utilization during the same period, as the proportion of household survey respondents who reported an illness or injury seeking care from a modern health care provider decreased by 25% [3]. several economic indicators can investigate the rarity of the economic crisis in 1998, e.g., inflation and interest rates. we are focusing on the inflation rate in the health sector due to increased morbidity in 1998 but low utilization of health facilities. the inflation and rising interest rates affect the decline in people’s purchasing power, slowing the economy and pushing it toward a recession [4]. after more than twenty years, there was no economic crisis as severe as in 1998 based on inflation and interest rates (fig. 1). moreover, very few experts research the joint return period between inflation and interest rates during the 1998 economic crisis in indonesia using copula. figure 1. inflation rates in the health category and bank of indonesia (bi) interest rates one popular approach to identifying the relationship between two random variables is copula-based joint distribution [5]. copula provides a simple way to construct the joint distribution of two variables [6-7]. therefore, we can calculate the joint probability of a specific event, e.g., the 1998 economic crisis [8]. moreover, we also can estimate the rarity and joint return period of this specific event using the joint probability [9]. this study uses copula-based joint distribution to observe the relationship between inflation and interest rates from 1990 to 2019. we estimate the joint return period of the 1998 economic crisis from the joint distribution. therefore, the result can assess the severity of the 1998 economic crisis and estimate when similar events will occur again. we divide the data into three periods, i.e., before the 1998 crisis, after the 1998 crisis, and after the 2008 crisis. using the same approach, we calculate and compare the joint distribution of these three periods. 2. materials and methods 2.1 materials we use two types of data, i.e., inflation and interest rates. we use interest rates in the health category from the ministry of trade republic of indonesia in 1990-2019 (https://statistik.kemendag.go.id/inflation), while the interest rate of time deposit in rupiah obtained from the bank of indonesia in 1990-2019 (https://www.bi.go.id/seki/). the data has an annual period and shows the highest value is in 1998 (fig. 1). to estimate the joint return period of the 1998 economic crisis, we use all data from 1990 to 2019. after that, we divide the data into three periods, i.e., 1990-1998, 1999-2008, and 2009-2019. https://statistik.kemendag.go.id/inflation https://www.bi.go.id/seki/tabel/tabel1_28.xls jurnal matematika mantik vol. 8, no. 1, june 2022, pp.10-17 12 2.2 copula function let 𝑋1 and 𝑋2 are representing inflation and interest rates, respectively. for bivariate cases (𝑋1,𝑋2), the copula function links the multivariate distribution 𝐹𝑋(𝑥1,𝑥2) to their univariate marginal distributions 𝐹1(𝑥1) and 𝐹2(𝑥2), given by ( )  1 2 1 1 2 2, ( ), ( )xf x x c f x f x= (1) where 𝐶:[0,1] × [0,1] → [0,1] called as copula function [10][11]. by differencing the left and right-hand sides, we get the joint probability density function between 𝑋1 and 𝑋2, i.e., ( )    1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 , ( ), ( ) ( ), ( ) ( ) ( ) x f x x c f x f x c f x f x f x f x x x   = =     (2) where 𝑐 is called a copula density function, while 𝑓1 and 𝑓2 are the probability density functions of 𝑋1 and 𝑋2 respectively [12][13][14]. to capture different upper and lower tail behaviour, we use copula functions from the archimedean family (table 1), such as gumbel, frank, and clayton [15][16][17]. the clayton (gumbel) copula has lower (upper) tail dependence, but the frank copula has no tail dependence. table 1. the selected bivariate copula function and their properties copula name copula function parameter range tail dependence lower upper clayton 1/ 1 2 ( 1)u u   − − − + − 0  2 −1/𝜃 0 gumbel 1/ 1 2 exp[ ( ) ]w w    − + , ln( ) i i w u= − 1  0 2 − 2 1/𝜃 frank 1 2 1 1 ln 1 ( 1) w w e −   − +  −  , 1 ui i w e − = − 0  0 0 𝑢𝑖 are the transformed variables 𝑢𝑖 = 𝐹𝑖(𝑥𝑖) for 𝑖 = 1,2 2.3 estimating the copula parameters to estimate the parameters of copulas, we use the inference of functions for margins (ifm) method [18][19] and the fittest copula chosen based on root mean squared error (rmse), akaike’s information criterion (aic), and kolmogorov-smirnov error (kse) [20]. in principle, ifm is a two-step method to estimate the copula parameters. the first step of ifm is to estimate the marginal distributions of each variable. in this paper, we use lognormal and generalized extreme value distribution to fit the marginal distribution of data. the probability density function (pdf) and the cumulative distribution function (cdf) of the lognormal (ln) distribution are given by ( ) ( ) 2 2 ln1 | , exp , 0 22 ln x f x x x       − − =      (3) and ( ) ln | , ln x f x     −  =     (4) where φ(𝑥) is the cdf of standard normal distribution, 𝜇 is the location parameter, and 𝜎 is the scale parameter [21]. meanwhile, the pdf and cdf of the generalized extreme value (gev) distribution are given by ( ) 1 1 1 1 | , , exp 1 1 k k gev x x f x k k k        − − −  − −    = − + +           (5) and m. k. najib, s. nurdiati, and f. fiqri rarity of joint probability between interest and inflation rates in the 1998 economic crisis in indonesia and their comparison over three time periods 13 ( ) 1 | , , exp 1 k gev x f x k k     −  −  = − +       (6) for 𝑥 ∈ 𝑅, and 𝑘 ≠ 0 [22][23]. we employed the anderson-darling statistical test with a 5% significance level to test the goodness-of-fit of distribution to the actual data [24]. using these marginal distributions, we fit the copula parameter by maximizing the log of the likelihood function, i.e., ( ) ( ) 1 1 1 2 2 2 1 1 1 1 2 2 2 1 ˆ ˆ ˆarg max ln arg max ln ( ; ), ( ; ); ˆ ˆarg max ln ( ; ), ( ; ); n t t x t n t t x t l c f x f x c f x f x        = = = = =   (7) where �̂� is the estimate of marginal distribution parameters and 𝜃 is the estimate of the copula parameter [25]. 2.4 joint return period from the joint distribution, we can estimate the probability (𝑃) of a condition exceeding a certain critical multivariate threshold. in this case, we use inflation and interest rates at the 1998 economic crisis as a threshold. therefore, the probability of two events coinciding as severe than in 1998 is defined by [26] ( )( ) 1 ( ) ( ) ( ), ( )and x y x yp p x x y y f x f y c f x f y=   = − − + (8) where 𝑥 and 𝑦 are inflation and interest rates in 1998, respectively. for the limited data, we can estimate the 95% confidence interval for 𝑃, i.e., 𝑃 ± 𝜀 with (1 ) 2 p p n  − = (9) where 𝑁 is the sample size of data [27]. since we use annual data, the joint return period (in years) is defined by 1/𝑃 [9]. 3. results and discussion 3.1 fitting processes table 2 shows the fitting result of the marginal distributions of the inflation and interest rates. after that, the copula parameters are estimated (table 3) using the fittest marginal distribution. the statistical result shows that the chosen marginal distribution (table 2) all passed the anderson-darling test with a 5% significance level, and the fittest copula is the gumbel copula. the gumbel copula is part of the extreme copula family and has an upper tail dependency (table 1). using the parameter value, the upper tail dependency between inflation and interest rates is 1/2.16412 2 0.6224u = − = (10) and the lower tail dependency is equal to zero. table 2. the fitting result of the marginal distribution of the inflation rate and the interest rate. data fittest distribution p-value the inflation rate gev (𝑘 = 0.5253,𝜇 = 4.3886,𝜎 = 2.0184) 0.9856 the interest rate lognormal (𝜇 = 2.3997,𝜎 = 0.4529) 0.7626 jurnal matematika mantik vol. 8, no. 1, june 2022, pp.10-17 14 table 3. the fitting result of the copula parameters between the inflation rate and the interest rate. copula 𝜃 kse rmse aic gumbel 2.1641 0.12790 0.052504 -21.512 frank 7.0248 0.13826 0.057042 -21.463 clayton 1.6339 0.12823 0.058021 -15.910 3.2 joint probability density function using the fittest parameter, we got the marginal distributions (eq. 11 and 12) and copula function (eq. 13), i.e., 1 0.5253 1 1 1 4.3886 ( ) exp 1 3.8424 x f x −  −  = − +       (11) ( ) 22 2 ln 2.3997 0.4529 f x x − =      (12) 2.1641 2.1641 1/ 2.1641 1 2 1 2 ( , ) exp[ ([ ln( )] [ ln( )] ) ] x c u u u u= − − + − (13) where 𝑢1 = 𝐹1(𝑥1) and 𝑢2 = 𝐹2(𝑥2). therefore, we obtained the joint distribution between the inflation and interest rates using eq. 1, i.e.,        ( ) 1 2 1 1 2 2 1/ 2.1641 2.1641 2.1641 1 1 2 2 ( , ) ( ), ( ) exp ln ( ) ln ( ) x f x x c f x f x f x f x =   = − − + −    (14) where φ(𝑥) is the cumulative distribution function (cdf) of standard normal distribution. we need first the copula density function to construct the joint probability density function. by differencing the copula function to 𝑢1 and 𝑢2, we got the copula density function, i.e., ( ) ( )                    1/ 2.1641 2.1641 2.1641 1 2 1 2 1 2 1 2 1 2 1 2.1641 2.1641 2.1641 1 2 1 2 1 2.1641 2.1641 2.1641 1 2 2.1641 2.164 1 2 , , exp ln( ) ln( ) 1 1 exp ln( ) ln( ) 1 ln( ) ln( ) ln( ) ln( ) c u u c u u u u u u u u u u u u t u u u u       = = − − + −          = − − + −       − + − + −     − + −      1 2 1 1.1641 1.16412.1641 1 2 ln( ) ln( )u u −  − −     (15) where 𝑢1 = 𝐹1(𝑥1) and 𝑢2 = 𝐹2(𝑥2). using eq. 2, the joint probability density function between the inflation and interest rates is calculated and visualized using a contour plot (fig. 2). the figure shows that the peaks of the joint probability between inflation and interest rates are 3.3 and 7.5, respectively. meaning that during the last three decades, the value of inflation and interest rates mostly appears around 3.3 and 7.5. however, within those 30 years, there is an outlier data in 1998, which shows inflation and interest rates up to 86.14 and 28.75. still, the probability of this event is minuscule which will be discussed in the following subchapter. m. k. najib, s. nurdiati, and f. fiqri rarity of joint probability between interest and inflation rates in the 1998 economic crisis in indonesia and their comparison over three time periods 15 figure 2. joint probability density function between inflation and interest rates 3.3 joint return period in 1998 the dark blue box on the right-top (fig. 2) shows the hazard area when inflation and interest rates are simultaneously higher than in the 1998 economic crisis. using eq. 8, the probability of this area can calculate, i.e.,   ( 86.14 28.75) 1 (86.14) (28.75) (86.14), (28.75) 0.26% and x y x x y p p x y f f c f f =   = − − + = (15) thus, the return period of the hazard area is 1/0.0026 years or about 389 years. this means that the probability of inflation and interest rates higher than in 1998 simultaneously is 1/389 years. using eq. 9, the 95% confident interval of 𝑃𝐴𝑁𝐷 is [0,0.0211] so the 95% confident interval of return period is [47, ∞] years. this means that the condition of inflation and interest rates is worse than in 1998 is very rare, and there is a possibility that the 1998 economic crisis will not happen again. a wide 95% confidence interval indicates that the joint return period is uncertain due to many factors affecting inflation and interest rates. 3.4 joint return period over three periods we calculate the joint probability density function between inflation and interest rates by three different periods using the same approach (fig. 3). figure 3. joint probability density function between inflation and interest rates by three different periods fig. 3 shows that the joint probability density between inflation and interest rates decreased over three periods. from the first period (1990-1998), the joint probability density function in the second period (1999-2008) decreased the inflation rate but did not experience a significant decrease in the interest rate. on the other hand, in the last decade, the joint probability density function has decreased significantly in terms of inflation and interest rates. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.10-17 16 4. conclusion the ifm method shows that the gumbel copula is the fittest bivariate copula to construct the joint distribution between inflation and interest rates. from the joint distribution, the joint return period of inflation and interest rates higher than in 1998 simultaneously is 389 years with a 95% confident interval [47, ∞]. however, the result is uncertain due to many factors affecting inflation and interest rates. after dividing the data into three different periods, the joint probability density function has decreased significantly in terms of inflation and interest rates in the last decade compared to the previous two decades. acknowledgement the author would like to thank the department of mathematics, ipb university, for the support. this article is the second winner of the modelling competition presented at the summer course on mathematical modelling in life and material sciences, august 7-28, 2021, jointly hosted by the department of mathematics and physics at ipb 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[26] m. h. afshar, a. ü. şorman, f. tosunoğlu, b. bulut, m. t. yilmaz, and a. danandeh mehr, “climate change impact assessment on mild and extreme drought events using copulas over ankara, turkey,” theor. appl. climatol., vol. 141, no. 3–4, pp. 1045–1055, 2020, doi: 10.1007/s00704-020-03257-6. [27] r. link, t. b. wild, a. c. snyder, m. i. hejazi, and c. r. vernon, “100 years of data is not enough to establish reliable drought thresholds,” j. hydrol. x, vol. 7, 2020, doi: 10.1016/j.hydroa.2020.100052. m. fariz fadillah mardianto, rosyida w. ulya, almira s. syamsudin analysis of society satisfaction of the e-toll system in indonesia based on structural equation model contact: m. fariz fadillah mardianto, m.fariz.fadillah.m@fst.unair.ac.id departement of mathematics, universitas airlangga, surabaya, jawa timur 60115, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.115-123 analysis of society satisfaction of the e-toll system in indonesia based on structural equation model m. fariz fadillah mardianto1*, rosyida widadina ulya2, almira sophie syamsudin3 1,2,3 department of mathematics, universitas airlangga, surabaya, indonesia article history: received jan 19, 2021 revised, jun 5, 2021 accepted, oct 31, 2021 kata kunci: structural equation modelling, kepuasan masyarakat, e-toll system, inovasi, perkembangan infrastruktur abstrak. implementasi penggunaan e-toll system memiliki beberapa evaluasi dari masyarakat. evaluasi tersebut mempengaruhi kepuasan masyarakat terhadap e-toll system. kepuasan masyarakat penting untuk diketahui agar pemerintah dapat mengevaluasi faktor apa yang kurang dari e-toll system agar pelayanannya lebih baik. dalam penelitian ini metode yang digunakan dalam analisis kepuasan adalah structural equation modeling (sem). penelitian ini menggunakan data berdasarkan hasil penyebaran kuesioner secara online yang telah diuji validitas dan reliabilitasnya. pada penelitian ini, dimensi kualitas layanan dinilai dari unsur kehandalan, daya tanggap, jaminan, empati dan bukti fisik. berdasarkan hasil kajian, masyarakat merasa puas dengan penggunaan sistem e-toll di jalan tol. namun terdapat dua dari lima dimensi kualitas layanan yang sangat signifikan yaitu dimensi keandalan dan jaminan. untuk itu pemerintah perlu lebih memperhatikan keandalan dan memberikan jaminan terhadap e-toll system. keywords: structural equation modelling, society satisfaction, e-toll system, innovation, infrastructure development abstract. the implementation of the use of the e-toll system has several evaluations from the community. this evaluation affects community satisfaction with the e-toll system. it is important to know public satisfaction so that the government can evaluate what factors are lacking from the e-toll system so that the service is better. in this study, the method used in the satisfaction analysis is structural equation modeling (sem). this study uses data based on the results of online questionnaires that have been tested for validity and reliability. in this study, the dimensions of service quality were assessed from the elements of reliability, responsiveness, assurance, empathy and physical evidence. based on the results of the study, the community is satisfied with the use of the e-toll system on toll roads. however, there are two of the five dimensions of service quality that are very significant, namely the dimensions of reliability and assurance. for this reason, the government needs to pay more attention to reliability and provide guarantees for the e-toll system. how to cite: m. f. f. mardianto, r.w. ulya, and a. s. syamsudin, “analysis of society satisfaction of the e-toll system in indonesia based on structural equation model”, j. mat. mantik, vol. 7, no. 2, pp. 115-123, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 115-123 issn: 2527-3159 (print) 2527-3167 (online) mailto:m.fariz.fadillah.m@fst.unair.ac.id https://doi.org/10.15642/mantik.2021.7.2.115-123 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol 7, no 2, october 2021, pp. 115-123 116 1. introduction indonesia, which is geographically very broad, has challenges in the field of transportation and infrastructure in the context of easy access. in facilitating the access of indonesia in transportation, distribution of goods such as cloth, food and shelter, the government carries out massive development in the infrastructure sector to foster competitiveness between regions. particularly, infrastructure development which is the focus of the current government is in building toll roads. in java, president jokowi built the trans java toll road that connects merak to banyuwangi along 1,150 km. in addition, the government also built the trans sumatra toll road, samarinda toll road, the first toll road in north sulawesi and in several other regions in indonesia. the development of toll road development is accompanied by the development of artificial intelligence (ai). ai is the theory and development of computer systems able to perform tasks normally requiring human intelligence, such as visual perception, speech recognition, decision-making, and translation between languages. ai in its implementation in the toll road system is a noncash payment tool that is commonly known by the public with the e-toll. with the use of electronic systems, it is expected that toll road services will be faster, thus toll road users are more comfortable in traveling. another reason for implementing transactions using the e-toll is to follow the flow of modernization where other countries have also begun to convert to non-cash payments. in the development of e-toll there are positive and negative responses from society. positive response is that the use of e-toll is more practical, then there is no need to bring a lot of cash and more time efficiency because when paying tolls the society only needs to attach the card to the transaction machine at the toll gate and it does not take time to wait for change, then the long queue at the gate tolls can be avoided. another advantage of using e-toll is that e-toll users often get promos such as cashback or discounts. while the negative response felt by the society is for some people to use the e-toll is a new and troublesome thing rather than using cash. in addition to this response there is another impact, namely the pruning of human labour in the toll road system. positive and negative responses reflect the satisfaction felt by the society. society satisfaction is important to know, thus the government can improve the welfare of the society and evaluate what factors are lacking in the toll road system then services on the toll road system can develop better. therefore, the appropriate method for conducting satisfaction analysis is the structural equation modelling (sem) method. structural equation modelling (sem) is an effective model testing and improving method that enables theoretical models to be used as a whole and that can be explained which models are based on statistical dependence [1]. sem can be viewed as a combination of factor analysis and regression or path analysis. this method is in accordance with the aim of the research, which is to understand how the service quality relates to the toll road system with society satisfaction. researchers who have studied sem are [2] about knowledge transfer in problem based learning teaching methods in software engineering education and the result is the study concludes that the framework is suitable for pbl teaching method in sem education. therefore, the future work will concentrate on identifying the relationship of the seci model in pbl teaching method for sem education, [3] the sem is used as the statistical technique in this study because it can analyse all the factors simultaneously. the outcome is significant direct effects of quality factors towards knowledge work productivity, [4] the partial least square (pls) technique was used to analyse the data. this technique has the ability to predict the theoretical model. since the main objective of this study is to investigate the determinant factors of flood victims’ knowledge-sharing behaviour, pls was selected as the analysis technique. smart pls 2.0 software was used to analyse the measurements and the structural model, and [5] this study makes use of partial least square (pls) technique for analysing the data. pls is able to predict the relationship among factors in the constructed model. therefore, it is appropriate for this m. fariz fadillah mardianto, rosyida w. ulya, almira s. syamsudin analysis of society satisfaction of the e-toll system in indonesia based on structural equation model 117 study because this study determines to predict the factors influencing the fitness of social media technology with information sharing during disaster, and particularly flood. this study aimed to understand the effect of service quality on society satisfaction in the e-toll system. in addition, this study is expected to be useful to provide recommendations and evaluations to the indonesian government and related parties in order to improve the quality of artificial intelligence services on toll roads which were assessed by society satisfaction. this study was also useful to provide insight into statistical analysis of service issues in the toll road system. sem comprises a statistical technique to test hypotheses on the relationships among observed and latent variables [6–9]. sem is used to measure the direct effects of structural models to predict the significant relationship among the factors of interactive persuasive learning among elderly [10–12]. a two-step model building approach was used to analyse the two conceptually distinct models: the measurement model followed by the structural model. the fit and construct validity of the proposed measurement model was first tested and once a satisfactory measurement was obtained, the structural paths of the sem were estimated [13]. the evaluation of the measurement models and structural models was done using maximum likelihood estimation [14,15]. 2. methods 2.1. data source this study was conducted by distributing an online questionnaire by using google form. survey-based research requires a very definitive population. the population is a complete set of research objects and a focus for the research [16]. the population is difned as society in java, the population size amounts 148,173,100 people according to indonesia statistics in 2018. this research uses a combination of techniques. this combination is convenient sampling with a slovin formula. convenience sampling is best suited and the most efficient for this research due to the lack of resources [17]. while it could be argued that the results from this sampling is less objective [18]. based on the slovin formula [19] using 10% as a margin of error, it is obtained that the sample size is 100 people. after the data was obtained, a reliability test and validity test of the questionnaire were conducted. after fulfilling the requirements contained in the reliability test and validity test then continued to analyse the data using the structural equation model (sem) method with the amos application. the data used was society satisfaction data on the use of ai in the toll road system in indonesia, e-toll which was assessed from the level of service quality of the etoll system. the endogenous variable in this study was society satisfaction with the use of e-tolls in indonesia, while the exogenous variable in this study was the quality of e-toll services in indonesia. service quality was divided into five dimensions, which were reliability, responsiveness, guarantee, empathy, physical evidence [20]. society satisfaction and dimensions of service quality were further divided into several factors which would be illustrated in table 1. jurnal matematika mantik vol 7, no 2, october 2021, pp. 115-123 118 table 1. definition of factors on variables variables factors reliability (x1) the easiness to top up balance on e-toll cards. (x1.1) e-toll card was not error easily. (x1.2) the accuracy of detecting costs to be paid. (x1.3) the accuracy of detecting the distance travelled. (x1.4) the accuracy to detect the vehicle type. (x1.5) the accuracy of balanced information. (x1.6) responsiveness (x2) speed response of toll engines during the transaction process. (x2.1) the toll gate officer responds when a problem occurs. (x2.2) the toll gate officer provides the right service when a problem occurs. (x2.3) guarantee (x3) the e-toll system can be trusted by consumers. (x3.1) there is an officer who controls the e-toll system. (x3.2) there is a sense of security for consumers. (x3.3) empathy (x4) the toll officer serves 24 hours. (x4.1) officers prioritize user interests in e-toll transactions. (x4.2) physical evidence (x5) card quality is good and not easily damaged. (x5.1) attractive card design. (x5.2) the distance of the e-toll engine with vehicles is easily accessible. (x5.3) society satisfaction (y) services in accordance with procedures. (y1) e-toll users feel safe and comfortable when transacting using the e-toll machine. (y2) 2.2. data analysis procedure the data analysis model in this study was divided into: a. descriptive analysis in this study descriptive statistical analysis of the respondent data was carried out. data descriptions of respondents included the purpose of respondents who used toll roads, vehicle classes that were often used by respondents to pass through toll roads and how to top-up balances that were often carried out by respondents. b. conducted hypothesis test 𝐻0 : there is no correlation between society's satisfaction with the use of e-toll in indonesia with the dimensions of service quality of artificial intelligence (ai) on the toll road system. 𝐻1 : there is correlation between society satisfaction with e-toll use in indonesia with dimensions of service quality of artificial intelligence (ai) on the toll road system. hypothesis testing was processed using the structural equation modelling (sem) method with amos applications stated above. analysis techniques in sem were consisted of five steps: 1) define hypothesis to be tested 2) define latent and manifest variable and relation between variables 3) arranging path diagram, like in figure 1(a) and figure 1(b) 4) perform reliability test on the latent variable 5) perform significance test against each loading factor (𝜆) 6) perform correlation analysis of latent variable to calculate gamma (𝛾) 7) evaluate goodness of fit 8) interpret the results m. fariz fadillah mardianto, rosyida w. ulya, almira s. syamsudin analysis of society satisfaction of the e-toll system in indonesia based on structural equation model 119 figure 1(a). path diagram 1 in amos figure 1(b). path diagram 2 in amos 3. result and discussions this study used primary data as many as 100 respondents obtained through distributing questionnaires regarding society satisfaction with the ai system on the toll road system, where satisfaction was assessed from the level of service quality of the e-toll system itself. the general description of the data obtained was displayed in the pie chart as follows: figure 2. pie chart the purpose of the society is using the toll road according to figure 2, it could be seen that out of 100 respondents there were 51% of respondents used toll roads because they want to shorten travel time, as many as 34% of respondents used toll roads because free of traffic, 5% of respondents aimed to save bpok or ease the burden of the government through road user participation and the rest had the aim to speed up distribution of the goods being sent. figure 3. pie chart the way top up 5% 5% 5% 34% 51% the purpose of the society is using the toll road the rest have the aim to speed up distribution of goods being sent ease the burden of the government through the participation of road users save bok free traffic shorten the travel time 78% 5% 8% 3% 6% pie chart the way top up minimarket others m-banking/sms-banking coming diretcly to the bank toll gates jurnal matematika mantik vol 7, no 2, october 2021, pp. 115-123 120 according to figure 3, it could be seen that out of 100 respondents, there were 78% of respondents who top up the balance through the minimarket. 8% of respondents top up through m-banking or sms banking, 6% through toll gates and the rest by coming directly to the bank or other. table 2. reliability test output cronbach’s alpha cronbach’s alpha based on standardized items n of items 0.927 0.931 19 the data met the requirement of reliability test when cronbach's alpha was > 0.6. the reliability test in this study was conducted by using the using the application of opensource software (oss) r resulting in the output in table 2 as follows: the cronbach's alpha value in this research data was 0.927, thus it could be said that the data met the reliability test conditions where cronbach's alpha (0.927) > 0.6. then, it indicated that the questionnaire of this research data could be trusted as a data collection tool and was able to reveal actual information in the field. table 3. validity test output variables corrected item-total correlation x1.1 0.491 x1.2 0.292 x1.3 0.405 x1.4 0.688 x1.5 0.655 x1.6 0.711 y1 0.732 x2.1 0.670 x2.2 0.648 x2.3 0.663 x3.1 0.691 x3.2 0.635 x3.3 0.790 y2 0.686 x4.1 0.610 x4.2 0.712 x5.1 0.685 x5.2 0.532 x5.3 0.485 table 4. dimension of satisfaction dimensions p-value information satisfaction  responsiveness 0.175 insignificant satisfaction  empathy 0.644 insignificant satisfaction  physical evidence 0.724 insignificant satisfaction  guarantee 0.000 significant satisfaction  reliability 0.015 significant validity tests could be conducted in many ways. in this study, the validity test used application of (oss) r and compared the validity test output with r table, free degrees of 98 and alpha 0.1 obtained a value of 0.1654, resulting in the output in table 3. data was said to be valid if the value of corrected item-total correlation was > 0.1654. in this m. fariz fadillah mardianto, rosyida w. ulya, almira s. syamsudin analysis of society satisfaction of the e-toll system in indonesia based on structural equation model 121 research data had been said to be valid because all values of corrected item-total correlation was > 0.1654. then it could be said that the questionnaire provided was relevant to the goal. based on table 4 the variables that influenced society satisfaction were the guarantee and reliability variables because the p-value values of the two variables were <0.05. the responsiveness variable was not very influential on satisfaction because the majority of people had agreed that transactions in the e-toll system were very fast or it could be said that machine responsiveness did not require a long time in the transaction. the physical evidence variable also had little effect on satisfaction because the majority of the society also agreed that the e-toll card was not easily damaged and the e-toll card design also attracted attention. the last variable that was less influential on satisfaction was empathy. because in this study, what was examined was an artificial intelligence system thus empathy from a system could not be felt in other words non-human systems that could have a sense of giving hospitality to customers. table 5. significant value of each indicator influence p-value information satisfaction  guarantee 0.000 significant satisfaction  reliability 0.008 significant y1  satisfaction 0.000 significant y2  satisfaction 0.000 significant x1.1  reliability 0.000 significant x1.2  reliability 0.003 significant x1.3  reliability 0.000 significant x1.4  reliability 0.000 significant x1.5  reliability 0.000 significant x1.6  reliability 0.000 significant x3.3  guarantee 0.000 significant x3.2  guarantee 0.000 significant x3.1  guarantee 0.000 significant table 6. goodness of fit criterion results cut off value model evaluation 𝜒2 50,763 101,879 appropriate probability 0.070 ≥ 0.05 appropriate rmsea 0.100 ≤ 0.08 appropriate gfi 0.950 ≥ 0.90 appropriate agfi 0.923 ≥ 0.90 appropriate tli 0.920 ≥ 0.90 appropriate pnfi 0.957 ≥ 0.90 appropriate after eliminating the insignificant dimensions of the output, it could be seen in table 5 that the dimensions of reliability and guarantee of p-value were less than alpha (0.05), which was equal to 0.000 and 0.008. thus, the right decision was to reject h0. this indicated that the dimensions of reliability and guarantee were very significant for the level of society satisfaction in the e-toll system. it can be supported based on table 6 for model evaluation has satisfied goodness of fit criterion. recommendations that could be given to the government and related parties was to improve the reliability and guarantee dimensions of the e-toll system. increasing the reliability dimension could be provided by providing top up outlets in each toll gate and rest area, always evaluating the e-toll system, thus the error rate decreases. whereas the jurnal matematika mantik vol 7, no 2, october 2021, pp. 115-123 122 dimension of guarantee could be conducted by guaranteeing the suitability of transaction costs with bills. the guarantee of the cost of an appropriate billing transaction would make the society not hesitate with the e-toll. in addition, the toll road system could be equipped with an ai system to refuel, drink and food machines, automatic car wash machines and automatic tow trucks. the evaluation that could be given was to increase the sensitivity of the machine to the e-toll card and broadly the convenience of ai could be used in various fields. in addition, the implementation of a payment system on the toll road could be carried out using one payment instrument, which meant that all transaction activities along the toll road used a practical payment instrument, the e-toll. 4. conclusion in this study, society satisfaction with ai used on toll roads was based on service quality divided into five dimensions. based on the results of the study, the society was satisfied with the use of ai in the toll road system. however, there were two of the five dimensions of service quality that were very significant, namely the dimensions of reliability and guarantee. on the results of the sem analyse it was concluded that the dimensions of reliability and guarantee affected society’s satisfaction. reliability affected society’s satisfaction because the main function of ai was to facilitate human activity, if the system was not reliable then satisfaction was not achieved. whereas the guarantee affected society’s satisfaction because the e-toll system was a payment system so that the security of funds needed by the society was needed thus the society did not doubt the etoll. therefore, increasing the reliability and guarantee of the ai system on toll roads should be continuously improved and updated to increase society satisfaction. the advice given was that it was required to carry out periodic evaluations and development by the government and related parties in the e-toll system, then the problems were resolved faster. in addition, the researchers suggested further research with a larger number of samples. references [1] r. larry, structural equation modeling (sem) concepts, applications, and misconceptions. new york: nova science publishers, 2015. [2] m. ahmad, a. zainol, n. m. darus, z. marzuki matt, f. baharom, and m. y. shafiz affendi, “knowledge transfer in problem-based learning teaching method in software engineering education: a measurement model,” arpn j. eng. appl. sci., vol. 10, no. 3, pp. 1486–1493, 2015. [3] m. z. yusoff, m. mahmuddin, and m. ahmad, “a knowledge work productivity conceptual model for software development process in sme,” arpn j. eng. appl. sci., vol. 10, no. 3, pp. 1123–1130, 2015. [4] m. ahmad, n. m. zani, and k. f. hashim, “knowledge sharing behavior among flood victims in malaysia,” arpn j. eng. appl. sci., vol. 10, no. 3, pp. 968–976, 2015. [5] k. f. hashim, s. h. ishak, and m. ahmad, “a study on social media application as a tool to share information during flood disaster,” arpn j. eng. appl. sci., vol. 10, no. 3, pp. 959–967, 2015. [6] r. hoyle, automated structural equation modeling strategies. 2016. 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[15] a. n. zulkifli, m. ahmad, j. a. abu bakar, r. c. mat, and n. m. noor, “examining the influence of interactive persuasive learning among elderly,” arpn j. eng. appl. sci., vol. 10, no. 3, pp. 1145–1153, 2015. [16] a. banerjee and s. chaudhury, “statistics without tears: populations and samples,” ind. psychiatry j., vol. 19, no. 1, 2010, doi: 10.4103/0972-6748.77642. [17] s. k. thompson, sampling, third. new jersey: john wiley & sons, inc., publication, 2005. [18] n. showkat and h. parveen, “in-depth interview: quadrant-i (e-text). ”august 2017. [19] j. tejada and j. punzalan, “on the misuse of slovin’s formula,” philipp. stat., vol. 61, no. 1, pp. 129–136, 2012. [20] r. lupiyoadi, manajemen pemasaran jasa: teori dan praktik. jakarta: pt. salemba emban patria, 2001 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: budi priyo prawoto, budiprawoto@unesa.ac.id department of mathematics, state university of surabaya, surabaya, east java, indonesia. the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.18-27 dynamics of sars-cov-2 spread model with vaccine administration and use of masks khofifah ichawati1, budi priyo prawoto1* 1department of mathematics, universitas negeri surabaya, indonesia article history: received mar 17, 2022 revised, may 5, 2022 accepted, may 31, 2022 kata kunci: sars-cov-2, vaksin, penggunaan masker, model seir, pemodelan matematika abstrak. penelitian ini bertujuan untuk mengonstruksi dan mengetahui dinamika model matematika penyebaran sars-cov2 dengan adanya pemberian vaksin dan penggunaan masker. pengonstruksian model dalam penelitian ini menggunakan model seir yang telah dimodifikasi dengan beberapa tahapan yaitu melakukan studi literatur mengenai pemodelan matematika pada virus sars-cov-2, menyusun asumsi awal, membuat diagram kompartemen, mengonstruksi model matematika, menentukan titik kesetimbangan, menentukan bilangan reproduksi dasar, melakukan analisis kestabilan dan sinkronisasi hasil analisa dengan melakukan simulasi numerik. didapatkan 2 titik kesetimbangan yaitu titik kesetimbangan bebas penyakit dan titik kesetimbangan endemik. dengan menggunakan bilangan reproduksi dasar, didapatkan syarat kestabilan untuk titik bebas penyakit dan juga titik endemik. ketika titik bebas penyakit stabil maka sars-cov-2 akan hilang dari populasi, sedangkan ketika titik bebas penyakit tidak stabil maka sars-cov-2 akan terus ada didalam populasi. keywords: sars-cov-2, vaccines, use of masks, sier model, mathematical modeling abstract. the purpose of this study was to construct and determine the dynamics of the mathematical model of the reach of sars-cov-2 with the provision of vaccines and the use of masks. in this study, the modified seir model was used with the stages of conducting a literature study on mathematical modeling of the sars-cov-2 virus, compiling initial assumptions, making compartment diagrams, constructing mathematical models, determining equilibrium points, determining basic reproduction numbers, conducting stability analysis and synchronization of analysis results by performing numerical simulations. in this study, two equilibrium points were obtained the disease-free equilibrium point and the endemic equilibrium point. using the basic reproduction number, we get the stability conditions for the disease-free point and the endemic point. when the disease-free point is stable, sars-cov-2 will disappear from the population, while when the disease-free point unstable, sars-cov-2 will be exist’s in the population. how to cite: k. ichawati and b. p. prawoto, “dynamics of sars-cov-2 spread model with vaccine administration and use of masks”, j. mat. mantik, vol. 8, no. 1, pp. 18-27, jun. 2022. jurnal matematika mantik vol. x, no. x, mmyy, pp. issn: 2527-3159 (print)2527-3167 (online) jurnal matematika mantik volume 8, no. 1, june 2022, pp. 18-27 issn: 2527-3159 (print)2527-3167 (online) mailto:budiprawoto@unesa.ac.id http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 khofifah ichawati and budi priyo prawoto dynamics of sars-cov-2 spread model with vaccine administration and use of masks 19 1. introduction coronaviruses are a group of viruses that infect animals and can attack the human respiratory system. in 2002 and 2012, respectively, the coronavirus named severe acute respiratory syndrome coronavirus (sars-cov) emerged with bats as natural hosts [1] and the middle east respiratory syndrome coronavirus (mers-cov), which were transmitted from camels to humans [2]. this has become a new health problem in the 21st century [3]. at the end of 2019, the coronavirus called severe acute respiratory syndrome coronavirus-2 (sars-cov-2) was detected in wuhan city, china. the sars-cov-2 virus is the virus that causes the covid-19 disease outbreak and has spread rapidly throughout the world [4]. coronavirus is an infectious disease and can spread quickly [5]. according to who, the spread of the sars-cov-2 virus occurs when interaction or contact with people with covid-19. when a person with covid-19 coughs, talks, sneezes, sings, or breathes, the virus spreads from the mouth and nose in the form of small particles or liquids. this virus can also spread in places with poor ventilation and also crowded. this can happen due to the aerosol can be suspended in the air. in addition, the transmission of covid-19 can also occur when someone touches the surface of an object that has been contaminated by the virus [6]. the use of masks in the community is one of the practical efforts in preventing the spread of the sars-cov-2. this is because masks work in 2 ways. the first is to prevent people infected with covid-19 from transmitting the virus by blocking breathing containing the virus into the air. second, masks protect people who are not infected by covid-19 by being a barrier for liquids or small particles containing the virus from sticking to an open nose and mouth. masks can also filter particulate matter from inhaled air [7]. based on a systematic review of 172 studies conducted by [8], the use of masks can protect a person from infection with the sars-cov-2. vaccination is another form of the government effort to suppress the spread of the sars-cov-2. vaccines contain antigens that can stimulate the body's immune system to produce an immune response [9]. according to [10], through clinical trials, the covid-19 vaccine has shown good immunogenicity with different levels of effectiveness in each vaccine variant. so that it can help suppress the spread of the sars-cov-2. according to [11] the seir model is the most commonly used mathematical algorithm to describe the diffusion of epidemic diseases, for example to characterize the covid-19 epidemic[12]. the seir model is applied to cases of disease that have a long incubation period, so that the effect of incubation on the spread of disease can be analyzed[13]. in various other research articles, many discuss the mathematical model of the spread of disease using the seir model. an article entitled mathematical model for mers-cov disease transmission with medical mask usage and vaccination by [14]. this article discusses the model for spreading the mers-cov virus using masks and vaccinations using the seir model. in addition, an article entitled stability analysis and numerical simulation of seir model for pandemic covid-19 spread in indonesia by [15]. therefore, in this article, the author discusses the spread of covid-19 in indonesia using the seir model. based on this description, the authors are interested in discussing the model for the space of sars-cov-2 with masks and vaccinations and adding the assumption of re-infection. re-infection is possible when recovered individuals ignore the use of masks even though they have been vaccinated. the purpose of this study was to construct and determine the mathematical model of the spread of sars-cov-2 by administering vaccines and using masks. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.18-27 20 2. method this research article refers to a research article conducted by [14] with the following research design are : a. literature study on mathematical modeling of the sars-cov-2 virus using masks and vaccines. b. preparation of initial assumptions. c. compile a compartment diagram based on assumptions. d. construct mathematical models based on compartment diagrams. e. determine the equilibrium point. f. an equilibrium point is a point that will not change with time. in other words, when time 𝑡 goes to infinity, the value of that point will not change. systematically the definition can be written as follows: g. definition : �̅� ∈ ℝ𝑛 point is said to be the equilibrium point of �̇�(𝑡) = 𝑓(𝑥, 𝑡) if it satisfies 𝑓(�̅�) = 0. [16] h. determine the basic reproduction number (r0). the basic reproduction number (r0) is used to measure the potential for disease transmission[17]. by determining the stability conditions, r0 can analyze the equilibrium point. when r0 < 1, then the number of infected populations is decreasing. this means that the disease will disappear from the population. but when r0 > 1, the number of infected individuals is increasing. this means that the disease will be exist in the population[18]. in order to get r0 we have to look at the dominant eigenvalues from the next generation matrix [19], as follows: 𝐾 = 𝐹𝑉−1 𝐹 and 𝑉 is the 𝑛𝑥𝑛 matrix with 𝐹 = [ 𝜕𝐹𝑖(𝑥1) 𝜕𝑦𝑗 ] dan 𝑉 = [ 𝜕𝑉𝑖(𝑥2) 𝜕𝑦𝑗 ] 𝑥1 : laten population 𝑥2 : infected population i. perform stability analysis. j. numerical simulation using matlab program. 3. results and discussion 3.1 mathematical model this study uses the seir model (susceptible, exposed, infected, recovered), modified into five subpopulations. a subpopulation of susceptible individuals not wearing a mask (s1), a subpopulation of susceptible individuals using a mask (s2), a subpopulation of latent individuals (e) where the individual has been infected but cannot transmit the virus because in the case of covid-19 there is an incubation period, a subpopulation of infected individuals (i) and the cured subpopulation (r). the initial assumptions used in the model are as follows: a. the population is assumed to be closed. b. the number of births and the number of deaths is assumed to be the same per unit of time at the rate 𝜇.this means that the population is constant. c. death due to the disease is negligible. death that occurs in each population is natural death. d. the population is homogeneous. this means that each individual has the same possibility to make contact with other individuals. e. vulnerable individuals will be vaccinated with a vaccination proportion is 𝜌. f. unvaccinated newborn will enter s1(t) at rate (1 − 𝜌)𝜇. khofifah ichawati and budi priyo prawoto dynamics of sars-cov-2 spread model with vaccine administration and use of masks 21 g. the virus cannot infect vulnerable individuals who wear masks at the awareness rate of wearing masks 𝑢. h. individuals in compartment s2 will be placed into compartment s1 if they are no longer wearing the mask at a rate of (1 − 𝑢). i. transmission of the virus occurs when susceptible individuals contact infected individuals, either directly or indirectly at a rate of 𝛽. j. individuals infected with the virus in the latent subpopulation (e) will enter the infected individual (i) at a rate of 𝛿. k. individuals infected with the virus can recover from the disease at a rapid rate 𝜎. l. individuals who have recovered from the disease but have a low awareness of using masks can be re-infected when they encounter infected individuals at a rate of 𝛽𝑅𝐼 𝑁 (1 − 𝑢). based on these assumptions, a compartment diagram can be formed, as follows: figure 1. compartment diagram of the sars-cov-2 seir model with vaccine administration and use of masks then based on the compartment diagram obtained a system of differential equations, as follows: 𝑑𝑆1 𝑑𝑡 = (1 − 𝜌)𝜇𝑁 + (1 − 𝑢)𝑆2 − (𝜇 + 𝑢)𝑆1 − 𝛽𝑆1𝐼 𝑁 (1) 𝑑𝑆2 𝑑𝑡 = 𝑢𝑆1 − (𝜇 + (1 − 𝑢))𝑆2 (2) 𝑑𝐸 𝑑𝑡 = 𝛽𝑆1𝐼 𝑁 + 𝛽𝑅𝐼(1−𝑢) 𝑁 − (𝜇 + 𝛿)𝐸 (3) 𝑑𝐼 𝑑𝑡 = 𝛿𝐸 − (𝜇 + 𝜎)𝐼 (4) 𝑑𝑅 𝑑𝑡 = 𝜎𝐼 + 𝜌𝜇𝑁 − 𝜇𝑅 − 𝛽𝑅𝐼(1−𝑢) 𝑁 (5) table 1. variable description variable description s1 number of vulnerable individuals not wearing masks s2 number of vulnerable individuals wearing masks e number of latent individuals i number of infected individuals r number of recovered individuals 𝑢𝑆1 s1 e i r s2 𝜇𝑆1 𝜇𝐸 𝜇𝐼 𝜇𝑆2 (1 − 𝑢)𝑆2 (1 − 𝜌)𝜇𝑁 𝛽𝑆1𝐼 𝑁 𝛿𝐸 𝜎𝐼 𝜌𝜇𝑁 𝜇𝑅 𝛽𝑅𝐼 (1 − 𝑢) 𝑁 jurnal matematika mantik vol. 8, no. 1, june 2022, pp.18-27 22 since n is constant, it is possible to do scaling on each subpopulation to the total population in order to simplify the system. the proportion of the number of individuals in each subpopulation is stated as follows: 𝑠1 = 𝑆1 𝑁 , 𝑠2 = 𝑆2 𝑁 , 𝑒 = 𝐸 𝑁 , 𝑖 = 𝐼 𝑁 , 𝑟 = 𝑅 𝑁 then the system can be changed to the following: 𝑑𝑠1 𝑑𝑡 = (1 − 𝜌)𝜇 + (1 − 𝑢)𝑠2 − (𝜇 + 𝑢)𝑠1 − 𝛽𝑠1𝑖 (6) 𝑑𝑠2 𝑑𝑡 = 𝑢𝑠1 − (𝜇 + (1 − 𝑢))𝑠2 (7) 𝑑𝑒 𝑑𝑡 = 𝛽𝑠1𝑖 + 𝛽𝑟𝑖(1 − 𝑢) − (𝜇 + 𝛿)𝑒 (8) 𝑑𝑖 𝑑𝑡 = 𝛿𝑒 − (𝜇 + 𝜎)𝑖 (9) 𝑑𝑟 𝑑𝑡 = 𝜎𝑖 + 𝜌𝜇 − 𝜇𝑟 − 𝛽𝑟𝑖(1 − 𝑢) (10) table 2. parameter description parameter description 𝜇 birth and death rates 𝜌 coverage rate of vaccination 𝑢 the rate of awareness of the use of masks 𝛽 the rate of transmission when in contact with an infected individual 𝛿 transmission rate from compartment e to i 𝜎 cure rate 3.2 equilibrium point equating to zero in equation (4) – (8), the equilibrium point is found [16]. we get two equilibrium points from the model, namely the disease-free equilibrium points and the endemic equilibrium point. a. disease-free equilibrium point the disease-free state is fulfilled when no individual is infected with the virus or it can be said i = 0. if 𝐸0 is a disease-free equilibrium point, then 𝐸0 = {𝑠1 = (1 − 𝜌)(𝜇 + 1 − 𝑢) 𝜇 + 1 , 𝑠2 = (1 − 𝜌)𝑢 𝜇 + 1 , 𝑒 = 0, 𝑖 = 0, 𝑟 = 𝜌} b. endemic equilibrium point if 𝐸1 is an endemic equilibrium point, then 𝐸1 = {𝑠1 = 𝑠1 ∗, 𝑠2 = 𝑠2 ∗, 𝑒 = 𝑒∗, 𝑖 = 𝑖∗, 𝑟 = 𝑟∗} because the shape of the endemic equilibrium point is complex, it cannot be written in the article 3.3 basic reproduction number the basic reproduction number (r0) is obtained using the next generation matrix [17][19]. known: 𝑑𝑒 𝑑𝑡 = 𝛽𝑠1𝑖 + 𝛽𝑟𝑖(1 − 𝑢) − (𝜇 + 𝛿)𝑒 𝑑𝑖 𝑑𝑡 = 𝛿𝑒 − (𝜇 + 𝜎)𝑖 khofifah ichawati and budi priyo prawoto dynamics of sars-cov-2 spread model with vaccine administration and use of masks 23 so that f= ( 𝛽𝑠1𝑖 + 𝛽𝑟𝑖(1 − 𝑢) 0 ) and v= ( (𝜇 + 𝛿)𝑒 −𝛿𝑒 + (𝜇 + 𝜎)𝑖 ) then the jacobian matrix is 𝐹 = [ 0 𝛽𝑠1 + 𝛽𝑟(1 − 𝑢) 0 0 ] and 𝑉 = [ 𝜇 + 𝛿 0 −𝛿 𝜇 + 𝜎 ] 𝑉−1 = 1 (𝜇 + 𝛿)(𝜇 + 𝜎) [ 𝜇 + 𝜎 0 𝛿 𝜇 + 𝛿 ] = [ 1 𝜇 + 𝛿 0 𝛿 (𝜇 + 𝛿)(𝜇 + 𝜎) 1 𝜇 + 𝜎] so that it is obtained 𝐾 = 𝐹 ∙ 𝑉−1 = [ 0 𝛽𝑠1 + 𝛽𝑟(1 − 𝑢) 0 0 ] [ 1 𝜇+𝛿 0 𝛿 (𝜇+𝛿)(𝜇+𝜎) 1 𝜇+𝜎 ] = [ 𝛿(𝛽𝑠1 + 𝛽𝑟(1 − 𝑢)) (𝜇 + 𝛿)(𝜇 + 𝜎) 𝛽𝑠1 + 𝛽𝑟(1 − 𝑢) 𝜇 + 𝜎 0 0 ] the substitution of the disease-free equilibrium point is 𝑠1 = (1−𝜌)(𝜇+1−𝑢) 𝜇+1 and 𝑟 = 𝜌, we get: 𝐾 = [ 𝛿(𝛽 (1−𝜌)(𝜇+1−𝑢) 𝜇+1 +𝛽𝜌(1−𝑢)) (𝜇+𝛿)(𝜇+𝜎) (𝛽 (1−𝜌)(𝜇+1−𝑢) 𝜇+1 +𝛽𝜌(1−𝑢)) (𝜇+𝜎) 0 0 ] the eigenvalues of k are obtained, namely 1. 𝜆1 = 𝛿𝛽((1−𝜌)(𝜇+1−𝑢)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) 2. 𝜆2 = 0 then the basic reproduction number is obtained as follows: r0 = 𝛿𝛽((1−𝜌)(𝜇+1−𝑢)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) 3.4 stability analysis based on the theory presented by [18] when r0 < 1 will cause the virus to disappear from the population or go to the point of being free of disease. therefore, the stability analysis is carried out using the basic reproduction number, and r0 < 1 will be used to determine the condition for the stability of the disease-free equilibrium point: r0 = 𝜹𝜷((𝟏−𝝆)(𝝁+𝟏−𝒖)+𝝆(𝟏−𝒖)(𝝁+𝟏)) (𝝁+𝟏)(𝝁+𝜹)(𝝁+𝝈) < 𝟏 so that it is obtained 𝛿𝛽((1−𝜌)(𝜇+1−𝑢)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) < 1 ⟺ 𝛿𝛽((1 − 𝜌)(𝜇 + 1 − 𝑢) + 𝜌(1 − 𝑢)(𝜇 + 1)) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) ⟺ 𝛿𝛽(𝜇 − 𝜌𝜇 + 1 − 𝜌 − 𝑢 + 𝑢𝜌 + 𝜌𝜇 + 𝜌 − 𝑢𝜌𝜇 − 𝑢𝜌) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) ⟺ 𝛿𝛽(𝜇 + 1 − 𝑢 − 𝑢𝜌𝜇) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) ⟺ 𝛿𝛽(𝜇 + 1 + 𝑢(−1 − 𝜌𝜇) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) ⟺ 𝛿𝛽(𝜇 + 1) + 𝛿𝛽𝑢(−1 − 𝜌𝜇) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) jurnal matematika mantik vol. 8, no. 1, june 2022, pp.18-27 24 ⟺ 𝛿𝛽𝑢(−1 − 𝜌𝜇) < (𝜇 + 1)(𝜇 + 𝛿)(𝜇 + 𝜎) − 𝛿𝛽(𝜇 + 1) ⟺ 𝑢 > (𝜇+1)(𝜇+𝛿)(𝜇+𝜎)−𝛿𝛽(𝜇+1) 𝛿𝛽(−1−𝜌𝜇) , because −1 − 𝜌𝜇 < 0 . so, when the value of 𝒖 > (𝝁+𝟏)(𝝁+𝜹)(𝝁+𝝈)−𝜹𝜷(𝝁+𝟏) 𝜹𝜷(−𝟏−𝝆𝝁) , r0 < 𝟏 is obtained. this means that the disease-free point is stable so that the sars-cov virus will disappear from the population. when we take the value of 𝒖 < (𝝁+𝟏)(𝝁+𝜹)(𝝁+𝝈)−𝜹𝜷(𝝁+𝟏) 𝜹𝜷(−𝟏−𝝆𝝁) , r0 > 𝟏 is obtained. this means that the disease-free point is unstable, so the sars-cov-2 virus will be exist in the population. 3.5 numerical simulation the simulation of the dynamics of the sars-cov-2 distribution model with the provision of vaccines and masks was carried out using the matlab program. simulations were carried out at disease-free points and endemic points, as follows: table 3. parameter value and source parameter parameter value source 𝜇 0.0125000 [20] 𝜌 0.6000000 assumption 𝛽 0.0400000 assumption 𝛿 0.0714258 [15] 𝜎 0.0000667 [15] with the initial value used is s1(0) = 0.20, s2(0) = 0.25, e(0) = 0.12 , i(0) = 0.25 , r(0) = 0.18 . a. disease-free equilibrium point the disease-free equilibrium point will be stable when the value of 𝑢 > (𝜇+1)(𝜇+𝛿)(𝜇+𝜎)−𝛿𝛽(𝜇+1) 𝛿𝛽(−1−𝜌𝜇) = 0.62. so here will be compared three parameter values as follows table 4. assumptions parameter values of u at the disease-free equilibrium point 𝒖𝟏 𝒖𝟐 𝒖𝟑 parameter value 0.65 0.75 0.85 figure 2. graph of disease-free equilibrium point with parameter values 𝑢1, 𝑢2, 𝑢3. (in order from left) based on figures 2, it can be seen that the population of infected individuals is closer to zero when the value 𝑢3 or 𝑢 = 0.85 is taken. this means that the greater the value of u taken, the faster the population of infected individuals will be exhausted. khofifah ichawati and budi priyo prawoto dynamics of sars-cov-2 spread model with vaccine administration and use of masks 25 when given the values of 𝑢1 = 0.65, 𝑢2 = 0.75, and 𝑢3 = 0.85 in the simulation of disease-free equilibrium point, we get r0 are 0.913, 0.656, and 0.399 respectively, the three results meet r0 < 1. b. endemic equilibrium point the endemic point will be stable when the value of 𝑢 < (𝜇+1)(𝜇+𝛿)(𝜇+𝜎)−𝛿𝛽(𝜇+1) 𝛿𝛽(−1−𝜌𝜇) = 0.62. so here will be compared three parameter values as follows: table 5. assumptions parameter values of u at the endemic equilibrium point 𝒖𝟏 𝒖𝟐 𝒖𝟑 parameter value 0.55 0.40 0.20 figure 3. graph of the endemic equilibrium point with parameter values 𝑢1, 𝑢2, 𝑢3. (in order from left) based on figure 3, in order from left it can be seen that by taking the value of 𝑢1 = 0.55 obtained stable at the number of infected subpopulations of 0.1230, 𝑢1 = 0.40 obtained stable at the number of infected subpopulations of 0.3018 and 𝑢1 = 0.20 obtained stable at the number of infected subpopulations of 0.4366. this shows that by taking a smaller value, the value of the infected individual or population will be even greater. when given the values of 𝑢1 = 0.55, 𝑢2 = 0.40, dan 𝑢3 = 0.20 in the simulation of endemic equilibrium point, we get r0 are 1.170, 1.556, and 2.071 respectively, the three results meet r0 > 1. on the results of [14], the value of 𝛽, 𝛿, 𝑢1, 𝑢2 and 𝜌 has changed to obtain the r0 > 1. however, in this article researchers have considered only one parameter namely 𝑢 or the awareness rate of using a mask with a threshold 𝑢 < (𝜇+1)(𝜇+𝛿)(𝜇+𝜎)−𝛿𝛽(𝜇+1) 𝛿𝛽(𝜌𝜇−1) so that the resulting mathematical model will be easier to use in determining sars-cov-2 virus control policies. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.18-27 26 4. conclusions based on the initial assumptions and the compartment diagram that has been compiled, a mathematical model of the spread of sars-cov-2 in the presence of vaccines and the use of masks is obtained as follows: 𝑑𝑆1 𝑑𝑡 = (1 − 𝜌)𝜇𝑁 + (1 − 𝑢)𝑆2 − (𝜇 + 𝑢)𝑆1 − 𝛽𝑆1𝐼 𝑁 𝑑𝑆2 𝑑𝑡 = 𝑢𝑆1 − (𝜇 + (1 − 𝑢))𝑆2 𝑑𝐸 𝑑𝑡 = 𝛽𝑆1𝐼 𝑁 + 𝛽𝑅𝐼(1 − 𝑢) 𝑁 − (𝜇 + 𝛿)𝐸 𝑑𝐼 𝑑𝑡 = 𝛿𝐸 − (𝜇 + 𝜎)𝐼 𝑑𝑅 𝑑𝑡 = 𝜎𝐼 + 𝜌𝜇𝑁 − 𝜇𝑅 − 𝛽𝑅𝐼(1 − 𝑢) 𝑁 the basic reproduction number (r0) is used to perform stability analysis and the basic reproduction number is r0= 𝛿𝛽((𝜇+1−𝑢)(1−𝜌)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) . when the value 𝑢 > 𝛿𝛽((𝜇+1−𝑢)(1−𝜌)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) taken, r0 < 1 is obtained. this means that the disease-free point is stable so that the sars-cov-2 virus will disappear from the population. when the value 𝑢 < 𝛿𝛽((𝜇+1−𝑢)(1−𝜌)+𝜌(1−𝑢)(𝜇+1)) (𝜇+1)(𝜇+𝛿)(𝜇+𝜎) taken, r0 < 1 is obtained. this means that the disease-free point is unstable so that the sars-cov-2 virus will continue to exist in the population the decreasing awareness of using masks will cause the number of infected individuals to become higher. thus, awareness of the use of masks significantly affects the control of the sars-cov-2 virus. for further researchers, it is recommended to add the assumption of death due to disease because in the case of covid-19, death caused by the disease will probably occur. researchers can also use new parameter values to simulate the model. references [1] n. s. zhong et al., “epidemiology and cause of severe acute respiratory syndrome.pdf,” lancet(london,england), vol. 362, no. 9393, pp. 1353–1358, 2003. 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[20] m. manaqib, i. fauziah, e. hartati, m. manaqib, i. fauziah, and e. hartati, “model matematika penyebaran covid-19 dengan penggunaan masker kesehatan dan karantina,” jambura j. biomath, vol. 2, no. 2, pp. 68–79, 2021. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: h. gunarto, gunarto@apu.ac.jp ritsumeikan asia-pacific university, 1–1 jumonjibaru, beppu 875 palindromes in some smarandache-type functions h. gunarto1, s.m.s. islam2 and a.a.k. majumdar3 1ritsumeikan asia-pacific university, beppu, oita, japan 2hajee danesh university of science & technology, dinajpur, bangladesh article history: received jan 16, 2022 revised, may 24, 2022 accepted, may 30, 2022 kata kunci: fungsi smarandache, fungsi sandorsmarandache, palindrom. abstrak. target riset dan tujuan utama dari makalah ini adalah untuk menyelidiki sifat palindrom (urutan angka yang sama jika dibaca dari arah depan/maju maupun dari arah belakang/mundur) yang terdapat dalam tiga fungsi aritmatika tipe smarandache; yaitu: fungsi smarandache s(n), fungsi pseudo smarandache z(n), dan fungsi sandor-smarandache ss(n). abstract. the objective of this paper is to investigate for palindromes in three smarandache type arithmetic functions, namely, the smarandache function s(n), the pseudo smarandache function z(n), and the sandor-smarandache function ss(n). issn: 2527-3159 (print) 2527-3167 (online) 3 ritsumeikan asia-pacific university, beppu, oita, japan sandorsmarandache function keywords: smarandache function, , palindromes. – 8577, oita, japan. the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.1-9 jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 1-9 am and a.a.k. majumdar, “p how to cite: h. gunarto, s.m.s. isl alindromes in some smarandache-type functions”, j. mat. mantik, vol. 8, no. 1, pp. 1-9, may 2022 mailto:gunarto@apu.ac.jp https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 2 1. introduction a palindromic number is one that reads the same forwards and backwards. examples of palindromes are 1, 11, 101, 111 121, …, of which the first three are prime, the third and the fourth ones are composite. in general, in concatenation form, a palindrome n has one of the following two forms: n = x1 x2 … xn … x2 x1 or n = x1 x2 … xn xn … x2 x1, (1) where x1{1, 2, …, 9} and xk {0, 1, 2, …, 9} for all 2  k  n. possibly, the study of palindromic numbers in smarandache-type arithmetical functions was initiated by ashbacher [1], who made extensive search up to 100,000. he reports his findings, together with some unsolved questions and conjectures. ibstedt [2] has studied palindromic primes, and has given a list of all prime palindromes up to 9–digit numbers, found on computers. the following result is due to him. lemma 1.1. a palindrome with an even number of digits is a composite number. proof. a palindrome n with 2n digits has the form n = x1x2 … xn xn … x2x1; x1  0, 0  xi  9 for each i = 1, 2, …, n. (2) in decimal representation, n can be written as n = x1(10 2n–1 + 1) + 10x2(10 2n–3 + 1) + 102x3(10 2n–5 + 1) + …+ 10 n–1xn(10 + 1). (3) now, it can be proved by induction on k that 102 k – 1 + 1 is divisible by 11 for any integer k  1. the proof is as follows: the result is clearly true for k = 1. so, we assume that 102 k – 1 + 1 is divisible by 11 for some integer k. now, since 102 k + 1 + 1 = 102(102 k – 1 + 1) – (102 – 1) (4) it follows that the result is true for k + 1. this shows that n is divisible by 11. a consequence of lemma 1.1 is that, 11 is the only two–digit palindromic prime; moreover, higher digit palindromic primes must have odd number of digits, as has already been pointed out by ibstedt. ibstedt [2] also introduced the concept of extended palindromes, which are palindromes of the form (1), where x1, x2, …, xn are all natural numbers, of which at least one is greater than or equal to 10. in this paper, we study the palindromic numbers in three commonly known smarandache functions, namely, the smarandache function s(n), the pseudo smarandache function z(n), and the sandor-smarandache function ss(n). we further study the roles of the palindromic primes in the first two functions. we consider the three cases separately in the next three sections. we conclude the paper with some remarks in the final section 5. 2. palindromes in s(n) the smarandache function, s(.) : + z → +z ( +z being the set of all positive integers) is defined by s(n) = min {m : m  0, n divides m!}, n  1, (5) with s(1) = 1. though explicit closed-form expressions for s(n) are not available, we have the following two results (see, for example, ashbacher [3]). jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 1-9 h. gunarto, s.m.s. islam and a.a.k. majumdar palindromes in some smarandache-type functions 3 lemma 2.1. let 1 2 21 ... k k α α α n p p p= be the (unique) representation of n in terms of its prime factors p1, p2, …, pk. then,   1 2 1 2 ( ) ( ), ( ), ( ) k k s n max s p s p ..., s p .=   (6) lemma 2.2. for any prime p  2, and any integer k  1, s(p k ) = p for some integer   1; (7) with s(p) = p, s(p p ) = p2. (8) in view of the above two lemmas, we see that, in order to search for palindromic numbers n for which s(n) is also palindromic, it is sufficient to restrict our attention to palindromic primes whose powers and multiples are also palindromic. for example, s(11k) = 11, s(101k) = 101 for all 1  k  9, (9) s(k.112) = 22, s(k.113) = 33 for 1  k  3, s(k.1012) = 202, s(k.1013) = 303 for 1  k  3, s(1014) = 404. now, given any integer m  1, it is always possible to find a number n, for example, n = m!, such that s(n) = m. thus, the function s(.) is onto. clearly, s(.) is not a bijective, since, for example, s(3) = 3 = s(6). let s( k)(n) be the k–fold composition of s(n) with itself, that is, o ( ) ( ) ( )( ).o o .... k k factors s n s s s n= (10) note that, if p is a palindromic prime then for any integer n with s(n) < p such that np is also a palindrome, we have s(np) = p. (11) in fact, in such a case, for any integer k  1, s( k)(np) = p. (12) the reader is referred to liu [4] for a brief review of s(n) till 2015. more results are given in majumdar [5, 6]. a list of values of s(n) for n = 1, 2, …, 4800 is given in ibstedt [7]. we have made use of the values listed in [7] in some of the examples above. 3. palindromes in z(n) the pseudo smarandache function, z(.) : + z → + z , introduced by kashihara [8], is defined by z(n) = min {m : m  1, n divides 2 ( 1)m m + }, n  1. (13) explicit expressions for z(n) are available only for certain cases. a brief review of z(n) till 2015 is given in liu [9]. the following results are due to majumdar [5]; in some cases, only the relevant parts are given. lemma 3.1. if p  3 is a prime and n  1 is an integer, then  1 2 1( ) 2 1 p , if n divides p z np p, if n divides p − − = + lemma 3.2. if p  3 is a prime and n is an integer not divisible by p, then jurnal matematika mantik vol. 8, no. 1, may 2022, pp. 1-9 4 2 2 2 2 2 1 2 ( 1) ( ) 2 ( 1) p , if n divides p z np p , if n divides p  − − =  + lemma 3.3. let p  3 be a prime such that 4 divides p + 1. then, z(2p k ) = p k for any odd integer k  3. lemma 3.4. let p  5 be a prime such that 3 divides p + 1. then, z(3p k ) = p k for any odd integer k  3. lemma 3.5. let p  3 be a prime such that 8 divides 3p + 1. then, z(4p) = 3p. lemma 3.6. let p  5 be a prime such that 5 divides 2p + 1. then, z(5p) = 2p. lemma 3.7. let p  3 be a prime. then, 12 1 (6 ) 3 4 3 1 p, if divides p z p p, if divides p + =  + lemma 3.8. if p  3 with p  7 is a prime, then 2 7 2 1 (7 ) 3 7 3 1 p, if divides p z p p, if divides p + =  + lemma 3.9. for any prime p  3, 3 16 3 1 (8 ) 5 16 5 1 7 16 7 1 p, if divides p z p p, if divides p p, if divides p +  = +  + lemma 3.10. for any prime p  5, 1 18 ( 1) 18 ( 1) 2 1 9 (2 1) (9 ) 2p 9 (2 1) 9 (4 1)4 1 9 (4 1)4 p , if divides p p, if divides p p , if divides p z p , if divides p if divides pp , if divides pp, − −  +  − − =  +  −−  + lemma 3.11. for any prime p  3 with p  5, 3 20 7 (10 ) 4 20 9 5 20 3 p, if divides p z p p, if divides p p, if divides p +  = +  − lemma 3.12. for any prime p  3 with p  11, 2 11 2 1 3 11 3 1 (11 ) 4 11 4 1 5 11 5 1 p, if divides p p, if divides p z p p, if divides p p, if divides p +  + =  +  + lemma 3.13. if p  3 is a prime, then 3 8 3 1 (12 ) 7 24 7 1 p, if divides p z p p, if divides p + =  + lemma 3.14. for any prime p  2, h. gunarto, s.m.s. islam and a.a.k. majumdar palindromes in some smarandache-type functions 5 2 13 2 1 3 13 3 1 (13 ) 4 13 4 1 5 13 5 1 6 13 6 1 p, if divides p p, if divides p z p p, if divides p p, if divides p p, if divides p +  +  = +  +  + from lemma 3.1 – lemma 3.14, we see that, if p is a palindromic prime, then we may find an integer n such that np is also a palindrome with z(np) = kp, where k  1 is an integer such that kp is also a palindrome. thus, for example, if p = 11 (which is the only 2–digit palindromic prime), then by virtue of lemma 3.1, lemma 3.7 and lemma 3.10, each of z(22), z(33), z(66) and z(99) is palindrome with z(22) = 11 = z(33) = z(66), z(99) = 44. now, 113 = 1331 is a palindrome, and so by virtue of lemma 3.3 and lemma 3.4, both z(2.113) = z(2662) and z(3.113) = z(3993) are palindromes with z(2662) = 1331= z(3993). again, with the smallest 3–digit palindromic prime p = 101, we can form several palindromic z(n). by lemma 3.1, lemma 3.5, lemma 3.7, lemma 3.8, lemma 3.9, lemma 3.10 and lemma 3.12, each of z(303), z(404), z(606), z(707), z(808), z(909) and z(1111) is a palindrome with z(303) = 101, z(404) = 303 = z(606) = z(808), z(707) = 202, z(909) = 404, z(1111) = 505. since 1013 = 1030301 is also a palindrome, by lemma 3.3 and lemma 3.4, z(2.1013) = z(2060602) = 1030301 = z(3090903) = z(3.1013). now, given any integer m  1, it is always possible to find a number n, for example, 2 ( 1)m m n + = such that z(n) = m. thus, the function z(.) is onto. however, z(.) is not bijective, since, for example, z(2) = 3 = z(6). let z( k)(n) be the k–fold composition of z(n) with itself, that is, o ( ) ( ) ( )( ).o o .... k k factors z n z z z n= (14) the following question has been raised by ashbacher [1]: given a palindromic number n, what is the maximum number of times one can perform the functional compositions z(n), z(2)(n), …, z( k)(n), …, and obtain a palindrome each time? ashbacher [1], using a computer program, searched for palindromic numbers n such that z(n) are also palindromes, in the range 10  n  10000, and found that, of the 189 palindromic n, z(n) was palindromic only for 37 values of n. over the same range, there is only one n, namely, n = 909, with z(n), z(2)(n) and z(3)(n) all palindromes: z(909) = 404, z (2)(909) = z(404) = 303, z(3)(909) = z(303) = 101. extending the range to 10  n  100000, only one n = 86868 was found such that z(n), z(2)(n), z(3)(n) and z(4)(n) all palindromes, with z(86868) = 17271, z(2)(86868) = z(17271) = 2222, z(3)(86868) = z(2222) = 1111, z(4)(86868) = z(1111) = 505. jurnal matematika mantik vol. 8, no. 1, may 2022, pp. 1-9 6 it is interesting to observe here that, in the above case, all the palindromes, except 86868, are multiples of the palindromic prime 101: 1111 = 11101, 17271 = 3319101, 86868 = 223219127. the number 86868 shows that palindromes can result from non-palindromic primes as well. for the reader who is interested in more detail on the pseudo smarandache function, z(n), we refer to majumdar [5, 6]. a list of the values of z(n) for n = 1, 2, …, 5000 is given in gunarto and majumdar [10]. some of the values of z(n), given above, are taken from [10]. another problem of interest is as follows: is there any palindrome n such that s(n) = z(n)? to answer the question, we first state the following result, a proof of which may be found in majumdar [5]. lemma 3.15. the solution of the equation s(n) = z(n) is n = tp, where p is a prime and t is an integer such that t divides (p – 1)!, and z(tp) = p. trivial examples are given below: s(22) = 11 = z(22), s(33) = 11 = z(33), s(66) = 11 =z(66). by virtue of lemma 3.1, we see that s(2p) = p = z(2p) for any odd prime p such that 4 divides p + 1, s(3p) = p = z(3p) for any prime p ( 5) such that 4 divides p + 1. thus, for example, s(262) = 131 = z(262), s(393) = 131 = z(393). 4. sandor-smarandache function the smarandache-type arithmetic function, denoted by ss(n), posed by sandor [11], and called the sandor-smarandache function, is defined as follows: ( ) = max : 1 1, divides n ss n k k n n , k      −      n  5 (15) where by convention, ss(1) = 1, ss(2) = 1, ss(3) = 1, ss(4) = 1, ss(6) = 1. the following result is due to sandor [11]. lemma 4.1. if n (  3) is an odd integer, then ss(n) = n – 2. it can, in fact, be proved that ss(n) = n – 2 if and only if n is an odd integer. some results related to the sandor-smarandache function are given in majumdar [6]. the function was later studied in more detail by islam and majumdar [12], islam, gunarto and majumdar [13], and more recently by [14] and [15]. the following result is also known about the sandor-smarandache function. lemma 4.2. ss(n) = n – 3 if and only if n is an even integer not divisible by 3. we now prove the following result in connection with palindromes in ss(n). lemma 4.3. there is an infinite number of integers n such that both n and ss(n) are palindromic numbers. h. gunarto, s.m.s. islam and a.a.k. majumdar palindromes in some smarandache-type functions 7 proof : consider the palindromic number .... 1 2 99 9 9(10 10 ... + 1); 2. k k k factors k − − = + +  (16) since the sum on the right-hand side is 10 k – 1, it follows that the integer n, defined by n  n(k) = (10 k – 1) + 2 = 10 k + 1, is palindromic and odd. therefore, by lemma 4.1, .... ( ) 99 9 ; 2. k factors ss n k=  thus, for example, from lemma 4.3, we see that, ss(101) = 99, ss(1001) = 999, ss(10001) = 9999, … . so far, these are the only known examples where both n and ss(n) are palindromes. note that, 101 is prime, 1001 is composite (which, by virtue of lemma 1.1, is divisible by 11), 10001 is prime, 100001 is composite, and so on. let ss( k)(n) be the k–fold composition of ss(n) with itself, that is, o ( ) ( ) ( )( ).o o .... k k factors ss n ss ss ss n= then, we have the following lemma. lemma 4.4. for any integer n (  1), ss( k)(2n + 1) = 2(n – k) + 1 for any 1  k  n. proof : the proof is by induction on k. the result is clearly true for k = 1 (by virtue of lemma 4.1). so, we assume the validity of the result for some k. but then ss( k+1)(2n + 1) = ss(ss(2n + 1) = ss(2(n – k) + 1) = 2(n – k) – 1, (17) which shows the validity of the result for k + 1. lemma 4.4 shows that, it is possible to find a palindrome n such that ss(n) is also a palindrome, by choosing an appropriate k. thus, for example, ss(2)(101) = 99 is a palindrome. 5. some remarks starting with the palindromic prime 101, we observe that 101k is an extended palindrome for any k = 10, 12, 13, …, 98, with s(101 k) = 101. but what happens if we apply z(.) on these numbers? from lemma 3.11, we see that z(1010) is not a palindrome. lemma 3.13 shows that z(1212) = 303 is a palindrome, but by lemma 3.14, z(1313) is not a palindrome. a limited search shows that, z(1515), z(1717), z(1818), z(1919), z(2323), z(2424), z(2727), z(2929), z(3030), z(3838), z(3939) and z(4545) are all palindromes, with z(1515) = 404 = z(1818) = z(2727) = z(3030) = z(4545), z(1717) = 101, z(1919) = 303 = z(2424) = z(3838), z(2323) = 505, z(2929) = 202, z(3939) = 909, and z(3232), z(4343), z(4646), z(4747), z(4848) and z(4949) are extended palindromes, with z(3232) = 1919 = z(4848), z(4343) = 2020 = z(4747), z(4646) = 2323, z(4949) = 1616. jurnal matematika mantik vol. 8, no. 1, may 2022, pp. 1-9 8 from z(9229) = 3355, we see that n may be a palindrome with z(n) being an extended palindrome. earls [16] has introduced the concept of the recursive palindromic smarandache values (rpsv) : an rpsv is a natural number n such that s(n) is a palindrome, and deleting successively the rightmost digits of n and applying s(.) at each step gives a palindrome. an example of an rpsv is n = 1514384, with s(1514384) = 94649, s(151438) = 373, s(15143) = 797, s(1514) = 757, s(151) = 151. it turns out that the rpsvs are finite with n = 1514384 being the largest such number. earls [16] then raises the following question: open problem 5.1: what is the sequence of rpsvs when the leftmost digits are repeatedly deleted? is the resulting sequence finite? in connection with the sandor-smarandache function ss(n), we have shown in lemma 4.1 how to construct a palindromic integer n such that ss(n) is also a palindrome. now, we state the following open problem. open problem 5.2: is there any other palindromic number n such that ss(n) is also a palindrome? the proof of lemma 4.3 gives the explicit form of the integer n such that both n and ss(n) are palindromes. note that, in view of lemma 4.1, this is the only possible case when n is odd. also, in view of lemma 4.1 and lemma 4.2, the following problem remains open. open problem 5.3: find ss(n) when n is even and divisible by 3, that is, n is of the form n = 6m. 6. conclusion for some smarandache-type functions s(.), z(.), ss(.), our conclusion is as follows. in smarandache function s(.), if p is a palindromic prime then for any integer n with s(n) < p then np is also a palindrome, or s(np) = p (lemma 2.2). for the pseudo smarandache function z(.), if p is a palindromic prime, then we may find an integer n such that np is a palindrome and z(np) = kp is also a palindrome, where k  1. for example, if p = 11 (2–digit palindromic prime), then z(22), z(33), z(66) and z(99) is also palindrome i.e. z(22) = z(33) = z(66)=11, and z(99) = 44 (lemma 3.1 – lemma 3.14). and finally for the sandor-smarandache function ss(.), it is possible to find a palindrome n such that ss(n) is also a palindrome, by choosing an appropriate k factors. 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[16] j. earls, “recursive palindromic smarandache values”, scientia magna, 1(2), 176 – 178, 2005. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: marsidi, marsidiarin@gmail.com department of mathematics education, universitas pgri argopuro jember, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.78-88 developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph marsidi universitas pgri argopuro jember, indonesia article history: received aug 8, 2022 revised, nov 9, 2022 accepted, nov 30, 2022 kata kunci: graf siklus, pewarnaan pelangi titik anti ajaib, nilai pewarnaan pelangi titik anti ajaib, modifikasi affine cipher, kriptosistem. abstrak. pelabelan sisi pada graf 𝐺 adalah fungsi 𝑔 dari himpunan sisi dari graf 𝐺 ke bilangan asli pertama sampai dengan kardinalitas himpunan sisi. graf 𝐺 mempunyai pewarnaan pelangi titik anti ajaib jika untuk setiap dua titik terdapat suatu lintasan dengan warna berbeda dari semua titik dalam. warna titik dari graf 𝐺 ditentukan oleh bobot titik. bobot titik pada graf 𝐺 diperoleh dengan menjumlahkan semua label sisi yang terkait dengan simpul tersebut. nilai pewarnaan pelangi titik anti ajaib dari graf 𝐺, dilambangkan dengan 𝑟𝑣𝑎𝑐(𝐺) adalah jumlah minimum warna yang diinduksi oleh pewarnaan pelangi titik anti ajaib. dalam makalah ini, kami menentukan batas atas nilai pewarnaan pelangi titik anti ajaib (𝑟𝑣𝑎𝑐) pada graf siklus (𝐶𝑛) dan mengembangkan modifikasi affine cipher dari pewarnaan pelangi titik anti ajaib untuk membuat kriptosistem yang aman. keywords: cycle graph, rainbow vertex antimagic coloring, rainbow vertex antimagic connection number, modified affine cipher, cryptosystem. abstract. an edge labeling of graph 𝐺 is a function 𝑔 from the edge set of graph 𝐺 to the first natural numbers up to the number of the edge set. graph 𝐺 admits a rainbow vertex antimagic coloring if, for any two vertices, there is a path with different colors of all internal vertices. the vertex color of graph 𝐺 is assigned by vertex weight. the vertex weight of graph 𝐺 is obtained by summing all edge labels that incident with that vertex. the rainbow vertex antimagic connection number of graph 𝐺, denoted by 𝑟𝑣𝑎𝑐(𝐺) is the smallest number of different colors induced by rainbow vertex antimagic coloring. in this research, we determine the upper bound of the rainbow vertex antimagic connection number (𝑟𝑣𝑎𝑐) on a cycle graph (𝐶𝑛) and create a secured cryptosystem using a modified affine cipher based on rainbow vertex antimagic coloring. how to cite: marsidi, “developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph”, j. mat. mantik, vol. 8, no. 2, pp. 78-88, october 2022 jurnal matematika mantik vol. 8, no. 2, october 2022, 78-88 issn: 2527-3159 (print) 2527-3167 (online) mailto:sadiq.taha@uod.ac https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 marsidi developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph 79 1. introduction graph theory is one of the studies of mathematics that has applications in the field of mathematics. one of the popular graph theory studies is coloring theory. coloring theory is developed from a four-color theorem. the theorem states that each map can be colored using four colors, so that adjacent areas do not have the same color. the development of graph coloring theory covers vertex coloring, edge coloring, face coloring, 𝒓-dynamic coloring, and rainbow coloring. in this paper, we discuss rainbow vertex antimagic coloring which begins with the following definition. the graph that is discussed in this study is a simple and connected graph, definitively it can be seen in [1]. the concept of rainbow connection has several interesting variants, one of them is rainbow vertex-connection. it was introduced by krivelevich and yuster [2]. let 𝐺 be a graph and 𝑐: 𝑉(𝐺) → {1,2,⋯,𝑘} be a vertex 𝑘coloring, for some 𝑘 ∈ 𝑁. a path 𝑃 in 𝐺 with a vertex 𝑘-coloring is said to be a rainbow vertex-path, if all internal vertices of 𝑃 have distinct colors. the graph 𝐺 is said to be a rainbow vertex connected, if for any two vertices 𝑢 and 𝑣 in 𝑉(𝐺) there is a rainbow vertex-path. a vertex 𝑘-coloring of 𝐺 is said rainbow vertex coloring, if 𝐺 rainbow vertex connected under 𝑐. the rainbow vertex connection number, denoted by 𝑟𝑣𝑐(𝐺), is the smallest positive integer k such that 𝐺 has rainbow vertex k-coloring. krivelevich and yuster also gave the lower bound for a graph 𝐺, namely 𝑟𝑣𝑐(𝐺) ≥ 𝑑𝑖𝑎𝑚(𝐺) – 1, where 𝑑𝑖𝑎𝑚(𝐺) is the diameter of the graph 𝐺 [3]. several other concepts as a result of developing the concept of rainbow vertex coloring can be seen in [4][5][6][7][8][9][10]. septory et al. [11] developed rainbow coloring into rainbow antimagic coloring, while marsidi et al. developed rainbow vertex coloring into rainbow vertex antimagic coloring. a formal definition related to rainbow vertex antimagic coloring can be seen in [12]. in this paper, we determine the upper bound of the rainbow vertex antimagic connection number of cycle graph (𝑟𝑣𝑎𝑐(𝐶𝑛)). some other results on rainbow antimagic coloring of graphs can be seen in antimagic labeling, rainbow coloring, rainbow antimagic coloring, and rainbow antimagic connection number[13][14]. furthermore, we apply the concept of rainbow vertex antimagic coloring of graphs for developing a secured cryptosystem. cryptography is a common approach for maintaining the secrecy of information dispersed across an insecure network [15]. a cryptosystem is a set of cryptographic algorithms used to implement a specific security service, such as confidentiality (encryption). the cryptosystem is a set of cryptographic techniques and infrastructure used to provide information security services. the cryptosystem is also known as a cipher system. the cryptosystem is typically formed by three algorithms: one for key generation, one for encryption, and one for decryption. the term cipher (or cypher) refers to a pair of algorithms, one for encryption and one for decryption. as a result, when the key generation algorithm is important, the term cryptosystem is most generally used. the term cryptosystem is commonly used to refer to public key techniques; however, for symmetric key techniques, both "cipher" and "cryptosystem" are used. the strength of cryptography protocols relies on the encryption-decryption keys management: how to protect the keys from disclose to unauthorized parties [16]. the affine cipher and rainbow vertex antimagic coloring concepts are combined in this article to form the modified affine cipher. the affine cipher is a monoalphabetic substitution cipher in which each letter of an alphabet is given a numeric representation, encrypted with a simple mathematical function, and then converted back to a letter. because of the formula, each letter encrypts to and from another letter, implying that the cipher is basically a regular substitution cipher with a structured which letter goes to which. as a result, it suffers from the same issues as all substitution ciphers. each letter is encoded using the function, where denotes the length of the shift. as encryption and jurnal matematika mantik vol 8, no 2, october 2022, pp. 78-88 80 decryption keys, we use labeling of rainbow vertex antimagic coloring and rainbow vertex antimagic connection number. 2. preliminaries in determining the exact value of rainbow vertex antimagic connection number of graph 𝐺 (𝑟𝑣𝑎𝑐(𝐺)), a lower bound is needed. the lower bound of 𝑟𝑣𝑎𝑐(𝐺) has been found by marsidi, et al. for more details see [12]. the lower bound of 𝑟𝑣𝑎𝑐(𝐺) that found by marsidi, et al is as follows. remark 1 [16] let 𝐺 be a connected graph, 𝑟𝑣𝑎𝑐(𝐺) ≥ 𝑟𝑣𝑐(𝐺). 2.1 cryptosystem a simple model of a cryptographic system that ensures the confidentiality of transmitted data is illustrated in figure 1. the illustration below depicts this fundamental model. figure 1. basic model of cryptosystem. before the plaintext encryption process is carried out, it must be converted first with the affine cipher method. the plaintext conversion rules with affine cipher are presented in table 1. table 1. the conversion rules on affine cipher a b c d e f g h i j 0 1 2 3 4 5 6 7 8 9 k l m n o p q r s t 10 11 12 13 14 15 16 17 18 19 u v w x y z 20 21 22 23 24 25 components of a cryptosystem the various components of a basic cryptosystem are as follows. a) plaintext. it is the data that must be secured during transmission. b) encryption algorithm. given a plaintext and an encryption key, it is a mathematical procedure that creates a ciphertext. it is a cryptographic technique that converts plaintext into ciphertext using an encryption key. marsidi developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph 81 c) ciphertext. it is the scrambled version of the plaintext generated by the encryption algorithm when a specific encryption key is used. d) decryption algorithm. it is a mathematical process that creates a unique plaintext for each ciphertext and decryption key combination. it is a cryptographic method that takes in ciphertext and a decryption key and outputs plaintext. because it reverses the encryption technique, the decryption algorithm is closely related to it. e) encryption key. the sender knows what it is. the sender inserts the encryption key and plaintext into the encryption algorithm to generate the ciphertext. f) decryption key. to the receiver, it is a known value. the encryption key and the decryption key are related, but not always identical. the receiver inputs the decryption key and the ciphertext into the decryption algorithm to retrieve the plaintext. 2.2 affine cipher an 𝑚-letter alphabet's letters are first mapped to integers in the range 0,⋯,𝑚 − 1. the number that corresponds to each plaintext letter is then converted into another integer that corresponds to a ciphertext letter using modular arithmetic. the encryption function of a single letter is 𝐸(𝑥) = (𝑎𝑥 + 𝑏) mod 𝑚 where modulus 𝑚 is the alphabet size and 𝑎 and 𝑏 are the cipher keys. the value of 𝑎 must be chosen so that 𝑎 and 𝑚 are coprime. the decryption of a function is 𝐷(𝑥) = 𝑎−1(𝑥 − 𝑏) mod 𝑚 where 𝑎−1 is the modular multiplicative inverse of 𝑎 modulo 𝑚. i.e., it satisfies the equation 1 = 𝑎𝑎−1 mod 𝑚 the multiplicative inverse of 𝑎 exists if 𝑎 and 𝑚 are coprime. without the constraint on 𝑎, the decryption may be impossible. as shown below, the decryption function is the inverse of the encryption function. 𝐷(𝐸(𝑥)) = 𝑎−1(𝐸(𝑥) − 𝑏) mod 𝑚 = 𝑎−1(((𝑎(𝑥) + 𝑏) mod 𝑚) − 𝑏) 𝑚𝑜𝑑 𝑚 = 𝑎−1(𝑎(𝑥) + 𝑏 − 𝑏) mod 𝑚 = 𝑎−1𝑎𝑥 mod 𝑚 = 𝑥 mod 𝑚 3. results and discussion this research produces the upper bound of rainbow vertex antimagic connection number of cycle graph that has not been found by previous researchers, so the results of this study are new. in addition, researchers develop applications in cryptosystems that are encrypted and decrypted using the resulting rainbow vertex antimagic connection number of cycle graph. https://en.wikipedia.org/wiki/modular_multiplicative_inverse https://en.wikipedia.org/wiki/modular_arithmetic jurnal matematika mantik vol 8, no 2, october 2022, pp. 78-88 82 3.1 rainbow vertex antimagic coloring of cycle graph theorem 2 if 𝐶𝑛 is a cycle graph with orders 𝑛 and 𝑛 ≥ 3, then 𝑟𝑣𝑎𝑐(𝐶𝑛) ≤ { 𝑛 2 , 𝑖𝑓 𝑛 ≡ 2(𝑚𝑜𝑑 4) 𝑛 2 + 1, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 4) ⌊ 𝑛 2 ⌋ + 2, 𝑖𝑓 𝑛 ≡ 1(𝑚𝑜𝑑 4) ⌊ 𝑛 2 ⌋ + 1, 𝑖𝑓 𝑛 ≡ 3(𝑚𝑜𝑑 4) proof. let 𝐶𝑛 be a cycle graph with vertex set 𝑉(𝐶𝑛) = {𝑥𝑖:1 ≤ 𝑖 ≤ 𝑛} and edge set 𝐸(𝐶𝑛) = {𝑒𝑛 = 𝑥1𝑥𝑛; 𝑒𝑖 = 𝑥𝑖𝑥𝑖+1:1 ≤ 𝑖 ≤ 𝑛 − 1}. the diameter of 𝐶𝑛 is ⌊ 𝑛 2 ⌋. we divide into four cases to prove the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛) as follows. case 1. for 𝑛 ≡ 2(𝑚𝑜𝑑 4) the edge labels in the following functions are constructed to illustrate the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛). for 1 ≤ 𝑖 ≤ 𝑛 2 , we have 𝑔(𝑒𝑖) = { 2𝑖 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖, 𝑖 ≡ 0(𝑚𝑜𝑑 2) for 𝑛 2 + 1 ≤ 𝑖 ≤ 𝑛, we have 𝑔(𝑒𝑖) = { 2𝑖 − 𝑛, 𝑖 ≡ 0(𝑚𝑜𝑑 2) 2𝑖 − 𝑛 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) we have the following vertex weights based on the edge labels above. 𝑤(𝑥1) = 𝑤( 𝑛 2 + 1) = 𝑛 + 1 𝑤(𝑥𝑖) = 4𝑖 − 3:2 ≤ 𝑖 ≤ 𝑛 2 𝑤(𝑥𝑖) = 4𝑖 − 2𝑛 − 3: 𝑛 2 + 2 ≤ 𝑖 ≤ 𝑛 from the vertex weight above, we can determine the different weight in the table 2. table 2. the different weight on the vertices of 𝐶𝑛. 𝒊 𝒘(𝒊) 𝒊 𝒘(𝒊) 1 𝑛 + 1 𝑛 2 + 1 𝑛 + 1 2 5 𝑛 2 + 2 5 3 9 𝑛 2 + 3 9 4 13 𝑛 2 + 4 13 5 17 𝑛 2 + 5 17 ⋮ ⋮ ⋮ ⋮ 𝑛 2 2𝑛 − 3 𝑛 2𝑛 − 3 marsidi developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph 83 we know that the number of different weights of 𝐶𝑛 is 𝑛 2 . it concludes that the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛) is 𝑛 2 . furthermore, we prove that every pair of 𝐶𝑛 vertices have rainbow vertex antimagic coloring. suppose that 𝑣 ∈ 𝑉(𝐶𝑛), table 3 shows the rainbow vertex path in accordance with the vertex weight. table 3. the path of rainbow vertex on cycle of order 𝑛. case 𝒖 𝒗 rainbow vertex coloring of 𝒖 − 𝒗 path 𝑖 < 𝑗;𝑗 − 𝑖 ≤ 𝑛 2 𝑣𝑖 𝑣𝑗 𝑣𝑖,𝑣𝑖+1,𝑣𝑖+2, 𝑣𝑖+3, ⋯,𝑣𝑗−1, 𝑣𝑗 𝑖 < 𝑗;𝑗 − 𝑖 > 𝑛 2 𝑣𝑖 𝑣𝑗 𝑣𝑗,𝑣𝑗+1 ,𝑣𝑗+2,⋯,𝑣𝑛 ,𝑣1,⋯,𝑣𝑖 case 2. for 𝑛 ≡ 0(𝑚𝑜𝑑 4) the edge labels in the following functions are constructed to illustrate the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛). for 1 ≤ 𝑖 ≤ 𝑛 2 , we have 𝑔(𝑒𝑖) = { 2𝑖 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖, 𝑖 ≡ 0(𝑚𝑜𝑑 2) for 𝑛 2 + 1 ≤ 𝑖 ≤ 𝑛, we have 𝑔(𝑒𝑖) = { 2𝑖 − 𝑛, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖 − 𝑛 − 1, 𝑖 ≡ 0(𝑚𝑜𝑑 2) we have the following vertex weights based on the edge labels above. 𝑤(𝑥1) = 𝑛 𝑤(𝑥𝑖) = 4𝑖 − 3:2 ≤ 𝑖 ≤ 𝑛 2 𝑤( 𝑛 2 + 1) = 𝑛 + 2 𝑤(𝑥𝑖) = 4𝑖 − 2𝑛 − 3: 𝑛 2 + 2 ≤ 𝑖 ≤ 𝑛 from the vertex weight above, we can determine the different weight in the table 4. table 4. the different weight on the vertices of 𝐶𝑛. 𝒊 𝒘(𝒊) 𝒊 𝒘(𝒊) 1 𝑛 𝑛 2 + 1 𝑛 + 2 2 5 𝑛 2 + 2 5 3 9 𝑛 2 + 3 9 4 13 𝑛 2 + 4 13 5 17 𝑛 2 + 5 17 ⋮ ⋮ ⋮ ⋮ 𝑛 2 2𝑛 − 3 𝑛 2𝑛 − 3 jurnal matematika mantik vol 8, no 2, october 2022, pp. 78-88 84 we know that the number of different weights of 𝐶𝑛 is 𝑛 2 + 1. it concludes that the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛) is 𝑛 2 + 1. furthermore, we prove that every pair of 𝐶𝑛 vertices have rainbow vertex antimagic coloring. suppose that 𝑣 ∈ 𝑉(𝐶𝑛), table 5 shows the rainbow vertex path in accordance with the vertex weight. table 5. the path of rainbow vertex on cycle of order 𝑛. case 𝒖 𝒗 rainbow vertex coloring of 𝒖 − 𝒗 path 𝑖 < 𝑗;𝑗 − 𝑖 ≤ 𝑛 2 𝑣𝑖 𝑣𝑗 𝑣𝑖,𝑣𝑖+1,𝑣𝑖+2, 𝑣𝑖+3, ⋯,𝑣𝑗−1, 𝑣𝑗 𝑖 < 𝑗;𝑗 − 𝑖 > 𝑛 2 𝑣𝑖 𝑣𝑗 𝑣𝑗,𝑣𝑗+1 ,𝑣𝑗+2,⋯,𝑣𝑛 ,𝑣1,⋯,𝑣𝑖 case 3. for 𝑛 ≡ 1(𝑚𝑜𝑑 4) the edge labels in the following functions are constructed to illustrate the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛). for 1 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉, we have 𝑔(𝑒𝑖) = { 2𝑖 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖, 𝑖 ≡ 0(𝑚𝑜𝑑 2) 𝑔(𝑒 ⌈ 𝑛 2 ⌉ ) = 𝑛 for ⌈ 𝑛 2 ⌉ + 1 ≤ 𝑖 ≤ 𝑛, we have 𝑔(𝑒𝑖) = { 2𝑖 − 𝑛 − 1, 𝑖 ≡ 0(𝑚𝑜𝑑 2) 2𝑖 − 𝑛 − 2, 𝑖 ≡ 1(𝑚𝑜𝑑 2) we have the following vertex weights based on the edge labels above. 𝑤(𝑥1) = 𝑛 − 1 𝑤(𝑥𝑖) = 4𝑖 − 3:2 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉ 𝑤(⌈ 𝑛 2 ⌉ + 1) = 𝑛 + 2 𝑤(𝑥𝑖) = 4𝑖 − 2𝑛 − 5: ⌈ 𝑛 2 ⌉ + 2 ≤ 𝑖 ≤ 𝑛 from the vertex weight above, we can determine the different weight in the table 6. table 6. the different weight on the vertices of 𝐶𝑛. 𝒊 𝒘(𝒊) 𝒊 𝒘(𝒊) 1 𝑛-1 ⌈ 𝑛 2 ⌉ + 1 𝑛 + 2 2 5 ⌈ 𝑛 2 ⌉ + 2 5 3 9 ⌈ 𝑛 2 ⌉ + 3 9 4 13 ⌈ 𝑛 2 ⌉ + 4 13 5 17 ⌈ 𝑛 2 ⌉ + 5 17 ⋮ ⋮ ⋮ ⋮ ⌈ 𝑛 2 ⌉ 2𝑛 − 1 𝑛 2𝑛 − 5 marsidi developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph 85 we know that the number of different weights of 𝐶𝑛 is ⌊ 𝑛 2 ⌋ + 2. it concludes that the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛) is ⌊ 𝑛 2 ⌋ + 2. furthermore, we prove that every pair of 𝐶𝑛 vertices have rainbow vertex antimagic coloring. suppose that 𝑣 ∈ 𝑉(𝐶𝑛), table 7 shows the rainbow vertex path in accordance with the vertex weight. table 7. the path of rainbow vertex on cycle of order 𝑛. case 𝒖 𝒗 rainbow vertex coloring of 𝒖 − 𝒗 path 𝑖 < 𝑗;𝑗 − 𝑖 ≤ ⌈ 𝑛 2 ⌉ 𝑣𝑖 𝑣𝑗 𝑣𝑖,𝑣𝑖+1,𝑣𝑖+2, 𝑣𝑖+3, ⋯,𝑣𝑗−1, 𝑣𝑗 𝑖 < 𝑗;𝑗 − 𝑖 > ⌈ 𝑛 2 ⌉ 𝑣𝑖 𝑣𝑗 𝑣𝑗,𝑣𝑗+1 ,𝑣𝑗+2,⋯,𝑣𝑛 ,𝑣1,⋯,𝑣𝑖 case 4. for 𝑛 ≡ 3(𝑚𝑜𝑑 4) the edge labels in the following functions are constructed to illustrate the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛). for 1 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉, we have 𝑔(𝑒𝑖) = { 2𝑖 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖, 𝑖 ≡ 0(𝑚𝑜𝑑 2) 𝑔(𝑒 ⌈ 𝑛 2 ⌉ ) = 𝑛 for ⌈ 𝑛 2 ⌉ + 1 ≤ 𝑖 ≤ 𝑛, we have 𝑔(𝑒𝑖) = { 2𝑖 − 𝑛 − 1, 𝑖 ≡ 1(𝑚𝑜𝑑 2) 2𝑖 − 𝑛 − 2, 𝑖 ≡ 0(𝑚𝑜𝑑 2) we have the following vertex weights based on the edge labels above. 𝑤(𝑥1) = 𝑛 𝑤(𝑥𝑖) = 4𝑖 − 3:2 ≤ 𝑖 ≤ ⌈ 𝑛 2 ⌉ − 1 𝑤(⌈ 𝑛 2 ⌉) = 2𝑛 − 2 𝑤(⌈ 𝑛 2 ⌉ + 1) = 𝑛 + 2 𝑤(𝑥𝑖) = 4𝑖 − 2𝑛 − 5: ⌈ 𝑛 2 ⌉ + 2 ≤ 𝑖 ≤ 𝑛 from the vertex weight above, we can determine the different weight in the table 8. table 8. the different weight on the vertices of 𝐶𝑛. 𝒊 𝒘(𝒊) 𝒊 𝒘(𝒊) 1 𝑛 ⌈ 𝑛 2 ⌉ + 1 𝑛 + 2 2 5 ⌈ 𝑛 2 ⌉ + 2 5 3 9 ⌈ 𝑛 2 ⌉ + 3 9 4 13 ⌈ 𝑛 2 ⌉ + 4 13 5 17 ⌈ 𝑛 2 ⌉ + 5 17 ⋮ ⋮ ⋮ ⋮ ⌈ 𝑛 2 ⌉ 2𝑛 − 2 𝑛 2𝑛 − 5 jurnal matematika mantik vol 8, no 2, october 2022, pp. 78-88 86 we know that the number of different weights of 𝐶𝑛 is ⌊ 𝑛 2 ⌋ + 1. it concludes that the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛) is ⌊ 𝑛 2 ⌋ + 1. furthermore, we prove that every pair of 𝐶𝑛 vertices have rainbow vertex antimagic coloring. suppose that 𝑣 ∈ 𝑉(𝐶𝑛), table 9 shows the rainbow vertex path in accordance with the vertex weight. table 9. the path of rainbow vertex on cycle of order 𝑛. case 𝒖 𝒗 rainbow vertex coloring of 𝒖 − 𝒗 path 𝑖 < 𝑗;𝑗 − 𝑖 ≤ ⌈ 𝑛 2 ⌉ 𝑣𝑖 𝑣𝑗 𝑣𝑖,𝑣𝑖+1,𝑣𝑖+2, 𝑣𝑖+3, ⋯,𝑣𝑗−1, 𝑣𝑗 𝑖 < 𝑗;𝑗 − 𝑖 > ⌈ 𝑛 2 ⌉ 𝑣𝑖 𝑣𝑗 𝑣𝑗,𝑣𝑗+1 ,𝑣𝑗+2,⋯,𝑣𝑛 ,𝑣1,⋯,𝑣𝑖 as a result, the vertex coloring on 𝐶𝑛 is rainbow vertex antimagic coloring. thus, we obtain 𝑟𝑣𝑎𝑐(𝐶𝑛) ≤ { 𝑛 2 , 𝑖𝑓 𝑛 ≡ 2(𝑚𝑜𝑑 4) 𝑛 2 + 1, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 4) ⌊ 𝑛 2 ⌋ + 2, 𝑖𝑓 𝑛 ≡ 1(𝑚𝑜𝑑 4) ⌊ 𝑛 2 ⌋ + 1, 𝑖𝑓 𝑛 ≡ 3(𝑚𝑜𝑑 4) . ∎ 3.2 application the encryption and decryption process can implement the result of labeling and the chromatic number of rainbow vertex coloring, which is used as the key for the encryption and decryption process. we developed a new algorithm to generate encryption and decryption keys with rainbow vertex antimagic chromatic numbers and labels. algorithm 1. role of extra key a) define 𝑓 as the function of graph element labels of graph 𝐺. b) if 𝑓 is a bijective function, do 3, and bring it back to 1 otherwise. c) define 𝑏 as the rainbow vertex antimagic chromatic number. d) to use the vertex weight, define 𝑧𝑖 1:1 ≤ 𝑖 ≤ 𝑛, where 𝑛 is the number of vertexes. e) add 𝑧𝑖 1 and arrange the sequence according to the vertex notation 𝑥𝑖: 1 ≤ 𝑖 ≤ 𝑛, where 𝑛 is the number of vertices. f) to use the edge label, define 𝑧𝑗 2:1 ≤ 𝑗 ≤ 𝑚, where 𝑚 is the number of edges. g) add 𝑧𝑗 2 and arrange the sequence according to the edge notation 𝑒𝑗:1 ≤ 𝑗 ≤ 𝑚. h) set 𝑘 as element of the 𝑧𝑖 1 and 𝑧𝑗 2 sequence. in algorithm 1, it will use the process of developing the key in the encryption and decryption process. algorithm 2 developed the affine cipher based on the results of algorithm 1, which includes an encryption and description process. marsidi developing a secure cryptosystem with rainbow vertex antimagic coloring of cycle graph 87 algorithm 2. modified affine cipher a) given that the plaintext and its conversion with affine cipher 𝑃𝑖: 1 ≤ 𝑖 ≤ 𝑛. b) compute the ciphertext using equation 1 and compute the plaintext blocks using equation 2. φ𝑖 = ((𝑃𝑖 + 𝐾1𝑖) + 𝐾2𝑖) 𝑚𝑜𝑑 26 (1) 𝑃𝑖 = ((φ𝑖 − 𝐾2𝑖) − 𝐾1𝑖) 𝑚𝑜𝑑 26 (2) where 𝑃𝑖, 𝐾1𝑖, 𝐾2𝑖, and φ𝑖 are the 𝑖-th of conversion plaintext, key sequence, and conversion ciphertext, respectively. note that, for 𝑛 = 1, φ𝑛−1 is a null vector. for an illustration how the algorithms are working, we give the following examples. given that a plaintext 𝑃 = 𝑈𝑁𝐼𝑃𝐴𝑅𝐽𝐸𝑀𝐵𝐸𝑅, by mean the two algorithms above we have a ciphertext φ = 𝐻𝑊𝑊𝐾𝐴𝑌𝑍𝑀𝐵𝑉𝐹𝑋. the cryptosystem process can be described in the following tables. table 9. encryption process. plaintext 𝑼 𝑵 𝑰 𝑷 𝑨 𝑹 𝑱 𝑬 𝑴 𝑩 𝑬 𝑹 𝑃𝑖 20 13 8 15 0 17 9 4 12 1 4 17 𝐾1𝑖 12 5 9 13 17 21 14 5 9 13 17 21 𝑃𝑖 + 𝐾1𝑖 32 18 17 28 17 38 23 9 21 14 21 38 𝐾2𝑖 1 4 5 8 9 12 2 3 6 7 10 11 (𝑃𝑖 + 𝐾1𝑖) + 𝐾2𝑖 33 22 22 36 26 50 25 12 27 21 31 49 φ𝑖 7 22 22 10 0 24 25 12 1 21 5 23 ciphertext 𝑯 𝑾 𝑾 𝑲 𝑨 𝒀 𝒁 𝑴 𝑩 𝑽 𝑭 x table 10. decryption process. ciphertext 𝑯 𝑾 𝑾 𝑲 𝑨 𝒀 𝒁 𝑴 𝑩 𝑽 𝑭 x φ𝑖 7 22 22 10 0 24 25 12 1 21 5 23 𝐾2𝑖 1 4 5 8 9 12 2 3 6 7 10 11 φ𝑖 − 𝐾2𝑖 6 18 17 2 -9 12 23 9 -5 14 -5 12 𝐾1𝑖 12 5 9 13 17 21 14 5 9 13 17 21 (φ𝑖 − 𝐾2𝑖) − 𝐾1𝑖 -6 13 8 -11 -26 -9 9 4 -14 1 -22 -9 𝑃𝑖 20 13 8 15 0 17 9 4 12 1 4 17 plaintext 𝑼 𝑵 𝑰 𝑷 𝑨 𝑹 𝑱 𝑬 𝑴 𝑩 𝑬 𝑹 4. conclusions we have determined the upper bound of 𝑟𝑣𝑎𝑐(𝐶𝑛). since determining the 𝑟𝑣𝑎𝑐 of a graph is considered a non-deterministic polynomial time-complete problem, the exact value of any graph 𝐺 remains unsolved. as a result, we offer the following open problems. a) find the exact value of the cycle graph's rainbow vertex antimagic connection number. b) find the exact value of any graph's rainbow vertex antimagic connection number. jurnal matematika mantik vol 8, no 2, october 2022, pp. 78-88 88 5. acknowledgments we would like to thank the department of mathematics education, universitas pgri argopuro jember, cgant university of jember in 2022, and the reviewers that helped us finish this research. references [1] g. chartrand, l. lesniak, and p. zhang, graphs & digraphs, fifth edition. 2010. 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[8] m. s. hasan, slamin, dafik, i. h. agustin, and r. alfarisi, “on the total rainbow connection of the wheel related graphs,” 2018. [9] p. heggernes, d. issac, j. lauri, p. t. lima, and e. j. van leeuwen, “rainbow vertex coloring bipartite graphs and chordal graphs,” leibniz int. proc. informatics, lipics, vol. 117, no. 83, pp. 1–13, 2018, doi: 10.4230/lipics.mfcs.2018.83. [10] dafik, slamin, and a. muharromah, “on the ( strong ) rainbow vertex connection of graphs resulting from edge comb product,” 2018. [11] b. j. septory, m. i. utoyo, dafik, b. sulistiyono, and i. h. agustin, “on rainbow antimagic coloring of special graphs,” j. phys. conf. ser., vol. 1836, no. 1, 2021, doi: 10.1088/1742-6596/1836/1/012016. [12] marsidi, i. h. agustin, dafik, and e. y. kurniawati, “on rainbow vertex antimagic coloring of graphs: a new notion,” cauchy – j. mat. murni dan apl., vol. 7, no. 1, pp. 64–72, 2021. [13] dafik, f. susanto, r. alfarisi, i. h. agustin, and m. venkatachalam, “on rainbow antimagic coloring of graphs,” adv. math. model. appl., vol. 6, no. 3, pp. 278–291, 2021. [14] 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: 10.1088/1742-6596/1832/1/012016. [15] a. c. prihandoko, d. dafik, and i. h. agustin, “implementation of super hantimagic total graph on establishing stream cipher,” indones. j. comb., vol. 3, no. 1, p. 14, 2019, doi: 10.19184/ijc.2019.3.1.2. [16] a. c. prihandoko, dafik, and i. h. agustin, “stream-keys generation based on graph labeling for strengthening vigenere encryption,” int. j. electr. comput. eng., vol. 12, no. 4, pp. 3960–3969, 2022, doi: 10.11591/ijece.v12i4.pp39603969. open access proceedings journal of physics: conference series how to cite: m. ikbal and riskawati, “dynamics of predator-prey model interaction with harvesting effort”, j. mat. mantik, vol. 6, no. 2, pp.93-103, october 2020. jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 93-103 issn: 2527-3159 (print) 2527-3167 (online) dynamics of predator-prey model interaction with harvesting effort muhammad ikbal1, riskawati2 1universitas muslim maros, ibbal@umma.ac.id 2universitas muslim maros, riskawati@umma.ac.id doi: https://doi.org/10.15642/mantik.2020.6.2.93-103 abstrak. di dalam penelitian ini, kami mempelajari dan mengonstruksi suatu dinamika model mangsa-pemangsa. kami memasukkan unsur kompetisi intraspesifik pada kedua pemangsa. kami menformulasikan fungsi respon holling tipe i pada masing-masing pemangsa. kami menganggap semua populasi bernilai ekonomis sehingga dapat dipanen. kami menganalisis solusi positifnya, keeksisan titik keseimbangannya, dan kestabilan pada titik-titik keseimbangannya itu. kondisi kestabilan lokalnya kami peroleh dengan pendekatan kriteria routh-hurwitz. kami juga mensimulasikan model tersebut. penelitian ini bisa dikembangkan dengan formulasi fungsi respon yang berbeda dan pengoptimalan pemanenan. kata kunci: mangsa-pemangsa; intraspesifik; pemanenan; routh-hurwitz abstract. in this research, we study and construct a dynamic prey-predator model. we include an element of intraspecific competition in both predators. we formulated the holling type i response function for each predator. we consider all populations to be of economic value so that they can be harvested. we analyze the positive solution, the existence of the equilibrium points, and the stability of the balance points. we obtained the local stability condition by using the routh-hurwitz criterion approach. we also simulate the model. this research can be developed with different response function formulations and harvest optimization. keywords: prey-predator; intraspecific; harvesting; routh-hurwitz http://u.lipi.go.id/1458103791 mailto:ibbal@umma.ac.id mailto:riskawati@umma.ac.id jurnal matematika mantik volume 6, no. 2, october 2020, pp. 93-103 94 1. introduction mathematical models can be used in observing individual behavior, population dynamics and population linkages in a system. mathematical models can also be used in determining a policy. mathematical modeling in the field of ecology is very interesting to study considering the many factors that affect the growth and life of living populations and the balance of organisms. the process of dynamics of organisms can be modeled mathematically by using differential equations involving continuous time or discrete time. one of the mathematical models used to explain this natural phenomenon is the preypredator population model. competition between predators and harvesting factors in populations is very important in the discipline of ecology. many researchers can evoke interesting things from behavioral dynamics in population ecosystems. by combining the two aspects above, namely the aspects of competition between predators and harvesting, population dynamics can be expressed in a model. one of the policies related to the use of living things is harvesting. intraspecific competition factors are also interesting to study. intraspecific competition is competition between predators in competing for prey. this is another factor in population dynamics that can affect the stability of a system. there are many researchers who model prey-predator interactions. [1] examined the resistance of predators in the prey-predator model system with non-periodic solutions. [2] discusses the dynamics of the prey-predator with diseased predators. [3] in his journal discussed global dynamics of a prey-predator model with antipredator behavior and two predators. [4] discuss the dynamics of the prey-predator model by quadratic harvesting. research from [5] discusses global dynamics and control of predator prey models with holling type iii response functions. [6] examined the effect of harvesting and competition between predators in the prey-predator model. [7] discuss a model of interaction of three species in one habitat. [8] discuss the complex dynamics of a three-species food chain model with holling type iii response functions. many previous researchers have examined prey-predator population models. we examine prey-predator population models with respect to intraspecific competition for predators and considering the economic value of all populations. our study constructs the factors influencing prey-predator population dynamics as investigated by previous researchers, but we add intraspecific competition and harvesting factors simultaneously to all three populations. 2. assumptions and model in this model, there is an interference between predators as modeled by other researchers [1], [6], [8], [9]. there are researchers who studied the intraspecific competition coefficient [9]. researchers frequently use the holling-type i response function [3], [5], [8]. the response function is used by researchers in their models [10]– [13]. researchers also use the harvesting rate [4], [6], [14], [15]. the assumptions used are: • the prey growth rate uses the logistical growth rate, • predators compete with each other for prey, • all of predators uses the holling type i response function for predation, • there is an intraspecific competition for each predator. • all of population have interest economic values. the model is formulated as follows: 𝑑𝑃 𝑑𝑡 = 𝑟𝑃 (1 − 𝑃 𝐾 ) − 𝛼1𝑃𝐻1 − 𝛼2𝑃𝐻2 − 𝑞1𝐸1𝑃 (1) m. ikbal, riskawati dynamics of predator-prey model interaction with harvesting effort 95 𝑑𝐻1 𝑑𝑡 = 𝑒1𝛼1𝑃𝐻1 − 𝑔1𝐻1 2 − 𝛽1𝐻1𝐻2 − 𝑑1𝐻1 − 𝑞2𝐸2𝐻1 𝑑𝐻2 𝑑𝑡 = 𝑒2𝛼2𝑃𝐻2 − 𝑔2𝐻2 2 − 𝛽2𝐻1𝐻2 − 𝑑2𝐻2 − 𝑞3𝐸3𝐻2 with initial condition p(0) ≥ 0, h1(0) ≥ 0, h2(0) ≥ 0. (2) the first predator (h1) and second predator (h2) are assumed to have direct access to prey (p). the effect of disturbance in the growth rate of competitors is assumed to be proportional to the density of the predator population with β 1 and β 2 respectively given disturbance rates. the parameter α1 and α2 represent predator rates for predator species h1 and h2 respectively. the parameter g1 and g2 represent coefficient intraspecific competition for two predators h1 and h2 respectively, here d1 and d2 are their mortality rate. the parameter e1 and e2 are the predator’s conversion efficiency. however, the predation functions of the two predators were made different one following a holling type i response and the other following a holling type ii response. besides experiencing a reduction due to the predation function, the prey population grew logistically with r as the intrinsic growth rate and k as the holding capacity. the parameter q 1 , q 2 , and q 3 are the catchability coefficient of susceptible prey, first predator, and second predator, respectively. the parameter e1, e2, and e3 are the harvesting effort prey, first predator, and second predator, respectively. 3. equilibrium points and stability analysis 3.1. equilibrium points equilibrium points of the system (1) are given below: • the trivial equilibrium point 𝑇0= (0, 0, 0), • the predator free equilibrium point 𝑇1 = (𝐾 − 𝐾𝑞 1 𝐸1 𝑟 , 0, 0), • the h1-free boundary equilibrium state 𝑇2 = ( 𝐾(𝑞 3 𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔 2 , 0, 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔 2 ) • the h2-free boundary equilibrium state 𝑇3 = ( 𝐾(𝑞2𝐸2𝛼1 + 𝛼1𝑑1 + 𝑟𝑔1 − 𝑞1𝑔1𝐸1) 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 , 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 , 0) • the interior equilibrium point 𝑇4 = (𝑃 ∗, 𝐻1 ∗ , 𝐻2 ∗), where p * = 𝐾(𝑞1𝐸1𝑔1𝑔2 + 𝑞2𝐸2𝛼2𝛽2 + 𝑞3𝐸3𝛼1𝛽1 + 𝑑1𝛼2𝛽2 + 𝑑2𝛼1𝛽1 + 𝑟𝛽1𝛽2) 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 − 𝐾(𝑞1𝐸1𝛽1𝛽2 + 𝑞2𝐸2𝛼1𝑔2 + 𝑞3𝐸3𝛼2𝑔1 + 𝑑1𝛼1𝑔2 + 𝑑2𝛼2𝑔1 + 𝑟𝑔1𝑔2) 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 h1 * = 𝐾𝑒1𝛼1𝑞1𝐸1𝑔2 + 𝐾𝑒2𝛼2 2𝑞2𝐸2 + 𝐾𝑑1𝑒2𝛼2 2 + 𝐾𝑟𝑒2𝛼2𝛽1 + 𝑔2𝑟𝑞2𝐸2 + 𝑟𝑑1𝑔2 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 − (𝐾𝑒1𝛼1𝛼2𝑞3𝐸3 + 𝐾𝑒2𝛼2𝛽1𝑞1𝐸1 + 𝐾𝑒1𝛼1𝛼2𝑑2 + 𝐾𝑟𝑒1𝛼1𝑔2 + 𝑟𝛽1 𝑞3𝐸3 + 𝑟𝑑2𝛽1) 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 h2 * = 𝐾𝑒1𝛼1𝑞1𝐸1𝛽2 + 𝐾𝑒2𝛼1𝛼2𝑞2𝐸2 + 𝐾𝑑1𝑒2𝛼1𝛼2 + 𝐾𝑟𝑒2𝛼2𝑔1 + 𝛽2𝑟𝑞2𝐸2 + 𝑟𝑑1𝛽2 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 − (𝐾𝑒1𝛼1 2𝑞3𝐸3 + 𝐾𝑒2𝛼2𝑔1𝑞1𝐸1 + 𝐾𝑒1𝛼1 2𝑑2 + 𝐾𝑟𝑒1𝛼1𝛽2 + 𝑟𝑔1𝑞3𝐸3 + 𝑟𝑑2𝑔1) 𝐾𝑒1𝛼1𝛼2𝛽2 + 𝐾𝑒2𝛼1𝛼2𝛽1 + 𝑟𝛽1𝛽2 − 𝐾𝑒1𝛼1 2𝑔2 − 𝐾𝑒2𝛼2 2𝑔1 − 𝑟𝑔1𝑔2 jurnal matematika mantik volume 6, no. 2, october 2020, pp. 93-103 96 3.2. stability analysis the stability analysis equilibrium point of the system (1) is studied and determined. the point 𝑇0 is trivial equilibrium point. jacobian matrix of the model system (1) is 𝐽 = [ 𝐽11 𝐽12 𝐽13 𝐽21 𝐽22 𝐽23 𝐽31 𝐽32 𝐽33 ] (3) where, 𝐽11 = 𝑟 − 2𝑟𝑃 𝐾 − 𝛼1𝐻1 − 𝛼2𝐻2 − 𝑞1𝐸1 𝐽12 = −𝛼1𝑃 𝐽13 = −𝛼2𝑃 𝐽21 = 𝑒1𝛼1𝐻1 𝐽22 = 𝑒1𝛼1𝑃 − 2𝑔1𝐻1 − 𝛽1𝐻2 − 𝑑1 − 𝑞2𝐸2 𝐽23 = −𝛽1𝐻1 𝐽31 = 𝑒2𝛼2𝐻2 𝐽32 = −𝛽2𝐻2 𝐽33 = 𝑒2𝛼2𝑃 − 2𝑔2𝐻2 − 𝛽2𝐻1 − 𝑑2 − 𝑞3𝐸3 theorem 1. equilibrium point 𝑇1 local stable if 𝑟 > 𝑞1𝐸1, 𝑞2𝐸2 + 𝑑1 > 𝐾𝑒1𝛼1(𝑟−𝑞1𝐸1) 𝑟 , and 𝑞3𝐸3 + 𝑑2 > 𝐾𝑒2𝛼2(𝑟−𝑞1𝐸1) 𝑟 . proof. the result of substitution equilibrium point t1 to jacobian matrix (3) 𝐽(𝑇1) = [ 𝐽11 1 𝐽12 1 𝐽13 1 𝐽21 1 𝐽22 1 𝐽23 1 𝐽31 1 𝐽32 1 𝐽33 1 ] (4) where 𝐽11 1 = 𝑞1𝐸1 − 𝑟 𝐽12 1 = − 𝛼1𝐾(𝑟 − 𝑞1𝐸1) 𝑟 𝐽13 1 = − 𝛼2𝐾(𝑟 − 𝑞1𝐸1) 𝑟 𝐽21 1 = 0 𝐽22 1 = 𝐾𝑒1𝛼1(𝑟 − 𝑞1𝐸1) 𝑟 − 𝑞2𝐸2 − 𝑑1 𝐽23 1 = 0 𝐽31 1 = 0 𝐽32 1 = 0 𝐽33 2 = 𝐾𝑒2𝛼2(𝑟 − 𝑞1𝐸1) 𝑟 − 𝑞3𝐸3 − 𝑑2 characteristic equation matrix 𝐽(𝑇1) is (𝜆 − 𝑞1𝐸1 + 𝑟) (𝜆 − 𝐾𝑒1𝛼1(𝑟 − 𝑞1𝐸1) 𝑟 + 𝑞2𝐸2 + 𝑑1) (𝜆 − 𝐾𝑒2𝛼2(𝑟 − 𝑞1𝐸1) 𝑟 + 𝑞3𝐸3 + 𝑑2) = 0 (5) the roots of the equation (5) is negative if 𝑟 > 𝑞1𝐸1, 𝑞2𝐸2 + 𝑑1 > 𝐾𝑒1𝛼1(𝑟−𝑞1𝐸1) 𝑟 , and 𝑞3𝐸3 + 𝑑2 > 𝐾𝑒2𝛼2(𝑟−𝑞1𝐸1) 𝑟 . m. ikbal, riskawati dynamics of predator-prey model interaction with harvesting effort 97 theorem 2. equilibrium point 𝑇2 local stable if 𝐽11 2 < 0, 𝐽22 2 < 0, 𝐽33 2 < 0, 𝐽11 2 𝐽22 2 + 𝐽11 2 𝐽33 2 + 𝐽22 2 𝐽33 2 > 𝐽13 2 𝐽31 2 and 𝐽13 2 𝐽31 2 𝐽22 2 > 𝐽11 2 𝐽22 2 𝐽33 2 . proof. the result of substitution equilibrium point t2 to jacobian matrix (3) 𝐽(𝑇2) = [ 𝐽11 2 𝐽12 2 𝐽13 2 𝐽21 2 𝐽22 2 𝐽23 2 𝐽31 2 𝐽32 2 𝐽33 2 ] (6) where 𝐽11 2 =𝑟 − 2𝑟(𝑞3𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 −𝛼2 [ 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] − 𝑞1𝐸1 𝐽12 2 = −𝛼1 [ 𝐾(𝑞3𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] 𝐽13 2 = −𝛼2 [ 𝐾(𝑞3𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] 𝐽21 2 = 0 𝐽22 2 = 𝑒1𝛼1 [ 𝐾(𝑞3𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] −𝛽1 [ 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] − 𝑑1 − 𝑞2𝐸2 𝐽23 2 = 0 𝐽31 2 = 𝑒2𝛼2 [ 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] 𝐽32 2 = −𝛽2 [ 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] 𝐽33 2 = 𝑒2𝛼2 [ 𝐾(𝑞3𝐸3𝛼2 + 𝛼2𝑑2 + 𝑟𝑔2 − 𝑞1𝑔2𝐸1) 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] −2𝑔2 [ 𝐾𝑟𝑒2𝛼2 − 𝑟𝑞3𝐸3 − 𝑑2𝑟 − 𝐾𝑒2𝛼2𝑞1𝐸1 𝐾𝑒2𝛼2 2 + 𝑟𝑔2 ] − 𝑑2 − 𝑞3𝐸3 characteristic equation matrix 𝐽(𝑇2) is 𝜆3 + 𝐴1𝜆 2 + 𝐴2𝜆 + 𝐴3 = 0 (7) where, 𝐴1 = −(𝐽11 2 + 𝐽22 2 + 𝐽33 2 ) 𝐴2 = 𝐽11 2 𝐽22 2 + 𝐽11 2 𝐽33 2 + 𝐽22 2 𝐽33 2 − 𝐽13 2 𝐽31 2 𝐴3 = 𝐽13 2 𝐽31 2 𝐽22 2 − 𝐽11 2 𝐽22 2 𝐽33 2 the roots of the equation (7) is negative if 𝐽11 2 < 0, 𝐽22 2 < 0, 𝐽33 2 < 0, 𝐽11 2 𝐽22 2 + 𝐽11 2 𝐽33 2 + 𝐽22 2 𝐽33 2 > 𝐽13 2 𝐽31 2 , 𝐽13 2 𝐽31 2 𝐽22 2 > 𝐽11 2 𝐽22 2 𝐽33 2 and a1a2 > a3 theorem 3. equilibrium point 𝑇3 local stable if 𝐽11 3 + 𝐽22 3 + 𝐽33 3 < 0, 𝐽11 3 𝐽22 3 + 𝐽11 3 𝐽33 3 + 𝐽22 3 𝐽33 3 > 𝐽12 3 𝐽21 3 , 𝐽12 3 𝐽21 3 𝐽33 3 > 𝐽11 3 𝐽22 3 𝐽33 3 , and 𝐵1𝐵2 > 𝐵3 proof. the result of substitution equilibrium point 𝑇3 to jacobian matrix (3) jurnal matematika mantik volume 6, no. 2, october 2020, pp. 93-103 98 𝐽(𝑇3) = [ 𝐽11 3 𝐽12 3 𝐽13 3 𝐽21 3 𝐽22 3 𝐽23 3 𝐽31 3 𝐽32 3 𝐽33 3 ] (8) where 𝐽11 3 = 𝑟 − 2𝑟 [ (𝑞2𝐸2𝛼1+𝛼1𝑑1+𝑟𝑔1−𝑞1 𝑔1𝐸1) 𝐾𝑒1𝛼1 2+𝑟𝑔1 ] − 𝛼1 [ 𝐾𝑟𝑒1𝛼1−𝑟𝑞2𝐸2−𝑑1𝑟−𝐾𝑒1𝛼1𝑞1 𝐸1 𝐾𝑒1𝛼1 2+𝑟𝑔1 ] − 𝑞1𝐸1 𝐽12 3 = − 𝛼1 [ (𝑞2𝐸2𝛼1 + 𝛼1𝑑1 + 𝑟𝑔1 − 𝑞1𝑔1𝐸1) 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] 𝐽13 3 = −𝛼2 [ (𝑞2𝐸2𝛼1 + 𝛼1𝑑1 + 𝑟𝑔1 − 𝑞1𝑔1𝐸1) 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] 𝐽21 3 = 𝑒1𝛼1 [ 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] 𝐽22 3 = 𝑒1𝛼1𝑃 − 2𝑔1 [ 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] −𝛽1 [ 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] − 𝑑1 − 𝑞2𝐸2 𝐽23 3 = −𝛽1 [ 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] 𝐽31 3 = 0 𝐽32 3 = 0 𝐽33 3 = 𝑒2𝛼2 [ 𝐾(𝑞2𝐸2𝛼1 + 𝛼1𝑑1 + 𝑟𝑔1 − 𝑞1𝑔1𝐸1) 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] −𝛽2 [ 𝐾𝑟𝑒1𝛼1 − 𝑟𝑞2𝐸2 − 𝑑1𝑟 − 𝐾𝑒1𝛼1𝑞1𝐸1 𝐾𝑒1𝛼1 2 + 𝑟𝑔1 ] − 𝑑2 − 𝑞3𝐸3 characteristics equation matrix 𝐽(𝑇3) is 𝜆3 + 𝐵1𝜆 2 + 𝐵2𝜆 + 𝐵3 = 0 (9) where, 𝐵1 = −(𝐽11 3 + 𝐽22 3 + 𝐽33 3 ) 𝐵2 = 𝐽11 3 𝐽22 3 + 𝐽11 3 𝐽33 3 + 𝐽22 3 𝐽33 3 − 𝐽12 3 𝐽21 3 𝐵3 = 𝐽12 3 𝐽21 3 𝐽33 3 − 𝐽11 3 𝐽22 3 𝐽33 3 . to ensure the stability of model system with equilibrium point 𝑇3, the point must qualify of the routh-hurtwiz criteria. the equation (9) have negative roots if 𝐽11 3 + 𝐽22 3 + 𝐽33 3 < 0, 𝐽11 3 𝐽22 3 + 𝐽11 3 𝐽33 3 + 𝐽22 3 𝐽33 3 > 𝐽12 3 𝐽21 3 , 𝐽12 3 𝐽21 3 𝐽33 3 > 𝐽11 3 𝐽22 3 𝐽33 3 , and 𝐵1𝐵2 > 𝐵3. theorem 4 equilibrium point 𝑇4 local stable if 𝐽11 4 + 𝐽22 4 + 𝐽33 4 < 0, 𝐽11 4 𝐽22 4 + 𝐽11 4 𝐽33 4 + 𝐽22 4 𝐽33 4 > 𝐽12 4 𝐽21 4 + 𝐽13 4 𝐽31 4 + 𝐽23 4 𝐽32 4 , 𝐽11 4 𝐽23 4 𝐽32 4 + 𝐽12 4 𝐽21 4 𝐽33 4 + 𝐽13 4 𝐽31 4 𝐽22 4 > 𝐽11 4 𝐽22 4 𝐽33 4 + 𝐽12 4 𝐽23 4 𝐽31 4 + 𝐽13 4 𝐽32 4 𝐽21 4 , and 𝐶1𝐶2 > 𝐶3. proof. the result of substitution equilibrium point t4 to jacobian matrix (3) 𝐽(𝑇4) = [ 𝐽11 4 𝐽12 4 𝐽13 4 𝐽21 4 𝐽22 4 𝐽23 4 𝐽31 4 𝐽32 4 𝐽33 4 ] (10) where m. ikbal, riskawati dynamics of predator-prey model interaction with harvesting effort 99 𝐽11 4 = 𝑟 − 2𝑟𝑃∗ 𝐾 − 𝛼1𝐻1 ∗ − 𝛼2𝐻2 ∗ − 𝑞1𝐸1 𝐽12 4 = −𝛼1𝑃 ∗ 𝐽13 4 = −𝛼2𝑃 ∗ 𝐽21 4 = 𝑒1𝛼1𝐻1 ∗ 𝐽22 4 = 𝑒1𝛼1𝑃 ∗ − 2𝑔1𝐻1 ∗ − 𝛽1𝐻2 ∗ − 𝑑1 − 𝑞2𝐸2 𝐽23 4 = −𝛽1𝐻1 ∗ 𝐽31 4 = 𝑒2𝛼2𝐻2 ∗ 𝐽32 4 = −𝛽2𝐻2 ∗ 𝐽33 4 = 𝑒2𝛼2𝑃 ∗ − 2𝑔2𝐻2 − 𝛽2𝐻1 ∗ − 𝑑2 − 𝑞3𝐸3 characteristics equation matrix j(t4) is 𝜆3 + 𝐶1𝜆 2 + 𝐶2𝜆 + 𝐶3 = 0 (11) where, 𝐶1 = −(𝐽11 4 + 𝐽22 4 + 𝐽33 4 ) 𝐶2 = 𝐽11 4 𝐽22 4 + 𝐽11 4 𝐽33 4 + 𝐽22 4 𝐽33 4 − 𝐽12 4 𝐽21 4 − 𝐽13 4 𝐽31 4 − 𝐽23 4 𝐽32 4 𝐶3 = 𝐽11 4 𝐽23 4 𝐽32 4 + 𝐽12 4 𝐽21 4 𝐽33 4 + 𝐽13 4 𝐽31 4 𝐽22 4 − 𝐽11 4 𝐽22 4 𝐽33 4 − 𝐽12 4 𝐽23 4 𝐽31 4 − 𝐽13 4 𝐽32 4 𝐽21 4 . to ensure the stability of model system with equilibrium point e4, the point must qualify of the routh-hurtwiz criteria. the equation (11) have negative roots if 𝐽11 4 + 𝐽22 4 + 𝐽33 4 < 0, 𝐽11 4 𝐽22 4 + 𝐽11 4 𝐽33 4 + 𝐽22 4 𝐽33 4 > 𝐽12 4 𝐽21 4 + 𝐽13 4 𝐽31 4 + 𝐽23 4 𝐽32 4 , 𝐽11 4 𝐽23 4 𝐽32 4 + 𝐽12 4 𝐽21 4 𝐽33 4 + 𝐽13 4 𝐽31 4 𝐽22 4 > 𝐽11 4 𝐽22 4 𝐽33 4 + 𝐽12 4 𝐽23 4 𝐽31 4 + 𝐽13 4 𝐽32 4 𝐽21 4 , and 𝐶1𝐶2 > 𝐶3. 4. numerical simulation in this section we simulated the model with some parameter values. the parameter values was adopted from literature [3], [6], [8], [9], [11], [15]. we try simulated the model with some condition. the first condition with parameter with e1 = 0.28773096, e2 = 0.24093140 and e3 = 0.13141604. the second condition with parameter e1 = 0.2, e2 = 0.4 and e3 = 0.3. the third condition with e1 = 0.2, e2 = 0.3 and e3 = 0.4. to see the system is in a stable state, a numerical simulation is performed with parameter estimates according to the following table 1. table 1. parameter values parameter values r 1 k 100 α1 0.21 α2 0.274 e1 1 e2 1 g 1 0.1 g 2 0.2 β 1 0.05 β 2 0.06 d1 0.05 d2 0.06 𝑞1 1 𝑞2 1 𝑞3 1 𝐸1 0.5 jurnal matematika mantik volume 6, no. 2, october 2020, pp. 93-103 100 (a) (c) (b) (b) parameter values 𝐸2 0.5 𝐸3 0.5 with the parameter values in table 1, the simulation results are given nonnegative equilibrium points: t0 = (0, 0, 0) t1 = (50, 0, 0), t2 = (3.288183092, 0, 1.704810836), t3 = (3.669623060, 2.206208426 , 0), t4 = (3.149229952, 0.4190122146, 1.388741370). figure 1. numerical simulation model. (a) prey population density; (b) first predator population density; (c) second predator population density m. ikbal, riskawati dynamics of predator-prey model interaction with harvesting effort 101 figure 2. time series of the model with e1 = 0.5, e2 = 0.5 and e3 = 0.5 figure 3. time series of the model with e1 = 0.2, e2 = 0.4 and e3 = 0.3 figure 4. time series of the model with e1 = 0.2, e2 = 0.3 and e3 = 0.4 jurnal matematika mantik volume 6, no. 2, october 2020, pp. 93-103 102 figure 1 shows the existence of each population. with a combination of the parameters presented, the food supply of predators is more than the predators themselves. the system is expected to last a long time with a combination of these parameters. figure 2 shows the simulation results with given parameters with the same parameter (e1 = 0.5, e2 = 0.5 and e3 = 0.5) of harvest effort for all populations. in this simulation, the system is stable and lasts a long time with the same harvesting effort conditions. to demonstrate system dynamics, we simulate models with varying combinations of harvest effort parameters. by changing the number of harvesting business parameters to e1 = 0.2, e2 = 0.4 and e3 = 0.3, the simulation plot will look different. figure 3 shows that the number of the first predator population is reduced compared to other populations because the number of harvesting efforts is also the highest. with different parameter numbers (e1 = 0.2, e2 = 0.3 and e3 = 0.4), figure 4 shows the same symptoms, the second predator population is the smallest because the number of harvesting efforts is the largest. apart from the stability of the system, the policy in harvesting in this study is a matter of focus. in this study, harvesting also determines the stability of a system, if the harvesting of a population changes and does not match the harvest rates in other populations, it will cause system stability disturbances. it can be seen clearly in the simulation that produces dynamic graphs in figures 2, 3, and 4, the population density of one predator is sometimes more than other predators, and vice versa. the results of this study are intended to show in general that harvesting efforts have a large enough impact on population sustainability and also have an impact on the system. harvesting effort can interfere with the growth and activity of predators. we look visually in the image with the selection of different harvesting efforts. population that is harvested in large numbers to other populations will decrease in population. this research can be developed by considering other factors that make a system dynamic. for example, further researchers can optimize harvesting efforts so that economic benefits can be clearly measured. 5. conclusions in this section we will make conclusions of this research. this study focuses on the dynamics of prey-predator populations with harvesting effort for all of population. there are 5 non-negative equilibrium point of the system. the interior point is 𝑻𝟒. the equilibrium point 𝑻𝟒 stabel if 𝑱𝟏𝟏 𝟒 + 𝑱𝟐𝟐 𝟒 + 𝑱𝟑𝟑 𝟒 < 𝟎, 𝑱𝟏𝟏 𝟒 𝑱𝟐𝟐 𝟒 + 𝑱𝟏𝟏 𝟒 𝑱𝟑𝟑 𝟒 + 𝑱𝟐𝟐 𝟒 𝑱𝟑𝟑 𝟒 > 𝑱𝟏𝟐 𝟒 𝑱𝟐𝟏 𝟒 + 𝑱𝟏𝟑 𝟒 𝑱𝟑𝟏 𝟒 + 𝑱𝟐𝟑 𝟒 𝑱𝟑𝟐 𝟒 , 𝑱𝟏𝟏 𝟒 𝑱𝟐𝟑 𝟒 𝑱𝟑𝟐 𝟒 + 𝑱𝟏𝟐 𝟒 𝑱𝟐𝟏 𝟒 𝑱𝟑𝟑 𝟒 + 𝑱𝟏𝟑 𝟒 𝑱𝟑𝟏 𝟒 𝑱𝟐𝟐 𝟒 > 𝑱𝟏𝟏 𝟒 𝑱𝟐𝟐 𝟒 𝑱𝟑𝟑 𝟒 + 𝑱𝟏𝟐 𝟒 𝑱𝟐𝟑 𝟒 𝑱𝟑𝟏 𝟒 + 𝑱𝟏𝟑 𝟒 𝑱𝟑𝟐 𝟒 𝑱𝟐𝟏 𝟒 , and 𝑪𝟏𝑪𝟐 > 𝑪𝟑. harvesting effort have impact to system. harvesting efforts will reduce the population size so that it can affect the stability of the dynamics of the prey-predator population system. the system will be stable and exist if we have control quantity of the harvesting effort. the system will be stable and exist if we have control’s the harvesting effort. 6. acknowledgment thanks to friends who helped this research and the directorate of research and community service deputy for strengthening research and development, the ministry of research and technology/ national research and innovation agency in indonesia which has funded this research. references [1] j. alebraheem and y. abu-hasan, “persistence of predators in a two predators-one m. ikbal, riskawati dynamics of predator-prey model interaction with harvesting effort 103 prey model with non-periodic solution,” appl. math. sci., vol. 6, no. 17–20, 2012. [2] k. pada das, “a mathematical study of a predator-prey dynamics with disease in predator,” isrn appl. math., vol. 2011, 2011, doi: 10.5402/2011/807486. [3] c. li, y. zhang, j. xu, and y. zhou, “global dynamics of a prey-predator model with antipredator behavior and two predators,” discret. dyn. nat. soc., vol. 2019, 2019, doi: 10.1155/2019/3586508. [4] r. p. gupta and p. chandra, “dynamical properties of a prey-predator-scavenger model with quadratic harvesting,” commun. nonlinear sci. numer. simul., vol. 49, 2017, doi: 10.1016/j.cnsns.2017.01.026. [5] t. k. kar and h. matsuda, “global dynamics and controllability of a harvested preypredator system with holling type iii functional response,” nonlinear anal. hybrid syst., vol. 1, no. 1, 2007, doi: 10.1016/j.nahs.2006.03.002. 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[10] a. rojas-palma and e. gonzález-olivares, “optimal harvesting in a predator-prey model with allee effect and sigmoid functional response,” appl. math. model., vol. 36, no. 5, 2012, doi: 10.1016/j.apm.2011.07.081. [11] a. chatterjee and s. pal, “interspecies competition between prey and two different predators with holling iv functional response in diffusive system,” comput. math. with appl., vol. 71, no. 2, 2016, doi: 10.1016/j.camwa.2015.12.022. [12] r. d. parshad, e. quansah, k. black, r. k. upadhyay, s. k. tiwari, and n. kumari, “long time dynamics of a three-species food chain model with allee effect in the top predator,” comput. math. with appl., vol. 71, no. 2, 2016, doi: 10.1016/j.camwa.2015.12.015. [13] c. ji, d. jiang, and n. shi, “a note on a predator-prey model with modified lesliegower and holling-type ii schemes with stochastic perturbation,” j. math. anal. appl., vol. 377, no. 1, 2011, doi: 10.1016/j.jmaa.2010.11.008. [14] x. y. meng, n. n. qin, and h. f. huo, “dynamics analysis of a predator–prey system with harvesting prey and disease in prey species,” j. biol. dyn., vol. 12, no. 1, 2018, doi: 10.1080/17513758.2018.1454515. [15] t. k. kar and s. k. chattopadhyay, “a dynamic reaction model of a prey-predator system with stage-structure for predator,” mod. appl. sci., vol. 4, no. 5, 2010, doi: 10.5539/mas.v4n5p183. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: fadilah akbar, fadilakbar783@gmail.com department of mathematics, uin sunan ampel surabaya, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.53-62 grouping of the second wave of covid-19 infection areas in east java province using k-means clustering fadilah akbar1, edo leonardo dekapriyo1, agoes moh. moefad2, achmad murtafi haris2 1department of mathematics, uin sunan ampel surabaya, indonesia 2department of communication studies, uin sunan ampel surabaya, indonesia article history: received aug 4, 2021 revised, jun 9, 2022 accepted, jun 30, 2022 kata kunci: covid-19, k-means clustering, pengelompokan, jawa timur, wilayah infeksi abstrak. serangan pandemi covid-19 gelombang kedua yang terjadi pada bulan juni 2021 mengakibatkan meningkatnya jumlah korban jiwa secara drastis. hal ini diakibatkan karena berkurangnya tanggung jawab masyarakat, rasa untuk saling menjaga, dan rasa untuk saling melindungi sehingga menyebabkan melonggarnya protokol kesehatan yang diterapkan. perlu adanya mitigasi yang tepat untuk menangani pandemi wabah penyakit covid-19 ini secara tepat, salah satunya adalah dengan menangani kasus pada setiap wilayah dengan tingkat keparahan dari tinggi ke rendah. dengan pengelompokkan tersebut, dapat memberikan keputusan wilayah mana saja yang harus diutamakan dalam mitigasi pandemi covid-19 ini. untuk pengelompokkan setiap wilayah dapat digunakan metode k-means clustering. dari cluster tersebut terdapat satu wilayah dengan tingkat keparahan tinggi, delapan wilayah dengan tingkat keparahan sedang, dan 29 wilayah dengan tingkat keparahan rendah. keywords: covid-19, k-means clustering, grouping, east java, infection areas abstract. the second wave of covid-19 pandemic attacks that occurred in june 2021 resulted in a drastic number of fatalities. this is due to reduced community responsibility, a sense of caring for each other, and a sense of mutual protection, resulting in loosening of the health protocols implemented. there needs to be proper mitigation to handle the covid-19 pandemic properly, one of which is by handling cases in each region with a low level of severity. with this grouping, it can provide a decision on which areas must be available in mitigating the covid-19 pandemic. for grouping each region, the kmeans clustering method can be used. from these clusters, there is one area with a severity level, eight areas with moderate severity, and 29 areas with low severity. how to cite: f. akbar, e. l. dekapriyo, a. m. moefad, and a. m. harris, “grouping of the second wave of covid-19 infection area in east java province using k-means clustering”, j. mat. mantik, vol. 8, no. 1, pp. 53-62, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.53-62 issn: 2527-3159 (print) 2527-3167 (online) mailto:fadilakbar783@gmail.com http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 1, june 2022, pp.53-62 54 1. introduction the coronavirus disease 2019 (covid-19) pandemic has been around for more than two years and has become a threat to all human life, until now covid-19 still occupies dangerous diseases throughout the world, one of which is indonesia. as of july 2021, reported cases of people who were positively exposed and suffering from covid-19 reached 193 million cases where the death toll was 4.14 million worldwide [1]. meanwhile, in indonesia, reported cases of people who were positively exposed and suffered from covid19 reached 3.03 million, with the number of patients who had been declared cured of 2.39 million, and the number of victims who died as many as 79,032 million people [2]. last june 2021 became the beginning of the second wave of covid-19 attacks in indonesia [3]. figure 1. the rate of development of the number of covid-19 cases in indonesia [4] attention to the graph in figure 1, after the first peak of cases in february fell, it appears that in mid-may 2021 there was a spike in the number of cases, this is the sign of the start of the second wave of the covid-19 pandemic. this second attack can occur as a result of reduced community responsibility, mutual care, and a sense of mutual protection, causing a loosening of the health protocols applied [5]. in addition, there are other factors that can cause this second wave of attacks, namely mutating the covid-19 virus which causes the virus to become stronger, has a more severe impact on victims, and accelerates infection or transmission [6]. conditions like this if not observed and get the right treatment, will result in more and more cases which will cause a large number of victims, the need for targeted mitigation and appropriate methods to accelerate the abatement of the covid-19 pandemic. in early july, the indonesian government imposed an emergency restriction on community activities or better known as the ppkm. this ppkm is implemented simultaneously for all regions on the islands of java and bali, this is due to the high number of positive cases from the second wave of covid-19 attacks that occurred in java and bali [7]. the centre of the case of the second wave of covid-19 attacks is the province of dki. jakarta as the capital and economic centre in indonesia, and the province of east java as the second largest metropolis in indonesia. especially for the east java region which was once the province with the highest cases in indonesia, especially the city of surabaya which was once the black zone (worst) [8]. this statement is also reinforced by research conducted by solichin and khairunnisa, where from the research conducted, it is said that the province of east java is the province with the highest cases in indonesia, although for now starting on july 25, 2021 for the province of east java itself ranks fourth with cases most as shown in picture 2 [9]. from the problems f. akbar, e. l. dekapriyo, a. m. moefad, and a. m. harris grouping of the second wave of covid-19 infection area in east java province using k-means clustering 55 that have been presented, it is necessary to classify which areas are vulnerable areas affected by the second wave of covid-19 attacks in east java province. figure 2. covid-19 cases in indonesia [4] one method that can be used to classify which areas are affected in east java province is to use clustering, for this reason, the use of clustering itself has been widely used to classify covid-19 cases around the world [10]. therefore, a grouping is needed that focuses on areas with a high number of cases to suppress the spike in cases and can minimize the number of victims, one of which is the grouping of areas affected by the second wave of covid-19 in the east java province [11]. in determining the right mitigation strategy for handling covid-19, it is necessary to know the severity of infection in each region to determine the priority of handling, one of which is is by clustering [12]. there are various types of clustering methods that can be used, but in this study, k-means clustering will be used. based on several previous studies, the main variables that are often used to be processed with k-means clustering are three, these variables are used as a benchmark for the severity of a case, and the variables are the number of infected people, recovered patients, and patients who died from covid-19 [13], [14]. the severity of outbreak cases from an area can be determine based on the number of infected cases, from the number of infected cases and then compared with the number of instances recovered and died in a city [15]. the number of infected cases who recovered if the ratio were more significant than the number of infected cases who died, then grouping with k-means would produce a group with low severity. there are several clustering methods available, but when compared to other clustering methods, k-means clustering is a clustering method with a higher accuracy level with a standard deviation of 2.2219, when compared to the standard deviation obtained from clustering with fcm of 2.7592. , it can be said that k-means gives better results [16], [17]. k-means clustering is generally often used to group an area affected by a pandemic or group something with a disease topic [18]. this method is also quite effective and efficient in classifying covid-19 cases, in a study conducted by zubair et al, an analysis was carried out on how efficient this method was, and it was proven to provide good results and a high level of accuracy [19]. in addition, k-means clustering can also be used to determine the source of the spread of where this virus came from, so using this method will provide more accurate results and can also provide a more complete analysis. there have been several previous studies that have been conducted discussing the grouping of provinces affected by the covid-19 pandemic, one of which was research conducted by dahlan a., et al. in this study, provinces across indonesia jurnal matematika mantik vol. 8, no. 1, june 2022, pp.53-62 56 were divided into three categories of outbreak severity. the method used to classify each province in this study using k-means clustering, dki. jakarta as the nation's capital has exceptions which will automatically be included in the area with the highest severity level, in addition, it was found that there are five other provinces classified in category one with the highest level of severity [20]. with high severity, then the other 28 provinces were classified in category two with moderate severity, while category three only contained dki. jakarta province [21]. then there is a study conducted by diah ns and irma y. in this study discusses the division based on the severity of covid-19 in all provinces throughout indonesia, the method used to classify is k-means clustering with the division of three categories, namely provinces with high levels of low severity, moderate severity, and high severity. the results obtained are 27 provinces fall into the category of areas with low severity, for the category with areas with moderate severity there are five provinces, while for the category with areas with high severity there are two provinces, where the two provinces are dki jakarta and east java. next, the researcher re-categorizes dki. jakarta and east java provinces into two, dki. jakarta province is categorized as the area with the worst impact of covid-19, followed by east java province [22]. next is the research conducted by fitria v., and yasmin e. f., which in this study discusses the division based on the severity of covid-19 in each province throughout indonesia into seven categories. the category with the lowest severity starts from category seven and continues up to category one with the highest severity. the method used for grouping provinces in this study uses k-means clustering. from this study, it was found that 10 provinces were included in category seven, 11 provinces were included in category six, five provinces were included in category five, five provinces were included in category four, two provinces were included in category three, one province was included in category two, and one province fall into category one. the province for category one is dki. jakarta, the province for category two is east java, and the two provinces for category three are central java and west java, this makes the island of java the region with the severity of covid-19 infection in indonesia [23]. and for the last literature is research conducted by ali m., which in this study discusses the distribution based on the level of spread of covid-19 cases in all districts or cities in central java province. the grouping is divided into three categories, the method used for grouping in this study is k-means clustering. from this study, the results obtained are 33 districts or cities which are included in category one, one city is included in category two, and one district is included in category three. this study does not classify the severity of each region, but only analyzes how good the clustering results are based on the similarity of characteristics, the distance between clusters, and the centroid [24]. from the four previous studies that have been carried out, in this study the k-means clustering method will be used to group each district or city in the province of east java. the ease and accuracy of k-means clustering is very suitable for use in this study. the categories will be divided into three categories, the determination of the number of 3 clusters is based on the level of accuracy obtained from several previous studies that have been described previously, where better cluster results are obtained when grouped into 3 clusters. namely for category one to be an area with a high severity level, category two to be an area with a moderate severity level, and category three to be an area with a low severity level. it is hoped that this research will provide a map of the distribution of the spread of covid-19 so that it can provide the right mitigation to be able to suppress the high number of cases. f. akbar, e. l. dekapriyo, a. m. moefad, and a. m. harris grouping of the second wave of covid-19 infection area in east java province using k-means clustering 57 2. methods in this research, of course, requires a method to run it in a structured, sequential, and correct manner. the methods used, if briefly described, are presented in a flowchart in figure 3 below. the following is an overview of this research: figure 3. research flowchart 2.1 data source the data used in this study is secondary data. data usage in july 2021 which can be accessed via http://infocovid19.jatimprov.go.id/index.php/data. table 1. sample data on the distribution of covid-19 in east java province no district / city active patient recovery patient patient died 1 dist. nganjuk 1538 7242 512 2 dist. trenggalek 697 4587 633 3 batu city 390 1821 180 ⋮ ⋮ ⋮ ⋮ ⋮ 38 dist. banyuwangi 839 8406 1144 2.2 k-means clustering clustering is the process of grouping objects based on data according to certain desired characteristics or partitioning data sets into subsets [25]. the cluster method is divided into two, namely hierarchical and non-hierarchical methods. the hierarchical method is done by grouping data that have the closest similarity, while non-hierarchical is used for large data and must determine the number of clusters to be formed [14]. many stages are used for cluster analysis, including the k-means method. k-means aims to partition data into two or more groups. the following are the steps in classifying data using the k-means method [26]: a) determine the number of 𝑘 as the number of groups formed. b) calculate the centre point using the formula : ∑ 𝑥𝑖 𝑛 𝑖=1 𝑛 (1) where : 𝑥𝑖 : data–𝑖 𝑛 : there are many observations in the group c) calculating the distance of each observation to each centroid of each group, namely the euclidian distance. 𝑑(𝑥, 𝑦) = ||𝑥 − 𝑦|| = √∑ (𝑥𝑖 − 𝑦𝑖 ) 2 𝑛 𝑖=0 (2) where : 𝑥𝑖 : data 𝑥 on 𝑖 observation 𝑛 : many observations 𝑦𝑖 : data 𝑦 on 𝑖 observation http://infocovid19.jatimprov.go.id/index.php/data jurnal matematika mantik vol. 8, no. 1, june 2022, pp.53-62 58 d) group each data based on the closest distance to the centroid. e) determine the position of the new centre point by calculating the average of the data in the same centre. the application of the k-means clustering method, each object is calculated at close range based on the characteristics it has with a predetermined cluster centre. if the smallest distance between objects and each cluster is a member of the closest cluster. data processing is carried out using r software. the following research steps are carried out, among others: a) covid-19 data collection in east java province based on district/city. b) perform the calculation of the k-means method as previous explanation. c) grouping the area from the calculation of the distance between objects that have the same characteristics. d) coloring the area based on the highest to lowest clustering results. 3. results and discussion based on the covid-19 data that has been obtained, the next step is to group each region using k-means clustering, the following results are obtained as shown in figure 4 below: figure 4. plot of k-means clustering results based on the results of the k-means clustering before, we can conclude in the form of a table as below: f. akbar, e. l. dekapriyo, a. m. moefad, and a. m. harris grouping of the second wave of covid-19 infection area in east java province using k-means clustering 59 table 2. results of grouping with clustering no. district / city cluster status no. district / city cluster status 1 dist. nganjuk 2 medium 20 malang city 2 medium 2 dist. trenggalek 3 low 21 dist. sampang 3 low 3 batu city 3 low 22 surabaya city 1 high 4 dist. kediri 2 low 23 mojokerto city 3 low 5 dist. blitar 3 low 24 dist. madiun 3 low 6 blitar city 3 low 25 dist. magetan 3 low 7 madiun city 3 low 26 dist. gresik 2 medium 8 dist. jember 2 medium 27 dist. bangkalan 3 low 9 dist. lumajang 3 low 28 dist. probolinggo 3 low 10 dist. situbondo 3 low 29 dist. tuban 3 low 11 dist. pasuruan 3 low 30 dist. lamongan 3 low 12 dist. sumenep 3 low 31 probolinggo city 3 low 13 dist. mojokerto 3 low 32 dist. bojonegoro 3 low 14 dist. pacitan 3 low 33 dist. bondowoso 3 low 15 pasuruan city 3 low 34 dist. jombang 2 sedang 16 dist. ponorogo 3 low 35 kediri city 3 low 17 dist. sidoarjo 2 medium 36 dist. malang 3 low 18 dist. tulungagung 3 low 37 dist. ngawi 3 low 19 dist. pamekasan 3 low 38 dist. banyuwangi 2 medium based on the results of the clustering of regions with clustering above, it appears that the city of surabaya enters cluster 1, where the status of the severity of the surabaya is high and has resulted in the surabaya being the city with the highest cases in the province of east java. then there are eight areas that are included in cluster 2 where in this group are areas with moderate severity due to covid-19, these areas are: dist. nganjuk, dist. kediri, dist. jember, dist. sidoarjo, malang city, dist. gresik, dist. jombang, and dist. banyuwangi. as for the areas that are included in cluster 3 where in this group are areas with low severity due to covid-19 totalling 29, these areas are: dist. trenggalek, batu city, kabu. blitar, blitar city, madiun city, dist. lumajang, dist. situbondo, dist. pasuruan, dist. sumenep, dist. mojokerto, dist. pacitan, pasuruan city, dist. ponorogo, dist. tulungagung, dist. pamekasan, dist. sampang, mojokerto city, dist. madiun, dist. magetan, dist. bangkalan, dist. probolinggo, dist. tuban, dist. lamongan, probolinggo city, dist. bojonegoro, dist. bondowoso, kediri city, dist. malang, and dist. ngawi. in this case, surabaya should be prioritized in taking mitigation measures to prevent the spread of covid-19 infection in east java province. given the small size of the surabayara area with a high population density, it is appropriate that surabaya deserves to be included in cluster 1 with a high severity level, to be prioritized in carrying out appropriate treatment. 4. conclusion based on the results obtained from the grouping of districts/cities based on the severity of the impact of the spread and transmission of covid-19, there is one high-affected area, eight moderately-affected areas, and 29 low-affected areas. when depicted on a map, the map of districts/cities impacting the second wave of the covid-19 pandemic attack is as follows: jurnal matematika mantik vol. 8, no. 1, june 2022, pp.53-62 60 figure 5. map of the distribution of areas affected by covid-19 east java from the map shown above, the areas marked in red must be implemented to prevent transmission and control activities that are not strictly necessary and strictly enforced. because if it is not handled quickly, the area around the red zone will also increase and cause the yellow and green zones to increase as well. then for the area with yellow colour, it is an area with an alert condition, this condition is not too high in severity but still has to follow the health protocols that have been recommended by the government and medical personnel, if the health protocol is not implemented, it will result in this yellow zone becoming a red zone, very dangerous for the wider community. the region. meanwhile, the green zone does not need to be too strict in handling cases in that area, but it must be handled seriously and until there are no active cases in the area. suggestions for future research may be adding variables that influence other than active patients, recovered patients, and patients who died. then it is possible that the number of affected area groupings can be increased and the results can also be compared if using three groups with groups >3. in addition, it is also possible to use other cluster methods such as fuzzi c-means clustering (fcm), hierarchical clustering, or gath-geva clustering to obtain more diverse regional grouping results. in addition, it is also possible to use other cluster methods such as fuzzy c-means clustering (fcm), hierarchical clustering, or gath-geva clustering to obtain more diverse regional grouping results. this is because in research related to clustering areas with case studies of the severity of covid19 infection, it is still rare to compare the results of clustering with the methods described. comparing the results of clustering with several methods simultaneously, can provide more information on decisions taken in the future which clustering method to use, or it can also be developed by combining two methods at once to get more optimal results. acknowledgement the researcher realizes that in completing this research 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[26] m. s. yana, l. setiawan, e. m. ulfa, and a. rusyana, “penerapan metode k-means dalam pengelompokan wilayah menurut intensitas kejadian bencana alam di indonesia tahun 2013-2018,” j. data anal., vol. 1, no. 2, pp. 93–102, 2018, doi: 10.24815/jda.v1i2.12584. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: m. fariz fadillah mardianto, m.fariz.fadillah.m@fst.unair.ac.id department of mathematics, universitas airlangga, surabaya 60115, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.86-95 contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator m. fariz fadillah mardianto1*, reynaldy aries ariyanto2, raka andriawan3, devayanti anugerahing husada4 1,2,3,4department of mathematics, universitas airlangga, surabaya, indonesia article history: received nov 11, 2020 revised may 24, 2021 accepted jun 11, 2021 kata kunci: estimator deret fourier, regresi nonparametrik, manajemen limbah, suroboyo bus abstrak. sampah plastik merupakan permasalahan yang hampir ada di seluruh negara. permasalahan ini muncul karena kurangnya fasilitas yang dapat menangani sampah plastik. suroboyo bus merupakan inovasi baru untuk masalah tersebut karena suroboyo bus menggunakan plastik botol sebagai alat pembayaran. tujuan dari penelitian ini adalah untuk memprediksi persentase kontribusi suroboyo bus dalam penanganan sampah plastik. estimator deret fourier memiliki kinerja yang baik untuk pemodelan data dengan pola tren musiman. penelitian ini membahas dua pendekatan seri fourier. perbedaan antara pendekatan tersebut adalah dimasukkannya fungsi phi (π) ke dalam model. hasil penelitian menunjukkan goodness of fit dengan fungsi π sebesar 99,96% untuk r2 dan 0,08% untuk mape sedangkan goodness of fit tanpa fungsi π adalah 100% untuk r2 dan 0,07% untuk mape. kesimpulannya model deret fourier tanpa fungsi π lebih baik karena model deret fourier tanpa fungsi π lebih memenuhi kriteria goodness of fit dibandingkan model deret fourier dengan fungsi π. keywords: fourier series estimator, nonparametric regression, waste management, suroboyo bus abstract. plastic waste is a problem that almost exists in all countries. this problem arises because of the lack of facilities that can handle the plastic waste. suroboyo bus is an innovation for this problem because suroboyo bus uses plastic bottles as payment. the purpose of this research is to predict the percentage contribution of suroboyo bus in handling plastic waste. the fourier series estimator performs well for data modeling with seasonal trend patterns. this paper examines two approaches to the fourier series. the difference between the approaches is the inclusion of the phi (π) function in the model. the result shows the goodness of fit criterion model with π function are for and 0,08% for mape whereas the fit criterion model without π function is 100% for and 0,07% for mape. in conclusion, the fourier series model without the π function is better because the fourier series model without the π function is more satisfy the goodness of fit criteria than the fourier series model with the π. how to cite: m. f. f. mardianto, r. a. ariyanto, r. andriawan, and d. a. husada, “contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator”, j. mat. mantik, vol. 7, no. 1, pp. 86-95, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 86-95 issn: 2527-3159 (print) 2527-3167 (online) mailto:m.fariz.fadillah.m@fst.unair.ac.id https://doi.org/10.15642/mantik.2021.7.1.86-95 http://u.lipi.go.id/1458103791 m. fariz fadillah mardianto, r. aries ariyanto, raka andriawan, and devayanti a. husada, contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator 87 1. introduction the issue of waste treatment is a crucial problem. in 2016, total garbage in indonesia was 65.200.000 tons with population of 261.115.456 people [1]. population in indonesia is predicted to increase over time and predicted to increase the amount of waste. according to the central bureau of statistics indonesia, the production of waste happens massively in town. surabaya’s city produces 9,896.78 waste every day, and jakarta’s city produces 7.164,53 volume of waste every day. if the waste problem is not solved, it will negatively affect water pollution, land, and air. some of the effects of the waste problem in indonesia are the average quality pollution with heavy iron, emission of greenhouse effect from waste in 2014, and 30,26% to the emission of word greenhouse effect and it cause 1,805 floods in 2016-2017 [1]. because of that, it is needed to reduce waste effectively. according to presidential decree number 97, released in 2017, indonesia’s government has the target to reduce waste by 30% and handle waste 70% from indonesia’s waste. it is still difficult because the amount of waste will increase in a row with an increase in indonesia’s population. waste management is matter that mentioned in sdgs for responsible consumption and production category. in order to increase waste management performance, one of the innovations that are applied is suroboyo bus. suroboyo bus is public transportation with a payment system using garbage. people can use the suroboyo bus for 2 hours for every ride by exchange three plastic bottles of 1,5 l or five plastic bottles of 500 ml or ten plastic glass of 240 ml. department of transportation says people who like the suroboyo bus increase 15% every day and maximumly increase in the weekend. surabaya’s government says that suroboyo bus can reduce plastic waste in surabaya and make surabaya’s people not littering. it is necessary to evaluate its function to solve the waste problem. prior research has already reviewed the mechanism of suroboyo bus both as a means of public transportation and waste management effort [2]. other previous research discusses the innovativeness and the implementation of the suroboyo bus, which concludes that the suroboyo bus is a good innovation as public transportation and its implementation is quite reasonable that positively accepted by the civilians [3]. however, until now, there is no further research of plastic waste management in suroboyo bus. as already mentioned, plastic waste is a means of payment used to enjoy this facility. in order to support the study of the suroboyo bus, it should be research-related contributions to the suroboyo bus in plastic waste management in surabaya. the method to evaluate suroboyo bus is predicting the contribution of suroboyo bus. the contribution of suroboyo bus is measured as the total of waste loaded by suroboyo bus divided by estimated total waste production in surabaya. nonparametric regression with fourier series estimation can be used to predict the contribution of the suroboyo bus. the regression model can be divided into three types: parametric, semiparametric, and nonparametric. the error term is assumed to follow specific distribution probabilities in parametric regression. this assumption is not required in nonparametric regression. nonparametric regression uses a smoothing technique to obtain an estimate of the observation value. because of that, nonparametric regression has high flexibility in approaching the pattern in data observation [4]. one of the methods in nonparametric regression is the fourier series that uses the trigonometric function. a parameter, namely λ, determines the smoothing of the regression curve in the fourier series. in general, nonparametric regression with fourier series estimation is used in data with unknown patterns and has a seasonal trend. this paper discusses two approaches to the fourier series. the difference between those approaches is the inclusion of the phi (π) function in the model. the performance of the models is tested and compared on a case study. compare both of model fourier jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 86-95 88 series; it is used data contribution of suroboyo bus. the case study involves suroboyo bus, which accepts bottle plastic as payment. the goal of the study is to predict the percentage of suroboyo bus contribution to handling plastic waste. the best model of the fourier series is determined with smaller mape and larger determination coefficient. 2. nonparametric regression based on fourier series estimator 2.1 estimator with phi (π) function fourier series is a trigonometric function which has show sine and cosine curve. fourier series is best used on-trend seasonal data [5-8]. based on takezawa [5], a regression model for observation data (𝑡𝑟 , 𝑦𝑟 ) can be written 𝑦𝑟 = 𝑚(𝑡𝑟) + 𝜀𝑟, 𝑟 = 1,2, … , 𝑛 (1) distribution of is normal with mean 0 and variant 1. assumed that 𝑚(𝑡𝑟) ∈ 𝐿2 [a,b] which is hilbert's room, so that 𝑚(𝑡𝑟) can be stated as linear combination with the element base from 𝐿2 [𝑎, 𝑏]. if {𝑥𝑗 }𝑗=1 ∞ is complete orthonomic system from 𝐿2 [𝑎, 𝑏] so 𝑚(𝑡𝑟) can be stated 𝑚(𝑡𝑟) = ∑ 𝛽𝑗 𝑥𝑗 (𝑡𝑟) ∞ 𝑗=1 (2) according to eubank [4], if 𝛽𝑗 is a scalar, then equation (1) became. 𝑦𝑟 = ∑ 𝛽𝑗 𝑥𝑗 (𝑡𝑟) ∞ 𝑗=1 (3) observation data is time function and periodic, then to estimate function equation (2) is used linear model which has sine function and cosine function [4]. complex exponential function can be stated in sine function and cosine function. complex exponential function can be written: 𝑥𝑗 (𝑡𝑟) = 𝑒 2𝜋𝑖𝑗𝑡𝑟 (4) substitution equation (4) to equation (2) and become as follows, �̂�(𝑡) = ∑ �̂�𝑗 𝑒 2𝜋𝑖𝑗𝑡𝑟𝜆 𝑗=−𝜆 (5) with 𝑒𝑖𝑥 = cos(𝑥) + 𝑖 sin (𝑥) and 𝑒−𝑖𝑥 = cos(𝑥) − sin (𝑥) fourier series estimator for �̂�(𝑡𝑟) is as follows �̂�(𝑡𝑟) = ∑ �̂�𝑗 𝑒 2𝜋𝑖𝑗𝑡𝑟𝜆 𝑗=−𝜆 �̂�(𝑡𝑟) = ∑ �̂�𝑗 𝑒 2𝜋𝑖𝑗𝑡𝑟 + �̂�(−𝑗)𝑒 2𝜋𝑖 (−𝑗)𝑡𝑟𝜆 𝑗=−𝜆 �̂�(𝑡𝑟) = �̂�0 + ∑ (�̂�𝑗 𝑒 2𝜋𝑖𝑗𝑡𝑟 + �̂�(−𝑗)𝑒 2𝜋𝑖(−𝑗)𝑡𝑟 )𝜆𝑗=1 �̂�(𝑡𝑟) = �̂�0 ∑ [�̂�𝑗 (cos(2𝜋𝑗𝑡𝑟 ) + 𝑖 sin(2𝜋𝑗𝑡𝑟 )) + �̂�(−𝑗)(cos(2𝜋𝑗𝑡𝑟) − 𝜆 𝑗=1 𝑖 sin (2𝜋𝑗𝑡𝑟 ))] �̂�(𝑡𝑟) = �̂�0 + ∑ [(�̂�𝑗 + �̂�(−𝑗)) cos(2𝜋𝑗𝑡𝑟) − 𝑖(�̂�𝑗 + �̂�(−𝑗)) sin(2𝜋𝑗𝑡𝑟 )] 𝜆 𝑗=1 �̂�(𝑡𝑟) = �̂�0 + ∑ [𝑎𝑗 cos(2𝜋𝑗𝑡𝑟) + 𝑏𝑗 sin(2𝜋𝑗𝑡𝑟)] 𝜆 𝑗=1 (6) where, 𝑎𝑗 = 2 𝑛 ∑ 𝑦𝑟 cos (2𝜋𝑡𝑟 ) 𝑛 𝑟=1 𝑏𝑗 = 2 𝑛 ∑ 𝑦𝑟 sin (2𝜋𝑡𝑟) 𝑛 𝑟=1 𝑡𝑟 = 𝑡𝑖−1 𝑛 m. fariz fadillah mardianto, r. aries ariyanto, raka andriawan, and devayanti a. husada, contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator 89 subtitute equation (5) into equation (1) and become. 𝑦𝑟 = �̂�0 + ∑ [𝑎𝑗 cos(2𝜋𝑡𝑟 ) + 𝑏𝑗 sin(2𝜋𝑡𝑟 )] + 𝜀𝑟 𝜆 𝑗=1 (7) with estimator for regression curve is given as follows �̂�𝑟 = �̂�0 + ∑ [�̂�𝑗 cos(2𝜋𝑗𝑡𝑟 ) + �̂�𝑗 sin(2𝜋𝑗𝑡𝑟 )] 𝜆 𝑗=1 (8) 2.2. estimator without phi (π) function in some conditions, the form of regression’s curve can not be determined. it means some pattern can not be determined with some model of the parametric curve because it will produce high error and variance. if data assumed that the form of regression curve is unknown, it is suggested to use a nonparametric regression approach. consider paired data, equation of nonparametric regression based on mardianto et al., [7] given as follows: 𝑦𝑖 = 𝑔(𝑡𝑖) + 𝜀𝑖 (9) with 𝑦𝑖 is respon variable, 𝑡𝑖 is predictor variable for nonparametric regression and 𝜀𝑖 is error of model with normal distribution. form of 𝑔(𝑡𝑖) function can not be determined and assumed with nonparametric regression function. error of nonparametric regression assumed that identical, independent, and have normal distribution with 0 mean, and σ2 varians [8]. if 𝑔(𝑡) is function which integrable and differentiable on interval [α, α +2l], so representation of fourier series on that interval related with 𝑔(𝑡) that have trigonometric component sine and cosine given as follows : 𝑔(𝑡) = 𝑎0 2 + ∑ (𝑎𝑛 cos(𝜆 ∗𝑡) + 𝑏𝑛 sin(𝜆 ∗𝑡))∞𝑛=1 (10) where 𝜆∗ ≈ 𝑛𝜋 𝐿 ; 𝑛 = 1, 2, 3, … as for fourier coefficient determine with formula : 𝑎𝑛 = 1 𝐿 ∫ 𝑔(𝑡) cos(𝜆∗𝑡) 𝑑𝑡 𝑎+2𝐿 𝑎 (11) 𝑏𝑛 = 1 𝐿 ∫ 𝑔(𝑡) sin(𝜆∗𝑡) 𝑑𝑡 𝑎+2𝐿 𝑎 (12) if 𝑔(𝑡) is even function, so fourier coefficient 𝑏𝑛=0. therefore, the fourier series is called cosine fourier series. if 𝑔(𝑡) can be integrable and differentiable on interval [0, l], so that cosine fourier series can be written 𝑔(𝑡) = 𝑎0 2 + ∑ 𝑎𝑛 cos (𝜆 ∗𝑡) ∞𝑛=1 (13) where 𝜆∗ ≈ 𝑛𝜋 𝐿 ; 𝑛 = 1, 2, 3, … according to biederman et al., [9] for fourier coefficient can be determined with formula: 𝑎0 = 2 𝐿 ∫ 𝑔(𝑡) 𝐿 0 𝑑𝑡 ∶ 𝑎𝑛 = 2 𝐿 ∫ 𝑔(𝑡) cos(𝜆∗𝑡) 𝑑𝑡 𝐿 0 (14) jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 86-95 90 if 𝑔(𝑡) is odd function, so fourier coefficient 𝑎𝑛= 0. therefore the fourier series called sine fourier. if 𝑔(𝑡) integrable and differentiable on interval [0, l], so sine fourier series given as follows : 𝑔(𝑡) = 𝑎0 2 + ∑ 𝑏𝑛 sin (𝜆 ∗𝑡)∞𝑛=1 (15) with 𝜆∗ ≈ 𝑛𝜋 𝐿 ; 𝑛 = 1, 2, 3, … as for fourier coefficient determine with formula : 𝑎0 = 2 𝐿 ∫ 𝑔(𝑡) 𝐿 0 𝑑𝑡 (16) 𝑏𝑛 = 2 𝐿 ∫ 𝑔(𝑡) sin (𝜆∗𝑡) 𝐿 0 𝑑𝑡 (17) because it has periodic formula and can approach trend data, fourier series can be used in nonparametric regression curve. based on bilodeau [10], by adjustment fourier series formula constructed fourier series in equation (10) with add trend function 𝑔(𝑡𝑖) = 𝑎0 2 + 𝜔𝑡𝑖 + ∑ 𝑎𝑘 cos(𝑘𝑡𝑖) + 𝑏𝑘 sin(𝑘𝑡𝑖) 𝜆 𝑘=1 (18) therefore, equation of nonparametric regression approach with fourier series estimator for paired data (𝑡𝑖, 𝑦𝑖 ) on equation (9) can be written as 𝑦𝑖 = 𝑎0 2 + 𝜔𝑡𝑖 + ∑ (𝑎𝑘 cos(𝑘𝑡𝑖 ) + 𝑏𝑘 sin (𝑘𝑡𝑖 )) 𝜆 𝑘=1 + 𝜀𝑖 (19) with 𝑎0, 𝜔, 𝑎𝑘 are regression parameters that, 𝑘 = 1,2, … , λ is oscillation parameter. value of regression parameter estimator in vector form can be determined according to optimation method with least square (ls) approach.the estimation form for nonparametric regression can be obtained as follows: �̂�𝑖 = �̂�0 2 + �̂�𝑡𝑖 + ∑ (�̂�𝑘 cos(𝑘𝑡𝑖 ) + �̂�𝑘 sin (𝑘𝑡𝑖 )) 𝜆 𝑘=1 (20) 2.3. selection of oscillation parameters the selection of λ values must be carried out optimally. determination of optimal λ can use generalized cross validation (gcv) method. based on [15, 16], formula of gcv can be written as follows: 𝐺𝐶𝑉(λ) = 𝑀𝑆𝐸(λ) (𝑁−1𝑡𝑟𝑎𝑐𝑒(𝑰−𝑨(λ)))2 (21) where 𝑀𝑆𝐸(λ) = 𝑁−1𝒚𝑻(𝑰 − 𝑨(λ)) 𝑇 𝑾(𝑰 − 𝑨(λ))𝒚 (22) the gcv value depends on mean square error (mse) value because the numerator of gcv formula is the mse formula. measurement of goodness model is determined by the value of the determination coefficient (r2) which shows the percentage contribution of the predictor variable to the response variable. the best model that can be used for prediction needs to pass the goodness of criteria. the goodness of criteria is the smallest gcv value for an optimal oscillation m. fariz fadillah mardianto, r. aries ariyanto, raka andriawan, and devayanti a. husada, contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator 91 parameter, the smallest mse value, and the enormous determination coefficient value [15, 16]. 2.4. measure of goodness of model the goodness of model can be measured by mean absolute percentage error (mape). mape is calculated using the absolute error in each period divided by the observed value that is evident for that period [13]. the formula of mape can be written: 𝑀𝐴𝑃𝐸 = ∑ |𝑌𝑖−�̅�𝑖| 𝑌𝑖 𝑛 𝑖=1 𝑛 𝑥100% (23) the goodness of model also can be measured by the determination coefficient (r2). the determination coefficients determine how much influence between independent variable and dependent variable [19]. formula of r2 can be written 𝑅2 = 1 − ∑ (𝑌𝑖−�̂�𝑖) 2𝑛 𝑖=1 ∑ (𝑌𝑖−�̅�) 2𝑛 𝑖=1 (24) 3. research method 3.1. data source data collected from surabaya government office that handling sanitary and green open space in surabaya city. the obtained data is a time-series data of contribution of suroboyo bus in waste management measured as the total of waste loaded by suroboyo bus divided by estimated total waste production in surabaya. the data start from may 1, 2018 to january 31, 2019. it uses data from 2018 as in sample and data from 2019 as out sample. 3.2. data analysis procedure the procedure in data analysis that is related to estimation contribution of suroboyo bus with nonparametric regression used complete fourier series estimator for both models from equation (7), and equation (19) was given as follows: a. study literature related to the contribution of suroboyo bus and its relationship to the related predictor variables. b. perform descriptive statistics for each variable based on minimum, maximum, and average values. c. determine the gcv value for each model of nonparametric regression with fourier series estimation that is used based on data. d. choose the smallest gcv value for every fourier series used to determine oscillation parameter optimally, and calculate mape using equation (23) and calculate using equation (24). e. comparing two fourier series estimators to determine the best model based on the smallest mape value and the largest r2. 4. result and discussion suroboyo bus was public transportation with a payment system using garbage. every day, from suroboyo bus payment system, waste is collected and distributed in a garbage bank. the contribution of suroboyo bus from may 1, 2018, until january 31, 2019, can be explained in figure 1. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 86-95 92 figure 1. graph of contribution of suroboyo bus from figure 1, in may 2018 until december 2018, it increased slowly. the highest contribution of suroboyo bus in october 2018 with 1,1%, and the lowest contribution in june 2018 with 0% (no operation). the data had an increased trend pattern and had a repetitive pattern. so, the data had a seasonal pattern, and the fourier series estimator was suitable to estimate the contribution percentage of suroboyo bus with nonparametric regression using the fourier series estimator. the results of the optimal gcv value that be calculated from r software used training data are presented in table 1. table 1. gcv value model λ gcv with π function 38 0,01836 without π function 122 5,1994x10-13 based on the result in table 1, the minimum gcv value for nonparametric regression using fourier series with π function was 0,01836 with λ equal to 38 was chosen. for nonparametric regression using the fourier series without π function, the minimum gcv value was 5,1994 x 10-13 with equal to 122 was chosen. for nonparametric regression using fourier estimation with π model’s, obtained model fourier series in nonparametric regression as follows �̂�𝑖 = �̂�0 + �̂�1 cos(2𝜋𝑡𝑟 ) + �̂�1 sin(2𝜋𝑡𝑟 ) + ⋯ + �̂�38 cos(76𝜋𝑡𝑟 ) + �̂�38 sin(76𝜋𝑡𝑟) (25) where 𝑡𝑟 = 𝑡𝑖−1 𝑛 based on the results of, the parameter values in the nonparametric regression using fourier series estimator with π model’s can be written as follows �̂�𝑖 = 0,33079 − 0,05716 cos(2𝜋𝑡𝑟) − 0,076012 sin(2𝜋𝑡𝑟 ) − ⋯ − 0,02365 sin(76𝜋𝑡𝑟 ) (26) for nonparametric regression using fourier series without π, obtained model as follows: 0 0.2 0.4 0.6 0.8 1 1.2 t h e c o n tr ib u ti o n o f s u ro b o y o b u s (% ) date file:///c:/users/asus/appdata/roaming/microsoft/word/figure.docx m. fariz fadillah mardianto, r. aries ariyanto, raka andriawan, and devayanti a. husada, contribution analysis of “suroboyo bus” in waste management based on two form of complete fourier series estimator 93 �̂�𝑖 = �̂�0 2 + �̂�𝑡𝑖 + �̂�1 cos(𝑡𝑖) + ⋯ + �̂�122 cos(122𝑡𝑖) + �̂�1 sin(𝑡𝑖 ) + ⋯ + �̂�122 sin(𝑡𝑖 ) (27) based on the results, the parameter values in the nonparametric regression using fourier series estimator without π function model’s can be written as follows �̂�𝑖 = 0,40293 − 0,00115𝑡𝑖 + 0,001581 cos(𝑡𝑖 ) + ⋯ + 0,02863 sin(𝑡𝑖 ) + ⋯ + 0,0033 sin(122𝑡𝑖 ) (28) figure 3 showed comparison between data estimation using fourier series estimation without π function, and data estimation using fourier series estimation with π function. figure 3. graph of comparison between actual data and result based on figure 3, estimation used fourier series estimator without π function closed to actual data than used fourier series estimator with π function. it means fourier series estimator without π function was better than fourier series estimator with π function for data because fourier series estimator without π function was smoother than fourier series estimator with π function. it was seen from graph for a model that closed to actual data. r2, and mape from fourier series estimator with π function in a row was 99,96% for r2, 0,08% for mape. for fourier series estimator without π function, the result of r2, mape in a row was 100% for r2, and 0,07% for mape. based on the result, fourier series estimator without π function was better than fourier series estimator without π function to estimate contribution percentage of suroboyo bus because fourier series estimator without π functions was more flexible than fourier series estimator with π functions. 5. conclusions suroboyo bus had a chance to solve the plastic waste problem in a big city because the suroboyo bus can increase waste management percentage. from both models, fourier series estimator without π function was more suitable than fourier series estimator with π -0.5 0 0.5 1 1.5 2 2.5 3 3.5 1 9 1 7 2 5 3 3 4 1 4 9 5 7 6 5 7 3 8 1 8 9 9 7 1 0 5 1 1 3 1 2 1 1 2 9 1 3 7 1 4 5 1 5 3 1 6 1 1 6 9 1 7 7 1 8 5 1 9 3 2 0 1 2 0 9 2 1 7 2 2 5 2 3 3 2 4 1 t h e c o n t r ib u t io n o f s u r o b o y o b u s (% ) period actual data estimation without π function estimation with π function file:///c:/users/asus/appdata/roaming/microsoft/word/figure.docx jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 86-95 94 function because mape’s fourier series estimator without π function was smaller than fourier series estimator with π function. furthermore, determination coefficient’s fourier series estimator without π function was higher than the fourier series estimator with π function. however, if we used the fourier series estimator without π function, it had an extended model because optimum λ in the fourier series estimator without π function was equal to 122. this value was enormous, so it would difficult to interpret the model. fourier series estimator with π function was more parsimony than fourier series estimator without π function because optimum λ in fourier series estimator with π function was smaller than fourier series estimator without π function. there is still low research of suroboyo bus as a research object, especially about its ability to handle plastic bottle waste and waste management flow after the waste has been collected. therefore, it is necessary to have further research to become input and feedback for related services, particularly the department of transportation and also surabaya government office that handling sanitary and green open space in surabaya city. references [1] central bureau of statistic indonesia and ministry of environment indonesia, environment statistics of indonesia, jakarta: central bureau of statistic, 2018 [2] kurniawan, a.a., & prabawati, i, implementasi suroboyo bus di dinas 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matematika dan aplikasi online internet pembelajaran contact: widya eka pranata, widyapranata983@gmail.com department of mathematics, universitas negeri gorontalo, kota gorontalo, gorontalo 96128, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.31-40 implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation nurwan1, widya eka pranata2* , muhammad rezky friesta payu3, nisky imansyah yahya4 1,2,3,4department of mathematics, universitas negeri gorontalo, gorontalo, indonesia article history: received sep 28, 2020 revised feb 19, 2021 accepted may 30, 2021 kata kunci: rute terpendek, jadwal optimal, algoritma dijkstra, algoritma welch-powell abstrak. penelitian ini membahas tentang penerapan algoritma dijkstra dan algoritma welch-powell pada masalah transportasi bus kampus. tujuan peneitian ini adalah untuk menentukan rute terpendek dan jadwal optimal untuk jalur trasportasi bus kampus ung. dalam menentukan rute terpendek, setiap persimpangan direpresentasikan sebagai simpul dan jalur yang dilalui direpresentasikan sebagai sisi. lintasan terpendek diperoleh 𝑉1 − 𝑉2 − 𝑉5 − 𝑉8 − 𝑉9 − 𝑉10 − 𝑉13 − 𝑉16. dalam menetukan jadwal optimal, jumlah bus merepresentasikan simpul dan waktu merepresentasikan sisi yang menghubungkan setiap simpul. jadwal optimal bus dimulai pukul 06.30 pagi sampai pukul 17.00 sore. setiap bus mendapatkan 4 (empat) sesi keberangkatan dan 4 (empat) sesi kepulangan dengan waktu tempuh masing-masing sesi 60 menit. keywords: shortest route, optimal schedule, dijkstra algorithm, welch-powell algorithm abstract. this research deals with applying the dijkstra algorithm and welch-powell algorithm to on-campus bus transportation problems. this research aims to determine the optimal solution of campus bus transportation routes in the shortest routes and schedules. in determining the shortest route, each intersection represented as a node and the path described as the sides. the shortest path obtained 𝑉1 − 𝑉2 − 𝑉5 − 𝑉8 − 𝑉9 − 𝑉10 − 𝑉13 − 𝑉16. in determining the optimal schedule, the number of buses represents the vertices, and the time expresses the side that connects each node. the optimal program of the bus starts from 06.30 am to 5.00 pm. every bus gets four sessions of departure and four sessions return with travel time each session is 60 minutes. how to cite: nurwan, w. e. pranata, m. r. f. payu, and n. i. yahya, “implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation”, j. mat. mantik, vol. 7, no. 1, pp. 31-40, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 31-40 issn: 2527-3159 (print) 2527-3167 (online) mailto:widyapranata983@gmail.com https://doi.org/10.15642/mantik.2021.7.1.31-40 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 31-40 32 1. introduction the development of a new campus of gorontalo state university (ung) at bone bolango regency is approximately 15 km from the center of gorontalo city. campus transfer impacts academic activities or activities as many as 4 (four) faculties, with about 8643 students. the ung campus transfer from the center of gorontalo city to bone bolango regency impacts transportation availability. the availability of transport is crucial for students to support activities in the new ung campus. transportation equipment that serves the route of gorontalo city to the new campus bone bolango is a bus provided by the gorontalo porvinsi transportation agency and the bone bolango regency transportation agency. gorontalo provincial government prepares public transportation services in the form of bus rapid transit (brt). brt is a mass transportation facility for the people, including ung students, to the ung bone bolango campus, even with a limited number. brt does not prioritize the needs of ung students. therefore, it has an impact on the delay of students in eroding lectures. the issue of distance and the discovery of optimal routes is the most crucial thing in transportation problems when students go to bone bolango's new campus. this situation is to reduce student delays in participating in academic activities. the above conditions required a solution to get the optimal bus route that serves students from the center of gorontalo city to the new campus of ung bone bolango regency and the optimal schedule of companies or transportation service providers. transportation problems can be solved using graph theory to describe the pain to make it easier to solve. one of the ideas developed in graph theory is coloring. there are three kinds of color in graph theory: vertex coloring, face coloring, and region color [1]. dijkstra's algorithm can also be called a greedy algorithm. it is one of the algorithms used to solve the shortest path. it does not have a negative cost [2]. the optimal route is completed using the dijkstra algorithm to get the optimal bus schedule used welch-powell algorithm. the working principle of algorithm dijkstra searches for the two smallest trajectories, so this algorithm is advantageous in determining the shortest course from one point to another [3]. dijkstra algorithms often search for the shortest routes, using nodes on a simple road network [3]. the dijkstra algorithm's use to determine the shortest route of a graph will result in the best route, namely selecting and analyzing the unselected node's weight, then selecting the node with the most negligible weight [4]. the dijkstra algorithm's application in determining the shortest route, among others, finds an effective route to avoid traffic jams during rush hour [5]. dijkstra algorithm is used to calculate the closest distance from one point to the museum chosen to be the destination [6]. implementation of dijkstra algorithms on urban rail transit networks [7]. determination of the shortest route with using dijkstra's algorithm on the path school bus [8]. one of the concepts of graphs to solve transportation scheduling problems is the concept of graph coloring. graph coloring is the coloration represented by the sorted number [9] [10]. use by coloring vertices based on the highest degree of all vertices [11]. the welch-powell algorithm invented by welch and powell is very useful in scheduling. the application of graph coloring uses the welch-powell algorithm in determining student guidance schedules [12]. another study about applying the dijkstra algorithm was carried for selecting the route to reduce traffic congestion in purwokerto. this study aims to solve congestion by determining alternative routes that are more effective and efficient. application of dijkstra's algorithm utilizing determine the most negligible weight of each road segment. from this research, the rider can choose alternative routes to avoid congestion [13]. they are searching for the shortest route with dijkstra's algorithm. this research aims to simulate shortest path search using the dijkstra algorithm to help find the shortest route [14]. application of dijkstra's algorithm in the bus route search application trans nurwan, widya e. pranata, muhammad r. f. payu, nisky i. yahya implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation 33 semarang. this researcher proposes a digital application solution to search for trans semarang bus routes using the dijkstra algorithm [15]. several studies related to dijkstra's algorithm in transportation problems only focus on determining the shortest route without optimal scheduling. to optimize students' transportation routes to the ung bone bolango campus and the opposite, the researchers solved two problems transportation: shortest route and the optimal schedule. therefore, the researcher uses two different algorithms, namely the dijkstra algorithm and the welch powell algorithm. the background presented above is needed for optimal bus transportation that serves students from campus 1 in gorontalo city to bone bolango campus. researchers applied dijkstra algorithms to determine the shortest routes and welch-powell algorithms to design schedules. this study will find the shortest route and planned bus departure schedule from campus 1 to bone bolango campus and scheduled return from bone bolango campus to campus 1 gorontalo city. 2. methods this study aims to find the optimal route solution of the bus by using the dijkstra algorithm and set the bus optimal schedule solution by using the welch-powell algorithm with the following stages: a. take a screenshot on google maps in the form of an image. b. determines several routes from campus 1 to the bone bolango campus. c. specify a starting point. d. specify a destination point. e. specify multiple intersections as nodes in the graph. f. create a straight line from node to node as a side on a graph. g. we create a weighted graph by connecting the vertices using the contents and giving weight according to the distance. h. analysis of data using dijkstra and welch-powell algorithms. i. make conclusions. 3. results and discussion 3.1 bus shortest route to ung bone bolango campus researchers take screenshots on google maps based on previous research methods and represented them in the figure's form. the resulting image is then graphed, as shown in figure 1. figure 1. weighted graph jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 31-40 34 figure 1 is a route that a bus can take from the starting point or departure point located at campus 1 ung gorontalo city to the endpoint located at the campus ung bone bolango. the route in this study represents the side that connects each node. we can see the definition of nodes in figure 1 in table 1. table 1. image copyright getty images image caption the 1st graph no node name (intersection) 1 𝑉1 campus 1 ung 2 𝑉2 the intersection of three sentra media (jendral sudirman street–pangeran hidayat street) 3 𝑉3 the intersection of four junior high school 6 gorontalo (jendral sudirman street– jaksa agung suprapto street– arif rahman hakim street) 4 𝑉4 the intersection of four darul muhtadin mosque (arif rahman hakim street –prof jhon aryo katili street – bj. pola isa street) 5 𝑉5 the intersection of four public health center (pangeran hidayat street – rusli datau street – prof. aryo katili street – bj. pola isa street) 6 𝑉6 the intersection of four baiturahim mosques (nani wartabone street– raja eyato street– sultan botutihe steer) 7 𝑉7 the intersection of four moodu market (sultan botutihe street– aloei saboe steer– matolodula street) 8 𝑉8 the intersection of three ubm (bj. pola isa street–aloei saboe streettinaloga street) 9 𝑉9 the intersection of three tinaloga gas station (tinaloga steer–toto tengah street) 10 𝑉10 the intersection of 4 bypass kabila (toto tengah street–b.j habibie street– sabes street– noho hudji street) 11 𝑉11 the intersection of three police office of kabila (pasar minggu streettapa kabila street) 12 𝑉12 the intersection of three al munawarah mosque (pasar minggu street– muh. van gobel street) 13 𝑉13 the intersection of four darul muhaimin mosque ( b.j habibie street– muh. van gobel street – el madinah road street) 14 𝑉14 the intersection of three indomaret (pasar minggu street–jembatan merah street) 15 𝑉15 the intersection of three adipura monument of bone bolango (jembatan merah street–b.j habibie street) 16 𝑉16 ung bone bolango campus table 1 data is used to determine the shortest trajectory using the dijkstra algorithm with steps: 1). node label with 𝜆(𝑠) = 0, and for each 𝑣 node in 𝐺 other than 𝑠, 𝑣 node label with 𝜆(𝑣) = ∞. 2). suppose 𝑢 ∈ 𝑇 with a minimum 𝜆(𝑢). 3). if 𝑢 = 𝑡, stop, then the shortest trajectory from s to t is 𝜆(𝑡). 4). for each side 𝑒 = 𝑢𝑣, 𝑣 ∈ 𝑇; replace label 𝑣 with 𝜆(𝑣) = minimum 𝜆(𝑣), 𝜆(𝑢) + w(e). 5). 𝑇 = 𝑇 − 𝑢, and return to iteration 2 [16]. the shortest trajectory problem in all knot pairs is to find the shortest trajectory between knot pairs. 𝑉𝑖 , 𝑉𝑗 ∈ 𝑣 in such a way that 𝑖 ≠ 𝑗 [17]. matrices of agility formed from graph weights with steps: 1). the distance of 𝑉𝑖 node with 𝑉𝑗 if there is a connecting side, then in writing with the weight, 2) 0 if the 𝑉𝑖 node is connected to 𝑉𝑖 and 3) if no side connects the 𝑉𝑖 node with 𝑉𝑗 after looking at the steps above then fill the matrix of agility with the weight on the graph. the results of graph representations weighted into the matrices of the neighboring show in table 2. nurwan, widya e. pranata, muhammad r. f. payu, nisky i. yahya implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation 35 table 2. representation of graphs in the matrices of neighboring 𝑽 𝑽𝟏 𝑽𝟐 𝑽𝟑 𝑽𝟒 𝑽𝟓 𝑽𝟔 𝑽𝟕 𝑽𝟖 𝑽𝟗 𝑽𝟏𝟎 𝑽𝟏𝟏 𝑽𝟏𝟐 𝑽𝟏𝟑 𝑽𝟏𝟒 𝑽𝟏𝟓 𝑽𝟏𝟔 𝑽𝟏 0𝑣1 1𝑣1 6𝑣1 ∞ ∞ 18𝑣1 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟐 1𝑣2 0𝑣2 ∞ ∞ 17𝑣2 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟑 6𝑣3 ∞ 0𝑣3 19𝑣3 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟒 ∞ ∞ 19𝑣4 0𝑣4 14𝑣4 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟓 ∞ 17𝑣5 14𝑣5 0𝑣5 ∞ ∞ ∞ 23𝑣5 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟔 18𝑣6 ∞ ∞ ∞ ∞ 0𝑣6 17𝑣6 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟕 ∞ ∞ ∞ ∞ ∞ 17𝑣7 0𝑣7 29𝑣7 ∞ ∞ 21𝑣7 ∞ ∞ ∞ ∞ ∞ 𝑽𝟖 ∞ ∞ ∞ ∞ 23𝑣8 ∞ 29𝑣8 0𝑣8 3.5𝑣8 ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟗 ∞ ∞ ∞ ∞ ∞ ∞ ∞ 3.5𝑣9 0𝑣9 11𝑣9 ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟏𝟎 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 11𝑣10 0𝑣10 24𝑣10 ∞ 24𝑣10 ∞ ∞ ∞ 𝑽𝟏𝟏 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 24𝑣11 0𝑣11 22𝑣11 ∞ 22𝑣11 ∞ ∞ 𝑽𝟏𝟐 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 22𝑣12 0𝑣12 22𝑣12 ∞ ∞ ∞ 𝑽𝟏𝟑 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 24𝑣13 ∞ ∞ 0𝑣13 ∞ ∞ 20𝑣13 𝑽𝟏𝟒 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 22𝑣14 ∞ ∞ 0𝑣14 22𝑣14 ∞ 𝑽𝟏𝟓 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 22𝑣15 0𝑣15 2𝑣15 𝑽𝟏𝟔 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 20𝑣16 14 20𝑣16 0𝑣16 weight change of each iteration based on algorithm dijkstra's steps obtained the shortest route from each node 𝑉𝑖 to the 𝑉𝑗 as in table 3. table 3. results of the shortest route in the matrices of neighbouring 𝑽 𝑽𝟏 𝑽𝟐 𝑽𝟑 𝑽𝟒 𝑽𝟓 𝑽𝟔 𝑽𝟕 𝑽𝟖 𝑽𝟗 𝑽𝟏𝟎 𝑽𝟏𝟏 𝑽𝟏𝟐 𝑽𝟏𝟑 𝑽𝟏𝟒 𝑽𝟏𝟓 𝑽𝟏𝟔 𝑽𝟏 0𝑣1 1𝑣1 6𝑣1 ∞ ∞ 18𝑣1 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟐 1𝑣1 6𝑣1 ∞ 18𝑣2 18𝑣1 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟑 6𝑣1 25𝑣3 18𝑣2 18𝑣1 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟒 25𝑣3 18𝑣2 18𝑣1 ∞ 41𝑣5 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟓 25𝑣3 18𝑣1 35𝑣4 41𝑣5 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟔 25𝑣3 35𝑣6 41𝑣5 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 𝑽𝟕 35𝑣6 41𝑣5 ∞ ∞ 56𝑣7 ∞ ∞ ∞ ∞ ∞ 𝑽𝟖 41𝑣5 44.5𝑣8 ∞ 56𝑣7 ∞ ∞ ∞ ∞ ∞ 𝑽𝟗 44.5𝑣8 55.5𝑣9 56𝑣7 ∞ ∞ ∞ ∞ ∞ 𝑽𝟏𝟎 55.5𝑣9 56𝑣7 ∞ 79.5𝑣10 ∞ ∞ ∞ 𝑽𝟏𝟏 56𝑣7 78𝑣11 79.5𝑣10 ∞ ∞ ∞ 𝑽𝟏𝟐 78𝑣11 79.5𝑣10 96𝑣12 ∞ ∞ 𝑽𝟏𝟑 79.5𝑣10 96𝑣12 ∞ ∞ 𝑽𝟏𝟒 96𝑣12 118𝑣14 99.5𝑣13 𝑽𝟏𝟓 101.5𝑣16 99.5𝑣13 𝑽𝟏𝟔 101.5𝑣16 representation of the neighbouring matrix in table 3 results in the shortest route of the bus with the following details: a. the 𝑉1 is connected to 𝑉1, 𝑉3 , 𝑉6 with a weight of 𝑉1 − 𝑉2 = 1 , 𝑉1 − 𝑉3 = 6, 𝑉1 − 𝑉6 = 18. because 𝑉2 the smallest weight choose and colour, it then lowered the 𝑉2. b. 𝑉2 connected with 𝑉1 and 𝑉5 with a 𝑉2 − 𝑉5 = 17. added with colored weight (17 + 1 = 18). because 𝑉3 the smallest weight, then the 𝑉3 selected and coloured, then lowered the 𝑉3. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 31-40 36 c. 𝑉3 connected with 𝑉1 and 𝑉4 nodes with a weight of 𝑉3 − 𝑉4 = 19 added with a coloured weight (19+6 = 25). furthermore, two nodes have the same weight (smallest), namely 𝑉5 and𝑉6, selected 𝑉5, then lowered the 𝑉5. d. 𝑉5 connected with 𝑉2 , 𝑉4, 𝑉8 with a weight of 𝑉5 − 𝑉4 = 14, added with a coloured weight (14 +18 = 32). because the weight is greater than the previous weight, the previous weight is lowered. next 𝑉5 −𝑉8 = 23 then add with the coloured weight (23+18 = 41). because 𝑉6 the smallest weight, then select and colour, then lowered the 𝑉6. e. 𝑉6 connected with 𝑉1 and 𝑉7 nodes with a 𝑉6 − 𝑉7 = 17, then add with the coloured weight (17 + 18 = 35). because 𝑉4 smallest weight choose and colour then lowered the 𝑉4. f. 𝑉4 connected with the 𝑉3 and 𝑉5 because the 𝑉3 and 𝑉5 already coloured, then further lose the smallest weight. since 𝑉7 smallest weight choose and colour, it is also lowered 𝑉7. g. 𝑉7 connected with 𝑉8 and 𝑉11 with a weight of 𝑉7 − 𝑉8 = 29, then add with the coloured weight (29 +35 = 64) because the result is 64, greater than the previous weight, then lower the previous weight. next 𝑉7− 𝑉11 = 21 then add with the colored weight (21 + 35 = 56). since the smallest weighted v8 select and colour, the next lowered 𝑉8. h. 𝑉8 is connected to 𝑉7 and 𝑉9 with a weight of 𝑉8 −𝑉9 = 3.5, then add with the coloured weight (3.5 + 41 = 44.5). because 𝑉9 smallest weight choose and colour then lowered the 𝑉9. i. 𝑉9 is connected to 𝑉8, and 𝑉10 with a weight of 𝑉9 − 𝑉10 = 11, then add with the coloured weight (11 + 44.5 = 55.5). since v_10 smallest weight choose and colour, the knot is further lowered 𝑉10. j. 𝑉10 connected with 𝑉9, 𝑉11, 𝑉13 with a weight of 𝑉10 − 𝑉11= 24 then add with the weight already coloured (24 + 55.5 = 79.5) because the result is 79.5, greater than the previous weight, then lower the previous weight. next 𝑉10 − 𝑉13 = 24 then add with the colored weight (24 + 55.5 = 79.5). because 𝑉11smallest weight choose and colour then lowered the 𝑉11. k. 𝑉11 connected with 𝑉7, 𝑉10, 𝑉12 with a weight of 𝑉11 − 𝑉12 = 22, then added with a coloured weight (22 + 56 = 78). because 𝑉12 smallest weight choose and colour then lowered the v_12. l. 𝑉12 is connected with 𝑉11, 𝑉13, 𝑉14 with a weight of 𝑉12 − 𝑉13= 22, then add with the coloured weight (22 + 78 = 100) because the result is 100, greater than the previous weight, then lower the previous weight. next 𝑉12 −v_14 = 18 then add the colored weight (18+78 = 96). because 𝑉13 smallest weight choose and colour then lowered the 𝑉13. m. 𝑉13 is connected with 𝑉10, 𝑉12, 𝑉16 with a weight of 𝑉13 − 𝑉16 = 20 then add with the colored weight (20 + 79.5 = 99.5). because 𝑉14 smallest weight choose and colour then lowered the 𝑉14. n. 𝑉14 is connected 𝑉12, 𝑉15 with a weight of 𝑉14 − 𝑉15 = 22 then 27 add with colored weight (22 + 96 = 118). because of the smallest weighted v16 select and colour, the next lowered 𝑉16. o. 𝑉16 is connected to the 𝑉13, and 𝑉15 with a weight of 𝑉16 −𝑉15 = 2, then add with the coloured weight (2+99.5 = 101.5) because the result is 101.5, smaller than the previous weight, then choose and colour. next, i lowered the 𝑉15. p. 𝑉15 connected to the 𝑉14 and 𝑉16 because the 𝑉14 and 𝑉16 has been coloured, then it is finished. figure 2 the shortest route a bus can take from campus 1 ung to bone bolango's new campus is from v_1 to v_16, or vice versa obtained the shortest route begins 𝑉1 − 𝑉2 − 𝑉5 − 𝑉8 − 𝑉9 − 𝑉10 − 𝑉13 − 𝑉16 or campus 1 ung the intersection of three sentra nurwan, widya e. pranata, muhammad r. f. payu, nisky i. yahya implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation 37 media (jendral sudirman street–pangeran hidayat street) the intersection of four public health center (pangeran hidayat street – rusli datau street – prof. aryo katili street – bj. pola isa street) the intersection of three ubm campus (bj. pola isa street–aloei saboe street-tinaloga street) the intersection of three tinaloga gas station (tinaloga steer–toto tengah street) the intersection of four bypass kabila (toto tengah street–b.j habibie street– sabes street– noho hudji street) the intersection of four darul muhaimin mosque ( b.j habibie street– muh. van gobel street – el madinah road street) – ung bone bolango campus. figure 2. bus shortest route to ung bone bolango campus 3.2 bus schedule to ung bone bolango campus the welch-powell algorithm is required to color the graph's vertices based on the highest degree of all its nodes. from the data available, there are four buses in operation [18]. these four buses run back and there, represented in a graph. the data of the number of buses be defined as a node on the graph. there are no specific labelling rules, labelled with an index to distinguish which buses operate back and away. suppose each node with 𝑉𝑎,𝑏 with the following description: 1. a: a bus, a ∈ [1, 4] 2. b: return or departure (1 for departure, 2 for return) figure 3. representation of bus data in graph figure 4. graph representation of bus schedules jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 31-40 38 each node in figure 3 is then connected and forms aside. the side represents the bus's operational time, and each bus does not operate simultaneously. the arrangement of the vertices' location establishes a circle to make it easier to draw a straight line for the relationship of each node. the node in figure 4 represents the bus, and the side is a representation of time. node 𝑉1.1, 𝑉2.1, 𝑉3.1, 𝑉4.1 is the bus departure node from campus 1 to campus ung bone bolango and node 𝑉1.2, 𝑉2.2, 𝑉3.2, 𝑉4.2 is the node from campus ung bone bolango to campus 1. what can see degrees on each node in table 4. table 4. vertices by degree no node neighbors degrees 1 𝑉1.1 𝑉1.2, 𝑉2.1, 𝑉3.1, 𝑉4.1 4 2 𝑉1.2 𝑉1.1, 𝑉2.2, 𝑉3.2, 𝑉4.2 4 3 𝑉2.1 𝑉1.1, 𝑉2.2, 𝑉3.1, 𝑉4.1 4 4 𝑉2.2 𝑉1.2, 𝑉2.1, 𝑉3.2, 𝑉4.2 4 5 𝑉3.1 𝑉3.2, 𝑉1.1, 𝑉2.1, 𝑉4.1 4 6 𝑉3.2 𝑉3.1, 𝑉1.2, 𝑉2.2, 𝑉4.2 4 7 𝑉4.1 𝑉4.2, 𝑉1.1, 𝑉2.1, 𝑉3.1 4 8 𝑉4.2 𝑉4.1, 𝑉1.2, 𝑉2.2, 𝑉3.2 4 table 4 shows the same degree on each node, then coloured by not giving the same colour to the neighbouring nodes as in figure 5. by doing the steps of the welch-powell algorithm, knot colouring is obtained as in figure 6. figure 5. the colouration of vertices in graph figure 6. final result of knot coloring in figure 6, four kinds of coloring are obtained. next, group the buses by chromatic numbers. a graph has a chromatic number denoted by 𝜒(𝐺) [19] [20]. zero graphs have a chromatic number of 𝜒(𝐺) = 1, while to color, a complete graph is required n color fruits because all points are interconnected [19] [21]. the chromatic number of the bus schedule representation graph is 𝜒(𝐺) = 4 which means each bus schedule for return or departure is at least four sessions. we can see the results of graph coloring for the campus bus schedule in table 5. table 5. bus departure and return schedule to/from ung bone bolango campus no departure schedule schedule of return time bus time bus 1 06.30 bus 1 07.30 bus 1 2 07.30 bus 2 08.30 bus 2 3 08.00 bus 3 09.00 bus 3 nurwan, widya e. pranata, muhammad r. f. payu, nisky i. yahya implementation of dijkstra algorithm and welch-powell algorithm for optimal solution of campus bus transportation 39 no departure schedule schedule of return time bus time bus 4 08.30 bus 4 09.30 bus 4 5 09.00 bus 1 10.00 bus 1 6 09.30 bus 2 11.00 bus 2 7 10.00 bus 3 12.30 bus 3 8 11.00 bus 4 13.00 bus 4 9 12.30 bus 1 13.30 bus 1 10 13.00 bus 2 14.00 bus 2 11 13.30 bus 3 14.30 bus 3 12 14.00 bus 4 15.00 bus 4 13 14.30 bus 1 15.30 bus 1 14 15.00 bus 2 16.00 bus 2 15 15.30 bus 3 16.30 bus 3 16 16.00 bus 4 17.00 bus 4 table 5 the bus departure schedule from campus 1 to ung bone bolango campus consists of 16 departure time sessions with departure time starting from 06:30 and ending at 16:00 with a delay of 60 minutes each departure time. the bus return schedule from ung bone bolango campus to campus 1 consists of 16 sessions by the same bus. the bus departure schedule from ung bone bolango campus to campus starts at 07.30 am and ends at 05.00 pm. the overall departure schedule consists of 32 departure sessions. each bus gets eight departure sessions per day with a minimum session break of 60 minutes, including travel time. 4. conclusions the dijkstra algorithm's calculation obtained the shortest bus route from campus 1 to campus ung bone bolango is the trajectory 𝑉1−𝑉2−𝑉5−𝑉8−𝑉9−𝑉10−𝑉13−𝑉16 with a distance of 9.95 km. the bus passed by the bus is campus 1 ung → the intersection of 3 sentra media → the intersection of 4 public health center → the intersection of 3 ubm campus → the intersection of 3 tinaloga gas station → the intersection of 4 bypass kabila → intersection of 4 darul muhaimin mosque → ung bone bolango campus. the coloring results using the welch-powell algorithm obtained the number of colors on the bus schedule's color to ung bone bolango in four colors. the bus departure and return schedule are 16 sessions each, and the bus operating time starts from 06.30 am to 05.00 pm. each bus gets four departure sessions and four return sessions 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[19] g. chartrand, l. lesniak, and p. zhang, graphs and digraphs. california: pacific grove california, 2016. [20] r. balakrishnan and k. ranganathan, a textbook of graph theory, second edi. springer science+business media new york, 2021. [21] a. m. soimah and n. s. m. mussafi, “pewarnaan simpul dengan algoritma welch-powell pada traffic light di yogyakarta,” j. fourier, vol. 2, no. 2, p. 73, 2013. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: abdulloh jaelani, abdjae@fst.unair.ac.id department of mathematics, faculty of science and technology, universitas airlangga, surabaya, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.36-44 relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces pragasto aji hendro puadi1*, eridani1, abdulloh jaelani1* 1department of mathematics, universitas airlangga, surabaya, indonesia article history: received apr 26, 2022 revised, jun 5, 2022 accepted, jun 28, 2022 kata kunci: ruang barisan, ruang morrey ruang morrey tipe lemah abstrak. ruang morrey merupakan ruang yang cukup penting dan banyak dikaji dalam banyak cabang matematika. ruang morrey tipe lemah merupakan ruang berorde semu dan memiliki sifat dasar yang sama dengan ruang barisan morrey.pada artikel ini diselidiki sifat elementer antara ruang barisan morrey dan ruang barisan morrey tipe lemah. selanjutnya ditunjukkan hubungannya antara ruang barisan morrey dan ruang barisan morrey tipe lemah dengan ruang barisan. berdasarkan hasil pembahasan diperoleh sifat elementer ruang barisan morrey ℓ𝑝 ⊂ ℓ𝑞 𝑝 dengan 1 ≤ 𝑝 ≤ 𝑞 < ∞ dan ℓ𝑞 𝑝2 ⊆ ℓ𝑞 𝑝1 dengan 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. sifat elementer dari ruang barisan morrey tipe lemah adalah ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 dengan 1 ≤ 𝑝 ≤ 𝑞 < ∞, 𝜔ℓ𝑞 𝑝2 ⊆ 𝜔ℓ𝑞 𝑝1 dengan 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ dan ruang barisan morrey tipe lemah merupakan ruang quasi norma. lebih lanjut hubungan ruang barisan morrey, ruang barisan morrey lemah dan ruang barisan adalah ℓ𝑝 ⊂ ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 . keywords: sequence spaces, morrey spaces, weak type morrey spaces, abstract. morrey space is a space that important spaces and widely studied in many branches of mathematics. weak type morrey spaces is a quasinormed spaces and have alike elementary properties with morrey sequence spaces. this article investigates some properties the elementary properties of morrey sequence spaces and weak type morrey sequence space. next is show relation morrey sequence spaces and weak type morrey sequence space with sequence spaces. based on the results of the discussion, it was obtained that the elementary properties of morrey sequence spaces is ℓ𝑝 ⊂ ℓ𝑞 𝑝 with 1 ≤ 𝑝 ≤ 𝑞 < ∞ and ℓ𝑞 𝑝2 ⊆ ℓ𝑞 𝑝1 with 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. the elementary properties of weak type morrey sequence spaces 𝜔ℓ𝑞 𝑝 is ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 with 1 ≤ 𝑝 ≤ 𝑞 < ∞, 𝜔ℓ𝑞 𝑝2 ⊆ 𝜔ℓ𝑞 𝑝1 with 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ and weak type morrey sequence spaces is quasinormed space. furthermore, the relation of morrey sequence spaces, weak type morrey sequence spaces and sequence spaces is ℓ𝑝 ⊂ ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 how to cite: p. a. h. puadi, eridani, & a. jaelani, “relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces”. j. mat mantik, vol. 8, no. 1, pp. 36-44, jun. 2022. jurnal matematika mantik vol. 8, no. 1, june 2022, pp. 36-44 issn: 2527-3159 (print) 2527-3167 (online) mailto:abdjae@fst.unair.ac.id https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 p. a. h. puadi, eridani, & a. jaelani relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces 37 1. introduction morrey spaces were first introduced by charles bardfield morrey jr. (1907-1984) on 1938. morrey spaces became an important space in many branches of mathematics even though it was first discovered to solve partial differential equations, now there are hundreds of articles and journals that discuss morrey space to take up the recent development of morrey spaces [1], [2]. on 2016, it is defined that morrey sequence space is denoted by ℓ𝑞 𝑝 with 1 ≤ 𝑝 ≤ 𝑞 < ∞ is a set of all real sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ that holds ‖𝑥‖ ℓ𝑞 𝑝 = sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 < ∞ with 𝑁 ∈ ℕ, 𝑆𝑁 = {−𝑁, −(𝑁 − 1), … , 0, … , 𝑁 − 1, 𝑁} and |𝑆𝑁 | denote the cardinality of 𝑆𝑁 [3], [4]. sequence spaces denote by ℓ 𝑝 = ℓ𝑝 𝑝 with 𝑝 = 𝑞 is a set of all sequences that holds ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ ℤ < ∞ and 𝑝 as parameter [5]–[7] . morrey sequence space ℓ𝑞 𝑝 has a very close relation with sequence space ℓ𝑝. one of the morrey sequence space’s elementary properties is ‖𝑥‖ ℓ𝑞 𝑝 ≤ ‖𝑥‖ ℓ 𝑝 for all 𝑥 ∈ ℓ𝑝, hence ℓ𝑝 ⊆ ℓ𝑞 𝑝 for 1 ≤ 𝑝 ≤ 𝑞 < ∞. in other words, morrey sequence space ℓ𝑞 𝑝 is an extension of sequence space ℓ𝑝. morrey sequence space ℓ𝑞 𝑝 can be more extend to weak type morrey sequence space ωℓ𝑞 𝑝 [8]–[10]. weak type morrey sequence spaces is a set of all real sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ that holds ‖𝑥‖ω ℓ𝑞 𝑝 = sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 < ∞. weak type morrey sequence spaces is a quasinormed spaces and have alike elementary properties with morrey sequence spaces. based on the description above, we will discuss about morrey sequence spaces’ elementary properties and weak type morrey sequence spaces and also the relation between sequence space ℓp, morrey sequence space ℓ𝑞 𝑝 dan weak type morrey sequence space ωℓ𝑞 𝑝 . definition [11] quasinorm ‖∙ ‖ on a vector space 𝑉 over a field ℝ is a function from 𝑉 to [0, ∞) and holds (i) ‖𝑢‖ = 0 if and only if 𝑢 = 0 (ii) ‖𝑟𝑢‖ = |𝑟|‖𝑢‖ for every 𝑟 ∈ ℝ and 𝑢 ∈ 𝑉 (iii) there exists 𝐶 ≥ 1 so that if 𝑢, 𝑣 ∈ 𝑉 then ‖𝑢 + 𝑣‖ ≤ 𝐶(‖𝑢‖ + ‖𝑣‖) if ‖ ‖ is a quasinorm and (𝑉, ‖∙ ‖) is a quasinormed space. theorem [12] (minkowski inequality) if 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 𝑝 ≥ 1 then (∑ |𝑥𝑘 + 𝑦𝑘 | 𝑝𝑛 𝑘=1 ) 1/𝑝 ≤ (∑ |𝑥𝑘 | 𝑝𝑛 𝑘=1 ) 1/𝑝 + (∑ |𝑦𝑘 | 𝑝𝑛 𝑘=1 ) 1/𝑝. theorem [12] if 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 0 < 𝑝 < 1 then ∑ |𝑥𝑘 + 𝑦𝑘 | 𝑝𝑛 𝑘=1 ≤ ∑ |𝑥𝑘 | 𝑝𝑛 𝑘=1 + ∑ |𝑦𝑘 | 𝑝𝑛 𝑘=1 . theorem [12] (hölder inequality) if 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 𝑝, 𝑞 is an exponent conjugation then ∑ |𝑥𝑘 𝑦𝑘 | 𝑛 𝑘=1 ≤ (∑ |𝑥𝑘 | 𝑝𝑛 𝑘=1 ) 1/𝑝(∑ |𝑦𝑘 | 𝑞𝑛 𝑘=1 ) 1/𝑞 . jurnal matematika mantik vol. 8, no. 1, june 2022, pp.36-44 38 definition [12] suppose that 𝑝 ∈ (0, ∞). ℓ𝑝 is a set of all sequence 𝑥 ∶ ℕ → ℝ that holds ∑ |𝑥𝑘 | 𝑝∞ 𝑖=1 convergent or ℓ𝑝 = {𝑥 = 〈𝑥𝑛 〉 ∶ ∑ |𝑥𝑘 | 𝑝∞ 𝑘=1 < ∞}. but on this article, ℓ𝑝 sequence space is a set of all sequence 𝑥 ∶ ℤ → ℝ that holds ∑ |𝑥𝑘 | 𝑝 𝑘∈ℤ = ∑ |𝑥𝑘 | 𝑝∞ −∞ < ∞. definition [3] suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞, 𝑁 ∈ ℕ, 𝑆𝑁 = {−𝑁, −(𝑁 − 1), … , 0, … , 𝑁 − 1, 𝑁} and |𝑆𝑁 | = 2𝑁 + 1 denote the cardinality of 𝑆𝑁. let ℓ𝑞 𝑝 denote morrey sequence space is a set of all sequence 〈𝑥𝑘 〉𝑘∈ℤ that holds ‖𝑥‖ ℓ𝑞 𝑝 = sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 < ∞. definition [3] suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞. let 𝜔ℓ𝑞 𝑝 denote weak type morrey sequence space is a set of all sequence 〈𝑥𝑘 〉𝑘∈ℤ that holds ‖𝑥‖ω ℓ𝑞 𝑝 = sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 < ∞. 2. methods we used the properties of definition quasinorm, minkowski inequality, hölder inequality, and characteristics of norm on the set of real [4], [13]–[15] to obtain properties of morrey sequence spaces, weak type morrey sequence space, and relation morrey sequence spaces and weak type morrey sequence space with sequence spaces. 3. results and discussion proposition morrey sequence space ℓ𝑞 𝑝 is a vector space over ℝ. proof : suppose that 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ, 𝑦 = 〈𝑦𝑘 〉𝑘∈ℤ is an element of ℓ𝑞 𝑝 . for every 𝑎 ∈ ℝ, ‖𝑎𝑥‖ ℓ𝑞 𝑝 = sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑎𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 = |𝑎| sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 = |𝑎|‖𝑥‖ ℓ𝑞 𝑝 < ∞ then 𝑎𝑥 ∈ ℓ𝑞 𝑝 . with minkowski inequality we got (∑ |𝑥𝑘 + 𝑦𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 + (∑ |𝑦𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 + |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑦𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ‖𝑥 + 𝑦‖ ℓ𝑞 𝑝 ≤ ‖𝑥‖ ℓ𝑞 𝑝 + ‖𝑦‖ ℓ𝑞 𝑝 . if ‖𝑥‖ ℓ𝑞 𝑝 < ∞ and ‖𝑦‖ ℓ𝑞 𝑝 < ∞, then ‖𝑥 + 𝑦‖ ℓ𝑞 𝑝 < ∞ so that 𝑥 + 𝑦 ∈ ℓ𝑞 𝑝 . ℓ𝑞 𝑝 is closed under the operation (+). let 𝑧 ∈ ℓ𝑞 𝑝 then (i) 𝑥 + 𝑦 = 〈𝑥𝑘 + 𝑦𝑘 〉𝑘∈ℤ = 〈𝑦𝑘 + 𝑥𝑘 〉𝑘∈ℤ = 𝑦 + 𝑥 for every 𝑥, 𝑦 ∈ ℓ𝑞 𝑝 p. a. h. puadi, eridani, & a. jaelani relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces 39 (ii) 𝑥 + (𝑦 + 𝑧) = (𝑥 + 𝑦) + 𝑧 for every 𝑥, 𝑦, 𝑧 ∈ ℓ𝑞 𝑝 (iii) there exists 0 = 〈… ,0,0,0, … . 〉, then 0 + 𝑥 = 𝑥 + 0 for every 𝑥 ∈ ℓ𝑞 𝑝 (iv) there exists −𝑥 = 〈−𝑥𝑘 〉𝑘∈ℤ, then (−𝑥) + 𝑥 = 𝑥 + (−𝑥) = 0 for every 𝑥 ∈ ℓ𝑞 𝑝 (v) let 𝑏 ∈ ℝ. 𝑎(𝑥 + 𝑦) = 𝑎𝑥 + 𝑎𝑦 (𝑎 + 𝑏)𝑥 = 𝑎𝑥 + 𝑏𝑥 (𝑎𝑏)𝑥 = 𝑎(𝑏𝑥) for every 𝑥 ∈ ℓ𝑞 𝑝 (vi) there exists 1 ∈ ℝ so that 1𝑥 = 𝑥 for every 𝑥 ∈ ℓ𝑞 𝑝 thus ℓ𝑞 𝑝 is a vector space over ℝ.∎ example: suppose that 1 ≤ 𝑝 < 𝑞 < ∞. a sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 = |𝑘| −𝑞/𝑝 for 𝑘 ≠ 0 and 𝑥𝑘 = 0 for 𝑘 = 0. for every 𝑁 ∈ ℕ, ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 = 2 ∑ 1 𝑘𝑞 <𝑁𝑘=1 2 ∑ 1 𝑘𝑞 ∞ 𝑘=1 . (1) ∑ 1 𝑘𝑞 ∞ 𝑘=1 = 1 + ( 1 2𝑞 + 1 3𝑞 ) + ( 1 4𝑞 + 1 5𝑞 + 1 6𝑞 + 1 7𝑞 ) + ⋯ < 1 + 2 2𝑞 + 4 4𝑞 + 8 8𝑞 + ⋯ = 1 + 1 2𝑞−1 + 1 4𝑞−1 + 1 8𝑞−1 + ⋯ if 𝑞 > 1 then 0 < 1 2𝑞−1 < 1 implies that ∑ 1 𝑘𝑞 ∞ 𝑘=1 < 1 1− 1 2𝑞−1 . (2) from (1) and (2) obtained |𝑆𝑁 | 𝑝 𝑞 −1 ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 = ( 1 |𝑆𝑁| 1− 𝑝 𝑞 ) ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 < ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 < 2 1− 1 2𝑞−1 (|𝑆𝑁| 𝑝 𝑞 −1 ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 < ( 2𝑞 2𝑞−1−1 ) 1/𝑞 sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ ( 2𝑞 2𝑞−1−1 ) 1/𝑞 . thus 𝑥 ∈ ℓ𝑞 𝑝 . ∎ theorem suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞. ‖𝑥‖ ℓ𝑞 𝑝 ≤ ‖𝑥‖ℓp for every 𝑥 ∈ ℓ 𝑝. proof : for all 𝑁 ∈ ℕ, it’s easy to see that 0 < |𝑆𝑁 | 1 𝑞 − 1 𝑝 ≤ 1 then |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 and sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ sup 𝑁 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ‖𝑥‖ ℓ𝑞 𝑝 ≤ ‖𝑥‖ℓp . ∎ jurnal matematika mantik vol. 8, no. 1, june 2022, pp.36-44 40 example : let 1 ≤ 𝑝 < 𝑞 < ∞ and a sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 = |𝑘| −1/𝑞 for 𝑘 ≠ 0 and 𝑥𝑘 = 0 for 𝑘 = 0. if 0 < 𝑝 𝑞 < 1 then ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ ℤ = 2 ∑ 1 𝑘𝑝/𝑞 ∞ 𝑘=1 ≥ 2 ∑ 1 𝑘 ∞ 𝑘=1 . we know that ∑ 1 𝑘 ∞ 𝑘=1 is harmonic series then it’s divergent and ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 divergent thus 𝑥 ∉ ℓ𝑝. for all 𝑁 ∈ ℕ, ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 = ∑ 1 |𝑘|𝑝/𝑞 𝑘 ∈ 𝑆𝑁,𝑘≠0 = 2 ∑ 1 𝑘𝑝/𝑞 𝑁 𝑘=1 . for every 𝑝 and 𝑞 with 1 ≤ 𝑝 < 𝑞 < ∞, function 𝑦 = 𝑥−𝑝/𝑞 has 𝑦′ < 0 and 𝑦′′ > 0 on 𝑥 ≥ 1. figure 1. area of partition below 𝑦 = 𝑥−𝑝/𝑞 we know that the sum of areas of partitions equal to ∑ 1 𝑘𝑝/𝑞 𝑁 𝑘=2 and ∑ 1 𝑘𝑝/𝑞 𝑁 𝑘=2 ≤ ∑ 1 𝑘𝑝/𝑞 𝑁 𝑘=1 = 1 + ∑ 1 𝑘𝑝/𝑞 𝑁 𝑘=2 ≤ 1 + ∫ 1 𝑥𝑝/𝑞 𝑑𝑥 𝑁 1 ≤ 1 − 𝑞 𝑞−𝑝 + 𝑞 𝑞−𝑝 𝑁 1− 𝑝 𝑞 . hence, |𝑆𝑁 | 𝑝 𝑞 −1 ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ≤ 2 ( 1− 𝑞 𝑞−𝑝 + 𝑞 𝑞−𝑝 𝑁 1− 𝑝 𝑞 (2𝑁+1) 1− 𝑝 𝑞 ) and lim 𝑁→∞ 2 ( 1− 𝑞 𝑞−𝑝 + 𝑞 𝑞−𝑝 𝑁 1− 𝑝 𝑞 (2𝑁+1) 1− 𝑝 𝑞 ) = 2 ( 𝑞 𝑞−𝑝 2 𝑝 𝑞 −1 ) = 𝑞 𝑞−𝑝 2 𝑝 𝑞 |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ (2 ( 1− 𝑞 𝑞−𝑝 + 𝑞 𝑞−𝑝 𝑁 1− 𝑝 𝑞 (2𝑁+1) 1− 𝑝 𝑞 )) 1/𝑝 sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘 ∈ 𝑆𝑁 ) 1/𝑝 ≤ 21/𝑝 ( 𝑞 𝑞−𝑝 ) 1/𝑝 thus 𝑥 ∈ ℓ𝑞 𝑝 .∎ theorem suppose that 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. for all 𝑥 ∈ ℓ𝑞 𝑝2 applies that p. a. h. puadi, eridani, & a. jaelani relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces 41 ‖𝑥‖ ℓ𝑞 𝑝1 ≤ ‖𝑥‖ ℓq p2 . proof : by hölder inequality, we have ∑ |𝑥𝑘 | 𝑝1 𝑘∈𝑆𝑁 ≤ (∑ |𝑥𝑘 | 𝑝2 𝑘∈𝑆𝑁 ) 𝑝1 𝑝2 (∑ 1𝑘∈𝑆𝑁 ) 1− 𝑝1 𝑝2 ( 1 |𝑆𝑁| ∑ |𝑥𝑘 | 𝑝1 𝑘∈𝑆𝑁 ) 1 𝑝1 ≤ ( 1 |𝑆𝑁| ∑ |𝑥𝑘 | 𝑝2 𝑘∈𝑆𝑁 ) 1 𝑝2 |𝑆𝑁 | 1 𝑞 − 1 𝑝1 (∑ |𝑥𝑘 | 𝑝1 𝑘∈𝑆𝑁 ) 1 𝑝1 ≤ |𝑆𝑁 | 1 𝑞 − 1 𝑝2 (∑ |𝑥𝑘 | 𝑝2 𝑘∈𝑆𝑁 ) 1 𝑝2 ‖𝑥‖ ℓ𝑞 𝑝1 ≤ ‖𝑥‖ ℓq p2 . ∎ from the above theorem, we have ℓ𝑞 𝑝2 ⊆ ℓ𝑞 𝑝1 on 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ but we can’t approve ℓ𝑞 𝑝2 ⊂ ℓ𝑞 𝑝1 yet [3]. theorem if 1 ≤ 𝑝 ≤ 𝑞 < ∞ then ‖𝑥‖𝜔ℓ𝑞 𝑝 ≤ ‖𝑥‖ℓ𝑞 𝑝 for every 𝑥 ∈ ℓ𝑞 𝑝 . proof: |𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 = |𝑆𝑁 | 1 𝑞 − 1 𝑝|𝛾𝑝{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 = |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ 𝛾𝑝𝑘∈𝑆𝑁 ,|𝑥𝑘|>𝛾 ) 1/𝑝 ≤ |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑥𝑘 | 𝑝 𝑘∈𝑆𝑁,|𝑥𝑘|>𝛾 ) 1/𝑝 |𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 ≤ |𝑆𝑁 | 1 𝑞 − 1 𝑝(∑ |𝑥𝑘 | 𝑝 𝑘∈𝑆𝑁 ) 1/𝑝 supremum over 𝑁 ∈ ℕ and 𝛾 > 0 on the above inequality, we have sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 ≤ sup 𝑁 |𝑆𝑁 | 1 𝑞 − 1 𝑝 (∑ |𝑥𝑘 | 𝑝 𝑘∈𝑆𝑁 ) 1/𝑝 ‖𝑥‖𝜔ℓ𝑞 𝑝 ≤ ‖𝑥‖ℓ𝑞 𝑝 . thus ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 ∎. example : suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞. a sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 = |𝑘| −1/𝑝 for 𝑘 ≠ 0 and 𝑥𝑘 = 0 for 𝑘 = 0. ∑ |𝑥𝑘 | 𝑝 𝑘 ∈ ℤ = 2 ∑ 1 𝑘 ∞ 𝑘=1 . we know that ∑ 1 𝑘 ∞ 𝑘=1 is divergent then 𝑥 ∉ ℓ 𝑝 but for any 𝛾 > 0, 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑘| −1/𝑝 > 𝛾}| 1/𝑝 = 2𝛾|{𝑘 ∈ ℕ ∶ 1 ≤ 𝑘 < 𝑁, 𝑘−1/𝑝 > 𝛾}| 1/𝑝 and 𝑘−1/𝑝 > 𝛾 ⟹ 𝑘−1 > 𝛾𝑝 ⟹ 𝑘 < 1 𝛾𝑝 then 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑘| −1/𝑝 > 𝛾}| 1/𝑝 < 2𝛾 ( 1 𝛾𝑝 ) 1/𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑘| −1/𝑝 > 𝛾}| 1/𝑝 < 2𝛾 ( 1 𝛾 ) = 2 then 𝑥 ∈ 𝜔ℓ𝑝 𝑝 and ℓ𝑝 ⊂ 𝜔ℓ𝑝 𝑝 = 𝜔ℓ𝑝. theorem if 1 ≤ 𝑝 ≤ 𝑞 < ∞ then ‖ ‖𝜔ℓ𝑞 𝑝 is a quasinorm and (𝜔ℓ𝑞 𝑝 , ‖ ‖𝜔ℓ𝑞 𝑝 ) is a quasinormed space. jurnal matematika mantik vol. 8, no. 1, june 2022, pp.36-44 42 proof: from the definition, we know that ‖𝑥‖𝜔ℓ𝑞 𝑝 ≥ 0 for all 𝑥 ∈ 𝜔ℓ𝑞 𝑝 . if 𝑥 = 0 (𝑥𝑘 = 0 for all 𝑘 ∈ ℤ) then {𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾} is an empty space and |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| = 0, |𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 = 0 sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝 = 0 ‖𝑥‖𝜔ℓ𝑞 𝑝 = 0. if ‖𝑥‖𝜔ℓ𝑞 𝑝 = 0 then |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| = 0 for all 𝑁 ∈ ℕ and 𝛾 > 0, hence for every 𝑘 ∈ ℤ applies that 0 ≤ |𝑥𝑘 | ≤ 𝛾 ⟹𝑥𝑘 = 0 untuk setiap 𝑘 ∈ ℤ (𝑥 = 0). let 𝑥 ∈ 𝜔ℓ𝑞 𝑝 and 𝑟 = 0 then ‖𝑟𝑥‖𝜔ℓ𝑞 𝑝 = |𝑟|‖𝑥‖𝜔ℓ𝑞 𝑝 = 0. if 𝑟 ≠ 0 then applies that ‖𝑟𝑥‖𝜔ℓ𝑞 𝑝 = sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑟𝑥𝑘 | > 𝛾}| 1/𝑝 = sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝛾 |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾 |𝑟| }| 1/𝑝 let 𝛾 |𝑟| = 𝑎 ⟹ 𝛾 = |𝑟|𝑎 then ‖𝑟𝑥‖𝜔ℓ𝑞 𝑝 = sup 𝑁∈ℕ,𝑎>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 |𝑟|𝑎|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝑎}| 1/𝑝 = |𝑟| sup 𝑁∈ℕ,𝑎>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝 𝑎|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝑎}| 1/𝑝 = |𝑟|‖𝑥‖𝜔ℓ𝑞 𝑝 . let 𝑥, 𝑦 ∈ 𝜔ℓ𝑞 𝑝 . for any 𝑁 ∈ ℕ, if 𝑘 ∈ 𝑆𝑁 dan |𝑥𝑘 | ≤ |𝑦𝑘 | then |𝑥𝑘 + 𝑦𝑘 | ≤ |𝑥𝑘 | + |𝑦𝑘 | ≤ 2|𝑦𝑘 | else if 𝑘 ∈ 𝑆𝑁 dan |𝑥𝑘 | > |𝑦𝑘 | then |𝑥𝑘 + 𝑦𝑘 | ≤ |𝑥𝑘 | + |𝑦𝑘 | ≤ 2|𝑥𝑘 |. thus {𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 + 𝑦𝑘 | > 𝛾} ⊆ {𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | + |𝑦𝑘 | > 𝛾} ⊆ {𝑘 ∈ 𝑆𝑁 ∶ 2|𝑥𝑘 | > 𝛾} ∪ {𝑘 ∈ 𝑆𝑁 ∶ 2|𝑦𝑘 | > 𝛾}. for any 𝛾 > 0 dan 𝑁 ∈ ℕ, |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 + 𝑦𝑘 | > 𝛾}| ≤ |{𝑘 ∈ 𝑆𝑁 ∶ 2|𝑥𝑘 | > 𝛾}| + |{𝑘 ∈ 𝑆𝑁 ∶ 2|𝑦𝑘 | > 𝛾}| multiply both sides by (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾) 𝑝 applies that (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾) 𝑝 |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 + 𝑦𝑘 | > 𝛾}| ≤ (|𝑆𝑁| 1 𝑞 − 1 𝑝𝛾) 𝑝 |{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾 2 }| + (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾) 𝑝 |{𝑘 ∈ 𝑆𝑁 ∶ |𝑦𝑘 | > 𝛾 2 }| let 𝛾 2 = 𝜎 ⟹ 𝛾 = 2𝜎 |𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 + 𝑦𝑘 | > 𝛾}| 1/𝑝 ≤ 2 ((|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎) 𝑝 |{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}| + (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎) 𝑝 |{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|) 1/𝑝 . if 𝑝 > 1 ⟹ 1 𝑝 < 1then |𝑆𝑁 | 1 𝑞 − 1 𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 + 𝑦𝑘 | > 𝛾}| 1/𝑝 p. a. h. puadi, eridani, & a. jaelani relation of morrey sequence spaces, weak type morrey sequence spaces, and sequence spaces 43 ≤ 2 ((|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎) 𝑝 |{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}| + (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎) 𝑝 |{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|) 1/𝑝 ≤ 2 (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}|) 1/𝑝 + 2 (|𝑆𝑁 | 1 𝑞 − 1 𝑝𝜎|{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|) 1/𝑝 taking supremum over 𝛾 > 0 dan 𝑁 ∈ ℕ from the above inequality, we get ‖𝑥 + 𝑦‖𝜔ℓ𝑞 𝑝 ≤ 2‖𝑥‖𝜔ℓ𝑞 𝑝 + 2‖𝑦‖𝜔ℓ𝑞 𝑝 = 2 (‖𝑥‖𝜔ℓ𝑞 𝑝 + ‖𝑦‖𝜔ℓ𝑞 𝑝 ). thus ‖ ‖𝜔ℓ𝑞 𝑝 is a quasinorm and (𝜔ℓ𝑞 𝑝 , ‖ ‖𝜔ℓ𝑞 𝑝 ) is a quasinormed space.∎ theorem if 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 then ‖𝑥‖𝜔ℓ𝑞 𝑝1 ≤ ‖𝑥‖𝜔ℓ𝑞 𝑝2 for all 𝑥 ∈ 𝜔ℓ𝑞 𝑝2 . proof : from the definition, for all 𝑥 ∈ ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 and any 𝑁 ∈ ℕ. |𝑆𝑁 | 1 𝑞 − 1 𝑝2 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝2 ≤ sup 𝑁∈ℕ,𝛾>0 |𝑆𝑁 | 1 𝑞 − 1 𝑝2 𝛾|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝛾}| 1/𝑝2 = ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 . and it is equivalent to 𝛾 ≤ |𝑆𝑁| 1 𝑝2 − 1 𝑞 |{𝑘∈𝑆𝑁∶|𝑥𝑘|>𝛾}| 1/𝑝2 ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 |𝑆𝑁 | 1 𝑞 − 1 𝑝1 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝1 ≤ |𝑆𝑁 | 1 𝑝2 − 1 𝑝1 |{𝑘∈𝑆𝑁 ∶|𝑥𝑘|>𝛾}| 1 𝑝2 − 1 𝑝1 ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 |𝑆𝑁 | 1 𝑞 − 1 𝑝1 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}| 1/𝑝1 ≤ ( |{𝑘∈𝑆𝑁∶|𝑥𝑘|>𝛾}| |𝑆𝑁| ) 1 𝑝1 − 1 𝑝2 ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 ≤ ‖𝑥‖ 𝜔ℓ𝑞 𝑝2 taking supremum over 𝑁 ∈ ℕ and 𝛾 > 0 on the above inequality, we get ‖𝑥‖ 𝜔ℓ𝑞 𝑝1 ≤ ‖𝑥‖𝜔ℓ𝑞 𝑝2 . ∎ from the above inequality we have 𝜔ℓ𝑞 𝑝2 ⊆ 𝜔ℓ𝑞 𝑝1 . 4. conclusions based on the results and discussion, we obtained some conclusions: 1) there are some morrey sequence space’s elementary properties : i. if 1 ≤ 𝑝 ≤ 𝑞 < ∞ then for all 𝑥 ∈ ℓ𝑝, we have ‖𝑥‖ ℓ𝑞 𝑝 ≤ ‖𝑥‖ℓp and ℓ 𝑝 ⊂ ℓ𝑞 𝑝 . ii. if 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ then for all 𝑥 ∈ ℓ𝑞 𝑝2 , we have ‖𝑥‖ ℓ𝑞 𝑝1 ≤ ‖𝑥‖ ℓq p2 and ℓ𝑞 𝑝2 ⊆ ℓ𝑞 𝑝1 . 2) there are some weak type morrey sequence space’s elementary properties : i. if 1 ≤ p ≤ q < ∞ then for all x ∈ ℓq p , we have ‖x‖ωℓq p ≤ ‖x‖ℓq p and ℓq p ⊆ ωℓq p . ii. if 1 ≤ p ≤ q < ∞ then ‖ ‖ωℓq p is a quasinorm and (ωℓq p , ‖ ‖ωℓq p ) is a quasinormed space. iii. if 1 ≤ p1 ≤ p2 ≤ q < ∞ then for all x ∈ ωℓq p2, we have ‖x‖ ωℓq p1 ≤ ‖x‖ωℓq p2 and 𝜔ℓ𝑞 𝑝2 ⊆ 𝜔ℓ𝑞 𝑝1 . jurnal matematika mantik vol. 8, no. 1, june 2022, pp.36-44 44 3) based on the results in points 1 and 2 it can be concluded that ℓ𝑝 sequence space is a subset of morrey sequence space and morrey sequence space is a subset of weak type morrey sequence space or equivalent to ℓ𝑝 ⊂ ℓ𝑞 𝑝 ⊆ 𝜔ℓ𝑞 𝑝 . references [1] y. sawano, h. gunawan, v. guliyev, and h. tanaka, “morrey spaces and related function spaces,” journal of function spaces, vol. 2014. pp. 1–2, 2014, doi: 10.1155/2014/867192. 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[15] a. a. masta, s. fatimah, and m. taqiyuddin, “third version of weak orlicz– morrey spaces and its inclusion properties,” indones. j. sci. technol., vol. 4, no. 2, pp. 257–262, 2019, doi: 10.17509/ijost.v4i2.18182. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm contact: dian c. rini novitasari, diancrini@uinsby.ac.id department of mathematics, uin sunan ampel surabaya, surabaya 60237, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.74-85 image x-ray classification for covid-19 detection using glcm-elm vivin umrotul m. maksum1, dian c. rini novitasari2*, abdulloh hamid3 1,2,3department of mathematics, uin sunan ampel surabaya, surabaya, indonesia article history: received feb 10, 2019 revised may 21, 2021 accepted jun 3, 2021 kata kunci: covid-19, citra xray, cad, glcm, elm abstrak. covid-19 merupakan penyakit atau virus yang baru-baru ini menyebar hampir di seluruh dunia. penyakit ini juga telah memakan banyak korban karena virus ini terkenal mematikan. pemeriksaan dapat dilakukan menggunakan chest x-ray karena biaya yang dikeluarkan untuk chest x-ray lebih murah dibandingkan dengan tes swab dan pcr. pada penelitian ini data yang digunakan adalah data citra chest xray. citra chest x-ray dapat diidentifikasi menggunakan computeraided diagnosis system dengan memanfaatkan klasifikasi machine learning. langkah awal yang dilakukan adalah tahap preprocessing, serta ekstraksi fitur menggunakan gray level co-occurrence matrix (glcm). hasil dari proses ekstraksi fitur tersebut akan digunakan pada tahap klasifikasi. proses klasifikasi yang digunakan adalah extreme learning machine (elm). extreme learning machine (elm) merupakan salah satu jaringan saraf tiruan dengan umpan maju (feedforward) yang mana memiliki satu lapisan tersembunyi yang disebut dengan single hidden layer feedforward neural networks (slfns). pada penelitian ini hasil yang diperoleh dengan ekstraksi fitur glcm dan klasifikasi menggunakan elm menghasilkan akurasi terbaik sebesar 91.21%, sensitifitas 100%, dan spesifisitas 91% pada rotasi 135° menggunakan fungsi aktivasi sin dengan percobaan node hidden 15. keywords: covid-19, x-ray image, cad, glcm, elm abstract. covid-19 is a disease or virus that has recently spread worldwide. the disease has also taken many casualties because the virus is notoriously deadly. an examination can be carried out using a chest x-ray because it costs cheaper compared to swab and pcr tests. the data used in this study was chest x-ray image data. chest x-ray images can be identified using computer-aided diagnosis by utilizing machine learning classification. the first step was the preprocessing stage and feature extraction using the gray level co-occurrence matrix (glcm). the result of the feature extraction was then used at the classification stage. the classification process used was extreme learning machine (elm). extreme learning machine (elm) is one of the artificial neural networks with advanced feedforward which has one hidden layer called single hidden layer feedforward neural networks (slfns). the results obtained by glcm feature extraction and classification using elm achieved the best accuracy of 91.21%, the sensitivity of 100%, and the specificity of 91% at 135° rotation using linear activation function with 15 hidden nodes. how to cite: v. u. m. maksum, d. c. r. novitasari, and a. hamid,“image x-ray classification for covid19 detection using gclm-elm”, j. mat. mantik, vol. 7, no. 1, pp. 74-85, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 74-85 issn: 2527-3159 (print) 2527-3167 (online) mailto:diancrini@uinsby.ac.id https://doi.org/10.15642/mantik.2021.7.1.74-85 http://u.lipi.go.id/1458103791 vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm 75 1. introduction the disease discussed by people around the world that was first discovered in wuhan, hubei province, china which was reported to the world health organization (who) on december 31, 2019 is an outbreak of acute respiratory infection (pneumonia) caused by the new corona virus severe acute respiratory syndrome coronavirus 2 (sars-cov-2). on february 12, 2020, who officially declared the disease coronavirus disease 2019 (covid-19). covid-19 is an acute infectious disease that attacks the respiratory system [1-2]. the most common symptoms of patients infected with covid-19 are fever, myalgia or fatigue, dry cough, headache, hemoptysis, diarrhea, and the patients slowly experience severe shortness of breath [3-4]. covid-19 has a very fast growth rate but can be suppressed to balance medical care capabilities. thus the death rate can be lowered [5]. several ways have been carried out to reduce the increased rate of people with covid19 diseases rapidly rising. one way is by swab or polymerase chain reaction (pcr) tests [6]. however, both of the tests require considerable time and cost. therefore, the latest way to detect covid-19 disease is by using x-ray. x-rays are considered capable of describing the condition of patients infected with covid-19 and can also be a clinical diagnostic tool [7]. the first step in detecting covid-19 based on x-ray images is pre-processing image data [8]. in the pre-processing stage, feature extraction from x-ray images is carried out using the gray level co-occurrence matrix (glcm) method. glcm is a second-order statistical method that calculates the relationship between two pixels in an image with degrees of grey. in previous research studies, glcm has a good performance in extracting image features, and it was implemented to extract colposcopy image features before carrying out the classification process. the best accuracy results in the classification process are 95%, which shows that the features extracted using the glcm algorithm can represent images well [9]. the classification process can be done by several methods, one of which is the extreme learning machine (elm) method. elm is a feedforward neural network with a hidden layer with hidden node parameters randomly generated and the weights calculated analytically [10]. the classification using elm has good performance with a relatively short computation time [11]. based on previous research and the above explanation, this study classifies x-ray images using the glcm and elm algorithms as feature extraction and classification to detect the presence of covid-19. this research is expected to be implemented so that patients exposed to covid-19 receive immediate and appropriate treatment. 2. preliminaries 2.1 corona virus disease 2019 (covid-19) corona virus disease 2019 (covid-19) is an infectious disease caused by a new virus and is known as severe acute respiratory syndrome coronavirus 2 (sars-cov-2) [12]. the disease attacks on the chest so it is almost similar to acute pneumonia [13]. the virus was first discovered in china on december 31, 2019 and on february 12, 2020, the world health organization (who) officially declared the disease covid-19 [14]. covid-19 is an animal-borne disease, but it can spread to humans. the most common symptoms of covid-19 are fever, fatigue, and dry cough [15]. some people may experience pain, nasal congestion, runny nose, headache, phlegm cough, sore throat, shortness of breath, and diarrhea [16]. these symptoms usually occur gradually, but some patients infected with covid-19 do not show these symptoms and do not feel any problems with their bodies. people who are susceptible to covid-19 are those who have a history of diseases related to the heart and blood vessels. covid-19 cannot be seen only with the naked eye, but the disease can be seen using chest x-ray. since covid-19 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 74-85 76 attacks the epithelial cells lining the respiratory tract, it can be analyzed using chest x-ray [17]. the following are the results of the covid-19 chest x-ray, as shown in figure 1. figure 1. covid-19 chest x-ray 2.2 gray level co-occurrence matrix (glcm) gray level co-occurrence matrix (glcm) is a very popular feature-based feature extraction method [18-19]. glcm determines the texture relationship between pixels that have the same degree of grayness so that statistical features or features are obtained in an image [20]. glcm is stored in the form of an 𝑖 × 𝑗 × 𝑛 matrix, where n is a glcm number of different rotation directions [21]. the rotation direction is formed by four shifting directions of 0°,45°,90° and 135°. there are several extraction features commonly used for glcm extraction as shown from equation(1) to equation (4) [22] : a. contrast contrast is a variation of the local intensity value in the texture matrix. the lower texture contrast means the neighboring pixels on the matrix have similar local intensity values and vice versa. the following is an equation to calculate the value of the contrast. 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = ∑ ∑ 𝑃𝑎,𝑏 (𝑎 − 𝑏) 2 𝑏𝑎 (1) where 𝑃𝑎,𝑏 represents the pixel values in row a and column b in the co-currency matrix. b. homogeneity homogeneity is the degree of homogeneity of the repetition of textures. 𝐻𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑖𝑡𝑦 = ∑ ∑( 𝑃𝑎,𝑏 1 + (𝑎 − 𝑏)2 ) 𝑏𝑎 (2) c. energy energy is a measure of the uniformity of texture. the high value of energy represents a high degree of texture uniformity in an image. the following is an equation to calculate the value of energy. 𝐸𝑛𝑒𝑟𝑔𝑦 = ∑ ∑(𝑃𝑎,𝑏) 2 𝑏𝑎 (3) d. correlation correlation is a measure of the degree of linear relationship of the grayness of one pixel relative to another. 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = ∑ ∑ 𝑃𝑎,𝑏 (𝑎 − 𝜇𝑖 )(𝑏 − 𝜇𝑗 ) √𝜎𝑎 2𝜎𝑏 2𝑏𝑎 (4) where, 𝜇𝑎 : ∑ ∑ 𝑎. 𝑃𝑎,𝑏𝑏𝑎 𝜇𝑏 : ∑ ∑ 𝑏. 𝑃𝑎,𝑏𝑏𝑎 𝜎𝑎 : √∑ ∑ (𝑎 − 𝜇𝑎) 2𝑃𝑎,𝑏𝑏𝑎 𝜎𝑏 : √∑ ∑ (𝑏 − 𝜇𝑏) 2𝑃𝑎,𝑏𝑏𝑎 vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm 77 2.3 k-fold cross validation k-fold cross validation or also called rotation estimation is a technique to minimize bias by random sampling of training and testing data [23]. one technique of k-fold cross validation is to estimate how accurate a model is when executed. in addition, the technique of k-fold cross validation breaks the data into k parts of the same size. training and testing are conducted as many times as k times. the first experiment is a subset of d1 data into testing data and the other subset of data into training data, the second experiment is a subset of d2 data into testing data and a subset of d1, d3 data up to a subset of dk data into training data [24]. the illustration of the k-fold cross validation is shown in figure 2 figure 2. data partition using k-fold cross validation 2.4 extreme learning machine (elm) extreme learning machine (elm) is a feedforward neural network with one hidden layer or commonly referred to as single hidden layer feedforward neural networks (slfns) [25]. in elm algorithms, hidden node learning parameters including weight input and bias input can be randomly assigned. for network output weights can be determined analytically using simple general inverse operations [26]. the advantage of using elm is efficient training without time-consuming learning for the process. in addition, on a universal approach, elm capabilities have been successfully applied in many real-world applications, such as classification and regression issues [11]. elm has a mathematical model that differs from backpropagation with a simpler and more effective model [27]. elm neural network model with neuron inputs, hidden layer neurons, and activation function 𝑔(𝑥). purpose 𝑛 = [𝑥1, 𝑥2, 𝑥3, ⋯ , 𝑥𝑛] with 𝑥𝑛 is the input value on the network, 𝐻 is a matrix connecting the input layer and hidden layer, then the matrix has a size 𝑛 × 𝑚. the determination of the values of the matrix elements is done randomly which can be formulated as follows [11]: ∑ 𝛽𝑖 𝑚 𝑖=1 𝑔(𝑤𝑖𝑥𝑗 + 𝑏𝑗 ) = 𝑌 (5) where, 𝑌 : predicted data, 𝛽𝑖 : weight that connect hidden nodes and output nodes with 𝛽𝑖 = (𝛽1, 𝛽2, ⋯ , 𝛽𝑚 ), 𝑤𝑖 : the weight that connects the hidden node to the i and the input nodes with 𝑤𝑖 = (𝑤1𝑖, 𝑤2𝑖, ⋯ , 𝑤𝑝𝑖) 𝑥𝑗 : vector input data with 𝑥𝑗 = 𝑥𝑗(𝑡−1), 𝑥𝑗(𝑡−2), ⋯ , 𝑥𝑗(𝑡−𝑝) 𝑗 : numbers 1, 2, ⋯ , 𝑚 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 74-85 78 𝑏𝑖 : bias weight to the i hidden node with 𝑏𝑖 = (𝑏1, 𝑏2, ⋯ , 𝑏𝑚) equation (5) can be converted into a matrix as equation (6) 𝐻𝛽 = 𝑌 (6) where, 𝐻 = [ 𝑔(𝑤1𝑥1 + 𝑏1) ⋯ 𝑔(𝑤𝑚𝑥1 + 𝑏𝑚) ⋮ ⋱ ⋮ 𝑔(𝑤𝑛𝑥𝑛,1 + 𝑏𝑛) ⋯ 𝑔(𝑤𝑚,𝑛𝑥𝑛 + 𝑏𝑚) ], dan 𝛽 = [ 𝛽1 ⋮ 𝛽𝑚 ] 𝛽 the matrix of the output weights and t is the matrix of the target. in elm, weights and bias values are determined randomly. so that the output weight associated with the hidden layer can be determined by equation [28]: 𝛽 = 𝐻+ 𝑌 (7) where 𝐻+ is a matrix that uses moore penrose pseudo inverse theory which has the best generalization results with fast computation time. in the elm architecture described above, each node on the input is connected to a node in the hidden layer using the activation function 𝑔(𝑤𝑚𝑥𝑛 + 𝑏𝑚 ) thus resulting in an h matrix of many nodes × lots of data [29]. 2.4 confusion matrix confusion matrix with a size of 2 × 2 associated with the classifier shows the results of classification and actual data where tp (true positive) is a class which is actually positive and predicted positive, so class labels are predicted correctly. and tn (true negative) is a class of actually negative and predicted negative, so class labels are predicted correctly. whereas, fp (false positive) is a class of actually negative but predicted positive, so class labels are predicted wrong, and fn (false negative) is a class of actually positive but predicted negative, so class labels are predicted wrong [30]. this can be seen in table 1 as follows: table 1. confusion matrix predicted value actual value true false true true positive (tp) false positive (fp) false false negative (fn) true negative (tn) from table 2.1, the level of accuracy, sensitivity, and specificity of an algorithm model can be calculated using equation 8, 9, and 10 [18] . accuracy = 𝑇𝑃+𝑇𝑁 𝑇𝑃+𝑇𝑁+𝐹𝑃+𝐹𝑁 × 100 (8) sensitivity = 𝑇𝑃 𝑇𝑃 + 𝐹𝑁 × 100 (9) specificity = 𝑇𝑁 𝑇𝑁 + 𝐹𝑃 × 100 (10) 3. research method 3.1 data this study used chest x-ray image data consisting of 1031 data derived from [31]. table 2 shows the data used in this study. vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm 79 table 2. research data details data type data format number of data covid-19 jpeg 119 non covid-19 jpeg 912 the data was taken on 03 april 2020 from [32-35]. the data increased every week so that when retrieving at different times the number of data increased. 3.2 research type this type of research is applied research. in this research report, a system is created to check whether a person is suffering from covid-19 by using elm method. figure 3 is the flowchart for the detection of covid-19. figure 3. covid-19 detection using glcm-elm flowchart 4. results and discussion experiments of k = 5 were conducted in this study using hidden nodes of 5, 10, 15, 20, 25, 30 and 35, then experiments were also conducted using several activation functions, that is sigmoid activation function, linear, sin, and radial basis. the results on the experiments using the hidden nodes of 5, 10, 15, 20, 25, 30 and 35, as well as those of the activation functions consisting of sigmoid, linear, sin and radial basis using four rotation directions of 0°,45°,90° and 135° are shown in the following table 3, table 4, table 5 and table 6. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 74-85 80 table 3. covid-19 and non covid-19 classification using glcm-elm with 0° direction hidden nodes activation function accuracy sensitivity specificity 5 sigmoid 88.35 50.00 89.10 linear 88.83 100.00 88.78 sin 88.83 66.67 89.16 radial basis 87.86 33.33 88.67 10 sigmoid 88.84 50.00 89.60 linear 87.86 100.00 88.72 sin 88.34 40.00 89.55 radial basis 88.83 50.00 90.00 15 sigmoid 91.26 85.71 91.46 linear 88.83 50.00 89.22 sin 89.86 71.42 90.5 radial basis 89.37 75.00 89.66 20 sigmoid 91.26 80.00 91.83 linear 88.83 100.00 88.78 sin 90.29 75.00 90.90 radial basis 89.80 66.67 90.86 25 sigmoid 88.84 54.55 90.77 linear 88.84 100.00 88.78 sin 87.86 42.86 89.45 radial basis 87.86 44.44 89.85 30 sigmoid 91.26 77.78 91.88 linear 88.84 66.67 89.16 sin 91.75 73.33 93.19 radial basis 90.29 64.28 92.19 35 sigmoid 91.75 81.82 92.31 linear 88.44 50.00 88.72 sin 89.81 58.82 92.59 radial basis 88.35 50.00 91.15 table 4. covid-19 and non covid-19 classification results using glcm–elm with 45° direction hidden nodes activation function accuracy sensitivity specificity 5 sigmoid 89.32 100.00 89.21 linear 89.32 75.00 89.6 sin 88.84 66.67 89.16 radial basis 88.83 100.00 88.78 10 sigmoid 87.37 34.00 89.00 linear 88.35 50.00 89.10 sin 87.86 43.00 89.45 radial basis 87.37 34.00 89.00 15 sigmoid 90.34 75.00 90.95 linear 88.40 50.00 88.78 sin 90.82 77.78 91.41 radial basis 91.30 80.00 91.87 20 sigmoid 92.72 83.33 93.30 linear 89.81 100.00 89.71 sin 92.23 76.92 93.23 radial basis 91.75 80.00 92.35 25 sigmoid 90.82 66.67 92.70 linear 89.37 100.00 89.27 sin 92.27 75.00 93.72 radial basis 91.30 71.43 92.75 vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm 81 table 4. covid-19 and non covid-19 classification results using glcm–elm with 45° direction (continued) hidden nodes activation function accuracy sensitivity specificity 30 sigmoid 90.78 72.72 91.79 linear 89.32 100.00 89.22 sin 90.78 72.73 91.79 radial basis 90.29 70.00 91.32 35 sigmoid 90.78 72.73 91.80 linear 89.32 100.00 89.22 sin 90.78 72.73 91.79 radial basis 90.29 70.00 91.33 table 5. covid-19 and non covid-19 classification results using glcm-elm 90° direction hidden nodes activation function accuracy sensitivity specificity 5 sigmoid 89.80 100.00 89.65 linear 89.32 75.00 89.60 sin 88.40 50.00 88.78 radial basis 90.24 100.00 90.00 10 sigmoid 89.32 66.67 90.00 linear 90.29 100.00 90.00 sin 90.29 75.00 90.90 radial basis 88.40 50.00 88.78 15 sigmoid 91.75 88.9 91.88 linear 89.32 100.00 89.21 sin 90.29 83.40 90.50 radial basis 90.30 75.00 90.90 20 sigmoid 89.81 80.00 90.04 linear 89.32 100.00 89.22 sin 89.81 80.00 90.04 radial basis 89.81 80.00 90.04 25 sigmoid 89.80 63.64 91.28 linear 88.84 100.00 88.78 sin 89.32 58.33 91.24 radial basis 89.32 62.50 90.90 30 sigmoid 88.84 54.55 90.78 linear 88.35 50.00 88.73 sin 88.84 55.56 90.36 radial basis 88.84 54.55 90.77 35 sigmoid 92.72 80.00 93.72 linear 88.84 66.68 89.16 sin 93.20 85.71 93.75 radial basis 92.72 80.00 93.72 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 74-85 82 table 6. covid-19 and non covid-19 classification results using glcm-elm with 135° direction hidden nodes activation function accuracy sensitivity specificity 5 sigmoid 88.89 100.00 88.84 linear 89.37 100.00 89.27 sin 88.83 100.00 88.89 radial basis 89.32 75.00 89.60 10 sigmoid 88.83 57.14 89.95 linear 88.35 50.00 89.10 sin 89.32 100.00 89.21 radial basis 89.80 100.00 89.65 15 sigmoid 90.73 100.00 90.54 linear 89.75 100.00 89.66 sin 91.21 100.00 91.00 radial basis 89.80 66.67 90.86 20 sigmoid 92.23 81.82 92.82 linear 89.32 100.00 89.27 sin 92.23 81.82 92.82 radial basis 90.78 70.00 91.84 25 sigmoid 89.86 58.82 92.63 linear 88.89 60.00 89.60 sin 91.30 71.43 92.75 radial basis 90.82 64.71 93.16 30 sigmoid 90.29 62.50 92.63 linear 87.38 33.33 89.00 sin 90.78 66.67 92.67 radial basis 91.75 76.92 92.75 35 sigmoid 90.78 77.78 91.37 linear 89.32 100.00 89.22 sin 90.78 77.78 91.37 radial basis 90.78 77.78 91.37 the best results are seen from the values of sensitivity, accuracy, and specificity. in this covid-19 detection system, the best sensitivity value is to avoid misdiagnosis that patients with covid-19 disease are detected incorrectly as non covid-19. table 3 represents the test results at 0°, using the linear activation function and 25 hidden nodes, and give the sensitivity, accuracy, and specificity values are 100%, 88.84%, and 88.78%, respectively. table 4 represents the test results at 45°, using the linear activation function and 20 hidden nodes, and give sensitivity, accuracy, and specificity values are 100%, 89.81%, and 89.71%, respectively. glcm feature extraction with an angle of 90° using linear activation function and 10 hidden nodes obtained sensitivity, accuracy, and specificity values of 100%, 90.29%, and 90%. meanwhile, in table 6, the feature extraction at an angle of 135° with 15 hidden nodes and the activation function sin obtained the best sensitivity, accuracy, and specificity results of 100%, 91.21%, and 91%. the best results from each trial, based on the feature extraction direction, were obtained using the activation function parameters sin, 10 hidden nodes, and the feature extraction direction of 135°, namely with sensitivity, accuracy, and specificity values of 100%, 91.21%, and 91%. future research is expected to increase the sensitivity and accuracy with other feature extraction methods such as gray level run-length matrix (glrlm). in a study conducted by [34], glrlm had better accuracy quality than the glcm method. the future work is by applying elm development methods, namely the kernel extreme learning machine (k-elm) [35] and multi-layer extreme learning machine (mllem) [36]. vivin umrotul m. maksum, dian c. rini novitasari, abdulloh hamid image x-ray classification for covid-19 detection using gclm-elm 83 5. conclusions the feature extraction process using glcm method can analyze the images of covid-19 and non covid-19. based on the extraction results of 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[36] s. ding, n. zhang, x. xu, l. guo, and j. zhang, “deep extreme learning machine and its application in eeg classification,” math. probl. eng., vol. 2015, 2015. contact: evita purnaningrum, purnaningrum@unipasby.ac.id department of management, universitas pgri adi buana surabaya, surabaya, jawa timur 60234, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.20-30 optimization of stock portfolios using goal programming based on the kalman-filter method fauziyah1, evita purnaningrum2* 1department of accounting, universitas pgri adi buana surabaya, surabaya, indonesia 2 department of management, universitas pgri adi buana surabaya, surabaya, indonesia article history: received oct 3, 2020 revised may 8, 2021 accepted may 30, 2021 kata kunci: harga saham, portofolio, goal programming, kalman filter, estimasi abstrak. pengembangan investasi saham jangka panjang dilakukan dengan cara optimasi portofolio. pemilihan saham untuk portofolio bukan saja berdasarkan harga saham yang bernilai tinggi tetapi juga memperhatikan fluktuasinya. estimasi fluktuasi harga saham di masa yang akan datang secara tidak langsung memberikan dampak bagi pembentukan portofolio yang akan datang. penelitian ini telah mengimplementasikan metode kalman filter untuk memperoleh hasil estimasi terbaik dari berbagai harga saham dengan tingkat akurasi yang tinggi. hasil tersebut selanjutnya digunakan untuk membentuk portofolio saham dengan basis goal programming. penelitian ini telah membandingkan hasil optimasi dengan nilai riil harga saham. hasil yang didapat, goal programming berbasis kalman filter lebih efektif untuk memprediksi portofolio masa depan dibandingkan dengan metode goal programming dengan selisih return sebesar rp 178.039.848. hal ini menunjukkan bahwa optimasi dengan pemrograman tujuan berbasis kalman filter dapat digunakan sebagai alat untuk menentukan portofolio saham yang akan datang. keywords: stock price, portfolio, goal programming, kalman filter, estimation abstract. long-term stock investment development is carried out by means of portfolio optimization. selection of stocks for portfolios is based not only on high-value stock prices but also on their fluctuations. estimation of future stock price fluctuations has an indirect impact on future portfolio formation. this research has implemented the kalman filter method to obtain the best estimation results from various stock prices with high accuracy. the results are then used to form a stock portfolio based on goal programming. this study has compared the optimization results with the real value of stock prices. the results obtained, kalman filter-based goal programming is more effective for predicting future portfolios compared to the goal programming method with a return difference of rp. 178,039,848. this suggests that optimization with the kalman filter-based objective programming can be used as a tool to determine future stock portfolios. how to cite: fauziyah and e. purnaningrum, “optimization of stock portfolios using goal programming based on the kalman-filter method”, j. mat. mantik, vol. 7, no. 1, pp. 20-30, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 20-30 issn: 2527-3159 (print) 2527-3167 (online) mailto:purnaningrum@unipasby.ac.id https://doi.org/10.15642/mantik.2021.7.1.20-30 http://u.lipi.go.id/1458103791 fauziyah and evita purnaningrum, optimization of stock portfolios using goal programming based on the kalman-filter method 21 1. introduction optimization of the stock portfolio is needed to maximize returns and reduce risk in investing. a collection of various stock prices is used to form a portfolio, in other words, the movement of stock values affects optimization. however, future portfolio formation strategies need to be planned and measured for effectiveness and efficiency to minimize the risk level in the future. because stock prices fluctuate, stochastic techniques are needed to estimate the stock price. stock forecasting is very important in the decision-making process, especially in the financial sector. forecasting can be used to monitor future stock price movements. therefore, forecasting will provide a better basis for planning and decision making. in addition, mathematical models have been applied to solve challenging problems in certain fields. the application of mathematical models needs to be supported by effective and efficient computations. the combination of mathematical and statistical theories with modern economic theory describes the solution to problems that exist in the field of economics. there have been many problem topics raised in capital market research with discussions in various scientific fields such as statistical journals, applied mathematics economics, estimation, banking, etc. several studies that take the topic of the stock market are ranking of cement companies on the tehran stock exchange [1], value-at-risk prediction on karachi stocks using the bayesian method [2], examining exchange rate responses to changes in stock prices in oecd countries [3], comparing the performance of ols bias correction estimator with nasdaq prediction [4], combining the vector autoregressive method and wavelet transform (modwt) to determine the effect of global stock market spillover on african stocks [5], analyzing the condition of the russian stock market after the 1998 crisis [6], developing methodology to predict daily stocks by combining the three prediction models tested in istanbul [7], evaluating two models to estimate the value at risk of the return of srocoi shares in iran [8], optimization of portfolios using a polynomial objective programming model [9], the impact of a pandemic for stock prices [10] and estimation stock prices in indonesia after the pandemic [11]. this research focused on portfolio optimization on the jii stock index (jakarta islamic index). portfolio optimization was the process by which the optimal portfolio (distribution of assets) was selected according to some objective measure, with the caveat that the associated risks must also be minimized [12]. portfolio optimization was concerned with maximizing the expected return from a series of investments and minimizing the associated risks, such as stock market volatility [13]. instead, one must consider how assets affect the risk and return of the entire portfolio. based on this portfolio optimization theory, it was possible to formulate a multi-objective problem that simultaneously maximizes returns and minimizes risk [14]. although various mathematical models have been developed to optimize portfolios such as [15]–[18], there have been studies that use the development of goal programming, one of which is [19]–[23]. in this paper, we have applied the kalman filter method based on gbm (general brownian motion) to predict future stock values [24], the estimation results have been applied as the basis for forming a stock portfolio using the goal programming method [25]. in other words, this research has presented the development of a new model of the goal programming method, namely the combination of the kalman filter and goal programming. as an additional note, this portfolio has been designed based on sharia principles, namely stocks that are included in the jii (jakarta islamic index), so that investors can get the maximum portfolio without worrying about being mixed with usury and providing solutions for estimating optimal future ownership portfolios. the remainder of this paper is organized as follows: section 2 discusses the formulation of methods for obtaining optimal portfolio objectives. section 3 reports the results of the proposed model. finally, the conclusions are summarized in section 4. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 20-30 22 2. methods this research uses mathematical methods to predict sharia future stock portfolio values. the value of the portfolio is compared with portfolio values that have been optimized by the goal programming method. figure 1 shows the flow of research design: figure 1. research design the data used in this research is the closing price of monthly jii stocks from january 2012-may 2017. the data consists of 30 stocks that are included in jii stocks in the period of december 2016-may 2017. there are three new shares included in the jii calculation, namely adhi karya (adhi), aneka tambang (antm) and hanson international (myrx). table 1. sharia stocks in jakarta islamic index period of december 2016-may 2017 [26] no. the shares name code no. the shares name code 1 astra agro lestari tbk. aali 16 mitra keluarga karyasehat tbk. mika 2 adhi karya (persero) tbk. adhi 17 hanson international tbk. myrx 3 adaro energy tbk. adro 18 perusahaan gas negara (persero) tbk pgas 4 akr corporindo tbk. akra 19 tambang batubara bukit asam (persero) tbk ptba 5 aneka tambang (persero) tbk. antm 20 pp (persero) tbk ptpp 6 astra international tbk. asii 21 pakuwon jati tbk pwon 7 bumi serpong damai tbk. bsde 22 siloam international hospitals tbk. silo 8 indofood cbp sukses makmur tbk. icbp 23 semen indonesia (persero) tbk smgr 9 vale indonesia tbk. inco 24 summarecon agung, tbk smra 10 indofood sukses makmur tbk. indf 25 sawit sumbermas sarana tbk. ssms 11 indocement tunggal prakarsa tbk. intp 26 telekomunikasi indonesia (persero) tbk tlkm 12 kalbe farma tbk. klbf 27 traktor bersatu tbk untr 13 lippo karawaci tbk. lpkr 28 unilever indonesia tbk unvr 14 matahari department store tbk. lppf 29 wijaya karya (persero) tbk wika 15 pp london sumatra indonesia tbk. lsip 30 waskita karya (persero) tbk. wskt the stock opening price is used as the next stock price prediction using mathematical models, namely geometric brownian motion [27] and the kalman filter method. this method has been developed for estimation, one of which is in the research of evita et al. [28]. if xt is a stochastic process that meet the following stochastic differential equations [29]: dxt = μxtdt + σxtdwt (1) the equation is called geometric brownian motion (gbm), with μ, σ is constant, and dwt is an increment from wiener process or increment from brownian motion. the gbm equation can be applied to the stock price movement model into the following equation: fauziyah and evita purnaningrum, optimization of stock portfolios using goal programming based on the kalman-filter method 23 dst = μst dt + σstdwt (2) μ is the direction of the stock price which is usually observed for a long time. σ2 is the volatility of the st stock price, the kalman filter algorithm for stock predictions using table 2. table 1. kalman filter algorithm algorithm: kalman filter implementation initial value μ, σ, ∆t, q, r set k = 1 system model: st = (1 + μ∆t + σ(wt − wt−1))st−1 + wk observation: zt = hksk + vk predictive: state estimation ŝ (k|k − 1) = (1 + μ∆t + σ(wt − wt−1))st−1|t−1 covariance error p̂ (k|k − 1) = (1 + μ∆t + σ(wt − wt−1))pt−1|t−1(1 + μ∆t + σ(wt − wt−1)) + qk update: pre-fit residual ŷk = zk − hkŝ(k|k − 1) covariance covk = rk + hkp̂(k|k − 1)hk t kalman gain kk = p̂(k|k − 1)hk tcovk −1 update state estimate ŝ (k|k) = ŝ(k|k − 1) + kkŷk covariance estimate pk|k = (i − kkhk)pk|k−1 measurement post-fit residual ŷk|k = zk − hkŝ(k|k) set 𝑘 = 𝑘 + 1 and repeat the prediction step then the estimation results from kalman filter are used for portfolio optimization using goal programming with the following objectives and constraints [25]: table 2. constraints and objective function constraint function and objective function goal programming model maximizing all funds for investment formation of portfolios using 100% funds (assumed investment funds are 100.000.000) maximizing portfolio returns expectations of investors can get more than the minimum return value on the capital market minimizing risk the portfolio risk is as minimal as possible so the portfolio beta is assumed 0 ≤ 𝛽𝑖 ≤ 1. 30 3 3 1 30 4 4 1 100 0 100 100.000.000 i i i i i i i i y p d d and y p d d   − + = − + = + − = + − =   limited allocation of stocks assumed that the limit of each stock is a maximum of 15% of the funds 4 4 100 15% 100.000.000, where 1, 2,..., 30 i i i i y p d d i − + + + + − =  = the simulation in goal programming is that one has an investment fund of 100.000.000 rupiahs used to form a portfolio with constraints following table 3. the maximum funds owned are 100.000.000, and we assume a maximum value of each stock of 15% of the total funds. the calculation of the expected return (profits expected by investors in the future) uses the equation [30] 30 1 1 1 100 + 100.000.000 i i i y p d d − + = + = 30 2 2 1 100 ( ) 100.000.000 i i i m i e r y p d d r − + = + − = jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 20-30 24 𝐸(𝑅) = ∑ 𝑅𝑖 𝑛 𝑖=1 𝑛−1 (3) 𝐸(𝑅) is expected return, 𝑛 is number of periods of stock calculation, and 𝑅𝑖 is realized return at the i-time which is calculated from comparing the difference between 𝑛 and 𝑛 − 1 stock prices compared to the 𝑛 − 1 stock price. while beta stocks are calculated based on comparing the number of realized returns and the expected return on stock prices with market prices (stock index). stock calculation is calculated using the equation: 𝛽 = ∑ (𝑅𝑖−𝐸(𝑅))(𝑅𝑚𝑛−𝐸(𝑅𝑚)) 𝑛 𝑖=1 ∑ (𝑅𝑚𝑛−𝐸(𝑅𝑚)) 𝑛 𝑖=1 2 (4) 𝛽 is a stock beta, 𝑅𝑚𝑛 is a market return at the n-time. 𝑅𝑚 is a market return. the stock market in this study is jii. the estimated kalman filter data is used as input for the expected return and beta stock calculations. rmse (root mean square error) is calculated using the following formula 𝑅𝑀𝑆𝐸 = √ (𝑒𝑟𝑟𝑜𝑟)2 𝑛 (5) 3. results and discussion data on monthly closing prices of sharia stocks from january 2012 to may 2017 are used as data for estimation using the kalman filter method. the kalman filter method estimates 30 stock prices in jii. investors can utilize this because it only requires one mathematical model, unlike other time series methods where each stock index requires one model. this is evidenced by calculating the value of rmse (root mean square error) from the estimated kalman filter results of 30 sharia stocks showing a relatively small value. the rmse (equation (5)) result diagram of each stock can be seen in figure 2. figure 2. percentage of rmse for thirty islamic stocks the rmse results showed approaches to 0 or very small, that’s means the model has good performance if used as a stock price prediction model. the average rmse of stocks at jii is 0,0552 or equal to 5,52%, this is smaller than the estimation results from the research conducted by previous researchers. in other words, the geometric brownian mathematical model with the kalman filter approach can predict the value of stock prices on stock indexes at one time. this is beneficial for investors, especially sharia stocks as an effective way to estimate stock prices. figure 2 shows that 77% of stocks have rmse value of less than 0,05, while only 23% of the rmse value is more than 0,05. investors could choose to invest in sharia stocks on stocks that have rmse of less than 0,05. after fauziyah and evita purnaningrum, optimization of stock portfolios using goal programming based on the kalman-filter method 25 getting the estimated stock price, the researchers conducted an analysis of the portfolio to make it easier for investors to calculate their portfolio and determine the amount of wealth that was purchased to obtain the optimal value. based on table 4, there is a comparison of the estimated value and the actual value of each sharia stock in the jii index in may 2017. from these results, it can be seen that the most striking comparison is in the stock price with a high value. this means if investors want to do investment planning, invest in share which has a low stock price. this reduces the risk of returning returns. table 3 comparing closing price and estimation no code close estimate no code close estimate 1 aali 14.300 14.300,04223 16 mika 2.040 2.039,99991 2 adhi 2.350 2.349,99992 17 myrx 123 122,99949 3 adro 1.520 1.520,01781 18 pgas 2.400 2.400,00175 4 akra 6.625 6.625,00702 19 ptba 2.180 2.179,99638 5 antm 775 774,99974 20 ptpp 3.130 3.129,99270 6 asii 8.750 8.749,99258 21 pwon 610 610,00256 7 bsde 1.810 1.809,99637 22 silo 10.733,90039 10.733,86004 8 icbp 8.700 8.700,01750 23 smgr 9.450 9.449,97729 9 inco 1.905 1.905,02877 24 smra 1.320 1.319,98700 10 indf 8.750 8.749,99258 25 ssms 1.790 1.790,00114 11 intp 18.500 18.500,00140 26 tlkm 4.350 4.349,99664 12 klbf 1.540 1.539,99767 27 untr 27.775 27.775,00572 13 lpkr 680 679,99912 28 unvr 46.175 46.174,93269 14 lppf 15.100 15.099,86690 29 wika 2.290 2.289,99618 15 lsip 1.525 1.524,99990 30 wskt 2.380 2.379,98815 the gap estimate data and real data can be seen in figure 3. a positive error indicates that the estimation result was smaller than the actual price. a negative error value indicates the estimation result has a value higher than the actual value. investors will get a lower return if they choose a stock that has a negative error. otherwise, a positive error means that the stock price is higher than the estimated stock. investors will get a higher return. figure 3. gap values from original data and their estimates jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 20-30 26 the estimation results are used as additional data when making stock portfolios in sharia stocks. portfolio determination uses the goal programming method which aims to minimize deviations from the objective function. the determination of the portfolio in this research assumed the investors have funds of idr 100.000.000,00 and want to invest a maximum of idr 15.000.000,00 per stock. the selection of the right stock will affect the profits and risks obtained in the future. criteria in determining stocks are maximizing funds owned by investors, maximizing return value, minimizing risk, and limiting investment in each stock. investment limitations on each stock are intended so that novice investors, especially in sharia stocks, not invest in only one stock but various stocks. this is to avoid and minimize risk because the value of stocks is volatile. we calculate the expected return and beta of the stocks of each sharia stock incorporated in jii (equation (3) and (4)). the results of the value acquisition of 𝐸(𝑅) dan 𝛽 for sharia stock can be seen in figure 4. in the graph, it could be seen that not all stocks have 𝐸(𝑅) values, but it was 𝛽 positive. the biggest 𝐸(𝑅) value is in wskt stock code at 0,033 or 3,3%, this means that investors will get an estimated return on the future at 3,3%. the average expected return of the 30 stocks in jii is 0,0079 or 0,79%. from the graph, there is a value that has a negative 𝐸(𝑅), so it is recommended that investors avoid investing in negative stocks. the value of β in each stock has a value above zero, which means that the movement of stocks is in line with the movement of the market index or jii. investors should choose stocks that have an 𝛽 value between 0 and 1, because the volatility of stocks is smaller than the index. the average 𝛽 is 0,592 or 59,2%, this indicates that if the market index increases by 1%, it means that the average islamic stock incorporated in it increases by 0,5%. figure 4. value of expected return and beta’s stocks from data estimation after determining 𝐸(𝑅) and beta stock, then enter the data into the goal programming model. the amount of expected return obtained from stocks in the portfolio has more value than the minimum capital market return which is equal to -0,087. goal programming is used to get the number of lots in each stock. comparison of expected return, beta stock, and number of lots per stock obtained from portfolio optimization can be seen in figure 5 and figure 6. fauziyah and evita purnaningrum, optimization of stock portfolios using goal programming based on the kalman-filter method 27 figure 5 amount of lot vs expected return the highest number of lots is 96 lots, namely the stocks with adro code with expected return of 0,0099 or 0,99%. 𝐸(𝑅) is 0.2% higher than the overall expected return. that is, if we invest 96 slots on adro stocks, we expect a stock return of 0,99%. if the current period of adro stocks is idr 1.520,02, that means 96 lots (1 lot = 100 stocks) that we invest in adro in the next month will get a profit of idr 145.512,23. however, there are two stocks in the portfolio, although the expected return is negative, namely stocks with the code pgas with 𝐸(𝑅) of -1,32% and smgr code of -1%. this is still a reasonable stage because the value is better than the lowest return from the capital market, equal to 8.7%. if the sum of many optimization results compared with beta stocks, the average number of lines of a lot that is obtained is a stock that has a beta value of the stock at the top of 0,50. two stocks with the most lots are adro and klbf which have beta values of 0,88 and 1,12. the beta stock is used as a reference for comparing the sensitivity of stock prices with the market index in this case, jii. if adro has a stock beta of 0,88, then if jii increases by 2%, adro experiences an increase in stock price of 1,76%. figure 6 amount of lot vs beta’s stocks the results of portfolio optimization and how many rupiahs must be invested in jii’s sharia stocks are presented in table 5. the investment limit for each stock is idr 15.000.000,00 per stock. the highest investment in adro stocks is idr 14.592.170,93, with the highest number of lots, 96 lots. while the lowest value of investment, which is on bsde stocks, only buys 1 lot from this stock, with the expected return is 0,52. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 20-30 28 table 4. portfolio results of implementation of the kalman filter-based on goal programming methods no code amount of lot total no code amount of lot total 1 aali 4 rp 5.720.016.89 12 mika 56 rp 11.423.999.51 2 adhi 1 rp 234.999.99 13 myrx 24 rp 295.198.77 3 adro 96 rp 14.592.170.93 14 pgas 7 rp 1.680.001.22 4 antm 43 rp 3.332.498.89 15 pwon 8 rp 488.002.04 5 bsde 1 rp 180.999.64 16 silo 8 rp 8.587.088.03 6 icbp 16 rp 13.920.028,00 17 smgr 5 rp 4.724.988.65 7 inco 21 rp 4.000.560,42 18 smra 2 rp 263.997.40 8 indf 8 rp 6.999.994.06 19 ssms 13 rp 2.327.001.49 9 klbf 79 rp 12.165.981.58 20 tlkm 4 rp 1.739.998.66 10 lppf 2 rp 3.019.973.38 21 untr 1 rp 2.777.500.57 11 lsip 10 rp 1.524.999.89 total rp 100.000.000,00 the amount of investment in the estimate is calculated using the estimated value of the share price and properly spend an investment fund of 100.000.000 idr. furthermore, by using the number of lots from the estimation we calculate the return of shares in the month after june 2017 to obtain a return of 100.883.039,8 idr, this shows that we get a return of 883.039,8 idr in the following month. whereas if we use goal programming only get a return of 100.705.000. this proves that the kalman filter-based goal programming has a good performance for predicting and deciding on future portfolios and can be used as a monthly plan for one year with a return difference of 178.039.848 idr. portfolio results obtained from this research are better if compared with the research that has been done previously with the use of optimization without estimation [25]. to further this research could be developed with other models such as machine learning [31] and other data usage such as google trends [10], [32], [33]. 4. conclusions based on the analysis carried out by the researcher, the mathematical goal programming method based on the kalman filter can be used as a method of future portfolio planning. this method is easy to apply to reduce anxiety for investors, especially investors who want to invest in jii stocks. if we compare it with the value of goal programming without estimation, this method distributes investment funds to only a few shares whose value is close to the maximum value of funds for each share of 15.000.000. meanwhile, if we use goal programming based on the kalman filter, this method divides the stock funds into some company. this means we get less risk if one of the shares goes down in price. hanson international tbk, which is a new stock in the december 2016 period, in both methods was chosen to be included in the portfolio. goal programming provided the largest funding at hanson international tbk (myrx) of 1.220 lots. this is quite risky considering myrx is a new stock on the jii index. besides the above, this research still needs to be improved in portfolio estimation and selection accuracy. the rmse obtained still has seven stocks that have rmse above 5%. this situation can be increased by doing an iteration at the time of estimation. for further research, it can add simulations to investment funds of 100 million but depends on users who want to invest. this method can be developed and formed by an application to make it easier for users to form future portfolios. fauziyah and evita purnaningrum, optimization of stock portfolios using goal programming based on the kalman-filter method 29 acknowledgement this research is a research grant from novice lecturers from kemenristekdikti with a letter number 041/sp2h/lt/k7/km/2018 references [1] a. mansory, n. ebrahimi, and m. ramazani, “ranking of companies considering topsis-dea approach methods (evidence from cement industry in tehran stock exchange),” pakistan j. stat. oper. res., 2014. 