research article enhanced simulated annealing for solving...

Research ArticleEnhanced Simulated Annealing for Solving AggregateProduction Planning

Mohd Rizam Abu Bakar1 Abdul Jabbar Khudhur Bakheet2 Farah Kamil3

Bayda Atiya Kalaf1 Iraq T Abbas1 and Lee Lai Soon1

1Department of Mathematics Faculty of Science Universiti Putra Malaysia Malaysia2Department of Statistics Faculty of Administration and Economics University of Baghdad Baghdad Iraq3Department of Mechanical and Manufacturing Engineering Faculty of Engineering Universiti Putra Malaysia Malaysia

Correspondence should be addressed to Bayda Atiya Kalaf hbama75yahoocom

Received 28 March 2016 Accepted 11 July 2016

Academic Editor Sergii V Kavun

Copyright copy 2016 Mohd Rizam Abu Bakar et alThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Simulated annealing (SA) has been an effective means that can address difficulties related to optimisation problems SA isnow a common discipline for research with several productive applications such as production planning Due to the fact thataggregate production planning (APP) is one of the most considerable problems in production planning in this paper we presentmultiobjective linear programming model for APP and optimised by SA During the course of optimising for the APP problem ituncovered that the capability of SA was inadequate and its performance was substandard particularly for a sizable controlled APPproblem with many decision variables and plenty of constraints Since this algorithm works sequentially then the current state willgenerate only one in next state that will make the search slower and the drawback is that the searchmay fall in local minimumwhichrepresents the best solution in only part of the solution space In order to enhance its performance and alleviate the deficiencies inthe problem solving a modified SA (MSA) is proposed We attempt to augment the search space by starting with 119873 + 1 solutionsinstead of one solution To analyse and investigate the operations of the MSA with the standard SA and harmony search (HS) thereal performance of an industrial company and simulation are made for evaluationThe results show that compared to SA and HSMSA offers better quality solutions with regard to convergence and accuracy

1 Introduction

Aggregate production planning (APP) is considered as animportant technique in operations management Other ele-mental methods that are closely related to this includedcapacity requirements planning (CRP) master productionscheduling (MPS) and material requirements planning(MRP) APP refers to a medium-term resource and capacityplanning that ascertains optimum levels of the inventoryworkforce subcontracting production and backlog to meetthe inconsistent demands and requirements with limitedcapacity and supply over a definite period of time that spansfrom 3 to 18 months Also APP is implemented to resolveproblems that involve collective decisions It finds the totalcapacity level in industries for a certain period while notcalculating the amount of produce from each individualstock-keeping unit The extent of details estimated makes

APP a useful instrument for analysing decisions with a tran-sitional time frame where determining levels of productionby stock-keeping unit becomes too early while arrangingfor additional capacity becomes too late The objective ofAPP is to fulfill the expected changing requirements anddemands in a cost-effective way through a definite periodTheusual expenses include the costs of inventory productionbacklog subcontracting hiring layoff backorder over timeand regular time

The APP problem has been researched widely since itwas first conceptualised in the 1955 study by Holt et al [1]They proposed the HMMS rule Bergstrom and Smith [2]generalized the HMMS approach to a multiproduct formu-lation Bitran and Yanasse [3] investigated the problems ofascertaining production plans under stochastic demands overa set of time periods Nam and Logendran [4] evaluated APPmodels and categorised them into near optimal and optimal

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1679315 9 pageshttpdxdoiorg10115520161679315

2 Mathematical Problems in Engineering

APP along with the other mathematical optimisation modelsindicated that linear programming has the most pervasiveacknowledgment Silva Filho [5] developed a stochasticoptimisationmodel with the limitations of production stockand workforce to explain a multiproduct and multiperiodAPP problem APP was described by Fung et al (2003) [6] asa method to ascertain inventory production and workforcelevels required to meet all market demands MirzapourAl-E-Hashem et al [7] formulated a robust multiproductmultiperiod multiobjective and multisite APP that handlescontradictions in relation to total expenses of supply chaincustomer assistance levels and efficiency of workers duringmedium-term planning in an uncertain environment

A mixed-integer linear programming model was pro-posed by Zhang et al [8] for APP problems related to theexpansion of production capacity Jamalnia and Feili [9]introduced a distinct hybrid event simulation and systemdynamics method to simulate and replicate APP problemsThe prime purpose of their study was to find out theefficiency of APP strategies with regard to the total profitTonelli et al [10] suggested an optimisation method toconfront aggregate planning difficulties of a hybrid modelproduction setting They assessed the planning problem of areal-world assembly manufacturing system and tackled themodel flexibility challenge A mixed-integer multiobjectivenonlinear programming model was proposed by Gholamianet al [11] for scheduling aggregate production in an SC underuncertainty demand The model resolves the problem witha fuzzy multiobjective optimisation approach The suggestedapproach changes the nonlinear model to a linear one andthen alters the fuzzy model to the parametric deterministicmodel

