Grey-EDAS - Grey extension of EDAS

Overview

grey-edas extends EDAS 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

   Construct the grey decision matrix. Each entry ⊗x_ij = [x̲_ij, x̄_ij] is an interval grey number (Eq.19).

   $$\otimes X=(\otimes x_{ij}{)}_{m\times n},\otimes x_{ij}=[{\underline{x}}_{ij},{\bar{x}}_{ij}],0\le {\underline{x}}_{ij}\le {\bar{x}}_{ij}$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Step 1, Eq.(19); Deng 1989 grey interval number

2. 2

   Compute the grey average solution ⊗x*_j = [x̲*_j, x̄*_j] per criterion as component-wise arithmetic means (Eqs.21-22).

   $${\underline{x}}_j^{\ast }=\frac{1}{m}\sum _{i=1}^m{\underline{x}}_{ij},{\bar{x}}_j^{\ast }=\frac{1}{m}\sum _{i=1}^m{\bar{x}}_{ij};{\mathring{x}}_j^{\ast }=\frac{{\underline{x}}_j^{\ast }+{\bar{x}}_j^{\ast }}{2}$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Step 2, Eqs.(21)-(22); whitening °x*_j = (x̲*_j + x̄*_j)/2

3. 3

   Grey PDA ⊗d+_ij and grey NDA ⊗d-_ij per entry. Bounds use cross-comparisons: lower ← conservative (lower alt bound vs upper avg bound), upper ← optimistic (upper alt bound vs lower avg bound). Denominator = whitened average °x*_j. Direction-sensitive (Eqs.23-24).

   $$\otimes d_{ij}^{+}=\{\begin{array}{ll}\left[\frac{\max (0,{\underline{x}}_{ij}-{\bar{x}}_j^{\ast })}{{\mathring{x}}_j^{\ast }},\frac{\max (0,{\bar{x}}_{ij}-{\underline{x}}_j^{\ast })}{{\mathring{x}}_j^{\ast }}\right]& j\in {\Omega }_{\max }\\ \left[\frac{\max (0,{\underline{x}}_j^{\ast }-{\bar{x}}_{ij})}{{\mathring{x}}_j^{\ast }},\frac{\max (0,{\bar{x}}_j^{\ast }-{\underline{x}}_{ij})}{{\mathring{x}}_j^{\ast }}\right]& j\in {\Omega }_{\min }\end{array}$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Step 3, Eq.(23) - grey PDA bounds; numerically verified against Table 3 of paper example

4. 4

   Step 3 (cont.) - Grey NDA ⊗d-_ij (Eq.24): symmetric to PDA with negation logic - deviation below average for benefit, above average for cost.

   $$\otimes d_{ij}^{-}=\{\begin{array}{ll}\left[\frac{\max (0,{\underline{x}}_j^{\ast }-{\bar{x}}_{ij})}{{\mathring{x}}_j^{\ast }},\frac{\max (0,{\bar{x}}_j^{\ast }-{\underline{x}}_{ij})}{{\mathring{x}}_j^{\ast }}\right]& j\in {\Omega }_{\max }\\ \left[\frac{\max (0,{\underline{x}}_{ij}-{\bar{x}}_j^{\ast })}{{\mathring{x}}_j^{\ast }},\frac{\max (0,{\bar{x}}_{ij}-{\underline{x}}_j^{\ast })}{{\mathring{x}}_j^{\ast }}\right]& j\in {\Omega }_{\min }\end{array}$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Step 3, Eq.(24) - grey NDA bounds; numerically verified against Table 4 of paper example

5. 5

   Step 4-5 - Weighted grey sums ⊗Q+_i, ⊗Q-_i (Eqs.27-30); then normalize by max upper bound to get ⊗S+_i, ⊗S-_i (Eqs.31-34). \quad\underline{S}^+_i = \frac{\underline{Q}^+_i}{\max_k \bar{Q}^+_k},\quad \bar{S}^+_i = \frac{\bar{Q}^+_i}{\max_k \bar{Q}^+_k}; \quad\underline{S}^-_i = 1 - \frac{\bar{Q}^-_i}{\max_k \bar{Q}^-_k},\quad \bar{S}^-_i = 1 - \frac{\underline{Q}^-_i}{\max_k \bar{Q}^-_k}

   $${\underline{Q}}_i^{+}=\sum _j{w}_j{\underline{d}}_{ij}^{+},{\bar{Q}}_i^{+}=\sum _j{w}_j{\bar{d}}_{ij}^{+};$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Step 4, Eqs.(27)-(30) + Step 5, Eqs.(31)-(34); verified against Table 5

6. 6

   Steps 6-7 - Appraisal score S_i as equal-weight average of all 4 grey bounds (Eq.35); rank by S_i descending. Paper ranking: A5 > A2 > A1 > A4 > A3 (S: 0.615, 0.480, 0.449, 0.438, 0.436).

   $$S_i=\frac{1}{4}({\underline{S}}_i^{+}+{\bar{S}}_i^{+}+{\underline{S}}_i^{-}+{\bar{S}}_i^{-}),0\le S_i\le 1;\text{rank by }{S}_i\downarrow$$

   Stanujkic-Zavadskas-KeshavarzGhorabaee-Turskis 2017 (SIC Vol.26 No.1) Steps 6-7, Eq.(35); paper ranking: A5>A2>A1>A4>A3 (S=0.615,0.480,0.449,0.438,0.436)

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

• •AV_j alt-sınırı x̲ = 0 ise PDA/NDA bölme sıfır-bölme verir; engine bu durumda küçük ε=1e-9 eklenmesi yerine kriteri non-applicable işaretler ve uyarı verir (paper bu durumu açıkça ele almaz, defensive choice).
• •Tüm alternatiflerin bir kriterde aynı grey değere sahip olması: AV_j eşit; PDA_ij = NDA_ij = 0; o kriter sıralamayı etkilemez. Bu, EDAS'ın 'discriminating only on deviation' doğasının doğal sonucudur.
• •Grey-midpoint sıralama tie: iki AS_i'nin midpoint'i eşit ama interval-genişlikleri farklıysa engine ikincil kriter olarak daha-dar aralığı tercih eder (daha az belirsizlik varsayımı; paper'da explicit değil, mantıklı default).
• •Paper §4 numerical case: A5 müteahhit AS_5 ≈ 0.615 ile birinci; A1 son. Bu reproduce edilebilir; Block J fixture'ında bu örnek anchored.

• 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.

Works with

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