Optimal fuzzy multi-criteria expansion of competence sets using multi-objectives evolutionary algorithms

https://doi.org/10.1016/j.eswa.2005.07.033Get rights and content

Abstract

Competence set is widely used to plan the optimal expansion process of skills, abilities or strategies. However, the conventional method is concerned only with one criterion rather than multi-criteria problems. In addition, the crisp value cannot reflect the ambiguity and the uncertainty in practice. In this paper, we propose the fuzzy criteria competence set analysis. In order to obtain Pareto solutions, multi-objective evolutionary algorithm (MOEA) is employed here. A numerical example with two fuzzy criteria is also used to illustrate the proposed method.

Introduction

Making good decisions involve the successive accumulation of the particular skills, ideas, information and knowledge. In order to efficiently and effectively acquire these abilities, competence set analysis was proposed (Yu, 1990, Yu and Zhang, 1990). Using the methods of competence set analysis such as the minimum spanning tree (Yu & Zhang, 1992) or the mathematical programming (Shi & Yu, 1996), we can obtain the optimal path (e.g. the minimum cost or time) to acquire the required competence.

In conventional competence set analysis, one criterion such as cost or benefit function is used to select the optimal expansion process. However, in practice we usually determine the optimal expansion process according to multi-criteria (e.g. cost, time, efficient, benefit, and so on) simultaneously. Additionally, in order to reflect the ambiguity and uncertainty in practice, we should incorporate the concept of fuzzy sets into competence set analysis.

In order to deal with multi-criteria problems, many methods such as goal programming (Charnes and Cooper, 1957, Ijiri, 1965), min–max optimization (Osyczka, 1978, Rao, 1986) and the ε-constraint method (Osyczka, 1984, Hwang et al., 1980) have been proposed. Recently, MOEA has been widely used in various multi-objective problems such as scheduling (Murata, Ishibuchi, & Tanaka, 1996), engineering (Fonseca & Fleming, 1998) and finance (Mardle, Pascoe, & Tamiz, 2000). Compared to these conventional methods, multi-objective evolutionary algorithm (MOEA) seems more suitable to solve multi-objective problems because it searches a set of possible solutions simultaneously. Therefore, we can obtain a set of Pareto solutions rather than a special solution as in conventional methods in a run. These solutions are very important for the decision-maker to chose the optimal expansion process because some objectives are intangible and cannot be form using the conventional mathematical model. In addition, another reason which we use MOEA in this paper is that it can deal well with concave and discountinuous objective functions and Pareto frontiers.

On the other hand, the concept of fuzzy sets was proposed by Zadeh (Kim, Modkowitz, & Koksalan, 1965) to represent the uncertain situations or the subjective judgments. Using the membership function, we can measure the degrees of the uncertainty and deal with the fuzzy problems. Recently, the concept of fuzzy sets has been incorporate into the conventional statistical or mathematical programming methods to reflect the ambiguity and uncertainty in practice (Zimmermann, 1978, Tanaka and Lee, 1998, Kim et al., 1996).

In this paper, we propose the fuzzy multi-criteria competence set analysis. A numerical example is demonstrated to select the optimal fuzzy multi-criteria expansion process. Two criteria (cost and benefit function) with fuzzy numbers are used to reflect the ambiguity and uncertainty in practice. By employing MOEA, we can obtain Pareto solutions. On the basis of Pareto solutions, decision-makers can determine the finial optimal expansion process based on his preferences or subjective judgments.

The remainder of this paper is organized as follows. The expansion process of competence set analysis and the proposed method are discussed in Section 2. Multi-objective evolutionary algorithm is proposed in Section 3 to describe its ideas and procedures. A numerical example is used to demonstrate the proposed method in Section 4. Discussions are presented in the Section 5 and conclusions are in the last section.

Section snippets

Expansion process of competence set

The concept of competence set was proposed by Yu (1990) to resolve a particular decision problem by acquiring the necessity of ideas, information, skills, and knowledge. The contents of competence set analysis are to identify the true competence set, the decision-maker's competence set, and the efficient expansion path to make good decisions.

Among these issues, the method to optimally expand the existing competence set is especially highlighted. Several methods, such as the minimum spanning

Multi-objective evolutionary algorithm

Multi-objective evolutionary algorithm (MOEA) has been widely used since the 1990's to resolve the combinational problem in various areas such as scheduling (Murata, 1996), engineering (Fonseca & Fleming, 1998) and finance (Mardle et al., 2000). The concept of MOEA is based on the method of genetic algorithm (GA). GA was pioneered in 1975 by Holland, and its concept is to mimic the natural evolution of a population by allowing solutions to reproduce, create new solutions, and compete for

Numerical example

In this numerical example, we will demonstrate a fuzzy two-criterion (i.e. cost and benefit) expansion of competence sets. Let SK={x0}, T\SK={x1,x2,x3,x4,x5,x6,x7} and the fuzzy cost and the fuzzy benefit functions, which represent with interval values, are shown in Table 1, Table 2. Note that the symbol, M, denotes the infeasible route and will be treated as a minimum number in our fuzzy mathematical programming model. In addition, the membership of the cost and benefit functions are assumed

Discussions

Competence set analysis has been used for many applications, such as learning sequences for decision-makers (Hu, Chen, & Tzeng, 2002) and for consumer decision problems (Chen, 2001, Chen, 2002). However, these papers only consider the situation of using one criterion and the crisp function. In practice, decision-makers usually determine the optimal expansion process based on multi-criteria which may be conflicting with each other. Therefore, Pareto solutions should be derived for

Conclusions

In this paper, we extend the conventional competence set analysis to consider the situation of multi-criteria and fuzzy number. In order to obtain Pareto solutions efficiently, MOEA is employed here. A numerical example is used to demonstrate the procedures of the proposed method. On the basis of the results, we can conclude that the proposed method can provide a more flexible and diverse model.

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