Fractional transportation problem with fuzzy parameters (original) (raw)
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T he fractional programming is a generalization of linear programming where the objective function is a ratio of two linear functions. Similarly, in fractional transportation problem the objective is to optimize the ratio of two cost functions or damage functions or demand functions. As the ratio of two functions is considered, the fractional programming models become more suitable for real life problems. Keeping in view the complexities associated with real life transportation problem like vagueness and uncertainty involved with the parameters. The implementation of fuzzy techniques can be very useful. Therefore, in this article a Fully Fuzzy Multi-objective Fractional Transportation Problem (FFMOFTP) is considered. All the coefficients of the parameters, demands and supplies are considered as fuzzy numbers. The purpose of using fuzzy numbers is to deal with the uncertainties and vagueness associated with the parameters. Two cases are considered, one with triangular and other with ...
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This paper deals with the transportation problem with additional impurity restrictions where costs are not deterministic numbers but imprecise ones. Here, the elements of the cost matrix are subnormal fuzzy intervals with strictly increasing linear membership functions. By the Max-Min criterion suggested by Bellman and Zadeh [7], the fuzzy transportation problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a linear fractional programming problem. This fractional programming problem is solved by the method given by Kanti Swarup [12].
A Solution Procedure to Solve Multi objective Fractional Transportation Problem
In decision making process if the objective function is ratio of two linear functions and objective function is to be optimized. For example one may be interested to know the ratio of total cost to total time required for transportation. This ratio is an objective function which is fractional objective function. When there are several such fractional objectives to be optimized simultaneously then the problem becomes multi objective fractional programming problem (MOFLPP). Initially Hungarian mathematician BelaMartos constructed such type of problem and named it as hyperbolic programming problem. Same problem in general referred as Linear Fractional Programming Problem. Fractional programming problem can be converted into linear programming problem (LPP) by using variable transformation given by Charnes and Cooper. Then it can be solved by Simplex Method for Linear Programming Problem.. In this paper we propose to solve multi objective fractional transportation problem. Initially will solve each of the transportation problem as single objective and then using Taylor series approach expand each of the problem about its optimal solution and ignoring second and higher order error terms each of the objective is converted into linear one. Then the problem reduces to MOLTPP. Evaluate each of the objectives at every optimal solution and obtain evaluation matrix. Define hyperbolic membership function using best and worst values of objective function with reference to evaluation matrix. These membership functions are fuzzy functions Compromise solution is obtained using weighted a.m. of hyperbolic membership functions and also weights quadratic mean of hyperbolic membership functions. Propose o solve problem at the end to explain the procedure.
Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment
Applied Mathematics, 2015
This paper is comprised of the modeling and optimization of a multi objective linear programming problem in fuzzy environment in which some goals are fractional and some are linear. Here, we present a new approach for its solution by using α-cut of fuzzy numbers. In this proposed method, we first define membership function for goals by introducing non-deviational variables for each of objective functions with effective use of α-cut intervals to deal with uncertain parameters being represented by fuzzy numbers. In the optimization process the under deviational variables are minimized for finding a most satisfactory solution. The developed method has also been implemented on a problem for illustration and comparison.