Multi objective linear programming problem (MOLPP) is one of the popular

Transcription

1 CHAPTER 5 FUZZY MULTI OBJECTIVE LINEAR PROGRAMMING PROBLEM 5.1 INTRODUCTION Multi objective linear programming problem (MOLPP) is one of the popular methods to deal with complex and ill - structured decision making. The decision maker is no longer forced to restrict his consideration to one main aspect but can take into account different points of view. When formulating a MOLPP, various factors of the real world system should be reflected in the description of the objective functions and the constraints. Most of the real world decision making problems usually involves multiple, noncommensurables and conflicting objectives which should be considered simultaneously. For example, a transportation problem might require the minimization of total transportation cost and minimization of total transit time of the goods being shipped and a production problem may require the meeting of demand while minimizing the use of a particular resource and maximizing the profit. The consideration of many objectives accomplishes three major improvements in problem solving. 1. Multi objective programming promotes more appropriate roles for the participants in decision making processes. 2. A wider range of alternatives is usually identified when a multi-objective methodology is employed. 59

2 60 3. Decision maker s perception of a problem will be more realistic if many objectives are considered. Many researchers have developed various algorithms for solving MOLPP. The basic approach to solve MOLPP is to determine a solution that represents an acceptable trade-off or compromise between the objectives, or to determine such set of solution and allow the decision maker to choose among them. We cannot improve all the objective functions at the same time because improvement of one objective function may result in worsening of at least one of the other objective functions. Many business decisions can be modeled as multiple objective linear programming problems. When formulating a MOLPP, objective functions and constraints involve many parameters which are not known precisely. If the coefficients involved in the objective and constraint functions are imprecise in nature and are interpreted as fuzzy numbers, then the resulting problem is referred to as a fully fuzzy Multi objective linear programming problem (FFMOLPP). FFMOLPP has become a larger area of research interest. Multi-objective with imprecise parameters make the problem difficult to solve with the traditional approaches. Various kinds of FFMOLPP models have been proposed to deal with different decision making situations that involve fuzzy values in objective function parameters, constraints parameters, or goals. Buckley and Leonard [83] applied fuzzy Monto Carlo method to generate a approximate solution for FFMOLPP. Cengiz Kahraman and Ihsan Kaya [84] has discussed about mathematical modeling of FFMOLPP with an application. Recently Jayalakshmi and Pandian [86] proposed a new method in which the FFMOLPP is transformed into an equivalent FLPP problem to obtain the fuzzy

3 61 efficient solution. Ritika Chopra and Ratnash R [89] developed an interactive method for solving fuzzy multi objective problems.pattnaik [90] introduced a robusts ranking technique for defuzzifying the fuzzy parameters of FFMOLPP and made a sensitive analysis. Sophia Porchelvi et al [91] extended the concept for solving a Multi objective fuzzy variable linear programming problem using ranking functions. Hence,most of the existing methods for solving fuzzy multi objective linear programming problems are based on parametric programming problem,goal programming approach and by interactive approaches. This chapter is devoted for finding a fuzzy pareto optimal solution to fully fuzzy multi objective linear programming problem.we have used the fuzzy ranking method given in Equation (2.2) and the fuzzy arithmetic operations introduced by Ming Ma[76] mentioned in chapter-2. This chapter begins with an introduction that briefly describes about fuzzy multi objective linear programming problem. Section 5.2 introduces the fully fuzzy multi objective linear programming problem and related results. Section 5.3 provides a new algorithm to find fuzzy pareto optimal solution to the given FFMOLPP. Section 5.4 discuss a numerical example and a comparison study to illustrate the theory developed in this chapter. 5.2 MATHEMATICAL FORMULATION Let F (R) be set of all triangular fuzzy numbers. The mathematical formulation of fuzzy multi objective linear programming problem with triangular fuzzy numbers is

4 62 defined as follows: subject to max Z ( Z 1, Z 2,..., Z N ) T, where Z i = c ij x j, i = 1, 2,..., N ã ij x j b i for all i = 1, 2,..., m 0 ã ij x j b i for all i = m 0 + 1, m 0 + 2,..., m (5.1) and x j 0 for all j = 1, 2,..., n where ã ij, c ij, x j, b i F (R), i = 1, 2,..., m and j = 1, 2,..., n. If ã ij, c ij, x j and bi are represented by location index number, left fuzziness index function and right fuzziness index function respectively, then the above problem can be rewritten as follows: max Z ( Z 1, Z 2,..., Z N ) where Z 1 = (c 1j ) 0, (c 1j ), (c 1j ) (x j ) 0, (x j ), (x j ) Z 2 = (c 2j ) 0, (c 2j ), (c 2j ) (x j ) 0, (x j ), (x j ) (5.2). Z k = (c kj ) 0, (c kj ), (c kj ) (x j ) 0, (x j ), (x j ) (a ij ) 0, (a ij ), (a ij ) (x j ) 0, (x j ), (x j ) (b i ) 0, (b i ), (b i ) for all i = 1, 2,..., m 0 (a ij ) 0, (a ij ), (a ij ) (x j ) 0, (x j ), (x j ) (b i ) 0, (b i ), (b i ) for all i = m 0 + 1,..., m (x j ) 0, (x j ), (x j ) 0 for all j = 1, 2,..., n.

