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MCDM Methods

Step-by-Step Algorithms

AHP Method

Analytic Hierarchy Process

1977 Thomas L. Saaty 6 Steps

The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decisions, developed by Thomas L. Saaty in 1977. It uses pairwise comparisons to derive priority scales, making it one of the most widely used multi-criteria decision making (MCDM) methods. AHP is applied in fields ranging from project selection and supplier evaluation to healthcare prioritization and environmental planning.

Video Demonstration

Step-by-Step Algorithm

Calculated cell
Source cells in formula
1

Construct the pairwise comparison matrix

Create a pairwise comparison matrix \(A=[a_{ij}]\) of size \(n \times n\), where each element represents the relative importance of criterion \(i\) compared to criterion \(j\).

\[ A=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \]

Properties:

    \(a_{ii}=1\) , \(a_{ji}=1/a_{ij}\) , \(a_{ij}>0\)

1=Equal importance, 3=Moderate, 5=Strong, 7=Very strong, 9=Extreme importance. Values 2, 4, 6, 8 are intermediate judgments.

Criterion$C_1$$C_2$$C_3$
$C_1$135
$C_2$0.333312
$C_3$0.20000.50001
2

Normalize the pairwise comparison matrix

Calculate the sum of each column.

Formula
\[N_{ij}=\frac{a_{ij}}{\sum_{i=1}^{n}a_{ij}}=\frac{a_{ij}}{S_j}\]

The normalized matrix is :

\[N=[n_{ij}]\]

Column Sums ($S_j$):

Sum$C_1$$C_2$$C_3$
$S_j$1.53334.50008.0000

Normalized Matrix ($N$):

Criterion$C_1$$C_2$$C_3$
$C_1$0.65220.66670.6250
$C_2$0.21740.22220.2500
$C_3$0.13040.11110.1250
3

Compute the priority vector

Calculate the average value of each row of the normalized matrix. The resulting vector represents the relative priority (weight) of each criterion.

Formula
\[w_i=\frac{1}{n} \sum_{j=1}^{n} N_{ij}\] \[W=\begin{bmatrix} w_1\\ w_2\\ \vdots\\ w_n \end{bmatrix}\]

The priority vector W represents the relative weights of each criterion. Higher wi means criterion i is more important in the decision model.

CriterionPriority Vector ($w_i$)
$C_1$0.6480
$C_2$0.2299
$C_3$0.1222
4

Compute the weighted priority vector

Multiply the original comparison matrix by the priority vector.

Formula
\[WP=A\times W\] or : \[WP_i=\sum_{j=1}^{n} a_{ij}w_j\]
CriterionWeighted Priority ($WP_i$)
$C_1$1.9487
$C_2$0.6903
$C_3$0.3668
5

Calculate the consistency ratio

Formula

Compute the Consistency Vector :

\[CV_i=\frac{WP_i}{w_i}\]

Maximum Eigenvalue :

\[\lambda_{\max}=\frac{1}{n} \sum_{i=1}^{n} CV_i\]

Consistency Index (CI) :

\[CI=\frac{\lambda_{\max}-n}{n-1}\]

Consistency Ratio (CR) :

\[CR=\frac{CI}{RI}\]

where :

  • RI=Random Index.
  • If CR < 0.10, the judgments are considered consistent.
  • If CR ≥ 0.10, revise the pairwise comparison matrix.

RI values: n=3→0.58, n=4→0.90, n=5→1.12, n=6→1.24, n=7→1.32, n=8→1.41, n=9→1.45, n=10→1.49

Consistency Vector ($CV_i$):

Criterion$CV_i$
$C_1$3.0074
$C_2$3.0030
$C_3$3.0020

Consistency Metrics:

MetricValue
$\lambda_{\max}$3.0041
CI0.0021
CR (RI=0.58)0.0035
Result: $CR=0.0035 < 0.10$. The level of consistency is acceptable.
6

Determine the final weights of criteria

Formula

If the consistency ratio satisfies the acceptable threshold (CR < 0.10), the priority vector becomes the final weight vector.

\[ Weight_i=w_i \]

The weights satisfy :

\[\sum_{i=1}^{n}w_i=1\]

The final weight vector is

\[W=\left[ w_1,w_2,\ldots,w_n \right]^T\]

These weights are then used in subsequent MCDM methods (e.g., TOPSIS, VIKOR, WASPAS, COPRAS, MOORA, or other ranking methods).

CriterionFinal Weight ($w_i$)
$C_1$0.6480
$C_2$0.2299
$C_3$0.1222
Sum1.0000

AHP Algorithm Flowchart