HF-AHP: Hesitant Fuzzy AHP (AHP-Hesitant Group Decision Making via HMPM)

Overview

Hesitant multiplicative pairwise comparison (HMPR): linear-programming prioritisation (HMPM). Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Hesitant multiplicative pairwise comparison (HMPR): linear-programming prioritisation (HMPM)
• •Preserves hesitant uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from classical AHP (Belton-Gear 1983; Saaty 1990); HMPM additionally inherits LP-sensitivity to deviation parameter t)
• •Assumes: Domain experts available and willing to express hesitancy as discrete value sets
• •Assumes: Pairwise comparisons drawn from Saaty's 1-9 ratio scale [1/9, 9]
• •Assumes: Reciprocity z_ij^(ρ(l)) · z_ji^(ρ(l)) = 1 holds for the ρ-th paired elements

Assumptions to verify

• •Domain experts available and willing to express hesitancy as discrete value sets
• •Pairwise comparisons drawn from Saaty's 1-9 ratio scale [1/9, 9]
• •Reciprocity z_ij^(ρ(l)) · z_ji^(ρ(l)) = 1 holds for the ρ-th paired elements

When not to use

• •Experts can give precise crisp judgments → use classical AHP
• •Judgment uncertainty is additive on [0,1] rather than multiplicative on [1/9, 9] → use Torra-style HF preference relations / HF-AHP variants on [0,1] (separate manifest)
• •Number of HME elements per cell is very large (LP scales as Σ|z_ij|)

Edge cases

• •When the problem has a goal-criterion-alternative hierarchy, run F1-F4 once at the criterion level (yielding criterion weights w_c) and once per criterion at the alternative level (yielding alternativ

