Application of Analytical Hierarchy Process (AHP) in the selection of a
flexible production system
SARA SREBRENKOSKA, ALEKSANDRA APOSTOLOVA, MISKO DZIDROV
DEJAN KRSTEV
Faculty of Mechanical Engineering
Goce Delcev University, Stip, North Macedonia
Krste Misirkov, 10A, 2000, Stip
NORTH MACEDONIA
Abstract: - The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex
decisions, based on mathematics and psychology. It has application in group decision-making and is used
worldwide in a wide variety of decision-making situations. Rather than prescribing a "correct" decision, AHP
helps decision-makers choose the decision that best fits their goal and their understanding of the problem. The
technique provides a comprehensive and rational framework for structuring a decision problem, representing and
quantifying its elements, relating those elements to general objectives, and evaluating alternative solutions. Given
each specific situation, making the right decisions is probably one of the most difficult challenges for managers.
The obtained results allow the manager to evaluate the employees in an objective way and make an objective
decision for their promotion. This tool not only supports and qualifies decisions, but also allows managers to
justify their choices as well as simulate possible outcomes.
Key-Words: - Analytic, decision, choice, technique, process.
Received: June 11, 2022. Revised: August 12, 2023. Accepted: September 25, 2023. Published: November 6, 2023.
1 Introduction
The terms problem, decision, decision-making,
decision-making process, decision-making, etc., are not
only used in everyday life but also in modern business
and professional work, and at the same time, their true
meaning is not always known. The identification and
analysis of the problem to be solved, the determination
of the possible solutions to the problem, and the criteria
according to which the possible solutions are evaluated,
i.e. the alternatives and the choice of the best possible
solution is a process of making a decision - DO (English
Decision Making - DM), i.e. decision-making process,
and as a result of the decision-making process, the
decision emerges. It represents the very choice of the
best, from the most possible alternative solutions to the
problem. [1]
The analytical hierarchy process has application in
group decision-making and is used worldwide in a wide
variety of decision-making situations in areas such as
government, business, industry, health, shipbuilding,
and education. [2]
The AHP method is one of the possible solutions for
the construction and application of a multi-criterion
decision-making system. It was developed in the 70s of
the last century in the USA. During the past decades, it
has been the subject of much methodological research
and it has been used with success in solving many
practical problems. [3-9]
The founder of the AHP method is Thomas Saaty,
who worked out the methodological foundations of this
concept as a professor at the Wharton Business School
in Philadelphia in the early 70s. Wider interest in the
method came in the 80s after the publication of the
publication The Analytic Hierarchy Process by the
renowned publisher MCGRAW -Hill. It seems that the
method began to be popularized and spread from the
establishment of the foundations of systems theory, as
well as from the attempts to develop and provide a
formal description of one of the basic characteristics of
this system, which L. Bertalanffy already calls
"hierarchical order". [3-9]
Saaty [6] describes seven basic pillars of the AHP
method, which are the following:
• ratio scales, proportional and normalized ratio
scales.
• mutual comparison of pairs.
• sensitivity of the basic right eigenvector.
• clustering and using pivots to scale.
• Synthesis of the created one-dimensional scale
of relationships that represent the total result.
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Sara Srebrenkoska, Aleksandra Apostolova,
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• Rank retention and vice versa.
• integrating group reasoning i.e., evaluate.
Using a ratio scale for comparisons from the
perspective of the result helps us unify the
multidimensionality of the problem into a single
dimension.
1.1 Steps in applying the AHP method
When applying the AHP method, five steps are set:
Step 1: Define the problem and the criteria.
Step 2: Define the alternatives.
Step 3: Prioritize the criteria and alternatives using
pairwise comparison.
Step 4: Check for consistency between pairwise
comparisons.
Step 5: Evaluate the relative weights of the pairwise
comparisons and obtain the calculated total priorities for
the alternatives. [3-9]
2 Problem Formulation
To show the essence of the AHP model, a concrete
example model for the evaluation of different flexible
production systems is presented. The first step of AHP
consists of developing a hierarchical structure of the
estimation problem. In this case, the objective is the
selection of the best flexible manufacturing system.
Fig.1 Hierarchical structure of the problem
The criteria are production machine, material
handling and storage systems, control computers, quality
of produced parts, quantity of produced parts and
production cost, and the alternatives are the selection of
the best FPS: sequential FPS, random FPS and
specific/dedicated FPS.
According to Fig. 1, in this case, the first level, that
is, at the top of the goal, is the selection of the most
favorable flexible production system.
On the second level, the next level is the criteria,
i.e.:
K1 - production machine,
K2 - systems for handling and storing materials,
K3- control computers,
K4- quality of manufactured parts,
K5- quantity of produced parts and
K6- production cost.
Finally, at the third or last level (alternative) of the
hierarchy, are the three different types of flexible
production systems, i.e., FPS1, FPS2 and FPS3,
(hereafter denoted as A1, A2 and A3), which must be
evaluated, and compared and chosen the best among
them.
