A hybrid GSA-GA algorithm for constrained optimization problems

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Nov 20, 2018 - School of Mathematics, Thapar Institute of Engineering ... tuned up with the gravitational search algorithm and then each solution is upgraded ...
Information Sciences 478 (2019) 499–523

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A hybrid GSA-GA algorithm for constrained optimization problems Harish Garg1 School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University Patiala-147004, Punjab, India

a r t i c l e

i n f o

Article history: Received 24 March 2018 Revised 13 November 2018 Accepted 17 November 2018 Available online 20 November 2018 Keywords: Constrained optimization GSA-GA algorithm Engineering design problems Genetic algorithm Gravitational search algorithm

a b s t r a c t In this paper, a new hybrid GSA-GA algorithm is presented for the constraint nonlinear optimization problems with mixed variables. In it, firstly the solution of the algorithm is tuned up with the gravitational search algorithm and then each solution is upgraded with the genetic operators such as selection, crossover, mutation. The performance of the algorithm is tested on the several benchmark design problems with different nature of the objectives, constraints and the decision variables. The obtained results from the proposed approach are compared with the several existing approaches result and found to be very profitable. Finally, obtained results are verified with some statistical testing. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Engineering design problems (EDPs) is one of the huge and challenging area in the field of designing the shape of the designs more accurately. In general, these EDPs considered as a nonlinear function which several constraints and treated as a nonlinear constraint optimization problems (COPs). To solve the COPs, mathematical programming and metaheuristic methods are the two groups widely used by the researchers. However, the mathematical programming methods (or gradientbased methods) have the faster rate of convergence with higher accuracy to solve such COPs but they needs the gradient information, the initial starting values and continuity of the function [27,38]. Due to the complexity of the system, it is difficult to compute the complex derivatives and provide the initial trial solution (which will affect the iteration size also) and hence researchers are depending on metaheuristic algorithms (MHAs) to solve the COPs [20]. MHAs are intense methods and have shown remarkable results to solve the COPs in which they utilize the probabilistic transition rules rather than deterministic rules to upgrade the solutions. In EDPs, researchers have applied the MHAs such as “differential evolution” (DE) [34,50], “harmony search” (HS) [31], “genetic algorithm” (GA) [15], “particle swarm optimization” (PSO) [20], “cuckoo search” (CS) [16], “Firefly algorithm” (FA) [17], “artificial bee colony” (ABC) [1,18], “ant colony optimization” (ACO) [29], “Memetic algorithm” (MA) [6,7,36], “gravitational search algorithm” (GSA) [41], “Differential Evolution” (jDE) [4] and some other hybrid approaches [19] to solve the COPs. The two fundamental tasks for any MHAs are to find an efficient solution by updating their previous ones using operations. Secondly, it ensures that the algorithm can investigate the search space more proficiently as opposed to by randomization. Despite the fact that the MHAs reveals a huge execution in comparison with the traditional strategies, regardless they may have some shortcoming, for example, the lower robustness, premature convergence and their uneven exploration/exploitation drifts in some complex cases. In order

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https://doi.org/10.1016/j.ins.2018.11.041 0020-0255/© 2018 Elsevier Inc. All rights reserved.

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H. Garg / Information Sciences 478 (2019) 499–523

to diminish these, a couple of contemplations have moved to reproducing through using the hybridization techniques in tackling advancement issues. Out of the current MHAs, GA and GSA are the typical ones which have been applied effectively to solve the COPs. GA [15] is a versatile heuristic searching algorithm introduced on the evolutionary thoughts of selection and genetics. Under this algorithm, each particle is updated with the set of the selection, crossover and mutation operators to avoid the premature convergence. In spite of the fact that with low converging rate, GA is versatile, self-learning, and equipped for worldwide improvement. On the other hand, GSA [41] is one of the MHAs based on the Newtonian law of gravity and motion. In GSA, each solution is updated by moving stochastically. The advantages of robustness, effortlessness and flexibility of GSA make it conceivable to be applied broadly in various COPs. In spite of the fact that GSA simple to accomplish convergence, it is more likely becomes inactive. To enhance the effectiveness of the algorithm, some modifications like quantum-inspired GSA [37], opposition based GSA [45] are presented by the researchers. Although GSA gives some sufficient results, it is liable to diverge and trap into the local optima. To overcome this, some different forms of GSA in terms of the hybridization are presented such as neural network-GSA [46], GA-GSA [19], etc. Since, GA and GSA are the two widely used algorithms to solve the COPs. However, both these algorithm have their own advantages and drawbacks. For instance, in GA, an information for not selected individual is completely lost during the execution while GSA has the own memory. On the other hand, in GAs, the solutions are updated towards the optimal one by using generic operators while GSAs don’t have these operators. However, in practical applications, the particles during the execution of the algorithm may lose diversity rapidly due to parameter setting and implementation of the algorithm and hence lose its ability to explore the searching space. Thus, we conclude that the GSA algorithm works well at the early stage to get the solution while it has trapped down in the later stage. Therefore, there is a need to add some new operators into GSA to increase the efficiency in solving various nonlinear benchmark functions. In the present article, by combining the features of GA and GSA algorithms, we present a hybrid algorithm called as GSA-GA. In this algorithm, the performance of the GSA algorithm to find the optimal solution is improved by adding the genetic operators such as the selection, crossover and mutation. In other words, the exploration and exploitation abilities of the GSA algorithm is improved by embedding the genetic operators into it. The applicability of the proposed approach has been tested on the by using nine commonly used benchmark EDPs from the literature which contain multimodal, mixedvariable, integer COPs. From the computed results, it is verified that the proposed GSA-GA algorithm performs quite well as compared to the several existing algorithms and validate it by means of some statistical measures. The rest of the manuscript is summarized as follows. In Section 2, we present the entire notations and the description of the COPs. In Section 3, we elaborate the proposed GSA-GA algorithm along with brief overview of the GA and GSA algorithms. Section 4 describes the various benchmark COPs and their computational results are compared with the several existing methods. Finally, Section 5 concludes the work. 2. Nonlinear constraint optimization problems In this section, we briefly overview the nonlinear COPs. 2.1. Notations In this paper, we used the notations which are stated as below. Symbol

Description

Symbol

Description

f

Objective function

Mai

Active gravitational mass

n

Number of decision variable

M pi

Passive gravitational mass

y

= (y1 , y2 , . . . , yn ) be decision vector

N

No. of agents

gj

jth constraint

G(t )

Gravitational constant at time t

q

No. of constraints

T

Total number of iterations

S

Feasible search space

vi

Velocity of the ith agent

d

Dimension of the problem

acci

Acceleration of the ith agent

GAi

Current GA iteration

GSAi

Current GSA iteration

GANumMax

Maximum numbers of agent affected by GA

GSAmaxiter

Maximum iteration of GSA

GANumMin

Minimum numbers of agent affected by GA

γ

Decreasing rate of number of agents that affected by GA

GAPS

Population size of GA

GSAPS

Population size of GSA Last population size of GA

GAMinPS

First population size of GA

GAMaxPS

GAminiter

First iteration number of GA

GAmaxiter

Last iteration number of GA

δ

Increasing rate of GA population size

β

Increasing rate of GA maximum iteration number

H. Garg / Information Sciences 478 (2019) 499–523

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2.2. Constraint optimization problem The general mathematical model of the nonlinear COPs which contains several mixed decision variables is expressed as

Minimize f (y ) subject to g j (y )  0

;

yli  yi  yui

j = 1, 2, . . . , q ;

(1)

i = 1, 2, . . . , n

where y = (y1 , y2 , . . . , yn )T denotes the “decision solution vectors”; f is “the objective function”; yli and yui are the bounds of the ith variable and q is “the number of constraints”. Since, for COPs, penalty function is one of the most representative paradigm for handling the constraints. In this paper, we use the parameter-free penalty function [12] to transform the objective function f into the modified function F as follows



F (y ) =

f (y ) f w (y ) +

q 

; if y ∈ S g j (y ) ; if y ∈ /S

(2)

j=1

where S is the “feasible search space” and fw is the “worst feasible solution” in the population. 3. Proposed GSA-GA approach In this section, we present a hybrid GSA-GA algorithm by combining the features of GA and GSA algorithms. A brief description of these algorithms are summarized as below. 3.1. Genetic algorithm (GA) GA is a probabilistic search technique used in the computing to find the exact or near to exact optimal solution to search problems [15]. Inspired from Darwin’s theory of evolution, Fogel et al. [15] introduced the principle of GAs. It works by the evolutionary principles and chromosomal processing in natural genetics. In GAs, the solutions called as chromosomes are updated by using the selection, crossover and mutation operations to reach towards the optimal solution. Under this algorithm, the selection operation is applied to find the best candidate according to the fitness value and then the crossover and mutation operations are applied to update the solution. As a functioning characteristics of GAs, elitism provides a means of selecting the best individuals for reproduction and replacing their offspring in the population if and only if they are better than the weakest one. The procedure for implementing this algorithm is given in Algorithm 1. Algorithm 1 Pseudo code of Genetic algorithm (GA). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

Objective function: f (x ). Define Fitness F. Initialize population, crossover ( pc ) and mutation ( pm ) rates. Compute the fitness value of each individual. Elitist Population: preserve best individuals of each individual. repeat Generate new solution by crossover and mutation if pc >rand, Crossover; end if if pm >rand, Mutate; end if Accept the new solution if its fitness better. Select the current best for the next generation. until requirements are met

3.2. Gravitational search algorithm (GSA) The GSA [41] is a MHA inspired by Newton’s basic physical theory that states that “a force of attraction works between every particle in the universe and this force are directly proportional to the product of their masses and inversely proportional to the square of the distance between their positions”. In the GSA, agents are considered as objects which attract each other each by a gravity force. Each agent consists of position, mass, active and passive gravitational masses. According to Rashedi et al. [41], consider a swarm of N agents, the position of the ith agent (i = 1, 2, . . . , N ) is defined as

yi = (y1i , y2i , . . . , ydi , . . . , yni ), where ydi is the position of ith agent in dth dimension in the n− dimensional search space.

