island-based harmony search for optimization problems
TRANSCRIPT
Expert Systems with Applications 42 (2015) 2026–2035
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Expert Systems with Applications
journal homepage: www.elsevier .com/locate /eswa
Island-based harmony search for optimization problems
http://dx.doi.org/10.1016/j.eswa.2014.10.0080957-4174/� 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.E-mail addresses: [email protected] (M.A. Al-Betar), ma.awadallah@alaqsa.
edu.ps (M.A. Awadallah), [email protected] (A.T. Khader), [email protected] (Z.A. Abdalkareem).
Mohammed Azmi Al-Betar a,⇑, Mohammed A. Awadallah c, Ahamad Tajudin Khader b,Zahraa Adnan Abdalkareem d
a Department of Information Technology, Al-Huson University College, Al-Balqa Applied University, P.O. Box 50, Al-Huson, Irbid, Jordanb School of Computer Sciences, Universiti Sains Malaysia, 11800 Pinang, Malaysiac Faculty of Computer Science, Al-Aqsa University, P.O. Box 4051, Gaza, Palestined Department of Quality Assurance and Performance, College of Imam Azam University, P.O. Box 72002, Baghdad, Iraq
a r t i c l e i n f o
Article history:Available online 23 October 2014
Keywords:Harmony searchIsland modelStructured populationDiversity
a b s t r a c t
Harmony search (HS) algorithm is a recent meta-heuristic algorithm that mimics the musical improvisa-tion concepts. This algorithm has been widely used for solving optimization problems. Moreover, manymodifications in this algorithm have been carried out in order to improve the performance of the search.Island model is a structured population mechanism used in evolutionary algorithms to preserve thediversity of the population and thus improve the performance. In this paper, the island model conceptsare embedded into the main framework of HS algorithm to improve its convergence properties where thenew method is refer to as island HS (iHS). In the proposed method, the individuals in population are dis-tributed into separate sub-population named (islands). Then the breeding loop is separately involved ineach island. After specific generations, a number of individuals run an exchange through a process calledmigration. This process is performed to keep the diversity of population and to allow islands to interactwith each other. The experimental result using a set of benchmark function shows that the island modelcontext is crucial to the performance of iHS. Finally the sensitivity analysis and the comparative study foriHS prove the efficiency of the island model.
� 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Harmony search (HS) algorithm, a recent Evolutionary Algo-rithm (EA), was proposed by Geem, Kim, and Loganathan (2001)to emulate the musical phenomena of the improvisation process.In musical rehearsal, a group of musicians play the tunes of theirmusical tools, practice after practice to formulate a pleasing har-mony. Analogously in optimization, a set of variables, taken selec-tive values, iteration by iteration, to formulate most probably anoptimal solution. The set of successful stories introduced by adapt-ing HS algorithm to a wide variety of optimization problems getscredit from the emergence of the tremendous research tendencyto the domain. Some examples that adopted HS solutions includeEngineering, timetabling, nurse rostering, space allocations, bioin-formatics, image processing (Abual-Rub, Al-Betar, Abdullah, &Khader, 2012; Al-Betar & Khader, 2012; Al-Betar, Khader, &Zaman, 2012b; Alkareem, Venkat, Al-Betar, & Khader, 2012;Awadallah, Khader, Al-Betar, & Bolaji, 2013, 2012; Geem, Yang, &
Tseng, 2013; Landa-Torres, Manjarres, Salcedo-Sanz, Del Ser, &Gil-Lopez, 2013), and many others as recorded in Manjarres et al.(2013).
The main merits of HS over other optimization methods aresummarized as follows: a novel stochastic derivative is embeddedwithin the HS (Geem, 2008); it needs less mathematical require-ments which iteratively generate a new solution after manipulat-ing all existing solutions (Mahdavi, Fesanghary, & Damangir,2007). Put simply, it is simple, flexible, adaptable, general, andscalable (Al-Betar, Khader, Geem, Doush, & Awadallah, 2013b).However, the performance of HS has continuously attractedresearcher attention account for the optimization problems combi-natorial nature (Alia & Mandava, 2011). Therefore, the HS theoryhas been improved by either replacing, adding, tailoring its opera-tors or hybridizing HS with other effective algorithms (Al-Betar,Khader, & Doush, 2014; Awadallah, Khader, Al-Betar, & Bolaji,2014; Maheri & Narimani, 2014; Zhao, Suganthan, Pan, & FatihTasgetiren, 2011). Furthermore, adaptive parameters of HS havebeen also studied (Geem & Sim, 2010; Gholizadeh & Barzegar,2013). The majority of improvements in HS performance haveadjusted the process of its operators to cope with the ‘‘survival ofthe fittest’’ principle of natural selection (Al-Betar, Doush,Khader, & Awadallah, 2012a; Xiang, An, Li, He, & Zhang, 2014).
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One of the shortcomings of simple HS algorithm is its inabilityto maintain diversity of population due to genetic drift (Al-Betar,Khader, Awadallah, Alawan, & Zaqaibeh, 2013a). The structuredpopulation mechanisms has recently captured the EA researcherto improve the performance by means of maintaining the diversityamong the population members during the search, thus the conicalpremature convergence can be avoided. In structured population,instead of all the other solutions in the population being treatedas potential mates as in panmictic populations, only those thatare in the same neighborhood can share their information andinteract (Tomassini, 2005). The island model is the most popularstructured population EA method that divides the whole individu-als of a large panmictic population into smaller independent sub-groups called ‘‘islands’’ (Kushida, Hara, Takahama, & Kido, 2013).The EA method is run to each subgroup independently and theinteraction between subgroups is achieved through the migrationprocess which will be periodically performed. Consequently, theisland EA can preserve the population diversity and can be imple-mented into parallel machines.
In this paper, the concepts of island model have been embeddedwith the framework of HS algorithm in a bid to improve thediversity of population concepts and trigger the new ‘‘ island HSalgorithm (iHS)’’. In iHS, the individuals in the Harmony Memory(HM) are divided into several islands. The individuals on eachisland are independently evolved using HS operators. The islandsinteract using a migration process depending on a random ringtopology. The IEEE-CEC2005 mathematical optimization functionshave been used for evaluation purposes. The results suggest thatincorporating the island model within the HS framework preservesthe population diversity and therefore the performance is directlyimproved.