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[33] h. k. nafah and e. purnaningrum, “penggunaan big data melalui analisis google trends untuk mengetahui perspektif pariwisata indonesia di mata dunia,” snhrp, vol. 3, pp. 430–436, 2021. r. y. masiha, s. t. abdulazeez, and d. s. saeed investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case contact: sadeq taha abdulazeez, sadeq.taha@uod.ac departement of mathematics college of basic education, university of duhok, duhok, iraq the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.155-164 investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case rogash younis masiha 1, sadeq taha abdulazeez*2, dindar saeed saeed3 1department of statistics, van yüzüncü yıl üniversitesi, van, turkey 2department of mathematics, college of basic education, university of duhok, duhok, iraq 3department of mathematics, van yüzüncü yıl üniversitesi, van, turkey article history: received, may 5, 2021 revised, oct 17, 2021 accepted, oct 31, 2021 kata kunci: ardl, kointegrasi, kausalitas, pertumbuhan ekonomi, sumber daya energi. abstrak. energi memberikan peran yang sangat penting bagi perekonomian. dalam konteks ini, banyak negara telah melakukan penelitian untuk meneliti bagaimana energi mempengaruhi ekonomi mereka. selain itu, hubungan antara energi dan pertumbuhan ekonomi merupakan indikator penting dalam memandu kebijakan ekonomi. dalam penelitian ini pengaruh produksi energi dari sumber fosil dan produksi energi dari sumber listrik tenaga air terhadap pertumbuhan ekonomi (pdb) untuk irak dianalisis dengan uji kointegrasi ardl. data irak yang digunakan dalam penelitian ini diambil dari alamat web resmi bank dunia antara tahun 1971-2018. hubungan yang signifikan dan positif telah ditemukan antara sumber daya energi yang dibahas dalam studi dan pertumbuhan ekonomi. selain itu, menurut analisis kausalitas toda-yamamoto, ditemukan hubungan kausalitas dari penggunaan sumber daya energi fosil terhadap pertumbuhan ekonomi. demikian pula, hubungan kausalitas telah ditemukan dari penggunaan sumber daya energi hidroelektrik terhadap pertumbuhan ekonomi. keywords: ardl, cointegration, causality, economic growth, energy resources. abstract. energy is an important input for economies. in this context, many countries have conducted studies to examine how energy affects their economies. in addition, the relationship between energy and economic growth is an important indicator in guiding economic policies. in this study, the effect of energy production from fossil sources (epfs) and energy production from hydroelectric sources (ephs) on economic growth (gdp) for iraq was analyzed with the ardl cointegration test. the data of iraq used in the study were taken from the official web address of the world bank and covers the years between 1971-2018. a significant and positive relationship has been found between the energy resources discussed in the study and economic growth. in addition, according to the toda-yamamoto causality analysis, a causality relationship from the use of fossil energy resources to economic growth was found. likewise, a causality relationship has been found from the use of hydroelectric energy resources to economic growth. how to cite: r. y. masiha, s. t. abdulazeez, and d. s. saeed, “investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case”, j. mat. mantik, vol. 7, no. 2, pp. 155-164, october 2021. jurnal matematika mantik vol. 7, no. 2, october 2021, pp. 155-164 issn: 2527-3159 (print) 2527-3167 (online) mailto:sadeq.taha@uod.ac http://u.lipi.go.id/1458103791 jurnal matematika mantik vol 7, no 2, october 2021, pp. 155-164 156 1. introduction access to energy is critical to people's well-being, economic development, and poverty reduction. ensuring everyone has adequate access to energy is an ongoing and increasingly important challenge to global development efforts. hence, the difficulty of keeping the balance between development and the environment requires that they have access to a sufficient amount of sustainable energy sources while having a sufficiently high standard of living. however, the environmental impact of our energy systems is also of great importance. historically and today's energy systems have been based on fossil fuels (coal, oil and natural gas). these produce carbon dioxide and other greenhouse gases which are the main drivers of global climate change. if we want to meet global climate targets and avoid the dangers of climate change, the world needs to take a deep-rooted and global review of its energy resources. hydroelectric energy means the production of electrical energy by using flowing water. hydroelectric energy, such as geothermal energy, biomass energy, wind energy and solar energy, is also a renewable, sustainable energy source. in this context, hydroelectric power plants called hepps are established. electricity can be generated at all times thanks to the continuous flow of water. the amount of energy produced varies according to the strength of the flowing water. the higher the water flowing from the river, the greater the amount of energy produced. expressed as environmentally friendly during their activities, hepps can cause great damage to the environment during the construction process. during the construction phase, the stream to be built on it is drained in another direction with canals, and during this process, damage to the surrounding forests is in question. the life of the creatures living in the stream built on it is intervened, in this case, it causes the death of those creatures. in addition, although the hepp causes serious damage to the area where it is located, it is a very costly system. the aim of this study is to determine the effect of fossil and hydroelectric energy production on economic growth in iraq with the ardl boundary test. the relationship between economic growth and energy consumption has been tested using different methods in the literature. the first of these methods; are studies based on production function. however, the weak point of the production function-based studies is that while growth encourages energy use due to the loud correlation between energy consumption and economic growth, it points to the conclusion that energy use may not be necessary for economic growth [1]. one of the other methods used is causality analysis. the first study in which the method was used in [2]. in this study, the relationship between energy consumption and gdp was investigated by using sim's causality test for the usa between 1950-1970. in the study, it was decided that there is a one-way positive causality from gross domestic product to energy consumption. the results obtained in the studies in which the relationship between production factors were tested also differ. while brendt and wood in [3] concluded that there is a substitution relationship between these two production factors in econometric studies on the determination of the relationship between energy and capital, the author in [4] results differ in terms of cross-section and time series in parallel with the series used, but in the long run energy and they concluded that there is a substitution relationship between the capital and a more complementary relationship in the short run. the authors in [5] studied, between the years 1960-2003 industry for turkey, and total residential energy consumption, industrial added value and annual real gdp data was used, cointegration and granger causality tests were conducted. as a result of the study, it was concluded that there is a neutral relationship between total energy consumption and real gdp and industrial energy consumption and industrial value added. r. y. masiha, s. t. abdulazeez, and d. s. saeed investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case 157 the aim of this study is to examine the relationship between economic growth and energy produced from fossil fuels and hydroelectricity for iraq with ardl bounds test. for this purpose, after the necessary data were obtained, analyzes were made and the results were examined in detail. the materials and methods used for the study and the results are given in detail below. 2. methods in practice, the effect of energy production from fossil sources (epfs) and energy production from hydroelectric sources (ephs) on economic growth (gdp) for iraq was analyzed with the ardl cointegration test. this data set belonging to iraq and between the years 1971-2018 was obtained from the internet address of the world bank and the necessary analyzes were made with eviews 9 package program. the methods used in the analysis are detailed under the headings below. 2.1. stationary tests in time series if the mean and variance do not change over time in a time series, it is accepted as stationary. if a time series satisfies the stationary condition, it is stated that in the long run, this time series fluctuates around the average and tends to return to the average. in cases where the effect of a one’s unit shock applied to the series is temporary, series that are stationary tend to return to the mean [6]. the presence of unit root in variables means that the series cannot be stationary. it was determined that the analyzes performed with non-stationary data did not give reliable results and caused a relationship called spurious regression [7]. in order to solve the pseudo-regression problem, it is one of the many different methods recommended to regress these differences by taking the differences of these series instead of these non-stationary series [8]. in this study, using augmented dickey-fuller (adf) and phillips-perron (pp) unit root tests, it has been tried to determine whether there is unit root in the series. 2.2. augmented dickey fuller unit root test (adf) autocorrelation problem was ignored in the unit root test developed in [9]. later, dickey and fuller in [10], the unit root test assumed that the error terms in the model were autocorrelated, and the lagged terms of the dependent variable were included in the model to solve the autocorrelation problem. in [20] the relationship between petroleum price and real exchange rate was examined by adf unit root test, johansen-juselius cointegration test and granger causality analysis. the authors in [10] used the critical values they developed for the unit root test in the unit root test (adf), which they expanded on. they used criteria such as the schwarz information criterion (sic) or the akaike information criterion (aic) to decide the appropriate number of delayed terms in the extended test. while aic gives stronger results in finite samples, sic gives more reliable results in large samples. in order to overcome the autocorrelation problem, equations with ar (p) process have been developed in the adf unit root test. 𝑌𝑡 = 𝑌𝑡−1 + 𝑍𝑡 𝑡 = 2,3, … 𝑛 (1) 𝑍𝑡 = 𝜃1𝑍𝑡−1 + 𝜃2𝑍𝑡−2+. . . +𝜃𝑝𝑍𝑡−𝑝 + 𝜀𝑡 (2) 𝑌𝑡 = 𝜌𝑌𝑡−1 + ∑ 𝜃𝑖 (𝑌𝑡−𝑖 − 𝑌𝑡−1−𝑖 ) + 𝜀𝑖 𝑝 𝑖=1 (3) 𝑌𝑡 = 𝛼 + 𝛽 [𝑡 − 1 2 (𝑛 − 𝑝 + 1)] + 𝜌𝑌𝑡+𝑝−1 + ∑ 𝜃𝑖 𝑍𝑡+𝑝−𝑖 + 𝜀𝑡+𝑝 𝑝 𝑖=1 (4) h0: 𝜌 =1 or δ=0 (the series unit contains roots, so the series is not stationary) h1: 𝜌 <1 or δ<0 (the serial unit does not contain a root, series is stationary). jurnal matematika mantik vol 7, no 2, october 2021, pp. 155-164 158 2.3. phillips perron unit root test phillips and perron in [11] introduced a non-parametric test that corrects the autocorrelation between error terms. in this non-parametric test, models are created using the autoregressive-moving average process (arma). phillips and perron (1988) is a unit root test developed against the weakness of df and adf tests in the stationary analysis of time series. this test gives stronger results than df and adf unit root tests in the stationary analysis of trend time series. phillips perron test is shown by equation (5) or (6). 𝑦𝑡 = �̂� + �̂�𝑦𝑡−1 + 𝜀�̂� (5) 𝑦𝑡 = �̃� + �̃� (𝑡 − 1 2 𝑇) + �̃�𝑦𝑡−1 + 𝜀�̃� (6) here, t is the number of observations, 𝜀 is the error term, and 𝜇, 𝛼 and 𝛽 are the least squares regression coefficients. h0: 𝜌=1 or δ=0 (the series unit contains roots, so the series is not stationary) h1: 𝜌<1 or δ<0 (the serial unit does not contain a root, series is stationary). 2.4. cointegration test the number of studies investigating the possible relationships between economic time series has been increasing in recent years. cointegration analysis is used to reveal these relationships. these analyzes are widely used in econometrics and form the basis of time series analysis. cointegration analysis method was developed by granger and engle [12,13]. it has been widely used since its development and has become very popular today. the authors in [13] demonstrated that an analysis with non-stationary time series may not reflect the real relationship, in other words, the relationship may be false. the existence of a long-term relationship between variables and their common stochastic trend is defined as cointegration. it is stated that in such a situation, they cannot act independently from each other [14]. 2.5. the distributed delay autoregressive model (ardl) boundary test approach one of the important advantages of the ardl approach is that some variables become level stationary i (0) and value variables become stationary i (1) at the first difference, in other words, variables that are differently integrated are used to test whether they are integrated in the long run. the ardl approach based on the least squares (ols) method, which peseran and shin [15] and peseran, shin and smith [16] have introduced to the literature, is used to explain the dynamic (autoregressive) relationship structure between variables. in the regression analysis using time series, if the model includes not only the current values of the independent variables but also the delayed values, this model is called the distributed lag model. if the model contains one or more delayed values of the dependent variable among its independent variables, this model is called a cascading model. these two models are shown by equations (7) and (8), respectively. 𝑌𝑡 = 𝛼 + 𝛽0𝑋𝑡 + 𝛽1𝑋𝑡−1 + 𝛽2𝑋𝑡−2 + 𝜀𝑡 (7) 𝑌𝑡 = 𝛼 + 𝛽𝑋𝑡 + 𝑦𝑌𝑡−1 + 𝜀𝑡 (8) hypotheses for determining the cointegration relationship in ardl bounds test approach: 𝐻0: 𝑦1 = 𝑦2 =. . . = 0, there is no cointegration relationship, 𝐻1: 𝑦1 ≠ 𝑦2 ≠. . . ≠ 0, there is a cointegration relationship. thus, when the calculated f statistic is greater than the upper bound critical value, the 𝐻0 hypothesis is rejected and it is said that there is cointegration between the variables, while it can be said that there is no cointegration between the variables by accepting the 𝐻0hypothesis when the lower bound is less than the critical value. if the r. y. masiha, s. t. abdulazeez, and d. s. saeed investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case 159 calculated f statistic is between the lower and upper bound critical values, a decision cannot be taken about cointegration. 2.6. toda-yamamoto causality test the granger causality test is generally used in empirical studies in the literature. however, in order to use this test, the series must be stationary or integrated in the same degree. but it is possible to come up with causality between degree-integrated series as well. the reason for choosing the toda-yamamoto test is that the variables in the model are not required to be stationary to the same degree. in the analysis of toda – yamamoto [17] the standard vector autoregressive model (var) was first established based on the levels of the series, regardless of their degree of integration. in the following steps, the degree of the var model is artificially changed by adding the actual degree (k) to the maximum degree of integration (dmax) (k + dmax). however, the coefficients of the terms added to the model later are ignored. in this causality procedure, there is a condition that the maximum degree of integration (dmax) should not exceed the true degree (k) of the var model [18]. in this respect, it can be stated that the todayamamoto causality test gives more consistent results compared to the granger causality test. causality analysis developed by toda and yamamoto [17] shows an improvement over the granger test, which uses a standard asymptotic distribution test statistic on standard granger causality analysis [19]. 3. analysis and findings adf and pp unit root tests were used to analyze the stationary of the variables, the analysis results of adf are given in table 1 and the analysis results for pp are given in table 2. table 1. adf unit root test results variables i(0) i(1) constant constant+trend constant constant+trend t-bar p-value t-bar p-value t-bar p-value t-bar p-value gdp -2.134 0.004 -3.201 0.005 epfs -2.370 0.230 -3.416 0.396 -2.251 0.001 -3.203 0.001 ephs -2.478 0.413 -3.378 0.401 -2.301 0.001 -3.004 0.020 p<0.05 adf unit root test results for both fixed and constant + trend models by taking the level (i (0)) and first order differences of the variables (i (1)) are given in table 1. when table 1 is examined, it can be said that the 𝐻0 hypothesis is accepted for the epfs and ephs series in both the fixed and the constant + trend model at the level (p> 0.05) and thus these variables are not stationary at the 5% significance level, that is, they contain unit root. for gdp, it is seen that the series is stationary in level (p <0.05). for gdp, the 𝐻0 hypothesis is rejected and the series does not contain a unit root. in order to stabilize the non-stationary epfs and ephs series, their first order differences are taken. after taking the first order differences, the variables were analyzed according to both the fixed and the fixed + trend model. 𝐻0 hypothesis was rejected for epfs and ephs variables in both models (p <0.05). thus, after taking the first-order differences of these variables, it can be said that at the 5% significance level, they are stabilized, that is, they do not contain unit roots. jurnal matematika mantik vol 7, no 2, october 2021, pp. 155-164 160 table 2. pp unit root test results variables i(0) i(1) constant constant+trend constant constant+trend t-bar p-value t-bar p-value t-bar p-value t-bar p-value gdp -2.214 0.001 -3.127 0.002 epfs -2.235 0.410 -3.631 0.381 -2.617 0.001 -3.410 0.001 ephs -2.271 0.398 -3.701 0.390 -2.304 0.001 -3.301 0.002 p<0.05 pp unit root test results for both fixed and constant + trend models by taking the level (i (0)) and first order differences of the variables (i (1)) are given in table 2. table 2 shows the t statistics and probability values calculated separately for three variables whose level and first-order differences are taken using both fixed and fixed + trend models. according to table 2, the 𝐻0 hypothesis is accepted for both models at the level (p> 0.05) and thus, it can be said that epfs and ephs are not stationary at the 5% significance level, that is, they contain unit root. for gdp, it is seen that the series is stationary in level (p <0.05). for gdp, the 𝐻0 hypothesis is rejected and the series does not contain a unit root. necessary analyzes were made by taking the first order differences of the non-stationary epfs and ephs series and these series were made stationary. according to the analysis results, the 𝐻0 hypothesis was rejected for both the fixed and the fixed + trend model for the variables of epfs and ephs (p <0.05). thus, after taking the first order differences of these variables, it can be said that they are stationary at the 5% significance level and according to the pp unit root test, that is, they do not contain unit roots. the use of ardl cointegration analysis is more suitable for cointegration analysis due to the fact that the variables discussed in the study are stationary at different levels, as stated in the literature. table 3. ardl cointegration bound test number of independent variables (k) f statistic value significance level critical values lower limit upper limit 2 19.4256 1% 1.83 2.903 5% 2.13 3.59 10% 2.61 3.63 whether there is cointegration between variables at 1%, 5% and 10% significance level is shown in table 3. as seen in table 3, the calculated f statistic value is greater than the upper limit critical value at the 5% significance level. therefore, it is determined that there is cointegration between variables by accepting the 𝐻1 hypothesis. after determining a long-term relationship between variables with the f test, the parameters of this relationship were estimated with the ardl model based on the least squares (ols) method and the results are given in table 4. table 4. ardl (1, 2, 2) model variables coefficients standard error tstatistic probability value (p) constant (c) 0.017893 0.001478 6.571362 0.031 gdp (-1) 0.103647 0.078312 -2.741693 0.013 epfs (-1) 0.136402 0.032147 -2.317895 0.025 epfs (-2) 0.112365 0.036415 -2.017852 0.031 ephs (-1) 0.127853 0.063219 -2.378901 0.019 ephs (-2) 0.107524 0.001436 -2.368710 0.021 p<0.05 r. y. masiha, s. t. abdulazeez, and d. s. saeed investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case 161 table 4 gives the values of the variables in the ardl (1, 2, 2) model. as seen in table 4, t all three variables have significant and positive coefficients (p <0.05). one of the important elements that should not be ignored in the analyzes made with the ardl model is the basic assumptions of the ekk. the results of the basic assumptions of ekk are given in table 5. table 5. ardl diagnostic tests diagnostic tests test statistics probability value (p) 𝑅2 0.750136 adjusted 𝑅2 0.716520 fstatistics 12.143026 0.001 breush-godfrey lm test 0.540367 0.348 arch test 2.390172 0.281 jargue-bera normality test 0.493075 0.432 ramsey-reset test 1.801637 0.601 the basic test results for the basic assumptions of the ekk are given in table 5. the coefficient of determination (𝑅2) expressed as a percentage varies between 0 and 1 and shows how much of the variance in the dependent variable is explained by the independent variables. thus, it is seen that approximately 75% of the gdp is disclosed by epfs and ephs. if the model was generalized with the corrected 𝑅2 and obtained from the model population, approximately 71% of the variation in gdp would have been explained by epfs and ephs. the changing variance problem is tested with the breushgodfrey lm test. when the breush-godfrey lm test probability value is greater than its critical value, it is assumed that there is no variance problem. according to the breushgodfrey lm test probability value (p> 0.05) in table 5, it can be said that there is no variance problem. whether there is autocorrelation in the estimated model is determined by arch test. autocorrelation is assumed when the probability value of the arch test is greater than the critical value. according to the arch test probability value in table 5 (p> 0.05), it was determined that there was no autocorrelation. the jargue-bera normality test tests whether the errors have a normal distribution or not. when the probability value of the jargue-bera normality test is greater than the critical value, the errors are considered to have a normal distribution. according to table 5 (p> 0.05), it is observed that the errors have a normal distribution. the ramsey-reset test analyzes whether there is a model building error or not. when the ramsey-reset test probability value is greater than the critical value, it is concluded that there is no modeling error. according to the ramsey-reset test probability value in table 5 (p> 0.05), it was determined that there was no modeling error. table 6. long term ardl cointegration results variables coefficients standard error t p constant 0.108349 0.003621 5.104562 0.023 epfs 0.118632 0.027369 -2.214690 0.002 ephs 0.012470 0.014785 -2.163147 0.002 table 6 shows the values of the parameters calculated with the long-term ardl model. in this way, the state of the long-term relationship between variables can be determined. in the study, gdp shows the dependent variable, and epfs and ephs show the independent variables. according to table 6, a positive and significant (p <0.05) relationship was determined between gdp and epfs and ephs. in addition, a one-unit increase in epfs causes an increase of 0.118 units in gdp and a one-unit increase in jurnal matematika mantik vol 7, no 2, october 2021, pp. 155-164 162 ephs causes an increase of 0.012 units in gdp. thus, when comparing the effects of epfs and ephs variables on gdp, it can be said that for iraq, the effect of epfs is greater. the stability of the ardl model was investigated by determining whether there is any structural break in the variables. for this purpose, cusum and cusumq graphs that exploit backward error term squares and investigate structural breakage in variables. in cusum and cusumsq graphs, if the variables are within the critical limits, it is determined that the ardl model is stable and thus the model coefficients are stable. -12 -8 -4 0 4 8 12 06 07 08 09 10 11 12 13 14 15 16 17 18 cusum 5% significance -0.4 0.0 0.4 0.8 1.2 1.6 06 07 08 09 10 11 12 13 14 15 16 17 18 cusum of squares 5% significance figure 1. cusum and cusumq results figure 1 shows the stability of the estimated ardl model. when the cusum and cusumsq graphs were examined, it was determined that the variables were between the critical limits at the 5% significance level. thus, it was observed that there was no structural break in the variables and the long-term coefficients calculated by the ardl boundary test were stable. after performing the ardl cointegration test, toda-yamamoto causality test was used to determine the causality direction among variables. first, the appropriate lag length was determined in the var model, and then the toda-yamamoto causality test was performed. table 7. selection of the lag length of the var model lag lr fpe aic sc hq 1 41.01547* 2915.2713* 19.06270* 23.28041* 21.03715* 2 49.34782 3124.016 21.10419 27.07614 25.00143 3 51.04861 3361.179 21.50793 29.10083 25.70391 4 53.07126 3641.061 22.16820 30.00731 26.00617 as seen in table 7; lr test statistics, fpe (final prediction error), aic (akaike information criterion), sic (schwarz information criterion) and hq (hannan-quinn information criterion) statistics were obtained as 1 (*). it can be said that the series do not have varying variance, serial correlation problem, and have normal distribution, because all values provide the same optimum delay. the results of the toda-yamamoto causality test applied after determining the optimum lag length are given in detail in table 8 below. r. y. masiha, s. t. abdulazeez, and d. s. saeed investigation of the relationship between economic growth and use of fossil and hydroelectric energy resources by ardl boundary test: 1971-2018 iraq case 163 table 8. toda-yamamoto causality test results causality direction test statistics value p-probability value gdp → epfs 2.731 0.091 epfs → gdp 2.045 0.021 gdp → ephs 2.731 0.004 ephs → gdp 2.976 0.001 as seen in table 8, it is seen that gdp does not cause epfs (p>0.05) but epfs causes gdp (p<0.05). in this case, it is seen that there is a causality relationship from epfs to gdp. it is seen that gdp causes ephs (p≤0.05) in the same way that ephs causes gdp (p≤0.05). in this case, there is a bidirectional causality relationship between gdp and ephs. 4. conclusion new econometric techniques are used every day in the modeling and testing of economic theories. more realistic results are obtained by making economic analysis using these techniques. it is expected that many economic variables exhibit asymmetrical behaviors in economic theory. therefore, it is thought that the relationships between economic variables can be modeled correctly by using nonlinear methods. energy use varies according to the energy resources of countries and their state of development. it is stated in the literature that there is a tendency to use renewable energy resources especially in developed countries. in addition to the damage caused by fossilbased energy types to the environment, their depletion is among the important reasons affecting the orientation towards renewable energy sources. iraq is a country rich in oil resources. it is also in the middle level human development category according to the 2020 human development index report. in the light of this information, the fact that fossil resources are generally used in iraq in terms of energy production supports the studies in the literature. the aim of this study is to determine the effect of fossil and hydroelectric energy production on economic growth in iraq with the ardl boundary test. in the study, gdp shows the dependent variable, and epfs and ephs show the independent variables. according to long term ardl cointegration results, a positive and significant (p <0.05) relationship was determined between gdp and epfs and ephs. in addition, a one-unit increase in epfs causes an increase of 0.118 units in gdp and a one-unit increase in ephs causes an increase of 0.012 units in gdp. thus, when comparing the effects of epfs and ephs variables on gdp, it can be said that for iraq, the effect of epfs is greater. according to toda-yamamoto causality test results, gdp does not cause epfs (p>0.05) but epfs causes gdp (p<0.05). in this case, it is seen that there is a causality relationship from epfs to gdp. it is seen that gdp causes ephs (p≤0.05) in the same way that ephs causes gdp (p≤0.05). in this case, there is a bidirectional causality relationship between gdp and ephs. it is in the analysis and findings section that the study has similar results with the studies in the literature. especially in the energy consumption of iraq, it is seen that the types of energy it has have a high rate. among these energy types, fossil fuel energy use has the highest rate. seeking new energy alternatives due to the damage caused by fossil fuel energy types to the environment and their depletion will result in healthier results. new policies should be developed especially for the use of renewable energy sources and investments should be made in these energy types. the fact that the study has not been studied or studied less with this method for iraq makes the study different. in addition, we believe that this study will contribute to the literature for iraq. jurnal matematika mantik vol 7, no 2, october 2021, pp. 155-164 164 references [1] d. i. stern, “energy and growth in the usa: a multivariate approach”, energy economics, 15, 137-150, 1993. 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[19] j. squalli, “electricity consumption and economic growth: bounds and causality analyses of opec members”, energy economics, 29(6), 1192-1205, 2007. [20] d. saeed et al., “the relationship between petroleum price and real exchange rate: an example of iraq”, general letters in mathematics, 11(1) (2021), 12-17 https://doi.org/10.31559/glm2021.11.1.3 https://doi.org/10.31559/glm2021.11.1.3 yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control contact: yuyun monita, yuyunmonita@gmail.com department of mathematics, uin sunan ampel surabaya, surabaya, jawa timur 60234, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.96-106 control of the spread of tb-hiv/aids coinfection using optimal control yuyun monita 1* , putroue keumala intan 2 1,2 department of mathematics, uin sunan ampel surabaya, indonesia article history: received nov 24, 2020 revised may 25, 2021 accepted jun 30, 2021 kata kunci: koinfeksi tb-hiv/aids, pontryagin, rungekutta orde keempat. abstrak. suatu keadaan dimana seorang individu terkena penyakit tb sekaligus hiv/aids di dalam tubuhnya dinamakan koinfeksi tb-hiv/aids. penelitian ini bertujuan untuk meminimumkan populasi koinfeksi tb-hiv/aids dengan pengeluran biaya pengobatan yang minimum, hal tersebut berarti meminimumkan fungsi objektif ( ) atau fungsi tujuan. pada penelitian ini dilakukan modifikasi model dengan penambahan populasi pengobatan untuk penderita hiv dengan arv ( ). jumlah populasi yang digunakan berjumlah 11 kelas dengan penggunaan tiga buah kontrol meliputi pengobatan untuk individu dengan tb laten ( ), tb aktif ( ), dan hiv ( ). setelah dilakukan simulasi numerik menggunakan metode forward backward runge-kutta orde keempat diperoleh hasil bahwa skenario 7 menjadi skenario terbaik dalam mengendalikan penyebaran koinfeksi tb-hiv/aids karena menghasilkan nilai paling minimum yaitu sebesar 1401,44. hal tersebut berarti dengan memberikan pengobatan pada individu dengan tb laten, tb aktif, dan hiv secara bersama-sama dapat mengurangi populasi koinfeksi tb-hiv/aids dengan biaya pengobatan yang minimum. keywords: tb-hiv/aids coinfection, pontryagin, fourth order rungekutta. abstract. the condition in which an individual is affected by tb and hiv/aids in his body is called a tb-hiv/aids coinfection. this research aims to minimize the populations of tb-hiv/aids coinfection with a minimum expenditure on medical expenses, that means minimizing the objective’s function ( ) or purpose function. in this research, modification of the model was carried out by adding the treatment population for hiv patients with arv ( ). the population used was 11 classes with the use of three controls including treatment for individuals with latent tb ( ), active tb ( ), and hiv ( ). after performing numerical simulation using the forward backward fourth order runge-kutta, the results show that scenario 7 is the best scenario in controlling the spread of tbhiv/aids coinfection because it resulted a minimum value of 1401,44. this means that providing the treatment for individuals with latent tb, active tb, and hiv in tandem can reduce the populations of tb-hiv/aids coinfection in the minimum treatment cost. how to cite: y. monita and p. k. intan, “control of the spread of tb-hiv/aids coinfection using optimal control”, j. mat. mantik, vol. 7, no. 1, pp. 96-106, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control 97 1. introduction aids (acquired immune deficiency syndrome) is an infectious disease in which the human immune system is decreased or damaged by hiv (human immunodeficiency virus) infection. hiv will attack the immune system so that the body is unable to fight infections due to weak immunity [1]. the immunity of people with hiv/aids (odha) decreases because the number of cluster differential four cells decreases. the joint united nations programme on hiv and aids (unaids) stated that in 2018 there were 37,9 million people in the world living with hiv, of which 640.000 occurred in indonesia [2], [3]. odha are more likely to get opportunistic infection (oi), diseases that attacks a person because his immune system is weak [4], [5]. the spread of hiv in the body can be suppressed by performing antiretroviral (arv) therapy (art). arv can inhibit the replication process of hiv in the body [1]. hiv/aids can be transmitted through sexual intercourse, blood, pre-ejaculation, vaginal fluids semen and by mothers to babies who are conceived or breastfed [6]. one-third of odha are infected with tuberculosis (tb), even tb is the most common oi and the main cause of death in odha [7], [8]. tb is an infectious disease caused by the mycobacterium tuberculosis (mtb) or tubercle bacilli that can attack some parts of the body such as muscles, bones, joints and most commonly the lungs [9], [10]. in 2017 there were 10 million people in the world infected with tb, of which 420.000 occurred in indonesia [8], [11]. the stage of tb disease starts from latent tb to active tb. the development of latent or exposed tb into active tb can occur at any time. people with latent tb have mtb in their body but cannot transmit it to other people, while people with active tb are able to transmit it to other. tb transmission occurs through the air, namely when an active tb patient cough or sneeze, mtb will come out of the lungs in the form of droplet nuclei that can last several hours in the air and cannot be seen with the naked eye. treatment of latent tb has an important role in reducing the risk of latent tb developing into active tb [7], [9], [12]–[16]. tb-hiv coinfection is a condition in which an individual is infected by both mtb and hiv [13]. when an individual is coinfected with tb-hiv, hiv is able to accelerate activation of tb and tb can increase the rate of progression from hiv to aids [17]. therefore, an appropriate strategy is needed to control the spread of tb-hiv/aids coinfection. these efforts can be done by using mathematical modeling. mathematical modeling has an important role in analyzing the control of a disease with optimal control [12]. in general, the optimal control problem is the same as the optimization, which is the process to get the best solution in a problem. optimization in optimal control theory is a function that can optimize (max/min) performance index or objective function [18]. objective function is a target that is to be achieved from an optimal control problem [19]. several studies on controlling the spread of disease with optimal control have been carried out, among them sukokarlinda [1] has analyzed controlling the spread of hiv by applying one control in the form of giving arv drug. gao and huang [20] have analyzed the control for tb by giving the vaccine and two other treatments. bhunu,et al [21] have analyzed and performed optimal control for the spread of tb-hiv/aids coinfection by using 10 populations and 3 types of controls in the form of treatment for individuals with latent tb , active tb , and aids , each with static or constant values. bhunu et al's research has been developed by several studies including rayhan [13] and tanvi and aggarwal [22]. rayhan uses dynamic control in the form of treatments for individuals with latent tb , active tb and aids , thereby causing changes in the rate of control over time. his research aims to minimize the number of people with tbhiv/aids infected. based on these objectives, the objective function established only involves the infected tb-hiv/aids population. meanwhile, tanvi and aggarwal added the aids treatment population so that the research involved 11 populations. the methods used in both studies are pontryagin minimum principle for optimal control problems and runge-kutta (rk) namely fourth order runge-kutta (rk4) for numerical solutions. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 98 the rk4 is one of the most frequently used numerical methods for solving differential equation problems because of its small truncation errors, resulting in a good approximate solution [23]. this was supported by subhashini and srividhya [24] who stated that the rk method had a small error value in solving an equation. and puspitasari, et al [25] stated that the rk4 method is better than the adam-bashfort moulton method. in this study, the authors will develop rayhan's research by modifying the model for the spread of tb-hiv/aids coinfection in the form of increasing the class of the treatment population for individuals with hiv ( ) using arv so that the population to be used is 11 classes. this is done because arv administration can be done when the people is infected with hiv. giving arv can reduce the spread of hiv because arv can inhibit the replication process of hiv in the body so that it can reduce the progression of hiv to aids [1]. furthermore, the objective function will be expanded by involving the tb-hiv/aids exposed population, so that the objective function to be used includes the exposed to and infected with tb-hiv/aids populations. this is in accordance with the aim of this study, namely to control the spread of tb-hiv/aids coinfection by reducing the number of tb-hiv/aids coinfection people, namely those tb-hiv exposed, tb-aids exposed, tb-hiv infected or tb-aids infected. and, there will also be addition trial scenarios aimed at obtain the most optimal strategy from the new model. 2. tb-hiv/aids coinfection model the population used is 11 classes including population class of susceptible , exposed to tb , infected with tb , recovered from tb , infected with hiv , infected with aids , exposed to tb-hiv , exposed to tb-aids , infected with tb-hiv , infected with tb-aids , treatment of hiv as shown in figure 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) with the initial conditions provided: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and unspecified for ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). ( ) (2) ( ( ( ))) (3) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control 99 figure 1. modified compartment diagram of tb-hiv/aids coinfection spread it is assumed that individuals born enter the susceptible class at the rate. susceptible individuals can contract hiv due to direct contact with hiv infected individuals at the rate thus enter into the hiv infected class. hiv in the body can develop into aids at the rate thus enter into the aids infected class. hiv and aids infected individuals can each contract tb due to direct contact with active tb individuals at the and the rate thus enter into the tb-hiv exposed and tb-aids exposed class. tb-hiv exposed individuals can progress their latent tb to the active tb naturally at the rate and re-infection at the rate thus enter into the tb-hiv infected class. in addition, tb-hiv exposed individuals can progress their hiv into aids at the rate thus enter into the tb-aids exposed class. tb-aids exposed individuals can naturally progress their latent tb into active tb at the rate and re-infection at the rate thus enter into the tb-aids infected class. tb-hiv infected individuals can progress their hiv into aids at the rate thus enter into the tb-aids infected class. susceptible individuals can contract tb due to direct contact with active tb individuals at the rate thus enter into the tb exposed class. individuals exposed with tb can become naturally active tb at the rate and re-infection at the rate thus enter into the tb infected class. individuals infected with tb can recover naturally from the disease at the rate and enter into the recovered from tb class. individuals exposed to tb and infected with tb can contract hiv due to direct contact with hiv infected individuals at the rate and the rate thus enter into the tb-hiv exposed class and tb-hiv infected class, with being a modified parameter that describes active tb individuals as more susceptible to hiv compared to latent tb individuals. each population experiences death naturally at the rate. individuals infected with tb can die from the disease at the rate. individuals with aids can die from the disease at the rate. individuals infected with tb-aids can die from the disease at the rate. the model is equipped with three treatments: the treatment of latent tb individuals using chemoprophylaxis at the rate, active tb individuals using several drugs at the rate and hiv individuals using arv at the rate. hiv infected individuals who are given treatment enter into the hiv treatment class. in equation (3), is the rate of hiv infection, describes individuals at the aids stage as more susceptible to transmitting hiv compared to individuals infected with hiv without showing symptoms of aids. is a modified parameter that describes transmission by individuals at the tb-aids exposed stage greater than individuals at the aids stage. then for describes that individuals at the tbaids infected stage are easier to transmit hiv than at the tb-aids exposed stage. in equation (2), is the rate of individuals exposed to tb due to direct contact with tb infected individuals and states the average contact rate. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 100 3. research design 3.1 solve optimal control problem if defined the control vector ⃗ ( ) ( ( ) ( ) ( )) and state vector ( ) ( ( ) ( ) ( )) at the time interval , the steps taken are: [26], [27] a. shape the hamiltonian ( ( ) ( ) ( ) ) ( ( ) ( ) ) ( ) ( ( ) ( ) ). b. minimize against all ( ) using , so get optimal ( ). c. substitute ( ) into to get optimal hamiltonian function. d. solve equation of state with ̇ and co-state with ̇ . the boundary conditions are given by the initial and final condition (transverse) used is ( ) . e. substitute the result of step d. into the expression in step b. to get optimal control. 3.2 discretization and simulation ( ) ( ) ( ( ) ( ( )) ( ( )) ( )) (4) ( ) ( ) ( ( ) ( ( )) ( ( )) ( )) (5) discretization of the state as ( ) ( ( ) ( )) using forward sweep (4) and co-state as ( ) ( ( ) ( )) using backward sweep (5) with [ ]. then create a program to get a simulation with the following steps:[18], [22], [28], [29] a. divide the interval by subintervals, so that the vector state ( ), co-state ⃗⃗ ( ) and control ⃗ ( ). b. use the initial value for ( ) and the to finish the state with forward sweep. c. use the final value for ( ) and the and values resulting from step b. to complete the co-state with backward sweep. d. update the by subtituting the new and values in the ( ) characterizations. e. performing convergence checks using | | |∑ | | | . where is the new , while old is the old value and is a constant of positive real value. if the value of each convergence test in the last iteration is positive, then the iteration stops and uses that value as the optimal value. but if not, then go back to step c. 4. result and discussions 4.1 optimal control problem based on the purpose in this study, namely minimizing the spread of tb-hiv/aids coinfection, it will be minimized the number of populations of tb-hiv/aids coinfection either exposed to tb-hiv, exposed to tb-aids, infected with tb-hiv or infected with tb-aids. therefore, objective function (6) was formed which means to minimize the number of tb-hiv/aids coinfection individuals with minimum medical expenses. ∫ ( ) (6) based on the objective function (6) and the function of constraints or state equations (1), the hamiltonian function is obtained: ∑ (7) yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control 101 where m is the co-state and g is the state. optimal control and co-state equations obtained: a. optimal conditions retrieved ( ( ( ) ( ) ( ) )) retrieved ( ( ( ) ( ) ( ) )) retrieved ( ( ( ) ( ) ( ) )) b. adjoin/co-state the acquisition of adjoin equations is ̇ , where and final conditions ( ) for . 4.2 numerical simulations the data used include the values of the initial conditions as = 9080, = 2080, = 3540, = 0, = 1500, = 420, = 1095, = 325, = 137, = 29, = 10 and the parameters shown in table 1 with [ ] , and and are worth 50, 80, and 100 [13], [21], [22], [30]. 7 scenarios trial shown in table 2 will be used to obtain the best strategy in minimizing the spread of tbhiv/aids coinfection. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) figure 2. population of (a) exposed to tb, (b) infected with tb, (c) recovered from tb, (d) infected with hiv, (e) infected with aids, (f) exposed to tb-hiv, (g) exposed to tbaids, (h) infected with tb-hiv, (i) infected with tb-aids and (j) treatment of hiv. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 102 table 1. parameters values no. parameters value 1. 0,029 2. 0,02 3. 3 4. 1,1 5. 1,05 6. 1,02 7. 0,1 8. 0,102 9. 0,25 10. 1,03 11. 1,2 12. 1,2 13. 0,000113 14. 0,00017 15. 0,0002 16. 0,71 17. 1,07 18. 1,101 19. 0,1 20. 0,333 21. 0,35 22. 0,15 23. 0,2 24. 0,0001 the simulation results obtained for the tb-hiv/aids coinfection populations include, figure 2 (f) shows the exposed to tb-hiv population without and with treatment. the figure shows that each scenario graph is lower than the no-control graph, this shows that treatment in each scenario may result in a reduced the number of tb-hiv exposed individuals. the scenario that results in the least number of tb-hiv exposed individuals is scenario 7 whereby by administering treatment for individuals with latent tb, active tb and hiv can reduce the population of exposed to tb-hiv to 3 people. figure 2 (g) shows the exposed to tb-aids population without and with treatment. treatment in each scenario resulted in the number of tb-aids exposed individuals being reduced as shown in each scenario graph decreased compared to the no-control graph. the scenario that results in the least number of tb-aids exposed individuals is scenario 7 whereby by administering treatment for individuals with latent tb, active tb and hiv can reduce the population of tb-aids exposed to 1 people. figure 2 (h) shows the infected with tb-hiv population without and with treatment. the figure shows that each scenario graph is lower than the no-control graph, this shows that treatment in each scenario may result in a reduced the number of tb-hiv infected individuals. the best scenario for reducing the population of infected with tb-hiv is scenario 6 whereby by administering treatment for individuals with active tb and hiv can eliminate the tbhiv infected population. figure 2 (i) shows the infected with tb-aids population without and with treatment. treatment in each scenario resulted in the number of tbaids infected individuals being reduced as shown in each scenario graph decreased compared to the no-control graph. the best scenario for reducing the population of infected with tb-aids is scenario 7 whereby by administering treatment for individuals with latent tb, active tb and hiv can eliminate the tb-aids infected population. the simulation results obtained for the tb populations include, figure 2 (a) shows the exposed to tb population without and with treatment. the figure shows that each scenario graph is lower than the no-control graph, this shows that treatment in each yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control 103 scenario may result in a reduced the number of tb exposed individuals. the scenario that results in the least number of tb exposed individuals is scenario 4 whereby by administering treatment for individuals with latent tb and active tb can reduce the population of tb exposed to 25 people. figure 2 (b) shows the infected with active tb population without and with treatment. treatment in each scenario resulted in the number of active tb individuals being reduced as shown in each scenario graph decreased compared to the no-control graph. the scenario that results in the least number of active tb individuals is scenario 4 whereby by administering treatment for individuals with latent tb and active tb can reduce the population of active tb to 1 people. figure 2 (c) shows the recovered from tb population without and with treatment. scenario graphs 1, 4, 5 and 7 experienced an increase compared to the no-control graph, this shows that giving treatment in these four scenarios can result in the number of recovered from tb individuals increased compared to without treatment. while the scenario graphs 2, 3 and 6 experienced an decrease compared to the no-control graph. this can happen because in these three scenarios tb treatment is done only for active tb individuals without treatment for latent tb individuals, so the recovered from tb class will only get additions from active tb individuals who are cured due to . therefore, the scenario graphs 2, 3 and 6 in the tb exposed population are higher than the other four scenarios that provide treatment for tb exposed individuals as shown in figure 2 (a). the scenario that results in the largest number of recovered from tb individuals is scenario 1 whereby by treating individuals with latent tb can result in as many as 2377 people recovered from tb. the simulation results obtained for the hiv/aids populations include, figure 2 (d) shows the infected with hiv population without and with treatment. scenario graphs 3, 5, 6 and 7 decrease compared to the no-control graph, this shows that giving treatment in these four scenarios can result in a reduced number of hiv infected individuals compared to without treatment. this can happen because each of these four scenarios is given treatment for hiv infected individuals using arv, resulted in hiv treatment class with arv increased in these four scenarios. meanwhile, scenario graphs 1, 2 and 4 increased compared to the no-control graph. this indicates that the number of hiv infected individuals in scenarios 1, 2 and 4 increases compared to the absence of treatment that occurs because in these three scenarios there is no treatment for hiv infected individuals. the scenario that results in the least number of hiv infected individuals is scenario 3 whereby by administering treatment for individuals with hiv infected can reduce the population of infected with hiv to 27 people. figure 2 (e) shows the affected with aids population without and with treatment. scenario graphs 3, 5, 6 and 7 are lower than the no-control graph. this is related to the previous discussion, in which in these four scenarios experienced a decrease in the number of hiv infected individuals which resulted in a decrease in the population of individuals with aids. while in the scenario graphs 1, 2 and 4 are higher than the no-control graph. this can happen because of the increasing hiv infected population, resulting in an increase in the aids population. the scenario that results in the least number of individuals affected with aids is scenario 3 whereby by administering treatment for individuals with hiv infected can reduce the population of infected with aids to 23 people. figure 2 (j) shows the hiv treatment population without and with treatment. scenario graphs 3, 5, 6 and 7 are higher than the no-control graph. this can happen because each of these four scenarios, treatment for people with hiv is carried out, causing the number of individuals in the hiv treatment class to increase. while the scenario graphs 1, 2 and 4 did not experience differences with no-control graph because in these three scenarios no treatment was given to people with hiv. the scenario that results in the largest number of individuals on hiv treatment is scenario 6 whereby by administering treatment for individuals with active tb and hiv infected can increase the number of individuals in hiv treatment to 2229 people. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 104 (a) (b) (c) (d) (e) (f) (g) figure 3. change control on (a) scenario 1, (b) scenario 2, (c) scenario 3, (d) scenario 4, (e) scenario 5, (f) scenario 6 and (g) scenario 7 table 2 shows the results of the objective function (6). the scenario with the largest objective function value is generated by scenario 2 with 10697,09 while the smallest objective function value is generated by scenario 7 with 1401,44. the control graphs , and in each scenario decrease as shown in figure 3. this can happen because the number of sufferers decreases over time, causing the rate of treatments also to decrease. table 2. scenario trial and objective function results scenario description objective function 1 on 2910,31 2 on 10697,09 3 on 3027,08 4 on 2017,46 5 on 1559,21 6 on 2893,22 7 on 1401,44 table 3 shows the final results of each population in the modified model showing a better value than the previous model. the final number of people with disease decreased more when using the modified model than using the previous model. this shows that using the modified model can further reduce the number of tb-hiv/aids coinfected patiens. table 3. comparison of the best scenario results class population initial final previous model modified model 2080 1785 94 354 57 3 0 899 2073 1500 1205 217 420 375 99 1095 428 3 325 149 1 137 24 0 29 20 0 yuyun monita and putroue keumala intan control of the spread of tb-hiv/aids coinfection using optimal control 105 5. conclusion after obtained the results as described above, it can be concluded that in this study the best scenario is a scenario that desires the smallest objective function value of to control the spread of tb-hiv/aids coinfection with the modified model is scenario 7 namely the use of , , and . this indicates that by treating individuals with latent tb, active tb, and hiv infected in tandem, it will reduce the populations of tbhiv/aids coinfection more quickly with smaller treatment costs. for further research can use the original and up-to-date data to see the realization of the modified model. then, can try using runge-kutta method with a higher order. and can add new controls in the form of the use of contraceptives such as condoms during sexual intercourse, thus forming a new model that may be more effective. references [1] w. sukokarlinda, “analisis dan kontrol optimal pada model penyebaran virus hiv dalam tubuh manusia,” skripsi, 2012. [2] unaids, “fact sheet global aids update 2019,” unaids, pp. 1–6, 2019. [3] unaids, “country indonesia,” unaids, 2019. [online]. available: https://www.unaids.org/en/regionscountries/countries/indonesia. [accessed: 25sep-2019]. [4] a. h. m. zeth, a. h. asdie, a. g. mukti, and j. mansoden, “perilaku dan risiko penyakit hiv-aids di masyarakat papua studi pengembangan model lokal kebijakan hiv-aids,” j. manaj. pelayanan kesehat., vol. 13, no. 4, pp. 206–219, 2010. [5] s. murni, c. w. green, s. djauzi, a. setiyanto, and s. okta, hidup dengan hiv/aids. jakarta: spiritia, 2009. [6] a. khan, j. f. g. aguilar, t. s. khan, and h. khan, “stability analysis and numerical solutions of fractional order hiv/aids model,” chaos, solitons and fractals, vol. 122, pp. 119–128, 2019. [7] a. rusli, “koinfeksi hiv & tb,” 2014. [8] who, global tuberculosis report. who (world health organization), 2018. [9] cdc, “latent tb infection and tb disease,” 2016. [online]. available: https://www.cdc.gov/tb/topic/basics/tbinfectiondisease.htm. [accessed: 19-sep2019]. [10] setiawan, “kontrol optimal penyebaran tuberkulosis dengan exogenous reinfection,” indonesia university, 2012. [11] i. marlina, tuberkulosis, vol. 2, no. 1. jakarta: kemenkes ri, 2018. [12] j. nainggolan, “kontrol pengobatan optimal pada model penyebaran tuberkulosis tipe seit,” e-jurnal mat., vol. 6, no. 2, p. 137, 2017. [13] s. n. rayhan, “kontrol optimum penyebaran koinfeksi penyakit tuberkulosis dan hiv/aids,” intitut pertanian bogor (ipb), 2017. [14] cdc, core curriculum on tuberculosis, sixth edit. (cdc) centers for disease control and prevention, 2013. [15] e. michael and r. c. spear, modelling parasite transmission and control. new york, usa: springer science & business media, 2010. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 96-106 106 [16] who, the health academy avoiding tuberculosis. switzerland: geneva, 2004. [17] bolarin and omatola, “a mathematical analysis of hiv/tb co-infection model,” appl. math., vol. 6, no. 4, pp. 65–72, 2006. [18] g. r. rose, numerical methods for solving optimal control problems. university of tennessee, 2015. [19] e. r. hidayah, “kontrol optimal model pertumbuhan tumor dengan imunoterapi,” skripsi, 2016. [20] d. peng gao and n. jing huang, “optimal control analysis of a tuberculosis model,” appl. math. model., vol. 58, pp. 47–64, 2018. [21] c. . bhunu, w. garira, and z. mukandavire, “modelling hiv/aids and tuberculosis coinfection,” bull. math. biol., vol. 71, pp. 1745–1780, 2009. [22] tanvi and r. aggarwal, “dynamics of hiv-tb co-infection with detection as optimal intervention strategy,” int. j. non-linear machanics, vol. 516, pp. 280– 307, 2019. [23] huzaimah, “metode analitik dan metode runge-kutta orde 4 dalam penyelesaian persamaan getaran pegas teredam,” universitas islam negeri maulana malik ibrahim, malang, 2016. [24] subhashini and srividhya, “comparison of several numerical algorithms with the use of predictor and corrector for solving ode,” int. j. trend sci. res. dev., vol. 3, pp. 1057–1060, 2019. [25] puspitasari, intan, a. sutrisno, t. ruby, and m. ansori, “pembandingan metode runge-kutta orde 4 dan metode adam-bashfort moulton dalam penyelesaian model pertumbuhan uang yang diinvestasikan,” in prosiding seminar nasional metode kuantitatif 2017, 2010, pp. 328–340. [26] a. chiang, fundamental method of mathematical economics fourth edition. new york: the mcgraw-hill, 2005. [27] d. s. naidu, optimal control systems. new york: crc press, 2002. [28] f. puspitasari, “masalah kontrol optimal pada model penyebaran penyakit demam berdarah dengan pengaruh musim,” institut teknologi bandung, 2017. [29] s. a. hardiyanti, “kontrol optimal sistem perawatan produksi dengan memperhatikan kerusakan produk dan tingkat diskon,” institut teknologi sepuluh nopember, surabaya, 2016. [30] fatmawati and h. tasman, “an optimal treatment control of tb-hiv coinfection,” int. j. math. math. sci., 2016. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 contact: sadeq taha abdulazeez, sadiq.taha@uod.ac department of mathematics, college of basic education, university of duhok, duhok, iraq the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.9-19 a new computational method based on integral transform for solving linear and nonlinear fractional systems diyar hashim malo1, rogash younis masiha2, muhammad amin sadiq murad3, sadeq taha abdulazeez4* 1,3department of mathematics, college of sciences, university of duhok, duhok, iraq 2department of statistics, van yüzüncü yıl üniversitesi, van, turkey 4department of mathematics, college of basic education, university of duhok, duhok, iraq article history: received mar 16, 2021 revised, apr 30, 2021 accepted, may 24, 2021 kata kunci: sistem fraksional stiff, persamaan diferensial, ehpm, metode kernel hilbert space, solusi aproksimasi abstrak. pada artikel ini, metode elzaki homotopy perturbation (ehpm) diterapkan untuk menyelesaikan sistem fraksional stiff. metode ehpm adalah kombinasi dari modifikasi transformasi integral laplace yang disebut transformasi elzaki dan metode gangguan homotopi. metode yang diusulkan telah diterapkan pada beberapa contoh sistem fraksional stiff linier dan nonlinier. hasil yang diperoleh dengan metode ini dibandingkan dengan hasil yang diperoleh dengan metode kernel hilbert space (khsm) menunjukkan bahwa metode ehpm merupakan metode yang efektif dan akurat untuk menyelesaikan sistem persamaan diferensial orde fraksional. keywords: fractional stiff systems, differential equations, elzaki homotopy perturbation methods, kernel hilbert space method, approximate solutions abstract. in this article, the elzaki homotopy perturbation method is applied to solve fractional stiff systems. the elzaki homotopy perturbation method (ehpm) is a combination of a modified laplace integral transform called the elzaki transform and the homotopy perturbation method. the proposed method is applied for some examples of linear and nonlinear fractional stiff systems. the results obtained by the current method were compared with the results obtained by the kernel hilbert space khsm method. the obtained result reveals that the elzaki homotopy perturbation method is an effective and accurate technique to solve the systems of differential equations of fractional order. how to cite: d. h. malo, r. y. masiha, m. a. s. murad, and s. t. abdulazeez, “a new computational method based integral transform for solving linear and nonlinear fractional systems”, j. mat. mantik, vol. 7, no. 1, pp. 9-19, may. 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 issn: 2527-3159 (print) 2527-3167 (online) mailto:sadiq.taha@uod.ac https://doi.org/10.15642/mantik.2021.7.1.9-19 https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 10 1. introduction many scientific disciplines can be modeled by initial value problems of fractional order which lead to a better understanding in characterizing the natural phenomena in various fields of science such as engineering, biology, economy, computer science, and physics. the exact and analytical solutions of fractional order differential equations are difficult to be found, thus the numerical methods are used to investigate the solutions of differential equations of fractional orders. the proposed system at first was highlighted by hirschfelder and curtiss [1]. then stiff systems were studied by many researchers in different branches [2]–[4]. several numerical methods have been developed to obtain the analytical and approximate solutions of stiff systems of integer and fractional orders, such as adomian decomposition method [5], homotopy perturbation method [6], homotopy analysis method [7], variation iteration method [8], rational homotopy perturbation method[9], block method [10], laplace adomian decomposition method and modified decomposition method [11], multistage bernstein method[12], and fractional power series method [13]. homotopy perturbation method (hpm) is an important semi-analytic technique for solving differential equations [14]–[16] and an efficient technique to study the different types of nonlinear functional equations. volterra-fredholm nonlinear systems were solved by hpm [17], hyperbolic pdes [18], zakharov-kuznetsov[19], and system of nonlinear differential equations[20]. so far, many modified techniques have been described to solve linear and nonlinear differential equations of integer and fractional orders. those of which are laplace homotopy perturbation method, laplace adomian decomposition method, and recently, elzaki homotopy transformation perturbation method is used to solve a range of problems such as a family of fisher’s equation [21], spatial diffusion of biological population [22], nonlinear oscillators [23], system of linear and nonlinear pdes of fractional orders [24], time-fractional navier–stokes equations [25], and an algorithm for solving the burgers– huxley equation using the elzaki transform [26] elzaki integral transform is a modification of the laplace and sumudu transforms which was invented by tariq [27], elzaki transformation is an efficient and powerful technique that has found the exact solutions to several differential equations which cannot be solved by sumudu transform [28]. elzaki integral equation is a powerful and efficient technique that has been used to solve many differential equations of integer and fractional orders [25][29]–[33]. the objective of this paper is to expand the applications of ehpm and illustrate the efficiency of the proposed method, thus we consider the stiff systems fractional ordinary differential equations: 𝐷𝜇𝑖 𝑧𝑖(𝑡) + 𝐹𝑖(𝑡, 𝑧1(𝑡), 𝑧2(𝑡), … , 𝑧𝑛 (𝑡)) = 𝑓𝑖 (𝑡), 𝑧𝑖 (𝑡0) = 𝑎𝑖,0, 𝑚 − 1 < 𝜇𝑖 < 𝑚 , 𝑚 ∈ 𝑁. (1) 2. preliminaries in this section, we introduce some definitions and properties of fractional calculus and elzaki transform which are used in this article. definition 2.1. [4] a real valued function 𝑔(𝑦), 𝑦 > 0 is belongs to the space ∁𝜎, 𝜎 ∈ 𝑅 if there exists at least a real number 𝑑 > 𝜎, such that 𝑔(𝑦) = 𝑦 𝑑 𝑔1(𝑦) where 𝑔1(𝑦) ∈ ∁(0, ∞), and it is said to be in the space ∁𝜎 𝑛 if 𝑔𝑛 ∈ 𝑅𝜎 , 𝑛 ∈ 𝑁. definition 2.2. [34] the function 𝑓(𝑢) is called r-l fractional integral of order ∝≥ 0 if it is defined as: 𝐽𝛼 𝑓(𝑢) = 1 𝛤(𝛼) ∫ 𝑢 0 (𝑢 − 𝑡)𝛼−1𝑓(𝑡)𝑑𝑡, 𝛼 > 0, 𝑡 > 0. in particular 𝐽0𝑓(𝑢) = 𝑓(𝑢). for 𝜃 ≥ 0 and 𝜗 ≥ −1, some properties of the operator 𝐽𝛼 diyar h. malo, rogas y. masiha, muhammad a. s. murad, and sadeq t. abdulazeez a new computational method based integral transform for solving linear and nonlinear fractional systems 11 1. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝛼+𝜃 𝑓(𝑢), 2. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝜃 𝐽𝛼 𝑓(𝑢), 3. 𝐽𝛼 𝑥𝜗 = 𝛤(𝜗+1) 𝛤(𝛼+𝜗+1) 𝑥𝛼+𝜗. definition 2.3. [34] the function 𝑓 ∈ 𝐶−1 𝑛 , 𝑛 ∈ 𝑁 is called caputo fractional derivative if it is defined as 𝐷𝛼 𝑓(𝑢) = 1 𝛤(𝑛−𝛼) ∫ 𝑢 0 (𝑢 − 𝑡)𝑛−𝛼−1𝑓 𝑛(𝑡)𝑑𝑡, 𝑛 − 1 < 𝛼 ≤ 𝑛. definition 2.4. [27] the elzaki-transform of the function 𝑓(𝑢) is defined as: 𝐸[𝑓(𝑢)] = 𝑇(𝑣) = 𝑣 ∫ ∞ 0 𝑓(𝑢)𝑒 −𝑢 𝑣 𝑑𝑢 𝑢 > 0. suppose that 𝑓 is piecewise continuous, then we can calculate 𝐸 [ 𝜕𝑓 𝜕𝑥 ] as follows: 𝐸 [ 𝜕𝑓(𝑥,𝑢) 𝜕𝑥 ] = ∫ ∞ 0 𝑣𝑒 −𝑢 𝑣 𝜕𝑓(𝑥,𝑢) 𝜕𝑥 𝑑𝑢 = 𝜕 𝜕𝑥 ∫ ∞ 0 𝑣𝑒 −𝑢 𝑣 𝑓(𝑥, 𝑢)𝑑𝑢 = 𝜕 𝜕𝑥 𝑇(𝑥, 𝑣), similarly, we can have: 𝐸 [ 𝜕2𝑓(𝑥, 𝑢) 𝜕𝑥2 ] = 𝑑2𝑇(𝑥, 𝑢) 𝑑𝑥2 . assume that 𝜕𝑓 𝜕𝑢 = ℎ, then we have: 𝐸 [ 𝜕2𝑓(𝑥, 𝑢) 𝜕𝑢2 ] = 𝐸 [ 𝜕ℎ(𝑥, 𝑢) 𝜕𝑢 ] = 1 𝑣 𝐸[ℎ(𝑥, 𝑢)] − 𝑣ℎ(𝑥, 0) 𝐸 [ 𝜕2𝑓(𝑥, 𝑢) 𝜕𝑢2 ] = 𝑇(𝑥, 𝑢) 𝑣 2 − 𝑓(𝑥, 0) − 𝑣 𝜕𝑓 𝜕𝑢 (𝑥, 0) by mathematical induction one can extend this result to the 𝑛𝑡ℎpartial derivative as follows: 𝐸 [ 𝜕𝑛𝑓(𝑥,𝑢) 𝜕𝑢𝑛 ] = 𝑇(𝑥,𝑡) 𝑣𝑛 − ∑𝑛−1𝑖=0 𝑣 2−𝑛+𝑖 𝜕 𝑖𝑓(𝑥,0) 𝜕𝑢𝑖 . (2) 3. elzaki homotopy perturbation method analysis assume the following system of nonlinear differential equations 𝛭𝑖(𝑧𝑖) + 𝛮𝑖 (𝑧𝑖) = 𝑔𝑖 (𝑡), 𝑡 ∈ 𝛬, 𝑖 = 1,2, … , 𝑛 (3) where 𝑧𝑖 and 𝑔𝑖(𝑡) are sought and known functions respectively, 𝛭𝑖 and 𝛮𝑖 are linear and nonlinear operators respectively, where: 𝛮𝑖(𝑧𝑖) = ∑ ∞ 𝑘=0 𝐴𝑖𝑘 , where 𝐴𝑖𝑘 represents the adomian polynomial as follows: 𝐴𝑖𝑘 = 1 𝑚! 𝑑𝑚 𝑑𝜏𝑚 (𝛮𝑖 (∑ ∞ 𝑘=0 𝑧1𝑘𝜏 𝑚, … , ∑ ∞ 𝑘=0 𝑧𝑛𝑘𝜏 𝑚 ) , now, we assume the following homotopy и𝑖 (ϒ𝑖 , 𝑝) = (1 − 𝑝)(𝛭𝑖(ϒ𝑖 ) − 𝑧𝑖,0) + 𝑝(𝛭𝑖(ϒ𝑖 ) + 𝛮𝑖(ϒ𝑖 ) − 𝑔𝑖), or equivalently, и𝑖 (ϒ𝑖 , 𝑝) = 𝛭𝑖(ϒ𝑖 ) − 𝑧𝑖,0 + 𝑝𝑧𝑖,0 + 𝑝(𝛮𝑖(ϒ𝑖 ) − 𝑔𝑖), (4) since 𝛭𝑖(ϒ𝑖 ) − 𝑧𝑖,0 = 0 and 𝛭𝑖(ϒ𝑖 ) + 𝛮𝑖(ϒ𝑖 ) − 𝑔𝑖 = 0 which are equivalent to the operator equations и𝑖(𝑧, 0) = 0 and и𝑖(𝑧, 1) = 0 respectively, also 𝑝 ∈ [0,1] is a homotopy parameter, 𝑧𝑖,0 are initial approximations and ϒ𝑖 : (𝑡, 𝑝): 𝛬 × [0,1] → 𝑅. now, we apply the elzaki transform on (4), we obtain 𝐸[𝛭𝑖(ϒ𝑖 ) − 𝑧𝑖,0 + 𝑝𝑧𝑖,0 + 𝑝(𝛮𝑖 (ϒ𝑖 ) − 𝑔𝑖 )] = 0 by differential property of of elzaki transform, we get 1 𝑣𝜇 𝐸[ϒ𝑖 ] − 𝑣 2−𝜇 ϒ𝑖,0 − 𝑣 3−𝜇 ϒ′ 𝑖,0 − ⋯ − 𝑣 𝑛+1−𝜇 ϒ𝑖,0 (𝑛−1) = 𝐸[𝑧𝑖,0 − 𝑝𝑧𝑖,0 + 𝑝(𝛮𝑖(ϒ𝑖 ) − 𝑔𝑖)] jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 12 applying the inverse elzaki transform, we obtain ϒ𝑖 = 𝐸 −1 [𝑣 2ϒ𝑖,0 + 𝑣 3ϒ′𝑖,0 + ⋯ + 𝑣 𝑛+1ϒ𝑖,0 (𝑛−1) ] + 𝐸−1 [𝑣 𝜇 𝐸[𝑧𝑖,0 − 𝑝𝑧𝑖,0 + 𝑝(𝛮𝑖 (ϒ𝑖) − 𝑔𝑖)]]. (5) by hpm, the following series represent the solution of equation (5), where 𝑝 ∈ [0,1]. ϒ𝑖 (𝑡) = ∑ ∞ 𝑗=0 𝑝 𝑗 ϒ𝑖,𝑗 , 𝑖 = 1,2, … , 𝑛 (6) substituting (6) in (5), we obtain ∑ ∞ 𝑛=0 𝑝𝑛ϒ𝑖,𝑛 = 𝐸 −1 [𝑣 2ϒ𝑖,0 + 𝑣 3ϒ′𝑖,0 + ⋯ + 𝑣 𝑛+1ϒ𝑖,0 (𝑛−1) ] + 𝐸−1 [𝑣 𝜇 𝐸 [𝑧𝑖,0 − 𝑝𝑧𝑖,0 + 𝑝 (𝛮𝑖 (∑ ∞ 𝑛=0 𝑝𝑛ϒ𝑖,𝑛) − 𝑔𝑖)]] now, by comparing the coefficients of the same powers of the embedded parameter 𝑝 , leads to 𝑝0 ∶ ϒ𝑖,0 = 𝐸 −1 [𝑣 2ϒ𝑖,0 + 𝑣 3ϒ′ 𝑖,0 + ⋯ + 𝑣 𝑛+1ϒ𝑖,0 (𝑛−1) + 𝐸[𝑧0]] 𝑝1 ∶ ϒ𝑖,1 = 𝐸 −1 [𝑣 𝜇 𝐸[𝛮𝑖 (ϒ𝑖,0) − 𝑧𝑖,0 − 𝑔𝑖]] 𝑝2 ∶ ϒ𝑖,2 = 𝐸 −1 [𝑣 𝜇 𝐸[𝛮𝑖(ϒ𝑖,0, ϒ𝑖,1)]] 𝑝3 ∶ ϒ𝑖,3 = 𝐸 −1 [𝑣 𝜇 𝐸[𝛮𝑖 (ϒ𝑖,0, ϒ𝑖,1, ϒ𝑖,2)]] ⋮ 𝑝 𝑗 ∶ ϒ𝑖,𝑗 = 𝐸 −1 [𝑣 𝜇 𝐸[𝛮𝑖(ϒ𝑖,0, ϒ𝑖,1, ϒ𝑖,2, … , ϒ𝑖,𝑗−1 )]] ⋮ assume that ϒ𝑖,0 = 𝛽𝑖,0, ϒ ′ 𝑖,0 = 𝛽𝑖,1, … , ϒ𝑖,0 (𝑛−1) = 𝛽𝑖,𝑛−1 are the initial approximations, where 𝑖 = 1, 2, … , 𝑛. finally, the solution of (3) is obtained by the following series: 𝑧𝑖(𝑡) = ϒ𝑖 = ϒ𝑖,0 + ϒ𝑖,1 + ϒ𝑖,2 + ⋯ +. (7) the convergence and uniqueness of the series (7) is discussed in [21]. 4. applications of elzaki homotopy perturbation method the numerical patterns are employed to confirm the efficiency and accuracy of the elzaki homotopy transform perturbation technique for solving the fractional stiff systems. for this purpose, at the first example, the comparison of error analysis with the kernel hilbert space technique [13] [35] is made, at the second example, the exact solution of the proposed problem is achieved where 𝛽 = 1, and at the third example the comparison between the proposed method and (adm and mldm [11]) is made. example 4.1 consider the following linear stiff system of fractional order 0 < 𝛽 ≤ 1 𝐷𝑡 𝛽 𝑧(𝑡) = −𝑧(𝑡) + 95𝑤(𝑡), 𝐷𝑡 𝛽 𝑤(𝑡) = −𝑧(𝑡) − 97𝑤(𝑡), (8) with the initial conditions 𝑧(0) = 1, 𝑤(0) = 1. the exact solution of the system when 𝛽 = 1 is 𝑧(𝑡) = 95 47 𝑒−2𝑡 − 48 47 𝑒−96𝑡 , 𝑤(𝑡) = 48 47 𝑒−96𝑡 − 1 47 𝑒−2𝑡 . to solve system (8) by the ehpm, we use the following homotopy 𝐷𝑡 𝛽 𝑍(𝑡) − 𝑧0(𝑡) + 𝑝(𝑧0(𝑡) + 𝑍(𝑡) − 95𝑊 (𝑡)) = 0 𝐷𝑡 𝛽 𝑊(𝑡) − 𝑤0(𝑡) + 𝑝(𝑤0(𝑡) + 𝑍(𝑡) + 97𝑊 (𝑡)) = 0. applying the elzaki transform, we have diyar h. malo, rogas y. masiha, muhammad a. s. murad, and sadeq t. abdulazeez a new computational method based integral transform for solving linear and nonlinear fractional systems 13 𝐸 [𝐷𝑡 𝛽 𝑍(𝑡) − 𝑧0(𝑡) + 𝑝(𝑧0(𝑡) + 𝑍(𝑡) − 95𝑊(𝑡))] = 0 𝐸 [𝐷𝑡 𝛽 𝑊(𝑡) − 𝑤0(𝑡) + 𝑝(𝑤0(𝑡) + 𝑍(𝑡) + 97𝑊 (𝑡))] = 0. using the differential property of elzaki transform, we obtain 1 𝑣 𝛽 𝐸[𝑍(𝑡)] = 𝑣 2−𝛽 𝑍0(𝑡) + 𝐸[𝑧0(𝑡) − 𝑝(𝑧0(𝑡) + 𝑍(𝑡) − 95𝑊(𝑡))] 1 𝑣𝛽 𝐸[𝑊(𝑡)] = 𝑣 2−𝛽𝑊0(𝑡) + 𝐸[𝑤0(𝑡) − 𝑝(𝑧0(𝑡) + 𝑍(𝑡) + 97𝑊 (𝑡))]. applying inverse elzaki transform, we have 𝑍(𝑡) = 𝐸−1 [𝑣 2𝑍0(𝑡) + 𝑣 𝛽𝐸[𝑧0(𝑡) − 𝑝(𝑧0(𝑡) + 𝑍(𝑡) − 95𝑊(𝑡))]] 𝑊(𝑡) = 𝐸−1 [𝑣 2𝑊0(𝑡) + 𝑣 𝛽 𝐸[𝑤0(𝑡) − 𝑝(𝑧0(𝑡) + 𝑍(𝑡) + 97𝑊(𝑡))]]. (9) here, the following form is the solution of equation (9): 𝑍(𝑡) = 𝑍0(𝑡) + 𝑝𝑍1(𝑡) + 𝑝 2𝑍2(𝑡) + ⋯, 𝑊(𝑡) = 𝑊0(𝑡) + 𝑝𝑊1(𝑡) + 𝑝 2𝑊2(𝑡) + ⋯, (10) substituting (10) in to (9) and collecting the coefficients of equivalent powers of embedded parameter 𝑝 , we obtain the following: 𝑝0 ∶ {𝑍0(𝑡) = 𝐸 −1 [𝑣 2𝑍0(0) + 𝑣 𝛽𝐸[𝑧0(𝑡)]] , 𝑊0(𝑡) = 𝐸 −1 [𝑣 2𝑊0(0) + 𝑣 𝛽𝐸[𝑤0(𝑡)]], 𝑝1 ∶ {𝑍1(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸[(𝑧0(𝑡) + 𝑍0(𝑡) − 95𝑊0(𝑡))]] , 𝑊1(𝑡) = 𝐸−1 [−𝑣 𝛽𝐸[(𝑧0(𝑡) + 𝑍0(𝑡) + 97𝑊0(𝑡))]], 𝑝2 ∶ {𝑍2(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸[(𝑍1(𝑡) − 95𝑊1(𝑡))]] , 𝑊2(𝑡) = 𝐸 −1 [−𝑣 𝛽 𝐸[(𝑍1(𝑡) + 97𝑊1(𝑡))]], 𝑝3 ∶ {𝑍3(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸[(𝑍2(𝑡) − 95𝑊2(𝑡))]] , 𝑊3(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸[(𝑍2(𝑡) + 97𝑊2(𝑡))]], ⋮ here, we set 𝑍0(0) = 𝑧0(𝑡) = 1 and 𝑊0(0) = 𝑤0(𝑡) = 1. thus, the above equations lead to the following results: 𝑍0(𝑡) = 1 + 𝑡𝛽 𝛤(𝛽+1) , 𝑊0(𝑡) = 1 + 𝑡𝛽 𝛤(𝛽+1) , 𝑍1(𝑡) = 93 𝛤(𝛽+1) 𝑡𝛽 + 94 𝛤(2𝛽+1) 𝑡 2𝛽 , 𝑊1(𝑡) = − 99 𝛤(𝛽+1) 𝑡𝛽 − 98 𝛤(2𝛽+1) 𝑡 2𝛽 , 𝑍2(𝑡) = − 9498 𝛤(𝛽+1) 𝑡 2𝛽 − 9404 𝛤(3𝛽+1) 𝑡 3𝛽 , 𝑊2(𝑡) = 9510 𝛤(2𝛽+1) 𝑡 2𝛽 + 9412 𝛤(3𝛽+1) 𝑡 3𝛽 𝑍3(𝑡) = 9129486 𝛤(3𝛽+1) 𝑡 3𝛽 + 9035446 𝛤(4𝛽+1) 𝑡 4𝛽 , 𝑊3(𝑡) = − 912972 𝛤(3𝛽+1) 𝑡 3𝛽 − 903560 𝛤(4𝛽+1) 𝑡 4𝛽 , thus, the solution of system (8) using the series (7) is 𝑧(𝑡) = 1 + 94 𝛤(𝛽 + 1) 𝑡𝛽 − 9404 𝛤(2𝛽 + 1) 𝑡 2𝛽 + 903544 𝛤(3𝛽 + 1) 𝑡 3𝛽 …, 𝑤(𝑡) = 1 − 98 𝛤(𝛽 + 1) 𝑡𝛽 + 9412 𝛤(2𝛽 + 1) 𝑡 2𝛽 − 903560 𝛤(3𝛽 + 1) 𝑡 3𝛽 …, jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 14 figure 1. the behavior of ehpm solution of system (8) for different values of 𝛽 and exact solution. figure 1: illustrates the behavior of the ehpm solution of system (8) for different values of 𝛽 and exact solution. it can be seen that there is a good agreement between the exact solution and the approximate solution using ehpm, especially when 𝛽 = 1. table 1. the absolute error of kernel hilbert space method khsm and ehpm of (8) 𝒕 𝒛(𝒕)ehpm 𝒛(𝒕)khsm 𝒘(𝒕)ehpm 𝒘(𝒕)khsm 0.00 0 0 0 0 0.5 3.0163443 × 10−15 4.76706570 × 10−4 3.0163483 × 10−15 4.44439990 × 10−4 0.1 5.0178941 × 10−14 4.11985309 × 10−5 3.4785609 × 10−13 1.37132721 × 10−5 0.15 6.4189477 × 10−11 2.37714435 × 10−5 1.1162905 × 10−10 4.49704615 × 10−7 0.2 7.3684661 × 10−9 1.99138385 × 10−5 1.2169510 × 10−9 2.1210257 × 10−7 0.25 3.4096780 × 10−7 1.67357026 × 10−5 3.6738962 × 10−7 1.76192995 × 10−7 table 1: shows the comparison between the absolute errors of the approximate solution of the proposed technique at different values of 𝑡 when 𝛽 = 1 for 20 order approximation and the approximate solution of khsm. the results illustrate that the proposed technique is superior to khsm and an efficient method for investigating the solution of proposed fractional systems. diyar h. malo, rogas y. masiha, muhammad a. s. murad, and sadeq t. abdulazeez a new computational method based integral transform for solving linear and nonlinear fractional systems 15 example 4.2: consider the following non-linear stiff system of fractional order 𝐷𝑡 𝛽 𝑧(𝑡) = −(𝜌−1 + 2)𝑧(𝑡) + 𝜌−1𝑤 2(𝑡), 0 < 𝛽 ≤ 1, 𝐷𝑡 𝛽 𝑤(𝑡) = 𝑧(𝑡) − 𝑤(𝑡) − 𝑤 2(𝑡), 𝑡 ∈ [0,2] (11) with the initial conditions 𝑧(0) = 1, 𝑤(0) = 1. the exact solution of the system when 𝛽 = 1 is 𝑧(𝑡) = 𝑒−2𝑡 , 𝑤(𝑡) = 𝑒𝑡 . to solve system (11) by the ehpm, we use the following homotopy 𝐷𝑡 𝛽 𝑍(𝑡) − 𝑧0(𝑡) + 𝑝(𝑧0(𝑡) + (𝜌 −1 + 2)𝑍(𝑡) − 𝜌−1𝑊2(𝑡)) = 0 𝐷𝑡 𝛽 𝑊(𝑡) − 𝑤0(𝑡) + 𝑝(𝑤0(𝑡) − 𝑍(𝑡) + 𝑊(𝑡) + 𝑊 2 (𝑡)) = 0. applying the elzaki transform, we have 𝐸 [𝐷𝑡 𝛽 𝑍(𝑡) − 𝑧0(𝑡) + 𝑝(𝑧0(𝑡) + (𝜌 −1 + 2)𝑍(𝑡) − 𝜌−1𝑊2(𝑡))] = 0 𝐸 [𝐷𝑡 𝛽 𝑊(𝑡) − 𝑤0(𝑡) + 𝑝(𝑤0(𝑡) − 𝑍(𝑡) + 𝑊(𝑡) + 𝑊 2(𝑡))] = 0. using the differential property of elzaki transform, we obtain 1 𝑣 𝛽 𝐸[𝑍(𝑡)] = 𝑣 2−𝛽 𝑍0(𝑡) + 𝐸[𝑧0(𝑡) − 𝑝(𝑧0(𝑡) + (𝜌 −1 + 2)𝑍(𝑡) − 𝜌−1𝑊2(𝑡))] 1 𝑣𝛽 𝐸[𝑊(𝑡)] = 𝑣 2−𝛽𝑊0(𝑡) + 𝐸[𝑤0(𝑡) − 𝑝(𝑤0(𝑡) − 𝑍(𝑡) + 𝑊(𝑡) + 𝑊 2 (𝑡))]. applying inverse elzaki transform, we have 𝑍(𝑡) = 𝐸−1 [𝑣 2𝑍0(𝑡) + 𝑣 𝛽𝐸[𝑧0(𝑡) − 𝑝(𝑧0(𝑡) + (𝜌 −1 + 2)𝑍(𝑡) − 𝜌−1𝑊2(𝑡))]] 𝑊(𝑡) = 𝐸−1 [𝑣 2𝑊0(𝑡) + 𝑣 𝛽 𝐸[𝑤0(𝑡) − 𝑝(𝑤0(𝑡) − 𝑍(𝑡) + 𝑊(𝑡) + 𝑊 2(𝑡))]]. (12) here, the following form is the solution of system (11): 𝑍(𝑡) = 𝑍0(𝑡) + 𝑝𝑍1(𝑡) + 𝑝 2𝑍2(𝑡) + ⋯, 𝑊(𝑡) = 𝑊0(𝑡) + 𝑝𝑊1(𝑡) + 𝑝 2𝑊2(𝑡) + ⋯. (13) substituting (13) in to (12) and collecting the coefficients of equivalent powers of embedded parameter 𝑝 , we obtain the following: 𝑝0 ∶ {𝑍0(𝑡) = 𝐸 −1 [𝑣 2𝑍0(0) + 𝑣 𝛽𝐸[𝑧0(𝑡)]] , 𝑊0(𝑡) = 𝐸 −1 [𝑣 2𝑊0(0) + 𝑣 𝛽𝐸[𝑤0(𝑡)]], 𝑝1 ∶ {𝑍1(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸[𝑧0(𝑡) + (𝜌 −1 + 2)𝑍0(𝑡) − 𝜌 −1𝑊0 2(𝑡)]] , 𝑊1(𝑡) = 𝐸−1 [−𝑣 𝛽𝐸[𝑤0(𝑡) − 𝑍0(𝑡) + 𝑊0(𝑡) + 𝑊0 2(𝑡)]], 𝑝 𝑗 : { 𝑍2(𝑡) = 𝐸−1 [−𝑣 𝛽𝐸 [(𝜌−1 + 2)𝑍𝑗−1(𝑡) − 𝜌−1 ∑ 𝑗−1 𝑙=0 𝑊𝑗 (𝑡)𝑊𝑗−𝑙−1(𝑡)]] , 𝑊2(𝑡) = 𝐸 −1 [−𝑣 𝛽𝐸 [−𝑍𝑗−1(𝑡) + 𝑊𝑗−1(𝑡) + ∑ 𝑗−1 𝑙=0 𝑊𝑗 (𝑡)𝑊𝑗−𝑙−1(𝑡)]] , 𝑗 = 2,3, …, ⋮ here, we set 𝑍0(0) = 𝑧0(𝑡) = 1 and 𝑊0(0) = 𝑤0(𝑡) = 1. thus, the above equations lead the following results. 𝑍0(𝑡) = 1 + 𝑡𝛽 𝛤(𝛽 + 1) , jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 9-19 16 𝑊0(𝑡) = 1 + 𝑡𝛽 𝛤(𝛽 + 1) , 𝑍1(𝑡) = −3 𝑡𝛽 𝛤(𝛽 + 1) + (𝜌−1 − 2) 𝑡 2𝛽 𝛤(2𝛽 + 1) + 𝜌−1𝛤(2𝛽 + 1) 𝑡 3𝛽 𝛤 2(𝛽 + 1)𝛤(3𝛽 + 1) , 𝑊1(𝑡) = −2 𝑡𝛽 𝛤(𝛽 + 1) − 2 𝑡 2𝛽 𝛤(2𝛽 + 1) − 𝛤(2𝛽 + 1)𝑡 3𝛽 𝛤 2(𝛽 + 1)𝛤(3𝛽 + 1) , 𝑍2(𝑡) = (6 − 𝜌 −1) 𝑡 2𝛽 𝛤(2𝛽 + 1) − ( 1 𝜌2 + 4𝜌−1 + 4𝜌−1𝛤(2𝛽 + 1) 𝛤 2(𝛽 + 1) − 4) 𝑡 3𝛽 𝛤(3𝛽 + 1) − ( 1 𝜌2 + 4𝜌−1 + 4𝜌−1𝛤(3𝛽 + 1)𝛤(𝛽 + 1) 𝛤 2(2𝛽 + 1) ) 𝛤(2𝛽 + 1) 𝑡 4𝛽 𝛤 2(𝛽 + 1)𝛤(4𝛽 + 1) − 2𝜌−1𝛤(2𝛽 + 1)𝛤(4𝛽 + 1) 𝑡 5𝛽 𝛤 3(𝛽 + 1)𝛤(3𝛽 + 1)𝛤(5𝛽 + 1) , 𝑊2(𝑡) = 3 𝑡 2𝛽 𝛤(2𝛽 + 1) + (𝜌−1 + 4 + 4𝛤(2𝛽 + 1) 𝛤 2(𝛽 + 1) ) 𝑡 3𝛽 𝛤(3𝛽 + 1) + (2𝜌−1 + 6 + 4𝛤(3𝛽 + 1) 𝛤 2(𝛽 + 1)𝛤(2𝛽 + 1) ) 𝑡 4𝛽 𝛤(4𝛽 + 1) + 2𝛤(2𝛽 + 1)𝛤(4𝛽 + 1)𝑡 5𝛽 𝛤 3(𝛽 + 1)𝛤(3𝛽 + 1)𝛤(5𝛽 + 1) , ⋮ thus, the solution of system (11) using the series (7) is 𝑧(𝑡) = 𝑍0(𝑡) + 𝑍1(𝑡) + 𝑍2(𝑡) + ⋯, 𝑤(𝑡) = 𝑊0(𝑡) + 𝑊1(𝑡) + 𝑊2(𝑡) + ⋯, 𝑧(𝑡) = 1 − 2𝑡𝛽 𝛤(𝛽 + 1) + 4𝑡 2𝛽 𝛤(2𝛽 + 1) + 𝜌−1𝛤(2𝛽 + 1) 𝑡 3𝛽 𝛤 2(𝛽 + 1)𝛤(3𝛽 + 1) − ( 1 𝜌2 + 4𝜌−1 + 4𝜌−1𝛤(2𝛽 + 1) 𝛤 2(𝛽 + 1) − 4) 𝑡 3𝛽 𝛤(3𝛽 + 1) − ( 1 𝜌2 + 4𝜌−1 + 4𝜌−1𝛤(3𝛽 + 1)𝛤(𝛽 + 1) 𝛤 2(2𝛽 + 1) ) 𝛤(2𝛽 + 1) 𝑡 4𝛽 𝛤 2(𝛽 + 1)𝛤(4𝛽 + 1) − 2𝜌−1𝛤(2𝛽 + 1)𝛤(4𝛽 + 1) 𝑡 5𝛽 𝛤 3(𝛽 + 1)𝛤(3𝛽 + 1)𝛤(5𝛽 + 1) + ⋯, 𝑤(𝑡) = 1 − 𝑡𝛽 𝛤(𝛽 + 1) + 𝑡 2𝛽 𝛤(2𝛽 + 1) − 𝛤(2𝛽 + 1)𝑡 3𝛽 𝛤 2(𝛽 + 1)𝛤(3𝛽 + 1) + (𝜌−1 + 4 + 4𝛤(2𝛽 + 1) 𝛤 2(𝛽 + 1) ) 𝑡 3𝛽 𝛤(3𝛽 + 1) + (2𝜌−1 + 6 + 4𝛤(3𝛽 + 1) 𝛤 2(𝛽 + 1)𝛤(2𝛽 + 1) ) 𝑡 4𝛽 𝛤(4𝛽 + 1) + 2𝛤(2𝛽 + 1)𝛤(4𝛽 + 1)𝑡 5𝛽 𝛤 3(𝛽 + 1)𝛤(3𝛽 + 1)𝛤(5𝛽 + 1) + ⋯. here, setting 𝛽 = 1 and follow the above solution, the following results are obtained: 𝑍0(𝑡) = 1 + 𝑡, 𝑊0(𝑡) = 1 + 𝑡 𝑍1(𝑡) = −3𝑡 + (𝜌 −1 − 2) 𝑡2 2! + 2 𝜌−1 𝑡 3 3! , 𝑊1(𝑡) = −2𝑡 − 2 𝑡 2 − 2𝑡 3 3! , 𝑍2(𝑡) = (6 − 𝜌 −1) 𝑡 2 2 − ( 1 𝜌2 + 4𝜌−1 + 8𝜌−1 − 4) 𝑡 3 3! − ( 2 𝜌2 + 20𝜌−1) 𝑡 4 4! − 2𝜌−1 𝑡 5 15 , 𝑊2(𝑡) = 3 𝑡 2 2! + (𝜌−1 + 12) 𝑡 3 3! + (2𝜌−1 + 3 4 ) 𝑡 4𝛽 4! + 2 𝑡 5 15 , diyar h. malo, rogas y. masiha, muhammad a. s. murad, and sadeq t. abdulazeez a new computational method based integral transform for solving linear and nonlinear fractional systems 17 one can express the above values in a series after finding the other terms of the solution 𝑧(𝑡) = 1 − 2(𝑡 − 𝑡 2) − 8 6 𝑡 3 − 4 6 𝑡 4 + ⋯ = ∑ ∞ 𝑘=0 (2𝑡)𝑘 (−1)𝑘 𝑘! = 𝑒 −2𝑡 𝑤(𝑡) = 1 − 𝑡 + 𝑡 2 2 − 𝑡 3 6 + 𝑡 4 24 + ⋯ = ∑ ∞ 𝑘=0 (𝑡)𝑘 (−1)𝑘 𝑘! = 𝑒−𝑡 , which is the exact solution of system (11). table 2. the absolute error of kernel hilbert space method khsm and ehpm of (8) 𝑡 𝑧(𝑡)ehpm 𝑧(𝑡)khsm 𝑤(𝑡)ehpm 𝑤(𝑡)khsm 0.00 0 0 0 1.2 × 10−6 0.4 5.55 × 10−17 1.20 × 10−6 0 2.47 × 10−2 0.8 5.55 × 10−16 1.28 × 10−6 0 1.19 × 10−1 1.2 1.70 × 10−12 9.10 × 10−7 5.55 × 10−17 2.62 × 10−1 1.6 6.90 × 10−10 5.59 × 10−7 0 4.38 × 10−1 2.0 7.30 × 10−8 3.18 × 10−7 5.56 × 10−16 1.2 × 10−6 table 2: illustrates the comparison between the absolute errors of the numerical solution of the proposed technique at different values of 𝑡 where 𝛽 = 1 for 20 terms iterations and the solution obtained by khsm. it is clear that the ehpm technique is more accurate and converges faster than the khsm technique. 5. conclusions in this work, a new computational technique namely elzaki homotopy perturbation method are constructed to solve linear and nonlinear systems of differential equation of fractional order. the construction scheme of the current method is discussed and implemented to show the performance of the computational method for this problem. the results obtained showed that the proposed method is a powerful and efficient method for solving linear and nonlinear stiff systems of fractional orders and compared the obtained results of the current method with the results obtained by the other methods. table 1 and table 2 observed that the suggested technique is superior to kernel hilbert space method khsm. references [1] c. f. curtiss and j. o. hirschfelder, “integration of stiff equations.,” proc. natl. acad. sci. u. s. a., vol. 38, no. 3, pp. 235–243, mar. 1952, doi: 10.1073/pnas.38.3.235. 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[35] o. abu arqub and m. al-smadi, “numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and dirichlet boundary conditions,” numer. methods partial differ. equ., vol. 34, no. 5, pp. 1577–1597, 2018, doi: 10.1002/num.22209. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 83 beberapa metode pada masalah pemrograman stokastik ihda hasbiyati1, hasriati2 matematika fmipa universitas riau1, ihdahasbiyati@gmail.com1 matematika fmipa universitas riau2, hasriati.hasri@gmail.com2 doi:https://doi.org/10.15642/mantik.2017.3.2.83-86 abstrak masalah pemrograman stokastik adalah masalah pemrograman matematis (linier, integer, mixed integer, dan nonlinier) dengan elemen stokastik berada dalam data. untuk mendapatkan solusi layak dan optimal dengan datanya stokastik dibutuhkan beberapa metode. metode-metode yang bisa diterapkan dalam masalah pemrograman stokastik diantaranya metode dekomposisi l-shape dan metode lagrange. setiap metode dapat menentukan solusi optimal untuk menyelesaikan masalah pemrograman stokastik. kata kunci: masalah pemrograman stokastik, optimisasi robust, chance-constrained, recourse-based stochastic. abstract stochastic programming problem is mathematical problem (linear, integer, mixed integer, and nonlinier) with stochastic element lies data. to get reasonable solution and optimal with its stochastic data is needed several method. applicable method in trouble stochastic programming are l-shape decomposition and lagrange decomposition. each method can determine optimal solution to troubleshoots stochastic programming keywords: stochastic programming problems, robust optimization, chance-constrained programming, recourse stochastic programming 1. pendahuluan pemograman stokastik adalah pemograman matematik dimana semua data yang tergabung ke dalam fungsi tujuan dan fungsi kendalanya berbentuk ketidakpastian. ketidakpastian ini dicirikan dengan distribusi probabilitas pada parameternya. walaupun ketidakpastian terdefinisi secara tepat, namun dalam prakteknya harus disusun secara terperinci dengan beberapa scenario sebagai akibat yang mungkin dari data, juga dalam spesifikasi dan ketepatan distribusi gabungan peluang. masalah pemrograman stokastik berbeda dengan masalah optimasi deterministik yang dirumuskan dengan parameter-parameter yang diketahui, dalam masalah di dunia nyata selalu memasukkan beberapa parameter-parameter yang tidak diketahui. contohnya pada suatu perusahaan gas yang mempunyai rencana untuk dua tahun, pada tahun pertama perusahaan gas membeli gas dari pasar, pengiriman kepada konsumen dilakukan segera dan juga dilakukan penyimpanan untuk tahun berikutnya. variabelvariabel yang mempengaruhi keputusan untuk mencadangkan penyimpanan atau membeli dari pasar diantaranya: berapa banyak gas yang dibeli untuk pengiriman, berapa banyak gas yang dibeli untuk penyimpanan, berapa banyak gas yang diambil dari penyimpanan dan pembelian untuk konsumen. keputusan bergantung kepada harga gas pada tahun pertama dan tahun kedua, biaya penyimpanan, ukuran kapasitas penyimpanan dan permintaan pada setiap periode.aplikasi lain dari pemrograman stokastik diantaranya adalah pada perencanaan produksi, penjadwalan, pembuatan rute, pengalokasian, kontrol dan manajemen lingkungan, finansial dan lain-lain. beberapa penelitian yang telah dilakukan jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 84 peneliti terhadap pemrograman stokastik diantaranya, oleh julia l. higle pada tahun 1991 yang menggunakan dekomposisi benders dengan variabel random pada fungsi tujuan, padatahun 1997 andrzej ruszczynski meneliti metode dekomposisi dengan menggunakan metode dekomposisi stokastik, pada tahun 2000 x. chen meneliti metode newton dengan fungsi tujuannya membutuhkan ekpektasi dan tidak mempunyai turunan, pada tahun 2006 jeff linderoth meneliti metode sampling dengan menggunakan metode dekomposisi untuk pemrograman stokastik duatahap dengan recourse. masalah pemrograman stokastik dapat dibedakan menjadi dua, yaitu masalah pemrograman stokastik linier dan masalah pemrograman stokastik nonlinier. masalah pemrograman stokastik linier adalah masalah pemrograman stokastik dengan fungsi tujuan dan fungsi kendalanya adalah fungsi-fungsi linier, sedangkan masalah pemrograman stokastik nonlinier adalah masalah pemrograman stokastik dengan salah satu atau keduanya dari fungsi tujuan dan fungsi kendalanya berbentuk fungsi nonlinier. untuk masing-masing masalah pemrograman stokastik baik linier ataupun nonlinier menggunakan pendekatan metode yang berbeda-beda. metode pendekatan yang dilakukan berasal dari metode-metode yang dipakai untuk menyelesaikan masalah pemrograman matematik (linier dan nonlinier), diantaranya: 1. untuk masalah pemrograman linier digunakan metode simpleks dan metode dekomposisi, metode dekomposisi dual. 2. untuk masalah pemrograman nonlinier digunakan metode cutting-plane, metode descent, metode penalty dan metode lagrange. tulisan ini merupakan hasil kajian dari beberapa metode penyelesaian masalah pemrograman matematis yang dipakai untuk masalah pemrograman stokastik. 2. kajian teori sebelum membahas masalah pemrograman stokastik, terlebih dahulu dibahas perubahan bentuk model dari masalah pemrograman matematik menjadi masalah pemrograman stokastik. misalkan masalah pemrograman linier sebagai berikut min{𝑐1𝑥1 + 𝑐2𝑥2 + ⋯+ 𝑐𝑛𝑥𝑛} kendala, 𝑎11𝑥1 + 𝑎12𝑥2 + ⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1 𝑎21𝑥1 + 𝑎22𝑥2 + ⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2 ⋮ 𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 + ⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚 𝑥1,𝑥2,⋯,𝑥𝑛 ≥ 0 } (1) dengan menggunakan notasi matriks persamaan (1) dapat dituliskan kembali sebagai berikut min𝑐𝑇𝑥 𝑘𝑒𝑛𝑑𝑎𝑙𝑎 𝐴𝑥 = 𝑏 𝑥 ≥ 0 } (2) persamaan (2) dapat dituliskan kembali dalam bentuk umum sebagai berikut min𝑔0(𝑥) 𝑘𝑒𝑛𝑑𝑎𝑙𝑎,𝑔𝑖(𝑥) ≤ 0,𝑖 = 1,⋯,𝑚 𝑥 ∈ 𝑋 ⊂ ℝ𝑛 } (3) himpunan 𝑋 mempunyai sifat yang sama dengan fungsi 𝑔𝑖:ℝ 𝑛 → ℝ,𝑖 = 0,⋯,𝑚. fungsi 𝑔𝑖 dan himpunan 𝑋 disebut pemrogramam linier atau nonlinier apabila: 1. linier, jika 𝑋 adalah himpunan konvek dan fungsi 𝑔𝑖:ℝ 𝑛 → ℝ,𝑖 = 0,⋯,𝑚 linier. 2. nonliner, jika paling sedikit satu dari fungsi 𝑔𝑖:ℝ 𝑛 → ℝ,𝑖 = 0,⋯,𝑚 nonlinier atau 𝑋 bukan himpunan konvek, dengan ketentuan sebagai berikut a. konveks, jika 𝑋 ∩ {𝑥|𝑔𝑖(𝑥) ≤ 0,𝑖 = 1,⋯,𝑚} adalah himpunan konveks dan 𝑔0 adalah fungsi konveks, b. nonkonveks, jika tidak ada dari 𝑋 ∩ {𝑥|𝑔𝑖(𝑥) ≤ 0,𝑖 = 1,⋯,𝑚} yang bukan konveks atau fungsi objektif 𝑔0 bukan konveks. persamaan (3) dapat dituliskan dalam bentuk masalah pemrograman stokastik dengan menambahkan parameter random 𝜉 sebagai berikut min𝑔0(𝑥,𝜉) 𝑘𝑒𝑛𝑑𝑎𝑙𝑎,𝑔𝑖(𝑥,𝜉) ≤ 0,𝑖 = 1,⋯,𝑚, 𝑥𝜖𝑋 ⊂ ℝ𝑛 } (4) pada persamaan (4) ditambahkan suatu fungsi recourse untuk memastikan bahwa setiap pemilihan vektor random 𝜉 menjamin tidak jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 85 terjadinya pelanggaran yang dapat mengakibatkan biaya ekstra perunit, sehingga persamaan (4) dapat dituliskan kembali sebagai berikut min𝑓0(𝑥,𝜉) = 𝑔0(𝑥,𝜉) + 𝑄(𝑥,𝜉) 𝑘𝑒𝑛𝑑𝑎𝑙𝑎,𝑔𝑖(𝑥,𝜉) ≤ 0,𝑖 = 1,⋯,𝑚, 𝑥𝜖𝑋 ⊂ ℝ𝑛 } (5) fungsi recourse pada persamaan (5) yaitu 𝑄(𝑥,𝜉) = 𝑚𝑖𝑛𝑦{𝑞 𝑇𝑦|𝑊𝑦 ≥ 𝑔 +(𝑥,𝜉),𝑦 ∈ 𝑌} dengan 𝑔+(𝑥,𝜉) = (𝑔1 +(𝑥,𝜉),⋯,𝑔𝑚 +(𝑥,𝜉)), 𝑦 ∈ 𝑌 ⊂ ℝ�̅�adalah vektor recourse dengan {𝑦|𝑦 ≥ 0}, 𝑊 adalah matriks berukuran 𝑚 × �̅�, dan 𝑞 ∈ ℝ�̅� adalah vektor unit biaya. 3. hasil dan pembahasan metode yang dipakai untuk meyelesaikan masalah pemrograman matematik (linier dan nonlinier) seperti metode simpleks, metode dekomposisi, metode dekomposisi dual, metode cutting-plane, metode descent, metode penalty dan metode lagrange dapat dilihat pada [3], [4], dan [5]. metode-metode yang dipakai pada masalah pemrograman matematik dapat juga digunakan pada masalah pemrograman stokastik karena pada dasarnya pemrograman stokastik adalah pemrograman matematik. salah satu pengembangan dari metode pada pemrograman matematik yaitu metode lagrange yang dipakai untuk masalah pemrograman nonlinier stokastik, yang diberikan sebagai berikut, misalkan program nonlinier stokastik dengan recourse diberikan sebagai berikut: min 𝑥𝜖𝑋 �̂�0 (𝑥) + εξ1 𝒬1(x, ξ1) fungsi recourse: kendala𝑐1(𝑥,𝑦1,𝜉1) = 0 dan untuk 𝑡 = 2,⋯,𝑇 − 1, berulang dipunyai 𝒬𝑡(𝑥,𝑦1,⋯,𝑦𝑡−1,𝜉1,⋯,𝜉𝑡) = min 𝑦𝑡 𝑞𝑡(𝑥,𝑦1,⋯,𝑦𝑡−1,𝜉1,⋯,𝜉𝑡) + 𝐸𝜉𝑡+1𝒬𝑡+1(𝑥,𝑦1,⋯,𝑦𝑡−1,𝜉1,⋯,𝜉𝑡,𝜉𝑡+1) s.t.𝑐1(𝑥,𝑦1,⋯,𝑦𝑡,𝜉1,⋯,𝜉𝑡) = 0 (6) 𝒬𝑇 = 0,𝑥 ∈ ℜ 𝑛0 adalah vektor deterministik, 𝜉𝑖 adalah realisasi dari vektor random 𝜉𝑖 ⋅ 𝑦𝑖 ∈ ℜ 𝑛𝑖 yang merupakan vektor keputusan padatahap kei, yang dibangun secara rekursif oleh 𝑥,𝑦1,⋯,𝑦𝑖−1 dan 𝜉1,⋯,𝜉𝑖, dalam hal ini𝑦𝑖(𝑥,𝑦1,⋯,𝑦𝑖−1,𝜉1,⋯,𝜉𝑖)direpresentasikan secaraaktual. �̂�0 dan 𝑐0 adalah fungsi bernilai riil pada ℜ𝑛0. 𝑐𝑡 random karena berelasi dengan 𝜉1,⋯,𝜉𝑡. untuk vektor random diskrit 𝜉 = (𝜉1,⋯,𝜉𝑇−1), jika 𝑐𝑡 mempunyai realisasi hingga 𝑐𝑖𝑡(𝑖 = 1,⋯,𝑆𝑡), maka semua 𝑐𝑡𝑖 bentuk fungsi konstrain pada tahap 𝑡, model pada persamaan (6) merujuk kepada model yang ada pada [1] dan [2]. proses penyelesaian masalah pemrograman nonlinear stokastik pada persamaan (6) dimulai dengan barisan “iterasi major”, pada setiap barisan iterasi major, konstrain nonlinier dilinierisasi pada beberapa titik dasar 𝑥𝑘 dan ketaklinieran didampingkan dengan fungsi objektif dengan pengali lagrange, kemudian definisikan, ))(()(),( ~ kkkk xxxjxfxxf  , dimana 𝐽𝑘 = [𝐽(𝑥𝑘)]𝑖𝑗 = 𝜕𝑓 𝑖/𝜕𝑥𝑗 adalah matriks jacobian dari turunan parsial pertama dari fungsi konstrain. selanjutnya selesaikan sub-problem linier berikut untuk iterasi major ke-k, sehingga diperoleh minimize xy ) ~ () ~ ( 2 1 ) ~ ()(),,,,( ffffffydxfxyxl tt k t kk   s.t )( 11 kkkk xfxjbyaxj  222 byaxa  (7) u y x l        fungsi objektif pada persamaan (7) adalah modifikasi lagrangian augmented, dimana parameter penalty 𝜌 mempertinggi sifat konvergen dari estimasi awal derajat optimum paling jauh. estimasi pengali lagrangian 𝜆𝑘 diperoleh sebagai nilai optimum pada solusi subproblem sebelumnya. sepanjang pendekatan barisan iterasi major, optimum (diukur dengan pertukaran relatif pada estimasi berurut dari 𝜆𝑘 dan degree untuk konstrain nonlinier yang dipenuhi pada 𝑥𝑘), parameter penalty 𝜌direduksi menjadi nol dan kuadrat nilai konvergen dari subproblem dipenuhi untuk mencapai kondisi optimum. 2 1 1 1 1 1 1 2 1 1 2 ˆ ˆ ˆ( , ) min ( , , ) ( , , , ) y q x q x y e q x y       jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 86 metode lain pada pemrograman matematis yang dapat dikembangkan pada pemrograman stokastik yaitu metode dekomposisi dual yang dikembangkan menjadi metode dekomposisi lshape. proses dimulai dengan membuat bentuk dual dari masalah pemrograman pada persamaan (2) menjadi sebagai berikut max𝑏𝑇𝑢 dengan kendala 𝐴𝑇𝑢 ≤ 𝑐 dan 𝑢 ≥ 0 selanjutnya diberikan fungsi recouse sebagai berikut: 𝑄(𝑥,𝜉) = 𝑚𝑖𝑛{𝑞(𝜉)𝑇𝑦|𝑊𝑦 = ℎ(𝜉) − 𝑇(𝜉)𝑥,𝑦 ≥ 0} dengan ℎ(𝜉) = ℎ0 + 𝐻𝜉 = ℎ0 + ∑ ℎ𝑖𝜉𝑖𝑖 adalah vector yang bergantung pada vector random 𝜉. 𝑇(𝜉) = 𝑇0 + ∑ 𝑇𝑖𝜉𝑖𝑖 adalah matriks deterministic dan 𝑞(𝜉) = 𝑞0 + ∑ 𝑞𝑖𝜉𝑖𝑖 . selanjutnya periksa ℎ(𝜉) − 𝑇(𝜉)𝑥 sehingga masalah pemrograman stokastik feasible (layak). solusi optimal diperoleh apabila dipenuhi kondisi ℎ(𝜉) − 𝑇(𝜉)𝑥 = ℎ0 − 𝑇0𝑥, artinya 𝑄(𝑥,𝜉) linier dan konkaf pada 𝜉. 4. penutup masalah pemrograman stokastik adalah juga masalah pemrograman matematis yang melibatkan parameter ketidakpastian. metodemetode yang dipakai pada masalah pemrograman matematik juga dapat digunakan pada masalah pemrograman stokastik dengan melakukan pengembangan. masing-masing metode mempunyai syarat kondisi optimal yang berbedabeda sesuai dengan bentuk model yang digunakan. referensi [1] ihda hasbiyati. simple technique of projected lagrange for a class of multi-stage stochastic nonlinear programs. global journal of technology & optimization. volume 6. issue 3. (2015). [2] ihda hasbiyati. a projected lagrangian approach for a class of multi-stage stochastic nonlinear programs. prociding conferenceof the international federation of operational research societies (ifors). melbourne convention and exhibition centre, melbourne, australia. (2011) [3] peter kall, dan janos mayer. stochastic linear programming, spriger, new york.(2005) [4] peter kall, dan stein w. wallace. stochastic programming, springer, new york.(1994) [5] m.s. bazaraa and c.m. shetty..nonlinear programming, theory and algorithms, (2nd ed.), john wiley & sons, newyork. (1993) . paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 44 penentuan harga opsi asia dengan metode monte carlo surya amami pramuditya 1 fkip, universitas swadaya gunung djati 1, amamisurya@fkip-unswagati.ac.id 1 abstrak opsi adalah kontrak antara holder dan writer dimana writer memberikan hak (bukan kewajiban) kepada holder untuk membeli atau menjual aset dari writer dengan harga tertentu (strike price) dan pada waktu yang telah ditentukan dimasa datang (maturity time). opsi asia termasuk pada opsi path dependent. artinya payoff opsi asia tidak hanya bergantung pada harga saham saat maturity time saja, tetapi merupakan rata-rata harga saham selama masa jatuh temponya dan disimbolkan 𝐴 (average). monte carlo pada dasarnya digunakan sebagai prosedur numerik untuk menaksir nilai ekspektasi pricing product derivative. teknik yang digunakan adalah monte carlo standar dan reduksi varians. hasilnya diperoleh harga opsi asia call dan put untuk kedua teknik dengan selang kepercayaan 95%. teknik reduksi varians terlihat lebih cepat memperkecil selang kepercayaan 95% dibandingkan metode standar. kata kunci: opsi,, asia, monte carlo abstract option is a contract between a holder and a writer in which the writer grants the rights (not obligations) to the holder to buy or sell the assets of the writer at a certain price (strike price) at maturity time. asian options are included in the dependent path option. this means that asia's payoff option depends not only on the stock price at maturity time, but it is the average stock price during its maturity and symbolized a (average). monte carlo is basically used as a numerical procedure to estimate the expected value of pricing product derivatives. the techniques used are the standard monte carlo and variance reduction. the result obtained the asia call option price and put for both techniques with 95% confidence interval. the variance reduction technique looks faster reducing 95% confidence interval than standard method. keyword: option, asian, monte carlo 1. pendahuluan hull [2] mendefinisikan opsi sebagai kontrak antara holder dan writer dimana writer memberikan hak (bukan kewajiban) kepada holder untuk membeli atau menjual suatu aset dari writer dengan harga tertentu (strike atau exercise price) dan pada waktu yang telah ditentukan dimasa datang (expiry date atau maturity time) [4][5]. salah satu jenis opsi adalah opsi asia. opsi asia termasuk pada opsi path dependent [4]. artinya payoff opsi asia tidak hanya bergantung pada harga saham saat maturity time saja. di sini payoff opsi asia merupakan rata-rata harga saham selama masa jatuh temponya dan disimbolkan 𝐴 (average). metode monte carlo pada dasarnya digunakan sebagai prosedur numerik untuk menaksir nilai ekspektasi dari suatu peubah acak sehingga metoda ini dapat digunakan untuk permasalahan pricing product derivative jika direpresentasikan sebagai nilai ekspektasinya. prosedur simulasi melibatkan generating dari peubah acak dengan suatu fungsi kepadatan dan dengan menggunakan law of large number maka rata-rata dari nilai ini dapat dinyatakan sebagai penaksir ekspektasi peubah acak tersebut. penelitian ini bertujuan untuk mencari payoff harga opsi asia call dan put fixed strike dan jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 45 average strike disertai selang kepercayaan 95%. selanjutnya, dicari selang kepercayaan terkecil melalui metode monte carlo standard dan reduksi varians. 2. simulasi monte carlo misalkan x peubah acak dengan ekspektasi e(x) = a dan var (x) = 𝑏2 yang nilainya belum diketahui. misalkan 𝑋1, 𝑋2, … , 𝑋𝑀 adalah barisan peubah acak yang berdistribusi identik dengan x, maka penaksir tak bias untuk a [6][3] adalah 𝑎𝑀 = 1 𝑀 ∑ 𝑋𝑖 𝑀 𝑖=1 (1) dan penaksir tak bias untuk 𝑏2 adalah 𝑏𝑀 2 = 1 1 − 𝑀 ∑(𝑋𝑖 − 𝑎𝑀) 2 (2) 𝑀 𝑖=1 berdasarkan teorema limit pusat untuk 𝑀 → ∞ berlaku ∑ 𝑋𝑖 𝑀 𝑖=1 − 𝑀𝑎 𝑏√𝑀 ~𝑁(0,1) atau ∑ 𝑋𝑖 𝑀 𝑖=1 ~𝑁(𝑀𝑎, 𝑏2𝑀) sehingga, 𝑎𝑀 − 𝑎 = 1 𝑀 ∑ 𝑋𝑖 𝑀 𝑖=1 − 𝑎~𝑁(0, 𝑏 2 𝑀 ⁄ ) 𝑎𝑀 − 𝑎 𝑏√𝑀 ~𝑁(0,1) akan didapatkan taksiran interval untuk a. perhatikan 𝑃 (| 𝑎𝑀 − 𝑎 𝑏√𝑀 | ≤ 1.96) = 0.95 𝑃 (−1.96 ≤ 𝑎𝑀 − 𝑎 𝑏√𝑀 ≤ 1.96) = 0.95 𝑃 (𝑎𝑀 − 1.96 𝑏 √𝑀 ≤ 𝑎 ≤ 𝑎𝑀 + 1.96 𝑏 √𝑀 ) = 0.95 di sini 𝑏 √𝑀 merupakan standard error , dengan mengambil 𝑏 ≈ 𝑏𝑀, maka 𝑃 (𝑎𝑀 − 1.96 𝑏𝑀 √𝑀 ≤ 𝑎 ≤ 𝑎𝑀 + 1.96 𝑏𝑀 √𝑀 ) = 0.95 (3) sehingga diperoleh selang kepercayaan 95 % untuk a adalah [𝑎𝑀 − 1.96 𝑏𝑀 √𝑀 , 𝑎𝑀 + 1.96 𝑏 √𝑀 ]. agar akurasi selang lebih akurat dapat diperoleh malalui dua cara yaitu : 1. memperbesar simulasi m, tetapi hal ini memberikan waktu komputasi yang lama. 2. mengecilkan 𝑏𝑀 atau mereduksi variansi dengan menggunakan kontrol variat. 2.1 teknik reduksi variansi dengan kontrol variat taksiran selang akan semakin akurat jika lebar dari selang tersebut semakin sempit/kecil, lebar selang kepercayaan dapat dipersempit dengan cara memperbanyak sampel (menambah jumlah simulasi). namun cara ini cukup menyulitkan karena faktor √𝑀. sebagai contoh, untuk mendapatkan selang kepercayaan yang lebih akurat, yaitu menyusutkan selang kepercayaan dengan faktor 10 membutuhkan sampel seratus kali lebih banyak dari semula. cara lain yang dapat dilakukan adalah memperkecil standar deviasi (𝑏𝑀 ) yang berarti memperkecil variansi [1]. ide dari teknik ini dalah mengganti 𝑋𝑖 dengan barisan peubah acak yang lain yang juga identik dengan mean sama dengan 𝐸(𝑋𝑖 ) namun dengan variansi yang lebih kecil. misalkan 𝜃 = 𝐸(𝑋) ingin ditaksir dengan simulasi monte carlo. andaikan ada peubah acak lain, selain x yaitu y dengan mean 𝐸(𝑌) = 𝜇𝑌, kemudian akan ditunjukkan 𝑣𝑎𝑟 (𝑋) > 𝑣𝑎𝑟(𝑌). tulis peubah acak 𝑍 = 𝑋 + 𝑐(𝑌 − 𝜇𝑌) (4) maka nilai ekspektasi dari z adalah 𝐸(𝑋 + 𝑐(𝑌 − 𝜇𝑌)) = 𝐸(𝑋) + 𝑐𝐸(𝑌 − 𝜇𝑌 ) = 𝜃 + 𝑐𝐸(𝑌 − 𝜇𝑌 ) = 𝜃 sedangakan variansinya 𝑣𝑎𝑟(𝑋 + 𝑐(𝑌 − 𝜇𝑌)) = 𝑣𝑎𝑟 (𝑋 + 𝑐𝑌) = 𝑣𝑎𝑟 (𝑋) + 𝑣𝑎𝑟 (𝑌) + 2𝑐𝑜𝑣(𝑋, 𝑐𝑌) = 𝑣𝑎𝑟 (𝑋) + 𝑐2𝑣𝑎𝑟(𝑌) + 2𝑐 𝑐𝑜𝑣(𝑋, 𝑌) (5) pilih y sedemikian rupa sehingga 𝑐𝑜𝑣(𝑋, 𝑌) ≠ 0. karena 𝑣𝑎𝑟 (𝑋), 𝑣𝑎𝑟(𝑌), 𝐶𝑜𝑣(𝑋, 𝑌) diketahui, maka 𝑣𝑎𝑟 (𝑍) = 𝑓(𝑐) yaitu fungsi kuadrat dalam c. minimumkan ruas kanan pada persamaan (5) terhadap c diperoleh 𝑐∗ = −𝑐𝑜𝑣(𝑋, 𝑌) 𝑣𝑎𝑟 (𝑌) (6) usahakan 𝑐𝑜𝑣 (𝑌, 𝑉) positif sehingga 𝑐∗ negatif. dengan mengsubsitusikan persaman (6) ke persamaan (5) diperoleh jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 46 𝑣𝑎𝑟(𝑋 + 𝑐∗(𝑌 − 𝜇𝑌 )) = 𝑣𝑎𝑟(𝑋) − 𝑐𝑜𝑣 2(𝑋, 𝑌) 𝑣𝑎𝑟(𝑌) diperoleh reduksi variansi 𝑣𝑎𝑟(𝑍) 𝑣𝑎𝑟(𝑌) = 1 − 𝑐𝑜𝑣 2(𝑋, 𝑌) 𝑣𝑎𝑟(𝑌)𝑣𝑎𝑟(𝑋) = 1 − 𝑐𝑜𝑟𝑟2 (𝑋, 𝑌) selanjutnya pilih 𝑌 = ∑ 𝑆(𝑛)𝑖 𝑀 𝑖=1 dan lakukan 𝑘 simulasi untuk 𝑋 dan 𝑌 diperoleh 𝑋𝑖 dan 𝑌𝑖 (𝑖 = 1,2, … , 𝑘). tulis �̅� = 1 𝑘 ∑ 𝑋𝑖 𝑘 𝑖=1 �̅� = 1 𝑘 ∑ 𝑌𝑖 𝑘 𝑖=1 𝑐𝑜�̂�(𝑋, 𝑌) = 1 𝑘 − 1 ∑(𝑋𝑖 − �̅�)(𝑌 − �̅�) 𝑘 𝑖=1 𝑣𝑎�̂� (𝑋, 𝑌) = 1 𝑘 − 1 ∑( 𝑘 𝑖=1 𝑌𝑖 − �̅�) peubah acak pembanding memiliki mean sampel 𝜃 = 1 𝑘 ∑( 𝑘 𝑖=1 𝑋𝑖 + �̂� ∗(𝑌 − 𝜇𝑌 )) 2.2 model harga saham misalkan model pergerakan harga saham [3] [2] [4] [5] adalah 𝑆(𝑇) = 𝑆0𝑒 (𝑟− 1 2 𝜎2)𝑇+𝜎√𝑇𝑧 dengan 𝑍~𝑁(0,1) (7) serta = 1 𝑁 , dimana 𝑁 merupakan banyak hari kerja dalam 1 tahun. selanjutnya model saham ini menghasilakan ekspektasi dari peubah acak [4][5] 𝐶 = 𝑒−𝑟𝑇 𝑚𝑎𝑘𝑠{𝑆0𝑒 (𝑟− 1 2 𝜎2 )𝑇+𝜎√𝑇𝑧 − 𝐾, 0} yaitu nilai opsi call eropa saat 𝑇 dihitung di 𝑡 = 0. 2.3 opsi asia payoff dari asian option ditentukan dari nilai rata-rata untuk tiap kasusnya [2][6], yaitu:  average price asian call (fixed strike price) 𝐶(𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 ( 1 𝑇 ∫ 𝑆(𝜏)𝑑𝜏 − 𝐾, 0 𝑇 0 )  average price asian put ((fixed strike price)) 𝑃(𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 (𝐸 − 1 𝑇 ∫ 𝑆(𝜏)𝑑𝜏, 0 𝑇 0 )  average strike price asian call 𝐶(𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 (𝑆(𝑇) − 1 𝑇 ∫ 𝑆(𝜏)𝑑𝜏, 0 𝑇 0 )  average strike price asian put 𝑃(𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 ( 1 𝑇 ∫ 𝑆(𝜏)𝑑𝜏 − 𝑆(𝑇),0 𝑇 0 ) pandang ∫ 𝑆 𝑇 0 (𝜏)𝑑𝜏 ≈ ∆𝑡 ∑ 𝑆𝑗 𝑁 𝑗=1 (8) dimana n adalah banyaknya partisi dan ingat bahwa 𝑇 = 𝑁∆𝑇, maka ∫ 𝑆 𝑇 0 (𝜏)𝑑𝜏 = 1 𝑁∆𝑡 ∆𝑡 ∑ 𝑆𝑗 𝑁 𝑗=1 = 1 𝑁 ∑ 𝑆𝑗 𝑁 𝑗=1 (9) sehingga penggunaan monte carlo untuk asian call option memiliki payoff [3] 𝐶𝑖 (𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 ( 1 𝑁 ∑ 𝑆𝑗 𝑁 𝑗=1 − 𝐾, 0) (10) atau 𝐶𝑖(𝑆, 𝑇) = 𝑚𝑎𝑘𝑠 (𝑆(𝑇) − 1 𝑁 ∑ 𝑆𝑗 𝑁 𝑗=1 , 0) (11) untuk 𝑖 = 1,2, … , 𝑀. berdasarkan persamaan (9), persamaan (10) dan (11) dapat dituliskan 𝑉𝑐𝑝𝑟𝑖𝑐𝑒 = 𝑒𝑥𝑝 (−𝑟 𝑛 𝑁 ) ( 1 𝑛 ∑ 𝑆𝑗 𝑛 𝑗=1 − 𝐾, 0) + (12) atau 𝑉𝑐𝑠𝑡𝑟𝑖𝑘𝑒 = 𝑒𝑥𝑝 (−𝑟 𝑛 𝑁 ) (𝑆𝑇 − 1 𝑛 ∑ 𝑆𝑗 𝑛 𝑗=1 , 0) + (13) jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 47 2.4 algoritma metode monte carlo berikut merupakan algoritma menentukan harga opsi asia dengan metode monte carlo [6]. tabel 1. harga saham input: 𝑆0, 𝐾, 𝑟, 𝜎, 𝑛, 𝑁, 𝑀 bangkitkan faktor acak 𝑧(𝑀, 𝑛) hitung 𝐸(𝑋) = 𝑚𝑢 = (𝑟 − 1 2 𝜎2) 𝑁⁄ 𝑉𝑎𝑟(𝑋) = 𝑑𝑒𝑣 = 𝜎 √𝑁⁄ untuk 𝑖 = 1,2, . . , 𝑀 dan 𝑗 = 1,2, . . , 𝑛  untuk 𝑗 = 1 , 𝑥(𝑖, 𝑗) = 𝑚𝑢 + 𝑑𝑒𝑣 ∗ 𝑧(𝑖, 𝑗)  untuk 𝑗 = 2, . . , 𝑛 , 𝑥(𝑖, 𝑗) = 𝑥(𝑖, 𝑗 − 1) + (𝑚𝑢 + 𝑑𝑒𝑣 ∗ 𝑧(𝑖, 𝑗)) harga saham 𝑆𝑏(𝑖, 𝑗) = 𝑆0 ∗ exp (𝑥(𝑖, 𝑗)) tabel 2. opsi asia bangun harga saham 𝑆𝑏 hitung 𝐴 mean 𝑆𝑏 sebagai harga saham selama [0, 𝑇] hitung 𝑆𝑛 yaitu harga saham saat maturity time hitung payoff jika mc standar, maka jika opsi call, maka 𝑉𝑝𝑟𝑖𝑐𝑒 = exp(−𝑟 ∗ 𝑛 𝑁⁄ ) ∗ 𝑚𝑎𝑥(𝐴 − 𝐾, 0) lainnya, 𝑉𝑝𝑟𝑖𝑐𝑒 = exp(−𝑟 ∗ 𝑛 𝑁⁄ ) ∗ 𝑚𝑎𝑥(𝐾 − 𝐴, 0) hitung mean dan standar deviasi dari 𝑉𝑝𝑟𝑖𝑐𝑒, serta selang kepercayaan 95% lainnya, (kontrol variat) pilih 𝑌 = ∑ 𝑆𝑖 𝑛 𝑖=1 jika opsi call, maka hitung 𝑋 = exp(−𝑟 ∗ 𝑛 𝑁⁄ ) ∗ 𝑚𝑎𝑥(𝐴 − 𝐾, 0) 𝐶𝑜𝑣(𝑋, 𝑌) 𝐶𝑜𝑟𝑟(𝑋, 𝑌) 𝑐∗ 𝑍 = 𝑋 + 𝑐∗(𝑌 − 𝜇𝑌 ) hitung mean dan standar deviasi dari 𝑍, serta selang kepercayaan 95% lainnya, 𝑉𝑝𝑟𝑖𝑐𝑒 = exp(−𝑟 ∗ 𝑛 𝑁⁄ ) ∗ 𝑚𝑎𝑥(𝐾 − 𝐴, 0) 𝐶𝑜𝑣(𝑋, 𝑌) 𝐶𝑜𝑟𝑟(𝑋, 𝑌) 𝑐∗ 𝑍 = 𝑋 + 𝑐∗(𝑌 − 𝜇𝑌 ) hitung mean dan standar deviasi dari 𝑍, serta selang kepercayaan 95% hitung rasio (reduksi) 3. hasil dan pembahasan untuk menentukan selang kepercayaan serta harga opsi call dan put asia digunakan program matlab. program ini menggunakan data fiktif dengan 𝑟 = 6%; 𝜎 = 0.3; 𝑇 = 1; 𝑆0 = 15; 𝑁 = 252; 𝑛 = 100. adapun 𝑟 adalah suku bunga, 𝜎 adalah volatilitas, 𝑇 adalah waktu satu tahun kerja, 𝑆0 adalah harga saham awal, 𝑁 adalah waktu hari kerja dan 𝑛 adalah partisi waktu. tabel 3. selang kepercayaan opsi call asia monte carlo standar k=9 m average price average strike 10 4.9872 6.3177 -0.2050 0.6875 100 5.9094 6.5492 0.5043 0.9211 1000 5.8887 6.0831 0.7038 0.8465 10000 6.0156 6.0804 0.7269 0.7719 tabel 4. harga opsi call asia monte carlo standar k=9 m fixed strike average strike 10 5.6525 0.2413 100 6.2293 0.7127 1000 5.9859 0.7751 10000 6.0480 0.7494 berdasarkan tabel 3 dan tabel 4 di atas, semakin besar langkah m, maka selang kepercayaan 95% semakin kecil, sehingga taksiran harga opsi call asia untuk fixed strike maupun average strike semakin baik. tabel 5. selang kepercayaan opsi call asia monte carlo reduksi varians k=9 m average price average strike 10 7.1584 7.1458 0.2549 1.6207 100 5.7575 5.7575 0.4718 0.8147 1000 6.0332 6.0332 0.6740 0.7982 10000 6.0314 6.0314 0.7220 0.7600 tabel 6. harga opsi call asia monte carlo reduksi varians k=9 m fixed strike average strike 10 7.1584 0.9378 100 5.7575 0.6432 1000 6.0332 0.7361 10000 6.0314 0.7410 berdasarkan tabel 5 dan tabel 6 di atas, semakin besar langkah m, maka selang kepercayaan 95% semakin kecil, sehingga taksiran harga opsi call asia untuk fixed strike maupun jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 48 average strike semakin baik. teknik reduksi varians terlihat lebih cepat memperkecil selang kepercayaan 95% dibandingkan metode standar. tabel 7. selang kepercayaan opsi put asia monte carlo standar k=17 m average price average strike 10 0.4797 2.0382 0.1392 1.0961 100 1.7892 2.3478 0.4318 0.7439 1000 1.8052 1.9740 0.5287 0.6294 10000 1.9049 1.9594 0.5406 0.5716 tabel 8. harga opsi put asia monte carlo standar k=17 m fixed strike average strike 10 1.2589 0.6176 100 2.0685 0.5879 1000 1.8896 0.5791 10000 1.9322 0.5561 berdasarkan tabel 7 dan tabel 8 di atas, semakin besar langkah m, maka selang kepercayaan 95% semakin kecil, sehingga taksiran harga opsi put asia untuk fixed strike maupun average strike semakin baik. tabel 9. selang kepercayaan opsi put asia monte carlo reduksi varians k=17 m average price average strike 10 1.4303 1.6182 0.5807 1.1102 100 1.9654 2.0924 0.4797 0.7850 1000 1.9720 2.0158 0.5212 0.6124 10000 1.8924 1.9078 0.5453 0.5744 tabel 10. harga opsi put asia monte carlo reduksi varians k=17 m fixed strike average strike 10 1.5243 0.8455 100 2.0289 0.6324 1000 1.9939 0.5668 10000 1.9001 0.5599 berdasarkan tabel 9 dan tabel 10 di atas, semakin besar langkah m, maka selang kepercayaan 95% semakin kecil, sehingga taksiran harga opsi put asia untuk fixed strike maupun average strike semakin baik. teknik reduksi varians terlihat lebih cepat memperkecil selang kepercayaan 95% dibandingkan metode standar 4. kesimpulan semakin besar banyaknya langkah m, maka semakin memperkecil jarak selang kepercayaan 95%, untuk menaksir harga opsi call maupun opsi put. dengan menggunakan teknik reduksi varians terlihat lebih cepat memperkecil selang kepercayaan 95% dibandingkan metode standar. daftar pustaka [1] fu, m. c., madan, d. b., & wang, t. pricing continuous asian options: a comparison of monte carlo and laplace transform inversion methods. journal of computational finance, 2(2), (1999). 49-74. [2] hull, j.c., options, futures, and other derivatives (eighth edition). pearson, england. (2012). [3] podlozhnyuk, v., & harris, m. monte carlo option pricing. cuda sdk. (2008). [4] pramuditya, s.a., & sidarto, k. a. penentuan harga opsi asia dengan model binomial dipercepat. repository fkip unswagati. (2013). [5] pramuditya, s.a. perbandingan metode binomial dan metode black-scholes dalam penentuan harga opsi. sainsmat, 5(1). (2016). [6] seydel, r., tools for computational finance. springer-verlag, berlin. (2002). paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 101 distribusi batik madura melalui penerapan generalized vehicle routing problem (gvrp) faisol1, masdukil makruf2 jurusan matematika fmipa universitas islam madura1, faisol.munif@gmail.com 1 jurusan teknik informatika ft universitas islam madura 2, masdukil.makruf @gmail.com2 doi:https://doi.org/10.15642/mantik.2017.3.2.101-104 abstrak proses distribusi produk merupakan suatu upaya untuk menyampaikan suatu produk ketangan konsumen dengan sistem yang terencana dan terprogram. metode cluster merupakan pengelompokan lokasi pasar yang terdekat, selanjutnya dilakukan analisa stentang lokasi fasilitas potensial melalui central of gravity. gvrp (generalized vehicle routing problem) salah satu algoritma dalam metode cluster. dalam gvrp ini dideskripsikan penentuan rute untuk meminimalkan biaya distribusi yang diperlukan. gvrp adalah generalisasi dari vrp, sehingga titik dari graf dipartisi menjadi beberapa set titik tertentu, yang disebut cluster [2]. pada penelitian ini dilakukan modifikasi model gvrp untuk kasus kendaraan multi kapasitas yang dapat menentukan rute dan meminimalkan biaya distribusi. diambil kasus pada ud. damai asih untuk penyebaran batik tulis madura ke 25 kabupaten di jawa timur. dari hasil running menggunakan matlab 7.8.0 didapatkan efesiensi biaya distribusi sebesar 8.71% dari biaya awal sebelum dilakukan pengclusteran berdasarkan jarak dan kapasitas maksimal mobil sebesar rp. 6,969,480.00. setelah dilakukan pengclusteran berdasarkan jarak dan kapasitas maksimal mobil diperoleh biaya sebesar rp. 6.365.500.00. nilai efesiensi paling tinggi diperoleh pada cluster empat, sedangkan nilai efesiensi terendah diperoleh pada cluster delapan. adanya efesieinsi biaya dikarenakan jarak tempuh yang berbeda pada proses pengclusteran. kata kunci : gvrp, batik tulis madura, distribusi produk abstract product distribution process is an effort to convey a product of consumer handlebar with planned and programmed system. cluster method is a grouping of the nearest market location, then analyzed the location of potential facilities through central of gravity. gvrp (generalized vehicle routing problem) is one of the algorithms in the cluster method [1]. in the gvrp describes the route determination to minimize the required distribution costs. gvrp is a generalization of vrp, so the point of the graph is partitioned into several sets of specific points, called clusters [2]. in this research, modification of gvrp model for multi-capacity vehicle case can determine the route and minimize the cost of distribution. taken case on ud. damai asih for the distribution of madura write batik to 25 districts in east java. from the results of running using matlab 7.8.0 obtained the efficiency of the distribution cost of 8.71% of the initial cost before doing the clustering based on distance and maximum capacity of the car of rp. 6,969,480.00. after the filtering based on the distance and maximum capacity of the car obtained a cost of rp. 6.365.500.00. the highest value of efficiency is obtained in cluster four, while the lowest efficiency value is obtained in cluster eight. the existence of cost efficiency is due to the different mileage in the clustering process. keywords: gvrp, madura write batik, product distribution jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 102 1. pendahuluan batik merupakan warisan budaya dengan keunikan ragam corak dan memiliki interpretasi makna. ciri khas motif batik dipengaruhi lingkungan sehingga masing-masing daerah penghasil batik mempunyai karakteristik tersendiri. selain bernilai seni tinggi, batik menjadi komoditi yang memberikan peluang bisnis menjanjikan. tiga kabupaten di madura antara lain bangkalan, madura dan sumenep sebagai penghasil batik madura dengan karakter yang kuat dan motif warna yang berani seperti merah, kuning, hijau muda. distribusi batik madura tidak hanya mencakup wilayah madura saja akan tetapi mencakup berbagai wilayah di indonesia. permasalahan distribusi barang menjadi permasalahan utama dalam dunia bisnis. proses distribusi produk merupakan suatu upaya untuk menyampaikan suatu produk ke tangan konsumen dengan sistem yang terencana dan terprogram. efektifitas dan efisiensi distribusi barang diperlukan untuk optimalisasi waktu waktu dan biaya. salah satu jalan yang dapat dilakukan untuk membuat rancangan pendistribusian barang adalah dengan membuat pemodelan jalur distribusi dengan menggunakan bantuan pemrograman atau simulasi dengan melakukan pemodelan matematika. pemodelan matematika sebagaimana yang dimaksud tentunya tidak hanya dilakukan untuk masalah pendistribusian barang dan jasa, akan tetapi juga pada aspek yang lain. sebagai contoh terapan pemodelan matematika untuk mitigasi bencana [6], peramalan cuaca laut [8] dan lain sebagainya. selain pemodelan matematika, hal yang diperlukan adalah suatu metode pengambilan keputusan yang dapat mendukung proses eksekusi model matematka yang sudah dibuat. banyak model pengambilan keputusan yang dapat dipergunakan sepeti metode analytical hierarchy process [7], metode cluster, decision tree; metode operation research: linear programming, queuing theory, network analysis (ie. cpm). ataupun dengan bantuan komputer seperti: information system, expert system, dss, eis. metode cluster digunakan dalam pengambilan keputusan untuk memperoleh distribusi yang optimal. metode cluster merupakan pengelompokan lokasi pasar yang terdekat, selanjutnya dilakukan analisis tentang lokasi fasilitas potensial melalui central of gravity. gvrp (generalized vehicle routing problem) salah satu algoritma dalam metode cluster [1]. dalam gvrp ini dideskripsikan penentuan rute untuk meminimalkan biaya distribusi yang diperlukan. gvrp adalah generalisasi dari vrp, sehingga titik dari graf dipartisi menjadi beberapa set titik tertentu, yang disebut cluster [2]. masalah yang terkait dengan penentuan rute optimal untuk kendaraan dari satu atau beberapa stasiun ke himpunan lokasi / pelanggan, tergantung pada berbagai kendala, seperti kapasitas kendaraan, lampu lalu lintas, panjang rute, waktu tempuh, yang dikenal sebagai masalah rute kendaraan (vrp, vehicle routing problem). masalah-masalah ini memiliki kepentingan ekonomi yang signifikan karena banyak aplikasi praktis di bidang distribusi, koleksi dan logistik. penentuan rute bagi produsen batik selama ini masih bersifat manual yang hanya berdasarkan intuisi dan perkiraan, kebiasaan dan subyektifitas orang-orang berpengala-man. pada penelitian ini akan dikonstruksi model matematika untuk meminimalkan total biaya distribusi batik ke konsumen. formulasi model akan dikonstruksi menggunakan variasi dari model gvrp [3] dengan menambah batasan armada kendaraan yang di gunakan memiliki kapasitas dan biaya. dengan tujuan menganalisis masalah distribusi batik madura ke beberapa kabupaten di jawa timur. dengan mengembangkan metode gvrp untuk menyelesaikan masalah rute perjalanan kendaraan. sehingga diperoleh rekomendasi mengenai rute kendaraan sehingga dapat meminimalkan biaya distribusi. 2. metode penelitian dalam penelitian ini, terdapat beberapa langkah yang dilakukan yaitu: langkah 1: studi literatur pada tahap ini dikaji teori-teori dasar yang mendukung pembahasan masalah dengan mengumpulkan beberapa referensi yang mendukung penulisan ini, baik dari buku, jurnal, maupun internet khususnya tentang gvrp. langkah 2: membangun model pada tahap ini dilakukan formulasi kondisi yang ada kedalam model matematis. model yang dibangun merupakan variasi dari model gvrp dengan armada kendaraan multi kapasitas. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 103 langkah 3: memsimulasikan model untuk mempermudah penyelesaian terhadap model gvrp yang telah dibangun, maka dilakukan penerapan model gvrp terhadap penetuan rute kendaraan pada permasalahan distribusi batik tulis madura. selanjutnya disimulasikan dalam bentuk grafik dengan menginput konstanta-kontanta sesuai model gvrp terhadap penetuan rute kendaraan. langkah 4: penarikan kesimpulan pada tahap ini dilakukan penarikan kesimpulan dari hasil simulasi terkait dengan masalah distribusi batik tulis madura. 3. hasil dan pembahasan 3.1 data jarak pamekasan ke 25 kota data jarak ini merupakan jarak kota pamekasan ke 25 kota di jawa timur yang merupakan tujuan dari produk batik tulis madura. data yang didapat dengan menggunakan aplikasi google map ini disajikan pada gambar 1 berikut. gambar 1. jarak pamekasan ke 25 kota 3.2 model gvrp model gvrp diformulasikan sebagai berikut [4]: meminimumkan ∑ ∑ 𝐶𝑖𝑗 𝑋𝑖𝑗 𝑛 𝑗=1 𝑛 𝑖=1 dengan fungsi pembatas (kendala) sebagai berikut [5]: 1. mendefinisikan bahwa untuk setiap cluster selain 𝑉0, hanya ada satu kendaraan yang menuju ke beberapa titik lain pada cluster lain ∑ ∑ 𝑋𝑖𝑗 = 1, 𝑝 ≠ 0, 𝑝 ∈ 𝐾 𝑗∈𝑉\𝑉𝑝𝑖∈𝑉𝑝 ∑ ∑ 𝑋𝑖𝑗 = 1, 𝑝 ≠ 0, 𝑝 ∈ 𝐾 𝑗∈𝑉𝑝𝑖∈𝑉\𝑉𝑝 2. harus ada maksimum m kendaraan yang keluar dari dan m kendaraan yang masuk ke kota asal (stasiun) ∑ 𝑋0𝑖 𝑛 𝑖=1 ≤ 𝑚 ∑ 𝑋𝑖0 𝑛 𝑖=1 ≤ 𝑚 3. setiap kendaran yang masuk dan keluar harus sama untuk setiap cluster ∑ 𝑋𝑖𝑗 𝑖∈𝑉\𝑉𝑝 = ∑ 𝑋𝑖𝑗 𝑖∈𝑉\𝑉𝑝 , 𝑗 ∈ 𝑉𝑝 , 𝑝 ∈ 𝐾 4. lintasan dari tiap cluster 𝑉𝑝 ke cluster 𝑉𝑟 𝑤𝑝𝑟 = ∑ ∑ 𝑋𝑖𝑗 𝑗∈𝑉𝑟𝑖∈𝑉𝑝 5. pembatasan kapasitas 𝑞𝑟 ≤ 𝑄 𝑞𝑟 = ∑ 𝑑𝑖𝑖∈𝑉𝑟 , 𝑟 ∈ 𝐾 6. beban kendaraan setelah meninggalkan cluster 𝑢𝑝 − ∑ 𝑞𝑟 𝑊𝑝𝑟 ≤ 𝑄 𝑟∈𝐾,𝑟≠𝑝 , 𝑝 ≠ 0 , 𝑝 ∈ 𝐾 pada model yang telah dibentuk keputusannya adalah untuk merancang sebuah rute yang meminimalkan biaya distribusi maka perhitungan manual dilakukan dengan mencoba semua kemungkinan dari variabel tersebut. untuk perhitungan manual ini digunakan data dua puluh lima kabupaten yang dipilih berdasarkan kapasitas maksimum sebuah mobil yang masuk ke kabupaten tersebut. kapasitas maksimum sebuah mobil adalah 6000 buah batik. untuk mendapatkan rute yang optimal diperlukan tahap solusi awal. tahap inisialisasi dilakukan untuk mendapatkan rute awal. langkah pengerjaan tahap insialisasi pada tabel 1 berikut. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 104 tabel 1. solusi awal untuk perhitungan manual kemudian 25 kabupaten akan di cluster menjadi 8 cluster sesuai dengan jarak dan kapasitas maksimal mobil yang dapat memasuki wilayah kabupaten yang ada di jawa timur. dari pengclusteran yang telah dilakukan berdasarkan jarak dan kapasitas maksimal mobil dengan jumlah kendaraan sebanyak empat buah dan kapasitas maksimal mobil untuk batik sebanyak enam ribu buah diperoleh hasil disajikan pada tabel 2 berikut tabel 2. pengclusteran di wilayah jawa timur setelah dilakukan perbaikan rute kemudian dilakukan perhitungan ulang menggunakan software matlabi maka dihasilkan selisih rata-rata dari perhitungan adalah 8,71% dari semula seperti disajikan dalam tabel 3 berikut. tabel 3. prosentase penurunan biaya jumlah biaya sebelum dilakukan perbaikan rute sebesar rp. 6,969,480.00. kemudian setelah dilakukan perbaikan rute menggunakan gvrp, jumlah biaya menjadi rp. 6,365,500.00 atau ratarata turun 8.71%. 4. kesimpulan dalam penelitian ini disimpulkan sebagai berikut: 1. jumlah cluster optimal dari 25 kabupaten di jawa timur adalah 8 cluster. 2. dari hasil modifikasi model pada gvrp menghasilkan efisiensi biaya sebesar 8.71% yaitu menjadi rp. 6,365,500.00 dari awal sebelum dilakukan perbaikan rute sebesar rp. 6,969,480.00. cluster 7 mempunyai nilai efisiensi paling tinggi, sedangkan cluster 6 mempunyai nilai efisiensi terendah. referensi [1] r. baldacci, e. bartolini, g. laporte. (2008), “some applications of the generalized vehicle routing problem”, le cahiers du gerad, g2008-82. [2] g.ghiani, g.improta. (2000), “an efficient transformation of the generalized vehicle routing problem “european journal of operational research, 122 (2000) 11-17 [3] petrica c. pop, i. kara, a. h. marc. (2012), “new mathematical models of the generalized vehicle routing problem and extensions”, applied mathematical modelling, 36 (2012) 97– 107 [4] p.c. pop, o. matei and h. valean. (2011),” an e_cient soft computing approach to the generalized vehicle routing problem”, advances in intelligent and soft computing, 87 (2011), pp. 281-289. [5] i. kara, t. bektas. (2003), “ integer linear programming formulation of the generalized vehicle routing problem”, in: proc. of the 5th euro/informs joint international meeting. [6] a lubab, ah asyhar, m hafiyusholeh, dc rini, y farida. volcanic ash flow modelling as an early warning system to national disaster (kelud eruption 2014). journal of theoretical and applied information technology (2016) 86 (3), 472 [7] m hafiyusholeh, ah asyhar, r komaria, “aplikasi metode nilai eigen dalam analytical hierarchy process untuk memilih tempat kerja”. jurnal matematika "mantik", 2015 [8] adyanti, d.a., asyhar, a.h., novitasari, d.c., lubab, a., and hafiyusholeh, m. “forecasts marine weather on java sea using hybrid methods: ts-anfis. proceeding 2017 4 th international conference on electrical, computer science and informatics. 19-21 september 2017, yogyakarta, indonesia. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 39 ketunggalan titik tetap di ruang dislocated quasi b-metrik pada pemetaan siklik malahayati 1 matematika, uin sunan kalijaga, malahayati_01@yahoo.co.id1 abstrak ruang dislocated quasi b-metrik (ruang metrik-dqb) pertama kali diperkenalkan oleh klin-eam dan suanoom pada tahun 2015. mereka berhasil membuktikan sifat ketunggalan titik tetap di ruang metrik-dqb pada pemetaan siklik dengan syarat memenuhi kondisi kontraksi banach siklik. selanjutnya, pada tahun 2016 dolicanin dkk menunjukkan bahwa sifat ketunggalan titik tetap di ruang metrik-dqb dapat dibuktikan tanpa mensyaratkan pemetaan tersebut memenuhi kondisi kontraksi banach siklik. kedua pernyataan tersebut dibuktikan ekuivalen dalam paper ini. kata kunci: ruang dislocated quasi b-metrik, pemetaan siklik, titik tetap. abstract the quasi b-metric dislocated space (dqb-metric space) was first introduced by klin-eam and suanoom in 2015. they had been proven the uniqueness of the fixed point in the dqb-metric space on cyclic mapping that provide the cyclic banach contraction conditions. furthermore, in 2016 dolicanin et al showed that the fixed point singularity properties in the dqb-metric space can be proven without requiring the mapping to satisfy the cyclic metrics banach contraction conditions. both statements are proved equivalent in this paper. keywords: dislocated quasi b-metric space, cyclic mapping, fixed point 1. pendahuluan konsep titik tetap pertama kali digunakan oleh seorang matematikawan perancis yang bernama poincare di tahun 1895 sampai 1900. pada tahun 1910, brouwer membuktikan bahwa terdapat paling sedikit satu titik tetap untuk pemetaan kontinu polyhedral di ruang berdimensi berhingga, dan inilah yang membuka banyak peluang penelitian mengenai teori titik tetap. khususnya pada tahun 1922, seorang matematikawan polandia yang bernama banach memberikan suatu inovasi penting yang dikenal dengan prinsip pemetaan kontraksi banach dengan menggunakan metode iterasi picard. hasil dari pembuktian tersebut telah menjadi aset penting untuk matematika terapan, karena aplikasi dari teori tersebut berperan besar pada berbagai cabang ilmu matematika yang meliputi persamaan diferensial, persamaan integral dan bidang ilmu matematika lainnya, terutama yang melibatkan logika pemrograman dan teknik elektronik. terutama dalam beberapa dekade terakhir, dengan perkembangan komputer banyak orang memiliki berbagai aplikasi dengan memanfaatkan berbagai metode iterasi untuk mendekati titik tetap dan oleh karena itu banyak para peneliti membuat terobosan dan berusaha untuk menyempurnakan teori-teori yang sudah ada. konsep ruang metrik quasi (quasimetric spaces) diperkenalkan oleh wilson [13] pada tahun 1931 dan di tahun 2000 hitzler dan seda [14] memperkenalkan ruang metrik terasing (dislocated metric spaces), selanjutnya zeyada [15] mengembangkan hasil yang telah diperoleh oleh wilson, hitzler dan seda dengan memperkenalkan ruang metrik quasi terasing (dislocated quasi metric spaces). sedangkan di tahun 1989, bakhtin [24] memperkenalkan ruang b-metrik (b-metric spaces) selanjutnya jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 40 czerwik [25] membuat hasil dari bakhtin tersebut lebih dikenal dan mengembangkannya di tahun 1998. sampai saat ini, telah banyak peneliti yang mengembangkan konsep ruang bmetrik, diantaranya ruang quasi b-metrik, ruang seperti b-metrik (b-metric like spaces) dan ruang seperti quasi b-metrik (quasi -b-metric like spaces). belakangan pembahasan yang sedang menarik dan terus mengalami kemajuan adalah mengenai ruang dislocated quasi b-metrik (dislocated quasi b-metric spaces) yang diperkenalkan oleh klin-eam dkk [2] pada tahun 2015, selanjutnya ruang ini di singkat ruang metrik dqb untuk mempermudah penulisan. sebelum membahas lebih jauh tentang ruang metrik dqb, perlu diingat kembali definisi pemetaan siklik. definisi 1.1. diberikan himpunan-himpunan 𝐴 dan 𝐵 masing-masing adalah himpunan bagian tak kosong dari ruang metrik (𝑋, 𝑑) dan 𝑇: 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵. pemetaan 𝑇 disebut pemetaan siklik apabila 𝑇(𝐴) ⊆ 𝐵 dan 𝑇(𝐵) ⊆ 𝐴. berikut ini diberikan definisi ruang metrik dqb. definisi 1.2 diberikan himpunan tak kosong 𝑋 dan suatu konstanta 𝑠 ≥ 1. fungsi 𝑑: 𝑋 × 𝑋 → [0, ∞) memenuhi kondisi: (𝑑1). jika 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) = 0 maka 𝑥 = 𝑦, untuk semua 𝑥, 𝑦 ∈ 𝑋; dan (𝑑2). 𝑑(𝑥, 𝑦) ≤ 𝑠 {𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)}, untuk semua 𝑥, 𝑦, 𝑧 ∈ 𝑋; pasangan (𝑋, 𝑑) disebut ruang metrik dislocated quasi b-metrik (dislocated quasi b-metric spaces) atau disingkat ruang metrik dqb dan bilangan 𝑠 disebut koofisien dari ruang metrik dqb (𝑋, 𝑑). berikut ini diberikan contoh ruang metrik dqb. contoh 1.3 diberikan 𝑋 = ℝ dan didefinisikan 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + |𝑥| 𝑛 + |𝑦| 𝑚 , 𝑛, 𝑚 ∈ ℕ\{1}, 𝑛 ≠ 𝑚. maka (𝑋, 𝑑) merupakan ruang metrik dqb dengan koefisien 𝑠 = 2. bukti: diambil sembarang 𝑥, 𝑦, 𝑧 ∈ ℝ. (𝑑1). jika 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) = 0 maka diperoleh 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + |𝑥| 𝑛 + |𝑦| 𝑚 = 0 maka |𝑥 − 𝑦|2 = 0, |𝑥| 𝑛 = 0, dan |𝑦| 𝑚 = 0 berarti 𝑥 = 𝑦. (𝑑2). akan dibuktikan: 𝑑(𝑥, 𝑦) ≤ 𝑠 {𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)}. perhatikan bahwa: 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + |𝑥| 𝑛 + |𝑦| 𝑚 = |𝑥 − 𝑧 + 𝑧 − 𝑦|2 + |𝑥| 𝑛 + |𝑦| 𝑚 ≤ (|𝑥 − 𝑧| + |𝑧 − 𝑦|)2 + |𝑥| 𝑛 + |𝑦| 𝑚 ≤ 2(|𝑥 − 𝑧|2 + |𝑧 − 𝑦|2) + |𝑥| 𝑛 + |𝑦| 𝑚 + |𝑧| 𝑚 + |𝑧| 𝑛 ≤ 2 {(|𝑥 − 𝑧|2 + |𝑥| 𝑛 + |𝑧| 𝑚 ) + (|𝑧 − 𝑦|2 + |𝑧| 𝑛 + |𝑦| 𝑚 )} = 2{𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)} dengan memilih 𝑠 = 2, maka pernyataan terbukti.∎ selanjutnya akan diberikan definisi barisan konvergen dan barisan cauchy di ruang metrik dqb. definisi 1.4 diberikan ruang metrik dqb (𝑋, 𝑑) dan barisan {𝑥𝑛 } di (𝑋, 𝑑). a. barisan {𝑥𝑛 } dikatakan konvergen-dqb (dislocated quasi b-converges) ke 𝑥 ∈ 𝑋 apabila 𝑙𝑖𝑚 𝑛→∞ 𝑑(𝑥𝑛, 𝑥) = 0 = 𝑙𝑖𝑚 𝑛→∞ 𝑑(𝑥, 𝑥𝑛 ) dalam hal ini, 𝑥 disebut limit-dqb barisan {𝑥𝑛 } dan ditulis (𝑥𝑛 → 𝑥) b. barisan {𝑥𝑛 } disebut barisan cauchy-dqb apabila berlaku 𝑙𝑖𝑚 𝑚,𝑛→∞ 𝑑(𝑥𝑛 , 𝑥𝑚 ) = 0 = 𝑙𝑖𝑚 𝑚,𝑛→∞ 𝑑(𝑥𝑚 , 𝑥𝑛) c. ruang metrik-dqb (𝑋, 𝑑) dikatakan lengkap apabila setiap barisan cauchy-dqb merupakan barisan konvergen-dqb di dalam 𝑋. sebelum membahas teorema–teorema utama dalam paper ini, berikut diberikan terlebih dahulu definisi kontraksi banach siklik-dqb. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 41 definisi 1.5 diberikan ruang metrik dqb (𝑋, 𝑑), dengan 𝐴, 𝐵 ⊆ 𝑋. pemetaan siklik 𝑇: 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 dikatakan kontraksi banach siklik-dqb (dqb-cyclic-banach contraction) apabila terdapat 𝑘 ∈ [0, 1 𝑠 ) , 𝑠 ≥ 1 sehingga berlaku 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘 𝑑(𝑥, 𝑦) untuk setiap 𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵. 2. pembahasan pada bagian ini akan dibahas teorema ketunggalan titik tetap di ruang metrik dqb, sebelumnya diberikan lemma yang akan digunakan dalam membuktikan teorema tersebut. lemma 2.1. diberikan ruang metrik dqb lengkap (𝑋, 𝑑) dan {𝐴𝑖 }𝑖=1 𝑝 keluarga himpunan bagian tertutup tak kosong di 𝑋. jika pemetaan 𝑇: ⋃ 𝐴𝑖 𝑝 𝑖=1 → ⋃ 𝐴𝑖 𝑝 𝑖=1 memenuhi kondisi: 1) 𝑇(𝐴𝑖) ⊆ 𝐴𝑖+1, untuk setiap 1 ≤ 𝑖 ≤ 𝑝 dengan 𝐴𝑝+1 = 𝐴1; 2) terdapat 𝑘 ∈ [0, 1 𝑠 ) sehingga untuk setiap 𝑥 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 , berlaku 𝑑(𝑇 2𝑥, 𝑇𝑥) ≤ 𝑘 𝑑(𝑇𝑥, 𝑥), 𝑑(𝑇𝑥, 𝑇 2𝑥) ≤ 𝑘 𝑑(𝑥, 𝑇𝑥) (2.1) maka ⋂ 𝐴𝑖 𝑝 𝑖=1 ≠ ∅. bukti: jika 𝑘 = 0, maka untuk setiap 𝑥 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 berdasarkan pertidaksamaan (2.1), diperoleh: 𝑑(𝑇 2𝑥, 𝑇𝑥) ≤ 0. 𝑑(𝑇𝑥, 𝑥), dan 𝑑(𝑇𝑥, 𝑇 2𝑥) ≤ 0. 𝑑(𝑥, 𝑇𝑥) yaitu, 𝑑(𝑇 2𝑥, 𝑇𝑥) = 𝑑(𝑇𝑥, 𝑇 2𝑥) = 0 selanjutnya, karena (𝑋, 𝑑) merupakan ruang metrik dqb maka berdasarkan definisi ruang metrik dqb (d1) didapat 𝑇 2𝑥 = 𝑇𝑥 atau dengan kata lain, 𝑇(𝑇𝑥) = 𝑇𝑥. hal ini berarti bahwa 𝑇𝑥 adalah titik tetap untuk 𝑇. disisi lain, diketahui bahwa 𝑇(𝐴𝑖) ⊆ 𝐴𝑖+1, untuk setiap 1 ≤ 𝑖 ≤ 𝑝 dengan 𝐴𝑝+1 = 𝐴1; karena 𝑥 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 , maka diperoleh 𝑇𝑥 ∈ ⋂ 𝐴𝑖 𝑝 𝑖=1 . jadi, untuk 𝑘 = 0, maka ⋂ 𝐴𝑖 𝑝 𝑖=1 ≠ ∅. selanjutnya, untuk 𝑘 ∈ (0, 1 𝑠 ) dan untuk 𝑥 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 , dengan menggunakan pertidaksamaan (2.1), diperoleh: 𝑑(𝑇 𝑛+1𝑥, 𝑇 𝑛𝑥) ≤ 𝑘. 𝑑(𝑇 𝑛𝑥, 𝑇 𝑛−1𝑥) ≤ 𝑘2. 𝑑(𝑇 𝑛−1𝑥, 𝑇 𝑛−2𝑥) ≤ ⋯ ≤ 𝑘𝑛. 𝑑(𝑇𝑥, 𝑥) (2.2) selain itu diperoleh pula: 𝑑(𝑇 𝑛𝑥, 𝑇 𝑛+1𝑥) ≤ 𝑘. 𝑑(𝑇 𝑛−1𝑥, 𝑇 𝑛𝑥) ≤ 𝑘2. 𝑑(𝑇 𝑛−2𝑥, 𝑇 𝑛−1𝑥) ≤ ⋯ ≤ 𝑘𝑛. 𝑑(𝑥, 𝑇𝑥) (2.3) dari pertidaksamaan (2.2) dan (2.3) didapat: lim 𝑛,𝑚→∞ 𝑑(𝑇 𝑛𝑥, 𝑇 𝑚𝑥) = 0, dan lim 𝑛,𝑚→∞ 𝑑(𝑇 𝑚𝑥, 𝑇 𝑛𝑥) = 0 dengan kata lain, {𝑇 𝑛𝑥} merupakan barisan cauchy-dqb dalam ruang metrik dqb (⋃ 𝐴𝑖 𝑝 𝑖=1 , 𝑑). perhatikan bahwa ⋃ 𝐴𝑖 𝑝 𝑖=1 ⊆ 𝑋 dan 𝑋 lengkap maka (⋃ 𝐴𝑖 𝑝 𝑖=1 , 𝑑) juga lengkap. oleh karena itu {𝑇 𝑛𝑥} konvergen-dqb ke suatu 𝑧 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 . selain itu, dari kondisi 1) karena 𝑇 𝑛𝑥 ⊆ 𝐴𝑖 , untuk setiap 𝑖 = 1,2, ⋯ , 𝑝 dan karena 𝐴𝑖 tertutup untuk setiap 𝑖 = 1,2, ⋯ , 𝑝, maka 𝑧 ∈ ⋂ 𝐴𝑖 𝑝 𝑖=1 dengan demikian ⋂ 𝐴𝑖 𝑝 𝑖=1 ≠ ∅.∎ berikut ini diberikan teorema yang telah dibahas oleh klin-eam dan suanoom[2], mengenai ketunggalan titik tetap di ruang metrik dqb lengkap yang mensyaratkan pemetaannya memenuhhi kondisi kontraksi banach siklik dqb. teorema 2.2 diberikan ruang metrik dqb lengkap (𝑋, 𝑑) dan himpunan-himpunan tertutup 𝐴, 𝐵 ⊆ 𝑋. jika pemetaan siklik 𝑇 memenuhi kondisi kontraksi banach siklik dqb, maka 𝑇 mempunyai titik tetap tunggal di 𝐴 ∩ 𝐵. selanjutnya diberikan teorema yang telah dibahas oleh dolicanin,dkk [1] yang kemudian dibuktikan ekuivalen dengan teorema 2.2 diatas. teorema 2.3 diberikan ruang metrik dqb lengkap (𝑋, 𝑑) dan bilangan 𝑠 ≥ 1. apabila pemetaan 𝑇: 𝑋 → 𝑋 memenuhi kondisi 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘 𝑑(𝑥, 𝑦) jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 42 untuk setiap 𝑥, 𝑦 ∈ 𝑋, dan untuk suatu 𝑘 ∈ [0, 1 𝑠 ), maka 𝑇 mempunyai titik tetap tunggal di 𝑋. teorema 2.4. teorema 2.2 jika dan hanya jika teorema 2.3 bukti: (⇒) dibentuk himpunan 𝐴𝑖 = 𝑋 untuk setiap 𝑖 = 1,2, ⋯ , 𝑝 seperti yang diketahui pada teorema 2.2. maka jelas teorema 2.3 terbukti. (⇐) ambil sebarang 𝑥 ∈ ⋃ 𝐴𝑖 𝑝 𝑖=1 , maka dari lemma 2.1, barisan {𝑇 𝑛𝑥} adalah barisan cauchy dqb sehingga ⋂ 𝐴𝑖 𝑝 𝑖=1 ≠ ∅. selanjutnya, karena (⋂ 𝐴𝑖 𝑝 𝑖=1 , 𝑑) adalah ruang metrik dqb lengkap dan 𝑇 di (⋂ 𝐴𝑖 𝑝 𝑖=1 , 𝑑) sehingga 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘 𝑑(𝑥, 𝑦), untuk setiap 𝑥, 𝑦 ∈ ⋂ 𝐴𝑖 𝑝 𝑖=1 maka berdasarkan teorema 2.3 𝑇 mempunyai titik tetap tunggal di ⋂ 𝐴𝑖 𝑝 𝑖=1 . dengan kata lain terbukti teorema 2.2.∎ berikut ini diberikan contoh sebagai ilustrasi dari teorema 2.4 diatas. contoh 2.5 diberikan 𝑋 = [−1,1] dan 𝑇𝑥 = − 𝑥 5 . apabila 𝐴 = [−1,0] dan 𝐵 = [0,1] kemudian didefinisikan fungsi 𝑑: 𝑋 × 𝑋 → [0, ∞) dengan 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|2 + |𝑥| 10 + |𝑦| 11 maka (𝑋, 𝑑) merupakan ruang metrik dqb dengan koefisien 𝑠 = 2. dan 𝑇 mempunyai titik tetap tunggal di 𝐴 ∩ 𝐵. bukti: berdasarkan hipotesa, jelas bahwa 𝑋 = 𝐴 ∪ 𝐵 dan 𝑇(𝐴) ⊆ 𝐵, 𝑇(𝐵) ⊆ 𝐴. akan dibuktikan 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘𝑑(𝑥, 𝑦). diambil sembarang 𝑥, 𝑦 ∈ 𝑋 𝑑(𝑇𝑥, 𝑇𝑦) = 𝑑 (− 𝑥 5 , − 𝑦 5 ) = |− 𝑥 5 + 𝑦 5 | 2 + |𝑥| 50 + |𝑦| 55 = 1 5 ( 1 5 |𝑥 − 𝑦|2 + |𝑥| 10 + |𝑦| 11 ) ≤ 1 5 (|𝑥 − 𝑦|2 + |𝑥| 10 + |𝑦| 11 ) ≤ 𝑘 𝑑(𝑥, 𝑦) dengan 𝑘 ∈ [ 1 5 , 1 2 ] ⊆ [0, 1 𝑠 ]. dengan demikian semua kondisi pada teorema 2.3 dipenuhhi, hal ini berarti 𝑇 mempunyai titik tetap tunggal di 𝑋. oleh karena itu, berdasarkan teorema 2.4, 𝑇 mempunyai titik tetap tunggal di 𝐴 ∩ 𝐵. karena 𝐴 ∩ 𝐵 = {0}, maka 0 adalah titik tetap tunggal yang dimaksud.∎ 3. penutup berdasarkan pembahasan yang telah dilakukan, dapat disimpulkan bahwa ketunggalan titik tetap di ruang metrik-dqb lengkap pada suatu pemetaan dapat dibuktikan tanpa mensyaratkan pemetaan tersebut memenuhi kondisi kontraksi banach siklik. referensi [1] dolic'anin-dekic', d. t. 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pgri jombang1, rakhmanahlia.stkipjb@gmail.com1 stkip pgri jombang2, ririn_febriyanti00@yahoo.com2 doi:https://doi.org/10.15642/mantik.2017.3.2.51-56 abstrak keberadaan transportasi umum yang ideal masih menjadi cita-cita banyak kota di indonesia, termasuk kota jombang. tingginya konsumsi masyarakat jombang terhadap kendaraan bermotor, membuat macet jalanan pada daerah sekolah karena kebutuhan menjemput anak ketika jam pulang sekolah. pada penelitian ini disusun sebuah graf berarah dari 3 trayek angkutan perdesaan untuk mendapatkan desain model penjadwalan menggunakan aljabar max-plus. dari hasil analisis terhadap model diperoleh periode keberangkatan angkutan perdesaan ( l ) adalah 10 menit sekali, dengan dua interval waktu keberangkatan awal. jadwal keberangkatan yang disusun memungkinkan angkutan perdesaan selalu ada di setiap titik pertemuan selama jam pulang sekolah. kata kunci: jadwal keberangkatan, angkutan perdesaan, aljabar max-plus abstract the existence of ideal public transportation is still a dream of many cities in indonesia, including the city of jombang. the high consumption of jombang people to motor vehicles, causing traffic jams in the school area. in this research, a directed graph from 3 rural transit routes to get the design of scheduling model using max-plus algebra. from the analysis results to the model obtained the period of departure of rural transport is 10 minutes once, with two initial departure time interval. arranged departure schedules allow rural transport always at every meeting point during school hours. keywords: departure schedule, rural transportation, max-plus algebra 1. pendahuluan transportasi merupakan permasalahan yang selalu dihadapi oleh banyak kota di indonesia terutama kota-kota yang telah maju ataupun kota yang sedang berkembang, baik di bidang transportasi perkotaan maupun transportasi regional antar kota [1]. fenomena yang terjadi di banyak kota besar adalah tuntutan ekonomi yang perpedoman pada pendidikan masyarakat. tingginya nilai yang diperoleh seseorang di sebuah sekolah menjadi modal utama seseorang dalam mencari pekerjaan. tuntutan inilah yang membuat masyarakat berlomba-lomba untuk menyekolahkan putra putri mereka di sekolah yang terbaik di lingkungannya. tak sedikit masyarakat yang mengupayakan sekolah bahkan diluar lingkungannya atau dengan kata lain lebih jauh dari tempat tinggalnya selama mereka masih mampu mengantar putra putri mereka sampai di sekolah. memusatnya sekolah tervaforit di kota jombang membuat peningkatan volume kendaraan sering terjadi pada daerah persekitaran sekolah utamanya pada jam pulang sekolah. menumpuknya kendaraan wali murid di sekitaran sekolah untuk menjemput putraputri mereka menyebabkan banyak jalan di jombang padat. ini merupakan fenomena yang jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 52 biasa namun berbahaya, karena penumpukan kendaraan di sekitar sekolah membuat masyarakat menjadi lengah tentang peraturan berlalu lintas. banyaknya kendaraan bermotor dan ketidakhati-hatian menambah jumlah kecelakaan yang sering terjadi di jombang dan melibatkan siswa sma/smp yang belum memiliki sim. terciptanya sistem transportasi yang dapat menjamin pergerakan manusia dan/atau barang secara lancar, aman, cepat, murah, dan nyaman merupakan tujuan utama pembangunan dalam sektor transportasi [2]. bagitu juga tujuan pembangunan sektor transportasi di kabupaten jombang. usaha yang telah dilakukan diantaranya adalah mentertibkan armada angkutan pedesaan yang keluar masuk terminal kepuhsari dan pembangunan infrastruktur seperti halte di beberapa titik ramai. pada penelitian sebelumnya, nahlia telah mengkaji rancangan trayek baru di kota jombang untuk mengatasi permasalahan peningkatan volume kendaraan di jam pulang sekolah [3]. berdasarkan hasil yang diperoleh, yaitu sebuah desain penjadwalan, diperoleh sebuah skenario waktu keberangkatan yang cukup cepat dengan asumsi kendaraan yang digunakan adalah bus sekolah (bus khusus untuk pelajar). rute yang diberikan pun menjangkau seluruh titik ramai di kota jombang. namun, rancangan tersebut masih dinilai sulit direalisasikan mengingat pengadaan bus sekolah bukanlah sesuatu yang mudah. pada penelitian ini, digunakan 3 trayek angkutan pedesaan yang dinilai dalam kondisi yang optimal untuk dioperasikan di wilayah jombang [4] sehingga harapannya solusi penjadwalan dapat lebih mengoptimalkan kinerja angkutan pedesaan di ketiga trayek tersebut. 2. transportasi umum di kota jombang sebagai daerah yang strategis, yaitu berada pada jalur utama yang menghubungan antar kota di jawa timur, jombang sebenarnya telah memiliki infrastruktur pendukung transportasi yang laik. namun, keberadaan angkutan pedesaan sebagai satu-satunya transportasi umum yang menghubungkan antar daerah di jombang bahkan antar kabupaten di sekitar jombang dirasa semakin menyusut dari tahun ke tahun. dari 25 trayek angkutan pedesaan yang beroperasi di jombang [5], 6 diantaranya sudah tidak beroperasi sejak tahun 2000, sedangkan trayek lainnya masih beroperasi meskipun jumlah armadanya semakin sedikit. menyusutnya jumlah armada ini dikarenakan rendahnya minat masyarakat akan angkutan pedesaan yang beroperasi di jombang. tentu saja ada banyak alasan mengapa minat masyarakat menjadi sangat rendah terhadap angkutan pedesaan ini. worldbank dan dirjenhubdat mengeluarkan standart yang harus dipenuhi untuk kinerja angkutan umum diantaranya adalah waktu perjalanan pergi pulang, frekuensi kendaraan, faktor muat, selisih waktu (headway), jumlah trip dan jarak tempuh per kendaraan per hari, waktu siklus (cycle time), jumlah penumpang per kendaraan per hari, jumlah kendaraan dan waktu sirkulasi [4]. pada penelitian ini akan disusun rancangan model penjadwalan angkutan pedesaan di jombang yang meliputi 3 trayek yang dilalui oleh angkutan pedesaan dengan kode b, c dan g. ketiga trayek ini digunakan karena berdasarkan penelitian oleh mubarok, ketiga angkutan pedesaan inilah yang memenuhi kriteria cukup pada penilaian kinerjanya. selanjutnya untuk mempermudah menyusun rancangan model desain penjadwalan, dari ketiga trayek disusun sebuah graph berarah sederhana sebagaimana ditunjukkan oleh gambar 1 berikut ini. gambar 1. graph berarah dari ketiga trayek tabel 1. waktu tempuh dan alokasi jumlah armada jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 53 dari gambar 1, diperoleh 3 titik pertemuan yaitu titik b (jalan gus dur), titik d (ploso) dan titik g (halte sambong). ketiga titik ini memungkinkan penumpang untuk berpindah dari satu trayek ke trayek lainnya. pada penelitian ini bobot graph ditentukan berdasarkan waktu tempuh antara masingmasing titik. data lebih lengkap disajikan pada tabel 1 berikut ini. var ro ut e from to trav eling time (min utes) the num ber of arma da x1 1 terminal (a) jl. gus dur (b) 15 2 x2 1 jl. gus dur (b) pasar gudo (c) 30 3 x9 1 pasar gudo (c) halte sambong (g) 30 3 x10 1 halte sambong (g) terminal (a) 20 2 x1 2 terminal (a) jl. gus dur (b) 15 2 x3 2 jl. gus dur (b) ploso (d) 30 3 x4 2 ploso (d) ngusikan (e) 20 2 x7 2 ngusikan (e) ploso (d) 15 2 x8 2 ploso (d) halte sambong (g) 20 2 x10 2 halte sambong (g) terminal (a) 20 2 x1 3 terminal (a) jl. gus dur (b) 15 2 x3 3 jl. gus dur (b) ploso (d) 30 3 x5 3 ploso (d) tapen (f) 20 2 x6 3 tapen (f) ploso (d) 20 2 x8 3 ploso (d) halte sambong (g) 20 2 x10 3 halte sambong (g) terminal (a) 20 2 3. aljabar max-plus definisi 1. definisi aljabar max-plus[6]     diberikan dengan adalah himpunansemua bilangan real dan . pada didefinisikan operasi berikut : , , max , dan . def def def def r r r r x y r x y x y x y x y                himpunan matriks di dalam aljabar maxplus dinyatakan dalam max n mr  . untuk n n didefinisikan  1,2,3,..., def n n . elemen dari matriks max n ma r  pada baris kei dan kolom kej dinyatakan dengan ija untuk dani n j m  . dalam hal ini matriks a ditulis sebagai 1,1 1,2 1, 2,1 2,2 2, ,1 ,2 , . m m n n n m a a a a a a a a a a              ada kalanya elemen ija juga dinotasikan sebagai   , , , . i j a i n j m  penjumlahan matriks max, n ma b r  dinotasikan oleh a b didefinisikan oleh     , ,, , ,max , untuk dan i j i ji j i j i j a b a b a b i n j m       untuk matriks max maxdan n p p ma r b r   perkalian matriks a b didefinisikan sebagai     , ,, 1 , ,max , untuk dan p i k k ji j k i k k j k p a b a b a b i n j m          misalkan matriks max n na r  , suatu graph berarah dari matriks a adalah    ,a e vg . graph  ag mempunyai n titik, himpunan semua titik dari  ag dinyatakan oleh v . suatu garis dari titik j ke titik i ada bila ija  , garis ini dinotasikan jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 54 oleh  ,j i . himpunan semua garis dari graph  ag dinotasikan oleh e. bobot dari garis  ,j i adalah nilai dari ija yang dinotasikan oleh   , max, .i jw j i a r  bila ,i ja  , maka garis  ,j i tidak ada [7]. algoritma untuk menentukan nilai eigen dan vektor eigen dari matriks max n na r  dilakukan secara berulang dari bentuk persamaan linear      1 1 1 m p p x k a x k p       dengan m adalah jumlah maksimum armada di setiap trayek. teorema 2.[7] bila untuk sebarang keadaan awal  0x  sistem persamaan (*) memenuhi    x p c x q  untuk beberapa bilangan bulat danp q dengan 0p q  dan beberapa bilangan real c , maka   lim t k x k k         dengan c p q    . selanjutnya  adalah suatu nilai eigen dari matriks a dengan vektor eigen diberikan oleh      1 1 p q p q i i x q i          v . berdasarkan teorema 2, menginspirasi suatu algoritma untuk mendapatkan nilai eigen sekaligus vektor eigen dari suatu matriks persegi a yang dikenal dengan algoritma power [5], yaitu sebagai berikut: 1. mulai dari sebarang vektor awal  0x  . 2. iterasi persamaan (*) sampai ada bilangan bulat 0p q  dan bilangan real c sehingga suatu perilaku periodik terjadi, yaitu    x p c x q  . 3. hitung nilai eigen c p q    . 4. hitung vektor eigen      1 1 p q p q i i x q i          v . 4. model penjadwalan transportasi umum di kota jombang sebelum disusun model maka terlebih dahulu akan dilakukan sinkronisasi keberangkatan armada angkutan pedesaan di masing-masing variabel. tujuan sinkronisasi adalah untuk mendapatkan rancangan waktu tunggu yang dibutuhkan agar rancangan model yang akan disusun mewakili keadaan yang dimodelkan. aturan sinkronisasi yang digunakan adalah sebagai berikut: 1. trayek 1. a. keberangkatan armada ke-(k+1) dari a menuju ke b menunggu kedatangan armada ke-(k-1) dari g menuju ke a. b. keberangkatan armada ke-(k+1) dari b menuju ke c menunggu kedatangan armada ke-(k-1) dari a menuju b. c. keberangkatan armada ke-(k+1) dari c menuju g menunggu kedatangan armada ke-(k-2) dari b menuju ke c. d. keberangkatan armada ke-(k+1) dari g menuju ke a menunggu kedatangan armada ke-(k) dari c menuju g dan kedatangan armada ke-(k-2) dari d menuju ke g. 2. trayek 2. a. keberangkatan armada ke-(k+1) dari a menuju ke b menunggu kedatangan armada ke-(k-1) dari g menuju ke a. b. keberangkatan armada ke-(k+1) dari b menuju ke d menunggu kedatangan armada ke-(k-1) dari a menuju ke b. c. keberangkatan armada ke-(k+1) dari d menuju ke e menunggu kedatangan armada ke-(k-2) dari b menuju ke d dan kedatangan armada ke-(k-1) dari e menuju ke d serta kedatangan armada ke-(k-1) dari f menuju ke d. d. keberangkatan armada ke-(k+1) dari e menuju ke d menunggu kedatangan armada ke-(k-1) dari d menuju ke e. e. keberangkatan armada ke-(k+1) dari d menuju ke g menunggu kedatangan armada ke-(k-2) dari b menuju ke d dan kedatangan armada ke-(k-1) dari e menuju jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 55 ke d serta kedatangan armada ke-(k-1) dari f menuju ke d. f. keberangkatan armada ke-(k+1) dari g menuju ke a menunggu kedatangan armada ke-(k-2) dari c menuju ke g dan kedatangan armada ke-(k-1) dari d menuju ke g. 3. trayek 3. a. keberangkatan armada ke-(k+1) dari a menuju ke b menunggu kedatangan armada ke-(k-1) dari g menuju ke a. b. keberangkatan armada ke-(k+1) dari b menuju ke d menunggu kedatangan armada ke-(k-1) dari a menuju ke b. c. keberangkatan armada ke-(k+1) dari d menuju ke f menunggu kedatangan armada ke-(k-2) dari b menuju ke d dan kedatangan armada ke-(k-1) dari e menuju ke d serta kedatangan armada ke-(k-1) dari f menuju ke d. d. keberangkatan armada ke-(k+1) dari f menuju ke d menunggu kedatangan armada ke-(k-1) dari d menuju ke f. e. keberangkatan armada ke-(k+1) dari d menuju ke g menunggu kedatangan armada ke-(k-2) dari b menuju ke d dan kedatangan armada ke-(k-1) dari e menuju ke d serta kedatangan armada ke-(k-1) dari f menuju ke d. f. keberangkatan armada ke-(k+1) dari g menuju ke a menunggu kedatangan armada ke-(k-1) dari c menuju ke g dan kedatangan d menuju ke g. berdasarkan aturan sinkronisasi seperti yang telah diuraikan, maka dapat dikonstruksi model keseluruhan trayek angkutan pedesaan sebagai berikut:                                                               1 10 2 1 3 1 4 3 6 7 5 3 6 7 6 5 7 4 8 3 6 7 9 2 10 9 1 20 1 1 15 1 1 15 1 1 30 2 20 1 15 1 1 30 2 20 1 15 1 1 20 1 1 15 1 1 30 2 20 1 15 1 1 30 2 1 30 2 20 x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k x k                                                               8 1x k  berdasarkan tabel 1 jumlah variabel adalah 10 variabel dan jumlah armada maksimum adalah 3 armada pada setiap trayek, maka 10 dan 3n m  . sehingga ada 3 buah matriks pa dengan  1,2,3p  dan masingmasing berukuran 10 10 . matriks  1 10,10a  karena tidak ada pemberangkatan yang menunggu kedatangan armada yang berangkat kedua, kemudian matriks 2a dan 3a adalah 2 20 15 15 20 15 20 15 20 15 20 15 20 a                                                                                                               3 30 30 30 30 30 a                                                                                                                      jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 56 maka diperoleh matriks representatif yang menggambarkan sinkronisasi dari ketiga trayek angkutan pedesaan yang memenuhi      1 1 1 m p p x k a x k p       dengan matriks a adalah sebagai berikut:         1 2 3 10,10 10,10 10,10 10,10 a a a a e e               selanjutnya dengan bantuan gui maxplus pada software scilab, dengan menerapkan teorema 2, diperoleh nilai eigen 10l = . interpretasi dari nilai eigen ini adalah bahwasanya periode keberangkatan atau waktu tunggu angkutan perdesaan di masing-masing titik adalah setiap 10 menit sekali. selanjutnya karena vektor eigen yang diperoleh dapat dibagi ke dalam dua interval waktu keberangkatan, maka terdapat dua jadwal keberangkatan bagi seluruh armada angkutan perdesaan di ketiga trayek tersebut. jadwal keberangkatan angkutan perdesaan interval pertama dengan waktu keberangkatan awal menit ke-0 yaitu angkutan pedesaan 2 3 7 9, , danx x x x sedangkan untuk interval kedua dengan waktu keberangkatan awal menit ke-5 yaitu angkutan perdesaan 1 4 5 6 8 10, , , , danx x x x x x . 5. kesimpulan hasil penelitian menunjukkan bahwa aljabar max-plus dapat diterapkan pada permasalahan penjadwalan armada angkutan perdesaan di jombang, yaitu trayek b,c dan g, dan menghasilkan waktu tunggu selama 10 menit disetiap titik. selanjutnya interval waktu yang diperoleh dibagi menjadi dua waktu keberangkatan yang dapat menjamin keberadaan angkutan umum disetiap titik pada jam pulang sekolah. 6. penghargaan ucapan terima kasih kami sampaikan kepada direktorat riset dan pengabdian masyarakat, direktorat jenderal penguatan riset dan pengembangan, kementerian riset, teknologi, dan pendidikan tinggi atas dana yang diberikan untuk mendukung pelaksanaan penelitian ini. sesuai dengan kontrak penelitian, nomor:086/sp2h/p/k7/km/2016, penelitian ini merupakan penelitian yang lolos pembiayaan pada skem penelitian dosen pemula (pdp) tahun pendanaan 2017. referensi [1] j. r. joesoef, a. prasetia, and u. gajayana, “di provinsi jawa timur media trend vol 11 no . 1 maret 2016 , hal 1-19,” vol. 11, no. 1, pp. 1–19, 2016. [2] d. e. kinerja and p. sektoral, “laporan akhir,” 2012. [3] n. rakhmawati, “study of school bus planningby using max-plus interval algebra,” pp. 252–255, 2015. [4] r. m. mubarok, “analisis kinerja angkutan umum kabupaten jombang,” 2013. [5] aminah. s, “transportasi publik dan aksesibilitas masyarakat perkotaan”, 2004. [6] molnárová. m, “generalized matrix period in max-plus algebra”, 404, 345– 366. https://doi.org/10.1016/j.laa.2005.02.033, 2005. [7] fahim. k, et all, “aplikasi aljabar maxplus pada pemodelan dan penjadwalan busway yang diintegrasikan dengan kereta api komuter”, 1(1), 1–6, 2013. https://doi.org/10.1016/j.laa.2005.02.033 contact: faraj y. ishak faraj.ishak@uod.ac department of statistics, collage of administration & economics university of duhok, iraq the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.89-98 jurnal matematika mantik vol. 8, no. 2, november 2022, pp.89-98 issn: 2527-3159 (print) 2527-3167 (online) mixed boundary value problem for nonlinear fractional volterra integral equation faraj y. ishak university of duhok iraq, faraj.ishak@uod.ac article history: received sep 9, 2022 revised, dec 9, 2022 accepted, dec 31, 2022 kata kunci: persamaan volterra, teorema titik tetap krasnoselskii, prinsip kontraksi banach, teori leray-schauder degree. abstrak. artikel ini menyajikan hasil penelitian tentang eksistensi dari solusi persamaan integral fraksional nonlinier tipe volterra dengan kondisi batas campuran, beberapa hipotesis yang diperlukan telah dikembangkan untuk membuktikan keberadaan solusi persamaan yang diusulkan. teorema krasnoselskii, prinsip banach contraction dan teori derajat leray-schauder adalah teorema dasar yang digunakan di sini untuk mencari solusi hasil. dalam artikel ini juga diberikan contoh sederhana dari penerapan hasil persamaannya. keywords: volterra equation, krasnoselskii theorem, banach contraction principle, lerayschauder degree theory. abstract. in this paper we present the existence of solutions for a nonlinear fractional integral equation of volterra type with mixed boundary conditions, some necessary hypotheses have been developed to prove the existence of solutions to the proposed equation. krasnoselskii theorem, banach contraction principle and leray-schauder degree theory are the basic theorems used here to find the results. a simple example of application of the main result is presented. how to cite: f. y. ishak, “mixed boundary value problem for nonlinear fractional volterra integral equation”, j. mat. mantik, vol. 8, no. 2, pp. 89-98, december 2022. https://doi.org/10.15642/mantik.2021.7.1.9-19 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 89-98 90 1. introduction fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order, the fractional calculus may be considered an old and yet novel topic. recently, fractional differential equations have been of great interest. this is because of both the intensive development of the theory of fractional calculus itself and its applications in various sciences, such as physics, mechanics, chemistry, engineering.[1],[2],[3]. integral equations appear in many engineering and scientific fields, such as unsteady aerodynamics, viscoelasticity, fluid dynamics, lots of population growth systems, neural network analysis, mathematical analysis of particle diffusion in an unstable fluid, heat conduction in memory resources, transmission lines, population dynamics theory, nuclear reactors, inheritance systems. for details, see [4], [5], [6], [7], [8]. boundary value problems have different applications. in addition to the previously mentioned fields, this type of problem appears in chemical engineering sciences, models of electromagnetic systems and thermoelectric theory. for more detailed information on boundary conditions, see [9], [10]. for more details on local and non-local boundary conditions, see [11],[12],[13],[14],[15],[16]. feng, zhang and yang [13] in 2011 studied the existence and multiplicity solution to the nonlocal boundary value problem, fixed-point theorems in the cone were the main tool to prove the solutions. in 2014 nyamoradi and alaei [17] employed the guo – krasnoselskii fixed point theorem in a cone to study the existence of solution to a new fractional nonlocal mixed boundary value problem. in 2022 ishak [18] investigated the existence solution for a fractional bvp of the first sort with hadamard type and three-point boundary conditions using krasnoselskii zabriko theorem and banach contraction principle. it is also well known that fixed-point theorems have been applied to various boundary value problems to show the existence of solutions; for example, see [3],[5]. however, this researcher’s remains not enough compared to the broad applications of this type of equations. the aim of this paper is to fill this gap. in this study we will investigate the existence and uniqueness solutions for the boundary value problem: 𝑫 𝒄 𝜶𝒙(𝒕) = 𝒈(𝒕) + ∫ 𝝍(𝒕, 𝒔)𝝓(𝒕, 𝒔, 𝒙(𝒔))𝒅𝒔 𝒕 −∞ 𝑥(0) = 𝑎𝑥(𝛾) + 𝑏, 𝑥 ́(0) = ω , 0 ≤ 𝑡 ≤ 1 , 1 < 𝛼 ≤ 2 (1) where 𝑫 𝒄 indicates the caputo fractional operator 𝝓: [𝟎, 𝟏]𝘅[𝟎, 𝟏]𝘅𝐗 → 𝐗 is a given continuous function in banach space (𝐗 ,...) and 𝐂 = 𝐂([𝟎, 𝟏], 𝐗) is banach space of all continuous functions from [0,1]→ 𝐗 endowed with the norm denoted by ‖. ‖, 𝒂 , 𝒃, 𝛚 [𝟎, +∞), 𝝍: [𝟎, 𝟏]𝘅[𝟎, 𝟏] → 𝐗 is a given kernel , 𝒈: [𝟎, 𝟏] → [𝟎, +∞) is known continuous function. 2. preliminaries in this section we will mention some basic definitions in fractional calculus. definition 2.1: [19] the fractional integral of order q is defined by 𝐼𝑞 𝑓(𝑡) = 1 γ(𝑞) ∫ (𝑡 − 𝑠)𝑞−1 𝑡 0 𝑓(𝑠)𝑑𝑠, 𝑞 > 0 provided the integral exists. faraj y. ishak mixed boundary value problem for nonlinear fractional volterra integral equation 91 definition 2.2: [19] the fractional derivative of order q is defined by 𝐷𝑞 𝑓(𝑡) = 1 γ(𝑛 − 𝑞) ( 𝑑 𝑑𝑡 ) 𝑛 ∫ (𝑡 − 𝑠)𝑛−𝑞−1 𝑡 0 𝑓(𝑠)𝑑𝑠, 𝑛 − 1 < 𝑞 ≤ 𝑛, 𝑞 > 0, provided the right-hand side is pointwise defined on (0, +∞). lemma 2.1: [18] for α, β > 0, then the following relation hold: 𝐷𝛼 𝑡𝛽 = γ(𝛽 + 1) γ(𝛽 + 1 − 𝛼) 𝑡𝛽−𝛼−1, 𝛽 > 𝑛 𝑎𝑛𝑑 𝐷𝛼 𝑡𝑘 = 0, 𝑘 = 0,1, … , 𝑛 − 1 lemma 2.2: [18] let 𝛼 > 0, then the differential equation 𝑐 𝐷0+ 𝛼 𝑥(𝑡) = 0 has a unique solution 𝑥(𝑡) = 𝑐0 + 𝑐1𝑡 + ⋯ 𝑐𝑛−1𝑡 𝑛−1, 𝑐𝑖 ∈ 𝑅, 𝑖 = 1,2, … , 𝑛, where 𝑛 − 1 < 𝛼 ≤ 𝑛 in view of lemma 2.2, it follows that 𝐼𝑞 𝑐 𝐷𝑞 𝑥(𝑡) = 𝑥(𝑡) + 𝑐0 + 𝑐1𝑡 + ⋯ 𝑐𝑛−1𝑡 𝑛−1, 𝑐𝑖 ∈ 𝑅, 𝑖 = 1,2, … , 𝑛 theorem 2.1: [5] (krasnoselskii fixed point theorem) let m be a closed convex and nonempty subset of a banach space x. let a, b be the operators such that (i) ax + by ∈ m whenever x, y ∈ m (ii) a is compact and continuous (iii) b is a contraction mapping. then there exists z ∈ m such that z = az + bz. theorem 2.2: (arzela -ascoli theorem) let 𝛺 be a compact hausdorff metric space. then 𝑀 ⊂ 𝐶(𝛺) is relatively compact ⟺ 𝑀 is uniformly bounded and uniformly equicontinuous. lemma 2.3: given 𝑓 ∈ 𝐶(0,1) ∩ 𝐿(0,1), the unique solution of (1) is: 𝑥(𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 𝑓(𝑠)𝑑𝑠 + 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) 𝑓(𝑠)𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 + 𝜔𝑡 (2) where, 𝑓(𝑡) = 𝑔(𝑡) + ∫ 𝜓(𝑡, 𝑠)𝜙(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠 𝑡 −∞ proof: in view of lemma (2.2) the fractional differential equation (1) is equivalent to the integral equation: 𝑥(𝑡) = 𝐼0+ 𝛼 𝑓(𝑡) + 𝑐0 + 𝑐1𝑡 𝑥(𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 𝑓(𝑠)𝑑𝑠 + 𝑐0 + 𝑐1𝑡 where 𝑐0, 𝑐1 ∈ 𝑅. from the boundary conditions (1), we have 𝑐1 = 𝜔 and 𝑐0 = 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) 𝑓(𝑠)𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 89-98 92 by substituting 𝑐0, 𝑐1 in 𝑥(𝑡), the proof will be completed . we will need the following hypotheses: (h1) for 𝐾 ∈ 𝑅+ the inequality holds: ‖𝜙(𝑡, 𝑠, 𝑥2) − 𝜙(𝑡, 𝑠, 𝑥1)‖ ≤ 𝐾‖𝑥2 − 𝑥1‖, for all 𝑡, 𝑠 ∈ [0,1], 𝑥1, 𝑥2 ∈ 𝑋 (h2) ‖𝜙(𝑡, 𝑠, 𝑥)‖ ≤ 𝐿 for all 𝑡, 𝑠, 𝑥 ∈ [0,1]𝗑[0,1]𝗑x, l ∈ 𝑅+ further ‖𝑔(𝑡)‖ ≤ 𝜉 , ‖𝜓(𝑡, 𝑠)‖ ≤ 𝛿𝑒 −𝜆(𝑡−𝑠) for all 𝜉, 𝛿, 𝜆 ∈ 𝑅+ 3. main result proves of theorems of existence and uniqueness solution for equation (1) will be given in this section. theorem 3.1: suppose that 𝜙: [0,1]𝗑[0,1]𝗑 x → x is continuous and fulfilled h1and h2, if: 𝑎𝛿 𝐾𝛾𝛼 𝜆(1 − 𝑎)γ(𝛼 + 1) < 1 (3) then equation (1) has at least one solution. proof: let 𝜑𝑟 = {𝑥 ∈ 𝐶: ‖𝑥‖ ≤ 𝑟} where: 𝑎𝛾𝛼 (𝜆𝜉 + 𝛿𝐿) 𝜆(1 − 𝑎)γ(𝛼 + 1) + 𝜆𝜉 + 𝛿𝐿 𝜆γ(𝛼 + 1) + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 + 𝜔 ≤ 𝑟 define tow mapping f, g on 𝜑𝑟 s.t. (𝐹𝑥)(𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 𝑓(𝑠)𝑑𝑠 (𝐺𝑥)(𝑡) = 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) 𝑓(𝑠)𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 + 𝜔𝑡 for 𝑥, 𝑦 ∈ 𝜑𝑟, by (h2) we obtain: ‖(𝐹𝑥)(𝑡) + (𝐺𝑥)(𝑡)‖ ≤ ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) ‖𝑔(𝑠)‖𝑑𝑠 + 𝑡 0 ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) (∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ 𝑡 0 ‖𝜙(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠)𝑑𝑠 + | 𝑎 1−𝑎 | ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) ‖𝑔(𝑠)‖𝑑𝑠 + 𝑎 1−𝑎 𝛾 0 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) ( 𝛾 0 ∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠)𝑑𝑠 + | 𝑎𝜔𝛾+𝑏 1−𝑎 | + |𝜔𝑡| ≤ 𝜉 γ(𝛼+1) + 𝛿 𝜆 ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝐿𝑑𝑠 𝑡 0 + 𝑎𝜉𝛾𝛼 (1−𝑎)γ(𝛼+1) + 𝑎𝛿 𝜆(1−𝑎) ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) 𝐿𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔𝑡 ≤ 𝜉 γ(𝛼+1) + 𝛿𝐿 𝜆γ(𝛼+1) + 𝑎𝜉𝛾𝛼 (1−𝑎)γ(𝛼+1) + 𝑎𝛿𝐿𝛾𝛼 𝜆(1−𝑎)γ(𝛼+1) + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔 ≤ 𝑎𝛾𝛼(𝜆𝜉+𝛿𝐿) 𝜆(1−𝑎)γ(𝛼+1) + 𝜆𝜉+𝛿𝐿 𝜆γ(𝛼+1) + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔 ≤ 𝑟 which means 𝐹𝑥 + 𝐺𝑥 ∈ 𝜑𝑟 . faraj y. ishak mixed boundary value problem for nonlinear fractional volterra integral equation 93 since f is continuous because ϕ is continuous, we have to prove that 𝐹 is compact. (𝐹𝑥)(t) is uniformly bounded on 𝜑𝑟 , as: ‖(𝐹𝑥)(𝑡)‖ ≤ 𝜆𝜉 + 𝛿𝐿 𝜆γ(𝛼 + 1) since 𝜙 is bounded on [0,1]𝗑[0,1]𝗑𝜑𝑟 let: 𝜙𝑚𝑎𝑥 = 𝑠𝑢𝑝⏟ 𝑠,𝑡,𝑥(𝑡)∈[0,1]𝗑[0,1]𝗑𝜑𝑟 ‖𝜙(𝑡, 𝑠, 𝑥(𝑡))‖ then for 𝑡1, 𝑡2 ∈ [0,1] we get ‖(𝐹𝑥)(𝑡2) − (𝐹𝑥)(𝑡1)‖ ≤ 1 γ(𝛼) ∫ ‖((𝑡2 − 𝑠) 𝛼−1 − (𝑡1 − 𝑠) 𝛼−1)𝑓(𝑠)𝑑𝑠‖ 𝑡1 0 + 1 γ(𝛼) ∫ ‖(𝑡2 − 𝑠) 𝛼−1𝑓(𝑠)𝑑𝑠‖ 𝑡2 𝑡1 ≤ 1 γ(𝛼) ∫ ‖((𝑡2 − 𝑠) 𝛼−1 − (𝑡1 − 𝑠) 𝛼−1) (𝑔(𝑠) + 𝑡1 0 ∫ 𝜓(𝑡, 𝑠)𝜙(𝑡, 𝑠, 𝑥(𝑠)) 𝑡 −∞ 𝑑𝑠)‖ 𝑑𝑠 + 1 γ(𝛼) ∫ ‖(𝑡2 − 𝑠) 𝛼−1 (𝑔(𝑠) + 𝑡2 𝑡1 ∫ 𝜓(𝑡, 𝑠)𝜙(𝑡, 𝑠, 𝑥(𝑠)) 𝑡 −∞ 𝑑𝑠)‖ 𝑑𝑠 ≤ 𝜉|(𝑡2 − 𝑡1) 𝛼 − 𝑡2 𝛼 | γ(𝛼 + 1) + 𝛿𝜙𝑚𝑎𝑥 |(𝑡2 − 𝑡1) 𝛼 − 𝑡2 𝛼 | 𝜆γ(𝛼 + 1) + 𝜉𝑡1 𝛼 γ(𝛼 + 1) + 𝛿𝜙𝑚𝑎𝑥 𝑡1 𝛼 𝜆γ(𝛼 + 1) − 𝜉|(𝑡2 − 𝑡1) 𝛼 | γ(𝛼 + 1) − 𝛿𝜙𝑚𝑎𝑥 |(𝑡2 − 𝑡1) 𝛼 | 𝜆γ(𝛼 + 1) ≤ ( 𝜉 γ(𝛼 + 1) + 𝛿𝜙𝑚𝑎𝑥 𝜆γ(𝛼 + 1) )|𝑡1 𝛼 − 𝑡2 𝛼 | which is independent of 𝑥 there for (𝐹𝑥)(𝑡 ) is relatively compact on 𝜑𝑟 ,by arzelaascoli’s theorem (𝐹𝑥)(𝑡 ) is compact in 𝜑𝑟 . for 𝑥, 𝑦 ∈ 𝜑𝑟 and 𝑡 ∈ [0,1], by h1 we have: ‖(𝐺𝑥)(𝑡) − (𝐺𝑦)(𝑡)‖ ≤ 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) ( 𝛾 0 ∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠)𝑑𝑠 − 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) ( 𝛾 0 ∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠)𝑑𝑠 ≤ 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) ( 𝛾 0 ∫ 𝛿𝑒 −𝜆(𝑡−𝑠)‖𝜙(𝑡, 𝑠, 𝑥(𝑠)) − 𝜙(𝑡, 𝑠, 𝑦(𝑠))‖ 𝑡 −∞ 𝑑𝑠)𝑑𝑠 ≤ 𝑎𝛿 𝜆(1−𝑎) ∫ (𝛾−𝑠)𝛼−1𝐾 γ(𝛼) ‖𝑥 − 𝑦‖𝑑𝑠 𝛾 0 ≤ 𝑎𝛿 𝐾𝛾𝛼 𝜆(1−𝑎)γ(𝛼+1) ‖𝑥 − 𝑦‖ (4) jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 89-98 94 it follows from (4) that (𝐺𝑥)(𝑡) is contraction mapping, this completes the prove. theorem 3.2: suppose that 𝜙: [0,1]𝗑[0,1]𝗑x → x is continuous and fulfilled (h1). if: ω = 𝛿𝐾 + 𝑎𝛿𝐾(𝛾𝛼 − 1) 𝜆(1 − 𝑎)𝛤(𝛼 + 1) < 1 (5) then equation (1) has a unique solution. proof: let φ: 𝐶 → 𝐶 is defined as: (φ𝑥)(𝑡) = ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 𝑓(𝑠)𝑑𝑠 + 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) 𝑓(𝑠)𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔𝑡, 𝑡 ∈ [0,1] let 𝑠𝑢𝑝𝑡∈[0,1]|𝜙(𝑡, 𝑠, 0)| = 𝑀 and choose: 𝑟 ≥ 𝜆𝜉+𝛿𝑀 (1−𝛺)λγ(𝛼+1) + 𝑎𝛾𝛼(𝜆𝜉+𝛿𝑀) (1−𝛺)λ𝑎γ(𝛼+1) + 𝑎𝜔𝛾+𝑏 (1−𝛺)(1−𝑎) + 𝜔𝑡 (1−𝛺) (6) it is claimed that φ𝜑𝑟 ⊂ 𝜑𝑟 where: 𝜑𝑟 = {𝑥 ∈ 𝐶: ‖𝑥‖ ≤ 𝑟} in fact, for 𝑥 ∈ 𝜑𝑟 , by (4), (5) and h1 we obtain: ‖(φ𝑥)(𝑡)‖ ≤ ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 ‖𝑔(𝑠)‖𝑑𝑠 + ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠) 𝑑𝑠 + 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) ‖𝑔(𝑠)‖𝑑𝑠 𝛾 0 + 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) (∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠) 𝑑𝑠 𝛾 0 + | 𝑎𝜔𝛾+𝑏 1−𝑎 | + |𝜔𝑡| ≤ 𝜉 𝛤(𝛼+1) + ∫ (𝑡−𝑠)𝛼−1 𝛤(𝛼) ∫ 𝛿𝑒 −𝜆 (𝑡−𝑠) 𝑡 −∞ (‖𝜙(𝑡, 𝑠, 𝑥(𝑠)) − 𝜙(𝑡, 𝑠, 0)‖ + 𝑡 0 ‖𝜙(𝑡, 𝑠, 0)‖)𝑑𝑠 𝑑𝑠 + 𝑎𝜉𝛾𝛼 (1−𝑎)𝛤(𝛼+1) + 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 𝛤(𝛼) (∫ 𝛿𝑒 −𝜆 (𝑡−𝑠) 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠)) − 𝛾 0 𝜙(𝑡, 𝑠, 0)‖ + ‖𝜙(𝑡, 𝑠, 0)‖𝑑𝑠)𝑑𝑠 + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔𝑡 ≤ 𝜉 γ(𝛼+1) + 𝑟𝛿𝐾 λγ(𝛼+1) + 𝛿𝑀 λγ(𝛼+1) + 𝑎𝜉𝛾𝛼 (1−𝑎)γ(𝛼+1) + 𝑟𝑎𝛿𝐾𝛾𝛼 𝜆(1−𝑎)𝛤(𝛼+1) + 𝑎𝛿𝑀𝛾𝛼 𝜆(1−𝑎)𝛤(𝛼+1) + 𝑎𝜔𝛾+𝑏 1−𝑎 + 𝜔𝑡 ≤ 𝑟 ( 𝛿𝐾 + 𝑎𝛿𝐾(𝛾𝛼 − 1) 𝜆(1 − 𝑎)𝛤(𝛼 + 1) ) + 𝜆𝜉 + 𝛿𝑀 λγ(𝛼 + 1) + 𝑎𝛾𝛼 (𝜆𝜉 + 𝛿𝑀) 𝜆(1 − 𝑎)𝛤(𝛼 + 1) + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 + 𝜔𝑡 ≤ 𝑟𝛺 + (1 − 𝛺)𝑟 = 𝑟 now we have to prove that the function φ is contraction. for 𝑥, 𝑦 ∈ 𝐶 and 𝑡 ∈ [0,1], by (5) and h1 we have: ‖(φ𝑥)(𝑡) − (φ𝑦)(𝑡)‖ faraj y. ishak mixed boundary value problem for nonlinear fractional volterra integral equation 95 ≤ ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) (∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠)) − 𝜙(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠) 𝑡 0 𝑑𝑠 + 𝑎 1−𝑎 ∫ (𝛾−𝑠)𝛼−1 γ(𝛼) (∫ ‖𝜓(𝑡, 𝑠)‖ 𝑡 −∞ ‖𝜙(𝑡, 𝑠, 𝑥(𝑠)) − 𝜙(𝑡, 𝑠, 𝑦(𝑠))‖𝑑𝑠)𝑑𝑠 𝛾 0 ≤ ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝐾‖𝑥 − 𝑦‖(∫ 𝛿𝑒 −𝜆(𝑡−𝑠) 𝑡 −∞ 𝑑𝑠 𝑡 0 + 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) 𝐾‖𝑥 − 𝑦‖(∫ 𝛿𝑒 −𝜆(𝑡−𝑠) 𝑡 −∞ 𝑑𝑠)𝑑𝑠 𝛾 0 ≤ 𝛿𝐾‖𝑥 − 𝑦‖ λγ(𝛼 + 1) + 𝑎𝛿𝛾𝛼 𝐾‖𝑥 − 𝑦‖ 𝜆(1 − 𝑎)γ(𝛼 + 1) ≤ 𝛿𝐾 + 𝑎𝛿𝐾(𝛾𝛼 − 1) 𝜆(1 − 𝑎)𝛤(𝛼 + 1) ‖𝑥 − 𝑦‖ ≤ ω‖𝑥 − 𝑦‖ ω < 1 ensure that (φ𝑥)(𝑡) is contractive. therefor the conclusion of the theorem follows from the contraction mapping principle. theorem 3.3: let 𝜙: [0, 𝑇] × [0, 𝑇] × 𝑅 → 𝑅 , and let 𝑘 ∈ 𝑅 such that 0 ≤ 𝑘 < 1 ω , where ω = 𝑡𝛼 γ(𝛼 + 1) + 𝑎𝛾𝛼 (1 − 𝑎)γ(𝛼 + 1) and m > 0 such that |𝜙(𝑡, 𝑠, 𝑥(𝑡))| ≤ 𝑘|𝑥| + 𝑀 for all 𝑡 ∈ [0, 𝑇], 𝑥 ∈ 𝑅 then problem (1) has at least one solution. proof: define an operator ψ: λ → λ as: ψ(𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 𝑓(𝑠)𝑑𝑠 + 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) 𝑓(𝑠)𝑑𝑠 𝛾 0 + 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 + 𝜔𝑡 where λ = 𝐶([0,1], 𝑅) denote to the bansch space of all continuous functions from [0,1] → 𝑅 endowed with the norm defined by ‖𝑥‖ = sup{|𝑥(𝑡)|, 𝑡 ∈ [0,1]}. let us define a fixed-point problem by: 𝑥 = ψ𝑥 (7) now we need to prove the existence of at least one solution 𝑥 ∈ {0, 𝑇] satisfying (7). define a ball ℬ𝑟 ⊂ 𝐶[0, 𝑇] with 𝑟 > 0 as: ℬ𝑟 = {𝑥 ∈ 𝐶[0, 𝑇]: max 𝑡∈[0,𝑇] |𝑥(𝑡)| < 𝑟} where 𝑟 well be given later, then it’s enough to show that ℬ𝑟̅̅̅̅ → 𝐶[0, 𝑇]satisfies: 𝑥 ≠ 𝜎ψ𝑥, ∀𝑥 ∈ 𝜕ℬ𝑟 𝑎𝑛𝑑 ∀𝜎 ∈ [0, 𝑇] (8) let us define 𝐻(𝜎, 𝑥) = 𝜎ψ𝑥, 𝑥 ∈ 𝐶(ℝ), 𝜎 ∈ [0, 𝑇] then by arzesla’-ascoli theorem ℎ𝜎 (𝑥) = 𝑥 − 𝐻(𝜎, 𝑥) = 𝑥 − 𝜎ψ𝑥 is completely continuous if (8) is true then the following jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 89-98 96 leray-schauder degree are well define and by the homotopy invariance of topological degree it follows that: deg(ℎ𝜎 , ℬ𝑟 , 0) = deg(𝐼 − 𝜎ψ𝑥, ℬ𝑟 , 0) = deg(ℎ1, ℬ𝑟 , 0) = deg(ℎ0, ℬ𝑟 , 0) = deg(𝐼, ℬ𝑟 , 0) = 1 ≠ 0,0 ∈ ℬ𝑟. where i denote the unit operator, by non-zero property of the leray-schauder degree ℎ1(𝑥) = 𝑥 − 𝜎ψ𝑥 = 0 for at least one 𝑥 ∈ ℬ𝑟 in order to prove (8) we assume that 𝑥 = 𝜎ψ𝑥 for some 𝜎 ∈ [0, 𝑇]and for all 𝑡 ∈ [0, 𝑇] such that: |𝑥(𝑡)| = |𝜎ψ𝑥 | = ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 (|𝑔(𝑡)| + ∫ |𝜓(𝑡, 𝑠)||𝜙(𝑡, 𝑠, 𝑥(𝑠))|𝑑𝑠 𝑡 −∞ )𝑑𝑠 + 𝑎 1 − 𝑎 ∫ (𝛾 − 𝑠)𝛼−1 γ(𝛼) (|𝑔(𝑡)| + ∫ |𝜓(𝑡, 𝑠)||𝜙(𝑡, 𝑠, 𝑥(𝑠))|𝑑𝑠 𝑡 −∞ )𝑑𝑠 𝛾 0 + | 𝑎𝜔𝛾 + 𝑏 1 − 𝑎 | + |𝜔𝑡 | ≤ (𝜉 + 𝛿(𝑘|𝑥| + 𝑀 𝜆 )( 𝑡𝛼 γ(𝛼 + 1) + 𝑎𝛾𝛼 (1 − 𝑎)γ(𝛼 + 1) ) + 𝑀1 ≤ (𝜉 + 𝛿(𝑘|𝑥| + 𝑀 𝜆 )ω + 𝑀1 which on taking norm (𝑠𝑢𝑝𝑡∈[0,𝑇]|𝑥| = ‖𝑥‖) and solving for ‖𝑥‖ yields: ‖𝑥‖ = ξω(mσ + 1) + 𝑀1 𝜎(𝜎 − ω𝛿𝑘) where 𝑀1 = | 𝑎𝜔𝛾+𝑏 1−𝑎 | + |𝜔𝑡 | , letting 𝑟 = ξω(mσ + 1) + 𝑀1 𝜎(𝜎 − ω𝛿𝑘) + 1 (8) hold, this completes the proof. example: consider the following fractional integrodifferential equation 𝐷3 2⁄ 𝑥(𝑡) = sin(𝑡) + ∫ (2𝑡 + 𝑠)(𝑡 + 𝑥(𝑠))𝑑𝑠 𝑡 −∞ 𝑥(0) = 0.5𝑥(1.25) + 3, �́�(0) = 1.5 (9) comparing (9) and (1), we see that 𝛼 = 3 2⁄ , 𝑔(𝑡) = sin(𝑡) , 𝜓(𝑡, 𝑠) = 2𝑡 + 𝑠, 𝜙(𝑡, 𝑠, 𝑥(𝑡)) = 𝑡 + 𝑥(𝑠), 𝛾 = 1.25, 𝑎 = 0.5, 𝑏 = 3, 𝜔 = 1.5. if we choose 𝜉 = 1, 𝛿 = 1, 𝜆 = 5 , 𝐾 = 2, then h1 holds, and 𝛿𝐾 + 𝑎𝛿𝐾(𝛾𝛼 − 1) 𝜆(1 − 𝑎)𝛤(𝛼 + 1) = (1)(2) + (0.5)(1)(2)(1.253 2⁄ − 1) (5)(1 − 0.5)( 1.3293) < 1 that is, (5) holds. thus, by theorem 3.2, equation (9) has a unique solution. faraj y. ishak mixed boundary value 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[19] z. wang, “existence and uniqueness of solutions for a nonlinear fractional integrodifferential equation with three-point fractional boundary conditions,” vol. 1, 2016. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 23 pemodelan matematika infiltrasi air pada saluran irigasi alur muhammad manaqib matematika uin syarif hidayatullah jakarta, muhammad.manaqib@uinjkt.ac.id abstrak air merupakan salah satu kebutuhan utama dalam kegiatan pertanian, karena tanpa air yang cukup tanaman pertanian tidak akan berproduksi optimal. cara untuk mencukupi air pada tanaman pertanian adalah dengan irigasi. salah satu metode irigasi yang banyak digunakan pada pertanian di dunia adalah metode irigasi alur (furrow). air masuk/infiltrasi ke dalam tanah dari dasar alur dan dinding alur menuju daerah perakaran tanaman. kekompleksan proses infiltrasi air dalam tanah membuat analasis infiltrasi dengan percobaan labolatorium sulit dilakukan dan memerlukan biaya yang besar. alternatif yang dapat digunakan adalah dengan pemodelan matematika. makalah ini membahas tentang pemodelan matematika infiltrasi air pada saluran irigasi alur berbentuk trapesium. model matematika ini berberbentuk masalah syarat batas (msb) dengan domain sebuah penampang melintang saluran irigai alur yang tertutup dan terbatas. persamaan pengaturnya diperoleh dari persamaan richard, yang selanjutnya ditransformasikan menggunakan tranformasi kirchoff dan variabel tak berdimensi menjadi persamaan helmholtz termodifikasi. sedangkan syarat batasnya berbentuk syarat batas campuran neuman dan robin. kata kunci : irigasi alur, infiltrasi, persamaan helmholtz termodifikasi. abstract water is one of the main necessity of agricultural activities, because without enough water agricultural crops will not be produced optimally. the way to insufficient water in agricultural crops is irrigation. one of the irrigation methods which is used on agriculture in the world is furrow irrigation method. water gets into the soil from the bottom of the furrow and furrow’s wall towards the root zone of the plants. the complexity of the water infiltration process in the ground makes infiltration analysis by laboratory experiment difficult to do and needs substantial cost. the alternative way which can do is with mathematical modeling. this paper discusses about mathematical modeling of water infiltration in furrow irrigation channel trapezoidal in shape. this mathematical modeling is shaped boundary condition problem with a cross section of a closed and limited line of irrigation. governing equation obtanined from richard equation which then transformed using kirchoff transformation and non dimensional variable into the modified helmholtz equation. while, the boundary condition is shaped mixture neuman and robin boundary condition. keywords: furrow irrigation, infiltration, modified helmholtz equation. 1. pendahuluan air adalah kebutuhan kebutuhan pokok manusia. sekitar 70% dari permukaan bumi adalah air, tetapi bukan berarti persediaan air untuk kebutuhan manusia berlimpah, karena 97,5% air tersebut adalah air laut yang tidak bisa langsung digunakan, perlu diolah dengan teknologi tinggi untuk dapat digunakan. sisanya 2,5% berupa air tawar yang 99,7%-nya terdapat di dalam perut bumi dan hanya 0,3%-nya yang berada di permukaan [1]. hal tersebut mengindikasikan bahwa air yang dapat digunakan oleh manusia untuk kebutuhan domestik seperti minum, mandi, mencuci, serta kebutuhan untuk pertanian dan industri sangat mailto:muhammad.manaqib@uinjkt.ac.id1 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 24 terbatas. oleh karena itu, dalam menggunakan air harus bijak dan hemat. terdapat tiga sektor utama pengguna air dunia, yaitu sektor pertanian sebesar 71%, sektor domestik 18%, dan sektor industri 8% [6]. sektor pertanian memiliki porsi terbesar dalam penggunaan air. hal tersebut dikarenakan jumlah penduduk dunia yang terus bertambah, akibatnya kebutuhan bahan makanan juga semakin meningkat. besarnya kebutuhan air untuk pertanian juga mengindikasikan bahwa efisiensi penggunaan air pada sektor ini cukup rendah [6]. diperlukan peningkatan efisiensi penggunaan air pada sektor pertanian agar ketersediaan air dunia dapat dijaga. salah satu kebutuhan utama dalam kegiatan pertanian adalah air, karena tanpa air yang cukup tanaman pertanian tidak akan berproduksi optimal, bahkan mati kekurangan air. salah satu cara mencukupi air pada tanaman pertanian adalah dengan irigasi. menurut [11] irigasi adalah usaha penyediaan, pengaturan, dan pembuangan air irigasi untuk menunjang pertanian yang jenisnya meliputi irigasi permukaan, irigasi rawa, irigasi air bawah tanah, irigasi pompa, dan irigasi tambak. salah satu jenis irigasi permukaan yang cukup efisien dalam penggunaan air adalah metode irigasi alur (furrow). air masuk/infiltrasi ke dalam tanah dari dasar alur dan dinding alur menuju daerah perakaran tanaman. salah satu hal yang penting pada sistem irigasi alur adalah distribusi air dalam tanah. permasalahan yang muncul adalah air yang tidak sampai pada daerah perakaran tanaman maupun kandungan air pada daerah perakaran yang kurang ataupun berlebih. jika tanaman kekurangan air maka akan mati, sedangkan jika berlebih akan terjadi pemborosan penggunaan air ataupun mati untuk jenis tanaman yang tidak tahan air berlebih. diketahui bahwa tanah yang terletak dekat dengan alur lebih banyak mengandung air daripada tanah yang jauh dari alur. selain itu proses infiltrasi air dalam tanah juga melibatkan berubahan keadaan dan kandungan air dalam tanah. hal ini mengindikasikan bahwa proses infiltrasi bukan masalah yang sedehana akan tetapi cukup kompleks. kekompleksan proses infiltrasi air dalam tanah membuat analasis infiltrasi dengan percobaan labolatorium sangatlah sulit. selain itu, penelitian dengan percobaan labolatorium cukup mahal karena peralatan yang mahal dan memerlukan banyak waktu karena data yang harus diperoleh secara reguler. alternatif yang dapat digunakan adalah dengan pemodelan matematika, yang selanjutnya dapat dianalisis proses infiltrasinya [13]. 2. kajian teori 2.1 hukum darcy hukum perembesan air di dalam tanah atau infiltrasi air dalam tanah pertamakali di dipelajari oleh henry darcy seorang ilmuwan dari perancis pada tahun 1856 [8]. hukum darcy menyatakan bahwa flux air q (berdimensi l/t) adalah sebanding dengan hydraulic head gradient, ∇𝐻. secra matematis hukum darcy dinyatakan dalam persamaan berikut q k h  (1) dengan k adalah hydraulic conductivity. tanda negatif pada (1) mengindikasikan bahwa aliran air yang melewati tanah kecepatannya akan berkurang. 2.2 persamaan richard hukum darcy memberikan model matematika infiltrasi air dalam media berporous yang jenuh air. selanjutnya dari hukum darcy tersebut l.a. richard mengembangkannya menjadi model matematika infiltrasi air dalam media berporous yang tidak jenuh air atau begantung pada waktu. richard mengembangkan hukum darcy dengan mengubah hydraulic conductivity menjadi fungsi dari suction potential dan fungsi dari kadar air [8], sehingga diperoleh ( ) ,q k h  (2) ( ) .q k h  (3) suction potential ( ) adalah potensial dari gaya yang timbul dari interasi antara tanah dan air. sedangkan moisture content ( ) adalah perbandingan antara berat air dengan berat butir tanah. diperhatikan bahwa yang akan diamati adalah infiltrasi air dalam tanah sehingga ruang di atas permukaan tanah tidak diperhatikan. oleh karena itu digunakan sistem koordinat oxyz dan dipandang sumbu-z berarah kebawah bernilai positif. didefinisikan hydraulik head sebagai energi per unit berat. berdasarkan sistem koordinat yang digunakan didefinisikan hydraulik head h z  (4) substitusi (4) ke (3) diperoleh diperoleh, jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 25 ( z) ( z) ( ) ( ) 1 q k x z k x z                             i j i j (5) misalkan u dan v berturut-turut adalah komponen flux horisontal dan komponen flux vertikal, maka ( ) , ( ) ( )u k v k k x z             (6) didefinisiksn flux normal pada sebuah permukaan dengan vektor normal 1 2( , )n nn yang berarah keluar adalah 1 2 1 2( ) 1 f u n v n k n n x z                  (7) hukum kekekalan massa pada aliran fluida menyatakan bahwa perubahan volume fluida terhadap waktu sama dengan perubahan aliran flux terhadap jarak [8]. hukum kekekalan massa jika diterapkan pada aliran air dalam tanah maka diperoleh, perubahan kandungan air dalam tanah terhadap waktu sama dengan perubahan flux terhadap jarak. secara matematis dapat dituliskan q t    . (8) diketahui bahwa flux yang masuk melewati tanah adalah lebih besar dari pada flux yang keluar melewati tanah, sehingga gradien dari flux benilai negatif. selanjutnya dari (2) dan (5), serta gradien flux yang negatif diperoleh, ( ) 1 ( ) ( ) ( ) , k t x z k k k x x z z z                                                  i j (9) dengan ( )k  adalah hydraulic conductivity yang berdimensi l/t dan  adalah suction potential yang berdimensi l. persamaan (9) inilah yang disebut sebagai persamaan richard yang memrepresentasikan perpindahan air berdimensi dua dalam tanah tidak jenuh. 3. pembahasan 3.1 persamaan pengatur persamaan richard berbentuk persamaan diferensial non linear yang penyelesaiannya sulit dicari, sehingga diperlukan transformasi untuk mengubah (9) kebentuk persamaan yang lebih mudah diselesaikan. persamaan richard akan ditransformasikan menjadi persamaan helmhotz yang berbentuk persamaan diferensial linear. prosedur transformasinya pertama, digunakan transformasi yang diberikan oleh kirchhoff, dilanjutkan transformasi menggunakan model exponensial konduktifitas hidraulik yang diberikan oleh garner, selanjutnya digunakan variabel tak berdimensi [4], dan terakhir digunakan transformasi yang diberikan oleh batu. 1. transformasi kirchhoff menggunakan rumus ( )k s ds     (10) dengan adalah (matric flux potential) (mfp). digunakan (13) maka diperoleh dank k x x z z            (11) 2. model eksponensial dari konduktifitas hidraulik didefinisikan oleh [7], 0 , 0k k e    (12) dengan  adalah sebuah parameter dan 0k adalah konduktifitas hidraulik pada tanah jenuh. diperhatikan bahwa dari (10) dan (12) dapat diperoleh 0 sk e ds k         (13) turunkan (13) terhadap z, maka diperoleh k z z z             . (14) substitusikan (11) dan (14) ke persamaan richard (9) diperoleh 2 2 2 2 t x x z z z x z z                                       (15) penelitian ini akan membahas infiltrasi air pada saluran irigasi jenuh, artinya untuk perubahan waktu yang terjadi, kondisi infiltrasi air tersebut tetap atau tidak bergantung waktu. akibatnya, t   pada (15) dapat diabaikan, sehingga diperoleh 2 2 2 2 2x z z z                 (16) selanjutnya, untuk tranformasi komponen vertikal dan horisontal flux (6), serta flux normal (7) adalah sebagai berikut, jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 26 1 2 , . u v x z f n n x z                    (17) (18) 3. ttransformasi ke bentuk persamaan yang bervariabel tidak berdimensi, yaitu menggunakan variabel tak berdimensi berikut [4]: 0 0 0 0 , , , 2 2 2 2 2 , , . x x z z v l u u v v f f v l v l v l                 (19) dengan 0v adalah flux awal dan l adalah setengah panjang saluran irigasi. menggunakan variabel-variabel tidak berdimensi (19), maka akan diperoleh 0 0 2 . 2 v lz z z z z v lx x x x x                           (20) (21) selanjutnya, dari persamaan (20) dan persamaan (21) dapat diperoleh, 22 2 0 0 2 2 22 2 0 0 2 2 , 2 4 . 2 4 v l v l z z z z v l v l x x x x                                       (22) (23) substitusikan (20)-(23) ke (16), (17), dan (18), diperoleh 2 2 2 2 2 0, z x z            (24) komponen horisontal dan vertikal flux tidak berdimensi dan 2 ,u v x z        (25) serta untuk flux normal tidak berdimensi, 1 2 1 22 . f un vn n n x z             (26) 4. transformasi oleh batu [4] dengan memisalkan .ze (27) berdasarkan persamaan (27) dapat diperoleh turunan pertama dan kedua  terhadap x dan y sebagai berikut, 2 2 2 2 2 2 2 2 , , , 2 z z z z e e x x z x e e x x z z z                                         (28) substitusikan (28) ke persamaan (24), diperoleh 2 2 2 2 2 2 0 (29)ze x z z x                               yang dapat disederhanakan menjadi 2 2 2 2 . x z          (30) persamaan (30) inilah yang disebut sebagai persamaan helmholtz termodifikasi, yang merepresentasikan infiltrasi air pada saluran irigasi jenuh. selanjutnya flux normal tidak berdimensi (26) dengan transformasi (27) menjadi 2 zf e n n          (31) selanjutnya, berdasarkan persamaan (31) dapat diperoleh definisi turunan normal, 2 . zn e f n       (32) akhirnya, diperoleh (30) berupa persamaan helmholtz termodifikasi dengan variabel tak berdimensi  . persamaan (30) dapat diselesaikan dengan metode numerik, salah satunya dengan meb atau drbem. setelah diperoleh nilai  , selanjutnya dapat ditranformasikan kembali ke bentuk awal yaitu 0 0 1 ln ze v l k           . (33) sehingga diperoleh  sebagai suction potential pada infiltrasi saluran irigasi. penelitian ini menggunakan nilai suction potential dan water content ( ) untuk analisis infiltrasi air pada saluran irigasi. berikut diberikan hubungan antara suction potential dan water content [14],     1 1 m s r rn               (34) dengan r dan s berturut-turut adalah residual water content dan saturated water content. nilai ,m dan n adalah parameter yang bergantung pada jenis tanah. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 27 3.2 syarat batas bentuk saluran irigasi alur yang diteliti adalah trapesium, karena bentuk ini lazim digunakan oleh para petani. asumsi-asumsi yang digunakan pada saluran adalah [13] 1. lebar saluran irigasi memiliki lebar yang sama dan memiliki panjang yang cukup panjang, sehingga dalam model ini panjang saluran irigasi diabaikan. 2. saluran irigasi berbentuk trapesium dengan panjang penampang permukaan saluran sebesar 2l. 3. jarak antar titik tengah dua buah saluran yang berdekatan adalah 2(l+d). 4. saluran selalu penuh dengan air. 5. pengaruh dari saluran irigasi lain diabaikan. 6. laju infiltrasi air/besar flux masuk pada permukaan saluran irigasi adalah konstan, yakni sebesar 0v . 7. tidak ada aliran air yang masuk kecuali dari saluran. sesuai dengan asumsi-asumsi tersebut maka koordinat yang digunakan adalah oxyz dengan o adalah pusat saluran dan oz menyatakan kedalaman yang bernilai positif. berikut diberikan gambar 1 sebagai ilustrasi penjelasan di atas. gambar 1. bentuk geometris saluaran irigasi trapesium berdasarkan sifat simetris pada model ini, seperti terlihat pada gambar 1 maka cukup dianalisis pada sebuah daerah 0 x l d   dan 0.z  sehingga dapat didefinisikan domain pada model ini adalah sebuah daerah semi takterbatas 0 x l d   dan 0,z  yang selanjutnya domain ini dimisalkan r [10]. berikut diberikan domain r beserta flux yang ada pada domain tersebut. gambar 2. domain masalah infiltrasi saluran irigasi alur selanjutnya karena tidak ada aliran air yang masuk kecuali pada permukaan saluran maka 0f  pada 0,x  ,x l d  dan 0z  . diasumsikan bahwa pada kedalaman yang takberhingga laju dari mfp mendekati nol, sehingga diperoleh 0 x    dan 0 z    untuk .z  berdasarkan uraian tersebut maka dapat dirumuskan syarat batas sebagai berikut. 0, pada permukaan saluran,f v (35) 2l 0, untuk x l+ddan z=0,f     (36) 0, untukx=l+ddanz 0,f   (37) 3l 0, untukx=0 dan z , 2 f    (38) dan 0, 0, untuk0 x l+d,danz= . x z          (39) berdasarkan sistem koordinat yang digunakan, dapat didefinisikan vektor normal yang berarah keluar domain r pada batas domain sebagai berikut.  , , padapermukaan saluran, 1 2n n n (40)   2l , ,untuk x l+ddanz=0,     n 0 1 (41)  , ,untukx=l+ddanz 0, n 1 0 (42)   3l , ,untukx=0 dan z , 2   n 1 0 (43)  , ,untuk0 x l+ddanz= .   n 0 1 (44) jadi, diperoleh msb (16) dengan syarat batasnya (35)-(39) sebagai model matematika infiltrasi air dalam saluran irisasi alur yang dinyatakan dalam mfp. berikut diberikan gambar 3, sebagai ilustrasi msb tersebut beserta vektor normal pada batas domainnya. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 28 gambar 3.msb infiltrasi irigasi alur dalam mfp seperti pada pembahasan sebelumnya setelah model infiltrasi air dalam bentuk mfp diperoleh, selanjutnya ditransformasikan dalam variabel tak berdimensi (19). oleh karena itu, syarat batas (35)-(39) juga akan ditransformasikan dalam variabel tak berdimensi, hasilnya adalah sebagai berikut. 2 padapermukaan saluran,f l    (45)  0,untuk , 0 2 l f x l d z         (46)  0,untuk dan 0, 2 f x l d z      (47) 3 0,untuk 0 dan , 4 l f x z      (48)   0, 0, untuk0 dan . 2 x z x l d z            (49) persamaan (45)-(49) telah memberikan nilai dari f atau , x z     pada batas domainnya. dari nilai tersebut akan digunakan untuk membentuk syarat batas yang berbentuk syarat batas neuman dan robin dengan memanfaatkan (31), (32) dan vektor normal (40)-(44), diperoleh 2 2 ,padapermukaan,zn e n l         (50)  ,untuk , =0, 2 l x l d z n            (51)  0,untuk dan 0, 2 x l d z n        (52) 3 0,untuk 0dan , 4 l x z n         (53)  ,untuk0 , . 2 x l d z n           (54) domain dalam model ini adalah daerah semi tak berhingga r, sedangakan penyelesaian msb dengan domain semi tak terbatas sulit dilakukan. sehingga perlu dilakukan pembatasan domain dengan mengambil z c , untuk suatu c bilangan real positif pada syarat batas (54)[12]. selanjutnya dapat didefinisikan ruas garis 1 2 3 4, , ,c c c c ,dan 5,c 1 :permukaan dari sal ,uranc (55)  2 : danz=0, 2 l c x l d       (56)  3 : dan0 z c, 2 c x l d      (57) 4 3 l : 0dan z c, 4 c x      (58)  5 : 0 danz=c. 2 c x l d     (59) misalkan 1 2 3 4 5c c c c c c     , maka diperoleh msb model matematika infiltrasi air dalam saluran irigasi alur dalam variabel tak berdimensi dengan domain r yang tertutup dan terbatas oleh kurva c sebagai berikut.       2 2 2 2 , , , , x z x z x z x z          (60) dengan syarat batas, 2 1 2 ,padac ,zn e n l         (61) 2,padac , n      (62) 30,padac , n    (63) 40,padac , n    (64) 5,padac . n      (65) secara sistematis msb tersebut dapat digambarkan sebagai berikut. gambar 4. msb dengan domain r yang tertutup dan terbatas oleh kurva c jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 29 4. penutup model matematika infiltrasi air pada saluran irigasi alur berbentuk msb dengan syarat batas campuran neuman dan robin. sedangkan persamaan pengaturnya berbentuk persamaan helmhotz termodifikasi. referensi [1] ali, m.h., 2010, fundamentals of irrigation and on farm, spinger, new york. [2] amoozegar,et al., 1984, design nomographs for trickle irrigation system, j of irr.and drainage eng., 110, pp. 107-120. [3] ang, whye-teong, 2007, a beginners course in boundary element methods, universal publishers, florida. [4] batu, v, 1978, steady ilfiltration from single and periodik strip source, soil sci.soc. am. j, 42, pp. 544-549. [5] bos, m.g., ell., 2008, water requirements for irrigation and environment, spinger, new york. [6] burton, martin, 2010, irrigation management : principles and practices, cabi, wallingford. [7] gardner, w.r., 1958, some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, soil science, 85: 228-232. [8] hillel, daniel, 2004, introduction to environmental soil physics, universal elsevier science, san diego. [9] katsikadelis, jhon t., 2002, boundary element : theory and applications, elsevier science, oxford. [10] lobo, maria, 2008, dual reciprocity boundary element methods for the solution of a class of ilfiltration problems, faculty of engineering, coputer and mathematical sciences, university of adelaide, doctor disertation. [11] republik indonesia, 2006, peraturan pemerintah tentang irigasi, sekretariat negara, jakarta. [12] solekhudin, imam dan ang, k.c., 2012, suction potential and water absorption from periodic chanel in different types of homogeneous soil, electronic jurnal of boundary element, 10, pp. 42-55. [13] solekhudin, imam, 2013, dual reciprocity boundary element methods for water ilfiltration problems in irrigation, national institude of education, nangyang technological university, doctor dissertation. [14] warrick, a.w., 2002, soil physics companion, crc press, washington d.c. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 96 pemodelan matematika perpindahan panas konveksi campuran (mixed convection) pada pelat horizontal leli deswita matematika fmipa universitas riau pekanbaru, deswital@yahoo.com doi:https://doi.org/10.15642/mantik.2017.3.2.96-100 abstrak penelitian ini mengkaji dan menganalisa model matematika perpindahan panas konveksi campuran (mixed convection) pada pelat horizontal. analisa aliran perpindahan panas ini menggunakan model dalam bentuk sistem persamaan diferensial parsial nonlinear berorde dua yang berdimensi. selanjutnya persamaan ini diturunkan terlebih dahulu ke bentuk persamaan tidak berdimensi, kemudian dirobah ke bentuk sistem persamaan diferensial biasa yang nonlinear, dengan menggunakan similarity transformation. sistem persamaan diferensial biasa yang nonlinear ini, diselesaikan dengan menggunakan metode finite-difference scheme, juga dengan program matematika dengan softwer matlab. hasil yang didapat dari program ini untuk menentukan koefisien pekali geseran kulit (skin friction coefficient)  0f  suhu permukaan (wall tempearture) θ(0), profil kecepatan (velocity profiles)  f  dan profil suhu (temperature profiles)  . kata kunci: finite difference sheme; konveksi campuran; pelat horizontal abstract this study examines and analyzes mathematical model of mixed convection in horizontal plate. the heat transfer uses the model of a two dimensional nonlinear partial differential equations system. then, this equation is derived first into the dimensionless equation form, and then it is changed into system of nonlinear ordinary differential equations form using similarity transformation. this system of nonlinear ordinary differential equations is solved by using the finite-difference scheme method, also with the mathematics program with software matlab. the results obtained form this program is to determine skin friction coefficient  0f  , wall temperature  0 velocity profiles  f  and temperature profiles  . keywords: finite difference sheme; mixed convection; horizontal plate 1. pendahuluan proses perpindahan panas secara konveksi merupakan suatu fenomena yang hanya terjadi di suatu permukaan, dimana keadaan permukaan dan kedudukan permukaan akan mempengaruhi proses konveksi [8]. proses perpindahan panas terjadi serentak dalam proses konveksi, disebabkan dalam proses konveksi fluida menjadi pengantar panas, sehingga besarnya kecepatan fluida yang mempengaruhi banyaknya panas yang disebarkan [14]. perbedaan suhu diantara fluida dengan persekitaran telah menyebabkan proses konveksi terjadi [1]. pemanasan akan menyebabkan fluida menjadi kurang padat mailto:deswital@yahoo.com jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 97 (dense) dan bergerak ke atas, manakala yang lebih padat bergerak ke bawah. proses ini akan berakhir apabila kestabilan panas tercapai. konsep dasar utama yang harus dipahami dalam penelitian yang akan dilakukan ini adalah masalah konsep seperti yang telah dikembangkan oleh [3] dalam penelitian tentang solusi kesamaan untuk aliran syarat batas konveksi campuran pada pelat horizontal permeable (similarity solution for mixed convection of boundary laminer flow on permeable horizontal plate). begitu juga dengan penelitian-penelitian yang telah dilakukan oleh [5]. dimana para peneliti menghasilkan beberapa parameter penting seperti parameter apungan, parameter mikrokutub dan bilangan prandtl dengan dua solusi. selanjutnya [4] dalam penelitian tentang solusi kesamaan untuk aliran syarat batas konveksi bebas pada pelat horizontal (similarity solution of free convection boundary layer flow on a horizontal plate with variable wall temperature). penelitian tersebut adalah berupa kajian secara teori tentang pengaruh apungan terhadap masalah syarat batas (boundary condition) aliran yang disebabkan oleh pemanasan atau pendinginan (heat transfer). dalam penelitian selanjutnya pakar-pakar penelitian melakukan kajian tentang aliran konveksi campuran syarat batas pada baji ( permeable walls). aliran mantap berlapis (laminar steady flow) pada permungkaan benda berbentuk baji dengan menunjukkan penggunaan secara teori syarat batas dan bilangan prandtl. kemudian [10] telah mengkaji konveksi campurann pada aliran fluida micropolar, parameter mikrokutub. selanjutnya [4] telah mengkaji aliran fluida pada pelat horizontal tentang masalah konveksi bebas dan seterusnya. deswital mengkaji masalah aliran fluida pada pelat horizontal tentang konveksi campuran (similarity solution for mixed convection boundary layer flow over a permeable horizontal flat plate) [3]. selain daripada masalah pembentukan lapisan batas yang diselesaikan dengan metoda finite-differenc scheme, terdapat banyak lagi peneliti-peneliti lain mengkaji menggunakan metoda ini antara lain adalah [5], [6], [7], [10], [12], [13], [15] dan [16]. penelitian ini menganalisa aliran perpindahan panas terhadap pelat horizontal dengan menggunakan persamaan diferensial parsial dengan syarat batas dan konveksi campuran pada pelat horizontal hansen [9]. 2. formulasi matematika telah dibicarakan di atas bahwa penelitian ini menggunakan sistem persamaan diferensial parsial nonlinear dengan masalah model matematika perpindahan panas konveksi campuran (mixed conection) dengan syarat batas pada pelat horizontal, adapun bentuk model matematika yang digunakan dapat di tulis sebagai berikut 0      y v x u (1) (2)     ttsg y p     1 0 (3) (4) dengan syarat batas, 0u ,  xvv w ,     kxqyt wy  0 , pada 0y  xuu  ,  tt ,  pp apabila .y (5) didefinisikan parameter dalam bentuk sebagai berikut : lxx  ,  lyy 2/1 re ,  ,re 0 2/1 u      , 0 uxuxu     ,re 0 21 uvxv ww    2 0 uppp   ,    ,re 21lqkttt wo  ,0uuu      .0www qxqxq  (6) dengan lu 0re  adalah nomor reynolds, wo q merupakan characteristik heat flux , sedangkan l adalah characteristik length scale dan 0u adalah characteristik velocity. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 98 substitusikan persamaan (6) ke persamaan (1)-(5) sehingga diperoleh sebagai berikut: 0      y v x u (7) , 2 2 y u x p yx u u              (8) ,0 t y p     (9) . pr 1 2 2 y t y t x t u          (10) dengan syarat batas (6) menjadi 0u ,  xvv w ,     kxqyt wy  0 , pada 0y (11)  xuu  , 0t , 0p apabila .y dengan r p adalah nomor prandtl dan  adalah konstanta konveksi campuran (mixed convection) atau disebut juga parameter keapungan (buoyancy parameter), didefenisikan sebagai berikut . 25 e r gr s (12) dimana   23 0  lklqggr w  merupakan nomor grashof. selanjutnya chellappa dan singh [2] mendefinisikan bentuk persamaan sebagai berikut:    , 1  fx    , 2  pxp     ,3  xt    ,4 yx  (13) dengan  adalah fungsi strim yang didefinisikan sebagai berikut, , y u     . x v     (14) persamaan (13) disubstitusikan ke persamaan (14), selanjutnya disubstitusikan kepersamaan (7)-(11) beserta persamaan (12), kemudian diperoleh persamaan diferensial biasa sebagai berikut, ,0 2 1 2 2 1 2      p n npfnff n f  (15) ,p (16) .0 2 15 2 1 pr 1       f n f n (17) dengan syarat batas (11) dihasilkan   ,00 f   ,0 0ff    10  (18)   ,1f   ,0   0p . 3. hasil dan pembahasan persamaan (15)–(17) dengan syarat batas (18) diselesaikan secara numerik untuk beberapa nilai parameter 0f (suction), nomor prantal rp , parameter n dan  adalah konstanta konveksi campuran atau disebut juga parameter keapungan (buoyancy parameter) dengan menggunakan matoda finite difference schem seperti yang telah dijelaskan dalam keller [11] penyelesaian secara numerik telah menghasilkan pekali geseran kulit (skin friction coefficient)  0f  , suhu permukaan  0 , profil kecepatan (velocity profiles)  f  dan profil suhu (temperature profiles)    . dalam kajian ini nilai n yang diteliti adalah 0.01 dan bebrapa nilai parameter 0f (suction), nilai nomor prandtl yang digunakan adalah 7,0 r p (udara). hasil yang didapati dari kajian ini dapat dilihat pada tabel 3.1, untuk beberapa nilai 0f , dengan nilai n = 0,01 dan r p = 0,7 untuk nilai kritikal parameter c . perubahan pekali geseran kulit (skin friction coefficient)  0f  terhadap  dapat dilihat pada gamabar 3.2 dengan beberapa nilai parameter 0f untuk n = 0,01 dan r p = 0,7. nilai-nilai parameter 0f yang dinyatakan pada gambar 3.2, untuk nilai 00 f (suction), dan 00 f (impermeable plate). perubahan suhu permungkaan  0 tehadap  dapat dilihat pada gambar 3.3. untuk nilai n = 0,01 dan r p = 0,7. selanjutnya pada gambar 3.2, jika nilai 0f meningkat, didapati bahwa pekali geseran kulit (skin friction coefficient)  0f  meningkat. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 99 gambar 3.4 menunjukkan profil kecepatan (velocity profiles)  f  untuk nilai n = 0,01 nilai 7,0 r p dan untuk beberapa nilai parameter 0f seperti yang dinyatakan pada gambar 3.4, untuk 0 0 f (sedutan) dan 00 f (impermeable plate). garis penuh artinya cabang atas dan garis putus-putus artinya cabang bawah. gambar teresbut menunjukkan, untuk nilai 0f meningkat untuk cabang atas didapati profil kecepatan berkurang dan ketebalan lapisan batas semangkin berkurang. gambar 3.5 menunjukkan profil suhu (temperature profiles)    untuk nilai n = 0,01 nilai 7,0 r p dan untuk beberapa nilai parameter 0 f , seperti yang dinyatakan pada gambar 3.5, untuk 00 f (sedutan) dan 00 f (impermeable plate). garis penuh artinya cabang atas dan garis putus-putus artinya cabang bawah. gambar teresbut menunjukkan, untuk nilai 0f meningkat untuk cabang atas didapati profil kecepatan berkurang dan ketebalan lapisan batas semangkin berkurang. tabel 3.1. nilai c untuk beberapa nilai n dan 7,0 r p n f0 c  0,01 0 -0,0020 0,1 -0,0054 0.,2 -0,0103 gambar 3.2. pekali geseran kulit (skin friction coefficient)  0f  sebagai fungsi  untuk beberapa nilai 0f apabila 7,0rp dan n = 0,01 gambar 3.3. suhu permukaan (wall temperature)  0 sebagai fungsi  untuk beberapa nilai 0f apabila 7,0 r p dan n = 0,01. gambar 3.4. profil kecepatan (velocity profiles)  f  untuk beberapa nilai 0f apabila 7,0 r p dan n = 0,01 gambar 3.5. profil suhu (temperature profiles)    untuk beberapa nilai 0f apabila 7,0 r p dan n = 0,01. -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8  f" (0 ) pr = 0.7 n = 0.01 f 0 = 0, 0.1, 0.2, 0.3 1st solution 2nd solution -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 5 10 15 20 25 30 35   (0 ) f 0 = 0.3 f 0 = 0.2 f 0 = 0.1 f 0 = 0 pr = 0.07 n = 0.01 1st solution 2nd solution 0 5 10 15 20 25 -0.2 0 0.2 0.4 0.6 0.8 1 1.2  f' (  ) pr = 0.7 n = 0.01 1st solution 2nd solution f 0 = 0, 0.1, 0.2, 0.3 f 0 = 0, 0.1, 0.2, 0.3 0 5 10 15 20 25 0 5 10 15 20 25 30 35   (  ) pr = 0.7 n = 0.01 1st solution 2nd solution f 0 = 0.3, 0.2, 0.1, 0 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 100 4. kesimpulan model matematika perpindahan panas konveksi campuran (mixed convection) pada pelat horizontal menggunakan model sistem persamaan diferensial parsial nonlinear berorde dua, kemudian dirobah kebentuk persamaan defenrensial biasa nonlinear dengan menggunakan metode finite-difference scheme dan program matematika softwer matlab. hasil yang didapat dari program ini untuk menentukan koefisien pekali geseran kulit (skin friction coefficient)  0f  suhu permukaan (wall tempearture)  0 , profil kecepatan (velocity profiles)  f  dan profil suhu (temperature profiles)    referensi [1] cebeci, t. & brandshaw, p. 1988. physical and computation aspects of convective heat transfer. new york: springer-verlag. [2] chellappa, ak. p. singh. 1989. possible similarity formulations for laminar free convection on a semi-infinite horizontal plate, int. j. engng. scei. 27: 161-167. [3] deswita, l., nazar. r., ishak. a., ahmad. r. & pop. i. 2010. similarity solution for mixed convection boundary layer flow over a permeable horizontal flat plate. applied mathematics and computation 217: 2619-2630. [4] deswita. l., nazar. r., ahmad. r. ishak. a. & pop. i. 2009. similary solution of free convection boundary layer flow on a horizontal plate with variable wall tempe-rature. european journal of scien tific research issn 1450216x vol. 2: 188-1985. [5] fadzilah m. ali, roslindanazar, norihan m. arifin & ioan pop. 2014. mixed con-vection stagnation-point flow on a vertical stretching sheetwith external magnetic field. applied mathematics & mechanics-english edition 35(2): 155-166. [6] fernandez-feria, r., ortega-casanova, j. 2014. a pseudospectral based method of lines for solving integro differential boundary layer equations. application to the mixed con-vection over a heated horizontal plate, applied mathematics and computation. 242, pages 388396 [7] fernandez-feria, del pino, c and fernandegutiérrez , a., view affiliations. separation in the mixed convection boundary layer radial flow over a constanta temperature horizontal plate, physics of fluids 26, 103603. [8] ghebart, b., jaluria, y., mahajan, r.l, sammakia, b. buoyancy induced flows and transport, hemisphere, new york, 1988. [9] hansen, a.g. similarity analyses of boundary value problems in engineering, prentice hal inc, new jersey,1964. [10] kartini ahmad, roslindanazar & ioan pop. 2012. mixed convection in laminar film flow of a micropolar fluid. interna-tional communi cations in heat and mass transfer 39(1): 36-39. [11] keller, h.b. 1970. a new differenc scheme for parabolic promlems. dalam: bramble, j numerical soluctions of partical defferen-tial equations. new york. achademic [12] rajesh sharma, anuar ishak, roslinda nazar & ioan pop. 2014. boundary layer flow & heat transfer over a permeable exponentially shrinking sheet in the presence of thermal radiation and partial slip. journal of applied fluid mech. 7(1): 125-134. [13]rashad, a. m., chamkha, a. j, modather. 2013. mixed convection boundary-layer flow past a horizontal circular cylinder embedded in a poros medium filled with ananofluid under convective boundary condition. november volume 86, pages 380-388. [14] stewartson, k. 1958. on free convection from a horizontal plate, j. app. math. phys. (zamp) 276-281. [15]tham l, nazar r, pop i. 2013. mixed convection boundary layer flow past a horizontal circular cylinder embedded in a porous medium saturated by a nano-fluid: brinkman model. j porous med 16: 445–457 [16] tham l, nazar r, 2014. mixed convec-ion flow about a solid sphere embedded in a porous medium filled with a nano-fluid. sains malaysia 41: 1643-1649 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 57 analisis risk asset portfolio berbasis reward to variability pada saham syariah di indonesia menggunakan nonlinear programming noor saif muhammad mussafi 1 program studi matematika uin sunan kalijaga1, noor.mussafi@uin-suka.ac.id1 doi:https://doi.org/10.15642/mantik.2017.3.2.57-64 abstrak penelitian ini berupaya menganalisis pengembangan metode optimasi saham syariah menggunakan nonlinier programming dalam rangka memberikan alternatif portofolio optimal sebagai referensi dalam meningkatkan kualitas pasar modal syariah di indonesia. desain penelitian yang digunakan adalah deskriptif kualitatif dengan menyajikan suatu data histori saham syariah dalam periode tertentu yang dianalisis dan dimodelkan untuk kemudian dicari solusinya. data dalam penelitian ini adalah informasi harga saham syariah yang tergabung dalam jakarta islamic index (jii). data yang terseleksi dengan analisis reward to variability (rval) kemudian dianalisis menggunakan teori-teori dalam matematika keuangan dan dikembangkan menggunakan quadratic programming. hasil penelitian ini adalah rumusan langkah sistematis memaksimalkan tingkat keuntungan dan meminimalkan tingkat risiko investasi saham syariah yang tergabung dalam jii pada domain waktu januari 2015–desember 2016. penelitian ini juga menyimpulkan bahwa dengan metode tersebut dapat diketahui proporsi dana yang dapat diinvestasikan pada lima emiten terbaik. pada sampel yang diambil, untuk tingkat keuntungan yang diharapkan sebesar 5,5% hingga 7,5%, maka seorang investor disarankan untuk menanamkan sahamnya berturut-berturut kepada akra, icbp, ptpp, tlkm, dan wskt rata-rata sebesar 29,74%; 13,42%; 18,14%; 29,58%; dan 9,1% dengan risiko antara 0.028301593% hingga 0.029386615%. kata kunci: saham syariah; quadratic proramming; tingkat risiko; tingkat keuntungan; rval. abstract this research seeks to analyze the development of syariah stock optimization method using nonlinear programming in order to provide an optimal portfolio as a reference in improving the quality of syariah capital market in indonesia. the research design used is descriptive qualitative by presenting a shariah stock history data within a certain period that is analyzed and modeled for later sought solving. the data in this research is the stock price information syariah incorporated in the jakarta islamic index (jii). selected data with reward to variability (rval) were then analyzed using theories in financial mathematics and developed using quadratic programming. the result of this research is systematic step formulation to maximize profit level and minimize risk level of syariah share investment incorporated in jii in january 2015-december 2016 time domain. this research also concludes that by this method can be known the proportion of funds that can be invested in five best issuers. in the sample taken, for the expected profit level of 5.5% to 7.5%, then an investor is advised to embed its shares consecutively to akra, icbp, ptpp, tlkm and wskt on average of 29.74% ; 13.42%; 18.14%; 29.58%; and 9.1% with risk between 0.028301593% to 0.029386615%. keywords: syariah stock; quadratic proramming; risk; return; rval. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 58 1. pendahuluan investasi menurut islam merupakan kegiatan muamalah yang sangat dianjurkan, karena dengan berinvestasi harta atau aset yang dimiliki seseorang menjadi produktif sehingga mampu mendatangkan manfaat bagi orang lain, dengan syarat penerapannya berpedoman pada prinsipprinsip syariah. bahkan dalam al quran (surat yusuf: 47-48) telah dijelaskan bagaimana seorang nabi yusuf ‘alaihissalam telah melakukan terobosan luar biasa dalam berinvestasi melalui tiga tahapan yaitu penanaman berturut-turut, penyimpanan sebagian hasil tanaman (bahanpangan) untuk mengantisipasi masa paceklik, dan penghematan dalam konsumsi [11]. semenjak pt. bursa efek jakarta (bej) menerbitkan daftar reksadana, saham, dan obligasi syariah dalam jakarta islamic index (jii) pada tahun 2000 yang ditindaklanjuti dengan nota kesepahaman antara bapepam dengan dewan syariah nasional majelis ulama indonesia (dsn-mui) tentang prinsip pasar modal syariah, pasar modal syariah mengalami perkembangan cukup signifikan. analis danareksa sekuritas [9] mengatakan kinerja indeks saham syariah mengungguli kinerja indeks harga gabungan (ihsg) pada awal hingga pertengahan tahun 2016. ada tiga faktor penyebab peningkatan kinerja saham syariah yaitu (1) saham-saham syariah biasanya termasuk good corporate governance (gcg) artinya memiliki kualitas faktor pengawasan yang jauh lebih baik dibandingkan saham konvensional, (2) dari sisi sektor, saham syariah banyak terdiri dari sektor konsumer dan infrastruktur yang notabene mendorong kinerja indeks, dan (3) investor syariah biasanya membeli saham untuk investasi, bukan semata-mata trading sehingga kenaikannya lebih stabil. seperti sifat investasi pada umumnya, terdapat 2 (dua) hal mendasar yang selalu melekat yaitu tingkat keuntungan (return) dan risiko (risk) yang akan dihadapi. keuntungan dan risiko mempunyai hubungan yang kuat dan linear, yaitu jika risiko tinggi maka keuntungan juga akan tinggi, atau sebaliknya [6]. investasi saham pada pasar modal merupakan investasi yang memilki risiko tinggi sehingga jika tidak berhati-hati dapat memungkinkan terjadinya kebangkrutan. untuk itu dalam upaya mencegah masalah tersebut perlu adanya manajemen risiko. salah satu cara dalam manajemen risiko adalah tidak menempatkan investasi hanya pada satu saham saja tetapi melakukan diversifikasi dengan membentuk portofolio saham. dalam membentuk portofolio saham, pertanyaan terbuka dan signifikan bagi setiap investor adalah bagaimana menentukan proporsi dana yang diinvestasikan untuk setiap saham pada suatu portofolio, sehingga keuntungan yang dihasilkan semaksimal mungkin dan risiko yang diambil seminimal mungkin. oleh karena itu, tujuan penelitian ini menganalisis portofolio berbasis reward to variability (rval) menggunakan prinsip nonlinear programming. selanjutnya pendekatan tersebut diterapkan pada aset risiko saham syariah di bursa efek indonesia (bei). 2. pasar modal syariah di indonesia pada tanggal 3 juli 2000, pt bursa efek indonesia bekerjasama dengan pt danareksa invesment management (dim) meluncurkan indeks saham yang dibuat berdasarkan syariah islam, yaitu jakarta islamic index (jii). indeks ini diharapkan menjadi tolak ukur kinerja sahamsaham yang berbasis syariah serta untuk lebih mengembangkan pasar modal syariah. jii terdiri atas 30 saham yang terpilih dari saham-saham yang sesuai dengan syariah islam dan pemilihan sahamnya dilakukan oleh bappepam-lk bekerjasama dengan dewan syariah nasional (dsn) melalui 2 tahap, yaitu seleksi syariah dan seleksi nilai volume transaksi [1]. dalam proses transaksi investasi saham dipengaruhi salah satunya oleh bi rate sebagai bank sentral. bi rate adalah suku bunga kebijakan yang mencerminkan sikap atau stance kebijakan moneter yang ditetapkan oleh bank indonesia dan diumumkan kepada publik. 3. matematika keuangan dalam matematika keuangan dikenal beberapa istilah penting diantaranya yaitu keuntungan, risiko, rataan aritmatik, rataan geometrik, variance, covariance, volatility, dan reward to variability (rval). konsep dasar tersebut berkorelasi langsung dengan istilahjurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 59 istilah dalam dunia keuangan khususnya bursa efek. 3.1 keuntungan dan risiko hal mendasar dalam keputusan investasi adalah tingkat keuntungan yang diharapkan dan risiko [12]. keuntungan merupakan hasil (tingkat pengembalian) yang diperoleh sebagai akibat dari investasi yang dilakukan, secara matematis dapat digunakan pendekatan geometris [8]. pendekatan ini menggunakan kaidah logaritma rasio harga dengan asumsi dividen 𝐷𝑡 adalah nol, secara matematis dinyatakan sebagai berikut: 𝑟𝑡 = 𝑙𝑛 𝐼𝑡 + 𝐷𝑡 𝐼𝑡−1 (1) risiko (risk) adalah tingkat ketidakpastian akan terjadinya sesuatu atau tidak terwujudnya suatu tujuan, pada kurun atau periode waktu tertentu [2]. perhitungan risiko juga dapat dilakukan menggunakan standar deviasi [5] dengan formula: 2 2 1 1 ( ) ( 1) n n i i i i n x x s n n        (2) 3.2 rataan aritmatik, rataan geometrik, variance, covariance, dan volatility dari sekumpulan data emiten dapat dikalkulasi rataan aritmatik tingkat keuntungan 𝑟𝑖𝑡 tiap aset 𝑖 pada periode 𝑡 menggunakan �̅�𝑖 = 1 𝑇 ∑ 𝑟𝑖𝑡 𝑇 𝑡=1 (3) secara konsep rataan geometrik berbeda dengan rataan aritmatik [7]. rataan geometrik merupakan rata-rata keuntungan tahunan yang bersifat konstan dan diaplikasikan pada tahun 𝑡 = 0 hingga 𝑡 = 𝑇 − 1. formula rataan geometrik 𝜇𝑖 adalah 𝜇𝑖 = (∏(1 + 𝑟𝑖𝑡 ) 𝑇 𝑡=1 ) 1 𝑇 − 1 (4) variance 𝜎2 pada variabel (berobot) berganda merupakan kombinasi linear dari covariance dengan bobot tiap aset 𝑤𝑖 dan nilai harapan pada tiap aset 𝑋𝑖 memiliki pola sebagai berikut: 𝜎2 = 𝑣𝑎𝑟 (∑ 𝑤𝑖 𝑋𝑖 𝑛 𝑖=1 ) = 𝑐𝑜𝑣 (∑ 𝑤𝑖 𝑋𝑖 𝑛 𝑖=1 , ∑ 𝑤𝑗 𝑋𝑗 𝑛 𝑗=1 ) = ∑ ∑ 𝑤𝑖 𝑤𝑗 𝑐𝑜𝑣( 𝑛 𝑗=1 𝑛 𝑖=1 𝑋𝑖 , 𝑋𝑗 ) = ∑ 𝑤𝑖 2 𝑣𝑎𝑟(𝑋𝑖 ) 𝑛 𝑖=1 + 2 ∑ ∑ 𝑤𝑖 𝑤𝑗 𝑐𝑜𝑣( 𝑛 𝑗=𝑖+1 𝑛 𝑖=1 𝑋𝑖 , 𝑋𝑗 ) (5) adapun covariance matrix dari dua sebarang aset pada periode t, tingkat keuntungan 𝑟𝑖𝑡 , 𝑟𝑖𝑡 dan rataan aritmatik �̅�𝑖 , �̅�𝑗 dapat dinyatakan sebagai berikut: 𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 1 𝑇 ∑(𝑟𝑖𝑡 − 𝑇 𝑡=1 �̅�𝑖 )(𝑟𝑗𝑡 − �̅�𝑗 ) (6) volatility merupakan perhitungan variasi harga instrumen keuangan berbasis waktu yang berkorelasi dengan standar deviasi. volatility 𝜎 tingkat keuntungan pada tiap aset i secara matematis [4] dinyatakan sebagai 𝜎𝑖 = √𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) (7) 3.3 reward to variability (rval) secara khusus kinerja portofolio dapat diukur dengan sharpe measure atau disebut dengan reward to variability (rval). menurut [7], teknik ini diperkenalkan pertamakali oleh william f. sharpe pada 1966 (perhatikan formula 8). misal ptr merupakan rata-rata return total portofolio dalam periode tertentu, rate bi merupakan aset bebas risiko yang ditetapkan bank indonesia, dan  adalah standard deviasi, maka jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 60 p rate p tr bi rval    (8) 4. teori optimasi dan matlab optimasi secara matematis berarti meminimalkan atau memaksimalkan fungsi tujuan dari beberapa variabel keputusan dengan kendala tertentu [10]. permasalahan optimasi dalam penelitian ini dibatasi pada optimasi diskrit dengan batasan tertentu. program kuadrat memungkinkan memiliki satu atau lebih kendala dalam bentuk persamaan ataupun pertidaksamaan [13]. bentuk umum dari program kuadrat adalah: min𝑥 1 2 𝑥𝑇 𝑄𝑥 + 𝑐𝑇 𝑥 s.s. 𝐴𝑥 = 𝑏 , 𝐶𝑥 ≥ 𝑑 , 𝑥 ≥ 0 (9) matriks 𝐴, 𝐶 ∈ ℝ𝑚𝑥𝑛 dan tiga vektor 𝑏 ∈ ℝ𝑚, 𝑑 ∈ ℝ𝑚, dan 𝑐 ∈ ℝ𝑛 diketahui. adapun 𝑄 merupakan matriks simetris (𝑄𝑖𝑗 = 𝑄𝑗𝑖 ) karena 𝑥𝑇 𝑄𝑥 = 1 2 𝑥𝑇 (𝑄 + 𝑄𝑇 )𝑥. tujuan akhir (9) adalah meminimalkan koefisien vektor 𝑥 pada fungsi tujuan kuadrat 1 2 𝑥𝑇 𝑄𝑥 + 𝑐𝑇 𝑥. program kuadrat dapat digunakan untuk meminimalkan risiko suatu saham yang diinvestasikan pada pasar modal dengan kendala jumlah dana yang diinvestasikan dengan acuan nilai covariance pada suatu data keuntungan [4]. saat ini permasalahan optimasi tidak hanya dapat diselesaikan secara manual namun juga dapat diselesaikan menggunakan bantuan beberapa software, salah satunya adalah matlab. portofolio optimal markowitz mvo pada prinsipnya menggunakan model program kuadrat, yaitu meminimalkan fungsi kuadrat terhadap satu atau lebih fungsi kendala dalam bentuk persamaan ataupun pertidaksamaan. dalam optimization toolbox matlab [3], salah satu sintak yang dapat digunakan dalam menyelesaikan program kuadrat yaitu 𝑥 = quadprog(𝑄, 𝑐, 𝐴, 𝑏, 𝐴𝑒𝑞, 𝑏𝑒𝑞) dimana 𝑄, 𝐴, dan 𝐴𝑒𝑞 adalah matriks sedangkan 𝑐, 𝑏, dan 𝑏𝑒𝑞 merupakan vektor. sintak 𝑥 = quadprog(𝑄, 𝑐, 𝐴, 𝑏, 𝐴𝑒𝑞, 𝑏𝑒𝑞) berarti vektor 𝑥 meminimalkan fungsi kuadrat 1 2 𝑥𝑇 𝑄𝑥 + 𝑐𝑇 𝑥 terhadap dua fungsi kendala yaitu pertidaksamaan 𝐴𝑥 ≤ 𝑏 dan persamaan 𝐴𝑒𝑞. 𝑥 = 𝑏𝑒𝑞. 5. hasil dan pembahasan metode optimasi nonlinear programming merupakan salah satu pendekatan manajemen portofolio untuk mendapatkan hasil optimal. untuk melakukan analisis risk asset portfolio menggunakan metode nonlinear programming diperlukan beberapa langkah sistematis dalam ranah statistik dan pemodelan (lihat gambar 1). gambar 1 flowchart penelitian investasi syariah di pasar modal indonesia identik dengan jakarta islamic index (jii) yang terdiri dari 30 saham syariah yang tercatat di bursa efek indonesia (bei) sebagaimana terlihat pada tabel 1. semua peserta yang tergabung dalam jii tersebut telah memenuhi kriteria syariah yang ditetapkan oleh pt. danareksa invesment management (dim). tabel 1. 30 perusahaan emiten saham syariah no. kode saham nama saham 1 aali astra agro lestari tbk. 2 adhi adhi karya tbk. 3 adro adaro energy tbk. 4 akra akr corporindo tbk. 5 antm aneka tambang tbk. 6 asii astra international tbk. 7 bsde bumi serpong damai tbk. 8 icbp indofood cbp sukses tbk. 9 inco vale indonesia tbk. 10 indf indofoof sukses makmur tbk. 11 intp indocement tunggal prakasa tbk. 12 klbf kalbe farma tbk. 13 lpkr lippo karawaci tbk. 14 lppf semen indonesia (persero) tbk. 15 lsip london sumatra indonesia tbk. 16 mika mitra keluarga karyasehat tbk. 17 myrx hanson internasional tbk. 18 pgas perusahaan gas negara tbk. 19 ptba tambang batubara bukit asam tbk 20 ptpp pembangunan perumahan tbk. 21 pwon pakuwon jati tbk. sorting portofolio tiap emiten berbasis rval aplikasi statistika matematika keuangan pemodelan quadratic programming analisis pemodelan menggunakan matlab proporsi dana beserta resiko dan return jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 61 22 silo siloam international hospital tbk. 23 smgr semen indonesia tbk. 24 smra summarecon agung tbk. 25 ssms sawit sumbermas sarana tbk. 26 tlkm telekomunikasi indonesia tbk. 27 untr united tractors tbk. 28 unvr unilever indonesia tbk. 29 wika wijaya karya tbk. 30 wskt waskita karya tbk. pergerakan harga saham dari 30 perusahaan emiten saham syariah pada kurun waktu januari 2015-desember 2016 dapat diunduh pada [14]. tabel 2. rekapitulasi rval berdasarkan urutan positif terbesar no kode saham return resiko rval 1 tlkm 0,000674409 0,016098664 -0,00105 2 unvr 0,000360023 0,01839513 -0,00121 3 akra 0,000587106 0,019726625 -0,00129 4 wskt 0,00117845 0,02000722 -0,0013 5 icbp 0,000541993 0,019937375 -0,00131 6 ptpp 0,000225039 0,020120157 -0,00132 7 asii 0,000228079 0,022562515 -0,00148 8 indf 0,000126139 0,022676586 -0,00149 9 untr 0,000419695 0,026141486 -0,00171 10 myrx 0,00047359 0,026700548 -0,00175 11 pwon 9,23604e-05 0,026571898 -0,00175 12 lppf 4,77805e-05 0,027122928 -0,00179 13 antm 3,96577e-05 0,030118297 -0,00199 14 ptba 2,05109e-05 0,030392115 -0,00201 15 adro 0,000996861 0,032972075 -0,00214 selanjutnya dilakukan sorting portofolio harian tiap emiten saham syariah selama periode januari 2015-desember 2016 dengan mempertimbangkan risiko (standar deviasi), return (tingkat keuntungan) bernilai positif, dan analisis rval dengan bi rate 6,6% berturut-turut menggunakan formula (1), (2), dan (8) sehingga diperoleh data 15 emiten sebagaimana pada tabel 2 yang diurutkan berdasarkan nilai rval terbesar. hal ini dilakukan karena semakin tinggi nilai rval maka akan semakin baik kinerja saham [7]. bila ditinjau secara grafis (gambar 2), terdapat hubungan antara performa return dan risiko terhadap rval, yaitu pada 6 emiten dengan rval terbaik berturut-turut tlkm, unvr, akra, wskt, icbp, dan ptpp. dalam grafik tersebut (ditandai dengan pelabelan nama emiten) terlihat posisi keenam saham tersebut berada pada area kanan bawah yang berarti high return low risk. gambar 2 topologi 6 emiten dengan rval terbesar tabel 3 menunjukkan hasil kalkulasi rataan aritmatik tingkat keuntungan 𝑟𝑖𝑡 tiap saham 𝑖 pada periode 𝑡 dan rataan geometrisnya (lihat (3) dan (4)). kemudian menggunakan formula (5) dan (6) dapat ditentukan covariance matrix atau matriks yang unsur-unsurnya berupa variance dan covariance dari enam variabel/saham berturutturut yaitu akra, icbp, ptpp, tlkm, unvr, dan wskt (tabel 4). tabel 3. rataan aritmatik dan rataan geometris rate of return (6 saham terbaik) akra icbp ptpp tlkm unvr wskt i r 0,000597715 0,00056 0,000227 0,000740614 0,004438108 0,001271307 i  0,000593798 0,000555 0,00022 0,000737094 0,004253044 0,001265167 tabel 4. covariance matrix 6 saham akra icbp ptpp tlkm unvr wskt akra 0,0000078429 0,0000017703 -0,0000002053 0,0000005441 -0,0000051125 -1,6838e-06 icbp 0,0000017703 0,0000098097 0,0000036495 0,0000023484 0,0000067100 4,3967e-06 ptpp -0,0000002053 0,0000036495 0,0000125603 -0,0000007600 0,0000052675 6,01781e-06 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 62 tlkm 0,0000005441 0,0000023484 -0,0000007600 0,0000070373 0,0000106337 2,22524e-06 unvr -0,0000051125 0,0000067100 0,0000052675 0,0000106337 0,0003723178 -4,4953e-06 wskt -1,68377e-06 4,3967e-06 6,01781e-06 2,22524e-06 -4,49528e-06 1,2306e-05 langkah berikutnya adalah pemodelan quadratic programming (program kuadrat) dalam konteks optimisasi portofolio dengan fungsi tujuan covariance matrix dan fungsi kendala rate of return. misalkan variabel saham akra, icbp, ptpp, tlkm, unvr, dan wskt berturut-turut dinotasikan dengan 𝑥𝐴, 𝑥𝐵 , 𝑥𝐶 , 𝑥𝐷 , 𝑥𝐸 , dan 𝑥𝐹, maka dapat disusun pemodelan matematika berikut ini 𝐦𝐢𝐧𝒙𝑨,𝑩,𝑪,𝑫,𝑬,𝑭 0.0000078429 𝑥𝐴 2 + (2 × 0.0000017703 𝑥𝐴𝑥𝐵 ) + (2 × −0.0000002053 𝑥𝐴 𝑥𝐶 ) +(2 × 0.0000005441 𝑥𝐴𝑥𝐷 ) + (2 × −0.0000051125 𝑥𝐴 𝑥𝐸 ) + (2 × −1.68377𝐸 − 06 𝑥𝐴𝑥𝐹 ) +0.0000098097 𝑥𝐵 2 + (2 × 0.0000036495 𝑥𝐵 𝑥𝐶 ) + (2 × 0.0000023484 𝑥𝐵 𝑥𝐷 ) + (2 × 0.0000067100 𝑥𝐵 𝑥𝐸 ) + (2 × 4.3967𝐸 − 06 𝑥𝐵 𝑥𝐹 ) + 0.0000125603 𝑥𝐶 2 + +(2 × −0.00000076 𝑥𝐶 𝑥𝐷 ) + (2 × 0.0000052675 𝑥𝐶 𝑥𝐸 ) + (2 × 6.01781𝐸 − 06 𝑥𝐶 𝑥𝐹 ) + 0.0000070373 𝑥𝐷 2 + (2 × 0.0000106337 𝑥𝐷 𝑥𝐸 ) + (2 × 2.22524𝐸 − 06 𝑥𝐷 𝑥𝐹 ) + 0.0003723178 𝑥𝐸 2 + (2 × −4.49528𝐸 − 06 𝑥𝐸 𝑥𝐹 ) + 1.2306𝐸 − 05 𝑥𝐹 2 sedemikian sehingga 0.000597715 𝑥𝐴 + 0.000560385 𝑥𝐵 + 0.000226514 𝑥𝐶 + 0.000740614 𝑥𝐷 + 0.004438108 𝑥𝐸 + 0.001271307 𝑥𝐹 ≥ 𝑅 𝑥𝐴 + 𝑥𝐵 + 𝑥𝐶 + 𝑥𝐷 + 𝑥𝐸 + 𝑥𝐹 = 1 𝑥𝐴 , 𝑥𝐵 , 𝑥𝐶 , 𝑥𝐷 , 𝑥𝐸 , 𝑥𝐹 ≥ 0 (10) solusi dari program kuadrat (10) yang selanjutnya disebut sebagai portofolio efisien diperoleh dengan menentukan nilai return investasi 𝑅 pada interval 0,055 ≤ 𝑅 ≤ 0,075 dengan kenaikan 0,0025. untuk menemukan solusi program kuadrat tersebut dapat digunakan software matlab versi r2010a [3]. hal tersebut dilakukan dengan memasukkan variabel 𝑄, 𝑓, 𝐴, 𝑏, 𝐴𝑒𝑞, 𝑏𝑒𝑞 dan diikuti dengan sintak program kuadrat 𝑥 = quadprog(𝑄, 𝑓, 𝐴, 𝑏, 𝐴𝑒𝑞, 𝑏𝑒𝑞). sintak quadprog pada matlab >> q=[1.56858e-05 1.7703e-06 -2.05327e-07 5.44074e07 -5.11247e-06 -1.68377e-06 1.7703e-06 1.96193e-05 3.64949e-06 2.34843e-06 6.71005e-06 4.3967e-06 -2.05327e-07 3.64949e-06 2.51206e-05 -7.59997e075.26752e-06 6.01781e-06 5.44074e-07 2.34843e06-7.59997e-07 1.40746e-05 1.06337e-05 2.22524e-06 -5.11247e-06 6.71005e-06 5.26752e-06 1.06337e05 0.000744636 -4.49528e-06 -1.68377e-06 4.3967e066.01781e-06 2.22524e-06 -4.49528e-06 2.4612e-05]; >> a=[0.000597715 0.000560385 0.000226514 0.000740614 0.004438108 0.001271307 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 0 0 0 0 0]; >> b=[0.00055 0 0 0 0 0]; >> c=[0 0 0 0 0 0]; >> aeq=[1 1 1 1 1 1; 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 0 0 0 0 0]; >> beq=[1;0;0;0;0;0]; >> options=optimset('largescale','off'); >> x=quadprog(q,c,a,b,aeq,beq) x = 0.2947 0.1477 0.2135 0.3050 -0.0104 0.0495 hasil pemrograman pada matlab tersebut, utamanya pada nilai x, menunjukkan proporsi atau bobot masing-masing dana yang akan diinvestasikan dalam prosen. diantara keenam saham tersebut ada satu saham yang memiliki bobot negatif -0,0104 (1,04%) yaitu saham unvr. dengan demikian perlu dilakukan perhitungan ulang optimasi pada portofolio dengan menghilangkan saham berbobot negatif, sehingga diperoleh portofolio baru dengan 5 (lima) saham berturut-turut akra, icbp, ptpp, tlkm, dan wskt. dengan cara yang sama seperti pada analisis optimasi pada 6 saham, maka dilakukan identifikasi rataan aritmatik, rataan geometris rate of return, covariance matrix tiap saham, dan pemodelan program kuadrat baru dengan 5 variabel. output dari pemrograman matlab terhadap model program kuadrat yang baru dapat disajikan dalam tabel 5 yang menunjukkan proporsi dana yang diinvestasikan jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 63 (prosen) pada kelima saham syariah terpilih yaitu akra, icbp, ptpp, tlkm, dan wskt. di samping itu, dengan dengan menentukan nilai return investasi 𝑅 pada interval 0,055 ≤ 𝑅 ≤ 0,075 dan menggunakan formula (5) dapat dihitung pula variance dari masing-masing proporsi saham dengan mempertimbangkan nilai covariance (lihat tabel 5). gambar 3 grafik rasio antara variance dan return variance pada tabel di atas menunjukkan tingkat risiko sedangkan r adalah tingkat keuntungan. pada umumnya tiap perubahan tingkat keuntungan menghasilkan proporsi saham yang berbeda, namun dalam kasus ini diperoleh tren proporsi yang sama atau konstan setelah mencapai interval 0,0675 ≤ 𝑅 ≤ 0,075 (lihat gambar 3). tabel 5. rekapitulasi proporsi tiap saham (%) dengan r dan variance tertentu r akra icbp ptpp tlkm wskt variance 5,5 29,82 15,49 24,3 29,48 0,91 0,029386615 5,75 29,79 14,84 22,35 29,51 3,51 0,028909367 6 29,77 14,18 20,4 29,54 6,1 0,02856281 6,25 29,75 13,53 18,45 29,58 8,7 0,028371985 6,5 29,72 12,87 16,51 29,61 11,29 0,028301593 6,75 29,71 12,48 15,33 29,63 12,86 0,028335619 7 29,71 12,48 15,33 29,63 12,86 0,028335619 7,25 29,71 12,48 15,33 29,63 12,86 0,028335619 7,5 29,71 12,48 15,33 29,63 12,86 0,028335619 baik tabel 5 maupun gambar 3 memberikan informasi kepada calon investor dalam berinvestasi di saham syariah jii. sebagai gambaran jika calon investor menghendaki tingkat keuntungan mencapai 6,75% maka proporsi dana (dalam prosen) yang dianjurkan, berdasarkan analisis nonlinier programming tersebut, untuk diinvestasikan kepada lima emiten saham syariah akra, icbp, ptpp, tlkm, dan wskt berturut-turut adalah 29,71%, 12,48%, 15,33%, 29,63%, dan 12,86%. adapun tingkat risiko yang akan ditanggung oleh investor tersebut adalah 0,028335619%. sehingga secara umum tren proporsi dana tertinggi direkomendasikan diberikan kepada akra sedangkan proporsi dana terendah diberikan kepada icbp atau wskt. grafik di atas juga mencerminkan aspek umum investasi yang menyatakan high risk high return. 6. kesimpulan proses seleksi portofolio saham syariah optimal dapat dilakukan dengan menghubungkan return-risiko dengan analisis rval. saham atau aset terpilih selanjutnya dapat dianalisis menggunakan pemodelan program kuadrat untuk menghasilkan bobot atau proporsi dana sebagai rekomendasi bagi para calon investor saham syariah. ucapan terimakasih penulis mengucapkan terimaksih kepada lembaga penelitian dan pengabdian masyarakat (lppm) uin sunan kalijaga atas skema pendanaan hibah penelitian rintisan pada tahun 2017. tidak lupa juga ucapan terimakasih kepada ketua program studi matematika atas dukungan jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 64 kegiatan penelitian ini dan pak mohammad farhan qudratullah atas diskusi kajian statistik terutama terkait dengan risk analysis. referensi [1] ayub, m., understanding islamic finance: a-z keuangan islam, jakarta: pt, gramedia pustaka utama (2009). [2] batuparan d.s., 2000, bei news: mengapa risk management? edisi 4, jakarta: bursa efek indonesia (bei). [3] brandimarte p., numerical methods in finance and economics : a matlab-based introduction, edisi kedua, john wiley and sons inc,, hoboken, new jersey (2006). [4] cornuejols g., and tuetuencue r., optimization methods in finance, cambridge university press (2007). [5] david ruppert, statistics and finance: an introduction, springer-verlag new york, llc (2004). [6] fahmi i., dan hadi y.i., teori portofolio dan analisis investasi, teori dan soal jawab, bandung: penerbit alfabeta (2009). [7] jogiyanto h,, 2013, teori portofolio dan analisis investasi, cetakan ketiga, bpfe yogyakarta (2013). [8] jorion p., value at risk : the new benchmark for managing financial risk, mcgraw-hill, new york (2002). [9] lucky bayu purnomo, diunduh dari http://investasi,kontan,co,id/news/indekssaham-syariah-mengungguli-ihsg pada tanggal 1 juni 2017. [10] mussafi, optimisasi portofolio risiko menggunakan model markowitz mvo, jurnal admathedu, issn: 2088-687x, vol. 1 no. 1, fkip uad yogyakarta (2011). [11] shihab m. quraish, tafsir al-misbah: pesan, kesan dan keserasian al-qur’an, vol. 6, jakarta: lentera hati (2002). [12] tandelilin, e., portofolio dan investasi: teori dan aplikasi, yogyakarta: penerbit kanisius (2010). [13] wolsey l.,a., integer programming, john wiley and sons, new york (1988). [14] ____________, harga saham bursa efek indonesia yang tergabung dalam jakarta islamic index, diambil dari https://finance,yahoo,com/ pada tanggal 1 juli 2017. http://investasi,kontan,co,id/news/indeks-saham-syariah-mengungguli-ihsg http://investasi,kontan,co,id/news/indeks-saham-syariah-mengungguli-ihsg https://finance.yahoo.com/ jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 30 pemodelan status usaha (pengusaha dan pekerja/karyawan) menggunakan regresi logistik multilevel eko yulian1, gandhi pawitan prodi magister statistika universitas padjadjaran ,bandung. okeyulian@gmail.com1 abstract the level of a country's economy is directly proportional to the number of entrepreneurs in the country. according to the world bank standard number of entrepreneurs, the ideal of a country is at least 4% of the total population. based on data from the indonesian young entrepreneurs association (hipmi), the number of entrepreneurs in indonesia is only about 1.5%. of course not easy to achieve the ideal number of bank standards-based world that is 4%. this study aims to determine what factors are driving someone in determining a career as an entrepreneur or not (worker / employee). the data used is the adult population survey (aps) in 2013 conducted by the global entrepreneurship monitor (gem). survey conducted on 16 provinces, 51 districts / cities and 176 subdistricts. data generated hierarchical modeling that will be performed using multilevel logistic regression. the variables studied were the state variable effort (y), variable knowent (x1), variable opport (x2), variable suskill (x3), variable fearfail (x4), the variable gender (x5) at level 1 and the variable sub-district at level 2. the analysis showed that the logistic regression model 2-level produce a better model than the ordinary logistic regression model. based on modeling results we concluded that all predictor variables (knowent, opport, suskill, fearfail, gender, etc.) affect the status of one's business. keyword : enterpreneur , logistic regression, multilevel model abstrak tingkat perekonomian suatu negara berbanding lurus dengan jumlah pengusaha/enterpreneur di dalam negara tersebut. menurut standar bank dunia jumlah pengusaha/enterpreneur ideal suatu negara adalah paling sedikit 4% dari total jumlah penduduk. berdasarkan data dari himpunan pengusaha muda indonesia (hipmi), jumlah pengusaha di indonesia baru sekitar 1,5 %. tentu saja bukan hal yang mudah untuk mencapai jumlah ideal berdasarkan standar bank dunia yaitu 4%. penelitian ini bertujuan untuk mengetahui faktor-faktor apa saja yang mendorong seseorang dalam menentukan karir sebagai pengusaha atau bukan (pekerja/karyawan). data yang digunakan adalah data adult population survey (aps) tahun 2013 yang dilakukan oleh global enterpreneurship monitor (gem). survey dilakukan terhadap 16 provinsi , 51 kabupaten/kota dan 176 kecamatan. data yang dihasilkan bersifat hierarki sehingga akan dilakukan pemodelan menggunakan regresi logistik multilevel. variabel yang diteliti adalah variabel status usaha (y), variabel knowent (x1), variabel opport(x2), variabel suskill (x3), variabel fearfail (x4) ,variabel gender (x5) pada level 1 dan variabel kecamatan pada level 2. hasil analisis menunjukkan bahwa model regresi logistik 2level menghasilkan model yang lebih baik daripada model regresi logistik biasa. berdasarkan hasil pemodelan diperoleh kesimpulan bahwa semua variabel prediktor (knowent, opport, suskill, fearfail, gender) berpengaruh terhadap status usaha seseorang. kata kunci : pengusaha/wirausaha/enterpreneur , regresi logistik, model multilevel jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 31 1. pendahuluan indonesia adalah salah satu negara besar di asia, hal ini bisa dilihat dari luas wilayah yang mencapai 1,905 juta km², dengan jumlah penduduk berdasarkan sensus penduduk bps tahun 2010 yang mencapai 237 juta jiwa tentu menjadi modal penting bagi kita untuk bisa bersaing di kancah internasional. tingkat kemapanan suatu negara bisa dilihat darai keadaan ekonomi di negara tersebut. memang indonesia belum bisa bersaing dengan negaranegara eropa atau amerika, atau bahkan dengan negara sesama asia seperti jepang dan korea, akan tetapi dengan melimpahnya sumber daya baik sumber daya alam dan sumber daya manusia kita harus tetap optimis bisa menyusul ketertinggalan khususnya di sektor ekonomi. ketika kita berbicara tentang ekonomi suatu bangsa maka hal itu berkaitan erat dengan masalah kewirausahaan atau juga dikenal dengan enterpreneurship. david mcclelland mengatakan bahwa suatu bangsa bisa mencapai kemakmuran jika jumlah entrepreneur paling sedikit 2% dari total penduduknya, sedangkan menurut standar bank dunia minimal adalah 4% dari total penduduk. menurut himpunan pengusaha muda indonesia (hipmi), jumlah entrepreneur yang ada di indonesia hanya sekitar 1,5 % saja dari sekitar 237 juta penduduk. ini artinya kita masih kekurangan sekitar minimal 0,5% untuk mencapai batas minimal yang telah disebutkan oleh david mcclelland dan kekurangan sekitar 2,5% menurut standar bank dunia. jika kita bandingkan dengan negaranegara asia tenggara lainnya semisal singapura yang memiliki pengusaha sekitar 7%, malaysia 6% dan thailand 3% memang kita masih tertinggal. akan tetapi hal ini sedikit dimaklumi karena jumlah serta luas wilayah indonesia lebih besar dibandingkan ketiga negara tersebut. tidak mudah memang untuk mencapai ratio yang telah disebutkan di atas, karena banyak faktor yang mendorong penduduk indonesia untuk menentukan karirnya, apakah menjadi seorang wirausahawan(enterpreneur) atau menjadi pekerja/karyawan. adapun tujuan dari penelitian ini adalah untuk mengetahui faktor-faktor yang mendorong seseorang menjadi enterpreneur. 2. tinjauan pustaka 2.1 wirausaha para ahli memiliki pendapat dan penafsiran yang berbeda tentang definisi dari wirausaha. berikut adalah pengertian wirausaha menurut beberapa sumber.  pengusaha atau wirausahawan (entrepreneur) merupakan seorang yang menciptakan sebuah usaha atau bisnis yang diharapkan dengan risiko dan ketidakpastian untuk memperoleh keuntungan dan mengembangkan bisnis dengan cara membuka kesempatan [1].  entrepreneurial is an innovator and individual developing something unique and new (wirausaha adalah seorang penemu dan individu yang membangun sesuatu yang unik dan baru) [2].  wirausaha adalah pengusaha yang mampu mengelola sumber-sumber daya yang dimiliki secara ekonomis (efektif dan efisien) dan tingkat produktivitas yang rendah menjadi tinggi [3].  wirausaha adalah orang yang mengelola, mengorganisasikan, dan berani menanggung segala resiko untuk menciptakan peluang usaha dan usaha baru [4]. dari beberapa definisi di atas bisa kita simpulkan bahwa seorang wirausaha adalah seseorang yang membuka lapangan pekerjaan untuk yang lain, dimana usaha tersebut bisa dimiliki oleh satu orang atau lebih. dalam penelitian ini seorang wirausahawan dilihat dari variabel q2a (are you, alone or with others, currently the owner of a business you help manage, self-employed, or selling any goods or services to others?) 2.2 kesempatan / peluang / opportunity peluang dalam bahasa inggris adalah opportunity yang berarti kesempatan yang muncul dari sebuah kejadian atau momen. tentu saja hal ini menjadi faktor penting yang mendorong seseorang untuk menjadi enterpreneur atau tidak. peluang usaha adalah kesempatan atau waktu yang tepat yang seharusnya di ambil atau dimanfaatkan bagi seseorang wirausahawan mendapat keuntungan. banyak peluang yang di jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 32 siasiakan, sehingga berlalu begitu saja karena tidak semua orang dapat melihat peluang dan yang melihatpun belum tentu berani memanfaatkan peluang tersebut. hanya seorang wirausahawan yang dapat berpikir kriatif serta berani mengambil risiko itulah yang dengan tanggap dan cepat memanfaatkan peluang. 2.3 skill dan pengalaman keterampilan berarti mengembangkan pengetahuan yang didapatkan melalui training dan pengalaman dengan melaksanakan beberapa tugas. baik wirausahawan maupun pekerja/karyawan tentu saja haruslah memiliki skill atau keterampilan dalam menjalankan kegiatanya. faktor skill dalam bisnis juga bisa mempengaruhi seseorang untuk memutuskan apakah memilih menjadi wirausahawan atau karyawan dalam pilihan karirnya. pengetahuan yang dimiliki, akan sangat bermanfaat jika didukung dengan skill atau keahlian yang dimiliki. peluang usaha yang dibangun menggunakan skill, akan lebih cepat berkembang dibandingkan perusahaan yang tidak dilandasi dengan skil. selain skill, faktor pengalaman juga merupakan faktor penting dalam seseorang untuk menentukan pilihan karirnya. bagi seseorang yang berpengalaman dalam bisnis tentu menjadi wirausahawan adalah pilihan yang menarik sekaligus menantang. 2.4 rasa takut gagal (mental usaha) rasa takut gagal dalam memulai bisnis pastilah ada pada setiap orang, dan memang hal itu wajar. masalahnya adalah bagaimana mengelola rasa takut itu agar tidak meruntuhkan ekspektasi dalam berwirausaha. tentu saja hal itu berkaitan erat dengan mental usaha yang berani mengambil resiko. mereka yang memiliki mental bisnis yang kuat cenderung akan menjadikan wirausaha sebagai pilihan karirnya, sebaliknya mereka yang memiliki rasa takut gagal yang besar cenderung memilih karyawan sebagai pilihan karirnya. 2.5 jenis kelamin di zaman yang semakin berkembang tentunya berpengaruh terhadap cara berpikir dan cara pandang seseorang terhadap segala sesuatu tidak terkecuali terhadap pekerjaan antara lakilaki dan wanita.. sekarang banyak kita temukan para pengusaha yang berjenis kelamin perempuan, walaupun memang jumlahnya masih lebih sedikit dibandingkan laki-laki. hal ini bisa disebabkan karena laki-laki adalah tulang punggung keluarga. dalam penelitian yang dilakukan oleh yuhendri l.v tahun 2015 tentang “perbedaan minat berwirausaha mahasiswa ditinjau dari jenis kelamin” [5] diperoleh kesimpulan bahwa ada perbedaan minat antara mahasiswa laki-laki dan perempuan, dimana minat mahaiswa laki-laki untuk berwirausaha lebih tinggi dibandingkan dengan mahasiswa perempuan. 2.6 relasi bisnis ketika seseorang memiliki kenalan seorang wirausahawan sedikit banyak akan berpengaruh terhadap keputusannya dalam menentukan karir, idealnya ketika kita berteman dengan seorang wirausahawan maka kita minimal memiliki niat untuk menjadi seperti mereka. dalam penelitian ini akan dilihat apakah faktor ini menentukan sesorang untuk berkarir sebagai pengusaha atau pekerja/karyawan. 2.7 model regresi model regresi merupakan model yang digunakan untuk melihat hubungan antara satu variabel tak bebas (variabel dependen), atau disebut juga dengan variabel respon, dengan beberapa variabel bebas (variabel independen), atau disebut juga dengan variabel penjelas . model regresi linear adalah model yang menunjukkan hubungan yang linear antara variabel respon dan variabel penjelas. terdapat dua persamaan model dalam model regresi linear, yaitu regresi linier sederhana (simple regression) dan regresi linear berganda (multiple regression). jika persamaan model regresi hanya terdiri dari satu variabel bebas, maka model tersebut disebut dengan regresi linear sederhana. sedangkan, jika persamaan model regresi terdapat lebih dari satu variabel bebas, maka model tersebut disebut dengan regresi linear berganda. secara umum, model regresi linear sebagai berikut. 𝑌 = 𝛽0 + 𝛽1𝑋1 + ⋯ + 𝛽𝑘𝑋𝑘 + 𝜀𝑖 (1) dimana : y : variabel respon xi : variabel penjelas εi : error βi : parameter atau disebut juga dengan koefisien regresi. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 33 2.8 model regresi multilevel model multilevel merupakan sebuah model yang digunakan pada data berjenjang (hierarchy). data berjenjang seringkali ditemukan pada penelitianpenelitian survei dimana unit-unit analisisnya berasal dari kelompok-kelompok (cluster), atau data yang diambil melalui penarikan sampel bertahap (cluster sampling). misalnya, dalam pengambilan sampel menggunakan metode sampling satu tahap (single stage cluster sampling), dimana unit-unit sampling yang berasal dari kelompok diperhitungkan keberadaanya dalam analisis, sehingga dalam hal ini model yang sesuai adalah model multilevel. unit-unit sampling yang ada dalam kelompok disebut level rendah dan kelompokkelompok disebut level tinggi. banyaknya unitunit analisis dalam kelompok bisa sama atau berbeda untuk setiap kelompok. suatu model regresi multilevel yang sederhana hanya terdiri dari 2 level. misalkan, diberikan data dalam j kelompok dan jumlah yang berbeda dari individu nj dalam setiap kelompok. dalam individu (level 1), terdapat variabel tak bebas/respon yij dan variabel bebas/penjelas xij, serta pada level kelompok terdapat variabel bebas zj. sehingga, terdapat persamaan model regresi terpisah pada setiap kelompok. model regresi 2 level dengan satu variabel bebas sebagai berikut. 𝑌𝑖𝑗 = 𝛽0𝑗 + 𝛽1𝑗 𝑋1𝑗 + 𝜀𝑗 (2) dimana : indeks i menyatakan individu pada tingkat level 2 ke-j (i = 1, 2, ..., nj) indeks j menyatakan tingkat level 2 (j = 1, 2, ..., m) dalam persamaan (2), β0 adalah intersep dalam regresi klasik, β1 adalah slope regresi untuk variabel penjelas x, dan εi merupakan residual error. koefisien regresi β dengan j untuk kelompok, dimana mengindikasi bahwa koefisien regresi mungkin bervariasi setiap kelompok. variasi dalam koefisien regresi βj dimodelkan dengan variabel penjelas dan residual acak pada level kelompok, sehingga diperoleh persamaan sebagai berikut. 𝛽0𝑗 = 𝛽00 + 𝛽01𝑍𝑗 + 𝑢0𝑗 (3) 𝛽1𝑗 = 𝛽10 + 𝛽11𝑍𝑗 + 𝑢1𝑗 (4) persamaan (2) disebut sebagai model pada level 1 dan persamaan (3) (4) disebut sebagai model pada level 2. selanjutnya, persamaan (3) dan (4) disubstitusikan ke dalam persamaan (2), sehingga diperoleh model regresi multilevel sebagai berikut. 𝑌𝑖𝑗 = 𝛽00 + 𝛽01𝑍𝑗 + 𝑢0𝑗 + (𝛽10 + 𝛽11𝑍𝑗 + 𝑢1𝑗 )𝑋1𝑗 + 𝜀𝑗 (5) 2.9 regresi logistik biner regresi logistik digunakan untuk mencari hubungan variabel dependen (y) 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 independen (x) yang bersifat kontinu atau kategorik (agresti, 2007). salah satu regresi logistik yang paling sederhana digunakan adalah regresi logistik biner. regresi logistik biner merupakan suatu metode analisis data yang digunakan untuk mencari hubungan antara variabel dependen (y) yang bersifat biner atau dikotomus dengan variabel independen (x) yang bersifat polikotomus data variabel dependen yang digunakan dalam regresi logistik biner adalah data dengan skala nominal dengan hanya berupa 2 kategori yaitu “sukses” atau “gagal” misalnya: ya-tidak, benar-salah, hidup-mati, hadir-absen, laki-wanita, dan seterusnya. sedangkan data variabel independen dapat berupa data dengan skala ordinal (seringkali digunakan pada kasus-kasus/penelitian sosial kemasyarakatan) ataupun data dengan skala rasio (seringkali dijumpai pada penelitian industri). outcome dari variabel dependen y terdiri dari 2 kategori yaitu “sukses” dan “gagal” yang dinotasikan dengan y=1 (sukses) dan y=0 (gagal). dalam keadaan demikian, variabel y mengikuti distribusi bernoulli untuk setiap observasi tunggal. fungsi probabilitas untuk setiap observasi diberikan sebagai berikut jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 34 𝑓(𝑌) = 𝑝 𝑦(1 − 𝑝)1−𝑦 model regresi logistik biner dapat dituliskan sebagai berikut 𝑙𝑜𝑔 ( 𝜋 1−𝜋 ) = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + ⋯ + 𝛽𝑘𝑋𝑘 = 𝑿 𝑻𝜷 (6) 2.10 model regresi multilevel respon biner model regresi multilevel dengan variabel respon (dependen) berupa data biner atau hanya terdiri dari dua kategori, maka estimasi parameter bisa dilakukan menggunakan estimasi maksimum likelihood dengan pendekatan suatu metode tertentu (goldstein, 1999) [6]. dalam model regresi, apabila variabel respon berupa biner atau dichotomous biasanya digunakan model regresi logistik yang dalam estimasi parameternya harus menggunakan suatu fungsi penghubung (link function). hal tersebut juga sama diterapkan dalam model multilevel. apabila variabel respon berdistribusi binomial dengan parameter proporsi (πij), maka fungsi penghubung yang digunakan adalah logit (log{π/(1-π)}) sehingga modelnya disebut dengan model logistik . secara umum, model 2 level dengan respon biner dapat dituliskan sebagai 𝑙𝑜𝑔 ( 𝜋𝑖𝑗 1 − 𝜋𝑖𝑗 ) = 𝛽0 + 𝛽1𝑋1𝑖 + 𝑢𝑗 (7) dimana uj merupakan efek acak pada level 2, tanpa uj, persamaan di atas akan menjadi model regresi logistik standar. model di atas seringkali dideskripsikan sebagai alternatif dalam literatur pada model multilevel dari persamaan berikut: 𝑙𝑜𝑔 ( 𝜋𝑖𝑗 1 − 𝜋𝑖𝑗 ) = 𝛽0𝑗 + 𝛽1𝑋1𝑖 + 𝜀𝑖 (model level 1) dan 𝛽0𝑗 = 𝛽0 + 𝑢𝑗 (model level 2) secara umum rumusan matematis untuk untuk model random intercept dua level dengan respon biner adalah sebagai berikut : 𝑙𝑜𝑔 ( 𝜋𝑖𝑗 1 − 𝜋𝑖𝑗 ) = 𝛽0 + 𝛽1𝑋1𝑖 + 𝑢𝑗 + 𝜀𝑖 (8) dimana uj adalah error pada level 2 3. metodologi penelitian 3.1 sumber data data yang digunakan adalah data sekunder yaitu bersumber dari survey adult population survey (aps) tahun 2013 yang dilakukan oleh global enterpreneurship monitor (gem). survey dilakukan terhadap 16 provinsi , 51 kabupaten/kota dan 176 kecamatan. tujuan dari survey aps yang diakukan oleh gem adalah untuk mempelajari kebiasaan dan perilaku individu yang berhubungan dengan kegiatan enterpreneurship. target responden adalah mereka yang berusia 18 – 64 tahun yang berasal dari setiap lapisan masyarakat yang dilakukan dengan metode sampling multistage sampling. adapun jumlah responden adalah 4500 responden. fokus pada tulisan ini adalah responden yang merupakan pekerja dan pengusaha. 3.2 variabel penelitian tidak semua variabel pada data aps digunakan pada penelitian ini, variabelvariabel yang akan digunakan dalam analisis yaitu variabel respon (y) : status usaha (1= ya , 2 = tidak) (diperoleh dari variabel q2a) variabel prediktor (x) : x1 : knowent x2 : opport x3 : suskill x4 : fearfail x5 : gender secara struktur variabel yang digunakan adalah sebagai berikut tabel 1. struktur variabel jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 35 3.3 langkah analisis adapun langkah-langkah yang dilakukan untuk menganalisis faktor-faktor yang mendorong seseorang menjadi wirausaha dengan menggunakan regresi logistik multilevel adalah sebagai berikut : 1. memisahkan responden apakah bekerja atau tidak, setelah mendapat data responden yang bekerja kemudian dipisahkan apakah merupakan pengusaha atau bukan (pekerja) 2. melakukan analisis deskriptif terhadap data 3. mengestimasi parameter dalam model 2 level dengan respon biner dengan menggunakan software spss dan r 4. membuat model analisis regresi logistik tanpa melibatkan efek kecamatan 5. membuat model analisis regresi logistik dengan melibatkan efek dari level 2 yaitu kecamatan 6. memilih model terbaik dengan menggunakan ukuran deviance 7. membuat kesimpulan 4. analisis dan pembahasan analisis deskriptif karakteristik data akan dianalisis dengan menggunakan statistik deskriptif baik menggunakan tabel kontingensi atau dengan menggunakan grafik.  variabel “status usaha” seperti yang telah disebutkan pada langkah analisis bahwa hal pertama yang dilakukan adalah memisahkan data berdasarkan status bekerja (bekerja atau tidak) berikut adalah hasilnya tabel 2. deskriptif status bekerja terlihat bahwa jumlah responden yang bekerja sebanyak 3427 dari 4500 orang yang disurvey. kemudian dari 3427 orang yang bekerja tersebut dipisahkan lagi berdasarkan status pengusaha atau bukan, hal ini bisa dilakukan dengan menggunakan variabel “q2a tabel 3. deskriptif status usaha q2a. are you, alone or with others, currently the owner of a business you help manage, self-employed, or selling any goods or services to others? frequency percent valid percent cumulative percent valid no 1468 42,8 42,8 42,8 yes 1959 57,2 57,2 100,0 total 3427 100,0 100,0 terlihat bahwa dari tabel 3 di atas jumlah responden yang merupakan wirausaha adalah 1959 (57,2%) lebih banyak daripada mereka yang merupakan pekerja yaitu sebanyak 1468 orang atau sekitar 42,8%. hal ini adalah kabar yang cukup menggembirakan walaupun belum mencapai rasio yang ideal. analisis regresi logistik respon biner level 1 uji multikolineritas variabel prediktor dikatakan independen satu sama lain bisa dilihat dari nilai vif. dikatakan independen jika nilai vif < 10. dengan menggunakan spss diperoleh nilai vif untuk masing-masing variabel x sebagai berikut x1 =1.1 , x2 = 1.2, x3 = 1.2, x4 = 1.01 dan x5 = 1.004. diperoleh nilai vif untuk semua variabel x < 10 sehingga variabel penjelas independen. uji estimasi parameter simultan uji ini bertujuan untuk melihat apakah variabelvariabel prediktor (x) yang dilibatkan berpengaruh terhadap variabel respon (y). h0 : β1 = β2 = β3 = β4 = β5 h1 : minimal ada satu βi ≠ 0 ; i = 1,2,3,4,5 α = 0,05 statistik uji chisquare df sig. step 1 step 589,972 5 ,000 block 589,972 5 ,000 model 589,972 5 ,000 frequency percent valid percent cumulative percent work:f-t, p-t 3427 76,2 76,4 76,4 not working 835 18,6 18,6 95,1 retired students 221 4,9 4,9 100,0 total 4483 99,6 100,0 m issing m issing, cannot classify 17 ,4 4500 100,0 gemwork3. gem harmonized work status: 3 categories valid total jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 36 h0 ditolak jika nilai chi-square > chi square tabel atau nilai sig < α. dari output di atas diperoleh nilai sig = 0 < α, sehingga h0 ditolak yang berarti minimal salah satu variabel prediktor dalam hal ini “knowent”,”opport”,”suskill”,”fearfail” dan “gender” memiliki pengaruh yang signifikan terhadap status usaha. uji estimasi parameter parsial setelah dilakukan uji simultan selanjutnya akan dilakukan uji estimasi parameter secara parsial yang bertujuan untuk melihat apakah ada variabel yang tidak signifikan terhadap variabel respon. tabel 4. koefisien regresi logistik variabel (x) koefisien s.e. df sig. knowent 0,793 ,091 1 ,000 opport -0,482 ,081 1 ,000 suskill -1,028 ,085 1 ,000 fearfail 0,484 ,077 1 ,000 gender -0,685 ,078 1 ,000 constant 1,267 ,089 1 ,000 uji hipotesis parsial sebagai berikut h0 : β = 0 (variabel tidak signifikan) h1 : βi ≠ 0 (variabel signifikan) α = 0,05 statistik uji p-value (sig) ho ditolak jika nilai p-value (sig) < α = 0,05 dari hasil di atas diperoleh bahwa semua variabel memiliki nilai p-value =0 < α, sehingga bisa disimpulkan bahwa kelima variabel prediktor memiliki pengaruh yang signifikan terhadap variabel respon. membentuk model regresi logistik setelah dilakukan uji estimasi parameter simultan dan parsial, langkah selanjutnya adalah membentuk model. dari hasil uji parsial diperoleh informasi bahwa variabel “knowent” (x1), “opport (x2)”, “suskill (x3) ”, “fearfail (x4) ” dan “ gender (x5)” berpengaruh signifikan terhadap “status usaha” sehingga model yang terbentuk adalah sebagai berikut 𝒍𝒐𝒈 ( 𝝅 𝟏 − 𝝅 ) = 𝟏, 𝟐𝟔𝟕 − 𝟎, 𝟕𝟗𝟑𝑿𝟏(𝟏) − 𝟎, 𝟒𝟖𝟐𝑿𝟐(𝟏) − 𝟏, 𝟎𝟐𝟖𝑿𝟑(𝟏) + 𝟎, 𝟒𝟖𝟒𝑿𝟒(𝟏) − 𝟎, 𝟔𝟖𝟓𝑿𝟓(𝟐) hasil analisis regresi logistik pada level individu (level 1) memiliki nilai deviance sebesar 4052,56. analisis regresi multilevel respon biner untuk random intercept analisis regresi multilevel yang digunakan pada penelitian ini adalah analisis regresi 2 level dengan random intersep, yaitu pemodelan dengan melibatkan variabel kecamatan tanpa mengikutsertakan variabel prediktor pada level 2 . dengan menggunakan software r diperoleh estimasi parameter regresi 2 level sebagai berikut : tabel 5. koefisien regresi logistik 2-level estimate std.error p-value (intercept) 1.771 0.163 2.00e-16 knowent (yes) -0.994 0.122 4.21e-16 opport (yes) -0.629 0.104 1.25e-09 suskill (yes) -1.421 0.112 2.00e-16 fearfail (yes) 0.622 0.102 9.21e-10 gender (female) -0.776 0.090 2.00e-16 akan dilakukan uji estimasi parameter secara parsial yang bertujuan untuk melihat apakah ada variabel yang tidak signifikan terhadap variabel respon. uji hipotesis parsial sebagai berikut h0 : β = 0 (variabel tidak signifikan) h1 : βi ≠ 0 (variabel signifikan) α = 0,05 statistik uji p-value (sig) ho ditolak jika nilai p-value (sig) < α = 0,05 dari hasil di atas diperoleh bahwa semua variabel memiliki nilai p-value ≈ 0 < α, sehingga bisa disimpulkan bahwa kelima variabel prediktor memiliki pengaruh yang signifikan terhadap variabel respon. membentuk model regresi logistik 2-level dari hasil uji parsial diperoleh informasi bahwa variabel “knowent”, “opport (x1)”, “suskill (x2) ”, “fearfail (x3) ” dan “ gender (x4)” jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 37 berpengaruh signifikan terhadap “status usaha” sehingga model yang terbentuk adalah sebagai berikut 𝒍𝒐𝒈 ( 𝝅 𝟏 − 𝝅 ) = 𝟏, 𝟕𝟕𝟏 − 𝟎, 𝟗𝟗𝟒 𝑿𝟏(𝟏) − 𝟎, 𝟔𝟐𝟗 𝑿𝟐(𝟏) − 𝟏, 𝟒𝟐𝟏 𝑿𝟑(𝟏) + 𝟎, 𝟔𝟐𝟐 𝑿𝟒(𝟏) − 𝟎, 𝟕𝟕𝟔 𝑿𝟓(𝟐) hasil analisis regresi logistik pada level 2 memiliki nilai deviance sebesar 3620. pemilihan model terbaik dari hasil analisis yang telah diperoleh dua model pada analisis regresi logistik dan regresi logistik 2-level, dimana kedua model tersebut sama-sama mengandung lima variabel prediktor (x) yang signifikan. untuk menentukan model terbaik bisa dilakukan dengan cara membandingkan nilai deviance dari kedua model. model yang memiliki nilai deviance yang kecil adalah model terbaik. berikut adalah nilai deviance dari kedua model tabel 6. deviance model model deviance regresi logistik biner 4052,56 regresi logistik 2-level dengan random intersep 3620 dari tabel 6 di atas terlihat bahwa regresi logistik 2-level dengan random intersep memiliki nilai deviance yanng lebih kecil dibandingkan dengan nilai deviance dari model regresi logistik biner. sehingga dapat disimpulkan bahwa model regresi logistik 2-level dengan random intersep adalah model yang terbaik sehingga model regresi untuk variabel “status usaha” adalah 𝒍𝒐𝒈 ( 𝝅 𝟏 − 𝝅 ) = 𝟏, 𝟕𝟕𝟏 − 𝟎, 𝟗𝟗𝟒 𝑿𝟏(𝟏) − 𝟎, 𝟔𝟐𝟗 𝑿𝟐(𝟏) − 𝟏, 𝟒𝟐𝟏 𝑿𝟑(𝟏) + 𝟎, 𝟔𝟐𝟐 𝑿𝟒(𝟏) − 𝟎, 𝟕𝟕𝟔 𝑿𝟓(𝟐) interpretasi model berbeda dengan interpretasi pada regresi linier, interpretasi pada regresi logistik sedikit rumit karena nilai y yang dihasilkan adalah nilai peluang. akan lebih mudah jika kita melakukan interpretasi terhadap nilai dari odds ratio (or). adapun nilai or = exp (koefisisen model). berikut adalah nilai or untuk masing-masing variabel tabel 7. odds ratio koefisien x 0.994 0.629 1.421 0.622 0.776 exp (koefisein) 2.7 1.88 4.14 1.86 2.17  knowent (“tidak” sebagai refrence category) seseorang yang memiliki kenalan wirausahawan memiliki peluang 2,7 kali lebih besar untuk menjadi seorang pengusaha daripada seseorang yang tidak memiliki kenalan.  opport (“tidak” sebagai refrence category) seseorang yang merasa ada kesempatan untuk berwirausaha memiliki peluang 1,88 kali lebih besar untuk menjadi seorang pengusaha daripada seseorang yang tidak  suskill (“tidak” sebagai refrence category) seseorang yang memiliki skil dan pengalaman memiliki peluang 4,14 kali lebih besar untuk menjadi seorang pengusaha daripada seseorang yang tidak memiliki skil dan pengalaman  fearfail (ya” sebagai refrence category) seseorang yang tidak memiliki rasa takut gagal akan berpeluang 1,86 kali lebih besar untuk menjadi seorang pengusaha daripada seseorang yang memiliki rasa takut gagal dalam memulai bisnis  gender (“perempuan” sebagai refrence category) seseorang yang berjenis kelamin laki-laki memiliki peluang 2,17 kali lebih besar untuk menjadi seorang pengusaha daripada seseorang yang berjenis kelamin perempuan 5. kesimpulan  model yang dihasilkan oleh model regresi logistik multilevel (dalam penelitian ini 2-level) memberikan model yang lebih baik dibandingkan dengan regresi logistik biasa. hal ini bisa dilihat dari nilai deviance yang dihasilkan dari kedua model, dimana model regresli logistik 2-level menghasilkan nilai deviance yang lebih kecil dibandingkan dengan nilai deviance yang dihasilkan oleh model regresi logistik biasa. jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 38  dengan menggunakan model regresi logistik 2level diperoleh bahwa faktor-faktor yang mempengaruhi keputusan untuk menjadi wirausahawan atau tidak adalah “knowent” (x1) / kenal dengan seorang enterpreneur,“opport”(x2)/ merasa ada kesempatan berbisnis, “suskill”(x3)/skil dan pengalaman, “fearfail”(x4)/rasa takut gagal saat memulai bisnis dan “gender”(x5)/jenis kelamin. selain itu faktor kecamatan ternyata juga berpengaruh terhadap keputusan seseorang dalam menentukan pilihan karir sebagai pengusaha atau sebagai pekerja/karyawan. daftar pustaka [1] sumardi, k, menakar jiwa wirausaha mahasiswa teknik mesin angkatan 2005, jurnal pendidikan teknologi kejuruan, iv(10) (pebruari 2007) [2] richard cantillon (1755), diambil dari http://learning.enggar.net/materipengajaran/pengertian-wirausaha/ pada tanggal, 10 april 2017 [3] j.b say (1803), diambil dari http://learning.enggar.net/materipengajaran/pengertian-wirausaha/ pada tanggal, 10 april 2017 [4] dan stein dan jhon f.burgess (1993), diambil dari http://learning.enggar.net/materipengajaran/pengertian-wirausaha/ pada tanggal, 10 april 2017 [5] yuhendri l.v, “perbedaan minat berwirausaha mahasiswa ditinjau dari jenis kelamin”, seminar nasional ekonomi manajemen dan akuntansi (snema) fakultas ekonomi universitas negeri padang tahun 2015, padang [6] goldstein, h. multilevel statistical models 2nd ed, london (1995) [7] antika, s. f, tugas akhir, jurusan statistika fakultas matematika dan ilmu pengetahuan alam, institut teknologi sepuluh nopember, surabaya, 2011. [8] poedjiati, s. a, tugas akhir, jurusan statistika fakultas matematika dan ilmu pengetahuan alam, institut teknologi sepuluh nopember, surabaya, 2009. [9] hox, j. j, multilevel analysis : techniques and applications, london : lawrence erlbaum associates publishers (2002). http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ http://learning.enggar.net/materi-pengajaran/pengertian-wirausaha/ jurnal jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 13 perbandingan keakuratan hasil peramalan produksi bawang merah metode holt-winters dengan singular spectrum analysis (ssa) yogo aryo jatmiko1, rini luciani rahayu2, gumgum darmawan3 prodi magister statistika terapan, universitas padjajaran, bandung1,2,3 yj29289@gmail.com1, al.ashry1211@gmail.com2, gumstat@gmail.com3 abstrak metode holt-winters digunakan untuk memodelkan data dengan pola musiman, baik mengandung trend maupun tidak. terdapat dua metode peramalan dalam singular spectrum analysis (ssa), yaitu metode rekuren (r-forecasting) dan metode vektor (v-forecasting). metode rekuren melakukan kontinuasi secara langsung (dengan bantuan lrf), sedangkan metode vektor berhubungan dengan l-continuation. perbedaan metode tentunya memberikan perbedaan dalam keakuratan hasil ramalan. untuk melihat perbedaan antara ketiga metode tersebut dilakukan dengan melihat perbandingan keakuratan dan keandalan hasil ramalan. untuk mengukur ketepatan peramalan digunakan mean absolute percentage error (mape) dan untuk mengukur keandalan hasil peramalan dilakukan dengan tracking signal. aplikasi dilakukan pada produksi bawang merah indonesia periode januari 2006-desember 2015. peramalan kedua metode di ssa menggunakan window length l=39 dan grouping r=8. dengan nilai α = 0.1, β= 0.001 dan γ=0.5, metode holtwinters additive memberikan hasil yang lebih baik dengan mape 13,469% dibanding metode ssa. kata kunci: holt-winters, mape, r-forecasting, ssa, v-forecasting abstract the holt-winters method is used to model data with seasonal patterns, whether trends or not. there are two methods of forecasting in singular spectrum analysis (ssa), namely recurrent method (r-forecasting) and vector method (v-forecasting). the recurrent method performs continuous continuation (with the help of lrf), whereas the vector method corresponds to the lcontinuation. different methods of course make a difference in the accuracy of forecast results. to see the difference between the three methods is done by looking at the comparison of accuracy and reliability of forecast results. to measure the accuracy of forecasting used mean absolute percentage error (mape) and to measure the reliability of forecasting results is done by tracking signal. applications are done on indonesian red onion production from january 2006 to december 2015. forecasting of both methods in ssa uses window length l = 39 and grouping r = 8. with α = 0.1, β = 0.001 and γ = 0.5, holt-winters additive method gives better result with mape 13,469% than ssa method. keywords: holt-winters, mape, r-forecasting, ssa, v-forecasting 1. pendahuluan bawang merah merupakan salah satu komoditas penting bagi masyarakat indonesia dan memiliki nilai ekonomis yang cukup tinggi. selain untuk konsumsi, bawang merah juga merupakan salah satu komoditas ekspor. bawang merah tidak hanya diekspor dalam bentuk sayuran segar, tetapi juga setelah diolah menjadi produk bawang goreng [1]. berdasarkan data food and agriculture organization (fao) tahun 2010-2015, indonesia menempati urutan keempat sebagai negara eksportir bawang merah di dunia setelah new zealand, perancis dan belanda. berdasarkan data susenas badan pusat statistik (bps), konsumsi bawang merah secara nasional per kapita per tahun pada maret 2015 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 14 sekitar 2,64 kilogram. sedangkan sebulan sekitar 0,22 kilogram, dan per minggu 52 gram per kapita. hal tersebut menyebabkan permintaan akan bawang merah terus meningkat seiring dengan perkembangan jumlah penduduk. data bappenas menunjukkan permintaan bawang merah pada tahun 2012 mencapai 904 ribu ton mengalami peningkatan pada tahun 2015 menjadi 963,4 ribu ton. hal ini menyatakan bahwa masyarakat indonesia tidak terlepas akan kebutuhan bawang merah setiap harinya, dikarenakan bawang merah merupakan penyedap pokok bagi pangan di indonesia. dari keterangan di atas diketahui bahwa bawang merah merupakan komoditas yang penting untuk diteliti perkembangannya. untuk meneliti perkembangan produksi bawang merah dari waktu ke waktu dapat digunakan metode holt-winters dan metode singular spectrum analysis (ssa). metode holt-winters dapat digunakan untuk data time series yang mengandung trend dan musiman [15]. metode ini mempunyai dua metode yakni metode perkalian musiman (multiplicative seasonal method) dan metode penambahan musiman (additive seasonal method). model multiplicative digunakan apabila terdapat kecenderungan atau tanda bahwa pola musiman bergantung pada ukuran data. dengan kata lain, pola musiman membesar seiring meningkatnya ukuran data. sedangkan model additive digunakan jika kecenderungan tersebut tidak terjadi. ssa adalah teknik analisis deret waktu dan peramalan yang menggabungkan unsur analisis klasik time series, multivariate statistics, multivariate geometric, dynamical systems, dan signal processing. terdapat dua metode peramalan dalam ssa, yaitu metode rekuren (r-forecasting) dan metode vektor (vforecasting). metode rekuren adalah metode dasar yang sering digunakan karena relatif lebih mudah [2]. metode vektor merupakan hasil modifikasi dari metode rekuren. perbedaan antara r-forecasting dan v-forecasting adalah peramalan dengan r-forecasting melakukan kontinuasi secara langsung (dengan bantuan lrf), sedangkan peramalan dengan vforecasting berhubungan dengan lcontinuation. hal ini menyebabkan dalam approximate continuation-nya biasanya memberikan hasil yang berbeda [2][7]. dengan kelebihan dan kemudahan metode holt-winters dan ssa, penulis menggunakan metode peramalan holt-winters model additive dan metode r-forecasting dan v-forecating dalam ssa untuk meramalkan produksi bawang merah, kemudian hasil ketiga metode tersebut akan dibandingkan dengan mengukur ketepatan peramalannya dengan menggunakan mape. 2. kajian teori 2.1 metode holt-winters metode holt-winters ini digunakan untuk mengatasi permasalahan adanya trend dan indikasi musiman [15]. metode ini merupakan gabungan dari metode holt dan metode winters [3]. titik berat metode ini adalah pada nilai level (α), kemiringan slope (β), maupun efek musiman (γ). parameter nilai level (α), kemiringan slope (β), maupun efek musiman (γ) berada diantara nilai 0 dan 1. nilai-nilai yang mendekati 0 berarti bahwa pengaruh pembobot relatif kecil pada nilai pengamatan terbaru ketika membuat perkiraan nilai-nilai masa depan [4]. peramalan dengan metode ini pada umumnya tidak selalu harus memenuhi kaidahkaidah deret waktu seperti signifikansi autokorelasi dan stasioneritas. 