k-means-clustering based evolutionary algorithm for multi … · algorithm, genetic algorithm(ga),...

Appl. Math. Inf. Sci.11, No. 6, 1681-1692 (2017) 1681

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.18576/amis/110615

K-means-Clustering Based Evolutionary Algorithm forMulti-objective Resource Allocation ProblemsA. A. Mousa, M. A. El-Shorbagy∗and M. A. Farag

Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menoufia University, Egypt.

Received: 24 Apr. 2015, Revised: 10 May 2017, Accepted: 14 May 2017Published online: 1 Nov. 2017

Abstract: The process of finding the optimal allocation of limited resources to a number of tasks for optimizing multiple objectives iscalled multi-objective resource allocation problem (MORAP). This paper presents K-means-clustering based on one of the evolutionaryalgorithm, genetic algorithm(GA), to solve MORAP. Using the K-means-clustering algorithm to divide the population toa specificnumber of sub-populations each of them with dynamic size. Therefore, different operators of GA (crossover&mutation) can beimplemented to each subpopulation instead of applied the same GA operators to the all population. The aim of dynamic clusteringis to preserve and introduce diversity into solutions, instead of the solutions becoming similar each other. Two problems taken fromthe literature are used to compare the performance of the proposed algorithm with the competing algorithms. Moreover, an exampleof optimum utilization of human resource in reclamation of derelict land in Toshka-Egypt is solved by our approach. The results ofdifferent test problems have showed the superiority of our algorithm to solve MORAP.

Keywords: Dalgaard-Strulik model, energy, economic growth, time delay, limit cycle

1 Introduction

Resource allocation is part of resource management. It isemployed to allocate the available amount ofresources economically. In management sciences,resource allocation is the scheduling of resources andactivities while taking into consideration both theresource availability and the project time in an economicway [1]. MORAP is the procedure of allocating amountof resources among the various tasks in order to meet theexpected objectives. Resources may be capital,man-power, raw materials or anything else in limitedsupply that can be used to achieve the objectives. Theobjectives may be minimizing costs, maximizing profits,or accomplishing the best possible quality or anythingelse [2]. MORAP has a variety of applications such as:

1.Assignment marketing resources [3]: thecharacteristics key of marketing managers is to takethe decision to allocate scarce marketing resources,e.g., retail shelf-space or merchandise inventories,selling hours, advertising dollars.

2.Stochastic network systems [4]: for getting theoptimum performance of the network by distributing

both resource and duration of activities in properdirection.

3.Scheduling allocation of workers in the manual laborenvironment [5]: for assigning the workers into thejobs to minimize human cost, to reduce productionduration and to control production overwork.

4.Portfolio optimization [6]: for making optimizedportfolios on allocating funds to stocks or bonds tomaximize profit for a given level of risk, or to reducerisk for a target rate of return.

5.Health care resource allocation [7]: Discounting costsand health benefits in cost-effectiveness analysis forthe health care resource allocation.

6.Traffic accidents sanitary [8]: creating new protocolsand applications to improve assistance in trafficaccidents.

7.Energy resources allocation [9,10]: satisfyingoptimization model for sustainable gas resourcesallocation

8.Dynamic PERT networks [11]: developing amulti-criteria model for the resource allocationproblem in a dynamic PERT network.

There are infinite variety of applications can behandled this way [12,13].

∗ Corresponding author e-mail:[email protected]