The remaining part of this paper is arranged as follows:The background to the harmony search and to island model ispresented in Section 2. The proposed iHS is thoroughly explainedin Section 4. Experiments and comparative results are presentedin Section 5. Finally, the a conclusion and some useful researchindications are given in Section 6.
2. Background
The global optimization problems are normally formulated interms of objective function as follows:
minff ðxÞjx 2 Xg;
where f ðxÞ is the objective function; x ¼ fxiji ¼ 1; . . . ;Ng is the set ofdecision variables. X ¼ fX iji ¼ 1; . . . ;Ng is the possible value rangefor each decision variable, where Xi 2 ½LBi;UBi�, where LBi and UBi
are the lower and upper bounds for the decision variable xi respec-tively and N is the number of decision variables.
2.1. Harmony search (HS) algorithm
HS has five main procedural steps summarized in Algorithm 1and described as follows:
Step 1: Initialize the parameters. The parameters of the HSalgorithm required to solve the optimization problem arespecified in this step: the Harmony Memory ConsiderationRate (HMCR) which determines the rate of selecting thevalue from the memory; the Harmony Memory Size(HMS) is similar to the population size in other EAs; PitchAdjustment Rate (PAR) determines the probability of localimprovement; the fret width (FW), determines the dis-tance of adjustment, and Number of Improvisations (NI)or number of iterations.
Step 2: Initialize the harmony memory. The harmony memory(HM) is a repository of the population individuals,HM ¼ x1; x2; . . . ; xHMS
� �T, of size HMS. In this step, theseindividuals are randomly generated as follows:xj
i ¼ LBiþðUBi� LBiÞ�Uð0;1Þ; 8i¼ 1;2; . . . ;N and 8j ¼ 1;2;. . . ;HMS, and Uð0;1Þ generate a uniform random numberbetween 0 and 1.
Step 3: Improvise a new harmony. A new harmony vector isgenerated, x0 ¼ ðx01; x02; . . . ; x0NÞ, based on three operators:(1) memory consideration (MC), (2) pitch adjustment(PA), and (2) random consideration (RC). The threeoperators assign a value for each decision variable x0i inthe new harmony as formulated in Eq. (1).
x0i x0i 2 x1
i ;x2i ; . . . ;x
HMSi
� �w:p: HMCR�ð1�PARÞ fMCg
x0i¼x0iþUð�1;1Þ�FW w:p: HMCR�PAR fPAgx0i 2Xi w:p: 1 - HMCR fRCg
8><>:
ð1Þ
: Update the harmony memory. The new harmony vector,
Step 4x0 ¼ ðx01; x02; . . . ; x0NÞ, replaces the worst harmony xworststored in HM if better.Step 5: Check the stop criterion. Step 3 and step 4 of HS algorithm
are repeated until the stop criterion (Normally it dependson NI) is met.
Algorithm 1. Harmony search algorithm
Set HMCR, PAR, NI, HMS, FW.
xji ¼ LBi þ ðUBi � LBiÞ � Uð0;1Þ; 8i ¼ 1;2; . . . ;N and8j ¼ 1;2; . . . ;HMS
Calculate (f ðxjÞ), 8j ¼ ð1;2; . . . ;HMSÞitr ¼ 0while (itr 6 NI) do
x0 ¼ /for i ¼ 1; . . . ;N doif ðUð0;1Þ 6 HMCRÞ then
x0i 2 fx1i ; x
2i ; . . . ; xHMS
i g {memory consideration}if ðUð0;1Þ 6 PARÞ then
x0i ¼ x0i þ Uð�1;1Þ � FW {pitch adjustment}end if
elsex0i¼ LBiþðUBi�LBiÞ�Uð0;1Þ {random consideration}
end ifend forif ðf ðx0Þ < f ðxworstÞÞ then
Include x0 to the HM.Exclude xworst from HM.
end ifitr ¼ itr þ 1
end while
3. Island model concepts
Island model is the most popular non-panmictic EA model intro-duced by Corcoran and Wainwright (1994). In island model, thetotal population is divided into sub-populations (i.e., islands). Eachsubpopulation independently runs a standard sequential EA(Tomassini, 2005). Periodically, the islands interact using migra-tion process which is responsible for sending and receiving certainindividuals across islands controlled by migration rate andmigration frequency. If there is no migration, an island model isnothing more than a set of separate runs and thus migration is veryimportant (Skolicki & De Jong, 2005). The migration process is
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normally performed using a migration topology which determinesthe feasible path of sending and receiving individuals acrossislands based on migration policy that determines which individu-als are to be exchange among islands.
To incorporate the island model in any EA context, severalparameters must be considered:
1. Number of islands refers to the number of sub-populations.2. Island size which normally has either the same size in homoge-
neous island model or variable size in heterogeneous islandmodel (Tomassini, 2005).
3. Migration rate determines the number or the percentage of sentand received individuals among islands.
4. Migration frequency determines the number of generationsbetween migrations.
Indeed, migration can be synchronous or asynchronous due tothe time of exchanging individuals among islands. If the individu-als are exchanged across islands at the same time, the migrationprocess is known to be synchronous, otherwise, it is known to beasynchronous.
Several migration topologies have been introduced to specify thefeasible paths of sending and receiving the individuals amongislands (Rucinski, Izzo, & Biscani, 2010). Based on migration fre-quency, a specific number of individuals determined by migrationrate are sent to the neighboring island using such migrationtopology. Ring, mesh and star are but few examples of migrationtopologies used in the context of island model. The migration topol-ogies in general have a high impact into the performance of islandEA. Normally, the researchers distinguish between two categoriesof migration topologies: (i) static where the feasible paths betweenthe neighboring islands are specified before the breeding step andremain static during the evolution (e.g., RING, MESH, STAR). (ii)Dynamic migration topology where the feasible paths among theisland are dynamically changed in every migration process (e.g.,RANDOM RING, RANDOM MESH, and RANDOM STAR) (Tomassini,2005). For example, Fig. 1 shows the RANDOM RING topology wherethe islands relate to each others as a unidirectional-connectedgraph. The edge between each two islands represents the feasiblepath between an island and its neighboring island. Note that in ringtopology, each island has only one neighboring island.
