DHF-VIKOR: Dual Hesitant Fuzzy extension of VIKOR

Overview

Dual Hesitant outranking/ranking: Dual Hesitant Fuzzy Element (DHFE: h(x) membership set, g(x) non-membership set). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Dual Hesitant outranking/ranking: Dual Hesitant Fuzzy Element (DHFE: h(x) membership set, g(x) non-membership set)
• •Preserves hesitant 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: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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

When not to use

• •Crisp data sufficient: use base VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'DHF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Hatalı: 'DHF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'DHF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: DHF-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: DHF-VIKOR'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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 '/'. Formül: E = (E_{ij})_{m\times n},\ E_{ij} = \langle\{h_{ij}^{1},\dots,h_{ij}^{K}\},\{g_{ij}^{1},\dots,g_{ij}^{K}\}\rangle;\ \sum_{k=1}^{K}\lambda_{k}=1;\ \sum_{j=t+1}^{n}\omega_{j}=d,\ \sum_{j=1}^{t}\omega_{j}=1-d Anchor: An et al. 2025 §3.4 Steps 1-3
2. 2.Adım 2 (F2): 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. Formül: \bar{E}_{ij} = \lambda\otimes E_{ij} = \langle\{\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}\}\rangle Anchor: An et al. 2025 §3.4 Step 4, Eq.(17)
3. 3.Adım 3 (F3): 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. Formül: E_{j}^{+} = \begin{cases}\langle\{1\},\{0\}\rangle & C_{j}\text{ benefit}\\ \langle\{0\},\{1\}\rangle & C_{j}\text{ cost}\end{cases},\quad E_{j}^{-} = \begin{cases}\langle\{0\},\{1\}\rangle & C_{j}\text{ benefit}\\ \langle\{1\},\{0\}\rangle & C_{j}\text{ cost}\end{cases} Anchor: An et al. 2025 §3.4 Step 5
4. 4.Adım 4 (F4): 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. Formül: y_{i}^{+}(\bar{\omega}) = \sum_{j=1}^{t}\omega_{j}\,d(\bar{E}_{ij},E_{j}^{+}),\quad y_{i}^{-}(\bar{\omega}) = \sum_{j=1}^{t}\omega_{j}\,d(\bar{E}_{ij},E_{j}^{-}) Anchor: An et al. 2025 §3.4 Step 6, Eqs.(10)-(11)
5. 5.Adım 5 (F5): 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). Formül: \min\left\{\bar{y}(\bar{\omega}),\bar{z}(\bar{\omega})\right\} \;\Rightarrow\; \min\left\{\frac{\alpha}{m}\sum_{j=1}^{t}\bigl(y_{j}^{+}(\bar{\omega})-y_{j}^{-}(\bar{\omega})\bigr) + (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}\geq 0 Anchor: An et al. 2025 §3.3 Eqs.(12)-(16), §3.4 Step 7
6. 6.Adım 6 (F6): 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. Formül: S_{i} = \sum_{j=1}^{n}\omega_{j}\,\dfrac{d(\bar{E}_{ij},E_{j}^{+})}{d(E_{j}^{+},E_{j}^{-})},\quad R_{i} = \max_{j}\left[\omega_{j}\,\dfrac{d(\bar{E}_{ij},E_{j}^{+})}{d(E_{j}^{+},E_{j}^{-})}\right] Anchor: An et al. 2025 §3.4 Step 8, Eqs.(18)-(19)
7. 7.Adım 7 (F7): 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. Formül: Q_{i} = \beta\,\dfrac{S_{i}-S^{-}}{S^{+}-S^{-}} + (1-\beta)\,\dfrac{R_{i}-R^{-}}{R^{+}-R^{-}},\ S^{+}=\max_{i}S_{i},\ S^{-}=\min_{i}S_{i},\ R^{+}=\max_{i}R_{i},\ R^{-}=\min_{i}R_{i};\ \text{rank descending in } Q_{i} Anchor: An et al. 2025 §3.4 Steps 9-10, Eq.(20)

Commonly paired with

• •n_a + DHF-VIKOR (common)

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