Fuzzy multi objective transportation problem evolutionary algorithm approach

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1 Journal of Physics: Conference Series PPER OPEN CCESS Fuzzy multi objective transportation problem evolutionary algorithm approach To cite this article: T Karthy and K Ganesan 08 J. Phys.: Conf. Ser View the article online for updates and enhancements. This content was downloaded from IP address on /07/08 at 08:48

2 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 Fuzzy multi objective transportation problem evolutionary algorithm approach T Karthy and K Ganesan ssistant Professor, Department of Mathematics, SRM Institute of science and technology, Kattanulathur, Tamilnadu, India Professor, Department of Mathematics, SRM Institute of science and technology, Kattanulathur, Tamilnadu, India E.mail : arthy.t@tr.srmuniv.ac.in bstract: This paper deals with fuzzy multi objective transportation problem. n fuzzy optimal compromise solution is obtained by using Fuzzy Genetic lgorithm. 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 tas of finding one or more fuzzy optimal solution is nown 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 [] in 965. Zimmerman [9] applied the fuzzy programming techniques to solve multi objective linear programming problems. C. Vayalashmi [3] solved the bi objective transportation problem using genetic algorithm and represented it by bipartite graphs. Waiel F. bd 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 maer s preference. The rest of the paper is organized as follows: In section, 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.. PRELIMINRIES Content from this wor may be used under the terms of the Creative Commons ttribution 3.0 licence. ny further distribution of this wor must maintain attribution to the authors) and the title of the wor, journal citation and DOI. Published under licence by Ltd

3 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 Definition.: fuzzy set defined on the set of real numbers R is said to be a fuzzy number, if μ : R 0, has the following characteristics: its membership function [ ] i) μ is convex. ii) μ is normal. iii) is upper semi continuous iv) sup ) is bounded in R. Definition.: fuzzy number is a triangular fuzzy number denoted by = a, a, a ) a a a are real numbers and its membership function μ x ) where,, 3 μ x a, a x a a a a3 x x =, a x a a3 a 0,otherwise ) Definition.3: triangular fuzzy number = a, a, a ) ), )) of function of ), ) = a a 3 3 is given below and FR) can also be represented as a pair ar a r for 0 r which satisfies the following requirements: i) ar ) is a bounded monotonic increasing left continuous function. ii) a r) is a bounded monotonic decreasing left continuous function. iii) ar ) a r),0 r.. Raning of Triangular Fuzzy Numbers = a, a, a F R), the raning function R : F R) R by graded mean is defined by For every ) 3 3 a+ a + a3 R ) =.For any two triangular fuzzy number = a, a, a3) and B = b, b, b3) 4 in FR). We have the following comparison : ) i Bif andonlyif R R B. ii) Bif andonlyif R R B. iii) Bif andonlyif R = R B. iv) B 0 if andonlyif R R B = 0.

4 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 and. rithmetic Operations: In particular for any two fuzzy numbers = a, a, a ) B = b, b, b3), we define: ) i ddition ii) Subtraction : + B = a+ b, a + b, a3+ b3) : B = a b, a b, a3 b3) iii) Multiplication : * B = min ab, ab 3, a3b, a3b3), ab,max ab, ab 3, a3b, a3b3)) iv) Division : / B = min a / b, a / b, a / b, a / b ), a / b,max a / b, a / b, a / b, a / b )) FUZZY MULTI OBJECTIVE TRNSPORTTION PROBLEM 3. Mathematical formulation of Fuzzy Transportation Problem Consider a fuzzy multi objective transportation problem with m sources and n destinations. Let a i a i 0) be the fuzzy availability at source i and b j b j 0) be the fuzzy requirement at destination j. Let c be the fuzzy unit transportation cost from source i to destination j. Let x denote the number of fuzzy units to be transported from source i to destination j. Now the problem is to determine a feasible way of transporting which minimizes the total fuzzy transportation cost. Minimize z x) = c x i= j= subject to x = a, i =,,... m m i= m x = b, j =,,...,n j and x 0,for all i, j. n j= i n { } on both z x) and where ) ), ),..., z x = z x z x z x) is a vector of fuzzy objective functions and the superscript c are used to indicate the number of fuzzy objective functions. Without loss of a b i j c i j and a b. generality, it is assumed in the paper that 0, 0,,, 0,, i j i j i j Definition 3.: If the fuzzy objective functions are said to be conflicting, then there exists a fuzzy pareto optimal solution. Definition 3.: 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. Definition 3.3: Fuzzy Pareto efficiency or fuzzy pareto optimality is a state of allocation of resources in which it is impossible to mae anyone individual better off without maing atleast one individual worse off. 3