5 63 Practically, it is difficult to reach fuzzy optima for all objective function subject to the given constraints in problem Equation(5.1) and Equation(5.2). Definition 5.1. x X is said to be a fuzzy feasible solution of problem Equation(5.2) if it satisfy the given constraints. Definition 5.2. x is said to be a complete fuzzy optimal solution for Equation(5.2) if there exists x X such that Z k ( x ) Z k ( x), k = 1, 2,..., N for all x X. Since the objective functions conflict with each other, a complete fuzzy optimal solution that simultaneously maximize all the multiple fuzzy objective functions doesn t exist always. Thus instead of complete fuzzy optimal solution, a new solution concept called fuzzy pareto optimality is introduced for FFMOLPP. Definition 5.3. x is said to be a fuzzy pareto optimal solution for Equation(5.2) if there doesn t exist another x X such that Zk ( x) Z k ( x ) for all k, (k = 1, 2,..., N) and Z s ( x) Z s ( x ) for at least one s. Definition 5.4. [47] An fuzzy ideal solution of Equation(5.2) is defined as the vector, whose components are composed by maximum value of each fuzzy objective function under the given constraints, Z = [ Z 1, Z 2,..., Z k] = [max Z 1, max Z 2,..., max Z N ]. (5.3) Definition 5.5. A fuzzy negative ideal solution of Equation(5.2) is defined as the vector, whose components are composed by minimum value of each fuzzy objective function under the given constraints, Z = [ Z 1, Z 2,..., Z k ] = [min Z 1, min Z 2,..., min Z N ]. (5.4)

6 64 Definition 5.6. The membership of each fuzzy objective function s satisfaction degree is defined as follows: 1 if Zk (x) Z k μ k (x) = 1 Z k Z k (x) Z k Z k if Z k < Z k (x) Z k k = 1, 2,..., N. (5.5) 0 if Zk (x) Z k 5.3 A NEW ALGORITHM FOR FUZZY MULTI OBJECTIVE LINEAR PROGRAMMING PROBLEM An algorithm for fuzzy multi objective linear programming problem to obtain the fuzzy pareto optimal solution is presented below: Step 1: Represent each fuzzy data ã = (a 1, a 2, a 3 ) in terms of ã = (a 0, a, a ). Step 2: Find the fuzzy ideal and fuzzy negative ideal solution of each single fuzzy objective function subject to the given constraints using fuzzy version of simplex algorithm for the fully fuzzy linear programming problem proposed by Mohanaselvi and Ganesan [85]. Step 3: Define the membership of each fuzzy objective function s satisfaction degree using Equation (5.5).

7 65 Step 4: Now, solve the following model for obtaining the fuzzy pareto optimal solution of FMOLPP: max λ subject to λ μ k (x), k = 1, 2,..., N ã ij x j b i for all i = 1, 2,..., m 0 ã ij x j b i for all i = m 0 + 1, m 0 + 2,..., m (5.6) λ [0, 1] and x j 0 for all j = 1, 2,..., n. Theorem 5.1. If there exists only one optimal solution (λ, x ) for FFMOLPP Equation(5.6), then x is a fuzzy pareto optimal solution to Equation(5.2). Proof. Suppose x is not a fuzzy Pareto-optimal solution for problem Equation(5.2), then there exists a fuzzy Pareto-optimal solution x. Then, for every k, we have Z k ( x ) Z k ( x ) and Z i ( x ) Z i ( x ), i = 1, 2,..., N. That is for every k, we have μ k ( x ) μ k ( x ) and i [1, N], μ i ( x ) > μ i ( x ). So λ = λ(x ) = min{μ 1 (x ), μ 2 (x ),..., μ N (x )} min{μ 1 (x ), μ 2 (x ),..., μ N (x )} = λ(x ) λ = λ This shows that x is not the only optimal solution of Equation(5.6), which is a contradiction. This contradiction proves that x is a fuzzy Pareto-optimal solution to Equation(5.2).

