Modelling and Simulation in Engineering
Volume 2018 (2018), Article ID 4945157, 14 pages
https://doi.org/10.1155/2018/4945157

Research Article

Application and Development of Enhanced Chaotic Grasshopper Optimization Algorithms

Akash Saxena,¹ Shalini Shekhawat,¹ and Rajesh Kumar²

¹Swami Keshvanand Institute of Technology, Jaipur 302017, India
²Malaviya National Institute of Technology, Jaipur 302017, India

Correspondence should be addressed to Akash Saxena; aakash.saxena@hotmail.com

Received 30 December 2017; Revised 23 March 2018; Accepted 8 April 2018; Published 23 May 2018

Academic Editor: Gaetano Sequenzia

Copyright © 2018 Akash Saxena et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In recent years, metaheuristic algorithms have revolutionized the world with their better problem solving capacity. Any metaheuristic algorithm has two phases: exploration and exploitation. The ability of the algorithm to solve a difficult optimization problem depends upon the efficacy of these two phases. These two phases are tied with a bridging mechanism, which plays an important role. This paper presents an application of chaotic maps to improve the bridging mechanism of Grasshopper Optimisation Algorithm (GOA) by embedding 10 different maps. This experiment evolves 10 different chaotic variants of GOA, and they are named as Enhanced Chaotic Grasshopper Optimization Algorithms (ECGOAs). The performance of these variants is tested over ten shifted and biased unimodal and multimodal benchmark functions. Further, the applications of these variants have been evaluated on three-bar truss design problem and frequency-modulated sound synthesis parameter estimation problem. Results reveal that the chaotic mechanism enhances the performance of GOA. Further, the results of the Wilcoxon rank sum test also establish the efficacy of the proposed variants.

1. Introduction

Optimization is a term which refers to the selection of the best option amongst the given set of alternatives. Examples of optimization processes are everywhere such as in business, human resource management, challenging engineering design problems, transportation, profit making propositions, and industrial applications. Optimization can be done for the maximization of any proposition or minimization of any proposition. In engineering problems particularly, the use of maximization is for efficiency maximization, classification accuracy maximization, and revenue or profit maximization, and on the other hand, minimization can be performed for cost, loss, risk, and execution time of any engineering process. Apart from these classifications of optimization, another classification of the optimization problem can be done on the basis of constraints. An optimization problem without any constraints is called unconstrained optimization; similarly another type is constrained optimization with linear and nonlinear constraints. Another classification can be done on the basis of the objective of the optimization; when an optimization problem aims towards a single objective, it is called the single objective optimization problem, and similarly when it aims towards multiobjectives, the same is called the multiobjective optimization problem [1]. A recent trend is to employ metaheuristic optimization algorithms to solve challenging problems of the real world. The term refers to problem-independent higher level heuristic mechanism [2]. In recent years, applications of metaheuristic algorithms in engineering problems have been reported. The successful and effective implementation of these algorithms on real applications has attracted the attention of researchers to work in this direction.

The metaheuristic optimization approaches can be subdivided into three categories:

(1)Evolutionary computing-based algorithms [3–5](2)Physics law-based algorithms [6–8](3)Swarm intelligence-based algorithms [9–16]

Evolutionary-based algorithms are based on natural evolution due to environmental pressure [2]. These algorithms employ selection or mixing criterion to generate an optimal solution set which possesses higher fitness values. Basic virtues of these algorithms are of stochastic nature, incorporating crossover and termination operators for hybridizing the solutions and enhancing the fitness value. A few examples of these algorithms are Genetic Algorithm [4] and Evolution Strategy and Evolutionary Programming [5]. Another class of algorithms is the algorithms which are inspired from physics and based on the laws of fundamental physics. A few examples of these algorithms are Gravitational Search Algorithm [14], Big Bang-Big Crunch Algorithm [7], and Black hole Algorithm [8].

The third category is based on swarm intelligence methods, where the cognitive and social behavior of the natural swarms like birds and school of fish is mimicked in the form of simulation. The most famous algorithm in this category is Particle Swarm Optimization (PSO), which works on the philosophy “Follow the Leader” [9]. Other examples of these algorithms are Bat Algorithm [10], Firefly Algorithm [11], and Cuckoo Search Algorithm [12]. A recently published swarm algorithm, which became popular nowadays, is Grey Wolf Optimizer (GWO), the algorithm that mimics the hunting behavior of grey wolves and is a fine example of the compliance of the social hierarchy of the wolf pack during searching, attacking, and hunting phases. A novel algorithm based on crow behavior named as Crow Search Algorithm (CSA) has been proposed [14]. CSA mimics the behavior of crow to store their excess food in hiding places and retrieve it when it is needed. Similarly, Ant Lion Optimizer Algorithm [15] and Grasshopper Optimisation Algorithm [16] are also a good example of social mimicry of the natural swarms.

The applications of swarm algorithms are very well reported in the literature and in many design problems, namely, Automatic Generation Control [17], Unit Commitment [18, 19], Feature Selection [20], and Ambient Air Quality Classification [21].

Many optimization algorithms have employed chaotic sequences over the random walk (random numbers generation) due to the fact that the random walk not always implements the global search well. Thus in some cases, the algorithm development is based on chaotic variables instead of random variables, and these algorithms are called chaotic algorithms [22–26]. A chaotic Firefly Algorithm was proposed in [22]. In this work, attractive movement of fireflies was simulated with ten chaotic maps. Chaotic sequences are used for parameters and in the Chaotic GWO approach [27]. Chaos enhanced Accelerated Particle Swarm Optimization (CAPSO) which was proposed by Gandomi et al. [25]. An attraction parameter was tuned with normalized chaotic maps in that work. Chaotic Bat Algorithm was proposed in [23], and the tuning of the crucial parameter of this algorithm was done with the help of chaotic maps. Different ten chaotic maps were employed in gravitational search algorithm in [28].

