qR-ARAS: q-Rung Orthopair extension of ARAS

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

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

Edge cases

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

Common pitfalls

• •Hatalı: 'QR-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid q-Rung Orthopair Fuzzy numbers/tuples
• •Hatalı: 'QR-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'QR-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: QR-ARAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: QR-ARAS'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: 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. Formül: \hat{a}_{ij}=\begin{cases}(\mu_{ij},\nu_{ij}) & j\in\Omega_b\\(\nu_{ij},\mu_{ij}) & j\in\Omega_c\end{cases} Anchor: tex Step 1 (Cost Adjustment); Yager 2017
2. 2.Adım 2 (F2): Step 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. Formül: \hat{a}_{0j}=\bigl(\max_i\hat\mu_{ij},\ \min_i\hat\nu_{ij}\bigr),\quad j=1,\ldots,n Anchor: tex Step 2 (Optimal Row); Yager 2017
3. 3.Adım 3 (F3): Step 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. Formül: \tilde{S}_i=\text{q-ROFWA}_w(\hat{a}_{i1},\ldots,\hat{a}_{in})=\Bigl(\bigl(1-\prod_{j=1}^{n}(1-\hat\mu_{ij}^{q})^{w_j}\bigr)^{1/q},\ \prod_{j=1}^{n}\hat\nu_{ij}^{w_j}\Bigr),\ i=0,1,\ldots,m Anchor: Liu & Wang 2018, Definition 8 (q-ROFWA)
4. 4.Adım 4 (F4): Step 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. Formül: S_i=M(\tilde{S}_i)=\dfrac{1+\mu_i^{q}-\nu_i^{q}}{2}\in[0,1],\ i=0,1,\ldots,m Anchor: Erdebilli et al. 2023, Eq.(1) M(α) score; Liu & Wang 2018 score function
5. 5.Adım 5 (F5): Step 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. Formül: K_i=\dfrac{S_i}{S_0}\in[0,1];\quad \text{rank}=\operatorname{argsort}_{\downarrow}(K_1,\ldots,K_m) Anchor: Zavadskas & Turskis 2010, Eq.(5): utility ratio applied to q-ROF aggregated scores

Commonly paired with

• •n_a + QR-ARAS (common)

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