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

Overview

Priority vector entries sum to 1 and indicate relative importance. The Group Satisfaction Index λ measures how well the priority vector satisfies the disjunctive consistency relation w_i/w_j ∈ z_ij (Eq.(5)). λ → 1 = fully consistent under at least one element-selection per cell; λ near 0 or negative = serious divergence, re-elicit judgments. Choose deviation parameter t = 1 unless you have a reason to tighten or loosen the V-shape (paper shows w is invariant under t > 0; only λ scales).

Output

Weight, higher is better

Data

Hesitant, expert input required

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

Water resources allocation, supplier selection under expert disagreement, infrastructure prioritisation, policy ranking with hesitant judgments

How it works

1. 1

   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.

   $$Z=(z_{ij}{)}_{n\times n},z_{ij}=\{z_{ij}^{(l)}\mid l=1,\dots ,|z_{ij}|\}\subset [1/9,9];z_{ij}^{(\rho (l))}\cdot z_{ji}^{(\rho (l))}=1,z_{ii}=1;z_{ij}^{(\rho (l))}<z_{ij}^{(\rho (l+1))},i<j$$

   Zhu-Xu 2014, p.795 Def.2 + Eqs.(3)-(4)

2. 2

   (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)).

   $$\frac{w_i}{w_j}=z_{ij}^{(1)}\text{or}\dots \text{or}{z}_{ij}^{(|z_{ij}|)};R_{ij}^{(l)}(w)=w_i-w_j\cdot z_{ij}^{(l)}\cong 0,i<j,l=1,\dots ,|z_{ij}|$$

   Zhu-Xu 2014, p.795 Eq.(5) + p.796 Eq.(9)

3. 3

   (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)).

   $$m_{ij}^{(l)}(w)=\{\begin{array}{ll}1-\frac{R_{ij}^{(l)}(w)}{t},& R_{ij}^{(l)}(w)\ge 0\\ 1+\frac{R_{ij}^{(l)}(w)}{t},& R_{ij}^{(l)}(w)<0\end{array};\lambda =\max \min \{m_{ij}^{(l)}(w)|i<j,l=1,\dots ,|z_{ij}|\}$$

   Zhu-Xu 2014, p.796 Eqs.(10)-(12) + Fig.1

4. 4

   (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*.

   $$\begin{gathered} \begin{array}{rl}\max & \lambda \\ \text{s.t.}& \underset{l=1}{\overset{|z_{ij}|}{\bigvee }}[t\lambda +(w_i-w_j{z}_{ij}^{(l)})\le t\wedge t\lambda -(w_i-w_j{z}_{ij}^{(l)})\le t],i<j\\ & \sum _{i=1}^n{w}_i=1,w_i\ge 0,i=1,\dots ,n\end{array} \\ \text{MILP form:}t\lambda +(w_i-w_j{z}_{ij}^{(l)})\le t+M(1-y_{ij}^{(l)}),t\lambda -(w_i-w_j{z}_{ij}^{(l)})\le t+M(1-y_{ij}^{(l)}),\sum _l{y}_{ij}^{(l)}=1,y_{ij}^{(l)}\in \{0,1\} \end{gathered}$$

   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

   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.

   $$\text{GSI}={\lambda }^{\ast }\in (-\infty ,1];{\lambda }^{\ast }\to 1⟺\text{HMPR fully consistent under Eq.(5)}$$

   Zhu-Xu 2014, p.796 ¶ after Eq.(15) + p.796 §4.1 'Group satisfaction index'

6. 6

   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].

   $$B_i=\sum _{r=1}^{n_{\text{crit}}}{A}_r[i]\cdot w_c[r],i=1,\dots ,n_{\text{alt}}$$

   Zhu-Xu 2014, p.798 §5 (water-conservancy example) + p.799 Table 7

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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

• Inserting HME elements out of ascending order violates Eq.(4) ordering; the LP still solves but the canonical reciprocity interpretation is lost.

• Using GHMA / GHMWA aggregation (Eqs.(17)-(18)) instead of HMPM produces different priorities - paper §4.3 ¶3 explicitly contrasts the two: HMPM picks values that satisfy consistency, while aggregation operators average over the entire HME. Don't mix.

• Choosing t too small can drive λ negative even for consistent matrices (Zhu-Xu 2014 Table 2, t=0.05 → λ=−0.8082). Negative λ is not a failure mode for w but can confuse users; report t alongside λ.

Works with

How to cite