Rough-ARAS: Rough extension of ARAS

Overview

Rough outranking/ranking: Rough number (lower approximation L, upper approximation U). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Rough outranking/ranking: Rough number (lower approximation L, upper approximation U)
• •Preserves rough 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 Rough 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 Rough 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 ROUGH-ARAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'ROUGH-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Rough numbers/tuples
• •Hatalı: 'ROUGH-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'ROUGH-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: ROUGH-ARAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: ROUGH-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: Augment matrix with optimal alternative A_0 (best per criterion). Formül: X_{0j}^{IRN} = \begin{cases}\max_i \mathrm{IRN}(x_{ij}) & j\in\Omega_{+}\\\min_i \mathrm{IRN}(x_{ij}) & j\in\Omega_{-}\end{cases},\quad \mathrm{IRN}(x_{ij}) = [RN(I'_{li}), RN(I'_{ui})] Anchor: Daoud Ben Amor-Moalla Frikha-Martínez López 2021 (DASA, DOI:10.1109/DASA53625.2021.9681928) Step 7; ARAS structure per Zavadskas-Turskis 2010 Eq.(1)
2. 2.Adım 2 (F2): Step 2: Linear-sum normalisation (cost criteria inverted first). Formül: \bar{x}_{ij}^{IRN} = \mathrm{IRN}(x_{ij}) \oslash \sum_{k=0}^{m} \mathrm{IRN}(x_{kj}),\ j\in\Omega_{+};\quad j\in\Omega_{-}: \mathrm{IRN}(x_{ij}) \leftarrow \bigl[1/\mathrm{RN}(I'_{ui}),\ 1/\mathrm{RN}(I'_{li})\bigr]\ \text{(IRN inverse: swap bounds to preserve }L\le U\text{; then normalize as benefit)} Anchor: Daoud Ben Amor et al. 2021 (DASA, DOI:10.1109/DASA53625.2021.9681928) Step 7; IRN inverse [1/U,1/L] per DEV-1 fix
3. 3.Adım 3 (F3): Step 3: Weighted normalised matrix d_ij = w_j · x̄_ij. Formül: d_{ij}^{IRN} = w_j \otimes \bar{x}_{ij}^{IRN} = [w_j \cdot \mathrm{RN}(I'_{li}), w_j \cdot \mathrm{RN}(I'_{ui})] Anchor: Daoud Ben Amor et al. 2021 (DASA) Step 8; ARAS weighting Eq.(3) applied to IRN-ELH matrix
4. 4.Adım 4 (F4): Step 4: Optimality function S_i = Σ d_ij (including S_0 row). Formül: S_i^{IRN} = \bigoplus_{j=1}^{n} d_{ij}^{IRN} = \left[\sum_j RN_l(d_{ij}), \sum_j RN_u(d_{ij})\right],\quad i=0,1,\ldots,m Anchor: Daoud Ben Amor et al. 2021 (DASA) Step 9; ARAS optimality function Eq.(4) in IRN domain
5. 5.Adım 5 (F5): Step 5: Utility K_i = S_i / S_0 and descending ranking. Formül: K_i = \mathrm{defuzz}(S_i^{IRN} \oslash S_0^{IRN}),\quad \mathrm{defuzz}([RN_l, RN_u]) = (\bar{RN}_l + \bar{RN}_u)/2;\quad \mathrm{rank\ by\ }K_i\downarrow Anchor: Daoud Ben Amor et al. 2021 (DASA) Step 9; ARAS utility degree Eq.(5) + IRN defuzzification; ranking: A3>A4>A2>A1 in paper example

Commonly paired with

• •n_a + ROUGH-ARAS (common)

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