L2T-TODIM: Linguistic extension of L2T-TODIM

Overview

Linguistic outranking/ranking: 2-Tuple Linguistic Variable (2TL: (s_i, α)). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Linguistic outranking/ranking: 2-Tuple Linguistic Variable (2TL: (s_i, α))
• •Preserves linguistic_2tuple 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 Linguistic 2-Tuple 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 Linguistic 2-Tuple 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 L2T-TODIM-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'L2T-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Linguistic 2-Tuple numbers/tuples
• •Hatalı: 'L2T-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'L2T-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: L2T-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: L2T-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: Construct the 2-tuple linguistic decision matrix x̃_ij = (s_ij, α_ij); define criteria weights w_j. Formül: \tilde{D} = [(s_{ij}, \alpha_{ij})]_{m \times n};\ \Delta^{-1}(s_{i}, \alpha) = i + \alpha Anchor: Report §3.3 Steps 1-2; Qi 2021 ITL-TODIM
2. 2.Adım 2 (F2): Step 2: Dominance degree Φ_j(A_i,A_k) using 2-tuple distance d²((s_ij,α_ij),(s_kj,α_kj)) and prospect-theory loss aversion θ (typical θ=2.25). Formül: \Phi_{j}(A_{i},A_{k}) = \begin{cases} \sqrt{w_{j}\cdot d^{2}((s_{ij},\alpha_{ij}),(s_{kj},\alpha_{kj}))/\sum_{j} w_{j}} & \Delta^{-1}(s_{ij},\alpha_{ij}) > \Delta^{-1}(s_{kj},\alpha_{kj})\\ 0 & \text{equal}\\ -\dfrac{1}{\theta}\sqrt{(\sum_{j} w_{j})\cdot d^{2}((s_{ij},\alpha_{ij}),(s_{kj},\alpha_{kj}))/w_{j}} & \Delta^{-1}(s_{ij},\alpha_{ij}) < \Delta^{-1}(s_{kj},\alpha_{kj})\end{cases} Anchor: Report §3.3 Formula 1: dominance degree
3. 3.Adım 3 (F3): Step 3: Overall dominance δ(A_i,A_k) = Σ_j Φ_j(A_i,A_k). Formül: \delta(A_{i},A_{k}) = \sum_{j=1}^{n} \Phi_{j}(A_{i},A_{k}) Anchor: Report §3.3 Formula 2: overall dominance
4. 4.Adım 4 (F4): Step 4: Overall prospect value ξ_i ∈ [0,1] normalised over Σ_k δ(A_i,A_k); descending rank. Formül: \xi_{i} = \dfrac{\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})};\ \ \text{rank} = \text{argsort}_{\text{desc}}(\xi_{i}) Anchor: Report §3.3 Formula 3: overall prospect value

Commonly paired with

• •n_a + L2T-TODIM (common)

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