Grey-MARCOS - Grey extension of MARCOS

Overview

grey-marcos extends MARCOS 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

   Normalize grey matrix ⊗v_ij; whiten v̂_ij=½(v_ij^L+v_ij^U). Grey AI: v̂_j^AI=max_i(v̂_ij) benefit; Grey AAI: v̂_j^AAI=min_i(v̂_ij).

   $$\begin{gathered} {\hat{v}}_{ij}=\frac{1}{2}(v_{ij}^L+v_{ij}^U); \\ {\hat{v}}_j^{\mathrm{AI}}=\underset{i}{\max }{\hat{v}}_{ij}(\text{benefit}); \\ {\hat{v}}_j^{\mathrm{AAI}}=\underset{i}{\min }{\hat{v}}_{ij} \end{gathered}$$

   Stević et al. 2020 §MARCOS; Deng 1989

2. 2

   Normalize by AI: n_ij = v̂_ij / v̂_j^AI.

   $$n_{ij}={\hat{v}}_{ij}/{\hat{v}}_j^{\mathrm{AI}}$$

   Stević et al. 2020 §step 3

3. 3

   Weighted sum S_i = Σ_j w_j·n_ij. Compute S_AI=1 and S_AAI = Σ_j w_j·(v̂_j^AAI/v̂_j^AI).

   $$\begin{gathered} S_i=\sum _j{w}_j{n}_{ij}; \\ {S}_{\mathrm{AI}}=1; \\ {S}_{\mathrm{AAI}}=\sum _j{w}_j({\hat{v}}_j^{\mathrm{AAI}}/{\hat{v}}_j^{\mathrm{AI}}) \end{gathered}$$

   Stević et al. 2020 §steps 4-5

4. 4

   Utility degrees K_i+ = S_i/S_AI = S_i; K_i- = S_i/S_AAI. Composite f(K_i). Rank descending.

   $$\begin{gathered} K_i^{+}=S_i; \\ {K}_i^{-}=S_i/S_{\mathrm{AAI}}; \\ f(K_i)=\frac{K_i^{+}+K_i^{-}}{1+(1-f(K_i^{+}))/f(K_i^{+})+(1-f(K_i^{-}))/f(K_i^{-})} \\ \text{where }f(K_i^{+})=K_i^{+}/(K_i^{+}+K_i^{-}); \\ \text{rank descending} \end{gathered}$$

   Stević et al. 2020 §composite function

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

How to cite