Research Article Multiobjective Optimization Method Based on …downloads.hindawi.com/journals/jam/2015/165601.pdf · 2019-07-31 · Research Article Multiobjective Optimization Method

Research ArticleMultiobjective Optimization Method Based on AdaptiveParameter Harmony Search Algorithm

P Sabarinath1 M R Thansekhar1 and R Saravanan2

1KLN College of Engineering Pottapalayam 630611 India2Sri Krishna College of Technology Coimbatore 641 042 India

Correspondence should be addressed to P Sabarinath jananisabarigmailcom

Received 17 October 2014 Revised 8 January 2015 Accepted 8 January 2015

Academic Editor Zong Woo Geem

Copyright copy 2015 P Sabarinath 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 present trend in industries is to improve the techniques currently used in design and manufacture of products in orderto meet the challenges of the competitive market The crucial task nowadays is to find the optimal design and machiningparameters so as to minimize the production costs Design optimization involves more numbers of design variables with multipleand conflicting objectives subjected to complex nonlinear constraints The complexity of optimal design of machine elementscreates the requirement for increasingly effective algorithms Solving a nonlinear multiobjective optimization problem requiressignificant computing effort From the literature it is evident that metaheuristic algorithms are performing better in dealing withmultiobjective optimization In this paper we extend the recently developed parameter adaptive harmony search algorithm tosolve multiobjective design optimization problems using the weighted sum approach To determine the best weightage set for thisanalysis a performance index based on least average error is used to determine the index of each weightage set The proposedapproach is applied to solve a biobjective design optimization of disc brake problem and a newly formulated biobjective designoptimization of helical spring problem The results reveal that the proposed approach is performing better than other algorithms

1 Introduction

Engineering design optimization has recently received lot ofattention from designers so as to produce better designs Itsaves time cost and energy involved Engineering designoptimization often deals with many design objectives undernonlinear and complex constraints Also the design problemis subjected to constraints limited by cost weight andmaterial properties like strength design specifications andavailability of resources The design objectives are oftenconflicting in nature and hence finding the true Paretooptimal front is difficult Conventionally numerous math-ematical methods such as linear nonlinear dynamic andgeometric programming have been developed in the past tosolve engineering design optimization problems But thesemethods have many drawbacks and no single method isfound suitable for solving all types of engineering designoptimization problems

Even for a single objective design optimization prob-lem involving large number of nonlinear design variables

arriving at a global best solution is not an easy task Theshortcoming of mathematical methods gives way for meta-heuristics algorithm to solve engineering design optimizationproblems Metaheuristics algorithms have proven to be pow-erful in solving this kind of optimization since they combinerules and randomness to mimic natural phenomena suchas biological systems (Genetic Algorithm) animal behavior(ant algorithm tabu search) and physical annealing process(Simulated Annealing) GA is a popular metaheuristics algo-rithm which has been broadly applied to solve various designoptimization problems and proven to be successful in findingthe global best than traditional methods These include thedesign optimization of machine elements such as gears [1]gearbox [2] journal bearing [3] magnetic thrust bearing [4]rolling element bearing [5] and automotive wheel bearingunit [6] Swarm intelligence is another important conceptin many recent metaheuristics algorithms such as particleswarm optimization firefly algorithm artificial bee colonyalgorithm and cuckoo search

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 165601 12 pageshttpdxdoiorg1011552015165601

2 Journal of Applied Mathematics

Particle swarm optimization mimics the group behaviorof birds and fish schools [7] Sabarinath et al [8] haveapplied PSO algorithm for solving optimal design of beltpulley system Yang [9] developed firefly algorithm thatimitates the flashing behavior of tropic firefly swarms Yangand Deb [10] also proposed cuckoo search algorithm basedon the behavior of cuckoo birds Recently Geem et al[11] proposed a new harmony search (HS) that draws itsmotivation not from a biological or physical process likemostother metaheuristic optimization algorithms but from anarty one that is the improvisation process ofmusicians in thehunt for amagnificent harmonyGeemet al [11] explained thesimilarity between musical performance and optimizationthat explains the strength and language of HSThe attempt tofind the harmony inmusic is similar to finding the optimalityin an optimization process and the musicianrsquos improvisationsare similar to local and global search schemes in optimiza-tion techniques HS algorithm uses random search basedon harmony memory considering rate (HMCR) and thepitch adjusting rate (PAR) instead of gradient search Eventhough the basic algorithm is successful in solving varietyof problems still there is a growing diversity of modified HSalgorithms that try to find improved performance RecentlyKumar et al [12] had proposed a newversion ofHS algorithmnamely parameter adaptive harmony search (PAHS) to solvestandard benchmark functions in single objective optimiza-tion successfully This PAHS algorithm is extended to solvemultiobjective design optimization problems by weightedsum approach In this weighted sum PAHS approach aperformance index based on least average error [13] is usedto evaluate the performance of each weightage set

2 Literature Review

21 PreviousWork onDisc BrakeOptimization Theproposedmultiobjective disc brake optimization problem was intro-duced by Osyczka and Kundu [21] The authors used themodified distance method in genetic algorithm to solve thedisc brake problem and compared their results with that ofa plain stochastic method Ray and Liew [22] used a swarmmetaphor approach in which a new optimization algorithmbased on behavioural concepts similar to real swarm wasproposed to solve the same problem Yıldız et al [23] usedhybrid robust genetic algorithm combining Taguchirsquosmethodand genetic algorithmThe combination of genetic algorithmwith robust parameter design through a smaller populationof individuals resulted in a solution that lead to betterparameter values for design optimization problems L16orthogonal arrays were considered to design the experimentsfor this problemThe optimal levels of the design parameterswere found using ANOVA with respect to the effects ofparameters on the objectives and constraints Yıldız [24] usedhybrid method combining immune algorithm with a hillclimbing local search algorithm for solving complex real-world optimization problems The results of the proposedhybrid approach for multiobjective disc brake problem werecompared with the previous solutions reported in literature

Yang and Deb [25] used multiobjective cuckoo search(MOCS) algorithm for solving this disc brake problem byconsidering two objective functions The authors extendedthe original cuckoo search for single objective optimizationby Yang and Deb for multiobjective optimization by modify-ing the first and third rule of three idealized rules of originalcuckoo search The same author [26] used multiobjectivefirefly algorithm (MOFA) for solving this disc brake problemBy extending the basic ideas of FA Yang et al developedmultiobjective firefly algorithm (MOFA) Yang et al [27]successfully extended a flower algorithm for single objectiveoptimization to solve multiobjective design problems Theauthor solved the biobjective disc brake problem using mul-tiobjective flower pollination algorithm (MOFPA) Reynoso-Meza et al [28] used the evaluation of design concepts and theanalysis of multiple Pareto fronts in multicriteria decision-making using level diagrams They addressed multiobjectivedesign optimization problem of disc brake by considering thefriction surfaces as 4 and 6 to obtain Pareto fronts

22 Previous Work on Helical Compression Spring Optimiza-tion In single objective spring design optimization problemtwo different cases were proposed with objective function ofvolume minimization as case I and weight minimization ascase II This single objective design problem had been solvedusing many optimization algorithms as two different casesIn this section we will look into a few reviews on case Iof a compression spring design problem since the proposedmultiobjective helical compression spring optimization prob-lem is the conversion of well-known standard benchmarkproblem of single objective design optimization that isvolume minimization of helical compression spring (case I)For both the cases the design variables are common Thethree design variables are the wire diameter 119889 = 119909