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The gravitational force acting on ith agent due to jth agent at tth iteration is defined as

Fidj (t ) = G(t )

 M pi (t ) × Ma j (t )  d y j (t ) − ydi (t ) Ri j + ∈

(3)

where Maj is the “active gravitational mass” related to agent j, Mpi is the “passive gravitational mass” related to agent i, ∈ is very small constant, Rij is the “Euclidean distance between the two agents i and j”. The gravitational constant G(t) is reduced with time to control the search accuracy and given by

G(t ) = G0 e−α T t

(4)

Here, α and G0 are user defined “descending coefficient” and “initial value”, respectively, and T is the “total number of iterations”. Thus, the total force acting on ith agent is given by

Fid (t ) =

Kbest 

rand j · Fidj (t )

(5)

j=1 j=i

where “Kbest” is the set of first K agents with better fitness, randj is a random number between 0 and 1. Therefore, by law of motion, the acceleration, accdi (t ), of the agent i at time t in dth dimension is given by:

accdi (t ) =

Fid (t ) , Mii (t )

d = 1, 2, . . . , n

(6)

where Mii (t) is the inertia mass. The velocity and position update equations of the ith agent at t iteration in the dth dimension are given below

vdi (t + 1 ) = randi · vdi (t ) + accdi (t )

(7)

ydi (t + 1 ) = ydi (t ) + vdi (t + 1 )

(8)

where vdi (t ), vdi (t + 1 ) and ydi (t ), ydi (t + 1 ) are the velocities and positions of agent yi at the t and t + 1 time respectively and d dimension. To sum up, the exhaustive procedure of the GSA algorithm has summarized in Algorithm 2. Algorithm 2 Pseudo code of Gravitational Search Algorithm (GSA). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:

Objective function: f (x ), x = (y1 , y2 , . . . , yd ); Set the constant α , G0 and maxiter for the algorithm. Initialize population. repeat Update G by using Eq. (4). Calculate the acceleration of each particle using Eq. (6). Calculate particle velocity by Eq. (7). Calculate particle position by Eq. (8). Evaluate the fitness function. until requirements are met.

3.3. Hybrid GSA-GA algorithm Due to the merits of GSA such as robustness, adaptability and simplicity, it is widely used to solve COPs. Moreover, GSA experiences the inalienable hindrances of trapping in local minima and the moderate convergence rates that decrease the solution quality. On the other hand, in GAs, the solutions are updated by using operations such as selection, crossover and mutation to avoid the premature convergence. Keeping the merits of both algorithms, a new hybrid algorithm named as GSA-GA is proposed to solve the COPs. In it, the performance of GSA algorithm is improved by adding the crossover and mutation operators of GA in terms of solution quality. The algorithm starts with the initialization phase where the agents of the decision variables are constructed randomly between the bounds of the variables. After the initialization phase, each agent of the particles is updated with the mechanism of the GSA algorithm towards the optimal solution. For it, compute the fitness “fit” value of each population and the gravitational force by using Eq. (3). During the updating, masses of all agents at time t are computed by

mi (t ) =

fiti (t ) − worst(t ) best(t ) − worst(t )

(9)

H. Garg / Information Sciences 478 (2019) 499–523

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where best(t) and worst(t) are “the best” and “the worst” fitness value of the objective function at the time t. Assume that M pi = Ma j = Mii = Mi for i = 1, 2, . . . , N, and are defined as

mi (t ) . N  mi (t )

Mi (t ) =

(10)

i=1

The velocity and position of all agents are updated by Eqs. (7) and (8) respectively. Therefore after an iteration, a new population of GSA algorithm is formed and apply GA to update some selected agents by using the generic operators. The two critical questions would be raised: How many and which agents should be selected from the population of GSA to update? As the size of the GSA population is huge so to save the time, we select the agents that are affected by GA by using the number GANum , defined in Eq. (11), and then GA algorithm is applied on each agent separately.

GANum = GANumMax −



γ GSAi (GANumMax − GANumMin ) GSAmaxiter

(11)

where GSAi represent the “current GSA iteration”, GSAmaxiter denote the “maximum iteration of GSA”, GANumMax and GANumMin represents the maximum and minimum numbers of agents that are effected by GA. Based on this number, the agent of the population (xk ) is selected for GA updating is lies between the first and “MaxIndex” where “Max-Index” is defined as



Max-Index = GANum + 1 −



GSAi (GSAPS − GANum ) GSAmaxiter

(12)

where, GSAPS represent the population size of GSA. After selecting the agents for updating, new agents are creating by applying the genetic operators such as selection, crossover and mutation. However, the best solution is preserved by performing an elitism with the equation



xk+1 =

xk yk

; if fitness value improves at xk ; otherwise

(13)

Finally, the population size as well as the iteration number for the GA algorithm changes with the change of the iteration of GSA to diminish time and memory usage which are defined by the following relations:

GAPS = GAminPS +



GAiter = GAminiter +

δ GSAi (GAmaxPS − GAminPS ) GSAmaxiter



β GSAi (GAmaxiter − GAminiter ) GSAmaxiter

(14)

(15)

where GAminPS , GAmaxPS be the first and last population size in GA. On the other hand, GAmaxiter , GAminiter be the first and last iteration number of GA and δ , β represents the increasing rate of GA population size and maximum iteration respectively. From these equations, it is clearly seen that the population size and iteration number are increases with the algorithm iterations. This complete procedure has been repeated until we get the desired met. To sum up, the flowchart of the proposed algorithm is given in Fig. 1. 4. Illustrative examples In this section, to evaluate the performance of the proposed algorithm, nine well-known benchmark structural EDPs are used. For it, different models with some linear and nonlinear constraints have been taken from the literature and compared their performance with the proposed algorithm. The population size of the algorithm is set to be 20 × d where d is the dimension of the problem. Algorithm of each problem is repeated 30 times to eliminate the stochastic discrepancy after randomly generated population. The experiment is implemented in Matlab (MathWorks). The other parameters are set to be GSA parameters: The constants ∈ , G0 and α are set to 10−100 , 100 and 20 respectively, while the initial values of other variables are zero. GA parameters: One-point crossover with rate 0.9, mutation rate is 0.01, roulette wheel selection is employed to update the chromosomes. GSA-GA parameters: The randomly chosen control parameters are γ = 2, δ = 15, β = 15, GAMinPS = 10, GAMinIter = 10, GANumMin = 1, GANumMax = 20. 4.1. Himmelblau’s nonlinear optimization problem The one of a well-known benchmark non-linear COP is Himmelblau’s problem which was originally proposed by Himmelblau [24]. The task of this problem is to find the decision vector Y = [y1 , y2 , y3 , y4 , y5 ] which minimize the function f stated in the optimization model as

Minimize f (Y ) = 5.3578547y23 + 0.8356891y1 y5 + 37.293239y1 − 40792.141

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H. Garg / Information Sciences 478 (2019) 499–523

Fig. 1. Flow chart of GSA-GA algorithm.

H. Garg / Information Sciences 478 (2019) 499–523

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Fig. 2. Pressure vessel design problem.

s.t.

