A new approach for solving cost minimization balanced transportation problem under uncertainty
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1 J Transp Secur (214) 7: DOI 1.17/s A new approach for solving cost minimization balanced transportation problem under uncertainty Sandeep Singh & Gourav Gupta Received: 21 July 214 / Accepted: 5 August 214 / Published online: 18 September 214 # Springer Science+Business Media New York 214 Abstract In the literature, there are several methods for solving fuzzy transportation problems and finding the fuzzy optimal values. In this paper a new method is proposed for finding an optimal solution for fuzzy transportation problem. The proposed method always gives a fuzzy optimal value without disturbance of degeneracy condition. This requires least computational work to reach optimality as compared to the existing methods available in the literature. Keywords Balanced fuzzy transportation problem. Degeneracy. Cost minimization Introduction In conventional transportation problems, it is assumed that decision maker is sure about the precise values of transportation cost, availability and demand of the product. In real world applications, all the parameters of the transportation problems may not be known precisely due to uncontrollable factors. This type of imprecise data is not always well represented by random variable selected from a probability distribution. These imprecise data may be represented by fuzzy numbers. The idea of fuzzy set was introduced by Zadeh (Zadeh 1965) in Bellmann and Zadeh (Bellmannand and Zadeh 197) proposed the concept of decision making in fuzzy environment. After this pioneering work many authors have studied fuzzy linear programming problem techniques such as S. C. Fang (Fang et al. 1999), H. Rommelfanger (Rommelfanger et al. 1989) andh. Tanaka et al. (1984) etc. S. Singh (*): G. Gupta School of Mathematics and Computer Applications, Thapar University, Patiala, Punjab, India sandychahal143@gmail.com G. Gupta gourav_gupta333@rediff.com
2 34 S. Singh, G. Gupta To solve a crisp transportation problem Taha (Taha 26) uses tabular methods such as Northwest Corner Rule, Least Cost Method and Vogel s Approximation Method (Reinfeld and Vogel, (Reinfeld and Vogel 1958)). These methods use techniques of a Linear programming problem. They differ only in steps of carrying out optimality conditions. Fuzzy transportation problem is a transportation problem whose decision parameters are fuzzy numbers. The objective of the fuzzy transportation problem is to determine the transportation schedule that minimizes the total fuzzy transportation cost while satisfying the availability and requirement limits. Ringuest and Rink (Ringuest and Rink 1987) use a linear programming technique to solve fuzzy transportation problem. Geetha and Nair (Geetha and Nair 1994) formulate a stochastic version of time minimizing transportation problem. Chanas and Kuchta (Chanas and Kuchta 1996) discuss transportation problem with fuzzy cost coefficients and transform problem into a bi-criterion transportation problem; nevertheless, produce only crisp solution. Liu and Kao (Liu and Kao 24) develop a procedure to solve fuzzy transportation problem based on Zadeh s extension principle. Pandian and Natarajan (Pandian and Natarajan 21a; Pandian and Natarajan 21b) propose an algorithm namely, fuzzy zero point method for finding an optimal solution to a fuzzy transportation problem. Fuzzy zero point method consider optimal solution as a trapezoidal fuzzy number. Techniques to solve unbalanced transportation problems in imprecise environment has been given by Rani and Gulati (Rani and Gulati 214a, 214b). In this paper, a new approach to find fuzzy optimal value of fuzzy transportation problem without converting to