A Solution Proposal to the Interval Fractional Transportation Problem

Transcription

1 Appl. Math. Inf. Sci. 6, No. 3, (2012) 567 Applied Matheatics & Inforation Sciences An International Journal A Solution Proposal to the Interval Fractional Transportation Proble Nuran GÜZE 1, Ýbrahi EMIROGU 2, Fatih TAPCI 2, Copkun GUER 2 and Mustafa SYVRY 2 1 Yildiz Technical University, Faculty of Science and iterature, Matheatical Departent, Davutpasa, Esenler, Istanbul, Turkey 2 Yildiz Technical University, Matheatical Engineering Departent, Davutpasa, Esenler, Istanbul, Turkey Received: Dec. 12, 2011; Revised Feb. 13, 2012; Accepted March 6, 2012 Published online: 1 Sep Abstract: In real world applications, frequently ay be faced up with the fractional transportation proble that these cost and preference paraeters of the fractional objective ay not be known in precise anner which are changed in the interval each other. In this study, fractional transportation proble with interval coefficient is transfored to a classical transportation proble by expanding the order 1 st Taylor polynoial series with ulti variables. Keywords: Transportation Proble, inear Fractional Prograing, Interval Coefficients. 1. Introduction In the odelling the real world probles like financial and, corporate planning, production planning, arketing and edia selection, university planning and student adission, health care and hospital planning, air force aintenance units, bank branches, etc. frequently. We are faced up with a decision to optiise profit/cost, inventory/sales, etc. respect to soe constraints (ai and Hwang 1996). Since linear fractional prograing proble approach offers ore efficient ethod than linear prograing Proble (PP), any researches has been working on inear Fractional Prograing Proble (FPP) intensively. In the literature [1-4], different approaches appear in solving different odels of the FPP. When soe of studies present solution ethods (ai and Hwang, 1996), (Charnes and Cooper,1962), (Zionts,1968), (Chakraborty and Gupta,2002), others have concentrated on applications (Musteanu and Rado,1960), (Gilore and Goory,1963) (Sengupta et all., 2001). While the traditional (classical) transportation proble considers transportation of the product fro source to destination with a linear objective function on the other hand several other approaches exist for the linear transportation proble with a single or ulti objective function, Kanti Swarup studied the optial (axiu) ratio of the inear function subject to a set of linear constraints and non negativity conditions on the variables [5], Dorina Moante has also presented a solution to a three diensional proble with an objective function which is the ratio of the linear functions [7], Guzel, N. and Sivri, M have presented a solution to Multi objective a linear fractional prograing proble by using Taylor series expansion[8], C.S. Raakrishnan presented an optial and near optial initial solution in the balanced and unbalanced transportation proble by using Vogel approxiation Method [6], Sivri M et all, propose an optial or near optial initial solution and optiality condition(by using [5]) for transportation proble with the linear fractional objective [11]. The ai of this paper is to introduce two solution procedures for the fractional transportation proble with interval coefficients. And an illustrative exaple is given to explain these procedures. 2. The Structure of the Fractional Transportation Proble Consider a fractional transportation proble with supply and n deand, in that a i > 0 units are supplied by supply i th and b j > 0 units are required by deand j th. There is a fractional objective function that unit shipping Corresponding author: e-ail: nguzel@yildiz.edu.tr, sivri@yildiz.edu.tr

