IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-10, NO. 1I, NOVEMER 1980 Linguistic Approach to Decisionmaking with Fuzzy Sets RICHARD M. TONG AND PIERO P. BONISSONE, MEMBER, IEEE Abstract-A technique for making linguistic decsions is presented. Fuzzy sets are assued to be an appropriate way of dealng with uncer- tainty, and it b therefore cncluded that decisions taken on the basis of such infomation must themselves be fuzzy. It b inappriate then to present the decision in numencal form; a statement in natural angug Is much better. For brevity only a single-stage multlabute decsion prob- lem is considered. Solutions to such problems are shown using ideas in linguisc approximation and truth qualiction. An extensive example illuminates the basic ideas and techniques. I. INTRODUCTION THE APPLICATION of fuzzy set theory to the prob- lem of making a decision when only incomplete or uncertain information is available has been the subject of much research over the last decade (see Kickert [7] for a recent review). The basic premise behind this work is that there are situations where it is more natural to handle uncertainty by fuzzy set theory than by probability the- ory. While we agree with this, we do feel that most published work does not go far enough in its utilization of the fuzzy approach. In this paper we present a technique for fuzzy de- cisionmaking that is based on linguistic approximation and truth qualification. The main feature, and advantage, of our approach is that it generates a linguistic assessment of the decision, thus making explicit the subjective nature of any choice that is made using fuzzy information. In Section II of the paper we discuss multichoice deci- sion problems in which information about the "suitability" of the alternatives is given by a set of fuzzy sets. We show how to generate a single fuzzy set that aggregates all the suitability information and how this may be interpreted as the basic fuzzy decision. Section III of the paper is con- cerned with the development of ideas in linguistic ap- proximation and truth qualification. Following Zadeh [14], we introduce the concept of a truth-qualified proposition in natural language and show how this may be made the Manuscript received January 7, 1980; revised July 8, 1980. This work was supported in part by a NATO Postdoctoral Research Fellowship from the Science Research Council of Great Britain (RMT) and also in part by Naval Electronic System Command Contract N00039-78-C-0013 (PPB). R. M. Tong was with the Department of Electrical and Computer Sciences, University of California, Berkeley 94720. He is now with Advanced Information and Decision Systems, 201 San Antonio Circle, Suite 286, Mountain View, CA 94040. P. P. Bonissone is with the Corporate Research and Development Department, General Electric Company, P.O. Box 43, Building 37-579, Schenectady, NY 12301. 10 I 50 100 Fig. 1. Simple suitability sets. basis for a linguistic decision. Finally, in Section IV we present a rather extensive example of an investment deci- sion problem in which the ratings are linguistic rather than numerical. This example is designed both to illustrate our technique and to give further insight into the proper- ties of the linguistic approach. II. A MULTICHOICE DECISION PROBLEM We assume that the basic problem is to choose between a set of alternatives, i= {ai: i= 1,- - * , m}, given some fuzzy information about the "suitability" of each of them. We also assume that this information is given as a set of fuzzy sets, 5= {Si: i= 1,- , M}, where each of the S, is defined by a membership function that maps the real line onto the closed interval [0, 1]. Suitability is simply interpreted as a measure of the ability of an alternative to meet our decision criteria. The concept arises quite naturally in this kind of problem and is essentially a fuzzification of the idea of a rating. Thus, for example, the 5 could be fuzzy expected utilities in the manner of Watson [9], the fuzzy values described by Efstathiou and Rajkovic [5], or they might be computed from linguistic assessments of the alternatives with respect to the decision criteria. (We present an example of the latter in Section IV.) Given this statement of the problem, we have to select the preferred alternative on the basis of 5 and then generate a linguistic statement about our decision. To help illustrate the selection procedure, consider the simple ex- ample of choosing between four alternatives given S1, S2' S3, and S4 as shown in Fig. 1. It is ambiguous as to which alternative we should choose, but intuitively we would prefer either a3 or a4. To help us decide we shall introduce a concept of dominance that is closely related to Zadeh's [10] definition of separa- tion. The separation a between two convex fuzzy sets A 0018-9472/80/1100-0716$00.75 (©1980 IEEE 716 Authorized licensed use limited to: UNIVERSIDAD DE GRANADA. Downloaded on March 9, 2009 at 13:27 from IEEE Xplore. Restrictions apply.