Improved F-parameter Mountain Gazelle Optimizer (IFMGO): A Comparative Analysis on Engineering Design Problems

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 10 Issue: 07 | July 2023 www.irjet.net p-ISSN: 2395-0072
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Improved F-parameter Mountain Gazelle Optimizer (IFMGO): A
Comparative Analysis on Engineering Design Problems
Abdul-Fatawu Seini Yussif1, Toufic Seini2
1 Department of Electrical and Electronic Engineering, Kwame Nkrumah University of Science and Technology,
Kumasi, Ghana
2Department of Physical Sciences, University for Development Studies, Ghana
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - This paper presents a comparativestudyofthree
metaheuristic algorithms: the Improved F-parameter
Mountain Gazelle Optimizer (IFMGO), the Mountain Gazelle
Optimizer (MGO), and the Particle Swarm Optimization(PSO)
algorithm, applied to a selection of challenging engineering
design problems. IFMGO, an advanced version of MGO,
demonstrates enhanced exploration and exploitation
capabilities owing to its inspirationfromthesocialbehavior of
mountain gazelles. The algorithms were implemented in the
MATLAB environment and evaluated on diverse engineering
design problems, includingthePressureVesselDesignProblem
(PVDP), the Spring Design Problem(SDP), theThree-barTruss
Design Problem (TTDP), the Cantilever Beam Design Problem
(CBDP), and the Welded Beam Design Problem (WBDP). The
primary objective is to investigate if IFMGO’s improvements
over MGO would lead to superior performance in solving
engineering optimization problems. Our experimental results
demonstrate that IFMGO indeed outperforms MGO across all
the engineering design problems considered. Furthermore,
IFMGO showcasescompetitiveperformancewhencomparedto
the well-established PSO algorithm, a testament to its efficacy
as a tool for handling intricate engineering design challenges.
Key Words: Algorithm, optimization, mountain gazelle,
engineering design problems, metaheuristic algorithm.
1.INTRODUCTION
This In the pursuit of optimizingcomplexengineeringdesign
problems, metaheuristic algorithms have emerged as
promising tools that can efficiently handle non-linear, multi-
objective optimization challenges [1][2]. Among these
algorithms, the Improved F-parameter Mountain Gazelle
Optimizer (IFMGO) [3] presents a significant advancement
over its predecessor, the Mountain GazelleOptimizer(MGO)
[4]. This paper aims to investigate and compare the
performance of IFMGO, MGO, and Particle Swarm
Optimization (PSO) on a set of diverse engineering design
problems [3][4][5].
Engineering design optimization plays a pivotal role in
various industries, including aerospace, mechanical, civil,
and structural engineering, among others [6][7]. The main
objective is to find the optimal designparametersthatsatisfy
multiple objectives while considering a range of constraints.
However, this task oftenpresentsa formidablechallengedue
to the presence of conflicting and competing objectives,
coupled with the high dimensionality and non-linearity of
the design space.
The IFMGO algorithm demonstrates superior exploration
and exploitation capabilities in comparisonto MGO, whichis
based on the social intelligence of mountain gazelles in the
wildlife [4][8]. The enhancements introduced in the IFMGO
aimed to address certain limitations present in the MGO,
making it more adept attacklingcomplex,multi-dimensional
engineering optimization problems.
To ascertain the performance of IFMGO in comparison to
MGO and PSO, these algorithms have been implementedand
tested using MATLAB software,a widely-adoptedandrobust
computational environment. The choice of engineering
design problems for evaluation includes the PressureVessel
Design Problem, the String Design Problem, the Three-bar
Truss Design Problem, the Cantilever BeamDesignProblem,
and the Welded Beam Design Problem [9][10][11]. These
problems are well-known benchmarks in the field of
engineering optimization, covering a diverse range of
complexities and dimensions.
Initial results from our experimentation demonstrated that
the IFMGO algorithm exhibits remarkable superiority over
MGO in all the engineering design problems considered.
Moreover, IFMGO demonstrates competitive performance
compared to the well-established PSO algorithm. The
objective of this paper is to shed light on the strengths and
weaknesses of these algorithms, providing valuable insights
for researchers and practitioners seeking efficient
optimization strategies for engineering design tasks.
The subsequent sections of this paper will delve into the
detailed methodology employed, the mathematical
formulation of the IFMGO algorithm,theexperimental setup,
and comprehensive analyses of the obtained results.Finally,
the implication of the findings in the context of engineering
design optimization would be discussed and concludedwith
recommendations for future research avenues in the realm
of metaheuristic algorithms.
This study, therefore, seeks to contribute to the growing
body of knowledge in the field of engineering optimization
and further establish thesignificanceoftheIFMGOalgorithm
as a powerful tool for tackling complex engineering design
problems.

