CODAS Method
Combinative Distance-Based Assessment
CODAS (Combinative Distance-Based Assessment) is an MCDM method introduced by Ghorabaee et al. in 2016. It evaluates alternatives based on their Euclidean distance from the negative-ideal solution, with the Taxicab (Manhattan) distance used as a secondary measure.
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 |
Normalized decision matrix
For benefit criteria :
For cost criteria :
| Alternative | $C_1$ (B) | $C_2$ (C) | $C_3$ (B) |
|---|---|---|---|
| $A_1$ | 0.4000 | 0.7500 | 0.8571 |
| $A_2$ | 0.6000 | 0.6000 | 1.0000 |
| $A_3$ | 1.0000 | 1.0000 | 1.0000 |
Weighted normalized decision matrix
The normalized decision matrix is multiplied by the corresponding criterion weights.
The normalized decision matrix is multiplied by the corresponding criterion weights.
$$ v_{ij}=w_j r_{ij} $$Where \(w_j\) represents the weight of criterion \(j\), and
$$ \sum_{j=1}^{n}w_j=1. $$| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2912 | 0.1507 | 0.0610 |
| $A_2$ | 0.4367 | 0.1205 | 0.0712 |
| $A_3$ | 0.7279 | 0.2009 | 0.0712 |
Negative ideal solution
The negative ideal solution is obtained by taking the minimum value of each criterion from the weighted normalized decision matrix.
The negative ideal solution is obtained by taking the minimum value of each criterion from the weighted normalized decision matrix.
Thus, the negative ideal solution is :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| NIS ($ns_j$) | 0.2912 | 0.1205 | 0.0610 |
Euclidean and Manhattan distances
The Euclidean distance of alternative \(i\) from the negative-ideal solution is calculated as :
Taxicab (Manhattan) distance :
| Alternative | $E_i$ | $T_i$ | Threshold check |
|---|---|---|---|
| $A_1$ | 0.0302 | 0.0302 | Compared in Step 6 |
| $A_2$ | 0.1459 | 0.1557 | Compared in Step 6 |
| $A_3$ | 0.4442 | 0.5273 | Compared in Step 6 |
Relative assessment matrix
The relative assessment of alternative $i$ over alternative $k$ is calculated using the Euclidean and Taxicab distances with a threshold function.
The relative assessment of alternative \(i\) over alternative \(k\) is calculated using the Euclidean and Taxicab distances.
$$ h_{ik}=\left( E_i-E_k \right) + \left( \psi(E_i-E_k) \right) \left( T_i-T_k \right) $$where \(\psi\) is a threshold function defined as :
$$ \psi(x)=\begin{cases} 1, & |x|\geq\tau,\\ 0, & |x|<\tau \end{cases} $$ \(\tau\) is the threshold parameter that can be set by the decision maker. It is suggested to set this parameter at a value between 0.01 and 0.05.Using threshold parameter $\tau=0.02$.
| $h_{ik}$ | $A_1$ | $A_2$ | $A_3$ |
|---|---|---|---|
| $A_1$ | 0.0000 | -0.2412 | -0.9111 |
| $A_2$ | 0.2412 | 0.0000 | -0.6699 |
| $A_3$ | 0.9111 | 0.6699 | 0.0000 |
All $|E_i - E_k|$ values exceed $\tau=0.02$, so $\psi=1$ for every off-diagonal cell. The matrix is skew-symmetric: $h_{ik}=-h_{ki}$.
Preference value and ranking
The assessment score of each alternative is calculated by summing its relative assessment values.
The assessment score of each alternative is calculated by taking the average of its relative assessment values.
The alternatives are ranked according to their assessment scores \(H_i\).
| Alternative | Score ($H_i$) |
|---|---|
| $A_1$ | -1.1523 |
| $A_2$ | -0.4287 |
| $A_3$ | 1.5810 |
Final ranking: $A_3 \succ A_2 \succ A_1$