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Ranking

qR-CoCoSo - q-Rung Orthopair extension of COCOSO

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

Yager, R. R.2017doi:10.1109/TFUZZ.2016.2604005 ↗

Overview

QR-CoCoSo (Peng-Huang 2020 / Kuvvetli 2023). Each q-ROFN cell is reduced to a crisp score r_ij = μ^q − ν^q − ln(1 + π^q) that explicitly penalises hesitancy (Step 1). Scores are min-max normalised per criterion to [0, 1] with benefit/cost direction handled here (Step 2). Two compromise sequences are computed: S_i (weighted sum) and P_i (weighted-power sum). Three appraisal scores k_a (additive ratio), k_b (sum of min-ratios), k_c (balanced λ-ratio, default λ = 0.5) combine S and P. Final k_i = ∛(k_a·k_b·k_c) + (k_a+k_b+k_c)/3 is ranked descending. q parameter is analyst-specified (paper default q = 3, λ = 0.5).

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
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); λ ∈ [0,1] (default 0.5)
1
Her hücre için π = (1 − μ^q − ν^q)^{1/q}, sonra r_ij = μ^q − ν^q − ln(1+π^q). Yüksek-hesitancy cezası burada işler
2
Fayda: r'_ij = (r_ij−r_j^-)/(r_j^+−r_j^-); Maliyet: r'_ij = (r_j^+−r_ij)/(r_j^+−r_j^-). Maliyet yönü BURADA işlenir, F1'de q-ROFN tümleyeniyle değil - yoksa çift-flip
3
S_i = Σ w_j·r'_ij (ağırlıklı toplam, TOPSIS skoru); P_i = Σ (r'_ij)^{w_j} (ağırlıklı üs-toplam, WPM skoru). İki farklı agregat semantiği
4
k_a = (S_i+P_i)/Σ(S_k+P_k) toplamsal oran; k_b = S_i/minS_k + P_i/minP_k worst-relative; k_c = (λS_i+(1−λ)P_i)/(λmaxS_k+(1−λ)maxP_k) λ-dengeli max-relative
5
k_i = ∛(k_a·k_b·k_c) + (k_a+k_b+k_c)/3 - geometrik ortalama + aritmetik ortalama hibridi. Azalan sıralama final ranking
7
Anchor (Kuvvetli 2023 Adana benzin istasyonu, 5 lokasyon × 10 kriter × 3 DM, q=3): MAGDM Steps 1-3 q-ROFWA ile DM agregasyonu, sonra single-DM çekirdek Steps 4-8 (bu manifest). Final ranking A2 ≻ A5 ≻ A3 ≻ A4 ≻ A1. Manifest J fixture sentetik 3×3 strict-dominance, expected [A1, A2, A3]

Fits when / Look elsewhere when

Fits when

•Wider expressivity than IFS/PFS - q-ROFS covers strict-dominance regions both methods reject (μ=0.9, ν=0.6 violates PFS but is valid q-ROFS at q≥3)
•Three-score compromise (k_a/k_b/k_c) reduces single-metric ranking volatility
•Hesitancy penalty in Peng-Huang score explicitly rewards low-π high-clarity assessments - discriminates assessment quality, not just μ−ν gap

Look elsewhere when

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

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

Limitations

•Score-reduction at Step 1 collapses q-ROFN to crisp; full fuzzy uncertainty (μ, ν) variability lost downstream
•k_c parameter λ has no canonical principled value beyond Yazdani 2019 default 0.5 - Kuvvetli 2023 Tablo 10 confirms low sensitivity on strict-dominance regimes, but mid-rank ties can swap under mixed-dominance
•MAGDM aggregation across DMs is external - single-DM kernel cannot natively handle expert disagreement

Edge cases and pitfalls

•Tüm r_ij eşit bir sütunda (degenerate min-max): r_j^+ = r_j^- olunca F2 sıfıra bölme - set r'_ij = 1 (uninformative criterion, eşit katkı)
•P_i sıfır terimi: min-max sonrası en az bir hücre = 0; (0)^{w_j} = 0, P_i bu kriter için katkısız (matematik doğru, Yazdani 2019 Eq.3 zero-protection içermez)
•k_i'de eşitlik: önce S_i+P_i büyüklüğüne bak, sonra alt-id determinist argsort
•λ uç değerleri: λ=1 → pure S_i/maxS oranı; λ=0 → pure P_i/maxP oranı; ikisi de geçerli. Mid-rank tie swap riski mixed-dominance rejimlerinde

Value-space violation: every input cell must satisfy μ_ij^q + ν_ij^q ≤ 1 with the same q across the matrix. The hesitancy π = (1 − μ^q − ν^q)^{1/q} is needed for Step 1 - illegal entries produce complex-valued π.
Score function choice matters: this manifest uses Peng-Huang 2020 r_ij = μ^q − ν^q − ln(1 + π^q) (with hesitancy penalty, per Kuvvetli 2023 Eq.9). Simpler s(α) = μ^q − ν^q (no penalty) or M(α) = (1+μ^q−ν^q)/2 produce DIFFERENT rankings because Step 1 outputs feed nonlinearly into k_a/k_b/k_c. Switching the score function is a methodological deviation - document explicitly.
After Step 2 min-max, at least one cell in each criterion column equals 0 (the worst). If 0 appears in P_i = Σ (r'_ij)^{w_j} with a fractional weight, (0)^{w_j} = 0 and may zero-out a row. This is mathematically correct CoCoSo behaviour but can produce extreme P_i contrasts; report Step 2 output to users.
λ parameter in k_c: paper default 0.5 (balanced). λ closer to 1 emphasises additive component S_i; closer to 0 emphasises multiplicative P_i. Kuvvetli 2023 sensitivity analysis (Tablo 10) shows λ has minimal effect on ranking under the strict-dominance regime.