Migration policy is one of the key aspects on migration process(Araujo & Merelo, 2011). When the migration frequency is reached,
Fig. 1. Random ri
the migration rate determines the number of migrant individualsto be sent and received based on migration topology. The migra-tion policy is another process in migration responsible for selectingthe migrant individuals to be exchanged among islands (Kushidaet al., 2013). Normally, some researchers introduced a migrationpolicy either based on greedy step or randomly selection. Forexample, the BEST-WORST migration policy is used to send thebest individuals from one island to its neighboring island by meansof replacing the worst individuals (Skolicki & De Jong, 2005).However, the RANDOM-RANDOM migration policy sends randomindividuals from one island to its neighboring island by replacingother random individuals so as not increase the selection pressure(Skolicki & De Jong, 2005).
The main reason of why the island model improves the perfor-mance of EA is that it observes the population diversity by meansof tracking different trajectory through the search space(Tomassini, 2005). Indeed, using multiple island provides a chanceto unfit individuals to be evolved to better individuals and thus theprobability of finding global optima might be increased (Tomassini,2005). The additional benefit of island model is that the probabilityof saving computation time when implementing it to parallel hard-ware is significantly improved (Eiben & Smith, 2003; Whitley,Rana, & Heckendorn, 1997).
There are several EA-based methods which have incorporatedthe island model concepts in order to preserve the diversity ofthe population during the search and establish a right balancebetween exploration and exploitation (Skolicki, 2005; Whitleyet al., 1997). Island based Genetic Algorithm (Lardeux & Goëffon,2010; Rahman, Sl�ezak, & Wróblewski, 2005; Skolicki & De Jong,2004; Whitley et al., 1997), Island based Differential Evolution(Kushida et al., 2013; Thein, 2014), Island based Ant Colony(Michel & Middendorf, 1998), and Island based Particle SwarmOptimization (Romero & Cotta, 2005) are but few examples ofmethods that successfully incorporate their frameworks withisland model concepts. Furthermore, the sensitivity of island-basedmethods to their parameters (number of islands, migration intervaland migration frequency) have been well studied to show theireffect on the convergence and their optimal values to improvethe performance (Skolicki & De Jong, 2005; Tomassini, 2005). Inmany cases, the optimal migration topology and policy is studiedand shown a direct impact on the performance of the island-basedmodels (Cantú-Paz, 1998; Fernández, Tomassini, & Vanneschi,2003; Rucinski et al., 2010; Tomassini, 2005).
ng topology.
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4. Island-based harmony search for optimization
Island model is the popular structured population strategy inEAs to partition a large panmictic population into several sub-populations called ‘‘ islands’’ (Tomassini, 2005). The main benefitof structuring population is to preserve the diversity of the popula-tion based on a kind of locality model known as ‘‘isolation bydistance’’ (Muhlenbein, 1991).
Harmony search (HS) algorithm is a recent EA algorithm thatconsiders the whole panmictic population during the breedingstep. It initially begins with a provisional population of sizeHMS. Iteratively, it considers the whole individuals stored in HMto generate a new harmony. That new harmony replaces theworst individual stored in HM, if better. This process is repeatedas many as NI is reached. As other EAs, HS algorithm suffers fromthe problem of fast convergence due to diversity losses. Thus thechronic premature convergence might occur (Al-Betar et al.,2013a).
Recently, structured population ideas have been investigatedfor HS algorithm to improve its diversity aspects. Cellular automatconcepts have been embedded into the framework of HS to comeup with a new version called Cellular HS (cHS) (Al-Betar et al.,2013a). The population has been arranged in the form of two-dimensional toroidal grid. At each breeding step, the new harmonyis generated by interacting with the neighboring individuals. Thus,a high level of diversity is preserved during evolutions. In anotherstudy by Pan, Suganthan, Liang, and Tasgetiren (2010), a local-bestharmony search algorithm with dynamic sub populations (DLHS)was proposed to solve global optimization problem. The idea inDLHS is to split the whole individuals of HM into small portionsthat named it sub-HM and run the evolution process indepen-dently for each sub-HM. After these divisions, the informationare exchanged among solution among sub-HMs using a periodicregrouping schedule. This method is suggested in order to preservethe diversity of the population in HM and to enhance the accuracyof the solution.
In order to embed the island model concepts into HS algorithm,the individuals of total HMS stored in HM are divided into In islandsof size Is where
PIni¼1 Is ¼ HMS. In Island-based harmony search
(iHS), the improvisation process and the updated HM process areseparately performed synchronously for each island. After a num-ber of iterations determined by migration frequency (Fm), themigration process is run to transfer a number of individualsdetermined by migration rate (Rm) among islands. The electiveindividuals to be transferred are based on the BEST-WORST migra-tion policy where the best individuals will be sent from one islandto the replaces the worst individuals in the neighboring island. Themigration topology (i.e., the RANDOM RING migration topology) isused in order to determine feasible migration paths. The frame-work of iHS has been flowcharted in Fig. 2. In the following steps,the details of iHS concepts are explained.
Step 1: Initialize the parameters. In this step, parameters of HS(i.e., HMS, HMCR, PAR, NI, and FW) are initialized. Theproblem parameters such as the number of decision vari-ables N, the feasible range of each decision variable½LBi;UBi�, and the objective function f ðxiÞ are defined inthis step. In addition, three new parameters related tothe island model shall be defined in this step: In whichreflects the number of islands, migration frequency (Fm)which determines the time to transfer the individualsacross islands, and migration rate (Rm) which determinesthe number of migrant individuals to be exchangedamong islands. Note that the island size Is is calculatedin this study on the bases that all islands have the samesize where Is ¼ HMS=In.
Step2: Initialize HM. The process of generating individuals iniHS is similar to what is done in the original HS algorithmpresented in Section 2.1.
Step 3: Divide HM into islands. The individuals in HM are dividedinto In islands of size Is. Practically, an individual vectorI ¼ ðI1; I2; . . . ; IHMSÞ is proposed in the partitioning processto determine where to allocate each individual to a cer-tain island. Note that the value range of Ij 2 ð1;2; . . . ; InÞand j 2 ð1;2; . . . ;HMSÞ. Algorithm 2 pseudo-codes thedividing process of HM individuals into In islands by aprocedure named ‘‘Divide-HM()’’. As a result of this step,a set of islands divided from whole HM is establishedwhere HM=(HM1;HM2; . . . HMIn ). Note that the HMi isthe sub-harmony memory of island i.
