PIV Method
Proximity Indexed Value
PIV (Proximity Indexed Value) is a multi-criteria decision-making (MCDM) method that ranks alternatives based on their weighted proximity (deviation) from ideal values. It is computationally simple and specifically designed to minimize the rank reversal problem. Lower overall proximity index values indicate better alternatives.
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
The normalized value is calculated as :
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.3244 | 0.5657 | 0.5185 |
| $A_2$ | 0.4867 | 0.7071 | 0.6049 |
| $A_3$ | 0.8111 | 0.4243 | 0.6049 |
Construct the weighted normalized decision matrix
Multiply each normalized value by the corresponding criterion weight:
The weighted normalized matrix is :
Where $w_j$ is the weight of criterion $j$.
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ (B) | $C_2$ (C) | $C_3$ (B) |
|---|---|---|---|
| $A_1$ | 0.2361 | 0.1136 | 0.0369 |
| $A_2$ | 0.3543 | 0.1421 | 0.0431 |
| $A_3$ | 0.5904 | 0.0852 | 0.0431 |
Weighted proximity index
Calculate the proximity value according to the type of criterion :
For benefit criteria :
For cost criteria :
Where $v_{ij}$ is the weighted normalized value of alternative $A_i$, $v_{\max}$ is the maximum weighted normalized value, and $v_{\min}$ is the minimum weighted normalized value.
| Alternative | $C_1$ ($v_{\max}=0.5904$) | $C_2$ ($v_{\min}=0.0852$) | $C_3$ ($v_{\max}=0.0431$) |
|---|---|---|---|
| $A_1$ | 0.3543 | 0.0284 | 0.0062 |
| $A_2$ | 0.2361 | 0.0569 | 0.0000 |
| $A_3$ | 0.0000 | 0.0000 | 0.0000 |
Overall proximity value and ranking
Sum the proximity values across all criteria :
Rank the alternatives according to their overall proximity values in ascending order :
| Alternative | Proximity Value ($d_i$) |
|---|---|
| $A_1$ | 0.3889 |
| $A_2$ | 0.2930 |
| $A_3$ | 0.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$