a classical transportation problem. The proposed method avoids the degeneracy and gives the optimal solution faster than others existing algorithms for the given fuzzy transportation problem. Preliminaries In this section, basic definitions, arithmetic operations and ranking functions are reviewed (Liou and Wang 1992; Kaufmann and Gupta 1991). A. Basic definitions In this section some basic definitions are reviewed (Kaufmann and Gupta 1991). Definition 1 The characteristic function μ A of a crisp set A X assigns a value either or 1 to each member in X. This function can be generalized to a function μea such that the value assigned to the universal set X fall within a specified range [, 1] i.e., μea ðþ x : X f; 1g. The assigned value indicates the membership grade of the element in the set A. The function μea is called the membership function and the set ea ¼ ; x X g defined by μea for each x X is called a fuzzy set. n x; μea ðþ x
3 A new approach for solving cost minimization balanced 341 Definition 2 A fuzzy set ea, defined on the universal set of real number R,issaidtobea trapezoidal fuzzy number if its membership function has the following characteristics: 1. μea ðþ: x X f; 1g is continuous. 2. μea ðþ¼ x for all x (,c] [d, ). 3. Is strictly increasing on [c, a] and strictly decreasing on [b, d]. 4. μ A (x)=1 for all x [a,b]. B. Arithmetic operations In this subsection, arithmetic operations between two trapezoidal fuzzy numbers, defined on universal set of real numbers R, are reviewed [25]. Let ea 1 ¼ ða 1 ; b 1 ; c 1 ; d 1 Þ and ea 2 ¼ ða 2 ; b 2 ; c 2 ; d 2 Þ be two trapezoidal fuzzy numbers, then ðþea i 1 ea 2 ¼ ða 1 þ a 2 ; b 1 þ b 2 ; c 1 þ c 2 ; d 1 þ d 2 Þ: ðiiþ ea 1 ΘeA 2 ¼ ða 1 d 2 ; b 1 c 2 ; c 1 b 2 ; d 1 a 2 Þ ðiiiþλ ea 2 ¼ ðλa 1 ; λb 1 ; λc 1 ; λd 1 Þ λ > : ðλd 1 ; λc 1 ; λb 1 ; λa 1 Þ λ < C. Ranking function A convenient method for comparing fuzzy numbers is by using ranking function(nehietal.24; Noora and Karami 28). A ranking function R:F(R) R, wheref(r) set of all fuzzy numbers defined on set of real numbers defined on set of real numbers, maps each fuzzy number into a real number. Let ea and eb be two fuzzy numbers, then ðþea i R eb if R ea R eb ðiiþea > eb if R ea R ¼ eb > R eb : ðiiiþea ¼eB if R ea R Fuzzy transportation problem In the literature (Chanas and Kuchta 1996; Kaufmann and Gupta 1988) for solving such type of transportation problems the costs are represented as normal fuzzy numbers. The fuzzy transportation problems, in which a decision maker is uncertain about the precise values of transportation cost from the i th source
4 342 S. Singh, G. Gupta Table 1 Tabular form of fuzzy cost transportation problem D 1 D 2 D n Availability (a i ) S 1 ec 11 ec 12 ec 1n a 1 S 2 ec 21 ec 22 ec 2n a 2 S m ec m1 ec m2 ec mn a m Demand (b j ) b 1 b 2 b n to the j th destination, but sure about the supply and demand of the product, can be formulated as follows: m Minimize X i¼1 Xn ec ij x ij Subject to X n x ij a i ; i ¼ 1; 2; ; m; X m j¼1 j¼1 i¼1 x ij b j ; j ¼ 1; 2; ; n; x ij : ðp1þ where, a i : total availability at the i th source b j : total demand at the j th destination x ij : total amount of product to be supplied from i th source to j th destination ec ij : unit fuzzy cost of transportation of unit product from i th source to j th destination The formulation (P1) can be transformed into the tabular form as given under as in Table 1. Proposed method The steps of the proposed methods are given below: Step 1: The given fuzzy transportation problem is transformed into the tabular form. Step 2: Reduce the obtained fuzzy matrix in row reduce form by subtracting the entry which has the least rank from each entries of its corresponding row of fuzzy transportation problem. Step 3: Calculate ep ij for each cell which has zero rank of fuzzy cost. Where ep ij ¼ Sum of fuzzy costs of nearest adjacent sides of zero rank entry No: of added fuzzy cost having non zero rank : Table 2 Balanced fuzzy cost transportation problem D 1 D 2 D 3 D 4 Availability (a i ) S 1 (1,3,4,8) (2,4,6,12) (3,5,8,16) (3,5,8,16) 4 S 2 (2,4,6,12) (3,5,8,16) (2,4,6,12) (2,5,7,14) 6 S 3 (2,3,5,1) (2,5,7,14) (2,4,6,12) (3,5,8,16) 5 Demand (b j )
