TOPSIS Method
Technique for Order Preference by Similarity to Ideal Solution
TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is a multi-criteria decision-making method that ranks alternatives based on their geometric distance to an ideal solution. The best alternative is the one closest to the Positive Ideal Solution (PIS) and farthest from the Negative Ideal Solution (NIS)
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 :
\(r_{ij}\) is the normalized performance value of alternative \(A_i\) for criterion \(C_j\).
| 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
The normalized decision matrix is multiplied by the corresponding criterion weights.
where \(w_j\) represents the weight of criterion \(j\), and
Using weights: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2362 | 0.1137 | 0.0369 |
| $A_2$ | 0.3542 | 0.1421 | 0.0431 |
| $A_3$ | 0.5904 | 0.0852 | 0.0431 |
Determine the positive and negative ideal solutions
Determine the positive ideal solution (PIS) and negative ideal solution (NIS) from the weighted normalized matrix.
Positive Ideal Solution (PIS) :
Negative Ideal Solution (NIS) :
\(J^+\) is the set of benefit criteria and \(J^-\) is the set of cost criteria.
C₁ and C₃ are benefit ($J^+$, use max for PIS, min for NIS). C₂ is cost ($J^-$, use min for PIS, max for NIS).
| Solution | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| PIS ($v_j^+$) | 0.5904 | 0.0852 | 0.0431 |
| NIS ($v_j^-$) | 0.2362 | 0.1421 | 0.0369 |
Separation measures, closeness coefficient and ranking
Calculate the Euclidean distance of each alternative from the Positive Ideal Solution.
A smaller \(D_i^+\) means that alternative \(A_i\) is closer to the ideal solution.
Calculate the Euclidean distance of each alternative from the Negative Ideal Solution.
A larger \(D_i^-\) means that alternative \(A_i\) is farther from the negative ideal solution.
Calculate the relative closeness coefficient of each alternative to the Positive Ideal Solution.
The coefficient normally satisfies :
A larger \(C_i^*\) indicates that the alternative is closer to the PIS and farther from the NIS.
Rank the alternatives according to their relative closeness coefficients in descending order.
| Alternative | $D_i^+$ | $D_i^-$ | $C_i^*$ |
|---|---|---|---|
| $A_1$ | 0.3554 | 0.0284 | 0.0740 |
| $A_2$ | 0.2430 | 0.1182 | 0.3272 |
| $A_3$ | 0.0000 | 0.3588 | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$