Fuzzy AHP - Fuzzy extension of the Analytic Hierarchy Process

Overview

Fuzzy AHP elicits criterion weights from triangular fuzzy pairwise judgments. Engine default implements Chang (1992, 1996) extent analysis: synthetic extent S_i for each row, degree of possibility V(S_i ≥ S_j) for each pair, then crisp priority via the minimum possibility values normalized. Switch variant via D.variant parameter to use Buckley 1985 (geometric mean), Van Laarhoven-Pedrycz 1983 (logarithmic least squares), or Cheng 1996 (entropy-based).

Output

Weight, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct the fuzzy pairwise comparison matrix à = [ã_ij] using the TFN linguistic conversion scale (Table 4.1 in Yazdi 2022). Diagonal entries ã_ii = (1,1,1); reciprocal ã_ji = (1/u_ij, 1/m_ij, 1/l_ij).

   $$\tilde{A}=[{\tilde{a}}_{ij}{]}_{n\times n},{\tilde{a}}_{ij}=(l_{ij},m_{ij},u_{ij}),{\tilde{a}}_{ji}=\left(\frac{1}{u_{ij}},\frac{1}{m_{ij}},\frac{1}{l_{ij}}\right),{\tilde{a}}_{ii}=(1,1,1)$$

   Yazdi 2022 Ch4 §4.7 Step 1, Eq.(4.1)-(4.2)

2. 2

   (Group decision, E experts) Aggregate individual pairwise matrices using TFN geometric mean. Skip this step when a single pre-aggregated matrix is provided.

   $${\tilde{a}}_{ij}={\left({\tilde{a}}_{ij}^1\otimes {\tilde{a}}_{ij}^2\otimes \cdots \otimes {\tilde{a}}_{ij}^E\right)}^{1/E}$$

   Yazdi 2022 Ch4 §4.7 Step 2, Eq.(4.3); Buckley 1985

3. 3

   Compute the fuzzy geometric mean of each row: r̃_i = (ã_i1 ⊗ ã_i2 ⊗ … ⊗ ã_in)^{1/n}. Component-wise: (∏_j l_ij)^{1/n}, (∏_j m_ij)^{1/n}, (∏_j u_ij)^{1/n}.

   $${\tilde{r}}_i={\left({\tilde{a}}_{i1}\otimes {\tilde{a}}_{i2}\otimes \cdots \otimes {\tilde{a}}_{in}\right)}^{1/n}$$

   Yazdi 2022 Ch4 §4.7 Step 3, Eq.(4.4); Buckley 1985 §2

4. 4

   Compute fuzzy weight for each criterion/alternative: w̃_i = r̃_i ⊗ (r̃_1 ⊕ r̃_2 ⊕ … ⊕ r̃_n)^{-1}. TFN inverse of sum: (1/U_sum, 1/M_sum, 1/L_sum).

   $${\tilde{w}}_i={\tilde{r}}_i\otimes {\left({\tilde{r}}_1\oplus {\tilde{r}}_2\oplus \cdots \oplus {\tilde{r}}_n\right)}^{-1}$$

   Yazdi 2022 Ch4 §4.7 Step 4, Eq.(4.5); Buckley 1985 §3

5. 5

   Defuzzify each fuzzy weight by Centre of Area (COA): w_i = (Lw_i + Mw_i + Uw_i) / 3, then normalize so that ∑ w_i = 1. Higher weight = higher priority.

   $$w_i=\frac{L{\tilde{w}}_i+M{\tilde{w}}_i+U{\tilde{w}}_i}{3},w_i\leftarrow \frac{w_i}{{\sum }_{k=1}^n{w}_k}$$

   Yazdi 2022 Ch4 §4.7 Step 5, Eq.(4.6); COA defuzzification

Fits when / Look elsewhere when

Assumptions to verify

• Experts can articulate preferences in Fuzzy (Triangular) linguistic variables
• Pairwise comparisons satisfy consistency requirements of underlying method

Edge cases and pitfalls

• •when a single pre-aggregated matrix is provided.

• Chang's extent analysis can yield zero weight for non-trivial criteria when one row's synthetic extent fully dominates others (cf. Wang-Luo-Hua 2008 EJOR 186:735). Cross-check against Buckley 1985 geometric mean if any weight is zero.

• Defuzzification of weights before normalization changes the ranking - keep the normalization step on the crisp d'(C_i) values, not on TFN extents.

Works with

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