3 Problem Solution
Based on the established hierarchical structure and
based on the preferences set by the decision maker, an
evaluation matrix is formed, i.e., a matrix of comparison
pairs (Table 1.).
The next step is to calculate an eigenvector
corresponding to the eigenvalues. The values for the
eigenvector are given in Table 2.
To obtain the values of the eigenvector, which is also
called the priority vector, the procedure is as follows:
first, we obtain the elements of the newly formed matrix
by dividing the elements of the reciprocal matrix (Table
2) by the sum of the corresponding column, i.e., the value
of element K11
is obtained as follows: 1/15=0.066667.
To obtain the final eigenvector, i.e., the weights of the
criteria according to which the ranking will be
performed, it is necessary to calculate the arithmetic
mean for each row of the normalized matrix (Table 2),
i.e., the sum of each row is divided by the number of
elements:
(1)
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Table 1. Evaluation matrix/comparison of criteria
Table 2. Normalized matrix/weight values
The values of the remaining eigenvectors are
identical to the previously described procedure. After
obtaining the values for the weights of each criterion, we
need to perform a consistency check on the comparisons.
In doing so, we search for the largest eigenvalue of the
corresponding matrix. For this purpose, we find the
values of the average matrix, i.e. the matrix of the sum
of the weights. We do this by multiplying each row of
the initial evaluation, that is:
(2)
The remaining values (Table 3) from the average matrix
are obtained identically .
Table 3. Average matrix matrix of the sum of the weights
K1
K2
K3
K4
K5
K6
K1
1
0.33
0.25
0.5
0.5
0.33
K2
3
1
0.2
0.33
0.5
0.5
K3
4
5
1
0.5
1
0.5
K4
2
3
2
1
1
0.33
K5
2
2
1
1
1
1
K6
3
2
2
3
1
1
SUM
15
13.33
6.45
6.33
5
3.66
K1
K2
K3
K4
K5
K6
ABSOLUTE
WEIGHTS
K1
0.06
0.02
0.03876
0.078989
0.1
0.090164
0.0665559
K2
0.2
0.07
0.031008
0.052133
0.1
0.136612
0.09912854
K3
0.26
0.37
0.155039
0.078989
0.2
0.136612
0.20206669
K4
0.1
0.22
0.310078
0.157978
0.2
0.090164
0.18610149
K5
0.13
0.15
0.155039
0.157978
0.2
0.273224
0.17826859
K6
0.2
0.150
0.310078
0.473934
0.2
0.273224
0.26787879
K1
K2
K3
K4
K5
K6
SUM OF
WEIGHTS
K1
0.066556
0.032712
0.050517
0.093051
0.089134
0.0884
0.420370033
K2
0.199668
0.099129
0.040413
0.061413
0.089134
0.133939
0.623696767
K3
0.266224
0.495643
0.202067
0.093051
0.178269
0.133939
1.369191726
K4
0.133112
0.297386
0.404133
0.186101
0.178269
0.0884
1.287400887
K5
0.133112
0.198257
0.202067
0.186101
0.178269
0.267879
1.165684442
K6
0.199668
0.198257
0.404133
0.558304
0.178269
0.267879
1.806510017
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The next step is the determination of own values. We do
this by dividing the values from the average matrix by
the weights from the normalized matrix:
(3)
Table 4. Eigenvalues λ
Next, we find the largest eigenvalue as follows, sum all
the obtained eigenvalues and divide by the number of
elements λmax.
After finding the largest eigenvalue, the next step is to
find the consistency index. We do that through the
following formula: CI
(4)
The next and the last step is finding the consistency ratio:
CR
(5)
From this it follows that the ratio of consistency is
acceptable and that our estimates are accepted.
After all the necessary calculations have been
performed, the final ranking (Figure 2.) can be
displayed for the importance of the criteria according to
which the selection will be made.
Fig.2. Ranking criteria according to importance
In the following, only the results of the research are
given. The method of their calculation is identical to the
one that was previously explained in detail.
Table 5. Average matrix matrix of the sum of the
weights according to K1 - production machine
According to the production machine criterion, the best
ranked is FPS1.
Table 6. Average matrix matrix of the sum of the
weights according to K2 - the criterion systems for
handling and storing materials
According to the material handling and storage systems
criterion, the best ranked is FPS2.
Table 7. Average matrix matrix of the sum of the
weights according to K3 - control computers
According to the controller computer’s criterion, FPS3 is
the best-ranked.
Table 8. Average matrix matrix of the sum of the
weights according to K4 - quantity of produced parts
According to the quantity of produced parts criterion,
FPS2 ranks best.
Table 9. Average matrix matrix of the sum of the
weights according to K5 - quality of manufactured parts
According to the criterion of the quality of the
manufactured parts, FPS1 is ranked best.