2. METHOD
Improve F-parameter Mountain Gazelle Optimizer
(IFMGO):
The Improved F-parameter Mountain Gazelle Optimizer
(IFMGO) is an enhanced version of the Mountain Gazelle
Optimizer (MGO) for more efficient performance in solving
complex optimization problems [3]. The concept of this
algorithm originated from mimicking the social life of
mountain gazelles in the wildlifethat includedbachelormale
herds (BMH), maternity herds (MH), territorial and solitary
males (TSM), and the migration pattern of gazelles in search
of food (MSF) [4]. The mathematical modeling of the IFMGO
algorithm is presented as follows.
Mathematical Modelling of IFMGO
Territorial Solitary Male (TSM):
The adult male gazelles’ mechanism of protecting their
territories against intruders is mathematically modeled in
equation (1).
 
 
1 2
| |
gazelle r
TSM male ri BH ri X t F Cof
       (1)
Where;
ri1 and ri2: are random integers of either 1 or 2.
malegazelle: is the position vector of the best male
gazelle so far.
The values of BH, F, and Cofr are determined using equations
(2), (3), and (4).
 
1 2 3
, .....
N
ra pr
BH X r M r ra N
     (2)
The value of Xra is a random solution (young male) in the
range of ra, and that of Mpr is the average number of search
agents. The value of N is the number of gazelles, and r1 and r2
are random values from a range of (0, 1).
 
(1, ) exp
F randn d Iter
   (3)
Where d represents the size of the problem dimension
determined using a standard distribution. The Iter and
MaxIter respectively represent the iteration count and the
maximum iterations.
 
 
 
       
 
3
2
4
2
3 4 4 3
1 ,
,
,
cos 2 ,
i
a r
a N D
Cof
r D
N D N D r N D
 




 

    

(4)
Where;
r3 and r4: represent random values within the range (0,
1).
N2, N3, and N4: are set of randomly generated values
with the size of the problem function.
The value of a is determined using equation (5) below at
every iteration.
1
1
a Iter
MaxIter

 
    
 
(5)
Maternity Herd (MH):
The intelligence behind themother gazelle’sactofprotecting
its offspring is mathematically modeled in equation (6).
   
1, 3 4 1,
r gazelle rand r
MH BH Cof ri male ri X Cof
       (6)
Where;
Xrand: represents a vector position of a gazelle randomly
selected from the population.
ri3 and ri4: are integers randomly chosen from (1, 2).
Bachelor Male Herds (BMH):
In part of the development process of the male gazelles, the
young adult male ones create their territories and try
winning female gazelles to join them. This behavior is
modeled as equation (7).
 
   
5 6
gazelle r
BMH X t D ri male ri BH Cof
       (7)
Where;
X(t): is the position vector of the gazelle in the current
iteration.
ri5, and ri6: are integers randomly from (1, 2).
r6: is a randomly selected value from range (0 1).
The value of D is determined using equation (8) below.
   
