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Grey Wolf Optimizer(爱思唯尔出版)

Grey Wolf

Optimizer

Seyedali Mirjalili a ,?,Seyed Mohammad Mirjalili b ,Andrew Lewis a

a School of Information and Communication Technology,Grif?th University,Nathan Campus,Brisbane QLD 4111,Australia

Department of Electrical Engineering,Faculty of Electrical and Computer Engineering,Shahid Beheshti University,G.C.1983963113,Tehran,Iran

a r t i c l e i n f o Article history:

Received 27June 2013

Received in revised form 18October 2013Accepted 11December 2013

Available online 21January 2014Keywords:Optimization

Optimization techniques Heuristic algorithm Metaheuristics

Constrained optimization GWO

a b s t r a c t

This work proposes a new meta-heuristic called Grey Wolf Optimizer (GWO)inspired by grey wolves (Canis lupus).The GWO algorithm mimics the leadership hierarchy and hunting mechanism of grey wolves in nature.Four types of grey wolves such as alpha,beta,delta,and omega are employed for sim-ulating the leadership hierarchy.In addition,the three main steps of hunting,searching for prey,encir-cling prey,and attacking prey,are implemented.The algorithm is then benchmarked on 29well-known test functions,and the results are veri?ed by a comparative study with Particle Swarm Optimization (PSO),Gravitational Search Algorithm (GSA),Differential Evolution (DE),Evolutionary Programming (EP),and Evolution Strategy (ES).The results show that the GWO algorithm is able to provide very com-petitive results compared to these well-known meta-heuristics.The paper also considers solving three classical engineering design problems (tension/compression spring,welded beam,and pressure vessel designs)and presents a real application of the proposed method in the ?eld of optical engineering.The results of the classical engineering design problems and real application prove that the proposed algo-rithm is applicable to challenging problems with unknown search spaces.

1.Introduction

Meta-heuristic optimization techniques have become very pop-ular over the last two decades.Surprisingly,some of them such as Genetic Algorithm (GA)[1],Ant Colony Optimization (ACO)[2],and Particle Swarm Optimization (PSO)[3]are fairly well-known among not only computer scientists but also scientists from differ-ent ?elds.In addition to the huge number of theoretical works,such optimization techniques have been applied in various ?elds of study.There is a question here as to why meta-heuristics have become remarkably common.The answer to this question can be summarized into four main reasons:simplicity,?exibility,deriva-tion-free mechanism,and local optima avoidance.

First,meta-heuristics are fairly simple.They have been mostly inspired by very simple concepts.The inspirations are typically re-lated to physical phenomena,animals’behaviors,or evolutionary concepts.The simplicity allows computer scientists to simulate dif-ferent natural concepts,propose new meta-heuristics,hybridize two or more meta-heuristics,or improve the current meta-heuris-tics.Moreover,the simplicity assists other scientists to learn meta-heuristics quickly and apply them to their problems.

Second,?exibility refers to the applicability of meta-heuristics to different problems without any special changes in the structure of the algorithm.Meta-heuristics are readily applicable to different problems since they mostly assume problems as black boxes.In other words,only the input(s)and output(s)of a system are impor-tant for a meta-heuristic.So,all a designer needs is to know how to represent his/her problem for meta-heuristics.

Third,the majority of meta-heuristics have derivation-free mechanisms.In contrast to gradient-based optimization ap-proaches,meta-heuristics optimize problems stochastically.The optimization process starts with random solution(s),and there is no need to calculate the derivative of search spaces to ?nd the opti-mum.This makes meta-heuristics highly suitable for real problems with expensive or unknown derivative information.

Finally,meta-heuristics have superior abilities to avoid local op-tima compared to conventional optimization techniques.This is due to the stochastic nature of meta-heuristics which allow them to avoid stagnation in local solutions and search the entire search space extensively.The search space of real problems is usually un-known and very complex with a massive number of local optima,so meta-heuristics are good options for optimizing these challeng-ing real problems.

The No Free Lunch (NFL)theorem [4]is worth mentioning here.This theorem has logically proved that there is no meta-heuristic best suited for solving all optimization problems.In other words,a particular meta-heuristic may show very promising results on a set of problems,but the same algorithm may show poor perfor-mance on a different set of problems.Obviously,NFL makes this ?eld of study highly active which results in enhancing current ap-proaches and proposing new meta-heuristics every year.This also

?Corresponding author.Tel.:+61434555738.

E-mail addresses:seyedali.mirjalili@grif?https://www.wendangku.net/doc/0b8024798.html,.au (S.Mirjalili),mohammad.smm@https://www.wendangku.net/doc/0b8024798.html, (S.M.Mirjalili),a.lewis@grif?https://www.wendangku.net/doc/0b8024798.html,.au (A.Lewis).

motivates our attempts to develop a new meta-heuristic with inspiration from grey wolves.

Generally speaking,meta-heuristics can be divided into two main classes:single-solution-based and population-based.In the former class(Simulated Annealing[5]for instance)the search pro-cess starts with one candidate solution.This single candidate solu-tion is then improved over the course of iterations.Population-based meta-heuristics,however,perform the optimization using a set of solutions(population).In this case the search process starts with a random initial population(multiple solutions),and this population is enhanced over the course of iterations.Population-based meta-heuristics have some advantages compared to single solution-based algorithms:

Multiple candidate solutions share information about the search space which results in sudden jumps toward the prom-ising part of search space.

Multiple candidate solutions assist each other to avoid locally optimal solutions.

Population-based meta-heuristics generally have greater explo-ration compared to single solution-based algorithms.

One of the interesting branches of the population-based meta-heuristics is Swarm Intelligence(SI).The concepts of SI was?rst proposed in1993[6].According to Bonabeau et al.[1],SI is‘‘The emergent collective intelligence of groups of simple agents’’.The inspi-rations of SI techniques originate mostly from natural colonies,?ock,herds,and schools.Some of the most popular SI techniques are ACO[2],PSO[3],and Arti?cial Bee Colony(ABC)[7].A compre-hensive literature review of the SI algorithms is provided in the next section.Some of the advantages of SI algorithms are:

SI algorithms preserve information about the search space over the course of iteration,whereas Evolutionary Algorithms(EA) discard the information of the previous generations.

SI algorithms often utilize memory to save the best solution obtained so far.

SI algorithms usually have fewer parameters to adjust.

SI algorithms have less operators compared to evolutionary approaches(crossover,mutation,elitism,and so on).

SI algorithms are easy to implement.

Regardless of the differences between the meta-heuristics,a common feature is the division of the search process into two phases:exploration and exploitation[8–12].The exploration phase refers to the process of investigating the promising area(s)of the search space as broadly as possible.An algorithm needs to have sto-chastic operators to randomly and globally search the search space in order to support this phase.However,exploitation refers to the lo-cal search capability around the promising regions obtained in the exploration phase.Finding a proper balance between these two phases is considered a challenging task due to the stochastic nature of meta-heuristics.This work proposes a new SI technique with inspiration from the social hierarchy and hunting behavior of grey wolf packs.The rest of the paper is organized as follows: Section2presents a literature review of SI techniques.Section3 outlines the proposed GWO algorithm.The results and discussion of benchmark functions,semi-real problems,and a real application are presented in Sections4-6,respectively.Finally,Section7con-cludes the work and suggests some directions for future studies.