5 A new approach for solving cost minimization balanced 343 Table 3 Row reduced fuzzy cost transportation problem D 1 D 2 D 3 D 4 Availability (a i ) S 1 ( 7,-1,1,7) ( 6,,3,11) ( 5,1,5,15) ( 5,1,5,15) 4 S 2 ( 1, 2,2,1) ( 9, 1,4,14) ( 1, 2,2,1) ( 1, 1,3,12) 6 S 3 ( 8, 2,2,8) ( 8,,4,12) ( 8, 1,3,1) ( 7,,5,14) 5 Demand (b j ) Step 4: Allocate the cell (i, j) for which rank of ep ij i.e. R ep ij is maximum. If two or more ep ij s have same values then allocate that cell which has least cost among all cells for which ep ij s are equal. Again, if costs of these cells are equal then randomly allocate that cell for which a i b j. Step 5: Check whether the resultant matrix is in row reduces form else repeats Step 2. Step 6: Repeat Step 3 to Step 5 until all the allocations are made and we get the fuzzy optimal solution of fuzzy transportation problem. Numerical example Consider a balanced fuzzy transportation problem having three sources and four destinations in which only unit cost of product taken as a fuzzy numbers as given in Table 2. Using Step 2, Table 2 is converted into Table 3. The rank of fuzzy cost of (1,1), (2,1), (2,3) and (3,1) is zero. Therefore the value of ep ij s corresponding to these cells are ep 11 ¼ ð 16; 2; 5; 21Þ;eP 21 ¼ ð 24; 4; 7; 29Þ;eP 23 ¼ ¼ ð 8; 1=2; 15=4; 51=4Þ;eP 31 ¼ ð 18; 2; 6; 22Þ ð 32; 2; 15; 51 Þ 4 Table 4 Resultant fuzzy transportation problem after 1st allocation D 2 D 3 D 4 Availability (a i ) S 1 ( 17,-3,3,17) ( 16, 2,5,21) ( 16, 2,5,21) 2 S 2 ( 9, 1,4,14) ( 1, 2,2,1) ( 1, 1,3,12) 6 S 3 ( 2, 5,3,18) ( 18, 4,4,18) ( 17, 3,6,22) 5 Demand (b j ) 3 5 5
6 344 S. Singh, G. Gupta Table 5 Resultant fuzzy transportation problem after 2nd allocation D 2 D 3 D 4 Availability (a i ) S 2 ( 9, 1,4,14) ( 1, 2,2,1) ( 1, 1,3,12) 6 S 3 ( 2, 5,3,18) ( 18, 4,4,18) ( 17, 3,6,22) 5 Demand (b j ) Since, rank of all ep ij s has equal values i.e. 2, therefore by using Step 4, (1,1) cell is allocated and by using Step 5, resultant matrix is shown in Table 4. Here, the rank of fuzzy cost of (1,2), (2,2) and (3,2) is zero and the rank of ep 12 has maximum value i.e. 2, therefore by using Step 4, (1,2) cell is allocated and by using Step 5, resultant matrix is shown in Table 5. Now, the rank of fuzzy cost of (2,3) and (3,3) is zero and the rank of ep 23 and ep 33 has equal value i.e. 1.5, therefore by using Step 4, (2,3) cell is allocated and by using Step 5, resultant matrix is shown in Table 6. Now, the rank of fuzzy cost of (2,4) and (3,2) is zero and the rank of ep 24 and ep 32 has equal value i.e. 1, therefore by using Step 4, (3,2) cell is allocated. The optimal solution of problem, given in Table 1, is given under as, x 11 ¼ 2; x 12 ¼ 2; x 23 ¼ 5; x 24 ¼ 1; x 32 ¼ 1; x 34 ¼ 4 The fuzzy optimal value is 2 ð1; 3; 4; 8Þþ2 ð2; 4; 6; 12Þþ5 ð2; 4; 6; 12Þþ1 ð2; 5; 7; 14Þþ1 ð2; 5; 7; 14Þþ4 ð3; 5; 8; 16Þ ¼ ð32; 64; 98; 196Þ: Conclusion Thus the proposed method provides an optimal value of the objective function for the balanced fuzzy transportation problem. This method straight forward and very easy to understand. The proposed method requires less computational work as compared to the existing methods. Table 6 Resultant fuzzy transportation problem after 3rd allocation D 2 D 4 Availability (a i ) S 2 ( 21, 4,5,24) ( 22, 4,4,22) 1 S 3 ( 38, 8,8,38) ( 35, 6,11,42) 5 Demand (b j ) 1 5
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