REGIME Method
Regime Method
REGIME (Regime Method) is a multi-criteria decision-making (MCDM) method based on pairwise comparison of alternatives across criteria. It is particularly suitable when decision information is ordinal or qualitative, while it can also handle quantitative information.
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 |
Superiority index
Compare two alternatives \(A_f\) and \(A_l\) criterion by criterion. The superiority index identifies the criteria for which alternative \(A_f\) is better than alternative \(A_l\).
For a benefit criterion, a higher value is preferred. Therefore, alternative \(A_f\) is superior to \(A_l\) when :
For a cost criterion, a lower value is preferred. Therefore, alternative \(A_f\) is superior to \(A_l\) when :
The set \(\widetilde{E}_{fl}\) contains the criteria on which \(A_f\) has a better performance than \(A_l\).
| $A_f \backslash A_l$ | $A_1$ | $A_2$ | $A_3$ |
|---|---|---|---|
| $A_1$ | — | C₂ | ∅ |
| $A_2$ | C₁ C₃ | — | C₃ |
| $A_3$ | C₁ C₂ C₃ | C₁ C₂ C₃ | — |
Superiority identifier
The superiority identifier represents the total weight of the criteria in the superiority index.
\(w_j\) is the weight of criterion \(C_j\). Therefore, \(\widehat{E}_{fl}\) measures the importance of the criteria where \(A_f\) is superior to \(A_l\).
| $A_f \backslash A_l$ | $A_1$ | $A_2$ | $A_3$ |
|---|---|---|---|
| $A_1$ | — | 0.2009 | 0.0000 |
| $A_2$ | 0.7991 | — | 0.0712 |
| $A_3$ | 1.0000 | 1.0000 | — |
Impacts matrix
Construct the impacts matrix by expressing the performance of each alternative for every criterion in ordinal form.
\(r_{ij}\) represents the ordinal impact of alternative \(A_i\) with respect to criterion \(C_j\).
Convert the original performance values into ordinal ranks for each criterion. This matrix shows the relative standing of alternatives on every criterion.
Lower ordinal value=better performance (order 1=best). Tied alternatives receive the same rank. For cost criterion C₂, lower original value → lower ordinal (better).
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 3 | 2 | 3 |
| $A_2$ | 2 | 3 | 1 |
| $A_3$ | 1 | 1 | 1 |
REGIME matrix
Compare alternatives \(A_f\) and \(A_l\) for every criterion. The REGIME matrix records whether \(A_f\) is better, equal, or worse than \(A_l\).
\(+1\) means that \(A_f\) is superior to \(A_l\), \(0\) means equal performance, and \(-1\) means that \(A_f\) is inferior to \(A_l\).
| Pair ($A_f$ vs $A_l$) | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ vs $A_2$ | −1 | +1 | −1 |
| $A_1$ vs $A_3$ | −1 | −1 | −1 |
| $A_2$ vs $A_1$ | +1 | −1 | +1 |
| $A_2$ vs $A_3$ | −1 | −1 | 0 |
| $A_3$ vs $A_1$ | +1 | +1 | +1 |
| $A_3$ vs $A_2$ | +1 | +1 | 0 |
Guide ndex
The guide index is the weighted sum of the elements of the REGIME vector.
A positive guide index indicates an overall preference for \(A_f\) over \(A_l\), while a negative value indicates a preference for \(A_l\) over \(A_f\).
| $A_f \backslash A_l$ | $A_1$ | $A_2$ | $A_3$ |
|---|---|---|---|
| $A_1$ | — | −0.5982 | −1.0000 |
| $A_2$ | 0.5982 | — | −0.9288 |
| $A_3$ | 1.0000 | 0.9288 | — |
Ranking
1. Ranking of Alternatives by Superiority Identifier
The first ranking technique uses the superiority identifier \(\widehat{E}_{fl}\). The REGIME method compares the superiority identifiers of two alternatives.
The difference between the two superiority identifiers determines which alternative is superior.
A positive difference indicates that alternative \(A_f\) is superior to \(A_l\). A negative difference indicates that \(A_l\) is superior to \(A_f\). A zero difference indicates indifference between the two alternatives.
2. Ranking of Alternatives by guide index
The second ranking technique uses the Guide Index \(E_{fl}\). It is calculated by multiplying the REGIME comparison value for each criterion by its corresponding criterion weight and then summing over all criteria.
where \(E'_{fl,j}\) represents the comparison of alternative \(A_f\) with alternative \(A_l\) for criterion \(C_j\), and \(w_j\) is the weight of criterion \(C_j\).
The guide index is calculated for each pair of alternatives. The resulting guide indices are then used to obtain the final ranking of the alternatives.
A positive guide index indicates that \(A_f\) is preferred to \(A_l\), while a negative guide index indicates that \(A_l\) is preferred to \(A_f\).
Technique 1: Superiority identifier differences
| Pair | $\widehat{E}_{fl} - \widehat{E}_{lf}$ | Result |
|---|---|---|
| $A_1$ vs $A_2$ | −0.5982 | $A_2 >A_1$ |
| $A_1$ vs $A_3$ | −1.0000 | $A_3 >A_1$ |
| $A_2$ vs $A_3$ | −0.9288 | $A_3 >A_2$ |
Technique 2: Guide index
| Pair | $E'_{fl}$ | Result |
|---|---|---|
| $A_1$ vs $A_2$ | −0.5982 | $A_2 >A_1$ |
| $A_1$ vs $A_3$ | −1.0000 | $A_3 >A_1$ |
| $A_2$ vs $A_3$ | −0.9288 | $A_3 >A_2$ |
Final ranking: $A_3 \succ A_2 \succ A_1$