5 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 Definition 3.4: If the fuzzy compromise solution satisfies the decision maer's preferences, then the solution is called the fuzzy preferred compromise solution. 4. Fuzzy Genetic lgorithm Fuzzy Genetic lgorithm consists of mainly three steps:. Fuzzy Selection. 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 FVM to obtain the initial fuzzy basic feasible solution. Fuzzy Crossover: There are different types of fuzzy crossover namely. 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.. 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 and Fuzzy child. 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 G can come to better fuzzy solution by using mutation. 5. Numerical Example Consider the balanced fuzzy multi objective transportation problem [7] 0,,),,3) = 0,,) 8,9,0) 7,8,9) 8,9,0) 6,7,8) 6,7,8),3,4) 3,4,5) 3,4,5) 5,6,7) =,3,4) 3,4,5) 4,5,6) 7,8,9) 5,6,7),,3),3,4) 3,4,5) 8,9,0) 9,0,) 4,5,6) 0,,) Fuzzy supplies: a = 0,3,5), a = 4,6,9), a 3 = 4,6,7) Fuzzy demands: b =,4,5), b =0,,), b 3 =,5,7), b 4 =4,5,7) Initial llocation 4

6 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 0,,3) 0,,) = 0,,) 0,0,0),,0) 0,0,0) 0,0,0) 4,5,7) ;,5,7) 0,0,0) =,4,5) 0,,) 0,3,5) 0,0,0),,) 0,0,0) 0,,0) 4,5,7) Fuzzy Parent : Z = 8, 53, 04), Z = 44, 9, 67) Fuzzy Parent : Z = 4, 7, 7), Z = 4, 56, 0) Fuzzy Single Point Crossover 0,,3) 0,,) = 0,,) 0,0,0),4,7) 0,0,0) ; 0,,0) 4,5,7) = 0,3,5) 0,,),,0) 0,0,0) 0,0,0) 0,3,5) 0,0,0) 4,,),5,7) 0,,) Fuzzy child : Z = 4,5, 93), Z = 44,9,53) Fuzzy child : Z =, 69, 43), Z = 6, 78, 73) Fuzzy Mutation 0,,3) 0,,) =,,) 0,0,0) 0,0,0),4,7) ;,5,7),,0) 0,0,0) 0,,) = 0,3,5) 0,0,0),,0) 0,0,0) 0,0,0) 0,,3) 0,0,0) 4,3,4),5,7) 0,0,0) Fuzzy child 3: Z =, 48, 86), Z = 30,84,55) Fuzzy child 4: Z = 8, 5, 0), Z = 44, 85, 55) fter Fuzzy Mutation, we have the better fuzzy optimal solution: Z =, 48, 86), Z = 0,,3) 0,,) 30,84,55) whose allocation is,,) 0,0,0) 0,0,0),4,7),5,7),,0) Conclusion: For the Bi objective fuzzy transportation problem solved by fuzzy Genetic lgorithm, the fuzzy compromised solution obtained is Z =, 48, 86), Z = 30,84,55) References: []. bdullah Kona, David W Coit and lice E Smith 006 Multi objective optimization using genetic algorithms: tutorial, Reliability Engineering and System Safety 9 pp

7 National Conference on Mathematical Techniques and its pplications NCMT 8) IOP Conf. Series: Journal of Physics: Conf. Series ) 0004 doi :0.088/ /000//0004 []. R Bellman and L Zadeh 970 Decision maing in a fuzzy environment, Management Sci. 7B4) pp 4-64 [3]. K Bharathi and C Vayalashmi 06 Optimization of Multi Objective Transportation Problem using Evolutionary lgorithms [4]. M P Biswal 99 Fuzzy programming technique to solve multi objective geometric programming problems, Fuzzy sets and Systems, 5 pp 67-7 [5]. D E Goldberg 989 Genetic lgorithms in search, Optimization and Machine Learning ddison Wesley Publishing company [6]. Kalyanmoy Deb 00 Multi objective optimization using Evolutionary lgorithm, John wiley & Sons, New Yor, US [7]. T Leelavathy and K Ganesan 06 Solution pproach to Multi Objective Fuzzy Transportation Problem, Global Journal of Pure and pplied Mathematics,, number, pp [8]. Waiel F.bd El- Wahed 00 multi objective transportation problem under fuzziness, Fuzzy sets and Systems 7 pp 7-33 [9]. H J Zimmermann 978 Fuzzy Programming and linear programming with several objective functions, Fuzzy Sets and Systems pp