VIKOR - VlseKriterijumska Optimizacija I Kompromisno Resenje (Multicriteria Optimisation and Compromise Solution)

Overview

Q ∈ [0,1] when all alternatives differ in both S and R. A lower Q is preferred. Read the result in two layers: (1) Q-ranking gives a linear order; (2) the compromise set tells which alternatives are practically indistinguishable from the top one - a single best alternative is proposed ONLY when both C1 (advantage of at least 1/(m-1) over the runner-up) AND C2 (top alternative is also best in S or R) hold simultaneously.

Output

utility, lower is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Compromise solutions, engineering design selection

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).

   $$S_i=\sum _{j=1}^n{w}_j\frac{f_j^{\ast }-x_{ij}}{f_j^{\ast }-f_j^{-}},R_i=\underset{j}{\max }\left[w_j\frac{f_j^{\ast }-x_{ij}}{f_j^{\ast }-f_j^{-}}\right]$$

   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.

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

   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)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

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)}.

• Reporting only the Q-best alternative when C1 fails - VIKOR's defining feature is the compromise SET, not a single winner.

• Mixing up f_j* (best) and f_j^- (worst) for cost criteria. For benefit criteria f_j* = max, f_j^- = min; for cost criteria the assignment flips.

• Choosing v=0 (pure max-regret) or v=1 (pure utility) collapses VIKOR to its limit cases - only v ∈ (0,1) gives the compromise blend Opricovic intended; v=0.5 is the canonical default.

Works with

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