In recent times with the influence of sophisticatedmodeling methods invented and increasing the number ofassumptions the APP problem has turned into a complicatedand large-scale problem In the research community thereis a tendency to resolve large and complex problems by theuse ofmodern heuristic optimisationmethodsThis ismainlydue to the time-consuming and unsuitability of classicaltechniques in many circumstances

A fuzzy multiobjective APP model was introduced byBaykasoglu and Gocken [12] along with a direct solutiontechnique based on ranking orders of fuzzy numbers andtabu search Ramazanian and Modares [13] presented agoal programming multiobjective model for a multiperiodmultistep and multiproduct APP problem in the cementsector The model was modified as a nonlinear programmingmodel with a single objective An extended objective functionmethod and a suggested PSO variant whose inertia weightwas established as a function were used to resolve it Acomparison of the simulation with GA and PSO confirmedthat PSO showsmore acceptable results thanGA To simulatethe process of human decision-making Wang and Fang [14]implemented a genetic algorithm (GA) based technique withfuzzy logic In place of finding the most optimal solutionthis procedure looks for a group of inaccurate solutions thatyields results within satisfactory levels Subsequently a finalsolution was selected by examining a convex combination ofthese solutions

Kumar andHaq [15] resolved anAPPproblemby utilisinggenetic algorithm (GA) ant colony algorithm (AGA) andhybrid genetic-ant colony algorithm (HGA) Based on theresults HGA and GA exhibited relatively good performance

Aungkulanon et al [16] considered various hybridisationtechniques of the harmony search algorithm to explain thefuzzy programming method for several objectives to theAPP problems To resolve the multiobjective APP problemLuangpaiboon and Aungkulanon [17] recommended hybridmetaheuristics of the hunting search (HuSIHSA) and firefly(FAIHSA) procedures based on the enhanced harmonysearch algorithm A mathematical model was suggested byKaveh and Dalfard [18] that took into consideration the timeand expenses for preventative upkeep of APP problems andused a simulated annealing algorithm to solve this In spiteof this they manage single objective only Wang and Yeh [19]introduced the concept of subparticles to the update rules ofPSO to relieve the lack of resources to change the PSO planso that the restrained integer linear programming model canbe implemented to solve real-world APP problems Howeverall these presented approaches are generality concentratedon the solution algorithm but not on a general model Themajority of models in the APP are relevant to single productsingle stage systems and even single objective they are notcompatible with production systems [20]

A simulated annealing algorithm (SAA) has not beenfunctional in solving multiobjective APP problems eventhough most applications made use of metaheuristic algo-rithms on APP In this paper we first suggest a conven-tional multiobjective linear programming model for APPThroughout the implementation of SA to the APP problemsit was observed that SA has an inadequacy with respect tobig APP problems that have plenty of decision variablesTo improve the SA efficiency for the production planningsystem a modified SA is introduced that can resolve APPproblems with multiobjective linear programming modelsince a large number of companies intend to fulfill more thanone objective while creating a response and flexibility Theobjective of themodel is to decrease the aggregate expenses ofproduction and workforce and simulation and real-life datawere used to verify the effectiveness of this method

The organisation of the paper is as follows Section 2gives an overview of harmony search Section 3 includessome essential theories of simulated annealing algorithmSection 4 introduces the scheme of modifying the simulatedannealing algorithm Section 5 explains the multiobjectivelinear programming model of the APP Section 6 discussesthe trials and comparisons for the altered SA with standardHS and SA Section 7 provides a few conclusions and offerssome discussions

2 Multiobjective Linear Programming Model

We proposed mathematical model for APP problem andassumed that an industrial companymanufacturing produces119899 types of products to fulfill market demand over planningtime horizon119879We considered two objective functions in thispaper tominimize production costs andminimizeworkforcecosts We will describe them below

Mathematical Problems in Engineering 3

21 Objective Function

Minimize Production Costs Consider the following

min1198851=

119873

sum

119899=1

119879

sum

119905=1

119888119899119905119875119899119905

+ 119894119899119905119868119899119905 (1)