1 the mean

coil diameter 119863 = 1199092 and the number of active coils 119873 =

1199093 But the data type of design variables objective function

and constraints of these two cases are different Sandgren[15] used integer programming to solve this problem Chenand Tsao [16] used simple genetic algorithm to minimizethe volume of the helical compression spring Wu and Chow[17] used metagenetic parameter in GA to solve the sameproblem Another improved version of GA called geneASwas used by Deb and Goyal [29] to address this problemThe same problem was solved by discrete version of PSOby Kennedy and Eberhart [20] and by using differentialevolution algorithm by Lampinen and Zelinka [19] Guo etal [18] used swarm intelligence to optimize the design ofhelical compression spring Datta and Figueira [30] used real-integer-discrete-coded PSO for solving this case I problemHe et al [31] used an improved version of PSO to solve thiscase I spring problem Deb et al [32] proposed NSGA IIalgorithm for solving some biobjective design optimizationof mechanical components He used case I spring problemand converted it to a biobjective problem by adding onemore objective of minimizing stress induced in the springIn this work we have converted the case I spring probleminto a biobjective problem by adding one more objective ofmaximizing the strain energy stored in the spring In all works

Journal of Applied Mathematics 3

related with case I spring problem the previous researchersused FPS unit system But in this work all data are convertedinto SI units and the results of this proposed multiobjectivespring problem are presented in SI units

The main objectives of this work are (i) multiobjectivedesign optimization of the disc brake and compression springusing the weighted sum approach of parameter adaptiveharmony search algorithm and (ii) to demonstrate the effec-tiveness of this proposed algorithm for multiobjective opti-mization of machine components So far the weighted sumapproach of parameter adaptive harmony search algorithmhas not been tried or experimented for the multiobjectiveoptimization of machine components In this paper abil-ity of the algorithm is demonstrated using a performanceindex based on least average error The optimization resultsobtained by using this proposed algorithm are comparedwiththose obtained by previous researchers using other methods

23 PreviousWorks on HS Algorithm HS algorithm is widelyused in solving various optimization problems such as pipenetwork design [33] design of coffer dam drainage pipes[34] water distribution networks [35] satellite heat pipedesign [36] design of steel frame [37] power economicload dispatch [38] economic power dispatch [39] powerflow problem [40] vehicle routing [41] orienteering [42]robot application [43] and data clustering [44] Mahdavi etal [45] proposed improved HS algorithm by changing theparameters of HS algorithm dynamically HS algorithm hasbeen implemented bymany researchers for the past few yearsto solve optimization problems inmany fields of engineeringscience and technology To increase the performance of HSalgorithm several improvements were done periodically inthe past From the literature it is observed that mainly twoimprovements had been considered First improvement isin terms of tuning the parameters of HS algorithm and theother one is in terms of hybridizing the components of HSalgorithm with other metaheuristic algorithms In this workwe are concentrating only with the first improvement

Mahdavi et al [45] first attempted an improvement inHS algorithm by proposing a new variant of HS known asimproved HS (IHS) algorithm They changed the variantssuch as pitch adjusting rate (PAR) and bandwidth (BW)dynamically with generations Linear increase in PAR andexponential decrease in BW were allowed between prespeci-fied minimum and maximum range

Kong et al [46] proposed an adaptive HS (AHS) algo-rithm to adjust PAR and BW In this approach PAR waschanged dynamically in response to objective function valueswhile BW was tuned for each variable Omran and Mahdavi[47] proposed a global best HS algorithm inspired by PSOalgorithm to improve the performance of HS algorithm Thedifficulty in finding the lower and upper bounds of BW wassolved by using global best particle which is the fittest particlein the swarm in terms of objective function values thanother particles A self-adaptive mechanism for selecting BWwas proposed by Das et al [48] known as explorative HSalgorithm BW was recomputed for each iteration while theother parameters were kept fixedThe other improvements in

Step 1 initialization of parameters decision variables and their possible ranges harmony memory size (HMS) range of harmony

Step 2 perform the random selection of solution vectors so as to initialize a harmony memory (HM)

Step 3 improvise a new harmony vector from the HM based onthe linear form of HMCR and the exponential form of PAR

Step 4 check whether the new harmony vector is better than the

existing harmony vectors in the HM

Update HM

Step 5 stopping criterion satisfied

End

No

No

Yes

Yes

memory considering rate (HMCRmin HMCRmax ) pitch adjustingrate (PARmin PARmax) and range of bandwidth (BWmin BWmax )

Figure 1 Flow chart of the proposed PAHS optimization approach

HS variants were reported in the literature [14 41 44 48ndash55]In all the above cases only PAR and BW values were changedwhereas HMCR was kept fixed This fixed value of HMCR isthe key factor which prevents to get global optimal solutionIn order to overcome this difficulty a parameter adaptiveharmony search (PAHS) algorithm was recently proposedby Kumar et al [12] A detailed description of the proposedparameter adaptive harmony search (PAHS) algorithm canbe seen from [12] However for the sake of easiness a briefintroduction and a step by step computational procedurefor implementing this algorithm are given in the followingsection A flow chart showing the Steps 1 to 5 of the proposedPAHS approach is given in Figure 1

3 Parameter Adaptive Harmony Search(PAHS) Algorithm

Theharmony search algorithm (HS) is one of the most recentmetaheuristic optimization algorithms analogous to music


improvisation process where the musicians improvise thepitches of their instruments to obtain better harmony Mah-davi et al [45] further improved HS in the form of improvedharmony search algorithm (IHS) wherein the parametersare changed dynamically Harmony memory considerationrate (HMCR) and pitch adjustment rate (PAR) are the twoimportant parameters that largely affect the performance ofHS algorithm While searching for global best solution wemake use of HMCR which in turns requires proper tuningfailing which the solution may stuck with local optima PARplays a vital role as far as the local search is concernedDuringimprovisation process the fixed value of these parametersmay lead to a delay in convergence because of the absenceof proper balance of global and local search capabilities Theonly solution for this issue may be obtained by changing bothHMCR and PAR dynamicallyThe above issue is addressed inthe newly proposed algorithm named as parameter adaptiveharmony search algorithm (PAHS) by Kumar et al [12] InPAHS a dynamic change in PAR and HMCR values wereproposed consequently by modifying improvisation step ofIHS The value of HMCR is initially kept small in order toexplore the entire search space to a larger extent and later onthe value of HMCR is increased to confine the search spaceHM (local search) only Similarly the value of PAR is initiallykept high in order tomodify the solutions either stored inHMor from feasible range Finally PAR value is reduced to obtainbetter solutions Based on the modifications stated above (1)in linear form forHMCRand (2) in exponential form for PARhad been proposed as given below

HMCR (gn) = HMCRmin +(HMCRmax minusHMCRmin)

NItimes gn

(1)

PAR (gn)

= PARmax sdot 119890 (ln (PARminPARmax)

NItimes gn)

(2)

where HMCR(gn) is the harmony memory considerationrate for generation gn and HMCRmin and HMCRmax are theminimum and maximum values of harmony memory con-sideration rate respectively Similarly PARmin and PARmaxare the minimum and maximum values of pitch adjustmentrate respectively NI represents the maximum number ofimprovisations and gn represents generation number Thebandwidth is given as

BW (gn) = BWmax sdot 119890 (ln (BWminBWmax)

NItimes gn) (3)

where BW(gn) is the bandwidth for generation gn andBWmin and BWmax are the minimum and maximum valuesof bandwidth respectively

They have considered four different cases wherein besidesthe linear change two parameters have been allowed expo-nential change also PAHSrsquos variants were applied on fifteenstandard benchmark test functions They have compared theresults of PAHS with the other versions of HS as proposedby Geem et al [11] Mahdavi et al [45] and Kong et al

