research article a hybrid multiobjective differential

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 841780 15 pageshttpdxdoiorg1011552013841780

Research ArticleA Hybrid Multiobjective Differential EvolutionAlgorithm and Its Application to the Optimization ofGrinding and Classification

Yalin Wang1 Xiaofang Chen1 Weihua Gui1 Chunhua Yang1

Lou Caccetta2 and Honglei Xu2

1 School of Information Science and Engineering Central South University Changsha 410083 China2Department of Mathematics amp Statistics Curtin University Perth WA 6845 Australia

Correspondence should be addressed to Xiaofang Chen xiaofangchencsueducn

Received 19 July 2013 Accepted 5 September 2013

Academic Editor Dewei Li

Copyright copy 2013 Yalin Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The grinding-classification is the prerequisite process for full recovery of the nonrenewable minerals with both production qualityand quantity objectives concerned Its natural formulation is a constrainedmultiobjective optimization problem of complex expres-sion since the process is composed of one grinding machine and two classification machines In this paper a hybrid differentialevolution (DE) algorithm with multi-population is proposed Some infeasible solutions with better performance are allowed tobe saved and they participate randomly in the evolution In order to exploit the meaningful infeasible solutions a functionallypartitionedmulti-populationmechanism is designed to find an optimal solution from all possible directions Meanwhile a simplexmethod for local search is inserted into the evolution process to enhance the searching strategy in the optimization processSimulation results from the test of some benchmark problems indicate that the proposed algorithm tends to converge quicklyand effectively to the Pareto frontier with better distribution Finally the proposed algorithm is applied to solve a multiobjectiveoptimization model of a grinding and classification process Based on the technique for order performance by similarity to idealsolution (TOPSIS) the satisfactory solution is obtained by using a decision-making method for multiple attributes

1 Introduction

Grinding-classification is an important prerequisite processfor most mineral processing plants The grinding processreduces the particle size of raw ores and is usually followed byclassification to separate them into different sizes Grinding-classification operation is required to produce pulp withsuitable concentration and fineness for flotation The pulpquality will directly influence the subsequent flotation effi-ciency and recovery of valuable metals from tailings In orderto improve economic efficiency and energy consumption theprocess optimization objectives include product quality andoutput yields Under certain mineral source conditions theobjectives are decided by a series of operation variables suchas the solid flow of feed ore to ball mill the steel ball fillingrate and the flow rates of water added to the first and the

second classifier recycles To solve the optimization modelof productsrsquo output and quality in the grinding-classificationprocess is of great significance to improve the technicaland economic specifications and it has been a continuousendeavor of the scientists and engineers [1ndash3]

Grinding-classification is an energy-intensive processinfluenced bymany interacting factorswithmutual restraintsThe goals of grinding-classification optimization problemare decided by multiple constrained input control variablesof nonlinear relationships So the optimization model ofgrinding-classification operation is a complex constrainedmultiobjective optimization problem (CMOP) Generallyconstrained multiobjective problems are so difficult to besolved that the constraint handling techniques and multi-objective optimization methods need to be combined foroptimization

2 Journal of Applied Mathematics

Multiobjective optimization problems (MOPs) in thecase of traditional optimization methods are often handledby aggregating multiple objectives into a single scalar objec-tive through weighting factors MOPs have a set of equallygood (nondominating) solutions instead of a single onecalled a Pareto optimum which was introduced by Edge-worth in 1881 [4] and later generalized by Pareto in 1896[5] The practical MOPs are often implicated in series ofequations functions or procedures with complicated con-straints Therefore the evolutionary algorithms are attractiveapproaches for low requirements onmathematical expression[6] Since the mid 1980s there has been a growing interestin solving MOPs using evolutionary approaches [7ndash10]One of the most successful evolutionary algorithms for theoptimization of continuous space functions is the differentialevolution (DE) [11] DE is simple and efficiently converges tothe global optimum in most cases [12 13] Its efficiency hasbeen proven [14] in many application fields such as patternrecognition [15] and mechanical engineering [16]

There have been many improvements for DE to solveMOPs Abbass [17] firstly provided a Pareto DE (PDE) algo-rithm for MOPs in which DE was employed to create newsolutions and only the nondominated solutions were kept asthe basis for the next generation Madavan [18] developed aPareto differential evolution approach (PDEA) in which newsolutions were created by DE and kept in an auxiliary popu-lation Xue et al [19] introduced multiobjective differentialevolution (MODE) and used Pareto-based ranking assign-ment and crowding distancemetric but in a differentmannerfrom PDEA Robic and Filipi [20] also adopting Pareto-based ranking assignment and crowding distance metricdeveloped a DE for multiobjective optimization (DEMO)with a different population update strategy and achievedgood results Huang et al [21] extended the self-adaptive DE(SADE) to solve MOPs by a so called multiobjective self-adaptive DE (MOSADE) They further extended MOSADEby using objectivewise learning strategies [22] Adeyemo andOtieno [23] provided multiobjective differential evolutionalgorithm (MDEA) In MDEA a new solution was generatedby DE variant and compared with target solution If itdominates the target solution then it was added to the newpopulation otherwise a target solution was added

On the other hand single-objective constrained opti-mization problems have been studied intensively in the pastyears [24ndash28] Different constraint handling techniques havebeen proposed to solve constrained optimization problemsMichalewicz and Schoenauer [29] divided constraints han-dling methods used in evolutionary algorithms into fourcategories preserving feasibility of solutions penalty func-tions separating the feasible and infeasible solutions andhybrid methods The differences among these methods arehow to deal with the infeasible individuals throughout thesearch phases Currently the penalty functionmethod ismostwidely used and this algorithm strongly depends on thechoice of the penalty parameter

Although the multiobjective optimization and the con-straint handling problem have received lots of contributionrespectively the CMOPs are still difficult to be solved inpractice Coello and Christiansen [30] proposed a simple

approach to solve CMOPs by ignoring any solution that vio-lates any of the assigned constraints Deb et al [8] proposeda constrained multiobjective algorithm based on the conceptof constrained domination which is also known as superi-ority of the feasible solution Woldesenbet et al [31] intro-duced a constraint handling technique based on adaptivepenalty functions and distance measures by extending thecorresponding version for the single-objective constrainedoptimization

In the MOP of grinding and classification process thedefinitions of Pareto solutions Pareto frontier and Paretodominance are in consistency with the classic definitionsClearly the Pareto frontier is amapping of the Pareto-optimalsolutions to the objective space In the minimization sensegeneral constrained MOPs can be formulated as follows

min 119865 (119883) = min [1198911(119883) 119891

2(119883) 119891

119903(119883)]

st 119892119894(119883) le 0 (119894 = 1 2 119901)

ℎ119895(119883) = 0 (119895 = 119901 + 1 119902)

119909119896isin [119909119896min 119909119896max] (119896 = 1 2 119899)

(1)

where 119865(119883) is the objective vector 119883 = (1199091 119909

119899) isin R119899 is

a parameter vector 119892119894(119883) is the 119894th inequality constraint

and ℎ119895(119883) is the 119895th equality constraint 119909

119896min and 119909119896max

are respectively the lower and upper bounds of the decisionvariable 119909

119896

In this paper based on the specific industrial backgroundof continuous bauxite grinding-classification operation anew hybrid DE algorithm is proposed to solve complex con-strained multiobjective optimization problems Firstly ahybrid DE algorithm for MOPs with simplex method (SM-DEMO) is designed to overcome the problems of globalperformance degradation and being trapped in local opti-mum Then for the MOPs with complicated constraints theproposed algorithm is formed by combining SM-DEMO andfunctional partitioned multi-population In this method theconstruction of penalty functions is not required and themeaningful infeasible solutions are fully utilized

The remainder of the paper is structured as followsSection 2 describes the SM-DEMO algorithm for uncon-strained cases The proposed algorithm of multipopulationfor constrained MOPs is given in Section 3 with verificationof performance by benchmark testing results Section 4describes the model of productsrsquo output and quality in thegrinding-classification process in detail and the applicationof the proposed algorithm in the optimizationmodel Finallythe conclusions based on the present study are drawn inSection 5

2 SM-DEMO Algorithm forUnconstrained MOPs

In order to efficiently solve multiobjective optimizationproblem and find the approximately complete and nearop-timal Pareto frontier we proposed a hybrid DE algorithmfor unconstrained multiobjective optimization with simplexmethod