c© 2017 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.18576/amis/110615

1682 A. A. Mousa et al.: K-means-clustering based evolutionary algorithm...

Traditionally, MORAP have been solved usingmethods in operations research [14-20].For instance, in[14] Zhao et al. proposed an integer programmingapproach for integrated resource assignment and jobscheduling for a multiple job-agents-system. While, in[15] a branch-and-bound approach is presented foractivity scheduling of a system of the PERT/CPM varietywhere the project duration is minimized. Furthermore, in[16] Lai and Li developed a new procedure based ondynamic programming for solving the multi-criteriaresource allocation problem. In addition, Bretthauer andShetty [17] presented a new algorithm to solve theresource allocation problem with nonlinearity, which isdefined as the optimization of a convex function over aconvex constraint subject to bounded integer variables.Firstly the authors introduce a pegging algorithm forcontinuous variable problem, and then implement thepegging method in a branch and bound algorithm forsolving the integer variable problem. A multi-dimensionalfairness procedure is proposed for auction-basedmulti-attribute resource allocation which faces theproblem in a more holistic way, taking into accountpriorities throughout all of the auction process [20]. Acomprehensive survey of the state of the art in resourceallocation problem can be found in [13]. But none ofthese methods is computationally tractable for anyreal-life application size, thus rendering them impracticalfor real-life application [21].

For MORAPs, evolutionary algorithms (EAs)methods provide more realistic techniques for finding asolution. There has been an increasing interest in studyingEAs for optimization problems due to its importance inreal life applications. EAs are conceptually different fromthe traditional programming techniques; where thesemethods are incorporate certain biological, molecular,and neurological phenomena. The reasons for theirpopularity are as follows [22]: (1) EAs do not require anyderivative information, (2) EAs are very simple toimplement, (3) EAs are flexible with having awide-spread applicability. Also, one of the basicadvantages of EAs is that can be find a high qualitysolutions with reasonable computational times.

Recently, several EAs have been proposed to solveMORAPs including GA [23,24,25,26], simulatedannealing (SA) [27], ant colony optimization (ACO) [28,29], particle swarm optimization (PSO) [30,31,32],variable neighborhood search [33] and cuckoo searchalgorithm (CSA) [34]. In [24], the authors proposed amulti-objective hybrid GA approach based on themultistage decision making model to get efficiently a setof non-dominated solutions. In [26], a two-stageprocedure is proposed to MORAP: firstly, NSGA-II isapplied to obtain the non-dominated solution set;secondly, entropy weight and TOPSIS (technique fororder performance by similarity to ideal solution) methodare implemented to determine the best compromise

solution. In [27], a hybrid SA technique is presented to acomplex energy resource management problem includinga large number of resources. In [29], a modified versionof ACO is proposed in order to get a set of non-dominatedsolution; where the algorithm efficiency is increased byincreasing the ants learning. In [31], a hybrid PSO isproposed; where a hill-climbing heuristic is embeddedinto the PSO for speeding the convergence. Also, in [32],the authors proposed a modified binary PSO algorithm forsolving the MORAP. In [33], variable neighborhoodsearch is used to solve the MORAP with two importantbut conflicting objectives: maximization of efficiency andminimization of cost. Finally, in [34] a penalty basedCSA is proposed to find the optimal solution of reliabilityallocation problems.

Clustering is the process of dividing the data togroups that has similar objects. Each group is calledcluster contains the objects that are similar betweenthemselves in the same cluster, and different to the objectsin the other cluters [35]. There are many clusteringtechniques [36]. The K-means-clustering is the mostcommonly used clustering algorithm due to its simplicityand accuracy [37]. Several clustering algorithms havebeen proposed such as: iterative self-organizing dataanalysis technique [38], clustering large applicationsbased up on randomized search [39], parallel-cluster [40],density-based spatial clustering of applications with noise[41] and balanced iterative reducing and clustering usinghierarchies [42].

This paper presents K-means-clustering based ongenetic algorithm (GA) to solve MORAP.K-means-clustering algorithm divided the population ofGA to a specific number of sub-populations each of themwith a dynamic size. So, different GA operators can beimplemented to each subpopulation instead of applied thesame GA operators to the all population. In addition, thedynamic clustering is preserve and introduce diversityinto solutions, instead of the solutions becoming similareach other. Test problems are used to compare theperformance of the proposed algorithm with thecompeting algorithms. The results of different testproblems have demonstrate the superiority of ouralgorithm to solve MORAP.

This paper is structured as follows: section 2describes the formulation of the multi-objective resourceallocation problem (MORAP). In section 3, geneticalgorithm procedures is displayed. In section 4, clusteringtechnique is introduced. The proposed approach isexplained in details in section 5. While, numericalsimulation is presented in section 6. Finally, conclusion ofthe paper is presented in section 7.

