Population based optimization algorithms improvement using the predictive particles

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 3, June 2020, pp. 3261~3274
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i3.pp3261-3274  3261
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
Population based optimization algorithms improvement using
the predictive particles
M. M. H. Elroby1
, S. F. Mekhamer2
, H. E. A. Talaat3
, and M. A. Moustafa. Hassan4
1
Electrical Engineering Department, Faculty of Engineering, Ain Shams University, Egypt
2,3
Electrical Engineering Department, Future University, Egypt
4
Electrical Engineering Department, Cairo University, Egypt
Article Info ABSTRACT
Article history:
Received Jun 12, 2019
Revised Dec 2, 2019
Accepted Dec 11, 2019
A new efficient improvement, called Predictive Particle Modification (PPM),
is proposed in this paper. This modification makes the particle look to
the near area before moving toward the best solution of the group.
This modification can be applied to any population algorithm. The basic
philosophy of PPM is explained in detail. To evaluate the performance of
PPM, it is applied to Particle Swarm Optimization (PSO) algorithm and
Teaching Learning Based Optimization (TLBO) algorithm then tested using
23 standard benchmark functions. The effectiveness of these modifications
are compared with the other unmodified population optimization algorithms
based on the best solution, average solution, and convergence rate.
Keywords:
Optimization
Particle Swarm Optimization
Population optimization
Predictive particle
Teaching Learning Based Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
M. M. H. Elroby,
Electrical Engineering Department,
Faculty of Engineering,
Ain Shams University, Egypt.
Email: mousaelroby@yahoo.com
1. INTRODUCTION
Recently, many Meta heuristic optimization algorithms have been developed. These include Particle
Swarm Optimization (PSO) [1-5], Genetic Algorithm (GA) [6-9] , Deferential Evolution (DE) [10],
Ant Colony (AC) [11], Gravitational Search algorithm (GSA) [12], Sine Cosine Algorithm (SCA) [13-15],
Hybrid PSOGSA Algorithm [16], Adaptive SCA integrated with particle swarm [17], and Teaching Learning
Based Optimization (TLBO) [18-20]. The same goal for them is to find the global optimum. In order to do
this, a heuristic algorithm should be equipped with two main characteristics to ensure finding global
optimum. These two major characteristics are exploration and exploitation. Exploration is the ability to
search whole parts of the space whereas exploitation is the convergence ability to the best solution. The goal
of all Meta heuristic optimization algorithms is to balance the ability of exploitation and exploration in order
to find global optimum. According to [21], exploitation and exploration in evolutionary computing are not
clear due to lake of a generally accepted perception. In other hand, with strengthening one ability, the other
will weaken and vice versa. Because of the above-mentioned points, the existing Meta heuristic optimization
algorithms are capable of solving finite set of problems. It has been proved that there is no algorithm, which
can perform general enough to solve all optimization problems [22]. Many hydride optimization algorithms
are to balance the overall exploration and exploitation ability.
In this study, the proposed modification increases the exploration and make the particle look to
the surrounding space before affected by the best solution. The proposed modification can be applied to any
population optimization algorithms. The PSO is one of the widely used population algorithms due to its
simplicity, convergence speed, and ability of searching global optimum. Recently TLBO is a new efficient
optimization method combine between teaching and learning phases. For the reasons listed above this

