PSI Method
Preference Selection Index
PSI (Preference Selection Index) is a simple MCDM method developed to select and rank alternatives without requiring subjective criterion weights. It uses statistical variation in the normalized decision matrix to determine the relative preference of criteria and then calculates a Preference Selection Index for each alternative.
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 |
Normalize the decision matrix
For benefit criteria :
For cost criteria :
C₁ and C₃ are benefit criteria. C₂ is a cost criterion.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.4000 | 0.7500 | 0.8571 |
| $A_2$ | 0.6000 | 0.6000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Average performance value
Calculate the mean of the normalized values for each criterion :
\(\overline{r_j}\) represents the average normalized performance of criterion \(C_j\).
The average is computed across all $m=3$ alternatives for each criterion.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Average ($\overline{r_j}$) | 0.6667 | 0.7833 | 0.9524 |
Calculate the Preference Variation Value
Calculate the preference variation for each criterion :
\(PV_j\) measures the variation of the normalized values around their mean. A smaller variation indicates that the criterion has less discriminatory variation.
Each $PV_j$ uses all normalized values in column $j$ and their mean $\overline{r_j}$.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Pref. Variation ($PV_j$) | 0.1867 | 0.0817 | 0.0136 |
Calculate the deviation in preference variation
Calculate the deviation associated with each criterion :
The deviation represents the remaining preference after accounting for the preference variation.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Deviation ($\varPhi_j$) | 0.8133 | 0.9183 | 0.9864 |
Calculate the overall preference value
Normalize the deviation values to obtain the overall preference value for each criterion :
The overall preference values satisfy :
\(W_j\) acts as the criterion's objective preference value and is obtained without assigning subjective weights.
Weights are determined objectively — each $W_j$ depends on all $\varPhi$ values.
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Weight ($W_j$) | 0.2992 | 0.3379 | 0.3629 |
Preference selection index and ranking
Calculate the PSI score for each alternative :
\(I_i\) combines the normalized performance of alternative \(A_i\) with the objective preference value of each criterion.
Rank the alternatives according to their PSI values in descending order :
| Alternative | PSI Score ($I_i$) |
|---|---|
| $A_1$ | 0.6842 |
| $A_2$ | 0.7452 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$