2.2 metode singular spectrum analysis (ssa) singular spectrum analysis (ssa) dikenal sebagai metode yang powerful untuk analisis deret waktu. ssa adalah teknik analisis deret waktu dan peramalan yang menggabungkan unsur analisis klasik time series, multivariate statistics, multivariate geometric, dynamical systems, dan signal processing [8]. tujuan utama dari ssa yaitu menguraikan serial aslinya menjadi sejumlah kecil komponen yang dapat diidentifikasi seperti tren, periodik, dan noise, kemudian diikuti oleh rekontruksi dari serial aslinya [5]. ssa merupakan sebuah metode yang sangat berguna untuk memecahkan masalah: (i) menemukan tren dari resolusi yang berbeda; (ii) smoothing; (iii) ekstraksi komponen musiman; (iv) ekstraksi simultan untuk siklus dengan periode kecil dan besar; (v) ekstraksi perioditas dengan amplitudo yang bervariasi; (vi) ekstraksi simultan untuk tren dan perioditas yang kompleks; (vii) mendeteksi change-point [2]. memecahkan ketujuh masalah tersebut adalah kapabilitas dasar dari ssa. untuk mencapai kapabilitas jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 15 tersebut, tidak perlu mempertimbangkan model parametrik dari time series. 3. metode penelitian 3.1 metode holt-winters additif model musiman aditif cocok untuk prediksi data time series yang dimana amplitudo atau ketinggian pola musimannya tidak tergantung pada rata-rata level atau ukuran data [6]. level: 𝑆𝑡 = 𝛼(𝑌𝑡 − 𝐼𝑡−𝐿 ) + (1 − 𝛼)(𝑆𝑡−1 + 𝑏𝑡−1) trend: 𝑏𝑡 = 𝛽(𝑆𝑡 − 𝑆𝑡−1) + (1 − 𝛽)𝑏𝑡−1 seasonal: 𝐼𝑡 = 𝛾(𝑌𝑡 − 𝑆𝑡 ) + (1 − 𝛾)𝐼𝑡−𝐿 forecast: 𝐹𝑡+𝑚 = 𝑆𝑡 + 𝑏𝑡 𝑚 + 𝐼𝑡−𝐿+𝑚 3.2 algoritma ssa 3.2.1 dekomposisi pada dekomposisi terdapat dua tahap yaitu embedding dan singular value decomposition (svd). parameter yang memiliki peran penting dalam dekomposisi adalah window length (l) [11]. a. embedding langkah pertama dalam ssa adalah embedding, dimana data deret waktu 𝐹 = (𝑓0, 𝑓1, … , 𝑓𝑁−1) dengan panjang n dan tidak terdapat data missing ditransformasi ke dalam matriks lintasan berukuran l x k. matriks lintasan ini merupakan matriks dimana semua elemen pada anti diagonalnya bernilai sama. 𝑋𝑖𝑗 = [ 𝑓0 𝑓1 ⋮ 𝑓𝐿−1 𝑓1 𝑓2 ⋮ 𝑓𝐿 … … ⋱ … 𝑓𝐾−1 𝑓𝐾 ⋮ 𝑓𝑁−1 ] (1) pada tahap ini diperlukan penentuan parameter window length dengan syarat 2 < 𝐿 < 𝑁 2 . dalam penentuan window length ini harus dipertimbangkan kemungkinan bahwa data memiliki komponen periodik. sampai saat ini penentuan window length masih dengan cara trial and error. konsep dasar pada tahap embedding ini adalah melakukan pemetaan yang mentransfer data deret waktu f satu dimensi ke dalam multi dimensi 𝐗𝟏, 𝐗𝟐, … , 𝐗𝑲 sehingga didapatkan output sebuah matriks yaitu matriks hankel dimana semua elemen pada anti diagonalnya bernilai sama. b. singular value decomposition (svd) langkah kedua dalam dekomposisi adalah membuat singular value decomposition (svd) dari matriks lintasan. misalkan 𝜆1, … , 𝜆𝐿 adalah eigenvalue dari matriks s dimana 𝐒 = xxt dengan urutan menurut 𝜆1 ≥ ⋯ ≥ 𝜆𝐿 ≥ 0 dan 𝑈1, … , 𝑈𝐿 adalah eigenvector dari masingmasing eigenvalue. rank dari matriks x dapat ditunjukkan dengan d = max{𝑖, 𝜆𝑖 > 0}. jika dinotasikan 𝑽𝒊 = 𝑿𝑻𝑼𝒊 √𝝀𝒊 untuk i = 1,..., d, maka svd dari matriks lintasan adalah sebagai berikut: 𝐗 = 𝑿𝟏 + 𝑿𝟐 + ⋯ + 𝑿𝒅 = 𝑈1√𝜆1𝑉1 𝑇 + 𝑈2√𝜆2𝑉2 𝑇 + ⋯ + 𝑈𝑑 √𝜆𝑑 𝑉𝑑 𝑇 = ∑ 𝑈𝑖√𝜆𝑖 𝑉𝑖 𝑇𝒅 𝒊=𝟏 (2) matriks x terbentuk dari eigenvector ui, singular value √𝜆𝑖 dan komponen utama 𝑉𝑖 𝑇 . ketiga elemen pembentuk svd ini disebut dengan eigentriple. konsep dasar pada tahap ini adalah mendapatkan barisan matriks dari matriks s, dimana pada masing-masing matriks dalam barisan tersebut mengandung eigenvector ui, singular value √𝜆𝑖 dan komponen utama 𝑉𝑖 𝑇 yang menggambarkan karakteristik pada masing-masing matriks dalam barisan tersebut. 3.2.2 rekontruksi dalam tahap rekontruksi terdapat dua langkah yang harus dilakukan, yaitu langkah grouping kemudian dilanjutkan dengan pembentukan deret rekontruksi berdasarkan hasil yang diperoleh pada langkah diagonal averaging. a. grouping pada langkah akan dilakukan pengelompokan hasil dekomposisi matriks lintasan yang berukuran l x k dengan tujuan untuk memisahkan kompononen aditif svd ke dalam beberapa sub kelompok, yaitu tren, musiman, periodik dan noise. setelah itu menjumlahkan matriks dalam setiap kelompok. hasil yang diperoleh berupa representasi dari matriks lintasan sebagai jumlah dari beberapa matriks resultan. matriks lintasan 𝑿𝒊 akan dipartisi ke dalam m subset disjoin 𝐼1, 𝐼2, … , 𝐼𝑚 . misalkan 𝐼 = jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 16 {𝑖1, 𝑖2, … , 𝑖𝑝} adalah matriks resultan 𝑿𝑰 dengan indeks 𝑖1, 𝑖2, … , 𝑖𝑝 sesuai dengan kelompok 𝐼 yang dapat didefiniskan sebagai 𝑿𝑰 = 𝑿𝒊𝟏 + 𝑿𝒊𝟐 + ⋯ + 𝑿𝒊𝒑. kemudian 𝑿𝑰 disesuaikan dengan kelompok 𝑰 = {𝐼1, 𝐼2, … , 𝐼𝑚}. maka, 𝐗 = 𝑿𝟏 + 𝑿𝟐 + ⋯ + 𝑿𝒅 dapat diekspansi menjadi x = 𝑿𝑰𝟏 + 𝑿𝑰𝟐 + ⋯ + 𝑿𝑰𝒎. pada tahap singular value decomposition telah didapatkan eigenvalue yang menggambarkan karakteristik untuk setiap kolom pada matriks 𝑺 = 𝑋𝑋𝑇 . eigenvalue ini direpresentasikan oleh eigenvector. oleh karena itu bahan dasar pengelompokan pada tahap grouping adalah eigenvector. eigenvector pada masing-masing kolom pada matriks lintasan akan dilakukan ekstraksi terhadap pola komponen seriesnya. ekstrasi pertama dilakukan terhadap pola tren, kemudian dilakukan ekstraksi dilakukan terhadap pola musiman dengan menggunakan analisis spektral. eigenvector yang mengikuti pola musiman dengan periode kurang dari 12 maka digolongkan pada kelompok musiman, sedangkan eigenvecktor yang megikuti pola musiman dengan periode lebih dari 12 akan digolongkan pada kelompok siklik. kemudian eigevector yang tidak mengikuti musiman atau siklik atau tren akan digolongkan sebagai noise [9]. metode untuk mengekstra tren pada ssa diantaranya dengan menggunakan pendekatan naïve [8][9]. tujuan dari pendekatan naive ini untuk mengkonstruksi tren dari beberapa komponen awal singular value decomposition. hal ini dikarenakan eigenvalue merupakan kontribusi dari komponen singular value decomposition yang bersesuaian ke dalam bentuk time series. tren biasanya mencirikan bentuk time series, dimana eigenvalue dari tren lebih besar dibandingkan dengan yang lainnya, yang berarti bahwa tren adalah komponen dengan nomor urutan yang terkecil. dalam mengekstraksi pola musiman digunakan metode analisis spektral. metode spektral ini dapat digunakan untuk mendeteksi masing-masing eigenvector apakah memiliki pola musiman atau tidak. jika masing-masing eigenvector memiliki pola musiman kemudian ditentukan perioditas musimannya. kelompok eigenvector yang memiliki periode yang sama akan dikelompokkan menjadi satu kelompok. b. diagonal averaging setelah melakukan grouping, tahap selanjutnya akan dilakukan transformasi dari hasil pengelompokan 𝑿𝑰𝒊 ke dalam deret baru dengan panjang n. tujuan dari tahap ini adalah mendapatkan singular value dari komponenkomponen yang telah dipisahkan, kemudian akan digunakan dalam peramalan. hasil pada tahap ini merupakan matriks f. 𝐅 = [ 𝑓11 𝑓21 ⋮ 𝑓𝐿 𝑓21 𝑓22 ⋮ 𝑓𝐿+1 … … ⋱ … 𝑓𝐾 𝑓𝐾+1 ⋮ 𝑓𝑁 ] (3) diagonal average dirumuskan sebagai berikut: 1 𝑘 ∑ 𝑓𝑚,𝑘−𝑚+1 ∗𝑘 𝑚=1 untuk 1 ≤ 𝑘 ≤ 𝐿 ∗ 𝑔𝑘 = 1 𝐿∗−1 ∑ 𝑓𝑚,𝑘−𝑚+1 ∗𝐿∗−1 𝑚=1 untuk 𝐿 ∗ < 𝑘 ≤ 𝐾∗ + 1 1 𝑁−𝑘+1 ∑ 𝑓𝑚,𝑘−𝑚+1 ∗𝑁−𝐾∗+1 𝑚=𝑘−𝐾+1 untuk𝐾 ∗ + 1 < 𝑘 ≤ 𝑁 dimana 𝐿∗ = min(𝐿, 𝐾) dan 𝐾∗ = max(𝐿, 𝐾). persamaan diatas jika diaplikasikan ke dalam matriks resultan 𝑿𝒊𝒎 akan membentuk deret �̃�(𝑘) = (�̃�1 (𝑘) , … , �̃�𝑁 (𝑘) ). oleh karena itu, deret asli akan didekomposisi menjadi jumlah dari m deret. 𝑦𝑛 = ∑ �̃�𝑛(𝑘) 𝑚 𝑘=1 (4) 3.3 peramalan ssa terdapat dua metode peramalan dalam ssa, yaitu metode rekuren (r-forecasting) dan metode vektor (v-forecasting). metode rekuren adalah metode dasar yang sering digunakan karena relatif lebih mudah [2]. metode vektor merupakan hasil modifikasi dari metode rekuren. dalam peramalan ssa, model dibangun dengan bantuan linear rekuren formula (lrf). bentuk polynomial sebagai berikut: 𝑥𝑖 +𝑑 = ∑ 𝑟𝑘 𝑥𝑖+𝑑−𝑘 𝑑 𝑘=1 untuk, 1 ≤ 𝑖 ≤ 𝑁 − 𝑑 perbedaan antara r-forecasting dan vforecasting adalah peramalan dengan rforecasting melakukan kontinuasi secara langsung (dengan bantuan lrf), sedangkan peramalan dengan v-forecasting berhubungan jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 17 dengan l-continuation. hal ini menyebabkan dalam approximate continuation-nya biasanya memberikan hasil yang berbeda [2]. kedua metode peramalan tersebut, memiliki dua tahapan umum yaitu, diagonal averaging dan continuation. pada metode rforecasting, diagonal averaging digunakan untuk memperoleh rekontrusksi dan continuation dilakukan dengan lrf. sedangkan pada v-forecasting, kedua tahap digunakan dalam urutan kebalik, yaitu peramalan vektor dilakukan terlebih dahulu, kemudian diagonal averaging memberikan nilai ramalan. untuk mendapatkan m periode ke depan, metode v-forecasting menggunakan prosedur m+l-1 langkah. tujuannya untuk melihat kesesuaian variasi di bawah m langkah, sehingga metode ini memiliki l-1 langkah tambahan. a. algoritma peramalan metode rekuren ssa (r-forecasting) algoritma peramalan dengan rforecasting adalah sebagai berikut: 1. penaksiran koefisien lrf (𝑟1, 𝑟2, … , 𝑟𝑑 ) digunakan eigenvector yang diperoleh dari langkah singular value decomposition. dengan 𝑃 = (𝑝1, 𝑝2, … , 𝑝𝐿−1, 𝑝𝐿) 𝑇 ,𝑃𝑉 = (𝑝1, 𝑝2, … , 𝑝𝐿−1) 𝑇 , 𝜋𝑖 komponen terakhir dari vektor (𝑝1, 𝑝2, … , 𝑝𝐿−1, 𝑝𝐿), dan 𝑣 2 = ∑ 𝜋𝑖 2𝐿−1 𝑖=1 maka koefisien lrf (vektor r) dapat dihitung dengan persamaan: ℜ = (𝑟𝑙−1, … , 𝑟1) = 1 1−𝑣2 ∑ 𝜋𝑖 𝑃𝑖 𝑉𝐿−1 𝑖=1 (6) 2. dalam metode rekuren ini, deret waktu yang digunakan adalah deret hasil rekontruksi yang diperoleh dari hasil diagonal averaging, kemudian ditentukan m buah titik baru untuk ramalan. sehingga akan terbentuk deret hasil peramalan, yaitu 𝐺𝑁+𝑀 = (𝑔1, … , 𝑔𝑁+𝑀) berdasarkan rumus dibawah ini: �̃�𝑖 untuk i = 0, ..., n 𝑔𝑖 = ∑ 𝑟𝑖 𝑔𝑖−1 𝐿−1 𝑗=1 untuk i = n + 1, ..., n+m dimana 𝑔𝑁+1, … , 𝑔𝑁+𝑀 adalah hasil ramalan dari ssa. b. algoritma peramalan metode vektor ssa (v-forecasting) metode vektor merupakan modifikasi dari metode rekuren. algoritma peramalannya adalah sebagai berikut: 1. setelah tahapan svd pada algoritma awal, matriks lintasan telah membentuk matriks hankel h, tahap selanjutnya adalah menghitung matriks 𝚷. matriks 𝚷 adalah matriks operator linier dari proyeksi ortogonal ℝ𝐿−1 → ℒ𝑟 ∇. 𝚷 = 𝐯∇(𝐯∇) 𝑇 + (1 − 𝑣 2)ℜℜ𝑇 (7) dengan 𝐯∇ = [𝑃1 ∇, … , 𝑃𝑟 ∇ ] 2. hitung operator linier untuk r-forecasting 𝑃(𝑣)𝑋 = [ 𝚷𝑋∆ ℜ𝑇 𝑋∆ ] (8) 𝑋∆ merupakan vektor yang berisi l-1 komponen akhir dari vektor x 3. hitung nilai 𝑍𝑖 �̂�𝑖 untuk i= 1,..., k 𝑍𝑖 = 𝑃 (𝑣)𝑍𝑖−1 untuk i= k+1,..., k+m+l-1 4. hitung diagonal averaging dari matriks 𝐙 = [𝐙𝟏, … , 𝐙𝐊+𝐌+𝐋−𝟏 ]. diagonal averaging yang diperoleh memiliki deret 𝑔0, … , 𝑔𝑁+𝑀+𝐿−1. 5. m nilai baru hasil peramalan metode vektor adalah 𝑔𝑁, … , 𝑔𝑁+𝑀+1. 3.4 mean absolute percentage error (mape) salah satu ukuran untuk membandingkan akurasi ramalan adalah mape, yang dirumuskan sebagai berikut: 𝑀𝐴𝑃𝐸 = ( 1 𝐻 ∑ | 𝑒ℎ 𝑍𝑇+ℎ | 𝐻 ℎ=1 ) = ( 1 𝐻 ∑ | 𝑍𝑇+ℎ−�̂�𝑇+ℎ 𝑍𝑇+ℎ |𝐻ℎ=1 ) × 100% (9) dimana: h: banyaknya observasi yang diramalakan (outsample) t: banyaknya observasi yang diuji (insample) jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 18 gambar 1. skema prosedur penelitian 3.5 tracking signal tracking signal merupakan ukuran toleransi yang dapat digunakan untuk menentukan kemungkinan digunakannya hasil peramalan tersebut yang memperkirakan apabila pola dasar berubah. nilai-nilai tracking signal berada diluar batas yang diterima, yaitu ± 5 maka model peramalan harus ditinjau kembali dan akan dipertimbangkan model baru [10]. dengan perhitungan sebagai berikut: 𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = ∑ 𝑒𝑛 𝑛 1 ∑ |𝑒𝑛| 𝑛 𝑛 1 (10) 4. hasil dan pembahasan pada bagian ini akan dilakukan peramalan produksi bawang merah indonesia dengan metode holt-winters dan singular spectrum analysis. pada perhitungannya menggunakan program r versi 3.3.2. peramalan dalam ssa dilakukan dalam 2 metode yaitu r-forecasting dan v-forecasting kemudian ketiga metode tersebut dibandingkan ketepatan dan keandalan peramalannya. untuk mengukur ketepatan peramalannya digunakan mean absolute percentage error (mape) dan untuk mengukur keandalannya dengan tracking signal. gambar 2. plot dan acf produksi bawang merah indonesia januari 2006-desember 2015 4.1 identifikasi data berdasarkan plot data produksi bawang merah indonesia dari bulan januari 2006 hingga desember 2015, produksi bawang merah di indonesia memiliki fluktuasi yang tajam [12], [13], [14]. hal ini terlihat dari turun naiknya produksi bawang merah dengan ekstrim. pada plot ini pola musiman tidak nampak dengan jelas, namun bila dilihat dari acfnya nampak jelas bahwa data produksi bawang memiliki pola musiman. selain dengan melihat plot dan acf, identifikasi data musiman dilakukan dengan analisis spektral. dari hasil perhitungan analisis spektral menunjukkan bahwa data memiliki pola musiman dengan periode 12. periode inilah yang akan digunakan dalam perhitungan ssa. 4.2 peramalan produksi bawang merah indonesia dengan metode holt-winters di dalam metode holt-winters terdapat tiga parameter, yaitu level (α), trend (β), dan efek musiman (γ). dalam penelitian ini akan digunakan metode holt-winters additif. dengan menggunakan software r 3.3.2 diperoleh nilai α = 0.1, β= 0.001 dan γ=0.5 yang memiliki mape terkecil. pada gambar 3 terlihat bahwa nilai fitted dan nilai observed (aktual) hampir mendekati dan mengikuti pola yang sama. hal ini menunjukkan metode holt-winters layak digunakan untuk data produksi bawang merah indonesia. data identifikasi pembentukan matriks haenkel singluar value decompositio n (svd) grouping diagonal averaging evaluasi peramalan tidak tidak peramalan ya ya ukuran ketepatan peramalan? uji keandalan peramalan? decompositio n reconstructio n jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 19 gambar 3. holt-winters method for produksi bawang merah indonesia 4.3 peramalan produksi bawang merah indonesia dengan singular spectrum analysis dalam proses singular spectrum analysis terdapat dua langkah, yaitu dekomposisi dan rekontruksi. dekomposisi terdiri embedding dan singular value decomposition (svd). sedangkan tahap rekontruksi terdiri dari grouping dan diagonal averaging. pada proses embedding menentukan nilai window length (l), dengan 2 < 𝐿 < 𝑁 2 . melalui trial dan error dilakukan pemilihan nilai l dengan sofware r dengan nilai mape minimum. nilai l=39 memiliki nilai mape minimum sehingga diperoleh matriks lintasan yaitu matriks hankel c berdimensi 39 x 39, seperti berikut : 𝐶39𝑥39 = [ 𝐶0 𝐶1 ⋮ 𝐶38 𝐶1 𝐶2 ⋮ … … … ⋱ 𝐶75 𝐶38 ⋮ 𝐶75 𝐶76 ] 𝐶39𝑥39 = [ 48884 40983 ⋮ 39997 40983 34691 ⋮ … … … ⋱ 104847 39997 ⋮ 104847 92480 ] selanjutnya mendapat nilai k=n-l+1, sehingga pada proses singular value decomposition (svd) akan membuat matriks dengan ordo lxk. salah satu unsur pada tahap ini adalah adanya nilai singular value yang merupakan akar kuadrat eigenvalue. gambar 4. plot eigenvalue tahap selanjutnya, yaitu svd yang akan menghasilkan 39 eigentriple dari matriks 𝐒 = 𝐗𝑿𝑻. eigentriple ini terdiri dari singular value, eigenvector, dan principal component. nilai eigentriple ini digunakan untuk memisahkan komponen, sehingga komponen ini dapat dikelompokan. hal ini dapat dilihat pada gambar 4. berdasarkan gambar 4, dapat diketahui banyaknya grouping, langkah selanjutnya adalah melakukan identifikasi komponenkomponen. berdasarkan gambar 5, eigenvector pertama memiliki pola tren, eigenvector 2 dan 3 diidentifikasi sebagai pola siklik. eigenvector 4-21 menunjukkan pola musiman, sedangkan sisanya dianggap noise. untuk melakukan identifikasi visual komponen periodik, tidak dapat ditentukan hanya dengan melalui plot saja, namun akan dilakukan melalui pengujian musiman dengan analisis spektral. dari informasi eigentriple yang terbentuk, dapat disimpulkan bahwa terdapat 8 group yaitu trend, siklis, season1, season2, season3, season 4, dan season 5, serta noise. untuk melihat keterpisahan komponen-komponen, dapat dilihat pada gambar 5. dari gambar 6, besarnya korelasi ditunjukkan oleh gradasi warna dari warna muda hingga tua. semakin tua warnanya semakin tinggi korelasinya. dari matriks diatas terlihat bahwa tren, siklik dan season 2 tidak berkorelasi dengan komponen lainnya, season 4 dan season 5 menunjukkan adanya korelasi antar komponen, namun korelasi tersebut tidak terlalu dalam. sehingga, group yang terbentuk dianggap sudah baik. component norms index n o r m s 10 5̂.0 10 5̂.5 10 6̂.0 10 6̂.5 0 10 20 30 40 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 20 gambar 5. eigenvectors gambar 6. matrik w-correlation 4.4 perbandingan peramalan dengan metode holt-winters, metode rekurren ssa dan metode vektor ssa gambar 7 menyajikan grafik outsample dan hasil ramalan produksi bawang merah indonesia untuk 12 bulan ke depan dengan menggunakan metode r-forecasting dan vforecasting. outsample dengan garis hitam, hasil ramalan metode holt-winters dengan garis warna hijau, hasil ramalan r-forecasting dengan garis merah dan hasil ramalan vforecasting dengan garis biru. dari gambar tersebut, terlihat bahwa peramalan produksi bawang merah indonesia untuk 12 bulan ke depan dengan metode holt-winters dan metode rekuren (r-forecasting) serta metode vektor (vforecasting), terdapat sedikit perbedaan. jika dibandingkan dengan data outsample, hasil ramalan metode holt-winters lebih mendekati nilai dari outsample. gambar 7. outsample, hasil ramalan holt-winters, r-forecasting dan v-forecasting selanjutnya akan dilihat keakuratan hasil peramalan dengan menggunakan nilai mape. berdasarkan tabel 1, menunjukkan bahwa hasil keakuratan peramalan dari metode holt-winters sebesar 13,469%. sedangkan metode rekuren sebesar 15,625 % dan metode vektor sebesar 14,295 %. kriteria mape menurut [10] yaitu, (i) <10 % peramalan sangat akurat, (ii) 10-20 % peramalan akurat, (iii) 20-50% peramalan cukup akurat, (iv) <50% kurang akurat. menurut mape kriteria lewis, ketiga metode peramalan tersebut, termasuk ke dalam kategori peramalan akurat. namun, hasil pengujian menunjukkan metode holt-winters memiliki mape lebih kecil dibandingkan metode rekuren dan metode vektor. dengan demikian, peramalan produksi bawang merah indonesia lebih akurat jika menggunakan metode holtwinters. tabel 1. keakuratan hasil ramalan metode peramalan mape rekuren (r-forecasting) 15.625 vektor (v-forecasting) 14.295 holt-winters 13.469 dari gambar 8 menunjukkan tracking signal hasil peramalan produksi bawang merah 12 bulan terakhir dengan menggunakan metode holt-winters, metode r-forecasting dan metode v-forecasting. besarnya nilai-nilai tracking signal dari 12 periode waktu yang diramalkan berada pada batas toleransi yang bisa diterima, yaitu ±5 (bovas dan ledolter, 1983). ini w-correlation matrix trend siklik season1 season2 season3 season4 season5 trend siklik season1 season2 season3 season4 season5 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 21 menunjukkan bahwa metode peramalan masih bisa digunakan untuk meramalkan m waktu ke depan. gambar 8. tracking signal holt-winters, rforecasting dan v-forecasting berdasarkan tabel 2, dengan metode holtwinters produksi bawang merah pada tahun 2016 akan mencapai puncaknya pada bulan januari 2016. metode tersebut meramalkan bulan maret 2016 saat bawang merah mengalami produksi terendah dalam setahun. pada bulan maret produksi bawang merah mengalami kekurangan sebesar 18 ribu ton. namun pada bulan januari mengalami surplus sebesar 53 ribu ton. selain bulan maret, pada bulan februari dan desember juga mengalami kekurangan masing-masing sekitar 9 ribu ton dan 4 ribu ton, namun pada bulan-bulan yang lain mengalami surplus, sehingga kebutuhan bawang merah untuk satu tahun masih bisa terpenuhi. tabel 2. hasil ramalan produksi bawang merah tahun 2016 dengan metode holt-winters bulan hasil ramalan (dalam ton) januari 133752.63 februari 71609.98 maret 62206.55 april 83141.05 mei 91047.23 juni 116373.16 juli 112312.77 agustus 118085.74 september 98538.70 oktober 97892.98 november 82314.93 desember 76112.33 5. kesimpulan berdasarkan fenomena data produksi bawang merah indonesia dapat disimpulkan bahwa data produksi bawang merah indonesia merupakan data musiman dengan periode 12. hasil trial dan error menghasilkan window length = 39. hasil mape menunjukkan bahwa metode r-forecasting dan v-forecasting serta holt-winters menghasilkan peramalan dengan kategori akurat, namun holt-winters lebih akurat dibandingkan r-forecasting dan v-forecasting. uji keandalan melalui tracking signal menunjukkan bahwa metode peramalan masih bisa digunakan untuk meramalkan m waktu ke depan. produksi bawang merah pada tahun 2016 akan mencapai puncaknya pada bulan januari 2016 dan pada bulan maret 2016 saat bawang merah mengalami produksi terendah dalam setahun. pada bulan maret produksi bawang merah mengalami kekurangan sebesar 18 ribu ton, sedangkan pada bulan januari mengalami surplus sebesar 53 ribu ton. selain bulan maret, pada bulan februari dan desember juga mengalami kekurangan masing-masing sekitar 9 ribu ton dan 4 ribu ton, namun pada bulan-bulan yang lain mengalami surplus, sehingga kebutuhan bawang merah untuk satu tahun masih bisa terpenuhi. referensi [1] latarang, burhanuddin, dkk. pertumbuhan dan hasil bawang merah (allium ascalonicum l.) pada berbagai dosis pupuk kandang, journal agroland 13(3), september (2006) 265-269 [2] golyandina n., nekrutkin, v., zhigljavsky a. analysis of time series structure: ssa and related techniques. chapman & hall/crc (2001) [3] croux, c., gelper, s. & fried, r. computational aspects of robust holt-winters smoothing based on m-estimation. appl math (2008) 53: 163. [4] coghlan, avril. a little book of r for time series release 0.2. parasite genomics group, wellcome trust sanger institute, cambridge, u.k. (2017) [5] hassani, hossein. singular spectrum analysis: methodology and comparison. journal of data science 5 (2007) 239-257 jurnal matematika “mantik” vol. 03 no. 01. mei 2017. issn: 2527-3159 e-issn: 2527-3167 22 [6] montgomery, d. c, introduction to time series analysis and forecasting, 2nd edition, 2008 [7] sakinah, a.m.. perbandingan stabilitas hasil peramalan suhu dengan r-forecasting dan v-forecasting ssa untuk long-horizon. tesis. universitas padjadjaran, bandung. 2012. [8] golyandina n., zhigljavsky a.. singular spectrum analysis for time series. new york: springer (2013) [9] darmawan, g., hendrawati t., arisanti r., model auto singular spectrum untuk meramalkan kejadian banjir di bandung dan sekitarnya. prosiding seminar nasional matematika dan pendidikan matematika uny 2015. november (2015) 457-462, yogyakarta [10] amalia, s.n. peramalan singular spectrum analysis dengan missing data. tesis. universitas padjadjaran, bandung. 2016. [11] darmawan, g.identifikasi pola data curah hujan pada proses grouping dalam metode singular spectrum analysis. prosiding seminar nasional matematika dan pendidikan matematika uny 2016. november (2016) 127-132, yogyakarta [12] kementerian pertanian. outlook bawang merah 2013. diambil dari http://epublikasi.setjen.pertanian.go.id/arsipoutlook/260-outlook-komoditas-bawangmerah-2013 pada tanggal 5 maret 2017 [13] kementerian pertanian. outlook bawang merah 2015. diambil dari http://epublikasi.setjen.pertanian.go.id/arsipoutlook/76-outlook-hortikultura/356outlook-bawang-merah-2015 pada tanggal 5 maret 2017 [14] kementerian pertanian. outlook bawang merah 2016. diambil dari http://epublikasi.setjen.pertanian.go.id/arsipoutlook/76-outlook-hortikultura/426outlook-bawang-merah-2016 pada tanggal 5 maret 2017 [15] suwandi, adi, dkk.. peramalan data time series dengan metode penghalusan eksponensial holt-winter. jurnal universitas hassanudin, makasar. 2014. http://epublikasi.setjen.pertanian.go.id/arsip-outlook/260-outlook-komoditas-bawang-merah-2013 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/260-outlook-komoditas-bawang-merah-2013 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/260-outlook-komoditas-bawang-merah-2013 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/356-outlook-bawang-merah-2015 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/356-outlook-bawang-merah-2015 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/356-outlook-bawang-merah-2015 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/426-outlook-bawang-merah-2016 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/426-outlook-bawang-merah-2016 http://epublikasi.setjen.pertanian.go.id/arsip-outlook/76-outlook-hortikultura/426-outlook-bawang-merah-2016 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 aplikasi jaringan bayes pada pembuatan butir soal tes wahyu hartono1, tonah2 1universitas swadaya gunung jati, wahyuhartono@unswagati.ac.id 2universitas swadaya gunung jati, tonahlabibah@gmail.com doi:https://doi.org/10.15642/mantik.2018.4.1.49-52 abstrak mata kuliah kalkulus diferensial merupakan mata kuliah yang penting karena merupakan materi prasyarat pada sebagian besar mata kuliah pada tingkat berikutnya. dari pengalaman peneliti, sebagian besar mahasiswa belum dapat menguasai materimateri prasyarat dari mata kuliah tersebut. kondisi tersebut akan menghambat proses belajar mengajar. pengetahuan terhadap kemampuan awal mahasiswa akan berguna untuk menerapkan model pembelajaran yang sesuai. penelitian ini memaparkan aplikasi jaringan bayes pada pembuatan butir soal fixed test dan adaptive test terkait mata kuliah kalkulus diferensial. metode penelitian yang digunakan adalah metode eksperimen. sampel yang digunakan adalah mahasiswa program studi pendidikan matematika sebanyak 98 mahasiswa yang pernah mendapatkan materi kalkulus diferensial. hasil penelitian menunjukkan bahwa kinerja desain adaptive test dalam memprediksi kemampuan mahasiswa lebih baik daripada desain fixed test terutama setelah soal kelima. kinerja dari fixed test yg soalnya diurutkan dari mudah ke sukar lebih baik daripada desain fixed test lainnya. hasil penelitian ini akan dijadikan input dalam membuat soal tes diagnosa yang berguna untuk memetakan/memprediksi kemampuan awal serta mengevaluasi kemampuan mahasiswa pada mata kuliah kalkulus diferensial. saran untuk pengembangan penelitian selanjutnya adalah membuat desain soal fixed test yang kemampuan diagnosanya setara dengan desain adaptive test. kata kunci: fixed test, adaptive test, jaringan bayes abstract the course of differential calculus is essential because it is a prerequisite material in most classes at the next level. from experience, most of the students have not been able to master the prerequisite topic. these conditions will disrupt the teaching and learning process. information about the students' initial knowledge will be useful for applying appropriate learning models. this research describes bayes network application on the manufacture of items about the fixed and adaptive test related to differential calculus courses. the research method is an experiment. the sample used is the students of mathematics education program as many as 98 students who already finish differential calculus course. the results showed that the performance of adaptive test design in predicting student ability is better than fixed test design, especially after the fifth question. the performance of the fixed test items sorted from easy to difficult is better than other fixed test designs. this study is useful for making diagnostic test questions in mapping/predicting students' initial knowledge as well as evaluating their abilities. the suggestion for further research is to make the performance of fixed test design is equivalent to adaptive test in diagnostic capability. keywords: fixed test, adaptive test, bayes network 1. pendahuluan mata kuliah kalkulus diferensial yang diajarkan pada tahun pertama perkuliahan merupakan mata kuliah yang penting untuk mahasiswa program studi pendidikan matematika karena merupakan materi prasyarat pada sebagian besar mata kuliah ditingkat jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 50 berikutnya. untuk menguasai kalkulus diferensial juga diperlukan penguasaan keahlian/skill materi tertentu sehingga perlu dibuat suatu alat diagnosa yang dapat memetakan kemampuan mahasiswa secara cepat dan tepat. hasil dari pemetaan tersebut dapat dijadikan acuan pembuatan bahan ajar ataupun program belajar yang unik untuk setiap mahasiswa. menurut vomlel [7], dalam merancang suatu tes, biasanya dibentuk sebuah himpunan yang terdiri dari keahlian (skill), kemampuan (ability), miskonsepsi, dan lain-lain serta sebuah himpunan yang berisi bank soal, tugas, dan lain-lain. misalkan 𝑆 = {𝑆1, 𝑆2,⋯ ,𝑆𝑘} menotasikan skill, ability, miskonsepsi, dan lain-lain serta 𝜒 = {𝑋1,𝑋2,⋯ ,𝑋𝑚} menotasikan bank soal, tugas, dan lain-lain. untuk mempermudah selanjutnya 𝑆 disebut skills dan 𝜒 disebut pertanyaan. perancang tes harus menentukan skill mana yang secara langsung berhubungan dengan setiap pertanyaan. hubungan tersebut seringkali bersifat probabilitas, khususnya pada soal tes berbentuk pilihan ganda. salah satu pendekatan dalam merancang suatu tes adalah menyusun serangkaian pertanyaan yang mengakomodasi seluruh skills yang akan diuji. pendekatan tersebut sering disebut sebagai fixed test. pendekatan lainnya adalah menyusun soal tes yang optimal untuk masing-masing peserta tes. setelah suatu pertanyaan dijawab oleh peserta tes, sistem akan memilih pertanyaan selanjutnya berdasarkan jawaban dari pertanyaan sebelumnya. karena pendekatan tersebut menggunakan komputer dalam penyajian tes nya, maka sering disebut computer adaptive testing (cat), mislevy [6]. tes yang secara otomatis menyajikan pertanyaan berdasarkan jawaban dari pertanyaan sebelumnya disebut sebagai adaptive tests. jaringan bayes (bayesian networks) adalah model yang populer dari keluarga model berbentuk grafik, lauritzen [4]. beberapa peneliti telah mengaplikasikan jaringan bayes untuk tes pendidikan dan pelatihan. millan et al. [5] menggunakan jaringan bayes untuk memodelkan siswa pada computerized adaptive tests. mislevy et al. [6] meneliti bagaimana parameter numerik dari model probabilitas dapat ditaksir. almond et al. [1] meneliti model untuk tabel-tabel probabilitas bersyarat pada penilaian pendidikan. dalam karya ilmiah yang ditulis oleh conati et al. [2] aplikasi menarik dari jaringan bayes untuk memodelkan siswa dalam melatih kemampuan pemecahan masalah telah disajikan. berdasarkan uraian pada paparan di atas, rumusan masalah penelitian ini adalah sebagai berikut. bagaimana membuat adaptive dan fixed test dari materi kalkulus diferensial menggunakan bayesian network? bagaimana tingkat ketelitian dari adaptive dan fixed test dalam menduga skill mahasiswa terkait materi kalkulus diferensial? berdasarkan masalah penelitian yang telah dirumuskan, maka tujuan penelitian ini secara khusus adalah untuk membuat adaptive test dan fixed test dari materi kalkulus diferensial menggunakan bayesian network serta membandingkan tingkat ketelitiannya dalam menduga skills mahasiswa. 2. metode penelitian penelitian ini merupakan penelitian eksperimen dengan empat tahapan. penelitian diawali dengan penyusunan soal tes kalkulus diferensial berbentuk essay berdasarkan skill yang diukur kemudian dilakukan validasi. setelah soal tes yang dibuat valid, selanjutnya dilakukan pelaksanaan tes terhadap 150 orang mahasiswa yang pernah mendapatkan mata kuliah kalkulus diferensial. hasil tes selanjutnya dinilai/diteliti learning obstacle nya untuk kemudian dikodekan sebagai dasar membangun model siswa. dari 150 sampel, kemudian kami hilangkan sampel yang tidak memiliki skill dan jawabannya salah semua sehingga tersisa 98 sampel. pada tahap kedua, kami membangun model yang menggambarkan hubungan antara skill siswa dan soal tes. akan digunakan model probabilistik 𝑃(𝑺, 𝑿) yang direpresentasikan oleh jaringan bayes untuk memodelkan masalah tersebut. selanjutnya kami memasukan nilai-nilai peluang bersyarat yang diperoleh dari data sampel sebanyak 98 mahasiswa ke software hugin explorer untuk menghitung peluang bayesnya. nilai peluang bayes yang diperoleh kami gunakan untuk mengklasifikasikan soal tes menjadi mudah, sedang, dan sukar. pengkategorian tersebut berguna untuk mendesain fixed test dan jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 51 adaptive test. pada tahap ketiga, dari model yang telah dibangun, akan digunakan untuk membandingkan empat metode desain tes menurut: a. fixed test dimana pertanyaannya pada urutan sesuai dengan saat tes awal diberikan. b. fixed test dimana pertanyaannya diberikan dengan urutan yang terbalik dari tes awal. c. adaptive test dimana pertanyaan yang diberikan sesuai dengan jawaban dari pertanyaan sebelumnya. soal pertama dipilih pada posisi tengah setelah seluruh soal diurutkan dari yang mudah ke yang sukar. d. fixed test dimana pertanyaannya diberikan dengan urutan dari yang mudah ke yang sukar. kesimpulan yang akan diperoleh adalah berupa desain tes yang paling optimal dan akurat untuk memprediksi skill siswa. 3. hasil dan pembahasan gambar 1 berikut ini adalah nilai-nilai peluang bayes yang diperoleh setelah nilai-nilai peluang bersyarat dari hasil ms. excel diinput ke software hugin explorer [3]. nilai peluang bayes tersebut akan digunakan untuk mendesain tes diagnosa berdasarkan tingkat kesukaran soal. semakin besar nilai tingkat kesukaran, maka semakin mudah soal tersebut. berdasarkan nilai tingkat kesukaran tersebut akan disusun fixed test dan adaptive test. gambar 1 nilai peluang bayes dari setiap node pada tahap memeriksa hasil jawaban mahasiswa, kami telah mengidentifikasi skill/kemampuan matematis yang dimiliki oleh setiap siswa yang dijadikan sebagai sampel penelitian, sehingga kami bisa memprediksi skill yang dimiliki siswa dengan cara membandingkannya dengan jawaban setiap pertanyaan tes. kami menggunakan software mathematica 11 untuk mempercepat proses perbandingan tersebut. jika semua prediksi skill dari desain fixed test dan adaptive test di plotkan kedalam satu bidang, maka akan diperoleh hasil seperti pada gambar 2. pada gambar 2 terlihat bahwa kinerja adaptive test lebih baik daripada semua metode desain soal fixed test terutama setelah soal kelima. metode fixed test yang urutan soalnya dari mudah ke sukar lebih baik daripada metode desain soal fixed test lainnya. desain soal fixed test yang kami rancang hanya mampu memprediksi sekitar 80% skill yang dimiliki siswa, lebih kecil jika dibandingkan dengan desain adaptive test dengan kemempuan memprediksi di atas 90%. untuk memprediksi sekitar 80% skill siswa, desain adaptive test hanya memerlukan 6 butir soal, jauh lebih sedikit dibandingkan desain soal fixed test yang memerlukan 12 butir soal. prosedur desain fixed test yang kemampuan prediksinya dapat menyamai kemampuan prediksi desain adaptive test sangatlah menarik untuk ditemukan. berbagai temuan dari penelitian ini akan digunakan untuk mencari prosedur desain fixed test tersebut. dalam penelitian ini juga terungkap bahwa soal diagnosa yang kami buat memiliki tingkat kesukaran dari sedang ke sukar. saat mendesain tes diagnosa sebaiknya diperhatikan sebarannya agar soal diagnosa dapat mewakili semua level kemampuan siswa. banyaknya soal diagnosa juga perlu ditambah agar keseluruhan soal diagnosa dapat memprediksi sebesar mungkin skill siswa. gambar 2 perbandingan metode desain tes 4. kesimpulan temuan penelitian ini menunjukkan bahwa kinerja adaptive test lebih baik daripada jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 52 semua metode desain soal fixed test terutama setelah soal kelima. metode fixed test yang urutan soalnya dari mudah ke sukar lebih baik daripada metode desain soal fixed test lainnya. desain soal fixed test mampu memprediksi sekitar 80% skill yang dimiliki siswa, lebih kecil dibandingkan dengan desain adaptive test dengan nilai di atas 90%. referensi [1] almond et al., models for conditional probability tables in educational assessment, in proc. of the 2001 conference on ai and statistics. society for ai and statistics. [2] conati, cristina et al., on-line student modeling for coached problem solving using bayesian networks. in anthony jameson, cecile paris, and carlo tasso, editors, proc. of the sixth int. conf. on user modeling (um97), chia laguna, sardinia, italy (1997) pages 231–242. springer verlag. [3] hugin explorer. 2017. ver. 6.0. computer software. http://www.hugin.com. [4] lauritzen, steffenl,graphical models. clarendon press, oxford. (1996) [5] millan, eva d an pérez-de-la-cruz, josé luis, a bayesian diagnostic algorithm for student modeling and its evaluation. user modeling and useradapted interaction,12 (2–3): (2002) 281–330. [6] mislevy, robert j et al., computerized adaptive testing: a primer. mahwah, n. j., lawrence erlbaum associates, second edition. (2000) [7] vomlel, jiri, bayesian networks in educational testing. international journal of uncertainty, fuzziness and knowledge based systems, vol. 12, supplementary issue 12(004) pp. 83-100. a draft version. http://www.hugin.com/ http://staff.utia.cas.cz/vomlel/ijufks-draft.pdf jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 53 text mining dengan k-means clustering pada tema lgbt dalam arsip tweet masyarakat kota bandung eko yulian1 pusdiklat badan pusat statistik jakarta selatan1, okeyulian@gmail.com1 doi:https://doi.org/10.15642/mantik.2018.4.1.53-58 abstrak gerakan lgbt berkembang cepat melalui media sosial sehingga ide-ide lgbt dapat dengan leluasa dikemukakan. tweeter merupakan salah satu media yang seringkali digunakan untuk tujuan tersebut. komentar-komentar atau “cuitan” tentang lgbt di twitter tentu banyak jumlahnya. banyaknya informasi yang ada di dunia maya membuat upaya-upaya pengembangan terhadap penggalian informasi dari basis data daring semakin pesat, salah satunya text mining. salah satu teknik statistika yang bisa digunakan untuk memanfaatkan hasil dari text mining adalah clustering. clustering yang digunakan pada penelitian ini adalah k-means clustering. penelitian ini menggunakan 5 cluster untuk mengelompokkan komentar-komentar di twitter yang berhubungan dengan lgbt di kota bandung. dari lima cluster yang dibentuk pada proses kmeans diperoleh bahwa kecenderungan cuitan pengguna tweeter kota bandung terkait lgbt secara umum masih berhubungan dangan perspektif religi yang ditandai dengan kemunculan kata agama yang sangat sering. kata kunci : k-means clustering, lgbt, text mining abstract the movement of lgbt is growing rapidly through social media so that lgbt ideas can be freely expressed. the tweeter is one of the media that is often used for that purpose. comments or "cuitan" about lgbt on twitter certainly many in number. the amount of information available in cyberspace makes development efforts to extract information from online databases rapidly, one of which is text mining. one of the statistical techniques that can be used to utilize the results of text mining is clustering. clustering used in this study is k-means clustering. this study uses 5 clusters to group comments on the twitter associated with lgbt in the city of bandung. of the five clusters formed in the k-means process, it is found that the tendency of tuet tweeter users of lgbt related bands in general, is still related to the religious perspective which is marked by the emergence of the word religion very often. keyword : k-means clustering, lgbt, text mining mailto:okeyulian@gmail.com 54 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 1. pendahuluan lgbt atau glbt adalah akronim dari "lesbian, gay, biseksual, dan transgender". istilah ini mulai sering digunakan tahun 1990-an sebagai pengganti frasa yang lebih dulu populer, "komunitas gay", karena lebih mewakili kelompokkelompok yang telah disebutkan. kaum lgbt merupakan salah satu kelompok orang yang memiliki orientasi seksual sebagai homoseksual atau penyuka sesama jenis yang terjadi pada kaum pria [1]. lebih lanjut kaum homoseksual didominasi oleh kaum lakilaki karena beberapa faktor seperti kelainan genetika dan faktor sosial seperti lingkungan yang memang mendukung untuk terbentuknya kaum tersebut atau karena terjadinya trauma dalam hubungan seksualitasnya [1]. pemberitaan lgbt di indonesia mulai marak di indonesia setelah mahkamah agung amerika serikat melegalkan pernikahan sesama jenis pada 26 juni 2015. sejak saat itu, muncullah pemberitaan di media massa pada akhir tahun 2015 bahwa telah terjadi pernikahan sesama jenis di indonesia. lgbt merupakan salah satu isu penting yang saat ini sedang marak diberitakan oleh beberapa media bahkan menjadi bahan diskusi oleh beberapa pakar di indonesia. contohnya majalah gatra edisi 4-10 februari 2016 [2] yang memberitakan isu “arus lgbt masuk kampus di indonesia”, bandung sebagai salah satu kota pelajar sekaligus metropolitan yang menjadi tujuan belajar bagi para mahasiswa dan wisata unggulan di indonesia tidak lepas sasaran perubahan pola sosial budaya yang begitu cepat termasuk kemunculan komunitaskomunitas lgbt. berdasarkan catatan badan kesatuan bangsa, perlindungan dan pemberdayaan masyarakat (bkppm) kota bandung, untuk sekitaran kota saja ada sekitar 6.000 warga yang menjadi bagian komunitas lgbt. hal ini tentu menjadi suatu kekhawatiran di kalangan masyarakat bandung yang notabene masih dikenal religius. gerakan lgbt berkembang cepat melalui media sosial, dengannya ide-ide lgbt dapat dengan leluasa dikemukakan. esensi pesan berkenaan dengan pilihan hidup lgbt dapat tersampaikan, namun tanpa melibatkan eksistensi aslinya. tweeter merupakan salah satu media yang seringkali digunakan untuk tujuan tersebut. komentar-komentar atau “cuitan” tentang lgbt di twitter tentu banyak jumlahnya. hartanto [3] pernah meneliti pengelompokkan komentar-komentar tentang lgbt di twitter, hasilnya terdapat 7 kelompok besar komentar. adapun tujuan dari penelitian ini adalah mengelompokkan komentar-komentar di twitter tentang lgbt di kota bandung ke dalam cluster cluster yang dapat dibedakan. pengelompokan yang dihasilkan dapat menjadi petunjuk dasar terkait stigma masyarakat bandung terkait berkembangnya lgbt. clustering yang digunakan pada penelitian ini adalah kmeans clustering. 2. tinjauan pustaka 2.1 twitter twitter didirikan oleh jack dorsey pada maret 2006. pada platform media sosial ini, pengguna tak terdaftar hanya bisa membaca kicauan sedangkan pengguna terdaftar bisa menulis kicauan melalui graphical user interface (gui) situs, pesan singkat (sms), atau melalui berbagai aplikasi dari perangkat seluler. perkembangan jumlah pengguna twitter meningkat dengan sangat cepat hingga dapat meraih popularitas di seluruh dunia. hingga januari 2013, tercatat sudah ada lebih dari 500 juta pengguna terdaftar. popularitas twitter yang makin meningkat menyebabkan layanan ini dimanfaatkan untuk berbagai keperluan diantaranya kampanye politik, pembentukan opini, sarana belajar, dan sebagai media komunikasi darurat. twitter juga dihadapkan pada berbagai masalah dan kontroversi seperti masalah keamanan, pendidikan (bullying), dan privasi pengguna [3]. https://id.wikipedia.org/w/index.php?title=sistem_komunikasi_darurat&action=edit&redlink=1 https://id.wikipedia.org/w/index.php?title=sistem_komunikasi_darurat&action=edit&redlink=1 https://id.wikipedia.org/w/index.php?title=sistem_komunikasi_darurat&action=edit&redlink=1 https://id.wikipedia.org/wiki/twitter#keamanan_dan_privasi https://id.wikipedia.org/wiki/twitter#keamanan_dan_privasi https://id.wikipedia.org/wiki/twitter#keamanan_dan_privasi https://id.wikipedia.org/wiki/twitter#keamanan_dan_privasi 55 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 2.2 text mining banyaknya informasi yang ada di dunia maya membuat upaya-upaya pengembangan terhadap penggalian informasi dari basis data daring semakin pesat, salah satunya text mining. text mining, yang juga disebut sebagai teks data mining (tdm) atau knowledge discovery in text (kdt), secara khusus dikembangkan untuk proses ekstraksi informasi dari dokumen-dokumen teks tak terstruktur (unstructured). text mining memiliki definisi menambang data berupa teks di mana sumber data biasanya didapatkan dari dokumen dan tujuanya adalah untuk mencari kata-kata yang dapat mewakili isi dari dokumen sehingga dapat dilakukan analis is keterhubungan antar dokumen [4]. text mining mencoba memecahkan masalah kelebihan informasi (information overload) dengan menggunakan teknikteknik dari bidang ilmu yang terkait. text mining dapat dipandang sebagai perluasan dari data mining atau knowledge discovery in database (kdd), yang bertujuan untuk menemukan polapola menarik dari basis data berskala besar. tahapan text mining tahapan text mining yang paling umum dilakukan adalah sebagai berikut gambar 1. tahapan text mining 2.3 term document matrix (tdm) term document matrix (tdm) adalah suatu matriks yang menggambarkan frekuensi kata yang terjadi dalam kumpulan dokumen. dalam matriks tdm, banyaknya baris menggambarkan banyaknya dokumen sedangkan banyaknya kolom menggambarkan banyaknya kata [5]. 2.4 term frequency (tf) tf (term frequency) adalah frekuensi dari kemunculan sebuah istilah dalam dokumen yang bersangkutan. semakin besar jumlah kemunculan suatu term (tf tinggi) dalam dokumen, semakin besar pula bobotnya atau akan memberikan nilai kesesuaian yang semakin besar [5]. 2.5 idf (inverse document frequency) idf merupakan sebuah perhitungan dari bagaimana term didistribusikan secara luas pada koleksi dokumen yang bersangkutan. idf menunjukkan hubungan ketersediaan sebuah istilah dalam seluruh dokumen. semakin sedikit jumlah dokumen yang mengandung term yang dimaksud, maka nilai idf semakin besar. idf dihitung dengan [5]: idfj = log (d / dfj ) (1) di mana: d : jumlah dokumen df : jumlah dokumen yang mengandung term (tj). selanjutnya untuk menghitung bobot (w) digunakan formula sebagai berikut: wij = tfij × idfj (2) di mana : wij : adalah bobot term (tj) terhadap dokumen (di) tfij : jumlah kemunculan term (tj) dalam dokumen (di) 2.6 k-means cluster clustering adalah salah satu metode yang dapat digunakan untuk mengeksplorasi distribusi dan pola data. 56 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 pola-pola dalam suatu cluster akan memiliki kesamaan ciri/sifat dibandingkan pola-pola dalam cluster yang lainnya. clustering bermanfaat untuk melakukan analisis pola-pola data, mengelompokkan, dan membuat keputusan . ada beberapa algoritma dalam clustering, salah satu diantaranya adalah algoritma k-means. k-means ditemukan oleh beberapa orang yaitu lloyd (1957, 1982), forgey (1965), friedman dan rubin (1967), mcqueen (1967) . ide dari clustering pertama kali ditemukan oleh lloyd pada tahun 1957, namun hal tersebut baru dipublikasi pada tahun 1982. pada tahun 1965, forgey juga mempublikasi teknik yang sama sehingga terkadang dikenal sebagai lloyd-forgy pada beberapa sumber. secara umum langkah-langkah dalam algoritma k-means clustering dapat diilustrasikan sebagai berikut [6]: gambar 2. algoritma k-means clustering berikut adalah algoritma dari metode kmeans [8]: a) masukkan data yang akan diklaster. b) tentukan jumlah klaster. c) ambil sebarang data sebanyak jumlah klaster secara acak sebagai pusat klaster (sentroid). d) hitung jarak antara data dengan pusat klaster, dengan menggunakan persamaan : 𝐷(𝑖, 𝑗) = √(𝑋1𝑖 − 𝑋1𝑗) 2 + ⋯ + (𝑋𝑘𝑖 − 𝑋𝑘𝑗) 2 (3) dimana : 𝐷(𝑖, 𝑗) = jarak data ke 𝑖 ke pusat klaster 𝑗 𝑋𝑘𝑖 = data ke 𝑖 pada atribut ke 𝑘 𝑋𝑘𝑗 = titik pusat ke 𝑗 pada atribut ke 𝑘 e) hitung kembali pusat klaster dengan keanggotaan klaster yang baru jika pusat klaster tidak berubah maka proses klaster telah selesai, jika belum maka ulangi langkah ke (d) sampai pusat klaster tidak berubah lagi. 3. sumber data data yang digunakan pada penelitian ini adalah data-data pada komentarkomentar yang ada di twitter yang mengandung kata lgbt di kota bandung selama 10 hari yaitu pada tanggal 17 – 26 desember 2017. 4. analisis dan pembahasan 4.1 persiapan data penelitian ini menggunakan program r. jumlah dokumen/komentar yang diperoleh yaitu sebanyak 701 dokumen/komentar. tentu saja data yang diperoleh tidak bisa langsung digunakan, akan tetapi harus melalui tahapan cleaning data dengan menggunakan tahapan-tahapan text mining yang telah disebutkan pada pembahasan sebelumnya yaitu tokenizing, filtering dan stemming. tahapan tagging tidak dilakukan karena tahapan ini hanya bisa dilakukan pada dokumen yang berbahasa inggris. setelah melalui tahapan cleaning data diperoleh sebanyak 691 dokumen yang siap untuk dianalisis menggunakan k-means clustering. 4.2 term document matrix tdm dan term frequency-invers document frequency dari proses yang dilakukan diperoleh matriks tdm berikut: 57 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 gambar 3. matriks tdm dan matriks pembobotan tf-idf gambar 4. matriks bobot tf-idf 4.3 k-means cluster penelitian ini menggunakan nilai k=5. dari lima cluster yang ditentukan tersebut setelah dilakukan pemrosesan dengan paket program r diperoleh informasi bahwa cluster satu yang terbentuk beranggotakan 51 sampel dokumen. cluster dua diperoleh 89 sampel, cluster tiga terdapat 80 sampel, cluster empat 419 sampel, dan cluster lima 34 sampel. informasi lain yang diperoleh adalah jumlah kuadrat (sum square) pada masing-masing cluster. jumlah kuadrat dalam cluster di cluster satu sebesar 45,069; cluster dua sebesar 84,64; cluster tiga 75,781; cluster empat 414,736; dan cluster lima 30,175. ada kesulitan yang dihadapi dalam menganalisis text berbahasa indonesia. corpus bahasa indonesia yang lengkap masih sulit diperoleh selain itu seringkali komentar-komentar atau cutan-cuitan dalam twitter tidak menggunakan bahasa indonesia baku. kendala ini menyebabkan hasil yang diperoleh dari data mining yang dilakukan mesin masih belum bisa seakurat yang diinginkan. jika dibuat plot, pengelompokan yang terbentuk sebagai berikut: gambar 5. plot cluster k-means berikutnya dapat dilihat pola sebaran kata dalam setiap cluster dengan menggunakan wordcloud di bawah ini: cluster 1 cluster 2 cluster 3 cluster 4 cluster 5 gambar 6. wordcloud pada cluster 58 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 dari plot-plot tersebut dapat dilihat bahwa centroid pada masing-masing cluster berbeda-beda. pusat perhatian cluster satu pada kata manusia, agama, dan kaum; cluster kedua lebih pada kata nikah, dan kata lobang; cluster tiga kata kaum, adzab dan penyakit; cluster empat pada kata agama dan negara; dan yang terakhir dihubungkan dengan kata israel dan dpr. untuk mendapatkan gambaran yang lebih jelas tentang isi dari masingmasing dapat pula dikaji beberapa dokumen yang masuk dalam suatu cluster. misalnya, untuk cluster satu jika dikaji lebih dalam didapati bahwa artikel-artikel di dalamnya lebih banyak berbicara tentang hukum perilaku lgbt menurut sudut pandang agama islam. cluster ke-dua lebih kepada masalah pembahasan moral dan cluster 3 pada opini tentang dampak perilaku lgbt. 5. kesimpulan dari tahapan-tahapan yang telah dilewati diperoleh gambaran bahwa penggunaan k-mean clustering dapat digunakan untuk pembentukan cluster kata pada arsip-arsip dokumen yang digunakan. sayangnya, kendala yang dihadapi pada saat proses text mining yang tidak sempurna memfilter kata-kata dalam bahasa indonesia yang digunakan pengguna tweeter dalam arsip komentar menyebabkan cluster yang terbentuk masing mengandung kata-kata yang tidak begitu penting. dari lima cluster yang dibentuk pada proses k-means diperoleh bahwa kecenderungan cuitan pengguna tweeter kota bandung terkait lgbt secara umum masih berhubungan dangan perspektif religi. kemunculan kata agama yang sangat sering menyebabkan asosiasi terhadap kata tersebut cukup besar. referensi [1] sinyo, anakku bertanya tentang lgbt, pt elex media komputindo, (2014). [2] gatra, melawan aksi lgbt di kampus, (2016). [3] alim, s, analysis of tweets related to cyberbullying: exploring information diffusion and advice available for cyberbullying victims. international [4] journal of cyber behavior, psychology and learning, (2015). [5] prasetyo, eko, data mining konsep dan aplikasi menggunakan matlab, penerbit andi yogyakarta (2012). [6] srihari, retrieval by content, diambil darihttp://www.cedar.buffalo.edu/~srihari/ cse626/lecture-slides, pada tanggal 6 februari 2018 [7] hastuti, n. f., saptono, r., &suryani, e., pemanfaatan metode k-means clustering dalam penentuan penerima beasiswa,jurnal informatika, (2012). [8] febrianti, f., hafiyusholeh, m., & asyhar, a.h., perbandingan pengklusteran data iris menggunakan metode k-means dan fuzzy c-means, jurnal matematika mantik, 2 (1), pp. 7-13 (2016) http://www.cedar.buffalo.edu/~srihari/cse626/lecture-slides http://www.cedar.buffalo.edu/~srihari/cse626/lecture-slides paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 1 optimisasi perencanaan produksi pupuk menggunakan firefly algorithm dinita rahmalia1, awawin mustana rohmah2 universitas islam darul ulum lamongan1, dinitarahmalia@gmail.com1 universitas islam darul ulum lamongan2, awawin.emer@gmail.com2 doi:https://doi.org/10.15642/mantik.2018.4.1.1-6 abstrak di indonesia, terdapat banyak petani sebagai matapencaharian karena tanah yang subur untuk pertanian serta kebutuhan akan pangan. perencanaan produksi merupakan bagian penting dalam mengelola biaya yang dikeluarkan perusahaan. pada perencanaan produksi, terdapat kendala yang harus dipenuhi misal : jumlah produksi, jumlah pekerja, dan jumlah pengadaan. pada penelitian sebelumnya, optimisasi berkendala telah diselesaikan menggunakan metode eksak maupun metode heuristik. pada penelitian ini, model optimisasi perencanaan produksi akan diselesaikan menggunakan firefly algorithm (fa). cara kerja fa menyerupai perilaku kunang-kunang. salah satu perilaku kunang-kunang yang digunakan adalah kunang-kunang yang kurang cerah akan mendekati kunang-kunang yang lebih cerah. hasil simulasi menunjukkan bahwa metode fa dapat menemukan pendekatan solusi optimum pada perencanaan produksi yaitu biaya produksi, biaya pekerja, dan biaya pengadaan yang memenuhi kendala jumlah produksi, jumlah pekerja, dan jumlah pengadaan. kata kunci: optimisasi berkendala, perencanaan produksi, firefly algorithm abstract in indonesia, there are many farmers as a livelihood because of fertile soil for agriculture and the demand for food. production planning is the important part of managing cost spent by the company. in production planning, there are many constraints which have to be satisfied such as the number of productions, the number of workers, and the number of inventory. in previous research, constrained optimizations have been solved by exact method or heuristic method. in this research, production planning optimization will be solved by firefly algorithm (fa). fa works as a behavior of firefly. one of firefly behavior used is less bright firefly will move toward brighter firefly. the simulation results show that fa method can find an approaching optimal solution of production planning like production cost, worker cost, and inventory holding cost satisfying the constraints of the number of productions, workers, and inventory. keywords: constrained optimization, production planning, firefly algorithm 1. pendahuluan di indonesia, terdapat banyak petani sebagai matapencaharian karena tanah yang subur untuk pertanian serta kebutuhan akan pangan. karena permintaan produksi pertanian, perusahaan yang bergerak pada bidang pertanian, yaitu perusahaan produksi pupuk memproduksi pupuk untuk didistribusikan pada petani. pada proses ini terdapat biaya yang ditimbulkan seperti biaya produksi dan perusahaan harus membuat perencanaan untuk mengendalikan pendapatan, biaya, dan produksi. perencanaan produksi merupakan bagian penting dalam mengelola biaya yang dikeluarkan perusahaan. pada perencanaan produksi, terdapat kendala yang harus dipenuhi misal : jumlah produksi, jumlah pekerja, dan jumlah pengadaan. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 2 dari model perencanaan produksi tersebut, dapat dibentuk optimisasi berkendala. pada penelitian sebelumnya, optimisasi berkendala telah diselesaikan menggunakan metode eksak [1][8] maupun metode heuristik [7] seperti genetic algorithm (ga) [5], particle swarm optimization (pso) [4], ant colony optimization (aco) [3], artificial bee colony (abc) [6]. pada penelitian ini, model optimisasi perencanaan produksi akan diselesaikan menggunakan firefly algorithm (fa). firefly algorithm (fa) ditemukan oleh xin-she yang pada tahun 2008. cara kerja fa menyerupai perilaku kunang-kunang dalam berkomunikasi, mencari mangsa dan menemukan pasangan menggunakan cahaya yang dipancarkan. salah satu perilaku kunang-kunang yang digunakan adalah kunang-kunang yang kurang cerah akan mendekati kunang-kunang yang lebih cerah sehingga dalam hal ini posisi kunang-kunang direpresentasikan solusi dan tingkat kecerahan direpresentasikan sebagai nilai fitness [11]. hasil simulasi menunjukkan bahwa metode fa dapat menemukan pendekatan solusi optimum pada perencanaan produksi yaitu biaya produksi, biaya pekerja, dan biaya pengadaan yang memenuhi kendala jumlah produksi, jumlah pekerja, dan jumlah pengadaan. 2. metode metode yang digunaka dalam penelitian ini menggunakan firefly algorithm (fa) pada masalah optimisasi perencanaan produksi. optimisasi perencanaan produksi merupakan salah satu optimisasi berkendala sehingga pada fa diperlukan modifikasi pada posisi kunangkunang baik pada tahap inisialisasi maupun tahap optimisasi supaya posisi kunang-kunang memenuhi semua kendala yang ada. 2.1 model optimisasi perencanaan produksi secara umum, model optimisasi perencanaan produksi adalah sebagai berikut [9] : 1 min t p w h l i t t t t t t t t t t t c p c w c h c l c i = + + + + (1) dengan kendala : t t t p n w , 1, 2,...,t t= (2) 1t t t t w w h l − = + − , 1, 2,...,t t= (3) 1t t t t i i p d − = + − , 1, 2,...,t t= (4) , , , , 0 t t t t t p w h l i  (5) model optimisasi perencanaan produksi dapat dijelaskan sebagai berikut : t d : jumlah unit yang diminta dalam periode t t n : jumlah unit yang diproduksi oleh setiap pekerja dalam periode t p t c : biaya produksi per unit dalam periode t w t c : biaya pekerja dalam periode t h t c : biaya pekerja yang masuk dalam periode t l t c : biaya pekerja yang keluar (lay off) dalam periode t i t c : biaya pengadaan dalam periode t t p : jumlah unit yang diproduksi dalam periode t t w : jumlah pekerja dalam periode t t h : jumlah pekerja yang masuk dalam periode t t l : jumlah pekerja yang keluar (lay off) dalam periode t t i : jumlah pengadaan dalam periode t 2.2 firefly algorithm firefly algorithm (fa) ditemukan oleh xinshe yang pada tahun 2008. cara kerja fa menyerupai perilaku kunang-kunang dalam berkomunikasi, mencari mangsa dan menemukan pasangan menggunakan cahaya yang dipancarkan. dalam fa, daya tarik (attractiveness) kunang-kunang ditentukan oleh tingkat kecerahan yang berhubungan dengan fungsi objective. tingkat kecerahan dari kunang-kunang di lokasi x dapat ditentukan sebagai ( )f x , dimana ( )f x adalah fungsi objective. tetapi, jika daya tarik  adalah relative maka ditentukan oleh kunang-kunang yang lain dan terdapat jarak ij r antara kunang-kunang i dan kunang-kunang j perilaku kunang-kunang dapat dijelaskan sebagai berikut [10] [11] : 1. kunang-kunang bersifat unisex. kunangkunang memiliki ketertarikan pada kunang-kunang yang lain tanpa memandang jenis kelamin. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 3 2. daya tarik bersifat proporsional pada tingkat kecerahan kunang-kunang. kunang-kunang yang kurang cerah akan mendekati kunang-kunang yang lebih cerah. 3. tingkat kecerahan kunang-kunang dipengaruhi oleh nilai fungsi objective. perilaku kunang-kunang tersebut dapat dirancang sebagai fa yaitu [2] : 1. inisialisasi populasi kunang-kunang , 1, 2,...max i x i pop= dan hitung nilai fitness ( ), 1, 2,...max i f x i pop= 2. tentukan kunang-kunang terbaik dalam populasi beserta posisinya ( )min arg min ( ), 1, 2,..., maxi i i f x i pop = (6) ( ) min arg min ( ), 1, 2,..., max i i i x x f x i pop = (7) 3. lakukan iterasi berikut : for 1: maxi pop= for 1: maxj pop= if ( ( ) ( )) j i f x f x a. hitung jarak (distance) antara kunangkunang i dan kunang-kunang j 2 1 ( ) t i j i j ij t t t r x x x x = = − = − (8) b. hitung fungsi daya tarik (attractiveness) kunang-kunang 0 ijre    −  (9) c. tentukan 1 ( ) 2 i u rand= − , dengan ~ (0,1)rand u d. update perpindahan kunang-kunang i (1 ) i i j i x x x u  − + + (10) end end end a. tentukan min 1 ( ) 2i u rand= − , dengan ~ (0,1)rand u b. update perpindahan kunang-kunang terbaik min min min i i i x x u + (11) 4. ulangi langkah 3 sampai aturan pemberhentian tercapai. 2.3 fa pada optimisasi perencanaan produksi representasi posisi kunang-kunang yang digunakan pada optimisasi perencanaan produksi dapat dikonstruksi seperti persamaan (12) 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ... ... ...( , , , , ) ... ... t t i t t t p p p p w w w w h h h hx p w h l i l l l l i i i i        = =         (12) dimana t p adalah jumlah unit yang diproduksi dalam periode t , t w adalah jumlah pekerja dalam periode t , t h adalah jumlah pekerja yang masuk dalam periode t , t l adalah jumlah pekerja yang keluar (lay off) dalam periode t , t i adalah jumlah pengadaan dalam periode t . dalam fa, terdapat tahap inisialisasi yang harus memenuhi kendala (2)-(5). oleh karena itu, tahap inisilisasi dapat dikonstruksi sebagai berikut : (max , max , max )initialization h l pop for 1: maxi pop= 1. bangkitkan ~ (0, max ) i t h u h and ~ (0, max ) i t l u l , 1, 2,...,t t= 2. tetapkan 0 i w for 1:t t= 1 i i i i t t t t w w h l − = + − end 3. for 1:t t= i i t t t p n w end 4. tetapkan 0 i i for 1:t t= 1 i i i t t t t i i p d − = + − end end jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 4 langkah-langkah fa pada optimisasi perencanaan produksi supaya memenuhi kendala (2)-(5) adalah sebagai berikut : 1. inisialisasi populasi kunang-kunang dengan (max , max , max )initialization h l pop dan hitung nilai fitness ( ), 1, 2,...max i f x i pop= 2. tentukan kunang-kunang terbaik dalam populasi beserta posisinya ( )min arg min ( ), 1, 2,..., maxi i i f x i pop = (13) ( ) min arg min ( ), 1, 2,..., max i i i x x f x i pop = (14) 3. lakukan iterasi berikut : for 1: maxi pop= for 1: maxj pop= if ( ( ) ( )) j i f x f x a. hitung jarak (distance) antara kunangkunang i dan kunang-kunang j 2 1 ( ) t i j i j ij t t t r x x x x = = − = − (15) b. hitung fungsi daya tarik (attractiveness) kunang-kunang 0 ijre    −  (16) c. tentukan 1 ( ) 2 i u rand= − , dengan ~ (0,1)rand u d. update perpindahan kunang-kunang i (1 ) i i j i h h h u  − + + (17) (1 ) i i j i l l l u  − + + (18) end end e. tetapkan 0 i w for 1:t t= 1 i i i i t t t t w w h l − = + − end f. for 1:t t= i i t t t p n w end g. tetapkan 0 i i for 1:t t= 1 i i i t t t t i i p d − = + − end end a. tentukan min 1 ( ) 2i u rand= − , dengan ~ (0,1)rand u b. update perpindahan kunang-kunang terbaik min min min i i i h h u + (19) min min min i i i l l u + (20) c. tetapkan min 0 i w for 1:t t= min min min min 1 i i i i t t t t w w h l − = + − end d. for 1:t t= min min i i t t t p n w end e. tetapkan min 0 i i for 1:t t= min min min 1 i i i t t t t i i p d − = + − end 4. ulangi langkah 3 sampai aturan pemberhentian tercapai. 3. hasil simulasi data yang digunakan dalam penelitian ini diambil dari salah satu perusahaan pupuk di gresik, jawa timur selama tahun 2011-2016. tabel 1 menunjukkan biaya per unit, tabel 2 menunjukkan jumlah permintaan setiap periode, tabel 3 menunjukkan unit yang diproduksi untuk setiap pekerja. simulasi dilakukan menggunakan fa dengan parameter yang diberikan seperti tabel 4. tabel 1. biaya per unit (dalam juta rupiah) t 2011 2012 2013 2014 2015 2016 p t c 4,00 4,54 5,17 5,33 5,71 5,33 w t c 7 7 7 7 7 7 h t c 4 4 4 4 4 4 l t c 2 2 2 2 2 2 i t c 2,46 3,83 4,34 4,03 4,63 4,26 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 5 tabel 2. jumlah permintaan (dalam ton) periode t jumlah permintaan 2011 4328630 2012 5008571 2013 5409669 2014 5375396 2015 5546783 2016 5357118 tabel 3. jumlah unit yang diproduksi per pekerja (dalam ton/pekerja) periode t jumlah unit/pekerja 2011 1060 2012 1271 2013 1249 2014 1266 2015 1285 2016 1292 tabel 4. parameter firefly algorithm parameter nilai populasi maksimum 20 iterasi maksimum 100 0  1  5  0,1 hasil simulasi ditunjukkan seperti gambar 1. pada iterasi awal, posisi kunang-kunang dipilih secara acak. pada proses optimisasi, tingkat kecerahan kunang-kunang diupdate sehingga menghasilkan nilai fitness yang turun atau tingkat kecerahan kunang-kunang bertambah seiring bertambahnya iterasi. dari gambar 1, terlihat bahwa biaya minimum sebagai nilai fitness yang dikeluarkan perusahaan dalam tahun 2011-2016 adalah 2,21×108 juta rupiah. tabel 5 dan tabel 6 menunjukkan solusi optimal dari optimisasi perencanaan produksi yang terdiri dari jumlah unit yang diproduksi dalam periode t , jumlah pekerja dalam periode t , jumlah pekerja yang masuk dalam periode t , jumlah pekerja yang keluar dalam periode t , dan jumlah pengadaan dalam periode t . gambar 1 hasil simulasi firefly algorithm pada optimisasi perencanaan produksi tabel 5. solusi optimal pada pekerja periode t t h t l t w 2011 128 281 3194 2012 152 286 3060 2013 286 208 3138 2014 217 256 3099 2015 275 245 3129 2016 307 162 3274 tabel 6. solusi optimal pada produksi dan pengadaan periode t t p t i 2011 2845724 10722261 2012 3173632 8887322 2013 3268910 6746563 2014 3457387 4828554 2015 3258592 2540363 2016 3406271 589516 4. kesimpulan metode fa dapat menyelesaikan masalah optimisasi perencanaan produksi. optimisasi perencanaan produksi merupakan salah satu optimisasi dengan kendala jumlah produksi, jumlah pekerja, dan jumlah pengadaan sehingga pada fa diperlukan modifikasi pada posisi kunang-kunang baik pada tahap inisialisasi maupun tahap optimisasi supaya posisi kunangkunang memenuhi semua kendala yang ada. hasil simulasi menunjukkan bahwa metode fa dapat menemukan pendekatan solusi optimum pada perencanaan produksi yaitu biaya produksi, biaya pekerja, dan biaya pengadaan yang memenuhi kendala jumlah produksi, jumlah pekerja, dan jumlah pengadaan. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 6 referensi [1] hillier, f.s., lieberman, g.j., introduction to operations research. mc graw hill. (2001) [2] lu kas ik, s., zak, s., firefly algorithm for continuous constrained optimization tasks. [3] r ahm ali a, d., estimation of exponential smoothing parameter on pesticide characteristic forecast using ant colony optimization (aco), eksakta: jurnal ilmu-ilmu mipa vol. 18 no. 1 pp. 56-63 (2018) [4] r ahm ali a, d., particle swarm optimizationgenetic algorithm (psoga) on linear transportation problem. aip conference proceeding. (2017) (020030)1-12. surabaya. [5] r ahm ali a, d., perbandingan metode analitik dan metode heuristik pada optimisasi masalah transportasi distribusi semen. prosiding seminar nasional matematika dan pembelajarannya 2016. (2016) 1164-1172. 13 agustus, malang. [6] r ahm ali a, d., herlambang, t., optimisasi masalah transportasi distribusi semen menggunakan algoritma artificial bee colony, multitek indonesia vol. 11 no. 2 (2018) [7] rao, s.s., engineering optimization: theory and practice. john wiley and sons. (2009) [8] taha, h.a., operation research: an introduction. pearson prentice hall. (2007) [9] tec ha ro o ngru en gk ij , b., prakasvudhisarn, c., yenradee, p., a pso based goal programming approach to aggregate planning of production, workforce, and pricing strategy [10] u d aiyak um ar , k.c, chandrasekaran, m., application of firefly algorithm in job shop scheduling problem for minimization of makespan, procedia engineering 97(2014)1798-1807 [11] yang, x.s., cui, z., xiao, r., swarm intelligence and bio-inspired computation. elsevier, inc. (2013) m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 137 penerapan non-linier support vector machine pada penggunaan alat kontrasepsi di provinsi maluku utara muhamad budiman johra universitas padjadjaran, muhamad.budiman@bps.go.id doi:https://doi.org/10.15642/mantik.2018.4.2.137-142 abstrak sasaran strategis dari badan kependudukan dan keluarga berencana nasional (bkkbn) adalah menurunkan laju pertumbuhan penduduk karena laju pertumbuhan penduduk yang tinggi menyebabkan kuantitas penduduk yang tinggi pula. menurut departemen kesehatan (2013) perempuan kawin usia 15-49 tahun yang tidak ber-kb sebagian besar di wilayah indonesia timur yang salah satunya maluku utara. menurut bkkbn provinsi maluku utara, angka kelahiran meningkat dari 57,4 menjadi 57,9. hal ini terjadi karena banyak peserta kb yang mengalami ketidakberlangsungan (drop out), kegagalan dan efek samping alat kontrasepsi, kebutuhan ber-kb yang tidak terlayani yakni 9,1 pada 2007 menjadi 8,5 pada 2012 dengan target 5 pada 2014. sehingga penting untuk mengetahui faktor determinan yang mempengaruhi wanita untuk menggunakan alat kontrasepsi. terdapat beberapa metode dalam klasifikasi, salah satunya adalah support vector machine (svm). svm memiliki kelebihan dibanding metode klasifikasi lainnya karena support vector machine tidak hanya meminimalkan error pada trainset, tetapi juga memiliki kemampuan generalisasi yang tinggi. hal ini tercermin pada pemilihan margin yang maksimal. penelitian ini menunjukkan support vector machine dapat menggambarkan keputusan wanita usia subur untuk menggunakan alat kontrasepsi atau tidak. kernel terbaik pada penelitian ini adalah kernel radial basis dengan cost 1 dan gamma 0.14286 kata kunci: nonlinier support vector machine, alat kontrasepsi, kernel trick, apparent error rate abstract the objective of bkkbn is to reduce the rate of population growth because the high population growth rate causes a high population quantity as well. according to the departemen kesehatan ri (2013), married women aged 15-49 years who don't use contraception mostly in eastern indonesia, one of them is provinsi maluku utara. according to bkkbn provinsi maluku utara, the birth rate increased from 57.4 to 57.9. this happens because many kb participants are drop out, contraceptive failure and side effects, the need for family planning is served 9.1 in 2007 to 8.5 in 2012 with a target of 5 in 2014. therefore, it important to know determinant factors that affect women to use contraceptives. there are several methods in the classification, one of which is the support vector machine (svm). svm has advantages over other classification methods because the support vector machine not only minimizes errors in the train set but also has a high generalization capability. this is reflected in maximal margin selection. this study shows the support vector machine can describe the decision of women to use contraception or not. the best kernel in this study is a radial base kernel with cost 1 and gamma 0.14286. keywords: nonlinier support vector contraceptive, kernel trick, apparent error rate 1. pendahuluan kereta api merupakan transportasi umum yang banyak digunakan oleh masyarakat baik dalam menunjang kegiatan sehari-hari terkait pekerjaan dan aktivitas harian lainnya yang bersifat rutin maupun sebagai solusi transportasi jarak jauh yang bersifat insidental. kereta api sebagai transportasi umum memiliki beberapa keunggulan diantaranya memiliki waktu mailto:muhamad.budiman@bps.go.id jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 138 tempuh yang lebih dapat diprediksi karena terhindar dari kemacetan jalan raya dan juga mampu melayani penumpang yang jauh lebih banyak dalam sekali perjalanan dibandingkan dengan sarana transportasi umum darat lainnya. penjadwalan dan perencanaan kapasitas kereta api terkait erat dengan tingkat kepuasan penggunanya dan tingkat keuntungan usaha dari penyelenggara jasa layanan kereta api tersebut. lai dan barkan [1] menyatakan bahwa manajemen kapasitas yang efektif merupakan kunci sukses dari penyelenggara jasa layanan kereta api akan tetapi hal ini tidaklah mudah. layanan kereta api yang over-capacity akan membuat pengguna kecewa dan beralih pada moda transportasi lain selain itu juga membuat penyelenggara layanan kereta api kehilangan potensi pendapatan dari para calon penumpang yang tidak tertampung. di lain pihak, under-capacity akan membuat penyelenggara layanan kereta api menanggung beban tambahan akibat gerbong yang tidak terisi. oleh karena itu, diperlukan sebuah peramalan yang cukup tepat dalam memperkirakan jumlah penumpang kereta api sehingga dapat dilakukan penyesuaian kapasitas layanan sesuai kebutuhan. back-propagation neural network (bpnn) merupakan teknik klasifikasi dan peramalan yang paling populer menggunakan supervised learning neural network. akan tetapi, menurut chen dan su [2], teknik bpnn terbilang lambat konvergensinya dan memiliki tendensi dapat terjebak dalam lokal minima. di lain pihak, menurut riedmiller (1993) dalam [2] resilient back-propagation (rprop) merupakan teknik yang memiliki konvergensi cepat dan masih menjaga akurasinya. penelitian ini bertujuan untuk melakukan peramalan penumpang kereta api menggunakan teknik rprop sehingga dapat dilakukan penyesuaian kapasitas layanan sesuai kebutuhan dengan harapan dapat meningkatkan kepuasan pengguna layanan sekaligus tingkat keuntungan dari penyelenggara jasa layanan kereta api. 2. tinjauan pustaka 2.1 faktor determinan penggunaan alat kontrasepsi faktor-faktor yang mempengaruhi pemilihan metode kontrasepsi adalah: faktor pasangan: pengetahuan, umur, gaya hidup, frekuensi senggama, jumlah keluarga yang diinginkan, pengalaman dengan kontrasepsi lalu, sikap wanita, dan sikap kepriaan. faktor kesehatan: satus kesehatan, riwayat haid, riwayat keluarga, pemeriksaan fisik, dan pemeriksaan panggul faktor metode kontrasepsi: efektivitas, efek samping minor, kerugian, komplikasi-komplikasi yang potensial, dan biaya umur dalam hubungannya dengan pemakaian kb berperan sebagai faktor intrinsik. umur secara umum berperan terhadap organ, struktur biokimia maupun hormonal terhadap tubuh seorang wanita yang secara tidak langsung mempengaruhi terhadap pemilihan terhadap pemilihan alat kontrasepsi. jaminan kesehatan masyarakat (jamkesmas) adalah kebijakan yang sangat efektif untuk mewujudkan keadilan dan kesejahteraan rakyat dan meningkatkan aksesibilitas masyarakat miskin terhadap pelayanan kesehatan yang tersedia. jamkesmas diharapkan dapat mempercepat pencapaian sasaran pembangunan kesehatan dan peningkatan derajat kesehatan yang optimal. sasaran jamkesmas adalah seluruh masyarakat miskin, sangat miskin, dan mendekati miskin yang diperkirakan jumlahnya mencapai 76,4 juta [4]. sumber dana jamkesmas adalah anggaran pendapatan dan belanja negara (apbn) departemen kesehatan. dengan adanya jamkesmas, keluarga miskin akan mendapatkan pelayanan kb secara cumacuma baik obat maupun alat kontrasepsi. program ini dimaksudkan agar keluarga miskin tidak kesulitan dalam mengakses program kb, karena bila pertambahan penduduk tidak dapat dikendalikan, maka beban pembangunan akan bertambah. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 139 menurut bouge dalam lucas (1990) menyatakan bahwa pendidikan menunjukkan pengaruh yang lebih kuat terhadap fertilitas daripada variabel lain [5]. pendidikan merupakan salah satu faktor yang sangat menentukan pengetahuan dan persepsi seseorang terhadap pentingnya sesuatu hal, termasuk pentingnya keikutsertaan dalam kb. ini disebabkan seseorang yang berpendidikan tinggi akan lebih luas pandangannya dan lebih mudah menerima ide dan tata cara kehidupan baru. penelitian menunjukkan bahwa tingkat pendidikan yang dimiliki mempunyai pengaruh yang kuat pada perilaku reproduksi dan penggunaan alat kontrasepsi. berdasarkan survei demografi dan kesehatan indonesia (sdki) 2002-2003, pemakaian alat kontrasepsi meningkat sejalan dengan tingkat pendidikan. sebesar 45% wanita yang tidak sekolah menggunakan cara kontrasepsi modern, sedangkan wanita berpendidikan menengah atau lebih tinggi yang menggunakan cara kontrasepsi modern sebanyak 58%. semakin tinggi tingkat pendidikan wanita maka akan sangat tinggi pula kesadaran. 3. metode peneltian 3.1 support vector machine (svm) svm adalah suatu teknik yang dikembangkan oleh vapnik pada tahun 1995 untuk melakukan prediksi, baik dalam kasus klasifikasi maupun regresi. svm termasuk dalam kelas supervised learning. secara teoritisk svm dikembangkan untuk masalah klasifikasi dengan dua kelas sebagai pemisah antara dua kelas pada input space [6]. setiap data latih dinyatakan oleh (𝑥𝑖 , 𝑦𝑖) dengan i=1,2,…,n, dan 𝑥𝑖 = {𝑥𝑖1, 𝑥𝑖2, … , 𝑥𝑖 𝑞} 𝑇 merupakan atribut set untuk data ke i. untuk 𝑦𝑖 ∈ {−1, +1} menyatakan label kelas. hyperplane klasifikasi linear svm seperti pada gambar diatas dinotasikan sebagai 𝑤. 𝑥𝑖 + 𝑏 = 0 (1) w dan b adalah parameter model 𝑤. 𝑥𝑖 merupakan inner-product antara w dan 𝑥𝑖 data 𝑥𝑖 yang masuk ke dalam kelas -1 adalah data yang memnuhi pertidaksamaan 𝑤. 𝑥𝑖 + 𝑏 ≤ −1 (2) data 𝑥𝑖 yang masuk ke dalam kelas +1 adalah data yang memenuhi pertidaksamaan 𝑤. 𝑥𝑖 + 𝑏 ≥ −1 (3) persamaan diatas dapat digambarkan pada gambar 1: gambar 1. model linier support vector machine dengan mengurangkan persamaan (2) terhadap (3) akan didapatkan 𝑤(𝑥𝑏 − 𝑥𝑎 )=2 (4) dimana 𝑥𝑏 − 𝑥𝑎 adalah vector paralel di posisi hyperplane. margin hyperplane diberikan oleh jarak antara dua hyperplane dari dua kelas tersebut. notasi diatas diringkas menjadi ‖𝑤‖𝑥𝑑 = 2 atau 𝑑 = 2 ‖𝑤‖ (5) 3.2 hyperplane svm klasifikasi kelas data pada svm pada persamaan (2) dan (3) dapat digabungkan dengan notasi : 𝑦𝑖 (𝒘. 𝒙𝒊 + 𝑏) ≥ 1, 𝑖 = 1,2, … , 𝑁 (6) margin optimal dihitung dengan memaksimalkan jarak antara hyperplane dan data terdekat. jarak ini dirumuskan dengan persamaan (3.10) (‖𝒘‖ adalah vector bobot 𝑤). selanjutnya masalah ini diformulasikan ke dalam problem quadratic programming (qp) dengan meminimalkan invers persamaan (3.10), 1 2 ‖𝒘‖2 , di bawah konstrain (syarat), sebagai berikut: minimalkan: 1 2 ‖𝒘‖2 syarat: 𝑦𝑖 (𝒘. 𝒙𝒊 + 𝑏) ≥ 1, 𝑖 = 1,2, … , 𝑁 optimalisasi ini dapat diselesaikan dengan lagrange multiplier: 𝐿𝑝 = 1 2 ‖𝒘‖2 − ∑ 𝛼𝑖. 𝑁 𝑖=1 𝑦𝑖 (𝒘. 𝒙𝒊 + 𝑏) 𝛼𝑖 adalah lagrange multiplier yang berkorespondensi dengan 𝑥𝑖 . nilai 𝛼𝑖 adalah nol atau positif. masalah optimasi di atas jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 140 masih sulit diiselesaikan karena banyaknya parameter (𝑤, 𝑏 dan 𝛼𝑖 ). untuk menyederhanakannya, persamaan optimasi lagrange harus ditransformasi ke dalam fungsi lagrange multiplier itu sendiri (disebut dualitas masalah). persamaan lagrange multiplier (3.13) dapat dijabarkan menjadi: 𝐿𝑝 = 1 2 ‖𝒘‖2 − (∑ 𝛼𝑖 𝑁 𝑖=1 𝑦𝑖 (𝒘. 𝒙𝒊) − 𝑏 ∑ 𝛼𝑖 𝑁 𝑖=1 𝑦𝑖 + ∑ 𝛼𝑖 𝑁 𝑖=1 ) syarat optimal ada dalam suku ketiga di ruas kanan dalam persamaan dan memaksa suku ini menjadi sama dengan 0. dengan mengganti 𝑤 dan suku ‖𝑤‖2 = 𝒘𝒊 . 𝒘𝒋 , maka persamaan di atas akan berubah menjadi dualitas lagrange multiplier berupa 𝐿𝑑 dan didapatkan: maksimalkan: 𝐿𝑑 = ∑ 𝛼𝑖 𝑁 𝑖=1 − 1 2 ∑ 𝛼𝑖 𝑖,𝑗 𝛼𝑗 𝑦𝑖 𝑦𝑗 𝑥𝑖 𝑥𝑗 𝒙𝒊𝒙𝒋 merupakan dot-product dua data dalam data latih. 3.3 non-linier support vector machine pada umumnya data dalam dunia nyata (real world) jarang yang bersifat linier separable, kebanyakan bersifat nonlinier. untuk menyelesaikan problem non-linier, svm dimodifikasi dengan memasukkan fungsi kernel. dalam non-linier svm [7] pertama-tama data x dipetakan oleh fungsi 𝛷(𝑥) ke ruang vector yang berdimensi lebih tinggi. pada ruang vector yang baru ini, hyperplane yang memisahkan kedua class dapat dikonstruksikan. hal ini dapat digambarkan sebagai berikut: gambar 2. model non-linier support vector machine pemetaan ini dilakukan dengan menjaga topologi data, dalam artian dua data yang berjarak dekat pada input space akan berjarak dekat juga pada feature space, sebaliknya dua data yang berjarak jauh pada input space juga akan berjarak jauh pada feature space. kernel trick memberikan berbagai kemudahan, karena dalam proses pembelajaran svm, untuk menentukan support vector, kita hanya cukup mengetahui fungsi kernel yang dipakai dan tidak perlu mengetahui wujud dari fungsi nonlinier 𝛷. pada penelitian ini fungsi kernel yang akan dipakai adalah kernel radial basis, polynomial, linier dan sigmoid. 3.4 kernel trick pada mulanya teknik machine learning dikembangkan dengan asumsi kelinearan, sehingga algoritma yang dihasilkan terbatas untuk kasus-kasus yang linear saja. sedangkan untuk domain data secara real sangat jarang ditemukan data yang bersifat linier. untuk menyelesaikan problem data yang tidak linier, svm dapat dimodifikasi dengan memasukkan kernel trick. kernel trick memberikan berbagai kemudahan, karena dalam proses pembelajaran svm, untuk menentukan support vector, maka cukup dengan mengetahui fungsi kernel yang dipakai, dan tidak perlu mengetahui wujud dari fungsi non-linear. kita dapat memahami fungsi kernel dengan menggunakan aljabar pada composition rule. yang menarik dari aljabar composition rule adalah kemudahan dalam memahami fungsi kernel pada feature space. misalnya kita memiliki dua kernel yang valid yaitu 𝑘𝑎 dan 𝑘𝑏 . jika kita membentuk kernel baru dengan 𝑘(𝑥, 𝑣) = 𝑘𝑎(𝑥, 𝑣) + 𝑘𝑏 (𝑥, 𝑣) kernel akan menjadi valid. 3.5 sumber data data yang digunakan adalah data jumlah penumpang kereta api yang dikumpulkan oleh badan pusat statistik dari pt kereta api indonesia dan pt. kai commuter jabodetabek. periode data dimulai dari januari 2006 hingga april 2018. set data yang digunakan merupakan level bulanan sebanyak 148 observasi. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 141 3.6 tahapan penelitian langkah-langkah dalam penelitian ini mencakup: 1. persiapan data, terdiri dari: a. pemeriksaan data untuk melihat apakah terdapat data hilang (missing) atau tidak, dan b. pembagian data menjadi data training dan data testing. 2. pembentukan model peramalan dengan resilient back-propagation (rprop) neural network, dengan tahapan: a. penentuan parameter, seperti jumlah hidden nodes, fungsi aktivasi/transfer yang akan digunakan, dan sebagainya. b. proses pelatihan jaringan syaraf tiruan (neural network training) menggunakan data training dan parameter-parameter yang sudah ditetapkan sebelumnya. c. pemilihan parameter yang menghasilkan nilai mape terkecil. d. pembentukan model peramalan dengan nilai parameter terpilih. 3. peramalan menggunakan model yang dibentuk pada tahap sebelumnya. 4. uji performa (validasi) hasil peramalan dengan menggunakan nilai mape. 4. hasil dan pembahasan 4.1 persiapan data penelitian ini menggunakan tujuh variabel prediktor, dimana enam diantaranya merupakan variabel kategorik. penjelasan mengenai variabel prediktor dapat dilihat pada tabel dibawah: tabel 1. variabel prediktor no variabel prediktor kategorisasi 1 umur 1: umur 15-19 2: umur 20-24 3: umur 25-29 4: umur 30-34 5: umur 35-39 6: umur 40-44 7: umur 45-49 2 umur kawin pertama 1: umur 15-19 2: umur 20-24 3: umur 25-29 4: umur 30-34 5: umur 35-39 6: umur 40-44 7: umur 45-49 3 status pekerjaan 1: bekerja 2: tidak bekerja 4 jumlah anak data kontinyu 5 jaminan kesehatan 1: memiliki 2: tidak memiliki 6 ijazah terakhir yang dimiliki 1: tidak punya ijazah 2: sekolah dasar 3: sekolah lanjutan pertama 4: sekolah lanjutan atas 5: pendidikan tinggi 7 pengeluaran perkapita 1: < 1.000.000 2: 1.000.001-2.000.000 3: 2.000.001-3.000.000 4: 3.000.001-4.000.000 5: 4.000.001-5.000.000 6: > 5.000.000 prosedur yang digunakan pada evaluasi klasifikasi pada penelitian ini adalah apparent error rate (aper). nilai aper menunjukkan proporsi kesalahan sampel klasifikasi oleh fungsi klasifikasi (jhonson et al, 1992). penjelasan mengenai konsep aper dapat lebih jelas diketahui melalui tabel error klasifikasi: tabel 2. tabel error klasifikasi actual predict true false true 𝑛11 𝑛12 false 𝑛21 𝑛22 dimana nilai aper adalah 𝐴𝑃𝐸𝑅 = 𝑛12 + 𝑛21 𝑛11 + 𝑛12 + 𝑛21 + 𝑛22 nilai aper terendah akan digunakan untuk memmodelkan pemelihan penggunaan alat kontrasepsi. penelitian ini menggunakan support vector machine dengan 75 persen dan 80 persen data training dengan empat jenis kernel yaitu kernel linier, polynomial, radial basis dan sigmoid. data diolah dengan menggunakan software r dengan package e1071 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 142 berikut ditampilkan hasil pengolahan data: tabel 3. support vector machine dengan 1-aper tertinggi 75% training data kernel cost gamma degree 1aper linear 1 0.14286 68,55 polynomial 1 0.14286 3 70,79 radial basis 1 0.14286 73,76 sigmoid 1 0.14286 55,32 80% training data kernel cost gamma degree 1aper linear 1 0.14286 69,66 polynomial 1 0.14286 3 71,36 radial basis 1 0.14286 73,83 sigmoid 1 0.14286 55,57 model terbaik yang didapatkan pada penelitian ini adalah dengan data training 80 persen, kernel radial basis dengan cost 1 dan gamma 0.14286. nilai aper yang dihasilkan adalah senilai 26,17 persen. nilai 1-aper yang sudah bernilai diatas 70 persen dapat menunjukkan bahwa model svm sudah dapat menggambarkan pemilihan perempuan usia subur untuk menggunakan alat kontrasepsi atau tidak. 5. kesimpulan support vector machine dapat menggambarkan keputusan wanita usia subur untuk menggunakan alat kontrasepsi atau tidak. kernel terbaik pada penelitian ini adalah kernel radial basis dengan cost 1 dan gamma 0.14286 referensi [1] tobing, m., kolibu, f., rumayar, a., hubungan antara faktor-faktor yang mempengaruhi penggunaan kb di wilayah kerja puskesmas kapitu kecamatan amurang barat, jurnal ilmiah farmasi unsrat vol.4 no 4 (2015) [2] departemen kesehatan ri. buletin jendela data dan informasi kesehatan. departemen kesehatan ri (2013) [3]..www.beritamalukuonline.com/2015/04/ank a-kelahiran-di-malut-tinggi [4] kusumaningrum, r., skripsi, hubungan antara faktor-faktor yang mempengaruhi pemilihan jenis kontrasepsi yang digunakan pada pasangan usia subur, universitas diponogoro, indonesia, 2009 [5] lucas, d., mc donald, p., young, e., pengantar kependudukan, gajah mada university press (1990) [6] boser, b. e., guyon, i. m., vapnik, v. n., a training algorithm for optimal margin classifiers. proceedings of the fifth annual workshop on computational learning theory – colt’92.p.144 [7] n. mussafi, “analisis risk asset portfolio berbasis reward to variability pada saham syariah di indonesia menggunakan nonlinear programming”, mantik, vol. 3, no. 2, pp. 57-64, oct. 2017. http://www.beritamalukuonline.com/2015/04/anka-kelahiran-di-malut-tinggi http://www.beritamalukuonline.com/2015/04/anka-kelahiran-di-malut-tinggi paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 105 analisis faktor-faktor yang mempengaruhi sektor moneter berdasarkan jumlah uang yang beredar pada statistik ekonomi dan keuangan indonesia (seki) dian c. rini novitasari uin sunan ampel surabaya, diancrini@uinsby.ac.id doi:https://doi.org/10.15642/mantik.2017.3.2.105-111 . abstrak salah satu dampak krisis ekonomi yang terjadi di indonesia dipengaruhi oleh sektor moneter yang menggerakkan ekonomi makro ke dalam perekonomian masyarakat seperti nilai tukar mata uang dan kebijakan suku bunga yang diberlakukan di seluruh bank di indonesia. analisis faktor-faktor yang mempengaruhi jumlah uang beredar merupakan salah satu langkah yang dapat digunakan untuk menganalisa statistik ekonomi dan kondisi keuangan indonesia. analisis yang dilakukan dimulai dengan mencari persamaan regresi menggunakan model regresi nonlinier dari beberapa faktor yang digunakan seperti aktivitas luar negeri, tagihan kepada pemerintah pusat, tagihan kepada sektor swasta, dan nilai produk domestik bruto (pdb). uji lain juga dilakukan seperti uji normalitas data, uji heteroskedastisitas, dan uji otokorelasi untuk membaca karaketeristik data yang digunakan dalam kurun waktu 1999-2016 menggunakan spss dan selanjutnya dianalisis. dimana hasil yang diperoleh pada regresi terhadap variabel independent pemerintah pusat tidak memiliki pengaruh yang signifikan, pada uji normalitas data menunjukkan data berdistribusi normal, tidak terjadi heterorkedastisitas, serta data tidak mengandung otokorelasi. kata kunci: sektor moneter, spss, model regresi nonlinier abstract one of the impacts of the economic crisis that occurred in indonesia was influenced by the monetary sector that moved the macroeconomy into the economy of the society such as currency rate of exchange and rate of interest policies applied in all banks in indonesia. analysis of the factors that influence the money supply is one step that can be used to analyze economic statistics and the financial condition of indonesia. the analysis begins with finding the regression equation using a nonlinear regression model of several factors used such as foreign activities, bills to the central government, bills to the private sector, and the value of gross domestic product (gdp). other tests were also carried out such as the data normality test, heteroscedasticity test, and autocorrelation test to read data characteristics used in the 1999-2016 period using spss and then analyzed. where the results obtained in the regression of the central government independent variables do not have a significant effect, the data normality test shows that the data is normally distributed, heteroscedasticity does not occur, and the data does not contain autocorrelation. keywords: monetary sector, spss, nonlinear regression 1. pendahuluan indonesia menjadi salah satu negara yang mengalami krisis ekonomi yang sangat parah pada tahun 1997 jika dibandingkan dengan krisis ekonomi beberapa negara yang pernah terjadi sebelumnya [1]. kondisi ini menjadi sangat parah ketika industri perbankan nasional juga menyebabkan perekonomian masyarakat luas kedalam pertumbuhan ekonomi yang berjalan lambat. kejadian ini merupakan salah satu permasalahan ekonomi makro yang dialami indonesia terhadap krisis moneter yang berdampak langsung pada banyaknya permintaan uang dan menyebabkan terjadinya inflasi hingga jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 106 diikuti oleh kondisi lain seperti naik turunnya suku bunga deposito dan kredit yang berdampak pada besarnya dana dan kredit yang diberikan. salah satu faktor utama yang memepengaruhi tolak ukur pada kemajuan suatu negara adalah pertumbuhan ekonomi. pertumbuhan ekonomi yang dialami indonesia pada tahun 1997 merupakan kondisi terpuruk perekonomian indonesia yang berjalan lambat. suatu negara dikatakan maju jika kondisi perekonomian negara tersebut bersifat stabil dam jika kondisi perekonomian negara tersebut tidak dapat meningkat maka negara tersebut belum dapat dikatakan sebagai negara yang maju. oleh karena itu kegiatan perekonomian negara diharapkan terus meningkat sebagai upaya untuk mendukung pertumbuhan ekonomi negara [2]. sektor moneter pada ekonomi makro seperti tingkat inflasi, nilai kurs, dll menjadi penggerak utama dalam meningkatkan perekonomian indonesia. inflasi dalam perspektif ekonomi merupakan suatu fenomena moneter negara yang menyebabkan terjadinya gejolak ekonomi akibat naik dan turunnya tingkat inflasi yang berpengaruh terhadap pertumbuhan ekonomi, neraca perdagangan internasional, nilai utang piutang antarnegara, tingkat bunga, tabungan, domestik, pengangguran, dan kesejahteraan masyarakat [3]. satu sisi dari faktor yang mempengaruhi tingkat naik dan turunnya inflasi dapat ditinjau dari segi banyaknya jumlah uang yang beredar yang berpengaruh pada sektor ekonomi makro. beberapa penelitian yang mengkaji tentang pertumbuhan ekonomi telah banyak dilakukan, seperti yang dilakukan oleh [4] mengkaji tentang “analisis pengaruh investasi, operasi moneter dan zis terhadap pertumbuhan ekonomi indonesia” dengan beberapa variabel yang digunakan seperti reksadana syariah, jumlah zakat, pdb, dll menggunakan analisis regresi linier dengan data time series dalam kurun waktu januari 2013-desmber 2015. hasil penelitian tersebut menunjukkan 97,2% variabel reksadana syariah, reksadana konvensional, fasbis, zis dan pdb periode sebelumnya mempengaruhi pdb riil indonesia sebagai indikator pertumbuhan ekonomi periode 2013-2015. sedangkan pada penelitian ini akan dilakukan suatu analisis terhadap faktor-faktor yang mempengaruhi sektor moneter (jumlah uang beredar) terhadap statistik ekonomi dan keuangan indonesia (seki) menggunakan metode regresi non linier model kubik yang diselesaikan menggunakan spss dan hasilnya akan dianalisis. uji-uji lain juga akan dilakukan untuk menganalisis data-data yang digunakan seperti uji normalitas data, uji heteroskedastisitas, dan uji otokorelasi. 2. tinjauan pustaka 2.1. definisi uang beredar pengertian uang dapat dilihat dari sisi kegunaan atau fungsinya sebagaimana yang digunakan dalam kehidupan sehari-hari bagi manusia. bank indonesia mendefinisikan uang sebagai benda yang digunakan untuk alat tukar, satuan hitung, alat penyimpan nilai, dan ukuran pembayaran yang tertunda [5]. nilai satuan pada masing-masing negara akan berbeda dan akan berlaku sebagai alat tukar yang sah dengan tingkat nominal masing-masing. uang beredar berkaitan sangat erat dengan otoritas moneter yang merupakan suatu alat pembayaran yang sah berupa uang kartal, yaitu uang kertas dan uang logam yang dikeluarkan dan diedarkan oleh bank indonesia kepada masyarakat luas. kebijakan dalam mengedarkan dan mengeluarkan uang merupakan wewenang dari sistem moneter, yaitu bank sentral dan bank umum. jika bank sentral bertugas mengeluaran dan mengedarkan uang kartal, maka bank umum adalah lembaga yang dapat menciptakan uang berupa uang giral dan uang kuasi. kebijakan sistem moneter dalam mengeluarkan dan mengedarkan uang berkewajiban kepada sektor swasta domestik yaitu masyarakat, badan usaha, lembaga, dll. hal ini menyebabkan kewajiban dalam sistem moneter terhadap sektor swasta domestik dapat didefinisikan sebagai uang beredar [5]. 2.2. sektor moneter sektor moneter yang dimaksud dalam penelitian ini adalah faktor-faktor yang mempengaruhi jumlah uang beredar meliputi aktivitas luar negeri, aktivitas dalam negeri, dan aktivitas lainnya [5]. pengaruh banyaknya jumlah uang yang beredar tidak lepas dari kebijakan moneter yang terjadi. kebijakan moneter merupakan pengendalian besaran untuk mencapai perkembangan dalam kegiatan perekonomian yang diinginkan [6]. perkembangan kegiatan jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 107 perekonomian di stabilitas ekonomi makro dicerminkan oleh stabilitas inflasi, membaiknya output riil hingga cukup luasnya kesempatan kerja yang tersedia bagi masyarakat. pertumbuhan ekonomi sangat dipengaruhi oleh pengaturan penciptaan uang atau jumlah uang yang beredar dimana kebijakan moneter dalam mengatur jumlah uang yang beredar terbagi menjadi 2 jenis yaitu [6] : 1. kebijakan moneter ekspansif merupakan salah satu upaya pemerintah dalam meningkatkan jumlah uang beredar pada bank sentral. 2. kebijakan moneter kontraktif merupakan kebalikan dari kebijakan moneter ekspansif, yaitu upaya pemerintah dalam mengurangi jumlah uang beredar pada bank sentral. 2.3. root mean square error pada berbagai penelitian, produk domestik bruto (pdb) digunakan sebagai parameter dalam mempengaruhi baik buruknya pertumbuhan ekonoomi suatu negara serta sebagai tolak ukur kesejahteraan masyarakat. nilai produk domestik bruto (pdb) ini menjadi salah satu faktor yang mempengaruhi jumlah uang beredar pada sektor ekonomi moneter. pdb indonesia memberikan prediksi optimistik yang meningkat dari tahun ke tahun. berdasarkan laporan badan pusat statistik, secara kumulatif pada tahun 2009 pdb tumbuh sebesar 5,8% dan pada tahun 2010 sebesar 5,9% serta cadangan devisa dapat mencapai usd 94,7 miliar dan nilai ekspor mencapai usd sebesar 150 miliar. angka pertumbuhan tersebut menunjukkan perubahan ekonomi yang semakin baik [7]. nilai produk domestik bruto dibedakan menjadi produk domestik bruto (pdb) menurut harga konstan, dan produk domestik bruto (pdb) menurut lapangan usaha. 2.4. regresi nonlinier mode kubik regresi nonlinier merupakan suatu metode penyelesaian persamaan regresi dalam statistika yang digunakan untuk mendapatkan model nonlinier yang menyatakan hubungan variabel dependent dengan variabel independent. pada dasarnya yang membedakan linier dan tidak linier adalah pada garis yang menjadi acuan dimana suatu keadaan linier merupakan kondisi dimana data-data yang digunakan terletak atau tersebar pada suatu garis lurus namun jika tidak linier maka data tersebut tidak terletak pada garis lurus melainkan pada garis lengkung. regresi nonlinier dapat digunakan untuk mengestimasi model hubungan variabel dependent dengan variabel independent dalam bentuk nonlinier dengan keakuratan yang lebih baik daripada regresi linier, hal ini dikarenakan dalam mengestimasi model digunakan iterasi algoritma. model regresi nonlinier kubik dapat dituliskan kedalam persamaan berikut, 𝑌 = 𝛽0 + 𝛽1x + 𝛽2𝑋 2 + 𝛽3𝑋 3 dimana pengujian ini dilakukan menggunakan bantuan spss yang akan dianalisis menggunakan tabel outputan dari hasil spss baik itu nilai signifikansi maupun nilai keakuratan. 3. metode penelitian adapun algoritma penyelesaian penelitian ini ditunjukkan pada gambar 1 berikut. gambar 1. alur penelitian 3.1. data data yang digunakan dalam penelitian ini merupakan data-data parameter yang digunakan dalam menganalisis faktor-faktor yang mempengaruhi jumlah uang beredar seperti aktivitas luar negeri, tagihan kepada pemerintah pusat, tagihan kepada sektor swasta, dan nilai produk domestik bruto (pdb). data tersebut diperoleh dari web site bank indonesia pada statistik ekonomi dan keuangan indonesia (seki). rentan waktu yang digunakan berada pada kurun waktu 1999-2016 yang terhitung selama 19 tahun. karena satuan yang digunakan dalam data tersebut sama, yaitu milyar maka untuk melakukan uji-uji data tidak harus jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 108 dinormalisasi sehingga kita dapat langsung mengujinya dengan spss untuk selanjutnya dianalisis. contoh data yang digunakan dalam penelitian ini disajikan pada tabel 1. 3.2. uji normalitas uji normalitas digunakan untuk melihat apakah data yang digunakan berdistribusi normal atau tidak. apabila data yang digunakan tidak berdistribusi normal maka data harus dinormalisasi terlebih dahulu untuk dilakukan pengujian selanjutnya. karena apabila data yang digunakan tidak berdistribusi normal, maka hasil output yang didapat juga tidak akan baik yang berakibat pada hasil mining yang didapatkan. uji normalitas data yang digunakan dala penelitian ini menggunakan uji normalitas komogorov-smirnov dengan bantuan spss. 3.3 uji heteroskedastisitas uji asumsi klasik heteroskedastisitas bertujuan untuk menguji apakah dalam model regresi terjadi ketidaksamaan varians dari nilai residual satu pengamatan ke pengamatan lain. heteroskedastisitas akan terjadi pada suatu model ketika varians yang dihasilkan tidak konstan, sehingga koefisien regresi dari model tidak konsisten. beberapa contoh penyebab perubahan nilai varian yang berpengaruh pada homoskedastisitas residualnya: 1. adanya pengaruh kurva dari pengalaman 2. adanya peningkatan dalam perekonomian 3. adanya peningkatan dalam teknik pengambilan data beberapa metode yang dapat digunakan untuk melakukan pengujian pada heteroskedastisitas data diantaranya menggunakan metode analisis grafik, dan metode statistik seperti metode glejser, metode park, metode rank spearman, dll. pengambilan kesimpulan dari hasil pengujian heteroskedastisitas data menggunakan metode glejser dapat dilakukan dengan melihat koefisien 𝛽1 dari persamaan regresi 𝑌1 = 𝛽0 + 𝛽1𝑋1 + 𝑣𝑖 (2) koefisien 𝛽1 akan diuji dengan uji t dimana bila nilai t hitung < t tabel pada taraf signifikansi tertentu dan df=n-k, maka ho diterima, yang berarti tidak terdapat hubungan yang signifikan antara residual dengan variabel penjelasnya, atau dengan kata lain tidak terdapat masalah heteroskedastisitas di dalam model [1]. tabel 1. data tahun jumlah uang yg beredar aktivitas luar negeri pemerintah pusat tagihan sektor swasta produk domestik bruto (pdb) 1999 646205,00 129096,00 397257,00 233714,00 275351,60 2000 747028,00 210733,00 520317,00 280566,00 366143,23 2001 844053,00 233975,00 529706,00 310816,00 416775,08 2002 883908,00 250696,00 510351,00 366407,00 462081,86 2003 955692,00 271820,00 479013,00 442741,00 503299,30 2004 1033877,00 253260,00 500318,33 605926,55 599478,20 2005 1202762,00 301573,00 495685,72 733182,94 758474,90 2006 1382493,00 401710,00 507336,71 821648,82 873403,00 2007 1649662,00 509843,00 507120,01 1005738,99 1035418,90 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 109 4. hasil dan pembahasan hasil spss yang dilakukan diterapkan pada masing-masing variabel yang digunakan. dimana dalam pengujian tersebut variabel independent dianalisis dilakukan secara satu per satu sehingga diperoleh hasil yang ditunjukkan pada tabel 2. berdasarkan hasil tabel 2, model regresi nonlinnier kubik yang dapat dibentuk adalah sebagai berikut: 𝑌 = 𝛽0 + 𝛽1x + 𝛽2𝑋 2 + 𝛽3𝑋 3 𝑌 = 360133,917 + 2,284x − 0,0000001738𝑋2 + 0,0000000000009796𝑋3 berdasarkan hasil tabel 3, maka model regresi nonlinnier kubik yang dapat dibentuk adalah sebagai berikut: 𝑌 = 𝛽0 + 𝛽1x + 𝛽2𝑋 2 + 𝛽3𝑋 3 𝑌 = −2139314,687 + 18,568x − 4,020𝑋2 + −0,00000000004153𝑋3 berdasarkan hasil tabel 4, maka model regresi nonlinnier kubik yang dapat dibentuk adalah sebagai berikut: 𝑌 = 𝛽0 + 𝛽1x + 𝛽2𝑋 2 + 𝛽3𝑋 3 𝑌 = 320484,872 + 1,482x − 0,0000002031𝑋2 + 0,00000000000002812𝑋3 berdasarkan hasil tabel 5, maka model regresi nonlinnier kubik yang dapat dibentuk adalah sebagai berikut: 𝑌 = 𝛽0 + 𝛽1x + 𝛽2𝑋 2 + 𝛽3𝑋 3 𝑌 = 505242,823 + 1,091x + −0,0000001665𝑋2 + 0,000000000000008837𝑋3 berdasarkan tabel 6 diatas dapat disimpulkan dari nilai signifikansinya, dimana nilai sig dari msing-masing variabel independet lebih dari 0,05 yaitu untuk aktivitas luar negeri sebesar 0,782, pemerintah pusat 0,927, tagihan sektor swasta 0,717, dan produk domestik bruto sebesar 1,417. sehingga dapat disimpulkan bahwa data populasi yang digunakan berdistribusi normal. sedangkan pada uji heteroskedastisitas dapat diketahui bahwa pada model regresi tidak terjadi gelaja heteroskedastisitas yang ditunjukkan oleh nilai sig dari masing-masing variabel yang bernilai lebih dari 0,05. pada tabel 7 menunjukkan nilai durbin watson sebesar 1,923. pengambilan kesimpulan dari uji ini memerlukan nilai bantu yang diperoleh dari tabel durbin watson yaitu nilai dl dan du dengan k=4 dan n=18. berdasarkan tabel durbin watson diperoleh nilai dl=0.8204 dan du=1.8719, sehingga 4du=2,1281 sedangkan nilai 4-dl=. 3,1796 karena nilai durbin watsonnya adalah 1,923 maka terletak antara nilai du dan nilai 4-du yang mengartikan bahwa tidak terjadi otokorelasi. tabel 2. coefficients unstandardized coefficients standardized coefficients t sig. b std. error beta aktivitas luar negeri 2,284 1,555 ,620 1,469 ,164 aktivitas luar negeri ** 2 -1,738e-007 ,000 -,066 -,072 ,944 aktivitas luar negeri ** 3 9,796e-013 ,000 ,457 . . (constant) 360133,917 266955,195 1,349 ,199 tabel 3. excluded terms beta in t sig. partial correlation minimum tolerance pemerintah pusat ** 2a -4,202 -,030 ,977 -,008 ,000 a. the tolerance limit for entering variables is reached. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 110 tabel 4. coefficients unstandardized coefficients standardized coefficients t sig. b std. error beta tagihan sektor swasta 1,482 ,147 1,366 10,060 ,000 tagihan sektor swasta ** 2 -2,031e-007 ,000 -,787 -2,387 ,032 tagihan sektor swasta ** 3 2,812e-014 ,000 ,428 . . (constant) 320484,872 62465,954 5,131 ,000 tabel 5. coefficients unstandardized coefficients standardized coefficients t sig. b std. error beta produk domestik bruto (pdb) 1,091 ,240 3,546 4,536 ,000 produk domestik bruto (pdb) ** 2 -1,665e-007 ,000 -6,216 -3,673 ,003 produk domestik bruto (pdb) ** 3 8,837e-015 ,000 3,753 . . (constant) 505242,823 168294,436 3,002 ,010 tabel 6. one-sample kolmogorov-smirnov test aktivitas luar negeri pemerintah pusat tagihan sektor swasta produk domestik bruto (pdb) n 18 18 18 18 normal parametersa,b mean 620597,125556 455938,3972 1579295,406667 4188789,515000 std. deviation 386521,1651110 60833,69691 1311931,5437935 4627309,9505792 most extreme differences absolute ,184 ,219 ,169 ,334 positive ,184 ,130 ,169 ,334 negative -,125 -,219 -,153 -,199 kolmogorov-smirnov z ,782 ,927 ,717 1,417 asymp. sig. (2-tailed) ,574 ,356 ,683 ,036 a. test distribution is normal. b. calculated from data. tabel 7. model summaryb model r r square adjusted r square std. error of the estimate durbin-watson 1 1,000a 1,000 ,999 34047,8503647 1,923 a. predictors: (constant), produk domestik bruto (pdb), pemerintah pusat , aktivitas luar negeri, tagihan sektor swasta b. dependent variable: jumlah uang yg beredar jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 111 5. kesimpulan berdasarkan uji-uji yang dilakukan diatas, dapat disimpulkan bahwa model regresi yang dihasilkan dari model regresi nonlinier kubik menunjukkan bahwa variabel independent yang tidak berpengaruh signifikan adalah variabel pemerintah pusat yang ditunjkkan pada tabel anova dengan nilai signifikansi 0,524 yaitu lebih dari 0,05. sehingga variabel independent yang tidak memiliki pengaruh signifikan adalah variabel pemerintah pusat. sedangkan pada uji-uji yang lain menunjukkan bahwa data yang digunakan bersistribusi normal, tidak mengandung heteroskedastisitas, dan tidak mengandung otokorelasi. 6. daftar pustaka [1] hasan, i. analisis faktor-faktor yang mempengaruhi jumlah uang beredar di indonesia periode 1985 2005," in skripsi, surakarta, universitas sebelas maret, 2009, p. 2. [2] machtra, c dan fachruddin, f. analisis efek kebijakan moneter terhadap output di indonesia. jurnal ekonomi dan kebijakan publik, vol. 3, no. 1, p. 2, 2016. [3] utami, a. t. dan soebagiyo, d. nilai tukar, "penentu inflasi di indonesia; jumlah uang beredar, ataukah cadangan devisa? jurnal ekonomi dan studi pembangunan, vol. 14, no. 2, p. 2, 2013. [4] tambunan, k. analisis pengaruh investasi, operasi moneter dan zis terhadap pertumbuhan ekonomi indonesia. attawassuth, vol. 1, no. 1, p. 19, 2016. [5] bank indonesia. uang: pengertian, penciptaan, dan peranannya dalam perekonomian. pusat pendidikan dan studi kebanksentralan (ppsk) bi, jakarta, 2002. [6] oktaviani, i. pengaruh kebijakan moneter syariah terhadap indeks pengaruh kebijakan moneter syariah terhadap indeks tahun 20112016. in skripsi. jakarta. universitas islam negeri syarif hidayatullah, 2017, p. 30. [7] hayati, s.r. "peran perbankan syariah terhadap pertumbuhan ekonomi indonesia," indoislamika, vol. 4, no. 1, p. 9, 2014. [8] helmi, i, "perbandingan penduga ordinary least squares (ols) dan generalized least squares (gls) pada model regresi linier dengan regresor bersifat stokastik dan galat model berautokorelasi," jurnal matematika unand, vol. 3, no. 4, p. 2. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 42 pemodelan akreditasi smk di provinsi banten dengan menggunakan logika fuzzy metode mamdani syamsuri1, indiana marethi2 universitas sultan ageng tirtayasa1, syamsuri@untirta.ac.id1 universitas sultan ageng tirtayasa2, indianamarethi@untirta.ac.id2 doi:https://doi.org/10.15642/mantik.2018.4.1.42-48 abstrak artikel ini bertujuan memodelkan akreditasi smk di provinsi banten yang diakreditasi selama 2009-2011 menggunakan logika fuzzy dengan metode mamdani. data yang digunakan diperoleh dari badan akreditasi provinsi banten – sekolah/madrasah (baps/m) sebnyak 275 program keahlian pada smk yang diakreditasi oleh bap-s/m provinsi banten periode 2009-2011. dalam memodelkan akreditasi dengan menggunakan logika fuzzy ini mengasumsikan bahwa : (1) standar isi, standar proses, standar kompetensi lulusan, dan standar penilaian memiliki korelasi yang kuat, sehingga hanya satu standar saja yang mewakili, yaitu : standar proses, (2) standar pendidik dan tenaga kependidikan, serta standar pengelolaan berkorelasi kuat, sehingga diambil satu standar saja yang mewakili, yaitu standar pendidik dan tenaga kependidikan, dan (3) standar sarana dan prasarana serta standar pembiayaan memiliki korelasi yang kuat, sehingga hanya satu standar saja yang mewakili, yaitu : standar sarana dan prasarana. model yang diperoleh dapat digunakan dalam memprediksi hasil akreditasi suatu smk dengan hanya melihat dari nilai standar proses, nilai standar pendidik, dan nilai standar sarana. model yang dihasilkan memiliki tingkat ketepatan sekitar 68 %. kata kunci : akreditasi sekolah, model, logika fuzzy, metode mamdani abstract this article aims to describe an accreditation model of vocational schools in banten province that accredited for 2009-2011 using method of mamdani of fuzzy logic. the data used were obtained from banten accreditation board for schools/madrasah (bap-s/m), 275 expertise in vocational programs are accredited by the bap-s/m banten during 20092011. in the accreditation model using fuzzy logic assumes that: (1) there are strong correlation among content standards, process standards, competency standards, and assessment standards, so that we use score of process standards in modelling, (2) standard educators and staff, as well as management standard strongly correlated, so that we choose educators, and (3) standards of infrastructure and financing have strong correlation, so that only one representing one standard, namely : standard of infrastructure. the model can be used in predicting the outcome of a vocational accreditation by just looking scores from the process standard, educators standard, and infrastructures standard. the resulting models have about 68% accuracy rate. keywords: accreditation of vocational schools, modelling, fuzzy logic, mamdani method jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 43 1. pendahuluan akreditasi sekolah adalah proses penilaian secara komprehensif terhadap kelayakan satuan atau program pendidikan, yang hasilnya diwujudkan dalam bentuk sertifikat pengakuan dan peringkat kelayakan yang dikeluarkan oleh suatu lembaga yang mandiri dan profesional [1]. selain itu, akreditasi merupakan bentuk asesmen dalam penyelenggaraan pendidikan yang penting. tidak hanya dalam penyelenggaraan pendidikan, akreditasi juga banyak diterapkan pada pendidikan dan pelatihan keteknikan [2]. bahkan di negara cina, dikembangkan model daete dalam mengevaluasi pendidikan berkelanjutan [3]. salah satu bentuk pemodelan matematika yang telah digunakan dalam penelitian pendidikan ialah menggunakan logika fuzzy ([4], [5]). di indonesia, penggunaan instrumen akreditasi yang komprehensif dikembangkan berdasarkan standar yang mengacu pada standar nasional pendidikan (snp). hal ini didasarkan pada peraturan pemerintah nomor 19 tahun 2005 yang memuat kriteria minimal tentang komponen pendidikan. seperti dinyatakan pada pasal 1 ayat (1) bahwa snp adalah kriteria minimal tentang sistem pendidikan di seluruh wilayah hukum negara kesatuan republik indonesia. oleh karena itu, snp harus dijadikan acuan guna memetakan secara utuh profil kualitas sekolah/madrasah. di dalam pasal 2 ayat (1), lingkup snp meliputi: (1) standar isi, (2) standar proses, (3) standar kompetensi lulusan, (4) standar pendidik dan tenaga kependidikan, (5) standar sarana dan prasarana, (6) standar pengelolaan, (7) standar pembiayaan, dan (8) standar penilaian pendidikan. instrumen akreditasi smk/mak disusun berdasarkan delapan komponen yang mengacu pada standar nasional pendidikan. instrumen akreditasi smk/mak memuat 185 butir pernyataan, masing-masing memiliki bobot butir yang berbeda-beda tergantung dukungannya terhadap pembelajaran bermutu. bobot butir pernyataan terendah diberikan bobot 1, dan tertinggi diberikan bobot 4. definisi operasional bobot butir adalah sebagai berikut. bobot 1 adalah bobot minimal untuk mendukung fungsi komponen dalam proses pembelajaran agar dapat berlangsung. o bobot 2 adalah bobot yang mendukung fungsi komponen tersebut dalam proses pembelajaran yang layak. o bobot 3 adalah bobot yang mendukung fungsi komponen tersebut dalam proses pembelajaran yang baik. o bobot 4 adalah bobot maksimal yang mendukung fungsi komponen tersebut dalam proses pembelajaran yang sangat baik. instrumen akreditasi smk/mak untuk masing-masing standar seperti ditunjukkan pada tabel berikut. tabel 1 : bobot komponen instrumen akreditasi smk/mak no. komponen akreditasi nomor butir jumlah butir jumlah skor butir maksimum 1 standar isi 1 ⎯ 18 18 54 2 standar proses 19 ⎯ 31 13 43 3 standar kompetensi lulusan 32 ⎯ 62 31 96 4 standar pendidik dan tendik 63 ⎯ 87 25 81 5 standar sarana dan prasarana 88 ⎯ 112 25 81 6 standar pengelolaan 113 ⎯ 138 26 80 7 standar pembiayaan 139 ⎯ 164 26 83 8 standar penilaian pendidikan 165 ⎯ 185 21 65 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 44 tabel 2. skor dan nilai tiap standar akreditasi n o. komponen akreditasi jumlah skor minimum (nilai standar minimum) jumlah skor maksimum (nilai standar maksimum) 1 standar isi 18 (33) 54 (100) 2 standar proses 13 (30) 43 (100) 3 standar kompetensi lulusan 31 (32) 96 (100) 4 standar pendidik dan tendik 25 (31) 81 (100) 5 standar sarana dan prasarana 25 (31) 81 (100) 6 standar pengelolaan 26 (33) 80 (100) 7 standar pembiayaan 26 (31) 83 (100) 8 standar penilaian pendidikan 21 (32) 65 (100) sekolah/madrasah memperoleh peringkat akreditasi sebagai berikut: 1. peringkat akreditasi a (sangat baik), jika memperoleh nilai akhir akreditasi (na) sebesar 86 sampai dengan 100, atau 86 < na < 100. 2. peringkat akreditasi b (baik), jika memperoleh nilai akhir akreditasi sebesar 71 sampai dengan 85, atau 71 < na < 85. 3. peringkat akreditasi c (cukup baik), jika memperoleh nilai akhir akreditasi sebesar 56 sampai dengan 70, atau 56 < na < 70. artikel ini bertujuan untuk memodelkan akreditasi smk di provinsi banten yang diakreditasi selama 2009-2011 menggunakan logika fuzzy dengan metode mamdani. 2. pemodelan 2.1 data data diperoleh dari badan akreditasi provinsi banten – sekolah/madrasah (baps/m) sebnyak 275 program keahlian pada smk yang diakreditasi oleh bap-s/m provinsi banten periode 2009-2011. 2.2 asumsi dalam pemodelan dalam memodelkan akreditasi dengan menggunakan logika fuzzy ini mengasumsikan bahwa: 1. standar isi, standar proses, standar kompetensi lulusan, dan standar penilaian memiliki korelasi yang kuat, sehingga hanya satu standar saja yanng mewakili. dalam hal ini diambil standar proses. 2. standar pendidik dan tenaga kependidikan, serta standar pengelolaan berkorelasi kuat, sehingga diambil satu standar saja yang mewakili, yaitu standar pendidik dan tenaga kependidikan. 3. standar sarana dan prasarana serta standar pembiayaan memiiki korelasi yang kuat, sehingga hanya satu standar saja yanng mewakili. dalam hal ini diambil standar sarana dan prasarana. gambar 1. keterkaitan antar standar dalam akreditasi sekolah [1] 2.3 variabel dalam model fuzzy dalam model ini dibuat 3 variabel fuzzy, yaitu : 1. nilai standar proses (x), pada variabel ini didefinisikan 2 himpunan fuzzy, yaitu : tinggi dan rendah. berdasarkan tabel 2, bahwa nilai minimum dari x ialah 30, dan nilai maksimum dari x adalah 100, maka fungsi derajat keangotaan dari himpunan fuzzy rendah dan jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 45 himpunan fuzzy tinggi sebagai berikut : μ rendah (x)= { 1 , jika x≤30 90-x 90-30 , jika 30<x≤90 0 , jika x>90 μ tinggi (x)= { 0 , jika x≤30 x-30 90-30 , jika 30<x≤90 1 , jika x>90 2. nilai standar pendidik dan tenaga kependidikan (y), pada variabel ini didefinisikan 2 himpunan fuzzy, yaitu : baik dan buruk. berdasarkan tabel 2, bahwa nilai minimum dari y ialah 31, dan nilai maksimum dari y adalah 100, maka fungsi derajat keangotaan dari himpunan fuzzy baik dan himpunan fuzzy buruk sebagai berikut : μ buruk (y)= { 1 , jika y≤31 95-y 95-31 , jika 31<y≤95 0 , jika y>95 μ baik (y)= { 0 , jika y≤31 y-31 95-31 , jika 31<y≤95 1 , jika y>95 3. nilai standar sarana dan prasarana (z), pada variabel ini didefinisikan 2 himpunan fuzzy, yaitu : bagus dan jelek. berdasarkan tabel 2, bahwa nilai minimum dari z ialah 31, dan nilai maksimum dari z adalah 100, maka fungsi derajat keangotaan dari himpunan fuzzy bagus dan himpunan fuzzy jelek sebagai berikut : μ jelek (z)= { 1 , jika z≤31 96-z 96-31 , jika 31<z≤96 0 , jika z>96 μ bagus (z)= { 0 , jika z≤31 z-31 96-31 , jika 31<z≤96 1 , jika z>96 4. nilai akreditasi (a), pada variabel ini didefinisikan 3 himpunan fuzzy, yaitu : a, b dan c. berdasarkan kriteria pemeringkatan akreditasi, maka fungsi derajat keangotaan dari himpunan fuzzy a, himpunan fuzzy b dan himpunan fuzzy c sebagai berikut : μ a (a)= { 0 , jika a≤78 a-78 86-78 , jika 78<a≤86 1 , jika a>86 μ b (a)= { 0 , jika a≤70 atau a≥86 a-70 78-70 , jika 70<a<78 86-a 86-78 , jika 78<a<86 μ c (a)= { 1 , jika a≤70 78-a 78-70 , jika 70<a<78 0 , jika a≥78 3. pembahasan 3.1 aturan fuzzy menurut wang [6], variabel-variabel yang telah didefinisikan harus dibentuk aturan inferensi fuzzy. oleh karena itu, beberapa kemungkinan aturan pada inferensi fuzzy pada kasus tersebut. tabel 3. hasil dari aturan-aturan yang terbentuk pada inferensi fuzzy atur an nilai standar proses (x) nilai standar pendidi k (y) nilai stand ar saran a (z) nilai akreditasi r1 tinggi baik bagus a r2 tinggi baik jelek b r3 tinggi buruk bagus b r4 tinggi buruk jelek b r5 rendah baik bagus b r6 rendah baik jelek b r7 rendah buruk bagus b r8 rendah buruk jelek c 3.2 contoh kasus misalkan smk negeri 1 rangkasbitung program teknik komputer jaringan memiliki nilai standar proses (x) sebesar 87, nilai standar pendidik (y) sebesar 81, dan nilai standar sarana (z) sebesar 91. masuk ke kategori akreditasi apakah sekolah tersebut? jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 46 penyelesaian berdasarkan model fuzzy di atas. 1. menghitung derajat keanggotaan dari masing-masing himpunan fuzzy dari tiap-tiap variabel a. variabel x μ rendah (87)= 90-85 90-30 = 5 60 =0.05 μ tinggi (87)= 87-30 90-30 = 55 60 =0.95 b. variabel y μ buruk (81)= 95-75 95-31 =0.22 μ baik (81)= 81-31 95-31 =0.78 c. variabel z μ jelek (91)= 96-91 96-31 =0.08 μ bagus (91)= 91-31 96-31 =0.92 2. penerapan aturan fuzzy [r1] jika x tinggi dan y baik dan z bagus, maka a adalah a α-predikat 1 = μ tinggi (x)∩μ baik (y)∩μ bagus (z) = min [μ tinggi (x),μ baik (y),μ bagus (z)] = min [μ tinggi (87),μ baik (81),μ bagus (91)] = min[0.95, 0.78, 0.92]=0.78 [r2] jika x tinggi dan y baik dan z jelek, maka a adalah b α-predikat 2 =μ tinggi (x)∩μ baik (y)∩μ jelek (z) = min [μ tinggi (x),μ baik (y),μ jelek (z)] = min [μ tinggi (87),μ baik (81),μ jelek (91)] = min[0.95, 0. 78, 0.08]=0.08 [r3] jika x tinggi dan y buruk dan z bagus, maka a adalah b α-predikat 3 =μ tinggi (x)∩μ buruk (y)∩μ bagus (z) = min [μ tinggi (x),μ buruk (y),μ bagus (z)] = min [μ tinggi (87),μ buruk (81),μ bagus (91)] =min[0.95, 0. 22, 0.92]=0.22 [r4] jika x tinggi dan y buruk dan z jelek, maka a adalah b αpredikat 4 =μ tinggi (x)∩μ buruk (y)∩μ jelek (z) = min [μ tinggi (x),μ buruk (y),μ jelek (z)] = min [μ tinggi (87),μ buruk (81),μ jelek (91)] = min[0.95, 0. 22, 0.08]=0.08 [r5] jika x rendah dan y baik dan z bagus, maka a adalah b α-predikat 5 =μ rendah (x)∩μ baik (y)∩μ bagus (z) = min [μ rendah (x),μ baik (y),μ bagus (z)] = min [μ rendah (87),μ baik (81),μ bagus (91)] = min[0.05, 0. 78, 0.92]=0.05 [r6] jika x rendah dan y baik dan z jelek, maka a adalah b α-predikat 6 =μ rendah (x)∩μ baik (y)∩μ jelek (z) = min[μ rendah (x),μ baik (y),μ jelek (z)] = min[μ rendah (87),μ baik (81),μ jelek (91)] = min[0.05, 0. 78, 0.08]=0.05 [r7] jika x rendah dan y buruk dan z bagus, maka a adalah c α-predikat 7 =μ rendah (x)∩μ buruk (y)∩μ bagus (z) = min [μ rendah (x),μ buruk (y),μ bagus (z)] = min [μ rendah (87),μ buruk (81),μ bagus (91)] = min[0.05, 0. 22, 0.08] = 0.05 [r8] jika x rendah dan y buruk dan z jelek, maka a adalah c α-predikat 8 = μ rendah (x)∩μ buruk (y)∩μ jelek (z) = min[μ rendah (x),μ buruk (y),μ jelek (z)] = min[μ rendah (87),μ buruk (81),μ jelek (91)] = min[0.05, 0. 22, 0.08] = 0.05 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 47 kemudian dari [r1]-[r8] dilakukan komposisi dari untuk semua aturan, sehingga didapatkan gabungan himpunan fuzzy untuk semua aturan, yaitu: gambar 2. grafik gabungan dari hasil r1 – r8 3. defuzzifikasi defuzzifikasi adalah mengubah fuzzy output menjadi crisp value berdasarkan fungsi keanggotaan yang telah ditentukan. dalam hal ini akan digunakan metode defuzzifikasi centroid untuk masukan data yang berbentuk diskrit. dari gambar 2 diperoleh : perhitungan momen ∫ 0.05a da=0.025a2| 0 70 70 0 =122.5 ∫ a-70 8 a da= 1 24 a335 8 a2| 70 71 71 70 =4.42 ∫ 0.08a da=0.04a2| 73 81 79 71 =48 ∫ a-78 8 a da= 1 24 a339 8 a2| 79 86 86 79 =328.42 ∫ 0.78a da=0.39a2| 86 100 100 86 =1015.56 perhitungan luas daerah ∫ 0.05 da=0.05 a|0 70 70 0 =0.05(70 )=3.5 ∫ a-70 8 da= 1 16 a270 8 a | 70 71 71 70 =0.0625 ∫ 0.36 da=0.36a | 71 79 79 71 =2.88 ∫ a-78 8 da= 1 16 a278 8 a | 79 86 86 79 =3.9375 ∫ 0.64 da=0.64a | 86 100 100 86 =8.96 diperoleh titik pusat, a*= total momen total luas daerah = 122.5+4.42+48+328.42+1015.56 3.5+0.0625+2.88+3.9375+8.96 = 1518.893 19.34 =78.54 himpunan fuzzy a, himpunan fuzzy b dan himpunan fuzzy c sebagai berikut : μ a (78.54)=0.04, μ b (78.547)=0.96, dan μ c (78.54)=0 . karena μ b paling besar, maka dapat dikategorikan bahwa akreditasi smkn 1 rangkasbitung masuk akreditasi b. selain metode centroid, dibandingkan juga dengan mengigunakan metode defuzzifikasi cog (center of gravity) untuk masukan data yang berbentuk diskrit. dengan metode cog dan berdasari gambar 10 diperoleh: 𝑎∗ = ∑𝑎𝜇(𝑎) ∑𝜇(𝑎) = 191.75 2.3 = 83.37 himpunan fuzzy a, himpunan fuzzy b dan himpunan fuzzy c sebagai berikut : μ a (83.37)=0.67, μ b (83.37)=0.33, dan μ c (83.37)=0 . karena μ a paling besar, maka dapat dikategorikan bahwa akreditasi smkn 1 rangkasbitung masuk akreditasi a. 4. kesesuaian model selanjutnya, dengan membandingkan tingkat kesesuian antara data asli dari bap provinsi banten dengan keluaran model dilakukan perhitungandan diperoleh bahwa dari 275 program studi yang diakreditasi hanya 187 yang sesuai antara hasil dari bap 86 70 0.08 100 0.78 71 79 0.05 0.22 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 48 dengan keluaran model. jadi tingkat ketepatan model baru sekitar 68 %. 4. kesimpulan model yang diperoleh dapat digunakan dalam memprediksi hasil akreditasi suatu smk dengan hanya melihat dari nilai standar proses, nilai standar pendidik, dan nilai standar sarana. model yang dihasilkan memiliki tingkat ketepatan sekitar 68 %. 5. ucapan terimakasih ucapan terimakasih disampaikan penulis kepada badan akreditasi sekolah provinsi banten yang memberikan data hasil akreditasi smk provinsi banten selama periode 2009-2011. referensi [1] ban-sm. kebijakan akreditasi sekolah/ madrasah. badan akreditasi nasional sekolah/madrasah. (2009). [2] patil, a & codner, g. accreditation of engineering education: review, observations and proposal for global accreditation, european journal of engineering education 32(6) (2007) 639-651. [3] dongcheng hu. use of the daete model in evaluating continuing education institutions in china, international journal of continuing education & lifelong learning. 2 (2) (2010) 1-10. [4] jörg, ton. thinking in complexity about learning and education: a programmatic view, an international journal of complexity & education 6 (1) (2009) 1-22. [5] nykänen, o. inducing fuzzy models for student classification, journal of educational technology & society 9 (2) (2006) 223-234. [6] wang, li-xin. a course in fuzzy systems and control. prentice-hall international, inc. (1997). http://web.b.ebscohost.com/ehost/viewarticle?data=dgjymppp44rp2%2fdv0%2bnjisfk5ie45pfirqm3sbkk63nn5kx95uxxjl6sru6tqk5jspa0uq6uueizls5lporweezp33vy3%2b2g59q7ra%2bpr02yq7rlt6%2bkhn%2fk5vxj5kr84lpjgoac8nnls79mpnfsvbartk60qrbrpnztiuvx8lxk6%2bqe8tv2jaaa&hid=127 http://web.b.ebscohost.com/ehost/viewarticle?data=dgjymppp44rp2%2fdv0%2bnjisfk5ie45pfirqm3sbkk63nn5kx95uxxjl6sru6tqk5jspa0uq6uueizls5lporweezp33vy3%2b2g59q7ra%2bpr02yq7rlt6%2bkhn%2fk5vxj5kr84lpjgoac8nnls79mpnfsvbartk60qrbrpnztiuvx8lxk6%2bqe8tv2jaaa&hid=127 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 16 pemodelan data return saham pt. bank republik indonesia dengan self-exciting threshold autoregressive dan algoritma genetika maulida nurhidayati1 iain ponorogo1, nurhidayatimaulida@gmail.com1 doi:https://doi.org/10.15642/mantik.2018.4.1.16-21 abstrak model deret waktu nonlinier adalah model deret waktu yang diterapkan pada data-data yang memiliki pola nonlinier. salah satu model deret waktu nonlinier adalah model self-exciting threshold autoregressive (setar). model setar adalah model deret waktu yang pemodelan dari datanya dilakukan dengan membagi data menjadi beberapa regime dengan masing-masing regime mengikuti suatu model autoregressive (ar). pembagian regime tersebut didasarkan pada nilai delay dan threshold dari data itu sendiri. banyaknya parameter pada model setar mengakibatkan proses pencarian menghasilkan model setar yang belum optimum. berdasarkan temuan tersebut, pada penelitian ini digunakan algoritma genetika untuk menghasilkan model setar terbaik yang merupakan model setar yang optimum. pada penelitian ini dilakukan pemodelan data simulasi setar dan data return saham bank rakyat indonesia (bri). metode yang digunakan untuk memodelkan data tersebut adalah grid search (gs) dan algoritma genetika (ga). hasil analisis pada data simulasi setar menunjukkan bahwa metode ga memberikan hasil pemodelan yang lebih baik dibandingkan metode gs. nilai aic motode ga untuk jumlah data 200 sebesar 3,976178 yang lebih kecil dari aic metode gs sebesar 1,361723. untuk jumlah data 500 nilai aic metode ga juga lebih kecil dari aic metode gs. pada data return saham bri, metode ga juga memberikan hasil pemodelan lebih baik dibandingkan dengan gs. hal ini ditandai dengan nilai aic metode ga sebesar 11147,66 kurang dari -11146,26 yang merupakan aic metode gs. jadi, hasil analisis pada data simulasi model setar dan return saham bri menunjukkan bahwa metode ga memberikan hasil pemodelan yang lebih baik dibandingkan dengan metode gs berdasarkan nilai aic yang dihasilkan. kata kunci: nonlinier, setar, grid search, algoritma genetika, return saham abstract nonlinear time series model is a time series model applied to data that has the nonlinear pattern. one of the nonlinear time series models is self-exciting threshold autoregressive (setar). the setar model is a time series model that data modeling is done by dividing data into multiple regimes, whereas each regime following an autoregressive (ar) model. the division of the regime based on the score of the delay and threshold of the data itself. the number of setar model parameters not only resulted from the best model search process but also resulted in a setar model that is not yet optimum. based on these findings, this study used genetic algorithm (ga) to produce the best and optimum setar model. in this research, using setar simulation data modeling and return data of bank rakyat indonesia (bri) were performed. the method used to model the data is grid search (gs) and genetic algorithm (ga). the result of analysis of setar simulation data shows that ga method gives better modeling result than gs method. the ga motive aic value for the amount of 200 data is -3.976178 which is smaller than the aic gs method of 1.361723. for the amount of data of 500 aic values, ga method is also smaller than aic gs method. in bri stock return data, ga method also gives better modeling result compared to gs. it is marked by the ga aic method value of -11147.66 less than -11146.26 which is the aic method of gs. thus, the result of analysis of setar model simulation data and bri stock return shows that ga method gives better modeling result compared to gs method based on generated aic value. key words: nonlinear, setar, grid search, genetic algorithm, stock return jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 17 1. pendahuluan model dari data deret waktu secara umum terdiri dari model deret waktu linier dan model deret waktu nonlinier. model deret waktu linier adalah model deret waktu yang menggambarkan suatu data deret waktu yang membentuk suatu pola linier. sedangkan model deret waktu nonlinier adalah model deret waktu yang menggambarkan suatu data deret waktu yang membentuk pola tidak linier. pada data-data keuangan seperti data return saham dan inflasi biasanya menunjukkan fenomena kluster volatilitas, yaitu periode dimana data menunjukkan perubahan yang bergantian untuk periode yang panjang dan diikuti periode yang menunjukkan keadaan yang stabil [1]. keadaan ini membuat data menjadi tidak stabil dan pemodelan data tersebut mulai didekati dengan menggunakan model deret waktu nonlinier. kelebihan dari model ini dapat menangkap fenomena yang tidak dapat ditangkap oleh model deret waktu linier. salah satu model deret waktu nonlinier yang dapat digunakan adalah model self-exiting threshold autoregressive (setar). model setar adalah kasus khusus dari model threshold autoregressive (tar) ketika threshold yang digunakan diambil dari nilai lag pada data deret waktu itu sendiri[2]. prinsip dasar dari model setar adalah membagi data menjadi beberapa regime berdasarkan suatu nilai delay dan threshold. masing-masing regime yang terbentuk mengikuti suatu model autoregressive(ar) lokal. model setar terbaik yang diperoleh memiliki masalah dalam penentuan parameterparameternya. yaitu nilai threshold, delay, serta order ar untuk masing-masing regime. banyaknya parameter yang dicari mengakibatkan proses pencarian model terbaik terkadang menghasilkan model setar yang belum optimum. berdasarkan masalah tersebut pada tahun 2002, [3] menggunakan algoritma genetika (ga) untuk mengestimasi model threshold autoregressive dengan kesimpulan bahwa algoritma genetika secara simultan dapat memilih subset yang sesuai untuk parameter delay dan threshold pada model tar. hal ini dapat mengurangi waktu pencarian yang sangat besar dibandingkan dengan penggunaan metode estimasi tradisional. keuntungan lain dari penggunaan algoritma genetika ini terletak pada fleksibilitas untuk mendapatkan hasil yang paling dekat dengan cara evolusi [3]. algoritma genetika (genetic algorithm atau biasa disingkat ga) adalah suatu teknik pencarian berorientasi target yang biasa diterapkan pada proses optimasi mencari nilainilai ekstrim universal. pada algoritma genetika, teknik pencarian dilakukan sekaligus atas sejumlah solusi yang dikenal dengan populasi. operasi-operasi yang digunakan pada algoritma genetika antara lain: seleksi, crossover dan mutasi. algoritma genetika menggunakan prosedur pencarian yang didasarkan pada nilai fungsi tujuan, tidak ada pemakaian gradien atau teknik kalkulus. algoritma genetika memiliki beberapa keunggulan dalam proses optimasi antara lain: (1) algoritma bekerja pada kumpulan solusi; (2) algoritma genetika mencari berdasarkan populasi dari solusi, bukan hanya satu solusi; dan (3) algoritma genetika menggunakan informasi fitness yang ingin dicari, bukan turunan atau pengetahuan khusus lainnya [4]. berdasarkan temuan yang telah dipaparkan diatas, pada penelitian ini disajikan pemodelan data simulasi model setar dan data return saham bank rakyat indonesia (bri) dengan menggunakan model self-exciting threshold autoregressive dengan metode untuk pemodelannya menggunakan algoritma genetika (ga) dan grid search (gs). 2. tinjauan pustaka 2.1 proses deret waktu proses deret waktu dapat dituliskan sebagai berikut , , , 0 0 0 0 0 0 ... t i t i i j t i t j i j k t i t j t k i i j i j k z a a a a a a         − − − − − − = = = = = = = + + +   (1) jika proses t z pada persamaan (1) hanya terdiri dari suku pertama maka proses yang terjadi adalah proses linier. jika proses t z pada persamaan (1) terdiri dari suku pertama dan suku lain maka proses yang terjadi merupakan suatu proses nonlinier [5]. 2.2 model threshold autoregressive model threshold autoregressive (tar) pertama kali diusulkan oleh tong dan lim pada tahun 1980 [6]. tar adalah alternatif model untuk mendeskripsikan deret waktu periodik. model ini dimotivasi oleh beberapa karakteristik nonlinier yang biasa ditemukan dalam kehidupan sehari-hari seperti adanya asimetri jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 18 dalam pola turun dan naik suatu proses, fenomena lompatan, serta frekuensi ketergantungan amplitudo yang tidak dapat ditangkap oleh model deret waktu linier[6]. model tar mengunakan threshold untuk meningkatkan pendekatan linier pada modelnya. model tar dapat dituliskan sebagai berikut 0 0 1 1 p p t d t i t i i t i t i i z z z z i a       − − − = =   −  = + + + +         (2) dengan t z adalah data deret waktu, d adalah parameter delay,  adalah parameter lokasi (threshold),  adalah parameter skala, dan (.)i adalah fungsi penghalus yang dapat berupa suatu fungsi logistik, eksponensial, maupun suatu fungsi indikator [5]. 2.3 model self-exciting threshold autoregressive model self-exciting threshold autoregressive (setar) adalah kasus khusus dari model threshold autoregressive (tar) ketika threshold yang digunakan diambil dari nilai lag pada data deret waktu itu sendiri [2]. suatu proses mengikuti suatu proses setar jika mengikuti model deret waktu berikut 0, , , 1 ; jp t j i j t i t j t jd i a jika rz z z  − − = = + +  (3) dengan 1, 2,...,j k= , d adalah bilangan bulat positif dan merupakan parameter delay, ,t j a adalah barisan peubah acak yang identik, independen, dan mengikuti distribusi tertentu dengan mean nol dan varian [7]. berdasarkan persamaan (3) dapat dibentuk model 1 2 (2, , )setar p p yang merupakan model setar dengan 2 regime dimana 1 p menunjukkan order ar pada regime bawah sedangkan 2 p menunjukkan order ar pada regime atas. bentuk persamaan model 1 2 (2, , )setar p p dapat dilihat pada persamaan (4) dengan d adalah delay dan 1 r adalah threshold. 1 2 0,1 ,1 ,1 1 1 0,2 ,2 ,2 1 1 , , p i t i t t d i t p i t i t t d i z a jika z r z z a jika z r     − − = − − =  + +   =   + +     (4) 2.4 kriteria pemilihan model terbaik akaike’s information criteria(aic) adalah suatu kriteria pemilihan model terbaik yang diperkenalkan oleh akaike dengan mempertimbangkan banyaknya parameter dalam model. semakin kecil nilai aic yang diperoleh semakin baik model yang digunakan. kriteria aic dapat dirumuskan sebagai berikut  ( ) 2 ln 2 ln 2 aic m maksimumlikelihood m sse n m n = − +   = +    (5) dengan m adalah banyak parameter dalam model [5]. 2.5 algoritma genetika (ga) algoritma genetika pertama kali dikembangkan oleh john holland dari universitas michigan pada tahun 1975. john holland mengatakan bahwa setiap masalah yang berbentuk adaptasi (alami maupun buatan) dapat diformulasikan dalam terminologi genetika. algoritma genetika adalah simulasi dari proses evolusi darwin dan operasi genetika atas kromosom [8]. algoritma genetika masuk dalam kelompok evolutionary algorithm. algoritma genetika didasarkan pada genetika dan seleksi alam. elemen-elemen ini yang dipakai dalam prosedur algoritma genetika. algoritma genetika banyak dipakai dalam menyelesaikan masalah kombinatorial seperti tsp (traveling salesman problem), crew scheduling untuk airline hingga masalah kontrol. algoritma genetika merupakan temuan penting dalam bidang optimasi, dimana suatu algoritma diciptakan dengan meniru mekanisme evolusi dalam perkembangan makluk hidup. dalam algoritma genetika prosedur pencarian hanya didasarkan pada nilai fungsi tujuan, tidak ada pemakaian gradient atau teknik kalkulus [9]. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 19 3. metode penelitian langkah-langkah analisis dalam penelitian ini dijabarkan dalam 2 tahap. tahap pertama adalah studi simulasi model setar. pada tahap ini, pencarian model setar terbaik dilakukan menggunakan metode grid search (gs) dan algoritma genetika (ga). untuk tahap kedua adalah mengaplikasikan metode gs dan algoritma genetika untuk pemodelan data return saham bank rakyat indonesia. data return saham yang digunakan dari 3 januari 2011 sampai dengan 30 desember 2016: 4. hasil dan pembahasan 4.1 simulasi model setar studi simulasi model setar dilakukan untuk membangkitkan model setar. nilai parameter model setar serta threshold yang digunakan dapat dilihat pada persamaan (6). model setar yang digunakan merupakan model setar dengan delay=1 dan merupakan model yang simulasikan oleh barogona, battaglia dan cucina [10]. banyaknya data yang dibangkitkan 200 dan 500. 1 2 ,1 1 3 ,2 1 1, 2 0, 7 jika 0 0, 8 jika 0 t t t t t t t t z z a z z z a z − − − − − − − +  =  +  (6) sebelum melakukan pemodelan dengan algoritma genetika perlu didefinisian fitness yang digunakan. hal ini dikarenakan nilai fitness yang sangat mempengaruhi hasil yang akan diperoleh. selain fitness, parameterparameter seperti peluang crossover, mutasi, banyak generasi, dan panjang kromosom yang digunakan juga harus ditentukan. nilai dari masing-masing parameter yang digunakan pada pemodelan dengan algoritma genetika dapat dilihat pada tabel 1. tabel 1. parameter algoritma genetika (ga) untuk model setar parameter nilai popsize 200 pc 0,8 pm 0,1 maksimum iterasi 200 fitness (aic) pemodelan data dengan setar dilakukan dengan beberapa tahapan. tahapan pertama adalah melakukan uji nonlinieritas terasvirta. uji ini dilakukan sebagai syarat suatu data dapat dianalisis dengan menggunakan model setar. tabel 2. uji nonlinieritas terasvirta dan white simulasi model subset setar n terasvirta f p-value 200 13,6304 2,856e-06 500 40,3998 2,2e-16 berdasarkan hasil yang ditunjukkan pada tabel 2 diketahui bahwa hasil simulasi model setar memenuhi sifat nonlinieritas karena nilai pada model tersebut lebih dari 3,04175. gambar 1. plot acf dan pacf data simulasi setelah diketahui bahwa data memiliki sifat nonlinier, langkah selanjutnya adalah menentukan orde maksimul dari model. orde maksimum dapat dilihat dari hasil plot pacf yang dibuat. plot pacf pada gambar 1 menunjukkan bahwa data cuts off pada lag 3 sehingga pendefinisian orde awal untuk identifikasi model dilakukan dengan mengambil nilai d adalah 1, 2, atau 3 dengan 1 2 3dan 3p p= = tabel 3. identifikasi model setar dengan metode gs dan ga n metode d p1 p2 threshold aic 200 gs 1 2 3 0,062794647 1,361723 ga 1 2 [3] 0,06279465 -3,976178 500 gs 1 2 3 0,01980077 40,34689 ga 1 2 [3] 0,01980077 33,7398 hasil identifikasi model setar dengan metode gs dan ga ditunjukkan pada tabel 3. berdasarkan hasil yang ditunjukkan pada tabel 3, metode gs dan ga memiliki hasil yang sama jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 20 untuk orde 1 , ,d p dan threshold. tetapi terdapat perbedaan untuk orde 2 p yang dihasilkan. metode gs memberikan hasil 2 3p = sedangkan metode ga memberikan hasil 2 [3]p = . berdasarkan hasil tersebut, metode ga memberikan hasil identifikasi yang lebih sesuai karena model yang terbentuk sesuai dengan model yang disimulasikan serta nilai aic yang dihasilkan dengan ga lebih kecil dibandingkan dengan metode gs. 4.2 aplikasi pada data return saham sebelum melakukan pemodelan pada data return saham bri perlu dilakukan pengujian nonlinieritas dari. pengujian ini dilakukan untuk membuktikan secara statistik bahwa data yang dianalisis benar merupakan suatu data yang memiliki sifat nonlinier. pengujian dilakukan dengan menggunakan metode terasvirta dan diperoleh hasil sebesar 9,5459 dengan p-value = 0,008456. karena p-value=0,008456<0,05 dapat dibuat suatu kesimpulan bahwa data return saham bri mengikuti sifat nonlinier sehingga medel setar dapat diterapkan pada data return saham tersebut. gambar 2. plot deret waktu dan pacf data return saham bri gambar 2 menunjukkan plot deret waktu dan plot pacf. plot pacf yang terbentuk menunjukkan bahwa data cuts off pada lag 4 sehingga pemodelan data return saham selanjutnya didekati dengan menggunakan orde tertinggi adalah 4. berdasarkan hal tersebut, pendefinisian orde untuk identifikasi model dilakukan dengan mengambil nilai d adalah 1, 2, 3, atau 4 dengan 1 2 4 dan 4p p= = . tabel 4. identifikasi data return saham dengan metode gs dan ga metode d p1 p2 threshold aic ga 2 2 [4] 0,006741573 -11147,66 metode d p1 p2 threshold aic gs 2 2 4 0,006667 -11146,26 tabel 4 memberikan hasil identifikasi model setar pada data return saham bri. hasil yang diperoleh menunjukkan nilai aic dari metode ga lebih kecil dibandingkan dengan metode gs. berdasarkan hasil tersebut, selanjutnya dilakukan estimasi pada model dan diperoleh model setar dengan identifikasi model dengan menggunakan algoritma genetika sebagai persamaan (7) berikut 4 ,1 2 1 3 ,2 2 0, 0008396 0, 07369 0, 00291 0,1883 >0,006741573 0,000, 08637 6741573 t t t t t t t t z a jika z z z z a jika z − − − − − − − +  =   − + +− model terbaik pada persamaan (7) menunjukkan bahwa suatu data return saham saat ini akan masuk regime 1 ketika return saham pada 2 hari sebelumnya kurang dari 0,006741573 atau dapat ditulisakan sebagai berikut 2 0,006741573 t z −  . apabila data return saham pada 2 hari sebelumnya lebih dari 0,006741573 maka data return saham saat ini masuk pada regime 2. nilai forecast untuk data pada regime 2 dipengaruhi oleh data return saham 1 hari sebelumnya dan 2 hari sebelumnya masingmasing sebesar 0,07369 dan -0,08637. artinya return saham pada regime 1 akan mengalami kenaikan sebesar 0,07369 rupiah ketika data return saham pada 1 hari sebelumnya naik sebesar 1 rupiah dengan asumsi bahwa data return saham pada 2 hari sebelumnya adalah tetap. nilai forecast untuk data pada regime 1 dipengaruhi oleh data return saham 4 hari sebelumnya sebesar -0,1883. artinya return saham pada regime 2 akan mengalami penurunan sebesar 0,1883 rupiah ketika data return saham pada 1 hari sebelumnya naik sebesar 1 rupiah dengan asumsi bahwa faktor yang lain tetap. 5. simpulan dan saran pemodelan data simulasi setar yang dilakukan dengan algoritma genetika (ga) memberikan hasil pemodelan yang lebih baik dibandingkan dengan metode grid search (gs). metode ga memberikan nilai aic yang lebih kecil dibandingkan metode gs dengan hasil identifikasi yang lebih sesuai dengan simulasi jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 21 model yang dibangkitkan. pemodelan data return saham bri dengan metode ga juga memberikan hasil aic lebih kecil dibandingkan dengan menggunakan metode gs. pada penelitian ini dibatasi regime yang digunakan adalah 2 sehingga pada penelitian selanjutnya dapat digunakan regime lebih dari 2. selain itu, dapat juga dicoba menggunakan model time series nonlinier lain seperti logistic smooth transtition autoregressive (lstar) maupun exponential smooth transtition autoregressive (estar). referensi [1] d. n. gujarati dan d. c. porter, dasar-dasar ekonometrika, 5 ed. jakarta: salemba empat, 2013. [2] p. h. franses dan d. van dijk, nonlinear time series models in empirical finance. new york: cambridge university press, 2003. [3] b. wu dan c.-l. chang, “using genetic algorithms to parameters (d;r) estimation for threshold autoregressive models,” comput. stat. data anal., vol. 38, hlm. 315–330, 2002. [4] m. sawaka, genetic algoritms and fuzzy multiobjective optimization. boston: kluwer academic publishers, 2002. [5] w. w. s. wei, time series analysis univariate and multivariate methods. new york: pearson, 2006. [6] h. tong dan k. s. lim, “threshold autoregression, limit cycles and cyclical data,” j. r. stat. soc. ser. b methodol., vol. 42, no. 3, hlm. 245–292, 1980. [7] r. s. tsay, “testing and modeling threshold autoregressive process,” j. am. stat. assoc., vol. 84, no. 405, hlm. 231–240. [8] d. sri kusuma dan h. purnomo, penyelesaian masalah optimasi dengan teknik-teknik heuristik. yogyakarta: graha ilmu, 2005. [9] b. santosa dan p. willy, metode metaheuristik, konsep dan implementasi. surabaya: guna widya, 2011. [10] r. barogana, f. battaglia, dan d. cucina, “estimating threshold subset autoregressive moving-average models by genetic algorithms,” metron int. j. stat., vol. lxii, hlm. 39–61, 2004. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 87 pelabelan harmonis ganjil pada kelas graf baru hasil operasi cartesian product fery firmansah1, muhammad ridlo yuwono2 pend. matematika, universitas widya dharma klaten1, fery.firmansah004@gmail.com1 pend. matematika, universitas widya dharma klaten2, ridloyuwono90@gmail.com2 doi:https://doi.org/10.15642/mantik.2017.3.2.87-95 abstrak kelas graf yang mempunyai sifat pelabelan harmonis ganjil disebut sebagai graf harmonis ganjil. graf jaring adalah graf yang diperoleh dari operasi cartesian product dari dua graf lintasan. kontruksi graf ular jaring terinspirasi dari definisi graf ular dengan mengganti graf lingkaran dengan graf jaring. pada makalah ini akan ditunjukkan bahwa graf ular jaring memenuhi sifat pelabelan harmonis ganjil sedemikian sehingga graf ular jaring adalah graf harmonis ganjil. pada bagian akhir akan ditunjukkan juga bahwa gabungan graf ular jaring merupakan graf harmonis ganjil. kata kunci: cartesian product, gabungan graf, graf harmonis ganjil, graf ular jaring abstract graph class which has the characteristic of odd harmonious labeling is called as odd harmonious graph. net graph is a graph which is gained by using operation cartesian product of two line graphs. the construction of snake-net graph is inspired by the definition of snake graph replacing the round graph to net graph. in this paper, the study will show that snake-net graph fulfill the characteristic of odd harmonious graph in such a way snake-net graph is the odd harmonious graph. in the end of this paper, it is also shown that the union of snake-net graph is also called as the odd harmonious graph. keywords: cartesian product, union graph, odd harmonious graph, snake-net graph 1. pendahuluan pelabelan graf merupakan salah satu topik dari matematika kombinatorik yang berkembang sangat pesat akhir-akhir ini. selain berkembang secara teori, pelabelan graf juga telah banyak diaplikasikan dalam berbagai bidang keilmuan. pada mulanya pelabelan graf diperkenalkan oleh sedlacek tahun 1963 yang terinspirasi dari kasus bujur sangkar ajaib yang disebut pelabelan ajaib. rosa tahun 1970, menulis makalah tentang pelabelan graceful yang digunakan untuk menyelesaikan kasus dekomposisi dari graf lengkap. sampai tahun 2016, hasil tentang penelitian pelabelan graf sudah banyak dipubliksikan dalam berbagai jurnal dan makalah yang dikumpulkan dan diperbaharui secara teratur oleh gallian. gallian (2016) telah merangkum kurang lebih 2000 jurnal pelabelan graf dari seluruh peneliti dunia dan dari 2000 jurnal tersebut diperoleh kurang lebih 200 kelas graf baru beserta pelabelannya. selain pelabelan ajaib dan pelabelan graceful, masih terdapat terdapat jenis pelabelan yang lain diantaranya adalah pelabelan harmonis yang mailto:fery.firmansah004@gmail.com mailto:ridloyuwono90@gmail.com jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 88 diperkenalkan oleh graham dan sloane pada tahun 1980 dan pelabelan harmonis ganjil yang diperkenalkan oleh liang dan bai pada tahun 2009. liang dan bai [10] telah menunjukkan sifat-sifat khusus yang dimiliki oleh graf harmonis ganjil, serta beberapa kelas graf yang memenuhi pelabelan harmonis ganjil. hasil penelitian yang relevan dengan penelitian ini diantaranya dapat dilihat di abdel-aal [1]; alyani, et. al. [2]; firmansah & sugeng [3]; firmansah [4]; firmansah & syaifuddin [5]; firmansah & yuwono [6]; jeyanthi dan philo [7], jeyanthi, et. al. [8], saputri et. al. [11]; dan vaidya & shah [12]. pada makalah ini akan diberikan kontruksi kelas graf baru hasil operasi cartesian product dari graf lintasan 𝑃𝑚 dan 𝑃𝑛 yaitu graf jaring 𝐿𝑚,𝑛 = 𝑃𝑚 × 𝑃𝑛. untuk 𝑚 = 𝑛 = 3 diperoleh graf jaring 𝐿3,3 yang akan membentuk graf ular jaring 𝑘𝐿3,3. selanjutnya akan dibuktikan bahwa graf ular jaring 𝑘𝐿3,3 memenuhi sifat-sifat pelabelan harmonis ganjil sehingga graf ular jaring 𝑘𝐿3,3 adalah keluarga baru dari graf harmonis ganjil. dibagian akhir akan ditunjukkan bahwa gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 juga merupakan keluarga dari graf harmonis ganjil. 2. kajian teori misalkan ℤ𝑛 adalah himpunan bilangan bulat modulo 𝑛. simbol [𝑎,𝑏] didefinsikan oleh {𝑥|𝑥 ∈ ℤ,𝑎 ≤ 𝑥 ≤ 𝑏} dan [𝑎,𝑏]𝑑 didefinsikan oleh {𝑥|𝑥 ∈ ℤ,𝑎 ≤ 𝑥 ≤ 𝑏,𝑥 ≡ 𝑎(𝑚𝑜𝑑 𝑑)}. graf 𝐺(𝑝,𝑞) adalah graf 𝐺 dengan order 𝑝 = |𝑉(𝐺)| simpul dan size 𝑞 = |𝐸(𝐺)| busur. definisi 2.1 [9] graf 𝐺(𝑝,𝑞) dikatakan graf harmonis jika terdapat fungsi injektif 𝑓:𝑉(𝐺) → 𝑍𝑞 sedemikian sehingga menginduksi 𝑓∗:𝐸(𝐺) → ℤ𝑞 yang didefinisikan oleh 𝑓∗(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣))(𝑚𝑜𝑑 𝑞) yang bersifat bijektif, dan 𝑓 adalah pelabelan harmonis dari graf 𝐺(𝑝,𝑞). definisi 2.2 [10] graf 𝐺(𝑝,𝑞) dikatakan graf harmonis ganjil jika terdapat fungsi injektif 𝑓:𝑉(𝐺) → [0,2𝑞 − 1] = {0,1,2,3,…,2𝑞 − 1} sedemikian sehingga menginduksi fungsi 𝑓∗:𝐸(𝐺) → [1,2𝑞 − 1]2 = {1,3,5,7,…,2𝑞 − 1} yang didefinisikan oleh 𝑓∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) yang bersifat bijektif, dan 𝑓 adalah pelabelan harmonis ganjil dari graf 𝐺(𝑝,𝑞). definisi 2.3 [10] graf ular 𝑘𝐶4 dengan 𝑘 ≥ 1 adalah graf terhubung dengan 𝑘 blok yang memiliki titik potong blok berupa lintasan dan setiap 𝑘 blok isomorfik dengan graf ular 𝐶4 3. hasil penelitian dan pembahasan 3.1 pelabelan harmonis ganjil pada graf ular jaring pada bagian ini diberikan definisi dari graf ular jaring 𝑘𝐿3,3 yang terinspirasi dari graf ular 𝑘𝐶4 yaitu dengan mengganti graf lingkaran 𝐶4 dengan graf jaring 𝐿3,3. definisi 3.1 graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf terhubung dengan 𝑘 blok yang memiliki titik potong blok berupa lintasan dan setiap 𝑘 blok isomorfik dengan graf jaring 𝐿3,3. berikut pada gambar 1 diberikan notasi simpul, notasi busur dan kontruksi dari graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 89 0 u 1 u 2 u k u 2 1 1 v 2 1 v 1 2 v 2 2 v 1 2 k v 2 2 k v 1 1 w 2 1 w 1 k w 2 k w 12 k u 1 12 k v 2 12 k v 22 k u gambar 1 notasi simpul, notasi busur dan kontruksi dari graf ular jaring 𝒌𝑳𝟑,𝟑 berdasarkan kontruksi pada gambar 1 diperoleh definisi himpunan simpul dan himpunan busur dari graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 sebagai berikut: 𝑉(𝑘𝐿3,3) = {𝑢𝑖|0 ≤ 𝑖 ≤ 2𝑘} ∪ {𝑣𝑖 𝑗|1 ≤ 𝑖 ≤ 2𝑘, 𝑗 = 1,2} ∪ {𝑤𝑖 𝑗|1 ≤ 𝑖 ≤ 𝑘, 𝑗 = 1,2} dan 𝐸(𝑘𝐿3,3) = {𝑢𝑖𝑣(𝑖+1) 𝑗|0 ≤ 𝑖 ≤ 2𝑘 − 1, 𝑗 = 1,2} ∪ {𝑣𝑖 𝑗𝑢𝑖|1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2} ∪ {𝑣(2𝑖−1) 𝑗𝑤 𝑖 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2} ∪ {𝑤𝑖 𝑗𝑣(2𝑖) 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2} selanjutnya diberikan sifat yang menyatakan bahwa graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil yang dinyatakan dalam teorema 3.2. teorema 3.2 graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil. bukti. berdasarkan himpunan himpunan simpul dan himpunan busur dari graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 diperoleh order 𝑝 = |𝑉(𝑘𝐿3,3)| = 9𝑘 − 1 dan size 𝑞 = |𝐸(𝑘𝐿3,3)| = 12𝑘. berikut didefinisikan fungsi pelabelan simpul 𝑓: 𝑉(𝑘𝐿3,3) → {0,1,2,3…,24𝑘 − 1} 𝑓(𝑢𝑖) = 4𝑖,0 ≤ 𝑖 ≤ 2𝑘 (1.1) 𝑓(𝑣𝑖 𝑗) = 4𝑖 + 2𝑗 − 5,1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (1.2) 𝑓(𝑤𝑖 𝑗) = 24𝑘 − 16𝑖 − 4𝑗 + 14, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (1.3) berdasarkan fungsi pelabelan simpul 𝑓 pada persamaan (1.1), (1.2), dan (1.3) diperoleh himpunan simpul 𝑓(𝑉(𝑘𝐿3,3)) setelah dilabel sebagai berikut. 𝑓(𝑉(𝑘𝐿3,3)) = {0,4,8,12,…,8𝑘} ∪ {1,5,9,…,8𝑘 − 3,3,7,11,…,8𝑘 − 1} ∪ { 24𝑘 − 6,24𝑘 − 22,24𝑘 − 38,…,8𝑘 + 10, 24𝑘 − 10,24𝑘 − 26,24𝑘 − 42,…,8𝑘 + 6 } = {0,1,3,4,5,7,8,9,11,…,8𝑘 − 3,8𝑘 − 1,8𝑘,8𝑘 + 6,8𝑘 + 10,…,24𝑘 − 10,24𝑘 − 6} = {0,1,3,4,…,24𝑘 − 6} terlihat bahwa fungsi pelabelan simpul 𝑓 memberikan label yang berbeda pada setiap simpul dan memenuhi 𝑓(𝑉(𝑘𝐿3,3)) = {0,1,3,4,…,24𝑘 − 6} ⊆ {0,1,2,3…,24𝑘 − 1} sedemikian sehingga fungsi pelabelan simpul 𝑓 memenuhi sifat pemetaan injektif. langkah selanjutnya akan ditunjukan bahwa fungsi pelabelan busur 𝑓∗ memenuhi sifat pemetaan bijektif. berdasarkan definisi 𝑓∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) maka fungsi pelabelan simpul 𝑓 akan menginduksi pelabelan busur 𝑓∗:𝐸(𝑘𝐿3,3) → {1,3,5,7,…,24𝑘 − 1}, sehingga didapatkan kontruksi fungsi pelabelan busur 𝑓∗ sebagai berikut: 𝑓∗(𝑢𝑖𝑣(𝑖+1) 𝑗) = 8𝑖 + 2𝑗 − 1, 0 ≤ 𝑖 ≤ 2𝑘 − 1,𝑗 = 1,2 (1.4) 𝑓∗(𝑣𝑖 𝑗𝑢𝑖) = 8𝑖 + 2𝑗 − 5, 1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (1.5) 𝑓∗(𝑣(2𝑖−1) 𝑗𝑤 𝑖 𝑗 ) = 24𝑘 − 8𝑖 − 2𝑗 + 5, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (1.6) 𝑓∗ (𝑤𝑖 𝑗𝑣(2𝑖) 𝑗 ) = 24𝑘 − 8𝑖 − 2𝑗 + 9, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (1.7) berdasarkan fungsi pelabelan busur 𝑓∗ pada persamaan (1.4), (1.5), (1.6) dan (1.7) diperoleh himpunan busur 𝑓∗ (𝐸(𝑘𝐿3,3)) setelah dilabel sebagai berikut; jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 90 𝑓∗ (𝐸(𝑘𝐿3,3)) = {1,9,17,25,…,16𝑘 − 7,3,11,19,27,…,16𝑘 − 5 } ∪ {5,13,21,29,…,16𝑘 − 3,7,15,23,31,…,16𝑘 − 1} ∪ { 24𝑘 − 5,24𝑘 − 13,24𝑘 − 21,24𝑘 − 29,…,16𝑘 + 3, 24𝑘 − 7,24𝑘 − 15,24𝑘 − 23,24𝑘 − 31,…,16𝑘 + 1 } ∪ { 24𝑘 − 1,24𝑘 − 9,24𝑘 − 17,24𝑘 − 25,…,16𝑘 + 7, 24𝑘 − 3,24𝑘 − 11,24𝑘 − 19,24𝑘 − 27,…,16𝑘 + 5 } = { 1,3,5,7,9,13,15,17,19,21,23,25,27,29,…,16𝑘 − 7,16𝑘 − 5,16𝑘 − 3,16𝑘 − 1 16𝑘 + 1,16𝑘 + 3,6𝑘 + 5,16𝑘 + 7,…,24𝑘 − 31,24𝑘 − 29,24𝑘 − 27,24𝑘 − 25, 24𝑘 − 23,24𝑘 − 21,24𝑘 − 19,24𝑘 − 17,24𝑘 − 15,24𝑘 − 13,24𝑘 − 11,24𝑘 − 9, 24𝑘 − 7,24𝑘 − 5,24𝑘 − 3,24𝑘 − 1 } = {1,3,5,7,…,24𝑘 − 1 }. terlihat bahwa fungsi pelabelan busur 𝑓∗ memberikan label yang berbeda pada setiap busur dan 𝑓∗ (𝐸(𝑘𝐿3,3)) = {1,3,5,7,…,24𝑘 − 1} sedemikian sehingga fungsi pelabelan busur 𝑓∗ memenuhi sifat pemetaan bijektif. akibatnya graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil. ∎ contoh 3.3 diberikan contoh pelabelan harmonis ganjil pada graf ular jaring 7𝐿3,3 pada gambar 2. 0 1 3 5 7 9 11 13 15 17 19 21 23 4 8 12 16 20 24 162 146 130 126142158 56 55 53 51 49 47 45 43 41 39 37 35 33 52 48 44 40 36 32 62 78 94 988266 28 2527 2931 110 114 gambar 2 pelabelan harmonis ganjil pada graf ular jaring 𝟕𝑳𝟑,𝟑 3.2 pelabelan harmonis ganjil pad gabungan graf ular jaring hasil penelitian selanjutnya adalah pelabelan harmonis ganjil pada gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1. berikut diberikan definisi dari gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 yang dinyatakan dalam definisi 3.4. definisi 3.4 graf 3,33,3 klkl  dengan 1k adalah gabungan dua graf ular jaring 3,3kl dengan 1k . notasi simpul, notasi busur dan kontruksi dari gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 diberikan pada gambar 4 sebagai berikut. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 91 0 u 1 u 2 u k u 2 1 1 v 2 1 v 1 2 v 2 2 v 1 2 k v 2 2 k v 1 1 w 2 1 w 1 k w 2 k w 12 k u 1 12 k v 2 12 k v 22 k u     3,33,3 lklk  0 x 1 x 2 x k x 2 1 1 y 2 1 y 1 2 y 2 2 y 1 2 k y 2 2 k y 1 1 z 2 1 z 1 k z 2 k z 12 k x 1 12 k y 2 12 k y 22 k x gambar 3 notasi simpul, notasi busur dan kontruksi dari gabungan graf ular jaring 𝒌𝑳𝟑,𝟑 ∪ 𝒌𝑳𝟑,𝟑 berdasarkan kontruksi pada gambar 4, diperoleh definisi himpunan simpul dan himpunan busur dari gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 sebagai berikut. 𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3) = {𝑢𝑖|0 ≤ 𝑖 ≤ 2𝑘} ∪ {𝑣𝑖 𝑗|1 ≤ 𝑖 ≤ 2𝑘, 𝑗 = 1,2} ∪ {𝑤𝑖 𝑗|1 ≤ 𝑖 ≤ 𝑘, 𝑗 = 1,2} ∪ {𝑥𝑖|0 ≤ 𝑖 ≤ 2𝑘} ∪ {𝑦𝑖 𝑗|1 ≤ 𝑖 ≤ 2𝑘, 𝑗 = 1,2} ∪ {𝑧𝑖 𝑗|1 ≤ 𝑖 ≤ 𝑘, 𝑗 = 1,2} dan 𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3) = {𝑢𝑖𝑣(𝑖+1) 𝑗|0 ≤ 𝑖 ≤ 2𝑘 − 1, 𝑗 = 1,2} ∪ {𝑣𝑖 𝑗𝑢𝑖|1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2} ∪ {𝑣(2𝑖−1) 𝑗𝑤 𝑖 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2} ∪ {𝑤𝑖 𝑗𝑣 (2𝑖) 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2} ∪ {𝑥𝑖𝑦(𝑖+1) 𝑗|0 ≤ 𝑖 ≤ 2𝑘 − 1, 𝑗 = 1,2} ∪ {𝑦𝑖 𝑗𝑥𝑖|1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2} ∪ {𝑦(2𝑖−1) 𝑗𝑧 𝑖 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2} ∪ {𝑧𝑖 𝑗𝑦 (2𝑖) 𝑗 |1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2}. pada teorema 3.2 diperoleh bahwa graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil, maka berikut diberikan sifat yang menyatakan bahwa gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 memenuhi fungsi pelabelan harmonis ganjil sedemikian sehingga gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil. teorema 3.5 gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil. bukti. berdasarkan himpunan himpunan simpul dan himpunan busur dari gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 diperoleh order 𝑝 = |𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)| = 18𝑘 − 2 dan size 𝑞 = |𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)| = 24𝑘. berikut didefinisikan fungsi pelabelan simpul 𝑓: 𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3) → {0,1,2,3…,48𝑘 − 1} 𝑓(𝑢𝑖) = 4𝑖,0 ≤ 𝑖 ≤ 2𝑘 (2.1) 𝑓(𝑣𝑖 𝑗) = 4𝑖 + 2𝑗 − 5,1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (2.2) 𝑓(𝑤𝑖 𝑗) = 24𝑘 − 16𝑖 − 4𝑗 + 14, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.3) 𝑓(𝑥𝑖) = 4𝑖 + 2,0 ≤ 𝑖 ≤ 2𝑘 (2.4) 𝑓(𝑦𝑖 𝑗) = 24𝑘 + 4𝑖 + 2𝑗 − 7, 1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (2.5) 𝑓(𝑧𝑖 𝑗) = 24𝑘 − 16𝑖 − 4𝑗 + 16, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.6) berdasarkan fungsi pelabelan simpul 𝑓 pada persamaan (2.1), (2.2), (2.3), (2.4), (2.5), dan (2.6) jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 92 diperoleh himpunan simpul 𝑓(𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) setelah dilabel sebagai berikut. 𝑓(𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) = {0,4,8,12,…,8𝑘} ∪ {1,5,9,…,8𝑘 − 3,3,7,11,…,8𝑘 − 1} ∪ { 24𝑘 − 6,24𝑘 − 22,24𝑘 − 38,…,8𝑘 + 10, 24𝑘 − 10,24𝑘 − 26,24𝑘 − 42,…,8𝑘 + 6 } ∪ {2,6,10,14,…,8𝑘 + 2} ∪ { 24𝑘 − 1,24𝑘 + 3,24𝑘 + 7,24𝑘 + 11,…,32𝑘 − 5 24𝑘 + 1,24𝑘 + 5,24𝑘 + 9,24𝑘 + 13,…,32𝑘 − 3 } ∪ { 24𝑘 − 4,24𝑘 − 20,24𝑘 − 36,24𝑘 − 52,…,8𝑘 + 12 24𝑘 − 8,24𝑘 − 24,24𝑘 − 40,24𝑘 − 56,…,8𝑘 + 8 } = { 0,1,2,3,4,5,6,7,8,9,…,8𝑘 − 3,8𝑘 − 1,8𝑘,8𝑘 + 2,8𝑘 + 6,8𝑘 + 8,8𝑘 + 10,…, 24𝑘 − 10,24𝑘 − 8,24𝑘 − 6,24𝑘 − 4,24𝑘 − 1,24𝑘 + 1,…,32𝑘 − 5,32𝑘 − 3 } = {0,1,2,3,4,…,32𝑘 − 3} terlihat bahwa fungsi 𝑓 memberikan label yang berbeda pada setiap simpul dan 𝑓(𝑉(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) = {0,1,2,3,4,…,32𝑘 − 3} ⊆ {0,1,2,3…,48𝑘 − 1} sedemikian sehingga fungsi pelabelan simpul 𝑓 memenuhi sifat pemetaan injektif. setelah menunjukan fungsi pelabelan simpul 𝑓 memenuhi pemetaan injektif, selanjutnya akan ditunjukan bahwa fungsi pelabelan busur 𝑓∗ memenuhi pemetaan bijektif. fungsi pelabelan 𝑓 akan menginduksi pelabelan 𝑓∗:𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3) → {1,3,5,7,…,48𝑘 − 1} yang diperoleh dari 𝑓∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣), sehingga didapatkan fungsi pelabelan busur sebagai berikut: 𝑓∗(𝑢𝑖𝑣(𝑖+1) 𝑗) = 8𝑖 + 2𝑗 − 1, 0 ≤ 𝑖 ≤ 2𝑘 − 1,𝑗 = 1,2 (2.7) 𝑓∗(𝑣𝑖 𝑗𝑢𝑖) = 8𝑖 + 2𝑗 − 5, 1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (2.8) 𝑓∗(𝑣(2𝑖−1) 𝑗𝑤 𝑖 𝑗 ) = 24𝑘 − 8𝑖 − 2𝑗 + 5, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.9) 𝑓∗ (𝑤𝑖 𝑗𝑣(2𝑖) 𝑗 ) = 24𝑘 − 8𝑖 − 2𝑗 + 9, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.10) 𝑓∗(𝑥𝑖𝑦(𝑖+1) 𝑗) = 24𝑘 + 8𝑖 + 2𝑗 − 1, 0 ≤ 𝑖 ≤ 2𝑘 − 1,𝑗 = 1,2 (2.11) 𝑓∗(𝑦𝑖 𝑗𝑥𝑖) = 24𝑘 + 8𝑖 + 2𝑗 − 5, 1 ≤ 𝑖 ≤ 2𝑘,𝑗 = 1,2 (2.12) 𝑓∗(𝑦(2𝑖−1) 𝑗𝑧 𝑖 𝑗 ) = 48𝑘 − 8𝑖 − 2𝑗 + 5, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.13) 𝑓∗ (𝑧𝑖 𝑗𝑦 (2𝑖) 𝑗 ) = 48𝑘 − 8𝑖 − 2𝑗 + 9, 1 ≤ 𝑖 ≤ 𝑘,𝑗 = 1,2 (2.14) berdasarkan fungsi pelabelan busur 𝑓∗ pada persamaan (2.7), (2.8), (2.9), (2.10), (2.11), (2.12), (2.13) dan (2.14) diperoleh himpunan busur 𝑓∗ (𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) setelah dilabel sebagai berikut. 𝑓∗ (𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) = {1,9,17,25,…,16𝑘 − 7,3,11,19,27,…,16𝑘 − 5 } ∪ {5,13,21,29,…,16𝑘 − 3,7,15,23,31,…,16𝑘 − 1} ∪ { 24𝑘 − 5,24𝑘 − 13,24𝑘 − 21,24𝑘 − 29,…,16𝑘 + 3, 24𝑘 − 7,24𝑘 − 15,24𝑘 − 23,24𝑘 − 31,…,16𝑘 + 1 } ∪ { 24𝑘 − 1,24𝑘 − 9,24𝑘 − 17,24𝑘 − 25,…,16𝑘 + 7, 24𝑘 − 3,24𝑘 − 11,24𝑘 − 19,24𝑘 − 27,…,16𝑘 + 5 } ∪ { 24𝑘 + 1,24𝑘 + 9,24𝑘 + 17,24𝑘 + 25,…,40𝑘 − 7, 24𝑘 + 3,24𝑘 + 11,24𝑘 + 19,24𝑘 + 27,…,40𝑘 − 5 } jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 93 ∪ { 24𝑘 + 5,24𝑘 + 13,24𝑘 + 21,24𝑘 + 29,…,40𝑘 − 3, 24𝑘 + 7,24𝑘 + 15,24𝑘 + 23,24𝑘 + 31,…,40𝑘 − 1 } ∪ { 48𝑘 − 5,48𝑘 − 13,48𝑘 − 21,48𝑘 − 29,…,40𝑘 + 3, 48𝑘 − 7,48𝑘 − 15,48𝑘 − 23,48𝑘 − 31,…,40𝑘 + 1 } ∪ { 48𝑘 − 1,48𝑘 − 9,48𝑘 − 17,48𝑘 − 25,…,40𝑘 + 7, 48𝑘 − 3,48𝑘 − 11,48𝑘 − 19,48𝑘 − 27,…,40𝑘 + 5 } = { 1,3,5,7,9,13,15,17,19,21,23,25,27,29,…,16𝑘 − 7,16𝑘 − 5,16𝑘 − 3,16𝑘 − 1 16𝑘 + 1,16𝑘 + 3,6𝑘 + 5,16𝑘 + 7,…,24𝑘 − 31,24𝑘 − 29,24𝑘 − 27,24𝑘 − 25, 24𝑘 − 23,24𝑘 − 21,24𝑘 − 19,24𝑘 − 17,24𝑘 − 15,24𝑘 − 13,24𝑘 − 11,24𝑘 − 9, 24𝑘 − 7,24𝑘 − 5,24𝑘 − 3,24𝑘 − 1, 24𝑘 + 1,24𝑘 + 3,24𝑘 + 5,24𝑘 + 7,24𝑘 + 9,24𝑘 + 11,24𝑘 + 13,24𝑘 + 15, 24𝑘 + 17,24𝑘 + 19,24𝑘 + 21,24𝑘 + 23,24𝑘 + 25,24𝑘 + 27,24𝑘 + 29, 24𝑘 + 31,…,40𝑘 − 7,40𝑘 − 5,40𝑘 − 3,40𝑘 − 1, 40𝑘 + 1,40𝑘 + 3,40𝑘 + 5,40𝑘 + 7,…,48𝑘 − 31,48𝑘 − 29,48𝑘 − 27, 48𝑘 − 25,48𝑘 − 23,48𝑘 − 21,48𝑘 − 19,48𝑘 − 17,48𝑘 − 15,48𝑘 − 13, 48𝑘 − 11,48𝑘 − 9,48𝑘 − 7,48𝑘 − 5,48𝑘 − 3,48𝑘 − 1 } = {1,3,5,7,…,48𝑘 − 1 }. terlihat bahwa fungsi 𝑓∗ memberikan label yang berbeda pada setiap busur dan 𝑓∗ (𝐸(𝑘𝐿3,3 ∪ 𝑘𝐿3,3)) = {1,3,5,7,…,48𝑘 − 1} sehingga fungsi pelabelan busur 𝑓 ∗ memenuhi sifat pemetaan bijektif. telah ditunjukkan bahwa fungsi pelabelan simpul 𝑓 memenuhi pemetaan injektif sedemikian sehingga menginduksi fungsi pelabelan busur 𝑓∗ yang bijektif. akibatnya gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 adalah graf harmonis ganjil ∎ contoh 3.6 berikut pada gambar 4 diberikan contoh pelabelan harmonis ganjil dari gabungan graf ular jaring 7𝐿3,3 ∪ 7𝐿3,3 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 94 gambar 4 pelabelan harmonis ganjil pada gabungan graf ular jaring 𝟕𝑳𝟑,𝟑 ∪ 𝟕𝑳𝟑,𝟑 4. penutup pada makalah ini telah didapatkan kelas graf baru yang merupakan hasil operasi cartesian product yang memenuhi sifat-sifat pelabelan harmonis ganjil yaitu graf ular jaring 𝑘𝐿3,3 dengan 𝑘 ≥ 1 dan gabungan graf ular jaring 𝑘𝐿3,3 ∪ 𝑘𝐿3,3 dengan 𝑘 ≥ 1 sedemikian sehingga diperoleh keluarga baru dari graf harmonis ganjil. hasil penelitian ini bisa dilanjutkan untuk graf ular jaring 𝑘𝐿𝑚,𝑛 dengan 𝑚 ≥ 1,𝑛 ≥ 1,𝑘 ≥ 1 dan gabungannya 𝑘𝐿𝑚,𝑛 ∪ 𝑘𝐿𝑚,𝑛 dengan 𝑚 ≥ 1,𝑛 ≥ 1,𝑘 ≥ 1. ucapan terima kasih penulis mengucapkan terima kasih kepada ristek dikti yang telah memberi dukungan financial terhadap penelitian ini melalui hibah penelitian dosen pemula (pdp) tahun 2017. referensi [1] abdel-aal, m. e. new families of odd harmonious graphs. international journal of soft computing, mathematics and control, 3(1), (2014) 1-13. [2] alyani, f., firmansah, f., giyarti, w., dan sugeng, k. a. the odd harmonious labeling of kcn-snake graphs for spesific values of n, that is, for n = 4 and n = 8. proceeding indoms international conference on mathemathics and its applications, ugm dan indoms. (2013) 225-230. 6-7 november, yogyakarta. [3] firmansah, f., dan sugeng, k. a. pelabelan harmonis ganjil pada graf kincir angin belanda dan gabungan graf kincir angin belanda. magistra, no 94 th. xxvii, issn 0215-9511, (2015) 56-92. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 95 [4] firmansah, f. 2016. pelabelan harmonis ganjil pada gabungan graf ular dan graf ular berlipat. proceeding konferensi nasional matematika dan pembelajarannya (knpmp 1) ums. (2016) 809-818. 12 maret, surakarta. [5] firmansah, f. dan syaifuddin, m. w. pelabelan harmonis ganjil pada graf kincir angin double quadrilateral. proceeding seminar nasional matematika dan pendidikan matematika, fkip uny, (2016) 53-58. 5 november, yogyakarta. [6] firmansah, f. dan yuwono, m. r. odd harmonius labeling on pleated of the dutch windmill graphs. cauchy – jurnal matematika murni dan aplikasi, 4(4). (2017) 161-166. p-issn: 2086-0382, e-issn: 24773344. [7] jeyanthi, p. dan philo, s. odd harmonious labeling of some cycle related graphs. proyecciones journal of matematics, 35(1). (2016) 85-98. [8] jeyanthi, p., philo, s. dan sugeng, k.a. odd harmonious labeling of some new families of graphs. sut journal of mathematics. 51(2). (2015) 53-65. [9] gallian, j. a. a dynamic survey of graph labeling. the electronic journal of combinatorics, 18. (2016) #ds6. [10] liang, z., dan bai, z. on the odd harmonious graphs with applications, j. appl. math. comput., 29, (2009) 105-116. [11] saputri, g. a., sugeng, k. a., dan froncek, d. the odd harmonious labeling of dumbbell and generalized prims graphs, akce int, j. graphs comb., 10(2), (2013) 221-228. [12] vaidya, s. k., dan shah, n. h. some new odd harmonious graphs. international journal of mathematics and soft computing, 1(1), (2011) 9-16. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 65 ketercapaian dan keterkontrolan sistem deskriptor diskrit linier positif yulia retno sari universitas putra indonesia “yptk” padang, jalan raya lubuk begalung, padang yuliaretnosari2012@gmail.com doi:https://doi.org/10.15642/mantik.2017.3.2.65-73 abstrak sistem deskriptor diskrit positif telah banyak digunakan dalam pemodelan bidang ekonomi, teknik, kimia dan sebagainya. dalam penelitian ini dikaji tentang syarat perlu dan syarat cukup agar sistem deskriptor diskrit positif adalah tercapai positif dan terkontrol positif. selain itu, juga dikaji tentang syarat perlu dan syarat cukup yang menjamin agar sistem diskrit (e, a, b) ≥ 0 terkontrol null. dengan metode aljabar linier dan invers drazin, dalam penelitian ini dibuktikan beberapa teorema agar sistem deskriptor diskrit (e, a, b) ≥ 0 tercapai positif, terkontrol positif dan terkontrol null. selain itu, diberikan contoh sebagai ilustrasi untuk memperkuat keberlakuan teorema yang telah dibuktikan. kata kunci : invers drazin, sistem deskriptor diskrit positif, matriks non negatif, matriks nilpoten, sifat ketercapaian. abstract a positive discrete descriptor system has been widely used in modeling economics, engineering, chemistry and others. in this research, we studied the necessary conditions and sufficient conditions for a positive discrete descriptors system is achieved positive and controlled postively. in addition, it is also studied on sufficient terms and conditions which ensure that discrete systems (e, a, b) ≥ 0 are null controlled. by using linear algebraic method and inverse drazin, this research has proved several theorems for discrete descriptors system (e, a, b) ≥ 0 achieved positive, controlled positively and controlled null. in addition, examples are given as illustrations to reinforce the validity of the proven theorems. key words : inverse drazin, discrete positive descriptor system, non negative matrix, nilpotent matrix, achieved properties. 1. pendahuluan diberikan suatu sistem persamaan beda linier (linear difference equations) sebagai berikut : 𝐸𝐱(𝑘 + 1) = 𝐴𝐱(𝑘) + 𝐵𝐮(𝑘), 𝑘 ∈ ℤ+ (1) dengan 𝐸, 𝐴 ∈ ℝ𝑛×𝑛, dan 𝐵 ∈ ℝ𝑛×𝑚. dalam sistem (1), 𝐱 ∈ ℝ𝑛 menyatakan vektor state (keadaan) dan 𝐮 ∈ ℝm menyatakan vektor input (kontrol). notasi ℝ𝑛×𝑚 menyatakan himpunan matriks-matriks riil berukuran 𝑛 × 𝑚, ℝ𝑛 menyatakan himpunan vektor berdimensi n dan ℤ+ menyatakan himpunan bilangan bulat non negatif. dalam [5], sistem (1) dikatakan sebagai sistem deskriptor diskrit. jika 𝐸 adalah matriks non singular, maka solusi dari sistem (1) adalah 𝐱(𝑘) = (𝐸−1𝐴)𝑘𝐱(0) + ∑(𝐸−1𝐴)𝑘−𝑖−1(𝐸−1𝐵)𝐮(𝑖) 𝑘−1 𝑖 =0 (2) untuk 𝐸 singular, sistem (1) mungkin tidak mempunyai solusi. hal ini disebabkan adanya mailto:yuliaretnosari2012@gmail.com jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 66 kondisi awal yang tidak dapat memberikan solusi untuk sistem (1). kondisi awal yang dapat memberikan solusi untuk sistem (1) disebut sebagai kondisi awal yang konsisten. dalam [9] dinyatakan bahwa sistem (1) mempunyai solusi tunggal jika untuk suatu kondisi awal yang konsisten 𝐱(0) berlaku 𝑑𝑒𝑡(𝜆𝐸 − 𝐴) ≠ 0 untuk suatu 𝜆 ∈ ℂ. jika kondisi ini terpenuhi, maka solusi sistem (1) diberikan sebagai berikut: 𝐱(𝑘) = (�̅�𝐷�̅�)𝑘�̅�𝐷�̅�𝐱(0) + ∑ �̅�𝐷 (�̅�𝐷�̅�)𝑘−𝑖−1 𝑘−1 𝑖=0 �̅�𝐮(𝑖) −(𝐼 − �̅��̅�𝐷) ∑(�̅�𝐷 �̅�)𝑖 �̅�𝐷 �̅�𝐮(𝑘 + 𝑖) 𝑞−1 𝑖=0 dengan �̅� = (𝜆𝐸 − 𝐴)−1𝐸, �̅� = (𝜆𝐸 − 𝐴)−1𝐴, �̅� = (𝜆𝐸 − 𝐴)−1𝐵 dan q adalah indeks dari matriks �̅�. dalam hal 𝑑𝑒𝑡(𝜆𝐸 − 𝐴) ≠ 0 untuk suatu 𝜆 ∈ ℂ, sistem (1) disebut sebagai sistem deskriptor diskrit regular. perlu diperhatikan bahwa solusi 𝐱(𝑘) untuk sistem (1) dapat bernilai negatif ataupun non negatif. solusi 𝐱(𝑘) dikatakan non negatif jika 𝐱(𝑘) ≥ 0 dan dikatakan negatif jika 𝐱(𝑘) < 0. jika solusi 𝐱(𝑘) untuk sistem (1) adalah non negatif, maka sistem (1) dikatakan sistem deskriptor diskrit positif [3]. untuk selanjutnya sistem deskriptor diskrit positif dapat ditulis dengan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0. salah satu isu penting tentang sistem deskriptor diskrit positif adalah masalah ketercapaian positif dan keterkontrolan positif. [3] telah mendefinisikan tentang ketercapaian positif dan keterkontrolan positif sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0. untuk sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0, suatu keadaan 𝐰 ∈ ℝ+ 𝑛 dikatakan tercapai positif jika terdapat 𝑘 ∈ ℤ+ dan suatu barisan kontrol 𝐮(𝑗) ≥ 0, 𝑗 = 0,1, … , 𝑘 + 𝑞 − 1, yang membawa keadaan 𝐱(0) = 𝟎 kepada keadaan 𝐰 pada waktu k. sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dikatakan tercapai positif jika untuk setiap 𝐰 ∈ ℝ+ 𝑛 adalah tercapai positif. selain itu, sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dikatakan terkontrol positif jika untuk sebarang 𝐱𝟎, 𝐱𝐟 ∈ ℝ+ 𝑛 , terdapat 𝑘 ∈ ℤ+ dan suatu barisan kontrol 𝐮(𝑗) ≥ 0, 𝑗 = 0,1, … , 𝑘 + 𝑞 − 1, yang membawa keadaan 𝐱(0) = 𝐱𝟎 kepada keadaan 𝐱(𝑘) = 𝐱𝐟. untuk selanjutnya, sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 yang tercapai positif disebut tercapai dan yang terkontrol positif disebut terkontrol. penelitian ini membicarakan syarat perlu dan cukup untuk ketercapaian dan keterkontrolan sistem deskriptor diskrit linier positif. berdasarkan uraian dari latar belakang, maka yang menjadi permasalahan dalam penelitian ini adalah : 1. syarat apakah yang harus dipenuhi oleh sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 agar tercapai. 2. syarat apakah yang harus dipenuhi oleh sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 agar terkontrol. kajian penelitian ini bertujuan untuk membuktikan syarat yang menjamin agar sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai dan terkontrol. penelitian ini diharapkan dapat memperluas wawasan penulis serta pembaca pada umumnya dan diharapkan dapat memberikan konstribusi kepada pembaca agar lebih memahami pembuktian tentang ketercapaian dan keterkontrolan untuk sistem deskriptor diskrit linier positif. 2. kajian teori 2.1 teori matriks matriks didefinisikan sebagai susunan bilangan-bilangan di dalam baris dan kolom yang membentuk jajaran empat persegi panjang [1]. suatu matriks 𝐴 = [𝑎𝑖𝑗 ]𝑖,𝑗=1 𝑛 ∈ ℝ𝑛×𝑛, a dikatakan non negatif dinotasikan 𝐴 ≥ 0, jika 𝑎𝑖𝑗 ≥ 0, ∀𝑖, 𝑗 = 1,2, ⋯ , 𝑛 dan a dikatakan positif, dinotasikan 𝐴 > 0, jika 𝑎𝑖𝑗 > 0, ∀𝑖, 𝑗 = 1,2, ⋯ , 𝑛. suatu vektor 𝐱 ∈ ℝ𝑛 dikatakan non negatif jika setiap komponennya non negatif, yakni 𝑥𝑖 ≥ 0, 𝑖 = 1, … , 𝑛 . jika 𝐱 non negatif maka ditulis 𝐱 ≥ 0 atau 𝐱 ∈ ℝ+ 𝑛 , dengan ℝ+ 𝑛 menyatakan himpunan ℝ𝑛 yang setiap komponennya adalah non negatif. untuk vektor 𝐱 yang positif dapat didefinisikan dengan cara yang sama. definisi 2.1. matriks persegi a disebut matriks nilpoten jika 𝐴𝑛 = 0 dan 𝐴𝑛−1 ≠ 0 dengan 𝑛 ∈ ℤ+ terkecil. bilangan tersebut didefinisikan sebagai indeks nilpotensi dari matriks a. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 67 dalam [1] dinyatakan bahwa jika a dan b adalah matriks-matriks sedemikian sehingga 𝐴𝐵 = 𝐵𝐴, maka a dan b didefinisikan sebagai dua matriks yang komutatif. definisi 2.2 misalkan 𝐸, 𝐴 ∈ ℝ𝑚𝑥𝑛. pasangan matriks (𝐸, 𝐴) dikatakan regular jika 𝑚 = 𝑛 dan det(𝜆𝐸 − 𝐴) ≠ 0 untuk suatu 𝜆 ∈ ℂ. jika berlaku sebaliknya maka pasangan matriks (𝐸, 𝐴) dikatakan non regular. definisi 2.3 dimensi ruang baris atau ruang kolom matriks a disebut rank dari a dan ditulis 𝑟𝑎𝑛𝑘 (𝐴). teorema 2.4 misalkan terdapat matriks 𝐴 ∈ ℝ𝑚𝑥𝑛. maka dimensi kernel (ruang penyelesaian dari 𝐴𝑥 = 0) adalah 𝑛 − 𝑟𝑎𝑛𝑘(𝐴). definisi 2.5 misalkan terdapat matriks 𝐴 ∈ ℝ𝑚𝑥𝑛. maka image a disimbolkan dengan 𝐼𝑚(𝐴), didefinisikan sebagai ruang peta dari a yaitu 𝐼𝑚(𝐴) = {𝒘 ∈ ℝ𝑚 |∃𝒙 ∈ ℝ𝑛 ∋ 𝒘 = 𝐴𝒙}. definisi 2.6 misalkan terdapat matriks 𝐴 ∈ ℝ𝑛𝑥𝑛, suatu vektor 𝑥 ∈ ℝ𝑛, 𝑥 ≠ 0 dikatakan vektor eigen (eigenvector) dari a jika 𝐴𝒙 adalah kelipatan skalar dari x, yakni 𝐴𝒙 = 𝜆𝒙 (3) untuk suatu skalar 𝜆. skalar 𝜆 dinamakan nilai eigen (eigenvalue) dari a. nilai 𝜆 pada (3) merupakan akar dari polinomial karakteristik : det(𝜆𝐼 − 𝐴) = 0. teorema cayley-hamilton [14] menyatakan bahwa jika polinomial karakteristik dari matriks a adalah 𝑝(𝜆) = 𝑎0 + 𝑎1𝜆 + 𝑎2𝜆 2 + ⋯ + 𝑎𝑛−1𝜆 𝑛−1 + 𝜆𝑛 , maka 𝑝(𝐴) = 𝑎0𝐼 + 𝑎1𝐴 + 𝑎2𝐴 2 + ⋯ + 𝑎𝑛−1𝐴 𝑛−1 + 𝐴𝑛 = 0 berikut ini akan disajikan beberapa hal penting mengenai invers drazin dari suatu matriks 𝐴𝑛𝑥𝑛 yang diambil dari [9]. invers drazin berguna untuk mencari solusi sistem deskriptor diskrit. definisi 2.7 misalkan 𝐴 ∈ ℝ𝑛𝑥𝑛. indeks dari matriks a, ditulis 𝑖𝑛𝑑(𝐴), didefinisikan sebagai bilangan bulat non negatif terkecil q sedemikian sehingga 𝑟𝑎𝑛𝑘(𝐴𝑞) = 𝑟𝑎𝑛𝑘(𝐴𝑞+1). definisi 2.8 misalkan 𝐴 ∈ ℝ𝑛𝑥𝑛. invers drazin dari a, ditulis 𝐴𝐷, adalah suatu matriks yang memenuhi tiga syarat berikut : 1. 𝐴𝐴𝐷 = 𝐴𝐷 𝐴, 2. 𝐴𝐷 𝐴𝐴𝐷 = 𝐴𝐷 , 3. 𝐴𝐷 𝐴𝑞+1 = 𝐴𝑞, dimana q merupakan indeks dari a. invers drazin 𝐴𝐷 dari suatu matriks persegi a selalu ada dan tunggal [4]. jika a adalah matriks non singular, maka invers klasik 𝐴−1 memenuhi sifat invers drazin seperti yang diberikan dalam definisi (2.8). dalam hal ini 𝐴𝐷 = 𝐴−1. berikut ini akan dipaparkan proses untuk menentukan invers drazin dari suatu matriks persegi. misalkan 𝐴 ∈ ℝ𝑛𝑥𝑛 mempunyai nilai eigen nol dengan multiplisitas aljabar 1 dan nilai eigen berbeda 𝜆𝑖 dengan multiplisitas aljabar 𝑛𝑖 , 𝑖 = 1,2, … , 𝑟. jika 𝑚 = 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑟, maka 𝑚 + 1 = 𝑛. berdasarkan teorema cayley-hamilton, invers drazin 𝐴𝐷 dapat ditulis sebagai polinomial dalam a. perhatikan polinomial berikut : 𝑝(𝜆) = 𝜆𝑙 (𝑎0 + 𝑎1𝜆 + ⋯ + 𝑎𝑚−1𝜆 𝑚−1) (4) koefisien 𝑎0, 𝑎1, … , 𝑎𝑚−1 pada (4) dapat ditentukan dengan menyelesaikan sistem persamaan berikut : 1 𝜆𝑖 𝑝(𝜆𝑖 ) −1 𝜆𝑖 2 = 𝑝 ′(𝜆𝑖) (5) ⋮ (−1)𝑛𝑖−1(𝑛𝑖−1)! (𝜆𝑖) 𝑛𝑖 = 𝑝(𝑛𝑖−1)(𝜆𝑖), untuk 𝑖 = 1, 2, … , 𝑟 lema berikut dapat digunakan untuk menghitung invers drazin dari suatu matriks persegi. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 68 lema 2.9 jika 𝑝(𝜆) didefinisikan oleh (4) dan (5), maka : 𝐴𝐷 = 𝑝(𝐴). (6) sebagai ilustrasi dari lema 2.9, perhatikan contoh berikut : 𝐴 = [ 2 4 1 4 6 5 5 4 0 −1 −1 −2 −1 0 −3 −3 ] nilai eigen dari a adalah 0, 0, 1 dan 1. untuk matriks a diatas, 𝑝(𝜆) = 𝜆2 (𝑎0 + 𝑎1𝜆). (7) dengan menggunakan (5), diperoleh : 1 = 𝑎0 + 𝑎1, −1 = 2𝑎0 + 3𝑎1. (8) solusi dari (8) adalah 𝑎0 = 4 dan 𝑎1 = −3. jadi, berdasarkan lema 2.9, 𝐴𝐷 = 𝐴2(4𝐼 − 3𝐴) = [ 3 8 2 7 11 11 9 9 −1 −3 −1 −3 −4 −4 −4 −4 ] [ 4 0 0 4 0 0 0 0 0 0 0 0 4 0 0 4 ] − [ 6 12 3 12 18 15 15 12 0 −3 −3 −6 −3 0 −9 −9 ] = [ 3 −1 2 1 2 2 3 3 −1 0 −1 0 −1 −1 −1 −1 ] lema 2.10 misalkan 𝐴, 𝐵 ∈ ℂ𝑛𝑥𝑛, 1. jika 𝐴𝐵 = 𝐵𝐴, maka 𝐴𝐵𝐷 = 𝐵𝐷 𝐴, 𝐵𝐴𝐷 = 𝐴𝐷 𝐵, 𝐴𝐷 𝐵𝐷 = 𝐵𝐷 𝐴𝐷 . 2. jika 𝐴𝐵 = 𝐵𝐴 dan 𝑘𝑒𝑟𝐴⋂𝑘𝑒𝑟𝐵 = {𝟎}, maka (𝐼 − 𝐴𝐴𝐷 )𝐵𝐵𝐷 = 1 − 𝐴𝐴𝐷 . 2.2 ruang vektor pada bagian ini akan dibicarakan konsep vektor yang digunakan pada pembahasan. definisi 2.11 misalkan 𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧 adalah vektor dan 𝑟1, 𝑟2, … , 𝑟𝑛 skalar maka vektor, 𝑤 = 𝑟1𝐯𝟏 + 𝑟2𝐯𝟐 + ⋯ + 𝑟𝑛𝐯𝐧 (9) adalah kombinasi linier dari 𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧. himpunan semua kombinasi linier dari 𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧 dikatakan membangun 𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧 dan dinotasikan sebagai 𝑠𝑝𝑎𝑛 {𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧}. 𝑆𝑝𝑎𝑛 {𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧} = {𝑟1𝐯𝟏 + 𝑟2𝐯𝟐 + ⋯ + 𝑟𝑛𝐯𝐧|𝑟1, 𝑟2, … , 𝑟𝑛 𝑎𝑑𝑎𝑙𝑎ℎ 𝑠𝑘𝑎𝑙𝑎𝑟}. definisi 2.12 misalkan vektor 𝐯𝟏, 𝐯𝟐, … , 𝐯𝐧 dikatakan bebas linier jika 𝑟1, 𝑟2, … , 𝑟𝑛 adalah skalar dan 𝑟1𝐯𝟏 + 𝑟2𝐯𝟐 + ⋯ + 𝑟𝑛𝐯𝐧 = 0 (10) hanya dipenuhi oleh 𝑟1 = 0, 𝑟2 = 0, … , 𝑟𝑛 = 0. definisi 2.13 ruang vektor v dikatakan hasil tambah langsung dari subruang 𝑊1, 𝑊2, … , 𝑊𝑛, jika 1. 𝑉 = ∑ 𝑊𝑖 𝑛 𝑖=1 dan 2. 𝑊𝑗 ∩ (∑ 𝑊𝑖 𝑛 𝑖=1 𝑖≠𝑗 ) = {𝟎} ; untuk semua 𝑗 = 1,2, … , 𝑛. hasil tambah langsung ditulis dengan notasi, 𝑉 = 𝑊1 ⊕ 𝑊2 ⊕ … ⊕ 𝑊𝑛 atau 𝑉 = 𝑛 ⊕ 𝑖 = 1 𝑊𝑖 . (11) 2.3 proyeksi definisi 2.14 jika 𝑇: 𝑉 → 𝑊 adalah sebuah fungsi yang memetakan sebuah ruang vektor v kesebuah ruang vektor w, maka t disebut sebagai transformasi linier (linier transformasi) dari v ke w, jika semua vektor u dan w pada v dan semua skalar c, 1. 𝑇(𝐮 + 𝐯) = 𝑇(𝐮) + 𝑇(𝐯). 2. 𝑇(𝑐𝐮) = 𝑐𝑇(𝐮) dalam kasus khusus dimana 𝑉 = 𝑊, transformasi linier 𝑇: 𝑉 → 𝑉 disebut sebagai operator linier (linier operator) pada v. teorema 2.15 misalkan 𝐺: 𝐹𝑛 → 𝐹𝑚 adalah transformasi linier. maka terdapat sebuah matriks a ukuran 𝑚 × 𝑛 sedemikian sehingga 𝐺 = 𝑇𝐴. definisi 2.16 misalkan 𝑉 = 𝑆 ⊕ 𝑆𝑐 dengan 𝑆𝑐 adalah komplemen dari s. pemetaan 𝑃: 𝑉 → 𝑉 yang didefinisikan sebagai 𝑃(𝒔 + 𝒔𝒄) = 𝒔, dengan 𝒔 ∈ 𝑆 dan 𝒔𝒄 ∈ 𝑆𝑐 disebut proyeksi pada s sepanjang 𝑆𝑐 . jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 69 teorema 2.17 misalkan p adalah suatu proyeksi pada s sepanjang 𝑆𝑐 maka 1. 𝐼𝑚(𝑃) = 𝑆 dan ker(𝑃) = 𝑆𝑐 , 2. 𝑉 = 𝐼𝑚(𝑃) ⊕ ker (𝑃). teorema 2.18 suatu operator linier p adalah proyeksi jika dan hanya jika 𝑃2 = 𝑃. 2.4 solusi sistem deskriptor diskrit perhatikan kembali sistem deskriptor diskrit (1) dengan kondisi awal 𝐱(0). jika entrientri dari a, b dan e tidak bergantung terhadap waktu, maka sistem pada persamaan (1) disebut time-invariant. sebaliknya, jika entri-entri dari a, b dan e bergantung terhadap waktu, maka sistem (1) disebut time-varying. asumsikan bahwa det(𝜆𝐸 − 𝐴) ≠ 0, 𝑢𝑛𝑡𝑢𝑘 𝑠𝑢𝑎𝑡𝑢 𝜆 ∈ ℂ. (12) dengan mengalikan kedua ruas (1) dengan (𝜆𝐸 − 𝐴)−1, diperoleh : 𝐱(k + 1) = a̅𝐱(k) + b̅𝐮(k), (13) dengan �̅� = (𝜆𝐸 − 𝐴)−1𝐸, �̅� = (𝜆𝐸 − 𝐴)−1𝐴, dan �̅� = (𝜆𝐸 − 𝐴)−1𝐵. (14) lema 2.19 untuk matriks �̅� dan �̅� yang didefinisikan dalam (14) berlaku, 1. �̅��̅� = �̅�𝐴,̅ 2. ker(�̅�) ∩ ker(�̅�) = {𝟎} bukti. 1. berdasarkan persamaan (14), diperoleh 𝜆�̅� − �̅� = 𝜆(𝜆𝐸 − 𝐴)−1𝐸 − (𝜆 − 𝐴)−1𝐴 = (𝜆𝐸 − 𝐴)−1(𝜆𝐸 − 𝐴) = 𝐼, atau dapat ditulis �̅� = 𝜆�̅� − 𝐼. akibatnya, �̅��̅� = (𝜆�̅� − 𝐼)�̅� = �̅�(𝜆�̅� − 𝐼) = �̅��̅�. 2. misalkan bahwa 𝐱 ∈ ker �̅� ∩ ker �̅�. maka �̅�𝐱 = 𝟎 dan �̅�𝐱 = 𝟎, dan (𝜆𝐸 − 𝐴)𝐱 = 𝟎. karena 𝜆�̅� − �̅� = 𝐼, maka 𝐱 = 𝟎. ∎ teorema 2.20 solusi dari (13) dengan kondisi awal 𝐱(0) yang konsisten adalah 𝐱(𝑘) = (�̅�𝐷 �̅�)𝑘�̅�𝐷 �̅�𝐱(0) + ∑ �̅�𝐷 (�̅�𝐷 �̅�)𝑘−𝑖−1�̅�𝐮(𝑖)𝑘−1𝑖=0 − (−�̅��̅�𝐷 ) ∑ (�̅��̅�𝐷 )𝑖 𝑞−1 𝑖=0 �̅� 𝐷 �̅�𝐮(𝑘 + 1), (15) dengan q adalah indeks dari matriks �̅�. bukti. misalkan �̅� dan �̅� didefinisikan seperti dalam (14). berdasarkan lema 2.19 (1) berlaku �̅��̅� = �̅��̅�. akibatnya, menurut lema 2.10 (1) diperoleh �̅��̅�𝐷 = �̅�𝐷 �̅�. selanjutnya, �̅�𝐱(𝑘 + 1) = (�̅�𝐷 �̅�)𝑘+1�̅�𝐱(0) + ∑(�̅�𝐷 �̅� 𝑘 𝑖=0 )𝑘−𝑖�̅��̅�𝐷 �̅�𝐮(𝑖) − (𝐼 − �̅��̅�𝐷 ) ∑(�̅��̅�𝐷 )𝑖+1 𝑞−1 𝑖=0 �̅�𝐮(𝑘 + 𝑖 + 1) dan �̅�𝐱(𝑘) = (�̅�𝐷 �̅�)𝑘+1�̅�𝐱(0) + ∑(�̅�𝐷 �̅�)𝑘−𝑖�̅�𝐮(𝑖) 𝑘−1 𝑖=0 − (𝐼 − �̅��̅�𝐷 ) ∑(�̅��̅�𝐷 )𝑖+1 𝑞−1 𝑖=0 �̅�𝐮(𝑘 + 𝑖). berdasarkan lema 2.19 berlaku �̅��̅� = �̅��̅� dan ker(�̅�) ∩ ker(�̅�) = {𝟎}, akibatnya menurut lema 2.10 (2) diperoleh (𝐼 − �̅��̅�𝐷 )�̅��̅�𝐷 = 𝐼 − �̅��̅�𝐷 , sehingga (𝐼 − �̅��̅�𝐷 )(�̅��̅�𝐷 )𝑞 = (𝐼 − �̅��̅�𝐷 )�̅��̅�𝐷 (�̅��̅�𝐷)𝑞 = (𝐼 − �̅��̅�𝐷 )�̅��̅�𝐷 �̅�𝑞(�̅�𝐷 )𝑞 = �̅��̅�𝐷�̅�𝑞(�̅�𝐷 )𝑞 − �̅��̅�𝐷 �̅��̅�𝐷�̅�𝑞 (�̅�𝐷)𝑞 = �̅��̅�𝐷�̅�𝑞(�̅�𝐷 )𝑞 − �̅��̅�𝐷 �̅��̅�𝐷�̅�𝐷 �̅�𝑞+1(�̅�𝐷 )𝑞 = �̅��̅�𝐷�̅�𝑞(�̅�𝐷 )𝑞 − �̅��̅�𝐷 �̅�𝐷 �̅�𝑞+1(�̅�𝐷 )𝑞 = �̅��̅�𝐷�̅�𝑞(�̅�𝐷 )𝑞 − �̅��̅�𝐷 �̅�𝑞 (�̅�𝐷 )𝑞 = 0. selanjutnya, untuk membuktikan bahwa (15) merupakan solusi dari (13), akan ditunjukkan �̅�𝐱(𝑘 + 1) − �̅�𝐱(𝑘) = �̅�𝐮(𝑘). �̅�𝐱(𝑘 + 1) − �̅�𝐱(𝑘) = �̅��̅�𝐷 �̅�𝐮(𝑘) − (𝐼 − �̅��̅�𝐷 ) ∑ [(�̅��̅�𝐷 )𝑖+1�̅�𝐮(𝑘 + 𝑖 + 1) + 𝑞−1 𝑖=0 (�̅��̅�𝐷 )𝑖 �̅�𝐮(𝑘 + 𝑖)] = �̅��̅�𝐷 �̅�𝐮(𝑘)(𝐼 − �̅��̅�𝐷 ) ∑ [(�̅��̅�𝐷 )𝑖�̅�𝐮(𝑘 + 𝑞−1 𝑖=0 𝑖) − (�̅��̅�𝐷 )𝑖+1 �̅�𝐮(𝑘 + 𝑖 + 1)] jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 70 =�̅��̅�𝐷 �̅�𝐮(𝑘) + (𝐼 − �̅��̅�𝐷 )[(�̅�𝐮 (𝑘) − (�̅��̅�𝐷 )�̅�𝐮(𝑘 + 1)) + ((�̅��̅�𝐷 )�̅�𝐮(𝑘 + 1) − (�̅��̅�𝐷 )2�̅�𝐮(𝑘 + 2)) + ⋯ + ((�̅��̅�𝐷 )𝑞−2�̅�𝐮(𝑘 + 𝑞 − 2) − (�̅��̅�𝐷 )𝑞−1�̅�𝐮(𝑘 + 𝑞 − 1)) + ((�̅��̅�𝐷 )𝑞−1�̅�𝐮(𝑘 + 𝑞 − 1) − (�̅��̅�𝐷 )𝑞�̅�𝐮(𝑘 + 𝑞))] = �̅��̅�𝐷 �̅�𝐮(𝑘) + (𝐼 − �̅��̅�𝐷 )[�̅�𝐮 (𝑘) − (�̅��̅�𝐷 )𝑞�̅�𝐮(𝑘 + 𝑞)] = �̅��̅�𝐷 �̅�𝐮(𝑘) + (𝐼 − �̅��̅�𝐷 )�̅�𝐮 (𝑘) − (𝐼 − �̅��̅�𝐷 )(�̅��̅�𝐷 )𝑞�̅�𝐮(𝑘 + 𝑞) = �̅��̅�𝐷 �̅�𝐮(𝑘) + �̅�𝐮 (𝑘) − �̅��̅�𝐷 �̅�𝐮(𝑘) = �̅�𝐮 (𝑘) jadi, solusi (15) memenuhi sistem (13). ∎ 2.5 sistem deskriptor diskrit positif teorema 2.21 untuk sistem deskriptor diskrit (e, a, b), asumsikan bahwa eed ≥ 0, ea = ae dan ker e ∩ ker a = {𝟎}. sistem diskrit (e, a, b) ≥ 0 jika dan hanya jika eda ≥ 0, edb ≥ 0 dan (i − ede)(ead) i adb ≤ 0 dengan q adalah indeks dari e. bukti. (⇒) misalkan bahwa sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0. karena 𝐸𝐴 = 𝐴𝐸 dan ker 𝐸 ∩ ker 𝐴 = {𝟎}, untuk setiap kondisi awal yang konsisten 𝐱(0) ≥ 𝟎 dan untuk setiap kontrol non negatif, berlaku 𝐱(𝑘) ≥ 𝟎 ∀𝑘 ∈ ℤ+, 𝐱(𝑘) = (𝐸𝐷 𝐴)𝑘𝐸𝐷 𝐸𝐱(0) + ∑ 𝐸𝐷 (𝐸𝐷 𝐴)𝑘−𝑖−1 𝑘−1 𝑖 =0 𝐵𝐮(𝑖) − (𝐼 − 𝐸𝐸𝐷 ) ∑(𝐸𝐴𝐷 )𝑖𝐴𝐷 𝐵𝐮(𝑘 𝑞−1 𝑖=0 + 𝑖) akan dibuktikan bahwa 𝐸𝐷 𝐴 ≥ 0. selanjutnya, karena 𝐸𝐸𝐷 ≥ 0, maka 𝐱(0) = 𝐸𝐸𝐷 𝐞𝑖 ≥ 𝟎, 𝑖 = 1,2, ⋯ , 𝑛, merupakan kondisi awal yang konsisten, dengan 𝐞𝑖 ∈ ℝ+ 𝑛 adalah vektor satuan ke-i. dengan menggunakan kontrol 𝐮(𝑗) = 𝟎, 𝑗 = 0,1, ⋯ , 𝑘 + 𝑞 − 1, maka pada 𝑘 = 1, 𝐱(1) = (𝐸𝐷 𝐴)1𝐸𝐷 𝐸𝐱(0) = 𝐸𝐷 𝐴𝐸𝐷 𝐸𝐸𝐸𝐷 𝐞𝑖 karena 𝐸𝐴 = 𝐴𝐸, maka berdasarkan lema 2.10 diperoleh 𝐸𝐷 𝐴 = 𝐴𝐸𝐷. dengan menggunakan definisi 2.8 yaitu 𝐸𝐷 𝐸 = 𝐸𝐸𝐷 dan 𝐸𝐷 𝐸𝐸𝐷 = 𝐸𝐷, diperoleh : 𝐱(1) = 𝐸𝐷 𝐴𝐸𝐸𝐷 𝐸𝐸𝐷 𝐞𝑖 = 𝐸𝐷 𝐴𝐸𝐸𝐷 𝐞𝑖 = 𝐴𝐸𝐷 𝐸𝐸𝐷 𝐞𝑖 = 𝐴𝐸𝐷 𝐞𝑖 . karena 𝐱(1) ≥ 𝟎 dan 𝐞𝑖 ≥ 𝟎 untuk setiap 𝑖 = 1,2, ⋯ , 𝑛, maka 𝐸𝐷 𝐴 ≥ 0. berikutnya akan dibuktikan bahwa 𝐸𝐷 𝐵 ≥ 0. ambil 𝐱(0) = 𝟎, 𝐮(𝑗) = 𝟎, 𝑗 = 1,2, ⋯ , 𝑞, dan 𝐮(0) = 𝐞𝑖 ∈ ℝ+ 𝑚 dengan q adalah indeks dari 𝐸. pada 𝑘 = 1, 𝐱(1) = 𝐸𝐷 (𝐸𝐷 𝐴)0𝐵𝐮(0) = 𝐸𝐷 𝐵𝒆𝑖 . karena 𝐱(1) ≥ 𝟎 dan 𝐞𝑖 ≥ 𝟎 untuk setiap 𝑖 = 1,2, ⋯ , 𝑚, maka 𝐸𝐷 𝐵 ≥ 0. akhirnya, akan dibuktikan bahwa (𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )𝑖𝐴𝐷 𝐵 ≤ 0 , ∀𝑖 = 0,1, ⋯ , 𝑞 − 1 dengan q adalah indeks dari 𝐸. dengan mengambil 𝐱(0) = 𝟎, 𝐮(𝑘) = 𝐞𝑖 ∈ ℝ+ 𝑚 , dan 𝐮(𝑗) = 𝟎, 𝑗 ≠ 𝑘, 𝑗 = 0,1, ⋯ , 𝑘 + 𝑞 − 1, diperoleh 𝐱(𝑘) = −(𝐼 − 𝐸𝐷 𝐸)𝐴𝐷 𝐵𝐞𝑖. karena 𝐱(𝑘) ≥ 𝟎 dan 𝐞𝑖 ≥ 𝟎 untuk setiap 𝑖 = 1,2, ⋯ , 𝑚, maka (𝐼 − 𝐸𝐷 𝐸)𝐴𝐷 𝐵 ≤ 0. dengan mengambil 𝐮(𝑘 + ℎ) = 𝐞𝑖 untuk setiap 𝑖 = 1,2, ⋯ , 𝑚 dan 𝐮(𝑗) = 𝟎, 𝑗 ≠ 𝑘 + ℎ, 𝑗 = 0, ⋯ , 𝑘 + 𝑞 − 1, maka diperoleh : 𝐱(𝑘) = −(𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )ℎ𝐴𝐷𝐵𝒆𝑖 . karena 𝐱(𝑘) ≥ 𝟎 dan 𝐞𝑖 ≥ 𝟎 untuk setiap 𝑖 = 1,2, ⋯ , 𝑚, maka (𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )ℎ 𝐴𝐷 𝐵 ≤ 0, untuk setiap ℎ = 1, 2, … , 𝑞 − 1. (⇐) misalkan 𝐸𝐷 𝐴 ≥ 0, 𝐸𝐷 𝐵 ≥ 0 dan (𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )𝑖𝐴𝐷 𝐵 ≤ 0, untuk setiap 𝑖 = 0,1, ⋯ , 𝑞 − 1 dengan q adalah indeks dari 𝐸. akibatnya, solusi dari sistem deskriptor diskrit (𝐸, 𝐴, 𝐵) adalah non negatif, 𝐱(𝑘) ≥ 𝟎, untuk setiap kontrol 𝐮(𝑗) ≥ 𝟎, 𝑗 = 1,2, ⋯ , 𝑘 + 𝑞 + 1, 𝑘 ∈ ℤ+, sehingga sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0.∎ teorema 2.22. misalkan sistem diskrit (e, a, b) ≥ 0 dengan eed ≥ 0 dan ea = ae. sistem (e, a, b) adalah terkontrol null jika dan hanya jika eda adalah suatu matriks nilpoten. bukti. (⟸) misalkan 𝐸𝐷 𝐴 adalah suatu matriks nilpoten dengan indeks nilpotensi l. akan dibuktikan bahwa sistem (𝐸, 𝐴, 𝐵) terkontrol null. karena indeks nilpotensi dari matriks 𝐸𝐷 𝐴 adalah l, pilih 𝑘 ≥ 𝑙 dan barisan kontrol 𝐮(𝑗) = jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 71 𝟎, 𝑗 = 0,1, … , 𝑘 + 𝑞 − 1, maka 𝐱(𝑘) = 𝟎. jadi, sistem (𝐸, 𝐴, 𝐵) adalah terkontrol null. (⟹) misalkan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah terkontrol null maka untuk sebarang 𝐱𝟎 ∈ ℝ+ 𝑛 ada 𝑘 ∈ ℤ+ dan pilih barisan kontrol 𝐮(𝑗) = 𝟎, 𝑗 = 0,1, … , 𝑘 + 𝑞 − 1 sedemikian sehingga 𝟎 = (𝐸𝐷 𝐴)𝑘𝐸𝐷 𝐸𝐱𝟎 (16) jika diambil 𝐱𝟎 = 𝐸 𝐷 𝐸 𝐞𝑖 dengan 𝐞𝑖 adalah vektor satuan ke-i, maka persamaan 16 dapat ditulis menjadi (𝐸𝐷 𝐴)𝑘 𝐸𝐷 𝐸 𝐸𝐷 𝐸𝐞𝑖 = (𝐸 𝐷 𝐴)𝑘 𝐸𝐷 𝐸𝐞𝑖 = (𝐸𝐷 )𝑘𝐴𝑘𝐸𝐷 𝐸𝐞𝑖 = (𝐸𝐷 )𝑘−1𝐸𝐷 𝐴𝑘𝐸𝐷 𝐸𝐞𝑖 = (𝐸𝐷 )𝑘−1𝐴𝑘𝐸𝐷 𝐸𝐸𝐷 𝐞𝑖 = (𝐸𝐷 )𝑘−1𝐴𝑘𝐸𝐷 𝐞𝑖 = (𝐸𝐷 )𝑘−1𝐸𝐷 𝐴𝑘𝐞𝑖 = (𝐸𝐷 )𝑘 𝐴𝑘𝐞𝑖 = (𝐸𝐷 𝐴)𝑘 𝐞𝑖 = 𝟎. untuk setiap 𝑖 = 1,2, … , 𝑛. karena (𝐸𝐷 𝐴)𝑘 𝐞𝑖 = 𝟎 maka terdapat 𝑙 ∈ ℝ+ 𝑛 sedemikian sehingga (𝐸𝐷 𝐴)𝑙 = 0, yaitu 𝐸𝐷 𝐴 adalah matriks nilpoten. ∎ 3. hasil dan pembahasan misalkan ℛ𝑘(𝐸, 𝐴, 𝐵) menyatakan himpunan keadaan tercapai dalam waktu 𝑘, yaitu ℛ𝑘(𝐸, 𝐴, 𝐵) = {𝐱(𝑘)|𝐱(𝑘)tercapai pada waktu 𝑘} perhatikan bahwa, ℛ𝑘(𝐸, 𝐴, 𝐵) dibangun oleh kolom-kolom dari sub matriks non negatif berikut, [𝐸𝐷 𝐵|𝐸𝐷 𝐸𝐷 𝐴𝐵|… |𝐸𝐷 (𝐸𝐷 𝐴)𝑘−1𝐵|−(𝐼 − 𝐸𝐷 𝐸)𝐴𝐷𝐵|… |−(𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )𝑞−1𝐴𝐷 𝐵] oleh karena itu, ℛ𝑘(𝐸, 𝐴, 𝐵) dapat juga ditulis sebagai berikut, ℛ𝑘(𝐸, 𝐴, 𝐵) = 〈𝐸𝐷 𝐵, 𝐸𝐷 𝐸𝐷 𝐴𝐵, … , 𝐸𝐷 (𝐸𝐷 𝐴)𝑘−1𝐵, −(𝐼 − 𝐸𝐷 𝐸)𝐴𝐷 𝐵, … , −(−𝐸𝐷 𝐸)(𝐸𝐴𝐷 )𝑞−1𝐴𝐷 𝐵〉 (17) selanjutnya, misalkan ℛ∞ (𝐸, 𝐴, 𝐵) menyatakan himpunan keadaan tercapai dalam waktu hingga, maka ℛ∞(𝐸, 𝐴, 𝐵) = ⋃ ℛ𝑘(𝐸, 𝐴, 𝐵) ∞ 𝑘=1 = {𝐱|𝐱 tercapai dalam waktu hingga}. lema berikut memberikan syarat perlu dan cukup untuk ketercapaian sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0. lema 3.1 sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai jika dan hanya jika ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 . bukti (⇒)misalkan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai. akan dibuktikan bahwa ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 . ambil 𝐱(𝑗) ∈ ℛ∞(𝐸, 𝐴, 𝐵) maka 𝐱(𝑗) ∈ ℛ𝑗 (𝐸, 𝐴, 𝐵). ini bermakna bahwa 𝐱(𝑗) tercapai dalam waktu j sehingga 𝐱(𝑗) ∈ ℝ+ 𝑛 . jadi ℛ∞(𝐸, 𝐴, 𝐵) ⊆ ℝ+ 𝑛 . berikutnya, ambil sebarang 𝐱(𝑗) ∈ ℝ+ 𝑛 . akan dibuktikan 𝐱(𝑗) ∈ ℛ∞(𝐸, 𝐴, 𝐵). karena sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0, maka untuk setiap keadaan awal 𝐱𝟎 ∈ ℝ+ 𝑛 , terdapat 𝐮(𝑗) ≥ 𝟎, 𝑗 = 0,1, … , 𝑘 + 𝑞 − 1 sedemikian sehingga 𝐱(𝑗) ≥ 𝟎, ∀𝑘 ∈ ℤ+. ini menunjukkan bahwa setiap 𝐱(𝑗) ∈ ℝ+ 𝑛 tercapai dari keadaan 0. sehingga 𝐱(𝑗) ∈ ℛ𝑗 (𝐸, 𝐴, 𝐵) untuk suatu j. jadi 𝐱(𝑗) ∈ ℛ∞(𝐸, 𝐴, 𝐵). (⇐) misalkan ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 , akan dibuktikan bahwa sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai. telah diketahui ℛ∞(𝐸, 𝐴, 𝐵) = ⋃ ℛ𝑘(𝐸, 𝐴, 𝐵) ∞ 𝑘=1 mendefenisikan bahwa x tercapai dalam waktu hingga. maka jelas, sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai. ∎ teorema 3.2 sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dengan 𝐸𝐸𝐷 ≥ 0 dan 𝐸𝐴 = 𝐴𝐸 adalah tercapai jika dan hanya jika, ℱ∞(𝐸, 𝐴, 𝐵) = 𝐼𝑚(𝐸𝐸 𝐷 ) dan ℬ(𝐸, 𝐴, 𝐵) = 𝑘𝑒𝑟(𝐸𝐸𝐷 ). jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 72 bukti dari lema 3.1, diketahui bahwa sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 tercapai jika dan hanya jika ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 . dari klaim diketahui bahwa ℛ𝑘(𝐸, 𝐴, 𝐵) = 𝐹𝑘(𝐸, 𝐴, 𝐵) ⊕ β(𝐸, 𝐴, 𝐵) akibatnya ℝ+ 𝑛 = ℛ∞(𝐸, 𝐴, 𝐵) = ⋃ ℛ𝑘(𝐸, 𝐴, 𝐵) ∞ 𝑘=1 = ⋃ 𝐹𝑘(𝐸, 𝐴, 𝐵) ∞ 𝑘=1 ⊕ β(𝐸, 𝐴, 𝐵) = 𝐹∞(𝐸, 𝐴, 𝐵) ⊕ β(𝐸, 𝐴, 𝐵) = 𝐼𝑚 (𝐸𝐸𝐷 ) ⊕ 𝑘𝑒𝑟(𝐸𝐸𝐷 ) sehingga sistem deskret (𝐸, 𝐴, 𝐵) ≥ 0 tercapai jika dan hanya jika 𝐹∞(𝐸, 𝐴, 𝐵) = 𝐼𝑚(𝐸𝐸 𝐷 ) dan β (𝐸, 𝐴, 𝐵) = 𝑘𝑒𝑟(𝐸𝐸𝐷 ). ∎ dari teorema 2.22 dan 3.1 dapat disimpulkan, teorema 3.3 diberikan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dengan 𝐸𝐸𝐷 ≥ 0 dan 𝐸𝐴 = 𝐴𝐸, maka sistem adalah terkontrol jika dan hanya jika ℛ∞(e, a, b) = ℝ+ n dan eda adalah matriks nilpoten. bukti (⇒) misalkan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dengan 𝐸𝐸𝐷 ≥ 0 dan 𝐸𝐴 = 𝐴𝐸 adalah terkontrol. akan dibuktikan ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 dan 𝐸𝐷 𝐴 adalah matriks nilpoten. karena sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 terkontrol maka untuk setiap 𝐱𝐟 ∈ ℝ+ 𝑛 tercapai dari sebarang keadaan awal 𝐱𝟎 ∈ ℝ+ 𝑛 khususnya 𝐱𝟎 = 𝟎. akibatnya sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai. berdasarkan lema 3.1 maka ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 selanjutnya, keterkontrolan dari sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 juga berakibat sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 terkontrol null. berdasarkan teorema 2.22, maka 𝐸𝐷 𝐴 adalah matriks nilpoten. (⇐) misalkan ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 dan 𝐸𝐷 𝐴 adalah suatu matriks nilpoten dengan indeks nilpotensi l. akan dibuktikan sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 terkontrol. ambil 𝐱𝟎, 𝐱𝐟 ∈ ℝ+ 𝑛 sebarang. karena 𝐱𝐟 ∈ ℝ+ 𝑛 maka 𝐱𝐟 ∈ ℛ∞(𝐸, 𝐴, 𝐵) yang berarti bahwa 𝐱𝐟 ∈ ℝ𝑘(𝐸, 𝐴, 𝐵) untuk suatu k, akibatnya ada 𝐮(𝑗) ≥ 𝟎, 𝑗 = 0,1,2, … , 𝑘 + 𝑞 − 1 sedemikian sehingga 𝐱𝐟 dibangun oleh kolom-kolom matriks ℛ𝑘 (𝐸, 𝐴, 𝐵) = 〈𝐸𝐷 𝐵, 𝐸𝐷 𝐸𝐷 𝐴𝐵, … , 𝐸𝐷 (𝐸𝐷 𝐴)𝑘−1𝐵, −(𝐼 − 𝐸𝐷 𝐸)𝐴𝐷 𝐵, … , −(𝐼 − 𝐸𝐷 𝐸)(𝐸𝐴𝐷 )𝑞−1𝐴𝐷 𝐵〉. karena 𝐸𝐷 𝐴 adalah suatu matriks nilpoten dengan indeks nilpotensi l, pilih 𝑘 ∈ ℤ+ dengan 𝑘 ≥ 𝑙, maka akibatnya 𝐱𝐟 = ∑ 𝐸 𝐷 (𝐸𝐷 )𝑘−𝑖−1𝐵𝐮(𝑖) − (𝐼 𝑘−1 𝑖=0 − 𝐸𝐸𝐷 ) ∑(𝐸𝐴𝐷 )𝑖 𝐴𝐷 𝐵𝐮(𝑘 𝑞−1 𝑖=0 + 𝑖). (18) fakta 3.2 memperlihatkan bahwa keadaan 𝐱𝐟 tercapai disebabkan karena adanya 𝐱𝟎 ∈ ℝ+ 𝑛 sehingga sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 terkontrol.∎ 4. kesimpulan berdasarkan uraian dari hasil dan pembahasan, dapat diberikan beberapa kesimpulan sebagai berikut: 1. suatu sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 adalah tercapai jika dan hanya jika, ℛ∞(𝐸, 𝐴, 𝐵) = ℝ+ 𝑛 . 2. sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dengan 𝐸𝐸𝐷 ≥ 0 dan 𝐸𝐴 = 𝐴𝐸 adalah tercapai jika dan hanya jika, ℱ∞(𝐸, 𝐴, 𝐵) = 𝐼𝑚(𝐸𝐸 𝐷 ) dan ℬ(𝐸, 𝐴, 𝐵) = 𝑘𝑒𝑟(𝐸𝐸𝐷 ). 3. sistem diskrit (𝐸, 𝐴, 𝐵) ≥ 0 dengan 𝐸𝐸𝐷 ≥ 0 dan 𝐸𝐴 = 𝐴𝐸 adalah terkontrol jika dan hanya jika tercapai dan terkontrol null. referensi [1] anton, h. aljabar linier elementer edisi kedelapan-jilid 1. erlangga. jakarta (2004) jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 73 [2] arifin, a. aljabar linear edisi kedua, penerbit itb. bandung (200) [3] bru, r., coll, c., sanchez, e. structural properties of positive linear timeinvariant difference-algebraic equations. lin. alg. appl. vol. 349 pp: 1-10 (2002) [4] campbell, s.l., meyer. c.d. and j.r. nicholas. applications of the drazin invers to linier systems of differential equations with singular constant coefficients. siam j. appl. math. vol. 31 no. 3 pp: 411-425 (1979) [5] canto, b., coll, c., sanchez, e. positive solutions of a discrete-time descriptor system. international journal of systems science. vol. 39 no. 1 pp: 81-88 (2008) [6] gantmacher, f.r. the theory of matrices. vol 1. ams chelsea publishing. rhode island (2000) [7] hoffman, k., kunze, r. linear algebra second edition. new jersey (2006) [8] jacob, bill. linear algebra. w. h. freeman and company. new york (1990) [9] kaczorek. t. linier control systems. vol. 1. research studies press ltd. england (1992) [10] noutsos, d., tsatsomeros, m.j. reachability and holdability of nonnegative states. siam journal on matrix analysis and applications. vol. 30 pp: 700-712 (2008) [11] rahmalina, widdya. thesis, sistem deskriptor diskrit positif, department of mathematics, faculty of mathematics and natural sciences, universitas andalas, padang, 2011 [12] retno sari, yulia. sistem deskriptor diskrit linier positif yang terkontrol null. journal proceeding seminar nasional matematika stkip pgri. padang, 2015 [13] roman, s. advance linear algebra. springer. new york, (1992) [14] serre, denis. matrices theory and application. second edition. springer. france (2010) m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 90 peramalan jumlah penumpang kereta api di indonesia dengan resilient back-propagation (rprop) neural network mertha endah ervina 1, rini silvi 2, intaniah ratna nur wisisono 3 prodi magister statistika terapan, universitas padjajaran1,2,3 merthiez@gmail.com1, rinisilvi@stis.ac.id2, rnwintaniah@gmail.com3 doi:https://doi.org/10.15642/mantik.2018.4.2.90-99 abstrak penjadwalan kereta api mempengaruhi tingkat kepuasan konsumen dan tingkat keuntungan dari penyelenggara jasa layanan kereta api. metode prediksi back-propagation neural network (bpnn) terbilang lambat konvergensinya, maka dari itu, penelitian ini menggunakan resilient back-propagation (rprop) karena memiliki teknik konvergensi yang cepat dan tingkat akurasi yang tinggi. model yang dihasilkan adalah model untuk jabodetabek, jawa (non-jabodetabek), sumatera, dan indonesia. dari hasil analisis data yang dilakukan, dapat disimpulkan bahwa performa model neural network dengan resilient backpropagation (rprop) yang dibentuk dari data training memberikan tingkat akurasi prediksi yang sangat baik/akurat dengan nilai mean absolute percentage error (mape) kurang dari 10% untuk masing-masing model. kemudian dilakukan peramalan untuk 12 bulan kedepan dan hasilnya dibandingkan dengan data testing, rprop memberikan tingkat akurasi peramalan yang sangat tinggi dengan nilai mape dibawah 10%. secara lengkap, nilai mape untuk masing-masing peramalan jumlah penumpang ka adalah 7.50% untuk jabodetabek, 5.89% untuk jawa (non-jabodetabek), 5.36% untuk sumatera, dan 4.80% untuk indonesia. artinya, empat arsitektur neural network dengan rprop dapat digunakan untuk kasus ini dengan hasil peramalan yang sangat akurat. kata kunci: resilient back-propagation (rprop), peramalan, penumpang kereta api, nnfor abstract train scheduling affects the level of customer satisfaction and profitability of the train service provider. the prediction method of back-propagation neural network (bpnn) has relatively slow convergence. therefore, this study uses resilient back-propagation (rprop) because it has a more fast convergence and high accuracy. the model produced is a model for jabodetabek, java (non-jabodetabek), sumatra, and indonesia. from the results of data analysis conducted, it can be concluded that the performance of neural network model with resilient back-propagation (rprop) formed from training data gives very accurate prediction accuracy level with mean absolute percentage error (mape) less than 10% for each model. then forecasting for the next 12 months conducted and the results compared with the data testing, rprop provides a very high forecasting accuracy with mape value below 10%. the mape value for each forecasting the number of rail passengers is 7.50% for jabodetabek, 5.89% for java (non-jabodetabek), 5.36% for sumatra and 4.80% for indonesia. that is, four neural network architectures with rprop can be used for this case with very accurate forecasting results. keywords: resilient back-propagation (rprop), forecasting, rail passengers, nnfor jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 91 1. pendahuluan kereta api merupakan transportasi umum yang banyak digunakan oleh masyarakat baik dalam menunjang kegiatan sehari-hari terkait pekerjaan dan aktivitas harian lainnya yang bersifat rutin maupun sebagai solusi transportasi jarak jauh yang bersifat insidental. kereta api sebagai transportasi umum memiliki beberapa keunggulan diantaranya memiliki waktu tempuh yang lebih dapat diprediksi karena terhindar dari kemacetan jalan raya dan juga mampu melayani penumpang yang jauh lebih banyak dalam sekali perjalanan dibandingkan dengan sarana transportasi umum darat lainnya. penjadwalan dan perencanaan kapasitas kereta api terkait erat dengan tingkat kepuasan penggunanya dan tingkat keuntungan usaha dari penyelenggara jasa layanan kereta api tersebut. lai dan barkan [1] menyatakan bahwa manajemen kapasitas yang efektif merupakan kunci sukses dari penyelenggara jasa layanan kereta api akan tetapi hal ini tidaklah mudah. layanan kereta api yang over-capacity akan membuat pengguna kecewa dan beralih pada moda transportasi lain selain itu juga membuat penyelenggara layanan kereta api kehilangan potensi pendapatan dari para calon penumpang yang tidak tertampung. di lain pihak, under-capacity akan membuat penyelenggara layanan kereta api menanggung beban tambahan akibat gerbong yang tidak terisi. oleh karena itu, diperlukan sebuah peramalan yang cukup tepat dalam memperkirakan jumlah penumpang kereta api sehingga dapat dilakukan penyesuaian kapasitas layanan sesuai kebutuhan. back-propagation neural network (bpnn) merupakan teknik klasifikasi dan peramalan yang paling populer menggunakan supervised learning neural network. akan tetapi, menurut chen dan su [2], teknik bpnn terbilang lambat konvergensinya dan memiliki tendensi dapat terjebak dalam lokal minima. di lain pihak, menurut riedmiller (1993) dalam [2] resilient back-propagation (rprop) merupakan teknik yang memiliki konvergensi cepat dan masih menjaga akurasinya. penelitian ini bertujuan untuk melakukan peramalan penumpang kereta api menggunakan teknik rprop sehingga dapat dilakukan penyesuaian kapasitas layanan sesuai kebutuhan dengan harapan dapat meningkatkan kepuasan pengguna layanan sekaligus tingkat keuntungan dari penyelenggara jasa layanan kereta api. 2. tinjauan pustaka 2.1 back-propagation neural network jaringan saraf tiruan (artificial neural network) menurut russel dan novig (2013) dalam [3] adalah sebuah sistem yang terdiri atas sekelompok unit pemroses yang dimodelkan untuk pemrosesan informasi yang meniru cara kerja sistem saraf biologis seperti jaringan saraf manusia. pemodelan ini didasari oleh kemampuan otak manusia dalam mengorganisir neuron sehingga mampu mengenali pola secara efektif [4]. namun, pemodelan ann jauh lebih sederhana dibanding otak manusia yang sebenarnya. banyak sistem otak manusia yang harus disimplifikasi agar dapat dimodelkan ke dalam dunia komputer. arsitektur neural network dapat dilihat pada gambar 1 [5]. gambar 1 arsitektur neural network back-propagation merupakan salah satu algoritma pembelajaran dalam ann. algoritma ini memiliki dua tahap perhitungan, yaitu perhitungan maju yang dilakukan untuk menghitung error antara output ann dengan target yang diinginkan. tahap kedua adalah perhitungan mundur jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 92 yang menggunakan error yang telah didapatkan untuk memberbaiki bobot pada semua neuron yang ada. 2.2 resilent back-propagation (rprop) resilient back-propagation atau biasa disingkat rprop dibuat oleh martin riedmiller dan heinrich braun pada tahun 1992 [6]. algoritma ini tidak jauh berbeda dengan algortima back-propagation dan merupakan perbaikan dari algortima backpropagation. menurut mccaffrey [7], ada dua keunggulan algortima rprop dibandingkan algoritma pendahulunya (back-propagation). pertama, proses pelatihan yang lebih cepat, dan yang kedua, rprop tidak mengharuskan untuk menentukan nilai parameter apa pun dalam perhitungannya. sementara, algoritma backpropagation memerlukan parameter rasio pembelajaran (learning rate), dan biasanya juga memerlukan parameter momentum. kerugian algoritma ini yaitu implementasinya yang cukup rumit. mathworks (1999) dalam [8] menyatakan, secara sederhana algoritma ini menggunakan tanda turunan untuk menentukan arah perbaikan bobot-bobot. besarnya perubahan setiap bobot ditentukan oleh suatu faktor yang diatur pada parameter yang disebut delt_inc dan delt_dec. apabila gradien fungsi error berubah tanda dari satu iterasi ke iterasi berikutnya, maka bobot akan berkurang sebesar delt_dec. sebaliknya apabila gradien error tidak berubah tanda dari satu iterasi ke iterasi berikutnya, maka bobot akan berkurang sebesar delt_inc. apabila gradien error sama dengan nol maka perubahan sama dengan perubahan bobot sebelumnya. pada awal iterasi, besarnya perubahan bobot diinisalisasikan dengan parameter delta0. besarnya perubahan tidak boleh melebihi batas maksimum yang terdapat pada parameter deltamax, apabila perubahan bobot melebihi batas maksimum perubahan bobot, maka perubahan bobot akan ditentukan sama dengan maksimum perubahan bobot. algoritma rprop membuat dua perubahan signifikan [7], pertama, rprop tidak menggunakan besarnya gradien untuk menentukan perubahan bobot tapi hanya menggunakan tanda dari gradien. gradien adalah kumpulan semua turunan parsial untuk semua bobot dan bias dari neural network. kedua, rprop mempertahankan perubahan bobot terpisah untuk setiap bobot dan bias, dan menyesuaikan perubahan ini selama pelatihan. 2.3 fungsi aktivasi dalam neural network, bagian yang paling penting adalah fungsi aktivasinya atau seringkali disebut juga dengan threshold function maupun transfer function. karakteristik fungsi aktivasi bpnn harus bersifat kontinu, differentiable, dan tidak turun secara monoton [9]. fungsi aktivasi merupakan fungsi matematis yang berguna untuk membatasi dan menentukan jangkauan output suatu neuron [10]. fungsi aktivasi yang digunakan pada penelitian ini adalah: 1. sigmoid biner (logistik) fungsi aktivasi sigmoid biner memiliki nilai output pada range nol sampai satu [0,1] dan didefinisikan sebagai berikut: x e xf − + = 1 1 )( (1) gambar 2 fungsi aktivasi sigmoid biner 2. fungsi hyperbolic tangent (tanh) output dari fungsi ini memiliki range antara satu sampai minus satu [-1,1] dan didefinisikan sebagai berikut: xx xx ee ee xf − − + − =)( (2) gambar 3 fungsi aktivasi hyperbolic tangent -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 93 2.4 mape mape atau mean absolute percetage error adalah ukuran akurasi dari suatu prediksi atau suatu peramalan. dalam penelitian ini mape digunakan untuk memilih model rprop neural network terbaik. berikut rumusan dari mape:  = −       = n t at ftat n mape 1 %100 (3) dimana, at: jumlah penumpang ka pada periode ke-t, ft: hasil prediksi/peramalan jumlah penumpang ka pada periode ke-t, n : jumlah observasi. interpretasi dari nilai mape menurut lewis (1982:40) dalam [11] yaitu: <10 % : peramalan sangat akurat, 10%-20% : peramalan akurat, 20%-50% : peramalan cukup akurat, >50% : peramalan tidak akurat 3. metode peneltian 3.1 sumber data data yang digunakan adalah data jumlah penumpang kereta api yang dikumpulkan oleh badan pusat statistik dari pt kereta api indonesia dan pt. kai commuter jabodetabek. periode data dimulai dari januari 2006 hingga april 2018. set data yang digunakan merupakan level bulanan sebanyak 148 observasi. 3.2 tahapan penelitian langkah-langkah dalam penelitian ini mencakup: 1. persiapan data, terdiri dari: a. pemeriksaan data untuk melihat apakah terdapat data hilang (missing) atau tidak, dan b. pembagian data menjadi data training dan data testing. 2. pembentukan model peramalan dengan resilient back-propagation (rprop) neural network, dengan tahapan: a. penentuan parameter, seperti jumlah hidden nodes, fungsi aktivasi/transfer yang akan digunakan, dan sebagainya. b. proses pelatihan jaringan syaraf tiruan (neural network training) menggunakan data training dan parameter-parameter yang sudah ditetapkan sebelumnya. c. pemilihan parameter yang menghasilkan nilai mape terkecil. d. pembentukan model peramalan dengan nilai parameter terpilih. 3. peramalan menggunakan model yang dibentuk pada tahap sebelumnya. 4. uji performa (validasi) hasil peramalan dengan menggunakan nilai mape. 4. hasil dan pembahasan 4.1 persiapan data data yang diperoleh untuk penelitian ini ada sebanyak 148 record untuk masingmasing wilayah kereta api (ka), dimana 136 diantaranya (periode januari 2006 hingga april 2017) digunakan sebagai data training pada tahap pembentukan model, dan 12 lainnya (periode mei 2017 hingga april 2018) digunakan sebagai data testing untuk menguji performa dari hasil peramalan. gambar 4 menunjukkan pola dari jumlah penumpang kereta api periode januari 2006 hingga april 2018. gambar 4 plot time series jumlah penumpang kereta api berdasarkan wilayah kereta api 4.2 pembentukan model peramalan model peramalan yang perlu dibentuk pada tahap ini adalah empat model yang terdiri dari model peramalan untuk jumlah penumpang ka wilayah jabodetabek, untuk jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 94 jumlah penumpang ka wilayah nonjabodetabek (jawa), untuk jumlah penumpang ka wilayah sumatera, dan untuk jumlah penumpang ka keseluruhan (indonesia). pembentukan model diawali dengan proses pelatihan jaringan syaraf tiruan (neural network) menggunakan data training. pada proses ini akan dilakukan pelatihan dengan arsitektur/struktur neural network yang berbeda-beda. arsitektur neural network terdiri dari input layer, hidden layer, dan output layer. beberapa parameter yang digunakan dalam hidden layer diantaranya jumlah hidden nodes dan fungsi aktivasi. jumlah hidden nodes yang digunakan mulai dari lima hidden nodes hingga 15 hidden nodes, sedangkan fungsi aktivasi yang digunakan yaitu logistic (sigmoid biner) dan hyperbolic tangent (tanh). sementara itu, stepmax/epoch ditetapkan 100000, jumlah pengulangan untuk pelatihan rprop neural network ditetapkan sebanyak 20 kali dan target error (threshold) ditetapkan 0,01. proses pelatihan menggunakan kombinasi dari seluruh parameter tersebut, sehingga menghasilkan 22 kandidat model untuk masing-masing wilayah ka atau total 88 kandidat model. hasil pelatihan rprop neural network menggunakan package “nnfor” dan package “neuralnet” pada program r disajikan pada tabel 1-4. tabel 1. hasil pelatihan untuk data jumlah penumpang ka wilayah jabodetabek no. hidden nodes fungsi aktivasi mse mape (%) 1 5 logistic 12348,44 0,58 2 5 tanh 17808,37 0,71 … … … … … 19 14 logistic 576,14 0,10 20 14 tanh 927,66 0,12 21 15 logistic 786,31 0,11 22 15 tanh 938,14 0,12 tabel 2. hasil pelatihan untuk data jumlah penumpang ka wilayah nonjabodetabek (jawa) no. hidden nodes fungsi aktivasi mse mape (%) 1 5 logistic 7813,40 1,10 2 5 tanh 7079,56 1,02 … … … … … 19 14 logistic 382,10 0,16 20 14 tanh 363,55 0,16 21 15 logistic 343,44 0,16 22 15 tanh 291.98 0.14 tabel 3. hasil pelatihan untuk data jumlah penumpang ka wilayah sumatera no. hidden nodes fungsi aktivasi mse mape (%) 1 5 logistic 273.09 2.20 2 5 tanh 225.25 2.28 … … … … … 19 14 logistic 8.24 0.28 20 14 tanh 20.11 0.41 21 15 logistic 23.12 0.39 22 15 tanh 15.00 0.37 tabel 4. hasil pelatihan untuk data jumlah penumpang ka indonesia no. hidden nodes fungsi aktivasi mse mape (%) 1 5 logistic 191055.52 1.65 2 5 tanh 204172.29 1.74 … … … … … 17 13 logistic 67480.44 0.68 18 13 tanh 101058.01 0.91 … … … … … 22 15 tanh 86200.03 0.80 pada tabel tersebut terlihat bahwa perubahan jumlah hidden nodes mempengaruhi hasil prediksi, makin banyak jumlah hidden nodes nilai mse dan mape dari hasil prediksi cenderung makin kecil. baris yang diberi warna abu-abu pada tabel adalah arsitektur rprop neural network dengan nilai mape dan mse paling kecil (minimum) untuk masing-masing data jumlah penumpang ka. berikut keempat arsitektur rprop neural network dimaksud: jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 95 1. untuk data jumlah penumpang ka wilayah jabodetabek, nilai mape dan mse minimum ditemukan pada arsitektur neural network 17-14-1 dengan fungsi transfer yang digunakan adalah logistic. input neuron yang digunakan pada arsitektur tersebut adalah 11 variabel seasonal dummy dan data pada lag 1, lag 2, lag 5, lag 7, lag 10, dan lag 12. pada gambar 5-8, variabel seasonal dummy ditandai dengan warna pink. gambar 5 plot multi layer perceptron (mlp) untuk data penumpang ka wilayah jabodetabek 2. untuk data jumlah penumpang ka wilayah non-jabodetabek (jawa), nilai mape dan mse minimum ditemukan pada arsitektur neural network 16-15-1 dengan fungsi transfer yang digunakan adalah hyperbolic tangent. input neuron yang digunakan pada arsitektur tersebut adalah 11 variabel seasonal dummy dan data pada lag 1, lag 2, lag 3, lag 4, dan lag 11. gambar 6 plot multi layer perceptron (mlp) untuk data penumpang ka wilayah non-jabodetabek (jawa) 3. untuk data jumlah penumpang ka wilayah sumatera, nilai mape dan mse minimum ditemukan pada arsitektur neural network 16-14-1 dengan fungsi transfer yang digunakan adalah logistic. input neuron yang digunakan pada arsitektur tersebut adalah 11 variabel seasonal dummy dan data pada lag 1, lag 2, lag 3, lag 4, dan lag 12. gambar 7 plot multi layer perceptron (mlp) untuk data penumpang ka wilayah jabodetabek 4. untuk data jumlah penumpang ka keseluruhan (indonesia), nilai mape dan mse minimum ditemukan pada arsitektur neural network 13-13-1 dengan fungsi transfer yang digunakan adalah logistic. input neuron yang digunakan pada arsitektur tersebut adalah 11 variabel seasonal dummy dan data pada lag 1 dan lag 2. gambar 8 plot multi layer perceptron (mlp) untuk data penumpang ka indonesia jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 96 gambar 9 plot data aktual dan prediksi jumlah penumpang kereta api periode januari 2006 – april 2017 nilai mape dibawah 10% menunjukkan kemampuan prediksi yang sangat baik (sangat akurat), hal tersebut dapat terlihat pada gambar 9. nilai prediksi (garis merah) hampir berimpitan sempurna dengan nilai aktual (garis biru). 4.3 peramalan (forecasting) dengan menggunakan arsitektur rprop neural network terbaik hasil tahap pembentukan model, dilakukan peramalan untuk 12 bulan kedepan dari periode terakhir data training. peramalan dilakukan untuk wilayah ka jabodetabek, wilayah ka non-jabodetabek (jawa), wilayah ka sumatera, dan untuk total keseluruhan wilayah ka. nilai aktual dan ramalan jumlah penumpang ka secara visual dapat dilihat pada gambar 10-13. gambar 10 plot data aktual dan ramalan penumpang ka wilayah jabodetabek gambar 11 plot data aktual dan ramalan penumpang ka wilayah non-jabodetabek (jawa) jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 97 gambar 12 plot data aktual dan ramalan penumpang ka wilayah sumatera gambar 13 plot data aktual dan ramalan penumpang ka indonesia uji performa hasil peramalan pada penelitian ini merupakan tahapan menghitung akurasi dari hasil peramalan menggunakan mean absolute percentage error (mape). semakin kecil nilai mape, maka akurasi suatu peramalan semakin tinggi atau semakin akurat. suatu peramalan dikatakan sangat akurat apabila memiliki nilai mape dibawah 10%. nilai mape hasil peramalan wilayah ka jabodetabek, wilayah ka non-jabodetabek (jawa), wilayah ka sumatera, dan nilai mape hasil peramalan total keseluruhan wilayah (indonesia) berturut-turut adalah 7.50%, 5.89%, 5.36%, dan 4.80%. hal tersebut menunjukkan empat arsitektur neural network yang digunakan untuk masingmasing wilayah kereta api (ka) menghasilkan peramalan yang sangat akurat. hasil peramalan jumlah penumpang ka untuk masing-masing wilayah ka periode mei 2017 sampai april 2018 dapat dilihat pada lampiran. 5. kesimpulan berdasarkan analisis yang dilakukan pada data jumlah penumpang kereta api, dapat disimpulkan bahwa: 1. performa model neural network dengan resilient back-propagation (rprop) yang dibentuk dari data training dan divalidasi dengan data testing memberikan tingkat akurasi prediksi serta tingkat akurasi peramalan yang sangat baik dengan nilai mean absolute percentage error (mape) kurang dari 10% untuk masing-masing model. 2. nilai mape untuk masing-masing peramalan jumlah penumpang ka adalah: untuk wilayah jabodetabek memberikan nilai mape sebesar 7.50%, untuk non jabodetabek (jawa) memberikan nilai mape sebesar 5.89%. sedangkan untuk wilayah sumatra memberikan nilai mape sebesar 5.36%. sementara itu, hasil peramalan total keseluruhan wilayah (indonesia) memberikan nilai mape sebesar 4.80%. artinya, empat arsitektur neural network dengan rprop dapat digunakan untuk kasus ini dengan hasil peramalan yang sangat akurat. referensi [1] y.-c. (rex) lai dan c. p. l. barkan, “enhanced parametric railway capacity evaluation tool,” transportation research record: journal of the transportation research board, vol. 2117, no. 1, hlm. 33–40, jan 2009. [2] c.-s. chen dan s.-l. su, “resilient backpropagation neural network for approximation 2-d gdop,” dipresentasikan pada imecs 2010, hong kong, 2010, vol. 2. [3] e. r. zulvima, “visualisasi dari klasterisasi dan peramalan kualitas udara kota surabaya menggunakan metode klasterisasi k-means dan jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 98 peramalan artificial neural network,” skripsi, institut teknologi sepuluh nopember, surabaya, 2017. [4] suyanto, data mining untuk klasifikasi dan klasterisasi data. informatika, 2017. [5] d. anderson dan g. mcneill, “artificial neural networks technology,” kaman sciences corporation, new york, a dacs state-of-the-art report, agu 1992. [6] wikipedia, “rprop,” wikipedia, 2016. [daring]. tersedia pada: https://en.wikipedia.org/wiki/rprop. [7] j. mccaffrey, “how to use resilient back propagation to train neural networks,” visual studio magazine, 09mar-2015. [daring]. tersedia pada: https://visualstudiomagazine.com/articles /2015/03/01/resilient-backpropagation.aspx? [8] n. susanto, “pengembangan model jaringan syaraf tiruan resilient backpropagation untuk identifikasi pembicara dengan praproses mfcc,” skripsi, institut pertanian bogor, bogor, 2007. [9] l. fausett, fundamentals of neural network: architectures, algorithms, and application. new jersey: prentice hall, 1994. [10] s. s. haykin, neural networks: a comprehensive foundation, 2 ed. new jersey: prentice hall, 1999. [11] j. j. m. moreno, a. p. pol, a. s. abad, dan b. c. blasco, “using the r-mape index as a resistant measure of forecast accuracy,” psicothema, vol. 25, no. 4, hlm. 500–506, nov 2013. [12] amrin, “peramalan tingkat inflasi indonesia menggunakan neural network backpropagation berbasis metode time series,” jurnal techno nusa mandiri, vol. 11, no. 2, hlm. 129–136, 2014. [13] m. fajar, “peramalan produksi cabai rawit dengan neural network,” 2017. [14] s. fritsch, f. guenther, m. suling, dan s. m. mueller, package ‘neuralnet.’ 2016. [15] g. a. saputro dan m. asri, anggaran perusahaan, 3 ed. yogyakarta: bpfe, 2000. [16] j. heizer dan b. render, manajemen operasi buku 1, 9 ed. jakarta: salemba empat, 2009. [17] b. karlik dan a. v. olgac, “performance analysis of various activation functions in generalized mlp architectures of neural networks,” international journal of artificial intelligence and expert systems (ijae), vol. 1, no. 4, hlm. 111– 122, 2011. [18] n. kourentzes, package ‘nnfor.’ 2017. [19] w. s. mcculloch dan w. pitts, “a logical calculus of the ideas immanent in nervous activity,” bulletin of mathematical biophysics, vol. 5, no. 4, hlm. 115–133, des 1943. [20] m. a. razak dan e. riksakomara, “peramalan jumlah produksi ikan dengan menggunakan backpropagation neural network (studi kasus: uptd pelabuhan perikanan banjarmasin),” jurnal teknik its, vol. 6, no. 1, hlm. 142– 148, 2017. [21] w. saputra, tulus, m. zarlis, r. w. sembiring, dan d. hartama, “analysis resilient algorithm on artificial neural network backpropagation,” dalam journal of physics: conference series, 2017, vol. 930, hlm. 012035. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 99 lampiran tabel 5 hasil peramalan jumlah penumpang kereta api (ribu orang) wilayah bulan nilai aktual nilai ramalan residual mape (%) jabodetabek mei 2017 27385.00 26425.18 959.82 7.50 juni 2017 24432.00 25583.31 -1151.31 juli 2017 27016.00 23296.11 3719.89 agustus 2017 27679.00 24692.44 2986.56 sept. 2017 26158.00 24875.94 1282.06 oktober 2017 28765.00 26098.91 2666.09 november 2017 28246.00 25379.16 2866.84 desember 2017 29059.00 26235.71 2823.29 januari 2018 28075.00 26012.92 2062.08 februari 2018 25362.00 24619.66 742.34 maret 2018 29223.00 27014.23 2208.77 apri 2018 28943.00 27401.47 1541.53 nonjabodetabek (jawa) mei 2017 5772.00 6768.36 -996.36 5.89 juni 2017 5749.00 6219.55 -470.55 juli 2017 6653.00 7016.80 -363.80 agustus 2017 5576.00 5992.58 -416.58 sept. 2017 5763.00 6224.38 -461.38 oktober 2017 5733.00 5817.42 -84.42 november 2017 5552.00 5706.04 -154.04 desember 2017 7081.00 7085.75 -4.75 januari 2018 6032.00 5943.28 88.72 februari 2018 5359.00 5471.85 -112.85 maret 2018 6049.00 6740.41 -691.41 apri 2018 6193.00 6498.28 -305.28 sumatera mei 2017 588.00 569.85 18.15 5.36 juni 2017 542.00 517.83 24.17 juli 2017 641.00 630.36 10.64 agustus 2017 536.00 486.68 49.32 sept. 2017 577.00 577.36 -0.36 oktober 2017 572.00 547.20 24.80 november 2017 563.00 463.49 99.51 desember 2017 667.00 619.33 47.67 januari 2018 610.00 580.12 29.88 februari 2018 557.00 543.35 13.65 maret 2018 603.00 605.99 -2.99 apri 2018 619.00 563.87 55.13 total mei 2017 33745.00 32990.29 754.71 4.80 juni 2017 30723.00 32644.87 -1921.87 juli 2017 34310.00 32745.96 1564.04 agustus 2017 33791.00 33101.98 689.02 sept. 2017 32498.00 32662.95 -164.95 oktober 2017 35070.00 32964.11 2105.89 november 2017 34361.00 32526.74 1834.26 desember 2017 36807.00 33071.01 3735.99 januari 2018 34717.00 32474.73 2242.27 februari 2018 31278.00 30932.04 345.96 maret 2018 35875.00 33910.57 1964.43 apri 2018 35755.00 33082.79 2672.21 m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 143 penyelesaian sistem persamaan linier fully fuzzy menggunakan metode dekomposisi nilai singular (svd) corry corazon marzuki1, agustian2, dewi hariati3, junitis afmilda4, nurul husna5, putra nanda6 jurusan matematika, fakultas sains dan teknologi, uin sultan syarif kasim riau1,2,3,4,5,6, jl. hr. soebrantas no. 155 simpang baru, panam, pekanbaru, 28293 email: corry@uin-suska.ac.id1, agustian2196@gmail.com2, dewih961002@gmail.com3, junitisafmilda21@gmail.com4, husnanurul2411@gmail.com5, putrananda581@gmail.com6 doi:https://doi.org/10.15642/mantik.2018.4.1.143-149 abstrak sistem persamaan linear dapat dibentuk ke dalam persamaan matriks ax = b. konstanta dalam persamaan linear dapat pula berupa bilangan fuzzy dan semua parameternya dalam bilangan fuzzy yang dikenal dengan istilah sistem persamaan linear fully fuzzy. metode singular value decomposition (svd) merupakan suatu metode yang mendekomposisikan suatu matriks a menjadi tiga komponen matriks usvh. metode svd dapat digunakan untuk mencari solusi dari sistem persamaan linear fully fuzzy yang konsisten maupun sistem persamaan linear fully fuzzy yang tidak konsisten. solusi yang diperoleh dari sistem persamaan linear fully fuzzy yang konsisten dengan menggunakan svd adalah solusi tunggal dan banyak solusi. sedangkan, solusi yang diperoleh dari sistem persamaan linear fully fuzzy yang tidak konsisten dengan menggunakan svd adalah solusi pendekatan terbaik. kata kunci: fuzzy, sistem persamaan linier fully fuzzy, singular value decomposition (svd) abstract linear equation system can be arranged into the ax = b matrix equation. constants in linear can also contain fuzzy numbers and all their parameters in fuzzy numbers known as fully fuzzy linear equation systems. singular value decomposition (svd) is a method that decomposes an a matrix into three components of the usvh. the svd method can be used to find a solution to the fully fuzzy fully linear equation system that is also an inconsistent fully fuzzy linear equation system. the solution obtained from a fully fuzzy linear equation system that is consistent using svd is a single solution and many solutions. whereas, the solution obtained from a fully fuzzy linear equation system that is inconsistent using svd is the best approach solution. keywords: fuzzy, fully fuzzy linear equation system, singular value decomposition (svd) 1. pendahuluan sistem persamaan linier merupakan kumpulan persamaan linier yang saling berhubungan untuk mencari nilai variabel yang memenuhi semua persamaan linier tersebut. sistem persamaan linier biasanya terdiri atas 𝑚 persamaan dan 𝑛 variabel. sistem persamaan linier dapat ditulis dalam bentuk persamaan matriks 𝐴𝑋 = 𝐵, dengan semua entri-entri di dalam 𝐴 dan 𝐵 adalah bilangan riil. seiring perkembangan ilmu matematika, konstanta dalam sistem persamaan linier dapat berupa bilangan fuzzy dan dapat diselesaikan dengan menggunakan metode yang sama. sistem persamaan linier dengan konstanta berupa bilangan fuzzy disebut sistem persamaan linier fuzzy. bentuk persamaan linier fuzzy mailto:corry@uin-suska.ac.id mailto:agustian2196@gmail.com2 mailto:dewih961002@gmail.com3 mailto:junitisafmilda21@gmail.com4 mailto:husnanurul2411@gmail.com5 mailto:putrananda581@gmail.com6 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 144 seperti sistem persamaan linier biasa, perbedaannya terletak pada unsur 𝐵. unsur 𝐵 dalam sistem persamaan linier fuzzy merupakan bentuk parameter yang berada pada interval tertentu. selain itu, dikenal juga sistem persamaan linier fully fuzzy. sistem persamaan fully fuzzy merupakan persamaan matriks 𝐴𝑋 = 𝐵 dengan 𝐴 adalah matriks fuzzy dan 𝑋, 𝐵 adalah bilangan fuzzy. gourav gupta melakukan penelitian pada tahun 2010 tentang penyelesaian sistem persamaan linier fully fuzzy menggunakan metode langsung (metode invers matriks, aturan cramer, dan metode dekomposisi lu) dan metode iterasi (metode gauss jacobi dan gauss seidel) dengan judul penelitian “some methods for solving fully fuzzy linear system of equations”. beberapa metode yang dapat digunakan untuk menyelesaikan sistem persamaan linear salah satunya adalalah menggunakan analisis svd. analisis svd merupakan suatu teknik yang melibatkan pemfaktoran 𝐴 ke dalam hasil kali 𝑈𝑆𝑉𝑇, dengan 𝑈,𝑆,𝑉 adalah matriks bujur sangkar dan semua entri diluar diagonal dari matriks 𝑆 adalah nol. sedangkan vektor kolom dari matriks 𝑈 dan 𝑉 adalah ortonormal. kelebihan metode analisis svd dalam menyelesaikan sistem persamaan linear yaitu, solusi dari sistem persamaan linear tetap dapat dicari meskipun sistem persamaan linear tersebut tidak mempunyai pemecahan, dalam hal ini solusi yang diperoleh adalah solusi pendekatan terbaik [1]. 2. tinjauan pustaka 2.1 back-propagation neural network sistem persamaan linier adalah sekumpulan persamaan linier yang terdiri dari persamaan 𝐿1, 𝐿2, … , 𝐿𝑚, dengan 𝑛 variabel yang tidak diketahui yaitu 𝑥1, 𝑥2, … , 𝑥𝑛 yang dapat dinyatakan dalam bentuk: 𝑎11𝑥1 + 𝑎12𝑥2 + ⋯+ 𝑎1𝑛𝑥𝑛 = 𝑏1 𝑎21𝑥1 + 𝑎22𝑥2 + ⋯+ 𝑎2𝑛𝑥𝑛 = 𝑏2 ⋮ ⋮ ⋮ ⋮ 𝑎𝑚1𝑥1 + 𝑎𝑚2𝑥2 + ⋯+ 𝑎𝑚𝑛𝑥𝑛 = 𝑏𝑚 dengan 𝑎11, 𝑎12, …, 𝑎𝑚𝑛 dan 𝑏1, 𝑏2, …, 𝑏𝑚 adalah konstanta-konstanta bilangan riil. menyelesaikan suatu sistem persamaan linier adalah mencari nilai variabel-variabel yang memenuhi sistem persamaan linier tersebut. sistem persamaan linier dikatakan konsisten jika memiliki satu atau banyak solusi sedangkan tidak konsisten jika tidak mempunyai solusi penyelesaian. sistem persamaan linier dapat dinyatakan dalam bentuk matriks seperti berikut [2]. [ 𝑎11 𝑎21 𝑎12 𝑎22 ⋯ 𝑎1𝑛 𝑎2𝑛 ⋮ ⋱ ⋮ 𝑎𝑚1 𝑎𝑚2 ⋯ 𝑎𝑚𝑛 ] [ 𝑥1 𝑥2 ⋮ 𝑥𝑛 ] = [ 𝑏1 𝑏2 ⋮ 𝑏𝑚 ] atau 𝐴𝑥 = 𝑏 dengan 𝐴 = [ 𝑎11 𝑎21 𝑎12 𝑎22 ⋯ 𝑎1𝑛 𝑎2𝑛 ⋮ ⋱ ⋮ 𝑎𝑚1 𝑎𝑚2 ⋯ 𝑎𝑚𝑛 ] dan 𝑏 = [ 𝑏1 𝑏2 ⋮ 𝑏𝑚 ]. 2.2 resilent back-propagation (rprop) fuzzy dapat diartikan kabur atau semu. himpunan fuzzy pertama kali dibahas oleh lotfi a. zadeh 1965 himpunan fuzzy merupakan kumpulan dari entri-entri dengan satu rangkaian tingkat keanggotaan. secara fungsional himpunan fuzzy disajikan dalam bentuk persamaan matematis sehingga untuk mengetahui derajat keanggotaan dari masing-masing elemen memerlukan perhitungan. definisi 2: misalkan 𝑋 adalah suatu himpunan semesta, kemudian himpunan bagian fuzzy 𝑈 dari 𝑋 adalah himpunan bagian dari 𝑋 yang keanggotaannya didefinisikan melalui fungsi keanggotaan sebagai berikut: 𝜇�̃�: 𝑋 → [0,1] berdasarkan definisi tersebut maka himpunan fuzzy 𝑈 dalam himpunan semesta 𝑥, ditulis dalam bentuk: 𝑈 = { (𝑥,𝜇�̃�𝑥)|𝑥 ∈ 𝑋} dengan (𝑥,𝜇�̃�𝑥), menyatakan elemen 𝑥 yang mempunyai derajat keanggotaan 𝜇�̃�𝑥 . jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 145 himpunan bilangan fuzzy dinamakan bilangan fuzzy segitiga jika fungsi keanggotaannya sebagai berikut [2]: 𝜇𝑢𝑥 = { 1 − 𝑚 − 𝑥 𝛼 𝑢𝑛𝑡𝑢𝑘 𝑚 − 𝑎 < 𝑥 ≤ 𝑚 𝑑𝑎𝑛 𝑎 > 0 1 − 𝑥 − 𝑚 𝛽 𝑢𝑛𝑡𝑢𝑘 𝑚 ≤ 𝑥 < 𝑚 + 𝛽 𝑑𝑎𝑛 𝛽 > 0 0 𝑢𝑛𝑡𝑢𝑘 𝑥 ≤ 𝑚 − 𝑎 𝑎𝑡𝑎𝑢 𝑥 ≥ 𝑚 + 𝛽 2.3 fungsi aktivasi sistem persamaan linear fully fuzzy dapat ditulis menjadi bentuk perkalian matriks fuzzy. sistem persamaan linier fully fuzzy merupakan sebuah sistem persamaan linier yang semua parameternya dalam bentuk fuzzy [4]. definisi 3: matriks �̃�= (�̃�𝑖𝑗) disebut dengan matriks fuzzy, jika setiap elemen �̃� adalah bilangan fuzzy. sebuah matriks fuzzy �̃� bernilai positif yang dinotasikan dengan �̃� > 0, jika setiap elemen positif. kita dapat mengatakan matriks fuzzy 𝑛 × 𝑛 �̃� = (�̃�𝑖𝑗)𝑛×𝑛 , yang mana �̃�𝑖𝑗 = (𝑎𝑖𝑗,𝛼𝑖𝑗,𝛽𝑖𝑗) , dengan notasi baru �̃� = (𝐴,𝑀,𝑁) dimana 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛 .𝑀 = (𝑚𝑖𝑗)𝑛×𝑛 dan 𝑁 = (𝑛𝑖𝑗)𝑛×𝑛 adalah matriks tegas. definisi 4 dua bilangan fuzzy dengan matriks �̃� = (𝑎1,𝑎2,𝑎3) dan �̃� = (𝑏1,𝑏2,𝑏3) dikatakan sama, jika dan hanya jika 𝑎1 = 𝑏1, 𝑎2 = 𝑏2 dan 𝑎3 = 𝑏3. definisi 5 (jika �̃� = (𝑎1,𝑎2,𝑎3) > 0, �̃� = (𝑏1,𝑏2,𝑏3) > 0, maka: �̃� ⊗ �̃� = (𝑎1,𝑎2,𝑎3) ⊗ (𝑏1,𝑏2,𝑏3) ≅ (𝑎1𝑏1,𝑏1𝑎2 + 𝑎1𝑏2,𝑏1𝑎3 + 𝑎3𝑏1). definisi 6 misalkan sistem persamaan linear fuzzy 𝑛 × 𝑛 sebagai berikut: (�̃�11 ⊗ �̃�1) ⊕ (�̃�12 ⊗ �̃�2) ⊕ … ⊕ (�̃�1𝑛 ⊗ �̃�𝑛) = �̃�1 (�̃�21 ⊗ �̃�1) ⊕ (�̃�22 ⊗ �̃�2) ⊕ … ⊕ (�̃�2𝑛 ⊗ �̃�𝑛) = �̃�2 ⋮ ⋮ ⋮ (�̃�𝑛 1 ⊗ �̃�1) ⊕ (�̃�𝑛2 ⊗ �̃�2) ⊕ … ⊕ (�̃�𝑛𝑛 ⊗ �̃�𝑛) = �̃�𝑛 bentuk matriks dari persamaan diatas adalah: �̃� ⊗ �̃� = �̃� dari bentuk diatas dapat diartikan bahwa matriks koefisien semua parameternya dalam bentuk bilangan fuzzy [5]. dimana matriks koefisien �̃�= (�̃�𝑖𝑗) ,1 ≤ 𝑖 , 𝑗 ≤ 𝑛 adalah matriks fuzzy 𝑛 × 𝑛 dan �̃�𝑗, �̃�𝑗 ∈ 𝐹(𝑅), dimana 𝐹(𝑅) adalah himpunan bilangan fuzzy segitiga. sistem ini disebut sistem linear fully fuzzy. solusi sistem persamaan linear fully fuzzy �̃� ⊗ �̃� = �̃�, diperoleh dari tiga sistem persamaan linear berikut: 𝐴𝑥 = 𝑏 𝐴𝑦 + 𝑀𝑥 = 𝑔 𝐴𝑧 + 𝑁𝑥 = ℎ (1) diasumsikan bahwa adalah sebuah matriks nonsingular maka diperoleh solusi sebagai berikut: 𝐴𝑥 = 𝑏 ⟹ 𝑥 = 𝐴−1𝑏 𝐴𝑦 + 𝑀𝑥 = 𝑔 ⟹ y = 𝐴−1(𝑔 − 𝑀𝑥) 𝐴𝑧 + 𝑁𝑥 = ℎ ⟹ 𝑧 = 𝐴−1(ℎ − 𝑁𝑥) 2.4 mape misalkan 𝑉adalah ruang hasil kali dalam. vektor-vektor 𝑢,𝑣 ∈ 𝑉 dan 𝑢 dikatakan ortogonal terhadap 𝑣 jika 〈𝑢,𝑣〉 = 0. berikut akan diberikan definisi tentang ortogonal. definisi 7: vektor 𝑢,𝑣 ∈ 𝑅𝑛 dikatakan ortogonal jika dan hanya jika 〈𝑢,𝑣〉 = 0. berikut akan diberikan teorema basis ortonormal. teorema 8 jika 𝑆= 𝑣1, 𝑣2,…, 𝑣𝑛 adalah basis ortonormal untuk ruang hasil kali dalam 𝑉 , dan 𝑢 adalah sebarang vektor dalam 𝑉, maka [1] 𝑢 = 〈𝑢,𝑣1〉𝑣1 + 〈𝑢,𝑣2〉𝑣2 + ⋯ + 〈𝑢,𝑣𝑛〉𝑣𝑛 2.5 nilai eigen dan vektor eigen untuk mencari nilai eigen matriks 𝐴 yang berukuran 𝑛 × 𝑛 maka kita menuliskannya sebagai berikut: 𝐴𝑥 = 𝜆𝑥 atau 𝐴𝑥 − 𝜆i 𝑥 = 0 dan persamaan di atas akan mempunyai penyelesaian jika |𝐴 − 𝜆i| = 0 persamaan di atas disebut sebagai persamaan karakteristik 𝐴. mencari nilai eigen berarti menghitung determinan tersebut sehingga diperoleh nilai-nilai 𝜆 . jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 146 berikut akan diberikan definisi tentang nilai eigen dan vektor eigen. definisi 9: jika 𝐴 adalah matriks 𝑛x𝑛 , maka vektor tak nol 𝑥 di dalam 𝑅𝑛 dinamakan vektor eigen dari 𝐴 jika 𝐴𝑥 adalah kelipatan skalar dari 𝑥 , yaitu: 𝐴𝑥 = 𝜆𝑥, untuk suatu skalar 𝜆 . skalar 𝜆 disebut nilai eigen dari 𝐴 dan 𝑥 dikatakan vektor eigen yang bersesuaian dengan 𝜆[6]. 2.6 metode singular value decomposition (svd) singular value decomposition (svd) adalah suatu metode yang mendekomposisikan matriks 𝐴 menjadi tiga komponen yaitu 𝑈𝑆𝑉𝑇, yang mana salah satu dari matriks tersebut entrinya merupakan nilai singular dari matriks 𝐴 [7]. berikut ini akan diberikan penjelasan tentang matriks 𝑈, 𝑆, dan 𝑉: matriks 𝑆 adalah matriks 𝑆 disebut matriks nilai singular dari 𝐴 karena entri diagonal dari matriks 𝑆 diisi dengan nilai singular dari 𝐴 sedangkan entri selain diagonalnya adalah nol. matriks 𝑆 berukuran 𝑚𝑥𝑛 dan mempunyai bentuk: 𝑏 = 𝑝𝑟𝑜𝑦𝑅(𝐴)𝑏 sehingga menurut persamaan (3) diperoleh persamaan: 𝑏 = ∑ 〈𝑏,𝑢𝑘〉 𝑟 𝑘=1 𝑢𝑘 karena 𝑢𝑘 = 1 𝜎𝑘 𝐴𝑣𝑘, maka 𝑏 = ∑ 〈𝑏,𝑢𝑘〉 𝑟 𝑘=1 𝐴𝑣𝑘 𝜎𝑘 operasi matriks bersifat linier, maka persamaan diatas dapat ditulis menjadi: 𝑏 = 𝐴∑ 〈𝑏,𝑢𝑘〉 𝑟 𝑘=1 𝑣𝑘 𝜎𝑘 (4) dengan membandingkan persamaan (4) dengan persamaan (2), didapatkan 𝑥 = ∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑟 𝑘=1 𝑣𝑘 (5) yang merupakan solusi dari spl fully fuzzy pada persamaan (1). tetapi, nilai solusi dari sistem linier bergantung pada ruang nol dari matriks 𝐴 yaitu 𝑁(𝐴). sehingga ada dua subkasus, yaitu: a. jika 𝑁(𝐴) = {0}, maka sistem persamaan linear fully fuzzy mempunyai satu solusi atau solusi tunggal, yang mana solusinya diberikan oleh persamaan (5). b. jika 𝑁(𝐴) ≠ {0} , maka sistem persamaan linear fully fuzzy mempunyai banyak solusi. solusinya diberikan oleh: 𝑥𝑖𝑛𝑓 = ∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑟 𝑘=1 𝑣𝑘 + ∑ 𝜇𝑘𝑣𝑘 𝑛 𝑘=𝑟+1 (6) yang diperoleh dari: setiap solusi umum dari spl dapat dinyatakan dengan 𝑋 = 𝑥 + 𝑥𝑛, dimana 𝑥𝑛 ∈ 𝑁(𝐴). pada subkasus a, 𝑁(𝐴) = {0} sehingga 𝑋 = 𝑥. namun karena pada kasus 𝑁(𝐴) ≠ {0}, maka terdapat titik 𝑥𝑛 ∈ 𝑁(𝐴) sedikimikian sehingga 𝐴𝑥𝑁 = 0. jadi, solusi umum untuk kasus ini adalah 𝑋 = 𝑥 + 𝑥𝑁, atau disini dinotasikan dengan 𝑥𝑖𝑛𝑓 = 𝑥 + 𝑥𝑛 (7) dengan demikian, untuk setiap titiktitiknya berlaku 𝐴(𝑥𝑖𝑛𝑓) = 𝐴(𝑥 + 𝑥𝑛) = 𝐴𝑥 + 𝐴𝑥𝑛 = 𝑏 + 0 = 𝑏. setiap titik-titik 𝑥𝑁 dapat dinyatakan sebagai kombinasi linier dari vektor basis. karena {𝑣𝑟+1,𝑣𝑟+2,…,𝑣𝑛} merupakan basis untuk 𝑁(𝐴), maka 𝑥𝑁 dapat dinyatakan dengan 𝑥𝑛 = ∑ 𝜇𝑘𝑣𝑘 𝑛 𝑘=𝑟+1 (8) sebelumnya telah diketahui dari persamaan (5), sehingga 𝑥𝑖𝑛𝑓 = 𝑥 + 𝑥𝑛 dapat dinyatakan dengan 𝑥𝑖𝑛𝑓 = ∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑟 𝑘=1 𝑣𝑘 + ∑ 𝜇𝑘𝑣𝑘 𝑛 𝑘=𝑟+1 kasus untuk 𝑏 ∉ 𝑅(𝐴) pada kasus ini sistem tidak mempunyai solusi, sehingga hanya bisa dihitung pendekatan terbaik dari solusinya. dalam hal ini, solusi pendekatan terbaik tersebut adalah vektor 𝑥𝑟 sehingga 𝐴𝑥𝑟 = 𝑏𝑟 dimana 𝑏𝑟 dalam 𝑅(𝐴), dan 𝑏𝑟 adalah vektor yang terdekat dengan 𝑏. solusi pendekatan terbaik pada kasus ini diberikan juga oleh persamaan (4), yaitu: 𝑥𝑟 = ∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑟 𝑘=1 𝑣𝑘 (9) jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 147 𝑥𝑟 disebut sebagai solusi pendekatan terbaik, artinya jika 𝐴𝑥𝑟 = 𝑏𝑟, maka 𝑏𝑟 adalah vektor di 𝑅(𝐴) yang terdekat dengan 𝑏. sehingga vektor (𝑏 − 𝑏𝑟) akan tegak lurus dengan setiap vektor di 𝑅(𝐴) termasuk vektor yang merentang 𝑅(𝐴) yaitu vektor –vektor 𝑢𝑖 dengan 1 ≤ 𝑖 ≤ 𝑟, 𝑢𝑖 adalah vektor yang ortonormal, maka berlaku: 〈(𝑏 − 𝑏𝑟),𝑢𝑖〉 = 〈(𝑏 − 𝐴𝑥𝑟),𝑢𝑖〉 = 〈(𝑏 − 𝐴(∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑟 𝑘=1 𝑣𝑘)),𝑢𝑖〉 = 〈𝑏,𝑢𝑖〉 − 〈𝑏,𝑢𝑖〉 = 0 hal ini menunjukkan bahwa (𝑏 − 𝐴𝑥𝑟) adalah tegak lurus dengan setiap vektor di 𝑅(𝐴) dan persamaan (10) merupakan solusi pendekatan terbaik [8]. contoh 1 diberikan sistem persamaan linear fully fuzzy sebagai berikut: (19,1,1)⨂(𝑥1, 𝑦1, 𝑧1)⨁(12,1.5,1.5) ⨂(𝑥2, 𝑦2,𝑧2)⨁(6,0.5,0.2)⨂(𝑥3, 𝑦3, 𝑧3) = (1897,427.7,536.2) (2,0.1,0.1)⨂(𝑥1, 𝑦1, 𝑧1)⨁(4,0.1,0.4) ⨂(𝑥2, 𝑦2,𝑧2)⨁(15,0.2,0.2)⨂(𝑥3, 𝑦3, 𝑧3) = (434.5,76.2,109.3) (2,0.1,0.2)⨂(𝑥1, 𝑦1, 𝑧1)⨁(2,0.1,0.3) ⨂(𝑥2, 𝑦2,𝑧2)⨁(4.5,0.1,0.1)⨂(𝑥3, 𝑦3, 𝑧3) = (535.5,88.3,131.9) carilah solusi dari spl fully fuzzy diatas. penyelesaian: 1. mengubah bentuk persamaan ke dalam matriks �̃�⨂�̃� = �̃� dimana �̃� = (𝐴,𝑀,𝑁) dan �̃� = (𝑏,𝑔,ℎ), dengan: 𝐴 = [ 19 12 6 2 4 1.5 2 2 4.5 ] 𝑀 = [ 1 1.5 0.5 0.1 0.1 0.2 0.1 0.1 0.1 ] 𝑁 = [ 1 1.5 0.2 0.1 0.4 0.2 0.2 0.3 0.1 ] 𝑏 = [ 1897 434.5 535.5 ] 𝑔 = [ 427.7 76.2 88.3 ] ℎ = [ 536.2 109.3 131.9 ] selanjutnya ubah matriks tersebut ke dalam bentuk sistem persamaan linear pada persamaan (1) sebagai berikut: 𝐴𝑥 = 𝑏 [ 19 12 6 2 4 1.5 2 2 4.5 ][ 𝑥1 𝑥2 𝑥3 ] = [ 1897 434.5 535.5 ] (10) 𝐴𝑦 + 𝑀𝑥 = 𝑔 [ 19 12 6 2 4 1.5 2 2 4.5 ][ 𝑦1 𝑦2 𝑦3 ] + [ 1 1.5 0.5 0.1 0.1 0.2 0.1 0.1 0.1 ][ 𝑥1 𝑥2 𝑥3 ] = [ 427.7 76.2 88.3 ] (11) 𝐴𝑧 + 𝑁𝑥 = ℎ [ 19 12 6 2 4 1.5 2 2 4.5 ][ 𝑧1 𝑧2 𝑧3 ] + [ 1 1.5 0.2 0.1 0.4 0.2 0.2 0.3 0.1 ][ 𝑥1 𝑥2 𝑥3 ] = [ 536.2 109.3 131.9 ] (12) dari persamaan (10) maka persamaan yang terbentuk adalah sebagai berikut: 19𝑥1 + 12𝑥2 + 6𝑥3 = 1897 2𝑥1 + 4𝑥2 + 1.5𝑥3 = 434.5 2𝑥1 + 2𝑥2 + 4.5𝑥3 = 535.5 2. mencari nilai eigen dan vektor eigen didapat nilai-nilai eigen dari 𝐴𝐻𝐴 adalah 𝜆1 = 4.4403, 𝜆2 = 14.0393 dan 𝜆3 = 573.0204 didapat vektor eigen untuk 𝜆1 = 4.4403, yaitu: 𝑥1 = [0.4401 −0.8372 0.3229] didapat vektor eigen untuk 𝜆2 = 14.0393, yaitu: 𝑥2 = [−0.4099 0.1325 0.9024] didapat vektor eigen untuk 𝜆3 = 573,0204, yaitu: 𝑥3 = [0.7960 0.5295 0.2852] jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 148 3. mendekomposisikan matriks 𝐴 menjadi tiga komponen matriks 𝑈𝑆𝑉𝐻 a. menyusun matriks 𝑆 nilai singular dari matriks 𝐴, yaitu: 𝜎1 = √𝜆1 = √4.4403 = 2.1072 𝜎2 = √𝜆2 = √14.0393 = 3.7469 𝜎3 = √𝜆3 = √573.0204 = 23.9378 matriks σ yang terbentuk adalah: σ = [ 2.1072 0 0 0 3.7469 0 0 0 23.9378 ] maka didapat matriks singular 𝑆, yaitu: 𝑆 = [ 2.1072 0 0 0 3.7469 0 0 0 23.9378 ] b. menyusun matriks 𝑉 dengan persamaan 𝑣𝑖 = 1 ‖𝑥𝑖‖ 𝑥𝑖 𝑉 = [ 0.4389 −0.4099 0.7977 −0.8349 0.1325 0.5307 0.3220 0.9024 0.2859 ] c. menyusun matriks 𝑈 dengan persamaan 𝑢𝑖 = 1 𝜎1 𝐴𝑣𝑖 𝑈 = [ 0.1197 −0.2092 0.9710 −0.9391 0.2839 0.1732 0.3118 0.9357 0.1647 ] sehingga bentuk svd dari matriks 𝐴 adalah: 𝐴 = 𝑈𝑆𝑉𝐻 𝐴 = [ 17.8112 12.0209 6.0192 1.7936 3.9921 1.5074 1.7991 2.0083 4.5025 ] 4. menentukan basis-basis ortonormal untuk 𝑅(𝐴),𝑅(𝐴𝐻),𝑁(𝐴) dan 𝑁(𝐴𝐻) a. untuk basis 𝑅(𝐴) adalah {𝑢1,𝑢2} = {[ 0.1197 −0.9391 0.3118 ],[ −0.2092 0.2839 0.9357 ]} b. untuk basis 𝑅(𝐴𝐻) adalah {𝑣1,𝑣2} = {[ 0.4389 −0.8349 0.3220 ],[ −0.4099 0.1325 0.9024 ]} c. untuk basis 𝑁(𝐴) adalah {𝑣3} = {[ 0.7977 0.5307 0.2859 ]} d. untuk basis 𝑁(𝐴𝐻) adalah {𝑢3} = {[ 0.9710 0.1732 0.1647 ]} 5. menentukan apakah 𝑏 sama dengan proyeksi 𝑏 pada 𝑅(𝐴) 𝑝𝑟𝑜𝑦𝑅(𝐴)𝑏 = ∑〈𝑏,𝑢𝑘〉𝑢𝑘 3 𝑘=1 = 〈𝑏,𝑢1〉𝑢1 + 〈𝑏,𝑢2〉𝑢2 + 〈𝑏,𝑢3〉𝑢3 = [ −1.6757 13.1466 −4.3650 ] + [ −47.6075 64.6070 212.9368 ] + [ 1947.2434 347.3353 330.2894 ] = [ 1897,9602 425,0889 538,8612 ] berdasarkan perhitungan tersebut diperoleh 𝑝𝑟𝑜𝑦𝑅(𝐴)𝑏 ≠ 𝑏 = [ 1897 434.5 535,5 ]. karena 𝑝𝑟𝑜𝑦𝑅(𝐴)𝑏 ≠ 𝑏, berarti 𝑏 ∉ 𝑅(𝐴). hal tersebut menandakan spl ini tidak mempunyai solusi. akan tetapi solusi pendekatan terbaiknya dapat dicari, yaitu: 𝑥𝑟 = ∑ 〈𝑏,𝑢𝑘〉 𝜎𝑘 𝑣𝑘 3 𝑘=1 = 〈𝑏,𝑢1〉 𝜎1 𝑣1 + 〈𝑏,𝑢2〉 𝜎2 𝑣2 + 〈𝑏,𝑢3〉 𝜎3 𝑣3 = [ −2.9158 5.5466 −2.1392 ] + [ −24.8954 8.0475 54.8076 ] + [ 66.8277 44.4596 23.9514 ] = [ 39.0165 58.0537 76.6198 ] jadi solusi pendekatan terbaik dari spl ini adalah: 𝑥1 = 39.0165, 𝑥2 = 58.0537 dan 𝑥3 = 76.6198. dengan cara yang sama kita dapat memperoleh solusi terbaik untuk nilai 𝑦1 = 7.5421, 𝑦2 = 8.8263 dan 𝑦3 = 10.3688 serta nilai 𝑧1 = 13,7228, 𝑧2 = 4,0627, dan 𝑧3 = 14,2515. referensi jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 149 [1] haidir ahmad, irdam dan lucia ratnasari. “menyelesaikan sistem persamaan linear menggunakan analisis svd”. jurnal matematika vol. 13, no.1;40-45. 2010.. [2] rosalina. f, farida. y, and hamid. a. “metode logika fuzzy sebagai evaluasi distribusi daya listrik berdasarkan beban puncak pembangkit tenaga listrik”, mantik, vol. 2, no. 1, : 22-29.2016. [3] howard, anton. “elementary linear algebra”, eighth edition. john wiley, new york. 2000. [4] kholifah. penyelesaian sistem persamaan linear fully fuzzy menggunakan metode gauss saidel. pekanbaru: skripsi jurusan matematika universitas islam negeri sultan syarif kasim riau. 2013. [5] k. jaikumar and s. sunantha. “sst decomposition methot for solving fully fuzzy linear systems”.int. j. industrial mathematics. vol. 5, no. 4. 2013. [6] sutojo, t. dkk. “teori dan aplikasi aljabar linear dan matriks”. andi: yogyakarta. 2010. [7] marni, sabrina indah. penyelesaian sistem persamaan linear fuzzy menggunakan metode dekomposisi nilai singular (svd): skripsi jurusan matematika universitas islam negeri sultan syarif kasim riau. 2013. [8] marzuki, corry corazon dan herawati. “penyelesaian sistem persamaan linear fully fuzzy menggunakan iterasi jacobi”. vol. 1, no.1, issn: 2460 -4542, januari 2015, pekanbaru. (2015). paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 68 identifikasi citra daging ayam berformalin menggunakan metode fitur tekstur dan k-nearest neighbor (k-nn) abstrak penelitian bertujuan untuk membuat sistem identifikasi daging ayam segar untuk mendeteksi perbedaan antara daging ayam berformalin dan tidak berformalin berdasarkan citra daging ayam mentah. metode ekstraksi ciri yang digunakan adalah metode fitur tekstur yang termasuk dalam metode statistik dimana dalam perhitungan statistiknya menggunakan distribusi derajat keabuan (histogram) dengan mengukur tingkat kekontrasan, granularitas, dan kekasaran suatu daerah dari hubungan ketetanggaan antar piksel di dalam citra kemudian dilakukan ekstraksi fitur, hasil ekstraksi fitur kemudian diklasifikasikan oleh k-nn. dengan klasifikasi menggunakan k-nn diperoleh hasil akurasi klasifikasi yang tinggi. metode k-nn merupakan metode yang sangat baik dalam menangani masalah pengenalan pola-pola komplek dalam bentuk pelatihan data dan kalibrasi pengolahan, berdasarkan literatur metode yang sangat cepat dan tinggi akurat lebih dari metode lain. gambar pengamatan akan dilakukan pada beragam jarak antara kamera smartphone dan sampel daging ayam. kata kunci: ayam, fitur tekstur, k-nn abstract the research aimed to create a fresh chicken meat identification system to detect differences between formalin and non-formalin chicken meat based on the image of raw chicken meat. feature extraction method used is the feature texture method which is included in the statistical method where the statistical calculation uses a gray degree distribution (histogram) by measuring the level of contrast, granularity, and roughness of an area from the neighboring relationships between pixels in the image then feature extraction, results feature extraction is then classified by k-nn. with the classification using k-nn results obtained high classification accuracy. the k-nn method is a very good method of dealing with the problem of recognizing complex patterns in the form of data training and processing calibration, based on very fast and high accurate literature methods more than other methods. observation images will be carried out at various distances between the smartphone camera and chicken meat samples. keywords: chicken, texture feature, k-nn 1. pendahuluan ayam adalah hewan unggas yang paling umum di seluruh dunia, dan telah diternakkan dan dikonsumsi sudah selama ribuan tahun lalu. manfaat makan daging ayam bagi kesehatan jelas sangat tinggi,karena daging ayam mengandung protein tinggi. tingginya kebutuhan masyarakat akan daging ayam setiap harinya menyebabkan banyak para pedagang daging ayam mencoba mengoplos daging segar dengan daging yang sudah rusak untuk memperoleh keuntungan yang lebih besar walaupun dengan cara yang tidak dibenarkan atau tidak halal, hal ini tentu sangat merugikan konsumen. kebanyakan masyarakat awam terutama ibu-ibu rumah tangga tidak mengetahui daging ayam mentah yang telah terkontaminasi formalin dan daging ayam mentah yang doi: https://doi.org/10.15642/mantik.2018.4.1.68-74 faris mushlihul amin universitas islam negeri sunan ampel surabaya1, faris@uinsby.ac.id jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 tidak mengandung formalin sehingga setelah dikonsumsi akan mengakibatkan penyakit yang dapat menyebabkan kematian. selama ini untuk mengetahui sebuah produk makanan mengandung zat kimia berbahaya atau tidaknya seperti formalin, hanya dilakukan dengan pengujian di laboratorium kimia. sampel produk makanan yang diduga mengandung zat kimia berbahaya seperti formalin akan dibawa ke laboratorium terlebih dahulu karena proses pengujiannya tidak dapat dilakukan di tempat produk makanan itu diproduksi atau dipasarkan. sehingga hasil dari pengujian sampel makanan tersebut tidak dapat kita ketahui secara cepat, karena harus menunggu hasil yang dikeluarkan oleh petugas laboratorium kimia. penelitian untuk membuktikan produk makanan yang berformalin maupun tidak berformalin telah dilakukan dengan berbagai objek dan metode. metode pendeteksian yang banyak digunakan adalah metode pengolahan citra dan metode jaringan saraf tiruan, dari metode tersebut dapat diketahui nilai acuan untuk mengetahui makanan tersebut mengandung formalin atau tidak formalin. untuk mengenali obyek dalam citra dibutuhkan parameter parameter yang mencirikan obyek tersebut. ciri yang dapat digunakan untuk membedakan obyek satu dengan obyek lainnya di antaranya adalah ciri bentuk, ciri ukuran, ciri geometri, ciri tekstur, dan ciri warna. tujuan yang ingin dicapai dalam penelitian ini adalah untuk membangun aplikasi berbasis matlab yang dapat membedakan antara daging ayam berformalin dan tidak berformalin dengan menerapkan metode fitur tekstur dan k-nn menggunakan matlab r2014a. adapun manfaat yang ingin didapatkan dari penelitian ini adalah hasil penelitian ini diharapkan dapat memberikan kontribusi dan masukan bagi siapa saja yang membutuhkan informasi yang berhubungan dengan judul penelitian ini.dan mengetahui bagaimana proses penerapan pengolahan citra digital dan metode fitur tekstur dan k-nn, serta membantu masyarakat atau para konsumen untuk dapat membedakan daging ayam berformalin atau tidak mengandung formalin. 2. tinjauan pustaka 2.1. penelitian terdahulu penelitian sejenis ini sebelumnya pernah dilakukan oleh beberapa peneliti sebagai berikut : penelitian yang dilakukan oleh arie qur’ania dkk yang berjudul analisis tekstur dan ekstraksi fitur warna untuk klasifikasi apel berbasis citra betujuan untuk mengklasifikasi jenis apel. metode yang digukan dalam penelitian tersebut yakni k-nearest neighbor (k-nn) dengan nilai parameter k yang digunakan yakni k=1 sampai k=3. ekstraksi fitur yang digunakan yakni fitur tekstur dan fitur warna rgb(red, green, dan blue). hasil penelitian menunjukkan tingkat akurasi sebesar 93,33% untuk fitur homogenitas, 73,33% untuk fitur tekstur dan 100% untuk fitur rgb[1]. penelitian selanjutnya dilakuan oleh retno nugroho whidhiasih dkk, dalam penelitiannya yang berjudul klasifikasi buah belimbing berdasarkan citra redgreen-blue menggunakan knn dan lda. penelitian tersebut membandingkan metode klasifikasi k-nearest neigbourhood (knn) dan linear discriminant analysis (lda) dengan variabel r-g dan r-g-b dari citra buah belimbing untuk memprediksi tingkat kemanisan buah belimbing[2] penelitian serupa dilakukan oleh oky dwi nurhayati dalam penelitiannya yakni sistem analisis tekstur secara statistik orde pertama untuk mengenali jenis telur ayam biasa dan telur ayam omega-3. metode pengolahan citra yang digunakan pada penelitian tersebut meliputi pengubahan nilai keabuan, peningkatan kontrast citra, penapisan dengan menggunakan filter gaussian, ekualisasi histogram, segmentasi thresholding, dan klasifikasi dengan pendekatan statistik orde pertama dapat 69 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 digunakan sebagai sarana analisis untuk membedakan jenis telur ayam omega-3 maupun telur ayam biasa[3]. sedangkan menurut yufika agustyani, sri setyaningsih, arie qur’ania(2014) dalam penelitiannya model deteksi kandungan formalin pada ikan dengan citra (hsv) menggunakan k-nearest neighbor mengatakan identifikasi ikan yang mengandung formalin. tujuan dari penelitian ini adalah merancang dan membangun model yang mampu mendeteksi kandungan formalin pada ikan melalui citra ikan tersebut[1] 2.2. landasan teori 1. ayam ayam merupakan salah satu ternak unggas yang sudah tidak asing lagi dikalangan masyarakat. ayam yang digunakan oleh masyarakat untuk diolah biasanya adalah ayam potong, untuk memilih daging ayam segar yang biasa perlu diperhatikan beberapa hal, yaitu warna daging yang putih kekuningan, warna lemak yang putih kekuningan dan merata di bawah kulit, memiliki bau yang segar, kekenyalan yang elastis dan tidak ada tanda-tanda memar atau tanda lain yang mencurigakan. daging ayam termasuk mengandung gizi yang tinggi, selain dari proteinnya juga daging ayam mengandung lemak. protein pada ayam yaitu 18,2g sedangkan lemaknya berkisar 25,0 g. 2. formalin formalin adalah salah satu zat kimia yang berbahaya, yang penggunaan dilarang oleh pemerintah. di dalam formalin biasanya ditambahkan metanol hingga 15% sebagai pengawet. dan atas dasar inilah formalin digunakan sebagai pengawet bahan makanan, baik dalam bentuk olahan ataupun segar seperti daging ayam yang masih segar. selain itu formalin juga dikenal sebagai bahan pembunuh hama dan banyak digunakan dalam industri. penggunaan formalin untuk mengawetkan makanan sesungguh nya telah dilarang sejak tahun 1982. pemerintah juga telah mengeluarkan dua peraturan untuk mengatur penggunaan bahan kimia ini. yaitu peraturan menteri kesehatan nomor 472 tahun 1996 tentang pengamanan bahan berbahaya bagi kesehatan, dan keputu-san menteri perindustrian dan perdagangan nomor 254 tahun2000 tentang tata niaga impor dan peredaran bahan berbahaya tertentu. formalin dan rodamin termasuk dalam kategori bahan berbahaya tersebut yang tenggunaan-nya harus diawasi secara ketat. 2.3. teori dasar 1.citra citra (image) adalah gambar pada bidang dwimatra (dua dimensi). ditinjau dari sudut pandang matematis, citra merupakan fungsi menerus (continue) dari intensitas cahaya pada bidang dwimatra. citra sebagai keluaran dari suatu sistem perekaman data dapat bersifat optik berupa foto, analog berupa sinyal video seperti gambar pada monitor televisi dan digital yang dapat langsung disimpan pada suatu pita magnetik. 2. pengolahan citra pengolahan citra merupakan kegiatan memperbaiki kualitas citra agar mudah di interpretasi oleh manusia / mesin (komputer). inputannya adalah citra dan lebih baik daripada citra masukan → misal citra warnanya kurang tajam, kabur (blurring), mengandung noise (misal bintik-bintik putih), dll sehingga perlu ada pemrosesan untuk memperbaiki citra karena citra tersebut menjadi sulit diinterpretasikan karena informasi yang disampaikan menjadi berkurang. 3. ekstrasi ciri ekstraksi ciri statistik analisa tekstur lazim dimanfaatkan sebagai proses antara untuk melakukan klasifikasi dan interpretasi citra. suatu proses klasifikasi citra berbasis analisis tekstur pada umum –nya membutuhkan ekstraksi ciri, yang dapat terbagi dalam empat macam metode yaitu: statistical, geometri, model-based, dan signal processing[4]. 70 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 4. ekstraksi ciri fitur tekstur ekstraksi ciri tekstur analisa tekstur lazim dimanfaatkan sebagai proses antara untuk melakukan klasifikasi dan interpretasi citra. suatu proses klasifikasi citra berbasis analisis tekstur pada umumnya membutuhkan ekstraksi ciri, yang dapat terbagi dalam empat macam metode yaitu: 1. geometri metode ini digunakan untuk mendeskripsikan atau mejelaskan elemen-elemen tekstur dan penempatan kaidah untuk menjelaskan organisasi spasisal diantara elemen-elemen. 2. model-based metode ini biasanya berdasarkan pada sebuah gambar dari sebuah model gambar. base model dapat digunakan untuk menjelaskan dan menkombinasikan tekstur. 3. signal processing metode ini berdasarkan pada analisis frekuensi pada sebuah gambar.ektraksi ciri dapat diilustrasikan ke dalam dua bentuk yaitu histogram citra dan matrik. 5. fitur tekstur statistik fitur pertama yang dihitung secara statistis adalah rerata intensitas. komponen fitur ini dihitung berdasar persamaan 𝒎 = ∑ 𝒊 . 𝒑(𝒊)𝑳−𝟏𝒊=𝟎 (13.1) dalam hal ini, i adalah aras keabuan pada citra f dan p(i) menyatakan probabilitas kemunculan i dan l menyatakan nilai aras keabuan tertinggi.rumus di atas akan menghasilkan rerata kecerahan objek. fitur kedua berupa deviasi standar. perhitungannya sebagai berikut: 𝝈 = √∑ (𝒊 − 𝒎)𝟐𝒑(𝒊)𝑳−𝟏𝒊=𝟏 (13.2) dalam hal ini, 2 dinamakan varians atau momen orde dua ternormalisasi karena p(i) merupakan fungsi peluang. fitur ini memberikan ukuran kekontrasan. fitur skewness merupakan ukuran ketidaksimetrisan terhadap rerata intensitas.definisinya : 𝒔𝒌𝒆𝒘𝒏𝒆𝒔𝒔 = ∑ (𝒊 − 𝒎)𝟑𝒑(𝒊)𝑳−𝟏𝒊=𝟏 (13.3) skewness sering disebut sebagai momen orde tiga ternormalisasi.nilai negatif menyatakan bahwa distribusi kecerahan condong ke kiri terhadap rerata dan nilai positif menyatakan bahwa distribusi kecerahan condong ke kanan terhadap rerata.dalam praktik, nilai skewness dibagi dengan (l-1)2 supaya ternormalisasi. deskriptor energi adalah ukuran yang menyatakan distribusi intensitas piksel terhadap jangkauan aras keabuan. definisinya sebagai berikut: 𝒆𝒏𝒆𝒓𝒈𝒊 = ∑ [𝒑(𝒊)]𝟐𝑳−𝟏𝒊=𝟎 (13.4) citra yang seragam dengan satu nilai aras keabuan akan memiliki nilai energi yang maksimum, yaitu sebesar 1. secara umum, citra dengan sedikit aras keabuan akan memiliki energi yang lebih tinggi daripada yang memiliki banyak nilai aras keabuan. energi sering disebut sebagai keseragaman. entropi mengindikasikan kompleksitas citra. perhitungannya sebagai berikut: 𝒆𝒏𝒕𝒓𝒐𝒑𝒊 = − ∑ 𝒑(𝒊)𝑳−𝟏𝒊=𝟎 𝐥𝐨𝐠𝟐 (𝒑(𝒊)) (13.5) semakin tinggi nilai entropi, semakin kompleks citra tersebut.perlu diketahui, entropi dan energi ber kecenderungan berkebalikan. entropi juga merepresentasikan jumlah informasi yang terkandung di dalam sebaran data. properti kehalusan biasa disertakan untuk mengukur tingkat kehalusan/kekasaran intensitas pada citra. definisinya sebagai berikut: 𝑹 = 𝟏 − 𝟏 𝟏+𝝈𝟐 (13.6) pada rumus di atas,  adalah deviasi standar.berdasarkan rumus di atas, nilai r yang rendah menunjukkan bahwa citra memiliki intensitas yang kasar. perlu 71 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 diketahui, di dalam menghitung kehalusan, varians perlu dinormalisasi sehingga nilainya berada dalam jangkauan [0 1] dengan cara membaginya dengan (l-1)2. 6. k-nearest neighbor k-nearest neighbor (knn) adalah suatu metode yang menggunakan algoritma supervised dimana hasil dari query instance yang baru diklasifikan berdasarkan mayoritas dari kategori pada knn. tujuan dari algoritma ini adalah mengklasifikasikan obyek baru berdasarkan atribut dan training sample. algoritma metode knn sangatlah sederhana, bekerja berdasarkan jarak terpendek dari query instance ke training sample untuk menentukan knn-nya. training sample diproyeksikan ke ruang berdimensi banyak, dimana masingmasing dimensi merepresentasikan fitur dari data. ruang ini dibagi menjadi bagian-bagian berdasarkan klasifikasi training sample. sebuah titik pada ruang ini ditandai kelac c jika kelas c merupakan klasifikasi yang paling banyak ditemui pada k buah tetangga terdekat dari titik tersebut. dekat atau jauhnya tetangga biasanya dihitung berdasarkan euclidean distance yang direpresentasikan sebagai berikut : 3. metode penelitian 1. pengumpulan data data yang digunakan dalam penelitian ini berupa citra daging ayam yang nantinya digunakan untuk data training dan data testing. objek dari penelitian ini adalah daging ayam segar dan daging ayam berformalin dengan ukuran 1024 piksel x 1024 piksel yang disimpan dalam bentuk jpg. proses pengambilan daging ayam berformalin dan tidak berformalin masing-masing berjumlah 50 citra sehingga didapatkan total jumlah citra daging ayam tersebut adalah 100 citra yang selanjutnya akan dibagi lagi menjadi dua yang tidak berformalin yaitu 10 buah citra untuk proses pelatihan, sedangkan selebihnya adalah 40 buah citra daging ayam mentah akan dipakai dalam proses pengujian. pada tahap berikutnya adalah mengambil data citra daging ayam. pengambilan data dilaku-kan dengan menggunakan camera smartphone. untuk hari pertama diambil 50 data citra mata daging ayam, sampel daging ayam yang di-fillet dipotong-potong berukuran ± 4 cm x 3 cm sebanyak 50 potong dengan ketebalan ± 0,5 mm. potongan sampel tersebut ditimbang ± 20 g, dan dibiarkan dalam suhu ruang selama ± 1 hari. proses berikutnya adalah memberikan formalin daging ayam yang dilakukan setelah pengambilan data pada hari kedua. daging ayam yang telah dibiarkan dalam suhu ruang direndam dalam larutan formalin dengan konsentrasi 5% selama ± 6 jam. 2. rancangan sistem pada subbab ini akan dijelaskan mengenai disain aplikasi dari sistem untuk implementasi metode pengolahan citra untuk melihat dagingayam yang tidak mengandung formalin dan yang mengandung formalin dari gambar atau citra. algoritma yang digunakan dalam sistem yang akan digambar menggunakan diagram alir atau flowchart. 72 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 gambar 3.1arsitektur sistem 4. hasil penelitian akurasi adalah tingkat ketepatan antara informasi yang diminta oleh pengguna dengan jawaban yang diberikan oleh sistem. sedangkan recall adalah tingkat keberhasilan sistem dalam menemukan kembali sebuah informasi.sedangkan di “dunia lain” seperti dunia statistika dikenal juga istilah accuray. hasil akurasi yang diperoleh confution matrix non formalin formalin prediksi non formalin 14 3 17 prediksi formalin 1 12 13 15 15 30 nilai akurasinya sebesar = (14+12) / (14+3+1+12)=26 / 30 = 0,8667 dipersentasekan 86,67% form ini berfungsi sebagai pengujian citra data uji(testing data) citra daging ayam. citra data uji juga ekstraksi ciri tekstur dan ciri warna yang kemudian hasil dari ekstraksi ciri warna tersebut akan dihitung jarak terdekat dengan data training yang telah ditentukan sebelumnya dengan menggunkan algoritma k-nearest neigbors. pada menu ini terdapat dua tombol yakni tombol pilih citra dan uji hasil. sebelum memilih citra yang akan diuji, pengguna diwajibkan mengisi nilai k yang akan digunakan pada proses klasifikasi algoritma k-nearest neighbors. tombol pilih citra berfungsi untuk memilih citra yang akan diuji terhadap data latih. tombol uji hasilberfungsi menguji data citra yang telah dipilih terhadap data latih dengan algoritmak-nearest neighbors apakah termasuk kategori formalin atau non formalin. hasil klasifikasi akan ditampilkan dalam bentuk dialog. 5. kesimpulan penelitian ini mengidentifikasi daging ayam berformalin dan tidak berformalin berdasarkan fitur tekstur. metode ekstraksi fitur tekstur yang digunakan adalah rerata intensitas, deviasi, skewness, energi, entropi, smoothness, meanr, meang, meanb, devisiasi r, devisiasi g, devisiasi b, skewness r, skewness g. hasil penelitian menunjukkan bahwa akurasi rata-rata knn dalam mengidentifikasi daging ayam berformalin dan tidak berformalin sebesar 86,67%. mulai dataset preprocessing pemisahan objek citra daging ayam dengan background ekstraksi ciri ciri tekstur : rerata intensitas, deviasi standar, skewness, energi, entropi, dan kehalusan. ciri warna rgb : rerata, deviasi standar, skewness, dan kurtosis klasifikasi k-nearest neighbor output : hasil klasifikasi selesai 73 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 6. daftar pustaka [1] qurania, arie. analisis tekstur dan estraksi fitur warna untuk klasifikasi apel berbasis citra. pakuan bogor : lokakarya komputasi dalam sains dan teknologi nuklir, 2012. 296-304. [2] whidhiasih, retno nugroho. klasifikasi buah belimbing berdasarkan citra red-green-blue menggunakan knn dan lda. : jurnal penelitian ilmu komputer, 2013. 29-35. [3] nurhayati, oky dwi. sistem analisis tekstur secara statistik orde pertama untuk mengenali jenis telur ayam biasa dan telur ayam omega-3. : jsiskom (jurnal sistem komputer), 2015. 22523456. [4] asmara, rosa andrie. identifikasi kesegaran daging sapi berdasarkan citranya dengan ekstraksi fitur warna dan teksturnya menggunakan metode gray level coccurrence matrix. malang : sentia, 2017. 20852347. 74 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 74 peramalan indeks harga konsumen dengan metode singular spectral analysis (ssa) dan seasonal autoregressive integrated moving average (sarima) deltha airuzsh lubis1, muhamad budiman johra2, gumgum darmawan3 magister statistika terapan, universitas padjajaran, jl. dipati ukur no.35, bandung 40132 delthalubis@gmail.com1, muhamadbudiman8271@gmail.com2, gumstat@gmail.com3 doi:https://doi.org/10.15642/mantik.2017.3.2.74-82 abstrak indeks harga konsumen (ihk) merupakan indikator yang digunakan untuk mengukur inflasi maupun deflasi dari sekelompok barang dan jasa secara umum. peramalan ihk menjadi penting sebagai deteksi dini dalam menghadapi lonjakan harga. penelitian ini menggunakan metode ssa dan sarima. sarima merupakan model parametrik yang membutuhkan berbagai asumsi sedangkan ssa merupakan teknik nonparametrik yang bebas dari berbagai asumsi namun kedua metode tersebut mensyaratkan adanya pola musiman pada data. berdasarkan hasil penelitian, metode ssa dengan length window (l) sebesar 24 dan grouping sebanyak 4 (1 kelompok trend dan 3 kelompok musiman) dan model sarima berorde (0,1,1)(0,1,1)6 merupakan model yang tepat dan andal dalam peramalan ihk kota padangsidempuan. peramalan ihk kota padangsidimpuan untuk 5 bulan ke depan dengan metode ssa dan sarima (0,1,1)(0,1,1)6 menunjukkan pola tren yang cenderung meningkat tetapi peramalan pada bulan ke-5 dengan metode ssa menunjukkan lonjakan nilai ihk yang tinggi atau akan tejadi inflasi yang tinggi. kata kunci: arima, ihk seasonal, singular spectral analysis abstract consumer price index (cpi) are the indicators used to measure the inflation and deflation of a group of goods and services in general. forecasting cpi to be important as early detection in facing price hikes. this study uses the ssa and sarima. sarima a parametric model that requires various assumptions while ssa is a nonparametric technique that is free from a variety of assumptions, but both methods require seasonal patterns in the data. based on the research results, methods of ssa with length window(l) of 24 and a grouping of 4 (1 group of seasonal and 3 groups of ternds) and sarima models of order (0,1,1), (0,1,1) 6 is the most accurate and reliable models in forecasting cpi to the value padangsidempuan city. forecasting cpi padangsidimpuan city for the next 5 months with ssa method and sarima (0,1,1), (0,1,1) 6 shows the pattern of trend is likely to increase but forecasting the 5th month with ssa method showed a surge in the value of cpi high or high inflation will occurs. keywords: arima, cpi, seasonal, singular spectral analysis 1. pendahuluan indeks harga konsumen (ihk) merupakan nilai indeks yang menggambarkan rata-rata perubahan harga dari suatu paket barang dan jasa yang dikonsumsi oleh rumah tangga dalam kurun waktu tertentu. perubahan ihk dari waktu ke waktu menggambarkan inflasi maupun deflasi dari barang dan jasa secara umum. data ihk secara resmi dirilis oleh pemerintah melalui badan pusat statistik (bps) setiap bulannya yang mencakup 65 kota di indonesia. kebutuhan barang dan jasa pada musimmusim tertentu seperti idul fitri, tahun ajaran baru dan sebagainya selalu melonjak sehingga menyebabkan peningkatan harga pada musimmailto:delthalubis@gmail.com1 mailto:muhamadbudiman8271@gmail.com2 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 75 musim tersebut. keadaan seperti ini akan mempengaruhi perubahan nilai ihk dan membentuk pola musiman di setiap tahunnya. dalam melakukan analisis deret waktu pada data ihk sebaiknya melibatkan efek musiman tersebut, diantaranya dengan menggunakan metode ssa dan sarima. pemodelan ssa merupakan metode nonparametrik sehingga lebih fleksibel dalam penggunaannya dan terhindar dari berbagai asumsi seperti stasioneritas, independensi dan normalitas residual sebagaimana pada model sarima yang parametrik [1]. selain itu, ssa efektif dalam meminimalisir eror [2]. dalam metode ssa tidak ada model yang dibangun seperti model sarima sehingga parameter pada ssa berupa panjang window (l) dan banyaknya grouping (r). kota padangsidempuan merupakan salah satu kota yang berada di bagian paling selatan di provinsi sumatera utara yang menjadi jalur perdagangan yang penting bagi beberapa kabupaten di sekitarnya. kota ini memiliki jarak yang paling jauh ke ibukota provinsi yaitu 448 km bila dibandingkan kota lain yang menjadi kota penghitung inflasi seperti sibolga (382 km), pematangsiantar (128 km), dan medan (0 km) [3]. peramalan ihk menjadi penting untuk melihat bagaimana perkembangan ihk di kota padangsidempuan, sehingga dapat menjadi deteksi dini dalam mengatasi lonjakan harga yang mungkin terjadi. 2. kajian teori 2.1 singular spectrum analysis (ssa) ssa merupakan teknik analisis data deret waktu yang terkini dan powerful dalam menggabungkan analisis deret waktu, statistik multivariat, geometrik multivariat, sistem dinamis dan proses signal. penerapan ssa kian meluas dari bidang matematika dan fisika ke bidang ekonomi dan matematika keuangan, dari ilmu metrologi dan oseanologi ke ilmu sosial dan riset pemasaran [4]. ssa dapat mengatasi beberapa masalah seperti, menemukan trend pada resolusi yang berbeda; smoothing; mengekstrak komponen musiman; mengekstrak secara simultan pola siklis; mengekstrak periodesitas dengan amplitudo yang beragam; menemukan struktur data pada data deret waktu yang pendek dan mendeteksi perubahan titik [4]. 