Two mechanisms: diversification and intensification are essential parts of any swarm algorithm. The initial phase of any swarm algorithm started with random search; usually this process swifts and holds responsibility to search every possible direction of the search space, and thus, the process is random in nature. On the other hand, the intensification process is strategic. The outcome of this process is specific and is treated as the solution of the problem. It is empirical to say that speed of these processes is different. Every algorithm employs a bridging mechanism to maintain a good amount of trade-off between these two processes. Some algorithms use different operators and different models of random walks in different phases, and in short, these operators/mechanisms help the algorithm to maintain a fair balance between these two processes. This paper investigates the impact of different chaotic sequences on the bridging mechanism of the GOA, by evaluating the performance of the proposed variants on standard benchmark functions and real applications. 10 different chaotic sequences are embedded with the parameter , and the careful observation is presented. Following research objectives are framed for this work:

(1)To employ 10 different chaotic maps through normalized function to propose chaotic variants of GOA. These variants are developed on the basis of adaptive, chaotic, and monotonically decreasing parameter .(2)To conduct a nonparametric Wilcoxon rank sum test for observing the efficacy of the chaotic variants with the GOA by observing values.(3)To apply these variants on three-bar truss design and parameter estimation for frequency-modulated sound waves and compare the performance with the other contemporary algorithms.

Remaining part of the paper is organized as follows: in Section 2, brief details of different chaotic maps are incorporated. In Section 3, an overview of GOA is presented. The development of chaotic variants is explained in Section 4. Simulation results on benchmark problems and engineering optimization problems are presented in Section 5. Last but not the least, major conclusions of this study have been presented in Conclusion.

2. Chaotic Map

In this section, the definitions of different chaotic maps are presented. Table 1 shows the definition range and names of the chaotic maps. These maps have also been studied in the approaches [22, 28]. The shape of these maps with starting point is shown in Figure 1.

Table 1: Definition of chaotic maps [28].

Name of map
Equation
Range

Chebyshev

Circle

Gauss

Iterative

Logistic

Piecewise

Sine

Singer

Sinusoidal

Tent

Figure 1: Chaotic maps.

3. Grasshopper Optimisation Algorithm: An Overview

Grasshopper Optimisation Algorithm (GOA) [16] is a recently proposed naturally inspired algorithm, which is based on one of the largest swarms of all creatures. As grasshoppers are herbivores, they cause severe damage to crops. The swarming behavior of a grasshopper depends on both nymphs and adults. The nymph moves on rolling on the ground and feeds on succulents and soft plants. An adult grasshopper can jump high in search of food and therefore have a larger area to explore. As a result, both type of movements are observed, that is, slow movement and abrupt movement of large range which represents exploration and exploitation. The mathematical frame work presented in [16] has been presented here. The swarming behavior of the grasshopper is represented mathematically as

where is the position of the grasshopper, is the social interaction, is the gravity force in the grasshopper, and is the wind advection.

The social interaction is given as

where is the distance between the and grasshopper and is a unit vector from the grasshopper to the grasshopper. Function implies the social forces which can be given mathematically as

where is the intensity of attraction and is the attractive length scale. In the search of food, grasshoppers create three types of regions in terms of social interaction known as the comfort zone, repulsion region, and attraction region. When the distance is larger between grasshoppers, then function “” is not able to apply strong forces. To resolve this, the component in (1) is given as

where is the gravitational constant and represents a unity vector towards the center of Earth. The component is calculated as

where is the constant drift and is a unity vector in the direction of wind. Substituting values of , , and in (1), we get

where is given by (3) and is the number of grasshoppers. A revised form of this formula can be used to solve optimization problem:

where is the upper bound in the dimension, is the lower bound in the dimension, is the value of the dimension in the target, and is the decreasing coefficient to shrink the comfort zone, repulsive zone, and attraction zone. It is assumed that the wind direction is always towards a target. In the process of searching food, nymphs move on rolling on the ground and adults move on jumping in the air, creating both the cases exploration and exploitation. One can balance both of these two by decreasing the parameter in (8) proportionally to the number of iteration. This can be calculated as

where is the maximum value, is the minimum value, indicates the current iteration, and is the maximum number of iterations. The application of GOA has been observed in many engineering optimization problems [29, 30].

4. Development of Enhanced Chaotic Grasshopper Optimization Algorithms

This section presents philosophy and chronological development of ECGOAs. In this work, we have obeyed the philosophy of GOA and decreased the parameter () in due course of iterations. However, with the inculcation of the different chaotic sequences in the comfort zone reduction parameter “,” the diversification virtue of the GOA enhances till the last iteration. To develop the variants, a normalization function is employed to distribute the sequences between maximum and minimum bias before it can be biased with the parameter . The mathematical expression for this function at any iteration can be given as

where denotes the maximum iteration. The normalized chaotic sequence can be given as per following equation:

where is the value of chaotic sequence computed as per Table 1.