Works with

Commonly takes its weights from

n_a

How to cite

Yager, R. R. (2017). Generalized orthopair fuzzy sets. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2016.2604005

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

QR-COCOSO

Overview

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

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); λ ∈ [0,1] (default 0.5)

Her hücre için π = (1 − μ^q − ν^q)^{1/q}, sonra r_ij = μ^q − ν^q − ln(1+π^q). Yüksek-hesitancy cezası burada işler

Fayda: r'_ij = (r_ij−r_j^-)/(r_j^+−r_j^-); Maliyet: r'_ij = (r_j^+−r_ij)/(r_j^+−r_j^-). Maliyet yönü BURADA işlenir, F1'de q-ROFN tümleyeniyle değil - yoksa çift-flip

S_i = Σ w_j·r'_ij (ağırlıklı toplam, TOPSIS skoru); P_i = Σ (r'_ij)^{w_j} (ağırlıklı üs-toplam, WPM skoru). İki farklı agregat semantiği

k_a = (S_i+P_i)/Σ(S_k+P_k) toplamsal oran; k_b = S_i/minS_k + P_i/minP_k worst-relative; k_c = (λS_i+(1−λ)P_i)/(λmaxS_k+(1−λ)maxP_k) λ-dengeli max-relative

k_i = ∛(k_a·k_b·k_c) + (k_a+k_b+k_c)/3 - geometrik ortalama + aritmetik ortalama hibridi. Azalan sıralama final ranking

Anchor (Kuvvetli 2023 Adana benzin istasyonu, 5 lokasyon × 10 kriter × 3 DM, q=3): MAGDM Steps 1-3 q-ROFWA ile DM agregasyonu, sonra single-DM çekirdek Steps 4-8 (bu manifest). Final ranking A2 ≻ A5 ≻ A3 ≻ A4 ≻ A1. Manifest J fixture sentetik 3×3 strict-dominance, expected [A1, A2, A3]

Fits when / Look elsewhere when

Fits when

•Wider expressivity than IFS/PFS - q-ROFS covers strict-dominance regions both methods reject (μ=0.9, ν=0.6 violates PFS but is valid q-ROFS at q≥3)
•Three-score compromise (k_a/k_b/k_c) reduces single-metric ranking volatility
•Hesitancy penalty in Peng-Huang score explicitly rewards low-π high-clarity assessments - discriminates assessment quality, not just μ−ν gap

Look elsewhere when

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

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

Limitations

•Score-reduction at Step 1 collapses q-ROFN to crisp; full fuzzy uncertainty (μ, ν) variability lost downstream
•k_c parameter λ has no canonical principled value beyond Yazdani 2019 default 0.5 - Kuvvetli 2023 Tablo 10 confirms low sensitivity on strict-dominance regimes, but mid-rank ties can swap under mixed-dominance
•MAGDM aggregation across DMs is external - single-DM kernel cannot natively handle expert disagreement

Edge cases and pitfalls

•Tüm r_ij eşit bir sütunda (degenerate min-max): r_j^+ = r_j^- olunca F2 sıfıra bölme - set r'_ij = 1 (uninformative criterion, eşit katkı)

•P_i sıfır terimi: min-max sonrası en az bir hücre = 0; (0)^{w_j} = 0, P_i bu kriter için katkısız (matematik doğru, Yazdani 2019 Eq.3 zero-protection içermez)

•k_i'de eşitlik: önce S_i+P_i büyüklüğüne bak, sonra alt-id determinist argsort

•λ uç değerleri: λ=1 → pure S_i/maxS oranı; λ=0 → pure P_i/maxP oranı; ikisi de geçerli. Mid-rank tie swap riski mixed-dominance rejimlerinde

Value-space violation: every input cell must satisfy μ_ij^q + ν_ij^q ≤ 1 with the same q across the matrix. The hesitancy π = (1 − μ^q − ν^q)^{1/q} is needed for Step 1 - illegal entries produce complex-valued π.

Score function choice matters: this manifest uses Peng-Huang 2020 r_ij = μ^q − ν^q − ln(1 + π^q) (with hesitancy penalty, per Kuvvetli 2023 Eq.9). Simpler s(α) = μ^q − ν^q (no penalty) or M(α) = (1+μ^q−ν^q)/2 produce DIFFERENT rankings because Step 1 outputs feed nonlinearly into k_a/k_b/k_c. Switching the score function is a methodological deviation - document explicitly.

After Step 2 min-max, at least one cell in each criterion column equals 0 (the worst). If 0 appears in P_i = Σ (r'_ij)^{w_j} with a fractional weight, (0)^{w_j} = 0 and may zero-out a row. This is mathematically correct CoCoSo behaviour but can produce extreme P_i contrasts; report Step 2 output to users.

λ parameter in k_c: paper default 0.5 (balanced). λ closer to 1 emphasises additive component S_i; closer to 0 emphasises multiplicative P_i. Kuvvetli 2023 sensitivity analysis (Tablo 10) shows λ has minimal effect on ranking under the strict-dominance regime.