Trapezoidal interval type-2 fuzzy set extension of Analytic Hierarchy Process

A trapezoidal interval type-2 fuzzy set (IT2FS) extension of the Analytic Hierarchy Process is introduced in this paper. The pairwise comparison matrix with trapezoidal IT2FSs which represent the linguistic relative importance judged by experts is constructed. The trapezoidal interval type-2 fuzzy weights are elicited. First, the logarithmic regression function is defined with respect to the pairwise comparison matrix of the criteria. Then, the sets of linear systems corresponding to the end points of the upper and lower membership functions are formulated by minimizing the logarithmic regression function. The non-normalized weights are solved. Some theorems are developed to show that the sets of linear systems are all dependent. Accordingly, the final normalized weights can be obtained. A quadratic programming model for finding the optimum normalized weights is constructed in the sense of minimizing the fuzziness of the interval type-2 fuzzy weights. Analogously, if the pairwise comparison are given for the alternatives, then the final normalized trapezoidal IT2FS rates are also obtained by the same mathematical analysis. Analogously, if the pairwise comparisons are for the alternatives, then the final normalized IT2FS rates are also obtained. We propose two approaches to find the aggregation of the IT2FS weights and rates. A numerical example about selecting the best professor is examined.