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MCDM Methods

Step-by-Step Algorithms

COBRA Method

Comprehensive Distance-Based Ranking

2022 Krstić et al. 9 Steps

COBRA (Comprehensive Distance-Based Ranking) is a distance-based MCDM method proposed by Krstić et al. in 2022. It ranks alternatives by evaluating their comprehensive distances from the positive ideal, negative ideal, and average solutions, providing a robust and reliable decision-making framework.

Video Demonstration

Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

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$102030
$A_2$152535
$A_3$251535
Criterion type: C₁ Benefit C₂ Benefit C₃ Benefit
Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)
2

Normalize the decision matrix

Create the normalized decision matrix by dividing each element by the maximum value in its column.

Formula

Create the normalized decision matrix in the following way :

$$ r_{ij}=\frac{x_{ij}} {\displaystyle\max_i(x_{ij})}, \qquad i=1,2,\ldots,m,\; j=1,2,\ldots,n. $$

The normalized decision matrix is

$$ R=[r_{ij}]=\begin{bmatrix} r_{11} & r_{12} & \cdots & r_{1n}\\ r_{21} & r_{22} & \cdots & r_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ r_{m1} & r_{m2} & \cdots & r_{mn} \end{bmatrix}. $$
Alternative$C_1$$C_2$$C_3$
$A_1$0.40000.80000.8571
$A_2$0.60001.00001.0000
$A_3$1.00000.60001.0000
3

Construct the weighted normalized decision matrix

Multiply each normalized value by the corresponding criterion weight.

Formula

The weighted normalized decision matrix is obtained by multiplying each normalized value by the corresponding criterion weight.

$$ v_{ij}=w_j\,r_{ij}, \qquad i=1,2,\ldots,m, \quad j=1,2,\ldots,n, $$

Where

  • \(w_j\) is the weight of criterion \(j\),
  • \(r_{ij}\) is the normalized value of alternative \(i\) under criterion \(j\).

The weighted normalized decision matrix is

$$ V=[v_{ij}]=\begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1n}\\ v_{21} & v_{22} & \cdots & v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} & v_{m2} & \cdots & v_{mn} \end{bmatrix}. $$
Alternative$C_1$$C_2$$C_3$
$A_1$0.29120.16070.0610
$A_2$0.43670.20090.0712
$A_3$0.72790.12050.0712
4

Positive ideal solution, negative ideal solution and average solution

After constructing the weighted normalized decision matrix $V=[v_{ij}]$, determine the Positive Ideal Solution (PIS), Negative Ideal Solution (NIS) and Average Solution (AS).

Formula

Positive ideal solution (PIS)

For each criterion, the positive ideal solution is defined as

$$ A^{+}=\left\{ v_{1}^{+}, v_{2}^{+}, \ldots, v_{n}^{+} \right\}, $$ $$ v_{j}^{+}=\begin{cases} \displaystyle\max_i(v_{ij}), & \text{Benefit criterion},\\[8pt] \displaystyle\min_i(v_{ij}), & \text{Cost criterion}. \end{cases} $$

Negative ideal solution (NIS)

For each criterion, the negative ideal solution is defined as

$$ A^{-}=\left\{ v_{1}^{-}, v_{2}^{-}, \ldots, v_{n}^{-} \right\}, $$ $$ v_{j}^{-}=\begin{cases} \displaystyle\min_i(v_{ij}), & \text{Benefit criterion},\\[8pt] \displaystyle\max_i(v_{ij}), & \text{Cost criterion}. \end{cases} $$

Average solution (AS)

The average solution is obtained by calculating the arithmetic mean of each criterion.

$$ A^{0}=\left\{ v_{1}^{0}, v_{2}^{0}, \ldots, v_{n}^{0} \right\}, $$ $$ v_{j}^{0}=\frac{1}{m} \sum_{i=1}^{m} v_{ij}, \qquad j=1,2,\ldots,n. $$
Metric$C_1$$C_2$$C_3$
PIS ($v_j^+$)0.72790.20090.0712
NIS ($v_j^-$)0.29120.12050.0610
AS ($v_j^0$)0.48530.16070.0678
5

Euclidean distances

Calculate the Euclidean distances from each alternative to PIS, NIS, AS⁺ and AS⁻.

