Z-VIKOR - Z-Number extension of VIKOR

Overview

z-vikor extends VIKOR to handle Z-Number uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Z-Number (Z = (A, B): restriction A, reliability B; both fuzzy) algebra. The final scores are defuzzified via convert to regular fuzzy: Ã = B·A, then centroid before ranking.

Output

utility, lower is better

Data

Z-Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Z-Number MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct Z-number decision matrix Z_ij = (A_ij, B_ij); convert via Ã'(x) = √α · μ_A(x) with α = ∫μ_B(x)dx (Kang 2012) or apply the Shen 2018 comprehensive weighted Z-distance directly. Determine best Z_j^* and worst Z_j^- per criterion direction in the converted space.

   $$Z_{ij}=(A_{ij},B_{ij});{\tilde{A}}^{\prime }(x)=\sqrt{\alpha }\cdot {\mu }_A(x),\alpha =\int {\mu }_B(x)dx;Z_j^{\ast }=\arg \max /\underset{i}{\min }{Z}_{ij},Z_j^{-}=\arg \min /\underset{i}{\max }{Z}_{ij}$$

   Report §5.2 Formulas 1-2; Shen 2018 Z-VIKOR

2. 2

   Apply classical VIKOR utility S_i and regret R_i on the converted Z-values: S_i = Σ_j w_j·d(Z_j^*, Z_ij)/d(Z_j^*, Z_j^-) (L_1-metric); R_i = max_j [w_j · d(Z_j^*, Z_ij)/d(Z_j^*, Z_j^-)] (L_∞-metric), with d the comprehensive Z-distance.

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

   Report §5.2 Formula 2 - utility and regret with Z-distance

3. 3

   VIKOR index Q_i as convex combination of normalised S and R, with compromise coefficient v (classical VIKOR aggregation).

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

   Report §5.2 Formula 3 - 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 Z-Number 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 Z = (A, B) where A restricts values, B indicates reliability of A; both TFNs typically before computation.

• Defuzzification method affects ranking: convert to regular fuzzy: Ã = B·A, then centroid is the canonical choice but alternatives exist.

• If B (reliability) is identical in every cell, the sqrt(alpha) constant cancels exactly in the f*/f- and S_i/R_i ratios; ranking AND scores stay unchanged. This is a mathematical property, not a defect. B changes the result only when reliability differs between alternatives.

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