L2T-TOPSIS - Linguistic extension of L2T-TOPSIS

Overview

l2t-topsis extends L2T-TOPSIS 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

   Define linguistic term set S = {s_0,…,s_q} and construct the 2-tuple decision matrix where each x̃_ij = (s_i, α_ij), α ∈ [-0.5, 0.5).

   $$S=\{s_0,s_1,\dots ,s_q\};\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n},\Delta (\beta )=(s_i,\alpha ),i=\text{round}(\beta ),\alpha =\beta -i;{\Delta }^{-1}(s_i,\alpha )=i+\alpha$$

   Report §3.1 Steps 1-2; Herrera-Martínez 2000 Δ/Δ⁻¹

2. 2

   Weighted 2-tuple aggregation per criterion via Δ⁻¹.

   $${\bar{x}}_j=\Delta \left(\sum _{i=1}^m{w}_i\cdot {\Delta }^{-1}(s_{ij},{\alpha }_{ij})\right)$$

   Report §3.1 Formula 4 - weighted 2-tuple aggregation

3. 3

   2-Tuple Linguistic Positive Ideal Solution (TLPIS) A⁺ and Negative Ideal Solution (TLNIS) A⁻.

   $$A^{+}=\{(s_1^{+},{\alpha }_1^{+}),\dots ,(s_n^{+},{\alpha }_n^{+})\};A^{-}=\{(s_1^{-},{\alpha }_1^{-}),\dots ,(s_n^{-},{\alpha }_n^{-})\}$$

   Report §3.1 Formulas 5-6 - TLPIS / TLNIS

4. 4

   Separation measures using Δ⁻¹-based 2-tuple distance d((s_i,α_i),(s_j,α_j)) = |Δ⁻¹(s_i,α_i) − Δ⁻¹(s_j,α_j)|.

   $$d((s_i,{\alpha }_i),(s_j,{\alpha }_j))=|{\Delta }^{-1}(s_i,{\alpha }_i)-{\Delta }^{-1}(s_j,{\alpha }_j)|;D_i^{+}=\sqrt{\sum _{j=1}^n{d}^2((r_{ij},{\alpha }_{ij}),(s_j^{+},{\alpha }_j^{+}))},D_i^{-}=\sqrt{\sum _{j=1}^n{d}^2((r_{ij},{\alpha }_{ij}),(s_j^{-},{\alpha }_j^{-}))}$$

   Report §3.1 Formulas 3 & 7 - distance & separation

5. 5

   Relative closeness RC_i = D_i⁻ / (D_i⁺ + D_i⁻); descending rank.

   $$R{C}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}},0\le R{C}_i\le 1;\text{rank}={\text{argsort}}_{\text{desc}}(R{C}_i)$$

   Report §3.1 Formula 8 - relative closeness

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

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