Decision Making Theories and Methods Based on Interval-Valued Intuitionistic Fuzzy Sets by Shuping Wan & Jiuying Dong

Author:Shuping Wan & Jiuying Dong
Language: eng
Format: epub
ISBN: 9789811515217
Publisher: Springer Singapore

(5.37)

There are several methods to solve a multiple objective programming. Here, we apply membership function-based linear sum approach to solving Eq. (5.37). Suppose that and are the maximum and minimum values of the single objective function ignoring the other objectives in Eq. (5.37). The corresponding linear membership function of the objective function is defined as follows:

(5.38)

Then, Eq. (5.37) can be solved by the following linear programming model:

(5.39)

where is the weight of objective such that and . If the importance of different objectives is equal, the objective weights can take equal value, i.e., . It is easily seen that Eq. (5.39) is a linear programming model. Using the simplex method to solve Eq. (5.39), the attribute weight vector can be derived.

Remark 5.3

We establish a multi-objective IVIF programming model that maximizes the collective overall IVIF ratings of all alternatives to determine optimal weights of attributes. Moreover, this model is transformed into the equivalent linear programming model by using membership function-based linear sum approach. However, Jin et al. [8] constructed a programming model that minimizes the entropy values of alternatives, Zhang and Xu [41] presented a multi-choice linear programming model that maximizing the closeness indices. Wan et al. [14] established a multi-objective interval programming model based on the comprehensive values of alternatives, Based on the above analysis, the major difference is that the proposed model is a multi-objective IVIF programming model in which the objective is an IVIFN, which is well-suited to express the hesitancy degree inherent in DMs’ judgments, while the objectives in methods [8, 14, 41] are crisp values or interval numbers which cannot reflect the hesitancy degrees inherent in DMs’ judgments. Thus, the proposed model in this chapter may lead to much less information loss than that of methods [8, 14, 41] in the determination of the attribute weights.

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