VIKOR

Overview

VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) finds the compromise solution closest to the ideal by combining L1 (Manhattan) and L∞ (Chebyshev) distance measures. A lower Q score means a better compromise.

For each trial, Tunny Dashboard provides:

Output	Description
Utility measure S	Overall gap from the ideal (smaller is better)
Regret measure R	Gap on the worst objective (smaller is better)
Compromise score Q	S and R combined (smaller is a better compromise)
Display score	1 − Q; higher is better (shown in the UI for readability)
Ranking	Trials ordered by Q ascending (closest compromise first)
Best / worst values	The best and worst value of each objective across valid trials
Compromise set	The compromise solutions that satisfy conditions C1/C2

Algorithm

Inputs

Variable	Description
f_ij	Value of trial i for objective j
w_j	Weight for objective j (normalized, Σ = 1)
v	Strategy weight (default 0.5)

Weight normalization: if the weights do not sum to 1, they are divided by their sum when it is positive and finite; otherwise (zero, negative, or NaN sum) uniform weights are used instead.

Step 1: Best and Worst Values

For each objective j:

$f_j^* = \text{best value across all trials} \quad (\min \text{ if minimize, } \max \text{ if maximize})$

$f_j^- = \text{worst value across all trials} \quad (\max \text{ if minimize, } \min \text{ if maximize})$

Step 2: S and R Values

$S_i = \sum_j w_j \cdot \frac{f_j^* - f_{ij}}{f_j^* - f_j^-}$

$R_i = \max_j \left[ w_j \cdot \frac{f_j^* - f_{ij}}{f_j^* - f_j^-} \right]$

S_i (utility): sum of weighted gaps — lower is better overall
R_i (regret): maximum weighted gap — lower means even the worst criterion is acceptable

When f*_j = f-_j (all trials identical for that objective), the contribution is 0.

Step 3: S, S-, R, R-

$S^* = \min_i S_i, \quad S^- = \max_i S_i$

$R^* = \min_i R_i, \quad R^- = \max_i R_i$

Step 4: Q Score

$Q_i = v \cdot \frac{S_i - S^$

v value	Emphasis	Meaning
v > 0.5	S (utility)	Maximize overall consensus
v = 0.5	Balanced	Balance utility and regret
v < 0.5	R (regret)	Minimize worst-case objective

Zero-division guard: if S- = S*, the first term is 0; if R- = R*, the second term is 0.

Step 5: Ranking

Sort trials by Q ascending — lower Q is the better compromise solution.

Step 6: Compromise Acceptance Conditions (C1/C2)

Whether the top-ranked solution $A^{(1)}$ can be accepted as the single compromise solution is decided by the two conditions of Opricovic & Tzeng (2004):

C1 (acceptable advantage):

$Q(A^{(2)}) - Q(A^{(1)}) \geq DQ, \qquad DQ = \frac{1}{J - 1}$

where $J$ is the number of valid trials (all objectives finite).

C2 (acceptable stability in decision making): $A^{(1)}$ is also ranked first (ties allowed) by S or by R.

Compromise set:

C1 and C2 both hold → $A^{(1)}$ is the single compromise solution
Only C2 fails → $A^{(1)}$ and $A^{(2)}$
C1 fails → all $A^{(1)}, \dots, A^{(k)}$ with $Q(A^{(k)}) - Q(A^{(1)}) < DQ$

Edge cases: $J = 1$ yields the single valid trial; $J = 0$ yields an empty set.

In Tunny Dashboard: the compromise set is shown with a ★ marker in the MCDM Scatter Chart 2D / 3D widgets (and in the click-detail modal).

Numerical Example

3 trials × 2 objectives, both minimized, weights $w = [0.5, 0.5]$ , $v = 0.5$ :

Trial	Obj 1	Obj 2
0	1	2
1	3	1
2	2	2

Step 1 — best/worst: Obj 1: $f_1^* = 1$ , $f_1^- = 3$ , range = 2. Obj 2: $f_2^* = 1$ , $f_2^- = 2$ , range = 1.

Step 2 — S and R:

Trial	contrib1	contrib2	S	R
0	$0.5 \times (1-1)/2 = 0$	$0.5 \times (1-2)/1 = 0.5$	0.5	0.5
1	$0.5 \times (3-1)/2 = 0.5$	$0.5 \times (1-1)/1 = 0$	0.5	0.5
2	$0.5 \times (2-1)/2 = 0.25$	$0.5 \times (1-2)/1 = 0.5$	0.75	0.5

Step 3: $S^* = 0.5$ , $S^- = 0.75$ , $R^* = 0.5$ , $R^- = 0.5$ (tied).

Step 4 — Q: since $R^* = R^-$ , the second term is 0 for every trial:

$Q_0 = 0.5 \times (0.5 - 0.5)/0.25 + 0.5 \times 0 = 0$
$Q_1 = 0.5 \times (0.5 - 0.5)/0.25 + 0.5 \times 0 = 0$
$Q_2 = 0.5 \times (0.75 - 0.5)/0.25 + 0.5 \times 0 = 0.5$

Step 5 — ranking: $Q = [0, 0, 0.5]$ → trials 0 and 1 tie for first, trial 2 is last.

Comparison with TOPSIS

Aspect	TOPSIS	VIKOR
Distance measure	Euclidean (L2)	Manhattan (L1) + Chebyshev (L∞)
Ranking order	Score descending	Q ascending
Strategy param	None	v (utility vs. regret)
Zero-div guard	score = 0.5	contribution = 0
Best for	Overall similarity	Compromise / balance-focused decisions

Complexity

O(m × n + m log m) — under 100 ms for 50,000 trials × 4 objectives.

When to Use

Minimize worst-case objective (max-regret)?   → VIKOR with v < 0.5
Maximize overall consensus across objectives? → VIKOR with v > 0.5
Want a parameter-free intuitive [0,1] score?  → TOPSIS

References

Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems. Faculty of Civil Engineering, Belgrade.
Opricovic, S., & Tzeng, G.-H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2), 445–455. https://doi.org/10.1016/S0377-2217(03)00020-1