Abstract
In a transportation problem generally a single criterion of minimizing the total transportation cost is considered but in certain practical situations two or more objectives are relevant. For example, the objectives may be minimization of total cost, consumption of certain scarce resources such as energy, total deterioration of goods during transportation etc. Clearly, this problem can be solved using any of the multiobjective linear programming techniques, but the computational efforts needed would be prohibitive in many cases. In this paper, The Bi-objective transportation problem, where only objectives are considered as fuzzy. We apply the fuzzy programming technique with hyperbolic membership function to solve a biobjective transportation problem as vector minimum problem.
Highlights
The transportation problem (TP) can be formulated as a linear programming problem, where the constraints have a special structure [1]
In most real world cases transportation problems can be formulated as multi-objective problems [2, 3]
The problem was solved by the Linear Interactive and Discrete Optimization (LINDO) Software The optimal solution is presented as follows: Xmn+1 = 1.351464, X11 = 3.785216, X12 = 3.0, X13 = 1.214784, X21 = 7.214784, X23 = 11.785216, X33 = 1.0, X34 = 16.0, and λ = 0.937 Transportation cost Z1 = 160. 8591, Deterioration of goods Z2 = 193
Summary
The transportation problem (TP) can be formulated as a linear programming problem, where the constraints have a special structure [1]. Leberling [5] used a special- type nonlinear (hyperbolic) membership function for the vector maximum linear programming problem. Biswal and Biswas [7] used the fuzzy programming technique with some non-linear (hyperbolic and exponential) membership functions to solve a multi-objective transportation problem. A penalty cij and dij are associated with transportation of a unit of the product from sources i to destination j. A variable Xij represents the unknown quantity to be transported from origin Oi to destination Dj. In the real would, transportation problems are not all-single objective type. Let X1* = {x1ij}, X2* = {x2ij}, be the optimum solutions for Z1, Z2 different single objective transportation problem. The X1*, X2* are the individual optimal solutions and each of these are used to determine the values of other individual objectives, the pay off matrix is developed as:. Has the following properties: 1. It is strictly decreasing function
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