[46] The authors also applied this PAHS to data clusteringproblem From their results it is evident that the PAHS-3 thatis linear change in HMCR and exponential change in PARhas outperformed the results of HS IHS AHS and othervariants of HS PAHS-3 also provided better results in bothdimensional and noisy environments Hence in this workPAHS-3 is used to solve multiobjective design optimizationproblems using weighted sum approach

31 Computational Procedure of PAHS

Step 1 (initialization) In this step the optimization problemand the algorithmic parameters are initialized In this paperthe minimization of objective function is considered for thetwo design optimization scenarios

Theoptimization problem is stated asminimize119891(119909) suchthat 119909 = (119909

1 1199092 119909

119899) is the decision variable set 119899 is

the number of decision variables UB119894and LB

119894are the upper

and lower bounds for decision variable 119909119894 respectively The

parameters of the PAHS algorithm are harmony memorysize (HMS) range for harmony memory consideration rate(HMCRmin HMCRmax) range for pitch adjustment rate(PARmin PARmax) range for distance bandwidth (BWminBWmax) and number of improvisation (NI)

Step 2 (initialize harmony memory (HM)) Solution vectorsare randomly generated for the size of HM that is (HMS) inorder to fill the harmony memory Next sorting of harmonymemory is done based on the values of the objective function119891(119909)

Step 3 (generation of new harmony) Using the proceduregiven below a new harmony vector 1199091 = (119909

1

1 1199092 119909

119899) is

generated

For each 119894 isin [1 119899] doHMCR = HMCR(gn)lowast Proposed equation (1)lowastPAR = PAR(gn)lowast Proposed equation (2)lowastBW = BW(gn)lowast(3)lowastif 119880(0 1) le HMCR thenlowastmemory consideration lowastbegin1199091015840

119894= 119909119894

119897 where 119897 sim 119880(1 HMS)if 119880(0 1) le PAR thenlowast pitch adjustment lowastbegin1199091015840

119894= 1199091015840

119894plusmn BW times Rand Rand sim 119880(0 1)

endifelselowast random selection lowast1199091015840

119894= LB119894+ (UB

119894minus LB119894) times Rand

endifdone

Step 4 (update harmony memory) Evaluate the newly gen-erated harmony vector 1199091015840 = (119909

1015840

1 1199091015840

2 119909

1015840

119873) in terms of

objective function value It is compared with that of the initial


harmonymemoryThe better harmony vector is added in theharmony memory and the worst harmony vector is droppedfrom the harmony memory This step is the deciding stageof the algorithm whether a new harmony vector is to beincluded in the harmony memory or not

Step 5 (check for the termination criterion) If the maximumnumber of improvisation step (termination criterion) isreached computation is terminated and the best harmonyvector is returned by the algorithm Otherwise repeat Steps3 and 4 till the best harmony vector is obtained

For solving the two design optimization problems ofmachine elements (disc brake and compression coil spring)explained in Section 4 using weighted sum approach ofPAHS the harmony memory size (HMS) is set to 20 andthe maximum generations (set as stopping condition) areset to 200 generations The harmony memory considerationrate (HMCR) is varied in the range of 07 to 099 while thepitch adjustment rate (PAR) is varied in the range of 001 to099 The bandwidth (BW) is varied in the range of 0001 to120(UB-LB) Static penalty method is applied for handlingthe constraints A total of 4000 fitness function evaluationswere made with this optimization approach in each runTo study the performance of proposed parameter adaptiveharmony search algorithm constrained optimization of inputparameters to minimize the objective functions was solvedand the best results obtained through the mentioned opti-mization approach in 40 trials were compared with thosereported in the literature

32Weight BasedMultiobjective Optimization Approach Theeasiest way of implementing the multiobjective optimizationis done through weighted sum method In this approach theparameter adaptive harmony search algorithm is modified tofacilitate multiobjective design optimization The implemen-tation of weighted sum approach by assigning suitable weightvalues to represent the preference in choice is explained byMarler and Arora in multiobjective optimization (MOO)[56] The implementation of weighted sum approach to arti-ficial bee colony algorithm is addressed in Hemamalini andSimon [57] where the multiobjective optimization is treatedas composite objective function The composite objectivefunction is expressed as follows

119891 (119909) = 11990811198911(119909) + 119908

21198912(119909) (4)

where 1199081and 119908

2are the assigned weights The relationship

between the variables 1199081and 119908

2is given by 119908

2= 1 minus 119908

1

where 1199081is chosen in the range of [0-1]

Min max normalization is performed for the objectivefunctions so as to assess the fitness of the composite objectivefunction The normalization is done for scaling the objectivefunctions within a specified range The following normaliza-tion formula is used [58]

fit1015840119894=

fit119894minusmin (fitoverall) times 120597

max (fitoverall) minusmin (fitoverall) times 120597 (5)

where fit119894represents the fitness to be normalized and fitoverall

represents the overall fitness 120597 is taken as 0999 in order toavoid zeroes during normalization process

33 Selection of Weightage Value and Performance Index Ananalysis is performed by changing the weight values 119908

1from

1 to 0 with a step size of 01 so as to assign the properweightage values in (4) It is done to see the importance ofeach weightage set (119908set) towards the objective function Aperformance index based on the least average error (LAE)formulated by Naidu et al [13] is used to evaluate theperformance of each weightage set The LAE is evaluated by

LAE = ((1198651119908set 119894

minusmin (1198651overall)

min (1198651overall)

)

+(1198652119908set 119894



)) times (2)minus1

(6)

where 1198651119908set 119894

represents fitness 1 value and 1198652119908set 119894

representsfitness 2 value at weightage set 119868 (where 119868 = 1 2 11)1198651overall and 1198652overall represent the overall fitness 1 and fitness

2 values The performance index based on LAE (120578LAE) isexpressed as

120578LAE =1

LAE (7)

4 Problem Formulation

41 Multiobjective Disc Brake Optimization Problem Themultiobjective disc brake optimization problemwas solved byOsyczka and Kundu [21] using plain stochastic method andgenetic algorithms for optimization of disc brake problemThey have shown that genetic algorithm was giving betterresults compared with that of plain stochastic method Theobjectives of the problem are to minimize the mass of thebrake and to minimize the stopping time The disc brakeoptimization model has four variables (as shown in Figure 2)that are

(1) 119877119894 inner radius of the discs in mm = 119909

1

(2) 119877119900 outer radius of the discs in mm = 119909

2

(3) F engaging force in N = 1199093

(4) n number of the friction surfaces (integer) = 1199094

The objective functions and constraints of the disc brakedesign optimization model provided by Osyczka and Kundu[21] are defined as follows

Objective Functions

Mass of the brake

1198911(119909) = 49 times 10

minus5(1199092

2minus 1199092

1) (1199094minus 1) in kg (8)

Stopping time

1198912(119909) =

982 times 106(1199092

2minus 1199092

1)

11990931199094(11990932minus 11990931)

in s (9)

The constraints are as follows


Ri

Ro

F

A120575

Figure 2 The disc brake design problem

(1) Side constraints 55 le 1199091le 80 75 le 119909

2le 110 1000 le

1199093le 3000 2 le 119909

4le 20

(2) Geometric constraints

(i) Minimum distance between radii

1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)

(ii) Maximum length of the brake

1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)

(3) Behaviour constraints

(i) Pressure constraint

1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)

(ii) Temperature constraint

1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)

(iii) Generated torque constraint

1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)

42 Compression Coil Spring Design Case I This probleminvolves the design of a coil spring which is to support aconstant axial compressive load It is required to minimizethe wire volume of the spring so that it can support a givenload without failure At the same time it is desired to storemaximum strain energy possible So these two objectivesare conflicting in nature As shown in Figure 3 there arethree design variables the real-valued outside diameter of thespring (119863) the integer-valued number of spring coils (119873)and the discrete-valued spring wire diameter (119889)