2.Literature review

Meta-heuristics may be classi?ed into three main classes: evolutionary,physics-based,and SI algorithms.EAs are usually inspired by the concepts of evolution in nature.The most popular algorithm in this branch is GA.This algorithm was proposed by Holland in1992[13]and simulates Darwnian evolution concepts. The engineering applications of GA were extensively investigated by Goldberg[14].Generally speaking,the optimization is done by evolving an initial random solution in EAs.Each new population is created by the combination and mutation of the individuals in the previous generation.Since the best individuals have higher probability of participating in generating the new population,the new population is likely to be better than the previous genera-tion(s).This can guarantee that the initial random population is optimized over the course of generations.Some of the EAs are Dif-ferential Evolution(DE)[15],Evolutionary Programing(EP)[16,17], and Evolution Strategy(ES)[18,19],Genetic Programming(GP) [20],and Biogeography-Based Optimizer(BBO)[21].

As an example,the BBO algorithm was?rst proposed by Simon in2008[21].The basic idea of this algorithm has been inspired by biogeography which refers to the study of biological organisms in terms of geographical distribution(over time and space).The case studies might include different islands,lands,or even continents over decades,centuries,or millennia.In this?eld of study different ecosystems(habitats or territories)are investigated for?nding the relations between different species(habitants)in terms of immi-gration,emigration,and mutation.The evolution of ecosystems (considering different kinds of species such as predator and prey) over migration and mutation to reach a stable situation was the main inspiration of the BBO algorithm.

The second main branch of meta-heuristics is physics-based techniques.Such optimization algorithms typically mimic physical rules.Some of the most popular algorithms are Gravitational Local Search(GLSA)[22],Big-Bang Big-Crunch(BBBC)[23],Gravitational Search Algorithm(GSA)[24],Charged System Search(CSS)[25], Central Force Optimization(CFO)[26],Arti?cial Chemical Reaction Optimization Algorithm(ACROA)[27],Black Hole(BH)[28]algo-rithm,Ray Optimization(RO)[29]algorithm,Small-World Optimi-zation Algorithm(SWOA)[30],Galaxy-based Search Algorithm (GbSA)[31],and Curved Space Optimization(CSO)[32].The mech-anism of these algorithms is different from EAs,in that a random set of search agents communicate and move throughout search space according to physical rules.This movement is implemented, for example,using gravitational force,ray casting,electromagnetic force,inertia force,weights,and so on.

For example,the BBBC algorithm was inspired by the big bang and big crunch theories.The search agents of BBBC are scattered from a point in random directions in a search space according to the principles of the big bang theory.They search randomly and then gather in a?nal point(the best point obtained so far)accord-ing to the principles of the big crunch theory.GSA is another phys-ics-based algorithm.The basic physical theory from which GSA is inspired is Newton’s law of universal gravitation.The GSA algo-rithm performs search by employing a collection of agents that have masses proportional to the value of a?tness function.During iteration,the masses are attracted to each other by the gravita-tional forces between them.The heavier the mass,the bigger the attractive force.Therefore,the heaviest mass,which is possibly close to the global optimum,attracts the other masses in propor-tion to their distances.

The third subclass of meta-heuristics is the SI methods.These algorithms mostly mimic the social behavior of swarms,herds,?ocks,or schools of creatures in nature.The mechanism is almost similar to physics-based algorithm,but the search agents navigate using the simulated collective and social intelligence of creatures. The most popular SI technique is PSO.The PSO algorithm was pro-posed by Kennedy and Eberhart[3]and inspired from the social behavior of birds?ocking.The PSO algorithm employs multiple particles that chase the position of the best particle and their

S.Mirjalili et al./Advances in Engineering Software69(2014)46–6147

own best positions obtained so far.In other words,a particle is moved considering its own best solution as well as the best solu-tion the swarm has obtained.

Another popular SI algorithm is ACO,proposed by Dorigo et al. in2006[2].This algorithm was inspired by the social behavior of ants in an ant colony.In fact,the social intelligence of ants in?nd-ing the shortest path between the nest and a source of food is the main inspiration of ACO.A pheromone matrix is evolved over the course of iteration by the candidate solutions.The ABC is another popular algorithm,mimicking the collective behavior of bees in ?nding food sources.There are three types of bees in ABS:scout, onlooker,and employed bees.The scout bees are responsible for exploring the search space,whereas onlooker and employed bees exploit the promising solutions found by scout bees.Finally,the Bat-inspired Algorithm(BA),inspired by the echolocation behavior of bats,has been proposed recently[33].There are many types of bats in the nature.They are different in terms of size and weight, but they all have quite similar behaviors when navigating and hunting.Bats utilize natural sonar in order to do this.The two main characteristics of bats when?nding prey have been adopted in designing the BA algorithm.Bats tend to decrease the loudness and increase the rate of emitted ultrasonic sound when they chase prey.This behavior has been mathematically modeled for the BA algorithm.The rest of the SI techniques proposed so far are as follows:

Marriage in Honey Bees Optimization Algorithm(MBO)in2001

[34].

Arti?cial Fish-Swarm Algorithm(AFSA)in2003[35].

Termite Algorithm in2005[36].

Wasp Swarm Algorithm in2007[37].

Monkey Search in2007[38].

Bee Collecting Pollen Algorithm(BCPA)in2008[39].

Cuckoo Search(CS)in2009[40].

Dolphin Partner Optimization(DPO)in2009[41].

Fire?y Algorithm(FA)in2010[42].

Bird Mating Optimizer(BMO)in2012[43].

Krill Herd(KH)in2012[44].

Fruit?y Optimization Algorithm(FOA)in2012[45].

This list shows that there are many SI techniques proposed so far,many of them inspired by hunting and search behaviors.To the best of our knowledge,however,there is no SI technique in the literature mimicking the leadership hierarchy of grey wolves, well known for their pack hunting.This motivated our attempt to mathematically model the social behavior of grey wolves,pro-pose a new SI algorithm inspired by grey wolves,and investigate its abilities in solving benchmark and real problems.

3.Grey Wolf Optimizer(GWO)

In this section the inspiration of the proposed method is?rst discussed.Then,the mathematical model is provided.

3.1.Inspiration

Grey wolf(Canis lupus)belongs to Canidae family.Grey wolves are considered as apex predators,meaning that they are at the top of the food chain.Grey wolves mostly prefer to live in a pack.The group size is5–12on average.Of particular interest is that they have a very strict social dominant hierarchy as shown in Fig.1.

The leaders are a male and a female,called alphas.The alpha is mostly responsible for making decisions about hunting,sleeping place,time to wake,and so on.The alpha’s decisions are dictated to the pack.However,some kind of democratic behavior has also been observed,in which an alpha follows the other wolves in the pack.In gatherings,the entire pack acknowledges the alpha by holding their tails down.The alpha wolf is also called the dominant wolf since his/her orders should be followed by the pack[46].The alpha wolves are only allowed to mate in the pack.Interestingly, the alpha is not necessarily the strongest member of the pack but the best in terms of managing the pack.This shows that the organization and discipline of a pack is much more important than its strength.

The second level in the hierarchy of grey wolves is beta.The be-tas are subordinate wolves that help the alpha in decision-making or other pack activities.The beta wolf can be either male or female, and he/she is probably the best candidate to be the alpha in case one of the alpha wolves passes away or becomes very old.The beta wolf should respect the alpha,but commands the other lower-level wolves as well.It plays the role of an advisor to the alpha and dis-cipliner for the pack.The beta reinforces the alpha’s commands throughout the pack and gives feedback to the alpha.

The lowest ranking grey wolf is omega.The omega plays the role of scapegoat.Omega wolves always have to submit to all the other dominant wolves.They are the last wolves that are allowed to eat.It may seem the omega is not an important individual in the pack,but it has been observed that the whole pack face internal ?ghting and problems in case of losing the omega.This is due to the venting of violence and frustration of all wolves by the ome-ga(s).This assists satisfying the entire pack and maintaining the dominance structure.In some cases the omega is also the babysit-ters in the pack.