8 NUMERICAL EXAMPLE Example 5.1. Consider an example discussed by Buckley et al [83] max Z 1 (4, 5, 6) x 1 + (2, 3, 4) x 2 max Z 2 (1, 2, 3) x 1 + (6, 8, 10) x 2 (5.7) such that (0, 1, 2) x 1 + (3, 4, 5) x 2 (95, 100, 105) (2, 3, 4) x 1 + (1, 2, 3) x 2 (140, 150, 160) (3, 5, 7) x 1 + (2, 3, 4) x 2 (180, 200, 220) (1, 2, 3) x 1 + (6, 8, 10) x 2 (70, 75, 80) x 1, x 2 0. Representing the triangular fuzzy numbers in terms of the left and the right index function, the above problem becomes, max Z 1 (5, 1 r, 1 r) x 1 + (3, 1 r, 1 r) x 2 (5.8) max Z 2 (2, 1 r, 1 r) x 1 + (8, 2 2r, 2 2r) x 2 such that (1, 1 r, 1 r) x 1 + (4, 1 r, 1 r) x 2 (100, 5 5r, 5 5r) (3, 1 r, 1 r) x 1 + (2, 1 r, 1 r) x 2 (150, 10 10r, 10 10r) (5, 1 r, 1 r) x 1 + (4, 1 r, 1 r) x 2 (200, 20 20r, 20 20r) (2, 1 r, 1 r) x 1 + (8, 1 r, 1 r) x 2 (75, 5 5r, 5 5r) x 1, x 2 0, r [0, 1].

9 67 Using fuzzy version of simplex algorithm proposed by Mohanaselvi and Ganesan [85], the fuzzy ideal and fuzzy negative ideal solution of each single fuzzy objective function subject to the given constraints is given by: Table 5.1: Fuzzy ideal and fuzzy negative ideal solution of FFMOLPP max min Z 1 (250, 20 20r, 20 20r) (200, 20 20r, 20 20r) Z 2 (200, 20 20r, 20 20r) (80, 20 20r, 20 20r) By defining the membership function of each objective function Z 1 and Z 2 s satisfaction degree and solving the following problem, such that max λ (5.9) (5, 1 r, 1 r) x 1 +(3, 1 r, 1 r) x 2 (50, 20 20r, 20 20r)λ (200, 20 20r, 20 20r) (2, 1 r, 1 r) x 1 +(8, 2 2r, 2 2r) x 2 (120, 20 20r, 20 20r)λ (80, 20 20r, 20 20r) (1, 1 r, 1 r) x 1 + (4, 1 r, 1 r) x 2 (100, 5 5r, 5 5r) (3, 1 r, 1 r) x 1 + (2, 1 r, 1 r) x 2 (150, 10 10r, 10 10r) (5, 1 r, 1 r) x 1 + (4, 1 r, 1 r) x 2 (200, 20 20r, 20 20r) (2, 1 r, 1 r) x 1 + (8, 1 r, 1 r) x 2 (75, 5 5r, 5 5r) λ [0, 1], r [0, 1] and x j 0.

10 68 Solving the above problem we have the following fuzzy Pareto-optimal solution: λ = , x 1 = (41.07, 20 20r, 20 20r), x 2 = (13.39, 20 20r, 20 20r) and the fuzzy objective function value will be max Z 1 = (245.5, 20 20r, 20 20r) and max Z 2 = (189.29, 20 20r, 20 20r). Table 5.2: Comparison of proposed method with Buckley s method max Z 1 max Z 2 Proposed Z1 = (245.5, 20 20r, 20 20r) Z2 = (189.29, 20 20r, 20 20r) method = (225.5, 245.5, 265.5) = (169.29, , ) Buckley s Method (183.43, , ) (197.7, , ) The largest interval of objective function Z 1 is in the interval [200, 250] and Z 2 is in the interval [80, 200]. By the proposed method, when the decision maker chooses r = 1, the fuzzy objective function Z 1 = and Z 2 = will lie within this interval. Also the fuzzy pareto optimal solution obtained by using the proposed method is having less spread which reduces the vagueness in the solution and helps the decision maker to make proper decision. Graphical representation of fuzzy pareto optimal solution to fully fuzzy multi objective linear programming problem is presented in Figure 5.1.

11 69 ~ Z (x) 1 max ~ Z (x) 1 Fuzzy Pareto Optimal Solution Feasible domain 0 max ~ Z (x) 2 ~ Z (x) 2 Figure 5.1: Graphical representation of fuzzy pareto optimal solution 5.5 CONCLUSION In this chapter, a new algorithm for solving fully fuzzy multi objective linear programming problem is proposed. Using the proposed algorithm the fuzzy pareto optimal solution for the FFMOLPP is obtained.a numerical example discussed by Buckley, J.J., et al.[83] is solved using the proposed method without converting the given problem to crisp equivalent problem and a comparitive study has been made. Further, the fuzzy pareto optimal solution obtained by the proposed algorithm is better than the existing methods since the spreads of the solution are less. Also the proposed method provides a set of solutions with various levels of satisfaction degree to the decision maker to make better decision depending upon the situation. Since the final decisions taken by the decision maker are always crisp, our methodology assists in the choice of these crisp decisions among the fuzzy solutions.