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modification has been applied to PSO and TLBO. The organization of this paper is as follows: Section 2
describes the standard PSO and its exploration problem. Section 3 describes the standard TLBO.
The proposed modification is presented in Section 4. Section 5 describes the results of the proposed
modification. Section 6 concludes this research.
2. THE STANDARD PARTICLE SWARM OPTIMIZATION
2.1. Particle Swarm Optimization Algorithm
PSO is a population computation algorithm, which is proposed by Kennedy and Eberhart [1].
The PSO was inspired from social behavior of bird flocking. It uses a number of particles, which fly, around
the search space. All particles try to find best solution. Meanwhile, they all look at the best particle in their
paths. In other words, particles consider their own best solutions and the best solution has found so far.
Each particle in PSO should consider the current position, the distance to pbest, the current velocity, and
the distance to global best (gbest) to modify its position. PSO was modeled as follow [1]:
𝑣𝑖
𝑡+1
= 𝑤𝑣𝑖
𝑡
+ 𝑐1 × 𝑟𝑎𝑛𝑑 × ( 𝑝𝑏𝑒𝑠𝑡𝑖
𝑡
− 𝑥𝑖
𝑡
) + 𝑐2 × 𝑟𝑎𝑛𝑑 × ( 𝑔𝑏𝑒𝑠𝑡𝑡
− 𝑥𝑖
𝑡
) (1)
𝑥𝑖
𝑡+1
= 𝑥𝑖
𝑡
+ 𝑣𝑖
𝑡+1
(2)
where vi
t+1
is the velocity of particle i at iteration t,
w is a weighting function,
cj is a weighting factor,
rand is a random number between 0 and 1,
xi
t
is the current position of particle i at iteration t,
pbesti is the pbest of agent i at iteration t,
gbest is the best solution so far.
The first part of (1), 𝑤𝑣𝑖
𝑡
provides exploration ability for PSO. The second and third parts,
𝑐1× 𝑟𝑎𝑛𝑑 × (𝑝𝑏𝑒𝑠𝑡 − 𝑥𝑖
𝑡
) and, 𝑐1× 𝑟𝑎𝑛𝑑 × (𝑝𝑏𝑒𝑠𝑡 − 𝑥𝑖
𝑡
) represent private thinking and collaboration of
particles respectively [23, 24]. The PSO is initialized with randomly placing the particles in a problem space.
In each iteration, the particles velocities are calculated using (1). After velocities calculating, the position of
particle can be calculated as (2). This process will continue until meeting an end criterion.
2.1.1. PSO Exploration Problem
The first part of (1), 𝑤𝑣𝑖
𝑡
provides PSO exploration ability. When the algorithm is started,
the velocity is initialized with zero value. Thus from Equation 1, the Global Best Particle (GBP)
(i.e. P1 in Figure 1 (a)) remains in its place until the best global solution is changed by a new particle.
This means the global best particle cannot explore near area because it is not exited by any particle.
In addition, particles that arrive from another places (P2 - P5) to the place of the global best solution with
a certain velocity after a number of iteration may be damped before reaching the optimal solution as shown in
Figure 1 (b). This phenomenon will be treated using PPM in Section 2.
(a) (b)
Figure 1.Particles at initial and final iteration, (a) initial iteration, (b) final iteration

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Population based optimization algorithms improvement… (M. M. H. Elroby)
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3. THE STANDARD TEACHING LEARNING BASED OPTIMIZATION
The TLBO method is based on the effect of the teacher on the learners. The teacher is considered as
a global best learned person (𝑔𝑏𝑒𝑠𝑡 𝑡
) who shares his knowledge with the learners. The process of TLBO is
divided to two phase. The first phase consists of the ‘Teacher Phase’ and the second phase consists of
the ‘Learner Phase’. The ‘Teacher Phase’ means learning from the teacher and the ‘Learner Phase’ means
learning through the interaction between learners. TLBO was modeled as follows [18]:
3.1. Teacher Phase
A learner learns from teacher by moving its mean to teacher value. Learner modification is
expressed as:
3.2. Learner Phase
A learner learns new something if the other learner has better knowledge than him. Learner
modification is expressed as:
4. PREDICTIVE PARTICLE
The main idea of the PPM based on that each iteration the particle should look at its near area and
see if it have a value best than the GBP or not. If it have value better than GBP, it will be the GBP. The PPM
can remedy non-exiting GBP (P1 in Figure 1 (a)) and not wait until excitation from another particle.
In addition, it can improve the vision of the particle before movement toward GBP and overcome the jump
over narrow area leaving goloabal solution.
Consider the initial values of the particles P1 to P5, which are shown in Figure 2. In the next
iteration, these particles will move toward P1 (as it is the GBP at this moment) and take positions P1, P2 to
P5. In addition, the P3 may jump to P3 without converge to gbest especially when the fitness function have
narrow area with high deep value. In addition, the P1 still in its position as it is GBP. These phenomena can
be treated if the particle try to find a best solution (target) from near area before move to GBP as shown in
Figure 3. This can be done using the numerical gradient with a definite target. Assume the fitness function
(F) is a linear function near the particle position in matrix form:
Figure 2. Particles movement
𝑭𝒐𝒓 𝑖 = 1 ∶ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑒𝑎𝑟𝑛𝑒𝑎𝑟𝑠
𝑅𝑎𝑛𝑑𝑜𝑚𝑙𝑦 𝑠𝑒𝑙𝑒𝑐𝑡 𝑡𝑤𝑜 𝑙𝑒𝑎𝑟𝑛𝑒𝑟𝑠 𝑋𝑖
𝑡
𝑎𝑛𝑑 𝑋𝑗
𝑡
, 𝑤ℎ𝑒𝑟𝑒 𝑖 ≠ 𝑗
𝑰𝒇 𝑓 (𝑋𝑖
𝑡
) < 𝑓 (𝑋𝑗
𝑡
)
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑟𝑎𝑛𝑑(𝑋𝑖
𝑡
− 𝑋𝑗
𝑡
)
𝑬𝒍𝒔𝒆
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑟𝑎𝑛𝑑(𝑋𝑗
𝑡
− 𝑋𝑖
𝑡
)
𝑬𝒏𝒅
𝐴𝑐𝑐𝑒𝑝𝑡 𝑋𝑖
𝑡+1
𝑖𝑓 𝑖𝑡 𝑔𝑖𝑣𝑒𝑠 𝑎 𝑏𝑒𝑡𝑡𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒.
𝑬𝒏𝒅
𝑇𝐹 = 𝑟𝑜𝑢𝑛𝑑[1 + 𝑟𝑎𝑛𝑑(0, 1)]
𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑡
= 𝑟(𝑔𝑏𝑒𝑠𝑡 𝑡
− 𝑇𝐹 𝑀 𝑡
)
Where 𝑀 𝑡
is the mean of the learner and ‘𝑔𝑏𝑒𝑠𝑡 𝑡
’ is the global best (the teacher) at any iteration 𝑡.
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑡
𝑡+1
𝑬𝒏𝒅

