qR-TOPSIS - q-Rung Orthopair extension of TOPSIS

Overview

QR-TOPSIS lifts crisp TOPSIS (Hwang-Yoon 1981) into q-Rung Orthopair Fuzzy arithmetic. Weighted q-ROF matrix is built via λα operation (Liu-Wang 2018). PIS/NIS are extracted coordinate-wise per criterion direction. Separation from ideals uses standard q-ROF Minkowski/Euclidean distance with (μ^q, ν^q, π^q) components (Du 2018 style). Final ranking is by closeness coefficient CC_i = d⁻/(d⁺+d⁻), descending. No defuzzification step - distances are crisp scalars produced by the q-ROF distance function.

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

   Construct the q-ROF decision matrix and convert cost criteria to benefit via (μ, ν) → (ν, μ) swap.

   $$\mathbf{D}=(({\mu }_{ij},{\nu }_{ij}){)}_{m\times n},{\mu }_{ij}^q+{\nu }_{ij}^q\le 1;{\tilde{a}}_{ij}^c=({\nu }_{ij},{\mu }_{ij})\text{for cost}j$$

   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

   Weighted q-ROF decision matrix via q-ROF scalar-multiplication (λα operation).

   $${\tilde{v}}_{ij}=((1-(1-{\mu }_{ij}^q{)}^{w_j}{)}^{1/q},{\nu }_{ij}^{w_j})$$

   Liu & Wang 2018 Eq.(6) λα operation; Pinar & Boran 2020 Eq.(58).

3. 3

   q-ROF positive-ideal A⁺ and negative-ideal A⁻ per criterion direction (computed on weighted matrix).

   $$A_j^{+}=(\underset{i}{\max }{\mu }_{ij}^v,\underset{i}{\min }{\nu }_{ij}^v),A_j^{-}=(\underset{i}{\min }{\mu }_{ij}^v,\underset{i}{\max }{\nu }_{ij}^v)\text{after cost-swap (benefit)}$$

   Pinar & Boran 2020 Eqs.(59)-(63).

4. 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.

   $$d_i^{\pm }=\sqrt{\sum _{j=1}^n{w}_j\frac{({\mu }_{ij}^{vq}-{\mu }_j^{\pm q}{)}^2+({\nu }_{ij}^{vq}-{\nu }_j^{\pm q}{)}^2+({\pi }_{ij}^{vq}-{\pi }_j^{\pm q}{)}^2}{2}}$$

   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

   Relative closeness coefficient; rank by CC_i descending.

   $$C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},0\le C{C}_i\le 1$$

   Pinar & Boran 2020 Eq.(65); structurally identical to Hwang & Yoon 1981 closeness coefficient lifted to q-ROF distances.

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

• 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.

• Distance variant choice affects ranking and CC magnitudes. This manifest uses standard q-ROF Euclidean (Du 2018, p=2 case). If a paper specifies Pinar-Boran 2020 novel k-parameter distance (Eq.64), results will differ - use a separate manifest or override L.deviations_from_seminal.

• Cost criteria: this manifest applies (μ, ν) → (ν, μ) swap at Step 1 (Liu-Wang convention). Some implementations (e.g. Pinar-Boran 2020) keep original (μ, ν) and flip min/max inside PIS/NIS construction - both are mathematically equivalent.

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