Vikor

Vlse Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is a multi-attribute decision analysis method used to determine the best compromise solution. It is located in SPSSAU -> Comprehensive Evaluation -> VIKOR.

SPSSAU Operations

To conduct the analysis, drag the positive or negative indicators into the right-hand box and click 'Start'. SPSSAU includes four parameters:

Indicator Weight: If not set, all indicators are assigned equal weight. If set, SPSSAU normalizes the weights automatically.

Normalization: Selected by default, it applies sum of squares normalization to the data.

Lambda: Default value is 0.5, with options ranging from 0 to 1 in increments of 0.1.

Save Process Values: When selected, SPSSAU saves the 'Sum of optimal scheme distances', 'Maximum value of optimal scheme distance', and the Vikor Index (Q) as new titles, such as "Vikor_S_****", "Vikor_R_****", and "Vikor_Q_****".

SPSSAU Data Format

If performing a VIKOR analysis, the evaluation object should be placed in a separate column. If not specified, SPSSAU automatically labels them as Item 1, Item 2, Item 3, etc. Each indicator (positive or negative) occupies one column.

Algorithm

1.Construct the Data Matrix

Construct a matrix X = [x_ij], where x_ij represents the value of scheme i under indicator j.

2.Normalization

If 'Normalization' is selected, SPSSAU normalizes the data using: $$\frac{x_i}{\sqrt{\sum _i^m}{x}_i^2}$$

3.Determine Positive and Negative Ideal Solutions

Calculate the positive ideal solution (r⁺) and the negative ideal solution (r^-) of each indicatorsss:
For positive indicators: $$r^{+}:max(x),r^{-}:min(x)$$ For negative indicators: $$r^{+}:min(x),r^{-}:max(x)$$

4.Calculate the Distances to the Optimal Solution

For each scheme i, compute: $$s=W_j\times (\frac{r^{+}-x}{r^{+}-r^{-}})$$ If no weights are specified, SPSSAU assigns W_j =1 / m, where m is the number of evaluation indicators.

5.Calculate Sum of optimal scheme distances and Maximum value of optimal scheme distance

The formulas are as follows: $$S=\sum _i^m{s}^i$$ $$R=max(s)$$ At the same time, SPSSAU also outputs four indicators: S⁺ (optimal value of S), S^- (worst value of S), R⁺ (optimal value of R), and R^- (worst value of R) $$S^{+}=min(S)$$ $$S^{-}=max(S)$$ $$R^{+}=min(R)$$ $$R^{-}=max(R)$$

6.Calculate the ViKOR Index Q

The formula for Q is: $$Q=lamba\times \frac{S-S^{+}}{S^{-}-S^{+}}+(1-lamba)\times \frac{R-R^{+}}{R^{-}-R^{+}}$$ lamba: the decision-maker's strategy weight, typically set to 0.5 but adjustable by the researcher.

7.Rank

Rank solutions based on Q_i, where lower values indicate better solutions. Following these steps, the VIKOR method effectively provides a compromise solution for multi-attribute decision-making problems.