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Applied Soft Computing 13 (2013) 2960–2969

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing

j ourna l ho me p age: www.elsev ier .com/ l ocate /asoc

hybrid evolutionary multiobjective optimization strategy for the dynamicower supply problem in magnesia grain manufacturing

eijian Konga,∗, Tianyou Chaia, Shengxiang Yangb, Jinliang Dinga

State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, 110004 Shenyang, ChinaDepartment of Information Systems and Computing, Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom

r t i c l e i n f o

rticle history:eceived 31 December 2011eceived in revised form 25 February 2012ccepted 28 February 2012vailable online 13 March 2012

eywords:ynamic multiobjective optimizationybrid strategyultiobjective evolutionary algorithm

ase-based reasoning

a b s t r a c t

The supply trajectory of electric power for submerged arc magnesia furnace determines the yields andgrade of magnesia grain during the manufacture process. As the two production targets (i.e., the yields andthe grade of magnesia grain) are conflicting and the process is subject to changing conditions, the supplyof electric power needs to be dynamically optimized to track the moving Pareto optimal set with time. Ahybrid evolutionary multiobjective optimization strategy is proposed to address the dynamic multiobjec-tive optimization problem. The hybrid strategy is based on two techniques. The first one uses case-basedreasoning to immediately generate good solutions to adjust the power supply once the environmentchanges, and then apply a multiobjective evolutionary algorithm to accurately solve the problem. Thesecond one is to learn the case solutions to guide and promote the search of the evolutionary algo-rithm, and the best solutions found by the evolutionary algorithm can be used to update the case library

anufacture process of magnesia grain to improve the accuracy of case-based reasoning in the following process. Due to the effectiveness ofmutual promotion, the hybrid strategy can continuously adapt and search in dynamic environments.Two prominent multiobjective evolutionary algorithms are integrated into the hybrid strategy to solvethe dynamic multiobjective power supply optimization problem. The results from a series of experi-ments show that the proposed hybrid algorithms perform better than their component multiobjectiveevolutionary algorithms for the tested problems.

. Introduction

Magnesia grain is an important type of refractory material,hich is widely utilized to produce nozzles of rocket engine, insula-

ion wall of nuclear reactor and crucibles in metallurgical industries1–4]. The manufacture process of magnesia grain is to gener-te electric arcs to heat raw material up with an adjustable largeurrent till finishing the meltdown, decarburization, dehydrationnd other decontaminating reactions. While the work conditionshange, the electric current needs to be adjusted in real time touarantee the completion of the above reactions against interrup-ion. An outdated or improper current will reduce the yields andower the grade of magnesia grain in a new work condition [5].

oreover, in determining the current, a Pareto optimal set [6–8]s required to represent the tradeoff between the two productionbjectives. Therefore, the electric power supply problem addressed

ere is a dynamic multiobjective optimization problem (DMOP).

For the moment, many enterprises in China manually deter-ine the electric current without considering the influence of

∗ Corresponding author. Tel.: +86 13897964033.E-mail address: weijian.kong9@gmail.com (W. Kong).

568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2012.02.025

© 2012 Elsevier B.V. All rights reserved.

changing environments. Hence, the yields and grade of magnesiagrain are very low. Wu et al. [9,10] proposed a current decisionmethod using a case-based reasoning technique, which eliminatesthe unreliability of manual operation, but cannot achieve optimiza-tion of the production objectives. There is, so far, no further studythat addressed the dynamic multiobjective electric power supplyoptimization problem.

For DMOPs, a optimization algorithm needs not only to find awide-spread and diverse Pareto Front, but also track the moving ofPareto Front with time [11]. In comparison with mathematical pro-gramming, evolutionary algorithms (EAs) have some advantages.First, EAs can find a number of Pareto optimal solutions in a singlerun since they are population-based [12]; second, if the diversityof the population is maintained properly, there are usually someindividuals in the population that perform not bad in a changingenvironment. Third, EAs can quickly provide a approximate opti-mum that meets the requirements of many industrial processes.

Researchers have developed several approaches to applyingEAs for DMOPs. One approach is to devise or adopt a new evo-

lutionary model that is devoted to track the moving of ParetoFront in dynamic environments. For example, Goh and Tan [11]proposed a new coevolutionary paradigm that hybridizes compet-itive and cooperative mechanisms to quickly converge to adapt

Computing 13 (2013) 2960–2969 2961

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Table 1Identification rules of operating modes.

