PL-TODIM: Probabilistic Linguistic extension of TODIM

Overview

Probabilistic Linguistic outranking/ranking: Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Probabilistic Linguistic outranking/ranking: Probabilistic Linguistic Term Set (PLTS: {L_k|p_k})
• •Preserves linguistic_probabilistic 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 Probabilistic Linguistic 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 Probabilistic Linguistic 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 PL-TODIM-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'PL-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Probabilistic Linguistic numbers/tuples
• •Hatalı: 'PL-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'PL-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: PL-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PL-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 PLTS decision matrix L_ij(p_ij) = {L_ij^(k)(p_ij^(k)) | k=1,…,#L_ij} on linguistic term set S={s_0,…,s_g}; normalise so Σ_k p_ij^(k)=1 and PLTSs in the same criterion have equal length. Define weights w_j and reference r = arg max_j w_j; relative weights w_jr = w_j / w_r. Formül: L_{ij}(p_{ij}) = \{L_{ij}^{(k)}(p_{ij}^{(k)}) \mid k=1,\dots,\#L_{ij}\},\ \sum_{k} p_{ij}^{(k)} = 1;\ \ r = \arg\max_{j} w_{j},\ w_{jr} = w_{j}/w_{r} Anchor: Report §4.3 Steps 1-2; Liu 2017 PL-TODIM
2. 2.Adım 2 (F2): Step 2: Pairwise dominance degree Φ_j(A_i,A_k) using PLTS distance d(L_ij(p_ij), L_kj(p_kj)) and prospect-theory loss aversion θ (typical θ=2.25); 3 cases based on expected value comparison E(L_ij) vs E(L_kj). Formül: \Phi_{j}(A_{i},A_{k}) = \begin{cases} \sqrt{\dfrac{w_{jr}\, d(L_{ij}(p_{ij}), L_{kj}(p_{kj}))}{\sum_{j'} w_{j'r}}} & E(L_{ij}) > E(L_{kj})\\ 0 & E(L_{ij}) = E(L_{kj})\\ -\dfrac{1}{\theta}\sqrt{\dfrac{(\sum_{j'} w_{j'r})\, d(L_{ij}(p_{ij}), L_{kj}(p_{kj}))}{w_{jr}}} & E(L_{ij}) < E(L_{kj})\end{cases} Anchor: Report §4.3 Formula 1: pairwise dominance
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 §4.3: overall dominance (same structure as P-TODIM)
4. 4.Adım 4 (F4): Step 4: Overall prospect value ξ_i ∈ [0,1] normalised over Σ_k δ(A_i,A_k); rank in descending order. 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 §4.3: overall prospect value (same structure as P-TODIM)

Commonly paired with

• •n_a + PL-TODIM (common)

How to cite