2.2. seasonal autoregressive integrated moving average (sarima) model arima yang melibatkan efek musiman didalamnya disebut juga dengan model sarima. secara umum, model sarima ditulis dengan persamaan berikut [6]: φ𝑃 𝐵 𝑆 𝜙𝑝(𝐵)(1 − 𝐵) 𝑑 (1 − 𝐵𝑆 )𝐷 𝑍𝑡 = 𝜃𝑞(𝐵)θ𝑄(𝐵 𝑆 )𝑎𝑡 (1) dimana: 𝜙𝑝(𝐵) : ar non seasonal φ𝑃 : ar seasonal (1 − 𝐵)𝑑 : differencing non seasonal (1 − 𝐵𝑆 )𝐷 : differencing seasonal 𝜃𝑞(𝐵) : ma non seasonal θ𝑄(𝐵 𝑆 ) : ma seasonal 2.3. penelitian terdahulu peramalan data ihk cukup banyak dilakukan dengan menggunakan model arima, salah satunya peramalan ihk bangladesh dengan model arima (1,1,1)(1,0,1)12 diperoleh akurasi peramalan berdasarkan nilai mean average percentage error (mape) sekitar 0,08 persen [5]. sementara itu, perbandingan metode ssa dan sarima oleh [4] untuk meramalkan tingkat kematian bulanan akibat kecelakaan di usa menunjukkan bahwa metode ssa memiliki akurasi paling baik dibandingkan metode sarima. hal tersebut dapat dilihat dari nilai mean relative average error (mrae) untuk ssa sekitar 2 persen dan sarima sekitar 6 persen. 3. metode penelitian 3.1. pengecekan pola musiman tahapan dalam pengujian pola musiman yaitu [7]: 1. tentukan koefisien pembeda (d) 2. jika nilai d > 0.5 maka lakukan pembedaan terlebih dahulu 3. tentukan nilai frekuensi fourier 𝜔𝑘 = 2𝜋𝑘 𝑁 , 𝑑𝑒𝑛𝑔𝑎𝑛 𝑘 = 1,2, … , 𝑁 2 (2) 4. tentukan nilai periode dari k kp  *2  (3) dari setiap k  jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 76 5. tentukan nilai periodogram (nilai periodogram akan menghasilkan nilai 𝛼 dan 𝛽 6. tentukan hipotesis penelitian yaitu h0: 𝛼 = 𝛽 (ada pola musiman) h1: 𝛼 ≠ 𝛽 (tidak ada pola musiman) statistik uji: 𝐹 = (𝑛−3)(𝛼𝑘 2+𝛽𝑘 2) 2 ∑ (𝛼𝑘 2+𝛽𝑘 2 ) 𝑛/2 𝑗−1 𝑗≠≠𝑘 (4) 𝑑𝑒𝑛𝑔𝑎𝑛 𝑗 = 1,2, … , 𝑛−1 2 𝑑𝑎𝑛 𝑘 = 𝑛 2 kriteria uji : tolak ho jika fhitung > ftabel(2,n-3;α) dengan taraf signifikansi 𝛼. 3.2. tahapan ssa tahapan dasar ssa terdiri dari 2 tahap, yaitu: dekomposisi dan rekonstruksi. tahap dekomposisi terdiri dari proses embedding dan svd. embedding mendekomposisi data time series awal ke dalam matriks lintasan yang akan membentuk pola tren, musiman, komponen bulanan, dan eror sesuai dengan nilai singularnya. tahap rekonstruksi mencakup pengelompokan yang berdasar dari dekomposisi matriks lintasan dan proses diagonal averaging untuk merkonstruksi data time series baru dari pengelompokan yang sudah dilakukan sebelumnya.  dekomposisi embedding pada tahap embedding, data time series awal x=(x1, x2, …, xn} diubah ke dalam bentuk matriks lintasan. matriks lintasan yang dibentuk adalah: 𝑻𝒊,𝒋 = ( 𝑥1 𝑥2 ⋯ ⋮ ⋮ ⋱ 𝑥𝐾 𝑥𝐾+1 ⋯ 𝑥𝐿 ⋮ 𝑥𝑁 ) (5) matriks lintasan tx berdimensi l x k dimana l adalah window length dengan 2 < l <n/2 dan k = n-l+1 singular value decomposition (svd) pada tahap ini, tx didekomposisi menjadi tx=udv’ yang disebut dengan tirple eigen. atau dapat ditulis menjadi: 𝑻𝑿 = 𝑻𝟏 + ⋯ + 𝑻𝒅 = 𝑼𝟏√𝝀𝟏𝑽𝟏 𝑻 + ⋯ + 𝑼𝒅√𝝀𝒅𝑽𝒅 𝑻 = ∑ 𝑼𝒊√𝝀𝒊𝑽𝒊 𝑻𝒅 𝒊=𝟏 (6)  rekonstruksi grouping pada tahap ini, matriks ti yang berdimensi lxk diekstrak ke dalam pola tren, musiman, komponen bulanan dan eror. tahap pengelompokan merupakan partisi himpunan dari indeks {i, …, d} ke dalam kelompok himpunan disjoin m dari i = {i1, …, im}. jadi ti berkorespondensi dengan kelompok i = {i1, …, im}. 𝑇𝐼𝑖 adalah jumlah dari tj dimana j ϵ ii. sehingga tx dapat dijabarkan menjadi: 𝑻𝑿 = 𝑻𝟏 + ⋯ + 𝑻𝑳 = 𝑻𝑰𝟏 + ⋯ + 𝑻𝑰𝒎 (7)  diagonal averaging pada diagonal averaging matriks ti yang telah dikelompokkan akan ditransformasi menjadi data deret berkala yang baru dengan panjang n dengan ketentuan berikut: �̃�𝑖,𝑗 1 𝑠 − 1 ∑ 𝑥𝑗,𝑠−𝑗 𝑠−1 𝑗=1 untuk 2 ≤ s ≤ l (8) 1 𝐿 ∑ 𝑥𝑗,𝑠−𝑗 𝐿 𝑗 =1 untuk l + 1 ≤ s ≤ k + 1 (9) 1 𝑁 − 𝑠 + 2 ∑ 𝑥𝑗,𝑠−𝑗 𝑁−𝑠+2 𝑗=𝑠−𝐾 untuk k+2 ≤ s ≤ n + 1 (10) 3.3. peramalan ssa peramalan ssa pada penelitian ini menggunakan model linear reccurance relations (lrr) bentuk polynomial yang dapat dituliskan dalam bentuk berikut: 𝑦𝑖+𝑑 = ∑ 𝑎𝑘𝑦𝑖+𝑑−𝑘 ∶ 1 ≤ 𝑖 ≤ 𝑁 − 𝑑 𝑑 𝑘=1 (11) koefisien lrf aj diperoleh dari persamaan: ℜ = (𝑎𝐿−1, 𝑎𝐿−2, … , 𝑎2, 𝑎1) 𝑇 = 1 1−𝑣2 ∑ 𝜋𝑖 𝑟 𝑖=1 𝑈 ∇ (12) 3.4. sarima sarima merupakan model parametrik sehingga diperlukan beberapa uji asumsi yang harus dipenuhi, diantaranya: a. stasioneritas. data deret waktu disebut stasioner jika nilai rata-rata dan varians konstan di sepanjang waktu. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 77 prosedur uji stasioneritas dalam mean dengan uji augmented dickey fuller (adf): h0 : ρ = 0 (terdapat unit roots, variabel z tidak stasioner) h1 : ρ ≠ 0 (tidak terdapat unit roots, variabel z stasioner) statistik uji: 𝜏 = �̂� 𝑠𝑡𝑑(𝜌)̂ (13) dan hasil statistik hitungnya dibandingkan dengan tabel 𝜏𝛼 b. uji normalitas. pengujian kenormalan dapat dilakukan dengan uji shapiro wilk. hipotesis: h0 : residual berdistibusi normal h1 : residual tidak berdistribusi normal c. uji white noise. suatu model bersifat white noise artinya residual dari model tersebut telah memenuhi asumsi variasi residual homogen serta antar residual tidak berkorelasi. hipotesis: h0 : 𝜌1 = 𝜌2 = ⋯ = 𝜌𝑘 = 0 h1 : minimal ada satu 𝜌𝑖 sama dengan nol; i = 1.2...., k statistik uji (uji ljung-box): 𝑄 = 𝑛(𝑛 + 2) ∑ �̂�𝑘 2 𝑛−𝑘 𝐾 𝑘=1 ; 𝑛 > 𝑘 (14) wilayah kritis: 𝑄 > 𝜒(𝛼;𝐾−𝑝−𝑞) 2 dengan: k : lag maksimum n : jumlah data (observasi) k : lag ke-k p, q : order dari arma (p,q) �̂�𝑘 : autokorelasi residual lag ke-k 3.5. ukuran ketepatan peramalan mape merupakan ukuran yang dipakai untuk mengetahui persentase penyimpangan hasil peramalan. jika 𝑋𝑖 adalah data aktual untuk periode i dan 𝐹𝑖adalah hasil peramalan untuk periode yang sama, maka mape dapat dihitung dengan formula berikut: 𝑀𝐴𝑃𝐸 = 1 𝑛 ∑ |( 𝑋𝑖−𝐹𝑖 𝑋𝑖 ) 𝑥100%|𝑛𝑖=1 (15) interpretasi dari hasil perhitungan mape yaitu [8]: < 10% : highly accurate forecasting 10%-20% : good forecasting 20%-50% : reasonable forecasting >50% : weak and inaccurate predictability 3.6. ukuran keandalan peramalan tracking signal adalah ukuran toleransi yang digunakan untuk menentukan sampai periode ke berapa peramalan dapat dilakukan. jika nilai-nilai tracking signal berada diuar batas yang dapat diterima yaitu ±5 maka model peramalan harus ditinjau kembali [9]. tracking signal dapat dihitung dengan formula: 𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = ∑ 𝑒𝑛 𝑛 1 ∑ |𝑒𝑛| 𝑛 𝑛 1 (16) 4. pembahasan hasil 4.1. gambaran ihk kota padangsidempuan penelitian ini menggunakan data ihk kota padangsidempuan periode bulanan mulai januari 2008 sampai november 2016 yang diperoleh dari web bps kota padangsidempuan [10]. perkembangan ihk kota padangsidempuan pada januari 2008 sampai november 2016 yang disajikan pada gambar 1 menunjukkan pola tren dan juga terjadi volatilitas nilai indeks di setiap bulannya. fenomena ini mengindikasikan bahwa indeks harga sepanjang tahun 2008 hingga akhir 2016 cenderung terus meningkat serta terjadi fluktuasi yang disebabkan adanya perubahan harga komoditas. perubahan harga tersebut disebabkan oleh adanya intervensi pemerintah, adanya efek musiman seperti bulan ramadhan, idul fitri, tahun ajaran baru dan faktor lainnya. keadaan ihk kota padangsidempuan sebelum periode juni 2012 menunjukkan nilai indeks di bawah 100, hal ini mengindikasikan bahwa kondisi harga barang dan jasa secara umum di kota padangsidempuan sebelum periode juni 2012 lebih murah dibandingkan setelah juli 2012. begitu pula sebaliknya, data ihk setelah juli 2012 menunjukkan nilai indeks di atas 100 yang mengindikasikan harga barang dan jasa secara umum lebih mahal dibandingkan keadaan tahun 2012 yang merupakan dasar. gambar 1. perkembangan ihk kota padang sidempuan tahun 2008-2016 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 78 4.2. pengujian pola musiman pada gambar 1 perkembangan data ihk kota padangsidempuan memiliki pola tren yang jelas, akan tetapi untuk melihat pola musiman perlu dilakukan pengujian secara statistik. pengujian dengan menggunakan formula (1), (2) dan (3) serta dengan bantuan software r diperoleh hasil bahwa data ihk kota padang sidempuan mengikuti pola musiman dengan periode 6 bulanan atau semesteran. 4.3. singular spectral analysis 4.3.1. embedding pada tahap ini akan diperoleh matriks lintasan untuk kemudian didekomposisi sesuai dengan nilai singularnya. namun sebelumnya, harus ditentukan terlebih dahulu window length (l)-nya. penentuan nilai window length (l) dilakukan dengan tracking dalam dua tahap yaitu tahap pertama menghitung mape terkecil pada window length nilai puluhan, kemudian setelah mendapat window length terbaik pada tahap satu maka dilakukan tracking tahap dua dengan mencari nilai satuan lima keatas dan lima kebawah. tabel 1. tracking tahap i windows length l l=10 l=20 l=30 l=40 l=50 mape 2,302 0,917 0,949 1,529 1,997 dari tabel 1 dapat terlihat bahwa window length dengan nilai 20 menunjukkan nilai mape terkecil, yaitu 0,917%. tracking window length dilakukan lagi dengan mencari nilai satuan terkecil diantara nilai 20. proses tracking window length tahap ii dapat dilihat pada tabel berikut. tabel 2. tracking tahap ii windows length l mape l mape 15 1,257 21 1,118 16 1,764 22 0,925 17 1,861 23 1,002 18 1,475 24 0,770 19 1,202 25 0,958 tabel 2 menunjukkan nilai mape antara windows length 21-25 pada proses tracking signal tahap ii. dari tabel 2 tersebut diperoleh bahwa window length dengan l=24 menunjukkan nilai mape terkecil sehingga dapat dikatakan bahwa dalam penelitian ini window length yang terbaik adalah 24 dan matriks lintasan yang terbentuk berukuran 24x78, yaitu: | 78,23 80,32 … 87.74 80,32 80,91 … 89.15 ⋮ 87.74 ⋱ 89.15 ⋱ ⋮ … 121,04 | 4.3.2. svd gambar 2. plot eigen value pada tahap svd terbentuk eigen value dan eigen vector dari matriks lintasan. eigen value yang diperoleh disajikan dalam gambar 2 berikut. dari gambar 2 dapat dilihat bahwa nilai eigen value untuk l1 = 4346,124 merupakan nilai terbesar yang berarti yang berarti eigenvalue dari l1 memberikan pengaruh terbesar dari komponen deret waktu terhadap karakteristik data dibandingkan dengan yang lainnya. hal ini dapat juga diartikan bahwa pengaruh komponen pertama sangat besar pada pembentukan rekonstruksi sinyal. eigenvalue selanjutnya cukup jauh berbeda dengan yang pertama dan cenderung menurun. dari penurunan ini dapat diketahui dua hal, yaitu komponen periodik dan noise. komponen periodik dengan periodesasi yang berbeda akan memiliki nilai singular yang berdekatan [4]. ini terjadi pada nilai singular 1 hingga 15. sedangkan penurunan perlahan (eigenvalue 16 sampai 24) menunjukkan adanya noise. 4.3.3. grouping proses pengelompokan masih bersifat subjektif yang dilakukan berdasarkan plot berpasangan dari eigen vector. plot yang berbentuk simetris mengindikasikan pola trend dan musiman sedangkan plot yang berbentuk random menunjukkan noise. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 79 tre n se ason 1 se as on 2 se ason 3 1 78.23 80.609 -1.121 -0.418 -0.440 78.63 -0.400 2 80.32 81.004 -0.753 -0.336 0.200 80.115 0.205 3 80.91 81.365 -0.489 -0.316 0.198 80.758 0.152 … … … … … … … 99 121.51 122.187 -1.227 0.152 0.219 121.194 0.316 100 120.67 122.654 1.731 -0.008 0.167 121.082 -0.412 101 121.04 123.130 -1.970 0.342 -0.613 120.889 0.151 re konstruks i re sidualt ihk aktual diagonal ave raging gambar 3. plot berpasangan eigen vector berdasarkan gambar 3, terlihat bahwa pasangan eigenvector 1-15 adalah gabungan pola tren dengan pola periodik. komponen trend dan periodik tidak dapat digabungkan menjadi satu kelompok karena akan terdapat noise. komponen trend menjadi tidak halus karena masih mengandung komponen periodik [11]. grouping dilakukan sampai periode ke-15. hal ini dapat disimpulkan dari pola sudah tidak dapat diidentifikasi lagi (berpola acak) berdasarkan identifikasi diatas, dengan window lenght l = 24 terbentuk sebanyak 4 grup yang terdiri dari 1 group trend dan 3 group musiman. komponen yang terhimpun dalam masingmasing group disajikan dalam tabel 3. tabel 3. pengelompokan komponen grup komponen eigentriple 1 trend 1 2 season 1 2-6 3 season 2 6-9 4 season 3 10-15 berdasarkan group yang telah terbentuk akan dilakukan pengecekan weak seperability antar group untuk memastikan bahwa antar group sudah tidak ada korelasi yang kuat. pada gambar 4 tampak bahwa hubungan antar group cukup lemah yang ditandai dengan korelasi antar group yang terarsir abu-abu. gambar 4. plot matriks korelasi antar group 4.3.4. diagonal averaging tabel 4. proses perhitungan diagonal averaging pada tahap ini, hasil dari ekspansi matriks berdasarkan proses grouping dijumlahkan untuk dihitung sesuai perhitungan diagonal averaging agar memperoleh deret baru. perhitungan diagonal averaging mengunakan hasil penjumlahan matriks ekspansi yang menghasilkan deret baru. proses perhitungannya dapat dilihat pada tabel 4 berikut. 4.4. peramalan ssa koefisien linear recurrent formula (aj) yang diperoleh dari persamaan (12) yaitu: tabel 5. koefisien linear recurrent formula (aj) no. aj no. aj no. aj 1 -0.065 9 0.023 17 -0.059 2 -0.073 10 0.072 18 0.054 3 -0.032 11 -0.118 19 -0.013 4 0.215 12 -0.075 20 0.013 5 -0.021 13 0.126 21 0.003 6 -0.216 14 0.000 22 -0.250 7 -0.022 15 0.044 23 0.302 8 0.007 16 -0.058 jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 80 dengan menggunakan persamaan (11) dan koefisien pada tabel 5 akan diperoleh hasil peramalannya. 4.5. sarima pada tahap awal, dilakukan pengecekan stasioneritas data dengan uji adf dan diperoleh nilai p-value = 0.357. sehingga disimpulkan data tidak stasioner. selanjutnya data akan didiffrencing pada d=1 agar data stasioner dalam mean. setelah data di-diffrencing maka data ihk sudah stasioner, hal ini dapat dibuktikan jika dilakukan uji adf pada data d=1 maka diperoleh nilai p-value = 0.01 yang berarti data sudah stasioner. selanjutnya, dilakukan plot acf dan pacf untuk identifikasi awal orde arima. gambar 5. plot acf dan pacf plot acf pada gambar 5 menunjukkan cut off terjadi pada lag ke-3 yang berarti orde ma bernilai p = 3, sedangkan pada plot pacf terjadi cutoff pada lag ke-2 yang berarti orde ar bernilai q = 2. sehingga identifikasi model sarima sementara adalah (2,1,3)(2,1,3). selanjutnya akan dilakukan uji coba model sarima beberapa orde dengan software r untuk memperoleh model terbaik dan memenuhi uji asumsi. tabel 6. hasil simulasi sarima dan uji asumsi model sarima aic signifikan koef. model white noise residual normal (2,1,3) (2,1,3)6 245.9 ya ya tidak (1,1,1) (0, 1,1)6 243.3 tidak ya ya (0,1,1) (0, 1,1)6 243.0 ya ya ya (0,1,2) (0, 1,1)6 242.8 tidak ya ya (0,1,3) (0, 1,1)6 244.8 tidak ya ya (0,1,2) (0, 1,2)6 244.7 ya ya ya (0,1,1) (0, 1,2)6 244.9 ya ya ya (0,1,1) (1,1,2)6 264.1 ya ya ya dari tabel 6 diatas, diperoleh model terbaik yang memiliki nilai aic terkecil dan memenuhi semua uji asumsi yaitu model sarima (0,1,1)(0,1,1)6. model sarima (0,1,1)(0,1,1)6 dapat dituliskan dalam persamaan berikut: (1 − 𝐵)(1 − 𝐵6)𝑌𝑡 = (1 − 0.2671 𝐵)(1 − 𝐵6)𝑎𝑡 . selanjutnya, dengan model tersebut dapat dilakukan permalan ihk untuk beberapa bulan ke depan. 4.6. evaluasi hasil peramalan ssa dengan model sarima dari hasil analisis dan pembahasan diatas, tingkat ketepatan dan kestabilan peramalan pada metode ssa dan sarima disajikan dalam tabel 7. nilai mape dari kedua metode sangat kecil (dibawah 10 persen) sehingga bisa dikatakan peramalan dengan ssa dan sarima (0,1,1)(0,1,1)6 tingkat akurasinya sangat tinggi. hal ini sejalan dengan penelitian yang dilakukan sebelumnya oleh [4] bahwa metode ssa dan sarima sama-sama memiliki akurasi yang sangat tinggi dalam meramalkan jumlah kematian akibat kecelakaan di usa. sementara itu, berdasarkan nilai tracking signal diketahui bahwa permalan ihk kota padangsidempuan baik dengan metode ssa maupun sarima stabil pada peramalan hingga 5 bulan ke depan saja karena nilai tracking signal yang diperoleh antara -5 sampai +5 di setiap bulannya, sedangkan pada bulan ke-6 nilai tracking signal-nya sudah lebih besar dari 5 dimana hal ini mengindikasikan bahwa peramalan sudah tidak stabil pada bulan ke-6. jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 81 tabel 7. tracking signal metode peramalan ssa dan sarima periode ssa forecasting sarima (0,1,1)(0,1,1)6 resid. tracking signal resid. tracking signal des/ 2016 -1.179 -1.000 -0.699 -1 jan/ 2017 -0.098 -2.000 0.063 -1.668 feb/ 2017 -0.825 -3.000 -0.581 -2.717 mar/ 2017 0.209 -3.278 0.504 -1.543 apr/ 2017 -0.605 -4.285 1.050 0.581 mei/ 2017 -2.812 -5.563 1.560 2.552 mape 0,770 0,559 tabel 8. perbandingan hasil peramalan data ihk kota padangsidempuan dengan metode ssa dan sarima periode peramalan metode peramalan ssa forecasting sarima (0,1,1)(0,1,1)6 des/ 2016 122.499 122.0194 jan/ 2017 123.328 123.1667 feb/ 2017 123.555 123.3114 mar/ 2017 123.541 123.2455 apr/ 2017 124.975 123.3203 berdasarkan tabel 8 diatas, tampak bahwa hasil peramalan ihk kota padangsidempuan menunjukkan pola yang terus meningkat selama 5 bulan ke depan atau dapat juga dikatakan akan terjadi inflasi untuk 5 bulan ke depan. hasil peramalan pada metode ssa menunjukkan akan terjadi inflasi yang cukup tinggi pada april 2017 sementara itu pada metode sarima menunjukkan inflasi yang terjadi selama lima bulan ke depan cenderung tidak begitu melonjak tinggi. 5. kesimpulan berdasarkan hasil analisis yang dilakukan, dapat disimpulkan bahwa: a. dengan singular spectral analysis data ihk kota padang sidempuan memiliki length window (l) sebesar 24 dan group sebanyak 4 (1 kelompok trend dan 3 kelompok musiman) dengan periode musiman terjadi di setiap semester b. ketepatan hasil peramalan data ihk kota padang sidempuan dengan metode ssa dan sarima (0,1,1)(0,1,1)6 sama-sama berada pada kategori highly accurate c. keandalan hasil peramalan data ihk kota padang sidempuan dengan metode ssa dan sarima (0,1,1)(0,1,1)6 sama-sama stabil/ andal untuk meramalkan ihk 5 bulan ke depan d. peramalan data ihk kota padang sidempuan dengan metode ssa serta sarima (0,1,1)(0,1,1)6 memiliki nilai yang tidak begitu jauh berbeda selama 4 bulan pertama. namun peramalan pada bulan ke-5 dengan metode ssa menunjukkan peningkatan nilai ihk yang cukup tinggi atau terjadi inflasi yang cukup tinggi. e. peramalan ihk baik dengan metode ssa maupun sarima sama-sama memiliki pola trend atau akan terjadi inflasi selama 5 bulan ke depan daftar pustaka [1] darmawan,g, dkk, perbandingan peramalan pada model singular spectrum analysis, studi kasus : curah hujan kota bandung dan sekitarnya, seminar nasional universitas muhammadiyah purwokerto, 2015 [2] no kang myung, singular spectrum analysis, graduate thesis, university of california, los angeles, 2009 [3] bps sumatera utara, sumatera utara dalam angka 2016. medan: bps sumut, (2016) [4] hassani, h., singular spectrum analysis: methodology and comparison, journal of data science 5(2007), 239-257 [5] akhter tahsina, short-term forecasting inflation of inflation in bangladesh with seasonal arima processes, munich personal repec archive no. 43729, 2013, diakses melalui https://mpra.ub.unimuenchen.de/43729/1/mpra_paper_43729. pdf pada 15 februari 2017 [6] wei, w.w.s, time series analysis univariate and multivariate methods, 2nd edition, pearson education inc. (2006) [7] darmawan,g, identifikasi pola data curah hujan pada proses grouping dalam metode singular spectrum analysis. seminar nasional pendidikan matematika 2016 https://mpra.ub.uni-muenchen.de/43729/1/mpra_paper_43729.pdf https://mpra.ub.uni-muenchen.de/43729/1/mpra_paper_43729.pdf https://mpra.ub.uni-muenchen.de/43729/1/mpra_paper_43729.pdf jurnal matematika “mantik” edisi: oktober 2017. vol. 03 no. 02 issn: 2527-3159 e-issn: 2527-3167 82 [8] lewis, c.d, industrial and business forecasting methods, butterworths (1982) [9] abraham, bovas and johannes ledolter, statistical methods for forecasting, wiley (1983) [10] www.padangsidimpuankota.bps.go.id diakses pada 1 februari 2017 [11] sakinah, a. m., perbandingan stabilitas hasil peramalan suhu dengan r-forecasting dan v-forecasting ssa untuk long horizon, tesis, departemen statistika, fmipa unpad (2013) http://www.padangsidimpuankota.bps.go.id/ jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 22 analisis cluster dengan data outlier menggunakan centroid linkage dan k-means clustering untuk pengelompokan indikator hiv/aids di indonesia rini silvi prodi magister statistika terapan,universitas padjajaran, rinisilvi@stis.ac.id doi:https://doi.org/10.15642/mantik.2018.4.1.22-31 abstrak analisis klaster adalah salah satu metode yang digunakan untuk mengelompokkan obyek atau pengamatan yang didasarkan atas kemiripannya. obyek yang berada dalam satu kelompok memiliki kemiripan satu sama lain. pada penelitian ini analisis klaster yang digunakan adalah k-means dengan centroid linkage. k-means adalah salah satu metode clustering non-hierarki yang sederhana dan mudah diimplementasikan. sedangkan, centroid linkage adalah metode klaster hierarki yang dapat digunakan pada data yang mengandung outlier, dimana outlier bisa membuat data yang diolah tidak mencerminkan gambaran sebenarnya. untuk memudahkan, outlier seringkali dibuang, padahal acapkali outlier mengandung informasi penting. hiv/aids adalah salah satu tantangan serius terhadap kesehatan masyarakat dunia, karena hiv/aids merupakan penyakit menular yang menyerang sistem kekebalan tubuh sehingga penderita mengalami penurunan ketahanan tubuh secara terus-menerus yang berujung pada kematian. data indikator hiv/aids di indonesia mengandung outlier. penelitian ini menggunakan gap statistik untuk menentukan jumlah klaster ideal yang mengelompokkan propinsi berdasarkan indikator hiv/aids sedemikian hingga terbagi menjadi 7 klaster. dari perbandingan rasio sw/sb, centroid linkage lebih homogen dibandingkan k-means. dengan custering, diharapkan pemerintah dapat mengambil kebijakan berdasarkan indikator-indikator dominan yang terdapat pada masing-masing klaster. kata kunci : clustering, centroid linkage, k-means abstract cluster analysis is a method to group data (objects) or observations based on their similarities. objects that become members of a group have similarities among them. cluster analyses used in this research are k-means clustering and centroid linkage clustering. k-means clustering, which falls under non-hierarchical cluster analysis, is a simple and easy to implement method. on the other hand, centroid linkage clustering, which belongs to hierarchical cluster analysis, is useful in handling outliers by preventing them skewing the cluster analysis. to keep it simple, outliers are often removed even though outliers often contain important information. hiv/aids is a serious challenge for global public health since hiv/aids is an infectious disease attacking body’s immune system that in turn lowering the ability to fight infections which in the end causing death. hiv/aids indicators data in indonesia contain outliers. this research uses gap statistic to define the number of clusters based on hiv/aids indicators that groups indonesia provinces into 7 clusters. by comparing sw/sb ratio, centroid linkage clustering is more homogenous than k-means clustering. using clustering, the government shall be able to create a better policy for fighting hiv/aids based on the dominant indicators in each cluster. keyword : clustering, centroid linkage, k-means mailto:rinisilvi@stis.ac.id 23 1. pendahuluan hiv/aids merupakan penyakit menular yang disebabkan oleh infeksi human immunodeficiency virus yang menyerang sistem kekebalan tubuh sehingga tidak dapat melawan infeksi dan berujung pada kematian. penyakit ini merupakan masalah dan tantangan serius terhadap kesehatan masyarakat di dunia. di indonesia, sebagian besar infeksi baru diperkirakan terjadi pada beberapa sub-populasi berisiko tinggi dengan prevalensi > 5% [1]. risiko penularan hiv/aids tidak hanya terbatas pada sub-populasi yang berperilaku risiko tinggi, tetapi juga dapat menular pada pasangan atau bahkan anaknya. pada tahun 2016, kasus baru infeksi hiv meningkat 33,4% dibandingkan tahun sebelumnya. terdapat 69,3% kasus baru infeksi hiv pada kelompok umur 25-49, sementara 63,3% penderita adalah laki-laki. rasio hiv/aids antara laki-laki dan perempuan tercatat pada kisaran 2:1 [2]. pada tahun 2016, jumlah kasus baru infeksi hiv terbanyak adalah jawa timur, kemudian diikuti oleh jakarta, jawa barat, jawa tengah, dan papua. akan tetapi, persentase kasus baru infeksi hiv terbesar adalah papua, jika dibagi dengan jumlah penduduknya. sementara, gorontalo menempati urutan terakhir propinsi dengan jumlah kasus baru infeksi hiv terbanyak di indonesia. perbedaan jumlah kasus baru hiv tampak begitu nyata di beberapa propinsi. contohnya dki jakarta dengan tingkat prevalensi 58,56 padahal dengan tingkat kemiskinan paling kecil yaitu sebesar 3,73. meskipun tingkat pendidikan di gorontalo termasuk rendah, akan tetapi tingkat prevalensi kasus baru hiv hanya sekitar 0.6. tingkat prevalensi disini dihitung berdasarkan jumlah kasus baru infeksi hiv suatu daerah dibandingkan setiap seratus ribu penduduk. terdapat beberapa indikator yang mempengaruhi hiv/aids yaitu tingkat penggunaan kontrasepsi (kondom), jumlah dokter/tenaga medis, proporsi muslim, tingkat kesuburan remaja, dan rata-rata lama sekolah [3]. sementara menurut [4], hiv dipengaruhi oleh indikator seperti kurangnya pendidikan, kemiskinan, seks bebas, dan kehidupan malam meskipun pada penelitiannya lebih difokuskan kepada perempuan. ada beberapa indikator yang dapat mempengaruhi prevalensi hiv yang digunakan disini. pada indikator tersebut teridentifikasi beberapa data outlier. hal ini bisa mempengaruhi hasil kesimpulan dari penelitian. outlier dapat menghasilkan output yang tidak sesuai dengan gambaran yang sebenarnya, termasuk dalam hal pengklasteran indikator prevalensi hiv. maka dari itu, penelitian ini perlu menggunakan metode klaster yang dapat menangani pengaruh keberadaan outlier tersebut. metode yang dimaksud adalah kmeans clustering. tujuan dari penelitian ini adalah untuk mengumpulkan propinsi ke dalam suatu kelompok sedemikian hingga dapat dibedakan menurut karakteristik indikator dari prevalensi hiv. dengan adanya kelompok-kelompok berdasarkan indikator-indikator tersebut, diharapkan pemerintah dapat membuat kebijakan yang tepat untuk pencegahan dini menyebarnya hiv/aids. 2. tinjauan pustaka 2.1 analisis klaster analisis klaster adalah suatu teknik statistik yang bertujuan untuk mengelompokkan obyek ke dalam suatu kelompok sedemikian sehingga obyek yang berada dalam satu kelompok akan memiliki kesamaan yang tinggi dibandingkan dengan obyek yang berada di kelompok lain [5]. dengan kata lain tujuan dari analisis cluster adalah pengklasifikasian obyek-obyek berdasar-kan similaritas diantaranya dan menghim-pun data menjadi beberapa kelompok [16]. ada dua metode dalam analisis klaster yaitu metode hierarki dan metode non hierarki. menurut [6], metode non hierarki umumnya digunakan jika jumlah satuan pengamatan besar dan jumlah klaster tidak ditentukan sebelumnya. salah satu metode non hierarki adalah metode k-means. ini adalah metode non hirarki yang paling banyak digunakan. algoritma k-means mudah diimplementasikan dan juga mudah diadaptasi sehingga menjadikannya lebih populer dalam hal pengelompokan. pada teknik k-means, biasanya peneliti sudah terlebih dahulu menentukan banyaknya klaster yang akan dibentuk. metode hierarki merupakan metode pengelompokkan yang terstruktur dan bertahap berdasarkan kemiripan sifat antar obyek. kemiripan sifat tersebut dapat ditentukan dari kedekatan jarak euclidean atau jarak mahalanobis. 24 jarak euclidean digunakan jika tidak terjadi korelasi. jarak euclidean dirumuskan sebagai berikut: ( ) ( ) nlxyxyd l k kk ,...,3,2,1;, 1 2 =−=  = dimana: ( )xyd , :kuadrat jarak euclid antar obyek y dengan obyek pada x. k y : nilai dari obyek y pada variabel ke-k k x : nilai dari obyek x pada variabel ke-k jarak mahalanobis digunakan jika data terjadi korelasi. jarak mahalanobis antara dua sampel x dan y dari suatu variabel acak didefinisikan sebagai berikut [5] : ( ) ( ) −−= − xyxyxyd t smahalanobi 1 ),( dengan  adalah suatu matriks varians kovarians. dalam metode klaster hierarki terdapat beberapa metode penghitungan jarak yang dapat digunakan, antara lain metode pautan tunggal (single linkage), metode pautan lengkap (complete linkage), metode pautan rataan (average linkage), metode ward, dan metode centroid linkage. 2.2 multikolinearitas multikolinearitas adalah adanya hubungan linear yang sempurna atau pasti di antara beberapa atau semua variabel [7]. multikolinearitas berkenaan dengan terdapatnya lebih dari satu hubungan linear pasti. untuk mengetahui adanya multikolinearitas salah satunya adalah dengan menghitung nilai variance inflation factor (vif) dengan rumus: 𝑉𝐼𝐹𝑗 = 1 1 − 𝑅𝑗 2 menurut [7], terjadinya multikolinearitas apabila nilai (𝑉𝐼𝐹𝑗) ≥ 10. jika terindikasi terjadi multikolinearitas maka harus dilakukan tindakan perbaikan multikolinearitas. 3. metode penelitian 3.1 sumber data dan variabel penelitian penelitian ini menggunakan data sekunder tahun 2016 yang diperoleh dari badan pusat statistik republik indonesia (bps ri) kemudian diolah dengan sofware r versi 3.4.3. data prevalensi kasus baru infeksi hiv digunakan sebagai data pembanding dalam pengelompokkan klaster. variabel indikator yang digunakan menjadi faktor risiko prevalensi hiv yaitu persentase penduduk miskin (x1) [8], tingkat pengangguran terbuka (x2) [9], jumlah puskesmas (x3) [10] dan rasio penduduk 15 tahun ke atas yang masih berpendidikan rendah (x4) [11], dan persentase pasangan usia subur yang memakai alat kontrasepsi berupa kondom (x5) [2]. pendidikan rendah diasumsikan untuk yang berumur 15 tahun keatas tapi tidak punya ijazah, atau hanya sampai tamatan sd dan smp. alat kontrasepsi yang digunakan difokuskan pada kondom karena alat kontrasepsi yang lainnya hanya digunakan untuk mencegah kehamilan tetapi tidak bisa mengurangi penularan hiv/aids. unit observasi pada penelitian ini adalah seluruh propinsi di indonesia. 3.2 tahapan penelitian 3.2.1 standardisasi data proses standardisasi dilakukan apabila di antara variabel-variabel yang diteliti terdapat perbedaan ukuran satuan yang besar. perbedaan satuan yang mencolok dapat mengakibatkan perhitungan pada analisis klaster menjadi tidak valid. oleh karena itu, perlu dilakukan proses standardisasi dengan melakukan transformasi pada data asli sebelum dianalisis lebih lanjut. 3.2.2 deteksi outlier dan multikolinearitas analisis klaster pada hakekatnya adalah teknik algoritma, bukan alat inferensi statistik. oleh sebab itu persyaratan seperti distribusi data yang harus normal (di analisis statistik lainnya) ataupun hubungan linier antar variabel tidak menjadi syarat dalam analisis klaster. namun demikian, karena data yang diolah dalam analisis klaster biasanya hanya sebagian kecil dari populasi, maka agar hasilnya bisa digeneralisasi, data yang diolah sebaiknya mencerminkan gambaran umum atau bersifat representatif. oleh sebab itu, outliers tetap harus dihilangkan dari sampel agar hasilnya tidak bias. deteksi outlier digunakan untuk mencari data yang berbeda dengan mayoritas data yang lain. walaupun memiliki perilaku yang berbeda dengan mayoritas data yang lain dan sering dianggap noise, tetapi outlier sering kali mengandung informasi yang sangat berguna. tidak semua data yang mengandung outlier bisa 25 ditransformasi karena kasus data yang berbedabeda. akan tetapi, dengan menggunakan metode centroid linkage, outlier tidak berpengaruh secara signifikan. selain itu, data yang digunakan seharusnya tidak berkorelasi, dengan kata lain sebaiknya tidak ada multikolinieritas. alasannya adalah di dalam analisis klaster setiap variabel diberi bobot yang sama dalam perhitungan jarak. manakala beberapa varibel saling berkorelasi, korelasi tersebut akan menyebabkan pembobotan yang tidak berimbang sehingga akan mempengaruhi hasil analisis [12]. 3.2.3 penentuan jumlah klaster optimum penentuan jumlah cluster optimum dilakukan dengan menggunakan gap statistik pada r. gap statistik bertujuan untuk menentukan jumlah klaster lebih konstan dibandingkan pengukuran lainnya. jarak obyek berpasangan dalam klaster dihitung dengan rumus: 𝐷𝑟 = ∑ 𝑑𝑖𝑖` 𝑖,𝑖`𝜖𝐶𝑟 dimana d adalah jarak euclidean kuadrat. kemudian hitung jumlah kuadrat di dalam klaster menggunakan rumus: 𝑊𝑘 = ∑ 1 2𝑛𝑟 𝐷𝑟 𝑘 𝑟=1 nilai gap didapatkan dengan mengestimasi jumlah klaster optimum pendekatan standardisasi 𝑊𝑘: 𝐺𝑎𝑝𝑛(𝑘) = 𝐸𝑛 ∗{log(𝑊𝑘)} − log(𝑊𝑘) dimana 𝐸𝑛 ∗ adalah nilai ekspektasi dari distribusi jumlah sampel. kriteria banyak cluster optimum diberikan oleh nilai gap statistik (k) yang paling tinggi, atau yang pertama kali mengindikasi kenaikan gap yang minimum jika gap selalu naik [13]. setelah penentuan klaster optimum, maka akan dibandingkan pengklasteran dengan menggunakan dua metode, yaitu metode centroid linkage dan metode k-means. 3.2.4 metode klaster k-means metode k-means adalah metode non hierarki yang paling banyak digunakan dalam pengklasteran. algoritma k-means mudah diimplementasikan. pada metode ini, peneliti menentukan sendiri jumlah klaster yang akan dibentuk. peneliti mengelompokkan entitas ke dalam k-kelompok, biasanya dilakukan secara acak. pada masing-masing kelompok dihitung rata-ratanya. hitung jarak setiap entitas terhadap pusat masing-masing kelompok (rata-rata kelompoknya). masing-masing objek dialokasikan ke klaster terdekat dengan pusatnya. update keanggotaan setiap entitas berdasarkan jarak terdekat dengan pusat kelompok dan ditentukan kembali pusat klaster yang baru. proses pangalokasian obyek kembali dilakukan. suatu obyek dapat berpindah ke klaster lain bila obyek tersebut lebih dekat ke pusat klaster tersebut. proses ini dilakukan secara berulang sampai tidak ada lagi entitas yang berpindah kelompok. 3.2.5 metode klaster centroid linkage centroid linkage adalah rata-rata semua obyek dalam klaster. jarak antara dua klaster adalah jarak antar centroid klaster tersebut. klaster centroid adalah nilai tengah observasi pada variabel dalam suatu set variabel cluster. dengan metode ini, setiap terjadi klaster baru segera terjadi perhitungan ulang centroid sampai terbentuk klaster yang tetap [5]. keuntungan dari metode ini adalah outlier tidak berpengaruh secara signifikan, jika dibandingkan dengan metode lain. jarak antara dua klaster didefinisikan sebagai berikut: ( ) 21)( , xxdd wuv = centorid klaster baru yang terbentuk didapat dengan rumus: 21 2211 nn xnxn x + + = dimana: 21 nn = : banyaknya obyek. centroid adalah rata-rata dari semua anggota dalam klaster tersebut. pada saat obyek digabungkan maka centroid baru dihitung, sehingga setiap kali ada penambahan anggota, centroid akan berubah pula [14]. 3.2.6 langkah-langkah k-means clustering langkah-langkah k-means: 1. menentukan k sebagai jumlah klaster yang diinginkan 2. mengalokasikan data ke dalam cluster secara acak 3. menentukan pusat klaster dari data yang ada pada masing-masing klaster dengan persamaan: 𝐶𝑘𝑗 = 𝑥1𝑗 + 𝑥2𝑗 + ⋯+ 𝑥𝑛𝑗 𝑛 26 dimana: 𝐶𝑘𝑗 : pusat cluster ke-k pada variabel ke j (j=1, 2, …, p) 𝑛 ∶ banyak data pada cluster ke-k 4. menentukan jarak setiap obyek dengan setiap centroid dengan perhitungan jarak setiap obyek dengan setiap centroid menggunakan jarak euclidean 5. menghitung fungsi obyektif dengan formula: 𝑙 = ∑ ∑ 𝑎𝑖𝑗𝑑(𝑥𝑖,𝐶𝑘𝑗) 2𝑘 𝑗=1 𝑛 𝑖=1 6. mengalokasikan masing-masing data ke centroid/rata-rata terdekat. 7. mengulangi kembali langkah 3-6 sampai tidak ada lagi perpindahan obyek atau tidak ada perubahan pada fungsi obyektifnya. 3.2.7 langkah-langkah centroid linkage clustering langkah-langkah centroid linkage clustering: 1. membuat k klaster. masing-masing individu atau unit observasi jadi kelompok. kemudian dibuat matrik jaraknya (dari i ke kelompoknya), dengan rumus: 𝐷 = {𝑑𝑖𝑘} 2. mencari jarak terkecil dari pasangan klaster, yaitu 𝑑𝑢𝑣 (jarak klaster u dan klaster v) 3. menggabungkan klaster u dan klaster v. kemudian update matriks jaraknya. 4. ulangi langkah 2 & 3 sebanyak n-1 kali. catat nilai jarak untuk setiap terjadi penggabungan klaster 5. tentukan nilai cut off untuk menentukan klaster terbentuk. lakukan dengan membuat dendogram, dan tentukan cut off jumlah klaster 6. pemberian nama klaster berdasarkan profiling, yaitu melihat karakteristik klaster terbentuk secara rata-rata. 3.2.8 penentuan metode terbaik tahap evaluasi dapat dilakukan dengan melakukan analisis klaster dengan ukuran jarak atau metode klaster yang berbeda kemudian dibandingkan hasilnya [7]. pemilihan metode yang menghasilkan kualitas pengelompokan terbaik dilakukan dengan memperhatikan nilai rasio rata-rata simpangan baku dalam klaster terhadap simpangan baku antar klaster [15]. rata-rata simpangan baku di dalam klaster (sw) dinyatakan dengan: 𝑠𝑤 = 1 𝑐 ∑ 𝑠𝑘 𝑐 𝑘=1 simpangan baku antar klaster (sb) dinyatakan sebagai: 𝑠𝑏 = [ 1 𝑐 − 1 ∑(�̅�𝑘 − �̅�) 2 𝑐 𝑘=1 ] 1 2 dimana c adalah jumlah klaster, sk merupakan simpangan baku di dalam klaster kek. �̅�𝑘 sebagai rata-rata klaster ke-kdan �̅� adalah rata-rata dari semua klaster. semakin kecil nilai 𝑠𝑤 dan semakin besar nilai 𝑠𝑏 maka metode tersebut memiliki kinerja yang baik, artinya memiliki homogenitas yang tinggi. metode yang dipilih adalah yang memberikan nilai rasio 𝑠𝑤/𝑠𝑏 terkecil. 4. hasil dan pembahasan 4.1 jumlah klaster optimum sebelum melakukan analisis lebih lanjut terlebih dahulu dilakukan standardisasi data, karena terdapat perbedaan satuan yang mencolok sehingga dapat mengakibatkan perhitungan pada analisis klaster menjadi tidak valid. transformasi dilakukan menggunakan r gui 3.4.3. pemeriksaan awal yang dilakukan adalah melihat apakah variabel yang digunakan terdapat outlier atau tidak. pada data ini terdapat beberapa provinsi yang outlier, diantaranya yaitu jawa barat, gorontalo, dan jawa timur, jawa tengah, serta beberapa daerah lainnya. pada gambar 1, terdapat 12 titik outlier. untuk menangani masalah outlier, tanpa melakukan perubahan pada data, perlu dibandingkan metode terbaik dari beberapa metode yang ada. gambar 1. pendeteksian outlier mendeteksi multikolinearitas dengan menggunakan vif dengan hasil masing-masing 27 yaitu 1,28 (x1), 1,38 (x2), 1,32 (x3), 1,52 (x4), dan 1,18 (x5). semua variabel menunjukkan nilai vif sangat kecil (dibawah 2) yang artinya variabel yang digunakan tidak mengandung multikolinearitas. tahap berikutnya adalah pemilihan jumlah klaster optimum dengan menggunakan pendekatan gap statistik. pada penelitian ini, gap statistik menunjukkan jumlah klaster optimum adalah sebanyak 7, yaitu pada titik yang pertama kali perbedaan gap minimum karena secara praktis tidak ideal memilih titik tertinggi mengingat hasil gap statistik yang semakin meningkat hingga jumlah klaster maksimal. gambar 2. jumlah klaster (k) berdasarkan gap statistik pada gambar 2 terlihat bahwa terdapat penurunan yang curam dari klaster 1 ke klaster 2. kemudian terjadi kenaikan hingga ujung klaster yang dimungkinkan. dapat diperhatikan bawah dari klaster 7 ke 8 terdapat kenaikan yang landai, tidak seperti kenaikan sebelumnya yang curam. sehingga diambil jumlah klaster sebanyak 7, yaitu pada saat pertama kali kenaikan menjadi lebih landai. 4.2 metode hierarchical centroid hasil dendogram dengan menggunakan 7 klaster terbentuk dapat dilihat pada gambar 3 di bawah ini. gambar 3. dendogram 7 klaster dengan centroid linkage kelompok yang terbentuk: klaster 1: aceh, sumatera utara, sumatera barat, riau, jambi, sumatera selatan, bengkulu, lampung, bangka belitung, diy, banten, bali, ntb, kalimantan barat, kalimantan tengah, kalimantan selatan, kalimantan timur, kalimantan utara, sulawesi utara, sulawesi tengah, sulawesi selatan, sulawesi tenggara, sulawesi barat, maluku, maluku utara, papua barat. klaster 2: kepulauan riau, dan dki jakarta. klaster 3: jawa barat. klaster 4: jawa tengah dan jawa timur. klaster 5: nusa tenggara timur klaster 6: gorontalo klaster 7: papua 4.3 metode k-means dari gambar terlihat dengan jelas anggota dari masing-masing klaster dari warna dan bentuk titik masing-masing klaster. 28 gambar 4. clusplot 7 klaster dengan k-means hasil gambar 4, diketahui gorontalo membentuk klaster sendiri terpisah dari kelompok besar seperti halnya klaster daerah khusus ibukota (dki) jakarta dan kepulauan riau (kepri). klaster lain yang terpisah dari kelompok besar yaitu, klaster jawa barat, jawa tengah dan jawa timur. pembagian klaster 7 tertuang dalam tabel 1 dibawah ini, dengan anggota sebagai berikut: tabel 1. anggota klaster k-means 4.4 penentuan metode terbaik dengan membandingkan rasio sw/sb dapat diketahui metode mana yang memiliki kualitas ketepatan kelompok yang lebih baik. hasil proses pengelompokkan dengan centroid linkage dan k-means disajikan pada tabel 2. tabel 2. rasio sw/sb untuk metode terbaik banyak cluster centroid linkage k-means (1) (2) (3) 7 0,067307 0,112232 berdasarkan tabel 2, terlihat bahwa nilai rasio sw/sb menggunakan metode k-means adalah sebesar 0,112232. rasio sw/sb menggunakan metode centroid linkage menghasilkan angka 0,067307 dan lebih kecil dari k-means. centroid linkage menghasilkan kelompok yang lebih homogen sehingga nilai rasio yang dihasilkan lebih kecil. artinya metode centroid linkage memiliki kualitas ketepatan kelompok yang lebih baik dibandingkan kmeans. 4.5 profiling cluster kelima variabel yang digunakan dalam penelitian ini menunjukkan indikator yang digunakan menjadi faktor risiko hiv/aids di indonesia. tabel 3. profiling variabel dominan dan tidak dominan terhadap hiv/aids klaster pertama, tidak ada indikator dominan yang mempengaruhi hiv/aids di daerah tersebut. hal ini mungkin disebabkan oleh adanya indikator lain yang belum digunakan dalam penelitian ini, seperti indikator kehidupan malam, jumlah pekerja seks komersial (psk), jumlah panti pijat masingmasing daerah, tingkat religius seperti yang dituliskan oleh [3], dan indikator lainnya. klaster kedua, adalah kelompok daerah dengan tingkat kemiskinan rendah dan proporsi umur 15 tahun ke atas dengan pendidikan diatas jenjang smp (tidak berpendidikan rendah). klaster ini terdiri dari dki jakarta dan kepulauan riau. klaster ketiga, adalah daerah proporsi pasangan usia subur (pus) dengan kontrasepsi kondom paling kecil, akan tetapi jumlah puskesmas yang terdapat di daerah ini juga paling banyak. daerah ini juga memiliki tingkat pengangguran terbuka (tpt) terbesar. klaster ini terdiri dari 1 propinsi saja yaitu jawa barat. klaster keempat, adalah kelompok proporsi pasangan usia subur (pus) dengan kontrasepsi kondom relatif kecil, akan tetapi jumlah puskesmas yang terdapat di daerah ini juga termasuk banyak. klaster ini terdiri dari propinsi jawa tengah dan jawa timur. klaster 1 klaster 2 klaster 3 klaster 4 klaster 5 klaster 6 klaster 7 aceh gorontalo jabar sumsel sumut sumbar diy maluku jateng bengkulu riau jambi bali papbar jatim lampung kepri babel sultra ntb dki kalbar ntt banten kalteng sulteng kaltim kalsel sulbar sulut kaltara papua sulsel malut 29 klaster kelima, adalah daerah dengan proporsi pasangan usia subur (pus) dengan kontrasepsi kondom relatif kecil, akan tetapi jumlah puskesmas yang terdapat di daerah ini termasuk sedikit. klaster ini adalah daerah nusa tenggara timur. klaster keenam, adalah daerah dengan proporsi pasangan usia subur (pus) dengan kontrasepsi kondom paling banyak, sehingga klaster ini termasuk daerah dengan jumlah hiv/aids terkecil, yaitu daerah gorontalo. klaster ketujuh, adalah daerah dengan persentase penduduk miskin paling besar dan pendidikan paling rendah. yaitu daerah papua. 5. kesimpulan hasil penelitian ini memberikan kesimpulan bahwa untuk data yang memiliki outlier, metode pengklasteran menggunakan centroid linkage lebih memberikan hasil yang sesuai dengan keadaan dibandingkan dengan metode k-means. metode k-means lebih heterogen dalam hal ini. dengan metode centroid linkage, outlier tidak mempengaruhi klaster analisis dan tidak mengubah hasil dari interpretasi data. dari 34 propinsi yang ada di indonesia, terdapat 7 klaster berdasarkan indikator yang menyebabkan terjadinya hiv/aids. klaster 1: aceh, sumatera utara, sumatera barat, riau, jambi, sumatera selatan, bengkulu, lampung, bangka belitung, diy, banten, bali, ntb, kalimantan barat, kalimantan tengah, kalimantan selatan, kalimantan timur, kalimantan utara, sulawesi utara, sulawesi tengah, sulawesi selatan, sulawesi tenggara, sulawesi barat, maluku, maluku utara, papua barat. klaster 2: kepulauan riau, dan dki jakarta. klaster 3: jawa barat. klaster 4: jawa tengah dan jawa timur. klaster 5: nusa tenggara timur klaster 6: gorontalo klaster 7: papua referensi [1] kementerian kesehatan ri, survei terpadu biologis dan perilaku, jakarta: kementerian kesehatan, (2011) [2] kementrian kesehatan ri, profil kesehatan indonesia tahun 2016, jakarta: kementrian kesehatan ri, (2017) [3] mondal, m., & shitan, m., factors affecting the hiv/aids epidemic: an ecological analysis of global data, african health sciences, 13(2) pp 301– 310, (2013) [4] singh, r. k., & patra, s., what factors are responsible for higher prevalence of hiv infection among urban women than rural women in tanzania?, ethiopian journal of health sciences, 25(4), pp 321– 328, (2015) [5] rahmawati, l., analisis kelompok dengan menggunakan metode hierarki untuk pengelompokan kabupaten/kota di jawa timur berdasar indikator kesehatan, jurnal matematika vol.1 no.2 universitas negeri malang, (2012) [6] anderberg, m., cluster analysis for applications, academic press.inc., (1973) [7] ningrat, d.r., analisis cluster dengan algoritma k-means dan fuzzy c-means clustering untuk pengelompokan data obligasi korporasi, jurnal gaussian vol.5 no.4 universitas diponegoro, 2016. [8] badan pusat statistik (bps), indikator pembangunan berkelanjutan 2017: jumlah penduduk miskin, (2017) [9] badan pusat statistik (bps), statistik indonesia 2017: tingkat pengangguran terbuka, (2017) [10] badan pusat statistik (bps), statistik indonesia 2017: jumlah puskesmas, (2017) [11] badan pusat statistik (bps), statistik kesejahteraan rakyat 2017: status pendidikan tertinggi, (2017) [12] puspitasari, m., pengelompokan kabupaten / kota berdasarkan faktorfaktor yang mempengaruhi kemiskinan di jawa tengah menggunakan metode ward dan average linkage, jurnal matematika vol. 5 no. 6 universitas negeri yogyakarta, (2016) [13] tibshirani, r., walther, g., & hastie, t., estimating the number of clusters in a data set via the gap statistic, journal of royal statistical society vol. 63 issue 2., (2001) [14] laeli, s., analisis cluster dengan average linkage method dan ward’s method untuk data responden nasabah asuransi jiwa unit link, s1 thesis, universitas negeri yogyakarta, indonesia, (2014) [15] purnamasari, s.b., pemilihan cluster optimum pada fuzzy c-means (studi kasus: pengelompokan kabupaten/kota di http://www.web.stanford.edu/~hastie/papers/gap.pdf http://www.web.stanford.edu/~hastie/papers/gap.pdf http://www.web.stanford.edu/~hastie/papers/gap.pdf 30 jawa tengah berdasarkan indikator indeks pembangunan manusia), jurnal gaussian vol.3 no.3 universitas diponegoro, (2014) [16] lailiyah, s. dan hafiyusholeh, m., perbandingan antara metode k-means clustering dengan gath-geva clustering, jurnal matematika mantik, 1(2), mei 2016. pp. 26-37 31 lampiran tabel 3. prevalensi penderita hiv per100.000 penduduk menurut propinsi tahun 2016 provinsi rasio penderita hiv*100.000 provinsi rasio penderita hiv*100.000 aceh 1.37 nusa tenggara barat 3.59 sumatera utara 13.41 nusa tenggara timur 9.36 sumatera barat 7.53 kalimantan barat 10.80 riau 12.64 kalimantan tengah 5.53 jambi 6.22 kalimantan selatan 11.19 sumatera selatan 4.24 kalimantan timur 23.22 bengkulu 6.04 kalimantan utara 24.46 lampung 4.64 sulawesi utara 16.78 kep. bangka belitung 9.63 sulawesi tengah 5.37 kepulauan riau 51.13 sulawesi selatan 11.54 dki jakarta 58.56 sulawesi tenggara 5.25 jawa barat 11.54 gorontalo 0.61 jawa tengah 11.85 sulawesi barat 1.68 di yogyakarta 19.78 maluku 36.20 jawa timur 16.67 maluku utara 10.12 banten 8.95 papua barat 59.32 bali 56.36 papua 120.53 m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 116 sebarang pembangun subgrup siklik dari suatu grup ( ),n +z indra bayu muktyas1 dan samsul arifin2 program studi pendidikan matematika stkip surya1,2 indrabayu.muktyas@stkipsurya.ac.id1, samsul.arifin@stkipsurya.ac.id2 doi: https://doi.org/10.15642/mantik.2018.4.1.116-121 abstrak ( ),n +z adalah grup himpunan bilangan bulat modulo n dengan suatu operasi penjumlahan modulo n. suatu subgrup siklik adalah subgrup yang dibangun oleh satu buah elemen dari suatu grup. pada grup ( ),n +z , sebarang subgrup siklik di dalamnya dapat ditentukan melalui pembangun yang merupakan faktor dari n. tujuan dari penelitian ini adalah untuk menentukan sebarang pembangun subgrup siklik dari suatu grup ( ),n +z dengan menggunakan bantuan pemrograman python. hasil penelitian menunjukkan bahwa dengan menggunakan python, untuk sebarang subgrup siklik, dapat ditentukan sebarang pembangunnya. kata kunci: grup ( ),n +z , subgrup siklik, pembangun, python abstract ( ),n +z is a group of the integer modulo n with an addition operation. a cyclic subgroup is a subgroup that is generated by one element of a group. in group ( ),n +z , any cyclic subgroup can be determined through a generator which is a factor of n. the aim of this article is to get all generator of the cyclic subgroup of a group ( ),n +z using python. the result of our study shows that by using python, for any cyclic subgroup of ( ),n +z , we can get all their generator. keywords: group ( ),n +z , cyclic subgroup, generator, python 1. pendahuluan suatu grup adalah himpunan g dengan operasi biner *, sedemikian hingga berlaku tertutup, asosiatif, terdapat elemen identitas dan setiap elemen memiliki invers. jika suatu grup memiliki sifat * *a b b a= ,untuk setiap elemen a dan b, maka dikatakan bahwa grup tersebut komutatif. sifat-sifat dasar mengenai grup dapat dipelajari di [1], [3] dan [5]. grup siklik adalah suatu grup yang setiap elemennya dapat ditulis sebagai perpangkatan dari setiap unsur tetap pada grup tersebut. karakteristik dari grup siklik dapat dilihat di [4], [7] dan [8]. subgrup adalah subhimpunan h di dalam grup g yang juga merupakan grup dengan operasi yang sama di g. untuk suatu a elemen grup g, dapat dibentuk subhimpunan s berisi semua elemen g yang merupakan hasil perpangkatan dari elemen a. subhimpunan s tersebut membentuk subgroup di g, dan disebut subgroup siklik yang dibangun oleh a. ingat bahwa setiap grup siklik adalah grup komutatif dan subgrup dari suatu grup siklik juga siklik. himpunan semua bilangan bulat modulo n, dinotasikan dengan nz , adalah suatu grup terhadap operasi penjumlahan modulo. grup ini sangat penting dalam mempelajari teori grup, karena banyak konsep dalam teori grup yang menggunakannya sebagai contoh. grup ( ),n +z dikonstruksi dengan menggunakan algoritma mailto:indrabayu.muktyas@stkipsurya.ac.id mailto:samsul.arifin@stkipsurya.ac.id2 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 117 pembagian pada himpunan semua bilangan bulat z . proses pembentukan nz ini dapat dipelajari di [2]. python adalah bahasa pemrograman yang multiguna dan mudah untuk dipelajari. python juga dapat berjalan di berbagai platform system operasi, seperti windows (lihat [9]), linux, mac os, android (lihat [10]), dan lain lain. dalam tulisan ini akan dikaji penentuan semua subgroup siklik dari grup ( ),n +z dengan menggunakan bantuan pemrograman python. 2. tinjauan pustaka 2.1. pengertian grup nz di sesi ini akan dikaji mengenai konstruksi dari grup ( ),n +z . dalam tulisan ini diasumsikan bahwa semua grup bersifat komutatif. berikut diberikan pengertian dari grup. definisi 2.1. fraleigh [1]. diberikan himpunan tidak kosong g yang dilengkapi dengan operasi “ ”. himpunan g disebut grup terhadap operasi “ ” jika memenuhi empat aksioma berikut ini: 1. ( ),a b g a b g    2. ( )( ) ( ), ,a b c g a b c a b c    =   3. ( )( )e g a g a e e a a     =  = 4. ( )( )1 1 1a g a g a a a a e− − −     =  = dalam [7], telah dibuktikan bahwa elemen identitas dan elemen invers dari suatu grup adalah tunggal, berlaku sifat kanselasi dan berlaku sifat socks-shoes. selanjutnya akan diberikan pengertian mengenai suatu grup khusus yang sangat penting dalam mempelajari teori grup, sebab banyak konsep dalam teori grup yang menggunakannya sebagai contoh. grup tersebut dikonstruksi menggunakan algoritma pembagian pada himpunan semua bilangan bulat z . diberikan suatu a z dan bilangan bulat positif nz . dengan menggunakan algoritma pembagian pada bilangan bulat, maka terdapat dengan tunggal ,q r z sedemikian hingga berlaku a qn r= + dengan 0 1r n  − . bilangan bulat q disebut dengan hasil bagi (quotient) dan bilangan bulat r disebut dengan sisa (residu). sisa pembagian r dinotasikan dengan modr a n= . selanjutnya, diperkenalkan konsep mengenai kongruensi pada bilangan bulat sebagai berikut. misalkan diberikan bilangan bulat ,a bz dan bilangan bulat positif nz . bilangan bulat a dikatakan kongruen b modulo n jika n membagi habis a b− , ditulis dengan ( )moda b n . dapat dibuktikan bahwa kongruensi modulo n merupakan relasi ekuivalensi. akibatnya, pada z terpecah menjadi kelas-kelas yang saling asing. untuk suatu a z , dapat dibentuk kelas yang memuat a, yaitu ( ) moda x x a n=  z . secara umum, jika diberikan bilangan bulat positif nz , maka relasi ekuivalensi kongruen modulo n mempunyai n partisi yang saling asing pada z , yaitu 0 , 1 , 2 , ..., dan 1n − . dibentuk nz adalah himpunan semua kelas yang didapatkan dari relasi ekuivalensi kongruen modulo n, yaitu  0, 1, 2,..., 1n n= −z . pada himpunan nz didefinisikan operasi penjumlahan “+”, yaitu untuk setiap , na b z didefinisikan :a b a b+ = + . dapat ditunjukkan bahwa ( ),n +z merupakan grup komutatif komutatif dengan elemen identitasnya adalah 0 nz . 2.2. pengertian subgrup siklik dalam sesi ini akan dikaji mengenai subgrup siklik. berikut adalah definisi dari grup siklik dan pembangun dari suatu grup. definisi 2.1. adkins [6]. suatu grup ( ),g  disebut grup siklik jika terdapat g g sedemikian hingga untuk setiap a g dapat dinyatakan sebagai n a g= , untuk suatu n . elemen g g tersebut dinamakan dengan elemen pembangun atau jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 118 generator dari g, dan g dikatakan grup siklik yang dibangun oleh g, dinotasikan dengan g g= . dari definisi grup siklik di atas, dapat dilihat bahwa grup siklik g yang dibangun oleh g g , yaitu g g= dapat dinyatakan sebagai  ng g n= z . lebih lanjut, setiap grup siklik adalah grup komutatif (lihat[1]). perhatikan contoh berikut. grup ( )72 , +z merupakan grup siklik yang dibangun oleh 72 1z , yaitu 72 1=z . di lain pihak, grup ( )72 , +z ternyata juga dibangun oleh 7211z , yaitu 72 11=z . misalkan diberikan grup ( ),g  dan himpunan bagian tidak kosong h g . ingat bahwa himpunan h disebut subgrup dari g jika h juga merupakan grup terhadap operasi biner “ ” yang sama pada grup g, dinotasikan dengan h g .(lihat [3]). rotman menjelaskan dalam [4] bahwa suatu subhimpunan dari suatu grup dapat diuji apakah merupakan subgroup atau bukan, yaitu misalkan diberikan grup ( ),g  dan h suatu subhimpunan tidak kosong dari g, maka h subgrup dari g jika dan hanya jika ( ) 1,a b h a b h−    . selanjutnya akan dibuktikan bahwa untuk suatu elemen g g , dapat dibentuk subhimpunan yang dibangun oleh g dan membentuk subgrup di g. teorema 2.2. dummit [2]. diberikan grup g dan g g , maka  ng g n=  merupakan subgrup dari g. selanjutnya, g disebut dengan subgrup siklik dari g yang dibangun oleh g. bukti: jelas bahwa g g dan g tidak kosong, sebab 0 g e g=  . selanjutnya diambil sebarang ,a b g , maka m a g= dan n b g= , untuk suatu ,m nz , sehingga m n− z . selanjutnya, perhatikan bahwa ( ) 1 1 m n m n m n ab g g g g g g − − − − = = =  . dengan demikian, terbukti bahwa g subgrup dari g. ingat bahwa order suatu subgrup adalah banyaknya elemen dari subgrup tersebut. perhatikan contoh berikut. contoh 2.3. diberikan grup  106 0,1, 2,...,105=z terhadap operasi penjumlahan modulo 106. himpunan  0, 2,...,104s = adalah subgrup dari ( )106 , +z yang dibangun oleh 1062z dengan order 53. misalkan diberikan sebarang ,n k z . perhatikan bahwa ( ), 1fpb n k = bermakna bahwa faktor persekutuan terbesar antara bilangan n dan k adalah 1. berikut adalah lemma yang mengatakan bahwa pembangun dari grup n z adalah bilangan k sedemikian hingga berlaku ( ), 1fpb n k = , serta subgrup-subgrup di grup n z adalah subhimpunan yang dibangun oleh faktor-faktor dari n. lemma 2.4. gallian [7]. a) suatu nk z adalah pembangun dari nz jika dan hanya jika ( ), 1fpb n k = . b) di grup nz , untuk setiap pembagi positif k dari n , himpunan n k adalah subgrup tunggal di nz dengan order k. lebih lanjut, hanya n k subgrup-subgrup di nz . lemma tersebut menjamin bahwa semua subgroup siklik di grup nz adalah subhimpunan yang dibangun oleh semua faktor dari n. hal ini yang akan menjadi dasar dari penentuan semua jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 119 subgrup siklik di pemrograman menggunakan python. perhatikan contoh berikut. contoh 2.5. misalkan diberikan grup  70 0,1, 2,..., 69=z . ingat bahwa faktor dari n adalah {1, 2, 5, 7, 10, 14, 35, 70}, sehingga diperoleh subgrupsubgrup di ( )70 , +z adalah:                 1 0,1,..., 69 70 2 0, 2,..., 68 35 5 0, 5,..., 65 14 7 0, 7,..., 63 10 10 0,10,..., 60 7 14 0,14,..., 56 5 35 0, 35 70 70 0 70 order order order order order order order order = = = = = = = = elemen pembangun suatu subgrup siklik tidak tunggal. perhatikan kembali contoh berikut. misalkan diberikan grup ( )100 , +z . subgrup siklik yang dibangun oleh 10010z adalah    10 10 0,10, 20,..., 90n n=  =z . subgrup siklik yang dibangun oleh 10011z adalah sebagai berikut,  11 11n n=  =z    0 1 2 3 10011 0,11 11,11 22,11 33,... 0,1, 2,..., 99 .= = = = = = z dapat dilihat bahwa 10011 = z , yaitu 10011z merupakan elemen pembangun dari grup siklik ( )100 , +z . di lain pihak grup ( )100 , +z juga dibangun oleh 1, yaitu 100 1=z . 3. metodologi penelitian 3.1. python dalam sesi ini akan dikaji pemrograman dalam menentukan semua subgroup siklik dari grup nz dengan menggunakan python 2.7.14, yang merupakan hasil utama dari tulisan ini. hal-hal yang menjadi dasar dalam pembuatan program adalah lemma 3.4. di atas. berikut adalah tampilan pemrograman yang digunakan dalam tulisan ini. lagi = "y" while lagi == "y": from functools import reduce def factors(n): return set(reduce(list.__add__, ([i, n//i] for i in range(1, int(pow(n, 0.5) + 1)) if n % i == 0))) print "================================= =========" print "menentukan semua subgrup siklik di grup zn" n = input("masukkan n:") print "z_%d : %s" %(n, range(n)) faktor = factors(n) listfaktor = list(faktor) listfaktor.sort() print "faktor-faktor dari",n," = %s" %(listfaktor) print "------------------------------------------" print "semua subgrup siklik di z_",n,":" for a in listfaktor: bangunan = [] for i in range(n): hasil = a*i%n if hasil not in bangunan: bangunan.append(hasil) bangunan.sort() print "<",a,"> =", bangunan, ", ---> orde", len(bangunan) quotient = [] for i in range(n): hasil2 = [ (i+b)%n for b in bangunan ] hasil2.sort() if hasil2 not in quotient: quotient.append(hasil2) lagi = raw_input("mau lagi? (y atau t):") berikut adalah tampilan program menggunakan python. dan contoh keluaran dari program di atas, yaitu untuk grup ( )72 , +z dan ( )108 , +z untuk versi os windows. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 120 4. hasil berikut adalah tampilan program menggunakan python. dan contoh keluaran dari program di atas, yaitu untuk grup ( )72 , +z dan ( )108 , +z untuk versi os windows. tampilan keluaran berikut akan menutup sesi ini. a) untuk grup ( )72 , +z ================================== ======== menentukan semua subgrup siklik di grup zn ----------------------------------------- masukkan n:72 z_72 : [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] faktor-faktor dari 72 = [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72] ----------------------------------------- semua subgrup siklik di z_ 72 : < 1 > = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] ---> orde 72 < 2 > = [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70] ---> orde 36 < 3 > = [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69] --> orde 24 < 4 > = [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68] ---> orde 18 < 6 > = [0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66] ---> orde 12 < 8 > = [0, 8, 16, 24, 32, 40, 48, 56, 64] ---> orde 9 < 9 > = [0, 9, 18, 27, 36, 45, 54, 63] ---> orde 8 < 12 > = [0, 12, 24, 36, 48, 60] ---> orde 6 < 18 > = [0, 18, 36, 54] ---> orde 4 < 24 > = [0, 24, 48] ---> orde 3 < 36 > = [0, 36] ---> orde 2 < 72 > = [0] ---> orde 1 mau lagi? (y atau t): b) untuk grup ( )108 , +z ================================== ======== menentukan semua subgrup siklik di grup zn ----------------------------------------- masukkan n:72 z_72 : [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] ================================== ======== menentukan semua subgrup siklik di grup zn ----------------------------------------- masukkan n:108 z_108 : [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107] faktor-faktor dari 108 = [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108] ----------------------------------------- semua subgrup siklik di z_ 108 : jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 121 < 1 > = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107] ---> orde 108 < 2 > = [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106] ---> orde 54 < 3 > = [0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105] ---> orde 36 < 4 > = [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104] ---> orde 27 < 6 > = [0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102] ---> orde 18 < 9 > = [0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99] ---> orde 12 < 12 > = [0, 12, 24, 36, 48, 60, 72, 84, 96] ---> orde 9 < 18 > = [0, 18, 36, 54, 72, 90] ---> orde 6 < 27 > = [0, 27, 54, 81] ---> orde 4 < 36 > = [0, 36, 72] ---> orde 3 < 54 > = [0, 54] ---> orde 2 < 108 > = [0] ---> orde 1 mau lagi? (y atau t): referensi [1] a first course in abstract algebra, sixth edition, john b. fraleigh, addison-wesley, new york, 2000 [2] abstract algebra, 3rd edition, david s. dummit, 2004 [3] abstract algebra, 3rd edition, herstein i., prenntice, 1996 [4] advanced modern algebra, rotman, j. j., prentice hall, new york, 2003 [5] algebra, a graduate course, i. martin isaacs, 1994 [6] algebra: an approach via module theory, adkins weintraub, 1992 [7] contemporary abstract algebra, 9th edition, j.a.gallian, usa, 2017 [8] fundamentals of abstract algebra, d.s. malik, john n. moderson and m.k. sen, usa, 1997 [9] python. (2018, 30 april) https://www.python.org/ [10] google. (2018, 30 april) https://play.google.com/store/apps/detai ls?id=org.qpython.qpy&hl=en https://www.python.org/ https://play.google.com/store/apps/details?id=org.qpython.qpy&hl=en https://play.google.com/store/apps/details?id=org.qpython.qpy&hl=en m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 75 regresi nonparametrik dengan pendekatan deret fourier pada data debit air sungai citarum intaniah ratna nur wisisono1, ade irma nurwahidah2, yudhie andriyana3, neneng sunengsih4 departemen statistika, fakultas mipa, universitas padjadjaran jl. dipati ukur no 35, bandung, jawa barat, indonesia1,2,3,4 rnwintaniah@gmail.com1, adeirmanurwahidah@gmail.com2, y.andriyana@unpad.ac.id3, neneng@unpad.ac.id4 doi:https://doi.org/10.15642/mantik.2018.4.2.75-82 abstrak debit air sungai adalah salah satu faktor yang mempengaruhi terjadinya banjir. besaran debit air bervariasi dari waktu ke waktu sehingga dibutuhkan pemodelan untuk mengetahui resiko banjir. analisis yang sering digunakan untuk pemodelan adalah analisis regresi. pemodelan regresi dapat dilakukan dengan tiga pendekatan yaitu pendekatan parametrik, semiparametrik, dan nonparametrik. pemodelan parametrik data debit air bisa menggunakan arima box jenkins. salah satu pendekatan nonparametrik yang dikembangkan adalah menggunakan deret fourier. regresi nonparametrik deret fourier menghasilkan kurva sinus cosinus, sehingga sebaran data yang berulang sangat sesuai didekati menggunakan deret fourier. estimasi deret fourier dapat menggunakan ols (ordinary least square). dalam regresi nonparametrik deret fourier tingkat kemulusan fungsinya ditentukan oleh bandwidthnya (k). penentuan bandwidth optimal dapat menggunakan metode gcv (generalized cross validation). dari hasil penghitungan didapat jumlah k optimal adalah 16. adapun rsquare yang dihasilkan adalah 0.7295 yang berarti bahwa 72,95% total variansi dalam variabel y (debit) dapat dijelaskan oleh model regresi nonparametrik deret fourier. model regresi deret fourier memberikan nilai rmse sebesar 50,51 lebih kecil dibanding nilai rmse arima(1,0,0) yaitu 83,10 sehingga dapat disimpulkan bahwa regresi nonparametrik deret fourier lebih baik dalam memodelkan debit air sungai citarum. kata kunci: fourier, arima, debit air, nonparametrik abstract river discharge is one of the factors that affect the occurrence of floods. it varies over time and hence we need to predict the flood risk. since the plot of the data changes periodically showing a sines and cosines pattern, a nonparametric technique using fourier series approach may be interesting to be applied. fourier series can be estimated using ols (ordinary least square). in a fourier series, nonparametric regression the level of subtlety of its function is determined by their bandwidth (k). optimal bandwidth determined using the gcv (generalized cross validation) method. from the calculation results, we have optimal bandwidth which is equal to 16 with r2 is 0.7295 which means that 72.95% of the total variance in the river discharge variable can be explained by the fourier series nonparametric regression model. comparing to a classical time series technique, arima box jenkins, we obtained arima (1,0,0) with rmse 83.10 while using fourier series approach generate a smaller rmse 50.51. keyword: fourier, arima, river discharge, nonparametric mailto:rnwintaniah@gmail.com mailto:adeirmanurwahidah@gmail.com2 mailto:y.andriyana@unpad.ac.id3 mailto:neneng@unpad.ac.id jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 76 1. pendahuluan citarum adalah sungai terbesar dan terpanjang di provinsi jawa barat. sungai ini sangat mempengaruhi kehidupan masyarakat disekitarnya. pemanfaatan sungai citarum sangat bervariasi dari hulu hingga hilir baik untuk memenuhi kebutuhan rumah tangga, irigasi, pertanian, peternakan maupun industri. banjir di kabupaten bandung adalah sejarah yang terus berulang. sejak puluhan tahun bahkan ratusan tahun lalu, kawasan hulu dari sungai citarum ini terus mengalami banjir. derasnya luapan sungai citarum sulit untuk dibendung. meskipun pernah ada proyek besar untuk normalisasi citarum dengan memotong aliran sungai yang berkelok dan mengalirkannya ke waduk jatiluhur di purwakarta, namun tetap saja ketika hujan deras, citarum pasti meluap dan memicu banjir di kabupaten bandung. salah satu faktor yang mempengaruhi terjadinya banjir adalah debit air sungai. debit air sungai adalah tinggi permukaan air sungai yang terukur oleh alat ukur permukaan air sungai [5]. debit air sungai dalam hidrologi adalah tinggi permukaan air sungai yang terukur oleh alat ukur permukaan air sungai. dalam sistem satuan si besaran debit dinyatakan dalam satuan meter kubik per detik (m3/detik). salah satu analisis statistik yang digunakan untuk pemodelan adalah analisis regresi. pendekatan regresi dapat dilakukan dengan tiga pendekatan yaitu pendekatan parametrik, semiparametrik, dan nonparametrik. data debit air sungai adalah data deret waktu karena diukur dari waktu ke waktu (dalam hal ini setiap bulan) sehingga dalam analisisnya perlu digunakan metode analisis untuk data deret waktu. arima box jenkins adalah metode analisis data deret waktu parametrik yang paling sering digunakan [11]. salah satu pendekatan lain, pendekatan nonparametrik yang dapat digunakan dalam analisis data deret waktu adalah deret fourier. deret fourier adalah deret yang digunakan dalam bidang rekayasa. deret ini pertama kali ditemukan oleh seorang ilmuan perancis jean-baptiste joseph fourier (17681830). deret ini juga dikenal sebagai deret dalam bentuk sinus dan cosinus yang digunakan untuk mem-presentasikan fungsifungsi periodik secara umum sehingga sebaran data yang berulang/periodik sebagaimana terlihat pada pola data debit air sungai citarum sangat sesuai didekati menggunakan deret ini. penelitian sebelumnya tentang pendekatan deret fourier dalam analisis data deret waktu nonparametrik antara lain; model regresi nonparametrik dengan pendekatan deret fourier pada pola data curah hujan di kota semarang oleh nurjanah dkk [6], model regresi nonparametrik dengan pendekatan deret fourier pada kasus tingkat pengangguran terbuka di jawa timur oleh prahutama [8], kemudian tripena (2007) mengkaji estimator deret fourier pada regresi nonparametrik [9]. sedangkan untuk regresi semi-parametrik menggunakan deret fourier telah dikembangkan oleh asrini [1]. estimasi deret fourier dapat menggunakan ols (ordinary least square). dalam regresi nonparametrik deret fourier tingkat kemulusan fungsinya ditentukan oleh bandwidth-nya. penentuan bandwidth optimal dapat menggunakan metode gcv (generalized cross validation) [10]. bandwidth yang optimal akan memberikan kurva yang mulus, variasi yang rendah dan bias yang besar. dari nilai gcv yang minimum maka akan didapatkan bandwith (k) yang optimal. penelitian ini memodelkan debit air sungai citarum menggunakan regresi nonparametrik dengan pendekatan deret fourier dan akan dibandingkan dengan hasil pemodelan arima box jenkins. 2. tinjauan pustaka 2.1 metode regresi metode regresi adalah suatu metode statistik yang sering kali digunakan untuk menyelidiki dan memodelkan hubungan antara variabel respon 𝑌 dan variabel prediktor 𝑋. misalnya diberikan himpunan data {(𝑋𝑖 , 𝑌𝑖 )}, 𝑖 = 1, … , 𝑛 secara umum hubungan antara 𝑌 dan 𝑋 dapat ditulis sebagai berikut: 𝑌𝑖 = 𝑚(𝑋𝑖 ) + 𝑖 (1) jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 77 dengan 𝑚(𝑋𝑖 ) merupakan suatu fungsi regresi yang belum diketahui dan akan ditaksir, dan 𝑖 adalah suatu variabel acak yang menggambarkan variasi 𝑌 di sekitar 𝑚(𝑥). penaksiran fungsi regresi dapat dilakukan dengan dua cara yaitu secara parametrik dan nonparametrik. pada regresi parametrik digunakan bentuk fungsi parametrik tertentu sebagai 𝑚(𝑥). fungsi 𝑚(𝑥) digambarkan oleh sejumlah parameter yang harus ditaksir. dalam regresi parametrik terdapat beberapa asumsi terkait dengan model, sehingga diperlukan pemeriksaan terhadap asumsiasumsi tersebut. pada regresi nonparametrik, fungsi regresi 𝑚(𝑥) ditaksir tanpa referensi bentuk kurva tertentu. cara ini lebih fleksibel karena tidak memerlukan informasi apapun tentang sebaran data. dengan suatu mekanisme matematis tertentu 𝑚(𝑥) akan mengikuti bentuk data. 2.2 regresi nonparametrik deret fourier salah satu model regresi dengan pendekatan nonparametrik yang dapat digunakan untuk menduga kurva regresi adalah regresi nonparametrik deret fourier. deret fourier merupakan polinomial trigonometri yang mempunyai fleksibilitas, sehingga dapat menyesuaikan diri secara efektif terhadap sifat lokal data. deret fourier baik digunakan untuk menjelaskan kurva yang menunjukkan gelombang sinus dan cosinus. diberikan data berpasangan sebagai berikut (𝑦𝑗 𝑡𝑖𝑗 ) dengan 𝑗 = 1, 2, 3, … , 𝑛 menyatakan banyaknya pengamatan dan 𝑖 = 1, 2, 3, … , 𝑝 menyatakan banyaknya variabel bebas. karena hanya terdapat satu variabel bebas, model regresi nonparametriknya sebagai berikut: 𝒚𝒋 = 𝑿𝒋𝜷 + 𝜺 (2) jika 𝑦 = 𝑓(𝑡) + maka 𝑓(𝑡) = [𝑓(𝑡1)𝑓(𝑡2)𝑓(𝑡3) … 𝑓(𝑡𝑛 )] 𝑓(𝑡) merupakan kurva yang tidak diketahui bentuknya maka 𝑓(𝑡) didekati dengan menggunakan deret fourier. sebelum pembentukan model fourier, variabel t ditransformasi terlebih dahulu menjadi: 𝑡 = 2𝜋(𝑡 − min(𝑡))/(max(𝑡) − min(𝑡)) (3) dengan model fourier sebagai berikut [2]: 𝑦 = 𝛼0 + 𝛼1𝑡 + 𝛼2𝑡 2 + ∑ 𝜆𝑘 sin(𝑡𝑘) + 𝐾 𝑘=1 𝛿𝑘 cos (𝑘𝑡) + 𝜺 (4) 2.3 pemilihan bandwith dalam regresi nonparametrik deret fourier pemilihan bandwith sangat penting, karena berpengaruh pada model regresi nonparametrik deret fourier yang akan dipilih. ada 2 strategi untuk memilih bandwith yang baik. strategi pertama adalah memilih banyaknya bandwith yang relatif sedikit, sedangkan strategi yang kedua adalah sebaliknya, yakni menggunakan bandwith yang relatif banyak. diantara kedua strategi tersebut, strategi kedua lebih banyak digunakan pada model yang sangat memperhatikan pola matematis yang ada pada data. sedangkan strategi pertama, lebih mengarah pada alasan kesederhanaan model. pemilihan bandwidth yang terlalu kecil akan menghasilkan kurva yang undersmoothing yaitu sangat kasar dan sangat fluktuatif, dan sebaliknya bandwidth yang terlalu lebar akan menghasilkan kurva yang over-smoothing yaitu sangat mulus, tetapi tidak sesuai dengan pola data [4]. penentuan lokasi bandwith yang berbeda akan menghasilkan model regresi nonparametrik deret fourier yang berbeda pula. lokasi bandwith tersebut akan berpengaruh terhadap nilai kriteria dari model regresi nonparametrik deret fourier yang dibentuk. salah satu metode pemilihan titik bandwith yang optimal adalah generalized cross validation (gcv). model regresi nonparametrik deret fourier yang sesuai berkaitan dengan titik bandwith yang optimal didapat dari nilai gcv minimum. fungsi gcv didefinisikan sebagai [10]: 𝐺𝐶𝑉(𝐾) = 𝑀𝑆𝐸(𝐾) (𝑛−1𝑡𝑟𝑎𝑐𝑒(𝑰 − 𝑨))2 (5) jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 78 𝑀𝑆𝐸(𝐾) = ∑ (𝑦𝑖 − 𝑦�̂�) 2𝑛 𝑖=1 𝑛 (6) dengan 𝑦𝑖 = variabel respon 𝑦�̂� = hasil dugaan dari 𝑦𝑖 𝑛 = banyaknya pengamatan 2.4 metode box-jenkins metode box-jenkins merupakan metode yang digunakan untuk analisis dan peramalan data kurva waktu (time series). metode ini dapat digunakan pada data seret waktu yang stasioner dan terdiri dari tiga langkah yaitu identifikasi model, pendugaan parameter, dan diagnostik model [7]. identifikasi model merupakan tahap untuk menentukan model-model sementara, yaitu dengan menentukan nilai p, q dan d. penentuan nilai-nilai tersebut dilakukan dengan mengamati grafik fungsi acf (korelogram) dan pacf (korelogram parsial). nilai p (ordo proses ar) dapat ditentukan dengan melihat nilai pada grafik fungsi pacf dan nilai q (ordo proses ma) dapat ditentukan dengan melihat nilai pada grafik fungsi acf. setelah identifikasi model tahap selanjutnya adalah pendugaan parameter. pendugaan parameter bertujuan untuk menentukan apakah parameter sudah layak digunakan dalam model. pendugaan parameter dapat dilakukan dengan menggunakan beberapa metode, yaitu metode momen, kuadrat terkecil dan kemungkinan maksimum (likelihood). tahap ketiga yaitu diagnostik model dilakukan untuk melihat model yang relevan dengan data. pada tahap ini model harus dicek kelayakannya dengan melihat sifat sisaan dari sisi kenormalan dan kebebasannya. secara umum pengecekan kebebasan sisaan model dapat dilakukan dengan menggunakan uji q modifikasi boxpierce (ljung-box). selanjutnya yaitu pengecekan pada kenormalan sisaan dengan melakukan uji shapiro-wilk normality. jika nilai-p yang dihasilkan > α, maka dapat disimpulkan bahwa sisaan telah memenuhi asumsi kenormalan sisaan. setelah semua proses dalam metode box-jenkins dilakukan tahap berikutnya adalah melakukan overfitting model yaitu membandingkan model dengan model lain yang berbeda satu ordo di atasnya. hal yang dibandingkan pada overfitting adalah signifikasi parameter, pemenuhan asumsi sisaan, dan akaike’s information criterion (aic). jika dalam proses overfitting didapatkan model yang relevan dengan data, maka langkah terakhir adalah proses peramalan. peramalan merupakan proses untuk menentukan data beberapa periode waktu kedepan dari titik waktu ke-t. 2.5 kriteria model terbaik akaike's information criterion (aic) ditemukan oleh akaike dimana kriteria ini dapat digunakan untuk melakukan perbandingan model pada data yang sama. aic adalah metode yang digunakan untuk mendapatkan model terbaik. aic didefinisikan sebagai: 𝐴𝐼𝐶 = −2𝑙(𝑦; 𝛼) + 2 𝑡𝑟𝑎𝑐𝑒(𝐻) (6) dimana 𝑙(𝑦; 𝛼) adalah fungsi likelihood. suatu model dikatakan model terbaik jika memiliki nilai aic terkecil. sebagai tambahan, eubank menyebutkan bahwa kinerja dari hasil pendugaan model regresi dapat dilihat dari nilai root mean square error (rmse) dimana diformulasikan sebagai berikut [3]: 𝑅𝑀𝑆𝐸 = √ 1 𝑛 ∑(𝑦𝑖 − �̂�𝑖 ) 2 𝑛 𝑖=1 (7) selanjutnya untuk mengetahui kualitas suatu model, kita dapat menggunakan koefisien determinasi (r2). nilai r2 menunjukkan kemampuan model dalam menjelaskan variabilitas data. semakin tinggi nilai r2 maka semakin baik kualitas dari suatu model. nilai r2 diperoleh dengan rumus: 𝑅2 = 1 − ∑ (𝑦𝑖 − �̂�𝑖 ) 2𝑛 𝑖=1 ∑ (𝑦𝑖 − �̅�𝑖 ) 2𝑛 𝑖=1 (8) 3. metodologi penelitian 3.1 sumber data dan variabel penelitian data yang digunakan dalam penelitian ini adalah data sekunder yang bersumber jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 79 dari bbws citarum (balai besar wilayah sungai citarum) periode januari 2009 – desember 2016 yang merupakan data bulanan. variabel respon (y) yang digunakan dalam penelitian ini adalah debit air sungai tertinggi sedangkan variabel bebas (x) yang digunakan pada penelitian ini adalah waktu. 3.2 tahapan penelitian langkah – langkah analisis dalam penelitian ini adalah sebagai berikut : 1. membuat plot time series debit air sungai 2. memilih jumlah bandwith yang optimal berdasarkan nilai dari gcv 3. mendapatkan model regresi nonparametrik deret fourier dengan bandwith yang optimum 4. menghitung nilai 𝑅2, aic, dan rmse dari model regresi nonparametrik deret fourier 5. melakukan pemodelan arima box jenkins untuk data debit 6. menghitung nilai rmse dari model arima box jenkins 7. membandingkan nilai rmse dari model regresi nonparametrik deret fourier dengan model arima 8. menarik kesimpulan 4. hasil dan pembahasan 4.1 deskriptif statistik berikut ini adalah deskriptif dari data debit air sungai tabel 1. deskriptif data debit air sungai (m3/detik) min. 5,25 median 107,72 mean 129,91 max. 549,35 stdev 97,62 berdasarkan tabel 1 diperoleh informasi bahwa debit air sungai memiliki nilai tertinggi sebesar 549,35 m3/detik dan nilai terendah sebesar 5,25 m3/detik. dimana nilai rata-ratanya adalah 129,91 m3/detik dan mediannya 107,72 m3/detik. 4.2 plot debit air sungai terhadap waktu pola hubungan antara waktu sebagai variabel prediktor terhadap debit air sungai sebagai variabel respon terlihat pada gambar 1. dari gambar terlihat adanya pola periodik, dengan kata lain ada pola seperti sinus cosinus. sehingga diharapkan regresi nonparametrik deret fourier akan dapat memodelkan data debit ini dengan baik. gambar 1. time plot debit air sungai 4.3 pemodelan regresi nonparametrik deret fourier regresi nonparametrik deret fourier adalah salah satu metode yang digunakan untuk menaksir kurva regresi nonparametrik. model nonparametrik deret fourier dengan titik bandwiths digunakan untuk menggambarkan pola periodik. metode yang digunakan dalam menaksir parameter nonparametrik deret fourier adalah metode kuadrat terkecil (ordinary least square). kriteria yang harus diperhatikan dalam membentuk model regresi nonparametrik deret fourier adalah menentukan bandwith (k) untuk model regresi. nilai k merupakan bilangan bulat positif. penentuan k optimal menggunakan kriteria gcv. adapun hasil yang didapat dari setiap k yang dicobakan adalah sebagai berikut: jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 80 tabel 2. nilai gcv untuk setiap k nilai k gcv nilai k gcv 1 9429,19 11 7465,17 2 9214,37 12 7331,25 3 9618,46 13 7005,44 4 9162,12 14 7066,98 5 9538,69 15 6769,36 6 9996,96 16 6317,50 7 10340,79 17 6670,39 8 7742,79 18 7141,09 9 7578,16 19 7496,61 10 7961,14 20 8033,92 berdasarkan tabel 2 terlihat nilai k optimum adalah 16 dengan nilai gcv sebesar 6371,50. selain melihat nilai gcv dilihat juga nilai r2 dan mse untuk lebih meyakinkan bahwa k optimum adalah 16. tabel 3. nilai r2 dan mse untuk setiap k nilai k r2 mse 1 0,10 8472,56 2 0,16 7919,60 3 0,16 7919,60 4 0,24 7182,76 5 0,24 7182,76 6 0,24 7182,76 7 0,24 7182,76 8 0,47 4981,23 9 0,51 4625,34 10 0,51 4625,34 11 0,57 4083,32 12 0,60 3787,34 13 0,64 3412,26 14 0,64 3412,26 15 0,69 2915,32 16 0,73 2550,72 17 0,73 2550,72 18 0,73 2550,72 berdasarkan nilai r2 dan mse untuk setiap k juga didapatkan nilai k optimum 16 dengan r2 sebesar 72,95% dan mse sebesar 2550,72. meskipun untuk k > 16 didapatkan nilai r2 dan mse yang sama dengan k = 16, dipilih k optimum 16 berdasarkan prinsip parsimoni, dipilih model yang lebih sederhana. untuk nilai k = 16 ada 35 parameter yang harus diduga. setelah didapatkan nilai k optimum, selanjutnya dilakukan pendugaan parameter. nilai dugaan parameter regresi nonparametrik deret fourier untuk data debit sungai citarum ditampilkan dalam tabel berikut: tabel 4. nilai dugaan parameter (statistik) untuk k=16 statistik nilai statistik nilai 𝛼0 -829,61 𝜆16 -26,95 𝛼1 905,30 𝛿1 586,32 𝛼2 -143,19 𝛿2 116,48 𝜆1 45,03 𝛿3 71,76 𝜆2 36,31 𝛿4 26,95 𝜆3 5,67 𝛿5 21,97 𝜆4 -31,92 𝛿6 23,17 𝜆5 5,09 𝛿7 14,14 𝜆6 0,54 𝛿8 52,61 𝜆7 -8,48 𝛿9 22,98 𝜆8 51,82 𝛿10 -2,76 𝜆9 15,12 𝛿11 -15,05 𝜆10 6,63 𝛿12 6,92 𝜆11 28,05 𝛿13 -4,55 𝜆12 -24,56 𝛿14 -1,50 𝜆13 -27,05 𝛿15 10,89 𝜆14 18,78 𝛿16 -5,78 𝜆15 22,79 model regresi nonparametrik deret fourier untuk menggambarkan hubungan debit air sungai dan waktu dengan jumlah bandwith 16 jika dituliskan persamaan regresinya adalah sebagai berikut: 𝑓(𝑥) = −829.61 + 905.3 𝑡 − 143.19 𝑡2 +45.03 sin 𝑡 + 36.31 sin 2𝑡 +5.66 sin 3𝑡 − 31.92 sin 4t +5.08 sin 5𝑡 + 0.54 sin 6𝑡 −8.48 sin 7𝑡 + 51.82 sin 8𝑡 +15.11 sin 9𝑡 + 6.63 sin 10𝑡 +28.05 sin 11𝑡 − 24.56 sin 12𝑡 −27.05 sin 13𝑡 + 18.78 sin 14𝑡 +22.79 sin 15𝑡 − 26.94 sin 16𝑡 +586.32 cos 𝑡 + 116.48 cos 2𝑡 +71.76 cos 3𝑡 + 26.95 cos 4𝑡 +21.97 cos 5𝑡 + 23.17 cos 6𝑡 +14.14 cos 7𝑡 + 52.62 cos 8𝑡 +22.98 cos 9𝑡 − 2.76 cos 10𝑡 −15.05 cos 11𝑡 + 6.92 cos 12𝑡 −4.55 cos 13𝑡 − 1.5 cos 14𝑡 +10.88 cos 15𝑡 − 5.78 cos 16𝑡 untuk melihat seberapa baik model yang dibentuk dapat dengan membandingkan jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 81 pola plot antara data debit dengan nilai dugaannya. gambar 2. plot model regresi nonparametrik deret fourier k 16 berdasarkan gambar 2 terlihat nilai dugaan dari model cukup menggambarkan pola dari data debit, meskipun ada beberapa titik yang tidak terjangkau. model regresi nonparametrik deret fourier dengan k = 16 menghasilkan model dengan r2 = 72,95%. dapat diartikan bahwa 72,95% total variansi dalam variabel y (debit) dapat dijelaskan oleh model regresi yang terbentuk. 4.4 pemodelan arima box-jenkins referensi langkah pertama dalam pemodelan arima box jenkins adalah pemeriksaan kestasioneran data. kestasioneran data dapat dilihat dari pola acf dan uji augmented dickey fuller. gambar 3. plot acf debit air sungai berdasarkan gambar 3 terlihat pola plot acf data terlihat pola sinusoidal serta hasil uji augmented dickey fuller memberikan p-value sebesar 0,01< 𝛼=0,05 dimana hipotesisnya adalah: h0 = data tidak stasioner h1 = data stasioner. sehingga disimpulkan data stasioner. selanjutnya dilakukan identifikasi model arima dari plot acf dan pacf. gambar 4. plot acf dan pacf debit air sungai berdasarkan gambar 4 dipilih kandidat model arima(1,0,1), arima(1,0,0), dan arima(0,0,1). dari ketiga kandidat model didapat nilai aic untuk arima(1,0,1) sebesar 1125,44 arima(1,0,0) sebesar 1125,39 dan arima(0,0,1) sebesar 1126,49. karena nilai aic model kedua lebih kecil maka model terbaik dari kandidat model adalah arima(1,0,0). 4.5 perbandingan model perbandingan antara model regresi nonparametrik deret fourier dan model arima menggunakan kriteria rmse ditampilkan pada tabel 5. tabel 5. perbandingan nilai rmse model rmse deret fourier 50,51 arima(1,0,0) 83,10 dari tabel 5 terlihat model regresi nonparametrik deret fourier memiliki nilai rmse yang lebih kecil dibandingkan model arima(1,0,0). jadi model terbaik diantara kedua model tersebut adalah model regresi nonparametrik deret fourier. 5. kesimpulan 1) dengan menggunakan nilai gcv pada model regresi nonparametrik deret fourier diperoleh jumlah bandwith yang optimum adalah 16. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 82 2) model regresi nonparametrik deret fourier yang digunakan untuk menggambarkan hubungan debit air sungai dan waktu pada penelitian ini adalah sebagai berikut: 𝑓(𝑥) = −829.61 + 905.3 𝑡 − 143.19 𝑡2 +45.03 sin 𝑡 + 36.31 sin 2𝑡 +5.66 sin 3𝑡 − 31.92 sin 4t +5.08 sin 5𝑡 + 0.54 sin 6𝑡 −8.48 sin 7𝑡 + 51.82 sin 8𝑡 +15.11 sin 9𝑡 + 6.63 sin 10𝑡 +28.05 sin 11𝑡 − 24.56 sin 12𝑡 −27.05 sin 13𝑡 + 18.78 sin 14𝑡 +22.79 sin 15𝑡 − 26.94 sin 16𝑡 +586.32 cos 𝑡 + 116.48 cos 2𝑡 +71.76 cos 3𝑡 + 26.95 cos 4𝑡 +21.97 cos 5𝑡 + 23.17 cos 6𝑡 +14.14 cos 7𝑡 + 52.62 cos 8𝑡 +22.98 cos 9𝑡 − 2.76 cos 10𝑡 −15.05 cos 11𝑡 + 6.92 cos 12𝑡 −4.55 cos 13𝑡 − 1.5 cos 14𝑡 +10.88 cos 15𝑡 − 5.78 cos 16𝑡 dalam memodelkan data debit air sungai dalam penelitian ini, model regresi nonparametrik deret fourier lebih baik dibandingkan model arima (1,0,0) referensi [1] asrini, luh juni. “regresi parametrik deret fourier”, prosiding seminar nasional fmipa universitas negeri surabaya, (2012) 77-80, 24 november, surabaya. [2] bilodeau, m. “fourier smoother and additive models”, the canadian journal of statistics, 3, (1992) 257-259. [3] eubank, r., nonparametric regression and spline smoothing. new york: marcel dekker. (1999). [4] hardle, wolfgang. 1994. applied nonparametric regression. springerverlag. berlin. (1994). [5] mulyana. pemodelan debit air sungai studi kasus das cikapundung. makalah, disampaikan pada lokakarya sistem informasi pengelolaan das: inisiatif pengembangan infrastruktur data. ipb. (2007). 5 september, bogor. [6] nurjanah, fatmawati dkk. model regresi nonparametrik dengan pendekatan deret fourier pada pola data curah hujan di kota semarang. universitas muhammadiyah semarang. semarang. (2015). [7] pankratz, a. forecasting with univariate box-jenkins models: concepts and cases. john wiley and sons, new york. (1983). [8] prahutama, alan. model regresi nonparametrik dengan pendekatan deret fourier pada kasus tingkat pengangguran terbuka di jawa timur. prosiding seminar nasional statistika. universitas diponegoro. (2013). semarang. [9] tripena, a. tesis. estimator deret fourier dalam regresi nonparametrik, its, surabaya. 2007. [10] wu, h. dan zhang, j.t. nonparametric regression methods for longitudinal data analysis, a john-wiley and sons inc. publication, new jersey. (2006). [11] ulinnuha, nurissaidah dan farida, yuniar “prediksi cuaca kota surabaya menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter”, jurnal matematika mantik, vol. 4, no. 1, hal. 59-67, mei 2018 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 32 analisis angka harapan lama sekolah di indonesia timur menggunakan weighted least squares regression arifin m. kahar prodi magister statistika terapan, universitas padjajaran, ipinputera@gmail.com doi:https://doi.org/10.15642/mantik.2018.4.1.32-41 abstrak indeks pembangunan manusia (ipm) merupakan salah satu data dan informasi yang digunakan oleh pemerintah daerah untuk mengukur pencapaian pembangunan manusia dengan sejumlah komponen dasar kualitas hidup yaitu angka harapan hidup yang mewakili dimensi kesehatan, angka harapan lama sekolah (hls) dan rata-rata lama sekolah (rls) mewakili dimensi pendidikan, dan rata-rata pengeluaran per kapita disesuaikan yang mewakili dimensi hidup layak. ipm khususnya di wilayah indonesia timur pada tiga tahun terakhir terus meningkat namun angkanya selalu berada di bawah angka nasional. salah satu dimensi yang masih rendah pencapaiannya adalah dimensi pendidikan. hls merupakan salah satu indikator pada dimensi pendidikan yang masih rendah pencapaiannya. untuk itu, penelitian ini ingin mengetahui pengaruh persentase penduduk miskin, produk domestik regional bruto (pdrb) per kapita, angka partisipasi murni (apm) smp, dan rasio fasilitas pendidikan terhadap hls di wilayah indonesia timur pada 2016. dengan metode weighted least squares (wls) diperoleh kesimpulan bahwa keempat variabel prediktor berpengaruh signifikan terhadap hls di indonesia timur. kata kunci : angka harapan lama sekolah, heteroskedastisitas, weighted least squares abstract human development index (hdi) is one of the data and information used by the local government to measure the achievement of human development with the basic components of quality of life that is life expectancy that represents health dimension, expected years of schooling (eys) and mean years of schooling (mys) represents the educational dimension, and purchasing power parity that represents decent living dimension. hdi especially in eastern indonesia in the last three years has continued to increase but the figure is always below from the national figure even left behind if compared with west indonesia. one dimension that is still low achievement is the educational dimension. eys is one of the indicators on the educational dimension that is still a low achievement. therefore, this research would like to know the influence of percentage of poor people, gross regional domestic product (grdp) per capita, net enrollment rate (ner) of junior high school, and a ratio of educational facilities to eys in eastern indonesia. using weighted least squares (wls) method concluded that the four predictor variables used were able to influence eys in eastern indonesia. keywords: expected years of schooling, heteroscedasticity, weighted least squares mailto:ipinputera@gmail.com 33 1. pendahuluan secara sederhana pembangunan dapat dimaknai sebagai usaha atau proses untuk melakukan perubahan ke arah yang lebih baik. pada pelaksanaannya pembangunan bersifat multi dimensi dan memiliki berbagai kompleksitas masalah. proses pembangunan terjadi di semua aspek kehidupan masyarakat, baik aspek ekonomi, politik, sosial, maupun budaya. sejak diberlakukannya otonomi daerah di indonesia, pemerintah daerah diberikan wewenang untuk mengelola daerahnya masing-masing. hal tersebut berdampak pada meningkatnya kebutuhan akan data atau informasi yang lebih detail mengenai keadaan suatu daerah. data tersebut selain berguna untuk mengetahui dan mengevaluasi hasil pembangunan juga digunakan sebagai acuan dalam merumuskan kebijakan pembangunan di tingkat daerah. indeks pembangunan manusia (ipm) merupakan salah satu data dan informasi yang digunakan oleh pemerintah daerah untuk mengukur pencapaian pembangunan manusia dengan sejumlah komponen dasar kualitas hidup. komponen yang digunakan dalam perhitungan ipm adalah angka harapan hidup yang mewakili dimensi kesehatan, angka harapan lama sekolah (hls) dan rata-rata lama sekolah (rls) mewakili dimensi pendidikan, dan rata-rata pengeluaran per kapita yang disesuaikan mewakili capaian pembangunan untuk hidup layak. data ipm yang dikeluarkan bps khususnya di wilayah timur indonesia pada tiga tahun terakhir terus meningkat namun angkanya selalu berada di bawah angka nasional. pada 2016, ipm indonesia sudah mencapai 70,18 namun angka ipm maluku, maluku utara, papua barat dan papua belum mampu menyentuh angka tersebut. salah satu dimensi penyebab ipm pada keempat provinsi tersebut rendah adalah dimensi pendidikan yang diukur dari hls dan rls [1]. data bps menunjukkan bahwa hls dan rls maluku dan maluku utara sudah berada di atas angka nasional. akan tetapi, papua dan papua barat masih jauh tertinggal. hasil ini memberi gambaran bahwa masih terdapat perbedaan angka yang cukup jauh dengan angka nasional terutama di papua dan papua barat. secara nasional, penduduk indonesia usia tujuh tahun ke atas berpotensi menempuh pendidikan hingga diploma i pada 2016. hal ini dapat dilihat dari angka hls yang mencapai 12,72 tahun (12,72 ≈13 tahun = sd 6 tahun + smp 3 tahun + sma 3 tahun + perguruan tinggi 1 tahun). pada tahun yang sama, hls wilayah timur indonesia khususnya di papua barat mencapai 12,26 tahun. artinya pada tahun tersebut penduduk usia 7 tahun ke atas di provinsi tersebut berpotensi menempuh pendidikan hingga tamat sma (12 tahun=sd 6 tahun + smp 3 tahun + sma 3 tahun). sementara hls di papua baru mencapai 10,23 tahun, artinya pada tahun tersebut penduduk usia 7 tahun ke atas di provinsi ini berpotensi menempuh pendidikan hingga jenjang sma kelas i (10 tahun = sd 6 tahun + smp 3 tahun + sma 1 tahun). perbedaan pembangunan pendidikan tampak begitu nyata antara wilayah barat dengan timur indonesia. meskipun maluku dan maluku utara memiliki hls yang sudah berada di atas angka nasional. akan tetapi keempat provinsi di wilayah timur indonesia ini masih sangat jauh tertinggal jika dibandingkan dengan wilayah barat indonesia seperti daerah istimewa yogyakarta. 2. tinjauan pustaka 2.1 angka harapan lama sekolah hls merupakan salah satu output yang dapat digunakan untuk memotret pemerataan pembangunan pendidikan di indonesia. karena hls mengukur kesempatan pendidikan seorang penduduk di mulai pada usia tujuh tahun. secara sederhana, hls dapat didefinisikan sebagai angka partisipasi sekolah menurut umur tunggal. hls merupakan indikator yang menggambarkan lamanya sekolah (dalam tahun) yang diharapkan akan dirasakan oleh anak pada umur tertentu di masa mendatang. angka ini diperoleh dengan 34 cara membagi banyaknya partisipasi sekolah penduduk pada usia a pada tahun t dengan jumlah penduduk yang bersekolah pada usia a pada tahun t. sebagai catatan indikator ini dianggap peka dalam menggambarkan variasi antar provinsi. menurut united nation development programme (undp), hls dihitung dengan cara sebagai berikut [2]: 𝐻𝐿𝑆𝑎 𝑡 = ∑ 𝐸𝑖 𝑡 𝑃𝑖 𝑡 𝑡 𝑖=𝑎 dimana: 𝐻𝐿𝑆𝑎 𝑡 : harapan lama sekolah pada usia a dan pada tahun t 𝐸𝑖 𝑡 : partisipasi sekolah penduduk usia i pada tahun t 𝑃𝑖 𝑡 : populasi penduduk usia i yang bersekolah pada tahun t 𝑖 : usia (a, a+1, ..., n) salah satu tujuan sustainable develompment goals (sdgs) yaitu tujuan keempat adalah menjamin kualitas pendidikan yang adil dan inklusif serta meningkatkan kesempatan belajar seumur hidup untuk semua. pada target 4b, dinyatakan bahwa memastikan semua anak perempuan dan anak laki-laki memiliki akses ke pengembangan anak usia dini yang setara, perawatan, dan pendidikan anak usia dini sehingga mereka siap untuk pendidikan dasar. pada target ini, diharapkan angka kelulusan baik sd, smp, maupun sma ditingkatkan. secara langsung, ketika target ini dicapai maka angka hls dan rls yang merupakan dua indikator penghitungan ipm akan ikut meningkat [3]. akan tetapi, kondisi saat ini dapat dikatakan bahwa target 4b pada sdgs akan sulit tercapai untuk wilayah timur indonesia dalam waktu dekat. belum meratanya pembangunan pendidikan di indonesia menjadi salah satu penyebab target tersebut tidak tercapai. menurut darisandi, beberapa hal yang menjadi penyebab belum meratanya pendidikan di indonesia diantaranya adalah [4]: 1. perbedaan tingkat sosial pernyataan world development report bahwa pendidikan adalah kunci untuk menciptakan, menyerap, dan menyebarluaskan pengetahuan. namun akses terhadap pendidikan tidak tersebar secara merata dan golongan miskin paling sedikit mendapat bagian. kasus ini dapat ditemukan di indonesia yang pendidikannya belum merata antara masyarakat miskin dan golongan masyarakat menengah ke atas. 2. keadaan geografis secara geografis, wilayah indonesia yang cukup luas sebagai negara kepulauan ternyata menjadi salah satu penghambat pemerataan pembangunan pendidikan. hal tersebut berakibat bahwa pembangunan pendidikan tidak dapat terlaksana dengan optimal khususnya di daerah indonesia timur. ketimpangan pembangunan pendidikan antara satu wilayah dengan wilayah yang lain sangat terlihat sekali, baik secara fisik maupun non fisik. 3. sebaran sekolah tidak merata sebagian besar pendirian lembaga pendidikan masih berada dan berorientasi di wilayah perkotaan, sedangkan minat untuk membangun lembaga pendidikan di daerah pedesaan masih sangat kurang. kemudian pembangunan sekolah yang hanya terpusat di wilayah barat khususnya pulau jawa membuat sebaran sekolah menjadi tidak merata. padahal dengan kebutuhan pendidikan yang sangat besar di indonesia timur seharusnya di prioritaskan pembangunan yang cukup besar pula. 2.2 ordinary least squares (ols) bentuk umum model regresi linier berganda dengan k variabel bebas adalah sebagai berikut [5]: 𝑌𝑖 = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + ⋯ +𝛽𝑘𝑋𝑘 + 𝜀𝑖 (1) dengan: • 𝑌𝑖 adalah variabel respon untuk pengamatan ke-i, untuk i = 1, 2, …, n. • 𝛽0, 𝛽1, 𝛽2, … , 𝛽𝑘 adalah parameter • 𝑋0, 𝑋1, 𝑋2, … , 𝑋𝑘 adalah variabel prediktor • 𝜀𝑖 adalah sisa (error) untuk pengamatan ke-i. dalam notasi matriks, persamaan (1) dapat dutulis sebagai berikut: 𝒀 = 𝑿𝜷 + 𝜺 (2) 35 dimana: 𝒀 = [ 𝑌1 𝑌2 ⋮ 𝑌𝑛 ], 𝑿 = [ 𝑋11 𝑋12 ⋯ 𝑋1𝑘 𝑋21 𝑋22 ⋯ 𝑋2𝑘 ⋮ ⋮ ⋱ ⋮ 𝑋𝑛1 𝑋𝑛2 ⋯ 𝑋𝑛𝑘 ], 𝜺 = [ 𝜀1 𝜀2 ⋮ 𝜀𝑛 ] dengan: • 𝒀 adalah vektor variabel tidak bebas berukuran n x 1. • 𝑿 adalah matriks variabel bebas berukuran n x (k+1). • 𝜷 adalah vektor parameter berukuran (k+1) x 1. • 𝜺 adalah vektor error berukuran n x 1. bentuk umum model regresi taksiran adalah sebagai berikut: �̂�𝑖 = �̂�0 + �̂�1𝑋1 + �̂�2𝑋2 + ⋯ . +�̂�𝑘 𝑋𝑘 (3) 2.3 deteksi heteroskedastisitas menurut gujarati [5] asumsi-asumsi pada model regresi linier berganda adalah sebagai berikut: a. model regresinya adalah linier dalam parameter. b. nilai rata-rata dari error adalah nol. c. variansi dari error adalah konstan (homoskedastitisitas). d. tidak terjadi autokorelasi pada error. e. tidak terjadi multikolinieritas pada variabel bebas. f. error berdistribusi normal. salah satu pelanggaran asumsi yang sering terjadi pada model ols adalah asumsi variansi dari error tidak konstan (heteroskedastisitas). dampak adanya heteroskedastisitas adalah walaupun estimator ols masih linier dan tidak bias, tetapi tidak lagi mempunyai variansi yang minimum dan menyebabkan perhitungan standard error metode ols tidak bisa dipercaya kebenarannya. selain itu interval estimasi maupun pengujian hipotesis yang didasarkan pada distribusi t maupun f tidak bisa lagi dipercaya untuk evaluasi hasil regresi. akibat dari dampak heteroskedastisitas tersebut menyebabkan estimator ols tidak menghasilkan estimator yang best linear unbiased estimator (blue) dan hanya menghasilkan estimator ols yang linear unbiased estimator (lue). selanjutnya untuk mendeteksi adanya heteroskedastisitas pada model ols dapat menggunakan uji goldfeld-quandt. berikut adalah tahapan pengujiannya [6]: a. asumsi: • n  2(k); k adalah banyaknya variabel bebas. • error berdistribusi normal. b. hipotesis: h0 : ei homoskedastisitas h1 : ei heteroskedastisitas c. prosedur: • urutkan nilai-nilai xi • untuk n besar maka hilangkan c pengamatan yang ditengah-tengah sehingga terdapat 2 bagian yang sama masing-masing ½(n-c) bagian nilai x yang kecil dan ½(n-c) bagian nilai x yang besar. • dengan ols taksir secara terpisah setiap bagian kemudian hitung masing-masing residunya i sehingga diperoleh i 2 kecil dan i 2 besar. • hitung statistik ujinya: 𝐹 = ∑ 𝜀𝑖𝐵𝑒𝑠𝑎𝑟 2𝑛2 𝑖=1 ∑ 𝜀𝑖𝐾𝑒𝑐𝑖𝑙 2𝑛1 𝑗=1 ~𝐹(1/2(𝑛−𝑐)−(𝑘+1), 1/2(𝑛−𝑐)−(𝑘+1)) dimana, n : banyaknya pengamatan, untuk n > 30, maka c optimum =n/4 c : banyaknya pengamatan yang ditengah-tengah k+1 : banyaknya parameter yang ditaksir. untuk k>1, maka pilih salah satu x yang di urutkan. d. keputusan: terdapat homoskedastisitas jika nilai f ~1, jika fhitung > ftabel atau p-value < α maka h0 ditolak, artinya terdapat heteroskedastisitas. 36 2.4 weighted least squares (wls) salah satu cara untuk mengatasi pelanggaran asumsi homoskedastisitas adalah menggunakan metode wls. metode ini merupakan salah satu metode penaksiran yang digunakan ketika error tidak saling berkorelasi namun memiliki varians yang sama. ketika  adalah matriks diagonal, maka dapat diinterpretasikan bahwa error tidak saling berkorelasi namun varians error tidak homogen. jika=diag (√1/𝑊𝑖,…, √1/𝑊𝑛 ), maka kita dapat meregresikan √wi x𝑖 pada √wi y𝑖 (kolom satuan pada matriks x perlu diganti dengan √wi) [7]. misalkan: e(i 2)=i 2 dan ∑ 𝑤𝑛𝑖=1 𝑖 e𝑖 2 = ∑ 𝑤𝑛𝑖=1 𝑖 (𝑌𝑖 − 𝛽0 − 𝛽1𝑋𝑖 ) 2 maka akan diperoleh taksiran untuk parameter 1 dan varians (1) sebagai berikut: �̂�1 = (∑ 𝑤𝑖)(∑ 𝑤𝑖𝑥𝑖𝑦𝑖)−(∑ 𝑤𝑖𝑥𝑖)(∑ 𝑤𝑖𝑦𝑖) 𝑛 𝑖=1 𝑛 𝑖=1 𝑛 𝑖=1 𝑛 𝑖=1 (∑ 𝑤𝑖) 𝑛 𝑖=1 (∑ 𝑤𝑖𝑥𝑖 2)𝑛𝑖=1 −(∑ 𝑤𝑖𝑥𝑖 𝑛 𝑖=1 ) 2 𝑣𝑎𝑟(�̂�1) = ∑ 𝑤𝑖 𝑛 𝑖=1 (∑ 𝑤𝑖) 𝑛 𝑖=1 (∑ 𝑤𝑖𝑥𝑖 2)𝑛𝑖=1 −(∑ 𝑤𝑖𝑥𝑖 𝑛 𝑖=1 ) 2 dengan: 𝑤𝑖 = 1 𝜎𝑖 2 penentuan 𝑊𝑖 juga dapat dilakukan dengan beberapa cara diantaranya adalah [8]: • jika varians error proporsional terhadap variabel respon (𝑣𝑎𝑟 (𝑖 ) ∝ 𝑌𝑖 ) maka disarankan 𝑊𝑖 = 1 𝑌𝑖 . • jika 𝑌𝑖 adalah rata-rata dari 𝑛𝑖 observasi dengan 𝑣𝑎𝑟 (y𝑖 ) ∝ 𝑣𝑎𝑟 (𝑖 ) ∝ 𝜎𝑖 2 𝑛𝑖 , disarankan 𝑊𝑖 = 𝑛𝑖. • jika varians error proporsional terhadap prediktor (𝑣𝑎𝑟 (𝑖 ) ∝ 𝑋𝑖 ) maka disarankan 𝑊𝑖 = 1 𝑋𝑖 . • jika varians error proporsional terhadap prediktor kuadrat (𝑣𝑎𝑟 (𝑖 ) ∝ 𝑋𝑖 2) maka disarankan 𝑊𝑖 = 1 √xi . • jika varians error proporsional terhadap [𝐸(𝑌𝑖 )] 2 maka disarankan 𝑊𝑖 = 1 𝑌�̂� . 3. metode penelitian 3.1 sumber data dan variabel penelitian penelitian ini menggunakan data sekunder tahun 2016 yang diperoleh dari badan pusat statistik republik indonesia (bps ri) kemudian diolah dengan sofware r seri 3.4.3. adapun data yang digunakan adalah angka harapan lama sekolah sebagai variabel repon serta variabel prediktornya adalah persentase penduduk miskin (x1), pdrb per kapita (dalam juta rupiah, x2), apm smp (x3) dan rasio fasilitas pendidikan per 10.000 penduduk (x4). unit observasi pada penelitian ini adalah seluruh kabupaten/kota di indonesia timur. 3.2 tahapan penelitian tahapan penelitian yang dilakukan adalah sebagai berikut: 1. melakukan uji linearitas dalam parameter, 2. melakukan pemodelan dengan metode ordinary least squares (ols), 3. melakukan uji asumsi klasik (normalitas, homogenitas, dan multikolinearitas), 4. melakukan pemodelan dengan metode weighted least squares (wls) untuk mengatasi pelanggaran asumsi homoskedastisitas, 5. melakukan uji asumsi normalitas dan multikolinearitas untuk model wls, 6. melakukan uji parameter model wls, 7. analisis pengaruh variabel prediktor terhadap variabel respon. 4. hasil dan pembahasan sebelum melakukan analisis lebih lanjut terlebih dahulu diuji apakah variabel yang digunakan memenuhi asumsi linearitas dalam parameter atau tidak. asumsi ini dapat menggunakan pendekatan grafis untuk mendeteksi apakah variabel prediktor yang digunakan memiliki hubungan yang linear atau tidak dengan variabel responnya. berikut adalah bentuk dari scatter plot untuk keempat variabel prediktor dengan variabel respon. 37 gambar 1. scatter plot variabel respon dengan variabel prediktor pada gambar 1 terlihat bahwa keempat variabel prediktor yang akan digunakan di dalam model memiliki hubungan yang linear. variabel prediktor x2, x3 dan x4 memiliki hubungan linear yang positif. artinya apabila semakin besar nilai x maka nilai y juga semakin besar. sebaliknya untuk variabel x1, semakin besar nilai x1 maka nilai y akan semakin menurun atau memiliki hubungan linear yang negatif. 4.1 pengujian asumsi model ols normalitas hasil pengujian normalitas menggunakan uji shapiro wilks dengan nilai p-value yang diperoleh sebesar 0.113, lebih besar dari α=0.05. hasil ini memberikan kesimpulan bahwa model ols berasal dari sampel berdistribusi normal. homoskedastisitas melalui uji goldfeld-quandt diperoleh nilai p-value sebesar 0.000138, lebih kecil dari α=0.05. dari hasil ini diputuskan bahwa tidak cukup bukti untuk mengatakan varians error adalah homogen sehingga dapat disimpulkan bahwa variansi error pada model adalah tidak konstan atau terdapat heteroskedastisitas. multikolinearitas variance inflation factor (vif) yang digunakan sebagai alat deteksi multikolinearitas menunjukkan nilai untuk keempat variabel prediktor sebagai berikut x1=2,72; x2=1,79; x3=1,14; dan x4=2,51. keempat nilai ini lebih kecil dari 10 sehingga dapat disimpulkan bahwa tidak terdapat multikolinearitas pada variabel prediktor. 4.2 penanganan heteroskedastisitas karena model ols yang terbentuk terdapat heteroskedastisitas, maka langkah selanjutnya adalah mengatasi pelanggaran asumsi tersebut menggunakan metode wls. langkah awal pada metode wls adalah menentukan pembobot yaitu dengan melihat pola yang ditunjukkan error terhadap variabel bebas. berdasarkan pola tersebut maka pada penelitian ini akan menggunakan beberapa pembobot diantaranya adalah 1/x1, 1/√x2, 1/x3, 1/x4. berikut hasil pendeteksian heteroskedastisitas dengan uji goldfeldquandt: tabel 1. hasil uji goldfeld-quandt (gq) dengan metode wls pembobot nilai gq p-value (1) (2) (3) 1/√x1 1.062 0.440 1/x2 12.588 0.000 1/√x3 10.174 0.000 1/x4 20.926 0.000 berdasarkan hasil pada tabel 1 di atas disimpulkan bahwa pembobot 1/√x1 yang dapat memenuhi asumsi homoskedastisitas karena nilai p-value lebih besar dari α=0.05. 4.3 pengujian asumsi model wls normalitas hasil uji shapiro wilks diperoleh nilai w sebesar 0.9794 dan p-value sebesar 0.371. karena nilai p-value lebih besar dari α=0.05 maka dapat disimpulkan bahwa model wls berasal dari sampel yang berdistribusi normal. multikolinearitas nilai vif yang diperoleh untuk model wls dengan pembobot 1/√x1 adalah x1=5.52; x2=1.67; x3=6.28; dan x4=5.36. keempat nilai ini lebih kecil dari 10 sehingga dapat disimpulkan bahwa 38 tidak terdapat multikolinearitas pada variabel prediktor. 4.4 pengujian parameter model wls statistik uji f statistik uji f dapat digunakan untuk mengetahui secara serentak seluruh variabel prediktor yang digunakan signifikan di dalam model. berikut hipotesis dan kriteria penolakannya: hipotesis: h0 : βi = 0 h1 : minimal ada satu βi ≠ 0 keputusan: h0 ditolak jika fhitung > fα;5;58 = 2,37 atau pvalue < α. nilai fhitung yang dihasilkan pada model dengan pembobot 1/√x1 adalah 926.3 dengan p-value sebesar 0.000. karena nilai p-value kurang dari α=0.05 maka dapat disimpulkan bahwa secara serentak seluruh variabel prediktor yang digunakan signifikan di dalam model. koefisien determinasi (r2 adjusted) terkoreksi yang dihasilkan pada uji serentak model wls adalah 0.9866. artinya, sebesar 98,66 persen variansi keragaman data hls mampu dijelaskan oleh keempat variabel prediktor di dalam model. statistik uji t statistik uji t digunakan untuk mengetahui secara parsial variabel prediktor yang digunakan signifikan atau tidak di dalam model. berikut hipotesis dan kriteria penolakannya: hipotesis: h0 : βi = 0 (tidak ada pengaruh peubah xi terhadap y) h1 : βi≠ 0 (ada pengaruh peubah xi terhadap y) keputusan: h0 ditolak jika |thitung| > 𝑡𝛼 2 ,58=2,00172 atau p-value < α. tabel 2. estimasi parameter regresi variabel ko efisien se thitung pvalue (1) (2) (3) (4) (5) c 7.646 1.396 5.477 0.000 x1 -0.052 0.022 -2.416 0.019 variabel ko efisien se thitung pvalue x2 0.007 0.003 2.064 0.044 x3 0.065 0.014 4.719 0.000 x4 0.016 0.006 2.676 0.009 hasil pengolahan pada tabel 2 menunjukkan bahwa nilai absolut thitung yang dihasilkan pada model dengan pembobot 1/√x1 untuk semua variabel prediktor lebih besar dari 2,00172 atau pvalue kurang dari 0.05 sehingga dapat disimpulkan bahwa secara parsial seluruh variabel prediktor yang digunakan signifikan di dalam model. 4.5 analisis pengaruh variabel prediktor terhadap hls hasil penelitian menunjukkan bahwa persentase penduduk miskin, pdrb per kapita, apm smp dan rasio fasilitas pendidikan per 10.000 penduduk berpengaruh signifikan terhadap angka harapan lama sekolah di indonesia timur pada 2016. model akhir yang terbentuk dengan metode wls adalah sebagai berikut: 𝐻𝐿𝑆 = 7.646 − 0.052 𝑀𝑖𝑠𝑘𝑖𝑛 + 0.007 𝑃𝐷𝑅𝐵 + 0.065 𝐴𝑃𝑀 + 0.016 𝐹𝑎𝑠𝑑𝑖𝑘 (4) persentase penduduk miskin persamaan (4) memperlihatkan bahwa terdapat hubungan yang negatif antara persentase penduduk miskin (x1) dengan hls. dapat dilihat bahwa koefisien regresi pada variabel persentase penduduk miskin (x1) bernilai negatif. jika diinterpretasikan, maka setiap peningkatan persentase penduduk miskin sebesar 1 persen akan diikuti dengan penurunan harapan lama sekolah sebesar 0.052 tahun dengan asumsi variabel lain dianggap tidak berubah. sebaliknya jika terjadi penurunan penduduk miskin sebesar 1 persen akan meningkatkan harapan lama sekolah sebesar 0.052 tahun. pada 2016, bps melaporkan bahwa penduduk miskin yang tidak tamat atau tidak pernah mengenyam pendidikan sd mencapai 54,70 persen di papua, 23,21 39 persen di papua barat, 17, 71 persen di maluku dan 29,73 persen di maluku utara. jika dilihat pada tingkat kabupaten/kota, terdapat 15 kabupaten di papua memiliki persentase penduduk miskin yang tidak pernah sekolah atau tidak tamat sd lebih dari 40 persen, papua barat terdapat 2 kabupaten, maluku utara terdapat 1 kabupaten, dan maluku tidak ada. kondisi ini memberi gambaran bahwa tingkat pendidikan yang rendah begitu dekat dengan kemiskinan [9]. selain tingkat pendidikan, bps juga melaporkan bahwa pada tahun yang sama, belum semua penduduk miskin di indonesia timur mengakses pendidikan formal. angka partisipasi sekolah (aps) penduduk miskin untuk usia 7-12 tahun dan 13-15 tahun pada beberapa kabupaten di indonesia timur masih rendah. di papua, aps usia 7-12 tahun di kabupaten puncak dan nduga adalah yang terendah yaitu 46,44 persen dan 47,20 persen. sementara aps usia 13-15 tahun terendah di kabupaten puncak (10,03 persen), kabupaten deiyai (38,50 persen), kabupaten yalimo (40,36 persen), dan kabupaten intan jaya (41,30 persen). aps penduduk miskin usia 7-12 tahun dan 1315 tahun untuk papua barat, maluku dan maluku utara telah mencapai lebih dari 70 persen pada seluruh kabupaten/kota. keadaan ini mengindikasikan bahwa pembangunan pendidikan untuk beberapa kabupaten di indonesia timur masih tertinggal. pdrb per kapita pada persamaan (4) juga dapat dilihat hubungan yang positif antara variabel pdrb per kapita dengan hls karena koefisien regresinya bernilai positif. jika diinterpretasikan, maka setiap peningkatan pdrb per kapita sebesar rp1.000.000,00 maka akan diikuti dengan peningkatan angka hls sebesar 0.007 tahun dengan asumsi variabel lain dianggap tidak berubah. jika pdrb per kapita semakin besar maka hls juga akan semakin besar dan berlaku juga sebaliknya. hal ini sejalan dengan data yang dipublikasikan oleh bps pada 2016 menunjukkan bahwa pdrb per kapita pada beberapa kabupaten di papua tergolong rendah dan memiliki hls terendah. sebaliknya untuk kabupaten/kota yang memiliki pdrb per kapita tinggi cenderung memiliki hls yang juga tinggi. apm smp dari persamaan (4) dapat dilihat bahwa nilai koefisien regresi pada variabel apm smp (x3) positif, menunjukkan terdapat hubungan yang positif antara variabel apm smp dengan hls. jika diinterpretasikan, maka setiap peningkatan apm smp sebesar 1 persen maka akan diikuti dengan peningkatan angka hls sebesar 0.065 tahun dengan asumsi variabel lain dianggap tidak berubah. apm smp adalah variabel yang berhubungan langsung dengan hls. hal ini dapat dilihat dari lebih besar pengaruhnya terhadap angka hls dibandingkan dengan tiga variabel lainnya. apm smp pada 2016, untuk beberapa kabupaten kota di papua berbanding lurus dengan angka hls. umumnya, kabupaten/kota yang memiliki apm smp 50 persen ke atas memiliki angka hls 10 tahun ke atas. sebaliknya, terdapat beberapa kabupaten yang memiliki apm smp di bawah 50 persen dan memiliki angka hls di bawah 10 tahun. rasio fasilitas pendidikan dari persamaan (4), terdapat hubungan yang positif antara variabel rasio fasilitas pendidikan per 10.000 penduduk (x4) dengan hls, dapat dilihat dari koefisien regresinya yang bernilai positif. jika diinterpretasikan, maka setiap peningkatan rasio fasilitas pendidikan per 10.000 penduduk (x4) sebesar 1 unit maka akan diikuti dengan peningkatan angka harapan lama sekolah sebesar 0.016 tahun dengan asumsi variabel lain dianggap tidak berubah. ditinjau dari fasilitas pendidikan, tiga provinsi di indonesia timur yaitu maluku, maluku utara, dan papua barat memiliki pencapaian yang lebih baik dibandingkan dengan papua. 40 kabupaten/kota pada ketiga provinsi ini memiliki rasio fasilitas pendidikan yang sebagain besar mencapai 40 unit per 10.000 penduduk atau lebih dan memiliki angka hls cukup besar. sedangkan di papua terdapat sejumlah kabupaten yaitu intan jaya, nduga, puncak jaya, tolikara, lanny jaya, dogiayai, dan yalimo yang memiliki rasio rasio fasilitas pendidikan kurang dari 40 unit per 10.000 penduduk dan memiliki hls rendah. 5. kesimpulan hasil penelitian ini memberikan kesimpulan bahwa persentase penduduk miskin, pdrb per kapita, apm smp dan rasio fasilitas pendidikan per 10.000 penduduk berpengaruh signifikan terhadap angka harapan lama sekolah di indonesia timur pada 2016. secara umum, provinsi papua menyimpan persoalan yang paling banyak ditinjau dari keempat variabel prediktor dan angka hls dibandingkan dengan tiga provinsi lainnya. penyediaan sarana dan prasaran pendidikan yang memadai adalah langkah utama yang harus dilaksanakan noleh pemerintah baik pusat maupun daerah karena fasilitas pendidikan memegang peranan penting dalam meningkatkan angka harapan lama sekolah di indonesia timur. jika fasilitas pendidikan tersedia dengan memadai salah satunya untuk jenjang smp terutama untuk wilayah-wilayah terisolir dan terpencil di papua maka perlahan akan memperbaiki apm smp. jika amp smp terkoreksi naik maka akan meningkatkan angka harapan lama sekolah. langkah selanjutnya adalah menurunkan angka kemiskinan. penduduk yang hidup di bawah garis kemiskinan akan susah memperoleh pendidikan yang layak. jika mereka dikeluarkan dari lingkaran kemiskinan maka mereka akan berpeluang untuk memperoleh pendidikan formal yang lebih baik. referensi [1] badan pusat statistik, diambil dari www.bps.go.id/subject/26/indekspembangunanmanusia.html#subjekviewtab3, pada tanggal 18 januari 2018. [2] badan pusat statistik (bps) kabupaten halmahera selatan, indeks pembangunan manusia kabupaten halmahera selatan 2015, bps kabupaten halmahera selatan, labuha. (2016). [3] badan pusat statistik (bps), indeks pembangunan manusia 2016, badan pusat statistik, jakarta. (2017). [4] darisandi, roby. diambil dari www.academia.edu/7310798/pemerataan_ pendidikan_untuk_wilayah_indonesia_ti mur, pada tanggal 18 januari 2018. [5] gujarati, n.d, basic econometrics, 4th edition. mcgraw-hill companies, inc, new york. (2003). [6] goldfeld, s.m., quandt, r. e., some tests for homoscedasticity. journal of the american statistical association. 60 (310), (june 1965) 539–547. [7] jaya, i.g.n.m, diktat kuliah analisis regresi, departemen statistika universitas padjajaran. (2016). [8] setyaningsih, y.d, noeryanti, penggunaan metode weighted least square untuk mengatasi masalah heteroskedastisitas dalam analisis regresi (studi kasus pada data balita gizi buruk tahun 2014 di provinsi jawa tengah), jurnal statistika industri dan komputasi (2017), 2(1), 5158. [9] badan pusat statistik, data dan informasi kemiskinan kabupaten/kota 2016, badan pusat statistik, jakarta. (2017). http://www.bps.go.id/subject/26/indeks-pembangunan-manusia.html#subjekviewtab3 http://www.bps.go.id/subject/26/indeks-pembangunan-manusia.html#subjekviewtab3 http://www.bps.go.id/subject/26/indeks-pembangunan-manusia.html#subjekviewtab3 http://www.academia.edu/7310798/pemerataan_pendidikan_untuk_wilayah_indonesia_timur http://www.academia.edu/7310798/pemerataan_pendidikan_untuk_wilayah_indonesia_timur http://www.academia.edu/7310798/pemerataan_pendidikan_untuk_wilayah_indonesia_timur 41 lampiran tabel 3. ipm beberapa provinsi dan indonesia, 2014-2016 provinsi/ indonesia ipm/ hls/ rls 2014 2015 2016 (1) (2) (3) (4) (5) maluku ipm 66.74 67.05 67.60 hls 13.53 13.56 13.73 rls 9.15 9.16 9.27 maluku utara ipm 65.18 65.91 66.63 hls 12.72 13.10 13.45 rls 8.34 8.37 8.52 papua barat ipm 61.28 61.73 62.21 hls 11.87 12.06 12.26 rls 6.96 7.01 7.06 papua ipm 56.75 57.25 58.05 hls 9.94 9.95 10.23 rls 5.76 5.99 6.15 diy ipm 76.81 77.59 78.38 hls 14.85 15.03 15.23 rls 8.84 9.00 9.12 indonesia ipm 68.90 69.55 70.18 hls 12.39 12.55 12.72 rls 7.73 7.84 7.95 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran contact: muhammad amin sadiq murad muhammad.murad@uod. department of mathematics, college of sciences, university of duhok, duhok, iraq the article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.124-138 a new analytical modeling for fractional telegraph equation arising in electromagnetic muhammad amin sadiq murad1*, mudhafar hamed hamadamen2 1*department of mathematics, college of sciences, university of duhok, iraq 2department of mathematics, college of education, salahaddin university, iraq article history: received oct 26, 2022 revised dec 23, 2022 accepted dec 31, 2022 kata kunci: persamaan telegraf pecahan, metode iterasi variasi, transformasi integral elzaki, polinomial he, metode perturbasi homotopi. abstrak. pada artikel ini, metode iterasi variasi he (vim) dan transformasi integral elzaki diusulkan untuk menyelesaikan persamaan telegraf fraksional linier dan nonlinier yang muncul dalam elektromagnetik. caputo sense digunakan untuk mendeskripsikan fractional derivatives. salah satu keuntungan dari teknik ini adalah tidak perlu menghitung pengali lagrange dengan menghitung integrasi dalam relasi perulangan atau dengan mengambil teorema konvolusi. selanjutnya, untuk mengurangi istilah komputasi nonlinier, polinomial adomian diidentifikasi dengan homotopy perturbation method (hpm). metode yang diusulkan diterapkan pada beberapa contoh persamaan telegraf fraksional linier dan nonlinier. solusi yang diperoleh dengan teknik komputasi baru menunjukkan bahwa metode ini efisien dan memfasilitasi proses penyelesaian time fractional differential equations. keywords: fractional telegraph equations, variation iteration method, elzaki integral transform, he’s polynomial, homotopy perturbation method. abstract. in this article, he’s variation iteration method (vim) and elzaki integral transform are proposed to analyze the time-fractional telegraph equations arising in electromagnetics. the caputo sense is used to describe fractional derivatives. one of the advantages of this technique is that there is neither need to compute the lagrange multiplier by calculating the integration in recurrence relations or via taking the convolution theorem. further, to decrease nonlinear computational terms, the adomian polynomial is identified with the homotopy perturbation method (hpm). the proposed method is applied to some examples of linear and nonlinear fractional telegraph equations. the solutions obtained by the new computational technique indicate that this method is efficient and facilitates the process of solving time fractional differential equations. how to cite: m. a. s. murad and m. h. hamadamen, “a new analytical modeling for fractional telegraph equation arising in electromagnetic”, j. mat. mantik, vol. 8, no. 2, pp. 124-138, december 2022. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 125 1. introduction differential equations of fractional orders can be used to simulate many scientific disciplines, which improves our understanding of how to characterize natural occurrences in a variety of scientific disciplines like engineering, electronics, biology, business, computer science, and physics. indeed, in the improvement of fractional calculus, many scientists such as bernoulli, liouville, euler, l’hopital, and wallis greatly contributed to this area of research. the numerical solutions are used to investigate the solutions of differential equations of fractional and integer orders, because the exact solutions of differential equations are quite difficult to be found. telegraph equations have applied to many problems in different fields of science which developed by heaviside in 1880. the difference and time are described on electric transmissions with current and voltage by telegraph equation, also the proposed equation is applied for investigating the wave propagation in the cable transmission and electric signals, and it is also applied in the field of telephone lines, wireless signals, and radio frequency [1]. telegraph equations of fractional orders have been solved, using various numerical and analytical methods, the adomian technique [2], homotopy perturbation technique[3], laplace decomposition combined with hpm [4], modified adomian decomposition method (madm) [5], and reduced differential transform technique [6]. the vim employed to study the solution of the proposed model and obtained the same result as obtained by (adm) with fewer computations [7], and the hyperbolic telegraph equation is analyzed by chebyshev tau technique [8]. the researcher inokuti was the first who study the vim [9][10], while the lagrange multiplier was difficult to be identified. then, variation iteration method developed by chinese mathematician he [11], and was applied by many researchers, see [12][13][14][15]. the homotopy perturbation method (hpm) is another crucial method which is employed to solve pdes [16][17][18]. the solution of voltera-fredhom is studied by hpm[19], also the hyperbolic pdes and many other pdes were solved by hpm, see [20][21][22]. in the last decade, different methods have been developed to analyze the solution of pdes of fractional orders[23][24]. recently, elzaki homotopy transformation perturbation method is employed to solve a class of models such, see [25][26][27]. the elzaki transform was proposed by the jordanian mathematician tarig elzaki [28], and this transform has been applied on many models to acquire their solution, see [29][30][31][32][33][34][35]. in this paper, the elzaki transform with a new method of vim combined with the homotopy perturbation method is utilized to study the solution of time fractional telegraph equation. the object of the present work is to extend the implementations of evim and show the accuracy of the suggested technique. therefore, the fractional telegraph equation is considered. nanoelectromechanical systems are playing an enormous rule in the area of sensing and actuating. however, nonlinearity effects negatively on the nanoelectromechanical systems devise. the nonlinear vibration systems have complex behaviors that are characterized by noise, instability in response, and bifurcation phenomena. therefore, controlling the nonlinear vibrations of nanoelectromechanical systems is essential to obtain stable vibrations. 𝜕𝛽𝑟 𝜕𝑥𝛽 + 𝐺 𝜕𝛼𝑤 𝜕𝑡 𝛼 + 𝐻𝑤 = 0 (1) 𝜕𝛽𝑤 𝜕𝑥𝛽 + 𝐿 𝜕𝛼𝑟 𝜕𝑡 𝛼 + 𝑅𝑟 = 0 (2) differentiate the equation (1) with respect to 𝑡 and (2) with respect to 𝑥, then solving the system, the following equation is obtained jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 126 𝜕2𝛽𝑤 𝜕𝑥2𝛽 + 𝑅 [−𝐺 𝜕𝛼𝑤 𝜕𝑡 𝛼 − 𝐻𝑤] + 𝐿 [−𝐺 𝜕2𝛼𝑤 𝜕𝑡 2𝛼 − 𝐻 𝜕𝛼𝑤 𝜕𝑡 𝛼 ] = 0 (3) assume that = 𝑅 𝐿 , 𝜖 = 𝐻 𝐺 , 𝛿 2 = 1 𝐿𝐺 , substituting these values in the equation (3), we obtain equation (4) is telegraph equation which arises in electromagnetic waves. 𝜕2𝛼𝑤 𝜕𝑡 2𝛼 + ( + 𝜖) 𝜕𝛼𝑤 𝜕𝑡 𝛼 + 𝜖𝑤 = 𝛿 2 𝜕2𝛽𝑤 𝜕𝑥2𝛽 (4) 2. preliminaries here, several basic definitions and characteristics of fractional calculus and the proposed transform are given. definition2.1.[36] a function g(y), y > 0 is considered to be a real valued function and belong to the space ∁σ, σ ∈ r . assume that the real number d > σ, such that g(y) = ydg1(y) where g1(y) ∈ ∁(0, ∞), and it is said to be in the space ∁σ n if gn ∈ 𝑅σ, n ∈ n. definition2.2.[37] the function f(u) is called riemann-liouvill fractional integral of order 𝛼 > 0 if it defines as: 𝐽𝛼 𝑓(𝑢) = 1 𝛤(𝛼) ∫ (𝑢 − 𝑡)𝛼−1 𝑢 0 𝑓(𝑡)𝑑𝑡 , 𝑡 > 0. in particular 𝐽0𝑓(𝑢) = 𝑓(𝑢). for θ ≥ 0 and ϑ ≥ −1, we have the following properties: 1. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝛼+𝜃 𝑓(𝑢), 2. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝜃 𝐽𝛼 𝑓(𝑢), 3. 𝐽𝛼 𝑥𝜗 = 𝛤(𝜗+1) 𝛤(𝛼+𝜗+1) 𝑥𝛼+𝜗. definition2.3.[37] assume that function f ∈ c−1 n , n ∈ n. the function f is called caputo fractional derivative and defined by 𝐷𝛼 𝑓(𝑢) = 1 𝛤(𝑛 − 𝛼) ∫ (𝑢 − 𝑡)𝑛−𝛼−1 𝑢 0 𝑓 𝑛(𝑡)𝑑𝑡, 𝑛 − 1 < 𝛼 ≤ 𝑛. definition2.4.[28] the function f(u) is called elzaki-transform if defined as follows: 𝐸[𝑓(𝑢)] = 𝑇(𝑣) = 𝑣 ∫ 𝑓(𝑢)𝑒 −𝑢 𝑣 𝑑𝑢 ∞ 0 𝑢 > 0. assume that f is piecewise continuous, then elzaki transform of the caputo derivative 𝐸 [ 𝜕𝑛𝑓(𝑥,𝑡) 𝜕𝑢𝑛 ] = 𝑇(𝑥,𝑣) 𝑣𝑛 − ∑ 𝑣 2−𝑛+𝑖𝑛−1𝑖=0 𝜕𝑖𝑓(𝑥,0) 𝜕𝑢𝑖 . (5) the caputo fractional derivative of laplace transform is defined as follows 𝐿(𝐷𝑥 𝛼 𝑔(𝑥, 𝑢)) = 𝑠𝛼 𝐺(𝑠) − ∑ 𝑠𝛼−1−𝑖 𝑔(𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛𝑛−1𝑖=0 . (6) where 𝐺(𝑠) represents the laplace transform of 𝑔(𝑥). theorem 2.1 let 𝐵 = {𝑓(𝑥, 𝑢)| 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑀, 𝑚1, 𝑚2 > 0 𝑠. 𝑡|𝑓(𝑥, 𝑢)| < 𝑀𝑒 |𝑢| 𝑚𝑗 , 𝑢 ∈ (−1)𝑗 × [0, ∞)} and let 𝑓(𝑥, 𝑢) ∈ 𝐵. the elzak transform 𝑇(𝑣) of 𝑓(𝑢)is 𝑇(𝑣) = 𝑣𝐺 ( 1 𝑣 ), where 𝐺(𝑠) is the laplace transform of 𝑔(𝑥). muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 127 theorem 2.2 assume 𝑇(𝑣) is the elzaki transform of the function 𝑓(𝑥, 𝑢). thus 𝐸(𝐷𝑢 𝛼 𝑓(𝑥, 𝑢)) = 𝑇(𝑣) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛 𝑛−1 𝑖=0 proof: by theorem 1 𝐸{𝐷𝛼 𝑓(𝑥, 𝑢), 𝑣} = 𝑣𝐿 {𝐷𝛼 𝑓(𝑥, 𝑢), 1 𝑣 }. using equation (6), we obtain 𝐸{𝐷𝛼 𝑓(𝑥, 𝑢), 𝑣} = 𝑣 𝑣 𝛼 𝐺 ( 1 𝑣 ) − 𝑣 ∑ ( 1 𝑣 ) 𝛼−𝑖−1 𝑓 (𝑖)(𝑥, 0) 𝑛−1 𝑖=0 𝑣𝐺 ( 1 𝑣 ) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛 𝑛−1 𝑖=0 = 𝑇(𝑣) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0). 𝑛−1 𝑖=0 3. applications of hetm recently, the lagrange multiplier is introduced in new manners [38][39][40]. in this work, the elzaki transform is used and multiply it by lagrange multiplier in order to obtain the recurrence relation that is restricted in order to define the lagrange multiplier. to avoid the convolution terms and integral evaluations, we use this technique. due to the limitations of elzaki transform on nonlinear parts, the hpm is used to decrease the computations. the innovative and modified scheme is constructed as follows: taking the elzaki transform of the proposed model and multiplying it via the lagrange multiplier to obtain the recurrence relation that identifies the lagrange multiplier, via variation approach. the adomin polynomial is used to evaluate the nonlinear terms, and then the well-known hpm is used to find the series solution of the proposed problem. 𝑅𝑢 − 𝑁𝑢 − 𝑘 = 0. taking the elzaki transform, the following relation is obtained 𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘] = 0. now, we take the lagrange multiplier 𝜇(𝑣), 𝜇(𝑣){𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘]} = 0. here, we can have the following recurrence relation 𝑈𝑗+1(𝑣) = 𝑈𝑗 (𝑣) + 𝜇(𝑣){𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘]}. (7) the recurrence relation represents the modified elzaki variation; we apply the optimal condition using the following relation to introduce the lagrange multiplier 𝜇(𝑣) 𝜌𝑈𝑗+1(𝑣) 𝜌𝑈𝑗 (𝑣) = 0. here, the inverse elzaki transform is applied on (7) to achieve the solution of equation (4) 𝑢𝑗+1(𝑣) = 𝑢𝑗 (𝑣) + 𝐸 −1[𝜇(𝑣){𝐸[𝑅𝑢𝑗 ] − 𝐸[𝐴𝑗 + 𝑘]}]. 𝑗 = 0,1,2,3, … where 𝐴𝑗 represents the adomian polynomial as follows: 𝐴𝑗 = 1 𝑗! 𝑑𝑗 𝑑𝜏𝑗 (𝑁(∑ 𝑢𝑗 𝜏 𝑗 )).∞𝑗=0 (8) jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 128 finally, to investigate the series approximate solution, the hpm is considered by equating the powers of the embedded parameter 𝑝. 4. homotopy perturbation in this portion, we study the concept of hpm for the solution of our problem. consider the following differential equation 𝑅𝑢 − 𝑁𝑢 = 𝑘, (9) where 𝑘 be a source term, 𝑅 is linear term and 𝑁 is the non-linear term, and 𝑢 the solution function. by the homtopy theory 𝐻(𝑤, 𝑝), 𝐻(𝑤, 𝑝): 𝑅 × [0,1] → 𝑅 which satisfies the following: 𝐻(𝑤, 𝑝) = (1 − 𝑝)[𝑅(𝑤) − 𝑅(𝑤0)] + 𝑝[𝑅(𝑤) − 𝑁(𝑤) − 𝑘] = 0, simple calculations, we obtain 𝑅(𝑤) − 𝑝𝑁(𝑤) = 𝑘, (10) the parameter 𝑝 ∈ [0,1], 𝑢0 is the initial term of (4), and 𝑤 is the homotopy function with 𝑅(𝑢0) = 𝑘. since, 𝑤 can be written as: 𝑤 = lim 𝑝→1 (𝑤0 + 𝑝𝑤1 + 𝑝 2𝑤2 + ⋯ ). (11) using (10) and (11), we have 𝑤0 + 𝑝𝑤1 + 𝑝 2𝑤2 + 𝑝 3𝑤3 … = 𝑘 + 𝑝𝑁(𝑤). equating the powers of 𝑝 can be written as follows: 𝑝0: 𝑤0 = 𝑘, 𝑝1: 𝑤1 = 𝑁(𝑤0), 𝑝2: 𝑤2 = 𝑤1𝑁 ′(𝑤0), 𝑝3: 𝑤3 = 𝑁 ′(𝑤0) + 𝑤1 2𝑁′′(𝑤0) 2 , ⋮ finally, as 𝑝 approach to 1, the following series solution is the solution of (4): 𝑢 = 𝑤0 + 𝑤1 + 𝑤2 + 𝑤3 … . (12) indeed, the convergence of the solution (12) is studied in[41] [42]. 5. applications the approximate patterns are employed to show the importance of the new method for solving the time fractional telegraph equation. here, the following models of time fractional differential equations is given: example 4.1 consider the following linear telegraph equation of fractional order 𝜕2𝑧 𝜕𝑥2 = 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧, 𝑡 ≥ 0, 0 < 𝛼 ≤ 1. with the initial conditions 𝑧(𝑥, 0) = 𝑒 𝑥 , 𝑧𝑡 (𝑥, 0) = −2𝑒 𝑥. the exact solution of equation (13) is: 𝑧(𝑥, 𝑡) = 𝑒 𝑥−2𝑡 . taking the elzaki transform of equation (13) 𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ] = 0. (13) muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 129 now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ] = 0. the recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕2𝛼𝑧 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧 𝜕𝑡 𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ]. (14) taking the variation of the above equation and using elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,0) 𝜕𝑡 + 𝐸 [2 𝜕𝛼�̂�𝑗 𝜕𝑡 𝛼 + �̂�𝑗 − 𝜕2�̂�𝑗 𝜕𝑥2 ]}. (15) here, �̂�𝑗 = �̂�𝑗 (𝑥, 0) = �̂�𝑗 (𝑥, 0) are restricted variables, it means that 𝜌�̂�𝑗 (𝑥, 0) = 𝜌�̂�𝑗 (𝑥, 0) = 0 and since 𝑍𝑗+1(𝑥,0) 𝑍𝑗(𝑥,0) = 0. substituting restricted variables in equation (15), gives 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 1 𝑣 2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑣). therefore, the lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . substituting the lagrange multiplier in equation (14), we obtain 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 𝑧 − 𝜕2𝑧𝑗 𝜕𝑥2 ]. applying elzaki inverse, we have 𝑧𝑗+1(𝑥, 𝑣) = 𝑧𝑗 (𝑥, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝑧𝑗 − 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 ]] since 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 = 0, 𝑗 = 0,1,2, …, to obtain he’s polynomial the homotopy perturbation method is utilized 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧𝑗 𝜕𝑡𝛼 + 𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 ]] = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [(2 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 𝑧0 − 𝜕2𝑧0 𝜕𝑥2 ) + 𝑝 (2 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 𝑧1 − 𝜕2𝑧1 𝜕𝑥2 ) + 𝑝2 (2 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 𝑧2 − 𝜕2𝑧2 𝜕𝑥2 ) + 𝑝3 (2 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 𝑧3 − 𝜕2𝑧3 𝜕𝑥 2 )]] equating the highest powers of 𝑝 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 𝑧0 − 𝜕2𝑧0 𝜕𝑥2 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 𝑧1 − 𝜕2𝑧1 𝜕𝑥2 ]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 𝑧2 − 𝜕2𝑧2 𝜕𝑥2 ]] therefore, we obtain 𝑧0 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 . jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 130 since by theorem 2, we have 𝑧1 = −𝐸 −1[−𝑣𝛼+34𝑒 𝑥 ], 𝑧1 = 4𝑒 𝑥 𝑡𝛼+1 𝛤(𝛼 + 2) . similarly, 𝑧2 = −8𝑒 𝑥 𝑡2𝛼+1 𝛤(2𝛼 + 2) , 𝑧3 = 16𝑒 𝑥 𝑡3𝛼+1 𝛤(3𝛼 + 2) , ⋯. one can expressed these results in a series such as: 𝑧 = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 + 4𝑒 𝑥 𝑡𝛼+1 𝛤(𝛼 + 2) − 8𝑒 𝑥 𝑡2𝛼+1 𝛤(2𝛼 + 2) + 16𝑒 𝑥 𝑡3𝛼+1 𝛤(3𝛼 + 2) − ⋯, when 𝛼 = 1, the hetm solution for equation (13) is 𝑧 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 + 4𝑒 𝑥 𝑡2 2! − 8𝑒 𝑥 𝑡3 3! + 16𝑒 𝑥 𝑡4 4! − ⋯. 𝑧 = 𝑒 𝑥−2𝑡 . thus, the obtained result using hetm can compute to the exact solution 𝑧 = 𝑒 𝑥−2𝑡 , when 𝛼 = 1 . figure 1. (a) exact solution and (b) hetm solution of 𝑧(𝑥, 𝑡) of equation (13) at 𝛼 = 1. the hetm solution of 𝑧(𝑥, 𝑡) of equation (13) at (c) 𝛼 = 0.8 and (d) 𝛼 = 0.2. muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 131 example 4.2 consider the following time fractional telegraph equation 𝜕2𝑧 𝜕𝑥2 + 𝜕2𝑧 𝜕𝑦2 = 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧, 𝑡 ≥ 0,0 < 𝛼 ≤ 1, (16) with the initial conditions 𝑧(𝑥, 𝑦, 0) = 𝑒 𝑥+𝑦 , 𝑧𝑡 (𝑥, 𝑦, 0) = −3𝑒 𝑥+𝑦. the exact solution of the equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦−3𝑡 . taking the elzaki transform of equation (16) 𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧 − 𝜕2𝑧 𝜕𝑥2 − 𝜕2𝑧 𝜕𝑦2 ] = 0. now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧 − 𝜕2𝑧 𝜕𝑥2 − 𝜕2𝑧 𝜕𝑦2 ] = 0. the recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]. (17) taking the variation of the above equation and using elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑦,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 𝑦, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 + 𝐸 [ 𝜕2𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 2𝛼 + 3 𝜕𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 𝛼 + 2�̂�𝑗 (𝑥, 𝑦, 0) − 𝜕2�̂�𝑗(𝑥,𝑦,0) 𝜕𝑥2 − 𝜕2�̂�𝑗(𝑥,𝑦,0) 𝜕𝑦2 ]}. (18) the variables �̂�𝑗 = �̂�𝑗 (𝑥, 𝑦, 0) = �̂�𝑗 (𝑥, 𝑦, 0) are restricted variable, since 𝜌�̂�𝑗 (𝑥, 𝑦, 0) = 𝜌�̂�𝑗 (𝑥, 𝑦, ,0) = 0 and �̂�𝑗+1(𝑥,𝑦,0) �̂�𝑗(𝑥,𝑦,0) = 0. substituting restricted variables in equation (18), gives 𝜌𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣) + 1 𝑣 2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣). therefore, the lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . substituting the lagrange multiplier in equation (17), we obtain 𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝑍𝑗 (𝑥, 𝑦, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]. applying elzaki inverse, we get 𝑧𝑗+1(𝑥, 𝑦, 𝑣) = 𝑧𝑗 (𝑥, 𝑦, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]] since 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 = 0, 𝑗 = 0,1,2, … to obtain he’s polynomial, we apply hpm 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑦, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧𝑗 𝜕𝑡𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]] jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 132 = 𝑧𝑗 (𝑥, 𝑦, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [(3 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 2𝑧0 − 𝜕2𝑧0 𝜕𝑥2 − 𝜕2𝑧0 𝜕𝑦2 ) + 𝑝 (3 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 2𝑧1 − 𝜕2𝑧1 𝜕𝑥2 − 𝜕2𝑧1 𝜕𝑦2 ) + 𝑝2 (3 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 2𝑧2 − 𝜕2𝑧2 𝜕𝑥2 − 𝜕2𝑧2 𝜕𝑦2 ) + 𝑝3 (3 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 2𝑧3 − 𝜕2𝑧3 𝜕𝑥2 − 𝜕2𝑧3 𝜕𝑦2 )]]. equating the highest powers of 𝑝 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑦, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑦, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 2𝑧0 − 𝜕2𝑧0 𝜕𝑥2 − 𝜕2𝑧0 𝜕𝑦2 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 2𝑧1 − 𝜕2𝑧1 𝜕𝑥2 − 𝜕2𝑧1 𝜕𝑦2 ]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 2𝑧2 − 𝜕2𝑧2 𝜕𝑥2 − 𝜕2𝑧2 𝜕𝑦2 ]] 𝑝4 ∶ 𝑧4 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 2𝑧3 − 𝜕2𝑧3 𝜕𝑥2 − 𝜕2𝑧3 𝜕𝑦2 ]]. therefore, we obtain 𝑧0 = 𝑒 𝑥+𝑦 − 3𝑡𝑒 𝑥+𝑦 . since by theorem 2.2, we have 𝑧1 = −𝐸 −1[−𝑣𝛼+39𝑒 𝑥+𝑦 ], 𝑧1 = 9𝑒 𝑥+𝑦 𝑡𝛼+1 𝛤(𝛼 + 2) . similarly, 𝑧2 = −27𝑒 𝑥+𝑦 𝑡2𝛼+1 𝛤(2𝛼 + 2) , 𝑧3 = 81𝑒 𝑥+𝑦 𝑡3𝛼+1 𝛤(3𝛼 + 2) , … here, the hetm for equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦 (1 − 3𝑡 + 9𝑡𝛼+1 𝛤(𝛼 + 2) − 27𝑡2𝛼+1 𝛤(2𝛼 + 2) + 81𝑡3𝛼+1 𝛤(3𝛼 + 2) − ⋯ ), when 𝛼 = 1, the hetm for equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦 − 3𝑡𝑒 𝑥+𝑦 + 9𝑒 𝑥+𝑦 𝑡2 2! − 27𝑒 𝑥+𝑦 𝑡3 3! + 81𝑒 𝑥+𝑦 𝑡4 4! − ⋯. 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦−3𝑡 . thus, exact solution of model (15) is obtained when 𝛼 = 1 . example 4.3 consider the following time fractional telegraph equation 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 = 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝑧2(𝑥, 𝑡) + 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡), 𝑡, 𝑥 ≥ 0, 1 < 𝛼 ≤ 2, (19) with the initial terms 𝑧(𝑥, 0) = 𝑥, 𝑧𝑡 (𝑥, 0) = 𝑥. muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 133 appling the elzaki transform of model (19), we obtain 𝐸 [ 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 + 𝑧2(𝑥, 𝑡) − 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡)] = 0. now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 + 𝑧2(𝑥, 𝑡) − 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡)] = 0. the recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 − 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 − 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 + 𝑧𝑗 2(𝑥, 𝑡) − 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]. (20) taking the variation of the above equation and using elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,0) 𝜕𝑡 − 𝐸 [ 𝜕2�̂�𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕�̂�𝑗(𝑥,𝑡) 𝜕𝑡 − �̂�𝑗 2(𝑥, 𝑡) + 𝑥�̂�𝑗 (𝑥, 𝑡)�̂�𝑗 𝑥 (𝑥, 𝑡)]}. (21) 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 1 𝑣2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑣). the variables �̂�𝑗 = �̂�𝑗 (𝑥, 0) = �̂�𝑗 (𝑥, 0) are restricted variables, since 𝜌�̂�𝑗 (𝑥, 0) = 𝜌�̂�𝑗 (𝑥, 0) = 0 and 𝜌𝑍𝑗+1(𝑥,𝑣) 𝜌𝑍𝑗(𝑥,𝑣) = 0. therefore, the lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . substituting the lagrange multiplier in (20), the following relation is acquired: 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝑧𝑗 2(𝑥, 𝑡) + 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]. applying elzaki inverse, we get 𝑧𝑗+1(𝑥, 𝑣) = 𝑧𝑗 (𝑥, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝑧𝑗 2(𝑥, 𝑡) + 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]]. since 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 = 0, 𝑗 = 0,1,2,3 … to get he’s polynomial, we apply hpm 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣2𝛼 𝐸 [ 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝐴𝑗 + 𝑥𝐵𝑗 ]], (22) where 𝐴𝑗 and 𝐵𝑗are the adomian polynomials of (𝑧0, 𝑧1, 𝑧2, 𝑧3 … ), we use (8) to calculate the adomian polynomials: 𝐴0 = 𝑧0 2, 𝐵0 = 𝑧0𝑧0𝑥 , 𝐴1 = 2𝑧0𝑧1, 𝐵1 = 𝑧0𝑧1𝑥 + 𝑧0𝑥 𝑧1 , 𝐴2 = 2𝑧0𝑧2 + 𝑧1 2, 𝐵2 = 𝑧0𝑧2𝑥 + 𝑧1𝑧1𝑥 + 𝑧2𝑧0𝑥 , 𝐴3 = 2𝑧0𝑧3 + 2𝑧1𝑧2, 𝐵3 = 𝑧3𝑧0𝑥 + 𝑧2𝑧1𝑥 + 𝑧1𝑧2𝑥 + 𝑧0𝑧3 𝑥 , table 1: the numerical and exact solutions at various values of 𝛼 and 𝑡 for equation (19), where 𝑥 = 0.5. jurnal matematika mantik vol. 8, no. 2, december 2022, pp. 124-138 134 table 1. demonstrate the comparison of exact and approximate solutions of (19) for different values of 𝑡 and 𝛼 using hetm. it is clear that the value 𝛼 = 2 using hetm gives almost the exact solution of model (19). 𝑡 exact solution 𝛼 = 2 𝛼 = 1.90 𝛼 = 1.80 0.0 0.5 0.5 0.5 0.5 0.5 0.8243606355 0.8243606355 0.8377559990 0.8531832410 1.0 1.359140914 1.359140914 1.398076434 1.439803212 1.5 2.240844535 2.240844535 2.312171661 2.385449643 2.0 3.694528050 3.694528049 3.803171996 3.911485870 equating the highest powers of 𝑝, and substituting the adomian polynomials in (22), leads 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧0 𝜕𝑡 + 𝜕2𝑧0 𝜕𝑥2 − 𝑧0 2 + 𝑥𝑧0𝑧0𝑥 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧1 𝜕𝑡 + 𝜕2𝑧1 𝜕𝑥2 − 2𝑧0𝑧1 + 𝑥(𝑧0𝑧1𝑥 + 𝑧0𝑥 𝑧1)]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧2 𝜕𝑡 + 𝜕2𝑧2 𝜕𝑥2 − 2𝑧0𝑧2 − 𝑧1 2 + 𝑥(𝑧0𝑧2𝑥 + 𝑧1𝑧1𝑥 + 𝑧2𝑧0𝑥 )]] therefore, we obtain 𝑧0 = 𝑥(1 + 𝑡), 𝑧1 = 𝑥𝑡 𝛼 𝛤(𝛼+1) , 𝑧2 = 𝑥𝑡 𝛼+1 𝛤(𝛼+2) , 𝑧3 = 𝑥𝑡 𝛼+2 𝛤(𝛼+3) , …. here, the hetm solution for equation (19) is 𝑧 = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧 = 𝑥 (1 + 𝑡 + 𝑡𝛼 𝛤(𝛼 + 1) + 𝑡𝛼+1 𝛤(𝛼 + 2) + 𝑡𝛼+2 𝛤(𝛼 + 3) + ⋯ ), when 𝛼 = 2, the hetm solution for equation (19) is 𝑧 = 𝑥 (1 + 𝑡 + 𝑡2 2! + 𝑡3 3! + 𝑡4 4! + ⋯ ). 𝑧 = 𝑥𝑒 𝑡 . thus, the exact solution of equation (19) is obtained when 𝛼 = 2 . figure 2. (a) the error plot of 𝑧(𝑥, 𝑡) of equation (19) at 𝛼 = 2. muhammad amin sadiq murad, mudhafar hamed mamadamen a new analytical modeling for fractional telegraph equation arising in electromagnetic 135 in this paper, several models of time fractional telegraph equations are studied using a novel numerical approach. the present outcomes are compared with the analytic solutions via tables and illustrative graphs. table 1. illustrates the comparison between the approximate solutions acquired via the proposed technique for various orders of fractional derivative 𝛼 with the exact solution. in figure 1. graph (a) the exact solution is given, graph (b) the solution of hetm for 𝛼 = 2 is given, and graphs (c) and (d) the solutions of hetm for 𝛼 = 0.8 and 𝛼 = 0.2 are given, respectively. finally, the error plot of equation (19) is given in figure 2. as a result, it can be observed that there is an excellent agreement between the present results and the exact solution. 6. conclusion in this work, a novel computational method called elzaki integral transform combined with a new technique of he’s variation iteration technique to investigate the solution of linear and nonlinear telegraph equations of fractional orders. the caputo sense is used to describe the fractional derivatives. this method is implemented on the several models of telegraph equations; the exact and approximate solutions are obtained for each model. the advantage of the proposed technique is that for defining the lagrange multiplier, there is no need to integration or convolution theorem in recurrence relation. because of the limitations of elzaki transform on nonlinear parts, the hpm is used to reduce the computations. finally, the present results show the accuracy of the novel computational method according to the obtained results. in future, the proposed method can be used to investigate the solutions of the differential equations. references [1] h. khan, r. shah, p. kumam, d. baleanu, and m. arif, “an efficient analytical technique, for the solution of fractional-order telegraph equations,” mathematics, vol. 7, no. 5, pp. 1–19, 2019, doi: 10.3390/math7050426. 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[42] m. turkyilmazoglu, “convergence of the homotopy perturbation method,” int. j. nonlinear sci. numer. simul., vol. 12, no. 1–8, pp. 9–14, 2011, doi: 10.1515/ijnsns.2011.02.y paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 150 pemodelan kriminal di jawa timur dengan metode geographically weighted regression (gwr) imanudin nurhuda1, i gede nyoman mindra jaya2 universitas padjadjaran1, imanudin.n@gmail.com1 universitas padjadjaran2, mindra@unpad.ac.id2 doi:https://doi.org/10.15642/mantik.2018.4.1.150-158 abstrak kriminalitas merupakan segala macam bentuk perbuatan yang merugikan baik secara ekonomis atau psikologis juga melanggar hukum yang berlaku di negara indonesia serta norma-norma sosial dan agama, sedangkan data kriminalitas adalah jumlah kasus yang terlapor pada instansi kepolisian. semakin tinggi jumlah pelapor maka jumlah kriminalitas di wilayah tersebut semakin tinggi. semakin besar resiko yang dimiliki masyarakat menggambarkan semakin tidak amannya suatu daerah. penelitian ini bertujuan untuk memperoleh model terbaik yang mempengaruhi tindak kejahatan atau kriminalitas di jawa timur. jumlah tindak kejahatan dalam penelitian ini dibatasi pada jumlah kasus pencurian baik pencurian biasa, pencurian dengan kekerasan, pencurian dengan pemberatan, dan pencurian kendaraan bermotor. pada penelitian ini digunakan model geographically weighted regression (gwr) karena metode ini cukup efektif dalam mengestimasi data yang memiliki spatial heterogenity (ketidakseragaman dalam lokasi/spasial). pada dasarnya, parameter model dalam gwr dapat dihitung pada lokasi pengamatan dengan variabel dependen dengan satu atau lebih variabel independen yang telah diukur di tempat-tempat yang lokasinya diketahui, dimana tindak kriminal pada penelitian yang dilakukan yaitu di wilayah jawa timur melibatkan efek heterogenitas spasial, dengan fungsi pembobot fixed kernel. hasil penelitian menunjukan bahwa variabelvariabel yang mempengaruhi tingkat kriminalitas di provinsi jawa timur adalah kepadatan penduduk, pertumbuhan ekonomi, gini rasio dan kemiskinan. kata kunci: geographically weigthed regression (gwr), spasial, kriminalitas abstract criminality constitutes all kinds of actions that are economically and psychologically harmful in violation of the law applicable in the state of indonesia as well as social and religious norms, while the criminal data is the number of cases reported to the police institution. the higher the number of complainants the higher the number of criminals in the region. the greater the risk the community represents the more insecure a region is. this study aims to obtain the best model affecting crime or crime in east java. the number of crimes in this study is limited to the number of theft cases (whether ordinary theft, theft by force, theft with theft, and the theft of motor vehicles). in this study, we use the geographically weighted regression (gwr) model because this method is quite effective in estimating data that has spatial heterogeneity (uniformity in location / spatial). in essence, the model parameters in gwr can be calculated at the observation location with the dependent variable and one or more independent variables that have been measured at the sites where the location is known, where criminal acts in the research conducted in east java involves the effects of spatial heterogeneity, with fixed kernel weighting function. the results showed that the variables affecting criminality in east java province are population density, economic growth, gini ratio, and poverty. keywords: geographically weigthed regression (gwr), spatial, criminalitas 1. pendahuluan kriminalitas adalah segala sesuatu perbuatan yang melanggar hukum dan melanggar norma-norma sosial, sehingga masyarakat menentangnya. hasil penelitian crime and punishment, menggunakan data kriminalitas amerika serikat (as), mengungkapkan bahwa individu yang rasional akan melakukan tindakan ilegal berdasarkan analisis biaya manfaat dan jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 151 diformulasikan dalam crime economic model (cem) [1] [2]. sedangkan penelitan [3] mengembangkan model penelitian [2]. lebih lanjut dengan mempertimbangkan opportunity cost dan menguji hubungan antara tingkat kriminalitas dengan variabel sosio ekonomi. [4] juga menjelaskan fenomena kriminalitas dengan mempertimbangkan perilaku individu konsumen, yaitu memaksimumkan utilitas. teori lain menunjukan semakin tinggi angka kriminalitas menunjukkan semakin banyak tindak kejahatan pada masyarakat yang merupakan indikasi bahwa masyarakat merasa semakin tidak aman disisi lain statistik dan indikator yang biasa digunakan untuk mengukur rasa aman masyarakat merupakan indikator negatif, misalnya jumlah angka kejahatan (crime total), jumlah orang yang berisiko terkena tindak kejahatan (crime rate) setiap 100.000 penduduk. semakin tinggi angka kriminalitas menunjukkan semakin banyak tindak kejahatan pada masyarakat yang merupakan indikasi bahwa masyarakat merasa semakin tidak aman. menurut hawaii dept of the attorney general (united states of america, 1984), menyimpulkan pada penelitiannya, ada hubungan positif antara kepadatan penduduk dengan tingkat kejahatan untuk kota new york city. makalah ini bertujuan untuk memaparkan analisis spasial geographically weighted regression (gwr) pada tingkat kejahatan di provinsi jawa timur dengan menggunakan data dari publikasi bps “statistik politik dan keamanan provinsi jawa timur 2016”. matriks pembobot yang digunakan adalah pembobot fix kernell. penelitian ini sangat penting menggunakan analisis spasial geographically weighted regression (gwr) karena dalam data melibatkan unsur spasial (lokasi) sehingga lokasi tersebut harus di pertimbangkan yang di fokuskan melihat spasial kewilayahan kota-kota yang ada dalam provinsi di jawa timur yang terdiri dari 38 kabupaten / kota. 2. tinjauan pustaka 2.1. spasial data cross section analisis regresi adalah suatu analisis yang dilakukan terhadap dua variabel yaitu variabel independen (prediktor) dan variabel dependen (respon), istilah regresi pertama kali diperkenalkan oleh francis galton pada 1886 [5]. model regresi dirumuskan sebagai berikut: 𝑦𝑖 = 𝛽0 + ∑ 𝛽𝑘𝑥𝑖𝑘 + 𝜀𝑖 𝑝 𝑘=1 (1) dimana: 𝑦i : variabel respon ke-i (dengan 𝑖 = 1,2, … 𝑛, dimana 𝑛 merupakan banyaknya observasi. 𝑥𝑖𝑘: variabel penjelas ke-k pada amatan kei (dengan 𝑘 = 1,2, …, 𝑝, dimana 𝑝 merupakan banyaknya variabel penjelas) 𝛽k : koefisien regresi sebagai parameter 𝜀𝑖 : error yang diasumsikan berdistribusi normal dengan rata-rata nol dan varians konstan (𝜀𝑖~(0, 𝜎2)). dalam bentuk matrik: y = xβ + ε (2) atau               +                             =               ppnpn p p n xx xx xx y y y             2 1 1 0 1 221 111 2 1 1 1 1 (3) analisis spasial adalah analisis yang digunakan untuk mendapatkan informasi pengamatan yang dipengaruhi efek ruang atau lokasi. dimana data spasial mengindikasikan terdapat ketergantungan antara pengukuran data dengan lokasi pengamatan atau sering disebut efek spasial [6]. model umum regresi spasial dengan menggunakan data cross section dapat dituliskan sebagai berikut [6]: 𝑌 = 𝜌𝑊1𝑦 + 𝑥𝛽 + 𝑢 (4) dimana, 𝑢 = 𝜆𝑊2 + 𝜀 (5) jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 152 𝑦 = (𝐼𝑁 − 𝜌𝑊1) −1𝑋𝛽 + (𝐼𝑁 − 𝜌𝑊1) −1(𝐼𝑁 − 𝜌𝑊) −1𝜀 (6) ε~(0, σ2in) dimana y adalah vektor variabel respon berukuran n x 1, w adalah matriks pembobot berukuran n x n, 𝛾 adalah koefisien regresi spasial lag eksogenus, 𝜌 adalah koefisien spasial lag dari variabel respon, x adalah matriks variabel predictor berukuran n x (k + 1), 𝜷 adalah vektor parameter koefisien spasial lag, 𝜆 adalah koefisien spasial autoregressive error yang bernilai |𝜆|< 1, u adalah vektor error pada sem berukuran n x 1, n adalah banyaknya amatan atau lokasi, k adalah banyaknya variabel predictor. sedangkan menurut [7] data spasial adalah data yang memiliki referensi ruang kebumian (georeference) berupa informasi lokasi dan dilengkapi dengan data atribut untuk tiap unit spasialnya., efek spasial dapat dibagi menjadi dua bagian yaitu autokorelasi spasial dan heterogenitas spasial, dengan penjelasan sebagai berikut: a. autokorelasi spasial autokorelasi spasial adalah jika sebaran data pada wilayah penelitian tidak memiliki pola-pola hubungan antara amatan yang berdekatan, maka data disebut tidak memiliki autokorelasi spasial [8] [9] [10] [11]. pendeteksian ketergantungan spasial dapat dilakukan menggunakan indeks moran. b. heterogenitas spasial dalam pemodelan regresi klasik, heteroskedastisitas mengacu pada kesamaan varian [12] yang dinyatakan sebagai salah satu konsekuensi dari heterogenitas. sedangkan [13] menyebut heterogenitas spasial sebagai suatu kondisi dimana suatu model regresi global tidak dapat menjelaskan hubungan antara variabel-variabel dikarenakan karak-teristik antar wilayah amatan yang bervariasi secara spasial. mendeteksi adanya heterogenitas spasial pada data dapat dilakukan dengan uji breusch-pagan [6]. 2.2. geographically weighted regression (gwr) model geographically weighted regression (gwr) adalah salah satu metode yang digunakan untuk mengestimasi data yang memiliki spatial heterogeneity (keragaman spasial) dengan estimasi menggunakan weighted least square [14]. pada model gwr, diasumsikan bahwa masing-masing lokasi pengamatan memiliki koordinat spasial dalam satu wilayah terregional. koordinat spasial pada lokasi pengamatan ke-i dilambangkan dengan (ui, vi). persamaan umum gwr [14] adalah sebagai berikut: 𝑦𝑖 = 𝛽0(𝑢𝑖, 𝑣𝑖 ) + ∑ 𝛽𝑖𝑘𝑥𝑖𝑘 (𝑢𝑖, 𝑣𝑖 ) + 𝜀𝑖 𝑝 𝑘=1 (7) atau yi = 𝛽 (𝑢𝑖,) + 𝜺 (8) dimana: yi adalah nilai observasi variabel respon pada lokasi ke-i, xik adalah nilai observasi variabel prediktor k pada lokasi ke-i, 𝛽𝑖𝑘 adalah koefisien regresi predictor ke k, (𝑢𝑖, 𝑣𝑖) menyatakan lokasi sumber ke-i berdasarkan koordinat lintang dan bujur, 𝜀𝑖 adalah residu pada lokasi ke-i dengan asumsi distribusi normal 𝑁(0, 𝜎2𝐼)), dan i matrik identitas. 2.3. matrik bobot matriks pembobot/penimbang spasial (w) dapat diperoleh berdasarkan informasi jarak dari ketetanggaan (neighborhood) atau jarak antara satu lokasi dengan lokasi yang lain. elemen matriks pembobot gwr yaitu wij ditentukan berdasarkan kedekatan titik regresi i dengan titik pengamatan j. titik pengamatan yang lebih dekat ke titik regresi diberi bobot lebih besar daripada titik pengamatan yang lebih jauh. fungsi pembobot yang digunakan dalam penelitian ini adalah fungsi fixed exponential kernel. untuk sampel yang sudah cukup teratur sebarannya dalam daerah penelitian, maka fungsi pembobot jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 153 yang dengan bandwidth tetap (fixed) merupakan pilihan yang tepat untuk penaksiran parameter. secara matematis, fungsi pembobot fixed exponential kernel [14]: 𝑤𝑖𝑗 (𝑢𝑖 , 𝑣𝑖 ) = 𝑒𝑥𝑝 (− 𝑑𝑖𝑗 𝑏 ) (9) dimana 𝑑𝑖𝑗 = √(𝑢𝑖 − 𝑢𝑗 )2+ (𝑣𝑖 − 𝑣𝑗 )2 merupakan jarak euclidean antara lokasi (𝑢i, 𝑣𝑖 ) dengan lokasi (𝑢𝑗 , 𝑣𝑗 ), dan b merupakan bandwidth yang fixed atau bandwidth yang sama yang digunakan untuk setiap lokasi. jika nilai 𝑖 = 𝑗, maka pembobot pada lokasi tersebut adalah 1. fixed kernel memiliki bandwith yang sama pada setiap titik lokasi pengamatan.terdapat berbagai jenis fungsi kernel tetap yang digunakan dalam gwr adalah: 1. kernell gaussian 𝑤𝑖𝑗 (𝑢𝑖, 𝑣𝑖 ) = 𝑒𝑥𝑝 (− 1 2 ( 𝑑𝑖𝑗 𝑏 ) 2 ) (10) 2. kernell bisquare 𝑊𝑗 (𝑢𝑖, 𝑣𝑖) = { (1 − ( 𝑑𝑖𝑗 𝑏 ) 2 ) 2 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 ≤ 𝑏 0 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 > 𝑏 } (11) 3. kernell tricube 𝑊𝑗 (𝑢𝑖, 𝑣𝑖) = { (1 − ( 𝑑𝑖𝑗 𝑏 ) 3 ) 3 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 ≤ 𝑏 0 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 > 𝑏 } (12) 4. adaptive bisquare 𝑊𝑗 (𝑢𝑖, 𝑣𝑖) = { (1 − ( 𝑑𝑖𝑗 𝑏 ) 2 ) 2 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 ≤ 𝑏 0 𝑢𝑛𝑡𝑢𝑘, 𝑑𝑖𝑗 > 𝑏 } (13) 𝑏 adalah parameter nonnegatif yang diketahui dan biasanya disebut bandwidth (parameter penghalus). kernel gaussian, bisquare, tricube adalah contoh fixed kernel, artinya kernel tersebut memiliki bandwidth yang sama pada masing-masing lokasi pengamatan. sedangkan adaptive bisquare memiliki bandwidth yang berbeda pada masing-masing lokai pengamatan. 2.4. pemilihan bandwidth optimum sebuah titik yang berada dalam radius lingkaran tersebut masih dianggap memiliki pengaruh. bandwidth, bandwidth dapat dianalogikan sebagai radius suatu lingkaran, sehingga merupakan pengontrol keseimbangan antara kesesuaian kurva terhadap data dan kemulusan data. pemilihan bandwidth yang optimum menjadi salah satu hal yang penting karena akan mempengaruhi ketepatan hasil estimasi [14]. metode yang dapat digunakan untuk pemilihan bandwidth salahsatunya adalah cross validation pada seluruh lokasi, dengan rumus: 𝐶𝑉 = ∑ [𝑦𝑖 − �̂�≠𝑖(𝑏)] 𝑛 𝑖=1 (14) 𝑦 ̂≠𝑖 adalah nilai dugaan 𝑦𝑖 pada pengamatan lokasi ke-i dengan bandwidth tertentu dihilangkan dari proses prediksi. bandwidth optimum (ℎ) akan diperoleh dengan proses iterasi sampai diperoleh cv yang minimum [14]. 2.5. multikolinearitas multikolinieritas disebabkan oleh adanya minimal satu kolom dari matriks x yang merupakan kombinasi linier dari kolom lainnya, atau dengan kata lain ada korelasi di antara variabel-variabel penjelas [5]. pada model gwr, multikolinieritas dideteksi dengan nilai vif yang dihitung di tiap lokasi ke-i yang disebut dengan vif lokal, sehingga diketahui multikolinieritas variabel pada masing-masing wilayah. nilai vif local dirumuskan sebagai berikut: 𝑉𝐼𝐹𝑘(𝑖) = 1 1−𝑅𝑘 2(𝑖) (15) dengan 𝑅𝑘 2 (𝑖) adalah koefisien determinasi antara 𝑥𝑘 dengan peubah penjelas lainnya untuk tiap lokasi ke-i [5]. 2.6. tahapan pengolahan dan analisis langkah – langkah analisis dalam penelitian ini adalah sebagai berikut : jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 154 1. mendeskripsikan tingkat kriminalitas pencurian di jawa timur dengan memetakan lokasinya. informasi yang ingin digali adalah pola sebaran wilayah, nilai variabel dan daerahnya. 2. mengidentifikasi pola hubungan antara variabel dengan membentuk regresi linear atau ols. 3. menetapkan matriks pembobot spasial (w). 4. memeriksa apakah ada efek spasial dengan uji moran’s i dan lagrange multiplier (lm) 5. melakukan pengujuan heterogenitas 6. melakukan pemodelan gwr. a. melakukan pemilihan bandwidth optimum dengan menggunakan kriteria cross validation (cv) minimum. b. membentuk matriks pembobot (i) w untuk setiap lokasi pengamatan dengan fungsi fixed exponential kernel sesuai persamaan. c. melakukan estimasi parameter model gwr berdasarkan matriks pembobot yang diperoleh sebelumnya sehingga didapatkan model yang bersifat lokal untuk setiap lokasi. 7. menarik kesimpulan 3. sumber data sumber data dalam penelitian ini adalah data sekunder yang di peroleh dari beberapa publikasi yang dilakukan oleh bps jawa timur. unit penelitian yang diteliti adalah 38 kabupaten/kota yang berada pada wilayah provinsi jawa timur. sedangkan variabel penelitian yang digunakan dalam penelitian adalah sebagai berikut: tabel 1. variabel penelitian variabel uraian y x1 x2 x3 crime rate tingkat pengangguran terbuka pengeluaran perkapita pertumbuhan ekonomi variabel uraian x4 x5 x6 x7 kepadatan penduduk angka harapan sekolah gini rasio jumlah penduduk miskin 4. hasil dan pembahasan secara deskriptif kondisi jawa timur pada tahun 2015, tingkat kriminalitas pencurian di jawa timur sangat bervariasi dan beranekaragam di setiap wilayah kabupaten atau kota di jawa timur. sedangkan untuk tingkat kriminalitas provinsi jawa timur adalah sebesar 19,64 artinya secara rata-rata di provinsi jawa timur terdapat 20 orang yang menjadi korban kasus kriminalitas pada setiap 100.000 orang penduduk. tindak kriminal tertinggi berada pada kabupaten bojonegoro dengan jumlah kasus 123 kasus, pada posisi kedua kabupaten blitar dengan angka kasus 28 kasus, dan pada tempat ketiga adalah kota blitar dengan jumlah kasus sebanyak 25 kasus, dalam gambar berikut di perlihatkan peta kejahatan di jawa timur: gambar 1 peta kriminal jawa timur tingkat kriminaltias terrendah terdapat di kabupaten pacitan 6 pencurian per 100.000 penduduk, dan tingkat kriminalitas pencurian tertinggi terjadi di kota bojonegoro dengan 123 pencurian per 100.000 penduduk. 4.1. model regresi sebelum membentuk model gwr langkah pertama yang harus dilakukan diantaranya adalah membentuk model regresi global tindak kriminal dengan 7 (tujuh) variabel penjelas. dimana dalam penelitian ini model yang digunakan jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 155 model ordinary least squrae (ols). model dugaan regresi linier berganda (full model) yang terbentuk adalah: �̂�=−114.387-2.624x1-0.006x2+ 1.417x3 +0.012x4+0.107x5-72.255x6-0.143x7 dari model terlihat x1, x2, x6 dan x7 bertanda negative yang berarti dengan kenaikan nilai x1, x2, x6 dan x7 tersebut akan menurunkan nilai y nya sedangkan untuk x3, x4 dan x5 bertanda positif yang berarti dengan kenaikan nilai x3, x4 dan x5 akan menaikan nilai y. berikut adalah tabel hasil estimasi parameter dari model regresinya: tabel 2. hasil analisis regresi variabel nilai�̂� t p-value intercept x1 x2 x3 x4 x5 x6 x7 114.387 -2.624 -0.006 1.417 0.012 0.107 -72.255 -0.143 1.31 -1.05 -2.23 1.08 4.58 0.06 -0.52 -2.43 0.199 0.299 0.033 0.285 7.5e-05 0.987 0.603 0.021 hasil analisis menunjukan nilai p value sebesar 2.946e-05 artinya model ols signifikan,model regresi global memiliki 𝑅2 sebesar 0.5535 yang menunjukkan bahwa variasi variabel penjelas yang ada mampu menjelaskan 53,35 persen dari variasi persentase tindak kriminal di jawa timur. dalam model yang berpengaruh secara signifikan diperlihatkan pada variabel pendapatan perkapita(x2), kepadatan penduduk(x4) dan jumlah penduduk miskin(x7). multikolinieritas pada model regresi global ditunjukkan dengan nilai vif sebagai beikut: tabel 3. hasil uji vif variabel x1 x2 x3 x4 x5 x6 x7 vif 1.8 3.4 1.1 3.4 3.3 2.1 2.0 dari tabel terlihat bahwa nilai setiap variabel x kurang dari 10, yang artinya dalam data tidak terdapat multikolinearitas. 4.2. pengujian dependensi dan heterogenitas spasial analisis dependensi atau autokorelasi dan heterogenitas dilakukan sebelum melakukan pengujian gwr, dimana dalam pengujian pada data tindak kriminal di jawa timur, di peroleh hasil uji heterogenitas bruce pagan sebesar 16.977 dan, p-value sebesar 0.01754 sehingga dapat disimpulkan bahwa pada taraf nyata 5%, model regresi global mengandung efek keragaman spasial. dalam pembuktian adanya efek spasial pada data kriminal di provinsi jawa timur dianalisis dengan menggunakan analisis moran’s i. hasil perhitungan nilai moran’s i secara umum memperlihatkan adanya keterkaitan spasial kriminal di jawa timur suatu wilayah dengan wilayah lainnya. hasil pengujian menunjukkan nilai moran’s i yang signifikan dan bernilai positif seperti yang ditunjukkan dengan moran’s i yang bernilai positif sebesar 0.0110 dan signifikan menandakan bahwa tingginya persentase kriminal di jawa timur di suatu wilayah memberikan pengaruh terhadap tingginya kriminal di jawa di wilayah sekitarnya, dan begitupun sebaliknya sehingga analisis dilanjutkan dengan pemodelan geographically weighted regression (gwr). 4.3. model geographically weighted regression (gwr) pemilihan bandwidth optimum diperoleh melalui teknik cross validation (cv). nilai bandwidth optimum yang diperoleh adalah 0.4382554. nilai bandwidth sebesar 0.4382554, sehingga hal ini menunjukkan bahwa wilayah atuto korelsi yang masih berada di sekitar 0.4382554 derajat dari titik lokasi pengamatan masih memberikan pengaruh terhadap nilai persentase tindak kriminal di lokasi pengamatan tersebut. hasil analisis data multikolinieritas lokal yang menunjukan bahwa semua nilai berada dibawah 10, sehingga disimpulkan tidak terjadi multikolinieritas pada beberapa semua variabel prediktor. untuk mengatasi analisisnya cukup menggunakan (geographically weighted regression (gwr). jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 156 tabel 4. ringkasan hasil estimasi parameter model gwr variabel nilai �̂� minimum mean maksimum intercept x1 x2 x3 x4 x5 x6 x7 -1.524 -0.259 -0.000 0.033 0.000 -0.241 -9.074 -0.009 2.544 -0.121 -0.000 0.052 0.000 0.009 -3.096 -0.005 6.936 -0.028 0.000 0.109 0.000 0.262 9.719 -0.002 tabel 4 diatas menginformasikan peubah x3, dan x4,memiliki koefisien dugaan parameter yang bernilai positif pada seluruh wilayah pengamatan. jika dilihat secara menyeluruh, dugaan parameter pada model gwr lebih baik. hasil model dugaan gwr untuk seluruh wilayah pengamatan. sedangkan variabel penjelas yang memiliki hubungan bervariasi antara negatif dan positif dengan variabel respon di beberapa lokasi pengamatan. nilai estimasi parameter model gwr untuk masing-masing lokasi pengamtan dengan variabel respon dalam model gwr adalah x1, x2, x3,, x5, x6 dan x7. hasil model gwr negative semua wilayah pada koefisien parameternya x6 nilai residual sum of square dari model gwr adalah sebesar 3.1, dimana nilai ini lebih kecil dari residual sum of square model regresi global yaitu 13,3. koefisien determinasi (𝑅2) dari model gwr adalah sebesar 0,91, lebih tinggi daripada 𝑅2 model regresi global. hal ini menunjukkan bahwa variasi variabel penjelas pada model gwr mampu menjelaskan 91% persen dari variasi persentase kriminal di provinsi jawa timur. tabel 5. perbandingan aic regresi aic ols gwr 86.21 36.50 jika dilihat dari nilai aic perbandingan antara ols dan gwr pada tabel diatas, maka diperoleh aic regresi ols sebesar 86.21 lebih besar jika dibaningkan dengan regresi gwr yaitu sebesar 36,50. hal ini menunjukan bahwa model gwr lebih cocok dibandingkan regresi ols diterapkan untuk data kriminal di provinsi jawa timur. tabel 6. variabel penelitian variabel t hitung signifikansi y x1 x2 x3 x4 x5 x6 x7 -3.918 -0.798 0.964 3.557 8.549 -0.785 0.661 2.457 tidak signifikan tidak signifikan tidak signifikan signifikan signifikan tidak signifikan signifikan signifikan tabel 5 terlihat variabel-variabel yang digunakan dalam penelitian terdapat nilai yang tidak signifikan yaitu di tunjukan pada variabel tingkat pengangguran terbuka(x1), pendapatan perkapita(x2), dan angka harapan sekolah(x5) sedangkan variabel yang signifikan mempengaruhi kriminalitas di provinsi jawa timur, ditunjukan pada variabel kepadatan penduduk(x3) pertumbuhan ekonomi(x4), gini rasio(x6) dan kemiskinan(x7). 5. kesimpulan a. data yang digunakan adalah data kriminal di provinsi jawa timur dalam pemodelan terdapat pengaruh spasial sehingga dilakukan pemodelan gwr dengan perhitungan pembobot menggunakan fixed kernel gaussian. b. berdasarkan analisis pemodelan kriminal di jawa timur dapat menggunakan pemodelan geographically weighted regression (gwr) terlihat dari nilai vif lokalnya. c. model geographically weighted regression (gwr) lebih baik daripada model regresi linier berganda dalam memodelkan kasus kriminal di provinsi jawa timur. hal tersebut di lihat dari nilai aic model gwr lebih kecil dari aic regresi biasa. jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 157 d. secara umum terdapat dua faktor yang secara signifikan (𝛼=5%) mempengaruhi jumlah kriminalitas di jawa timur yaitu pertumbuhan ekonomi(x2), kepadatan penduduk(x4), gini rasio(x6) dan kemiskinan(x7) referensi [1] kartono, patologi sosial, jakarta: raja grafindo persada, (1999). 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[9] miron, spatial autocorrelation in regression analysis: a beginner’s guide, in spatial statistics and models. theory and decision library (an international series in the philosophy and methodology of the social and behavioral sciences), dordrecht, springer, pp. 201-222 (1984). [10] jaya ignm, "estimation and testing for moran spatial and spatiotemporal index," working paper, pp. 1-10, (2018). [11] jaya ignm, h. folmer, b. n. ruchjana, f. kristiani and a. yudhie, modeling of infectious diseases: a core research topic for the next hundred years, in regional research frontiers vol. 2 methodological advances, regional systems modeling and open sciences, usa, springer international publishing, pp. 239-254 (2017). [12] dutilleul p. and legendre p., spatial heterogeneity against heteroscedasticity: an ecological paradigm versus a statistical concept, oikos, vol. 66, no. 1, pp. 152-171, (1993). [13] brunsdon c., fotheringham s. and charlton m., geographically weighted regression-modelling spatial nonstationarity, journal of the royal statistical society. series d (the statistician), vol. 47, no. 3, pp. 431-443, (1998). [14] fotheringham s., brunsdon c. and charlton m., geographically weighted regression: the analysis of spatially varying relationships, usa: wiley, (2002). jurnal matematika “mantik” 2018. vol. 4 no. 2 issn: 2527-3159 e-issn: 2527-3167 158 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 59 prediksi cuaca kota surabaya menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter nurissaidah ulinnuha1, yuniar farida2 universitas islam negeri sunan ampel surabaya1, nuris.ulinnuha@uinsby.ac.id 1 universitas islam negeri sunan ampel surabaya2, yuniar_farida@uinsby.ac.id 2 doi:https://doi.org/10.15642/mantik.2018.4.1.59-67 . abstrak perubahan kondisi musim di indonesia menimbulkan banyak bencana seperti tanah longsor, banjir, angin puting beliung bahkan hujan es. kondisi cuaca ekstrem yang terjadi ada baiknya harus tetap di waspadai untuk mengantisispasi berbagai kemungkinan yang terjadi dan untuk mengurangi serta meminimalkan dampak yang merugikan masyarakat. perancangan sistem prediksi cuaca pada penelitian ini menggunakan model autoregressive integrated moving average arima box jenkins dan kalman filter dengan tujuan dapat meprediksi cuaca kota surabaya yang semakin ekstrem di akhir tahun 2017. pada penelitian ini prediksi difokuskan pada variabel kelembapan, suhu, dan kecepatan angin dengan hasil prediksi 5 hari kedepan. adapun prediksi cuaca kota surabaya menggunakan metode arima – kalman filter didapatkan goal eror terkecil (eror mape) masing masing sebesar 0.000389 untuk prediksi kelembapan, 0.000705 untuk prediksi suhu, dan 0.0169 untuk prediksi kecepatan angin. kata kunci: prediksi cuaca, arima, kalman filter, polinomial abstract season changes conditions in indonesia cause many disasters such as landslides, floods and whirlwinds and even hail. extreme weather conditions that occur, it is better to remain alert to anticipate the various possibilities that occur and to reduce and minimize the impact that can harm the people. the design of weather prediction system in this research using autoregressive integrated moving average arima box jenkins model and kalman filter with the aim to predict the increasingly extreme weather of surabaya city at the end of 2017. in this research, weather prediction focused on humidity, temperature, and velocity wind with results 5 days later. the prediction of surabaya city weather using arima method kalman filter obtained the smallest error goal (error mape) of 0.000014 each for the prediction of humidity, 0.000037 for temperature prediction, and 0.0123 for wind speed prediction. keywords: weather prediction, arima, kalman filter, polynomial 1. pendahuluan pada awal tahun 2017, indonesia telah mengalami cuaca ekstrem yang disertai dengan hujan deras. hal ini disebabkan oleh anomali yang cukup signifikan yang terjadi di sejumlah daerah di indonesia khusunya di wilayah samudra hindia pada akhir tahun 2016. adanya anomali yang ekstrem ini membuat musim di indonesia berubah. seperti musim kemarau yang biasanya terjadi pada bulan april sampai dengan bulan september, berubah menjadi musim penghujan. faktor penyebab cuaca ekstrem yang terjadi tidak hanya dari anomali saja, melainkan di sebabkan faktor monsun, pertumbuhan awan konvektif lokal yang signifikan, meningkatnya spl (suhu permukaan laut), tingginya kelembapan udara, gaya coriolis dan dipole mode [1]. hal ini membuat kondisi cuaca regional diperkirakan tetap mengalami cuaca ekstrem sampai akhir tahun 2017. mailto:nuris.ulinnuha@uinsby.ac.id mailto:yuniar_farida@uinsby.ac.id jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 60 perubahan kondisi musim ini menimbulkan banyak bencana seperti tanah longsor, banjir, angin puting beliung bahkan hujan es. tidak hanya itu, cuaca ekstrem ini juga memengaruhi harga sejumlah kebutuhan pokok masyarakat karena mengganggu produksi serta distribusi sehingga mendorong kenaikan harga dan kelangkaan stok. kondisi cuaca ekstrem yang terjadi ada baiknya harus tetap di waspadai untuk mengantisispasi berbagai kemungkinan yang terjadi dan untuk mengurangi serta meminimalkan dampak yang merugikan. oleh karena itu, diperlukan informasi cuaca yang lebih cepat, tepat, dan terperinci. bahkan beberapa pihak lain menuntut tersedianya prediksi atau bahkan ramalan mengenai kondisi atmosfer dengan rentang waktu bulanan, harian, jam, bahkan dalam waktu menit. badan meteorologi, klimatologi, dan geofisika (bmkg) sebagai lembaga pengamat cuaca hadir untuk memfasilitasi tersedianya informasi yang dapat diakses oleh masyarakat umum. namun jika dilihat, prakiraan yang dilakukan bmkg masih perlu ditingkatkan dari segi keakuratannya, karena pada dasarnya para forecaster (peramal cuaca) menganalisis hasil prediksi secara subyektif [1] [2] [3]. oleh karena itu, perlunya merancang prediksi cuaca untuk meminimalkan dampak yang akan terjadi. penelitian mengenai prediksi cuaca telah banyak dilakukan dan dipublikasikan dengan berbagai macam metode. beberapa diantaranya adalah penelitian yang dilakukan oleh arief hanifan p. menggunakan media komunikasi short message service [4], metode fuzzy takagi sugeno yang digunakan dalam penelitian nur wakhid habibulloh [5], penelitian oleh saarika sharma yang menggunakan fuzzy time series model [6], prediksi cuaca dengan jaringan syaraf tiruan yang dilakukan kumar abhishek [2] dan deasy adyanti yang mengimplementasikan time series-anfis dalam memprediksi cuaca maritim di laut jawa [1]. perancangan sistem prediksi cuaca pada penelitian ini menggunakan model autoregressive integrated moving average arima box jenkins yang mana metode tersebut cocok dalam masalah forecasting [7] dan kalman filter merupakan salah satu metode stokastik pengembangan model peramalan statistik autoregresive yang recursive dalam mengintegrasikan data pengamatan terbaru ke dalam model untuk memperbaharui (update) prediksi sebelumnya dan melanjutkan prediksi ke periode yang akan datang [8]. berdasarkan kondisi tersebut, maka peneliti melakukan penelitian tentang prediksi cuaca kota surabaya menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter dengan harapan mampu menghasilkan sistem prediksi yang lebih efektif dan efisien khususnya di wilayah surabaya. 2. kerangka teori 2.1 cuaca cuaca adalah fenomena-fenomena yang saat ini sedang terjadi di atmosfer bumi [9]. menurut arifin, cuaca merupakan suatu kondisi udara di suatu tempat pada waktu yang relatif singkat, yang dinyatakan dengan nilai berbagai parameter seperti suhu, tekanan udara, kecepatan angin, kelembaban udara, dan berbagai fenomena atmosfer lainnya. pemilihan metode untuk menentukan kondisi cuaca adalah kegiatan yang akhir-akhir ini sering dilakukan oleh beberapa peneliti atmosfer atau cuaca [10]. 2.2 parameter cuaca angin, kelembapan udara, tekanan udara, suhu udara, dan penyinaran matahari merupakan unsur-unsur cuaca yang sangat penting. berikut adalah definisi-definisi dari unsur-unsur cuaca. a. suhu suhu udara adalah keadaan panas atau dinginnya udara. alat untuk mengukur suhu udara atau derajat panas disebut thermometer [11]. faktor-faktor yang memengaruhi tinggi rendahnya suhu udara di muka bumi adalah lamanya penyinaran matahari, sudut datang sinar matahari, relief permukaan bumi, banyak sedikitnya awan, dan perbedaan letak lintang [11]. b. kelembapan udara kelembapan udara terbagi menjadi dua, yaitu kelembapan udara absolut dan kelembapan udara relatif. kelembapan udara absolut adalah banyaknya uap air yang terdapat di udara pada suatu tempat sedangkan kelembapan udara relatif adalah perbandingan jumlah uap air dalam udara dengan jumlah uap air maksimum yang dapat dikandung oleh udara tersebut dalam suhu yang sama [11]. c. kecepatan angin angin adalah udara yang bergerak dari tempat yang bertekanan tinggi ke daerah yang bertekanan rendah. sifat angin dipengaruhi oleh tiga hal yaitu kekuatan angin, arah angin, dan kecepatan angin. atmosfer udara mengikuti rotasi bumi setiap harinya, hal ini menyababkan, molekul-molekul udara juga jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 61 mempunyai kecepatan yang arah geraknya menuju ke arah timur (menyesuaikan rotasi). kecepatan gerak angin ini disebut sebagai kecepatan linear yang apabila semakin kecil kecepatan linearnya semakin mendekati ke arah kutub [11]. 2.3 autoregressive integrated moving average box jenkins model arima (autoregressive integrated moving average) dikembangkan oleh george box dan gwilyn jenkins pada tahun 1976 [7]. metode ini sangat baik untuk prediksi jangka pendek, dan tidak disarankan untuk prediksi jangka panjang karena hasil ketepatan prediksinya kurang baik. arima merupakan metode yang menggunakan data-data masa lalu maupun data sekarang sebagai variabel dependen untuk menghasilkan prediksi jangka pendek yang akurat. arima adalah salah satu metode stokastik yang sangat bermanfaat untuk membangkitkan proses (data) deret waktu dimana setiap kejadian saling berkorelasi [7]. arima sangat ketat terhadap asumsi (data dan residual white noise) dan digunakan untuk data yang berpola linear. secara harfiah, model arima merupakan gabungan antara model ar (autoregressive) dan model ma (moving average). model arima terdiri dari tiga langkah dasar, yaitu tahap identifikasi, tahap penaksiran dan pengujian, dan pemeriksaan diagnostik. selanjutnya model arima dapat digunakan untuk melakukan prediksi jika model yang diperoleh memadai [7]. secara umum model arima (box-jenkins) dirumuskan dengan notasi sebagai berikut: arima (p,d,q) [7] dimana: p: menunjukkan orde / derajat autoregressive (ar). d: menunjukkan orde / derajat differencing (pembedaan). q: menunjukkan orde / derajat moving average (ma). a. model autoregressive (ar) model autoregressive adalah model yang variabel dependennya dipengaruhi oleh variabel dependen itu sendiri pada periodeperiode dan waktu-waktu sebelumnya [7]. secara umum model autoregressive (ar) dengan ordo p (ar(p)) atau model arima (p,0,0) mempunyai bentuk sebagai berikut: yt = 0+1yt-1 + 2yt-2+ … +pyt-p +et (2.1) dimana: yt = deret waktu stasioner yt-1, yt-2,….,, yt-p = variabel respon pada masingmasing selang waktu t 1, t 2,…, t – p. nilai y berperan sebagai variabel bebas. 0= suatu konstanta p = parameter autoregresif ke-p et = galat pada saat t yang mewakili dampak variabelvariabel yang tidak dijelaskan oleh model. dari model ar (yang diberi notasi p) ditentukan oleh jumlah periode variable dependen yang masuk dalam model. b. model ma (moving average) secara umum model moving average ordo q (ma(q)) atau arima (0,0,q) mempunyai bentuk sebagai berikut [7] : yt = 0+et 1et-1 2et-2 … -pet-q (2.2) dimana: yt : deret waktu stasioner 0 : konstanta 1,…,q :parameter-parameter moving average yang menunjukkan bobot. et q : nilai kesalahan pada saat t – k. c. model arma (autoregressive moving average) model yang memuat kedua proses ar dan ma disebut model arma. bentuk umum model ini adalah 𝑌𝑡 = 𝛾0 + 𝜕1𝑌𝑡−1 + 𝜕2𝑌𝑡−2 + ⋯ + 𝜕𝑛𝑌𝑡−𝑃 − 𝜆1𝑒𝑡−1 − 𝜆2𝑒𝑡−2 − 𝜆𝑛𝑒𝑡−𝑞 (2.3) dimana yt adalah deret waktu stasioner dan et adalah galat. jika model menggunakan dua lag dependen dan tiga lag residual, model itu dilambangkan dengan arma. dan jika menambahkan proses stasioner data, model arma yang ada menjadi model umum arima (p,d,q) [7]. 2.4 kalman filter kalman filter adalah suatu metode estimasi yang optimal. komponen dasar dari metode kalman filter adalah persamaan pengukuran dan persamaan transisi [8]. data pengukuran digunakan untuk memperbaiki hasil estimasi. secara umum algoritma kalman filter untuk sistem dinamik linear waktu diskrit adalah: model sistem dan model pengukuran: 𝑥𝑘+1 = 𝐴𝑘 𝑥𝑘 + 𝐵𝑘𝑢𝑘 + 𝐺𝑘𝑤𝑘 (2.4) 𝑍𝑘 = 𝐻𝑘𝑥𝑘 + 𝑣𝑘 (2.5) 𝑥0~(�̅�0, 𝑃𝑥0),𝑤𝑘~(0,𝑄𝑘), 𝑣𝑘~(0, 𝑅𝑘) inisialisasi: 𝑃(0) = 𝑃𝑥0 ,𝑥 0 = �̅�0 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 62 tahap prediksi: estimasi: 𝑥−𝑘 + 1 = 𝐴𝑘 𝑥𝑘 + 𝐵𝑘𝑢𝑘 (2.6) kovarians eror: 𝑃𝑘− + 1 = 𝐴𝑘𝑃𝑘𝐴𝑘𝑇 + 𝐺𝑘𝑄𝑘𝐺𝑘𝑇 (2.7) tahap koreksi: kalman gain: 𝐾𝑘 + 1 = 𝑃−𝑘+1 𝐻 𝑇 𝑘+1(𝐻𝑘+1𝑃 − 𝑘+1 𝐻 𝑇 𝑘+1 + 𝑅𝑘+1) −1 (2.8) kovarians eror: 𝑃𝑘+1 = [𝐼 − 𝐾𝑘+1𝐻𝑘+1]𝑃 − 𝑘+1 (2.9) estimasi: 𝑥 𝑘+1 = 𝑥 − 𝑘+1 + 𝐾𝑘+1[𝑧𝑘+1 − 𝐻𝑘+1𝑥 − 𝑘+1] (2.10) dengan: 𝑥𝑘 ∶ variabel keadaan system pada waktu 𝑘 yang nilai estimasi awalnya adalah �̅�0 dan kovarian awal 𝑃𝑥0 𝑢𝑘 ∶ variabel input deterministik pada waktu 𝑘 𝑤𝑘 ∶ noise pada pengukuran dengan mean sama dengan nol dan kovariansi 𝑄𝑘 𝑧𝑘 ∶ variabel pengukuran 𝐻 ∶ matriks pengukuran 𝑣𝑘 ∶ noise pada pengukuran dengan mean sama dengan nol dan kovarian 𝑅𝑘 3. metode penelitian penelitian tentang peramalan cuaca kota surabaya menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter termasuk kedalam jenis penelitian terapan yang dilihat dari aspek fungsinya. solusi dari hasil prediksi ini bertujuan sebagai pemecahan masalah yang akan diimplementasikan pada kondisi yang ada di lapangan. penelitian terapan juga dapat diartikan sebagai suatu tindakan aplikatif untuk pemecahan masalah tertentu. a. pengumpulan dan analisis data data diambil dari pengamatan secara sinoptik dan data rekam alat automatic weather system (aws) dengan jumlah data sebanyak 735 data pada dari bulan september 2015 sampai awal september 2017 yang diakumulasi dalam rentan waktu setiap hari selama dua tahun terakhir. data ini berjumlah 735 data per hari yang terbagi menjadi dua yaitu data training dan data testing. data training yang akan digunakan sejumlah 715 data dan data testing yang akan digunakan sejumlah 20 data. data tersebut diperoleh dari stasiun meteorologi bmkg maritim perak ii surabaya yang selanjutnya diproses menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter. data berupa data suhu, kelembapan udara, dan kecepatan angin selama dua tahun terakhir di wilayah kota surabaya. analisis data ini dilakukan dengan menggunakan data suhu, kelembapan udara, dan kecepatan angin pada dua tahun sebelumnya untuk memprediksi satu bulan setelahnya. b. validasi dan metode analisa data validasi data dalam penelitian ini menggunakan metode keakuratan yaitu mape. sedangkan metode analisa data pada penelitian ini dilakukan dengan beberapa tahapan yaitu : pengumpulan data, pada penelitian ini data yang digunakan merupakan data sekunder, yaitu suhu, kelembapan udara, dan kecepatan angin di kota surabaya dalam dua tahun terakhir, ploting data dengan tujuan melihat stasioner dan tidaknya suatu data, identifikasi model untuk melihat model yang dapat dibangun, uji model dengan tujuan mendapatkan model terbaik, dan model terbaik nantinya digunakan dalam prediksi. gambar 1 merupakan diagram alur dari metode arima. gambar 1. skema box jenkins skema box jenkins menjelaskan bahwa tahap-tahap melakukan peramalan adalah: jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 63 • rumuskan kelompok model yang umum. dalam merumuskan model yang tepat untuk arima dilakukan dengan mengidentifikasi model melalui partial autocorrelation dengan harapan menemukan model yang paling sesuai untuk prediksi. • dari serangkaian model yang didapatkan, tahap selanjutnya adalah melakukan penetapan model untuk sementara. • setelah penetapan model langkah selanjutnya adalah dilakukan penaksiran terhadap parameter-parameter dalam model agar mengetahui kelayakan model yang telah dibuat. • setelah model dapat dikatakan layak atau sesuai, langkah terakhir adalah melakukan peramalan. kalman filter berkaitan dengan pengembangan model peramalan statistik autoregresive yang recursive dalam mengintegrasikan data pengamatan terbaru ke dalam model untuk memperbaharui (update) prediksi sebelumnya dan melanjutkan prediksi ke periode yang akan datang. sedangkan metode arima merupakan bagian dari time series untuk memprediksi karena dapat menghasilkan suatu model yang akurat yang mewakili pola masa lalu dan masa depan dari suatu data time series, di mana polanya bisa random, seasonal, trend, atau kombinasi pola-pola tersebut. pada tahapan ini, hasil model peramalan analisis time series dari unsur-unsur cuaca di kota surabaya dapat dinyatakan sebagai parameter dan akan dilakukan pendekatan yang didasarkan pada koreksi dari bias prakiraan dalam penggunaan kalman filter. selanjutnya akan difokuskan pada studi parameter. diberikan polinomial [11] : 𝑦𝑖0 = 𝑎0,𝑖 + 𝑎1,𝑖𝑚𝑖 + ⋯ + 𝑎𝑛−1,𝑖 𝑚𝑖 𝑛−1 + 𝜀𝑖 dengan: 𝑦𝑖 ∶ error dari selisih data aktual dan data prediksi arima ke-i 𝑎𝑗, 𝑖 ∶ koefisien atau parameter yang harus diestimasi oleh kalman filter, dengan 𝑗 = 0,1,… ,𝑛 − 1 𝑚𝑖 ∶ data ke𝑖 𝜀𝑖 ∶ konstanta gambar 2 merupakan diagram alir dari kalman filter untuk prediksi cuaca. gambar 2. flowchart kalman filter 4. hasil dan pembahasan pada penelitian ini data yang digunakan merupakan data sekunder, yaitu data kelembapan udara, suhu, dan kecepatan angin selama dua tahun terakhir di wilayah kota surabaya pada tahun terakhir sejumlah 735 data. 4.1. arima adapun langkah pertama pada anlisis runtun waktu model arima adalah melihat kestasioneran data seperti pada gambar 3. gambar 3. data kelembapan yang belum stasioner pada unsur kelembapan, suhu dan kecepatan angin data belum stasioner dalam varian maupun dalam mean, sehingga perlu dilakukan differencing. pada penelitian ini proses differencing dilakukan sebanyak satu kali dan estimasi model yang digunakan dalam arima (p,d,q) adalah kombinasi dari differencing 1. berdasarkan identifikasi data tersebut, dapat dilakukan pendugaan terhadap model peramalan, yaitu arima (1,1,1), arima (0,1,1), arima (1,1,0). dari estimasi parameter model arima (1,1,1), arima (0,1,1), arima (1,1,0) untuk variabel kelembapan, suhu dan kecepatan angin di representasikan pada tabel 1. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 64 tabel 1. estimasi parameter model pada variabel kelembapan, suhu dan kecepatan angin v a r model rmse mape mae t k (1,1,1) sig 0.049 3.933 3.840 2.937 11.323(ar) 43.966(ma) sig 0.000(ar) 0.000(ma) (0,1,1) sig. 0.000 4.088 3.982 3.053 13.935(ma) sig. 0.000(ma) (1,1,0) sig. 0.000 4.195 4.080 3.130 -7.754(ar) sig. 0.000(ar) s (1,1,1) sig. 0.285 0.839 2.169 0.621 9.010(ar) 41.367(ma) sig. 0.000(ar) 0.000(ma) (0,1,1) sig. 0.000 0.872 2.228 0.637 19.730(ma) sig. 0.000(ma) (1,1,0) sig. 0.000 0.910 2.221 0.634 -8.349(ar) sig. 0.000(ar) a (1,1,1) sig. 0.000 0.157 28.034 1.191 2.238(ar) 59.558(ma) sig. 0.026(ar) 0.000(ma) (0,1,1) sig. 0.000 1.580 28.273 1.200 58.337(ma) sig. 0.000(ma) (1,1,0) sig. 0.000 1.861 31.715 1.374 -12.408(ar) sig. 0.000(ar) keterangan: var : variabel k : kelembapan s : suhu a : angin dari tabel 1, menunjukkan bahwa untuk signifikansi model tidak menggunakan uji tstudent melainkan dengan melakukan uji signifikansi error. hal ini dikarenakan jumlah data yang terlalu besar. dan untuk uji white noise digunakan dengan menggunakan garis barlet. pada tabel 2 menujukkan hanya variabel suhu dengan parameter arima (1,1,1) yang sudah signifikan dan white noise. akan tetapi untuk variabel kelembapan dan kecepatan angin belum mendapatkan model yang optimal sehingga digunakan metode expert modeler. pada tabel 2, diperoleh model untuk variabel kelembapan dengan parameter arima (0,1,7) dan kecepatan angin dengan parameter arima (2,1,10) adalah yang paling optimal. tabel 2. uji asumsi signifikan dan white noise var model t sig white k (1,1,1) sig 0.049 11.323(ar) 43.966(ma) sig 0.000(ar) 0.000(ma) ya tidak (0,1,1) sig. 0.000 13.935(ma) sig. 0.000(ma) ya tidak (1,1,0) sig. 0.000 -7.754(ar) sig. 0.000(ar) ya tidak (0,1,7) 2.058(ma lag 7) sig. 0.040 ya ya s (1,1,1) sig. 0.285 9.010(ar) 41.367(ma) sig. 0.000(ar) 0.000(ma) ya ya (0,1,1) sig. 0.000 19.730(ma) sig. 0.000(ma) ya tidak (1,1,0) sig. 0.000 -8.349(ar) sig. 0.000(ar) ya tidak a (1,1,1) sig. 0.000 2.238(ar) 59.558(ma) sig. 0.026(ar) 0.000(ma) ya tidak (0,1,1) sig. 0.000 58.337(ma) sig. 0.000(ma) ya tidak (1,1,0) sig. 0.000 -12.408(ar) sig. 0.000(ar) ya tidak (2,1,10) sig. 0.000 -21.543(ar) lag 2 4.001(ma) lag 10 sig. 0.000(ar) 0.000(ma) ya ya tabel 2 menunjukkan bahwa model arima (0,1,7) untuk variabel kelembapan, model arima (1,1,1) untuk variabel suhu, dan model arima (2,1,10) untuk variabel kecepatan angin adalah model arima yang paling optimal, dan didapatkan persamaan model sebagai berikut: jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 65 model arima (0,1,7) 𝑌𝑡 = 0.004 + 𝑌𝑡−1 − 0.434𝑒𝑡−1 − 0.185𝑒𝑡−2 − 0.016𝑒𝑡−3 − 0.114𝑒𝑡−4 − 0.068𝑒𝑡−5 + 0.053𝑒𝑡−6 − 0.075𝑒𝑡−7 model arima (1,1,1) 𝑌𝑡 = 1.401𝑌𝑡−1 − 0.401𝑌𝑡−2 − 0.897𝑒𝑡−1 model arima (2,1,10) 𝑌𝑡 = −0.003 + 1.004𝑌𝑡−1 − 0.004𝑌𝑡−2 + 0.998𝑌𝑡−3 − 0.824𝑒𝑡−1 + 0.946𝑒𝑡−2 − 0.858𝑒𝑡−3 + 0.031𝑒𝑡−4 − 0.192𝑒𝑡−5 + 0.183𝑒𝑡−6 − 0.161𝑒𝑡−7 + 0.094𝑒𝑡−8 − 0.035𝑒𝑡−9 − 0.027𝑒𝑡−10 4.2. kalman filter mape arima (0,1,7) pada prediksi kelembapan sebesar 3.790, arima (1,1,1) pada prediksi suhu sebesar 2.169, dan arima (2,1,10) pada prediksi kecepatan angin sebesar 26.775 untuk 5 hari kedepan. pada penelitian ini estimasi dilakukan untuk menghasilkan peramalan yang lebih mendekati nilai aktual dan simulasi algoritma kalman filter dilakukan dengan menggunakan software matlab r2013a. karena dalam estimasi ini akan mengambil polinomial dengan 𝑛 = 2 sehingga untuk 𝑦𝑖 0 = 𝑎0 + 𝑎1𝑚𝑖 dengan 𝑥(𝑡𝑖) = [ 𝑎0,𝑖 𝑎1,𝑖 ] dan 𝐻𝑖 = [1 𝑚𝑖] dimana 𝑚𝑖 merupakan data ke-i. selain itu juga akan ditentukan beberapa sebagai nilai awal yaitu: a : sebagai matriks sistem. n : sebagai masukkan banyak iterasi yang diinginkan. q : sebagai matriks kovarians. rk = r sebagai matriks kovarians. 𝑎00 : sebagai masukkan nilai awal 𝑎00. 𝑎01 : sebagai masukkan nilai awal 𝑎01. setelah mempunyai nilai awal maka untuk selanjutnya akan mencari nilai dari noise dengan random yang berdistribusi normal. dari menjalankan perangkat lunak yang ditentukan yakni sebanyak n kali maka proses prediksi dan koreksi pada algoritma filter kalman akan berulang atau looping sebanyak n kali. model sistem 𝐵𝑈 ∶ 𝑋𝑘+1 = 𝐴𝑘𝑋𝑘 + 𝐵𝑘𝑈𝑘 + 𝐺𝑘𝑤𝑘 [ 𝑎0,𝑖 𝑎1,𝑖 ] 𝑘+1 = [ 1 0 0 1 ][ 𝑎0,𝑖 𝑎1,𝑖 ] + 𝑤𝑘 model pengukuran 𝐵𝑈 ∶ 𝑧𝑘 = 𝐻𝑘𝑋𝑘 + 𝑣𝑘 𝑦𝑘 0 = [1 𝑚𝑖][ 𝑎0 𝑎1 ] 𝑘 + 𝑣𝑘 pada tahap prediksi �̂�𝑘 − = 𝐴�̂�𝑘−1 + 𝑤𝑘 dengan �̂�0 = [ 79.77 82.1 ] untuk variabel kelembapan, ,�̂�0 = [ 28 27.73 ] untuk variabel suhu, dan �̂�0 = [ 6.62 5.07 ] untuk variabel kecepatan angin, lalu 𝐴 = [ 1 0 0 1 ] dan 𝑤𝑘didapat secara random yang berdistribusi normal. 𝑃𝑘 − = 𝐴𝑃𝑘−1𝐴 𝑇 + 𝑄𝑘 dengan 𝑃0 = [ 1 0 0 1 ] dan 𝑄𝑘 = [ 1 0 0 1 ] dan 𝑄 = 0.1 dan 𝑅 = 0.001 pada tahap koreksi untuk mendapatkan nilai dari kalman gain akan menggunakan hasil perhitungan pada tahap prediksi. 𝐾𝑘 = 𝑃𝑘 −𝐻𝑇(𝐻𝑃𝑘 −𝐻𝑇 + 𝑅)−1 dengan r = 1. dan untuk mendapatkan nilai koreksi dari �̂�𝑘 akan juga menggunakan nilai �̂�𝑘 − yang telah didapatkan pada tahap sebelumnya. �̂�𝑘 = �̂�𝑘 − + 𝐾𝑘(𝑧𝑘 − 𝐻�̂�𝑘 −) dengan 𝑧𝑘 merupakan identik dengan 𝑦𝑘 0 yang di dapatkan dari bias atau selisih antara data dengan hasil forcast ke-k. lalu untuk menghitung nilai kovarians error pada tahap koreksi ini akan menggunakan nilai dari 𝑃𝑘− yang didapatkan dari perhitungan di tahap prediksi. 𝑃𝑘 = (𝐼 − 𝐾𝑘𝐻)𝑃𝑘 − dengan i merupakan matriks identitas. setelah mengestimasi dengan menggunakan polinomial 𝑛 = 2 didapatkan hasil prediksi arima dan arima kalman – filter yang direpresentasikan pada tabel 3 s/d 5 dan grafik hasil prediksi kelembapan, suhu, dan kecepatan angin pada gambar 4 s/d gambar 6. tabel 3. prediksi kelembapan dengan arima dan arima kalman filter kelembapan tanggal data prediksi arima prediksi arima-kf 1-9-2017 64 65.11 64.00 2-9-2017 66 66.29 65.99 3-9-2017 64 66.49 64.00 4-9-2017 70 66.48 69.99 5-9-2017 66 67.15 66.00 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 66 tabel 3. prediksi suhu dengan arima dan arima kalman filter suhu tanggal data prediksi arima prediksi arima-kf 1-9-2017 27.7 28.63 27.701 2-9-2017 27.9 28.48 27.89 3-9-2017 28.8 28.42 28.79 4-9-2017 28.8 28.40 28.79 5-9-2017 29 28.39 28.99 tabel 3. prediksi kecepatan angin dengan arima dan arima kalman filter kecepatan angin tanggal data prediksi arima prediksi arima-kf 1-9-2017 5 4.617 5.0011 2-9-2017 4 4.697 4.0012 3-9-2017 7 5.128 6.9988 4-9-2017 6 5.117 6.0003 5-9-2017 5 4.795 5.0004 gambar 4. hasil simulasi data kelembapan dengan kalman filter dari gambar 4 s/d 6 dapat diamati bahwa kondisi prediksi cuaca menggunakan arimakalman filter paling sesuai untuk mengestimasi nilai polinomial yang berguna untuk memperbaiki hasil prediksi dari arima. selanjutnya, metode yang digunakan untuk mengetahui keakuratan dari prediksi unsur-unsur cuaca meliputi kelembapan, suhu dan kecepatan angin adalah dengan goal error mean absolute percentage error (mape). hasil prediksi menggunakan arima – kalman filter didapatkan mape semakin kecil yang direpresentasikan pada tabel 3. gambar 5. hasil simulasi data suhu dengan kalman filter gambar 6. hasil simulasi data kecepatan angin dengan kalman filter tabel 3. nilai mape variabel cuaca no model variabel nilai mape 1. arima (0,1,7) kelembapan 3.790 2. arima (1,1,1) suhu 2.169 3. arima (2,1,10) k. angin 26.775 4. arima – kf kelembapan 0.000389 5. arima – kf suhu 0.000705 6. arima – kf k. angin 0.0169 5. simpulan dari hasil penelitian prediksi cuaca kota surabaya menggunakan autoregressive integrated moving average (arima) box jenkins dan kalman filter dapat diambil kesimpulan sebagai berikut: a. proses pelatihan data menggunakan arima – kalman filter didapatkan nilai eror goal terkecil (eror mape) masing masing sebesar 0.000389 untuk prediksi kelembapan, 0.000705 untuk prediksi suhu, dan 0.0169 untuk prediksi kecepatan angin. b. penggunaan algoritma kalman filter mempunyai pengaruh baik terhadap perbaikan prediksi cuaca yang meliputi prediksi kelembapan, prediksi suhu dan prediksi kecepatan angin. hal ini dapat jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 67 dilihat dari nilai eror goal mape yang lebih kecil. 6. referensi [1] d. a. adyanti, a. h. asyhar, d. c. r. novitasari, a. lubab and m. hafiyusholeh, "forecast marine weather on java sea using hybrid methods : time series anfis," 2017 4th international conference on electrical engineering, computer science and informatics (eecsi). doi:10.1109/eecsi 2017.8239162, pp. 1-6, 2017. [2] k. abhishek, m. singh, s. ghosh and a. anand, "weather forecasting model using artifical neural network," sciverse sciencedirect, pp. 311-318, 2012. [3] m. k. anshari, s. arifin and a. rahmadiansah, "perancangan prediktor cuaca maritim berbasis logika fuzzy menggunakan user interface android," jurnal teknik pomits, vol. 2, no. 2, p. 2, 2013. [4] a. h. p, s. arifin and a. s. aisyah, "perancangan sistem informasi cuaca maritim untuk para nelayan jawa timur dengan media komunikasi short message service," p. 1, 2010. [5] n. w. habibullah, s. arifin and b. l. widijiantoro, "perancangan sistem prediktor cuaca maritim dengan menggunakan metode fuzzy takagi sugeno," jurnal teknik pomits, vol. 1, no. 1, p. 1, 2012. [6] s. sharma and m. chouhan, "a review : fuzzy time series model for forecasting," international journal of advances in science and technology (ijast), vol. 2, no. 3, p. 1, 2014. [7] d. n. samsiah, "analisis data runtun waktu menggunakan model arima (p,d,q)," skripsi uin sunan kalijaga, yogyakarta, 2008. [8] p. febritasari, "estimasi inflasi wilayah kerja kpwbi malang menggunakan arima-filter kalman dan var-filter kalman," institut teknologi sepuluh nopember surabaya, surabaya, 2016. [9] s. p. endar, "perancangan aplikasi perkiraan cuaca wilayah yogyakarta berbasis android," 2013. [10] a. c. pratama and a. a. s. syamsul, "perancangan model adaptive neuro fuzzy inference system untuk memprediksi cuaca maritim," p. 1, 2008. [11] c. m. regariana, atmosfer (cuaca dan iklim), jakarta, 2009. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 7 pemodelan tingkat okupansi penumpang kereta api dari surabaya dengan metode s-sur (spatial-seeminglyunrelated regression) kuzairi1, anwari2, m. fariz fadillah mardianto3 1program studi matematika, universitas islam madura, kuzairi@fmipa.uim.ac.id 2 program sistem informasi, universitas islam madura, anwari@ft.uim.ac.id 3 program studi statistika, universitas airlangga, m.fariz.fadillah.m@fst.unair.ac.,id doi:https://doi.org/10.15642/mantik.2018.4.1.7-15 abstrak kereta api merupakan sarana transportasi yang terdiri dari kelas ekonomi, bisnis atau ekonomi plus, dan eksekutif. tingkat okupansi dari masing-masing kelas untuk jurusan yang sama juga berbeda. tingkat okupansi penumpang kereta api yang berangkat dari surabaya menarik untuk diteliti karena ruang lingkup asal penumpang lebih luas daripada penumpang kereta di jabodetabek. asal penumpang kereta api di surabaya tidak hanya penumpang yang berasal atau memiliki kepentingan di kota surabaya saja melainkan kabupaten dan kota disekitarnya, sampai pulau madura. dalam penelitian ini dilakukan pemodelan tingkat okupansi penumpang kereta api untuk tiap kelas berdasarkan faktor-faktor yang berpengaruh terhadap tingkat okupansi semua kereta api lintas kota yang berangkat dari stasiun surabaya gubeng, dan pasar turi menggunakan metode spatialseemmingly unrelated regression (s-sur). metode s-sur digunakan karena mampu mengakomodasi efek spasial pada seluruh pengamatan. penelitian ini terdiri atas 12 pengamatan rute tujuan dari surabaya, 8 prediktor, dan 3 respon yang saling berkorelasi spasial berdasarkan pengujian morans i. hasilnya adalah prediktor yang berpengaruh signifikan terhadap tingkat okupansi penumpang kereta api untuk semua kelas merupakan prediktor yang terkait dengan kependudukan yaitu proporsi rata-rata jumlah penduduk, jumlah wisatawan, jumlah tenaga kerja dan jumlah penduduk musiman di sekitar dareah yang disinggahi. kata kunci: s-sur, regresi spasial, tingkat okupansi penumpang, kereta api, surabaya abstrct train is a popular transportation consist of economy, business or economy plus, and executive class. the occupation rate from every class different. the occupation rate from passengers who depart from surabaya is interesting to be explored because the scope of passengers is larger than passengers in jabodetabek. the train passengers in surabaya is not only people who have business in surabaya but also from the nearby region and madura island. in this research, we make the model based on factors that affect the occupation rate for all of the passengers in every training class who depart from surabaya gubeng, and surabaya pasar turi station using spatial-seemingly unrelated regression (s-sur). s-sur is used because can accommodate spatial effect for all of the observation. there are 12 routes that become the observations, 8 predictors, and 3 responses that each all have spatial correlation based on morans i test. the result is predictors that give significant effect is proportion of average from the number of people, the number of tourists, the number of labor, and the number of urban people around the area that be stoped off. keywords : s-sur, spatial regression, the occupation rate of passengers, train, surabaya jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 8 1. pendahuluan salah satu sektor transportasi yang dikelola oleh pemerintah melalui unit badan usaha milik negara (bumn) adalah kereta api. kereta api merupakan sarana transportasi yang populer di pulau jawa, pulau dengan mobilitas penduduk tertinggi di indonesia. transportasi yang dikelola penuh oleh unit bumn pt. kereta api indonesia (pt kai) persero memiliki tiga kelas kereta api antar kota yaitu ekonomi, bisnis, dan eksekutif. masing-masing kelas mempunyai segmentasi penumpang dan kualitas pelayanan serta kecepatan waktu tempuh yang berbeda[1] tingkat keterisian tempat duduk atau okupansi dari masing-masing kelas untuk jurusan yang sama juga berbeda. tingkat okupansi penumpang kereta api ekonomi belum tentu paling tinggi dibandingkan kereta api kelas bisnis, dan eksekutif. hal tersebut dapat terjadi sebaliknya. dalam penelitian ini penulis ingin mengkaji faktor-faktor eksternal yang berpengaruh terhadap tingkat okupansi penumpang kereta api kelas ekonomi, bisnis, dan eksekutif untuk semua kereta api lintas kota yang berangkat dari stasiun gubeng, dan pasar turi surabaya. surabaya sebagai kota terbesar kedua di indonesia memiliki tingkat mobilitas penduduk yang tinggi. sedikitnya terdapat 35 tujuan yang dilayani oleh kereta api kelas ekonomi, bisnis, dan eksekutif dari dua stasiun keberangkatan di surabaya, dengan jumlah lebih dari 40 rangkaian kereta [7]. tingkat okupansi penumpang kereta api yang berangkat dari surabaya menarik untuk diteliti karena cakupan asal penumpang lebih luas daripada penumpang kereta di jabodetabek. asal penumpang kereta api di surabaya tidak hanya penumpang yang berasal atau memiliki kepentingan di kota surabaya saja melainkan kabupaten dan kota di sekitarnya seperti sidoarjo, gresik, kabupaten dan kota mojokerto, lamongan, sampai pulau madura yang terdiri atas empat kabupaten. beberapa penelitian tentang perkeretaapian di surabaya pernah dilakukan diantaranya mukminin dan zainul [10] menganalisis kepuasan penumpang terhadap fasilitas dan layanan pt kai daops xiii surabaya [10]. penelitian-penelitian [6], [13], [14]dan penelitian lain hanya sebatas meramalkan jumlah penumpang di salah satu rangkaian kereta api saja. sampai saat ini belum ada penelitian yang menganalisis dan memodelkan faktor-faktor yang berpengaruh terhadap tingkat okupansi penumpang kereta api dalam lingkup yang lebih besar. metode yang tepat untuk menentukan dan memodelkan faktor-faktor yang berpengaruh terhadap tingkat okupansi penumpang kereta api kelas ekonomi, bisnis, dan eksekutif untuk semua kereta api lintas kota yang berangkat dari stasiun gubeng, dan pasar turi di surabaya adalah spatial-seemmingly unrelated regression (s-sur) yang merupakan model sur berbasis spasial.berdasarkan [9] keunggulan ananlisis data spasial adalah mengakomodasi informasi lokasi dan keterhubungan antar wilayah [9].dalam hal ini terdapat jaringan jalan berupa rel yang terhubung sehingga membuat jumlah kereta api yang melintas di wilayah yang berdekatan saling berpengaruh satu sama lain. tujuan dari peneltian ini adalah mendapatkan model s-sur yang terdiri dari faktor pengaruh perkembangan tingkat okupansi penumpang kereta api ekonomi, bisnis, dan eksekutif dari surabaya untuk perencanaan. 2. tinjauan pustaka berikut diberikan penjelasan singkat terkait konsep fundamental yang digunakan dalam menerapkan metode s-sur untuk faktor pengaruh tingkat okupansi penumpang kereta api di surabaya. 2.1. seemingly unrelated regression seemingly unrelated regression (sur) merupakan sebuah pengembangan dari model regresi linear multivariat yang terdiri dari beberapa persamaan regresi, dimana setiap persamaan memiliki variabel respon yang berbeda dan dimungkinkan memiliki himpunan variabel prediktor yang berbeda-beda pula [11]. secara umum model sur untuk m buah persamaan dimana masing-masing persamaan jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 9 terdiri dari pj variabel prediktor dapat ditulis sebagai berikut: m ipm im pm imm immm i jipjijpjijjijjji ipipiiii ipipiiii mm jj xxxy xxxy xxxy xxxy     +++++= +++++= +++++= +++++= ,2,21,10 ,2,21,10 2,2,22,2221,221202 1,1,12,1121,111101 22 11       (1) dimana 𝑖 = 1, 2, . . . , 𝑛 dan 𝑗 = 1, 2, … , 𝑚 persamaan (1) jika ditulis dalam bentuk matriksdengan bentuk sebagai berikut: persamaan (2) secara umum dapat ditulis sebagai berikut: y = x  +  y merupakan vektor berukuran 𝑚𝑛 × 1, x merupakan matriks ukuran 𝑚𝑛 × ∑ 𝑝𝑗 𝑚 𝑗=1 , β adalah vektor parameter berukuran ∑ 𝑝𝑗 × 𝑚 𝑗=1 1, dan merupakan vektor error berukuran 𝑚𝑛 × 1. asumsi yang harus dipenuhi pada model sur adalah sebagai berikut: a. 0ε =)( je untuk 𝑗 = 1, 2, . . . , 𝑚 b.      =  = kjuntuk kjuntuk e jk k t j 0 i εε  )( dengan 𝑗 = 1, 2, . . . , 𝑚 dan 𝑘 = 1, 2, … , 𝑚 c. xj ,untuk𝑗 = 1, 2, . . . , 𝑚 merupakan fixedvariable. 2.2. spatial-seemingly unrelated regression pemodelan spatial-seemingly unrelated regression (s-sur) pada dasarnya memiliki kesamaan spesifikasi dengan model sur yang ditambahkan efek spasial pada setiap persamaanya [11]. istilah s-sur diperkenalkan pertama kali oleh [12] dengan mengacu pada kasus model space-time.. karakteristik dari pendekatan ini adalah adanya heterogenitas yang terbatas, sehingga koefisien regresi diasumsikan sama untuk setiap individu.estimasi parameter s-sur dilakukan berdasarkan hasil dari metode weighted least square (wls) dengan pembobot spasial atau berdasarkan metodel maximum likelihood estimator (mle) dengan pembobot spasial. prosedur yang dilakukan terkait analisis data dengan menggunakan s-sur adalah a. penentuan bobot spasial penentuan bobot spasial atau w yang berguna untuk pengujian dependensi spasial, dan estimasi spasial untuk parameter s-sur. ada banyak jenis pembobot spasial. dalam hal ini pembobot spasial yang digunakan adalah pembobot customize. pembobot ini merupakan pembobot yang disusun tidak hanya memperhatikan faktor persinggungan antar wilayah tetapi juga mempertimbangkan faktor kedekatan ekonomi, transportasi, sosial, infrastruktur, ataupun faktor lainnya. b. dependensi spasial dependensi spasial didefinisikan sebagai adanya hubungan fungsional antara apa yang terjadi pada satu titik dalam ruang dan apa yang terjadi di tempat lain [2]. besarnya dependensi spasial, dapat dilihat dengan indeks morans i yang dirumuskan sebagai berikut: εε wεε t t i = (3) untuk matriks pembobot yang belum distandartkan, indeks morans i didapatkan dengan mengalikan persamaan (3) dengan n/s, n merupakan banyak pengamatan, s adalah faktor standardisasi yang merupakan jumlahan dari seluruh elemen matriks pembobot yang belum distandartkan. untuk melihat apakah besarnya dependensi spasial (ij) signifikan pada data, dilakukan dengan pengujian pada indeks morans i dengan hipotesis sebagai berikut: h0 : ij = 0 (tidak terdapat dependensi spasial)             +                         =             m 2 1 m 2 1 m 2 m 2 1 ε ε ε β β β x00 0x0 00x y y y       1 (2) jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 10 h1 : ij ≠ 0 (terdapat dependensi spasial) statistik uji yang digunakan pada pengujian signifikansi morans i adalah sebagai berikut:   2/1)var(/)( jjj iieiz −= (4) h0 ditolak jika z > zα/2. nilai dari indeks morans i besarnya antara -1 sampai 1. jika ij> e(ij) maka data memiliki autokorelasi positif dan jika ij< e(ij) maka data memiliki autokorelasi negatif. c. heterogenitas spasial adanya heterogenitas spasial pada data, dapat dilihat dengan melakukan uji breushpagan dengan hipotesis sebagai berikut: h0: 22 2 2 1 n  ===  (homoskedastisitas) h1: paling tidak ada satu 22 ji   (heteroskedastisitas) statistik uji yang digunakan pada uji breushpagan (bp) menurut anselin (2016) adalah sebagai berikut: bp = (1/2) ft z (zt z)-1 zt f (5) nilai statistik uji bp asimtotik dengan distribusi 2 )1( +jp  , sehingga h0 ditolak jika bp > 2 )1( +jp  atau p-value kurang dari α. untuk kasus adanya dependensi spasial maka statistik uji breush-pagan pada persamaan (5) ditambahkan dengan ((εt w ε)/σ2)2 /tr(wt w+w2). 3. metode penelitian tahapan pemodelan tingkat okupansi penumpang kereta api dari surabaya dengan menggunakan pendekatan s-sur adalah sebagai berikut: 1. melakukan analisis statistika deskriptif untuk mengetahui karakteristik data pada masing – masing persoalan atau variabel. 2. menentukan pembobot spasial. 3. melakukan pengujian dependensi spasial 4. melakukan pengujian heterogenitas spasial. 5. menentukan nilai estimator untuk parameter s-sur, dan melakukan pengujian signifikansi secara individu. 6. menentukan nilai parameter s-sur baru berdasarkan prediktor yang signifikan. 7. menentukan ukuran kebaikan model. adapun variabel yang digunakan dalam penelitian ini disajikan pada tabel 1. tabel 1. variabel penelitian simbol variabel satuan y1 tingkat okupansi penumpang kelas ekonomi dari surabaya persen dari kapasitas semua gerbong ekonomi y2 tingkat okupansi penumpang kelas bisnis dari surabaya persen dari kapasitas semua gerbong bisnis y3 tingkat okupansi penumpang kelas eksekutif dari surabaya persen dari kapasitas semua gerbong eksekutif x1 rata-rata jumlah penduduk di sekitar daerah yang disinggahi persen dari penduduk indonesia x2 rata-rata jumlah wisatawan domestik di sekitar daerah yang disinggahi persen dari wisatawan domestik di indonesia x3 rata-rata jumlah tenaga kerja di sekitar daerah yang disinggahi persen dari jumlah tenaga kerja di indonesia x4 rata-rata jumlah penduduk musiman di sekitar daerah yang disinggahi persen dari penduduk indonesia x5 rata-rata rasio kepemilikan kendaraan bermotor di sekitar daerah yang disinggahi persen dari rasio kepemilikan kendaraan bermotor di indonesia x6 jumlah rangkaian kereta api yang melayani dalam semua kelas persen dari keseluruhan rangkaian kereta api yang ada di indonesia x7 jumlah total tiket yang disediakan untuk semua kelas persen dari kuota tiket tiap rangkaian x8 jumlah armada bus rute sejenis yang melayani dari surabaya persen dari seluruh armada bus antar kota yang mempunyai trayek di terminal purabaya jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 11 4. pembahasan berikut merupakan pembahasan yang berisi hasil dari penelitian ini. 4.1. statistika deskriptif statistika deskriptif digunakan untuk mengetahui karakteristik dari data yang akan dianalisis dengan menggunakan metode statistika lebih lanjut. analisis statistika deskriptif dilakukan untuk masing – masing variabel respon. ukuran statistika deskriptif yang digunakan yaitu ukuran nilai minimum, maksimum, dan rata-rata berdasarkan proporsi. gambar 1. rata-rata proporsi tingkat okupansi kereta api ekonomi dari surabaya gambar 2. rata-rata proporsi tingkat okupansi kereta api bisnis dari surabaya dalam penelitian ini terdapat 12 pengamatan yaitu kota – kota yang menjadi tempat tujuan kereta api dari surabaya. kota – kota tersebut adalah jakarta, semarang, bandung, solo, yogyakarta, purwokerto, cirebon, malang, madiun, jember, bojonegoro, dan banyuwangi. kota – kota tersebut dilalui kereta api untuk kelas ekonomi, bisnis, dan eksekutif dari surabaya. kota – kota tersebut merupakan kota penting di pulau jawa sebagai pusat mobilitas dan ekonomi tingkat regional sampai nasional. selain kereta api, terdapat transportasi lain seperti bus yang berangkat dari surabaya setiap hari. gambar 1 menyajikan rata-rata proporsi tingkat okupansi kereta api ekonomi dari surabaya. berdasarkan gambar 1, penumpang yang berangkat ke malang dari surabaya dengan kereta api ekonomi memiliki rata-rata proporsi tertinggi. penumpang yang berangkat ke jember dari surabaya dengan kereta api ekonomi memiliki rata-rata proporsi terendah. gambar 2 menyajikan rata-rata proporsi tingkat okupansi kereta api bisnis dari surabaya. gambar 2 menyajikan rata-rata proporsi tingkat okupansi kereta api bisnis dari surabaya. berdasarkan gambar 2, penumpang yang berangkat ke yogyakarta dan banyuwangi dari surabaya dengan kereta api bisnis memiliki rata-rata proporsi tertinggi pertama dan kedua dengan selisih tidak terlalu jauh. penumpang yang berangkat ke bojonegoro dari surabaya dengan kereta api bisnis memiliki rata-rata proporsi terendah. gambar 3 menyajikan rata-rata proporsi tingkat okupansi kereta api eksekutif dari surabaya. berdasarkan gambar 2, penumpang yang berangkat ke bandung dan jakarta dari surabaya dengan kereta api eksekutif memiliki rata-rata proporsi tertinggi pertama dan kedua dengan selisih tidak terlalu jauh. penumpang yang berangkat ke bojonegoro dari surabaya dengan kereta api eksekutif memiliki rata-rata proporsi terendah. 4,67 1,81 2,78 2,26 4,65 4,05 3,29 5,2 4,32 8,03 1,15 2,43 0 1 2 3 4 5 6 7 8 9 ja ka rt a b a n d u n g c ir e b o n p u rw o ke rt o s e m a ra n g y o g ya ka rt a s o lo b o jo n e g o ro m a d iu n m a la n g je m b e r b a n yu w a n g i 3,44 9,41 10,25 1,07 5,8 15,76 12,33 0,59 5,27 3,87 11,38 14,07 0 2 4 6 8 10 12 14 16 18 ja ka rt a b a n d u n g c ir e b o n p u rw o ke rt o s e m a ra n g y o g ya ka rt a s o lo b o jo n e g o ro m a d iu n m a la n g je m b e r b a n yu w a n g i jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 12 gambar 3. rata-rata proporsi tingkat okupansi kereta api eksekutif dari surabaya berdasarkan analisis tersebut, tampak bahwa untuk kereta api kelas ekonomi memiliki segmentasi penumpang jarak dekat. kereta api kelas bisnis memiliki segmentasi penumpang jarak menengah, dan kereta api kelas eksekutif memiliki segmentasi penumpang jarak jauh. ketiga respon saling berkorelasi berdasarkan hasil pengujian bartlett sphericity sebagai berikut hipotesis : 𝐻0: antar variabel respon tak berkorelasi 𝐻1: antar variabel respon berkorelasi statistik uji : 𝜒ℎ𝑖𝑡𝑢𝑛𝑔 2 = − {𝑛 − 1 − 2𝑞+5 6 } 𝑙𝑛|𝐑| (6) dengan matriks korelasi r=( 1 0,4523 0,2667 0,4523 1 0,5823 0,2667 0,5823 1 ) (7) daerah penolakan : jika 𝜒ℎ𝑖𝑡𝑢𝑛𝑔 2 > 𝜒𝑡𝑎𝑏𝑒𝑙 2 = 𝜒 𝛼, 1 2 𝑞(𝑞−1) 2 maka tolak 𝐻0 sehingga antar variabel respon berkorelasi. berdasarkan persamaan (6) dengan 𝑛 = 12, 𝑞 = 3 dan r dari (7), diperolah 𝜒ℎ𝑖𝑡𝑢𝑛𝑔 2 =8,894>𝜒𝑡𝑎𝑏𝑒𝑙 2 = 7,815, maka tolak 𝐻0 sehingga antar variabel respon berkorelasi. 4.2. pemodelan s-sur berikut hasil analisis data dengan menggunakan s-sur. 4.2.1. penentuan pembobot spasial pemobobot spasial yang digunakan dalam penelitian ini adalah pembobot customize. pembobot customizememiliki kelebihan dibandingkan pembobot lainnya. pembobot ini merupakan pembobot yang disusun tidak hanya memperhatikan faktor persinggungan antar wilayah tetapi juga mempertimbangkan faktor kedekatan hubungan. antar wilayah memiliki kedekatan hubungan tingkat okupansi penumpang kereta api. misalkan jika pada hari libur, tingkat okupansi penumpang kereta api ke suatu kota tinggi, maka tingkat okupansi penumpang ke kota lain juga tinggi. untuk kasus tersebut diberikan nilai pembobot 1. secara lengkap, bobot diberikan dalam tabel 2 berikut: tabel 2.pembobotcustomized w 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 4.2.2. pengujian dependensi spasial pengujian morans i dilakukan untuk menentukan dependensi spasial antar wilayah. persamaan (3) digunakan untuk menentukan nilai-nilai kolom 2 pada tabel 3. persamaan (4) digunakan untuk menentukan nilai-nilai kolom 3 pada tabel 3. tabel 3 disajikan sebagai berikut: dengan 𝛼 = 0,05, maka 𝑍𝛼/2 = 1,96, dan𝐼0 = 2,156. terlihat bahwa seluruh variabel dependen maupun independen menghasilkan 10,8711,14 8,75 3,05 7,57 9,02 8,31 0,32 4,584,46 5,12 6,59 0 2 4 6 8 10 12 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 13 nilai 𝑍(𝐼) < 𝑍𝛼/2 yang artinya gagal tolak 𝐻0. ini mengindikasikan bahwa tidak ada dependensi secara spasial antar daerah. namun demikian semua nilai morans i bernilai lebih besar dari 𝐼0 yang artinya semua variabel dependen maupun independen mempunyai nilai yang mirip dan cenderung dapat membentuk kelompok. tabel 3. hasil perhitungan morans i variabel morans i z(i) y1 0,127 2,748 y2 0,144 2,623 y3 0,190 2,879 x1 0,087 2,923 x2 0,161 2,417 x3 0,176 2,633 x4 0,029 3,632 x5 0,123 2,805 x6 0,055 2,231 x7 0,125 2,677 x8 0,178 2,505 4.2.3. pengujuan heterogenitas spasial adanya heterogenitas spasial pada data, dapat dilihat dengan melakukan uji breushpagan dengan hipotesis sebagai berikut: h0: 2 12 2 2 2 1  ===  (homoskedastisitas) h1: paling tidak ada satu 22 ji   (heteroskedastisitas) statistik uji yang digunakan pada uji breush-pagan(bp) menggunakan persamaan (5). secara komputasi dihasilkan nilai bp test = 11,564 dan nilai p-value = 0,172 kurang dari α = 0,05 sehingga h0gagalditolak. jadi terdapat homoskedastisitas secara spasial. 4.2.4. hasil estimasi parameter s-sur dari hasil perhitungan secara komputasi, tabel 4 menyajikan hasil estimasi untuk parameter model s-sur beserta ukuran yang digunakan untuk pengujian signifikansi parameter. tabel 4. hasil estimasi dan signifikansi parameter models-sur respon prediktor nilai para meter pvalue signifikan/ tidak y1 intercept 1,7717 0,0492 signifikan x1 10,3383 0,0463 signifikan x2 1,5015 0,0619 signifikan x3 2,2742 0,0748 signifikan x4 6,6751 0,0704 signifikan x5 -15,2313 0,5312 tidak x6 7,3282 0,7166 tidak x7 -4,5746 0,9471 tidak x8 2,5606 0,8896 tidak y2 intercept -0,8099 0,0797 signifikan x1 28,2835 0,0153 signifikan x2 9,7531 0.0500 signifikan x3 10,0560 0,0295 signifikan x4 47,5345 0.0823 signifikan x5 -64,1814 0,0841 signifikan x6 -3,6481 0,8852 tidak x7 92,9923 0,3189 tidak x8 13,2154 0,5778 tidak y3 intercept -0,3754 0,0644 signifikan x1 15,8025 0,0181 signifikan x2 3,1951 0,0233 signifikan x3 6,1462 0,0444 signifikan x4 13,2939 0,0643 signifikan x5 -27,3302 0,0187 signifikan x6 -2,8822 0,6578 tidak x7 75,1773 0,0225 signifikan x8 4,8883 0,4288 tidak nilai log likelihood 9,4062 nilai r2 0,8541 nilai aic 36,8124 keputusan signifikan atau tidak diperoleh dengan membandingkan nilai p-value dan taraf signifikansi α = 0,1. jika p-value < 0,1, maka nilai parameter signifikan. berdasarkan hasil tersebut nilai parameter yang semuanya signifikan diperoleh untuk parameter prediktor x1, x2, x3 dan x4. sama dengan konsep regresi linear, dilakukan analisis s-sur untuk mendapatkan model baru dengan menggunakan prediktor yang nilai parameternya signifikan. prediktor yang nilai parameternya tidak signifikan tidak dilibatkan dalam pembentukan model baru. 4.2.5. hasil akhir estimasi s-sur estimasi parameter pada tahap ini hanya dilakukan dengan melibatkan prediktor – prediktor yang nilai parameter pada tabel 4 signifikan untuk semua respon. dari hasil perhitungan secara komputasi, tabel 5 jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 14 menyajikan hasil estimasi untuk parameter model s-sur baru beserta ukuran yang digunakan untuk pengujian signifikansi parameter. tabel 5. hasil estimasi dan signifikansi parameter model s-sur baru var. respon prediktor nilai parameter thitung pvalue kesimpulan (α=5%) parameter spasial y1 intercept 3,3682 3,1933 0,0015 signifikan x1 0,4283 -1,007 0,0351 signifikan 0,0131 x2 0,1545 0,1845 0,0459 signifikan 0,0093 x3 0,8743 0,5379 0,0316 signifikan 0,0047 x4 0,5144 0,7927 0,0453 signifikan 0,0035 y2 intercept 4,9542 2,1875 0,0466 signifikan x1 0,9119 -0,534 0,0216 signifikan 0,0107 x2 0,9031 2,1601 0,0368 signifikan 0,0066 x3 0,1364 0,0468 0,0179 signifikan 0,0023 x4 0,0713 0,2295 0,0238 signifikan 0,0058 y3 intercept 3,3467 3,3725 0,0120 signifikan x1 0,9694 0,4932 0,0406 signifikan 0,0115 x2 0,1111 1,4199 0,0202 signifikan 0,0087 x3 0,3514 2,0357 0,0028 signifikan 0,0064 x4 0,7025 1,1589 0,0286 signifikan 0,0038 nilai log likelihood 14,673 nilai r2 0,8776 nilai aic 11,6277 tabel 5 memperlihatkan bawa semua nilai parameter untuk prediktor yang dilibatkan signifikan berdasarkan indikator yang digunakan yaitu nilai p-value < α = 0,1. selain itu dilihat dari ukuran kebaikannya seperti r2 dan aic, estimasi s-sur yang baru lebih baik. 5. kesimpulan pemodelan tingkat okupansi penumpang kereta api untuk tiap kelas berdasarkan faktorfaktor yang berpengaruh terhadap tingkat okupansi semua kereta api lintas kota yang berangkat dari stasiun surabaya gubeng, dan pasar turi menggunakan metode spatialseemmingly unrelated regression (s-sur). hasilnya adalah prediktor yang berpengaruh signifikan terhadap tingkat okupansi penumpang kereta api untuk semua kelas merupakan prediktor yang terkait dengan kependudukan yaitu proporsi rata-rata jumlah penduduk, jumlah wisatawan, jumlah tenaga kerja dan jumlah penduduk musiman di sekitar dareah yang disinggahi. referensi [1] rozi, f. (2011), analisis pengaruh kualitas pelayanan pt kai terhadap pengaruh konsumen. tugas akhir uin malang, malang. [2] anselin, l. (2016), “estimation and testing in the spatial seemingly unrelated regression model”, geoda center, vol. 1, hal. 1-13. [3] badan pusat statistik. (2015), statistik transportasi jawa timur 2015. bps jawa timur, surabaya. [4] baltagi, b.h., dan pirotte, a. (2010), “seemingly unrelated regressions with spatial error components”, policy research paper, vol. 125, hal. 1-22. [5] biana, l. (2010), “manajemen strategi pt kai”, http://lellykelilingdunia. blogspot.com/2010/01/manajemen-strategi-ptkereta-api.html, diakses tang-gal 9 september 2015. [6] ditago, a.p. (2011), analisis peramalan jumlah penumpang kereta api penataran tujuan surabaya-malang. tugas akhir its, surabaya. [7] kereta api indonesia. (2015), jadwal perjalanan kereta api 2015. pt kereta api indonesia (persero), bandung. [8] kereta api indonesia. (2015), laporan tahunan pt.kai persero 2015. pt kereta api indonesia (persero), jakarta. [9] liu, y. dan jarrett, d. (2007), “spatial statistical modeling of traffic accidents”, makalah, _________ [10] mukminin, dan zainul, e. (2013), analisis kepuasan penumpang terhadap fasilitas dan pelayanan pt. kereta api indonesia: studi pada pt. kereta api indonesia daops xiii surabaya. tesis uin sunan ampel, surabaya. [11] mur, j., dan f. lópez. (2010), spatial sur models: specification, testing and selection. research projectuniversity of zaragoza, zaragoza. jurnal matematika “mantik” edisi: mei 2018. vol. 04 no. 01 issn: 2527-3159 e-issn: 2527-3167 15 [12] anselin, l. (1988), “a test for spatial autocorrelation in seemingly unrelated regressions”, economics letters, vol. 28, hal. 335-341. [13] widodo, e. (2015), peramalan jumlah penumpang kereta api trayek surabayamalang-blitar dan blitar-malang-surabaya dii dapos viii surabaya. tugas akhir its, surabaya. [14] wulandari, r. (2012), peramalan jumlah permintaan tiket kereta api di stasiun gubeng surabaya dengan metode boxjenkins. tugas akhir its, surabaya. [15] wang, x. dan kockelman, k., (2007), “specification and estimation of a spatially and temporally autocor-related seemingly unrelated regres-sion model: application to crash rates in china”, journal of transportation, vol.34, hal. 281-300. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 122 peramalan kebutuhan daya listrik menggunakan model arima dan fungsi transfer (studi kasus: pt. pln (persero) area sumbawa) mikhratunnisa1, tri susilawati2 universitas teknologi sumbawa1, mikhratunnisa@uts.ac.id1 universitas teknologi sumbawa2, tri.susilawati@uts.ac.id2 doi:https://doi.org/10.15642/mantik.2018.4.1.122-127 abstrak energi adalah salah satu kebutuhan dasar manusia. salah satu sumber energi yang paling vital adalah energi listrik. kebutuhan listrik di nusa tenggara barat (ntb) terus meningkat dari tahun ke tahun seiring dengan perkembangan penduduk dan perekonomian ntb, khususnya di kabupaten sumbawa. untuk memenuhi kebutuhan pelanggan akan energi listrik, maka dibutuhkan penyediaan dan penyaluran kebutuhan energi listrik yang memadai. oleh karena itu, diperlukan suatu cara yang tepat dalam penyesuain jumlah kapasitas listrik agar sesuai dengan permintaan pelanggan. salah satu cara yang dapat dilakukan adalah meramalkan/memprediksi kebutuhan daya listrik. peramalan dapat dilakukan dengan menggunakan model arima dan fungsi transfer. hasil penelitian menunjukkan bahwa dengan menggunakan model arima diperkirakan kebutuhan daya listrik pada tahun 2018 mengalami peningkatan sebesar 18,21% dari satu tahun sebelumnya, sedangkan menggunakan model fungsi transfer diperkirakan mengalami peningkatan sebesar 18,18% dari satu tahun sebelumnya. kata kunci: daya listrik, arima, fungsi transfer abstract energy is one of the basic need of human being. one of the vital energy is electricity. the need of electricity in ntb is increase along with the citizen economic development in ntb especially in sumbawa regency. therefore, there is a need for the right way in adjusting the amount of electrical capacity to match customer demand. one way that can be done is to forecast/ predict the need for electricity. the forecast can be used by using the arima and transfer function models. the results of the study show that using the arima model is estimated to require electricity in 2018 experienced an increase of 18,21% from the previous year, while using the transfer function model is estimated to increase by 18,18% from the previous year. key words: electricity power, arima, transfer function 1. pendahuluan energi adalah salah satu kebutuhan dasar manusia. salah satu sumber energi yang paling vital adalah energi listrik. listrik adalah jenis energi yang menopang kelangsungan hidup manusia, khususnya di sektor industri [1]. energi listrik merupakan salah satu kebutuhan masyarakat yang sangat penting dan sebagai sumber daya ekonomis yang paling utama yang dibutuhkan dalam berbagai kegiatan. dalam waktu yang akan datang kebutuhan listrik akan terus meningkat seiring dengan adanya peningkatan dan perkembangan baik dari jumlah penduduk, jumlah investasi, dan perkembangan teknologi [2]. kabupaten sumbawa adalah salah satu kabupaten/kota di provinsi nusa tenggara barat (ntb) yang terletak di pulau sumbawa. kebutuhan listrik di nusa tenggara barat (ntb) terus meningkat dari tahun ke tahun seiring dengan perkembangan penduduk dan jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 123 perekonomian ntb, khususnya di kabupaten sumbawa [3]. untuk memenuhi kebutuhan pelanggan akan energi listrik, maka dibutuhkan penyediaan dan penyaluran energi listrik yang memadai baik dari segi teknis maupun ekonomisnya. oleh karena itu, diperlukan suatu cara yang tepat dalam penyesuaian jumlah kapasitas listrik agar sesuai dengan permintaan pelanggan. salah satu cara yang dapat dilakukan adalah meramalkan/memprediksi kebutuhan daya listrik, yakni dengan menggunakan model arima dan model fungsi transfer. model arima merupakan salah satu model time series yang populer dan banyak digunakan. model arima adalah model time series univariat, sedangkan fungsi transfer merupakan model time series multivariat. model arima hanya menghubungkan time series berdasarkan nilainilai sebelumnya dari time series tersebut. pada model fungsi transfer, selain menghubungkan suatu time series dengan nilai-nilai sebelumnya dari time series tersebut, juga menghubungkan suatu time series dengan time series lainnya [4]. sehingga dengan penelitian ini diharapkan mendapatkan model yang sesuai untuk meramalkan/memprediksi kebutuhan daya listrik pada periode yang akan datang. 2. tinjauan pustaka 2.1 model arima model arima (p,d,q) merupakan gabungan dari model ar (p) dan ma (q), dengan pola data non stasioner dan differencing orde d. bentuk model arima (p,d,q) sebagai berikut [5]: ( )( ) ( ) tqt d p abybb  =−1 (1) dimana p adalah orde dari model ar, q adalah orde dari model ma, d adalah orde differencing, dan ( ) ( ) ( ) ( )qqq p pp bbbb bbbb   −−−−= −−−−= ...1 ...1 2 21 2 21 generalisasi model arma untuk pola data musiman ditulis sebagai arima (p,d,q) (p,d,q)s, yaitu ( ) ( )( ) ( ) ( ) ( ) tsqqt dsds pp abbybbbb 11 =−−  dimana s adalah periode musiman, ( ) ( )pspsssp bbbb −−−−= ...1 221 ( ) ( )qsqsssq bbbb −−−−= ...1 221 model arima memiliki beberapa asumsi residual yang harus dipenuhi. pengujian asumsi sering disebut diagnostic checking. asumsi tersebut adalah residual bersifat white noise dan berdistribusi normal. model arima yang telah memenuhi asumsi tersebut diklasifikasikan sebagai model yang baik [5]. karakteristik model ar, ma, dan arma berdasarkan plot acf dan pacf dari data yang telah stasioner sebagai berikut. tabel 1. karakteristik acf dan pacf untuk proses stasioner proses acf pacf ar (p) dies down secara eksponensial atau sinusoidal cut off pada lag ke-p ma (q) cut off pada lag ke-q dies down secara eksponensial atau sinusoidal arma (p, q) dies down secara eksponensial atau sinusoidal dies down secara eksponensial atau sinusoidal 2.2 model fungsi transfer bentuk umum model fungsi transfer single input (model fungsi transfer dengan satu variabel input) sebagai berikut [6]. ( ) ttt nxbvcy ++= (2) dimana ( ) ( ) ( ) b r s b b b bv   = ( ) ( ) tt a b b n   = ( ) sss bbbb  ++++= ... 2 210 ( ) rrr bbbb  ++++= ...1 2 21 ( ) qq bbbb  −−−−= ...1 2 11 ( ) pp bbbb  −−−−= ...1 2 11 sehingga model fungsi transfer single input ditulis sebagai berikut ( ) ( ) ( ) ( ) tt b r s t a b b xb b b cy     ++= (3) jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 124 3. metode penelitian adapun tahapan peramalan menggunakan model arima yaitu identifikasi model arima, estimasi dan pengujian signifikasi parameter model arima, diagnostic checking, dan peramalan. adapun tahapan peramalan menggunakan model fungsi transfer sebagai berikut: a. persiapan deret input dan output b. prewhitening deret input dan output c. perhitungan korelasi silang deret input dan output yang telah melalui proses prewhitening d. penentuan orde b, r, s model fungsi transfer e. perhitungan bobot respon impuls f. penentuan model deret noise g. estimasi parameter model fungsi transfer h. diagnostic checking model fungsi transfer i. peramalan data yang digunakan dalam penelitian ini adalah data sekunder yang diperoleh dari pt. pln (persero) area sumbawa. data yang digunakan dalam penelitian ini adalah data penggunaan daya listrik dan jumlah pelanggan pln bulan november 2012 – desember 2017. 4. hasil dan pembahasan data yang digunakan dalam peramalan menggunakan model arima adalah data penggunaan daya listrik. tahap awal pada identifikasi model arima yaitu pemeriksaan stasioneritas data baik dalam mean maupun varians. data penggunaan daya listrik bulan november 2012 – desember 2017 diperlihatkan pada plot time series berikut. 60544842363024181261 170000000 160000000 150000000 140000000 130000000 120000000 110000000 100000000 90000000 index y t time series plot of yt gambar 1. plot time series daya listrik gambar 1 menujukkan bahwa data belum stasioner dalam mean karena adanya pola trend, sehingga diperlukan proses differencing. sebelum dilakukan proses differencing, perlu dilakukan pemeriksaan stasioneritas data dalam varian. 5,02,50,0-2,5-5,0 1400000 1300000 1200000 1100000 1000000 lambda s t d e v lower c l upper c l limit estimate -0,21 lower c l -1,64 upper c l 1,26 rounded value 0,00 (using 95.0% confidence) lambda box-cox plot of yt gambar 2. transformasi box-cox data daya listrik hasil pemeriksaan stasioneritas data dalam varians menggunakan transformasi box-cox (gambar 2) menunjukkan bahwa data belum stasioner dalam varians, hal ini dapat dilihat pada rounded value sebesar 0,00. sehingga perlu dilakukan transformasi agar data menjadi stasioner dalam varians. transformasi yang sesuai adalah transformasi ln. selanjutnya dilakukan proses differencing agar data stasioner dalam mean. proses differencing yang sesuai adalah differencing orde-2. berikut plot time series setelah transformasi dan differencing. 60544842363024181261 0,025 0,020 0,015 0,010 0,005 0,000 -0,005 -0,010 -0,015 index y t 2 time series plot of yt-2 gambar 3. plot time series daya listrik (d=2) gambar 3 menunjukkan bahwa data telah stasioner dalam mean. pengujian stasioneritas dalam mean dapat dilakukan dengan menggunakan uji augmented dickey-fuller unit root. hasil pengujian sebagai berikut. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 125 gambar 4. uji augmented dickey-fuller unit root gambar 4 menunjukkan bahwa data telah stasioner dalam mean, karena memiliki p-value <  ( )%5= . selanjutnya adalah membentukan model arima berdasarkan plot acf dan pacf. gambar 5. plot acf dan pacf daya listrik (d=2) berdasarkan gambar 5 diduga beberapa model arima yang terbentuk yaitu arima (1,2,0), arima (0,2,1) dan arima (1,2,1). kemudian dilakukan estimasi parameter dan pengujian signifikansi parameter dari model tersebut. tabel 2. hasil estimasi parameter dan pengujian signifikansi parameter model model parameter estimasi p-value arima (1,2,0) 1  -0,524 0,000 arima (0,2,1) 1  0,699 0,000 arima (1,2,1) 1 1   -0,149 0,627 0,534 0,004 %5= berdasarkan pengujian signifikansi parameter model diperoleh bahwa parameter model arima (1,2,0) dan arima (0,2,1) signifikan. kemudian dilakukan diagnostic checking terhadap model tersebut. hasil diagnostic checking diperoleh bahwa model arima (0,2,1) telah memenuhi asumsi residual white noise. maka model arima (0,2,1) merupakan model terbaik. model arima (0,2,1) dapat ditulis sebagai berikut: ( ) ( ) tt abyb 1 *2 11 −=− 1 * 2 * 1 * 699,02 −−− −+−= ttttt aayyy dimana tt yy ln * = kemudian dilakukan peramalan kebutuhan daya listrik tahun 2018 menggunakan model arima (0,2,1), dengan hasil peramalan sebagai berikut: tabel 3. hasil peramalan 12 bulan kedepan bulan peramalan januari 2018 172446914 februari 2018 175543951 maret 2018 178696608 april 2018 181924077 mei 2018 185191318 juni 2018 188517236 juli 2018 191922077 agustus 2018 195368876 september 2018 198877577 oktober 2018 202469537 november 2018 206105761 desember 2018 209807290 berdasarkan hasil peramalan menggunakan model arima bahwa kebutuhan daya listrik pada tahun 2018 diperkirakan mengalami peningkatan yakni sebesar 18,21% dari satu tahun sebelumnya. selanjutnya dilakukan peramalan menggunakan model fungsi transfer. data yang digunakan dalam peramalan kebutuhan daya listrik menggunakan model fungsi transfer yaitu data penggunaan daya listrik sebagai deret output (y) dan jumlah pelanggan pln sebagai deret input (x). tahap awal dalam identifikasi model fungsi transfer yaitu mempersiapkan deret input dan output, dalam hal ini ditentukan model arima deret input dan output. 60544842363024181261 150000 140000 130000 120000 110000 100000 90000 index p e la n g g a n time series plot of pelanggan gambar 6. plot time series deret input berdasarkan plot time series pada gambar 6 diperoleh bahwa data deret input belum stasioner. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 126 sehingga diperlukan proses differencing agar data menjadi stasioner. hasil differencing yang sesuai yaitu differencing orde-2. 60544842363024181261 2000 1500 1000 500 0 -500 -1000 index x t 2 time series plot of xt-2 gambar 7. plot time series deret input (d=2) kemudian dilakukan pemodelan arima deret input berdasarkan plot acf dan pacf, diperoleh beberapa kemungkinan model yang terbentuk yaitu arima (0,2,1), arima (1,2,0), arima (2,2,0), arima (1,2,1) dan arima (2,2,1). berdasarkan hasil pengujian signifikasi parameter model dan diagnostic checking diperoleh model terbaik yaitu model arima (0,2,1) karena telah memenuhi asumsi residual white noise dan beridistribusi normal. selanjutanya dilakukan prewhitening deret input dan output. hasil prewhitening deret input dan output dengan model arima (0,2,1) sebagai berikut. 1880,0 −+= ttt x  dan 1880,0 −+= ttt y  dimana t x dan t y adalah hasil differencing dari tx dan ty . selanjutnya penentuan orde b, r dan s model fungsi transfer berdasarkan plot korelasi silang antara deret input dan output yang telah melalui proses prewhitening, diperoleh orde b = 0, r = 0, dan s = 0. kemudian dilakukan pehitungan bobot respon impuls dan identifikasi model arima deret noise. diperoleh model deret noise yang sesuai dan telah memenuhi asumsi (white noise dan berdistribusi normal) yaitu model arima (1,1,0). berdasarkan orde b, r, s dan model deret noise diperoleh model fungsi transfer sebagai berikut ( ) ( ) ttt tttttt axx xxyyyy +− −−+−+= −− −−−− 2101 102111 )(   model yang diperoleh tersebut sudah sesuai dan dapat digunakan untuk peramalan, karena parameter model telah signifikan dan memenuhi asumsi white noise (uji korelasi silang dan uji autokorelasi). sehingga model tersebut dapat ditulis sebagai berikut ( ) ( ) ttt tttttt axx xxyyyy +− −−+−−= −− −−−− 21 1211 736,370 805,603)(614,0 hasil peramalan kebutuhan daya listrik tahun 2018 menggunakan model fungsi transfer sebagai berikut. tabel 4. hasil peramalan 12 bulan kedepan bulan ramalan januari 171213124 februari 176541333 maret 178226054 april 183428947 mei 184993329 juni 190083038 juli 191538613 agustus 196526103 september 197883293 oktober 202778470 november 204046694 desember 208858512 berdasarkan hasil peramalan menggunakan model model fungsi transfer bahwa kebutuhan daya listrik pada tahun 2018 diperkirakan mengalami peningkatan yakni sebesar 18,18% dari satu tahun sebelumnya. 5. kesimpulan berdasarkan hasil penelitian diperoleh bahwa dengan menggunakan model arima diperkirakan kebutuhan daya listrik pada tahun 2018 mengalami peningkatansebesar 18,21% dari satu tahun sebelumnya. sedangkan menggunakan fungsi transfer diperkirakan kebutuhan daya listrik mengalami peningkatan sebesar 18,18% dari satu tahun sebelumnya. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 127 referensi [1] alfita, r., mamluah, d., ulum, m., nahari, r.v, implementation of fuzzy sugeno method for power efficiency, international journal of advanced engineering research and science (ijaers). vol. 4. no. 9. (2017) 1-5. [2] wahid, a., junaidi., arsyad, m.i, analisis kapasistas dan kebutuhan daya listrik untuk menghemat penggunaan energi listrik di fakultas teknik universitas tanjungpura. jurnal teknik elektro universitas tanjungoura. vol 2, no 1. (2014). [3] bappeda, “nusa tenggara barat dalam angka 2012”, penerbit katalog bps, provinsi ntb, (2012). [4] arumugam, p. and anithakumari, v, seasonal time series and transfer function modelling for natural rubber forecasting in india. international journal of computer trends and technology (ijctt). vol. 4. no. 5. (2013) 13661370. [5] wei, w.w.s, time series analysis: univariate and multivariate method. canada. addisonwesley pub. inc. (2006). [6] liu, l.m. time series analysis and forecasting. scientific computing associates corp. (2006). m.e.ervina_rprop_mantik jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 110 analisis faktor angka kematian ibu dan bayi di provinsi jawa tengah dengan menggunakan regresi bivariat poisson mutiara widhika a1, a’yunin sofro2 prodi matematika, universitas negeri surabaya1,2, mutiarawidhika@gmail.com1, ayuninsofro@unesa.ac.id2 doi:https://doi.org/10.15642/mantik.2018.4.2.110-115 abstrak kematian ibu dan bayi merupakan dua perihal yang saling berkorelasi, hal ini disebabkan saat masa kehamilan plasenta ibu menyalurkan gizi kepada janinnya sehingga bayi yang dilahirkan dipengaruhi oleh kondisi ibunya. jawa tengah adalah provinsi yang memiliki angka kematian ibu dan bayi cukup besar di indonesia. dengan demikian, pada kasus ini diperlukan penelitian guna menganalisa faktor-faktor yang mempengaruhi kematian ibu dan bayi menggunakan metode regresi bivariat poisson (rbp). rbp adalah metode yang tepat untuk memodelkan korelasi dua data yang berdistribusi poisson. penelitian ini menghasilkan tiga model. model pertama adalah angka kematian ibu memiliki beberapa faktor signifikan, diantaranya ibu hamil yang melaksanakan program k1dan k4, vitamin a untuk ibu nifas, pemberian tablet fe untuk ibu hamil, dan komplikasi ditangani kebidanan. model kedua adalah faktor angka kematian bayi antara lain, ibu hamil yang melaksanakan program k4, pertolongan oleh tim kesehatan, ibu nifas mendapat vit. a, ibu hamil memperoleh tablet fe, komplikasi ditangani kebidanan, dan pengguna kb aktif. model terakhir melibatkan kematian ibu dan bayi. faktor yang signifikan yaitu ibu hamil melaksanakan program k1, ibu hamil melaksanakan program k4, pemberian vitamin a pada ibu nifas, dan peserta kb aktif. kata kunci : kematian ibu, kematian bayi, rbp. abstract maternal and infant mortality are two correlated subjects, because during pregnancy the mother's placenta distributes nutrients to the fetus so the baby born is affected by the condition of his mother. central java has significant maternal and neonatal mortality rates in indonesia. in this case, need a research to analyze the factors that influence maternal and infant mortality using bivariate poisson regression (bpr) method. bpr is the right method because it can reconfirm two data that are correlated with poisson distribution. this study produced three models. the first model is the maternal mortality rate has several significant factors, including pregnant women implementing the k1 and k4 program, vitamin a to postpartum mothers, pregnant women getting fe tablets, and midwifery handle complications. the second model is the infant deaths that have factors pregnant women implementing the k4 program, helped assistance by medical team, postpartum mothers receiving vitamin a, pregnant women getting fe tablets, complications handled by midwifery, and kb participants. the final model involves maternal and infant mortality. significant factors are pregnant women implementing the k1 program, pregnant women implementing the k4 program, giving vitamin a to postpartum mothers, and kb participants. keywords: maternal death, infant death, pbr. 1. pendahuluan angka kematian ibu dan bayi di indonesia masih cukup tinggi yaitu 4.912 untuk kasus kematian ibu dan 4.361.072 untuk kasus kematian bayi. sustainable development goals (sdgs) 2015-2030 mempunyai tujuan target penurunan angka kematian ibu (aki) dengan angka 70 setiap 100.000 kelahiran hidup, dan angka kematian bayi (akb) 12 per 1.000 kelahiran hidup [2]. mailto:mutiarawidhika@gmail.com mailto:ayuninsofro@unesa.ac.id jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 111 angka kematian bayi (akb) adalah angka kematian bayi yang berusia 0 sampai 11 bulan per 1.000 kelahiran hidup, akb di provinsi jawa tengah tahun 2016 sebesar 99,9 per 1.000 kelahiran hidup. sedangkan angka kematian ibu (aki) merupakan angka kematian ibu di tahun dan daerah tertentu per 100.000 kelahiran hidup. pada kasus ini, aki di provinsi jawa tengah sebesar 109,65 per 100.000 kelahiran hidup tahun 2016 [3]. kematian ibu dan bayi adalah dua hal yang berkorelasi satu sama lain, hal ini disebabkan saat masa kehamilan plasenta ibu menyalurkan gizi kepada janinnya sehingga bayi yang dilahirkan dipengaruhi oleh kondisi ibunya.. oleh sebab itu, agar tindak lanjut yang akan diberikan mendapatkan hasil yang sesuai, maka perlu dilakukan sebuah penelitian mengenai jumlah kematian ibu dan bayi dengan menyertakan faktor-faktor yang saling berpengaruh terhadap keduanya [8]. regresi bivariat poisson (rbp) merupakan metode regresi untuk memodelkan dua data yang saling berkorelasi. penelitian sebelumnya oleh nina fauziah rachmah pada tahun 2014 dengan metode ini diperoleh kesimpulan bahwa metode rbp dapat memperoleh model terbaik terhadap dua data yang saling berkorelasi dengan berdisitribusi poisson [7]. berdasarkan latar belakang tersebut, maka pada penelitian ini diterapkan metode rbp untuk memperoleh model dari jumlah kematian ibu dan bayi di provinsi jawa tengah pada tahun 2016, dan menganalisa faktor-faktor yang berpengaruh terhadap angka kematian ibu dan bayi dengan data yang berasal dari dinas kesehatan provinsi jawa tengah pada 2016. 2. tinjauan pustaka 2.1 distribusi poisson distribusi poisson adalah distribusi banyak hasil percobaan yang terjadi dalam suatu interval waktu tertentu atau di suatu daerah tertentu. distribusi probabilitas variabel random poisson 𝑌, merupakan jumlah perolehan yang terjadi pada interval waktu atau wilayah tertentu, disimbolkan dengan 𝑡, [9]: 𝑃(𝑦; 𝜆𝑡 ) = 𝑒−𝜆𝑡 (𝜆𝑡) 𝑦 𝑦! , 𝑦 = 0, 1, 2, …; 𝜆 > 0 (1) (1) dengan 𝑃(𝑦; 𝜆𝑡 ) probabilitas distribusi poissona, 𝜆𝑡 = rata-rata banyak peristiwa yang terjadi per daerah atau satuan waktu tertentu, 𝑦 adalah banyak kejadian 𝑒 nilai konstan (2,71828…), nilai pemusatan data (mean) dan sebaran data (varians) pada distribusi poisson bernilai sama yaitu 𝜆. 2.2 distribusi bivariat poisson distribusi bivariat poisson merupakan penggabungan dari dua variabel random, dengan masing-masing dari variabelnya adalah variabel random yang berdistribusi poisson [6]. misal 𝑊0, 𝑊1, dan 𝑊2 adalah variabel random berdistribusi poisson dengan parameter 𝜆0, 𝜆1, dan 𝜆2. dengan variabel random seperti berikut [6] 𝑌1 = 𝑊1 + 𝑊0 𝑌2 = 𝑊2 + 𝑊0 fungsi dari distribusi bivariat poisson adalah: 𝑃(𝑦1, 𝑦2) = 𝑒 −(𝜆1+𝜆2+𝜆0) ∑ 𝜆1𝑖 𝑦1−𝑘𝜆2𝑖 𝑦2−𝑘𝜆0 𝑘 (𝑦1−𝑘)!(𝑦2−𝑘)!𝑘! min(𝑦1,𝑦2) 𝑘=0 (2) dengan, 𝑦1, 𝑦2 = 0,1,2, …a 2.3 regresi poisson regresi poisson merupakan suatu regresi yang dapat menunjukkan hubungan antara variabel-variabel respon (𝑦) dengan variabel predictor 𝑥, dengan variabel responnya berdistribusi poisson [1, 10]. model regresi poisson adalah: 𝑦~𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜆) 𝜆𝑖 = exp( 𝒙𝑖 𝑇 𝜷 ) (3) dengan 𝜆 adalah nilai rata-rata jumlah kejadian yang terjadi dalam interval waktu tertentu. sedangkan 𝒙 adalah variabel prediktor yang dinotasikan dalam bentuk matriks sebagai berikut: 𝒙 = [1 𝑥1 𝑥2 ⋯ 𝑥𝑘 ] 𝑇 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 112 adapun 𝜷 merupakan parameter regresi poisson: 𝜷 = [𝛽0 𝛽1 𝛽2 ⋯ 𝛽𝑘 ] 𝑇 2.4 regresi bivariat poisson regresi bivariat poisson merupakan metode untuk memperoleh model dari sepasang data yang mempunyai korelasi dengan beberapa variabel prediktornya dan berdistribusi poisson. model regresi bivariat poisson adalah [5]: (𝑌1𝑖 , 𝑌2𝑖 )~𝑃𝑜𝑖𝑠𝑠𝑜𝑛𝐵𝑖𝑣𝑎𝑟𝑖𝑎𝑡𝑒(𝜆1𝑖 , 𝜆2𝑖 , 𝜆0𝑖 )𝑎 log(𝜆𝑘𝑖 ) = 𝒘𝑘𝑖 𝜷𝑘 , k=1,2,3,… (4) dengan 𝑘 menyatakan variabel predictor, 𝑖 = 1,2,3,…, n, menyatakan banyak percobaan, 𝒘𝑘𝑖 menyatakan vektor variabel explanatory untuk observasi kei yang digunakan untuk memodelkan 𝜆𝑘𝑖, dan 𝜷𝑘 menyatakan vektor korespondensi dari koefisien regresi. 2.5 uji parameter berdasarkan model yang telat didapatkan, langkah berikutnya perlu dilakukan uji variabel untuk menunjukkan apakah variabel-variabel prediktor pada model tersebut memiliki hubungan yang signifikan dengan variabel responnya. uji yg pertama dilakukan yaitu uji serentak, untuk mengetahui kesignifikanan koefisien 𝛽 terhadap variabel responnya dengan serentak, dengan hipotesis uji sseperti berikut [4]: 𝐻0 : 𝛽1 = 𝛽2 = ⋯ = 𝛽𝑝 = 0 𝐻1 : paling sedikit koefisien 𝛽𝑗 ≠ 0, 𝑗 = 1, 2, … , 𝑝 statistik uji 𝐺 sebagai berikut: 𝐺 = −2 ln ( ( 𝑛1 𝑛 ) 𝑛1 ( 𝑛0 𝑛 ) 𝑛0 ∑ 𝜋�̂� 𝑦𝑖 (1−𝜋�̂�) (1−𝑦𝑖)𝑁 𝑖=1 ) (5) statistik uji g mengikuti distribusi chisquare dan dinyatakan 𝐻0 ditolak jika 𝐺 > 𝜒2 (𝛼,𝑝) atau 𝑝 − 𝑣𝑎𝑙𝑢𝑒 < 𝛼 [4]. apabila tolak 𝐻0, maka berikutnya melakukan uji parsial. hipotesis uji parsial, seperti berikut [4]: 𝐻0 : 𝛽𝑗 = 0 𝐻1 : 𝛽𝑗 ≠ 0 dengan 𝑗 = 1, 2, … , 𝑝 statistik sampel uji wald, seperti berikut: 𝑊 = 𝛽�̂� 𝑆�̂�(𝛽�̂�) (6) statistik uji 𝑊 mengikuti distribusi normal baku, yaitu tolak 𝐻0 jika nilai |𝑊| > 𝑍(𝛼/2) atau 𝑝 − 𝑣𝑎𝑙𝑢𝑒 < 𝛼 [4]. 2.6 kematian ibu dan bayi kematian ibu dan bayi adalah dua hal yang saling berkorelasi. hal ini disebabkan karena ada pemberian fasilitas pelayanan kesehatan yang sama kepada ibu hamil dan bayinya yang akan dilahirkan. terdapat beberapa pemicu mengenai tingginya angka kematian ibu dan bayi yang baru lahir sebagaimana yang dinyatakan oleh kusumaningtyas [11], yakni kualitas pelayanan kesehatan, sistem rujukan kesehatan, implementasi jaminan kesehatan nasional, dan kebijakan pemerintah daerah yang berkaitan dengan kesehatan. selain faktor tersebut, terdapat juga faktor budaya yang berkembang dalam masyarakat tertentu, yang mana keputusan untuk mendapatkan pertolongan di rumah sakit atau tidak bergantung pada keputusan suami. dengan kata lain, ada semacam ketimpangan gender yang terjadi di sebagian masyarakat indonesia. faktor lain yang juga berpengaruh adalah latar belakang pendidikan, sosial ekonomi keluarga, lingkungan masyarakat dan politik. 3. metode peneltian 3.1 sumber data data yang digunakan dalam penelitian ini merupakan data yang berasal dari profil kesehatan provinsi jawa tengah tahun 2016 dengan jumlah pengamatan sebanyak 35 kabupaten/kota yang terdiri dari 29 kabupaten dan 6 kota. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 113 3.2 variabel penelitian variabel respon (𝑦) untuk penelitian ini adalah jumlah kematian ibu dan bayi. sedangkan variabel prediktor (𝑥) terdiri dari 7 variabel yang dipilih berdasarkan indikator program kia dan four pillars of safe motherhood dari dinas kesehatan provinsi jawa tengah tahun 2016. variabel penelitian tersebut ditabelkan pada tabel 1. tabel 1. variabel penelitian kode variabel 𝑦1 kasus angka kematian ibu 𝑦2 kasus angka kematian bayi 𝑥1 persentase ibu hamil yang melaksanakan program k1 𝑥2 persentase ibu hamil yang melaksanakan program k4 𝑥3 persentase pertolongan persalinan oleh tenaga kesehatan 𝑥4 persentase pemberian vit. a pada ibu nifas 𝑥5 persentase ibu hamil yang mendapatkan tablet fe 𝑥6 persentase komplikasi kebidanan ditangani 𝑥7 persentase kb aktif 3.3 tahapan penelitian penelitian ini menggunakan beberapa tahapan yaitu: 1. melakukan studi literatur, 2. menyusun asumsi dan batasan masalah, 3. melaukan pemodelan dengan menggunakan regresi bivariat poisson, 4. mencari faktor-faktor yang mempengaruhi secara signifikan untuk kematan ibu, bayi, dan keduanya berdasarkan nilai parameter yang signifikan, 5. melakukan uji serentak semua variabel prediktor terhadap variabel respon dan, 6. melakukan uji parsial masing-masing variabel prediktor terhadap variabel respon, 7. melakukan interpretasi hasil 8. menulis kesimpulan 4. hasil dan pembahasan hubungan antar variabel respon (y) yaitu jumlah kematian ibu dan bayi dapat dilihat melalui nilai koefisien korelasi variabel angka kematian ibu dan angka kematian bayi. berdasarkan hasil analisis disimpulkan bahwa terdapat korelasi (hubungan yang erat) antara angka kematian ibu dan angka kematian bayi di provinsi jawa tengah pada tahun 2016. dari hasil analisis dengan program software sas studio, diperoleh nilai 𝐺ℎ𝑖𝑡𝑢𝑛𝑔 adalah 195,57; dan nilai 𝜒0,05;7 2 = 2,17. dengan diperolehnya nilai 𝐺ℎ𝑖𝑡𝑢𝑛𝑔 > 𝜒0,05;7 2 maka didapat kesimpulan bahwa 𝐻0 ditolak dan 𝐻1 diterima, dimana: 𝐻0 : 𝛽1 = 𝛽2 = ⋯ = 𝛽𝑝 = 0 𝐻1 : paling sedikit koefisien 𝛽𝑗 ≠ 0, j= 1, 2, … , 𝑝 dengan artian sedikitnya ada satu variabel yang berpengaruh terhadap model. adapun model regresi bivariat poisson untuk jumlah kematian ibu adalah 𝑦1 = exp(−11.0454 − 0.06598𝑥1 + 0.03257𝑥2 − 0.00349𝑥3 + 0.02774𝑥4 + 0.003730𝑥5 − 0.01389𝑥6 + 0.006600𝑥7) setelah model regresi bivariat poisson diperoleh, selanjutnya dilakukan pengujian secara parsial terhadap masing-masing parameter untuk mengetahui variabel prediktor mana sajakah yang berpengaruh secara signifikan terhadap angka kematian ibu dan bayi. tabel 2. estimasi parameter model regresi bivariat poisson 𝑦1 untuk kematian ibu par estimasi t value pr > |t| alpha 𝛽10 -11.0454 -8.15 <.0001 0.05 𝛽11 -0.06598 -6.72 <.0001 0.05 𝛽12 0.03257 5.58 <.0001 0.05 𝛽13 -0.00349 -0.35 0.7282 0.05 𝛽14 0.02774 6.53 <.0001 0.05 𝛽15 0.00373 2.31 0.0267 0.05 𝛽16 -0.01389 -3.71 0.0007 0.05 𝛽17 0.0066 1.26 0.2154 0.05 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 114 berdasarkan tabel 2 diperoleh model 𝑦1 atau variabel yang berpengaruh signifikan terhadap angka kematian ibu yaitu variabel ibu hamil yang melaksanakan program k1 (𝛽11), ibu hamil yang melaksanakan program k4 (𝛽12), ibu nifas yang mendapatkan vitamin a (𝛽14), ibu hamil yang mendapat tablet fe (𝛽15), dan komplikasi kebidanan ditangani (𝛽16). variabel-variabel tersebut dinyatakan bepengaruh signifikan karena diperoleh nilai pr > |t| yang kurang dari alpha (0,05). dengan cara yang sama didapatkan model regresi bivariat poisson untuk jumlah kematian bayi adalah 𝑦2 = exp(−8.8726 − 0.00478𝑥1 + 0.009156𝑥2 − 0.08092𝑥3 + 0.04254𝑥4 − 0.01608𝑥5 − 0.03661𝑥6 + 0.05751𝑥7) tabel 3. estimasi parameter model regresi bivariat poisson 𝑦2 untuk kematian bayi par estimasi t value pr > |t| alpha 𝛽20 -8.8726 -14.57 <.0001 0.05 𝛽21 -0.00478 -0.99 0.3299 0.05 𝛽22 0.009156 3.48 0.0014 0.05 𝛽23 -0.08092 -18.72 <.0001 0.05 𝛽24 0.04254 22.15 <.0001 0.05 𝛽25 -0.01608 -24.87 <.0001 0.05 𝛽26 -0.03661 -20.28 <.0001 0.05 𝛽27 0.05751 21.51 <.0001 0.05 diperoleh model 𝑦2 atau variabel yang berpengaruh signifikan terhadap jumlah kematian bayi yaitu variabel ibu hamil melaksanakan program k4 (𝛽22), persalinan oleh tenaga kesehatan (𝛽23), pemberian vit a pada ibu nifas (𝛽24), ibu hamil mendapat tablet fe (𝛽25), komplikasi kebidanan ditangani (𝛽26), dan peserta kb aktif (𝛽27). variabel-variabel tersebut dinyatakan bepengaruh signifikan karena diperoleh nilai pr > |t| yang kurang dari alpha (0,05). sedangkan model regresi bivariat poisson dari persamaan kedua variable respon adalah 𝜆0 = exp(−12.0412 − 0.08850𝑥1 + 0.04266𝑥2 + 0.00071𝑥3 + 0.02793𝑥4 − 0.00056𝑥5 − 0.00631𝑥6 + 0.03426𝑥7) untuk melihatmanakah diantara variabelvariabel yang berpengaruh secara signifikan terhadap angka kematian ibu dan bayi, maka dilakukan uji secara parsial dan diperoleh hasil sebagaimana pada tabl 4 sebagai berikut. tabel 4. estimasi parameter model regresi bivariat poisson 𝜆0 untuk kedua var. respon par estimasi t value pr > |t| alph a 𝛽00 -12.0412 -9.19 <.0001 0.05 𝛽01 -0.0885 -9.45 <.0001 0.05 𝛽02 0.04266 7.56 <.0001 0.05 𝛽03 0.000071 0.01 0.9942 0.05 𝛽04 0.02793 6.88 <.0001 0.05 𝛽05 -0.00056 -0.36 0.7196 0.05 𝛽06 -0.00631 -1.74 0.0904 0.05 𝛽07 0.03426 6.45 <.0001 0.05 berdasarkan tabel 4 tersebut, maka diperoleh model 𝜆0 yang berpengaruh signifikan terhadap kedua variabel respon adalah variabel ibu hamil yang melaksanakan program k1 (𝛽01), ibu hamil yang melaksanakan program k4 (𝛽02), pemberian vit a pada ibu nifas (𝛽04), dan peserta kb aktif (𝛽07). variabel-variabel tersebut dinyatakan bepengaruh signifikan karena diperoleh nilai pr > |t| yang kurang dari alpha (0,05). 5. penutup 5.1 simpulan berdasarkan pembahasan dan analisis yang telah usai dilakukan, maka diperoleh model regresi bivariat poisson seperti berikut: model pertama adalah angka kematian ibu memiliki beberapa faktor signifikan, diantaranya ibu hamil yang melaksanakan program k1 dan k4, vitamin a untuk ibu nifas, pemberian tablet fe untuk jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 115 ibu hamil, dan komplikasi ditangani kebidanan. model kedua adalah faktor angka kematian bayi antara lain, ibu hamil yang melaksanakan program k4, pertolongan oleh tim kesehatan, ibu nifas mendapat vit. a, ibu hamil memperoleh tablet fe, komplikasi ditangani kebidanan, dan pengguna kb aktif. model terakhir, variabel yang berpengaruh signifikan pada jumlah kematian ibu dan bayi yaitu variabel ibu hamil melaksanakan program k1, persentase ibu hamil melaksanakan program k4, persentasi pemberian vitamin a pada ibu nifas, dan persentase peserta kb aktif. 5.2 saran saran kepada peneliti berikutnya, peneliti dapat menggunakan metode lain untuk mengembangkan penelitian ini dan untuk menyelesaikan penaksiran parameter regresi bivariat poisson. peneliti berikutnya juga dapat menangani adanya kasus overdispersi, sehingga memungkinkan untuk diperoleh model yang lebih baik. selain itu dapat juga menambahkan faktor-faktor lain dari penyebab kematian ibu dan bayi. referensi [1] agresti, categorical data analysis. new york: john wiley & sons, inc. 1990 [2] badan perencanaan pembangunan nasional. laporan pencapaian tujuan pembangunan milenium di indonesia. jakarta. 2012. [3] dinas kesehatan provinsi jawa tengah, profil kesehatan provinsi jawa tengah 2016. semarang. 2016. [4] hosmer, d. w & lemeshow, s., applied logistic regression. new jersey: john wiley & sons, inc. 2000. [5] karlis, d., & ntzoufras, i., bivariate poisson and diagonal inflated bivariate poisson regression models in r. journal of statistical software. 2005. [6] kawamura, kazutomo, the structure of bivariate poisson distribution. kodai math. sem. rep. 25 (1973), no. 2, 246-256. doi:10.2996/kmj/1138846776. [7] rachmah, nina f, pemodelan jumlah kematian ibu dan jumlah kematian bayi di provinsi jawa timur menggunakan bivariate poisson regression. surabaya: its surabaya. 2014. [8] the world health report, world health organization. who: geneva. 2004. [9] walpole, re & myers, ilmu peluang dan statistika untuk insinyur dan ilmuwan. bandung: itb bandung. 1995. [10] kahar, arifin m, analisis angka harapan lama sekolah di indonesia timur menggunakan weighted least squares regression, jurnal matematika mantik, vol. 4, no. 1, hal. 32-41, may 2018. [11] kusumaningtyas, s. angka kematian ibu dan bayi di indonesia tinggi, riset ungkap sebabnya. diakses pada tanggal, 12 juni 2018 melalui, https://sains.kompas.com/read/2018/03/2 8/203300723/angka-kematian-ibu-danbayi-di-indonesia-tinggi-riset-ungkapsebabnya. https://sains.kompas.com/read/2018/03/28/203300723/angka-kematian-ibu-dan-bayi-di-indonesia-tinggi-riset-ungkap-sebabnya https://sains.kompas.com/read/2018/03/28/203300723/angka-kematian-ibu-dan-bayi-di-indonesia-tinggi-riset-ungkap-sebabnya https://sains.kompas.com/read/2018/03/28/203300723/angka-kematian-ibu-dan-bayi-di-indonesia-tinggi-riset-ungkap-sebabnya https://sains.kompas.com/read/2018/03/28/203300723/angka-kematian-ibu-dan-bayi-di-indonesia-tinggi-riset-ungkap-sebabnya paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 83 klasifikasi alzheimer dan non alzheimer menggunakan fuzzy c-mean, gray level cooccurrence matrix dan support vector machine dian c. r. novitasari1, wahyu t. puspitasari2, putri wulandari3, a. z. foeady4, m. fahrur rozi5 universitas islam negeri sunan ampel surabaya1,2,3,4,5, diancrini@uinsby.ac.id1 doi:https://doi.org/10.15642/mantik.2018.4.1.83-89 abstrak berdasarkan piagam alzheimer, dua sampai tiga juta kasus demensia oleh penyakit alzheimer terjadi setiap tahun. penderita penyakit alzheimer mengalami gangguan memori dan kognitif secara progresif selama 3 sampai 9 tahun. penderita mengalami kebingungan dalam memahami pertanyaan serta memiliki urutan memori yang kacau, sehingga dapat mengganggu aktivitas sehari-hari dan apabila dibiarkan akan menyebabkan kematian. klasifikasi yang dilalukan berdasarkan data magnetic resonance imaging (mri) penyakit alzheimer dan non alzheimer menggunakan support vector machine (svm). segmentasi fitur data menggunakan fuzzy c-means (fcm) dan ekstraksi fitur menggunakan gray level co-occurrence matrix (glcm) memperoleh hasil akurasi yang baik sebesar 93,33 % kata kunci: alzheimer, fuzzy c-means, gray level co-occurence matrix, support vector machine abstract based on the alzheimer's charter, 2-3 million cases of dementia by alzheimer's disease occur every year. people with alzheimer's disease experience memory and cognitive disorders progressively for 3 to 9 years. patients experience confusion in understanding the question and have a chaotic sequence of memory, which can interfere with daily activities and unchecked well, it cause death. the classification system is based on alzheimer's and non-alzheimer's disease magnetic resonance imaging (mri) using support vector machine (svm). the feature data segmentation using fuzzy c-means (fcm) and feature extraction using gray level co-occurrence matrix (glcm) and give accuracy result of 93.33%. keywords: alzheimer, fuzzy c-means, gray level co-occurence matrix, support vector machine 1. pendahuluan seseorang yang telah memasuki usia lanjut seringkali mengalami gangguan ingatan. gangguan ingatan pada usia lanjut disebabkan kerena syaraf pusat mengalami degenerasi pada syaraf pusat. gangguan ingatan yang terjadi disebut demensia atau alzheimer. sebagian besar penderita alzheimer mengalami gangguan memori, perubahan kepribadian, suasana hati dan perilaku, bermasalah dalam interaksi [1]. seseorang yang terkena alzheimer akan mengalami gangguan secara bertahap. rata rata gangguan penurunan akan dialami selama tiga sampai sembilan tahun [2]. teknologi yang semakin maju memungkinkan untuk melakukan deteksi penyakit alzheimer dapat dilakukan dengan mengambil gambar melalui proses magnetic resonance imaging (mri). hasil dari proses mri menghasilkan gambar yang dapat dilihat untuk deteksi beberapa penyakit. menggunakan gambar mri deteksi penyakit alzheimer dapat dilakukan dengan melakukan perhitungan numerik. perhitungan numerik dilakukan dengan mengambil nilai piksel pada gambar mri pasien. penggunaan gambar mri sering dilakukan untuk proses pemeriksaan berbagai macam penyakit pada tubuh. pada penelitian sebelumnya gambar mri juga digunakan jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 84 untuk melakukan deteksi kanker prostat [3]. selain itu penelitian yang lain citra mri digunakan juga untuk melihat kandungan lemak pada sumsum tulang belakang [4]. dari beberapa penelitian tersebut menghasilkan hasil yang maksimal dengan menggunakan citra mri sebagai citra masukan yang dapat digunakan sebagai acuan. menggunakan hasil mri, deteksi alzheimer dapat dilakukan dengan melakukan segmentasi pada gambar. segementasi dilakukan untuk membantu pengambilan fitur fitur yang diinginkan dan membuang background yang dapat mengganggu proses klasifikasi penyakit alzheimer. pada penelitian sebelumnya segmentasi optik disk dilakukan dengan menggunakan metode fuzzy c-means [5]. selain itu pada penelitian sebelumnya, metode fuzzy c-means juga digunakan untuk segmentasi gambar mri [6]. menggunakan algoritma fuzzy c-means gambar mri akan memisahkan antara fitur yang diperlukan ataupun fitur yang tidak diperlukan untuk proses selanjutnya. fuzzy c-means juga dapat digunakan untuk melakukan segmentasi pada citra mammogram untuk melakukan deteksi kanker payudara [7]. penelitian sebelumnya menyatakan fuzzy c-means dapat dapat digunakan secara optimal untuk melakukan segmentasi pada sebuah citra [8]. melalui algoritma fuzzy c-means gambar mri akan memisahkan antara fitur yang diperlukan ataupun fitur yang tidak diperlukan untuk proses selanjutnya. pengambilan fitur alzheimer dapat diambil dengan melakukan ekstraksi fitur. hasil dari proses segmentasi fuzzy c-means dapat dilakukan ekstraksi fitur menggunakan metode gray level co-occurrence matrix (glcm). pada penelitian sebelumnya glcm digunakan untuk pengenalan karakteristik suatu partikel [9]. selain itu pada penelitian sebelumnya, glcm digunakan untuk ekstraksi fitur pada gambar yang memiliki motif berbeda beda [10]. glcm juga digunakan pada analisis tekstur pada kulit [11]. pada penelitian yang sudah dilakukan, menyatakan glcm merupakan salah satu metode yang dapat digunakan untuk melakukan ekstraksi fitur pada suatu data [12]. gray level co-occurrence matrix digunakan untuk pengambilan fitur yang dapat digunakan sebagai acuan dalam proses klasifikasi. nilai fitur yang dihasilkan pada proses glcm dapat digunakan untuk klasifikasi pada gambar yang diuji. pada penelitian sebelumnya klasifikasi dilakukan menggunakan support vector machine (svm) sebagai metode untuk melakukan entifikasi jenis kanker [13]. klasifikasi menggunakan support vector machine (svm) juga digunakan untuk melakukan klasifikasi pada kanker payudara [14]. dari beberapa penelitian yang sudah dilakukan, klasifikasi dari gambar penyakit otak dapat menggunakan support vector machine (svm). melihat dari beberapa penelitian yang sudah dilakukan dan permasalahan yang ada, penelitian ini bertujuan untuk melakukan klasifikasi gambar mri penderita alzheimer dan gambar mri yang bukan penderita alzheimer. untuk segmentasi gambar dilakukan dengan menggunakan fuzzy cmeans dan ekstraksi fitur menggunakan gray level co-occurrence matrix, sedangkan klasifikasi dilakukan dengan menggunakan metode svm. 2. tinjauan pustaka 2.1 fuzzy c-means segmentation fuzzy c-means merupakan salah satu metode yang dapat digunakan untuk melakukan clustering atau pengelompokan suatu data. penggunaan clustering yang dimiliki oleh fuzzy c-means dapat juga digunakan untuk pengelompokan nilai piksel pada citra atau serig disebut segmentasi citra. segmentasi menggunakan fuzzy c-means dilakukan dengan menghitung nilai keanggotaan pada setiap pikselnya. metode ini akan digunakan jiga pada suatu bagian citra memiliki dua atau lebih kelompok atau kelas. data piksel pada citra akan diubah menjadi beberapa kelas yang memiliki derajat keanggotaan yang berbeda beda. nilai keanggotan mempunyai rentang antara nol sampai satu [15]. beberapa langkah yang digunakan untuk segmentasi menggunakan fuzzy c-means yaitu: jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 85 1. menyiapkan data yang akan dikelaskan dengan membentuk suatu matriks yang mempunyai ukuran 𝑀 𝑥 𝑁, dengan m sebagai banyak data dan n merupakan banyak variabel yang digunakan. 