Grey-TODIM: Grey extension of TODIM

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

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

Edge cases

• •if v̂_ij>v̂_kj, 0 if equal, -(1/θ)√(Σw̄_jr·d_j/w̄_jr) if v̂_ij<v̂_kj. θ=1.

Common pitfalls

• •Hatalı: 'GREY-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Grey numbers/tuples
• •Hatalı: 'GREY-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'GREY-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: GREY-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: GREY-TODIM'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; whiten v̂_ij=½(v_ij^L+v_ij^U). Formül: \hat{v}_{ij}=\tfrac{1}{2}(v_{ij}^L+v_{ij}^U) Anchor: Gomes-Lima 1992; Deng 1989
2. 2.Adım 2 (F2): Step 2: Relative weights: w̄_jr = w_j/w_r, where w_r = max_j(w_j). Formül: \overline{w}_{jr}=w_j/w_r,\quad w_r=\max_j w_j Anchor: Gomes-Lima 1992 §relative weights
3. 3.Adım 3 (F3): Step 3: Dominance contributions: d_j(i,k)=|v̂_ij - v̂_kj|. φ_j(A_i,A_k)= +√(w̄_jr·d_j/Σw̄_jr) if v̂_ij>v̂_kj, 0 if equal, -(1/θ)√(Σw̄_jr·d_j/w̄_jr) if v̂_ij<v̂_kj. θ=1. Formül: d_j(i,k)=|\hat{v}_{ij}-\hat{v}_{kj}|;\\ \phi_j(A_i,A_k)=\begin{cases}+\sqrt{\overline{w}_{jr}d_j(i,k)/\textstyle\sum_j\overline{w}_{jr}}&\hat{v}_{ij}>\hat{v}_{kj}\\0&\hat{v}_{ij}=\hat{v}_{kj}\\-(1/\theta)\sqrt{\textstyle\sum_j\overline{w}_{jr}d_j(i,k)/\overline{w}_{jr}}&\hat{v}_{ij}<\hat{v}_{kj}\end{cases};\quad\theta=1 Anchor: Gomes-Lima 1992 §TODIM; Wang-Wang 2014 §grey TODIM
4. 4.Adım 4 (F4): Step 4: Overall dominance: δ(A_i,A_k) = Σ_j φ_j(A_i,A_k). Score: ξ_i = Σ_k δ(A_i,A_k). Normalize and rank descending. Formül: \delta(A_i,A_k)=\sum_j\phi_j(A_i,A_k);\\ \xi_i=\frac{\sum_k\delta(A_i,A_k)-\min_k\sum_k\delta(A_k,\cdot)}{\max_k\sum_k\delta(A_k,\cdot)-\min_k\sum_k\delta(A_k,\cdot)};\\ \text{rank descending} Anchor: Gomes-Lima 1992 §global dominance

Commonly paired with

• •n_a + GREY-TODIM (common)

How to cite