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Figure 3. Initial and target of the particle
F = AX + b (3)
Using numerical gradient method:
Xnew = Xold − R ∗
dF
dX
(4)
where
𝑑𝐹
𝑑𝑋
= [
∆𝐹/∆𝑥1
∆𝐹/∆𝑥 𝑛
] = 𝐴′
Xnew is the new postion of the particle in column form
Xold is the current position of the particle
R is the step size
∆𝐹/∆𝑥𝑖 is calculated numerically near 𝑋 𝑜𝑙𝑑 by change only 𝑥𝑖
From (3)∶
F 𝑜𝑙𝑑 = AX 𝑜𝑙𝑑 + b (5)
F 𝑛𝑒𝑤 = AX 𝑛𝑒𝑤 + b (6)
From (5) and (6) by substraction:
Xnew = −
Fold − Fnew
A
+ Xold (7)
where
Fold is the current fitinenss value
Fnew is the new fitinenss value
From (4) and (7).
R =
Fold − Fnew
A ∗
dF
dX
=
Fold − Fnew
(
dF
dX
)′ ∗
dF
dX
(8)
Xnew = Xold −
Fold − Fnew
(
dF
dX
)′ ∗
dF
dX
∗ dF/dX (9)

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If Fi is the current fitness value of the particle and Ft is the target fitness of the particle (less than gbest
value). It is nice to dived search steps to N steps as follows:
Assume dist = Fi − Ft (10)
for each step
∆X =
dist/N
(
dF
dX
)′ ∗
dF
dX
∗
dF
dX (11)
Xnew = Xold − ∆X (12)
The complete PPM algorithm before moving to GBP is shown in Table 1. In addition, the Modified PSO
(MPSO) and Modified TLBO (MTLBO) are shown in Table 2 and Table 3 respectively.
Table 1. Gradient algorithm
Set particle gradient parameter:
𝐹𝑡 < gbest
𝑋 𝑜𝑙𝑑 = current position of particle
𝑑𝑖𝑠𝑡 = 𝐹𝑖 − 𝐹𝑡
𝑖𝑛𝑡𝑖𝑎𝑙𝑖𝑧𝑒 Vtemp = 0
Execute gradiant algorithm:
For N step
𝑋 𝑛𝑒𝑤 = 𝑋 𝑜𝑙𝑑 − ∆𝑋 according to (12)
𝑋 𝑛𝑒𝑤 =max(𝑋 𝑛𝑒𝑤, xmin);
𝑋 𝑛𝑒𝑤 = min(𝑋 𝑛𝑒𝑤, xmax);
If F(𝑋 𝑛𝑒𝑤 ) < F(𝑋 𝑛𝑒𝑤 )
𝑋𝑡𝑒𝑚𝑝 = 𝑋 𝑛𝑒𝑤
Vtemp=∆𝑋
Else
𝑋𝑡𝑒𝑚𝑝 = 𝑋𝑡 − 2 ∗ Vtemp
𝑋𝑡𝑒𝑚𝑝=max(𝑋𝑡𝑒𝑚𝑝, xmin);
𝑋𝑡𝑒𝑚𝑝= min(𝑋𝑡𝑒𝑚𝑝, xmax);
Vtemp=Vtemp
End
End
Update particle position :
If 𝐹(𝑋𝑡𝑒𝑚𝑝) < 𝑔𝑏𝑒𝑠𝑡
𝑥𝑖
𝑡+1
= 𝑋𝑡𝑒𝑚𝑝
𝑣𝑖
𝑡+1
=Vtemp
End
Table 2. Modified PSO
For each particle
initialize particle
End
Choose the particle with the best fitness value
of all the particles as the gbest
Do
For each particle
Update particle position according to
𝑣𝑖
𝑡+1
= 𝑤𝑣𝑖
𝑡
+ 𝑐2 × 𝑟𝑎𝑛𝑑 × (𝑔𝑏𝑒𝑠𝑡 − 𝑥𝑖
𝑡
)
𝑥𝑖
𝑡+1
= 𝑥𝑖
𝑡
+ 𝑣𝑖
𝑡+1
gradient algorithm as shown in Table 1
End
For each particle
Calculate fitness value
If the fitness value is better than the best
fitness value (pbest) in history set current
value as the new pbest
End
Choose the particle with the best fitness value
of all the particles as the gbest
While maximum iterations or minimum error
criteria is not attained