If a wolf is not an alpha,beta,or omega,he/she is called subor-dinate(or delta in some references).Delta wolves have to submit to alphas and betas,but they dominate the omega.Scouts,senti-nels,elders,hunters,and caretakers belong to this category.Scouts are responsible for watching the boundaries of the territory and warning the pack in case of any danger.Sentinels protect and guar-antee the safety of the pack.Elders are the experienced wolves who used to be alpha or beta.Hunters help the alphas and betas when hunting prey and providing food for the pack.Finally,the caretak-ers are responsible for caring for the weak,ill,and wounded wolves in the pack.

In addition to the social hierarchy of wolves,group hunting is another interesting social behavior of grey wolves.According to Muro et al.[47]the main phases of grey wolf hunting are as follows:

Tracking,chasing,and approaching the prey.

Pursuing,encircling,and harassing the prey until it stops moving.

Attack towards the prey.

These steps are shown in Fig.2.

In this work this hunting technique and the social hierarchy of grey wolves are mathematically modeled in order to design GWO and perform

optimization.

48S.Mirjalili et al./Advances in Engineering Software69(2014)46–61

3.2.Mathematical model and algorithm

In this subsection the mathematical models of the social hierar-chy,tracking,encircling,and attacking prey are provided.Then the GWO algorithm is outlined.

3.2.1.Social hierarchy

In order to mathematically model the social hierarchy of wolves when designing GWO,we consider the?ttest solution as the alpha (a).Consequently,the second and third best solutions are named beta(b)and delta(d)respectively.The rest of the candidate solu-tions are assumed to be omega(x).In the GWO algorithm the hunting(optimization)is guided by a,b,and d.The x wolves fol-low these three wolves.

3.2.2.Encircling prey

As mentioned above,grey wolves encircle prey during the hunt. In order to mathematically model encircling behavior the follow-ing equations are proposed:

~D?j~Cá~X

etTà~XetTje3:1T

~Xett1T?~X

etTà~Aá~De3:2Twhere t indicates the current iteration,~A and~C are coef?cient vec-tors,~X p is the position vector of the prey,and~X indicates the posi-tion vector of a grey wolf.

The vectors~A and~C are calculated as follows:

~A?2~aá~r

à~ae3:3T

~C?2á~r

e3:4T

points illustrated in Fig.3.So a grey wolf can update its position in-

side the space around the prey in any random location by using

Eqs.(3.1)and(3.2).

The same concept can be extended to a search space with n

dimensions,and the grey wolves will move in hyper-cubes(or hy-

per-spheres)around the best solution obtained so far.

3.2.3.Hunting

Grey wolves have the ability to recognize the location of prey

and encircle them.The hunt is usually guided by the alpha.The

beta and delta might also participate in hunting occasionally.How-

ever,in an abstract search space we have no idea about the loca-

tion of the optimum(prey).In order to mathematically simulate

the hunting behavior of grey wolves,we suppose that the alpha

(best candidate solution)beta,and delta have better knowledge

about the potential location of prey.Therefore,we save the?rst

three best solutions obtained so far and oblige the other search

agents(including the omegas)to update their positions according

to the position of the best search agents.The following formulas

are proposed in this regard.

a?j

á~X aà~X j;~D b?j~C2á~X bà~X j;~D d?j~C3á~X dà~X je3:5T

?~X aà~A1áe~D aT;~X2?~X bà~A2áe~D bT;~X3?~X dà~A3áe~D dTe3:6T

~Xett1T?

t~X2t~X3

e3:7T

Fig.4shows how a search agent updates its position according to

alpha,beta,and delta in a2D search space.It can be observed that

the?nal position would be in a random place within a circle which

is de?ned by the positions of alpha,beta,and delta in the search grey wolves:(A)chasing,approaching,and tracking prey(B–D)pursuiting,harassing,and encircling(E)stationary

S.Mirjalili et al./Advances in Engineering Software69(2014)46–6149

between its current position and the position of the prey.Fig.5(a)shows that |A|<1forces the wolves to attack towards the prey.With the operators proposed so far,the GWO algorithm allows its search agents to update their position based on the location of the alpha,beta,and delta;and attack towards the prey.However,the GWO algorithm is prone to stagnation in local solutions with these operators.It is true that the encircling mechanism proposed shows exploration to some extent,but GWO needs more operators to emphasize exploration.

3.2.5.Search for prey (exploration)

Grey wolves mostly search according to the position of the al-pha,beta,and delta.They diverge from each other to search for prey and converge to attack prey.In order to mathematically mod-el divergence,we utilize ~A with random values greater than 1or less than -1to oblige the search agent to diverge from the prey.This emphasizes exploration and allows the GWO algorithm to search globally.Fig.5(b)also shows that |A|>1forces the grey wolves to diverge from the prey to hopefully ?nd a ?tter prey.

Another component of GWO that favors exploration is ~C .As may

be seen in Eq.(3.4),the ~C vector contains random values in [0,2].

This component provides random weights for prey in order to sto-chastically emphasize (C >1)or deemphasize (C <1)the effect of prey in de?ning the distance in Eq.(3.1).This assists GWO to show

(X,Y)

(X*,Y*)

(X*,Y)

(X,Y*)

(X,Y*-Y)

(X*-X,Y)

(X*,Y*-Y)

(X*-X,Y*-Y)

(X*-X,Y*)

(X,Y,Z)

(X*,Y*,Z*)

(X,Y*-Y,Z*-Z)

(X*-X,Y,Z*-Z)

(X*,Y*-Y,Z*-Z)(X*-X,Y*-Y,Z-Z*)(X*-X,Y*,Z*-Z)

(X,Y*,Z)

(X,Y*-Y,Z)

(X,Y*,Z*)

(X,Y,Z*)

(X*,Y*,Z*-Z)

(X,Y*,Z*-Z)

(X*,Y,Z*-Z)

(X,Y,Z*-Z)

(X*-X,Y,Z*)

(X*-X,Y,Z)

(X*,Y,Z)

(X,Y,Z*)

(X*,Y,Z*)

(a) (b)

Fig.3.2D and 3D possible next locations.

D delta

D alpha

a 1

a 3

C 3

C 1

or any other hunters Estimated position of the

S.Mirjalili et al./Advances in Engineering Software 69(2014)46–61

a more random behavior throughout optimization,favoring explo-ration and local optima avoidance.It is worth mentioning here that C is not linearly decreased in contrast to A .We deliberately require C to provide random values at all times in order to emphasize exploration not only during initial iterations but also ?nal itera-tions.This component is very helpful in case of local optima stag-nation,especially in the ?nal iterations.

The C vector can be also considered as the effect of obstacles to approaching prey in nature.Generally speaking,the obstacles in nature appear in the hunting paths of wolves and in fact prevent them from quickly and conveniently approaching prey.This is ex-actly what the vector C does.Depending on the position of a wolf,it can randomly give the prey a weight and make it harder and far-ther to reach for wolves,or vice versa.

To sum up,the search process starts with creating a random population of grey wolves (candidate solutions)in the GWO algo-rithm.Over the course of iterations,alpha,beta,and delta wolves estimate the probable position of the prey.Each candidate solution updates its distance from the prey.The parameter a is decreased from 2to 0in order to emphasize exploration and exploitation,respectively.Candidate solutions tend to diverge from the prey when j ~A j >1and converge towards the prey when j ~A j <1.Finally,the GWO algorithm is terminated by the satisfaction of an end criterion.

The pseudo code of the GWO algorithm is presented in Fig.6.To see how GWO is theoretically able to solve optimization problems,some points may be noted:

The proposed social hierarchy assists GWO to save the best solutions obtained so far over the course of iteration.

