Grey-TOPSIS - Grey extension of TOPSIS

Overview

grey-topsis extends TOPSIS 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

   2 - Construct grey decision matrix ⊗a_ij=[a_ij^L, a_ij^U] and normalize: benefit ⊗v_ij=[a_ij^L/max_i(a_ij^U), a_ij^U/max_i(a_ij^U)]; cost ⊗v_ij=[min_i(a_ij^L)/a_ij^U, min_i(a_ij^L)/a_ij^L].

   $$\begin{gathered} \otimes a_{ij}=[a_{ij}^L,a_{ij}^U]; \\ \text{benefit: }\otimes v_{ij}=[\frac{a_{ij}^L}{{\max }_k{a}_{kj}^U},\frac{a_{ij}^U}{{\max }_k{a}_{kj}^U}];\text{cost: }\otimes v_{ij}=[\frac{{\min }_k{a}_{kj}^L}{a_{ij}^U},\frac{{\min }_k{a}_{kj}^L}{a_{ij}^L}] \end{gathered}$$

   Zavadskas-Turskis-Bagocius 2015, §Grey-TOPSIS; Deng 1989

2. 2

   Step 3 - Weighted normalised grey matrix: ⊗ṽ_ij = w_j · ⊗v_ij = [w_j·v_ij^L, w_j·v_ij^U]. Then whiten: v̂_ij = ½(ṽ_ij^L + ṽ_ij^U).

   $$\begin{gathered} \otimes {\tilde{v}}_{ij}=w_j\cdot \otimes v_{ij}=[w_j{v}_{ij}^L,w_j{v}_{ij}^U]; \\ {\hat{\tilde{v}}}_{ij}=\frac{1}{2}(w_j{v}_{ij}^L+w_j{v}_{ij}^U) \end{gathered}$$

   Zavadskas-Turskis-Bagocius 2015; Liu & Lin 2006 §whitenisation

3. 3

   Step 4 - Grey PIS/NIS: ⊗A+ = {max_i(ṽ_ij^L), max_i(ṽ_ij^U)} per j; ⊗A- = {min_i(ṽ_ij^L), min_i(ṽ_ij^U)} per j. Whitenised: v̂_j+ = max_i(v̂_ij); v̂_j- = min_i(v̂_ij).

   $$\begin{gathered} {\hat{v}}_j^{+}=\underset{i}{\max }{\hat{\tilde{v}}}_{ij}(\text{benefit}); \\ {\hat{v}}_j^{-}=\underset{i}{\min }{\hat{\tilde{v}}}_{ij}(\text{benefit}) \end{gathered}$$

   Zavadskas-Turskis-Bagocius 2015; Hwang-Yoon 1981 §ideal solutions

4. 4

   Step 5 - Grey Euclidean separation measures using whitenised distances: d_i+ = √Σ_j w_j(v̂_ij - v̂_j+)²; d_i- = √Σ_j w_j(v̂_ij - v̂_j-)².

   $$\begin{gathered} d_i^{+}=\sqrt{\sum _{j=1}^n{w}_j({\hat{\tilde{v}}}_{ij}-{\hat{v}}_j^{+}{)}^2}; \\ {d}_i^{-}=\sqrt{\sum _{j=1}^n{w}_j({\hat{\tilde{v}}}_{ij}-{\hat{v}}_j^{-}{)}^2} \end{gathered}$$

   Liu & Lin 2006 §grey distance; Zavadskas-Turskis-Bagocius 2015

5. 5

   Step 6 - Closeness coefficient CC_i = d_i- / (d_i+ + d_i-) ∈ [0,1]. Rank descending.

   $$\begin{gathered} C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},C{C}_i\in [0,1]; \\ \text{rank descending} \end{gathered}$$

   Hwang-Yoon 1981 §closeness coefficient; Zavadskas-Turskis-Bagocius 2015

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

• 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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