DHF-VIKOR - VIKOR yönteminin Çift Kararsız Bulanık (DHF) uzantısı

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

Adım 1–3 (An §3.4) — Alternatif kümesi A={A_1,…,A_m} ve öznitelik kümesi C={C_1,…,C_n}; DM ağırlık vektörü λ=(λ_1,…,λ_K), Σλ_k=1; K adet öznel öznitelik ağırlık vektörü ω̄^k; üst liderin t bilinmeyenli kısmi ağırlık vektörü ω (bilinmeyenler 1-d toplar); DHF karar matrisi E=(E_{ij})_{m×n}.

$E = (E_{i j})_{m \times n}, E_{i j} = ⟨ {h_{i j}^{1}, \dots, h_{i j}^{K}}, {g_{i j}^{1}, \dots, g_{i j}^{K}} ⟩; \sum_{k = 1}^{K} λ_{k} = 1; \sum_{j = t + 1}^{n} ω_{j} = d, \sum_{j = 1}^{t} ω_{j} = 1 - d$

LaTeX
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
Adım 4 (An Eq.17) — DM-ağırlıklı DHF karar matrisi Ē; her h_{ij}^k ve g_{ij}^k bileşeni ilgili λ_k ile çarpılır.

${\bar{E}}_{i j} = λ \otimes E_{i j} = ⟨ {λ_{1} h_{i j}^{1}, λ_{2} h_{i j}^{2}, \dots, λ_{K} h_{i j}^{K}}, {λ_{1} g_{i j}^{1}, λ_{2} g_{i j}^{2}, \dots, λ_{K} g_{i j}^{K}} ⟩$

LaTeX
\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
Adım 5 — Yöne-duyarlı DHF pozitif ideal (E_j^+) ve negatif ideal (E_j^-): fayda kriterleri için E_j^+=⟨{1},{0}⟩, E_j^-=⟨{0},{1}⟩; maliyet kriterleri için tersi.

$E_{j}^{+} = {\begin{matrix} ⟨ {1}, {0} ⟩ & C_{j} benefit \\ ⟨ {0}, {1} ⟩ & C_{j} cost \end{matrix}, E_{j}^{-} = {\begin{matrix} ⟨ {0}, {1} ⟩ & C_{j} benefit \\ ⟨ {1}, {0} ⟩ & C_{j} cost \end{matrix}$

LaTeX
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}
Adım 6 (An Eq.10–11) — Her A_i için ağırlıklı sapma toplamları y_i^+(ω̄) (PIS'e) ve y_i^-(ω̄) (NIS'e); hibrit DHF Minkowski/Hamming/Euclidean mesafesi kullanılır.

$y_{i}^{+} (\bar{ω}) = \sum_{j = 1}^{t} ω_{j} d ({\bar{E}}_{i j}, E_{j}^{+}), y_{i}^{-} (\bar{ω}) = \sum_{j = 1}^{t} ω_{j} d ({\bar{E}}_{i j}, E_{j}^{-})$

LaTeX
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}^{-})
Adım 7 (An Eq.12–16) — Eq.(16) iki-amaçlı eniyilemeyi çöz; nesnel + öznel ω̄'yi ticari katsayı α ile birleştir; kısıt Σ ω_j = 1-d, ω_j ≥ 0. Lagrange çarpanları ile çözülür.

$min {\bar{y} (\bar{ω}), \bar{z} (\bar{ω})} \Rightarrow min {\frac{α}{m} \sum_{j = 1}^{t} (y_{j}^{+} (\bar{ω}) - y_{j}^{-} (\bar{ω})) + (1 - α) \sum_{k = 1}^{K} λ_{k} \sum_{j = 1}^{t} (ω_{j} - ω_{k j})^{2}} s.t. \sum_{j = 1}^{t} ω_{j} = 1 - d, ω_{j} \geq 0$

LaTeX
\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
Adım 8 (An Eq.18–19) — Grup faydası S_i ve bireysel pişmanlık R_i; Adım 7'deki kapsamlı ω kullanılır.

$S_{i} = \sum_{j = 1}^{n} ω_{j} \frac{d ({\bar{E}}_{i j}, E_{j}^{+})}{d (E_{j}^{+}, E_{j}^{-})}, R_{i} = {max}_{j} [ω_{j} \frac{d ({\bar{E}}_{i j}, E_{j}^{+})}{d (E_{j}^{+}, E_{j}^{-})}]$

LaTeX
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]
Adım 9–10 (An Eq.20) — Toplam VIKOR skoru Q_i ve azalan Q sıralaması (paper konvansiyonu; klasik VIKOR'dan ters yönde). En yüksek Q en iyidir.

$Q_{i} = β \frac{S_{i} - S^{-}}{S^{+} - S^{-}} + (1 - β) \frac{R_{i} - R^{-}}{R^{+} - R^{-}}, S^{+} = {max}_{i} S_{i}, S^{-} = {min}_{i} S_{i}, R^{+} = {max}_{i} R_{i}, R^{-} = {min}_{i} R_{i}; rank descending in Q_{i}$

LaTeX
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}

Sezgi

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

Sonucu okuma: 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.

Varsayımlar

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

Ne zaman kullanılmaz

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

Sınırlılıklar

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

Sık yapılan hatalar

Sıralama yönü karışıklığı: An et al. 2025 Q'da AZALAN sıralar (en yüksek Q en iyidir). Klasik VIKOR (düşük Q = iyi) ile karıştırma; varsayılan paper konvansiyonu olmalı.
λ-çarpmasından önce DHFE geçerliliği: γ^++η^+≤1; '/' slotları max'tan dışlanır.
Mesafe seçimi (Hamming/Euclidean/Minkowski) S/R/Q'yu etkiler. An §4.4 p=1; duyarlılık için p'yi değiştir.

Hesap adımları ve dayanakları

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 '/'.

Dayanak: An et al. 2025 §3.4 Steps 1-3
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.

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

Dayanak: An et al. 2025 §3.4 Step 5
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.

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

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

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

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