Fuzzy analytic hierarchy process for pentagonal fuzzy numbers and its application in sustainable supplier selection

Different fuzzy analytic hierarchy process (FAHP) methods are typically developed for conventional fuzzy numbers (FNs), mainly triangular and trapezoidal FNs. The ascending and descending parts of triangular and trapezoidal membership functions (MFs) are straight lines. Therefore, if the value of a member increases by the same values, its membership degrees in ascending (descending) part increase (decrease) equally. These MFs do not necessarily reflect the experts' preferences. Consider two different cases; in the first one, the product quality of supplier A compared to supplier B increases from 2 to 3 times, and in the second case, this quality increases from 8 to 9 times. Triangular and trapezoidal membership degrees in both cases increase equally. However, the quality advantage of supplier A increases by 50% in the first case, while it increases by 12.5% in the second case. Therefore, the experts' preferences may increase differently for these cases that cannot be reflected accurately by triangular and trapezoidal MFs. To overcome this flaw, we develop four FAHP methods for pentagonal FNs (PFNs). We also use an indirect method to defuzzify the fuzzy local weights. Indirect defuzzification is a simple method equivalent to the center of gravity (COG) method and needs low computations. Numerical examples are used to illustrate the proposed methods. A real example of ranking suppliers in a sustainable framework based on pentagonal fuzzy preferences is also presented. To extract crisp local weights from pentagonal fuzzy comparison matrices, we use the pentagonal FAHP methods presented in this study.