Formula

Euclidean distance from positive ideal solution

$$ d_{E}(\mathrm{PIS})_{i}=\sqrt{ \sum_{j=1}^{n} \left( v_{ij}-v_{j}^{+} \right)^2}, \qquad i=1,2,\ldots,m. $$

Euclidean distance from negative ideal solution

$$ d_{E}(\mathrm{NIS})_{i}=\sqrt{ \sum_{j=1}^{n} \left( v_{ij}-v_{j}^{-} \right)^2}, \qquad i=1,2,\ldots,m. $$

Euclidean distance from positive average solution

$$ d_{E}(\mathrm{AS})_{i}^{+}=\sqrt{ \sum_{j=1}^{n} \tau_{ij}^{+} \left( v_{ij}-v_{j}^{0} \right)^2}, \qquad i=1,2,\ldots,m. $$

Euclidean distance from negative average solution

$$ d_{E}(\mathrm{AS})_{i}^{-}=\sqrt{ \sum_{j=1}^{n} \tau_{ij}^{-} \left( v_{ij}-v_{j}^{0} \right)^2}, \qquad i=1,2,\ldots,m. $$ $$ \tau_{ij}^{+}=\begin{cases} 1, & \text{if} v_{j}^{0} < v_{ij} ,\\ 0, & \text{if} v_{j}^{0} >v_{ij} . \end{cases} $$ $$ \tau_{ij}^{-}=\begin{cases} 1, & \text{if} v_{j}^{0} >v_{ij},\\ 0, & \text{if} v_{j}^{0} < v_{ij}. \end{cases} $$

τ indicator values (based on $v_{ij}$ vs $v_j^0$):

Alt$\tau_1^+$$\tau_2^+$$\tau_3^+$$\tau_1^-$$\tau_2^-$$\tau_3^-$
$A_1$000101
$A_2$011100
$A_3$101010
Alternative$d_E(\text{PIS})$$d_E(\text{NIS})$$d_E(\text{AS})^+$$d_E(\text{AS})^-$
$A_1$0.43870.04020.00000.1942
$A_2$0.29120.16660.04030.0486
$A_3$0.08040.43680.24260.0402
6

Manhattan distances

Calculate the Manhattan distances from each alternative to PIS, NIS, AS⁺ and AS⁻.

Formula

Manhattan distance from positive ideal solution

$$ d_{T}(\mathrm{PIS})_{i}=\sum_{j=1}^{n} \left| v_{ij}-v_{j}^{+} \right|, \qquad i=1,2,\ldots,m. $$

Manhattan distance from negative ideal solution

$$ d_{T}(\mathrm{NIS})_{i}=\sum_{j=1}^{n} \left| v_{ij}-v_{j}^{-} \right|, \qquad i=1,2,\ldots,m. $$

Manhattan distance from positive average solution

$$ d_{T}(\mathrm{AS})_{i}^{+}=\sum_{j=1}^{n} \tau_{ij}^{+} \left| v_{ij}-v_{j}^{0} \right|, \qquad i=1,2,\ldots,m. $$

Manhattan distance from negative average solution

$$ d_{T}(\mathrm{AS})_{i}^{-}=\sum_{j=1}^{n} \tau_{ij}^{-} \left| v_{ij}-v_{j}^{0} \right|, \qquad i=1,2,\ldots,m. $$ $$ \tau_{ij}^{+}=\begin{cases} 1, & \text{if} v_{j}^{0} < v_{ij} ,\\ 0, & \text{if} v_{j}^{0} >v_{ij}. \end{cases} $$ $$ \tau_{ij}^{-}=\begin{cases} 1, & \text{if} v_{j}^{0} >v_{ij},\\ 0, & \text{if} v_{j}^{0} < v_{ij}. \end{cases} $$
Alternative$d_T(\text{PIS})$$d_T(\text{NIS})$$d_T(\text{AS})^+$$d_T(\text{AS})^-$
$A_1$0.48710.04020.00000.2009
$A_2$0.29120.23610.04360.0486
$A_3$0.08040.44690.24600.0402
7

Correction coefficient

The correction coefficient is calculated as the difference between the maximum and minimum Euclidean distances for each type.