А1
А2
А3
SUM OF
WEIGHTS
А1
0.411111
0.522222
0.327778
1.261111111
А2
0.205556
0.261111
0.327778
0.794444444
А3
0.411111
0.261111
0.327778
1
λ1
λ2
λ3
λ4
λ5
λ6
SUM
λMAX
6.3
6.2
6.7
6.9
6.5
6.7
39.6
6.597
A1
A2
A3
SUM OF
WEIGHTS
A1
0.328296
2.397525
0.04115
0.922323586
A2
0.045961
0.342504
2.962807
1.117090778
A3
2.626365
0.037675
0.329201
0.997747009
A1
A2
A3
SUM OF
WEIGHTS
A1
0.106014
0.085796
0.126799
0.318609891
A2
0.318043
0.259989
0.209219
0.787250374
A3
0.530071
0.779966
0.633997
1.944034375
A1
A2
A3
SUM OF
WEIGHTS
A1
0.334032
0.262603
0.422284
1.018919388
A2
0.668064
0.525207
0.422284
1.615554665
A3
0.110231
0.173318
0.140761
0.424310118
A1
A2
A3
SUM OF
WEIGHTS
A1
0.724174
0.5775453
0.966598
2.268317325
A2
0.101384
0.0825065
0.063795
0.247686292
A3
0.144835
0.2475194
0.19332
0.585673814
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Table 10. Average matrix matrix of the sum of the
weights according to K6 - cost of the production
A1
A2
A3
SUM OF
WEIGHTS
A1
0.106014
0.085796
0.126799
0.318609891
A2
0.318043
0.259989
0.209219
0.787250374
A3
0.530071
0.779966
0.633997
1.944034375
According to the cost of production criterion, FPS3
ranks best.
After receiving the weights for the importance of each
flexible production system in relation to each criterion,
they are summed up and the final ranking of each of them
is obtained.
The subtotal of each flexible production system
offered is given in Table 11.
Table 11. Subtotal of FPS
The final ranking, and thus the result of the research
conducted to rank the most favorable flexible production
system that would be used by manufacturing enterprises,
is given in Table 12 and Figure 3.
Table 12. Final rank of the best FPS
Fig. 3. Final ranking of the best flexible manufacturing
system
After the detailed analysis and assessment in this case,
the most favorable flexible production system is the third
one, that is, a dedicated or specific flexible production
system.
4 Conclusion
Production at FPS is largely automated, reducing overall
labor costs. Systems typically consist of three main
functions: a central control computer, production
machinery, and material handling systems that allow the
system to remain operational. These systems have a great
impact on the future of production, making them an
indispensable tool for many companies in the future.
Using robots, computer numerical control machines and
other automation technologies, FPS can significantly
improve production efficiency and reduce labor costs.
After detailed analysis and evaluation, the third FPS3,
i.e., the dedicated or specific flexible production system,
was obtained as the most favorable flexible production
system.
References:
[1] T. L. Saaty, The Analytic Hierarchy Process,
McGraw-Hill Book Co., N.Y., 1980.
[2] T.L.Saat.Fundamentals of Decision Making and
Priority Theory with the Analytic Hierarchy
Process, RWS Publications, Pittsburgh, 2000,
Vol.6
[3] J. Warren. Uncertainties in the Analytic
Hierarchy Proces, 2004
[4] T.A.Hamdy. Operations Research, Prentice
Hall, 2006.
[5] M. Zeleny. Multiple citeria decision making,
McGraw-Hill Book Comy, New York, 1982.
[6] T. L. Saaty. Fundamentals of Decision Making,
RWS Publications, Pittsburgh, PA, 1994.
[7] T. L. Saaty. Decision Making with Dependence
and Feedback, RWS Publications, Pittsburgh,
PA, 1996.
[8] T. L. Saaty and M. OzdeOzdemir.
Negativeorities in the analytic hierarchy process,
2006.
[9] S.T.Foster and G.LaCava,.The Analytical
Hierarchy Process: A Step-by-Step Approach.
A3 -DEDICATED FPS
0.413
A1 -SEQUENTIAL FPS
0.301
A2 -RANDOM FPS
0.286
K1
K2
K3
K4
K5
K6
INTERS
UM OF
EACH
FPS
A
1
0.41
1
0.32
8
0.10
6
0.33
4
0.72
4
0.10
6
0.301
A
2
0.26
1
0.34
2
0.25
9
0.52
5
0.08
2
0.25
9
0.286
A
3
0.32
7
0.32
9
0.63
3
0.14
0
0.19
3
0.63
3
0.413
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Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
- Sara Srebrenkoska, Aleksandra Apostolova
organized and executed the experiment and the
optimization.
- Misko Dzidrov carried out the conceptualization,
- data curation, formal analysis and methodology.
- Dejan Krstev was responsible for the statistics and the
visualization.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare that
are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_U
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