Rules Antecedents Conclusions

Rule 1 dIA/dt ≥ ˛1 and dIB/dt ≥ ˛1 and t < T1 j = 1Rule 2 dIA/dt < ˛2 and dIC/dt ≥ ˇ2 and T1 < t < T2 j = 2Rule 3 dI /dt < ˛ and dI /dt ≥ ˇ and T < t < T j = j

W. Kong et al. / Applied Soft

o the changing environment. Zhang et al. [13] adopted artificialmmunity system to solve DMOPs, which has an excellent diversity-

aintenance mechanism. Bingul [14] proposed a adaptive geneticlgorithm (GA) to solve DMOPs, which aggregates multiple objec-ives to a single objective by rule-based weight set method.

Another approach is to incorporate some dynamic adaptationtrategies into tradition EAs. In single-objective dynamic opti-ization, some strategies have been proposed to enhance the

erformance of EAs. For example, random immigrants have beensed in GAs, which maintain the diversity of the population by

nserting some randomly generated individuals in each generation15,16]. The analogous schemes are hyper mutation [17] and re-nitialization of a part of population when the environment changes18]. Besides, some researchers had proposed multi-populationchemes, such as self-organizing scouts [19], shifting balance GA20] and multinational GA [21], which group individuals into sev-ral cooperating sub-populations that search in different regions,espectively. A so-called memory scheme was proposed to storeseful information (such as best solutions) from previous optimiza-ion, and retrieved them in an appropriate environment [22,23].evertheless, the above schemes were all proposed to address a

ingle-objective DOPs.Deb et al. [24] incorporated a scheme similar to random immi-

rants in the improved non-dominance sorting GA (NSGA2) toddress the dynamic multiobjective hydrothermal power schedul-ng problem. Zeng et al. [25] devised an orthogonal design methodo improve the traditional multiobjective EAs for DMOPs. Someynamic multiobjective optimization algorithms have been applied

n semi-batch epoxy polymerization process [26] and temperatureontrol [27]. More comprehensive discussions on DMOPs can beound in Ref. [28].

Case-based reasoning (CBR) is a kind of problem solving tech-ique based on expertise, which was proposed by Roger Schank

or the first time in 1982 [29]. CBR retains a successful experiencen a case library, and retrieves and reuses it in a similar problemater. CBR has been applied in many optimization problems suchs production scheduling [30], automated timetabling [31], prod-ct configuration [32] and marketing plans [33]. Some researchersroposed hybrid strategies that combine CBR and meta-heuristiclgorithm to address hard optimization problems. For example, CBRas used to tune the parameter of tabu search for different opti-ization models [34], or select the heuristic that has worked well

n previous similar situations [35]. Besides, GA was also introducedn CBR system to improve the level of matching [36].

The electric power supply problem during the manufacturingrocess of magnesia grain is addressed as a DMOP in this paper. Itequires an algorithm not only track the moving of Pareto Front, butlso timely respond to the changing environment. For DMOPs, mul-iobjective EAs (MOEAs) can find an approximate Pareto optimalet in a higher efficiency than mathematical programming, and candapt to the changing environment via maintaining the diversity ofopulation. Nevertheless, it is time-consuming to re-evaluate theopulation and evolve it in a new environment, especially for aomputationally complex optimization model. Therefore, a hybridtrategy that combines MOEAs and CBR is proposed to address theMOP in this paper, and the hybrid algorithms instantiated from

he hybrid strategy are called CBR-MOEAs correspondingly.In the proposed hybrid strategy, a CBR system is used to imme-

iately provide approximate solutions based on the expertise oruccessful experiences once the environment changes, and thenOEAs are used to improve the accuracy of the CBR solutions.

population initialization scheme based on the CBR solutions is

roposed to promote the search process of MOEAs. After evolu-ion, the precise solutions of MOEAs can be used to modify thease library to improve the performance of CBR in the followingrocess.

A 2 C 2 1 2 k

Rule n ˇn−1 ≤ dIB/dt < ˇn and �n−1 ≤ dIC/dt < �n

and Tn−1 < t < Tn

j = 6

The rest of the paper is organized as follows. Section 2 introducesthe electric power supply optimization problem. The proposedhybrid strategy is presented in Section 3. In Section 4, a suitof experiments is conducted to validate the effectiveness of theproposed CBR-MOEA for the electric power supply optimizationproblem.

2. Electric power supply optimization problem

In this section, we will first give a general definition of DMOPs,and then introduce the electric power supply optimization problemand elaborate its specific features and challenges to optimizationalgorithms.

2.1. Definition of DMOPs

Without loss of generality, we assume that each objective func-tion in a DMOP is a maximization problem. A DMOP can then bedefined as follows:

maxx ∈ Xn

f(x, t) = {f1(x, t), f2(x, t), . . . , fk(x, t)}s.t. g(x, t) > 0, h(x, t) = 0

(1)

where x is the vector of decision variables, and Xn is the n-dimension decision space. fi(x,t)(i ∈ {1, . . ., k}) denotes the ithobjective function to be maximized with respect to time t, and kis the number of objectives. The functions g(x, t) and h(x, t) rep-resent the set of inequality and equality constrains, respectively,which define the feasible region.

