WPM Method
Weighted Product Model
WPM (Weighted Product Model) is a multiplicative MCDM method that evaluates alternatives by multiplying criterion performance ratios, with each ratio raised to the corresponding criterion weight.
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
Normalize the criterion values so that all criteria become comparable and a higher normalized value indicates better performance.
For benefit criteria :
For cost criteria :
\(r_{ij}\) is the normalized value of alternative \(A_i\) for criterion \(C_j\).
C₁ and C₃ are benefit criteria (divide by max). C₂ is a cost criterion (divide min by value).
| Alternative | $C_1$ (Benefit) | $C_2$ (Cost) | $C_3$ (Benefit) |
|---|---|---|---|
| $A_1$ | 0.4000 | 0.7500 | 0.8571 |
| $A_2$ | 0.6000 | 0.6000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Calculate the weighted normalized decision matrix
Raise each normalized criterion value to the power of its corresponding criterion weight :
where \(w_j\) is the weight of criterion \(C_j\).
The resulting matrix is the weighted normalized decision matrix:
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.5132 | 0.9439 | 0.9891 |
| $A_2$ | 0.6893 | 0.9025 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
WPM score and ranking
\(Q_i^{WPM}\) is the WPM score of alternative \(A_i\). The multiplicative structure means that the performance of all criteria contributes to the final score.
Rank the alternatives according to their WPM scores in descending order.
| Alternative | WPM Score ($Q_i^{WPM}$) |
|---|---|
| $A_1$ | 0.4787 |
| $A_2$ | 0.6221 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$