0 ≤ g1 (Y ) ≤ 92 90 ≤ g2 (Y ) ≤ 110 20 ≤ g3 (Y ) ≤ 25

where

g1 (Y ) = 85.334407 + 0.0056858y2 y5 + 0.0006262y1 y4 − 0.0022053y3 y5 g2 (Y ) = 80.51249 + 0.0071317y2 y5 + 0.0029955y1 y2 − 0.0021813y23 g3 (Y ) = 9.300961 + 0.0047026y3 y5 + 0.0012547y1 y3 + 0.0019085y3 y4 78 ≤ y1 ≤ 102 ;

33 ≤ y2 ≤ 45 ;

27 ≤ y3 , y4 , y5 ≤ 45

Previously, this problem was solved by several researchers [12,14,16,18,20,22,24,32] using different methods, while some of the researchers [9,10,18,20,47] solved their modified version (named as Version II) in which entry 0.0 0 06262 in the first constraint is modified to 0.0 0 026. Here, we solved both the versions with the proposed GSA-GA approach and compared their results with these existing approaches in Table 1. The obtained optimal design variables by the proposed algorithm under version I is Y = [77.960999, 32.991971, 29.9921261, 45.00, 36.7941348] while for version II is Y = [77.961, 32.99948, 27.0728355, 45.00, 44.973943] with objective function values −30668.004045 and −31027.6407622 respectively. From the table, it is concluded that results obtained by the proposed approach show superior to the existing ones. Furthermore, we also observed that the solution given by authors in [10,22,47] are infeasible as it violated g1 constraint. On the other hand, the best, the worst and the average of the solution obtained after 30 independent runs are summarized in Table 2 and conclude that the proposed GSA-GA algorithm work well over the existing approaches and able to find the best solution more accurately. 4.2. Pressure vessel design problem In this problem, consider a cylindrical vessel which is mounted by hemispherical balls on both ends and is combined by two longitudinal welds to formulate a cylinder, shown in Fig. 2 [27]. In it, the objective is to minimize the overall cost of the pressure vessel by considering the four decision variables namely “thickness of the pressure vessel (Ts )”, “thickness of the head (Th )”, “inner radius of the vessel (R)”, and “length of the vessel without heads (L)”. Therefore, the optimization model for this design problem with decision vector Y = [Ts , Th , R, L] = [y1 , y2 , y3 , y4 ], is given as follows:

Minimize f (Y ) = 0.6224y1 y3 y4 + 1.7781y2 y23 + 3.1661y21 y4 + 19.84y21 y3 s.t.

g1 (Y ) = −y1 + 0.0193y3 ≤ 0 g2 (Y ) = −y2 + 0.00954y3 ≤ 0 4 g3 (Y ) = −π y23 y4 − π y33 + 1, 296, 0 0 0 ≤ 0 3 g4 (Y ) = y4 − 240 ≤ 0

In the literature, two different version of this problem has been considered which are defined as: Region I: The bound of the variables are

1 × 0.0625 ≤ y1 , y2 ≤ 99 × 0.0625 ;

10 ≤ y3 , y4 ≤ 200

Region II: In the region I, if bound y4 ≤ 200 is used then the g4 (Y) constraint is automatically satisfied. Thus, for more elaborated way, the researchers [14,17,18,31] have extended the upper limit of y4 to 240 i.e., 10 ≤ y4 ≤ 240. Therefore, under this region II, we have bound of the variables are

1 × 0.0625 ≤ y1 , y2 ≤ 99 × 0.0625 ;

10 ≤ y3 ≤ 200 ;

10 ≤ y4 ≤ 240

The results obtained for these two regions are computed by the proposed algorithm and compare with the results gave by authors [1,5,8,10,11,16,18,20–22,27–29,33,34,44] for region I and authors [14,17,18,31] for region II in Table 3. From this

506

Table 1 Optimal results for Himmelblau’s nonlinear optimization problem (NA means not available).

I

II

a

Design variables

MPId

Constraints

Methods

y1

y2

y3

y4

y5

f(Y)

0 ≤ g1 ≤ 92

90 ≤ g2 ≤ 110

20 ≤ g3 ≤ 25

Himmelblau [24] Deb [12] He et al. [22] Dimopoulos [14] Gandomi et al. [16] Mehta and Dasgupta [32] Garg [18] Garg [20] Present study Shi and Eberhart [47] Coello [9] Coello [10] Garg [18] Garg [20] Present study

NA NA 78.00 78.00 78.00 78.00 78.00 78.00 77.960999 78.00 78.5958 78.0495 78.00 78.00 77.961

NA NA 33.00 33.00 33.00 33.00 33.00 33.00 32.991971 33.00 33.01 33.007 33.00 33.00 32.99948

NA NA 29.995256 29.995256 29.99616 29.995256 29.99516951 29.9951741 29.9921261 27.07099 27.6460 27.081 27.07097927 27.0709505 27.0728355

NA NA 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00

NA NA 36.7758129 36.775813 36.77605 36.775813 36.77574688 36.7757340 36.7941348 44.969 45.0 0 0 0 44.94 44.96902388 44.9691668 44.973943

−30373.9490 −30665.539 −30665.539 −30665.54 −30665.233 −30665.538741 −30665.566806 −30665.56614 −30668.004045 −31025.561 −30810.359 −31020.859 −31025.575692 −31025.57471 -31027.6407622

NA NA 93.28536a 92.0 0 0 0 0 91.99996 NA 91.999998 91.99999 91.999701 93.28533a 91.956402 93.28381a 91.999974 91.99999 91.99970

NA NA 100.40478 98.84050 98.84067 NA 98.840473 98.84047 94.912286 100.40473 100.54511 100.40786 100.404731 100.40476 97.204181

NA NA 20.0 0 0 0 0 20.0 0 0 0 0 20.0 0 03 NA 20.0 0 0 0 0 20.0 0 0 20.0 0 0 19.99997a 20.251919 20.00191 20.0 0 0 0 02 20.0 0 0 20.00

Violate constraints;

b

f ( proposed ) Infeasible solution; d Maximum possible improvement(MPI) = f (otherf)(−other )

−0.96812% -0.00804% -b −0.00804% −0.00904% −0.00804% −0.00795% −0.00795% -b −0.70522% -b -0.00666% -0.00666%

H. Garg / Information Sciences 478 (2019) 499–523

Version

H. Garg / Information Sciences 478 (2019) 499–523

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Table 2 Statistical results for the Himmelblau’s problem (NA means not available). Version

Methods

Best

Median

Mean

Worst

Std

I

Deb [12] He et al. [22] Dimopoulos [14] Gandomi et al. [16] Mehta and Dasgupta [32] Garg [18] Garg [20] Present study Shi and Eberhart [47] Coello [9] Coello [10] Garg [18] Garg [20] Present study

−30665.537 −30665.539 −30665.54 −30665.2327 −30665.538741 −30665.56680546 -30665.566141 −30668.004045 -31025.561 -30810.359 −31020.859 −31025.57569195 −31025.574717 −31027.6407622

−30665.535 NA NA NA NA −30665.49388961 -30665.487911 −30667.51076 NA NA −31017.21369 −31025.5612911 −31025.561141 −31026.28126

NA −30643.989 NA NA NA −30665.40461198 -30665.398373 −30667.17208 NA NA −30984.240703 −31025.55841263 −31025.557816 −31026.07246

−29846.654 NA NA NA NA −30664.62469625 -30664.624696 −30665.32463 NA NA −30792.407737 −31025.49205458 −31025.492054 −31025.38705

NA 70.043 NA 11.6231 NA 0.2383866 0.24084 0.23158 NA NA 73.633536 0.0153528 0.01526 0.01803

II

Table 3 Comparison of the best solution for pressure vessel design problem found by different methods. Region

I

II

d

Design variables

Sandgren [44] Kannan and Kramer [27] Coello [10] Coello and Montes [11] He and Wang [21] Montes and Coello [33] Kaveh and Talatahari [28] Kaveh and Talatahari [29] Cagnina et al. [5] Coelho [8] He et al. [22] Montes et al. [34] Gandomi et al. [16] Akay and Karaboga [1] Garg [18] Present study Dimopoulos [14] Mahdavi et al. [31] Gandomi et al. [17] Garg [18] Present study

Cost

y1

y2

y3

y4

f(Y)

1.1250 0 0 1.1250 0 0 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.812500 0.7781977 0.77819652 0.75 0.75 0.75 0.72759583 0.727610046

0.6250 0 0 0.6250 0 0 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.437500 0.3846656 0.3846644 0.375 0.375 0.375 0.359655288 0.359658023

47.70 0 0 0 0 58.2910 0 0 40.323900 42.097398 42.091266 42.098087 42.103566 42.098353 42.098445 42.098400 42.098445 42.098446 42.0984456 42.098446 40.32105455 40.3210580446 38.86010 38.86010 38.86010 37.69913599 37.70 0 0 0238

117.7010 0 0 43.690 0 0 0 20 0.0 0 0 0 0 0 176.654050 176.746500 176.640518 176.573220 176.637751 176.6365950 176.637200 176.6365950 176.6360470 176.6365958 176.636596 199.9802367 199.9799646 221.36549 221.36553 221.36547 239.999805 239.983428

8129.1036 7198.0428 6288.7445 6059.946 6061.0777 6059.7456 6059.0925 6059.7258 6059.714335 6059.7208 6059.7143 6059.701660 6059.7143348 6059.714339 5885.4032828 5885.38533633 5850.38306 5849.76169 5850.38306 5804.4486708 5804.4048008

MPId

27.60105% 18.23631% 6.41399% 2.88056% 2.89870% 2.87735% 2.86688% 2.87704% 2.87685% 2.87696% 2.87685% 2.87665% 2.87685% 2.87685% 0.0 0 030% 0.78590% 0.77536% 0.78590% 0.0 0 076%

f ( proposed ) Maximum Possible improvement(MPI) = f (otherf)(−other )

analysis, it is seen that the searching capacity of the GSA-GA algorithm is far better than the existing approaches. The statistical simulation results after 30 runs are shown in Table 4 which further reveals that the standard deviation of the proposed GSA-GA approach is less and average is better than the existing studies. The optimal decision variables obtained from GSA-GA algorithm under region I and region II are

Y = [0.77819652, 0.3846644, 40.3210580446, 199.9799646] and

Y = [0.727610046, 0.359658023, 37.70 0 0 0238, 239.983428] with objective function values f (Y ) = 5885.38533633 and 5804.4048008 respectively. 4.3. Welded beam design problem The originality of this benchmark constrained optimization problem is taken from [40], in which the task is to find the fabricating cost by taking “set-up”, “welding labor”, and “material” costs of the welded beam. The structure of this design problem is shown in Fig. 3 which consists of four decision variables namely, “the thickness of the weld (h)”, “length of the welded joint (l)”, “width of the beam (t)” and “the thickness of the beam (b) ”. In the literature, several researchers solved this problem by varying the number of the constraints. In this study, we have solved all the version of this problem by our proposed algorithm.