The proposed encircling mechanism de?nes a circle-shaped neighborhood around the solutions which can be extended to higher dimensions as a hyper-sphere.

The random parameters A and C assist candidate solutions to have hyper-spheres with different random radii.

The proposed hunting method allows candidate solutions to locate the probable position of the prey.

Exploration and exploitation are guaranteed by the adaptive values of a and A .

The adaptive values of parameters a and A allow GWO to smoothly transition between exploration and exploitation.

With decreasing A ,half of the iterations are devoted to explora-tion (|A |P 1)and the other half are dedicated to exploitation (|A |<1).

The GWO has only two main parameters to be adjusted (a and C ).

There are possibilities to integrate mutation and other evolu-tionary operators to mimic the whole life cycle of grey wolves.

However,we have kept the GWO algorithm as simple as possible with the fewest operators to be adjusted.Such mechanisms are recommended for future work.The source codes of this algorithm can be found in https://www.wendangku.net/doc/0b8024798.html,/GWO.html and https://www.wendangku.net/doc/0b8024798.html,.au/matlabcentral/?leexchange/44974.4.Results and discussion

In this section the GWO algorithm is benchmarked on 29bench-mark functions.The ?rst 23benchmark functions are the classical functions utilized by many researchers [16,48–51,82].Despite the simplicity,we have chosen these test functions to be able to compare our results to those of the current meta-heuristics.These benchmark functions are listed in Tables 1–3where Dim indicates dimension of the function,Range is the boundary of the function’s search space,and f min is the optimum.The other test beds that we have chosen are six composite benchmark functions from a CEC 2005special ses-sion [52].These benchmark functions are the shifted,rotated,ex-panded,and combined variants of the classical functions which offer the greatest complexity among the current benchmark func-tions [53].Tables 4lists the CEC 2005test functions,where Dim indi-cates dimension of the function,Range is the boundary of the function’s search space,and f min is the optimum.Figs.7–10illustrate the 2D versions of the benchmark functions used.

Generally speaking,the benchmark functions used are minimi-zation functions and can be divided into four groups:unimodal,multimodal,?xed-dimension multimodal,and composite func-tions.Note that a detailed descriptions of the composite bench-mark functions are available in the CEC 2005technical report [52].The GWO algorithm was run 30times on each benchmark func-tion.The statistical results (average and standard deviation)are re-ported in Tables 5–8.For verifying the results,the GWO algorithm is compared to PSO [3]as an SI-based technique and GSA [24]as a physics-based algorithm.In addition,the GWO algorithm is com-pared with three EAs:DE [15],Fast Evolutionary Programing (FEP)[16],and Evolution Strategy with Covariance Matrix Adapta-tion (CMA-ES)[18].4.1.Exploitation analysis

According to the results of Table 5,GWO is able to provide very competitive results.This algorithm outperforms all others in F 1,F 2,and F 7.It may be noted that the unimodal functions are suitable for benchmarking exploitation.Therefore,these results show the superior performance of GWO in terms of exploiting the optimum.This is due to the proposed exploitation operators previously discussed.

4.2.Exploration analysis

In contrast to the unimodal functions,multimodal functions have many local optima with the number increasing exponentially with dimension.This makes them suitable for benchmarking

the

Table 1

Unimodal benchmark functions.Function

Dim Range f min f 1ex T?P n i ?1x

i 30[à100,100]0f 2ex T?P n i ?1j x i j tQ

n i ?1j x i j

[à10,10]0f 3ex T?P n i ?1eP i j à1x j T

30[à100,100]0f 4ex T?max i fj x i j ;16i 6n g

30[à100,100]0f 5ex T?P n à1i ?1?100ex i t1àx 2i T2

tex i à1T2

30[à30,30]0f 6ex T?P n

i ?1e?x i t0:5 T2

30[à100,100]0f 7ex T?P n i ?1ix 4

i trandom ?0;1T

[à1.28,1.28]

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exploration ability of an algorithm.According to the results of Tables 6and 7,GWO is able to provide very competitive results on the multimodal benchmark functions as well.This algorithm outperforms PSO and GSA on the majority of the multimodal func-tions.Moreover,GWO shows very competitive results compare to DE and FEP;and outperforms them occasionally.These results show that the GWO algorithm has merit in terms of exploration.

4.3.Local minima avoidance

The fourth class of benchmark functions employed includes composite functions,generally very challenging test beds for meta-heuristic algorithms.So,exploration and exploitation can be simultaneously benchmarked by the composite functions.Moreover,the local optima avoidance of an algorithm can be examined due to the massive number of local optima in such test functions.According to Table 8,GWO outperforms all others on half of the composite benchmark functions.The GWO algorithm also provides very competitive results on the remaining composite benchmark functions.This demonstrates that GWO shows a good balance between exploration and exploitation that results in high local optima avoidance.This superior capability is due to the adap-tive value of A .As mentioned above,half of the iterations are de-voted to exploration (|A |P 1)and the rest to exploitation (|A |<1).This mechanism assists GWO to provide very good explo-ration,local minima avoidance,and exploitation simultaneously.

4.4.Convergence behavior analysis

In this subsection the convergence behavior of GWO is investi-gated.According to Berg et al.[54],there should be abrupt changes in the movement of search agents over the initial steps of optimi-zation.This assists a meta-heuristic to explore the search space extensively.Then,these changes should be reduced to emphasize exploitation at the end of optimization.In order to observe the con-vergence behavior of the GWO algorithm,the search history and trajectory of the ?rst search agent in its ?rst dimension are illus-trated in Fig.11.The animated versions of this ?gure can be found in Supplementary Materials .Note that the benchmark functions are shifted in this section,and we used six search agents to ?nd the optima.

The second column of Fig.11depicts the search history of the search agents.It may be observed that the search agents of GWO tend to extensively search promising regions of the search spaces and exploit the best one.In addition,the fourth column of Fig.11shows the trajectory of the ?rst particle,in which changes of the ?rst search agent in its ?rst dimension can be observed.It can be seen that there are abrupt changes in the initial steps of iterations which are decreased gradually over the course of iterations.According to Berg et al.[54],this behavior can guarantee that a SI algorithm eventually convergences to a point in search space.To sum up,the results verify the performance of the GWO algo-rithm in solving various benchmark functions compared to well-known meta-heuristics.To further investigate the performance of

Table 2

Multimodal benchmark functions.

Function

Dim Range f min

F 8ex T?P

n i ?1àx i sin e???????j x i j p T30[à500,500]à418.9829?5F 9ex T?P n i ?1?x 2i à10cos e2p x i Tt10 30[à5.12,5.12]0F 10ex T?à20exp à0:2???????????????????1n P n i ?1x 2i q àexp 1n

P n i ?1cos e2p x i Tàát20te 30[à32,32]0F 11ex T?14000P n i ?1x 2i àQ n i ?1cos x i ?i

p t130[à600,600]0F 12ex T?p n f 10sin ep y 1Tt

P n à1i ?1ey i à1T2?1t10sin 2ep y i t1T tey n à1T2

g tP n i ?1u ex i ;10;100;4T30

[à50,50]

y i ?1t

x i t1

u ex i ;a ;k ;m T?k ex i àa Tm

x i >a 0àa

k eàx i àa Tm x i <àa

F 13ex T?0:1f sin 2e3p x 1TtP n i ?1ex i à1T2?1tsin 2e3p x i t1T tex n à1T2

?1tsin 2e2p x n T g tP n i ?1u ex i ;5;100;4T

30[à50,50]0F 14ex T?àP n i ?1sin ex i Tásin i :x

2i p 2m

;m ?10

30[0,p ]à4.687F 15ex T?e àP n

i ?1ex i =b T2m

à2e àP n i ?1x 2i h i áQ n i ?1cos 2

x i ;m ?5

30[à20,20]à1F 16ex T?f?P n i ?1sin 2ex i T àexp eàP n i ?1x 2i Tg áexp ?àP n i ?1sin 2

???????j x i j p 30

[à10,10]

à1

Table 3

Fixed-dimension multimodal benchmark functions.Function Dim

Range f min F 14ex T?

