Fuzzy α-discounting method for multi-criteria decision-making

Abstract

The α-Discounting Method was developed to be an alternative to and extension of the Analytical Hierarchy Process (AHP) to solve multi-criteria decision-making (MCDM) problems with non-commensurable and conflicting criteria. In contrast to the AHP, this method works not only for pairwise comparisons but also for n-wise comparisons if relative importance of criteria can be expressed in a system of linear homogenous equations. This method also has a comparative advantage as it can transform those MCDM problems, classified as inconsistent by the AHP, into a consistent form. This study briefly compares the two methods and then develops the Fuzzy α-Discounting Method for Multi-Criteria Decision Making (Fα-DM MCDM). Two illustrative fuzzy MCDM problems from the literature have been solved to show how the Fα-DM MCDM works.

Keywords:

• MCDM
• α-Discounting Method
• consistency in fuzzy decision problems
• fuzzy set theory

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Secondary variables are used to indicate the values of the main variables in a decision-making problem with linear preference statements. Assume that in a linear decision-making problem with three criteria – x, y, and z – the degrees of importance assigned to criteria x and y, as main variables, can be indicated through z, a secondary variable, based on the transitivity rule. An example from Smarandache (2010) shows this as follows: Suppose that C1 (x) is four times as important as C2 (y); C2 (y) is three times as important as C3 (z); and C3 (z) is one-twelfth as important as C1 (x). The linear homogeneous system associated to this decision-making problem is: x = 4y; y = 3z; z = x/12. Using z as secondary variable, the weights to be assigned to main variables – x and y – can be obtained easily. Solving this homogeneous linear system, we get its general solution that we set as a vector [12z 3z z], where z can be any real number (z is considered a secondary variable, while x = 12z and y = 3z are main variables).