508

H. Garg / Information Sciences 478 (2019) 499–523 Table 4 Statistical results of different methods for pressure vessel (NA means not available). Region

Method

Best

Mean

Worst

Std Dev

Median

I

Sandgren [44] Kannan and Kramer [27] Coello [10] Coello and Montes [11] He and Wang [21] Montes and Coello [33] Kaveh and Talatahari [29] Kaveh and Talatahari [28] Gandomi et al. [16] Cagnina et al. [5] Coelho [8] He et al. [22] Akay and Karaboga [1] Garg [18] Present study Dimopoulos [14] Mahdavi et al. [31] Gandomi et al. [17] Garg [18] Present study

8129.1036 7198.0428 6288.7445 6059.9463 6061.0777 6059.7456 6059.7258 6059.0925 6059.714 6059.714335 6059.7208 6059.7143 6059.714339 5885.403282 5885.3853363 5850.38306 5849.7617 5850.38306 5804.44867 5804.4048008

N/A N/A 6293.8432 6177.2533 6147.1332 6850.0049 6081.7812 6075.2567 6447.7360 6092.0498 6440.3786 6289.92881 6245.308144 5887.557024 5884.24637 N/A N/A 5937.33790 5805.47391 5806.596206

N/A N/A 6308.1497 6469.3220 6363.8041 7332.8798 6150.1289 6135.3336 6495.3470 NA 7544.4925 NA NA 5895.126804 5884.462541 N/A N/A 6258.96825 5811.977127 5808.16968

N/A N/A 7.4133 130.9297 86.4545 426.0 0 0 0 67.2418 41.6825 502.693 12.1725 448.4711 305.78 205 2.745290 0.50281 N/A N/A 164.54747 1.411462 1.028072

NA NA NA NA NA NA NA NA NA NA 6257.5943 NA NA 5886.149289 5884.58128 NA NA NA 5805.073797 5806.7764199

II

Fig. 3. Welded beam design problem.

Version I: Under the decision variables Y = (h, l, t, b) = (y1 , y2 , y3 , y4 ), an optimization model is constructed as

Minimize f (Y ) = 1.10471y21 y2 + 0.04811y3 y4 (14 + y2 ) s.t.

τ (Y ) − τmax ≤ 0 g2 (Y ) = σ (Y ) − σmax ≤ 0 g3 (Y ) = y1 − y4 ≤ 0 g4 (Y ) = 0.125 − y1 ≤ 0 g5 (Y ) = δ (Y ) − 0.25 ≤ 0 g6 (Y ) = P − Pc (Y ) ≤ 0 g1 (Y ) =

0.1 ≤ y1 ≤ 2 ;

0.1 ≤ y2 ≤ 10 ;

0.1 ≤ y3 ≤ 10

;

0.1 ≤ y4 ≤ 2

where τ max is the “allowable shear stress of the weld (= 13600 psi)”, σ the “normal stress in the beam”, σ max is the “allowable normal stress for the beam material (= 30 0 0 0 psi)”, Pc the “bar buckling load”, P the “load (= 60 0 0lb)”, δ the “beam end deflection” and τ is the “shear stress in the weld”. The shear stress τ is given as



τ= where

τ12 + 2τ1 τ2 

y2 M = P L+ 2



y  2

2R

+ τ22



;

;

τ1 = √

y1 y2 y22 J=2 √ + 2 12

P 2y1 y2

;

τ2 =

 y + y  1 3 2 2

MR , J

H. Garg / Information Sciences 478 (2019) 499–523

509

Table 5 Comparison of the best solution for Welded beam found by different methods (NA means not available). Version

Design variables

I

Ragsdell and Phillips [38] Rao [40] Deb [12] Ray and Liew [42] Hwang and He [26] Mehta and Dasgupta [32] Garg [18] Present study Coello [10] Coello and Montes [11] He and Wang [21] Dimopoulos [14] Mahdavi et al. [31] Montes et al. [34] Montes and Coello [33] Cagnina et al. [5] Kaveh and Talatahari [28] Kaveh and Talatahari [29] Gandomi et al. [17] Mehta and Dasgupta [32] Akay and Karaboga [1] Garg [18] Present study

II

a

Violate g1 and g6 constraints;

R=

δ

y22 + 4

y + y  1 3 2 2

b

4P L = Ey33 y4

;

G = 12 × 106 psi,

y3

y4

f(Y)

MPId

0.245500 0.245500 NA 0.244438276 0.223100 0.24436895 0.24436198 0.24436822 0.208800 0.205986 0.202369 0.2015 0.20573 0.205730 0.199742 0.205729 0.205729 0.205700 0.2015 0.20572885 0.205730 0.2057245 0.20572943

6.1960 0 0 6.1960 0 0 NA 6.2379672340 1.5815 6.21860635 6.21767407 6.21754641 3.420500 3.471328 3.544214 3.5620 3.47049 3.470489 3.612060 3.470488 3.469875 3.471131 3.562 3.47050567 3.470489 3.25325369 3.253123897

8.2730 0 0 8.2730 0 0 NA 8.2885761430 12.84680 8.29147256 8.29163558 8.29147808 8.997500 9.020224 9.048210 9.041398 9.03662 9.036624 9.037500 9.036624 9.036805 9.036683 9.0414 9.03662392 9.036624 9.03664438 9.03662392

0.245500 0.245500 NA 0.2445661820 0.2245 0.24436895 0.24436883 0.24436894 0.210 0 0 0 0.206480 0.205723 0.205706 0.20573 0.205730 0.206082 0.205729 0.205765 0.205731 0.2057 0.20572964 0.205730 0.20572999 0.20572964

2.385937 2.3860 2.38119 2.3854347 2.25a 2.3811341 2.38099617 2.38095970 1.748309 1.728226 1.728024 1.731186 1.7248 1.724852 1.73730 1.724852 1.724849 1.724918 1.73121 1.724855 1.724852 1.69526388 1.695247383

0.20861% 0.21124% 0.00967% 0.18760% -b 0.00732% 0.00153%

σ=

;



Pc =

y2

3.03503% 1.90824% 1.89677% 2.07595% 1.71339% 1.71636% 2.42057% 1.71636% 1.71619% 1.72012% 2.07731% 1.71653% 1.71636% 0.0 0 097%

f ( proposed ) Infeasible solution; d Maximum possible improvement(MPI) = f (otherf)(−other )

4.013

3

y1

6P L y4 y23

EGy23 y64 36

L2

E = 30 × 106 psi,

 y3 1− 2L



 E 4G

P = 60 0 0lb,

L = 14in

This problem has been solved by the researchers [12,18,26,32,38,40,42] using different algorithms. A comparison of the present GSA-GA approach with respect to these existing approaches is given in Table 5 while their statistical measures are summarized in Table 6. From these tables, the best solution obtained is Y = (0.24436822, 6.21754641, 8.29147808, 0.24436894) with optimal cost f (Y ) = 2.38095970 and is found to be superior than all of the existing approaches. Also, from the standard deviation of the proposed approach reveals that it is more consistent than the existing approaches. Version II: In the literature, we also observed that several other researchers have solved this design problem in some different version. In it, they have considered one more constraint g7 which is defined as

g7 (Y ) = 0.10471y21 + 0.04811y3 y4 (14 + y2 ) − 5 ≤ 0 along with above defined g1 − g6 constraints. The other terms which are differ from version I and are associated with this version are



J=2

y + y  √ y2 1 3 2 2y 1 y 2 2 + 4 2











4.013E Pc =

L2

y23 y64 36

y3 1− 2L



;

δ=

6P L3 Ey33 y4

;

