@1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Solving Fuzzy-Parametric Linear Programming Problems Iden Hassan Dept. of Mathematics / College of science for woman / University of Baghdad Nabeel H. Saeed Dept. of Mathematics/College of Education for Pure Science (Ibn AL-Haitham) University of Baghdad Received in: 19 June 2012 , Accepted in: 12 September 2012 Abstract The fuzzy sets theory has been applied in many fields, such as operations research, control theory and management sciences, etc. In particular, an application of this theory in decision making problem is linear programming problems with fuzzy technological coefficients numbers, as well as studying the parametric linear programming problems in the case of changes in the objective function. In this paper presenting a new procedure which connects and makes link between fuzzy linear programming problem with fuzzy technological coefficients numbers and parametric linear programming problem with change in coefficients of the objective function, then develop a numerical example illustrates the steps of solution to this kind of problems. Keywords: Fuzzy linear programming; parametric linear programming; fuzzy- parametric linear programming. 303 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Introduction The linear programming is one of the important tools to help decision- maker to take the best decision, but it needs to a certain date (exact) and this is not realistic because the environmental conditions often provide us data which are uncertain. Therefore, the concept of fuzzy set theory was used to solve the linear programming problems when uncertain date by Bellman and Zadeh[1]. Tanaka [2] hired this concept to solve the fuzzy linear programming with multiple goals. Followed by Nagoita[3] formulated the fuzzy linear programming problems. The issue of parametric linear programming problems studied by Saaty and Gass[4] after determining the feasible solution and the values of parameters, its could be possible to generate all solutions with keeping to optimal solution. In our paper representing idea (proposed) to make link between fuzzy linear programming and parametric linear programming in the case of one vehicle. We consider in fuzzy linear programming the technological coefficients are fuzzy numbers, but in parametric linear programming the change will be happened in coefficients of objective function. The paper is outlined as follows: Introduction in section1. Linear programming problem in section 2. Fuzzy linear programming problems with technological coefficients are fuzzy number in section 3. In section 4 we studied the parametric linear programming problems with change in coefficients of objective function. But in section 5 we studied a new proposed which call it fuzzy- parametric linear programming problems. After that in section 6 we examined the application of fuzzy- parametric linear programming to concrete example. I) Linear Programming Problems [5]: Let X= (x1, x2, …, xn) be (n) variables and (m) constraints, then the linear programming problems is defined as follows: Maximize (minimize) Z= )1...(0Xb,)(xPCX n 1j jj         ≥≥≤∑ = Where: Pj: The original basic columns in the matrix A, where Ph= aih, i= 1, 2,…, m and h ∈{1,2,…n} xj: Decision variables. b: Right hand side values. II) Fuzzy Linear Programming Problems: If any coefficient for linear programming problems is uncertain value, then it is called fuzzy linear programming problems and its formis defined as: Maximize (minimize) Z= )...