The instantaneous value of the chaotic sequence embedded parameter for ECGOA will be given as per the following equation:

By using (8)–(10), one can easily get the value of the chaotic sequence for any of the chaotic maps given in Table 1. For example, we briefly present here (12)–(15) for piecewise map (ECGOA6). Further, in this paper, 10 different chaotic sequences are embedded with the mapping through a normalized function to the parameter in GOA. In classical GOA, this parameter act as a bridging mechanism for the exploration and exploitation phase over the whole course of iterations. In the initial phase, the search agents take large steps to explore the search space in effective manner, and in later case, these steps are reduced with the help of linear decrement in the parameter . In this work, the focus is on this linear variation with different chaotic sequences embedded through a normalized function. The major motivation to perform this experiment is to seek the possibility of better exploration and exploitation by introducing the chaotic sequences in each iteration. In GOA, this parameter decreases linearly, which means that algorithm either performs diversification (exploration) or intensification (exploitation). In this work, the authors have changed the parameter chaotically so that the exploration virtue can be kept alive in the final steps of iterations.

For justification, an implementation of a logistic chaotic map with the abovementioned procedure is shown in Figure 2. On the basis of this mathematical procedure, 10 different variants of chaotic algorithms are proposed here which are named as Enhanced Chaotic Grasshopper Optimization Algorithms (ECGOAs): the different variants are ECGOA1 with Chebyshev map, ECGOA2 with Circle map, ECGOA3 with Gauss map, ECGOA4 with Iterative map, ECGOA5 with Logistic map, ECGOA6 with Piecewise map, ECGOA7 with Sine map, ECGOA8 with Singer map, ECGOA9 with Sinusoidal map, and ECGOA10 with Tent map.

 Case 1: ,

 Case 2: ,

 Case 3: ,

 Case 4: ,

(a)

(b)

(c)

Figure 2: Development of chaotic bridging mechanism.

Parameter “” is an important parameter of GOA and used twice in (7), and the inner “c” contributes to shrink the attraction and repulsion zones between grasshoppers. This effect is analogous to the exploitation phase mechanism. However, with the increment in the iteration counter, outer reduces the search and helps algorithm to converge. For balancing the intensification and diversification processes in GOA, parameter decreases linearly with every passing iteration. The comfort of grasshoppers is reduced with every iteration by varying the parameter from 1 to zero linearly. However, in the proposed ECGOAs, chaotic sequence changes the boundary of the comfort zone randomly in monotonically decreasing trend. This mechanism assists the search agents to release themselves from the local minima trap. The transition from the diversification phase to the intensification phase can be achieved slowly with the employment of a different chaotic sequences-enabled adaptive approach. This change makes parameter “” adaptive and random concurrently. The values for and are considered as and 0.2, respectively. In the following section, benchmarking of these variants and application of these variants on two engineering problems are investigated.

5. Simulation and Results

The testing of the optimization problem on some known functions is the best way to showcase the efficacy of the algorithm. Some of the essential characteristics of these functions are that the functions should be multimodal or unimodal in nature, the function should be nonseparable, and moreover, the functions should lag in the global structure. By keeping these virtues in consideration, benchmarking of the variants is done on five unimodal and five multimodal shifted and biased benchmark functions [31–33]. In the standard benchmark functions, the minima lies at zero; however, in multimodal functions, multioptima (local) can exist. To make the problem harder, shift and bias have been provided to the functions so that the robustness of variants can be tested. Figure 3 shows the 2D version of these functions and definition, and other relevant details of these functions are given in Table 2.

(a)

(b)

Figure 3: Shifted and biased benchmark functions. (a) Unimodal and (b) multimodal test functions.

Table 2: Benchmark functions [28–31].

Function
Dimension
Range
Minimum value

Unimodal benchmark function

30
[−100, 100]
−50

30
[−10, 10]
−50

30
[−100, 100]
−50

30
[−30, 30]
−50

30
[−100, 100]
−80

Multimodal benchmark function

30
[−500, 500]
−418.9829  (32)

30
[5.12, 5.12]
−50

30
[−32, 32]
−80

30
[−600, 600]
−80

30
[−50, 50]
−80

where

The results of the proposed variants on the unimodal functions are shown in Tables 3 and 4 for 30 dimensions and 50 dimensions; similarly the results on the multimodal benchmark problems are shown in Tables 5 and 6 for 30 and 50 dimensions, respectively. For making the analysis meaningful, four different statistical parameters, namely, standard deviation (SD), maximum value (Max), minimum value (Min), and Mean value parameters are calculated. The stopping criterion for these variants along with GOA is the maximum iteration which is set to 500. Each variant is tested on all the ten benchmark functions, and the results are averaged over 20 independent runs. The following subsection presents the results of unimodal benchmark functions.

Table 3: Results and comparison of ECGOAs with GOA (30-D) on unimodal functions.

