Discussion on Inexact Optimal Solution under Fuzzy Environment
Abstract
The purpose of this paper is to explain that a convex combination of several partial solutions using a new criterion which did not solve the fuzzy problem. The main feature of this paper is twofold. First, we prepare a simple optimization problem to explain the previously proposed approach that considered criteria separately. Consequently, we conclude, previous results are partial solutions. Secondly, we study the same optimization problem as the one that appears in a previously published paper. After his new criterion is added, we solve the new fuzzy optimization problem to demonstrate that the previous solution is not the optimal one. Hence, the previously proposed approach is questionable and then his assertion of the meaninglessness of the exact optimal solution for the fuzzy problem cannot be treated as a valid statement. At last, we cite a paper that had referred to the questionable approach which had been improved by another published article to support our argument.
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Introduction
Zadeh [1] introduced the concept of fuzzy set theory, and then Bellman and Zadeh [2] proposed the decision-making problem in a fuzzy environment. Zimmermann [3] applied the fuzzy set theory with proper membership functions to solve linear programming problems with several objective functions. Fuzzy set theory not only provided a mathematical way of representing imprecision or vagueness but also fuzzy linear programming has been applied to many practical areas such as production planning, resources allocation, transportation problems, and so on. Based on Zimmermann’s method, the fuzzy objective functions, and fuzzy constraints and transferred to crisps ones by membership functions with the max-min operator. Hence, a unique exact optimal solution with the highest membership degree is derived. However, Wang [5] assumed that the exact optimal solution may not be desired by the decision-maker, owing to there being another criterion that is preferred by the decision-maker. He first found a family of inexact solutions, with an acceptable membership degree, that is obtained by a genetic algorithm with mutation along the weighted gradient direction. These solutions developed a convex set in the neighborhood of the exact optimal solution. Next, under the new criterion, he considered human-computer interaction to derive his fuzzy optimal solution. There are ten papers (e.g., Tang and Wang [6]; He et al. [8]; Ma et al. [9]; Tang et al.[10]; Van Hop [11]; Chiu et al. [13]; Lu et al. [14]; Baykasoglu and Gocken [15]) which have referred to Wang [5] in their references. However, none of them discovered the questionable results of Wang [5] that will be explained and revised in this paper. In this paper, we will carefully examine Wang’s approach. After we add the new criterion preferred by the decisionmaker into the fuzzy optimization problem, the result is that Wang’s method will derive an unwarranted solution. We will demonstrate that his approach has a severe logic fault and that his assertion of “exact optimal solution in the fuzzy environment is meaningless” should be treated as invalid.
Conclusion
This paper has carefully examined Wang’s [5] approach to solve a fuzzy optimization problem when a new criterion preferred by the decision-maker is added to the fuzzy optimization problem. The consequence is that Wang’s method derives an unwarranted solution. Furthermore, it demonstrates that Wang’s [5] approach has a severe logic fault, and did not solve the fuzzy problem under the new conditions. Moreover, we have shown how, when the new criterion is added to Wang’s example, one can derive the correct optimal solution. At last, we recall that Van Hop [12] referred to Wang [5] to develop his solution process to solve linear programming problems under fuzziness and randomness environment. Chou et al. [16] pointed out that there are questionable results in the solution process of Van Hop [12] and then Chou et al. [16] provided a revision for Van Hop [12].