2. menentukan nilai eror (e), banyak kelas (c), bobot (w), awal iterasi dan banyaknya iterasi (t), dan fungsi objektif awal (p). 3. membuat bilangan acak sebagai awal elemen matriks partisi dengan ukuran 𝑀𝑥𝐶. 4. menghitung pusat kelompok menggunakan persamaan 1 sebagai berikut. 𝑣𝑘𝑗 = ∑ 𝜇𝑖𝑘 𝑤 𝑥𝑖𝑗 𝑚 𝑖=1 ∑ 𝜇𝑖𝑘 𝑤𝑚 𝑖=1 (1) dengan : 𝑣𝑘𝑗 = pusat 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 ke k untuk atribut ke j 𝜇𝑖𝑘 = derajat keanggotaan untuk data sampel ke i pasa 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 ke k 𝑥𝑖𝑗 = 𝑑𝑎𝑡𝑎 𝑘𝑒 𝑖, 𝑎𝑡𝑟𝑖𝑏𝑢𝑡 𝑘𝑒 𝑗 5. menghitung fungsi objektif pada iterasi ke t dengan persamaan 2 sebagai berikut. 𝑃𝑡 = ∑ ∑ ([∑(𝑥𝑖𝑗 − 𝑣𝑘𝑗 ) 2 𝑛 𝑗=1 ] (𝜇𝑖𝑘 ) 𝑤) 𝑐 𝑘=1 𝑚 𝑖=1 (2) 6. menghitung perubahan matriks partisi 𝜇𝑖𝑘 = [∑ (𝑥𝑖𝑗 − 𝑣𝑘𝑗 ) 2𝑐 𝑗=1 ] 1 𝑤−1 ∑ [∑ (𝑥𝑖𝑗 − 𝑣𝑘𝑗 ) 2𝑐 𝑗=1 ] 1 𝑤−1𝑐 𝑘=1 (3) 2.2 gray level co-occurrence matrix gray level co-occurrence matrix (glcm) merupakan salah satu metode ekstraksi fitur yang menggunakan histogram orde kedua dari tingkat keabuan [16]. pengambilan fitur didasarkan pada dua parameter, yaitu jarak dan sudut, dimana jarak adalah selisih piksel yang digunakan untuk the second order statistics, dan sudut yang terbentuk antara pasangan piksel. dalam metode glcm, orientasi sudut dinyatakan dalam derajat. orientasi sudut dibagi menjadi 4 arah sudut yang berbeda dengan interval 45°, yaitu 0°, 45°,90°, 135°[17]. misalkan f(a,b) adalah citra dengan ukuran na dan nb yang memiliki piksel dengan kemungkinan l level dan p adalah vektor arah offset spasial. glcm (i,j) didefinisikan sebagai jumlah piksel (j) yang terjadi pada offset r terhadap piksel (i) yang dapat dinyatakan ssebagai berikut 𝐺𝐿𝐶𝑀(𝑖, 𝑗) = {(𝑥_1, 𝑦_1 ), (𝑥_2, 𝑦_2)} (4) dimana offset r dapat berupa sudut atau jarak, j ∈ 1,...,l, dan i ∈ 1,...,l. matriks co occurrence digunakan untuk mendapatkan fitur dari citra. beberapa besaran yang diusulkan harlick untuk mendapatkan fitur dari glcm yaitu angular second moment (asm), kontras, inverse difference moment (idm), energi, korelasi [18]. 2.2.1 angular second moment (asm) asm juga dikenal sebagai keseragaman. asm berhubungan dengan energi, dimana energi merupakan jumlah kuadrat elemenelemen glcm angular second moment untuk mengukur homogenitas. nilai tertinggi dicapai ketika gambar memiliki homogenitas yang sangat baik yaitu saat elemen glcm semuanya sama [19]. asm dihitung dengan menggunakan rumus pada persamaan 6 berikut. 𝐴𝑆𝑀 = ∑ ∑(𝐺𝐿𝐶𝑀(𝑖, 𝑗))2 𝑗𝑖 (5) 2.2.2 kontras kontras merupakan ukuran variasi aras keabuan piksel citra. kontras dihitung dengan menggunakan persamaan 6 sebagai berikut. 𝐶𝑜𝑛𝑡𝑟𝑎𝑠 = ∑ ∑(𝑖 − 𝑗)2𝐺𝐿𝐶𝑀(𝑖, 𝑗) 𝑗𝑖 (6) 2.2.3 invers difference moment (idm) idm memiliki nilai tinggi saat tingkat keabuan lokal sama dan balikan glcm tinggi. idm dihitung dengan menggunakan persamaan 7 sebagi berikut. 𝐼𝐷𝑀 = ∑ ∑ 𝐺𝐿𝐶𝑀(𝑖, 𝑗) 1 + (𝑖, 𝑗)2 𝑗𝑖 (7) 2.2.4 entropy jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 86 entropy merupakan pengukuran keacakan aras keabuan di dalam citra. mencapai nilai tertinggi ketika elemen-elemen glcm memiliki nilai yang relatif sama dan memiliki nilai rendah apabila elemen-elemen glcm mendekati 0 atau 1. entropi dihitung dengan menggunakan rumus [20]. 𝐸𝑛𝑡𝑟𝑜𝑝𝑦 = − ∑ ∑ 𝐺𝐿𝐶𝑀(𝑖, 𝑗) log 𝐺𝐿𝐶𝑀(𝑖, 𝑗) 𝑗𝑖 (8) 2.2.5 korelasi korelasi digunakan menghitung ketergantungan linear tingkat keabuan dari piksel tetangga. untuk mendapatkan nilai korelasi dapat menggunakan rumus berikut : 𝐶𝑜𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = ∑ ∑ (𝑖𝑗)𝐺𝐿𝐶𝑀(𝑖, 𝑗) − 𝜇𝑖 ′𝜇𝑗 ′𝑗𝑖 𝜎𝑖 ′𝜎𝑗 ′ (9) 2.3 support vector machine (svm) support vector machine merupakan salah satu metode yang digunakan untuk melakukan klasifikasi suatu data. support vector machine akan melakuka klasifikasi data dengan membagi data menjadi beberapa daerah sesuai denga banyaknya klasifikasi yang diinginkan. untuk melakukan pembagian data, metode support vector machine menggunakan hyperplane sebagai garis bagi sebagai garis pemisah antara kelompok pada kelas satu dengan kelas yang lain. pembuatan garis bagi atau hyperplane dapat dibentuk menggunakan beberapa fungsi kernel yang ada pada metode support vector machine. beberapa fungsi kernel yang dapat digunakan untuk membangkitkan suatu hyperplane yaitu kernel gaussian, linear, ataupun polynomial [21]. pada penelitian ini fungsi kernel yang digunakan menggunakan fungsi polinomial. pemilihan fungsi polinomial karena pada penelitian ini hasil support vector machine menunjukkan hasil yang optimal menggunakan fungsi kernel polinomal. adapun fungsi kernel polinomial dibentuk menggunakan persamaan 10. 𝐾(𝑥, 𝑦) = (𝑥. 𝑦)𝑑 (10) 3. penulisan tabel dan gambar 3.1 jenis penelitian pada penelitian mengenai identifikasi penyakit alzhaimer menggunakan metode glcm dan svm termasuk kedalam jenis penelitian aplikatif karena data input dan output yang digunakan dalam penelitian ini berupa data numerik, dan hasil dari penelitian ini betujuan sebagai alternatif diagnosa penyakit alzheimer. 3.2 pengumpulan data data yang digunakan dalam penelitian ini merupakan data axial otak mri yang akan digunakan untuk mendapatkan karakteristiknya dengan menggunakan metode dwt. terdapat 95 data citra axial otak mri. data tersebut diperoleh dari alzheimer’s disease neuroimageing initiative (adni) dan e-health laboratory. 3.3 pengolahan data tahap–tahap yang dilakukan dalam identifikasi penyakit alzheimer yaitu, preprocessing, segmentasi fitur, ekstraksi fitur, dan pengklasifikasian. pada penelitian ini, data akan diklasifikasikan menjadi dua jenis yaitu alzheimer dan non-alzheimer. tahap pertama, dilakukan preprocessing yang befungsi untuk mempermudah proses dalam pengolahan citra, karena tidak semua data citra memiliki kualitas yang baik sehingga kontras dan pencahayaan terkadang tidak merata. selain itu, terdapat noise yang berbeda dari masing-masing data citra. pada tahap pre-processing, data otak mri merupakan citra grayscale. selanjutnya dilakukan peningkatan cahaya dengan mengubah histogram yang bertujuan untuk menormalisasi citra. citra yang telah diproses akan digunakan sebagai masukan dalam segmentasi fitur menggunakan fuzzy c-mean (fcm). jumlah cluster yang dimasukan sebanyak tiga sehingga diperoleh tiga fitur yaitu gray matter (gm), white matter (wm), dan cerebrospinal fluid (csf). dari tiga fitur tersebut, fitur yang diambil hanya fitur gm. tahap ketiga adalah ekstraksi fitur. dalam penelitian ini, ekstraksi fitur dengan metode glcm menggunakan derajat ketetanggan 0o. dari setiap sub-band diambil jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 87 beberapa fitur yaitu kontras, korelasi, energi, dan homogenitas. fitur tersebut akan digunakan dalam tahapan klasifikasi. dalam proses klasifikasi terdapat beberapa step yaitu menentukan parameter ciri, memilih kernel mana yang cocok dengan data. melakukan pelatihan dan melakukan pengujian data hingga mendapatkan hasil klasifikasi. dalam penelitian ini menggunakan kernel polinomial. skema tahap-tahap identifikasi alzheimer ditunjukan dalam gambar. 1. gambar 1 flowchat identifikasi alzheimer 4. hasil dan pembahasan untuk mengidentifikasi penyakit alzheimer dilakukan menggunakan metode fcm, glcm dan svm dengan fungsi secara berurutan untuk segmentasi, ekstrasi fitur, dan klasifikasi. sebelum dilakukan segmentasi diperlukan preprocessing citra yang bertujuan untuk memperbaiki kualtitas citra. pada tahap preprocessing, citra diperbaiki menggunakan histogram equalization. proses histogram equalization digunakan untuk memperoleh penyebaran histogram derajat keabuan citra yang merata dan relatif sama. hasil histogram equalization tersebut akan disegmentasi fitur berupa gray matter (gm), white matter (wm) dab cerebrospinal fluid (csf). fitur yang diambil untuk proses ekstraksi fitur hanya gm. pada gambar 3 ditunjukkan citra hasil preprocessing dan segmentasi fitur. setelah didapatkan hasil segmentasi kemudian dilakukan proses ekstraksi fitur menggunakan metode glcm. proses ekstraksi fitur dilakukan untuk mendapatan ciri statistik pada citra. pada penelitian ini menggunakan glcm dengan derajat ketetanggan 0° diambil ciri statistik yang berupa kontras, korelasi, energi, homogenitas. sampel hasil ekstraksi fitur menggunakan glcm ditampilkan pada tabel 1. (a) (b) (c) (d) (e) gambar 3 (a) barin mri (b) histogram equalization (c) gray matter (d) cererospinl fluid (e) white matter selanjutnya akan dilakukan klasifikasi menggunakan svm. fitur-fitur yang telah diperoleh pada proses ekstraksi fitur akan digunakan sebagai parameter dalam klasifikasi. pada proses klasifikasi dilakukan melalui dua tahap yaitu tahap training dan tahap testing. tahap training merupakan tahap pembuatan model, sedangkan tahap tesing merupakan tahap pengujian keakuratan model. data training yang digunakan sebanyak 50 data dan data uji yang digunakan sebanyak 45 data yang kemudian akan diklasifikasi kedalam dua kelas yaitu alzheimer dan non-alzheimer. proses klasifikasi dilakukan dengan menggunakan kernel polinomial. tabel 1. sampel hasil ekstraksi fitur menggunakan glcm. data ke kontras korelasi energi homogen iti 1 0,608802 0,929677 0,700024 0,944456 2 0,830997 0,915252 0,662213 0,938277 3 0,682498 0,916594 0,720156 0,947706 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 88 4 0,660632 0,924824 0,705951 0,947408 5 0,710567 0,925752 0,686143 0,945724 6 0,733653 0,921226 0,690052 0,946255 7 0,757863 0,926632 0,662385 0,942731 8 0,788584 0,930982 0,636891 0,940234 9 0,802268 0,921175 0,626319 0,935237 10 0,815404 0,92574 0,629073 0,936168 dari hasil klasifikasi tersebut, terdapat 45 data testing, 14 data benar terklasifikasi nonalzheimer dan 28 data benar terklasifikasi alzheimer sehingga diperoleh akurasi sebesar 93,333% dengan tingkat sensitivitas 93,333% dan spesifisitas 93,333%. dari hasil tersebut menunjukkan bahwa metode ini mampu mengenali masing-masing kelas dengan sangat baik. sehingga metode ini cocok digunakan untuk mengidentifikasi alzheimer. 5. kesimpulan berdasarkan percobaan yang telah dilakukan, identifikasi penyakit alzheimer menggunakan fuzzy c-means untuk segmentasi fitur, gray level co-occurrence matrix untuk proses ekstraksi fitur sedangkan untuk klasifikasi menggunakan support vector machine diperoleh hasil terbaik dengan nilai akurasi sebesar 93,33%. berdasarkan hasil yang telah diperoleh, dapat disimpulkan bahwa metode tersebut sangat cocok digunakan untuk indentifikasi penyakit alzheimer dengan tingkat sensitifitas 93.33% dan spesifitas 93.33%. referensi [1] n. gharaibeh and a. a. kheshman, “automated detection of alzheimer disease using region growing technique and artificial neural network,” vol. 7, no. 5, pp. 204–208, 2013. 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[21] jiawei han and micheline kamber, data mining, second. san francisco: morgan kaufmann publisher, 2006. how to cite: a. mardhaningsih, “a note on the partition dimension of thorn of fan graph”, mantik, vol. 5, no. 1, pp. 45-49, may 2019. a note on the partition dimension of thorn of fan graph auli mardhaningsih andalas university, aulimardhaningsih@gmail.com doi: https://doi.org/10.15642/mantik.2019.5.1.45-49 abstrak: misalkan g adalah suatu graf terhubung.himpunan titikv(g) di partisi menjadi k buah partisi s1, s2,…, sk yang saling lepas. notasikan π = {s1, s2, ..., sk}.maka representasi v ∈v(g) terhadap phi didefenisikan : r(v|π)=(d(v,s1),d(v,s2),...,d(v,sk)), jika untuk setiap dua titik yang berbeda 𝑢, 𝑣 ∈ v(g) berlaku r(u|π) = r(v|π), maka π dikatakan partisi penyelesaian dari graf g. graf kipas diperoleh dari operasi graf hasil tambah k1+pn. graf kipas dinotasikan dengan 𝐹1,𝑛 untuk n ≥ 2. graf thorn untuk graf kipas diperoleh dengan cara menambahkan daun sebanyak li kesetiap titik di graf kipas, dinotasikan dengan 𝑇ℎ(𝐹1,𝑛, 𝑙1, 𝑙2, … , 𝑙𝑛+1). pada tulisan ini, akan dibahas tentang dimensi partisi graf thorn dari graf kipas f1,nuntuk n = 2, 3,4. kata kunci: partisi penyelesaian, dimensi partisi, graf kipas, graf thorn abstract: let 𝐺 = (𝑉, 𝐸) be a connected graph and 𝑆 ⊆ 𝑉(𝐺). for a vertex 𝑣 ∈ 𝑉(𝐺) and an ordered k-partition π = {𝑆1, 𝑆2, … , 𝑆𝑘 } of 𝑉(𝐺), the presentation of 𝑣 concerning π is the k-vector 𝑟(𝑣|π) = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), … , 𝑑(𝑣, 𝑆𝑘 )), where 𝑑(𝑣, 𝑆𝑖 ) denotes the distance between 𝑣 and 𝑆𝑖 for 𝑖 ∈ {1,2, … , 𝑛}. the k-partition π is said to be resolving if for every two vertices 𝑢, 𝑣 ∈ 𝑉(𝐺), the representation 𝑟(𝑢|π) ≠ 𝑟(𝑣|π). the minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). let 𝑉(𝐺) = {𝑥1, 𝑥2, … , 𝑥𝑛}. let 𝑙1, 𝑙2, … , 𝑙𝑛 be non-negative integer, 𝑙𝑖 ≥ 1,for 𝑖 ∈ {1,2, … , 𝑛}. the thorn of 𝐺, with parameters 𝑙1, 𝑙2, … , 𝑙𝑛 is obtained by attaching 𝑙𝑖 vertices of degree one to the vertex 𝑥𝑖, denoted by 𝑇ℎ(𝐺, 𝑙1, 𝑙2, … , 𝑙𝑛 ). in this paper, we determine the partition dimension of 𝑇ℎ(𝐺, 𝑙1, 𝑙2, … , 𝑙𝑛 )where 𝐺 ≃ 𝐹1,𝑛, the fan on n+1 vertices, for 𝑛 = 2,3,4. keywords: resolving partition, partition dimension, fan, thorn graph jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 45-49 issn: 2527-3159 (print) 2527-3167 (online) jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 45-49 46 1. introduction let 𝐺 = (𝑉, 𝐸) be an arbitrary connected graph. [1] defined the partition dimension as follows. let 𝑢 and 𝑣 be two vertices in 𝑉(𝐺). the distance 𝑑(𝑢, 𝑣) is the length of the shortest path between 𝑢 and 𝑣in 𝐺. for an ordered set π = {𝑆1, 𝑆2, … , 𝑆𝑘} of vertices in a connected graph 𝐺 and a vertex 𝑣of 𝐺, the k-vector 𝑟(𝑣|π) = (𝑑(𝑣, 𝑆1), 𝑑(𝑣, 𝑆2), … , 𝑑(𝑣, 𝑆𝑘)), is the presentation of 𝑣 with respect to π. the minimum k for which there is a resolving k-partition of 𝑉(𝐺) is called the partition dimension of 𝐺, denoted by 𝑝𝑑(𝐺). all notation in graph theory needed in this paper refers to [2]. stated the following theorem. theorem 1.1. [2] let 𝐺 be a connected graph on n vertices, 𝑛 ≥ 2. then 𝑝𝑑(𝐺) = 2 if and only if 𝐺 ≃ 𝑃𝑛. in the same paper, chartrand et al. [2] also gave the necessary condition in partitioning the set of vertices as follows. lemma 1.2. [2] suppose that π is the resolving partition of 𝑉(𝐺)and 𝑢, 𝑣 ∈ 𝑉(𝐺). if 𝑑(𝑢, 𝑤) = 𝑑(𝑣, 𝑤) for every vertex 𝑤 ∈ 𝑉(𝐺)\{𝑢, 𝑣} then 𝑢 and 𝑣 belong to a different class of π. 2. main results the fan 𝐹1,𝑛 on 𝑛 + 1 vertices is defined as the graph constructed by joining 𝐾1 and 𝑃𝑛, denoted by 𝐾1 + 𝑃𝑛 where 𝐾1 is the complete graph on 1 vertex and 𝑃𝑛 is a path on 𝑛 vertices, for 𝑛 ≥ 2. the vertex set and edge set of 𝐹1,𝑛 are as follows. 𝑉(𝐹1,𝑛) = {𝑥𝑖 |1 ≤ 𝑖 ≤ 𝑛 + 1}, 𝐸(𝐹1,𝑛) = {𝑥1𝑥𝑡 |1 ≤ 𝑡 ≤ 𝑛} ∪ {𝑥𝑠 𝑥𝑠+1|1 ≤ 𝑠 ≤ 𝑛 − 1}. 𝑙1, 𝑙2, … , 𝑙𝑛+1be some positive integer. the thorn graph of 𝐹1,𝑛 is obtained by adding 𝑙𝑖 leaves to vertex 𝑥𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 + 1, denoted by 𝑇ℎ(𝐹1,𝑛, 𝑙1, 𝑙2, … , 𝑙𝑛+1). the construction of thorn graph is taken from [3]. the vertex set and edge set of 𝐻 ≃ 𝑇ℎ(𝐹1,𝑛, 𝑙1, 𝑙2, … , 𝑙𝑛+1) are as follows. 𝑉(𝐻) = {𝑥𝑖 |1 ≤ 𝑖 ≤ 𝑛 + 1} ∪ {𝑥𝑖𝑗 |1 ≤ 𝑖 ≤ 𝑛 + 1, 1 ≤ 𝑗 ≤ 𝑙𝑖 }, and 𝐸(𝐻) = {𝑥1𝑥𝑡 |1 ≤ 𝑡 ≤ 𝑛} ∪ {𝑥𝑠 𝑥𝑠+1|1 ≤ 𝑠 ≤ 𝑛 − 1} ∪ {𝑥𝑖 𝑥𝑖𝑗 |1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑛}. in theorem 2.1 we determine the partition dimension of 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3)for 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3. theorem 2.1. let 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3) be thorn of fan 𝐹1,2with 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3. denote 𝑙𝑚𝑎𝑥 = max {𝑙1, 𝑙2, 𝑙3}. the partition dimension of 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3) is 𝑝𝑑(𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3)) = { 3, 𝑓𝑜𝑟 𝑙𝑚𝑎𝑥 = 1, 2 𝑜𝑟 3 𝑙𝑚𝑎𝑥 , 𝑓𝑜𝑟 𝑙𝑚𝑎𝑥 ≥ 4 a. mardhaningsih a note on the partition dimension of thorn of fan graph 47 figure 1. 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3) proof. the proof is divided into two cases. case 1. 1 ≤ 𝑙𝑚𝑎𝑥 ≤ 3. let 𝐻1 ≃ 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3), with 1 ≤ 𝑙𝑚𝑎𝑥 ≤ 3. because 𝐻1 ≠ 𝑃𝑛 then from theorem 1.1, it is obtained that 𝑝𝑑(𝐻1) ≥ 3. next, it will be shown that 𝑝𝑑(𝐻1) ≤ 3 by constructing three ordered partitions. note that from lemma 1.2, every leaf at the vertex 𝑥𝑖 must be on a different partition. therefore, we define π = {𝑆1, 𝑆2, 𝑆3}, where 𝑆𝑖 = {𝑥𝑖 , 𝑥𝑘𝑖 |1 ≤ 𝑖 ≤ 3, 1 ≤ 𝑘 ≤ 3}, because of 𝑑(𝑣, 𝑆𝑖) = 0 while 𝑑(𝑢, 𝑆𝑖) ≠ 0 for 𝑣 ∈ 𝑆𝑖 and 𝑢 ∉ 𝑆𝑖 , it is clear that every two vertices in different partitions have different representations. therefore, it is sufficient to check the representations of two vertices in the same partition. because of 𝑑(𝑥𝑘𝑖 , 𝑆𝑗 ) = 𝑑(𝑥𝑖 , 𝑆𝑗 ) + 1for 𝑖 ≠ 𝑗, 1 ≤ 𝑖, 𝑗 ≤ 3, then 𝑟(𝑥𝑘𝑖 |π) ≠ 𝑟(𝑥𝑖 |π). thus, we have that 𝑝𝑑(𝐻1) ≤ 3. case 2. 𝑙𝑚𝑎𝑥 ≥ 4. let 𝐻2 ≃ 𝑇ℎ(𝐹1,2, 𝑙1, 𝑙2, 𝑙3), with 𝑙𝑚𝑎𝑥 ≥ 4. let 𝑙𝑚𝑎𝑥 = 𝑚 and suppose that 𝑝𝑑(𝐻2) = 𝑚 − 1. then we have π = {𝑆1, 𝑆2, … , 𝑆𝑚−1}. thus there are at least two vertices, namely 𝑥1𝑝 and 𝑥1𝑞 , in the same partition, for 1 ≤ 𝑝, 𝑞 ≤ 𝑚. but from lemma 1.2, 𝑥1𝑝 and 𝑥1𝑞 must be placed in different partitions. therefore, |π| ≥ 𝑚, a contradiction. next, we construct π = {𝑆1, 𝑆2, … , 𝑆𝑚}, where 𝑆𝑖 = {𝑥𝑖 , 𝑥𝑘𝑖 |1 ≤ 𝑖 ≤ 3, 1 ≤ 𝑘 ≤ 3}, 𝑆𝑗 = {𝑥𝑘𝑗 |1 ≤ 𝑘 ≤ 3, 4 ≤ 𝑗 ≤ 𝑙𝑚𝑎𝑥 }, because of 𝑑(𝑥𝑘𝑖 , 𝑆𝑗 ) = 𝑑(𝑥𝑖 , 𝑆𝑗 ) + 1for 𝑖 ≠ 𝑗, 1 ≤ 𝑖, 𝑗 ≤ 𝑙𝑚𝑎𝑥 , then 𝑟(𝑥𝑘𝑖 |π) ≠ 𝑟(𝑥𝑖 |π). next, because of 𝑑(𝑥𝑘𝑖 , 𝑆𝑗 ) ≠ 𝑑(𝑥𝑙𝑖 , 𝑆𝑗 ) + 1for 𝑘 ≠ 𝑙, 1 ≤ 𝑘, 𝑙 ≤ 𝑙𝑚𝑎𝑥 , it is clear that 𝑟(𝑥𝑘𝑖 |π) ≠ 𝑟(𝑥𝑙𝑖 |π). therefore, we have 𝑝𝑑(𝐻2) ≤ 𝑙𝑚𝑎𝑥 . ∎ in theorem 2.2 we determine the partition dimension of 𝑇ℎ(𝐹1,3, 𝑙1, 𝑙2, 𝑙3, 𝑙4)for 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3,4. theorem 2.2. let 𝑇ℎ(𝐹1,3, 𝑙1, 𝑙2, 𝑙3, 𝑙4) be a thorn of fan 𝐹1,3with 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3,4. denote 𝑙𝑚𝑎𝑥 = max {𝑙1, 𝑙2, 𝑙3, 𝑙4}. let 𝑥𝑙𝑖 be the vertex in 𝐹1,3 with 𝑙𝑖 leaves, and |𝑥𝑙𝑚𝑎𝑥 | be the number of vertices with 𝑙𝑚𝑎𝑥 leaves. the partition dimension of 𝑇ℎ(𝐹1,3, 𝑙1, 𝑙2, 𝑙3, 𝑙4) is jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 45-49 48 figure2. 𝑇ℎ(𝐹1,3, 𝑙1, 𝑙2, 𝑙3, 𝑙4) proof. the proof is similar to the proof of theorem 2.1 ∎ in theorem 2.3 we determine the partition dimension of 𝑇ℎ(𝐹1,4, 𝑙1, 𝑙2, 𝑙3, 𝑙4, 𝑙5)for 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3,4,5. theorem 2.3. let 𝑇ℎ(𝐹1,4, 𝑙1, 𝑙2, 𝑙3, 𝑙4, 𝑙5) be a thorn of fan 𝐹1,4with 𝑙𝑖 ≥ 1, 𝑖 ∈ 1,2,3,4,5. denote 𝑙𝑚𝑎𝑥 = max {𝑙1, 𝑙2, 𝑙3, 𝑙4, 𝑙5}. let 𝑥𝑙𝑖 be the vertex in 𝐹1,4 with 𝑙𝑖 leaves, and |𝑥𝑙𝑚𝑎𝑥 | be the number of vertices with 𝑙𝑚𝑎𝑥 leaves. the partition dimension of 𝑇ℎ(𝐹1,3, 𝑙1, 𝑙2, 𝑙3, 𝑙4, 𝑙5) is a. mardhaningsih a note on the partition dimension of thorn of fan graph 49 figure 3. 𝑇ℎ(𝐹1,4, 𝑙1, 𝑙2, 𝑙3, 𝑙4, 𝑙5) proof. the proof is similar to the proof of theorem 2.1 and theorem 2.2. references [1] g. chartrand, s. e and z. p, "the partition dimension of a graph," aequationes math, pp. 45-54, 2000. [2] j. a. bondy and u. murty, graph theory with applications, london, 1976. [3] a. "partition dimension of amalgamation," bulletin of mathematics, pp. 161-167, 2012. 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[16] i. tomescu, "discrepancies between metric dimension of a connected graph," discrete math, pp. 5026-5031, 2008. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran aplikasi algoritma classify-by-sequence untuk penilaian kredit pada bank y mohammad iqbal 1 jurusan matematika, fmipa-institut teknologi sepuluh nopember 1 iqbalmohammad.math@gmail.com 1 abstrak dalam penilaian kredit, suatu bank dalam memutuskan pelanggannya aman atau tidak untuk diberikan kredit harus melihat pada beberapa faktor. beberapa faktor tersebut dapat diperoleh dari data historis pelanggan, dari data historis pelanggan dapat dilakukan analisa dalam penerimaan atau penolakan pemberian kredit kepada pelanggan. analisa penilaian kredit dapat dilakukan dengan menggunakan teknik data mining. salah satu teknik data mining yang dapat digunakan adalah klasifikasi. pada penelitian ini, dilakukan klasifikasi pelanggan yang dikategorikan menjadi kredit aman atau kredit macet dan data historis pelanggan yang digunakan adalah data temporal maka digunakan algoitma classify by sequence. simulasi yang dilakukan variasi minimum support dan jumlah data training dan testingnyadan hasil yang diperoleh pada data training 70% dengan minimum support krang dari sama dengan 0.2 diperoleh akurasi 100%, coverage 43% dan covacc 1.44 . kata kunci:data mining; data temporal; klasifikasi; sequential; algoritma cbs; penilaian kredit 1. pendahuluan saat ini seseorang sangat membutuhkan adanya kredit untuk memenuhi kebutuhannya yang diberikan suatu bank atau penyedia jasa kredit lainnya. begitu pula, pengusaha menginginkan usahanya untuk dibangun lebih baik lagi pasti membutuhkan bantuan kredit yang diberikan oleh bank atau penyedia jasa kredit lainnya. namun untuk memberikan kredit suatu bank atau penyedia jasa kredit lain tidak secara langsung memberikan kepada seseorang ataupun pengusaha agar tidak mengalami kerugian besar jika terjadi kredit macet. untuk menentukan seseorang atau pengusaha diberikan kredit, bank dan penyedia jasa kredit lain harus melakukan penilaian kredit atau credit scoring terhadap data historis suatu pelanggan atau usaha. penilaian kredit adalah nilai numerik yang ditentukan oleh model statistik berdasarkan histori dari kebiasan kredit dengan memprediksi kedekatan/kesamaan dari pemberi kredit di masa yang akan datang [1]. dari penilaian kredit seorang pelanggan atau pengusaha tersebut yang digunakan bank atau penyedia jasa kredit dalam menentukan pemberian kredit kepada debitur. pembentukan model statistik untuk penilaian kredit dapat dilakukan dengan berbagai cara. salah satu teknik yang dapat digunakan adalah data mining. data mining merupakan salah satu teknik yang ada dalam bidang ilmu komputer yang perkembangannya sangat pesat. tujuan dari data mining adalah untuk menemukan informasi yang sangat berguna dari database yang sangat besar [2]. terdapat beberapa teknik dalam data mining yaitu klasifikasi, aturan asosiasi, pola sekuen, klastering. klasifikasi adalah tugas pembelajaran fungsi target f yang memetakan setiap himpunan atribut x pada label kelas yang telah didefinisikan sebelumnya [3]. teknik klasifikasi diantaranya adalah decision tree, bayesian network, neural network, dan lain – lain. beberapa metode data mining yang telah diaplikasikan untuk penilaian kredit antara lain dengan metode decision network [4], k-nn [5] dan feature selection [6]. pada penelitian ini, digunakan algoritma classify by sequence yang mengintegrasikan teknik klasifikasi dan sekuen [2]. algoritma ini mampu bekerja pada data temporal dan data historis yang digunakan merupakan data historis suatu usaha dalam mengajukan kredit dan termasuk data temporal. data historis tersebut terdiri atas plafond kredit, suku bunga atau rate, maksimum kredit, produk kredit, usaha kredit dan jangka waktu serta kelas yang dikategorikan menjadi dua buah kelas yaitu kredit aman atau kredit macet. tata penulisan pada makalah ini antara lain : pendahuluan, penjelaan mengenai penilaian kredit, algoritma classify by sequence, evaluasi hasil penelitian dan penarikan kesimpulan. 2. penilaian kredit kredit merupakan salah satu mekanisme pembayaran yang sangat umum di masyarakat. fungsi pokok kredit yaitu untuk memenuhi pelayanan terhadap kebutuhan masyarakat dalam rangka memperlancar perdagangan, produksi dan jasa – jasa bahkan konsumsi yang kesemuanya itu meningkatkan kesejahteraan manusia [3]. seiring berjalannya waktu pengajuan permohonan kredit(debitur) semakin meningkat akan tetapi terdapat beberapa kredit macet diantaranya sehingga dapat menyebabkan kerugian kepada kreditur, sehingga para pemberi kredit (kreditur) perlu berhati – hati dalam menentukan apakah seorang debitur aman untuk diberikan kredit atau tidak. dalam menentukan pemberian kredit terhadap debitur para kreditur menggunakan penilaian kredit. penilaian kredit adalah suatu alat yang melibatkan model statistic untuk mengevaluasi seluruh informasi yang tersedia dengan objektif dalam pengambilan keputusan kredit [3]. seperti yang telah dijelaskan sebelumnya, untuk membentuk model statistic tersebut dapat digunakan teknik yang ada pada data mining dan pada penelitian ini teknik data mining yang digunakan adalah algoritma classify by sequence. 3. algoritma classify by sequence algoritma classify by sequence terdiri atas dua tahap algoritma antara lain algoritma classifiable sequence pattern dan algoritma classifier builder [2]. pada algoritma classifiable sequence pattern dibentuk atau ditentukan suatu pola dari dataset yang diperoleh yang memenuhi minimum support yang telah diberikan. pola yang telah diperoleh merupakan barisan yang mungkin terklasifikasi dengan baik karena memiliki frekuensi yang lebih besar dari minimum support yang disebut classifiable sequence pattern /csp. pada gambar 1 ditunjukkan diagram alir dari algoritma classifiable sequence pattern gambar 1. diagram alir algoritma classifiable sequence pattern sedangkan pada algoritma classifier builder dilakukan proses pembentukan model dari csp – csp yang telah diperoleh yang natinya start end input : min_supp, dataset untuk setiap kelas pada dataset dibentuk csp kondisi fk > min_sup ya tidak digunakan sebagai prediksi pada data testing untuk menentukan apakah debitur dapat dikategorikan kredit aman atau macet. pada gambar 2 ditunjukkan diagram alir dari algoritma classifier builder. gambar 2. diagram alir algoritma classifier builder 4. evaluasi hasil penelitian pada penelitian ini digunakan data historis kredit yang diperoleh dari bank y dengan pemohon kredit merupkan sektor usaha. data yang diambil adalah sektor usaha, produk kredit, suku bunga, maksimum kredit, plafond kredit, jangka waktu suku bunga dan kategori kredit dalam satu bulan yang dapat dilihat pada tabel 1. tabel 1. contoh dataset historis debitur plafond kredit produk kredit maksimum kredit suku bunga jangka waktu sektor usaha kelas 1,000,000,000 kmk bni usaha kecil idr 1,000,000,000 13.5 12 bangunan jalan jembatan dan landasan 1 750,000,000 kmk bni usaha kecil idr 750,000,000 14 3 perdagangan eceran bahan bakar dan minyak pelumas 1 1,250,000,000 kmk bni usaha kecil idr 1,250,000,000 14 12 jasa kegiatan lainnya 1 7,555,034,251 ki bni usaha kecil idr 7,555,034,251 14 82 konstruksi perumahan sederhana lainnya tipe s.d. 21 1 1,200,000,000 ki bni usaha kecil idr 1,200,000,000 14 60 jasa kegiatan lainnya 1 4,700,000,000 ki bni usaha kecil idr 4,700,000,000 14 24 jasa pelayanan bongkar muat barang 1 5,000,000,000 ki bni usaha kecil idr 5,000,000,000 14 24 angkutan laut domestik 1 2,500,000,000 tl bni usaha kecil idr 2,500,000,000 14 24 angkutan laut domestik 1 4,000,000,000 kmk bni usaha kecil idr 4,000,000,000 13.25 12 perdagangan eceran komoditi lainnya (bukan makanan, minuman, atau tembakau) 1 600,000,000 kmk bni usaha kecil idr 600,000,000 15 12 perdagangan eceran komoditi lainnya (bukan makanan, minuman, atau tembakau) 1 1,000,000,000 kmk bni usaha kecil idr 1,000,000,000 14 12 perdagangan besar dalam negeri hasil perikanan 1 proses transformasi data digunakan dengan mengubah data numeric menjadi data kategorial dengan menggunakan metode normalisasi data yaitu (1) dengan x merupakan dataset dan k merupakan jumlah kategorial. pada tabel 2 diberikan contoh hasil transformasi dataset. simulasi yang telah dilakukan adalah dengan melakukan variasi jumlah data training 70%, start end input : csp set, class csp terkategori dalam kelas k class isexist? ya tidak perhitungan frekuensi class terbesar 80% dan 90% dari jumlah dataset yang diberikan serta variasi minimum support. diberikan persamaan akurasi, coverage dan covacc yang merupakan gabungan antara akurasi dan coverage antara lain : (2) (3) (4) berikut hasil simulasi yang telah dilakukan tabel 3. simulasi dengan data training 70% minimum support akurasi coverage covacc 0.9 0 0 0 0.8 0 0 0 0.7 0 0 0 0.6 0 0 0 0.5 0 0 0 0.4 0 0 0 0.3 0 0 0 0.2 100 42.8064 1.4281 0.1 100 42.8064 1.4281 tabel 4. simulasi dengan data training 80% minimum support akurasi coverage covacc 0.9 0 0 0 0.8 0 0 0 0.7 0 0 0 0.6 0 0 0 0.5 0 0 0 0.4 0 0 0 0.3 3.7267 25.0389 0.0466 0.2 100 25.0389 1.2504 0.1 100 25.0389 1.2504 tabel 5. simulasi dengan data training 90% minimum support akurasi coverage covacc 0.9 0 0 0 0.8 0 0 0 0.7 0 0 0 0.6 0 0 0 0.5 0 0 0 0.4 0 0 0 0.3 0 0 0 0.2 1.25 11.0497 0.0139 0.1 100 11.0497 1.1105 tabel 2. contoh transformasi data atribut dataset plafond kredit produk kredit maksimum kredit suku bunga jangka waktu sektor usaha kelas 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'perdagangan eceran bahan bakar dan minyak pelumas' 'kredit aman' 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'jasa kegiatan lainnya' 'kredit aman' 'plafond kredit normal' 'ki bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'konstruksi perumahan sederhana lainnya tipe s.d. 21' 'kredit aman' 'plafond kredit rendah' 'ki bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'jasa kegiatan lainnya' 'kredit aman' 'plafond kredit rendah' 'ki bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'jasa pelayanan bongkar muat barang' 'kredit aman' 'plafond kredit normal' 'ki bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'angkutan laut domestik' 'kredit aman' 'plafond kredit rendah' 'tl bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'angkutan laut domestik' 'kredit aman' 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'perdagangan eceran komoditi lainnya (bukan makanan, minuman, atau tembakau)' 'kredit aman' 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'perdagangan eceran komoditi lainnya (bukan makanan, minuman, atau tembakau)' 'kredit aman' 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'perdagangan besar dalam negeri hasil perikanan' 'kredit aman' 'plafond kredit rendah' 'kmk bni usaha kecil idr' 'maksimum kredit rendah' 'suku bunga rendah' 'jangka waktu cepat' 'konstruksi gedung lainnya' 'kredit aman' dari tabel 3 sampai tabel 5 ditunjukkan bahwa terdapat akurasi, coverage dan covacc yang nilainya nol yang memiliki arti bahwa algoritma classify by sequence tidak dapat menghasilkan rue atau modelminimum support yang lebih dari 0.3 dan hasil akurasi, coverage dan covacc yang lebih baik didapat pada saat data training 70% dan data testing 30%. contoh hasil rule atau model yang dibangkitkan pada data training 70% dan minimum support 0.2 sebagai berikut if 'tl bni usaha kecil idr' then 'kredit aman' if 'plafond kredit rendah' and 'tl bni usaha kecil idr' then 'kredit aman' if 'plafond kredit rendah' and 'ki bni usaha kecil idr' and 'suku bunga rendah' and 'jangka waktu cepat' then 'kredit macet' 5. kesimpulan dari hasil simulasi dapat disimpulkan bahwa algoritma classify by sequence mampu untuk membuat model dari data historis debitur sebagai penilaian kredit yang dapat digunakan oleh kreditur. dari simulasi menggunakan data training 70% dari dataset keseluruhan diperoleh model atau rule yang digunakan pada data testing sehingga diperoleh coverage sekitar 43% yang artinya model yang dihasilkan hanya mampu mengcover tidak sampai separuh data akan tetapi akurasi 100% artinya ketepatan model yang dihasilkan sangat akurat. referensi [1] mac, f. understanding credit scoring instructor guide. module 6 credit smart ® . (2013) [2] tseng, v. s., lee, c. h. , effective temporal data classification by integrating sequential pattern mining and probabilistic induction, journal of expert system with applications 36 p p (2009) 9524-9532. [3] mitsa, c., temporal data mining, a chapman & hall/crc., new york. (2010). [4] yogi, y. w., praktiko, f. r., vivianne a. s., evaluasi pemohon kredit mobil di pt “x” dengna menggunakan teknik data mining decision tree , simposium nasional rapi viii p p (2009) i42-i49. [5] nugroho, a., kusrini, arief, r. , sistem pendukung keputusan kredit usaha rakyat pt. bank rakyat indonesia unit kaliangkrik magelang, citec journal vol 2 no 1 p p (2015) 1-15. [6] koutanaei, f. n., sajedi, h., khanbabaei, m., a hybrid data mining model of feature selection algorithms and ensemble learning classifiers for credit scoring, journal of retailing and consumer services 27 p p (2015) 11-23. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: m. fajar, “an application of hybrid forecasting singular spectrum analysis– extreme learning machine method in foreign tourists forecasting”, mantik, vol. 5, no. 2, pp. 60-68, october 2019. an application of hybrid forecasting singular spectrum analysis – extreme learning machine method in foreign tourists forecasting muhammad fajar bps-statistics indonesia, mfajar@bps.go.id doi: https://doi.org/10.15642/mantik.2019.5.2.60-68 abstrak: wisman adalah salah satu indikator untuk melihat perkembangan pariwisata. perkembangan pariwisata mempunyai andil penting bagi perekonomian karena pariwisata merupakan booster peningkatan devisa, menciptakan peluang usaha, dan membuka kesempatan kerja. sebagai bahan input untuk strategi dan program pariwisata adalah prediksi terhadap jumlah wisman di masa depan yang diperoleh dari peramalan. dalam paper ini, penulis menggunakan metode hybrid singular spectrum analysis – extreme learning machine untuk meramalkan jumlah wisman. data yang digunakan dalam penelitian adalah jumlah wisman yang bersumber dari badan pusat statistik. hasil penelitian ini bahwa kemampuan metode hybrid ssa-elm sangat baik dalam meramalkan jumlah wisman. hal tersebut ditunjukkan oleh nilai mape sebesar 4.91 persen, dengan out sample sebanyak delapan observasi. kata kunci: wisata mancanegara, singular spectrum analysis, extreme learning machine abstract: international tourism is one indicator of measuring tourism development. tourism development is important for the national economy since tourism could boost foreign exchange, create business opportunities, and provide employment opportunities. the prediction of foreign tourist numbers in the future obtained from forecasting is used as an input parameter for strategy and tourism programs planning. in this paper, the hybrid singular spectrum analysis – extreme learning machine (ssa-elm) is used to forecast the number of foreign tourists. data used is the number of foreign tourists january 1980 december 2017 taken from badan pusat statistik (statistics indonesia). the result of this research concludes that hybrid ssa-elm performance is very good at forecasting the number of foreign tourists. it is shown by the mape value of 4.91 percent with eight observations out a sample. keywords: foreign tourist, singular spectrum analysis, extreme learning machine jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 60-68 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 m. fajar an application of hybrid forecasting singular spectrum analysis–extreme learning machine method in foreign tourists forecasting 61 1. introduction indonesia is a country that has a lot of mesmerizing landscapes, natural resources, and diverse cultures. these are tourist attractions that could be optimally utilized to advance the national economy. tourism development is important for the national economy since tourism could boost foreign exchange, create business opportunities, and provide employment opportunities. one indicator of measuring tourism development is the number of foreign tourists to visit. the number of foreign tourists visiting indonesia from january 1980 to december 2017 is visually presented in fig. 1.1. it could be seen that in this period, the number of foreign tourists visiting indonesia shows an increasing trend each year. the implication for indonesia as the host is that the strategy of infrastructural development is needed to avoid the decreasing number of foreign tourists and to lower the negative impacts to the environment caused by the increasing number of foreign tourists visit. such as environment based transportation vehicles, hotels, recreation facilities, etc. figure 1.1. the number of foreign tourists january 1980 – december 2017 therefore, one input for the strategy is the prediction number of foreign tourists obtained from forecasting, specifically, time series forecasting. time series forecasting is a quantitative method used to analyze a series of data collected in time order using the right technique. the result could be used as a reference to forecast the value of the series in the future [9]. the development of forecasting methods is increasingly rapid and complex as advances in the development of computing technology. the interesting thing from the time series method development is the reconstruction of hybrid time series forecasting method, a time series constructed from two different types of forecasting method [1] – [5]. in this paper, singular spectrum analysis and extreme learning machine techniques are combined to forecast the number of foreign tourists visiting indonesia. extreme learning machine is an exclusive example of feed-forward neural network in the form of ffnn with only one hidden layer, where the singular spectrum analysis – ffnn method has been developed before [5]. time w is m an t 1980 1990 2000 2010 0 20 00 00 60 00 00 10 00 00 0 14 00 00 0 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 60-68 62 2. data data used in this research is the number of foreign tourists visiting indonesia from january 1980 until august 2018. train data (in the sample) covers the number of foreign tourists in january 1980 until august 2018. meanwhile, test data (out sample) covers the number of foreign tourists from january 2018 until august 2018. 3. method 3.1 singular spectrum analysis (ssa) ssa is a non-parametric time series method based on multivariate statistics principle. ssa decomposes time series additively into several independent components. these components could be identified as trend, periodic, quasiperiodic, and noise component. ssa procedure consists of four steps, they are [6][10]: step 1. embedding given an 𝑥1,𝑥2,…, 𝑥𝑇 time series, choose an even number 𝐿, where 𝐿 parameter is the window length defined as 2 < 𝐿 < 𝑇 2⁄ , and 𝐾 = 𝑇 − 𝐿 + 1. the cross matrix is: 𝑿 = (𝑋1,…,𝑋𝑇) = ( 𝑥1 𝑥2 𝑥2 𝑥3 ⋯ 𝑥𝐾 ⋯ 𝑥𝐾+1 ⋮ ⋮ 𝑥𝐿 𝑥𝐿+1 ⋱ ⋮ ⋯ 𝑥𝑇 ) the cross matrix proves to be a hankel matrix, which means every element in the main anti diagonal has the same value. thus, 𝑿 could be assumed as multivariate data with l characteristic and k observations so that the covariance matrix is 𝑺 = 𝑿𝑿′ with the dimension of 𝐿 × 𝐿. step 2. singular value decomposition (svd) suppose that 𝑺 has eigenvalue and eigenvector 𝜆𝑖 and 𝑈𝑖, respectively. where 𝜆1 ≥ 𝜆2 ≥ ⋯ ≥ 𝜆𝐿 and 𝑈1,…,𝑈𝐿. thus, obtained svd from 𝑿 as follows: 𝑿 = 𝐸1 + 𝐸2 + ⋯+ 𝐸𝑑 (1) where 𝐸𝑖 = √𝜆𝑖𝑈𝑖𝑉𝑖 ′ , 𝑖 = 1,2,…,𝑑, 𝐸𝑖 is the main component, 𝑑 is the number of eigenvalue 𝜆𝑖, and 𝑉𝑖 = 𝑿 ′ 𝑈𝑖 √𝜆𝑖⁄ . step 3. grouping in this step, 𝑿 is additively grouped into subgroups based on patterns that form a time series. they are trend, periodic, quasi-periodic, and noise component. partition the index set {1,2,…,𝑑} into several groups 𝐼1, 𝐼2,…,𝐼𝑛, then correspond matrix 𝑿𝐼 into group 𝐼 = {𝑖1, 𝑖2,…, 𝑖𝑏} which is defined as: m. fajar an application of hybrid forecasting singular spectrum analysis–extreme learning machine method in foreign tourists forecasting 63 𝑿𝐼 = 𝐸𝑖1 + 𝐸𝑖2 + ⋯+ 𝐸𝑖𝑏 (2) thus, the decomposition represents as: 𝑿 = 𝑿𝐼1 + 𝑿𝐼2 + ⋯+ 𝑿𝐼𝑛 (3) with 𝑿𝐼𝑗(𝑗 = 1,2,…,𝑛) is reconstructed component (rc). 𝑿𝐼 component contribution measured with corresponding eigenvalue contribution: ∑ 𝜆𝑖𝑖∈𝐼 ∑ 𝜆𝑖 𝑑 𝑖=1⁄ . using the close frequency range from the main components is based on the study of the grouping process using auto grouping [11]. main components with relatively close frequency ranges are grouped into one reconstructed component. soon, until several reconstructed components are formed. step 4. reconstruction in this last step, 𝑿𝐼𝑗 is transformed into a new time series with t observations obtained from diagonal averaging or hankelization. suppose that 𝒀 is a matrix with 𝐿 × 𝐾 dimensions and has 𝑦𝑖𝑗,1 ≤ 𝑖 ≤ 𝐿,1 ≤ 𝑗 ≤ 𝐾 elements. then, 𝐿 ∗ = min(𝐿,𝐾),𝐾∗ = max(𝐿,𝐾),and 𝑇 = 𝐿 + 𝐾 − 1. then, 𝑦𝑖𝑗 ∗ = 𝑦𝑖𝑗 if 𝐿 < 𝐾 and 𝑦𝑖𝑗 ∗ = 𝑦𝑗𝑖 if 𝐿 > 𝐾. matrix 𝒀 transferred into 𝑦1,𝑦2,…,𝑦𝑇 series with using the following formula: 𝑦𝑘 = { 1 𝑘 ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝑘 𝑚=1 ,1 ≤ 𝑘 ≤ 𝐿∗ 1 𝐿∗ ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝐿 𝑚=1 ,𝐿∗ ≤ 𝑘 ≤ 𝐾∗ 1 𝑇 − 𝑘 + 1 ∑ 𝑦𝑚,𝑘−𝑚+1 ∗ 𝑇−𝐾∗+1 𝑚=𝑘−𝐾∗+1 ,𝐾∗ ≤ 𝑘 ≤ 𝑇 (4) diagonal averaging on equation (4) is applied to every matrix component 𝑿𝐼𝑗 on equation (3) resulting a �̃�(𝑘) = (�̌�1 (𝑘) , �̌�2 (𝑘) ,…, �̌�𝑇 (𝑘) ) series. thus, 𝑥1,𝑥2,…,𝑥𝑇 series is decomposed into an addition of reconstructed m series: 𝑥𝑡 = ∑�̌�𝑡 (𝑘) 𝑚 𝑘=1 , 𝑡 = 1,2,…,𝑇 (5) ssa forecasting used in this research is ssa recurrent, with estimating min-norm lrr (linear recurrence relationship) coefficient. the lrr coefficient is calculated with the following algorithm: 1. input: matrix 𝐏 = [𝑃1:…:𝑃𝑟], 𝐏 is a matrix composed of 𝑃𝑖 eigenvector from svd step, where select a group of r (1 ≤ r ≤ l) eigenvectors 𝑃1,…,𝑃𝑟. suppose that 𝐏 is a 𝐏 that the last row is removed, and 𝐏 is a 𝐏 that the first row is removed. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 60-68 64 2. for every 𝑃𝑖 vector column from 𝐏, calculate 𝜋𝑖, where 𝜋𝑖 is the last component from 𝑃𝑖 , and 𝑃𝒊 is a 𝑃𝑖 that the last component is removed. 3. calculate: 𝑣2 = ∑ 𝜋𝑖 2𝑟 𝑖=1 . if 𝑣 2 = 1, then stop with a warning message “verticality coefficient equals 1.” 4. calculate the min-norm lrr coefficient (ℛ): ℛ = 1 1 − 𝑣2 ∑𝜋𝑖𝑃𝒊 𝑟 𝑖=1 5. from point (4) obtained: ℛ = (𝛼𝐿−1 …𝛼1) ′. 6. then, calculate the forecasting value with: �̂�𝑛 = ∑𝛼𝑖�̃�𝑛−1 𝐿−1 𝑖=1 , 𝑛 = 𝑇 + 1,…. ,𝑇 + ℎ 3.2 extreme learning machine (elm) extreme learning machine is a learning scheme of feedforward neural network for single-hidden layer feedforward neural networks (slfn). elm could adaptively set the number of nodes and randomly chooses the input weight w and bias 𝒃𝑖 on a hidden layer. hidden layer weight is obtained by using the least square method [12]. suppose that a slfn training process with 𝐾 hidden nodes and an activation vector function 𝒈(�̌�) = (𝑔1(�̌�),𝑔2(�̌�),…,𝑔𝐾(�̌�)) for 𝑁 samples (�̌�𝑖,𝒑𝑖) learning process, with �̌�𝑖 = [�̌�𝑖1, �̌�𝑖2,…, �̌�𝑖𝑛] ′ and 𝒑𝑖 = [𝑝𝑖1,𝑝𝑖2,…,𝑝𝑖𝑛] ′ is performed. if slfn could approx 𝑁 samples without any error (zero error), thus: ∑‖𝒚𝑗 − 𝒑𝑗‖ 𝑁 𝑗=1 = 0, (6) with 𝒚𝑗 is actual output value of slfn. there are also parameter 𝜷𝑖 = [𝛽𝑖1,…,𝛽𝑖𝕞] ′,𝒘𝑖 = [𝑤𝑖1,…,𝑤𝑖𝕞]′ and 𝑏𝑖 which are interconnected in: ∑𝛽𝑖𝑔𝑖(𝒘𝑖�̌�𝑗 + 𝑏𝑖) 𝐾 𝑖=1 = 𝒑𝑗, 𝑗 = 1,…,𝑁,𝑖 = 1,…,𝐾 (7) 𝒘𝑖 is a weight vector connecting the 𝑖-th hidden node and the input node, 𝜷𝑖 is a weight vector connecting the 𝑖-th hidden node and the output node, and 𝑏𝑖 is the threshold of the 𝑖-th hidden node. equation (7) could be expressed as: 𝐇𝛃 = 𝐓 (8) with 𝐇 = {ℎ𝑖𝑗} is the output matrix of the hidden layer, ℎ𝑖𝑗 = 𝑔(𝑤𝑗�̌�𝑖 + 𝑏𝑗) represents the output of 𝑗-th hidden neuron corresponding with �̌�𝑖, 𝛃 = m. fajar an application of hybrid forecasting singular spectrum analysis–extreme learning machine method in foreign tourists forecasting 65 [𝜷1,…,𝜷𝐾] is the weight output matrix, and 𝐓 = [𝒑1,…,𝒑𝑁]′ is the target matrix. the output weight (weight connecting the hidden layer and the output) is obtained from finding the solution of least square from the linear system given. the solution of the linear system (8) is: �̂� = 𝐇+𝐓 (9) with 𝐇+ is moore-penrose generalized inverse matrix of 𝐇. the solution of equation (8) is unique and has the shortest distance compared to other solutions. as mentioned in the reference [12], elm tends to give a generally good performance along with the increasing of the learning speed using the moorepenrose generalized inverse method. elm algorithm could be summarized into four steps [12], they are: 1. define the number of hidden nodes (𝐾), then randomly choose the initial value of 𝛽𝑖 and 𝑏𝑖. 2. calculate matrix 𝐇. 3. based on equation (9), calculate the weight output �̂�. 4. then, the forecasting result �̂� is calculated with: �̂� = 𝐇 �̂� (10) 3.3 hybrid singular spectrum analysis – extreme learning machine in this section, this paper proposes the hybrid ssa – elm forecasting method as follows: 1. time series is decomposed into main components by using ssa method. 2. the main components obtained from (1) could be defined as trend, periodic, quasi-periodic, and noise component. 3. the reconstructed component is formed by adding up several main components based on the frequency range closeness. 4. elm is applied to every reconstructed component thus, elm architecture is different for every reconstructed component. 5. the final result of hybrid ssa-elm forecasting is an addition of forecasting results from several elm architectures using equation (10). steps above are visually presented in picture 3.1. 3.4 forecasting accuracy forecasting accuracy of the test data (out sample) in this research uses mape (mean absolute percentage error) formulated as follows: mape = 1 𝑣 − 𝑇 ∑ | 𝐹𝑡 − 𝐴𝑡 𝐴𝑡 × 100%| 𝑣 𝑡=𝑇+1 (11) with 𝐹𝑡 is the 𝑡-th forecasting result value, and 𝐴𝑡 is the 𝑡-th actual value. mape characteristics: (1) if the mape < 10%, then the hybrid ssa – elm method performance is very good, (2) if the mape value is in the range of 10% 20%, then the hybrid ssa – elm method forecasting performance is good, (3) if the jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 60-68 66 mape in the range of 20% 50%, then the hybrid ssa – elm method forecasting performance is adequate, and (4) if the mape > 50%, then the hybrid ssa – elm method forecasting performance is bad. ssa decomposition 1st reconstruction component 2nd reconstruction component m-th reconstruction component 1st architecture of elm original time series 2nd architecture of elm m-th architecture of elm . . . . . . forecasted of 1st elm architecture . . . start summation result from forecasted of elm architecture result of hybrid forecasting ssa-elm finish forecasted of 2nd elm architecture forecasted of m-th elm architecture figure 3.1. hybrid ssa – elm method flowchart 4. results and discussion in the process of hybrid ssa – elm forecasting, the first step is decomposing foreign tourists data with ssa. in ssa, defining the value of l in this paper is the number of train data (528 observations) divided by two. thus l is 264. based on svd (singular value decomposition), 264 components are obtained from ssa process of train data, but only the first ten components are picked because with only these first ten components could explain the variation of foreign tourists 99.38 percent to be analyzed further. figure 4.1. (a). variation of the first ten components, (b). data scree plot grouping is performed by looking at the similarities of 10 components plot patterns that indirectly indicate the similarities of the components. ten components are grouped into ten groups. group 1 consists of the 1st components, m. fajar an application of hybrid forecasting singular spectrum analysis–extreme learning machine method in foreign tourists forecasting 67 group 2 consists of the 2nd components, group 3 consists of the 3rd components, and so on. table 4.1 presents the forecasting result of ssa-elm, ssa dan elm methods in test data forecasting. ssa-elm has the lowest mape (4.91 percent) compared to ssa (28 percent) and elm (9.07 percent). based on mape characteristics, ssa-elm and elm methods’ performance is very good. meanwhile, ssa performance is adequate. table 4.1. the forecasting result of hybrid ssa-elm, ssa, and elm the year 2018 the actual number of foreign tourists forecast result mape month: ssa-elm ssa elm ssaelm ssa elm january 1100677 1215770.91 925266.00 1141727.56 10.46 15.94 3.73 february 1201001 1218719.07 929846.08 1057204.11 1.48 22.58 11.97 march 1363339 1277141.99 935022.03 1093649.66 6.32 31.42 19.78 april 1300277 1322031.55 941383.50 1205315.22 1.67 27.60 7.30 may 1242588 1321390.12 944461.27 1182573.77 6.34 23.99 4.83 june 1318094 1347513.63 948271.13 1178043.33 2.23 28.06 10.63 july 1540549 1424376.79 954890.47 1404689.88 7.54 38.02 8.82 august 1510764 1461883.81 960299.96 1427398.44 3.24 36.44 5.52 average 4.91 28.00 9.07 source: author 5. conclusions based on the previous discussion, it could be concluded that hybrid ssaelm performance is very good in forecasting the number of foreign tourists. it is shown by the mape value of 4.91 percent with eight observations out the sample. references [1] c.h. aladag, e. egrioglu, and c. kadilar, “improvement in forecasting accuracy using the hybrid model of arfima and feed forward neural network american,” journal of intelligent systems, vol.2, no.2, pp. 12-17, 2012. [2] d. rahmani, “a forecasting algorithm for singular spectrum analysis based on bootstrap linear recurrent formula coefficients,” international journal of energy and statistics, vol.2, no.4, pp. 287-299, 2014. [3] m. fajar, “perbandingan kinerja peramalan pertumbuhan ekonomi indonesia antara arma, ffnn dan hybrid arma-ffnn,” 2016. doi:10.13140/rg.2.2.34924.36483. [4] m. fajar, “meningkatkan akurasi peramalan dengan menggunakan metode hybrid singular spectrum analysis-multilayer perceptron neural networks,” 2018. doi: 10.13140/rg.2.2.32839.60320. [5] m. fajar, “perbandingan kinerja peramalan antara metode hybrid singular spectrum analysis-multilayer perceptrons neural network dan hybrid singular spectrum analysis-feed forward neural network pada data inflasi,” jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 60-68 68 2018. doi: 10.13140/rg.2.2.10312.98561. [6] n. golyandina, v. nektrutkin, and a. zhiglovsky, analysis of time series: ssa and related techniques. chapman and hall/crc, 2001. [7] r. siregar, d. prariesa, and g. darmawan, “aplikasi metode singular spectral analysis (ssa) dalam peramalan pertumbuhan ekonomi indonesia tahun 2017”, mantik, vol. 3, no. 1, pp. 5-12, oct. 2017 [8] y. jatmiko, r. rahayu, and g. darmawan, “perbandingan keakuratan hasil peramalan produksi bawang merah metode holt-winters dengan singular spectrum analysis (ssa)”, mantik, vol. 3, no. 1, pp. 13-22, oct. 2017. [9] d. lubis, m. johra, and g. darmawan, “peramalan indeks harga konsumen dengan metode singular spectral analysis (ssa) dan seasonal autoregressive integrated moving average (sarima)”, mantik, vol. 3, no. 2, pp. 74-82, oct. 2017. [10] th. alexandrv, and n. golyandina, “automatic extraction and forecast of time series cyclic components within the framework of ssa,” . in proceedings of the 5th st.petersburg, 2005. [11] w. makridakis, and macgee, metode dan aplikasi peramalan. binarupa aksara, 1999. [12] s. ding, h. zhao, y. zhang, x. xu, and r. nie, “extreme learning machine: algorithm, theory and applications,” artificial intelligence review, vol.44, no.1 pp. 103-115, 2013. jurnal jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 128 analisis regresi kuantil b-splines monoton naik pada hubungan rata-rata lama sekolah dan pengeluaran rumah tangga per kapita di provinsi yogyakarta yogo aryo jatmiko badan pusat statistik, jakarta yj29289@gmail.com doi:https://doi.org/10.15642/mantik.2018.4.1.128-136 abstrak masalah multidimensional di berbagai negara yang selalu menjadi perhatian pemerintah adalah masalah kemiskinan, tidak terkecuali indonesia. kemiskinan sering kali dikaitkan dengan sektor pendidikan dikarenakan fungsi pendidikan sebagai driving force atau daya penggerak transformasi masyarakat untuk memutus rantai kemiskinan. pola hubungan kemiskinan dan sektor pendidikan dapat dilihat dari hubungan antara tingkat pendidikan (rata-rata lama sekolah) dan tingkat kemiskinan (pengeluaran rumah tangga per kapita). di yogyakarta masih merupakan provinsi dengan persentase kemiskinan terbesar di pulau jawa walaupun menunjukkan tren penurunan sejak tahun 2007. penelitian ini bertujuan melihat hubungan antara tingkat pendidikan (rata-rata lama sekolah) dan tingkat kemiskinan (pengeluaran rumah tangga per kapita) di provinsi di yogyakarta tahun 2016. model yang sesuai untuk menentukan karakteristik rumah tangga adalah regresi kuantil dengan metode b-splines monoton naik yang menghubungkan antara rata-rata lama sekolah dengan pengeluaran rumah tangga per kapita. hasil estimasi berdasarkan model regresi kuantil dengan b-splines monoton naik diperoleh bahwa rumah tangga dengan tingkat pendidikan terendah dikatakan sebagai rumah tangga sangat miskin jika pengeluaran per kapita sebulan kurang dari 322.205 rupiah dan dikatakan sebagai rumah tangga miskin jika pengeluaran per kapita sebulan antara 322.205 rupiah sampai dengan 426.666 rupiah. sedangkan, rumah tangga dengan tingkat pendidikan tertinggi dikatakan sebagai rumah tangga sangat miskin jika pengeluaran per kapita sebulan kurang dari 3.410.965 rupiah dan dikatakan sebagai rumah tangga miskin jika pengeluaran per kapita sebulan antara 3.410.965 rupiah sampai dengan 4.676.718 rupiah. kata kunci: regresi kuantil, b-splines, rata-rata lama sekolah, pengeluaran rumah tangga per kapita abstract the multidimensional problem in various countries that is always become the government's attention is the problem of poverty, indonesia is no exception. poverty is often associated with the education sector due to the function of education as a driving force of the transformation of society to break the chain of poverty. the pattern of relations between poverty and the education sector can be seen from the relationship between the level of education (mean years of schooling) and poverty level (per capita household expenditure). di yogyakarta is still the province with the largest percentage of poverty on the java island despite showing a downward trend since 2007. this study aims to look at the relationship between the level of education (mean years of schooling) and poverty level (per capita household expenditure) in di yogyakarta province 2016. the model that is suitable for determining household characteristics is quantile regression with the increased monotone b-splines method that links the mean years of schooling and per capita household expenditure. estimation results based on the quantile regression model with increased monotone b-splines method found that households with the lowest education level are said to be very poor households if monthly per capita expenditure is less than 322,205 rupiah and is said to be a poor household if monthly per capita expenditure is between 322,205 rupiah to 426,666 rupiah. meanwhile, households with the highest level of education are said to be very poor households if monthly per capita expenditure is less than 3,410,965 rupiahs and is said to be a poor household if monthly per capita expenditure is between 3,410,965 rupiahs up to 4,676,718 rupiahs. keywords: quantile regression, b-splines, mean years of schooling, per capita household expenditure jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 129 1. pendahuluan kemiskinan dilihat sebagai ketidakmampuan memenuhi kebutuhan dasar makanan dan bukan makanan dari sisi ekonomi yang diukur dari sisi pengeluaran [1]. tingkat kemampuan ekonomi masyarakat dan daya beli masyarakat merupakan cerminan dari tingkat kesejahteraan masyarakat yang ditunjukkan dari pengeluaran untuk kebutuhan konsumsi makanan dan bukan makanan. kemampuan memenuhi kebutuhan hidup menunjukkan daya beli masyarakat yang tinggi dan selanjutnya berdampak pada meningkatnya kesejahteraan masyarakat. masalah multidimensional di berbagai negara yang selalu menjadi perhatian pemerintah adalah masalah kemiskinan, tidak terkecuali indonesia. kemiskinan sering kali dikaitkan dengan sektor pendidikan dikarenakan fungsi pendidikan sebagai driving force atau daya penggerak transformasi masyarakat untuk memutus rantai kemiskinan. pendidikan membantu menurunkan kemiskinan melalui efeknya pada produktivitas tenaga kerja dan melalui jalur manfaat sosial, maka pendidikan menjadi suatu tujuan pembangunan yang penting bagi bangsa [2]. dengan pendidikan, peluang kerja menjadi lebih terbuka dan upah yang didapat juga akan lebih tinggi karena pendidikan merupakan sarana untuk memperoleh wawasan, ilmu pengetahuan dan meningkatkan keterampilan. pendidikan seseorang merupakan salah satu determinan konsumsi per kapita [3]. rata-rata lama sekolah yang menunjukkan tingkat pendidikan masyarakat mampu menurunkan tingkat kemiskinan di indonesia [4]. masyarakat yang berpendidikan tinggi akan mempunyai keterampilan dan keahlian, sehingga dapat meningkatkan produktivitasnya. peningkatan produktivitas akan meningkatkan output perusahaan, peningkatan upah pekerja, peningkatan daya beli masyarakat sehingga akan mengurangi kemiskinan. pendidikan khususnya peningkatan jumlah tahun belajar merupakan suatu syarat untuk tahap dari pembangunan ekonomi [5]. semakin tinggi pendidikan seseorang, maka kualitas sumber daya manusia (sdm) juga bertambah baik sehingga produktivitas semakin tinggi. tentunya semakin tinggi produktivitas, akan meningkatkan penghasilan serta pengeluaran. seseorang dengan pendidikan yang lebih tinggi biasanya memiliki akses yang lebih besar untuk mendapatkan pekerjaan dengan bayaran lebih tinggi, dibandingkan dengan individu dengan tingkat pendidikan lebih rendah [6]. penduduk miskin akan mendapat kesempatan yang lebih baik untuk keluar dari status miskin di masa depan dengan pendidikan yang memadai [7]. hal ini sejalan dengan yang dikemukakan oleh [8] yaitu kemiskinan akan berkurang apabila investasi pendidikan dilakukan secara merata, termasuk pada masyarakat yang berpenghasilan rendah. pada bulan maret tahun 2016, angka kemiskinan di indonesia yang dikeluarkan oleh badan pusat statistik (bps) tercatat sebesar 10,86 persen atau sekitar 28 juta jiwa. lima puluh empat persen dari jumlah penduduk miskin tersebut berada di pulau jawa. di yogyakarta masih merupakan provinsi dengan persentase kemiskinan terbesar di pulau jawa yaitu sebesar 13,34 persen walaupun menunjukkan tren penurunan sejak tahun 2007. penelitian ini akan melihat hubungan antara tingkat pendidikan (rata-rata lama sekolah) dan tingkat kemiskinan (pengeluaran per kapita rumah tangga) di provinsi di yogyakarta tahun 2016. rata-rata lama sekolah adalah jumlah tahun belajar penduduk usia 15 tahun ke atas yang telah diselesaikan dalam pendidikan formal (tidak termasuk tahun yang mengulang). untuk level rumah tangga, keterkaitan antara tingkat pendidikan (ratarata lama sekolah) dengan tingkat kemiskinan (pengeluaran rumah tangga) dapat ditunjukkan berdasarkan suatu model regresi. metode penaksiran model regresi yang diasumsikan mengikuti bentuk persamaan regresi tertentu seperti linier, kuadratik dan yang lainnya disebut regresi parametrik. akan tetapi peneliti seringkali menemui kesulitan dalam menentukan model hanya melalui sebaran data, sehingga metode yang digunakan adalah regresi nonparametrik. salah satu teknik penaksiran dalam regresi nonparametrik adalah bjurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 130 splines. b-splines merupakan teknik penaksiran pada fitting kurva regresi yang memperhitungkan penghalusan. metode regresi nonparametrik yang dapat digunakan untuk menduga kurva regresi antara lain dengan menggunakan pendekatan regresi spline. regresi spline adalah pendekatan ke arah plot data dengan tetap memperhitungkan kemulusan kurva. spline adalah penduga yang diperoleh dengan meminimumkan penalized least square, yaitu kriteria pendugaan yang menggabungkan goodness-of-fit dengan kemulusan kurva, dimana kedua ukuran ini diatur oleh suatu parameter pemulusan. pemilihan model optimal (terbaik) menggunakan kriteria mean squared error (mse) atau generalized cross validation (gcv) dan taksiran parameter menggunakan metode kuadrat terkecil. 2. tinjauan pustaka 2.1 sumber data data pada penelitian ini menggunakan data yang bersumber dari hasil survei sosial ekonomi nasional (susenas) semester i tahun 2016 di provinsi di yogyakarta. set data yang digunakan merupakan data rumah tangga sebanyak 3.662 rumah tangga. variabel yang digunakan adalah variabel pengeluaran per kapita sebagai variabel respon dan variabel rata-rata lama sekolah per kapita sebagai variabel prediktor. 2.2 variabel penelitian pengeluaran rumah tangga merupakan biaya yang dikeluarkan untuk dikonsumsi semua anggota rumah tangga selama sebulan, yang terdiri dari konsumsi makanan dan non makanan, tanpa melihat asal barang serta terbatas pada konsumsi untuk keperluan usaha atau yang diberikan kepada pihak lain [1]. pengeluaran rumah tangga perkapita merupakan pengeluaran rumah tangga dibagi banyaknya anggota rumah dalam suatu rumah tangga atau dengan kata lain rata-rata pengeluaran rumah tangga untuk setiap anggota rumah tangga. rata-rata lama sekolah (rls) merupakan jumlah tahun yang digunakan penduduk dalam menjalani pendidikan formal. penduduk yang dicakup dalam perhitungan rls adalah penduduk yang berusia 15 tahun ke atas. rata-rata lama sekolah per kapita merupakan rata-rata lama sekolah dari seluruh anggota rumah tangga yang berusia 15 tahun ke atas pada suatu rumah tangga dibagi dengan banyaknya anggota rumah tangga tersebut. rls dihitung dengan formula sebagai berikut [9]: 𝑅𝐿𝑆 = 1 𝑃15+ ∑ 𝐿𝑆𝑖 𝑃15+ 𝑖=1 .… …….(1) di mana: p15+= jumlah penduduk usia 15 tahun ke atas lsi = lama sekolah penduduk ke-i. = a. tidak pernah sekolah = 1 b. masih sekolah di sd sampai dengan s1 = konversi ijazah terakhir + kelas terakhir – 1 c. masih sekolah di s2/s3 = konversi ijazah terakhir d. tidak bersekolah lagi dan tamat di kelas terakhir = konversi ijazah terakhir e. tidak bersekolah lagi dan tidak tamat di kelas terakhir = konversi ijazah terakhir + kelas terakhir lama sekolah penduduk berusia 15 tahun ke atas di jenjang pendidikan terakhir yang telah ditamatkan menggunakan konversi berikut (lihat tabel 1): tabel 1. konversi pendidikan tertinggi yang ditamatkan no. pendidikan tertinggi yang ditamatkan konversi (tahun) (1) (2) (3) 1. tidak/ belum pernah sekolah 0 2. sd/ sederajat 6 3. smp/ sederajat 9 4. sma/ sederajat 12 5. diploma i 13 6. diploma ii 14 7. akademi/ diploma iii 15 8. diploma iv/ sarjana (s1) 16 9. master (s2) 18 10. doktor (s3) 21 sumber: bps, 2011 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 131 2.3 regresi kuantil smoothing b-splines fungsi obyektif kuantil untuk smoothing b-splines dalam bentuk persamaan linier yaitu: min {�̂�𝑻𝒖 + �̂�𝑻𝒗 | �̂�𝜶 + 𝒖 − 𝒗 = �̂�, (𝒖, 𝒗 ∈ ℝ+ (𝒏+𝒖))}…..… ……. …(2) dimana u v adalah vektor positif dan negatif bagian dari residual regresi. �̂�(𝑛+𝑢)×1 = ( 𝒘 𝟏𝒖×𝟏 )………… ……(3) dengan 𝒘 = (𝜌𝜏 (𝑧1), … , 𝜌𝜏 (𝑧𝑛 )) 𝑇 adalah vektor penimbang �̂� = ( 𝒀 𝟎𝒖×𝟏 )……...… …………..(4) �̂�(𝑛+𝑢)×1 merupakan vektor respon pseudo dengan 𝒀 = (𝑦1, … , 𝑦𝑛 ) 𝑇 �̂� = ( 𝑩 𝝀𝑪 )……… ………(5) �̂� ((𝑛+𝑢)×𝑚) merupakan matriks design pseudo dengan: 𝑩𝒏×𝒎 = [ 𝐵1(𝑥1; 𝑣) 𝐵2(𝑥1; 𝑣) 𝐵1(𝑥2; 𝑣) 𝐵2(𝑥2; 𝑣) … 𝐵𝑚 (𝑥1; 𝑣) … 𝐵𝑚 (𝑥2; 𝑣) ⋮ ⋮ 𝐵1(𝑥𝑛; 𝑣) 𝐵2(𝑥𝑛 ; 𝑣) ⋱ ⋮ … 𝐵𝑚 (𝑥𝑛 ; 𝑣) ] fungsi obyektifnya: �̂�𝑻𝒖 + �̂�𝑻𝒗 fungsi kendalanya: �̂�𝜶 + 𝒖 − 𝒗 = �̂� 2.4 pemilihan parameter penghalus dan knot kriteria pemilihan parameter penghalus (λ) yang paling optimum menggunakan nilai schawrz information criterion (sic) terkecil [10], dengan formulasi: 𝑆𝐼𝐶(𝜆) = log ( 1 𝑛 ∑ 𝜌𝜏(𝑦𝑖 − 𝑛 𝑖=1 ∑ �̂�𝑗 𝐵𝑗 (𝑥𝑖 ; 𝑣) 𝑚 𝑗=1 )) + 1 2 𝜌𝜆 log (𝑛) 𝑛 …(6) dimana pλ merupakan jumlah dari residual nol untuk model yang fit. jumlah knot untuk regresi kuantil smoothing b-splines adalah 20 knot dimana lokasinya dipilih berdasarkan nilai unik dari variabel x [10]. titik knot ke-u(tu) diperoleh dari: 𝑡𝑢 = 𝑘𝑢𝑎𝑛𝑡𝑖𝑙 𝑘𝑒 − ( 𝑢 20 ) 𝑑𝑎𝑟𝑖 𝑛𝑖𝑙𝑎𝑖 𝑢𝑛𝑡𝑢𝑘 𝑣𝑎𝑟𝑖𝑎𝑏𝑒𝑙 𝑋; 𝑢 = 1,2, … ,20 (7) 2.5 fungsi kendala monoton pada program linier untuk regresi kuantil penambahan fungsi kendala monoton naik atau turun saat melakukan penaksiran parameter adalah untuk memberikan efek penghalus pada kurva suatu model regresi. kriteria pengecekan kendala monoton pada fungsi obyektif regresi kuantil pada smoothing b-splines yaitu: hα > 0, untuk fungsi monoton naik hα < 0, untuk fungsi monoton turun dimana: 𝑯 = [ 𝐵′1(𝑡𝑣; 𝑣) … 𝐵 ′ 𝑚 (𝑡𝑣 ; 𝑣) ⋮ ⋱ ⋮ 𝐵′1(𝑡𝑚; 𝑣) … 𝐵 ′ 𝑚 (𝑡𝑚−1; 𝑣) ] (8) atau �̂�𝒙 > 0, untuk fungsi monoton naik �̂�𝒙 < 0, untuk fungsi monoton turun dimana: �̂� = ( 𝐇 𝟎((𝒖+𝟐)×𝟐(𝒏+𝒖)) )…… .(9) 3. metode penelitian langkah-langkah dalam penelitian ini mencakup: 1. membuat scatter plot antar variabel respon dan variabel bebas 2. melakukan spesifikasi model berdasarkan scatter plot, dalam hal ini digunakan pendekatan fungsi b-splines 3. pengecekan outlier pada hasil scatter plot dan bila terdapat outlier maka digunakan kuantil. pengecekan outlier dapat juga dengan melihat distribusi eror dengan rata-rata sebagai ukuran pemusatan data pada fungsi b-splines 4. menentukan constrained dari hubungan kedua variabel, apakah monoton baik, monoton turun 5. menentukan jumlah knot dan parameter smoothing (λ). pada makalah ini digunakan fungsi smoothing b-splines dengan jumlah knot yang digunakan 20 knot, dan parameter smoothing (λ) ditentuan berdasarkan nilai schawrz information criterion (sic) yang terkecil jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 132 6. menaksir kurva regresi kuantil berdasarkan nilai parameter penghalus (λ) yang optimal pada beberapa titik kuantil, yaitu pada τ = 0,2; 0,4; 0,6; 0,8 7. menaksir selang kepercayaan untuk kurva regresi kuantil dengan metode langsung. 3. pembahasan hasil bagian ini akan menjelaskan keterkaitan antara variabel rata-rata lama sekolah dan rata-rata pengeluaran per kapita rumah tangga di provinsi di yogyakarta tahun 2016 yang dimodelkan dengan metode cobs. alasan digunakan metode cobs adalah plot data (gambar 1) yang menunjukkan pola yang tidak dapat dispesifikasi secara jelas namun mempunyai kecenderungan meningkat, sehingga akan lebih baik pemodelan dilakukan secara nonparametrik. adapun yang dimaksud dengan “constrained” di sini adalah asumsi bahwa hubungan kedua data adalah pola cenderung meningkat yang lebih lanjut disebut dengan “increase constrained”. hal ini dapat diartikan bahwa rata-rata pengeluaran rumah tangga per kapita meningkat seiring dengan rata-rata lama sekolah anggota rumah tangga. (a) (b) gambar 1. plot data pengeluaran rumah tangga per kapita dan rata-rata lama sekolah (a) plot asli (b) plot transformasi selain fenomena ketidakberaturannya pola data, fenomena berikutnya adalah terdapatnya data outlier (gambar 2) dari residual model b-splines yang disajikan dalam regresi mean. dengan demikian untuk menangkap fenomena keberadaan outlier residual tersebut, maka diterapkan analisis regresi kuantil dalam menaksir parameter model. gambar 2. boxplot residual dari model b-splines lebih lanjut, untuk membagi rumah tangga ke dalam kelompok-kelompok dengan karakteristik yang hampir sama berdasarkan rata-rata lama sekolah anggota rumah tangga dan pengeluaran rumah tangga per kapita maka akan diterapkan empat buah pemodelan regresi kuantil dengan batasan yakni kuantil ke-0,2, kuantil ke-0,4, kuantil ke-0,6 dan kuantil ke-0,8 [11]. dalam pemodelan regresi kuantil, setiap kuantil memiliki jumlah dan lokasi titik-titik knot (lihat tabel 2.) tabel 2. titik-titik knot pada masingmasing kuantil titik ke t1 t2 t3 t4 t5 knot 0,00 2,25 3,50 4,67 5,60 titik ke t6 t7 t8 t9 t10 knot 6,40 7,17 7,80 8,57 9,25 titik ke t11 t12 t13 t14 t15 knot 9,86 10,57 11,33 11,83 12,67 titik ke t16 t17 t18 t19 t20 knot 13,50 14,33 15,20 16,67 21,00 sumber: susenas maret 2016 bps, diolah tabel 2 menunjukkan titik-titik knot yang dihitung dengan metode kuantil dari nilai unik variabel rata-rata lama sekolah sebanyak 174 nilai. sementara itu, parameter penghalus kurva (λ) optimum pada masingmasing kuantil memiliki nilai yang berbedabeda sebagaimana disajikan pada gambar 3. jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 133 (a) kuantil (τ) = 0,2 (b) kuantil (τ) = 0,4 (c) kuantil (τ) = 0,6 (d) kuantil (τ) = 0,8 gambar 3. parameter penghalus (λ) optimum berdasarkan nilai sic terkecil pada masing-masing kuantil. gambar 3 (a) menunjukkan pada kuantil ke-0,2 diperoleh parameter penghalus (λ) optimum sebesar 4.97 dengan sic minimum sebesar -1,8019. gambar 3 (b) menunjukkan pada kuantil ke-0,4 diperoleh parameter penghalus (λ) optimum sebesar 37.3 dengan sic minimum sebesar -1,4268. gambar 3 (c) menunjukkan pada kuantil ke0,6 diperoleh parameter penghalus (λ) optimum sebesar 73 dengan sic minimum sebesar -1,3919. gambar 3 (d) menunjukkan pada kuantil ke-0,8 diperoleh parameter penghalus (λ) optimum sebesar 37,3 dengan sic minimum sebesar -1,6959. berbekal titik-titik knot dan parameter penghalus (λ) optimum yang telah diperoleh pada masing-masing kuantil, maka kurva regresi kuantil yang berbasis metode cobs untuk smoothing b-splines linier dengan asumsi monoton naik (increase constrain) disajikan pada gambar 4. gambar 4. kurva regresi kuantil dengan cobs gambar 4 menunjukkan estimasi pengeluaran rumah tangga per kapita berdasarkan rata-rata lama sekolah anggota rumah tangga pada kuantil ke-0,2, kuantil ke0,4, kuantil ke-0,6 dan kuantil ke-0,8. pada kuantil ke-0,6 dan kuantil ke-0,8 terlihat bahwa pada rata-rata lama sekolah sekitar tujuh tahun ke atas secara drastis meningkatkan pengeluaran per kapita. estimasi pengeluaran rumah tangga per kapita tersebut diperoleh dari model regresi kuantil yang memiliki koefisien (α) sebagaimana disajikan pada tabel 3. tabel 3. koefisien regresi kuantil bsplines linier pada kuantil ke-0,2, ke-0,4, ke-0,6 dan ke-0,8 knot ke-i (ti) koefisien (α) τ = 0,2 τ = 0,4 τ = 0,6 τ = 0,8 (1) (2) (3) (4) (5) 1 12,6829 12,9638 13,2692 13,7414 2 12,6829 12,9708 13,3160 13,7878 3 12,6829 12,9747 13,3421 13,8135 4 12,6829 12,9783 13,3664 13,8376 5 12,7152 13,0045 13,3858 13,8568 6 12,7429 13,0271 13,4024 13,8733 7 12,7695 13,0487 13,4184 13,8892 8 12,7914 13,0665 13,4316 13,9022 9 12,8180 13,1050 13,4485 13,9180 10 12,8850 13,1949 13,5784 14,0208 11 12,9450 13,2755 13,6948 14,1130 12 13,0149 13,4391 13,8304 14,2203 13 13,0897 13,6144 13,9755 14,3352 14 13,1389 13,7296 14,0710 14,4107 15 13,4338 13,9233 14,2314 14,5377 16 13,7252 14,1147 14,3645 14,6631 17 14,0166 14,3060 14,4976 14,7885 18 14,1478 14,4180 14,6372 14,9200 19 14,3279 14,6073 14,8730 15,1422 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 134 20 15,0425 15,3581 15,8082 16,0233 sumber: susenas maret 2016 bps, diolah tabel 3 menunjukkan nilai koefisien pada masing-masing kuantil yang cenderung naik, hal inilah yang menjadikan kurva regresi kuantil pada gambar 4 cenderung naik. dengan demikian adanya hubungan antara rata-rata lama sekolah anggota rumah tangga dan pengeluaran rumah tangga per kapita yang cenderung meningkat sejalan dengan asumsi (increase constrain). selanjutnya akan dihitung estimasi pengeluaran per kapita pada beberapa ratarata lama sekolah seseorang yang menunjukkan jenjang pendidikan tertentu. beberapa jenjang pendidikan yang digunakan dalam estimasi pengeluaran per kapita tersebut meliputi 0 tahun (tidak/belum pernah sekolah), 6 tahun (sd/sederajat), 9 tahun (smp/sederajat), 12 tahun (sma/sederajat), 13 tahun (diploma i/ii), 15 tahun (akademi/diploma iii), 16 tahun (diploma iv/sarjana s1), 18 tahun (master/sarjana s2) dan 21 tahun (sarjana s3). tabel 4 menginformasikan bahwa pada kuantil ke-0,2 anggota rumah tangga dengan rata-rata lama sekolah 0 tahun memiliki pengeluaran per kapita sebesar 322.205 rupiah, sementara itu anggota rumah tangga dengan rata-rata lama sekolah 6 tahun memiliki pengeluaran per kapita 337.396 rupiah, demikian seterusnya hingga anggota rumah tangga dengan rata-rata lama sekolah 21 tahun memiliki pengeluaran per kapita sebesar 3.410.965 rupiah. pada kuantil ke-0,4 menunjukkan anggota yang tidak/belum pernah sekolah memiliki pengeluaran per kapita sebesar 426.666 rupiah, anggota rumah tangga dengan rata-rata lama sekolah 6 tahun akan memiliki pengeluaran per kapita sebesar 449.455 rupiah, demikian seterusnya hingga pada anggota rumah tangga dengan rata-rata lama sekolah 21 tahun memiliki pengeluaran per kapita sebesar 4.676.718 rupiah. pada kuantil ke-0,6 menunjukkan anggota yang tidak/belum pernah sekolah memiliki pengeluaran per kapita sebesar 579.083 rupiah, anggota rumah tangga dengan rata-rata lama sekolah 6 tahun akan memiliki pengeluaran per kapita sebesar 656.112 rupiah, demikian seterusnya hingga pada anggota rumah tangga dengan rata-rata lama sekolah 21 tahun memiliki pengeluaran per kapita sebesar 7.334.840 rupiah. pada kuantil ke-0,8 menunjukkan anggota yang tidak/belum pernah sekolah memiliki pengeluaran per kapita sebesar 928.555 rupiah, anggota rumah tangga dengan ratarata lama sekolah 6 tahun akan memiliki pengeluaran per kapita sebesar 1.050.794 rupiah, demikian seterusnya hingga pada anggota rumah tangga dengan rata-rata lama sekolah 21 tahun memiliki pengeluaran per kapita sebesar 9.095.333 rupiah (lihat tabel 4). tabel 4. nilai estimasi pengeluaran rumah tangga berdasarkan rata-rata lama sekolah pada kuantil ke-0,2, ke-0,4, ke-0,6 dan ke-0,8 ratarata lama sekolah (tahun) estimasi pengeluaran per kapita (rp) τ = 0,2 τ = 0,4 τ = 0,6 τ = 0,8 (1) (2) (3) (4) (5) 0 322.205,3 426.665,6 579.082,8 928.554,8 6 337.395,9 449.455,0 656.112,0 1.050.794,5 9 384.754,1 520.132,3 752.112,9 1.182.351,8 12 539.621,9 954.385,9 1.333.644,9 1.860.593,8 13 766.577,1 1.201.850,7 1.598.030,4 2.164.145,8 15 1.352.709,0 1.780.285,7 2.202.482,8 2.927.901,4 16 1.537.741,4 2.024.960,6 2.585.688,9 3.405.582,4 18 1.964.911,5 2.619.812,9 3.563.721,7 4.607.457,1 21 3.410.964,9 4.676.717,7 7.334.840,2 9.095.332,7 sumber: susenas maret 2016 bps, diolah berdasarkan estimasi pengeluaran per kapita pada masing-masing kuantil akan ditentukan klasifikasi rumah tangga berdasarkan rata-rata lama sekolah dan pengeluaran per kapita rumah tangga di provinsi di yogyakarta tahun 2016. adapun kriteria klasifikasi pada setiap rata-rata lama sekolah adalah sebagai berikut: rumah tangga dengan rata-rata lama sekolah anggota rumah tangganya 0 tahun diklasifikasikan: a) “sangat miskin” jika pengeluaran per kapitanya kurang dari 322.205,27 rupiah; b) “miskin” jika pengeluaran per kapitanya antara 322.205,27 sampai dengan 426.665,57 rupiah; c) “menengah” jika pengeluaran per kapitanya antara 426.665,57 sampai dengan 579.082,84 rupiah; jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 135 d) “kaya” jika pengeluaran per kapitanya antara 579.082,84 sampai dengan 928.554,80 rupiah; dan e) “sangat kaya” jika pengeluaran per kapitanya di atas 928.554,80 rupiah; sedangkan rumah tangga dengan ratarata lama sekolah anggota rumah tangganya 21 tahun diklasifikasikan: a) “sangat miskin” jika pengeluaran per kapitanya kurang dari 3.410.964,9 rupiah; b) “miskin” jika pengeluaran per kapitanya antara 3.410.964,9 sampai dengan 4.676.717,7 rupiah; c) “menengah” jika pengeluaran per kapitanya antara 4.676.717,7 sampai dengan 7.334.840,2 rupiah; d) “kaya” jika pengeluaran per kapitanya antara 7.334.840,2 sampai dengan 9.095.332,7 rupiah; dan e) “sangat kaya” jika pengeluaran per kapitanya di atas 9.095.332,7 rupiah; dengan menggunakan model regresi kuantil dapat ditentukan nilai selang interval pengeluaran per kapita pada masing-masing kuantil yang disajikan pada gambar 5. gambar 5 menunjukkan nilai selang interval pengeluaran per kapita dan rata-rata lama sekolah pada taraf signifikansi 95 persen. secara umum hubungan rata-rata lama sekolah dan pengeluaran rumah tangga per kapita pada masing-masing kuantil menunjukkan hubungan yang searah, artinya semakin tinggi rata-rata lama sekolah seseorang maka pengeluaran per kapita juga akan semakin tinggi. seluruh batas bawah dan batas atas menunjukkan kecenderungan meningkat baik dari rata-rata lama sekolah paling rendah (0 tahun) ke rata-rata lama sekolah yang paling tinggi (21 tahun) maupun kecenderungan meningkat dari kuantil ke-0,2 ke kuantil ke0,8. hal ini menunjukkan adanya asumsi increase constrained pada data rata-rata lama sekolah yang secara signifikan berpengaruh terhadap peningkatan pengeluaran per kapita rumah tangga. 4. kesimpulan beberapa kesimpulan yang dapat diambil dari penelitian ini adalah sebagai berikut: 1) model regresi kuantil yang terbentuk dengan metode cobs terbagi menjadi empat yaitu kuantil ke-0,2, kuantil ke0,4, kuantil ke-0,6 dan kuantil ke-0,8. seluruh koefisien pada model regresi menunjukkan kecenderungan yang meningkat baik dari knot terkecil ke knot terbesar dan dari kuantil terkecil hingga kuantil terbesar; 2) estimasi pengeluaran rumah tangga per kapita yang diperoleh dari model regresi kuantil dengan metode cobs menunjukkan kecenderungan peningkatan baik dari knot terkecil ke knot terbesar dan dari kuantil terkecil hingga kuantil terbesar; 3) berdasarkan nilai estimasi pengeluaran rumah tangga per kapita dan rata-rata lama sekolah dapat ditentukan klasifikasi rumah tangga menurut level ekonomi sangat miskin, miskin, menengah, kaya dan sangat kaya. berdasarkan klasifikasi yang terbentuk dapat dikatakan bahwa semakin tinggi rata-rata lama sekolah anggota rumah tangga akan semakin (a) kuantil (τ) = 0,2 (b) kuantil (τ) = 0,4 (c) kuantil (τ) = 0,6 (d) kuantil (τ) = 0,8 gambar 5. selang interval kurva regresi kuantil dengan cobs untuk masing-masing kuantil: (a) kuantil ke-0,2 ; (b) kuantil ke-0,4 ; (c) kuantil ke-0,6 ; (d) kuantil ke-0,8 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 136 berpengaruh meningkatkan pengeluaran per kapita. referensi [1] bps. data dan informasi kemiskinan kabupaten kota tahun 2016. jakarta: bps (2016) [2] world bank. era baru dalam pengentasan kemiskinan di indonesia (ikhtisar). the world bank office jakarta: jakarta (2005) [3] andersson, magnus & engvall, anders & kokko, ari. (2006). determinants of poverty in lao pdr. the european institute of japanese studies, eijs working paper series. [4] suparno. analisis pertumbuhan ekonomi dan pengurangan kemiskinan: studi pro poor growth policy di indonesia. tesis. s2 ie-ipb. bogor (2010) [5] frankel, j.a. (1997) regional trading blocs in the world economic system. institute for international economics, washington dc. (1997). http://dx.doi.org/10.7208/chicago/9780226 260228.001.0001 [6] bureau of labor statistics. a profile of the working poor 2011. federal publication. washington, dc: cornell university ilr school. (2013) [7] wirawan, i. m. t, arka, s. analisis pengaruh pendidikan, pdrb per kapita, dan tingkat pengangguran terhadap jumlah penduduk miskin provinsi bali. e-jurnal ep unud, vol. 4, no. 5, mei (2015) (pp. 348 607) [8] mankiw, n. g., romer, d., weil, d.n. a contribution to the empirics of economic growth. the quarterly journal of economics, (1992) pp. 407-437. doi: 10.3386/w3541 [9] bps. rata-rata lama sekolah (mys). https:// sirusa.bps.go.id/ index.php? r= indicator / view&id=11. diakses pada 9 oktober 2018. [10] he, x dan ng, p. cobs: quatitatively constraint smoothing via linier programming. computational statistics. (1999) 14:315-337 [11] hudoyo, l.p. pemodelan hubungan antara rata-rata lama sekolah dan pengeluaran rumah tangga menggunakan constrained b-splines (cobs) pada regresi kuantil. (2017). tesis: universitas padjajaran http://dx.doi.org/10.7208/chicago/9780226260228.001.0001 http://dx.doi.org/10.7208/chicago/9780226260228.001.0001 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: m. iqbal, c. wulandari, w. yunanto, and g. sari, “mining non-zero-rare sequential patterns on activity recognition”, mantik, vol. 5, no. 1, pp. 1-9, may 2019. mining non-zero-rare sequential patterns on activity recognition mohammad iqbal1, chandrawati putri wulandari2,a, wawan yunanto3, and ghaluh indah permata sari2,b institut teknologi sepuluh nopember, iqbal@matematika.its.ac.id1 national taiwan university of science and technology, d10301809@mail.ntust.edu.tw2,a, ghaluhips@gmail.com2,b politeknik caltex riau, wawan@pcr.ac.id3 doi: https://doi.org/10.15642/mantik.2019.5.1.1-9 abstrak: penemuan pola langka aktivitas manusia yang diperoleh dari sensor gerak yang aktif dapat memberikan informasi yang tidak biasa untuk memberitahukan seseorang dalam keadaan yang berbahaya. penelitian ini bertujuan untuk mengenali aktivitas manusia yang langka menggunakan teknik penambangan pola non-zero-rare sekuensial. pola tersebut harus muncul pada barisan sensor yang aktif dan jumlah kemunculannya tidak melebihi ambang batas kemunculan yang telah ditentukan sebelumnya. penelitian ini mengusulkan sebuah algoritma untuk menambang pola nonzero-rare aktivitas manusia yang disebut mining multi-class non-zero-rare sequential patterns (mmrsp). hasil eksperimen menunjukkan bahwa pola non-zero-rare aktivitas manusia mampu menangkap aktivitas yang tidak biasa. selanjtunya, mmrsp bekerja dengan baik berdasarkan hasil nilai precision dari aktivitas yang jarang. kata kunci: pola sekuensial, pola langka, pengenalan aktivitas. multi-kelas abstract: discovering rare human activity patterns—from triggered motion sensors deliver peculiar information to notify people about hazard situations. this study aims to recognize rare human activities using mining non-zero-rare sequential patterns technique. in particular, this study mines the triggered motion sensor sequences to obtain non-zero-rare human activity patterns—the patterns which most occur in the motion sensor sequences and the occurrence numbers are less than the pre-defined occurrence threshold. this study proposes an algorithm to mine non-zero-rare pattern on human activity recognition called mining multi-class non-zero-rare sequential patterns (mmrsp). the experimental result showed that non-zero-rare human activity patterns succeed to capture the unusual activity. furthermore, the mmrsp performed well according to the precision value of rare activities. keywords: sequential patterns, rare patterns, activity recognition, multi-class. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 1-9 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 1-9 2 1. introduction discovering human activity patterns can help human life in better ways. we further plan to build a pleasant and safe living place by installing motion sensors in a house or apartment. in specific, we capture resident daily motions when the sensors are triggered. by collecting the triggered sensors sequences, we attempt to generate the triggered sensors patterns which can be used to understand the resident activities. however, understanding human activities from motion sensors are difficult and still being developed by finding its useful patterns. machine learning and data mining are preferable techniques to uncover these useful patterns. kasteren et al. [1] used hidden markov model (hmm) to classify the sensor sequences into several resident activities. some other machine learning methods, such as naïve bayes, conditional random forest (crf), and support vector machines (svm) was utilized in cook et al. [2]. however, they did not present the generated model that clearly describes what is happening on the triggered sensors, lie what we may directly interpret in this paper. therefore, this study employs sequential pattern mining to offer useful information about the patterns in 𝒔 → 𝑎 form, where 𝒔 and 𝑎 denote a sensor sequence and a human activity, respectively. iqbal and pao [3], and mukhlash et al. [4] studied several human activities types that can be considered as a typical pattern. this study focuses on one type of rare patterns, so-called non-zero-rare patterns [5,6]. the goal of this study is to provide a rare human activity pattern which can inform the resident in hazard condition. we argue that the generated patterns may not represent the whole activities, if they only consider the occurrence pattern in the whole activity sequences, since the dataset may have several activities as the labels. thus, we present a non-zero-rare human activity pattern as a subsequence, which must occur in sequences of one activity with the occurrence number is less than a pre-defined occurrence threshold. we propose an algorithm to mine non-zero-rare human activity pattern called mining multi-class non-zero-rare sequential patterns (mmrsp). based on the mmrsp, we obtain a non-zero pattern for each activity that differs from the previous works [5,6]. as far as we are aware, there is a limited number of research that discusses rare patterns on human activity recognition. the organization of this paper describes as follows: section 2 explains about mining sequential patterns techniques, especially on human activity recognition. the explanation about the non-zero-rare human activity pattern and the mining technique of the proposed method will be presented in section 3. in section 4 and section 5, we discuss the experimental results and conclude the discussion, consecutively. 2. related works like the abovementioned, there are two previous works on human activity recognition using sequential pattern mining. in [3], a distinguishing subsequence on the multi-class classification proposed to recognize a distinguish sensor subsequence that is frequent in one activity sequences yet rarely in other sequences based on two support thresholds. they extended the idea in [7] using one-vs-all strategy. mukhlash et al. [4] introduced a periodic human activity pattern. thus we know about regular activity in a certain time interval using fp-growth prefix-span and fuzzy theory to discretize the time interval. also, there are several studies on human activity recognition using sequential pattern mining. a sensor pattern that significantly distinguishes from one to other activities was studied in [8]. furthermore, frequent pattern mining based on multiple order temporal information was performed in [9], and weighted frequent patterns mining was proposed by [10] to adapt with classification task. we could say that those studies still focusing on typical activity patterns. concerning about environmental safety, we also need to take a rare pattern into m. iqbal, c. p. wulandari, w. yunanto, and g. i. p. sari mining non-zero-rare sequential patterns on activity recognition 3 our consideration to deliver a quick alert that may put the resident may in the unpredicted situation. this study discusses a non-zero-rare pattern based on [5,6] into human activity recognition. since there are more than two activities, we extend the definition of the nonzero-rare pattern on the multi-class case. also, we propose an algorithm to mine the nonzero-rare pattern on multi-class based on [11]. 3. mining multi-class non-zero-rare sequential patterns on human activity recognition in this section, we define the non-zero-rare-pattern on human activity recognition and describe an algorithm to mine the patterns. 3.1 multi-class non-zero-rare sequential patterns first, we define a non-zero-rare human activity pattern. assume that we have pairs of motion sensor sequences and human activity set 𝐷 = {(𝒔𝑖, 𝑎𝑖)|𝒔𝑖 ∈ 𝑆, 𝑎𝑖 ∈ 𝐴, 1 ≤ 𝑖 ≤ |𝐷|}. a set of motion sensor sequences 𝑆 = {𝒔𝑗 |𝒔𝑗 = (𝑠𝑗1 , 𝑠𝑗2 , ⋯ , 𝑠𝑗𝑛 ), 𝑠𝑗𝑛 ∈ 𝑅 × 𝑀, 1 ≤ 𝑗 ≤ 𝑛 ≤ |𝑆|}, is collected when the sensors are triggered during 𝑡𝑡ℎ-time intervals with the sequence 𝒔𝑖 which a result of cartesian product between a set of motion sensor types 𝑀 = {𝑚1, 𝑚2, ⋯ , 𝑚|𝑀|} and a set of sensor location 𝑅 = {𝑟1, 𝑟2, ⋯ , 𝑟|𝑅|}. for each time interval, a triggered sensor sequence belongs to a certain human activity label 𝑎𝑘 ∈ 𝐴, where 𝐴 = {𝑎1, 𝑎2, ⋯ , 𝑎|𝐴|} is a set of human activity labels. let 𝒔𝑚 = (𝑠𝑚1 , 𝑠𝑚2 , ⋯ , 𝑠𝑚ℓ ) be a subsequence of 𝒔𝑗 (𝒔𝑚 ⊆ 𝒔𝑗 , 1 ≤ 𝑚ℓ ≤ 𝑗 ≤ |𝒔𝑗 |) and 𝑠𝑢𝑝𝑝(𝒔𝑚, 𝐷𝑎𝑘 ) = |{𝒔𝑗≼𝒔𝑚|𝒔𝑗∈𝐷𝑎𝑘 }| |𝑆𝑎𝑘 | be a relative support value of the subsequence 𝒔𝑚 in 𝐷𝑎𝑘 , which can be used to extract rare patterns w.r.t. a certain activity label. thus, we define 𝒔𝑚 as a non-zero-rare sequential pattern for each activity below. definition 3.1. (a non-zero-rare human activity pattern) given a pre-defined support threshold 𝛾, and a set of pairs of sensor sequence and human activity label 𝐷, a subsequence 𝒔𝑚 is a non-zero-rare pattern on human activity label 𝑎𝑘 or (𝒔𝑚 → 𝑎𝑘) if and only if 𝒔𝑚 satisfies 𝑠𝑢𝑝𝑝(𝒔𝑚, 𝐷𝑎𝑘 ) > 0 and 𝑠𝑢𝑝𝑝(𝒔𝑚, 𝐷𝑎𝑘 ) < 𝛾. in this study, we can say a pattern 𝒔𝑚 w.r.t. an activity label 𝑎𝑘 as a rule (𝒔𝑚 → 𝑎𝑘). the rule form can be employed into the classification task. furthermore, this study meets a multi-class classification problem since |𝐴| > 2. we may use a traditional way, i.e., binary classification strategies to fit with a multi-class problem. there are two general binary classification strategies on the multi-class problem, such as one-vs-all (ova), and one-vs-one (ovo). both strategies may have better performance in some cases—accuracy result, but they are computationally expensive since both of them need to compare each positive class with the negative classes. consequently, we extend definition 3.1. by checking whether the maximum support value of 𝒔𝑚 for each activity, label is less than a pre-defined support threshold 𝛾 and greater than 0. definition 3.2. (a multi-class non-zero-rare human activity pattern) given a pre-defined support threshold 𝛾, and a set of pairs of sensor sequence and human activity label 𝐷, a subsequence 𝒔𝑚 is a non-zero-rare pattern on human activity label 𝑎𝑘 if and only if 𝒔𝑚 satisfies 0 < max 𝑘 {𝑠𝑢𝑝𝑝(𝒔𝑚, 𝑆𝑎𝑘 )|∀𝑎𝑘 ∈ 𝐴} = 𝑠𝑢𝑝𝑝(𝒔𝑚, 𝐷𝑎𝑘 ) < 𝛾. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 1-9 4 table 1. an example of motion sensor sequences and its activity label set. event motion sensor sequence activity label 𝑡1 𝑚4, 𝑚1, 𝑚2, 𝑚3 𝑎1 𝑡2 𝑚1, 𝑚3, 𝑚2, 𝑚3 𝑎1 𝑡3 𝑚1, 𝑚2, 𝑚3 𝑎1 𝑡4 𝑚1, 𝑚3, 𝑚2 𝑎2 𝑡5 𝑚1, 𝑚3 𝑎2 example 3.1. assume we have a dataset 𝐷 as shown in table 1. and a pre-defined support threshold 𝛾 = 4 5 . a subsequence 𝒔𝑚 = (𝑚1, 𝑚2, 𝑚3) is a non-zero-rare pattern for activity 𝑎1 since 0 < max(𝑠𝑢𝑝𝑝(𝒔𝑚, 𝑆𝑎𝑘 ), ∀𝑎𝑘 ∈ {𝑎1, 𝑎2}) = 𝑠𝑢𝑝𝑝(𝒔𝑚, 𝑆𝑎1 ) = 2 3 < 𝛾. according to definition 3.2., we do not need to perform two stages the classifier from the generated pattern, which is the basic procedures of mining sequential pattern techniques for classification case. therefore, we can have an efficient algorithm as we directly build the rules (not only the patterns). in the next section, we will explain how to mine multiclass non-zero-rare human activity patterns. 3.2 mining non-zero-rare sequential patterns algorithm this study presents an algorithm to mine a multi-class non-zero-rare pattern called mining multi-class non-zero-rare sequential patterns (mmrsp). the mmrsp consists of two main stages, (1) rare sequential patterns builder (rspb) and (2) classification unseen sequence (cus). algorithm 1 presents the detailed procedure of rspb to build a non-zero-rare pattern based on a training set (𝐿), where (𝐿 ⊆ 𝐷). by the frequent subsequence mining techniques used in [11,12], we first generate a motion sensor type subsequences tree (lines 5,11). then, the support values of each pattern in each activity sequences are calculated (line 7). the maximum support value is being checked to decide whether it is a non-zero-rare pattern (lines 8-9). later, its activity label becomes the class or be placed in the consequent part if the subsequence is a non-zero-rare pattern. additionally, we do not append the leaves node according to the max-pruning strategy in [7]. otherwise, a single sensor type 𝑚 from a set 𝑀 is added into the left node until all the possible candidate subsequences no longer satisfying the frequent conditions [12]. algorithm 1. (rare sequential patterns builder) input: a training set (𝐿), a pre-defined support threshold (𝛾), a set of activity labels (𝐴), and a sequence (𝒄) procedure rspb (𝐿, 𝐴, 𝛾, 𝒄) 1. 𝑆𝑚 = ∅; 2. for each 𝑎𝑘 ∈ 𝐴 do 3. 𝐷𝑎𝑘 = {(𝒔𝑗 → 𝑎𝑘 )|𝒔𝑗 ∈ 𝑆}; 4. for each 𝑚 ∈ 𝑅 × 𝑀 do 5. if (𝒄 ∘ 𝑚) ⋣ 𝑆𝑚 then 6. 𝒎𝒄 = 𝒄 ∘ 𝑚; 7. count 𝑠𝑢𝑝𝑝(𝒎𝒄, 𝐷𝑎𝑘 ); 8. if 0 < 𝑠𝑢𝑝𝑝(𝒎𝒄, 𝐷𝑎𝑘 ) < 𝛾then 9. 𝑆𝑚 = 𝑆𝑚 ⋃(𝒎𝒄 → 𝑎𝑘 ) ; 10. else if 𝑠𝑢𝑝𝑝(𝒎𝒄, 𝐷𝑎𝑘 ) ≥ 𝛾 m. iqbal, c. p. wulandari, w. yunanto, and g. i. p. sari mining non-zero-rare sequential patterns on activity recognition 5 algorithm 1. (rare sequential patterns builder) 11. rspb (𝐿, 𝐴, 𝛾, 𝒄); 12. end if 13. end if 14. end if 15. end for 16. end for output : 𝑆𝑚 = {(𝒔𝑚 → 𝑎𝑘 )} is a non-zerorare activity rule set after we hold the non-zero-rare patterns, the next stage is performing the cus in algorithm 2. in this stage, the activity label of unseen motion sensor sequences is being predicted. a simple way to predict is calculating the similarity between the generated nonzero-rare patterns with the unseen sequence by using cosine similarity. an activity label which has the highest support value is considered as the prediction result. in particular, we realize that the same cosine similarity values may be found or all the cosine similarity values are 0. these conditions indicate that there are no similar sensor types in 𝑃 with the generated non-zero-rare patterns. hence, we restrain two conditions as follows: (i) an activity label is selected randomly if the cosine similarity values are the same, and (ii) a default activity label is built in the early stage based on the maximum number of activity label in 𝑆𝑚 to provide an activity prediction result when the cosine value is 0. the detailed procedure of cus is presented in algorithm 2. since this study focuses on rare pattern performance in classification, we employ a precision formula to evaluate our mmrsp performance. the precision formula is denoted by: 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛(%) = 𝑡𝑝 𝑡𝑛 + 𝑓𝑝 ⋅ 100% (1) where 𝑡𝑝 is a number of true positive, 𝑡𝑛 is a number of true negative and 𝑓𝑝 is a number of false positive. algorithm 2. (classification unseen sequences) input: a testing set (𝑃), and a set of non-zero-rare activity rules (𝑆𝑚 ) procedure cus (𝑆𝑚 , 𝑃) 1. 𝑎𝑑 =defaultclass(𝑆𝑚 ); 2. for each 𝒑 ∈ 𝑃 do 3. 𝑎𝑝𝑖 = argmax𝑖∈𝑆𝑚 {cos(∠(𝒑, (𝒔𝑚 → 𝑎𝑚 )))} ; 4. if |𝑎𝑝𝑖 | > 1 do 5. 𝑎𝑝𝑚 =rand(𝑎𝑝𝑖 ); 6. else if |𝑎𝑝𝑖 | = 0 do 7. 𝑎𝑝𝑚 = 𝑎𝑑 ; 8. else 9. count++; 10. end if 11. end if 12. end for output: 𝐴ℎ = {𝑎𝑝𝑚 |1 ≤ 𝑚 ≤ |𝑃|} is a set of activity labels prediction jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 1-9 6 in the next section, we will describe the dataset information and the performance evaluation of our proposed algorithm on mining non-zero-rare pattern. furthermore, we analyze the generated non-zero-rare human activity patterns. 4. experimental results we now discuss the experimental results which are started by the dataset description. 4.1 dataset descriptions to aim our goals, we perform the dataset from [1] on our proposed algorithm. the dataset contains recorded sensors in three apartments—their called house a, b, and c. in this study, the dataset from house a is being used. in the ‘house a’ dataset, it comprises 14 sensors that installed in three rooms. as the sensors are triggered, 10 activities that annotated by bluetooth within 25 days were recorded. figure 1. the transformation phase on human activity sequences in specific, we discretize the sensory data with a different time interval δ𝑡 = 60𝑠. as a result, we have around 42000 human activity events data. to fit in our algorithms, the dataset needs to be transformed into the form of a sequence. it can be done by taking one discretization event as one sequence with the activity label as the last item of the sequence. we provide an example of the transformation process in figure 1. figure 2. a human activity sequences form m. iqbal, c. p. wulandari, w. yunanto, and g. i. p. sari mining non-zero-rare sequential patterns on activity recognition 7 furthermore, a training set consists of 200 sequences with the maximum length is 3752 and there are 15 distinct motion sensor labels in a set of sequences 𝑆 and 8 activity labels, such that brush teeth, get drink, go to bed, leave house, prepare breakfast, prepare dinner, take a shower, and use toilet. a testing set consists around 200 sequences with the maximum length is 3752 that we observe as the unseen sequences. we give an example of human activity sequences forms in figure 2. 4.2 evaluation this study simulated on a range of 𝛾-threshold into [0.01,1] since we found that the scale of the dataset is quite small over the number of distinct motion sensor label in 𝑆. based on 𝛾 = 0.05, we extracted 10 non-zero-rare patterns only for use toilet, such as: freezer → use toilet—motion sensor in freezer is being triggered and recognized as use toilet, groceries cupboard → use toilet—motion sensor in groceries cupboard is triggered when the resident use toilet, etc. (see in figure 3). these patterns are categorized as unusual resident activities when they still use the toilet. also, we obtain 13 non-zero-rare human activity patterns only to go to bed. the patterns are dishwasher → go to bed, plates cup board → go to bed, etc. figure 3. non-zero-rare human activity pattern when 𝛾 = 0.01 interestingly, we found that the number of generated non-zero-rare patterns are different for each support thresholds 𝛾 value (it is depicted in fig 4). in this case, each 𝛾 value built a different particular activity that contains a different number of non-zero-rare patterns. additionally, a support threshold 𝛾 can be represented as a particular activity event. as another viewpoint, the resident notifies that there is an unusual activity during the event and in place(s), which the sensors are triggered. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 1-9 8 figure 4. the number of generated non-zero-rare human activity patterns vs the support threshold values. in addition, we test the generated non-zero-rare human activity patterns to predict the unseen sequences based on precision values. the overall precision result is 87.5%. 5. conclusion and future works as the precision result, the generated non-zero-rare human activity patterns can discover the unusual events during the residents do their activity. this can be used as an alert for the residents. even though the mmrsp is well-performed, we still need to discuss the phenome that each support threshold give us different rare event only for a particular activity. thus, we will obtain properties to explain the relation between support threshold and activity events in the future references [1] t. l. m. van kasteren, g. englebienne and b. j. a. kröso, “human activity recognition from wireless sensor network data: benchmark and software,” in activity recognition in pervasive intelligent, 2010, pp. 165-185 [2] d. j. cook, c. krishnan, and p. rashidi, “activity discovery and activity recognition: a new partnership,” ieee trans. cybernetics, 2013, vol. 43, pp. 820– 828. 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[12] r. agrawal, and r. srinkant, “mining sequential patterns,” in international conference on data engineering, 1995, pp. 3-14. paper title (use style: paper title) how to cite: m. fajar, “the estimation of production function and technical efficiency shallot farming”, mantik, vol. 5, no. 1, pp. 50-59, may 2019. estimation of production function and technical efficiency shallot farming muhammad fajar1, oktya putri gitaningtyas2, muhammad muhtoni3, purwaning dhahari4 badan pusat statistik, mfajar@bps.go.id1 badan pusat statistik, oktyaputri@bps.go.id2 badan pusat statistik, muhtoni@bps.go.id3 badan pusat statistik, dhahari@bps.go.id4 doi: https://doi.org/10.15642/mantik.2019.5.1.50-59 abstrak: bawang merah merupakan salah satu komoditas potensi hortikultura. tujuan studi ini adalah untuk mengestimasi fungsi produksi dan efisiensi usaha tani bawang merah. metode yang digunakan dalam penelitian adalah estimasi fungsi produksi dengan menggunakan stochastic frontier. data yang digunakan dalam penelitian ini adalah produksi bawang merah (kg), luas panen (m2), tenaga kerja yang digunakan (hok), penggunaan benih (kg), pupuk (kg), pestisida yang digunakan (kg), bersumber dari shr2014 yang dilakukan badan pusat statistik. dalam proses estimasi seluruh variabel ditransformasi logaritma natural. hasil penelitian menunjukan bahwa estimasi fungsi produksi bawang merah baik untuk musim kemarau maupun musim hujan dengan variabel independen meliputi luas panen, tenaga kerja, benih, pupuk, dan pestisida signifikan dalam model sehingga model yang terbentuk valid untuk digunakan lebih lanjut. rata-rata efisiensi teknis usaha tani bawang merah pada musim kemarau dan hujan masing-masing sebesar 0.6626 dan 0.6627 yang berarti secara umum usaha tani bawang merah di indonesia tidak efisien dari sisi teknis. artinya, terdapat indikasi bahwa teknologi pengolahan optimal input produksi dalam usaha belum dilakukan secara optimal. kata kunci: fungsi produksi, efisiensi teknis, bawang merah. abstract: shallot is one of the potential horticultural commodities. the purpose of this study is to estimate the production function and efficiency of shallot farming. the method used in the study is the estimation of production functions using stochastic frontier. the data used in this study were shallot production (kg), harvested area (m2), labor used (hok), use of seeds (kg), fertilizer (kg), pesticides used (kg), sourced from shr2014 which conducted by the central statistics agency. in the estimation process, all variables are transformed by natural logarithms. the results showed that the estimation of the function of shallot production for both the dry season and the wet season with independent variables included harvested area, labor, seeds, fertilizers, and significant pesticides in the model, so that formed model was valid for further use. the average technical efficiency of shallot farming in the dry and wet season is 0.6626 and 0.6627, respectively, which means that in general, shallot farming in indonesia is not efficient on the technical side. that is, there are indications that the optimal processing technology of production inputs in the business has not been carried out optimally. keywords: production function, technical efficiency, shallots. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 50-59 issn: 2527-3159 (print) 2527-3167 (online) mailto:mfajar@bps.go.id m. fajar, oktya pg, muhtoni, purwaning the estimation of production function and technical efficiency shallot farming 51 1. introduction the horticulture sub-sector is one of the agricultural sub-sectors, which includes fruit, vegetables, medicinal plants, and ornamental plants of a very diverse type compared to the food crop subsector and plantations. however, the contribution of the horticulture subsector seen from indonesia's gross domestic product (gdp) during 2014 2018 was only in the range of 1.49 percent, smaller than the contribution of the food crops and plantation subsectors (figure 1). this shows that the horticulture sub-sector still has not contributed significantly to the overall economy. figure 1. the contribution of food crops, horticulture, and estate crops subsectors to gdp for the 2014-2018 period shallot is one of the potential horticultural commodities. from 2013 to 2017, the shallot production index tended to increase (figure 2). this indicates that the trend of commodity production during the observation period has increased. the increase in shallot production cannot be separated from the efforts of the farmers. however, it is not yet known whether the efforts made by farmers have been efficient or not efficient. therefore, in this study, the authors estimated the shallot production function using the stochastic frontier model so that from the model we can obtain the magnitude of shallot farming efficiency in indonesia, both during the dry season and the wet season. source: agricultural indicators, 2017 figure 2. the growth of indonesian shallot production index 2. research methods 2.1. cobb-douglas stochastics frontier a stochastic frontier function is formulated as follows: 𝑦𝑖 ∗ = 𝑓(𝒙𝒊; 𝜷) exp 𝜀𝑖 (1) jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 50-59 52 in equation (1) there is a component 𝑓(𝒙𝒊; 𝜷) which is assumed to be the cobb-douglas function. then equation (1) is transformed into a natural logarithm as follows: 𝑦𝑖 = 𝜷′𝒙𝑖 + 𝜀𝑖 (2) in equation (2) 𝒙𝒊 is production input vector (𝑗 × 1) . 𝑗 is the number of independent variables (production factors) and 𝑦𝑖 ∗ is the production produced by the observation (business unit) i . the component 𝜀𝑖 is described as a component 𝑣𝑖 and 𝑢𝑖 by specifying that: 𝜀𝑖 = 𝑣𝑖 + 𝑢𝑖 and 𝑖 states the observation (business unit) i. according to reference [6] addition (𝑣𝑖 + 𝑢𝑖) reflecting technical efficiency and 𝑢𝑖 reflect the effects of technical inefficiencies and 𝑢𝑖 ≤ 0. 𝑢𝑖 following a normal truncated distribution (𝑢𝑖~𝑁 +(𝜇, 𝜎𝑢 2)) and 𝑣𝑖 assumed to first follow a normal distribution (𝑣𝑖 ~𝑁(0, 𝜎𝑣 2))[10]. according to reference [3] equation (2) has the log-likelihood function and the first derivative of the log-likelihood function as follows: ln 𝐿 = 𝑛 ln √2 √𝜋 + 𝑛 ln 𝜎−1 + ∑ ln 1 − 𝐹∗((𝑦𝑖 − 𝜷 ′𝒙𝑖)𝜆𝜎 −1) 𝑛 𝑖=1 − 1 2𝜎2 ∑(𝑦𝑖 − 𝜷 ′𝒙𝑖) 2 𝑛 𝑖=1 (3) 𝜕 ln 𝐿 𝜕𝜎2 = − 𝑛 2𝜎 2 + 1 2𝜎4 ∑(𝑦𝑖 − 𝜷 ′𝒙𝑖) 2 𝑛 𝑖 =1 + 𝜆 2𝜎3 ∑ 𝑓𝑖 ∗ (1 − 𝐹𝑖 ∗) (𝑦𝑖 − 𝜷 ′𝒙𝑖) 𝑛 𝑖=1 (4) 𝜕 ln 𝐿 𝜕𝜆 = − 1 𝜎 ∑ 𝑓𝑖 ∗ (1 − 𝐹𝑖 ∗) (𝑦𝑖 − 𝜷 ′𝒙𝑖) 𝑛 𝑖=1 (5) 𝜕 ln 𝐿 𝜕𝜷 = 1 𝜎2 ∑(𝑦𝑖 − 𝜷 ′𝒙𝑖)𝒙𝒊 𝑛 𝑖=1 + 𝜆 𝜎 ∑ 𝑓𝑖 ∗ (1 − 𝐹𝑖 ∗) (𝑦𝑖 − 𝜷 ′𝒙𝑖 )𝒙𝒊 𝑛 𝑖=1 (6) by: n is the number of observation samples, 𝜎2 = 𝜎𝑢 2 + 𝜎𝑣 2 , 𝜆 = 𝜎𝑢 2 𝜎𝑣 2⁄ ,𝛾 = 𝜎𝑢 2 𝜎2⁄ 𝑓 ∗(. ) declare standard normal pdf[𝑓𝑖 ∗ = 𝑓 ∗((𝑦𝑖 − 𝜷 ′𝒙𝑖)𝜆𝜎 −1)], and 𝐹∗(. ) declare the default normal cdf [𝐹𝑖 ∗ = 𝐹∗((𝑦𝑖 − 𝜷 ′𝒙𝑖)𝜆𝜎 −1)]. in equation (4) it becomes maximum when the component: ∑ 𝑓𝑖 ∗ (1 − 𝐹𝑖 ∗) (𝑦𝑖 − 𝜷 ′𝒙𝑖) = 0 𝑛 𝑖=1 , enter these results into equation (3) so that: 𝜕 ln 𝐿 𝜕𝜎2 = − 𝑛 2𝜎2 + 1 2𝜎4 ∑(𝑦𝑖 − 𝜷 ′𝒙𝑖 ) 2 𝑛 𝑖=1 , (7) from equation (6) obtained: �̂�2 = 1 𝑛 ∑(𝑦𝑖 − 𝜷 ′𝒙𝑖 ) 2 𝑛 𝑖=1 . (8) the determination of β estimation cannot be direct as in the determination �̂�2 however �̂�2 depends on β and estimation 𝜆 depends on β and �̂� 2, so to get a solution in equations (4), (5), and (6) that must implement nonlinear optimization so that numerical optimization is needed involving an iterative process where the value 𝜎2 > 0, 𝜷 > 𝟎, and 𝜆 > 0. in this study to find solutions in equations (4), (5), and (6), so the authors use package npsf. m. fajar, oktya pg, muhtoni, purwaning the estimation of production function and technical efficiency shallot farming 53 package npsf is used to get parameter estimates of stochastics frontier equation (2) and technical efficiency. 2.2. technical efficiency (te) technical efficiency will be achieved if the entrepreneur can allocate production factors in such a way that high results can be achieved [4]. technical efficiency is defined as producing more, with the same input or producing the same number of outputs with fewer inputs [9]. technical efficiency requires or requires a production process that can take advantage of fewer inputs to produce output in the same amount [1]. so the use of production factors is said to be technically efficient if the production factors used to produce maximum production. technical efficiency is formulated as follows: 𝑇𝐸𝑖 = 𝑦𝑖 ∗ 𝑦𝑖 ∗̂ (9) with �̂�𝑖 is the potential output (fitted value) by observation ke-i. equation (9) state the technical efficiency of each observation so that it needs to be represented as an average. the technical efficiency mean is formulated: 𝑇𝐸 = 𝐸[exp(−𝑢𝑖|𝜀𝑖)] (10) if the value of technical efficiency is greater than 0.7, then a farm can be said to be quite efficient [8], and if the value of technical efficiency is more than 0.8, it can be said to be efficient [1]. 2.3. data source the data used in this study are secondary data and cross sections, namely data on the results of the cost structure of strategic horticultural commodities 2014 (st2013shr.s 2014), statistic indonesia. the input-output variables are from st2013-shr. the one used is 𝑦 ∗: shallot production (kg), 𝑥1: harvested area (m 2), 𝑥2: labor used (hok), 𝑥3: seed use (kg), 𝑥4: fertilizer (kg), 𝑥5: pesticides used (kg). in the estimation process, all variables are transformed by natural logarithms. 3. results and discussion 3.1. general description of shallot farming in indonesia in 2014 3.1.1. harvested area in figures 3 and 4, it is generally seen that the island of java has the largest harvested area compared to other islands. the potential for extensive shallot harvest in the dry season is in the provinces of central java, east java, and west java. in the dry season, the total harvested area in central java is 72,980.35 hectares, east java is 39,732.48 hectares, and west java is 13,179.99 hectares. while in the wet season, the largest harvested area is in the provinces of central java, east java, and south sulawesi. the total harvested area in the wet season in central java was 53,607.74 hectares, east java was 11,929.38 hectares, and south sulawesi was 6,296.67 hectares. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 50-59 54 figure 3. distribution of shallot harvests in the dry season in indonesia, 2014 figure 4. distribution of shallot harvests in the wet season in indonesia, 2014 3.1.2. production in line with the pattern of the wide distribution of harvests in indonesia, the highest distribution of shallot production is on the island of java which is spread in the province of central java, followed by east java, then west java. there is no difference in the distribution pattern of shallot production in the dry season and wet season (see figure 5 and figure 6). figure 5. distribution of shallot production in the dry season in indonesia, 2014 figure 6. distribution of shallot production in the wet season in indonesia, 2014 other input distributions such as seeds, fertilizers, pesticides, and labor force follow a broad distribution pattern of harvest and production. dry season wet season seed m. fajar, oktya pg, muhtoni, purwaning the estimation of production function and technical efficiency shallot farming 55 dry season wet season fertilizer pesticide figure 7. distribution of shallot seeds, fertilizers, and pesticides in the dry season and wet season in indonesia, 2014 3.2. shallot cost structure based on figure 7, it can be seen that in both seasons, shallot farmers spend more on costs for seeds, labor, and other expenses. in the dry season, expenditures for seeds amounted to 22.85 million rupiahs (35.39 percent), labor wages amounted to 20.18 million rupiahs (31.27 percent), and other expenses amounted to 9.9 million rupiahs (15.36 percent). other expenses are dominated by land expenditures of 6.8 million rupiahs (68.89 percent), the remainder is for expenses other than land (business equipment, interest on loans, indirect taxes, etc.). figure 7. cost structure of shallot farming during the wet season, expenditure on seeds is greater when compared to the dry season. this is because, in the wet season, the land becomes more humid than during the dry season, allowing pests and diseases to develop quickly. as a result, many shallot plants cannot survive, so insertion is necessary (planting new shallots on damaged land) that need more seeds. therefore, to reduce seed expenditure in the wet season, farmers can use shallot varieties that can survive in the wet season. seed expenditure in the wet jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 50-59 56 season was 31.68 million rupiahs (43.89 percent), labor wages amounted to 20.69 million rupiahs (28.68 percent), and other expenditures amounting to 7.53 million rupiahs (10.42 percent). as with other expenditures in the dry season, other expenses in the wet season are dominated by the land expenditure of 5.18 million rupiahs (68.79 percent), the rest for expenditure other than land. 3.3. estimated results of production function parameters stochastic frontier cobb douglas the estimated parameters of the stochastic frontier production function are divided into two, namely the production function during the dry season and the wet season. the following are the results of parameter estimates of the function of shallots production in the dry season. table 1. results of function parameter estimate stochastic frontier cobb douglass shallot in the dry season variable dry season coefficient z p>|z| intercept 0.5701 7.58 0.0000 log (harvested area) 0.5056 37.43 0.0000 log (labor) 0.2649 22.46 0.0000 log (seed) 0.0952 15.82 0.0000 log (fertilizer) 0.1588 22.17 0.0000 log (pesticide) 0.0370 7.50 0.0000 log likelihood -5911.0601 table 2. results of function parameter estimate stochastic frontier cobb douglass shallot in the wet season variable wet season coefficient z p>|z| intercept 0.2580 2.30 0.0217 log (harvested area) 0.6002 31.29 0.0000 log (labor) 0.1014 6.39 0.0000 log (seed) 0.1043 11.07 0.0000 log (fertilizer) 0.1599 17.44 0.0000 log (pesticide) 0.0537 10.02 0.0000 log likelihood -2759.7659 table 1 and table 2 provide information that in both the dry season and the wet season, production inputs, namely harvested area, labor, seeds, fertilizers, and pesticides have a statistically significant effect on shallot production at the five percent significance level. the estimation results produce an input coefficient marked positive. the coefficient on the model (2) is interpreted as the elasticity value of the production input. if the elasticity value is specified according to the type of input, then the value of labor elasticity in the dry season is greater than in the wet season. meanwhile, the value of the elasticity of the area of harvest, seeds, fertilizers, and pesticides in the wet season is greater than in the dry season. the greatest elasticity value of the five production inputs, both in the dry season and wet season, is the harvest elasticity. based on the results of parameter estimation, the production function equation is obtained stochastic frontier cobb-douglass shallots in the dry season as follows: 𝑙𝑛 𝑃𝑟𝑜𝑑_𝑀𝐾̂ = 0,5701 + 0,5056 × 𝑙𝑛𝐻𝑎𝑟𝑣𝑒𝑠𝑡𝑒𝑑𝐴𝑟𝑒𝑎𝑖 + 0,2649 × 𝑙𝑛𝐿𝑎𝑏𝑜𝑟 + 0,0952 × 𝑙𝑛𝑆𝑒𝑒𝑑𝑖 + 0,1588 × 𝑙𝑛𝐹𝑒𝑟𝑡𝑖𝑙𝑖𝑧𝑒𝑟𝑖 + 0,037 × 𝑙𝑛𝑃𝑒𝑠𝑡𝑖𝑐𝑖𝑑𝑒𝑖 (11) m. fajar, oktya pg, muhtoni, purwaning the estimation of production function and technical efficiency shallot farming 57 variable harvested area has a coefficient of 0.5056, which means that each addition of harvested area by 1 percent of the harvested area (assuming other inputs remain) can increase production by 0.5056 percent. when compared with other input variable coefficients, the harvested area has the largest coefficient, so it becomes the main influence of increasing production. the wider the area planted is then harvested, the greater the amount of production obtained. furthermore, labor (hok) has a coefficient of 0.2649. the results of this study indicate that each addition of 1 percent of the workforce assuming other inputs is fixed, it will increase production by 0.2649 percent. in the dry season, the workforce is the second variable that has a large contribution to the production of shallots. the use of intensive labor, both paid labor and unpaid labor can provide benefits in the production process. the seed coefficient of 0.0952 indicates that with the assumption that other inputs are fixed, each addition of 1 percent will increase production by 0.0952 percent. the selection of the right seed varieties by soil conditions and quality seeds can increase the productivity of shallots. the results showed that the amount of fertilizer used had a positive and significant effect on shallot production with a coefficient of 0.1588. this means that every 1 percent increase in fertilizer, assuming other inputs will continue to increase shallot production by 0.1588 percent. current conditions, the most use of fertilizer for shallots is npk fertilizer (22.14 percent) and manure/compost (17.32 percent). according to david prasatya et al., the use of npk fertilizer and manure with appropriate doses can increase nitrogen uptake in shallots. pesticide variable has a coefficient of 0.037, which means that every 1 percent addition of pesticides will increase shallot production by 0.037 percent assuming other inputs remain. the use of pesticides must follow a certain dose rather, it can protect plants from pests without damaging the organic elements of the soil. 𝑙𝑛𝑃𝑟𝑜𝑑_𝑀𝐻̂ = 0,2580 + 0,6002 × 𝑙𝑛𝐻𝑎𝑟𝑣𝑒𝑠𝑡𝑒𝑑𝐴𝑟𝑒𝑎𝑖 + 0,1014 × 𝑙𝑛𝐿𝑎𝑏𝑜𝑟𝑖 + 0,1043 × 𝑙𝑛𝑆𝑒𝑒𝑑𝑖 × 0,1599 × 𝑙𝑛𝐹𝑒𝑟𝑡𝑖𝑙𝑖𝑧𝑒𝑟𝑖 + 0,0537 × 𝑙𝑛𝑃𝑒𝑠𝑡𝑖𝑐𝑖𝑑𝑒𝑖 (12) in the wet season, the harvested area variable has a coefficient of 0.6002, which means that each addition of a harvested area of 1 percent of the harvested area (assuming other inputs remain) can increase production by 0.6002 percent. similar to the production function equation in the dry season, when compared with the other input variable coefficients, the harvested area has the largest coefficient, so that it becomes the main influence of increasing production. labor (hok) has a coefficient of 0.1014. the results of the study show that each addition of 1 percent of the workforce assuming other inputs is fixed, it will increase production by 0.1014 percent. the seed coefficient of 0.1043 shows that assuming other inputs are fixed, each addition of 1 percent will increase production by 0.1043 percent. the selection of shallot seed varieties in the wet season in addition to paying attention to soil conditions, but also must consider the conditions of soil moisture. the results showed that the amount of fertilizer used had a positive and significant effect on shallot production with a coefficient of 0.1599. this means that every addition of 1 percent fertilizer with the assumption that other inputs will continue to increase shallot production by 0.1599 percent. current conditions, the most use of fertilizer used in the wet season for shallots is the same as fertilizer use in the dry season, namely npk fertilizer (25.08 percent) and manure/compost (13.95 percent). in the wet season, fertilizer has a second important role after the harvested area increases the production of shallots. therefore, the selection of the right fertilizer must be considered by farmers in the shallot business. pesticide variables have a coefficient of 0.0537, which means that every 1 percent addition of pesticides will increase shallot production by 0.0537 percent assuming other inputs remain. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 50-59 58 3.4. shallot technical efficiency value the following is a table of descriptive statistics on the technical efficiency of shallots in the dry season and the wet season. table 3. results of estimation of the technical efficiency of shallots in the dry season and the wet season descriptive technical efficiency dry season wet season mean 0.6626 0.6627 median 0.7212 0.7202 minimum 0.0106 0.0188 maksimum 0.9320 0.9385 standar deviasi 0.1772 0.1836 n 6.289 3.135 based on the table, the technical efficiency of shallot farming in the dry and wet season is 0.6626 and 0.6627, respectively. there is no significant difference in the average technical efficiency in both seasons. the coefficient of 0.6626 (in the dry season) and 0.6627 (in the wet season) shows that households are only able to achieve the realization of production of 66.26 percent or 66.27 percent of the input owned. based on the specified criteria [8], information can be obtained that in general, the shallot farming in indonesia is not efficient on the technical side. that is, there are indications that the optimal processing technology of production inputs in the business has not been carried out optimally. to examine what factors cause technically inefficient production of shallots, further research needs to be done. 4. conclusion based on the discussion it can be concluded that the estimation of the function of shallot production, both for the dry season and 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[13] w. nicholson, mikroekonomi intermediate dan aplikasinya. erlangga, 2002. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran 17 | p a g e analisis optimal penjualan petis madura menggunakan metode topsis sumiatun dan kuzairi jurusan matematika, fmipa, universitas islam madura (uim) jl. bettet no. 04, pamekasan, madura 60111 indonesia e-mail: kuzairi81@gmail.com e-mail: sumiatun181@gmail.com abstrak petis merupakan jenis makanan yang digemari orang banyak, selain dijadikan bahan untuk usaha yang ditekuni, petis juga bemanfaat untuk bahan pendamping makanan rigan seperti gorengan, uniknya petis madura ini awet dan dijual sampai keluar madura. dalam penelitian ini metode yang digunakan adalah technique for order preference by similarity to ideal solution (topsis). metode topsis adalah salah satu metode yang digunakan untuk menyelesaikan masalah multi attribute decision making (madm). metode topsis didasarkan pada konsep dimana alternatif terpilih yang terbaik tidak hanya memiliki jarak terpendek dari solusi ideal positif, namun juga memiliki jarak terpanjang dari solusi ideal negatif. metode topsis memiliki beberapa kelebihan, diantaranya konsepnya yang sederhana dan mudah dipahami, komputasinya efisien, dan memiliki kemampuan untuk mengukur kinerja relatif dari alternatif-alternatif keputusan dalam bentuk matematis yang sederhana. kata kunci: petis, pemasaran, sistem pendukung keputusan, topsis 1. pendahuluan perekonomian di indonesia saat ini memasuki ajang persaingan yang sangat ketat dalam berbagai sektor yang ada. hal ini mendorong dunia usaha untuk meningkatkan efisiensi kerja dan mutu di bidang usaha yang dikelolanya untuk meningkatkan mutu di bidang usaha yang dikelola perusahaan harus tetap menjaga persediaan yang cukup, agar kegiatan operasi produksi dapat berjalan dengan lancar dan efisien. setiap perusahaan selalu memerlukan persediaan, tanpa adanya persediaan para pengusaha akan dihadapkan pada resiko bahwa perusahaannya pada suatu waktu tidak dapat memenuhi permintaan pelanggannya. saat ini pabrik petis masih menggunakan cara manual untuk mengoptimalkan penjualan petis madura. sehingga pengolahan data kurang efektif membutuhkan waktu yang relatif lama dan sering terjadi subjektifitas dari para pengambil keputusan.untuk mempermudah para pemilik petis dalam mengoptimalkan petis madura, sedangkan dengan menggunakan metode topsis yaitu alternatif yang baik tidak hanya memiliki jarak terpendek dari solusi ideal positif tetapi juga memilki jarak terpanjang dari solusi ideal negatif.[2] penelitian terkait sistem pendukung keputusan berdasarkan telaah, belum ada penelitian yang menggunakan metode topsis dalam proses analisis optimal penjualan petis madura yang menggunakan metode topsis. 2. uraian penelitian 2.1 petis petis adalah komponen dalam masakan indonesia yang dibuat dari produk sampingan pengolahan makanan berkuah (biasanya dari pindang, kupang, atau udang) yang dipanasi hingga cairan kuah menjadi kental seperti saus mailto:kuzairi81@gmail.com mailto:sumiatun181@gmail.com http://id.wikipedia.org/wiki/masakan_indonesia http://id.wikipedia.org/wiki/masakan_indonesia http://id.wikipedia.org/wiki/pengolahan_makanan http://id.wikipedia.org/wiki/pindang http://id.wikipedia.org/wiki/kupang_%28makanan%29 http://id.wikipedia.org/wiki/udang http://id.wikipedia.org/wiki/saus 18 | p a g e yang lebih padat. dalam pengolahan selanjutnya, petis ditambah karamel gula batok. 2.1 metode topsis (technique for order preference by similarty to ideal solution) technique for order preference by similarity to ideal solution (topsis) didasarkan pada konsep dimana alternatif terpilih yang terbaik tidak hanya memiliki jarak terpendek dari solusi ideal positif, namun juga memiliki jarak terpanjang dari solusi ideal negatif. [1] langkah-langkah penyelesaian masalah madm dengan topsis: a. membuat matriks keputusan yang ternormalisasi. b. membuat matriks keputusan yang ternormalisasi terbobot. c. menentukan matriks solusi ideal positif & matriks solusi ideal negatif. d. menentukan jarak antara nilai setiap alternatif dengan matriks solusi ideal positif & matriks solusi ideal negatif. e. menentukan nilai preferensi untuk setiap alternatif. topsis membutuhkan rating kinerja setiap alternatif pada setiap kriteria yang ternormalisasi, yaitu: √∑ (1) dengan , dan solusi ideal positif dan solusi ideal negatif dapat ditentukan berdasarkan rating bobot ternormalisasi sebagai: (2) dengan , dan (3) (4) dengan { { jarak antara alternatif dengan solusi ideal positif dirumuskan sebagai: √∑ ( ) (5) jarak antara alternatif dengan solusi ideal negatif dirumuskan sebagai: √∑ ( ) (6) nilai preferensi untuk setiap alternatif : (7) (2.13) nilai yang lebih besar menunjukkan bahwa alternatif lebih dipilih.[1] 3. penerapan metode topsis ke pemasaran petis madura 3.1 data penelitian daftar pemasaran petis madura sebagai bahan yang dinilai terdapat pada tabel 3.1 dan untuk kriteria penilaian yang digunakan oleh pihak pemilik untuk melakukan evaluasi kinerja pemasaran disajikan dalam tabel 3.2. selanjutnya keempat kecamatan disebut dengan alternatif. tabel 3.1 data alternatif pemasaran petis madura no. nama 1 pamekasan 2 sumenep 3 sampang 4 bangkalan sumber : empat kecamatan di kota madura tabel 3.2 kriteria yang digunakan dalam proses optimal pemasaran petis no. kriteria 1 pemilik mengambil per toples untuk pemasaran petis. 2 pemilik mengambil per kg untuk pemasaran petis. 3 pemilik memanfaatkan tempat untuk lancarnya pemasaran petis. sumber: petis madura 2014 konang a. bulan pertama perta-tama, dihitung terlebih dahulu, matriks keputusan berdasarkan persamaan (1).[3] matriks ternomalisas r: r [ ] http://id.wikipedia.org/wiki/karamel http://id.wikipedia.org/w/index.php?title=gula_batok&action=edit&redlink=1 19 | p a g e selanjutnya dihitung perkalian antara bobot dengan nilai setiap atribut untuk membentuk matriks (y) berdasarkan persamaan (2).[1] matrik (y): [ ] solusi ideal positif dihitung berdasarkan persamaan (3) sebagai berikut:[3] { } { } { } { } solusi ideal positif dihitung berdasarkan persamaan (4) sebagai berikut:[3] { } { } { } { } jarak antara nilai terbobot setiap alternatif terhadap solusi ideal positif , dihitung berdasarkan persamaan (5) sebagai berikut:[1] √ √ √ √ jarak antara nilai terbobot setiap alternatif terhadap solusi ideal positif , dihitung berdasarkan persamaan (6) sebagai berikut:[1] √ √ √ √ kedekatan setiap alternatif terhadap solusi ideal sehingga dihitung berdasarkan persamaan (7) sebagai berikut;[1] sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.3. tabel 3.3. nilai prefernsi dan rangking preferensi nilai rangking 1 0 3 2 2 b. bulan kedua pada bulan kedua sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.4. tabel 3.4. nilai prefernsi dan rangking preverensi nilai rangking 1 0 2 3 0,4304 4 c. bulan ketiga pada bulan ketiga sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.5. tabel 3.5. nilai prefernsi dan rangking preverensi nilai rangking 20 | p a g e 1 0 4 2 3 d. bulan keempat pada bulan keempat sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.6. tabel 3.6. nilai prefernsi dan rangking preverensi nilai rangking 1 0 4 3 2 e. bulan ke lima pada bulan kedua sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.7. tabel 3.7. nilai prefernsi dan rangking preverensi nilai rangking 1 0 4 3 2 f. bulan ke enam pada bulan keenam sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.8. tabel 3.8. nilai prefernsi dan rangking preverensi nilai rangking 1 0 3 2 2 g. bulan ke tujuh pada bulan tujuh sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.9. tabel 3.9. nilai prefernsi dan rangking preverensi nilai rangking 1 0 4 3 2 h. bulan ke delapan pada bulan tujuh sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.9. tabel 3.9. nilai prefernsi dan rangking preverensi nilai rangking 1 0 3 2 2 i. bulan kesembilan pada bulan sembilan sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.11. tabel 3.11. nilai prefernsi dan rangking preverensi nilai rangking 1 21 | p a g e 0 3 2 2 j. bulan kesepuluh pada bulan kesepuluh sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.12. tabel 3.12. nilai prefernsi dan rangking preverensi nilai rangking 1 0 3 2 2 k. bulan kesebelas pada bulan kesebelas sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.13. tabel 3.13. nilai prefernsi dan rangking preverensi nilai rangking 1 0 3 2 2 l. bulan keduabelas pada bulan keduabelas sama caranya dengan menggunakan persamaan (1) sampai (7) sehingga dengan menggunakan software matlab diperoleh hasil petis madura dengan metode topsis dari hasil nilai akhir pada tabel 3.14. tabel 3.4. nilai prefernsi dan rangking preverensi nilai rangking 1 4 2 3 4. kesimpulan/ringkasan dari hasil penelitian di atas diperoleh bahwa penerapan metode topsis dapat digunakan menyelesaikan permasalahan berdasarkan kriteria-kriteria yang berbeda pada tiap pemasaran yang ada. untuk pengembangan lebih lanjut dapat diteliti permasalahan ini dalam bentuk dua dimensi atau dibandingkan dengan metode yang lain seperti anp, atau yang lain, dan bisa dilanjutkan menggunakan fuzy topsis. ucapan terima kasih penulis s.n. mengucapkan terima kasih kepada dekan mipa uim, kajur matematika uim, dosen-dosen serta beberapa mahasiswa matematika uim yang telah memberikan dukungan baik secara finansial (materiil) maupun moril dalam pengembangan penelitian ini. daftar pustaka [1] badriah, “fuzzy multi atriibete decision making,” (2014). [2] fairus a himi, dkk. “analisa validitas penerimaan biasiswa menggunakan anp dan topsis,”(2015) [3] kusumadewi, dkk. “fuzzy multi-atribute decision making (madm”). graha ilmu, yogyakarta (2006). jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 100 analisis strategi penjualan stok spare part di pt fajar mas murni surabaya ida purwanti1,yuniar farida2 jurusan matematika, fakultas sains dan teknologi, uin sunan ampel surabaya1,2 email : idapurwanti344@yahoo.co.id1, yuniar_farida@uinsby.ac.id2 doi:https://doi.org/10.15642/mantik.2018.4.1.100-109 abstrak penelitian ini menggunakan studi kasus pada pt fajar mas murni surabaya yang bertujuan untuk mengklasifikasikan stok item spare part dengan pendekatan matriks bcg guna menentukan strategi penjualan stok item spare part. hasil klasifikasi dengan matriks bcg diperoleh 6,60% item termasuk dalam kuadran stars yang menyumbang omzet 80,67% bagi perusahaan; 15,57% item termasuk dalam kuadran question marks yang menyumbang omzet 14,95% bagi perusahaan; 71,70% item termasuk dalam kuadran dogs yang menyumbang omzet hanya 1,87% bagi perusahaan; 6,13% item termasuk dalam kuadran cash cows yang menyumbang omzet 2,50% bagi perusahaan. analisis strategi yang sebaiknya dilakukan adalah: (1) pada kuadran stars sebaiknya dilakukan forecasting penjualan untuk menjaga kontinuitas stok spare part, (2) pada kuadran question marks sebaiknya dilakukan peningkatan nilai penjualan dengan menjual item secara diskon agar dapat menghabiskan stok, (3) pada kuadran dogs sebaiknya lebih meningkatkan kegiatan promosi dari item-item tersebut, (4) pada kuadran cash cows sebaiknya mempertahankan nilai penjualannya. selanjutnya dilakukan forecasting penjualan pada item stok spare part kuadran stars pada tahun 2018 agar tetap terjaga kontinuitasnya dengan menggunakan metode trend (t) rata-rata bergerak dengan variasi siklus (c), variasi musim (s), dan indeks gerak tak beraturan (i), diperoleh nilai mape sebesar 23%. jika hanya menggunakan metode trend saja, diperoleh mape yang lebih besar, yakni 27%. kata kunci: pengendalian persediaan, matriks bcg, forecasting, metode trend dengan rata-rata bergerak abstract this research uses a case study at pt fajar mas murni surabaya which aims to classify item of spare parts inventory with a bcg matrix approach to determine sales strategy of spare part inventory. the classification results with bcg matrix obtained 6,60% items included in quadrant stars which contributed 80,67% turnover for the company; 15,57% items included in the question marks quadrant which contributed 14,95% turnover for the company; 71,70% items included in the quadrant dogs which contributed only 1,87% turnover for the company; and 6,13% is included in the cash cows quadrant which contributes a 2,50% turnover for the company. the strategy analysis that should be carried out (1) on the stars quadrant is to forecast the sales to maintain the continuity of spare part inventory, (2) on the question marks quadrant is to develop the sales by selling items at a discount so that can increase sales volume and can spend inventory, (3) on the dogs quadrant is to further enhance the promotion activities of these items, (4) on the cash cows quadrant is to maintain sales. then sales forecasting is carried out on stars quadrant spare parts inventory in 2018 so that continuity is maintained by using the trend (t) moving average ratio method with cycle variation (c), season variation (s), and irregular movement (i), which obtained mape value of 23%. if only using trend, it obtained greater mape value of 27%. keywords: inventory control, bcg matrix, forecasting, trend with moving average mailto:idapurwanti344@yahoo.co.id1 mailto:yuniar_farida@uinsby.ac.id2 jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 101 1. pendahuluan persediaan merupakan sumber utama penghasilan bagi perusahaan karena di dalam persediaan terdapat barang-barang dagangan yang dapat dijual sehingga menghasilkan laba bagi perusahaan. dalam perusahaan dagang, persediaan adalah kumpulan barang yang disimpan dengan tujuan untuk dijual kembali tanpa mengubah apapun dari barang itu sendiri. dalam persediaan, permasalahan yang krusial adalah adanya ketidaksesuaian antara jumlah barang yang disediakan dengan jumlah barang yang terjual, sehingga menimbulkan biaya penyimpanan yang berpotensi mengurangi laba perusahaan. oleh karena itu, diperlukan pengendalian persediaan agar proses perencanaan terhadap pengadaan suatu barang dapat ditentukan seoptimal mungkin. salah satu perusahaan dagang yang berkembang adalah pt fajar mas murni (fmm) surabaya, sebuah perusahaan yang bergerak di bidang distributor penjualan alat-alat industri, konstruksi, pertambangan, dan laboratorium kesehatan. selama ini pengendalian persediaan di pt fmm surabaya masih belum menggunakan metode yang ilmiah dan tepat. peneliti mencoba menerapkan pendekatan matriks bcg (boston consulting group) sebagai metode dalam inventory management di pt fmm surabaya agar tidak terjadi kelebihan stok maupun kekurangan stok agar dapat meminimalisir biaya yang ditimbulkan dari adanya persediaan tersebut. adapun penelitian-penelitian yang berkaitan dengan pengendalian persediaan diantaranya adalah analisis pengendalian perusahaan menggunakan pendekatan music 3d (multi unit spares inventory control-three dimensional approach) pada warehouse di pt semen indonesia (persero) tbk pabrik tuban [1], analisis portopolio produk pada pt. asuransi umum bumiputeramuda 1967 cabang lampung menggunakan matrik boston consulting group (bcg) [2], analisis pengendalian persediaan bahan baku ikan tuna pada cv. golden kk [3] , analisis matrik boston consulting group (bcg) terhadap portofolio produk guna perencanaan strategi pemasaran dalam menghadapi persaingan [4], dan analisis matriks boston consulting grup (bcg) pada sepeda motor merek honda (studi kasus pada pt. astra honda motor tahun 2013) [5]. pengendalian persediaan dapat didekati dengan beberapa metode seperti pada penelitianpenelitian sebelumnya yang pernah dilakukan. pendekatan dengan matriks bcg yang dilakukan terhadap pengendalian persediaan masih belum pernah diterapkan namun karena tujuan akhirnya adalah mengoptimalkan penjualan persediaan spare part maka matriks bcg ini dimodifikasi dan diterapkan sebagai metode untuk menganalisa strategi penjualan stok spare part sebagai bagian dari pengendalian persediaan. dengan menggunakan pendekatan matriks bcg, persediaan dikelompokkan menjadi 4 kelompok, yaitu kelompok stars, question marks, dogs, dan cash cows. 2. tinjauan pustaka 2.1 persediaaan persediaan (inventory) adalah sumber daya ekonomi yang perlu diadakan dan dipelihara untuk menunjang kelancaran produksi. sumber daya ekonomi tersebut dapat berupa kapasitas produksi, tenaga kerja, tenaga ahli, modal kerja, waktu yang tersedia, bahan baku, produk jadi, barang sedang dalam proses pengerjaan, serta bahan penolong [6]. dalam perusahaan dagang, persediaan merupakan item yang dimiliki perusahaan yang tersimpan di gudang guna untuk dijual dalam jangka waktu tertentu. persediaan timbul karena kuantitas item yang dibeli oleh perusahaan lebih besar dari kuantitas item yang terjual. persediaan yang berlebihan akan menimbulkan biaya yang besar bagi perusahaan sehingga perusahaan perlu untuk melakukan pengendalian persediaan agar biaya yang timbul dapat diminimalisir. 2.2 pengendalian persediaan pengendalian persediaan merupakan proses pengelolaan persediaan guna untuk menjaga keseimbangan antara jumlah persediaan dengan biaya persediaan yang merupakan faktor penunjang dalam produktivitas. salah satu tujuan adanya pengendalian persediaan adalah untuk mengoptimalkan persediaan agar perusahaan tidak kehabisan stok maupun kelebihan stok jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 102 serta mengoptimalkan biaya pengadaan persediaan [7]. 2.3 produk spare part pt fmm surabaya memiliki banyak produk untuk dijual yang dikelompokkan menjadi 14 grup. namun, pada penelitian ini produk yang dikaji adalah produk pada grup spp-asg yaitu spare part compressor. spare part merupakan suatu produk yang terdiri dari lebih dari satu komponen yang membentuk satu kesatuan dan memiliki fungsi tertentu. sedangkan compressor adalah alat yang digunakan untuk memasukkan, mengirim, dan menyediakan udara dengan tekanan tinggi [8]. sehingga, spare part compressor merupakan suatu produk yang terdiri lebih dari satu komponen yang berfungsi untuk memasukkan, mengirim, dan menyediakan udara dengan tekanan tinggi. 2.4 matriks boston consulting group matriks bcg merupakan perencanaan portofolio model yang dikembangkan oleh bruce henderson yang berasal dari boston consulting group pada tahun 1970. boston consulting group merupakan perusahaan konsultan manajemen swasta yang berkecimpung dalam hal perkembangan pangsa pasar di boston. secara umum, matriks bcg digunakan untuk mengelola portofolio bisnis dengan mempertimbangkan posisi pangsa pasar relatif dan tingkat pertumbuhan instansi yang dapat membantu perusahaan dalam menganalisis unit bisnis atau lini produk [9]. ide dari matriks bcg ini adalah setiap bisnis korporasi dapat dievaluasi dan diplot ke dalam sebuah matriks berukuran 2 × 2, sehingga dalam penelitian ini menggunakan pertimbangan posisi quantity out dan price untuk membantu perusahaan dalam menganalisis unit bisnis atau lini produk [10]. berikut gambaran dari matriks bcg: gambar 1 matriks bcg tujuan utama dari matriks bcg adalah untuk mengetahui item manakah yang layak mendapatkan perhatian khusus dan dukungan dana agar dapat bertahan dan menjadi kontributor terhadap kinerja perusahaan dalam jangka panjang. metode analisis matriks bcg dapat membantu untuk mengetahui posisi instansi berdasarkan pada kombinasi dari quantity out dan price terhadap pesaing dalam 4 kelompok, yaitu [2]: 1. stars (bintang) kategori stars menggambarkan kondisi perusahaan yang penjualannya melesat sehingga kategori ini memiliki peluang terbaik dalam jangka panjang dalam hal pertumbuhan dan profit bagi perusahaan [10].item yang masuk dalam kategori ini merupakan item yang memiliki quantity out (penjualan) tinggi dan price yang tinggi pula. pada dasarnya, item-item tersebut dapat memberikan investasi yang besar bagi perusahaan sehingga dapat mempertahankan dan memperkuat posisi dominan perusahaan serta dapat memberikan arus kas yang positif. 2. cash cows (sapi perah) kategori cash cows menggambarkan kondisi perusahaan yang mengalami quantity out yang tinggi, tetapi memiliki price yang rendah. kategori ini dinamakan cash cows karena menghasilkan kas yang lebih rendah dari yang dibutuhkan, sehingga perusahaan seringkali “diperah” [9]. item yang masuk dalam kategori ini harus dikelola dengan baik guna mempertahankan posisi dalam jangka panjang. 3. dogs (anjing) kategori dogs menggambarkan kondisi perusahaan yang memiliki quantity out dan price rendah, sehingga mengakibatkan laba quantity out low high l o w h i g h p r i c e question marks dogs stars cash cows jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 103 yang diterima perusahaan sangat kecil [9]. kategori ini memerlukan beberapa investasi karena sdm dan sumber cost nya sangat rendah yang disebabkan oleh posisi internal dan eksternalnya lemah. bisnis seperti ini seringkali dilikuidasi, di-divestasi, atau dipangkas dengan retrenchment. retrenchment merupakan solusi yang terbaik dalam kategori ini karena banyak dogs yang muncul kembali setelah pemangkasan biaya dan aset secara besar-besaran menjadi bisnis yang dapat bertahan dan menguntungkan bagi perusahaan. 4. question marks (tanda tanya kondisi question marks menggambarkan keadaan perusahaan yang memiliki quantity out yang rendah tetapi memiliki price yang tinggi. secara umum, kategori ini membutuhkan cost yang tinggi tapi cash in yang dihasilkan rendah [9]. untuk mengatasi hal tersebut maka kategori ini lebih mudah ditingkatkan penjualannya. hal tersebut dikarenakan item yang masuk dalam kategori ini bisa masuk dalam kategori stars. apabila strategi tersebut dilaksanakan maka akan terjadi cash out dalam jangka pendek untuk melunasi penjualan dengan harapan akan terjadi hal sebaliknya yaitu terjadi cash in di kemudian hari. kategori ini dinamakan question marks karena pihak perusahaan harus memberikan keputusan apakah akan memperkuat item-nya dengan menjalankan strategi ataukah item tersebut dibuang [2]. terdapat 2 sumbu yang digunakan dalam matriks bcg, yaitu: a. sumbu vertikal, merupakan pembatasan terhadap price. price yang dimaksud dalam penelitian ini adalah harga pokok pembelian dari masing-masing item. harga pokok pembelian ini akan dievaluasi dan dibagi menjadi 2 kelompok, yaitu sebagai berikut: 1. high (tinggi) dengan menghitung rata-rata harga pokok pembelian pada persamaan berikut : 𝑝𝑟𝑖𝑐𝑒̅̅ ̅̅ ̅̅ ̅ = ∑ 𝐻𝑃𝑃 𝑛 (1) dimana : 𝑝𝑟𝑖𝑐𝑒̅̅ ̅̅ ̅̅ ̅ = price rata-rata ∑ 𝐻𝑃𝑃 = jumlah harga pokok pembelian seluruh item 𝑛 = jumlah seluruh item spare part untuk menentukan item mana yang tergolong price tinggi, dengan cara mencari item yang price-nya lebih dari nilai price rata-rata. 2. low (rendah) dilakukan dengan mencari item yang price-nya kurang dari nilai price ratarata. b. sumbu horizontal, merupakan pembatasan terhadap quantity out. quantity out yang dimaksud dalam penelitian ini adalah jumlah unit yang terjual dari masing-masing item. quantity out ini akan dievaluasi dan dibagi menjadi 2 kelompok, yaitu sebagai berikut : 1. high (tinggi) dengan menghitung rata-rata quantity out yang disajikan dalam persamaan berikut : 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑢𝑡̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = ∑ 𝑢𝑛𝑖𝑡 𝑡𝑒𝑟𝑗𝑢𝑎𝑙 𝑛 (2) dengan : 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑢𝑡̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅=penjualan rata-rata 𝑛= jumlah seluruh item spare part untuk menentukan item yang tergolong dalam quantity out tinggi, dengan cara mencari item yang memiliki quantity out lebih dari quantity out rata-rata. 2. low (rendah) dilakukan dengan mencari item yang quantity out nya kurang dari quantity out rata-rata. 2.5 peramalan (forecasting) penjualan melakukan penjualan terhadap semua stok item yang disimpan dalam gudang (inventory) merupakan persoalan yang harus dihadapi oleh semua perusahan dagang, termasuk pt fmm surabaya. bagaimanapun, stok item dalam gudang akan menimbulkan holding cost yang bisa mengurangi profit perusahaan. oleh karena itu, diperlukan pendekatan yang tepat untuk mengoptimalkan penjualan persediaan (stok) item di gudang melalui peramalan (forecasting), yakni ilmu yang digunakan untuk mengetahui kejadian di masa mendatang. dengan forecasting, perusahaan dapat mengetahui berapa banyak unit dari masing-masing item yang harus di stok dalam gudang jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 104 2.6 deret berkala terdapat 4 komponen dalam deret berkala, yaitu t (trend), s (variasi musim), c (variasi siklus) dan i (indeks gerak tak beraturan) yang diringkas dalam persamaan berikut [11]: 𝑌 = 𝑇 × 𝑆 × 𝐶 × 𝐼 (3) komponen yang pertama yaitu t (trend). trend merupakan data yang memiliki kecenderungan gerakan meningkat atau menurun dari waktu ke waktu dalam jangka panjang. pada penelitian ini digunakan metode kuadrat kecil. 2.6.1 metode least square (kuadrat terkecil) metode kuadrat terkecil merupakan salah satu metode analisis trend untuk melakukan peramalan dengan menentukan garis trend yang mempunyai jumlah paling kecil dari kuadrat selisih data aktual dengan data pada garis trend. berikut persamaan utuk mendapatkan garis trend [11]: 𝑌′ = 𝑎 + 𝑏𝑋 (4) dimana : 𝑌′ =nilai trend hasil prediksi 𝑎 =nilai konstanta 𝑏 =nilai kemiringan 𝑋 =nilai periode tahun untuk memperoleh nilai 𝑎 dan 𝑏 dapat menggunakan persamaan berikut : 𝑎 = ∑ 𝑌 𝑛 𝑏 = ∑ 𝑋𝑌 ∑ 𝑋 2 (5) komponen yang kedua yaitu s (variasi musim). variasi musim merupakan fluktuasi dalam musim-musim tertentu. dalam penelitian ini menggunakan metode rasio rata-rata bergerak. 2.6.2 metode rata-rata bergerak pada metode rasio rata-rata bergerak (ratio to moving average method) dilakukan dengan membuat rata-rata bergerak selama periode tertentu. nilai periode (n) tergantung pada kondisi pengaruh fluktuasi musiman, bisa 2,3, 4, atau 12. persamaan dari metode rasio rata-rata bergerak yaitu [11]: 𝑆 = 𝑛𝑖𝑙𝑎𝑖 𝑟𝑎𝑠𝑖𝑜 × 𝑓𝑎𝑘𝑡𝑜𝑟 𝑘𝑜𝑟𝑒𝑘𝑠𝑖 (6) dimana : nilai rasio = data aktual/data rata-rata bergerak faktor koreksi = (100× 𝑛)/jumlahrata-rata rasio selama 𝑛 komponen yang ketiga yaitu c (siklus). dalam metode analisis variasi siklus dapat diperoleh dengan menggunakan indeks siklus. 2.6.3 indeks siklus siklus merupakan perubahan naik dan turun dalam suatu periode dan berulang pada periode lain. jika y, t, dan s telah diketahui maka ci dapat dicari dengan persamaan berikut [11]: 𝑌 𝑆 = 𝑇 × 𝐶 × 𝐼 (7) dimana 𝑇 × 𝐶 × 𝐼 menunjukkan data normal. untuk memperoleh faktor siklus maka unsur t (trend) dikeluarkan dari data normal. sehingga faktor siklus menjadi : 𝐶𝐼 = 𝑇𝐶𝐼 𝑇 (8) komponen yang keempat adalah i (gerak tak beraturan). metode analisis gerak tak beraturan dapat diperoleh dengan mencari indeks gerak tak beraturan. 2.6.4 indeks gerak tak beraturan gerak tak beraturan (irregular movement) merupakan perubahan kenaikan dan penurunan yang tidak beraturan baik dari waktu dan lama siklusnya. untuk mendapatkan indeks gerak beraturan, dapat dicari dengan membagi faktor siklus (ci) dengan (c) atau dapat disajikan dalam persamaan berikut [11]: 𝐼 = 𝐶𝐼 𝐶 (9) 2.7. mape mape merupakan persentase kesalahan hasil prediksi atau peramalan terhadap nilai aktual selama periode tertentu. mape dinyatakan dalam persamaan berikut [11, 12]: 𝑀𝐴𝑃𝐸 = ( 100 𝑛 ) ∑ |𝑃0 − 𝑃1 𝑃0 | (10) dengan : 𝑃0 = data aktual pada periode tertentu 𝑃1 = hasil prediksi pada periode tertentu 𝑛 = jumlah periode peramalan jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 105 3. metode penelitian 3.1 data a. untuk melakukan klasifikasi persediaan dengan pendekatan matriks bcg digunakan data harga pokok pembelian dan jumlah item yang terjual pada tahun 2017. data item tersebut merupakan data persediaan spare part compressor grup spp-asg sebanyak 212 item di pt fmm surabaya. b. data yang digunakan untuk forecasting adalah data item stock pada kuadran stars (14 item) mulai tahun 2009 hingga tahun 2017. 3.2 langkah-langkah pengolahan data a. menghitung total penjualan setiap item. kemudian menghitung ratarata price dan rata-rata quantity out setiap item. b. mengelompokkan item pada price rendah atau tinggi. price yang nilainya diatas rata-rata price maka dikelompokkan pada price tinggi, sedangkan ketika nilainya dibawah rata-rata price maka dikelompokkan pada price rendah. c. mengelompokkan item pada quantity out rendah atau tinggi. quantity out yang nilainya diatas rata-rata quantity out maka dikelompokkan pada quantity out tinggi, sedangkan ketika nilainya dibawah rata-rata quantity out maka dikelompokkan pada quantity out rendah. d. mengklasifikasikan item-item pada 4 kuadran, yaitu kuadran stars, question marks, dogs, dan cash cows. e. melakukan forecasting pada item stars menggunakan metode trend (t) rata-rata bergerak dengan variasi siklus (c), variasi musim (s), dan indeks gerak tak beraturan (i). 4 hasil dan pembahasan 4.1 klasifikasi dan analisis strategi penjualan stok spare part berdasarkan matriks bcg terdapat 4 kuadran, yaitu stars, question marks (?), dogs dan cash cows. berikut kategori masingmasing kuadran: tabel 1.kategori matriks bcg kuadran quantity out price stars tinggi tinggi (?) rendah tinggi dogs rendah rendah cash cows tinggi rendah dengan menggunakan persamaan (1) diperoleh rata-rata price sebesar 𝑅𝑝 5.498.706.51 ≈ 𝑅𝑝 5.000.000, item yang memiliki price lebih dari rp. 5.000.000,masuk dalam golongan tinggi, sedangkan item yang price -nya kurang dari rp. 5.000.000,masuk dalam golongan rendah. dengan menggunakan persamaan (2) rata-rata quantity out sebanyak 9,04717 ≈ 9. item yang memiliki quantity out lebih dari 9 masuk dalam golongan tinggi, sedangkan item yang quantity out-nya kurang dari 9 masuk dalam golongan rendah. hubungan price dan quantity out dapat digambarkan dalam matriks bcg seperti ditunjukkan pada gambar 3. gambar 3 gambaran matriks bcg pt fajar mas murni surabaya berdasarkan gambar di atas, diperoleh klasifikasi item dalam 4 kelompok, yaitu: 5juta 9 0 864 12ribu 164juta quantity out low high l o w h i g h p r i c e question marks dogs stars cash cows jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 106 4.1.1 kuadran stars pada mulanya, terdapat 6 item spare part yang tergolong dalam kuadran i (stars). pada kelompok stars, item-item tersebut berada pada posisi quantity out (penjualan) yang tinggi dan price (harga pokok pembelian) yang tinggi pula, yakni penjualannya lebih dari 9 unit dalam 1 tahun dan harga pokok pembeliannya lebih dari rp 5.000.000. selanjutnya, dilakukan analisa kembali pada total penjualan masing-masing item, dengan menghitung rata-rata penjualannya. diperoleh rata-rata total penjualan sebesar rp 29.463.392.90 ≈ rp 29.000.000. item yang memiliki nilai total penjualan lebih dari 𝑅𝑝 29.000.000 digolongkan dalam total penjualan tinggi, sedangkan item yang memiliki nilai total penjualan kurang dari 𝑅𝑝 29.000.000 digolongkan dalam total penjualan rendah. sehingga, terdapat tambahan 8 item lagi yang dapat masuk ke dalam kelompok stars. mulanya, 8 item tersebut masuk dalam kelompok cash cows namun karena item tersebut memiliki total penjualan yang tinggi yaitu lebih dari 𝑅𝑝 29.000.000 maka 8 item tersebut dapat tergolong menjadi kelompok stars. oleh karena itu, jumlah keseluruhan item spare part yang tergolong dalam kelompok stars sebanyak 14 item yang ditunjukkan pada tabel 2. tabel 2. kelompok stars no. part number out total penjualan 1 38459582 846 rp 3.353.350.638,24 2 54509435 23 rp 407.851.468,42 3 22219174 22 rp 306.562.073,84 4 54509427 17 rp 129.661.821,83 5 39863857 13 rp 92.336.668,19 6 67731158 12 rp 60.708.004,68 7 39903281 215 rp 200.134.123,85 8 39911631 112 rp 132.236.016,64 9 54601513 38 rp 99.393.725,68 10 39708466 132 rp 75.115.659,96 11 54749247 52 rp 64.705.755,40 12 22089551 20 rp 48.362.085,80 13 89237903 34 rp 35.111.932,26 14 39911615 49 rp 34.268.409,21 dari 6,6% item tersebut dapat disimpulkan bahwa kelompok stars menyumbang 80,67% penjualan bagi perusahaan. secara umum, kuadran stars ini memiliki margin yang kecil namun omzet penjualannya besar karena rata-rata item-item tersebut dijual dengan diskon dengan tujuan untuk mengejar volume penjualan. sehingga, kontinuitas stok item tersebut perlu dijaga agar tidak kehabisan stok. karena kuadran stars memiliki total penjualan yang besar bagi perusahaan, maka item-item pada kuadran ini harus dipertahankan. salah satu alternatif strategi untuk mempertahankan kuadran stars ini adalah dengan melakukan forecasting penjualan persediaan agar tetap menjaga kontinuitas stok item. 4.1.2. kuadran question marks terdapat 33 item spare part yang tergolong dalam kuadran ii (question marks) yang ditunjukkan pada tabel 3. tabel 3. kelompok question marks no. part number out total penjualan 1 39433743 plt 3 rp 490.079.640,63 2 39433743 7 rp 285.879.790,35 3 92722750 5 rp 31.890.772,90 4 39817655 2 rp 29.631.003,28 5 43074947 1 rp 29.349.036,40 6 22110399 1 rp 29.197.779,07 7 39807532 2 rp 27.868.964,22 8 22699706 1 rp 9.889.346,02 9 00446575 0 rp 10 42447177 0 rp . . . . . . . . . . . . 33 39844113 0 rp pada kuadran question marks, item-item tersebut berada pada posisi quantity out yang rendah dan price yang tinggi. dimana penjualannya kurang dari 9 unit dalam 1 tahun dan harga pokok pembeliannya lebih dari rp 5.000.000. sebanyak 15,57% item masuk pada kuadran question marks menyumbang 14,95% penjualan bagi perusahaan. item yang termasuk dalam kuadran ini merupakan item yang memiliki resiko tinggi untuk distok karena telah tersubstitusi oleh produk lain dan juga memiliki price yang tinggi sehingga penjualanna menjadi rendah. alternatif strategi yang sebaiknya dilakukan adalah mengembangkan nilai penjualan dengan menjual diskon agar terjadi peningkatan volume penjualan dan dapat menghabiskan stok. 4.1.3 kuadran dogs terdapat 152 item spare part yang tergolong dalam kuadran iii (dogs). pada kelompok dogs, item-item tersebut berada jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 107 pada posisi quantity out yang rendah dan price yang rendah pula. dimana penjualannya kurang dari 9 unit dalam 1 tahun dan harga pokok pembeliannya kurang dari rp 5.000.000. sebanyak 71,70% item masuk pada kuadran dogs dan hanya menyumbang 1,87% penjualan bagi perusahaan. alternatif strategi yang sebaiknya dilakukan adalah lebih meningkatkan kegiatan promosi agar nilai penjualan item tersebut meningkat. 4.1.4 kuadran cash cows terdapat 21 item spare part yang tergolong dalam kuadran iv (cash cows). pada kelompok sapi perah, item-item tersebut berada pada posisi quantity out yang tinggi dan price yang rendah. pada kuadran sapi perah memiliki penjualan yang lebih dari 9 unit dalam 1 tahun dan harga pokok pembeliannya kurang dari rp 5.000.000. kemudian dilakukan analisa kembali pada total penjualan masing-masing item dengan menggunakan persamaan (1). item yang memiliki nilai total penjualan lebih dari 𝑅𝑝 29.000.000 digolongkan dalam total penjualan tinggi, sedangkan item yang memiliki nilai total penjualan kurang dari 𝑅𝑝 29.000.000 digolongkan dalam total penjualan rendah. terdapat 8 item cash cows yang dapat masuk pada kelompok stars karena item tersebut memiliki total penjualan yang tinggi yaitu lebih dari 𝑅𝑝 29.000.000. oleh karena itu, jumlah keseluruhan item spare part yang tergolong dalam kelompok cash cows sebanyak 13 item yaitu sebagai berikut: tabel 4. kelompok cash cows no. part number out total penjualan 1 37952355 28 rp 28.850.378,76 2 22334155 10 rp 26.929.326,10 3 46866331 15 rp 23.247.761,40 4 54672654 35 rp 16.694.706,70 5 39907175 25 rp 11.869.352,00 6 22203095 14 rp 11.054.323,56 7 37952264 11 rp 8.053.531,86 8 39155478 32 rp 7.237.623,68 9 32012957 9 rp 5.664.898,44 10 39588470 11 rp 4.802.033,50 11 89295976 10 rp 4.816.236,80 12 67500892 9 rp 3.976.906,68 13 39194915 12 rp 2.846.740,56 sebanyak 6,13% item masuk pada kuadran cash cows menyumbang 2,50% penjualan bagi perusahaan. pada cash cows ini, perusahaan berusaha mendapatkan margin yang besar karena para pelanggan sudah banyak meng-order item-item tersebut karena harga jual item tersebut murah. sehingga alternatif strategi yang sebaiknya dilakukan adalah dengan mempertahankan nilai penjualan. 4.1.5 diagram hasil matriks bcg dari pembahasan di atas, diperoleh diagram akhir matriks bcg pt fajar mas murni surabaya ketika menggunakan kombinasi antara quantity out dan price yang disajikan dalam gambar 4. gambar 4 hasil akhir klasifikasi matriks bcg berdasarkan pendekatan matriks bcg, diperoleh persentase akhir item 6,60% yang termasuk dalam kuadran stars, kuadran question marks sebesar 15,57%, kuadran dogs sebesar 71,70% dan pada kuadran cash cows diperoleh persentase item sebesar 6,13% sebagaimana ditunjukkan pada gambar 5. gambar 5 persentase item akhir jumlah item = 33 total penjualan = rp 5.039.798.384 jumlah item = 13 total penjualan = rp 933.786.333 jumlah item = 14 question marks stars jumlah item = 152 total penjualan = rp 116.610.758 dogs cash cows total penjualan = rp 156.043.820 9 0 864 164 juta h i g h p r i c e 5 juta l o w 12 ribu high low quantity out 6.60% 15.57% 71.70% 6.13% persentase item akhir stars ? dogs cash cows jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 108 sedangkan diagram persentase penjualan matriks bcg dapat disajikan pada gambar 6. gambar 6 persentase penjualan akhir berdasarkan pendekatan matriks bcg, kuadran stars menyumbang 80,67% omzet penjualan, kuadran question marks menyumbang 14,95% omzet penjualan, kuadran dogs hanya menyumbang omzet penjualan sebesar 1,87% dan pada kuadran cash cows menyumbang 2,50% omzet penjualan bagi perusahaan. 4.2 forecasting penjualan pada penelitian ini dilakukan peramalan penjualan item stok kuadran stars di tahun 2018 dengan menggunakan metode analisis variasi siklus yang menggabungkan 4 komponen yaitu trend, musim, siklus, dan indeks gerak tak beraturan. langkah awal dalam melakukan forecasting penjualan yaitu mencari nilai trend (t) dengan metode kuadrat terkecil seperti pada persamaan (4) diperoleh 𝑌′ = 347,667 − 0,181𝑋, dengan memasukkan nilai x sesuai periode waktu yang akan diramalkan sehingga diperoleh hasil pramalan penjualan tahun 2018 pada item dengan part number 38459582 sebesar 1.034,856 ≈ 1.035. kemudian variasi musim dihitung dengan metode rasio rata-rata bergerak seperti pada persamaan (6) sehingga diperoleh indeks musim (s) kuartalan yang dikalikan dengan faktor koreksi sebagai berikut: indeks triwulan i = 1,13687156 × 0,996441 = 1,1328225101 indeks triwulan ii = 1,00814955 × 0,996441 = 1,004561253 indeks triwulan iii = 0,86569491 × 0,996441 = 0,862613646 selanjutnya variasi siklus dihitung dengan menggunakan indeks siklus (c) dalam persamaan (7) dan (8) diperoleh (0,8529611+1,1447773+1,1901608)/3 = 1,0626331, kemudian gerak tak beraturan (i) dihitung dengan persamaan (9) diperoleh (0,9812825+0,9779907+1,0739391)/3 = 1,0110708. setelah diperoleh nilai trend, indeks musim, indeks siklus, dan indeks gerak tak beraturan (tsci) maka langkah yang terakhir yaitu menggabungkan 4 komponen tersebut dengan persamaan (3) sehingga diperoleh prediksi penjualan tahun 2018 pada item dengan part number 38459582 sebesar 1.111,899 ≈ 1.112. hasil peramalan penjualan dari masing-masing item kuadran stars disajikan pada tabel 5. tabel 5. hasil prediksi penjualan tahun 2018 no. part number hasil prediksi 2018 (t) hasil prediksi 2018 (tsci) 1 38459582 1035 1112 2 54509435 138 141 3 22219174 103 90 4 54509427 91 54 5 39863857 50 20 6 67731158 19 14 7 39903281 15 10 8 39911631 12 13 9 54601513 6 4 10 39708466 33 35 11 54749247 60 63 12 22089551 24 13 13 89237903 10 6 14 39911615 46 44 dari tabel 5 terlihat bahwa hasil peramalan penjualan ketika hanya menggunakan metode trend dengan variasi 4 komponen (tsci) tidak jauh berbeda dengan hanya menggunakan metode trend saja. namun, hasil peramalan ketika menggunakan 4 komponen (tsci) menghasilkan nilai mape (pers. 10) sebesar 23% yang lebih kecil daripada ketika hanya menggunakan metode trend saja, yaitu 27%. 80,67% 14,95% 1,87% 2,50% persentase penjualan akhir stars ? dogs jurnal matematika “mantik” oktober 2018. vol. 04 no. 02 issn: 2527-3159 e-issn: 2527-3167 109 5. simpulan berdasarkan pendekatan matriks bcg, diperoleh klasifikasi persediaan item spare part menjadi 4 kuadran yaitu kuadran stars terdapat 14 item, kuadran question marks terdapat 33 item, kuadran dogs terdapat 152 item, dan kuadran cash cows terdapat 13 item. analisis strategi yang sebaiknya dilakukan di tiap kuadran adalah: (1) pada kuadran stars sebaiknya dilakukan forecasting penjualan untuk menjaga kontinuitas stok spare part, (2) pada kuadran question marks sebaiknya dilakukan peningkatan nilai penjualan dengan menjual item secara diskon agar dapat menghabiskan stok, (3) pada kuadran dogs sebaiknya lebih meningkatkan kegiatan promosi dari item-item tersebut, (4) pada kuadran cash cows sebaiknya mempertahankan nilai penjualannya. hasil peramalan atau forecasting penjualan stok spare part dalam kurun waktu 1 tahun mendatang yaitu tahun 2018 pada kuadran stars dilakukan dengan metode trend rasio ratarata bergerak dengan variasi musim, siklus, dan gerak tak beraturan karena menghasilkan nilai kesalahan mape yang lebih kecil (23%) daripada hanya dengan menggunakan metode trend saja (27%). referensi [1] d. janari, m. m. rahman dan a. r. anugerah, “analisis pengendalian perusahaan menggunakan pendekatan music 3d (multi unit spares inventory control-three dimensional approach) pada warehouse di pt semen indonesia (persero) tbk pabrik tuban,” teknoin, vol. 22, no. 4, pp. 261-268, 2016. [2] m. y. s. barusman dan s. gunardi, “analisis portopolio produk pada pt. asuransi umum bumiputeramuda 1967 cabang lampung menggunakan matrik boston consulting group (bcg),” jurnal manajemen dan bisnis, p. 1, 2014. [3] m. c. tuerah, “analisis pengendalian persediaan bahan baku ikan tuna pada cv. golden kk,” emba, vol. 2, no. 4, pp. 524-536, 2014. [4] w. wahyuandari, “analisis matrik boston consulting group (bcg) terhadap portofolio produk guna perencanaan strategi pemasaran dalam menghadapi persaingan,” jurnal universitas tulungagung bonorowo, vol. 1, no. 1, pp. 88-104, 2013. [5] y. s. putra, “analisis matriks boston consulting grup (bcg) pada sepeda motor merek honda (studi kasus pada pt. astra honda motor tahun 2013),” among makarti, vol. 7, no. 13, pp. 48-71, 2014. [6] m. haming dan m. nurnajamuddin, manajemen produksi modern, jakarta: pt bumi aksara, 2017. [7] a. wibisono, “penerapan analisis abc dalam pengendalian persediaan produk furniture pada java furniture, wonosari, klaten,” universitas sebelas maret, surakarta, 2009. [8] aditiya, “makalah kompresor,” universitas sarjanawiyata tamansiswa yogyakarta, yogyakarta, 2003. [9] r. p. suci, esensi manajemen strategi, sidoarjo: zifatama, 2015. [10] hery, manajemen strategik, jakarta: pt grasindo, 2018. [11] suharyadi dan purwanto, statistika untuk ekonomi dan keuangan modern, jakarta: salemba empat, 2017. [12] r. l. r. d. g. d. yogo aryo jatmiko, perbandingan keakuratan hasil peramalan produksi bawang merah metode holt-winters dengan singular spectrum analysis (ssa)”, mantik, 2017. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran m. ridho and d. devianto bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability how to cite: m. ridho and d. devianto, “bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability”, mantik, vol. 5, no. 1, pp. 10-18, may 2019. bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability muhammad ridho1 and dodi devianto2 andalas university, dho22.mhd@gmail.com1 andalas university, ddevianto@sci.unand.ac.id2 doi: https://doi.org/10.15642/mantik.2019.5.1.10-18 abstrak: tujuan dari penelitian ini adalah untuk mengetahui faktor-faktor yang mempengaruhi tingkat kemampuan wirausaha pada daerah wisata di nagari salayo sumatera barat. dalam hal ini, tingkat kemampuan wirausaha menjadi variabel respon dengan skala ordinal yang terdiri dari empat kategori, yaitu rendah, sedang, tinggi dan sangat tinggi. sedangkan, variabel prediktor terdiri dari empat faktor sosio-demografi yaitu jenis kelamin, tingkat pendidikan, kelompok usia dan pekerjaan serta lima variabel motivasi wirausaha. untuk menentukan variabel prediktor yang mempengaruhi variabel respon digunakan regresi logistik ordinal dengan pendugaan bootstrap. hasil dari penelitian ini menunjukkan terdapat dua variabel prediktor yang secara signifikan mempengaruhi variabel respon, yaitu motif wirausaha dan motif sosial dengan hit ratio sebesar 61,667. sehingga dapat disimpulkan bahwa model yang terbentuk dapat digunakan untuk menentukan tingkat kemampuan wirausaha pada daerah wisata. kata kunci: regresi logistik ordinal, pendekatan bootstrap, kemampuan wirausaha abstract: the purpose of this study is to determine the factors that affect the level of entrepreneurial capability in tourism of rural area in nagari salayo of west sumatra. the level of entrepreneurial capability is the response variable in this study with an ordinal scale consisting of four categories; they are lower, middle, high, or very high. whereas the predictor variables consist of 4 sociodemographic factor variables, they are gender, education level, age group, and occupation, and also five entrepreneurial motivation variables. to determine the predictor variables that are significantly affecting response variables, an ordinal logistic regression with a bootstrap estimation is executed. the study’s result shows two predictor variables that affect the response variable significantly; they are the entrepreneurial motive and social motive with the hit ratio of 61,667%. with that result, the model formed by bootstrapping logistic regression can determine the level of entrepreneurial capability in tourism of the rural area. keywords: ordinal logistic regression, bootstrap estimation, entrepreneurial capability jurnal matematika mantik vol. 5, no. 1, may 2019, 10-18 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 m. ridho and d. devianto bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability 11 1. introduction tourism in indonesia is currently growing rapidly, along with the increase in visits by domestic and foreign tourists. this extraordinary tourism potential which owned by indonesia can be a mainstay to raise society’s lives. cultural village is one type of business that immediately captivates many regions, one of which is solok regency. solok as one of the tourist destinations in west sumatra, indonesia seeks to encourage increased tourist visits to this area by developing a cultural or traditional village program in the rural areas of nagari salayo at west sumatra province. the entrepreneur in tourism certainly needs serious attention so that they can successfully invite tourists to come. the arrival of tourists can grow society’ economic, which is marked by the growing number of business communities. there is some evidence about regional development being community entrepreneurs with research surveys on tourism motivation [1] entrepreneurial motivation [2], travel motivation and value [3], sme performance [4] and also investigations about the effects of motivation on entrepreneurial capability. the entrepreneurial ability influences the entrepreneurial process [5]. this means that the success rate of an entrepreneurial is directly proportional to the level of the entrepreneur's ability to run a business. the entrepreneurs who have a high level of achievement of initial motivation also have high entrepreneurial abilities [6]. there are specific factors that influence entrepreneurial capability, which is solved using panel bootstrap analysis [5], exploratory factor analysis and confirmatory factor analysis [7] logistic regression biner [8], logistic regression ordinal [9]. this study examines factors which affect the level of entrepreneurial capability in tourist areas in nagari salayo west sumatera, which are grouped into four levels. ordinal regression analysis with bootstrap estimation is used to predict socio-demographic factors and entrepreneurial motives that significantly affect the level of entrepreneurial capability. the confident interval of bootstrap estimation is used to test the accuration of ordinal regression. the results of this study can be considered in the development of creative economics and local resources to support the sustainability of tourism village programs in west sumatra. 2. literature review a data analysis used to find the relationship between the response variable (y) with one or more predictor variables (x) is called the regression method [10]. this method aims to get a suitable model to describe the relationship between response variables and set of predictor variables in the most simple and best way. meanwhile, logistic regression is a regression used to analyze the relationship between nominal or ordinal scale with two or more categories variables with a set of continuous or categorical predictor variables [11]. logistic regression equation used from the approximate form of probability function 𝜋(𝑥) = e(𝑌|𝑥) as the following equation [10] )( exp1 )( exp )( 1 1 xβα xβα xπ ++ + =  () then to simplify equation (1), a logit transform is performed as: xβα x x xg 1 )( 1 )( ln)( +=        − =   () furthermore, an ordinal logistic regression is a regression analysis used to describe the relationship between response variables with predictor variables, where the response variable is ordinal scale with more than two categories. the cumulative opportunity 𝑃(𝑌 ≤ 𝑟|𝑥) is defined as: jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 10-18 12 )( exp1 )( exp )( p 1 1   = = ++ + == t k kkr t k kkr r x x xry     () where x is the vector value of the predictor variable with t categories [11]. the estimation of the regression parameter is done by describing it using the logit transformation of p(𝑌 ≤ 𝑟|𝑥).         −  = )(y p1 )(y p ln)(y plogit xr xr xr () by substituting equation (3) to equation (4), obtained  = += t k kkr xxr 1 )(y plogit  () if there are r response categories, then the cumulative opportunities of the response in the equation below: )( exp1 )( exp )( )1(y p 11 11 1   = = ++ + == t k kk t k kk x x xx    () )( exp1 )( exp )( )2(y p 12 12 2   = = ++ + == t k kk t k kk x x xx    () )( exp1 )( exp )( )(y p 1 1   = = ++ + == t k kkr t k kkr r x x xxr    () based on the cumulative opportunities in the above equation, we obtain the opportunity for each category of the response as follows )( exp1 )( exp )1(y p 11 11   = = ++ + == t k kk t k kk x x x   () )( exp1 )( exp )( exp1 )( exp )2(y p 11 11 12 12     = = = = ++ + − ++ + == t k kk t k kk t k kk t k kk x x x x x     ()  )( exp1 )( exp )( exp1 )( exp )(y p 11 11 1 1     =− =− = = ++ + − ++ + == t k kkr t k kkr t k kkr t k kkr x x x x xr     () the estimation of ordinal logistic regression model parameters is used the maximum likelihood estimator (mle). in the mle method, maximizing a joint probability density function is performed to estimate the regression parameter, which is also called the likelihood function. in ordinal logistic regression analysis, the response of each observation is assumed to be spread according to the multinomial distribution. if the response variable y~multinomial (y1, y2, ∙∙∙, yr; p1, p2, ∙∙∙, pr), then the function likelihood for the response y is m. ridho and d. devianto bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability 13  = = n i y ir y i y i riii xxx 1 21 ])()()([β),(α l 21   () by doing natural logarithm of the equation (12), we obtain a log-likelihood function as follows  = +++= n i irriiiii xyxyxy 1 2211 )]}([ ln)]([ ln)]([ ln{β)),(α (lln   () a maximum ln-likelihood can be obtained by differentiating l (α,β) against α and β and by equating to zero. the first derivative solution of the ln-likelihood function is a nonlinear function so that a numerical method is needed to obtain the parameter estimation, one of them is newton-raphson method [11]. 3. data and research methods the data used in this study come from the questionnaire that was distributed to 60 beginner entrepreneurs. the questionnaire measures for motivation adopted from [12] and the capability of entrepreneurship measurement are extracted from [13]. there are two variables that are used, they are response variables (y) and predictor variables (x). the level of entrepreneurial capability becomes the response variable that divided into four categories, they are y = 1 if the entrepreneurial capability is low, y = 2 if the entrepreneurial capability is middle, y = 3 if the entrepreneurial capability is high, or y = 4 if the entrepreneurial capability is very high whereas the predictor variables consisted of 4 variables derived from socio-demographic factors and five variables of entrepreneurial motivation. the predictor variables are (1) gender (x1): entrepreneurs’ genders are divided into two categories, x1 = 1 if the entrepreneur is male or x1 = 2 if the entrepreneur is female; (2) education level (x2): entrepreneurs’ education levels are divided into 3 categories, x2 = 1 if the entrepreneur does not attend/primary education graduate, x2 = 2 if the entrepreneur attends medium school graduate, or x2 = 3 if the entrepreneur is a high school graduated; (3) age group (x3): entrepreneurs’ age groups are divided into 5 categories, x3 = 1 if the entrepreneur is 20-30 years old, x3 = 2 if the entrepreneur is 3040 years old, x3 = 3 if the entrepreneur is 40-50 years old, x3 = 4 if the entrepreneur is 50-60 years old, or x3 = 5 if the entrepreneur is more than 60 years old; (4) occupation (x4) entrepreneurs’ occupations are divided into 3 categories, x4 = 1 if the entrepreneur is an enterpriser, x4 = 2 if the entrepreneur is a farmer, or x4 = 3 if the entrepreneur is not an enterpriser or a farmer; (5) entrepreneurial motive (x5); working motive (x6); (7) social motive (x7); (8) individual motive (x8); and (9) economy motive (x9). the data analysis steps undertaken in this study are as follows: 1) providing the statistics for the level of entrepreneurial capability variable and predictor variables. 2) establishing the best-guessed model of ordinal logistic regression by involving all predictor variables that affect the level of entrepreneurial capability. 3) testing the overall parameter significances of the ordinal logistic regression model simultaneously this test aims to investigate the significance of the coefficient β of the response variable simultaneously. the hypothesis of this test is: h0: β1 = β2 = ∙∙∙ = βr = 0 h1: there is βk ≠ 0; k = 1, 2, ..., r the statistic test used is a g-test or likelihood ratio test. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 10-18 14       −= m l l 0ln2g  () where l0 is a likelihood of the model containing constants only and lm is the likelihood of the model containing the variables. h0 is rejected at a significant level α if g > χ2α,df with degrees of freedom is r or p-value < α. [10] 4) conducting a partial test between predictor variables and response variables this test is done if the simultaneous test occurs rejection h0 and the purpose of this test is to determine predictor variables that have a significant effect on the response variable. the tested hypothesis is: h0: βk = 0 h1: βk ≠ 0 this test is performed for each k = 1, 2, ..., r. the statistic test used is wald's test. )ˆ( ˆ w k k k se   =  () where se (βk) is the standard error of the regression coefficient of k and βk is the expected regression coefficient of k. h0 is rejected at a significant level of α if wk > χ 2 α,r or p-value < α. [10] 5) perform bootstrap estimate of standard error the algorithm of bootstrap estimate according to [14] is a) select b independent bootstrap samples x*1, x*2, ∙∙∙, x*b, each consisting of n data values drawn with replacement from x, b) evaluate the bootstrap replication corresponding to each bootstrap sample, )()(ˆ b xsb  =  () where b = 1, 2, ∙∙∙, b c) estimate the standard error sef(θ) by the sample standard deviation of the b replications 2 1 })](ˆ)(ˆ[ 1 1 {ˆ 1 2**  = − − = b b b b b es   () where  = − = b b bb 1 1 )(ˆ)(ˆ  6) creating the best ordinal logistic regression models that loading the significant variables in the partial test only. 7) finding and interpreting the odds ratio for each significant predictor variable. 8) conducting test accuracy model this test is done to determine the measure of data prediction accuracy to know how accurate the model that is formed. the accuracy of the model formed is determined by a measure called the hit ratio, which is defined as follows: %100 nsobservatio of total correctly classified are that objectsmany ratiohit = () 4. results and discussion in this study, we use the ordinal logistic regression analysis with bootstrap estimation. the response variable is the level of entrepreneurial capability that divided into four categories, namely low, middle, high, or very high. the data used are 60 responses on questionnaires filled by beginner entrepreneurs in tourism of rural area in nagari salayo of west sumatra. m. ridho and d. devianto bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability 15 figure 1. the level of entrepreneurial capability the graphic in figure 1 shows that most beginner entrepreneurs’ capability has been high. characteristics of beginner entrepreneurs can be known through descriptive statistics as follows. these analytics are used to find a simple view of the data or the beginner entrepreneurs from all sides. table 1. characteristics of beginner entrepreneurs variable sum percentage gender male 24 40,000% female 36 60,000% education level not attend/primary school graduate 2 3,333% medium school graduate 38 63,333% high school graduate 20 33,333% age group < 30 8 13,333% 31 – 40 14 23,333% 41 – 50 13 21,667% 51 – 60 14 23,333% > 60 11 18,333% occupation entrepriser 15 25,000% farmer 15 25,000% other 30 50,000% the characteristics of beginner entrepreneurs are shown in table 1. the proportion of female beginner entrepreneurs is more than male beginner entrepreneurs with 60,000%. meanwhile, the beginner entrepreneurs with medium education have the highest percentage than other education levels, with 63,333%. in contrast, the beginner entrepreneurs’ age groups are almost the same, and half of the beginner entrepreneurs are not an enterpriser or a farmer. after knowing the characteristics of beginner entrepreneurs, the next step is establishing the best-guessed model of ordinal logistic regression by involving all predictor variables and testing the overall parameter significances of the ordinal logistic regression model simultaneously to find out the presence or absence of predictor variables that affect the response variables significantly, the data are tested simultaneously using the g statistic test or likelihood ratio test. table 2. model fitting information model -2 log likelihood chi-square df sig, intercept only 158,345 final 82,795 75,550 14 0,000 from table 2, we obtain that the test value g > χ20.05, 14 and p-value < 0.05. the rejection of h0 happens because the g test is bigger than χ 2 table. this means that the 15% 25% 40% 20% the level of entrepreneurial capability low middle high very high jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 10-18 16 response variable is significantly affected at least by one predictor variable. the test is continued with the partial test by using the wald test because h0 is rejected. table 3. parameter estimates variable estimate wald df sig, entrepreneurial motive (x5) 0,288 4,453 1 0,035 social motive (x7) 0,527 21,882 1 0,000 the significance value of all predictor variables is less than α. therefore, we can conclude that the variable entrepreneurial motive (x5) and social motive (x7) affect the response variable. to increase the confidence interval of the model, bootstrap estimate of the standard error is performed. using b = 200 bootstrap sample, each consists of n = 60 observations, the ordinal logistic regression with bootstrap estimation of standard error approach yield as the following result: table 4. bootstrap for parameter estimates predictor estimate std error 95% confidence interval (ci) ordinal logistic regression bootstrap logistic regression lower upper lower upper constant (1) 12,530 3,668 7,128 17,933 7,114 23,101 constant (2) 15,017 4,196 9,111 20,923 9,509 26,360 constant (3) 18,161 4,519 11,705 24,616 12,466 31,461 entrepreneurial motive (x5) 0,288 0,134 0,021 0,556 0,019 0,630 social motive (x7) 0,527 0,179 0,306 0,747 0,302 0,983 the confidence interval of each predictor variables in table 4 show increasing. the confidence interval length for an entrepreneurial motive variable increase from 0,535 to 0,611, and a social motive variable, the confidence interval increase from 0,441 to 0,681. the best ordinal logistic regression models: 751 0,5270,28812,530)( xxx ++= (19) 752 0,5270,28815,017)( xxx ++= (20) 753 0,5270,28818,161)( xxx ++= (21) the next step is to interpret how the significant variables affect the level of entrepreneurial capability. the value of odds ratio can be used to interpret the effect. the value can be seen in the following table: table 5. value odds ratio variable estimate exp (β) entrepreneurial motive (x5) 0,288 1,334 social motive (x7) 0,527 1,694 base on table 5, the odds ratio for entrepreneurial motive is 1,334. this means that each increase of 1 unit entrepreneurial motive will increase the risk of 1,334 times the entrepreneurial capability will be very high. on the other hand, the risk posed by an increase of 1 unit variable social motive is 1,694 times the entrepreneurial capability will be very high because the value of the odds ratio is 1,694. m. ridho and d. devianto bootstrap logistic regression on determining factors affecting the level of entrepreneurial capability 17 the last step is to calculate the exact value of the classification between the actual value and the predicted value obtained from the model that has been formed. the model predictions on response categorize eight low entrepreneurial capability, three middle entrepreneurial capability, 17 high entrepreneurial capability, and nine very high entrepreneurial capability correctly. for more details, this classification mistake can be seen in the following table: table 6. accuracy of classification observation prediction classification accuracy percentage low middle high very high low 8 1 0 0 88,889% middle 1 3 11 0 20,000% high 2 5 17 0 70,833% very high 0 0 3 9 75,000% overall percentage 5,369% 2,013% 11,409% 6,040% 61,667% from table 6, the accuracy of the classification of the model is 61,667%. it means the model can predict the data of 61,667% correctly, which means that the resulting model is good enough and feasible used to predict the response variable 5. conclusion this study explores the development of entrepreneurial capability in tourism of rural area based on local resources and creative economy in nagari salayo of west sumatra. more than half of new entrepreneurs involved in this study are female with 60,000%, and most of them are medium school graduate. while the new entrepreneurs’ age is averaged, and half of them are not an enterprise or a farmer. this study’s results show two variables positively have a significant effect on the level of entrepreneurial capability; they are an entrepreneurial motive and social motive. meanwhile, the variables, gender, education level, age group, occupation, working motive, individual motive, and economy motive do not significantly influence the level of entrepreneurial capability. the variable social motive gives the highest effect on the level of entrepreneurial capability with odd ratio 1,694. this means that each increase of 1 unit of social motive will increase the risk 1,694 times the capability of new entrepreneurs will be very high. while the variable entrepreneurial motive gives 1,334 times affect the level of entrepreneurial capability. higher variable social and entrepreneurial motive will lead to higher entrepreneurial capability level. based on the hit ratio, the model with the significant predictor variable gives 61,667% model accuracy. thus, it can be concluded that the model that is formed is feasibly used to determine the factors that affect the level of entrepreneurial capability in tourism of rural area in nagari salayo of west sumatra. references [1] m. li, h. zhang, and l. a. cai, “a subcultural analysis of tourism motivations,” journal of hospitality & tourism research, vol. 40, no. 1, pp. 85–113, 2013, https://doi.org/10.1177/1096348013491601. 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[6] n. rametse, t. m. nganunu, m. j. ding, and p. arenius, “entrepreneurial motivations and capabilities of migrant entrepreneurs in australia,” international migration, vol. 56, no. 4, pp. 217–234, 2018, https://doi.org/10.1111/imig.12452. [7] r. w. lv, c. lai, and j. liu, “entrepreneurial capability scale and new venture performance: the moderating role of entrepreneurship education,” national social science foundation of china, 2015. [8] c. petchprapunkul, n. koojaroenprasit, r. pitipanya, and s. pleehajinda, “using logistic regression to identify the entrepreneurial capability of thai agricultural cooperatives members on their farm management,” cultural and religious studies, vol 4, no. 11, pp. 668–676, 2016, https://doi.org/10.17265/2328-2177/2016.11.002. [9] m. j. silva, and j. leitão, “what determines the entrepreneurial innovative capability of portuguese industrial firms?,” munich personal repec archive paper, no. 5216, 2007. [10] d. w. hosmer and s. lemeshow, applied logistic regression, new york: john wiley & sons, inc., 2000. [11] a. agresti, categorical data analysis, new york: john wiley & sons inc, 1990. [12] v. vijaya and t. j. kamalanabhan, “a scale to assess entrepreneurial motivation,” journal of entrepreneurship, vol. 7, no. 2, pp. 183–197, 1998, https://doi.org/10.1177/097135579800700204. [13] j. a. m. leon, and m. gergievski, psychology of entrepreneurship: research and education, madrid: universidad nacional de education a distancia, 2007. [14] b. efron and r. j. tibshirani, an introduction of the bootstrap, new york: chapman and hall/crc, 1993. 37 | p a g e pengukuran maturity level control objective ke-5 domain delivery and support : ensuring system security menggunakan framework cobit 4.1 andhy permadi prodi sistem informasi universitas islam negeri sunan ampel surabaya, jl. ahmad yani 117 surabaya andhy@uinsby.ac.id abstrak penelitian ini berfokus pada tata usaha sebagai pusat pengolahan data elektronik (pde) di smkn 24 dengan tujuan untuk menilai pengendalian umum apakah mampu memenuhi tujuannya dan melakukan pemetaan terhadap tahap audit ti beserta keamanannya yang kemudian diaplikasikan pada sebuah organisasi yang bergerak dibidang pendidikan, yaitu smkn 24 jakarta untuk melihat kinerja ti yang ada. kerangka kerja yang digunakan sebagai acuan adalah framework cobit 4.1 pada control objective ke-5 dari domain delivery and support (ds), ensuring system security (memastikan keamanan sistem) untuk menjamin integritas informasi di pde. bukti (evidence) digunakan untuk menentukan apakah data yang diaudit sesuai dengan kriteria dan tujuan audit dengan cara : peninjauan terhadap struktur organisasi, wawancara kepada personel yang tepat dan pengisian kuesioner. penentuan maturity level (tingkat kedewasaan), merupakan bagian dari pengujian kepatutan terhadap aktivitas yang ada atau dilakukan di tiap proses ti berdasarkan kerangka kerja cobit sesuai dengan tingkatan levelnya. kata kunci: audit ti, cobit 4.1, domain delivery and support (ds), ensuring system security. abstract this research is focused on administration as electronic data processing centerat smkn 24 jakarta, in order to assess whether it is able to meet its goals and to perform mapping toward it audit and its security.framework used as a reference is cobit 4.1 on control objective no.5 from domaindelivery and support (ds), ensuring security system to ensure the integrity of information at pde. evidence is used to determine whether the data to be audited is in accordance with the criteria and the purpose of audit by: organizational structurereview, appropriate personnel interview and questionnaire filling.the determination of maturity level is part of proprietytesting on the existing activitiesor executed in each of it process, based on cobit framework inaptly its level. keywords: audit ti, cobit 4.1, domain delivery and support (ds), ensuring system security. 1. pendahuluan sekolah menengah kejuruan negeri 24 jakarta beralamat di jalan bambu hitam, kelurahan bambu apus, kecamatan cipayung, jakarta timur. memiliki dua kelompok keahlian yaitu, pariwisata (akomodasi perhotelan, jasa boga, busana butik), dan teknologi informasi & komunikasi (rekayasa perangkat lunak). berdirinya program keahlian rpl pada tahun 2004 memberikan banyak pengaruh terhadap penerapan ict bagi proses kegiatan belajar mengajar (kbm), diantaranya :  pengisian perangkat pembelajaran seperti, silabus dan rencana pelaksanaan pembelajaran (rpp) sudah ada softcopy-nya, sehingga setiap guru mengisi perangkat pembelajaran sesuai dengan jurusannya dari masing-masing kelas dikomputer sekolah atau notebook pribadi.  penggunaan media pembelajaran berbasis it, dengan menggunakan notebook yang berisi modul mata pelajaran, dan dengan menggunakan proyektor untuk menampilkannya dilayar, sehingga bisa dilihat seluruh siswa dikelas.  semua manajemen sekolah di tata usaha (tu) sudah meninggalkan mesin tik (manual), menjadi pengolahan data elektronik (pde) yang terintegrasi. saat ini smkn 24 memiliki dua lab. komputer, satu lab. kkpi, dan sebuah server yang berisi 38 | p a g e beberapa database sekolah. meningkatnya pengelolaan teknologi informasi (ti) pada kegiatan sekolah, maka harus ada audit sistem informasi atau teknologi informasi, yang berfokus pada keamanan sistem dan manajemen data di tu bidang pde, untuk menilai apakah pengendalian umum mampu memenuhi tujuannya. metodologi yang digunakan diantaranya adalah wawancara dan pengisian kuesioner terhadap user yang kesehariannya mengelola pde di smkn 24 ini dan dengan menggunakan framework cobit 4.1 diharapkan mendapatkan evidence, namun tidak semua langkah yang ada didalam framework tersebut dilaksanakan keseluruhannya, dengan alasan mengurangi pengulangan aktivitas, maka tetap berpegang pada aturan-aturan yang bersifat umum yang telah ditetapkan oleh it assurance guide [3]. 2. metode penelitian cobit dianggap sebagai kerangka kerja yang tepat untuk dipakai dalam melakukan proses audit pengelolaan ti yang ada di pde tata usaha (tu) smkn 24, karena cobit menyediakan standar dalam kerangka kerja domain yang terdiri dari sekumpulan proses ti yang merepresentasikan aktivitas yang dapat dikendalikan dan terstruktur. sehingga cocok diterapkan di tu yang berfokus pada tata kelola ti-nya, saat ini masih sebagai kontrol dari proses bisnis. aktivitas teknologi informasi dalam cobit didefinisikan kedalam model proses yang generik dan dikelompokkan dalam 4 domain dan 34 high level control objectives. framework cobit secara keseluruhan dapat dilihat pada gambar berikut. melalui gambar tersebut dapat dilihat model proses cobit yang terdiri dari 4 domain dan 34 macam proses [1]. gambar 2.1. ilustrasi konsep cobit framework [2] 2.1 teknik pengumpulan data dengan mengimplementasikan framework cobit 4.1 pada control objective ke-5 dari domain delivery and support (ds), ensuring system security (memastikan keamanan sistem) [5] untuk menjamin integritas informasi di pde tu smkn 24 dan dengan beberapa metode sebagai berikut : a) wawancara dan diskusi wawancara dilakukan dengan mengadakan tanya jawab secara langsung kepada pihak yang berwenang yaitu ibu sri nuryani, ba, selaku kabag. tata usaha (tu) sekolah menengah kejuruan negeri 24 jakarta. b) observasi observasi dilakukan dengan mengamati keseharian para pegawai di lingkungan tu bidang pde sekolah menengah kejuruan negeri 24 jakarta. c) dokumentasi dokumentasi merupakan pengumpulan data dan pencarian data yang mendukung permasalahan dengan jalan menyalin laporan-laporan, dan catatan-catatan yang berkaitan dengan masalah yang dibahas termasuk mengisi checklist berdasarkan hasil wawancara. 2.2 identifikasi resiko keamanan identifikasi resiko keamanan dibuat dalam bentuk tabel yakni tabel 1.1. tabel 1.1 identifikasi resiko keamanan 39 | p a g e tabel 1.2 checklist yang digunakan dalam wawancara 40 | p a g e 3. hasil dan pembahasan hasil observasi lapangan dan wawancara yang dilakukan oleh peneliti adalah pengelolaan teknologi informasi pada aspek ds 5 terkait dengan memastikan system keamanan ti yaitu: 3.1 penerapan sistem keamanan pada sistem informasi manajemen sekolah dilakukan dengan cara, user dalam menggunakan hak aksesnya dibatasi dengan security matrix dimana semua pengguna komputer menerima otorisasinya berdasarkan role. role yang diberikan disesuaikan dengan kebutuhan pengguna sesuai dengan job description tiap-tiap pengguna komputer. role ini dimaksudkan untuk membatasi apa saja yang dapat dilakukan oleh programprogram tersebut. pengaksesan sistem aplikasihanya dapat dilakukan oleh orangorang yang terotorisasi dan diberi wewenang untuk mengakses. misalnya, data apa yang dapat diakses, data mana yang hanya dapat dilihat, ditambah, diubah atau dihapus, apabila pengaturan role ini tidak tepat, maka akan banyak pihak-pihak yang tidak berwenang dapat mengakses data tertentu, sehingga jika itu terjadi maka keyakinan akan integritas data akan menjadi berkurang dan juga akan terjadi banyak perubahanperubahan data yang tidak diinginkan. 3.2 setiap user login menggunakan password, dengan kombinasi angka dan huruf, password yang dimasukan tidak terlihat dan secara otomatisakan lock user apabila terjadi 3 kali kesalahan login yang dilakukan oleh user , sistem aplikasi menampilkan pesan jika verifikasi login tidak valid dan yang dapat membuka kembali lock user adalah administrator sehingga dengan adanya pembatasan sistem kesalahan dalam penginputan login akses ini akan mempersulit bagi orang-orang yang tidak memiliki otoritas untuk mengakses ke sistem aplikasi. penggunaan password bertujuan untuk mencegah kepada pihakpihak yang tidak mempunyai hak akses atas aplikasi dan data-data dalam pde. 3.3 untuk melindungi akses dari luar, digunakan vpn (virtual private network), dimana untuk login lewat internet, maka user akan memasukkan password untuk melakukan akses. 3.4 untuk mengantisipasi perkembangan virus telah dipasanganti virus pada setiap komputer yang update signature nya setiap ada virus baru, virus internasional dipergunakan antivirus kespersky 2010 yang update secara otomatis dan virus lokal dipergunakan antivirus smadav pro yang update secara otomatis ketikat erhubung internet. 3.5 untuk keamanan aset-aset fisik, dalam tata usaha (tu) disediakan 2 buah pemadam kebakaran yang diletakkan masing-masing pada bagian pde dan bagian kepegawaian, hal ini dilakukan jika sewaktu-waktu ada kebakaran yang mungkin disebabkan oleh hubungan arus listrik pendek atau akibat yang lain. 3.6 bidang pde beroperasi 8 jam setiap harinya (kecuali hari minggu/hari libur), dimonitoring seorang administrator yang memiliki staff ti sebanyak tiga orang yang kompeten dibidangnya, sehingga komputer pde selalu ada pengawasan. 3.7 pde menggunakan uninterupable power supply (ups) yang digunakan untuk menstabilkan tegangan listrik, ups ini juga berfungsi sebagai pengamanan data apabila listrik mati mendadak, ups dapat bertahan kurang lebih 3 jam, sehingga dalam jangka waktu tersebut administrator dapat melakukan back up data sekolah untuk disimpan di tempat yang lebih aman dan melakukan shut down sesuai dengan prosedur. 3.8 untuk pencegahan kerusakan perangkat keras, pde melakukan kontrak pemeliharaan dengan pihak ketiga dimana pekerjaan pemeliharaan dilakukan oleh pihak ketiga 2 kali setahun yaitu bulan november dan bulan april yang pelaksanaannya ditetapkan oleh sekolah, apabila hardware ada masalah 41 | p a g e pihak ketiga menyediakan pelayanan 24 jam dan pihak ketiga menjamin dapat menyelesaikan perbaikan masing-masing hardware dalam jangka waktu paling lama 3x24 jam, sedangkan untuk software, perusahaan membeli software yang berlisensi sehingga jelas legalitasnya. berikut tabel hasil kuesioner ds5-menjamin keamanan sistem. tabel 1.3 level 0 – non existent tabel 1.4 level 1 – initial ad-hoc tabel 1.5 level 2 – repeatable but intuitive tabel 1.6 level 3 – define process tabel 1.7 level 4 – manage and measurable tabel 1.8 level 5 – optimised 42 | p a g e tabel 1.9 penentuan maturity level (tingkat kedewasaan) dari hasil perhitungan checklist seperti yang terlihat dari tabel diatas. sekolah menengah kejuruan negeri 24 jakarta termasuk dalam kategori maturity level 2 yaitu repeatable but intuitive. level 2 (repeatable but intuitive) adalah ketika tanggung jawab dan penanggung jawab keamanan ti ditentukan dalam koordinator keamanan ti, walaupun manajemen otoritasnya terbatas. kesadaran akan kebutuhan keamanan terpecah dan terbatas. walaupun informasi terkait dengan keamanan diproduksi oleh sistem, namun tidak dianalisis. layanan dari pihak ketiga mungkin tidak memenuhi kebutuhan keamanan perusahaan secara spesifik. kebijakan keamanan sedang dikembangkan tetapi keahlian dan perakatan tidak mencukupi. pelaporan keamanan ti tidak lengkap, cenderung membingungkan atau tidak berhubungan. pelatihan keamanan tersedia namun dilakukan umumnya karena inisiatif individu. keamanan ti terutama terlihat sebagai tanggung jawab dan area ti sementara bisnis tidak melihat keamanan ti dalam areanya. 4. kesimpulaan keamanan teknologi informasi di pengolahan data elektronik (pde) smk negeri 24 jakarta berdasarkan domain delivery and support (ds) control objective ensuring system security telah mencapai maturity level 2 (repeatable but intuitive). untuk level sekolah, ini sudah cukup termasuk kategori cukup optimal, hal ini dapat dilihat dari adanya pembatasan hak akses user yang didasarkan pada job description masingmasing pegawai, setiap user login menggunakan password dengan kombinasi angka dan huruf, password yang dimasukan tidak terlihat dan secara otomatis akan lock user apabila terjadi 3 kali kesalahan login yang dilakukan oleh user [4]. daftar pustaka [1] sarno, riyanarto. 2009. audit sistem & teknologi informasi. edisi pertama. surabaya: its press. [2] it governance institute. 2007. cobit 4.1. usa: it governance institute. [3] it assurance guide: using cobit, chicago, 2007. [4] mcleod, r dan schell, g.p.2007. management information system. prentice hall. [5] it governance institute. cobit 4.0: chicago, 2007. 43 | p a g e 6 | p a g e aplikasi metode nilai eigen dalam analytical hierarchy process untuk memilih tempat kerja moh. hafiyusholeh 1 , ahmad hanif asyhar 2 , dan ririn komaria 3 1,2 prodi matematika uin sunan ampel surabaya 3 balai latihan kerja jombang email: hafiyusholeh@uinsby.ac.id abstrak dalam fakta kehidupan, kita seringkali dihadapkan pada suatu permasalahan yang cukup kompleks sehingga diperlukan banyak kriteria sebagai bahan pertimbangan dalam menentukan pilihan atau keputusan. dalam kondisi semacam itu, adanya berbagai macam kriteria, ditambah lagi dengan ketidaksempurnaan informasi seringkali menyulitkan dalam membuat keputusan.salah satu solusi yang memungkinkan adalah dengan analytical hierarchy process (ahp). pada penelitian ini dikaji salah satu metode untuk menentukan vector prioritas dalam ahp dengan menggunakan nilai eigen. selain itu terapan metode nilai eigen juga dibahas untuk memberikan alternatif pilihan tempat bekerja bagi siswa smk negeri i jombang yang telah memiliki kerjasama dengan berbagai badan usaha diantaranya pt. sai mojokerto, pt. jai pasuruan, pt hwt surabaya, bpr surasari hutama bangil dan western digital malaysia. kata kunci: nilai eigen, vector eigen, ahp 1. pendahuluan matematika adalah ilmu dasar yang dapat digunakan sebagai alat bantu memecahkan masalah dalam b e r b a g a i bidang ilmu. supatmono [11] berpendapat bahwa “mathematics is queen and servant of science”. peran matematika sebagai ratu, maksudnya ialah perkembangan matematika tidak tergantung pada ilmu-ilmu lain. banyak cabang matematika yang dulu biasa disebut matematika murni, dikembangkan oleh bebarapa matematikawan yang mencintai dan belajar matematika hanya sebagai hobi atau kegemaran tanpa mempedulikan fungsi dan manfaatnya untuk ilmu-ilmu yang lain. dengan semakin berkembangnya teknologi, banyak cabang matematika murni yang ternyata di kemudian hari bisa diterapkan dalam berbagai ilmu pengetahuan dan teknologi mutakhir. sedangkan peran matematika sebagai pelayan, matematika adalah ilmu dasar yang mendasari dan melayani berbagai ilmu pengetahuan yang lain. tidak mengherankan apabila dalam fungsinya sebagai pelayan ilmu yang lain, matematika muncul di ilmu kimia, fisika, biologi, astronomi, psikologi, ekonomi dan masih banyak yang lain. pada bidang ekonomi, mathematics is a tool to solve the problem of economy [6]. kalimat ini menyiratkan bahwa matematika adalah suatu alat yang digunakan untuk memecahkan permasalahan-permasalahan ekonomi. matematika merupakan alat untuk menyederhanakan penyajian dan pemahaman masalah-masalah ekonomi dengan menggunakan bahasa dan simbol-simbol matematik suatu masalah menjadi lebih sederhana untuk disajikan, dipahami dan dianalisa. inti dari masalah ekonomi adalah kelangkaan (scarcity). kelangkaan seringkali terjadi pada lapangan 7 | p a g e pekerjaan. hal yang bisa dilakukan adalah membuat atau menciptakan lapangan pekerjaan dan mencari lapangan pekerjaan. kedua hal tersebut tentu tidak mudah, sebagai contoh untuk mendapatkan pekerjaan di suatu badan usaha, tentu banyak hal yang harus dipertimbangkan. kesulitan pemilihan badan usaha sebagaimana yang dimaksud banyak dialami oleh siswa yang berorientasi pada pekerjaan setelah mereka lulus sekolah. hal ini juga di alami oleh siswa smk negeri 1 jombang. lulusan sekolah ini diharapkan bisa siap kerja sesuai dengan ilmu yang dipelajari. namun demikian, siswa merasa kesulitan dalam memutuskan pemilihan badan usaha sebagai tempat kerja karena ada beberapa kriteria yang harus dipertimbangkan diantaranya jenjang karir yang sesuai dengan minat dan potensi siswa, tingkat gaji yang ditawarkan badan usaha bervariasi, lokasi dan fasilitas yang ditawarkan badan usaha juga berbeda. dalam fakta kehidupan, kita seringkali dihadapkan pada suatu permasalahan yang cukup kompleks sehingga diperlukan banyak kriteria sebagai bahan pertimbangan. sebagai contoh, keunggulan apa yang kita tawarkan dalam mendidik siswa? mana yang harus kita prioritaskan antara membangun jalan, tempat olah raga, dan pusat perbelanjaan? atau untuk kasus yang saat ini terjadi di kampus kita, mana yang perlu diprioritaskan antara membangun gedung, pemenuhan kualitas dan kuantitas dosen, pengadaan laboratorium, tempat parkir, ataukah perbaikan system manajemen di tingkat internal?. begitujuga pada saat memilih badan usaha bagi siswa smk negeri i jombang, mereka dihadapkan pada berbagai kriteria yang perlu dipertimbangkan. dalam kondisi semacam itu, adanya berbagai macam kriteria, ditambah lagi dengan ketidakpastian atau ketidaksempurnaan informasi seringkali menyulitkan pembuat keputusan.salah satu solusi yang memungkinkan untuk membantu pembuatan keputusan adalah analytical hierarchy process (ahp). ahp dikembangkan oleh dr. thomas l. saaty dari wharton school of business pada tahun 1970-an untuk mengorganisasikan informasi dan judgment dalam memilih alternatif yang paling disukai. model pendukung keputusan ini akan menguraikan masalah multi faktor yang kompleks menjadi suatu hirarki. menurut saaty, hirarki didefinisikan sebagai suatu representasi dari sebuah permasalahan yang kompleks dalam suatu struktur multi level dimana level pertama adalah tujuan, yang diikuti level faktor, kriteria, sub kriteria, dan seterusnya ke bawah hingga level terakhir yaitu alternatif [8]. dengan demikian sebuah hirarki dapat digunakan untuk mendekomposisi permasalahan yang kompleks, sehingga permasalahan yang ada akan tampak lebih terstruktur dan sistematis. dalam pembuatan keputusan, konsep tentang prioritas merupakan hal yang sangat esensial, ini tidak hanya terkait dengan bagaimana prioritas dari suatu keputusan itu dibangun, tetapi juga mempertimbangkan metode apa yang digunakan untuk mendapatkan prioritas dengan tingkat kekonsistenan yang dapat dipertanggungjawabkan. terdapat beberapa metode untuk mendapatkan vektor prioritas diataranya adalah dengan menggunakan eigenvalue method [9], least square method (lsm) [1], [2], chi square method (x 2 m) [12], singular value decomposition method (svdm) [4], dan lain sebagainya. matriks perbandingan berpasangan sebagai dasar untuk mengkonstruksi 8 | p a g e keputusan akan memiliki nilai eigen yang sama dengan ukuran matriks, jika matriks tersebut konsisten ber-rank satu. tetapi pada kenyataannya, jika matriks tersebut dikenakan gangguan, maka akan merubah sifat ideal dari keputusan. nilai eigen matriks tersebut berubah sebagai akibat berubahnya unsur-unsur matriks yang konsisten [5]. pada kondisi seperti ini, nilai eigen maksimal dapat digunakan sebagai pendekatan untuk mendapatkan vector prioritas [3]. pada penelitian ini akan dikaji metode nilai eigen untuk menentukan vector prioritas dalam ahp yang berguna dalam menentukan keputusan utama yang perlu diambil oleh pembuat keputusan. selain itu, dibahas pula terapan metode tersebut untuk membantu siswa kelas xii smk negeri i jombang dalam menentukan tempat bekerja di badan usaha yang sudah bekerjasama dengan sekolah. badan usaha yang dimaksud adalah pt. sai mojokerto, pt. jai pasuruan, pt hwt surabaya, bpr surasari hutama bangil dan western digital malaysia. 2. kajian pustaka analytical hierarchy process (ahp) analytical hierarchy process (ahp) dikembangkan pertamakali oleh dr. thomas l. saaty dari wharton school of business pada tahun 1970an. dengan menggunakan ahp, persoalan yang kompleks dapat disederhanakan dan dipercepat proses pengambilan keputusannya. prinsip kerja ahp adalah penyederhanaan suatu persoalan yang kompleks, tidak terstruktur, menjadi bagian-bagian yang lebih sederhana, serta menata dalam suatu hirarki. secara grafis, persoalan keputusan ahp dapat dikonstruksikan sebagai diagram bertingkat, yang dimulai dengan tujuan (goal), lalau kriteria sebagai level pertama, subkriteria dan akhirnya alternatif. ahp memungkinkan pengguna untuk memberikan nilai bobot relatif dari suatu kriteria majemuk (alternatif majemuk terhadap suatu kriteria) secara intuitif, yaitu dengan melakukan perbandingan berpasangan (pairwise comparisons). gambar 2.1. struktur hirarki agar diperoleh skala yang bermanfaat ketika membandingkan dua unsur, seseorang yang akan memberikan jawaban memerlukan pemahaman yang baik tentang unsur-unsur yang dibandingkan dan relevansinya terhadap kriteria atau tujuan yang akan dicapai. adapaun skala dasar yang digunakan untuk membandingkan unsur-unsur yang ada oleh saaty [9] dibuat dalam tabel skala perbandingan sebagai berikut; tabel 2.1 skala penilaian perbandingan pasangan nilai keterangan 1 3 5 7 9 2, 4, 6, 8 kriteria/alternatif a sama pentingnya dengan kriteria/alternatif b a sedikit lebih penting dari b a jelas lebih penting dari b a sangat jelas lebih penting dari b a mutlak lebih penting dari b apabila ragu-ragu antara dua nilai yang berdekatan kebalikan jika untuk aktivitas i mendapat satu angkat dibanding dengan aktivitas j, maka j mempunyai nilai kebalikannya 9 | p a g e dengan i matriks positif definisi: suatu matriks a berordo n x n dengan entri bilangan real disebut taknegatif jika aij  0 untuk setiap i dan j, dan disebut positif jika aij > 0 untuk setiap i dan j [7]. dalam teorema perron, “jika a adalah matriks positif berorde n x n, maka a memiliki nilai eigen real positif r dengan sifat-sifat sebagai berikut: (i) r adalah akar sederhana dari persamaan karakteristik. (ii) r memiliki vektor eigen positif (iii) jika  adalah sembarang nilai eigen lainnya dari a, maka l l < r teorema perron diperlukan sebagai jaminan bahwa nilai eigen maksimal senantiasa mempunyai nilai positif. karena matriks yang digunakan untuk mengkonstruksi vector prioritas dalam ahp adalah matriks perbandingan berpasangan (pairwise comparison) yang positif, maka berdasarkan teorema perron, terdapat vektor eigen yang berkorespondensi dengan nilai eigen maksimal yang mempunyai nilai positif juga [13] 3. metode penelitian penelitian ini dilakukan untuk mengkaji nilai eigen dalam matriks positif sebagai alat untuk menentukan vector prioritas dalam ahp. setelah diperoleh teorema untuk menjamin kekonsistenan dari vector eigen, konsep tersebut akan diterapkan untuk membantu siswa dalam memilih badan usaha tempat bekerja. penelitian ini dilaksanakan dengan memberikan kuisioner kepada responden untuk mengetahui perbandingan masingmasing kriteria atau alternatif. data tersebut berupa perbandingan berpasangan skala 1 – 9. data-data yang terkumpul tersebut selanjutnya diolah dengan nilai eigen pada analytical hierarchy process (ahp) dengan terlebih dahulu diuji konsistensi rasionya (cr) 4. pembahasan nilai eigen untuk menentukan peringkat dengan memperhatikan kajian pustaka yang telah disajikan, untuk setiap kriteria dan alternatif, perlu dilakukan perbandingan berpasangan (pairwise comparisons). nilai-nilai perbandingan relatif kemudian diolah untuk menentukan peringkat relatif dari seluruh alternatif. baik kriteria kualitatif maupun kriteria kuantitatif, dapat dibandingkan sesuai dengan judgment (penilaian) yang telah ditentukan untuk menghasilkan bobot dan prioritas. bobot atau prioritas dihitung dengan manipulasi matriks. misalkan terdapat n objek yang dinotasikan dengan yang akan dinilai tingkat kepentingannya, maka hasil perbandingan secara berpasangan elemen-elemen operasi tersebut akan membentuk matriks perbandingan. tabel 2.2 matriks perbandingan berpasangan bila diketahui bahwa nilai perbandingan elemen terhadap elemen adalah , maka secara teoritis matriks 10 | p a g e tersebut berciri positif reciprocal (berkebalikan) yakni ⁄ . bobot yang dicari dinyatakan dalam vektor . nilai menyatakan bobot kriteria terhadap keseluruhan set kriteria pada sub sistem tersebut. jika mewakili derajat kepentingan terhadap faktor dan menyatakan kepentingan dari faktor terhadap faktor , maka agar keputusan menjadi konsisten, kepentingan terhadap faktor harus sama dengan atau jika untuk semua maka matriks tersebut konsisten. untuk suatu matriks konsisten dengan vektor , maka elemen dapat ditulis menjadi: (1) jadi matriks konsisten adalah: dari persamaan (1) ⁄ diperoleh selanjutnya berdasarkan persamaan (3) dengan demikian untuk matriks pairwise comparison yang konsisten menjadi: ∑ ∑ ∑ ∑ ∑ ⏟ jadi ∑ ∑ persamaan di atas ekuivalen dengan bentuk persamaan matriks di bawah ini: (7) dalam teori matriks, adalah vektor eigen dari matriks a dengan nilai eigen n. dalam aljabar linier, semua nilai eigen adalah nol kecuali satu yang kemudian disebut dengan . dalam bentuk persamaan matriks dapat ditulis sebagai berikut: [ ] [ ] [ ] karena a merupakan matriks positf yang reciprocal, yaitu ⁄ dan untuk semua nilai , berlaku ∑ sayangnya, dalam kasus umum nilai-nilai ⁄ tidak dapat diberikan secara tepat. nilai-nilai ⁄ hanya bisa ditaksir. sehingga permasalahan kita sekarang menjadi dengan adalah nilai eigen terbesar dari a. pada prakteknya, nilai yang digunakan untuk mengkonstruksi vektor prioritas akan lebih besar daripada 11 | p a g e ukuran matriks a sebagaimana yang diuraikan pada teorema berikut. teorema 1. misalkan . jika merupakan vektor tak nol di c sehingga maka . bukti secara lengkap dapat dilihat di hafiyusholeh [5]. indikator terhadap konsistensi diukur melalui indeks konsistensi [9], yang didefinisikan sebagai: dengan = nilai eigen maksimum dan n = ukuran matriks indeks konsistensi (ci); matriks random dengan skala penilaian 9 (1 sampai dengan 9) beserta kebalikannya sebagai indeks random (ri) dapat dilihat di tabel 2.3 sebagai berikut: tabel 2.3 nilai indeks random ukuran matriks 1,2 3 4 5 6 7 8 9 ri 0,00 0,58 0,90 1,12 1,24 1,32 1,41 1,45 ukuran matriks 10 11 12 13 14 15 ri 1,49 1,51 1,48 1,56 1,57 1,59 perbandingan antaran ci dan ri untuk suatu matriks didefinisikan sebagai rasio konsistensi (cr). untuk model ahp, matriks perbandingan dapat diterima jika nilai rasio konsistensi kurang dari sama dengan 10% [10] penerapan metode nilai eigen untuk menentukan pemilihan tempan bekerja di bu secara khusus penetapan urutan prioritas tempat bekerja yang dapat dijadikan sebagai alternatif dalam menentukan keputusan melalui metode nilai eigen dalam analytical hierarchy process (ahp) dapat disajikan sebagai berikut. perhitungan faktor pembobotan hirarki untuk semua kriteria langkah pertama yang dilakukan peneliti dalam mengolah data adalah menyajikan data ke dalam matriks perbandingan berpasangan. data tersebut diperoleh dari kuesioner yang kemudian dianalisis. data yang akan dianalisis dan dibandingkan terdiri dari beberapa kriteria yaitu gaji, karir, fasilitas, dan lokasi tempat bekerja dari badan usaha yang akan dilamar. data hasil analisis preferensi gabungan menunjukkan bahwa: kriteria gaji sama penting dengan kriteria karir, kriteria gaji 3 kali lebih penting dari kriteria fasilitas dan kriteria gaji 5 kali lebih penting dari kriteria lokasi. kriteria karir 5 kali lebih penting dari kriteria fasilitas dan kriteria karir 4 kali lebih penting dari kriteria lokasi. sedangkan kriteria fasilitas 3 kali lebih penting dari kriteria lokasi. oleh karena itu matriks perbandingan berpasangan hasil preferensi di atas adalah: tabel 3.1 matriks faktor pembobotan hirarki untuk semua kriteria gaji karir fasilitas lokasi gaji 1 1 3 5 karir 1 1 5 4 fasilitas 1/3 1/5 1 3 lokasi 1/5 1/4 1/3 1 langkah kedua, peneliti akan menghitung nilai eigen dan vektor eigen dengan cara normalisasi matriks, yaitu unsur-unsur pada tiap kolom dibagi dengan jumlah kolom yang bersangkutan, akan diperoleh bobot 12 | p a g e relatif yang dinormalkan. nilai vektor eigen yang dinormalkan dihasilkan dari rata-rata bobot relatif untuk setiap baris. hasilnya dapat dilihat pada tabel 3.2 tabel 3.2 matriks faktor pembobotan hirarki untuk semua kriteria yang dinormalkan dan vektor eigen gaji karir fasilitas lokasi vektor eigen yg dinormalkan gaji 0,395 0,408 0,321 0,385 0,377 karir 0,395 0,408 0,536 0,308 0,412 fasilitas 0,131 0,082 0,107 0,231 0,138 lokasi 0,079 0,102 0,036 0,077 0,073 weighted sum vector diperoleh melalui hasil perkalian antara matriks asal dengan vector eigen yang dinormalkan [ ] [ ] [ ] langkah selanjutnya adalah menguji consistency vector (cv) dengan jalan membagi nilai tiap baris dengan nilai vektor yang bersangkutan, [ ] [ ] nilai rata-rata dari hasil pembagian tersebut merupakan nilai eigen maksimum. langkah ketiga, peneliti akan menguji konsistensi data. karena matriks berordo 4 (yakni terdiri dari empat kriteria), nilai indeks konsistensi yang diperoleh: untuk diperoleh cr = . karena berarti prefensi responden konsisten. dari hasil perhitungan pada tabel di atas menunjukkan bahwa: kriteria karir merupakan kriteria yang paling penting bagi siswa yang ingin bekerja di badan usaha dengan nilai bobot 0,412 atau 41,2%, berikutnya adalah kriteria gaji dengan nilai bobot 0,377 atau 37,7%, kemudian kriteria fasilitas dengan nilai bobot 0,138 atau 13,8% sedangkan kriteria lokasi dengan nilai bobot 0,073 atau 7,3%. perhitungan faktor evaluasi untuk kriteria gaji dengan langkah yang hampir sama, disusun matriks perbandingan berpasangan. perbandingan berpasangan untuk kriteria gaji pada 5 badan usaha yang telah memiliki ikatan kerja dengan smk negeri 1 jombang yaitu perbandingan berpasangan antara pt. sai (a) terhadap pt. jai (b), pt. hwt (c), bpr surasai (d), western d.m (e). perbandingan berpasangan antara pt. jai (b) terhadap pt. hwt (c), bpr surasai (d), western d.m (e) sampai pada perbandingan berpasangan antara bpr surasari (d) dengan western d.m (e). data hasil analisis preferensi gabungan untuk kriteria gaji disajikan dalam matriks resiprokal sebagai berikut: tabel 3.3. matriks faktor evaluasi untuk kriteria gaji a b c d e a 1 2 1 2 1/2 b 1/2 1 1 2 1 c 1 1 1 3 1 d 1/2 1/2 1/3 1 1/2 13 | p a g e e 2 1 1 2 1 selanjutnya dengan cara yang sama kita dapatkan vektor eigen untuk setiap kriteria sebagai berikut. tabel 3.4. matriks faktor faktor evaluasi untuk kriteria gaji dan vektor eigen yang dinormalkan a b c d e vektor eigen a 0,200 0,364 0,231 0,200 0,125 0,224 b 0,100 0,182 0,231 0,200 0,250 0,193 c 0,200 0,182 0,231 0,300 0,250 0,233 d 0,100 0,091 0,077 0,100 0,125 0,099 e 0,400 0,182 0,231 0,200 0,250 0,253 nilai eigen maksimum dari perhitungan di atas adalah dengan dan , maka didapat yang berarti prefensi responden konsisten. dari hasil perhitungan pada tabel di atas diperoleh urutan prioritas untuk kriteria gaji yakni western d.m menjadi prioritas pertama dengan nilai bobot 0,253 atau 25,3%, kemudian pt. hwt menjadi prioritas ke-2 dengan nilai bobot 0,233 atau 23,3%, pt. sai menjadi prioritas ke-3 dengan nilai bobot 0,224 atau 24,4% sedangkan pt. jai dan bpr surasari menjadi prioritas ke-4 dan ke-5 dengan nilai bobot masing-masing 0,193 atau 19,3% dan 0,099 atau 9,9%. perhitungan faktor evaluasi untuk kriteria karir, fasilitas, lokasi dan dengan langkah yang hampir sama, didapat nilai eigen dan vector eigen untuk kriteria karir, fasilitas, dan lokasi sebagai berikut: tabel 3.5 matriks faktor evaluasi untuk kriteria karir a b c d e a 1 2 2 2 1 b 1/2 1 2 1 2 c 1/2 1/2 1 1 1 d 1/2 1 1 1 3 e 1 1/2 1 1/3 1 tabel 3.6 matriks faktor faktor evaluasi untuk kriteria karir dan vektor eigen yang dinormalkan a b c d e vektor eigen dinormalkan a 0,286 0,400 0,286 0,375 0,125 0,294 b 0,143 0,200 0,286 0,188 0,250 0,213 c 0,143 0,100 0,143 0,188 0,125 0,140 d 0,143 0,200 0,143 0,188 0,375 0,210 e 0,286 0,100 0,143 0,062 0,125 0,143 nilai eigen maksimum untuk kriteria karir adalah dengan dan , diperoleh berarti prefensi responden konsisten. tabel 3.7. matriks faktor evaluasi untuk kriteria fasilitas a b c d e a 1 1 1/2 3 2 b 1 1 1/2 3 3 c 2 2 1 4 2 d 1/3 1/3 1/4 1 3 e 1/2 1/3 1/2 1/3 1 tabel 3.8. matriks faktor faktor evaluasi untuk kriteria fasilitas dan vektir eigen yang dinormalkan a b c d e vekt or eigen a 0,20 7 0,21 4 0,18 2 0,26 5 0,18 2 0,210 b 0,20 7 0,21 4 0,18 2 0,26 5 0,27 3 0,228 c 0,41 4 0,42 9 0,36 4 0,35 3 0,18 2 0,348 d 0,06 9 0,07 1 0,09 1 0,08 8 0,27 3 0,118 e 0,10 3 0,07 1 0,18 2 0,02 9 0,09 1 0,095 karena berarti prefensi responden konsisten. tabel 3.9 matriks faktor evaluasi untuk kriteria lokasi a b c d e 14 | p a g e a 1 5 4 5 4 b 1/5 1 2 3 3 c 1/4 ½ 1 4 4 d 1/5 1/3 ¼ 1 3 e 1/4 1/3 ¼ 1/3 1 tabel 3.10 matriks faktor faktor evaluasi untuk kriteria lokasi dan vektor eigen a b c d e vektor eigen a 0,526 0,698 0,533 0,375 0,267 0,480 b 0,105 0,140 0,267 0,225 0,200 0,187 c 0,132 0,070 0,133 0,300 0,267 0,180 d 0,105 0,046 0,033 0,075 0,200 0,092 e 0,132 0,046 0,033 0,025 0,067 0,061 indeks konsistensi yang diperoleh dari matriks tersebut adalah dengan . karena berarti prefensi responden tidak konsisten, sehingga peneliti harus mengambil data ulang. setelah dilakukan pengambilan data ulang diperoleh matriks berpasangan untuk kriteria lokasi sebagai berikut: tabel 3.11 matriks faktor evaluasi untuk kriteria lokasi a b c d e a 1 3 1 4 5 b 1/3 1 ½ 2 3 c 1 2 1 4 5 d 1/4 1/2 ¼ 1 3 e 1/5 1/3 1/5 1/3 1 tabel 3.12 matriks faktor faktor evaluasi untuk kriteria lokasi dan vektor eigen yang dinormalkan a b c d e vektor eigen a 0,359 0,439 0,339 0,353 0,294 0,357 b 0,120 0,146 0,169 0,176 0,176 0,158 c 0,359 0,293 0,339 0,353 0,294 0,328 d 0,090 0,073 0,085 0,088 0,176 0,102 e 0,072 0,049 0,068 0,029 0,059 0,055 kemudian nilai indeks konsistensi , dan berarti prefensi responden konsisten. setelah tahapan tersebut selesai dilakukan maka untuk menentukan prioritas global dari masing-masing kriteria berikut pilihan, maka dilakukan sentesa prioritas. sintesis prioritas sintesis prioritas merupakan tahapan akhir yang dilakukan dalam metode analytical hierarchy process (ahp). sintesis prioritas diperoleh melalui penjumlahan dari bobot yang diperoleh di setiap pilihan pada masing-masing kriteria, yaitu kriteria gaji, karir, fasilitas dan lokasi. bobot tersebut disajikan dalam tabel berikut. tabel 3.13 matriks faktor evaluasi masingmasing alternatif gaji karir fasilitas lokasi pt. sai 0,224 0,294 0,210 0,357 pt. jai 0,193 0,213 0,228 0,158 pt. hwt 0,233 0,140 0,348 0,328 bpr 0,099 0,210 0,118 0,102 western 0,253 0,143 0,095 0,055 untuk mencari prioritas global untuk masing-masing badan usaha adalah dengan cara mengkalikan faktor evaluasi masing-masing alternatif dengan faktor bobot diperoleh tabel 3.14 perhitungan prioritas global gaji karir fas lok total bobot bobot 0,377 0,412 0,138 0,073 1 pt sai 0,084 0,121 0,029 0,026 0,260 pt jai 0,072 0,087 0,031 0,011 0,203 pt hwt 0,087 0,057 0,048 0,024 0,217 bpr 0,037 0,086 0,016 0,007 0,147 west ern 0,095 0,059 0,013 0,004 0,171 15 | p a g e gambar 2.2 hasil pemrosesan ahp dari hasil di atas diketahui bahwa urutan prioritas badan usaha yang paling tepat dijadikan sebagai pilihan mulai dari prioritas 1, 2, dan seterusnya adalah sebagai berikut: prioritas ke nama tempat bekerja 1 pt. sai mojokerto 2 pt. hwt surabaya 3 pt . jai pasuruan 4 western digital malaysia 5 bpr surasai hutama bangil 5. simpulan pada prakteknya, nilai yang digunakan untuk mengkonstruksi vektor prioritas akan lebih besar daripada ukuran matriks. misalkan . jika merupakan vektor tak nol di c sehingga maka . indikator terhadap konsistensi diukur melalui indeks konsistensi dan rasio konsistensi (cr). matriks perbandingan dapat diterima jika nilai rasio konsistensi kurang dari atau sama dengan 10% . kriteria karir merupakan kriteria yang paling penting untuk siswa smk negeri 1 jombang dalam pemilihan tempat bekerja dengan nilai bobot 41,2%, berikutnya adalah kriteria gaji dengan nilai bobot 37,7% kemudian kriteria fasilitas dengan bobot 13,8% serta kriteria lokasi dengan bobot 7,3%. adapun urutan prioritas badan usaha yang bisa dijadikan sebagai acuan pemilihan tempat kerja adalah pt. sai mojokerto sebagai alternatif pertama dengan nilai bobot 26,07%, kemudian pt. hwt surabaya menjadi prioritas ke-2 dengan nilai bobot 21,72%, pt. jai pasuruan menjadi prioritas ke-3 dengan nilai bobot 20,34% sedangkan western d.m dan bpr surasari menjadi prioritas ke-4 dan ke-5 dengan nilai bobot masing-masing 17,14% dan 14,74%. referensi [1] baz´o ki, s, a method for solving lsm problems of small size in ahp, central european journal of operations research 1 1 p p (2003) 17-33. [2] baz´oki, s, (2008). solution of the least square method problem of pairwise comparison matrices, central european journal of operations research. [3] garminia, h., hafiyusholeh, moh. dan pudji astuti (2010). pengaruh gangguan pada perubahan prioritas dan indeks konsistensi matriks perbandingan berpasangan dalam analytical hierarchy process, jurnal matematika dan sains. desember 2010, vol. 15 nomor 3. halaman: 143. [4] gass, s.i., rapcs´ak, t., (2004). singular value decomposition in ahp, european journal of operations research 154 (2004) 573-584. [5] hafiyusholeh, moh. (2011). menentukan vektor prioritas dalam analytic hierarchy process (ahp) dengan 0.000 0.050 0.100 0.150 0.200 0.250 0.300 hasil pemrosesan ahp pt sai pt jai pt hwt bpr western 16 | p a g e metode nilai eigen. stkip pgri jombang [6] kunawangsih, tri dan anto pracoyo. 2006. aspek dasar ekonomi mikro. jakarta: pt. grasindo [7] leon, steven j. 2001. aljabar linier dan aplikasinya edisi ke-5. jakarta: erlangga [8] marimin. 2004. teknik dan aplikasi pengambilan keputusan kriteria majemuk. jakarta : pt. grasindo. [9] saaty, t.l. (1980), decision making for leaders, university of pittsburg. [10] saaty, t.l., (2002). decision-making with the ahp: why is principal eigenvector necessary, ejor 145 (2002) 85-91. [11] supatmono, catur. 2009. matematika asyik: asyik mengajarnya, asyik belajarnya. jakarta: pt. grasindo. [12] xu, z.s., (2000). generalized chi square method for the estimation of weights, jota 1 8 3 p p (2000) 183192. [13] hafiyusholeh, moh. (2009). pengaruh gangguan pada matriks pairwise comparison terhadap pembalikan dominasi dan konsistensi rasio dalam ahp. projek. bandung: itb paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran 31 | p a g e klasifikasi sinyal eeg menggunakan metode fuzzy c-means clustering (fcm) dan adaptive neighborhood modified backpropagation (anmbp) dian candra rini novitasari jurusan matematika universitas islam negeri sunan ampel, surabaya. email: diancrini@uinsby.ac.id abstrak instrumen eeg (electroencephalography) merupakan suatu instrumen yang digunakan sebagai perekam aktivitas otak dengan memperlihatkan gelombang otak. prinsip kerja eeg adalah dengan mendeteksi perubahan muatan secara tibatiba dari sel neuron yang ditandai dengan adanya interictal spike-and-wave pada hasil eeg (electroencephalogram). terdapat suatu data set sinyal eeg, direkam pada sukarelawan normal dan epilepsi. pada penelitian ini dengan menggunakan data tersebut akan dilakukan suatu sistem klasifikasi sinyal eeg dengan berdasar pada kondisi normal dan epilepsi. klasifikasi sinyal eeg menggunakan metode adaptive neighborhood base modified backpropagation (anmbp). hasil ekstraksi fitur dari sinyal eeg dengan menggunakan metode fuzzy c-means (fcm) clustering, dimana proses awalnya melalui dekomposisi wavelet menggunakan discrete wavelet transform (dwt) dengan level 2 didapatkan 3 koefisien wavelet kemudian pada masing masing koefisien tersebut di clustering menggunakan fcm dengan 2 cluster sehingga menghasilkan 6 fitur yang akan menjadi vektor fitur. dari vektor fitur tersebut digunakan sebagai inputan untuk dilakukan proses klasifikasi dengan menggunakan metode anmbp. hasil sistem sementara didapatkan recognition rate sebesar 74.37%. kata kunci: eeg, wavelet, fcm, backpropagation, modified. abstract eeg (electroencephalography) is an instrument used to records brain activity and shows some brain waves. the working principle is to detect changes in eeg sudden charge of neuron cells characterized by interictal spike-and-wave on the eeg. there is a data set of eeg signals, recorded from normal volunteers and epilepsy. by using the data, we will build classification of eeg signals system and the classification system based on the eeg signals, normal and epilepsy conditions. classification of eeg signals is using adaptive neighborhood base modified backpropagation (anmbp) method. the result of eeg signals feature extraction, using fuzzy c-means (fcm) clustering. using wavelet decomposition level 2 and using fcm with 2 cluster each sub-band of wavelet coefficients then the probability distribution of each cluster in each sub-band is calculated. the result of probability distribution in each cluster of each sub-band is the feature vectors and will be inputed to the classification process by using anmbp. the temporary result system gives the accuracy of 74.37%. keywords: eeg, wavelet, fcm, backpropagation, modified. 32 | p a g e 1. pendahuluan penyakit epilepsi adalah penyakit yang dapat terjadi pada siapun walaupun dari garis keturunan tidak ada yang pernah mengalami epilepsi ini. akan tetapi penyakit epilepsi tidak dapat menular. epilepsi ini merupakan sebuah gangguan yang terjadi di sistem syaraf otak manusia yang disebabkan adanya aktifitas kelompok sel neuron yang terlalu berlebihan hingga akhirnya terjadi berbagai reaksi pada penderitanya. epilepsi dapat terjadi karena lepasnya muatan listrik yang berlebihan dan mendadak pada otak sehingga penerimaan serta pengiriman impuls dari otak ke bagian-bagian lain dalam tubuh terganggu [10]. untuk mendiagnosis penyakit epilepsi menggunakan pemeriksaan eeg. karena instrumen electroencephalogram (eeg) dapat digunakan untuk perekaman yang menunjukkan aktivitas listrik otak, hal tersebut dapat memberikan suatu pengetahuan mengenai gangguan aktivitas otak. dalam konteks ini, rekaman eeg diukur dalam interval bebas kejang dari pasien epilepsi dianggap sebagai komponen penting untuk proses diagnosis atau prediksi [2][6]. meskipun terjadinya serangan epilepsi tampaknya tidak terduga [7], lebih banyak upaya yang difokuskan pada pengembangan model komputasi untuk deteksi otomatis debit epilepsi, yang kemudian dapat digunakan untuk memprediksi terjadinya kejang [2][3]. banyak sistem diagnosis yang digunakan untuk mengklasifikasikan sinyal eeg untuk mengklasifikasikan antara epilepsi dengan kondisi normal [6][7][8]. adaptive neighborhood base modified backpropagation (anmbp), merupakan suatu metode klasifikasi yang sangat handal [1][9]. anmbp ini memodifikasi metode backpropagation, dengan menggabungkan error linier dan non linier, adaptif learning rate serta neighborhood pada backpropagation. pada penelitian kali ini, akan dibuat suatu sistem klasifikasi, dimana menggabungkan antara discrete wavelete transform (dwt), fuzzy c-means clustering (fcm), dan perhitungan probabilitas sebagai proses ekstraksi fiturnya, dan menggunakan anmbp sebagai metode klasifikasinya. diharapkan dengan menggabungkan beberapa metode tersebut dapat menghasilkan sistem klasifikasi yang handal yaitu membedakan kondisi normal atau epilepsi dengan menggunakan data eeg. 2. data data sinyal eeg digital dapat diperoleh dari database yang tersedia di universitas bonn yang tersedia secara online dan dibuat oleh dr. ralph andrzejak dari pusat epilepsi di universitas bonn, jerman dan dapat di download atau di unduh dari link berikut: (http://epileptologie.bonn.de/cms/front_conten t.php?idcat=193&lang=3&changelang=3). selain itu, data sinyal eeg dalam bentuk digital dapat diperoleh di http://sccn.ucsd.edu/~arno/fam2data/publicly_ available_eeg_data.html. data sinyal eeg dari universitas bonn terdiri atas lima kelas dataset yaitu a, b, c, d, dan e. tiap dataset berisi 100 segmen eeg saluran tunggal dengan durasi selama 23.6 detik. setiap segmen dipilih dan dipotong dari rekaman eeg multichannel secara kontinyu setelah inspeksi artefak secara visual, misalnya gerakan mata atau aktivitas otot. sebelum dilakukan proses ektraksi fitur, sebelumnya dilakukan untuk preprocessing terhadap data yang akan digunakan dalam penelitian ini. proses preprocessing pertama adalah memecah data menjadi segmen-segmen dengan ukuran 256 data dari 2 data set sinyal eeg, yaitu set a dan set e. dari masingmasing kelas data, terdiri dari 100 segmen data dengan panjang masing-masing data 4097. data tiap segmen sebanyak 4097 tersebut kemudian dilakukan windowing (pemotongan) dengan panjang 256. hasil pemotongan sinyal didapatkan data yang lebih kecil sebanyak 4097/256 = 16 segmen. jumlah segmen data keseluruhan yang diperoleh dari pemotongan sinyal eeg set a dan set e didapatkan data sebanyak 2x100x16 = 3200 segmen data (ket: 2 merupakan kelas dari set a dan set e, 100 adalah jumlah masing-masing data awalnya, 16 merupakan segmen per 1 data awal). jadi 3200 segmen data inilah yang nanti akan digunakan sebagai data training dan data testing. 33 | p a g e 3. metodologi metode yang diterapkan dalam penelitian ini, dapat dilihat pada gambar flowchart pada gambar 1. gambar 1. diagram alur sistem proses klasifikasi sinyal eeg 3.1 ekstraksi fitur proses ektraksi fitur yang pertama dilakukan dengan menggunakan metode discrete wavelet transform (dwt), dimana proses awalnya dwt menguraikan sinyal ke dalam sub-band dengan menggunakan low-pass filtering dan high-pass filtering dari domain waktu sinyal. low-pass filtering menghasilkan aproksimasi dan high-pass filtering menghasilkan koefisien detil. dimana pada level selanjutnya dekomposisi dilakukan dengan menggunakan aproksimasi pada level sebelumnya. pada penelitian ini tipe wavelet yang digunakan adalah daubechies wavelet order 2 (db2), hal tersebut didapatkan dari hasil penelitian sebelumnya. secara garis besar, dekomposisi membagi panjang data menjadi 2, yaitu berdasarkan lowpass filtering dan high-pass filtering. dilakukan permisalan data mempunyai panjang 256, didekomposisikan pada level 1 berdasarkan lowpass filtering menghasilkan panjang data 128 yang di sebut aproksimasi 1 (a1) dan berdasarkan high-pass filtering menghasilkan panjang data 128 yang di sebut detil 1 (d1). untuk melakukan dekomposisi pada level 2 maka menggunakan aproksimasi 1 dengan panjang 128, didekomposisikan low-pass filtering menghasilkan panjang data 64 yang di sebut aproksimasi 2 (a2) dan berdasarkan lowpass filtering menghasilkan panjang data 64 yang disebut detil 2 (a2). gambar 2. proses dekomposisi sinyal eeg untuk proses selanjutnya, dilakukan clustering pada masing-masing koefisien wavelet menggunakan fcm dengan 2 cluster, jadi aproksimasi 2 dibagi ke dalam 2 cluster, detil 2 dibagi ke dalam 2 cluster, dan detil 1 dibagi ke dalam 2 cluster. fcm menggunakan fungsi keanggotaan untuk mencari kesamaan antara kumpulan data dengan center, menggunaan fungsi objektif dan mempartisi data masuk kedalam kluster-kluster hingga optimasi dari fungsi objektif tercapai. dari hasil clustering menggunakan fcm ini didapatkan 6 buah cluster, selanjutnya masingmasing cluster dihitung nilai probabilitasnya. proses ektraksi fitur yang terakhir yaitu perhitungan probabilitas dari masing-masing cluster, karena aproksimasi 2 dari proses fcm menghasilkan 2 cluster, untuk perhitungan probabilitasnya dimisalkan p1 dan p2, p1 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 1 dibagi dengan panjang data aproksimasi 2 dan p2 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 2 dibagi dengan panjang data aproksimasi 2. untuk detil 2, dari proses fcm menghasilkan 2 cluster untuk perhitungan probabilitasnya dimisalkan p3 dan p4, p3 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 1 dibagi dengan panjang data detil 2 dan p2 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 2 dibagi dengan panjang data detil 2. hal tersebut juga berlaku untuk detil 1, dimana menghasilkan dua nilai probabilitas sinyal eeg discrete wavelete transform fuzzy c-means clustering (fcm) adaptive neighborhood modified backpropagation(anmbp) hasil klasifikasi 34 | p a g e dimisalkan p5 dan p6, p5 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 1 dibagi dengan panjang data detil 1 dan p6 didapatkan dari jumlah koefisien wavelet yang masuk ke cluster 2 dibagi dengan panjang data detil 1. proses ekstraksi fitur secara keseluruhan dapat dilihat pada gambar berikut ini: gambar 3. contoh hasil ektraksi fitur 3.2 klasifikasi untuk klasifikasi metode yang digunakan adalah metode adaptive neighborhood modified backpropagation (anmbp). metode tersebut merupakan pengembangan dari metode backpropagation yaitu dengan menggabungkan dari penjumlahan error linier dan error non linier menggunakan adaptive parameter learning dan terdapat struktur neighborhood pada hidden layer. untuk mencari nilai net/jaringan digunakan rumus berikut: 1 1    s i n i s ji s j ywu (1) s ju s j y e uf s j     )1( 1 )( (2) dengan n menunjukkan jumlah neuron dan s ij w adalah bobot dari neuron ke i dari layer (s-1) ke neuron ke j dari layer s. untuk menghitung error (e) yang didapatkan dari penjumlahan dari kuadrat error linier dan non linier yang dihasilkan dari output.      n j n j s j s jp eee 1 1 2 2 2 1 )( 2 1 )( 2 1  (3) dimana λ adalah learning rate, error non-linier di simbolkan dengan e1 dan error linier disimbolkan dengan e2 yang diperoleh rumus: s j s j s j yde  1 (4) s j s j s j ulde  2 (5) )( 1 s j s j dfld   (6) simbol d adalah output yang diinginkan dan y adalah output yang dihasilkan dari sistem. sehingga perubahan bobot pada layer output menjadi seperti rumus berikut: ji s ji w e w     (7) 1 21 21              s i s js ji s j s j s js j s ji s ji s js js ji s js j s ji ye w u u y ew w u e w y ew   1 2 1' 1 )(   s i s j s i s j s j s ji yeyufew  (8) error linier dan non linier pada hidden layer (l) adalah: 111 1 ' 1 )( 1      l rj l ir l r n r l j weufe l ( 9) 11 2 1 1 2 1 ))(      l rj l r n r l j l j weufe l (10) sehingga perubahan bobot pada hidden layer adalah : 1 2 '1 1 )(   l i l j l j l i l j l ji yeufyew  (11) parameter learning μ dan μλ diganti dengan parameter adaptive η’ dan μ’       2 1 1' 2 1' ||)(|| |||| j s ij j eyuf e (12)       2 1 1' 2 1' ||)(|| |||| j s ij j eyuf e (13) dimana μ, λ adalah konstanta dengan nilai kecil positif dan ε konstanta dengan nilai kecil positif untuk menjamin adanya ketidakstabilan ketika error menuju 0. sehingga perubahan bobot yang terjadi pada layer output dan hidden layer menjadi : 35 | p a g e 1 2 '1' 1 ' )(   s i s j s i s j s j s ji yeyufew  (14) 1 2 '1' 1 ' )(   l i l j l i l j l j l ji yeyufew  (15) )()()1( twtwtw  (19) perubahan bobot hanya dilakukan pada bobot di node-node layer hidden dan node layer output yang merupakan neighborhood dari neuron-neuron yang terpilih di layer hidden. neighborhood terlihat seperti gambar 4, dimana diilustrasikan dengan garis putus-putus. gambar 4. struktur jaringan dengan neighborhood 4. pembahasan hasil dari proses ektraksi fitur yang menghasilkan 6 fitur yang kemudian dijadikan satu menjadi vektor fitur digunakan sebagai inputan pada klasifikasi menggunakan anmbp. contoh hasil ektraksi fitur dengan level dekomposisi waveletnya 2 dan klaster pada fcm ditentukan sebanyak 2 dapat dilihat pada gambar 5. gambar 5. tabel contoh hasil ekstraksi fitur setelah didapatkan hasil ekstraksi fitur, vektor fitur tersebut yang digunakan sebagai masukkan / inputan pada proses klasifikasi menggunakan anmbp. untuk hasil klasifikasi menggunakan anmbp strukturnya dibuat seperti gambar 6. gambar 6. jaringan anmbp yang dibangun dengan menggunakan jaringan diatas, tidak mutlak menggunakan keseluruhan jaringan, karena bersifat adaptif maka tergantung generate dari parameter yang digunakan, sehingga garis putus-putus diatas, menunjukkan contoh bahwa ada beberapa net yang tidak dipakai. untuk training (pelatihan) dan testing (pengujian), maka diambil sampel secara random dari 3200 data set. untuk kelas normal (set a) jumlah data setnya 1600, diambil 70% secara random, begitu pula dengan data set untuk kelas epilepsi (set e) yaitu sejumlah 1120 data set, sehingga total yang digunakan sebagai data training yaitu 2240. selain data training, atau sisa 30% dari data set digunakan sebagai data testing (pengujian) yaitu sebanyak 960 data set. struktur jaringan adalah input sejumlah 6 didapatkan dari hasil ekstraksi fitur, hidden layer menggunakan satu layer dengan 8 node, fungsi aktivasi yang digunakan adalah sigmoid biner. untuk mencapai hasil yang optimal perlu diperhatikan pemilihan nilai konstanta μ, λ (learning rate) dan ε pada adaptif learning rate. pemilihan nilai ini dilakukan secara heuristik. pada penelitian ini nilai konstanta μ = 0.3, λ = 0.000001, dan ε =0.1 serta bias = 0.5. dengan menggunakan parameter tersebut 36 | p a g e hasil satu kali training dan satu kali testing (uji data), dengan menggunakan parameter uji recognition rate berikut. recognition rate = , didapatkan yang teruji benar adalah 714 dari 960 data, sehingga didapatkan akurasi sebesar 74.37%. 5. kesimpulan dari sistem yang telah dibuat penulis masih jauh dari sempuna, dimana hasil recognition rate nya masih cukup rendah dibandingkan dua penelitian sebelumnya yaitu sebesar 74.37%. pada penelitian sebelumnya bahkan anmbp mampu sampai melebihi 90% untuk recognition rate-nya. untuk penelitian selanjutnya perlu dilakukan uji coba dalam beberapa hal, antara lain: 1. penggunakan jumlah level dekomposisi wavelet. 2. penggunaan jumlah kluster, untuk metode fcm. 3. parameter-parameter yang digunakan pada jaringan adaptive neighborhood base modified backpropagation (anmbp) seperti jumlah hidden layer, learning rate, dan lain sebagainya. penelitian ini masih jauh dari kata sempurna, karena masih dilakukan satu kali training dan satu kali testing (pengujian) maka akan terus dilakukan pengujian data dengan menggunakan parameter-parameter yang ada baik pada proses ektraksi fitur maupun klasifikasi yang ada. referensi [1]werdiningsih, indah. tranformasi wavelet dan adaptive neighborhood base modified backpropagation (anmbp) untuk klasifikasi data mamogram.jurnal scan vol ix no 2. 2014. [2] adeli h, zhou z, dadmehr n.”analysis of eeg records in an epileptic patient using wavelet transform”. j neurosci methods; 123(1):69–87. elsivier science ltd. 2003. [3] elif derya übeylï.”least squares support vector machine employing model-based methods coefficients for analysis of eeg signals”, expert systems with applications 37, 233–239. elsivier science ltd. 2010. [4] hekim, m., & orhan, u.“subtractive approach to fuzzy c-means clustering method”. journal of itu-d, 10(1). 2011. [5] orhan, u., hekim, m., & ozer, m. “epileptic seizure detection using probability distribution based on equal frequency discretization”. journal of medical systems. doi :10.1007/s10916-0119689-y. springler ltd. 2011. [6] subasi, a. eeg signal classification using wavelet feature extraction and a mixture of expert model. expert systems with applications, 32, 1084–1093. 2007. [7] subasi, a., & ercelebi, e. classification of eeg signals using neural network and logistic regression. computer methods and programs in biomedicine, 78, 87–99. elsivier science ltd. 2005. [8] novitasari, dian candra rini., klasifikasi sinyal eeg menggunakan metode fuzzy cmeans (fcm) clustering dan adaptive neuro fuzzy inference system (anfis). undergraduate thesis, department of information technology, faculty of information technology, institut teknologi sepuluh nopember, indonesia, 2013 [9] werdiningsih, indah. sistem diagnosis kanker payudara menggunakan wavelete transform dan modified backpropagation pada data mamogram. undergraduate thesis, department of information technology, faculty of information technology, institut teknologi sepuluh nopember, indonesia, 2011 [10] penyakit epilepsi. http://penyakitepilepsi.com/. diakses tanggal 17 agustus 2015. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 28-34 how to cite: b. r. a. febrilia, s. rahayu, and b. d. korida, “ordinal logistic regression analysis of factors affecting the length of student study”, mantik, vol. 5, no. 1, pp. 28-34, may 2019. ordinal logistic regression analysis of factors affecting the length of student study baiq rika ayu febrilia1, suning rahayu2, baiq dewi korida3 institut keguruan dan ilmu pendidikan mataram, rikafebrilia@ikipmataram.ac.id1 institut keguruan dan ilmu pendidikan mataram, suningrahayu1198@gmail.com2 institut keguruan dan ilmu pendidikan mataram, dewikorida12@gmail.com3 doi: https://doi.org/10.15642/mantik.2019.5.1.28-34 abstrak: lama waktu seorang mahasiswa untuk menyelesaikan masa studi menjadi ukuran prestasi mahasiswa tersebut dan kesuksesan program studinya. oleh karena lama masa studi cukup berpengaruh terhadap kualitas suatu program studi dan proses pembelajaran di dalamnya, maka perlu dilakukan studi lebih mendalam mengenai faktor-faktor yang mempengaruhi lama studi mahasiswa. penelitian ini bertujuan untuk memodelkan faktor-faktor yang mempengaruhi lama masa studi mahasiswa menggunakan regresi logistik ordinal. faktor-faktor tersebut adalah ipk mahasiswa dan jenis kelaminnya. data mengenai lama studi, ipk dan jenis kelamin diambil untuk mahasiswa program studi pendidikan matematika ikip mataram yang lulus pada tahun 2017 dan 2018. hasil penelitian menunjukkan bahwa kedua faktor berpengaruh signifikan terhadap lama masa studi mahasiswa. kata kunci: regresi logistik ordinal, masa studi abstract: the length of time a student completes the study period is a measure of the student's achievement and the success of his study program. because the duration of the study period is quite influential on the quality of a study program and the learning process in it, it is necessary to do a more in-depth study of the factors that influence the duration of student studies. this study aims to model the factors that influence the length of the study period of students using ordinal logistic regression. these factors are the student's gpa and gender. data on the length of study, gpa and gender were taken for students of the mathematics education department, ikip mataram, who graduated in 2017 and 2018. the results of the study showed that the two factors had a significant effect on the length of the study period. keywords: ordinal logistic regression, length of the study jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 28-34 issn: 2527-3159 (print)2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 b. r. a. febrilia, s. rahayu, b. d. korida ordinal logistic regression analysis of factors affecting the length of student study 29 1. introduction the duration of the student's study period is determined by how long a student can complete all his obligations in college until he is entitled to a bachelor's degree. the duration of the study period determines the success of learning at the college level, especially at the smallest unit level, especially on the study program [1]. the length of time a student completes the study period is also a measure of student achievement [2]. under normal conditions, students must be able to complete their studies at the college level for eight semesters [3]. thus, if there is a study program with an average length of the study period of more than eight semesters or four years, then the student achievement in the study program is not so good [4]. the duration of the study period is also one of the very important factors in determining the accreditation of study programs [5]. therefore, managers of study programs need to pay special attention to the factors that affect the length of time student’s study. there are many factors that can affect the length of time student’s study. two factors include the grade point average (gpa) and student gender. gpa is one of the determinants of one's study period [6] because the value of student achievement, in this case, the ip / gpa value, determines how many semester credit systems (sks) can be taken by the student. the higher the gp / gpa of a student, the student is allowed to take more credits. thus, students with high gp / gpa will be able to shorten their study period or graduate for less than four years. in terms of student gender, female students are known to pay more attention to their study process than male students. female students are more disciplined with time and they have sufficiently high motivation to be able to complete the study period on time. one of the results of the study also shows that female students' academic performance is better than male students [7]. likewise, for learning motivation, female students have higher learning motivation than male students [8]. therefore, gender is also a long-standing determinant of student studies. to determine whether gpa and gender are factors that significantly influence the length of a person's studies, further and more detailed statistical analysis is needed. a review of the factors that influence the length of the study period can be analyzed using regression analysis. regression analysis is a tool in statistics to show a causal relationship between two or more variables, where one variable acts as a response variable and another variable as a predictor variable so that one variable can be predicted through another variable [2]. regression analysis can be either linear regression or nonlinear regression. because the data used is categorical or ordinal data, then the regression analysis that can be used is ordinal logistic regression [9]. previous research shows that ordinal logistic regression analysis can provide information that the factors that influence the predicate of student graduation are faculty, gender, admission path, father's job, mother's work, and income [2]. using the same analysis, [3] indicates that the student graduation time is influenced by the department taken and its gender and provides a good model in explaining the relationship between the three variables. in contrast to previous research, this study will analyze the extent of the influence of gpa and student gender on the duration of student studies this study aims to describe the old characteristics of the student's study period, and the model is based on the factors that influence it. the results of this study are expected to contribute knowledge to managers of study programs regarding the study period of students as well as a reflection in improving the management system that is at the level of study programs. 2. ordinal logistic regression in the regression model, if the response variable is binary or dichotomous, then the analysis will use a logistic regression model, which in its parameter estimation must use a jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 28-34 30 connecting function [10]. ordinal logistic regression has the characteristics of response variables and predictors that have an ordinal scale. suppose that the 𝑌 response variable has 𝑘 categories with 𝑥𝑖 = (𝑥𝑖1, 𝑥𝑖2 , . . . , 𝑥𝑖𝑝 ) is a vector of predictor variables, 𝑝, at the value of the first observation with 𝑖 = 1,2, . . . , 𝑛. opportunities of the 𝑘-response variable with predictor 𝑋 are expressed as 𝑃[𝑌 ≤ 𝑘|𝑥] [11]. cumulative opportunities are as follows [12]. 𝑃[𝑌 ≤ 𝑘|𝑥] = 𝜋(𝑥) = exp(𝛽0𝑘 + ∑ 𝛽𝑟 𝑥𝑖𝑟 𝑝 𝑟=1 ) 1 + exp(𝛽0𝑘 + ∑ 𝛽𝑟 𝑥𝑖𝑟 𝑝 𝑟=1 ) the cumulative logit model is defined using the following formula. 𝑙𝑜𝑔𝑖𝑡 (𝑃[𝑌 ≤ 𝑘|𝑥]) = log ( 𝑃[𝑌 ≤ 𝑘|𝑥] 𝑃[𝑌 > 𝑘|𝑥] ) = log ( 𝑃[𝑌 ≤ 𝑘|𝑥] 1 − 𝑃[𝑌 ≤ 𝑘|𝑥] ) = 𝛽0𝑘 + ∑ 𝛽𝑟 𝑥𝑖𝑟 𝑝 𝑟=1 which are 𝑘 = 1,2, . . . , 𝐾 − 1 and 𝛽 are vector regression coefficients. 3. research methods this study uses secondary data obtained from the academic section of the mathematics education study program, ikip mataram. this type of research is a quantitative type with a case study of factors that influence the duration of study of undergraduate students in the mathematics education study program year passed 2017 and 2018 through an ordinal logistic regression approach. the variables used in this study were divided into two types, namely the duration of the study period as the response variable (𝑌) and gpa and gender as predictor variables (𝑋1 and 𝑋2). this response variable consists of three categories, namely: 1 = 𝑌 < 4 years 2 = 4 ≥ 𝑌 ≤ 4 years 6 months 3 = 𝑌 > 4 years 6 months. the 4 years 6 months is adjusted to the second period of graduation at the ikip mataram. this period is still assumed to be a normal category. each predictor variable that will be used in this study consists of several categories, such as the following. 𝑥1,𝑖 = student gpa, which is 𝑖: 1 = 𝐺𝑃𝐴 > 3,50 2 = 3,00 < 𝐺𝑃𝐴 ≤ 3,50 3 = 𝐺𝑃𝐴 ≤ 3,00 𝑥2,𝑖 = gender which is 𝑖: 1 = male 2 = female. b. r. a. febrilia, s. rahayu, b. d. korida ordinal logistic regression analysis of factors affecting the factors affecting the length of student study 31 this study consists of several steps. the first step is to make cross-tabulations to analyze student characteristics. the next step is followed by estimating the ordinal logistic regression parameter. then test the significance of the parameters simultaneously and partially. after that, the test of model suitability was followed by concluding the results of the study. 4. results and discussion before estimating parameters, the collected data were analyzed for characteristics based on the length of the study period on the gpa and length of study period by gender. table 1. characteristics based on the length of the study period and student gpa student gpa range length of study total 𝒀 < 𝟒 year 𝟒 ≥ 𝒀 ≤ 𝟒 years 6 months 𝒀 > 𝟒 years 6 months 𝐺𝑃𝐴 > 3,50 2 (28.6%) 3 (42.8%) 2 (28.6%) 7 (100%) 3,00 < 𝐺𝑃𝐴 ≤ 3,50 1 (0.95%) 42 (40%) 62 (59.05%) 105 (100%) 𝐺𝑃𝐴 ≤ 3,00 0 (0%) 0 (0%) 22 (100%) 22 (100%) based on table 1, information was obtained that the characteristics of students of mathematics education study program in the years of 2017 and 2018 had the most study period of more than 4 years 6 months and were dominated by graduates with gpa in the range of 3.00 to 3.50. this shows that the study period of mathematics education study program students in graduating 2017-2018 is quite long. this is possible due to a number of things such as, students having difficulty in receiving the given lecture material or the final project being worked on, students have external problems, such as family problems or lack of tuition fees which require them to work while studying. table 2. characteristics based on the length of the study period and student gender gender length of study total 𝒀 < 𝟒 year 𝒀 < 𝟒 year 𝒀 < 𝟒 year male 0 (0%) 8 (15,4%) 44 (84,6%) 52 (100%) female 3 (3,66%) 37 (45,12%) 42 (51,22%) 82 (100%) table 2 explains that the characteristics of the students of mathematics education study program in the year of 2017 and 2018 in terms of length of the study period and gender are more than those who studied for more than 4.6 years and were dominated by male students at that time. female students dominated more during the study period under 4 years or in the range of 4 to 4 years 6 months. this phenomenon is possible because usually female students pay more attention to the study process than male students. then the parameter estimation of the ordinal logistic regression model is performed using the help of spss software. the estimation results can be seen in more detail in table 3 below. table 3. parameter estimation of ordinal logistic regression model variable category coefficient (𝜷) standard error coefficient exp (𝜷) p-value length of study constant 1 -22,158 0,633 0,000 constant 2 -18,540 0,245 0,000 jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 28-34 32 variable category coefficient (𝜷) standard error coefficient exp (𝜷) p-value range of students gpa 𝐺𝑃𝐴 > 3,50 -20,349 0,907 1,454 0,000 3,00 < 𝐺𝑃𝐴 ≤ 3,50 -18,632 0,000 8,095 . 𝐺𝑃𝐴 ≤ 3,00 0 . 1 . gender male 1,455 0,467 4,284 0,002 female 0 . 1 . table 3 shows there are two constant values obtained in the model because there are three response variables that have coefficient values. this will produce two logit models. table 3 also shows the odds ratio (𝐸𝑥𝑝 (𝛽)) of students who graduate with a gpa from 3.00 to 3.50 is at 8,095. this shows that students with a gpa of 3.00 to 3.50 have an opportunity of 8.095 times greater than other gpa categories. based on gender, students with male gender have an odds ratio of 4.284 or in other words, male students have a chance of 4.284 times greater than women. next will be the suitability test of the model and the significance of the model both simultaneously and partially. model suitability test is needed to find out whether the model obtained is appropriate or not. the results of this test are presented in table 4. this test uses the null hypothesis (h0) and the alternative hypothesis (h1) as follows. h0 : model is fit h1 : model is not fit table 4. test goodness of fit model using deviance chi-square df p-value decision deviance 3,676 5 0,597 accepted h0 table 4 shows that the p-value = 0.597 > 0.05, then the decision is h0 accepted or it can be concluded that the ordinal logistic regression model obtained is appropriate. table 5. model test using likelihood ratio model 𝑮𝟐 chi-square df p-value decision intercept only 57,000 final 18,414 38,586 3 0,000 rejected h0 to test the meaning of β coefficients together, simultaneous testing was carried out on the ordinal logistic regression model. this test is carried out using the 𝐺 2 test or also known as the likelihood ratio test. based on table 4, it is known that the value of 𝐺 2 > 𝜒(0.05;3) = 7.815, which means that h0 is rejected or in other words, the β coefficient is significant for the ordinal logistic regression model. this result is also in accordance with the p-value = 0.000 < 0.05. partial testing using the wald test is intended to determine whether the variables in the ordinal logistic regression model are significant or not. this test uses the following statistical hypothesis. h0 : 𝛽𝑘 = 0 h1 : 𝛽𝑘 ≠ 0, 𝑘 = 1, 2 partial test results can be seen in table 3 of the sixth column through the p-value value. through table 3, it is known that all variables have variable values less than 0.05. these b. r. a. febrilia, s. rahayu, b. d. korida ordinal logistic regression analysis of factors affecting the factors affecting the length of student study 33 results indicate that these variables are significant. based on the results of data analysis, the logit model of the length of the study period in terms of student gpa and student gender as many as 2 (two) models, as follows. 𝐿𝑜𝑔𝑖𝑡 (𝑦1) = 𝑙𝑜𝑔 ( 𝑦1 1 − 𝑦1 ) = −22,158 + (−20,349𝑋1,1) + (1,455𝑋2,1) 𝐿𝑜𝑔𝑖𝑡 (𝑦2) = 𝑙𝑜𝑔 ( 𝑦2 1 − 𝑦2 ) = −18,540 + (−20,349𝑋1,1) + (1,455𝑋2,1) 5. conclusion based on the results of the research and discussion, it was found that the characteristics of the students of the mathematics education study program were dominated by graduates with a gpa from 3.00 to 3.50, but the length of the study period of students was more than 4 years 6 months. the length of the study period for more than 4 years and 6 months is dominated by male students. the test results simultaneously and partially indicate that the factors that affect the length of the study period are gpa and gender. references [1] z. zakariyah and i. zain, “analisis regresi logistik ordinal pada prestasi belajar lulusan mahasiswa di its berbasis skem,” j. sains dan seni its, vol. 4, no. 1, pp. 121–126, 2015. 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[12] a. agresti, categorical data analysis 2nd ed, vol. 45, no. 1. 