1500

tP 25

j ?1

j t

P 2

i ?1

ex i àa ij T6

à1

2[à65,65]1F 15ex T?P 11i ?1

a i àx 1e

b 2i tb i x 2Tb 2i tb i x 3tx 4

4[à5,5]0.00030F 16ex T?4x 21à2:1x 41t13x 61tx 1x 2à4x 22t4x 4

2[à5,5]à1.0316F 17ex T?x 2à5:1p 2x 21t5

p x 1à6

2t101à18p àácos x 1t102[à5,5]0.398F 18ex T??1tex 1tx 2t1T2e19à14x 1t3x 21à14x 2t6x 1x 2t3x 22T ??30te2x 1à3x 2T2?e18à32x 1t12x 21t48x 2à36x 1x 2t27x 22T

2[à2,2]3F 19ex T?àP 4i ?1c i exp eàP 3

j ?1a ij ex j àp ij T2T

3[1,3]à3.86F 20ex T?àP 4i ?1c i exp eàP 6j ?1a ij ex j àp ij T2

6[0,1]à3.32F 21ex T?àP 5i ?1?eX àa i TeX àa i TT

tc i

à14[0,10]à10.1532F 22ex T?àP 7i ?1?eX àa i TeX àa i TT

tc i

à14[0,10]à10.4028F 23ex T?àP 10i ?1?eX àa i TeX àa i TT

tc i

à14

[0,10]

à10.5363

S.Mirjalili et al./Advances in Engineering Software 69(2014)46–61

the proposed algorithm,three classical engineering design prob-lems and a real problem in optical engineering are employed in the following sections.The GWO algorithm is also compared with well-known techniques to con?rm its results.

5.GWO for classical engineering problems

In this section three constrained engineering design problems: tension/compression spring,welded beam,and pressure vessel designs,are employed.These problems have several equality and inequality constraints,so the GWO should be equipped with a con-straint handling method to be able to optimize constrained prob-lems as well.Generally speaking,constraint handling becomes very challenging when the?tness function directly affects the posi-tion updating of the search agents(GSA for instance).For the?tness independent algorithms,however,any kind of constraint handling can be employed without the need to modify the mechanism of the algorithm(GA and PSO for instance).Since the search agents of the proposed GWO algorithm update their positions with respect

Table4

Composite benchmark functions.

Function Dim Range f min

F24(CF1):

f1,f2,f3,...,f10=Sphere Function10[à5,5]0?,1;,2;,3;...;,10 ??1;1;1;...;1

[k1,k2,k3...,k10]=[5/100,5/100,5/100,...,5/100]

F25(CF2):

f1,f2,f3,...,f10=Griewank’s Function10[à5,5]0?,1;,2;,3;...;,10 ??1;1;1;...;1

[k1,k2,k3,...,k10]=[5/100,5/100,5/100,...,5/100]

F26(CF3):

f1,f2,f3,...,f10=Griewank’s Function10[à5,5]0?,1;,2;,3;...;,10 ??1;1;1;...;1

[k1,k2,k3,...,k10]=[1,1,1, (1)

F27(CF4):

f1,f2=Ackley’s Function10[à5,5]0 f3,f4=Rastrigin’s Function

f5,f6=Weierstras’s Function

f7,f8=Griewank’s Function

f9,f10=Sphere Function

?,1;,2;,3;...;,10 ??1;1;1;...;1

[k1,k2,k3,...,k10]=[5/32,5/32,1,1,5/0.5,5/0.5,5/100,5/100,5/100,5/100]

F28(CF5):

f1,f2=Rastrigin’s Function10[à5,5]0 f3,f4=Weierstras’s Function

f5,f6=Griewank’s Function

f7,f8=Ackley’s Function

f9,f10=Sphere Function

?,1;,2;,3;...;,10 ??1;1;1;...;1

[k1,k2,k3,...,k10]=[1/5,1/5,5/0.5,5/0.5,5/100,5/100,5/32,5/32,5/100,5/100]

f29(CF6):

f1,f2=Rastrigin’s Function10[à5,5]0 f3,f4=Weierstras’s Function

f5,f6=Griewank’s Function

f7,f8=Ackley’s Function

f9,f10=Sphere Function

?,1;,2;,3;...;,10 ??0:1;0:2;0:3;0:4;0:5;0:6;0:7;0:8;0:9;1

[k1,k2,k3,...,k10]=[0.1?1/5,0.2?1/5,0.3?5/0.5,0.4?5/0.5,0.5?5/100,0.6?5/100,0.7?5/32,0.8?5/32,0.9?5/100,1?5/100]

(F1)(F2)(F3) (F4)

(F5)(F6)(F7)

Fig.7.2-D versions of unimodal benchmark functions.

S.Mirjalili et al./Advances in Engineering Software69(2014)46–6153

to the alpha,beta,and delta locations,there is no direct relation be-tween the search agents and the?tness function.So the simplest constraint handling method,penalty functions,where search agents are assigned big objective function values if they violate any of the constraints,can be employed effectively to handle constraints in GWO.In this case,if the alpha,beta,or delta violate constraints,they

(F8)(F9)(F10) (F11)

(F12) (F13)

Fig.8.2-D versions of multimodal benchmark functions.

(F14) (F16)(F17)(F18)

Fig.9.2-D version of?xed-dimension multimodal benchmark functions.

(F24) (F25)(F26)

(F27) (F28)

(F29)

Fig.10.2-D versions of composite benchmark functions.

Table5

Results of unimodal benchmark functions.

F GWO PSO GSA DE FEP

Ave Std Ave Std Ave Std Ave Std Ave Std F1 6.59Eà28 6.34Eà050.0001360.000202 2.53Eà169.67Eà178.2Eà14 5.9Eà140.000570.00013 F27.18Eà170.0290140.0421440.0454210.0556550.194074 1.5Eà099.9Eà100.00810.00077 F3 3.29Eà0679.1495870.1256222.11924896.5347318.9559 6.8Eà117.4Eà110.0160.014 F4 5.61Eà07 1.315088 1.0864810.3170397.35487 1.741452000.30.5 F526.8125869.9049996.7183260.1155967.5430962.2253400 5.06 5.87 F60.8165790.0001260.0001028.28Eà05 2.5Eà16 1.74Eà160000 F70.0022130.1002860.1228540.0449570.0894410.043390.004630.00120.14150.3522 54S.Mirjalili et al./Advances in Engineering Software69(2014)46–61

are automatically replaced with a new search agent in the next iter-ation.Any kind of penalty function can readily be employed in order to penalize search agents based on their level of violation.In this case,if the penalty makes the alpha,beta,or delta less?t than any other wolves,it is automatically replaced with a new search agent in the next iteration.We used simple,scalar penalty functions for the rest of problems except the tension/compression spring design problem which uses a more complex penalty function.

