L2T-TODIM - Linguistic extension of L2T-TODIM

Overview

l2t-todim extends L2T-TODIM to handle Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using 2-Tuple Linguistic Variable (2TL: (s_i, α)) algebra. The final scores are defuzzified via Δ^{-1}(s_i, α) = i + α before ranking.

Output

utility, higher is better

Data

Linguistic 2-Tuple, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Linguistic 2-Tuple MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct the 2-tuple linguistic decision matrix x̃_ij = (s_ij, α_ij); define criteria weights w_j.

   $$\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n};{\Delta }^{-1}(s_i,\alpha )=i+\alpha$$

   Report §3.3 Steps 1-2; Qi 2021 ITL-TODIM

2. 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).

   $${\Phi }_j(A_i,A_k)=\{\begin{array}{ll}\sqrt{w_j\cdot d^2((s_{ij},{\alpha }_{ij}),(s_{kj},{\alpha }_{kj}))/{\sum }_j{w}_j}& {\Delta }^{-1}(s_{ij},{\alpha }_{ij})>{\Delta }^{-1}(s_{kj},{\alpha }_{kj})\\ 0& \text{equal}\\ -\frac{1}{\theta }\sqrt{({\sum }_j{w}_j)\cdot d^2((s_{ij},{\alpha }_{ij}),(s_{kj},{\alpha }_{kj}))/w_j}& {\Delta }^{-1}(s_{ij},{\alpha }_{ij})<{\Delta }^{-1}(s_{kj},{\alpha }_{kj})\end{array}$$

   Report §3.3 Formula 1 - dominance degree

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 §3.3 Formula 2 - overall dominance

4. 4

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

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

   Report §3.3 Formula 3 - overall prospect value

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy 2TL: s_i ∈ S (linguistic term set), α ∈ [-0.5,0.5) before computation.

• Defuzzification method affects ranking: Δ^{-1}(s_i, α) = i + α is the canonical choice but alternatives exist.

Works with

How to cite