Common pitfalls

• •Hatalı: 'HF-AHP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Domain experts available and willing to express hesitancy as discrete value sets
• •Hatalı: 'HF-AHP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Pairwise comparisons drawn from Saaty's 1-9 ratio scale [1/9, 9]
• •Hatalı: 'HF-AHP bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Reciprocity z_ij^(ρ(l)) · z_ji^(ρ(l)) = 1 holds for the ρ-th paired elements
• •Hatalı: HF-AHP'yi 'Experts can give precise crisp judgments → use classical AHP' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: HF-AHP'yi 'Judgment uncertainty is additive on [0,1] rather than multiplicative on [1/9, 9] → use Torra-style HF preference relatio' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: HF-AHP'yi 'Number of HME elements per cell is very large (LP scales as Σ|z_ij|)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1 (Zhu-Xu 2014, p.795 Def.2): Collect pairwise judgments as a hesitant multiplicative preference relation Z = (z_ij)_{n×n}, with z_ij ⊂ [1/9, 9] (Saaty 1-9 ratio scale), z_ii = {1}, ρ-th-element reciprocity z_ij^(ρ(l)) · z_ji^(ρ(l)) = 1, and strict ordering z_ij^(ρ(l)) < z_ij^(ρ(l+1)) for i<j. Formül: Z = (z_{ij})_{n \times n}, \quad z_{ij} = \{ z_{ij}^{(l)} \mid l = 1, \ldots, |z_{ij}| \} \subset [1/9, 9]; \quad z_{ij}^{(\rho(l))} \cdot z_{ji}^{(\rho(l))} = 1, \ z_{ii} = 1; \quad z_{ij}^{(\rho(l))} < z_{ij}^{(\rho(l+1))}, \ i < j Anchor: Zhu-Xu 2014, p.795 Def.2 + Eqs.(3)-(4)
2. 2.Adım 2 (F2): Step 2 (Zhu-Xu 2014, p.795-796 Eqs.(5)-(9)): State the consistency relation w_i/w_j = z_ij^(1) or … or z_ij^(|z_ij|) (Eq.(5)). For each pair (i,j) with i<j and each l ∈ {1,…,|z_ij|}, define the residual R_ij^(l)(w) = w_i − w_j · z_ij^(l), giving a disjunctive system of |z_ij| candidate equations per pair (Eq.(9)). Formül: \dfrac{w_{i}}{w_{j}} = z_{ij}^{(1)} \ \text{or} \ \ldots \ \text{or} \ z_{ij}^{(|z_{ij}|)}; \quad R_{ij}^{(l)}(w) = w_{i} - w_{j} \cdot z_{ij}^{(l)} \cong 0, \quad i<j, \ l = 1, \ldots, |z_{ij}| Anchor: Zhu-Xu 2014, p.795 Eq.(5) + p.796 Eq.(9)
3. 3.Adım 3 (F3): Step 3 (Zhu-Xu 2014, p.796 Eqs.(10)-(12)): Replace each crisp equation by a linear V-shaped membership function m_ij^(l)(w) on residual R_ij^(l)(w) with deviation parameter t. The overall satisfaction is the pointwise minimum λ = min_{i<j,l} m_ij^(l)(w), and HMPM maximises λ (Eq.(12)). Formül: m_{ij}^{(l)}(w) = \begin{cases} 1 - \dfrac{R_{ij}^{(l)}(w)}{t}, & R_{ij}^{(l)}(w) \ge 0 \\[4pt] 1 + \dfrac{R_{ij}^{(l)}(w)}{t}, & R_{ij}^{(l)}(w) < 0 \end{cases}; \quad \lambda = \max \, \min \bigl\{ m_{ij}^{(l)}(w) \,\big|\, i<j, \ l=1,\ldots,|z_{ij}| \bigr\} Anchor: Zhu-Xu 2014, p.796 Eqs.(10)-(12) + Fig.1
4. 4.Adım 4 (F4): Step 4 (Zhu-Xu 2014, p.796 Eqs.(14)-(15)): Reformulate the max-min program as the DISJUNCTIVE linear program in (λ, w_1, …, w_n). The 'or' inside Eq.(15) means for each pair (i,j), i<j, the priority vector only needs to satisfy the V-shape constraints for AT LEAST ONE l ∈ {1,…,|z_ij|}, not all l simultaneously. Two tractable implementations: (a) enumerate the Π_{i<j}|z_ij| candidate element selections, solve a conjunctive LP per combination, return the combination with the largest λ; (b) MILP with binary selector variables y_ij^(l) ∈ {0,1} (one selected per pair) and big-M relaxation of the unselected constraints. Output: priority vector w and group satisfaction index GSI = λ*. Formül: \begin{aligned} \max \quad & \lambda \\ \text{s.t.} \quad & \bigvee_{l=1}^{|z_{ij}|} \bigl[\, t \lambda + (w_{i} - w_{j} z_{ij}^{(l)}) \le t \ \wedge\ t \lambda - (w_{i} - w_{j} z_{ij}^{(l)}) \le t \,\bigr], \quad i<j \\ & \sum_{i=1}^{n} w_{i} = 1, \quad w_{i} \ge 0, \ i = 1, \ldots, n \end{aligned} \\ \text{MILP form:} \quad t\lambda + (w_i - w_j z_{ij}^{(l)}) \le t + M(1 - y_{ij}^{(l)}), \ t\lambda - (w_i - w_j z_{ij}^{(l)}) \le t + M(1 - y_{ij}^{(l)}), \ \sum_l y_{ij}^{(l)} = 1, \ y_{ij}^{(l)} \in \{0,1\} Anchor: Zhu-Xu 2014, p.796 Eqs.(14)-(15) (disjunctive LP form of HMPM); see also p.795 Eq.(5) which establishes the 'or' semantics
5. 5.Adım 5 (F5): Step 5 (Zhu-Xu 2014, p.796 §3 closing ¶ + §4.1): Interpret GSI = λ as the maximum overall membership-function value. λ → 1 means the HMPR is (almost) fully consistent following Eq.(5); λ near 0 or negative indicates strong divergence. HMPM focuses only on the values that satisfy Eq.(5)'s disjunction, not on aggregating the whole HME. Formül: \text{GSI} = \lambda^{*} \in (-\infty, 1]; \quad \lambda^{*} \to 1 \Longleftrightarrow \text{HMPR fully consistent under Eq.(5)} Anchor: Zhu-Xu 2014, p.796 ¶ after Eq.(15) + p.796 §4.1 'Group satisfaction index'
6. 6.Adım 6 (F6): Step 6 (Zhu-Xu 2014, p.798 §5, p.799 Table 7): Hierarchical synthesis. When the problem has a goal-criterion-alternative hierarchy, run F1-F4 once at the criterion level (yielding criterion weights w_c) and once per criterion at the alternative level (yielding alternative weights A_r for criterion r). Combine by weighted arithmetic average: global priority of alternative i = Σ_r A_r[i] · w_c[r]. Formül: B_{i} = \sum_{r=1}^{n_{\text{crit}}} A_{r}[i] \cdot w_{c}[r], \quad i = 1, \ldots, n_{\text{alt}} Anchor: Zhu-Xu 2014, p.798 §5 (water-conservancy example) + p.799 Table 7

Commonly paired with

• •HF-AHP + HF-TOPSIS (high)
• •HF-AHP + HF-VIKOR (medium)
• •HF-AHP + HF-WASPAS (medium)

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