Algorithm
Statistical parameters
F₁
F₂
F₃
F₄
F₅

ECGOA1
Max
4.72E + 03
7.441252
32384.27
5997.771
11547.22

SD
1189.661
20.48785
8687.739
1448.595
2648.273

Mean
9.54E + 02
−34.0542
13561.55
519.9767
1515.979

Min
−33.212
−50
3212.748
−23.8636
−49.6495

ECGOA2
Max
3.67E + 03
6.983289
27021.89
774.8997
7411.605

SD
1070.123
15.60652
7021.802
235.0562
2190.903

Mean
8.65E + 02
−33.8791
14337.61
67.54067
1770.953

Min
−49.6441
−50
3949.875
−22.9774
−46.6215

ECGOA3
Max
2.64E + 03
−10
23806.04
359978.7
4481.247

SD
806.0854
14.47655
6705.574
80425.93
1249.42

Mean
5.10E + 02
−32.2934
11977.49
18315.61
891.0015

Min
−49.7058
−50
1076.156
−23.1032
−45.3564

ECGOA4
Max
1.70E + 03
7.013903
22406.58
5997.795
3592.651

SD
536.6934
15.66702
5003.504
2352.145
855.4461

Mean
5.19E + 02
−30.1622
11631.74
1257.859
479.4171

Min
−44.7163
−50
4640.106
−22.0574
−44.2489

ECGOA5
Max
5.67E + 03
−10.9219
48674.12
359978.5
6140.426

SD
1268.831
14.3114
11345.15
80389.73
1753.316

Mean
8.38E + 02
−38.4507
15310.28
18486.14
1587.872

Min
−40.7524
−50
4158.884
−22.8343
−49.772

ECGOA6
Max
4.56E + 03
27.11865
34182.77
359978.7
9207.069

SD
1504.872
19.65367
7717.328
80359.56
2300.953

Mean
1.26E + 03
−35.3059
14001.46
18656.76
1582.863

Min
−49.5997
−50
3692.072
−22.1018
−49.5038

ECGOA7
Max
4.63E + 03
−30
25403.45
774.9623
9922.265

SD
1378.628
8.3955
5469.37
333.3362
2460.44

Mean
1.21E + 03
−44.166
13289.92
179.7019
1421.978

Min
−45.0917
−50
5563.239
−22.918
−49.0028

ECGOA8
Max
1.68E + 03
5.974824
29394.74
359978.8
1992.826

SD
480.1784
16.12038
5952.928
80354.38
486.393

Mean
3.60E + 02
−34.3295
12586.33
18679.3
411.1577

Min
−49.5353
−50
4302.044
−23.1203
−49.105

ECGOA9
Max
6.40E + 03
27.15447
35257.01
263688.6
8310.101

SD
1522.377
20.47155
8866.673
58719
1997.682

Mean
956.9384
−34.7665
15903.13
14423.59
1573.284

Min
−29.154
−50
2960.342
−22.7203
7.090624

ECGOA10
Max
7.43E + 03
6.999618
28524.88
3813.648
4410.677

SD
1720.647
15.64801
6605.129
1176.846
1183.068

Mean
1042.276
−35.9717
10828.61
360.635
1071.854

Min
−30.8355
−50
1853.837
−23.0794
41.99444

GOA [16]
Max
6.52E + 03
−10
23228.38
5997.962
3775.516

SD
1578.947
11.67419
5972.024
1329.84
1181.988

Mean
1276.143
−38.4728
12189.96
554.6474
898.7971

Min
−43.7533
−50
1178
−22.9119
−49.7339

Table 4: Results and comparison of ECGOAs with GOA (50-D) on unimodal functions.

Algorithm
Statistical parameters
F₁
F₂
F₃
F₄
F₅

ECGOA1
Max
2.91E + 04
32.18483
111797.5
3033130912
3.24E + 04

SD
4580.993499
24.20682
20978.46
3033130912
4894.9252

Mean
2.03E + 04
6.261021
62956.04
3033130912
21270.264

Min
12525.67662
−31.5255
30301.64
−1.53368951
13898.103

ECGOA2
Max
3.48E + 04
106.5421
103711.7
2154795033
29074.924

SD
5707.583776
45.64063
18934.69
486469119.6
5100.4422

Mean
2.26E + 04
14.83906
68327.38
138349178.8
19007.027

Min
15420.5153
−50
38549.12
−1.24711837
10868.143

ECGOA3
Max
2.85E + 04
69.54283
113569.7
1206254731
31566.385

SD
4356.900319
31.4644
18027.51
393145232.4
5251.3516

Mean
2.00E + 04
14.19818
63156.77
189714077.2
23529.016

Min
14758.56528
−32.9833
40865.46
−4.86134464
15436.59

ECGOA4
Max
3.22E + 04
90
104517.8
2346293923
35774.413

SD
4196.200051
33.22862
20597.45
669971386.6
5791.7485

Mean
2.10E + 04
7.25466
59656.25
261272209.6
22008.399

Min
15427.28384
−50
24785.01
−1.42685917
15822.568

ECGOA5
Max
3.15E + 04
106.4788
109516.7
2513564142
36989.27

SD
5903.244549
32.80914
23233.04
857624011.7
6199.7488

Mean
2.13E + 04
8.691083
64561.99
558000246.9
23826.641

Min
14246.34036
−50
30868.1
−2.94514106
13771.658

ECGOA6
Max
2.87E + 04
70
85934.37
2670344909
36744.052

SD
5121.771356
27.96234
15565.56
872997453.1
5968.3631

Mean
2.04E + 04
16.29936
57717.67
397263587.7
24972.077

Min
10043.07676
−50
25295.89
−1.43401992
16032.898

ECGOA7
Max
3.47E + 04
48.67325
95502.4
2284274161
32038.771

SD
6057.849926
27.39903
17070.54
829809398.2
4873.0274

Mean
2.29E + 04
4.364473
60573.92
417129473.9
21394.938

Min
9880.378913
−50
30157.06
−2.83394271
11814.998

ECGOA8
Max
3.15E + 04
47.85565
137403.3
2664314422
37842.546

SD
5462.667635
22.65879
26818.08
895484803.7
6337.3582

Mean
2.16E + 04
2.030997
67721.54
410479727.9
23106.754

Min
8900.984581
−31.5151
32203.72
−1.36190421
14299.857

ECGOA9
Max
3.45E + 04
106.6678
102734.8
2841946390
30316.017

SD
5138.922636
38.41419
21197.83
765285734.9
4717.0682

Mean
21536.60375
7.865827
56321.14
356879478.5
23646.126

Min
11155.93754
−50
26036.76
−1.46623175
13613.975

ECGOA10
Max
3.03E + 04
50
122011.5
2270930776
33079.78

SD
5296.153654
24.70026
18910.63
618827002.9
4896.2693

Mean
21341.72611
9.529229
69260.53
288246084.2
22333.503

Min
14745.05282
−30
36566.91
−1.44162177
13425.011

GOA [16]
Max
3.55E + 04
86.94913
84051.88
3011759319
29750.024

SD
5820.921429
37.35558
14837.63
925046677.7
4977.283

Mean
22165.10941
10.92968
56877.2
525116848.9
21473.229

Min
13433.88239
−46.8306
37391.19
−1.44722832
9884.174

Table 5: Results and comparison of ECGOAs with GOA (30-D) for multimodal functions.