E 4G

Under this version, researchers [1,5,10,11,14,17,18,21,28,29,31–34] solved this problem and their results along with the proposed GSA-GA approach result are summarized in Table 5 which reveals that proposed result is much better in comparison to other existing results. Further, the statistical measures of central tendencies corresponding to proposed as well as the existing approaches are represented in Table 6 obtained after 30 independent runs. It shows that mean of the cost is 1.69524738 with a standard deviation of 1.9787 × 10−8 . Also, the worst solution is much confi-

510

H. Garg / Information Sciences 478 (2019) 499–523 Table 6 Statistical results of different methods for welded beam design problem (NA means not available). Version

Method

Best

Mean

Worst

Std-dev

Median

I

Ragsdell and Phillips [38] Rao [40] Deb [12] Ray and Liew [42] Hwang and He [26] Mehta and Dasgupta [32] Garg [18] Present study Coello [10] Coello and Montes [11] Dimopoulos [14] He and Wang [21] Montes et al. [34] Montes and Coello [33] Cagnina et al. [5] Kaveh and Talatahari [29] Kaveh and Talatahari [28] Gandomi et al. [17] Mehta and Dasgupta [32] Akay and Karaboga [1] Garg [18] Present study

2.385937 2.3860 2.38119 2.3854347 2.25 2.381134 2.38099617 2.3809597 1.748309 1.728226 1.731186 1.728024 1.724852 1.737300 1.724852 1.724918 1.724849 1.7312065 1.724855 1.724852 1.69526388 1.69524738

NA NA NA 3.2551371 2.26 2.3811786 2.38108932 2.3794755 1.771973 1.792654 NA 1.748831 1.725 1.813290 2.0574 1.729752 1.727564 1.8786560 1.724865 1.741913 1.69530842 1.6952473

NA NA NA 6.3996785 2.28 2.3812614 2.38146999 2.380147 1.785835 1.993408 NA 1.782143 NA 1.994651 NA 1.775961 1.759522 2.3455793 1.72489 NA 1.69537060 1.6952473

NA NA NA 0.9590780 NA NA 1.01227×10−4 1.182265×10−4 0.011220 0.07471 NA 0.012926 1E-15 0.070500 0.2154 0.0 0920 0 0.008254 0.2677989 NA 0.031 2.83×10−5 1.978×10−8

NA NA NA 3.0025883 NA 2.3811641 2.38107233 2.3786824 NA NA NA NA NA NA NA NA NA NA 1.724861 NA 1.69530879 1.6952473

II

Fig. 4. Tension/compression string design problem.

dent than the existing approaches result. The best solution is f (Y ) = 1.695247383 corresponding to the decision vector Y = (0.20572943, 3.253123897, 9.03662392, 0.20572964) by proposed algorithm. 4.4. Tension string design problem In this design problem, the task is to minimize the weight of the tension string, shown in Fig. 4, subject to the several constraints in the form of the optimization problem which are stated as follows

Minimize f (Y ) = (y3 + 2 )y2 y21 s.t. g1 (Y ) = 1 − g2 (Y ) =

y32 y3 71785y41

≤0

4y22 − y1 y2

12566(

y2 y31



y41

)

+

1 −1≤0 5108y21

140.45y1 g3 (Y ) = 1 − ≤0 y22 y3 y1 + y2 g4 (Y ) = −1≤0 1.5 0.05 ≤ y1 ≤ 2 ; 0.25 ≤ y2 ≤ 1.3 ;

2 ≤ y3 ≤ 15

where Y = [y1 , y2 , y3 ] is the decision variable which represents mean coil diameters, wire diameter and the number of active coil respectively. Originally, this problem is solved by Belegundu [3] and after that several researchers [1,5,8,10,11,18,21,22,29,31,33,34,39,42,43,49,50] have solved this problem by using both mathematical and heuristic approaches. The optimal results for this problem by proposed GSA-GA algorithm and the existing approaches are reported in Table 7 which suggests that GSA-GA algorithm finds a better solution. Further, from this table it is observed that the

H. Garg / Information Sciences 478 (2019) 499–523

511

Table 7 Comparison of the best solution for tension/compression string design problem by different methods. Design variables

Belegundu [3] Coello [10] Ray and Saini [43] Coello and Montes [11] Ray and Liew [42] He et al. [22] Raj et al. [39] Tsai [49] Mahdavi et al. [31] Montes et al. [34] He and Wang [21] Cagnina et al. [5] Zhang et al. [50] Montes and Coello [33] Kaveh and Talatahari [29] Coelho [8] Akay and Karaboga [1] Garg [18] Present study a

Violate g1 constraint; f (other )− f ( proposed ) f (other )

c

y1

y2

y3

f(Y)

MPId

0.05 0.051480 0.050417 0.051989 0.0521602170 0.05169040 0.05386200 0.05168906 0.05115438 0.051688 0.051728 0.051583 0.0516890614 0.051643 0.051865 0.051515 0.051749 0.051689156131 0.051689294783

0.315900 0.351661 0.321532 0.363965 0.368158695 0.35674999 0.41128365 0.3567178 0.34987116 0.356692 0.357644 0.354190 0.3567177469 0.355360 0.361500 0.352529 0.358179 0.356720026419 0.356723362042

14.250 0 0 11.632201 13.979915 10.890522 10.6484422590 11.28712599 8.68437980 11.28896 12.0764321 11.290483 11.244543 11.438675 11.2889653382 11.397926 11.0 0 0 0 0 11.538862 11.203763 11.288831695483 11.288636144656

0.0128334 0.01270478 0.013060 0.0126810 0.01266924934 0.0126652812c 0.01274840 0.01266523 0.0126706 0.012665 0.0126747 0.012665 0.012665233 0.012698 0.0126432c 0.012665a 0.012665c 0.01266523278 0.01266523278

1.31039% 0.31128% 3.02272% 0.12434% 0.03170% -b 0.65237% −0.0 0 0 02% 0.04236% −0.00184% 0.07469% −0.00184% 0.0 0 0 0 0% 0.25805% -b -b -b 0.0 0 0 0 0%

Violate g2 constraint ;

b

Infeasible solution ;

d

Maximum possible improvement(MPI) =

Table 8 Statistical results of different methods for tension/compression string (NA means not available). Method

Best

Mean

Worst

Std Dev

Median

Belegundu [3] Coello [10] Ray and Saini [43] Coello and Montes [11] Ray and Liew [42] He et al. [22] He and Wang [21] Zhang et al. [50] Montes et al. [34] Montes and Coello [33] Cagnina et al. [5] Kaveh and Talatahari [29] Coelho [8] Akay and Karaboga [1] Garg [18] Present study

0.0128334 0.01270478 0.0130600 0.0126810 0.01266924934 0.0126652812 0.0126747 0.012665233 0.012665 0.012698 0.012665 0.0126432 0.012665 0.012665 0.0126652327883 0.01266523278

NA 0.01276920 0.015526 0.012742 0.012922669 0.01270233 0.012730 0.012669366 0.012666 0.013461 0.0131 0.012720 0.013524 0.012709 0.0126689724845 0.012665551685

NA 0.01282208 0.018992 0.012973 0.016717272 NA 0.012924 0.012738262 NA 0.164850 NA 0.012884 0.017759 NA 0.012710407729 0.012668306480

NA 3.9390×10−5 NA 5.90 0 0×10−5 5.92×10−4 4.12439×10−5 5.1985×10−5 1.25×10−5 2.0×10−6 9.6600×10−4 4.1×10−4 3.4888×10−5 0.001268 0.012813 9.429426×10−6 8.6519×10−7

NA 0.01275576 NA NA 0.012922669 NA NA NA NA NA NA NA 0.012957 NA 0.012665314728 0.01266555276

solution reported in [22,29] is infeasible. In addition to these, the statistical results of the proposed GSA-GA approach with existing approaches are summarized in Table 8 which suggests us that GSA-GA has emerged that there is a large consistency to get the optimal solution. 4.5. Three bar truss design The structure of this design problem is shown in Fig. 5, where the target is to minimize the volume subject to stress constraints on each side of the truss members. For it, the mathematical model is given as below:

 √  Minimize f (A1 , A2 ) = 2 2A1 + A2 × l √ 2A 1 + A 2 s.t. g1 = √ P−σ ≤0 2A21 + 2A1 A2 A2 g2 = √ P−σ ≤0 2A21 + 2A1 A2 1 P−σ ≤0 √ A1 + 2A2 0 ≤ A1 , A2 ≤ 1

g3 =

512

H. Garg / Information Sciences 478 (2019) 499–523

Fig. 5. Three bar truss design problem figure. Table 9 Comparison of the best solution for three-bar truss design problem found by different methods (NA means not available).

