L2T-CODAS - Linguistic extension of L2T-CODAS

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.

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

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.

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

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.

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

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.

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

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.

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