Grey-ARAS - Grey extension of ARAS

Overview

grey-aras extends ARAS to handle Grey uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Grey Interval Number (GIN: [x̲, x̄]) algebra. The final scores are defuzzified via whitenisation: (x̲ + x̄)/2 before ranking.

Output

utility, higher is better

Data

Grey Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

How it works

1. 1

   Form Grey Decision-Making Matrix (GDMM). Each entry ⊗x_{ij} = [x_{ij}^α, x_{ij}^γ] is a grey interval. Append optimal row A0: for benefit criteria ⊗x_{0j} = max_i ⊗x_{ij}; for cost criteria ⊗x_{0j} = min_i ⊗x_{ij}. \otimes x_{0j} = \begin{cases}\max_i \otimes x_{ij} & j\in J^{+}\\ \min_i \otimes x_{ij}^{*} & j\in J^{-}\end{cases}

   $$\otimes x_{ij}=[x_{ij}^{\alpha },x_{ij}^{\gamma }],x_{ij}^{\alpha }\le x_{ij}^{\gamma }$$

   Turskis & Zavadskas 2010, §2 Eqs.(1)-(7); §3 Eqs.(8)-(9). Grey operations: addition (α+α,γ+γ), multiplication (αα,γγ), inverse (1/γ,1/α).

2. 2

   Normalize GDMM. For cost criteria, first convert ⊗x*_ij = (⊗x_ij)^{-1} = [1/x_ij^γ, 1/x_ij^α]. Then for all criteria (benefit direction): ⊗x̄_ij = ⊗x_ij / Σ_{k=0}^m ⊗x_kj = [x_ij^α / Σ x_kj^γ, x_ij^γ / Σ x_kj^α]. \otimes\bar{x}_{ij} = \frac{\otimes x_{ij}}{\sum_{k=0}^{m}\otimes x_{kj}} = \left[\frac{x^{\alpha}_{ij}}{\sum_k x^{\gamma}_{kj}},\; \frac{x^{\gamma}_{ij}}{\sum_k x^{\alpha}_{kj}}\right]

   $$j\in J^{-}:\otimes x_{ij}^{\ast }=(\otimes x_{ij}{)}^{-1}=\left[\frac{1}{x_{ij}^{\gamma }},\frac{1}{x_{ij}^{\alpha }}\right]\text{[Eq.7 applied]}$$

   Turskis & Zavadskas 2010, §3 Eqs.(11)-(12). Grey division rule: [a/b]=[aα/bγ, aγ/bα] from Eq.(5).

3. 3

   Compute normalized-weighted matrix. Multiply each normalised grey entry by the grey criterion weight: ⊗x̂_ij = ⊗x̄_ij × ⊗w_j = [x̄_ij^α × w_j^α, x̄_ij^γ × w_j^γ]. Constraint: Σ w_j = 1. \otimes\hat{x}_{ij} = \otimes\bar{x}_{ij} \times \otimes w_j = \left[\bar{x}^{\alpha}_{ij} \cdot w^{\alpha}_j,\; \bar{x}^{\gamma}_{ij} \cdot w^{\gamma}_j\right]

   $$\sum _{j=1}^n{w}_j=1,w_j\in (0,1)$$

   Turskis & Zavadskas 2010, §3 Eqs.(13),(15). Grey multiplication: [a×b]=[aα×bα, aγ×bγ] from Eq.(4).

4. 4

   Compute grey optimality function ⊗S_i = Σ_j ⊗x̂_ij = [Σ x̂_ij^α, Σ x̂_ij^γ]. Then whitenise (defuzzify) via centre-of-area: S_i = (1/2)(S_i^α + S_i^γ). S_i = \tfrac{1}{2}\left(S_i^{\alpha} + S_i^{\gamma}\right) \quad\text{[Eq.17 - whitenisation]}

   $$\otimes S_i=\sum _{j=1}^n\otimes {\hat{x}}_{ij}=\left[\sum _j{\hat{x}}_{ij}^{\alpha },\sum _j{\hat{x}}_{ij}^{\gamma }\right]\text{[Eq.16]}$$

   Turskis & Zavadskas 2010, §3 Eqs.(16)-(17). Grey addition: Eq.(2).

5. 5

   Compute utility degree K_i = S_i / S_0 ∈ [0,1] comparing each alternative against the optimal A0. Rank alternatives by K_i descending.

   $$K_i=\frac{S_i}{S_0},K_0=1,0\le K_i\le 1\text{[Eq.18]}$$

   Turskis & Zavadskas 2010, §3 Eq.(18).

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Grey 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

• •GREY-ARAS kritik edge case'ler: (1) Negatif grey number (α<0) - paper Eq.4 çarpım sadece pozitif kriter değerleri için tanımlı; α<0 durumunda Eq.4 (b⁺>0) ve alternatif Eq (b⁺<0) ayırımı gerekir - engine pozitif grey number varsayar, negatif input için reddeder. (2) Dejenerasyon α=γ → white number → ARAS-G klasik ARAS'a iner; manifest bu durumda sonuç eşitliğini garanti eder. (3) Cost normalizasyonun iki-aşaması Eq.12 invert ⊗x_ij^* = [1/γ_ij, 1/α_ij] - α_ij=0 bölünme hatası, ε-clamp gerekir. (4) Centroid 2-term simetrik grey için aritmetik orta verir; asimetrik kullanım olmaz (grey interval [α,γ] tanımı gereği iki uçludur, üçüncü 'modal' nokta yoktur - FUZZY-ARAS centroid 3-term kullanır, GREY-ARAS 2-term - bu farkı manifestler arası çapraz audit'te DOĞRULAYIN).

• Value-space violation: ensure all entries satisfy GIN: x̲ ≤ x̄ (lower and upper bounds of interval) before computation.

• Defuzzification method affects ranking: whitenisation: (x̲ + x̄)/2 is the canonical choice but alternatives exist.

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