Hernández [23] Ray and Saini [43] Ray and Liew [42] Raj et al. [39] Tsai [49] Zhang et al. [50] Gandomi et al. [16] Present study a

Violate g1 constraint;

b

A1

A2

g1

g2

g3

f

MPId

0.788 0.795 0.7886210370 0.78976441 0.788 0.7886751359 0.78867 0.788676171219

0.408 0.395 0.4084013340 0.40517605 0.408 0.4082482868 0.40902 0.408245358456

NA −0.00169 −8.275×10−9 −7.084×10−9 0.0 0 082a −2.104×10−11 −0.0 0 029 −1.587×10−13

NA −0.26124 −1.46392765 −1.4675992 −0.2674 −1.46410161 −0.26853 −1.4641049

NA −0.74045 −0.536072358 −0.53240078 −0.73178 −0.5358983 −0.73176 −0.535895

263.9 264.3 263.8958466 263.89671 263.68 263.8958434 263.9716 263.8958433

0.00158% 0.15292% 0.0 0 0 0 0% 0.0 0 033% -b 0.0 0 0 0 0% 0.02870%

f ( proposed ) Infeasible solution; d Maximum possible improvement(MPI) = f (otherf)(−other )

Table 10 Statistically result of different methods for truss-bar problem (NA means not available). Method

Best

Mean

Worst

Std Dev

Median

Ray and Liew [42] Zhang et al. [50] Gandomi et al. [16] Present study

263.8958466 263.8958434 263.97156 263.8958433

263.9033 263.8958436 264.0669 263.8958437

263.96975 263.8958498 NA 263.8958459

1.26×10−2 9.72×10−7 0.0 0 0 09 5.34×10−7

263.8989 263.8958434 NA 263.8958436

where l =100cm, P = 2KN/cm2 , σ = 2KN/cm2 Previously, several researchers solved this benchmark problem using several techniques such as evolutionary computational technique [39], a swarm - like based approach [42,43], convexification strategies [49], cuckoo search [16], dynamic stochastic selection differential evolution [50]. Here, a proposed GSA-GA algorithm has been applied to solve the model and summarize their results along with these existing approaches in Table 9, while their statistical measures are reported in Table 10. From these tables, it is concluded that result obtained by proposed GSA-GA algorithm is far better than the existing and also standard deviation is very less after 30 independent runs. It has also be notified that solution proposed by Tsai [49] is infeasible as it violates the g1 constraint. The best results obtained for this problem by GSA-GA is f (Y ) = 263.895843377258 corresponding to Y = [A1 , A2 ] = [0.788676171219, 0.408245358456] and constraints [g1 (Y ), g2 (Y ), g3 (Y )] = [−1.587 × 10−13 , −1.4641049, −0.535895]. 4.6. Speed reducer design Speed reducer problem originated by [30] is one of the important benchmark design problems, shown in Fig. 6, in which the target is to minimize their total weight. In this problem, seven decision variable namely, “face width” (b), “module of teeth” (m), “number of teeth on pinion” (z), “length of shaft 1 between bearing” (l1 ), “length of shaft 2 between bearing” (l2 ), “diameter of shaft1” (d1 ) and “diameter of shaft2” (d2 ). The optimization model of this design problem, by considering the design variables Y = (b, m, z, l1 , l2 , d1 , d2 ) = (y1 , y2 , y3 , y4 , y5 , y6 , y7 ), is given as follows

Minimize f (Y ) = 0.785y1 y22 (3.3333y23 + 14.9334y3 − 43.0934 ) −1.508y1 (y26 + y27 ) + 7.477(y36 + y37 ) + 0.7854(y4 y26 + y5 y27 ) s.t.

g1 (Y ) = g3 (Y ) =

27 −1≤0 y1 y22 y3 1.93y34 y2 y3 y46

−1≤0

;

g2 (Y ) =

;

g4 (Y ) =

397.5 −1≤0 y1 y22 y23 1.93y35 y2 y3 y47

−1≤0

H. Garg / Information Sciences 478 (2019) 499–523

513

Fig. 6. Speed reducer design problem figure.

 g5 (Y ) =

 g6 (Y ) =

745y4 y2 y3

 2

+ 1.69 × 106

110y36

745y5 y2 y3



2

−1≤0

+ 157.5 × 106

85y37

−1≤0

y2 y3 5y2 − 1 ≤ 0 ; g8 (Y ) = −1≤0 40 y1 y1 1.5y6 + 1.9 g9 (Y ) = − 1 ≤ 0 ; g10 (Y ) = −1≤0 12y2 y4 1.1y7 + 1.9 g11 (Y ) = −1≤0 y5 g7 (Y ) =

where 2.6 ≤ y1 ≤ 3.6, 0.7 ≤ y2 ≤ 0.8, 17 ≤ y3 ≤ 28, 7.3 ≤ y4 ≤ 8.3, 7.8 ≤ y5 ≤ 8.3, 2.9 ≤ y6 ≤ 3.9, 5.0 ≤ y7 ≤ 5.5. Many researchers [2,5,16,30,34,35,42,43,50] have reported solution to this problem which are arranged in Table 11 along with the proposed GSA-GA algorithm solution. Their statistical measures of these approaches are summarized in Table 12 which suggest us that the solution reported by GSA-GA is superior to the previously reported solutions in the literature. However, the solution gave by the authors in [30,39,43] are infeasible as they violates the g6 constraints. 4.7. Tabular column design In this design problem, the objective is to design a uniform column of tabular section, with length (L) 250cm, at minimum cost which includes material and construction cost, so as to carry a compressive load P=2500 kgf shown in Fig. 7 [40]. The mean diameter (d) of the column is restricted between 2 and 14cm while thickness (t) lies in the range 0.2 - 0.8cm. The column is made of a material with a “yield stress” (σy = 500kg f /cm2 ), “a modulus of elasticity” (E = 0.85 × 106 kgf/cm2 ), and a “density (ρ = 0.0025kg f /cm2 ). Based on these configurations, the optimization model is formulated as

Minimize f (Y ) = 9.82dt + 2d P s.t. g1 (Y ) = −1≤0 π dt σy g2 (Y ) =

π

8P L2 −1≤0 (d 2 + t 2 )

3 Edt

2.0 −1≤0 d d g4 (Y ) = −1≤0 14 0.2 g5 (Y ) = −1≤0 t t g6 (Y ) = −1≤0 0.8 g3 (Y ) =

This design problem solved previously by the researchers [25,40] and [16]. But from their studies, it is observed that the best solutions reported by Hsu and Liu [25], Rao [40] are infeasible. On the other hand, if we employed our proposed GSAGA algorithm to solve this model, the best solution corresponding to it along with existing solutions are summarized in

514

Table 11 Comparison of the best solution for speed reducer problem found by different methods. Kuang et al. [30]

Ray and Saini [43]

Akhtar et al. [2]

Montes et al. [35]

Ray and Liew [42]

Raj et al. [39]

Montes et al. [34]

Cagnina et al. [5]

Zhang et al. [50]

Gandomi et al. [16]

y1 y2 y3 y4 y5 y6 y7 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 f(Y) MPId

3.6 0.7 17 7.3 7.8 3.4 5 −0.0996 −0.2203 −0.5279 −0.8769 −0.0433 0.1821a −0.7025 −0.0278 −0.5714 −0.0411 −0.0513 2876.11762 -b

3.514185 0.70 0 0 05 17 7.497343 7.8346 2.9018 5.0022 −0.0777 −0.2012 −0.0360 −0.8754 −0.4857 0.1805a −0.7025 −0.0040 −0.5816 −0.1660 −0.0552 2732.9006 -b

3.506122 0.70 0 0 06 17 7.549126 7.85933 3.365576 5.289773 −0.0755 −0.1994 −0.4562 −0.8994 −0.0132 −0.0017 −0.7025 −0.0017 −0.5826 −0.0796 −0.0179 3008.08 3.76791%

3.506163 0.700831 17 7.460181 7.962143 3.3629 5.3090 −0.0777 −0.2013 −0.4741 −0.8971 −0.0110 −0.0125 −0.7022 −0.0 0 06 −0.5831 −0.0691 −0.0279 3025.005 4.30633%

3.50 0 0 0681 0.70 0 0 0 0 01 17.0 0 0 0 7.32760205 7.71532175 3.35026702 5.28665450 −0.0739171 −0.1980 0 01 −0.9999967 −0.9999995 −0.6667294 −1.954 × 10−8 −0.7024999 −0.0 0 0 0 019 −0.5833325 −0.0548885 −2.333 × 10−7 2994.744241 3.33938%

3.50 0 0711 0.70 0 0 0 0 0 17 7.300 7.8207280 2.9001726 5.0 0 0 0 046 −0.0739341 −0.1980148 −0.9999941 −0.9999994 −0.4864499 0.1820623a −0.70250 0 0 −0.0 0 0 0203 −0.5833248 −0.1438001 −0.0537958 2724.05500 -b

3.50 0 010 0.70 0 0 17 7.300156 7.80 0 027 3.350221 5.286685 −0.073918 −0.198001 −0.499144 −0.901471 −0.0 0 0 0 05 −0.0 0 0 0 01 −0.702500 −0.0 0 0 0 03 −0.583332 −0.051345 −0.010856 2996.356689 3.39140%