5.1.Tension/compression spring design

The objective of this problem is to minimize the weight of a ten-sion/compression spring as illustrated in Fig.12[55–57].The min-imization process is subject to some constraints such as shear stress,surge frequency,and minimum de?ection.There are three variables in this problem:wire diameter(d),mean coil diameter (D),and the number of active coils(N).The mathematical formula-tion of this problem is as follows:

Consider~x??x1x2x3 ??dDN ;

Minimize fe~xT?ex3t2Tx2x2

;

Subject to g

1e~xT?1àx32x3

71785x4

60;

g 2e~xT?4x22àx1x2

12566ex2x3

àx4

5108x2

60;

g 2e~xT?4x22àx1x2

12566ex2x3

àx4

5108x2

60;

g 3e~xT?1à140:45x1

60;

e~xT?x1tx2à160;

Variable range0:056x162:00;

0:256x261:30;

2:006x3615:0

e5:1T

This problem has been tackled by both mathematical and heuristic approaches.Ha and Wang tried to solve this problem using PSO [58].The Evolution Strategy(ES)[59],GA[60],Harmony Search (HS)[61],and Differential Evolution(DE)[62]algorithms have also been employed as heuristic optimizers for this problem.The mathematical approaches that have been adopted to solve this problem are the numerical optimization technique(constraints correction at constant cost)[55]and mathematical optimization technique[56].The comparison of results of these techniques and GWO are provided in Table9.Note that we use a similar penalty function for GWO to perform a fair comparison[63].Table9 suggests that GWO?nds a design with the minimum weight for this problem.

5.2.Welded beam design

The objective of this problem is to minimize the fabrication cost of a welded beam as shown in Fig.13[60].The constraints are as follows:

Shear stress(s).

Bending stress in the beam(h).

Buckling load on the bar(P c).

End de?ection of the beam(d).

Side constraints.

This problem has four variables such as thickness of weld(h), length of attached part of bar(l),the height of the bar(t),and thick-ness of the bar(b).The mathematical formulation is as follows: Consider~x??x1x2x3x4 ??hltb ;

Minimizeef~xT?1:10471x2

x2t0:04811x3x4e14:0tx2T;

Subject to g

e~xT?se~xTàs max60;

e~xT?re~xTàr max60;

e~xT?de~xTàd max60;

e~xT?x1àx460;

e~xT?PàP ce~xT60;

e~xT?0:125àx160

e~xT?1:10471x2

t0:04811x3x4e14:0tx2Tà5:060

e5:2T

Variable range0:16x162;

0:16x2610;

0:16x3610;

0:16x462

Table6

Results of multimodal benchmark functions.

F GWO PSO GSA DE FEP

Ave Std Ave Std Ave Std Ave Std Ave Std

F8à6123.1à4087.44à4841.291152.814à2821.07493.0375à11080.1574.7à12554.552.6 F90.31052147.3561246.7042311.6293825.968417.47006869.238.80.0460.012 F10 1.06Eà130.0778350.2760150.509010.0620870.236289.7Eà08 4.2Eà080.0180.0021 F110.0044850.0066590.0092150.00772427.70154 5.040343000.0160.022 F120.0534380.0207340.0069170.026301 1.7996170.951147.9Eà158Eà159.2Eà06 3.6Eà06 F130.6544640.0044740.0066750.0089078.8990847.126241 5.1Eà14 4.8Eà140.000160.000073

Table7

Results of?xed-dimension multimodal benchmark functions.

F GWO PSO GSA DE FEP

Ave Std Ave Std Ave Std Ave Std Ave Std

F14 4.042493 4.252799 3.627168 2.560828 5.859838 3.8312990.998004 3.3Eà16 1.220.56 F150.0003370.0006250.0005770.0002220.0036730.001647 4.5Eà140.000330.00050.00032 F16à1.03163à1.03163à1.03163 6.25Eà16à1.03163 4.88Eà16à1.03163 3.1Eà13à1.03 4.9Eà07 F170.3978890.3978870.39788700.39788700.3978879.9Eà090.398 1.5Eà07 F18 3.00002833 1.33Eà153 4.17Eà1532Eà15 3.020.11 F19à3.86263à3.86278à3.86278 2.58Eà15à3.86278 2.29Eà15N/A N/Aà3.860.000014 F20à3.28654à3.25056à3.266340.060516à3.317780.023081N/A N/Aà3.270.059 F21à10.1514à9.14015à6.8651 3.019644à5.95512 3.737079à10.15320.0000025à5.52 1.59 F22à10.4015à8.58441à8.45653 3.087094à9.68447 2.014088à10.4029 3.9Eà07à5.53 2.12 F23à10.5343à8.55899à9.95291 1.782786à10.5364 2.6Eà15à10.5364 1.9Eà07à6.57 3.14

S.Mirjalili et al./Advances in Engineering Software69(2014)46–6155

where

s e~x T?????????????????????????????????????????????????

es 0T2t2s 0s 00x 2

tes 00T2q ;s 0?P ??2p x 1x 2

;s 00?MR ;M ?P eL tx 2

T;R ?

?????????????????????????

x 2

4tx 1tx 32àá2q ;J ?2???2p x 1x 2x 22tx 1t

x 3àá2

h i n o ;r e~x T?6PL x 4x

;d e~x T?6PL 3

Ex 23

x 4P c e~x T?

4:013E

??????

x 23x 6

436

q L 1àx

????E

;P ?6000lb ;L ?14in :;d max ?0:25in :;E ?30?16psi ;G ?12?106psi ;

in Fig.14.Both ends of the vessel are capped,and the head has a

hemi-spherical shape.There are four variables in this problem: Thickness of the shell (T s ). Thickness of the head (T h ). Inner radius (R ).

Length of the cylindrical section without considering the head (L ).

Table 8

Results of composite benchmark functions.F

GWO PSO GSA

DE CMA-ES Ave

Std Ave

Std Ave Std Ave Std Ave Std F 2443.8354469.8614610081.65 6.63E à17 2.78E à17 6.75E à02 1.11E à01100188.56F 2591.8008695.5518155.9113.176200.620267.7208728.7598.6277161.99151F 2661.4377668.68816172.0332.76918091.89366144.4119.401214.0674.181F 27123.1235163.9937314.320.06617082.32726324.8614.784616.4671.92F 28102.142981.2553683.45101.11200

47.1404510.789 2.604358.3168.26F 29

43.14261

84.48573

861.42

125.81

142.0906

88.87141

490.94

39.461

900.26

8.32E à02

Fig.11.Search history and trajectory of the ?rst particle in the ?rst dimension.

S.Mirjalili et al./Advances in Engineering Software 69(2014)46–61

This problem has also been popular among researchers and optimized in various studies.The heuristic methods that have been adopted to optimize this problem are:PSO[58],GA[57,60,69],ES [59],DE[62],and ACO[70].Mathematical methods used are aug-mented Lagrangian Multiplier[71]and branch-and-bound[72].The results of this problem are provided in Table11.According to this ta-ble,GWO is again able to?nd a design with the minimum cost.

In summary,the results on the three classical engineering prob-lems demonstrate that GWO shows high performance in solving challenging problems.This is again due to the operators that are designed to allow GWO to avoid local optima successfully and con-verge towards the optimum quickly.The next section probes the performance of the GWO algorithm in solving a recent real prob-lem in the?eld of optical engineering.

Fig.11(continued)

Fig.12.Tension/compression spring:(a)shematic,(b)stress heatmap(c)displacement heatmap.

6.Real application of GWO in optical engineering (optical buffer design)

The problem investigated in this section is called optical buffer design.In fact,an optical buffer is one of the main components of optical CPUs.The optical buffer slows the group velocity of light and allows the optical CPUs to process optical packets or adjust its timing.The most popular device to do this is a Photonic Crystal Waveguide (PCW).PCWs mostly have a lattice-shaped structure with a line defect in the middle.The radii of holes and shape of the line defect yield different slow light characteristics.Varying ra-dii and line defects provides different environments for refracting the light in the waveguide.The researchers in this ?eld try to manipulate the radii of holes and pins of line defect in order to achieve desirable optical buffering characteristics.There are also different types of PCW that are suitable for speci?c applications.In this section the structure of a PCW called a Bragg Slot PCW (BSPCW)is optimized by the GWO algorithm.This problem has several constraints,so we utilize the simplest constraint handling method for GWO in this section as well.