6
| ( ) | | | 2 1
gazelle
D X t male r
     (8)
Migration in Search of Food (MSF):
The foraging mechanism of mountain gazelles involves
roaming to search the green pasture of their choice. This
random movement is modeled in equation (9).
  7
MSF ub lb r lb
    (9)
lb and ub represent the lower searchboundaryandtheupper
search boundary respectively. The value of r7 is randomly
chosen from (0,1). The pseudocode is presented below:

Pseudocode of IFMGO Algorithm
Inputs: iteration counter (Iter), maximum iteration
(MaxIter), population size (N).
Output: gazelle’s position, and its fitness value
Initialize random gazelle populations, Xi(i=1, 2, …N)
Evaluate the fitness values of the population.
While (Iter < MaxIter), do
for (every gazelle, Xi) do
Calculate TSM using equation (1)
Calculate MH using equation (6)
Calculate BMH using equation (7)
Calculate MSF using equation (9)
Evaluate the fitness values of TSM, MH, BMH, and MSF.
End for
Output best gazelle, Xbest, and its fitness value.
End while
Engineering Design Problems:
To effectively assess the performance of these algorithms on
engineering design problems, some well-known standard
engineering design problems have been considered in this
study. They include the Pressure Vessel Design Problem
(PVDP), the Spring Design Problem (SDP), the Three-bar
Truss Design Problem (TTDP), the Cantilever Beam Design
Problem (CBDP), and the Welded Beam Design Problem
(WBDP) [12]. The details of eachengineeringdesignproblem
are presented as follows.
1. Pressure Vessel Design Problem (PVDP):
The PVDP is one of the standard engineering design
benchmark functions for validating optimization algorithms
developed for solving engineering problems [10]. The
problem involves determining the valuesoffourparameters:
thickness (x1), the thickness of the heads (x2), the inner
radius (x3), and the length of the cylindrical section (x4). The
main objective of this design problem is to minimize the
overall cost, subject to non-linear constraints of stress and
yield criteria. The optimization problem is written as in
equation (10).
2
1 3 4 2 3
min( ( )) 0.6224 1.7781
f x x x x x x
  (10)
Subject to:
1 1 3
2 2 3
2 3
4
3 3 4 3
3
3 4
( ) 0.0193 0
( ) 0.00954 0
( ) 1,296,000 0
( ) 240 0
g x x x
g x x x
g x x x x
g x x


   

    


    

   

(11)
And search bounds of: 0.0625  X1,X2  99 0.0625,10 X3,
and X4  200
2. Spring Design Problem (SDP):
The Spring DesignProblem(SDP)isacontinuousconstrained
design problem and the design is illustrated in Figure 1. The
objective of the problem is to minimize the volume of a coil
spring under a constant tension/compression load [11]. The
problem focuses on threedesignvariables.Themathematical
formulation is presented in Equation (12).
  2
3 2 1
min( ( )) 2
f x x x x
  (12)
Subject to:
 
3
2 3
1 4
1
2
2 1 2
2 2
3 4
1
2 1 1
1
3 2
2 3
2 1
4
( ) 1 0
71785
4 1
( ) 1 0
5108
12566
140.45
( ) 1 0
( ) 1 0
1.5
x x
q x
x
x x x
q x
x
x x x
x
q x
x x
x x
q x
  

   

  

  
(13)
The design upper and lower bounds for the variables are
given below:
1 2 3
2 15, 0.25 1.3, 0.05 2
x x x
     
Fig -1: Schematic Diagram of Spring Design Problem
3. Three-bar Truss Design Problem (TTDP):
The design of the three-bar problem seeks to minimize
the weight of the three-bar truss, which is illustrated in
Figure 2 [13]. The objective function is mathematically
formulated in Equation (14).
 
