FAHP Method

Fuzzy Analytic Hierarchy Process

Video Demonstration

Step-by-Step Algorithm

Calculated cell

Source cells in formula

Construct the fuzzy pairwise comparison matrix

Compare the criteria pairwise using linguistic judgments. Each judgment is represented by a triangular fuzzy number (TFN).

\[ \widetilde{A}=[\widetilde{a}_{ij}]_{n\times n} \]

\[ \widetilde{a}_{ij}=(l_{ij},m_{ij},u_{ij}) \]

$l_{ij}$, $m_{ij}$, and $u_{ij}$ represent the lower, modal, and upper values of the fuzzy judgment.

Criterion	$C_1$	$C_2$	$C_3$
$C_1$	(1, 1, 1)	(1, 2, 3)	(2, 3, 4)
$C_2$	(0.33, 0.5, 1)	(1, 1, 1)	(1, 1, 1)
$C_3$	(0.25, 0.33, 0.5)	(1, 1, 1)	(1, 1, 1)

Calculate the fuzzy geometric mean

Formula

Calculate the fuzzy geometric mean for each criterion by taking the geometric mean of all fuzzy judgments in its corresponding row.

\[ \widetilde{g}_i=\left( \prod_{j=1}^{n} \widetilde{a}_{ij} \right)^{\frac{1}{n}} \]

For triangular fuzzy numbers, the calculation can be expressed component-wise as:

\[ \widetilde{g}_i=\left( \left(\prod_{j=1}^{n}l_{ij}\right)^{1/n}, \left(\prod_{j=1}^{n}m_{ij}\right)^{1/n}, \left(\prod_{j=1}^{n}u_{ij}\right)^{1/n} \right) \]

$\widetilde{g}_i$ is the fuzzy geometric mean of criterion $C_i$.

Criterion	Fuzzy GM ($\widetilde{g}_i$)
$C_1$	(1.26, 1.82, 2.29)
$C_2$	(0.69, 0.79, 1.00)
$C_3$	(0.63, 0.69, 0.79)

Calculate the fuzzy weights

Formula

Obtain the fuzzy weight of each criterion by dividing its fuzzy geometric mean by the sum of all fuzzy geometric means.

\[ \widetilde{w}_i=\widetilde{g}_i \otimes \left( \sum_{k=1}^{n} \widetilde{g}_k \right)^{-1} \]

If $\widetilde{g}_i=(l_i,m_i,u_i)$, then :

\[ \widetilde{w}_i=\left( \frac{l_i}{\displaystyle\sum_{k=1}^{n}u_k}, \frac{m_i}{\displaystyle\sum_{k=1}^{n}m_k}, \frac{u_i}{\displaystyle\sum_{k=1}^{n}l_k} \right) \]

$\widetilde{w}_i$ represents the fuzzy importance of criterion $C_i$.

Criterion	Fuzzy Weight ($\widetilde{w}_i$)
$C_1$	(0.31, 0.55, 0.89)
$C_2$	(0.17, 0.24, 0.39)
$C_3$	(0.15, 0.21, 0.31)

Defuzzify the fuzzy weights

Formula

Convert each fuzzy weight into a single crisp value so that the relative importance of the criteria can be compared directly.

\[ w_i=\frac{ l_i+m_i+u_i}{3} \]

This centroid or average method uses the three components of the triangular fuzzy weight to obtain a crisp weight.

Criterion	Crisp Weight ($w_i$)
$C_1$	0.58
$C_2$	0.27
$C_3$	0.22

Calculate the normalised weights

Formula

Normalize the crisp weights so that the sum of all criterion weights equals one.

\[ W_i=\frac{w_i} {\displaystyle\sum_{k=1}^{n}w_k} \]

\[ \sum_{i=1}^{n}W_i=1 \]

$W_i$ is the final normalized weight of criterion $C_i$.

Criterion	Normalised Weight ($W_i$)
$C_1$	0.54
$C_2$	0.25
$C_3$	0.21
Sum	1.00

FAHP Algorithm Flowchart

START

Construct the fuzzy pairwise comparison matrix

Calculate the fuzzy geometric mean

Calculate the fuzzy weights

Defuzzify the fuzzy weights

Calculate the normalised weights

END