qR-CODAS - q-Rung Orthopair extension of CODAS

Overview

QR-CODAS extends CODAS to q-ROFN inputs. Step 1 applies q-ROFN complement for cost criteria and Liu-Wang 2018 q-ROFWA per-criterion weighting; q-ROFN tuples persist through Step 2 (coordinate-wise NIS). Step 3 reduces q-ROFN to crisp scalars via Du 2018 q-ROF Euclidean (E_i) and Hamming (T_i) distances from the NIS. Step 4 builds the relative assessment h_ik with the standard CODAS threshold function ψ (default τ = 0.02). Step 5 sums h_ik into AS_i and ranks descending. NO score-function defuzzification is performed - the distance step is the natural q-ROFN → crisp reduction in CODAS.

Output

utility, higher 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

   Karar matrisi m×n, hücreler (μ_ij, ν_ij) q-ROFN; ağırlıklar w_j (Σw_j=1); kriter yönleri max/min; q sabiti (default 3); τ eşiği (default 0.02)

2. 1

   Maliyet kriterleri için (μ,ν)↔(ν,μ); sonra ṽ_ij = ((1−(1−μ̂^q)^{w_j})^{1/q}, ν̂^{w_j}). q-ROFN'lar HÂLÂ fuzzy

3. 2

   α_j^- = (min_i μ̃_ij, max_i ν̃_ij). Worst q-ROFN her kriter için. Hesitancy π̃ buradan türetilir (1−μ̃^q−ν̃^q)^{1/q}

4. 3

   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; d_H = (|μ_1^q−μ_2^q|+|ν_1^q−ν_2^q|+|π_1^q−π_2^q|)/3. E_i = √Σw_j·d_E²(ṽ_ij, α_j^-); T_i = Σw_j·d_H(ṽ_ij, α_j^-). q-ROFN → crisp BURADA

5. 4

   h_ik = (E_i−E_k) + ψ(E_i−E_k)·(T_i−T_k); ψ(x)=1 eğer |x|≥τ, aksi 0; τ=0.02 default. T-bileşeni sadece Öklid eşik üstünde aktif

6. 5

   AS_i = Σ_k h_ik; azalan sıralama final ranking. Büyük AS = NIS'ten daha uzak = daha iyi

7. 7

   Anchor (Naz-Akram 2022 arc welding robot, 9 alt × 6 crit × 4 DM, q=3, τ=0.02, ϖ'=(0.19, 0.31, 0.17, 0.33)): MAGDM 4-DM Steps 1-3 2TLq-ROFWHM agregasyonu, sonra CODAS Steps 4-8 single-DM. Manifest J fixture sentetik 3×3 strict-dominance, expected [A1, A2, A3]

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

• •Tüm alternatifler bir sütunda identik (degenerate NIS): α_j^- = ṽ_ij her i için, d_E = d_H = 0, sütun katkısız (sabit kriter düşürme eşdeğeri)
• •Tüm E_i, T_i eşit (extreme degenerate): h_ik ≡ 0, AS_i ≡ 0; tie-breaker K.tie_handling alternative_id argsort
• •τ çok büyük (ψ ≡ 0): h_ik (E_i−E_k)'ye iner, sadece Öklid; Hamming kullanılmaz, dual-distance robustness kaybolur
• •Karma q yasak (E-2): Liu-Wang q-ROFWA q-üs-ortalama uniform q gerektirir; Du 2018 (μ^q, ν^q, π^q) koordinatları uniform q

• Value-space violation: every input cell must satisfy μ_ij^q + ν_ij^q ≤ 1 with a single q ≥ 1 across the matrix. The hesitancy π = (1 − μ^q − ν^q)^{1/q} enters BOTH d_E and d_H - illegal entries make π complex and break Step 3.

• Threshold τ in ψ: default 0.02 follows Keshavarz-Ghorabaee 2016 crisp CODAS, but q-ROF distances are typically smaller (bounded by hesitancy term). For q-ROFN inputs, an analyst-controlled scaling of τ (e.g., τ scaled by max-pairwise-E) may be needed to avoid trivial ψ ≡ 1.

• Cost direction handled at F1 via q-ROFN complement (μ, ν) ↔ (ν, μ), NOT via min-NIS-flip at F2. F2's NIS uses min on μ̃ and max on ν̃ uniformly for all post-F1 criteria. Applying both would double-flip the cost criterion.

• F2 NIS is coordinate-wise (min μ, max ν), NOT a tuple drawn from the matrix. The resulting α_j^- need not equal any actual alternative's q-ROFN at column j - it is the worst-corner extreme.

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