qR-EDAS - q-Rung Orthopair extension of EDAS

Overview

QR-EDAS follows Pattern B (Li-Wei-Lu 2019 + Liu-Wang 2018 + Keshavarz-Ghorabaee 2015 lineage): at F1 the q-ROFN matrix is reduced to crisp scores s_ij = μ_ij^q − ν_ij^q (Liu-Wang 2018 Def. 2.4). Cost criteria are handled at F1 by q-ROFN complement α^c=(ν,μ), equivalent to flipping the score sign. From F2 onwards the pipeline is identical to crisp EDAS (Keshavarz-Ghorabaee 2015): AV per criterion, PDA/NDA, weighted sums SP/SN, NSP/NSN normalisation, appraisal score AS=(NSP+NSN)/2. Rank descending by AS.

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

   Liu-Wang score reduction on q-ROFN decision matrix: s_ij = μ_ij^q − ν_ij^q ∈ [−1, 1]. Cost criteria use q-ROFN complement α^c = (ν, μ) before scoring (equivalently, score sign flips).

   $$s_{ij}={\mu }_{ij}^q-{\nu }_{ij}^q;({\mu }_{ij},{\nu }_{ij}{)}^c=({\nu }_{ij},{\mu }_{ij})\text{for}j\in {\Omega }_c$$

   Liu-Wang 2018, Def. 2.4 score function; QR-EDAS.tex §Algorithmic Steps, Step 1

2. 2

   Average solution per criterion on score values: AV_j = s̄_j = (1/m) Σ_i s_ij.

   $$A{V}_j={\bar{s}}_j=\frac{1}{m}\sum _{i=1}^m{s}_{ij}$$

   QR-EDAS.tex §Algorithmic Steps, Step 2; Keshavarz-Ghorabaee 2015 Eq.(3) adapted to score domain

3. 3

   Positive (PDA) and negative (NDA) distance from AV. Benefit criteria: PDA rewards above-average, NDA penalises below-average. Direction handled at F1 via complement so both branches use the benefit form here.

   $$PD{A}_{ij}=\frac{\max (0,s_{ij}-{\bar{s}}_j)}{|{\bar{s}}_j|},ND{A}_{ij}=\frac{\max (0,{\bar{s}}_j-s_{ij})}{|{\bar{s}}_j|}$$

   QR-EDAS.tex §Algorithmic Steps, Step 3; Keshavarz-Ghorabaee 2015 Eqs.(4)-(5)

4. 4

   Weighted PDA and NDA sums across criteria.

   $$S{P}_i=\sum _{j=1}^n{w}_{j}PD{A}_{ij};S{N}_i=\sum _{j=1}^n{w}_{j}ND{A}_{ij}$$

   QR-EDAS.tex §Algorithmic Steps, Step 4; Keshavarz-Ghorabaee 2015 Eqs.(6)-(7)

5. 5

   Normalise SP and SN by their per-method maxima.

   $$NS{P}_i=\frac{S{P}_i}{{\max }_{k}S{P}_k};NS{N}_i=1-\frac{S{N}_i}{{\max }_{k}S{N}_k}$$

   QR-EDAS.tex §Algorithmic Steps, Step 4 (normalisation); Keshavarz-Ghorabaee 2015 Eqs.(8)-(9)

6. 6

   Appraisal score AS_i ∈ [0, 1] and descending ranking.

   $$A{S}_i=\frac{1}{2}(NS{P}_i+NS{N}_i);\text{rank}={\mathrm{argsort}}_{\downarrow }(A{S}_1,\dots ,A{S}_m)$$

   QR-EDAS.tex §Algorithmic Steps, Step 4 (AS); Keshavarz-Ghorabaee 2015 Eq.(10)

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 the analyst-specified q (q=1→IFS; q=2→PFS; q=3→FFS). DecisionMind requires a single q across the entire matrix.

• Cost direction must be handled at F1 via q-ROFN complement (μ,ν)^c=(ν,μ), NOT applied to score sign later. Applying both double-flips and breaks ranking.

• Degenerate AV: if s̄_j = 0 for some criterion (alternatives split symmetrically), PDA/NDA division by |s̄_j| explodes. DecisionMind falls back to s̄_j → max(|s̄_j|, ε) with ε=1e-9, recorded in P.edge_case_policies.

• max SP = 0 or max SN = 0: NSP/NSN normalisation undefined. DecisionMind sets NSP_i=0 (no alternative exceeds AV) or NSN_i=1 (no alternative below AV), preserving the AS interpretation.

Works with

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