Extension Principle and Fuzzy Numbers

Abstract

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 attributes with the corresponding weights w₁, w₂, ..., w _m respectively. If the evaluation of the alternative A _i w.r.t. C _j is denoted by r _ij , then the overall evaluation of A _i may be computed as

$$r_i=\frac{\sum\limits_{j=1}^{m}w_jr_{ij}}{\sum\limits_{j=1}^{m}w_j}\qquad(i=1,2,\cdots,n).$$

and the final ranking of alternatives A₁,A₂, ..., A _n is based on the comparison of the real numbers r₁,r₂, ..., r _n . 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 r _ij or w _j (i = 1,2, ..., n; j = 1,2, ..., m) or both of them may be replaced by fuzzy numbers $\tilde{r}_{ij}$ and $\tilde{w}_{j}$ respectively. Accordingly, the overall evaluation of A _i becomes

$$\tilde{r}_i=\frac{\sum\limits_{j=1}^{m}\tilde{w}_j\tilde{r}_{ij}} {{\sum\limits_{j=1}^{m}\tilde{w}_j}}\qquad(i=1,2,\cdots,n).$$