TODIM Method
TOmada de Decisão Interativa e Multicritério – Interactive and Multicriteria Decision Making
TODIM (TOmada de Decisão Interativa e Multicritério) is a compensatory MCDM method based on Prospect Theory. It evaluates alternatives through pairwise comparisons and explicitly considers the decision-maker's risk perception, gains, losses, and loss aversion.
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 decision matrix to transform the original criterion values into comparable values. The normalization depends on whether the criterion is a benefit or a non-benefit criterion.
For benefit criteria :
For benefit criteria, a larger value of \(x_{ij}\) produces a larger normalized value \(r_{ij}\).
For cost criteria :
For non-benefit criteria, the reciprocal transformation ensures that a smaller original value receives a larger normalized value.
C₁ and C₃ are benefit criteria. C₂ is a cost criterion (uses reciprocal transformation).
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2000 | 0.3191 | 0.3000 |
| $A_2$ | 0.3000 | 0.2553 | 0.3500 |
| $A_3$ | 0.5000 | 0.4256 | 0.3500 |
Calculate relative weights
Select the criterion with the largest weight as the reference criterion \(C_r\).
Calculate the relative weight of every criterion with respect to the reference criterion.
Consequently, the relative weight of the reference criterion is equal to 1.
Reference criterion: $C_1$ with $w_1=0.7279=\max(w_j)$.
| Criterion | Weight $w_j$ | Relative $w_{jr}$ |
|---|---|---|
| $C_1$ (ref) | 0.7279 | 1.0000 |
| $C_2$ | 0.2009 | 0.2760 |
| $C_3$ | 0.0712 | 0.0978 |
| $\sum$ | 1.0000 | 1.3738 |
Calculate the dominance degree
Sum the partial dominance values over all criteria to obtain the dominance of \(A_i\) over \(A_t\).
The partial dominance of \(A_i\) over \(A_t\) under criterion \(C_j\) is calculated as :
\(\theta\) is the attenuation factor of losses (loss-aversion parameter). It controls the influence of losses in the TODIM value function.
| $\delta(A_i, A_t)$ | $A_1$ | $A_2$ | $A_3$ |
|---|---|---|---|
| $A_1$ | 0 | −1.0953 | −2.2081 |
| $A_2$ | −0.2342 | 0 | −1.4451 |
| $A_3$ | 0.6733 | 0.5666 | 0 |
Calculate the overall dominance degree
Sum the pairwise dominance values of each alternative against all alternatives and normalize the results to obtain the global dominance value.
\[ \xi(A_i)=\frac{ D_i-\displaystyle\min_i D_i}{ \displaystyle\max_i D_i-\displaystyle\min_i D_i} \]
The global dominance values are normalized to the interval \([0,1]\).
Rank the alternatives according to their global dominance values in descending order.
Overall dominance $D_i$:
| Alternative | $D_i$ |
|---|---|
| $A_1$ | −3.3034 |
| $A_2$ | −1.6793 |
| $A_3$ | 1.2399 |
| min $D_i$ | −3.3034 |
| max $D_i$ | 1.2399 |
Global dominance $\xi(A_i)$:
| Alternative | $\xi(A_i)$ |
|---|---|
| $A_1$ | 0.0000 |
| $A_2$ | 0.3574 |
| $A_3$ | 1.0000 |
Final ranking: $A_3 \succ A_2 \succ A_1$