The differential evolution with initialization crossoverand selection as in usual genetic algorithms uses a pertur-bation of two members as the mutation operator to produce

Journal of Applied Mathematics 3

a new individualThemutation operator of the DE algorithmis described as follows

Considering each target individual 119909119866119894 in the 119866th gener-

ation of size119873119901 a mutant individual 119909119866+1119894

is defined by

119909119866+1

119894= 119909119866

1199033+ 119865 (119909

119866

1199031minus 119909119866

1199032) (2)

where indexes 1199031 1199032 and 119903

3represent mutually different

integers that are different from 119894 and that are randomlygenerated over [1119873119901] and 119865 is the scaling factor

The simplex method proposed by Spendley Hext andHimsworth and later refined by Nelder andMead (NM) [32]is a derivative-free line-search method that is particularlydesigned for traditional unconstrained minimization scenar-ios Clearly NM method can be deemed as a direct line-search method of the steepest descent kind The ingredientsof the replacement process consist of four basic operationsreflection expansion contraction and shrinkage Throughthese operations the simplex can improve itself and approxi-mate to a local optimum point sequentially Furthermore thesimplex can vary its shape size and orientation to adapt itselfto the local contour of the objective function

21 Main Strategy of SM-DEMO The SM-DEMO algorithmis improved by the following three points compared with DE

211 Modified Selection Operation After traditional DE evo-lution the individual 119906119866+1

119894119895may violate the boundary con-

straints 119909max119894119895

and 119909min119894119895

119906119866+1119894119895

is replaced by new individual119908119866+1

119894119895being adjusted as follows

119908119866+1

119894119895

=

119909max119894119895

+ rand () lowast (119909max119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895gt 119909

max119894119895

)

119909min119894119895

+ rand () lowast (119909min119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895lt 119909

min119894119895

)

119906119866+1

119894119895 otherwise

(3)

The new population is combined with the existing parentpopulation to form a new set 119872119892 of bigger size than 119873119901A nondominated ranking of 119872119892 is performed and the 119873119901best individuals are selected This approach allows a globalnondomination checking between both the parent and thenew generation rather than only in the new generation asis done in other approaches whereas it requires additionalcomputational cost in sorting the combined

212 Nondominated Ranking Based on Euclidean DistanceThe solutions within each nondominated frontier that residein the less crowded region in the frontier are assigned a higherrank as the NSGA-II algorithm [8] developed by Deb et alindicated The crowding distance of the 119894th solution in itsfrontier (marked with solid circles) is the average side lengthof the cuboids (shown with a dashed box in Figure 1(a))The crowding-distance computation requires sorting thepopulation according to each objective function value inascending order of magnitude As shown in Figure 1 119860 and119862 are two solutions near 119861 in the same rank and 120590

0(119861) is the

crowding distance of the119861th solution traditionally calculatedas follows

1205900(119861) =

119899

sum

119895=1

10038161003816100381610038161003816119891119895(119860) minus 119891

119895(119862)

10038161003816100381610038161003816 (4)

where 119891119895(119860) 119891

119895(119862) are the objective vectors For each

objective function the boundary solutions (solutions withthe smallest and the largest function values) are assigned aninfinite distance value

A crowding-distance metric is used to estimate thedensity of solutions surrounding a particular solution in thepopulation and is obtained from the average distance of thetwo solutions on either side of the solution along each of theobjectives As shown in Figure 1119860119861119862 are the individuals ofthe generation on the same frontier and we easily know thatthe density in Figure 1(a) is better than that in Figure 1(b) Ifwe use (4) to calculate the crowding distance of 119862 we onlyknow that in Figure 1(a) it is better than in Figure 1(b) thecrowding distance of119862 in Figures 1(a) and 1(c) is equal whichis against the knowledge

To distinguish the mentioned situations we proposean improved crowding-distance metric based on Euclideandistance 119872 is the center point of the line 119860119861 119891

119895is the 119895th

objective vector and the crowding distance 120590(119861) is defined asfollows

120590 (119861) = |119860119862| minus |119861119872|

= radic

119899

sum

119895=1

[119891119895(119860) minus 119891

119895(119862)]2

minus radic

119899

sum

119895=1

119891119895(119861) minus

[119891119895(119860) + 119891

119895(119862)]

2

2

(5)

The crowded-comparison operator guides the selectionprocess at the various stages of the algorithm toward auniformly distributed Pareto-optimal frontier To carry outthe comparison we assume that every individual in thepopulation has two attributes (1) nondomination rank 119894rankand (2) crowding distance 119894distance Then a partial order isdefined as 119894 ≺ 119895 If 119894 ≺ 119895 that is between two solutionswith different nondomination ranks we prefer the solutionwith the lower (better) rank namely 119894rank ≺ 119895rank Otherwiseif both solutions belong to the same frontier that is 119894rank =

119895rank then we prefer the solution that is located in a lessercrowded region that is 119894distance ≻ 119895distance

213 Simplex Method for Local Search The simplex methodfor local search is mixed in the evolution process to enhancethe searching strategy in the optimization process The goalof integrating NM simplex method [32] and DE is to enrichthe population diversity and avoid being trapped in localminimum We apply NM simplex method operator to thepresent population when the number of iterations is greaterthan a particular value (like 119866max2) The individuals thatachieved the single extreme value in each objective functionaremarked as the initial vertex points of simplexmethodThe


A

B

C

f1

f2

(a)

A

B

C

f1

f2

(b)

A

B

C

f1

f2

(c)

A

B

C

M

f1

f2

(d)

Figure 1 Crowding-distance diagram

present population is updated according to simplex methoduntil the terminal conditions are satisfied

The computation steps of the algorithm are included inSection 32

22 Evaluation Criteria Unlike the single-objective opti-mization it is more complicated for solution quality eval-uation in the case of multiobjective optimization Many ofthe suggested methods can be summarized in two typesOne is to evaluate the convergence degree by computing theproximity between the solution frontier and the actual Paretofrontier The other is to evaluate the distribution degree ofthe solutions in objective space by computing the distancesamong the individuals Here we choose both methods toevaluate the performance of the SM-DEMO algorithm

(1) Convergence Evaluation Deb et al [8] proposed thismethod in 2002 It is described as follows

120574 =1

119876(

119876

sum

119894=1

min 1003817100381710038171003817119875lowastminus 119875FT

1003817100381710038171003817) (6)

where 120574 is the extent of convergence to a known of Pareto-optimal set 119875lowast is the obtained nondomination Pareto fron-tier 119875FT is the real nondomination Pareto frontier 119875lowast minus119875FTis the Euclidean distance of 119875lowast with 119875FT and119876 is the numberof obtained solutions

(2) Distribution Degree EvaluationThenonuniformity in thedistribution is measured by SP as follows

SP = radic 1

(119876 minus 1)

119876

sum

119894=1

(119889 minus 119889119894)2

(7)

where 119889119894is the Euclidean distance among consecutive solu-

tions in the obtained nondominated set of solutions andparameter 119889 is the average distance

23 Experimental Studies Four well-known benchmark testfunctions [33] are used here to compare the performance ofSM-DEMO with NSGA-II DEMOParent These four prob-lems are called ZDT2 ZDT3 ZDT4 and ZDT6 each has twoobjective functions We describe them in Table 1


Table 1 Test problems

Test problems Objective functionsmin119865(119883) = min[119891

1(119883) 119891

2(119883)]

Range of variable

ZDT21198911(119883) = 119909

1 1198912(119883) = 119892(119883)(1 minus (

1198911

119892(119883))

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT31198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892) minus (

1198911

119892) sin (10120587119891

1))

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT41198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892))

119892 (119883) = 1 + 10 (119899 minus 1) +

119899

sum

119894=2

(1199092

119894minus 10 cos (4120587119909

119894))

119899 = 10

0 le 1199091le 1

minus5 le 119909119894le 5 119894 = 2 119899

ZDT61198911(119883) = 1 minus exp (minus4119909

1) sin6 (6120587119909

1) 1198912(119883) = 119892(1 minus (

1198911

119892)

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

(119899 minus 1)025

119899 = 10

0 le 119909119894le 1

Table 2 The performance results of the each algorithm on the test function

Test function Algorithm 120574 SP

ZDT2DEMOParent 0005120 plusmn 0000312 0000630 plusmn 0000010

NSGA-2 0007120 plusmn 0000413 0000540 plusmn 0000940

SM-DEMO 0004013 plusmn 0000230 0000423 plusmn 0000011










The simulation is carried out under the environment ofIntel Pentium 4 CPU 306GHz 512MB memory WindowsXP Professional Matlab71 Initialization parameters are setas follows population size119873119901 = 100 scaling factor 119865 = 08cross rate 119862