2002. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 1 watermarking citra digital menggunakan metode discrete cosine transform aris fanani 1 , nurissaidah ulinnuha 2 uin sunanampel surabaya 1 , arisfa@uinsby.ac.id 1 uin sunanampel surabaya 2 , nuris.ulinnuha@uinsby.ac.id 2 abstrak pada paper ini, mengusulkan penyisipan watermark pada citra grayscale dan rgb dengan cara modulating relative size pada blok koefisien dct. watermark disisipkan pada blok-blok frekuensi menengah. modulating relative size pada blok koefisien dct diterapkan dengan menambahkan atau mengurangi blok-blok koefisien dct. dari uji coba yang dilakukan metode yang diusulkan menghasilkan watermark yang tahan terhadap berbagai serangan, serta kualitas yang baik dari citra yang disisipi watermark. kata kunci: watermarking, citra, dct abstract in this paper, we propose embedding the grayscale and rgb image by modulating relative size on the block dct coefficients. watermark is embedded in the intermediate frequency blocks. modulating relative to the block size dct coefficients applied by adding or subtracting blocks of dct coefficients. from experiments performed proposed methods produce watermarks that are resistant to a variety of attacks, as well as good quality of the watermark inserted image. keywords: watermarking, image, dct 1. pendahuluan pesatnya pertumbuhan teknologi semakin mempermudah penyebaran media digital kemajuan semacam ini telah menyebabkan maraknya tindakan seperti duplikasi dan penyebaran data secara ilegal, serta penyalahgunaan hak akan kekayaan dan intelektual (haki). berbagai upaya telah dilakukan untuk mengatasi masalah haki pada media digital, salah satunya dengan digital watermarking. watermarking merupakan proses penyembunyian informasi [1], untuk menunjukkan kepemilikan atau melacak penyalahgunaan hak cipta pada arsip digital seperti citra digital, audio, video [2]. digital watermarking harus memenuhi beberapa kriteria robustness, imperceptibility, dan security[3]. robustness berarti seberapa tangguh watermark dapat bertahan dari berbagai macam serangan untuk menghilangkan watermark seperti scaling, cropping dan compression. imperceptibility berhubungan dengan keberadaan watermark yang tidak boleh tampak oleh kasat mata manusia dan degradasi pada citra. security berarti watermark yang disisipkan tidak dapat terdeteksi dengan analaisa statistic atau metode lain. watermarking citra digital berdasarkan domain aplikasi dibagi menjadi jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 2 dua kelompok: domain spasial dan domain transformasi [4]. teknik watermarking pada domain spasial seperti least significant bit (lsb) pernah dilakukan oleh [5] yang menggabungkan discrete wavelet transform (dwt) dan lsb untuk aplikasi biometrik. metode ini memiliki keunggulan dalam kapasitas informasi yang disisipkan, tetapi mudah di dideteksi oleh beberapa program [6]. meskipun metode domain spasial terlihat sederhana, teknik watermarking pada domain transformasi lebih tangguh terhadap serangan daripada metode domain spasial. pada domain transformasi, teknik yang banyak digunakan adalah discrete cosine transform(dct). beberapa penelitian memperkenalkan berbagai teknik penyisipan pada domain dct, diantaranya metode dengan menggunakan strategi pemilihan block tepi, recursive matrix, dan kriteria pemilihan koefisien terbesar untuk menentukan lokasi penyisipan watermark [7]. saad al momen [8] mengusulkan penyisipan watermark citra grayscale pada 8x8 block koefisien dct, dengan perubahan koefisien dct pada frekuensi menengah dan frekuensi tinggi. akan tetapi, salah satu kelemahan dari penyisipan pada frekuensi yang berbeda adalah citra asli terdistorsi oleh noise karena watermark itu sendiri. pada penelitian ini, diusulkan penyisipan watermark pada citra grayscale dan rgb dengan cara modulating relative size koefisien dct, dimana perubahan hanya dilakukan pada frekuensi menengah. dari metode yang diusulkan bertujuan untuk mendapatkan citra terwatermark dengan sedikit distorsi dan tingkat keamanan yang baik. 2. watermarking citra digital menggunakan berdasarkanperubahan koefisien dct 2.1 transformasi dct discrete cosine transform (dct) merupakan dekomposisi sinyal yang mengkonversikan citra dari domain spasial ke dalam domain frekuensi. fungsi dua dimensi dct (2d dct)untuk matrik s berukuran n x n diberikan persamaan 1 [9]: ( ) ( ) ( ) ∑∑ ( ) [ ( ) ] [ ( ) ] ( ) dimana ( ) ( ) { √ √ setiap elemen s(i,j) dari hasil transformasi merupakan hasil kali dalam antara elemen-elemen matrik s dengan basis fungsi, dimana basis fungsi merupakan matrik berukuran n x n. setiap matrik basis dua dimensi (fungsi basis) merupakan hasil kali dari dua basis vektor satu dimensi. untuk n = n = 8, dapat dibentuk sebuah array 8x8 dari matrik basis-matrik basis 8x8. setiap basis matrik dapat dianggap sebagai sebuah citra. citra basis sejumlah 64 di dalam array seperti pada gambar 1. gambar 1. basis fungsi dct dua dimensi. sejumlah nilai matrik s dapat dikembalikan dari transformasi s(i,j) dengan menerapkan fungsi invers dct (idct) dua dimensi: jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 3 ( ) ∑ ∑ ( ) ( ) ( ) [ ( ) ] [ ( ) ] ( ) dimana ( ) ( ) { √ √ 2.2 proses penyisipan watermark proses penyisipan atau penanaman adalah proses menyisipkan suatu label watermark ke dalam cover host. terdapat dua jenis watermark yang digunakan dalam desain watermarking yaitu pembangkitan secara acak noise dan watermark citra logo yang memiliki arti. karena watermark citra logo lebih berisi informasi dan menyakinkan dibandingkan hasil keluaran biner, sehingga citra logo biner dipilih sebagai watermark. pada penelitian ini diusulkan penyisipan watermark dengan cara perubahan dua koefisien dct didalam satu block citra. proses penyisipan terdiri dari tiga langkah yaitu: transformasi, penyisipan dan invers transformasi seperti pada gambar 2. 2.1.1 transformasi cover host yang akan disisipi watermark terebih dahulu dibaca sebagai suatu matrik dua dimensi. matrik tersebut dibagi menjadi sejumlah blok yang telah ditentukan banyaknya (jumlah block adalah 8), dimana satu blok berisi sejumlah pixel. matrik yang telah dibentuk menjadi sejumlah blok kemudian ditransformasikan dengan menggunakan transformasi dct seperti dijelaskan pada persamaan 1. dari transformasi tersebut blok-blok pada matrik akan terbagi menjadi tiga daerah, yaitu daerah blok-blok dengan frekuensi rendah (fl), daerah blok-blok dengan frekuensi menengah (fm), dan daerah blok-blok dengan frekuensi tinggi (fh) seperti terlihat pada gambar 3. blok-blok yang dipilih sebagai daerah penyisipan watermark adalah blok-blok pada frekuensi menengah. alasan penanaman pada blok-blok frekuensi menengah karena penanaman pada daerah tersebut tidak akan merusak citra secara signifikan dibandingkan jika penanaman dilakukan pada blok frekuensi tinggi dan rendah. penanaman pada blok frekuensi rendah dan tinggi sangatlah tidak cocok, karena penglihatan manusia lebih peka terhadap perubahan cover host jika dilakukan modifikasi terhapat block frekuensi tinggi dan rendah. 2.1.2 penyisipan setelah menentukan blok frekuensi menengah, langkah selanjutnya adalah memilih dua blok pada blok frekuensi menegah. dua blok yang telah ditentukan sebagai area penanaman watermark selanjutnya dilakukan penukaran nilai koefisien dct antara keduanya dengan cara membandingkannya. sedangkan untuk menyisipkan suatu watermark ke dalam citra diperlukan parameter (k), dimana k merupakan bilangan yang menjadikan blokblok koefisien dct yang telah ditukar antara dua blok yang telah ditentukan sebagai area penanaman watermark memiliki selisih tertentu. pada citra rgb, channel blue yang akan dijadikan sebagai kandidat penyisipan watermark. karena mata manusia kurang sensitif terhadap modifikasi pada channel blue dari pada channel red dan channel green[9]. algoritma penyisipan seperti pada gambar 4. 2.1.3 invers transformasi proses selanjutnya dilakukan transformasi balik dengan menggunakan transformasi idct untuk mendapatkan citra terwatermark. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 4 2.2 proses pengekstrakanwatermark seperti ditunjukkan pada gambar 5, proses pengekstrakan watermark dari cover host terdiri dari dua proses, yaitu: gambar 2. proses penyisipan watermark gambar 3. pendefinisian blok pada dct gambar 4. algoritma penyisipan 2.2.1 transformasi dct proses awal pengekstrakan adalah citra terwatermark ditransformasikan dengan menggunakan transformasi dct. 2.2.2 pengekstrakan watermark setelah dilakukan transformasi dct pada citra terwatermark, akan terbentuk blok-blok dct. blok-blok koefisien dct pada dua blok frekuensi menengah yang digunakan sebagai tempat penyisipan dibandingkan untuk mendapatkan citra watermark. pada ekstraksi juga dibutuhkan citra watermark asli yang digunakan sebagai pembanding ukuran dalam pembentukan citra watermark hasil ekstraksi. proses pembandingan dua blok sesuai kriteria sebagai berikut: { ( ) ( ) ( ) ( ) (3 gambar 5. proses pengekstrakan watermark jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 5 3. skenario uji coba serangkaian uji coba dilakukan pada beberapa citra standart 512x512 citra berwarna dan 256x256 citra grayscale. watermark yang digunakan dalam percobaan adalah citra logo dengan ukuran 32x32. contoh cover host dan watermark seperti pada gambar 6. blok-blok koefisien dct pada frekuensi menengah (fm) yang dipilih sebagai tempat penyisipan watermark adalah blok(5,2) dan blok (4,3). semakin besar penentuan nilai parameter k yang digunakan akan mengakibatkan penurunan kualitas citra[10]. nilai k yang digunakan pada ujicoba adalah k= {0.5, 1, 2, 3, 5, 10, 20, 30, 40 dan 50} untuk mengukur kualitas citra pada watermarking, diperlukan alat ukur yang akan digunakan sebagai parameter. alat ukur tersebut adalah peak signal to noise ratio (psnr). parameter psnr menunjukkan perbandingan antara nilai maksimum dari sinyal yang diukur dengan besarnya derau yang berpengaruh pada sinyal tersebut [4], diukur dalam satuan desibel (db). pada penelitian ini, psnr digunakan untuk mengetahui kualitas cover host yang disisipi watermark. untuk menentukan psnr, terlebih dahulu harus diketahui nilai rata-rata kuadrat dari error (mean square error mse). semakin besar nilai psnr berarti semakin mirip citra terwatermark dengan cover host. hal ini juga berarti bahwa skema watermark semakin efektif. citra dengan nilai psnr >35 db dapat dikatakan memiliki kualitas yang baik [11]. persamaan 4 menunjukkan rumus psnr. ∑( ) (4) dimana iw, ih merupakan lebar dan tinggi citra terwatermark, ix.y nilai piksel cover host pada koordinat (x,y) dan i * x.y nilai piksel cover host yang ditukar pada koordinat (x,y). emax merupakan nilai maksimum piksel (yaitu, emax = 255 untuk 256 citra gray-level). ujicoba pertama yang dilakukan adalah memproses watermarking dan mengekstrak watermark untuk beberapa nilai paremeter k serta mencatat nilai psnr. sedangkan ujicoba kedua yang dilakukan adalah memberikan serangan terhadap citra terwatermark, dan kemudian watermark yang disisipkan di ekstrak. serangan ini berupa beberapa operasi citra untuk membuktikan seberapa efektif skema watermarking. gambar 6. (a) cover host berukuran 512x512 ; (b) cover host berukuran 256x256; (c) watermark 4. hasil uji coba tabel 1 menunjukkan hasil percobaan terhadap citra rgb lenna dengan logo watermark ”its”, dimana dilakukan proses watermarking dan ekstraksi watermark. dari tabel 1 terlihat bahwa untuk rentang k= (0.5, 1, 2, 3, 4, 5, 10, 20, 30, 40, 50) citra terwatermark dengan kualitas yang dapat diterima semuanya. karena tidak ada citra terwatermark dengan nilai psnr dibawah 35db. sedangkan pada proses ekstraksi watermark, nilai k berpengaruh terhadap watermark yang terekstrak. dari uji coba yang dilakukan untuk k ≥ 3, didapatkan watermark terekstrak mirip dengan watermark asli. serangkain serangan juga jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 6 dilakukan untuk mengetahui efektifitas skema watermarking. dari tabel 2 terlihat bahwa dengan beberapa serangan citra watermark dengan kasat mata dapat terekstrak hampir sama dengan citra asli watermark. tabel 1. nilai psnr dan watermark terekstak dengan nilai parameter k berbeda parameter k psnr (db) watermark terekstrak 0.5 62.73 1 62.67 2 62.61 3 62.36 4 62.07 5 61.62 10 59.81 20 57.55 30 56.18 40 54.98 50 53.97 tabel 2. nilai psnr dan watermark terekstak dengan nilai parameter k serangan citra terwatermark (k=3) psnr watermark terekstrak auto contras 25.38 auto level 25.38 cropping 14.29 clear 25.30 hp 16.74 tabel 3. nilai psnr dan watermark terekstak dengan nilai parameter k berbeda parameter k psnr (db) watermark terekstrak 0.5 44.53 1 44.52 2 44.51 3 44.46 4 44.41 5 44.31 tabel 4. nilai psnr dan watermark terekstak dengan nilai parameter k serangan citra terwatermark (k=4) psnr watermark terekstrak cropping 19.22 clear 18.78 tabel 3 menunjukkan hasil percobaan terhadap citra grayscale lenna dengan watermark ”aris”, dimana dilakukan proses watermarking ekstraksi watermark. sama seperti pada citra rgb, untuk k ≥ 3 mendapatkan watermark terekstrak hampir mirip dengan watermark asli. akan tetapi memiliki kualitas yang menurun. dari tabel 4, terlihat juga bahwa dengan beberapa serangan 5. penutup dari uji coba yang dilakukan dapat disimpulkan bahwa semakin besar nilai k yang digunakan akan menghasilkan kualitas citra terwatermark yang semakin menurun sehingga secara kasat mata tampak adanya perubahan pada cover host. metode yang diuslkan juga menghasilkan citra terwatermark yang tahan terhadap serangan. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 7 referensi [1] z. wei, j. dai, and j. li., 2006. “genetic watermarking based on dct domain techniques”. canadian conference on electrical and computer engineering, ccece 06: 2365 2368. [2] b. isac and v. santhi., 2011. “a study on digital image and video watermarking schemes using neural networks”. international journal of computer application, vol. 12: 1-6. [3] langelaar, g. setyawan, i. lagendijk, r.l., 2000. “watermarking digital image and video data”. ieee signal processing magazine, vol. 17: 20-43. [4] m. rafigh and m.e. moghaddam, 2010.“a robust evolutionary based digital image watermarking technique in dct domain”. 2010 seventh international conference on computer graphics, imaging and visualization, pp. 105-109. [5] g.-j. lee, e.-j. yoon, and k.-y. yoo, 2008. “a new lsb based digital watermarking scheme with random mapping function”. 2008 international symposium on ubiquitous multimedia computing, pp. 130-134. [6] m. habib, s. sarhan, and l. rajab, 2005 “a robust-fragile dual watermarking system in the dct domain”. springerverlag berlin heidelberg, pp. 548-553. [7] d. salomon, 2007 “data compression: the complete reference”, springer, forth edition. [8] al-momen, saad. e.goerge, loay, 2010. ”image hiding using magnitude modulation on the dct coefficients”. journal of applied computer science and mathematics, 8: 1-6. [9] p.-t. yu, h.-h. tsai, j.-s. lin, 2001. ”digital watermarking based on neural networks for color images”. signal process. 81 (3) : 663–671. [10] johnson, n.f. katezenbeisser, s.c, 1999. “a survey of steganographic techniques” in information techniques for steganography and digital watermarking, s.c. katezenbeisser et al., eds. northwood, ma: artec house. [11] w. na and w. yunjin, 2009 “a novel robust watermarking algorithm based on dwt and dct,” 2009 international conference on computational intelligence and security how to cite: s. suwanto, m. bisri, d. novitasari, and a. asyhar, “classification of eeg signals using fast fourier transform (fft) and adaptive neuro fuzzy inference system (anfis)”, mantik, vol. 5, no. 1, pp. 36-45, may 2019. jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 35-44 issn: 2527-3159 (print) 2527-3167 (online) classification of eeg signals using fast fourier transform (fft) and adaptive neuro-fuzzy inference system (anfis) dian c. r. novitasari1, suwanto2, m. hasan bisri3, ahmad hanif asyhar4 uin sunan ampel surabaya, diancrini@uinsby.ac.id1 uin sunan ampel, h72216070@uinsby.ac.id2 uin sunan ampel, bisri7940@gmail.com3 uin sunan ampel, hanif@uinsby.ac.id4 doi: https://doi.org/10.15642/mantik.2019.5.1.35-44 abstrak: epilepsi merupakan penyakit yang menyerang otak dan mengakibatkan seseorang mengalami kejang karena adanya gangguan system saraf pusat (neurologis) sehingga menyebabkan hilang kesadaran. aktifitas listrik otak direkam menggunakan uji sinyal eeg, karena dengan uji eeg dapat digunakan untuk mendiagnosa penyakit otak dan mental seperti epilepsi. tujuan yang hendak dicapai pada penelitian ini adalah agar dapat mengidentifikasi seseorang mengidap epilepsi atau tidak menggunakan metode fast fourier transform (fft) dan adaptive neuro fuzzy inference system (anfis) serta hasil tingkat akurasi, sensitivitas, dan presisi dari penggunaan metode tersebut. metode fft digunakan untuk mentransformasikan sinyal eeg yang semula berbasis waktu menjadi sinyal eeg berbasis frekuensi dan dilanjutkan dengan proses ekstraksi fitur untuk mengambil ciri setiap sinyal hasil pemfilteran menggunakan median, mean dan standar deviasi pada setiap sinyal eeg. hasil dari ektraksi fitur digunakan sebagai input pada proses pengelompokan berdasarkan ciri data (klasifikasi) menggunakan anfis. data sinyal eeg didapat dari database yang tersedia secara online dari pusat epilepsi universitas bonn, jerman. hasil sistem klasifikasi dengan dua kelas (normal-epilepsi) didapatkan akurasi, sensitivitas, dan presisi sebesar 100% dan sistem klasifikasi sinyal eeg menggunakan anfis dengan pembagian tiga kelas (normal-not seizure epilepsy-epilepsy) menghasilkan akurasi sebesar 89.33% sensitivitas sebesar 89.37% dan presisi sebesar 89.33%. kata kunci: epilepsi, eeg, ekstraksi fitur, klaifikasi. abstract: epilepsy is a disease that attacks the brain and results in seizures due to neurological disorders. the electrical activity of the brain recorded by the eeg signal test, because eeg test can be used to diagnose brain and mental diseases such as epilepsy. this study aims to identify whether a person has epilepsy or not along with the result of accurate, sensitivity, and precision rate using fast fourier transform (fft) and adaptive neuro-fuzzy inference system (anfis) method. the fft is used to transform eeg signals from time-based into frequency-based and continued with feature extraction to take characteristics from each filtering signal using the median, mean, and standard deviations of each eeg signal. the results of the feature extraction used for input on the category process based on characteristics data (classification) using anfis. eeg signal data is obtained from epilepsy center online database of bonn university, german. the results of the eeg signal classification system using anfis with two classes (normal-epilepsy) states accuracy, sensitivity, and precision of 100%. the classification systems with three class division (normal-not seizure epilepsy-epilepsy) resulted in an accuracy of 89.33% sensitivity of 89.37% and precision of 89.33%. keywords: epilepsy, eeg, feature extraction, classification. mailto:h72216070@uinsby.ac.id jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 35-44 36 1. introduction epilepsy comes from greek, namely "epilepsia" which means chronic general neurological disorders characterized by recurrent and temporary seizures. about 90% of people with epilepsy founded in developing countries. in indonesia, the number of epilepsy sufferers is increasing every year, in 2012 there were 1.8 million active epilepsy patients per 220 million people [1]. recently, epilepsy can be detected by looking at eeg signals. eeg signal measurement was considered easier and cheaper than measurement with another method so that the measurement of eeg signals is more desirable. eeg is a device capable of capturing spontaneous electrical activity from the brain obtained by capturing electrical signals from the neurons to neurons [2]. alternative detection using eeg signals can use machine learning assistance, and signal data processing and early identification can be done quickly. these alternatives can be used to find specific features that exist by searching for the main signals. research into the detection of epilepsy using eeg signals has been carried out by several methods. some of them were conducted by [3, 4] who implemented the fcm and anmbp methods that extracted features by clustering each signal using fcm then calculating each probability on each cluster and classifying it using anmbp. in other research [5] rational discrete short-time fourier transform is used, besides that [6] also conducted eeg signal classification research using empirical mode decomposition (emd) and adaptive neurofuzzy inference system (anfis). in the eeg signal classification process, it is necessary to do preprocessing with feature extraction, there are several methods used for feature extraction, including wavelets, emd, and fft. in the research [6] on eeg signal classification got 95% accuracy using emd as feature extraction, wavelet feature extraction in the study conducted by [7] found an average accuracy percentage of 87.424%. then in cough disease classification research was based on sound data signals using fft as a method for feature extraction by [8] gets the highest accuracy rate of 86.6667%. various methods for classifying data include back propagation by [9], which in the human voice signal pattern recognition study obtained an accuracy of 74%, then on the classification of types of cow leather using multilayer perceptron by [10] that found the classification accuracy rate reached 87.83%, then the research classification of employee performance by [11] using anfis obtained an accuracy value of 89%. based on existing conditions, in this study a classification system will be created with fft as a feature extraction method and anfis as a classification method, using these methods is expected to be able to distinguish with the best accuracy result in normal or epileptic condition and distinguish normal, epileptic not seizure, or epileptic seizure condition. 2. literature review 2.1 electroencephalography (eeg) on the human scalp, there is electricity generated from the flow of ions flowing in a group of neurons in the brain. this electrical activity can be recorded by electroencephalography (eeg). electrodes installed along the scalp to record brain activity in different positions. this eeg contains information about the state of the brain. most doctors use eeg as a tool for diagnosing a brain and psychiatric illness [12]. eeg plays an important role in brain research, especially for diagnosing and classifying neurological diseases such as epilepsy. recordings in patients with seizures, eeg signals show abnormal high amplitude. the object recorded is the bioelectric activity of neurons in the cortex cerebrum layer. the results of eeg recordings are brain wave signals that are in the voltage range of 5-100mv [13]. recording this eeg signal captured by placing electrodes on the d. c. r. novitasari, suwanto, m. h. bisri, and a. h. asyhar classification of eeg signals using fast fourier transform (fft) and adaptive neuro fuzzy inference system (anfis) 37 scalp with the high temporal resolution the eeg signal will make the response to any changes in brain activity faster [14]. 2.2 fast fourier transform (fft) fast fourier transform (fft) is a fourier transformation found by j. fourier in 1965, which is the development of the discrete fourier transform (dft) algorithm in 1822. the fft algorithm calculates transformation faster than dft because there is a reduction in the looping process. fft applied to the system to filter signals from the time domain to the frequency domain. fourier transforms mathematically expressed as contained in equation 1 [15, 16]. 𝑠(𝑓) = ∫ 𝑠(𝑡) ∾ −∾ 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑡 (1) where: s(f) = frequency domain signal s(t) = time domain signal 𝑒−𝑗2𝜋𝑓𝑡 = constant f = frequency t = time 2.3 adaptive neuro-fuzzy inference system (anfis) anfis (adaptive neuro-fuzzy inference system) is a hybrid algorithm which is a combination of the fuzzy inference system (fis) mechanism and the neural network method. anfis has two parameters, and there are premise parameters and consequent parameters. this hybrid training algorithm is carried out by step, forward, and backward [17, 18, 19]. figure 1 shows the anfis architecture. figure 1. anfis architecture layer 1: fuzzyfication in this layer, the formation of fuzzy sets will be carried out by using the membership function. there are several membership functions that can be used, including, bell, gaussian, trap, triangle, and others. the output in layer 1 stated as in equation 2, with i being data. 𝑂1𝑖 = µ𝐴𝑖(x) = 1 1 + | 𝑥 − 𝑐 𝑎 | 2 (2) jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 35-44 38 where: µai(x) = degree of membership x = data input c = mean a = standard deviation layer 2: product in this layer, synthesized information transmitting with layer 1 and multiplying all incoming signals and sending the product out. output in this layer expressed by equation 3: 𝑂2𝑖 = µ𝐴𝑖(x). µ𝐵𝑖(y) = 𝑊𝑖 (3) where: µai(x) = degree of membership µbi(y) = degree of membership w = firing strength layer 3: normalization the result of layer two normalized in layer 3. the output in layer 3 expressed in equation 4. 𝑂3𝑖 = 𝑊𝑖 𝑊1 + 𝑊2 = ŵ (4) where: w = firing strength ŵ = normalized firing strength layer 4: defuzzification output at layer 4 calculated by using the formula in equation 5. 𝑂4𝑖 = ŵ𝑖ŷ𝑖 = ŵ𝑖(𝑝𝑖 𝑥1 + 𝑞𝑖 𝑥2 + 𝑟𝑖 ) (5) where: ŵ = normalized firing strength ŷ = {p, q, r} = set of consequent parameters. layer 5: output in this layer, the result of layer 5 is calculated by equation 6 that sum all entries. 𝑂5𝑖 = σŵ𝑖 ŷ𝑖 = 𝛴ŵ𝑖ŷ𝑖 𝛴ŵ𝑖 (6) where: ŵ ŷ = results of layer 4 2.4 k-fold cross validation cross-validation also is called rotation estimation is a model validation technique used to assess how the results of statistical analysis will generalize independent data sets. this technique is used to make model predictions and estimate how accurate the predictive model is when it is carried out in practice. k fold cross validation is used to eliminate bias in data, where breaking data into k parts of data sets of the same size. the experiment was carried out several times. in the first experiment, the first parts were done as testing data, and the other parts were done as training data. in the second experiment, the second parts as test data and the other parts as training data, etc. [20]. k fold partition of data set represented in figure 2. d. c. r. novitasari, suwanto, m. h. bisri, and a. h. asyhar classification of eeg signals using fast fourier transform (fft) and adaptive neuro fuzzy inference system (anfis) 39 figure 2. k fold cross-validation of data set for each k trials, one fold is used for testing data and the remaining folds (k-1 fold) for training data, and the average error all of k trials is computed as follows equation 7. [21] 𝐸 = 1 𝐾 ∑ 𝐸𝑖 𝐾 1 (7) where: e = mse of fold k = number of fold 3. research methodology the research process needs to be initiated by conducting a literature study, especially prior research related to the topic, data collection and data processing, program making, and analysis of results. before data processing using the adaptive neuro-fuzzy inference system (anfis) method, preprocessing needs to be done. the method for this research represented as a flowchart in figure 3. figure 3. research flowchart the feature extraction process is done by changing the time domain from eeg signals to the frequency domain using the fft method. each eeg signal that has been in the frequency domain calculated the median value, mean, and standard deviation. these three variables become features and input to the classification process. the classification method uses the anfis method, which is a fuzzy inference system that is implemented in adaptive networks. adaptive neuro-fuzzy inference system (anfis) is a combination of the mechanism of fuzzy inference systems in neural network architecture. according to [22] the fuzzy inference system used is the first-order takagi sugeno kang (tsk) model with consideration of simplicity and ease of computing. optimal (small error)? start data input fft filter result feature extraction extraction result classification (anfis) classification analisys end yes no jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 35-44 40 the classification process using anfis has 5 steps, including fuzzification, product, normalization, defuzzification, and output. the fuzzification process is an input mapping into a fuzzy set to get a certain degree of membership from a classic set input transformation. the next process is product and normalization, which is the process of calculating the degree of activation, then calculating the normalized activation power by dividing each result from the second layer by the total number of w. in the fourth step, fuzzy results will be changed to the classic set (crisp), and the calculation in the fifth layer is done to get the value of the coefficient parameter and add up all the inputs from the fourth layer. 4. result before being classified, the original signal data as in figure 4, at the top is the eeg signal data for normal patients, the middle image is an image with epilepsy without seizures (epilepsy not seizure) and in the bottom picture is a signal image of patients suffering from epilepsy followed by seizures. figure 4. eeg signals (normal-epilepsy not seizure-epilepsy) the first step is to do preprocessing. in this study, the fast fourier transform (fft) method is used, for example, visualization of the results of signal filtering using fft as shown in figure 5. from figure 4 and figure 5, there are significant differences because in figure 5 the original signal is filtered using fft so the noise of the original signal was removed. d. c. r. novitasari, suwanto, m. h. bisri, and a. h. asyhar classification of eeg signals using fast fourier transform (fft) and adaptive neuro fuzzy inference system (anfis) 41 figure 5. filtering results of eeg signals fft result will be processed to the feature extraction process, this process produces three features data are used to input data on a classification process that is the mean, median, and standard deviation of the signal filtering results of each patient. the function of mean, median, and standard deviation in this study is to take characteristics of patient signal data to simplify the classification process. some examples of this feature extraction seen in table 1. table 1. extraction result (normal-epilepsy) median mean standard deviation class 0.0022 0.3544 2432.2163 normal 0.4809 0.9299 4270.7733 normal 0.0025 0.3656 2768.9500 normal 0.0015 0.3385 2449.7238 normal 0.0020 0.3601 27028.7744 epilepsy 0.0020 0.3361 28348.9893 epilepsy 0.0020 0.3648 20656.6976 epilepsy 0.0041 0.2082 9421.0647 epilepsy jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 35-44 42 table 2. extraction result (normalnon seizure epilepsy -epilepsy) the feature extraction results are the initial variables in the classification process by using adaptive neuro-fuzzy inference system (anfis). after obtaining the extract on each data feature, the results divided into two data, there are training data and testing data. the distribution of training and testing data are using the k-fold cross validation method. confusion matrix for the classification of two classes (normal-epilepsy) and three classes (normal-epilepsy not seizure-epilepsy) represented in table 3 and table 4. the results of the level of accuracy, sensitivity, and precision shown in table 5. table 3. confusion matrix (normal-epilepsy) class normal epilepsy normal 25 0 epilepsy 0 25 table 4. confusion matrix (normalnon seizure epilepsy-epilepsy) median mean standard deviation class 0.0022 0.3544 2432.2163 normal 0.4809 0.9299 4270.7733 normal 0.0025 0.3656 2768.9500 normal 0.0023 0.3186 2866.5607 epilepsy not seizure 0.0026 0.3946 2762.9526 epilepsy not seizure 0.0025 0.3219 3309.7655 epilepsy not seizure 0.0020 0.3601 27028.7744 epilepsy 0.0020 0.3361 28348.9893 epilepsy 0.0020 0.3648 20656.6976 epilepsy actual classification result data normal non seizure epilepsy epilepsy normal 23 2 0 non-seizure epilepsy 4 20 1 epilepsy 0 1 24 d. c. r. novitasari, suwanto, m. h. bisri, and a. h. asyhar classification of eeg signals using fast fourier transform (fft) and adaptive neuro fuzzy inference system (anfis) 43 table 5. classification result classification by using anfis method get good result with provide accuracy, sensitivity, and precision value. accuracy is the level of closeness between the results of classification with the actual value, sensitivity is possibility the results of classification being identificated correctly, and precission is level of correctly between the information requested and the classification results. the accuracy, sensitivity, and precision value for the classification of two classes (normal-epilepsy) of 100% and the results of the accuracy of 89.33%, sensitivity of 89.37%, and the precision 89.33% for the three classes (normal epilepsy not seizure -epilepsy). 5. conclusion based on the results of analysis and testing, it concluded that the level of accuracy, sensitivity, and precision classification of two classes (normal-epilepsy) are 100% and classification with three classes (normal-non seizure epilepsy-epilepsy) level of accuracy, sensitivity, and precision of 89.33%, 89.37%, and 89.33%. references [1] n. c. w. maryanti, "epilepsi dan budaya," buletin psikologi, vol. 24, no. 1, pp. 2231, 2016. 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[22] i. haimi, "peramalan beban listrik jangka pendek dengan menggunakan metode adaptive neuro fuzzy inference system (anfis)," uin sultan syarif kasim riau, pekanbaru, 2010. jurnal jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 38 klasifikasi menggunakan metode hybrid bayessian-neural network (studi kasus: identifikasi virus komputer) dian c. rini 1 , yuniar farida 2 , dwi puspitasari 3 uin sunan ampel, surabaya 1,2 , politeknik negeri malang 3 diancrini@uinsby.ac.id 1 , greatyuniar@gmail.com 2 , dwi_sti@yahoo.com 3 abstrak virus komputer merupakan suatu program yang menginfeksi komputer terutama pada saat komputer sedang beroperasi dan menjadi momok bagi pengguna komputer. virus komputer dapat menggandakan dirinya sendiri dan menyebar dengan cara menyisipkan dirinya pada program dan data lainnya. efek negatif virus komputer adalah memperbanyak dirinya sendiri, yang membuat sumber daya pada komputer terutama penggunaan memori menjadi berkurang secara signifikan. diperlukan suatu penangkal atau antivirus dalam mencegah penyebaran yang lebih jauh dalam sistem komputer. pada penelitian ini, dilakukan suatu identifikasi virus dengan menggabungkan dua metode yaitu naïve bayes classifier dengan neural network. fitur virus didapatkan dari mengkodekan ciri-ciri dari virus. untuk klasifikasi awal digunakan metode naïve bayes classifier untuk membagi dua jenis fitur, yaitu virus dan bukan virus. setelah masuk kedalam jenis virus, maka diklasifikasikan kedalam dua jenis virus yaitu trojan atau worm menggunakan salah satu metode neural network (perceptron). hasil sistem setelah dilakukan uji coba didapatkan recognition rate tertinggi yaitu sebesar 94.12%. kata kunci: virus komputer, naïve bayes classifier, neural network, perceptron abstract a computer virus is a program that infects computer, especially when the computer is running. a computer virus can reproduce itself and it spreads by inserting in the program and other data. the negative effect of a computer virus is they are multiplied, which makes computer resources, especially memory usage to be reduced significantly. in order to prevent further spread of the computer system an antidote or antiviral is needed. this study carries out an identification of the virus by combining the two methods, naïve bayes classifier and neural network. features of the virus is obtained by encoding the characteristics of the virus. for the initial classification bayesian method is used to divide the two types of features, namely the virus and not the virus. the virus is classified into two types, trojan or worm, by one of neural network (perceptron) methods. the results after testing the system gain recognition rate of 94.12%. keywords: virus computer, naïve bayes classifier, neural network, perceptron jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 39 1. pendahuluan virus komputer umumnya dapat merusak perangkat lunak komputer dan tidak dapat secara langsung merusak perangkat keras komputer tetapi dapat mengakibatkan kerusakan dengan cara memuat program yang memaksa over process ke perangkat tertentu. efek negatif virus komputer adalah memperbanyak dirinya sendiri, yang membuat sumber daya pada komputer (seperti penggunaan memori) menjadi berkurang secara signifikan. hampir 95% virus komputer berbasis sistem operasi windows. sisanya menyerang linux/gnu, mac, freebsd, os/2 ibm, dan sun operating system. virus yang ganas akan merusak perangkat keras.[1]. berbagai macam-macam jenis virus antara lain worm, trojan, spyware dan lain sebagainya. ditinjau dari segi kemampuan[2], kriteria virus dibagi menjadi 5 hal antara lain: 1. kemampuan untuk mendapatkan informasi 2. kemampuan memeriksa suatu program 3. kemampuan untuk menggandakan diri 4. kemampuan mengadakan manipulasi 5. kemampuan menyembunyikan diri penggolongan atau pengelompokkan terhadap virus dapat dilakukan dengan beberapa metode klasifikasi. beberapa metode yang sangat bagus menggunakan naïve bayes antara lain digunakan dalam klasifikasi deteksi email spam[3], klasifikasi sinyal eeg[4], klasifikasi kanker payudara [5] dan pattern recognition yang lainnya [6]. naïve bayes sangat cocok dan baik dalam binary classification. terdapat pula banyak metode klassifikasi, antara lain single layer perceptron, multi layer perceptron dan backpropagation. bahkan perceptron masih banyak digunakan dalam beberapa penelitian, antara lain klasifikasi sinyal otak (eeg)[7], handal juga dalam prediksi[8][9][10] dan pattern recognition[11][12]. perceptron sangat bagus dalam binary classification sehingga cocok apabila dilakukan klasifikasi kedalam dua kelas, pada penelitian ini, akan dilakukan suatu identifikasi virus berdasarkan kriteria yang ada. naïve bayes digunakan untuk mengklasifikasikan apakah fitur yang diinputkan masuk kedalam kelas virus atau bukan virus, kemudian jika masuk virus maka akan dilakukan klasifikasi menggunakan metode single layer perceptron kedalam dua kelas yaitu virus jenis worm atau jenis trojan. 2. metodologi metode yang diterapkan dalam penelitian ini, dapat dilihat pada gambar flowchart pada gambar 1. gambar 1. diagram alur sistem proses klasifikasi virus 2.1 ekstraksi fitur didapatkan data berdasarkan kriteria virus, yaitu mendapatkan informasi, memeriksa program, menggandakan diri, memanipulaso dan menyembunyikan diri (hidden) sesuai dengan gambar 2. gambar 2. data awal virus dari data yang ada perlu dikodekan untuk mempermudah dalam proses klasifikasiya. proses pengkodean dilakukan dengan menggunakan kode biner yaitu 0 dan 1. untuk parameter yang pertama yaitu mendapatkan informasi, jika mendapatkan maka dikodekan 1 dan jika diam dikodekan data virus pengkodean ciri virus naïve bayes classifier neural network (single layer perceptron) hasil klasifikasi jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 40 0. pada parameter kedua yaitu program, jika bersifat memeriksa program maka dikodekan 1 dan jika tidak memeriksa program maka dikodekan 0. parameter selanjutnya yaitu menggandakan diri, jika tidak ganda maka dikodekan 0 dan jika bersifat menggandakan diri maka dikodekan 1. pada parameter manipulasi apabila bersifat memanipulasi maka dikodekan 1 dan jika tidak dikodekan 0, begitu juga dengan parameter hidden, jika dapat melakukan menyembunyikan file maka dikodekan dengan 1 dan jika tidak dikodekan 0. berlaku pula dengan parameter yang lain, sehingga didapatkan hasil pengkodean pada tabel 1. tabel 1. data hasil pengkodean informasi program ganda diri manipulasi hidden virus 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 2.2 klasifikasi untuk proses klasifikasi terdapat dua metode yaitu naïve bayes dan single layer perceptron. 2.2.1 naïve bayes classifier dalam mengklasifikasikan vector fitur kedalam kelas virus atau bukan virus ini dengan menggunakan teknik naïve bayes classifier untuk menentukan peluang kemungkinan besar termasuk kedalam kelas virus atau bukan virus. metode naive bayes merupakan metode yang digunakan memprediksi probabilitas. sedangkan klasifikasi bayes adalah klasifikasi statistik yang dapat memprediksi kelas dari suatu anggota probabilitas. untuk klasifikasi bayes sederhana yang lebih dikenal sebagai naïve bayesian classifier dapat diasumsikan bahwa efek dari suatu nilai atribut sebuah kelas yang diberikan adalah bebas dari atribut-atribut lain. naïve bayes classifier merupakan sebuah metoda klasifikasi yang berakar pada teorema bayes. ciri utama dari naïve bayes classifier ini adalah asumsi yang sangat kuat (naif) akan independensi dari masing-masing kondisi / kejadian, dimana diasumsikan bahwa setiap atribut contoh (data sampel) bersifat saling lepas satu sama lain berdasarkan atribut kelas. naive bayes categorial adalah naive bayes dengan data statik berupa kategori atau merupakan data pasti, sehingga dalam pengerjaannya sudah didapatkan hasil yang pasti. naive bayes merupakan metode dengan rumus dasar bayesian, pada teorema bayes, bila terdapat dua kejadian yang terpisah (misalkan a dan b), maka teorema bayes sebagai berikut[3]: p (a|b) = (p(b|a) * p(a))/p(b) (1) peluang kejadian a sebagai b ditentukan dari peluang b saat a, peluang a, dan peluang b. representasi peluang seperti gambar 3. gambar 3. diagram venn probabilitas 2.2.2 single layer perceptron single layer perceptron merupakan salah satu metode klasifikasi neural network, dimana merupakan metode yang meniru sistem kerja dari jaringan syaraf atau yang biasa disebut jaringan syarat tiruan (jst). gambar 4. arsitektur single layer perceptron perceptron sendiri biasanya memiliki layer yaitu layer input dan output. layer input jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 41 menunjukkan informasi dasar yang kemudian digunakan dalam jaringan syaraf tiruan (neural network). aktifitas setiap unit-unit lapisan tersembunyi (hidden layer) ditentukan oleh aktifitas dari unit-unit input dan bobot dari koneksi antara unit-unit input dan unitunit lapisan tersembunyi. karakteristik dari unit-unit output tergantung dari aktifitas unitunit layer tersembunyi dan bobot antara unitunit layer tersembunyi dan unit-unit output. arsitektur single layer perceptron, seperti pada gambar 4, layer pertama merupakan input dari jaringan perceptron dapat direpresentasikan dengan x, yaitu x1 sampai dengan x ke n. kemudian diberikan bobot (weights) dan sebuah bias ditambahkan ke dalam jaringan dengan angka pembelajaran (learning rate). output yang dihasilkan adalah y. karena metode ini adalah supervised learning maka ada threshold yang harus dilewati. berikut adalah beberapa langkah-langkah keseluruhan metode single layer perceptron: 1. inisialisasi input, bias dan bobot. x1 xn = input ke 1 sampai n, (2) b = bias, α = angka pembelajaran (0<α≤1), wn = bobot masing-masing input, t = batas ambang / threshold 2. mengitung output, dengan rumus yin = σwn.xn + α.b (3) 3. output dalam bentuk biner dapat dinotasikan (y) adalah 1 jika 1 ≤ yin dan 0 jika yin ≤ 0 (4) 4. perubahan nilai bobot (w) dan bias (b) selama learning dinotasikan sebagai berikut: wbaru = wlama + x, jika y < t (5) wbaru = wlama x, jika y > t (6) wbaru = wlama , jika y = t (7) 3. pembahasan hasil aplikasi klasifikasi menggunakan metode hybrid bayessian-neural network untuk identifikasi virus ini menggunakan development tool visual basic 6.0. untuk klasifikasi, pertama dilakukan untuk menggolongkan data kedalam virus dan bukan virus. apabila, ada data baru mendapatkan informasi, menggandakan diri dan memeriksa program, apakah virus atau bukan. sebelumnya kita menggunakan data pada tabel 1 dan menghitung propabilitas pada masing-masing peluang kejadian menggunakan rumus 1. fakta: p(virus=ya)=4/6, p(virus=bukan)=2/6 p(menggandakan=ya|virus=ya)=1/2, p(menggandakan=ya|virus=bukan)=1/2 p(memeriksa-prog=ya|virus=ya)= ¾ p(memeriksa-prog=ya|virus=bukan)=1/2 hmap dari keadaan ini dapat dihitung dengan: probabilitas virus = ya, p(menggandakan=ya,memeriksaprog=ya|virus=ya)= {p(menggandakan=ya|virus=ya).p(memeriksa=ya|vir us=ya)}.p(virus=ya) ={(1/2).(3/4)}.(4/6) = ¼ probabilitas virus = tidak, p(menggandakan=ya,memeriksaprog=ya|virus=bukan )={p(menggandakan=ya|virus=bukan).p(memeriksa= ya|virus=bukan)}.p(virus=bukan)={(1/2).(1/2)}.(2/6) = 1/12 nilai probabilitas terhadap virus nilainya lebih besar dari probabilitas bukan virus, sehingga data yang dimasukkan tadi terdeteksi sebagai virus. telah diuj coba terhadap 67 data. dari 67 data, yang terklasifikasi kedalam virus sebanyak 52 dan bukan virus 25. data yang terklasifikasi masuk ke virus, akan digolongkan menjadi dua yaitu apakah trojan atau worm menggunakan single layer perceptron. antarmuka atau tampilan dari metode single layer perceptron menggunakan development tool visual basic 6.0, inputan dari x1 sampai dengan x5 sesuai dengan parameter input yang ada pada tabel 1 yaitu mendapatkan informasi, memeriksa program, menggandakan diri, memanipulasi data, dan menyembunyikan diri (hidden). di bangkitkan secara random pembobot antara 01, sesuai dengan jumlah input di inisialisasi jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 42 bobot berjumlah 5 dan ditambah satu bobot (w0) sebagai pengali dari bias. kemudian dihitung outputnya dan dilakukan pengupdate bobot sesuai dengan aturan pada rumus 5,6 dan 7. gambar 5. antarmuka single layer perceptron struktur jaringan yang digunakan sebagai training dan testing seperti gambar 5, telah dilakukan uji / testing dengan menggunakan parameter uji recognition rate berikut. recognition rate = , sebanyak 52 data, diambil secara random untuk data training dan testingnya, didapatkan seperti tabel 2 berikut. tabel 2. hasil uji training dan testing program data training data testing jumlah terklasifika benar recognition rate (%) 30 22 20 90,90 35 17 16 94,12 27 25 23 92,00 40 12 11 91,67 32 30 28 93,33 dari hasil uji coba 5 kali training dengan random dan perbedaan jumlah data training dan data testing, rata-rata recognition rate-nya diatas 90%, bahkan tertinggi dengan data training sejumlah 40, data testing 12 dengan 11 terklasifikasi benar sehingga recognition rate nya tinggi yaitu sebesar 94,12%. 4. kesimpulan dari sistem yang telah dibuat, pada tabel 2 menyatakan bahwa rata-rata recognition ratenya diatas 90%, sehingga metode ini sudah sangat mampu untuk mengenali pola atau dengan kata lain mampu mengidentifikasikan virus atau bukan virus, serta worm atau trojan. dari 5 kali uji coba training dan testing, didapatkan recognition rate sistem sebesar 94,12%, hal ini berarti metode hybrid ini mampu mengidentifikasi jenis virus, hal tersebut dikarenakan data yang digunakan biner (0 dan 1) dimana kedua metode yang di hybrid sangat mampu melakukan klasifikasi yang pemisahannya secara linear. penelitian ini masih belum bisa dikatakan sempurna, karena masih dilakukan 5 kali pengujian, maka akan terus dilakukan pengujian data dengan menggunakan parameter-parameter yang ada. referensi [1] dr. solomon's virus encyclopedia, 1995, isbn:1-897661-00-2, abstract at http://vx.netlux.org/lib/aas10.html (diakses tanggal 2 januari 2010) http://arykoerniawan.blogspot.co.id/2010/ 12/mengenal-dan-cara-mengatasi-virusdi.html (diakses tanggal 5 mei 2016) [2] anugroho, prasetyo., dkk. klasifikasi email spam dengan metode naive bayes classifier menggunakan java programing.tugas akhir. politeknik elektronika negeri surabaya.2008. [3] kang, hyohyeong and choi, seungjin. bayesian common spatial patterns for multi-subject eeg classification. neural networks, volume 57 (2014) pages 3950 [4] mersova, a.,et all. the differentiation of malignant and benign human breast tissue at surgical margins and biopsy using xray interaction data and bayesian classification. radiation physics and chemistry, volume 95 (2014) 210-213 [5] chaphalkara, n.b, et all. prediction of outcome of construction dispute claims using multilayer perceptron neural network model n.b.international journal of project management, volume 33, https://id.wikipedia.org/wiki/istimewa:sumber_buku/1897661002 http://vx.netlux.org/lib/aas10.html http://arykoerniawan.blogspot.co.id/2010/12/mengenal-dan-cara-mengatasi-virus-di.html http://arykoerniawan.blogspot.co.id/2010/12/mengenal-dan-cara-mengatasi-virus-di.html http://arykoerniawan.blogspot.co.id/2010/12/mengenal-dan-cara-mengatasi-virus-di.html jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 43 issue 8 (2015) 1827-1835 [6] orhana, umut, et all. eeg signals classification using the k-means clustering and a multilayer perceptron neural network model. expert systems with applications volume 38, issue 10 (2011) 13475–13481. [7] heidari, elham.,et all. accurate prediction of nanofluid viscosity using a multilayer perceptron artificial neural network (mlp-ann). chemometrics and intelligent laboratory systems,volume 155(2016)73-85 [8] dudek, grzegorz multilayer perceptron for gefcom2014 probabilistic electricity price forecasting. international journal of forecasting, volume 32 (2016) 10571060 [9] fan, xinghua., el all. chaotic characteristic identification for carbon price and an multi-layer perceptron network prediction model. expert systems with applications, volume 42, issue 8 (2015) 3945-3952 [10] isa, nor ashidi mat., et all. clusteredhybrid multilayer perceptron network for pattern recognition application. applied soft computing, volume 11 (2011) 1457-1466 [11] taner, danisman., et all. intelligent pixels of interest selection with application to facial expression recognition using multilayer perceptron. signal processing, volume 93, issue 6, (2013) 1547-1556 http://www.sciencedirect.com/science/journal/09574174/38/10 how to cite: z. ni’mah and y. farida, “multi unit spares inventory control-three dimensional (music 3d) approach to inventory management”, mantik, vol. 5, no. 1, pp. 19-27, may 2019. multi-unit spares inventory control – three dimensional (music 3d) approach to inventory control zaidatun ni’mah1 and yuniar farida2 department of mathematics, uin sunan ampel surabaya, zaidatunimahh@gmail.com1 department of mathematics, uin sunan ampel surabaya, yuniar_farida@uinsby.ac.id2 doi: https://doi.org/10.15642/mantik.2019.5.1.19-27 abstrak: pengendalian persediaan (inventory control) merupakan serangkaian usaha yang perlu dilakukan bagi setiap perusahaan untuk memaksimalkan keuntungan dari persediaan yang ada. dalam penelitian ini dilakukan pengendalian persediaan melalui pendekatan music 3d (multi unit spares inventory control-three dimensional approach) pada pt. fajar mas murni surabaya menggunakan tiga analisis yaitu abc analysis, sde analysis, dan fsn analysis. hasil abc analysis menunjukkan bahwa kategori a terdiri dari 6 item (3%) yang memberikan kontribusi 81% tehadap perusahaan, kategori b terdiri dari 16 item (8%) yang memberikan kontribusi 15% terhadap perusahaan, sedangkan kategori c terdiri dari 190 item (89%) yang memberikan kontribusi 4% terhadap perusahaan. hasil sde analysis menunjukkan bahwa terdapat 127 yang masuk kategori s (60% dari seluruh item), 43 item masuk dalam kategori d (20% dari seluruh item), sedangkan item yang masuk kategori e terdiri dari 42 item (20% dari seluruh item). hasil fsn analysis menujukkan bahwa terdapat 15 item yang masuk dalam kategori f (7% dari seluruh item), kategori s terdapat 41 item (19% dari seluruh item), dan 156 item masuk dalam kategori n (74% dari seluruh item). kata kunci: inventory control, abc analysis, sde analysis, fsn analysis, music 3d abstract: inventory control is a series of efforts that need to be done for each company to generate the maximum profit from existing inventory. in this study inventory control was conducted through the multi-unit spares inventory control – three dimensional (music 3d) approach at pt fajar mas murni surabaya using three analysis, namely abc analysis, sde analysis, and fsn analysis. the result of abc analysis show that category a consists of 6 items (3%) which contribute 81% to company income, category b consists of 16 items (8%) which contributes 15% to company income, while category c consists of 190 items (89%) which contribute 4% to company income. the result of sde analysis shows that category s consists of 127 items (60% of all items), category d consists of 43 items (20% of all items), while category e consist of 42 items (20% of all items). the result of fsn analysis show that category f consists of 15% (7% of all items), category s consists of 41 items (19% of all items) and category n consists of 156 items (74% of all items). keywords: inventory control, abc analysis, sde analysis, fsn analysis, music 3d jurnal matematika mantik vol. 5, no. 1, may 2019, 19-27 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 19-27 20 1. introduction in trading companies, inventory is an important problem that must be managed carefully. inventory occurs because the amount of goods purchased is greater than the number of goods sold so that it can potentially cause losses. on the other hand, inventories are economic resources that need to be maintained to support the continuity of sales. therefore inventory control is needed to minimize costs incurred. one of the developing trading companies is pt fajar mas murni. pt fajar mas murni is a trading company in the field of equipment in various sectors. so far, inventory control at pt fajar mas murni has not used a specific method that can control inventory in the company. the company has excess inventory on many items and sometimes has a shortage of inventory on certain items. of course, the cost of inventory caused is not small. so it is necessary to control inventory so that there is no excess or lack of inventory. inventory control can be approached by several methods including eoq (economic order quantity) and rop (re-order point) method, abc analysis, sde (scarce, difficult, and easy) analysis, and fsn (fast, slow, and non-moving) analysis. in this study, the author used three analyzes to control inventory, namely abc analysis, sde analysis, and fsn analysis which were combined in the music 3d (multi-unit spares inventory control – three dimensional) approach to obtain the optimal solution from each item group. abc analysis results are obtained based on total sales volume. sde analysis results are obtained based on the lead time, and fsn analysis is obtained based on the turn over of each item. studies related to inventory control include the analysis of the economic order quantity (eoq) method as an inventory evaluation at pt fajar mas murni batam [1], the application of eoq and rop method [2], application of abc analysis in controlling inventory of furniture products at java furniture wonosari klaten [3], the application abc analysis for inventory control of consumable items [4], classifying spare parts and its vendor selection models at the metering station at pt chevron pacific indonesia [5], analysis of spare parts classification using music 3d view of spares [6], the company control analysis using the music 3d (multi-unit spares inventory control – three dimensional) approach in the warehouse at pt semen indonesia (persero) tbk tuban [7]. the three analysis approach using music 3d is still not widely used, but the music 3d approach can produce a more accurate inventory control analysis because combining the three analysis is better than just using one analysis. 2. theoretical basis 2.1 inventory inventory is an economic resource that needs to be held and maintained to support smooth production. these economic resources can be in the form of production capacity, labor, experts, working capital, time available, raw materials, finished goods, items are in the process of being worked on, and auxiliary materials [8] [9] [10]. inventory occurs if the amount of goods purchased is greater than the amount used. according to [3] [11], the purpose of the inventory is: storing resources to provide good service to customers buffer inventory to anticipate inventory shortages and inventory stockout transit inventory to anticipate price fluctuations every year quantity discount seasonal inventory z. ni’mah and y. farida multi-unit spares inventory control – three dimensional (music 3d) approach to inventory control 21 based on the benefit above, the availability of inventory is important for each company, but if the quantity of inventory is excessive, it will cause losses due to inventory cost, so it is necessary to have inventory control. 2.2 inventory control inventory control is a series of activities that need to be carried out by each company so that there is no accumulation of goods in the warehouse [8] [12]. the factors that influence inventory control are as follows: total sales total sales are the total amount generated from the sale of each item within a certain period to obtain maximum profit to be able to support the development of the company. the greater the total value of the sales, the greater the profits earned by the company too. the total sales value is obtained based on the following equations (1) and (2). 𝑇 = 𝑛 × 𝐻𝑃𝑃(𝑗) (1) 𝑇 = 𝑛 × [𝐻𝑃𝑃(𝑏) + 𝑚𝑎𝑟𝑔𝑖𝑛] (2) where : 𝑇 = total sales 𝑛 = number of items sold 𝐻𝑃𝑃(𝑗) = cost of items sold 𝐻𝑃𝑃(𝑏) = cost of purchase lead time lead time is waiting time what the company needs, starting from ordering items until the item arrives. lead time is one of the important factors in inventory control because if the company does not take into account lead time on procurement of inventory in the warehouse, it will hamper the process of sending the item to the customer. this causes the waiting time of each item to be estimated even though there is still a risk of errors in the assessment. inventory turnover inventory turnover is the ratio to measure the efficiency of inventory management of items that can control the capital in inventory. the higher the ratio, getting better and increasingly showing efficient inventory management. turnover values can be obtained based on equation (4) [13]. 𝐼𝑇𝑂 = 𝑛 �̅�𝑝 (3) �̅�𝑝 = 𝑃(𝑜)+𝑃(𝑎) 2 (4) where : ito = inventory turnover 𝑛 = number of items sold �̅�𝑝 = average lot size 𝑃(𝑜) = initial inventory 𝑃(𝑎) = ending inventory based on these factors above, an approach is needed to manage inventory using music 3d (multi-unit spares inventory control – three dimensional). jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 19-27 22 2.3 music 3d music 3d is a method to classify item products of the company using threedimensional approach. according to [14] there are several types of item classifications that need to be considered in the music 3d approach shown in table 1. table 1. category of classification no. category criteria 1 abc analysis sales volume 2 sde (scarce, difficult, easy) analysis lead time 3 fsn (fast, slow, non-moving) analysis turn over 4 hml (high, medium, low) analysis the price of each item 5 ved (vital, essential, desirable) analysis critical level of each item 6 golf (govt, ordinary, local, foreign) analysis technical payment based on supplier location 7 sos (seasonal, off-seasonal) analysis seasonal items in this study, music 3d using three dimensions as a combination of three types of classification, namely abc analysis, sde analysis, and fsn analysis. here's the explanation: 2.3.1 abc analysis abc analysis is a method which classifies inventory into three groups based on the volume of sales in a particular period. abc analysis uses the concept of pareto law that says 80/20 means that 80% of a company's sales (company income) is generated by 20% of items [15]. the steps of the abc analysis method are [16]: collecting sales data (last 12 months) for each item. sales sort items from the largest to the smallest. calculating the percentage of sales of each item to total sales, then make a cumulative percentage. cumulative percentage calculation is obtained based on equation (5). %𝐾𝑢𝑚. (𝑚) = %𝑝𝑒𝑛𝑗. +(𝑚 − 1) (5) %𝑝𝑒𝑛𝑗. = 𝑇(𝑚) 𝑇(𝑛) × 100% (6) where : %𝐾𝑢𝑚. = cumulative percentage %𝑝𝑒𝑛𝑗. = percentage of total sales 𝑛 = number of items sold 𝑚 = 1,2,3,…,n 𝑇(𝑚) = total sales 𝑇(𝑛) = total sales of all items grouping of abc analysis is shown in table 2. table 2. criteria of abc analysis no. group %𝐾𝑢𝑚. 1 a up to 80% 2 b 81% – 96% 3 c >96% z. ni’mah and y. farida multi-unit spares inventory control – three dimensional (music 3d) approach to inventory control 23 2.3.2 sde analysis sde (scarce, difficult, and easy) analysis is based on the lead time of each item. lead time is the time needed to order items from suppliers until the items arrive at the company [7]. the criteria for each group are shown in table 3. table 3. criteria of sde analysis no. group criteria 1 s (scarce) >60 days 2 d (difficult) 31 – 60 days 3 e (easy) 1 – 30 days 2.3.3 fsn analysis fsn (fast, slow, and non-moving) analysis is a classification of items based on the turnover value of each item. inventory turnover shows how many times the item came out or replaced in a certain period. turnover value is obtained based on equation (3) [7]. the criteria for each group are shown in table 4. table 4. criteria of fsn analysis no. group turnover 1. f (fast moving) >4 times rotating in one year 2. s (slow moving) 1≤x≤4 times rotating in one year 3. n (non-moving) does not rotating in one year 3. research methods 3.1 data this study uses data obtained from pt fajar mas murni surabaya in 2017 precisely in the warehouse section. the data consists of 212 compressor spare part items that have sales price (price), the number of incoming items (in), the number of items coming out or total sales (out), the number of initial inventory (𝑷(𝒐)), the number of ending inventory (𝑷(𝒂)) and lead time of an item (lt). the data used are shown in table 5. table 5. data used no. part number descriptions of the items price in out 𝑃(𝑜) 𝑃(𝑎) lt (days) 1. 38459582 ssr ultra coolant (39433735) xxx 1013 846 5 172 89 2. 39433743 plt coolant-ultra,208 litre xxx 4 3 0 1 89 3. 54509435 element separator (89213011) xxx 23 23 3 3 99 4. 22219174 el.separator (54509500/39863873) xxx 25 22 2 5 99 5. 39433743 coolant ultra, 208 litre xxx 9 7 0 2 89 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 212. 39844113 filter, hi-dust 14 inch xxx 0 0 1 1 94 3.2 data processing steps a. abc analysis : sorting sales of each item from the largest to the smallest. making a cumulative percentage based on equation (5) and (6) classifying total sales of an item based on criteria on table 2 b. sde analysis : classifying lead time of item based on criteria on table 3 jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 19-27 24 c. fsn analysis : calculating turn over of each item based on equation (3) and (4). classifying turn over of item based on criteria on table 4. d. combining three analysis above into one using music 3d approach, then analyzes how to overcome or contribute to what policy analysis can be used as consideration for pt fajar mas murni surabaya. 4. result and discussion 212 data of spare parts items was obtained from the warehouse of pt fajar mas murni surabaya and processed into three classifications, namely abc analysis, sde analysis, and fsn analysis. the results of each classification are then combined into one in the music 3d method. here is a description of each analysis: 4.1 abc analysis result abc analysis is a method which classifies inventory into three groups based on the volume of sales in a particular period. from the results of the cumulative frequency calculation in equations (5) and (6), it produces three groups, namely, a, b and c. these results indicate that from 212 spare parts items totally, 6 items are in category a with a percentage of total items of 3% which contributes to sales of 81% of all items, 16 items included in category b with a percentage of total items of 8% which contributes to sales of 15% of all items, and 190 items included in category c with a percentage of total items 89% contributing sales of 4% of all items. the result of spare parts classification using the abc analysis shown in figure 1. figure 1. abc analysis pie chart based on these results, the items ssr ultra coolant with part number 38459582 have a very high total sales that each year these items provide 53.68% of the company's total sales, so it needs strict supervision and care to avoid losses. while 190 items included in category c means that for the next period inventory in the warehouse can be reduced and does not require intensive supervision. 4.2 sde analysis result sde analysis is based on the lead time of each item. the classification of sde analysis is obtained from the criteria shown in table 3. this result indicated that from 212 spare parts items totally, there are 127 items that included into the category s (scarce) or 60% of all items, 43 items in the category d (difficult) or 20% of all items, and 42 items included in the e (easy) category or 20% of all items. the result of spare parts classification using sde analysis shown in figure 2. figure 2. sde analysis pie chart 3%8% 89% the percentage of items that fall into category a the percentage of items that fall into category b the percentage of items that fall into category c 60% 20% 20% scarce(lead time >60 days) difficult (lead time 3060 days) easy (lead time < 30 days) z. ni’mah and y. farida multi-unit spares inventory control – three dimensional (music 3d) approach to inventory control 25 based on the results of sde analysis, item hose assembly has the highest lead time 256 days and do not have a total value of sales, this means that item not sold at all in one year. this results in a hose assembly item that does not require strict supervision. while the element separator item by part number 54509435 in the category of s (scarce) because it has a lead time of more than 60 days with 79 days. although the lead time is long enough, the item is an item that has a high total sales value for the company, so that the item needs to be closely monitored in terms of the quantity ordered. 4.3 fsn analysis result fsn analysis is a classification of items based on the turnover value of each item. the classification of fsn analysis is obtained from the criteria shown in table 4. this result indicated that from 212 spare part items totally, there are 15 items that included in category f (fast moving) or 7% of all items, 41 items are categorized as s (slow moving) or 19% of all items and 156 items included in the category n (non-moving) or 74% of all items. the result of spare parts classification using fsn analysis shown in figure 3. the results of the fsn analysis show that there are more than 50% of non-moving items. thus, in the next period, the company can reduce the lotsize of items included in the category of non-moving (n) to minimize storage costs. the items that have the highest turnover are the element air filter item which in one year the item rotates 22 times. although it has the highest turnover, the item does not have a high total sales, so in the abc analysis above, it is categorized as c. while items that need strict supervision are ultra coolant ssr items because, in addition to having a high turnover, the total sales owned are very high, which greatly supports the company's income. figure 3. fsn analysis pie chart 4.4 music 3d result music 3d is an approach method that combines three analyzes, they are abc, sde, and fsn. the result of music 3d approach for each item is shown in table 6. table 6. category of each items using music 3d approach no. part number descriptions of the items category 1. 38459582 ssr ultra coolant (39433735) adf 2. 39433743 plt coolant-ultra, 208 litre adf 3. 54509435 element separator (89213011) asf 4. 22219174 el.separator (54509500/39863873) aef 5. 39433743 coolant ultra, 208 litre adf ⋯ ⋯ ⋯ ⋯ 212. 39844113 filter, hi-dust 14 inch csn by using music 3d approach, policy analysis can be obtained that can be used as consideration for pt fajar mas murni surabaya. the policy analysis submitted to the company is shown in table 7. 74% 19% 7% non moving (turnover 0) slow moving (turnover 1-4 kali) fast moving (turnover > 4) jurnal matematika mantik vol. 5, no. 1, may 2019, pp. 19-27 26 table 7. the policy analysis to inventory control no. category quantity item the policy analysis to inventory control 1 asf 1 items that included in this category are items that should get strict supervision and maintenance for the warehouse. 2 adf 3 items that included in this category are items that should get fairly strict supervision and maintenance for the warehouse. 3 aef 2 items that included in this category are items that should get scheduled maintenance for the warehouse. 4 bsf 1 items that included in this category are items that should get routine supervision and maintenance but futures. 5 bss 2 items that included in this category are items that need to be scheduled for calculation using the eoq method without the need for direct supervision. 6 bsn 2 the company only makes an order if the item in a state of low inventory. 7 bds 2 items that are included in this category are an item that needs to be scheduling calculations using eoq method without the need for direct supervision. 8 bef 4 items that are included in this category an item that must be scheduled and performed calculations direct supervision. 9 bes 4 items that are included in this category an item that needs to be scheduling calculations using eoq method without the need for direct supervision. 10 ben 1 the company only orders if needed and if the item is in a low inventory condition. 11 csf 2 the company needs to plan an inventory schedule. 12 css 7 the company only needs to make an allowance in supervision and make an order if the item is in a low inventory condition. 13 csn 112 the company only orders if the item is in a low inventory condition 14 cdf 1 the company needs to plan an inventory schedule. 15 cds 9 the company only needs to make an allowance in supervision and make an order if the item is in a low inventory condition. 16 cdn 28 the company only orders if the item is very low inventory. 17 cef 1 the company needs to plan an inventory schedule. 18 ces 17 the company only needs to make an allowance in supervision and make an order if the item is in a low inventory condition. 19 cen 13 the company only orders if needed and if the item is in a very low inventory condition 5. conclusion inventory control of spare part items using abc analysis shows that category c (category of small total sales value) is more dominant, that is 190 items and contributes 4% to company income rather than categories a and b where both categories have a large total sales value. while the calculation with sde analysis shows that the s category (category whose item has a waiting time> 60 days) is more dominant, there are 127 items or 60% of all items than the d and e categories that have a waiting time of <60 days. then using fsn analysis, it shows that the n category (a category that is not rotating at all in one year) is more dominant, that is there are 156 items or 74% of all items rather than the f and s categories, namely the fast moving and slow categories moving. the three’s analysis approach (abc, sde, and fsn analysis) were combined into music 3d approach shows that there is one item that needs strict z. ni’mah and y. farida multi-unit spares inventory control – three dimensional (music 3d) approach to inventory control 27 supervision and maintenance, it is element separator with part number 54509435. this is because the separator element has a high selling price and a long order time of 79 days and is classified as a fast-moving item. references [1] f. d., “analisis metode economic order quantity sebagai evaluasi persediaan pada pt fajar mas murni batam.” politeknik negeri batam, batam, 2012. [2] t. lukmana and d. t. yulianti, “penerapan metode eoq dan rop,” j. tek. inform. dan sist. inf., vol. 1, no. e-issn : 2443-2229, pp. 271–279, 2015. [3] wibisono, “penerapan analisis abc dalam pengendalian persediaan produk furniture pada java furniture wonosari tuban.” universitas sebelas maret, surakarta, pp. 1–84, 2009. [4] t. wahyuni, “penggunaan analisis abc untuk pengendalian persediaan barang habis pakai,” j. vokasi indones., vol. 3, pp. 1–20, 2015. [5] h. w. n. i. vanany, “pengklasifikasian spare part dan model pemilihan vendornya pada metering station di pt . chevron,” no. november, pp. 1–10, 2013. [6] y. d. astanti, “analisis klasifikasi persediaan suku cadang menggunakan music-3d view of spares,” telematika, vol. 11, pp. 1–8, 2014. [7] a. r. anugerah, d. janari, and manzula maulida rahman, “analisis pengendalian perusahaan menggunakan pendekatan music 3d (multi unit spares invnetory control-three dimensional approach) pada warehouse di pt semen indonesia (prsero) tbk pabrik tuban,” no. november. universitas islam indonesia, yogyakarta, 2017. [8] h. m. n. mahfud, manajemen produksi modern, pertama. jakarta: pt. bumi aksara, 2007. [9] a. zaldiansyah, “perencanaan dan pengendalian persediaan spare part mesin di unit produksi i pt . petrokimia gresik menggunakan kebijakan can-order,” surakarta, 2012. [10] i. purwanti and y. farida, “analisis strategi penjualan stok spare part di pt fajar mas murni surabaya,” mat. “mantik,” vol. 04, no. 02, pp. 100–109, 2018. [11] z. yamit, manajemen persediaan. yogyakarta: ekonosia fe-vii, 1998. [12] a. meilani, “pengendalian persediaan spare part dan pengembangan dengan konsep 80-20 (analisis abc) pada auto 2000 cabang sutoyo malang,” 2014. 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[16] suparmi, konsep dasar statistika. jakarta: universitas terbuka, 2012. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 44 vektor prioritas dalam analytical hierarchy process (ahp) dengan metode nilai eigen moh. hafiyusholeh 1 , ahmad hanif asyhar 2 matematika uin sunanampel surabaya, hafiyusholeh@uinsby.ac.id 1 matematika uin sunanampel surabaya, hanif@uinsby.ac.id 2 abstrak pada penelitian ini dikaji metode nilai eigen yang digunakan untuk mengkonstruksivektor prioritas model pengambilan keputusan yang dikenal dengan analytical hierarchy process (ahp). ahp merupakan suatu metode pengambilan keputusan yang berdasarkan pada keragaman kriteria. melalui metode nilai eigen ini diperoleh , dengan adalah nilai eigen maksimum dan n adalah ukuran matriks. untuk membatasi apakah suatu keputusan yang telah diambil dengan ahp sudah valid atau belum, bisa diverifikasi dengan menggunakan indeks konsistensi. kata kunci : nilai eigen, vektor eigen, ahp, matriks konsisten abstract in this paper examines eigenvalues methods to construct models of decision-making priority vector as known as the analytical hierarchy process (ahp). ahp is a decision making method based on the diversity criteria. through this method obtained with is the maximum eigen value and n is the size of the matrix. decision taken by the ahp is valid or not, can be verified by using an index of consistency keywords: eigenvalues, eigenvectors, ahp, consistent matrix 1. pendahuluan setiap orang tentu terlibat dalam pengambilan suatu keputusan, entah secara kolektif maupun secara pribadi. namun demikian, pada tataran tertentu tidak semua orang mudah dalam mengambil suatu keputusan yang melibatkan banyak kriteria, ditambah lagi ketidakpastian atau ketidaksempurnaan informasi seringkali menyulitkan pembuat keputusan. salah satu solusi yang dapat digunakan untuk membantu dalam menetapkan suatu keputusan berbasis multi kriteria adalah dengan analytical hierarchy process (ahp). ahp dikembangkan oleh thomas l. saaty di wharton school of business, university of pennsylvania pada sekitar tahun 1970-an dan baru dipublikasikan pada tahun 1980 dalam bukunya yang berjudul analytical hierarchy process. seiring dengan perkembangan zaman, penerapan ahp telah meluas sebagai model alternatif untuk menyelasaikan bermacammacam masalah, seperti penentuan tempat kerja bagi siswa smk sebagaimana yang dilakukan oleh hafiyusholeh, dkk [11], ataupun terapan yang lain. hal ini dimungkinkan karena ahp cukup mengandalkan pada intuisi yang datang dari pengambil keputusan, yang cukup akan informasi dan memahami masalah keputusan yang dihadapi. dalam ahp, permasalahan yang kompleks dipecah menjadi beberapa unsur yang kemudian disusun berdasarkan hirarki. menurut saaty, definisi hirarki adalah suatu mailto:hafiyusholeh@uinsby.ac.id1 mailto:hanif@uinsby.ac.id jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 45 representasi dari sebuah permasalahan yang kompleks dalam suatu struktur tingkatan majemuk (multilevel) dengan tingkat pertama adalah tujuan, yang diikuti oleh ktiteria, subkriteria, dan seterusnya sampai pada tingkat terakhir [8]. salah satu tahapan terpenting dalam ahp adalah penilaian tingkat perbandingan (comparative judgment). prinsip ini berarti membuat penilaian tentang kepentingan relatif dua elemen pada suatu tingkat tertentu dalam kaitannya dengan tingkat di atasnya yang biasa disajikan dalam bentuk matriks pairwise comparison (pc). penilaian ini merupakan inti dari ahp karena hal tersebut akan berpengaruh terhadap prioritas masing-masing elemen. prioritas-prioritas tersebut pada akhirnya disajikan dalam bentuk vektor yang disebut dengan vektor prioritas. beberapa pendekatan untuk mendapatkan vektor prioritas dari matriks pc , antara lain metode nilai eigen (eigenvalue method) [5], metode chi square [10], metode kuadrat terkecil (lsm) [6] yang meminimumkan norm frobrenius dari matriks pc dengan taksirannya. metode lain disajikan oleh gass dan rapscak [7] yaitu dengan menggunakan metode singular value decomposition (svd). matriks pc konsisten ber-rank satu akan memiliki nilai eigen yang sama dengan ukuran matriks. pada kenyataannya, jika matriks pc tersebut dikenakan gangguan, maka akan merubah sifat ideal dari suatu keputusan. nilai eigen matriks tersebut berubah sebagai akibat berubahnya unsur-unsur matriks yang konsisten. pada kondisi tersebut, nilai eigen maksimal dapat digunakan sebagai pendekatan untuk mendapatkan vektor prioritas. vektor prioritas yang berkorespondensi dengan nilai eigen maksimal dikenal dengan istilah vektor eigen utama. dari teorema perron [3] diketahui bahwa nilai eigen maksimal yang berkorespondensi dengan vektor eigen utama merupakan akar sederhana (simple root). beberapa kajian dan analisis mengenai vektor eigen utama dari matriks pc telah dilakukan diantaranya oleh farkas [2] yang mengkonstruksi beberapa sifat pc. astuti dan garnadi [1] mengkaji nilai dan vektor eigen dari matriks pc yang terganggu dengan mentransformasikannya kedalam matriks 3x3 yang lebih fleksibel. garmenia, h., dkk [12] melakukan penelitian yang mengakji kekonsistenan matrik sebagai akibat dikenai gangguan. berdasarkan uraian di atas, pada penelitian ini akan dibahas vektor prioritas dalam analytical hierarchy process (ahp) dengan menggunakan metode nilai eigen. 2. kajian teori 2.1 matriks positif matriks positif merupakan matriks yang semua entri-entrinya berupa bilangan positif. dalam bahasa formal, misalkan diberikan matriks ( ) ( ). matriks a dikatakan positif, jika untuk setiap unsur di a, untuk setiap i,j = 1, 2, ..., n. yang dinotasikan dengan a > 0. matriks positif memiliki beberapa sifat yang terkait dengan nilai dan vektor eigen yang akan digunakan untuk membuktikan teorema perron. pada pembuktian teorema perron, digunakan pula teorema jordan yaitu pada saat membuktikan multiplisitas aljabar dari (a) adalah satu. teorema perron diperlukan sebagai jaminan atas nilai dan vektor eigen positif yang akan digunakan untuk mengkonstruksi vektor prioritas dalam ahp. beberapa sifat matriks positif yang akan digunakan untuk membuktikan teorema perron diantaranya diketengahkan dalam teorema berikut. teorema 2.1 misalkan ( ) ( ) dengan dan misalkan . 1. jika sehingga , maka ( ) 2. jika , maka ( ) 3. jika , maka ( ) bukti dari teorema tersebut dapat dilihat di [3]. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 46 lemma 2.1. misalkan ( ) ( ) dengan | | ( ) maka | | ( )| |. bukti: perhatikan bahwa ( )| | | || | | | | | | || | | | dengan demikian diperoleh ( )| | | |. misalkan | | ( )| | . untuk kasus diperoleh | | ( )| | untuk kasus , pandang | | ( ) | | misalkan | |, didapatkan ( ) . selanjutnya berdasarkan teorema 2.1.1 didapat ( ) ( ). hal ini kontradiksi. jadi harusnya yaitu | | ( )| | . beberapa teorema yang terkait dengan matriks positif selanjutnya dapat dirangkum dalam teorema perron sebagai berikut: teorema 2.2 (teorema perron). misalkan ( ) , maka 1. ( ) ; 2. ( ) adalah nilai eigen a; 3. terdapat dengan dan ( ) ; 4. ( ) merupakan nilai eigen sederhana dari a, yaitu ( ( )) ; 5. | | ( ) untuk setiap nilai eigen ( ); teorema perron tersebut diperlukan sebagai jaminan bahwa nilai eigen maksimal senantiasa mempunyai nilai positif. 2.2 analytic hierarchy process (ahp) secara umum terdapat tiga prinsip dalam menyelesaikan persoalan dengan ahp yakni decomposition, comparative judgment, dan synthesis of priority. setelah persoalan didefenisikan, maka perlu dilakukan decomposition yaitu memecah persoalan yang utuh menjadi unsur-unsur yang lebih sederhana. dengan kata lain, setelah persoalan didefinisikan, perlu dibuat struktur hirarki yang diawali dengan tujuan umum, dilanjutkan dengan kriteria-kriteria, kemudian sub-kriteria dengan kemungkinan alternatifalternatif pada tingkatan kriteria yang paling bawah. apabila ingin mendapatkan hasil yang akurat, pemecahan dapat juga dilakukakan terhadap unsur-unsur sampai tidak mungkin dilakukan pemecahan lagi. struktur hirarki keputusan dapat dapat dilihat pada gambar berikut gambar 1. struktur hirarki comparative judgment, prinsip ini berarti membuat penilaian tentang kepentingan relatif dua unsur pada suatu tingkat tertentu dalam kaitannya dengan tingkat di atasnya. penilaian ini merupakan inti dari ahp karena ia akan berpengaruh terhadap prioritas unsurunsur. hasil dan penilaian ini akan tampak lebih baik apabila disajikan dalam bentuk matriks pc. agar diperoleh skala yang bermanfaat ketika membandingkan dua unsur, seseorang yang akan memberikan jawaban memerlukan pemahaman yang baik tentang unsur-unsur yang dibandingkan dan relevansinya terhadap kriteria atau tujuan yang akan dicapai. adapun skala dasar yang digunakan untuk membandingkan unsur-unsur yang ada oleh saaty [5] dibuat tabel skala perbandingan sebagai berikut; jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 47 tabel 1. tabel skala perbandingan berpasangan synthesis of priority. dari setiap matriks pc yang telah dibuat kemudian dicari vektor eigennya untuk mendapatkan prioritas lokal. karena matriks pc terdapat pada setiap tingkat, maka untuk mendapatkan prioritas globalnya harus dilakukan sintesa di antara prioritas lokal. prosedur melakukan sintesa berbeda menurut bentuk hirarki. adapun tahapan proses pengambilan keputusan dengan ahp adalah sebagai berikut: 1. mendefinisikan masalah dan menentukan solusi yang diinginkan. 2. membuat struktur hirarki yang diawali dengan tujuan umum, dilanjutkan dengan sub tujuan, kriteria dan kemungkinan alternatif pada tingkatan kriteria yang paling bawah. 3. membuat matriks pc yang menggambarkan kontribusi relatif setiap unsur terhadap masing-masing tujuan atau kriteria yang setingkat di atasnya. 4. melakukan perbandingan berpasangan sehingga diperoleh ketetapan (judgment) seluruhnya sebanyak n x [(n-1)/2] buah, dengan n adalah banyaknya unsur yang dibandingkan. 5. menghitung prioritas dan menguji kekonsistenannya. 6. mengulangi langkah 3, 4, dan 5 untuk seluruh tingkat hirarki 3. pembahasan pada bagian ini akan dikaji salah satu metode untuk menentukan vector prioritas yang dikenal dengan metode nilai eigen. misalkan terdapat n obyek yang dinotasikan dengan yang akan dinilai tingkat kepentingannya dengan bobot pengaruh . apabila diketahui nilai perbandingan unsur terhadap unsur adalah , dengan adalah entri dari matriks pc a, dapat dibuat matriks pc sebagai berikut; matriks a merupakan matriks positif n x n yang reciprocal sehingga aji = 1/aij. matriks ini konsisten jika aij.ajk = aik untuk i, j, k = 1, 2, 3,… n. misalkan ( ) dan a adalah matrik pc yang konsisten dengan , dapat diselidiki bahwa yaitu [ ] [ ] [ ] dalam teori matriks, persamaan dipenuhi hanya jika w adalah vektor eigen dari a dengan nilai eigen n. dalam aljabar linear, semua nilai eigen adalah nol kecuali satu yang kemudian disebut dengan . karena a merupakan matriks positif yang reciprocal, yaitu dan untuk semua nilai , maka berlaku ∑ ( ) trace dari matriks a adalah jumlah unsurunsur diagonal matriks a. permasalahannya kemudian, dalam kasus umum nilai-nilai tidak dapat diberikan secara tepat. nilai-nilai hanya jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 48 dapat ditaksir sehingga permasalahannya sekarang adalah mendekati persamaan dengan ̂ ̂ dengan adalah nilai eigen terbesar dari matriks a. berkaitan dengan , teorema perron telah menjamin bahwa dari matriks positif akan mempunyai nilai positif. karena matriks , maka terdapat vector eigen positif yang berkaitan dengan . selain itu, merupakan nilai eigen simple dengan ( ) . dengan adanya sifat-sifat tersebut, maka nilai eigen dari matriks pc yang digunakan untuk mendapatkan vektor prioritas dalam ahp akan senantiasa bernilai positif. lebih lanjut vektor prioritas yang berkorespondensi dengannya akan bernilai positif pula. pada praktiknya, nilai yang digunakan untuk mengkonstruksi vector prioritas akan lebih besar dari ukuran matriks a sebagaimana yang diuraikan pada teorema berikut. teorema 3.1 misalkan ( ) ( ) . jika ̂ merupakan vektor tak nol di , sehingga ̂ ̂, maka bukti: karena ̂ merupakan vektor eigen yang berkorespondensi dengan , maka berlaku ( ) ̂ persamaan ke-i dari sistem persamaan linear tersebut dapat ditulis sebagai ∑ ̂ ̂ dengan kata lain ∑ ̂ ̂ selanjutnya akan diperoleh ∑ ∑ ̂ ̂ karena dan ̂ ̂ ̂ ̂ , dan dengan memperhatikan bahwa ( ) ̂ ̂ , maka akan diperoleh ∑ ∑ ( ̂ ̂ ̂ ̂ ) ∑ ∑ ( ( ̂ ̂ ) ̂ ̂ ) perhatikan bahwa ∑ ∑ ∑ ∑ ∑ ( ) ( ) ( ( )) [( ) ( ) ] [ ( ) [ ( ) ( )( )]] [ ( ) ] ( ) dengan demikian, ∑ ∑ ( ( ̂ ̂ ) ̂ ̂ ) berdasarkan uraian tersebut, dapat disimpulkan bahwa dan kesamaan hanya akan dipenuhi jika dan hanya jika untuk semua . jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 49 dengan kata lain, jika dekat dengan n, maka ̂ konsisten. indikator terhadap kekonsistenan diukur melalui indeks konsistensi (consistency index (ci)) yang didefinisikan sebagai batasan terhadap ketidakkonsistenan suatu matriks, oleh saaty [8] diukur dengan menggunakan rasio konsistensi (cr) yang diperoleh dari perbandingan antara indeks konsistensi dan random indeks (ri), random indeks tersebut bergantung pada ukuran suatu matriks yang dapat disajikan sebagai berikut tabel 2. tabel random indeks hasil penelitian dapat diterima jika nilai rasio konsistensi (cr) tidak lebih dari 10%. apabila nilai rasio konsistensi lebih dari 10%, maka penilaian yang dilakukan belum dianggap konsisten. 4. penutup ahp merupakan salah satu metode pengambilan keputusan yang melibatkan banyak kriteria. salah satu metode yang dapat digunakan untuk mengkonstruksi vektor prioritas adalah metode nilai eigen. dengan metode ini diperoleh nilai eigen terbesar akan lebih dari atau sama dengan n. untuk membatasi apakah suatu keputusan yang telah diambil dengan ahp sudah valid atau belum, bisa diverifikasi dengan menggunakan indeks konsistensi. referensi [1] pudji astuti, agah d. garnadi, on eigenvalues and eigenvectors of perturbed pairwise comparison matrices, fundamental research grant 2008 (2008). [2] andras farkas, the analysis of the principal eigenvector of pairwise comparison matrices, acta polytechnica hungarica 4 (2007). [3] a. horn, roger., a. johnson, charles., matrix analysis, cambridge university press, (1985). [4] r. fletcher, d. sorensen, an algorithmic derivation of the jordan canonical form, amer. math. montly, 90 pp (1983) 12-16. [5] saaty, t.l., decision making for leaders, university of pittsburg, (1980). [6] baz$\acute{o}$ki, s, solution of the least square method problem of pairwise comparison matrices, central european journal of operations research (2008). [7] gass, s.i., rapcs$\acute{a}$k, t., singular value decomposition in ahp, european journal of operations research 154 (2004) 573-584. [8] saaty, t.l., the analytic hierarchy process, university of pittsburgh (1988). [9] saaty, t.l., decision-making with the ahp: why is principal eigenvector necessary, ejor 145 (2002) 85-91. [10] xu, z.s., generalized chi square method for the estimation of weights, jota 183 pp (2000) 183-192 [11] hafiyusholeh, m., asyhar, a.h., komaria, r., aplikasi metode nilai eigen dalam analytical hierarchy process untuk memilih tempat kerja, jurnal matematika mantik vol. 1 no. 1 (2015) http://mantik.uinsby.ac.id/index.php/mant ik1/article/download/02-2015/3 [12] garminia, h., hafiyusholeh, moh. dan astuti, p. pengaruh gangguan pada perubahan prioritas dan indeks konsistensi matriks perbandingan berpasangan dalam analytical hierarchy process, jurnal matematika dan sains. desember 2010, vol. 15 no. 3. 143 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: i. maulana, “schanuel’s lemma in p-poor modules”, mantik, vol. 5, no. 2, pp. 66-82, october 2019. schanuel’s lemma in p-poor modules iqbal maulana universitas singaperbangsa karawang, hmiqbal1202@gmail.com doi: https://doi.org/10.15642/mantik.2019.5.2.76-82 abstrak: modul merupakan perumuman dari ruang vektor aljabar linier yaitu dengan memperumum lapangan skalarnya menjadi ring dengan elemen satuan. dalam teori modul terdapat konsep modul proyektif, yaitu suatu modul atas ring r yang proyektif relatif terhadap semua modul atas r selanjutnya, diperoleh fakta bahwa setiap modul atas r adalah modul proyektif relatif terhadap sebarang modul semisederhana atas r. jika p adalah suatu modul atas r yang proyektif relatif hanya terhadap semua modul semisederhana atas r saja, maka p disebut modul p-miskin. dalam pembahasan modul proyektif terdapat suatu lemma yang berkaitan dengan keekuivalenan dua buah modul k1 dan k2 dengan syarat terdapat dua buah modul proyektif p1 dan p2 sedemikian hingga 1 2k p isomorfik dengan 2 1k p . lemma tersebut dikenal sebagai lemma schanuel di modul proyektif. karena modul p-poor merupakan kasus khusus dari modul proyektif, maka pada tulisan ini akan dibahas tentang lemma schanuel di modul p-poor. kata kunci: modul proyektif, modul semisederhana, modul p-poor, lemma schanuel abstract: modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. in module theory there is a concept about projective module, i.e. a module over ring r in which it is projective module relative to all modules over ring r. next, there is the fact that every module over ring r is projective module relative to all semisimple modules over ring r. if p is a module over ring r which it’s projective relative only to all semisimple modules over ring r, then p is called p-poor module. in the discussion of the projective module, there is a lemma associated with the equivalence of two modules k1 and k2 provided that there are two projective modules p1 and p2 such that 1 2k p is isomorphic to 2 1k p . that lemma is known as schanuel’s lemma in projective modules. because the p-poor module is a special case of the projective module, then in this paper will be discussed about schanuel’s lemma in p-poor modules. keywords: projective module, semisimple module, p-poor module, schanuel’s lemma jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 76-82 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 i maulana schanuel’s lemma in p-poor modules 77 1. introduction let m and n are r-modules, i.e. modules over a ring r. in this paper, mod-r denotes the set of all right r-modules and ssmod-r the set of all semisimple right r-modules. an r-module is called a semisimple module if that module is a direct sum of simple modules [5]. a non-zero r-module is called a simple module if that module has no non-trivial submodules. in other words, its submodule is only {0} and himself. following [3], for any r-module m, 𝔓𝑟−1(m) = { n mod r − | m is n projective module} is called the projectivity domain of m. if 𝔓𝑟−1(m) = modr, then m is called a projective module. next, alahmadi et al. [1] which discuss poor-module become the initial idea of the emergence of p-poor module concept, which p-poor module is dual of poor-module. furthermore, this p-poor module is a special case of the projective module because the projectivity domain of p-poor only consists of all semisimple modules over ring r [2]. regarding the existence of the p-poor module, it was found that each ring has a p-poor module. as for the formation of the p-poor module, it was found that an r-module, which is the result of the direct sum of all cyclic modules over r is a p-poor module [2]. this paper is inspired by similar ideas and problems in [4][5], where there is a lemma introduced by stephen schanuel in 1958 and known as the schanuel’s lemma in projective modules. that lemma associated with the equivalence of two modules k1 and k2 provided that there are two projective modules p1 and p2 such that 𝐾1 ⊕𝑃2 is isomorphic to 𝐾2 ⊕𝑃1. the organization of this paper describes as follows: section 2 explains a basic theory about exact sequences of r-modules and semisimple module. the explanation about the schanuel’s lemma in projective modules and schanuel’s lemma in p-poor modules will be presented in section 3. in section 4, we conclude the discussion. 2. basic theory in this section, we define the external direct sum, the short exact sequence, the split exact sequence, and some properties of the semisimple module. 2.1 external direct sum before we define the external direct sum, will first be discussed about the direct product. definition 2.1. [3] the cartesian product ×𝐴 𝑋𝛼 of the sets {𝑋𝛼}𝛼∈𝐴 be the set of all a-tuple (𝑥𝛼)𝛼∈𝐴 such that 𝑥𝛼 ∈ 𝑋𝛼, for all 𝛼 ∈ 𝐴. if a is finite, 𝐴 = {1,…,𝑛} then be obtained ×𝐴 𝑋𝛼 = 𝑋1 ×…×𝑋𝑛 = {(𝑥1,…,𝑥𝑛)|𝑥𝑖 ∈ 𝑋𝑖, 𝑖 = 1,…,𝑛}. definition 2.2. [3] let {𝑀𝜆}𝜆∈λ be the set of r-modules. defined the operations in ×λ 𝑀𝜆, for every (𝑥𝜆)𝜆∈λ , (𝑦𝜆)𝜆∈λ ∈ ×λ 𝑀𝜆 and 𝑟 ∈ 𝑅 then (𝑥𝜆)𝜆∈λ +(𝑦𝜆)𝜆∈λ = (𝑥𝜆 +𝑦𝜆)𝜆∈λ and 𝑟(𝑥𝜆)𝜆∈λ = (𝑟𝑥𝜆)𝜆∈λ . next, the cartesian product ×λ 𝑀𝜆, together with the above operations is rmodules. furthermore, the module ×λ 𝑀𝜆 is said to be the direct product of {𝑀𝜆}𝜆∈λ and be written ∏λ𝑀𝜆. definition 2.3. [3] let {𝑀𝜆}𝜆∈λ be the set of r-modules. the external direct sum of {𝑀𝜆}𝜆∈λ is defined as ⨁ 𝑀𝜆λ = {𝑚 ∈ ∏λ𝑀𝜆 | 𝜋𝜆 (𝑚) ≠ 0 𝑓𝑜𝑟 𝜆 ∈ λ is finite}. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 76-82 78 2.2 exact sequences the concept of exact sequences of r-modules and r-module homomorphisms and their relation to direct summands is a useful tool to have available in the study of modules. we start by defining exact sequences of r-modules. definition 2.4. [6] let r be a ring. a sequence of r-modules m and r-module homomorphisms f … 𝑓𝑖−1 → 𝑀𝑖−1 𝑓𝑖 →𝑀𝑖 𝑓𝑖+1 → 𝑀𝑖+1 𝑓𝑖+2 → … (1) is said to be exact at 𝑀𝑖 if 𝐼𝑚(𝑓𝑖) = 𝐾𝑒𝑟(𝑓𝑖+1). the sequence is said to be exact if it is exact at each 𝑀𝑖. as particular cases of definition 2.1. note that if 𝑀, 𝑀1, and 𝑀2 are r-modules 1. 0→𝑀1 𝑓 →𝑀 is exact if and only if f is injective, 2. 𝑀 𝑔 →𝑀2 →0 is exact if and only if g is surjective, and 3. the sequence 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (2) is exact if and only if f is injective, g is surjective and 𝐼𝑚(𝑓) = 𝐾𝑒𝑟(𝑔). definition 2.5. [7] given a sequence of r-modules 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (3) 1. the sequence (3) is said to be a short exact sequence if it is exact. 2. the sequence (3) is said to be a split exact sequence (or just split) if it is exact and if 𝐼𝑚(𝑓) = 𝐾𝑒𝑟(𝑔) is a direct summand of m. next, in the following theorem will be given a characterization of split exact sequence. theorem 2.1. [7] if 0→𝑀1 𝑓 →𝑀 𝑔 →𝑀2 →0 (4) is a short exact sequence of r-modules, then the following are equivalent: 1. there exists a homomorphism 𝛼: 𝑀 → 𝑀1 such that 𝛼 ∘𝑓 = 𝑖𝑑𝑀1. 2. there exists a homomorphism 𝛽: 𝑀2 → 𝑀 such that 𝑔 ∘𝛽 = 𝑖𝑑𝑀2. 3. the sequence (4) is split exact. if these equivalent conditions hold then 𝑀 ≅ 𝐼𝑚(𝑓)⊕𝐾𝑒𝑟(𝛼) ≅ 𝐾𝑒𝑟(𝑔)⊕𝐼𝑚(𝛽) ≅ 𝑀1 ⊕𝑀2 2.3 semisimple module next theory is needed in the next discussion is a semisimple module and some of its properties. however, it will first be defined as a simple module. i maulana schanuel’s lemma in p-poor modules 79 definition 2.6. [3] a non-zero r-module m is called a simple module if m has no non-trivial submodules. in other words, the submodule of m is only {0} and m. definition 2.7. [6] an r-module m is called a semisimple module if m is a direct sum of simple modules. a semisimple module has some characterization which will be given in the following proposition. proposition 2.2. [6] for an r-module m, the following properties are equivalent: 1. m is a semisimple module. 2. every submodule of m is a direct summand. 3. every exact sequence 0 → 𝐾 → 𝑀 → 𝐿 → 0 splits, for each k and l are rmodules. 3. main results based on the previous introduction, we have that p-poor module is a special case of the projective module because the projectivity domain of p-poor only consists of all semisimple modules over ring r. in other words, r-modules p is ppoor if for every semisimple r-modules s satisfies for each epimorphism 𝑔 ∶ 𝑆 → 𝑁 and homomorphism 𝑓 ∶ 𝑃 → 𝑁 there exists a homomorphism ℎ ∶ 𝑃 → 𝑆 such that 𝑔 ∘ℎ = 𝑓 (i.e. the following diagram commute). therefore, before we explain schanuel’s lemma in p-poor modules, we will first discuss schanuel’s lemma in projective modules. 3.1 schanuel’s lemma in projective modules this lemma associated with the equivalence of two modules m1 and m2 provided that there are two projective modules p1 and p2 such that 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. furthermore, it will be discussed in the following lemma. lemma 3.1. [4] given the sequences of r-modules 0→𝑀1 𝑓1 →𝑃1 𝑔1 →𝑀→0 (5) 0→𝑀2 𝑓2 →𝑃2 𝑔2 →𝑀→0 (6) if (5) and (6) are exact with 𝑃1 and 𝑃2 are projective, then 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. proof. from r-modules 𝑃1 and 𝑃2 can be formed a direct sum 𝑃1 ⊕𝑃2. next, be formed 𝑋 = {(𝑝1,𝑝2) ∈ 𝑃1 ⊕𝑃2|𝑔1(𝑝1) = 𝑔2(𝑝2)}. clearly, 𝑋 ⊆ 𝑃1 ⊕𝑃2 and 𝑋 ≠ ∅ because (0,0) ∈ 𝑋. then, for each (𝑥1,𝑥2) and (𝑦1,𝑦2) in x and r in r, we s p jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 76-82 80 see that 𝑔1(𝑥1 +𝑦1) = 𝑔1(𝑥1)+𝑔1(𝑦1) = 𝑔2(𝑥2)+𝑔2(𝑦2) = 𝑔2(𝑥2 +𝑦2) and 𝑔1(𝑥1𝑟) = 𝑔1(𝑥1)𝑟 = 𝑔2(𝑥2)𝑟 = 𝑔2(𝑥2𝑟). so, we have (𝑥1 +𝑦1,𝑥2 +𝑦2) and (𝑥1𝑟,𝑥2𝑟) in x. in other words, x is submodule of 𝑃1 ⊕𝑃2. next, we see that 𝑔1 is epimorphism (surjective homomorphism) so that we have 𝑀 = 𝑔1(𝑃1). since 𝑔2 is also epimorphism, then for each 𝑔1(𝑃1) ∈ 𝑀 there exists 𝑝2 ∈ 𝑃2 such that 𝑔1(𝑝1) = 𝑔2(𝑝2). defined homomorphism 𝜋1 : 𝑋 → 𝑃1 with 𝜋1(𝑝1,𝑝2) = 𝑝1. then, we have 𝐾𝑒𝑟 (𝜋1) = {(𝑝1,𝑝2) | 𝜋1(𝑝1,𝑝2) = 0 } = {(𝑝1,𝑝2) | 𝑝1 = 0 } = {(0,𝑝2) | 𝑔2(𝑝2) = 0 } ≅ 𝐾𝑒𝑟 (𝑔2) = 𝐼′𝑚 (𝑓2) furthermore, based on the particular cases of definition 2.1, because (6) are exact, then 𝑓2 is monomorphism (injective homomorphism), and because 𝑓2 is injective, then we have 𝐼𝑚 (𝑓2) ≅ 𝑀2. as a result, we have 𝐾𝑒𝑟 (𝜋1) ≅ 𝑀2. next, can be formed a short exact sequence 0 → 𝑀2 → 𝑋 𝜋1 → 𝑃1 → 0 (7) since 𝑃1 is a projective module, there exists a homomorphism ℎ: 𝑃1 → 𝑋 such that 𝜋1 ∘ℎ = 𝑖𝑑𝑃1 , then the sequence (7) is split exact, and we have 𝑋 ≅ 𝑀2 ⊕𝑃1. furthermore in an analogous way, then can be formed a short exact sequence 0 → 𝑀1 → 𝑋 𝜋2 → 𝑃2 → 0 (8) and we have 𝑋 ≅ 𝑀1 ⊕𝑃2. therefore, we have 𝑀1 ⊕𝑃2 ≅ 𝑀2 ⊕𝑃1. 3.2 schanuel’s lemma in p-poor modules next, can be made schanuel’s lemma in p-poor modules, i.e. we replace sufficient conditions projective module in lemma 3.1. with p-poor module which it is also a semisimple module, or we call that module as a semisimple p-poor. this is because the p-poor module is a special case of the projective module, where the projectivity domain of p-poor only consists of all semisimple modules. therefore, need a certain condition is semisimple so that the concept of its projective module can be used in the p-poor module. lemma 3.2. given the sequences of r-modules 0→𝑀1 𝑓1 →𝑃1 𝑔1 →𝑀→0 (9) 0→𝑀2 𝑓2 →𝑃2 𝑔2 →𝑀→0 (10) if (9) and (10) are exact with 𝑃1 and 𝑃2 are semisimple p-poor modules, then 𝑀1 ⊕ 𝑃2 is isomorphic to 𝑀2 ⊕𝑃1. proof. from semisimple p-poor modules 𝑃1 and 𝑃2 , then we have 𝑃1 ⊕𝑃2 is also semisimple p-poor module. next, be formed 𝑊 = {(𝑝1,𝑝2) ∈ 𝑃1 ⊕ 𝑃2|𝑔1(𝑝1) = 𝑔2(𝑝2)}. clearly, w is a submodule of 𝑃1 ⊕𝑃2 because its proof is same with the proof of x is a submodule of 𝑃1 ⊕𝑃2 in lemma 3.1. furthermore, according to [3] because every submodule of a semisimple module is semisimple, then we have w is a semisimple module. next, we see that 𝑔1 is epimorphism (surjective homomorphism) so that we have 𝑀 = 𝑔1(𝑃1). since 𝑔2 is also epimorphism, then for each 𝑔1(𝑃1) ∈ 𝑀 there exists i maulana schanuel’s lemma in p-poor modules 81 𝑝2 ∈ 𝑃2 such that 𝑔1(𝑝1) = 𝑔2(𝑝2). defined homomorphism 𝜋1: 𝑊 → 𝑃1 with 𝜋1(𝑝1,𝑝2) = 𝑝1. then, we have 𝐾𝑒𝑟 (𝜋1) = {(𝑝1,𝑝2) | 𝜋1(𝑝1,𝑝2) = 0 } = {(𝑝1,𝑝2) | 𝑝1 = 0 } = {(0,𝑝2) | 𝑔2(𝑝2) = 0 } ≅ 𝐾𝑒𝑟 (𝑔2) = 𝐼𝑚 (𝑓2) furthermore, because 𝑓2 is monomorphism (injective homomorphism), then we have 𝐼𝑚 (𝑓2) ≅ 𝑀2. as a result, we have 𝐾𝑒𝑟 (𝜋1) ≅ 𝑀2. next, can be formed a short exact sequence 0 → 𝑀2 → 𝑊 𝜋1 → 𝑃1 → 0 (11) since 𝑃1 is a p-poor module (i.e. projective module which its projectivity domain only consists of all semisimple modules), then for semisimple module w there exists homomorphism ℎ: 𝑃1 → 𝑊 such that 𝜋1 ∘ℎ = 𝑖𝑑𝑃1. in other words, the sequence (11) is split exact and we have 𝑊 ≅ 𝑀2 ⊕𝑃1. furthermore in an analogous way, then can be formed a short exact sequence 0 → 𝑀1 → 𝑊 𝜋2 → 𝑃2 → 0 (12) and we have 𝑊 ≅ 𝑀1 ⊕𝑃2. therefore, we have 𝑀1 ⊕𝑃2 ≅ 𝑀2 ⊕𝑃1. 4. conclusion some properties which have sufficient conditions of the projective module can be modified by replacing the projective module into the p-poor module with certain additional conditions. the result of this research only discuss how to get schanuel's lemma in p-poor modules, i.e. with modify schanuel’s lemma in projective modules. its method is to replace sufficient conditions projective module on schanuel’s lemma in projective modules with a semisimple p-poor module. this is because the p-poor module is a special case of the projective module, where the projectivity domain of p-poor only consists of all semisimple modules. actually, this lemma also as an introduction of an equivalence relation in the p-poor module, i.e. modules 𝑀1 and 𝑀2 are equivalent if there exist semisimple p-poor modules 𝑃1 and 𝑃2 such that 𝑀1 ⊕𝑃2 is isomorphic to 𝑀2 ⊕𝑃1 . jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 76-82 82 references [1] a. n. alahmadi, m. alkan, and s. r. lopez-permouth, “poor modules: the opposite of injectivity,” glasgow mathematical journal 52a, pp. 7-17, 2010. [2] c. holston, s. r. lopez-permouth, and n. o. ertas, “rings whose modules have maximal or minimal projectivity domain,” journal of pure and applied algebra 216, pp. 673-678, 2012. [3] f. w. anderson and k. r. fuller, rings and categories of modules (second edition), new york, 1992. [4] i. kaplansky, “fields and rings (second edition),” chicago lectures in mathematics series, pp. 165-168, 1972. [5] f d lestari et al 2019 j. phys.: conf. ser. 1211 012053 [6] r. wisbauer, foundations of module and ring theory, germany, 1991. [7] w. a. adkins and s. h. weintraub, algebra an approach via module theory, new york, 1992. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 17 sistem pendukung keputusan penetapan tunjangan prestasi dengan menggunakan metode fuzzy-tsukamoto (studi kasus di pt.boxtime indonesia) yusron rijal 2) , yus amalia 1) s1/jurusan teknik informatika, stimik yadika bangil email : yusronrijal@stmik-yadika.ac.id, yus@mhs.stmik-yadika.ac.id abstract : the high quality of human resourches according the expertise and competence is needed for increasing factory productivity. according the employee’s expertise and competence, it’s required for giving appreciation for the working achievement for motivated to increase the working achievement. in awarding the decision of giving appreciation, in this case feat allowance required for the performance appraisal is expected to assist in delivering the right decisions and avoid subjectivity. employees deserve appreciation if the result of the performance evaluation meets the criteria specified by the company, the job performance, work quality, dicipline, responsibilities, absenteeism and konduite. decision support systems is apt to be applied in the decision giving appreciation. the decision support system performance allowance is using fuzzy tsukamoto. the result showed that the reasoning in the process of data input and output can analyze whether or not the provision of benefits achievement, this system can help accurately make decisions feasibility of providing allowances achievement with the accuracy of 100% and the amount of allowances achievement of 94.71% using the method of system testing the black box testing. keywords : decision support system, performance evaluation, feat allowance, fuzzy logic, fuzzy tsukamoto 1. pendahuluan pt. boxtime indonesia merupakan sebuah perusahaan yang bergerak dalam bidang pembuatan kotak / kemasan dan display untuk jam tangan dan perhiasan dengan kualitas ekspor. pt. boxtime indonesia merupakan persahaan tunggal di indonesia dengan jumlah karyawan mencapai 550 orang. pemberian apresiasi terhadap kinerja karyawan berupa pemberia tunjangan prestasi atau premi merupakan salah satu faktor yang dapat meningkatkan kualitas kinerja karyawan dalam suatu perusahaan. sistem yang digunakan untuk penilaian kinerja karyawan pada pt. boxtime indonesia saat ini masih bersifat manual dan belum secara maksimal memanfaatkan teknologi dalam mengembangkan proses bisnis, serta penigkatan efektifitas dalam pekerjaan mereka. hal ini disebabkan oleh sistem penilaian yang terbangun belum didasarkan pada kompetensi individu. selain itu proses penilaian membutuhkan waktu lama dan dokumentasi tidak teratur. untuk megatasi hal tersebut, maka akan dirancang sebuah sistem penilaian kinerja karyawan dengan menggunakan metode logika fuzzy tsukamoto untuk menentukan besarnya tunjangan prestasi atau premi yang akan diterima oleh masing – masing karyawan. logika fuzzy adalah teknologi berbasis aturan yang mengizinkan ketidakakuratan dan bahkan menggunakannya untuk menyelesaikan masalah yang belum pernah dipecahkan sebelumnya[1]. dengan mengekspresikan logika menggunakan beberapa ketidakakuratan yang sudah ditetapkan dengan cermat sebelumnya, logika fuzzy menjadi lebih dekat pada cara berfikir orang yang sebenarnya daripada aturan-aturan tradisional if-then [2]. himpunan fuzzy pertama kali dikembangkan pada tahun 1965 oleh prof. lofti a. zadeh dari california unerersity usa. logika fuzzy adalah suatu cara yang tepat memetakan suatu ruang input ke dalam ruang output [3]. proses-proses dalam fuzzy logic adalah fuzzification, penalaran (inferensi), dan defuzzifikasi [4]. mailto:yusronrijal@stmik-yadika.ac.id mailto:yus@mhs.stmik-yadika.ac.id jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 18 gambar 1.1 proses perhitungan metode fuzzy dalam sistem ini perhitungan keputusan pengajuan kredit akan menggunakan metode tsukamoto karena menurut analisis penulis metode tersebut layak dan tepat digunakan untuk menyelesaikan permasalahan di pt.boxtime indonesiar. pada metode penarikan kesimpulan samar tsukamoto, setiap konsekuen pada aturan yang berbentuk if-then harus direpresentasikan dengan suatu himpunan samar dengan fungsi keanggotaan yang monoton. sebagai hasilnya, output hasil penarikan kesimpulan (inference) dari tiaptiap aturan diberikan secara tegas (crisp) berdasarkan α-predikat (fire strength). hasil akhir diperoleh dengan menggunakan ratarata berbobot (weight average)[5]. metode penelitian a. disain sistem perancangan sistem ini adalah tahap awal dalam perancangan perangkat lunak, perancangan sistem ini dilakukan untuk mengetahui gambaran keseluruhan dari sistem. gambar 2.1 context diagram / dfd level 0 sistem pendukung keputusan penetapan tunjangan prestasi dengan menggunakan metode fuzzy – tsukamoto gambar 2.1 menjelaskan bahwa sistem pendukung keputusan penetapan tunjangan prestasi di pt.boxtime indonesia terdapat 3 external entity yaitu administrator, hrd, dan kepala bagian. sedangkan detail sub sistem dijelaskan melalui dfd level 1 pada gambar 2.2 gambar 2.2 dfd level 1sistem pendukung keputusan penetapan tunjangan prestasi dengan menggunakan metode fuzzy – tsukamoto aturan f uzz y informas i data karyawan data penilaian karyawan data user & hak aks es batas himpunan f uzz y informas i has il rekomendasi data karyawan 1 spk penetapan t unjangan prestas i + human resourches department(hrd) administrator kepala bag ian informas i has il rekomendasi data karyawan aturan f uzz y hak akses sistem hak akses sistem hak akses sistem hak akses sistem data aturan f uzz y data aturan f uzz y data has il rekomendasi informas i data karyawan data penilaian karyawan informas i data karyawan informas i data login all user data karyawan data login user kabag data login user hrd data login all user data user & hak aks es human res ourc hes department(hrd) administrator kepala bag ian 1 peng olahan data user & hak akses + 1 t _loginus er 2 sis tem log in us er + 3 peng olahan data karyawan + 2 t _master_kar yawan 5 sist em pendukung keput usan penet apan t unjangan prest asi 5 t _rekomendasi 6 peng olahan data aturan f uzz y + 6 t_aturan jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 19 sistem pendukung keputusan penerimaan pengajuan kredit dengan menggunakan metode fuzzy mamdani pada gambar 2.2 ini terdapat 5 sub sistem antara lain : sub sistem pengolahan data user & hak akses, sub sistem login user, sub sistem pengolahan data karyawan, sub sistem pengolahan data aturan fuzzy, dan sub sistem pendukung keputusan penetapan tunjangan prestasi. b. analisis variabel input dan variabel output berdasarkan hasil wawancara yang dilakukan dengan bagian hrd, didapatkan beberapa kriteria yang dapat dijadikan variabel input dan variabel output dalam himpunan fuzzy. berikut ini adalah variabel input dan variabel output yang akan digunakan dalam logika fuzzy. 1) variabel input terdapat 6 variabel input yang digunakan antara lain yaitu, prestasi kerja, mutu kerja, disiplin kerja, tanggungjawab, absesnsi dan konduite. a. variabel prestasi kerja pada variabel prestasi kerja dibagi menjadi 3 himpunan fuzzy, yaitu : kurang, cukup dan baik. tabel keanggotaan prestasi kerja ditunjukkan pada tabel 2.1. gambar 2.3 merupakan fungsi keanggotaan prestasi kerja. sedangkan persamaan 1, 2, dan 3 merupakan hasil dari pembentukan fungsi keanggotaan prestasi kerja. tabel 2.1 himpunan fuzzy untuk variabel prestasi kerja variabel domain himpunan fuzzy prestasi kerja 10-55 kurang 60-75 cukup 80-100 baik gambar 2.3 fungsi keanggotaan prestasi kerja [ ]{ ............... 1 [ ] { ( ) ( ) ( ) ( ) ............. 2 [ ]{ ( ) ............ 3 b. variabel mutu kerja pada variabel mutu kerja dibagi menjadi 3 himpunan fuzzy, yaitu : kurang, cukup dan baik. tabel keanggotaan mutu kerja ditunjukkan pada tabel 2.2. gambar 2.4 merupakan fungsi keanggotaan mutu kerja. sedangkan persamaan 5, 6, dan 7 merupakan hasil dari pembentukan fungsi keanggotaan mutu kerja. tabel 2.2 himpunan fuzzy untuk variabel mutu kerja variabel domain himpunan fuzzy muu kerja 10-55 kurang 60-75 cukup 80-100 baik gambar 2.4 fungsi keanggotaan mutu kerja [ ]{ ............... 5 [ ] { ( ) ( ) ( ) ( ) ........... 6 [ ]{ ( ) ............ 7 jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 20 c. varriabel disiplin kerja pada variabel disiplin kerja dibagi menjadi 3 himpunan fuzzy, yaitu : kurang, cukup dan baik. tabel keanggotaan disiplin kerja ditunjukkan pada tabel 2.3. gambar 2.5 merupakan fungsi keanggotaan disiplin kerja. sedangkan persamaan 8, 9, dan 10 merupakan hasil dari pembentukan fungsi keanggotaan disiplin kerja. tabel 2.3 himpunan fuzzy untuk variabel disiplin kerja variabel domain himpunan fuzzy disiplin kerja 10-55 kurang 60-75 cukup 80-100 baik gambar 2.5 fungsi keanggotaan disiplin kerja [ ]{ ............... 8 [ ] { ( ) ( ) ( ) ( ) ……………9 [ ]{ ( ) ................. 10 d. variabel tanggungjawab pada variabel tanggungjawab dibagi menjadi 2 himpunan fuzzy, yaitu : tidak tetap dan tetap. tabel keanggotaan tanggungjawab ditunjukkan pada tabel 2.4. gambar 2.6 merupakan fungsi keanggotaan tanggungjawab. sedangkan persamaan 11, 12 dan 13 merupakan hasil dari pembentukan fungsi keanggotaan tanggungjawab. tabel 2.4 himpunan fuzzy untuk variabel tanggungjawab variabel domain himpunan fuzzy tanggung jawab 10-55 kurang 60-75 cukup 80-100 baik gambar 2.6 fungsi keanggotaan tanggungjawab [ ]{ ............ 11 [ ] { ( ) ( ) ( ) ( ) ............. 12 [ ]{ ( ) ................. 13 e. variabel absensi pada variabel absensi dibagi menjadi 2 himpunan fuzzy, yaitu : kurang, cukup dan baik. tabel keanggotaan absensi ditunjukkan pada tabel 2.5. gambar 2.7 merupakan fungsi keanggotaan absensi. sedangkan persamaan 14, 15 dan 16 merupakan hasil dari pembentukan fungsi keanggotaan absensi. tabel 2.5 himpunan fuzzy untuk variabel absensi variabel domain himpunan fuzzy absensi 20-60 kurang 70-80 cukup 85-100 baik (2) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 21 gambar 2.7 fungsi keanggotaan absensi [ ]{ ............. 14 [ ] { ( ) ( ) ( ) ( ) .............. 15 [ ]{ ( ) .................. 16 f. variabel konduite pada variabel konduite dibagi menjadi 3 himpunan fuzzy, yaitu : rendah, sedang dan tinggi. tabel keanggotaan konduite ditunjukkan pada tabel 2.6. gambar 2.8 merupakan fungsi keanggotaan konduite. sedangkan persamaan 17, 18 dan 19 merupakan hasil dari pembentukan fungsi keanggotaan konduite. tabel 2.6 himpunan fuzzy untuk variabel konduite variabel domain himpunan fuzzy konduite 20-60 kurang 70-80 cukup 85-100 baik gambar 2.8 fungsi keanggotaan konduite [ ]{ ............ 17 [ ] { ( ) ( ) ( ) ( ) ............. 18 [ ]{ ( ) ................. 19 2) variabel output output dari sistem ini adalah rekomendasi penetapan tunjangan. nilai dari variabel rekomendasi penetapan tunjangan ini adalah tidak layak dan layak. tabel keanggotaan dapat dilihat pada tabel 2.8. gambar 2.10 merupakan fungsi keanggotaan variabel rekomendasi penetapan tunjangan sedangkan persamaan 1 dan 2 merupakan hasil dari pembentukan fungsi keanggotaan rekomendasi penetapan tunjangan. tabel 2.7 himpunan fuzzy untuk rekomendasi penetapan tunjangan variabel domain himpunan fuzzy keputusan 1 – 70 tidak layak >70 100 layak gambar 2.9 fungsi keanggotaan rekomendasi penetapan tunjangan [ ]{ ( ) ....................... 20 [ ]{ ( ) ................... .21 3. hasil dan pembahasan a. hasil penelitian ini menjelaskan bagaimana proses pemberian rekomendasi penetapan tunjangan prestasi yang akan diberikan oleh sistem pendukung keputusan ini melalui aplikasi. terdapat beberapa proses yang dilakukan pada jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 22 sistem ini yaitu fuzzifikasi, sistem inferensi dan defuzzifikasi. 1) fuzzifikasi fuzzifikasi yaitu proses pengkonversian input-input (yang berupa prestasi kerja, mutu kerja, disiplin kerja, tanggungjawab, absensi dan konduite) yang bersifat tegas (crips) ke dalam bentuk (fuzzy) variabel linguistik menggunakan fungsi keanggotaan tertentu[7]. pada proses ini terdiri dari variabel prestasi kerja, mutu kerja, displin kerja, tanggungjawab, absensi dan konduite yang diambil dari data sampel penilaian. prestasi kerja (pk) = 76 mutu kerja (mk) = 80 disiplin kerja (dk) = 78 tanggungjawab (tk) = 85 absensi (ak) = 75 konduite (kk) = 77 dari inputan data (crips) tersebut, kemudian dikelompokkan menjadi bentuk fuzzy variabel linguistik dengan fungsi tertentu pada proses fuzzifikasi yang akan ditunjukkan sebagai berikut. gambar 3.1 proses fuzzifikasi dari inputan data (crips) tersebut, kemudian dikelompokkan menjadi bentuk fuzzy variabel linguistik dengan fungsi tertentu yang akan derajat keanggotaan 0,8 pada variabel linguistik cukup dan 0,2 pada variabel linguistik baik, derajat keanggotaan 1 pada variabel linguistik baik, derajat keanggotaan 1 pada variabel linguistik baik, derajat keanggotaan 0.4 pada variabel linguistik cukup dan 0.6 pada variabel linguistik baik, derajat keanggotaan 1 pada variabel linguistik cukup dan 1 pada variabel linguistik cukup. 2) sistem inferensi sistem inferensi proses pengkonversian input-fuzzy (prestasi kerja, mutu kerka, disiplin kerja, tanggungjawab, absensi dan konduite) menggunakan aturan-aturan "ifthen" menjadi output-fuzzy (rekomendasi penetapan tunjangan prestasi)[8]. dalam proses inferensi ditentukan variabel output yang akan dijadikan untuk rekomendasi penetapan tunjangan prestasi. aturan fuzzy yang diperoleh dari aturan if prestasi kerja and mutu kerja and disiplin kerja and tanggungjawab and absensi and konduite then rekomendasi. tabel 3.1 aturan fuzzy total aturan fuzzy = 3 x 3 x 3 x 3 x 3 x 3 = 729 id_rule presrasi kerja mutu kerja disiplin tanggung jawab absensi konduite rekomendasi r-001 kurang kurang kurang kurang kurang kurang tidak layak r-002 kurang kurang kurang kurang kurang cukup tidak layak r-003 kurang kurang kurang kurang kurang baik tidak layak r-004 kurang kurang kurang kurang cukup kurang tidak layak r-005 kurang kurang kurang kurang cukup cukup tidak layak r-006 kurang kurang kurang kurang cukup baik tidak layak r-007 kurang kurang kurang kurang baik kurang tidak layak r-008 kurang kurang kurang kurang baik cukup tidak layak r-009 kurang kurang kurang kurang baik baik tidak layak jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 23 id_rule presrasi kerja mutu kerja disiplin tanggung jawab absensi konduite rekomendasi r-010 kurang kurang kurang cukup kurang kurang tidak layak .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... r-728 baik baik baik baik baik cukup layak r-729 baik baik baik baik baik baik layak dari aturan fuzzy tersebut kemudian dicocokan dengan hasil yang telah diperoleh dari proses fuzzifikasi, sehingga dari 729 aturan didapatkan 4 aturan yang sesuai fuzifikasi, hasil dari aturan yang didapat ditunjukkan pada proses inferensi seperti pada gambar 3.2. tabel 3.2 aturan yang sesuai fuzifikasi setelah mendapatkan aturan yang sesuai , maka langkah selanutnya adalah mengambil derajat keanggotaan minimum (alpha) dan nilai z dari nilai linguistik yang ada dari setiap aturan, seperti ditunjukkan pada gambar 3.3 tabel 3.3 nilai alpha dan z masing – masing variabel 3) defuzzifikasi langkah terakhir dalam proses ini adalah defuzzifikasi. defuzzifikasi yaitu proses pengkonversian output-fuzzy (rekomendasi penetapan tunjangan prestasi) dari sistem inferensi ke dalam bentuk tegas (crips) menggunakan fungsi keanggotaan menjadi sebuah nilai[9]. metode yang digunakan dalam defuzzifikasi ini adalah metode average. berikut adalah perhitungan defuzzyfikasi dengan metode average jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 24 (( ) ( ) ( ) ( )) ( ) z = (32,8 + 52,8 + 15,2 + 15,2 ) / 1,4 = 82,86 jadi nilai akhir setelah defuzzyfikasi adalah 82,86 dengan rekomendasi layak. untuk menentukan nilai tunjangan adalah dengan mengalikan hasil defuzzyfikasi dengan koefisien rp.100. hasi tunjangan yang didapat adalah rp.8,286 / hari. selanjutnya untuk mengetahui presentase layak, nilai 82,86 akan dimasukkan kedalam fungsi keanggotaan output layak, yaitu : ( ) ( ) ( ) jadi, rekomendasi keputusan kreditnya adalah layak dengan presentase sebesar 42.87%. gambar 3.2 hasil proses defuzzifikasi 2. pembahasan pembahasan pada penelitian ini akan membahas hasil perbandingan rekomendasi penetapan tunjangan prestasi data analisia evaluasi kinerja yang dilakukan oleh pihak perusahaan dengan rekomendasi yang diberikan oleh aplikasi sistem pendukung keputusan. data evaluasi kinerja karyawan untuk penetapan tunjangan prestasi sebanyak 10 data. tujuan dari perbandingan ini adalah untuk mengetahui tingkat akurasi uji reliabilitas dalam menentukan rekomendasi penetapan tunjangan prestasi yang ditunjukkan pada tabel dibawah ini. tabel 3.5 hasil perbandingan keputusan jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 25 berdasarkan tabel diatas, maka tingkat akurasi sistem dengan uji reliabilitas bahwa tingkat akurasi aplikasi sebesar 100%, tingkat akuransi output rekomendasi sistem sebesar 100%, sedangkat tingkat akuransi output tunjangan sebesar 94.71%. nilai tingkat akuransi output diperoleh dari pengurangan 100% dengan penjumlahan persentase error tsukamoto dibagi dengan banyaknya jumlah data sample yaitu sebesar 5.29%. 3. simpulan berdasarkan hasil penelitian dan perancangan sistem pendukung keputusan penetapan tunjangan prestasi dengan menggunakan metode fuzzy tsukamoto, dapat disimpulkan bahwa : 1. aplikasi telah berhasil dibangun secara terintegrasi berdasarkan perancangan sistem pendukung keputusan penetapan tunjangan prestasi dengan pendekatan metode fuzzy tsukamoto. 2. sistem pendukung keputusan penerimaan penetapan tunjangan prestasi telah mampu memberikan rekomendasi kelayakan karyawan dalam penerimaan tunjangan prestasi dengan tepat menggunakan metode fuzzy tsukamoto pada pt.boxtime indonesia dengan tingkat akurasi 100%. sistem pendukung keputusan penetapan tunjangan prestasi dengan metode fuzzy tsukamoto studi kasus pt.boxtime indonesia ini masih dapat dikembangkan lebih lanjut untuk mencapai tahap yang lebih tinggi dan kinerja sistem yang lebih baik. berikut adalah beberapa saran untuk pengembangan lebih lanjut : 1. diharapkan agar tampilan lebih menarik. 2. diharapkan dapat dikembangkan menjadi sistem yang bebasis clientserver. 3. diharapkan agar sistem dapat terintegrasi dengan sistem presensi dan sistem penggajian karyawan. daftar pustaka [1] laudon, kenneth c dan laudon, jane p. 2004. penterjemah erwin philippus, management information systems, managing the digital firm, eighth edition, yogyakarta : andi [2] laudon, kenneth c dan laudon, jane p. 2004. penterjemah erwin philippus, management information systems, managing the digital firm, eighth edition, yogyakarta : andi [3] kusumadewi, sri. 2011. artificial intelligence (teknik dan aplikasinya), yogyakarta : graha ilmu. [4] kusumadewi, sri. 2011. artificial intelligence (teknik dan aplikasinya), yogyakarta : graha ilmu. [5] kusumadewi, sri. 2011. artificial intelligence (teknik dan aplikasinya), yogyakarta : graha ilmu. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 40 pengklasteran lahan sawah di indonesia sebagai evaluasi ketersediaan produksi pangan menggunakan fuzzy c-means nur afifah1, dian c. rini2, ahmad lubab3 matematika, fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya e-mail: afifahaziec22@gmail.com1, diancrini@uinsby.ac.id2,ahmadlubab@uinsby.ac.id3 abstrak luas lahan sawah di indonesia semakin sempit dengan maraknya pembangunan perumahan dan gedung-gedung. hal ini berakibat pada ketersediaan produksi pangan yang semakin rendah dan harus mengimpor beras dari negara lain. dengan mengklasterkan lahan sawah dapat digunakan sebagai evaluasi untuk meningkatkan produksi pangan di indonesia sehingga kegiatan impor beras dapat terminimalisi. metode yang digunakan untuk mengelompokan lahan sawah adalah metode fuzzy c-means. implementasi program pada matlab dengan data training dan data testing. pada program fuzzy c-means tersebut menghasilkan tiga kelompok/cluster data, yaitu luas lahan sawah luas, sedang, dan sempit. hasil pengklusteran, wilayah yang paling berpotensi dalam produksi pangan dari lahan sawah adalah jawa timur, jawa tengah dan jawa barat. kata kunci: lahan sawah, evaluasi, fuzzy c-means abstract the number of rice field in indonesia is decreasing due to development of residential areas and buildings. consequently, it reduces foodstuff availability and government should import it from other. increasing food production and minimizing imported food can be started by clustering fields as an evaluation. this clustering is approached by fuzzy c-means. training and testing data are implemented on matlab and yield three categories, wide, medium and narrow field. moreover, the most potential field is east java, central java, and west java. keywords: field, evaluation, fuzzy c-means 1. pendahuluan dahulu pada masa orde baru kepemimpinan soeharto, indonesia mendapat sebutan sebagai lumbung padi, mengubah statusnya dan mencapai swasembada beras pada tahun 1980-an [1]. selain itu, indonesia menyandang sebagai negara produsen beras terbesar ke tiga di dunia [2] dalam kegiatan ekspor beras. hal ini dikarenakan luas lahan sawah di indonesia masih sangat luas untuk produksi tanaman padi. namun semakin bertambahnya tahun jumlah produksi beras menurun. salah satu penyebabnya adalah perkembangan luas lahan sawah dari tahun ketahun yang semakin menurun. data pada badan pusat statistik (2001), luas lahan sawah indonesia pada tahun 1993 + 8.500.000 ha, selanjutnya pada tahun 2000 (7 tahun) telah menyusut serius hingga tinggal 7.790.000 ha atau susutnya lahan 710.000 ha atau setiap tahunnya tanah sawah di indonesia menyusut 59,167 ha[3]. penurunan ini disebabkan oleh pengalihan fungsi dari lahan sawah untuk pembangunan pemukiman dan industri. sehingga penurunan tersebut berdampak pada berkurangnya ketersediaan produksi pangan. dan untuk menanggulangi ketersediaan tersebut, pemerintah menggencar impor beras jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 41 untuk memenuhi kebutuhan pangan dalam negeri. pengklasteran lahan sawah pada setiap provinsi dapat digunakan sebagai bahan evaluasi dalam meningkatkan ketersediaan produksi pangan pada suatu daerah dengan mengetahui luas lahan sawah yang ada di indonesia. dengan diimbangi teknologi, hal tersebut dapat minimalisasi kegiatan impor beras dalam pemenuhan kebutuhan pangan dalam negeri. fuzzy c-means adalah metode yang banyak digunakan untuk menyelesaikan permasalahan yang berhubungan dengan klaster/klasifikasi. diantaranya adalah digunakan untuk mengklasterkan varietas padi[4], diagnose penyakit jantung[5], analisa kepercayaan data berdasarkan metode klasifikasi[6]. sehingga untuk menyelesaikan permasalahan klaster lahan sawah pada penelitian ini dapat digunakan metode fuzzy cmeanss. metode fuzzy c-meansadalah suatu teknik pengclusteran data yang mana keberadaan tiap-tiap titik data dalam suatu cluster ditentukan oleh derajat keanggotaan[7]. sehingga dapat ditentukan cluster lahan sawah. 2. tinjauan pustaka teori himpunan fuzzy akan memberikan jawaban terhadap suatu permasalahan yang mengandung ketidakpastian[8]. salah satu bab dalam fuzzy adalah fuzzy clustering, fuzzy ini digunakan untuk mengklusterkan data. dalam clustering ini dibagi menjadi empat metode, yaitu metode fuzzy subtractive clustering, mountain, k-means dan c-means. fuzzy c-means (fcm) adalah suatu teknik pengklasteran data yang mana keberadaan tiaptiap data dalan suatu cluster ditentukan oleh nilai keanggotaan[9]. pada kondisi awal, pusat cluster masih belum akurat sehingga dibutuhkan perbaikan pusat cluster secara berulang hingga berada pada titik yang tepat. setiap data akan memiliki derajat keanggotaan untuk setiap clusternya. algoritma dari fuzzy c-means adalah sebagai berikut[10] : 1. input data yang akan di cluster x, berupa matriks berukuran n x m (n= jumlah sampel data, m= atribut setiap data). = data ke-i(i=1,2,..,n) , atribut kej(j=1,2,..,m) 2. tentukan :  jumlah cluster (c)  pangkat (w)  maksimum itarasi  error terkecil yang diharapkan  fungsi objektif awal (p0=0)  iterasi awal (t=1) 3. bangkitkan bilangan random sebagai elemen matriks partisi awal u. hitung umlah setiap kolom : 1 kemudian hitung: 2 4. hitung pusat cluster ke-k: vkj. dengan k=1,2,…,c dan j=1,2,..,m ∑ ∗ ∑ 3 5. hitung fungsi objektif pada iterasi ke-t, pt (yan, 1994): 4 6. hitung perubahan matriks partisi : ∑ ∑ ∑ 5 7. cek kondisi berhenti :  jika t>max iter maka berhenti  jika tidak, t=t+1, ulangi langkah ke-4 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 42 3. metode penelitian 3.1. studi literatur lahan yang diambil sebagai obyek penelitian adalah lahan sawah dari seluruh provinsi di dindonesia. data luas lahan sawah ini didapatkan dari badan pusat statistik (bps). data yang digunakan untuk penelitian ini adalah data luas lahan sawah dari tahun 2004 sampai 2013. 3.2. pengolahan data dari data lahan sawah tiap provinsi yang diperoleh sebanyak 34 data akan diklasterkan menjadi 3 tingkat luasan, yaitu tinggi, sedang, dan rendah. proses pengolahan data menjadi beberapa tahap. berikut adalah tahapan alur pengolahan data seperti pada gambar 2 berikut. pengklasteran ini dilakukan menggunakan algoritma fuzzy c-means dan aplikasi matlab dengan fungsi c-means. untuk perhitungan cluster menggunakan penentuan jumlah kluster 3, yaitu sempit, sedang, dan luas dengan pangkat berbobot 2 dan maksimal iterasi 100. setelah penentuan tersebut, dilakukan perhitungan matriks random dengan persamaan (1) yang dilanjutkan dengan menghitung fungsi objektif hingga algoritma selesai. namun pada implementasi program, data dirubah dalam bentuk matriks dan disimpan dengan format “.dat”. kemudian untuk mencari cluster fungsi yang digunakan pada program adalah : x=load('data.dat'); [center,u,objfcn]=fcm(x,3) fungsi center untuk mencari pusat cluster pada matriks random u dan kemudian dicari fungsi objektif. dalam pengelolaan data dalam matlab ada 2 macam data yang digunakan yaitu data gambar 1 sampel data studi literatur pengumpulan data perancangan metode fuzzy c-meanss implementasi program uji coba evaluasi gambar 2. alur olah data gambar 3. diagram alur proses clustering ya tidak jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 43 input dan data output. untuk mendapatkan data output dilakukan cluster terlebih dahulu sehingga didapatkan derajat keanggotaan setiap clusternya. data hasil cluster tersebut akan menjadi data output. dari kedua data tersebut digunakan untuk menghitung tingkat akurasi atau error dengan fungsi rmse. dari data input dan output juga diambil data sebagai training dan testing untuk pengujian. sehingga dapat diketahui error keseluruhan. data hasil clustering dengan error terkecil akan digunakan sebagai evaluasi dalam meningkatkan ketersediaan produksi pangan di indonesia. 4. hasil dan pembahasan implementasi program untuk penelitian ini terbagi menjadi dua, yaitu proses clustering dan proses pengujian. dalam proses clustering menggunakan data awal dan dihasilkan tiga cluster dengan derajat keanggotaannya. warna hijau menunjukkan cluster 1 yaitu sempit, warna merah menunjukkan cluster 2 yaitu sedang, dan warna kuning menunjukkan cluster 3 yaitu luas. proses clustering menghasilkan data output. dari data tersebut di ambil data training dan data testing yang kemudian dari kedua data tersebut digunakan untuk menghitung error. nilai pada cluster di atas tidak akan sama jika dilakukan cluster ulang. hal ini disebabkan oleh nilai matriks partisi u awal yang dibangkitkan secara random. namun perubahan dari nilai-nilai tersebut tidak terlalu signifikan karena tidak mempengaruhi keanggotaan cluster. hasil cluster terlihat pada gambar 4 dengan perbedaan warna untuk setiap clusternya. cluster yang berada paling atas merupakan pusat cluster terbesar yaitu lahan yang memiliki wilayah sawah yang luas. begitu pula untuk cluster yang berada di bawah merupakan wilayah dengan luas lahan sawah sempit. dari proses clustering dihasilkan data untuk tiap cluster dapat dilihat pada tabel 1. cluster data ke provinsi 1 3 sumatra barat 4 sumatra utara 5 jambi 7 bengkulu 9 kep. bangka belitung 10 kep. riau 11 dki jakarta 14 di yogyakarta 16 banten 17 bali 18 nusa tenggara barat 19 nusa tenggara timur 21 kalimantan tengah 23 kalimantan timur 24 kalimantan utara 25 sulawesi utara 26 sulawesi tengah 28 sulawesi tenggara 29 gorontalo 30 sulawesi barat 31 maluku 32 maluku utara gambar 4. hasil clustering tabel 1. data hasil clustering jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 44 33 papua barat 34 papua 2 1 aceh 2 sumatera utara 6 sumatera selatan 8 lampung 20 kalimantan barat 22 kalimantan selatan 27 sulawesi selatan 3 12 jawa barat 13 jawa tengah 15 jawa timur proses pengujian data training dan data testing dilakukan dengan beberapa kali percobaan untuk menghasilkan nilai dengan tingkat error terkecil. sehingga dari nilai ini bisa di terima untuk keakurasian data. dari data yang di testing dapat diketahui nilai error untuk setiap datanya. dan untuk mengetahui tingkat keakuratan digunakan fungsi rmse (root mean square error), didapatkan nilai error keseluruhan dari 5 percobaan seperti pada tabel 2. no data training data testing hasil error 1 24 10 2.72x10‐15 2 23 11 1.40x10‐14 3 27 7 7.44x10‐15 4 28 6 2.60x10‐14 5 25 9 1.93x10‐14 dari hasil percobaan diatas didapatkan nilai error terkecil yaitu pengujian yang dilakukan dengan 24 data training dan 10 data testing yang menghasilkan nilai error 0.00000000000000272. gambar 5. data training gambar 6. data testing gambar 7. checking error tabel 2. data hasil percobaan training dan testing jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 45 5. kesimpulan metode fuzzy c-means (fcm) dapat diimplementasikan dalam pengklasteran lahan sawah di indonesia. cluster yang dihasilkan dari proses clustering ada 3, yaitu lahan sawah sempit, sedang, dan luas. pengklusteran ini menggunakan 34 data. dari proses clustering dihasilkan data output sehingga dapat dilakukan proses training dan testing. dari 34 data input dan output dengan beberapa percobaan training dan testing didapatkan percobaan dengan nilai error terkecil yaitu 24 data training dan 10 data testing serta menghasilkan nilai error yang sangat kecil yaitu 0.00000000000000272. sehingga dari pengklasteran ini dapat menjadi sebagai bahan untuk evaluasi untuk pemerintah dalam meningkatkan ketersediaan produksi pangan sebagai minimasi impor beras, khususnya pemaksimalan pada lahan sawah yang luas yaitu daerah jawa, yaitu jawa barat, jawa tengah dan jawa timur. berdasarkan luas lahan, pada daerah lahan dengan luas sedang, pemerintah dan masyarakat harus bekerja sama menekan faktor yang membuat semakin menyempitnya lahan seperti maraknya pembangunan perumahan, pembangunan mall dan gedung bertingkat dan seharusnya pemerintah dan masyarakat bekerjasama dalam menambahkan dan menggunakan metode-metode baru dalam peningkatan produksi pertanian. referensi [1] khairunnisa. kompas, 12 april 2011. diakses pada 20 juni 2016. http://kompas.com [2] sawit, husein. 2006. indonesia dalam tatanan perubahan perdagangan beras dunia. jurnal. bogor.pusat analisis social ekonomi dan kebijakan pertanian [3] priyono.2011. alih fungsi lahan pertanian merupakan suatu kebutuhan atau tantangan. jurnal. bengkulu [4] nurjanah. farmadi, andi. 2014. implementasi metode fuzzy c-means pada sistem clustering data varietas padi. jurnal isjd [5] hammouda, khaled. a comparative studi of data clustering techniques. journal universitas of waterloo. ontario. canada [6] asyali, musa. alci, musa. reliability analysis of microarray data using fuzzy c-meansand normal mixture modeling based classification methods. jurnal bioinformatics vol.21 no.5 2005, pages 644-649 [7] kusumadewi, sri. purnomo, hadi. 2010. aplikasi logika fuzzy untuk pendukung keputusan eds.2.yogyakarta. graha ilmu [8] kusumadewi, sri. purnomo, hadi. 2010. aplikasi logika fuzzy untuk pendukung keputusan eds.2.yogyakarta. graha ilmu [9] kusumadewi, sri hartati. 2006. neuro fuzzy: integrasi sistem fuzzy dan jaringan syaraf. yogyakarta. graha ilmu [10] kusumadewi, sri. purnomo, hadi. 2010. aplikasi logika fuzzy untuk pendukung keputusan eds.2.yogyakarta. graha ilmu paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 1 metode historis untuk perhitungan value at risk pada model generalized autoregressive conditional heteroscedacity in mean alfi reny kusumaningtyas1, abdul aziz2 1mahasiswa jurusan matematika, uin maulana malik ibrahim malang 2dosen jurusan matematika, uin maulana malik ibrahim malang email: alfireny@gmail.com1, abdulaziz_uinmlg@yahoo.com2 abstrak investasi merupakan suatu komitmen penempatan data pada suatu atau beberapa objek investasi dengan harapan akan mendapatkan keuntungan di masa mendatang. motif utama investasi adalah mencari keuntungan atau laba dalam jumlah tertentu, namun di balik sisi baik terdapat satu sisi yang dapat merugikan atau yang disebut dengan risiko, untuk itu dibutuhkan suatu pengukuran risiko dimana metode value at risk (var) sangat populer digunakan secara luas oleh industri keuangan di seluruh dunia. tiga metode utama pada perhitungan var yaitu metode historis, metode parametrik dan metode monte carlo. sehingga, dipilih perhitungan var model garch-m dengan metode simulasi historis pada penutupan saham bank mandiri tbk tahun 2005-2010. penelitian ini bertujuan untuk mengetahui perhitungan var model garch-m melalui metode historis dan implementasi model garch-m pada perhitungan var melalui simulasi pada penutupan saham bank mandiri tbk. pendekatan metode historis merupakan model perhitungan nilai var yang ditentukan oleh nilai masa lalu (historis) atau return yang dihasilkan dengan melakukan simulasi (pengulangan) sebanyak data yang digunakan. langkah-langkah yang dilakukan yaitu menjelaskan metode historis pada estimasi var model garch-m dengan distribusi normal, kemudian mengaplikasikan model garch-m pada kasus kerugian yang diperoleh investor setelah menginvestasikan dana dengan bantuan software minitab, e-views dan matlab. kata kunci: var, metode historis, garch-m abstract investment is a commitment of the placement of the data on an object or a few investments with expectations will benefit in the future. the main motive is to seek investment gain or profit in a certain amount, but behind the good side there is one side that can harm or the risk of, for it required a measurement of risk where methods of value at risk (var) is very popular is widely used by the financial industry worldwide. three main method on calculation of var historical method, parametric method and monte carlo method. so, the selected calculation of var garch-m model with historical simulation method on bank mandiri tbk closing stock in 2005-2010. this research aims to know the calculation of var model garch-m through the historical method and implementation model garch-m on the computation of var via simulation on closing stock bank mandiri tbk. historical method approach is a model calculation of var is determined by the value of the past (historical) or return generated by simulation (repetition) of data used. the measures undertaken that explains the historical simulation method var models in the estimation of garch-m with a normal distribution, then apply garch-m in case of loss obtained by investors after investing with the help of minitab software, e-views software and matlab software. keywords: var, historical method, garch-m mailto:alfireny@gmail.com mailto:abdulaziz_uinmlg@yahoo.com jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 1 1. pendahuluan investasi merupakan suatu komitmen penempatan data pada satu atau beberapa objek investasi dengan harapan akan mendapatkan keuntungan di masa mendatang. keuntungan merupakan salah satu sisi yang baik dari investasi, namun di balik kebaikan tersebut terdapat risiko. risiko didefinisikan sebagai kondisi yang di dalamnya mengandung eksposur yang mungkin merugikan (gallati, 2003). menurut (mcmanus, j, 2004) memiliki pendekatan lain dengan membandingkan kesimpulan yang telah dilakukan oleh weigers (1998) dan gultch (1994) yaitu walaupun terdapat perbedaan dalam konteks, apa yang didefinisikan memiliki kesamaan yaitu ketidakpastian, kegagalan, dan kemalangan yang dapat memicu malapetaka dan kerugian. menurut (ferdiansyah, 2006) pengukuran risiko dengan metode value at risk (var) saat ini sangat populer digunakan secara luas oleh industri keuangan di seluruh dunia. untuk menghitung var ada tiga metode utama yaitu metode parametrik (disebut juga varianskovarians), metode monte carlo, dan metode historis. metode historis merupakan metode yang mudah diimplementasikan jika data historis pada faktor risiko telah dikumpulkan secara internal untuk nilai pasar harian dan metode ini menyederhanakan perhitungan dalam kasus portofolio yang mempunyai asset yang banyak dan periode sempit. jika diketahui atau memiliki database nilai historis masa lalu yang semakin banyak, maka hasil perhitungan nilai var yang dihasilkan akan semakin baik. 2. kajian teori 2.1 model generalized autoregressive conditional heteroscedasticity in mean (garch-m) menurut (jorion, p, 2001) model garch(p, q)-m dapat didefinisikan sebagai mana: 𝑌𝑡 = 𝛽0 + 𝛽1𝑌𝑡−1 + 𝛼1𝜎𝑡 2 + 𝜀𝑡 (1) 𝜎𝑡 2 = 𝛼0 + 𝛼1𝜎𝑡−1 2 + ⋯ + 𝛼𝑝𝜎𝑡−𝑝 2 + 𝛽1𝜀𝑡−1 2 + ⋯ + 𝛽𝑞 𝜀𝑡−𝑞 2 (2) dengan, 𝑌𝑡 |𝐹𝑡−1~𝑁(0, ℎ), 𝑡 = 1, 2, … 𝑇 dimana 𝛽1dan 𝛼1 adalah konstan. perumusan dari model garch-m pada persamaan (1) menyatakan bahwa ada serial korelasi dalam deret return𝑌𝑡 . untuk model garch(p,q)-m pada data return yang tidak mengandung model arma di dalamnya maka untuk persamaan model mean-nya menjadi: 𝑌𝑡 = 𝐶 + 𝜀𝑡 (3) 2.2 value at risk var merupakan suatu metode pengukuran risiko secara statistik yang memperkirakan maksimum yang mungkin terjadi atas suatu portofolio pada tingkat kepercayaan (level of confidence) tertentu. var dapat didefinisikan sebagai estimasi kerugian maksimum yang akan didapat selama periode waktu tertentu (jorion, p, 2001). var biasanya ditulis dalam bentuk var(𝛼) atau 𝑉𝑎𝑅(𝛼, 𝑇) yang menandakan bahwa var bergantung pada nilai 𝛼 dan t (dowd, k, 2002). apabila data diasumsikan berdistribusi normal, 𝛼 −quantile dari 𝑁(𝜇, 𝜎2) adalah: (mcnail, aj, dkk, 1967) 𝑄𝑢𝑎𝑛𝑡𝑖𝑙𝑒(𝛼) = 𝜇 + φ−1(𝛼)𝜎 dengan: 𝜇 : rata-rata pada data φ−1(𝛼) : nilai z-tabel 𝜎 : nilai volatilitas atau standart deviasi data maka estimasi var(𝛼) adalah: (mcnail, aj, dkk, 1967) 𝑉𝑎𝑟(𝛼) = 𝑊0 × {𝜇 + φ −1(𝛼)𝜎} dengan 𝑊0 adalah dana investasi awal saham oleh investor. 2.3 metode historis teknik perhitungan var bisa menggunakan metode historis, metode analitis, dan metode monte-carlo. metode historis jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 2 adalah suatu metode yang menggunakan asumsi bahwa kondisi perubahan harga pasar pada hari ini sampai esok hari adalah sama dengan kondisi perubahan harga pasar pada masa lalu (bruce dan zhi, ling, 1996). rumus yang dapat digunakan dalam menghitung var adalah sebagai berikut (jorion, p, 2001): 𝑉𝑎𝑅 = 𝑊0𝑅 ∗√𝑡 (6) dimana: 𝑉𝑎𝑅 : potensi kerugian maksimal 𝑊0 : dana investasi awal asset 𝑅∗ : nilai kuantil ke-𝛼 dari distribusi return √𝑡 : horizon waktu kelebihan metode historis mencakup pula nilai-nilai return pada saat kondisi pasar yang sedang mengalami gangguan atau tidak normal, seperti sedang terjadi crash. kelemahan metode ini adalah bahwa untuk keperluan analisis dan pengambilan keputusan melalui perhitungan var membutuhkan data return historis dengan rentang waktu yang panjang, sehingga memiliki potensi tidak relevan lagi dengan kondisi pasar terkini (jorion, p, 2001). 3. metode penelitian 3.1 pendekatan penelitian pendekatan yang digunakan pada penelitian ini adalah pendekatan literatur dan kuantitatif. pendekatan literatur digunakan dalam menganalisis model garch-m, dan untuk menentukan estimasi parameter dari model garch-m dengan menggunakan estimasi parameter dengan metode maximum likelihood (ml). studi kasus digunakan untuk mengaji kerugian yang diperoleh investor setelah menginvestasikan dananya. 3.2 jenis dan sumber data pada penelitian ini jenis data yang digunakan adalah data sekunder, karena peneliti tidak mendapatkan data secara langsung dari observasi, tetapi mendapatkannya dari penelitian sebelumnya evi sufianti pada tahun 2011 yang melakukan pengambilan data secara langsung sebanyak 254 dengan rentang data mingguan mulai dari tanggal 02 mei 2005 sampai dengan 20 september 2010 pada alamat berikut, http://finance.yahoo.com/q/hp?s=bmri.jk+his torical+prices. 3.3 metode analisis 1. menjelaskan metode historis pada estimasi var model garch-m dengan distribusi normal. 2. mengaplikasikan model garch-m pada kasus kerugian yang diperoleh investor setelah menginvestasikan dana, dengan langkah-langkah: a. menguji normalitas data log return dengan bantuan software minitab 14. b. mengidentifikasi model. c. memodelkan garch-m. d. menguji model. e. mengaplikasikan model garch-m pada var data harga saham penutupan dari bank mandiri tbk dengan metode historis. 4. hasil dan pembahasan 4.1 metode historis pada estimasi value at risk model garch-m dengan distribusi normal pada bab sebelumnya telah diketahui bentuk model garch(p,q)-m seperti persamaan (1), persamaan (2), dan persamaan (4) yaitu: 𝑌𝑡 = 𝐶 + 𝛼1𝑌𝑡−1 + 𝛽1𝜎𝑡 2 + 𝜀𝑡 𝜎𝑡 2 = 𝛼0 + 𝛼1𝜎𝑡−1 2 + ⋯ + 𝛼𝑝 𝜎𝑡−𝑝 2 + 𝛽1𝜀𝑡−1 2 + ⋯ + 𝛽𝑞 𝜀𝑡−𝑞 2 𝑅∗ = 𝜇 + φ−1(𝛼)𝜎 pada persamaan (6) telah diketahui rumus metode historis, maka substitusi persamaan (1), (2) dan (4) ke dalam persamaan (6), yaitu: 𝑉𝑎𝑅 = 𝑊0𝑅 ∗ √𝑡 = 𝑊0 (𝜇 + φ −1(𝛼)𝜎)√𝑡 = 𝑊0(𝐶 + 𝛼1𝑌𝑡−1 + 𝛽1𝜎𝑡 2 + 𝜀𝑡 + φ−1(𝛼)√𝛼0 + 𝛼1𝜎𝑡−1 2 + ⋯ + 𝛼𝑝𝜎𝑡−𝑝 2 + 𝛽1𝜀𝑡−1 2 + ⋯ + 𝛽𝑞 𝜀𝑡−𝑞 2 ) √𝑡 ( 7 ) http://finance.yahoo.com/q/hp?s=bmri.jk+historical+prices http://finance.yahoo.com/q/hp?s=bmri.jk+historical+prices jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 3 untuk data yang tidak mengandung model arma seperti pada subbab 2, maka metode historisnya menjadi: 𝑉𝑎𝑅 = 𝑊0𝑅 ∗ √𝑡 = 𝑊0 (𝜇 + φ −1(𝛼)𝜎)√𝑡 = 𝑊0(𝐶𝜀𝑡 + φ−1(𝛼)√𝛼0 + 𝛼1𝜎𝑡−1 2 + ⋯ + 𝛼𝑝 𝜎𝑡−𝑝 2 + 𝛽1𝜀𝑡−1 2 + ⋯ + 𝛽𝑞 𝜀𝑡−𝑞 2 ) √𝑡 (8) 4.2. analisis data a. uji stasioneritas pada data harga saham penutup bank mandiri, tbk. pergerakan data mengalami kenaikan dan penurunan setiap minggunya sehingga perlu dilakukan transformasi ke dalam bentuk return pada data sehingga nantinya data tersebut dapat stasioner, setelah itu barulah diuji stasioneritasnya. data time series yang nonstasioner akan ditransformasikan menjadi data yang stasioner dengan cara diubah ke dalam bentuk logaritma natural. hasil transformasi data ditunjukkan oleh trend analysis plot pada gambar 1 berikut. gambar 1. trend analysis plot data returnsaham penutup bank mandiri, tbk.menggunakan software minitab 14 b. uji normalitas setelah dilakukan transformasi data return yang kemudian diuji stasioneritasnya, maka langkah selanjutnya yakni dilakukan uji kenormalan data return tersebut. gambar 2. uji normalitas data return saham penutup bank mandiri, tbk. menggunakan software minitab 14 pada gambar 2 dapat dilihat bahwa data menyebar di sekitar garis diagonal dan mengikuti arah garis diagonal. lalu, dari nilai kolmogorovsmirnov (ks) sebesar 0,055, dan nilai p-value sebesar 0,066, maka perbandingan nilai p-value dengan 𝛼 yaitu 0,066 > 0,05, dari kedua pernyataan tersebut seperti pada subbab 2 dapat diambil kesimpulan bahwa 𝐻0 diterima. jadi return saham penutup bank mandiri, tbk. berdistribusi normal. 4.3 identifikasi model a. identifikasi model arma dengan acf dan pacf gambar 4. plot acf data return saham penutup bank mandiri, tbk. dari gambar 4 dan gambar 5 menunjukkan bahwa tidak terdapat cuts off maupun dies down sehingga kurang sesuai jika menggunakan model ar, ma, maupun arma. sehingga digunakan model arch/garch-m, karena dari gambar 1 menunjukkan bahwa data index r e t u r n 2502252001751501251007550251 0,2 0,1 0,0 -0,1 -0,2 -0,3 a ccuracy measures ma pe 101,984 ma d 0,051 msd 0,004 variable a ctual fits trend analysis plot for return linear trend model yt = 0,000614923 + 0,0000376117*t jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 4 return tersebut memiliki nilai variansi yang stasioner. gambar 5. plot pacf data return saham penutup bank mandiri, tbk. b. identifikasi model arch/garch dengan acf dan pacf pengujian keberadaan efek arch/garch terhadap sisaan data return yang dimodelkan ke dalam model 𝑌𝑡 = 𝐶 + 𝜀𝑡 dengan menggunakan uji ljung box q untuk sisaan kuadrat pada data return harga saham penutupan bank mandiri tbk, dengan hipotesis yang digunakan untuk menguji keberadaan efek arch/garch pada 𝜀𝑡 2 adalah sebagai berikut: 𝐻0 : tidak terdapat proses arch/garch 𝐻1 : terdapat proses arch/garch tabel 1: acf pada sisaan kuadrat dengan bantuan eviews, minitab 14 dan microsoft excel autocorelation function: sisaan kuadrat lag acf t lbq 𝜒𝑘 2(𝛼) p 1 0,218 2,33 12,223 3,841 0,000 2 0,211 1,54 23,745 5,991 0,000 3 0,059 0,82 24,643 7,818 0,000 4 0,271 0,14 43,723 9,488 0,000 5 0,201 -0,25 54,281 11,070 0,000 6 0,061 -0,65 55,257 12,592 0,000 7 0,043 -0,68 55,741 14,067 0,000 8 0,008 -0,71 55,756 15,507 0,000 9 -0,012 -0,72 55,793 16,919 0,000 10 -0,074 -0,51 57,271 18,307 0,000 11 -0,043 -0,48 57,777 19,675 0,000 karena 𝐿𝐵𝑄 > 𝜒𝑘 2 dan 𝛼 > 𝑝 − 𝑣𝑎𝑙𝑢𝑒, maka menolak 𝐻0 yang berarti terdapat proses arch/garch pada 𝜀𝑡 2. c. identifikasi model garch-m menggunakan metode maximum likelihood pendugaan parameter model garch(1,1)-m dengan bantuan software eviews yang menggunakan metode ml diperoleh hasil sebagai berikut: gambar 7. hasil analisis garch(1,1) dari gambar 7, dapat diasumsikan bahwa volatilitas data log return saham penutup bank mandiri, tbk. mengikuti model garch-m. hal tersebut dapat dilihat dari nilai probabilitasnya yang lebih kecil dari tingkat signifikansi 𝛼 = 5%, sehingga diperoleh model garch(1,1)-m sebagai berikut: 𝑌𝑡 = 0,004879 + 𝜀𝑡 (9) 𝜎𝑡 2 = 0,001579 + 0,231448𝜎𝑡−1 2 + 0,415142𝜀𝑡−1 2 (10) d. uji kesesuaian model pada plot acf untuk data return harga saham, menunjukkan bahwa tidak terdapat autokorelasi yang berbeda nyata untuk sisaan model garch-m yang dibakukan, sehingga jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 5 dapat dikatakan model garch-m sesuai memodelkan data return. kesesuaian model garch-m ditunjukkan dengan uji ljung box-q untuk sisaan model garch-m yang dibakukan seperti pada tabel 2 berikut. tabel 2. hasil uji ljung box q untuk sisaan yang dibakukan data return dengan bantuan eviews3 dan minitab 14 dari hasil tabel di atas ditunjukan nilai statistik q lebih kecil dibandingkan 𝜒(𝑘) 2 (𝛼 = 0,05) serta nilai p-value yang lebih besar dari 𝛼 = 0,05, maka tidak terdapat hubungan antar sisaan yang dibakukan sehingga model garchm sesuai untuk data return harga saham penutup bank mandiri, tbk. gambar 8 menunjukkan hasil simulasi data return yang pertama sampai ke-245 dan didapat rata-rata value at risk saham penutupan bank mandiri tbk dengan bantuan software matlab adalah sebesar 1.695847318994527e+07. jadi dapat disimpulkan bahwa dengan tingkat kepercayaan 95% yang berarti peluang terjadinya kerugian adalah hanya 5% dengan kemungkinan dari dana yang telah diinvestasikan pada saham penutupan bank mandiri tbk adalah sebesar rp. 16.958.473,-. e. perhitungan var dengan metode historis untuk hasil perhitungan var dengan metode historis di dapat dengan menggunakan bantuan software matlab, sebagaimana di dapat hasil sesuai dengan grafik seperti pada gambar 8. gambar 8. hasil var metode historis dengan bantuan software matlab 5 kesimpulan dari penelitian yang dilakukan dapat disimpulkan sebagai berikut. a. persamaan metode historis hasil perpaduan model garch-m pada perhitungan var, seperti berikut: 𝑉𝑎𝑅 = 𝑊0𝑅 ∗√𝑡 = 𝑊0 (𝜇 + φ −1(𝛼)𝜎)√𝑡 = 𝑊0 (𝐶 + 𝜀𝑡 + φ−1(𝛼)√𝛼0 + 𝛼1𝜎𝑡−1 2 + ⋯ + 𝛼𝑝𝜎𝑡−𝑝 2 + 𝛽1𝜀𝑡 −1 2 + ⋯ + 𝛽𝑞 𝜀𝑡−𝑞 2 )√𝑡 b. hasil perhitungan risiko oleh var dari uang yang diinvestasikan investor sebesar rp. 150.000.000,00 ke bank mandiri, tbk. dengan bantuan software matlab adalah dengan tingkat kepercayaan 95% yang berarti peluang terjadinya kerugian hanya 5% dengan kemungkinan kerugian maksimum sebesar rp. 16.958.473,-. referensi [1] bruce dan zhi, ling. (1996). historical method analysis. west sessex: john wiley & sons inc. [2] dowd, k. (2002). an introduction to market risk measurement. west sessex: john wiley & sons inc. [3] ferdiansyah, t. (2006). refleksi dan strategi penerapan manajemen risiko perbankan indonesia. jakarta: pt. elex media komputindo. lag acf t lbq     2 0.05 k    p 1 -0.033503 -0.13 0.02 3.841 0.713 2 -0.504175 -1.88 4.76 5.991 0.716 3 -0.175186 -0.53 5.39 7.818 0.416 4 0.024039 0.07 5.40 9.488 0.575 5 0.303986 0.91 7.70 11.070 0.353 6 0.029143 0.08 7.73 12.592 0.540 7 -0.248784 -0.70 9.71 14.067 0.306 8 0.124849 0.34 10.29 15.507 0.306 9 0.048787 0.13 10.40 16.919 0.389 10 -0.111786 -0.30 11.09 18.307 0.440 11 0.021281 0.06 11.13 19.675 0.450 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 6 [4] gallati, r. (2003). risk management & capital adequacy. new york: mcgraw-hill inc. [5] mcmanus, j. (2004). risk management. uk: churcill livingstone. [6] mcnail, aj, dkk. (1967). quantitative risk management. princenton and oxford: princenton university press. [7] jorion, p. (2001). value at risk. new york: mcgraw-hill inc. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 8 analisa ketunggalan titik tetap pada pemetaan kontraktif di ruang metrik lengkap dengan memanfaatkan jarak-w malahayati program studi matematika fakultas sains dan teknologi uin sunan kalijaga yogyakarta e-mail: malahayati_01@yahoo.co.id abstrak penelitian ini mengkaji tentang sifat titik tetap pada jarak-w diruang metrik lengkap. di dalam penelitian ini diberikan pula suatu contoh penggunaan sifat titik tetap berdasarkan sifat yang telah dibahas. kata kunci: ruang metrik lengkap, jarak-w, titik tetap 1. pendahuluan mempelajari matematika yang sesuai dengan paradigma ulul albab, tidak cukup hanya berbekal kemampuan intelektual semata, tetapi perlu didukung secara bersamaan dengan kemampuan emosional dan spiritual. pola pikir deduktif dan logis dalam matematika juga bergantung pada kemampuan intuitif dan imajinatif serta mengembangkan pendekatan rasionalis, empiris, dan logis. sebagaimana dalam firman allah swt dalam surat al-imran ayat 191 berikut:                                                       artinya : “(yaitu) orang-orang yang mengingat allah sambil berdiri atau duduk atau dalam keadan berbaring dan mereka memikirkan tentang penciptaan langit dan bumi (seraya berkata): "ya tuhan kami, tiadalah engkau menciptakan ini dengan sia-sia, maha suci engkau, maka peliharalah kami dari siksa neraka” (qs. al-imran: 191). sesuai dengan tujuan pembelajaran matematika yang melatih cara berpikir secara sistematis, logis, analis, kritis, maka di dalam matematika terdapat satu bidang yang dalam mempelajarinya dituntut untuk berfikir secara analitis, yaitu bidang analisis. analisis merupakan cabang matematika yang berkembang dari kalkulus. topik yang dibahas dalam analisis diantaranya adalah teori titik tetap (fixed point theory). titik tetap mempunyai peranan yang penting dalam analisis fungsional. banyak masalah matematis yang dapat dipecahkan dengan menggunakan prinsip titik tetap. beberapa diantaranya adalah masalah persamaan linear, persamaan diferensial biasa, persamaan integral, dan persamaan diferensial parsial. eksistensi titik tetap (fixed point ) untuk suatu fungsi telah banyak dikaji jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 9 oleh para ahli sebagai salah satu metode menyelesaikan problem matematika. pada tahun 1922, sebuah karya yang terkenal dan dihargai dalam bidang teori titik tetap untuk pemetaan kontraktif pada ruang metrik lengkap, berhasil dibuktikan oleh banach yang kemudian disebut dengan teori titik tetap banach. hasil dari pembuktian tersebut telah menjadi aset penting untuk matematika terapan, karena aplikasi dari teori tersebut berperan besar pada berbagai cabang ilmu matematika yang meliputi persamaan diferensial, persamaan integral, dan bidang ilmu matematika lainnya, terutama yang melibatkan logika pemrograman dan teknik elektronik. teori titik tetap itu sendiri merupakan gabungan yang menarik dari analisis, topologi, dan geometri. selama 50 tahun terakhir teori titik tetap diakui sangat maju dan penting sebagai alat dalam studi fenomena nonlinier. secara khusus teknik titik tetap telah diterapkan pada berbagai bidang seperti: biologi, kimia, ekonomi, teori permainan, dan fisika teorema titik tetap banach telah menarik banyak peneliti untuk terlibat dalam mempelajari dan mengeksplorasi teorema tersebut untuk mendapatkan hasil yang baru dalam pemetaan kontraktif menggunakan berbagai kondisi. pada tahun 1996 w.takahashi memperkenalkan teori titik tetap di ruang metrik dengan terlebih dahulu mendefinisikan sebuah jarak, yang selanjutnya disebut w-distance (jarak-w). teori tersebut telah menarik banyak peneliti untuk terlibat dalam mempelajari dan mengeksplorasi lebih jauh lagi, diantaranya razani, dkk (2009) telah meneliti sifat titik tetap untuk pemetaan kontraktif pada ruang bertipe integral dengan memanfaatkan jarakw. selain itu lakzian,dkk (2009) telah meneliti sifat titik tetap dalam ruang metrik cone dengan memanfaatkan jarak-w. berdasarkan teori yang diperkenalkan oleh takahashi dan beberapa hasil penelitian terdahulu tersebut, menarik untuk diteliti lebih lanjut mengenai sifat titik tetap di ruang metrik lengkap dengan memanfaatkan jarakw pada suatu pemetaan kontraktif. 2. teori dasar pada bagian ini akan diberikan beberapa pengertian dasar dan sifat yang merupakan konsep awal untuk dipahami agar mudah mengikuti pembahasan selanjutnya. pengertian-pengertian dan sifatsifat yang disajikan diadopsi dari beberapa literatur yang disebutkan pada daftar pustaka. berikut ini, diberikan beberapa teori dasar diawali dengan membahas konsep fungsi semikontinu, yang sangat diperlukan dalam pembahasan pada bagian selanjutnya. fungsi – fungsi yang dibicarakan bernilai real dan didefinisikan pada , dengan e himpunan bagian dari ruang metrik. sebelumnya disepakati terlebih dahulu bahwa setiap pengambilan infimum dan supremum dari suatu himpunan pada bagian ini, himpunan yang dimaksud merupakan himpunan bagian dari ̅, dengan ̅ * +. dalam mendefinisikan fungsi semikontinu diperlukan konsep limit atas dan limit bawah, oleh karena itu pada sub bab ini dimulai dengan menjelaskan konsep limit atas dan limit bawah beserta sifat-sifatnya. definisi 2.1 diberikan fungsi f yang didefinisikan pada dan . 1) limit atas (upper limit) fungsi f ketika x mendekati ditulis dengan ( ) dan didefinisikan ( ) * ( ) + dengan ( ) * ( ) ( ) + 2) limit bawah (lower limit) fungsi f ketika x mendekati ditulis dengan ( ) dan didefinisikan ( ) * ( ) + , dengan ( ) * ( ) ( ) + jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 10 pada definisi diatas, nilai limitnya selalu ada dan dapat bernilai berhingga, , atau . selanjutnya diberikan definisi fungsi semikontinu. definisi 2.2 diberikan fungsi f yang didefinisikan pada dan 1) fungsi dikatakan semikontinu atas (upper semicontinuous) di apabila ( ) ( ). selanjutnya, fungsi dikatakan semikontinu atas pada e apabila fungsi f semikontinu atas disetiap 2) fungsi dikatakan semikontinu bawah (lower semicontinuous) di apabila ( ) ( ) selanjutnya, fungsi dikatakan semikontinu bawah pada e apabila fungsi f semikontinu bawah disetiap 3) fungsi yang semikontinu atas atau semikontinu bawah dinamakan fungsi semikontinu. selanjutnya akan diberikan definisi tentang jarak-w pada ruang metrik definisi 2.3 (mohanta 2011:134) diberikan ruang metrik ( ), fungsi  : 0,p x x   dikatakan jarak-w pada x apabila memenuhi: 1. ( ) ( ) ( ) untuk setiap ; 2. ( )  0, merupakan fungsi lower semicontinous (lsc), apabila untuk setiap dan di maka ( ) ( ); 3. untuk setiap terdapat sedemikian sehingga ( ) dan ( ) berlaku ( ) untuk mempermudah memahami lebih lanjut tentang jarak-w perhatikan contoh berikut. contoh 2.4 (razani, dkk 2009:114) diberikan { }⋃ * + dan merupakan barisan turun, jika fungsi , ) didefinisikan sebagai berikut: jika maka ( ) dan apabila maka ( ) , dengan d adalah metrik atas dan ( ) adalah ruang metrik, selanjutnya didefinisikan ( ) , untuk setiap , maka fungsi p adalah jarak-w. bukti : (i) ambil sebarang akan ditunjukan bahwa ( ) ( ) ( ) . karena maka berlaku ( ) ( ) ( ) sehingga terbukti bahwa apabila untuk sebarang x,y,z x berlaku ( ) ( ) ( ). (ii) ambil sebarang akan ditunjukan bahwa fungsi ( ) , ) merupakan fungsi lower semicontinous (lsc). fungsi ( ) , ) dikatakan lsc jika memenuhi ( ) ( ) . ambil sebarang . perhatikan bahwa : ( ) ( ( ) ( ) = ( ) diperoleh : ( ) sehingga terbukti bahwa : ( ) ( ) dengan demikian terbukti benar bahwa ( ) adalah fungsi lower semicontinous . (iii)ambil sebarang , dipilih δ = 2  , dengan ( ) dan ( ) , karena maka berlaku ( ) dan ( ) . sehingga diperoleh : ( ) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 11 jadi terbukti bahwa apabila terdapat sebarang ε > 0 dipilih δ = 2  dengan ( ) dan ( ) , berlaku ( ) . karena memenuhi kondisi jarak-w, sehingga terbukti bahwa adalah jarak-w pada himpunan x. selanjutnya akan diberikan dua himpunan yang akan digunakan untuk membuktikan lemma-lemma penunjang dalam membuktikan sifat titik tetap untuk jarak-w pada ruang metrik. berikut ini adalah himpunan-himpunannya. definisi 2.5 (razani 2009:114) diberikan sebuah himpunan yang berisikan fungsi kontinu kanan yang tidak turun katakan dan untuk setiap ( ) . berikut adalah himpunannya, * , ) , )+ selanjutnya diberikan sebuah himpunan yang berisikan fungsi kontinu kanan yang tidak turun katakan dan untuk setiap berlaku ( ) . berikut himpunannya, * , ) , )+ perhatikan contoh berikut, agar mempermudah dalam memahami himpunan di atas. contoh 2.6 (razani 2009:115): diberikan barisan non-negatif * + dan * + , dengan * + adalah barisan yang turun tegas, konvergen ke 0 dan untuk setiap dengan , didefinisikan fungsi, , ) , ) dengan definisi fungsi sebagai berikut : ( ) { maka adalah anggota . bukti: akan ditunjukan bahwa , artinya akan ditunjukan adalah barisan tidak turun, kontinu kanan dan untuk setiap ( ) . a. akan ditunjukan adalah barisan tidak turun, artinya akan ditunjukan bahwa apabila maka ( ) ( ). ambil sebarang dan dengan , ) dan untuk dan , menggunakan kontraposisinya maka akan ditunjukan bahwa apabila ( ) ( ) maka , perhatikan bahwa, berdasarkan definisi fungsi ( ) di atas, diperoleh ( ) ( ) karena maka sehingga diperoleh, maka jadi terbukti bahwa , sehingga bernilai sama dengan apabila maka ( ) ( ). b. fungsi kontinu kanan, sebab : 1. ( ) 2. ( ) = 3. ( ) ( ) sehingga terbukti bahwa kontinu kanan. c. akan ditunjukan bahwa ( ) untuk setiap ambil sebarang , 1. jika , dan maka ( ) 2. jika dan maka, ( ) sehingga terbukti bahwa ( ) untuk setiap . jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 12 berdasarkan dari pembuktian (a),(b),(c) maka benar bahwa . kemudian diberikan beberapa lemma yang akan digunakan untuk membuktikan sifat titik tetap pada jarak-w di ruang metrik. lemma 2.7 (razani 2009:115) jika maka ( )( ) untuk setiap . bukti: diketahui , ambil sebarang barisan pada yaitu ( ) . dengan demikian berlaku bahwa: ( ) fungsi kontinu kanan, tidak turun untuk setiap berlaku ( )( ) ( ) fungsi kontinu kanan, tidak turun untuk setiap berlaku ( )( ) ( ) fungsi kontinu kanan, tidak turun untuk setiap berlaku ( )( ) sehingga diperoleh ( ) . selanjutnya ambil sebarang , { ( )( )} adalah barisan bilangan non-negatif yang turun. andaikan ( )( ) sehingga terdapat , sehingga ( )( ) , karena maka kontinu kanan, oleh karena itu ( ) ( ) untuk . dengan demikian ( ) . ingat kembali bahwa { ( )( )} adalah barisan bilangan non-negatif yang turun maka: ( )( ) * ( )( ) + sehingga terbukti bahwa lemma 2.8 (razani 2009:115) jika , * + , ) dan ( ) maka ( ) . bukti : andaikan ( ) sehingga terdapat dan * + berlaku ≥ >0 , karena berarti ( ) untuk setiap , oleh karena itu diperoleh kn a ≥ ( )>0 sehingga berlaku juga kn a ≥ ( )>0 selanjutnya diperoleh ( ) , sehingga diperoleh ( ) . dan ini kontrakdiksi dengan yang diketahui bahwa ( ) , sehingga terbukti bahwa ( ) lemma 2.9 (razani 2009:115) diberikan ruang metrik ( ) dan adalah jarak-w pada , apabila * + adalah barisan di sedemikian sehingga ( ) ( ) maka , selanjutnya jika ( ) ( ) maka bukti : ambil sebarang , karena ( ) sehingga terdapat sedemikian hingga ( ) untuk setiap , karena ( ) sehingga terdapat sedemikian hingga ( ) untuk setiap . dipilih * +. sehingga ( ) dan ( ) untuk setiap , karena p adalah jarak-w maka berdasarkan definisi jarak-w bagian (3) sehingga diperoleh ( ) , karena berlaku untuk sebarang didapat ( ) dan ( )  . selanjutnya diketahui ( ) sehingga ( ) dan diketahui ( ) maka ( ) , oleh karena itu berdasarkan lemma yang telah dibuktikan di atas maka berlaku jadi terbukti bahwa apabila ( ) ( ) maka , selanjutnya jika ( ) ( ) maka . lemma 2.10 (razani 2009:115) diberikan ruang metrik (x,d) dan p adalah jarak-w pada x . diketahui * + adalah barisan di x dan jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 13 * + adalah barisan di , ), dengan * + konvergen ke 0 sedemikian sehingga apabila terdapat ( ) untuk setiap dengan maka * + adalah barisan cauchy. bukti : akan ditunjukan * + adalah barisan cauchy. berikut akan ditunjukan bahwa untuk setiap terdapat sehingga untuk setiap berlaku ( ) . diketahui bahwa ( ) , dan karena * + konvergen ke 0 maka untuk sebarang terdapat sehingga untuk setiap berlaku selanjutnya karena ( ) perhatikan bahwa : ( ) oleh karena itu diperoleh : ( ) . dengan demikian terbukti bahwa * + adalah barisan cauchy. 3. pembahasana pada bagian ini akan dibuktikan sifat ketunggalan titik tetap pada pemetaan kontraktif di ruang metrik lengkap dengan memanfaatkan jarak-w. setelah pengertian jarak-w pada ruang metrik diberikan, selanjutnya akan dibahas sifat titik tetap pada jarak-w di ruang metrik lengkap dan diakhiri dengan diberikan contoh. teorema 3.1.(razani, 2009:115) diberikan p adalah jarak-w pada ruang metrik lengkap ( ) dengan dan . diketahui adalah ( ) – pemetaan kontraksi pada x untuk setiap berlaku: ( ( ( ) ( ))) ( ( ( ))) ) maka mempunyai titik tetap tunggal pada x, selanjutnya adalah titik tetap untuk setiap . bukti : teorema ini akan menunjukan bahwa mempunyai titik tetap, selanjutnya akan ditunjukan ketunggalan titik tetap di . untuk menunjukan bahwa mempunyai titik tetap maka terlebih dahulu akan ditunjukan bahwa ( ) ambil sebarang (fix), dibentuk ( ) dengan . diambil sebarang barisan pada yaitu * ( )+ { ( ( ( )))} , oleh karena itu berdasarkan lemma 2.7 berlaku: ( ) ( ( ( ))) dengan kata lain ( ( ( ))) untuk sebarang , maka: ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) oleh karena itu diperoleh ( ) selanjutnya berdasarkan lemma 2.8 diperoleh ( ) menggunakan cara yang sama diperoleh, ( ) kemudian akan ditunjukan bahwa ( ) andaikan ( ) artinya terdapat sedemikian sehingga untuk setiap ada sehingga ( ) oleh karena itu dipilih , diambil * + , dan * + dengan sedemikian sehingga ( ) karena ( ) artinya terdapat untuk setiap maka: ( ) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 14 untuk dengan demikian maka . asumsikan bahwa adalah minimal index yang mengakibatkan ( ) misalkan * + sehingga ( ) perhatikan bahwa : ( ) ( ) ( ) ( ) jadi untuk maka ( ) . hal tersebut kontradisi dengan (3.8) sehingga benar bahwa ( ) . kemudian dipilih kembali dan sehingga untuk maka ( ) maka untuk terdapat * + sedemikian sehingga : ( ) , karena dan , sehingga ( ) ( ( )) ( ( ( ))) ( ( ( ))) ( ( ( ))) berdasarkan uraian di atas maka diperoleh ( ) sehingga kita dapatkan : ( ) terjadi kontradiksi. dengan demikian benar bahwa ( ) berdasarkan pembuktian diatas dan dari lemma 2.9 maka * + adalah barisan cauchy. karena adalah ruang metrik lengkap, maka terdapat sedemikian sehingga . selanjutnya akan ditunjukan adalah titik tetap pada s, karena * + adalah barisan cauchy untuk setiap terdapat dengan maka ( ) , diketahui juga bahwa untuk dan ( ) adalah lsc oleh karena itu diperoleh ( ) ( ) ( ) ( ) ( ) ( ) dipilih dan , ( ) maka ( ( )) ( ) ( ( )) ( ) untuk diperoleh ( ( )) tetapi perhatikan bahwa ( ( )) ( ) ( ( )) karena ( ( )) maka diperoleh ( ( )) ( ) ( ( )) oleh karenanya ( ( )) (3.13) berdasarkan lemma 2.8 maka ( ) . sehingga terbukti bahwa adalah titik tetap pada . setelah mengetahui bahwa adalah titik tetap pada . selanjutnya akan ditunjukan ketunggalan titik tetap pada . misalkan dan merupakan dua titik tetap pada , sehingga perhatikan bahwa jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 15 ( ( )) ( ( ( ) ( ))) ( ( ( ))) karena mengakibatkan ( ) sehingga diperoleh ( ) kemudian perhatikan bahwa : ( ) ( ( ) ( )) ( ) karena mengakibatkan ( ) sehingga di dapat, ( ) hal ini mengakibatkan . sehingga terbukti bahwa titik tetap di adalah tunggal. setelah mengetahui dan membuktikan sifat titik tetap pada jarak-w di ruang metrik lengkap, selanjutnya akan diberikan contoh untuk mempermudah memahaminya. contoh 3.2 diberikan ruang metrik ( ) dan yang telah didefinisikan pada contoh 2.4 dan dengan pemetaan ( ) , ( ) . misalkan , ) , ) adalah fungsi yang kontinu dan turun tegas. diberikan seperti pada contoh 2.6 dengan ( ), selanjutnya diasumsikan ketaksaman berikut ini ( ) ( ) ( ) ( ) dengan ( ) ( ) ( ) ( ) akan ditunjukan ketunggalan titik tetap pada . sebelum membuktikan ketunggalan titik tetap pada akan ditunjukkan bahwa adalah pemetaan kontraksi. disebut pemetaan kontraksi apabila memenuhi persamaan ( ( ( ) ( ))) ( ( ( ))) untuk setiap . berikut adalah pembuktiannya. menggunakan definisi pada contoh 2.6 diperoleh ( ( )) ( ) untuk ( ) oleh karena itu perhatikan bahwa: ( ( ) ( )) ( ( )) ( ) ( ) sehingga diperoleh ( ( ) ( )) ( ) ( ) maka ( ( ) ( )) ( ) dengan kata lain terbukti bahwa adalah pemetaan kontraksi. selanjutnya berdasarkan teorema di atas maka adalah pemetaan kontraksi sehingga terbukti bahwa mempunyai titik tetap yang tunggal. 4. penutup berdasarkan hasil pembahasan dapat disimpulkan bahwa setiap pemetaan yang memenuhi kondisi*), serta terdefinisi pada jarak-w di ruang metrik lengkap mempunyai titik tetap yang tunggal. pembuktian sifat ketunggalan tersebut memanfaatkan sifat fungsi kontraksi serta sifat kelengkapan pada ruang metriknya. ucapan terima kasih: terima kasih penulis sampaikan kepada lp2m uin sunan kalijaga yang telah membantu membiayai penelitian ini. referensi [1] a. branchiari. a fixed point theorem for mapping satisfying a general contractive condition of integral type, internasional journal 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[2] a. latif and w. a. albar. fixed points result in complete metric space. demonstratio mathematica, xli (2008). 145-150. [3] agarwal, ravi p., donald d reagen dan d.r. sahu. 2009. fixed point theory for lipschitzian type mappings with applications. usa.spinger. [4] khamsi, mohammad a., and krik, william a. 2001. an introduction to metric spaces and fixed point theory. new york: john wiley & sons, inc. [5] o. kada. t. suzuki and w. takashashi. nonconvex minimization theorems and fixed point theorems in complete metric spaces. math. japonico 44 (1996), 381591. [6] p. vijayaraju, b.e. rhoades and r. mohanraj. a fixed point theorems for pair of maps satisfying a general contractive condition of integral type, internasional journal of mathematics and mathematical sciences 15 (2005). 2359-2364. [7] razani dkk. a fixed point theorem for w-distance. applied sciens, (2009). vol.11 pp.114-117. [8] shirali, staish and vasudeva, harkrishan l. 2006. metric spaces. london: springer-verlag. [9] t. suzuki. meir-keeiler contractions of integral type are still meir-keeiler constractions. internasional journal of mathematics and mathematical sciences aticle id 39281 (2007). 1-6. [10] t. suzuki and w. takahashi, fixed point theorems and characterizations of metric completness. topol. methods nonlinear anal., 8 (1996), 371-382 [11] t. suzuki. several fixed point theorems in complete metric space. yokohama math. j., 44 (1997), 61-72. [12] tuwankotta, johan matheus. 2012. analisis real a: teori ukuran dan integral. bandung: itb jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 7 perbandingan pengklusteran data iris menggunakan metode k-means dan fuzzy cmeans fitria febrianti1, moh. hafiyusholeh2, ahmad hanif asyhar3 fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya e-mail: fitriafebrianti09@gmail.com1, hafiyusholeh@uinsby.ac.id2, hanif@uinsby.ac.id3 abstrak indonesia dengan kekayaan alam yang melimpah, tentu memiliki banyak tanaman yang tak terhitung banyaknya. untuk mengklaster tanaman menjadi beberapa kelompok yang berbeda dapat menggunakan beberapa metode. salah satunya metodenya adalah k-means dan fuzzy c-means. akan tetapi, dua metode ini memiliki perbedaan. tidak hanya dari segi algoritma, akan tetapi dari segi perhitungan nilai root mean square error (rmse)-nya juga berbeda. untuk menghitung nilai rmse ada dua indikator yang diperlukan, yaitu data training dan data checking. dari pembahasan, metode fuzzy c-means memiliki tingkat rmse yang lebih kecil dibandingkan metode k-means yaitu pada 80 data training dan 70 data checking dengan nilai rmse 2,2122e-14. hal ini menunjukkan bahwa metode fuzzy c-means memiliki tingkat ketepatan yang lebih tinggi dibandingkan dengan metode k-means. kata kunci: data iris, logika fuzzy, fuzzy c-means, data mining, k-means abstract indonesia with abundant natural resources, certainly have a lot of plants are innumerable. to clasify the plants into different clusters can use several methods. methods used are k-means and fuzzy c-means. however, this methods have difference. not only in terms of algorithms, but in terms of value calculation on the root mean square error (rmse) also different. to calculate the value of rmse there are two indicators are required, namelt the training data and the checking data. of discussion, the fuzzy c-means method has rmse values smaller than the k-means method, namely on 80 training data and 70 checking data with rmse value 2,2122e-14. this indicates that the fuzzy c-means method has a higher level of accuracy than the k-means method. kata kunci: iris data, fuzzy logic, fuzzy c-means, mining data, k-means 1. pendahuluan indonesia merupakan negara yang kaya akan sumber daya alamnya, oleh karena itu indonesia memiliki begitu banyak ragam tumbuhan dan bunga yang tersebar diwilayah indonesia. dari sekian banyak tumbuhan di indonesia, hanya 20% yang sudah teridentifikasi [1]. pada umumnya, beberapa tanaman yang belum diidentifikasi diklaster atau dikelompokkan menjadi beberapa kelompok. pengklasteran atau pengelompokkan adalah pengelompokan objek atau kasus menjadi kelompokkelompok yang lebih kecil, dimana setiap kelompok berisi objek atau kasus yang mirip satu sama lain [2]. terdapat pengklasteran beberapa jenis bunga berdasarkan lebar jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 8 mahkota, panjang mahkota, lebar kelopak dan panjang kelopak yang sering disebut dengan data iris. data iris merupakan data dari 150 bunga yang diidentifikasi berdasarkan panjang mahkota, lebar mahkota, panjang kelopak dan lebar kelopak [3]. dari 150 data tersebut pada umumnya peneliti-peneliti sebelumnya mengelompokkan menjadi tiga kelompok bunga, yaitu iris setosa, iris virginica dan iris versi color [3][4][5]. untuk menguji metode pengklasteran banyak peneliti-peneliti sebelumnya yang menggunakan data iris, karena data iris merupakan data sederhana yang mudah didapat. ada beberapa metode yang dapat digunakan untuk mengelompokkan data menjadi beberapa kelompok data, diantaranya adalah dengan menggunakan salah satu cabang dari ilmu matematika, yaitu data mining dan logika fuzzy. data mining adalah adalah suatu istilah yang digunakan untuk menguraikan penemuan pengetahuan didalam daftar data. data mining merupakan proses yang menggunakan teknik statistik, matematika, kecerdasan buatan dan machine learning untuk mengekstrasi dan mengidentifikasi informasi yang bermanfaat dan pengetahuan yang terkait dari berbagai daftar data besar [6]. dalam data mining terdapat sebuah metode yang digunakan untuk mengklaster data menjadi kelompok-kelompok data, yaitu metode k-means. beberapa peneliti sebelumnya menggunakan metode k-means untuk mengklaster data karena dalam data mining metode k-means adalah metode pengklasteran yang mudah dipahami dengan algoritma yang cukup mudah [7][8][9]. selain data mining, terdapat cabang ilmu matematika yang mempunyai metode untuk mengklaster data yaitu logika fuzzy. logika fuzzy adalah salah satu cabang ilmu matematika yang mempelajari tentang logika kabur. dimana logika fuzzy ini memiliki rentang keanggotaan berkisar antara 0 dan 1, berbeda dengan logika klasik yang memiliki rentang keanggotan yang bernilai 0 atau 1[10]. dalam pengklasteran data, metode fuzzy c-means adalah salah satu metode yang digunakan dalam logika fuzzy. beberapa peneliti sebelumnya menggunakan metode fuzzy c-means dalam penelitiannya, seperti pengklasifikasian sinyal eeg [11][12], dan analisa klasifikasi status gizi[13]. dalam jurnal ini akan ditunjukkan perbandingan pengklasteran data iris dengan menggunakan metode k-means dan fuzzy cmeans dilihat dari root mean square error (rmse). root mean square error (rmse) adalah nilai rata-rata kuadrat dari perbedaan nilai estimasi dengan nilai observasi suatu data. semakin kecil nilai rmse maka data tersebut semakin valid. 2. tinjauan pustaka 2.1 data mining data mining merupakan proses yang menggunakan teknik statistik, perhitungan, kecerdasan buatan dan machine learning untuk mengekstrasi dan mengidentifikasi informasi yang bermanfaat dan pengetahuan yang terkait dari berbagai basis data besar [14]. dalam data mining terdapat sebuah metode yang digunakan untuk mengklaster data, yaitu k-means. metode k-means merupakan metode pengklasteran data mining yang sering digunakan peneliti untuk mengklaster data. dalam metode k-means, data-data yang memiliki karakteristik yang sama diklaster dalam satu kelompok dan data yang memiliki karakteristik yang berbeda dikelompokan dengan kelompok lain yang sesuai dengan karakteristik data tersebut, sehingga data yang berada dalam satu kelompok memiliki tingkat variasi yang kecil [9]. berikut adalah algoritma dari metode kmeans: (1) masukkan data yang akan diklaster. (2) tentukan jumlah klaster. (3) ambil sebarang data sebanyak jumlah klaster secara acak sebagai pusat klaster (sentroid). (4) hitung jarak antara data dengan pusat klaster, dengan menggunakan persamaan : , ⋯ 2.1.1 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 9 dimana : , = jarak data ke ke pusat klaster = data ke pada atribut ke = titik pusat ke pada atribut ke (5) hitung kembali pusat klaster dengan keanggotaan klaster yang baru (6) jika pusat klaster tidak berubah maka proses klaster telah selesai, jika belum maka ulangi langkah ke (4) sampai pusat klaster tidak berubah lagi. 2.2 logika fuzzy logika fuzzy pertama kali diperkenalkan oleh prof. lotfi a. zadeh pada tahun 1965. dalam banyak hal, logika fuzzy digunakan sebagai suatu cara untuk memetakan permasalahan dari input menuju ke output yang diharapkan. dalam logika fuzzy terdapat fuzzy clustering yang merupakan salah satu metode untuk menentukan klaster optimal dalam suatu ruang vektor yang didasarkan pada bentuk normal euclidian untuk jarak antar vektor[15]. dalam logika fuzzy terdapat metode yang sering digunakan untuk mengklaster data, yaitu metode fuzzy cmeans. fuzzy c-means adalah suatu metode pengklasteran data yang ditentukan oleh derajat keanggotaan. berikut adalah algoritma fuzzy c-means: 1. masukkan data yang akan diklaster, berupa matriks berukuran . 2. tentukan : a. jumlah klaster = c b. pangkat = w c. maksimum iterasi = maxiter; d. error terkecil yang diharapkan = e. fungsi objektif awal = 0 f. iterasi awal = 1 3. bangkitkan bilangan acak , dengan 1,2,…, ; 1,2,…, ; sebagai elemen-elemen matriks partisi awal . hitung jumlah setiap kolom: 2.2.1 dengan 1,2,…, hitung: 2.2.2 4. hitung pusat klaster ke : ∑ ∗ ∑ 2.2.3 dengan 1,2,…, ; dan 1,2,…, 5. hitung fungsi objektif pada iterasi ke , 2.2.4 6. hitung perubahan matriks partisi: ∑ ∑ ∑ 2.2.5 dengan 1,2,…, dan 1,2,…, 7. cek kondisi berhenti: a. jika: | | atau maka berhenti, b. jika tidak: 1, ulangi langkah ke-4 output yang dihasilkan dari fuzzy cmeans (fcm) merupakan deretan pusat klaster dan beberapa derajat keanggotaan untuk tiap-tiap titik data. 2.3 root mean square error root mean ssquare error (rmse) merupakan parameter yang digunakan untuk mengevaluasi nilai hasil dari pengukuran terhadap nilai sebenarnya atau nilai dianggap benar. semakin kecil nilai rmse, maka pengklasteran data semakin mendekati benar. secara umum, persamaan yang digunakan untuk menghitung nilai rmse adalah seperti pada persamaan 2.3.1 sebagai berikut. 2.3.1 dimana: , nilai perhitungan , nilai exact jumlah data jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 10 3 metode penelitian pada jurnal ini, pengklasteran data iris menggunakan dua metode, yaitu metode kmeans dan fuzzy c-means. seperti yang telah dijelaskan pada bab sebelumnya mengenai algoritma dua metode tersebut, terdapat perbedaan pada masing-masing algoritma. untuk lebih memahami perbedaan kedua algoritma tersebut, dapat dilihat dari flowchart algoritma k-means seperti pada gambar 3.1. gambar 3.1 algoritma k-means dalam pengklasteran data iris menggunakan metode k-means, hal yang pertama dilakukan adalah memasukkan data iris terlebih dahulu. setelah itu, tentukan jumlah klaster yang diharapkan. lalu tentukan pula titik pusat klaster yang secara acak diambil dari data. selanjutnya dengan menggunakan persamaan (2.1.1), hitung jarak data ke pusat klaster. setelah itu, kelompokkan data berdasarkan hasil minimum perhitungan jarak data kepusat klaster. lalu ulangi lagi langkah awal untuk mengecek apakah titik pusat klaster yang telah dihasilkan sudah tepat dengan mengambil sebarang data dari data baru hasil dari perhitungan jarak data ke pusat klaster. jika titik pusat klaster berubah maka kita ulangi lagi langkah-langkah sebelumnya sehingga titik pusat klaster tidak berubah. gambar 3.2 algoritma fuzzy c-means pada gambar 3.2 diatas menunjukkan algoritma pengklasteran data menggunakan metode fuzzy c-means. sebagai langkah awal yang perlu dilakukan adalah memasukkan data yang akan diklaster dalam bentuk matriks . lalu tentukan beberapa indikator yang tidak ya mulai tentukan jumlah klaster tentukan titik pusat klaster hitung jarak data ke pusat klaster kelompokkan data berdasarkan minimum jarak ke pusat klaster selesai pusat klaster masukkan data mulai data masukan bangkitkan bilangan random hitung pusat klaster hitung fungsi objektif hitung perubahan matriks partisi iterasi maksimal selesai ya ya tidak tidak nilai epsilon terpenuhi jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 11 diperlukan pada metode fuzzy c-means. setelah itu bangkitkan bilangan random dengan menggunakan persamaan 2.2.1. lalu, hitung pusat klaster dengan menggunakan persamaan 2.2.1. dari perhitungan pusat klaster, hitung fungsi objektif pada iterasi dengan menggunakan persamaan 2.2.4. setelah itu, hitung perubahan matriks partisi dengan menggunakan persamaan 2.2.5. lalu, cek kondisi berhenti dengan dilihat dari apakah nilai epsilon yang merupakan salah satu indicator telah terpenuhi atau tidak. jika sudah terpenuhi maka iterasi selesai, jika iterasi telah maksimal maka kondisi berhenti. perbandingan dari metode k-means dan fuzzy c-means tidak benrhenti pada algoritma perhitungannya, akan tetapi perbandingannya terlihat ketika dihitung nilai rmse-nya dengan menggunakan persamaan 2.3.1. 4 hasil dan pembahasan pada penelitian akan menjelaskan mengenai perbandingan pengklasteran data iris menggunakan metode k-means dan cmeans. akan tetapi, pembahasan ini akan akan direpresentasikan dengan menggunakan software matlab. pada matlab terdapat fungsi yang dapat digunakan untuk mengklaster data. pada metode k-means, sebelum mengklaster data menggunakan matlab, siapkan data berupa file (.dat). setelah itu, tentukan jumlah klaster yangdiharapkan. lalu, masukkan fungsi metode k-means pada matlab, seperti berikut: ketika program ini telah disimpan, maka ketika dijalankan akan menghasilkan kelompok-kelompok data. kelompokkelompok data tersebut dapat direpresentasikan menggunakan grafik/plot pada matlab, sehingga diperoleh sebaran data pada masing-masing klaster berdasarkan titik kedekatannya dengan pusat klaster, hal tersebut terlihat seperti pada gambar 4.1. gambar 4.1 pengklasteran iris menggunakan kmeans begitu pula metode fuzzy c-means, metode ini juga menggunakan fungsi pada matlab untuk menunjukkan kelompokkelompok data yang telah diklaster. adapun fungsi yang digunakan adalah sebagai berikut: ketika fungsi tersebut telah disimpan dan dijalankan akan diperoleh kelompokkelompok data. dan dapat juga ditampilkan dengan menggunakan grafik/plot, sehingga diperoleh hasil klasterisasi seperti pada gambar 4.2.   gambar 4.2 pengklasteran iris menggunakan fuzzy c-means x=load(‘datairis.dat’); jumlah_klaster=3; [idx,c]=kmeans(x,jumlah_klaster) x=load(‘datairis.dat’); jumlah_klaster=3; [center,u,objfcn]=fcm(x,jumlah_klaster) jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 12 untuk lebih terlihat perbandingan pengklasteran data iris dari kedua metode tersebut, hitung rmse dari data yang sudah diklaster. perhitungan rmse-pun bisa dilakukan menggunakan matlab. ada beberapa indikator yang harus disiapkan terlebih dahulu, yaitu data training dan data checking. data training lebih banyak dari data checking. tabel hasil rmse dari dua metode yang berbeda dan data yang sama dapat dilihat pada tabel 4.1 tabel 4.1 rmse k-means dan fuzzy c-means no data metode check train fcm k-means 1 27 123 0.0530 0.0728 2 35 115 0.0019 0.0608 3 40 160 0.0011 0.0072 4 44 106 0.0604 0.0705 5 60 90 2,2166e-5 0.1051 6 63 87 8,3924e-5 2,6578e-3 7 70 80 2,2122e-14 4,1188e-13 untuk lebih jelasnya. perbandingan rmse dari kedua data tersebut dapat direpresentasikan menggunakan grafik/plot, sehingga diperoleh seperti pada gambar 4.3. gambar 4.3 perbandingan rmse dari k-means dan fuzzy c-means dari grafik gambar 4.3, garis biru merepresentasikan hasil perhitungan rmse dari metode fuzzy c-means dan garis hijau merepresentasikan hasil perihtungan rmse dari metode k-means. 5 kesimpulan dari pembahasan yang telah disampaikan, dapat disimpulkan bahwasanya hasil pengklasteran data iris menggunakan metode k-means dan fuzzzy c-means berbeda. jika dilihat hasil perhitungan rmse dari kedua metode tersebut, menunjukkan bahwa metode fuzzy c-means memiliki nilai rmse yang lebih kecil dibandingkan dengan nilai rmse metode k-means. hal ini menunjukkan bahwa pengklasteran menggunakan metode fuzzy c-means lebih mendekati ketepatan (valid) dibandingkan dengan metode kmeans. penelitian ini masih jauh dari sempurna, masih perlu dilakukan penelitian dengan menggunakan data yang berbeda dan menggunakan lebih banyak data training dan checking lebih banyak untuk mendapatkan nilai rmse. 6 daftar pustaka [1] siregar, mustaid. jumlah spesies tumbuhan flora di indonesia, diambil dari http://www.lipi.go.id/, pada tanggal 28 juni 2016 [2] kuniawati, rizki taher dkk. pengelompokan kualitas udara ambien menurut kabupaten/kota di jawa tengah menggunakan analisis klaster. jurnal gaussian, vol 4 no 2 tahun 2015 : 393-402 [3] kadir, abdul. identifikasi tiga jenis bunga iris menggunakan anfis. [4] azmi, meri. komparasi metode jaringan syaraf tiruan som dan luq untuk mengidentifikasi data bunga iris. jurnal tek noif, vol 2 no 1, april 2014 [5] riyanto, hendrik puasa dkk. analisa dan implementasi fuzzy inference system pada hasil klasterisasi algoritma fuzzy subtractive jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 13 clustering. universitas telkom. 2010 [6] pane, dewi kartika. implementasi data mining pada penjualan produk elektronik dengan algoritma apriori (studi kasus : kreditplus). jurnal pelita informatika budi darma, vol iv no 3, agustus 2013 [7] narwati. pengelompokan mahasiswa menggunakan k-means. semarang: fakultas teknologi informasi unisbank. 2010 [8] rivani, edmira. aplikasi k-means cluster untuk pengelompokan provinsi berdasarkan produksi jagung, padi, kedelai dan kacang hijau. pusat pengkajian pengolahan data dan informasi, sekretaris jenderal dpr ri . jurusan statistika terapan, universitas padjajaran, bandung [9] ong, johan oscar. implementasi algoritma k-means clustering untuk menentukan strategi marketing president university. jurnal ilmiah teknik industri, vol 12, no 1, juni 2013 . [10] kusumadewi, sri dan purnomo, hari. aplikasi logika fuzzy untuk pendukung keputusan. edisi 2. yogyakarta. graha ilmu. 2010 [11] rini, dian c, klasifikasi sinyal eeg menggunakan metode fuzzy c-means (fcm) clustering dan adaptive neuro fuzzy inference system (anfis). undergraduate thesis, department of information technology, faculty of information technology, institut teknologi sepuluh nopember, indonesia. 2013 [12] rini, dian c, klasifikasi sinyal eeg menggunakan metode fuzzy c-means clustering (fcm) dan adaptive neighborhood modified backpropagation (anmbp). fakultas sains dan teknologi. universitas islam negeri sunan ampel surabaya. 2015. [13] sudirman, nerfita nikentari dan martaleli. analisa klasifikasi status gizi dengan metode fuzzy c-means menggunakan aplikasi berbasis android. jurusan informatika. universitas maritim raja ali haji. tanjung pinang [14]sutrisno, afriyudi wiyanto. penerapan data mining pada penjualan menggunakan metode clustering study kasus pt. indoamarco palembang. palembang: universitas bina darma. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran 3 how to cite: m. athoillah, “k-nearest neighbor for recognize handwritten arabic character”, mantik, vol. 5, no. 2, pp. 83-89, october 2019. k-nearest neighbor for recognize handwritten arabic character muhammad athoillah universitas pgri adi buana surabaya, athoillah.muhammad@gmail.com doi: https://doi.org/10.15642/mantik.2019.5.2.83-89 abstrak: pengenalan teks tulisan tangan adalah kemampuan sebuah sistem untuk mengenali tulisan tangan manusia dan mengubahnya menjadi teks digital. pengenalan teks tulisan tangan adalah bagian dari masalah klasifikasi, sehingga algoritma klasifikasi seperti nearest neighbor (nn) diperlukan untuk menyelesaikannya. algoritma nn adalah algoritma yang sederhana namun memberikan hasil yang baik. berbeda dengan algoritma lain yang biasanya ditentukan oleh beberapa kelas hipotesis, algoritma nn menemukan label pada titik uji tanpa mencari prediktor dalam beberapa kelas fungsi yang telah ditentukan. bahasa arab adalah salah satu bahasa terpenting di dunia. mengenali karakter bahasa arab sangat menarik untuk dijadikan bahan kajian, tidak hanya karena merupakan bahasa utama yang digunakan dalam agama islam tetapi juga karena jumlah penelitian tulisan tangan arab yang ada masih jauh jumlahnya bila dibandingkan dengan penelitian pengenalan tulisan latin atau cina. berdasarkan latar belakang tersebut, penelitian ini membangun sebuah sistem untuk mengenali karakter tulisan arab dari sebuah citra menggunakan algoritma nn. hasil penelitian menunjukkan bahwa metode yang diusulkan dapat mengenali karakter dengan sangat baik ditunjukkan melalui rata-rata presisi, recall dan akurasi yang tinggi. kata kunci: klasifikasi, karakter bahasa arab, nearest neighbor, pengenalan teks, tulisan tangan abstract: handwritten text recognition is the ability of a system to recognize human handwritten and convert it into digital text. handwritten text recognition is a form of classification problem, so a classification algorithm such as nearest neighbor (nn) is needed to solve it. nn algorithms is a simple algorithm yet provide a good result. in contrast with other algorithms that usually determined by some hypothesis class, nn algorithm finds out a label on any test point without searching for a predictor within some predefined class of functions. arabic is one of the most important languages in the world. recognizing arabic character is very interesting research, not only it is a primary language that used in islam but also because the number of this research is still far behind the number of recognizing handwritten latin or chinese research. due to that's the background, this framework built a system to recognize handwritten arabic character from an image dataset using the nn algorithm. the result showed that the proposed method could recognize the characters very well confirmed by its average of precision, recall and accuracy. keywords: arabic character, classification, handwritten, nearest neighbor, text recognition jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 83-89 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 83-89 84 1. introduction handwritten text recognition is the ability of a system to recognize human handwritten and convert it into digital text, this study was first conducted by t.l diamond in 1958 [1]. recently, the study of handwritten text recognition has been growing rapidly, the huge number of these researches is proved by the number of its publication such as sadkhan et al. in 2018 who recognized handwritten based on ann and wavelet transformation [2], iqbal and zafar studied about offline handwritten quranic text recognition in 2019 [3], nguyen et al. who recognized japanese handwritten using recurrent neural network in 2018 [4] and much more. handwritten recognition is divided into two categories, namely on-line and off-line recognition. the on-line recognition is a system that has the ability to recognize humanly handwritten directly from smartphones, personal digital assistant (pda) and others similar device, whilst the off-line recognition is a system that needs to scan any images first to recognize the human handwritten[5]. arabic is one of the most important languages in the world. since it is a primary language used in islam, recognizing arabic character had become a very interesting task. study of arabic text recognition was started by a. nazif in 1975. in his master's thesis, he built a system to recognize arabic characters based on extracting strokes called radicals and their positions. he used correlation among the templates of the character image and the radicals, a segmentation phase was included in the cursive text segment [6]. however, the number of recognizing handwritten arabic research is still far behind the number of recognizing handwritten latin or chinese research. this fact is based on the lack of arabic digital dictionaries and programming tools or the lack of public databases of handwritten arabic characters that found nowadays[5]. handwritten text recognition is a form of classification problem, so a classification algorithm is needed to solve it, one of that algorithm is the nearest neighbor (nn). nearest neighbor algorithms are simple algorithm but give a good result. the idea of this method is to memorize the training data and predict the label of any new instance on the basis of labels from its closest neighbours in the training data. in contrast with other algorithms that usually determined by some hypothesis class, nearest neighbor algorithm finds out a label on any test point without searching for a predictor within some predefined class of functions [7]. by default, the most function that is used to measure the distance between the data (neighbour) is euclidean. moreover, the parameter 𝑘 is added which decided how many neighbours will be chosen for knn algorithm. the suitable choice of 𝑘 has a significant impact on the result of the knn algorithm[8][9]. due to that's the background, this framework built a system to recognize handwritten arabic character scanned from an image dataset using k-nearest neighbor (knn) algorithm. the framework used madbase dataset provided by the american university in cairo. the result of this work was presented through the values of its precision, recall and accuracy while cross-validation method was used to validate the results. 2. related work k-nearest neighbor (k-nn) was first introduced in 1951 and 1952 by fix and hodges [10][11] and then further study conducted by cover and hart in 1967 [12]. m. athoillah k-nearest neighbor for recognize handwritten arabic character 85 the algorithm was inspired by the hypothesis that "things that look alike must be alike" (cover and hart). different with other classification algorithms, knn is a classification method based on data that were located closest to the object without searching for a predictor within some predefined class of functions to find out the label on test point[7]. figure 1. illustration of k-nn algorithm assume that dataset domain 𝑋 is given with metric function 𝑝, then the function that returns the distance between two elements of 𝑋 written as 𝑝: 𝑋 × 𝑋 → ℝ. for instance, if 𝑋 = ℝ𝑑 then 𝑝 can be euclidean distance 𝑝(𝑥, 𝑥′) = ‖𝑥 − 𝑥′‖ = ∑(𝑥𝑖 − 𝑥𝑖 ′)2 𝑑 𝑖=1 given a sequence of training data as 𝑆 = (𝑥1, 𝑦1), (𝑥2, 𝑦2), … ,( 𝑥𝑚, 𝑦𝑚). for each 𝑥 ∈ 𝑋, let 𝜋1(𝑥), 𝜋2(𝑥), … , 𝜋𝑚(𝑥) be a reordering of {1,2, … , 𝑚} according to their distance to 𝑥, 𝑝(𝑥, 𝑥𝑖 ). then, for all 𝑖 < 𝑚, 𝑝(𝑥, 𝑥𝜋𝑖(𝑥)) ≤ 𝑝(𝑥, 𝑥𝜋𝑖+1(𝑥)) for a number 𝑘, the binary classification of 𝑘 −nn rule is defined as follows: • input: sample of training 𝑆 = (𝑥1, 𝑦1), (𝑥2, 𝑦2), … ,( 𝑥𝑚 , 𝑦𝑚) • output: for every point 𝑥 ∈ 𝑋, return the majority label within {𝑦𝜋𝑖(𝑥) ; 𝑖 ≤ 𝑘} if 𝑘 = 1, then the 1-nn rule: ℎ𝑠 (𝑥) = 𝑦𝜋1(𝑥) since the output depends on the number of 𝑘, therefore the appropriate of 𝑘 has a significant impact on the result of the knn algorithm. 3. implementation this framework used a dataset from the american university in cairo called madbase which contain images of arabic handwritten numeral character from zero to nine-digit, the images used in this experiment amount to 500 in each object, which means there are 5000 images used in total. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 83-89 86 figure 2. sample of dataset the image dataset was divided in two-part that was dataset for training process which contains 90 percent of the total image, whilst the rest 10 percent was used for the testing process. this experiment used histogram of the images as input, the histogram was chosen because the colour features which distinguish each image is generally represented by the colour histogram[13]. k-nearest neighbor is a simple algorithm, the process of training for this algorithm only contain storing feature vectors and labels of the images, while in classification process, the unlabeled query point is assigned to the label of its k nearest neighbours. when used k-nn, the object was classified based on majority labels among the k nearest neighbours. if k=1, the object was classified as the class from the object nearest to it. if there are only two classes, k must be an odd integer. however, there was still bond when k is an odd integer in case performing multiclass classification. in addition, euclidean distance function was used to measure the distance between data. 4. result and discussion in this section, the performance of the system to recognize the arabic handwritten is delivered by computing its precision, recall and accuracy. since the result of the k-nn algorithm is greatly influenced by the value of 𝑘, this framework measured all the performance used 6 sorts of 𝑘, namely 1, 3, 5,10 15 and 20. in addition, k-fold cross-validation was applied to validate the result of the system, this process was executed by dividing data into ten batches (𝑘 = 10), which are nine batches for training and the rest batch for testing, then the experiment was repeated 𝑘 times under the condition that training data and testing data were always different during each process. the results are presented in the following table: table 1. the average result of system with 𝑘 = 1 (%) class/number precision recall accuracy zero 93,57 91,00 98,42 one 92,16 94,20 98,58 two 22,34 20,40 84,68 three 13,20 15,60 81,52 four 14,89 15,40 83,02 five 72,44 68,20 94,06 six 40,31 25,60 88,72 seven 17,22 18,60 82,70 eight 20,79 26,40 82,50 nine 24,27 21,60 85,20 average off all 42,99 41,71 88,24 m. athoillah k-nearest neighbor for recognize handwritten arabic character 87 table 2. the average result of system with 𝑘 = 3 (%) class/number precision recall accuracy zero 94,81 91,00 98,56 one 92,47 96,20 98,80 two 23,22 20,80 84,80 three 14,04 15,60 82,22 four 17,38 17,80 83,46 five 72,80 74,40 94,50 six 44,09 27,60 89,18 seven 18,31 19,20 83,28 eight 21,24 27,60 82,42 nine 26,53 23,40 85,50 average off all 44,26 43,36 88,58 table 3. the average result of system with 𝑘 = 5 (%) class/number precision recall accuracy zero 94,63 91,40 98,58 one 92,41 96,00 98,76 two 25,82 24,00 85,36 three 14,29 16,20 82,22 four 19,07 18,80 83,78 five 74,24 76,00 94,82 six 50,80 32,60 89,96 seven 20,45 19,80 83,94 eight 20,06 25,80 82,16 nine 28,45 26,00 85,74 average off all 45,75 44,51 88,84 table 4. the average result of system with 𝑘 = 10 (%) class/number precision recall accuracy zero 95,35 91,20 98,64 one 92,12 95,80 98,70 two 28,47 26,80 85,50 three 13,90 14,80 82,58 four 21,49 21,60 84,20 five 74,31 79,80 95,00 six 53,01 35,80 90,08 seven 23,35 20,80 85,10 eight 21,66 28,80 82,60 nine 32,77 29,40 86,56 average off all 47,07 46,16 89,16 table 5. the average result of system with 𝑘 = 15 (%) class/number precision recall accuracy zero 95,30 90,80 98,60 one 92,74 95,20 98,70 two 31,61 29,00 86,00 three 16,07 16,40 83,30 four 22,90 22,80 84,44 five 72,74 80,40 94,84 six 51,72 39,80 89,86 seven 23,58 19,20 85,30 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 83-89 88 class/number precision recall accuracy eight 22,22 28,60 82,64 nine 34,12 30,60 86,88 average off all 47,65 46,91 89,30 table 6. the average result of system with 𝑘 = 20 (%) class/number precision recall accuracy zero 95,79 89,20 98,50 one 93,41 94,60 98,72 two 34,07 30,60 86,10 three 17,63 17,40 83,32 four 22,78 23,20 84,54 five 72,12 80,60 94,74 six 49,60 41,60 89,64 seven 25,15 18,20 86,06 eight 23,55 28,80 82,92 nine 34,51 33,00 86,90 average off all 48,23 47,13 89,39 the result showed from table 1 until table 6 that k-nn could recognize the handwritten very well, proved by the lowest average of accuracy is 88,24% while the lowest average of its precision and recall are 42,99% and 41,71%. the highest performance belongs to the recognition of “number one” and “number zero”, this is because the shapes of them are the simplest and the most different from the other numbers which are only a straight vertical line for “number one” and only a dot for “number zero”. otherwise, the lowest performance belongs to “number three” with the highest point of precision is 17,63%, 17,40% for recall point and 83,32% for its accuracy, that is because “number three” has a shape very similar to “number two” so those characters are often misclassified. this framework results also showed that the higher value of 𝑘 provided the higher value of its accuracy, precision and recall. nevertheless, it cannot be concluded that the higher value of 𝑘 is always better because there is no clear evidence or definite guidance to determine the best value of 𝑘 to get the optimal result. in general, a higher value of 𝑘 makes it less sensitive to noise and cause smoother boundaries. as a result, it is almost impossible to choose the same best value of 𝑘 for different applications[13]. 5. conclusion this framework built a system that could recognize handwritten arabic character using k-nearest neighbour (k-nn). database from the american university in cairo called madbase was used in this framework that contains arabic handwritten numeral character images from zero to nine-digit with a total of 5000 images. since 𝑘 values have a lot of effect on the results, then this framework measured all the performance used six types of 𝑘 values, i.e. 1, 3, 5, 10, 15 and 20. relating to validation, k-fold cross-validation was applied to validate the result by dividing all data into ten batches then all of these batches were used for train and test process alternately. the result not only showed that the proposed method could recognize the characters very well confirmed by its average of precision, recall and accuracy but also showed that the higher 𝑘 values produced m. athoillah k-nearest neighbor for recognize handwritten arabic character 89 the higher performance. even so, it cannot be concluded that the higher 𝑘 values always present a better result because each data has its own characteristics. references [1] t. l. dimond, “devices for reading handwritten characters,” in papers and discussions presented at the december 9-13, 1957, eastern joint computer conference: computers with deadlines to meet, 1957, pp. 232–237. [2] s. b. sadkhan and s. f. jawad-smieee, “handwritten recognition based on hybrid ann and wavelet transformation,” in 2018 al-mansour international conference on new trends in computing, communication, and information technology (ntccit), 2018, pp. 76–80. [3] a. iqbal and a. zafar, “offline handwritten quranic text recognition: a research perspective,” in 2019 amity international conference on artificial intelligence (aicai), 2019, pp. 125–128. [4] h. t. nguyen, c. t. nguyen, and m. nakagawa, “online japanese handwriting recognizers using recurrent neural networks,” in 2018 16th international conference on frontiers in handwriting recognition (icfhr), 2018, pp. 435–440. [5] m. shatnawi, “off-line handwritten arabic character recognition: a survey,” in proceedings of the international conference on image processing, computer vision, and pattern recognition (ipcv), 2015, p. 52. [6] b. al-badr and s. a. mahmoud, “survey and bibliography of arabic optical text recognition,” signal processing, vol. 41, no. 1, pp. 49–77, 1995. [7] s. shalev-shwartz and s. ben-david, understanding machine learning: from theory to algorithms. cambridge university press, 2014. [8] z. zhang, “introduction to machine learning: k-nearest neighbors,” ann. transl. med., vol. 4, no. 11, 2016. [9] f. amin, “identifikasi citra daging ayam berformalin menggunakan metode fitur tekstur dan k-nearest neighbor (k-nn)”, mantik, vol. 4, no. 1, pp. 68-74, may 2018. [10] e. fix and j. l. hodges jr, “discriminatory analysis-nonparametric discrimination: consistency properties,” california univ berkeley, 1951. [11] e. fix and j. l. hodges jr, “discriminatory analysis-nonparametric discrimination: small sample performance,” california univ berkeley, 1952. [12] t. m. cover and p. hart, “nearest neighbor pattern classification,” ieee trans. inf. theory, vol. 13, no. 1, pp. 21–27, 1967. [13] b. lei, e.-l. tan, s. chen, d. ni, and t. wang, “saliency-driven image classification method based on histogram mining and image score,” pattern recognit., vol. 48, no. 8, pp. 2567–2580, 2015. [14] c.-m. ma, w.-s. yang, and b.-w. cheng, “how the parameters of k-nearest neighbor algorithm impact on the best classification accuracy: in case of parkinson dataset,” j. appl. sci., vol. 14, no. 2, pp. 171–176, 2014. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: marsidi and i. h. agustin, “the local antimagic on disjoint union of some family graphs”, mantik, vol. 5, no. 2, pp. 69-75, october 2019. the local antimagic on disjoint union of some family graphs marsidi1,2, ika hesti agustin1,3* cgant research group, university of jember, indonesia1 mathematics edu. depart. ikip pgri jember, jember, indonesia 2, marsidiarin@gmail.com2 mathematics depart. university of jember, jember, indonesia3, ikahesti.fmipa@unej.ac.id 3 doi: https://doi.org/10.15642/mantik.2019.5.2.69-75 abstrak: graf 𝐺 pada penelitian ini adalah nontrivial, berhingga, terhubung, sederhana, dan tidak terarah. misal u, 𝑣 adalah dua elemen di himpunan titik dan q adalah kardinalitas himpunan sisi di graf 𝐺, fungsi bijektif dari himpunan sisi dipetakan ke bilangan asli q disebut vertex local antimagic edge labeling jika untuk dua titik yang bertetangga 𝑣1 dan 𝑣2, bobot 𝑣1 tidak sama dengan bobot 𝑣2, dimana bobot 𝑣 (dinotasikan dengan 𝑤(𝑣)) adalah jumlah label sisi yang terkait dengan 𝑣. selanjutnya, setiap vertex local antimagic edge labeling menginduksi pewarnaan titik pada graf 𝐺 dimana 𝑤(𝑣) adalah warna pada titik 𝑣. bilangan kromatik vertex local antimagic edge labeling 𝜒𝑙𝑎(𝐺) adalah banyaknya warna minimum yang disebabkan oleh titik local antimagic pelabelan sisi pada graf 𝐺. dalam artikel ini, kita membahas tentang bilangan kromatik vertex local antimagic edge labeling pada graf disjoint union dari keluarga graf seperti lintasan, lingkaran, bintang, dan friendship, dan menentukan batas bawah dari vertex local antimagic edge labeling. nilai kromatik pada graf disjoint union pada penelitian ini mencapai batas bawah. kata kunci: pelabelan local antimagic, nilai kromatik local antimagic, graf disjoint union abstract: a graph 𝐺 in this paper is nontrivial, finite, connected, simple, and undirected. graph 𝐺 consists of a vertex set and edge set. let u,v be two elements in vertex set, and q is the cardinality of edge set in g, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices 𝑣1and 𝑣2, the weight of 𝑣1 is not equal with the weight of 𝑣2, where the weight of 𝑣 (denoted by 𝑤(𝑣)) is the sum of labels of edges that are incident to 𝑣. furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where 𝑤(𝑣) is the colour on the vertex 𝑣. the vertex local antimagic chromatic number 𝜒𝑙𝑎(𝐺) is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of 𝐺. in this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. the chromatic numbers of disjoint union graph in this paper attend the lower bound. keywords: local antimagic labelling, local antimagic chromatic number, disjoint union graphs jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 69-75 issn: 2527-3159 (print) 2527-3167 (online) mailto:marsidiarin@gmail.com2 mailto:ikahesti.fmipa@unej.ac.id http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 69-75 70 1. introduction any mathematical object involving points and connection between them called a graph. a graph 𝐺(𝑉,𝐸) consists of two sets 𝑉 and 𝐸. the elements of 𝑉 are called vertices, and the elements of 𝐸 called edges. the all graphs discussed in this paper are nontrivial, finite, connected, simple, and undirected [1] – [3]. graph labelling is the assignment of labels to the graph elements such as edges or vertices, or both, from the graph. the concept of antimagic labelling of a graph was introduced by hartsfield and ringel [4]. every component of the graph, such as vertex, edge, and face, has given the distinct label by a natural number. given a graph 𝐺(𝑉,𝐸)and q is the number of edges in g, a bijective function from the edges set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices 𝑣1and 𝑣2, 𝑤(𝑣1) ≠ 𝑤(𝑣2), where 𝑤(𝑣) is the sum of labels of edges that are incident to 𝑣. furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where 𝑤(𝑣) is the colour on the vertex 𝑣. the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of 𝐺 denoted by 𝜒𝑙𝑎(𝐺). if a graph 𝐺 has antimagic labelling, it is called antimagic. since been introduced by hartsfield and ringel [4], the topic has attracted a lot of attention, for details, see gallian [11]. one of well-known remark and theorem related to this is the following. remark 1.1. for any graph g, 𝜒𝑙𝑎(𝐺) ≥ 𝜒(𝐺) and the difference 𝜒𝑙𝑎(𝐺) − 𝜒(𝐺)can be arbitrarily large[4]. theorem 1.1. for any tree t with l leaves, 𝜒𝑙𝑎(𝑇) ≥ 𝑙 + 1 [5]. arumugam et al. [5] give a lower bound and upper bound of local antimagic vertex colouring of 𝐾1 + 𝐻 and also give the exact value of local antimagic vertex colouring. agustin et al. [6] studied the local edge antimagic colouring of graphs. the other results about local antimagic of graphs can be seen in [6] –[10]. thus, in this paper, we have found the chromatic number of vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship. before we present our results, we define a definition of disjoint union graph that we discuss in this paper. for any graph 𝐺, the graph ⋃ 𝐺𝑖=𝑚 denotes the disjoint union of 𝑚 copies of graph 𝐺. 2. main result in this section, we present our results by showing the vertex local antimagicchromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship in the following theorems. we also present the remark and theorem about the lower bound of the disjoint union of graphs. remark 2.1. for any disjoint union graph g, we have 𝜒𝑙𝑎(⋃ 𝐺𝑖=𝑚 ) ≥ 𝜒𝑙𝑎(𝐺). theorem 2.1. for any tree t with kpendant vertices, 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1 marsidi and i. h. agusti the local antimagic on disjoint union of some family graphs 71 proof. let 𝑔 be any local antimagic labelling of disjoint union of 𝑇, then the colouring induced by g. the colour of a pendant vertex v is the label on edge that is incident with 𝑣. thus, all the pendant vertices receive distinct colours. furthermore, for any non-pendant vertex incident with an edge 𝑒1 with 𝑓(𝑒1) = 𝑚, the colour assigned to 𝑤 is larger than 𝑚. since the number of colours in the colouring induced by 𝑔 is at least 𝑘 + 1, 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1. theorem 2.2. for the path 𝑃𝑛 with 𝑛 ≥ 3, then we have 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) = 2𝑚 + 1. proof. the ⋃ 𝑃𝑛𝑗=𝑚 is the disjoint union of path graph. the vertex set is 𝑉(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑣1 1,𝑣2 1,…,𝑣𝑖 𝑗 ,…,𝑣𝑛−1 𝑚 ,𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}} and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑣1 1𝑣2 1,𝑣2 1𝑣3 1 …,𝑣𝑖 𝑗 𝑣𝑖+1 𝑗 ,…,𝑣𝑛−2 𝑚 𝑣𝑛−1 𝑚 ,𝑣𝑛−1 𝑚 𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚}}. hence |𝑉(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚𝑛,|𝐸(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚(𝑛 − 1). because graph ⋃ 𝑃𝑛𝑗=𝑚 have 2𝑚 leaves, based on theorem 2.1, we have 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1, so the color needed for⋃ 𝑃𝑛𝑗=𝑚 is2𝑚 + 1. thus, we have𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≥ 2𝑚 + 1. furthermore, we will prove that the upper bound is 2𝑚 + 1, denoted by 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1. we define the labelling on the edges of 𝑃𝑛 by the following functions. 𝑓(𝑣𝑖 𝑗 𝑣𝑖+1 𝑗 ) = { 1 2 (𝑖 + 𝑗(𝑛 − 1) − 𝑛 + 1, 𝑖 ∈ {2,4,6,…,𝑛 − 1}; 𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 odd 𝑚(𝑛 − 1) + ( 𝑛 − 𝑖 2 ) − 𝑗( 𝑛 − 1 2 ), 𝑖 ∈ {1,3,5,…,𝑛 − 2}; 𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 odd ( 𝑖 2 − 1)𝑚 + 𝑗, 𝑖 ∈ {2,4,6,…,𝑛 − 2}; 𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 even 𝑚𝑛 − 𝑚( 𝑖 + 1 2 ) − 𝑗 + 1, 𝑖 ∈ {1,3,5,…,𝑛 − 1}; 𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 even base on the function f, we determine the set of vertex weight on pendant vertices as the following. 𝑊1 = {𝑓(𝑣1 𝑗 ; 𝑗 ∈ {1,2,3,…,𝑚}} ∪ {𝑓(𝑣𝑛 𝑗 ; 𝑗 ∈ {1,2,3,…,𝑚}} = {𝑚(𝑛 − 1),𝑚(𝑛 − 1) − ( 𝑛 − 1 2 ),…,(𝑚 + 1)( 𝑛 − 1 2 )} ∪ { 𝑛 − 1 2 ,𝑛 − 1,…, 𝑚(𝑛 − 1) 2 } the vertex weights on the two degrees vertices are 𝑊2 = 𝑚(𝑛 − 1) + 1 𝑊3 = 𝑚(𝑛 − 1) hence from the above, it easy to see that the different vertex weights are2𝑚 + 1, so 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1. therefore𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1 and 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≥ 2𝑚 + 1. it concludes that 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) = 2𝑚 + 1. the illustration of local antimagic labelling of ⋃ 𝑃𝑛𝑗=𝑚 can be seen in figure 1. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 69-75 72 figure 1. illustration of local antimagic labelling of ⋃ 𝑃7𝑚=5 theorem 2.3. for the cycle 𝐶𝑛 with 𝑛 ≥ 𝑡ℎ𝑟𝑒𝑒 𝑎𝑛𝑑 𝑛 is even integer, then we have 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) = 3. proof. the ⋃ 𝐶𝑛𝑗=𝑚 is the disjoint union of cycle graph.the vertex set 𝑉(⋃ 𝐶𝑛𝑗=𝑚 ) = {𝑣1 1,𝑣2 1,…,𝑣𝑖 𝑗 ,…,𝑣𝑛−1 𝑚 ,𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚} and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑒1 1,𝑒2 1,𝑒3 1,…,𝑒𝑛 1,…,𝑒𝑖 𝑗 ,…,𝑒𝑛−1 𝑚 ,𝑒𝑛 𝑚,𝑤ℎ𝑒𝑟𝑒 𝑒1 1,= 𝑣1 1𝑣2 1,𝑒2 1 = 𝑣2 1𝑣3 1,…,𝑒𝑛 1 = 𝑣𝑛 1𝑣1 1; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}}. hence |𝑉(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚𝑛,|𝐸(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚𝑛. we will prove that 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≥ 3. based on arumugam theorem [4], the local antimagic chromatic number of cycle graph is 𝜒𝑙𝑎(𝐶𝑛) = 3. furthermore, based on remark 2.1 we have 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑖=𝑚 ) ≥ 𝜒𝑙𝑎(𝐶𝑛) so we can see that 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≥ 3. furthermore, we will prove that the upper bound is 3, denoted 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≤ 3, by defining the labelling as follow. 𝑔(𝑒𝑖 𝑗 ) = { 1 2 (𝑖 + 1) + 1 2 (𝑗 − 1), 𝑓𝑜𝑟 𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚} 𝑛𝑚 + 1 − 𝑖 2 − 𝑛 2 (𝑗 − 1), 𝑓𝑜𝑟 𝑖 ∈ {2,4,6,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚} then we have the vertex weights as follow: 𝑊(𝑒𝑖 𝑗 ) = { 𝑛𝑚 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚} 𝑛𝑚 + 2,𝑓𝑜𝑟 𝑖 = 1,3,5,…,𝑛 − 1,𝑗 ∈ {1,2,3,…,𝑚} 𝑛𝑚 + 2 − 𝑛 2 ,𝑓𝑜𝑟 𝑖 = 𝑛,𝑗 ∈ {1,2,3,…,𝑚} hence from the function above, it easy to see that the vertex weight 𝑊(𝑒𝑖 𝑗 ) = {𝑛𝑚 + 1,𝑛𝑚 + 2,𝑛𝑚 + 2 − 𝑛 2 } contains three elements which induce a proper vertex colouring of ⋃ 𝐶𝑛𝑗=𝑚 . thus, it gives 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≤ 3. it concludes that 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) = 3. 17 18 1 2 3 4 5 6 2 7 8 9 10 11 12 13 14 15 16 19 20 21 22 23 24 25 26 27 28 29 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 30 30 30 30 30 30 30 30 30 30 27 24 21 18 3 6 2 9 12 15 marsidi and i. h. agusti the local antimagic on disjoint union of some family graphs 73 theorem 2.4. for the star 𝑆𝑛 with 𝑛 ≥ 3, then we have 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) = 𝑛𝑚 + 1, 𝑚 ≠ even integer when 𝑛 is an odd integer. proof. the ⋃ 𝑆𝑛𝑗=𝑚 is the disjoint union of star graph. the vertex set 𝑉(⋃ 𝑆𝑛𝑗=𝑚 ) = {𝐴,𝑣1 1,𝑣2 1,…,𝑣𝑖 𝑗 ,…,𝑣𝑛−1 𝑚 ,𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}} and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝐴𝑣1 1,𝐴𝑣2 1,𝐴𝑣3 1 …,,𝐴𝑣𝑖 𝑗 ,…,𝐴𝑣1 𝑚,𝐴𝑣2 𝑚,…,𝐴𝑣𝑛−1 𝑚 ,𝐴𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}}. hence |𝑉(⋃ 𝑆𝑛𝑗=𝑚 )| = 𝑚𝑛 + 𝑚, |𝐸(⋃ 𝑆𝑛𝑗=𝑚 )| = 𝑚𝑛. we will prove that 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1. graph ⋃ 𝑆𝑛𝑗=𝑚 have𝑛𝑚 leaves, based on theorem 2.1, we have 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1so the color needed for ⋃ 𝑆𝑛𝑗=𝑚 is 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1. furthermore, we will prove that the upper bound is 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 + 1, we define the labelling into two cases by the following. case 1. for 𝑛 even ℎ(𝐴𝑥𝑖 𝑗 ) = { 𝑚(𝑖 − 1) + 𝑗;𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚} 𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {2,4,6,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚} case 2. for 𝑛 odd ℎ(𝐴𝑥𝑖 𝑗 ) = { 𝑗 + 1 2 ;𝑖 = 1,𝑗 ∈ {1,3,5,…,𝑚} 𝑚 + 𝑗 + 1 2 ;𝑖 = 1,𝑗 ∈ {2,4,6,…,𝑚 − 1} 𝑗 2 + 𝑚;𝑖 = 2,𝑗 ∈ {2,4,6,…,𝑚 − 1} 3𝑚 + 𝑗 2 ;𝑖 = 2,𝑗 ∈ {1,3,5,…,𝑚} 3𝑚 + 1 − 𝑗;𝑗 ∈ {1,2,3,…,𝑚} 𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {4,6,8,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 𝑚(𝑖 − 3) − 𝑗 + 1;𝑖 ∈ {5,7,9,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚} based on the labelling above, the weight on each pendant vertex is the label of the edge which incident with pendant vertex. therefore, the number of vertex weight on the disjoint union of star graph is 𝑛𝑚. the weight of each centre vertex is as follow. 𝑊(𝐴) = 𝑛 2 (𝑛𝑚 + 1) because the number of pendant vertex is 𝑛𝑚, the number of different vertex weight on the disjoint union of star graph is 𝑛𝑚 + 1, so 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 + 1.therefore 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 + 1 and 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1, it concludes𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) = 𝑛𝑚 + 1 theorem 2.5. for the friendship ℱ𝑛 with 𝑛 ≥three and n is even integer, then we have 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) = 3. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 69-75 74 proof. the ⋃ ℱ𝑛𝑗=𝑚 is the disjoint union of friendship graph. the vertex set 𝑉(⋃ ℱ𝑛𝑗=𝑚 ) = {𝐴} ∪ {𝑢1 1,𝑢2 1,…,𝑢𝑖 𝑗 ,…,𝑢𝑛−1 𝑚 ,𝑢𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚}} ∪ {𝑣1 1,𝑣2 1,…,𝑣𝑖 𝑗 ,…,𝑣𝑛−1 𝑚 ,𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚}}and the edge set 𝐸(⋃ ℱ𝑛𝑗=𝑚 ) = {𝐴𝑢1 1,𝐴𝑢2 1,𝐴𝑢3 1 …,,𝐴𝑢𝑖 𝑗 ,…,𝐴𝑢1 𝑚,𝐴𝑢2 𝑚,…,𝐴𝑢𝑛−1 𝑚 ,𝐴𝑢𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚}} ∪ {𝐴𝑣1 1,𝐴𝑣2 1,𝐴𝑣3 1 …, ,𝐴𝑣𝑖 𝑗 ,…,𝐴𝑣1 𝑚,𝐴𝑣2 𝑚,…,𝐴𝑣𝑛−1 𝑚 ,𝐴𝑣𝑛 𝑚; 𝑖 ∈ {1,2,3,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚}} ∪ {𝑢1 1𝑣1 1,𝑢2 1𝑣2 1,…,𝑢𝑖 𝑗 𝑣𝑖 𝑗 ,…,𝑢𝑛−1 𝑚 𝑣𝑛−1 𝑚 ,𝑢𝑛 𝑚𝑣𝑛 𝑚; }.hence |𝑉(⋃ ℱ𝑛𝑗=𝑚 )| = 2𝑚𝑛 + 𝑚, |𝐸(⋃ ℱ𝑛𝑗=𝑚 )| = 3𝑚𝑛. we will prove that 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3. based on arumugam theorem [4], the local antimagic chromatic number of friendship graph is 𝜒𝑙𝑎(ℱ𝑛) = 3 . furthermore, based on remark 2.1 we have 𝜒𝑙𝑎(⋃ ℱ𝑛𝑖=𝑚 ) ≥ 𝜒𝑙𝑎(ℱ𝑛) so we can see that 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3. furthermore, we will prove that the upper bound is 3, denoted𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3, we define the labeling by the following. 𝑓(𝐴𝑢𝑖 𝑗 ) = { 𝑚(𝑖 − 1) + 𝑗;𝑖 ∈ {1,3,5,…,𝑛 − 1}; 𝑗 ∈ {1,2,3,…,𝑚} 𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {2,4,6,…,𝑛};𝑗 ∈ {1,2,3,…,𝑚} 𝑓(𝐴𝑣𝑖 𝑗 ) = { 𝑚(𝑖 − 1) + 𝑗 + 2𝑚𝑛 ;𝑖 ∈ {1,3,5,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 𝑚𝑖 − 𝑗 + 1 + 2𝑚𝑛;𝑖 ∈ {2,4,6,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚} 𝑓(𝑢𝑖 𝑗 𝑣𝑖 𝑗 ) = { 2𝑚𝑛 + 1 − 𝑚(𝑖 − 1) − 𝑗 ; 𝑖 ∈ {1,3,5,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 2𝑚𝑛 − 𝑚𝑖 + 𝑗;𝑖 ∈ {2,4,6,…,𝑛}; 𝑗 ∈ {1,2,3,…,𝑚} based on the labeling above, we have the vertex weights as follows: 𝑊1(𝐴) = 3𝑚𝑛 2 + 𝑛 𝑊2(𝑢𝑖 𝑗 ) = 2𝑚𝑛 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,2,3,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 𝑊3(𝑣𝑖 𝑗 ) = 4𝑚𝑛 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,2,3,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} hence from the above, it easy to see that the vertex weight 𝑊 = {3𝑚𝑛2 + 𝑛, 2𝑚𝑛 + 1,4𝑚𝑛 + 1} contains 3 element which induces a proper vertex coloring of ⋃ 𝐶𝑛𝑗=𝑚 , so𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3.therefore 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3 and 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3, it concludes𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) = 3. ∎ 3. concluding remarks all results in this paper are the vertex local antimagic chromatic number of disjoint union of the path, cycle, star, and friendship graph. the vertex local antimagic chromatic number of all graphs in this paper has attended the lower bound. open problem. find the vertex local antimagic chromatic number on disjoint union of cycle and friendship when the number of their vertices is odd, disjoint union of a star when the number of copies is even where the number of vertice sis odd, and also find the vertex local antimagic of disjoint union of another graph. marsidi and i. h. agusti the local antimagic on disjoint union of some family graphs 75 acknowledgement we gratefully acknowledge the support from cgant university of jember and ikip pgri jember of the 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, l. lesniak, and p. zhang, graphs & digraphs 6th edition. crc press: taylor & francis group, 2016 [3] d. zaenab, d. adyanti, a. fanani, and n. ulinnuha, “aplikasi graph coloring pada penjadwalan perkuliahan di fakultas sains dan teknologi uin sunan ampel surabaya”, mantik, vol. 2, no. 1, pp. 30-39, october 2016. [4] j. l. gross, j. yellen, and p. zhang, handbook of graph theory second edition. crc press taylor and francis group, 2014 [5] s. arumugam s, k. premalatha, m. baca and a. semanicova-fenovcikova, "local antimagic vertex colouring of a graph", graphs and combinatorics volume 33, issue 2, pp. 275-285, 2017 [6] i. h. agustin, dafik, m. hasan, r. alfarisi r, and r. m. prihandini, "local edge antimagic colouring of graphs". in far east journal of mathematical sciences, 102(9):1925-1941, 2017 [7] i. h. agustin, s. dafik, r. alfarisi, e.y. kurniawati, "the construction of super local edge antimagic total coloring by using an eavl technique accepted", 2017 [8] i. h. agustin, s. dafik, e. r. ermita, r. alfarisi, "on the total local edge super antimagicness of special graph and graph with pendant edge accepted, 2017 [9] e. y. kurniawati, i. h. agustin, dafik, and r. alfarisi. super local edge antimagic total colouring of {p}_{n}\vartriangleright h, journal of physics: conference series, volume 1008, issue 1, 2018 [10] e. y. kurniawati, i. h. agustin, dafik, r. alfarisi, and marsidi, "on the local edge antimagic total chromatic number of amalgamation of graphs", aip conference proceedings, 2018 [11] yung-ling lai and g. j. chang, "on the profile of the corona of two graphs, information processing letters, vol. 89, issue 6, pp. 287-292, 2004 https://link.springer.com/journal/373/33/2/page/1 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   14    aplikasi metode adams bashforth-moulton (abm) pada model penyakit kanker kuzairi1, tony yulianto2 , lilik safitri3 jurusan matematika, fakultas matematika dan ilmu pengetahuan alam, universitas islam madura (uim) jl. bettet no. 04, pamekasan, madura 60111 indonesia email: kuzairi81@gmail.com 1, liliksafitri@yahoo.co.id 2 abstrak kanker merupakan penyakit yang mematikan yang ditandai dengan pertumbuhan sel-sel yang abnormal, pertumbuhan tersebut berlangsung terus – menerus sehingga terbentuklah tumor. tumor dibagi menjadi dua bagian yaitu tumor jinak dan tumor ganas. tumor ganas merupakan istilah umum untuk penyakit kanker. penyakit kanker mempunyai model matematika berupa sistem persamaan differensial, untuk itu dibutuhkan sebuah metode untuk mendapatkan solusi dari sistem persamaan differensial tersebut. metode yang digunakan adalah metode numerik yakni metode adams bashforth moulton (abm) orde satu, dua, tiga, dan empat. dari hasil penelitian, dapat disimpulkan bahwa pada permasalahan model penyakit kanker, metode abm orde tiga lebih baik dibandingkan metode abm orde satu, orde dua, dan orde empat. hal ini dapat dilihat pada grafik simulasi menggunakan abm orde tiga, menunjukkan bahwa seiring bertambahnya waktu populasi sel kekebalan efektor dan populasi molekul efektor semakin meningkat kemudian stabil. populasi sel kekebalan efektor stabil pada angka 33.3336, sedangkan populasi molekul efektor stabil pada lingkup angka 33.333, dikatakan berada pada lingkup 33.333 karena perubahan populasi molekul efektor tidak dapat diketahui dengan pasti. sedangkan populasi sel kanker tetap bernilai 0 pada tiap iterasi (stabil) yakni tetap berada dalam kondisi bebas kanker. kata kunci: kanker, metode adams bashforth-moulton (abm), konvergensi, stabilitas, konsistensi abstract cancer is a deadly disease that is characterized by the growth of abnormal cells, the growth is ongoing, forming a tumor. tumors are divided into two parts, namely benign and malignant tumors. malignant tumors are a general term for cancer. the disease of cancer has a mathematical model in the form of a system of differential equations, for it required a method to obtain the solution of the system of differential equations. the method used is the method of numerical methods bashforth adams moulton (abm) order one, two, three, and four. from the results of this study concluded that the method abm order three better than the method abm first order, second order and fourth order at issue models of cancer, it can be seen in the graphic simulation using abm order three, it shows that increasing time population of immune effector cells and a population of effector molecules increased and then stabilized. the population of immune effector cells stabilized at 33.3336, while the population of the effector molecule is stable in the scope of the numbers 33,333, 33,333 are said to be in scope for changes in population effector molecule can not be known with certainty. while the population of cancer cells remains at 0 at each iteration (stable) remains in a state that is free of cancer. keywords: cancer, differential equation system, adams bashforth-moulton (abm) method, convergence, stability, consistency jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   15    1. pendahuluan kanker merupakan salah satu penyakit yang mematikan yang ditandai dengan pertumbuhan sel-sel abnormal dan membelah secara terus-menerus. pembelahan secara terus–menerus tersebut menjadi tidak terkendali dan membentuk sel-sel tumor. sel– sel tumor kemudian menginvasi bagian di sekitarnya dan bermetastase ke jaringan dan organ–organ lain yang ada di dalam tubuh. tumor dibagi menjadi dua golongan besar yaitu tumor jinak (benign) dan tumor ganas (malignant). kanker merupakan istilah umum untuk tumor ganas [6]. menurut data globocan (iarc) tahun 2012 diketahui bahwa kanker payudara merupakan penyakit kanker dengan persentase kasus baru (setelah dikontrol oleh umur) tertinggi, yaitu sebesar 43,3%, dan persentase kematian (setelah dikontrol oleh umur) akibat kanker payudara sebesar 12,9%. kanker paru–paru tidak hanya merupakan jenis kanker dengan kasus baru tertinggi dan penyebab utama kematian akibat kanker pada penduduk laki–laki, namun kanker paru juga memiliki persentase kasus baru cukup tinggi pada penduduk perempuan, yaitu sebesar 13,6% dan kematian akibat kanker paru–paru sebesar 11,1%. data globocan tersebut menunjukkan bahwa kasus baru dan kematian akibat kanker hati pada penduduk laki–laki maupun perempuan memiliki persentase yang hampir berimbang, sedangkan kanker payudara dan kanker prostat memiliki persentase kematian yang jauh lebih rendah dibandingkan dengan persentase kasus baru, sehingga jika penyakit kanker tersebut dapat dideteksi dan ditangani sejak dini maka kemungkinan sembuh akan lebih tinggi [3]. model matematika yang dapat digunakan untuk memodelkan penyakit kanker adalah model matematika kp yang dikembangkan oleh kirschner dan panneta [4]. model matematika kp termasuk model matematika non linear sehingga sulit diselesaikan dengan metode analitik. hal tersebut menyebabkan perlu adanya metode numerik yang digunakan untuk mempermudah penelitian ini. metode numerik adalah teknik yang digunakan untuk memformulasikan persoalan matematik sehingga dapat dipecahkan dengan operasi perhitungan/aritmetika biasa (tambah, kurang, kali, dan bagi). metode artinya cara, sedangkan numerik artinya angka. jadi metode numerik secara harfiah berarti cara berhitung dengan menggunakan angka–angka [5]. metode numerik yang digunakan pada penelitian ini adalah metode adams bashforth–moulton dan runge kutta. metode adams bashforth–moulton merupakan metode numerik yang memiliki banyak langkah (multi step) atau biasa disebut sebagai metode predictor-korektor karena dalam penyelesaiannya digunakan persamaan prediktor dan persamaan korektor. metode adams bashfoth-moulton dapat digunakan tanpa harus mencari turunan-turunan fungsinya terlebih dahulu, melainkan langsung menggunakan persamaan prediktorkorektor, karena metode ini banyak langkah (multi step), maka dibutuhkan beberapa solusi awal yang dapat diperoleh dari metode one step menggunakan metode runge–kutta [1]. metode runge–kutta adalah alternatif dari metode deret taylor yang tidak membutuhkan perhitungan turunan. metode ini berusaha mendapatkan derajat ketelitian yang lebih tinggi, dan sekaligus menghindarkan keperluan mencari turunan yang lebih tinggi dengan jalan mengevaluasi fungsi f x,y pada titik terpilih dalam setiap selang langkah. metode runge-kutta adalah metode pdb yang paling popular karena banyak dipakai dalam praktek [5]. 2. bahan dan metode 2.1 bahan dan alat dalam penelitian menggunakan windows 7 dan software pendukung komputasi yaitu matlab r2009a, jaringan wifi dan koneksi internet. 2.2 metode pada sub bab ini akan dijelaskan tentang metode yang digunakan dalam penelitian ini disertai dengan pustaka yang mendasari teori dalam penelitian ini, seperti penelitian sebelumnya, pengertian kanker dan metode adams bashforth–moulton (abm). adapun untuk langkah-langkah dalam penelitian ini dapat dilihat dalam gambar 1. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   16    gambar 1. flowchart penelitian 2.3 penelitian sebelumnya berdasarkan peneliti yanse [8] dengan penelitian yang berjudul “efektivitas metode adams bashforth-moulton order sembilan dalam menganalisis model penyebaran penyakit demam berdarah dengue (dbd)” menghasilkan formula metode adams bashforth–moulton order sembilan yang dapat digunakan sebagai acuan bagi peneliti lain untuk menemukan formula metode adams bashforth-moulton dengan order yang lebih tinggi. sedangkan untuk penelitian yang dilakukan oleh apriadi, prihandono, dan noviani [1] dengan penelitian yang berjudul “metode adams-bashforth-moulton dalam penyelesaian persamaan diferensial non linear” menyatakan bahwa solusi dari masalah nilai awal dengan menggunakan metode adams bashforth–moulton orde empat adalah berbentuk . langkah awal yang dilakukan adalah menggunakan metode runge–kutta orde empat untuk memperoleh tiga solusi awal , , dan pada titik sebelum . kemudian digunakan metode adams bashforth–moulton orde empat untuk memprediksi nilai . kemudian nilai prediksi tersebut diperbaiki menggunakan metode adams-moulton orde empat. 2.4 pengertian kanker penyakit kanker adalah penyakit yang timbul akibat pertumbuhan tidak normal sel jaringan tubuh yang berubah menjadi sel kanker, sedangkan tumor adalah kondisi dimana pertumbuhan sel tidak normal sehingga membentuk suatu lesi atau dalam banyak kasus, benjolan di tubuh. tumor terbagi menjadi dua, yaitu tumor jinak dan tumor ganas. tumor jinak memiliki ciri–ciri, yaitu tumbuh secara terbatas, memiliki selubung, tidak menyebar dan bila dioperasi dapat dikeluarkan secara utuh sehingga dapat sembuh sempurna, sedangkan tumor ganas memiliki ciri–ciri, yaitu dapat menyusup ke jaringan sekitarnya, dan sel kanker dapat ditemukan pada pertumbuhan tumor tersebut [3]. penyakit kanker merupakan penyebab kematian utama di seluruh dunia. pada tahun 2012, sekitar 8,2 juta kematian disebabkan oleh kanker. kanker paru–paru, hati, perut, kolorektal, dan kanker payudara adalah penyebab terbesar kematian akibat kanker setiap tahunnya. pada setiap tahunnya terdapat lebih dari 60% seseorang terjangkit kanker dan sekitar 70% terjadi kematian akibat kanker setiap tahunnya yang terjadi di afrika, asia, dan amerika tengah dan selatan. diperkirakan kasus kanker tahunan akan meningkat dari 14 juta pada 2012 menjadi 22 juta dalam dua dekade berikutnya, sehingga penanganan pengobatan kanker harus dilakukan dengan tepat [3]. 3. model matematika penyakit kanker berikut ini model yang dikembangkan kirschner dan panneta (1998) yang dikenal dengan model kp [4]. mula studi literatur  analisa metode adams bashforth‐moulton orde  satu sampai orde empat  aplikasi metode adams bashforth‐moulton (abm)   pada model penyakit kanker  simulasi  validasi hasil simulasi  penarikan kesimpulan  selesai  cek konvergensi, stabilitas, dan konsistensi  jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   17    (1) 1 (2) (3) 3.1 persamaan predictor metode adams bashforth–moulton a. persamaan predictor metode adams bashforth-moulton orde satu : b. persamaan predictor metode adams bashforth-moulton orde dua : 2 3 c. persamaan predictor metode adams bashforth-moulton orde tiga : 12 5 16 23 d. persamaan predictor metode adams bashforth-moulton orde empat : 24 9 37 59 55 3.2 persamaan corrector metode adams bashforth–moulton a. persamaan corrector metode adams bashforth-moulton orde satu : b. persamaan corrector metode adams bashforth-moulton orde dua : ( c. persamaan corrector metode adams bashforth-moulton orde tiga : 12 8 5 d. persamaan corrector metode adams bashforth-moulton orde empat : 24 5 19 9 3.3 konvergensi, stabilitas, dan konsistensi kesuksesan solusi numerik diukur berdasar kriteria konvergensi, konsistensi, serta stabilitas. konvergensi berhubungan dengan besarnya penyimpangan solusi pendekatan oleh metode numerik terhadap solusi eksak atau solusi analitik (closed form) [7]. konvergensi, konsistensi, dan stabilitas pada metode beda hingga akan dijelaskan sebagai berikut: a. konvergensi pada metode beda hingga kriteria konvergen dipahami sebagai kriteria dimana solusi metode beda hingga (tanpa hadirnya round off error) merupakan solusi pendekatan pdp, jika → 0 dan ∆ → 0. b. stabilitas pada metode beda hingga ada dua kriteria lain yang diasosiasikan dengan kriteria konvergen, yaitu: stabilitas dan konsistensi.kriteria stabilitas merupakan kondisi perlu dan cukup agar diperoleh solusi konvergen, sedang kriteria konsistensi merupakan kondisi ideal dimana solusi metode beda hingga sesuai dengan solusi pdp yang diharapkan. terminologi stabilitas menunjukkan karakteristik persamaan differensial tertentu jika ∆t→0 serta berhubungan dengan amplifikasi solusi selama proses komputasi. jika amplifikasi solusi semakin besar, maka proses komputasi akan divergen dan tidak memperoleh hasil (tidak konvergen). bisa jadi solusi divergen ini dipengaruhi oleh amplifikasi yang terlalu besar selama komputasi. di lain pihak, amplifikasi yang besar belum tentu tidak menghasilkan solusi konvergen. amplifikasi yang sangat besar menunjukkan bahwa stabilitas komputasi sangat rendah. c. konsistensi pada metode beda hingga terminologi konsistensi menunjukkan bahwa solusi dengan metode beda hingga merupakan pendekatan solusi pdp analitik seperti diharapkan, bukan solusi persamaan yang lain. jika → 0 dan ∆ → 0, maka solusi dengan metode beda hingga sama dengan solusi analitik pdp. pada umumnya solusi dengan metode beda hingga akan sesuai solusi pdp, sehingga kriteria konsistensi dengan sendirinya terpenuhi (taken for granted). jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   18    4. hasil dan pembahasan 4.1 algoritma penyelesaian metode adams bashforth-moulton (abm) berikut ini algoritma penyelesaian metode adams bashforth-moulton (abm) orde satu : 0,05 0 0,03 33,333 0,1245 , , , 1 0,001392844 (4) 1 0,18 0 1 10 0 1 33,333 0 10 0 0 (5) 0,1245 33,333 0 10 0 1 0,03 33,333 10 (6) substitusikan persamaan (4), (5), dan (6) ke dalam masing-masing metode ab (prediktor) berikut : 33,333 0,2 0,001392844 33,333 0,000278568 33,33327857 0 0,2 0 0 33,333 0,2 0,00001 33,333 0,000002 33,33302 setelah itu dicari nilai : , , dan , , , 0,05 0 0,03 33,33327857 0,1245 , , , 1 0,0013845 (7) , , , 1 0,18 0 1 10 0 1 33,33327857 0 10 0 0 (8) , , , 0,1245 33,33327857 0 10 0 1 0,03 33,33302 0,0000094 (9) substitusikan persamaan (7), (8), dan (9) ke dalam masing-masing metode am (korektor) berikut: 33,333 0,2 0,0013845 33,3332769 0 33,333 0,2 0,00000 33,33300188 dihitung galat relatif: |33,3332769 33,33327857| |33,3332769| 5.10 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   19    0 |33,33300188 33,33302| |33,33300188| 5,43 .10 dari hasil tersebut terlihat bahwa galat relatif lebih kecil dari kriteria pemberhentian 10 , sehingga tidak perlu dlakukan pemilihan beda , iterasi terus dilakukan sampai 10 iterasi, sedangkan untuk 2-10 dapat dilihat nilai , , dan pada grafik hasil simulasi 4.2 hasil simulasi berikut ini grafik solusi numerik menggunakan software matlab r2009a gambar 2 grafik populasi sel kekebalan efektor per 0,2 satuan waktu pada gambar 2 menunjukkan bahwa ketika menggunakan : 1. abm orde satu, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat ke titik maksimum yaitu t=2 satuan waktu. 2. abm orde dua, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat di awal yaitu pada t= 0,2 satuan waktu namun beberapa waktu kemudian populasi sel kekebalan efektor akan menurun menuju ke titik minimum t=2 satuan waktu. 3. abm orde tiga, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat sampai titik t=0,4 satuan waktu kemudian stabil sampai t=2 satuan waktu. 4. abm orde empat, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat sampai t=0,6 satuan waktu, kemudian stabil dan menurun di akhir waktu yaitu pada t=2 satuan waktu. gambar 3 grafik populasi sel kanker per 0,2 satuan waktu pada gambar 3 menunjukkan bahwa walaupun waktu bertambah tetapi populasi sel kanker tidak ada perubahan yakni tetap bernilai 0 (bebas kanker) yang artinya tidak ada populasi sel kanker ketika menggunakan abm orde satu, orde dua, orde tiga, dan orde empat. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   20    gambar 4 grafik populasi molekul efektor per 0,2 satuan waktu pada gambar 4 menunjukkan bahwa ketika menggunakan : 1. abm orde satu, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat ke titik maksimum yaitu t=2 satuan waktu. peningkatan ini terjadi pada tiap iterasi dengan peningkatan yang sangat kecil sehingga pada grafik tidak terlihat perubahannya. 2. abm orde dua, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat di awal yaitu pada t= 0,2 satuan waktu namun beberapa waktu kemudian populasi sel kekebalan efektor akan menurun menuju ke titik minimum t=2 satuan waktu. 3. abm orde tiga, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat sampai titik t=0,4 satuan waktu kemudian stabil sampai t=2 satuan waktu 4. abm orde empat, setiap bertambahnya waktu, populasi sel kekebalan efektor akan meningkat sampai t=0,6 satuan waktu, kemudian stabil dan menurun di akhir waktu yaitu pada t=2 satuan waktu. tampilan gambar 2 dengan gambar 4 terlihat sama, tetapi berbeda, perbedaannya terletak pada nilai sumbu dan nilai sumbu . pada gambar 2 menjelaskan bahwa nilai sumbu berbeda tiap titik, jelas bahwa nilai setiap titik itu berbeda pada suatu sumbu, tetapi tidak demikian pada gambar 4, pada gambar 4 memperlihatkan nilai pada semua titik sama yaitu 33,333 , hal ini terjadi karena pembulatan menggunakan software matlab r2009a sampai 4 digit angka dibelakang koma. sehingga dapat dikatakan bahwa walaupun pada nilai sumbu terlihat sama, tetapi ada perbedaan nilai dari . karena nilai setiap titik itu berbeda pada suatu sumbu. 3.3 validasi hasil simulasi validasi hasil simulasi yaitu dengan cara membandingkan hasil dari metode adams bashforth-moulton (abm) dengan nilai eksaknya yang diperoleh dari metode analitik. gambar 5 grafik populasi sel kanker , sel kanker , dan molekul efektor per 0,2 satuan waktu menggunakan metode analitik pada gambar 5 menunjukkan hasil analitik dalam 10 iterasi pada populasi sel kanker sebagai berikut: 33.3331, 33.3331, 33.3332, 33.3333, 33.3333, 33.3334, 33.3335, 33.3336, 33.3336, 33.3337, sedangkan pada populasi sel kanker tidak ada perubahan yakni tetap bermilai 0 dalam 10 iterasi tersebut, sedangkan pada populasi molekul efektor juga tidak ada perubahan yakni tetap berada dalam keadaan stabil yaitu 33,333. dapat disimpulkan bahwa hasil simulasi yang telah dilakukan valid karena solusi numerik menggunakan metode abm orde satu, dua, tiga, dan empat pada populasi sel kekebalan efektor (e) dan populasi molekul jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 06 issn: 2527-3159 e-issn: 2527-3167   21    efektor (c) mendekati solusi eksak, sedangkan pada populasi sel kanker (t) solusi numeriknya sama dengan solusi eksaknya. 4.3 cek konvergensi, kestabilan, dan konsistensi konvergensi digunakan sebagai parameter (alat estimasi) untuk memperkirakan bilamana problem yang dihadapi memiliki solusi atau jawab yang “mendekati solusi eksak”, dapat diterima dengan prosentase galat tertentu [2]. dapat disimpulkan bahwa solusi numerik di atas konvergen karena solusi numerik menggunakan metode abm mendekati solusi eksak dan memiliki prosentase galat yang relatif kecil. ada dua kriteria lain yang diasosiasikan dengan kriteria konvergen, yaitu stabilitas dan konsistensi. stabilitas adalah adalah kondisi perlu dan cukup agar diperoleh solusi konvergen [7], sedangkan konsistensi merupakan kondisi ideal dimana solusi dari metode adams bashfrorth-moulton (abm) sesuai dengan solusi pdb yang diharapkan. dapat disimpulkan bahwa solusi numerik di atas stabil karena solusi mumerik di atas konvergen, dan juga dapat dikatakan konsisten karena solusi numerik di atas mendekati solusi eksak. 5. simpulan dari penelitian di atas dapat disimpulkan bahwa metode abm orde tiga lebih baik dibandingkan metode abm orde satu, orde dua, dan orde empat pada permasalahan model penyakit kanker, hal ini dapat dilihat pada grafik simulasi menggunakan abm orde tiga, pada grafik simulasi abm orde tiga menunjukkan seiring bertambahnya waktu populasi sel kekebalan efektor dan populasi molekul efektor semakin meningkat kemudian stabil. populasi sel kekebalan efektor stabil pada angka 33.3336, sedangkan populasi molekul efektor stabil pada lingkup angka 33.333, dikatakan berada pada lingkup 33.333 karena perubahan populasi molekul efektor tidak dapat diketahui dengan pasti. sedangkan populasi sel kanker tetap bernilai 0 pada tiap iterasi (stabil) yakni tetap berada dalam kondisi bebas kanker. 6. daftar pustaka [1] apriadi, prihandono, b., & noviani, e. (2014). metode adams bashforthmoulton dalam penyelesaian persamaan diferensial non-linear. buletin ilmiah mat. stat. dan terapannya (bimaster), 107-116. [2] intellectual property of dr. ir. setijo bismo, dea., tgp-ftui. in modul. [3] kementerian kesehatan ri. (2015). info data dan informasi kementerian kesehatan ri. stop kanker, pp. 1-6. [4] lestari, d. (2013). model matematika terapi gen untuk perawatan penyakit kanker. yogyakarta : universitas negeri yogyakarta. [5] munir, r. (2010). metode numerik. bandung : informatika bandung. [6] pertamawati, l. (2013). aplikasi kendali optimum pada kemoterapi kanker. in skripsi. yogyakarta : universitas islam negeri sunan kalijaga. [7] rohemah, s. (2015). penyelesaian model perpindahan panas pada substrat asam nukleat menggunakan metode leapfrog. in skripsi. pamekasan : universitas islam madura. [8] yanse, n. m. (2012). efektivitas metode adams bashforth-moulton order sembilan dalam menganalisis model penyebaran penyakit demam berdarah dengue (dbd). in skripsi. jember : universitas jember. pengembangan model fuzzy project evaluation untuk analisis kelayakan finansial pendirian pabrik baru 22 | p a g e pengembangan model fuzzy project evaluation untuk analisis kelayakan finansial pendirian pabrik baru yuniar farida prodi matematika fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya email : yuniar.farida@uinsby.ac.id abstrak untuk rencana pembangunan suatu pabrik baru, aspek finansial merupakan aspek terpenting dalam evaluasi kelayakannya. dikatakan demikian, karena sekalipun aspek lain tergolong layak, jika studi aspek finansial memberikan hasil yang tidak layak, maka usulan proyek akan ditolak karena tidak memberikan manfaat ekonomi. dalam penelitian ini net present value (npv) digunakan sebagai metode evaluasi kelayakan finansial rencana pendirian pabrik pt. x. dalam perhitungan npv, salah satu faktor yang krusial adalah tarif diskonto atau discount rate yang berlaku pada masa pengembalian investasi suatu proyek. npv suatu proyek harus dihitung dengan discount rate konstan sampai masa pengembalian investasi, meski pada kenyataannya faktor – faktor yang mempengaruhi discount rate setiap tahun tidak selalu sama, akibatnya nilai npv menjadi samar (fuzzy). untuk mengatasi hal tersebut, maka dilakukan suatu pemodelan untuk mendekati nilai discount rate yang tepat. dalam penelitian ini discount rate dihitung berdasarkan nilai wacc (weighted average cost of capital) yang merupakan gabungan dari struktur modal, yaitu hutang dan ekuitas. untuk memperoleh nilai wacc yang tepat, dilakukan pendekatan dengan menggunakan triangular fuzzy number (tfn). adapun penggunaan fuzzy dilakukan karena wacc mengandung unsur ketidakpastian yang tinggi, yang bisa membuat perhitungan wacc dengan metode konvensional menjadi samar/kabur. dari hasil perhitungan menggunakan tfn, diperoleh nilai wacc sebesar 13.64 % dan menghasilkan npv sebesar 6,430,464,000,000. sedangkan nilai wacc deterministik yang dihasilkan evaluator sebesar 13.72 % dan menghasilkan npv sebesar 6,358,310,540,000. kata kunci : evaluasi kelayakan investasi/proyek, net present value (npv), discount rate, weighted average cost of capital (wacc), triangular fuzzy number (tfn) abstract in order to decide a new development of industry, financial feasibility is the most important factor to consider. the financial aspects will determine whether a project should be feasible or not. this study proposes a net present value (npv) method for financial evaluation of the development of new industry, in which the setting up of discount rate. long period of project is an uncertain and critical factor that is assume to be constant factor. in reality, discount rate is function of variables of time and consist of imprecise factors (fuzzy). base on this fact, this research construct a model for calculation of discount rate that allows one to consider the fuzzyness on it. to do so, the discount rate is computed based on weighted average cost of capital (wacc) that imply aggregation of capital structure, debts and equities. in order to obtain the precisely wacc, this study utilize triangular fuzzy number (tfn) model. the result of study conclude that wacc is 13.64% with the result of npv to 6,430,464,000,000 rupiahs. based on previous wacc (deterministic), its value is 13.72% with 6,358,310,540,000 rupiahs of npv. key words : invesment/project feasibility evaluation, net present value (npv), discount rate, weighted average cost of capital (wacc), triangular fuzzy number (tfn) mailto:yuniar.farida@uinsby.ac.id 23 | p a g e 1. pendahuluan investasi merupakan keputusan yang sangat beresiko, karena mengeluarkan uang pada saat sekarang dengan tujuan untuk mendapatkan keuntungan atau manfaat yang lebih besar di masa mendatang. investasi bisa dilakukan untuk bisnis maupun proyek. oleh karena dana yang dikeluarkan untuk investasi jumlahnya besar, sedangkan manfaatnya baru akan diterima di masa mendatang, maka selalu ada resiko. semakin jauh jarak antara waktu pelaksanaan investasi dan waktu pemulihan investasi, akan semakin besar pula resiko yang dihadapi. berbagai perubahan dapat terjadi yang mungkin saja besar pengaruhnya atas operasi proyek, seperti inflasi, perubahan tingkat suku bunga, perubahan nilai tukar valuta asing, persaingan global, kebijakan pemerintah dan perubahan cita rasa konsumen. untuk menghilangkan atau paling tidak memperkecil resiko serta untuk memastikan besarnya manfaat atau keuntungan yang diinginkan bisa diperoleh, maka diperlukan studi kelayakan. secara konvensional kelayakan finansial proyek publik maupun swasta dievaluasi berdasarkan kriteria ekonomis – finansial dengan rasio manfaat dan biaya (cost – benefit analysis), net present value (npv), tingkat pengembalian pendapatan (irr), pay back period, titik pulang pokok (bep) dan sebagainya. pada penelitian ini metode evaluasi kelayakan financial yang digunakan adalan net present value (npv). penggunaan alat evaluasi proyek tersebut digunakan mengingat kemudahan penerapannya dalam memberikan penilaian yang obyektif terhadap arus kas mendatang suatu proyek yang dinyatakan nilainya dalam waktu saat ini/sekarang, sehingga metode ini paling banyak digunakan dalam penilaian kelayakan suatu proyek [9]. dalam perhitungan npv, salah satu faktor yang krusial adalah tarif diskonto atau discount rate yang berlaku pada masa pengembalian investasi suatu proyek. npv suatu proyek harus dihitung dengan discount rate konstan sampai masa pengembalian investasi, meski pada kenyataannya faktor – faktor yang mempengaruhi discount rate setiap tahun tidak selalu sama, akibatnya nilai npv menjadi samar (fuzzy). untuk mengatasi hal tersebut, maka dilakukan suatu pemodelan untuk mendekati nilai discount rate yang tepat. dalam penelitian ini discount rate dihitung berdasarkan nilai wacc (weighted average cost of capital) yang merupakan gabungan dari struktur modal, yaitu hutang dan ekuitas. untuk memperoleh nilai wacc yang tepat, dilakukan pendekatan dengan menggunakan triangular fuzzy number (tfn). adapun penggunaan fuzzy dilakukan karena fuzzy mampu mengkuantifikasi pemikiran dan persepsi manusia yang kabur terhadap sesuatu, dimana dalam wacc terefleksi tingkat resiko yang mengandung unsur ketidakpastian yang tinggi, yang bisa membuat perhitungan wacc dengan alat evaluasi konvensional menjadi samar/kabur. 2. weighted average cost of capital (wacc) bila suatu proyek dibiayai dengan hutang, implikasinya adalah perusahaan telah menggunakan sebagian dari potensinya untuk mendapatkan hutang baru dengan biaya yang rendah. dalam perkembangan perusahaan pada tahun – tahun berikutnya, suatu saat perusahaan akan terpaksa menambah pemodalan ekuitas untuk menjaga agar ratio hutang jangan menjadi terlalu besar. karena itu, persahaan harus menganggap dirinya sebagai perusahaan berjalan dan biaya modalnya harus dihitung sebagai rata – rata tertimbang (gabungan) dari berbagai dana yang digunakannya yaitu hutang, saham preferen dan ekuitas saham biasa (common stock). modal rata – rata tertimbang (weighted average cost of capital = wacc) merupakan tarif diskonto (discount rate) yang digunakan untuk mendiskonto arus kas. wacc merefleksikan resiko usaha (business risk) dan kapasitas target hutang (target debt capacity) dari suatu perusahaan atau suatu proyek yang akan dinilai [2]. beberapa penelitian tentang penentuan discount rate mengarah pada konsep weighted average cost of capital (wacc), diantaranya dilakukan oleh brigham [3], reghavandra rau [13], machala [10], glenday [7], pinteris [12]. oleh karena itu, dalam penelitian ini discount rate didekati dengan konsep weighted average cost of capital (wacc). secara umum persamaan yang digunakan untuk estimasi wacc adalah : wacc = rd(1-t)(d/v) + re(e/v) (1) dimana : rd = biaya hutang (cost of debt) re = biaya ekuitas (cost of equity) t = tingkat pajak marjinal d = nilai pasar dari hutang 24 | p a g e e = nilai pasar dari ekuitas v = nilai pasar dari perusahaan (v = d + e) 2.1 estimasi biaya hutang (cost of debt) cost of debt / biaya hutang perusahaan menunjukkan biaya yang digunakan untuk melakukan pinjaman dana dari kreditor. untuk mengestimasi biaya hutang perusahaan, dibutuhkan informasi mengenai tingkat suku bunga yang sedang berlaku, resiko default (perusahaan yang berhutang tidak memenuhi kewajibannya) dan tingkat pajak marjinal. untuk mengestimasi besarnya biaya hutang perusahaan sebaiknya berdasarkan informasi pasar sebanyak mungkin [2]. 2.2 estimasi biaya ekuitas perusahaan mengestimasi biaya ekuitas suatu perusahaan adalah sangat sulit. kebalikan dari hutang, biaya ekuitas tidak bisa diobservasi dalam pasar. jenis ekuitas dalam struktur modal antara lain saham biasa. untuk estimasi saham biasa lebih sulit daripada estimasi obligasi dan saham preferen. hal tersebut dikarenakan pada saham biasa, pendapatan masa depan dan harga saham tidak konstan, selalu diharapkan tumbuh/berkembang. sedangkan pada obligasi dan saham preferen, bunga obligasi dan deviden saham preferen diketahui relatif pasti. pendekatan standar yang digunakan untuk menentukan biaya ekuitas adalah model penilaian aktiva modal (capital asset pricing model = capm) yang dirumuskan : re = rf + β(rm – rf) (2) nilai (rm – rf) adalah premi resiko pada rata – rata saham, sedangkan β adalah indeks resiko saham bersangkutan yang sedang dianalisa. 3. metode fuzzy teori fuzzy set yang pertama kali diperkenalkan oleh zadeh [17], telah dikembangkan untuk menyelesaikan permasalahan dimana deskripsi aktivitas, observasi dan penilaian adalah subyektif, tidak pasti dan tidak presisisi. kata “fuzzy” umumnya mengarah pada situasi dimana tidak ada batas dari aktivitas dan penilaian yang dapat didefinisikan secara tepat. perkembangan teori fuzzy selanjutnya diikuti dengan konsep algoritma fuzzy fuzzy decission making [18]. dalam dunia ekonomi, banyak terdapat unsur ketidakpastian (uncertainty), ketidaktepatan (imprecise) dan resiko yang semuanya bersifat fuzzy (samar). dalam analisis kelayakan investasi/proyek, keputusan analisis bahwa investasi/proyek itu layak ataun tidak adalah didasarkan pada hasil telaah proyeksi arus kas bersih sesudah pajak yang diestimasikan akan diterima di masa mendatang selama masa ekonomis proyek. oleh karena studi kelayakan berkenaan dengan waktu yang akan datang, sedangkan waktu yang akan datang tersebut penuh dengan ketidakpastian, maka meskipun hasil studi menyatakan program investasi itu layak, sesungguhnya hasil penilaian tersebut masih merupakan sebuah pendugaan dan tidak ada satu pun jaminan yang pasti bahwa arus kas yang akan datang itu benar – benar akan sama, atau akan lebih besar daripada yang dihitung saat ini. sehingga untuk mengakomodasi adanya unsur ketidakpastian dalam analisa ekonomi, fuzzy banyak digunakan sebagai alat pengambilan keputusan, termasuk pada kasus evaluasi kelayakan investasi/proyek. cengiz kahraman [9] menggunakan metode fuzzy benefit/cost ratio untuk evaluasi kelayakan proyek publik. selain itu, cedric lesage [4], yang menggunakan fuzzy untuk melakukan pendekatan terhadap discounted cash flow, yaitu npv dan irr, dimana dalam penelitian tersebut variabel yang di-fuzy-kan adalah cash flow. peneliti – peneliti lainnya yang menggunakan fuzzy sebagai alat pengambilan keputusan adalah ward, t.l. [16], christer carlsson and robert fuller [6], chiu, c.-y. and park, c.s. [5], boussabaine a.h. and elhag t. [1], peter majlender [11]. 