Minimize Workforce Costs Consider the following

min1198852=

119879

sum

119905=1

119908119905119882119905+ ℎ119905119867119905+ 119891119905119865119905+ 119900119905119874119905 (2)

22 Subject to Constraint Consider the following

119875119899119905

+ 119868119899(119905minus1)

minus 119868119899119905

= 119863119899119905 forall119899 forall119905

119865119905minus 119867119905+ 119882119905minus 119882119905minus1

= 0 forall119905

119874119905minus 119860119874

times 119882119905le 0 forall119905

119873

sum

119899=1

119872119899119875119899119905

minus 119860119877times 119882119905minus 119874119905le 0 forall119905

119867119905le 119867max

119865119905le 119865max

119867119905 119865119905119882119905are integer 119875

119899119905 119868119899119905 119874119905ge 0

(3)

3 Harmony Search Algorithm

Geem et al [21] first introduced the harmony search algo-rithm HSA HSA is believed to be an inspiration algorithmbased socially on local search attributes The concept of HSAhas originated from the natural pattern of musicians conductwhen they play their musical instruments or improvise themusic together

This creates a perfect state of harmony or a pleasurableharmony as governed by an aesthetic quality with the pitch ofevery musical device In a similar manner the optimisationtechnique searches for a globule solution as ascertained byan objective function through a series of values designatedto every decision variable In a musical orchestration the setof pitches from every musical instrument accomplishes theaesthetic evaluation The quality of harmony gets enrichedwith every practice Each style ofmusic is composed bymusi-cians out of specific instruments If all musical pitches createa perfect harmony then that musical experience remainsin every instrument players memory and the likelihood tocreate a good harmony the next time increases manyfold Ifevery player plays together with dissimilar notes then a newmusical harmony is composedThere are three rules of musi-cal improvisation playing a completely random pitch fromthe workable sound range playing a pitch from memory orplaying an adjoining pitch of a pitch from memory Thesetechniques rules are implemented in HSA and the same wasexplained to describe the process of HSA

The HSA process can be summed up as follows

Step 1 Initialize the harmony memory (HM) which containsHMS vectors generated randomly where 119909

119894= 119909119894119895 119894 =

1 2 HMS and 119895 = 1 2 119899 where n is the size of theproblem and the harmony memory (HM) matrix is filledwith HMS and 119891 represented the fitness function as follows

HM

=

[[[[[[[

[

11990911

11990912

11990913

sdot sdot sdot 1199091119899

119891 (1199091)

11990921

11990922

11990923


119891 (1199092)

119909HMS1

119909HMS2

119909HMS3

sdot sdot sdot 119909HMS119899

119891 (119909HMS119899

)

]]]]]]]

]

(4)

Step 2 Improvise a new harmony There are three rules forthis

(1) Harmony memory considering (HMC) rule for thisrule a new randomnumber 119903

1is generated within the

range [0 1]If 1199031lt HMCR then the first decision variable in the

new vector 1199091015840119894119895is chosen randomly from the values in

the current HM as follows

119909new119894119895

= 119909119894119895 119909119894119895

isin 1199091119895 1199092119895 119909HMS

119895

(5)

(2) Pitch adjusting rate (PAR) a new random number 1199032

is generated within the range [0 1]If 1199032

lt PAR then the pitch adjustment decisionvariable is calculated as follows

119909new119894119895

= 119909119894119895plusmn 1199033sdot BW (6)

where BW is a bandwidth factor which is used tocontrol the local search around the selected decisionvariable in the new vector

(3) Random initialization rule is as followsIf the condition 119903

1lt HMCR fails the new first

decision variable in the new vector 119909new119894119895

is generatedrandomly as follows

119909new119894119895

= 119897119894119895+ (119906119894119895minus 119897119894119895) 1199034 (7)

where 119897 119906 are the lower and upper bound for the givenproblem and 119903

3 1199034isin (0 1)

Step 3 Update harmony memory If improvised harmonyvector is better than the worst harmony replace the worstharmony in the HM form

if 119891 (1199091015840new

) lt 119891 (119909worst

) then 119909worst

= 119909new (8)

Step 4 Check the stopping criterion If the stopping criterionis satisfied the computation is terminated Otherwise repeatSteps 3 and 4

The steps of HSA are elucidated in Figure 1


Start

Set and initialize the parameters HMSHMCR PAR max number of iterations K

Generated HM vectors xijinitialize r1 r2 r3 r4 isin (0 1)

r1 lt HMCR

Yes

xnewij = xij

No

No

r2 lt PAR

Yes

xnewij = xnew

ij plusmn r3 BW

No

Yes

Update HM xworst = xnew

No Stopping conditioniteration = k

Yes

End

x998400ij = lij + (uij minus lij)r4

f(xnew) lt f(xworst)