2 Mathematical formulation of MORAP

The general form of the MORAP is as follows [23]:


Appl. Math. Inf. Sci.11, No. 6, 1681-1692 (2017) /www.naturalspublishing.com/Journals.asp 1683

Max Z1 (x1,x2, ...,xn) =n∑

k=1z1

k (xk)

Max Z2 (x1,x2, ...,xn) =n∑

k=1z2

k (xk)

· · · · · · · · · · · ·

Max Zq (x1,x2, ...,xn) =n∑

k=1zq

k (xk)

Subject to :n∑

k=1gk (xk)≤ S,

gk (xk)≥ 0,xk ≥ 0;

(1)

wherexk define (decision variables),S donates the(resources),gk (xk) represents the activities stages andZq (x1,x2, ...,xn) is theq-objective functions. The processof MORAP seeks to optimize theq-objective functionswhich subjected to some constraints of resource and findsan optimal allocation (decision variables) of limitedamount of resource to a number of tasks (activities stages)[23].

3 GA procedures

GAs operateson a population of candidate solutionscalled chromosome. To obtain optimality, allchromosomes (solutions) exchanges information witheach other by using operators Inspired from naturalgenetic to produce an improved solution [43,44]. Thesteps of GA is described below [24]:Step 0: Generate randomly the initial population.Step 1: Evaluate the values of theq-objective functions forevery chromosome.Step 2: Select the best chromosomes (parents).Step 3: Crossover is implemented on every pair selected ofchromosomes (parents) to generate two offspring.Step 4: Mutation is applied to every two offspring obtainedafter the crossover.Step 5: Check the stopping criterion, if it is not reached,go to Step 1, else, continue.Step 6: The most preferred solutions among thealternatives are selected by the DM.

The basic step of GA is illustrated in flowchart asshown in figure 1 (taken from [45]).

4 Clustering Technique

Clustering is the process to find groups of objects(clusters); where the objects in any group is similar toeach other and not-similar to the objects in other groups[35]. K-means-clustering approach [37] is one of thesimplest algorithms that using to solve clusteringproblems. The procedures of K-means-clustering aresimple and easy to classify a given data set to a certainnumber of clusters (k); which is already known. The mainidea of K-means-clustering is to define a centroid k to

‘

Fig. 1: Main flowchart of GA

each cluster. The better choice, for the centroids, is toplace them as far as possible be spaced out. The next stepis to associate each point in the given data set with thenearest centroid. The first step is completed when nopoint is pending i.e. an early groups was generated. Then,k new centroids are re-calculated as centers of theresulting clusters. Every point in the same data set isassociated with the nearest new centroid. These steps arerepeated i.e. the location of the centroids is changed stepby step until it do not move any more. The coordinates ofthe centroid point is defined as the average of thecoordinates of the points in the cluster. TheK-means-clustering algorithm can be described by thefollowing steps [46]:

Step 1: the number of desired clusters (k) is defined. Step2: the initial centroid of the cluster is chosen randomly.

Step 3: the sum of square error (the squared Euclideandistance) between each object and the centers of clustersis computed as follows:

Sum of squre error(SSE) =k

∑i=1

∑x∈Ci

dist2(mi,x) (2)

wherex is a point in clusterCi andmi is the centroid ofclusterC.

Step 4: each object is assigned to the nearest cluster.

Step 5: for each cluster, the new centroid is computed.

Step 6: steps 3 to 5 are repeated until the location of thecentroids are not change. Figure 2 illustrates the steps ofthe K-means procedure on dataset with three clusters inR2

[47].


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Fig. 2: K-means procedure on dataset with three clusters inR2.