119877= 06 maximum evolution generation 119866max =

250 and number of SM evolution iterations 119866SM = 100All of the three algorithms are real coded with equal

population size and equal maximum evolution generationEach algorithm independently runs 20 times for each testfunction Because we cannot get the real Pareto-optimal setwewill take 60 Pareto-optimal solutions obtained by the threealgorithms as a true Pareto-optimal solution set

We evaluated the algorithms based on the two perfor-mance indexes 120574 and SP Table 2 shows themean and varianceof 120574 and SP using three algorithms SM-DEMO NSGA-II and DEMOParent We can learn from Table 2 for theZDT2 function that all of the three algorithms have a good

performance while the SM-DEMO is slightly better than theother two algorithms In terms of convergences for ZDT3ZDT4 and ZDT6 which are more complex than ZDT2 SM-DEMO is significantly better thanDEMOParent andNSGA-II

Figure 2 shows a random running of SM-DEMO algo-rithm It is clear that SM-DEMO algorithm can produce agood approximation and a uniform distribution

3 Proposed Hybrid Algorithm for CMOP

The space of constrained multiobjective optimization prob-lem can be divided into the feasible solution space and theinfeasible solution space as shown in Figure 3 where 119878 isthe search space Ω is the feasible solution space and 119885 isthe infeasible solution space 119909

119894(119894 = 1 2 3 4) is the feasible

solution and 119910119894(119894 = 1 2 3 4) is the infeasible solution


SM-DEMOPareto frontier

0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6

Figure 2 SM-DEMO simulation curve

Assume that 119909lowast is the global optimal solution and 1199101is

the closest one to 119909lowast If the infeasible population 1199101is not

excluded by the evolution algorithm it is permitted to exploreboundary regions from new directions where the optimumis likely to be found

31 General Idea of the Proposed Algorithm Researchershave gradually realized the merit of infeasible solutions insearching for the global optimum in the feasible region Someinfeasible solutions with better performance are allowed to besaved Farmani et al [34] formulated a method to ensure thatinfeasible solutions with a slight violation become feasible inevolution Based on the constraints processing approach ofmultiobjective optimization problems the proposed hybridDE algorithm avoids constructing penalty function anddeleting meaningful infeasible solutions directly

Here the proposed algorithm will produce multiplegroups of functional partitions which include an evolution-ary population 119875119892 of size 119873119901 an intermediate population119872119892 to save feasible individuals an intermediate population119878119892 to save infeasible individuals a population 119875119891 to save theoptimal feasible solution found in the search process and apopulation 119875119888 to save the optimal infeasible solution Therelationship of multi-population is shown in Figure 4

With the description of (1) equality constraints are alwaystransformed into inequality constraints as |ℎ

119895(119883)| minus 120575 le 0

where 119895 = 119901 + 1 119902 and 120575 is a positive tolerance valueTo evaluate the infeasible solution the degree of constraintviolation of individual 119883 on the 119895th constraint is calculatedas follows

119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)


Feasible solutionspace Ω

Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast

Infeasible solutionspace Z

Figure 3 Distribution diagram of search space

Evolutionarypopulation

Pg

Intermediatepopulation

Mg

Optimal Randomly selected


SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1

g = g + 1Randomly selected

feasiblesolution Pf

Optimal infeasible

solution Pc

Figure 4The relationship diagram ofmultipopulation demonstrat-ing

The final constraint violation of each individual in thepopulation can be obtained by calculating the mean of thenormalized constraint violations

In order to take advantage of the infeasible solutionswith better performance we proposed the following adaptivedifferential mutation operator to generate individual varia-tion learning from the mutation operators in DErand-to-best1bin according to rules defined by Price et al [11]Considering each individual vector 119909119866

119894 a mutant individual

119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)

where 1199031and 1199032represent different integers and also different

from 119894 randomly generated over [1119873119901] 119865 is the scalingfactor 119909119866

119891119894(119894 = 1 2 119899) is randomly generated from 119875119891

119910119866

119894(119894 = 1 2 119899) is randomly generated from 119875119888 and 119877

119862is

the mutation factor as follows

119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)

where 1198771198880is the initial value of the variability factor const is

a small constant to ensure that the fractional is meaningfuland 120574119866 is defined as follows

120574119866=the number of infeasible solutions in 119875119892

119873119901 (11)

32 Framework of the Proposed Algorithm The proposedalgorithm is described as follows

Step 1 (initialization) Generate the population 119875119892 119875119891 and119875119888 of size 119873119901 119873119875

1 and 119873119875

2 Set the value of 119862

119877(crossover

probability)119866max (the number of function evaluations)119866SM(the iterative number of evolution by NM simplex method)119892 = 1 (the current generation number) and positive controlparameter for scaling the difference vectors 119865

1 1198652 Randomly

generate the parent population 119875119892 from the decision spaceSet the 119875119891 and 119875119888 and let the intermediate populations 119878119892and119872119892 be empty

Step 2 (DE reproduction) By (3) and (9) for mutationcrossover and selection an offspring 119878119892 is created Judge theconstraints of all individuals in 119875119892 In accordance with (8)we first calculate constraint violation degree 119881

119894(119883) of all of

the individuals If 119881119894(119883) = 0 the solution is feasible and

preserved to the intermediate set 119872119892 if 119881119894(119883) gt 0 the

solution is infeasible and preserved to the intermediate set 119878119892

Step 3 (simplex method) Apply NM simplex method oper-ator to the present population if 119892 ge 119866max2 Updatethe present population 119872119892 when the number of iterationsexceeds maximum iterations

Step 4 (119875119891 construction) Rank chromosomes in 119872119892 basedon (5) and generate the elitist population 119875119891 (the size is119873119901)from the ranked population119872119892

Step 5 (119875119888 construction) Add the chromosomes in 119878119892 withslight constraint violations to the 119875119888

Step 6 (mixing the population) Combine 119878119892with the existingparent population to form a new set119872119892 Remove the dupli-cate individuals in119872119892 and the existing parent population

Step 7 (evolution) Randomly choose chromosomes from119875119888 119875119892 and 119875119891 Use the adaptive differential mutation anduniform discrete crossover to obtain the offspring population119875119892 + 1

Step 8 (termination) If the stopping criterion ismet stop andoutput the best solution else go to Step 2

33 Experimental Study In this section we choose threeproblems CTP TNK and BNH as shown in Table 3 to testthe proposed method and compare the method with thecurrent CNSGA-II [35]


Table 3 Test functions

Testfunction

Objective functionmin119865(119883) = min[119891

1(119883) 119891

2(119883)]

Constraints Range ofvariable

CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00

Feasible solutionInfeasible solution

02 04 06 08 10f1

f2

Pareto-optimal frontier

(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2

Figure 5 CTP solution space

For CTP problem there are the six parameters 120579 119886 119887 119888119889 and 119890 that must be chosen in a way so that a portion ofthe unconstrained Pareto-optimal region is infeasible Eachconstraint is an implicit non-linear function of decisionvariables Thus it may be difficult to find a number ofsolutions on a non-linear constraint boundary We take twosets of values for six parameters in CTP problem which aredetermined as (1) CTP1 120579 = 01120587 119886 = 40 119887 = 05 119888 = 1119889 = 2 and 119890 = minus2 (2) CTP2 120579 = minus02120587 119886 = 02 119887 = 10119888 = 1 119889 = 6 and 119890 = 1 The Pareto frontiers the feasiblesolution spaces and the infeasible solution spaces are shownin Figure 5

The parameters are initialized as followsThe size of pop-ulation 119875119892 is119873119901 = 200 size of 119875119891 is119873119875

1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865

2are randomly generated

within [05 1] cross rate is 119862119877= 06 maximum evolution

generation is 119866max = 300 and number of SM evolutioniterations is 119866SM = 100 All of the proposed algorithmsand CNSGA-II are real coded with equal population size andequal maximum evolution generation Each algorithm runs20 times independently for each test function

Figure 6 shows the result of a random running of theproposed algorithm and NSGA-II the smooth curve ldquomdashrdquorepresents the Pareto frontier and ldquoXrdquo stands for the solutionachieved by the proposed algorithm or NSGA-II

