Fuzzy AHP: Fuzzy extension of the Analytic Hierarchy Process

Overview

Pairwise comparison with Triangular Fuzzy Number (TFN: l, m, u) judgments: umbrella over Van Laarhoven-Pedrycz 1983, Buckley 1985, Chang 1992 extent analysis, Cheng 1996 entropy-based. Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Pairwise comparison with Triangular Fuzzy Number (TFN: l, m, u) judgments: umbrella over Van Laarhoven-Pedrycz 1983, Buckley 1985, Chang 1992 extent analysis, Cheng 1996 entropy-based
• •Preserves fuzzy_TFN uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: Belton-Gear-1983 (crisp base); fuzzy variants inherit the issue)
• •Assumes: Experts can articulate preferences in Fuzzy (Triangular) linguistic variables
• •Assumes: Pairwise comparisons satisfy consistency requirements of underlying method

Assumptions to verify

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

When not to use

• •Crisp pairwise data available: use base AHP
• •Experts disagree fundamentally on scale interpretation

Edge cases

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

Common pitfalls

• •Hatalı: 'FUZZY-AHP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts can articulate preferences in Fuzzy (Triangular) linguistic variables
• •Hatalı: 'FUZZY-AHP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Pairwise comparisons satisfy consistency requirements of underlying method
• •Hatalı: FUZZY-AHP'yi 'Crisp pairwise data available' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-AHP'yi 'Experts disagree fundamentally on scale interpretation' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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). Formül: \tilde{A} = [\tilde{a}_{ij}]_{n\times n},\quad \tilde{a}_{ij}=(l_{ij},m_{ij},u_{ij}),\quad \tilde{a}_{ji}=\left(\tfrac{1}{u_{ij}},\tfrac{1}{m_{ij}},\tfrac{1}{l_{ij}}\right),\quad \tilde{a}_{ii}=(1,1,1) Anchor: Yazdi 2022 Ch4 §4.7 Step 1, Eq.(4.1)-(4.2)
2. 2.Adım 2 (F2): Step 2: (Group decision, E experts) Aggregate individual pairwise matrices using TFN geometric mean. Skip this step when a single pre-aggregated matrix is provided. Formül: \tilde{a}_{ij} = \left(\tilde{a}^{1}_{ij}\otimes\tilde{a}^{2}_{ij}\otimes\cdots\otimes\tilde{a}^{E}_{ij}\right)^{1/E} Anchor: Yazdi 2022 Ch4 §4.7 Step 2, Eq.(4.3); Buckley 1985
3. 3.Adım 3 (F3): Step 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}. Formül: \tilde{r}_{i} = \left(\tilde{a}_{i1}\otimes\tilde{a}_{i2}\otimes\cdots\otimes\tilde{a}_{in}\right)^{1/n} Anchor: Yazdi 2022 Ch4 §4.7 Step 3, Eq.(4.4); Buckley 1985 §2
4. 4.Adım 4 (F4): Step 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). Formül: \tilde{w}_{i} = \tilde{r}_{i}\otimes\left(\tilde{r}_{1}\oplus\tilde{r}_{2}\oplus\cdots\oplus\tilde{r}_{n}\right)^{-1} Anchor: Yazdi 2022 Ch4 §4.7 Step 4, Eq.(4.5); Buckley 1985 §3
5. 5.Adım 5 (F5): Step 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. Formül: w_{i} = \frac{L\tilde{w}_{i}+M\tilde{w}_{i}+U\tilde{w}_{i}}{3},\quad w_{i} \leftarrow \frac{w_{i}}{\sum_{k=1}^{n}w_{k}} Anchor: Yazdi 2022 Ch4 §4.7 Step 5, Eq.(4.6); COA defuzzification

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