3.50 0 0 0 0 0.7 17 7.3 7.800 3.350214 5.286683 −0.073915 −0.197998 −0.499172 −0.901471 −0.0 0 0 0 0 −5 × 10−16 −0.702500 −1 × 10−16 −0.583333 −0.051325 −0.010852 2996.348165 3.39112%

3.5 0.7 17 7.3 7.7153199115 3.3502146661 5.2866544650 −0.0739152 −0.1979985 −0.9999967 −0.9999995 −0.6668526 −0.0 0 0 0 0 0 0 −0.70250 0 0 −0.0 0 0 0 0 0 0 −0.5833333 −0.0513257 −0.0 0 0 0 0 0 0 2994.471066 3.33056%

3.5015 0.7 17.0 0 0 7.6050 7.8181 3.3520 5.2875 −0.0743 −0.1983 −0.4349 −0.9008 −0.0011 −0.0 0 04 −0.7025 −0.0 0 04 −0.5832 −0.0890 −0.0130 30 0 0.9810 3.54027%

violate constraint;

b

f ( proposed ) infeasible solution; d Maximum Possible improvement(MPI) = f (otherf)(−other )

Present study 3.499682 0.699934 16.999234 7.300392 7.799441 2.89971 5.286725 −0.073614 −0.197702 −0.107329 −0.901482 −0.486193 −0.0 0 0 023 −0.702541 −0.0 0 0 0 03 −0.583331 −0.143941 −0.010775 2894.73832

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a

Design variable

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515

Table 12 Statistical data for speed reducer design problem (NA means not available). Algorithms

Best

Median

Mean

Worst

Std-dev

Kuang et al. [30] Ray and Saini [43] Akhtar et al. [2] Montes et al. [35] Ray and Liew [42] Montes et al. [34] Cagnina et al. [5] Zhang et al. [50] Gandomi et al. [16] Present study

2876.117623 2732.9006 3008.08 3025.005 2994.744241 2996.356689 2996.348165 2994.471066 30 0 0.9810 2894.73832

NA NA NA NA 3001.758264 NA NA 2994.471066 NA 2894.97128

NA 2741.5642 3012.12 3088.7778 3001.7582264 2996.367220 2996.3482 2994.471066 3007.1997 2894.71248

NA 2757.8581 3028 3078.5918 3009.964736 NA NA 2994.471066 NA 2895.03219

NA NA NA NA 4.0091423 8.2×10−3 0.0 0 0 0 3.58×10−12 4.9634 4.96×10−4

Fig. 7. A tabular column design. Table 13 Comparison of the best solution for tabular column design problem found by different methods.

d t g1 g2 g3 g4 g5 g6 f(Y) MPId a

Hsu and Liu [25]

Rao [40]

Gandomi et al. [16]

Present study

5.4507 0.2920 −7.8 × 10−5 0.1317a -0.6331 -0.6107 −0.3151 −0.6350 25.5316 -b

5.4400 0.2930 −0.8579 0.0026a −0.8571 0 −0.7500 0 26.5323 0.00366%

5.45139 0.29196 −0.0241 −0.1095 −0.6331 −0.6106 −0.3150 −0.6351 26.5321 0.00291%

5.45115623 0.29196548 −0.9002 × 10−8 −0.7501 × 10−8 −0.633105 −0.610631 −0.314987 −0.635043 26.531328

Violate constraint;

f (other )− f ( proposed ) f (other )

b

Infeasible solution; d Maximum Possible improvement(MPI) =

Table 13. From this table, it is clearly seen that proposed solution is far better than the existing results. Also, to show their consistency, the statistical analysis results are summarized in Table 14 which reveals that the variation in the optimal results are pretty low as compared to approaches [16,25,40]. Therefore, the GSA-GA algorithm provides the best results. 4.8. Gear train design This problem is initially introduced by Sandgren [44] with the task to find the optimal number of teeth of the gearwheel with ranges 12–60 such that it will minimize the gear ratio cost. The structural diagram of this problem is shown in Fig. 8

516

H. Garg / Information Sciences 478 (2019) 499–523 Table 14 Statistical data for the tubular column problem (NA means not available). Algorithms

Best

Median

Mean

Worst

Std-dev

Gandomi et al. [16] Present

26.532170 26.531328

NA 26.531330

26.53504 26.531332

26.53972 26.55315

0.00193 3.94×10−4

Fig. 8. Geartear design problem. Table 15 Comparison of the best solution for gear train design problem found by different methods.

Td (y1 ) Tb (y2 ) Ta (y3 ) Tf (y4 ) Gear ratio f(Y)

Sandgren [44]

Kannan and Kramer [27]

Deb and Goyal [13]

Gandomi et al. [16]

Present study

18 22 45 60 0.146667 5.712 × 10−6

13 15 33 41 0.144124 2.146 × 10−8

19 16 49 43 0.144281 2.701 × 10−12

19 16 43 49 0.144281 2.701 × 10−12

19 16 43 49 0.14428096 2.7008571 × 10−12

Table 16 Statistical results of the gear train model (NA means not available). Algorithms Gandomi et al. [16] Present

Best

Median −12

2.7009 × 10 2.7008571 × 10−12

NA 9.9215795 × 10−10

Mean

Worst −9

1.9841 × 10 1.2149276 × 10−9

Std-dev −9

2.3576 × 10 3.2999231 × 10−9

3.5546 × 10−9 8.77 × 10−10

Fig. 9. Belleville spring design problem.

and the optimization model, with decision vector Y = (Td , Tb , Ta , T f ) = (y1 , y2 , y3 , y4 ), is formulated as follows:

Minimize f (Y ) =

 1 6.931

12 ≤ y1 , y2 , y3 , y4 ≤ 60 y y

;



y1 y2 y3 y4

2

yi s ∈ Z+

where gear ratio = y1 y2 . 3 4 Researchers [13,16,27,44] solved this model using different algorithms where their corresponding reported solutions are reported in Table 15. On the other hand, their statistical results are reported in Table 16 along with the results of the Gandomi et al. [16], which reveals that result proposed by GSA-GA algorithm is superior than it, as the standard deviation is pretty low. Therefore, the results reported by GSA-GA are significantly better than those reported by authors in [13,16,27,44].

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517

Fig. 10. Box-plot of objective function using the reported optimizers. F1: Himmelblau’s problem (Version-I); F2: Himmelblau’s problem (Version-II); F3: Pressure Vessel (Version-I); F4: Pressure Vessel (Version-II); F5: Welded beam (Version-I); F6: Welded beam (Version-II); F7: Compression string; F8: Three-bar truss; F9: Speed reducer ; F10: Tabular column.

518

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Fig. 10. Continued

H. Garg / Information Sciences 478 (2019) 499–523

Fig. 11. Optimization Process of the considered EDPs.

519

520

H. Garg / Information Sciences 478 (2019) 499–523 Table 17 Values of a and f(a) [9]. a f(a)

≤ 1.4 1

1.5 0.85

1.6 0.77

1.7 0.71

1.8 0.66

1.9 0.63

2.0 0.60

2.1 0.58

2.2 0.56

2.3 0.55

2.4 0.53

2.5 0.52

2.6 0.51

2.7 0.51

≥ 2.8 0.50

Table 18 Comparison of the best solution for Belleville spring design problem found by different methods.

a

Design variables

Coello [9]

Deb and Goyal [13]

Siddal [48]

Present study

y1 y2 y3 y4 g1 g2 g3 g4 g5 g6 g7 f(Y) MPId

0.208 0.200 8.751 11.067 2145.4109 39.75018 0.0 0 0 0 1.592 0.943 2.316 0.21364 2.121964 6.41212%

0.210 0.204 9.268 11.499 1988.370 197.7260 0.0040 1.5860 0.5110 2.2310 0.20856 2.16256 8.16897%

0.204 0.200 10.030 12.010 134.0816 −12.5328a 0.0 0 0 0 1.5960 0.0 0 0 0 1.9800 0.19899 1.978715 -b

0.204301567205 0.200161336075 9.966845654574 11.961230981067 39.365210212039 2.2401509532965 0.0 0 01613360754 1.5955370967192 0.0487690189329 1.9943853264934 0.1996375808543 1.985901142149

Violate constraint;

b

Infeasible solution;

f (other )− f ( proposed ) f (other )

d

Maximum Possible improvement(MPI) =

Table 19 Statistically result for the Belleville spring problem (NA means not available). Method

Best

Mean

Worst

Std-dev

Median

Siddal [48] Deb and Goyal [13] Coello [9] Present study

1.978715 2.16256 2.121964 1.9859011

NA NA NA 1.9979438

NA NA NA 2.0188231

NA NA NA 0.00782

NA NA NA 1.99622386

4.9. Belleville spring design In this design problem, the goal is to design a Belleville spring with minimum weight whose description is given in Siddal [48]. For it, the four decision variables namely “thickness of the spring” (t), “height of the spring”(h), “external diameter of the spring” (D), and “internal diameter of the spring” (Di ), shown in Fig. 9, are used to formulate a model with Y = (t, h, De , Di ) = (y1 , y2 , y3 , y4 ) as

Minimize f (Y ) = 0.07075π (y23 − y24 )t s.t.

g1 (Y ) = S −

  δ β y2 − max + γ y1 ≥ 0 2   δmax 3 y − y − δ y + y ( ) max 2 2 1 1 − Pmax ≥ 0 2

4E δmax (1 − μ2 )α y23

4E δ ( 1 − μ2 ) α y 3 g3 (Y ) = δl − δmax ≥ 0 g4 (Y ) = H − y2 − y1 ≥ 0 g5 (Y ) = Dmax − y3 ≥ 0 g6 (Y ) = y3 − y4 ≥ 0 g2 (Y ) =

2

y2 ≥0 y3 − y4 0.01 ≤ y1 ≤ 0.6 ; 0.05 ≤ y2 ≤ 0.5

g7 (Y ) = 0.3 −

;

5 ≤ y3 , y4 ≤ 15

where S (= 200kPsi) is the “allowable strength”, E(= 30 × 106 psi) is the “modulus of elasticity” for the spring material, μ(= 0.3) is the “Poisson’s ratio”, δ max (= 0.2in) is the “maximum deflection”, Pmax (= 5400lb) is the “maximum load acting on the spring”, H(=2in) is the “overall height of the spring” and Dmax (= 12.01in) is the “outside diameter of the spring”.





