qR-TOPSIS: q-Rung Orthopair extension of TOPSIS

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 TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for QR-TOPSIS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'QR-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid q-Rung Orthopair Fuzzy numbers/tuples
• •Hatalı: 'QR-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'QR-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: QR-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: QR-TOPSIS'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 (μ, ν) → (ν, μ) swap. Formül: \mathbf{D} = \bigl((\mu_{ij},\nu_{ij})\bigr)_{m\times n},\quad \mu_{ij}^{q}+\nu_{ij}^{q}\le 1;\quad \tilde{a}_{ij}^{c} = (\nu_{ij},\mu_{ij})\ \text{for cost}\ j Anchor: Yager 2017 Def 1 (q-ROFS); Liu & Wang 2018 Eq.(1) (basic operations); cost-swap convention from Liu & Wang 2018 §4 Eq.(26).
2. 2.Adım 2 (F2): Step 2: Weighted q-ROF decision matrix via q-ROF scalar-multiplication (λα operation). Formül: \tilde{v}_{ij} = \Bigl((1-(1-\mu_{ij}^{q})^{w_{j}})^{1/q},\ \nu_{ij}^{w_{j}}\Bigr) Anchor: Liu & Wang 2018 Eq.(6) λα operation; Pinar & Boran 2020 Eq.(58).
3. 3.Adım 3 (F3): Step 3: q-ROF positive-ideal A⁺ and negative-ideal A⁻ per criterion direction (computed on weighted matrix). Formül: A^{+}_{j} = \bigl(\max_{i}\mu_{ij}^{v},\ \min_{i}\nu_{ij}^{v}\bigr),\quad A^{-}_{j} = \bigl(\min_{i}\mu_{ij}^{v},\ \max_{i}\nu_{ij}^{v}\bigr)\ \text{after cost-swap (benefit)} Anchor: Pinar & Boran 2020 Eqs.(59)-(63).
4. 4.Adım 4 (F4): Step 4: Standard q-ROF Euclidean separation using (μ^q, ν^q, π^q) components, where π = (1−μ^q−ν^q)^{1/q}. Weighted across criteria. Formül: d_{i}^{\pm} = \sqrt{\sum_{j=1}^{n} w_{j}\,\frac{(\mu_{ij}^{vq}-\mu_{j}^{\pm q})^{2}+(\nu_{ij}^{vq}-\nu_{j}^{\pm q})^{2}+(\pi_{ij}^{vq}-\pi_{j}^{\pm q})^{2}}{2}} Anchor: Du 2018 Minkowski-type distance for q-ROFS (p=2 case); standard practice in q-ROF MCDM. NOT Pinar-Boran 2020 Eq.(64) novel distance: see L.deviations_from_seminal.
5. 5.Adım 5 (F5): Step 5: Relative closeness coefficient; rank by CC_i descending. Formül: CC_{i} = \dfrac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}},\quad 0 \le CC_{i} \le 1 Anchor: Pinar & Boran 2020 Eq.(65); structurally identical to Hwang & Yoon 1981 closeness coefficient lifted to q-ROF distances.

Commonly paired with

• •n_a + QR-TOPSIS (common)

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