qR-VIKOR: q-Rung Orthopair extension of VIKOR

Overview

q-Rung Orthopair outranking/ranking: q-Rung Orthopair Fuzzy Number (q-ROFN: μ, ν; μ^q+ν^q ≤ 1, q ≥ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: q-Rung Orthopair outranking/ranking: q-Rung Orthopair Fuzzy Number (q-ROFN: μ, ν; μ^q+ν^q ≤ 1, q ≥ 1)
• •Preserves q_rung_orthopair 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 q-Rung Orthopair 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 q-Rung Orthopair 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

• •default v = 0.5 for consensus).
• •If C1 fails, return the maximal prefix A^(1)..A^(M) such that Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'QR-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid q-Rung Orthopair Fuzzy numbers/tuples
• •Hatalı: 'QR-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'QR-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: QR-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: QR-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: Construct the q-ROF decision matrix and convert cost criteria to benefit via the (μ, ν) → (ν, μ) swap (q-ROFN complement). After this step every criterion is treated as benefit. Formül: \hat{a}_{ij} = \begin{cases}(\mu_{ij},\nu_{ij}) & j\in\Omega_{b}\\(\nu_{ij},\mu_{ij}) & j\in\Omega_{c}\end{cases},\quad \mu_{ij}^{q}+\nu_{ij}^{q}\le 1 Anchor: Yager 2017 Def 1 (q-ROFS); Liu & Wang 2018 Eq.(1) basic operations; cost complement (μ,ν)^c=(ν,μ): standard q-ROF MCDM convention, also stated in QR-VIKOR.tex §3 Step 1.
2. 2.Adım 2 (F2): Step 2: Score-based per-criterion q-ROF positive-ideal Q^PIS_j and negative-ideal Q^NIS_j: pick the ENTIRE q-ROFN from the alternative i that maximises (resp. minimises) the score function M(α) = (1 + μ^q − ν^q)/2 ∈ [0,1] for each criterion j (computed on the cost-adjusted matrix where all criteria are benefit). Note: PIS/NIS coordinates always come from a real alternative under this extraction. Formül: Q^{PIS}_{j} = \arg\max_{i}\,M(\alpha_{ij}),\quad Q^{NIS}_{j} = \arg\min_{i}\,M(\alpha_{ij}),\quad M(\alpha) = \dfrac{1 + \mu^{q} - \nu^{q}}{2} Anchor: Erdebilli et al. 2023 Eqs.(16)-(17); Erdebilli-Sıcakyüz 2024 Eqs.(15)-(16); Liu-Wang 2018 Def 2 (score function S(α)=μ^q−ν^q; M(α)=(1+S)/2 is the [0,1]-rescaled form used as PIS/NIS selector in both q-ROF VIKOR application anchors).
3. 3.Adım 3 (F3): Step 3: Normalised q-ROF gap f_ij defined as the ratio of two q-ROF Euclidean distances: distance from Q^PIS_j to the cell, normalised by the criterion range distance from Q^PIS_j to Q^NIS_j. Euclidean form: d_E(α_1, α_2) = √(((μ_1^q − μ_2^q)² + (ν_1^q − ν_2^q)² + (π_1^q − π_2^q)²) / 2), with π = (1 − μ^q − ν^q)^{1/q}. Formül: f_{ij} = \dfrac{d_{E}(Q^{PIS}_{j},\,\hat{a}_{ij})}{d_{E}(Q^{PIS}_{j},\,Q^{NIS}_{j})},\quad d_{E}(\alpha_{1},\alpha_{2}) = \sqrt{\dfrac{(\mu_{1}^{q}-\mu_{2}^{q})^{2}+(\nu_{1}^{q}-\nu_{2}^{q})^{2}+(\pi_{1}^{q}-\pi_{2}^{q})^{2}}{2}} Anchor: Erdebilli et al. 2023 Eq.(20) + Erdebilli-Sıcakyüz 2024 Eq.(19) (q-ROF Euclidean used in q-ROF VIKOR pipeline); underlying distance: Du 2018 (Minkowski p=2, q-ROF generalisation).
4. 4.Adım 4 (F4): Step 4: Group utility S_i (L_1 aggregation) and individual regret R_i (L_∞ aggregation) of weighted q-ROF gaps. Formül: S_{i} = \sum_{j=1}^{n} w_{j}\,f_{ij},\quad R_{i} = \max_{j}\bigl[w_{j}\,f_{ij}\bigr] Anchor: Erdebilli et al. 2023 Eqs.(18)-(19) + Erdebilli-Sıcakyüz 2024 Eqs.(17)-(18) (S_i = Σ w_j·d(a_ij,Q^PIS)/d(Q^PIS,Q^NIS), R_i = max_j w_j · same ratio). Structurally Opricovic 1998 L_1/L_∞ aggregation, lifted to q-ROF gap.
5. 5.Adım 5 (F5): Step 5: Compromise index Q_i: convex combination of normalised S and R, weighted by the mechanism coefficient v ∈ [0, 1] (default v = 0.5 for consensus). Formül: Q_{i} = v\,\dfrac{S_{i}-S^{*}}{S^{-}-S^{*}} + (1-v)\,\dfrac{R_{i}-R^{*}}{R^{-}-R^{*}},\quad S^{*}=\min_{i}S_{i},\ S^{-}=\max_{i}S_{i},\ R^{*}=\min_{i}R_{i},\ R^{-}=\max_{i}R_{i} Anchor: Erdebilli et al. 2023 Eq.(21) + Erdebilli-Sıcakyüz 2024 Eq.(20) (merit function Q^mf = v·(S−S*)/(S^−−S*) + (1−v)·(R−R*)/(R^−−R*), v=0.5 default).
6. 6.Adım 6 (F6): Step 6: Rank ASCENDING by Q_i (smaller Q is better). Propose A^(1) (the lowest-Q alternative) as the compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximal prefix A^(1)..A^(M) such that Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: DQ = \dfrac{1}{m-1};\quad C1:\ Q(A^{(2)})-Q(A^{(1)}) \ge DQ;\quad C2:\ A^{(1)}\ \text{is best in}\ S\ \text{or in}\ R Anchor: Classical Opricovic 1998 / Opricovic-Tzeng 2004 C1 (acceptable advantage, DQ=1/(m−1)) and C2 (acceptable stability) checks. NOTE: Erdebilli et al. 2023 §3.2 Step 5 and Erdebilli-Sıcakyüz 2024 Step 8 both state 'largest to smallest': this contradicts the merit-function normalisation in Eq.(21)/(20) which gives smaller Q = closer to ideal. We treat the paper wording as a shared paper-side error and follow ascending convention.

Commonly paired with

• •n_a + QR-VIKOR (common)

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