3.1 operasi dalam bilangan fuzzy bilangan fuzzy adalah sebuah fuzzy subset dari bilangan real, menyatakan pengembangan konsep rentang kepercayaan. berdasarkan definisi sebuah triangular fuzzy number (tfn) memiliki ciri-ciri dasar seperti dibawah ini : sebuah bilangan fuzzy ã pada  adalah tfn bila fungsi keanggotaannya ã(x): [0,1] adalah sama dengan :             lainnya uxm mxl muxu lmlx x a , , 0 ),/()( ),/()( ~ (3) dimana l dan u adalah batas bawah dan batas atas bilangan fuzzy ã, sedangkan m adalah nilai tengah. triangular fuzzy number (tfn) merupakan bentuk bilangan fuzzy yang sederhana namun merupakan pilihan yang baik untuk mendekati suatu konsep yang kurang jelas [15]. menurut sanches, pamplona & montevechi [14], 25 | p a g e triangular fuzzy number (tfn) ini merupakan bilangan fuzzy yang paling menarik dan sesuai digunakan untuk merepresentasikan data – data/informasi finansial. sehingga dalam penelitian ini menggunakan tfn. tfn dapat dinotasikan dengan ã = (l, m, u), dan berikut ini adalah hukum operasi dua tfn ã1 = (l1, m1, u1) dan ã2 = (l2, m2, u2). a. penjumlahan bilangan fuzzy        212121 22211121 ,, ,,,, ~~ uummll umlumlaa   (4) b. perkalian bilangan fuzzy        212121 22211121 ,, ,,,, ~~ uummll umlumlaa   (5) c. pengurangan bilangan fuzzy ɵ ã1 ɵ ã2 = (l1, m1, u1) ɵ (l2, m2, u2) = (l1–u2 ,m1–m2, u1– l2) (6) d. pembagian bilangan fuzzy  ã1  ã2 = (l1, m1, u1)  (l2, m2, u2) = (l1/u2, m1/m2, u1/l2) (7) untuk li > 0, mi > 0, ui > 0 e. inversi bilangan fuzzy ã1 -1 = (l1, m1, u1) -1 = (1/l1, 1/m1, 1/u1) (8) untuk li > 0, mi > 0, ui > 0 3.2 pengambilan keputusan pada kondisi yang bersifat fuzzy saat keputusan dilakukan dalam keadaan uncertain, maka keputusan dinyatakan dalam suatu rentang perkiraan. pada kondisi demikian, pengambil keputusan dapat menerima “sedikit pelanggaran” terhadap batasan yang ditentukan. “sedikit pelanggaran” dalam pemrograman fuzzy selanjutnya disebut dengan batas toleransi atau batas deviasi yang dinyatakan dalam rentang batas toleransi bawah sampai batas toleransi atas. besarnya tingkat penerimaan keputusan fuzzy, baik yang bergerak ke batas bawah maupun batas atas dari target, ditunjukkan oleh suatu fungsi keanggotaan yang dalam bentuk grafis dinyatakan sebagai berikut : gambar 2 fungsi keanggotaan )(xf i 3.3 defuzzifikasi input dari pross defuzzy adalah suatu himpunan fuzzy yang diperoleh dari komposisi aturan – aturan fuzzy, sedangkan output yang dihasilkan merupakan suatu bilangan pada domain himpunan fuzzy tersebut. defuzzifikasi digunakan untuk mendapatkan nilai crisp dari bilangan fuzzy. pada penelitian ini, defuzzifikasi dilakukan dengan menggunakan metode bnp (best nonfuzzy performance). penggunaan metoda bnp lebih sederhana dan praktis, tidak memerlukan preferensi evaluator. nilai bnp dari bilangan fuzzy i r ~ dapat diperoleh dengan persamaan berikut: bnpi = [(uri lri) + (mri lri)]/3 (9) dimana lri dan uri adalah batas bawah dan batas atas bilangan fuzzy i r ~ , sedangkan mri adalah nilai tengah. 4. studi kasus : proyek pendirian pabrik baru pt. x pt x merupakan perusahaan berskala nasional yang memproduksi semen. sejak oktober 1998, pt x. mempunyai kapasitas sebesar 17,2 juta ton semen per tahun dan menguasai + 45% pangsa pasar semen dalam negeri. saat ini perusahaan tersebut tengah merencanakan ekspansi produksi dengan mendirikan sebuah pabrik baru. adapun sumber modal yang digunakan oleh pt. x dalam mendanai proyek tersebut berasal dari : ekuitas perusahaan (self financing) sebesar 31.03% pinjaman dari bank komersial (commercial bank loan) sebesar 39.68 % kredit ekspor sebesar 29.29 % l m u 1 µ[c] gambar 1 triangular fuzzy number 0 )]([ xfi  1 0 min i f max i f i f ~ )(xf i 26 | p a g e sehingga wacc proyek pt x dipengaruhi oleh ketiga sumber modal yang digunakan dalam membiayai proyeknya. wacc dihitung berdasarkan persamaan (2) yang dikembangkan menjadi : wacc = % sf*re + % bl*rdbl*(1-tax) + %ke*rdke*(1-tax) (10) berikut ini data proyek pt x : tabel 1. data sumber modal proyek pt x data proyek self financing 31.03% commercial bank loan 39.68% kredit ekspor 29.29% rd project (bl) 16.00% rd project (ke) 15.00% tax 29.77% β 0.79 rm 21.76% rf 12.80% berdasarkan data diatas, maka biaya ekuitas dapat dihitung berdasarkan persamaan (2) sehingga : re = 12.8 % + 0.79*(21.76 % 12.8 %) = 19.91 % sedangkan nilai wacc berdasarkan persamaan 5.1 diatas adalah : wacc = (0.3103*0.1991) + (0.3968*0.16)* (1 – 0.2977) + (0.2929*0.15)*(1 – 0.2977) = 13.72 nilai wacc diatas merupakan nilai hasil perhitungan sebagaimana yang dilakukan evaluator dengan menggunakan nilai – nilai yang deterministik. 4.1 penentuan wacc dengan metode fuzzy dalam perumusan wacc (persamaan 10) diatas, ada beberapa parameter yang bersifat fuzzy, antara lain re (cost of equitas/biaya ekuitas), rd (cost of debt/biaya hutang), serta pajak (tax). parameter – parameter tersebut dikatakan mengandung unsur ketidakpastian (uncertainty) karena faktor-faktor yang mempengaruhinya (seperti tingkat suku bunga bank dan obligasi, serta nilai saham perusahaan maupun ihsg) bersifat fluktuatif, sehingga berpengaruh terhadap tingkat pengembalian investasi/proyek tersebut. deviasi disekitar nilai deterministik pada parameter – parameter pembentuk wacc yang diberikan oleh pengambil keputusan adalah : tabel 2 data nilai parameter pembentuk wacc beserta deviasinya data proyek deviasi rf 12.80% -10 % s/d 10 % β 0.79 -10 % s/d 10 % rm 21.76% -10 % s/d 10 % rdbl 16% -10 % s/d 5 % rdke 15% -10 % s/d 5 % pajak 29.77% -10 % s/d 5 % adanya deviasi pada parameter – parameter diatas, maka gambar fungsi keanggotaan untuk setiap parameter adalah sebagai berikut : gambar 3 fungsi keanggotaan )(xf i & deviasi pada parameter : (a) rf (b) β (c) rm (d) rdbl (e) rdke (f) pajak fungsi keanggotaan dari masing – masing parameter di atas berdasarkan persamaan (3) adalah :  fungsi keanggotaan tingkat bebas resiko/risk free rate (rf) :             lainnya uxm mxl x x x rf , , 0 ),8.1208.14/()08.14( ),52.118.12/()52.11(  (f) )]([ xfi  1 0 )(xfi 12.8 11.52 14.08 )]([ xfi  1 0 )(xfi 0.79 0.711 0.86 9 (a) (b) )]([ xfi  1 0 )(xfi 21.76 19.58 4 23.93 6 )]([ xfi  1 0 )(xfi 1 6 14.4 16.8 (c) (d) )]([ xfi  1 0 )(xfi 15 13.5 15.75 )]([ xfi  1 0 )(xfi 29.77 26.79 3 31.25 85 (e) 27 | p a g e  fungsi keanggotaan tingkat bebas resiko/risk free rate (rf) :             lainnya uxm mxl x x x , , 0 ),79.0869.0/()869.0( ),711.079.0/()711.0(   fungsi keanggotaan nilai rata-rata tingkat pengembalian ihsg selama periode tertentu (rm)             lainnya uxm mxl x x x rm , , 0 ),76.21936.23/()936.23( ),584.1976.21/()584.19(   fungsi keanggotaan biaya hutang/cost of debt yang berasal dari pinjaman bank komersial (rdbl)             lainnya uxm mxl x x x rdbl , , 0 ),168.16/()8.16( ),4.1416/()4.14(   fungsi keanggotaan biaya hutang/cost of debt yang berasal dari kredit ekspor (rdke) :             lainnya uxm mxl x x x rdke , , 0 ),1575.15/()75.15( ),5.1315/()5.13(   fungsi keanggotaan pajak (tax) :             lainnya uxm mxl x x x tax , , 0 ),77.292585.31/()2585.31( ),793.2677.29/()793.26(  berdasarkan gambar 3 diatas, maka parameter pembentuk wacc yang bersifat fuzzy adalah cost of equity dan cost of debt, dimana nilai cost of equity dihitung berdasarkan nilai tingkat bebas resiko (rf), slope antara ihsg dan ihs perusahaan serta rata – rata tingkat pengembalian ihs perusahaan. sedangkan cost of debt (rd) diperoleh dari interest rate pinjaman bank komersial maupun interest rate kredit ekspor, yang masing – masing dihitung setelah pajak. berikut ini perhitungan nilai fuzzy dari cost of equity dan cost of debt pada pinjaman bank dan kredit ekspor : a. cost of equity/biaya ekuitas bentuk persamaan fuzzy pada parameter biaya ekuitas adalah sebagai berikut :    fmfe rrrr ~~ ~~~   (11) dimana : ),,(~ rfrfrff umlr   08.14;8.12;52.11   uml ,, ~   869.0;79.0;711.0   rmrmrmm umlr ,,~   936.23;76.21;584.19 sehingga diperoleh :  645.22;878.19;254.17~  e r nilai real re diperoleh dengan melakukan defuzzifikasi menggunakan bnp (best nonfuzzy performance), berdasarkan persamaan (9) diperoleh : bnpre = [(22.645–17.254)+(19.878 – 17.254)]/3 + 17.254 = 19.9256 sehingga nilai crisp re adalah 19.9256 % atau 0.199256 b. cost of debt/biaya hutang dari pinjaman bank komersial (rdbl) bentuk persamaan fuzzy pada parameter biaya hutang yang berasal dari pinjaman bank komersial (rdbl) yang dihitung setelah pajak adalah sebagai berikut :      taxtaxtax rdblrdblrdbl umltax umlrdblrdbltaxrdblf ,,1 ,,)1(*(   (12)     312585.0;2977.0;26793.01 168.0;16.0;144.0  rdbl sehingga diperoleh :  1155.0;11237.0;1054.0rdbl nilai real rdbl diperoleh dengan melakukan defuzzifikasi menggunakan bnp (best nonfuzzy performance) : bnprdbl=[(0.1155–0.1054)+(0.11237–.1054)]/3 + 0.1054 = 0.11109 sehingga nilai crisp rdbl adalah 0.11109 atau 11.109% c. cost of debt/biaya hutang dari kredit ekspor (rdke) bentuk persamaan fuzzy pada parameter biaya hutang yang berasal dari kredit ekspor (rdke) yang dihitung setelah pajak adalah sebagai berikut :      taxtaxtax rdkerdkerdke umltax umlrdkerdketaxrdkef ,,1 ,,)1(*(   (13)     312585.0;2977.0;26793.01 1572.0;15.0;135.0  rdke sehingga diperoleh :  1083.0;1053.0;09883.0rdke 28 | p a g e nilai real rdke diperoleh dengan melakukan defuzzifikasi menggunakan bnp (best nonfuzzy performance) : bnprdke = [(0.1083–0.09883) + (0.1053– 0.09883)]/3 + 0.09883 = 0.10415 sehingga nilai crisp rdke adalah 0.10415 atau 10.415 % nilai – nilai fuzzy dari parameter – parameter diatas selanjutnya digunakan untuk menghitung nilai fuzzy wacc. nilai fuzzy wacc diperoleh berdasarkan persamaan :   rerere umlresfwaccf ,,)(   rdblrdblrdbl umlrdblbl ,, ),,( rdkerdkerdke umlrdkeke  (14) sehingga nilai wacc adalah : )22648.0;19878.0;17254.0(3103.0)( waccf )1155.0;11237.0;1054.0(3968.0  )1083.0,1053.0,0993.0(2929.0  sehingga diperoleh : )14782.0;13714.0;1243.0()( waccf nilai real wacc diperoleh dengan melakukan defuzzifikasi menggunakan bnp (best nonfuzzy performance) : bnpwacc = [(0.14782 0.1243)+(0.13714– 0.1243)]/3 + 0.1243 = 1364.146 sehingga diperoleh nilai crisp wacc sebesar 0.1364 atau 13.64 % nilai fuzzy wacc diatas ternyata sedikit berbeda (sedikit lebih rendah) dibandingkan perhitungan wacc deterministik yang ditetapkan oleh tim evaluator atau tim pengambil keputusan pt. x, yakni sebesar 13.72 %. 4.1.1 nilai keanggotaan bilangan fuzzy wacc nilai fuzzy yang dihasilkan merupakan representasi dari nilai – nilai yang diestimasikan pengembilan keputusan berdasarkan tingkat konfidensi tertentu. untuk mengetahui seberapa besar tingkat konfidensi pengambil keputusan terhadap nilai yang diestimasikan, maka dicari derajat keanggotaan dari nilai crips biaya hutang, biaya ekuitas dan wacc yang dihasilkan dari metode fuzzy. untuk mencari derajat/nilai kenggotaan fuzzy, digunakan persamaan (3). berikut ini hasil perhitungan derajat keanggotaan biaya hutang, biaya ekuitas dan wacc : tabel 3. nilai fuzzy biaya hutang, biaya ekuitas dan wacc beserta derajat keanggotaannya l m u x µ [x] µ [x] (nilai batas bawah) (nilai tengah) (nilai batas atas) (nilai crisp) pada l pada u re 0.17254 0.19878 0.22645 0.1993 0.983 rdbl 0.1054 0.11237 0.1155 0.1111 0.8164 rdke 0.09883 0.1053 0.1083 0.1042 0.8223 wacc 0.1243 0.13714 0.14782 0.1364 0.9435 dimana : µ [x] pada l = nilai keanggotaan parameter berada pada level l (rendah/batas bawah) µ [x] pada u = nilai keanggotaan parameter berada pada level u (tinggi/batas atas) dari tabel 3 diatas menunjukkan bahwa pada biaya hutang (bank loan dan kredit ekspor) menghasilkan nilai keanggotaan fuzzy yang besar (mendekati nilai 1) pada daerah batas bawah (nilai low), artinya pengambil keputusan (decision maker) memiliki tingkat konfidensi yang tinggi bahwa pada masa umur ekonomis proyek, kemungkinan terjadi penurunan nilai daripada kenaikan pada parameter tingkat suku bunga bank komersial dan bunga kredit ekspor. sementara untuk biaya ekuitas yang memiliki nilai keanggotaan besar (0.9828) pada daerah batas atas, disebabkan deviasi batas atas dan batas bawah pada nilai β adalah sama, artinya tingkat pengembalian pada ihsg dan ihs perusahaan sangat fluktuatif. sehingga pengambil keputusan cenderung konfiden terjadi kenaikan tingkat pengembalian ihsg dan ihs perusahaan. yang mengakibatkan biaya ekuitas menjadi lebih tinggi dari nilai deterministik yang diestimasi evaluator. namun meski biaya ekuitas menjadi lebih besar, tidak membuat wacc menjadi lebih besar karena proporsi/bobot biaya ekuitas lebih kecil (31.03 %) dibandingkan proporsi total biaya hutang (bank loan dan kredit ekspor) yang mencapai 68.97 %, dimana biaya hutang tersebut mengalami penurunan, sehingga pada akhirnya nilai wacc menjadi lebih rendah dari nilai deterministik yang dihasilkan evaluator, dengan nilai keanggotaan/tingkat konfidensi sebesar 0.9435. 4.2 net present value (npv) proyek pt x nilai npv proyek dihitung guna mendapatkan penilaian apakah suatu proyek/investasi layak untuk direalisasikan. npv proyek diperoleh dari proyeksi cash flow yang 29 | p a g e terjadi tiap tahun selama masa umur ekonomis proyek dikalikan dengan discount factor :    t t t i f dffnpv )1( * (15) dimana : df = discount fctor  ti  1 1 i = wacc tabel 4 berikut ini merupakan perhitungan npv berdasarkan proyeksi cash flow yang terjadi selama masa umur ekonomis proyek pendirian pabrik baru pt x, dengan discount rate berdasarkan nilai wacc sebesar 13.64 % : tabel 4. proyeksi cash flow dan npv proyek pt x dengan discount rate 13.64 % cash flow thn ke discount factor present value (in million) (2,975,580) 0 1 (2,975,580.04) 615,169 1 0.879961 541,324.44 795,791 2 0.774331 616,205.24 931,931 3 0.68138 634,999.78 960,900 4 0.599588 576,144.18 995,360 5 0.527614 525,165.30 1,030,096 6 0.464279 478,252.09 1,069,962 7 0.408547 437,130.07 1,112,526 8 0.35951 399,959.22 1,147,566 9 0.31635 363,033.32 1,203,159 10 0.278376 334,930.62 1,266,162 11 0.24496 310,159.01 1,334,244 12 0.215555 287,603.15 1,406,421 13 0.18968 266,769.89 1,479,289 14 0.166911 246,909.52 1,559,262 15 0.146875 229,016.64 1,554,744 16 0.129244 200,941.63 1,650,285 17 0.11373 187,686.65 1,741,405 18 0.100078 174,275.88 1,836,160 19 0.088065 161,700.47 16,790,182 20 0.077493 1,301,125.85 1,836,160 21 0.068191 125,209.61 16,790,182 22 0.060005 1,007,501.49 npv 6,430,464.00 dari tabel diatas memberikan hasil npv yang bernilai positif, yakni sebesar rp. 6,430,464,000,000 pada discount rate 13.64146 % dan rp. 6,358,310,540,000 pada discount rate 13.72 %. dengan demikian proyek pendirian pabrik baru pt. x, secara finansial dikatakan layak. 5. kesimpulan a) pemberian deviasi sebagai ketidakpastian dalam parameter pembentuk wacc pada proyek pt x., memberikan nilai wacc yang lebih rendah dibanding nilai deterministik yang diberikan evaluator. selisih nilai tersebut relatif kecil, sehingga nilai deterministik wacc relatif aman dipakai sebagai discount rate cash flow proyek pt x.. adapun nilai fuzzy wacc adalah sebesar 13.64 %, sedangkan nilai wacc deterministik yang ditetapkan evaluator adalah sebesar 13.72 %. b) adanya pertimbangan ketidakpastian dalam parameter pembentuk wacc memberikan gambaran terhadap kemungkinan perubahan kemampuan perusahaan dalam merealisasikan discounted cash flow yang diproyeksikan. jika nilai fuzzy wacc lebih rendah dari nilai deterministik yang ditentukan perusahaan, maka nilai wacc deterministik tersebut masih relatif aman digunakan sebagai discount rate dengan catatan bahwa perbedaan yang ada tidak terlalu besar. namun jika nilai fuzzy wacc yang dihasilkan lebih tinggi dari nilai wacc deterministik yang ditetapkan perusahaan, maka peerusahaan perlu mengevaluasi kembali nilai wacc yang dipakai, karena akan menurunkan discounted cash flow yang sudah diproyeksikan. c) wacc yang dihasilkan dengan metode fuzzy menghasilkan npv yang lebih besar dibanding npv dengan wacc deterministik. npv dengan fuzzy wacc menghasilkan rp. 6,430,464,000,000. sedangkan wacc deterministic yang ditetapkan evaluator menghasilkan npv sebesar rp. 6,358,310,540,000. artinya bahwa perusahaan berpeluang memperoleh pendapatan lebih besar dari yang sudah diproyeksikan. berdasarkan nilai npv diatas yang bernilai positif, maka proyek pendirian pabrik baru pt x., secara finansial dinyatakan layak. 30 | p a g e daftar pustaka [1] boussabaine a.h. and elhag t. : applying fuzzy techniques to cash ow analysis, constr. manag. econom., vol.17, no.6, pp (1999) 745-755. [2] brigham e., financial management : theory and practice, the dryden press (1985). [3] brigham e., financial leverage and use of the net present value investment criterion : a reexamination, financial management, vol. 14, pp. (1985) 48-52. [4] cedric lesage, discounted cash flow analysis : an interactive fuzzy arithmetic approach, european journal of economic and social systems, no.2, pp. (2001) 4968. [5] chiu, c.-y. and park, c.s., 'fuzzy cash flow analysis using present worth criterion', the engineering economist, vol. 39, no. 2, pp. (1994) 113-139. [6] christer carlsson & robert fuller, real option evaluation in fuzzy environment, proceeding of the international symposium of hungarian researchers on computational intelligence, pp. (2002) 69 – 77. [7] glenday g. & tham j., what weight in the wacc ?, terry sanford institute of public policy, (2003). [8] kahraman c., fuzzy versus probabilistic benefit/cost ratio analysis for public work projects, int. j. appl. math. comput. sci., vol. 11, no.3, pp. (2001) 705-718. [9] karsak, e. and tolga, e., fuzzy multicriteria decision-making procedure for evaluating advanced manufacturing system investments, international journal of production economics, vol. 69, pp. (2001) 49-64. [10] machala r., adjusting weighted average cost of capital in the discount rate of investment project, management, vol. 4, no 1-2, pp. (1999) 191-193. [11] peter majlender, strategic investment planning by using dynamic decision trees, proceeding of the 36th hawaii international conference of system sciences (hicss’03), (2002). [12] pinteris, notes on weighted average cost of capital (wacc), departement of finance, college of business university of illionis at urbana – champaign, (2003). [13] reghavendra rau, the weighted average cost of capital : an introduction, kraneert school, purdue university, (1997). [14] sanches, alexandre leme, pamplona, edson de o. e montevechi, jose arnaldo b., capital budgeting using triangular fuzzy number, v encuentro internacional de finanzas. santiago, chile, (2005). [15] sri kusuma dewi, hari purnomo, aplikasi logika fuzzy untuk pendukung keputusan, penerbit graha ilmu, (2004). [16] ward, t.l., 'fuzzy discounted cash flow analysis', in evans, g.w., karwowski, w., and wilhelm, m.r. applications of fuzzy set methodologies in industrial engineering pp. (1989) 91-102. [17] zadeh l.a. : fuzzy sets. | inf. contr., vol.8, pp. (1965) 338-353. [18] zadeh, l.a., fuzzy sets & systems, in: fox, j., ed., system theory. brooklyn, ny: polytechnic press, pp. (1972) 29-37. jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   22    metode logika fuzzy sebagai evaluasi distribusi daya listrik berdasarkan beban puncak pembangkit tenaga listrik fifi d. rosalina1, yuniar farida2, abdulloh hamid3 universitas islam negeri sunan ampel, surabaya, indonesia1, 2, 3. e-mail: fifid.rosalina@gmail.com1, greatyuniar@gmail.co.id2, doelhamid@uinsby.ac.id3 .  abstrak evaluasi beban puncak pada sistem tenaga listrik yang dibangkitkan sangat berpengaruh terhadap perkembangan ketersediaan tenaga listrik di berbagai provinsi. dengan meninjau beban puncak selama satu tahun, dapat diimplementasikan terhadap evaluasi pembangkitan energi listrik sebagai simulasi ketersediaan energi listrik untuk kedepannya. mengevaluasi beban puncak juga bergantung terhadap beberapa faktor seperti kapasitas terpasang, daya mampu, dan hasil produksi pada beberapa sistem pembangkit. hal tersebut dapat menjadi kontrol dari daya-daya yang dihasilkan pada masing-masing pembangkit listrik seperti plta, pltu, pltn, pltg, dan plts. metode logika fuzzy merupakan metode yang efektif yang dapat diaplikasikan untuk mengevaluasi beban puncak dengan tingkat keakuratan yang tinggi. dengan begitu pemenuhan akan tenaga listrik akan terpenuhi dengan tingkat keandalan yang diinginkan. dari evaluasi tersebut output yang dihasilkan dapat dijadikan sebagai kontrol untuk keamanan dari sistem pembangkit. dengan hasil yang diperoleh adalah tingkat error tertinggi mencapai 60% dan telah dilakukan training dan testing data sebanyak 4x untuk menguji parameter dari fungsi keanggotaan yang telah ditentukan dengan hasil recognize tertinggi sebesar 12,5%. kata kunci: beban puncak, evaluasi, logika fuzzy. abstract evaluation of peak load on the power system is raised very influential on the development of electric power availability in the various provinces. by reviewing the peak load for a year, can be implemented for the evaluation of power generation as a simulation of the electrical energy supply for the future. evaluating the peak load also depends on several factors such as installed capacity, power capacity, and production at some plants systems. it can be the control of the forces generated on each such plta, pltu, pltg, and plts. fuzzy logic method is an effective method that can be applied to evaluate peak loads with high accuracy. thus the fulfillment of the electricity will be met with the desired reliability level. the evaluation of the resulting output can be used as a control for the security of the power system. with the results obtained is the highest error rate reached 60%, and has done training and testing data is as much as 4x to test the parameters of the membership function has been determined by the highest recognize result of 12.5%. key word: peak load, evaluation, fuzzy logic. 1. pendahuluan indonesia sebagai salah satu negara yang memiliki jumlah penduduk terpadat dunia, tentu untuk menjadi negara yang makmur dibutuhkan sistem yang baik untuk menunjang kebutuhan akan warganya. tenaga pembangkit listrik salah satunya, dimana dibutuhkan ketersediaan energi yang tinggi dan lebih sesuai yang harus diproduksi dengan kapasitas yang sangat besar. untuk menjembatani permasalahan tersebut dengan mengevaluasi setiap produksi dalam pembangkitan energi listrik dan faktor-faktor yang mempengaruhi dapat dilakukan dengan menggunakan metode matematis. berbagai jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   23    penelitian terkait tentang kemajuan dalam bidang pembangkit tenaga listrik baik dalam ketersediaan energi, sistem transmisi, pengembangan sumber energi dan lain sebagainya sudah dilakukan dan hasilnya dapat dijadikan sebagai pengambilan kebijakan dari pihak tertentu. ketersediaan tenaga listrik di indonesia masih menjadi tugas penting dalam pelaksanaan perkembangannya, dimana di berbagai wilayah masih terdapat beberapa daerah yang belum terpasok sumber energi listrik sebagai pemenuhan kebutuhan hidup. namun, berbagai upaya telah dilakukan oleh pt. pln untuk memperbaiki ketersediaan tenaga listrik dengan membangun sarana-sarana untuk mengoptimalkan hasil produksi agar dapat terdistribusi ke semua daerah termasuk daerah terpencil dan daerah pedalaman. berdasarkan peraturan pemerintah nomor 14 tahun 2012 tentang kegiatan usaha penyediaan tenaga listrik yang menyatakan “ usaha penyediaan tenaga listrik untuk kepentingan umum dilaksanakan sesuai dengan rencana umum ketenagakerjaan dan rencana usaha penyediaan tenaga listrik (ruptl)”. dengan adanya rencana tersebut, sistem evaluasi berlaku sebagai metode yang efektif dalam penyelesaian permasalahan diatas termasuk efisiensi energi. karena kesulitan yang dihadapi selama pengujian jika harus dilakukan secara langsung, mengembangkan model evaluasi untuk memperkirakan beban puncak pada waktu yang akan datang berdasarkan parameter yang mempengaruhi akan selalu menarik untuk diteliti [1,2]. beberapa penelitian yang dilakukan dengan metode yang berbeda telah banyak dilakukan seperti prediksi beban listrik menggunakan kecerdasan buatan. diantaranya prediksi beban listrik di pulau bali menggunakan jaringan syaraf tiruan backpropagation [3] serta peramalan beban listrik jangka pendek pada sistem kelistrikan jawa timur dan bali menggunakan fuzzy time series [4]. salah satu metode yang dapat diimplementasikan untuk mengevaluasi suatu kejadian sebagai perbaikan kebijakan pada kejadian yang akan datang juga dapat dilakukan menggunakan metode logika fuzzy seperti fis mamdani, tsukamoto, sugeno, fuzzy c-means, clustering, dan analisis regresi linier. aplikasi logika fuzzy sebagai pendukung keputusan kini semakin diperlukan tatkala semakin banyak kondisi yang menuntut adanya keputusan yang tidak hanya bisa dijawab dengan ‘ya’ atau ‘tidak’[5]. dari aplikasi logika fuzzy tersebut, metode yang akan digunakan adalah metode mamdani karena memiliki keakuratan dan kemampuan meramal yang lebih baik. hasilnya akan dibandingkan dengan hasil yang terdapat pada pt. pln. parameter yang dijadikan sebagai dasar dari perhitungan tersebut meliputi: kapasitas terpasang, daya mampu, dan produksi. 2. tinjauan pustaka teori himpunan fuzzy diperkenalkan oleh zadeh (1965) untuk menangani konsepsi ketidakpastian akibat ketidaktepatan dan ketidakjelasan[6]. dalam aplikasinya fuzzy dapat di implementasikan terhadap data-data yang kurang valid atau data-data yang bersifat linguistik yang ditetapkan oleh fungsi keanggotaan. hal ini berbeda dengan perhitungan secara tradisional yang hanya menghendaki nilai interval antara 0 dan 1. gambar 1. sistem fuzzy dalam penyelesaiannya himpunan fuzzy mengaitkan 4 metode penyelesaian dimana di antaranya adalah fuzzifikasi, komposisi aturan, sistem inferensi, dan defuzzifikasi. representasi dari sistem fuzzy dapat diekspresikan pada gambar 1. 2.1 fuzzifikasi jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   24    fuzzifikasi merupakan suatu proses yang mengubah nilai crips menjadi nilai fuzzy yang diterapkan terhadap suatu fungsi keanggotaan yang telah ditentukan. fuzzifikasi dapat dipenuhi dengan salah satu bentuk kurva fungsi keanggotaan seperti linear, segitiga, trapesium, lonceng, bahu, dan gauss. bentuk fungsi keanggotaan yang paling umum digunakan adalah jenis linear, trapesium, dan segitiga[7-9]. variabelvariabel inputan seperti kapasitas terpasang, daya mampu, dan produksi diterapkan terhadap fungsi keanggotaan tersebut beserta variabel-variabel linguistik. variabel linguistik merupakan suatu penamaan grup yang mewakili suatu keadaan atau kondisi tertentu dengan menggunakan bahasa alami[10]. untuk menghasilkan nilai keanggotaan berdasarkan proses dari fungsi keanggotaan pada himpunan fuzzy yang menggunakan tipe segitiga dan trapesium dapat diselesaikan berdasarkan persamaan 1 dan persamaan 2 [11]: , ,0 (1) ,1, ,0 (2) dimana adalah suatu fungsi keanggotaan dari himpunan fuzzy, dan x merupakan nilai dari parameter output. sedangkan a, b, c, dan d adalah konstan. 2.2 komposisi aturan aturan dasar merupakan aplikasi dari sebuah fungsi implikasi yang membentuk pernyataan bersyarat if then yang terdiri dari premis dan kesimpulan atau akibat. banyaknya jumlah rule yang dibuat bergantung dari banyaknya variabel dan fungsi keanggotaan yang digunakan. rule yang dibuat akan di aplikasikan terhadap sistem inferensi fuzzy untuk mengetahui hasil keputusan. dalam penyelesaian pada sistem inferensi fuzzy mamdani digunakan fungsi implikasi min. namun pada dasarnya fungsi implikasi terdiri dari 3 metode, yaitu min, max, dan probabilistik or. untuk masingmasing metode, secara umum dapat dituliskan sebagai berikut[12]: a. metode max (maximum) max , (3) dengan: = nilai keanggotaan solusi fuzzy sampai aturan ke-i. = nilai keanggotaan konsekuen fuzzy aturan ke-i. b. metode additive (sum) min 1, (4) dengan: = nilai keanggotaan solusi fuzzy sampai aturan ke-i. = nilai keanggotaan konsekuen fuzzy aturan ke-i. c. metode probabilistik or ∗ (5) dengan: = nilai keanggotaan solusi fuzzy sampai aturan ke-i. = nilai keanggotaan konsekuen fuzzy aturan ke-i. 2.3 sistem inferensi sistem inferensi merupakan langkah yang memetakan input terhadap aturanaturan yang telah dibuat pada komposisi aturan. karena aturan yang dibuat antara satu dengan yang lain berbeda maka keputusan yang dihasilkan juga akan berbeda dengan menghasilkan pola yang berbeda pula. keputusan tersebut dapat dilihat dari daerah hasil yang telah dihasilkan berdasarkan aturan yang telah dibuat. dari daerah hasil atau pola yang dihasilkan akan diproses kembali meggunakan metode defuzzifikasi. 2.4 defuzzifikasi ketika pada proses yang pertama nilai crisp dipetakan terhadap fungsi keanggotaan sehingga nilai tersebut menjadi sebuah nilai yang difuzzikan, untuk mendapatkan nilai yang sesuai dengan harapan nilai tersebut harus diubah ke bentuk semula, nilai crisp. proses inilah yang dinamakan dengan defuzzifikasi. dalam penyelesaian nilai fuzzy ke nilai crisp dapat diselesaikan dengan beberapa metode diantaranya metode centroid, metode bisektor, metode min of maximum, metode largest of maximum, dan metode smallest jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   25    of maximum. solusi umum dari masingmasing metode dapat dirumuskan sebagai berikut[13]: a. metode centroid dilakukan dengan mengambil titik pusat ∗ untuk variabel kontinu (6) ∗ ∑ ∑ untuk variabel diskret (7) b. metode bisektor keputusan dalam mengubah nilai fuzzy ke nilai crisp ini juga melibatkan komposisi aturan yang telah dibuat yang selanjutnya ditentukan berdasarkan daerah output yang dihasilkan. secara umum dituliskan[14]: sedemikian sehingga (8) 3. metode penelitian penyelesaian pada evaluasi suatu beban puncak dalam penelitian ini, sistem fuzzy mamdani digunakan dengan 3 variabel inputan yang berkaitan seperti kapasitas terpasang, daya mampu, dan produksi serta dengan beberapa sample data yang akan diproses menggunakan metode mamdani. selain menggunakan perhitungan metode mamdani dalam tulisan ini juga menerapkan aplikasi yang tersedia pada matlab. hal ini akan dapat membantu dalam pengambilan keputusan berdasarkan visualisasi yang tersedia seperti rule, surface, dan fungsi keanggotaan. algoritma metode penelitian yang digunakan dapat dilihat pada gambar 2. 3.1 data 3.1.1 identifikasi variabel beberapa data yang digunakan dalam penelitian ini diperoleh dari data statistik pt.pln pada tahun 2013 dimana data tersebut diakumulasikan dari beberapa provinsi di seluruh indonesia. untuk melakukan prediksi beban puncak pada sistem pembangkit tenaga listrik dalam penelitian ini digunakan 3 variabel masukan dan 1 variabel keluaran. 3 variabel masukan yang digunakan dinilai sangat berpengaruh terhadap hasil evaluasi sistem pembangkit. 3 variabel masukan tersebut adalah kapasitas terpasang, daya mampu, dan produksi. sedangkan variabel keluaran adalah beban puncak. gambar 2. algoritma penyelesaian dari hasil keluaran tersebut dijadikan sebagai nilai beban puncak masing-masing daerah yang ditujukan untuk pengembangan sistemsistem pembangkit energi listrik dan sebagai sistem kontrol dari pembangkit itu sendiri. variabel tersebut dapat dilihat pada gambar 3. gambar 3. variabel. 3.1.2 pengumpulan data mengumpulkan data-data yang berkaitan dengan variabel masukan. dalam penelitian ini masukan yang digunakan adalah kapasitas terpasang, daya mampu, dan produksi. data yang digunakan diperoleh dari data statistik pt. pln (persero) yang di akumulasikan selama 1 tahun yang terhitung pada bulan januari sampai akhir desember jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   26    pada tahun 2013 dari berbagai provinsi di indonesia. data-data tersebut dapat dilihat pada tabel 1. tabel 1. sampling data provinsi kapasitas terpasang (mw) daya mampu (mw) produksi netto (gwh) 1. 113,39 76,08 548,3 2. 88,24 43,54 840,51 3. 221,39 156,47 2.107,09 4. 292,19 252,07 1.730,85 5. 195,61 99,82 809,42 6. 147,61 71,3 512,68 7. 152,59 103,07 1.181,96 8. 100,15 60,71 746,31 9. 171,04 129,47 1.330,07 10. 153,04 107,03 720,6 keterangan: 1. wilayah aceh 2. wilayah bangka belitung 3. wilayah kalimantan barat 4. sulawesi utara 5. wilayah maluku dan maluku utara 6. maluku 7. wilayah papua 8. papua 9. wilayah nusa tenggara barat 10. wilayah nusa tenggara timur 3.2 fuzzifikasi fungsi keanggotaan yang digunakan menggunakan representasi kurva bentuk trapesium. dalam hal ini variabel-variabel linguistik untuk masing-masing variabel digunakan 3 fungsi keanggotaan yaitu rendah, sedang dan tinggi. sebagai contoh ditentukan fungsi keanggotaan variabel kapasitas terpasang berdasarkan persamaan 2, diperoleh fungsi keanggotaan sebagai berikut: ,1, ,0 ,1, ,0 ,1, ,0 3.3 komposisi aturan karena terdapat 3 variabel masukan dan 1 variabel keluaran dengan 3 variabel linguistik pada fungsi keanggotaan jumlah rule yang didapat ada 27 rule yang membentuk sebuah premis dan konsekuen pada masing-masing rule. daftar 27 aturan tersebut dapat dilihat pada gambar 4 dengan menggunakan aplikasi matlab. gambar 4. komposisi aturan 3.4 sistem inferensi pada proses ini dipetakan nilai inputan untuk mendapatkan hasil pola sebagai keputusan dengan melibatkan rule-rule yang telah dibuat. dalam hal ini digunakan contoh inputan dengan kapasitas terpasang = 260 mw, daya mampu = 185 mw, dan produksi = 2900 gwh. 3.5 defuzzifikasi pada fuzzy metode mamdani untuk menyelesaikan permasalahan ini dipilih metode centroid dengan penyelesaian mengambil titik pusat pada pola yang dihasilkan pada sistem inferensi yang berdasarkan pada persamaan 7. ∗ ∑ ∑ untuk variabel diskret (7) 4. hasil dan pembahasan representasi apliksai matlab untuk menunjukkan hasil dari fungsi keanggotaan pada 3 variabel dapat dilihat pada gambar 5. setelah melalui 4 proses yang telah dilakukan pada aplikasi logika jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   27    fuzzy dengan memasukkan contoh inputan data gambar 5. fungsi keanggotaan kapasitas terpasang pada proses sistem inferensi, didapatkan hasil pada gambar 6 tersebut. hasil yang terdapat pada pola gambar 6 tersebut merupakan hasil yang dapat digunakan untuk evaluasi daya listrik terhadap sistem pembangkit tenaga listrik yang melibatkan 3 variabel masukan dengan masing-masing 3 fungsi keanggotaan hingga mendapatkan komposisi aturan sebanyak 27. suatu pola surface yang berwarna biru adalah sebagai hasilnya yang dapat dilihat pada gambar 7. gambar 6. sistem inferensi gambar 7. hasil tabel 2. hasil perbandingan no. fuzzy pt.pln error 1. 141 265 46.79% 2. 141 148 4.72% 3. 341 213  60.09%  4. 350 236,33  48.09% 5. 143 172,85 17.26% 6. 141 107,7 30.91% 7. 141 204,73  31.12% 8. 141 123,56 14.11% 9. 150 247  39.27%  10. 141 143  1.39% setelah memperoleh hasil yang diinginkan pada matlab berdasarkan data yang digunakan, selanjutnya kita bandingkan hasil beban puncak dengan menggunakan matlab dan hasil beban puncak yang terdapat pada pt. pln(persero) untuk mengetahui keakuratan sistem yang digunakan. berikut adalah perbandingan hasil beban puncak sistem fuzzy dan data statistik pada tabel 2. untuk menghitung error gunakan rumus pada persamaan 9 berikut: 100% (9) 5. training dan testing training dan testing pada dasarnya digunakan untuk melihat sejauh mana fungsi keanggotaan yang ditetapkan dapat digunakan pada data testing yang merupakan sebuah data yang berbeda dari data training. dalam hal ini digunakan aplikasi yang telah disediakan pada jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   28    matlab yaitu simulink yang dapat dilihat pada gambar 8, dari hasil output yang dihasilkan pada simulink perhatikan apakah hasilnya sama atau tidak pada output menggunakan toolbox dengan menginputkan data testing. gambar 8. simulink semakin banyak training dan testing yang dilakukan, maka kita dapat melihat recognize terbesar dari training dan testing data untuk melihat ketepatan parameter fungsi keanggotaan yang telah ditetapkan. dalam penelitian ini dilakukan taining dan testing data sebanyak 4x yang dapat dilihat pada tabel 3, dengan hasil recognize terbesar yaitu 12,5% pada 20 data training dan 8 data testing yang dapat dilihat pada tabel 4. tabel 3. training dan testing data data training data testing 20 8 18 10 23 13 15 9 hasil recognize ini diperoleh dari sisi banyaknya data testing yang dapat terdeteksi oleh simulink, dimana hasil simulink dan toolbox harus bernilai sama. dalam penelitian ini, setelah dihitung error menggunakan rumus 9, didapatkan hasil recognize seperti pada tabel 4 berikut. tabel 4. hasil recognize data testing 8 439,82 225,18 2.705,56 376 24,04 18,4 148,16 18,51 147,61 71,3 512,68 107,7 48 28,52 296,74 65,15 32,25 17,4 185,73 35 0,7 0,48 6,39 3,5 88,24 43,54 840,51 148 recognize : 12,5% 6. kesimpulan dalam penelitian ini model logika fuzzy menggunakan metode mamdani digunakan sebagai hasil evaluasi daya listrik pada sistem pembangkit listrik dengan menggunakan 3 faktor yang mempengaruhi yaitu kapasitas terpasang, daya mampu, dan produksi. namun, berdasarkan data pada tabel 2 diperoleh hasil aplikasi fuzzy yang menghasilkan nilai beban puncak pada 10 sampling data tersebut dengan hasil error tertinggi sebesar 60% dan terendah sebesar 1.39 %. berdasarkan algoritma penelitian pada gambar 2 karena hasil belum optimal atau masih terdapat error yang cukup tinggi, maka harus dilakukan perbaikan pada fungsi keanggotaannya yang telah ditetapkan untuk mendapatkan hasil yang lebih akurat. selain dilakukan menggunakan sistem fuzzy, melakukan training dan testing data juga berguna untuk melihat ketepatan dari parameter fungsi keanggotaan yang telah ditetapkan terhadap data testing yang dapat dilakukan menggunakan simulink. jurnal matematika “mantik”  edisi: oktober 2016. vol. 02 no. 01  issn: 2527-3159 e-issn: 2527-3167   29    dalam penelitian ini telah dilakukan training dan testing data yang dilakukan sebanyak 4x, dan didapatkan hasil recognize terbesar 12,5% yaitu pada 20 data training dan 8 data testing. karena hasil nilai recognize yang cukup rendah, hal tersebut juga dipengaruhi oleh parameter dari fungsi keanggotaan, karena itu parameter yang telah ditetapkan perlu untuk dievaluasi agar mendapatkan hasil yang lebih akurat. selain hal diatas, karena untuk mengevaluasi distribusi daya lisrik yang didasarkan pada beban puncak, maka akan dilakukan penelitian selanjutnya dengan manambahkan analisis time series pada proses awal pengolahan data untuk memprediksi beban puncak tersebut dikarenakan untuk memprediksi beban puncak harus memperhatikan atau menganalisis factor dari waktu seperti hari, minggu, bulan, dan tahun. referensi [1] majdi a, beiki m. evolving neural network using a genetic algorithm for predicting the deformation modulus of rock masses. int j rock mech min sci 2010;47(2):246–53. [2] beiki m, bashiri a, majdi a. genetic programming approach for estimating the deformation modulus of rock mass using sensitivity analysis by neural network. int j rock mech min sci 2010;47(7):1091–10103. [3] fitriyah, q., & istardi, d. (2012). prediksi beban listrik pulau bali dengan menggunakan metode backpropagasi. [4] handoko, b. (2013). peramalan beban listrik jangka pendek pada sistem kelistrikan jawa timur dan bali menggunakan fuzzy time series. [5] kusumadewi, sri dan purnomo, hari. 2013. aplikasi logika fuzzy. yogyakarta: graha ilmu [6] zadeh al. fuzzy sets. inf control 1965;8:338–53. [7] habibagahi g, katebi s. rockmass classification using fuzzy sets. iran j sci technol trans b 1996;20(3):273–84. [8] den hartog mh, babuska r, deketh hjr, grima ma, verhoef pnw, verbruggen hb. knowledge-based fuzzy model for performance prediction of a rockcutting trencher. int j approx reason 1997;16(1):43–66. [9] grima ma. neuro-fuzzy modelling in engineering geology. leiden: a.a. balkema publishers; 2000. [10] kusumadewi, sri dan purnomo, hari. 2013. aplikasi logika fuzzy untuk pendukung keputusan. graha ilmu: yogyakarta. [11] mohammad rezaei, mostafa asadizadeh, abbas majdi, mohammad farouq hossaini, prediction of representative deformation modulus of longwall panel roof rock strata using mamdani fuzzy system, international journal of mining science and technology 25 (2015) 23. [12] kusumadewi, sri dan purnomo, hari. 2013. aplikasi logika fuzzy untuk pendukung keputusan. graha ilmu: yogyakarta. [13] kusumadewi, sri dan purnomo, hari. 2013. aplikasi logika fuzzy untuk pendukung keputusan. graha ilmu: yogyakarta. [14] kusumadewi, sri dan purnomo, hari. 2013. aplikasi logika fuzzy untuk pendukung keputusan. graha ilmu: yogyakarta. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 penerapan metode exponential smoothing untuk peramalan jumlah klaim di bpjs kesehatan pamekasan faisol1, sitti aisah2 jurusan matematika, fakultas matematika dan ilmu pengetahuan alam, universitas islam madura e-mail: faisol.munif@gmail.com 1, aisah.ayy94@gmail.com 2 abstrak model time series merupakan model yang digunakan untuk memprediksi masa depan dengan menggunakan data masa lalu, salah satu contoh dari model time series adalah exponential smoothing. metode exponential smoothing adalah prosedur perbaikan yang dilakukan secara terus-menerus pada peramalan terhadap data yang terbaru. dalam penelitian ini metode exponential smoothing diterapkan untuk meramalkan jumlah klaim di bpjs kesehatan pamekasan dengan menggunakan data dari periode januari 2014 sampai desember 2015, langkahlangkah yang digunakan untuk memperoleh output dari penelitian ini terdapat 4 tahap, yaitu 1) identifikasi data, 2)pemodelan, 3) peramalan, 4) evaluasi hasil peramalan dengan rmse dan mape. berdasarkan metodologi penelitian tersebut, didapatkan hasil untuk periode 25 = 833,828 , periode 26 = 800,256 , periode 27 = 766,684 , periode 28 = 733,113, periode 29 = 699,541, dan periode 30 = 655, 970. nilai untuk rmse = 98,865 dan mape = 7,002. dalam kasus ini juga digunakan metode moving average untuk membandingkan hasil peramalan dengan metode double exponential smoothing. hasil peramalan untuk periode 25 = 899,208 , periode 26 = 885 ,792, periode 27 = 872,375, periode 28 = 858,958 , periode 29 = 845,542 , dan periode 30 = 832,125. nilai untuk rmse = 101,131 dan mape = 7,756. kedua metode sama – sama mempunyai kinerja sangat bagus karena nilai mape berada dibawah 10 %, tapi metode exponential smoothing memiliki nilai rmse dan mape yang lebih kecil dibandingkan dengan metode moving average. kata kunci: double exponential smoothing, bpjs kesehatan pamekasan abstract time series model is the model used to predict the future using past data, one example of a time series model is exponential smoothing. exponential smoothing method is a repair procedure performed continuously at forecasting the most recent data. in this study the exponential smoothing method is applied to predict the number of claims in the health bpjs pamekasan using data from the period january 2014 to december 2015, the measures used to obtain the output of this research there are four stages, namely 1) the identification of data, 2) modeling, 3) forecasting, 4) evaluation of forecasting results with rmse and mape. based on the research methodology, the result for the period 25 = 833.828, the 26 = 800.256, period 27 = 766.684, a period of 28 = 733.113, period 29 = 699.541, and the period of 30 = 655, 970. value for rmse = 98.865 and mape = 7.002, in this case the moving average method is also used to compare the results of forecasting with double exponential smoothing method. forecasting results for the period 25 = 899.208, the 26 = 885, 792, 27 = 872.375 period, a period of 28 = 858.958, period 29 = 845.542, and the period of 30 = 832.125. value for rmse = 101.131 and mape = 7.756. both methods together both have very good performance because the value of mape is below 10%, but the method of exponential smoothing has a value of rmse and mape are smaller than the moving average method. keyword: double exponential smoothing, bpjs kesehatan pamekasan jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 46 1. pendahuluan asuransi merupakan suatu bentuk manajemen potensi resiko dari hal-hal yang tak terduga seperti kerugian, kematian, kehilangan, kesehatan dan lain sebagainya. sehingga apabila terjadi suatu resiko yang tidak diinginkan terhadap dirinya, maka mereka memperoleh kompensasi atau tanggungan moneter sesuai dengan jenis asuransi yang diikutinya. asuransi memungkinkan seseorang untuk melindungi diri terhadap potensi kerugian yang signifikan dan kesulitan keuangan pada tingkat yang cukup terjangkau [1]. salah satu perusahaan asuransi dibidang kesehatan adalah bpjs kesehatan. bpjs kesehatan (badan penyelenggara jaminan sosial kesehatan) merupakan badan usaha milik negara yang ditugaskan khusus oleh pemerintah untuk menyelenggarakan jaminan pemeliharaan kesehatan. bpjs kesehatan bersama bpjs ketenaga kerjaan dahulu bernama jamsostek merupakan program pemerintah dalam kesatuan jaminan kesehatan nasional (jkn) yang diresmikan pada tanggal 31 desember 2013 [1]. untuk memprediksi jumlah pasien rawat inap di rumah sakit yang mengajukan klaim ke bpjs kesehatan, perlu adanya suatu metode khusus untuk mempermudah masalah tersebut, salah satunya dengan menggunakan peramalan matematika, yaitu metode exponential smoothing. metode exponential smoothing adalah prosedur perbaikan yang dilakukan secara terus-menerus pada peramalan terhadap data yang terbaru. metode ini menunjukkaan pembobotan menurun secara eksponensial pada data terdahulu. metode ini menunjukkaan pembobotan menurun secara eksponensial pada data terdahulu. metode ini merupakan metode peramalan yang cukup baik untuk peramalan jangka panjang dan jangka menengah, terutama pada tingkat operasional suatu perusahaan [2]. atas dasar tersebut metode exponential smoothing dipakai untuk meramalkan jumlah klaim di bpjs kesehatan pamekasan. 2. metode penelitian langkah 1 :studi literatur studi literatur yang dilakukan penulis yaitu dengan mempelajari buku-buku, jurnal, artikel, serta bahan-bahan dari internet lainnya yang berhubungan dengan metode peramalan khususnya metode exponential smoothing. langkah 2 : pengumpulan data data yang dikumpulkan dalam penelitian ini adalah data yang diperoleh dari kantor bpjs kesehatan pamekasan, yaitu data jumlah pasien rawat inap di rumah sakit umum daerah dr. h. slamet martodirdjo kabupaten pamekasan yang mengajukan klaim ke bpjs periode januari 2014 hingga desember 2015. langkah 3: pengolahan data dalam penelitian ini pengolahan data dilakukan denagan tahapan sebagai berikut: 1) identifikasi data untuk mengetahui pola data. data yang sudah ada diplot dengan menggunakan aplikasi minitab. setalah itu akan terlihat grafik yang dihasilkan, kemudian akan ditentukan penggunaan metode exponential smoothing yang tepat. 2) melakukan pemodelan dengan metode exponential smoothing yang tepat 3) melakukan peramalan dengan metode exponential smoothing 4) evauasi hasil peramalan dengan rmse dan mape untuk metode exponential smoothing 5) melakukan peramalan dengan metode moving average dan evauasi hasil peramalan dengan rmse dan mape. langkah 4: penarikan kesimpulan. data yang telah diidentifikasi kemudian dilakukan pemodelan, setelah itu dilakukan peramalan dan menghitung kesalahan peramalan, maka langkah selanjutnya adalah menarik kesimpulan dari hasil yang telah diperoleh. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 47 3. hasil dan pembahasan 3.1 identifikasi data sebelum proses peramalan, data yang ada dibagi menjadi dua kelompok, yaitu kelompok inisialisasi dan kelompok pengujian. kelompok inisialisasi digunakan untuk menentukan nilai taksiran awal, kelompok data ini terdiri dari data pada bulan januari 2014 sampai bulan oktober 2014 dengan n=10 observasi. pada kelompok pengujian digunakan dalam proses evaluasi data, yaitu rmse dan mape. kelompok data pengujian ini terdiri dari data bulan november 2014 sampai data bulan desember 2015, dengan n=14 observasi. tabel 1. data jumlah pasien tahun 2014 tabel 2. data jumlah pasien tahun 2015 no bulan pelayanan jumlah peserta 1 januari 1249 2 februari 1144 3 maret 1195 4 april 1188 5 mei 1166 6 juni 1048 7 juli 979 8 agustus 937 9 september 922 10 oktober 986 11 november 960 12 desember 863 data yang sudah ada tersebut diplot dalam bentuk grafik untuk mengetahui pola data yang ada, data tersbut diplot dengan menggunakan aplikasi minitab. plot data dengan minitab untuk mengetahui apakah data tersebut stationer atau tidak. kemudian akan ditentukan penggunaan metode exponential smoothing yang tepat. dan seperti gambar 1 berikut adalah hasil dari data yang sudah diplot. gambar 1: grafik tahun 2014-2015 dari hasil melakukan plot pada data yang ada, maka diketahui pola datanya tidak stasioner, hal ini ditunjukkan dari nilai rounded value = -0.50 sedangkan untuk mengetahui data tersebut stasioner maka nilai rounded value harus = 1. maka metode yang digunakan adalah double exponential smothing. 3.2 pemodelan dengan metode double exponential smoothing pada tahap ini data yang ada akan diimplementasikan ke dalam rumus double exponential smoothing. berikut adalah rumus double exponential smoothing. = + 1 (1) 1 (2) . (3) dimana: = peramalan untuk periode t. xt = nilai aktual = trend pada periode ke t α = parameter pertama perataan yaitu 0.27 = parameter kedua untuk pemulusan trend yaitu 0.45 = hasil peramalan ke m m = jumlah periode ke muka yang akan diramalkan no bulan pelayanan jumlah peserta 1 januari 1093 2 februari 1080 3 maret 1209 4 april 1157 5 mei 1367 6 juni 1188 7 juli 1080 8 agustus 1120 9 september 1279 10 oktober 1360 11 november 1012 12 desember 1193 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 48 3.3 peramalan dengan double exponential smoothing proses inisialisasi untuk exponential smoothing dari holt memerlukan dua taksiran, yaitu menentukan nilai , dan menentukan nilai trend . untuk yang pertama pilih , untuk taksiran nilai trend memerlukan taksiaran dari satu periode ke periode lainnya, salah satunya dengan menggunakan: 3 2 4 3 /3 = 1093 3 2 4 3 3 = =21,333 setelah nilai taksiran untuk dan diketahui, maka peramalan periode selanjutnya menggunakan rumus double exponential smoothing (persamaan (1) sampai (3)). peramalan untuk periode 2 adalah: 1 = 1093 + (21,333) (1) =1.114,333 untuk peramalan pada periode 3 yaitu: + 1 =0,27 (1080) + (1 0,27) (1093+21,333) =1.105,067 1 = 0,45 (1105,067 – 1093) + (1 – 0,45)(21,333) = 17,1618 1 =1105,067 + (17,1618) (1) =1122,225 begitu juga seterusnya sampai peramalan pada periode 24. ramalan untuk periode 26, 27, 28, 29, dan 30 dapat dihitung seperti: 2 = 867,399 + ( 33,572) (2) =800,255 3 = 867,399 + ( 33,572) (3) = 766,683 4 = 867,399 + ( 33,572) (4) =733,111 5 = 867,399 + ( 33,572) (5) = 699,539 6 = 867,399 + ( 33,572) (6) = 665,967 tabel 3. hasil perhitungan double exponential smoothing dengan excell periode data 1 1093 1093 21,333 2 1080 1114,333 1105,063 17,1618 3 1209 1122,225 1145,654 27,705 4 1157 1173,359 1168,942 25,717 5 1367 1194,660 1241,192 46,657 6 1188 1287,848 1260,889 34,525 7 1080 1295,414 1237,252 8,352 8 1120 1245,605 1211,691 -6,909 9 1279 1204,783 1224,821 2,109 10 1360 1226,930 1262,859 18,277 11 1012 1281,136 1208,469 -14,423 12 1193 1194,046 1193,763 -14,550 13 1249 1179,213 1198,056 -6,071 14 1144 1191,984 1179,029 -11,901 15 1195 1167,127 1174,653 -8,515 16 1188 1166,138 1172,041 -5,859 17 1166 1166,182 1166,133 -5,881 18 1048 1160,252 1129,944 -19,519 19 979 1110,425 1074,940 -35,487 20 937 1039,453 1011,790 -47,935 21 922 963,855 952,554 -53,021 22 986 899,533 922,879 -42,515 23 960 880,364 901,866 -32,839 24 863 869,026 867,399 -33,572 25 833,828 608,694 -134,88 26 800,256 27 766,684 28 733,113 29 699,541 30 665,970 berikut merupakan hasil plot antara data aktual dan data hasil peramalan menggunakan metode double exponential smoothing dapat terlihat jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 49 seperti pada gambar 2 grafik data actual dan hasil peramalan. gambar.2 grafik data aktual dan hasil peramalan 3.4 evaluasi hasil peramalan dengan rmse dan mape untuk metode exponential smoothing. rmse = ∑ / = ….. = , … , = , = √9774,202 = 98,865 mape = ∑ = ⋯ = , ⋯ , = , = 7,002 3.5 peramalan dengan rata – rata bergerak (moving average) pada kasus ini moving average yang digunakan adalah moving average dengan empat periode. 4 1157 1209 1080 1093 4 1134,75 4 1198 1230,25 1203,25 1134,75 4 1191,563 2 = 2(1198) 1191,563 = 1204,437 = 1198 1191,563 = 4,291 maka peramalan untuk periode 8: 1 = 1204,437 + 4,291 = 1208,728 begitu juga seterusnya sampai peramalan pada periode 24. untuk periode 26,27,28,29, dan 30 ramalannya menggunakan nilai terakhir dari dan (periode 24). 3.5 evaluasi hasil peramalan dengan rmse dan mape untuk moving average rmse = ∑ / = ….. = , …… , = , = 10227,516 = 101,131 mape = ∑ = ⋯ = , ⋯ , = , jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 50 = 7,756 mape = ⋯ = , ⋯ , = , = 7,756 tabel 4. hasil perhitungan moving average dengan excell peri ode data                     1 1093 2 1080 3 1209                4 1157 1,134, 750             5 1367 1,203, 250             6 1188 1,230, 250             7 1080 1,198, 000 1,191, 563 1,204, 438 4,29 2    8 1120 1,188, 750 1,205, 063 1,172, 438 10,8 75 1,208, 729 9 1279 1,166, 750 1,195, 938 1,137, 563 19,4 58 1,161, 563 10 1360 1,209, 750 1,190, 813 1,228, 688 12,6 25 1,118, 104 11 1012 1,192, 750 1,189, 500 1,196, 000 2,16 7 1,241, 313 12 1193 1,211, 000 1,195, 063 1,226, 938 10,6 25 1,198, 167 13 1249 1,203, 500 1,204, 250 1,202, 750 0,50 0 1,237, 563 14 1144 1,149, 500 1,189, 188 1,109, 813 26,4 58 1,202, 250 15 1195 1,195, 250 1,189, 813 1,200, 688 3,62 5 1,083, 354 16 1188 1,194, 000 1,185, 563 1,202, 438 5,62 5 1,204, 313 17 1166 1,173, 250 1,178, 000 1,168, 500 3,16 7 1,208, 063 18 1048 1,149, 250 1,177, 938 1,120, 563 19,1 25 1,165, 333 19 979 1,095, 1,152, 1,037, 1,101, 250 938 563 38,4 58 438 20 937 1,032, 500 1,112, 563 952,43 8 53,3 75 999,1 04 21 922 971,5 00 1,062, 125 880,87 5 60,4 17 899,0 63 22 986 956,0 00 1,013, 813 898,18 8 38,5 42 820,4 58 23 960 951,2 50 977,81 3 924,68 8 17,7 08 859,6 46 24 863 932,7 50 952,87 5 912,62 5 13,4 17 906,9 79 25                899,2 08 26                885,7 92 27                872,3 75 28                858,9 58 29                845,5 42 30                832,1 25 4. kesimpulan hasil dari eksperimen pada penelitian ini didapatkan beberapa kesimpulan sebagai berikut: 1. dengan menggunakan parameter = 0,27 dan parameter = 0,45, hasil peramalan dengan metode double exponential smoothing untuk periode 25 = 833,828 , periode 26 = 800,256 , periode 27 = 766,684 , periode 28 = 733,113 , periode 29 = 699,541 , dan periode 30 = 655, 970. nilai untuk rmse = 98,865 dan mape = 7,002. 2. hasil peramalan dengan menggunakan metode moving average untuk periode 25 = 899,208 , periode 26 = 885 ,792, periode 27 = 872,375, periode 28 = 858,958 , periode 29 = 845,542 , dan periode 30 = 832,125. nilai untuk rmse = 101,131 dan mape = 7,756. kedua metode sama – sama mempunyai kinerja sangat bagus karena nilai mape berada dibawah 10 %, tapi metode exponential smoothing memiliki nilai rmse dan mape yang lebih kecil dibandingkan dengan metode moving average. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 51 5. ucapan terima kasih penulis s.n. mengucapkan terima kasih kepada dekan mipa uim, kajur matematika uim, dosen-dosen serta beberapa mahasiswa matematika uim yang telah memberikan dukungan baik secara finansial (materiil) maupun moril dalam pengembangan penelitian ini. daftar pustaka [1] bpjs, a. (2014, 05 08). “bpjs kesehatan”. diambil kembali dari http://mail.bpjskesehatan.go.id/. (diakses tanggal 20 september 2016) [2] makridakis, s., & wheelwright, s. c. (1999). “metode dan aplikasi peramalan”. jakarta: erlangga. [3] raharja, a., dkk. (2010). “penerapan metode exponential smoothing untuk peramalan penggunaan waktu telepon di pt. telkomsel divre3 surabaya”. sisfo jurnal sistem informasi, 1-8. [4] terkini, madura. (2016, februari 08). “ribuan peserta bpjs nunggak iuran”. diambil kembali dari file:///d:/ribuan%20peserta% 20bpjs%20 nunggak%20iuran% 20_%20 berita%20 madura%20terkini.html. (diakses tanggal 14 maret 2016) [5] zainun, n., & majid, m. (2003). “low cost house demand predictor”. malaysia: universitas teknologi malaysia. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 30 aplikasi graph coloring pada penjadwalan perkuliahan di fakultas sains dan teknologi uin sunan ampel surabaya devi saidatuz z1, deasy alfiah a2, aris fanani3, nurissaidah ulinnuha4 universitas islam negeri sunan ampel1, 2, 3, 4 email:devisaidatuzzaenab@gmail.com1, deasy_alfiaadyanti@yahoo.co.id2, arisfa@uinsby.ac.id3 , nuris.ulinnuha@uinsby.ac.id4 abstrak dalam lingkungan akademik terdapat berbagai macam permasalahan salah satunya permasalahan penyusunan jadwal mata kuliah di tingkat universitas. permasalahan ini dipengaruhi oleh alokasi waktu (kesesuaian sks dengan masing-masing mata kuliah), ketersediaan ruang kuliah, dan banyaknya program studi. dalam penelitian penyusunan jadwal perkuliahan kali ini, parameter-parameter yang digunakan antara lain: mahasiswa semester dua di masing-masing program studi, jumlah program studi, ketersediaan ruang kuliah, waktu perkuliahan, jumlah sks sehari dari masing-masing program studi di fakultas sains dan teknologi. penelitian ini bertujuan sebagai alternatif dalam menyusun sistem penjadwalan ketika dalam fakultas terjadi ketidakseimbangan antara ruang perkuliahan, banyaknya program studi, dan banyaknya sks yang ditempuh masing-masing program studi. berdasarkan analisa graph coloring dan pewarnaan sisi pada kasus penjadwalan program studi fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya yang memiliki 6 program studi dan 5 ruang perkuliahan dengan tiga kali simulasi kombinasi dari data inputan, didapatkan penjadwalan yang tidak berubah (konsisten) dan hasil penjadwalannya tidak saling tumpang tindih/tabrakan, baik dalam penggunaan ruang perkuliahan, dan waktu perkuliahan. kata kunci: graph coloring, jadwal kuliah. pewarnaan sisi abstract in the academic environment there are various problems, one of problem is scheduling of courses at the university level. these problems are affected by the allocation of time (conformity of sks with each course), the availability of classrooms, and many study program. in the study preparation lecture schedules in this time, the parameters used, among others: second semester students in each study program, number of study program, availability of classrooms, lectures, number of credits a day of each study program at the faculty of science and technology. this research is aimed as an alternative in formulating scheduling system when the faculty there is an imbalance between the lecture hall, many study program, and the number of credits taken each study program. based on the analysis of graph coloring and edge coloring at case of scheduling study program faculty of science and technology state islamic university sunan ampel surabaya, which has 6 programs and 5 lecture room with three simulated combinations of input data, obtained scheduling unchanged (consistent) and results scheduling does not overlap / collision, both in the lecture room, and the time of the lecture. keywords : graph coloring, course’s schedule, edge coloring jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 30 1. pendahuluan graf adalah salah satu cabang ilmu matematika yang secara khusus merupakan suatu kajian dalam matematika diskrit [1]. graf digunakan untuk merepresentasikan objek-objek diskrit dan hubungan antara objek-objek tersebut [2]. representasi visual dari graf adalah dengan menyatakan objek dalam bentuk noktah, titik, atau dalam bentuk bulatan, sedangkan adanya garis adalah sebagai penghubung antara objekobjek tersebut [3]. pada tahun 1736, graf pertama kali digunakan untuk memecahkan masalah besar dan terkenal di eropa yakni masalah jembatan konigberg. ide untuk menyelesaikan permasalahan jembatan ini menggunakan graf pertama kali dicetus oleh l. euler, matematikawan asal swiss [6]. dua bagian terpenting dalam representasi graf adalah simpul (vertex) dan sisi (edge) [7]. sehingga dapat dikatakan bahwa graf adalah sebagai himpunan dari simpul dan sisi. karena graf disajikan dalam bentuk simpul (vertex) dan sisi (edge), hal ini yang menjadikan graf sebagai sebuah teori yang unik dikarenakan kesederhanaan pokok bahasan yang dipelajarinya. saat ini teori graf semakin berkembang dan menarik karena keunikan dan penerapannya. salah satu cabang yang biasa digunakan dalam memodelkan permasalahan adalah pewarnaan graf (graph coloring). pewarnaan graf adalah pemberian warna pada elemen graf yang akan dijadikan subjek untuk memahami suatu permasalahan. perwarnaan graf (graph coloring) sendiri terdiri dari tiga macam persoalan, yaitu pewarnaan titik (vertex), pewarnaan sisi (edge), dan pewarnaan wilayah (region) [4]. pewarnaan titik dalam graf , adalah pemberian warna yang berbeda untuk setiap titik atau sisi yang terhubung sedemikian hingga dua titik yang berdekatan atau dua sisi yang bertemu pada titik yang sama memiliki warna yang berbeda [2]. pewarnaan graf banyak sekali diterapkan dalam kehidupan sehari-hari diantaranya pewarnaan graf terhadap pewarnaan sebuah peta, pemetaan, penentuan frekuensi untuk radio, pencocokan pola, penjadwalan dan lain-lain. dalam lingkungan akademik terdapat berbagai macam permasalahan salah satunya permasalahan penyusunan jadwal mata kuliah di tingkat universitas. permasalahan ini dipengaruhi oleh alokasi waktu (kesesuaian sks dengan masing-masing mata kuliah), ketersediaan ruang kuliah, dan banyaknya kelas. salah satu fakultas yang mengalami masalah dalam penyusunan jadwal adalah fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya. permasalahan ini ada dikarenakan umur fakultas yang masih muda, keterbatasan dosen, keterbatasan ruang perkuliahan yang mengakibatkan ruang perkuliahan terkadang saling tumpang tindih, alokasi waktu perkuliahan yang tidak tentu mengakibatkan jadwal perkuliahan yang ada terkadang tidak sesuai sks yang ditentukan di masingmasing mata kuliah. padahal, penentuan ruang perkuliahan dan waktu perkuliahan merupakan elemen penting di dalam penyusunan jadwal perkuliahan di fakultas sains dan teknologi. oleh karena itu, solusi yang tepat agar pembelajaran dapat terlaksana secara efektif dan maksimal adalah pembuatan sistem penjadwalan perkuliahan. berdasarkan uraian tersebut, pada penelitian ini mengimplementasikan jurnal yang berjudul “aplikasi graph coloring pada penjadwalan perkuliahan di fakultas sains dan teknologi uin sunan ampel surabaya”. metode graph coloring merupakan alternatif metode yang tepat dalam sistem penyusunan jadwal, seperti dalam penelitian ataupun jurnal-jurnal sebelumnya graph coloring digunakan sebagai penjadwalan perkuliahan di program studi pendidikan matematika unwidha klaten [1], penyusunan jadwal perkuliahan di jurusan pendidikan matematika fmipa ung [5], implementasi algoritma greddy untuk melakukan graph coloring pada peta [6]. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 31 adapun dalam penelitian ini yang menjadi tujuan adalah sebagai alternatif untuk memudahkan penyusunan jadwal perkuliahan agar tidak saling tumpang tindih di fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya dengan menerapkan konsep graph coloring dan pewarnaan sisi. . 2. tinjauan pustaka 2.1 definisi graf graf g didefinisikan sebagai pasangan himpunan ( v, e ) dengan v adalah himpunan tidak kosong dari simpul-simpul ( vertices atau node ) dari g atau dan e adalah himpunan sisi ( edge atau arcs ) dari g yang menghubungkan sepasang simpul [2]. 2.2 jenis – jenis graf berdasarkan ada tidaknya gelang atau sisi ganda pada suatu graf, secara umum digolongkan menjadi dua jenis [2] yaitu: 1. graf sederhana ( simple graph ) graf sederhana adalah graf yang tidak mengandung gelang maupun sisi ganda. 2. graf tak sederhana ( unsimple – graph ) graf tak sederhana adalah graf yang mengandung sisi ganda atau gelang. ada dua macam graf tak sederhana yaitu: a. graf ganda ( multigraph ) b. graf semu ( pseudograph) berdasarkan orientasi arah pada sisi, secara umum digolongkan menjadi dua jenis [3] yaitu: 1. graf berarah ( directed graph ) graf yang sisinya mempunyai orientasi arah disebut graf berarah. 2. graf tak berarah ( undirected graph ) graf yang sisinya tidak mempunyai orientasi arah disebut graf tak berarah. berdasarkan jumlah titik pada suatu graf, secara umum digolongkan menjadi dua jens [3] yaitu: 1. graf berhingga 2. graf tak berhingga berdasarkan beberapa graf sederhana khusus, maka secara umum graf dibedakan menjadi [3] : 1. graf lengkap ( complete graph ) 2. graf lingkaran 3. graf teratur 4. graf bipartit (bipartite graph) 2.3 pewarnaan graf pewarnaan graf adalah metode pemberian warna pada vertex, edges, maupun wilayah dalam suatu graf. tujuan pemberian warna untuk mencari wilayah tetangga yang ada pada graf. pewarnaan graf terbagi menjadi 3 macam, yaitu pewarnaan sisi (edge colouring) [4], pewarnaan wilayah (region colouring), pewarnaan simpul (vertex coloring). pewarnaan sisi (edge colouring) merupakan pemberian warna pada setiap sisi pada graf sampai sisi-sisi yang saling berhubungan tidak memiliki warna yang sama. pewarnaan wilayah (region colouring) adalah pemberian warna pada setiap wilayah di graf sehingga tidak ada wilayah yang bersebelahan memiliki warna yang sama. pewarnan simpul (vertex colouring) adalah pemberian warna pada setiap simpul (vertex) dimana warna yang sama akan diberikan pada vertex yang saling bertetangga [4]. a)                 b)  c) gambar 1. a) pewarnaan titik b) pewarnaan wilayah c) pewarnaan sisi 2.4 penjadwalan perkuliahan jadwal adalah pembagian waktu berdasarkan rencana pengaturan kerja [5]. salah satu permasalahan dalam perguruan tinggi adalah penjadwalan perkuliahan. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 32 pada penelitian terdahulu, faktor-faktor yang memengaruhi penjadwalah perkuliahan adalah kelompok dari mahasiswa, kelompok dari pengajar, kelompok dari mata kuliah, dan kelompok dari waktu yang ditentukan [5]. sedangkan, untuk penelitian kali ini, faktor-faktor dari masalah penjadwalan perkuliahan adalah: 1. kelompok dari program studi 2. kelompok dari ruangan 3. kelompok dari slot waktu yang telah ditentukan 4. jumlah sks sehari dari masing-masing program studi aktivitas perkuliahan merupakan gabungan dari kelompok mahasiswa, dosen yang mengajar, dan mata kuliah yang diajarkan. permasalahan yang dihadapi dalam kasus ini adalah permasalahan penempatan beberapa aktivitas perkuliahan yang ada ke dalam dimensi slot waktu dan ruangan yang telah ditentukan sehingga diperoleh solusi yang paling optimal. 2.5 penerapan pewarnaan graf dalam penyusunan jadwal perkuliahan pengaplikasian metode pewarnaan graf dalam membuat jadwal adalah dengan menggambarkan graf yang menyatakan penjadwalan. pewarnaan tersebut memiliki ciri-cirinya masing-masing. untuk pewarnaan titik pada sebuah g, dua titik yang berbeda yang dihubungkan oleh sebuah sisi harus memiliki warna yang berbeda. sedangkan pewarnaan sisi pada sebuah g, sisi-sisi dari sebuah titik yang menghubungkan dengan titik-titik yang lain harus memiliki warna yang berbeda [5]. untuk memudahkan dalam penyusunan jadwal dengan pewarnaan graf, data-data yang behubungan dengan penjadwalan diklasifikasikan ke dalam beberapa himpunan. dalam penyusunan jadwal perkuliahan, faktor-faktor yang harus diperhatikan antara lain [5]: 1. jumlah program studi di fakultas sains dan teknologi 2. ketersediaan ruang kuliah 3. waktu perkuliahan 4. jumlah sks sehari dari masingmasing program studi untuk poin 1 dan 2 di dalam graf akan dinyatakan dalam bentuk himpunan titiktitik. sedangkan untuk poin 3 dan 4 di dalam graf akan dinyatakan dalam bentuk sisi. dari proses metode pewarnaan graf tersebut, dapat dibuat jadwal yang tepat sehingga jadwal dari ruang yang sama, dibuat tidak pada waktu yang sama selain itu dapat diatur jumlah sks dari masing-masing program studi. dari sebuah graf yang dibuat dengan pewarnaan sisi, dari warna sisi, dapat diketahui apabila sisi yang bewarna sama, menunjukkan waktu yang sama. namun apabila warna yang digunakan berbeda, maka jadwal tersebut tidak dibuat pada waktu yang sama. begitu juga untuk ruangannya. 3. metode penelitian metode penelitian yang digunakan dalam penelitian ini adalah studi literature. bukubuku yang dijadikan sebagai acuan pada penelitian ini antara lain matematika diskrit, graph and application. selain itu, penelitian ini menggunakan data-data dari fakultas sains dan teknologi seperti jumlah program studi, hari aktif perkuliahan dan ruang perkuliahan yang tersedia. selanjutnya literature utama dan literature pendukung dianalisis untuk mengetahui aplikasi pewarnaan graf pada penjadwalan perkuliahan di fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya. adapun tahap-tahap penelitian jika disajikan dalam bentuk bagan alir penelitian (flowchart) seperti pada gambar 2 berikut. data domain kodamain pewarnaan sisi mulai g r a p h c o l o r i n g jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 33 gambar 2. bagan alir penelitian langkah-langkah dari proses ini dijelaskan sebagai berikut: a. inputkan data mahasiswa fakultas sains dan teknologi. data nantinya dibagi menjadi dua bagian, yaitu data sebagai domain dan data sebagai kodomain.   b.tarik garis dari domain ke kodomain, kemudian sisi atau garis yang berhubungan diberi warna yang berbeda. lakukan perwanan sisi atau garis pada data yang diteliti sampai selesai.    4. hasil dan pembahasan 4.1 graf jadwal matakuliah ketersediaan ruang perkuliahan yang terbatas dibandingkan jumlah program studi yang ada di fakultas sains dan teknologi mengakibatkan kegiatan perkuliahan saling tumpang tindih. maka perlunya jadwal yang efektif dan efisien dalam hal ruang kuliah, jumlah program studi dan waktu perkuliahan. oleh karena itu, sebelum metode pewarnaan graf digunakan untuk membuat jadwal kuliah terlebih dahulu mempresentasikan komponen-komponen penjadwalan ke dalam graf. adapun penggambaran banyaknya program studi, banyaknya hari perkuliahan dan banyaknya ruang perkuliahan adalah sebagai berikut :     gambar 3. graf antara banyaknya program studi, banyaknya hari, dan banyaknya jumlah kelas keterangan: mtk = matematika bio = biologi ik = ilmu kelautan si = sistem informasi tl = teknik lingkungan arsk = arsitektur dari gambar graf di atas terlihat masingmasing program studi di semester ii, dan ruangan perkuliahan yang ditempati mahasiswa fakultas sains dan teknologi. 4.2 pewarnaan graf dalam penjadwalan pewarnaan yang digunakan pada graf penjadwalan di jurnal ini adalah pewarnaan sisi. pewarnaan sisi merupakan pemberian warna pada sisi-sisi suatu graf sedemikian sehingga setiap dua sisi yang bertemu pada titik yang sama mendapatkan warna yang berbeda [4]. kemudian banyaknya warna yang digunakan dalam mewarnai sisi dapat menentukan bilangan kromatiknya [2]. penjadwalan di ambil pada mahasiswa angkatan 2015 fakultas sains dan teknologi uin sunan ampel surabaya di semester genap. untuk membuat jadwal perkuliahan yang seefisien mungkin di mulai dari pewarnaan harian. 4.2.1 pewarnaan harian berikut ini merupakan representasi pejadwalan dengan pewarnaan harian berdasarkan 3 simulasi. a. simulasi 1       a) b) selesai jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 34 (c) (d) (e) gambar 4. graf antara banyaknya program studi dan banyaknya jumlah kelas di semua hari aktif perkuliahan pada simulasi 1 a) hari senin b) hari selasa c) rabu d) kamis e) jum’at b. simulasi 2 (a) (b) (c) (d) (e) gambar 5. graf antara banyaknya program studi dan banyaknya jumlah kelas di semua hari aktif perkuliahan pada simulasi 2. a) hari senin b) hari selasa c) rabu d) kamis e) jum’at c. simulasi 3 (a) (b) (c) (d) (e) gambar 6. graf antara banyaknya program studi dan banyaknya jumlah kelas di semua hari aktif perkuliahan pada simulasi 3. a) hari senin b) hari selasa c) rabu d) kamis e) jum’at jadi pewarnaan minimal yang mungkin terjadi adalah dua sehingga bilangan kromatiknya 2. perbedaan warna pada graf menyatakan bahwa dari programprogram studi dapat melaksanakan perkuliahan pada waktu yang sama, dengan warna biru sebagai warna 1 dan warna merah sebagai warna 2. graf antara banyaknya program studi dan banyaknya jumlah kelas di semua hari aktif perkuliahan direpresentasikan pada tabel 1. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 35 tabel 1. banyaknya program studi dan banyaknya jumlah kelas di semua hari aktif perkuliahan fakultas sains dan teknologi hari prodi ruangan w.1 w.2 w.1 w.2 senin mtk tl g1. 203 g1. 201 bio si g1. 201 g1. 203 ik sac. 201 si sac. 203 arsk g1. 202 selasa mtk tl g1. 203 g1. 201 bio g1. 201 ik sac. 201 si sac. 203 arsk g1. 202 rabu mtk tl g1. 203 g1. 201 bio g1. 201 ik sac. 201 si sac. 203 arsk g1. 202 kamis mtk tl g1. 203 g1. 201 bio arsk g1. 201 g1. 202 ik sac. 201 si sac. 203 arsk g1. 202 jumat mtk tl g1. 203 g1. 203 bio si g1. 201 g1. 201 ik sac. 201 si sac. 203 arsk g1. 202 keterangan: w.1 = warna 1 w.2 = warna 2 4.2.2 pewarnaan berdasarkan jam matakuliah berikut adalah pewarnaan graf dari setiap program studi fakultas sains dan teknologi dengan ruangan perkuliahan, jam matakuliah dan sks (sistem kredit semester) yang diberikan dengan 3 simulasi. a. simulasi 1 1) hari senin   gambar 7 graf jadwal kuliah fakultas sains dan teknologi pada hari senin 1) hari selasa       gambar 8 graf jadwal kuliah fakultas sains dan teknologi pada hari selasa 2) hari rabu      jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 36                          gambar 9 graf jadwal kuliah fakultas sains dan teknologi pada hari rabu 3) hari kamis        gambar 10 graf jadwal kuliah fakultas sains dan teknologi pada hari kamis 4) hari jumat       gambar 11 graf jadwal kuliah fakultas sains dan teknologi pada hari jum’at b. simulasi kedua 1) hari senin gambar 12 graf jadwal kuliah fakultas sains dan teknologi pada hari senin 2) hari selasa gambar 13 graf jadwal kuliah fakultas sains dan teknologi pada hari selasa 3) hari rabu gambar 14 graf jadwal kuliah fakultas sains dan teknologi pada hari rabu 4) hari kamis jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 37 gambar 15 graf jadwal kuliah fakultas sains dan teknologi pada hari kamis 5) hari jumat gambar 16 graf jadwal kuliah fakultas sains dan teknologi pada hari jum’at c. simulasi 3 1) hari senin gambar 17 graf jadwal kuliah fakultas sains dan teknologi pada hari senin 2) hari selasa gambar 18 graf jadwal kuliah fakultas sains dan teknologi pada hari selasa 3) hari rabu gambar 19 graf jadwal kuliah fakultas sains dan teknologi pada hari rabu 4) hari kamis           jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 38          gambar 20 graf jadwal kuliah fakultas sains dan teknologi pada hari kamis 5) hari jumat      gambar 21 graf jadwal kuliah fakultas sains dan teknologi pada hari jum’at pewarnaan sisi berdasarkan jam matakuliah pada masing-masing gambar graf di atas yaitu setiap sisinya tidak mempunyai hubungan dengan sisi yang lain. dengan demikian cukup diwarnai dengan satu warna saja. kemudian sisi yang berhubungan diberi warna yang berbeda, untuk pewarnaan sisi yang selanjutnya, pemberian warna sama dengan warna awal yang digunakan. berikut ini adalah graf jadwal kuliah dari program studi fakultas sains dan teknologi di setiap harinya di mulai dengan hari senin sampai jum’at berdasarkan ruang perkuliahan yang tersedia dan jumlah sks (sistem kredit semester) pada setiap program studi pada semester genap angkatan 2015 yang direpresentasikan pada tabel 2. tabel 2. penjadwalan matakuliah setiap program studi berdasarkan waktu dan sks hari jam prodi ruangan 2 sks 3 sks 5 sks s e n i n 07.45 -09.25 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 09.25 -11.55 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 12.30 -15.00 tl si g1. 201 g1. 203 s e l a s a 07.45 -09.25 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 09.25 -11.55 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 12.30 -15.00 tl g1. 203 r a b u 07. 45 -10.15 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 10. 15 -11.55 tl g1. 201 12.30 -14.10 mtk bio ik si arsk g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 k a m i s 07.45 -10.15 mtk bio ik si tl g1. 203 g1. 201 sac. 201 sac. 203 g1. 202 10.15 -11.55 tl arsk sac. 201 g1. 202 12.30 -15.00 tl g1. 202 12.30 -16.30 arsk sac. 201 j u m a t 07.45 -10.25 mtk bio ik g1. 203 g1. 201 sac. 201 13.00 -15.30 tl si g1. 203 5. simpulan permasalahan penyusunan jadwal mata kuliah di tingkat universitas dipengaruhi oleh parameter-parameter diantaranya alokasi waktu (kesesuaian sks dengan masingmasing mata kuliah), ketersediaan ruang kuliah, dan banyaknya program studi di semester genap . berdasarkan analisa graph coloring berupa pewarnaan sisi pada kasus penjadwalan program studi fakultas sains dan teknologi universitas islam negeri sunan ampel surabaya yang memiliki 6 program studi dan 5 ruang perkuliahan dengan tiga kali simulasi kombinasi dari data inputan, didapatkan penjadwalan yang tidak berubah (konsisten) dan hasil penjadwalannya didapatkan penjadwalan yang tidak saling tumpang tindih/tabrakan, jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 39 baik dalam penggunaan ruang perkuliahan, dan waktu perkuliahan. 6. referensi [1] tasari. aplikasi pewarnaan graf pada penjadwalan perkuliahan di program perkuliahan pendidikan matematika unwidha klaten. magistra (2014) no.82. issn.0215-9611. [2] munir, rinaldi. 2012. “matematika diskrit revisi ke-5”. bandung: informatika. [3] suryadi, didi, dan nanang priatna. “pengantar dasar teori graph”. [4] aldous, joan m. dan robin j.w.2000. “graphs and application”. london: springer. [5] yahya, n.i. dkk. penerapan konsep graf dalam penyusunan jadwal perkuliahan di jurusan pendidikan matematika fmipa ung. [6] ardiansyah. dkk. implementasi algoritma greedy untuk melakukan graph coloring : studi kasus peta propinsi jawa timur, jurnal informatika (2010) vol 4, no. 1. [7] andayani, sri dan endah wulan perwitasari. penentuan rute terpendek pengambilan sampah di kota merauke menggunakan algoritma dijkstra, seminar nasional teknologi informasi dan komunikasi terapan (semantik 2014) semarang (2014) isbn 979-26-0276-3. how to cite: n. atikah, “application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors”, mantik, vol. 5, no. 2, pp. 123-134, october 2019. application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors nur atikah universitas negeri malang, nur.atikah.fmipa@um.ac.id doi: https://doi.org/10.15642/mantik.2019.5.2.123-134 abstract: wilayah metropolitan malang adalah salah satu daerah di jawa timur yang merupakan tujuan wisata terkemuka di indonesia dengan kota wisata batu sebagai pusatnya. mengingat perkembangan pariwisata di malang, perlu dilakukan pengelompokan popularitas objek wisata sehingga dapat dijadikan referensi untuk pembuatan kebijakan oleh departemen pariwisata dan manajemen pariwisata. pada artikel ini, pengelompokan dianalisis dengan menggunakan metode pengelompokan algoritma expectation maximation (em). data yang digunakan adalah data sekunder yang diperoleh dari data bps, yaitu data banyak pengunjung wisata di malang raya. hasil pengelompokan popularitas objek wisata unggulan di malang didasarkan pada indikator jumlah pengunjung dibagi menjadi lima kelompok, ada kelompok 1: selecta; grup 2: balekambang, pemandian wendit dan wisata oleh-oleh brawijaya; grup3: museum angkut, coban rondo, museum satwa, taman jatim, bns, petik apel “makmur abadi dan agro kebun teh wonosari; grup 4: kusuma agro wisata, kampoeng kidz, air panas cangar, eco green park, taman hiburan predator, wana wisata coban rais, gunung banyak, t-shirt mahajaya & oleh-oleh, ngliyep dan bendungan selorejo; grup 5: vihara "dammadhipa arama", arung jeram "kaliwatu", arung jeram, wana wisata coban talun, pemandian tirta nirwana, pemandian air panas alam songgoriti, wahana air rafting, sahabat air rafting, petik apel mandiri, batu agro apel, kampung wisata. kata kunci: algoritma em, objek wisata, malang raya abstract: malang metropolitan area is one of the areas in east java which is a leading tourism destination in indonesia with batu tourism city (kota wisata batu) as the center. considering the development of tourism in malang, it is necessary to do a grouping of the popularity of tourism objects so that it can be used as a reference for making policy by the tourism department and tourism management. in this article, the grouping is analyzed by using the method of grouping the expectation maximation (em) algorithm. the data used is secondary data obtained from bps data, namely data of many tourism visitors in malang raya. the results of the grouping the popularity of leading tourism objects in malang are based on indicators of the number of visitors divided into five groups, there are group 1: selecta; group 2: balekambang, pemandian wendit and wisata oleh-oleh brawijaya; group3: museum angkut, coban rondo, museum satwa, jatim park, bns, petik apel “makmur abadi and agro kebun teh wonosari; group 4: kusuma agro wisata, kampoeng kidz, air panas cangar, eco green park, predator fun park, wana wisata coban rais, gunung banyak, mahajaya t-shirt & oleh-oleh, ngliyep and bendungan selorejo; group 5: vihara “dammadhipa arama”, rafting “kaliwatu”, batu rafting, wana wisata coban talun, pemandian tirta nirwana, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro apel, kampung wisata. keywords: em algorithm, tourism object, malang raya . jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 123-134 issn: 2 52 7 3 1 59 ( p r i n t ) 2 5 27 3 1 6 7 ( o n l i n e ) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 124 1. introduction nowadays, malang is transformed into the second-largest city in east java after surabaya, which is only 90 kilometres away. this has become one of the causes for malang city to grow as a trade and service area, education city and also as a city that has a creative economy industry. malang city continues to morph into a metropolitan city that has large magnets. malang is no longer just an apple city and a cool area at the foothill of bromo and semeru. geographically, malang city is very close even surrounded by malang regency and batu city, then known as malang raya. malang is a cool metropolitan city. therefore, it is reasonable if the metropolitan malang raya area is a leading tourist destination in indonesia with batu tourism city as the center. malang raya area is the second largest metropolitan in east java after gerbang kertosusila. the regency and city of culture and tourism office (disbudpar) revealed that the number of tourist visits in 2017 in malang regency reached 6.5 million people and malang city reached 4 million people. while based on data from the batu city culture and tourism office, the number of tourists was recorded at 4.7 million people. the high level of tourist visits to malang, which is a motivation for the malang city government, malang regency, and batu city continue to concentrate on boosting tourism to increase locally-generated revenue or pad (pendapatan asli daerah). various existing potentials are packed again more interesting to be able to boost the economy of the community [1]. in building a good tourism industry and developing existing tourism that is better in quality and can provide many positive influences for the development of economic conditions, a specific strategy is needed to achieve it. various important factors need to be seen and implemented in order to achieve a targeted and sustainable development and development plans, such as careful planning, effective strategies and objectives, revamping tourism objects, facilities, services to tourism promotion or marketing are important factors in supporting tourism development. tourist visitor data on each tourist attraction is the main and potential input for developing tourism. of course, not all attractions are developed because of limited government funds. thus, it is very necessary to be informed about tourism objects that are a priority of development. of course, tourism grouping is needed, which will make it easier for managers to repair facilities and infrastructure that can increase the number of tourists. statistically, grouping tourist objects can be done using one method in statistics, namely clustering. some studies on clustering include silvi grouping hivaids indicators in indonesia with centroid linkage and k-means clustering methods that contain data outliers [2]. in addition to centroid linkage and k-means clustering, another method of grouping is using the expectation-maximization (em) algorithm. the em algorithm is an algorithm that functions to find the estimated maximum likelihood value of the parameters in a probabilistic model [3]. the advantage of the em algorithm is that it can solve statistical problems such as estimating parameters for a combination of functions and parameters from incomplete data [4]. in this algorithm, there are two things that are used interchangeably, namely e-step that calculates the expectation value of likelihood including latent variables as if they exist, and mn. atikah application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors 125 step calculates the estimated value of ml from parameters by maximizing the expected value of likelihood which found in e-step. the renewal of this paper is the application of the expectation-maximization (em) method to find out the popularity of tourist objects in malang raya based on many visitors in 2018. some previous studies using grouping with the em algorithm include darwianto & sirait discussing implementation and clustering expectation-maximizationon algorithm analysis at the final assignment of telkom university. the research aims to facilitate users in finding information on a large enough document, namely by grouping or categorizing documents according to the similarity of documents [5]. soeyapto & johari examines the application of data mining for data on the number of vehicles using the expectationmaximization (em) algorithm in the palembang city dispenda [6]. the study aims to provide clearer information for the dispenda party and simplify the analysis of the increase in the number of vehicle data by looking at the grouping of the number of vehicles in an area [7]. 2. literature review 2.1 definition of multivariate analysis multivariate analysis is one of the statistical techniques applied to understand data structures in high dimensions. where are the intended variables these are interrelated with each other? according to santoso, multivariate analysis can be defined simply as a method of processing variables in large quantities to find their influence on an object simultaneously [8]. multivariate statistical analysis is a statistical method that allows conducting research on more than two variables simultaneously. where there is at least one dependent variable and more than one independent variable, and there is a correlation between one variable and another. as with other statistical analysis, the multivariate analysis also has types of data or scale data. the scale of the data used is of two kinds, namely metric data and non-metric data. metric data is data that is numerical or contains numbers and mathematical calculations can be carried out in it. metric data are also called numerical data or quantitative data. in this case, there are two kinds of metrics data, namely interval data and ratio data. while non-metric data is non-numeric data or also called qualitative data or categorical data. there are two types of nonmetric data, namely nominal data and ordinal data. 2.2 definition of cluster analysis cluster analysis is a multivariate technique whose purpose is to get object groupings by arranging objects into groups in such a way that in a group has maximum similarity (rencher, 2002) [9]. in general, there are two types of methods used for clustering, namely hierarchical and non-hierarchical methods. hierarchical cluster analysis is done by grouping two or more objects that have the closest and the same thing so that the levels between groups become clearly visible. in hierarchical cluster analysis, the grouping results are displayed in the form of dendograms, while in non-hierarchical cluster analysis clustering begins by first determining the cluster to be formed. one of these methods can be used in this study, which is a non-hierarchical method, namely the analysis of the expectation-maximization (em) algorithm. there are several things that need to jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 126 be emphasized in the em algorithm, namely, maximum likelihood estimation (mle), gaussian distribution, and expectation-maximization (em). 2.3 maximum likelihood estimation maximum likelihood estimation (mle) was introduced by r. a fisher in 1912. mle is usually used to estimate parameter values that a function has, such as mean, variance, and so on. bain and engelhardt define mle as follows [10]: for example, x1, x2, x3, ..., xn is a random sample of a population with density 𝑓 ( 𝑋𝑖 ; 𝜃 ) . then the likelihood function is defined as a joint density function after data {𝑥1, 𝑥2, ⋯ , 𝑥𝑛 } is obtained. so that the shared density function is seen as a function of the parameters {𝜃1, 𝜃2, ⋯ , 𝜃𝑛 } and expressed by: 𝐿(𝜃1,𝜃2, … , 𝜃𝑛 ) = ∏ 𝑓(𝑥𝑖 ; 𝜃) 𝑛 𝑖=1 thus, what mle wants is to maximize the value of the likelihood function to obtain an estimator value from {𝜃1, 𝜃2, ⋯ , 𝜃𝑛 } . this is very reasonable to understand because it is in accordance with determining the parameter estimator value that has the greatest chance. the steps to determine the estimator are in accordance with determining the optimal value in the differential calculation. if this likelihood function is differentiated, then the possible likelihood estimator is 𝜃 such that 𝜕𝐿(𝜃) 𝜕𝜃 = 0 to prove that 𝜃 really maximizes the likelihood function l(𝜃) it must be shown that: 𝜕2𝐿(𝜃) 𝜕2𝜃 < 0 in many cases (especially if the value of the likelihood function is very large) where differentiation is used, it will be easier to work on the logarithm of l(𝜃) which is log l(𝜃). of course, to determine the optimization of likelihood function remains the same as determining the maximum function of the logarithmic likelihood function, this is possible because the monotonous logarithm function rises at (0, ∞) which means that l(𝜃) has the same extreme, so to determine the maximum likelihood estimator from 𝜃 as follows: a. determine the likelihood function 𝐿(𝜃1,𝜃2, … , 𝜃𝑛 ) = ∏ 𝑓(𝑥𝑖 ; 𝜃) 𝑛 𝑖=1 b. forms a log-likelihood 𝑙 = log 𝐿(𝜃) c. determine the derivative of 𝑙 = log 𝐿(𝜃) against 𝜃 𝜕 log[𝐿(𝜃)] 𝜕𝜃 = 0 the completion of point 3 is the maximum likelihood estimator for 𝜃. d. determine the second derivative of 𝑙 = log 𝐿(𝜃) against 𝜃. if 𝜕2𝐿(�̂�) 𝜕2�̂� < 0, it will prove that 𝜃 really maximizes the likelihood function. n. atikah application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors 127 2.4 expectation maximation (em) algorithm the expectation-maximization (em) algorithm was first introduced by dempster, laird, and rubin in 1977. according to kusrini & lutfi, the expectation-maximization (em) algorithm is often used to find the maximum likelihood (ml) estimation of the parameters in a probabilistic model, where the model also depends on unknown latent variables [11]. in this algorithm, there are two things that are used interchangeably, namely e-step that calculates the expectation value of likelihood including latent variables as if they exist, and m-step calculates the estimated value of ml from parameters by maximizing the expected value of likelihood found in e-step. the expectation-maximization (em) algorithm has better properties than other approaches or methods. some of the advantages of the em algorithm compared to other approaches include [12]: a. the em algorithm is more numerically stable, wherein each iteration the log-likelihood rises. b. under general conditions, em algorithms converge to a reliable value. that is, by starting an arbitrary value θ(0) it will almost always converge to a local maximizer, except wrong in taking the initial value θ(0). c. the em algorithm tends to be easy to implement because it relies on calculating complete data. d. the em algorithms are easily programmed because they do not involve either integrals or derivatives of likelihood. e. the em algorithm only takes up a little hard disk space and memory on the computer because it doesn't use a matrix or its inverse in each iteration. f. the analysis is easier than other methods. g. taking into account the increase in monotonous likelihood in iterations, it is easy to monitor convergence and program errors. h. can be used to estimate the value of missing data. although there are many advantages of the em algorithm, there are weaknesses of the em algorithm, including: a. it does not provide a procedure for generating covariance matrix estimates from parameter estimators. b. the em algorithm can converge slowly, that is if there is too much incomplete information. c. the em algorithm does not guarantee that it will converge to a maximum global value if there are multiple maxima. the em algorithm is a process that is divided into two steps, there are: 1) expectation step (e-step) search for expectation values to the likelihood function based on observed variables. 2) maximization step (m-step) mle a search of parameters by maximizing likelihood expectations generated from e-step. searching for parameters generated from the m-step will be used again for the next e-step, and this step will be repeated until it gives a convergent value and is an estimator of a parameter. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 128 suppose there is a sample of 𝑛 items where 𝑛1of the item is observed while 𝑛2 = 𝑛 − 𝑛1 items are not observed. the observed items are denoted by 𝑋’ = ( 𝑋1 , 𝑋2 , … , 𝑋𝑛 ) and unobserved items are denoted ’ = ( 𝑍1 , 𝑍2 , … , 𝑍𝑛 ) ).. assume 𝑋𝑖𝑠 and 𝑍𝑗𝑠 are mutually independent and identically distributed variables (independent and identically distribution) with a probability density function 𝑓(𝑥|𝜃), where 𝜃 ∈ ω. assume 𝑋𝑖𝑠 and 𝑍𝑗𝑠 are mutually independent. denote a function of the density of the combined probability of x with 𝑔(𝑥|𝜃). then ℎ(𝑥, 𝑧| 𝜃) for the combined probability density function for observed and unobserved data. whereas 𝑘(𝑧| 𝜃, 𝑥) represents the conditional probability density function notation of the missing data to provide observed data. then it can be obtained 𝑘(𝑧| 𝜃, 𝑥) = ℎ(𝑥, 𝑧| 𝜃) 𝑔(𝑥|𝜃) = 0 the observed likelihood function of the data is 𝐿(𝜃|𝑥) = 𝑔(𝑥|𝜃) then the likelihood function for complete data is defined by 𝐿𝑐 (𝜃|𝑥, 𝑧) = ℎ(𝑥, 𝑧|𝜃) our goal is to maximize the likelihood function 𝐿(𝜃|𝑥) by using the complete likelihood function 𝐿𝑐 (𝜃|𝑥, 𝑧) in the process. use the equation 𝑘(𝑧| 𝜃, 𝑥), obtained log 𝐿(𝜃|𝑥) = ∫ log 𝐿(𝜃|𝑥) . 𝑘(𝑧|𝜃0, 𝑥)𝑑𝑧 log 𝐿(𝜃|𝑥) = ∫ log 𝑔(𝜃|𝑥) . 𝑘(𝑧|𝜃0, 𝑥)𝑑𝑧 log 𝐿(𝜃|𝑥) = ∫[log ℎ(𝑥, 𝑧|𝜃) − log 𝑘(𝑧|𝜃0, 𝑥)] . 𝑘(𝑧|𝜃0, 𝑥)𝑑𝑧 log 𝐿(𝜃|𝑥) = ∫ log ℎ(𝑥, 𝑧|𝜃)𝑘(𝑧|𝜃0, 𝑥)𝑑𝑧 − ∫ log 𝑘 (𝑧|𝜃, 𝑥). 𝑘(𝑧|𝜃, 𝑥) 𝑑𝑧 log 𝐿(𝜃|𝑥) = 𝐸𝜃0 [𝑙𝑜𝑔𝐿 𝑐 (𝜃|𝑥, 𝑧)|𝜃0, 𝑥] − 𝐸𝜃0 [𝑙𝑜𝑔𝑘(𝑍|𝜃, 𝑥)|𝜃0, 𝑥] where expectations are taken under the conditional probability density function of 𝑘(𝑧|𝜃0, 𝑥). then define the first part on the right side of the function above. 𝑄(𝜃|𝜃0, 𝑥) = 𝐸𝜃0 [𝑙𝑜𝑔𝐿 𝑐 (𝜃|𝑥, 𝑧)|𝜃0, 𝑥] the expectation defined by the q function is called e-step from the em algorithm. denoted 𝜃(0) estimation initials of θ, based on the observed likelihood function. then 𝜃(1) becomes the argument that maximizes 𝑄(𝜃|𝜃0, 𝑥). this is the first step to estimate θ then we define the em algorithm as follows. denoted 𝜃(𝑚) in estimating step m. then to estimate steps to (m+1): a. expectation step (e-step) 𝑄(𝜃|𝜃(𝑚), 𝑥) = 𝐸 �̂�(𝑚) [𝑙𝑜𝑔𝐿𝑐 (𝜃|𝑥, 𝑧)|𝜃 (𝑚), 𝑥] where expectations are taken from the conditional probability density function 𝑘(𝑧|𝜃 (𝑚), 𝑚) b. maximization step (m-step) 𝜃(𝑚+1) = 𝐴𝑟𝑔 max 𝑄( 𝜃|𝜃(𝑚), 𝑥) where, n. atikah application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors 129 𝑄(𝜃(𝑚+1)|𝜃(𝑚), 𝑥) ≥ 𝑄(𝜃(𝑚)|�̂�(𝑚), 𝑥) the following are some of the advantages of using the em algorithm [13]: a. the em algorithm is quite stable and easy to make the program. b. in general, the em algorithm has reliable convergence, means that it always converges almost to its local maximum point. c. requires a small storage capacity on the computer. d. can be used to estimate the value of lost data, because, in the em algorithm, there is a process of distributing incomplete data to complete data based on the conditional opportunity value. 3. research methods the approach in this research proposal is quantitative research. the quantitative approach aims to test the theory, build facts, show relationships between variables, provide statistical descriptions, estimate and forecast results. the type of this research used is the descriptive quantitative study of existing problems, modelling in the input and output systems in weka data mining applications, solving problems and interpreting them. this study aims to classify the level of popularity of tourist attractions in malang raya based on indicators of many tourist attractions using the expectation-maximization (em) algorithm. the steps to be carried out in this study are described as follows: a. input data into the weka data mining application 3.8. b. it is using clustering methods that are used to determine the value of the cluster to be processed. the method used is by grouping data that has been inputted from the data. c. calculate the value of the centroid cluster. determine the cluster value first if, after clustering, there is still data that changes then it is repeated again to the cluster iteration process. d. displays clustering grouping, aims to show the iterative process and class in the cluster with the number of records in the store name. 4. results and discussion in this study, the author will determine clustering into two groups, three groups, four groups and five groups. grouping with two groups consists of groups of tourist objects with high popularity and tourist groups with low popularity. grouping with three groups consists of groups of tourist objects with high levels of popularity, groups of tourist objects with moderate levels of popularity and groups of tourist objects with low levels of popularity. grouping with four groups consists of a group of tourist objects with a very high level of popularity, a group of tourist objects with a high level of popularity, a group of tourist objects with low popularity and tourist groups with very low levels of popularity. grouping with five groups consists of groups of tourist objects with a very high level of popularity, groups of tourist objects with a high level of popularity, groups of tourist objects with moderate levels of popularity, groups of tourists with low popularity and tourist groups with a high level of popularity very low. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 130 analysis with the expectation-maximization (em) algorithm is calculated using the help of weka software 3.8. the algorithm begins with the expectation step, namely initializing the initial value then iterating so that it reaches a convergent value. the results of the grouping are as follows: grouping with 2 groups based on the analysis with the em algorithm by using the help of weka 3.8 software, it is known that the iteration step carried out with two groups is 27 times so that the results obtained are group 1 which is included in the group of attractions with high popularity consisting of 22 attractions. group 2 which is included in the group of tourist objects with a low level of popularity consists of 18 attractions. the division of group members is as follows: group 1 : kusuma agro wisata, selecta, bns, museum satwa, jatim park, petik apel “makmur abadi”, air panas cangar, museum angkut, eco green park, predator fun park, wana wisata coban rais, pemandian tirta nirwana, wana wisata coban talun, gunung banyak, mahajaya t-shirt & oleh-oleh, wisata oleh-oleh brawijaya, agro kebun teh wonosari, bendungan selorejo balekambang, ngliyep, pemandian wendit and coban rondo. group 2 : vihara “dammadhipa arama”, rafting “kaliwatu”, kampoeng kidz, batu rafting, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro apel, kampung wisata kungkuk, desa wisata sumberejo, desa wisata bumiaji, mega star indonesia, wisata oleh-oleh deduwa, candi jago, sengkaling, pemandian dewi sri and candi kidal. grouping with 3 groups based on the analysis with the em algorithm by using the help of weka 3.8 software, it is known that the iteration step carried out with three groups is 18 times so that the results are group 1 which belongs to the group of tourist objects with a high level of popularity consisting of 5 attractions. group 2 is included in the group of tourist objects with a popularity level that is comprised of 17 attractions. group 3 which is included in the group of tourist objects with a low level of popularity consists of 18 attractions. the division of group members is as follows: group 1 : selecta, wisata oleh-oleh brawijaya, balekambang, pemandian wendit and coban rondo. group 2 : kusuma agro wisata, jatim park, air panas cangar, bns, petik apel “makmur abadi”, museum satwa, eco green park, museum angkut, predator fun park, gunung banyak, pemandian tirta nirwana, wana wisata coban talun, wana wisata coban rais, mahajaya t-shirt & oleh-oleh, agro kebun teh wonosari, ngliyep and bendungan selorejo. group 3 : vihara “dammadhipa arama”, rafting “kaliwatu”, kampoeng kidz, batu rafting, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro apel, kampung wisata kungkuk, desa wisata n. atikah application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors 131 sumberejo, desa wisata bumiaji, mega star indonesia, wisata oleh-oleh deduwa, candi jago, sengkaling, pemandian dewi sri and candi kidal. grouping with 4 groups the iteration step carried out with four groups is 29 times so that the results obtained are group 1 which belongs to the group of tourist objects with a very high level of popularity consisting of 6 attractions. group 2 which is included in the group of tourist objects with a high level of popularity consists of 5 attractions. group 3 which is included in the group of tourist objects with low popularity consists of 12 attractions. group 4 which is included in the group of attractions with a very low level of popularity consists of 17 attractions. the division of group members is as follows: group 1 : selecta, museum angkut, wisata oleh-oleh brawijaya, balekambang, pemandian wendit and coban rondo. group 2 : museum satwa, jatim park, bns, petik apel “makmur abadi and agro kebun teh wonosari. group 3 : kusuma agro wisata, kampoeng kidz, air panas cangar, eco green park, predator fun park, wana wisata coban rais, pemandian tirta nirwana, wana wisata coban talun, gunung banyak, mahajaya t-shirt & oleh-oleh, ngliyep and bendungan selorejo. group 4 : vihara “dammadhipa arama”, rafting “kaliwatu”, batu rafting, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro ape, kampung wisata kungkuk, desa wisata sumberejo, desa wisata bumiaji, mega star indonesia, wisata oleh-oleh deduwa, candi jago, sengkaling, pemandian dewi sri and candi kidal. grouping with 5 groups in grouping with these five groups, there were no iterations made, so that the results obtained were group 1 which was included in the group of tourist objects with a very high level of popularity consisting of 1 tourist attraction. group 2 which is included in the group of tourist objects with a high level of popularity consists of 3 attractions. group 3 is included in the tourist attraction group with a popularity level that consists of 7 attractions. group 4 which is included in the group of tourist objects with a low level of popularity consists of 10 attractions. group 5 which is included in the group of attractions with a very low level of popularity consists of 19 attractions. the division of group members is as follows: group 1 : selecta. group 2 : balekambang, pemandian wendit and wisata oleh-oleh brawijaya. group 3 : museum angkut, coban rondo, museum satwa, jatim park, bns, petik apel “makmur abadi and agro kebun teh wonosari. group 4 : kusuma agro wisata, kampoeng kidz, air panas cangar, eco green park, predator fun park, wana wisata coban rais, gunung banyak, mahajaya t-shirt & oleh-oleh, ngliyep and bendungan selorejo. group 5 : vihara “dammadhipa arama”, rafting “kaliwatu”, batu rafting, jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 132 wana wisata coban talun, pemandian tirta nirwana, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro ape, kampung wisata kungkuk, desa wisata sumberejo, desa wisata bumiaji, mega star indonesia, wisata oleh-oleh deduwa, candi jago, sengkaling, pemandian dewi sri and candi kidal. with the grouping of 5 groups, it was found that the iteration process had stopped (no more iterations). this shows that the grouping of attractions divided into five groups is the maximum result. selection of the best model in the expectation-maximization (em) algorithm, the best model represents the best data or model is indicated by the largest log-likelihood value. the following is the log-likelihood value of each cluster. tabel 4.1 log likelihood value number of clusters log-likelihood 2 -12,80051 3 -12,66082 4 -12,5927 5 -12,09286 based on table 4.1, it can be seen that testing with five groups has the largest log-likelihood value, so the best model is the expectation-maximization (em) algorithm testing with five groups that produce one tourist attraction with a very high level of popularity, three attractions with popularity the high, seven tourist objects with moderate popularity, ten attractions with low popularity and 19 tourist attractions with low popularity. 5. conclusion and recommendation the conclusions from this study are: (a) the application of the expectationmaximization (em) algorithm in grouping the popularity of leading tourist objects in malang raya based on indicators of many visitors was carried out using weka 3.8 software assistance. the number of groupings used was two groups, three groups, four groups and 5 groups. based on the log likelihood value, it was found that the group with 5 groups had the largest log-likelihood value, so the model suitable for this research was a model with 5 groups, namely a group of tourists with very high popularity, a group of tourists with high popularity, groups tourist attraction with a moderate level of popularity, a group of tourist objects with low popularity and tourist groups with very low levels of popularity, (b) the results of grouping the popularity of leading tourist objects in malang raya based on indicators of many visitors using the expectation-maximization (em) algorithm are as follows: group 1 : selecta. group 2 : balekambang, pemandian wendit and wisata oleh-oleh brawijaya. group 3 : museum angkut, coban rondo, museum satwa, jatim park, bns, petik apel “makmur abadi and agro kebun teh wonosari. group 4 : kusuma agro wisata, kampoeng kidz, air panas cangar, eco n. atikah application of expectation-maximization (em) algorithm in grouping popularity tourism objects in malang raya based on indicator of many visitors 133 green park, predator fun park, wana wisata coban rais, gunung banyak, mahajaya t-shirt & oleh-oleh, ngliyep and bendungan selorejo. group 5 : vihara “dammadhipa arama”, rafting “kaliwatu”, batu rafting, wana wisata coban talun, pemandian tirta nirwana, pemandian air panas alam songgoriti, wonderland waterpark, sahabat air rafting, petik apel mandiri, batu agro ape, kampung wisata kungkuk, desa wisata sumberejo, desa wisata bumiaji, mega star indonesia, wisata oleh-oleh deduwa, candi jago, sengkaling, pemandian dewi sri and candi kidal. based on the conclusions obtained, it appears that groups 4 and 5 are in the form of tourist objects that are related to low lands and are consumptive. whereas group 1, group 2, and group 3 are attractions related to water and the beauty of nature in both the highlands and waters (sea, waterfall). therefore, we need to recommend: (a) to the local government of malang raya to prioritize the improvement of services in tourism objects related to the waters (beaches, seas, and waterfalls) and to further explore the beauty of nature, (b) for other researchers, it is recommended to add indicators that influence the popularity of tourist objects in malang to represent the characteristics of each tourist attraction better, and (c) the use of the em estimation method is possible to obtain endless iterations. to minimize this occurrence, it is recommended to use other methods to group the popularity of attractions in malang raya. acknowledgement we gratefully acknowledge to faculty of mathematics and sciences of malang state university of the year 2018. references [1] e. kurniawan, “malang city government concentration on boosting tourism to increase pad”. 2017. https://malangtoday.net/malangraya/kota-malang/dongkrak-pariwisata-untuk-tingkatkan-pad/amp/ [2] r. silvi, “analisis cluster dengan data outlier menggunakan centroid linkage dan k-means clustering untuk pengelompokkan indikator hiv/aids di indonesia”, mantik, vol. 4, no. 1, pp. 22-31, may 2018. [3] s. borman, “the expectation maximization algorithm: a short tutorial”, july 2006. [4] t. a. kusuma and suparman. “algoritma expectation-maximization (em) untuk estimasi distribusi mixture”, jurnal konvergensi, vol. 4, no. 2 oktober 2014. [5] r. e. d. sirait, e. darwianto, and d. d. j. suwawi, “implementasi dan analisis algoritma clustering expectation–maximization (em) pada data tugas akhir universitas telkom”, e-proceeding of enginering, vol. 2, no. 2, agustus 2015. [6] i. johari, d. soeyapto, and mardiani, “penerapan data mining untuk data jumlah kendaraan menggunakan algoritma expectation maximization (em) pada dispenda kota palembang”, stmik mdp, 2015. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 123-134 134 [7] clustering, k. “implementation and analysis of clustering expectation maximization (em) algorithms on telkom university final project data”, vol. 2, no. 2, pp. 6711–6717, 2015. [8] s, santoso, “statistik multivariat: konsep dan aplikasi dengan spss”, jakarta. elex media komputindo, 2004. [9] a. c. rencher, “method of multivariate analysis (second edition)”, new york: john wiley and sons, inc. 2002. [10] l. j. bain and m. engelhardt, “introduction to probablity and mathematical statistcs”. california: duxbury press. 1992. [11] kusrini and e. t. luthfi, “algoritma data mining”, yogyakarta: andi, 2009. [12] h. glanz, h and l. carvalho, “an expectation-maximization algorithm for the matrix normal distribution with an application in remote sensing”. journal of multivariate analysis, vol. 167, pp. 31-48, september 2018, doi: 10.1016/j.jmva.2018.03.010. [13] g. j. mclachlan and t. krishnan, “the em algorithm and extensions”, john wiley & sons, hoboken, 2008, doi: 10.1002/9780470191613. contact: ari dwi hartanto, ari@ugm.ac.id department of mathematics, universitas gadjah mada, sleman, d.i. yogyakarta 55281, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.1-8 binary cyclic pearson codes ari dwi hartanto1*, al. sutjijana2 1,2department of mathematics, universitas gadjah mada, yogyakarta, indonesia article history: received sep 2, 2019 revised feb 4, 2021 accepted mar 3, 2021 kata kunci: jarak pearson, kode pearson, kode siklis abstrak. fenomena gain atau offset yang tidak terduga pada sistem komunikasi dan media penyimpan data modern seperti media penyimpanan berjenis optik (cd) dan memori non-volatile (flash) merupakan gangguan yang serius. permasalahan ini dapat ditangani dengan mengaplikasikan jarak pearson pada detektor error pada sistem tersebut karena jarak pearson menawarkan kekebalan terhadap gain dan offset yang tidak menentu. jarak ini hanya dapat digunakan pada suatu himpunan codewords tertentu, yaitu himpunan pearson/kode pearson. salah satu contoh kode pearson dapat ditemukan di kelas kode t-constrained. dalam paper ini, diberikan kode 2-constrained biner dengan sifat siklis. konstruksi kode ini diadopsi dari konstruksi pada kode siklis, akan tetapi kode yang dihasilkan tidak dapat dipandang sebagai kode siklik. keywords: pearson distance, pearson code, cyclic code abstract. the phenomena of unknown gain or offset on communication systems and modern storage such as optical data storage and non-volatile memory (flash) becomes a serious problem. this problem can be handled by pearson distance applied to the detector because it offers immunity to gain and offset mismatch. this distance can only be used for a specific set of codewords, called pearson codes. an interesting example of pearson code can be found in the t-constrained code class. in this paper, we present binary 2constrained codes with the cyclic property. the construction of this code is adopted from cyclic codes, but it cannot be considered as cyclic codes how to cite: a. d. hartanto and al. sutjijana, “binary cyclic pearson codes”, j. mat. mantik, vol. 7, no. 1, pp. 1-8, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 1-8 issn: 2527-3159 (print) 2527-3167 (online) mailto:ari@ugm.ac.id https://doi.org/10.15642/mantik.2021.7.1.1-8 http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 1-8 2 1. introduction based on [1] and [2], gain and/or offset mismatch often occur on modern storages and communication channels. in non-volatile memories, for instance: a flash memory, the data is stored in a floating gate. it can leak away from the floating gate and affects a shift of the offset. in optical disc media, the signal retrieved by the sensor depends on the dimensions of the written features and the quality of the light path. the quality of the light path usually occurs because of fingerprints and scratches on the disc surface. these lead to gain and offset mismatch of the retrieved signal. regarding this problem, weber et.al. [3] presented a system code which involves the person distance. pearson distance is resistant to gain and offset mismatch, so it can be an alternative to the euclidean distance for improving the error performance of noisy channels with unknown gain and offset. unfortunately, pearson distance cannot be used for arbitrary sets of codewords, but it can only be used for a pearson set/pearson code, a set of codewords with special properties. weber, swart, and immink [3] stated that one of the known pearson code is 𝑇(𝑛, 𝑞). this code is a member of the t-constrained code class which consists of sequences in which each of t pre-determined reference symbols appears at least once ([1], [3], [4]). let 𝐶 ⊆ 𝑄𝑛 be a q-ary code of length n, where 𝑄 = {0,1, ⋯ , 𝑞 − 1} is the code alphabet of size 𝑞 ≥ 2. the alphabet symbols in codebook c will not be treated as elements of 𝑍𝑞, but as just integers. the reader should be careful that this situation will be different when we discuss linear codes in this paper. weber, swart, and immink ([3], [5]) presented two coding procedures, i.e. fixed-tofixed (ff) and variable-to-fixed (vf) length coding schemes which are special codes of 𝑇(𝑛, 𝑞). we present another special code of 𝑇(𝑛, 𝑞), that is binary cyclic pearson codes. we focus on how to construct the pearson codes c in which the codewords have cyclic property, that is, if (𝑎0, 𝑎1, ⋯ , 𝑎𝑛−1) ∈ 𝐶 then (𝑎𝑛−1, 𝑎0, 𝑎1, ⋯ , 𝑎𝑛−2) ∈ 𝐶. our idea is taking the similar way of cyclic code construction (see [6] and [7]) for constructing a cyclic pearson code. cyclic code is one of the most important classes of linear codes. the cyclic codes have a rich algebraic structure and can be efficiently implemented using simple shift registers [8]. although the cyclic codes possess algebraic structure, we do not emphasize algebraic structure on the binary cyclic pearson codes. we only concern on the set of codewords that have cyclic property and satisfy the axioms of pearson code. we investigate whether it can be constructed and the simple encoding scheme with a generator matrix can be built in a similar way on the simple encoding scheme of cyclic codes. the rest of this paper is organized as follows. we start our presentation with a brief introduction to pearson distance, pearson codes, and t-constrained codes in section 2. in section 3, we give a brief exposition of the cyclic codes, and then present binary cyclic pearson codes. finally, we draw some conclusions in section 4. 2. preliminaries we first introduce a necessary notation used in this paper. for simplicity, we use the shorthand notation 𝛼𝒖 + 𝛽 = (𝛼𝑢1 + 𝛽, 𝛼𝑢2 + 𝛽, ⋯ , 𝛼𝑢𝑛 + 𝛽) as described in [3]. the bold letter indicates a vector that can mean either a message word or a codeword. suppose a codeword c is sent, and then a vector 𝒓 = 𝑎(𝒄 + 𝒗) + 𝑏 is received, where 𝑎, 𝑏 ∈ 𝑅, 𝑎 > 0, and 𝑣 = (𝑣1, ⋯ , 𝑣𝑛) , 𝑣𝑖 ∈ 𝑅. the real numbers a and b are called gain and offset, respectively, and the entries of vector 𝑣 are noise sample from a zero-mean gaussian distribution. the gain and offset will not affect sent codewords if we use pearson distance detection ([9], [10]). the following is the pearson distance that is a well-known concept in statistics. let 𝒙 be a vector in 𝑅𝑛. define: ari dwi hartanto, al. sutjijana binary cyclic pearson codes 3 𝒙 = 1 𝑛 ∑ 𝑥𝑖 𝑛 𝑖=1 , and 𝜎𝑥 2 = ∑(𝑥𝑖 − 𝒙) 2 𝑛 𝑖=1 . the pearson correlation coefficient of x and y is defined by 𝜌𝑥,𝑦 = ∑ (𝑥𝑖 − 𝒙) 𝑛 𝑖=1 (𝑦𝑖 − 𝒚) 𝜎𝑥 𝜎𝑦 . next, the pearson distance is defined as follow: 𝛿(𝒙, 𝒚) = 1 − 𝜌𝑥,𝑦 . to know the bound of 𝛿(𝒙, 𝒚), we first note that −1 ≤ 𝜌𝑥,𝑦 ≤ 1. this implies that 0 ≤ 𝛿(𝒙, 𝒚) ≤ 2. by nearest neighbour decoding, a received vector will be decoded to the codeword closest to it, with respect to pearson distance. if we have a received vector r, the minimum pearson distance detector outputs the codeword: 𝒄𝟎 =𝑎𝑟𝑔 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑐∈𝑆 𝛿(𝒓, 𝒄), where s is the codebook. we can observe that for all 𝛼, 𝛽 ∈ 𝑅, 𝛼 > 0, we have 𝜌 𝑥,𝛼𝑦+𝛽 = 𝜌𝑥,𝑦 . the equation implies that 𝛿(𝒙, 𝛼𝒚 + 𝛽) = 𝛿(𝒙, 𝒚). it means that the pearson distance is invariant under translation and scale. as a result, the minimum pearson distance detector is immune to gain and offset mismatch. however, this arises a weakness. we cannot use the minimum pearson distance detector for arbitrary codebooks because it cannot distinguish between vector x and 𝒚 = 𝛼𝒙 + 𝛽 where 𝛼, 𝛽 ∈ 𝑅, 𝛼 > 0, (𝛼, 𝛽) ≠ (1,0), or mathematically, it can be written as 𝛿(𝒓, 𝒙) = 𝛿(𝒓, 𝒚) if 𝒚 = 𝛼𝒙 + 𝛽, for all 𝛼, 𝛽 ∈ 𝑅 with 𝛼 > 0 and (𝛼, 𝛽) ≠ (1,0). because of this fact, our codebook c should has property that if 𝒄 ∈ 𝐶 then 𝛼𝒄 + 𝛽 ∉ 𝐶 for all 𝛼, 𝛽 ∈ 𝑅 with 𝛼 > 0 and (𝛼, 𝛽) ≠ (1,0). moreover, another property that has to be satisfied by codebook c is that vectors in the form of 𝒙 = (𝑎, 𝑎, ⋯ , 𝑎) is not allowed in the codebook c since the vectors will lead to an undefined pearson correlation coefficient. definition 1. let c be a nonempty subset of 𝑄𝑛. the set c is called pearson code (or pearson set) if it satisfies the following properties: a. if 𝒄 ∈ 𝐶 then 𝛼𝒄 + 𝛽 ∉ 𝐶 for all 𝛼, 𝛽 ∈ 𝑅 with 𝛼 > 0 and (𝛼, 𝛽) ≠ (1,0). b. 𝒙 = (𝑎, 𝑎, ⋯ , 𝑎) ∉ 𝐶 for all 𝑎 ∈ 𝑅. to have an example of pearson codes, we first give a description of t-constrained codes. let 𝑇 be an integer with 1 ≤ 𝑇 < 𝑞 and reference symbols 𝑎1, ⋯ , 𝑎𝑇 be element of 𝑄. a t-constrained code [1], denoted by 𝑆𝑞,𝑛(𝑎1, ⋯ , 𝑎𝑇 ), consist of all q-ary codewords of length n, (𝑥1, ⋯ , 𝑥𝑛 ), such that #{𝑖: 𝑥𝑖 = 𝑗} > 0 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑗 ∈ {𝑎1, ⋯ , 𝑎𝑇 }. note that not all t-constrained codes are pearson codes. for 𝑇 = 2, we can see that both 2constrained code 𝑆𝑞,𝑛(0,1) and 𝑆𝑞,𝑛(0, 𝑞 − 1) are pearson codes, but for 𝑞 ≥ 5, 𝑆𝑞,𝑛(0,2) is not a pearson code since it does not meet the first property of pearson code. we now focus to discuss the specific case, that is the t-constrained code 𝑆𝑞,𝑛(0,1). this code is pearson codes, and will be denoted by 𝑇(𝑛, 𝑞) as in [3]. weber, swart, and immink [3] proposed easy coding procedures, i.e., fixed-to-fixed (ff) and variable-to-fixed (vf) length coding schemes, denoted by 𝑇𝐹𝐹(𝑛, 𝑞) and 𝑇𝑉𝐹 (𝑛, 𝑞), respectively. the simple jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 1-8 4 ff scheme is to fill the first 𝑛 − 2 positions in the codeword c with information symbols and to fill the last two symbols for reference purpose, i.e., 𝑐𝑛−1 = 0 and 𝑐𝑛 = 1. for vf schemes, it can be described as follows. a. take 𝑛 − 2 information from the q-ary source, and then set these as (𝑐1, 𝑐2, ⋯ , 𝑐𝑛−2). b. if there exists 𝑖, 1 ≤ 𝑖 ≤ 𝑛 − 2, such that 𝑐𝑖 = 0, then choose 𝑐𝑛−1 to be a new information symbol, otherwise set 𝑐𝑛−1 = 0. c. if there exists 𝑖, 1 ≤ 𝑖 ≤ 𝑛 − 1, such that 𝑐𝑖 = 1, then choose 𝑐𝑛 to be a new information symbol, otherwise set 𝑐𝑛 = 1. the resulted codewords from both schemes are in 𝑇(𝑛, 𝑞) since it contains at least one symbol 0 and at least one symbol 1. 3. cyclic pearson codes in this section, we intend to construct cyclic pearson codes by using similar ways as construction of cyclic codes. let f be a finite field of order q, and 𝑉𝑛 (𝐹) = {(𝑥1𝑥2 ⋯ 𝑥𝑛 ) ∣ 𝑥𝑖 , ⋯ , 𝑥𝑛 ∈ 𝐹} be a vector space over f. a nonempty subset c of 𝑉𝑛(𝐹) is called cyclic code if it is a subspace and every (𝑎1𝑎2 ⋯ 𝑎𝑛−1𝑎𝑛) ∈ 𝐶 implies (𝑎𝑛𝑎1𝑎2 ⋯ 𝑎𝑛−1) ∈ 𝐶. cyclic codes can be constructed by ring polynomial approach. clearly, 𝑉𝑛(𝐹) is an abelian group under vector addition, but it is not a ring since we have not had a multiplication between any two vectors yet. in order to have multiplication operation such that 𝑉𝑛 (𝐹) is a ring, the easiest way is to associate vectors in 𝑉𝑛(𝐹) with polynomials in 𝐹[𝑥], that is, if 𝒂 = (𝑎0𝑎1 ⋯ 𝑎𝑛−1) ∈ 𝑉𝑛 (𝐹), then let 𝑎(𝑥) = 𝑎0 + 𝑎1 + ⋯ + 𝑎𝑛−1𝑥 𝑛−1. we select the polynomial 𝑓(𝑥) = 𝑥𝑛 − 1 ∈ 𝐹[𝑥] to have a quotient ring 𝐹[𝑥]/⟨𝑓(𝑥)⟩. for all 𝒂, 𝒃 ∈ 𝑉𝑛(𝐹), let 𝑎(𝑥)𝑏(𝑥) = 𝑣(𝑥) where 𝑣(𝑥) is the polynomial of least degree in the equivalence class [𝑎(𝑥)𝑏(𝑥)] of 𝐹[𝑥]/⟨𝑓(𝑥)⟩. note that 𝑣(𝑥) is the remainder polynomial when 𝑎(𝑥)𝑏(𝑥) is divided by 𝑓(𝑥), and it is a polynomial of degree at most 𝑛 − 1 over f, which is associated with an element of 𝑉𝑛(𝐹). in conclusion, now we have multiplication between any two vectors in 𝑉𝑛(𝐹) which is defined by this way, and with the association between the vectors and polynomials, we can essentially think of 𝑉𝑛 (𝐹) and 𝐹[𝑥]/⟨𝑓(𝑥)⟩ interchangeable. moreover, we can prove that 𝑉𝑛(𝐹) is a ring. to have cyclic property of vectors, the choice of the polynomial 𝑓(𝑥) = 𝑥𝑛 − 1 is most suitable. multiplication a polynomial 𝑣(𝑥) in 𝐹[𝑥]/⟨𝑓(𝑥)⟩ by x corresponds to a cyclic shift of v. this results in a fundamental theorem in cyclic codes presented as follows. a nonempty subset c of 𝑉𝑛 (𝐹) is a cyclic code if only if the set of polynomials i associated with c is an ideal in the ring 𝐹[𝑥]/⟨𝑥𝑛 − 1⟩. this property helps us in constructing cyclic codes. in fact, there is a unique monic polynomial of least degree that generates a nonzero ideal i of 𝐹[𝑥]/⟨𝑥𝑛 − 1⟩, i.e., a monic polynomial divisor of 𝑓(𝑥) = 𝑥𝑛 − 1. thus, there is a 1 − 1 correspondence between cyclic codes in 𝑉𝑛 (𝐹) and monic polynomials 𝑔(𝑥) ∈ 𝐹[𝑥] that divide 𝑓(𝑥). a monic polynomial 𝑔(𝑥) divisor of 𝑓(𝑥) = 𝑥𝑛 − 1 over f having degree 𝑛 − 𝑘 will become the generator for a cyclic code c of dimension k in 𝑉𝑛 (𝐹). the encoder encodes the message polynomial 𝑎(𝑥) (of degree less than or equal to 𝑘 − 1) to the codeword 𝑎(𝑥)𝑔(𝑥) ∈ 𝐹[𝑥]/⟨𝑓(𝑥)⟩. thus, the generator matrix for cyclic code c is given in terms of 𝑔(𝑥) by 𝐺 = [ 𝑔(𝑥) 𝑥𝑔(𝑥) ⋮ 𝑥𝑘−1𝑔(𝑥) ]. note that g is a matrix of 𝑘 × 𝑛 where the 𝑖𝑡ℎ row of g, 2 ≤ 𝑖 ≤ 𝑘, is the 𝑖 − 1 right shifts of the first row of g. ari dwi hartanto, al. sutjijana binary cyclic pearson codes 5 we now turn our discussion to pearson codes. for simplicity, sometimes we will write a vector in 𝑉𝑛(𝐹) or in codes c in the form 𝑎0𝑎1 ⋯ 𝑎𝑛 instead of (𝑎0, 𝑎1, ⋯ , 𝑎𝑛). we note that the cyclic codes c are not pearson codes since 00 ⋯ 0 belongs to c. we observe some nonempty subsets of cyclic code c such that it is pearson codes. in case our discussion is in pearson code, we treat the information symbols of codewords as real numbers instead of elements of a finite field. in order to find a nonempty subset of cyclic codes c such that it is a pearson code, clearly, we should eliminate the zero vector from c since it breaks the second axiom of the definition of pearson codes. we should observe whether there exist other elements having form 𝑎𝑎 ⋯ 𝑎 in c. let us see a (7,4)-cyclic code over 𝑍2 generated by 𝑔(𝑥) = 1 + 𝑥 + 𝑥 3. the generator matrix of code c is 𝐺 = [1 + 𝑥 + 𝑥3 𝑥 + 𝑥2 + 𝑥4 𝑥2 + 𝑥3 + 𝑥5 𝑥3 + 𝑥4 + 𝑥6 ] = [1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 ]. the message 𝒎 = 1011 is encoded to be codeword 𝒄 = 𝒎𝐺 = 1111111 ∈ 𝐶. such codeword must be omitted from c. it means that if our observation is in (𝑛, 𝑘)-cyclic codes, we have to make attention for messages from which codewords having form 𝑎𝑎 ⋯ 𝑎. we note that such messages are the polynomials which are quotient of division ℎ(𝑥) = 𝑟 + 𝑟𝑥 + ⋯ + 𝑟𝑥𝑛−1, where 𝑟 ∈ 𝐹, by the generator polynomial 𝑔(𝑥) in 𝐹[𝑥]/⟨𝑥𝑛 − 1⟩. clearly, if c is a codeword of (𝑛, 𝑘)-cyclic code 𝐶 over finite field f, then 𝛼𝑐 + (1𝐹, 1𝐹 , ⋯ , 1𝐹) must be element of c, for all 𝛼 ∈ 𝑁. this does not meet the first axiom of the definition of pearson codes. now we go on a specific case, that is (𝑛, 𝑘)-cyclic code 𝐶 over finite field 𝑍2. in this case, we have a good result. we note that 𝐶 − {00 ⋯ 0,11 ⋯ 1)} is a 2-constrained code 𝑆2,𝑛(0,1) if we treat the information symbols of codewords in c as real numbers. in conclusion, we obtain that the code 𝑃 = 𝐶 − {00 ⋯ 0,11 ⋯ 1} is a pearson code, and we call this code the binary cyclic pearson codes. for a simple encoding scheme, we intend to use the generator matrix, but we face a problem when the message word is the zero vector. we should give a special treatment for this message word. first, we need to limit n to an even positive integer. we correspond the zero vector with the codeword 1010 ⋯ 10. since 1010 ⋯ 10 is a codeword, 0101 ⋯ 01 must be considered as a codeword. we correspond 0101 ⋯ 01 with a message word 𝑎0𝑎1 ⋯ 𝑎𝑘−1 such that 𝑎0𝑎1 ⋯ 𝑎𝑘−1𝐺 = 111 ⋯ 1. unfortunately, we cannot guarantee that such a message word will exist. in order to guarantee the existence of such a message word, the polynomial ℎ(𝑥) = 1 + 𝑥 + ⋯ + 𝑥𝑛−1 must be divisible by the generator polynomial 𝑔(𝑥). by this condition, we have 𝑎0 + 𝑎1𝑥 + ⋯ + 𝑎𝑘−1𝑥 𝑘−1 is ℎ(𝑥)/𝑔(𝑥). next, we should be careful if 1010 ⋯ 10 belongs to (𝑛, 𝑘)-cyclic code c. it should not be allowed since we have corresponded 1010 ⋯ 10 with message word 000 ⋯ 0. table 1. some generator polynomials of binary cyclic pearson codes n k generator polynomial 2 1 𝑔(𝑥) = 1 + 𝑥 4 1 𝑔(𝑥) = (1 + 𝑥) 3 6 5 𝑔(𝑥) = 1 + 𝑥 3 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2) 1 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2)2 8 1 𝑔(𝑥) = (1 + 𝑥)7 10 9 𝑔(𝑥) = 1 + 𝑥 5 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4) 1 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4)2 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 1-8 6 table 1. some generator polynomials of binary cyclic pearson codes (continued) n k generator polynomial 12 9 𝑔(𝑥) = (1 + 𝑥)3 7 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2) 5 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)2 3 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)3 1 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)4 based on the explanation above, not all (𝑛, 𝑘)-cyclic codes over 𝑍2 can be used for constructing a binary cyclic pearson code. as summary, the following conditions must be held in order that (𝑛, 𝑘)-cyclic codes over 𝑍2 can be taken for constructing a binary cyclic pearson code: a. n is even. b. the polynomial ℎ(𝑥) = 1 + 𝑥 + ⋯ + 𝑥𝑛−1 is divisible by the generator polynomial 𝑔(𝑥). c. the polynomial 1 + 𝑥2 + 𝑥4 + ⋯ + 𝑥𝑛−2 is not divisible by the generator polynomial 𝑔(𝑥). by using magma (computational algebra system) (see [11]), we observe all possibility of (𝑛, 𝑘)-cyclic codes over 𝑍2, for 𝑛 = 2,4, ⋯ ,12, that meets three conditions above. table 1 gives all generator polynomials of binary cyclic pearson codes for 𝑛 = 2,4, ⋯ ,12. example 1. consider 𝑛 = 6, 𝑉6(𝑍2), and 𝑓(𝑥) = 𝑥 6 − 1. we take polynomial 𝑔(𝑥) = 1 + 𝑥 as a generator of the binary cyclic pearson code, so we have a generator matrix: 𝐺 = [1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 ]. we divide 1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 by 𝑔(𝑥), and we have 1 + 𝑥2 + 𝑥4 as a result. then, the message words 00000 and 10101 are encoded to the codewords 101010 and 010101, respectively. for all 𝒎 ∈ 𝑉5(𝑍2) − {00000,10101}, m is encoded to codeword 𝒎𝐺. therefore, the binary cyclic pearson code of length 𝑛 = 6 and with generator 𝑔(𝑥) = 1 + 𝑥 is 𝑃 = {𝒎𝐺 ∣ 𝒎 ∈ 𝑉5(𝑍2) − {00000,10101}} ∪ 𝑃0, where 𝑃0 = {101010,010101}. 4. conclusions a binary cyclic code of even length usually has a small hamming distance. some of the hamming distances of cyclic codes can be seen on table 2. a code with small distance can be said as a bad code because it can only detect and correct small errors. however, it does not mean that the code is always useless. as we explained above, an even length of code become one of the sufficient conditions for constructing a binary cyclic pearson code. therefore, some of cyclic codes of even length are needed for binary cyclic pearson codes. in this paper we have investigated binary cyclic pearson codes for 𝑛 = 2,4,6,8,10,12, and the result is that the binary cyclic pearson codes exist for each of these n. we have not yet investigated codes of even length generally. our conjecture is the binary cyclic pearson codes exist for all even positive integer n. ari dwi hartanto, al. sutjijana binary cyclic pearson codes 7 table 2. some generators of (𝑛, 𝑘, 𝑑)-cyclic codes and its compatibility for binary cyclic pearson codes (bcpc) n k generator polynomial d compatibility of generator for bcpc 2 1 𝑔(𝑥) = 1 + 𝑥 2 yes 4 3 𝑔(𝑥) = 1 + 𝑥 2 no 2 𝑔(𝑥) = (1 + 𝑥)2 2 no 1 𝑔(𝑥) = (1 + 𝑥)3 4 yes 6 5 𝑔(𝑥) = 1 + 𝑥 2 yes 4 𝑔(𝑥) = (1 + 𝑥)2 2 no 4 𝑔(𝑥) = 1 + 𝑥 + 𝑥 2 2 no 2 𝑔(𝑥) = (1 + 𝑥 + 𝑥 2)2 3 no 3 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2) 2 yes 2 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2) 4 no 1 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2)2 6 yes 8 7 𝑔(𝑥) = (1 + 𝑥) 2 no 6 𝑔(𝑥) = (1 + 𝑥)2 2 no 5 𝑔(𝑥) = (1 + 𝑥)3 2 no 4 𝑔(𝑥) = (1 + 𝑥)4 2 no 3 𝑔(𝑥) = (1 + 𝑥)5 4 no 2 𝑔(𝑥) = (1 + 𝑥)6 4 no 1 𝑔(𝑥) = (1 + 𝑥)7 8 yes 10 9 𝑔(𝑥) = 1 + 𝑥 2 yes 8 𝑔(𝑥) = (1 + 𝑥)2 2 no 6 𝑔(𝑥) = 1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4 2 no 2 𝑔(𝑥) = (1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4)2 5 no 5 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4) 2 yes 4 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4) 4 no 1 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4)2 10 yes 12 11 𝑔(𝑥) = 1 + 𝑥 2 no 10 𝑔(𝑥) = (1 + 𝑥)2 2 no 9 𝑔(𝑥) = (1 + 𝑥)3 2 yes 8 𝑔(𝑥) = (1 + 𝑥)4 2 no 10 𝑔(𝑥) = 1 + 𝑥 + 𝑥 2 2 no 8 𝑔(𝑥) = (1 + 𝑥 + 𝑥 2)2 2 no 6 𝑔(𝑥) = (1 + 𝑥 + 𝑥 2)3 3 no 4 𝑔(𝑥) = (1 + 𝑥 + 𝑥 2)4 3 no 9 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2) 2 no 7 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2)2 2 no 5 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2)3 4 no 3 𝑔(𝑥) = (1 + 𝑥)(1 + 𝑥 + 𝑥 2)4 6 no 8 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2) 2 no 6 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2)2 2 no jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 1-8 8 table 2. some generators of (𝑛, 𝑘, 𝑑)-cyclic codes and its compatibility for binary cyclic pearson codes (bcpc) (continued) n k generator polynomial d compatibility of generator for bcpc 12 4 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2)3 4 no 2 𝑔(𝑥) = (1 + 𝑥)2(1 + 𝑥 + 𝑥 2)4 6 no 7 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2) 4 yes 5 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)2 4 yes 3 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)3 4 yes 1 𝑔(𝑥) = (1 + 𝑥)3(1 + 𝑥 + 𝑥 2)4 12 yes acknowledgement we thanks to jos weber from delft university of technology (tu delft) for the discussion and his papers shared with us. references [1] k. a. s. immink and j. h. weber, “minimum pearson distance detection for multilevel channels with gain and/or offset mismatch,” ieee trans. inf. theory, vol. 60, no. 10, pp. 5966–5974, 2014, doi: 10.1109/tit.2014.2342744. [2] f. sala, k. a. s. immink, and l. dolecek, “error control schemes for modern flash memories,” ieee consum. electron., vol. 4, no. 1, pp. 66–73, 2015. [3] j. h. weber, k. a. s. immink, and s. r. blackburn, “pearson codes,” ieee trans. inf. theory, vol. 62, no. 1, pp. 131–135, 2016, doi: 10.1109/tit.2015.2490219. [4] k. a. s. immink and j. h. weber, “hybrid minimum pearson and euclidean distance detection,” ieee trans. commun., vol. 63, no. 9, pp. 3290–3298, 2015, doi: 10.1109/tcomm.2015.2458319. [5] j. h. weber, t. g. swart, and k. a. s. immink, “simple systematic pearson coding,” ieee int. symp. inf. theory proc., vol. 2016-augus, pp. 385–389, 2016, doi: 10.1109/isit.2016.7541326. [6] s. a. vanstone and p. c. oorschot, an introduction to error correcting codes with applications. springer us, 1989. [7] s. ling and c. xing, coding theory. cambridge university press, 2004. [8] a. betten, m. braun, h. fripertinger, a. kerber, a. kohnert, and a. wassermann, error-correcting linear codes. springer berlin heidelberg, 2006. [9] k. a. s. immink, “coding schemes for multi-level channels with unknown gain and/or offset,” ieee int. symp. inf. theory proc., pp. 709–713, 2013, doi: 10.1109/isit.2013.6620318. [10] k. a. s. immink, “coding schemes for multi-level flash memories that are intrinsically resistant against unknown gain and/or offset using reference symbols,” electron. lett., vol. 50, no. 1, pp. 20–22, 2014, doi: 10.1049/el.2013.3558. [11] magma handbook, ”http://magma.maths.usyd.edu.au/magma/handbook/ (accessed nov. 05, 2018). perbandingan antara metode k-means clustering dengan gath-geva clustering jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 26 perbandingan antara metode k-means clustering dengan gath-geva clustering (studi kasus pada volume ekspor non migas pakaian jadi) oleh : siti lailiyah 1) , moh. hafiyusholeh 2) 1) pendidikan matematika uin sunan ampel surabaya, siti03_math_its@yahoo.com 2) matematika uin sunan ampel surabaya, hafiyusholeh@uinsby.ac.id abstrak perdagangan luar negeri indonesia sedang ditata kembali format dan kinerjanya, agar pemerintah tidak membuat kesalahan dalam mengambil keputusan untuk meningkatkan ekspor non migas, maka pemerintah harus mampu memprediksi volume ekspor non migas. prediksi pada dasarnya merupakan suatu perkiraan tentang terjadinya suatu kejadian di waktu yang akan datang. salah satu cara yang dapat digunakan untuk memprediksi nilai ekspor tersebut adalah dengan k-means clustering dan gath-geva clustering. kemudian dibentuk fuzzy inference system (fis) untuk memperoleh hasil prediksi sehingga didapatkan error dan validasi hasil prediksi.berdasarkan hasil analisa rmse, cek maksimum dan cek minimum maka dapat disimpulkan bahwa metode gath-geva (gg) clustering lebih teliti dibandingkan dengan metode k-means clustering. kata kunci: sistem fuzzy, k-means clustering, gath-geva clustering. 1. pendahuluan perdagangan luar negeri indonesia sekarang ini sedang ditata kembali format dan kinerjanya ke kondisi awal dimasamasa sebelum krisis ekonomi melanda bangsa indonesia. meskipun belum pulih secara keseluruhan, kegiatan perdagangan luar negeri indonesia mulai menunjukkan kekuatannya kembali. sudah beberapa tahun terakhir ini hasil industri pakaian jadi indonesia banyak diminati oleh negara-negara asing khususnya amerika dan eropa karena dianggap cukup berkualitas dengan harga yang bersaing. bahkan dibeberapa negara, hasil industri pakaian jadi indonesia dikenakan kuota agar industri pakaian jadi negara tersebut tidak kalah bersaing dengan produk indonesia. agar pemerintah tidak membuat kesalahan dalam mengambil keputusan untuk meningkatkan ekspor non migas khususnya untuk sektor industri, maka pemerintah harus mampu melihat ke depan atau memprediksi volume ekspor non migas. cara yang dapat digunakan untuk memprediksi nilai ekspor diantaranya dengan metode k-means clustering, dan metode gath-geva clustering. beberapa penelitian yang terkait adalah yunianti [10], dalam “prediksi volume ekspor non migas dengan metode subtractive clustering” menyimpulkan bahwa pengolahan data dengan metode subtractive clustering menghasilkan jangkauan data dapat berpengaruh terhadap pembentukan cluster sehingga mempengaruhi hasil prediksi meskipun diberikan radius yang sama untuk data-data yang ada. penelitian yang lain, gunawan [7], dalam “perbandingan antara metode fuzzy subtractive clustering dengan metode gustafson kessel clustering (studi kasus data time series volume pinjaman kredit jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 27 gadai)” menyimpulkan bahwa hasil rmse, error maksimum dan error minimum metode gustafson kessel clustering lebih teliti dibandingkan dengan metode subtractive clustering. sebagai kelanjutan dari penelitian sebelumnya, maka pengkajian dari penelitian penelitian ini dilakukan dengan menganalisa secara khusus terhadap data volume ekspor non migas pada pakaian jadi menggunakan metode k-means clustering dan metode gath-geva clustering. perbandingan hasil analisis terhadap kedua algoritma metode tersebut akan menentukan metode yang lebih teliti untuk memprediksi volume ekspor non migas dengan nilai-nilai parameter dan besaran yang terkait permasalahan utama yang diangkat dalam penelitian ini adalah: bagaimana perbandingan antara metode k-means clustering dan gath-geva clustering untuk mendapatkan cluster optimal. dalam pembahasan ini, kajian permasalahan akan dibatasi dengan menentukan nilai-nilai parameter dan besaran yang terkait, antara lain: 1. jari-jari cluster. 2. indeks validasi masing-masing algoritma yaitu koefisien partisi (pc), entropi klasifikasi (ce), partisi indeks (sc), separasi indeks (s) dan jumlah kuadrat kesalahan (sse). 3. rmse (root means square error) dalam peramalan. tujuan dari penelitian ini adalah (1) dapat memprediksi volume ekspor non migas untuk pakaian jadi dengan metode kmeans clustering dan gath-geva clustering; (2) dapat mengetahui unjuk kerja dua algoritma pengklasteran diatas secara empiris. 2. teori dasar 1) sistem fuzzy sistem fuzzy terdiri dari himpunan fuzzy, fungsi keanggotaan, fuzzy inferensi sistem dan penalaran fuzzy. himpunan fuzzy a dalam semesta x, ditulis dengan ã dan didefinisikan oleh pasangan:   xxxxã  )(, ã  dengan ]1 ,0[: ã x adalah fungsi atau derajat keanggotaan dari himpunan fuzzy ã. fungsi keanggotaan (membership function) adalah suatu kurva yang menunjukkan pemetaan titik-titik input data ke dalam nilai keanggotaannya yang memiliki interval antara 0 dan 1. pada dasarnya nilai keanggotaan fuzzy dapat digambarkan melalui beberapa representasi yaitu: a. representasi linear ada 2 keadaan himpunan fuzzy yang linear. pertama, fungsi keanggotaan :          b x ; 1 b x a ; a)a)/(b(x a x ; 0 μ[x] kedua, fungsi keanggotaan :       b x ; 0 b x a ; a) x)/(b(b μ[x] b. representasi kurva segitiga fungsi ini diidentifikasi dengan 3 parameter. fungsi keanggotaan:          c x b ; b) x)/(c(b b x a ; a)a)/(b(x c x a x ; 0 μ[x] c. representasi kurva trapesium ada 4 parameter yang dapat digunakan yaitu [ a b c d ]. fungsi keanggotaan :             d x ; c) x)/(d(d c x b ; 1 b x a ; a)a)/(b(x d atau x a x ; 0 μ[x] jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 28 2) fuzzy inference system (fis) fuzzy inference system (fis) adalah sistem yang didasarkan pada konsep teori himpunan fuzzy, aturan fuzzy, dan penalaran fuzzy. secara umum, didalam logika fuzzy ada 5 langkah dalam melakukan penalaran, yaitu : 1. memasukkan input fuzzy. 2. mengaplikasikan operator fuzzy. 3. mengaplikasikan metode implikasi 4. mengkomposisi semua output 5. defuzzifikasi implikasi adalah proses pembentukan himpunan fuzzy pada konsekuen yang didasarkan pada anteseden. apabila sistem terdiri dari beberapa aturan, maka inferensi diperoleh dari kumpulan dan korelasi antar aturan. input dari proses defuzzifikasi adalah suatu himpunan fuzzy yang diperoleh dari komposisi aturan-aturan fuzzy, sedangkan output yang dihasilkan merupakan suatu bilangan pada domain himpunan fuzzy tersebut. 3) fuzzy clustering fuzzy clustering adalah salah satu teknik untuk menentukan cluster optimal dalam suatu ruang vektor yang didasarkan pada bentuk norma euclid untuk jarak antar vektor. suatu ukuran fuzzy menunjukkan derajat ke-fuzzy-an dari himpunan fuzzy. secara umum ukuran ke-fuzzy-an dapat ditulis sebagai suatu fungsi: rxpf )(: dengan p(x) adalah himpunan semua subset dari x. f(a) adalah suatu fungsi yang memetakan subset a ke karakteristik derajat ke-fuzzy-annya. dalam mengukur nilai ke-fuzzy-an, fungsi f harus mengikuti hal-hal sebagai berikut: 1. f(a) = 0 jika dan hanya jika a adalah himpunan crisp. 2. jika a b, maka f(a) f(b). disini, a b berarti b lebih fuzzy dibanding a. relasi ketajaman a b didefinisi-kan dengan:    ,xx ba   jika   5.0x a     ,xx ba   jika   5.0xb 3. f(a) akan mencapai maksimum jika dan hanya jika a benar-benar fuzzy secara maksimum. nilai fuzzy mak-simal biasanya terjadi pada saat   5.0x a  untuk setiap x. pada aplikasi fuzzy clustering, biasanya perlu dilakukan pre-processing terlebih dahulu. dengan demikian kita perlu melakukan normalisasi untuk suatu nilai u, menjadi u normal (ū) dengan rumus: minmax min uu uu u    dengan min u adalah nilai terkecil yang terukur dan max u adalah nilai terbesar yang terukur. setelah melakukan preprocessing, variabel-variabel yang relevan dapat segera dipilih. untuk sekumpulan data   n uuuu ..., , , 21  dapat dicari:  mean    n i i u n m 1 1  variansi   2 11 1      n i i mu n v  deviansi standar v  range minmax uus   koefisien korelasi                n i n i ii n i ii mymx mymx r 1 1 2 2 2 1 1 21 dengan m1 adalah mean dari x dan m2 adalah mean dari y. 4) analisis cluster tujuan dari analisis cluster adalah pengklasifikasian obyek-obyek berdasarjurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 29 kan similaritas diantaranya dan menghimpun data menjadi beberapa kelompok. data yang diberikan adalah data kuantitatif yang diperoleh dari hasil pengukuran berupa data runtun waktu (time series). setiap pengamatan terdiri dari n variabel, dikelompokkan dalam vektor baris pada dimensi-n   , ..., ,, 21 t knkkk xxxx  n k x r himpunan pada n pengamatan di-notasikan oleh  nkx k , 2, ,1| x dan disajikan dalam matriks n x n:              nnnn n n xxx xxx xxx     21 22221 11211 x (2.1) baris dari x disebut pola atau obyek, sedangkan kolomnya disebut features atau attributes dan x sendiri disebut matrik pola atau matrik data. data dapat dinyatakan sebagai cluster pada ruang geometri yang berbeda, ukuran dan kepadatannya, yang ditunjukkan pada gambar 2.1. cluster adalah kumpulan dari benda-benda yang memiliki banyak kesamaan satu sama lain daripada dengan anggota cluster yang lain. cluster dapat berbentuk bola (a), garis panjang atau lurus (b), berlubang (c) dan (d). gambar 2.1. cluster dengan bentuk dan dimensi yang berbeda dalam 2  . pada hard clustering, setiap obyek (data) hanya bisa menjadi anggota tepat satu cluster. sedangkan pada fuzzy clustering obyek-obyek dapat menjadi anggota dari lebih satu cluster dengan derajat keangggotaan yang berbeda. dengan demikian x mempunyai c himpunan bagian fuzzy. struktur dari partisi matrik :][ ik u               cnnn c c u .2.1. .22.21.2 .12.11.1        1. hard partition hard partisi dapat diartikan sebagai keluarga dari himpunan bagian  0,1|  ii acixa , yang meme-nuhi: xa i c i   1 (2.2) 0 ji aa , ,1 cji  (2.3) dimana, ai = himpunan bagian pada cluster ke-i. aj = himpunan bagian pada cluster ke-j. x = semesta pembicaraan. c = banyaknya himpunan bagian cluster. bentuk dari fungsi keanggotaannya: ,1 1   ia c i  (2.4) ,1 ji aa  cji 1 (2.5) ,10  ia  ci 1 (2.6) dimana, ia  = fungsi karakteristik dari himpunan bagian ai yang bernilai 0 dan 1. ja  = fungsi karakteristik dari himpunan bagian aj yang bernilai 0 dan 1. c = banyaknya himpunan bagian cluster. matrik n x c, ][ ik u  menggambarkan hard partisi jika hanya jika elemenelemennya memenuhi:  1 ,0 ij  , ni 1 , ck 1 (2.7)    c k ik 1 ,1 ni 1 , (2.8) ,0 1 n n i ik     ck 1 (2.9) dimana, ij  = fungsi karakteristik pada himpunan bagian ai dan aj. ik  = fungsi karakteristik pada xk. d ) c ) a b ) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 30 n = banyaknya cluster. c = banyaknya himpunan bagian cluster. definisi 2.1 (hard partition space) misal ],...,,[ 21 n xxxx  adalah himpunan berhingga dan misal  nc , ,3 ,2  . hard partition space pada x adalah himpunan kium ik nxc hc ,,1,0|{      n i ik c k ik kni 11 },0;,1  (2.10) 2. fuzzy partition pada fuzzy partisi ik  mencapai nilai riil [0, 1]. matriks n x c, ][ ik u  digambarkan sebagai fuzzy partisi. kondisi ini diberikan oleh: ]1,0[ ij  , ni 1 , ck 1 (2.11)    c k ik 1 ,1 ni 1 , (2.12) ,0 1 n n i ik     ck 1 (2.13) dimana, ij  = fungsi karakteristik pada himpunan bagian ai dan aj. ik  = fungsi karakteristik pada xk. n = banyaknya cluster. c = banyaknya himpunan bagian cluster. definisi 2.2 (fuzzy partition space) misal ],...,,[ 21 n xxxx  adalah himpunan berhingga dan misal  nc , ,3 ,2  . fuzzy partition space pada x adalah himpunan ;,],1,0[|{ kium ik nxc fc      n i ik c k ik kni 11 },0;,1  (2.14) 5) peramalan sering terdapat senjang waktu (time lag) antara kesadaran akan peristiwa atau kebutuhan mendatang dengan peristiwa itu sendiri. adanya waktu tenggang (time lead) ini merupakan alasan utama bagi perencanaan dan peramalan. perihal mendasar yang perlu diperhatikan dalam peramalan adalah bagaimana mengukur kesesuaian suatu metode peramalan tertentu untuk suatu kumpulan data yang diberikan. berbagai ketepatan ukuran peramalan (pemodelan) didefinisikan dengan mempertimbangkan ukuran-ukuran yang ada. ukuran-ukuran yang dimaksudkan diantaranya yaitu : 1. ukuran statistik. jika xt merupakan data aktual untuk periode t dan ft merupakan ramalan (atau nilai kecocokan/fitted value) untuk periode yang sama, maka kesalahannya didefinisikan sebagai berikut : ttt fxe  (2.15) jika terdapat nilai pengamatan ramalan untuk n periode waktu, maka akan terdapat n buah kesalahan dan ukuran statistik standar berikut dapat didefinisikan: nilai tengah kesalahan (mean error)    n t t n e me 1 (2.16) nilai tengah absolute (mean absolute error)    n t t n e mae 1 (2.17) jumlah kuadrat kesalahan (sum of squared error)    n t t esum 1 2 (2.18) nilai tengah kesalahan kuadrat (mean squared error)    n t t n e mse 1 2 (2.19) deviasi standar kesalahan (standard deviation of error)   )1( 2 n e sde t (2.20) 2. ukuran-ukuran relatif. tiga ukuran yang sering digunakan adalah : kesalahan persentase (percentage error)           t tt t x fx pe (2.21) nilai tengah kesalahan persentase (mean persentage error) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 31    n t t n pe mpe 1 (2.22) nilai tengah kesalahan persentase absolut (mean absolute persentage error) npemape n t t / 1    (2.23) 3. metode penelitian tahap-tahap yang digunakan dalam penelitian ini agar dapat mencapai tujuan penelitian adalah: 1. studi literatur pada tahap ini meliputi pencarian informasi dan pemahaman teoritis dari penerapan metode k-means clustering dan gath-geva clustering untuk memprediksi volume ekspor non migas pada pakaian jadi. 2. pengambilan data data yang diolah adalah data sekunder yang diambil dari badan pusat statistik (bps) mulai dari bulan januari 1987 sampai dengan bulan desember 2005, yaitu data ekspor non migas pada pakaian jadi dalam satuan berat (ton). 3. penyusunan algoritma k-means clustering metode k-means clusrtering dapat dirumuskan sebagai berikut:     c i ak ik i vx 1 2 |||| (3.1) dimana, k x = titik data ke-k pada cluster, ai = himpunan titik data dalam cluster ke-i, vi = mean titik data pada data cluster ke-i, dalam algoritma k-means, vi disebut sebagai cluster prototype, yaitu pusat cluster , 1 , ik i n k k i ax n x v i    (3.2) dimana, k x = titik data ke-k pada cluster, ai = himpunan titik data dalam cluster ke-i, ni = banyaknya titik-titik data pada cluster ke-i, hasil pengklasteran yang diperoleh selanjutnya dipergunakan pada model peramalan. 4. penyusunan algoritma gath-geva clustering metode gath-geva dapat dirumuskan sebagai berikut: i i ikik f vxd   )det( ),(  x         )()( 2 1 exp )(1)( l iki tl ik vxfvx  (3.5) dimana, k x = titik data ke-k pada cluster, vi = mean titik data pada data cluster ke-i, i f = matriks kovariansi fuzzy dari cluster ke-i, i  = probabilitas prior dari cluster terpilih, dengan matriks kovariansi fuzzy       n k ik n k t ikikik i vxvx f 1 1 )( )))(((      ci 1 (3.6) dan    n k iki n 1 1  (3.7) derajat keanggotaan ik  diinterpretasikan sebagai probabilitas posterior dari cluster ke-i yang terpilih. hasil pengklasteran yang diperoleh selanjutnya juga akan dipergunakan pada model peramalan. 5. validasi perbandingan terhadap unjuk kerja kedua algoritma pengklasteran di atas dilakukan dengan mengukur indeks validasi masing-masing algoritma: (a). koefisien partisi (pc): mengukur nilai overlapping antara klaster dengan     c i n j ij n cpc 1 1 2 )( 1 )(  (3.12) jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 32 dimana ij  adalah anggota dari titik data j dalam cluster ke-i. optimal jumlah cluster adalah nilai maksimum dari pc. (b). entropi klasifikasi (ce): mengukur ke-fuzzy-an partisi cluster dengan )log( 1 )( 1 1 ij c i n j ij n cce     (3.13) (c). partisi indeks (sc): merupakan rasio antara kekompakan dan separasi cluster-cluster, dengan          c i c k iki n j ij m ij vvn vx csc 1 1 2 1 2 |||| ||||)( )(  (3.14) nilai paling rendah dari sc menunjukkan partisi terbaik. (d).separasi indeks (s): separasi indeks menggunakan separasi jarak minimum untuk validasi partisinya, dengan 2 , 1 1 22 ||||min ||||)( )( ikki c i n j ijij vvn vx cs        (3.15) (e). jumlah kuadrat kesalahan (sse): jumlah simpangan ku-adrat antara realita dengan ha-sil prediksi (peramalan). 6. pengolahan data. langkah-langkahnya sebagai berikut: (a). penerapan metode pengklasteran pada data observasi. (b).penerapan masing-masing algoritma pada data yang diberikan untuk menentukan banyaknya cluster. (c). penerapan metode pengklasteran dalam peramalan pada data volume ekspor non migas pakaian jadi. (d). menghitung masing-masing mse , dengan menggunakan rumus:   n xx mse n i i    1 2 (3.16) (e). menghitung rmse, dengan menarik akar persamaan (3.16) sehingga didapat rumus untuk menghitung rmse sebagai berikut:   n xx rmse n i i    1 2 (3.17) 4. pembahasan dan hasil 4.1 penyajian data untuk menerapkan konsep tersebut ke dalam data real, diambil data volume ekspor non migas pada pakaian jadi yang dimulai bulan januari tahun 1987 sampai dengan bulan desember tahun 2005. perubahan waktu (interval waktu) data tersebut dalam satu bulan sehingga jumlah keseluruhannya adalah sejumlah 228 bulan. perubahan secara grafis volume ekspor non migas pakaian jadi dapat dilihat pada gambar sebagai berikut: gambar 4.1 volume ekspor nonmigas pakaian jadi 4.2 penentuan cluster ada beberapa cara yang dilakukan untuk menentukan cluster, diantaranya yaitu: 1. hard clustering menggunakan algoritma kmeans clustering. metode k-means clustering didasarkan pada pengalokasian setiap titik data kedalam satu dari c cluster yang meminimumkan jumlah kuadrat da-lam cluster. berikut ini hasil dari ploting algo-ritma kmeans: jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 33 gambar 4.2 kmeans volume ekspor nonmigas pakaian jadi dari gambar 4.2 diatas diperoleh jumlah cluster = 4, dengan pusat cluster sebagai berikut:                3792.08722.0 6845.06693.0 3536.04438.0 1200.01382.0 c dari hasil coba dengan merubah parameter c (jumlah cluster) maka didapatkan hasil sebagai berikut: tabel 4.1: hasil validasi kmeans c pc (partition coeffisient) ce (classification entropy) sc (partition index) s (separation index) 3 1 nan 0.7599 0.0056 4 1 nan 0.5672 0.0040 5 1 nan 0.5418 0.0040 untuk menentukan cluster optimal diperoleh saat pc bernilai maksimum dan sc bernilai minimum, karena pc yang bernilai tetap dan nilai sc yang se-makin berkurang maka tidak dapat ditentukan cluster optimal dari nilai pc dan sc. oleh karena itu dibutuhkan nilai validasi yang lainnya yaitu s yang ber-nilai tetap pada saat jumlah cluster = 4 se-hingga dapat disimpulkan bahwa cluster optimal terjadi pada saat jumlah cluster =4. 2. algoritma gath-geva clustering metode gath-geva didasarkan pada fungsi jarak dengan fuzzy maximum likelihood estimates. berikut ini hasil dari ploting algoritma ggclust gambar 4.3 ggclust volume ekspor nonmigas pakaian jadi dari gambar 4.3 diatas diperoleh jumlah cluster = 4, dengan pusat cluster sebagai berikut:                7358.06950.0 3761.06323.0 2425.02295.0 0869.01101.0 c dari hasil coba dengan merubah parameter c (jumlah cluster) maka didapatkan hasil sebagai berikut: tabel 4.2: hasil validasi ggclust c pc (part. coeffisient) ce (classification entropy) sc (partition index) s (separation index) 3 0.9583 0.0755 0.0038 2.5524 e-005 4 0.9607 nan nan nan 5 0.9459 nan nan nan untuk menentukan cluster optimal diperoleh saat pc bernilai maksimum dan sc bernilai minimum. dari tabel 4.2 diatas cluster optimalnya diperoleh pada saat c = 4 karena nilai pc pada saat jumlah cluster = 4 bernilai maksimum dan nilai sc bernilai minimum walaupun tidak dapat ditentukan nilainya. 4.3 perhitungan prediksi peramalan dengan metode k-means clustering 1. data-data yang digunakan data yang akan digunakan untuk memprediksi besar volume ekspor non jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 34 migas pada pakaian jadi berdasarkan periode waktu dua belas bulan dengan metode moving average atau metode ratarata bergerak. input sistemnya adalah permintaan ke(t-11), (t-10), (t-9), (t-8), (t7), (t-6), (t-5), (t-4), (t-3), (t-2), (t-1), (t), sedangkan output sistemnya adalah permintaan ke(t+1), dengan demikian terdapat rentang data dari 12 hingga 227 (216 data). dari data tersebut akan digunakan sebanyak 172 data untuk dicluster, sedangkan sisanya sebanyak 44 data akan digunakan sebagai data cek. 2. hasil clustering jika dilakukan dengan menggunakan influence range 0,50 maka akan diperoleh 4 cluster dengan matriks pusat cluster sebagai berikut:                3792.08722.0 6845.06693.0 3536.04438.0 1200.01382.0 c 3. hasil inferensi hasil inferensi dengan jari-jari = 0.50, accept ratio = 0.5 dan reject ratio = 0.15, seperti terlihat pada berikut ini: gambar 4.4: plot hasil inferensi peramalan terhadap realita tampilan error secara grafis terlihat pada gambar 4.5. error terbesar terjadi pada data inferensi yang ke-157 yaitu sebesar 14.6175, sedangkan error terkecil terjadi pada data inferensi yang ke-162 yaitu sebesar -14.3349. berdasarkan tabel 1 pada lampiran 3 didapatkan hasil validasi jumlah kuadrat kesalahan (sse) untuk data inferensi yaitu 3080.886294. gambar 4.5: grafik error hasil inferensi peramalan terhadap realita 4. hasil pengetesan pengetesan sistem fuzzy dengan data pengecekan berupa peramalan permintaan untuk data ke : 173 s/d 216. hasil pengetesan dengan jari-jari = 0.50, accept ratio = 0.5 dan reject ratio = 0.15, seperti terlihat pada berikut ini: gambar 4.6: plot hasil pengetesan peramalan terhadap realita tampilan error secara grafis terlihat pada gambar 4.7 error terbesar terjadi pada data pengetesan yang ke-43 yaitu sebesar 15.7584, sedangkan error terkecil terjadi pada data pengetesan yang ke-44 yaitu sebesar -9.9222. berdasarkan tabel 2 pada lampiran 3 didapatkan hasil validasi jumlah kuadrat kesalahan (sse) untuk data pemgetesan yaitu 563.9616241. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 35 gambar 4.7: grafik error hasil cek peramalan terhadap realita 4.4 perhitungan prediksi peramalan dengan metode gath-geva clustering jika digunakan metode gath-geva clustering dilakukan dengan menggunakan infuence range 0.50 maka akan diperoleh 4 cluster dengan matriks pusat cluster sebagai berikut:                7358.06950.0 3761.06323.0 2425.02295.0 0869.01101.0 c 1. hasil inferensi hasil inferensi dengan jari-jari = 0.50, accept ratio = 0.5 dan reject ratio = 0.15, (dapat dilihat pada tabel 3 lampiran 3) seperti terlihat pada berikut ini: gambar 4.8: plot hasil inferensi peramalan terhadap realita tampilan error secara grafis terlihat pada gambar 4.9. error terbesar terjadi pada data inferensi yang ke-134 yaitu sebesar 23.3592, sedangkan error terkecil terjadi pada data inferensi yang ke-151 yaitu sebesar -34.8103. berdasarkan tabel 3 pada lampiran 3 didapatkan hasil validasi jumlah kuadrat kesalahan (sse) untuk data inferensi yaitu 10869.45645. gambar 4.9: grafik error hasil inferensi peramalan terhadap realita 2. hasil pengetesan pengetesan sistem fuzzy dengan data pengecekan yang terdapat dalam tabel 2 (lampiran 1) berupa peramalan permintaan untuk data ke : 173 s/d 216. hasil pengetesan dengan jari-jari = 0.50, accept ratio = 0.5 dan reject ratio = 0.15, seperti terlihat pada berikut ini: gambar 4.10: plot hasil pengetesan peramalan terhadap realita tampilan error secara grafis terlihat pada gambar 4.11. error terbesar terjadi pada data pengetesan yang ke-134 yaitu sebesar 0.852e-013, sedangkan error terkecil terjadi pada data pengetesan yang ke-39 yaitu sebesar -0.2842 e-013. berdasarkan tabel 4 pada lampiran 3 didapatkan hasil validasi jumlah kuadrat kesalahan (sse) untuk data pengetesan yaitu 3.616627719e-013. jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 36 gambar 4.11: grafik error hasil cek peramalan terhadap realita berdasarkan hasil diatas k-means clustering pada data inferensi menghasilkan nilai rmse = 4.2624, error maksimum = 14.6175, dan error minimum = -14.3349, sedangkan pada data pengetesan kmeans clustering menghasilkan error maksimum = 15.7584, dan error minimum = -9.9222. kmeans clustering termasuk hard clustering, oleh karena itu perlu dilakukan normalisasi untuk membandingkan hasil tersebut dengan hasil gath-geva clustering yang termasuk fuzzy clustering sehingga tidak perlu dilakukan normalisasi. rumus penormalisasian berdasar-kan dasar teori diatas yaitu: minmax min uu uu u    dengan u adalah data hasil normalisasi, u adalah data sebelum dinormalisasikan, min u adalah nilai terkecil yang terukur dan max u adalah nilai terbesar yang terukur. diperoleh hasil normalisasi dari hasil kmeans clustering adalah:  hasil inferensi 40 5175.1 1.131.53 1.136175.14 max    error 0379375.0 40 4349.27 1.131.53 1.133349.14 min     error 6858725.0  hasil pengetesan 40 6584.2 1.131.53 1.137584.15 max    error 06646.0 40 0222.23 1.131.53 1.139222.9 min    error 575555.0 5. simpulan dan saran 5.1 simpulan berdasarkan hasil analisa dan pembahasan sebelumnya, maka dapat disimpulkan beberapa hal yaitu: 1. berdasarkan nilai validasi k-means clustering dihasilkan cluster optimal terjadi pada saat cluster = 4 dengan pusat cluster adalah:                3792.08722.0 6845.06693.0 3536.04438.0 1200.01382.0 c dan hasil validasi pada saat cluster = 4, diperoleh nilai pc (partition coeffisient) = 1, ce (classification entropy) = nan (bernilai sangat kecil sekali sehingga tidak dituliskan nilainya), sc (partition index) = 0.5672, s (separation index) = 0.0040 dan sse (jumlah kuadrat kesalahan) pada hasil inferensi = 20.09905621. sedangkan sse pada hasil pengetesan = 4.620844103. 2. berdasarkan nilai validasi gath-geva (gg) clustering dihasilkan cluster optimal terjadi pada saat cluster = 4 dengan pusat cluster adalah:                7358.06950.0 3761.06323.0 2425.02295.0 0869.01101.0 c dan hasil validasi pada saat cluster = 4, diperoleh nilai pc (partition coeffisient) = 0.9607, ce (classification entropy) = nan (bernilai sangat kecil sekali sehingga tidak dituliskan nilainya), sc (partition jurnal matematika “mantik” vol. 01 no. 02. mei 2016. issn: 2527-3159 e-issn: 2527-3167 37 index) = nan, s (separation index) = nan dan sse (jumlah kuadrat kesalahan) pada hasil inferensi = 10869.45645. sedangkan sse pada hasil pengetesan = 3.61662719e-013. 3. dengan jari-jari 0.5 dan jumlah cluster = 4, k-means clustering menghasilkan rmse = 4.2624, sedangkan gath-geva (gg) clustering menghasilkan rmse = 2.8668e-14. 4. dengan jumlah jari-jari dan cluster yang sama, k-means clustering menghasilkan error cek maksimum = 0.06646 dan error cek minimum = 0.575555, sedangkan gath-geva (gg) clustering menghasilkan error cek maksimum = 0.8527e-013, dan error cek minimum = -0.2842e-013. 5. berdasarkan hasil rmse, error cek maksimum dan error cek minimum, dapat disimpulkan bahwa gath-geva (gg) clustering lebih teliti dibandingkan dengan metode k-means clustering. 5.2 saran berdasarkan dari kesimpulan di atas maka dapat diberikan saran-saran sebagai berikut: 1) algoritma gg clustering dapat digunakan dalam pengklasteran data time series. 2) ada beberapa metode clustering yang lainnya yang masih perlu dianalisis untuk mengetahui tingkat ketelitannya. 3) masih perlu dianalisis untuk mengetahui tingkat ketelitiannya diantara kedua metode tersebut. 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[2] babuska, r., -, fuzzy clustering with applications in patter recognition and data-driven modeling; delft center for system and control, www.dscs.tudelft.nl/~babuska, 5 april 2006. [3] balasko, b.; j. abonyi; b. feil; -; fuzzy clustering and data analysis toolbox, departement of process engineering univ. of veszprem, hungary, www.fmt.vein.hu, 5 april 2006. [4] biro pusat statistik indonesia, , indikator ekonomi januari 1987desember 2005, jakarta, indonesia. [5] biro pusat statistik indonesia, 2001, ekspor indonesia menurut kode isic, cv nario sari, jakarta, indonesia. [6] chen, a; efstrations nikolaidis, 1999, comparison of probabilistic and fuzzy set methods for designing under uncertainty, american institute of aeronautics and astronaitics, vol. 99. [7] gunawan; 2006, perbandingan antara metode fuzzy subtractive clustering dengan metode gustafson kessel clustering. (studi kasus data time series volume pinjaman kredit gadai), tesis, matematika, fmipa-its. [8] kusumadewi, sri; 2002, analisis desain sistem fuzzy menggunakan toolbox matlab, graha ilmu, edisi pertama, cetakan pertama, yogyakarta. [9] makridas.s, steven c.w and victor e.m, 1999, metode dan aplikasi peramalan, erlangga, edisi kedua, jilid 1, cetakan keenam, jakarta. [10] yunianti, d, w; (2005), prediksi volume ekspor non-migas dengan metode subtractive clustering, skripsi, jurusan matematika, fmipaits. http://www.fmt.vein.hu/ paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: n. a. sudibyo, a. iswardani , and y. p. s. r. hidayat, “total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph”, jmm, vol. 6, no. 1, pp. 47-51, may 2020. total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph nugroho arif sudibyo1, ardymulya iswardani2, yohana putra surya rahmad hidayat3 1department of informatics engineering, universitas duta bangsa, nugroho_arif@udb.ac.id 2department of information systems, universitas duta bangsa, ardymulya@udb.ac.id 3smk negeri 2 kudus, yohan.artup@gmail.com doi: https://doi.org/10.15642/mantik.2020.6.1.47-51 abstrak: akan diselidiki pelabelan graf yang disebut total vertex irregularity strength (tvs(g)). tvs(g) adalah minimum 𝒌 yang graf tersebut memenuhi pelabelan 𝒌-total titik tidak teratur. pada makalah ini, akan ditentukan pelabelan total tak reguler titik dari graf disjoint union of ladder rung dan graf disjoint union of domino graph. kata kunci: graf; tvs; ladder rung; domino abstract: we investigate a graph labeling called the total vertex irregularity strength (tvs(g)). a tvs(g) is minimum 𝒌 for which graph has a vertex irregular total 𝒌-labeling. in this paper, we determine the total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph. keywords: graph; tvs; ladder rung; domino jurnal matematika mantik volume 6, no.1, may 2020, pp. 47-51 issn: 2527-3159 (print) 2527-3167 (online) mailto:yohan.artup@gmail.com http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 47-51 48 1. introduction in 1735, graph theory was first introduced by leonhard euler to solve the problem of the konigsberg bridge on the river pregel, russia [1]. graph labeling is an interesting topic in graph theory so that various types of labeling are researched and developed [2]. graph labeling is the assignment of labels to the graph elements such as edges or vertices, or both, from the graph [3]. graph labeling can be applied to various fields including transportation systems, communication systems, geographical navigation, radar, and also security systems. for example, the design of the code for radar signals and missiles is equivalent to labeling a complete graph, where each point is connected to one side which has a label that is always different. this side label describes the distance between points, while the point label is the position at the time the signal is sent [4]. for a graph 𝐺(𝑉,𝐸), a labeling 𝑓:𝑉(𝐺) ∪ 𝐸(𝐺) → {1,2,…,𝑘} to be a vertex irregular total 𝑘-labeling if for any two different vertices x and y, their weights satisfy 𝑤𝑡𝑓(𝑥) ≠ 𝑤𝑡𝑓(𝑦) (see [5], [6]). the total vertex irregularity strength is minimum 𝑘 for which graph has a vertex irregular total 𝑘-labeling (see [6], [7]). the total vertex irregularity strength problem has been investigated for trees [8], 𝐶𝑛 ∗2 𝐾𝑛 graph [9], regular graph [10], forest graph [11], disjoint union of sun graph [5], wheel related graphs [12], graph obtained of a star [13], trees with maximum degree five [14], and comb product of two cycles and two stars [15]. in this paper we answer the open problem proposed by baca, et.al [7]. in particular, we determine the total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph. 2. preliminaries ball and coxeter define the ladder graph 𝑛𝑃2, is 𝑛 copies of the path graph 𝑃2 [4]. the ladder rung graph can be depicted as in figure 1. figure 1. the ladder rung graph theorem 1. the total vertex irregularity strength of disjoint union of ladder rung graph is 𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥ 2. proof. the disjoint union of ladder rung graph 𝑡𝐿𝑛 has 2𝑛𝑡 vertices. the smallest 𝑤𝑡(𝑡𝐿𝑛) must be 2 and the largest 𝑤𝑡(𝑡𝐿𝑛) is at least 𝑛𝑡 + 1. because of every vertex has degree one, then 𝑡𝑣𝑠(𝑡𝐿𝑛) ≥ 𝑛𝑡 + 1. to show that 𝑡𝑣𝑠(𝑡𝐿𝑛) ≤ 𝑛𝑡 + 1. the label of vertices of 𝑡𝐿𝑛 are described in the following formulas: 𝜆(𝑢𝑖) = 𝑖, for 𝑖 ∈ [1,𝑛𝑡], 𝜆(𝑣𝑖) = 𝑖+1, for 𝑖 ∈ [1,𝑛𝑡]. then, the label of edges of 𝑡𝐿𝑛 are 1 u 2u nu 1v 2v nv n. a. sudibyo, a. iswardani , and y. p. s. r. hidayat, total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph 49 𝜆(𝑣𝑖𝑣𝑖) = 𝑖+1, for 𝑖 ∈ [1,𝑛𝑡]. the weights of vertices 𝑢𝑖 and 𝑣𝑖 of 𝑡𝐿𝑛 are: 𝑤𝑡(𝑢𝑖) = 2𝑖 + 1, for 𝑖 ∈ [1,𝑛𝑡], 𝑤𝑡(𝑣𝑖) = 2(𝑖 + 1), for 𝑖 ∈ [1,𝑛𝑡]. the weights calculated at vertices are distinct. so, 𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥ 2. 3. results the domino graph is (2,3)-grid graph [16]. the disjoint union of domino graph can be depicted as in figure 2. figure 2. the disjoint union of domino graph theorem 2. the total vertex irregularity strength of disjoint union of domino graph is 𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. proof. the disjoint union of ladder rung graph 𝑛𝐷 has 6𝑛 vertices. the smallest 𝑤𝑡(𝑛𝐷) must be 3 and the largest 𝑤𝑡(𝑛𝐷)at least 3𝑛 + 5. it easy to see that 𝑡𝑣𝑠(𝑛𝐷) ≥ 𝑛 + 1. to show that 𝑡𝑣𝑠(𝑛𝐷) ≤ 𝑛 + 1. the label of vertices of 𝑛𝐷 are: 𝜆(𝑢𝑖,1) = 𝑖, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,2) = 𝑖, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,3) = 2, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,4) = 2, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,5) = 𝑖, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,6) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛]. then, the label of edges of 𝑛𝐷 are 𝜆(𝑢𝑖,1𝑢𝑖,2) = 𝑖, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,1𝑢𝑖,3) = 𝑖, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,2𝑢𝑖,3) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,3𝑢𝑖,4) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,3𝑢𝑖,5) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,4𝑢𝑖,6) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛], 𝜆(𝑢𝑖,5𝑢𝑖,6) = 𝑖 + 1, for 𝑖 ∈ [1,𝑛]. the weights of vertices 𝑢𝑖 and 𝑣𝑖 of 𝑡𝐿𝑛 (respectively) are: jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 47-51 50 𝑤𝑡(𝑢𝑖,1) = 3𝑛, for 𝑖 ∈ [1,𝑛], 𝑤𝑡(𝑢𝑖,2) = 3𝑛 + 1, for 𝑖 ∈ [1,𝑛], 𝑤𝑡(𝑢𝑖,3) = 3𝑛 + 4, for 𝑖 ∈ [1,𝑛], 𝑤𝑡(𝑢𝑖,4) = 3𝑛 + 5, for 𝑖 ∈ [1,𝑛], 𝑤𝑡(𝑢𝑖,5) = 3𝑛 + 2, for 𝑖 ∈ [1,𝑛], 𝑤𝑡(𝑢𝑖,6) = 3(𝑛 + 1), for 𝑖 ∈ [1,𝑛]. the weights calculated at vertices are distinct. so, 𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. 4. conclusions the total vertex irregularity strength of disjoint union of ladder rung graph is 𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥ 2 and the total vertex irregularity strength of disjoint union of domino graph is 𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. references [1] n. a. sudibyo and s. komsatun, “pelabelan total tak reguler pada graf barbel,” j. math. math. educ., vol. 8, no. 1, pp. 16–19, 2018. [2] j. a. gallian, “a dynamic survey of graph labeling,” electron. j. comb., vol. 1, no. dynamicsurveys, 2018. [3] m. marsidi and i. h. agustin, “the local antimagic on disjoint union of some family graphs,” j. mat. “mantik,” vol. 5, no. 2, pp. 69–75, 2019, doi: 10.15642/mantik.2019.5.2.69-75. [4] n. a. sudibyo and s. komsatun, “pelabelan total tak reguler pada beberapa graf,” j. ilm. mat. dan pendidik. mat., vol. 10, no. 2, pp. 9–16, 2018. [5] slamin, dafik, and w. winnona, “total vertex irregularity strength of the disjoint union of sun graphs,” int. j. comb., vol. 2012, pp. 1–9, 2012, doi: 10.1155/2012/284383. [6] d. indriati, widodo, i. e. wijayanti, k. a. sugeng, m. bača, and a. semaničováfeňovčíková, “the total vertex irregularity strength of generalized helm graphs and prisms with outer pendant edges,” australas. j. comb., vol. 65, no. 1, pp. 14–26, 2016. [7] m. bača, s. jendrol’, m. miller, and j. ryan, “on irregular total labellings,” discrete math., vol. 307, no. 11–12, pp. 1378–1388, 2007, doi: 10.1016/j.disc.2005.11.075. [8] nurdin, e. t. baskoro, a. n. m. salman, and n. n. gaos, “on the total vertex irregularity strength of trees,” discrete math., vol. 310, no. 21, pp. 3043–3048, 2010, doi: 10.1016/j.disc.2010.06.041. [9] i. k. dewi, d. indriati, and t. a. kusmayadi, “on the total vertex irregularity strength of cn ∗2 kn graph,” j. phys. conf. ser., vol. 1211, no. 1, pp. 2–7, 2019, doi: 10.1088/1742-6596/1211/1/012008. [10] r. ramdani, a. n. m. salman, and h. assiyatun, “on the total irregularity strength of regular graphs,” j. math. fundam. sci., vol. 47, no. 3, pp. 281–295, 2015, doi: 10.5614/j.math.fund.sci.2015.47.3.6. [11] m. anholcer, m. karoński, and f. pfender, “total vertex irregularity strength of forests,” no. 1, pp. 1–15, 2011. [12] a. ahmad, k.m. awan, i. javaid, and slamin, “total vertex irregularity strength of wheel related graphs ∗,” australas. j. comb., vol. 51, pp. 147–156, 2011. [13] r. ramdani, a. n. m. salman, and h. assiyatun, “on the total edge and vertex irregularity strength of some graphs obtained from star,” vol. 25, no. 3, pp. 314– 324, 2019. n. a. sudibyo, a. iswardani , and y. p. s. r. hidayat, total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph 51 [14] susilawati, e. t. baskoro, and r. simanjuntak, “total vertex irregularity strength of trees with maximum degree five,” electron. j. graph theory appl., vol. 6, no. 2, pp. 250–257, 2018, doi: 10.5614/ejgta.2018.6.2.5. [15] r. ramdani, “on the total vertex irregularity strength of comb product of two cycles and two stars,” indones. j. comb., vol. 3, no. 2, p. 79, 2020, doi: 10.19184/ijc.2019.3.2.2. [16] e. w. weisstein, “domino graph.” [online]. available: http://mathworld.wolfram.com/dominograph.html. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: p. k. intan, “comparison of kernel function on support vector machine in classification of childbirth", mantik, vol. 5, no. 2, pp. 90-99, october 2019. comparison of kernel function on support vector machine in classification of childbirth putroue keumala intan uin sunan ampel surabaya, putroue@uinsby.ac.id doi: https://doi.org/10.15642/mantik.2019.5.2.90-99 abstrak: angka kematian ibu hamil saat proses melahirkan dapat diturunkan melalui upaya ketepatan tim medis dalam menentukan proses persalinan yang harus dijalani. pembelajaran menggunakan mesin dalam hal mengklasifikasikan proses persalinan bisa menjadi solusi bagi tim medis dalam menentukan proses persalinan. salah satu metode klasifikasi yang dapat digunakan adalah metode support vector machine (svm) yang mampu menentukan hyperplane yang akan membentuk decision boundary yang baik sehingga mampu mengklasifikasikan data dengan tepat. pada svm terdapat fungsi kernel yang berguna untuk menyelesaikan kasus klasifikasi non linier dengan cara mentransformasi data ke dimensi yang lebih tinggi. pada penelitian ini akan digunakan empat fungsi kernel; linier, radial basis function (rbf), polinomial, dan sigmoid pada proses klasifikasi proses persalinan guna mengetahui fungsi kernel yang mampu menghasilkan nilai akurasi tertinggi. berdasarkan penelitian yang telah dilakukan diperoleh bahwasanya nilai akurasi yang dihasilkan oleh svm dengan fungsi kernel linier lebih tinggi dibandingkan dengan tiga fungsi kernel yang lainnya. kata kunci: persalinan, svm, fungsi kernel abstract: the maternal mortality rate during childbirth can be reduced through the efforts of the medical team in determining the childbirth process that must be undertaken immediately. machine learning in terms of classifying childbirth can be a solution for the medical team in determining the childbirth process. one of the classification methods that can be used is the support vector machine (svm) method which is able to determine a hyperplane that will form a good decision boundary so that it is able to classify data appropriately. in svm, there is a kernel function that is useful for solving non-linear classification cases by transforming data to a higher dimension. in this study, four kernel functions will be used; linear, radial basis function (rbf), polynomial, and sigmoid in the classification process of childbirth in order to determine the kernel function that is capable of producing the highest accuracy value. based on research that has been done, it is obtained that the accuracy value generated by svm with linear kernel functions is higher than the other kernel functions. keywords: childbirth, svm, kernel functions jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 90-99 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 p. k. intan comparison of kernel function on support vector machine in classification of childbirth 91 1. introduction childbirth is a period of high risk for mother and baby, either by vaginal birth or cesarean section. some risks caused by childbirth are postpartum risks of cardiac arrest, wound hematoma hysterectomy, major puerperal infection, anaesthetic complications, venous thromboembolism and haemorrhage requiring hysterectomy[1]. cesarean section risks include the morbidity associated with any major abdominal surgical procedure such as anaesthesia accidents, damage to blood vessels, an accidental extension of the uterine incision, damage to the urinary bladder and other organs. the cesarean section procedure is a potent risk factor for respiratory distress syndrome (rds) in preterm infants and for other forms of respiratory distress in mature infants. rds are major causes of neonatal morbidity and mortality.[2] in order to reduce the risks caused by childbirth, several solutions are needed. one of the solutions that can be implemented is to use machine learning in the childbirth classification process, such as amin and ali's research on the performance evaluation of supervised machine learning classifiers for predicting healthcare operational decisions [3]. the most popular classification method is support vector machine (svm) which is capable of producing good accuracy values, as in the study conducted by qiong li, qinglin meng, et al. which compares svm models and different artificial neural network models to predict hourly cooling loads in buildings and svm results can achieve better accuracy and generalization [4]. in other research about comparison of support vector machine, neural network, and cart algorithms for the land-cover classification using limited training data points, these results indicated that svm's had superior generalization capability, particularly with respect to small training sample sizes and the overall accuracies for the svm algorithm were 91% (kappa = 0.77) and 64% (kappa = 0.34) for homogeneous and heterogeneous pixels [5], and in research on comparison of vector engine support and artificial neural network systems for drug / nondrug classification, svm yields 82% correct predictions and ann produces 80% correct predictions[6]. svm can also be applied in the field of medical science such as research conducted by rajani and selvi who applied the svm classifier for early detection of breast cancer and research on svm for diagnosis of diabetes mellitus.[7] svm method is able to determine the optimal hyperplane that will form a good decision boundary so that it can classify data appropriately. in svm on nonlinearly separable data, there are two solutions, the first is to make soft margin that is particularly adapted to noised data, and the second is to use a kernel function. the kernel function is used to protect the data points to higher dimensional space for better classification.[8][9] in this paper, we have developed a framework for the classification of childbirth using svm classifier and studied the effect of various kernel functions on classification accuracy and performance. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 90-99 92 2. antecedents 2.1 kernel function the kernel function is used to protect the data points to higher dimensional space to improve its ability to find the best hyperplane to separate the data points of different classes.[9] definition (kernel function) [10] a kernel is a function of 𝜅 for all 𝑥, 𝑧 ∈ 𝑋 will meet the condition 𝜅(𝑥, 𝑧) = 〈𝜙(𝑥). 𝜙(𝑧)〉, where 𝜙 is the mapping of inner product from 𝑋 to space with higher dimension 𝐹 𝜙: 𝑥 ↦ 𝜙(𝑥) ∈ 𝐹. some common kernel function:[11] a. linear the linear kernel function is defined as: 𝜅(𝒙𝒊, 𝒙𝒋 ) = 𝒙𝒊 𝑇 𝒙𝒋 (1) the linear kernel function is the simplest kernel function which is a dot product of two vectors. b. radial basis function (rbf) rbf can also be called a gaussian kernel function. rbf is defined as: 𝜅(𝒙𝒊, 𝒙𝒋 ) = exp (−𝛾‖𝒙𝒊 − 𝒙𝒋‖ 2 ) , 𝛾 > 0 (2) where 𝛾 is a positive parameter to set the distance c. polynomial the polynomial kernel function with has a degree𝑑, where 𝑟 and 𝑑 are the parameters defined as follows: 𝜅(𝒙𝒊, 𝒙𝒋 ) = (𝛾 𝒙𝒊 𝑇𝒙𝒋 + 𝑟) 𝑑 , 𝛾 > 0 (3) d. sigmoid the sigmoid kernel function is defined as: 𝜅(𝒙𝒊, 𝒙𝒋 ) = tanh(𝛾𝒙𝒊, 𝒙𝒋 + 𝒓) (4) where tanh(𝑎) = 2𝜎(𝑎) − 1, and 𝜎(𝑎) = 1 1+exp(𝑎) 2.2 support vector machine (svm) svm is a data mining method developed by boser, guyon, and vapnik and presented for the first time in 1992 [12]. the basic idea of this svm is to determine a hyperplane function in the form of a linear model that will form a decision boundary (db) by maximizing margins. margin is the distance between the hyperplane and the nearest data. the svm method is not only able to solve linear classification problems but also able to solve non-linear classification problems using the kernel trick concept. a. svm on linearly separable data for 𝑁 data set: {𝒙𝒊, 𝑡𝑖 }, 𝑖 = 1 … 𝑁 (5) p. k. intan comparison of kernel function on support vector machine in classification of childbirth 93 with 𝒙𝒊 = [𝑥1, 𝑥2, … , 𝑥𝑛] is a line vector with dimensions n and 𝑡𝑖 = {−1,1} is target value on each row vector. the data will be classified into two class, i.e. class 𝑅1 for target value 𝑡𝑖 = +1 and class 𝑅2 for target value 𝑡𝑖 = −1. svm uses a linear model as a hyperplane with a general form: 𝑦(𝒘) = 𝒘𝑇 𝒙 + 𝑏 (6) where x is the input vector, w is the weight parameter, and b is a bias. so with the hyperplane svm will classify the data into two classes 𝑅1 and 𝑅2 with each class will have a delimiter field parallel to the hyperplane as: 𝒘𝑇 𝒙𝒊 + 𝑏 ≥ 1, 𝑓𝑜𝑟 𝑡𝑖 = +1 (7) 𝒘𝑇 𝒙𝒊 + 𝑏 ≤ −1, 𝑓𝑜𝑟 𝑡𝑖 = −1 (8) or both of the delimiter fields are written in the following inequality: 𝑡𝑖 (𝒘 𝑇 𝒙𝒊 + 𝑏) − 1 ≥ 0 (9) the search for the best hyperplane in the svm method is done by maximizing margins, which is maximizing the value of 1 ‖𝒘‖ which is the same as minimizing the value of ‖𝒘‖2 can be formulated into the following optimization problem: [13] 𝑤,𝑏 arg 𝑚𝑖𝑛 1 2 ‖𝑤‖2 (10) 𝑠. 𝑡 𝑡𝑖(𝒘 𝑇 𝒙𝒊 + 𝑏) − 1 ≥ 0, 𝑖 = 1, … , 𝑁 (11) this problem will be more easily solved if it is changed into the lagrange function (primal problem), thus the optimization problem can be changed to: 𝐿(𝒘, 𝑏, 𝑎) = 1 2 ‖𝒘‖2 − ∑ 𝑎𝑖 𝑡𝑖(𝒘 𝑇 𝒙𝒊 + 𝑏) 𝑁 𝑖=1 + ∑ 𝑎𝑖 𝑁 𝑖−1 (12) with the addition of the lagrange multiplier 𝑎𝑖 ≥ 0. the dual problem is obtained as follows: 𝑎 arg 𝑚𝑎𝑥 𝐿(𝑎) = ∑ 𝑎𝑖 𝑁 𝑖−1 − 1 2 ∑ ∑ 𝑎𝑖 𝑡𝑖𝑎𝑗 𝑡𝑗 (𝒙𝒊 𝑻𝒙𝒋) 𝑁 𝑗=1 𝑁 𝑖=1 (13) 𝑠. 𝑡 ∑ 𝑎𝑖 𝑁 𝑖−1 𝑡𝑖 = 0, 𝑎𝑖 ≥ 0 (14) the optimization form above must meet the following the karush-kuhntucker (kkt) conditions [13]: 𝑎𝑖 ≥ 0 (15) 𝑡𝑖 (𝒘 𝑻𝒙𝑖 + 𝑏) − 1 ≥ 0 (16) 𝑎𝑖 (𝑡𝑖(𝒘 𝑻𝒙𝑖 + 𝑏) − 1) = 0 (17) from the kkt conditions, the optimization problem solution (13) is (𝑎∗, 𝒘∗, 𝑏∗) that satisfies 𝑎𝑖 ∗(𝑡𝑖(𝒘 ∗𝑻𝒙𝑖 + 𝑏 ∗) − 1) = 0 . so if there is training jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 90-99 94 data that has the value 𝑎𝑖 > 0 then 𝑡𝑖 (𝒘 𝑻𝒙𝑖 + 𝑏) = 1, it means that the data is a support vector while the rest have the value 𝑎𝑖 = 0 . thus the resulting decision function is only influenced by support vectors. after the problem solution is found (𝑎∗, 𝒘∗, 𝑏∗) then the class of the x test data can be determined based on value of the decision function: 𝑦(𝒙) = ∑ 𝑎𝑖 ∗ 𝑁𝑠 𝑖−1 𝑡𝑖𝒙𝒊 𝑻𝒙 + 𝑏∗ (18) where 𝒙𝒊 is a support vector, ns = number of support vector. b. svm on nonlinearly separable data for data that is not classified correctly, the svm model must be modified by adding the slack variable 𝜉𝑖 . hyperplane search by adding slack variables is also called a hyperplane soft margin as follows: 𝑤,𝑏,𝜉 arg 𝑚𝑖𝑛 1 2 ‖𝑤‖2 + 𝐶 (∑ 𝜉𝑖 𝑁 𝑖=1 ) (19) 𝑠. 𝑡 𝑡𝑖 (𝒘 𝑇 𝒙𝒊 + 𝑏) ≥ 1 − 𝜉𝑖 , 𝑖 = 1, … , 𝑁 (20) 𝜉𝑖 ≥ 0 , 𝑖 = 1, 2, … , 𝑛 (21) where c is the parameter that determines the amount of penalty due to errors in classification and the value of c is not obtained in the learning process but must be determined before learning. in addition to adding slack variables to deal with data that is not linearly classified, it is also necessary to transform the data into higher dimensions with kernel functions so that it can be linearly separated at higher dimensions. using the kernel trick, the 𝒙𝒊 data will be mapped with the function 𝜙(𝒙𝑖 ), and each product (𝒙𝑖 . 𝒙𝑗) will be calculated using 𝜅(𝒙𝑖 , 𝒙𝑗). thus the linear model used as a hyperplane is: 𝑦(𝒙) = 𝒘𝑇 𝜙(𝒙) + 𝑏 (22) the form of optimization issues from soft margin to: arg min 𝒘, 𝑏, 𝜉 1 2 ‖𝒘‖2 + 𝐶 ∑ 𝜉𝑖 𝑛 𝑖=1 (23) 𝑠. 𝑡 𝑡𝑖(𝒘 𝑇 𝜙(𝒙𝑖) + 𝑏) ≥ 1 − 𝜉𝑖 , 𝑖 = 1, 2, … , 𝑁 (24) 𝜉𝑖 ≥ 0 , 𝑖 = 1, 2, … , 𝑁 (25) by multiplying the lagrange multiplier 𝑎𝑖 ≥ 0 and 𝜇𝑖 ≥ 0 to the primal form of the optimization problem, so the lagrange function is obtained as follows: 𝐿(𝒘, 𝑏, 𝜉, 𝑎, 𝜇) = 1 2 ‖𝒘‖2 + 𝐶 ∑ 𝜉𝑖 𝑛 𝑖=1 + ∑ 𝑎𝑖{1 − 𝜉𝑖 − 𝑡𝑖 (𝒘 𝑇 𝜙(𝒙𝑖) + 𝑏)} 𝑛 𝑖=1 − ∑ 𝜇𝑖 𝜉𝑖 𝑛 𝑖=1 the dual problem form is obtained as follows: max 𝑎 ∑ 𝑎𝑖 𝑛 𝑖=1 − 1 2 ∑ ∑ 𝑎𝑖 𝑡𝑖 𝑎𝑗 𝑡𝑗 (𝜙(𝒙𝑖 ) 𝑇 𝜙(𝒙𝑗)) 𝑛 𝑗=1 𝑛 𝑖=1 (26) 𝑠. 𝑡 ∑ 𝑎𝑖 𝑡𝑖 𝑛 𝑖=1 = 0 p. k. intan comparison of kernel function on support vector machine in classification of childbirth 95 0 ≤ 𝑎𝑖 ≤ 𝐶 , 𝑖 = 1, 2, … , 𝑛 the dual form in equation (3.35) satisfies the kkt conditions as follows [12]: 𝑎𝑖 ≥ 0 (27) 𝜇𝑖 ≥ 0 (28) 𝑡𝑖(𝒘 𝑻𝜙(𝒙𝑖) + 𝑏) − 1 + 𝜉𝑖 ≥ 0 (29) 𝜉𝑖 ≥ 0 (30) 𝑎𝑖 {𝑡𝑖 (𝒘 𝑻𝜙(𝒙𝑖 ) + 𝑏) − 1 + 𝜉𝑖 } = 0 (31) 𝜇𝑖 𝜉𝑖 = 0 (32) from the kkt conditions, the optimization problem solution (26) is (𝑎∗, 𝒘∗, 𝑏∗) that satisfies 𝑎𝑖 ∗(𝑡𝑖(𝒘 ∗𝑻𝜙(𝒙𝑖) + 𝑏 ∗) − 1 + 𝜉𝑖 ) = 0 and 𝜇𝑖 𝜉𝑖 = 0. so if there is training data that has the value 𝑎𝑖 = 0, then 𝜇𝑖 = 𝐶 > 0 and 𝜉𝑖 = 0 results in 𝑡𝑖 (𝒘 𝑻𝜙(𝒙𝑖 ) + 𝑏) ≥ 0 then the data is correctly classified and not support vector. if there is training data that has the value 0 < 𝑎𝑖 < 𝐶, then 𝜇𝑖 > 0 and 𝜉𝑖 = 0 results in 𝑡𝑖 (𝒘 𝑻𝜙(𝒙𝑖 ) + 𝑏) ≥ 0 then the data is correctly classified and not support vector. whereas if there is training data that has the value 𝑎𝑖 = 𝐶, then 𝜇𝑖 = 0 and 𝜉𝑖 > 0 results in 𝜉𝑖 ≤ 1 or 𝜉𝑖 > 1, so 𝑡𝑖(𝒘 𝑻𝜙(𝒙𝑖) + 𝑏) ≥ 0 then the data is not properly classified. after the problem solution is found (𝑎∗, 𝒘∗, 𝑏∗) then the class of the x test data can be determined based on the value of the decision function: 𝑦(𝒙) = ∑ 𝑎𝑖 ∗𝑡𝑖 𝜅(𝒙𝑖, 𝒙𝑗) 𝑁𝑠 𝑖=1 + 𝑏∗ (33) where 𝒙𝒊 is a support vector, ns = number of support vectors. 3. methodology 3.1 tools used the main tools used for this analysis and study is python programming language, which is a free open source platform for machine learning. there are many packages available with standard implementations for various machine learning algorithms. the scikit-learn packages (reference) used in this study for the implementation of preprocessing, model selection and svm classification with four kernel functions [14]. 3.2 preprocessing normalization or scaling of each feature data highly recommended before being processed by svm, i.e. in the interval [-1, +1] [15]. preprocessing is done to avoid the domination of features with large values over features with small values and to avoid numerical difficulties during the calculation process. 3.3 model selection in almost all data mining methods, there will be parameters that cannot be determined in the learning process. determining the value of these parameters must be done before the learning process. determination of these parameters is called model selection. model selection aims to tune the hyperparameters of svm classification (the penalty parameter c and any kernel parameters) in order to jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 90-99 96 achieve the lowest test error, i.e. the lowest probability of misclassification of unseen test examples [16]. in svm, there two parameters that need to be determined parameters c and 𝛾. rows of parameter pairs (𝐶, 𝛾) are recommended as candidates is 𝐶 = 2−5, 2−3, … , 215 dan 𝛾 = 215, 213, … , 23 [11]. random search methods will be used in this study. according to bergstra and bengio (2012), random search method is a more efficient method than the grid search method, due to determine the value of the parameter that can produce the same accuracy value even better without having to try all possible parameter values in the range of a specified value [17]. 3.4 classification the childbirth data will be classified using svm with different kernel functions after obtaining the parameter values (𝐶, 𝛾) using the scikit learn package in python programming language. 3.5 classification evaluation a. evaluation procedure k fold cross validation is one procedure that is commonly used to estimate the performance of a model.[15] this product consists of 3 stages: divide data into k parts of the same size. k-1 part is used as training data, and one part is used as testing data. this process is done as k times as repetition for each different combination of testing data and training data so that the whole section will become testing data. the accuracy of each iteration is averaged to get an estimate of the final accuracy of the model. this study uses k = 5 or 5-fold cross-validation, so that for each experiment will use four subsets for training data, and one subset for data testing conducted five times trial for all possibilities. b. unit of evaluation size confusion matrix the confusion matrix is used to present the results of the k-fold cross validation as follows:[18] table 1. confusion matrix actual prediction -1 1 -1 true negative (tn) false positive (fp) 1 false negative (fn) true positive (tp) one unit of performance measurement based on confusion matrix is accuracy value which can be calculated by: 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 (34) roc curve to visualize the comparison results of two or more classification models, you can use the roc curve (receiver operating characteristics). the roc curve p. k. intan comparison of kernel function on support vector machine in classification of childbirth 97 is a two-dimensional graph with false positive (fp) as a horizontal line and true positive (tp) as a vertical line[18]. point (0.1) states that a perfect classification of all positive and negative cases with no fp value or fp = 0 and a high tp value or tp = 1. point (0,0) states that the classification predicts each case to be -1. point (1,1) states that the classification that predicts each case becomes 1. to evaluate the classification can be seen from the auc (area under the curve) value. the accuracy level of the auc (area under the curve) value in the classification is divided into five groups expressed in table 2:[18] table 2. accuracy levels of auc value in classifications auc interval value accuracy level 0.90 – 1.00 excellent classification 0.80 – 0.89 good classification 0.70 – 0.79 fair classification 0.60 – 0.69 poor classification 0.50 – 0.59 failure classification 4. experimental analysis this experiment used 304 data childbirth which is divided into two classes; vaginal birth 104 data and caesarean section 200 data. the data consists of 10 features; age, hypertension history, glucose disease, first pregnancy, fetal position, parturition history, number of fetuses, hip size, another disease, and ruptured amniotic fluid history. the results of model selection we got the parameters value 𝐶 = 2048,0 and 𝛾 = 8,6316745750310983𝑒 − 0.5, that will be used to whole kernel functions on support vector classification and specifically the polynomial kernel function will be used in 3rd degree. figure 1. roc curve for various kernel functions figure 1. shows the mean auc from 5fold validation for svm with four different kernel function. svm-linear, svm-rbf, and svm-sigmoid are jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 90-99 98 categorized in good classification because the auc values are at intervals of 0.80 0.89, but svm-polynomial is only categorized as a fair classification because its auc value is 0.70. based on the results of the auc value, it can be seen that svm-linear is the best model because its auc value is greater than the others. to confirm the conclusion that we need to see the results of the accuracy of the model. accuracy value of each svm model with different kernel functions can be seen below: table 3. accuracy value of each model no model tn fp fn tp accuracy value 1 svm-linear 75 29 24 176 0,83 2 svm-rbf 67 37 18 182 0,82 3 svm-polynomial 0 104 0 200 0,66 4 svm-sigmoid 78 26 43 157 0,77 table 4. shows the whole results of the confusion matrix from each classification with a different kernel function. svm-polynomial fails to predict class -1 data properly because the result of tn is 0, but it succeeds in predicting class 1 data perfectly. other models produce different class 1 and -1 data predictions so that different accuracy values are obtained. based on the results of the accuracy value of each model, we can find out the best model for classifying the childbirth. svm-linear can produce an accuracy value of 0,83, and this result is greater than the svm-rbf result of 0,82, svmpolynomial of 0,66, and svm-sigmoid of 0,77. it can be concluded that svmlinear method is the best model in classifying the childbirth. so the classification model that is most suitable for labour classification is svm which uses linear kernel functions. 5. conclusion based on experiments that have been done to see the effect of kernel function selection on the svm method on the accuracy of the analysis of the childbirth classification, it can be concluded that the svm-linear method is the best model in classifying the childbirth. this can be seen from the accuracy value produced by the svm-linear method that is 0.83, and this value is higher than the accuracy value generated by svm-rbf which is 0.82, svm-polynomial which is 0.66, and svm-sigmoid which is 0,77. in addition, the auc value produced by svm-linear is 0.85 and is greater than other methods, which shows good classification. references [1] s. liu et al., “maternal mortality and severe morbidity associated with lowrisk 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[12] b. e. boser, i. m. guyon, and v. n. vapnik, “training algorithm for optimal margin classifiers,” proc. fifth annu. acm work. comput. learn. theory, pp. 144–152, 1992. [13] christopher m bishop, pattern recognition and machine learning (information science and statistics). heidelberg: springer-verlag berlin, 2006. [14] “1.4. support vector machines — scikit-learn 0.21.3 documentation.” [online]. available: https://scikit-learn.org/stable/modules/svm.html#svmclassification. [accessed: 29-aug-2019]. [15] f. pedregosa fabianpedregosa et al., “scikit-learn: machine learning in python gaël varoquaux bertrand thirion vincent dubourg alexandre passos pedregosa, varoquaux, gramfort et al. matthieu perrot,” j. mach. learn. res., vol. 12, pp. 2825–2830, 2011. [16] c. gold and p. sollich, “model selection for support vector machine classification,” neurocomputing, vol. 55, no. 1–2, pp. 221–249, 2003. [17] j. bergstra and y. bengio, “random search for hyper-parameter optimization,” j. mach. learn. res., vol. 13, no. feb, pp. 281–305, 2012. [18] f. gorunescu, dana mining: concepts, models, and techniques. new york: springer us, 2011. how to cite: b. p. silalahi, n. fathiah, and p. t. supriyo, “use of ant colony optimization algorithm for determining traveling salesman problem routes", mantik, vol. 5, no. 2, pp. 100-111, october 2019. use of ant colony optimization algorithm for determining traveling salesman problem routes bib paruhum silalahi1, nurul fathiah2, prapto tri supriyo3 department of mathematics, ipb university, bibparuhum@gmail.com1 department of mathematics, ipb university, nurulfathiah393@gmail.com2 department of mathematics, ipb university, praptotrisupriyo@gmail.com3 doi: https://doi.org/10.15642/mantik.2019.5.2.100-111 abstrak: optimisasi koloni semut merupakan salah satu metode meta-heuristic yang digunakan untuk menyelesaikan masalah optimisasi kombinatorial yang cukup sulit. algoritma optimisasi koloni semut diinspirasi dari perilaku semut dalam dunia nyata untuk membangun jalur terpendek antara sumber makanan dan sarangnya. traveling salesman problem adalah suatu permasalahan dalam optimisasi. traveling salesman problem merupakan permasalahan untuk mencari jarak minimal dari node awal menuju node selanjutnya, dengan setiap node harus didatangi persis satu kali dan harus kembali ke node awal. traveling salesman problem merupakan suatu masalah yang nondeterministic polynomial-time complete. pada penelitian ini didiskusikan penyelesaian traveling salesman problem menggunakan algoritma optimisasi koloni semut dan juga menggunakan algoritma eksak. hasil penelitian menunjukkan bahwa semakin besar ukuran kasus traveling salesman problem, waktu eksekusi yang dibutuhkan semakin lama. kemudian diperoleh hasil bahwa waktu eksekusi yang dihasilkan optimisasi koloni semut jauh lebih cepat dibandingkan waktu eksekusi metode eksak. kata kunci: algoritma, optimisasi , optimisasi koloni semut, traveling salesman problem abstract: ant colony optimization is one of the meta-heuristic methods used to solve combinatorial optimization problems that are quite difficult. ant colony optimization algorithm is inspired by ant behaviour in the real world to build the shortest path between food sources and their nests. travelling salesman problem is a problem in optimization. travelling salesman problem is a problem to find the minimum distance from the initial node to the whole node with each node must be visited exactly once and must return to the initial node. travelling salesman problem is a non-deterministic polynomial-time complete problem. this research discusses the solution of the traveling salesman problem using the ant colony optimization algorithm and also using the exact algorithm. the results showed that the greater the size of the traveling salesman problem case, the longer the execution time required. the results also showed that the execution times of the ant colony optimization are much faster than the execution time of the exact method. keywords: algorithm, optimization, ant colony optimization, traveling salesman problem jurnal matematika mantik volume 5, nomor 2, october 2019, pp. 100-111 issn: 2527-3159 (print) 2527-3167 (online) mailto:bibparuhum@gmail.com3 http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 b. p. silalahi, n. fathiah, and p. t. supriyo use of ant colony optimization algorithm for determining traveling salesman problem routes 101 1. introduction optimization is the study of techniques for finding the maximum or minimum value of a problem. some examples of studies of optimization techniques are modifying of steps in the steepest descent method [1] [3], computational test of newton's method variance algorithm on nonlinear optimization problems without constraints [4], comparison of sensitivity analysis on linear optimization using optimal partition and optimal basis [5], elaborating a combination method for solving nonlinear equations [6], overview of newton's optimization method [7]. optimization is applied in many aspects of life such as generating star catalogue for satellite attitude navigation using density-based clustering [8], but many of them are applied to transportation and distribution systems [9] [11]. travelling salesman problem (tsp) is an example of problems in optimization. problems associated with tsp are often encountered in distribution systems. basically, tsp is a problem to find the minimum distance from the initial node to the next node with the rule that each node must be visited exactly once. furthermore, after all the nodes visited exactly once, they must return to the initial node. one of the characteristics of tsp is non-deterministic polynomialtime complete (npc) which implies that there is no optimal solution other than having to try all possible solutions [12]. the application of tsp in the real world is often used in the case of distributing goods and services. problems that are often encountered relating to distribution such as determining travel routes that minimize travel time, distance travelled, or operational costs. tsp can be solved by exact, heuristic or meta-heuristic methods. one of the meta-heuristic methods that can be used to solve the tsp is the ant colony optimization algorithm. some other meta-heuristic methods such as genetic algorithm [13], particle swarm optimization [14], firefly algorithm [15][16] and interior point method [17], [18]. ant colony optimization (aco) is adapted from the behaviour of the ant colony as a system for finding and collecting food. ants have the ability to find food effectively by finding the shortest path between the food source and the nest without using the sense of sight at the beginning of the ant, walking random looking for food. along the way, the ant will leave a trail called pheromones. after successfully obtaining food, the ant will return to the nest with a trail guide pheromone left behind. other ants will take food by following in the footsteps of pheromones. pheromones can evaporate. the longer the distance of a route, the longer the ants are on the route, so that the pheromones on the route will be reduced faster because it evaporates when compared to the shorter distance route. this causes the ants to tend to choose the shortest route [19]. aco is widely used as a solution to various optimization problems, one of which is tsp. this algorithm is compiled using distance data between locations, some ants that communicate with their colonies indirectly using the help of pheromones. 2. research method in this research, tsp cases will be solved using the integer linear programming (ilp) model and the ant colony optimization (aco) model. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 100-111 102 furthermore, the results of the two models will be compared specifically in the aspects of execution time and total travel distance. both models were executed using a laptop with the specifications of asus a 455ld i5 8gb ram. 3. results and discussion 3.1. travelling salesman problem tsp is a problem of determining the route when someone will come to all locations that have been determined. starting from the first location and then visiting all the locations that have been determined exactly once. then after that, the person/object must return to the first location. what we want to achieve is the minimum total travel distance [20]. tsp can be formulated as follows, for example: 𝑛 : number of nodes to be visited, 𝑐𝑖𝑗 : distance of node 𝑖 to node 𝑗, 𝑢𝑖 : additional variables when visiting node 𝑖. decision variable: 𝑥𝑖𝑗 = { 1, if there is a trip from node 𝑖 to node 𝑗 0, else the objective function of the tsp is to find a value so that the total distance traveled is minimum, that is: min 𝑍 = ∑ 𝑛 𝑖=1 ∑ 𝑛 j=1 𝑐𝑖𝑗 𝑥𝑖𝑗 . constraints: 1. nodes must be visited exactly once. ∑ 𝑥𝑖𝑗 𝑛 𝑖=1 = 1, ∀𝑗 = 1,2, … , 𝑛 ∑ 𝑥𝑖𝑗 𝑛 𝑗=1 = 1, ∀𝑖 = 1,2, … , 𝑛 2. no sub tour formed on the journey in the tsp. this means that there is only one tour from tsp. because returning to the initial location will form a cycle, 𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1, 𝑖 ≠ 𝑗, ∀𝑖 = 2, 3, … , 𝑛; ∀𝑗 = 2, 3, … , 𝑛; 𝑢𝑗 ≥ 0 . 3.2. ant colony optimization algorithm ant colony optimization (aco) is one of the meta-heuristic methods used to find a combinatorial optimization problem that is quite difficult. the aco algorithm is represented by ant behaviour in the real world to build the shortest path between food sources and their nests. each ant randomly starts its route from a node. each node is visited by ants to form a route, and this is done repeatedly. the ant selects the nodes to be visited using the probability function, based on the distance of the node, and the number of pheromones found on the side connecting the node. nodes that have smaller distances and have higher levels of pheromones will be more likely to be visited by ants [21]. nodes that have been visited by ants will be recorded in memory which we call by the name tabulist. this tabulist prevents ants from going to the nodes that have been visited. the tabulist will be full when all ants have visited all nodes. then the pheromone update rules are applied. calculation of the loss of b. p. silalahi, n. fathiah, and p. t. supriyo use of ant colony optimization algorithm for determining traveling salesman problem routes 103 pheromone levels on each side is done. for shorter routes, the loss of pheromone levels will be relatively longer compared to the longer routes. the process will be repeated until the tour reaches the maximum value or the system is in a stagnant situation, which is a situation where the system is no longer looking for other alternative solutions [22]. figure 1 shows a simple aco flow. 3.3. steps of ant colony optimization algorithm the flow of the aco algorithm is shown in figure 1. the steps taken are as follows. 1. data the initial data presented is only in the form of 𝑥 and 𝑦 coordinate points, so the distance between nodes must be calculated using the euclidian formula: 𝑑𝑖𝑗 = √(𝑥𝑖 − 𝑥𝑗 ) 2 + (𝑦𝑖 − 𝑦𝑗) 2 , where 𝑑𝑖𝑗 : the distance between node 𝑖 and node 𝑗 𝑥𝑖 : 𝑥 coordinate of node 𝑖, 𝑥𝑗 : 𝑥 coordinate of node 𝑗, 𝑦𝑖 : 𝑦 coordinate of node 𝑖, 𝑦𝑗 : 𝑦 coordinate of node 𝑗. 2. determine the next node the ant will pass by calculating the probability using the formula: 𝑃𝑖𝑗 𝑘 (𝑡) = [𝜏𝑖𝑗 (𝑡)] 𝛼 ∙ [𝜂𝑖𝑗 (𝑡)] 𝛽 ∑ [𝜏𝑖𝑗(𝑡)] 𝛼 ∙ [𝜂𝑖𝑗 (𝑡)] 𝛽 𝑗 𝜖 𝑁𝑖 𝑘 , if 𝑗 𝜖 𝑁𝑖 𝑘 where: 𝑃𝑖𝑗 𝑘 (𝑡) : the probability of ant 𝑘 run from node 𝑖 to node 𝑗 at time 𝑡, 𝜏𝑖𝑗(𝑡) : number of ant pheromones from node 𝑖 to node 𝑗 at time 𝑡, 𝜂𝑖𝑗 (𝑡) : distance inverse between node 𝑖 and node 𝑗 at time 𝑡, 𝑁𝑖 𝑘 : the set of points to be visited by ant 𝑘 which is at point 𝑖, 𝛼: ant pheromones parameter, 𝛽: distance control parameter, and informing the route, the next node selection criteria use the roulette wheel selection algorithm, with the following steps: a. determine one random value between 0 and 1, b. calculate the cumulative probability value 𝑃𝑖𝑗 𝑘 (𝑡), c. compare the random value in step a with each cumulative probability value 𝑃𝑖𝑗 𝑘 (𝑡) in step b, d. if the cumulative probability value 𝑃𝑖𝑗 𝑘 (𝑡) is greater than the random value, then it will be selected as the next node. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 100-111 104 figure 1. the flow of ant colony optimization algorithm. 3. repeat step 2 for each ant until all the nodes are selected and form a route. 4. calculate the total distance of each node that each ant passes. 5. choose the best route from all the routes produced, i.e. the route with the minimum total distance obtained in step 4. 6. update the pheromone with the following formula: 𝜏𝑖𝑗 (𝑡) = (1 − 𝜌)𝜏𝑖𝑗(𝑡) + δ𝜏𝑖𝑗(𝑡) with δ𝜏𝑖𝑗(𝑡) = { 1 𝐿𝑘 , if ant 𝑘 run from node 𝑖 to node 𝑗 0 , else where: 𝜏𝑖𝑗 (𝑡) : number of ant pheromones from node 𝑖 to node 𝑗 at time 𝑡, δ𝜏𝑖𝑗 (𝑡) : changes in pheromone values in the node (𝑖, 𝑗) made by ant 𝑘, 𝐿𝑘 : travel length of ant 𝑘, 𝜌 : global pheromone evaporation parameter with 0 < 𝜌 ≤ 1, pheromone updates are carried out with the aim of updating the amounts of pheromones left by ants as a result of evaporation. the number of pheromones that have been updated will be used as information for ants in the next iteration. 7. this algorithm is terminated if the stopping criteria have been reached, i.e. when the number of iterations has been reached. 8. return to step 2 if the stopping criteria have not been reached. b. p. silalahi, n. fathiah, and p. t. supriyo use of ant colony optimization algorithm for determining traveling salesman problem routes 105 3.4. solving by using integer linear programming (ilp) accomplishment by using ilp is carried out with the help of lingo 11.0 software. the data used are 𝑥 and 𝑦 coordinate which indicate the location of the nodes in cartesian coordinates. the data are hypothetic, obtained from a random function. the 𝑥 and 𝑦 coordinate data are presented in table 1. from the data in table 1, the distance between nodes is searched using the euclidian formula to produce a distance matrix. case i the results of execution using ilp for the case of 30 nodes required execution time of 1 minute 34 seconds with the number of iterations of 436.649 and obtained a minimum total travel distance 388.371 km. case ii the results of execution using ilp for the case of 35 nodes required 11 minutes 04 seconds of execution time with the number of iterations 3.351.017 and obtained a minimum total travel distance 427.584 km. case iii the results of execution using ilp for the case of 38 nodes required execution time of 6 hours 38 minutes 14 seconds with the number of iterations 142.214.825 and the result obtained a minimum total travel distance 430.251 km. 3.5. solving by using ant colony optimization algorithm in this work, the helping of a mathematics software is used to find the solution of the tsp using the aco algorithm. each case uses different parameter values α, β, and m (number of ants). the value of α is at the interval 0<α≤1, β>0, and m>0 [23]. the values of α, β, dan m, which can produce the smallest total approach distance will be presented. case i in case i, the experiments are carried out using parameters with values as presented in table 2. after conducting several experiments, the smallest total approach distance is 392.8014 km with execution time 11.836 s in 500 iterations. the route obtained is as shown in figure 2. case ii in case ii, the experiments are carried out using parameters with values, as shown in table 3. after conducting several experiments, the smallest total approach distance was 463.4509 km with an execution time 21.340 s in 700 iterations. figure 3 shows the route obtained. case iii in case iii, experiments will be conducted by using parameters with the values presented in table 4. after conducting several experiments, the smallest total approach distance is 464.7083 km with an execution time of 38.00 s in 800 iterations. the route obtained is presented in figure 4. table 1. the data in each case case i case ii case iii x y x y x y 81.91 37.47 81.91 37.47 81.91 37.47 83.16 49.92 83.16 49.92 83.16 49.92 jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 100-111 106 case i case ii case iii x y x y x y 84.76 40.37 84.76 40.37 84.76 40.37 75.27 63.61 75.27 63.61 75.27 63.61 38.10 28.80 38.10 28.80 38.10 28.80 23.28 95.00 23.28 95.00 23.28 95.00 77.49 15.56 77.49 15.56 77.49 15.56 39.22 22.32 39.22 22.32 39.22 22.32 62.88 32.60 62.88 32.60 62.88 32.60 36.63 65.29 36.63 65.29 36.63 65.29 17.02 41.47 17.02 41.47 17.02 41.47 89.50 30.82 89.50 30.82 89.50 30.82 17.02 52.23 17.02 52.23 17.02 52.23 13.25 100.12 13.25 100.12 13.25 100.12 73.20 52.75 73.20 52.75 73.20 52.75 34.08 89.16 34.08 89.16 34.08 89.16 55.29 90.36 55.29 90.36 55.29 90.36 41.42 25.66 41.42 25.66 41.42 25.66 53.31 90.58 53.31 90.58 53.31 90.58 42.66 31.08 42.66 31.08 42.66 31.08 54.49 56.85 54.49 56.85 54.49 56.85 99.85 60.28 99.85 60.28 99.85 60.28 63.39 100.56 63.39 100.56 63.39 100.56 21.86 77.04 21.86 77.04 21.86 77.04 35.09 36.91 35.09 36.91 35.09 36.91 56.86 61.57 56.86 61.57 56.86 61.57 83.89 52.24 83.89 52.24 83.89 52.24 53.04 56.22 53.04 56.22 53.04 56.22 34.06 37.48 34.06 37.48 34.06 37.48 66.57 25.84 66.57 25.84 66.57 25.84 34.56 20.48 34.56 20.48 63.63 63.48 63.63 63.48 56.74 22.22 56.74 22.22 15.90 62.84 15.90 62.84 88.86 79.06 88.86 79.06 73.48 82.00 85.58 55.79 90.92 54.49 table 2. the results of case i. the number of iterations 𝛼 𝛽 𝑚 total distance (in km) 300 1 5 50 425.1656 350 1 5 50 417.1519 400 1 5 50 407.1231 450 1 5 50 399.8285 500 0.9 5 50 420.6379 500 1 5 50 392.8014 b. p. silalahi, n. fathiah, and p. t. supriyo use of ant colony optimization algorithm for determining traveling salesman problem routes 107 figure 2. route approach in case i. table 3. the results of case ii. the number of iterations 𝛼 𝛽 𝑚 total distance (in km) 300 0.7 1 50 535.0748 300 1 5 50 497.4685 400 1 5 50 481.1290 500 1 5 50 472.6510 600 1 5 50 470.1542 700 1 5 50 463.4509 figure 3. route approach in case ii. jurnal matematika mantik vol. 5, no. 2, october 2019, pp. 100-111 108 table 4. the results of case iii. the number of iterations 𝛼 𝛽 𝑚 total distance (in km) 400 0.9 5 50 543.4537 400 0.9 5 70 494.852 500 0.9 5 70 487.6729 600 1 5 70 488.1252 700 0.9 5 70 481.1320 800 0.9 5 70 464.7083 figure 4. route approach in case iii. 3.6. results comparison ilp and aco execution times are shown in table 5. from the table, it can be seen that aco execution time is much faster than ilp execution time. for case i, aco execution time is eight times faster than ilp. in case ii, aco execution time is 31 times faster than ilp execution time. aco execution time for case iii is 629 times faster than ilp. table 6 shows the total distance produced by each method. ilp method produces optimal values whereas aco produces a value approach. in case i, the percentage of distance difference is 1%. while in case ii and case iii have a percentage difference of 8%. table 5. comparison of execution time. case execution time (hours: minutes: seconds) ilp aco i 0:01:34 0:0:11.836 ii 0:11:04 0:0:21.340 iii 6:38:14 0:0:38.000 b. p. silalahi, n. fathiah, and p. t. supriyo use of ant colony optimization algorithm for determining traveling salesman problem routes 109 table 6. comparison of the total distance. case total distance (in km) ilp aco i 388.371 392.8014 ii 427.584 463.4509 iii 430.251 464.7083 4. conclusion based on the results of this study, it can be seen that the larger the cases that must be solved, the longer the execution time required. it can also be seen that the execution time produced by aco is much faster than the execution time of the exact method. from the three cases, the percentage of distance difference of the first case is 1 %, while the two other cases have a percentage difference of 8%. references [1] f. fadhillah, b. p. silalahi, and m. ilyas, “modifikasi stepsize pada metode steepest descent dalam pengoptimuman fungsi: kasus fungsi kuadratik diagonal”, jurnal matematika dan aplikasinya, 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[23] m. dorigo and k. socha, “an introduction to ant colony optimization”, belgia (be), iridia-technical report series, april 2007 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 52    penarapan metode goal programming untuk mengoptimalkan beberapa tujuan pada perusahaan dengan kendala jam kerja, permintaan dan bahan baku fauziyah universitas pgri adi buana, basta.fauziyah@gmail.com abstrak suatu perusahaan memiliki beberapa tujuan (multi-objective) yang ingin dicapai, seperti memaksimalkan total nilai penjualan, memaksimalkan total produksi dan meminimalkan biaya produksi tanpa harus mengurangi kualitas produk. tujuan tersebut mengandung aspek yang berbeda sehingga sering tidak sejalan antara satu dengan yang lain. untuk memberikan solusi optimal yang merupakan titik temu dari beberapa tujuan yang telah ditetapkan, metode yang digunakan dalam penelitian ini adalah goal programming. penerapan metode goal programming dilakukan dengan bantuan software lindo. hasil perhitungan dengan metode goal programming dapat memaksimumkan total nilai penjualan dan total produksi dengan meminimumkan biaya. hasil penelitian menunjukkan bahwa tujuan yang telah ditetapkan tercapai secara optimal. the company has multiple objectives that want to be achieved, which are maximizing selling, maximizing production and minimizing the cost production without reducing the quality of the product. the objective contains different aspects and so often incompatible with each other. to provide an optimal solution with respect to those objectives of some predetermined goals, the applied method that used is goal programming. the goal programming show that it is run by using the lindo program. the result of the goal programming method can maximize the total selling and total production by minimize cost. the result showed that the define objectives achieved optimally. kata kunci : goal programming, optimasi, kendala 1. pendahuluan 1.1 latar belakang matematika merupakan salah satu bagian dari ilmu pengetahuan yang memiliki peranan penting dalam dunia teknologi dan perusahaan. banyak sekali permasalahan yang dapat dirumuskan (dimodelkan) dan dicari penyelesaiannya melalui perhitungan matematis. dalam hal ini dosen diwajibkan melakukan tri dharma perguruan tinggi, sehingga dituntut untuk menerapkan ilmu yang dimiliki sehingga dapat bermanfaat baik pada perusahaan maupun sektor lain sebagai aplikasi dari matematika. suatu perusahaan terkadang memiliki beberapa permasalahan yang sulit untuk dipecahkan dengan cara sederhana namun setiap perusahaan menginginkan penyelesaian yang tepat guna untuk mendapatkan hasil yang optimal tanpa harus coba-coba. di indonesia banyak terdapat jenis perusahaan, salah satunya adalah perusahaan industri kertas. keunggulan sistem manajemen dan distribusi yang telah diterapkan pada beberapa perusahaan menjadikan suatu perusahaan dapat berkembang pesat seiring dengan kebutuhan konsumen. namun perekonomian di indonesia tidak selalu berjalan mulus, sehingga secara tidak langsung mengganggu aktivitas perusahaan. berdasarkan chowdary & slomp, dalam membuat suatu perencanaan produksi terdapat tiga elemen yang perlu dipertimbangkan, yaitu konsumen, produk dan proses manufaktur ketiga elemen tersebut jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 53    merupakan masalah yang sangat kompleks yang harus dihadapi oleh setiap perusahaan industri. seluruh sumber daya yang dimiliki harus terkoordinir dengan baik sehingga dalam proses produksinya akan menghasilkan produk yang baik dan optimal. dalam menghasilkan produk, setiap perusahaan industri menginginkan kebijakan yang dapat menghasilkan keuntungan tanpa mengesampingkan kebijaksanaan lainnya seperti memaksimalkan total produk dengan keterbatasan bahan baku yang dimiliki, memaksimalkan barang produksi tanpa mengesampingkan jumlah permintaan yang ada di pasaran, dan meminimalkan biaya produksi tanpa mengurangi kualitas produksi. semua tujuan tersebut diharapkan dapat tercapai secara optimal, namun tidak menutup kemungkinan terpenuhi satu tujuan akan mengabaikan tujuan yang lain. hal ini sering terjadi mengingat sumber daya yang dimiliki terbatas, sehingga pemenuhan tujuan secara bersama-sama tidak mungkin tercapai. berdasarkan latar belakang di atas dan tujuan yang ingin dicapai lebih dari satu (multi-objective) maka metode yang digunakan dalam penulisan ini adalah goal programming. metode ini tepat digunakan dalam perencanaan produksi karena potensial untuk menyelesaikan aspek-aspek yang bertentangan antara elemen-elemen dalam perencanaan produksi, yaitu konsumen, produksi, dan proses manufaktur. 1.2 rumusan masalah adapun rumusan masalah yang akan dibahas adalah bagaimana membuat model matematika yang fisibel dengan menerapkan metode goal programming berdasarkan permasalahan yang ada untuk dapat memaksimalkan total nilai penjualan (memaksimalkan keuntungan), memaksimalkan total produksi (volum) dan meminimalkan biaya produksi (tanpa mengurangi kualitas produk). 1.3 tujuan penelitian tujuan yang ingin dicapai dalam penelitian ini adalah menyusun model matematika yang fisibel dengan menerapkan metode goal programming berdasarkan permasalahan yang ada untuk dapat memaksimalkan total nilai penjualan (memaksimalkan keuntungan), memaksimalkan total produksi (volum) dan meminimalkan biaya produksi (tanpa mengurangi kualitas produk). 1.4 manfaat penelitian hasil penelitian ini diharapkan dapat memberikan sumbangan pemikiran dalam perencanaan produksi untuk memaksimalkan total nilai penjualan (memaksimalkan keuntungan), memaksimalkan total produksi (volume) dan meminimalkan biaya produksi dengan menentukan model matematikanya 2. tinjauan pustaka 2.1 permasalahan optimasi dan program linear (linear programming) masalah optimasi merupakan masalah memaksimumkan atau meminimumkan sebuah besaran tertentu yang disebut tujuan objektif (objective) yang bergantung pada sejumlah berhingga variabel masukan (input variabels). variabel-variabel ini dapat independen maupun dependent melalui satu atau lebih kendala (constraints). persoalan optimasi merupakan persoalan mencari nilai numerik terbesar (maksimasi) atau terkecil (minimasi) yang mungkin dari sebuah fungsi dan sejumlah variabel tertentu. persoalan optimasi tersebut dapat diselesaikan dengan menggunakan program linear. program linear merupakan salah satu metode matematika yang berkarakteristik linear untuk menemukan suatu penyelesaian optimal dengan cara memaksimumkan atau meminimumkan fungsi tujuan terhadap susunan kendala [3]. karakteristik linear yang dimaksud adalah seluruh fungsi model matematika harus berupa kombinasi linear. secara umum persoalan program linear dengan variabel keputusan nxxxx ,...,,, 321 dapat dirumuskan dalam suatu model matematika sebagai berikut : (a) memaksimumkan fungsi tujuan     n j jj nnjj xc xcxcxcxc 1 2211 ... ... z (1) jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 54    dengan pembatasnya (fungsi kendala) 1 n ij j i j a x b   untuk 1,2,3,…, dan , ,…, 0 atau ,0jx dengan nj ...,,3,2,1 . (b) meminimumkan fungsi tujuan     n j jj nnjj xc xcxcxcxc 1 2211 ... ... z (2) dengan fungsi kendala 1 n ij j i j a x b   untuk 1,2,3,…, dan , ,…, 0 atau ,0jx dengan nj ...,,3,2,1 . keterangan: jx = variabel pengambilan keputusan ke – j. jc = koefisien fungsi tujuan ke – j. ib = kapasitas kendala ke – i. ija = koefisien fungsi kendala ke – i untuk variabel keputusan ke – j. 2.2 goal programming metode goal programming merupakan perluasan dari model program linear. goal programming diperkenalkan oleh charles dan cooper pada awal 1960. teknik ini disempurnakan oleh ijiri pada pertengahan 1960 dan penjelasan yang lengkap pada beberapa aplikasi dikembangkan oleh ignizo dan leen pada 1970. karena goal programming merupakan perluasan dari program linear sehingga seluruh asumsi, notasi, formula matematika, prosedur perumusan model dan penyelesaiannya tidak berbeda. perbedaan utamanya terletak pada struktur dan penggunaan fungsi tujuan. dalam program linear hanya mengandung satu fungsi tujuan sedangkan dalam goal programming terdapat satu atau beberapa gabungan fungsi tujuan. hal ini dapat dilakukan dengan mengekspresikan tujuan itu dalam bentuk sebuah kendala (goal constraint). memasukkan variabel simpangan (deviational variabel) dalam kendala tersebut untuk mencerminkan seberapa jauh tujuan itu tercapai dan menggabungkan variabel deviasional dalam fungsi tujuan. tabel 1. model matematika goal programming tipe fungsi kendala pada program linear model matematika goal programming variabel deviasio ner yang diminim umkan 1)(1 1)(1 1)(1 bxf bxf bxf    111)(1 111)(1 111)(1 bddxf bddxf bddxf       1,1 1 1 dd d d untuk metode goal programming paling sedikit memiliki tiga komponen yaitu: fungsi tujuan, kendala sasaran dan kendala nonnegatif. 2.3 bentuk umum model matematis metode goal programming secara umum model matematis metode goal programming dapat dirumuskan sebagai berikut : mencari  jxxxx ...,,, 21 yang meminimumkan fungsi tujuan                     kkkk ddgp ddgpddgp z ,....., ,...,,, 22221111 (3) fungsi kendala mbmdmdnxmnaxmaxma bddnxnaxaxa bddnxnaxaxa    ...2211 2222...222121 1111...212111  dan jx ,  id dan 0  id untuk i = 1, 2, ..., m dengan jx = variabel keputusan ke – j. ib = kapasitas kendala ke – i. ija = parameter fungsi kendala ke – i untuk variabel keputusan ke – j. = jumlah seluruh tingkat prioritas yang ada pada model. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 55      kkk ddg , = fungsi linear dari variabelvariabel keputusan. pk = prioritas yang sesuai dengan   kkk ddg , 3. metode penelitian penelitian ini menggunakan studi kasus pada pt gangsar jaya untuk penerapan metode goal programming. pt gangsar jaya merupakan salah satu perusahan industri kertas yang ada di mojokerto. data yang diambil terkait dengan pemaksimalan keuntungan, pemaksimalan total produksi, dan peminimalan biaya produksi. untuk menemukan solusi yang optimal dari permasalahan tersebut penulis menggunakan bantuan program komputer yaitu program lindo. untuk mencapai tujuan penelitian yang ditetapkan, disusun prosedur penelitian yang dijelaskan pada gambar 1. gambar 1 prosedur penelitian 4. analisi data dan pembahasan 4.1 hasil penelitian pt gangsar jaya merupakan anak cabang dari pt tjiwi kimia, sehingga proses produksi perusahaan ini hanya mengolah bahan baku yang berasal dari pt tjiwi kimia. produk jadi hasil olahan berupa buku tulis, buku gambar dan tas kertas akan dikirim lagi ke pt tjiwi kimia untuk didistribusikan ke konsumen. dalam proses produksi setiap jenis barang pada pt gangsar jaya perlakuanya berbeda. untuk produksi buku tulis, lembaran kertas plano dimasukan ke dalam mesin pemotong dan dipotong sesuai ukuran kemudian dirapikan (sorting process) dan diberi sampul (cover). setelah itu dimasukkan ke dalam mesin penjilidan dan yang terakhir proses pelipatan secara manual oleh manusia. untuk produksi buku gambar prosesnya hampir sama, yaitu dari lembaran kertas plano dipotong sesuai ukuran kemudian diberi sampul (cover) dan dilipat secara manual. setelah itu proses selanjutnya adalah penjilidan dan yang terakhir dirapikan (sorting process). sedangkan proses produksi pada tas kertas dimulai dari lembaran kertas plano yang dipotong sesuai ukuran kemudian dimasukkan pada mesin pembuatan pola, proses terakhir pelekatan masing-masing bagian dan diberi tali untuk pegangan, sehingga terbentuk tas yang dilakukan secara manual. secara keseluruhan data dari proses produksi dan kondisi pada pt gangsar jaya disajikan secara singkat melalui tabulasi berikut ini dengan perincian data untuk setiap seratus unit. 4.1.1 jenis produksi pt gangsar jaya dalam produksinya menghasilkan beberapa jenis produk. adapun data produk tersebut dijelaskan pada tabel 2. 4.1.2 data kebutuhan jam kerja kapasitas jam kerja setiap harinya 7 jam per mesin dan setelah waktu tersebut semua mesin yang ada sudah tidak beroperasi lagi. pt gangsar jaya memiliki dua mesin pemotong, dua mesin penjilidan, dan dua mesin pembuat pola. waktu pemakaian tiga studi pustaka pengamatan identifikasi masalah perumusan tujuan pengumpulan data mengidentifikasi kendala-kendala pada perumusan model matematika dengan menggunakan metode goal programming pengolahan data solusi kesimpulan dan saran jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 56    jenis mesin pt gangsar jayapada saat beroperasi dijelaskan tabel 3 dan tabel 4. tabel 2. jenis poduksi pt gangsar jaya no produk 1 buku gambar ukuran 2 buku tulis 3 tas kertas tabel 3. data kebutuhan jam kerja pt gangsar jaya no produk (tiap seratus unit) kebutuhan jam kerja (jam) 1 buku gambar 7,16 2 buku tulis 16,87 3 tas kertas 10,94 tabel 4. kebutuhan waktu pemakaian mesin pt gangsar jaya dalam jam no produk (tiap 100 unit) mesin pemotong mesin penjilid mesin pembuat pola 1 buku gambar 0,024 0,096 2 buku tulis 0,036 0,144 3 tas kertas 0,048 0,068 4.1.3 kebutuhan bahan baku kebutuhan bahan baku pada pembuatan produk yang dihasilkan pt gangsar jaya untuk setiap harinya dijelaskan pada tabel 5. tabel 5. kebutuhan bahan baku pt gangsar jaya no bahan baku jumlah bahan baku (kg) 1 plano untuk buku gambar 3.000 2 plano untuk buku tulis 3.000 3 plano untuk tas kertas 3.000 tabel 6. kapasitas produksi pt gangsar jaya no produk kapasitas produksi 1 buku gambar 19.500 2 buku tulis 16.500 3 tas kertas 90.000 4.1.4 kapasitas produksi data kapasitas produksi yang diperoleh setiap hari dijelaskan pada tabel 6. 4.1.5 estimasi permintaan berdasarkan dari data yang diperoleh, estimasi permintaan setiap hari dijelaskan pada tabel 7. tabel 7. data estimasi permintaan pt gangsar jaya no produk estimasi permintaan 1 buku gambar 16.500 2 buku tulis 12.000 3 tas kertas 80.000 4.1.6 data harga jual dan biaya produksi harga jual produk pt gangsar jaya ke pasaran dan biaya produksi tiap seratus produk dijelaskan pada tabel 8. tabel 8. harga jual dan biaya produksi pt gangsar jaya no nama produk harga jual (rp) biaya produksi(rp) 1 buku gambar 450.000 423.000 2 buku tulis 650.000 582.500 3 tas kertas ukuran s 600.000 552.500 4.1.7 data rencana anggaran anggaran merupakan ketetapan perusahaan yang harus dicapai setiap hari. detail rencana anggaran dijelaskan pada tabel 9. tabel 9. rencana anggaran pt gangsar jaya setiap hari no anggaran jumlah (rp) 1 total nilai penjualan 156.500.000,00 2 total produksi 108.500 3 biaya produksi 143.000.000 jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 57    4.2 analisi data disusun model matematika berdasarkan metode goal programming dengan data yang didapat pada sub bab 4.1 4.2.1 variabel keputusan terdapat tiga variabel keputusan yang digunakan dalam penelitian ini yang dijelaskan pada tabel 10. tabel 10. variabel keputusan pada produk yang dihasilkan no produk variabel keputusan 1 buku gambar 2 buku tulis 3 tas kertas 4.2.2 fungsi pencapaian tujuan dan model matematika fungsi pencapaian tujuan merupakan sekumpulan fungsi tujuan di setiap kendala, sedangkan model matematika yang dimaksud adalah sekumpulan dari fungsi kendala dan fungsi tujuan yang telah diformulasikan. capaian dalam penelitian ini adalah mencapai total nilai penjualan yang maksimal, mencapai jumlah produksi yang maksimal dan mencapai anggaran biaya yang tersedia. adapun fungsi pencapaian dan model matematika tersebut dengan tiga variabel keputusan, 26 variabel deviasional, 13 kendala dan empat prioritas akan dijelaskan pada subbab 4.2.2.1 dan 4.2.2.2. 4.2.2.1 minimize: 1 1 2 3 2 4 3 5 4 6 1 2 3 4 5 5 6 1 6 7 7 8 9 10 2 11 3 12 4 13 ( )z p p p p p p p p a a a a a a d d d d d d d d d d d d d d d d                                           (4) 4.2.2.2 kendala terdapat tiga kendala dalam permasalahan ini yaitu kendala jam kerja, kendala permintaan dan kendala bahan baku. 350094,1087,1616,7 11321   ddxxx 1 2 2 2 2 2 3 2 2 0,024 0,036 0,048 14; 14; 14; x x x d d d d d d                1 3 3 2 3 3 0,096 0,144 14; 14; x x d d d d           3 4 4 0,068 14;x d d    1 5 5 2 6 6 3 7 7 165; 120; 800; x x x d d d d d d                1 8 8 2 9 9 3 10 10 3.000; 3.000; 3.000; x x x d d d d d d                1 2 3 11 11 450 156.500.000 650 600 ; x x x d d       1 2 3 12 12 1.085;x x x d d      1 2 3 13 13 423 582,5 552,5 143.000.000; x x x d d       variabel buku gambar variabel buku tulis variabel tas kertas variabel deviasional menampung penyimpangan nilai di bawah sasaran variabel deviasional menampung penyimpangan nilai di atas sasaran 1,2,3,…,13 nilai , , , dan bukan merupakan nilai parameter tapi hanya menunjukkan bahwa prioritas lebih penting dari pada . dimana merupakan prioritas pertama dengan kendala berupa jam kerja, jumlah permintaan dan bahan baku, merupakan prioritas kedua dengan tujuan mencapai total nilai penjualan, merupakan prioritas ketiga dengan tujuan mencapai total produksi, dan yang merupakan prioritas terakhir dengan tujuan tercapainya anggaran yang tersedia. jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 58    4.3.pembahasan berdasarkan model matematika yang telah dibentuk dari data yang telah didapatkan pada pt gangsar jaya maka akan dicari solusi optimal dengan meminimalkan simpangan fungsi pencapaian pada kendala sistem maupun kendala tujuan. model matematika yang telah diperoleh sudah dalam bentuk kanonik, sehingga dapat tabel 11. solusi optimal prioritas sasaran hasil yang diinginkan hasil yang didapat keterangan i ii iii iv kendala sistem berupa a. jam kerja 1) tenaga kerja langsung 2) mesin pemotong 3) mesin penjilidan 4) mesin pembuat pola b. permintaan 1) buku gambar 2) buku tulis 3) tas kertas c. bahan baku 1) plano untuk buku gambar 2) plano untuk buku tulis 3) plano untuk tas kertas total nilai penjualan total produksi biaya produksi 3.500 14 14 14 16.500 12.000 80.000 3.000 3.000 3.000 156.500.000 108.500 143.000.000 3.361,52 13,056 13,68 13,651 16.500 12.000 80.300 3.000 3.000 3.000 156.650.000 108.800 142.585.000 tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai tercapai langsung diselesaikan dengan program lindo. program lindo menggunakan prinsip algoritma simplek untuk menghasilkan penyelesaian optimal, dari hasil running menggunakan program tersebut didapat penyelesaian optimal setelah iterasi ke-12 0 didapat setelah iterasi ke-12). hasil ini sesuai dengan penjelasan dari algoritma simplek, jika 0 untuk j maka peluang menurunkan nilai z pada iterasi berikutnya tidak ada dan dikatakan bahwa hasil telah optimal. dari perhitungan menggunakan program lindo, didapatkan hasil yang dijelaskan pada tabel 11. dari tabel 11 dapat diambil kesimpulan sebagai berikut : prioritas i : pada prioritas pertama target produksi untuk memenuhi permntaan konsumen dan persediaan bahan baku terpenuhi namun untuk jam kerja yang dihasilkan masih dibawah target yang ada. dari tabel 11 bahan baku yang dianggarkan , semuanya terpakai untuk produksi buku tulis, buku gambar dan tas kertas. produksi barang untuk memenuhi permintaan konsumen jumlahnya berlebih 300 buah pada produksi tas kertas sedangkan produk buku tulis dan buku gambar jumlahnya sama seperti permintaan konsumen. prioritas ii : pada prioritas kedua untuk mencapai total nilai penjualan pt gangsar jaya telah terpenuhi. hasil yang didapat dari metode goal programming lebih besar dari pada hasil yang diinginkan, besarnya seleisih antara hasil yang didapat dan hasil yang diinginkan untuk nilai penjualan adalah rp. 150.000,00 sehingga perusahaan mendapatkan keuntungan yang lebih. prioritas iii : pada prioritas ketiga, sasaran untuk mencapai total produksi telah terpenuhi. kombinasi produk hasil optimasi goal programming memiliki jumlah yang sama dengan estimasi permintaan untuk setiap jenis produknya, kecuali untuk hasil produksi tas kertas yang jurnal matematika “mantik” edisi: oktober 2016. vol. 02 no. 01 issn: 2527-3159 e-issn: 2527-3167 59    melebihi permintaan pasar yatu sebesar 300 buah. prioritas iv : pada prioritas yang terakhir, untuk mencapai target anggaran biaya yang tersedia telah terpenuhi. tidak hanya itu, anggaran biaya produksi yang didapat dengan metode goal programming lebih kecil bila dibandingkan dengan anggaran biaya produksi yang tersedia. besarnya perbedaan anggaran yang didapat dengan metode goal programming dan hasil yang diinginkan adalah rp 415.000,00. hal ini menyebabkan pt gangsar jaya dapat mengalokasikan sisa anggaran kebagian lain. berdasarkan tabel 11 dan dari uraian prioritas, dapat diketahui bahwa tujuan yang ingin dicapai pt gangsar jaya semua terpenuhi dengan tidak mengabaikan prioritas yang telah ditetapkan. dalam sehari pt gangsar jaya dapat: 1 memaksimalkan total nilai penjualan sebesar rp 156.650.000,00 2 memaksimalkan total produksi sebanyak 108.800 buah. 3 meminimalkan biaya produksi sebesar rp 142.585.000,00 dengan kombinasi produk sebagai berikut : a) buku gambar : 16.500 buah b) buku tulis : 12.000 buah c) tas kertas : 80.300 buah 5. simpulan berdasarkan tujuan dan permasalahan yang ada pada pt gangsar jaya serta hasil analisa dari penelitian yang dilakukan, maka dapat disimpulkan berikut: 1. dengan menggunakan model matematika goal programming, semua sasaran pada masing-masing prioritas terpenuhi namun nilai pencapaian yang telah didapat untuk kapasitas produksi dan jam kerja masih di bawah target yang ada. 2. perusahaan harus memproduksi barang pada kondisi yang sedang terjadi adalah sebagai berikut:  buku gambar : 16.500 buah  buku tulis : 12.000 buah  tas kertas : 80.300 buah dapat diketahui bahwa pt gangsar jaya dalam sehari dapat memaksimalkan total nilai penjualan yaitu sebesar rp 156.650.000; dapat memaksimalkan total produksi sebanyak 108.800 buah dengan kombinasi produk sama dengan permintaan pasar yang diinginkaan dan dapat meminimalkan biaya produksi sebesar rp 142.585.000 referensi [1] ravindran, phillips & solberg. operation research principles and practice. new york. john wiley & sons (2000) [2] sakawa, masathosi. fuzzy sets and interactive multiobjective optimization. new york: plenum press (1993) [3] siswanto. operation research jillid i. bagor: erlangga (2006) [4] spronk, jaap. interactive multiple goal programming. london. martinus nijhoff publishing (1981) [5] taha, hamdy. riset operasi suatu pengantar (edisi kelima) jilid 1. jakarta: binaputra aksara (1996) paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: suparna, “analysis of main components status of food security at village/sub-district level in yogyakarta special region”, jmm, vol. 6, no. 1, pp. 30-37, may 2020. analysis of main components status of food security at village/sub-district level in yogyakarta special region suparna badan pusat statistik provinsi d.i. yogyakarta, suparna@bps.go.id doi: https://doi.org/10.15642/mantik.2020.6.1.30-37 abstrak: pangan merupakan aspek pokok dari kebutuhan hidup manusia untuk menjamin keberlangsungan hidup individu maupun komunitas. perwujudan ketahanan pangan nasional dimulai dari pemenuhan pangan di wilayah terkecil yaitu desa/kelurahan. tujuan dari analisis dengan metode komponen utama ini adalah (1) mendeskripsikan komponen utama status ketahanan pangan pada tingkat desa/kelurahan; (2) mengelompokkan desa/kelurahan berdasarkan status ketahanan pangan di daerah istimewa yogyakarta. sumber data berasal dari data sekunder (podes 2018). dari analisis dihasilkan beberapa hal yakni: (1) komponen utama status ketahanan pangan pada tingkat desa/kelurahan di daerah istimewa yogyakarta ada lima, yaitu keterjangkauan wilayah/akses, ketersediaan pangan, kesehatan lingkungan, jaminan akses, dan pemanfaatan pangan; (2) tipologi desa/kelurahan berdasarkan status ketahanan pangan di daerah istimewa yogyakarta ada 4 yakni: (a) rawan pangan meliputi 55 desa/kelurahan (b) kurang tahan meliputi 169 desa/keluarahan; (c) tipologi 3 rentan tahan meliputi 170 desa; (d) tipologi 4 tahan pangan meliputi 44 desa. kata kunci: komponen utama; ketahanan pangan; daerah istimewa yogyakarta; desa abstract: food is a basic aspect of the needs of human life to ensure the survival of individuals and communities. the realization of national food security starts from the fulfillment of food in the smallest region, namely the village /sub-districts. the objectives of the analysis using the principal component method are (1) to describe the main components of the status of food security at the village/sub-districts level; (2) to grouping villages/sub-districts based on food security status in the yogyakarta special region. the data source of analysis comes from secondary data (podes 2018). from the analysis produced several things, namely: (1) the main components of the status of food security at the village/sub-district level in the yogyakarta special region there are five, namely affordability/access, food availability, environmental health, guaranteed access, and utilization of food; (2) village/sub-districts typology based on the status of food security in the yogyakarta special region, namely 4: (a) food insecurity covering 55 villages/sub-districts (b) less resistant to 169 villages/sub-districts; (c) typologies 3 are vulnerable to cover 170 villages/sub-districts; (d) foodresistant typology covering 44 villages/sub-districts. keywords: principal component; food security, yogyakarta special region; village jurnal matematika mantik vol. 6, no. 1, may 2020, pp. 30-37 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 suparna analysis of main components status of food security at village/sub-district level in yogyakarta special region 31 1. introduction food insecurity is still a global issue which is the main concern to date, not only in poor and developing countries but also in developed countries [1][2][3]. the importance of the problem of food insecurity is the main point discussed at the world level meeting set forth in the sustainable development goals (sdg’s). in developing countries, more than half of household income is used to meet their food needs and this causes a precarious situation if there are sudden price fluctuations that can push people into poverty and impede poverty alleviation efforts [4][5]. the food and agriculture organization estimates that 1.5 billion people in the world are affected by one or more forms of micronutrient deficiency[6]. iron deficiency in women of reproductive age is a form of micronutrient deficiency that can occur also in women who are overweight or well cared for. the high prevalence of each form of deficiency or malnutrition in the form of short children is found in 73 countries; lean children in 14 countries; overweight children in 29 countries; adult obesity in 101 countries; and anemia in women of reproductive age in 35 countries. the prevalence threshold that is considered high for short children is 20 percent or more; for thin children and overweight children, the threshold is 10 percent or more. indonesia is the only country that shows a high prevalence of these three forms of child malnutrition. yogyakarta special region is the province with the highest percentage of poor population in java, which is still 11.81 percent and still relies on the agricultural sector as the main axis of its economy. this is evident from the contribution of the agricultural sector in the order of the four major regional gross domestic product (grdp) and the proportion reached 9.78 percent in 2018 [7]. bearing in mind that food security as a human right and limited food production greatly affect the achievement of food security at the community level, it is necessary to further study food security at the village /sub-district level in order to give birth to efforts in achieving village/sub-district to become food self-sufficient. the embodiment of national food security starts from the fulfillment of food in the smallest region, namely villages as the basis of agricultural activities [8]. in addition, the village is also an entry point for the entry of various programs that support the realization of food security at the household level which cumulatively supports the realization of food security at the district /city, provincial, and national levels. food security is multidimensional, both in terms of supply and utilization, and regional levels [9][10][11]. there are many variables that explain the number, so we need indicators that explain the main dimensions of food security. the method used is mostly in the form of principal component analysis [12][13][14]. 2. methods 2.1. variable and component analysis achieving food security is a guarantee against the threat of hunger and malnutrition, both of which can slow down economic development [15][16]. thus poverty and food security are interrelated [17]. food system resilience is basically to ensure adequate and access to food for everyone. sufficiency that is meant is sufficiency in quantity and quality with access including economic and physical access [18][19]. there are three main dimensions of food security delivered by the food agricultural organization (fao), namely the availability, access and utilization of food. principal component analysis is a statistical analysis tool that aims to reduce the dimensions of the data by generating new variables (main components) which are linear combinations of the original variables so that the main component variances are maximum and the main components are mutually independent [20]. the principal component analysis model can be written with the following matrix. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 30-37 32 [ 𝑌1 𝑌… 𝑌𝑚 ] = [ 𝑎11 𝑎1.. 𝑎1𝑝 𝑎1… 𝑎… 𝑎…𝑝 𝑎𝑚1 𝑎𝑚… 𝑎𝑚𝑝 ] ⌈ 𝑋1 𝑋… 𝑋𝑝 ⌉ (1) in equation (1): y1 = the first major component, the component that has the largest variance y .. = the second main component and so on, the component that has the second largest variance and so on ym = the m-th main component, the component that has the m-th largest variance x1 = first origin variable x .. = second origin variable and so on xp = p-origin variable m = number of main components p = number of original variables. the main components do not correlate and have the same variation with the root characteristic of σ. the root characteristic of the diversity matrix σ is a variant of the main component y, so the diversity matrix of y is: ∑ = [ 1 0 0 0 … 0 0 0 𝑝 ] (2) the total diversity of origin variables will be the same as the total diversity explained by the main components, namely: ∑ 𝑣𝑎𝑟(𝑋𝑖) = 𝑝 𝑖=1 1 + … + 𝑝 = ∑ 𝑣𝑎𝑟(𝑌𝑖) 𝑚 𝑖=1 (3) depreciation of the dimensions of the original variables is done by taking a small number of components that are able to explain the largest part of the diversity of data. if the main component is taken as many as q component, where q <p, then the proportion of total diversity that can be explained by the i-th major component is: 𝑖 1+2+⋯+𝑝 (4) with i=1,2,…,p if the variable units covered are not the same, it is necessary to standardize before the principal component analysis is performed. due to the standardization of this data, the covariance matrix variance of the standardized data will be the same as the data correlation matrix before it is standardized and the magnitude of the total variance of the main components is equal to the number of origin variables (p). the number of main components (m) can be determined by various criteria, one of the criteria commonly used is to use the criterion of the magnitude of the main component variance. 2.2. research methods the method used in this study is the analysis of the main components using food security indicators that refer to the variables in the classification used in making food security and vulnerability atlas (fsva) maps in 2009 and 2015. typologies are formed based on the results of the main component scores, which will show the food security status of a village/kelurahan. suparna analysis of main components status of food security at village/sub-district level in yogyakarta special region 33 broadly speaking, the stages in analyzing the principal components are as follows: a. determine what variables are eligible for further analysis. through the information obtained based on the 2018 podes data collection, the variables considered relevant are: x1 = the ratio of food stalls / food stalls per 1.000 households x2 = ratio of shops / grocery stalls per 1.000 households x3 = percentage of agricultural land to total village/sub-district area x4 = percentage of population receiving jamkesmas (pbi) x5 = sktm ratio (certificate of incapacity) per 1.000 population x6 = percentage of poor households (heads of households) according to the village head/lurah x7 = non-electricity consumption ratio per 10.000 families x8 = index of access of village main roads is inadequate x9 = distance of village/village office to nearest main sub-district office x10 = index of drinking and bathing/washing water sources x11 = index of final disposal of feces and liquid waste x12 = ratio of health workers per 10.000 population x13 = ratio of malnutrition residents per 10.000 population x14 = active posyandu (integrated healthcare centre) ratio per 10.000 population x15 = ratio of slums to 10.000 families b. calculate the correlation with the bartlett test of spericity and msa (measure of sampling) measurements. c. extraction. the component extraction method used in this study is the pca (principal component analysis) method. determination of the number of components is based on the amount of eigen value of each component that appears. eigen value is the number of variants explained by each component. 3. result and discussion 3.1. test of data assumptions kaiser-meyer-olkin (kmo) of data adequacy test is conducted to determine whether the data to be used meets the requirements for component analysis or not, while the bartlett test shows whether there is correlation between variables. measure of sampling adequacy (msa) is also used to determine whether the variables are sufficient for further analysis. based on the test results obtained a kmo value of 0,744 or it can be said that the data is sufficient for further analysis (greater than 0.5). the results of the bartlett test are also significant, with a p-value smaller than the significance level (α) of 0,05, which means that there are correlations between variables. the results of the measure of sampling adequacy (msa) can be seen from the value of the anti-image correlation matrix. if the msa value is greater than 0,5 then the variable is sufficient for further analysis. if there is an msa value of initial variables less than 0,5 must be excluded one by one from the analysis, sorted from the variable whose msa value is the smallest and not used again in subsequent analyzes. 3.2. principal component analysis result by using the principal component analysis method, the communality value is obtained. the value of communalities formed shows how much diversity of variables can be explained by the components formed. based on the extraction value obtained from the results of the analysis of the main components shows that for the variable x1 = ratio of stalls/ food stalls per 1.000 households by 70,4 percent the diversity of these variables can be explained by the components formed, jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 30-37 34 while for the diversity of the variable x2 = ratio of shops/ grocery stalls per 1.000 households can be explained by the components formed by 76,1 percent. diversity x3 = percentage of agricultural land to total village/ sub-district area explained by the component formed by 55,2 percent, and so on up to 64,0 percent diversity x14 = the ratio of active posyandu per 10.000 population is explained by the component formed. finally, for diversity x15 = the ratio of slum families per 10.000 families is explained by the component formed by 42,6 percent. furthermore, it can be seen how many components are formed based on eigenvalues (eigenvalues). components which are stated to represent other variables are components that have an eigenvalue of more than 1 (one). the results of the eigenvalues of each component are listed in table 1. based on table 1. it can be seen that more than one eigenvalue there are 5 components. the first component has an eigenvalue of 2,935 and component five has an eigenvalue of 1,004. of the thirteen variables can be reduced and formed into five components in which the five components are able to explain 56,56 percent of total diversity. table 1. eigenvalues of each component total variance explained component eigen values total % of variance cumulative % 1 2,935 22,575 22,575 2 1,295 9,965 32,540 3 1,095 8,424 40,964 4 1,024 7,875 48,839 5 1,004 7,722 56,561 extraction method: principal component analysis. for component 1 which can explain the greatest variance can be perceived as a component of regional affordability / access, while in component 2 it can be described as food availability. component 3 can be called environmental health, component 4 is called guaranteed access, and component 5 is food use. this is formed based on the largest correlation of each variable to the existing components based on rotated components that are also obtained. the results of the analysis of the main components show that the high correlation variable with component 1, namely x3 = percentage of agricultural land to the total area of the village /sub-district has a correlation of 0,608; x6 = percentage of poor households according to the village / lurah head who has a correlation of 0,662; x8 = inadequate access road index for the main village has a correlation of 0,638; and x9 = distance of the village/village office to the nearest main sub-district office has a correlation of 0,607. high variable correlation with component 2, namely x1 = ratio of stalls / food stalls per 1.000 households has a correlation of 0,736; x2 = ratio of shops / grocery stalls per 1.000 households and x12 = ratio of health workers per 10.000 residents, each of which has a correlation of 0,860 and 0,430. likewise, component 3 consists of variables x11 = index of final disposal of sewage and liquid waste and x14 = ratio of active posyandu per 10.000 population. component 4 consists of x5 = sktm ratio per 1.000 inhabitants and x10 = index of drinking and bathing / washing water sources, while component 5 consists of x7 = ratio of non-electricity users per 10.000 families and x15 = ratio of slum families per 10.000 families. 3.3. village/sub-district typology the objects that are grouped are villages /sub-district throughout the yogyakarta special region using information on the results of the analysis of the main components. suparna analysis of main components status of food security at village/sub-district level in yogyakarta special region 35 village /sub-district grouping is done based on food security index which is the sum of scores of all main components with the same weight. the grouping process uses statistical considerations with group limits in the form of an average plus minus one standard deviation. the number of groups obtained was four groups. this is with consideration of the number of aspects of food security, amounting to four (food resistant, susceptible resistant, less resistant, and food insecurity). food resistance if the combined component score is more than the average plus one standard deviation, vulnerable to hold if the score is between the average added one standard deviation, less food resistance if the score is between the average minus one standard deviation, and food insecurity if the score smaller than the average minus one standard deviation. based on table 2 it can be shown that there are 55 villages/sub-district in the special region of yogyakarta (12,6%) which can be called a food insecure area. meanwhile, there are 44 regions (10%) of truly food-resistant villages/sub-district, while 39% of foodresistant and vulnerable villages /sub-district are vulnerable. if we pay attention per district/city, villages/sub-district that are food insecure, especially in gunungkidul regency, have 22 villages. then followed in bantul regency as many as 17 villages and in the city of yogyakarta as many as 14 subdistricts. for villages/sub-district that are categorized as food resistant, mainly in the regencies of gunungkidul, kulonprogo and sleman. table 2. typology of villages/sub-district based on food security status in yogyakarta special region, 2018 4. conclusions based on the results of the analysis that has been done there are several things that can be concluded, namely: a. analysis of the main components reduces from thirteen initial variables to five new components. component one describes area / access affordability, while component 2 can be described as food availability. component 3 can be called environmental health, component 4 is called guaranteed access, and component 5 is food use. b. the results of the typology of villages/ wards in the yogyakarta special region get 55 villages/ sub-district in the yogyakarta special region (12,6%) can be called a food insecure area. meanwhile, there are 44 regions (10%) of truly food-resistant villages/sub-district, while 39% of food-resistant and vulnerable villages/sub-district are vulnerable. if we pay attention per district/ city, villages/sub-district that are food insecure, especially in gunungkidul regency, have 22 villages. then followed in regency/city typology total food insecurity less resistant susceptible resistant food resistant 01 kulonprogo n 2 37 39 10 88 % 2,3% 42,0% 44,3% 11,4% 100,0% 02 bantul n 17 41 16 1 75 % 22,7% 54,7% 21,3% 1,3% 100,0% 03 gunungkidul n 22 53 49 20 144 % 15,3% 36,8% 34,0% 13,9% 100,0% 04 sleman n 0 19 57 10 86 % 0,0% 22,1% 66,3% 11,6% 100,0% 71 yogyakarta n 14 19 9 3 45 % 31,1% 42,2% 20,0% 6,7% 100,0% 34 d.i. yogyakarta n 55 169 170 44 438 % 12,6% 38,6% 38,8% 10,0% 100,0% jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 30-37 36 bantul regency as many as 17 villages and in the city of yogyakarta as many as 14 villages. for villages/kelurahan that are categorized as food resistant, mainly in the regencies of gunungkidul, kulonprogo and sleman. there are several things that can be suggested, namely: a. to increase the affordability of the area/access, it is necessary to optimize the existing irrigation channels or build irrigation channels in order to increase the area of irrigated land which in the end will increase the productivity of agricultural land. efforts are needed to reduce or even prevent the conversion of productive agricultural lands. as well as providing production incentives for farmers to have the motivation to continue producing. need better and serious attention to empowerment programs and increasing community income through comprehensive programs involving all relevant regional apparatus organizations. b. to increase food availability, it is necessary to increase the number of shops/grocery stalls and food stalls/food stalls so that people can easily access food. utilization of cooperatives or the establishment of shops as food providers so that they can reach all regions of yogyakarta special region. references [1] p. conceição, s. levine, m. lipton, and a. warren-rodríguez, “toward a food secure future: ensuring food security for sustainable human development in subsaharan africa,” food policy, vol. 60, pp. 1–9, 2016, doi: 10.1016/j.foodpol.2016.02.003. [2] b. häsler et al., “using participatory rural appraisal to investigate food production, nutrition and safety in the tanzanian dairy value chain,” glob. food sec., vol. 20, no. september 2018, pp. 122–131, 2019, doi: 10.1016/j.gfs.2019.01.006. [3] trussell trust, “the state of hunger: introduction to a study of poverty and food insecurity in the uk,” no. november, pp. 1–32, 2019. [4] a. ickowitz, b. powell, d. rowland, a. jones, and t. sunderland, “agricultural intensification, dietary diversity, and markets in the global food security narrative,” glob. food sec., vol. 20, no. november 2018, pp. 9–16, 2019, doi: 10.1016/j.gfs.2018.11.002. [5] p. v preckel and t. w. hertel, “poverty analysis using an international crosscountry demand system,” 2007. [6] fao, ifad, unicef, wfp, and who, food security and nutrition in the world the state of building climate resilience for food security and nutrition. 2018. [7] bps d.i. yogyakarta, “daerah istimewa yogyakarta dalam angka 2018,” p. 464, 2018. [8] k. nainggolan,”program akselerasi pemantapan ketahanan pangan berbasis pedesaan. 2006, [online]. available: http://pse.litbang_deptan.go.id/pdffiles/pros_kaman_06.pdf. [9] a. suryana, “menuju ketahanan pangan indonesia berkelanjutan 2025: tantangan dan penanganannya,” forum penelit. agro ekon., vol. 32, no. 2, p. 123, 2014, doi: 10.21082/fae.v32n2.2014.123-135. [10] k. thome, m. d. smith, k. daugherty, n. rada, c. christensen, and b. meade, “international food security, 2019-2029,” no. gfa-30, p. 65, 2019, [online]. available: www.ers.usda.gov. [11] c. béné et al., “understanding food systems drivers: a critical review of the literature,” glob. food sec., vol. 23, no. april, pp. 149–159, 2019, doi: 10.1016/j.gfs.2019.04.009. 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[14] m. g. thorpe, c. m. milte, d. crawford, and s. a. mcnaughton, “a comparison of the dietary patterns derived by principal component analysis and cluster analysis in older australians,” int. j. behav. nutr. phys. act., vol. 13, no. 1, pp. 1–14, 2016, doi: 10.1186/s12966-016-0353-2. [15] a. davies, “food security initiatives in nigeria: prospects and challenges.,” j. sustain. dev. africa, vol. 11, no. 1, pp. 186–202, 2009, [online]. available: http://www.cabdirect.org/abstracts/20103307624.html. [16] s. c. omar, “ensuring food security during the covid-19 pandemic,” no. april, 2020. [17] oecd, “agriculture: achieving greater food security,” march, 2015. [18] d. m. tendall et al., “food system resilience: defining the concept,” glob. food sec., vol. 6, pp. 17–23, 2015, doi: 10.1016/j.gfs.2015.08.001. [19] w. a. and m. h. amjad a.h., maqbool h.s., “assessing food insecurity trends and determinants by using mix methods in pakistan: evidence from household pooled data (20052014),” sarhad j. agric., vol. 35, no. i, pp. 87–101, 2019, doi: http://dx.doi.org/10.17582/journal.sja/2019/35.1.87.101. [20] w. härdle and l. simar, “applied multivariate statistical analysis,” appl. multivar. stat. anal., no. april, 2003, doi: 10.1007/978-3-662-05802-2. