WISP Method
Weighted Integrated Sum Product
WISP (Weighted Integrated Sum Product) is an MCDM method that integrates weighted-sum and weighted-product approaches and uses four utility measures to evaluate and rank alternatives. It was developed to provide a relatively simple procedure for combining benefit and cost criteria.
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
\(r_{ij}\) is the normalized performance of alternative \(A_i\) for criterion \(C_j\).
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.4000 | 0.8000 | 0.8571 |
| $A_2$ | 0.6000 | 1.0000 | 1.0000 |
| $A_3$ | 1.0000 | 0.6000 | 1.0000 |
Calculate the utility measures
WISP evaluates each alternative using four utility measures representing differences and ratios between the weighted sums and weighted products of benefit and cost criteria.
Weighted Sum Difference :
\(u_i^{sd}\) represents the difference between the weighted sum of benefit and cost criteria.
Weighted Product Difference :
\(u_i^{pd}\) represents the difference between the weighted products of benefit and cost criteria.
Weighted Sum Ratio :
\(u_i^{sr}\) represents the ratio between the weighted sum of benefit and cost criteria.
Weighted Product Ratio :
\(u_i^{pr}\) represents the ratio between the weighted product of benefit and cost criteria.
| Alternative | $u_i^{sd}$ | $u_i^{pd}$ | $u_i^{sr}$ | $u_i^{pr}$ |
|---|---|---|---|---|
| $A_1$ | 0.1915 | -0.1429 | 2.1917 | 0.1105 |
| $A_2$ | 0.3070 | -0.1698 | 2.5278 | 0.1548 |
| $A_3$ | 0.6786 | -0.0687 | 6.6315 | 0.4301 |
Recalculated utility measures values
Because the four utility measures can have different scales, they are recalculated to obtain comparable values in the interval $[0,1]$.
Recalculated weighted sum difference :
Recalculated weighted product difference :
Recalculated weighted sum ratio :
Recalculated weighted product ratio :
Max values: $\max u_i^{sd}=0.6786$, $\max u_i^{pd}=-0.0687$, $\max u_i^{sr}=6.6315$, $\max u_i^{pr}=0.4301$
| Alternative | $\overline{u}_i^{sd}$ | $\overline{u}_i^{pd}$ | $\overline{u}_i^{sr}$ | $\overline{u}_i^{pr}$ |
|---|---|---|---|---|
| $A_1$ | 0.7098 | 0.9203 | 0.4183 | 0.7765 |
| $A_2$ | 0.7786 | 0.8915 | 0.4623 | 0.8075 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
Overall utility and ranking
Calculate the overall utility of each alternative by taking the arithmetic mean of the four recalculated utility measures.
\(u_i\) is the final overall utility of alternative \(A_i\).
Rank the alternatives according to their overall utility values in descending order.
| Alternative | Utility ($u_i$) |
|---|---|
| $A_1$ | 0.7062 |
| $A_2$ | 0.7350 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$