Algorithm 2. Divide-HM()
for j ¼ 1 . . . HMS dorepeat
k ¼ Uð1 . . . InÞif (HMk 6 Is) then
Ij ¼ kend if
until (Ij is assigned to the island k)end for
: Improvise a new harmony for each island. In this step of
Step 4iHS, each island sequentially invokes the process ofimprovising a new harmony as flowcharted in Fig. 2.The new harmony is generated based on three operators:(i) Island memory consideration, (ii) Pitch adjustment,and (iii) Random consideration. Note that the pitch adjust-ment and random consideration are used as proposed bythe original HS algorithm presented in Section 2.1.Island memory consideration: Unlike the originalmemory consideration where the whole individualsstored in HM are considered to share their informationwith the new harmony, in island memory consideration,the first decision variable x01ðjÞ of the new harmony,x0ðjÞ ¼ ðx01ðjÞ; x02ðjÞ; . . . ; x0NðjÞÞ generated for island j israndomly assigned by a historical value as follows:x01ðjÞ 2 fx11ðjÞ; x21ðjÞ; . . . ; xIs
1 ðjÞg. The next values ðx02ðjÞ; x03ðjÞ;. . . ; x0NðjÞÞ are assigned in the same manner with aprobability of HMCR where HMCR 2 ½0;1�. Note thatxiðjÞ 2 HMj; 8j ¼ ð1;2; . . . ; InÞ ^ 8i ¼ ð1;2; . . . ; IsÞ.
Step 5: Update HM of each island. This step is also tailored to berelevant with iHS. When the new harmony x0ðjÞ of island jis generated (or improvised), it compares with worst indi-vidual xwostðjÞ stored in HMj. The replacement process willbe taken over when the fitness value of new harmonyf ðx0ðjÞÞ is better than the fitness value of the worst har-mony f ðxwostðjÞÞ where the xwostðjÞ ¼ x0ðjÞ.
Step 6: Migration process. At each predefined iterations specifiedby Fm, the migration of iHS is trigged in order to exchangethe individuals across islands. The RANDOM-RING migra-tion topology is used to connect the islands during themigration. In practice, the islands are randomly arrangedto formulate a unidirectional ring. Then, a number of indi-viduals determined by migration rate Rm are exchangedamong islands based on ‘‘BEST-WORST’’ migration policy.Formally, let HMj ¼ fx1ðjÞ; x2ðjÞ; . . . ; xIs ðjÞg and HMk ¼fx1ðkÞ; x2ðkÞ; . . . ; xIs ðkÞg be two islands j and k. LetRm ¼ 20% and Is ¼ 10. Thus the number of migrantsis Rm � Is ¼ 20%� 10 ¼ 2. Let HMj and HMk bean ordered list based on fitness value where
Fig. 2. Island-based harmony search framework.
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f ðx1ðjÞÞ 6 f ðx2ðjÞÞ 6 . . . 6 f ðxIs ðjÞÞ and Ik members orderedin the same manner. Suppose we have a uni-directed edgewhich goes from Ij to Ik. Therefore, the Best two individu-als in island j (i.e., x1ðjÞ; x2ðjÞ) replace the worst twoindividuals in the island k (i.e., xIs�1 ðkÞ; xIs ðkÞ). This processwill be repeated for the other islands in the ring.
Step 7: Stop condition. Steps 3–6 are repeated until the stoppingcriteria in the proposed method are met.
In Algorithm 3 shows the pseudo-codes of the proposed iHS.
Algorithm 3. Island harmony search algorithm
5. Experimental results
In order to analyze the convergence behavior of the proposediHS and to provide a comparison evaluation, this section is pre-sented including the below subsections summarized as follows:firstly, the characteristics of the test functions used are expressedin Section 5.1. The sensitivity analysis to the parameters of iHS isintroduced in Section 5.2. Subsequently, the comparative evalua-tion with a total of thirteen comparative methods using the sametest functions is provided in Section 5.3.
All the experiments are run using a computer with 2.66 IntelCore 2 Quad with 4 GB of RAM. The operating system used isMicrosoft windows Vista Enterprize Service Pack 1. The sourcecode is implemented using MATLAB Version 7.6.0.324 (R2008a).
5.1. Test functions
In order to evaluate the proposed iHS, 25 test functions with dif-ferent characteristics have been used (f 1–f 25) selected from the set
of test functions of IEEE-CEC2005 (Suganthan et al., 2005). A briefsummary of the test functions used in our experiments is providedin Table 1 where each test function is abbreviated and its variables’value range are recorded as appears in the original source(Suganthan et al., 2005). The dimension D of each solution vectorused in the experiments is also recorded together with the fitnessvalue of the optimal solution f ðx�Þ.
The set of IEEE-CEC2005 test functions (f 1–f 25) can be charac-terized as follows: the first five functions (f 1–f 5) are unimodaland shifted; the second 7 test functions (f 6–f 12) are basic multi-modal and shifted; the third two functions (f 13–f 14) are expandedmultimodal; and the fourth 11 functions (f 15–f 25) are hybridcomposition (i.e., All of them are non separable, rotated, andmultimodal functions containing a large number of local minima).The detailed principle of the IEEE-CEC2005 is given in (Suganthanet al., 2005).
5.2. Sensitivity analysis of iHS to its parameters
In this section, a sensitivity analysis of the proposed iHS to itsparameters is conducted in order to study its convergence behav-ior. Note the parameters of HS algorithm have been tuned in ourprevious work where cellular automata were incorporated for HSas structured population mechanism (Al-Betar et al., 2013a), thusthe tuned parameters are used in this study as follows:HMS = 100, HMCR = 0.98, PAR = 0.30, FW = 0.01, and NI = 100,000.The ultimate goal of the analysis presented here is to answer thequestion: What settings of island model parameters cause iHS tosucceed?
For the 25 test functions of IEEE-CEC2005, we follow the param-eters and conditions of the CEC competition (Suganthan et al.,2005) where the 25 repeated runs have been performed for eachtest function. The 25 runs have been summarized in terms of theaverage of the error (AE) of the best individual (i.e.,AE ¼ jf ðx�Þ � f ðxbestÞj). Note that x� is a given optimal solution whilethe xbest is the average best solution obtained in 25 runs.