It is obvious that the proposed algorithm returns abetter approximation to the true Pareto-optimal frontie anda distribution of higher uniformity We also evaluated the


40

35

30

25

20

15

10

050

Pareto frontierProposed algorithm

02 0301 04 05 07 0906 08 10

f2

f1

(a) CTP1 calculated by the proposed algorithm

CNSGA-IIPareto frontier

40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10

(b) CTP1 calculated by CNSGA-II

1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10

(c) CTP2 calculated by the proposed algorithm


1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10

(d) CTP2 calculated by CNSGA-II

50

40

30

20

10

00 50 100 150


f2

f1

(e) BNH calculated by the proposed algorithm

Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II

(f) BNH calculated by CNSGA-II

Figure 6 Continued


15

10

05

00 05 10 15


f2

f1

(g) TNK calculated by the proposed algorithm

Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II

(h) TNK calculated by CNSGA-II

Figure 6 The comparison chart of Pareto frontier

Table 4 The comparison of performance


CTP1 CNSGA-Π 0021317 plusmn 0000323 0873321 plusmn 008725

The proposed 0009836 plusmn 0000410 0567933 plusmn 001845



BNH CNSGA-Π 0014947 plusmn 0000632 0336941 plusmn 000917


TNK CNSGA-Π 0013235 plusmn 0000740 0464542 plusmn 000730


Feed water Water

Feed ore Ball mill OverflowMixing

PumpClassifierwater

additionFirst

classifierrecycle First-

overflow Spiral classifier 2Second-stage

to flotationSpiral classifier 1 Second classifier recycle(rough concentrate)

addition

pump

stageoverflow

Figure 7 Flow diagram of the grinding and classification process

algorithms based on the two criterions 120574 and SP as shownin Table 4 It can be observed from the data in Table 4that the proposed algorithm performs significantly betterthan the classical CNSGA-II algorithm in convergence anddistribution uniformityThe simulation results show that thisalgorithm can accurately converge to global Pareto solutionsand can maintain diversity of population

4 Optimization of Grinding andClassification Process

41 Bauxite Grinding and Classification Process Thegrindingand classification process is the key preparation for thebauxite mineral processing Here we consider a bauxitegrinding process in a certain mineral company with singlegrinding and two-stage classification as shown in Figure 7

The process consists in a grinding ball mill and twospiral classifiers First classifier recycle will be put back tothe ball mill for regrinding and the first-stage overflowwill be put into second spiral classifier after being mixedwith water the second classifier recycle will be prepared forBayer production as the rough concentrate and the second-stage overflow will be sent to the next flotation processThe production objectives are composed of the productionyields technically represented by the solid flow of feed oresince the process is nonstorable and the mineral processingquality represented by percentage of the small-size fractionsof mineral particles in the second-stage overflows


Table 5 Notations for the model of the grinding and classificationprocess

Notation Description

119901119894

Particle percentage of the 119894th size fraction inthe ball mill overflow

119891119895

Particle percentage of the 119894th size fraction inthe Feed ore

119862 Rate of the first classifier recycle119864119886119894 The efficiency of the first spiral classifier

120591 Themean residence time1198751 The internal concentration in ball mill119872MF The solid flow of feed ore1198821 The water addition of the first classifier recycle

1198822 The classifier water addition

119887119894119895 The breakage distribution function119878119894 The breakage rate function119889119894 The particle with the 119894th size

120572119888119894

The particle percentage of the 119894th size in thefirst classifier overflow

119889119900 The unit size of the particle1198752 First-stage overflow120601119861 Ball filling rate

119860119888 119861119888

Parameters of the first-stage classifier overflowsize fraction distribution

119864119886119894 The efficiency of the first spiral classifier

11988950119888

The particle size fraction after correctionseparation

119898 The separation accuracy119896 The intermix index

1198641198861015840

min 1198891015840

50119888 1198981015840 1198961015840 The corresponding key parameters to theefficiency of the second spiral classifier

119886 120572 120583 ΛFour key parameters to control the breakagerate function

119889min 119889max Theminimum and maximum particle sizes119860MF 119861MF Parameters of feed ore size fraction distribution

119865(119894)The cumulative particle percentage less thanthe 119894th size fraction in feed ore

1205721198881015840

119894

The particle percentage of the 119894th size fractionin the second classifier overflow

1198641198861015840

119894 The efficiency of the second spiral classifier

42 Predictive Model of the Grinding and Classification Pro-cess Here we establish the mathematical predictive modelof each unit process in the bauxite grinding and classificationprocess The notations of the indexes decision variables andparameters are listed in Table 5 These notations will be usedfor the model of the grinding and classification process

421 Ball Mill Circuit Model Here 119901119894is the particle per-

centage of 119894th size fraction in the ball mill overflow 119891119895is the

particle percentage of 119894th size fraction in feed ore rate of thefirst classifier recycle 119862 is known and 119864119886

119894is the efficiency of

the first spiral classifier According to a technical report offield investigation and study we have that

119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)

where 120591 is the mean residence time of minerals 1198751is the

internal concentration in ball mill and

1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)

where119872MF (119905ℎ) is the solid flow of feed ore1198821is the water

addition of the first classifier recycle and1198822is the classifier

water addition 119887119894119895is the breakage distribution function 119878

119894

is the breakage rate function and it satisfied the followingequation

119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)

where 119889119894is the particle with the 119894th size 119889

119900it is a unit when

per millimeter is a unit 119889119900= 1 119889

119894= 119894 (mm) and 119886 120572 120583

and Λ are four key parameters to control the breakage ratefunction

In a concrete grinding and classification process the ballmill size is fixed and the speed of ball mill is constantThrough data acquisition and testing of grinding and clas-sification steady-state loop the regression model between 119886120572 120583 Λ and condition variables size fraction distributioncan be established The input variables are ball filling rate120601119861 solid flow of feed ore 119872MF water addition of the first

classifier recycle1198821 and parameters of feed ore size fraction

distribution 119860MF 119861MF The regression model is

[119886 120572 120583 Λ]119879

=[[

[

11990911

sdot sdot sdot 1199091119895

d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)

The value of 119909119894119895(119894 = 1 2 3 4 119895 = 1 2 5) can be

obtained by the experimental data regression 119860MF 119861MF can


be obtained from feed ore size fraction distribution and 119865(119894)is the cumulative particle percentage less than the 119894th sizefraction in feed ore and it is represented as follows

119865 (119894) = 1 minus exp (minus119860MF119889119861MF119894

) (16)

422 Spiral Classifier Model 120572119888119894is the particle percentage

of the 119894th size fraction in the first classifier overflow and 119901119894

is the particle percentage of the 119894th size fraction in ball milloverflow The spiral classifier model is as follows

120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)

where 119864119886119894is the efficiency of the first spiral classifier and the

mechanism formula of 119864119886119894is shown as follows

119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]

+ 119864119886min sdot [1 minus (119889119894minus 119889min119889max

)]

119896

(18)

where 119889119894is the particle with the 119894th size 119889min and 119889max

represent maximum and minimum particle sizes 11988950119888

isthe particle size fraction after correction separation 119898 isseparation accuracy and 119896 is intermix index

Through data acquisition and testing of grinding andclassification steady-state loop the regressionmodel betweenclassification parameters and condition variables size frac-tion distribution can be established The input variablesinclude the solid flow of feed ore 119872MF the classifier wateraddition 119882

2 and the parameters of ball mill overflow size

fraction distribution 119860MF 119861MF The regression model isshown as follows

[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)

where the value of 119910119894119895(119894 = 1 2 3 4 119895 = 1 2 3 4) can be

obtained by data regressionThe first-stage overflow 119875

2calculation formula is as

follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)

Similarly we can get the second spiral classifier model asfollows

1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)

where 1205721198881015840119894is the particle percentage of the 119894th size fraction in

the second classifier overflow 120572119888119894is the particle percentage

of the 119894th size fraction in the first classifier overflow and

1198641198861015840

119894is the efficiency of the second spiral classifier The spiral

classifier model is as follows

1198641198861015840

119894= 1 minus exp[

[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896

1015840 are key parameters to theefficiency of the second spiral classifier Through data acqui-sition and testing of grinding and classification steady-stateloop the regression model between classification parametersand condition variables size fraction distribution can beestablished The input variables include solid flow of feedore119872MF and parameters of the first-stage classifier overflowsize fraction distribution 119860

