Generating a Set of Consistent Pairwise Comparison Test Matrices in
AHP using Particle Swarm Optimization
YOUSSEF KARKOUR1, CHAKIR TAJANI2∗, IDRISS KHATTABI3
1,2SMAD Team, Department of Mathematics
Polydisciplinary Faculty of Larache, Abdelmalek Essaâdi University
MOROCCO
3Department of Informatics
Polydisciplinary Faculty of Larache, Abdelmalek Essaâdi University
MOROCCO
Abstract: Pairwise Comparison Matrix (PCM) play a crucial role in multi-criteria decision-making, especially
within the framework of the Analytic Hierarchy Process (AHP). It is essential for an expert to systematically
assess and compare alternatives, considering various criteria as part of the decision-making process. The AHP
relies on consistency to ensure the accuracy of pairwise comparisons made by decision makers, increasing the
overall integrity of the decision-making process. The aim of this paper is to present an approach based on Particle
Swarm Optimization (PSO) as a metaheuristic method to generate a set of consistent pairwise comparison matri-
ces. Numerical results are presented with different sizes of matrices showing the effectiveness of our algorithm
in producing acceptable matrices for the benefit of expets.
Key-Words: - Pairwise Matrix, Analytic Hierarchy Process, Decision Theory, Consistency, Metaheuristics,
Particle Swarm Optimizition
Received: October 11, 2023. Revised: December 18, 2023. Accepted: February 19, 2024. Published: April 1, 2024.
1 Introduction
The Analytic Hierarchy Process (AHP) is a method
for multi-attribute decision-making. It was developed
in the last cemtry [1], [2]. It is widely used for mak-
ing decisions including, for instance, social, military,
managerial and many other fields, [3], [4], [5].
The main problem of the AHP method in its prac-
tical application is the judgment matrix which must
be constructed by the decision makers on the basis of
their experiences and knowledge. Due to the limita-
tions of experience and knowledge and the complex
nature of the decision problem, [6], [7], especially
when dealing with a large number of judgments, the
pairwise comparison matrix may be inconsistent.
To test whether the matrix is consistent, [8], sug-
gested the use of the consistency ratio CR. Indeed,
the pairwise comparison matrix could pass the consis-
tency test when the so-called consistency ratio CR <
0.1. However, in many cases, the judgment matrix
cannot pass this test to be acceptable and it has to be
adjusted.
The consistency of pairwise matrix has been a sub-
ject of many studies for several decades, [9], [10],
[11], [12]. However, some of the methods devel-
oped for revising inconsistent comparison matrices
are complicated and difficult to use, while others
struggle to preserve the original comparison informa-
tion as a new matrix must be constructed to replace
the original.
Metaheuristic methods such as Genetic Algo-
rithms (GAs), Particle Swarm Optimisation (PSO)
and others have been used to study the consistency
problem of the pairwise comparison matrix. GAs
were developed by [13]. Their inspiration is the the-
ory of evolution of Charles Darwin and natural bio-
logical processes, [14]. They entail multiple iterative
processes that allow for steady and iterative progress
towards optimal solutions. To fully realize the ben-
efits of genetic algorithms as fundamental problem-
solving approaches and improve their performance,
it is imperative to adapt and incorporate appropriate
genetic operators for addressing the specific problem
at hand, [15]. In [16], we propose an alternative ap-
proach to create consistent pairwise matrices. This
approach is based on GAs to define a set of consistent
pairwise matrices for different dimensions, to help ex-
perts in making appropriate judgment decisions.
Particle swarm optimization (PSO) is a population
based optimization algorithm inspired by the group
dynamics of flocks of birds or schools of fish. It was
proposed in [17] and was first intended for simulating
social behaviour. The idea behind PSO is that each
particle represents a possible solution to the problem,
and the particles move around in the search space ac-
cording to simple mathematical formulas, [18]. The
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particles movements are influenced by their own best
known positions and the best known positions of the
entire swarm.