The objective in this problem is to find the optimalcombination of the above three design parameters thatprovides minimum weight and maximum storage of energysubjected to the constraints

D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t

Figure 3 The compression coil spring design problem

Accordingly the biobjective optimization problem isformulated as below

Determine x = (119863119873 119889) (15)

To minimize the weight of the spring that is

1198651(119909) =

1205872

41198631198892(119873 + 2) (16)

and to maximize the strain energy stored that is

1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)

The supplied numerical data for the problem are given in SIunits in Table 1 along with the allowable discrete values of thespring wire diameter (119889) in Table 2

The present work aims to minimize the volume of thespring and maximize the energy stored Since the objectivesare conflicting in nature modification of second objectivethat is strain energy (SE) is done to get it converted forminimization

The objective functions are given below

Objective 1 = Volume of the helical spring

Objective 2 = 1

SE

(19)

5 Results and Discussion

The multiobjective PAHS algorithm has been applied forconstrained design optimization of input parameters of discbrake and helical compression spring problems The opti-mization process has been implemented usingMatlab 2009 torun on aPC compatiblewith Pentium IV a 32GHzprocessorand 2GB of RAM (Random Access Memory)

51 Disc Brake Problem The weightage set is varied in therange of 0 to 1 with a step size of 01 It is evident from theresults that single objective optimization is achieved whenthe weights are defined at their extreme values The optimaldesign parameters of disc brake at differentweights are shownin Table 3 When 119908

1= 1 and 119908

2= 0 minimizing the mass

of the brake is the effective objective function and it can beobserved that the mass of the brake is minimum but thestopping time value is large In the other extreme when119908

1=

0 and 1199082= 1 minimizing the stopping time of the brake is

the effective objective function in which the stopping time isminimum but the mass of the brake is largeThese results arein accordance with the literature discussed above By varyingthe weightage set with a step size of 01 we can generate 11possible combinations In order to find the best combinationof weightage set a performance index based on least averageerror (LAE) is used as explained in Section 32 The purposeof this performance index is to find the least possible errorfor both objective functions The first part of (5) is to findthe lowest error ratio for the first objective function that is

Table 1 Numerical data for spring problem

Parameter name Symbol Numerical valueMaximum working load 119865max 444822NMaximum allowable shear stress 119878 1303109MPaMaximum free length 119897max 3556mmMinimum wire diameter 119889min 508mmMaximum outside springdiameter 119863max 762mm

Preload compression force 119865119901 133447N

Maximum allowable deflectionunder preload 120575pm 1524mm

Deflection from preload positionto maximum load position 120575

1199083175mm

Shear modulus of the material 119866 7928932MPa

Table 2 Allowable wire diameters (discrete values of 119889)

Allowable wire diameters (mm)02286 02413 026416 029972 032512 033528 035560381 041148 043942 04572 0508 05842 063507112 08128 0889 10414 11938 13716 1600218288 2032 23368 2667 3048 3429 3759241148 44958 48768 52578 5715 61976 6680271882 77978 84074 91948 100076 111125 127

mass of the brake This is done by subtracting the minimumerror value (min119865

1overall) Every objective function value ofmass of the brake in Table 3 will be subtracted from thisparticular value and then the resulting value will be dividedby theminimumerror value (min119865

1overall) to obtain the ratioA negative answer would not appear through this methodsince we are subtracting the least value The same principleis applied to the second objective function that is stoppingtime of the brake The results for both the calculations arethen averaged to obtain a single index value Based on thehighest 120578LAE value observed in Table 3 the weightage set of1199081= 09 and 119908

2= 01 is chosen as the best combination

Figure 4 shows the plot of performance index based on LAE(120578LAE) calculated from the average results of 40 independenttrials each trial having 200 generations for the three differentweight sets It is evident from the plot that theweightage set of1199081= 09 and119908

2= 01 shows better performance with higher

120578LAE values The best combination is marked by bold font inTable 3

The statistical measures like mean best worst andstandard deviation are calculated from 40 independent trialsStatistical performances are given in Table 4

The results of the proposed PAHS approach are comparedwith the previous solutions reported in literature Yıldız [24]used Hybrid Immune-Hill Climbing algorithm for solvingthis problem Osyczka and Kundu [21] used distance methodin GA to solve this problem and compared their results withthat of the plain stochasticmethod Sabarinath et al [59] usedfast and elitist NSGA II algorithm for solving this disc brakeproblem Table 5 compares the results obtained by various


Table 3 Optimal design parameters of disc brake at different weights

1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883

4Mass of the brake (Kg) Stopping time (Sec) Comb fitness 120578LAE

1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732

Table 4 Statistical performance for disc brake design optimization

1198821

1198822

Best Worst Mean Standard deviation1 0 0127401474 0127433238 0127409073 812523119864 minus 06

09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200

Number of generations

Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01

Figure 4 Comparison of results of disc brake at different weights

methods including the proposed approach for solving thebiobjective disc brake problem It is evident from Table 5 thatthe results provided by the proposed PAHS approach are bet-ter than the results of Yıldız [24] andOsyczka andKundu [21]with less function evaluations Yıldız [24] also reported theextreme points (ie minimum value points for each separate

criterion) obtained from three methods in their paper Theextreme points reported for NSGA II algorithm is the sameas that of the proposed PAHS algorithm Comparison of theextreme points obtained by PAHS against other fourmethodsis given in Table 5

52 Compression Coil Spring Problem The weightage set isvaried in the range of 0 to 1 with a step size of 01 It is evidentfrom the results that single objective optimization is achievedwhen the weights are defined at their extreme values Theoptimal design parameters of spring at different weights areshown in Table 6 When 119908

1= 1 and 119908

2= 0 minimizing the

volume of the spring is the effective objective function and itcan be observed that the volume of the spring is minimumIn the other extreme when 119908

1= 0 and 119908

2= 1 maximizing

strain energy stored is the effective objective function inwhich the strain energy stored is maximum Here also theconcept of performance index based on least average error(LAE) as explained in disc brake problem is applied Based onthe highest 120578LAE value observed in Table 6 the weightage setof 1199081= 05 and 119908





Table 5 Comparison of results for disc brake problem

Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879

Plain stochastic method Min 1198911(119909) [626 835 29209 11] [179 277] [089 000 009 076 450]

Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]

Hybrid Immune-HillClimbing algorithm (HIHC)

Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]

Proposed PAHS Min 1198911(119909) [5795 7857 273672 2] [01274 1738] [06142 225 009 077 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]

Table 6 Optimal design parameters of compression coil spring at different weights

1198821

1198822

1198831(mm) 119883

2(mm) 119883

3Volume (mm3) Strain energy stored (J) Comb fitness 120578LAE

1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05

Figure 5 Comparison of results of biobjective helical springproblem



Since we are proposing a new biobjective designoptimization for spring problem by converting the standard

single objective benchmark problem we are unable tocompare the results as in the case of disc brake But when1199081= 1 and 119908

2= 0 minimizing the volume of the spring

is the effective objective function and thus it becomessingle objective benchmark problem The results obtainedfor single objective that is volume minimization for theproposed PAHS approach are compared with that of theresults reported in literature in Table 8 Since the values forcomparison are available in inches the result of the proposedPAHS algorithm is reported in inches as given in Table 8

For validating our results for biobjective spring problemthe same problem is solved usingNSGA II algorithmwith thefollowing values of the parameters of NSGA-II algorithm

Variable type = real variable population size = 50crossover probability = 08 real-parameter mutation prob-ability = 001 real-parameter SBX parameter = 10 real-parameter mutation parameter = 100 and total number ofgenerations = 100 The results are presented in Table 9 Itis evident from Table 9 that the results provided by theproposed PAHS approach are slightly better than the resultsof NSGA II algorithm

