Group decision making model and approach based on interval preference orderings

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Abstract

In group decision making under uncertainty, interval preference orderings as a type of simple uncertain preference structure, can be easily and conveniently used to express the experts’ evaluations over the considered alternatives. In this paper, we investigate group decision making problems with interval preference orderings on alternatives. We start by fusing all individual interval preference orderings given by the experts into the collective interval preference orderings through the uncertain additive weighted averaging operator. Then we establish a nonlinear programming model by minimizing the divergences between the individual uncertain preferences and the group’s opinions, from which we derive an exact formula to determine the experts’ relative importance weights. After that, we calculate the distances of the collective interval preference orderings to the positive and negative ideal solutions, respectively, based on which we use a TOPSIS based approach to rank and select the alternatives. All these results are also reduced to solve group decision making problems where the experts’ evaluations over the alternatives are expressed in exact preference orderings. A numerical analysis of our model and approach is finally carried out using two illustrative examples.

Highlights

► Establish the nonlinear programming models to derive the experts’ weights. ► Give a group decision making approach based on interval preference orderings. ► Give a group decision making approach based on exact preference orderings.

Introduction

Due to the various causes, such as the experts’ limited attention and information processing capabilities, time pressure and lack of data, etc. (Weber, 1987), in group decision making with multiple experts and a discrete set of alternatives, the experts usually provide their evaluations over the alternatives by means of uncertain preference representation structures, in which each evaluation value cannot be specified, but a value range can be obtained. Interval preference orderings, originally introduced by González-Pachón and Romero (2001) as a collection of positive integer ranges given for providing the possible order positions of a set of alternatives, is one of the simplest uncertain preference representation tools, and can be easily and conveniently used to express the experts’ uncertain evaluation values especially when considering incomparability as a possible attitude. Taking the voting system as an example, when asked to select the desirable person from a collection of candidates, the voter may find it is hard to make the final decision of determining who the best one is, but it is easy for him/her to rank top three candidates. Another example concerning the convenience of interval preference orderings was provided by Fan and Liu (2010). Suppose that a consumer wants to buy a car among four color cars and the preferences of him/her are as follows: The black one ranked top 2, the white one is top 3, the blue one is second or third and the yellow one is bottom 2. Such preferences can be represented easily and properly by interval preference orderings but cannot be depicted by any other structures. Consequently, group decision making problems with interval preference orderings on alternatives have been becoming more and more common in real life, and thus it is an important research topic, which has received attention from some researchers recently (Fan and Liu, 2010; González-Pachón and Romero, 2001; González-Pachón et al., 2003; Wang et al., 2005). In earlier work, González-Pachón and Romero (2001) proposed an approach based on an interval goal programming model to aggregate the individual interval preference orderings into the collective ones and then get the ranking ordinals of the alternatives. González-Pachón et al. (2003) used an interval goal programming model to solve the multi-criteria decision making problem with both interval preference orderings and pairwise comparison matrix. Wang et al. (2005) developed a linear programming method to generate an aggregated ranking from a set of ordinal rankings by estimating a utility interval for each alternative and every ordinal ranking. More recently, Fan and Liu (2010) gave a possibility degree formula to compare two interval preference orderings, and based on which and the collective expectation possibility degree matrix on pairwise comparisons of alternatives, an optimization model was built to solve group decision making problems with interval preference orderings. But we note that all the approaches in the above literature cannot be used to deal with group decision making problems where the relative importance information of the experts is unknown. Indeed, in the process of group decision making, many scholars have assumed that all the experts have the same importance (Jabeur et al., 2004). But this is sometimes unsuitable to depict the actual situations because a single expert cannot be expected to have sufficient knowledge to evaluate all aspects of the problem considered but on a part of the problem for which he/she is familiar (Weiss and Rao, 1987). Consequently, the individual experts do not always have the equal importance in the decision making process. To assign the weights to the experts is a tough task especially if the group is large as in the case of public policy decisions and when judgments are elicited through the use of questionnaires (Ramanathan and Ganesh, 1994). Up to now, many schemes have been proposed to determine the weights of experts within different contexts. Brock (1980) suggested a Nash bargaining based approach to estimate the relative importance information of the experts. By using the experts’ subjective opinions, Ramanathan and Ganesh (1994) proposed a simple and intuitively appealing eigenvector based method to intrinsically determine the weights of the experts. As to the judgments given by interval numbers, Yue (2011) employed the extended TOPSIS (Hwang and Yoon, 1981) (technique for order preference by similarity to ideal solution) method to determine the weight vector of the experts. How to determine the experts’ weights plays an important role in group decision making process, thus it is necessary to pay attention to this issue within the context of interval preference orderings, which is also the focus of this paper.

We organize the rest of the paper as follows. In Section 2, we first give a detailed description of group decision making problems with interval preference orderings. By using the uncertain additive weighted averaging operator, we aggregate all individual interval preference orderings given by the experts into the collective interval preference orderings, and then establish a nonlinear programming model by minimizing the divergences between the individual uncertain preferences and the group’s opinions, from which we derive an exact formula to determine the experts’ relative importance weights. After that, we use a TOPSIS based approach to rank and select the alternatives. All these results can be reduced to accommodate group decision making with exact preference orderings. Section 3 gives a numerical analysis of our model and approach using two illustrative examples. We end the paper with the concluding remarks in Section 4.

Section snippets

Group decision making model and approach based on interval preference orderings

Due to the limited knowledge and experience related to the problem domain and the inaccuracies and vagueness of people’s estimation and perception, it is hard for the experts to provide the exact preference rankings or ranking ordinals of alternatives. To solve this problem, González-Pachón and Romero (2001) originally introduced the notion of interval preference orderings. But they did not give the definition mathematically, which was complemented by Fan and Liu (2010).

Defintion 1

(Fan and Liu, 2010) Let Z

Examples

In the following, we use two practical examples of the alliance partner selection of a software company and the evaluation of key factors that influence the cooperation among enterprises to illustrate our model and approach.

Example 2

Eastsoft is one of the top five software companies in China (Fan and Liu, 2010). It offers a rich portfolio of businesses, mainly including industry solutions, product engineering solutions, and related software products and platform and services. It is dedicated to becoming

Concluding remarks

In this paper, we have focused on group decision making problems based on interval preference orderings or exact preference orderings, in which the weight information about the experts cannot be predefined. We have established a nonlinear programming model on the basis of minimizing the divergences between the individual interval preference orderings and the group’s opinions. The solution to the model is an exact formula used to derive the experts’ weights. Based on the collective interval

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