BSPCW structure was ?rst proposed by Caer et al.in 2011[73].The structure of BSPCWs is illustrated in Fig.15.The background slab is silicon with a refractive index equal to 3.48.The slot and There are two metrics for comparing the performance of slow light devices:Delay-Bandwidth Product (DBP)and Normalized DBP (NDBP),which are de?ned as follows [74]:

DBP ?D t áD f e6:1T

where D t indicates the delay and D f is the bandwidth of the slow light device.

In slow light devices the ultimate goal is to achieve maximum transmission delay of an optical pulse with highest PCW band-width.Obviously,D t should be increased in order to increase DBP.This is achieved by increasing the length of the device (L ).To compare devices with different lengths and operating frequen-cies,NDBP is a better choice [75]:

NDBP ?n g áD x =x 0

e6:2T

where ng is the average of the group index,D x is the normalized bandwidth,and x 0is the normalized central frequency of light wave.

Since NDBP has a direct relation to the group index (n g ),can be formulated as follows [76]:

n g ?

v g

d k d x

e6:3T

Table 9

Comparison of results for tension/compression spring design problem.Algorithm

Optimum variables Optimum weight d

D N

GWO 0.051690.35673711.288850.012666GSA

0.0502760.32368013.5254100.0127022PSO (Ha and Wang)0.0517280.35764411.2445430.0126747ES (Coello and Montes)0.0519890.36396510.8905220.0126810GA (Coello)

0.0514800.35166111.6322010.0127048HS (Mahdavi et al.)0.0511540.34987112.0764320.0126706DE (Huang et al.)0.0516090.35471411.4108310.0126702Mathematical optimization (Belegundu)

0.053396

0.399180

9.1854000

0.0127303

Constraint correction (Arora)

0.0500000.31590014.2500000.0128334

Fig.13.Structure of welded beam design (a)shematic (b)stress heatmap (c)displacement heatmap.

Table 10

Comparison results of the welded beam design problem.

Algorithm

Optimum variables Optimum cost h

b GWO 0.205676 3.4783779.036810.205778 1.72624GSA

0.182129 3.85697910.000000.202376 1.879952GA (Coello)N/A N/A N/A N/A 1.8245GA (Deb)N/A N/A N/A N/A 2.3800GA (Deb)0.2489 6.17308.17890.2533 2.4331HS (Lee and Geem)0.2442 6.22318.29150.2443 2.3807Random 0.4575 4.7313 5.08530.6600 4.1185Simplex 0.2792 5.62567.75120.2796 2.5307David 0.2434 6.25528.29150.2444 2.3841APPROX

0.2444

6.2189

8.2915

0.2444

2.3815

58S.Mirjalili et al./Advances in Engineering Software 69(2014)46–61

tion rage of ±10%[75].Detailed information about PCWs can be found in [77–80].

Finally,the problem is mathematically formulated for GWO as follows:

Consider :

~x ??x 1x 2x 3x 4x 5x 6x 7x 8 ?

R 2R 3R 4R 5l w h w l ??;

Maximize :

f e~x T?NDBP ?n g

D x x 0;

Subject to :max ej b 2ex TjT<106a =2p c 2;

x H max ex down band T;

k n >k nH !x Guided mode >x H ;k n

e6:5T

where :x H ?x ek nH T?x e1:1n g 0T;

x L ?x ek nL T?x e0:9n g 0T;

k n ?ka

D x ?x H àx L ;

a ?x 0?1550enm T;

Variable range :06x 1à560:5;

06x 661;

06x 7;861;

Note that we consider ?ve constraints for the GWO algorithm.The second to ?fth constraints avoid band mixing.To handle feasi-bility,we assign small negative objective function values (à100)to those search agents that violate the constraints.

The GWO algorithm was run 20times on this problem and the best results obtained are reported in Table 12.Note that the algo-rithm was run by 24CPUs on a Windows HPC cluster at Grif?th University.This table shows that there is a substantial,93%and 65%improvement in bandwidth (D k )and NDBP utilizing the GWO algorithm.

The photonic band structure of the BSPCW optimized is shown in Fig.16(a).In addition,the corresponded group index and opti-mized super cell are shown in Figs.16(b)and 17.These ?gures show that the optimized structure has a very good bandwidth without band mixing as well.This again demonstrated the high performance of the GWO algorithm in solving real problems.

This comprehensive study shows that the proposed GWO algo-rithm has merit among the current meta-heuristics.First,the

re-

Fig.14.Pressure vessel (a)shematic (b)stress heatmap (c)displacement heatmap.

Table 11

Comparison results for pressure vessel design problem.Algorithm

Optimum variables Optimum cost

T s

T h

GWO 0.8125000.43450042.089181176.7587316051.5639GSA

1.1250000.62500055.988659884.45420258538.8359PSO (He and Wang)0.8125000.4375004

2.091266176.7465006061.0777GA (Coello)

0.8125000.43450040.323900200.0000006288.7445GA (Coello and Montes)0.8125000.43750042.097398176.6540506059.9463GA (Deb and Gene)0.9375000.50000048.329000112.6790006410.3811ES (Montes and Coello)0.8125000.43750042.098087176.6405186059.7456DE (Huang et al.)

0.8125000.43750042.098411176.6376906059.7340ACO (Kaveh and Talataheri)0.8125000.43750042.103624176.5726566059.0888Lagrangian Multiplier (Kannan) 1.1250000.62500058.29100043.69000007198.0428Branch-bound (Sandgren)

1.125000

0.625000

47.700000

117.701000

8129.1036

Table 12

Structural parameters and calculation results.Structural parameter Wu et al .[81]GWO R 1–

0.33235a R 2–0.24952a R 3–0.26837a R 4–0.29498a R 5–0.34992a l –0.7437a W h –0.2014a W l

–0.60073a a (nm)430343 n

g 2319.6D k (nm)

17.633.9Order of magnitude of b 2(a/2p c 2)103103NDBP

0.26

0.43

sults of the unconstrained benchmark functions demonstrate the performance of the GWO algorithm in terms of exploration,exploi-tation,local optima avoidance,and convergence.Second,the re-sults of the classical engineering problems show the superior performance of the proposed algorithm in solving semi-real con-strained problems.Finally,the results of the optical buffer design problem show the ability of the GWO algorithm in solving the real problems.

7.Conclusion

This work proposed a novel SI optimization algorithm inspired by grey wolves.The proposed method mimicked the social hierar-chy and hunting behavior of grey wolves.Twenty nine test func-tions were employed in order to benchmark the performance of the proposed algorithm in terms of exploration,exploitation,local optima avoidance,and convergence.The results showed that GWO was able to provide highly competitive results compared to well-known heuristics such as PSO,GSA,DE,EP,and ES.First,the results on the unimodal functions showed the superior exploitation of the GWO algorithm.Second,the exploration ability of GWO was con-?rmed by the results on multimodal functions.Third,the results of the composite functions showed high local optima avoidance.Fi-nally,the convergence analysis of GWO con?rmed the convergence of this algorithm.

Moreover,the results of the engineering design problems also showed that the GWO algorithm has high performance in un-known,challenging search spaces.The GWO algorithm was?nally applied to a real problem in optical engineering.The results on this problem showed a substantial improvement of NDBP compared to current approaches,showing the applicability of the proposed algorithm in solving real problems.It may be noted that the results on semi-real and real problems also proved that GWO can show high performance not only on unconstrained problems but also on constrained problems.

