qR-ARAS - q-Rung Orthopair extension of ARAS

Overview

QR-ARAS extends ARAS to q-Rung Orthopair Fuzzy Numbers. Pipeline: (1) cost criteria are complemented via (μ,ν)→(ν,μ); (2) a fictitious optimal q-ROFN row A_0 is built coordinate-wise; (3) each row (including A_0) is aggregated by q-ROFWA into a single q-ROFN; (4) the q-ROFN is reduced to a crisp score S_i = (1+μ_i^q−ν_i^q)/2 ∈ [0,1]; (5) utility K_i = S_i/S_0 ∈ [0,1] is ranked descending. The q parameter (≥1) is analyst-specified; q=1 reduces to IF-ARAS, q=2 to PF-ARAS, q=3 to FF-ARAS.

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

   Cost-criterion adjustment via q-ROFN complement. For j ∈ Ω_c (cost): â_ij = (ν_ij, μ_ij). For benefit criteria â_ij = a_ij. All criteria are thus represented as 'larger-is-better' in q-ROFN space.

   $${\hat{a}}_{ij}=\{\begin{array}{ll}({\mu }_{ij},{\nu }_{ij})& j\in {\Omega }_b\\ ({\nu }_{ij},{\mu }_{ij})& j\in {\Omega }_c\end{array}$$

   tex Step 1 (Cost Adjustment); Yager 2017

2. 2

   Construct optimal q-ROFN row A_0 coordinate-wise on the cost-adjusted matrix: â_0j = (max_i μ̂_ij, min_i ν̂_ij). A_0 is the fictitious best alternative against which all real alternatives are compared.

   $${\hat{a}}_{0j}=(\underset{i}{\max }{\hat{\mu }}_{ij},\underset{i}{\min }{\hat{\nu }}_{ij}),j=1,\dots ,n$$

   tex Step 2 (Optimal Row); Yager 2017

3. 3

   q-ROFWA aggregation per alternative (including i=0): S̃_i = q-ROFWA_w(â_i1, ..., â_in) = ((1−∏_j(1−μ̂_ij^q)^{w_j})^{1/q}, ∏_j ν̂_ij^{w_j}). Produces one aggregate q-ROFN per row.

   $${\tilde{S}}_i={\text{q-ROFWA}}_w({\hat{a}}_{i1},\dots ,{\hat{a}}_{in})=\left(\right(1-\prod _{j=1}^n(1-{\hat{\mu }}_{ij}^q{)}^{w_j}{)}^{1/q},\prod _{j=1}^n{\hat{\nu }}_{ij}^{w_j}),i=0,1,\dots ,m$$

   Liu & Wang 2018, Definition 8 (q-ROFWA)

4. 4

   Score-function reduction: S_i = M(S̃_i) = (1 + μ_i^q − ν_i^q)/2 ∈ [0,1] where (μ_i, ν_i) are the components of S̃_i. The normalized score M is used (rather than s = μ^q−ν^q ∈ [−1,1]) to keep K_i in [0,1] and avoid sign issues in the ratio K_i = S_i/S_0.

   $$S_i=M({\tilde{S}}_i)=\frac{1+{\mu }_i^q-{\nu }_i^q}{2}\in [0,1],i=0,1,\dots ,m$$

   Erdebilli et al. 2023, Eq.(1) M(α) score; Liu & Wang 2018 score function

5. 5

   Utility degree K_i = S_i / S_0 ∈ [0,1] (since S_0 ≥ S_i by construction). Rank descending: larger K_i = better. K_0 ≡ 1 for the fictitious optimal alternative.

   $$K_i=\frac{S_i}{S_0}\in [0,1];\text{rank}={\mathrm{argsort}}_{\downarrow }(K_1,\dots ,K_m)$$

   Zavadskas & Turskis 2010, Eq.(5) - utility ratio applied to q-ROF aggregated scores

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 every input cell satisfies μ_ij^q + ν_ij^q ≤ 1 with the same q value across the matrix. Different q per cell breaks aggregation semantics.

• Aggregation operator choice matters: q-ROFWA (used here) is averaging-style; q-ROFWG would give different K_i values. Document the operator in any report.

• Score function choice: this manifest uses normalized M(α) = (1+μ^q−ν^q)/2 ∈ [0,1] to keep K_i = S_i/S_0 well-defined and ∈ [0,1]. Using raw s(α) = μ^q−ν^q ∈ [−1,1] can produce negative S_i, making the ratio K_i = S_i/S_0 meaningless when signs differ.

• A_0 is constructed coordinate-wise (max μ, min ν) on the cost-adjusted matrix; this is not necessarily the q-ROFN of any real alternative - it is a fictitious reference.

Works with

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