The aim of this paper is to propose an alternative
approach to create consistent pairwise matrices. This
approach is based on PSO in defining a set of consis-
tent pairwise matrices for different dimensions to aid
experts to define the appropriate judgment decision.
Thus, instead of determining a pairwise matrix that
can be inconsistent, especially in the case of high or-
der, and trying to modify it, the expert will have pair-
wise comparison matrices during the algorithm pro-
cess allowing the identification of the appropriate one.
The remaining of this paper is structured as fol-
lows. In the second section, we briefly describe
the different steps of the Analytic Hierarchy Process.
Section 3 is devoted to the pairwise comparison ma-
trix. In section 4, we present the particle swarm op-
timization approach to identify consistent judgment
matrices. In Section 5, numerical results are per-
formed with different sizes of matrices, showing how
effective and accurate is the proposed method to over-
come the problem of inconsistency in pairwise com-
parison matrix in the AHP method.
2 Overview of the AHP Method
The Analytic Hierarchy Process (AHP) serves as a
comprehensive theory of measurement, employed in
the generation of ratio scales through comparisons,
encompassing both discrete and continuous dimen-
sions. These comparative assessments can be drawn
from empirical measurements or a foundational scale
that captures the inherent hierarchical relationships
of preferences and sentiments. The AHP method ex-
hibits a specific focus on evaluating deviations from
consistency, measuring such deviations, and assess-
ing dependencies within and between the constituent
groups of its structural elements, [19]. The general
idea of the AHP method is showing in Figure 1.
Figure 1: Schema of AHP method
In recent years, AHP method is applied in different
fields such as economic, mathematic applications, re-
newable energy and decision-making problem. A sys-
tematic procedure exists for effective application of
AHP in decision-making, which can be summarized
in the following steps:
•Step 1 : Construction of a structural hierarchy
that highlights the objectives and identifies the
criteria and alternatives. Figure 1. illustrates the
hierarchical structure of the information, where,
the top level representing the goals, the second
level outlining the ranking criteria and the last
level consists of the alternatives, [19].
•Step 2 : Construction of Pairwise Comparison
Matrices (PCM) (Comparative Judgments) for
all the criteria and alternatives. The pairwise
comparison approach, inspired by the research
in [20], is employed. After constructing a hierar-
chy, the subsequent step involves establishing the
priorities of variables at each level through the
creation of comparison matrices for all variables
in relation to each other. This pairwise compar-
ison reveals the degree to which variable 0A0is
more favorable or important than variable 0B0.
Table 1 illustrates how an opinion scaling/pair-
wise comparison evaluation scaling from point
one to nine scaling (1-9) is used to quantify these
logical preferences.
•Step 3 : Weight determination via normaliza-
tion procedure involves calculating the weights
for criteria and the local weight of alternatives
based on the pairwise comparison matrices. For
each value in a column 0j0, it is divided by the
total of the values in that column. The sum of
values in each column is normalized to 1 in the
matrix.
AW =
a11
Pai1
a12
Pai2. . . a1n
Pain
a21
Pai1
a22
Pai2. . . a2n
Pain
.
.
..
.
..
.
..
.
.
.
.
..
.
..
.
..
.
.
an1
Pai1
an2
Pai2. . . ann
Pain
(1)
•Step 4 : Integration of weight and consistency
testing: Begin by synthesizing local weights to
derive global weights for the alternatives. The
eigenvector of matrix will then be computed. The
Consistency Index (CI)will be calculated by us-
ing (3) below. Subsequently, it is essential to ver-
ify the consistency judgment for the appropriate
value of 0n0using the Consistency Ratio (CR)to
ensure the consistency of the pairwise compari-
son matrix, as outlined in the representation (4).
•Step 5 : Assessing the CR : If CR is equal to
or less than 0.10 (10%), the consistency level is
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deemed satisfactory. However, if CR exceeds
0.10, it suggests the presence of significant in-
consistencies.