(20Xb,)(xP ~ CX n 1j jj         ≥≥≤∑ = Where: jP ~ : The original fuzzy columns in the matrix A Assumption 1: By using fuzzy logic which defined by L.A. Zadeh[6] the membership function of fuzzy technological coefficients are presented as: )3...( xAAif....o AAxAif...x)/AA(A Axif....1 (x)M t ttt A ~           ≤+ +<<−+ ≤ = Where: x∈R 304 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 At: The tolerance values of each element in the matrix A where A= aij and At = dij > 0. For defuzzification of the problem (2), we first fuzzify the objective function, by retailing it into two subproblems (Z1, Z2) as follows (only maximize): Maximize Z1= )4.(.. 0Xb,xPCX n 1j jj         ≥≤∑ = And Maximize Z2 = ...(5) 0Xb,x)P(PCX n 1j j t jj         ≥≤+∑ = t jP : The tolerance columns of original columns (matrix A t). Solving these two subproblems by using regular simplex method can obtain two different values Z1 and Z2 [7]. Then we can determine the upper and lower bound as Zu and z . Where: Zu= max. { Z1, Z2} … (6) And, Z = min. { Z1, Z2} … (7) The objective function takes values between Z, Zu while technological coefficients vary between (aij) and (aij+dij). In more certain the objective function takes the maximum value Zu when the technological coefficients at the lower value (aij), and the minimum value Z at the upper value (aij+dij). This means the membership function of the objective function G which is subset in Rn is defined as [7]: )(8... CXZif...1 ZCXZif)...Z)/(ZZ(CX ZCXif...0 (CX)M u uuG           ≤ <<−− ≤ =   And it illustrated in fig. (1). The fuzzy set of the i-th constraints Ci which is a subset in Rm, is defined by )(9... b)XA(Aif...1 )XA(AbAXif... XAX)/A(b AXbif...0 (X)M t tt Ci           ≤+ +<<− ≤ = And it illustrate in fig. (2). By using the definition of the fuzzy decisive proposed by Bellman and Zadeh[1] we have: MD(x) = Min {MG(x), Min (Mci(x)} … (10) In this case the optimal fuzzy decision is a solution of the problem: MinMax(x))MMax( 0 D 0 ≥≥ = xx {MG(x), Min (Mci(x)} … (11) 305 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Consequently, the problem (3) becomes to the following optimization problem [6]: Max λ MG(x) ≥ λ Mci(x) ≥ λ, i=1, 2, …m …(12) X ≥ 0, 0 ≤ λ ≤ 1. By using (8) and (9), the problem (12) can be written as: Max λ ( ) ( ) 1λ0 0,x m ..., 2, 1,i 0,bX AA 0CXZZZλ i t u ≤≤≥ =≤−+ ≤−+− λ  … (13) λ  zzzu +− )( its a linear function of λ then we called it a slope function and symbol as f(λ). Solving this problem by using bisection method to obtain λ* which is the optimal value of λ, such that the value of objective function (CX*) becomes greater than value of f(λ*), by other words the intersection of the objective function at λ* with the feasible solutions set is non- empty. The condition stopping of the bisection method is [8],[9]: 14)....( ελλ i1i <−+ Now, we can write the problem (2) with λ* as: Maximize (Min.) Z* = )15.(.. 0Xb,)( xPCX n 1j jj         ≥≥≤∑ = ∗∗∗∗ Where: ∗P : The original crisp columns, its elements aij+λ*dij are approximate. And: ....(16) PλPP tjjj ∗∗ += X*, Z* are symmetric to X, Z but at the λ* level. III) Parametric Linear Programming Changes in C [5], [10]: Parametric linear programming is an extension of the sensitivity optimal analysis. It investigates the effect of predetermined continuous variations in the objective function coefficients on the optimum solution. Let X = (x1, x2, …, xn) and define the parametric linear programming changes in C as: (17).... 