Fuzzy Set Theory
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This chapter reviews concepts from fuzzy set theory indispensable for the fuzzy extension of th e multi-criteria decision making methods based on pairwise comparison matrices. Trapezoidal and triangular fuzzy numbers and intervals, which are most often used for the fuzzy extension of pairwise comparison methods, are introduced here, and normalization of fuzzy vectors is studied. Standard fuzzy arithmetic, that is usually used for the fuzzy extension of pairwise comparison methods, is reviewed, and the difference between applying standard fuzzy arithmetic and simplified standard fuzzy arithmetic on computations with fuzzy numbers is illustrated graphically on numerical examples. It is shown on an example that standard fuzzy arithmetic is not able to cope with problems in which there are interactions between the operands (even when these problems seem to be very simple and their solutions are intuitive) and that constrained fuzzy arithmetic needs to be applied instead. Because there are interactions between the operands in arithmetic operations conducted in fuzzy pairwise comparison methods, it results to be necessary to apply constrained fuzzy arithmetic instead of standard fuzzy arithmetic in these methods. Therefore, this section provides a detailed introduction to constrained fuzzy arithmetic complemented by illustrative examples.