6 K−1 2 6 Assuming K = y3 /y4 then α = π ln , β = π ln K K K where a = y2 /y1 , and f(a) is defined in Table 17 [9].



K−1 ln K



6 − 1 , γ = π ln K



K−1 2



and the limiting deflection is δl = f (a )y2 ,

Table 20 Statistical test using MCT for considered optimizers. Comparing

Lower bound

Group mean

Upper bound

p-value

Function

Comparing

Lower bound

Group mean

Upper bound

p-value

F1

PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA

29.778 50.938 150.078 144.868 113.658 132.108 59.543 537.334 200.973 114.053 43.113 147.073 100.787 13.517 115.457 210.377 96.457 64.557 601.967 50.632 780.617 999.597 460.0 0 0 311.497 7.632 80.206 210.326 159.326 94.316 64.376

89.400 110.560 209.700 204.490 173.280 191.730 119.140 64.970 260.570 173.650 102.710 206.670 160.440 73.170 175.110 270.030 156.110 124.210 721.082 169.747 899.732 1118.712 579.115 430.612 58.740 139.710 269.830 218.830 153.820 123.880

149.021 170.181 269.321 264.111 232.901 251.351 178.736 124.566 320.166 233.246 162.306 266.266 220.092 132.822 234.762 329.682 215.762 183.862 840.197 288.862 1018.847 1237.827 698.229 549.727 118.243 199.213 329.333 278.333 213.323 183.383

1.82E - 14 5.55E - 16 9.47E - 16 1.84E - 17 1.99E - 18 2.08E - 19 5.78E - 19 4.44E - 15 0 0 5.46E - 20 0 4.16E - 20 1.14E - 3 4.77E - 10 0 1.12E - 09 1.30E - 06 0 7.59E - 05 0 0 0 4.08E - 12 5.18E - 09 2.08E - 10 0 0 3.31E - 12 7.61E - 13

F2

PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA PSO vs. GSA-GA ABC vs. GSA-GA CS vs. GSA-GA GA vs. GSA-GA GSA vs. GSA-GA jDE vs. GSA-GA

18.709 122.959 67.909 184.769 80.749 15.789 126.469 -26.010 59.089 185.009 17.029 43.729 73.136 -4.933 30.076 149.626 4.816 -17.703 157.452 59.832 101.652 183.452 3.992 119.892 50.489 -38.070 86.069 180.629 27.149 4.149

78.330 182.580 127.530 244.390 140.370 75.410 186.060 33.580 118.680 244.600 76.620 103.320 132.760 54.690 89.700 209.250 64.440 41.920 217.060 119.440 161.260 243.060 63.600 179.500 110.100 21.540 145.680 240.240 86.760 63.760

137.950 242.201 187.151 304.010 199.990 135.030 245.650 93.170 178.270 304.190 136.210 162.910 192.383 114.313 149.323 268.873 124.063 101.543 276.667 179.047 220.867 302.667 123.207 239.107 169.710 81.150 205.290 299.850 146.370 123.370

1.17E - 03 2.12E - 10 4.90E - 08 2.41E - 20 6.23E - 08 8.39E - 04 0 1.14E - 03 6.87E - 11 0 1.08E - 06 4.77E - 11 2.23E - 02 1.28E - 01 7.17E - 02 1.25E - 14 6.14E - 02 9.72E - 01 1.32E - 11 1.99E - 04 3.22E - 07 9.99E - 16 2.13E - 03 1.54E - 17 1.31E - 06 3.27E - 02 6.37E - 11 0 2.14E - 06 2.59E - 05

F3

F5

F7

F9

F4

F6

F8

F10

H. Garg / Information Sciences 478 (2019) 499–523

Function

F1: Himmelblau’s problem (Version-I); F2: Himmelblau’s problem (Version-II); F3: Pressure Vessel (Version-I); F4: Pressure Vessel (Version-II); F5: Welded beam (Version-I); F6: Welded beam (Version-II); F7: Compression string; F8: Three-bar truss; F9: Speed reducer ; F10: Tabular column

521

522

H. Garg / Information Sciences 478 (2019) 499–523

This problem was solved before by Siddal [48] using linear approximation, Deb and Goyal [13] by using GeneAS and Coello [9] by using evolutionary algorithms. The optimal results by the proposed GSA-GA algorithms and these existing algorithms are summarized in Table 18. Also, it is seen that the solution reported by Deb and Goyal [13] violates the second constraint. Finally, the statistically simulation results in terms of mean, median, worst, standard deviation are summarized in Table 19 and conclude that the GSA-GA algorithm provides the best results with pretty low standard deviation. 4.10. Statistical testing To analyze whether the outcomes gotten by the proposed approach are statistically significant or not, we perform the non-parametric statistical tests (NPST) which can be used to make the inferences about a population without requiring an assumption about the specific form of the population’s probability distribution. In it, we used two test including KruskalWallis H Test (KWT) and Multiple Comparison Test (MCT) in this research. These tests have been utilized here to perform a multiple comparison of means or other estimates. KWT is a rank-based NPST that plays out a non-parametric one-way ANOVA to test the null hypothesis that independent samples from at least two groups originate from distributions with equal estimates. On the other hand, the MCT performs a multiple comparison using one-way ANOVA results to determine which estimates (such as means, slopes, or intercepts) are significantly different. These test have been applied on the considered problems by comparing two by two methods for reported optimizers with the proposed GSA-GA algorithm in Table 20. In this table, first column represents the considered problem and the second column represent the indices of the two samples being compared. Columns 3rd to 5th are a lower bound, estimate, and upper bound for their true mean difference for 95% confidence interval. The sixth column represent the p-value for the test obtained by KWT with α = 0.05 corresponding to the null hypothesis that their mean difference is zero. The box-plot and MCT among the considered EDPs are summarized in Fig. 10. In this figure, the box-plot produces a box and whisker plot for the algorithm. The boxes have lines at the 1st, 2nd and 3rd quartile values. The whiskers have lines extending vertically from the boxes to indicate the degree of rest of the information. In the right side of the Fig. 10, a MCT is reported among the different optimizers where the red color lines signifies the methods which are statistically differ with the proposed GSA-GA method (the blue line represent GSA-GA algorithm). For instance, at F4 in Fig. 10, we observe that five groups (PSO, CS, GA, GSA, jDE) have mean ranks significantly different from GSA-GA. Further, the right (left) vertical lines around the GSA-GA (i.e., blue lines) represent that the border area for showing all those methods which are not statistically (and statistically) better than GSA-GA for a specific EDPs. Now in order to show the convergence performance of the proposed algorithm over the reported algorithm, the variation of the best solution for each specified EDPs are display in Fig. 11 for 200 iterations. From this, it is analyzed that if the iteration number is increasing, the GSA-GA can perform better and statistically significance than the other algorithms. 5. Conclusion In this manuscript, a new algorithm named as GSA-GA algorithm is presented to solve the COPs. In this approach, the advantages of both GSA and GA algorithms are embedded together by feeding the genetic operators into the GSA algorithm. For it, firstly GSA algorithm has been applied to find the solution of the problem and then during each epoch, the best solution is modified with the help of genetic operators to balance between the exploration and exploitation process. The performance of the proposed algorithm has been tested on nine well-known benchmark structural EDPs. The reported results are compared with the existing results present in the literature and showed that proposed results are more feasible and optimal. Further, non-parametric KWT and MCT statistical tests are executed to shows its statistically significant. Sensitivity analysis over the number of iteration parameter has been performed. Computational results of various EDPs along with statistical results and tests reveals that proposed algorithm work is more reliable to find the global solution. As future research, we will extend the approach to solve other benchmarks and COPs. References [1] B. Akay, D. 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