Algorithm
Statistical parameters
F₆
F₇
F₈
F₉
F₁₀

ECGOA1
Max
−8904.24
271.0372
−66.5449
−29.2767
3279130

SD
1254.676
51.23868
4.119399
13.23356
884472.8

Mean
−11029.5
146.0832
−78.6616
−51.4071
1238399

Min
−13826.5
66.54498
−80
−69.8688
−66.7927

ECGOA2
Max
−8832.74
229.4997
−66.7126
−28.9994
3958975

SD
1142.206
52.12368
2.975594
11.40269
1072203

Mean
−10745.6
145.3066
−79.171
−47.2001
1923464

Min
−13627.7
15.85951
−80
−69.4254
−68.3447

ECGOA3
Max
−8376
222.1649
−64.3491
−39.1735
2999948

SD
1527.962
47.29322
3.493484
10.75939
1038006

Mean
−11293.5
164.0275
−78.9705
−60.5568
1442602

Min
−14534.5
56.96255
−80
−78.9487
−66.9631

ECGOA4
Max
−8307.93
256.214
−78.3538
−19.0129
3999958

SD
1109.739
57.38026
0.506681
12.7704
1292593

Mean
−11006.9
141.5465
−79.6353
−51.0195
1278308

Min
−12818.6
53.7245
−80
−69.8415
−72.994

ECGOA5
Max
−8540.51
243.5495
−60.3452
−28.1305
4922217

SD
1220.617
48.63998
5.114423
13.8308
1398789

Mean
−11059.9
146.0103
−78.1109
−47.6149
1688204

Min
−13079.1
54.99126
−80
−69.8507
−75.9006

ECGOA6
Max
−8426.57
249.5214
−60.0117
−39.1139
2999975

SD
1345.302
44.48447
5.512268
11.35779
913600.2

Mean
−10663.5
147.7425
−78.1364
−54.092
758313.7

Min
−13323.2
72.3563
−80
−78.4344
−72.602

ECGOA7
Max
−7639.37
244.7659
−60.0035
−9.16557
3742472

SD
1637.129
40.17972
4.45385
15.04767
1044970

Mean
−10769.7
144.7726
−78.7532
−51.0586
1140919

Min
−15549.9
79.90368
−80
−69.5569
−73.8642

ECGOA8
Max
−8815.14
238.5191
−78.3538
−29.0018
3999953

SD
972.5037
39.69233
0.675579
11.76663
1258635

Mean
−11237
163.4965
−79.6707
−50.8158
1575727

Min
−12712
90.96168
−80
−69.9456
−68.8842

ECGOA9
Max
−7711.69
250.473
−78.3538
−19.8782
2999949

SD
1198.213
40.70405
0.675578
12.726
778483.5

Mean
−10681.1
167.9705
−79.6707
−41.31
1095574

Min
−12474.2
96.95393
−80
−69.4142
−64.8591

ECGOA10
Max
−8746.85
269.3226
−60.1722
−19.3277
5249683

SD
1190.82
41.32749
4.41648
12.70165
1582795

Mean
−10661.7
161.4352
−78.7616
−43.6748
1394908

Min
−13545.7
84.36102
−80
−69.0206
−61.9252

GOA [16]
Max
−9225.8
258.3774
−60.0393
−20.1744
3000918

SD
1292.871
50.35495
8.248127
12.85493
975411.2

Mean
−11237.6
161.2432
−75.4165
−57.6119
1122615

Min
−14022.9
76.48267
−80
−69.8943
−69.4996

Table 6: Results and comparison of ECGOAs with GOA (50-D) for multimodal functions.

