Fuzzy multi objective transportation problem – evolutionary algorithm approach

This paper deals with fuzzy multi objective transportation problem. An fuzzy optimal compromise solution is obtained by using Fuzzy Genetic Algorithm. A numerical example is provided to illustrate the methodology.


Introduction:
Fuzzy Transportation problem is a fuzzy optimization problem deals with transporting commodities from various sources to various destinations in such a way so that the total fuzzy transportation cost is minimum. When a fuzzy transportation problem involves more than one objective function the task of finding one or more fuzzy optimal solution is known as fuzzy multi objective transportation problem. For multiple conflicting fuzzy objectives, there cannot be a single fuzzy optimum solution which simultaneously optimizes all the fuzzy objectives. The resulting outcome is a set of fuzzy optimal solutions with varying degree of objective values. Hence it is better to compute the fuzzy compromise solution between two or more conflicting fuzzy objectives. In this article, we propose a fuzzy genetic algorithm approach for the solution of fuzzy multi objective transportation problems.
In real life situations, supply, demand and unit transportation cost are uncertain. Hence idea of fuzzy sets was introduced by Zadeh [2] in 1965. Zimmerman [9] applied the fuzzy programming techniques to solve multi objective linear programming problems. C. Vijayalakshmi [3] solved the bi objective transportation problem using genetic algorithm and represented it by bipartite graphs. Waiel F. Abd El-Wahed [8] applied fuzzy programming approach to determine the optimal compromise solution of a crisp multi objective transportation problem. For the balanced fuzzy multi objective transportation problem [7] T. leelavathy and et.al applied weighted sum of the objectives method and obtained the compromise solution by decision maker's preference.
The rest of the paper is organized as follows: In section 2, we have discussed the basic concepts of triangular fuzzy number and their arithmetic operations. In section 3, we introduce the fuzzy multi objective transportation problem with cost coefficients, supplies and demands as triangular fuzzy numbers. In section 4, we define the basic concepts of fuzzy genetic algorithm. In section 5, a numerical example is provided to illustrate the efficiency of the proposed methodology." 0,1 A R μ → has the following characteristics:

Ranking of Triangular Fuzzy Numbers
For every ( ) For any two triangular fuzzy number We have the following comparison :

i A B if and only if R A R B ii A B if and only if R A R B iii A B if and only if R A R B iv A B if and only if R A R B
ii Subtraction A B a b a b a b iii Multiplication A B  a b a b a b a b a b  a b a b a b a b iv

Definition 3.1:
If the fuzzy objective functions are said to be conflicting, then there exists a fuzzy pareto optimal solution.

Definition 3.2:
A fuzzy solution is called fuzzy non dominated, fuzzy pareto optimal, fuzzy pareto efficient or non inferior, if none of the fuzzy objective functions can be improved in value without degrading some of the other fuzzy objective values.

Fuzzy Genetic Algorithm
Fuzzy Genetic Algorithm consists of mainly three steps:

Fuzzy Selection 2. Fuzzy Crossover 3. Fuzzy Mutation
Fuzzy Selection: Of the three methods, Fuzzy North West corner rule, fuzzy least cost method, fuzzy Vogel's approximation method we select FVAM to obtain the initial fuzzy basic feasible solution." Fuzzy Crossover: There are different types of fuzzy crossover namely 1. "Fuzzy single point Crossover -One fuzzy crossover point is selected, fuzzy allocation from the beginning to the fuzzy crossover point is copied from the first fuzzy parent solution, the rest is copied from the other fuzzy parent solution." 2. "Fuzzy Two point Crossover -Two fuzzy crossover points are selected, fuzzy allocation from the beginning of the first fuzzy crossover point is copied from the first fuzzy parent, the part from the first to the second fuzzy crossover point is copied from the other fuzzy parent and the rest is copied from the first fuzzy parent again." 3." Fuzzy Uniform Crossover -Fuzzy allocations are randomly copied from the fuzzy first or from the fuzzy second parent." Initial basic fuzzy feasible solutions are considered as the fuzzy parent solutions. By using Fuzzy Crossover operator we generate a second generation population of Fuzzy solutions from those Fuzzy parent solutions and we obtain Fuzzy child 1and Fuzzy child 2.
In the problem illustrated, Fuzzy single point Crossover is used.
Fuzzy Mutation: Fuzzy Mutation alters one or more gene values in a chromosome from its initial state. In mutation, the fuzzy solution may change entirely from the previous fuzzy solution. Hence fuzzy GA can come to better fuzzy solution by using mutation.