25 Fuzzy AHP
Fuzzy logic applied to AHP takes into consideration the uncertainty inherent in a small number of expert opinions, uncertainty, and subjectivity. Since Saaty’s original development of AHP, multiple means of fuzzy AHP have emerged. Generally, fuzzy logic membership functions are applied to defined linguistic variables – Very High, High, Moderate, Low, Very Low. Membership functions capture the ‘space of truth’. For AHP, the linguistic variables are already defined. These are the 1 to 9 scale for pairwise comparison.
For the trapezoid membership function (see figure), the AHP is mapped as follows
- (1,1,1,1)
- (1,1.5,2.5,3)
- (2,2.5,3.5,4)
- (3,3.5,4.5,5)
- (4,4.5,5.5,6)
- (5,5.5,6.5,7)
- (6,6.5,7.5,8)
- (7,7.5,8.5,9)
- (8,8.5,9,9)
Considering the comparison of n alternatives within the context of a defined criterion, then, the fuzzy pairwise matrix would be constructed as
[latex]\tilde X = \begin{bmatrix} \tilde\chi_{11} & \tilde\chi_{12} & \cdots & \tilde\chi_{1n} \\ \tilde\chi_{21} & \tilde\chi_{22} & \cdots & \tilde\chi_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ \tilde\chi_{n1} & \tilde\chi_{n2} & \cdots & \tilde\chi_{nn} \end{bmatrix}[/latex]
where
[latex]\begin{split} & \tilde \chi_{ij} = (l_{ij},m_{ij},n_{ij},s_{ij}) \\ & \tilde \chi_{ji}^{-1} = (s_{ij}^{-1},n_{ij}^{-1},m_{ij}^{-1},l_{ij}^{-1}) \end{split}[/latex]
Fuzzy weights are then computed by taking the geometric mean for each row and then adding them. That is, each point in the membership function. The trapezoid membership function is then skewed in a manner reflecting the subjectivities in judgments. When the final, utility set(s) is obtained, the ‘crisp number’ is calculated from the center of mass of that member ship function.
[latex]\Large n = \frac{v_1 + 2v_2 + 2v_3 + v_4}{6}[/latex]
Excerpted from – Emma K. Redfoot, Kelley M. Verner, R. A. Borrelli (2022). Applying analytic hierarchy process to industrial process design in a nuclear renewable hybrid energy system. Progress in Nuclear Energy 145, 104083