0Xb,xPt)XC(CZMaximize n 1j jj         ≥≤′+= ∑ = Where: C′ : Variation profit vector ),....,,( 21 ncccC ′′′=′ . Let XBi, Bi, CBi(t) be the elements that define the optimal solution associated with critical value ti. Starting at t0 = 0 with B0 as its optimal basis. Next, the critical value ti+1 where (i = 0, 1, 2, …) and its optimal basis, if one exists, are determined. The changes in C can affect only the optimality of the problem, the current solution XBi = 1−iB b will remain optimal t ≥ ti so long the reduced profit, Zj(t) -cj(t), satisfies the following optimality condition: 306 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 (18).... column nonbasis jallfor0,)(CP)B(C)(C)(Z jj 1 Bjj ≥−=− − tttt ii The value of ti+1 equals the largest t in [ti, ti+1] that satisfies all the optimality conditions [5] . There is a function of t represents to the objective function in each interval. Such that [10]: (19)... ZZ)Z( ′+= tt Where: Z = CX … (20) And Z′= C′X … (21) The optimal solution for the entire range of t is summarized in table (1) [5]: To find all critical values which determine the intervals of t with the optimal solution in each alternative basic. IV) Fuzzy- Parametric Linear Programming Problems: Now, suggesting new procedure which connect between fuzzy linear programming with technological coefficients are uncertain and parametric linear programming with change in coefficients of objective function which will define it as follows: )22(...........................................................0Xb,xP ~ )X(CZMaximize n 1j jj         ≥≤′+= ∑ = tC The general idea of solving fuzzy- parametric linear programming problems is to start with the optimal solution at t = 0. Next, solving by fuzzy linear programming as in (II) to obtain λ* which make the problem in a crisp. Then, solving by using the optimality condition in (18) which become as: ......(23) 0)(CP)(B)(C)(C)(Z jj 1 Bjj ≥−=− ∗−∗∗ ∗ tttt ii For all j are non basic columns. Numerical Example: This example is found by the research which are as follows: Max Z = (3-6t) x1+ (2-2t) x2+ (5+5t) x3 Subject to: xj ≥ 0, dij = 0.1 for each aij ≠ 0 and dij = 0 otherwise. Solution: 1) Let t = 0 then, the problem become without parametric changes, i.e. it has turned into a fuzzy linear programming problem. 2) Retail the problem to two subproblems and solving by simplex method: 1600204.151 == uZandZ 3) Now, formulate the problem as in (13) we get: 30x4 ~ x1 ~ 60x2 ~ x3 ~ 40x1 ~ x2 ~ x1 ~ 21 31 321 ≤+ ≤+ ≤++ 307 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 max λ 100,, 30)1.04()1.01( 60)1.02()1.03( 40)1.01()1.02()1.01( 0)523(0204.1519796.8 321 21 31 321 321 ≤≤≥ ≤+++ ≤+++ ≤+++++ ≤++−+ λ λλ λλ λλλ λ andxxx xx xx xxx xxx To find the values of CX by simplex and the value of f(λ) for each level of λ. For λ = 0 implice Cx = 160 and f(0) = 151.0204, this means the intersection is non- empty. For λ= 1 implice CX= 151.0204 and f(1) =160, this means the intersection is empty. And by using the bysection method the first point is λ1 = 0.5 that implice CX=155.3836965 and f(0.5) =155.5102, this means the intersection is empty. Therefore, the next λ will be λ2= 0.25 non- empty. At the end we obtain ∗λ = 0.492954735 implice the optimal solution is .446945.1550 emptynonisCXZ −== ∗ Β ∗  )05522()(),,( 00 632 tttCimplicexxxX BB +−== ∗∗ ∗∗∗ , ∗∗∗ == 632 278354.29,527583.4 xandxx is not real value.           − − =−∗ 1011738.1975945.1 0487973.00 0249855.0487973.0 )( 10B The crisp original columns become as: ttt PPPPPPPPP 333222111 ,, ∗∗∗∗∗∗ +=+=+= λλλ or:           =           =           = ∗∗∗ 0 049295.2 049295.1 , 049295.4 0 049295.2 , 049295.1 049295.3 049295.1 321 PPP We apply the optimality condition in (23): )049971.049971.0975946.0975946.0())(( 10 0 ttBtC B +−−=−∗∗ 5,4,10)())(( 10 0 =≥−∗−∗∗ jfortCPBtC jjB To find the first critical value t1=1 this means the first interval is [0, 1]. And to find the alternative basic )( 1 ∗B as a second basic therefore P4 must enter and P2 is leaving the second basic. )0550()(),,( 11 634 ttCimplicexxxX BB +== ∗∗ ∗∗∗ And .,,2783546.29 643 valuerealnotarexxx ∗∗∗ = 308 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 391773.1465 31 == ∗∗ xZ           − =−∗ 100 0487973.00 0512025.01 )( 11B 1 1 ))(( 1 −∗ ∗ BtCB = (0 2.439865 + 2.439865t 0) 5,2,10)())(( 11 1 =≥−∗−∗∗ jfortCPBtC jjB To find the second critical value we apply the optimality condition in (23). But in this case t goes to infinite i.e. there is no second critical value, and the second interval is [1,∞) therefore, the parametric analysis ends. Now, we apply (19), (20), (21) to find z*(t): 336604.13752)( 320 =+−=′ ∗∗∗ xxZ Then: ttZ 336604.137446945.155)(*0 += And 391773.146)( * 1 1 =′=′ ∗ B XCZ Then: ttZ 391773.146391773.146)(1 += ∗ The summarized solution illustrate in table (2). References 1. Bellman, R.E. and Zadeh, L.A. (1970) "Decision making in a fuzzy environment", Management Science 17, B141-B164. 2. Tanaka, H., Okuda, T. and Asai, k. (1984) "on fuzzy mathematical programming", Journal Cybernetics, V.3, PP. 291-298. 3. Nagoita, C.V., (1981), "The Current Interest in Fuzzy Optimization", Fuzzy Set and Systems, V.6, PP.261-269. 4. Gass, S.I. and Saaty, T.L. (1955) "Parametric objective function (part2) Generalization", Opns. Res. Soc. Am, V.3, PP. 395-401. 5. Hamdy, A. T. (2012) "Operation Research An Introduction", Ninth Edition, Pearson Company, Boston, Dubai. 6. Zadeh, L.A. (1965) "fuzzy sets"; Information and control, V.8, PP. 338-353. 7. Rafail, N., Gasimov and kursat yenilme (2002) "solving fuzzy linear programming problems with linear membership functions", Turkish Journal of mathematics, V. 26, PP. 375- 396, © TUBITAK. 8. Curtis, F.G. and Patriclc, O.W. (1985) "Applied numerical analysis", Addison Wesley publishing company, New York. 9. Samule, D. C. and Carl de Boor (2008) "Elementary Numerical Analysis", Tata Mc Graw- Hill publishing company limited, New Delhi. 10. Gupta, P.K. and Hira, D.S. (2011) "Operation Research", S. chand, New York. 309 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Table (1) the Summarized Solution t X Z All intervals Value of variables in each interval Value of z in each interval Table (2) the Summarized Numerical Solution t ∗ 1x ∗ 2x ∗ 3x )(tZ ∗ 0 ≤ t ≤ 1 0 4.527583 29.278354 155.446945+137.336604 t 1≤ t < ∞ 0 0 29.278354 146.391773+146.391773t Figure (1) Membership functions of the objective function Figure (2) Membership functions of the constraints MG(CX) Zℓ Zu CX Mci(x) Ax (A+ At)x X 310 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 المعلمیة –حل مشاكل البرمجة الخطیة الضبابیة حسن حسینآیدن جامعة بغداد /كلیة العلوم للبنات /الریاضیات علوم قسم نبیل حمھ سعید جامعة بغداد / )ابن الھیثم( للعلوم الصرفة كلیة التربیة / الریاضیاتعلوم قسم 2012ایلول 12قبل البحث في: ، 2012حزیران 19استلم البحث في: الخالصة تطب��ق ف��ي الكثی��ر م��ن المج��االت، عل��ى س��بیل المث��ال:بحوث العملی��ات، ان نظری��ة المجموع��ات الض��بابیة یمك��ن ان نظریة السیطرة ، علم االدارة... الخ. ان احدى تطبیقات نظریة المجموعات الضبابیة ھي في مشاكل اتخاذ القرار حیث تم تطبیقھا في مشاكل البرمجة الخطیة م�ع ق�رار) باالض�افة ال�ى دراس�ة مش�اكل البرمج�ة الخطی�ة المعلمی�ة ف�ي اعداد ضبابیة للمعامالت التكنولوجیة (معامالت اتخ�اذ ال حالة التغیر في معامالت دالة الھدف. في ھذا البحث سنعرض اسلوب جدید یربط بین مشكلة البرمجة الخطی�ة الض�بابیة م�ع مع�امالت تكنولوجی�ة ض�بابیة ومش�كلة ن ثم صیاغة مثال عددي یوضح خطوات الحل لھذا النوع البرمجة الخطیة المعلمیة مع تغیرات في معامالت دالة الھدف، وم من المشاكل. معلمیة.-برمجة خطیة ضبابیة، برمجة خطیة معلمیة، برمجة خطیة ضبابیة :الكلمات المفتاحیة 311 | Mathematics