An extended TOPSIS for determining weights of decision makers with interval numbers

Abstract

In this paper, we develop a method for determining weights of decision makers under group decision environment, in which the each individual decision information is expressed by a matrix in interval numbers. We define the positive and negative ideal solutions of group decision, which are expressed by a matrix, respectively. The positive ideal solution is expressed by the average matrix of group decision and the negative ideal solution is maximum separation from positive ideal solution. The separation measures of each individual decision from the ideal solution and the relative closeness to the ideal solution are defined based on Euclidean distance. According to the relative closeness, we determine the weights of decision makers in accordance with the values of the relative closeness. Finally, we give an example for integrated assessment of air quality in Guangzhou during 16th Asian Olympic Games to illustrate in detail the calculation process of the developed approach.

Introduction

Multiple attribute decision making (MADM) occurs in a variety of actual situations, such as economic analysis, strategic planning, forecasting, medical diagnosis, venture capital and supply chain management. The increasing complexity of the socioeconomic environment makes it less and less possible for a single decision maker (DM) to consider all relevant aspects of a problem [1]. As a result, many decision making processes, in the real world, take place in group settings. Moving from single DM’s setting to group members’ setting would lead to a great deal of complexity of the analysis. For example, consider that these DMs usually come from different specialty fields, and thus each DM has his/her unique characteristics with regard to knowledge, skills, experience and personality, which implies that each DM usually has different influence in overall decision result, i.e., the weights of DMs are different. Therefore, how to determine the weights of DMs will be an interesting and important research topic.

At present, many methods have been proposed to determine the weights of DMs. French [2] proposed a method to determine the relative importance of the group’s members by using the influence relations, which may exist between the members. Theil [3] proposed a method based on the correlation concepts when the member’s inefficacy is measurable. Keeney and Kirkwood [4] and Keeney [5] suggested the use of the interpersonal comparison to determine the scales constant values in an additive and weighted social choice function. Bodily [6] and Mirkin [7] proposed two approaches which use the eigenvectors method to determine the relative importance of the group’s members. Brock [8] used a Nash bargaining based approach to estimate the weights of group members intrinsically. Ramanathan and Ganesh [9] proposed a simple and intuitively appealing eigenvector based method to intrinsically determine the weights of group members using their own subjective opinions. Martel and Ben Khélifa [10] proposed a method to determine the relative importance of group’s members by using individual outranking indexes. Van den Honert [11] used the REMBRANDT system (multiplicative AHP and associated SMART model) to quantify the decisional power vested in each member of a group, based on subjective assessments by the other group members. Jabeur and Martel [12] proposed a procedure which exploits the idea of Zeleny [13] to determine the relative importance coefficient of each member. By using the deviation measures between additive linguistic preference relations, Xu [14] gave some straightforward formulas to determine the weights of DMs.

Many of literatures mentioned above described the individual decision information by a multiplicative preference matrix. Until now there has been little investigation of the weights of DMs based on individual decision information, in which the attribute values are given as observations in nonnegative real numbers, and the DMs have their subjective preferences on alternatives.

By considering the fact that, in some cases, determining precisely the exact values of the attributes is difficult and that, as a result of this, their values are considered as intervals. Therefore, in this article, we shall discuss the weights of DMs based on technique for order performance by similarity to ideal solution (TOPSIS) [15] with interval numbers.

The rest of the paper is organized as follows: Section 2 reviews multiple attribute group decision making (MAGDM) and the TOPSIS technique, the basic idea and main contributions of the developed method in this paper are presented. The preliminaries, including comparing and ranking interval numbers, are given in Section 3. The developed approach and its algorithm to determine the weights of DMs are presented in Section 4. Section 5 makes two comparisons between the proposed method in this paper and the literature of other methods. In Section 6, we illustrate our proposed algorithmic method with an example. Conclusions appear in Section 7.