Linguistic measures based on fuzzy coincidence for reaching consensus in group decision making

Assuming a linguistic framework, a model for the consensus reaching problem in heterogeneous group decision making is proposed. This model contains two types of linguistic consensus measures: linguistic consensus degrees and linguistic proximities to guide the consensus reaching process. These measures evaluate the current consensus state on three levels of action: level of the pairs of alternatives, level of the alternatives, and level of the relation. They are based on a fuzzy characterization of the concept of coincidence, and they are obtained by means of several conjunction functions for handling linguistic weighted information, the LOWA operator for aggregating linguistic information, and linguistic quantifiers representing the concept of fuzzy majority. © 1997 Elsevier Science Inc. K E Y W O R D S : Linguistic modeling, group decision making, linguistic preference relations, consensus degrees. 1. I N T R O D U C T I O N Consensus or synthesis consists in combining a data set provided by different informat ion sources with a view to obtaining more elaborate Address correspondence to F. Herrera, Department of Computer Science and A.I., ETS de Ingenieda Inform~tica, University of Granada, 18071 Granada, Spain. Received May 1, 1996; accepted October 1, 1996. International Journal of Approximate Reasoning 1997; 16:309-334 @ 1997 Elsevier Science Inc. 0888-613X/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0888-613X(96)00121-1 310 F. Herrera et al. information [31, 32]. When the information sources provide imprecise information, the use of fuzzy set theory to deal with this type of information is most advisable. A usual situation, in the real world, which presents the appropriate characteristics to apply consensus theory and fuzzy set theory together, is the group decision making (GDM) situation. In a classical GDM situation there is a problem to solve, a set of possible solution alternatives, and a group of two or more experts, who express their opinions about the set of solution alternatives and attempt to reach a collective decision with the maximum possible consensus on this question: what is/are the best solution alternative(s) to the problem?. Many papers on consensus theory applied to GDM make use of Arrow's work [1] as a starting point and a basic guide. Arrow proposed a qualitative setting composed by a set of axioms, which any acceptable consensus tool for GDM should satisfy. Arrow's impossibility theorem was an important result thereof. According to this theorem, it is impossible to aggregate individual preferences into group preference in a completely rational way. This is a problem that disappears in a cardinal setting in a fuzzy context, on introducing preference intensities, which provide additional degrees of freedom to any aggregation model [13, 9]. In a fuzzy context, the application of consensus theory to GDM problems presents two ways to relate to different decision schemata [6]. The first way, called algebraic consensus, consists in establishing a group choice process which obtains a decision scheme as a solution to the GDM problem. The second way, called topologic consensus, consists in establishing a group consensus reaching process, which, guided by means of a measure of closeness among different decision schemata, called the consensus measure, attempts to achieve the maximum possible degree of consensus on solution alternative(s). Both consensus types may be combined in a resolution scheme (see Figure 1). Given that the set of experts initially have diverging opinions, firstly, topologic consensus is applied, and in each step, the degree of existing consensus among experts' opinions is measured. If the moderator thinks that the consensus degree is satisfactory, then algebraic consensus is applied in order to obtain a solution; otherwise, the experts are persuaded to update their opinions. In this way, a GDM process may be defined as a dynamic and iterative process, in which the experts, via the exchange of information and rational arguments, agree to update their opinions until they become sufficiently similar, and then the solution alternative(s) is/are obtained. Here, we shall focus our research on the topologic consensus. As was mentioned earlier, the topologic consensus is guided by means of a consensus measure. Assuming numerical preference relations for providing the experts' opinions, several authors introduced hard consensus measures varying between 0 (no consensus or partial agreement) and 1 (full Consensus in Group Decision Making 311