3 Pairwise Comparison Matrix
Pairwise comparison plays a crucial role in the realm
of multi-criteria decision analysis (MCDA), [21]. The
quantitative utilization of pairwise comparisons in de-
cision analysis was introduced by [22]. Subsequently,
we further developed the concept of pairwise compar-
ison, extending it into a widely adopted multi-criteria
decision-making approach known as the Analytic Hi-
erarchy Process (AHP) in [23].
The matrix A= (aij )records the pairwise com-
parisons between alternatives. Each entry aij rep-
resents the numerical answer to the question ’How
many times is alternative ibetter than alternative j?’.
Let R+to denote the set of positive numbers, Rn
+be
the set of positive vectors of size n, and Rn×n
+to be
the set of positive square matrices of size nwith all
elements greater than zero. The matrix of pairwise
comparison is defined as follows:
A=
1a12 . . . a1n
a21 1. . . a2n
.
.
..
.
.....
.
.
.
.
..
.
.....
.
.
an1an2. . . 1
(2)
Definition 1. The matrix A= (aij )∈Rn×n
+is said
to be pairwise comparison matrix if:
aij =1
aji
∀1≤i, j ≤n
The scale used for comparisons indicates the rela-
tive importance of one element over another one with
respect to a given attribute.
Table 1 presents a scale ranging from 1 to 9 (that
is, from least important to most important).
It should be noted that, if activity 0i0is assigned
one of the above numbers and is compared with ac-
tivity 0j0, then 0j0will have reciprocal values when
compared with 0i0.
Table 1.1-9 Pairwise comparison scale
Linguistic term Preference number
Equally important 1
Weakly more important 3
Strongly more important 5
Very strong important 7
Absolutely more important 9
Intermediate values 2,4,6,8
Let Ato denote the set of pairwise matrices and
An×ndenotes the set of pairwise matrices of size n,
respectively.
Definition 2. A= (aij )∈ An×nis consistent if :
aik =aij ×ajk ∀1≤i, j, k ≤n
Otherwise, it is said to be inconsistent.
The Theorem of Perron-Frobenius states that each
pairwise comparison matrix A∈ An×nhas a unique
positive weight vector w= (wi)that satisfies the
condition Aw=λmaxwand Pn
i=1 wi= 1, with λmax
is the maximal or Perron eigenvalue of the given
matrix A.
Definition 3. : Let A∈ An×nbe an pairwise com-
parison matrix of size n. Its Consistency Index (CI)
is given by :
CI =λmax −n
n−1(3)
So, CI = 0 ⇐⇒ λmax =n.
The consistency index (CI)is a useful measure for
determining the degree to which a pairwise compari-
son matrix deviates from consistency, [24]. It should
be noted that, in [25], the authors proposed a method
for establishing an upper bound on the value of CI
when the pairwise comparison matrix entries are ex-
pressed on a bounded scale.
We has recommended using a discrete scale for the
matrix elements, i.e., ∀1≤i, j ≤n:
aij ∈ {1
9,1
8,1
7, . . . , 1
2,2, . . . , 8,9}
We suggested obtaining a normalized measure of in-
consistency.
Definition 4. The random index (RI)of a pairwise
comparison matrix Ais provided to:
• Generate numerous pairwise comparison matri-
ces, draw each entry above the diagonal indepen-
dently and uniformly from the Saaty scale men-
tioned in Table 1.
• Calculate the CI for each pairwise comparison
matrix.
• Calculate the average of these values.
Several authors have published random indices.
These are based on simulation methods and the num-
ber of generated matrices used, see, [26]. The random
indices RInare reported in Table 2 for 1≤n≤10
as provided by [27], and validated by [28].
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Table 2.Random Consistency Index
n1 2 3 4 5 6 7 8 9 10
RIn0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49
Definition 5. Let A∈ An×n. The Consistency Ratio
(CR)of Ais defined by :
CR =CI
RIn
(4)
Definition 6. Let A∈ An×n. The matrix Ais con-
sistent enough to be accepted if CR < 0,1.
