PL-TODIM - Probabilistic Linguistic extension of TODIM

Overview

pl-todim extends TODIM to handle Probabilistic Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}) algebra. The final scores are defuzzified via expected linguistic value E = Σ p_k · index(L_k) before ranking.

Output

utility, higher is better

Data

Probabilistic Linguistic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Probabilistic Linguistic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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.

   $$L_{ij}(p_{ij})=\{L_{ij}^{(k)}(p_{ij}^{(k)})\mid k=1,\dots ,\#L_{ij}\},\sum _k{p}_{ij}^{(k)}=1;r=\arg \underset{j}{\max }{w}_j,w_{jr}=w_j/w_r$$

   Report §4.3 Steps 1-2; Liu 2017 PL-TODIM

2. 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).

   $${\Phi }_j(A_i,A_k)=\{\begin{array}{ll}\sqrt{\frac{w_{jr}d(L_{ij}(p_{ij}),L_{kj}(p_{kj}))}{{\sum }_{j^{\prime }}{w}_{j^{\prime }r}}}& E(L_{ij})>E(L_{kj})\\ 0& E(L_{ij})=E(L_{kj})\\ -\frac{1}{\theta }\sqrt{\frac{({\sum }_{j^{\prime }}{w}_{j^{\prime }r})d(L_{ij}(p_{ij}),L_{kj}(p_{kj}))}{w_{jr}}}& E(L_{ij})<E(L_{kj})\end{array}$$

   Report §4.3 Formula 1 - pairwise dominance

3. 3

   Overall dominance δ(A_i,A_k) = Σ_j Φ_j(A_i,A_k).

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\Phi }_j(A_i,A_k)$$

   Report §4.3 - overall dominance (same structure as P-TODIM)

4. 4

   Overall prospect value ξ_i ∈ [0,1] normalised over Σ_k δ(A_i,A_k); rank in descending order.

   $${\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{\min }_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}{{\max }_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)-{\min }_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)};\text{rank}={\text{argsort}}_{\text{desc}}({\xi }_i)$$

   Report §4.3 - overall prospect value (same structure as P-TODIM)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy PLTS: {L_k|p_k} where L_k is linguistic label, Σ p_k ≤ 1 before computation.

• Defuzzification method affects ranking: expected linguistic value E = Σ p_k · index(L_k) is the canonical choice but alternatives exist.

Works with

How to cite