119888 119861119888 which are solved by similar

equation to (20) The regression model is shown as follows

[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840

11sdot sdot sdot 1199101015840

1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)

where the value of1199101015840119894119895(119894 = 1 2 3 4 119895 = 1 2 3) can be obtained

by experimental data regression

43 Optimization Model of Grinding and Classification Pro-cess Two objective functions in the process are identifiedone is tomaximize output119891

1(119883) and the other is tomaximize

the small-size fractions (less than 0075mm fractions) in thesecond-stage overflow 119891

2(119883) It is also necessary to ensure

that the grinding productmeets all of other technical require-ments and the least disturbance in the following flotationcircuit As the constraints the feed load of the grinding circuit119872MF the steel ball filling rate 120601

119861 the first and the second

overflows 1198751and 119875

2 and the particle percentage of fine size

fraction in the first and the second classifier overflows 120572119888minus0075

and 1205721198881015840minus0075

should be within the user specified boundsThe operation variables are the solid flow of feed ore

119872MF water addition of the first classifier recycle 1198821 ball

filling rate 120601119861 and water addition of the second classifier119882

2

Based on all of the above grinding and classification processmultiobjective optimization model is as follows

max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)

st 119872MFmin le 119872MF le 119872MFmax120601119861min le 120601119861 le 120601119861max1198751min le 1198751 le 1198751max1198752min le 1198752 le 1198752max120572119888minus0075

ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)


Table 6 The optimization results calculated by the proposed algo-rithm

Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000

Optimization resultsThe raw data

f1(x)

f2(x)

()

89000

Figure 8 The comparison chart between optimization results andindustrial data

In (24) 119891(119872MF11988211198822 120601119861) implicates the model ofgrinding and classification represented by (12)ndash(23) Theproposed algorithm is applied to solve the problem and theoptimization results are shown in Table 6

With the practical process data from a grinding circuitof a mineral plant the simulation of this hybrid intelligentmethod adopted the same parameters on the variation infresh slurry feed velocity density particle size distributionand cyclone feed operating configurations

The comparison of production data and optimizationresults in Table 6 is shown in Figure 8 where ldquoXrdquo repre-sents the proposed algorithm optimization results and ldquoIrdquorepresents the original data collected from the field withoutoptimization of raw data According to the objectives thedata point closer to the upper right edge is more beneficialObviously the proposed optimization result is far betterthan the original data indicating the effectiveness of theoptimization approach

44 TOPSIS Method for Solution Selection The resolution ofamultiobjective optimization problemdoes not endwhen thePareto-optimal set is found In practical operational prob-lems a single solution must be selected TOPSIS [36] is auseful technique in dealing with multiattribute or multicrite-ria decision-making (MADMMCDM) problems in the realworld The standard TOPSIS method attempts to choosealternatives that simultaneously have the shortest distancefrom the positive-ideal solution and the farthest distancefrom the negative-ideal solution According to the TOPSISmethod the relative closeness coefficient is calculated andthe best solution in Table 6 is the solution number 10 as1198911(119883) = 89298 (119905ℎ) and 119891

2(119883) = 95763 () The

corresponding decision variables are 119872MF = 89298 (119905ℎ)120601119861= 32119882

1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions

Promoted by the requirements of engineering optimizationin complex practical processes of grinding and classificationwe proposed a hybrid multiobjective differential evolutionalgorithm with a few beneficial features integrated Firstly anarchiving mechanism for infeasible solutions is establishedwith functional partitioned multi-population which aims todirect the population to approach or land in the feasibleregion from different directions during evolution Secondlywe propose an infeasible constraint violations function toselect infeasible population with better performance sothat they are allowed to be saved and to participate inthe subsequent evolution Thirdly a nondominating rankingstrategy is designed to improve the crowding-distance sortingand return uniform distribution of Pareto solutions Finallythe simplex method is inserted in the differential evolutionprocess to purposefully enrich the diversity without excessivecomputation cost The advantage of the proposed algorithmis the exemption from constructing penalty function andthe preservation of meaningful infeasible solutions directlySimulation results on benchmarks indicate that the proposedalgorithm can converge quickly and effectively to the truePareto frontier with better distribution

Based on the investigated information about grinding cir-cuit process we established a multiobjective optimal modelwith equations frommechanism knowledge parameters rec-ognized by data regression and constraints of technicalrequirements The nonlinear multiobjective optimizationmodel is too complicated to be solved by traditional gradient-based algorithmsThe proposed hybrid differential evolutionalgorithm is applied and tested to achieve a Pareto solution


set It is proven to be valuable for operation decision makingin the industrial process and showed superiority to the oper-ation carried out in the production In fact many operatingparameters in complex processes are highly coupled andconflicting with each other The optimal operation of theentire production process is very difficult to obtain bymanualcalculation let alone the fluctuation situation of processconditions The application case indicates that the proposedmethod has good performance and is helpful to inspirefurther research on evolutionary methods for engineeringoptimization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61374156 61273187 and 61134006)the National Key Technology Research and DevelopmentProgram of the Ministry of Science and Technology ofChina (2012BAK09B04) the Science Fund for CreativeResearch Groups of the National Natural Science Foundationof China (61321003) the Fund for Doctor Station of theMinistry of Education (20110162130011 and 20100162120019)and theHunanProvince Science andTechnology PlanProject(2012CK4018)

References

[1] K Mitra and R Gopinath ldquoMultiobjective optimization ofan industrial grinding operation using elitist nondominatedsorting genetic algorithmrdquo Chemical Engineering Science vol59 no 2 pp 385ndash396 2004

[2] G Yu T Y Chai and X C Luo ldquoMultiobjective productionplanning optimization using hybrid evolutionary algorithmsfor mineral processingrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 4 pp 487ndash514 2011

[3] T Y Ma W H Gui Y L Wang and C H Yang ldquoDynamicoptimization control for grinding and classification processrdquoControl andDecision vol 27 no 2 pp 286ndash290 2012 (Chinese)

[4] F Y Edgeworth Mathematical Physics An Essay on the Appli-cation of Mathematics to the Moral Sciences C Kegan Paul ampCompany London UK 1881

[5] V Pareto Cours drsquoEconomie Politique F Rouge LausanneSwitzerland 1896

[6] X F Chen W H Gui Y L Wang and L Cen ldquoMulti-stepoptimal control of complex process a genetic programmingstrategy and its applicationrdquo Engineering Applications of Artifi-cial Intelligence vol 17 no 5 pp 491ndash500 2004

[7] C A C Coello ldquoEvolutionary multi-objective optimizationa historical view of the fieldrdquo IEEE Computational IntelligenceMagazine vol 1 no 1 pp 28ndash36 2006

[8] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[9] C M Fonesca and P J Fleming ldquoGenetic algorithms formulti objective optimization formulation discussion and gen-eralizationrdquo in Proceedings of the 5th International Conferenceon Genetic Algorithms pp 415ndash423 Morgan Kaufmann SanMateo Calif USA 1993

[10] S Sengupta S Das M Nasir A V Vasilakos and W PedryczldquoEnergy-efficient differentiated coverage of dynamic objectsusing an improved evolutionary multi-objective optimizationalgorithm with fuzzy-dominancerdquo in Proceedings of IEEECongress on Evolutionary Computation pp 1ndash8 IEEE PressBrisbane Australia 2012

[11] K V Price R M Storn and J A Lampinen DifferentialEvolution a Practical Approach to Global Optimization SpringerBerlin Germany 2005

[12] W Y Gong and Z H Cai ldquoAn improved multiobjective dif-ferential evolution based on Pareto-adaptive 120576-dominance andorthogonal designrdquo European Journal of Operational Researchvol 198 no 2 pp 576ndash601 2009

[13] X H Zeng W K Wong and S Y Leung ldquoAn operatorallocation optimization model for balancing control of thehybrid assembly lines using Pareto utility discrete differentialevolution algorithmrdquo Computers and Operations Research vol39 no 5 pp 1145ndash1159 2012

[14] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[15] A Al-Ani A Alsukker and R N Khushaba ldquoFeature subsetselection using differential evolution and a wheel based searchstrategyrdquo Swarm and Evolutionary Computation vol 9 pp 15ndash26 2013