The parameters related to the island model are intensively stud-ied to measure their effect on the convergence of the proposedmethod. The number of islands (In), the migration frequency (Fm),and migration rate (Rm) are interact in different convergence cases
Table 1The test functions.
abb Name Search range D f ðx�Þ
f 1 Shifted sphere function [�100,100] 10 �450f 2 Shifted Schwefels problem 1.2 [�100,100] 10 �450f 3 Shifted rotated high conditioned elliptic function [�100,100] 10 �450f 4 Shifted Schwefels problem 1.2 with noise in fitness [�100,100] 10 �450f 5 Schwefels problem 2.6 with global optimum on bounds [�100,100] 10 �310f 6 Shifted Rosenbrocks function [�100,100] 10 390f 7 Shifted rotated Griewanks function without bounds [0,600] 10 �180f 8 Shifted rotated Ackleys function with global optimum on bounds [�32,32] 10 �140f 9 Shifted Rastrigins function [�5,5] 10 �330f 10 Shifted rotated Rastrigins function [�5,5] 10 �330f 11 Shifted rotated Weierstrass function [�0.5,0.5] 10 90f 12 Schwefels problem 2.13 [�p;p] 10 �460f 13 Expanded extended Griewanks plus Rosenbrocks function (F8F2) [�5,5] 10 �130f 14 Shifted rotated expanded Scaffers F6 [�100,100] 10 �300f 15 hybrid composition function [�5,5] 10 120f 16 Rotated hybrid composition function [�5,5] 10 120f 17 Rotated hybrid composition function with noise in fitness [�5,5] 10 120f 18 Rotated hybrid composition function [�5,5] 10 10f 19 Rotated hybrid composition function with a narrow basin for the global optimum [�5,5] 10 10f 20 Rotated hybrid composition function with the global optimum on the bounds [�5,5] 10 10f 21 Rotated hybrid composition function [�5,5] 10 360f 22 Rotated hybrid composition function with high condition number matrix [�5,5] 10 360f 23 Non-continuous rotated hybrid composition function [�5,5] 10 360f 24 Rotated hybrid composition function [�5,5] 10 260f 25 Rotated hybrid composition function without bounds [2,5] 10 260
Table 2The nine experiment cases to evaluate the sensitivity of b-hill climbing to itsparameters.
Experiment cases In Fm Rm (%)
Case 1 2 100 20Case 2 5 100 20Case 3 10 100 20Case 4 10 50 20Case 5 10 100 20Case 6 10 500 20Case 7 10 500 10Case 8 10 500 20Case 9 10 500 30
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as shown in Table 2. The first three cases are designed to measurethe effect of In on the convergence. The second three convergencecases are designed to measure the impact of Fm, while the lastthree convergence cases are designed to measure the impact ofRm. In the next three sub-sections, the effect of each parameteron iHS algorithm is discussed.
5.2.1. Effect of island number (In) on iHS convergence behaviorTheoretically, based on some results produced for island GA, it
is assumed that the number of islands (In) and the island size (Is)are more important and are studied first, keeping the other param-eters fixed at reasonable values on the basis of previous empiricalknowledge (Cantu-Paz, 2000; Tomassini, 2005).
As aforementioned, the In with various (i.e., In = {2, 5, 10}) val-ues have been studied. Note that these values are chosen basedon the value of HMS which should be less. Table 3 summarizesthe results produced for the first three cases (Case 1, Case 2, andCase 3) for 25 functions of CEC-IEEE2005. The values in the tableis the AE produced for 25 runs. The best solutions produced arehighlighted in bold (lowest is best).
As apparent in Table 3, the higher the value of In is, the betterthe AE will be. In other words, when the value of In is small, theindividuals in the HM will not be sub-divided into several islandsand thus few search space regions will be simultaneously explored.However, a larger value of In leads to a better diffusion on the pop-ulation. Using a smaller number would probably cause too muchgranularity of each island. It should be emphasized that althoughincreasing the value of In improves the results produced, it alsoincreases the computational complexity of the algorithm. Forexample, if the value of In ¼ 5 then this mean that the iHS at eachiteration will repeat the evolution process five times. Therefore, itis recommended to run the proposed system in a parallel hardwareto control the computational complexity.
The size of island, it should be emphasized, is crucial in deter-mining the iHS behavior. For example, islands that are too smallwill be disadvantaged for a lake of diversity and they are proneuntimely converges. In contrast,islands which are too large willmake room for extra useless work (Tomassini, 2005).
5.2.2. Effect of migration frequency (Fm) on iHS convergence behaviorAfter the effect of In parameter on the behavior of iHS is studied,
the In ¼ 10 value that achieved the best results among the firstthree cases is selected to perform the other consecutive three cases(Case 4, Case 5, and Case 6). Note that the result obtained in Case 3is repeated in Case 5. In this section, the effect of the migrationfrequency (Fm) on the behavior of iHS is studied using Case 4,Case 5, Case 6. These three cases are carefully designed withvarying values of Fm (i.e., Fm ¼ 50, 100, and 500) as shown inTable 2. The value Fm ¼ 500 means the migration process will beperiodically invoked after each 500 evolutions. In the iHS,migration occurs every X generations and copies of the individualsthat make up the most fit 20% of the individuals of each island areallowed to migrate. The neighboring island receiving theseindividuals deletes the least fit bottom 20% of its own individualsbased on BEST-WORST migration policy. Migrations occurbetween all islands simultaneously. At each iteration a feasiblepath between each neighboring pairs of island is decided basedon RANDOM-RING migration topology.
Table 4 shows the results produced for the 25 IEEE-CEC2005functions using the above mentioned three cases. Again, the bestreported results are highlighted in bold. It is worth mentioningthat the best results produced are mostly obtained by Case 6 wherethe value of Fm ¼ 500. A pleasing explanation of this situation is
Table 3The effect of various values of In on the convergence behavior of iHS.
Problem Case 1 Case 2 Case 3In ¼ 2 In ¼ 5 In ¼ 10
f 1 6.35E � 08 1.04E � 08 1.00E � 09f 2 5.23E � 007 1.01E � 007 4.70E � 008f 3 5.22E + 004 8.89E + 003 2.34E + 003f 4 4.78E + 002 2.56E + 002 1.38E + 002f 5 1.12E + 004 1.11E + 004 1.11E + 004f 6 1.98E + 001 3.56E + 001 2.43E + 001f 7 1.68E + 000 1.57E + 000 1.18E + 000f 8 2.01E + 001 2.01E + 001 2.01E + 001f 9 1.96E � 007 3.31E � 008 1.14E � 008f 10 1.48E + 001 1.17E + 001 1.80E + 001f 11 6.87E + 000 6.78E + 000 5.88E + 000f 12 6.19E + 002 3.42E + 002 2.78E + 002f 13 3.40E � 001 3.66E � 001 3.45E � 001f 14 3.63E + 000 3.51E + 000 3.35E + 000f 15 1.79E + 002 3.45E + 001 4.00E + 000f 16 1.33E + 002 1.23E + 002 1.46E + 002f 17 1.33E + 002 1.27E + 002 1.40E + 002f 18 9.79E + 002 9.67E + 002 9.19E + 002f 19 9.76E + 002 9.48E + 002 9.01E + 002f 20 9.77E + 002 9.74E + 002 8.54E + 002f 21 9.91E + 002 1.01E + 003 8.44E + 002f 22 8.33E + 002 8.13E + 002 7.97E + 002f 23 1.19E + 003 1.03E + 003 1.01E + 003f 24 2.40E + 002 2.12E + 002 2.00E + 002f 25 2.84E + 002 2.52E + 002 2.00E + 002
Table 4The effect of various values of Fm on the convergence behavior of iHS.
