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MCDM Methods

Step-by-Step Algorithms

SD Method

Standard Deviation Weighting

2010 Rao and Patel 4 Steps

SD (Standard Deviation) method is an objective weighting method in MCDM. It assigns higher weights to criteria with greater variation among alternatives, because greater variation provides more information for distinguishing 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
2

Normalize the decision matrix

Formula

Normalize the decision matrix to eliminate differences in measurement units and make the criteria comparable.

For benefit criteria :

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

For cost criteria :

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

\(x_j^{\max}\) and \(x_j^{\min}\) are respectively the maximum and minimum values of criterion \(C_j\).

C₁ and C₃ are benefit criteria. C₂ is a cost criterion.

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 standard deviation

Formula

Calculate the arithmetic mean of the normalized values for each criterion.

\[ \bar{r}_j=\frac{1}{m} \sum_{i=1}^{m}r_{ij} \]

\(\bar{r}_j\) represents the mean normalized performance of criterion \(C_j\).

Calculate the standard deviation of each normalized criterion to measure its variation across alternatives.

\[ \sigma_j=\sqrt{ \frac{1}{m} \sum_{i=1}^{m} \left(r_{ij}-\bar{r}_j\right)^2} \]
Metric$C_1$$C_2$$C_3$
Mean ($\bar{r}_j$)0.44440.50000.6667
Std. Dev. ($\sigma_j$)0.41570.40820.4714
4

Weights of criteria

Formula

Normalize the standard deviation values to obtain the objective weight of each criterion.

\[ w_j=\frac{\sigma_j} {\displaystyle\sum_{j=1}^{n}\sigma_j} \]

The criterion weights satisfy :

\[ \sum_{j=1}^{n}w_j=1 \]

Criteria with greater standard deviation receive greater weights.

Metric$C_1$$C_2$$C_3$
Weight ($w_j$)0.32100.31510.3639
Computed weights: $w_3=0.3639 >w_1=0.3210 >w_2=0.3151$
Verification: $w_1 + w_2 + w_3=0.3210 + 0.3151 + 0.3639=1.0000$ ✓

SD Algorithm Flowchart