DHF-VIKOR - Dual Hesitant Fuzzy extension of VIKOR

Overview

DHF-VIKOR (An et al. 2025) is a MAGDM extension of VIKOR for K decision-makers with incomplete attribute weights. The pipeline (i) λ-aggregates per-DM DHFE judgements (Eq.17), (ii) sets direction-aware DHF PIS/NIS, (iii) computes hybrid DHF Minkowski distances (Eqs.7-9), (iv) solves a bi-objective programme (Eq.16) to derive a comprehensive subjective+objective weight vector ω̄ via trade-off α, (v) computes S_i, R_i, Q_i and ranks DESCENDING in Q_i (the highest Q_i is best - An et al.'s explicit convention; this inverts classical Opricovic-Tzeng VIKOR). Two tunables control the answer: α (subjective vs. objective weight emphasis) and β (group utility vs. individual regret emphasis); both default to 0.5.

Output

utility, lower is better

Data

Hesitant, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Dual Hesitant Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   3 (An §3.4) - Identify alternative set A={A_1,…,A_m}, attribute set C={C_1,…,C_n}; receive DM weight vector λ=(λ_1,…,λ_K) with Σλ_k=1, λ_k∈(0,1]; receive K subjective attribute weight vectors ω̄^k=(ω_{k1},…,ω_{kt}); receive leader's partial weight vector ω with t unknown entries summing to (1-d) and (n-t) known entries summing to d; assemble DHF decision matrix E=(E_{ij})_{m×n} where each E_{ij}=⟨{h_{ij}^1,…,h_{ij}^K},{g_{ij}^1,…,g_{ij}^K}⟩ holds K per-DM (h,g) pairs (Table 3); DM abstention is encoded by '/'.

   $$E=(E_{ij}{)}_{m\times n},E_{ij}=⟨\{h_{ij}^1,\dots ,h_{ij}^K\},\{g_{ij}^1,\dots ,g_{ij}^K\}⟩;\sum _{k=1}^K{\lambda }_k=1;\sum _{j=t+1}^n{\omega }_j=d,\sum _{j=1}^t{\omega }_j=1-d$$

   An et al. 2025 §3.4 Steps 1-3

2. 2

   Step 4 (An Eq.17) - Compute the DM-weighted DHF decision matrix Ē=(Ē_{ij})_{m×n} via the scalar-DHFE multiplication Ē_{ij}=λ⊗E_{ij}: each per-DM membership h_{ij}^k is scaled by λ_k, each per-DM non-membership g_{ij}^k is scaled by λ_k, preserving the K-tuple structure. Slot positions occupied by '/' (DM abstention) remain '/' after scaling.

   $${\bar{E}}_{ij}=\lambda \otimes E_{ij}=⟨\{{\lambda }_1{h}_{ij}^1,{\lambda }_2{h}_{ij}^2,\dots ,{\lambda }_K{h}_{ij}^K\},\{{\lambda }_1{g}_{ij}^1,{\lambda }_2{g}_{ij}^2,\dots ,{\lambda }_K{g}_{ij}^K\}⟩$$

   An et al. 2025 §3.4 Step 4, Eq.(17)

3. 3

   Step 5 (An §3.4) - Construct the direction-aware DHF positive ideal solution (DHFPIS) E_j^+ and negative ideal solution (DHFNIS) E_j^- per attribute: for benefit attributes (max) E_j^+=⟨{1},{0}⟩ and E_j^-=⟨{0},{1}⟩; for cost attributes (min) E_j^+=⟨{0},{1}⟩ and E_j^-=⟨{1},{0}⟩. The (PIS,NIS) pair acts as the universal reference point for distance-based group utility / individual regret.

   $$E_j^{+}=\{\begin{array}{ll}⟨\{1\},\{0\}⟩& C_j\text{ benefit}\\ ⟨\{0\},\{1\}⟩& C_j\text{ cost}\end{array},E_j^{-}=\{\begin{array}{ll}⟨\{0\},\{1\}⟩& C_j\text{ benefit}\\ ⟨\{1\},\{0\}⟩& C_j\text{ cost}\end{array}$$

   An et al. 2025 §3.4 Step 5

4. 4

   Step 6 (An Eqs.10-11) - Compute the sums of weighted deviations from DHF PIS y_i^+(ω̄) and from DHF NIS y_i^-(ω̄) for every alternative A_i, leaving ω as a symbolic vector (functions of the t unknown ω_j); the hybrid DHF Minkowski distance d(·,·) of Eq.(9) (or its Hamming p=1 / Euclidean p=2 specialisations Eqs.7-8) is used. A_i is preferable when y_i^+ is small and y_i^- is large.

