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MCDM Methods

Step-by-Step Algorithms

LOPCOW Method

Logarithmic Percentage Change-driven Objective Weighting

2022 Fatih Ecer and Dragan Pamučar 4 Steps

LOPCOW (Logarithmic Percentage Change-driven Objective Weighting) is an objective MCDM criteria-weighting method that determines criterion weights from the variability of the alternatives. It uses a logarithmic percentage-change measure, based on the relationship between the criterion's mean-square values and its standard deviation.

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
2

Normalize the decision matrix

Formula

For a cost criterion , where a lower value is preferred :

$$ r_{ij}=\frac{x_j^{\max}-x_{ij}} {x_j^{\max}-x_j^{\min}} $$

For a benefit criterion , where a higher value is preferred :

$$ r_{ij}=\frac{x_{ij}-x_j^{\min}} {x_j^{\max}-x_j^{\min}} $$

C₁: $x_1^{\max}=25,\;x_1^{\min}=10$ (benefit) · C₂: $x_2^{\max}=25,\;x_2^{\min}=15$ (cost) · C₃: $x_3^{\max}=35,\;x_3^{\min}=30$ (benefit)

Alternative$C_1$$C_2$$C_3$
$A_1$0.00000.50000.0000
$A_2$0.33330.00001.0000
$A_3$1.00001.00001.0000
3

Calculate the percentage value (PV)

Formula

Calculate the standard deviation of each normalized criterion :

$$ \sigma_j=\sqrt{ \frac{1}{m} \sum_{i=1}^{m} (r_{ij}-\bar r_j)^2} $$

Where \(\bar r_j\) is the mean value of criterion \(j\) :

$$ \bar r_j=\frac{1}{m} \sum_{i=1}^{m}r_{ij} $$

Calculate the mean-square value of each normalized criterion and express it as a percentage of its standard deviation :

$$ PV_j=\left| \ln \left( \frac{ \sqrt{ \displaystyle\frac{1}{m} \sum_{i=1}^{m}r_{ij}^{2}}}{ \sigma_j} \right) \right| \times100 $$

Where \(PV_j\) is the percentage value of criterion \(j\).

Metric$C_1$$C_2$$C_3$
Mean ($\bar{r}_j$)0.44440.50000.6667
Std. Dev. ($\sigma_j$)0.41570.40820.4714
RMS0.60860.64550.8165
$PV_j$38.1145.8154.93
4

Weights of criteria

Formula

Normalize the percentage values to obtain the final LOPCOW weights :

$$ w_j=\frac{PV_j} {\displaystyle\sum_{j=1}^{n}PV_j} $$

The resulting weights satisfy :

$$ \sum_{j=1}^{n}w_j=1 $$
CriterionLOPCOW Weight ($w_j$)
$C_1$0.2745
$C_2$0.3300
$C_3$0.3956
LOPCOW weights: $w_3=0.3956 >w_2=0.3300 >w_1=0.2745$
Check: $\sum_{j=1}^{3}w_j=0.2745 + 0.3300 + 0.3956=1.0001 \approx 1.0000\;\checkmark$

LOPCOW Algorithm Flowchart