1 2
min( ( )) 2 2
f x x x l
   (14)
Subject to:

1 2
1 2
1 1 2
2
2 2
1 1 2
3
2 1
2
( ) 0
2 2
( ) 0
2 2
1
( ) 0
2
x x
g x P
x x x
x
g x P
x x x
g x P
x x




  

  

  

(15)
Where;
2 2
1 2
0 , 1, 100 , 2 / , 2 /
x x l cm P KN cm KN cm

    
Fig -2: Three-bar Truss
4. Cantilever Beam Design Problem (CBDP):
Cantilever Beam Design is one of the widely used standard
engineering design problems for validating the performance
of nature-inspired optimization problems with the main aim
of developing to solve engineering optimization problems
[10]. The objective is to minimize the overall weight of the
cantilever beam with square cross sections. It is formulated
as shown in Equation (16).
1 2 3 4 5
min( ( )) 0.0624( )
f x x x x x x
     (16)
Subject to inequality:
3 3 3 3 3
1 2 3 4 5
61 37 19 7 1
( ) 1 0
g x
x x x x x
       (17)
The limits for the five design variables are:
0.01 100, 1,2,...,5.
i
x i
  
5. Welded Beam Design Problem (WBDP):
This is one of the several engineering design problems that
gain substantial consideration in validating optimization
algorithms. It is designed to minimizethecostbasedonshear
stress constraints, beams’ end deflection, bending stress in
the beam, and buckling load on the bar [9]. The main
objective is to design a welded beam withtheleastcostinput,
and the cost function is formulated as the objective function
shown in Equation (18).
2
1 2 3 4 2
min( ( )) 1.10471 0.04811 (14.0 )
f x x x x x x
   (18)
Subject to:
 
1
2
3 1 4
2
4 1 3 4 2
5 1
6
7
( ) ( ) 13000 0
( ) ( ) 30000 0
( ) 0
( ) 0.1047 0.04811 14 5 0
( ) 0.125 0
( ) ( ) 0.25 0
( ) 6000 ( ) 0
c
g x x
g x x
g x x x
g x x x x x
g x x
g x x
g x P x



  
  
  
    
  
  
  
(19)
Where:
2
2
' ' " " 2
2 2
'
1 2
"
2
2
2
1 3
2
2
2
1 3
2
1 2
2
4 3
3
3 4
3
3 3 4
( ) ( ) 2 ( )
6000
2
6000(14 )
4 2
( )
2 2
12 2
504000
( )
2.1952
( )
( ) 64746.022(1 0.0282346 )
x
R
x
c
x
x x
MR
J
M
x x
x
R
x x
x
J x x
x
x x
x
x x
P x X X X
    




  


 

 
   
 
 
 

 
 
 
 
 
 
 
 
 
 
 


 
(20)
Test Implementation:
To establish a fair test comparison of the algorithms on the
above detailed standardengineeringdesigntestproblems,all
the algorithms were coded in MATLAB environment
(MATLAB R2019a) on the samelaptop. For each engineering
design problem, each algorithm is used to solve it in
repetitions thirty (30) times, and the best results are
recorded. The results of all three algorithms (IFMGO, MGO,
and PSO) for each engineering design problem are recorded
and compared in tabular forms. The parameter settings for
the simulation are presented in Table 1.