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Table 3. Modified TLBO
For each particle
initialize particle
End
Choose the particle with the best fitness value of all the particles as the gbest
Do
1 ) Teacher phase
𝑇𝐹 = 𝑟𝑜𝑢𝑛𝑑[1 + 𝑟𝑎𝑛𝑑(0, 1)]
𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑡
= 𝑟(𝑔𝑏𝑒𝑠𝑡 𝑡
− 𝑇𝐹 𝑀 𝑡
)
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑡
gradient algorithm as shown in Table. 1 for i 𝐴𝑐𝑐𝑒𝑝𝑡 𝑋𝑖
𝑡+1
𝑬𝒏𝒅
2) learner phase
𝑅𝑎𝑛𝑑𝑜𝑚𝑙𝑦 𝑠𝑒𝑙𝑒𝑐𝑡 𝑡𝑤𝑜 𝑙𝑒𝑎𝑟𝑛𝑒𝑟𝑠 𝑋𝑖
𝑡
𝑎𝑛𝑑 𝑋𝑗
𝑡
, 𝑤ℎ𝑒𝑟𝑒 𝑖 ≠ 𝑗
gradient algorithm as shown in Table 1 for i and j
𝑰𝒇 𝑓 (𝑋𝑖
𝑡
) < 𝑓 (𝑋𝑗
𝑡
)
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑟𝑎𝑛𝑑(𝑋𝑖
𝑡
− 𝑋𝑗
𝑡
)
𝑬𝒍𝒔𝒆
𝑋𝑖
𝑡+1
= 𝑋𝑖
𝑡
+ 𝑟𝑎𝑛𝑑(𝑋𝑗
𝑡
− 𝑋𝑖
𝑡
)
𝑬𝒏𝒅
gradient algorithm as shown in Table 1 for i
𝑡+1
𝑬𝒏𝒅
Choose the particle with the best fitness value of all the particles as the gbest
While maximum iterations or minimum error criteria is not attained
5. EXPERIMENTAL RESULTS AND DISCUSSION
The standard PSO, PSOSGSA, SCA, TLBO, MPSO, and MTLBO with the parameter in
Table 4 [25-28] have executed 30 independent runs over each benchmark function for statistical analysis.
As shown in Table 5, MPSO and MTLBO outperformed all of the other algorithms with regard to the quality
of the solutions for all functions. In contrast, the other algorithms produced poor results on certain functions
and accurate results on others. This finding reflects the efficient performance of the MPSO and MTLBO in
comparison with the other unmodifeied algorithms. In addition, Figure 4 to Figure 11 show a comparison
between MPSO and MTLBO and all the other algorithms for the convergence rate for the fitness versus
the iterations. These figures show that MPSO and MTLBO outperforms all the other unmodifeied algorithms
in terms of the convergence speed with an accurate solutio
Table 4. Algorithms parameter
Algorithm Parameter
PSO C1=C2=2 wdamp=0.9
PSOGSA G0=1, C1=0.5, C2=1.5
SCA a = 2, r2=(2*pi)*rand , r3=2*rand, r4=rand
TLBO TF=randi([1 2])
MPSOA C1=C2=2, wdamp=0.9 , N=5
MTLBO G0=1, C1=0.5, C2=1.5, N=5
MaxVelocity=0.2*(VarMax-VarMin) , MinVelocity= —MaxVelocity
Table 5. Benchmark functions
Function n Range PSO PSOGSA SCA TLBO MPSO MTLBO

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Table 5. Benchmark functions (continue)
Function n Range PSO PSOGSA SCA TLBO MPSO MTLBO
Figure 4. Converge rate curves for F1 to F3

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Figure 4. Converge rate curves for F1 to F3 (continue)

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6. CONCLUSION
In this paper, the PPM has the advantage of powerful exploration. Thus, it was necessary to enhance
the population algorithms by merging it with PPM, which has the advantage of powerful exploitation.
Hence, the proposed modification improves the exploration quality and maintaining fast convergence.
PPM optimization was tested to find the optimal solution for standard mathematical functions, and results
demonstrated improvement in solution quality and convergence rate..
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