5 K-means-Clustering Based GeneticAlgorithm

In this paper, K-means-clustering based GA is proposedto solve MORAP; where the population of GA is dividedto a specific number of sub-populations by usingK-means-clustering technique. Each sub-population hascommon features. In addition, different GA operators areimplemented to each sub-population instead of applyingthe same GA operator to all population. The steps ofK-means-clustering based GA are described below:

5.1 Initialization

MORAP is reformulated as network model as in figure3.It is considered as multi-objective; where it aims tomaximize the total efficiency of the human resourceallocation problem and minimize the total cost. It isrequired to obtain the path between the two nodes sourcenode (s) and terminal node (t) that has the minimum totalcost and maximum efficiency [23].The path from S to T isa sequence of paths(s,x1m), (x1m,x2m), · · · (xN−1m, t).A path can be described as a sequence of nodes(s,x1m,x2m, · · · ,xN−1m, t); wherem = 0,1,2, · · · ,m andNis the number of stages. So, each chromosome isinitialized as sequence of nodes (path).

Fig. 3: Representation MORAP as network model

In this step, the population is initialized by generatingNpop strings; where the algorithm chose a randomelement from available number in each stage. Forexample, in the MORAP to allocate 10 workers in a set of4 jobs (4 stages) as in figure 4. MORAP aims todetermine a path in the 11 states (0→10 workers) and the4 stages to achieve the minimum costs and the maximumefficiency. Figure 5 shows the structure of thechromosome for 4 stages allocation path.

Fig. 4: Allocation path of 10 workers (states) in 4 jobs (stages)to minimize costs and maximize efficiency



Fig. 5: The structure of the chromosome for 4-stages allocationpath

5.2 Rejection of illegal individuals

The initialization step may be produced illegalindividuals, which do not satisfy the problem feasibleregion, i.e., if there areN workers needed to allocate inMjobs, the number of worker must equal toN [23]. So, weshould delete these individuals and replace them bygenerating legal individuals randomly.

5.3 Evaluation of non-dominated solutions

In multi-objective optimization problem (MOOP), there isno single solution optimizes the all objectives. But, thereexists a number of solutions (may be infinite). Thesesolutions are called non-dominated solutions, Paretooptimal solutions, Pareto efficient or noninferior [48]. So,to find these solutions, the population must be classifiedaccording to the definition of Pareto optimal solution.Definition (Pareto optimal solution): the solutionx∗ issaid to be a Pareto optimal solution for MOOP if there isno other feasible solutionx such that:f j(x)≤ f j(x∗) for allj = 1,2, ...,q and f j(x) < f j(x∗) for at least one objectivefunction f j.The image of the Pareto set (Pareto solutions) is calledPareto front. A population can be evaluated according tonon-domination criteria. Consider two solutionsx1 andx2for a problem having more than one objective functionq(q > 1). These solutions may have one of these twopossibilities: one of these solutions can be dominates theother or non-dominates the other. The solutionx1 is saidto dominate other solutionx2 if the two followingcondition are achieved (say the operator≺ denotes worseand≻ denotes better) [49]:

1.fq(x1) 6≺ fq(x2) for all q = 1, . . . ,q objectives.2.fq(x1)≻ fq(x2) for at least oneq ∈ {1,2, . . . ,q}.

The following procedures is used to find thenon-dominated set of solutions from a set of populationeach havingq objective function values(q > 1) [50].Step 0: Begin withm = 1.Step 1: For alln = 1,2, · · · ,NPOP and m 6= n, comparesolutionsxm andxn for domination using the previous twoconditions for allq objectives.Step 2: If for anyn, xm is dominated byxn, mark xm as‘dominated’, and it is inefficient.Step 3: Ifm = NPOP, go to Step 4, elsem = m+1and goto Step 1.

Step 4: All set of solutions that are not marked by‘dominated’ are the set of non-dominated solutions. Allnon-dominated solutions are constitute the Pareto(non-dominated) front on the population in a specificgeneration.

5.4 Selection

The aim of selection (parent selection stage) is todetermine the best individuals that are suited to be parentand pass their chromosomes to the next generation. Byother words, Better the fitness, the bigger chance to beselected [45]. It is intended to improve the quality of thepopulation; where it gives the high quality individuals abetter chance to copy in the next generation. Also, itdirects the search of GA in the direction of promisingregions in the search domain.