Algorithm
Statistical parameters
F₆
F₇
F₈
F₉
F₁₀

ECGOA1
Max
−13284.9
525.3938
−59.4202
1142.297
5.99E + 08

SD
1468.401
61.31428
0.128399
376.3986
179942439

Mean
−16012.5
441.7274
−59.6647
474.3385
130982138.6

Min
−18346
296.6225
−59.8477
80.21982
9945376.637

ECGOA2
Max
−13222.8
568.819
−59.4939
1225.204
518183195.7

SD
1284.353
59.19765
0.114828
380.9211
161215153.4

Mean
−15729.6
431.9621
−59.682
526.118
111311331.7

Min
−17918.7
321.537
−59.9695
92.68036
8548635.153

ECGOA3
Max
−12054.8
615.8589
−59.4566
1189.236
717878837.1

SD
1475.55
70.58729
0.119339
325.6841
234394118.5

Mean
−15601.9
455.3608
−59.6446
488.7296
203256596.5

Min
−18405
313.2195
−59.8967
109.8307
8852551.366

ECGOA4
Max
−12155.6
486.4432
−59.4112
1128.317
802793748.5

SD
1175.43
50.69295
0.12958
343.5583
260492546.8

Mean
−15259.2
410.6172
−59.6654
417.0988
194442446.2

Min
−16864.5
316.0931
−59.8891
77.3793
7577441.415

ECGOA5
Max
−13473.6
536.6068
−59.4432
1395.387
506974142.9

SD
1503.746
49.9885
0.14537
383.249
146280518.5

Mean
−15499.8
429.7536
−59.7203
523.472
101913643.5

Min
−18340.6
353.849
−60.0317
96.59476
8061427.882

ECGOA6
Max
−12379
561.8097
−59.4954
1562.137
622102603.5

SD
1635.44
75.40271
0.115495
390.2075
201123569.1

Mean
−15574.9
431.8302
−59.6764
381.6959
152537664.3

Min
−18784
278.6123
−59.9099
72.10621
5058058.625

ECGOA7
Max
−13522.3
611.5379
−59.5858
924.5556
1019814442

SD
1291.999
81.91863
0.083408
253.7802
296529381.7

Mean
−16657.1
442.2678
−59.7265
360.7756
207773478.8

Min
−19647.9
283.6656
−59.8738
112.8999
6189273.462

ECGOA8
Max
−12906.6
522.0567
−59.4512
1035.483
469635197.1

SD
1467.799
53.46532
0.15148
291.7955
156859406.9

Mean
−15172.3
422.7514
−59.6563
401.8186
111764679.8

Min
−18254.9
290.0732
−60.062
96.79589
5845757.794

ECGOA9
Max
−13639.7
543.6966
−59.3504
1138.901
434575957.9

SD
1417.122
80.27476
0.119305
367.9628
173595240.7

Mean
−15904.4
426.5996
−59.6111
457.5753
130924283.2

Min
−18462.9
262.1963
−59.8748
94.65694
6641433.378

ECGOA10
Max
−13343.5
580.2786
−59.3751
940.1151
601504311

SD
1218.772
56.89642
0.144239
286.1847
172390624.5

Mean
−15770
460.4834
−59.7354
336.8918
119948547.1

Min
−18108
365.7742
−60.151
96.49957
11133246.78

GOA [16]
Max
−13843.7
556.5219
−59.4587
1254.326
524047875.8

SD
1266.274
61.51136
0.142779
341.6776
125063391.3

Mean
−16059.7
446.7562
−59.6615
528.9932
60674853.38

Min
−18377.6
283.9794
−60.076
113.8766
8321958.122

5.1. Qualitative Results and Discussions of Unimodal Benchmark Problems

Unimodal functions are the functions which have no local minima, or in other words, they possess only one minima. These functions are suitable for benchmarking the exploitation quality and convergence speed of any algorithm. This section presents the results on unimodal benchmark problems for 30 dimensions and 50 dimensions.

5.1.1. Simulation Results of 30-D Unimodal Benchmark Problems

The chaotic variants are benchmarked for the exploitation of quality and convergence properties on unimodal benchmark functions. The results for 30 and 50 dimensions are shown in Tables 3 and 4, respectively. For the unimodal functions, that is, F₁, F₂, and F₃, it is observed that, as per maximum values obtained for each variant, the lowest maximum value of function 1 is for ECGOA8 (1.68E + 03), and the mean value and the standard deviation value for this variant are also lowest, that is, 3.60E + 02 and 480.17. The convergence properties of this variant for function 1 is shown in Figure 4. It is observed that convergence properties of this variant is superior to others. Similarly, for functions 2 and 3, the algorithms ECGOA7 and ECGOA4 have the minimum standard deviation values. The differences in the minimum, maximum, standard deviation, and mean values are very marginal with other variants. Similarly, for function 4, ECGOA2 performs better than other variants as the values of three statistical parameters out of four are minimum. For function 5, again ECGOA8 provides better results as per the obtained Max, SD, and Mean values. The solution of function 1 by ECGOA8 is shown in Figure 5.

Figure 4: Convergence curve for function 1 (30-D).

Figure 5: Results of ECGOA8 for unimodal function 1.

5.1.2. Simulation Results of 50-D Unimodal Benchmark Problems

Further the analysis is carried out on 50-D unimodal functions. The results of all the developed variants with GOA on unimodal benchmark functions are shown in Table 4. From the careful inspection of the results, it is observed that for function 1, ECGOA8 possesses minimum values of statistical parameter Min. However, other parameters, namely, Max and Mean are low for ECGOA3. For function 2, the values of SD and Mean are optimal for ECGOA8. Hence, it can be concluded that this variant outperforms others for this particular function. For functions 3, 4, and 5, ECGOA9, ECGOA1, and ECGOA2 possess optimal mean values. From this analysis, it can be concluded that the exploitation capability of GOA has been substantially improved by the chaotic comfort zone function adaptation.

5.2. Qualitative Results and Discussions on Multimodal Benchmark Problems

In this section, experiments are carried out on multimodal functions. The results for both 30-D and 50-D problems are shown in Tables 5 and 6. Multimodal functions are those functions which possess one global minima and can have several local minima. The nature of these functions is used to benchmark the exploration quality of the proposed variants. This benchmarking exhibits the ability of the variants to search a global optimum, in such a challenging environment, where the probability of getting trapped in a local optimum is high. The bias and shift in conventional multimodal benchmark problems make the functions more complex and suitable for benchmarking the variants for the real-world engineering problems.