Comparing to static optimization problems, the fitness function,design variables and/or environment conditions may change overtime in DOPs. Hence, the location of optimal solution for a DOP maybe moving with time, and the task of optimization algorithm is nolonger to locate an optimal solution but to track the moving optima.While addressing a DMOP, it is the Pareto optimal set instead of asingle solution that needs to be tracked over time, which brings abig challenge to the current EAs.

2.2. Manufacture process of magnesia

The power supply process is illustrated in Fig. 1. The productionprocesses data are detected by transducers, collected by transmit-ter, and then mined to identify the environmental parameters. Thepower supply optimization model will be adjusted on the basis ofthe environmental changes. A hybrid MOEA strategy is proposedto provide a new electric current set point for the automatic con-trol system in real time. By regulating actuator, automatic controlsystem can keep the electric current of furnaces in the set point.

Submerged arc magnesia furnaces adopt intermittent feed andexhaust, and uninterrupted charging with electricity. So the man-ufacture process oscillates among six operating modes of overburning, feed, under burning, exhaust, spraying and punching. Afurnace requires a different power supply trajectory for each oper-

ating mode. The above six operating modes are denoted as j = 1,j = 2, . . ., j = 6, which can be identified by the data of smelting time t,change rates of three-phase electrode currents dIA/dt, dIB/dt, dIC/dt.The identification rule of operating modes is shown in Table 1,

2962 W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969

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Fig. 1. Power sup

here jk = 1, 2, . . ., 6, n is the number of all rules, Tn, ˛n, ˇn, �n

re technological parameters.The boundary conditions also have an effect on electric power

upply. For example, it requires a large power input when theranularity and grade of raw materials increase. The granular-ty of raw materials is the diameter of material granules, whichften fluctuates between 10 mm and 60 mm. The grade of mate-ial is the percentage of magnesia in the material, which fluctuatesetween 75% and 95%. During the manufacturing process, the depthf molten pool will rise from 0 mm to 2500 mm, which also changeshe power demand of furnaces.

.3. The dynamic multiobjective power supply optimizationodel

The objectives of optimization model are the yields and gradef magnesia grain during the sampling period T, and the decisionariables are three phase electric currents of furnaces. As theres no available exact fundamental model to formulize the rela-ion between the optimization objectives and decision variablesill now, we develop a data based model by the least square sup-ort vector machine (LS-SVM) [37] that is a strong machine learningechnique and widely adopted in regression and classification prob-ems.

Here three-phase electric currents are denoted as i, the envi-onment parameters � = [ı1, ı2, ı3], where ı1 represents materialsranularity, ı2 represents the grade of material and ı3 representsool depth. The two objectives (yields and grade of magnesia grain)re denoted as Y and G, respectively. The electric power supplyptimization model can be described as follows:

Max : Y, Max : Gs.t. : Y = fj(i, �),

G = lj(i, �),ilj ≤ i ≤ iu

j ,

(2)

ocess of furnaces.

where j = {1, 2, . . ., 6} represents the current operating mode. ilj and

iuj represent the lower bound and upper bound of electric currents

of furnaces on the jth operating mode.Y = fj(·) and G = lj(·) are LS-SVM models that adopt the Radial

Basis Function (RBF) kernel function, and the parameter settingsfor training the model can be found in our previous study in Ref.[38]. Taking the LS-SVM model Y = f3(i, �) as an example, the modelis trained with 401 groups of data, and validated with 100 groups ofdata. The model validation results are shown in Fig. 2, and the modelprediction error has proved to be random noise via autocorrelationanalysis.

In dynamic environments, the power supply optimizationmodel is adjusted with time as shown in Fig. 3. In contrast withother DMOPs, there are some specific features in the electric powersupply optimization problem, which are described as follows:

(1) The manufacturing process oscillates among six operatingmodes, so the objective function also varies among six opti-mization models.

(2) While the operating mode changes, the solutions must beupdated to meet the new requirements in real time.

(3) The environment parameters fluctuate acutely with time, so theoptimal solutions often jump far away from the past solutions.Therefore, the traditional EAs that rely on exploiting of pastinformation may lose their effectiveness.

3. The proposed hybrid strategy

As mentioned above, the electric power supply optimizationproblem is a DMOP, in which the environmental conditions oftenchange acutely and a timely response is required. To address thisproblem, we propose a hybrid strategy that combines MOEAs andthe CBR technique to create a CBR-MOEA as the solver.