In this paper, the multiple objective functions iscombined into a scalar fitness solution as [51]:

f (x) = w1 f1(x)+ ...+wi fi(x)+ ...+wq fq(x); (3)

wherex is an individual,fi(x) is theith objective functionand wi is a weighting-vector forfi(x) such that:wi ≥ 0

for all i = 1, . . . ,q andq∑

i=1wi = 1. In general, each weight

can be determine randomly. For a MOOP withq objectivefunctions, we can assign a random real number to eachweight as follows:

wi =randi

q∑

i=1randi

, i = 1,2, ...,q; (4)

whererandi ∀ i = 1, ...,q are non-negative random realnumbers.

Finally, binary tournament selection [52] is used;where two chromosomes are chosen randomly and thebetter of the two chromosomes according to the scalarfitness is copied in the mating pool

5.5 K-means-Clustering Technique

In this step, as shown in figure 6, K separatedsub-populations with dynamic size is generated bydividing the population using K-means-ClusteringTechnique.

5.6 Genetic Algorithm (GA) Operators

In this step, different GA operators are applied tosub-populations instead of one GA operator applied to theall population; where the population is divided to many ofsub-populations with a dynamic size.




Fig. 6: Division of population into K sub-populations withdynamic size

5.6.1 Crossover operator

The crossover aims to exchange the information betweentwo individuals to produce two new offspring [53]. In ourstudy, we used some of crossover techniques; which areexplained briefly below.

A. One-point crossoverIn one-point-crossover operator, a crossover point is

selected randomly within an individual then the twoparent individuals at this point are interchanged toproduce two new offspring [54].

B. Two-point crossoverIn this crossover, two different cut-off points were

randomly selected. New offspring were obtained bytransmitting the zones, in the two parents, between thetwo cut points [54].

C. Uniform crossoverIn this method, a random binary string is generated

with the same-size of chromosome. Then relative genesunder this binary string between parents is exchanged,where parent strings exchange their bit at the positionwhere the corresponding position in random binary stringis 1. Otherwise, no exchange of bit is performed [55].

D. Cross crossoverIn such crossover, the new offspring are selected from

different parts of parent. Often the produced generation arevery different from their parent. A cut point is selected inthe middle point in the two parent. The left side gene of thecut point of the first parent are copied directly to right sideof the new first offspring. While, the remaining portion ofoffspring array are selected from left side gene of the cutpoint of the second parent. In order to produce the secondoffspring the parents are swapped in this process as in thispervious manner [56].

5.6.2 Mutation operator

Mutation operator is applied to all offspring in the newgeneration and used to modify some features ofchromosomes depending on a predetermined smallprobability value to produce new chromosomes [53]. Inthe following, a brief explanation of mutation techniquesthat used in our study:A. Twors mutation

In Twors mutation [57], two genes are randomlychosen to exchange of their position, as shown in figure 7.

Fig. 7: Twors mutation

B. One point mutationIn one point mutation, data at a particular point is

mutated (particular gene was randomly selected and thenit was replaced with a random state from the available set)[57].C. Reverse Sequence Mutation

In this mutation, a sequenceS is taken; which isbounded by two randomly chosen positionsi and j, suchthat i < j. Then, The arrange of the gene in this sequenceis reversed by the same method as in the previousoperation. Figure 8 shows the implementation of thismutation operator.

Fig. 8: Reverse Sequence Mutation

D. Centre inverse mutationAs shown in Figure 9, this mutation divided the parent

into two sections. All genes in every section are copied andthen inversely placed in the same section of a child [57].

Fig. 9: Centre inverse mutation



5.7 Combination stage

In this stage, to create a new population, allsub-populations are combined together, as shown inFigure 10.

