WSM Method
Weighted Sum Model
WSM (Weighted Sum Model) or SAW (Simple Additive Weighting) is one of the oldest and simplest multi-criteria decision-making (MCDM) methods. It calculates a weighted sum of normalized performance scores for each alternative. The alternative with the highest total score is ranked best. It is easy to understand and widely used as a baseline method.
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. 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 |
Calculate the weighted normalized decision matrix
Multiply each normalized value by the 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 | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2912 | 0.1507 | 0.0610 |
| $A_2$ | 0.4367 | 0.1205 | 0.0712 |
| $A_3$ | 0.7279 | 0.2009 | 0.0712 |
Total score and ranking
Calculate the overall score of each alternative by summing all its weighted normalized criterion values.
\(S_i\) is the overall WSM score of alternative \(A_i\).
Rank the alternatives according to their WSM scores in descending order.
| Alternative | WSM Score ($S_i$) |
|---|---|
| $A_1$ | 0.5029 |
| $A_2$ | 0.6285 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$