Extension Principle and Fuzzy Numbers
In the traditional multi-attribute decision analysis, there is a well-defined problem-solving model–the Simple Additive Weighting (SAW) method. This model can be formulated as follows. Let A1, A2, ..., A n be n alternatives and C1,C2, ..., C m m attribute
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Extension Principle and Fuzzy Numbers
In the traditional multi-attribute decision analysis, there is a well-defined problem-solving model–the Simple Additive Weighting (SAW) method. This model can be formulated as follows. Let A1 , A2 , . . . , An be n alternatives and C1 , C2 , . . . , Cm m attributes with the corresponding weights w1 , w2 , . . . , wm respectively. If the evaluation of the alternative Ai w.r.t. Cj is denoted by rij , then the overall evaluation of Ai may be computed as m
ri =
wj rij
j=1 m
j=1
(i = 1, 2, · · · , n). wj
and the final ranking of alternatives A1 , A2 , . . . , An is based on the comparison of the real numbers r1 , r2 , . . . , rn . In a real world problem, it is often difficult to give an evaluation for the weights or the attributes in a precise way. Sometimes it is more reasonable to say that ‘the evaluation is approximately 0.7’ than ‘the evaluation is exactly 0.7’. Such situation arises typically due to the lack of objective information or/and the use of subjective estimations. The traditional mathematical theory can give little help as to make a decision with such kind of imperfect knowledge. On the contrast, fuzzy set theory provides a strongly effective apparatus to deal with the problem. Firstly, imprecise descriptions can be modeled by means of fuzzy numbers. That is to say, the crisp numbers rij or wj (i = 1, 2, . . . , n; j = 1, 2, . . . , m) or both of them may be replaced by fuzzy numbers r˜ij and w ˜j respectively. Accordingly, the overall evaluation of Ai becomes m
r˜i =
w ˜j r˜ij
j=1 m
j=1
(i = 1, 2, · · · , n). w ˜j
X. Wang et al.: Mathematics of Fuzziness – Basic Issues, STUDFUZZ 245, pp. 113–151. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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4 Extension Principle and Fuzzy Numbers
Furthermore, operations like the weighted sum of fuzzy numbers can also be formulated in the framework of fuzzy set theory using the Zadeh’s extension principle. In this way, we can still evaluate the performance of alternatives although only imprecise data are available. This is significant particularly when modeling a system for which there is no way or it is simply not necessary to acquire the full knowledge of it. After r˜1 , r˜2 , · · · , r˜n , which are still fuzzy numbers, are obtained, the problem of how to rank them arises. Unlike in the crisp case, fuzzy numbers have no natural order. As a result, we have to design a way to compare the fuzzy numbers r˜1 , r˜2 , · · · , r˜n in order to rank the alternatives A1 , A2 , . . . , An . It is the so-called problem of ranking fuzzy numbers. The same problem is encountered in Fuzzy Decision Tree [2, 158], Fuzzy Analytic Hierarchy Process [19, 87, 145], Fuzzy Linear Programming [36, 125], even electrocardiological diagnostics [75] after the fuzzy information is involved and processed. It is because of its extensive applications that many efforts have been made to deal with the problem. Therefore, three tasks will be fulfilled in this chapter. The first is the introduction of the Zadeh’s extension principle.
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