In CBR-MOEA, CBR is used to provide relatively satisfactorysolutions as a transitional decision to respond to the change of envi-ronments, and MOEAs are used to improve the solution graduallyduring evolution process. Moreover, case solutions come from the

W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969 2963

0 50 100 150 200 250 300 350 400 4505

6

7

8

Sample pointsY

ield

s (t

on)

Train Set Regression Predict by LSSVM

originalpredict

0 10 20 30 40 50 60 70 80 90 1005

6

7

8

Sample points

Yie

lds

(ton

)Test Set Regression Predict by LSSVM

originalpredict

Fig. 2. The comparison of model data and original data.

Operating mode i dentification

based on rule r easonin g

Model switch

Y=f1(i, δ)G=l1(i, δ)

Y=f2(i, δ)G=l2(i, δ)

Y=f3(i, δ)G=l3(i, δ)

Y=f4(i, δ)G=l4(i, δ)

Y=f5(i, δ)G=l5(i, δ)

Y=f6(i, δ)G=l6(i, δ)

Tuning th eparameters δ

over burning feed exhaustunder burning material-spraying punching

Manufacturing processof magnesia grain

1 1

l ui i i

2 2

l ui i i 3 3

l ui i i 4 4

l ui i i

5 5

l ui i i 6 6

l ui i i

monitoring th eboundary conditions

Fig. 3. The adjustment mechanism of power supply optimization model in the changing environment.

Case

retainin g

Case

retrieval

Case revisal b ased onsolutions by MOEA

The dynamic multiobjectiveoptimization problem

Population

initialization b ased on

case s olutions

MOEA

Case

solutionsEnvironments

changed?

NO

YES

Case

reuse

Pareto

optimal se t

Generation=

Generation+1

Fig. 4. Flow chart o

Decision-maker

f CBR-MOEA.

2964 W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969

Table 2The case structure.

Environment parameters Solutions Performance

epmsmcflf

3

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3

es

t{iM

S

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s

s

w

3

natI

w

NO

YES

Prune casesolutions

?caseCS P>

Exploit the regionsnear to case solutions

Obtain casesolution set: CS

Combine populationof MOEA

Evaluate all individualsand select population

j, ı1, ı2, ı3 {ik} , k = 1, 2, . . ., K {[Yk Gk]} , k = 1, 2, . . ., K

xpertise or previous successful experiences, which imply someromising regions. Therefore, the regions are exploited to provideore good candidates as the initial population, which promotes the

earch process of MOEAs. After evolution, the solutions of MOEAsay be better than the case solutions, so they can be used to modify

ase library to improve the performance of CBR in next solving. Theow chart of CBR-MOEA is shown in Fig. 4. The detailed explanation

or each step of CBR-MOEA is presented as follows.

.1. Case presentation

A case contains the description of problem, corresponding solu-ions and the performance of solutions, so it can be expressedy a triple 〈environment parameters, solutions, estimation ofbjectives〉. The environment parameters consist of operating mode

and boundary conditions ı1, ı2, ı3. The case solutions are aet of currents

{ik

}, k = 1, 2, . . . , K , the performances of which,

.e., estimations of objectives are yields and grade of magnesia[Yk Gk]} , k = 1, 2, . . ., K. The case structure is shown in Table 2.

.2. Case retrieval

Case retrieval is to find some cases similar to the currentnvironment, which is measured by similarity. The definition ofimilarity is provided next.

Supposed the new problem M with environment parame-ers {j ı1 ı2 ı3}. The lth case Ml with environment parametersj(l) ı1(l) ı2(l) ı3(l)}, l = 1, 2, . . ., Lc, and Lc is the amount of all casesn case library. The similarity of new problem M and Ml denoted as

l can be formulated:

IMl = sim(j, j(l)) + 0.1 ×∑3

g=1�gsim(ıg, ıg(l))∑3g=1�g

, l = 1, 2, . . . , Lc

(3)

here �g, g = 1, 2, 3 is the weight of environment parameter that isrovided by field experts. sim(j, j(l)) is the operating mode similar-

ty, and sim(ıg, ıg(l)) is the boundary condition similarity. They cane respectively figured out by:

im(j, j(l)) ={

1, if j = j(l)0, if j /= j(l)

(4)

im(ıg, ıg(l)) = 1 − |ıg − ıg(l)|�ıg max

(5)

here �ıg max represents the fluctuation range of ıg.

.3. Population initialization based on case solutions

It is promising to mine more good candidates in the regionsear to the case solutions. Therefore, a population initializationpproach is proposed based on case solutions. The approach aimso exploit the promising regions under consideration of diversity.

ts flow chart is shown in Fig. 5. The main steps are presented next.

Step 1: Prune case solutions.If the number of retrieved case solutions nCS exceeds Pcase, we

ill adopt the following pruning operation:

Fig. 5. Flow chart of population initialization.