[16] G Y Li and M G Liu ldquoThe summary of differential evolutionalgorithm and its improvementsrdquo in Proceedings of the 3rdInternational Conference on Advanced Computer Theory andEngineering (ICACTE rsquo10) pp V3153ndashV3156 Chengdu ChinaAugust 2010

[17] H A Abbass ldquoThe self-adaptive Pareto differential evolutionalgorithmrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation pp 831ndash836 IEEE Press Honolulu Hawaii USAMay 2002

[18] N K Madavan ldquoMultiobjective optimization using a Paretodifferential evolution approachrdquo in Proceedings of the IEEECongress on Evolutionary Computation pp 1145ndash1150 IEEEPress Honolulu Hawaii USA May 2002

[19] F Xue A C Sanderson and R J Graves ldquoPareto-based multi-objective differential evolutionrdquo in Proceedings of the IEEECongress on Evolutionary Computation pp 862ndash869 IEEEPress Piscataway NJ USA December 2003

[20] T Robic and B Filipi ldquoDEMO differential evolution formultiobjective optimizationrdquo in Evolutionary Multi-CriterionOptimization pp 520ndash533 Springer Berlin Germany 2005

[21] V L Huang A K Qin P N Suganthan and M F TasgetirenldquoMulti-objective optimization based on self-adaptive differen-tial evolution algorithmrdquo in Proceedings of the IEEE Congresson Evolutionary Computation (CEC rsquo07) pp 3601ndash3608 IEEEPress Singapore September 2007

[22] V L Huang S Z Zhao R Mallipeddi and P N SuganthanldquoMulti-objective optimization using self-adaptive differentialevolution algorithmrdquo in Proceedings of the IEEE Congress onEvolutionary Computation (CEC rsquo09) pp 190ndash194 IEEE PressTrondheim Norway May 2009


[23] J A Adeyemo and F A O Otieno ldquoMulti-objective differentialevolution algorithm for solving engineering problemsrdquo Journalof Applied Sciences vol 9 no 20 pp 3652ndash3661 2009

[24] Y Wang and Z Cai ldquoCombining multiobjective optimizationwith differential evolution to solve constrained optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 16 no 1 pp 117ndash134 2012

[25] A Homaifar C X Qi and S H Lai ldquoConstrained optimizationvia genetic algorithmsrdquo Simulation vol 62 no 4 pp 242ndash2531994

[26] J A Joines and C R Houck ldquoOn the use of non-stationarypenalty functions to solve nonlinear constrained optimizationproblems with GArsquosrdquo in Proceedings of the 1st IEEE Conferenceon Evolutionary Computation pp 579ndash584 Orlando Fla USAJune 1994

[27] F Jimenez and J L Verdegay ldquoEvolutionary techniques forconstrained optimization problemsrdquo in Proceedings of the 7thEuropean Congress on Intelligent Techniques and Soft Computing(EUFIT rsquo99) Aachen Germany 1999

[28] C Sun J Zeng and J Pan ldquoAn improved vector particleswarm optimization for constrained optimization problemsrdquoInformation Sciences vol 181 no 6 pp 1153ndash1163 2011

[29] Z Michalewicz and M Schoenauer ldquoEvolutionary algorithmsfor constrained parameter optimization problemsrdquo Evolution-ary Computation vol 4 no 1 pp 1ndash32 1996

[30] C A C Coello and A D Christiansen ldquoMoses a multiob-jective optimization tool for engineering designrdquo EngineeringOptimization vol 31 no 1ndash3 pp 337ndash368 1999

[31] Y G Woldesenbet G G Yen and B G Tessema ldquoConstrainthandling in multiobjective evolutionary optimizationrdquo IEEETransactions on Evolutionary Computation vol 13 no 3 pp514ndash525 2009

[32] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquoThe Computer Journal vol 7 no 4 pp 308ndash3131965

[33] D A van Veldhuizen and G B Lamont ldquoMultiobjective evolu-tionary algorithms analyzing the state-of-the-artrdquo EvolutionaryComputation vol 8 no 2 pp 125ndash147 2000

[34] R Farmani D A Savic andG AWalters ldquoEvolutionarymulti-objective optimization in water distribution network designrdquoEngineering Optimization vol 37 no 2 pp 167ndash183 2005

[35] K Deb and T Goel Controlled Elitist Non-Dominated SortingGenetic Algorithms for Better Convergence Springer BerlinGermany 2001

[36] Y J Lai T Y Liu and C L Hwang ldquoTopsis for MODMrdquoEuropean Journal of Operational Research vol 76 no 3 pp486ndash500 1994

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Multiobjective optimization problems (MOPs) in thecase of traditional optimization methods are often handledby aggregating multiple objectives into a single scalar objec-tive through weighting factors MOPs have a set of equallygood (nondominating) solutions instead of a single onecalled a Pareto optimum which was introduced by Edge-worth in 1881 [4] and later generalized by Pareto in 1896[5] The practical MOPs are often implicated in series ofequations functions or procedures with complicated con-straints Therefore the evolutionary algorithms are attractiveapproaches for low requirements onmathematical expression[6] Since the mid 1980s there has been a growing interestin solving MOPs using evolutionary approaches [7ndash10]One of the most successful evolutionary algorithms for theoptimization of continuous space functions is the differentialevolution (DE) [11] DE is simple and efficiently converges tothe global optimum in most cases [12 13] Its efficiency hasbeen proven [14] in many application fields such as patternrecognition [15] and mechanical engineering [16]

There have been many improvements for DE to solveMOPs Abbass [17] firstly provided a Pareto DE (PDE) algo-rithm for MOPs in which DE was employed to create newsolutions and only the nondominated solutions were kept asthe basis for the next generation Madavan [18] developed aPareto differential evolution approach (PDEA) in which newsolutions were created by DE and kept in an auxiliary popu-lation Xue et al [19] introduced multiobjective differentialevolution (MODE) and used Pareto-based ranking assign-ment and crowding distancemetric but in a differentmannerfrom PDEA Robic and Filipi [20] also adopting Pareto-based ranking assignment and crowding distance metricdeveloped a DE for multiobjective optimization (DEMO)with a different population update strategy and achievedgood results Huang et al [21] extended the self-adaptive DE(SADE) to solve MOPs by a so called multiobjective self-adaptive DE (MOSADE) They further extended MOSADEby using objectivewise learning strategies [22] Adeyemo andOtieno [23] provided multiobjective differential evolutionalgorithm (MDEA) In MDEA a new solution was generatedby DE variant and compared with target solution If itdominates the target solution then it was added to the newpopulation otherwise a target solution was added

On the other hand single-objective constrained opti-mization problems have been studied intensively in the pastyears [24ndash28] Different constraint handling techniques havebeen proposed to solve constrained optimization problemsMichalewicz and Schoenauer [29] divided constraints han-dling methods used in evolutionary algorithms into fourcategories preserving feasibility of solutions penalty func-tions separating the feasible and infeasible solutions andhybrid methods The differences among these methods arehow to deal with the infeasible individuals throughout thesearch phases Currently the penalty functionmethod ismostwidely used and this algorithm strongly depends on thechoice of the penalty parameter

Although the multiobjective optimization and the con-straint handling problem have received lots of contributionrespectively the CMOPs are still difficult to be solved inpractice Coello and Christiansen [30] proposed a simple

approach to solve CMOPs by ignoring any solution that vio-lates any of the assigned constraints Deb et al [8] proposeda constrained multiobjective algorithm based on the conceptof constrained domination which is also known as superi-ority of the feasible solution Woldesenbet et al [31] intro-duced a constraint handling technique based on adaptivepenalty functions and distance measures by extending thecorresponding version for the single-objective constrainedoptimization

In the MOP of grinding and classification process thedefinitions of Pareto solutions Pareto frontier and Paretodominance are in consistency with the classic definitionsClearly the Pareto frontier is amapping of the Pareto-optimalsolutions to the objective space In the minimization sensegeneral constrained MOPs can be formulated as follows

min 119865 (119883) = min [1198911(119883) 119891

2(119883) 119891

119903(119883)]

st 119892119894(119883) le 0 (119894 = 1 2 119901)

ℎ119895(119883) = 0 (119895 = 119901 + 1 119902)

119909119896isin [119909119896min 119909119896max] (119896 = 1 2 119899)

