A soft computing method for multi-criteria decision making with dependence and feedback

Abstract

In this paper, the decision making problems with the dependence and the feedback effects are considered. Although the analytic network/hierarchy process (ANP/AHP) has been proposed to deal with the problems above, several problems make the method impractical. In this paper, we proposed the fuzzy decision maps (FDM), which incorporates the eigenvalue method, the fuzzy cognitive maps (FCM), and the weighting equation, to overcome the problem of preferential independent and the shortcomings of the ANP. In addition, two numerical examples are used to demonstrate the proposed method. On the basis of the numerical results, we can conclude that the proposed method can soundly deal with the decision making problems with the dependence and the feedback effects.

Introduction

Multi-criteria decision making (MCDM) involves determining the optimal alternative among multiple, conflicting, and interactive criteria [1]. Many methods, which are based on multiple attribute utility theory (MAUT), have been proposed (e.g. the weighted sum and the weighted product methods) to deal with the MCDM problems. The concept of MAUT is to aggregate all criteria to a specific unidimension which is called utility function to evaluate alternatives. Although many papers have been proposed to discuss the aggregation operator of MAUT [2], the main problem of MAUT is the assumption of preferential independence [3], [4].

On the assumption of preferential independence, it can be seen that the dependence and the feedback effects cannot be considered. However, the real-life situation usually emerges the dependence and the feedback effects simultaneously while making decisions. The analytic network process (ANP) was proposed in [5], [6] to overcome the problem of dependence and feedback among criteria or alternatives. The ANP is the general form of the analytic hierarchy process (AHP) [7], which has been used for multi-criteria decision making (MCDM), to release the restriction of hierarchical structure, and has been applied to project selection [8], [9], product planning, strategic decision [10], [11], and optimal scheduling [12].

The advantages of the ANP are that it is not only appropriate for both quantitative and qualitative data types, but it also can overcome the problem of interdependence and feedback among criteria. Although the ANP have been widely used in various applications, two main problems should be highlighted as follows. The first is the problem of comparison. In the ANP, the decision maker is asked to answer the question like “How much importance does a criterion have compared to another criterion with respect to our interests or preferences?” However, sometimes the questions are hard even for the expert to answer the question above due to some questions are anti-intuitive. We will highlight the problem again in Section 3. Furthermore, the key for the ANP is to determine the relationship structure among features in advance [9]. The different structure results in the different priorities. However, it is usually hard for the decision maker to give the true relationship structure by considering many criteria.

In this paper, we proposed the fuzzy decision maps (FDM), which incorporates the eigenvalue method, the fuzzy cognitive maps (FCM) [13], [14], and the weighting equation, to overcome the problem of preferential independent and the shortcomings of the ANP. Not only dependence effects but also feedback effects can be considered to derive the best alternative. Besides, two numerical examples are used to demonstrate the proposed method and compared with the ANP. On the basis of the numerical results, we can conclude that FDM can provide another method to deal with the structural MCDM problem.

The rest of this paper is organized as follows. In Section 2, we describe the contents of the analytic network process. Fuzzy decision maps are proposed in Section 3. Two numerical examples, which are used here to demonstrate the proposed method, are in Section 4. Discussions are presented in Section 5 and conclusions are in the last section.