IV-TODIM: Interval extension of TODIM

Overview

Interval outranking/ranking: Interval Number (IN: [a, b]). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Interval outranking/ranking: Interval Number (IN: [a, b])
• •Preserves interval 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 Interval 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 Interval 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

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

Common pitfalls

• •Hatalı: 'IV-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Interval numbers/tuples
• •Hatalı: 'IV-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'IV-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: IV-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IV-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): Formül: \tilde{a}_{ij}^c=([\nu^L_{ij},\nu^U_{ij}],[\mu^L_{ij},\mu^U_{ij}])\quad\text{for cost criteria} Anchor: IV-TODIM-base
2. 2.Adım 2 (F2): Formül: \bar{w}_{jr}=\frac{w_j}{w_r},\quad w_r=\max_j w_j Anchor: IV-TODIM-base
3. 3.Adım 3 (F3): Formül: \phi_j(A_i,A_k)=\begin{cases}\bar{w}_{jr}\sqrt{\frac{d(\tilde{a}_{ij},\tilde{a}_{kj})\sum_j\bar{w}_{jr}}{\bar{w}_{jr}}} & \text{gain}\\[4pt]-\frac{1}{\theta}\sqrt{\frac{d(\tilde{a}_{ij},\tilde{a}_{kj})\sum_j\bar{w}_{jr}}{\bar{w}_{jr}}} & \text{loss}\\0 & \text{tie}\end{cases} Anchor: IV-TODIM-base
4. 4.Adım 4 (F4): Formül: \delta(A_i,A_k)=\sum_{j=1}^n\phi_j(A_i,A_k),\quad d(\alpha_1,\alpha_2)=\sqrt{\tfrac{1}{4}[(\mu^L_1-\mu^L_2)^2+(\mu^U_1-\mu^U_2)^2+(\nu^L_1-\nu^L_2)^2+(\nu^U_1-\nu^U_2)^2]} Anchor: IV-TODIM-base
5. 5.Adım 5 (F5): Formül: \xi_i=\frac{\sum_k\delta(A_i,A_k)-\min_i\sum_k\delta(A_i,A_k)}{\max_i\sum_k\delta(A_i,A_k)-\min_i\sum_k\delta(A_i,A_k)},\quad\text{rank descending} Anchor: IV-TODIM-base

Commonly paired with

• •n_a + IV-TODIM (common)

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