Satisficing solutions of multi-objective fuzzy optimization problems using genetic algorithm
Graphical abstract
Highlights
► A multi-objective optimization problem with max-product fuzzy relation equations as constraints is presented. ► Pareto optimal solutions are obtained for solving such problems observing the role of non-convexity of the feasible domain of decision problem. ► Solutions are kept within feasible region during mutation and crossover operations. ► Test problems are developed for evaluating performance and determining satisficing decisions. ► In case of two objectives, weighting method is also applied to find the locus of optimal solutions.
Section snippets
Background and motivation
Max–min fuzzy relation equations were first studied by Sanchez [21] in 1976. Later on, Pedrycz [19] studied the max-product fuzzy relation equations. Different types of fuzzy relation equations have been studied since then [2], [4], [8], [13], [17], [20], [23]. Linear optimization problem with fuzzy relation equations as constraints was first considered by Fang and Li [5] with max–min composition and by Loetamonphong and Fang [14] with max-product composition. For both the compositions,
The problem
Consider the following system of fuzzy relation equations:where A = [aij]m×n, 0 ≤ aij ≤ 1, is a m × n dimensional fuzzy matrix, b = [b1, b2, …, bn], 0 ≤ bj ≤ 1, is a n-dimensional vector, and “∘ ” denotes the max-product composition of x and A. Let I = {1, 2, …, m} and J = {1, 2, …, n} be the index sets. Given fuzzy relation matrix A, output vector b, and relation (1), input vectors x = [x1, x2, …, xm] are decidable.
Let X(A, b) = {x ∈ [0, 1]m|x ∘ A = b} be the feasible domain of fuzzy relation Eq. (1). For any x1, x2 ∈ X,
Analysis and formation of genetic algorithm
A genetic algorithm (or GA) is a search technique used in computing to find true or approximate solutions to optimization and search problems. The scalar concept of “optimality” does not apply directly in the multi-objective setting. A useful replacement is the notion of Pareto optimality. In our case, the solution space is, in general, a non-convex set and the objective functions considered are not necessarily linear. Therefore, the only one consideration in deriving the satisficing solutions
Randomly generating max-product fuzzy relation equations
To obtain the measurement of performance of proposed genetic algorithm, test problems are considered by taking multi-objective objective functions and constructing a feasible problem by randomly generating a fuzzy matrix A and constructing vector b from A according to some criteria. The procedure of generating feasible fuzzy relation equations is as follows:1. Generate a m × n matrix A whose elements are random numbers from [0, 1]. 2. Generate vector b such that its jth element is a random number
Summary of the proposed genetic algorithm
The summary of the proposed method can be summarized as under:
- 1.
Initialize maximum number of generations, Gen = 1000.
- 2.
Set the current generation number c = 1.
- 3.
Generate initial population using Algorithm 1.
- 4.
Perform selection procedure and establish the efficient set ES by putting all efficient solutions of the initial population into it.
- 5.
Perform mutation procedure using Algorithm 3.
- 6.
Perform crossover operation using Algorithm 4.
- 7.
Again apply selection/reproduction procedure and update the efficient set ES.
- 8.
Linear weighting method
In case of multi-objective optimization, most practical problems require the simultaneous multiple objectives. In applications of optimization techniques, one way of obtaining the solution to such problems is by combining the objectives into a single one according to some utility function. To solve the multi objective problems, the linear weighting method is used. The linear weighting method [3], combines all the objectives into a single scalar parameterized objective function by using
Illustrations
In this section, implementation of the proposed genetic algorithm for multiple linear and nonlinear objective functions is discussed. We consider multi-objective linear and nonlinear optimization problems and randomly generated fuzzy relation equations with max-product composition to investigate the nature of the Pareto optimal solution set. Some of the test problems are from well known sources [15].
Conclusion
This paper discusses a multi-objective optimization problem with max-product fuzzy relation equations. The problem is simplified by finding out the fixed components for all the solutions. Genetic algorithm is used to derive the satisficing decisions of the problem by finding Pareto optimal solutions. The feasibility of solutions is always maintained during the crossover and mutation operations. The efficient set is constantly updated and finally hill climbing method is employed to further
Acknowledgements
Authors are thankful to CSIR for supporting this work under CSIR grant no. 25(0167)/08/EMR-II. Authors are grateful to all the reviewers for their valuable suggestions and comments.
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