Fig. 10: Combination stage

5.8 Archive Update (Update the archive ofnon-dominated solution).

The algorithm has an external archive of non-dominatedsolutions which is updated iteratively based on the conceptof non-domination in the state of presence new solutions.The main objective of this archive is to store a historicalrecord of the non-dominated solutions found through theprocess of search [58]. During the process of search, a setof non-dominated solutions is added to the archive whichupdated iteratively every generation. Algorithm 1 show theprocedure which is used to update the archive [59]. Themain idea of Algorithm 1 is to generate a new archive setA(k)in each iterationk, using the contents of the old archivesetA(k−1) and the current populationP(t).

Algorithm 1: Update the archive of non-dominatedsolution

1. k∆= 0

2. A(0) = 0

3. While terminate(A(k), k) = false do

4. k∆= k+1

5. p(k)∆= generate( ) {generate new search point}

6. A(k) ∆= update (A(k−1)

, p(k)) {update archive }7. End while8. Output : A(k)

6 Experimental results

In this section, the proposed algorithm is evaluated bytwo multi-objective resource allocation test problems

taken from the literature [23,24]. Moreover, the proposedalgorithm is applied to an engineering application [60] toverify the performance of it for solving MORAPs andillustrate its ability to handle engineering problems. Alltest problems have been solved on an Intel core I5, 2.6GHz processor. The proposed approach is coded usingMATLAB programming language. The parameter settingused for all runs are depicted in Table 1.

Table 1:parameters setting of the proposed algorithm

6.1 Test problem 1

Test problem 1 [23] of allocate 6 workers in a set of 4 jobsis tested by our approach. Table 2 provides the expectedcost and efficiency.

Table 2: The expected cost and efficiency of testproblem 1

Tables 3 and 4 present the efficient solution obtainedby our algorithm at k=1 and k=4, respectively. While,table 5 gives the results obtained by other approach (theeffective genetic algorithm [23]). Furthermore, figure 11shows the efficient solution by our algorithm at k=1, k=4and the effective genetic algorithm [23].

Table 3: Efficient solution obtained at k=1




Table 4: Efficient solution obtained at k=4

Table 5: Efficient solution of the effective geneticalgorithm approach [23]

Fig. 11: Efficient solution of test problem 1 obtained bythe proposed algorithm at k=1, k=4and the effective geneticalgorithm approach [23].

It is clear from previous comparison in test problem 1that the results obtained by the introduced algorithm atk=1, k=4 and the results obtained by effective geneticalgorithm is the same values.

6.2 Test problem 2

Test problem 2 24 of allocate 10 workers in a set of 4 jobsis solved by our approach. Table 6 provides the expectedcost and efficiency. Table 7 and table 8 presents theefficient solutions obtained by the introduced algorithm atk=1 and k=4, respectively. While, Table 9 presents the

results obtained by multistage decision-based GA [24].Furthermore, figure 12 shows the Pareto results of theproposed algorithm at different values of k and multistagedecision-based GA [24].

Table 6: The expected cost and efficiency of testproblem 2

Table 7: Efficient solutions of the proposed algorithmat k=1

Table 8: Efficient solutions of the proposed algorithmat k=4



Table 9: Efficient solutions of the multistage decision-based genetic algorithm [24]

As shown in figure 12, the results obtained byproposed algorithm with k=4 dominate the most of thatobtained by multistage decision-based GA. In addition,the result obtained by the proposed algorithm withclustering technique (i.e. k=4) dominate the resultsobtained by the algorithm without clustering data (i.e.,k=1). So, the proposed algorithm is more appropriate forusing to solve MORAPs.

Fig. 12: Simulation results of test problem 2 obtained by ouralgorithm at k=1, k=4 and the multistage decision-based geneticalgorithm [24].

6.3 Application

In this subsection, the proposed approach is applied tosolve an application of reclamation of derelict land inToshka-Egypt [60]. The aim is to illustrate the ability ofproposed approach to solve engineering applications. Inthis application 4 stages is considered; which affect theoptimization of the agriculture reclamation, includingland settlement, land planning, digging of canals andplant cultivation. Also, two objectives are considered:

maximizing benefit of reclamation and minimizing thecosts of agriculture reclamation. The annually data of 10staff are given in Table 10, where for each staff theminimum cost and maximum profit for the4 stages areprovided. Figure 13 shows the Pareto solutions obtainedby proposed approach at k=1, k=4 and that obtained bymulti-objective multipheromone ant colony optimizationapproach [60].