Theorem 7. [29], let A∈ An×n. The matrix Ais
consistent if and only if rank(A) = 1.
Theorem 8. [29], let A= (aij )∈ An×n.Ais
consistent, if and only if
λmax =n
Proof. Following the Theorem 7, if Ais consistent,
we have rank(A) = 1,
Also, all but one of its eignevalues are zero.
However, T race(A) = Pn
i=1 aii =n, and,
T race(A) = Pkλk=λmax,
Then, λmax =n.
Conversely, if λmax =n
nλmax =
n
X
i,j=1
aij
wj
wi
=n+X
1≤i<j≤naij
wj
wi
+aji
wi
wj
=n+X
1≤i<j≤nyij +1
yji
Since yij +1
yji ≥2, and nλmax =n2, equality is
obtained for yij = 1, i.e. aij =wi
wj.
Then, aij ajk =aik for all i, j and k, which shows
that Ais consistent.
Although it may seem strange to apply a crisp de-
cision rule to the fuzzy concept of ”large inconsis-
tency”, it is necessary to do so in order to ensure clar-
ity and objectivity, [30].
4 Approach based on particle swarm
optimization for pairwise matrix
4.1 Particle swarm optimization
Particle swarm optimization (PSO) is a more recent
evolutionary computational method compared to the
genetic algorithm and the evolutionary programming.
While PSO has some common properties of evolu-
tionary computation including randomly searching,
iteration time and so on, the classical PSO lacks
crossover and mutation operators. PSO mimics the
social dynamics of birds: Individual birds share infor-
mation about their position, velocity and fitness, and
then the behavior of the flock is affected in a way to
increase the probability of migration to regions with
high fitness.
Assuming that:
• The swarm, consists of Nparticles, in search
space with ndimensions, S⊆Rn.
• The position of the ith particle is an n-
dimensional vector xi= (xi1, xi2, . . . , xin)∈S.
• The velocity of this particle is also an n-
dimensional vector vi= (vi1, vi2, . . . , vin)∈S.
• The optimal previous position visited by the
ith particle is a point in S, denoted by pi=
(pi1, pi2, . . . , pin)∈S.
• The index of the particle that achieved the best
previous position among the entire swarm is de-
noted by g, and the iteration counter is denoted
by t.
Then, the following update equations are used to ma-
nipulate the standard PSO:
vid(t+ 1) =vid(t) + c1r1(pid(t)−xid(t))
+c2r2(pgd(t)−xid(t)) (5)
xid(t+ 1) = xid(t) + vid(t+ 1) (6)
where i= 1,2, . . . , N stands for the index of the
particle, d= 1,2, . . . , n indicates the dth component
of the particle, and c1and c2are some positive num-
bers representing cognitive and social parameters, re-
spectively.
The variables (r1, r2)∈[0,1] are uniformly dis-
tributed random numbers.
The standard algorithm for PSO method for an op-
timization problem is shown below :
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Algorithm 1: Particle Swarm Optimization
(PSO)
Input : J(x),N,c1,c2,ω,maxiter,
tolerance
Output: x∗, min J(x∗)
Initialize particle velocities viand positions
xifor i= 1, . . . , N
Initialize the best positions pi=xiand global
best position of the swarm pg
Evaluate J(xi)for each particle
Update piand pg
for k= 1 to maxiter do
for i= 1 to Ndo
Update velocity: vi=
ωvi+c1r1(pi−xi) + c2r2(pg−xi)
Update position: xi=xi+vi
Evaluate J(xi)
if J(xi)< J(pi)then
Update pi
if J(pi)< J(pg)then
Update pg
end
end
end
if |J(pg)−J(poldg)|< tolerance then
break
end
end
return pg,J(pg)
4.2 The proposed approach
4.2.1 Initialization
PSO begins with the generation of a random popula-
tion of potential solutions to the problem to be solved.
This is typically done by randomly generating a set of
particles in a way that each particle is a potential so-
lution represented as a set of particles.
