VIKOR Method
VIseKriterijumska Optimizacija I Kompromisno Resenje - Multicriteria Optimization and Compromise Solution
VIKOR (VIseKriterijumska Optimizacija I Kompromisno Rešenje) is an objective weighting method for MCDM, introduced by is a compromise-ranking MCDM method designed for decision problems involving conflicting and non-commensurable criteria. It identifies a compromise solution that provides a balance between maximum group utility and minimum individual regret.
Video Demonstration
Step-by-Step Algorithm
Form the decision matrix
Create the decision matrix with $m$ alternatives and $n$ criteria:
\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]
\(x_{ij}\) presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 10 | 20 | 30 |
| $A_2$ | 15 | 25 | 35 |
| $A_3$ | 25 | 15 | 35 |
Determine the best and worst values
Determine the best \(f_j^+\) and worst \(f_j^-\) values for each criterion.
For benefit criteria :
For cost criteria :
\(f_j^+\) represents the best performance and \(f_j^-\) represents the worst performance for criterion \(C_j\).
C₁ and C₃ are benefit criteria (best=max, worst=min). C₂ is a cost criterion (best=min, worst=max).
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Best ($f_j^+$) | 25 | 15 | 35 |
| Worst ($f_j^-$) | 10 | 25 | 30 |
Si and Ri values
Weighted distance for each criterion:
Sum the weighted distances over all criteria. This represents the overall or group utility loss of alternative \(A_i\).
Determine the maximum weighted distance among all criteria. This represents the maximum individual regret of alternative \(A_i\).
Weighted distances $d_{ij}$:
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.7279 | 0.1005 | 0.0712 |
| $A_2$ | 0.4853 | 0.2009 | 0.0000 |
| $A_3$ | 0.0000 | 0.0000 | 0.0000 |
Group utility $S_i$ and individual regret $R_i$:
| Alternative | $S_i$ | $R_i$ |
|---|---|---|
| $A_1$ | 0.8996 | 0.7279 |
| $A_2$ | 0.6862 | 0.4853 |
| $A_3$ | 0.0000 | 0.0000 |
S+ , S− , R+ and R−
Determine the best and worst values among all alternatives for the \(S_i\) and \(R_i\) measures.
| Measure | Best (+) | Worst (−) |
|---|---|---|
| $S$ | 0.0000 | 0.8996 |
| $R$ | 0.0000 | 0.7279 |
VIKOR index \(Q_i\) and ranking
Combine the group utility measure \(S_i\) and the individual regret measure \(R_i\) into the VIKOR index.
The parameter \(v\) represents the weight of the group utility , usually set to \(v=0.5\).
A smaller \(Q_i\) represents a better compromise solution.
Rank the alternatives according to \(S_i\), \(R_i\), and \(Q_i\), with the VIKOR index \(Q_i\) providing the main compromise ranking.
The alternative with the lowest \(Q_i\) is the preferred compromise alternative, subject to VIKOR's acceptable-advantage and stability conditions.
Using $v=0.5$ (equal weight for group utility and individual regret).
| Alternative | VIKOR Index ($Q_i$) |
|---|---|
| $A_1$ | 1.0000 |
| $A_2$ | 0.7149 |
| $A_3$ | 0.0000 |
Q-ranking: $A_3 \succ A_2 \succ A_1$
Acceptable advantage and acceptable stability
VIKOR does not select the best compromise solution solely from the numerical value of \(Q_i\). The leading alternative must satisfy the acceptable-advantage and acceptable-stability conditions.
Acceptable advantage :
The best alternative must have a sufficiently large advantage over the second-best alternative.
The threshold \(DQ\) is calculated as :
\(m\)=number of alternatives.
If the acceptable-advantage condition is not satisfied, a set of alternatives \(A^{(1)},...,A^{(k)}\) are proposed as the best choices while \(Q(A^{(k)})-Q(A^{(1)})\leq DQ\)
Acceptable Stability :
The alternative ranked first according to \(Q_i\) must also have a strong position according to at least one of the two measures : group utility \(S_i\) or individual regret \(R_i\).
If \(A^{(1)}\) is not ranked first by either \(S_i\) or \(R_i\), the acceptable stability condition is not satisfied. \(A^{(1)}\) and \(A^{(2)}\) are proposed as the best choices.
Acceptable advantage check:
| Item | Value | Status |
|---|---|---|
| DQ=1/(m−1) | 0.5000 | — |
| $Q(A^{(2)}) - Q(A^{(1)})$ | 0.7149 | ✓ 0.7149 >0.5000 |
Acceptable stability check — ranking comparison:
| Alternative | Q-rank | S-rank | R-rank | Stability |
|---|---|---|---|---|
| $A_3$ | 1 | 1 | 1 | ✓ Satisfied |
| $A_2$ | 2 | 2 | 2 | — |
| $A_1$ | 3 | 3 | 3 | — |
Acceptable stability: $A_3$ is rank 1 by both $S$ and $R$ ✓ Satisfied
Conclusion: $A_3$ is the best compromise solution.
VIKOR Algorithm Flowchart
Si and Ri values
VIKOR index Qi and ranking
A(1) is the best
rank on S or R
A(1) ,...,A(k) are proposed as the best choices while
Q(A(k) )-Q(A(1) ) ≤ DQ
Both alternatives A(1) and A(2) are proposed as the best choices
A(1) is the best choice