qR-VIKOR - q-Rung Orthopair extension of VIKOR

Overview

QR-VIKOR lifts classical VIKOR (Opricovic 1998 / Opricovic-Tzeng 2004) into q-Rung Orthopair Fuzzy arithmetic. Cost criteria are converted to benefit via q-ROFN complement (μ, ν) → (ν, μ). Per-criterion positive- and negative-ideals are extracted coordinate-wise (f_j* = (max μ, min ν), f_j^- = (min μ, max ν)). The crisp gap (f_j*−x_ij)/(f_j*−f_j^-) is replaced by the q-ROF Euclidean distance ratio d_E(f_j*, a_ij) / d_E(f_j*, f_j^-) (Du 2018 distance, p=2). Group utility S_i (L_1) and individual regret R_i (L_∞) aggregate the weighted gaps. The compromise index Q_i = v·(S−S*)/(S^−−S*) + (1−v)·(R−R*)/(R^−−R*) is ranked ASCENDING (smaller Q = better). The final compromise solution is post-processed via C1 (acceptable advantage) and C2 (acceptable stability) checks. No defuzzification step - the gap is already crisp once distance is taken.

Output

utility, lower is better

Data

Q-Rung Orthopair Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Q-Rung Orthopair Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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.

   $${\hat{a}}_{ij}=\{\begin{array}{ll}({\mu }_{ij},{\nu }_{ij})& j\in {\Omega }_b\\ ({\nu }_{ij},{\mu }_{ij})& j\in {\Omega }_c\end{array},{\mu }_{ij}^q+{\nu }_{ij}^q\le 1$$

   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

   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.

   $$Q_j^{PIS}=\arg \underset{i}{\max }M({\alpha }_{ij}),Q_j^{NIS}=\arg \underset{i}{\min }M({\alpha }_{ij}),M(\alpha )=\frac{1+{\mu }^q-{\nu }^q}{2}$$

   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

   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}.

   $$f_{ij}=\frac{d_E(Q_j^{PIS},{\hat{a}}_{ij})}{d_E(Q_j^{PIS},Q_j^{NIS})},d_E({\alpha }_1,{\alpha }_2)=\sqrt{\frac{({\mu }_1^q-{\mu }_2^q{)}^2+({\nu }_1^q-{\nu }_2^q{)}^2+({\pi }_1^q-{\pi }_2^q{)}^2}{2}}$$

   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

   Group utility S_i (L_1 aggregation) and individual regret R_i (L_∞ aggregation) of weighted q-ROF gaps.

   $$S_i=\sum _{j=1}^n{w}_j{f}_{ij},R_i=\underset{j}{\max }\left[w_j{f}_{ij}\right]$$

   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

   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).

   $$Q_i=v\frac{S_i-S^{\ast }}{S^{-}-S^{\ast }}+(1-v)\frac{R_i-R^{\ast }}{R^{-}-R^{\ast }},S^{\ast }=\underset{i}{\min }{S}_i,S^{-}=\underset{i}{\max }{S}_i,R^{\ast }=\underset{i}{\min }{R}_i,R^{-}=\underset{i}{\max }{R}_i$$

   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

   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)}.

   $$DQ=\frac{1}{m-1};C1:Q(A^{(2)})-Q(A^{(1)})\ge DQ;C2:A^{(1)}\text{is best in}S\text{or in}R$$

   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.

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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)}.

• Value-space violation: ensure all entries satisfy μ^q + ν^q ≤ 1 with q ≥ 1 BEFORE computation. q=1→IFS, q=2→PFS, q=3→FFS.

• PIS/NIS extraction variant choice. This manifest uses SCORE-BASED extraction Q^PIS_j = argmax_i M(α_ij) (entire q-ROFN of the alternative that maximises M(α)=(1+μ^q−ν^q)/2), matching both verified application anchors (Erdebilli 2023 Eqs.16-17, Erdebilli-Sıcakyüz 2024 Eqs.15-16). A coordinate-wise variant (f_j* = (max μ, min ν) per j) gives identical results on strictly-dominant data but differs on non-dominant data - see L.alternative_formulations_known.

• Cost criteria: this manifest applies q-ROFN complement (μ, ν) → (ν, μ) at Step 1. Some implementations keep original (μ, ν) and reverse min/max inside the f_j*/f_j^- extraction - mathematically equivalent for monotone distances.

• Ranking direction: Q_i is SMALLER-is-better (lower Q = closer to ideal). DO NOT confuse with TOPSIS closeness coefficient CC_i (higher is better). Both anchor papers (Erdebilli 2023 §3.2 / Erdebilli-Sıcakyüz 2024 Step 8) state 'largest to smallest' - this contradicts the merit-function normalisation Eq.(21)/(20) and is a shared paper-side wording error; follow classical Opricovic ascending convention.

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