L2T-VIKOR - Linguistic extension of L2T-VIKOR

Overview

l2t-vikor extends L2T-VIKOR 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, lower 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; each entry is (s_ij, α_ij). Determine best (s_j*, α_j*) and worst (s_j⁻, α_j⁻) per criterion (max for benefit, min for cost using Δ⁻¹).

   $$(s_j^{\ast },{\alpha }_j^{\ast })=\underset{i}{\max }(s_{ij},{\alpha }_{ij})\text{by}{\Delta }^{-1};(s_j^{-},{\alpha }_j^{-})=\underset{i}{\min }(s_{ij},{\alpha }_{ij})\text{by}{\Delta }^{-1}$$

   Report §3.2 Steps 1-2; Ju 2013 2TL-VIKOR

2. 2

   Utility measure S_i and regret measure R_i using 2-tuple distance d via Δ⁻¹.

   $$S_i=\sum _{j=1}^n{w}_j\frac{d((s_j^{\ast },{\alpha }_j^{\ast }),(s_{ij},{\alpha }_{ij}))}{d((s_j^{\ast },{\alpha }_j^{\ast }),(s_j^{-},{\alpha }_j^{-}))},R_i=\underset{j}{\max }\left[w_j\frac{d((s_j^{\ast },{\alpha }_j^{\ast }),(s_{ij},{\alpha }_{ij}))}{d((s_j^{\ast },{\alpha }_j^{\ast }),(s_j^{-},{\alpha }_j^{-}))}\right]$$

   Report §3.2 Formulas 2-3 - utility & regret

3. 3

   VIKOR index Q_i = v·(S_i − S*)/(S⁻ − S*) + (1−v)·(R_i − R*)/(R⁻ − R*); ascending sort over S, R, Q.

   $$Q_i=v\frac{S_i-S^{\ast }}{S^{-}-S^{\ast }}+(1-v)\frac{R_i-R^{\ast }}{R^{-}-R^{\ast }},S^{\ast }=\underset{i}{\min }{S}_i,S^{-}=\underset{i}{\max }{S}_i,R^{\ast }=\underset{i}{\min }{R}_i,R^{-}=\underset{i}{\max }{R}_i$$

   Report §3.2 Formula 4 - VIKOR index

4. 4

   Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

   $$DQ=\frac{1}{m-1};C1:Q(A^{(2)})-Q(A^{(1)})\ge DQ;C2:A^{(1)}\text{ is best in }S\text{ or in }R$$

   Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

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

• •If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

• 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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