Grey-MARCOS: Grey extension of MARCOS

Overview

Grey outranking/ranking: Grey Interval Number (GIN: [x̲, x̄]). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Grey outranking/ranking: Grey Interval Number (GIN: [x̲, x̄])
• •Preserves grey uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Grey numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

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

When not to use

• •Crisp data sufficient: use base MARCOS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for GREY-MARCOS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'GREY-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Grey numbers/tuples
• •Hatalı: 'GREY-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'GREY-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: GREY-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GREY-MARCOS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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). Formül: \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\\ \hat{v}_j^{\mathrm{AI}}=\max_i\hat{v}_{ij}\;(\text{benefit});\\ \hat{v}_j^{\mathrm{AAI}}=\min_i\hat{v}_{ij} Anchor: Stević et al. 2020 §MARCOS; Deng 1989
2. 2.Adım 2 (F2): Step 2: Normalize by AI: n_ij = v̂_ij / v̂_j^AI. Formül: n_{ij}=\hat{v}_{ij}/\hat{v}_j^{\mathrm{AI}} Anchor: Stević et al. 2020 §step 3
3. 3.Adım 3 (F3): Step 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). Formül: 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}}) Anchor: Stević et al. 2020 §steps 4-5
4. 4.Adım 4 (F4): Step 4: Utility degrees K_i+ = S_i/S_AI = S_i; K_i- = S_i/S_AAI. Composite f(K_i). Rank descending. Formül: 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} Anchor: Stević et al. 2020 §composite function

Commonly paired with

• •n_a + GREY-MARCOS (common)

How to cite