Fuzzy Analytic Hierarchy Process (FAHP)

Fuzzy Analytic Hierarchy Process (FAHP) is a multi-criteria decision-making method developed based on the traditional Analytic Hierarchy Process (AHP) combined with fuzzy set theory. It is located in SPSSAU -> Comprehensive Evaluation -> Fuzzy Analytic Hierarchy Process (FAHP).

SPSSAU Operations

After pasting or editing the complementary matrix, click "Start".

SPSSAU Data Format

For the FAHP data format, refer to the following details.

Algorithm

1.Fuzzy Judgment Matrix

In FAHP, when performing pairwise comparisons between factors, the relative importance of one factor over another is quantified, forming the fuzzy judgment matrix , A=(a_ij)_n×n which serves as the original input data matrix:

$$\left(\begin{array}{ccccc}0.5& a_{12}& a_{13}& \cdots & a_{1n}\\ a_{21}& 0.5& a_{23}& \cdots & a_{2n}\\ a_{31}& a_{32}& 0.5& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& a_{n3}& \cdots & 0.5\end{array}\right)$$

It has the following properties：

n*n matrix, with all values ranging between 0.1 and 0.9.The larger the number, the more important the factor.

a_{ii}=0.5,\mathrm{\forall }i\in N

a_{ij}+a_{ji}=1,a_{ij}\ge 0,\mathrm{\forall }i,j\in N(i\ne j)

2.Data Scale

Data Scale defines the relative importance between two factors, as detailed in the table below:

Data Scaling Table
Scale	Definition	Description
0.5	Equally Important	Two Elements are Equally Important
0.6	Slightly More Important	One Element is Slightly More Important than the Other
0.7	Significantly More Important	One Element is Significantly More Important than the Other
0.8	Much More Important	One Element is Much More Important than the Other
0.9	Extremely Important	One Element is Extremely More Important than the Other
0.1，0.2，0.3，0.4	Reverse Comparison	If Element ai is Compared with Element aj to Obtain Rij, then aj Compared with ai Yields Rji

3.Weight Calculation

For the fuzzy complementary judgment matrix, the formula for calculating matrix weights is as follows:

$$W_i=\frac{\sum _{j=\mathit{1}}^n{a}_{ij}+\frac{n}{\mathit{2}}-\mathit{1}}{n\mathit{(}n-\mathit{1}\mathit{)}},i\in N$$

4.Consistency Test

The weight values obtained using the above formula need to be tested for consistency. The compatibility of the fuzzy judgment matrix is used to verify its consistency. The weight vector of fuzzy judgment matrix A is derived using the above formula: \mathrm{W}=\mathit{(}{\mathrm{W}}_{\mathit{1}},{\mathrm{W}}_{\mathit{2}},\dots ,{\mathrm{W}}_{\mathit{n}}{\mathit{)}}^{\mathrm{T}},\sum _{i=\mathit{1}}^n{W}_i=1,W_i\ge 0\mathit{(}i\in N\mathit{)}

The characteristic matrix elements are constructed as:

$$W_{ij}=\frac{W_i}{W_i+W_j},\mathrm{\forall }i,j\in n$$

Thus, the characteristic matrix of judgment matrix A is:

$$W=\mathit{(}{W}_{ij}{\mathit{)}}_{n\times n}$$

For the decision-maker's attitude α, when the following condition is met:

$$I\mathit{(}A,W\mathit{)}=\frac{\mathit{1}}{n^2}\sum _{i=\mathit{1}}^n\sum _{j=\mathit{1}}^n|a_{ij}+W_{ji}-\mathit{1}|\le \alpha$$

The judgment matrix achieves satisfactory consistency. The smaller the value, the stricter the consistency requirement. Generally, α = 0.1.