PL-VIKOR - Probabilistic Linguistic extension of VIKOR

Overview

pl-vikor extends VIKOR to handle Probabilistic Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Probabilistic Linguistic Term Set (PLTS: {L_k|p_k}) algebra. The final scores are defuzzified via expected linguistic value E = Σ p_k · index(L_k) before ranking.

Output

utility, lower is better

Data

Probabilistic Linguistic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

How it works

1. 1

   Identify the best and worst PLTS values per criterion using expected value E(L(p))=Σ_k r(L^(k))·p^(k): L_j^* = arg max_i E(L_ij(p_ij)) (benefit) or arg min_i (cost); L_j^- = arg min_i (benefit) or arg max_i (cost).

   $$E(L(p))=\sum _{k=1}^{\#L}r(L^{(k)})\cdot p^{(k)};L_j^{\ast }(p_j^{\ast })=\arg \max /\underset{i}{\min }E(L_{ij}(p_{ij})),L_j^{-}(p_j^{-})=\arg \min /\underset{i}{\max }E(L_{ij}(p_{ij}))$$

   Report §4.2 Formula 1; Li 2021 PL-VIKOR

2. 2

   Compute utility S_i and regret R_i in PLTS form using normalised PLTS distance d(L_j^*(p_j^*), L_ij(p_ij)) / d(L_j^*(p_j^*), L_j^-(p_j^-)) weighted by w_j.

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

   Report §4.2 Formulas 2-3 - utility and regret

3. 3

   Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v (classical VIKOR aggregation; identical structure to crisp VIKOR with PLTS-derived S, R).

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

   Report §4.2 Formula 4 - VIKOR index (classical)

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 Probabilistic Linguistic 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 PLTS: {L_k|p_k} where L_k is linguistic label, Σ p_k ≤ 1 before computation.

• Defuzzification method affects ranking: expected linguistic value E = Σ p_k · index(L_k) is the canonical choice but alternatives exist.

Works with

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