Table-1: Parameter Settings for Simulation
Parameter Value
Population Size (N) 30
Maximum Iterations 1000
Number of Runs 30
The computer used for the simulations possesses the
following specifications as shown in Table 2.
Table-2: Specifications of Machine Used for Simulation
Specifications of Machine for Simulation
Type hp pavilion laptop computer
Processor AMD A8-6410 APU
Memory (RAM) 4.00 Gigabyte (3.43 GB usable)
Clock Speed 2.00 GHz
3. RESULTS AND DISCUSSION
This section presents the simulation results to show the
outcome of the experiment and establish a comprehensive
comparative performance analysis of thevariousalgorithms
on the engineering design problems considered. To present
the results concisely, it is presented under subsections
according to the various engineering design problems.
Pressure Vessel Design Problem (PVDP):
Results from the test of algorithms on the PVDP are
presented in Table 3. The original MGO algorithm produced
the worst results of 6108.9319.ThePSOalgorithmproduced
a much better result of 6055.7985. However, the IFMGO
algorithm, which is a modified versionofMGO, exceptionally
outperforms the PSO and producedthe bestsolutionvalue of
5897.7704. The result shows that themodificationproposed
in IFMGO has effectively improved the performance of the
algorithm in solving the Pressure Vessel Design Problem.
Table-3: Results of Algorithms on PVDP
Name of
Algorith
m
X1 X2 X3 X4 F(x)
PSO 0.87035
83
0.42841
47
45.096
29
142.72
09
6055.79
85
MGO 0.89477
89
0.44043
52
46.361
6
130.11
25
6108.93
19
IFMGO 0.78052
81
0.38600
1
40.424
8
198.79
9
5897.77
04
Spring Design Problem (SDP):
Table 4 contains the results of the algorithms on the Spring
Design Problem, and it shows a very competitive outcome
from all three algorithms. However, the MGO produced the
worst result of 0.014116, followed by the PSO with
0.013013, and the best result among the three algorithms of
0.012708 is produced by the IFMGO.Thisaswell showedthe
superior performance of IFMGO in handling the Spring
Design Problem.
Table-4: Results of Algorithms on SDP
Name of
Algorithm
X1 X2 X3 F(x)
PSO 0.05618 0.47469 6.6855 0.013013
MGO 0.060979 0.62414 4.0823 0.014116
IFMGO 0.0501768 0.321414 13.7041 0.012708
Three-bar Truss Design Problem (TTDP):
In the Three-bar Truss Design Problem (TTDP), all the
algorithms produced very close outcomes. However, the
IFMGO algorithm produced the best result of 263.8959, a
very small margin from that of the PSO algorithm of
263.8991. The original MGO algorithm produced the least
good results of 263.9041 as shown in Table 5.
Table-5: Results of Algorithms on TTDP
Name of
Algorithm
X1 X2 F(x)
PSO 0.79077 0.40234 263.8991
MGO 0.78534 0.41775 263.9041
IFMGO 0.78845 0.4089 263.8959
Cantilever Beam Design Problem (CBDP):
The results on the Cantilever Beam Design Problem (CBDP)
in Table 6 show that the IFMGO algorithm producedthe best
result with a value of 1.3400. However, both the original

MGO algorithm and the PSO algorithm produced very
competitive results of 1.3405 and 1.3403 respectively.
Table-6: Results of Algorithms on CBDP
Name of
Algorithm
X1 X2 X3 X4 X5 F(x)
PSO 6.00
9
5.266
3
4.454
4
3.563
7
2.185
5
1.340
3
MGO 6.03
54
5.412
4
4.396
8
3.477
4
2.160
5
1.340
5
IFMGO 6.01
29
5.308
9
4.496
1
3.503
1
2.152
7
1.340
0
Welded Beam Design Problem (WBDP):
On the Welded Beam Design Problem (WBDP), the PSO
algorithm produced 1.4829, the MGO algorithm produced
1.5766, and the IFMGO produced 1.473 as shown in Table 7.
Here, another competitive result was obtained with the best
produced by the IFMGO to show its superiority, followed by
the PSO algorithm, and finally the original MGO algorithm.
Table-7: Results of Algorithms on WBDP
Name of
Algorithm
X1 X2 X3 X4 F(x)
PSO 0.18001 2.5008 9.5847 0.18312 1.4829
MGO 0.21403 2.1015 8.8515 0.21443 1.5766
IFMGO 0.18298 2.4073 9.5818 0.18298 1.4730
4. CONCLUSIONS AND RECOMMENDATION
A comparative analysis of the performance of the PSO
algorithm, MGO algorithm, and IFMGO algorithm on
engineering design problems is conducted to assess the
performance of the IFMGO relative to the other two
algorithms. The IFMGO algorithm exceptionally performed
better than the MGO algorithm, and slightly better than the
PSO algorithm. By this performance, it is concluded that the
IFMGO is superior to the MGO and the PSO in solving
complex engineering optimization problems.
The IFMGO algorithm is therefore recommended for
adoption in the field of engineering for solving optimization
problems. For instance, optimizing the integration of
renewable energy and energy storage devices in electrical
distribution networks.
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