qR-COPRAS - q-Rung Orthopair extension of COPRAS

Overview

QR-COPRAS Pattern B: each q-ROFN cell is reduced to a crisp score s_ij = μ^q − ν^q at F1, shifted to [0, 2] via s̃ = s + 1 (so the cost denominator in F3 stays positive), column-sum normalised, and weighted. F2 splits the weighted row into benefit (S_i^+) and cost (S_i^-) sums. F3 computes Q_i = S_i^+ + (Σ_k S_k^-)/(S_i^- · Σ_k(1/S_k^-)) - the larger Q_i, the better. F4 reports N_i = Q_i / max Q × 100% as a utility-degree summary. Ranking is descending in Q_i (equivalently, in N_i). Cost direction is handled by the Ω_c partition at F2, NOT by q-ROFN complement at F1 (the s + 1 shift assumes scores are interpreted in their original sign).

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

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

• •All-benefit (Ω_c=∅): Q_i=S_i^+ fallback; ranking identical to q-ROF SAW under all-benefit inputs.
• •All-cost (Ω_b=∅): Q_i=(Σ_k S_k^-)/(S_i^- · Σ_k(1/S_k^-)); smaller cost wins.
• •S_k^-=0 for some k: engine adds ε=1e-12 to all S_k^- before harmonic mean to avoid division-by-zero.
• •Mixed q across cells: E-2 input check fails; Liu-Wang score is q-dependent, mixed q produces incomparable scores.

• Value-space violation: every input cell must satisfy μ_ij^q + ν_ij^q ≤ 1 with the same q across the matrix. Liu-Wang s(α) = μ^q − ν^q ∈ [−1, 1] is q-dependent, so mixed q produces incomparable scores.

• Score function choice: this manifest uses Liu-Wang 2018 s(α) = μ^q − ν^q (no hesitancy penalty). Peng-Huang 2020 hesitancy-penalty score r = μ^q − ν^q − ln(1 + π^q) is an alternative; results can DIFFER under hesitancy-heavy inputs.

• Q_i denominator: F3 requires Σ_k(1/S_k^-) and S_i^- · Σ_k(1/S_k^-). If any S_k^- = 0 (e.g., all-zero cost column after shift+normalisation), the formula divides by zero. The s + 1 shift guarantees ŝ_ij ≥ 0; degenerate-all-zero cost only occurs when ALL alternatives have μ^q = ν^q for every cost criterion. Engine should fall back to Q_i = S_i^+ in that edge case.

• No cost criteria (Ω_c = ∅): the harmonic-mean term in Q_i is undefined (sum over empty set). Engine should set Q_i = S_i^+ and proceed. Ranking is identical to SAW under all-benefit inputs.

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