`analytical_hierarchy_process`

Syntax


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analytical_hierarchy_process(A)
analytical_hierarchy_process(A, method)

Description


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W, CI, CR = analytical_hierarchy_process(A)

The Analytic Hierarchy Process is a mathematical model for decision-making problems developed by Thomas L. Saaty.

Computes the weight of each element according to pairwise comparison matrix A. The eigenvector of A corresponding to the maximum eigenvalue is chosen as the weight vector as default.

The consistency index and ratio are returned along with the weight vector.


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W, CI, CR = analytical_hierarchy_process(A, method)

method specifies the way how to compute the weight vector. Default is eigenvector.

Examples


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>>> from OpenHA.assessment.system import analytical_hierarchy_process
>>> import numpy as np
>>> A = np.array([1, 1/2, 4, 3, 3,
        2, 1, 7, 5, 5,
        1/4, 1/7, 1, 1/2, 1/3,
        1/3, 1/5, 2, 1, 1,
        1/3, 1/5, 3, 1, 1]).reshape((-5, 5))
>>> w, ci, cr = analytical_hierarchy_process(A)
>>> print(w, ci, cr)

[[0.26360349]
 [0.47583538]
 [0.0538146 ]
 [0.09806829]
 [0.10867824]] 0.018021102142554035 0.01609026977013753

Input Arguments

A —— The n-by-n pairwise comparison matrix.

method —— str, optional. It specifies the way how to compute the weights vector, specified as eigenvector (default), geometric_mean, or arithmetic_mean.

For example, assuming a comparison matrix

\begin{matrix} A = [\begin{matrix} a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}] \end{matrix}

eigenvector: The eigenvector corresponding to the maximum eigenvalue.
geometric_mean: the geometric mean of each row of matrix A.

That is,

\begin{matrix} A \Rightarrow [\begin{matrix} \sqrt{a_{1} \cdot a_{2}} \\ \sqrt{a_{3} \cdot a_{4}} \end{matrix}] \Rightarrow [\begin{matrix} \frac{\sqrt{a_{1} \cdot a_{2}}}{\sqrt{a_{1} \cdot a_{2}} + \sqrt{a_{3} \cdot a_{4}}} \\ \frac{\sqrt{a_{3} \cdot a_{4}}}{\sqrt{a_{1} \cdot a_{2}} + \sqrt{a_{3} \cdot a_{4}}} \end{matrix}] \end{matrix}

arithmetic_mean: the arithmetic mean of each row of matrix A.

\begin{matrix} A \Rightarrow [\begin{matrix} \frac{a_{1}}{a_{1} + a_{3}} & \frac{a_{2}}{a_{2} + a_{4}} \\ \frac{a_{3}}{a_{1} + a_{3}} & \frac{a_{4}}{a_{2} + a_{4}} \end{matrix}] \Rightarrow [\begin{matrix} \frac{1}{2} (\frac{a_{1}}{a_{1} + a_{3}} + \frac{a_{2}}{a_{2} + a_{4}}) \\ \frac{1}{2} (\frac{a_{3}}{a_{1} + a_{3}} + \frac{a_{4}}{a_{2} + a_{4}}) \end{matrix}] \end{matrix}

Properties of Arguments

Name of the parameters	Is optional?	Source, dialog or input port?
`A`	No	Dialog
`method`	Yes	Dialog

Output Arguments

A tuple (W, CI, CR), where W is the weight vector of length n, CI is the Consistency Index, and CR is the Consistency Ratio.

C I = \frac{λ_{max} - n}{n - 1} < \frac{σ^{2}}{2}

$\frac{\sigma^2}{2}$ provides an upper bound for the measure of consistency index.

$A$ $A$ $\lambda_{\max}=n$ .

$n$ $CI<\frac{\sigma^2}{2}$ $\lambda_{\max}$ $n$ $CI$ might not provide a meaningful measure of consistency.

$RI$ . A random sample of 500 pairwise reciprocal matrices is constructed. $RI$ .

n	1	2	3	4	5	6	6	8	9	10
RI	0.00	0.00	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

$RI$ for matrices up to order 10.

$CI$ $RI$ for the same matrix order. That is,

C R = \frac{C I}{R I}

If the consistency ratio is 0.10 or less, the decision-maker is not too inconsistent and the result obtained by the AHP is acceptable.

References

[1] G. H. Nguyen, "The Analytic Hierarchy Process: A Mathematical Model for Decision Making Problems" (2014). Senior Independent Study Theses. Paper 6054. https://openworks.wooster.edu/independentstudy/6054