Formula

Correction coefficient for PIS

$$ \sigma_{\mathrm{PIS}}=\max_i d_E(\mathrm{PIS})_i - \min_i d_E(\mathrm{PIS})_i $$

Correction coefficient for NIS

$$ \sigma_{\mathrm{NIS}}=\max_i d_E(\mathrm{NIS})_i - \min_i d_E(\mathrm{NIS})_i $$

Correction coefficient for positive average solution

$$ \sigma_{\mathrm{AS}}^{+}=\max_i d_E(\mathrm{AS})_i^{+} - \min_i d_E(\mathrm{AS})_i^{+} $$

Correction coefficient for negative average solution

$$ \sigma_{\mathrm{AS}}^{-}=\max_i d_E(\mathrm{AS})_i^{-} - \min_i d_E(\mathrm{AS})_i^{-} $$
Metric$\sigma_{\text{PIS}}$$\sigma_{\text{NIS}}$$\sigma_{\text{AS}}^+$$\sigma_{\text{AS}}^-$
Value0.35830.39660.24260.1540
8

Distance from PIS, NIS and AS

Compute the corrected distances by incorporating the correction coefficient, Euclidean distance and Manhattan distance.

Formula

Distance from positive ideal solution

$$ d(\mathrm{PIS})_{i}=d_{E}(\mathrm{PIS})_{i} + \sigma_{\mathrm{PIS}} \times d_{E}(\mathrm{PIS})_{i} \times d_{T}(\mathrm{PIS})_{i} $$

Distance from negative ideal solution

$$ d(\mathrm{NIS})_{i}=d_{E}(\mathrm{NIS})_{i} + \sigma_{\mathrm{NIS}} \times d_{E}(\mathrm{NIS})_{i} \times d_{T}(\mathrm{NIS})_{i} $$

Distance from positive average solution

$$ d(\mathrm{AS})_{i}^{+}=d_{E}(\mathrm{AS})_{i}^{+} + \sigma_{\mathrm{AS}}^{+} \times d_{E}(\mathrm{AS})_{i}^{+} \times d_{T}(\mathrm{AS})_{i}^{+} $$

Distance from negative average solution

$$ d(\mathrm{AS})_{i}^{-}=d_{E}(\mathrm{AS})_{i}^{-} + \sigma_{\mathrm{AS}}^{-} \times d_{E}(\mathrm{AS})_{i}^{-} \times d_{T}(\mathrm{AS})_{i}^{-} $$
Alternative$d(\text{PIS})$$d(\text{NIS})$$d(\text{AS})^+$$d(\text{AS})^-$
$A_1$0.51530.04080.00000.2002
$A_2$0.32160.18220.04070.0490
$A_3$0.08270.51420.25710.0404
9

Comprehensive distances and ranking

Compute the comprehensive distance for each alternative and rank them.

Formula

Comprehensive distance

$$ d_{C_i}=\frac{ d(\mathrm{PIS})_i - d(\mathrm{NIS})_i - d(\mathrm{AS})_i^{+} + d(\mathrm{AS})_i^{-}}{4} $$

Ranking

The alternatives are ranked according to their comprehensive distances. The ranking is in ascending order of \(d_{C_i}\).

$$ d_{C_i} < d_{C_k} \quad\Longrightarrow\quad A_i >A_k $$

A lower $d_{C_i}$ means the alternative is closer to PIS, farther from NIS, and above the average — indicating a better choice.

AlternativeComprehensive Distance ($d_{C_i}$)
$A_1$0.1687
$A_2$0.0369
$A_3$-0.1621
Comprehensive distances: $d_{C_3}=-0.1621 < d_{C_2}=0.0369 < d_{C_1}=0.1687$
Final ranking: $A_3 \succ A_2 \succ A_1$

COBRA Algorithm Flowchart