(1) Firstly eliminate the dominated solutions to obtain a nondom-inated set ND;

(2) If |ND| > Pcase, then select the case solutions by their distributionmetrics. The crowding method [39] is used to evaluate their dis-tribution performance on objective space. It needs to be addedthat some other techniques just as clustering [40] and adaptivegrid [41] can be also used to prune case solutions. Comparedwith the latter, crowding is parameterless and computation-simple.

Step 2: Exploit the regions near to case solutions.It needs to achieve the trade-off of exploitation and diversi-

fied population. A straightforward idea is that the region will beexploited in more chances if it is closer to some case solution.

Supposed we need mine out of Pexploit solutions near some casesolution ik, one of which, denoted as ik,s can be generated by theformula:

ik,s = unifrnd(A, B)

A = min

(iuj

− ilj

Pexploit − 1× (s − 1) + ilj, ik

)

B = max

(iuj

− ilj

Pexploit − 1× (s − 1) + ilj, ik

) (6)

where k = 1, 2, . . ., Pcase, s = 1, 2, . . ., Pexploit. The function of unifrnd(A,B) generates uniformly random number between A and B.

Step 3: Select initialized population.We can exploit Pcase × Pexploit solutions in all, together with Pcase

case solutions that are combined in the population of MOEA. Afterreevaluation in new environment, the new population is pruned bythe above operation to keep population size of MOEAs invariant.

3.4. Multiobjective evolutionary algorithms

Many MOEAs have been devised to address the multiobjectiveproblems. Among them, NSGA2 [39] and SPEA2 [42] are prominentand often are used to solve the practical problems in manufac-ture processes [43,44]. We also introduce NSGA2 and SPEA2 in theproposed hybrid framework to scheme out two algorithms of CBR-

NSGA2 and CBR-SPEA2 to solve DMOPs in manufacture process ofmagnesia grain. The procedures of two algorithms are shown inFigs. 6 and 7.

W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969 2965

0

2

0

( )

t

NSGA

Initialize Population Pt

terminals donot be metAssign fitness for P based on nondominated sortEvaluate fur ther individuals on the same front ba sed on crowd ingSelect individuals to cr eat pa rent po

=

proce durebegin

odelihw

1

pulationBinary tournament selection in parent populationRecombination an d mutation to cr eat of fspring populationCombine pa rent an d of fspring populationt t= +end

3

tcBf

T2{

f

c

tr

4

arphfm

1 1 22 33

( )

1:

( ) (1 1% ) (l) 501

c

case revisal

Select K solutions from results of MOEAbased on Pareto dominate and crowding

L LCreat a new case by the K solutionsRetain the case

l Lk l lif kbreak

procedure

begin

if

else

for

en

1

0

dense

l

l

kCombine the K solutions in case MPrune the solutions of M

kCreat a new case by the K solutionsInsert the case in case libraryCalculate the crowding of all cases on operating modesSelect the cases that are mostCalcu

d

if

end

if

dense

late the crowding of the selected cases on boundary conditionsEliminate the cases that is most

end

end

Fig. 6. The procedure of NSGA2.

.5. Case revisal based on solutions by MOEA

The MOEAs can evolve better results on the basis of case solu-ions. These solutions by MOEA can be used to supplement or revisease solutions to improve the performance of CBR in next solving.ut it needs to be noted that the case library size is restricted to Lc

or its real-time performance.Supposed the current environment parameters are {j ı1 ı2 ı3}.

he case library has L cases, and the library is denoted as {Ml} , l = 1,, . . ., L. The lth case Ml associates with environment parametersj(l) ı1(l) ı2(l) ı3(l)}. Case revisal procedure is shown in Fig. 8.

The crowding of case Ml on operating modes is calculated by theormula:

rowding o(Ml) = |{Mi|j(i) = j(l), i = 1, . . . , L}| (7)

The crowding of case Ml on boundary conditions is analogous tohe original crowding. While calculating, the three parameters areegarded as three objectives.

. Experimental results

In this section, firstly the parameter settings are discussed to find suitable suite of parameters for the proposed algorithms. Then aepeated running experiment is implemented to prove the mutualromotion mechanism between CBR and MOEA in the proposed

ybrid strategy. At last, CBR-NSGA2 and CBR-SPEA2 are validated

or power supply multiobjective optimization in dynamic environ-ents.

0

0

1

1

2

0

( )

t t

t t t

t

SPEA

Initiali ze Populatio n PCreat e empt y ext ernal set Et

terminals donot be metAssign fitn ess for P an d ECopy al l nondominated individuals of P an d E to EPrune E or supplement dominated indivi

+

+

=

proce durebegin

odelihw

1

1

1

t

t

dual s til l E is ful lPerform binary tournament selection in EApply crossov er an d mutation op eratorst t

+

+

= +end

Fig. 7. The procedure of SPEA2.