   $$y_i^{+}(\bar{\omega })=\sum _{j=1}^t{\omega }_{j}d({\bar{E}}_{ij},E_j^{+}),y_i^{-}(\bar{\omega })=\sum _{j=1}^t{\omega }_{j}d({\bar{E}}_{ij},E_j^{-})$$

   An et al. 2025 §3.4 Step 6, Eqs.(10)-(11)

5. 5

   Step 7 (An Eqs.12-16) - Solve the bi-objective optimisation Eq.(16) for the comprehensive attribute weight vector ω̄, which fuses the objective sub-model Eq.(13) (single-objective amalgamation of multi-objective Eq.(12) minimising Σ(y_j^+(ω̄)-y_j^-(ω̄)) over m alternatives) with the subjective sub-model Eq.(14) (minimise Σ_k λ_k Σ_j (ω_j-ω_{kj})^2 over K DMs) via trade-off coefficient α; subject to Σ_{j=1}^{t} ω_j = 1-d and ω_j ≥ 0. Solved by Lagrange multipliers (paper §4.4 closed-form for K=3).

   $$\min \left\{\bar{y}(\bar{\omega }),\bar{z}(\bar{\omega })\right\}\Rightarrow \min \left\{\frac{\alpha }{m}\sum _{j=1}^t(y_j^{+}(\bar{\omega })-y_j^{-}(\bar{\omega }))+(1-\alpha )\sum _{k=1}^K{\lambda }_k\sum _{j=1}^t({\omega }_j-{\omega }_{kj}{)}^2\right\}\text{s.t.}\sum _{j=1}^t{\omega }_j=1-d,{\omega }_j\ge 0$$

   An et al. 2025 §3.3 Eqs.(12)-(16), §3.4 Step 7

6. 6

   Step 8 (An Eqs.18-19) - Compute the group utility S_i (sum across attributes of normalised distance-to-PIS) and the individual regret R_i (maximum across attributes of normalised distance-to-PIS) for every alternative, using the comprehensive ω from Step 7.

   $$S_i=\sum _{j=1}^n{\omega }_j\frac{d({\bar{E}}_{ij},E_j^{+})}{d(E_j^{+},E_j^{-})},R_i=\underset{j}{\max }\left[{\omega }_j\frac{d({\bar{E}}_{ij},E_j^{+})}{d(E_j^{+},E_j^{-})}\right]$$

   An et al. 2025 §3.4 Step 8, Eqs.(18)-(19)

7. 7

   Step 9-10 (An Eq.20) - Compute the aggregate VIKOR score Q_i with compromise coefficient β∈[0,1] and rank alternatives in descending order of Q_i (this is An et al.'s explicit convention: with S^+=max_i S_i and R^+=max_i R_i the alternative attaining max Q is taken as best). NOTE: this convention differs from classical Opricovic-Tzeng VIKOR, where lower Q is better - kept verbatim for literature fidelity.

   $$Q_i=\beta \frac{S_i-S^{-}}{S^{+}-S^{-}}+(1-\beta )\frac{R_i-R^{-}}{R^{+}-R^{-}},S^{+}=\underset{i}{\max }{S}_i,S^{-}=\underset{i}{\min }{S}_i,R^{+}=\underset{i}{\max }{R}_i,R^{-}=\underset{i}{\min }{R}_i;\text{rank descending in }{Q}_i$$

   An et al. 2025 §3.4 Steps 9-10, Eq.(20)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• •when y_i^+ is small and y_i^- is large.

• Ranking-direction confusion: An et al. 2025 ranks DESCENDING in Q (highest Q = best) because S^+/R^+ in their normalisation are max (not min) of S_i/R_i. Do NOT silently swap to classical VIKOR (lower Q = better) - the engine must preserve An's convention; if a user prefers classical convention, expose it as a separate option, not the default.

• DHFE validity check before λ-multiplication: each original E_{ij}=⟨h,g⟩ must satisfy γ^++η^+≤1 (Zhu et al. 2012, Def.1); '/' DM-slots are excluded from max(h)/max(g) but the remaining elements must still pass the check.

• Distance choice (p=1 Hamming Eq.7, p=2 Euclidean Eq.8, general Minkowski Eq.9) affects S/R/Q. An et al. §4.4 uses p=1; for sensitivity, vary p and re-rank.

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