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MCDM Methods

Step-by-Step Algorithms

SWARA Method

Step-wise Weight Assessment Ratio Analysis

2010 Violeta Keršulienė, Edmundas Kazimierz Zavadskas and Zenonas Turskis 4 Steps

SWARA (Step-wise Weight Assessment Ratio Analysis) is a subjective criteria-weighting method in MCDM. It determines criterion weights from expert judgments about the relative importance of criteria, allowing experts to express how much more or less important one criterion is than the preceding criterion.

Video Demonstration

Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

Order the criteria

Arrange the criteria according to the decision-maker's perceived importance. Select a pivot criterion as the starting/reference criterion.

$$ C_1,C_2,\ldots,C_n $$

For each criterion \(C_j\), determine its relative importance compared with the previous criterion \(C_{j-1}\). This assessment is represented by \(S_j\).

$$ S_j=\frac{\text{importance of}C_j} {\text{importance of}C_{j-1}}, \qquad j=2,\ldots,n $$

Starting from the second criterion, assess its relative importance compared with the previous criterion :

$$ S_j=\begin{cases} >1, & \text{if} C_j \text{ is more important than} C_{j-1},\\=1, & \text{if} C_j \text{ is equally important},\\ <1, & \text{if} C_j \text{ is less important than} C_{j-1}. \end{cases} $$

For the first criterion :

$$ S_1=1 $$
CriterionRelative Importance (\(S_j\))
$C_1$1
$C_2$0.5
$C_3$1
$C_4$0.8

The criteria are ordered according to their perceived importance, and \(S_j\) expresses the relative importance of each criterion compared with the criterion immediately preceding it.

2

Calculate the coefficient \(k_j\)

Formula

Convert the comparative importance value \(S_j\) into the coefficient \(k_j\).

\[ k_j=\begin{cases} 1, & j=1,\\[6pt] 1+S_j, & j>1 \end{cases} \]

For the first criterion, \(k_1=1\). For every subsequent criterion, \(k_j\) is obtained by adding 1 to \(S_j\).

CriterionCoefficient (\(k_j\))
$C_1$1
$C_2$1.5
$C_3$2
$C_4$1.8

The coefficient \(k_j\) transforms the relative importance assessment into a value suitable for calculating the recalculated weight.

3

Initial weight

Formula

Calculate the recalculated or comparative weight of each criterion sequentially. The first criterion is assigned a value of 1.

\[ q_j=\begin{cases} 1, & j=1,\\[6pt] \dfrac{q_{j-1}}{k_j}, & j>1 \end{cases} \]

Each subsequent \(q_j\) is obtained by dividing the previous criterion's value \(q_{j-1}\) by its coefficient \(k_j\).

CriterionInitial Weight (\(q_j\))
$C_1$1
$C_2$0.6667
$C_3$0.3333
$C_4$0.1852

The \(q_j\) values accumulate the relative importance assessments throughout the sequence of criteria.

4

Weights of criteria

Formula

Normalize the initial weights to obtain the final SWARA weights :

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

The final weights satisfy :

$$ \sum_{j=1}^{n}w_j=1 $$
CriterionFinal Weight (\(w_j\))
$C_1$0.4576
$C_2$0.3051
$C_3$0.1525
$C_4$0.0848
Sum1.0000

SWARA Algorithm Flowchart