5.2.1. Simulation Results of 30-D Multimodal Benchmark Problems

Inspecting the results of the multimodal functions in Table 5, it is observed that ECGOA8 performs better as compared to other variants for function 6, as the value of the SD is low. It is evident to say that the values of these statistical parameters can be a meaningful indicator to judge the performance of variants. For function 6, the standard deviation value for ECGOA8 is 972.50 as compared with other variants. The standard deviation of GOA for this function is 1292.87. For function 7, this variant also performs better as compared to others as the standard deviation value for this variant is the lowest, that is, 39.69, and for GOA, it is 50.35. From the results, it can be concluded that ECGOA8 (Singer map-enabled chaotic mechanism) provides better results for first three multimodal functions. The convergence characteristics for function 7 are plotted in Figure 6. For function 8, again this variant shows promising results as the three out of 4 parameters are low as compared to other variants. For function 9, ECGOA3 performs better as the parameters associated with the judgement attain lower values. For function 10, ECGOA6 has least mean values. Hence, it can be concluded that for most of the functions these variants outperform GOA.

Figure 6: Convergence curve for multimodal function 7 (30-D).

5.2.2. Simulation Results of 50-D Multimodal Benchmark Problems

The results of this experiment are shown in Table 6. For function 6, the optimal value for parameter SD is attained for ECGOA4, and Mean and Min for ECGOA7; for function 9, the parameters Max, SD, and Mean attain optimal values for ECGOA7 variant. The optimal values of the statistical parameters are shown in boldface. Inspecting the results of this experiment on the multimodal functions, it is clearly evident that the exploration capability of the variants is enhanced substantially by employing the chaotic comfort zone functions. To judge the significance of the results, authors performed the Wilcoxon rank sum test [34] with 5% confidence interval. The test results ( values) are shown in Tables 7 and 8.

Table 7: Results of the Wilcoxon rank sum test on unimodal benchmark functions.

Function
F₁
F₂
F₃
F₄
F₅

ECGOA1
0.036048
0.169275
0.967635
0.6359
0.3683

ECGOA2
0.147847
0.013881
0.336915
N/A
0.0540

ECGOA3
0.881731
0.007431
0.989209
0.8817
0.9533

ECGOA4
0.228694
0.000986
N/A
0.1404
0.4755

ECGOA5
0.081032
0.166588
0.524987
0.4094
0.7590

ECGOA6
0.072045
0.088317
0.409356
0.9892
0.4340

ECGOA7
0.014364
N/A
0.310402
0.6554
0.9423

ECGOA8
N/A
0.04359
0.797197
0.5792
N/A

ECGOA9
0.15557
0.14484
0.163596
0.1719
0.4971

ECGOA10
0.126431
0.043738
0.490334
0.3793
0.9917

GOA [16]
0.014364
0.07045
0.755743
0.0036
0.0266

Table 8: Results of the Wilcoxon rank sum test on multimodal benchmark functions.

Function
F₆
F₇
F₈
F₉
F₁₀

ECGOA1
0.56
0.776391
0.27
0.03
0.08

ECGOA2
0.2
0.542772
0.73
8.36E − 04
8.36E − 04

ECGOA3
0.81
0.14042
0.0098
N/A
0.02

ECGOA4
0.59
0.655361
N/A
0.02
0.23

ECGOA5
0.88
0.507505
0.041
0.040
0.019

ECGOA6
0.22
0.71498
0.0114
0.0909
N/A

ECGOA7
0.19
0.126431
0.525
0.0337
0.085

ECGOA8
0.9892
N/A
0.2503
0.0207
0.02

ECGOA9
0.23
0.113551
0.285
5.85E − 06
0.0601

ECGOA10
0.14
0.163596
0.6554
1.99E − 04
0.081

GOA [16]
N/A
0.208454
0.239
0.0076
0.1719

5.2.3. Discussion

Parameter is an important parameter and acts as a bridging mechanism between exploration and exploitation phases. This parameter ensures the swift movement of grasshoppers from exploration phase to exploitation phase by reducing the comfort zone of grasshoppers. Chaotic mechanism not only enhances the exploitation phase by keeping alive the virtue of exploration till the last iteration but also adds random behavior with each iteration on the basis of different adaptive chaotic comfort zone function-enabled mechanisms. Further, the following section presents the statistical analysis of the performance of these variants.

5.3. Wilcoxon Rank Sum Test

To judge the significance of the results, the Wilcoxon rank sum test [34] is performed with 5% significance interval, and the values are obtained. The test results are shown in Tables 7 and 8. The term N/A indicates that the variant has outperformed over others and cannot be compared with itself. From the results of function 1 (Table 7), it is observed that there is a significant difference between ECGOA1 and ECGOA7 as compared with ECGOA8 as the values obtained for these variants are less than 0.05. Similarly, for function 3, ECGOA4 is the best algorithm; however, in this category, all the algorithms are statistically not different from each other as the values are greater than 0.05. It is empirical to observe that the lowest value other than ECGOA4 is for ECGOA9. For function 2, it is observed that ECGOA7 is the best performer and a significant difference exists between ECGOA2, 3, 4, 8, and 10 variants as the values are less than 0.05. The values which are less than 0.05 are highlighted in boldface and underlined.

Inspecting the results of Table 8 for multimodal functions, ECGOA8 is the best performer, and for functions 4 and 5, variants ECGOA2 and ECGOA3 are the second best performers as per the values. For function 6, it is observed that a significant difference exists between variants ECGOA3, ECGOA4, and ECGOA5. In most of the functions, chaotic variants outperform GOA and the variants are significantly different from each other. In the following section, the application of these variants on real-world problems and comparative performance of the variants with other contemporary algorithms is presented.

5.4. Application of the ECGOAs on Real Applications

Application of these variants on structural design-constrained optimization problem and parameter estimation problem is investigated in this section. The impact of the chaotic bridging mechanism in reducing the comfort zone of the grasshoppers that results in the better exploration and exploitation properties of GOA are evaluated with these applications.