Fig. 8. The procedure of case revisal.

4.1. Parameter settings

The parameters of CBR include K (the number of solutions in acase), Pcase (the largest amount of retrieved case solutions), Pexploit(the number of exploited solutions near each selected case solution)and Lc (the size of case library). The setting of K depends on the num-ber of objectives, because more solutions will be required to drawthe outline of Pareto Front in the optimization problems with moreobjectives. It is set to 10, the quintupling of the number of objectivesin this paper. Pcase should be no bigger than K, and it is also set to10, which will save some pruning operations. The parameter Pexploitcan be figured out from the equation: Pcase + Pcase × Pexploit = Pop, ofwhich Pop is population size of MOEAs. The equation is to guaranteethe share of CBR solutions in combined population is same as one ofMOEA population. If Pexploit has fractional part, it will be truncatedto an integer. The bigger Lc can cover more environment situationsbut makes solving by CBR more time-consuming. The time spent byCBR with various values of Lc is shown in Table 3. In this problem,Lc is set to 12,000.

We adopt the same settings of genetic operators as in NSGA2[39] and SPEA2 [42]. The parameter of population size needs to bedetermined by experiments, because a large population will result

in fewer evolution generations in dynamic environments. A wellsetting should achieve a trade-off of two sides.

Table 3The time spent by CBR with various values of Lc .

Lc

6000 12,000 20,000 24,000

Time (s) 0.54 1.43 3.47 5.11

2966 W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969

Table 4The settings of population size and maximum evolution generations.

Setting 1 Setting 2 Setting 3 Setting 4 Setting 5

Pop = 200 Pop = 100 Pop = 50 Pop = 20 Pop = 100Generation = 5 Generation = 10 Generation = 20 Generation = 50 Generation = 10

Table 5The winning and losing frequency of various settings for CBR-NSGA2.

Setting 1 Setting 2 Setting 3 Setting 4 Setting 5

Winning 201/1000 589/1000 117/1000 61/1000 32/1000Losing 286/1000 67/1000 97/1000 133/1000 417/1000

Table 6The winning and losing frequency of various settings for CBR-SPEA2.

Setting 1 Setting 2 Setting 3 Setting 4 Setting 5

afois1p

ifda

4

eiNesF

5.6 5.65 5. 7 5.75 5. 8 5.85 5. 997.794

97.796

97.798

97.8

97.802

97.804

97.806

97.808

97.81

97.812

97.814

Yields (ton)

Gra

de

The promotion of NSGA2

Run 1

Run 2

Run 3

Run 4

Run 5

Fig. 10. The evolutionary solutions in each run of CBR-NSGA2.

5.6 5.65 5. 7 5.75 5. 8 5.85 5. 9 5.9597.785

97.79

97.795

97.8

97.805

97.81

97.815

97.82

Yields (ton)

Gra

de

The promotion of CBR

Run 1Run 2Run 3Run 4Run 5

Fig. 11. The case solutions in each run of CBR-SPEA2.

Winning 151/1000 501/1000 217/1000 79/1000 52/1000Losing 196/1000 71/1000 91/1000 241/1000 401/1000

The CBR-NSGA2s and CBR-SPEA2s with different population sizere run for a maximum of 1000 function evaluations to test theirast-search performances in dynamic environments. The settingsf population size and maximum evolution generations are shownn Table 4. To guarantee fairness, the above algorithms use theame case solutions in experiments. The test problems set contain000 optimization models with randomly generated environmentarameters.

The statistic results derived by graphical comparison are shownn Tables 5 and 6. It can be concluded that the setting 2 has a biggestrequency to win other settings, and has a least frequency to beefeated by other settings. Therefore, the setting 2 is most reliablend the Pop are set to 100 for CBR-NSGA2 and CBR-SPEA2.

.2. A repeated running experiment

In the proposed hybrid framework, CBR and MOEAs can promptach other. The mutual effect helps CBR-MOEAs sustainablymprove themselves in dynamic environments. To prove that, CBR-SGA2 and CBR-SPEA2 are repeatedly run for a test problem and

ach run evolves for 10 generations. The results of CBR-NSGA2 arehown in Figs. 9 and 10. The results of CBR-SPEA2 are shown inigs. 11 and 12.

5.6 5.65 5. 7 5.75 5. 8 5.8597.785

97.79

97.795

97.8

97.805

97.81

97.815

97.82

Yields (ton)

Gra

de

The promotion of CBR

Run 1Run 2Run 3Run 4Run 5

Fig. 9. The case solutions in each run of CBR-NSGA2.

