Grey-VIKOR: Grey extension of VIKOR

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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'GREY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Grey numbers/tuples
• •Hatalı: 'GREY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'GREY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: GREY-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GREY-VIKOR'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 decision matrix; whiten: v̂_ij = ½(v_ij^L + v_ij^U). Compute best/worst whitenised values: v̂_j* = max_i(v̂_ij) benefit; v̂_j- = min_i(v̂_ij). Formül: \otimes v_{ij}=[v_{ij}^L,v_{ij}^U];\\ \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U);\\ \hat{v}_j^*=\max_i\hat{v}_{ij}\;(\text{benefit});\\ \hat{v}_j^-=\min_i\hat{v}_{ij} Anchor: Deng 1989 §whitenisation; Opricovic-Tzeng 2004 §VIKOR normalization
2. 2.Adım 2 (F2): Step 2-3: Normalised gap: f_ij = (v̂_j* - v̂_ij)/(v̂_j* - v̂_j-). Utility S_i = Σ_j w_j·f_ij; regret R_i = max_j(w_j·f_ij). Formül: f_{ij}=\frac{\hat{v}_j^*-\hat{v}_{ij}}{\hat{v}_j^*-\hat{v}_j^-};\\ S_i=\sum_{j=1}^n w_j f_{ij};\\ R_i=\max_j(w_j f_{ij}) Anchor: Opricovic 1998; Opricovic-Tzeng 2004 §2 Eqs.(3)-(4)
3. 3.Adım 3 (F3): Step 4-5: Compromise index Q_i = v·(S_i-S*)/(S⁻-S*) + (1-v)·(R_i-R*)/(R⁻-R*) with v=0.5. Rank ascending; check conditions C1 and C2. Formül: S^*=\min_i S_i,\ S^-=\max_i S_i;\ R^*=\min_i R_i,\ R^-=\max_i R_i;\\ Q_i=v\frac{S_i-S^*}{S^--S^*}+(1-v)\frac{R_i-R^*}{R^--R^*};\\ \text{rank ascending by }Q_i Anchor: Opricovic 1998; Opricovic-Tzeng 2004 §2 Eq.(6)

Commonly paired with

• •n_a + GREY-VIKOR (common)

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