Table 10: The average profit and cost of staff neededfor reclamation

Fig. 13: Simulation results of test problem 2 obtained by ouralgorithm at k=1, k=4 and the multistage decision-based geneticalgorithm [24].

It is clear that the Pareto solutions set obtained bymulti-objective multipheromone ant colony optimizationapproach is part of the Pareto solutions set obtained bythe algorithm with clustering data (i.e. k=4). In addition,the result obtained by the proposed algorithm withclustering technique (i.e. k=4) is dominant to the resultsobtained by the algorithm without clustering data (i.e.k=1).As shown in figure 13 the proposed algorithmoutperformed the multi-objective multipheromone antcolony optimization approach in both distribution andspread and able to find points that the other method failedto reach. So, we can say that the proposed algorithm is asuitable method to solve to solve real life MORAPs.




7 Conclusion

This paper presents K-means-clustering approach basedon one of the evolutionary algorithm, genetic algorithm(GA), to solve MORAP. K-means-clustering algorithm isused to divide the population to a specific number ofsub-populations, each of them with a dynamic size.Therefore, we can implement different operators of GA(crossover and mutation) to each subpopulation instead ofapplied the same operator to the all population. Theresults of different test problems have showed thesuperiority of our algorithm to solve MORAP.Finally, theproposed algorithm has the following benefits :

1.It integrates the powerful searching of GA anddiversityof dynamic clustering.

2.Incorporating the evolutionary algorithm withdynamic clustering, preserves the diversity of thesolutions and prevents it to be similar with each other

3.The results of different test problems have showed thesuperiority of it to solve MORAP

4.Due to its simplicity, it was demonstrated to be a goodtool to solve MORAPs.

In our future works, the following will be researched:

1.Solving larger scale examples to demonstrate theefficiency of our approach.

2.Updating our approach to can be applied to solve manyapplications of MORAP.

3.Using other clustering techniques to accelerate theconvergence property of the proposed approach andimprove the solution quality.

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A. A. Mosua receivedthe B.Sc. degree in electricalengineering from MenoufiaUniversity, Shebin El-Kom,Egypt, in 1997, M.Sc. andPh.D degrees in engineeringmathematics from Menoufiauniversity, in 2003 and 2006,respectively. He has beenwith the Basic Engineering

sciences, Menoufia University, since 1998. He is currentlyan associate professor in the Basic Engineering Sciences,Faculty of Engineering, Menoufia University and at Taifuniversity faculty of sciences. His research interests areevolutionary multiobjective optimization, multimodaloptimization, and optimization techniques applied topower systems.

M. A. El-Shorbagyreceived the B.Sc. degreein electrical engineering fromMenoufia University, ShebinEl-Kom, Egypt, in 2004,M.Sc. and Ph.D degreesin engineering mathematicsfrom Menoufia university, in2010 and 2013, respectively.He is currently an assistant

professor in Department of Basic Engineering Sciences,Faculty of Engineering, Menoufia University, ShebinEl-Kom, Egypt. His current research interests include:multiobjective optimization, swarm intelligence,numerical optimization and optimization for engineeringapplications.

M. A. Farag receivedthe B.Sc. degree in ElectricalEngineering and theM.Sc. degree in engineeringmathematics from Facultyof Engineering, MenoufiaUniversity, Shebin El-Kom,Egypt, in 2011 and 2016,respectively. She is workingtowards the Ph.D. degree in

the Department of Basic Engineering Sciences at thesame college. She is currently an associate Lecturer inDepartment of Basic Engineering Sciences, Faculty ofEngineering, Menoufia University, Shebin El-Kom,Egypt. Her current research interests include:evolutionary multiobjective optimization, SwarmIntelligence, Numerical optimization and evolutionaryalgorithms in general.


k-means-clustering based evolutionary algorithm for multi … · algorithm, genetic algorithm(ga),...

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