Multi-Criteria Decision Making And Reasoning Methods In An Intuitionistic Fuzzy Or Interval-Valued Fuzzy Environment

The concepts of intuitionistic fuzzy sets (vague sets) and interval-valued fuzzy sets are two extensions of Zadeh's fuzzy sets. Since they can be unified to L* — fuzzy sets in the sense of Goguen, where L* is a complete lattice, we call them IF sets (IFSs for short), i.e., the fuzzy sets that their membership functions take values in L*, where the "IF" denotes the words "intuitionistic fuzzy" or "interval-valued fuzzy". Because an IF set (IFS, for short) provides more choice manners for the attribute description of an object and has stronger ability to express uncertainty than an ordinary fuzzy set, it has gained extensive attention from the academic circles and the circles of engineering and technology. In this paper, we will study the problems of multi-criteria fuzzy decision making and fuzzy reasoning by means of IFSs and will establish a series of methods for solving above problems. This study not only develops and enriches the fundamental theory of IFSs but also provides a new idea for the applications of IFS theory. This paper contains the following four chapters.Chapter 1 solves Chen and Tan (1994)'s multi-criteria fuzzy decision-making problem in an IF environment and proposes a series of methods: score function and weighted score function methods. First, we introduce the concept of IF point operator by means of an IF operator defined by Atanassov, and analyze the unknown information of an element by using the point operator for finding and mining new useful information . Further, we define the score function and weighted score function for soving the multi-criteria decision-making problem in an IF environment. Because these score functions are based on the information mining, they effectively improve the existing methods. Finally, the relationship between new score functions and the existing score functions is discussed.In chapter 2, the inclusion degrees, similarity measures of IFSs and their applications to multi-criteria fuzzy decision making are discussed. Inclusion degree is a quantity describing that a set is contained by another set and is quantitative description of containment relation. It holds the uncertainty of the relation. The inclusion degree theory and the fuzzy set theory are the important tools in studying the uncertain knowledge. In this chapter, the inclusion degree of IFSs is defined and its some specific computing formulae are established by means of fuzzy implication operators. By defining the cardinality of IFSs we generalize some computingformulae for the inclusion degrees of fuzzy sets to IFSs. We also discuss the generation problem of the inclusion degrees of IFSs and give some generation methods. For the similarity measures followed with interest in the circles of the theory and applications, we mate discussion from the following three directions. Firstly, the strong similarity measures between IFSs are constructed by means of the inclusion degree of IFSs and a type of symmetric and non-decreasing function defined on unit interval. And we show that some formulae of similarity measures between fuzzy sets are the special cases of above strong similarity measures; Secondly, By making full use of the information reflected by an IFS, we define the similarity measures between IFSs by means of the distance measures. These new similarity measures effectively improve the existing formulae of similarity measures based on distance measures; Thirdly, considering the differences between scores and between accuracy degrees of IF values, we establish a formula of similarity measure between IF sets by means of the IF point operator defined in chapter 1. We also illustrate by an example that this formula has stronger discrimination. In the final part of this chapter, we extend the idea of TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and apply the inclusion degrees and the similarity measures to the multi-criteria fuzzy decision-making problem, respectively. This method find a new way for the wide application of IFSs.Chapter 3 discusses the IF reasoning methods on the basis of compositional rules. In recent years, it is because the IFSs can effectively solve the uncertainty problem that IF reasoning is followed with interest. At present, lots of methods for IF reasoning in literatures are based on the composition of triangular norms. In general, the compositional idea on the basis of triangular norms derives from the Zadeh's CRI (Compositional Rule of Inference) method. In this chapter, we make general discussion about the CRI methods for IF reasoning. Firstly, we discuss the IF t-norms and IF residual implications satisfying residuation principle. Some useful properties are given and several specific t-representable IF t-norms and IF residual implications are obtained by examples. This part is the foundation of this chapter and the next chapter. Because any general fuzzy reasoning can be transformed into the fundamental form: FMP(Fuzzy Modus Ponens) or FMT(Fuzzy Modus Tollens) by proper methods, we only discuss two most fundamental forms of reasoning in this chapter, and give the general forms of the CRI solutions for FMP problem and IFT problem in an IF environment, respectively. The reversibility conditions areattached importance in our discussion and the reversibility criteria are given. We also point out that Wu's two new methods for IF reasoning are the particular cases of the general formulas. At present, a lot of methods for reasoning are in the CRI framework. Therefore, The study in this chapter will be useful not only for real applications but also for the further improvement of the inference methods.In chapter 4, the fully implicational methods for fuzzy reasoning in an IF environment are studied. On one hand, the existence conditions of the fully implicational solutions are discussed. On the other hand, based on the IF residual implications satisfying residuation principle, we establish the fully implicational methods for IF fuzzy reasoning, i.e., a-triple I method for IFMP(IF Modus Ponens), a-triple I and a-triple I* methods for IFMT(IF Modus Tollens), and a-multiple I method for general IF reasoning. And At the same time we analyze the reversibility properties of these triple I, triple I* and multiple I methods. As one of the applications of the multiple I method for IF reasoning we discuss Chen and Tan's multi-criteria fuzzy decision-making problem, and therefore provide a theoretical foundation from another direction for the existing methods for multi-criteria fuzzy decision making. Wang's fully implicational methods for fuzzy reasoning improved the ordinary CRI method. The fully implicational methods for the basic and the general IF fuzzy reasoning obtained in this chapter also effectively improved some existing inference methods depending on the composition idea of CRI method.