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MCDM Methods

Step-by-Step Algorithms

CoCoSo Method

Combined Compromise Solution

2019 Yazdani et al. 8 Steps

CoCoSo (Combined Compromise Solution) is a Multi-Criteria Decision-Making (MCDM) method proposed by Yazdani et al. in 2019. It combines the ideas of Simple Additive Weighting (SAW) and Exponentially Weighted Product (EWP) to obtain a compromise ranking of alternatives.

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 Cost C₃ Benefit
Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)
2

Normalized decision matrix

The criteria are normalized using the formulas below.

Formula

For benefit criteria , the normalized value is calculated as :

$$ r_{ij}=\frac{ x_{ij}-\min_i x_{ij}}{ \max_i x_{ij}-\min_i x_{ij}} $$

For cost criteria :

$$ r_{ij}=\frac{ \max_i x_{ij}-x_{ij}}{ \max_i x_{ij}-\min_i x_{ij}} $$

where \(x_{ij}\) is the original performance value of alternative \(i\) with respect to criterion \(j\).

Alternative$C_1$ (B) $C_2$ (C) $C_3$ (B)
$A_1$0.00000.50000.0000
$A_2$0.33330.00001.0000
$A_3$1.00001.00001.0000
3

Weighted normalized decision matrix

The normalized decision matrix is multiplied by the corresponding criterion weights.

Formula

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$0.00000.10050.0000
$A_2$0.24260.00000.0712
$A_3$0.72790.20090.0712
4

$S_i$ values

The $S_i$ value is calculated as the sum of the weighted normalized values for each alternative.

Formula
$$ S_i=\sum_{j=1}^{n} v_{ij} $$

or equivalently :

$$ S_i=\sum_{j=1}^{n} w_j r_{ij}. $$
Alternative$S_i$
$A_1$0.1005
$A_2$0.3138
$A_3$1.0000
5

Exponentially weighted normalized decision matrix

Formula

The exponentially weighted normalized decision matrix is calculated by raising each normalized value to the power of its corresponding criterion weight.

$$ p_{ij}=r_{ij}^{w_j} $$
Alternative$C_1$$C_2$$C_3$
$A_1$0.00000.87010.0000
$A_2$0.44960.00001.0000
$A_3$1.00001.00001.0000
6

$P_i$ values

Formula

The \(P_i\) value is obtained by summing the exponentially weighted normalized values for each alternative.

$$ P_i=\sum_{j=1}^{n} p_{ij} $$

or equivalently :

$$ P_i=\sum_{j=1}^{n} r_{ij}^{w_j}. $$
Alternative$P_i$
$A_1$0.8701
$A_2$1.4496
$A_3$3.0000
7

$K_{ia}$, $K_{ib}$ and $K_{ic}$ values

Compute three aggregation strategies combining $S_i$ and $P_i$ scores.

Formula

$K_{ia}$:

$$ K_{ia}=\frac{P_i+S_i}{\displaystyle\sum_{i=1}^{m}(P_i+S_i)} $$

$K_{ib}$:

$$ K_{ib}=\frac{S_i}{\min_i S_i} + \frac{P_i}{\min_i P_i} $$

$K_{ic}$:

$$ K_{ic}=\frac{\lambda S_i+(1-\lambda)P_i}{\lambda\max_i S_i+(1-\lambda)\max_i P_i}, \qquad 0\leq\lambda\leq1. $$
Alternative$K_{ia}$$K_{ib}$$K_{ic}$
$A_1$0.14422.00000.2427
$A_2$0.26194.78940.4409
$A_3$0.594013.39811.0000
8

$K_i$ values and ranking

The final CoCoSo appraisal score $K_i$ is calculated by combining the three aggregation scores.

Formula
$$ K_i=\left( K_{ia} \times K_{ib} \times K_{ic} \right)^{1/3} + \frac{K_{ia}+K_{ib}+K_{ic}}{3} $$

Ranking

The alternatives are ranked in descending order of $K_i$.

$$ K_i >K_k \quad\Longrightarrow\quad A_i >A_k $$
AlternativeScore ($K_i$)
$A_1$1.2077
$A_2$2.6516
$A_3$6.9938
CoCoSo scores: $K_3=6.9938 >K_2=2.6516 >K_1=1.2077$
Final ranking: $A_3 \succ A_2 \succ A_1$

CoCoSo Algorithm Flowchart