For future work,we are going to develop binary and multi-objective versions of the GWO algorithm.

Appendix A.Supplementary material

Supplementary data associated with this article can be found,in the online version,at https://www.wendangku.net/doc/0b8024798.html,/10.1016/j.advengsoft.2013.

12.007.

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What’s journal publishing really about? Certainly it’s about a process, a well-established formalized system for communicating research results. But it’s not just about systems and technology. Most importantly, it’s about people and relationships. Journal publishing has stood the test of time since at least the mid-17th century when Denis de Sallo founded the Journal des S?avans and Henry Oldenburg founded the journal Philosophical Transactions. What has made journal publishing truly valuable and enduring is human communication, the comments and ideas exchanged between authors, editors, reviewers and publishers. Ultimately, journal publishing cannot be a purely mechanical exercise. To ensure research articles Mayur Amin deliver high-quality information and serve as building blocks for intellectual advancement and scientific discovery, researchers must submit their work to trusted and reputable journals which bring together experts – editors, advisory boards and reviewers – to see and comment on the work. To ensure we collectively achieve excellence and advancement, authors, reviewers and editors must deliberate and communicate regarding submitted papers and so improve their final versions, and publishers must ensure that high-quality literature is published and preserved for the future. As we publish journals, Elsevier provides the following valuable functions – unchanged since the days of Sallo and Oldenburg:■Registration: Date-stamping the research of a particular author to establish precedence ■Peer review: Employing a wholly independent peer-review process ■Dissemination: Broadcasting authors’ claims to peers and the media ■Archival record: Establishing a permanent record of authors’ findings While providing the four core services traditionally associated with scholarly publishing, Elsevier also offers value-adding services – especially relating to our electronic products. Today these services include helping define new disciplines and providing forums for their debate and discourse; establishing and actively managing editorial boards; and investing in new technologies and partnerships that make new and archived research more accessible to a broad range of users. Whether we’re discussing the core functions or the value-adding services of journal publishing, we can see that people and relationships are at the heart of the enterprise. Through dialog and collaboration, authors, editors and reviewers working with Elsevier help us ensure the integrity of content we publish, encourage and introduce innovation, and guard our collective intellectual heritage. In this fast-changing world, no one can rest on what has been accomplished thus far. Elsevier will continue to listen, learn, test and adapt to meet the changing needs of our customers and the scholarly and scientific communities we serve. If you’re considering publishing with Elsevier, I hope this pamphlet provides useful information. If you’re already among librarians, authors, reviewers and editors working with Elsevier, I thank you for your confidence in our company and invite your input on how we can serve you better. My appreciation to all colleagues who've contributed to this pamphlet. Kind regards, Mayur Amin, Senior Vice President, Global Academic & Customer Relations Department, Elsevier, Oxford, UK

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2 （二）润色流程 1、只有两人能看到您的稿件：总编和语言编辑。他们都签署了保密协议。 2、总编将审阅您的稿件，并会为您指定一位具有相关科学领域背景的语言编辑为您的稿件润色。您的稿件会在总编审阅完成后交付于您。您可跟踪查看都发生了哪些变化。 3、在五个工作日内，您将通过电子邮箱收到润色后的文档。（三）我们的编辑爱思唯尔语言润色编辑，全部由英语为母语，有优秀英语语言技能的博士学位获得者或博士研究生组成。为了满足我们的需求，他们接受广泛的培训并且经常参与绩效考评。所有的执行编辑都拥有学术背景。 ? 爱思唯尔语言润色编辑，服务于超过160个国家，覆盖世界所有的研究领域； ? 爱思唯尔拥有一个由上千名高质量、以英语为母语、有相关专业研究背景的编辑组成的编辑网络； ? 爱思唯尔编辑网络可以扩大到任何方向，完成来自学术界的任何需求； ? 客户满意调查显示出很高的满意度和推荐比率； ? 只有最优秀的人才会承担编辑工作，持续不断地分享经验，以保持很高的水准； ? 质量评估和用户体验调查是我们服务的核心。（四）字数统计准则 1、我们并不帮助删减您的手稿及彻底重写您的文字; 然而,我们的编辑可能会提醒您手稿中一些重复的可以安全删除的内容。 2、我们不会修改参考文献以及数据表格的内容, 所以计费时它们并不被包含在字数统计里。我如何得到最准确的字数统计？为了准确的计算字数，您应该备份您的论文，删除您不希望我们进行编辑的部分。如果您提交编辑请求但不想您的引用部分被编辑，您可以直接删除您的引文列表。您还可以

Elsevier投稿指南

Elsevier期刊网上投稿指南浙江工业大学图书馆信息咨询部编写 Elsevier是荷兰一家全球著名的学术期刊出版商，每年出版大量的农业和生物科学、化学和化工、临床医学、生命科学、计算机科学、地球科学、工程、能源和技术、环境科学、材料科学、航空航天、天文学、物理、数学、经济、商业、管理、社会科学、艺术和人文科学类的学术图书和期刊，出版的1800种期刊中大部分期刊是被SCI、SSCI、EI收录的核心期刊，其中SCI、SSCI收录期刊1,221种，EI收录期刊515种，社科类期刊数量为255种（SCI、SSCI收录期刊152种）、科技类期刊数量1,302种（SCI收录期刊1069种）是世界上公认的高品位学术期刊。 Elsevier期刊提供了在线投稿的作者服务（Author gateway），服务网址：https://www.wendangku.net/doc/0b8024798.html,，可以查看所有期刊的投稿要求、编委组成、投稿渠道等信息。 Elsevier以支持全球学术交流为宗旨，不向作者收取刊登论文的版面费（注：当作者希望在文章中刊登彩色图片，如仅为电子版本也免收版面费，但如果作者在纸本期刊中仍要求彩色插图，各期刊会酌情收取费用）。对于非英语国家和地区的作者，如果在英文撰稿方面存在语言困难，爱思唯尔向您建议专业的语言校对中心，他们可以帮助您进行语言校对和稿件加工。详细情况，可以点击https://www.wendangku.net/doc/0b8024798.html,/LanguageEditing.html，或者与作者支持部门联系（邮件地址：authorsupport@https://www.wendangku.net/doc/0b8024798.html,）。图1：Elsevier期刊投稿主页作者指南网上投稿向Elsevier期刊投稿，有几种情况：一是在作者服务网址（author gateway）上一次注册后，点击submit online to this journal按钮就可以对多种期刊直接进行网上投稿；二是有些刊不接受author gateway上注册的作者对该刊进行网上投稿，也就是说在点击submit online to

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ScienceDirect ——数据库培训
白琮博士
励德爱思唯尔信息技术(北京有限公司励德爱思唯尔信息技术北京)有限公司北京 2013年10月年月
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内容提要 1. 了解Elsevier 了解 -Elsevier出版社出版社 -Elsevier旗舰产品旗舰产品-ScienceDirect 旗舰产品
2. ScienceDirect的使用的使用
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ScienceDirect是什么是什么? 是什么
世界上最著名的科技和医学信息传递系统世界上最著名的科技和医学信息传递系统全球使用者超过 110万万从上线至今文章下载量超过 10亿亿每秒的全文下载数平均 36次次文章被接收15天文章被接收天后即出现在平台
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ScienceDirect的范围的范围: 的范围
出版文章的比例涉及到的学科领域
Environmental Sciences Sciences Earth
Social Sciences
26%
Others Others
26%
Elsevier
Maths & computer science
Life sciences
Physics
Chemistry & Chemical Engineering
Health sciences
Materials Science & Engineering
>10,000,000 篇的英文论文通过 Elsevier发表发表
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