Problem Case 4 Case 5 Case 6Fm ¼ 50 Fm ¼ 100 Fm ¼ 500
f 1 2.40E � 09 1.00E � 09 1.04E � 09f 2 4.70E � 008 4.13E � 008 5.17E � 008f 3 2.34E + 003 1.57E + 003 1.70E + 003f 4 1.38E + 002 1.22E + 002 1.04E + 002f 5 1.11E + 004 1.11E + 004 1.11E + 004f 6 2.43E + 001 1.99E + 001 3.22E + 001f 7 1.18E + 000 1.38E + 000 9.40E � 001f 8 2.01E + 001 2.01E + 001 2.01E + 001f 9 1.14E � 008 8.95E � 009 1.74E � 008f 10 1.80E + 001 1.50E + 001 2.01E + 001f 11 5.88E + 000 5.68E + 000 5.00E + 000f 12 2.78E + 002 2.63E + 002 1.13E + 002f 13 3.45E � 001 3.42E � 001 3.20E � 001f 14 3.35E + 000 3.40E + 000 3.31E + 000f 15 4.00E + 000 3.35E + 001 1.27E � 007f 16 1.46E + 002 1.33E + 002 1.43E + 002f 17 1.40E + 002 1.32E + 002 1.38E + 002f 18 9.67E + 002 9.26E + 002 7.93E + 002f 19 9.48E + 002 9.35E + 002 8.73E + 002f 20 9.74E + 002 9.16E + 002 9.03E + 002f 21 1.01E + 003 7.81E + 002 7.06E + 002f 22 8.13E + 002 8.06E + 002 7.96E + 002f 23 1.03E + 003 1.01E + 003 7.76E + 002f 24 2.12E + 002 2.00E + 002 2.00E + 002f 25 2.52E + 002 2.36E + 002 2.00E + 002
Table 5The effect of various values of Rm on the convergence behavior of iHS.
Problem Case 7 Case 8 Case 9Rm ¼ 10% Rm ¼ 20% Rm ¼ 30%
f 1 1.00E � 09 1.00E � 09 1.00E � 09f 2 5.17E � 008 4.70E � 008 4.99E � 008f 3 1.70E + 003 1.21E + 003 4.04E + 003f 4 1.04E + 002 7.35E + 001 5.96E + 001f 5 1.11E + 004 1.11E + 004 1.11E + 004f 6 3.22E + 001 2.10E + 001 2.27E + 001f 7 9.40E � 001 5.43E � 001 5.91E � 001f 8 2.01E + 001 2.01E + 001 2.01E + 001f 9 1.74E � 008 5.61E � 009 2.06E � 008f 10 2.01E + 001 2.29E + 001 2.10E + 001f 11 5.00E + 000 5.25E + 000 5.13E + 000f 12 1.13E + 002 2.49E + 000 1.75E + 002f 13 3.20E � 001 3.36E � 001 2.87E � 001f 14 3.31E + 000 3.42E + 000 3.38E + 000f 15 1.27E � 007 5.75E � 008 2.00E + 001f 16 1.43E + 002 1.55E + 002 1.66E + 002f 17 1.38E + 002 1.36E + 002 1.33E + 002f 18 7.93E + 002 8.74E + 002 8.94E + 002f 19 8.73E + 002 7.49E + 002 8.56E + 002f 20 9.03E + 002 8.05E + 002 9.06E + 002f 21 7.06E + 002 6.88E + 002 5.76E + 002f 22 7.96E + 002 7.77E + 002 7.90E + 002f 23 7.76E + 002 7.82E + 002 7.78E + 002f 24 2.00E + 002 2.00E + 002 2.00E + 002f 25 2.00E + 002 2.00E + 002 2.00E + 002
2032 M.A. Al-Betar et al. / Expert Systems with Applications 42 (2015) 2026–2035
that the search space of the functions in IEEE-CEC2005 functionsare very rugged and therefore it requires a deepest descent processto reach the local minima of the individuals of each island region.
Basically, if migration takes place at a large scale, then the iHSbehavior will be negatively affected while low scale migration givemore chances for the migrant to move with no effect on the targetindividuals of any island and so convergence will take longer tooccur.
5.2.3. Effect of migration rate (Rm) on iHS convergence behaviorAs aforementioned, two very basic and important parameters of
island models are migration frequency (Fm) and migration rate(Rm), denoting how frequently migrations occur and how manyindividuals migrate. The value of Fm ¼ 500 is selected from the pre-vious experiments to be used in the next three convergence caseswhich produced the best results. The last set of experiments (i.e.,Case 7, Case 8, and Case 9) is designed to measure the impact ofRm on the convergence behavior iHS. Three alternative values ofRm are rigourously studied (i.e., Rm ¼ 10%; 20%, and 30%) asshown in Table 2. The value of Rm ¼ 20% means that the numberof migrants are Is � 20% ¼ HMS=In � 20% ¼ 100=10� 20% ¼ 2.
Table 5 shows the results produced for the 25 IEEE-CEC2005functions using the above mentioned three cases. It is worth men-tioning that there is no indication of what the optimal value of Rm
that obtained the best results. To put it differently, the number ofmigrants did not affect the behavior of iHS as much. One couldexpect a little bit more sensitivity for small migration rate, whichhas not been observed till. Moreover, for very small Rm we havesometimes observed an increase in diversity in generations follow-ing a migration, which suggests that a single individual is able toact as a seed for changes in a stagnated island. Big value of Rm tendsto have a direct effect on diversity simply due to replacing a largernumber of individuals.
