Fuzzy Ranking Methods

Recall that in Chapter 2 we define the Multiple Attribute Decision Making (MADM) problem as: $$\text{D=}\begin{matrix} {{\text{A}}_{1}}\\ {{\text{A}}_{2}}\\ \vdots\\ {{\text{A}}_{\text{m}}}\\ \end{matrix}\text{ }\left[\begin{matrix} {{\text{X}}_{1}}\text{ }{{\text{X}}_{2}}\text{ }\cdots \text{ }{{\text{X}}_{\text{n}}}\\ {{\text{X}}_{11}}\text{ }{{\text{X}}_{12}}\text{ }\cdots \text{ }{{\text{X}}_{1\text{n}}}\\ {{\text{X}}_{21}}\text{ }{{\text{X}}_{22}}\text{ }\cdots \text{ }{{\text{X}}_{\text{2n}}}\\ \vdots \text{ }\\ {{\text{X}}_{\text{m1}}}\text{ }{{\text{X}}_{\text{m2}}}\text{ }\cdots \text{ }{{\text{X}}_{\text{mn}}}\\ \end{matrix}\right]$$$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{W} = ({\text{W}}_1 {\text{,W}}_2 {\text{,}} \ldots {\text{,W}}_{\text{n}} {\text{)}} $$ where A_i, i = 1, ..., m are possible courses of action (or alterna-tives); X_j, j = 1, ..., n are attributes with which alternative performances are measured; x_ij is the performance (or rating) of alternative A_i with respect to attribute X_j; w_j, j = 1,...,n are the relative importance of attributes.