6 Conclusion

In this paper a recently proposed parameter adaptive har-mony search algorithm based on the combinations of linearand exponential changes has been applied successfully to


Table 7 Statistical performance for compression coil spring optimization

1198821

1198822

Best Worst Mean Standard deviation1 0 435672821 435683547 435678357 0000654809 01 438983784 438995618 438989832 0000487508 02 408943587 408963241 408955897 0000297107 03 395730857 395749854 395740658 0000218406 04 419651857 419665267 419657364 0000365405 05 404044875 404052148 404049587 000023804 06 370319547 370326874 370320548 000032203 07 290733154 290746257 290739658 000048302 08 282401586 282416328 282408294 000036501 09 203590587 203608724 203600875 00005870 1 34145119864 minus 06 342654119864 minus 06 34196119864 minus 06 5826489119864 minus 06

Table 8 Comparison of optimal solutions for the compression coil spring design problem

Variablesfunctions Sandgren [15] Chen and Tsao[16]

Wu and Chow[17] Guo et al [18] Lampinen and

Zelinka [19]PSO of Kennedyand Eberhart [20]

ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559

Table 9 Optimum results of biobjective spring design optimization problem

Method Solution 1198831(mm) 119883

2(mm) 119883

3Volume (mm3) Strain energy stored (J)

NSGA II Min volume 71882 31122112 9 43645734 101435Max energy 77978 49408799 10 88954819 3256445

Proposed weighted sum PAHS Min volume 71882 310654446 9 43566263 100883Max energy 77978 5391658 8 808921 338523

solve multiobjective design optimization problems usingweighted sum approach In this weighted sum approacha performance index based on least average error (LAE)was used to find the performance of each weightage setBased on the higher LAE value optimumweightage selectionwas determined First a standard benchmark problem ofbiobjective disc brake optimization has been solved and theresults are compared with that of a plain stochastic methodsimple genetic algorithm HIHC and NSGA II Based onthe analysis carried out the weightage set of 119908

1= 09 and

1199082= 01 is determined as the best selection for disc brake

optimization problem It is evident from the results that thisapproach performs better than other algorithms Hence thisapproach is extended to solve a newly formulated biobjectivedesign optimization of spring problem and the results arepresented The optimal weightage set of 119908

1= 05 and 119908

2=

05 is determined as the best selection for spring optimizationproblem The result of spring problem is compared with theresults of NSGA II algorithm and the results are better thanNSGA II This weighted sum approach based on LAE can beextended to any metaheuristic algorithm for implementingmultiobjective optimization Also Pareto optimal solutionscan be found by using adaptive weighted sum method as afuture extension of the present problem

Conflict of Interests

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

References

[1] J L Marcelin ldquoGenetic optimisation of gearsrdquo InternationalJournal of Advanced Manufacturing Technology vol 17 no 12pp 910ndash915 2001

[2] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of MechanicalDesign vol 125 no 3 pp 609ndash619 2003

[3] H Hirani K Athre and S Biswas ldquoComprehensive designmethodology for an engine journal bearingrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 214 no 4 pp 401ndash412 2000

[4] J S Rao and R Tiwari ldquoOptimum design and analysis of thrustmagnetic bearings using multi objective genetic algorithmsrdquoInternational Journal for Computational Methods in EngineeringScience and Mechanics vol 9 no 4 pp 223ndash245 2008

[5] B R Rao and R Tiwari ldquoOptimum design of rolling elementbearings using genetic algorithmsrdquo Mechanism and MachineTheory vol 42 no 2 pp 233ndash250 2007

[6] D-H Choi and K-C Yoon ldquoA designmethod of an automotivewheel-bearing unit with discrete design variables using genetic


algorithmsrdquoTransactions of ASME Journal of Tribology vol 123no 1 pp 181ndash187 2001

[7] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[8] P Sabarinath M R Thansekhar and R Saravanan ldquoPerfor-mance evaluation of particle swarm optimization algorithm foroptimal design of belt pulley systemrdquo in Swarm EvolutionaryandMemetic Computing vol 8297 of Lecture Notes in ComputerScience pp 601ndash616 Springer Cham Switzerland 2013

[9] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[10] X S Yang and S Deb ldquoCuckoo Search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NaBIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009

[11] Z W Geem J-H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[12] V Kumar J K Chhabra and D Kumar ldquoParameter adaptiveharmony search algorithm for unimodal and multimodal opti-mization problemsrdquo Journal of Computational Science vol 5 no2 pp 144ndash155 2014

[13] K Naidu H Mokhlis and A H A Bakar ldquoMultiobjective opti-mization using weighted sum Artificial Bee Colony algorithmfor Load Frequency Controlrdquo International Journal of ElectricalPower amp Energy Systems vol 55 pp 657ndash667 2014

[14] A Kattan R Abdullah and R A Salam ldquoHarmony searchbased supervised training of artificial neural networksrdquo in Pro-ceedings of the International Conference on Intelligent SystemsModelling and Simulation (ISMS rsquo10) pp 105ndash110 LiverpoolUK January 2010

[15] E Sandgren ldquoNonlinear integer and discrete programmingin mechnical design optimizationrdquo ASME Transactions onMechanical Design vol 112 no 2 pp 223ndash229 1990

[16] J-L Chen andY-C Tsao ldquoOptimal design ofmachine elementsusing genetic algorithmsrdquo Journal of the Chinese Society ofMechanical Engineers vol 14 no 2 pp 193ndash199 1993

[17] S-J Wu and P-T Chow ldquoGenetic algorithms for nonlinearmixed discrete-integer optimization problems via meta-geneticparameter optimizationrdquo Engineering Optimization vol 24 no2 pp 137ndash159 1995

[18] C-X Guo J-S Hu B Ye and Y-J Cao ldquoSwarm intelligencefor mixed-variable design optimizationrdquo Journal of ZhejiangUniversity Science vol 5 no 7 pp 851ndash860 2004

[19] J Lampinen and I Zelinka ldquoMixed integer-discrete-continuousoptimization by differential evolutionrdquo in Proceedings of the 5thInternational Conference on Soft Computing pp 71ndash76 BrnoCzech Republic June 1999

[20] J Kennedy and R C Eberhart ldquoA discrete binary versionof the particle swarm algorithmrdquo in Proceedings of the IEEEInternational Conference on Systems Man and Cybernetics pp4104ndash4108 October 1997

[21] A Osyczka and S Kundu ldquoA modified distance method formulticriteria optimization using genetic algorithmsrdquo Comput-ers and Industrial Engineering vol 30 no 4 pp 871ndash882 1996

[22] T Ray and K M Liew ldquoA swarm metaphor for multiobjectivedesign optimizationrdquo Engineering Optimization vol 34 no 2pp 141ndash153 2002

[23] A R Yıldız N Ozturk N Kaya and F Ozturk ldquoHybrid multi-objective shape design optimization using Taguchirsquos methodand genetic algorithmrdquo Structural and Multidisciplinary Opti-mization vol 34 no 4 pp 317ndash332 2007

[24] A R Yıldız ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[25] X-S Yang and S Deb ldquoMulti objective cuckoo search for designoptimizationrdquo Computers and Operations Research vol 40 no6 pp 1616ndash1624 2013

[26] X-S Yang ldquoMultiobjective firefly algorithm for continuousoptimizationrdquo Engineering with Computers vol 29 no 2 pp175ndash184 2013

[27] X-S Yang M Karamanoglu and X Heb ldquoMulti-objectiveflower algorithm for optimizationrdquo Procedia Computer Sciencevol 18 pp 861ndash868 2013

[28] G Reynoso-Meza X Blasco J Sanchis and J M HerreroldquoComparison of design concepts in multi-criteria decision-making using level diagramsrdquo Information Sciences vol 221 pp124ndash141 2013