The representation (encoding) of the possible solu-
tions is a crucial step in the process of any metaheuris-
tic. Depending on the problem studied, we distin-
guish several types of coding, namely, matrix repre-
sentation, binary coding and real coding. Indeed, the
choice of representation influences the performance
of particle swarm optimization. In our case, the pair-
wise matrix, which is the objectif of the problem, can
be identified by:
n(n−1)/2 elements
from the set:
{2,3, ..., 9,1/2,1/3, ..., 1/9}
Thus, we have to randomly generate mmatrices, each
of them is encoded by a vector under the form:
(a1,2, ...., a1,n, a2,3, ..., a2,n, ........., an−1,n)
Example:
3 7 1/3 5 1/6 2
⇓
1 3 7 1/3
1/3 1 5 1/6
1/7 1/5 1 2
3 6 1/2 1
4.2.2 Principle of the algorithm
The proposed algorithm, as illustrated in Figure 2,
is based on an improved PSO to find an acceptable
pairwise matrix with different sizes, depending on the
studied application, to help experts to have a panel of
choices of acceptable matrices to define by their ex-
pertise the desired matrix or to complete an incom-
plete defined pairwise matrix.
The proposed approach is a combination of an
adapted PSO and the AHP method. PSO method aim
to exploit the space of possible solutions; however,
the AHP method will make it possible to evaluate the
acceptability of the matrices by calculating their con-
sistency ratio CR until a stopping test is satisfied. It
should be noted that following the desired results, two
stopping criteria are considered: The first one consists
of get CR < 0.1and the second one is the maximum
number of iterations.
Figure 2: Schema of the proposed approach
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4.2.3 The Algorithm
The proposed approach is based on PSO method to
propose a set of consistent matrices to experts for
choosing the appropriate one for the considered appli-
cation. The PSO is combined with the AHP method to
evaluate the consistency of the matrices by calculat-
ing the index CR. The different steps of the proposed
approach are described as follows:
Algorithm 2: PSO-based approach to iden-
tify consistent pairwise comparison matrix
1: Step 1 Define the number of criteria n,
2: Step 2 Initialization: random generation of
initial population V(0) of kvectors
(V(0)
i), i = 1, ..., k; of length n(n−1)/2 from
the set {2,3, ..., 9,1/2,1/3, ..., 1/9},
3: Step 3 Encoding: transform each (particle)
vector of V(0) = (V(0)
i)to a population of
pairwise matrices M(0) = (M(0)
i)with
i= 1, ..., k,
4: Step 4 Evaluation: calculate CR(i)for each
matrix M(0)
ifor i= 1, ..., k,
5: Step 5 Update particles position and particles
velocity using (5) and (6)
6: Step 6 Repeat the step 4 with M(1) replace
M(0),
7: Step 7 The process continue until a stopping test
is satisfied.
5 Numerical results and discussion
Several simulations were conducted to demonstrate
PSO’s ability and efficiency in identifying consistent
matrices with varying criteria, resulting in matrices of
different sizes. Therefore, the size of the population
as well as the maximum number of iterations were ad-
justed based on the matrix size to be identified.
The considered PSO parameters are described bel-
low:
•c1= 1.5 Cognitive Coefficient,
•c2= 1.5 Social coefficient,
•w= 0.7 Inertia weight.
To show the efficiency of the PSO and its ability
to provide experts with a number of acceptable matri-
ces (CR < 0.1) in a reasonable time, numerical ex-
periments are developed using an Intel (R) Core(TM)
i3-6006U CPU @ 2.00 GHz RAM 4.00 GB.
Figure 3 and Figure 4 show the evolution of CR
during the iterative process for different numbers of
criteria, namely 4, 5, 6, 8, 10, 12 and 15, using particle
swarm optimisation. These show that the proposed
algorithm achieves a better evolution of the CR in a
shorter period of time.
