Interval-Valued Fuzzy Multi-Criteria Decision-Making by Combining Analytic Hierarchy Process with Utility Representation Function

To encompass uncertainty and vagueness of information, the analytic hierarchy process (AHP) was often extended into fuzzy multi-criteria decision-making (FMCDM) under an uncertain environment. However, the extension of AHP was rarely constructed on interval-valued fuzzy numbers. Recently, interval-valued fuzzy numbers were utilized for decision-making to obtain more messages than others. For AHP extended under a fuzzy environment into fuzzy AHP, fuzzy computations are critical to derive priorities of pairwise comparison matrices. Although AHP’s approximate computations including the normalization of row arithmetic averages may be adopted to the fuzzy environment, the fuzzy extension of AHP is still complicated for division and multiplication of fuzzy numbers, especially interval-valued fuzzy numbers. To resolve complicated ties, a utility representation function of interval-valued fuzzy numbers in fuzzy AHP is used for yielding vectors consisting of priority representations of fuzzy pairwise comparison matrices on evaluation criteria based on objective, alternatives based on evaluation criteria, and more hierarchies. Then, sum product of multiplying the priority representation vectors is derived to form the utility representations of alternative performance indices, and alternative performance indices are represented by their corresponding utility representations. Therefore, FMCDM problems are easily solved by fuzzy AHP, i.e., combining AHP with the utility representation function under an interval-valued fuzzy environment.