5.4.1. Three-Bar Truss Design Problem

Three truss bar design problem is a well-known engineering design problem and has been used for benchmarking of many problems [14–16]. A schematic diagram of this problem is shown in Figure 7. The objective of this problem is to minimize the volume (X) by adjusting the cross-sectional area (x, y) as per (16) subject to the constraints [17–19]. This objective function is nonlinear in nature and possesses three nonlinear constraints which contain the stress parameter. For solving this optimization problem, the number of search agents (30) and maximum iteration count (500) are considered and kept constant for all the variants. Each algorithm is run for 20 times and results shown in Table 9 are averaged over these runs. The convergence curve of the problem is shown in Figure 8. The expression for the volume is given as

Figure 7: Three truss bar design problem.

Table 9: Results of three truss bar design problem.

Algorithm
Max
Mean
Min
SD

ECGOA1
266.5823
264.5394
263.8976
0.851265

ECGOA2
268.235
264.6162
263.8974
1.107169

ECGOA3
268.0221
264.5413
263.896
1.202524

ECGOA4
266.4968
264.3473
263.8961
0.76741

ECGOA5
265.6408
264.2707
263.8963
0.513983

ECGOA6
266.5492
264.4754
263.897
0.739352

ECGOA7
268.1783
264.3391
263.8971
0.936519

ECGOA8
265.4146
264.1322
263.8965
0.401496

ECGOA9
268.8148
264.3604
263.8962
1.108937

ECGOA10
266.0352
264.2575
263.8961
0.562701

GOA [16]
265.528
264.3357
263.3274
0.57364

Figure 8: Convergence curve for three truss bar design problem.

Various parameters for this optimization problem have been considered as , , and with variable range .

The results of this problem are shown in Table 9, and it is observed that, for this design problem also, ECGOA8 variant outperforms others as the values of the standard deviation and other statistical parameters are optimal as compared with other opponents. This variant exhibits better convergence properties; for the sake of clarification, the convergence curve is shown in Figure 8.

5.4.2. Parameter Estimation for Frequency-Modulated Sound Waves

Parameter estimation of the frequency-modulated synthesizer is a six-dimensional optimization problem and a part of FM sound wave synthesis. The problem is formulated as the parameter estimation for generation of the sound as per the target sound. The problem is complex and multimodal in nature. The minima for the objective function is at zero. The parameter vector is estimated through the optimization process [35]. The vector has 6 parameters as per the following equation:

The expressions for the estimated and the target sound waves are as follows:

The results of this problem are shown in Table 10, and the convergence of the variants along with GOA is shown in Figure 9. To solve this optimization problem, the maximum number of function evaluations and the number of search agents are set to be 30,000 and 30, respectively. The optimization results are averaged over 30 independent runs. It has been observed that the performance of these variants is competitive with some of the recently published approaches. ECGOA1 possesses the minimum SD parameter value as compared with others. From these applications, it can be concluded that the variants show the competitive performance not only on shifted and biased benchmark functions but also on the real applications. In the following section, the conclusions drawn from this study are presented.

Table 10: Error in objective function values for frequency-modulated sound waves synthesis [35].

Algorithm
Min
Max
Mean
SD

ECGOA1
14.048
26.720
20.748
2.754

ECGOA2
8.416
25.560
20.266
4.067

ECGOA3
11.407
27.283
20.652
4.335

ECGOA4
1.4E − 7
27.280
19.863
5.439

ECGOA5
8.416
27.354
20.708
4.771

ECGOA6
8.416
26.522
19.687
5.183

ECGOA7
10.177
25.913
18.830
4.596

ECGOA8
0.000
26.999
19.108
5.710

ECGOA9
11.549
27.123
20.896
4.064

ECGOA10
13.393
27.462
21.266
3.890

GOA [16]
8.416
26.745
20.110
4.673

CPSOH [36–38]
3.45
42.52
27.08
60.61

GWO [13]
1.9311
20.03
25.1633
5.9177

TRIBES-D [38–40]
2.22
22.24
14.68
4.57

CGSA [28]
8.4161
24.71
17.43
4.1609

G-CMA-ES [36, 38, 39]
3.326
55.09
38.75
16.77

Figure 9: Convergence curve for frequency synthesis parameter estimation problem.

6. Conclusion

Exploration and exploitation phases of a metaheuristic algorithm are connected with a bridging mechanism. The efficacy of this bridging mechanism is important to achieve better convergence characteristics, solution quality, and optimization performance. This paper focuses on this mechanism, and 10 chaotic bridging mechanisms have been proposed for GOA. Following are the major highlights of this work:

(1)10 different chaotic maps have been embedded with the conventional GOA parameter “”, and the chaotic mechanism has been proposed. The benefit of this mechanism is that it enables exploration phase till last iteration with chaotic properties.(2)Ten shifted and biased benchmark functions have been considered to benchmark the variants. The proposed variants have been evaluated on 30-dimensional and 50-dimensional benchmark problems. It has been observed that the mechanism which is enabled with the Singer chaotic map, that is, ECGOA8 is suited for unimodal and multimodal optimization problems.(3)Further the application of these variants on three truss bar design problem and parameter estimation of the frequency-modulated sound wave synthesis problem have also been investigated. It is observed that the performance of the developed variants is competitive with other contemporary algorithms. In some cases, variants outperform.(4)A nonparametric Wilcoxon rank sum test has been conducted, and the values have been obtained for all the ten functions. It has been concluded that variant ECGOA8 exhibits better results as compared with other opponents.

For further studies, it would be interesting to explore the application of different comfort zone reduction functions to improve the bridging mechanism between exploration and exploitation phases of GOA.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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