2 568 G. Rajchakit: A Solution Proposal to the Interval Fractional... cost c and unit preferring route d for transportation. et x denote the nuber of units to be transported fro supply i th to deand j th. The atheatical odel of the fractional transportation proble with interval coefficients that coefficients of a fractional objective function are interval of the real nubers in this work is stated as follows: c, c R x Z in [ d, d] R (2.1) x subject to j1 x a i i 1,..., i1 x b j, j 1,..., n where [ c, ] cr is an interval representing the uncertain cost and [ d, ] dr is an interval representing uncertain preference of route for the transportation proble. That is, c,d cr,dr are the lower and upper bounds for the unit shipping cost c and unit preferring route d to be transported fro supply i th to deand j th, respectively. 3. Solving the Proble Proposal 2.1 Since the proble 2.1 is a iniization proble, in the fractional objective function these given intervals [ c, ] cr and d, d R can be stated as follows: c, c R c + θ (c R c ), 0 θ 1 for i 1, [..., ;] j 1,..., n GrindEQ 2 d, d R d + λ (d R d ) 0 λ 1 for i 1,..., ; j 1,..., n GrindEQ 3 The proble 2.1 ay be restated as follows: { Z Min c + θ (c R c )x } d + λ (d R d )x 0, s.t j1 x a i i 1,..., i1 x b j, j 1,..., n θ, λ [0, 1], i, j c + θ (c R c )x d + λ (d R d )x > 0. (3.1) When the first Taylor polynoial for the objective function of proble 3.1 about X (k) (x (k) ), the last, θ(k), λ(k) for of 3.1 can be constructed as follows: Min Z (k) Z (x x (k) ) x X (0) + + Z θ (0) X Z λ (0) X (θ θ (k) ) (λ λ (k) ) s.t j1 x a i i 1,..., i1 x b j, j 1,..., n θ, λ [0, 1], i, j (3.2) An initial basic feasible solution X (0) (x (0), θ(0), λ(0) ) is obtained with the North-West Corner Rule of the classical transportation proble or obtained by using the other known ethods for transportation proble where θ (0) ( ),λ(0) are constant 0 θ (0) 1, 0 λ (0) 1 for i 1,..., ; j 1,..., n. If proble 3.2 for X (k) X (0) is solved, then the solution X 2.1 (x (1), θ2.1, λ2.1 )is obtained. And the objective function of 3.2 is rearranged for the new point X 2.1. In other words, the fractional objective function of proble 3.1 is again expanded to its first Taylor polynoial at the new point X 2.1. The solution of proble 3.2 with rearranged objective function is closer to the optial value. In order to have a better solution, the solutions { X (k) (x (k), θ(k) }, λ(k) ), (k 0, 1, 2, 3,...) of proble 3.2 can be continued until (x (k) (x (k+1), θ (k+1) (x (k), θ(k), λ (k+1), θ(k) ). A last obtained solution, λ(k) ) is a solution of proble 2.1 [18]. Proposal?? et us consider the proble 2.1 as follows: MinZ1 R c R x, MinZ1 C MaxZ 2 ( c + cr )x, 2 d x, MaxZ2 C ( d + dr 2 )x j1 x a i i 1,..., s.t. i1 x b j, j 1,..., n GrindEQ 6, λ(k) )

3 Appl. Math. Inf. Sci. 6, No. 3, (2012) / In order to propose a solution to proble??, it is axiized of the su of the linear ebership functions of Z R 1, Z C 1, Z 2 and Z C 2 under the constraints of?? where, µ(z R 1 ), µ(z C 1 ), µ(z 2 ) and µ(z C 2 ) are the ebership functions of Z R 1, Z C 1, Z 2 and Z C 2, respectively. By rearranging proble??, we have Max (µ(z1 R ) + µ(z1 C ) + µ(z2 ) + µ(z2 C )) j1 x a i i 1,..., Subject to i1 x b j, j 1,..., n j1 b GrindEQ 7 j 4. An Exaple We consider a fractional interval transportation proble with interval shipping costs and preferring routes, crisp supplies and deands. The proble has the following fractional transportation proble for: c 11 [1, 3] c 12 [2, 5] c 13 [0, 3] d 11 [3, 4] d 12 d 13 [3, 5] [4, 6] c 21 [1, 2] c 22 [3, 4] c 23 [1, 3] d 21 [5, 7] d 22 d 23 [1, 2] [5, 7] b 1 20 b 2 10 b 3 20 a 1 30 a 2 20 Since, the given proble is a balanced transportation proble. The given fractional transportation proble can be written as the following interval fractional linear prograing proble: ([1, 3]x 11 + [2, 5]x 12 + [0, 3]x 13 +[1, 2]x Z in 21 + [3, 4]x 22 + [1, 3]x 23 ) /([3, 4]x 11 + [3, 5]x 12 + [4, 6]x 13 +[5, 7]x 21 + [1, 2]x 22 + [5, 7]x 23 ) s.t. (3.3) Using?? and??, the given interval fractional transportation proble can be written as follows: θ 11, θ 21, θ 22, θ 23, λ 11, λ 12, λ 13, λ 21, λ 22, λ 23 [0, 1] (3.4) In order to achieve our goal, initial feasible solution values can be chosen as x 11 20, x 12 10, x 13 0, x 21 0, x 22 0, x 23 20, θ 11, θ 21, θ 22, θ 23, λ 11, λ 12, λ 13, λ 21, λ 22, λ 23 1 respect to the North West Corner Rule. And the objective function is: z 1 (1 + 2θ 11 )x 11 + (2 + 3θ 12 )x 12 +(0 + 3θ 13 )x 13 + (1 + θ 21 )x 21 +(3 + θ 22 )x 22 + (1 + 2θ 23 )x 23 z 2 (3 + λ 11 )x 11 + (3 + 2λ 12 )x 12 +(4 + 2λ 13 )x 13 + (5 + 2λ 21 ) x 21 + (1 + λ 22 )x 22 + (5 + 2λ 23 )x 23 Z in ( z1 z 2 )width0.12in, height0.19in, keepaspectratiofalse]iage1.eps width0.12in, height0.19in, keepaspectratiofalse]iage2.eps For the above solution, z 1 40 and z and the new objective function is After that, the objective function 3.5 is solved subject to constraints of??, and the below solution is obtained: x 11 0, x 12 10, x 13 20, x 21 20, x 22 0, x 23 0, θ 11 0, θ 12 0, θ 13 0, θ 21 0, θ 22 0, θ 23 0, λ 11 0, λ 12 1, λ 13 1, λ 21 1, λ 22 0, λ For the above solution,z 1 40 and z 2 310and the new objective function is: Consequently, the objective function 3.6 is solved subject to constraints of??, and the below solution is obtained: x 11 0, x 12 10, x 13 20, x 21 20, x 22 0, x 23 0, θ 11 0, θ 12 0, θ 13 0, θ 21 0, θ 22 0, θ 23 0, λ 11 0, λ 12 1, λ 13 1, λ 21 1, λ 22 0, λ 23 0, Since the solution is the sae as the solution obtained fro a previous step, the solution is the better solution of the given original proble. Now, If the sae proble is solved respect to proposal??, the given original proble is written as the following ultiple objective transportation proble: Min Z R 1 3x x x 13 +2x x x 23 Min Z C 1 2x x x x x x 23 (3.7) Max Z 2 3x x x 13 +5x 21 + x x 23 Max Z C 2 3.5x x x 13 +6x x x 23