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: a. nurhasanah, m. manaqib, and i. fauziah, “analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method”, j. mat. mantik, vol. 6, no. 1, pp. 52-65, may 2020. analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method ana nurhasanah1, muhammad manaqib2, irma fauziah3 1uin syarif hidayatullah jakarta, ana.nurhasanah14@mhs.uinjkt.ac.id 2uin syarif hidayatullah jakarta, muhammad.manaqib@uinjkt.ac.id 3uin syarif hidayatullah jakarta, irma.fauziah@uinjkt.ac.id doi: https://doi.org/10.15642/mantik.2020.6.1.52-65 abstrak: penelitian ini membahas tentang infiltrasi saluran irigasi alur dalam berbagai bentuk saluran irigasi pada jenis tanah homogen. model matematika untuk masalah infiltrasi adalah persamaan richard. persamaan richard ini kemudian ditransformasikan dengan menggunakan transformasi kirchhoff dan variabel tak berdimensi menjadi persamaan helmholtz termodifikasi. selanjutnya dengan menggunakan dual reciprocity boundary element method (drbem), solusi numerik dari persamaan helmholtz termodifikasi diperoleh. metode tersebut digunakan untuk menyelesaikan masalah infiltrasi pada saluran flat, non-flat tanpa impermeable dan non-flat dengan impermeable. nilai daya serap dan kadungan air yang paling besar terletak di bawah permukaan saluran. urutan bentuk saluran berdasarkan kandungan air yang paling banyak berturut-turut adalah non-flat channel tanpa impermeable, non-flat channel dengan impermeable dan saluran flat pada jenis tanah lakish clay. kata kunci: infiltrasi; saluran periodik, drbem. abstract: this research discusses the infiltration of furrow irrigation invarious forms of irrigation channels in homogeneous soils. the governing equation of the problems is a richard’s equation. this equation is transformed using a set of transformation including kirchhoff and dimensionless variables into helmholtz modified equations. furthermore with dual reciprocity boundary element method (drbem), numerical solution of modified helmholtz equation obtained. the proposed method is tested on problem involved infiltration from periodic flat channels, non-flat channels without impermeable and non-flat channels with impermeable. the greatest value of suction potential and water content is located below the channel surface. the most water consecutively is a non-flat channel without impermeable, non-flat channel with impermeable and flat channel on lakish clay soils. keywords: infiltration; periodic channels, drbem. jurnal matematika mantik vol. 6, no. 1, may 2020, pp. 52-65 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 53 1. introduction water access for using productivity such as agriculture and family business are very important to create livelihood opportunity, generate income, and contribute towards economical productivity. almost fifth of world population, along half billion people live at the place which encounter physical water crisis [1]. water availability originated of surface, water in the soil and source, which is year by year tend to decrease as result of environmental damage. on the other hand, water necessary time by time becomes increase as the result of increase in population and industrial growth [2]. water function in agriculture generally as irrigation or watering to fulfill plant needs, without good irrigation that the result of plant which managed by farmers will not achieve maximum result. regular water supply is expected can help the soil to tend water content stability as of infiltration process. one approach to find and explain solutions to problems that occur in the real world is by mathematical modeling. after the mathematical model is obtained it can be solved mathematically and can be reapplied in real problems [3]. along with the development of science, a lot of mathematical model has been done relating to infiltration in irrigation channels. one of them is research of i. solekhudin and k.c. ang [4], namely for irrigation channels in the form of flat, rectangular, semi-circular, and trapezoidal. mathematical models of water infiltration problems in channel irrigation channels are in the form of boundary conditions with a governing equation in the form of a modified helmholtz equation with the robin boundary conditions. one approach to solving the problem this time is to be able to use the dual reciprocity boundary element method (drbem) numerical method. drbem is a part or development of the boundary element method (bem). bem is a numerical method used to solve partial differential equations found in mathematical physics and engineering. such as laplace's equation, helmholtz's equation, diffusion convection equation, potential and viscous flow equations, electrostatic and electromagnetic equations, and elastostatics and elestodynamic linear equations [5]. there are several advantages of the boundary element method (bem) compared to other numerical methods, such as the finite element method (fem) and the finite difference method (fdm). here are some of the advantages [6]. • discretization is only done at the domain boundary, thus making numerical modeling with bem simple and reducing the number of collocation points needed. • modified bem can solve problems with unlimited domains. • meb is proven effective in calculating derivatives from field functions such as flux, voltage, pressure, and moment. meb can also resolve the concentration of forces and moments in the domain interior and domain boundaries. • using a set of collocation points located at the domain boundaries can be used to find solutions at all points in the domain. in contrast to fem and fdm the solution is obtained only at the collocation point. • bem can also solve problems with complicated domains, such as a crack. this research will discuss about the solution to the problem of water infiltration in various forms of furrow irrigation channels using the dual reciprocity boundary element method, to determine the characteristics of infiltration and the distribution characteristics of the water in the form of flat channel, non-flat channel without impermeable (rectangular channel, semi-circular channel, trapezoidal channel), and non-flat channel with impermeable (rectangular channel, trapezoidal channel) [7]. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 54 2. methods the steps to solve the boundary condition problem of the helmholtz equation with the dual reciprocity boundary element method (drbem) are [8]: • reciprocal relation reciprocal relation between the fundamental solution of the laplace equation and the solution of the helmholtz equation in the r. ( )2 φ φ φ . c r ds g k dxdy n n       − = −       (1) • boundary integral equations the form of the boundary integral equation obtained from reciprocal relations and domain modifications. 𝜆(𝜉, 𝜂)𝜙(𝜉, 𝜂) = ∫ (𝜙(𝑥, 𝑦) 𝜕φ(𝑥, 𝑦; 𝜉, 𝜂) 𝜕𝑛 − φ(𝑥, 𝑦; 𝜉, 𝜂) 𝜕𝜙(𝑥, 𝑦) 𝜕𝑛 ) c 𝑑𝑠 + ∬ φ(𝑥, 𝑦; 𝜉, 𝜂)(𝑔(𝑥, 𝑦) − 𝑘2𝜙(𝑥, 𝑦))𝑑𝑥𝑑𝑦 𝑅 with if if lies on a s 0, moo ( , ) 1 ( , ) , ( , ) 2 1, ( , th of i )f r c r c              =    • domain integral approach by using the radial basis function approach, an integral domain approach is obtained as follows. ∬ 𝛷 𝑅 (𝑥, 𝑦; 𝜉, 𝜂)(𝑔(𝑥, 𝑦) − 𝑘2𝜙(𝑥, 𝑦))𝑑𝑥𝑑𝑦 = ∑ [ ∑ 𝜔 𝑀 𝑚=1 (𝑎(𝑗), 𝑏(𝑗); 𝑎(𝑚), 𝑏(𝑚))𝜓(𝜉, 𝜂; 𝑎(𝑚), 𝑏(𝑚))] 𝑀 𝑗=1 [𝑔(𝑎(𝑗), 𝑏(𝑗)) − 𝑘2𝜙(𝑎(𝑗), 𝑏(𝑗))] (3) • boundary integral equations in the form of line integral ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( , ) ( , ) ( , ; , ) ( , ; , ) m m j j m m m m j m a b a b a b          = =   =       ( ) ( ) 2 ( ) ( ) ( , ) ( , ) j j j j g a b k a b − +  ( , ; , ) ( , ) ( , ) ( , ; , ) c x y x y x y x y ds n n          −      for ( , ) .r c    • completion of boundary integral equations substitute the collocation points to the boundary integral equation obtained by linear system. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) 1 ( , ) ( , ) ( , ) ( , ) n l n n n n nj j j j j j x y x y g x y k x y    + =  = − +  (2) (4) a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 55 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 ( , ) ( , ) n k k n n k k n n k f x y p f x y =  −  (5) for n=1,2,..., n+l with ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( , ; , ) ( , ; , ) n l nj j j m m m m m x y x y x y     + = =  ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) 2 1 1 ( , ) ( ) ( ) 4 k k n n n n c f x y ln x x y y ds  = − + − ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) 2 2 = 1 ( , ) ( ) ( ) . 4 k k n n n n c f x y ln x x y y ds n   − + −    • complete the linear system. next, by substituting the linear system solution into the boundary integral equation, an equation can be obtained that can be used to evaluate the partial differential equation solution at all points in the domain. 3. result and discussion 3.1. problem formulation this study uses drbem to solve water infiltration problems in various furrow irrigation channels on lakish clay soil types with 𝛼(𝑐𝑚−1) = 1.38 × 10−2, 𝐾0 (𝑐𝑚/𝑠) = 8.1 × 10 −5 𝜃𝑟 = 0.068 , 𝜃𝑠 = 0.38 and 𝑛 = 1.09 of [9] and [10]. the forms of channel irrigation that were studied were flat channel, non-flat channel without impermeable (rectangular channel, trapezoidal channel, semi-circular channel), and non-flat channel with impermeable (rectangular channel, trapezoidal channel). the assumptions used on each channel are: • the width of the irrigation channel has the same width and has a long enough length, so in this model the length of the irrigation channel is ignored. • irrigation channels in the form of flat channel, non-flat channel without impermeable (rectangular channel, trapezoidal channel, semi-circular channel), and non-flat channel with impermeable (rectangular channel, trapezoidal channel). • the distance between the midpoints of two adjacent channels is 2(𝐿 + 𝐷). • the channel is always full of water. • the influence of other irrigation channels is ignored. • the infiltration rate of water / large incoming fluxes on the surface of the irrigation channel is constant, which is equal to 𝑣0. there is no incoming water flow except from the channel. 3.2. mathematical models of infiltration in various forms of irrigation channels mathematical model of water infiltration in channel irrigation channels in the form of problem boundary conditions (msb) with the condition of a neumann and robin boundary. while the governing equation is in the form of a modified helmholtz equation jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 56 [11]. the equation governing the mathematical model of water infiltration comes from richard's equation [11]. 𝜕𝜃 𝜕𝑇 = −𝛻 (−𝐾(𝜃) [ 𝜕𝜓 𝜕𝑋 𝑖 + ( 𝜕𝜓 𝜕𝑍 − 1) 𝑗]) = 𝜕 𝜕𝑋 (𝐾(𝜃) 𝜕𝜓 𝜕𝑋 ) + 𝜕 𝜕𝑍 (𝐾(𝜃) ( 𝜕𝜓 𝜕𝑍 − 1)) = 𝜕 𝜕𝑋 (𝐾(𝜃) 𝜕𝜓 𝜕𝑋 ) + 𝜕 𝜕𝑍 (𝐾(𝜃) 𝜕𝜓 𝜕𝑍 ) − 𝜕𝐾(𝜃) 𝜕𝑍 . ( 1 ) richard's equation will be transformed into the helmholtz equation in the form of a linear differential equation. • kirchhoff's transformation uses the formula 𝛩 = ∫ 𝐾 𝜓 −∞ (𝑠)𝑑𝑠. • an exponential model of hydraulic conductivity is defined 𝐾 = 𝐾0𝑒 𝛼𝜓, 𝛼 > 0. • transform to the form of the dimensionless variable equation so that it is obtained: 𝑥 = 𝛼 2 𝑋, 𝑧 = 𝛼 2 𝑍, 𝛷 = 𝜋𝛩 𝑣0𝐿 , 𝑢 = 2𝜋 𝑣0𝛼𝐿 𝑈, 𝑣 = 2𝜋 𝑣0𝛼𝐿 𝑉, 𝑓 = 2𝜋 𝑣0𝛼𝐿 𝐹. so, we find 𝜕2𝜙 𝜕2𝑥 + 𝜕2𝜙 𝜕2𝑧 = 𝜙 ( 2 ) as a governing equation of mathematical models of water infiltration which includes the modified helmholtz equation. this research will discuss about the solution to the problem of water infiltration in various forms of channel irrigation channels by using the dual reciprocity boundary element method, to determine the characteristics of infiltration and the distribution characteristics of the water in the form of flat channel, non-flat channel without impermeable (rectangular channel, semi-circular channel, trapezoidal channel), and non-flat channel with impermeable (rectangular channel, trapezoidal channel), as shown in the figure 1. a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 57 (a) flat channel (b) rectangular channel (c) semi-circular channel (d) trapezoidal channel (e) rectangular with impermeable (f) trapezoidal with impermeable figure 1. geometry of furrow irrigation channels[4] furthermore, based on the shape of the channel and the assumptions about irrigation channels, the boundary conditions for the mathematical model of water infiltration on the channel irrigation can be obtained as follows [12] [13][14]. • flat channel 𝜕𝜙 𝜕𝑛 = 2𝜋 𝛼𝐿 − 𝜙 , for 0 and 0 2 x l z    = 𝜕𝜙 𝜕𝑛 = −𝜙 , for ( )and 0 2 2 l x l d z     + = 𝜕𝜙 𝜕𝑛 = 0 , for 0 and 0x z=  𝜕𝜙 𝜕𝑛 = 0 , for ( ) and 0 2 x l d z  = +  𝜕𝜙 𝜕𝑛 = −𝜙 2 , for 0 andx l d z    + =  • non-flat channel without impermeable 𝜕𝜙 𝜕𝑛 = 2𝜋 𝛼𝐿 𝑒 −𝑧 − 𝜙𝑛2 ,on the surface of the channel 𝜕𝜙 𝜕𝑛 = −𝜙 ,on the surface of the soil outside the channels 𝜕𝜙 𝜕𝑛 = 0 towards , 0 and 0x z=  𝜕𝜙 𝜕𝑛 = 0 , ( )and 0 2 towards x l d z  = +  𝜕𝜙 𝜕𝑛 = −𝜙 2 , towards 0 and x l d z    + =  jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 58 • non-flat channel with impermeable 𝜕𝜙 𝜕𝑛 = 2𝜋 𝛼𝐿 𝑒 −𝑧 − 𝜙𝑛2 , on the surface permeable channel 𝜕𝜙 𝜕𝑛 = 𝜙𝑛2 , on the surface impermeable channel 𝜕𝜙 𝜕𝑛 = −𝜙 ,on the surface of the soil outside the channels 𝜕𝜙 𝜕𝑛 = 0 , towards 0 and 0x z=  𝜕𝜙 𝜕𝑛 = 0 , towards ( ) and 0 2 x l d z  = +  𝜕𝜙 𝜕𝑛 = −𝜙 2 , towards 0 and x l d z    + =  3.3. settlement with drbem mathematical model of water infiltration in channel irrigation in the form of a modified helmholtz equation with boundary conditions of each channel shape. the model will be completed using drbem. the main step in solving the use of drbem is to form a linear system. however, before the step is conducted, the boundary integral equation needs to be formed. substitute boundary conditions to the integral equation of the helmholtz equation. after the boundary integral equations are obtained, a linear system can be formed by discretizing domain boundaries into a number of line segments and selecting a number of interior points. then the linear is converted into the ax = b matrix equation with x is a column vector containing unknown variables. figure 1. schematic representation of the program to implement the drbem [15] a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 59 the matlab implementation in this case is divided into three stages, namely the ones named pre-processing, processing, and post-processing, as shown in figure (2). after the matlab program implementation is complete, it is then used to solve the infiltration problem in various types of channels in one type of soil namely lakish clay. next, choose n= 200, i.e. the number of line segments for discretizing domain boundaries. n = 200 because the accuracy of the resulting numerical approach is quite good [15]. m values for each type of channel are different because the size of the dimensionless domain of each channel type is different depending on the shape of the channel. flat channel given m = 625, trapezoidal channel given m = 619, rectangular channel given m = 625, semi-circular channel given m = 593, trapezoidal channel with impermeable given m = 619, rectangular channel with impermeable given m = 625. each type of channel will evaluate the values of 𝜓 and 𝜃 in some values at several points along the line 𝑋 = 10 𝑐𝑚, 𝑋 = 30 𝑐𝑚, 𝑋 = 50 𝑐𝑚, 𝑋 = 70 𝑐𝑚, 𝑋 = 90 𝑐𝑚, for 0 ≤ 𝑍 ≤ 200 𝑐𝑚. (a) suction potential (𝛙) (b) water content (𝛉) figure 2. suction potential and water content toward 𝑿 = 𝟏𝟎 (a) suction potential (𝝍) (b) water content (𝜽) figure 3. suction potential dan water content toward 𝑿 = 𝟑𝟎 it is shown that figure 3 is a graph of values ψ and θ below the channel. the graphs ψ and θ are broken because there are points evaluated in the channel, so the graphs ψ and θ in the channel can be ignored. value of ψ and θ flat channel and non-flat without impermeable decreases with increasing depth and goes to the point of convergence. as for non-flat channels with impermeable values ψ and θ increase with increasing soil depth to the point of convergence due to impermeable layers in the channel. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 60 based on figure 4, the graphs ψ and θ are broken because there are points that are evaluated in the channel, so the graphs ψ and θ in the channel can be ignored. values ψ and θ at x = 30 go up at a certain depth and go down at a certain depth to the point of convergence. it was concluded that the value of ψ in the shallow position is greater than in the deep position. this is in accordance with the assumption that water enters from the channel and then water seeps into the lower and deeper soil below the channel. (a) suction potential (𝛙) (b) water content (𝛉) figure 4. suction potential and water content torward 𝑿 = 𝟓𝟎 (a) suction potential (𝛙) (b) water content (𝜽) figure 5. suction potential and water content toward 𝑿 = 𝟕𝟎 (a) suction potential (𝝍) (b) water content (𝜽) figure 6. suction potential and water content toward 𝑿 = 𝟗𝟎 based on figures 5, figure 6, and figure 7 it is clear that the values of ψ and θ along these lines are values of ψ and θ not below the surface of the channel. the values of ψ and θ in the flat channel decrease at a certain depth and non-flat without impermeable increases with increasing depth and go to the point of convergence. as for non-flat channels with a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 61 impermeable values ψ and θ rise to a certain depth to the point of convergence due to the impermeable layer on the channel. the value of ψ is directly proportional to the value of θ. however, the pattern of relationships for each type of channel is different, seen from different curves for each type of channel. the value of θ which increases with increasing depth indicates that the shallow water content is smaller than the deep position for the soil which is not below the surface of the canal. this is consistent with the assumption that no flow of water enters the ground surface. based on figure 3 to figure 7 there is no significant difference in each type of nonflat channel, but there is a remarkable difference in the flat channel with other channels, especially at ground level. the values of ψ and θ always decrease when away from the center of the channel surface. next to see the distribution pattern of the value of suction potential (ψ) and water content (θ) in the domain of each channel type. in this case the domain is a cross section of a channel on the ground with a width of 100 cm and a depth of 200 cm. the following is given figure 8 to figure 10 which are consecutive flat channel, rectangular channel, semi-circular channel, trapezoidal channel, trapezoidal with impermeable and rectangular with impermeable. the image on the left is the distribution of suction potential values (ψ) and the image on the right is the distribution of the value of water content (θ). (a) suction potential (𝛙) (b) water content (𝜽) figure 7. water distribution pattern on the flat channel based on figure 8 on the flat channel, it appears that the value of the greatest suction potential lies below the channel, the deeper the ground level the smaller the suction potential value. likewise, the value of the largest water content is located under the channel, the further from the center of the channel the smaller the value of water content. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 62 (a) suction potential (𝝍) (b) water content (𝛉) (a) suction potential (𝛙) (b) water content (𝛉) (a) suction potential (𝛙) (b) water content (𝛉) figure 8. water distribution pattern on non-flat channel without impermeable based on figure 9 on a non-flat channel consisting of rectangular, circular, and trapezoidal channels. it can be seen that the value of the suction potential spreads evenly below the channel, the closer to the center of the channel the greater the value of the suction potential. you can also see the value of the water content is spread but not flat, the greatest value of water content is located below the channel. a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 63 (a) suction potential (𝛙) (b) water content (𝛉) (a) suction potential (𝛙) (b) water content (𝛉) figure 9. water distribution pattern on non-flat channel with impermeable based on figure 10 on a non-flat channel with impermeable. it can be seen that the value of the greatest suction potential is located on the wall of the channel that is not closed, the closer to the center of the channel, the greater the value of suction potential. likewise, with the value of its water content, the largest value of water content is located on the wall of the channel that is not closed. the distribution pattern of suction potential and water content values is seen from the surface plot of the suction potential and water content values in the domain. based on figure 8 to figure 10 it can be seen that the greatest suction potential is located below the channel, while the smallest is at the ground surface which is far from the center of the channel. in addition, it can be seen that in the upper soil layers the farther away from the center of the channel the smaller the value of water content. 4. conclusions drbem can solve the problem of water infiltration in the channel irrigation channel in the form of boundary conditions with the governing equation is the modified helmholtz equation. the greatest suction potential and water content values are located below the surface of the channel, and the smallest suction potential and water content values are located far from the center point of the channel. suction potential value is directly proportional to the value of water content, the more waterpower is absorbed, the more jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 52-65 64 water content in the soil. channel sequences based on the water content that are most numerous are rectangular channel, trapezoidal channel, semi-circular channel, rectangular channel with impermeable, trapezoidal channel with impermeable, and flat channel. in addition, it can be concluded that the form of non-flat channel channels without impermeable has more water content in the lakish clay soil type. this study uses drbem to complete channel irrigation channel infiltration in various types of canals on lakish clay soil types without adding plant elements. future studies can add plant elements to agricultural land with different soil types and complete the irrigation of time-dependent channel irrigation channels. references [1] un world water assessment programme, the united nations world water development report 2015. unesco, 2015. [2] m. g. bos, r.a.l. kselik, r. allen, and d. molden, “water requirements for irrigation and the environment”, 2009, doi: 10.1007/978-1-4020-8948-0. [3] m. manaqib, i. fauziah, and m. mujiyanti, "mathematical model for mers-cov disease transmission with medical mask usage and vaccination", inprime: indonesian journal of pure and applied mathematics, vol. 1, no. 2, pp. 97-109, 2019. doi: 10.15408/inprime.v1i2.13553 [4] i. solekhudin and k.c. ang, "a dual-reciprocity boundary element method for steady infiltration problems", anziam journal, vol. 54, pp. 171-180, 2013. doi: 10.21914/anziamj.v54i0.5699 [5] c. pozrikidis, a practical guide to boundary element method with the software library bemlib. florida: chapman and hall/crr, 2002. [6] j. t. katsikadelis, boundary element: theory and application. oxford: elsevier, 2002. [7] m. i. azis, d. l. clements, and m. lobo, "a boundary elements method for steady infiltration from periodic channels", anziam journal, vol. 44, pp. c61-c78, 2003. doi: https://doi.org/10.21914/anziamj.v44i0.672. [8] m. manaqib and i. solekhudin, "dual reciprocity boundary element method untuk menyelesaikan masalah infiltrasi air pada saluran irigasi alur," in seminar matematika dan pendidikan matematika uny, yogyakarta, 2017. [9] e. bresler, "analysis of trickle irrigation with application to design problems", irrigation science, vol. 1, pp. 3-17, 1978. [10] a. w. warrick, soil physics companion. washington d.c.: crc press, 2002. [11] m. manaqib., “pemodelan matematika infiltrasi air pada saluran irigasi alur”, j. mat. mantik, vol. 3, no. 1, pp. 23-29, oct. 2017. [12] v. batu, “steady infiltration from single and periodic strip sources”, soil science society of america journal, vol. 42, pp. 544-549, 1978. doi:10.2136/sssaj1978.03615995004200040002x [13] m. lobo, d. l. clements, and n. widana, "infiltration from irrigation channels in a soil with impermeable inclusions," anziam journal, vol. 46, pp. c1055-c1068, 2005. [14] d. l. clements, m. lobo, and n. widana, "a hypersingular boundary integral equation for a class of a problems concerning infiltration from periodic channels", electronic journal of boundary element, vol. 5, no. 1, pp. 1-16, 2007. doi: https://doi.org/10.14713/ejbe.v5i1.779. https://doi.org/10.2136/sssaj1978.03615995004200040002x a. nurhasanah, m. manaqib, and i. fauziah analysis infiltration waters in various forms of irrigation channels by using dual reciprocity boundary element method 65 [15] i. solekhudin and k.c. ang, "suction potential and water absorption from periodic channels in different types of homogeneous soils," electronic jurnal of boundary element, vol. 10, no. 2, pp. 42-55, 2012. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: v. a. fitria and y. a. sapoetra, “stability analysis of finance system model with information effect”, jmm, vol. 6, no. 1, pp. 13-19, may 2020. stability analysis of finance system model with information effect vivi aida fitria1, yudistira arya sapoetra2 1institut teknologi dan bisnis asia malang, viviaidafitria@gmail.com 2institut teknologi dan bisnis asia malang, yuditstiraarya@gmail.com doi: https://doi.org/10.15642/mantik.2020.6.1.13-19 abstrak: partisipasi indonesia dalam berinvestasi di pasar modal masih sangat rendah, salah satu penyebabnya adalah kurangnya informasi. oleh karena penelitian ini membahas tentang analisis kestabilan pada sistem keuangan jika ditambah dengan pengaruh informasi. terdapat dua titik keseimbangan, yaitu titik tanpa suku bunga dan indeks harga instrument keuangan serta titik eksisnya suku bunga, tingkat permintaan investasi, indeks harga dan input control pengaruh informasi. hasil analisis kestabilan lokal kedua titik kesetimbangan tersebut adalah stabil dengan syarat tertentu. hasil simulasi numerik pada penelitian ini menunjukkan bahwa sesuai dengan hasil analisis. kata kunci: analisis kestabilan, sistem keuangan, informasi abstract: indonesia's participation in investing in the capital market is still very low, one of the causes is the lack of information. so this study discusses the analysis of stability in the financial system if influenced by information. we find that the model has two equilibrium point, that are point without interest rates and the price index of financial instruments and then the existing point of interest rates, the level of investment demand, price indexes and the influence of control input the information. the results of the local stability analysis of the equilibrium points are stable with certain conditions. the analytical result are confirmed by numerical simulations. keywords: stability analysis, finance system, information jurnal matematika mantik volume 6, no. 1, may 2020, pp. 13-19 issn: 2527-3159 (print) 2527-3167 (online) mailto:viviaidafitria@gmail.com1 mailto:yuditstiraarya@gmail.com2 http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 jurnal matematika mantik volume 6, issue 1, may 2020, pp. 13-19 14 1. introduction the financial system is a key factor of economic growth in a country. if the financial system does not function properly, it can be one of the reasons a country cannot develop. because with a good financial system, it encourage efficient capital allocation. capital allocation will be efficient if there is a channeling of funds from people who cannot use money productively to people who can use it productively. an efficient capital allocation is one of the causes of a stable financial system the stability of a financial system is influenced by several factors, including interest rates, price indexes and investment demand. a three-dimensional financial system model consisting of interest rates, price indices and investment demand was introduced by ma and chen in 2001 [1]. until now, researchers have continued to develop or modify financial system models. one modification is to provide input control. as done by [2], namely by adding input controls in the form of government policies which are influenced by the high and low price indexes. therefore, this research discusses the modification of the financial system model, namely by adding input controls in the form of information about financial literacy that affects the rate of investment demand. based on [3], only 500,000 indonesians become investors in the stock market. whereas the capital market has an important role in the economic development of a country because the capital market has two functions, namely the economic function and financial function [4]. with the capital market, individual investors and business entities can channel their excess funds to invest in the capital market, and entrepreneurs can obtain additional capital funds to expand their business networks from investors in the capital market [5]. according to [6], this is due to lack of information about financial literacy. in accordance with the national survey on financial literacy it shows that only 3.79% of indonesians understand well in the capital market. on the other hand, 93.79% are classified as ignorant about the capital market, meaning that for every 100 people, 94 do not know the capital market [3]. therefore, the purpose of this study is to analyze the stability of the financial system model if coupled with input controls in the form of information about financial literacy that affects the rate of investment demand. according to [7], the indonesia stock exchange (idx) always provides better education aimed at increasing the number of active investors in the capital market in indonesia, then the results of the analysis of this research are expected to provide an overview to the idx, how the stability of the financial system model with the information variable. in addition, this research also conducted a numerical simulation of a financial system model using matlab software 2. methods the method used in this research is literature study method, which contains theories that are relevant to research problems. the literature study method is useful for developing concepts or theories that form the basis of study in research [8]. while the stages in this study are: a. review and understand the continuous form of the financial system model b. modify the model by adding information influence c. analyzing the dynamic stability of a modified financial system model. d. the numerical scheme obtained is then simulated using the matlab software. e. conclusions. v. a. fitria and y. a. sapoetra stability analysis of finance system model with information effect 15 3. result and discussion 3.1 model formulation this section discusses modification of the financial system model by adding control input in the form of information. the financial system model that will be modified is from the model introduced by ma and chen in [1], namely: czxz xbyy zxayx −−= −−= +−=    2 1 )( (1) where 𝒙 is the interest rate, 𝒚 is the level of investment demand, 𝒛 is the price index of the financial instrument, 𝒂 is the total savings in the bank, 𝒃 is the investment cost, and c is the elasticity of demand. after that it was developed by [2] by adding input controls in the form of government policies. the following are the results of the modification of saputri, et al [2] : mzkudxyu uczxz xbyy zxayx −−−= +−−= −−= +−=     2 1 )( (2) where 𝒖 is the input control, 𝒅,𝒌 and 𝒎 are the appropriate amplitude [9]. furthermore, in this study the financial system model was modified by adding an input control (𝒖) in the form of the influence of information about financial literacy. this input control affects the rate of investment demand. the following are the results of modifications to the financial system model introduced by ma and chen: . 1 1 )( 0 2 uk by ky u czxz uxbyy zxayx − + = −−= +−−= +−=     (3) the influence function of information is introduced by [10]. the effect of information on financial literacy can increase the rate of investment demand, with is the rate of information growth and represents the saturation constant [11]. 3.2 stability analysis in this section, will determine the point of equilibrium in system (3). then determine the stability of each equilibrium point. analysis of the stability of the equilibrium point is done by linearizing the system (3) around each equilibrium point obtained. jurnal matematika mantik volume 6, issue 1, may 2020, pp. 13-19 16 system (3) has 2 points of equilibrium, it is ( )        + ++ * 0 * 2 0 22 0 2 0 1 ,0, 2 4 ,0 byk ky bk bkkk e and ( ) ( ) ( ) ( )        ++ + + ++ + +−− babcck ack c x c a babcck ack c b abe 0 * 0 2,1 1 ,, 1 , 1 1 on condition of existence ( ) ( ) 1 1 0  ++ + −+ babcck ack c b ab . from the two equilibrium points, an equilibrium analysis is carried out by first linearizing the system. to linearize system (3), a transformation must be made using perturbation theory [12], so that a jacobian system is obtained around a fixed point:               − + −− −− − = 0* * ** 0 )1( 0 001 102 01 k by k c bx xay j (4) the jacobi matrix is then used to analyze the stability of each model's equilibrium point. analyzing system stability (3) is done by determining the eigenvalues of matrix j. the eigenvalues of matrix j are obtained by solving the equation 𝒅𝒆𝒕(𝑱 − 𝝀𝑰) = 𝟎 [13], namely : 0 43 2 2 3 1 4 0 =++++ hhhhh  (5) with ( ) ( ) ( ) ( ) ( )2* * 2* 00 * 004 2* * 2* 0 2* 0 * 0 ** 00003 2* 2*** 0 * 0002 01 0 1 2 1 22 1 21 1 by kaykc xckabckycbkbkh by yack xkcxackabcyckcbyybkabkbckkbh by k xaccyykbyakabckcbbkh yackbh h + −− +++−= + −+ −++++−−−+++= + −++−−−+++++= −+++= = the equilibrium point of the autonomous system with the jacobi matrix (4) will be stable if all the roots of equation (5) are negative or 𝝀𝒊 < 𝟎, 𝒊 = 𝟏,𝟐,…,𝒏. the routhhurwitz criterion can be used to determine the sign of the eigenvalues matrix (4) by using the coefficients of the equation [14]. based on the routh-hurwitz criteria, so that the eigenvalue in equation (5) is negative then it must apply 0 4 h with v. a. fitria and y. a. sapoetra stability analysis of finance system model with information effect 17      −− −  0 0 0,,, 4 2 1 2 3321 321 4321 hhhhhh hhh hhhh (6) if condition (6) is fulfilled, system (3) is at a 0e and 2,1e fixed point will be stable. however, if the opposite condition is taken 0 1 h then the system (3) will be in an unstable condition. 3.3 numerical simulation in this section, a system solution (3) is simulated. to simulate the results of numerical analysis of models used the runge kutta order 4 method in matlab software. here is a numerical simulation for the equilibrium point. ( )        + ++ * 0 * 2 0 22 0 2 0 1 ,0, 2 4 ,0 byk ky bk bkkk e . in this simulation parameters are used 9,0=a ; 8,0=b ; 5,0=c ; 0001,0=k and 8,00 =k , so the point of equilibrium 0e is ( )0.00016;0;25.1;0 . the following is figure 1 which shows the phase portrait of the solution model (3) with initial values (10, 5, 1, 5). figure 1. graph of 0e stability point based on figure 1, the point 0e is stable, shown by a graph that leads to the equilibrium point. this proves that if the level of investment demand is influenced by information even though there is no influence of interest rates and the price index of financial instruments, the financial condition is stable with certain conditions. for point ( ) ( ) ( ) ( )        ++ + + ++ + +−− babcck ack c x c a babcck ack c b abe 0 * 0 2,1 1 ,, 1 , 1 1 of financial 0 50 100 150 -8 -6 -4 -2 0 2 4 6 8 10 time ra te interest rate level of investment demand the price index of the financial instrument input control jurnal matematika mantik volume 6, issue 1, may 2020, pp. 13-19 18 simulations is used 9,0=a ; 5,0=b ; 5,1=c ; 05,0=k and 13.00 =k parameter values, so the point of equilibrium is ( )0.338;496.0.;57.1;745.0 − . a phase portrait of the 2,1e equilibrium point is shown in figure 2, with initial values (10, 5, 1, 5). figure 2. graph of 2,1 e stability point the graph in figure 2 also goes to the point of equilibrium. therefore, the point is stable with certain conditions. this proves that the financial system model that is affected by interest rates, investment demand levels, changes in financial instrument price indexes and the addition of input controls in the form of information will be stable, provided that the stability conditions of the model are met. this is in accordance with research [15] which concludes that investment knowledge has a significant positive effect on student investment interests. 4. conclusions the results of the analysis of the stability of the financial system model if coupled with input controls in the form of information about financial literacy that affects the rate of investment demand is stable, both at a 0e fixed point and at a 2,1e fixed point provided 0,0,,, 3214321 − hhhhhhh and 0 4 2 1 2 3321 −− hhhhhh . therefore, if the financial system in indonesia adds control in the form of information about financial literacy, the condition of the financial system in indonesia will be stable. this research can be developed by adding a delay time on certain variables. given the conditions that cause the rate of change in the financial system has been delayed. for example, if there is a delay in the increase in interest rates in indonesia due to the adjustment of the united states (us) interest rates, it will only be done if inflation reaches 2 percent references [1] a. aram, “dynamic behavior of a nonlinear macro-financial system,” york university, 2014. [2] a. d. saputri, h. hariyanto, and m. s. winarko, “analisis bifurkasi hopf pada sistem keuangan dengan kontrol input,” j. sains dan seni its, vol. 7, no. 2, 2018, 0 50 100 150 -10 -8 -6 -4 -2 0 2 4 6 8 10 time ra te interest rate level of investment demand the price index of the financial instrument input control v. a. fitria and y. a. sapoetra stability analysis of finance system model with information effect 19 doi: 10.12962/j23373520.v7i2.30171. [3] g. s. djojohadikusumo, “indonesian national strategy for financial literacy,” no. 2, jakarta: financial services authority of the republic of indonesia, 2013. [4] f. muklis, “perkembangan dan tantangan pasar modal indonesia,” j. lemb. keuang. dan perbank., vol. 1, no. 1, 2016. [5] i. yuliana, investasi produk keuangan syariah. malang: uin-maliki press, 2010. [6] novi yushita amanita, “pentingnya literasi keuangan bagi pengelolaan keungan pribadi,” j. nominal, vol. vi, p. 11, 2017. [7] d. saputra, “pengaruh manfaat, modal, motivasi dan edukasi terhadap minat dalam berinvestasi di pasar modal,” futur. j. manaj. dan akunt., vol. 5, no. 2, p. 178 _ 190, 2018. [8] v. w. sujarweni, metodeologi penelitian. yogyakarta: pustaka baru perss, 2014. [9] subiono, “sistem linear dan kontrol optimal,” in 1, surabaya: institut teknologi sepuluh nopember, 2013. [10] a. kumar, p. k.srivastava, and y. takeuchi, “modeling the role of information and limited optimal treatment on disease prevalence,” j. theor. biol., vol. 414, pp. 103– 119, 2017. [11] r. w. wilda and m. a. imron, “sensitivity and stability analysis of a seir epidemic model with information,” vol. 9, no. 1, pp. 47–53, 2019. [12] j. r. chasnov, mathematical biology. hongkong: university of science and technology, 2009. [13] w. . kelley and a.c.peterson, the theory of differential equations: classical and qualitative. new york: springer, 2010. [14] j. d. murray, mathematical biology i: an introduction third edition. verlag berlin heidelberg: springer, 2002. [15] l. k. merawati and i. p. m. j. s. putra, “kemampuan pelatihan pasar modal memoderasi pengaruh pengetahuan investasi dan penghasilan pada minat berinvestasi mahasiswa,” j. ilm. akunt. dan bisnis, vol. 10, no. 2, pp. 105–118, 2015. how to cite: p. t. bantining n, b. surarso, and sutimin, “comparison between zero point and zero suffix methods in fuzzy transportation problems”, jmm, vol. 6, no. 1, pp. 38-46, may 2020. comparison between zero point and zero suffix methods in fuzzy transportation problems pukky t. b. ngastiti1, bayu surarso2, sutimin3 1department of mathematics, universitas billfath, tetralian@billfath.ac.id 2department of mathematics, universitas diponegoro, bayusurarso@yahoo.com 3department of mathematics, universitas diponegoro, sutimin@undip.ac.id doi: https://doi.org/10.15642/mantik.2020.6.1.38-46 abstrak: persoalan transportasi membahas tentang masalah pendistribusian suatu barang dari sejumlah sumber kepada sejumlah tujuan dengan tujuan meminumkan biaya pengangkutan. masalah transportasi fuzzy merupakan biaya transportasi, persediaan dan permintaan dengan kuantitas jumlah fuzzy. tujuan dari penelitian ini adalah untuk mempelajari komparasi teoritis dan numerik antara metode zero-point dan metode zero suffix dalam menentukan solusi optimal pada biaya pengangkutan barang. berdasarkan hasil komparasi didapatkan iterasi pada metode zero-point lebih besar dalam mencapai nilai optimal dibandingkan dengan metode zero suffix. berdasarkan hasil perbandingan teoritis dapat disimpulkan bahwa proses menggunakan metode zero-suffix lebih pendek dalam menentukan solusi optimal yaitu 6 langkah daripada metode zero-point yaitu 11 langkah. untuk mencapai nilai optimal menunjukkan bahwa, untuk metode zero suffix terjadi iterasi pada langkah keenam, namun untuk metode zero-point iterasi terjadi pada tahap kesembilan. hasil perbandingan numerik, kami menyimpulkan biaya distribusi dengan menggunakan dua metode adalah sama, berdasarkan permintaan dan penawaran diperoleh 7 kali iterasi dan 7 item alokasi untuk metode zero point, sedangkan 6 kali iterasi dan 7 item alokasi untuk metode zero suffix. kata kunci: metode zero-point; metode zero-suffix; masalah transportasi fuzzy abstract: transportation is discussing the problems of distribution items from a source to a destination with an aim to minimize transportation costs. the problem of fuzzy transport is the cost of transportation, supply, and demand with a quantity of fuzzy. the purpose of the research is a study of a comparison of theories from the zero-point method and the zero-suffix method in determining the optimal solution on cost transportation. based on the result of the theoretical comparison, it can be concluded that the process of using the zero-suffix method is shorter in determining an optimal solution in 6 steps than that of a zero-point method in 11 steps. for achieving the optimal value shows that for zero-suffix the method of occurrence iteration in the sixth step, but for the zero-point method the iteration occurs in the ninth step. the results in the numerical comparison we conclude the distribution cost using two methods is the same, based on the demand and supply obtained 7 times iteration and 7 items allocation for zero point method, while 6 times iteration and 7 items allocation for zero suffix method. keywords: zero-point method; zero-suffix method; fuzzy transportation problems jurnal matematika mantik vol. 6, no. 1, may 2020, pp. 38-46 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 p. t. bantining n, b. surarso, and sutimin comparison between zero point and zero suffix methods in fuzzy transportation problems 39 1. introduction many applications have in solving problems in everyday life. the parameters of the transportation problem consist of the amount of costs, supply, and demand [1]. conditions that occur in the field of transportation problems of the value of an item often occur uncertainty and tend to change from time to time. this happens because, lack of information data related to the value of these problems [2]. based on zadeh introduces theory fuzzy and problems fuzzy that have been studied in relation to daily life [3]-[4]. the problem of transportation fuzzy is the amount of costs, supply, and demand of value fuzzy [5]-[6]. one method used for transportation problems fuzzy is the zero-point method and the method zero suffix to find the optimal solution for transportation costs. based on research conducted by annie christi ms and kumari shoba. k gives the conclusion that using the robust ranking method with the zero-suffix method gets the optimal solution of the fuzzy problem accurately and effectively [7]. in the following year research chandrasekaran, s, et al concluded that the transportation problem fuzzy using a heptagon fuzzy number with the zero-suffix method obtained optimal solutions in solving problems fuzzy [8]. fegade, et al about the method, zero suffix it was concluded that the optimal solution by changing the problem fuzzy to crisp using the method robust ranking so that the total fuzzy cost becomes optimal and more effective based on the example of the problem fuzzy [9]. this research was supported by nirmala. g and anju r in their research concluded that the zero-suffix method produces an optimal solution with few iterations in transportation problems [10]. based on research on the zero point method by ismail mohideen, s and senthil kumar p conclude that using the method zero point in multiplication operations is better than the vogel's approximation method and the method modified distribution in the distribution problem fuzzy [11]. p. pandian and g. natarajan, it is concluded that the zero-point method provides the optimal value of the objective function with the number fuzzy trapezoid for transportation fuzzy [12]. subsequent research carried out by pukky tetralian, et al concluded that the results of research on cv. bintang elektrik grace associated with transportation problems fuzzy obtained optimal solutions that the zero point method and zero suffix method are the same but in terms of the number of iterations the method zero point is greater in achieving the optimal solution [13]. this is in line with samuel's research, edward concluded that the method was improved zero point is an efficient and better method than the vam, svam, gvam, rvam, bvam methods in achieving optimal solutions to the transportation problem fuzzy [14]. sharma gaurav et al in his research concluded that the method zero point is a symmetrical procedure for transportation problems that is easily applied and utilized for all types of transportation problems with objective functions in the form of maximum or minimum values, to make decisions when there are various types of logistical problems and provide optimal solutions to transportation problems [15]. l. sujatha, p. vinothini and r. jothilakshmi conclude that the procedure developed in this paper provides the optimal fuzzy solution and the optimal fuzzy objective value which are non-negative fuzzy numbers, hence the method developed, serve as an important tool for the decision maker while handling the transportation problem under fuzzy environment [16]. p. k senthil conclude that obtained an optimal solution to the type-2 iftp without using the basic feasible solution and the method of testing optimality. the main advantage of this method is that the obtained solution is always optimal, and it is not required to have (m + n – 1) allotted entries. in feature, the proposed method may be modified to find intuitionistic fuzzy optimal solution of solid intuitionistic fuzzy transportation problems and solid assignment problems with ifns [17]-[18]. the purpose of this research is to study the theoretical and numerical comparisons of transportation problems fuzzy with the zero-point method and zero suffix method. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 38-46 40 2. methods following are different theoretical and numerical comparisons obtained from the zeropoint method and zero suffix method. 2.1. methods north west corner the methods north west corner used to calculate solution a feasible of the transportation problem. following are the steps to get a solution feasible [19]. step 1: allocate and select cells in the upper left corner on transportation issues. this cell must be allocated as many units as possible. the unit must be the same as the minimum requirements between demand and available inventory. step 2: the next step is to adjust the number of requests and supplies in the allocated row and column. the first cell in the second row will only continue, if the first row's supply has run out. likewise, the next cell in the second column will also be continued with the condition that the request for the first cell has been fulfilled. step 3: for allocated cells, each supply of cells is equal to demand, the next allocation can be made in the next row or column. this procedure is repeated until the full allocation of the number of successful units according to the cells in need. 2.2. methods zero point [20] step 1: establish a transport table fuzzy transport problem fuzzy is then converted into transport table fuzzy balanced if it is not balanced. step 2: reduce each row in the transportation table fuzzy from the minimum row. step 3: reduce each column in the transportation table results fuzzy in step 2 of the minimum column. step 4: check if each column is fuzzy demand (𝑏𝑗 ) less than the total fuzzy inventory (𝑎𝑖 ) resulting from cost reduction in the zero-value column. also check if each line is fuzzy in inventory (𝑎𝑖 ) less than the number of columns fuzzy request (𝑏𝑗 ) where row reduction is zero(𝑐𝑖𝑗 = 0). if fulfilled, continue to step 7. if it is not met, then go to step 5. step 5: closing the minimum value with horizontal lines and vertical lines that are zero from the reduction results of the transportation table fuzzy so that some rows or columns that do not meet the requirements in step 4 are not closed. step 6: form a transportation improvement table fuzzy follows: a. find the smallest value from the reduction of the cost matrix fuzzy that is not covered by a line. b. subtract the smallest reduced cost to all costs that are not covered and add costs covered by two intersection lines, then return to step 4. step 7: selecting the reduction cell in the transportation table fuzzy that has the largest reduced cost, is called . if there are more than one cell, then choose just one. ij w ij w ij c ij c ),(  p. t. bantining n, b. surarso, and sutimin comparison between zero point and zero suffix methods in fuzzy transportation problems 41 step 8: select cells in the row or column in the reduced transportation table fuzzyin which the cost-reduction cell is zero cij = 0 and the maximum allocation to the cell. if the cell does not appear at the maximum value, find another maximum value so that the other maximum values will be met. if there is no value in the cell that appears, in the transportation table fuzzy reduced the reduction cost is zero. step 9: reform the transportation table fuzzy after it is deleted in the supply row and demand column. step 10: repeat steps 7 through 9 until inventory fuzzy and demand fuzzy are fulfilled. step 11: allocation produce solutions fuzzy on transportation fuzzy matters. 2.3. method zero suffix [21]-[22] step 1: build a transportation table. step 2: subtract each row entry from the transportation table from the corresponding minimum row after that subtract each column entry from the transportation table to the appropriate minimum column. step 3: in the cost matrix there will be at least one zero in each row and column, then look for suffix values that are denoted by s. s = add cost from the nearest side zero which is greater than zero/ additional cost step 4: select the maximum value of s, if it has one maximum value. if you have two or more of the same value then choose one, then the cost becomes the allocation of goods with due regard to demand and supply. step 5: after step 4, select the minimum {ai, bj} then allocate it to the transportation table. the resulting table must have at least one cost worth 0 in each row and column, otherwise repeat step 2. step 6: repeat step 3 through step 5 until the optimal cost is obtained. optimal cost is obtained if the column or row is saturated (suffix values = 0). 3. results and discussions 3.1. theoretical comparison of zero-point method and zero suffix method step 1 of the two methods is in the first step of the two methods forming the transportation table into a transportation fuzzy balanced, so zero suffix method and zeropoint method have the same steps in step 1. step 2 of the two methods is to reduce rows and columns by reducing the minimum value in zero suffix method while zero point method in step 2 only reduces rows, so in step 2 method zero suffix has occurred in column and row reduction. next step 3 of the two methods is the zero suffix method there will be at least one zero in rows and columns to get the suffix value (s) while zero point method reduces the column, so zero suffix method in step 3 looks for the value suffix and zero point only reduces column.   jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 38-46 42 step 4 of the two methods is zero suffix method with the maximum s value being the allocation of goods by taking into account the amount of supply and demand and zero point method checks each column with the condition that demand is less than inventory and checks the row with the condition that inventory is less than demand on the row and column zero value, so zero suffix method selects the maximum s value and zero point method checks the supply and demand requirements. while step 5 of the two methods is zero suffix method the minimum inventory or demand is selected and then allocated to the transportation table. while zero point method closes the minimum horizontal and vertical rows or columns that are not met in step 4 are not covered, so zero suffix method in step 5 will go to the iteration process and zero point method meets the requirements of step 4. step 6 of the two methods is zero suffix method is iterated to get the optimal value where all column or row values are zero. while zero point method still performs a transportation improvement table with several iterations until each column with the condition that demand is less than inventory and rows with the condition that inventory is less than demand on rows and columns has zero value, so that step 6 zero suffix method has occurred iterating and zero point method is still form a repair table. next step 7 to step 11 zero-point method is iterated to get the optimal solution value. 3.2. numerical comparison of the zero-point method and zero suffix method following is a numerical example of fuzzy transportation problem: table 1. transportation fuzzy full destination inventory source 1 2 3 4 5 1 (5,7,8,11) (1,6,7,12)122 (2,4,5,7) (2,5,7,9) (7,9,10,12) (20,35,45,60) 2 (5,8,9,12) (2,5,7,9) (1,6,7,12) (5,7,8,11) (5,8,9,12) (15,25,35,45) 3 (1,6,7,12) (5,8,9,12) (7,9,10,12) (1,6,7,12) (2,5,7,9) (10,15,25,30) 4 (2,5,7,9) (5,7,8,11) (5,7,8,11) (5,8,9,12) (1,6,7,12) (5,8,12,15) demand (15,25,25,45) (15,25,35,45) (8,14,16,22) (10,15,25,30) (2,4,6,8) from transportation problems fuzzy, then changed to the transportation problem crisp using robust ranking method [23]. 𝑅(�̃�) = ∫ (0,5)(𝛼𝛼 𝐿 , 1 0 𝛼𝛼 𝑈 )𝑑𝛼 where: 𝛼𝛼 𝐿 , 𝛼𝛼 𝑈 = {(𝑎2 − 𝑎1)𝛼 + 𝑎1, 𝑎4 − (𝑎4 − 𝑎3)𝛼} r(�̃�11) = r (5, 7, 8, 11) = ∫ (0,5)(2 1 0 𝛼 + 5 + 11 − 3𝛼)𝑑𝛼 = ∫ (−0,5𝛼 1 0 + 8)𝑑𝛼 = 7,75 so that: 𝑅(�̃�12) = 6,5, 𝑅(�̃�13) =4,5, 𝑅(�̃�14) = 5,75𝑅(�̃�15) =9,5 𝑅(�̃�21) = 8,5, 𝑅(�̃�22) =5,75, 𝑅(�̃�23) = 6,5, 𝑅(�̃�24) =7,75,𝑅(�̃�25) =8,5 to from p. t. bantining n, b. surarso, and sutimin comparison between zero point and zero suffix methods in fuzzy transportation problems 43 𝑅(�̃�31) = 6,5, 𝑅(�̃�32) =8,5, 𝑅(�̃�33) =9,5 𝑅(�̃�34) = 6,5𝑅(�̃�35) =5,75 𝑅(�̃�41) = 5,75, r(c̃42) =7,75, 𝑅(�̃�43) =7,75 ,𝑅(�̃�44) = 8,5𝑅(�̃�45) =5,75 inventory column: r(ã11) = 40r(ã12) =30, 𝑅(�̃�13) = 20, r(ã14) =10 request line: r(b̃11) = 30r(b̃12) =30, r(b̃13) =15, r(b̃14) = 20,r(b̃15) =5 next, enter table transportation crisp. table 2. transportation crisp destination inventory source 1 2 3 4 5 1 7,75 6,5 4,5 5,75 9,5 40 2 8,5 5,75 6,5 7,75 8,5 30 3 6,5 8,5 9,5 6,5 5,75 20 4 5,75 7,75 7,75 8,5 6,5 10 demand 30 30 15 20 5 from table 2, we get the total inventory and demand: ∑ �̃�𝑚𝑖 = 100 ∑ �̃�𝑛𝑗 = 100 can be concluded that ∑ �̃�𝑚𝑖 = ∑ �̃� 𝑛 𝑗 , so the transportation problem is balanced. next calculate the solution feasible using north west corner method table 3. solution feasible method north west corner destination inventory source 1 2 3 4 5 1 7,75 (30) 6,5 (10) 4,5 5,75 9,5 2 8,5 5,75 (20) 6,5 (10) 7,75 8,5 3 6,5 8,5 9,5 (5) 6,5 (15) 5,75 4 5,75 7,75 7,75 8,5 (5) 6,5 (5) thus, transportation total costs = 7,75(30) + 6,5(10) + 5,75(20) + 6,5(10) + 9,5(5) + 6,5(15) + 8,5(5) + 6,5(5) = 697,5. then the optimal solution will be calculated using zero-point method. from from to to jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 38-46 44 table 4. optimal solutions for zero-point method destination inventory source 1 2 3 4 5 1 7,75 (5) 6,5 4,5 (15) 5,75 (20) 9,5 40 2 8,5 5,75 (30) 6,5 7,75 8,5 30 3 6,5 (15) 8,5 9,5 6,5 5,75 (5) 20 4 5,75 (10) 7,75 7,75 8,5 6,5 10 request 30 30 15 20 5 allocation from the method is obtained zero-point, 𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20 𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5 𝑥41 = 10 so that the minimum transportation cost is to 𝑍 = ∑ ∑ 𝑐𝑖𝑗 𝑥𝑖𝑗 𝑛 𝑗 𝑚 𝑖 = 𝑐11(𝑥11) + 𝑐13(𝑥13) + 𝑐14(𝑥14) + 𝑐22(𝑥22) + 𝑐31(𝑥31) + 𝑐35(𝑥35) + 𝑐41(𝑥41) = 7,75(5) + 4,5(15) + 5,75(20) + 5,75(30) + 6,5(15) + 5,75(5) + 5,75(10) = 577,5 calculate the optimal solution using zero suffix method. table 5. optimal solutions to zero suffix method objective inventory source 1 2 3 4 5 1 7,75 (5) 6,5 4,5 (15) 5,75 (20) 9,5 40 2 8,5 5,75 (30) 6,5 7,75 8,5 30 3 6,5 (15) 8,5 9,5 6,5 5,75 (5) 20 4 5,75 (10) 7,75 7,75 8,5 6,5 10 requests 30 30 15 20 5 allocation is obtained from the method zero suffix: 𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20 𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5 𝑥41 = 10 thus, the minimum transportation cost is: 𝑍 = ∑ ∑ 𝑐𝑖𝑗 𝑥𝑖𝑗 𝑛 𝑗 𝑚 𝑖 = 𝑐11(𝑥11) + 𝑐13(𝑥13) + 𝑐14(𝑥14) + 𝑐22(𝑥22) + 𝑐31(𝑥31) + 𝑐35(𝑥35) + 𝑐41(𝑥41) = 7,75(5) + 4,5(15) + 5,75(20) + 5,75(30) + 6,5(15) + 5,75(5) + 5,75(10) = 577,5 from to to from p. t. bantining n, b. surarso, and sutimin comparison between zero point and zero suffix methods in fuzzy transportation problems 45 from the results of the comparison of zero-point methods and zero suffix methods then arranged into a table: table 6. comparison results of zero-point method and zero suffix method method name the number of iterations allocation optimum solution zero point method 7 𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20, 𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5, 𝑥41 = 10 577,5 zero suffix method 6 𝑥11 = 5, 𝑥13 = 15, 𝑥14 = 20, 𝑥22 = 30, 𝑥31 = 15, 𝑥35 = 5, 𝑥41 = 10 577,5 4. conclusions the results of the comparison of zero point and zero suffix methods concluded that in the sixth step the theoretical comparative zero suffix method, has happened iteration while the method in the ninth step just iterates so that zero point method is greater in achieving optimal values than zero suffix method. in numerical comparison the number of allocations and optimal solutions in both methods is equal, while the number of iterations of zero-point method is greater than the zero-suffix method. references [1] p. pandian and g. natarajan, “an appropriate method for real life fuzzy transportation problems”, international journal of information sciences and application, vol. 3, no. 2, pp. 127-134, 2011 [2] a. khoshnava and m.r. mozaffari, “fully fuzzy transportation problem”, journal of new researches in mathematics, vol 1, no. 3, pp 42-54, 2015 [3] sakawa, masatoshi. 1993. fuzzy sets and interactive multiobjective optimization. applied information technology. [4] susilo, frans. 2006. himpunan logika kabur. yogyakarta: graha ilmu. [5] a. thiruppathi and d. iranian, “an innovative method for finding optimal solution to assignment problems”, international journal of innovative research in science engineering and technology, vol 8, issue 8, pp. 7366 – 7370, 2015. [6] b. satheeskumar, g. nagalakshmi, r. nandhini, and t. nanthini, “a comparative study on zsm and lcm in fuzzy transportation problem”, global journal of pure and applied mathematics, vol 13, no 10, pp. 7081–7088, 2017. [7] m.s.a. christi and k.k shoba, “two stage fuzzy transportation problem using symmetric trapezoidal fuzzy numbers”, international journal of engineering inventions, vol. 4, no.11, pp. 7-10, 2015 [8] s. chandrasekaran, g. kokila, j. saju, “ranking of heptagon numbers using the zero suffix method”, international journal of science and research, vol. 4, no.5 pp. 2256-2257, 2013. 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2018 [22] m. babenko, g. paweł, k. tomasz, k. ignat, and s. tatiana, “computing minimal and maximal suffixes of a substring”, theoretical computer science, vol. 638, pp. 112–212, 2016 [23] b. srinivas and g. ganesan, “optimal solution for degeneracy fuzzy transportation problem using zero termination and robust ranking methods”, international journal of science and research, vol 4, issue 1, pp. 929-933, 2012 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: h. yozza, f. yanuar, i. rahmi, n. and p. alisya, “determining of risk factors of lowbirthweight babies in padang, west sumatra using logistic regression analysis”, j. mat. mantik, vol. 6, no. 2, pp. 135-141, october 2020. determining of risk factors of low-birth weight babies in padang, west sumatra using logistic regression analysis hazmira yozza1, ferra yanuar2, izzati rahmi3, nadya putri alisya4 universitas andalas padang, hazmirayozza@sci.unand.ac.id1 universitas andalas padang, ferrayanuar@sci.unand.ac.id2 universitas andalas padang, izzatirahmihg@gmail.com3 universitas andalas padang, nadyaalisya3011@gmail.com4 doi: https://doi.org/10.15642/mantik.2020.6.2.135-141 abstrak. angka kematian bayi adalah salah satu indikator yang digunakan untuk mengukur kualitas hidup dari suatu bangsa. world health organization (who) menyatakan bahwa salah satu penyebab utama dari kematian bayi adalah bayi yang dilahirkan dengan berat badan rendah (bblr). upaya untuk menurunkan kejadian bblr dapat dilakukan mulai pada masa prenatal dengan memonitor faktor resiko yang secara nyata mempengaruhi kejadian bblr tersebut. penelitian ini bertujuan untuk mengidentifikasikan faktor resiko yang secara nyata mempengaruhi kejadian bblr di kota padang, sumatera barat. analisis data dilakukan dengan menggunakan analisis regresi logistik terhadap data ibu melahirkan yang berdomisili di kota padang. disimpulkan bahwa, faktor yang berpengaruh nyata dari kejadian bblr adalah pertambahan berat badan ibu selama hamil, paritas, jarak dengan kelahiran sebelumnya, banyaknya masalah kesehatan selama hamil dan jenis kelamin bayi. kata kunci: bayi berat lahir rendah; faktor resiko; analisis regresi logistik abstract. infant mortality is one of the indicators used to measure the quality of life of a nation. the world health organization (who) stated that one of the main causes of infant mortality is the low birth weight (lbw). efforts to reduce the incidence of lbw can be done by monitoring risk factors that influence the occurrence of lbw in the prenatal phase. this study aims to identify factors that significantly influence the incidence of lbw babies in padang, west sumatra, indonesia. the analysis was carried out by using logistic regression analysis on the data of maternal births domiciled in padang, west sumatra, indonesia. it was concluded that variables that significantly affect the incidence of lbw are maternal weight, parity, distance from a previous birth, problems during pregnancy, and babies’ gender. keywords: low birth weight babies; risk factors; logistic regression analysis jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 135-141 issn: 2527-3159 (print) 2527-3167 (online) mailto:ferrayanuar@sci.unand.ac.id2 mailto:nadyaalisya3011@gmail.com http://u.lipi.go.id/1458103791 jurnal matematika mantik volume 6, no. 2, october 2020, pp. 135-141 136 1. introduction infant mortality is one indicator of the quality of health service and -broadly speakingthe quality of life of a nation. high infant mortality rate is related to the inadequate health services in a country. the world health organization (who) estimates 4,1 million babies in 2017 died before completing the first year of age. more than twothirds of these deaths occur in the early neonatal period [1]. generally, these occur due to low birth weight, that is commonly referred as the incidence of low birth weight (lbw). who defines lbw as weight at birth is less than 2500 grams. this continues as a globally health problem. who estimated that 15% 20% of all births, or more than 20 million births a year, worldwide are low birth weight. almost all cases (96.5%) occur in developing countries [2]. babies born with lbw have a risk of death 4 times higher than those with normal weight [3], based on the indonesian health and demographic survey (skdi) conducted by the national family planning board together with central bureau of statistics of republic of indonesia, the indonesian ministry of health, and usaid, in 2012, the neonatal mortality rate in indonesia was 20 per 1000 live births which tended to stagnate a decade earlier. in 2017, skdi noted that this number decreased to 15 deaths per 1000 births [4]. despite the decline, the infant mortality rate (imr) in indonesia is still very high when compared to imr in other asean countries. using data sourced from the world bank obtained from the website www.theworldbank.org, indonesia's imr in 2017 is 1.3 9.7 times of imr in other asean countries. high imr, both in early neonatal deaths and in infants less than a year old, 35% are due to lbw [5]. efforts to reduce perinatal death can be done by suppressing the number of births weighing less than 2500 grams and this can be done by providing supervision for all pregnant women and correcting the factors that affect fetal and neonatal safety [6]. in the initial stages, supervision can be carried out by identifying risk factors for lbw incidents and the causes of lbw incident increasement. it is necessary to evaluate policies related to health services for pregnant women and infants. many risk factors can affect the incidence of lbw. from a search conducted on several research results, there are factors that influence the occurrence of lbw including pregnant women at the age of less than 20 years or more than 35 years, the distance of pregnancy is too short (less than 2 years), mothers with previous lbw conditions, doing physical work several hours without rest, socioeconomic, poor nutritional status and smokers, drug users and alcohol. other factors which can also be suspected to influence are parity, the amount of weight gain during pregnancy, prenatal care and haemoglobin concentration [7], [8], [9], [10], [11]. in addition, several studies have shown that there is a fairly close relationship between the incidence of lbw and premature birth [12]. most of researcher conducted to determine influencing variables (also called risk factors) of lbw used bivariate analysis, most of them by using chi-square analysis. in this research, we used logistic regression analysis, where the influence of variables is determined simultaneously. in this method, the modeling is carried out by considering the relationship among predictors. in addition, by using this approach, the magnitude of the effect of a variable on lbw events can be determined. this research aims to identify risk factors which are significantly affected by the incident of the lbw in padang, west sumatra and its effect on the incident of the lbw baby. the study is limited to single births and to women who live in the city of padang, west sumatra. http://www.theworldbank.org/ hazmira yozza, ferra yanuar, izzati rahmi, nadya putri alisya determining of risk factors of low-birth weight babies in padang, west sumatra using logistic regression analysis 137 2. theoretical framework 2.1 low-birth weight baby low birth weight (lbw) babies are babies with birth weights of less than 2500 grams regardless of gestational period. birth weight is the weight of a baby weighed within 1 (one) hour after birth [13]. babies born with a weight of 2000 2499 grams are 10 times higher risk of dying than babies born with a body weight of 3000-3499 grams [8]. other research claimed that the risk is 4,1 times higher [3]. there are 3 forms of lbw namely: lbw premature babies, lbw small babies for pregnancy and lbw premature babies and small babies for pregnancy. 2.2 the logistic regression analysis regression analysis in statistics is a statistical method that aims to determine the causal relationship between response variables with one or more predictors. when the response variable is binary data that has only two possible outcomes, ordinary regression analysis can no longer be used to model the relationship between these variables. one approach that can be used in this condition is logistic regression analysis. logistic regression analysis is one of the regression analysis to model the causal relationship between one or more predictors (also called risk factors) and binary response variable that has only two possible outcome, i.e success and failure [14]. this method involves a logit transformation of response data. the logistic regression model that describes the relationship between response variable and k risk factors, 𝑋1, 𝑋2, … , 𝑋𝑘 is 𝑙𝑜𝑔𝑖𝑡(𝜋(𝒙)) = log ( 𝜋(𝒙) 1−𝜋(𝒙) ) = 𝛽0 + 𝛽1𝑋1𝑖 + 𝛽2𝑋2𝑖 + ⋯ + 𝛽𝑘 𝑋𝑘𝑖 + 𝜀𝑖 (1) where 𝜋(𝒙) denotes the probability of success (p(y=1) at specified value of 𝑋1, 𝑋2, … , 𝑋𝑘, 𝛽𝑖′𝑠 denotes model parameters. the parameter 𝛽𝑗 refers to the effect of xj, controlling the other x’s [14]. standard method to estimate the logistic regression parameters is the maximum likelihood estimation. it is based on finding the estimates that maximize likelihood for observed data under the chosen model [15]. interpretation of the effect of risk factors xj on response variables in logistic regression uses the odds ratio. for model (1), odds are defined as 𝜋(𝑥) 1−𝜋(𝑥) = 𝑒𝛽0+𝛽1𝑋1𝑖+𝛽2𝑋2𝑖+⋯+𝛽𝑘𝑋𝑘𝑖. (2) this relationship provides the interpretation of βj; for a risk factor, odds at x+1 is the equal to odds at x multiply by 𝑒𝛽𝑗 . ratio between odss at x+1 and odds at x is usually called odds ratio (or) and formulated as or=𝑒𝛽𝑗 . this ratio measures the influence of each risk factors that is the odds is multiply 𝑒𝛽𝑗 for every increase of xj and at fixed level of other x’s. when 𝛽𝑗 = 0, 𝑒 𝛽𝑗 = 1, the odds do not change as x change. this means that the probability of success is independent of x. in logistic regression, wald test uses to test the hyphotesis 𝐻0 ∶ 𝛽𝑗 = 0 or to assess the significance of an individual regression coeficient. this test based on wald statistic which is asymptotically distributed as a chisquare distribution [14]. 3. methods 3.1 population and sample population in this research is post-portum women and their babies in padang, west sumatra. sample is consist’s of postpartum mothers who were cured at several hospitals/maternity hospitals in padang, west sumatra. jurnal matematika mantik volume 6, no. 2, october 2020, pp. 135-141 138 3.2 data and variables this research used primary data collected from postpartum women in several health facilities in padang. observation is limited to mothers with single births. the response variable measured is the birth weight of the baby which is then used as the basis for grouping babies into two groups, namely: a. low birth weight group, namely babies with birth weights of less than 2500 grams b. normal birth weight group, i.e. babies with birth weights equal or more than 2500 grams predictors or risk factors that are assumed to influence the incidence of lbw babies are: a. mother age (x1) in years b. mother education (x2), is a categorical variable with categories: low education, middle education and higher education c. weight gain during pregnancy (x3), in kgs d. mother's pre-pregnancy weight (x4), in kgs e. mother hemoglobin level (x5), f. parity (x6), g. birth spacing to previous birth (x7), in years h. number of health problems during pregnancy (x8). this is a categorical variable. based on this variable, observation can be categorized into: no-problem, one problem, more than one problem. i. frequency of prenatal visit (x9) j. sex of the baby (x10). this is a categorical variable with male and female category. male baby was treated as reference category 3.3 data analysis data were analyzed using logistic regression analysis, a regression method that aims to model the relationship between risk factors and binary response that has only two possible categories, “success” and “failure”. in this research, the incidence of lbw is denoted as success. in the logistic regression analysis, the relationship between response variable and predictor variables is described by the logistic regression model (model (1)). the logistic regression model constructed to describe the relationship between the incidence of lbw and it’s assumed risk factors is: 𝑙𝑜𝑔𝑖𝑡(𝜋(𝒙)) = log ( 𝜋(𝒙) 1−𝜋(𝒙) ) = 𝛽0 + 𝛽1𝑋1𝑖 + 𝛽2𝑋2𝑖 + ⋯ + 𝛽10𝑋10𝑖 + 𝜀𝑖 (3) in model (3), 𝜋(𝑥) denotes the probability of having a lbw baby. at the first stage of the analysis, all risk factors were included in the model. based on the initial model, wald test was conducted to examine the significance of the effect of each risk factor on lbw event. in the next step, a new model is reconstructed by only involving variables that have a significant effect. finally, odds ratio resulted, 𝑒 𝛽⏞𝑗 is used to estimate risk factors’ effect on the incidence of lbw 4. result and discussion in this study, there were 10 variables that are assumed to influence lbw events, namely maternal age, maternal education, maternal weight gain during pregnancy, maternal weight before pregnancy, hemoglobin levels in maternal blood, parity, distance from previous births, number of health problems that occur during pregnancy, the number of prenatal visits and the sex of the baby. hazmira yozza, ferra yanuar, izzati rahmi, nadya putri alisya determining of risk factors of low-birth weight babies in padang, west sumatra using logistic regression analysis 139 by using logistic regression analysis conducted on mothers giving birth to a single baby who lives in padang, west sumatra, an initial model is obtained. following table shows the result of significance test of variables influence to lbw baby incidences. table 1. test results of the significance of independent variables variable label p-value x1 age 0,971 x2 maternal education z21 low education 0,950 z22 middle education 0,496 x3 weight gain 0,882 x4 maternal pre-pregnancy weight 0,048* x5 haemoglobin level 0,614 x6 parity 0,013 * x7 birth spacing to previous birth 0,010* x8 number of problems during pregnancy z81 1 problem 0,123 z82 > 1 problem 0,017* x9 prenatal visits 0,295 x10 sex of the baby 0,027* * variable significantly influence the incidence of lbw baby at α=5% based on the significance test conducted on a model involving all independent variables, it was concluded that, at significant level of α = 5%, only maternal weight before birth, parity, distance from previous birth, many problems during pregnancy, and the sex of the baby significantly affected the incidence of lbw. for many problems, the dummy variable that influences the lbw occurrence is a dummy variable for more than 1 problem. in the next step, we constructed new model by only involving these influential variables. maximum likelihood method was used to found estimated coefficients that maximize likelihood function. by using maximum likelihood estimation implemented at spss statistical software, this following model is produced. logit(𝑝𝑖) = 0,245 − 0,132𝑋4𝑖 − 2,992𝑋6𝑖 − 0,938𝑋7𝑖 + 5,415𝑋8𝑖 + 3,947𝑋10𝑖 (4) interpretation of the effect of each xj variable on lbw events is done through the odds ratio (or) value, which is obtained from the value of eβj . the odds ratio values for all independent variables entered into the second model are shown in the following table. tabel 2. odds ratio (or) variable label 𝑒 𝛽𝑗 x4 pre-pregnancy weight 0,877 x6 parity 0,050 x7 birth spacing to previous birth 0,392 z82 > 1 problem 24,754 x10 sex of the baby 5,783 from this table can be seen that odd ratio is less than 1 for three variables, i,e, prepregnancy weight, parity and birth spacing. it means that the increase of these variables has effect to reduce the risk to have lbw baby. the increase of number of problems during pregnancy has effect to increase the risk of having lbw baby the odd ratio of pre-pregnancy weight is 0,877, means that an increase in maternal weight before pregnancy by 1 kg will reduce the risk of lbw incidence 0.877 times. parity is the number of times that a womens has given birth to a fetus with a gestational age of 24 weeks or more, regardless of whether the child was born alive or stillborn. in this research, the odds ratio of parity (x6) is 0,050. it is concluded that the risk of lbw will drop to 0.050 times for subsequent child births. table 2 also shows that the odd ratio jurnal matematika mantik volume 6, no. 2, october 2020, pp. 135-141 140 of x7 (birth spacing to previous pregnancy) is 0,392. this term refers to time from one birth to the next pregnancy. from the odd ratio shown in the table 2, it is concluded that a 1 year increase in birth spacing from previous births will reduce the risk of giving birth to a lbw baby by 0.392 times it is also known that the risk of lbw will increase to 24,754 times when there is a problem during pregnancy. since male baby is treated as reference category, or = 5,783 means that the risk of lbw occurrence for female babies is greater 5,783 time compared to male babies. 5. conclusions initially, there were 10 factors assumed to influence the incidence of lbw babies in padang, west sumatra. after analyzing the data with logistic regression analysis, i.e. it can be concluded that the factors that significantly influence lbw events are maternal pre-pregnancy weight, parity, birth spacing to previous births, number problems during pregnancy, and sex of the baby. there is not enough evidence to conclude that maternal age and education, weight gain during pregnancy and number of antenatal visits influence the incidence of lbw babies. the increase of pre-pregnancy weight, parity, and birth spacing has an effect on reducing the risk of giving birth to lbw babies. on the contrary, the number of problems during pregnancy has an effect on increasing the risk of giving birth to lbw babies. the risk of giving birth to lbw babies is higher for female babies. references [1] the national development planning agency, "laporan pencapaian millenium development goals indonesia 2017 [2] world health nation, "global nutrition targets 2025: low birth weight policy brief," world health nation, geneva, july 2014. [3] i. p. sari, y. ardillah, and t. a. widyastuti, "the determinat of infant mortality in neonatal period," jurnal kesehatan masyarakat, vol. 12, no. 1, pp. 139-149, july 2016. [4] the national family planning board, "survei kesehatan dan demografi indonesia [in bahasa]," the national family planning board, jakarta, 2018. [5] s. djaja and s. soemantri, "the cause of neonatal death and the attributed health care system in indonesia mortality study of household health survey," bul.penel.kesehatan, vol. 31, no. 2, pp. 153-156., 2003. [6] direktorat kesehatan ri bina kesehatan masyarakat, "manajemen bayi berat lahir rendah (bblr) untuk bidan desa [in bahasa]," indonesia ministry of health, jakarta, 2008. [7] 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, 2019. [8] s. badshah, l. mason, k. mckelvie, l. payne, and p. j. lisboa, "risk factors for low birth weight in the public hospitals at peshawar nwfp-pakistan," biomed central, vol. 8, pp. 197-205, 2008. [9] s. leila, d. robab, and h. somaiasadat, "relationship between maternal haemoglobin concentration and neonatal birth weight," biomed, pp. 373-376, 2013. [10] tonasih and d. kumalasary, "faktor-faktor yang mempengaruhi berat bayi lahir hazmira yozza, ferra yanuar, izzati rahmi, nadya putri alisya determining of risk factors of low-birth weight babies in padang, west sumatra using logistic regression analysis 141 rendah (bblr) di puskesmas wilayah kesehatan harjamukti kota cirebon tahun 2018 [in bahasa]," jurnal riset kebidanan indonesia, vol. 2, no. 1, pp. 21-27, 2018. [11] h. s. joshi et al., "risk factors for low birth weight (lbw) babies and it's medico legal significance," j. indian acad. forensic. med., vol. 32, no. 2, pp. 212-215, 2016. [12] l. d. hailu and d. l. kebede, "determinant of low birth weight among deliveries at a referral hospital in northern ethiopia," biomed research international, vol. 2018, pp. 1-8, april 2018. [13] world health organization, "international statistical classification of disease and related health problems, tenth revision, 2nd edition," world health organization, geneva, 2004. [14] a. agresti, an introduction to categorical data analysis, 2nd ed. new jersey, usa: john wiley and son, 2007. [15] d. g. kleinbaum and m. klein, logistic regression: a self-learning text, 3rd ed. new york, usa: springer-verlag, 2010. s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization how to cite: s. tamaela, y. a. lesnussa, v. y. i. ilwaru, and a. m. balami, “analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization”, j. mat. mantik, vol. 6, no. 2, pp.104-112, october 2020. analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization stevanny tamaela1, yopi andry lesnussa2, v. y. i. ilwaru3, a. m. balami4 1universitas pattimura ambon, annytamaela@gmail.com 2universitas pattimura ambon, yopi_a_lesnussa@yahoo.com 3universitas pattimura ambon, vennilwaru007@gmail.com 4universitas pattimura ambon, abdulmalikbalami@gmail.com doi: https://doi.org/10.15642/mantik.2020.6.2.104-113 abstrak. peminatan peserta didik adalah suatu pembelajaran berbasis minat peserta didik sesuai kesempatan belajar yang ada dalam satuan pendidikan. penyelenggaraan pendidikan dalam satuan pendidikan di sma berdasarkan kurikulum 2013 terdapat program penentuan peminatan bagi peserta didik sma yang dilaksanakan di kelas x. peminatan dalam kurikulum 2013 di sma adalah kelompok peminatan ipa dan ips. penelitian ini menggunakan metode support vector machine (svm) dan metode simple additive weighting (saw) yang bertujuan untuk membandingkan tingkat keakuratan tiap metode dalam sistem pengambilan keputusan (spk) peminatan ipa dan ips pada sma negeri 1 ambon. dari hasil penelitian diperoleh hasil pemilihan peminatan dari metode saw berbeda dengan data riil, sedangkan hasil dari metode svm menunjukan hasil yang sama dengan pemilihan peminatan riil di sma negeri 1 ambon. kata kunci: kurikulum 2013; support vector machine; simple additive weighting abstract. the specialization of students is a learning based on the interests of students according to learning opportunities that exist in educational units. providing education in high school education units based on the 2013 curriculum there is a program for determining specialization for high school students held in class x. specialization in the 2013 curriculum in high schools is the specialization group for natural sciences and social sciences. this study uses the support vector machine (svm) method and the simple additive weighting (saw) method which aims to compare the accuracy of each method in decision making (spk) specialization program in the natural science and social sciences at sma negeri 1 ambon. from the research results, the results of the specialization selection from the saw method differ from the real data, while the results of the svm method show the same results as the selection of real specialization in sma negeri 1 ambon keywords: 2013 curriculum; support vector machine; simple additive weighting jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 104-113 issn: 2527-3159 (print) 2527-3167 (online) mailto:vennilwaru007@gmail.com3 http://u.lipi.go.id/1458103791 s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization 105 1. introduction the curriculum is the most important part of the national education system, whose existence has existed since the beginning of the existence of national education [1]. the curriculum is a plan about the formation of abilities and character of children based on a standard [2], 2013 curriculum is a curriculum that simplifies[3] and thematic-integrative, adding rainy hours to encourage students . one of the forms is class division specialization at the senior high school level [4]. specialization aims to provide opportunities for students to develop their interests in a group of subjects by their scientific interests in higher education[5] and develop their interest in a particular discipline or skill[6]. according to [6] decision-making system is the process of choosing between two or more alternative actions to achieve goals or objectives. several studies have concluded that various decisionmaking methods can determine reliable results. however, in this case, two methods will be applied including (1) the support vector machine (svm) algorithm, the svm method is a technique for making predictions, both in the case of classification and regression [7]. svm was introduced by boser, guyon, vanpik, and was first presented in 1992 [7] at the annual workshop on computational learning theory. the basic concept of svm is a harmony of computational theories that existed decades before, such as the hyperplane margin introduced by aronszajn in 1950 however, up until 1992 there had never been any attempt to assemble these components. the basic principle of svm is a linear classifier [8] and subsequently developed to work on non-linear problems by incorporating the concept of kernel tricks in high-dimensional space [9]. the main purpose of svm is to increase speed in training and testing so that svm can be used for large data (2) simple additive weighing (saw) method, the basic concept of the saw method is to find the weighted sum of the performance ratings for each alternative on all attributes [10][11]. saw can carry out a more precise assessment because it is based on criteria values [12] and weights that have been determined and can select the best alternative. larger values indicate that alternatives are preferred [13]. the simple additive weighting (saw) method is recommended to solve the selection problem in a multi-process decision-making system [14]. simple additive weight (saw) method is a method that is widely used in decision making that has many attributes [15]. in this paper, we will compare the accuracy of svm methods and saw methods in decision making system to determine the specialization in sma negeri 1 ambon. 2. methods the type of research used is a case study, by comparing the svm method with the saw method to determine the choice of interest in tenth grade students. the material used in this study is secondary data obtained from sma negeri 1 ambon. secondary data taken from sma negeri 1 ambon in the form of a value criterion as a measure of interest selection include the initial test scores in sma 1 or a comparative value to determine a major course in tenth grade students. the simple additive weighting (saw) work process flowchart can be seen in the following figure: jurnal matematika mantik volume 6, no. 2, october 2020, pp.104-113 106 figure 1. simple additive weighting (saw) work process flow chart in detail the workflow diagram of the support vector machine (svm) can be seen in the following figure: figure 2. the work process of support vector machine (svm) svm linear search for hyperplane with the largest margin, known as the 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) start enter data from each criterion on each alternative criteria weight make a decision matrix normalization of the decision matrix multiply the decision matrix by criteria weight preference for each alternative finish start divide data into training and testing yes svm classification if score > 0 no natural science (ipa) interest social science (ips) interest classification results finish s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization 107 where 𝑦𝑖 is the label vector support class 𝑋𝑖. 𝑋 𝑇 is a test tuple; 𝛼𝑖 and 𝑏0 are numerical parameters that are automatically determined by the svm algorithm, and l is the number of support vector. 3. results and discussion 3.1. saw method for this research method, the variables used for data are the national examination score (c1), entrance test score (c2), psychological value (c3). the amount of data in this study amounted to 50 data. the interest or major divide in 2 part, such as: natural sciences department (ipa) and social sciences department (ips). table 1. research data no ne score (rek) initial score value psychology score final score major/ interest 1 82 35 106.7 74.57 ipa 2 80.2 34 86.2 66.8 ips 3 80 34 100 71.33 ipa 4 81.5 34 94.4 69.97 ips 5 80 34 93.9 69.30 ips 6 85.5 33 111.8 76.77 ipa 7 84.5 33 111.9 76.47 ipa 8 80 33 86.1 66.37 ips 9 81.5 33 109.6 74.70 ipa 10 80.8 32 106.3 73.03 ipa 11 80 32 97 69.67 ips 12 83 32 106.7 73.90 ipa 13 82.5 32 92.6 69.03 ips 14 81.9 31 107.7 73.53 ipa 15 80.5 31 92.6 68.03 ips 16 80 31 98.7 69.90 ips 17 81.5 31 83.3 65.27 ips 18 80 31 91.3 67.43 ips 19 85.5 30 90.6 68.70 ips 20 84.5 30 101.3 71.93 ipa 21 80 30 89.7 66.57 ips 22 80 30 93.6 67.87 ips 23 83 30 88.3 67.10 ips 24 82.5 30 111.4 74.63 ipa 25 81.9 30 97.7 69.87 ips 26 80.5 30 109.6 73.37 ipa 27 80 30 104.5 71.50 ipa 28 81.5 30 100 70.50 ipa 29 83 30 91.1 68.03 ips 30 82.5 30 103.9 72.13 ipa 31 81.9 30 100 70.63 ipa 32 80.5 28 104.2 70.90 ipa 33 83.9 28 106 72.63 ipa jurnal matematika mantik volume 6, no. 2, october 2020, pp.104-113 108 no ne score (rek) initial score value psychology score final score major/ interest 34 84 28 92.9 68.30 ips 35 80.5 28 87.7 65.40 ips 36 80.4 28 86.9 65.10 ips 37 81.3 28 111.9 73.73 ipa 38 81.9 28 92.5 67.47 ips 39 81 28 86.8 65.27 ips 40 80.2 28 105 71.07 ipa 41 82 27 109.9 72.97 ipa 42 83.6 27 110.5 73.70 ipa 43 80 27 93.8 66.93 ips 44 82.5 27 94.9 68.13 ips 45 81.6 27 91.3 66.63 ips 46 85 27 106.6 72.87 ipa 47 82.9 27 106 71.97 ipa 48 83.5 27 94.7 68.40 ips 49 80 27 93.3 66.77 ips 50 80 27 111.1 72.70 ipa in this research process, the calculation process will be carried out using sample data from students who will register at sman 1 ambon, using the saw method. alternative data used in the study are in table 2. table 2. alternative data alternatives ai a1 natural sciences (ipa) a2 social sciences (ips) the criteria used to determine each alternative above are the national examination score, initial test score, and psychological test score. departmental target data is divided into 3 groups with the following conditions: department of natural sciences for a value of ≤ 85 with very clever provisions department of neutral for a value of 70 ≤ y ≤ 80 with the provisions of the average ministry of social sciences for low value of 70 with a low provision note: y = major based on subject criteria, it is determined that preference (w) is = {30, 30, 40}. after preference weights are determined, a matrix is made based on the previous weighting table. then, the x matrix is normalized, based on the equation of the saw method to obtain an r normalized matrix, then the normalized matrix r is multiplied by w which is the weight of the predetermined preference. s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization 109 table 3. c ++ software results table no rij un rij ta rij sp vij recommendation 1 0.75 1.00 0.67 79.17 ipa 2 0.75 1.00 0.33 65.83 ips 3 0.75 1.00 0.33 65.83 ipa* 4 0.75 1.00 0.33 65.83 ips 5 0.75 1.00 0.33 65.83 ips 6 1.00 1.00 1.00 100.00 ipa 7 1.00 1.00 1.00 100.00 ipa 8 0.75 1.00 0.33 65.83 ips 9 0.75 1.00 0.67 79.17 ipa 10 0.75 1.00 0.67 79.17 ipa 11 0.75 1.00 0.33 65.83 ips 12 0.75 1.00 0.67 79.17 ipa 13 0.75 1.00 0.33 65.83 ips 14 0.75 1.00 0.67 79.17 ipa 15 0.75 1.00 0.33 65.83 ips 16 0.75 1.00 0.33 65.83 ips 17 0.75 1.00 0.33 65.83 ips 18 0.75 1.00 0.33 65.83 ips 19 1.00 0.50 0.33 58.33 ips 20 1.00 0.50 0.33 58.33 ipa* 21 0,75 0.50 0.33 58.33 ips 22 0.75 0.50 0.33 58.33 ips 23 0.75 0.50 0.33 58.33 ips 24 0.75 0.50 1.00 77.50 ipa 25 0.75 0.50 0.33 50.83 ips 26 0.75 0.50 0.67 64.17 ipa 27 0.75 0.50 0.33 50.83 ipa 28 0.75 0.50 0.33 50.83 ipa* 29 0.75 0.50 0.33 50.83 ips 30 0.75 0.50 0.33 50.83 ipa* 31 0.75 0.50 0.33 50.83 ipa* 32 0.75 0.50 0.33 50.83 ipa* 33 0.75 0.50 0.67 64.17 ipa* 34 0.75 0.50 0.33 50.83 ips 35 0.75 0.50 0.33 50.83 ips 36 0.75 0.50 0.33 50.83 ips 37 0.75 0.50 1.00 77.50 ipa 38 0.75 0.50 0.33 50.83 ips 39 0.75 0.50 0.33 50.83 ips 40 0.75 0.50 0.33 50.83 ipa 41 0.75 0.50 0.67 64.17 ipa* 42 0.75 0.50 1.00 77.50 ipa jurnal matematika mantik volume 6, no. 2, october 2020, pp.104-113 110 no rij un rij ta rij sp vij recommendation 43 0.75 0.50 0.33 50.83 ips 44 0.75 0.50 0.33 50.83 ips 45 0.75 0.50 0.33 50.83 ips 46 1.00 0.50 0.67 71.67 ipa 47 0.75 0.50 0.67 64.17 ipa* 48 0.75 0.50 0.33 50.83 ips 49 0.75 0.50 0.33 50.83 ips 50 0.75 0.50 1.00 77.50 ipa according to table 3, 𝑅𝑖𝑗 un, 𝑅𝑖𝑗 ta, 𝑅𝑖𝑗 sp, and 𝑉𝑖𝑗 representing respectively, the weight of national examination, entrance test, psychological value and the final score. from the final score or final value will use to determine the decision making to choose the specialization in natural sciences department or social sciences department. 3.2. svm method for this research method, the variables used for data are the national examination score (x1), entrance test score (x2), psychological value (x3). while the target or output is class or major course (y). the data used are the same as table 1. this study uses the linear svm method with matlab software to conduct data analysis. a comparison of training and testing data used is 70:30 from each grouping that has been determined. the results of the software for grouping the training data are as follows: table 4. software 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 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 4, it can be seen the svm method classification shows the output with 100% accuracy for training data. and also, class 0 is a group of natural sciences department) and class 1 is a group of social sciences department. this can be seen from the comparison between the target data and the output target. svmstruct1, svmstruct1, s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization 111 svmstruct1 each representing national examination, entrance test, and psychological value. with hyperplan obtained from each class as follows: figure 3. hyperplane for svmstruct1 based on figure 3, 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 4. hyperplane for svmstruct2 based on figure 4, it can be explained that the data in svmstruct1 is correctly classified in class 0 (a group of natural sciences department) or class 1 (a group of social sciences department) while the support vector is a circle within the circle. figure 5. hyperplane for svmstruct3 jurnal matematika mantik volume 6, no. 2, october 2020, pp.104-113 112 based on figure 5, it can be explained that all svmstruct3 data is properly classified into class 0 (which is the natural science department) or class 1 (which is the social sciences department), while the support vector is the circle within the circle. by the purpose of this study to see the level of accuracy between the saw method and the svm method, so the results obtained namely the saw method are found. from the results in table 3 above it can be seen that 12 students can enter the natural science (ipa) class and 38 in the social science (ips) class. this shows that there are differences in decision criteria in the determination a major course according to the results of calculations with the saw method in table 3, with the results of student selection in the data in table 1. some of the different data can be seen in data 3, 20, 28, 30, 31, 32, 33, 41, 47 which are different from the actual data in table 1. from these different data, the results of the saw calculation are more suitable for the ips interest class, but in the real data, it is in the ipa interest class. as for the svm method, the results of the software output matched with the real data are seen that 100% of the output data is the same as the real data. this shows for the svm method of determining a major course carried out by the school according to the ability of children. 4. conclusions from the results and discussion can be conclude that, the results of research by the saw method there are some students who have low criteria but are in the interest or department of natural sciences and vice versa. however, based on the results of research with the svm method there are classifications of interest or majors that are different from the interests or classifications obtained are imperfect classifications. references [1] m. ali, “implementasi kurikulum pendidikan nasional 2013,” j. pedagog., vol. 2(2), pp. 49-60, 2013. [2] m. lestari, “implementation of citizenship character formation by the study of civic education on senior high school in the district of bantul,” e-civics, vol. 5 (6), 2016. [3] suyatmini, “implementasi kurikulum 2013 pada pelaksanaan pembelajaran akuntansi di sekolah menengah kejuruan,” j. pendidik. ilmu sos., vol. 20(1), pp.6068, 2017. [4] subandi, “pengembangan kurikulum 2013,” j. pendidik. dan pembelajaran dasar, vol. 1(1), pp. 18-36, 2014. [5] 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., vol. 9(3), 2012. [6] direktorat pembinaan sma, modul pelatihan implementasi kurikulum 2013 sma tahun 2018. pp. 1-62, 2018. [7] y. lin, h. tseng, and c. fuh, “using support vector machine,” image process., pp.123-130, 2003. [8] m. a. oskoei and h. hu, “support vector machine-based classification scheme for myoelectric control applied to upper limb,” ieee trans. biomed. eng., vol. 55(8), pp. 1956-1965, 2008, doi: 10.1109/tbme.2008.919734. [9] s. vijayakumar and s. wu, “sequential support vector classifiers and regression,” proceedings of international conference on soft computing (soco ‘99), vol. s. tamaela, y. a. lesnussa, v. y. i. ilwaru, a. m. balami, analysis of support vector machine (svm) method and simple additive weighting (saw) method in making decisions to choose specialization 113 1999(619), pp. 610-619, 1999. [10] m. elistri, j. wahyudi, and r. supardi, “penerapan metode saw dalam sistem pendukung keputusan pemilihan jurusan pada sekolah menengah atas negeri 8 seluma,” j. media infotama penerapan metod. saw…issn, vol. 10(2), pp. 18582680, 2014. [11] s. h. sahir, r. rosmawati, and k. minan, “simple additive weighting method to determining employee salary increase rate,” ijsrst, vol. 3(8), pp. 42-48, 2017. [12] e. roszkowska and d. kacprzak, “the fuzzy saw and fuzzy topsis procedures based on ordered fuzzy numbers,” inf. sci. (ny)., vol. 369, pp. 564-584, 2016, doi: 10.1016/j.ins.2016.07.044. [13] s. e. widodo sri, s. lutfi, and solikhin, “sistem pendukung keputusan penilaian kinerja karyawan menggunakan metode simple additive weighting (saw) pada pt. indonesia steel tube work,” sist. inf., 2014. [14] f. fitriani, “sistem pendukung keputusan penentuan jenis rambut manusia dengan menerapkan metode simple additive weighting (saw),” pelita inform. budi darma, 2015. [15] ratnasari and t. susilowati, “sistem pendukung keputusan kelayakan pengajuan kredit sepeda motor pada dealer tunas dwipa matra gadingrejo menggunakan metode saw,” stmik pringsewu lampung, pp. 442-448, 2016. contact: muhammad fajar, mfajar@bps.go.id badan pusat statistik-statistics indonesia, jakarta 10710, indonesia the article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.67-73 modeling of covid-19 epidemic growth curve in indonesia muhammad fajar1, wahyudi2 1,2badan pusat statistik-statistics indonesia, indonesia article history: received jun 26, 2020 revised mar 8, 2021 accepted may 30, 2021 kata kunci: pemodelan, logistik, gompertz, epidemi, covid-19 abstrak. tujuan studi ini adalah untuk melakukan pemodelan parametrik terhadap kurva pertumbuhan epidemi covid-19 sehingga dapat diperoleh nilai maksimum dan waktu pada titik tersebut dari kumulatif kasus covid-19. sumber data yang digunakan adalah data jumlah kumulatif kasus terkonfirmasi positif covid-19 yang berasal dari https://covid19.go.id/. metode yang digunakan dalam penelitian ini adalah fitting data dengan model logistik dan gompertz. hasil penelitian ini adalah (1) model logistik dan gompertz memiliki tingkat akurasi tinggi (konteks fitting data) dalam memodelkan kurva pertumbuhan epidemi covid-19, diindikasikan dari nilai r2 (koefisien determinasi) yang mencapai lebih dari 99%; (2) dari model logistik diperoleh bahwa estimasi jumlah kasus kumulatif maksimum pada akhir epidemi covid-19 adalah 7.714 kasus konfirmasi positif yang dicapai dalam waktu sekitar 82 hari (22 mei 2020) dari tanggal 2 maret 2020 ketika kasus positif covid-19 pertama diumumkan oleh pemerintah; dan (3) dari model gompertz diperoleh bahwa estimasi jumlah kasus kumulatif maksimum pada akhir epidemi covid-19 adalah 33.975 kasus konfirmasi positif yang dicapai dalam waktu sekitar 152 hari (30 juli 2020) dari tanggal 2 maret 2020. temuan ini dapat dijadikan bahan input pembuatan langkah pemerintah dalam mengendalikan penyebaran covid-19. keywords: modeling, logistic, gompertz, epidemic, covid-19 abstract. aim of this study is to make parametric modeling of the covid-19 epidemic growth curve so that the maximum value and time at that point can be obtained from the cumulative cases of covid-19. the data used in this study is the cumulative number of positive confirmed cases of covid-19 from https://covid19.go.id/. the method used in this study is fitting data with the logistic and gompertz models. result of this study are (1) the logistic and gompertz models are very fit in modeling the covid-19 epidemic growth curve, indicated from the value of r2 (coefficient of determination) which reaches more than 99%; (2) from the logistics model it is obtained that the estimated amount of the maximum cumulative case at the end of the covid-19 epidemic is 7,714 positive confirmed cases, achieved in about 82 days (may 22, 2020) from mar 2, 2020, when the first positive covid-19 case was announced by the government; and (3) from the gompertz model, it is obtained that the estimated maximum cumulative case at the end of the covid-19 epidemic is 33,975 positive confirmed cases, achieved in about 152 days (jul 30, 2020) from mar 2, 2020. the results of this study can be used as input to the government to take steps in controlling the spread of covid-19. how to cite: m. fajar and wahyudi, “modelling of covid-19 epidemic growth curve in indonesia”, j. mat. mantik, vol. 7, no. 1, pp. 67-73, may 2021. jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 67-73 issn: 2527-3159 (print) 2527-3167 (online) mailto:mfajar@bps.go.id https://doi.org/10.15642/mantik.2021.7.1.67-73 https://covid19.go.id/ https://covid19.go.id/ http://u.lipi.go.id/1458103791 jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 67-73 68 1. introduction novel coronavirus 2019 (covid-19) is an infectious disease caused by sars-cov2, one of the coronavirus types [5]. this virus can cause pneumonia, which is inflammation of the lung tissue causing interference with oxygen exchange, resulting from breathing, becoming congested. patients with covid-19 can experience fever, dry cough, and breathing difficulties. in indonesia, covid-19 has spread to 32 provinces. as of apr 12, 2020, the cumulative number of positive confirmed cases was recorded at 4,241 cases, with the details of as many as 3,509 patients in the treatment, 359 patients recovered, and 373 patients died [13]. a total of 80.22% of covid-19 positive cases in indonesia came from java island. dki jakarta is the province with the most covid-19 case found, with 2,044 cases of positive confirmed cases, 1,707 patients in the treatment, 142 patients cured, and 195 died [13]. figure 1. number of cumulative cases of covid-19 in indonesia, as of apr 12 2020 figure 1 visually presents the cumulative cases of covid-19 or also called the epidemic curve. when we look at figure 1, the cumulative cases of covid-19 continue to increase over time, and the pattern formed is nonlinear. generally, on the epidemic growth curve, the movement of the data moves nonlinearly to reach a point where the cumulative case number will not increase again (stable data movement) which is visually shaped in a horizontal line pattern. to explore more detailed information on these curves, modeling is needed. modeling in epidemiology [1,2,4,7,9,11,14,15,16] is commonly used to estimate parameters, such as how large the maximum cumulative case can be achieved by an epidemic [2,10] and the epidemic's growth rate and the point of time when the maximum cumulative case is reached. estimations of these parameters can be used as an input for the formation of a mitigation plan for the government. therefore, in this study, the authors modeled the covid-19 epidemic curve with a parametric model (analogy based on references [11,12,17]), namely the logistics and gompertz functions. considerations of using these parametric models include: (1) can predict the time point when reaching the maximum asymptotic peak point in the cumulative 0 500 1000 1500 2000 2500 3000 3500 4000 4500 3 /1 /2 0 2 0 3 /3 /2 0 2 0 3 /5 /2 0 2 0 3 /7 /2 0 2 0 3 /9 /2 0 2 0 3 /1 1 /2 0 2 0 3 /1 3 /2 0 2 0 3 /1 5 /2 0 2 0 3 /1 7 /2 0 2 0 3 /1 9 /2 0 2 0 3 /2 1 /2 0 2 0 3 /2 3 /2 0 2 0 3 /2 5 /2 0 2 0 3 /2 7 /2 0 2 0 3 /2 9 /2 0 2 0 3 /3 1 /2 0 2 0 4 /2 /2 0 2 0 4 /4 /2 0 2 0 4 /6 /2 0 2 0 4 /8 /2 0 2 0 4 /1 0 /2 0 2 0 4 /1 2 /2 0 2 0 m, fajar and wahyudi modelling of covid-19 epidemic growth curve in indonesia 69 data of a disease (in this case is covid-19), (2) can predict how large the maximum cumulative covid-19 case is achieved, (3) ease and calculation speed in parameters estimation process is easier and faster than generalized logistic model, and (4) unlike the sir model (susceptible, infectious, recovered) which is commonly used for studies of disease transmission (epidemics) [5, 14] and requires assumptions, among others [8]: (a) the population is constant, (b) the birth rate is equal to the death rate in the population, (c) changes in susceptible and infected individuals are proportional to the population, (d) infected individuals assumed to be able to recover with constant chance over time, (e) constant speed in terms of transmission and recovery, and (f) it is assumed that once the person has been infected and then has recovered, then that person will not be re-infected due to strong immunity, the logistic and gompertz models do not require these assumptions. the difference in the length of time in reaching the inflection points of the two models can be used as the basis for an optimistic and pessimistic scenario in controlling covid-19 with the assumption that there is no intervention, meaning that the growth process of the covid19 epidemic runs naturally without intervention. the implication is for stakeholders to manage the risk of an ongoing epidemic so that they can make the right steps. 2. methods 2.1 data source the data used in this study is daily data on the cumulative number of covid-19 positive confirmed cases in indonesia as of apr 12 2020, sourced from www.covid19.go.id [13]. 2.2 growth model the cumulative growth of covid-19 cases (by analogy as referenced [11, 12, 17]) indicates a phase, in which the growth rate starts from zero at t = 0, and then accelerates to 𝜇𝑚 within a certain time interval of 𝜆. when heading to the final phase of the epidemic, growth rates are decreasing to reaching zero (it means the rate of addition of new cases is very low so that the number is approaching zero even reaches zero, the implications of the cumulative case data plot will be horizontally shaped), and the number of final cumulative cases is achieved asymptotically (maximum, a). figure 1. epidemic growth curve time jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 67-73 70 in plotting1 data on the cumulative number of covid-19 cases, the authors used the parametric model as follows: a. logistics model [11,12,17] 𝑦𝑡 = 𝐴 1 + exp ( 4𝜇𝑚 𝐴 (𝜆 − 𝑡) + 2) , (1) where 𝑦𝑡 is the cumulative number of covid-19 cases at time t, 𝐴 is the maximum cumulative case at the end of the epidemic, 𝜇𝑚 is the maximum growth rate, 𝜆 is the time point when it reaches 𝜇𝑚 . when 𝑦𝑡 = 𝐴/2, the time needed is based on equation (1): 𝑡𝐴 2 ,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘 = 𝜆 + 𝐴 2𝜇𝑚 (2) from equation (2) it can be obtained the time needed to reach 𝐴 asymptotically is: 𝑡𝐴,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘 = 2𝑡𝐴 2 ,𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑘 (3) b. model gompertz [11,12,17] 𝑦𝑡 = 𝐴 exp (− exp ( 𝜇𝑚𝑒 𝐴 (𝜆 − 𝑡) + 1)) , (4) where 𝑒 is a natural number (2.718…). when 𝑦𝑡 = 𝐴/2, the time needed is based on equation (4): 𝑡𝐴 2 ,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧 = 𝜆 − 𝐴(ln(ln 2) − 1) 𝜇𝑚𝑒 (5) from equation (5) it can be obtained the time needed to reach 𝐴 asymptotically, is: 𝑡𝐴,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧 = 2𝑡𝐴 2 ,𝑔𝑜𝑚𝑝𝑒𝑟𝑡𝑧 (6) in these two models, there are parameters 𝐴, 𝜇𝑚, and 𝜆 which must be estimated. in the estimation process, this study uses r software with a nonlinear square estimation algorithm [3]. 3. results and discussions the following figure 2 is a visual presentation of the results of data plotting with the model proposed by the author in the previous section. 1 plotting is done based on the origin o. m, fajar and wahyudi modelling of covid-19 epidemic growth curve in indonesia 71 (a) (b) figure 2. (a) fitting the plot data with the logistic model, (b) fitting the plot data with the gompertz model notes: o states the actual data, states the fitting model, and states the slope of the model from figure 2, visually, both models are accurate in plotting data patterns. this is indicated by the value of r2 (coefficient of determination) generated by the logistic and gompertz models which are 99.571% and 99.789%, respectively. that is, variations of response variables (in this case the cumulative covid-19 case) can be explained by both models by more than 99%, while the remaining less than 1% are explained by other variables outside the model. by using the nonlinear least square estimation mechanism, the estimation results obtained are presented in table 1, as follows, table 1. the estimated results of the logistics model and gompertz which are visualized in figure 2. no model r2 𝑨 𝝁𝒎 𝝀 𝒕𝑨 𝟐 𝒕𝑨 1 logistic 99.571% 7,713.719 [6125.587, 9301.852] 253.860 [220.735, 286.984] 25.961 [24.213, 27.709] 41.154 82.307 2 gompertz 99.789% 33,975.144 [16247.870, 51702.420] 409.535 [269.645, 549.426] 34.356 [28.378, 40.335] 76.062 152.123 note: 95% confidence interval [lower limit, upper limit] based on table 1, there are few information that can be obtained, namely: (1) from the logistics model it is found that the estimated maximum cumulative cases at the end of the covid-19 epidemic is 7,714 positive confirmed cases achieved in about 82 days (may 22, 2020) from mar 2, 2020, when the first positive covid-19 case was announced by the government and the maximum growth rate achieved was the slope of the model, which was 253.860% (it means when the λ = 25.961 days from the start of the covid-19 epidemic began, then the maximum growth rate reached by the epidemic was 253.860%). (2) from the gompertz model, it is found that the estimated maximum cumulative cases at the end of the covid-19 epidemic is 33,975 positive confirmed cases achieved in about 152 days (jul 30, 2020) from mar 2, 2020 when the first positive covid-19 case was announced by the government and the maximum growth rate achieved is the slope of the model, which is 409.535% (that is, when the time λ = 34.356 days from the start of the covid-19 epidemic began, the maximum growth rate achieved by the epidemic is 409.535%). jurnal matematika mantik vol. 7, no. 1, may 2021, pp. 67-73 72 based on the results of the estimated maximum cumulative number and time2 needed to reach the maximum cumulative case from table 1, it can be considered that the logistic model is an optimistic scenario. in contrast, the gompertz model is a pessimistic scenario, because: (1) the estimated maximum cumulative case achieved by covid-19 from the logistic model is lower than the results of the gompertz model, and (2) the estimated time from the logistic model to reach the maximum cumulative case is almost twice as fast as the time obtained from the gompertz model. therefore, this information can be used as input for the covid-19 control policy. for example, according to an optimistic scenario (logistics model), it is predicted that the peak of the covid-19 epidemic occurred on may 22, 2020 (about 1 week before idul fitri) with a cumulative number of 7,714 cases, so the government has to do a lockdown policy in major cities throughout provinces in java (not only in dki jakarta) by prohibiting people returning home to celebrate lebaran and encouraging people to stay at home to reduce the number of positive cases does not reach the maximum point. if government authorities assume that the growth of the covid-19 epidemic follows the gompertz model, then the government only needs to implement social distancing instead of lockdown or quarantine. 4. conclusions based on the results of the discussion, it can be concluded that: (1) logistics and gompertz models are very fit in modeling the covid-19 epidemic growth curve, indicated from the value of r2 (coefficient of determination) which reaches more than 99%, (2) from the logistic model it is obtained that the estimation the maximum cumulative cases at the end of the covid-19 epidemic is 7,714 positive confirmed cases which are achieved in about 82 days (may 22, 2020) from mar 2, 2020, when the government announced the first positive covid-19 case. (3) from the gompertz model, it is obtained that the estimated maximum cumulative cases at the end of the covid-19 epidemic is 33,975 positive confirmed cases which are achieved in about 152 days (jul 30, 2020) from mar 2, 2020, when government announced the first positive covid-19 case. from this finding, the results of this study can be used as input to the government to take steps in controlling the spread of covid-19. references [1] ahmadi a, fadai y, shirani m, rahmani f. modeling and forecasting trend of covid-19 epidemic in iran. 2020. doi: https://doi.org/10.1101/2020.03.17.20037671. [2] arino j, brauer f, van den driessche p, watmough j, wu j. a final size relation for epidemic models. math biosci eng. 2007;4:159. doi: 10.3934/mbe.2007.4.159. [3] bates d m, watts d g. nonlinear regression analysis and its applications, wiley. 1988. doi: 10.1002/9780470316757. 2 the maximum cumulative number of cases and the length of time to reach the cumulative number of cases can be used as a proxy indicator of whether an attempt to control an outbreak is effective or ineffective. https://doi.org/10.1101/2020.03.17.20037671 m, fajar and wahyudi modelling of covid-19 epidemic growth curve in indonesia 73 [4] datolli g, di palma e, licciardi s, sabia e. a note on the evolution of covid-19 in italy. 2020. https://arxiv.org/pdf/2003.08684.pdf. [5] gorbalenya a e, baker s c, baric r s, de groot r j, drosten c, et al. 2020. severe acute respiratory syndrome-related coronavirus: the species and its viruses a statement of the coronavirus study group. biorxiv. 2020. doi: https://doi.org/10.1101/2020.02.07.937862. 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[17] zwietering m, jongenburger i, rombouts f, van ’t riet k. 1990). modeling of the bacterial growth curve. appl & env micr. 1990; 56:1875. doi: 10.1128/aem.56.6.1875-1881.1990. https://arxiv.org/pdf/2003.08684.pdf https://doi.org/10.1101/2020.02.07.937862 https://web.stanford.edu/~jhj1/teachingdocs/jones-on-r0.pdf https://web.stanford.edu/~jhj1/teachingdocs/jones-on-r0.pdf https://arxiv.org/ftp/arxiv/papers/2003/2003.05447.pdf http://econfin.massey.ac.nz/school/publicati-ons/discuss/2020/dp2004.pdf https://doi.org/10.1371/jou-rnal.pone.0178691 http://www.covid19.go.id/ paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: r. i. hasan, “existence, uniqueness, and stability solutions of non-linear system of integral equations”, j. mat. mantik, vol. 6, no. 2, pp. 76-82, october 2020. existence, uniqueness, and stability solutions of nonlinear system of integral equations rizgar issa hasan duhok polytechnic university, rizqar.issa@dpu.edu.krd doi: https://doi.org/10.15642/mantik.2020.6.2.76-82 abstrak. tujuan dari penelitian ini adalah mempelajari keberadaan, keunikan dan solusi stabilitas sistem baru nonlinier persamaan integral dengan menggunakan metode pendekatan picard (pendekatan berurutan) dan teorema titik tetap banach. studi tentang persamaan integral nonlinier semacam itu bersifat lebih umum dan mendorong kita untuk memperbaiki guna memperluas hasil butris. teorema tentang keberadaan dan keunikan solusi ditetapkan dalam beberapa kondisi yang diperlukan dan cukup pada domain tertutup dan terbatas (ruang kompak). kata kunci: metode pendekatan picard; teorema titik tetap banach; persamaan integral abstract. the aim of this work is to study the existence, uniqueness, and stability solutions of a new nonlinear system of integral equation by using picard approximation (successive approximation) method and banach fixed point theorem. the study of such nonlinear integral equations is more general and leads us to improve to extend the result of butris. theorems on the existence and uniqueness of a solution are established under some necessary and sufficient conditions on closed and bounded domains (compact spaces). keywords: picard approximation method; banach fixed point theorem; integral equation jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 76-82 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 rizgar issa hasan existence, uniqueness, and stability solutions of nonlinear system of integral equations 77 1. introduction integral equation has been arisen in many mathematical and engineering field, so that solving this kind of problems are more efficient and useful in many research branches. analytical solution of this kind of equation is not accessible in general form of equation and we can only get an exact solution only in special cases. but in industrial problems we have not spatial cases so that we try to solve this kind of equations numerically in general format. many numerical schemes are employed to give an approximate solution with sufficient accuracy [3,4,6,10]. many authors create and develop successive approximation method and banach fixed point theorem [1, 2,5,7,8,9] and schemes to investigate the solution of integral equations describing many applications in mathematical and engineering field. burris [1] has been used picard approximation (successive approximation) method and banach fixed point theorem. which were introduced by rama [6] to study the solution of volterra integral equation of the second kind which has the form: 𝑢(𝑡) = 𝑓(𝑡) + ∫ 𝐹 ( 𝑡 𝑎 𝑡, 𝑠)𝓊(𝑠 )𝑑𝑠. in this equation the functions f(t ) and f(t, s ) are continuous on the interval 0 ≤ t ≤ a and square region 0 ≤ t , s ≤ b. definition 1. let 0 { ( )} m m f t  = be a sequence of functions defined on a set. e ⊆ 𝑅1 we say that 0 { ( )} m m f t  = converges uniformly to the limit function f on e if , given  > 0 there exists a positive integer n such that : ( ) ( ) m f t f t− <  , ( ,m n t e  ) theorem1. if f is continuous on [ , ]a b and if ( ) ( ) x a f x f t dt=  , a x b  , then ( )f x is also continuous on [ , ]a b definition 2. let 𝑓 be a continuous function defined on a domain 𝐺 = {(𝑡, 𝑥): 𝑎 ≤ 𝑡 ≤ 𝑏, 𝑐 ≤ 𝑥 ≤ 𝑑} . then 𝑓 is said to satisfy a lipschitz condition in the variable 𝑥 on 𝐺, provided that a constant 𝐿 > 0 exists with the property that |𝑓(𝑡, 𝑥1) − 𝑓(𝑡, 𝑥2)| ≤ 𝐿|𝑥1 − 𝑥2| , for all (𝑡, 𝑥1), (𝑡, 𝑥2 ) ∈ 𝐺. the constant 𝐿 is called a lipschitz constant for 𝑓. definition 3. a solution 𝑥(𝑡) is said to be stable if for each > 0 , there exists a 𝛿 > 0 such that any solution �̅�(𝑡) which satisfies ‖�̅�(𝑡0) − 𝑥(𝑡0)‖ < 𝛿 for some 𝑡0 , also satisfies ‖�̅�(𝑡) − 𝑥(𝑡)‖ < for all 𝑡 ≥ 𝑡0 . definition 4. let e be a vector space a real-valued function ‖. ‖ of 𝐸 into 𝑅1 called a norm if satisfies: i. ‖𝑥‖ ≥ 0 for all 𝑥 ∈ 𝐸, ii. ‖𝑥‖ = 0 if and only if 𝑥 = 0, iii. ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖ for all 𝑥, 𝑦 ∈ 𝐸, iv. ‖𝛼 𝑥‖ = |𝛼| ‖𝑥‖ for all 𝑥 ∈ 𝐸 and 𝛼 ∈ 𝑅. definition 5. a linear space 𝐸 with a norm defined on it is called a normed space. definition 6. a normed linear space 𝐸 is called complete if every cauchy sequence in 𝐸 converges to an element in. definition 7. a complete normed linear space is a banach space. definition 8. if 𝑇 maps 𝐸 into itself and 𝑧 is a point of 𝐸 such that 𝑇𝑧 = 𝑧 , then 𝑧 is a fixed point of 𝑇 . definition 9. let (c [0, t], ‖. ‖ ) be a norm space if 𝑇 maps into itself we say that 𝑇 is a contraction mapping on c [0,t] if there exists 𝛼 ∈ 𝑅 with 0 < 𝛼 < 1 such that ‖𝑇𝑥 − 𝑇𝑦‖ ≤ 𝛼‖𝑥 − 𝑦‖, (𝑥, 𝑦) ∈ c [0, t]. jurnal matematika mantik volume 6, no. 2, october 2020, pp. 76-82 78 theorem 2. let 𝐸 be a banach space, if 𝑇 is a contraction mapping on 𝐸 then 𝑇 has one and only one fixed point in. (for the definitions and theorems see [1,5,6]). our work is extending some results of butris [1], by using the same method above. consider the following integral equation: 𝑢(𝑡) = 𝑢0 + ∫ 𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢 (𝑠))𝑑𝑠 + 𝑡 0 ∫ 𝐾 ( 𝑏(𝑡) 𝛼(𝑡) 𝑡, 𝑠)𝓊(𝑠)𝑑𝑠 (1) where t∈ 𝐷 ⊆ 𝑅1,d is a compact set. suppose the functions 𝑔(t, u), 𝑎(t) and 𝑏(t) are defined, continuous on the domain: 𝐷= {(t,s) : 0 ≤s≤ t ≤ b } (2) also the function 𝑔(t, u) and kernels f(t, s) and k(t,s) satisfied the following inequalities: ‖𝑔(𝑡, 𝓊)‖ ≤ m (3) ‖𝑔(𝑡, 𝓊1) – 𝑔(𝑡, 𝓊2)‖ ≤ 𝐿 ‖𝓊1 – 𝓊2‖ (4) ∫ 𝑏(𝑡) 𝛼(𝑡) ‖ k(t, s)‖ds ≤ kh , h = ‖𝑏(𝑡) − 𝑎(𝑡)‖ (5) ‖ f(t, s)‖ ≤ 𝑁 (6) where f(t, s) and k(t,s) are kernels of the integral equation (1) and m, l,k , 𝑁 > 0. define a sequence of functions {𝑢𝑚(𝑡)}𝑚=0 ∞ by 𝑢𝑚+1 (𝑡 )=𝓊0 + ∫ 𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢𝑚 (𝑠))𝑑𝑠 𝑡 0 + ∫ 𝐾(𝑡, 𝑠)𝑢𝑚(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) (7) with 𝓊0(0) = 𝓊0 , m =0, 1, 2, ... . also define a non-empty set as follows: 𝐷𝑓 = d (mnb + kh δ0), ‖𝑢0‖ ≤ δ0 (8) 2. existence and uniqueness of solution of (1) in this section, we study the existence and uniqueness of equation (1) by using picard approximation method (successive approximation). theorem 3. suppose that the integral equation (1) satisfying the inequalities (2), (3), (4), (5), (6), and relation (8). then there exists a sequence of functions (7) converges uniformly to the limit functions u = u(t) which is define by the integral equation 𝑢(𝑡) = 𝑢0 + ∫ 𝐹(𝑡, 𝑠) 𝑔 (𝑡, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾 (𝑡, 𝑠)𝑢(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 (9) which is a unique solution of (1). proof for m= 0 in (7), we have ‖𝑢1(𝑡) − 𝑢0‖ ≤ ∫ ‖𝑔(𝑠, 𝑢0)‖ 𝑡 0 ‖𝐹(𝑡, 𝑠)‖ 𝑑𝑠 + 𝐾ℎ 𝛿0 ≤ 𝑀𝑁𝑏 + 𝐾ℎ 𝛿0 so that ‖𝑢1(𝑡) − 𝑢0‖ ≤ 𝑀𝑁𝑏 + 𝐾ℎ 𝛿0 for all t∈ [0, 𝑏], i. e. 𝑢1(𝑡) ∈ 𝐷 for all and 𝑢0 𝜖 𝐷𝑓 . by mathematical induction, we can prove that 𝑢m(t) ∈ d , for all [0, 𝑏] and 𝑢0 𝜖 𝐷𝑓 , m = 1,2,3,.... that is ‖𝑢𝑚(𝑡) − 𝑢0‖ ≤ 𝑀𝑁𝑏 + 𝐾ℎ 𝛿0 (10) for all 𝑡𝜖 [0, 𝑏] and 𝑢0 𝜖 𝐷𝑓. next, we shall prove that (7) convergent uniformly on d. for m = 1 in (7), we have ‖𝑢2(𝑡) − 𝑢1(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)‖𝑢1(𝑡) − 𝑢0‖ for m = 2 in (8), we have rizgar issa hasan existence, uniqueness, and stability solutions of nonlinear system of integral equations 79 ‖𝑢3(𝑡) − 𝑢2(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ) 2 𝑜 ‖𝑢1(𝑡) − 𝑢0‖ and so on, by mathematical induction, we have ‖𝑢𝑚+1(𝑡) − 𝑢𝑚(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ) 𝑚 𝑜 ‖𝑢1(𝑡) − 𝑢0‖ suppose that 𝛿 =( 𝐿𝑁𝑏 + 𝐾ℎ) < 1. then ∑‖𝑢𝑚+1(𝑡) − 𝑢𝑚 (𝑡)‖ ≤ (1 + 𝛿 + 𝛿 2 + 𝛿 3 + ⋯ 𝛿 𝑚 + ⋯ ) ‖𝑢1(𝑡) − 𝑢0‖ ≤ 1 1 − 𝛿 ‖𝑢1 − 𝑢0‖ 𝑘 𝑖=1 therefore, the sequence of functions {𝑢𝑚(𝑡)}𝑚=0 ∞ converges uniformly on the domain d. now, we shall prove that 𝑢(t) ∈c(d). taking 𝑢(𝑡) = 𝑢0 + ∫ 𝐹 (𝑡, 𝑠)𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 and 𝑢𝑚(𝑡) = 𝑢0 + ∫ 𝐹 (𝑡, 𝑠) 𝑔(𝑠, 𝑢𝑚(𝑠)) 𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢𝑚−1(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 therefore, ‖𝑢 (𝑡) − 𝑢𝑚(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ) 𝑚 𝑜 ‖𝑢𝑚(𝑡) − 𝑢 (𝑡)‖ since {𝑢𝑚(𝑡)}𝑚=0 ∞ , is convergent uniformly, then lim 𝑚→∞ 𝑢𝑚(𝑡) = 𝑢(𝑡). i.e. ‖𝑢𝑚(𝑡) − 𝑢(𝑡)‖ ≤∈1 choosing ∈1= ∈ 𝐿𝑁𝑏+𝐾ℎ , we get ‖𝑢𝑚(𝑡) − 𝑢(𝑡)‖ < ∈ and hence u(t) ∈ c(d), for all 𝑡𝜖 [0, 𝑏] and 𝑢0 𝜖 𝐷𝑓 . finally, we shall prove that 𝑢(t) is a unique solution of (1). suppose that �̅�(𝑡) = 𝑢0 + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, �̅�(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)�̅�(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 is another solution of (1). for m = 1 in (8), we have ‖𝑢1(𝑡) − �̅�(𝑡)‖ ≤ (𝐿𝑁𝑏 + 𝐾ℎ)‖�̅�(𝑡) − 𝑢0‖ by mathematical induction, we have ‖𝑢𝑚(𝑡) − �̅�(𝑡)‖ ≤ 𝛿 𝑚 𝑜 ‖�̅�(𝑡) − 𝑢0‖ ≤ 1 1 − 𝛿 ‖𝑢1 − 𝑢0‖ since 𝛿 < 1, then lim𝑚→∞ 𝑢𝑚 = �̅�(𝑡) = 𝑢(𝑡) thus �̅�(t) = 𝑢(t) and hence 𝑢(t) is a unique solution of (1). 3. stability solution of integral equation (1) in this section we study the stability solution of (1) by the following theorem: theorem 4. if the inequalities (3), (4), (5) and relation (7) are satisfied and 𝑤(𝑡) is another solution of (1), then the solution u(t) is stable if satisfies the following inequality: ‖u(t) – ω(t)‖ <∈, for all t > 0 and ∈ > 0 where jurnal matematika mantik volume 6, no. 2, october 2020, pp. 76-82 80 𝜔(𝑡) = 𝜔0 + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, 𝜔(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝜔(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 proof. consider ‖𝑢(𝑡) – 𝜔(𝑡)‖ ≤ ‖𝑢0 − 𝜔0‖ + ∫ 𝐿𝑁 𝑡 0 ‖𝑢(𝑠) − 𝜔(𝑠)‖𝑑𝑠 + ∫ 𝑏(𝑡) 𝑎(𝑡) 𝐾‖𝑢(𝑠) − 𝜔(𝑠)‖ds ≤ ‖𝑢0 − 𝜔0‖ + ( 𝐿𝑁𝑏 +kh) ‖𝑢(𝑡) − 𝜔(𝑡)‖ so ‖𝑢(𝑡) – 𝜔(𝑡)‖ ≤ 1 1 − 𝛿 || 𝑢0 – 𝜔0|| putting ∈2 = ∈ || 𝑢0 – 𝜔0|| ( 1−𝛿) ,then ‖𝑢(𝑡) – 𝜔(𝑡)‖ ≤ ∈. so that the solution 𝑢(𝑡) is stable for all 𝑡 ≥ 0. 4. another method of a solution of the integral equation (1) in this section, we study the existence and uniqueness solution of (1) by using banach fixed point theorem. theorem 5. let all assumptions and conditions of theorem 4 be satisfied. then the solution u(t) is a unique of (1) . proof. let (c[0, b], ‖. ‖ ) be banach space. define a mapping t on c[0, b] by: t 𝑢(t) = 𝑢0 + ∫ 𝐹(𝑡, 𝑠)𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 since 𝑔(t,𝑢) and the kernels f(t, s) and k(t,s) are continuous on the domain(2),then ∫ 𝐹(𝑡, 𝑠)𝑑𝑔(𝑠, 𝑢(𝑢))𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) ) 𝑡 0 is also continuous on the same domain. i.e. t: c[0, b] → c[0, b]. now, we claim that t is a contraction mapping on c[0, b]. let 𝑢(t) and �̅�(𝑡) ∈ d, then ‖t𝑢(𝑡) − t�̅�(𝑡)‖ ≤ ∫ 𝐿𝑁 ‖𝑢(𝑠) − �̅�(𝑠)‖𝑑𝑠 𝑡 0 +∫ 𝐾‖𝑢(𝑠) − �̅�(𝑠)‖𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) ≤ 𝛿‖𝑢(𝑠) − �̅�(𝑠)‖ thus ‖t𝑢(𝑡) − t�̅�(𝑡)‖ ≤ 𝛿‖𝑢(𝑡) − �̅�(𝑡)‖ since 𝛿 < 1, therefore t is a contraction mapping on c[0, b]. then t𝑢(t) = 𝑢(t) and 𝑢(𝑡) = 𝑢0 + ∫ 𝐹(𝑡, 𝑠) 𝑔(𝑠, 𝑢(𝑠))𝑑𝑠 + ∫ 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠 𝑏(𝑡) 𝑎(𝑡) 𝑡 0 hence 𝑢(𝑡) is a unique continuous solution of (1). remark. the picard approximation method given global solution but banach fixed point theorem give us the local solution of the integral equation(1). 5. some examples only in successive approximation. example a. consider the following system of integral equation: u(t) = u0 + ∫ f(t, s)g(s, u (s))ds + t 0 ∫ k ( b(t) α(t) t, s)𝓊(s)ds with 𝑢0 = 0.1 , i = (0,2), s =3.22 f(t,s)=2.23ln(t+0.55), g(s,u(s))= u(s)+4.22s and rizgar issa hasan existence, uniqueness, and stability solutions of nonlinear system of integral equations 81 k(t,s)=2.23(2𝑡−1-6.23), a(t)=t, b(t)=t+h,h=0.01 figure 1. successive approximation of global solution of (1) example b. consider the same system of integral equation but local interval and another functions. u(t) = u0 + ∫ f(t, s)g(s, u (s))ds + t 0 ∫ k ( b(t) α(t) t, s)𝓊(s)ds with u0 = 0.1 , i= [0,2] ,s=3.22 and f(t,s)=5.23e−s 2 , g(s,u(s))= 𝑠2-3s k(t,s)= 5.23tan(t + 2) − 6.23, a(t)=t , b(t)=t+h ,h=0.001. figure 2. successive approximation of local solution of (1) remark. successive approximation using matlab are show in figure (a) and (b). jurnal matematika mantik volume 6, no. 2, october 2020, pp. 76-82 82 6. conclusions this paper provided the existence, uniqueness, and stability solution for non-linear system of integral equations. picard approximation (successive approximation) method and banach fixed point theorem have been used in this study which were introduced by [7]. thus, the non-linear integral equations that we have introduced in the study become more general and detailed than those introduced by butris [1]. 7. acknowledgement the author is grateful to the reviewers for their suggestions and prof. dr. raad n. butris for his scientific orientations, advices, and support references [1] butris, r. n., 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[10] tricomi, f. g., integral equations, turin university, turin, italy, 1965. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: n. khasanah and farikhin, “counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form”, jmm, vol. 6, no. 1, pp. 20-29, may 2020. counter example: the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form nur khasanah1, farikhin2 1uin walisongo semarang, khasanah.nur@walisongo.ac.id 2universitas diponegoro, farikhin@undip.ac.id doi: https://doi.org/10.15642/mantik.2020.6.1.20-29 abstrak: algoritma dalam perhitungan determinan matriks centrosymetric telah diperoleh sebelumnya. algoritma tersebut menunjukkan tahapan dalam perhitungan komputasi dari determinan yang efisien pada matriks centrosymmetric yaitu dengan melakukan perhitungan pada blok matriksnya saja. salah satu blok matriks yamng muncul pada algoritma tersebut adalah matriks hessenberg bawah. meskipun, bentuk matriks lainnya juga memungkinkan muncul pada perhitungan determianan matriks centrosymmetric. oleh sebab itu, artikel ini bertujuan untuk menunjukan kemungkinan kemunculan blok matriks centrosymmetric dan bagaimana algoritma yang telah diperoleh, diterapkan dalam menyelesaikan determinan berbagai jenis matriks centrosymmetric. beberapa contoh dari blok matriks yang berbeda pada determinanan matriks centrosymmetric diberikan pula. contoh tersebut sangat bermanfaat dalam pemahaman lanjut saat menggunakan algoritma ini dengan berbagai kasus yang berbeda. kata kunci: centrosymmetric; determinan; matriks blok; hessenberg bawah abstract: the algorithm for computing determinant of centrosymmetric matrix has been evaluated before. this algorithm shows the efficient computational determinant process on centrosymmetric matrix by working on block matrix only. one of block matrix at centrosymmetric matrix appearing on this algorithm is lower hessenberg form. however, the other block matrices may possibly appear as block matrix for centrosymmetric matrix’s determinant. therefore, this study is aimed to show the possible block matrices at centrosymmetric matrix and how the algorithm solve the centrosymmetric matrix’s determinant. some numerical examples for different cases of block matrices on determinant of centrosymmetric matrix are given also. these examples are useful for more understanding for applying the algorithm with different cases. keywords: centrosymmetric; determinant; block matrices; lower hessenberg jurnal matematika mantik volume 6, no. 1, may 2020, pp. 20-29 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 n. khasanah and farikhin counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form 21 1. introduction centrosymmetric matrix is the matrix with a special structure at its entries that is symmetry about its center. some special properties of centrosymmetric matrix about its special structure are observed before [1-9]. based on the role of orthogonal matrix and special structure of centrosymmetric matrix, the centrosymmetric matrix can be formed as block matrices. these block matrices can be used to construct an efficient algorithm on determinant. the study at [10-14] shows the computational process for inverse and determinant at special matrix, called centrosymmetric matrix. moreover, [15] also working at the analytical process for determinant of centrosymmetric matrix with lower hessenberg as block matrix. the result of the study is only computing block matrix at computational process for determinant centrosymmetric matrix having a half size of centrosymmetric’s size. however, this analytical process only aplicable at one example with lower hesssenberg at certain block matrix at centrosymmetric matrix. on the other side, based on its applications, the other block matrices also appear at centrosymmetric matrix. by the roles of centrosymmetric matrix [16], then the evaluation of previous algorithm at some block matrices is needed. the different block matrices that possibly appear are upper hessenberg, centrosymmetric, lower triangular, upper triangular, diagonal, tridiagonal matrix also appears on its. based on previous study, the algorithm of determinant centrosymmetric matrix can not be applied at general centrosymmetric matrix which will be evaluated at this paper. this study shows the possible different block matrices appear at this matrix and the different treatment for applying the algorithm of determinant of centrosymmetric matrix. 2. preliminaries in this section, there are some basic properties of centrosymmetric matrix before further discussion on determinant of centrosymmetric matrix. definition 1 [14,15]. the 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛 ∈ 𝑅𝑛×𝑛 is a centrosymmetric matrix, if 𝑎𝑖𝑗 = 𝑎𝑛−𝑖+1,𝑛−𝑗+1, 1 ≤ 𝑖 ≤ 𝑛,1 ≤ 𝑗 ≤ 𝑛 or equivalently 𝑱𝒏𝑨𝑱𝒏 = 𝑨, (1) with 𝑱𝒏 = (𝒆𝒏,𝒆𝒏−𝟏,⋯,𝒆𝟏) and 𝒆𝒊 is the unit vector with the i-th elements 1 and others 0. furthermore, based on the definition of centrosymmetric matrix, by using orthogonal matrix, the properties of this matrix is written as : lemma 2 [14,15]. the 𝑨 = (𝑎𝑖𝑗)𝑛×𝑛 ∈ 𝑅𝑛×𝑛(𝑛 = 2𝑚) is centrosymmetric matrix, if and only if 𝑨 has the form: 𝐴 = ( 𝐵 𝐽𝑚𝐶𝐽𝑚 𝐶 𝐽𝑚𝐵𝐽𝑚 ), and 𝑄𝑇𝐴𝑄 = ( 𝐵 −𝐽𝑚𝐶 0𝑚 0𝑚 𝐵 +𝐽𝑚𝐶 ) (2) where 𝑩 ∈ 𝑅𝑚×𝑚, 𝑪 ∈ 𝑅𝑚×𝑚 and 𝑄 = √2 2 ( 𝐼𝑚 𝐼𝑚 −𝐽𝑚 𝐽𝑚 ). proof. 𝑄𝑇𝐴𝑄 = √2 2 ( 𝐼𝑚 −𝐽𝑚 𝐼𝑚 𝐽𝑚 )( 𝐵 𝐽𝑚𝐶𝐽𝑚 𝐶 𝐽𝑚𝐵𝐽𝑚 ) √2 2 ( 𝐼𝑚 𝐼𝑚 −𝐽𝑚 𝐽𝑚 ) = ( 𝐵 − 𝐽𝑚𝐶 0𝑚 0𝑚 𝐵 +𝐽𝑚𝐶 ).∎ therefore, the computing determinant block matrix only, 𝑩−𝑱𝒎𝑪 and 𝑩+ 𝑱𝒎𝑪, is same as computing determinant of centrosymmetric matrix. jurnal matematika mantik volume 6, issue 1, may 2020, pp. 20-29 22 3. results and discussion 3.1 the algorithm of determinant of centrosymmetric matrix based on the algorithm of determinant of lower hessenberg matrix, this algorithm can be used to compute the determinant of centrosymmetric matrix with lower hessenberg as block matrices. the steps of determinant of centrosymmetric matrix with lower hessenberg form as block matrices are written as follows [14,15]. a. construct block centrosymmetric matrix by the lemma 2 and based on orthogonal matrix then the centrosymmetric matrix is written as block centrosymmetric matrix ( 𝑩−𝑱𝒎𝑪 𝟎𝒎 𝟎𝒎 𝑩+ 𝑱𝒎𝑪 ). it can be seen that centrosymmetric matrix has block matrices 𝑩−𝑱𝒎𝑪 and 𝑩+𝑱𝒎𝑪. b. construct block centrosymmetric matrix on hessenberg matrix by the previous explanation, centrosymmetric has block matrices 𝑩− 𝑱𝒎𝑪 and 𝑩+ 𝑱𝒎𝑪. these matrices have special form as lower hessenberg matrix. based on the definition of centrosymmetric matrix and lower hessenberg matrix, then the centrosymmetric matrix with lower hessenberg as block matrices has the form as follows. 𝑨 = ( 𝑩 𝑱𝒎𝑪𝑱𝒎 𝑪 𝑱𝒎𝑩𝑱𝒎 )                                 = −−− −−−−−− −− −− −−−−−− −−− 11122,1, 2122233,12,11,1 1,12,11,1,1,21,22221 1,2,1,,,11,11211 11121,1,1,1,2,1, 21221,2,2,11,12,11,1 1,12,13,1232221 1,2,1211 bbcc bbbccc bbbbcccc bbbbcccc ccccbbbb ccccbbbb cccbbb ccbb mm mmm mmmmmmmm mmmmmmmm mmmmmmmm mmmmmmmm mmm mm       then, the block matrices of this matrix are 𝑩 = ( 𝑏11 𝑏12 𝑏21 𝑏22 𝑏23 ⋮ ⋮ ⋱ ⋱ 𝑏𝑚−1,1 𝑏𝑚−1,2 ⋯ 𝑏𝑚−1,𝑚−1 𝑏𝑚−1,𝑚 𝑏𝑚,1 𝑏𝑚,2 ⋯ 𝑏𝑚,𝑚−1 𝑏𝑚,𝑚 ) and 𝑱𝒎𝑪 = ( 𝑐11 𝑐12 𝑐21 𝑐22 𝑐23 ⋮ ⋮ ⋱ ⋱ 𝑐𝑚−1,1 𝑐𝑚−1,2 ⋯ 𝑐𝑚−1,𝑚−1 𝑐𝑚−1,𝑚 𝑐𝑚,1 𝑐𝑚,2 ⋯ 𝑐𝑚,𝑚−1 𝑐𝑚,𝑚 ) n. khasanah and farikhin counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form 23 where both of them are hessenberg matrices. there is special form as block matrices on centrosymmetric matrix. this condition is useful on computing determinant of centrosymmetric matrix using determinant of lower hessenberg matrix. from lemma 2, there is an orthogonal matrix 𝑸 = √2 2 ( 𝑰𝒎 𝑰𝒎 −𝑱𝒎 𝑱𝒎 ), then 𝑸𝑻𝑨𝑸 = ( 𝑩−𝑱𝒎𝑪 𝟎𝒎 𝟎𝒎 𝑩+ 𝑱𝒎𝑪 ) = ( 𝑴 𝟎𝒎 𝟎𝒎 𝑵 ) (3) where 𝑴 = 𝑩−𝑱𝒎𝑪, 𝑵 = 𝑩+ 𝑱𝒎𝑪 and 𝑴,𝑵 are hessenberg matrices. by the same way [16] about the determinant of lower hessenberg matrix, then we can assume �̃�−𝟏 = ( 𝜶𝑴 𝑾𝑴 𝑟𝑀 𝜷𝑴 𝛵 ), �̃� −𝟏 = ( 𝜶𝑵 𝑾𝑵 𝑟𝑁 𝜷𝑵 𝛵 ) (4) and �̃� = ( 𝒆𝟏 𝛵 0 𝑴 𝒆𝒎 ), �̃� = ( 𝒆𝟏 𝛵 0 𝑵 𝒆𝒎 ). (5) therefore 𝑟𝑀,𝑟𝑁 ≠ 0, if 𝑟𝑀, 𝑟𝑁 = 0 than 𝜶𝑴,𝜶𝑵 = 0 since 𝑴𝜶𝑴 = 0 and 𝑵𝜶𝑵 = 0. it implies that �̃�−𝟏 and �̃�−𝟏 are nonsingular matrix, which are contradiction. c. compute determinant of centrosymmetric matrix the block of centrosymmetric matrix as explained before, it is formed as 𝑨 = 𝑸( 𝑩−𝑱𝒎𝑪 𝟎𝒎 𝟎𝒎 𝑩+𝑱𝒎𝑪 )𝑸𝛵 = 𝑸( 𝑴 𝟎𝒎 𝟎𝒎 𝑵 )𝑸𝛵 (6) then 𝑑𝑒𝑡(𝑨) = 𝑑𝑒𝑡(𝑸) ⋅𝑑𝑒𝑡( 𝑴 𝟎𝒎 𝟎𝒎 𝑵 )⋅𝑑𝑒𝑡(𝑸𝛵) = 𝑑𝑒𝑡(𝑴) ⋅𝑑𝑒𝑡(𝑵). (7) the theorem of determinant of centrosymmetric matrix with lower hessenberg as block matrices is obtained as follows. theorem 3 [14,15]. let 𝑨 is centrosymmetric matrix and it’s block matrices are 𝑴,𝑵as hessenberg matrices which are described before, then 𝑑𝑒𝑡(𝑨) = 𝑟𝑁 ⋅ 𝑟𝑀 ⋅∏ (𝑔𝑖,𝑖+1 ⋅ 𝑞𝑖,𝑖+1) 𝑚−1 𝑖=1 . (8) this step is the algorithm determinant of centrosymmetric matrix with lower hessenberg as block matrix. the other blocks matrices also appear on centrosymmetric matrix are upper hessenberg, centrosymmetric, lower triangular, upper triangular, diagonal, tridiagonal matrix. furthermore, some numerical examples of determinant of centrosymmetric matrix with different block matrices are given. 3.2 numerical experiences this part will show the different block matrices arising at some examples of centrosymmetric matrices by applying the previous algorithm. example 1. [14,15] given the following centrosymmetric matrix 𝑨 = ( 1 1 0 0 0 0 2 1 0 2 2 0 0 1 3 0 1 2 2 1 2 2 2 1 1 0 1 1 4 1 3 2 2 3 1 4 1 1 0 1 1 2 2 2 1 2 2 1 0 3 1 0 0 2 2 0 1 2 0 0 0 0 1 1) where 𝑩 = ( 1 1 0 0 0 2 2 0 1 2 2 1 1 0 1 1 )is lower hessenberg as jurnal matematika mantik volume 6, issue 1, may 2020, pp. 20-29 24 block matrix, 𝑪 = ( 2 3 1 4 1 2 2 2 0 3 1 0 1 2 0 0 ) and 𝑱𝟒 = ( 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ). therefore, 𝑴 = 𝑩−𝑱𝟒𝑪 = ( 0 −1 0 0 0 −1 1 0 0 0 0 −1 −1 −3 0 −3 ), 𝑵 = 𝑩+𝑱𝟒𝑪 = ( 2 3 0 0 0 5 3 0 2 4 4 3 3 3 2 5 ) are lower hessenberg matrices. based on the algorithm before, this step can be continued caused by both of block matrices are lower hessenberg form. determinant of this matrix can be solved with �̃� = ( 1 0 0 0 0 0 −1 0 0 0 0 −1 1 0 0 0 0 0 −1 0 −1 −3 0 −3 1) , �̃� = ( 1 0 0 0 0 2 3 0 0 0 0 5 3 0 0 2 4 4 3 0 3 3 2 5 1) . so, it can be founded �̃�−𝟏 = ( 1 0 0 0 0 0 −1 0 0 0 0 −1 1 0 0 0 0 0 −1 0 −1 −3 0 −3 1) and �̃�−𝟏 = ( 1 0 0 0 0 −0.6667 0.3333 0 0 0 1.1111 −0.5556 0.3333 0 0 −1.2593 0.2963 −0.4444 0.3333 0 3.0741 −1.3704 1.5556 −1.6667 1) . therefore, 𝑟𝑀 = 1,𝑟𝑁 = 3.0741 and ∏ (𝑔𝑖,𝑖+1) 4 𝑖=1 = (−1)(1)(−1) = 1, ∏ (𝑞𝑖,𝑖+1) 4 𝑖=1 = (3)(3)(3) = 27 then 𝑑𝑒𝑡(𝑯) = 𝑟𝑀 ⋅ 𝑟𝑁 ⋅ ∏(𝑔𝑖,𝑖+1 ⋅ 𝑞𝑖,𝑖+1) 𝑚−1 𝑖=1 = (1)(3.0741)(1)(27) = 83.0007. example 2. consider the centrosymmetric matrix 𝑨 = ( 2 3 1 4 1 1 0 1 1 2 2 2 1 2 2 1 0 3 1 0 0 2 2 0 1 2 0 0 0 0 1 1 1 1 0 0 0 0 2 1 0 2 2 0 0 1 3 0 1 2 2 1 2 2 2 1 1 0 1 1 4 1 3 2) with 𝑩 is not lower hessenberg as block matrix. by lemma 2, the matrix of 𝑨 is formed as block matrices 𝑩−𝑱𝟒𝑪 = 𝑴 = ( 1 3 0 3 0 0 0 1 0 1 −1 0 0 1 0 0 ), 𝑩+𝑱𝟒𝑪 = 𝑵 = ( 3 3 2 5 2 4 4 3 0 5 3 0 2 3 0 0 ) n. khasanah and farikhin counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form 25 and we have �̃� = ( 1 0 0 0 0 1 3 0 3 0 0 0 0 1 0 0 1 −1 0 0 0 1 0 0 1) , �̃� = ( 1 0 0 0 0 3 3 2 5 0 2 4 4 3 0 0 5 3 0 0 2 3 0 0 1) which are not lower triangular matrices form, then the algorithm is not suitable for this condition. this algorithm is stopped because of non lower hessenberg of its block matrices 𝑴 and 𝑵. example 3. consider centrosymmetric matrix 𝑨 = ( 1 1 0 0 4 1 3 2 0 2 2 0 2 2 2 1 1 2 2 1 0 1 3 0 1 0 1 1 0 0 2 1 1 2 0 0 1 1 0 1 0 3 1 0 1 2 2 1 1 2 2 2 0 2 2 0 2 3 1 4 0 0 1 1) has 𝑩 and 𝑪 are lower hessenberg as block matrices. thus, we have 𝑴 = ( −1 −2 −1 −4 −1 0 0 −2 1 −1 1 1 0 −2 1 1 ), 𝑵 = ( 3 4 1 4 1 4 4 2 1 5 3 1 2 2 1 1 ), then we have �̃� = ( 1 0 0 0 0 −1 −2 −1 −4 0 −1 0 0 −2 0 1 −1 1 1 0 0 −2 1 1 1) , �̃� = ( 1 0 0 0 0 3 4 1 4 0 1 4 4 2 0 1 5 3 1 0 2 2 1 1 1) which are not lower triangular matrices form, then the algorithm is not suitable for this condition caused by the lower hessenberg matrices appear on two block matrices 𝑩, 𝑪. example 4. given 𝑨 = ( 2.5 1.5 0 0 2.5 1.5 1 0 1.5 2.5 1.5 0 1.5 2.5 1.5 1 1 1.5 2.5 1.5 1 1.5 2.5 1.5 0 1 1.5 2.5 0 1 1.5 2.5 2.5 1.5 0 0 2.5 1.5 1 0 1.5 2.5 1.5 0 1.5 2.5 1.5 1 1 1.5 2.5 1.5 1 1.5 2.5 1.5 0 1 1.5 2.5 0 1 1.5 2.5) then 𝑴 = 𝑩−𝑱𝟒𝑪 = ( 2.5 0.5 −0.5 −2.5 0.5 1 −1 −0.5 −0.5 −1 1 0.5 −2.5 −0.5 0.5 2.5 ), 𝑵 = 𝑩+ 𝑱𝟒𝑪 = ( 2.5 2.5 2.5 2.5 2.5 4 4 2.5 2.5 4 4 2.5 2.5 2.5 2.5 2.5 ) are centrosymmetric matrices too. it can be seen that centrosymmetric matrix can appear on centrosymmetric matrix as block matrix. it seen that block matrices of matrix are 𝑩 = 𝑪 cause be zero on determinant and the algorithm is not suitable for this condition. jurnal matematika mantik volume 6, issue 1, may 2020, pp. 20-29 26 example 5. given 𝑨 = ( 1 2 3 4 1 0 4 3 2 1 2 3 0 1 0 4 3 2 1 2 4 0 1 0 4 3 2 1 3 4 0 1 1 0 4 3 1 2 3 4 0 1 0 4 2 1 2 3 4 0 1 0 3 2 1 2 3 4 0 1 4 3 2 1) , then 𝑴 = 𝑩−𝑱𝟒𝑪 = ( −2 −2 3 3 −2 1 1 3 3 1 1 −2 3 3 −2 −2 ), 𝑵 = 𝑩+ 𝑱𝟒𝑪 = ( 4 6 3 5 6 1 3 3 3 3 1 6 5 3 6 4 ) are centrosymmetric matrices too. it can be seen that centrosymmetric matrix can appear on centrosymmetric matrix as block matrix. it can be seen that block matrices of matrix are 𝑩 ≠ 𝑪 caused determinant is nonzero and the algorithm is not suitable for this condition. moreover, there no lower hessenberg form as block matrix at matrix 𝑩. example 6. consider the centrosymmetric matrix 𝑨 = ( 1 0 0 0 3 1 3 2 2 3 0 0 0 3 2 2 3 2 1 0 0 0 2 1 2 3 1 2 0 0 0 2 2 0 0 0 2 1 3 2 1 2 0 0 0 1 2 3 2 2 3 0 0 0 3 2 2 3 1 3 0 0 0 1) has tridiagonal matrix 𝑩 and 𝑪 as block matrices. by lemma 2, the matrix of 𝑨 is formed as block matrices 𝑩−𝑱𝟒𝑪 = 𝑴 = ( −1 −3 −1 −3 0 1 −3 0 2 0 1 0 0 3 1 2 ), 𝑩+𝑱𝟒𝑪 = 𝑵 = ( 3 3 1 3 4 5 3 0 4 4 1 0 4 3 1 2 ) and we have �̃� = ( 1 0 0 0 0 −1 −3 −1 −3 0 0 1 −3 0 0 2 0 1 0 0 0 3 1 2 1) , �̃� = ( 1 0 0 0 0 3 3 1 3 0 4 5 3 0 0 4 4 1 0 0 4 3 1 2 1) which are not lower triangular matrices form, then the algorithm is not suitable for this condition. the algorithm cannot be continued caused the block matrix is not lower hessenberg form. example 7. consider the centrosymmetric matrix 𝑨 = ( 1 2 3 2 3 0 0 0 0 3 2 3 1 3 0 0 0 0 1 1 3 2 2 0 0 0 0 2 2 2 1 2 2 1 2 2 2 0 0 0 0 2 2 3 1 1 0 0 0 0 3 1 3 2 3 0 0 0 0 3 2 3 2 1) where block matrices 𝑩 and 𝑪 are upper hessenberg n. khasanah and farikhin counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form 27 form. by lemma 2, the matrix of 𝑨 is formed as block matrices 𝑩−𝑱𝟒𝑪 = 𝑴 = ( 1 2 3 −1 0 3 −1 2 0 −2 −1 −2 −2 −1 −2 0 ), 𝑩+𝑱𝟒𝑪 = 𝑵 = ( 1 2 3 5 0 3 5 4 0 2 3 4 2 1 2 4 ) and we have �̃� = ( 1 0 0 0 0 1 2 3 −1 0 0 3 −1 2 0 0 −2 −1 −2 0 −2 −1 −2 0 1) , �̃� = ( 1 0 0 0 0 1 2 3 5 0 0 3 5 4 0 0 2 3 4 0 2 1 2 4 1) . which are not lower triangular matrices form, then the algorithm is not suitable for this condition. it happen caused by block matrix 𝑩 is not lower hessenberg matrix. example 8. consider the centrosymmetric matrix 𝑨 = ( 1 0 0 0 8 0 0 0 0 2 0 0 0 7 0 0 0 0 3 0 0 0 6 0 0 0 0 4 0 0 0 5 5 0 0 0 4 0 0 0 0 6 0 0 0 3 0 0 0 0 7 0 0 0 2 0 0 0 0 8 0 0 0 1) where 𝑩 and 𝑪 are diagonal matrices as block matrix. by lemma 2, the matrix of 𝑨 is formed as block matrices 𝑩−𝑱𝟒𝑪 = 𝑴 = ( 1 0 0 −8 0 2 −7 0 0 −6 3 0 −5 0 0 4 ), 𝑩+𝑱𝟒𝑪 = 𝑵 = ( 1 0 0 8 0 2 7 0 0 6 3 0 5 0 0 4 ) and we have �̃� = ( 1 0 0 0 0 1 0 0 −8 0 0 2 −7 0 0 0 −6 3 0 0 −5 0 0 4 1) , �̃� = ( 1 0 0 0 0 1 0 0 8 0 0 2 7 0 0 0 6 3 0 0 5 0 0 4 1) . which are not lower triangular matrices form, then the algorithm is not suitable for this condition. it cannot be continued because of lower hessenberg form does not appear as block matrix. example 9. consider the centrosymmetric matrix 𝑨 = ( 2 1 0 0 4 3 0 0 1 2 1 0 3 4 3 0 0 1 2 1 0 3 4 3 0 0 1 2 0 0 3 4 4 3 0 0 2 1 0 0 3 4 3 0 1 2 1 0 0 3 4 3 0 1 2 1 0 0 3 4 0 0 1 2) has 𝑩 and 𝑪are tridiagonal matrix as block matrices. by lemma 2, the matrix of 𝑨 is formed as block matrices jurnal matematika mantik volume 6, issue 1, may 2020, pp. 20-29 28 𝑩−𝑱𝟒𝑪 = 𝑴 = ( 2 1 −3 −4 1 −1 −3 −3 −3 −3 −1 1 −4 −3 1 2 ), 𝑩+𝑱𝟒𝑪 = 𝑵 = ( 2 1 3 4 1 5 5 3 3 5 5 1 4 3 1 2 ) and we have �̃� = ( 1 0 0 0 0 2 1 −3 −4 0 1 −1 −3 −3 0 −3 −3 −1 1 0 −4 −3 1 2 1) , �̃� = ( 1 0 0 0 0 2 1 3 4 0 1 5 5 3 0 3 5 5 1 0 4 3 1 2 1) . which are not lower triangular matrices form, then the algorithm is not suitable for this condition. 4. conclusions the algorithm of determinant centrosymmetric matrix is presented efficiently. this algorithm focusing with the centrosymmetric matrix with lower hessenberg matrix as block matrix. but there are some block matrices appear as block of centrosymmetric matrices such as upper hessenberg, centrosymmetric, lower triangular, upper triangular, diagonal, tridiagonal matrix. therefore, this algorithm is only on compute the determinant of centrosymmetric matrix with lower hessenberg as block matrix. for the other block matrices, this algorithm does not work well. references [1] zhong-yun liu, "some properties of centrosymmetric matrices", appl. math. comput. 141 pp. 297-306, 2003. [2] hongyi li, di zhao, fei dai and donglin su, "on the spectral radius of a nonnegative centrosymmetric matrix", appl. math. comput., 218 (9) pp. 4962-4966, 2012. [3] melman, "symmetric centrosymmetric matrix-vector multiplication", linear algebra and its appl., 320 pp.193-198, 2000. [4] iyad t. abu-jeib, "centrosymmetric matrices : properties and an alternative approach", canadian applied mathematics quarterly, 10 pp. 429-445, 2000. [5] alan l. andrew, "eigenvector of certain matrices", linear algebra and its appl., 7 pp. 151-162, 2000. [6] charles f van loan and joseph p vokt, "approximating matrices with multiple symmetries", siam j.matrix anal.appl., 36(3) pp.974-993, 2015. [7] dattatreya a.v. rao and k. venkata ramana, "on lu decomposition of a centrosymmetric matrix", information sciences, 63 pp. 3-10, 2000. [8] konrad burnik, "a structure-preserving qr factorization for centrosymmetric real matrices", linear algebra and its applications, 484 pp. 356-378, 2015. [9] gene h. golub and charles f. van loan, "matrix computations, third ed.", johns hopkins university press, baltimore and london, (1996). [10] tomohiro sogabe, "on a two-term recurrence for the determinant of a general matrix", appl. math. comput., 187 pp. 785-788, 2007. [11] mohamed elouafi and a.d. aiat hadj, "a new recursive algorithm for inverting hessenberg matrices", appl. math. comput., 214 pp. 497-499, 2009. [12] f bunger, “inverse, determinant, eigenvalues, and eigenvectors of real symmetric toeplitz matrices with linearly increasing entries”, linear algebra and its n. khasanah and farikhin counter example:the algorithm of determinant of centrosymmteric matrix based on lower hessenberg form 29 applications.459 pp.595-619, 2014. [13] mohamed elouafi and a.d. aiat hadj, "a new recursive algorithm for inverting hessenberg matrices", appl. math. comput., 214 pp. 497-499, 2009. [14] di zhao and hongyi li, "on the computation of inverse and determinant of a kind of special matrices", appl. math. comput., 250 pp. 721-726, 2015. [15] n khasanah, farikhin and b surarso, "the algorithm of determinant of centrosymmetric matrix based on lower hessenberg form", iop conf. series : journal of physics : conf. series 824 ,2017. [16] datta and morgera, "on the reducibility of centrosymmetric matrices-aplication in engineering problems", circuits system signal process., 8 (1) pp. 71-95, 1989. how to cite: s. harini, “identification covid-19 cases in indonesia with the double exponential smoothing method”, j. mat. mantik, vol. 6, no. 1, pp. 66-75, may 2020. identification covid-19 cases in indonesia with the double exponential smoothing method sri harini uin maulana malik ibrahim malang, sriharini@mat.uin-malang.ac.id doi: https://doi.org/10.15642/mantik.2020.6.1.66-75 abstrak: pendekatan time-series adalah metode yang digunakan untuk menganalisis serangkaian data dalam urutan waktu untuk memperkirakan nilai suatu seri di masa depan. dalam artikel ini akan di identifkasi kasus covid-19 di indonesia menggunakan metode double eksponensial smoothing. metode double eksponensial smoothing merupakan salah satu metode yang dapat digunakan untuk pengoptimalkan pendugaan dari model arima dengan parameter pemulusan α. data yang digunakan bersumber dari badan nasional penanggulangan bencana yang dirilis mulai 2 maret 2020. berdasarkan hasil pengujian pacf, acf dan estimasi parameter model arima pada kasus covid-19 di indonesia mengikuti model arima (0,1,1). kata kunci: time series; covid-19; double eksponensial smoothing; arima abstract: the time-series approach is a method used to analyze a series of data in a time sequence to estimate the value of a series in the future. this article will identification the covid-19 case model in indonesia using the double exponential smoothing method. the double exponential smoothing method is one method that can be used to optimize the estimation of the arima model with smoothing parameters α. the data used is sourced from the national disaster management agency which was released starting march 2, 2020. based on the results of pacf, acf, and estimated parameters of the arima model in the covid-19 case in indonesia following the arima model (0,1,1). keywords: time series; covid-19; double eksponensial smoothing; arima jurnal matematika mantik vol. 6, issue 1, may 2020, pp. 66-75 issn: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 s. harini identification covid-19 cases in indonesia with the double exponential smoothing method 67 1. introduction in early 2020, the world was rocked by the emergence of a coronavirus (covid-19) in wuhan city, hubei province, china. covid-19 is a new type of virus whose transmission from bats to humans causes upper respiratory infections such as middle east respiratory syndrome (mers-cov) and severe acute respiratory syndrome (sarscov). clinical manifestations of covid-19 usually appear within 2 days to 14 days after exposure to common signs and symptoms of infection including acute respiratory disorders such as fever, coughing, and shortness of breath. severe cases can cause pneumonia, acute respiratory syndrome, kidney failure, and even death [1] on december 31, 2019, the who china country office reported a case of pneumonia of unknown etiology in wuhan city, hubei province, china. on 7 january 2020, china identified pneumonia of unknown etiology as a new type of coronavirus (covid-19). the increase in the number of covid-19 cases has taken place quite quickly and has already spread to almost 208 countries in the world. as of april 7, 2020, based on data from worldometers, up to 3:30 pm west indonesia time, april 7, 2020, the total number of positive cases of covid-19 in the world had reached 1,379,175 patients. as many as 78,223 coronae positive patients worldwide have died and 294,149 people have successfully recovered from covid-19 disease [2]. in addition, if referring to data from csse johns hopkins university, until 3:30 pm west indonesia time, april 6, 2020, the total number of positive cases of covid-19 in the world was recorded at 1,277,962 patients with 69,527 covid-19 patients had died and 264,048 people were successfully recovered. the highest number of positive covid-19 cases in the world is in the united states, spain, italy, germany, and france. while most deaths occurred in italy, spain, and france [3] likewise in indonesia, as one of the countries affected by covid-19 based on data from the task force for the acceleration of handling covid-19 (bnpb) on april 7, 2020, which was renewed at 15.40 wib, the total number of corona positive cases reached 2,731 patients with the number of covid-positive patients 19 of those who are still undergoing treatment in indonesia are 2,090 people or 83.9 percent of the total cases. an increase also occurred in the covid-19 patient mortality rate until april 7, 2020, 221 positive covid19 patients have died. the covid-19 case fatality rate (cfr) in indonesia currently reaches 8.39 percent. while covid-19 positive patients who recovered increased to 204 people, equivalent to 7.7 percent of the total number of positive cases [4]. seeing the development of covid-19 case data updates that continue to increase and refer to previous research [5] to make patterns of bird flu spread from birds to humans in the form of mathematical models using the differential equation system. from the facts about bird flu, assumptions were formed which would later be used to make mathematical models. after the mathematical model is formed, then the stability of the model is searched, and the stability of the model is analyzed after that the model is simulated. [6] make a sir model for the spread of bird flu. the sir model is a mathematical model that explains the interaction between susceptible and infected bird population groups and susceptible, infected, and recovered human population groups. the equilibrium points of the model include disease-free equilibrium points and endemic equilibrium points. from the stability analysis, it is obtained that the stability of the balance point depends on the base reproduction number and the balance point is stable if the balance point is given population around the balance point. similar to pandemic influenza, rather than the other two coronaviruses [7]. in 1918, a significant proportion of the deaths were from pneumonia followed by influenza infection. the study proposed a model incorporating individual reaction, holiday effects as well as weather conditions (temperature in london, united kingdom), which successfully captured the multiple-wave feature in the influenza-associated mortality in london in this study, we followed the form of individual reaction and governmental action effects. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 66-75 68 establish the dynamics model of infectious diseases and time series models to predict the trend and short-term prediction of the transmission of covid-19, which will be conducive to the intervention and prevention of covid-19 by departments at all levels in mainland china and buy more time for clinical trials. [8] based on the transmission mechanism of covid-19 in the population and the implemented prevention and control measures, we establish the dynamic models of the six chambers, and establish the time series models based on different mathematical formulas according to the variation law of the original data. [9] proposes conceptual models for the covid-19 outbreak in wuhan with the consideration of individual behavioral reaction and governmental actions, e.g., holiday extension, travel restriction, hospitalization, and quarantine. we employe the estimates of these two key components from the 1918 influenza pandemic in london, united kingdom, incorporated zoonotic introductions and the emigration, and then compute future trends and the reporting ratio. the model is concise in structure, and it successfully captures the course of the covid-19 outbreak, and thus sheds light on understanding the trends of the outbreak. based on this background, this covid-19 case in indonesia will be predicted at this time using the time-series approach. time-series method is a quantitative method used to analyze a series of data collected in a time sequence and the results can be used as a reference to estimate the value of a series in the future [10]. 2. methods data in this research is taken from the tabulation of the indonesian national disaster management agency (bnpb) from 2 march until 7 april 2020 [4]. the exponential smoothing method and general autoregressive integrated moving average (arima) processes refer to [10] with the following equation: • determine the first smoothing value and determine the parameter 𝛼 𝑆𝑡 ′ = 𝛼𝑋 + (1 − 𝛼)𝑆𝑡−1 ′ (1) • determine the second smoothing value 𝑆𝑡 ′′ = 𝛼𝑆𝑡 ′ + (1 − 𝛼)𝑆𝑡−1 ′′ (2) • arima model for time series data arima model is stated as follows: ɸ(𝐵)(1 − 𝐵)𝑑𝑋𝑡 = ө(𝐵)𝑍𝑡 (3) this is an arima process of order (p,d,q). in general equation (3) can be approached using a regression model: 𝑦𝑡 = 𝛽0 + 𝛽1𝑦𝑡−1 + ⋯+ 𝛽𝑝𝑦𝑡−𝑝 + 𝛽1𝑒𝑡−1 + ⋯+ 𝛽𝑞𝑒𝑡−𝑞 + 𝜀𝑡 (4) 3. results and discussions the covid-19 pandemic case in indonesia has entered 1 month since 2 positive people were found on 2 march 2020. this condition continues to increase, with the latest data on 7 april 2020 recorded 2,738 people positive cases, 204 people recovery, and 221 people died. the covid-19 pandemic certainly has a multi-sectoral impact, whether in the fields of education, health, defense, worship, social activities, or community economic activities. to prevent the increasing number of covid-19 cases in indonesia, the government uses 4 regulations at once. among them, law 6/2018 on health quarantine, pp 21/2020 on large-scale social distancing (psbb), presidential decree 11/2020 on determination of public health emergency, and perppu 1/2020 on state financial policy and financial system stability. s. harini identification covid-19 cases in indonesia with the double exponential smoothing method 69 in addition, since the beginning, the government has also issued policies on social distancing, physical distancing, work from home (wfh) and large-social distancing (psbb) for all levels of society. the aim of this policy is to break the distribution chain of covid-19 in indonesia. however, this condition is still not in line with government expectations, where more and more people are infected with covid-19. table 1. covid-19 data from 2 march to 7 april 2020 (source [4]) date positive recovery death date positive recovery death 2 2 0 0 21 450 20 38 3 2 0 0 22 514 29 48 4 2 0 0 23 579 30 49 5 2 0 0 24 686 30 55 6 2 0 0 25 790 31 58 7 4 0 0 26 893 35 78 8 6 0 0 27 1046 46 87 9 19 0 0 28 1155 59 102 10 27 0 0 29 1285 64 114 11 34 2 0 30 1414 75 122 12 34 4 1 31 1528 81 136 13 69 5 4 1 1677 103 157 14 96 8 5 2 1790 112 170 15 117 8 5 3 1986 134 181 16 134 8 5 4 2092 150 191 17 172 9 5 5 2273 164 198 18 227 11 19 6 2491 192 209 19 309 15 25 7 2738 204 221 20 369 17 32 based on these data in table 1, we then plot the data as follows: figure 1. plot data covid-19 in indonesia (data processed by minitab 18, 2020) using the time-series model approach, the pattern of covid-19 data distribution behavior in indonesia (figure 1) shows an exponential distribution pattern, where the addition of positive cases of covid-19 increases significantly from everyday. this condition is also followed by a pattern of distribution of the number of people who recovered and died. as we know that in the time-series model the type of exponential distribution consists of a single exponential, double exponential, and winters' method. in this article, the author further analyzed using the three methods and the best model is the double exponential model. jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 66-75 70 figure 2. prediction positive covid-19 case in indonesia (data processed by minitab 18, 2020) by using a 10 percent error rate [figure 2], the best α parameter values are 0.659 and the best γ is 1.260. with a mean absolute percentage error (mape) of 8.998 percent. mape is the error value for each period divided by the actual observation value for that period. in positive covid-19 case, the mape value is smaller than the error rate at 10% (the exact model is generated). figure 1 shows that there are significant differences between positive cases, recovery, and death. where the decrease in data recovery and death is very far when compared with an increase in the amount of positive data. the increase in the number of people who were positive for covid-19, also directly affected the model of prediction patients who recovery and death (figure 3 and figure 4). figure 3. prediction recovery covid-19 case in indonesia (data processed by minitab 18, 2020) based on figure 3. using a 10 percent error rate, the estimated value of the parameter recovery patients at α is 0.660, γ is 0.627 and the mape of 14.179 percent. in the recovery of covid-19 cases, the mape value is greater than the error value set at 10 percent error rate. the still low cure rate of covid-19 patients is due to the still low awareness of the community to comply with government recommendations. s. harini identification covid-19 cases in indonesia with the double exponential smoothing method 71 figure 4. prediction death covid-19 case in indonesia (data processed by minitab 18, 2020) figure 4. using a 10 percent error rate, the estimated value of the parameter death patients at α is 0.9605, γ is 0.3407, and the mape of 13.95 percent. in the death of covid19 cases, the mape value is greater than the error value set at 10%. the high death rate of covid-19 patients due to the still limited number of health workers and supporting health facilities to prevent. after the parameter values, α and γ are obtained, the next step is the identification process partial autocorrelation function (pacf) and autocorrelation function (acf) from positive, recovery, and death data. pacf and acf tests are used to test the accuracy of the results of the double exponential smoothing model and a means of determining the stationarity of the variable and the lag lengths of the arima model. figure 5. pacf of residuals for covid-19 positive data (data processed by minitab 18, 2020) figure 6. acf of residuals for covid-19 positive data (data processed by minitab 18, 2020) 987654321 1 .0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1 .0 lag p ar ti al a u to co rr el at io n pacf of residuals for positive (with 5% significance limits for the partial autocorrelations) 987654321 1 .0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1 .0 lag a ut oc or re la ti on acf of residuals for positive (with 5% significance limits for the autocorrelations) jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 66-75 72 based on figures 5 and 6, pacf and acf plots of residuals for covid-19 positive data are obtained the lag time through pacf cuts off at lag one, and acf tails off slowly. in the time series model with the error probability (α) 5%, the graph follows the arima process (0,1,1) with the p-value ma 1 (0.1%) is smaller than α. the estimated results of the arima model as in table 2. table 2. final estimates of parameters model for covid-19 positive data type coef se coef t-value p-value ma 1 -0.592 0.161 -3.67 0.001 constant 77.4 14.0 5.51 0.000 differencing: 1 regular difference. number of observations: original series 37, after differencing 36. referring to equation (4), mathematically the arima model (0,1,1) in table 2 can be stated as follows: 𝑦𝑡 = 77.4 − 0.592𝑒𝑡−1 with the same steps as testing positive data, next step the identification process pacf and acf from recovery data. figure 7. pacf of residuals for covid-19 recovery data (data processed by minitab 18, 2020) figure 8. acf of residuals for covid-19 recovery data (data processed by minitab 18, 2020) same as covid-19 positive data in indonesia, figures 7 and 8 to show pacf and acf plots of residuals for covid-19 recovery data are obtained the lag time through pacf cuts off at lag one and acf tails off slowly. in the time series model with the error probability (α) 5%, the graph follows the arima process (0,1,1) with the p-value ma 1 (2.3%) is smaller than α. the estimated results of the arima model as in table 3. 987654321 1 .0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1 .0 lag pa rt ia l a ut oc or re la ti on pacf of residuals for recovery (with 5% significance limits for the partial autocorrelations) 987654321 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 co rr el at io n acf of residuals for recovery (with 5% significance limits for the autocorrelations) s. harini identification covid-19 cases in indonesia with the double exponential smoothing method 73 table 3. final estimates of parameters model for covid-19 recovery data type coef se coef t-value p-value ma 1 -0.384 0.161 -2.38 0.023 constant 5.61 1.52 3.68 0.001 differencing: 1 regular difference, number of observations: original series 37, after differencing 36. referring to equation (4), mathematically the arima model (0,1,1) in table 3 can be stated as follows: 𝑦𝑡 = 5.61 − 0.384𝑒𝑡−1 after positive and recovery data are analyzed, next the pacf and acf models of the data death will be shown in figure 9 and figure 10. figure 9. pacf of residuals for covid-19 death data (data processed by minitab 18, 2020) figure 10. acf of residuals for covid-19 death data (data processed by minitab 18, 2020). same as covid-19 positive and recovery data in indonesia, figures 9 and 10 to show pacf and acf plots of residuals for covid-19 death data are obtained the lag time through pacf cuts off at lag two and acf tails off slowly. in time series model with the error probability (α) 5%, the graph follows the arima process (0,1,1) with the p-value ma 1 (3.6%) is smaller than α. the estimated results of the arima model as in the table 4. 987654321 1 .0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1 .0 lag p ar ti al a u to co rr el at io n pacf of residuals for death (with 5% significance limits for the partial autocorrelations) 987654321 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 co rr el at io n acf of residuals for death (with 5% significance limits for the autocorrelations) jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 66-75 74 table 4. final estimates of parameters model for covid-19 death data type coef se coef t-value p-value ma 1 -0.351 0.161 -2.18 0.036 constant 6.14 1.28 4.81 0.000 differencing: 1 regular difference. number of observations: original series 37, after differencing 36 referring to equation (4), mathematically the arima model (0,1,1) in table 4 can be stated as follows: 𝑦𝑡 = 6.14 − 0.351𝑒𝑡−1 based on the results of predictions of covid-19 cases that occurred in indonesia with a double exponential model and the results of pacf, acf and estimated parameters model following the arima model (0,1,1) with the p-value ma 1 is smaller than α. the results of predictions of covid-19 cases that occurred in indonesia (positive, recovery, and death) showed a gap in the resulting distribution patterns. where the increase in the number of positive cases has not been offset by an increase in the number of patients who recovery and a decrease in the number of patients who died. this indicates that public behavior still does not comply with the rules set by the government (social distancing, large-scale social restrictions (psbb), mask use), the limited medical staff and the lack of standard equipment handling covid-19 is one of the causes of the low handling of healing from positive patients. 4. conclusions based on the analysis of data patterns of behavior of people who tested positive, recovery, and death with a time-series approach, it was found that the covid-19 distribution model in indonesia followed with the double exponential model. the results of pacf, acf, and estimated parameters model following the arima model (0,1,1). where are the results of predictions of covid-19 cases that occurred in indonesia (positive, recovery, and death) showed a gap in the resulting distribution patterns. references [1] y. yuliana, “corona virus diseases (covid-19): sebuah tinjauan literatur. wellness and healthy magazine, vol. 2, no. 1, pp. 187 – 192, february 2020. retrieved from 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[10] s. makridakis, metode dan aplikasi peramalan jilid 1 (edisi revisi). jakarta: binarupa aksara, 2003. https://doi.org/10.1016/s0140-6736(20)30260-9 https://doi.org/10.1016/j.ijid.2020.02.058 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: e. kurniadi, n. gusriani, and b. subartini, “duflo-moore operator for the square-integrable representation of the 2-dimensional affine lie group”, j. mat. mantik, vol. 6, no. 2, pp.114-122, october 2020. jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 114-122 issn: 2527-3159 (print) 2527-3167 (online) duflo-moore operator for the square-integrable representation of the 2-dimensional affine lie group edi kurniadi1, nurul gusriani2, betty subartini3 1universitas padjadjaran bandung, edi.kurniadi@unpad.ac.id 2universitas padjadjaran bandung, nurul.gusriani@unpad.ac.id 3universitas padjadjaran bandung, betty.subartini@unpad.ac.id doi: https://doi.org/10.15642/mantik.2020.6.2.114-122 abstrak. dalam artikel ini, dipelajari representasi quasi-regular dan representasi unitar tak tereduksi grup lie affine aff +(1) berdimensi dua. pertama, diberikan bukti lengkap dari hasil kerja fuhr tentang transformasi fourier untuk representasi quasi-regular dari aff +(1). kedua, ketika representasi dari grup lie affine aff +(1) adalah square-integrable maka dihitung operator duflomoore secara langsung tanpa menggunakan transformasi fourier seperti dalam hasil fuhr. kata kunci: grup lie affine; operator duflo-moore; representasi square-integrable. abstract. in this paper, we study the quasi-regular and the irreducible unitary representation of affine lie group aff +(1) of dimension two. first, we prove a sharpening of fuhr’s work of fourier transform of quasi-regular representation of aff +(1). the second, in such the representation of affine lie group aff +(1) is square-integrable then we compute its duflo-moore operator instead of using fourier transform as in führ’s work. keywords: affine lie group; duflo-moore operator; square-integrable representation. http://u.lipi.go.id/1458103791 e. kurniadi, n. gusriani, b. subartini duflo-moore operator for the square-interable representation of 2-dimensional affine lie group 115 1. introduction the current research about square-integrable representations of lie groups can be found, for instance in [1] and [2]. in the previous work, the notion of square-integrable representation of a lie group associating to wavelet transforms was introduced by grossmann, morlet, and paul (see [3]). particularly, they investigated the nice examples of a square-integrable representation of 𝑎𝑥 + 𝑏group, known as affine lie group aff(1) as can be seen in [4]. in the other hand, the research about 𝑎𝑥 + 𝑏-groups can also be found, for instance in [5] and [6]. it is well known that aff(1) is the exponential solvable lie group which is non unimodular group whose lie algebra of aff(1) is frobenius. other examples are parabolic subgroups which are fobenius as well (see [7] and[8]). but we thought that grossmann’s work is the best example for young researchers how to understand the square-integrable representations for case nonunimodular groups which is started from the aff(1) lie group. moreover, other examples of nonunimodular groups are lie groups whose lie algebras are 4-dimensional real frobenius lie algebras. kurniadi and ishi [9] showed that irreducible unitary representations of these lie groups are square-integrable representations and they wrote the duflo-moore operators in the terms of groups fourier transforms. many reseachers study affine lie algebras and the structure of affine for instance we see some results in [10], [11], [12], [13],[14], [15], [16], and [17]. in the other hand, in easier stage we can also study square-integrable representations for unimodular lie groups case. heisenberg lie groups of dimension 2𝑛 + 1 and filiform lie groups are in these types. in fact, the duflo-moore operators for square-integrable representations of unimodular groups are scalar multiple (see [18]). in current work, kurniadi in [19] proved that irreducible-unitary representation of lie group of 4dimensional standard filiform lie algebra is square-integrable and its duflo-moore operator is scalar multiple of identity which is equal to one. in this work, we shall give another alternative to compute the duflo-moore operator for square-integrable representation of aff +(1) by direct computations instead of forming in group fourier transform which was written in [18]. 2. preliminaries let aff +(1) be the 2-dimensional affine lie group whis is expressed as a semidirect product of the set of all real numbers ℝ and the set of all positive real numbers ℝ+. namely, we can write this group as aff +(1) ≔ ℝ ⋊ ℝ+. particularly, in this work we concentrate to aff +(1) which is the exponential solvable nonunimodular lie group. to make easier in computations we write aff +(1) in matrix terms. namely, we have aff +(1) ∋ ( 𝛼 𝛽 0 1 ) , 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ. (1) regarding this notations, we denote 𝑔(𝛼, 𝛽) ≔ ( 𝛼 𝛽 0 1 ), δ(𝛼) ≔ ( 𝛼 0 0 1 ), and ∇(𝛽) ≔ ( 1 𝛽 0 1 ). the lie algebra of aff +(1) is denoted by aff(1) whose basis is {𝑒1, 𝑒2}. the nonzero bracket of aff(1) is given by [𝑒1, 𝑒2] = 𝑒2. the lie algebra aff(1) is a frobenius lie algebra which has two open coadjoint orbits as follows (see [20]). ω±: = {(𝑎, 𝑏) ; 𝑎, 𝑏 ∈ ℝ, ± 𝑏 > 0}. (2) jurnal matematika mantik volume 6, no. 2, october 2020, pp.114-122 116 the representations of the affine lie group aff(1) can be realized on the hilbert space of all square-integrable functions l2(ℝ+). before doing that, let us mention here some basic notion of representation theory of lie groups corresponding to our research. definition 1 [21]. let π be a representation of a lie group g on the carrier space ℋ. π is said to be irreducible if π has no nontrivial π-invariant subspace ℋ0 in ℋ. moreover, π is said to be uintary if for each f ∈ ℋ and each g ∈ g ‖𝜋(𝑔)𝑓‖ = ‖𝑓‖. (3) proposition 2 [20]. the irreducible unitary representations of aff +(1) correponsding to open coadjoint orbit ω+ in eqs. (2) in the space l 2(ℝ+) is of the form 𝜋+(𝑔)𝑓(𝑥) = 𝑒 2𝜋𝑖𝛽𝑥 𝑓(𝛼𝑥), (4) where 𝑔 ≔ 𝑔(𝛼, 𝛽) ∈ aff +(1) , 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝑓 ∈ l 2(ℝ+). furthermore, the representation of affine lie group aff +(1) can be realized as a quasi-regular representations (see [18]). it is written in the formula as follows. 𝜋(𝑔(𝛼, 𝛽)) = 𝛼 − 1 2𝜓( 𝑥−𝛽 𝛼 ), 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝜓 ∈ l 2(ℝ+). (5) we are mainly interested in the square-integrable representation. let 𝜋 be an irreducible unitary representation of a lie group 𝐺 realized on the space ℋ and l2(𝐺) be the space of all square-integrable functions on 𝐺. for vector 𝑓1 ∈ ℋ, we define the operator on ℋ given by ℰ𝑓1 : ℋ ∋ 𝑓2 ↦ ℰ𝑓1 𝑓2 ∈ l 2(𝐺). (6) where ℰ𝑓1 𝑓2(𝑥) = 〈𝑓1|𝜋(𝑥)𝑓2〉. definition 3 [22].the irreducible unitary representation π of locally compact topological group g realized on a space ℋ is said to be square-interable if there exist two vectors f1, f2 ∈ ℋ − {0} such that ‖ℰ𝑓1 𝑓2‖ 2 = 〈𝑓1|𝜋(𝑥)𝑓2〉 = ∫ 𝑓1(𝑔)𝜋(𝑥)𝑓2(𝑔) ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ dμ(𝑔) 𝐺 < +∞. (7) in the other words, 〈f1|π(x)f2〉 ∈ l 2(g, μg) where μg is a measure on g. such vectors which satisfied eqs. (7) are called admissible vectors. duflo-moore state their results in the following theorem theorem 4 [23]. if π is square-integrable representations of locally compact group g realized on the space ℋ then there exists a positive selfadjoint operator cπ: ℋ → ℋ which is called the duflo-moore operator such that a. a vector ψ ∈ ℋ − {0} is admissible if and only if ψ is an element of domain of cπ. b. if f1, f2 ∈ ℋ and f3, f4 ∈ dom (cπ) then 〈ℰ𝑓1 𝑓3|ℰ𝑓2 𝑓4〉l2(𝐺,𝜇𝐺) = 〈𝑓1 |𝑓2〉ℋ 〈𝐶𝜋𝑓4|𝐶𝜋𝑓3〉ℋ . (8) e. kurniadi, n. gusriani, b. subartini duflo-moore operator for the square-interable representation of 2-dimensional affine lie group 117 2. methods in this research we apply the literature reviews method, particularly we focus on results in [18] and [20]. we obtain the quasi-regular representation of aff +(1) in fuhr’s work and we compute the fourier transform of its representation to determine the duflomoore operator. on the other hand, we also obtain the irreducible unitary representation of aff +(1) corressponding to open coadjoint orbits and we show that representatation is square-integrable. using direct computations, we obtain the duflo-moore operator for that representation. 3. results and discussion our results and discussion consist of two main part as follows. 3.1 the duflo-moore operator for the quasi-regular representation of 𝐀𝐟𝐟+(𝟏). the following statement can be deduced from [18] in page 30--31. however, we give a detail proof for its own interest. lemma 5 [18]. the fourier transform of quasi-regular representation π of aff +(1) as in eqs. (5) is of the form ℱ(𝜋(𝑔(𝛼, 𝛽))𝜓)(𝜉) = 𝛼 1 2𝑒−2𝜋𝑖𝜉𝛽 ℱ𝜓(𝛼𝜉). (9) proof. by direct computation we obtain ℱ(𝜋(𝑔(𝛼, 𝛽))𝜓)(𝜉) = ∫ 𝑒 −2𝜋𝑖𝜉𝑥 (𝜋(𝑔(𝛼, 𝛽))𝜓)(𝑥) 𝑑𝑥 ℝ = ∫ 𝑒−2𝜋𝑖𝜉𝑥 𝛼−1/2𝜓 ( 𝑥 − 𝛽 𝛼 ) 𝑑𝑥 ℝ = ∫ 𝑒−2𝜋𝑖𝜉(𝛼𝜂+𝛽)𝛼 −1/2𝜓(𝜂) 𝛼 𝑑𝜂 ℝ ( substituting 𝜂 = 𝑥−𝛽 𝛼 ) = ∫ 𝑒−2𝜋𝑖𝜉(𝛼𝜂)𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2𝜓(𝜂) 𝑑𝜂 ℝ = ∫ 𝑒−2𝜋𝑖(𝛼𝜉)𝜂𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2𝜓(𝜂) 𝑑𝜂 ℝ = 𝑒−2𝜋𝑖𝜉𝛽 𝛼1/2 ∫ 𝑒−2𝜋𝑖(𝛼𝜉)𝜂𝜓(𝜂) 𝑑𝜂 ℝ = 𝑒−2𝜋𝑖𝜉𝛽 𝛼 1 2ℱ𝜓(𝛼𝜉). ∎ proposition 6 [18]. the duflo-moore operator for quasi-regular representation π of aff +(1) as in eqs. (5) in the term of fourier transform can be written as follows. jurnal matematika mantik volume 6, no. 2, october 2020, pp.114-122 118 ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉 −1/2ℱ𝜓(𝜉). (10) proof. let 𝜓1 and 𝜓2 be elements of contiunuous functions space of compact support on aff +(1) denoted by 𝐶𝑐(aff +(1) ). using plancherel’s theorem and fubini’s theorem we obtain ∫ |〈𝜓1|𝜋(𝑔(𝛼, 𝛽))𝜓2〉| 2 𝑑𝛼 𝛼2 aff+(1) 𝑑𝛽 = ∫ |〈ℱ𝜓1|ℱ𝜋(𝑔(𝛼, 𝛽))𝜓2〉| 2 𝑑𝛼 𝛼2 aff+(1) 𝑑𝛽 = ∫ |∫ ℱ𝜓1(𝜉)𝑒 −2𝜋𝑖𝜉𝛽 𝛼1/2ℱ𝜓̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝛼𝜉) ℝ 𝑑𝜉| 2 𝑑𝛼 𝛼2 aff+(1) 𝑑𝛽 = ∫ |∫ ℱ𝜓1(𝜉)𝑒 2𝜋𝑖𝜉𝛽 𝛼1/2ℱ𝜓̅̅ ̅̅ (𝛼𝜉) ℝ 𝑑𝜉| 2 𝑑𝛼 𝛼2 aff+(1) 𝑑𝛽 = ∫ ∫ |∫ ℱ𝜓1(𝜉)𝑒 2𝜋𝑖𝜉𝛽ℱ𝜓̅̅ ̅̅ (𝛼𝜉) ℝ 𝑑𝜉| 2 ℝ+ 𝑑𝛼 𝛼 ℝ 𝑑𝛽 = ∫ ∫ |ℱ𝜏𝛼 (−𝛽)| 2 ℝ+ 𝑑𝛼 𝛼 ℝ 𝑑𝛽 ( 𝜏𝛼 (𝜉) = ℱ𝜓1(𝜉)ℱ𝜓̅̅ ̅̅ (𝛼𝜉) ) = ∫ ∫ |ℱ𝜓1(𝜉)ℱ𝜓̅̅ ̅̅ (𝛼𝜉)| 2 ℝ+ 𝑑𝛼 𝛼 ℝ 𝑑𝛽 = ∫ |ℱ𝜓1(𝜉)| 2 { ∫ |ℱ𝜓̅̅ ̅̅ (𝛼𝜉)| 2 ℝ+ 𝑑𝛼 𝛼 } ℝ 𝑑𝜉 = {∫ |ℱ𝜓1(𝜉)| 2𝑑𝜉 } { ∫ |ℱ𝜓̅̅ ̅̅ (𝛼𝜉)| 2 ℝ+ 𝑑𝛼 𝛼 } ℝ = { ∫ |ℱ𝜓1(𝜉)| 2𝑑𝜉 } {∫ |ℱ𝜓̅̅ ̅̅ (𝛼 ′)| 2 ℝ 𝑑𝛼′ 𝛼′ } ℝ ( 𝛼′ ≔ 𝛼𝜉 ). = ‖ℱ𝜓1‖ 2 {∫ |ℱ𝜓̅̅ ̅̅ (𝛼′)| 2 ℝ 𝑑𝛼′ 𝛼′ }. thus, from the latter equation we obtain the duflo-moore operator is equal to ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉 −1/2ℱ𝜓(𝜉) as desired. ∎ 3.2 the duflo-moore operator for the irreducible unitary representation of 𝐀𝐟𝐟+(𝟏) this session is the main result. first, we recall that the irreducible unitary representation of group aff +(1) in proposition 2 can be written in the following proposition e. kurniadi, n. gusriani, b. subartini duflo-moore operator for the square-interable representation of 2-dimensional affine lie group 119 proposition 7. the irreducible unitary representations of aff +(1) correponsding to open coadjoint orbit ω+ in eqs. (2) in the space l 2(ℝ+) is of the form 𝜋+(δ(𝛼))𝑓(𝑥) = 𝑓(𝛼𝑥), 𝜋+(∇(𝛽))𝑓(𝑥) = 𝑒 2𝜋𝑖𝛽𝑥 𝑓(𝑥), (11) where 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ and 𝑓 ∈ l 2(ℝ+). proof. let aff(1) be a lie algebra of aff +(1) whose basis is {𝑒1, 𝑒2}. we consider its dual space as aff(1)∗ ∋ ( 𝑎 ∗ 𝑏 ∗ ), where 𝑎, 𝑏 ∈ ℝ. moreover, let ( 𝛼 𝛽 0 1 ) , 𝛼 ∈ ℝ+, 𝛽 ∈ ℝ be an element of group affine aff +(1). we shall construct the irreducible unitary representation of aff +(1) corresponding to open coadjoint orbit ω+ ≔ {(𝑎, 𝑏) ; 𝑏 > 0}. to do that, fix a point 𝜏 ≔ 𝑒2 ∗ ∈ ω+ ⊂ aff(1) ∗ as a linear functional. for subalgebra ℵ ≔ 〈𝑒2〉 we have ℵ has maximal dimension and the value of linear functional 𝜏 on the commutator [ ℵ, ℵ] is given by 𝜏([ ℵ, ℵ]) = 0. therefore, ℵ is a polarization in aff(1). let ℵ⊥ be the orthogonal subspace. furthermore, since 𝜏 + ℵ⊥ is contained in ω+ then ℵ satisfies pukanszky condition. now we construct a 1-dimensional representation 𝜆𝜏 of ν ≔ exp ℵ as follows. 𝜆𝜏 (exp 𝑒) ≔ 𝑒 2𝜋𝑖〈𝜏|𝑒〉 = 𝑒2𝜋𝑖𝛽 , 𝑒 ≔ 𝛼𝑒1 + 𝛽𝑒2 , 𝜏 ∈ ω+. (12) we identify the coset aff +(1)/ν by ℝ+ and we obtain the section given by 𝑠: ℝ+ ∋ 𝑥 ↦ exp 𝑥𝑒1 ∈ aff +(1). (13) to obtain the explicit formula of the representation of aff +(1) we need to solve what we called the master equation 𝑠(𝑥)𝑔 = ℎ𝑠(𝑥, 𝑔)𝑠(𝑥𝑔), (𝑥 ∈ ℝ+, 𝑔 ∈ aff +(1), ℎ𝑠(𝑥, 𝑔) ∈ ν ). (14) using the basis {𝑒1, 𝑒2 } we solve the following master equations with respect to its basis: a. ( 𝑥 0 0 1 ) ( 𝛼 0 0 1 ) = ( 1 𝑢 0 1 ) ( 𝑦 0 0 1 ), by solving with respect to 𝑢 and 𝑦 we obtain 𝑦 = 𝛼𝑥. therefore, 𝜋+(δ(𝛼))𝑓(𝑥) = 𝑓(𝛼𝑥). we mention here that we apply a right action of aff +(1) in space l2(ℝ+). b. ( 𝑥 0 0 1 ) ( 1 𝛽 0 1 ) = ( 1 𝑢 0 1 ) ( 𝑦 0 0 1 ). in this case, we have 𝑦 = 𝑥 and 𝑢 = 𝛽𝑥. therefore, 𝜋+(∇(𝛽))𝑓(𝑥) = 𝑒 2𝜋𝑖𝛽𝑥 𝑓(𝑥) as desired. ∎ in the next section, we shall compute the duflo-moore operator for the representation of aff +(1) with respect to its right haar measure. the result of duflomoore operator for the representation of aff +(1) with respect left haar measure can be found in [24] pages 82-85. jurnal matematika mantik volume 6, no. 2, october 2020, pp.114-122 120 proposition 8. the duflo-moore operator for the irreducible unitary representation π+ of aff +(1) as written in eqs. (11) is of the form 𝐶𝜋+ 𝑓(δ(𝑥)) = 𝑥 −1/2𝑓(𝑥), ( 𝑓 ∈ l2(ℝ+), 𝑥 ∈ ℝ+ ) (15) proof. let 𝜗1 and 𝜗2 be elements in 𝐶𝑐(aff +(1)). using the right haar measure, we shall compute the integral ∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(δ(𝛼))𝜗2〉l2(ℝ+)| 2 𝑑𝛽 𝑑𝛼 𝛼 aff+(1) to do that, first we compute the following inner product. 〈𝜗1|𝜋+(∇(𝛽))𝜋+(δ(𝛼))𝜗2〉l2(ℝ+) = ∫ 𝜗1(𝑥)𝜋+(∇(𝛽))𝜋+(δ(𝛼))𝜗2 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ℝ+ (𝑥) 𝑑𝑥 𝑥 = ∫ 𝜗1(𝑥)𝜋+(δ(𝛼))𝑒 2𝜋𝑖𝛽𝑥 𝜗2 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ℝ+ (𝑥) 𝑑𝑥 𝑥 = ∫ 𝑒 −2𝜋𝑖𝛽𝑥 𝜗1(𝑥)𝜋+(δ(𝛼))𝜗2 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ℝ+ (𝑥) 𝑑𝑥 𝑥 . using plancherel’s theorem we have ∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(δ(𝛼))𝜗2〉l2(ℝ+)| 2 𝑑𝛽 = ℝ ∫ |𝑒−2𝜋𝑖𝛽𝑥 𝜗1(𝑥)𝜋+(δ(𝛼))𝜗2 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑥)| 2 ℝ+ 𝑑𝑥 𝑥2 = ∫ |𝜗1(𝑥)𝜋+(δ(𝛼))𝜗2 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ (𝑥)| 2 ℝ+ 𝑑𝑥 𝑥2 = ∫ |𝜗1(𝑥)𝜗2̅̅ ̅(𝛼𝑥)| 2 ℝ+ 𝑑𝑥 𝑥2 . therefore, using fubini’s theorem we obtain ∫ |〈𝜗1|𝜋+(∇(𝛽))𝜋+(δ(𝛼))𝜗2〉l2(ℝ+)| 2 𝑑𝛽 𝑑𝛼 𝛼 = ∫ |𝜗1(𝑥)| 2 { ∫ |𝜗2(𝛼𝑥)| 2 𝑑𝛼 𝛼 ℝ+ } ℝ+aff+(1) 𝑑𝑥 𝑥2 = ∫ |𝜗1(𝑥)| 2 𝑑𝑥 𝑥2 { ∫ |𝜗2(𝛼 ′)|2 𝑑𝛼′ 𝛼′ ℝ+ } ℝ+ (𝛼′ ≔ 𝛼𝑥) = ∫ |𝑥 −1/2𝜗1(𝑥)| 2 𝑑𝑥 𝑥 { ∫ |𝜗2(𝛼 ′)|2 𝑑𝛼′ 𝛼′ ℝ+ } ℝ+ = ∫ |𝑥−1/2𝜗1(𝑥)| 2 𝑑𝑥 𝑥 ℝ+ . ‖𝜗2‖ 2. e. kurniadi, n. gusriani, b. subartini duflo-moore operator for the square-interable representation of 2-dimensional affine lie group 121 therefore, the duflo-moore operator for the irreducible unitary representation of aff +(1) as written in eqs. (11) is of the form 𝐶𝜋+ 𝑓(δ(𝑥)) = 𝑥 −1/2𝑓(𝑥) as desired. ∎ 4. conclusions the duflo-moore operator for the representations of aff +(1) in this paper is considered in two cases. the first case, it is for the quasi-regular representation and written in the term of fourier transform. namely, we obtain ℱ(𝐶𝜋𝜓)(𝜉) = 𝜉 −1/2ℱ𝜓(𝜉) (see [18]). the second case, the duflo-moore operator is considered for irreducible unitary representation with respect to its right haar measure and we have 𝐶𝜋+ 𝑓(δ(𝑥)) = 𝑥−1/2𝑓(𝑥). on the other hand, the duflo-moore operator for a square-intergarble representation of aff +(1) with respect to its left haar measure can be seen in [24] pages 82-85. it is more interesting to compute the duflo-moore operator for the representation of higher dimension of affine lie groups. 5. acknowledgement the first author is very grateful to professor hideyuki ishi from osaka city university japan for generous support and guidance to the first author to study representation theory of lie groups during his study. we also thank to the university of padjadjaran who has funded the work through riset percepatan lektor kepala (rplk) year 2020 with contract number 1427/un6.3.1/lt/2020. references [1] k. grochenig and d. rottensteiner, “orthonormal bases in the orbit of squareintegrable representations of nilpoten lie groups,” j. funct. anal., vol. 275, no. 12, pp. 3338--3379, 2018. [2] a. . farashahi, “square-integrability of metaplectic wave-packet representations of l^2(r),” j. math. anal. appl., vol. 449, no. 1, 2017. [3] 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. [4] 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. [5] p. stachura, “on the quantum ax + b group,” j. geom. phys., vol. 73, pp. 125-149, 2013. [6] zeitlin,a.m, “unitary representations of a loop ax+b group, wiener measure and \gamma -function,” j. funct. anal., vol. 263, no. 3, pp. 529--548, 2012. [7] m. . dyer and g. . lehres, “parabolic subgroup orbits on finite root systems,” j. pure appl. algebr., vol. 222, no. 12, pp. 3849--3857, 2018. [8] m. calvez and et al, “conjugacy stability of parabolic subgroups of artin-tits groups of spherical type,” j. algebr., vol. 556, pp. 621--633, 2020. [9] e. kurniadi and h. ishi, “harmonic analysis for 4-dimensional real frobenius lie algebras,” in springer proceeding in mathematics & statistics, 2019. [10] w. rump, “affine stucture of decomposable solvable groups,” j. algebr., vol. 556, pp. 725--749, 2020. jurnal matematika mantik volume 6, no. 2, october 2020, pp.114-122 122 [11] h. li and q. wang, “trigonometric lie algebras, affine lie algebras, and vertex algebras,” j. adv. math., vol. 363, 2020. [12] j. . souza, “sufficient conditions for dispersiveness of invariant control affine system on the heisenberg group,” syst. &control lett., vol. 124, pp. 68--74, 2019. [13] e. marberg, “on some actions the 0-zero hecke monoids of affine symmetric groups,” j. comb. theory, vol. 161, pp. 178--219, 2019. [14] ayala,v, a. da silva, and m. ferreira, “affine and bilinear systems on lie groups,” syst. &control lett., vol. 117, pp. 23--29, 2018. [15] f. catino, i. colazzo, and p. stefanella, “regular subgroups of the affine group and asymmetric product of radical braces,” j. algebr., vol. 455, pp. 164--182, 2016. [16] d. burde and et al, “affine actions on lie groups and post-lie algebra structures,” j. linear algebr. its appl., vol. 437, no. 5, pp. 1250--1263, 2012. [17] h. kato, “low dimensional lie groups admitting left-invariant flat projective or affine structures,” j. differ. geom. its appl., vol. 30, no. 2, pp. 153--163, 2012. [18] h. fuhr, abstrac harmonic analysis of continuous wavelet transforms, lecture notes in mathematics. berlin: springer-verlag, 2005. [19] e. kurniadi, “on square-integrable representations of a lie group of 4dimensional standard filiform lie algebra,” cauchyjurnal mat. murni dan apl., 2020. [20] a. a. kirillov, “lectures on the orbit method, graduate studies in mathematics,” am. math. soc., vol. 64, 2004. [21] r. berndt, representation of linear groups. an introduction based on examples from physics and number theory. wiesbaden: vieweg, 2007. [22] p. aniello, g. cassinelli, e. de vito, and levrero,a., “square-integrability of induced representations of semidirect products,” rev.math.phys, vol. 10, pp. 301-313, 1998. [23] m. duflo and c. c. moore, “on the regular representation of a nonunimodular locally compact,” j. funct. anal., vol. 21, pp. 209–243, 1976. [24] e. kurniadi, “harmonic analysis for finite dimensional real frobenius lie algebras, ph.d thesis, ” nagoya university, 2019. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: f. y. ishak, “existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions”, jmm, vol. 6, no. 1, pp. 1-12, may 2020. existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions faraj y. ishak university of duhok iraq, faraj.ishak@uod.ac doi: https://doi.org/10.15642/mantik.2020.6.1.1-12 abstrak: artikel ini menyelidiki eksistensi, keunikan, dan solusi stabil dari sistem persamaan diferensial-diferensial volterra fraksional baru dengan kondisi batas fraksional dengan menggunakan teorema eksistensi dan keunikan. teorema tentang eksistensi dan keunikan dari solusi yang ditetapkan di bawah beberapa kondisi yang diperlukan dan cukup pada ruang kompak. contoh sederhana dari hasil aplikasi utama disajikan dalam artikel ini. kata kunci: keberadaan dan keunikan, stabilitas, fractional integrodifferential equations, masalah nilai batas, teorema eksistensi dan keunikan. abstract: this article investigates existence, uniqueness and stability solutions of new fractional volterra integro-differential equations system with fractional boundary conditions by using the existence and uniqueness theorem. theorems on existence and uniqueness of solution are established under some necessary and sufficient conditions on compact space. a simple example of application of the main results of this article is presented. keywords: existence and uniqueness, stability, fractional integrodifferential equations, boundary value problem, existence and uniqueness theorem. jurnal matematika mantik vol. 6, no. 1, may 2020, pp. 1-12 issn: 2527-3159 (print) 2527-3167 (online) mailto:faraj.ishak@uod.ac http://u.lipi.go.id/1458103791 jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 1-12 2 1. introduction gottfried leibniz and guilliaume l’hopital sparked initial curiosity into the theory of fractional calculus during a 1695 correspondence on the possible value and meaning of non-integer-order derivatives. by the late nineteenth century, the combined efforts of a number of mathematicians most notably liouville, grunwald, letnikov, and riemann produced a fairly solid theory of fractional calculus for functions of a real variable. though several viable fractional derivatives were proposed, the so-called riemannliouville and caputo derivatives are the two most-commonly used today. mathematicians have employed this fractional calculus in recent years to model and solve a variety of applied problems. indeed, as podlubney outlines in [1]. fractional differential equations have extensive applications in various fields of science and engineering. many phenomena in viscoelasticity, electrochemistry, control theory, porous media, electromagnetism, and other fields, can be modelled by fractional differential equations. we refer the reader to [2, 4] and references therein for some applications. fractional bvps defined on intervals have been studied by many authors. many results on the existence, uniqueness, multiplicity, and nonexistence of solutions for fractional differential equations subject to various boundary conditions (bcs) have been obtained; see for example [9,10,11,12,13,14,15,16]. the fractional difference calculus had its origin in the works by al-salam [6] and agarwal [7]. more recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional difference calculus were made, specifically, analogues of the integral and differential fractional operators properties such as the mittag-leffler function, the laplace transform, and taylor’s formula [3,5,8,17,18], just to mention some. butris and ishak [20], used both methods picard approximation and banach fixed point theorems for studying the existence and uniqueness solutions to the following fractional integral equations: 𝑢(𝑡) = 𝑓(𝑡)+ 1 γ(𝛼) ∫ (𝑡 −𝑠)𝛼−1𝐹(𝑡,𝑠,𝑢(𝑠),𝑤(𝑠))𝑑𝑠 𝑡 𝑎 𝑤(𝑡) = 𝑔(𝑡)+ 1 γ(𝛼) ∫ (𝑡 −𝑠)𝛼−1𝐺(𝑡,𝑠,𝑢(𝑠),𝑤(𝑠))𝑑𝑠 𝑏 𝑎 in this work our aim is to show the existence solutions of the system of integrodifferential equations 𝐷𝛼𝑥(𝑡)+ 𝑓(𝑡,𝑠, [𝜙 𝑥](𝑥)) = 0 𝐷𝛽𝑦(𝑡) +𝑔(𝑡,𝑠, [𝜑 𝑦](𝑦)) = 0 𝐷𝛼−1𝑥(0) = 0, 𝐷𝛼−1𝑥(1) = 𝑏1, 𝐷𝛽−1𝑦(0) = 0, 𝐷𝛽−1𝑦(1) = 𝑏2 where: [𝜙 𝑥](𝑥) = ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠))𝑑𝑠 𝑡 −∞ , [𝜑 𝑦](𝑦) = ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠))𝑑𝑠 𝑡 −∞ 1 < 𝛼,𝛽 ≤ 2,0 ≤ 𝑠 ≤ 𝑡 ≤ 1,𝑏1,𝑏2 ∈ 𝑅, 𝑥 ∈ 𝐷1 ⊆ 𝑅 𝑛 and 𝑦 ∈ 𝐷2 ⊆ 𝑅 𝑚,𝐷1 and 𝐷2 are compact domain, let the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) , 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) is defined and continuous on the domain: 𝐷 = {(𝑡,𝑠,𝑥,𝑦);𝑡,𝑠 𝜖 [0,1] ,𝑥 ∈ 𝐷1 ,𝑦 ∈ 𝐷2} (1.2) (1.1) f. y. ishak existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions 3 assume that the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)), 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)), and kernels 𝐾(𝑡,𝑠),𝐺(𝑡,𝑠) are satisfying the following inequalities: ‖𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡))‖ ≤ 𝑀1 , ‖𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡))‖ ≤ 𝑀2 (1.3) ‖𝐹(𝑡,𝑠,𝑥2,𝑦2)− 𝐹(𝑡,𝑠,𝑥1,𝑦1)‖ ≤ 𝐿1(‖𝑥2 −𝑥1‖+ ‖𝑦2 −𝑦1‖) (1.4) ‖𝐻(𝑡,𝑠,𝑥2,𝑦2)− 𝐻(𝑡,𝑠,𝑥1,𝑦1)‖ ≤ 𝐿2(‖𝑥2 − 𝑥1‖+ ‖𝑦2 −𝑦1‖) (1.5) ‖𝐾(𝑡,𝑠)‖ ≤ 𝛿1𝑒 −𝜆1(𝑡−𝑠) , ‖𝐺(𝑡,𝑠)‖ ≤ 𝛿2𝑒 −𝜆2(𝑡−𝑠) (1.6) where 𝑀1,𝑀2,𝐿1,𝐿2,𝜆1,𝜆2,𝛿1,𝛿2,are positive constants 𝑥1,𝑥2 ∈ 𝐷 1 𝑦1,𝑦2 ∈ 𝐷2 𝑡 ,𝑠 𝜖 [0,1] and ‖.‖ = max 𝑡∈[0,1] |. |, we defined non-empty sets as: 𝐷𝐹 = 𝐷1 − 𝑀1𝛿1(𝛼−1)+𝑏1𝜆1𝛼 𝜆1𝛤(𝛼+1) 𝐷𝐻 = 𝐷2 − 𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β 𝜆2γ(𝛽+1) as well as we suppose the maximum value of the following matrix: δ0 = ( 𝐿1𝛿1(𝛼−1) 𝜆1γ(𝛼+1) 𝐿1𝛿1(𝛼−1) 𝜆1γ(𝛼+1) 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) ) , less than one i.e. . 𝜆 𝑚𝑎𝑥(δ0) = 𝐿1𝛿1(𝛼−1) 𝜆1γ(𝛼+1) + 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) < 1 (1.8) define a sequence of functions {𝑥𝑚(𝑡,𝑥0)}𝑚=0 ∞ , {𝑦𝑚(𝑡,𝑦0)}𝑚=0 ∞ as: 𝑥𝑚+1(𝑡,𝑥0) = 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) 𝑦𝑚+1(𝑡,𝑦0) = 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) with 𝑥0 = 𝑥(0) = 0, 𝑦0 = 𝑦(0) = 0, m = 0,1,2… 2. preliminaries let us recall some basic definitions on fractional calculus, which can be found in the literature. definition 2.1 [19] assume that 𝑓(𝑥,𝑦) is defined on the set (𝑎,𝑏)x𝐺,𝐺 ⊂ 𝑅,𝑓(𝑥,𝑦) is said to satisfy lipschitz condition with respect to the second variable, if for all 𝑥 ∈ (𝑎,𝑏) and for any 𝑦1 ,𝑦2 ∈ 𝐺 |𝑓(𝑥,𝑦1)−𝑓(𝑥,𝑦2)| ≤ 𝜉|𝑦1 − 𝑦2| (1.9) (1.7) jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 1-12 4 where 𝜉 > 0 does not depend on 𝑥 ∈ (𝑎,𝑏). definition 2.2 [19] the riemann-liouville fractional integral of order q is defined by 𝐼𝑞𝑓(𝑡) = 1 γ(𝑞) ∫ (𝑡 − 𝑠)𝑞−1 𝑡 0 𝑓(𝑠)𝑑𝑠, 𝑞 > 0 provided the integral exists. definition 2.3 [19] the riemann-liouville fractional derivative of order q is defined by 𝐷𝑞𝑓(𝑡) = 1 γ(𝑛− 𝑞) ( 𝑑 𝑑𝑡 ) 𝑛 ∫ (𝑡 − 𝑠)𝑛−𝑞−1 𝑡 0 𝑓(𝑠)𝑑𝑠, 𝑛 −1 < 𝑞 ≤ 𝑛, 𝑞 > 0, provided the right-hand side is pointwise defined on (0,+∞). lemma 2.1 for 𝛼,𝛽 > 0, then the following relation hold: 𝐷𝛼𝑡𝛽 = γ(𝛽 + 1) γ(𝛽 +1 −𝛼) 𝑡𝛽−𝛼−1,𝛽 > 𝑛 𝑎𝑛𝑑 𝐷𝛼𝑡𝑘 = 0,𝑘 = 0,1,…,𝑛 −1 lemma 2.2 [3] the equality 𝐷0+ 𝛼 𝐼0+ 𝛼 𝑓(𝑡) = 𝑓(𝑡), 𝛼 > 0 holds for 𝑓 ∈ 𝐿1(0,1). lemma 2.3 let 𝛼,𝛽 > 0 and let f be a function defined on [0, 1]. then the following formulas hold: (i) (𝐼𝑞 𝛽 𝐼𝑞 𝛼𝑓)(𝑥) = (𝐼𝑞 𝛼+𝛽 𝑓)(𝑥) (ii) (𝐷𝑞 𝛼𝐼𝑞 𝛼𝑓)(𝑥) = 𝑓(𝑥) lemma 2.4 let 𝛼 > 0 and n be a positive integer. then, the following equality holds: (𝐼𝑞 𝛽 𝐷𝑞 𝛼𝑓)(𝑥) = (𝐷𝑞 𝑛𝐼𝑞 𝛼𝑓)(𝑥) −∑ 𝑥𝛼−𝑛+𝑘 γ(α+k−n+1) (𝐷𝑞 𝑘𝑓)(0)𝑛−1𝑘=0 lemma 2.5 a functions 𝑥(𝑡),𝑦(𝑡), are solution of system (1.1) if and only if 𝑥(𝑡),𝑦(𝑡) have the form: 𝑥 (𝑡) = 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) 𝑦 (𝑡) = 𝑡𝛽−1 𝛤(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛽−1 𝛤(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 𝛤(𝛽) proof: it follows from lemma 2.4 that the system of fractional differential equation in (1.1) is equivalent to the integral equations: 𝑥(𝑡) = −𝐼𝛼𝑓(𝑡)+ 𝑐1𝑡 𝛼−1 + 𝑐2𝑡 𝛼−2 𝑦(𝑡) = −𝐼𝛼𝑔(𝑡)+𝑑1𝑡 𝛼−1 +𝑑2𝑡 𝛼−2 (2.1) f. y. ishak existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions 5 where: 𝑓(𝑡) = ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠)) 𝑑𝑠 𝑡 −∞ , 𝑔(𝑡) = ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠 𝑡 −∞ , 𝑐1,𝑐2,𝑑1,𝑑2 ∈ 𝑅 from the boundary conditions of (1.1) we have 𝑐2 = 0,𝑑2 = 0 and: 𝑐1 = 𝑏1 γ(𝛼) + 1 γ(𝛼) ∫ 𝑓(𝑠)𝑑𝑠 1 0 , 𝑑1 = 𝑏2 γ(𝛽) + 1 γ(𝛽) ∫ 𝑔(𝑠)𝑑𝑠 1 0 which is complete the proof. 3. result and discussion in this section, the theorems of existence, uniqueness, and stability of a solution for system (1.1) will be given. theorem 3.1: let the right side of system (1.1) are defined and continuous on domain (1.2) .suppose that the vector functions 𝐹(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) , 𝐻(𝑡,𝑠,𝑥(𝑡),𝑦(𝑡)) are satisfying the inequalities (1.3)-(1.5) and the conditions (1.6)-(1.9).then there exist a sequences of functions (1.9) converges uniformly as 𝑚 → ∞ on domain (1.2) to the limit functions which satisfying integral equations (2.1) provided that: ‖𝑥∞(𝑡,𝑥0)− 𝑥0‖ ≤ 𝑀1𝛿1(𝛼−1)+𝜆1𝑏1𝛼 𝜆1γ(𝛼+1) ‖𝑦∞(𝑡,𝑥0)− 𝑦0‖ ≤ 𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β 𝜆2γ(𝛽+1) ( ‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖ ‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖ ) ≤ δ0 𝑚(𝐼 − δ0) −1φ0 for all 𝑚 ≥ 1 , 𝑡 ∈ [0,1] proof: by using the sequence of function (1.9) when m=0, we get: ‖𝑥1(𝑡,𝑥0)−𝑥0‖ ≤ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) ≤ 𝑡𝛼−1𝑀1 γ(𝛼) ∫ 1 0 ∫𝛿1𝑒 −𝜆1(𝑡−𝑠) 𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠)𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) ≤ 𝑀1𝛿1 𝜆1γ(𝛼) − 𝑀1𝛿1 𝜆1γ(𝛼 +1) + 𝑏1 γ(α) ≤ 𝑀1𝛿1(𝛼− 1)+ 𝑏1𝜆1𝛼 𝜆1𝛤(𝛼 + 1) and by the same we have ‖𝑦1(𝑡,𝑦0)− 𝑦0‖ ≤ 𝑀2𝛿2(𝛽− 1)+ 𝜆2𝑏2β 𝜆2γ(𝛽 +1) that is: 𝑥1(𝑡,𝑥0)𝜖𝐷1 𝑦1(𝑡,𝑦0)𝜖𝐷2, for all 𝑡𝜖[0,1],𝑥0 𝜖 𝐷𝐹,𝑦0 ∈ 𝐷𝐻 (3.1) jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 1-12 6 suppose that 𝑥𝑝(𝑡,𝑥0) 𝜖 𝐷1 ,𝑦𝑝(𝑡,𝑦0) 𝜖 𝐷2 for each 𝑥0 𝜖 𝐷𝐹 ,𝑦0 𝜖 𝐷𝐻 ,𝑝𝜖𝑍 + , t∈ [0,1] ,by mathematical induction we conclude that: 𝑥𝑚(𝑡,𝑥0) 𝜖 𝐷1 ,𝑦𝑚(𝑡,𝑦0) 𝜖 𝐷2 for each 𝑥0 𝜖 𝐷𝐹 ,𝑦0 𝜖 𝐷𝐻 , 𝑚 = 0,1,2,… to prove that the sequences (1.9) convergence uniformly in domain (1.2): ‖𝑥2(𝑡,𝑥0)−𝑥1(𝑡,𝑥0)‖ ≤ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))‖ 𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))‖𝑑𝑠)𝑑𝑠 𝑡 −∞ − 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠 𝑡 −∞ + ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠 𝑡 −∞ ≤ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠) ‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))− 𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖ 𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠) ‖𝐹(𝑡,𝑠,𝑥1 (𝑠),𝑦1 (𝑠))− 𝐹(𝑡,𝑠,𝑥0 (𝑠),𝑦0 (𝑠))‖𝑑𝑠)𝑑𝑠 𝑡 −∞ ≤ 𝐿1𝑡 𝛼−1 γ(𝛼) (‖𝑥1 −𝑥0‖ +‖𝑦1 −𝑦0‖)∫ 1 0 ∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠) 𝑑𝑠𝑑𝑠 𝑡 −∞ − 𝐿1(‖𝑥1 − 𝑥0‖+ ‖𝑦1 − 𝑦0‖)∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠) 𝑡 −∞ 𝑑𝑠)𝑑𝑠 ≤ 𝐿1𝛿1(𝛼𝑡 𝛼−1 −𝑡𝛼) 𝜆1γ(𝛼 +1) (‖𝑥1 − 𝑥0‖+ ‖𝑦1 −𝑦0‖) and by the same ‖𝑦2(𝑡,𝑦0)−𝑦1(𝑡,𝑦0)‖ ≤ 𝐿2𝛿2(𝛽𝑡 𝛽−1 − 𝑡𝛽) 𝜆2𝛤(𝛽 +1) −(‖𝑥1 −𝑥0‖ +‖𝑦1 − 𝑦0‖) by the mathematical induction the following inequalities hold: ‖𝑥𝑚+1(𝑡,𝑥0)−𝑥𝑚(𝑡,𝑥0)‖ ≤ 𝐿1𝛿1(𝛼𝑡 𝛼−1−𝑡𝛼) 𝜆1γ(𝛼+1) (‖𝑥𝑚 − 𝑥𝑚−1‖+ ‖𝑦𝑚 − 𝑦𝑚−1‖) ‖𝑦𝑚+1(𝑡,𝑦0)−𝑦𝑚(𝑡,𝑦0)‖ ≤ 𝐿2𝛿2(𝛽𝑡 𝛽−1−𝑡𝛽) 𝜆2γ(𝛽+1) (‖𝑥𝑚 − 𝑥𝑚−1‖+ ‖𝑦𝑚 −𝑦𝑚−1‖) rewrite (3.2) with vector form: ( ‖𝑥𝑚+1(𝑡,𝑥0)− 𝑥𝑚(𝑡,𝑥0)‖ ‖𝑦m+1(𝑡,𝑦0)− 𝑦m(𝑡,𝑦0)‖ ) ≤ ( 𝐿1𝛿1(𝛼𝑡 𝛼−1 − 𝑡𝛼) 𝜆1γ(𝛼 +1) 𝐿1𝛿1(𝛼𝑡 𝛼−1 −𝑡𝛼) 𝜆1γ(𝛼 +1) 𝐿2𝛿2(𝛽𝑡 𝛽−1 − 𝑡𝛽) 𝜆2γ(𝛽+1) 𝐿2𝛿2(𝛽𝑡 𝛽−1 −𝑡𝛽) 𝜆2γ(𝛽+ 1) ) ( ‖𝑥𝑚 − 𝑥𝑚−1‖ ‖𝑦𝑚 − 𝑦𝑚−1‖ ) that is: φ𝑚+1(𝑡,𝑥0,𝑦0) ≤ δ(𝑡)φ𝑚(𝑡,𝑥0,𝑦0) (3.3) where: (3.2) f. y. ishak existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions 7 δ(𝑡) = ( 𝐿1𝛿1(𝛼𝑡 𝛼−1−𝑡𝛼) 𝜆1γ(𝛼+1) 𝐿1𝛿1(𝛼𝑡 𝛼−1−𝑡𝛼) 𝜆1γ(𝛼+1) 𝐿2𝛿2(𝛽𝑡 𝛽−1−𝑡𝛽) 𝜆2γ(𝛽+1) 𝐿2𝛿2(𝛽𝑡 𝛽−1−𝑡𝛽) 𝜆2γ(𝛽+1) ),φ𝑚+1 = ( ‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖ ‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖ ) ,φ𝑚 = ( ‖𝑥𝑚 −𝑥𝑚−1‖ ‖𝑦𝑚 − 𝑦𝑚−1‖ ) take the maximum value for both sides of (3.3): φ 𝑚+1 ≤ δ0 φ𝑚 (3.4) where δ0 = max 𝑡𝜖[0,1] δ (𝑡), δ0 = ( 𝐿1𝛿1(𝛼−1) 𝜆1γ(𝛼+1) 𝐿1𝛿1(𝛼−1) 𝜆1γ(𝛼+1) 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) ) by repletion of (3.4) we obtain: φ𝑚+1 ≤ λ0 𝑚 φ1 φ1 ≤ ( 𝑀1𝛿1(𝛼−1)+𝜆1𝑏1𝛼 𝜆1γ(𝛼+1) 𝑀2𝛿2(𝛽−1)+𝜆2𝑏2β 𝜆2γ(𝛽+1) ),and also we get: ∑ φ𝑖 𝑚 𝑖=1 ≤ ∑ δ0 𝑖−1φ1 𝑚 𝑖=1 (3.5) since the matrix δ0 has eigenvalue 𝜆1 = 0, 𝜆2 = 𝜆𝑚𝑎𝑥(δ0) = 𝑀1𝛿1(𝛼 −1) +𝜆1𝑏1𝛼 𝜆1γ(𝛼+ 1) + 𝑀2𝛿2(𝛽 − 1)+ 𝜆2𝑏2β 𝜆2γ(𝛽 +1) < 1 the series (3.5) is uniformly convergent, i.e. lim 𝑚→∞ ∑ δ0 𝑖−1φ1 𝑚 𝑖=1 = ∑ δ0 𝑖−1φ1 ∞ 𝑖=1 = (𝐼 −δ0) −1φ1 (3.6) thus the limiting relation (3.6) signifies uniform convergence of sequences: {𝑥𝑚(𝑡,𝑥0)}𝑚=0 ∞ , {𝑦𝑚(𝑡,𝑦0)}𝑚=0 ∞ ,that is: lim 𝑚→∞ 𝑥𝑚(𝑡,𝑥0) = 𝑥 (𝑡,𝑥0) ,and lim 𝑚→∞ 𝑦𝑚(𝑡,𝑦0) = 𝑦 (𝑡,𝑦0) by all conditions and inequalities of the theorem the estimate ( ‖𝑥𝑚+1(𝑡,𝑥0) −𝑥𝑚(𝑡,𝑥0)‖ ‖𝑦m+1(𝑡,𝑦0)−𝑦m(𝑡,𝑦0)‖ ) ≤ δ0 𝑚(𝐼 − δ0) −1φ1 is hold for all m=1, 2, … to prove that 𝑥 (𝑡,𝑥0)𝜖𝐷1 and 𝑦(𝑡,𝑦0)𝜖𝐷2 we prove that: lim 𝑚→∞ ( 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) ) = jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 1-12 8 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) (3.7) lim 𝑚→∞ ( 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) = 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) (3.8) we have: ‖ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) − 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ + ∫ (𝑡 − 𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ − 𝑏1𝑡 𝛼−1 γ(α) ‖ ≤ 𝑡𝛼−1 γ(𝛼) ∫ ∫𝛿1𝑒 −𝜆1(𝑡−𝑠)‖𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))−𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))‖𝑑𝑠𝑑𝑠 𝑡 −∞ 1 0 −∫ (𝑡 −𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝛿1𝑒 −𝜆1(𝑡−𝑠)‖𝐹(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠)) 𝑡 −∞ −𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))‖𝑑𝑠)𝑑𝑠 ≤ 𝐿1𝛿1(𝛼𝑡 𝛼−1 − 𝑡𝛼) 𝜆1γ(𝛼+ 1) (‖𝑥𝑚 − 𝑥 ‖+ ‖𝑦𝑚 − 𝑦 ‖) and for the function y(t,𝑦0) we have ‖ 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ −∫ (𝑡 −𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥𝑚(𝑠),𝑦𝑚(𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) − 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ +∫ (𝑡 −𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ − 𝑏2𝑡 𝛽−1 γ(β) ‖ f. y. ishak existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions 9 ≤ 𝐿2𝛿2(𝛽𝑡 𝛽−1 − 𝑡𝛽) 𝜆2γ(𝛽+ 1) (‖𝑥𝑚 − 𝑥 ‖+ ‖𝑦𝑚 −𝑦 ‖) and since the sequences: {𝑥𝑚(𝑡,𝑥0)}𝑚=0 ∞ , {𝑦𝑚(𝑡,𝑦0)}𝑚=0 ∞ uniformly convergence to 𝑥 (𝑡,𝑥0), 𝑦 (𝑡,𝑦0) respectively on the interval [0,1] ,that is (3.7),(3.8) satisfies. theorem (3.2): if all conditions and assumptions of theorem (3.1) satisfied, then the functions 𝑥 (𝑡,𝑥0), 𝑦(𝑡,𝑦0) are unique solution for system (1.1) on domain (1.2). proof: let ( 𝑢 (𝑡,𝑢0) 𝑤 (𝑡,𝑤0) ) = ( 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) ) be another solution for system (1.1) then: ‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖ ≤ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))− 𝑡 −∞ 𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠𝑑𝑠− ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝐾(𝑡,𝑠)‖‖𝐹(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))− 𝑡 −∞ 𝐹(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠)𝑑𝑠 ≤ 𝐿1𝛿1(𝛼𝑡 𝛼−1 −𝑡𝛼) 𝜆1γ(𝛼 +1) (‖𝑥 − 𝑢 ‖ +‖𝑦 −𝑤 ‖) and ‖𝑦 (𝑡,𝑦0)−𝑤(𝑡,𝑤0)‖ ≤ 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ ‖𝐺(𝑡,𝑠)‖‖𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))− 𝑡 −∞ 𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠𝑑𝑠− ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ ‖𝐺(𝑡,𝑠)‖‖𝐻(𝑡,𝑠,𝑥 (𝑠),𝑦 (𝑠))− 𝑡 −∞ 𝐻(𝑡,𝑠,𝑢 (𝑠),𝑤 (𝑠))‖𝑑𝑠)𝑑𝑠 ≤ 𝐿2𝛿2(𝛽𝑡 𝛽−1 −𝑡𝛽) 𝜆2γ(𝛽 + 1) (‖𝑥 −𝑢 ‖+ ‖𝑦− 𝑤 ‖) rewrite in vector form: ( ‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖ ‖𝑦 (𝑡,𝑦0)−w(𝑡,𝑤0)‖ ) ≤ δ(𝑡)( ‖𝑥(𝑡)−𝑢(𝑡)‖ ‖𝑦 (𝑡)− 𝑤(𝑡)‖ ) (3.9) by take the maximum value for both sides of (3.9) and reputation it we get: ( ‖𝑥 (𝑡,𝑥0)−𝑢(𝑡,𝑢0)‖ ‖𝑦 (𝑡,𝑦0)−w(𝑡,𝑤0)‖ ) ≤ δ0 𝑚 ( ‖𝑥(𝑡)−𝑢(𝑡)‖ ‖𝑦 (𝑡)− 𝑢(𝑡)‖ ) (3.10) from (3.9) and condition (1.8) we have 𝐴0 𝑚 → 0 when 𝑚 → ∞ that is: 𝑥 (𝑡,𝑥0) = 𝑢(𝑡,𝑢0) and 𝑦 (𝑡,𝑦0)− w(𝑡,𝑤0) therefor 𝑥 (𝑡,𝑥0), 𝑦(𝑡,𝑦0) is a unique solution for system (1.1). jurnal matematika mantik volume 6, issue. 1, may 2020, pp. 1-12 10 theorem (3.3): under the hypothesis and conditions of theorem (3.1) if �̃�(𝑡,𝑥0), �̃�(𝑡,𝑦0) is any other solution of system (1.1), then the solution is stable if satisfies the inequality: ( ‖𝑥 (𝑡,𝑥0)−�̃�(𝑡,𝑥0)‖ ‖𝑦 (𝑡,𝑦0)−�̃�(𝑡,𝑦0)‖ ) ≤ ( 𝜖1 𝜖2 ) where: �̃�(𝑡,𝑥0) = 𝑡𝛼−1 γ(𝛼) ∫ 1 0 ∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠, �̃� (𝑠),�̃� (𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛼−1 γ(𝛼) 𝑡 0 (∫ 𝐾(𝑡,𝑠)𝐹(𝑡,𝑠, �̃� (𝑠), �̃� (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏1𝑡 𝛼−1 γ(α) �̃�(𝑡,𝑦0) = 𝑡𝛽−1 γ(𝛽) ∫ 1 0 ∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠, �̃� (𝑠), �̃�(𝑠))𝑑𝑠𝑑𝑠 𝑡 −∞ − ∫ (𝑡−𝑠)𝛽−1 γ(𝛽) 𝑡 0 (∫ 𝐺(𝑡,𝑠)𝐻(𝑡,𝑠, �̃� (𝑠), �̃� (𝑠))𝑑𝑠)𝑑𝑠 𝑡 −∞ + 𝑏2𝑡 𝛽−1 γ(β) proof: ‖𝑥(𝑡,𝑥0) −�̃�(𝑡,𝑥0)‖ ≤ 𝐿1𝛿1(𝛼𝑡 𝛼−1−𝑡𝛼) 𝜆1γ(𝛼+1) (‖𝑥 −�̃�‖+ ‖𝑦 −�̃�‖) ‖𝑦(𝑡,𝑦0)−�̃� (𝑡,𝑦0)‖ ≤ 𝐿2𝛿2(𝛽𝑡 𝛽−1−𝑡𝛽) 𝜆2𝛤(𝛽+1) (‖𝑥 −�̃�‖+ ‖𝑦− �̃� ‖) rewrite (3.11), (3.12) in victor form we get: ( ‖𝑥(𝑡,𝑥0)− �̃�(𝑡,𝑥0)‖ ‖𝑦(𝑡,𝑦0)−�̃� (𝑡,𝑦0)‖ ) ≤ δ(𝑡)( ‖𝑥(𝑡) −�̅�(𝑡)‖ ‖𝑦(𝑡) −�̅�(𝑡)‖ ) by condition (1.8) and for 𝜖1,𝜖2 ≥ 0 we have: ( ‖𝑥(𝑡,𝑥0)− �̃�(𝑡,𝑥0)‖ ‖𝑦(𝑡,𝑦0)− �̃� (𝑡,𝑦0)‖ ) ≤ ( 𝜖1 𝜖2 ) (3.13) by the definition of stability, we find that �̃�(𝑡,𝑥0), �̃� (𝑡,𝑦0) is stable solution for system (1.1) example (3.1): consider the following system of fractional integrodifferential equations: 𝐷4 3⁄ 𝑥(𝑡) = ∫ (4𝑒2𝑠 +1) 𝑡 −∞ 𝑥(𝑠) 𝑦(𝑠) 𝑑𝑠 𝐷3 2⁄ 𝑦(𝑡) = ∫ 3cos (2𝑠)(𝑦(𝑠) 𝑡 −∞ + 𝑠𝑖𝑛(𝑥(𝑠)))𝑑𝑠 𝐷1 3⁄ 𝑥(0) = 0, 𝐷1 3⁄ 𝑥(1) = 2, 𝐷1 2⁄ 𝑥(0) = 0, 𝐷1 2⁄ 𝑥(1) = 3 comparing (3.14) and (1.1) we see that, 𝛼 = 4 3⁄ ,𝛽 = 3 2⁄ ,𝑘(𝑡,𝑠) = (4𝑒2𝑠 + 1),𝐺(𝑡,𝑠) = 3cos (2𝑠) 𝐹(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠)) = 𝑥(𝑠) 𝑦(𝑠)⁄ ,𝐻(𝑡,𝑠,𝑥(𝑠),𝑦(𝑠)) = 𝑦(𝑠) +sin(𝑥(𝑠)) , if we choose 𝑀1 = 1, 𝑀2 = 1,𝐿1 = 2𝑀1,𝐿2 = 2𝑀2, 𝛿1 = 1,𝛿2 = 2,𝜆1 = 2,𝜆2 = 3,then (1.3)− (1.6) holds and ∶ (3.11) (3.12) (3.14) f. y. ishak existence solution for nonlinear system of fractional integrodifferential equations of volterra type with fractional boundary conditions 11 𝜆 𝑚𝑎𝑥(δ0) = 𝐿1𝛿1(𝛼 −1) 𝜆1γ(𝛼 + 1) + 𝐿2𝛿2(𝛽−1) 𝜆2γ(𝛽+1) = (2)(1)(4 3⁄ −1) (2)( 1.1906) + (2)(2)(3 2− 1)⁄ (3)( 1.3293) < 1 thus, by theorems (3.1) -(3.3), we obtain that (1.1) has a unique stability solution. 4. conclusions the article presented some existence and uniqueness results for a boundary value problem of fractional integro-differential system. the prove of the theorems based on two basic conditions (1.8), (3.1). the basic of fractional differentiation were used to find the solution formula. the idea of existence and uniqueness theorem is the basis for finding results. the present work can be extended to boundary value problem with nonlocal and nonseparated fractional boundary conditions. references [1] podlubny, i: fractional differential equations, mathematics in science and engineering. academic press, new york/london/toronto (1999) [2] r. hilfer, applications of fractional calculus in physics. world scientific,singapore (2000). 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[20] butris and ishak, “some result’s in the theory of fractional integral equations of volterra-fredholm types “, volume 9, no.1, january – february 2020 international journal of information systems and computer sciences https://doi.org/10.30534/ijiscs/2020. http://dx.doi.org/10.3770/j.issn:2095-2651.2016.01.010 http://dx.doi.org/10.3770/j.issn:2095-2651.2016.01.010 paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: m. febriyani faris, y. farida, d. candra, n. ulinnuha, and m. hafiyusholeh, “implementation of the open jackson queuing network to reduce waiting time”, j. mat. mantik, vol. 6, no. 2, pp. 83-92, october 2020. implementation of the open jackson queuing network to reduce waiting time monike febriyani faris1, yuniar farida2, dian c. r. novitasari3, nurissaidah ulinnuha4, moh. hafiyusholeh5 1uin sunan ampel surabaya, monikefebriyani@gmail.com 2uin sunan ampel surabaya, yuniar_farida@uinsby.ac.id 3uin sunan ampel surabaya, diancrini@uinsby.ac.id 4uin sunan ampel surabaya, nuris.ulinnuha@uinsby.ac.id 5uin sunan ampel surabaya, hafiyusholeh@uinsby.ac.id doi: https://doi.org/10.15642/mantik.2020.6.2.83-92 abstrak. menunggu cukup lama untuk mendapatkan pelayanan menjadi suatu hal yang umum dalam pelayanan rumah sakit. semakin banyak pasien yang menunggu, semakin banyak pasien yang pelayanannya tertunda, sehingga waktu menunggu semakin lama. dalam proses pelayanan kesehatan suatu rumah sakit, seorang pasien akan mengantri beberapa kali di lebih dari satu antrian dalam instalasi rawat jalan rumah sakit. penelitian ini menggunakan studi kasus pada sistem antrian instalasi rawat jalan rumah sakit, yang bertujuan untuk untuk meminimalisir waktu tunggu dan panjang antrian tiap workstation dengan menerapkan model jaringan antrian jackson terbuka. workstation yang diteliti adalah workstation registrasi, prekonsultasi dan konsultasi poli jantung hingga farmasi. pengumpulan data dilakukan selama 6 hari menghitung jumlah kedatangan dan keberangkatan setiap titik dengan interval 5 menit. dengan menerapkan model jaringan antrian jackson terbuka, diperoleh rekomendasi kepada pihak rumah sakit untuk menambah pegawai di titik registrasi menjadi sebanyak 4 loket, titik prekonsultasi poli jantung menjadi sebanyak 3 perawat dan 4 dokter sedangkan untuk farmasi menjadi sebanyak 7 pegawai. dengan penambahan personel tersebut, total waktu tunggu pasien dalam system antrian menjadi kurang lebih 12 menit/pasien. sehingga model antrian ini akan mengurangi waktu tunggu dalam system antrian yang semula ratarata 108 menit/pasien. kata kunci: antrian; jaringan antrian jackson terbuka; workstation abstract. waiting for service is a common thing in-hospital services. the more patients are waiting, the service delay increases, so waiting time in the queue gets longer. in health care in a hospital, a patient will queue several times in more than one queue in a hospital outpatient installation. the case study in this research is the queue system in the hospital's outpatient treatment, implementing an open jackson queueing network to minimize waiting time. the workstations examined in this study were the registration, pre-consultation, and cardiology poly consultation, and pharmacy. the data is carried out for six days, counting the number of arrivals and departures with each point at intervals of 5 minutes. applying the jackson open queue network model, a recommendation was obtained for the hospital to increase employees' numbers. the registration workstation must have four servers; a poly cardiology workstation had three nurses and four doctors, while for pharmacy, had seven employees. with this personnel's addition, patients' total waiting time in the queuing system is approximately 12 minutes/patient. so, it can reduce waiting times in the queueing system that was initially 108 minutes/patient. keywords: queue; an open jackson queueing network; workstation jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 83-92 issn: 2527-3159 (print) 2527-3167 (online) mailto:monikefebriyani@gmail.com mailto:yuniar_farida@uinsby.ac.id mailto:diancrini@uinsby.ac.id mailto:nuris.ulinnuha@uinsby.ac.id http://u.lipi.go.id/1458103791 jurnal matematika mantik volume 6, no. 2, october 2020, pp. 83-92 84 1. introduction cardiovascular is a type of disease related to the heart and blood vessels. based on who data from all deaths globally, 17.9 million people died due to cardiovascular disease, with 85% of these deaths caused by heart attacks [1]. more deaths are caused by heart attacks, making the ministry of health always urges the public to do periodic health checks. currently, almost all hospitals have cardiology poly, as in the hospital sidoarjo. the hospital of sidoarjo has many types of polyclinic in outpatient installations. cardiology poly was rarely deserted every day. too many patients make a lot of queues so that patients have to wait long. long queues occur not only in cardiology poly but also at the point of registration and hospital pharmacy. the length of the queue can make the quality of hospital services less useful. the queuing system is a set of individuals, servers, and a rule that regulates the arrival of individuals and services [2]. the queuing system is the process of birth-death of the population consisting of individuals waiting to be served and being served [3]. there are three components in the queue system: population arrival, queue, and service facilities [4] [5]. one solution to overcome long queues is implementing a queuing network model that can estimate the actual queue situation to be analyzed. a queueing network is a group workstation where individuals can move from one workstation to another workstation. implementing the queuing model can estimate the length of service time; therefore, implementing the queuing model can be identified and minimized queues [6]. there are two types of queue networks, open and closed queue network. in the available queue network, individuals who enter the queue network system can come from inside or outside the queue system, and patients can exit the queue network system [7]. the outpatient installation queue system of the hospital has an open queue network characteristic. a patient can come from outside or in the queuing system, and patients can get out of the queuing network system. several queueing network models, including feedforward queueing network, gordon-newell queueing network, bcmp queueing network, and jackson queueing network. feedforward queueing network is the most straightforward queueing network where the queueing system is not branched and feedforward. gordon-newell queueing network is a closed queueing network with service distribution using exponential distribution [8]. bcmp queueing network is a queueing network used for open, closed, or mixed queueing network with service discipline can be last come first serve (lcfs) [7]. the jackson queueing network is a queue where individuals can move from one workstation to another workstation before leaving the system. the individual arrives at the workstation in the form of poisson distribution, which has queue discipline first come first serve (fcfs) and service time of a workstation in the form of exponential distribution [9] [10]. patient transfer is the probability of patient movement to the next workstation after being served at the previous workstation with certain services [11]. the jackson queue network has a continuous nature in each workstation where for each queue is independent so that they can analyze each workstation separately [12]. the jackson queueing network has two types, open and closed queues. in an open queue network, individuals who enter the queue network system can come from inside or outside the queuing system, and patients can exit the queuing network system if they have finished receiving the service. open queue networks have been extensively studied by burke (1969), who reviewed three workstations [13]. as for the network, jackson closed is known as the gordon-newell queueing network [8]. this research will apply the open jackson queueing network, as shown in figure 1. m. febriyani faris, y. farida, d. candra, n. ulinnuha, m. hafiyusholeh implementation of the open jackson queuing network to reduce waiting time 85 figure 1. open jackson queuing network one of several studies related to the open jackson queueing network application is the research with the title “on the application of the open jackson queueing network,” which aims to minimize queue waiting times at university covenant hospital center by adopting the open jackson queuing network. the results of these studies provide recommendations for the addition of employees at each point of service facilities to reduce patient waiting time [9]. besides being implemented in hospital queues, research implements an open jackson queue network in the surabaya carnival vehicle queue system [11]. method jackson network queueing can also be implemented to queuing systems in fields packing goods in a textile company, whereby implementing the jackson queue network, a longer time in packaging, is 570 seconds faster [14]. based on some of these studies, it can be shown that implementing the jackson queue network can minimize the waiting time of individuals in a queue network. so in this study, by implementing an open jackson queue network model, it is hoped will reduce waiting time in queueing at cardiology outpatient hospital installation. this study implemented an open jackson queueing network because the hospital's outpatient installation queue system has an open jackson queue network characteristic. in this study, the queueing network jackson will be implemented in 4 workstations, including registration, pre-consultation, and consultation of cardiology poly and pharmacy. 2. methods 2.1. data the research was conducted at an outpatient installation for bpjs patients in sidoarjo hospital, observed in 4 workstations, including registration, cardiology poly preconsultation, cardiology poly consultation, and pharmacy. research time to check the hospital queue system and data collection was carried out for six working days. data was collected from monday, november 16th to saturday, november 21st, 2019. the data was collected at an outpatient installation by observing the queuing system and recording patients' numbers coming and leaving at four workstations. the number of patients coming and leaving each workstation recorded every time with an interval of 5 minutes, where each point observed for 4 hours from 07.00 to 11.00. furthermore, the researcher interviewed one of the hospital employees about the queuing system at the hospital. 2.2. data processing the steps of research and data processing are presented in the flow chart in figure 2. jurnal matematika mantik volume 6, no. 2, october 2020, pp. 83-92 86 figure 2. flowchart of research and data processing a. hospital queue model analysis 1) calculate the average arrival averages and departure averages first, calculate the average arrival rate (𝜆) and departure rate (𝜇) of each workstation using equations (1) and (2). 𝜆 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠 𝑡𝑖𝑚𝑒 (1) 𝜇 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠 𝑡𝑖𝑚𝑒 (2) 2) check the condition of the queue system then proceed to check whether the queue system's condition included steady-state or not using equation (3). condition steady state (𝜌) is the state of the system that does not depend on the initial event and the time elapsed. condition steady state of m/m/s queuing model with (s) is the number of servants than the system utility, 𝜌 = 𝜆 𝜇 < s (3) 3) calculate queuing system performance the queueing model in this study has the same character as the queue model (m/m/s): (fcfs/∞/∞), therefore, to calculate queue system performance using equation (4) to (7). queue model (m/m/s): (fcfs/∞/∞) is a queueing model with services without capacity limit. this model has the number of employees or server (s) more than one so that a maximum of individuals can be served simultaneously by the server as many as (s). the queuing discipline in this model includes fcfs, where first-come individuals will be first served. in this model, the arrival rates are the number of individuals/units of time (λ), and the departure rate is the number of individuals/units of time (µ) following the poisson distribution and service time ( 1 𝜇 ) following the exponential distribution [15]. queue performance for queuing model (m/m/s): (fcfs/∞/∞) is, 𝑃0 = {[∑ (𝜆 𝜇⁄ ) 𝑛 𝑛! 𝑠−1 𝑛=0 ]+ (𝜆 µ⁄ ) 𝑠 𝑠𝑟!(1− 𝜆 𝑠𝜇⁄ ) } −1 (4) 𝐿𝑞 = 𝜆µ(𝜆 µ⁄ ) 𝑠 (𝑠−1)!(𝑠𝜇−𝜆) 𝑃0 (5) m. febriyani faris, y. farida, d. candra, n. ulinnuha, m. hafiyusholeh implementation of the open jackson queuing network to reduce waiting time 87 𝐿𝑠 = 𝐿𝑞 + 𝜆 µ (6) 𝑊𝑠 = 𝐿𝑠 𝜆 (7) where: 𝑃0 = probability of no individuals in the queuing system 𝐿𝑞 = average number of individuals waiting in a queue (individuals/minutes) 𝐿𝑠 =average number of individuals waiting in the queuing system (individuals/minutes) 𝑊𝑠 = average waiting time in a queuing system b. calculation of open jackson queuing network some conditions are assumed in the jackson queuing network; including first individuals come from outside the queueing network with an arrival rate𝜆𝑖 following poisson distribution and all arrivals are independent. second, the individual service time of each queue following an exponential distribution. third, every individual who has been served at the workstation (i) will move to the workstation to (j) with (pij) probability. finally, service discipline following fcfs (first come first serve) [9]. 1) first, make a mathematical model of a linear equation using equations (8) and (9). 𝜆𝑗 = 𝑎𝑗 +∑ 𝜇𝑖𝑃𝑖,𝑗 𝑁 𝑖=1 , 1≤ i ≤ n (8) µ𝑖 = ∑ 𝜇𝑖𝑃𝑖,𝑗 𝑁 𝑗=1 , 1≤ j ≤ n (9) where, 𝑎𝑖 = external arrival rate to the workstation (i) 𝜆𝑖 = total arrival rate to the workstation (i) µ𝑖 = total departure rate to the workstation (i) 𝑃𝑖,𝑗 = probability of individual displacement from the workstation (i) to (j) n = total workstation make a mathematical model of a linear equation using equations (8) and (9) of each workstation and based on outpatient patients' network queue in figure 3. this linear system equation contains three variables. figure 3. flowchart of hospital queue network queue network diagram schemes formed into linear equations (8) and (9) illustrate the probability of patient movement. 𝜆1 = 𝑎1 + 0𝑃1,2 +0𝑃1,3 + 0𝑃2,3 +0𝑃3,4 +0𝑃3,𝑜𝑢𝑡 +𝜇4𝑃4,1 + 0𝑃4,𝑜𝑢𝑡 (10) 𝜆2 = 𝜇1𝑃1,2 + 0𝑃1,3 +0𝑃2,3 + 0𝑃3,4 +0𝑃3,𝑜𝑢𝑡 + 𝜇4𝑃4,1 +0𝑃4,𝑜𝑢𝑡 (11) 𝜆3 = 0𝑃1,2 + 𝜇1𝑃1,3 +𝜇2𝑃2,3 + 0𝑃3,4 +0𝑃3,𝑜𝑢𝑡 + 𝜇4𝑃4,1 +0𝑃4,𝑜𝑢𝑡 (12) jurnal matematika mantik volume 6, no. 2, october 2020, pp. 83-92 88 𝜆4 = 0𝑃1,2 + 0𝑃1,3 +0𝑃2,3 + 𝜇3𝑃3,4 + 0𝑃3,𝑜𝑢𝑡 + 𝜇4𝑃4,1 +0𝑃4,𝑜𝑢𝑡 (13) 𝜇1 = 𝜇1𝑃1,2 +𝜇1𝑃1,3 +0𝑃2,3 +0𝑃3,4 + 0𝑃3,𝑜𝑢𝑡 + 𝜇4𝑃4,1 + 0𝑃4,𝑜𝑢𝑡 (14) 𝜇2 = 0𝑃1,2 +0𝑃1,3 + 𝜇2𝑃2,3 +0𝑃3,4 +0𝑃3,𝑜𝑢𝑡 +𝜇4𝑃4,1 + 0𝑃4,𝑜𝑢𝑡 (15) 𝜇3 = 0𝑃1,2 +0𝑃1,3 + 0𝑃2,3 +𝜇3𝑃3,4 +𝜇3𝑃3,𝑜𝑢𝑡 +𝜇4𝑃4,1 + 0𝑃4,𝑜𝑢𝑡 (16) 𝜇4 = 0𝑃1,2 +0𝑃1,3 + 0𝑃2,3 +0𝑃3,4 +0𝑃3,𝑜𝑢𝑡 +𝜇4𝑃4,1 + 𝜇4𝑃4,𝑜𝑢𝑡 (17) 2) the second step is to determine the transition matrix jackson. workstations in the queueing network jackson form a schematic diagram illustrating the possibility of individual displacement from one workstation to another workstation. transition matrix jackson is a matrix that shows the probability of displacement. the average arrival and departure rates are known based on the data that has been collected so that the value that needs to be sought is p, the probability of patient movement. the equation will be formed into the matrix multiplication λ = µp then to find the value of the matrix p, the matrix µ is converted and multiplied by the matrix (λ). 3) finally, a linear equation of the new departure rate is: 𝜆𝑗 = ∑ 𝑃𝑖𝑗𝜇𝑖 𝑁 𝑖=1 , 1 ≤ i ≤ n (18) now that the probability of transfer from the workstation (i) to (j) is known, the next step is to form a mathematical model for the new departure rate. the total waiting time for patients in the queuing system must be faster than before the calculation. the new mathematical model uses the help of equations (10) to (13) and (18). by the new departure rate value, the performance of the queue system also changes. hence, it is necessary to check the queue performance's size so that it can be compared with the performance of the queue system before implementing the jackson queue network. 3. result and discussion the data in this study is the total of arrival and departure of patients at each observed workstation with calculation intervals every 5 minutes for 4 hours. in the queuing system, there is a situation called the busy period, where the number of patients has increased higher than the other times. the registration counter has a rush hour from 07.00-08.00, at the pharmacy point has a rush hour from 09.00-10.00, at the cardiology poly pre-consultation has a rush hour from 07.00-08.00, while at the doctor's consultation point has a rush hour 08.00-09.00. based on data analysis, the arrival and departure rate used for the next calculation step uses the arrival rate at rush hour. table 1. arrival and departure rates every minute no workstation arrival rate (λ) departure rate (µ) 1 registration 2.23056 0.91111 2 pre-consultation 0.70833 0.37500 3 consultation 0.57778 0.31390 4 pharmacy 2.30833 0.77500 source: processing primary data from observation 3.1. performance analysis of the initial queue model the hospital queue network system's characteristics follow the poisson distribution for arrival time during the exponential distribution for the distribution of service time. the applicable service discipline is that patients who come first are first served to apply the m. febriyani faris, y. farida, d. candra, n. ulinnuha, m. hafiyusholeh implementation of the open jackson queuing network to reduce waiting time 89 fcfs (first come first served) discipline. in contrast, the capacity of the patient arrival source system is unlimited. also, the servants who owned all workstations were more than 1. the registration workstation had 4 counters; the pharmacy point had 3 employees, the pre-consultation point in cardiology poly had 2 nurses and had 2 doctors on duty for the consultation point. all workstations are steady-state. based on the queuing system's analysis and calculation of the performance size of the queuing model shown by the number of patients waiting in the queuing system and the average number of patients waiting in the queuing system, the results of performance for each workstation are listed in table 2. table 2. initial queue model performance no workstation lq ls ws 1 registration 2.98811 5.43629 2.43718 2 pre-consultation 15.5968 17.4857 24.6857 3 consultation 10.19411 12.03482 20.82949 4 pharmacy 36.62870 139.60720 60.47965 total 108.432 based on the calculation of the performance initial queuing model above, it can be concluded that the average time spent by a patient to conduct a health check at the cardiology polyclinic of the hospital is 108 minutes/patient. 3.2. application of the jackson queue network a. probability of movement between workstations the first step of jackson's calculation is to calculate the probability of movement using equation (10) to equation (17), which describes individuals' movement between workstations. to make it easier to find probability values, the linear equation is converted into a matrix. ( 𝜆2 𝜆3 𝜆4 µ1 µ2 µ3 µ4) = ( µ1 0 0 µ1 0 0 0 0 µ1 0 µ1 0 0 0 0 µ2 0 0 µ2 0 0 0 0 µ3 0 0 µ3 0 0 0 0 0 0 µ3 0 0 0 0 0 0 0 µ4 0 0 0 0 0 0 µ4) = ( 𝑃1,2 𝑃1,3 𝑃2,3 𝑃3,4 𝑃3,𝑜𝑢𝑡 𝑃4,1 𝑃4,𝑜𝑢𝑡) ( 0.708 0.578 2.308 0.911 0.375 0.314 0.775) = ( 0.911 0 0 0.911 0 0 0 0 0.911 0 0.911 0 0 0 0 0.375 0 0 0.375 0 0 0 0 0.314 0 0 0.314 0 0 0 0 0 0 0.314 0 0 0 0 0 0 0 0.775 0 0 0 0 0 0 0.775) = ( 𝑃1,2 𝑃1,3 𝑃2,3 𝑃3,4 𝑃3,𝑜𝑢𝑡 𝑃4,1 𝑃4,𝑜𝑢𝑡) ( 𝑃1,2 𝑃1,3 𝑃2,3 𝑃3,4 𝑃3,𝑜𝑢𝑡 𝑃4,1 𝑃4,𝑜𝑢𝑡) = ( 0.911 0 0 0.911 0 0 0 0 0.911 0 0.911 0 0 0 0 0.375 0 0 0.375 0 0 0 0 0.314 0 0 0.314 0 0 0 0 0 0 0.314 0 0 0 0 0 0 0 0.775 0 0 0 0 0 0 0.775) −1 ( 0.708 0.578 2.308 0.911 0.375 0.314 0.775) = ( 0.78 0.22 1 0.99 0.01 0.5 0.5 ) jurnal matematika mantik volume 6, no. 2, october 2020, pp. 83-92 90 the determinant of matrix µ is 0, so that the next step is to find the inverse value using the moore-penrose pseudoinverse [16]. based on the result of the calculation, the probability of the patient moving from registration (point 1) to pre-consultation (point 2) is 0.78, while if going to the consultation (point 3), the probability is 0.22. the probability value of pre-consultation to the consultation is 1. the probability of consultation to the pharmacy (point 4) is 0.99 while going straight out of the consultation is 0.01. the probability value is used to calculate the new µ departure rate. b. new departure rate implementing the jackson queue network is to minimize patient waiting time at each workstation from the queueing network. to minimize patient waiting time is increasing the number of patients who leave the service. the arrival of individuals from outside or inside the system cannot be limited, but the number of individual leave from the queuing system can be increased so that the queues in the system can be reduced. the solution is by calculating the new µ departure rate from each workstation using a linear equation from equations (10) to (13). the linear equation is converted into a matrix to simplify the calculation. ( 0 𝑃1,2 𝑃1,3 0 0 0 𝑃2,3 0 0 0 0 𝑃3,4 𝑃4,1 0 0 0 )( µ1 µ2 µ3 µ4 ) = ( 𝜆1 𝜆2 𝜆3 𝜆4 ) ( µ1 µ2 µ3 µ4 ) = ( 3.662 2.835 3.649 7.299 ) based on the new departure rate results, the value of the new departure rate (µ) from registration is 3.662 if rounded up to 4. for pre-consultation departure rate change to 2.835 rounded up to 3 while for consultation workstation to 3.649 rounded up to 4 and for pharmacy departure rate rounded down to 7 employees. the new departure rate value explains that for workstations to produce as many patients as the new departure rate, the number of employees per workstation must also be as much as the new departure rate. therefore, it needs to add employees for the registration workstation are become 4 servers. cardiology poly is recommended to have 3 nurses and 4 doctors, while for pharmacy have 7 employees. 3.3. simulation and discussion the results of implementing the jackson queue network is recommended to add servers or employees. the registration workstation is recommended to have 4 servers. cardiology poly is recommended to have 3 nurses and 4 doctors, while for pharmacy have 7 employees. base on these recommendations, the performance of the new model queue is shown in table 3. table 3. performance of the new model queue no workstation employees / servers lq ls ws 1 registration 4 1.68829 4.73417 2.32241 2 pre-consultation 3 0.90705 2.91496 4.21524 3 consultation 4 0.22489 2.33356 4.13885 4 pharmacy 7 0.26650 2.10115 0.98025 total 11.65675 source: author’s calculation m. febriyani faris, y. farida, d. candra, n. ulinnuha, m. hafiyusholeh implementation of the open jackson queuing network to reduce waiting time 91 the queuing system's performance after implementing an open jackson queue network is simulated in figure 4 so that the queuing system can be easily understood. figure 4. simulation results by adding the number of employees based on the jackson queue network's recommendation, it was obtained that the total length of waiting time was approximately 12 minutes. so, it can reduce waiting times in the queueing system that was initially 108 minutes/patient. in this case study, a queueing model was produced that could reduce waiting times. still, it was not easy for hospitals to be able to implement the recommendations given by researchers. because of the addition of human resources will undoubtedly incur costs as well. for the next research, it should not only optimize through the queue model but also conduct a study related to the costs incurred as a result of the recommendation of the model compared to the economic benefits obtained (cost and benefit ratio analysis). so that the optimization will be more effectives and its recommendations can be applied. 4. conclusions implementing an open jackson queue network was recommended to add employees to each workstation. the registration workstation must have four servers, a poly cardiology workstation had three nurses, and four doctors, while for pharmacy, had seven employees. by adding the number of employees following the recommendations, the patient's total time in the queuing system is approximately 12 minutes per patient if this recommendation is implemented in the sidoarjo hospital. so, it can reduce waiting times in the queueing system that was initially 108 minutes/patient. references [1] who, “cardivascular disease,” who, 2016. [online]. available: origin.who.int/cardivascular_disease/en/. [2] a. rangkuti, 7 model riset operasi dan aplikasinya. surabaya: firstbox media, 2013. jurnal matematika mantik volume 6, no. 2, october 2020, pp. 83-92 92 [3] aminuddin, prinsip-prinsip riset operasi. jakarta: erlangga, 2005. [4] s. a. mangkona and i. murdifin, “implementation of queue model for measuring the effectiveness of suzuki car maintenance,” world j. bus. manag., vol. 3, no. june 2017, pp. 55–66, 2017. [5] h. a. taha, “operations research,” 8th ed., new york: pearson education, 2007. [6] c. skautis and t. boyle, “human enabled health care,” ca emerg. technol., pp. 1– 10, 2009. [7] f. baskett and s. unzvers, “open, closed, and mixed networks of queues with different classes of customers open, closed, and mixed networks of queues with different classes of customers,” j. assoc. comput. mach., vol. 22, no. april, pp. 248–260, 1975. [8] w. j. gordon and g. f. newell, “closed queuing systems with exponential servers,” oper. res., vol. 15, pp. 254–265, 1967. [9] a. e. owoloko, o. adeleke, and s. o. edeki, “on the application of the open jackson queuing network,” glob. j. pure appl. math., vol. 11, no. august 2016, pp. 2299–2313, 2015. [10] a. u. amri and i. endrayanto, “aplikasi model jaringan antrian jackson terbuka dengan studi kasus di unit gawat darurat,” universitas gajah mada, 2018. [11] s. w. djatmiko and l. m. w cahya, “analisis sistem antrian menggunakan metode jackson pada wahana outdoor suroboyo carniva,” semin. nas. sains dan teknol. umj, no. november, pp. 1–9, 2016. [12] k. sigman, “the stability of open queueing network,” stoch. process. a their appl., vol. 35, pp. 11–25, 1990. [13] p. j. burke, “the dependence of service in tandem m/m/s queues,” oper. res., vol. 17, pp. 754–755, 1969. [14] l. ratnasari, y. widiatama, and r. n. dewanti, “analisa antrian pengerjaan benang heat technology dengan metode jackson network di pt. kurabo manunggal textil industries,” teknologi, vol. 1, no. maret 2018, pp. 18–26, 2018. [15] h. a. taha, “riset operasi,” 5th ed., jakarta: binarupa aksara, 1997. [16] j. c. a. barata and m. s. hussein, “the moore-penrose pseudoinverse. a tutorial review of the theory,” pp. 1–23, 2011. paradigma baru pendidikan matematika dan aplikasi online internet pembelajaran how to cite: a. l. firdiansyah, “dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge”, j. mat. mantik, vol. 6, no. 2, pp. 123-134, october 2020. dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge adin lazuardy firdiansyah stai muhammadiyah probolinggo, adin.lazuardy@gmail.com doi: https://doi.org/10.15642/mantik.2020.6.2.123-134 abstrak. model mangsa-pemangsa dengan laju kejadian nonlinear dan perlindungan dimunculkan untuk menggambarkan perubahan prilaku pada mangsa yang sehat ketika jumlah mangsa yang terinfeksi meningkat, sedangkan pemangsa dapat memakan mangsa dengan mengakses perlindungan pada mangsa. oleh karena itu, analisis dinamik model mangsa-pemangsa dengan penyebaran penyakit yang dinotasikan dengan laju kejadian nonlinear dan perlindungan mangsa dibahas pada penilitian ini. hasil analisis ditemukan bahwa ada delapan titik kesetimbangan, dimana semua titik kesetimbangan tersebut stabil asimtotik secara lokal. selanjutnya, perlindungan mangsa juga dipertimbangkan pada penelitian ini. perlindungan mangsa memiliki pengaruh penting pada model. hal ini ditunjukkan dengan adanya perlindungan mangsa yang dapat mencegah kepunahan pada populasi mangsa. pada bagian akhir, simulasi numerik dilakukan untuk mengilustrasikan hasil analisis yang diperoleh. untuk penelitian berikutnya, model mangsa-pemangsa dapat diinvestigasi efek pemanenanya untuk kedua populasi. kata kunci: model mangsa-pemangsa; laju kejadian nonlinear; perlindungan; kestabilan abstract. a predator-prey system with nonlinear incidence rate and refuging in prey is proposed to describe behavior change of certain infected diseases on healthy prey when the number of infected prey is getting large, while predator can predate prey by accessing refuging in prey. therefore, this paper discusses the dynamics behavior predator-prey model with the spread of infected disease that is denoted by nonlinear incidence rate and adding prey refuge. we find the existence of eight nonnegative equilibrium in the model, which their local stability has been determined. furthermore, we also observe the prey refuge properties in the model. we find that prey refuge can prevent extinction in prey populations. in the end, some numerical solutions are carried out to illustrate our analytic results. for future work, we can investigate the harvesting effect in both populations, which is disease control in the predator-prey model with the spread of infected disease. keywords: predator-prey system; nonlinear incidence rate; refuge; stability jurnal matematika mantik vol. 6, no. 2, october 2020, pp. 123-134 issn: 2527-3159 (print) 2527-3167 (online) jurnal matematika mantik volume 6, no. 2, october 2020, pp. 123-134 124 1. introduction after lotka and volterra proposed a mathematical model that could represent the interactions of predators and prey, many researchers were interested to study the model. similarly, kermack and mckendrick [1] also proposed a sir (susceptible-infectiousrecovered) epidemic model that represented the infected disease transmission. after that, anderson and may [2] for the first time proposed the eco-epidemiology model and investigated invasion, persistence, and spread of infected disease in the model. since then, many researchers have studied the eco-epidemiology model. generally speaking, the ecoepidemiology model is divided into three cases. the first case is an only infected disease in prey as in [3]–[5]. the second case is an only infected disease in predator as in [6]. meanwhile, the third case is an infected disease in both populations as in [7]–[9]. in the eco-epidemiology model, there is a basic component that can represent the spread of infected disease. generally, it is denoted by the simple mass incidence rate, 𝛽𝑆𝐼, with 𝛽 means the infection rate. this incidence rate shows that the transmission disease with rate 𝛽𝑆𝐼, i.e. when the transmission disease increases significantly, then the number of infected population increases too [1]. in the model [3]–[8], the authors assume the simple mass incidence rate to represent the transmission disease in the model. in several cases, the simple mass incidence rate doesn’t produce appropriate results. when the amount of infected population increases significantly, the susceptible population tend to change their behavior to reduce contact with the infected population. therefore, capasso and serio [10] proposed the saturated incidence rate to describe the transmission disease. as done in [9], they investigated the local stability where the transmission disease in the eco-epidemiology model followed the saturated incidence rate. however, there is a nonlinear incidence rate that is suggested by [11]. they introduced nonlinear incidence rate, 𝛽𝐼𝑝 1+𝛼𝐼𝑞 (𝑝, 𝑞, 𝛼, 𝛽 were positive constants), where 𝛽𝐼𝑝 meant infection rate and 1 1+𝛼𝐼𝑞 meant inhibition rate from the behavior change on healthy population when infected population was getting large. many researcher uses this incidence rate by giving 𝑝, 𝑞, 𝛼 makes different values as in [12]–[14]. this becomes more rational because it includes the effect of tightness from infective individuals [12]. recently, the effect of disease in the eco-epidemiology model has become an important topic for many researchers. however, other components that can influence the dynamic of species interactions is the allee effect, habitat complexity, harvesting, and prey refuge. for the prey refuge effect, theoretical research provides a conclusion that it can influence stabilizing and destabilizing the predator-prey model and can avoid extinction in prey [15]–[17]. it can make the predator-prey model to form more realistic. motivated by previous research, we modify a model from [9] by including nonlinear incidence rate and adding the effect of prey refuge. at this time, several similar models have emerged, but the latest distinctive feature of our model is the inclusion of transmission disease in both populations and also the inclusion of prey refuge properties of the prey population. incorporation of prey refuge gives a factor which can be accessed by predator populations. under this adding effect, our model is more realistic and differs from the previous eco-epidemiology model. the model is analyzed to determine the local stability of its equilibrium points. moreover, several simulations are given to demonstrate the results of our analysis. 2. methods the method used for this research is the literature study which studies some previous research. it’s used to modify a model from [9]. the steps to solve this research are below: a. reviewing and studying the predator-prey model from previous research. adin lazuardy firdiansyah dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge 125 b. modifying the predator-prey model by including nonlinear incidence rate and adding the effect of prey refuge. c. analyzing the existence of equilibrium points and local stability in the modified model. d. performing numerical simulation by using the runge-kutta 4th order method as a numerical method to support our analysis results. e. making a conclusion based on the analysis results. 3. result and discussion 3.1 the formulation model our mathematical model contains two populations, namely the prey population and predator population. at time 𝑡, susceptible and infected prey are denoted by 𝑆(𝑡) and 𝐼(𝑡), respectively. meanwhile, susceptible and infected predator are denoted by 𝑌𝑆(𝑡) and 𝑌𝐼(𝑡), respectively. the following is several basic assumptions for our mathematical model: a. in the absence of predation and disease, the prey grows according to logistics with a growth rate 𝑟 (𝑟 > 0) and carrying capacity 𝐾 (𝐾 > 0). b. only one population that can reproduce, namely susceptible prey. c. the prey has one infection source like viruses or other sources. meanwhile, the predator can be infected due they eat the infected prey. d. the transmission disease follows the nonlinear incidence rate 𝛽𝑆𝐼 1+𝐼 , where 𝛽𝐼 means infection rate and 1 1+𝐼 means inhibition rate from behavior change on healthy prey, when infected prey is getting large. meanwhile, the transmission disease in predator follows the simple mass incidence rate (𝛾𝑌𝑆𝑌𝐼), where 𝛾 expressed infection rate. e. both of infected prey and predator isn’t recovered and no get immune. f. the infected predator can’t predate healthy prey. both of infected predator and susceptible predator can eat infected prey because it is easy to be predated by them. but, the ability to catch from an infected predator is lower than susceptible predator. g. all species have natural death rates and death rates due to infection. h. the functional response for predation of predator is the lotka-volterra type. i. the refuge protection can be denoted with (1 − 𝑚3)𝑆 for susceptible prey and (1 − 𝑚4)𝐼 for infected prey, where 𝑚3, 𝑚4 ∈ (0,1] and healthy prey is more agile than infected prey. the details of the model structure are shown in the schematic flow diagram as in figure 1. from the flow chart in figure 1, the mathematical model is presented as follows: 𝑑𝑆 𝑑𝑡 = 𝑟𝑆 (1 − 𝑆 + 𝐼 𝐾 ) − 𝛽𝑆𝐼 1 + 𝐼 − 𝑝1(1 − 𝑚3)𝑆𝑌𝑆 − 𝑎1𝑆, 𝑑𝐼 𝑑𝑡 = 𝛽𝑆𝐼 1 + 𝐼 − 𝑝2(1 − 𝑚4)𝐼𝑌𝑆 − 𝑝3(1 − 𝑚4)𝐼𝑌𝐼 − 𝑎2𝐼, 𝑑𝑌𝑆 𝑑𝑡 = 𝑎3(1 − 𝑚3)𝑆𝑌𝑆 + 𝑎4(1 − 𝑚4)𝐼𝑌𝑆 − 𝛾𝑌𝑆𝑌𝐼 − 𝑑3𝑌𝑆, 𝑑𝑌𝐼 𝑑𝑡 = 𝑎5(1 − 𝑚4)𝐼𝑌𝐼 + 𝛾𝑌𝑆𝑌𝐼 − 𝑎6𝑌𝐼. (1) with the initial condition as 𝑆(0) > 0, 𝐼(0) > 0, 𝑌𝑆(0) > 0, 𝑌𝐼(0) > 0. all parameters with their biological meaning are given as follows: 𝑎1 = 𝑚1 + 𝑑1, 𝑎2 = 𝑚2 + 𝑑2 + 𝑐, 𝑎3 = 𝑝1𝑞1, 𝑎4 = 𝑝2𝑞2, 𝑎5 = 𝑝3𝑞3, and 𝑎6 = 𝑑4 + 𝑑5 where 𝑚1 and 𝑚2 are migration rates for susceptible prey and infected prey, respectively. 𝑝1(𝑝2) and 𝑝3 are predation rate from healthy predator to healthy prey (to infected prey) and predation rate from infected predator to infected prey, respectively. 𝑞1(𝑞2) and 𝑞3 are conversion rate from healthy prey into healthy predator (infected predator) and conversion rate from infected prey into an infected jurnal matematika mantik volume 6, no. 2, october 2020, pp. 123-134 126 predator. 𝑑1(𝑑2) and 𝑑3(𝑑4) are natural death from susceptible prey (infected prey) and natural death from a susceptible predator (infected predator), respectively. 𝑐(𝑑5) is the death rate from infected prey (from an infected predator) due to infection from disease. we assume that all parameters are positive values. figure 1. schematic diagram of the model 3.2 equilibrium point of mathematical model we set the right-hand sides equal to zero. thus, we get possible equilibriums: a. the trivial equilibrium point is 𝐸0(0,0,0,0). this point always exists. b. the axial equilibrium point is 𝐸1(𝑆 (1), 0,0,0), where 𝑆(1) = 𝐾(𝑟−𝑎1) 𝑟 . it will exist when 𝑟 > 𝑎1. c. 𝐸2(𝑆 (2), 𝐼(2), 0,0) is the predator-free equilibrium point, where 𝑆(2) = (1 + 𝐼(2))(𝑟𝐾 − 𝑟𝐼(2) − 𝐾𝑎1) − 𝐾𝛽𝐼 (2) 𝑟(1 + 𝐼(2)) and 𝐼(2) is the positive root of quadratic equations 𝐴0(𝐼 (2)) 2 + 𝐴1𝐼 (2) + 𝐴2 = 0, where 𝐴0 = 𝑟(𝑎2 + 𝛽), 𝐴1 = 2𝑟𝑎2 − 𝛽𝑟𝐾 + 𝛽𝑟 + 𝛽𝑎1𝐾 + 𝛽 2𝐾, and 𝐴2 = 𝑟𝑎2 − 𝛽𝑟𝐾 + 𝛽𝑎1𝐾. the equilibrium 𝐸2 will exist if 𝑟 ≥ 𝛽𝑎1𝐾 𝛽𝐾−𝑎2 . d. 𝐸3 (𝑆 (3), 0, 𝑌𝑆 (3) , 0) is the disease-free equilibrium point, where 𝑆(3) = 𝑑3 𝑎3(1 − 𝑚3) and 𝑌𝑆 (3) = 𝐾(𝑟 − 𝑎1) − 𝑟𝑆 (3) 𝐾𝑝1(1 − 𝑚3) . the equilibrium point 𝐸3 will exist if 𝑟 > 𝑎1𝑎3𝐾(1−𝑚3) 𝑎3𝐾(1−𝑚3)−𝑑3 . e. 𝐸4 (𝑆 (4), 𝑎6 𝑎5(1−𝑚4) , 0, 𝑌𝐼 (4) ) is the healthy predator-free equilibrium point, where 𝑆 𝐼 𝑌𝑆 𝑌𝐼 𝛽𝑆𝐼 1 + 𝐼 𝑟𝑆 (1 − 𝑆 + 𝐼 𝐾 ) 𝑎1𝑆 𝑝1𝑆𝑌𝑆 𝑑3𝑌𝑆 𝑚3𝑆𝑌𝑆 𝑝2𝐼𝑌𝑆 𝑚4𝐼𝑌𝑆 𝑝3𝐼𝑌𝐼 𝛾𝑌𝑆𝑌𝐼 𝑎2𝐼 𝑚4𝐼𝑌𝐼 𝑎6𝑌𝐼 adin lazuardy firdiansyah dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge 127 𝑆(4) = (𝑎5(1 − 𝑚4) + 𝑎6)(𝑝3(1 − 𝑚4)𝑌𝐼 (4) + 𝑎2) 𝛽𝑎5(1 − 𝑚4) and 𝑌𝐼 (4) = 𝐶1 − 𝐶2 𝑟𝑝3(1 − 𝑚4)(𝑎5(1 − 𝑚4) + 𝑎6) 2 with 𝐶1 = 𝐾𝛽𝑎5(1 − 𝑚4)(𝑟 − 𝑎1)(𝑎5(1 − 𝑚4) + 𝑎6) and 𝐶2 = 𝑟𝑎2(𝑎5(1 − 𝑚4) + 𝑎6) 2 + 𝐾𝛽2𝑎5𝑎6(1 − 𝑚4) + 𝑟𝛽𝑎6(𝑎5(1 − 𝑚4) + 𝑎6). this equilibrium exists when 𝐶1 > 𝐶2 and 𝑟 > 𝑎1. f. 𝐸5 (𝑆 (5), 0, 𝑎6 𝛾 , 𝑌𝐼 (5) ) is the infected prey-free equilibrium point, where 𝑆(5) = 𝐾𝛾(𝑟 − 𝑎1) − 𝐾𝑝1𝑎6(1 − 𝑚3) 𝑟𝛾 and 𝑌𝐼 (5) = (1 − 𝑚3)𝑎3𝑆 (5) − 𝑑3 𝛾 . it will exist if 𝑟 > 𝐾𝑎3(1−𝑚3)(𝛾𝑎1+𝑝1𝑎6(1−𝑚3)) 𝛾(𝐾𝑎3(1−𝑚3)−𝑑3) . g. 𝐸6 ( 𝑑3−𝑎4(1−𝑚4)𝐼 (6) 𝑎3(1−𝑚3) , 𝐼(6), 𝑌𝑆 (6) , 0) is the infected predator-free equilibrium, where 𝑌𝑆 (6) = 𝛽𝑑3 − 𝛽𝑎4(1 − 𝑚4)𝐼 (6) − 𝑎2𝑎3(1 − 𝑚3)(1 + 𝐼 (6)) 𝑝2𝑎3(1 − 𝑚3)(1 − 𝑚4)(1 + 𝐼 (6)) and 𝐼(6) is the positive root of quadratic equations 𝐷0(𝐼 (6)) 2 + 𝐷1𝐼 (6) + 𝐷2 = 0 with 𝐷0 = 𝑟𝑝2(1 − 𝑚4)(𝑎3(1 − 𝑚3) − 𝑎4(1 − 𝑚4)), 𝐷1 = 𝐾𝑎3𝑝2(1 − 𝑚3)(1 − 𝑚4)(𝛽 + 𝑎1 − 𝑟) +𝑟𝑝2(1 − 𝑚4)(𝑑3 + 𝑎3(1 − 𝑚3) − 𝑎4(1 − 𝑚4)) −𝐾𝑝1(1 − 𝑚3)(𝑎2𝑎3(1 − 𝑚3) + 𝛽𝑎4(1 − 𝑚4)), 𝐷2 = 𝑟𝑝2(1 − 𝑚4)(𝑑3 − 𝐾𝑎3(1 − 𝑚3)) +𝐾𝑎3(1 − 𝑚3)(𝑎1𝑝2(1 − 𝑚4) − 𝑎2𝑝1(1 − 𝑚3)) + 𝐾𝛽𝑝1𝑑3(1 − 𝑚3). this equilibrium will exist if 𝐼(6) < 𝛽𝑑3−𝑎2𝑎3(1−𝑚3) 𝛽𝑎4(1−𝑚4)+𝑎2𝑎3(1−𝑚3) . h. 𝐸7 (𝑆 (7), 𝐼(7), 𝑎6−𝑎5(1−𝑚4)𝐼 (7) 𝛾 , 𝑎3(1−𝑚3)𝑆 (7)+𝑎4(1−𝑚4)𝐼 (7)−𝑑3 𝛾 ) is the interior equilibrium point, where 𝑆(7) = (1+𝐼(7))(𝐾𝑟𝛾−𝑎1𝛾𝐾−𝑟𝛾𝐼 (7)−𝐾𝑝1(1−𝑚3)(𝑎6−𝑎5(1−𝑚4)𝐼 (7)))−𝐾𝛾𝛽𝐼(7) 𝛾𝑟(1+𝐼(7)) and 𝐼(7) is the positive root of the cubic equation 𝑄0(𝐼 (7)) 3 + 𝑄1(𝐼 (7)) 2 + 𝑄2𝐼 (7) + 𝑄3 = 0, with 𝑄0 = 𝐾𝑎3𝑎5𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) 2 − 𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) jurnal matematika mantik volume 6, no. 2, october 2020, pp. 123-134 128 +𝛾𝑟𝑎4𝑝3(1 − 𝑚4) 2 − 𝛾𝑟𝑎5𝑝2(1 − 𝑚4) 2, 𝑄1 = 2𝛾𝑟𝑎4𝑝3(1 − 𝑚4) 2 − 2𝛾𝑟𝑎5𝑝2(1 − 𝑚4) 2 + 𝛾𝑟𝑎6𝑝2(1 − 𝑚4) + 𝛾 2𝑟𝑎2 −𝛾𝑟𝑑3𝑝3(1 − 𝑚4) + 2𝐾𝑎3𝑎5𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) 2 + 𝛽𝛾2𝑟 −𝐾𝑎3𝑎6𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) − 𝐾𝛾𝑎1𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) +𝐾𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) − 𝐾𝛽𝛾𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) −𝐾𝛽𝛾𝑎5𝑝1(1 − 𝑚3)(1 − 𝑚4) − 2𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4), 𝑄2 = 𝛾𝑟𝑎4𝑝3(1 − 𝑚4) 2 − 𝛾𝑟𝑎5𝑝2(1 − 𝑚4) 2 + 2𝛾𝑟𝑎6𝑝2(1 − 𝑚4) + 2𝛾 2𝑟𝑎2 −2𝛾𝑟𝑑3𝑝3(1 − 𝑚4) + 𝐾𝑎3𝑎5𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) 2 + 𝛽𝛾2𝑟 −2𝐾𝑎3𝑎6𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) − 2𝐾𝛾𝑎1𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) +2𝐾𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) − 𝐾𝛽𝛾𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) + 𝐾𝛽 2𝛾2 −𝐾𝛽𝛾𝑎5𝑝1(1 − 𝑚3)(1 − 𝑚4) + 𝐾𝛽𝛾𝑎6𝑝1(1 − 𝑚3) − 𝐾𝛽𝛾 2𝑟 + 𝐾𝛽𝛾2𝑎1 −𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4), 𝑄3 = 𝐾𝛾𝑟𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) − 𝐾𝛾𝑎1𝑎3𝑝3(1 − 𝑚3)(1 − 𝑚4) −𝐾𝑎3𝑎6𝑝1𝑝3(1 − 𝑚3) 2(1 − 𝑚4) + 𝐾𝛽𝛾𝑎6𝑝1(1 − 𝑚3) +𝛾𝑟𝑎6𝑝2(1 − 𝑚4) − 𝛾𝑟𝑑3𝑝3(1 − 𝑚4) + 𝐾𝛽𝛾 2𝑎1 − 𝐾𝛽𝛾 2𝑟 + 𝛾2𝑟𝑎2. the existing form of the positive root in the cubic equation can be determined by using cardan’s method as in [18]. 3.3 stability analysis to check the local stability of the equilibrium point, we have to determine the eigenvalues of the jacobian matrix. here, the jacobian matrix at the equilibrium point 𝐸∗(𝑆∗, 𝐼∗, 𝑌𝑆 ∗, 𝑌𝐼 ∗) is as follows: 𝐽∗ = [ 𝑢11 𝑢12 −𝑝1(1 − 𝑚3)𝑆 ∗ 0 𝛽𝐼∗ 1 + 𝐼∗ 𝑢22 −𝑝2(1 − 𝑚4)𝐼 ∗ −𝑝3(1 − 𝑚4)𝐼 ∗ 𝑎3(1 − 𝑚3)𝑌𝑆 ∗ 𝑎4(1 − 𝑚4)𝑌𝑆 ∗ 𝑢33 −𝛾𝑌𝑆 ∗ 0 𝑎5(1 − 𝑚4)𝑌𝐼 ∗ 𝛾𝑌𝐼 ∗ 𝑢44 ] , where 𝑢11 = = 𝑟 (1 − 𝑆∗ + 𝐼∗ 𝐾 ) − 𝑟𝑆∗ 𝐾 − 𝛽𝐼∗ 1 + 𝐼∗ − 𝑝1(1 − 𝑚3)𝑌𝑆 ∗ − 𝑎1, 𝑢12 = − 𝑟𝑆∗ 𝐾 − 𝛽𝑆∗ 1 + 𝐼∗ + 𝛽𝑆∗𝐼∗ (1 + 𝐼∗)2 , 𝑢22 = 𝛽𝑆∗ 1 + 𝐼∗ − 𝛽𝑆∗𝐼∗ (1 + 𝐼∗)2 − 𝑝2(1 − 𝑚4)𝑌𝑆 ∗ − 𝑝3(1 − 𝑚4)𝑌𝐼 ∗ − 𝑎2, 𝑢33 = 𝑎3(1 − 𝑚3)𝑆 ∗ + 𝑎4(1 − 𝑚4)𝐼 ∗ − 𝛾𝑌𝐼 ∗ − 𝑑3, 𝑢44 = 𝑎5(1 − 𝑚4)𝐼 ∗ + 𝛾𝑌𝐼 ∗ − 𝑎6. the characteristic equation of 𝐸0 is (𝑟 − 𝑎1 − 𝜆)(−𝑎2 − 𝜆)(−𝑑3 − 𝜆)(−𝑎6 − 𝜆) = 0, which has eigenvalues 𝜆1 = 𝑟 − 𝑎1; 𝜆2 = −𝑎2; 𝜆3 = −𝑑3; 𝜆4 = −𝑎6. thus, 𝐸0 is locally asymptotically stable if only if 𝑟 < 𝑎1. on point 𝐸1, the characteristic equation of 𝐸1 is (𝑎1 − 𝑟 − 𝜆)(𝑔1 − 𝜆)(𝑔2 − 𝜆)(−𝑎6 − 𝜆) = 0, with adin lazuardy firdiansyah dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge 129 𝑔1 = 𝛽𝐾(𝑟 − 𝑎1) 𝑟 − 𝑎2, 𝑔2 = 𝑎3𝐾(1 − 𝑚3)(𝑟 − 𝑎1) 𝑟 − 𝑑3. this point has eigenvalues 𝜆1 = 𝑎1 − 𝑟; 𝜆2 = 𝑔1; 𝜆3 = 𝑔2; 𝜆4 = −𝑎6. thus, 𝐸1 can be said the locally asymptotically stable if only if 𝑎1 < 𝑟 < 𝑚𝑖𝑛 { 𝛽𝐾𝑎1 𝛽𝐾−𝑎2 , 𝐾𝑎1𝑎3(1−𝑚3) 𝐾𝑎3(1−𝑚3)−𝑑3 }, where 𝛽𝐾 > 𝑎2 and 𝐾𝑎3(1 − 𝑚3) > 𝑑3. on point 𝐸2, the characteristic equation of 𝐸2 is (𝜆2 − (ℎ1 + ℎ2)𝜆 + (ℎ1ℎ2 − ℎ3ℎ2))(ℎ4 − 𝜆)(ℎ5 − 𝜆) = 0, with ℎ1 = 𝑟 (1 − 𝑆(2) + 𝐼(2) 𝐾 ) − 𝑟𝑆(2) 𝐾 − 𝛽𝐼(2) 1 + 𝐼(2) − 𝑎1, ℎ2 = 𝛽𝑆(2) 1 + 𝐼(2) − 𝛽𝑆(2)𝐼(2) (1 + 𝐼(2))2 − 𝑎2, ℎ3 = 𝛽𝐼(2) 1 + 𝐼(2) , ℎ4 = 𝑎3(1 − 𝑚3)𝑆 (2) + 𝑎4(1 − 𝑚4)𝐼 (2) − 𝑑3, ℎ5 = 𝑎5(1 − 𝑚4)𝐼 (2) − 𝑎6. eigenvalues on this point 𝐸2 is 𝜆1,2 = (ℎ1+ℎ2)±√(ℎ1+ℎ2) 2−4(ℎ1ℎ2−ℎ3ℎ2) 2 ; 𝜆3 = ℎ4; 𝜆4 = ℎ5. thus, the point 𝐸2 can be said the locally asymptotically stable if only if ℎ1 + ℎ2 < 0; ℎ1ℎ2 > ℎ3ℎ2; 𝑎5(1 − 𝑚4)𝐼 (2) < 𝑎6; 𝑎3(1 − 𝑚3)𝑆 (2) + 𝑎4(1 − 𝑚4)𝐼 (2) < 𝑑3. on point 𝐸3, we obtain the characteristic equation of 𝐸3 is (𝜆2 − (𝑛1 + 𝑛2)𝜆 + (𝑛1𝑛2 − 𝑛3𝑛4))(𝑛5 − 𝜆)(𝑛6 − 𝜆) = 0, with 𝑛1 = 𝑟 (1 − 𝑆(3) 𝐾 ) − 𝑟𝑆(3) 𝐾 − 𝑝1(1 − 𝑚3)𝑌𝑆 (3) − 𝑎1, 𝑛2 = 𝑎3(1 − 𝑚3)𝑆 (3) − 𝑑3, 𝑛3 = −𝑝1(1 − 𝑚3)𝑆 (3), 𝑛4 = 𝑎3(1 − 𝑚3)𝑌𝑆 (3) , 𝑛5 = 𝛽𝑆(3) − 𝑝2(1 − 𝑚4)𝑌𝑆 (3) − 𝑎2, 𝑛6 = 𝛾𝑌𝑆 (3) − 𝑎6, which has eigenvalues 𝜆1,2 = (𝑛1+𝑛2)±√(𝑛1+𝑛2) 2−4(𝑛1𝑛2−𝑛3𝑛4) 2 ; 𝜆3 = 𝑛5; 𝜆4 = 𝑛6. so, we can say that the point 𝐸3 is locally asymptotically stable if only if 𝑛1 + 𝑛2 < 0; 𝑛1𝑛2 > 𝑛3𝑛4; 𝛽𝑆 (3) − 𝑝2(1 − 𝑚4)𝑌𝑆 (3) < 𝑎2 and 𝛾𝑌𝑆 (3) < 𝑎6. on point 𝐸4, the characteristic equation of 𝐸4 is (𝑧1 − 𝜆)(𝜆 3 + 𝜑1𝜆 2 + 𝜑2𝜆 + 𝜑3) = 0. one eigenvalue is 𝜆1 = 𝑧1, where 𝑧1 = 𝑎3(1 − 𝑚3)𝑆 (4) + 𝑎4(1 − 𝑚4)𝐼 (4) − 𝛾𝑌𝐼 (4) − 𝑑3. other eigenvalues are obtained from the roots of cubic equations, which are obtained from the matrix 𝐽(𝐸4). jurnal matematika mantik volume 6, no. 2, october 2020, pp. 123-134 130 𝐽(𝐸4) = ( 𝑧2 𝑧3 0 𝛽𝐼(4) 1 + 𝐼(4) 𝑧4 −𝑝3(1 − 𝑚4)𝐼 (4) 0 𝑎3(1 − 𝑚4)𝑌𝐼 (4) 𝑎5(1 − 𝑚4)𝐼 (4) − 𝑎6) , with 𝑧2 = 𝑟 (1 − 𝑆(4) + 𝐼(4) 𝐾 ) − 𝑟𝑆(4) 𝐾 − 𝛽𝐼(4) 1 + 𝐼(4) − 𝑎1, 𝑧3 = − 𝑟𝑆(4) 𝐾 − 𝛽𝑆(4) 1 + 𝐼(4) + 𝛽𝑆(4)𝐼(4) (1 + 𝐼(4))2 , 𝑧4 = 𝛽𝑆(4) 1 + 𝐼(4) − 𝛽𝑆(4)𝐼(4) (1 + 𝐼(4))2 − 𝑝3(1 − 𝑚4)𝑌𝐼 (4) − 𝑎2. thus, the roots of the cubic equation are 𝜑1 = −𝑡𝑟 (𝐽(𝐸4)) ; 𝜑2 = 𝑀11 + 𝑀22 + 𝑀33; and 𝜑3 = − 𝑑𝑒𝑡 (𝐽(𝐸4)), where 𝑀𝑖𝑗 is minor of entry (𝐽(𝐸4)) 𝑖𝑗 . by using routh-hurwitz criteria, the point 𝐸4 is the locally asymptotically stable which can be provided with the following condition 𝜑1 > 0; 𝜑3 > 0; 𝜑1𝜑2 > 𝜑3; 𝑎3(1 − 𝑚3)𝑆 (4) + 𝑎4(1 − 𝑚4)𝐼 (4) < 𝛾𝑌𝐼 (4) + 𝑑3. on point 𝐸5, the characteristic equation of 𝐸5 is (𝑒1 − 𝜆)(𝜆 3 + 𝛿1𝜆 2 + 𝛿2𝜆 + 𝛿3) = 0. one eigenvalue is 𝜆1 = 𝑒1, where 𝑒1 = 𝛽𝑆 (5) − 𝑝2(1 − 𝑚4)𝑌𝑆 (5) − 𝑝3(1 − 𝑚4)𝑌𝐼 (5) − 𝑎2. meanwhile, three eigenvalues are the roots of cubic equations 𝜆3 + 𝛿1𝜆 2 + 𝛿2𝜆 + 𝛿3 = 0, which are given by matrix 𝐽(𝐸5). 𝐽(𝐸5) = ( 𝑒2 −𝑝1(1 − 𝑚3)𝑆 (5) 0 𝑎3(1 − 𝑚3)𝑌𝑆 (5) 𝑒3 −𝛾𝑌𝑆 (5) 0 𝛾𝑌𝐼 (5) 𝑒4 ), where 𝑒2 = 𝑟 (1 − 𝑆(5) 𝐾 ) − 𝑟𝑆(5) 𝐾 − 𝑝1(1 − 𝑚3)𝑌𝑆 (5) − 𝑎1, 𝑒3 = 𝑎3(1 − 𝑚3)𝑆 (5) − 𝛾𝑌𝐼 (5) − 𝑑3, 𝑒4 = 𝛾𝑌𝑆 (5) − 𝑎6. so, we obtain the roots of the cubic equation, i.e. 𝛿1 = −𝑡𝑟 (𝐽(𝐸5)) ; 𝛿2 = 𝑀11 + 𝑀22 + 𝑀33; and 𝛿3 = − 𝑑𝑒𝑡 (𝐽(𝐸5)) with 𝑀𝑖𝑗 is minor of entry (𝐽(𝐸5)) 𝑖𝑗 . by using routhhurwitz criteria, the point 𝐸5 can be said locally asymptotically stable if it satisfies the following condition, 𝛽𝑆(5) < 𝑝2(1 − 𝑚4)𝑌𝑆 (5) + 𝑝3(1 − 𝑚4)𝑌𝐼 (5) + 𝑎2; 𝛿1 > 0; 𝛿3 > 0; 𝛿1𝛿2 > 𝛿3. on point 𝐸6, the characteristic equation of 𝐸6 is (𝑙1 − 𝜆)(𝜆 3 + 𝜎1𝜆 2 + 𝜎2𝜆 + 𝜎3) = 0. one eigenvalue is 𝜆1 = 𝑙1, with 𝑙1 = 𝑎5(1 − 𝑚4)𝐼 (6) + 𝛾𝑌𝑆 (6) − 𝑎6. three eigenvalues are the roots of cubic equations, i.e. 𝜆3 + 𝜎1𝜆 2 + 𝜎2𝜆 + 𝜎3 = 0, which are given by matrix adin lazuardy firdiansyah dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge 131 𝐽(𝐸6). 𝐽(𝐸6) = ( 𝑙2 𝑙3 −𝑝1(1 − 𝑚3)𝑆 (6) 𝛽𝐼(6) 1 + 𝐼(6) 𝑙4 −𝑝2(1 − 𝑚4)𝐼 (6) 𝑎3(1 − 𝑚3)𝑌𝑆 (6) 𝑎4(1 − 𝑚4)𝑌𝑆 (6) 𝑙5 ) , where 𝑙2 = 𝑟 (1 − 𝑆(6) + 𝐼(6) 𝐾 ) − 𝑟𝑆(6) 𝐾 − 𝛽𝐼(6) 1 + 𝐼(6) − 𝑝1(1 − 𝑚3)𝑌𝑆 (6) − 𝑎1, 𝑙3 = − 𝑟𝑆(6) 𝐾 − 𝛽𝑆(6) 1 + 𝐼(6) + 𝛽𝑆(6)𝐼(6) (1 + 𝐼(6))2 , 𝑙4 = 𝛽𝑆(6) 1 + 𝐼(6) − 𝛽𝑆(6)𝐼(6) (1 + 𝐼(6))2 − 𝑝2(1 − 𝑚4)𝑌𝑆 (6) − 𝑎2, 𝑙5 = 𝑎3(1 − 𝑚3)𝑆 (6) + 𝑎4(1 − 𝑚4)𝐼 (6) − 𝑑3. thus, we obtained other corresponding eigenvalues, namely 𝜎1 = −𝑡𝑟 (𝐽(𝐸6)) ; 𝜎2 = 𝑀11 + 𝑀22 + 𝑀33; 𝜎3 = − 𝑑𝑒𝑡 (𝐽(𝐸6)), with a minor of entry (𝐽(𝐸6)) 𝑖𝑗 that is denoted by 𝑀𝑖𝑗. by using routh-hurwitz criteria, the point 𝐸6 can be said locally asymptotically stable if only if 𝑎5(1 − 𝑚4)𝐼 (6) + 𝛾𝑌𝑆 (6) < 𝑎6; 𝜎1 > 0; 𝜎3 > 0; 𝜎1𝜎2 > 𝜎3. on point 𝐸7, we have the characteristic equation of 𝐸7, which is obtained from the matrix 𝐽∗(𝐸7). meanwhile, 𝐽 ∗(𝐸7) is the jacobian matrix at 𝐸7. eigenvalues of 𝐸7 are the roots of the fourth-order equation, i.e. 𝜆4 + 𝜂1𝜆 3 + 𝜂2𝜆 2 + 𝜂3𝜆 + 𝜂4 = 0, with 𝜂1 = −𝑡𝑟(𝐽 ∗(𝐸7)), 𝜂2 = 𝜏11 + 𝜏22 + 𝜏33 + 11 + 22 + 𝜌11, 𝜂4 = −(𝑀11 + 𝑀22 + 𝑀33 + 𝑀44), 𝜂4 = 𝑑𝑒𝑡(𝐽 ∗(𝐸7)), where 𝜏( ) and 𝜌 are minor of order 2 from the remaining submatrix after 4th row and 4th column (after 3rd row and 3rd column) and after 2nd row and 2nd column are removed from 𝐽∗(𝐸7), respectively. meanwhile, 𝑀𝑖𝑗 is minor of order 3 for entry (𝐽 ∗(𝐸7))𝑖𝑗 . by using routh-hurwitz criteria, we get the condition for the stability of asymptotical locally at the point 𝐸7, namely 𝜂1 > 0; 𝜂1𝜂2 > 𝜂3; 𝜂3(𝜂2𝜂1 − 𝜂3) − 𝜂4𝜂1 2 > 0; 𝜂4 > 0. 3.4 numerical simulation in this section, we present some numerical solutions by using the runge-kutta 4th order as a numerical method to illustrate our analytical results. we use some parameters for the system (1), namely 𝑟 = 0.45, 𝐾 = 10, 𝛾 = 0.15, 𝑝1 = 0.6, 𝑝2 = 0.8, 𝑝3 = 0.45, 𝑎1 = 0.12, 𝑎2 = 0.16, 𝑎3 = 0.25, 𝑎4 = 0.35, 𝑎5 = 0.15, 𝑎6 = 0.16, 𝑑3 = 0.103, 𝛽 = 0.3333. case 1: we start with 𝑚3 = 0.157 and 𝑚4 = 0.108. the prey refuge is very small, which means that the number of prey outside the refuge is huge. all solutions are convergent to jurnal matematika mantik volume 6, no. 2, october 2020, pp. 123-134 132 𝐸3(0.4887,0,0.6090,0), see figure 2(a). this equilibrium is also locally asymptotically stable, see figure 2(b). it indicates that infected predator and infected prey are extinct. case 2: we consider 𝑚3 = 0.65 and 𝑚4 = 0.3. the prey refuge means that the number of prey outside prey refuge is less than the first case. all solutions with different values are convergent to equilibrium 𝐸5(2.3556,0,1.0667,0.6874), see figure 3(a). 𝐸5 is also locally asymptotically stable, see figure 3(b). it indicates that the infected prey is extinct. (a) (b) figure 2. numerical solution for case 1: (a) time graph; (b) the phase portrait (a) (b) figure 3. numerical solution for case 2: (a) time graph; (b) the phase portrait case 3: when the prey refuge increases to become 𝑚3 = 0.85 and 𝑚4 = 0.5, then it shows that amount of prey outside refuge is small. all trajectories with different values converge to the point of equilibrium 𝐸7(2.7641,0.4901,0.8216,0.5761), see figure 4(a), and is also locally asymptotically stable, see figure 4(b). it indicates that all populations exist. (a) (b) figure 4. numerical solution for case 3: (a) time graph; (b) the phase portrait to investigate the effect of refuge in prey populations, we can observe the dynamic behavior of the system (1) by using 𝑚3, 𝑚4 makes different values. figure 5 shows the adin lazuardy firdiansyah dynamics of infected predator-prey system with nonlinear incidence rate and prey in refuge 133 time graph of prey populations which means that prey refuge can prevent extinction in prey populations when 𝑚3, 𝑚4 values are getting large. figure 5. influence refuge in prey by using 𝑚3, 𝑚4 makes different values 4. conclusions in this paper, we have merged an eco-epidemiology model with transmission disease in both populations by using non-linear incidence rate in prey populations and also prey refuge proportional in prey populations. eight equilibrium points that exist under certain conditions and also that local stability for those points have been determined. we have noticed that prey refuge can avoid extinction in prey populations. for future work, we have to investigate the harvesting effect in both populations. it is used as a disease control in an eco-epidemiology model. references [1] w. o. kermack and a. g. mckendrick, “a contribution to the mathematical theory of epidemics,” proc. r. soc. london, vol. 115, no. 772, p. 22, 1927. 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