IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 3, JUNE 1999 255 Generalization of the Dempster–Shafer Theory: A Fuzzy-Valued Measure Caro Lucas and Babak Nadjar Araabi, Student Member, IEEE Abstract—The Dempster–Shafer theory (DST) may be consid- ered as a generalization of the probability theory, which assigns mass values to the subsets of the referential set and suggests an interval-valued probability measure. There have been several attempts for fuzzy generalization of the DST by assigning mass (probability) values to the fuzzy subsets of the referential set. The interval-valued probability measures thus obtained are not equivalent to the original fuzzy body of evidence. In this paper, a new generalization of the DST is put forward that gives a fuzzy-valued definition for the belief, plausibility, and probability functions over a finite referential set. These functions are all equivalent to one another and to the original fuzzy body of evidence. The advantage of the proposed model is shown in three application examples. It can be seen that the proposed generalization is capable of modeling the uncertainties in the real world and eliminate the need for extra preassumptions and preprocessing. Index Terms—Dempster–Shafer theory, fuzzy body of evidence, fuzzy generalization of the Dempster–Shafer theory, fuzzy set of consistent probability measures, fuzzy valued belief function, fuzzy valued plausibility function. I. INTRODUCTION A. Dempster–Shafer Theory T HE Dempster–Shafer theory (DST) could be considered as a generalization of the probability theory [1]–[3]. Consider a set-valued mapping : , where is the set of all nonfuzzy subsets of . The referential set is a finite set through this paper. Assume a probability measure over ; now, what can be said about a probability measure over that is induced by ? This is the basic question in [1], where Dempster shows that for each , belongs to the following interval: (1) Manuscript received January 30, 1997; revised Septembeer 15, 1998. C. Lucas is with the Department of Electrical and Computer Engineering, University of Tehran, Tehran 14395, Iran. B. Nadjar Araabi was with the Department of Electrical and Computer Engineering, University of Tehran, Tehran, 14395 Iran. He is now with the Department of Electrical Engineering, Texas A&M University, TX 77843 USA. Publisher Item Identifier S 1063-6706(99)04940-1. in which is any nonempty member of the range of and Range (2) About ten years later Shafer introduced his evidence theory and defined and functions. Consider a referential set ;a body of evidence is defined as follows [2]: (3) in which each is a focal element, and is the corre- sponding mass value. Evidence theory could be considered as a direct generalization of Bayesian statistics [3]. One may think of mass values as probability density values; but in evidence theory, mass values are assigned to the subsets of instead of the elements of ; so, it conveys a higher level of uncertainty and is capable of modeling both ignorance and indeterminism. Shafer defined the concepts of belief and plausibility as two measures over the subsets of in an axiomatic manner and then he showed that and with the following definitions were belief and plausibility functions (4) (5) Using the concept of M¨ obius inversion, Shafer proved a one- to-one correspondence between (3)–(5); i.e., if we have each one of function, function, or body of evidence, then we may build up the others two. Although Shafer’s approach was totally different from Dempster’s, he obtained the same results. So, when we refer to the DST we mean both approaches with their corresponding results and interpretations. In his capital study [1], Dempster had also introduced the set of consistent probability measures. Definition 1—Set of Consistent Probability Measures : Consider a body of evidence (3) over a referential set , is defined as 1063–6706/99$10.00  1999 IEEE