(1)

where 119865(119883) is the objective vector 119883 = (1199091 119909

119899) isin R119899 is

a parameter vector 119892119894(119883) is the 119894th inequality constraint

and ℎ119895(119883) is the 119895th equality constraint 119909

119896min and 119909119896max

are respectively the lower and upper bounds of the decisionvariable 119909

119896

In this paper based on the specific industrial backgroundof continuous bauxite grinding-classification operation anew hybrid DE algorithm is proposed to solve complex con-strained multiobjective optimization problems Firstly ahybrid DE algorithm for MOPs with simplex method (SM-DEMO) is designed to overcome the problems of globalperformance degradation and being trapped in local opti-mum Then for the MOPs with complicated constraints theproposed algorithm is formed by combining SM-DEMO andfunctional partitioned multi-population In this method theconstruction of penalty functions is not required and themeaningful infeasible solutions are fully utilized

The remainder of the paper is structured as followsSection 2 describes the SM-DEMO algorithm for uncon-strained cases The proposed algorithm of multipopulationfor constrained MOPs is given in Section 3 with verificationof performance by benchmark testing results Section 4describes the model of productsrsquo output and quality in thegrinding-classification process in detail and the applicationof the proposed algorithm in the optimizationmodel Finallythe conclusions based on the present study are drawn inSection 5

2 SM-DEMO Algorithm forUnconstrained MOPs

In order to efficiently solve multiobjective optimizationproblem and find the approximately complete and nearop-timal Pareto frontier we proposed a hybrid DE algorithmfor unconstrained multiobjective optimization with simplexmethod

The differential evolution with initialization crossoverand selection as in usual genetic algorithms uses a pertur-bation of two members as the mutation operator to produce





is defined by

119909119866+1

119894= 119909119866

1199033+ 119865 (119909

119866

1199031minus 119909119866

1199032) (2)








straints 119909max119894119895

and 119909min119894119895

119906119866+1119894119895



119908119866+1

119894119895

=

119909max119894119895

+ rand () lowast (119909max119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895gt 119909

max119894119895

)

119909min119894119895

+ rand () lowast (119909min119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895lt 119909

min119894119895

)

119906119866+1


(3)



0(119861) is the


1205900(119861) =

119899

sum

119895=1

10038161003816100381610038161003816119891119895(119860) minus 119891

119895(119862)

10038161003816100381610038161003816 (4)

where 119891119895(119860) 119891





119895is the 119895th


120590 (119861) = |119860119862| minus |119861119872|

= radic

119899

sum

119895=1

[119891119895(119860) minus 119891

119895(119862)]2

minus radic

119899

sum

119895=1

119891119895(119861) minus

[119891119895(119860) + 119891

119895(119862)]

2

2

(5)





A

B

C

f1

f2

(a)

A

B

C

f1

f2

(b)

A

B

C

f1

f2

(c)

A

B

C

M

f1

f2

(d)






120574 =1

119876(

119876

sum

119894=1


1003817100381710038171003817) (6)



SP = radic 1

(119876 minus 1)

119876

sum

119894=1

(119889 minus 119889119894)2

(7)







1(119883) 119891

2(119883)]

Range of variable

ZDT21198911(119883) = 119909

1 1198912(119883) = 119892(119883)(1 minus (

1198911

119892(119883))

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT31198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892) minus (

1198911

119892) sin (10120587119891

1))

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT41198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892))

119892 (119883) = 1 + 10 (119899 minus 1) +

119899

sum

119894=2

(1199092

119894minus 10 cos (4120587119909

119894))

119899 = 10

0 le 1199091le 1

minus5 le 119909119894le 5 119894 = 2 119899


1) sin6 (6120587119909

1) 1198912(119883) = 119892(1 minus (

1198911

119892)

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

(119899 minus 1)025

119899 = 10

0 le 119909119894le 1
























119894(119894 = 1 2 3 4) is the feasible




0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6








119895(119883)| minus 120575 le 0


119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)



Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













is defined by

119909119866+1

119894= 119909119866

1199033+ 119865 (119909

119866

1199031minus 119909119866

1199032) (2)








straints 119909max119894119895

and 119909min119894119895

119906119866+1119894119895



119908119866+1

119894119895

=

119909max119894119895

+ rand () lowast (119909max119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895gt 119909

max119894119895

)

119909min119894119895

+ rand () lowast (119909min119894119895

minus 119906119866+1

119894119895) if (119906119866+1

119894119895lt 119909

min119894119895

)

119906119866+1


(3)



0(119861) is the


1205900(119861) =

119899

sum

119895=1

10038161003816100381610038161003816119891119895(119860) minus 119891

119895(119862)

10038161003816100381610038161003816 (4)

where 119891119895(119860) 119891





119895is the 119895th


120590 (119861) = |119860119862| minus |119861119872|

= radic

119899

sum

119895=1

[119891119895(119860) minus 119891

119895(119862)]2

minus radic

119899

sum

119895=1

119891119895(119861) minus

[119891119895(119860) + 119891

119895(119862)]

2

2

(5)





A

B

C

f1

f2

(a)

A

B

C

f1

f2

(b)

A

B

C

f1

f2

(c)

A

B

C

M

f1

f2

(d)






120574 =1

119876(

119876

sum

119894=1


1003817100381710038171003817) (6)



SP = radic 1

(119876 minus 1)

119876

sum

119894=1

(119889 minus 119889119894)2

(7)







1(119883) 119891

2(119883)]

Range of variable

ZDT21198911(119883) = 119909

1 1198912(119883) = 119892(119883)(1 minus (

1198911

119892(119883))

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT31198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892) minus (

1198911

119892) sin (10120587119891

1))

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT41198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892))

119892 (119883) = 1 + 10 (119899 minus 1) +

119899

sum

119894=2

(1199092

119894minus 10 cos (4120587119909

119894))

119899 = 10

0 le 1199091le 1

minus5 le 119909119894le 5 119894 = 2 119899


1) sin6 (6120587119909

1) 1198912(119883) = 119892(1 minus (

1198911

119892)

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

(119899 minus 1)025

119899 = 10

0 le 119909119894le 1
























119894(119894 = 1 2 3 4) is the feasible




0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6








119895(119883)| minus 120575 le 0


119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)



Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

B

C

f1

f2

(a)

A

B

C

f1

f2

(b)

A

B

C

f1

f2

(c)

A

B

C

M

f1

f2

(d)






120574 =1

119876(

119876

sum

119894=1


1003817100381710038171003817) (6)



SP = radic 1

(119876 minus 1)

119876

sum

119894=1

(119889 minus 119889119894)2

(7)







1(119883) 119891

2(119883)]

Range of variable

ZDT21198911(119883) = 119909

1 1198912(119883) = 119892(119883)(1 minus (

1198911

119892(119883))

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT31198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892) minus (

1198911

119892) sin (10120587119891

1))

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT41198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892))

119892 (119883) = 1 + 10 (119899 minus 1) +

119899

sum

119894=2

(1199092

119894minus 10 cos (4120587119909

119894))

119899 = 10

0 le 1199091le 1

minus5 le 119909119894le 5 119894 = 2 119899


1) sin6 (6120587119909

1) 1198912(119883) = 119892(1 minus (

1198911

119892)

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

(119899 minus 1)025

119899 = 10

0 le 119909119894le 1
























119894(119894 = 1 2 3 4) is the feasible




0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6








119895(119883)| minus 120575 le 0


119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)



Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1(119883) 119891

2(119883)]

Range of variable

ZDT21198911(119883) = 119909

1 1198912(119883) = 119892(119883)(1 minus (

1198911

119892(119883))

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT31198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892) minus (

1198911

119892) sin (10120587119891

1))

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

119899 minus 1

119899 = 30

0 le 119909119894le 1

ZDT41198911(119883) = 119909

1 1198912(119883) = 119892(1 minus radic(

1198911

119892))

119892 (119883) = 1 + 10 (119899 minus 1) +

119899

sum

119894=2

(1199092

119894minus 10 cos (4120587119909

119894))

119899 = 10

0 le 1199091le 1

minus5 le 119909119894le 5 119894 = 2 119899


1) sin6 (6120587119909

1) 1198912(119883) = 119892(1 minus (

1198911

119892)

2

)

119892 (119883) = 1 + 9

119899

sum

119894=2

119909119894

(119899 minus 1)025

119899 = 10

0 le 119909119894le 1
























119894(119894 = 1 2 3 4) is the feasible




0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6








119895(119883)| minus 120575 le 0


119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)



Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1f2

(a) ZDT2

0

02

04

06

08

1

minus08

minus06

minus04

minus02

f2


0 01 02 03 04 05 06 07 08 09 1f1

(b) ZDT3


0 01 02 03 04 05 06 07 08 09 1f1

0

01

02

03

04

05

06

07

08

09

1

f2

(c) ZDT4

SM-DEMO

0

01

02

03

04

05

06

07

08

09

1

Pareto frontier

02 03 04 05 06 07 08 09 1f1

f2

(d) ZDT6








119895(119883)| minus 120575 le 0


119881119894(119883) =

max 0 119892119894(119883) (119894 = 1 2 119901)

max 0 10038161003816100381610038161003816ℎ119895 (119883)10038161003816100381610038161003816minus 120575 (119895 = 119901 + 1 119902)

(8)



Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Search space S

x2

x1y1

y4

y3

y2 x3

x4

xlowast




Pg


Mg



SgInfeasible

Feasibleindividuals

individuals

Randomlyselected

Evolution

g = g + 1

g = g + 1


feasiblesolution Pf

Optimal infeasible

solution Pc





119909119866+1

119894is defined by

119909119866+1

119894

= 119909119866

119894+ 1198651sdot (119909119866

1198911minus 119909119866

1199031)+ 1198652sdot (119909119866

1198912minus 119909119866

1199032) 119877

119862ge rand ( )

119909119866

119894+ 1198651sdot (119910119866

119894minus 119909119866

1199031)+ 1198652sdot (119909119866

1198911minus 119909119866

1199032) 119877

119862lt rand ( )

(9)




119910119866


119862is


119877119862= 1198771198880sdot120574119866+ const

120574119866minus1 + const (10)




119873119901 (11)



1 and 119873119875


119877(crossover


1 1198652 Randomly



119894(119883) of all of













Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Testfunction


1(119883) 119891

2(119883)]


CTP

1198911(119883) = 119909

1

1198912(119883) = exp(minus

1198911(119883)

119888 (119883))

times41 +

5

sum

119894=2

[119909119894

2minus 10 cos (2120587119909

119894)]

1198921(119883) = cos (120579) (119891

2(119883) minus 119890) minus sin (120579) 119891

1(119883)

1198922(119883) = 119886

1003816100381610038161003816sin 119887120587 [sin (120579) (1198912 (119883) minus 119890)+ cos (120579) 119891

1(119883)]119888

10038161003816100381610038161003816

119889

1198921(119883) ge 119892

2(119883)

0 le 1199091le 1

minus5 le 119909119894le 5

119894 = 2 3 4 5

BNH1198911(119883) = 4119909

1

2+ 41199092

2

1198912(119883) = (119909

1minus 5)2+ (1199092minus 5)2

1198921(119883) = (119909

1minus 5)2+ 1199092

2minus 25

1198922(119883) = minus(119909

1minus 8)2+ (1199092+ 3) + 77

1198921(119883) le 0 119892

2(119883) le 0

0 le 1199091le 5

0 le 1199092le 3

TNK1198911(119883) = 119909

1

1198912(119883) = 119909

2

1198921(119883) = minus119909

1

2minus 1199092

2+ 1

+01 cos(16 arctan(11990911199092

))

1198922(119883) = (119909

1minus 05)

2+ (1199092minus 05)

1198921(119883) le 0 119892

2(119883) minus 05 le 0

0 le 119909119894le 120587

119894 = 1 2

20

15

10

5

00


02 04 06 08 10f1

f2


(a) CTP1

20

15

10

05

0002 04 06 08 10



f1

f2

(b) CTP2




1= 150 size of 119875119888 is

1198731198752= 10 scaling factors 119865

1and 119865







40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40

35

30

25

20

15

10

050


02 0301 04 05 07 0906 08 10

f2

f1



40

35

30

25

20

15

10

05

f2

f1

0 02 0301 04 05 07 0906 08 10


1211100908070605040302

f2


f1

0 02 0301 04 05 07 0906 08 10



1211100908070605040302

f2

f1

0 02 0301 04 05 07 0906 08 10


50

40

30

20

10

00 50 100 150


f2

f1


Pareto frontier

50

40

30

20

10

00 50 100 150

f2

f1

CNSGA-II


Figure 6 Continued


15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










15

10

05

00 05 10 15


f2

f1


Pareto frontier

15

10

05

00 05 10 15

f2

f1

CNSGA-II













Feed water Water


PumpClassifierwater

additionFirst




addition

pump

stageoverflow









119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901119894


119891119895






120572119888119894



119860119888 119861119888



11988950119888



1198641198861015840

min 1198891015840





1205721198881015840

119894


1198641198861015840








119901119894(1 + 119862) =

119889119894119894119891119894+ sum119894minus1

119895=1119894gt1119889119894119895[119864119886119894119901119895(1 + 119862) + 119891

119895]

1 minus 119889119894119894119864119886119894

119889119894119895=

119890119895 119894 = 119895

119894minus1

sum

119896=119895

119888119894119896119888119895119896(119890119896minus 119890119894) 119894 gt 119895

119888119894119895=

minus

119895minus1

sum

119896=119894

119888119894119896119888119895119896 119894 lt 119895

1 119894 = 119895

1

119878119894minus 119878119895

119894minus1

sum

119896=119895

119878119896119887119894119896119888119896119895 119894 gt 119895

119890119895=

1

(1 + 05 sdot 120591119878119895) (1 + 025 sdot 120591119878

119895)2 120591 =

81198751

119872MF (12)



1198751=

119872MF (1 + 119862)

119872MF (1 + 119862) +1198821 + 03 (1198821 +1198822) (13)




119894


119878119894=

(119886(119889119894119889119900)120572

)

(1 + (119889119894120583)Λ

)

(14)




119894= 119894 (mm) and 119886 120572 120583





[119886 120572 120583 Λ]119879

=[[

[

11990911


d

1199091198941

sdot sdot sdot 119909119894119895

]]

]

sdot [1198821 120601119861 119860MF 119861MF 119872MF]119879

(15)






) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












) (16)




120572119888119894=

119901119894times (1 minus 119864119886

119894)

sum119894=1

(119901119894times (1 minus 119864119886

119894))times 100 (17)



119864119886119894= 1 minus exp [minus0693(

119889119894minus 119889min

11988950119888

minus 119889min)

119898

]


)]

119896

(18)







[119864119886min 11988950119888

119898 119896]119879

=[[

[

11991011


d

1199101198941

sdot sdot sdot 119910119894119895

]]

]

sdot [119872MF 1198822119860MP 119861MP]

119879

(19)




follows

1198752=

119872MF(119872MF +1198821 +1198822)

(20)


1205721198881015840

119894=

120572119888119894times (1 minus 119864119886

1015840

119894)

sum119894(120572119888119894times (1 minus 1198641198861015840))

times 100 (21)




1198641198861015840



1198641198861015840


[

minus0693(1198891015840

119894minus 1198891015840

min1198891015840

50119888minus 1198891015840

min)

1198981015840

]

]

+ 1198641198861015840

min sdot [1 minus (1198891015840

119894minus 1198891015840

min1198891015840max

)]

1198961015840

(22)

where 1198641198861015840min 1198891015840

50119888 1198981015840 and 119896




[1198641198861015840

min 1198891015840

5011988811989810158401198961015840]119879

=

[[[

[

1199101015840


1119895

d

1199101015840


1015840

119894119895

]]]

]

sdot [119872MF 119860119888119861119888]119879

(23)












and 1205721198881015840minus0075




2


max 119865 = max [1198911(119883) 119891

2(119883)]

1198911(119883) = 119872MF

1198912(119883) = 120572119888

1015840

minus0075

= 119891 (119872MF11988211198822 120601119861)


ge 120572119888min

1205721198881015840

minus0075ge 1205721198881015840

min

(24)



Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Number 1198911(119883) (th) 119891

2(119883) ()

1 92972 908142 91810 919003 92360 909234 90620 934605 91310 924946 90170 938007 89390 952008 89480 949329 91530 9240010 89298 9576311 89824 9454912 89800 9439513 89710 9461814 91170 9280015 91880 9184016 92190 9128117 89870 9409018 91000 9328519 91960 9152020 91124 93221

85

87

89

91

93

95

97

74000 79000 84000 94000


f1(x)

f2(x)

()

89000






2(119883) = 95763 () The


1= 75127 (119905ℎ) and119882

2= 16296 (119905ℎ)

5 Conclusions







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a hybrid multiobjective differential

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