[29] K Deb and M Goyal ldquoOptimizing engineering designs using acombined genetic searchrdquo in Proceedings of the 7th InternationalConference on Genetic Algorithms I T Back Ed pp 512ndash5281997

[30] D Datta and J R Figueira ldquoA real-integer-discrete-codedparticle swarm optimization for design problemsrdquo Applied SoftComputing Journal vol 11 no 4 pp 3625ndash3633 2011

[31] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[32] K Deb A Pratap and S Moitra ldquoMechanical componentdesign for multiple objectives using elitist non-dominatedsorting GArdquo Technical Report No 200002 Kanpur GeneticAlgorithms Laboratory (KanGAL) Indian Institute of Technol-ogy Kanpur India 2000

[33] ZWGeem J-H Kim andGV Loganathan ldquoHarmony searchoptimization application to pipe network designrdquo InternationalJournal of Modelling and Simulation vol 22 no 2 pp 125ndash1332002

[34] K R Paik J H Jeong and J H Kim ldquoUse of a harmony searchfor optimal design of coffer dam drainage pipesrdquo Journal of theKorean Society of Civil Engineers vol 21 no 2 pp 119ndash128 2001

[35] Z W Geem ldquoOptimal cost design of water distribution net-works using harmony searchrdquo Engineering Optimization vol38 no 3 pp 259ndash277 2006

[36] Z Geem and H Hwangbo ldquoApplication of harmony search tomulti-objective optimization for satellite heat pipe designrdquo inProceedings of the Us-Korea Conference on Science Technologyand Entrepreneurship pp 1ndash3 Citeseer Teaneck NJ USA 2006

[37] S O Degertekin ldquoOptimum design of steel frames usingharmony search algorithmrdquo Structural and MultidisciplinaryOptimization vol 36 no 4 pp 393ndash401 2008

[38] L D S Coelho and V C Mariani ldquoAn improved harmonysearch algorithm for power economic load dispatchrdquo EnergyConversion and Management vol 50 no 10 pp 2522ndash25262009

[39] V R Pandi B K Panigrahi M K Mallick A Abraham and SDas ldquoImproved harmony search for economic power dispatchrdquoin Proceedings of the 9th International Conference on HybridIntelligent Systems (HIS rsquo09) pp 403ndash408 Shenyang ChinaAugust 2009


[40] S Sivasubramani and K S Swarup ldquoMulti-objective harmonysearch algorithm for optimal power flow problemrdquo Interna-tional Journal of Electrical Power and Energy Systems vol 33no 3 pp 745ndash752 2011

[41] Z W Geem K S Lee and Y Park ldquoApplication of harmonysearch to vehicle routingrdquoAmerican Journal of Applied Sciencesvol 2 no 12 pp 1552ndash1557 2005

[42] Z W Geem C Tseng and Y Park ldquoHarmony search forgeneralized orienteering problem best touring in Chinardquo inAdvances in Natural Computation vol 3612 of Lecture Notes inComputer Science pp 741ndash750 Springer Berlin Germany 2005

[43] H Xu X Z Gao T Wang and K Xue ldquoHarmony searchoptimization algorithm application to a reconfigurable mobilerobot prototyperdquo in Recent Advances in Harmony Search Algo-rithm vol 270 of Studies in Computational Intelligence pp 11ndash22 Springer Berlin Germany 2010

[44] B Amiri L Hossain and S E Mosavi ldquoApplications ofharmony search algorithm on clusteringrdquo in Proceedings of theWorld Congress on Engineering and Computer Science pp 460ndash465 2010

[45] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

[46] Z Kong L Gao L Wang Y Ge and S Li ldquoOn an adaptiveharmony search algorithmrdquo International Journal of InnovativeComputing Information andControl vol 5 no 9 pp 2551ndash25602009

[47] M G H Omran and M Mahdavi ldquoGlobal-best harmonysearchrdquo Applied Mathematics and Computation vol 198 no 2pp 643ndash656 2008

[48] S Das A Mukhopadhyay A Roy A Abraham and B K Pan-igrahi ldquoExploratory power of the harmony search algorithmanalysis and improvements for global numerical optimizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 1 pp 89ndash106 2011

[49] OMAlia andRMandava ldquoThevariants of the harmony searchalgorithm an overviewrdquo Artificial Intelligence Review vol 36no 1 pp 49ndash68 2011

[50] M Mahdavi and H Abolhassani ldquoHarmony 119870-means algo-rithm for document clusteringrdquo Data Mining and KnowledgeDiscovery vol 18 no 3 pp 370ndash391 2009

[51] CWorasucheep ldquoA harmony search with adaptive pitch adjust-ment for continuous optimizationrdquo International Journal ofHybrid Information Technology vol 4 no 4 pp 13ndash24 2011

[52] X S Yang ldquoHarmony search as a metaheuristic algorithmrdquo inMusic Inspired Harmony Search Algorithm Theory and Appli-cations Z W Geem Ed pp 1ndash14 Springer Berlin Germany2009

[53] N Taherinejad ldquoHighly reliable harmony search algorithmrdquo inProceedings of the European Conference on Circuit Theory andDesign pp 818ndash822 August 2009

[54] P Chakraborty G G Roy S Das D Jain and A Abraham ldquoAnimproved harmony search algorithm with differential mutationoperatorrdquo Fundamenta Informaticae vol 95 no 4 pp 401ndash4262009

[55] C-M Wang and Y-F Huang ldquoSelf-adaptive harmony searchalgorithm for optimizationrdquo Expert Systems with Applicationsvol 37 no 4 pp 2826ndash2837 2010

[56] R T Marler and J S Arora ldquoThe weighted sum methodfor multi-objective optimization new insightsrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 853ndash862 2010

[57] S Hemamalini and S P Simon ldquoEconomicemission loaddispatch using artificial bee colony algorithmrdquo ACEEE Interna-tional Journal on Electrical and Power Engineering vol 1 no 2pp 27ndash33 2010

[58] Y K Jain and S K Bhandare ldquoMin max normalization baseddata perturbation method for privacy protectionrdquo InternationalJournal of Computer amp Communication Technology vol 2 2011

[59] P Sabarinath R Hariharasudhan M R Thansekhar andR Saravanan ldquoOptimal design of disc brake using NSGAII algorithmrdquo International Journal of Innovative Research inScience Engineering and Technology vol 3 no 3 pp 1400ndash14052014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Particle swarm optimization mimics the group behaviorof birds and fish schools [7] Sabarinath et al [8] haveapplied PSO algorithm for solving optimal design of beltpulley system Yang [9] developed firefly algorithm thatimitates the flashing behavior of tropic firefly swarms Yangand Deb [10] also proposed cuckoo search algorithm basedon the behavior of cuckoo birds Recently Geem et al[11] proposed a new harmony search (HS) that draws itsmotivation not from a biological or physical process likemostother metaheuristic optimization algorithms but from anarty one that is the improvisation process ofmusicians in thehunt for amagnificent harmonyGeemet al [11] explained thesimilarity between musical performance and optimizationthat explains the strength and language of HSThe attempt tofind the harmony inmusic is similar to finding the optimalityin an optimization process and the musicianrsquos improvisationsare similar to local and global search schemes in optimiza-tion techniques HS algorithm uses random search basedon harmony memory considering rate (HMCR) and thepitch adjusting rate (PAR) instead of gradient search Eventhough the basic algorithm is successful in solving varietyof problems still there is a growing diversity of modified HSalgorithms that try to find improved performance RecentlyKumar et al [12] had proposed a newversion ofHS algorithmnamely parameter adaptive harmony search (PAHS) to solvestandard benchmark functions in single objective optimiza-tion successfully This PAHS algorithm is extended to solvemultiobjective design optimization problems by weightedsum approach In this weighted sum PAHS approach aperformance index based on least average error [13] is usedto evaluate the performance of each weightage set