5.3. Comparison between iHS and other methods
To compare the proposed iHS algorithm with the other state-of-the-art methods, thirteen comparative methods are used.
The comparative methods also include cHS and HS algorithm(Al-Betar et al., 2013a). Note that cHS is a structured papulationmethod that incorporates the cellular automata context in HSalgorithm to preserve the population diversity. The keys to thecomparator methods are summarized in Table 6. Note thatthe most of these methods are the competition winners which areoften hybrid EA methods. Therefore, their results are very powerfulto the IEEE-CEC2005 test functions, as will be noted in Table 7.
The numbers reported in Table 7 refers to the best AE achievedby comparators for the IEEE-CEC2005 functions. The best results inthis Table have been highlighted in bold (lowest is best). It appears
M.A. Al-Betar et al. / Expert Systems with Applications 42 (2015) 2026–2035 2033
that the AE results produced by iHS are very competitive to thoseobtained by comparator methods. Interestingly, iHS is able to
Table 6Key to comparative methods.
Key Method name
BLX-GL50 Hybrid real-coded genetic algorithms with female and male differen
BLX-MA Adaptive local search parameters for real-coded memetic algorithmsCOEVO Real-parameter optimization using the mutation step co-evolutionDE Real-parameter optimization with differential evolutionDMS-L-PSO Dynamic multi-swarm particle swarm optimizer with local searchEDA Experimental results for the special session on real-parameter optim
EDAIPOP-CMA-ES A restart CMA evolution strategy with increasing population sizeK-PCX A population-based, steady-state procedure for real-parameter optimLR-CMA-ES Performance evaluation of an advanced local search evolutionary algL-SADE Self-adaptive differential evolution algorithmSPC-PNX Real-parameter optimization performance study on the CEC-2005 beHS Harmony search algorithmcHS Cellular harmony search algorithm
Table 7Average error rate obtained in CEC’2005 special session in dimension 10.
Algorithm f 1 f 2 f 3 f 4
BLX-GL50 1.00E � 009 1.00E � 009 5.71E + 002 1.00E � 009BLX-MA 1.00E � 009 1.00E � 009 4.77E + 004 2.00E � 008COEVO 1.00E � 009 1.00E � 009 1.00E � 009 1.00E � 009DE 1.00E � 009 1.00E � 009 1.94E � 006 1.00E � 009DMS-L-PSO 1.00E � 009 1.00E � 009 1.00E � 009 1.89E � 003EDA 1.00E � 009 1.00E � 009 2.12E + 001 1.00E � 009IPOP-CMA-ES 1.00E � 009 1.00E � 009 1.00E � 009 1.00E � 009K-PCX 1.00E � 009 1.00E � 009 4.15E � 001 7.94E � 007LR-CMA-ES 1.00E � 009 1.00E � 009 1.00E � 009 1.76E + 006L-SADE 1.00E � 009 1.00E � 009 1.67E � 005 1.42E � 005SPC-PNX 1.00E � 009 1.00E � 009 1.08E + 005 1.00E � 009HS 1.00E � 009 1.00E � 009 5.20E + 004 9.63E + 000cHS 1.00E � 009 1.00E � 009 1.49E + 007 4.66E + 003iHS 1.00E � 009 1.00E � 009 1.21E + 003 5.96E + 001
Algorithm f 10 f 11 f 12 f 13
BLX-GL50 4.97E + 000 2.33E + 000 4.07E + 002 7.50E � 001BLX-MA 5.64E + 000 4.56E + 000 7.43E + 001 7.74E � 001COEVO 2.68E + 001 9.03E + 000 6.05E + 002 1.14E + 000DE 1.25E + 001 8.47E � 001 3.17E + 001 9.77E � 001DMS-L-PSO 3.62E + 000 4.62E + 000 2.40E + 000 3.69E � 001EDA 5.29E + 000 3.94E + 000 4.42E + 002 1.84E + 000IPOP-CMA-ES 7.96E � 002 9.34E � 001 2.93E + 001 6.96E � 001K-PCX 2.39E � 001 6.65E + 000 1.49E + 002 6.53E � 001LR-CMA-ES 4.08E + 001 3.65E + 000 2.09E + 002 4.94E � 001L-SADE 4.97E + 000 4.89E + 000 4.50E � 007 2.20E � 001SPC-PNX 7.30E + 000 1.91E + 000 2.60E + 002 8.38E � 001HS 5.97E + 000 1.47E + 000 2.67E + 005 5.93E � 002cHS 6.17E + 001 9.74E + 000 1.28E + 004 3.90E + 000iHS 1.17E + 001 5.00E + 000 2.49E + 000 2.87E � 001
Algorithm f 19 f 20 f 21 f 22
BLX-GL50 4.49E + 002 4.46E + 002 6.89E + 002 7.59E + 002BLX-MA 7.63E + 002 8.00E + 002 7.22E + 002 6.71E + 002COEVO 8.45E + 002 8.63E + 002 6.35E + 002 7.79E + 002DE 4.20E + 002 4.60E + 002 4.92E + 002 7.18E + 002DMS-L-PSO 7.14E + 002 8.22E + 002 5.36E + 002 6.92E + 002EDA 5.64E + 002 6.52E + 002 4.84E + 002 7.71E + 002IPOP-CMA-ES 3.26E + 002 3.00E + 002 5.00E + 002 7.29E + 002K-PCX 7.51E + 002 8.13E + 002 1.05E + 003 6.59E + 002LR-CMA-ES 5.16E + 002 4.42E + 002 4.04E + 002 7.40E + 002L-SADE 7.05E + 002 7.13E + 002 4.64E + 002 7.35E + 002SPC-PNX 3.80E + 002 4.40E + 002 6.80E + 002 7.49E + 002HS 7.54E + 002 8.00E + 002 8.53E + 002 7.42E + 002cHS 1.09E + 003 1.08E + 003 1.09E + 003 9.00E + 002iHS 7.49E + 002 8.05E + 002 5.76E + 002 7.77E + 002
achieve three best results as done by others in the literature forf 1; f 2; and f 24 and yield best overall results for f 25.
