Fuzzy VIKOR - Fuzzy extension of VIKOR

Overview

fuzzy-vikor extends VIKOR to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

utility, lower is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Determine the best f_j* and worst f_j^- value of every criterion across alternatives, respecting benefit/cost direction.

   $$f_j^{\ast }=\{\begin{array}{ll}{\max }_i{x}_{ij}& j\in J\\ {\min }_i{x}_{ij}& j\in J^{\prime }\end{array},f_j^{-}=\{\begin{array}{ll}{\min }_i{x}_{ij}& j\in J\\ {\max }_i{x}_{ij}& j\in J^{\prime }\end{array}$$

   Opricovic & Tzeng 2004, §2 Eq.(3)

2. 2

   Compute the group utility S_i and the individual regret R_i. S aggregates weighted normalised regret (L_1-metric); R is the maximum weighted regret (L_∞-metric).

   $${\tilde{S}}_i=\sum _{j=1}^n{\tilde{w}}_j\otimes \left(\frac{{\tilde{f}}_j^{\ast }\ominus {\tilde{x}}_{ij}}{{\tilde{f}}_j^{\ast }\ominus {\tilde{f}}_j^{-}}\right)$$

   Opricovic & Tzeng 2004, §2 Eq.(4)

3. 3

   Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v.

   $${\tilde{Q}}_i=v\frac{{\tilde{S}}_i\ominus {\tilde{S}}^{\ast }}{{\tilde{S}}^{-}\ominus {\tilde{S}}^{\ast }}\oplus (1-v)\frac{{\tilde{R}}_i\ominus {\tilde{R}}^{\ast }}{{\tilde{R}}^{-}\ominus {\tilde{R}}^{\ast }}$$

   Opricovic & Tzeng 2004, §2 Eq.(5)

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 Fuzzy (Triangular) 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 TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

Works with

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