2 Literature Review

21 PreviousWork onDisc BrakeOptimization Theproposedmultiobjective disc brake optimization problem was intro-duced by Osyczka and Kundu [21] The authors used themodified distance method in genetic algorithm to solve thedisc brake problem and compared their results with that ofa plain stochastic method Ray and Liew [22] used a swarmmetaphor approach in which a new optimization algorithmbased on behavioural concepts similar to real swarm wasproposed to solve the same problem Yıldız et al [23] usedhybrid robust genetic algorithm combining Taguchirsquosmethodand genetic algorithmThe combination of genetic algorithmwith robust parameter design through a smaller populationof individuals resulted in a solution that lead to betterparameter values for design optimization problems L16orthogonal arrays were considered to design the experimentsfor this problemThe optimal levels of the design parameterswere found using ANOVA with respect to the effects ofparameters on the objectives and constraints Yıldız [24] usedhybrid method combining immune algorithm with a hillclimbing local search algorithm for solving complex real-world optimization problems The results of the proposedhybrid approach for multiobjective disc brake problem werecompared with the previous solutions reported in literature

Yang and Deb [25] used multiobjective cuckoo search(MOCS) algorithm for solving this disc brake problem byconsidering two objective functions The authors extendedthe original cuckoo search for single objective optimizationby Yang and Deb for multiobjective optimization by modify-ing the first and third rule of three idealized rules of originalcuckoo search The same author [26] used multiobjectivefirefly algorithm (MOFA) for solving this disc brake problemBy extending the basic ideas of FA Yang et al developedmultiobjective firefly algorithm (MOFA) Yang et al [27]successfully extended a flower algorithm for single objectiveoptimization to solve multiobjective design problems Theauthor solved the biobjective disc brake problem using mul-tiobjective flower pollination algorithm (MOFPA) Reynoso-Meza et al [28] used the evaluation of design concepts and theanalysis of multiple Pareto fronts in multicriteria decision-making using level diagrams They addressed multiobjectivedesign optimization problem of disc brake by considering thefriction surfaces as 4 and 6 to obtain Pareto fronts

22 Previous Work on Helical Compression Spring Optimiza-tion In single objective spring design optimization problemtwo different cases were proposed with objective function ofvolume minimization as case I and weight minimization ascase II This single objective design problem had been solvedusing many optimization algorithms as two different casesIn this section we will look into a few reviews on case Iof a compression spring design problem since the proposedmultiobjective helical compression spring optimization prob-lem is the conversion of well-known standard benchmarkproblem of single objective design optimization that isvolume minimization of helical compression spring (case I)For both the cases the design variables are common Thethree design variables are the wire diameter 119889 = 119909

1 the mean

coil diameter 119863 = 1199092 and the number of active coils 119873 =

1199093 But the data type of design variables objective function

and constraints of these two cases are different Sandgren[15] used integer programming to solve this problem Chenand Tsao [16] used simple genetic algorithm to minimizethe volume of the helical compression spring Wu and Chow[17] used metagenetic parameter in GA to solve the sameproblem Another improved version of GA called geneASwas used by Deb and Goyal [29] to address this problemThe same problem was solved by discrete version of PSOby Kennedy and Eberhart [20] and by using differentialevolution algorithm by Lampinen and Zelinka [19] Guo etal [18] used swarm intelligence to optimize the design ofhelical compression spring Datta and Figueira [30] used real-integer-discrete-coded PSO for solving this case I problemHe et al [31] used an improved version of PSO to solve thiscase I spring problem Deb et al [32] proposed NSGA IIalgorithm for solving some biobjective design optimizationof mechanical components He used case I spring problemand converted it to a biobjective problem by adding onemore objective of minimizing stress induced in the springIn this work we have converted the case I spring probleminto a biobjective problem by adding one more objective ofmaximizing the strain energy stored in the spring In all works












Update HM


End

No

No

Yes

Yes









NItimes gn

(1)

PAR (gn)


NItimes gn)

(2)



NItimes gn) (3)







1 1199092 119909



119894are the upper





1

1 1199092 119909

119899) is

generated


119894= 119909119894


119894= 1199091015840



119894= LB119894+ (UB


endifdone


1015840

1 1199091015840

2 119909

1015840

119873) in terms of







119891 (119909) = 11990811198911(119909) + 119908

21198912(119909) (4)

where 1199081and 119908



2is given by 119908

2= 1 minus 119908

1



fit1015840119894=






1from


LAE = ((1198651119908set 119894



)

+(1198652119908set 119894



)) times (2)minus1

(6)

where 1198651119908set 119894




120578LAE =1

LAE (7)




1


2




Objective Functions

Mass of the brake

1198911(119909) = 49 times 10

minus5(1199092

2minus 1199092

1) (1199094minus 1) in kg (8)

Stopping time

1198912(119909) =

982 times 106(1199092

2minus 1199092

1)

11990931199094(11990932minus 11990931)

in s (9)



Ri

Ro

F

A120575



2le 110 1000 le

1199093le 3000 2 le 119909

4le 20



1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)


1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)



1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)


1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)


1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)



D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t



Determine x = (119863119873 119889) (15)


1198651(119909) =

1205872

41198631198892(119873 + 2) (16)


1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Update HM


End

No

No

Yes

Yes









NItimes gn

(1)

PAR (gn)


NItimes gn)

(2)



NItimes gn) (3)







1 1199092 119909



119894are the upper





1

1 1199092 119909

119899) is

generated


119894= 119909119894


119894= 1199091015840



119894= LB119894+ (UB


endifdone


1015840

1 1199091015840

2 119909

1015840

119873) in terms of







119891 (119909) = 11990811198911(119909) + 119908

21198912(119909) (4)

where 1199081and 119908



2is given by 119908

2= 1 minus 119908

1



fit1015840119894=






1from


LAE = ((1198651119908set 119894



)

+(1198652119908set 119894



)) times (2)minus1

(6)

where 1198651119908set 119894




120578LAE =1

LAE (7)




1


2




Objective Functions

Mass of the brake

1198911(119909) = 49 times 10

minus5(1199092

2minus 1199092

1) (1199094minus 1) in kg (8)

Stopping time

1198912(119909) =

982 times 106(1199092

2minus 1199092

1)

11990931199094(11990932minus 11990931)

in s (9)



Ri

Ro

F

A120575



2le 110 1000 le

1199093le 3000 2 le 119909

4le 20



1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)


1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)



1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)


1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)


1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)



D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t



Determine x = (119863119873 119889) (15)


1198651(119909) =

1205872

41198631198892(119873 + 2) (16)


1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












NItimes gn

(1)

PAR (gn)


NItimes gn)

(2)



NItimes gn) (3)







1 1199092 119909



119894are the upper





1

1 1199092 119909

119899) is

generated


119894= 119909119894


119894= 1199091015840



119894= LB119894+ (UB


endifdone


1015840

1 1199091015840

2 119909

1015840

119873) in terms of







119891 (119909) = 11990811198911(119909) + 119908

21198912(119909) (4)

where 1199081and 119908



2is given by 119908

2= 1 minus 119908

1



fit1015840119894=






1from


LAE = ((1198651119908set 119894



)

+(1198652119908set 119894



)) times (2)minus1

(6)

where 1198651119908set 119894




120578LAE =1

LAE (7)




1


2




Objective Functions

Mass of the brake

1198911(119909) = 49 times 10

minus5(1199092

2minus 1199092

1) (1199094minus 1) in kg (8)

