L2T-CODAS - Linguistic extension of L2T-CODAS

Overview

l2t-codas extends L2T-CODAS 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) on linguistic term set S = {s_0,…,s_q}; define criteria weights w_j and direction sets J^+ (benefit) / J^- (cost). Δ⁻¹(s_i,α)=i+α is the canonical defuzzification.

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

   Report §3.4 Step 1; He 2019 P2TL-CODAS

2. 2

   Negative-Ideal Solution per criterion in 2-tuple space: (s_j^-, α_j^-) = min_i (s_ij, α_ij) for benefit criteria (max_i for cost), with the min/max taken in the Δ⁻¹ order.

   $$(s_j^{-},{\alpha }_j^{-})=\{\begin{array}{ll}{\min }_i(s_{ij},{\alpha }_{ij})& j\in J^{+}\\ {\max }_i(s_{ij},{\alpha }_{ij})& j\in J^{-}\end{array}$$

   Report §3.4 Formula 1 - negative ideal solution

3. 3

   Euclidean distance E_i and Taxicab distance T_i from the 2-tuple negative-ideal, using 2-tuple distance d((s_ij,α_ij),(s_j^-,α_j^-)) = |Δ⁻¹(s_ij,α_ij) − Δ⁻¹(s_j^-,α_j^-)| with criterion weights w_j.

   $$E_i=\sqrt{\sum _{j=1}^n{w}_j{d}^2((s_{ij},{\alpha }_{ij}),(s_j^{-},{\alpha }_j^{-}))},T_i=\sum _{j=1}^n{w}_{j}d((s_{ij},{\alpha }_{ij}),(s_j^{-},{\alpha }_j^{-}))$$

   Report §3.4 Formulas 2-3 - Euclidean and Taxicab distances

4. 4

   Relative assessment matrix h_ik via threshold ψ(τ) on |E_i − E_k|; combines Euclidean and Taxicab gaps when Euclidean gap is significant.

   $$h_{ik}=(E_i-E_k)+\psi (E_i-E_k)\cdot (T_i-T_k),\psi (z)=\{\begin{array}{ll}1& |z|\ge \tau \\ 0& |z|<\tau \end{array}$$

   Report §3.4 - relative assessment matrix

5. 5

   Final assessment score H_i = E_i + ψ·E_i·T_i (combined-distance form), or equivalently H_i = Σ_k h_ik; rank in descending order.

   $$H_i=E_i+\psi \cdot E_i\cdot T_i=\sum _{k=1}^m{h}_{ik};\text{rank}={\text{argsort}}_{\text{desc}}(H_i)$$

   Report §3.4 Formula 4 - assessment score

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

• •when Euclidean gap is significant.

• 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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