5.6 5.65 5. 7 5.75 5. 8 5.85 5. 9 5.9597.785

97.79

97.795

97.8

97.805

97.81

97.815

97.82

Yields (ton)

Gra

de

The promotion of SPEA2

Run 1Run 2Run 3Run 4Run 5

Fig. 12. The evolutionary solutions in each run of CBR-SPEA2.

W. Kong et al. / Applied Soft Computing 13 (2013) 2960–2969 2967

Table 7The response time by MOEA.

MOEA

cmCEpa

4o

4

ttrIwanawam

(

(

4

nMbttttt

s

TT

5.5 5.55 5. 6 5.65 5. 7 5.75 5. 897.77

97.775

97.78

97.785

97.79

97.795

97.8

97.805

97.81

97.815

97.82

Gra

de

Solutions by NSGA2 on each sampling point

Initial populationSampling point 1Sampling point 2Sampling point 3Sampling point 4Sampling point 5

4.3.3. Time-average performance testThe time-average coverage metrics (CS) have been shown in

Table 9, where the result in the ith row and the jth column

97.81

97.815

97.82

97.825Solutions by CBR−NSGA2 on each sampling point

ade

Initial populationSampling point 1Sampling point 2Sampling point 3Sampling point 4Sampling point 5

Pop = 50 Pop = 100 Pop = 150 Pop = 200

Time (s) 0.26 0.49 0.72 0.98

From the above results, in CBR-MOEAs, case solutions by CBRan enhance the MOEAs, and then they are revised by the opti-ized solutions by MOEAs. Through the cyclic mutual promotion,

BR-MOEAs can improve themselves in dynamic environments.ssentially, the hybrid strategy based on CBR retains the evolutionrocess of MOEA that is interrupted by the change of environment,nd continues it in a same or similar environment later.

.3. The validation of CBR-MOEAs for electric power supplyptimization

.3.1. Performance metricsFor the static MOPs, we are interested in the three aspects [45]:

he number of elements of the Pareto optimal set found at last,he distance of the produced by our algorithm with respect to theeal Pareto Front and the spread of the Pareto optimal set found.n DMOPs, the Pareto optimal set and Pareto Front may move

ith time, so a good algorithm should perform well in the time-verage metric. Besides, it is required to immediately provide newear optimal solutions once the environment changes because ancutely deteriorated power supply will be very harmful. Therefore,e are also focused on the response performance of the adopted

lgorithm. In the experiments, the adopted time-average perfor-ance metrics are defined in next.

1) The time-average coverage metric. As the real Pareto Front hasbeen not known, the coverage metric [45] is adopted to calcu-late the dominance ratio between two optimal sets that can beused to compare the convergence of two algorithms. The time-average coverage, notated as CS, is the average of dominanceratio in all sampling points.

2) The time-average spacing metric. The spacing metric is pro-posed by Schott [46], which can evaluate the distribution ofa Pareto optimal set, but it is affected by the range of objec-tive functions. Therefore each objective function is normalizedbefore calculating the spacing metric here. Similar to CS, thetime-average spacing metric, notated as SP, is the average ofspacing values in all sampling points.

.3.2. Response performance testWhile the environments change, CBR can immediately provide

ew solutions by retrieving the most similar historical case, whileOEA needs to reevaluate the population. The response time spent

y CBR depends on the number of environment parameters. Andhe time to reevaluate the population spent by MOEA depends onhe population size and the computation complexity of optimiza-ion function. For the problem addressed here, the time to respond

he change of environment by MOEA and CBR with different set-ings are shown in Tables 7 and 8, respectively.

It needs to be noted that CBR with Lc cases contains 10 × Lc

olutions in all that are corresponding to Lc kinds of environment

able 8he response time by CBR.

CBR

Lc = 6000 Lc = 12,000 Lc = 20,000 Lc = 240,00

Time (s) 0.54 1.43 3.47 5.11

Yields (ton)

Fig. 13. Solutions found by NSGA2 on each sampling point.

situations, so CBR can respond both timely and relatively properlyhowever environment changes. The population of MOEA inheritsthe past evolution information, so it has a well response per-formance in the slightly changing environment. MOEA will notrespond well if the environment changes acutely, such as operatingmodes vary or boundary conditions fluctuate by a large amplitude.

In Figs. 13 and 14, it shows the solutions found by NSGA2and CBR-NSGA2 on each sampling point after the operating modechanges from j = 4 to j = 1. Similarly, the results by SPEA2 and CBR-SPEA2 are shown in Figs. 15 and 16. In Figs. 13 and 15, the label“initial population” represents the population initialized partly byrandom and partly by inheriting the past population in NSGA2 andSPEA2. In Figs. 14 and 16, the label “initial population” representsthe initialized population generated by the proposed approach inthis paper. The label “Sampling k” represents the solutions by theabove algorithms on kth sampling point.