Stopping time

1198912(119909) =

982 times 106(1199092

2minus 1199092

1)

11990931199094(11990932minus 11990931)

in s (9)



Ri

Ro

F

A120575



2le 110 1000 le

1199093le 3000 2 le 119909

4le 20



1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)


1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)



1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)


1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)


1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)



D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t



Determine x = (119863119873 119889) (15)


1198651(119909) =

1205872

41198631198892(119873 + 2) (16)


1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119891 (119909) = 11990811198911(119909) + 119908

21198912(119909) (4)

where 1199081and 119908



2is given by 119908

2= 1 minus 119908

1



fit1015840119894=






1from


LAE = ((1198651119908set 119894



)

+(1198652119908set 119894



)) times (2)minus1

(6)

where 1198651119908set 119894




120578LAE =1

LAE (7)




1


2




Objective Functions

Mass of the brake

1198911(119909) = 49 times 10

minus5(1199092

2minus 1199092

1) (1199094minus 1) in kg (8)

Stopping time

1198912(119909) =

982 times 106(1199092

2minus 1199092

1)

11990931199094(11990932minus 11990931)

in s (9)



Ri

Ro

F

A120575



2le 110 1000 le

1199093le 3000 2 le 119909

4le 20



1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)


1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)



1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)


1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)


1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)



D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t



Determine x = (119863119873 119889) (15)


1198651(119909) =

1205872

41198631198892(119873 + 2) (16)


1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ri

Ro

F

A120575



2le 110 1000 le

1199093le 3000 2 le 119909

4le 20



1198921(119909) = (119909

2minus 1199091) minus 20 ge 0 (10)


1198922(119909) = 30 minus 25 (119909

4+ 1) ge 0 (11)



1198923(119909) = 04 minus

1199093

314 (11990922minus 11990921)ge 0 (12)


1198924(119909) = 1 minus

222 times 10minus31199093(1199093

2minus 1199093

1)

(11990922minus 11990921)2

ge 0 (13)


1198925(119909) =

266 times 10minus211990931199094(1199093

2minus 1199093

1)

(11990922minus 11990921)

minus 900 ge 0 (14)



D

d

F

i = N

i = 2

i = 1

Free

leng

th

Disp

lace

men

t



Determine x = (119863119873 119889) (15)


1198651(119909) =

1205872

41198631198892(119873 + 2) (16)


1198652(119909) =

1

2times 119875 times 120597

1198652(119909) =

1

2times 119875 times

81198751198633119873

1198661198894

1198652(119909) =

41198652

max1198633119873

1198661198894

Subjected to 1198921(119909) =

8119862119870119865max1205871198892

minus 119878 le 0

1198922(119909) = 119897 minus 119897max le 0

1198923(119909) = 119889min minus 119889 le 0

1198924(119909) = (119863 + 119889) minus 119863max le 0

1198925(119909) = 30 minus 119862 le 0

1198926(119909) = 120575

119901minus 120575119901119898

le 0

1198927(119909) = 120575

119901+119865max minus 119865119901

119896


+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 105 (119873 + 2) 119889 minus 119897 le 0

1198928(119909) = 120575

119908minus119865max minus 119865119901

119896le 0

(17)

where

119862 =119863

119889 119870 =

4119862 minus 1

4119862 minus 4+0615

119862

119896 =119866119889

81198731198623 120575119901=119865119901

119896

119897 =119865max119896

+ 105 (119873 + 2) 119889

(18)





Objective 2 = 1

SE

(19)




1= 1 and 119908



1=








1199083175mm















1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198821

1198822

1198831(mm) 119883

2(mm) 119883

3(N) 119883


1 0 55 75 26669 2 01274 212723 479119864 minus 05 0215715

09 01 799987 1000002 2997 4 05117 61905 0151448 039955108 02 799841 999841 3000 6 08539 41348 0239908 029855

07 03 799880 999887 3000 8 11792 31422 029341 0227969

06 04 799900 999901 3000 10 15099 25261 0321191 0180646

05 05 799885 999907 3000 11 17631 21963 0310603 0155044

04 06 799960 999964 2997 11 17641 21960 024995 0154951

03 07 799888 999942 3000 11 17631 21968 0189033 0155041

02 08 799934 1000140 2997 11 17686 21949 0128576 0154535

01 09 799970 1090948 2999 11 26382 20885 0095252 0101438

0 1 80 110 3000 11 27890 2071 1 0095732


1198821

1198822


09 01 1079576294 1079820227 1079606351 490341119864 minus 05

08 02 1509991256 1510078349 1510009705 179996119864 minus 05

07 03 1768104744 1768180817 1768123371 182142119864 minus 05

06 04 1916348799 1916466239 1916370027 206882119864 minus 05

05 05 1979631 1980991 1979855 0000272

04 06 2022804 2023961 2023146 0000294

03 07 2065902 2067994 2066419 0000422

02 08 2109074 2110377 2109351 00003

01 09 214321361 2143577898 2143269642 687429119864 minus 05

0 1 2071118 2072861 2071514 0000387

0

00

05

10

15

20

25

50 100 150 200


Perfo

rman

ce in

dex

base

d on

LA

E

W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 09 W2 = 01





1= 1 and 119908

2= 0 minimizing the


1= 0 and 119908

2= 1 maximizing







Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Method Minima 119883 = [119909119897 1199092 1199093 1199094]119879

119891(119909) = [1198911(119909) 119891

2(119909)]119879

119892(119909) = [1198921(119909) 119892

2(119909) 119892

5(119909)]119879


Min 1198912(119909) [704 1066 29484 11] [376 224] [161 000 025 086 2640]

GA method Min 1198911(119909) [658 861 29824 10] [166 287] [065 025 009 075 116]

Min 1198912(119909) [787 1083 29883 11] [325 211] [957 000 022 083 3400]


Min 1198911(119909) NA [0137 2587] NA

Min 1198912(119909) NA [2816 2083] NA

NSGA II Min 1198911(119909) [55 75 273672 2] [01274 1683] [0 225 0034 075 14122]

Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


Min 1198912(119909) [7999 10999 299999 11] [33459 2071] [10 0 023 0832 3612]


1198821

1198822

1198831(mm) 119883

2(mm) 119883


1 0 71882 3107 9 43567 100926 0000272 105056709 01 71882 31915 10 48776 119617 0107336 127323208 02 71882 364877 9 51118 1635858 0128889 207181107 03 77978 419044 7 56533 1392175 0239447 142522206 04 77978 271831 9 69942 1856148 0239112 168851805 05 77978 5392 8 80809 338443 0116524 277716704 06 84074 3796 12 92581 131135 0525667 084755903 07 84074 46352 10 96911 199013 0308506 117881302 08 91948 5646 10 141201 251755 0239791 083119201 09 91948 4884 18 203591 292851 0159556 05444990 1 77978 356234 24 138822 292851 0066173 0914756

0

0

50 100 150 200

10

20

minus10

minus20

minus30

minus40

minus50

minus60

Perfo

rman

ce in

dex

base

d on

LA

E


W1 = 1 W2 = 0

W1 = 0 W2 = 1

W1 = 05 W2 = 05










6 Conclusion




1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198821

1198822






ProposedPAHS

119863 (in) 1180701 12287 1227411 1223 1223042 1223047 122304142119873 10 9 9 9 9 9 9119889 (in) 0283 0283 0283 0283 0283 0283 02831198911(x) (in3) 27995 26709 26681 2659 265856 2658573 2658559



2(mm) 119883





1= 09 and



1= 05 and 119908

2=




References






































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Multiobjective Optimization Method Based on …downloads.hindawi.com/journals/jam/2015/165601.pdf · 2019-07-31 · Research Article Multiobjective Optimization Method