Reference
tiation García-Martínez and Lozano(2005)Molina et al. (2005)Posik (2005)Ronkkonen et al. (2005)Liang and Suganthan (2005)
ization at CEC 2005: a simple, continuous Yuan and Gallagher (2005)
Auger and Hansen (2005b)ization Sinha et al. (2005)orithm Auger and Hansen (2005a)
Qin and Suganthan (2005)nchmark with SPC-PNX Ballester et al. (2005)
Al-Betar et al. (2013a)Al-Betar et al. (2013a)
f 5 f 6 f 7 f 8 f 9
1.00E � 009 1.00E � 009 1.17E � 002 2.04E + 001 1.15E + 0002.12E � 002 1.49E + 000 1.97E � 001 2.02E + 001 4.38E � 0012.13E + 000 1.25E + 001 3.71E � 002 2.03E + 001 1.92E + 0011.00E � 009 1.59E � 001 1.46E � 001 2.04E + 001 9.55E � 0011.14E � 006 6.89E � 008 4.52E � 002 2.00E + 001 1.00E � 0091.00E � 009 4.18E � 002 4.20E � 001 2.03E + 001 5.42E + 0001.00E � 009 1.00E � 009 1.00E � 009 2.00E + 001 2.39E � 0014.85E + 001 4.78E � 001 2.31E � 001 2.00E + 001 1.19E � 0011.00E � 009 1.00E � 009 1.00E � 009 2.00E + 001 4.49E + 0011.23E � 002 1.20E � 008 1.99E � 002 2.00E + 001 1.00E � 0091.00E � 009 1.89E + 001 8.26E � 002 2.10E + 001 4.02E + 0001.40E � 003 5.42E � 002 1.82E � 001 2.02E + 001 1.00E � 0091.33E + 004 5.84E + 007 7.85E + 001 2.03E + 001 2.69E + 0011.11E + 004 1.98E + 001 5.43E � 001 2.01E + 001 5.61E � 009
f 14 f 15 f 16 f 17 f 18
2.17E + 000 4.00E + 002 9.35E + 001 1.09E + 002 4.20E + 0022.03E + 000 2.70E + 002 1.02E + 002 1.27E + 002 8.03E + 0023.71E + 000 2.94E + 002 1.77E + 002 2.12E + 002 9.02E + 0023.45E + 000 2.59E + 002 1.13E + 002 1.15E + 002 4.00E + 0022.36E + 000 4.85E + 000 9.48E + 001 1.10E + 002 7.61E + 0022.63E + 000 3.65E + 002 1.44E + 002 1.57E + 002 4.83E + 0023.01E + 000 2.28E + 002 9.13E + 001 1.23E + 002 3.32E + 0022.35E + 000 5.10E + 002 9.59E + 001 9.73E + 001 7.52E + 0024.01E + 000 2.11E + 002 1.05E + 002 5.49E + 002 4.97E + 0022.92E + 000 3.20E + 001 1.01E + 002 1.14E + 002 7.19E + 0023.05E + 000 2.54E + 002 1.10E + 002 1.19E + 002 4.40E + 0021.95E + 000 1.00E � 009 1.01E + 002 1.06E + 002 8.81E + 0023.97E + 000 3.476E + 02 2.73E + 002 2.89E + 002 8.85E + 0023.31E + 000 5.75E � 008 1.23E + 002 1.27E + 002 7.93E + 002
f 23 f 24 f 25
6.39E + 002 2.00E + 002 4.04E + 0029.27E + 002 2.24E + 002 3.96E + 0028.35E + 002 3.14E + 002 2.57E + 0025.72E + 002 2.00E + 002 9.23E + 0027.30E + 002 2.24E + 002 3.66E + 0026.41E + 002 2.00E + 002 3.73E + 0025.59E + 002 2.00E + 002 3.74E + 0021.06E + 003 4.06E + 002 4.06E + 0027.91E + 002 8.65E + 002 4.42E + 0026.64E + 002 2.00E + 002 3.76E + 0025.76E + 002 2.00E + 002 4.06E + 0025.59E + 002 2.36E + 002 3.29E + 0021.30E + 003 1.06E + 003 8.84E + 0027.76E + 002 2.00E + 002 2.00E + 002
2034 M.A. Al-Betar et al. / Expert Systems with Applications 42 (2015) 2026–2035
We are particulary interested in comparing the results pro-duced by HS and cHS with those produced by iHS. Note that thebest results among the three versions of HS algorithms are high-lighted in italic. Apparently, the iHS is able to produce better resultsthan those produced by cHS for almost all functions while domi-nating the results produced by HS algorithm in 12 out of 25 testfunctions. Indeed, the main motivation of proposing iHS is toensure a high-level of diversity, and thus, the iHS might require ahigher number of evolutions to hit better results for the testfunctions.
6. Conclusion and future work
This paper proposes a new version of HS algorithm called iHS.The island model concepts have been embedded within theoriginal framework of HS. The individuals in HM are divided intoseveral sup-groups called ‘‘islands’’. Each island runs theimprovisation process of new harmony and it depends on its HMindividuals. After such generations, the migration process withthe dynamic ring topology and BEST-WORST migration policy isperformed in order to exchange the individuals among islands.Three new parameters are added into iHS: Number of Island (In),migration frequency (Fm), and migration rate (Rm). The interactionof these parameters has been intensively studied through a set ofconvergence cases.
Experimentally, higher value of In leads to better resultsobtained due to better population diffusion. The number of indi-viduals (Is) in each island also has a significant impact on the pro-duced results. Islands that are too small will fail to provide enoughdiversity and will easily converge prematurely. On the other hand,if the islands are too large with respect to a single population thatsolves the problem, extra useless work will be performed.
The surprising observation from the experiments was that themigration frequency (Fm) was playing a much bigger role thanthe migration rate (Rm), especially for short frequency, as we wereregularly able to observe a strong decrease in the performance. TheiHS is, indeed, not quite sensitive to the Rm setting where no singlevalue is universally recommended for global optimization func-tions. However, using Rm with a small number of individuals givesa better choice for the iHS algorithm in order to reduce the memoryspace and computational time.
As structured population strategy of island model incorporatedwith HS algorithm came up with a new variation called iHS wereable to achieve good performance for global optimization func-tions, future work can be directed to:
� Investigate how to simplify the usage of iHS algorithm bydynamically tuning the island model parameters.� Investigate the effect of other migration topologies and policies
on the iHS algorithm.� Adapt iHS algorithm to real world data set.� Study the diversity measurements for iHS algorithm.� Apply the proposed iHS algorithm in parallel platforms.� Compare the performance of iHS algorithm with other free
search methods such as Artificial Bee Colony, Genetic Algo-rithm, Ant Colony Optimization, etc, using other difficult opti-mization problems.
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