4 570 G. Rajchakit: A Solution Proposal to the Interval Fractional... (1 + 2θ11)x11 + (2 + 3θ12)x12 + (0 + 3θ13)x13 + (1 + θ21)x21 + (3 + θ22)x22 + (1 + 2θ23)x23 z in{ } (3 + λ 11 )x 11 + (3 + 2λ 12 )x 12 + (4 + 2λ 13 )x 13 + (5 + 2λ 21 )x 21 + (1 + λ 22 )x 22 + (5 + 2λ 23 )x 23 Z (2) in {70(x 11 ) + 260(x 12 10) 160(x 13 20) + 30(x 21 20) + 650x 22 50x θ (θ 12 ) (θ 13 ) (θ 21 ) + 0(θ 22 ) + 0(θ 23 ) 0(λ 11 1) 800(λ 12 1) 1600(λ 13) 1600(λ 21) 0λ 22 0(λ 23 1)} (3.5) Z (3) in {190(x 11 ) + 420(x 12 10) 240(x 13 20) + 30(x 21 20) + 890x x θ (θ 12) (θ 13) + 800(θ 21) + 0(θ 22) + 0(θ 23) 0(λ 11 ) 800(λ 12 1) 1600(λ 13 1) 1600(λ 21 1) 0λ 22 0(λ 23 )} (3.6) Subject to (3.8) Maxiu and iniization values of the objective functions 3.7 under the constrains 3.8 are 150 Z1 R 170,95 Z1 C 115,160 Z2 210,195 Z2 C 260. Using the fuzzy approach their ebership functions are µ(z1 R ) 1 20 (3x x x 13 +2x x x ), µ(z C 1 ) 1 20 (2x x x x x x ), (3.9) µ( Z 2 ) 1 50 (3x x x 13 +5x 21 + x x ) µ( Z C 2 ) 1 65 (3.5x x x 13 +6x x x ). Then, Max(.13615x x 12 (3.10) x x x x ) x 0, i 1, 2; j 1, 2, 3. The solution of proble 3.10 is x 11 0, x 12 10, x 13 20, x 21 20, x 22 0, x Thus, the solution obtained with proposal 2.1 is equal to the solution obtained with proposal??. 5. Conclusion In this paper, two solution procedures are proposed for the interval fractional transportation proble. One of the is based on Taylor series approxiation [8,11], the other one is based on interval arithetic [10]. Also an illustrative exaple is given for explaining these approaches. The results obtained by these approaches are the sae. References [1] Ahlatcıoglu, M., Sivri M., An Efficient Solution Proposal To Interval Transportation Proble, Journal of Yıldız Technical Uni.,Vol.4, (1993), pp [2] Ahlatcıoglu, M., Sivri M., Raakrishnan Approach to Generalized Transportation Proble, Dokuz Eylül University, Journal of Faculty of Econoic and Adinistrative Sciences,Vol.5(1-2), (1990), pp [3] Ahlatcıoglu, M., Sivri M., Interval Transportation Proble and A Solution Proposal, Marara University, Journal of Sciences and Technology, Vol.14, (1998), pp [4] S. Chanas, W. Kolodziejczyk, A.Machaj, A fuzzy approach to the transportation proble, Fuzzy Sets and Systes, 13 (1984), [5] Swarup,K., inear Fractional Functionals Prograing, Operational Research, vol.13, No.3,(1984), pp [6] Raakrishnan, C.,S., An Iproveent to Gayal s Modified VAM for the unbalanced transportation Proble, J.Op.Res. Soc.Vol.39, No.6, , [7] Moanta, D., Soe Aspects On Solving a inear Fractional Transportation Proble, Journal of Applied Quantitative Methods, Vo.2, No.3,2007. [8] Guzel N., Sivri M., Taylor Series solution of Multi objective linear Fractional Prograing Proble, Trakya University J Sci, 6??: 80-87,2005.

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