Grey-TOPSIS: Grey extension of TOPSIS

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

Edge cases

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

Common pitfalls

• •Hatalı: 'GREY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Grey numbers/tuples
• •Hatalı: 'GREY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'GREY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: GREY-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GREY-TOPSIS'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 1-2: 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]. Formül: \otimes a_{ij}=[a_{ij}^L,a_{ij}^U];\\ \text{benefit: } \otimes v_{ij}=\Bigl[\frac{a_{ij}^L}{\max_k a_{kj}^U},\frac{a_{ij}^U}{\max_k a_{kj}^U}\Bigr]; \quad \text{cost: } \otimes v_{ij}=\Bigl[\frac{\min_k a_{kj}^L}{a_{ij}^U},\frac{\min_k a_{kj}^L}{a_{ij}^L}\Bigr] Anchor: Zavadskas-Turskis-Bagocius 2015, §Grey-TOPSIS; Deng 1989
2. 2.Adım 2 (F2): 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). Formül: \otimes\tilde{v}_{ij}=w_j\cdot\otimes v_{ij}=[w_j v_{ij}^L,\; w_j v_{ij}^U];\\ \hat{\tilde{v}}_{ij}=\tfrac{1}{2}(w_j v_{ij}^L+w_j v_{ij}^U) Anchor: Zavadskas-Turskis-Bagocius 2015; Liu & Lin 2006 §whitenisation
3. 3.Adım 3 (F3): 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). Formül: \hat{v}_j^+=\max_i\hat{\tilde{v}}_{ij}\;(\text{benefit});\\ \hat{v}_j^-=\min_i\hat{\tilde{v}}_{ij}\;(\text{benefit}) Anchor: Zavadskas-Turskis-Bagocius 2015; Hwang-Yoon 1981 §ideal solutions
4. 4.Adım 4 (F4): 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. Formül: 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} Anchor: Liu & Lin 2006 §grey distance; Zavadskas-Turskis-Bagocius 2015
5. 5.Adım 5 (F5): Step 6: Closeness coefficient CC_i = d_i- / (d_i+ + d_i-) ∈ [0,1]. Rank descending. Formül: CC_i=\frac{d_i^-}{d_i^++d_i^-},\quad CC_i\in[0,1];\\ \text{rank descending} Anchor: Hwang-Yoon 1981 §closeness coefficient; Zavadskas-Turskis-Bagocius 2015

Commonly paired with

• •n_a + GREY-TOPSIS (common)

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