From the above results, it is evident that the initialized pop-ulation based on case solutions has a better spread than one byrandom and inheriting. With the help of a good initial population,CBR-MOEAs can converge more rapidly and accurately.

5.4 5. 5 5. 6 5. 7 5. 8 5. 997.79

97.795

97.8

97.805

Yields (ton)

Gr

Fig. 14. Solutions found by CBR-NSGA2 on each sampling point.

2968 W. Kong et al. / Applied Soft Comp

5.5 5.55 5. 6 5.65 5. 7 5.75 5. 8 5.8597.77

97.775

97.78

97.785

97.79

97.795

97.8

97.805

97.81

97.815

97.82

Yields (ton)

Gra

de

Solutions by SPEA2 on each sampling points

Initial population

Sampling points 1

Sampling points 2

Sampling points 3

Sampling points 4

Sampling points 5

Fig. 15. Solutions found by SPEA2 on each sampling point.

5.5 5.55 5. 6 5.65 5. 7 5.75 5. 8 5.85 5. 9 5.9597.79

97.795

97.8

97.805

97.81

97.815

97.82

97.825

Yields (ton)

Gra

de

Solutions by CBR−SPEA2 on each sampling point

Initial populationSampling point 1Sampling point 2Sampling point 3Sampling point 4Sampling point 5

Fig. 16. Solutions found by CBR-SPEA2 on each sampling point.

Table 9The time-average coverage metrics between four algorithms.

CS NSGA2 SPEA2 CBR-NSGA2 CBR-SPEA2

NSGA2 1 0.6874 0.1578 0.2687SPEA2 0.4598 1 0.2018 0.2694

rth

aNptC

TT

CBR-NSGA2 0.8937 0.9357 1 0.6538CBR-SPEA2 0.7643 0.8649 0.5739 1

epresents the coverage metric value of the ith row algorithm tohe jth column algorithm. The time-average spacing metrics (SP)ave been shown in Table 10.

From the results of Table 9, the hybrid algorithms CBR-NSGA2nd CBR-SPEA2 achieve a clear better performance than a pure

SGA2 and SPEA2 in the time-average coverage metrics. NSGA2erforms a little better than SPEA2, and CBR-NSGA2 performs a lit-le better than CBR-SPEA2 too. In Table 10, it can be shown thatBR-NSGA2 and CBR-SPEA2 also perform better than NSGA2 and

able 10he time-average spacing metrics.

NSGA2 SPEA2 CBR-NSGA2 CBR-SPEA2

SP 0.1931 0.2266 0.0872 0.1193

[

[

[

[

uting 13 (2013) 2960–2969

SPEA2 in the time-average spacing metrics. Therefore, it can be con-cluded that CBR-NSGA2 or CBR-SPEA2 performs better than NSGA2and SPEA2 in the metrics of convergence and spacing in changingenvironments.

5. Conclusions

A CBR based hybrid strategy is proposed to enhance MOEAs toaddress the power supply optimization problem, a DMOP with arequirement of real-time response. The hybrid algorithms instan-tiated from the hybrid strategy, CBR-MOEAs can not only timelyrespond to the changing environment, but also perform better thanpure MOEAs in tracking the moving of Pareto optimal set.

In the experimental results, it shows a pure EA may meetwith some troubles while handling the multiobjective optimiza-tion problems in acute changing environments. The main reason isthe inherited information by population maybe lose effectivenessin an acute change of environment. Besides, the task of trackingthe Pareto Front also brings a challenge to EAs in ever-changingenvironments.

The hybrid strategy based on CBR remedies the defect by retriev-ing and reusing historical cases in dynamic environments. Theinitial case solutions come from the expertise or experiences offield experts, and then are continuously revised by the solutionsby MOEAs. Essentially, the CBR-MOEAs retain the evolution pro-cess of MOEA that is interrupted by the change of environment,and continue it in a same or similar environment later. Fromthe experimental results, in CBR-MOEAs, case solutions by CBRcan enhance the MOEAs to tracking the moving of Pareto opti-mal set, and then they are improved by the optimized solutionsby MOEAs. Through the cyclic mutual promotion, CBR-MOEAs cankeep improving themselves in dynamic environments.

Acknowledgments

We would like to express our sincere appreciation to the anony-mous reviewers for their insightful comments, which have greatlyaided us in improving the quality of the paper. This work wassupported in part by the National Nature Science Foundationof China under Grant 61020106003, 61134006 and 60904079,by the National Basic Research Program of China under Grant2009CB320601, and the 111 project (B08015).

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