Rough-ARAS - Rough extension of ARAS

Overview

ROUGH-ARAS (IRN-ELH-ARAS) extends ARAS to IRN uncertainty. Step 1 (F1): define optimal baseline X_0 via element-wise max/min. Step 2 (F2): normalise using ARAS sum-based formula; for cost criteria invert IRN as [1/U,1/L] (swap bounds) before summing. Step 3 (F3): weight normalised values with crisp or rough weights. Step 4 (F4): compute rough row sums S_i^{IRN}. Step 5 (F5): utility degree K_i = defuzz(S_i^{IRN} ⊘ S_0^{IRN}) - rough IRN division first, then midpoint defuzz. Rank descending by K_i.

Output

utility, higher is better

Data

Rough Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Rough MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Augment matrix with optimal alternative A_0 (best per criterion).

   $$X_{0j}^{IRN}=\{\begin{array}{ll}{\max }_i\mathrm{IRN}(x_{ij})& j\in {\Omega }_{+}\\ {\min }_i\mathrm{IRN}(x_{ij})& j\in {\Omega }_{-}\end{array},\mathrm{IRN}(x_{ij})=[RN(I_{li}^{\prime }),RN(I_{ui}^{\prime })]$$

   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

   Linear-sum normalisation (cost criteria inverted first).

   $${\bar{x}}_{ij}^{IRN}=\mathrm{IRN}(x_{ij})\oslash \sum _{k=0}^m\mathrm{IRN}(x_{kj}),j\in {\Omega }_{+};j\in {\Omega }_{-}:\mathrm{IRN}(x_{ij})\leftarrow [1/\mathrm{RN}(I_{ui}^{\prime }),1/\mathrm{RN}(I_{li}^{\prime })]\text{(IRN inverse: swap bounds to preserve }L\le U\text{; then normalize as benefit)}$$

   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

   Weighted normalised matrix d_ij = w_j · x̄_ij.

   $$d_{ij}^{IRN}=w_j\otimes {\bar{x}}_{ij}^{IRN}=[w_j\cdot \mathrm{RN}(I_{li}^{\prime }),w_j\cdot \mathrm{RN}(I_{ui}^{\prime })]$$

   Daoud Ben Amor et al. 2021 (DASA) Step 8; ARAS weighting Eq.(3) applied to IRN-ELH matrix

4. 4

   Optimality function S_i = Σ d_ij (including S_0 row).

   $$S_i^{IRN}=\underset{j=1}{\overset{n}{⨁}}{d}_{ij}^{IRN}=\left[\sum _{j}R{N}_l(d_{ij}),\sum _{j}R{N}_u(d_{ij})\right],i=0,1,\dots ,m$$

   Daoud Ben Amor et al. 2021 (DASA) Step 9; ARAS optimality function Eq.(4) in IRN domain

5. 5

   Utility K_i = S_i / S_0 and descending ranking.

   $$K_i=\mathrm{defuzz}(S_i^{IRN}\oslash S_0^{IRN}),\mathrm{defuzz}([R{N}_l,R{N}_u])=({\overline{RN}}_l+{\overline{RN}}_u)/2;\mathrm{rank}\mathrm{by}{K}_i\downarrow$$

   Daoud Ben Amor et al. 2021 (DASA) Step 9; ARAS utility degree Eq.(5) + IRN defuzzification; ranking: A3>A4>A2>A1 in paper example

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy Rough: L ≤ U; approximations defined by equivalence classes before computation.

• Defuzzification method affects ranking: midpoint (L+U)/2 is the canonical choice but alternatives exist.

Works with

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