Figure 3: Evolution of the CR during the iterative
process for different size of matrices
Table 3 and Table 4 present a comparison of PSO’s
performance in producing satisfactory pairwise ma-
trices. Specifically, we provide the necessary itera-
tion count to achieve a consistent pairwise matrix with
CR < 0.1. The results indicate that the PSO algo-
rithm suggested enables the generation of a suitable
pairwise matrix after a minimum number of iterations,
which can increase depending on the size of the ma-
trix. In particular, it necessitates fewer iterations to
fulfill the requested criteria. Furthermore, throughout
the iterative process, the CR progressively decreases,
thereby empowering experts to base their decisions on
data from previous iterations.
Table 5 shows the number of required iterations to
obtain the initial pairwise matrix with a CR < 0.1.
In all cases, the algorithm is initiated by inconsistent
pairwise matrices, which corresponds to randomly
generated matrices that exhibit higher individual con-
sistency with a CR greater than 0.1, except in the case
of matrix of size 4 where we manage to generate con-
sistent matrices at the first iteration. As a result, the
consistency rate of matrices starts at 0% for all ex-
amples examined and gradually increases after a few
iterations until a high consistency rate is achieved.
In other words, the experts will have a set contain-
ing an increasing number of consistent matrices as
the PSO method proceeds, according to their exper-
tise and discussions, to find an acceptable pairwise
consistent matrix, thereby avoiding inconsistent ma-
trices.
The aim of the research is to demonstrate the sig-
nificance and effectiveness of PSO in the genera-
tion of reliable pairwise matrices in accordance with
Saaty’s criteria. Thus, PSO offers a fascinating ap-
proach to assist specialists in determining the appro-
priate matrix for their study’s application.
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Figure 4: The performance of different criteria in the PSO algorithm for the production of a consistent pairwise
comparison matrix
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Table 3.Comparing the consistency of PSO process performance using different criteria.
Number of criteria(size of PCM) 4 5 6 8 10 12 15
Consistency Ration (CR)0.083 0.065 0.58 0.080 0.096 0.098 0.097
Number of Iteration 1 3 6 13 17 12 23
Table 4.Numerical experiment with a variety of matrices
Matrix size Population Initial Min_CR Initial M ax_CR first CR < 0.1Iterations
4 20 0.161 2.584 0.015 1
5 20 0.358 1.666 0.008 4
6 20 0.311 1.840 0.051 10
8 30 0.742 1.469 0.015 14
10 40 0.771 1.544 0.027 20
12 50 0.914 1.261 0.039 35
15 60 0.878 1.277 0.046 50
Table 5.Evolution of the percentage of consistent matrices according to different criteria
Criteria Min_CR M ax_CR % of matrices with CR < 0.1Iterations
0.083 2.597 2.5% 1
4 0.015 1.901 27.5% 3
0.002 0.596 90% 10
0.001 0.093 100% 50
0.216 2.013 0% 1
5 0.065 1.198 7.5% 3
0.031 0.711 30% 10
0.031 0.031 100% 50
0.474 1.320 0% 1
8 0.158 1.238 0% 3
0.059 0.494 15% 15
0.035 0.241 95% 50
0.710 1.280 0% 1
12 0.111 0.933 0% 10
0.040 0.259 82.5% 40
0.032 0.421 87.5% 50
0.863 1.208 0% 1
15 0.120 0.641 0% 15
0.078 0.731 40% 30
0.061 0.154 82.5% 50
6 Conclusion
In this study, particle swarm optimization (PSO) was
utilized as a metaheuristic approach to establish con-
sistent pairwise matrices when employing the AHP
method. The diverse outcomes achieved for matri-
ces of varying sizes illustrate that PSO is a viable
option to assist experts in selecting from a range of
consistent matrices or in defining incomplete pairwise
matrices. Additionally, the performance of PSO can
be enhanced to produce pairwise consistent matrices,
particularly for large matrices, within a reasonable
time frame.
Acknowledgment:
The authors would like to thank the editors and the
anonymous reviewers for their comments and
suggestions.
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