Decision Support
A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations

https://doi.org/10.1016/j.ejor.2005.03.026Get rights and content

Abstract

This paper proposes a goal programming approach to solve group decision-making (GDM) problems where the preference information on alternatives provided by decision makers is represented in two different formats, i.e. multiplicative preference relations and fuzzy preference relations. In order to narrow the gap between the collective opinion and each decision maker’s opinion, a linear goal programming model is constructed to integrate the two different formats of preference relations and to compute the collective ranking values of the alternatives. Thus, the ranking of alternatives or selection of the most desirable alternative(s) is obtained directly from the computed collective ranking values. A numerical example is also used to illustrate the applicability of the proposed approach.

Introduction

In group decision making (GDM) analysis, the preference information on alternatives provided by decision makers are often aggregated to form a collective opinion. Ranking of the alternatives or selection of the best alternative(s) is based on the derived collective opinions [16]. Several approaches have been proposed to solve GDM problems [1], [2], [8], [13], [15], [16], [19]. In these GDM approaches, the preference information provided by decision makers is represented in the same format. But in practical GDM applications, decision makers can participate in decision tasks at different time and various locations [25], they also have different cultural and educational backgrounds. Thus there is a need to provide different preference formats for them to express their preference [3], [11] in GMD. Commonly used preference formats include: preference orderings [3], [11], [22], utility values [3], [11], [24], multiplicative preference relations [3], [21], [25], fuzzy preference relations [11], [20], [23] and linguistic variables [2], [6], [12], [13], [14], etc. Research on GDM problems with different formats of preference information can be found in [3], [6], [9], [11], [26].

Delgado et al. [6] present an approach to solve GDM problems where the preference information on alternatives is represented in two formats, i.e., fuzzy preference relations and linguistic preference relations. A fusion operator is presented to combine the two preference relations and to develop a selection process for alternatives. Chiclana et al. [3] propose an approach to solve GDM problems with three preference formats, i.e., preference orderings, utility values and fuzzy preference relations, where the preference orderings and utility values are uniformed into the fuzzy preference relations. Then these uniformed representations are aggregated into a collective opinion based on the concept of fuzzy majority. The most desirable alternative is selected using two quantifier guided choice degrees of alternatives. Based on the work of [3], the internal consistence of the decision model is also studied by analyzing the transformation functions [5]. Herrera et al. [9] use three preference formats (i.e., preference orderings, utility values and multiplicative preference relations) to represent decision makers’ judgments on alternatives. Using the multiplicative preference relation as the uniform format, preference orderings and utility values on alternatives are uniformed. A fuzzy majority guided aggregation operator, i.e., the ordered weighted geometric operator [4], is used to aggregate the uniformed preference information. The selection of the most desirable alternative(s) is done using two choice degrees, i.e. quantifier guided dominance degree and quantifier guided non-dominance degree.

Current approaches [3], [6], [9] for GDM analysis support different preference formats, but their computational procedures [3], [6], [9], [26] are very complicated. Usually, they consist of three steps: (1) uniform the preference information given by decision makers through a transformation function, (2) aggregate the uniformed preference information into a collective one by means of the aggregation operators, and (3) rank alternatives or select the most desirable alternative(s) by the selection methods. Generally in the process of uniforming the preference information, if one format of preference information is transformed into another format by a transformation function, part of the original preference information may be lost or distorted. Sometimes, it can be difficult to transform preference information from one format into another (i.e. a uniformed format). For instance, if utility values on alternatives are transformed into a preference ordering on alternatives, the resultant one can hardly reflect the meaning of the original utility values. It is a very complicated process to transform a fuzzy preference relation into utility values. Current approaches also neglect to consider how to narrow the gap between the collective opinion and each decision maker’s opinions so as to reach consensus among decision makers.

This paper presents a new approach to simplify the procedure of GDM analysis with different preference formats while still maintaining the original preference information. For simplicity, we only consider a GDM problem with two formats of preference information, i.e., multiplicative preference relations and fuzzy preference relations. In the proposed approach, a linear goal programming model is constructed to integrate the two formats of preference relations and to assess the collective ranking values of alternatives so as to narrow the gap between the collective opinion and each decision maker’s opinion. Different from current approaches of uniforming different formats of preference information and aggregating multiple individual preferences into a collective ranking value by means of the aggregation operators, the proposed approach computes the ranking of alternatives or selection of the most desirable alternative(s) directly using the obtained collective ranking values.

This paper is organized as follows. Section 2 presents the GDM problem with multiplicative preference relations and fuzzy preference relations. Section 3 proposes a goal programming approach to integrate the two formats of preference relations and to obtain collective ranking values of alternatives. To illustrate the use of the proposed approach, a numerical example is presented in Section 4. Finally, Section 5 concludes the paper and discusses the future research.

Section snippets

Presentation of the problem

This section describes the GDM problem with multiplicative preference relations and fuzzy preference relations on alternatives.

Let X = {x1, x2 ,… , xn}(n  2) be a finite set of alternatives and E = {e1, e2 ,… , em}(m  2) be a finite set of decision makers. Let C = (c1,c2 ,… , cm)T be the weight vector of the decision makers, where h=1mch=1, ch  0, h = 1,  , m and ch denotes the important degree of decision maker eh and is usually determined by the decision maker. In GDM analysis, the alternatives x1, x2,  , xn need to

The proposed approach

Suppose the collective ranking value of alternative xi is wi, and wi is an unknown variable where i=1nwi=1, wi  0, i = 1,  , n. The collective ranking values, w1, w2,  , wn, are thought to be a final collective result on alternatives in the GDM analysis. In the following, the research problem is how to determine the collective ranking values of alternatives based on the two formats of preference relations about the alternatives provided by the decision makers.

In order to do that, we construct a

Illustrative example

In this section, an investment decision problem is used to illustrate the proposed approach. An investment company wishes to invest a sum of money in the best option [12]. There are four possible alternatives for the company to invest:

  • x1 is a car company,

  • x2 is a food company,

  • x3 is a computer company,

  • x4 is an arms company.

The investment company has a group of four consultancy departments:

  • e1 is the risk analysis department,

  • e2 is the growth analysis department,

  • e3 is the social-political analysis

Conclusions and discussion

This paper presents a new approach to solve GDM problems with two different formats of preference information, i.e. multiplicative preference relations and fuzzy preference relations. Based on a linear goal programming model, the proposed approach integrates the two formats of preference relations and computes the ranking values of alternatives. Comparing with the existing approaches [3], [6], [10], the proposed approach has the following distinct characteristics.

In the proposed approach, the

Acknowledgments

This research is partly supported by the Competitive Earmarked Research Grant of Hong Kong SAR (Project No: CityU1237/03E, CityU1146/02E), the National Science Foundation of China (NSFC, Project Nos. 70371050 and 70301008) and the Teaching and Research Award Program for Outstanding Young Teachers (TRAPOYT) in Higher Education Institutions of Ministry of Education, China.

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