WASPAS Method
Weighted Aggregated Sum Product Assessment
WASPAS (Weighted Aggregated Sum Product Assessment) is a hybrid multi-criteria decision-making method that combines the Weighted Sum Model (WSM) and the Weighted Product Model (WPM). It calculates a joint optimality criterion by aggregating additive and multiplicative utility functions, offering higher accuracy and ranking consistency than either method alone.
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 (non-benefit) criteria :
\(r_{ij}\) is the normalized performance 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 additive relative importance and multiplicative relative importance
The Weighted Sum Model (WSM) calculates the total relative importance of each alternative by summing the weighted normalized criterion values.
\(Q_i^{(1)}\) is the WSM score of alternative \(A_i\). A higher value indicates a better alternative.
The Weighted Product Model (WPM) calculates the multiplicative relative importance of each alternative. Each normalized criterion value is raised to the power of its criterion weight.
\(Q_i^{(2)}\) is the WPM score of alternative \(A_i\). A higher value indicates a better alternative.
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $Q_i^{(1)}$ (WSM) | $Q_i^{(2)}$ (WPM) |
|---|---|---|
| $A_1$ | 0.5029 | 0.4791 |
| $A_2$ | 0.6285 | 0.6222 |
| $A_3$ | 1.0000 | 1.0000 |
Qi value and ranking
WASPAS combines the additive WSM score and the multiplicative WPM score using the aggregation parameter \(\lambda\).
The parameter \(\lambda\) satisfies :
A commonly used value is :
When \(\lambda=0.5\), WSM and WPM have equal influence on the final WASPAS score.
\(\lambda=1\) → only WSM is considered.
\(\lambda=0\) → only WPM is considered.
\(\lambda=0.5\) → WSM and WPM contribute equally.
Using aggregation parameter: $\lambda=0.5$
| Alternative | WASPAS Score ($Q_i$) |
|---|---|
| $A_1$ | 0.4910 |
| $A_2$ | 0.6253 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$
WASPAS Algorithm Flowchart
Qi value and ranking