Figure 1 Flowchart for HS procedure

4 Simulated Annealing Algorithm

For complicated optimisation problems simulated annealing(SA) has been regarded as an effective measure Kirkpatricket al [22] introduced SA for the first time SA refers to a ran-dom search technique that makes use of a similarity betweenthe method through which a metal cools down and freezes toa crystalline structure (the annealing process) withminimumenergy and the search process for a minimum in a moreundefined systemThis is used as the base of an optimisationmethod that can be used for combinatorial problems Themethod has been extensively used to solve various problemsThe foremost benefits and strengths of SA over other searchtechniques are its flexibility and capability to achieve overalloptimality Conversely the main disadvantage of the methodis its extremely slow convergence in big problems In SA

first a primary solution is randomly created and a neighbouris found and this is accepted with a probability of min(1 119890(minus120575119891119879)

) where 119879 is the control parameter correspondingto the temperature of the physical analogy and is calledtemperature and 120575119891 is the cost difference [23] On gradualdecrease of temperature the algorithm congregates to theglobal minimum The operations of the SA are depicted inFigure 2

5 Modified Simulated Annealing Algorithm

Practitioners make extensive use of SA to solve NP-hardproblems But SA can get stuck at a local minimum andnot attain the global optimum because of its inadequatecapability and substandard performance specifically with alarge number of decision variables in APP problems It is also


p = e(minus(|f(x998400 )minusf(x)|)T)

Generation random number Z

NoP lt Z

YesYes

k = k + 1

Generate a new solution x998400 for x

No

Yes

x = x998400

Nok = m

Reduce the parameter T

NoStopping criteria

YesEnd

Start

Initialize solution x k = 0

set m and the temperature T

f(x998400) le f(x)

f(x) = f(x998400)

Tk+1 = 120572T 120572 = 095

Figure 2 Operations for SA

programmed to function sequentially and hence the currentstate will generate only one subsequent state This will slowdown the search In order to improve the problem solvingby the aforementioned SA a modified SA (MSA) is proposedto augment its performance and address the problem-solvinginsufficiencies We attempt to augment the search space bystarting with 119873 + 1 solutions instead of one solution Thesesolutions work in a parallel way for utilising the overall searcharea in short time The details are described in the followingsteps and the operations of the MSA are also illustrated inFigure 3

Step 1 Generate 119899 + 1 initial solutions namely 1199091 1199092

119909119899 119909119899+1

and set initial high temperature 119879

Step 2 Find the objective function for each 119909119894 119894 = 1 2

119899 119899 + 1

Step 3 Sort the solution such as 119891(1199091) le 119891(119909

2) le sdot sdot sdot le

119891(119909119899) le 119891(119909

119899+1) where 119891(119909

1) is the best solution (current

solution)

Step 4 Repeat this step 119898 times

(i) Generate a new solution 1199091015840

119894for each 119909

119894

(ii) If 119891(1199091015840

119894) le 119891(119909

119894) then 119909

119894= 1199091015840

119894and 119891(119909

119894) = 119891(119909

1015840

119894)

(iii) Else if 119901 = 119890(minus(|119891(119909

1015840

119894)minus119891(119909

119894)|)119879) and 119875 lt 119885 then 119909

119894= 1199091015840

119894

119891(119909) = 119891(1199091015840

119894) where 119885 is a random number isin (0 to 1)

and 119901 is Boltzmann probability factor

Step 5 Reduce the parameter 119879 according to 119879119896+1

= 120572119879 120572 =

095 Convergence criteria are satisfied and if not then go toStep 3 Concerning the stopping condition theory suggests afinal temperature equal to 0

6 Computational Study and Results

To assess the functioning of the MSA algorithm for the APPproblem two criteria are used APP problem from real-world(case study) and simulation Standard SA MSA and HS arecoded by MATLAB 1198772015119886 the prevailing algorithms (SAHS) are compared with the proposed algorithm


p = e(minus(|f(x998400119894 )minusf(x119894)|)T)


NoP lt Z

Yes

k = k + 1

Generate a new solution x998400i for each x

No

Yes

xi = x0i

Nok = m


NoStopping criteriaYes

Yes

End

Start



Find f(xi) forallif(x1) le f(x2) le middot middot middot le f(xn+1)

f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095

Figure 3 Operation for modified SA

61 Case Study In this section the General Company forVegetable Oils is used as a case study to demonstrate the pro-posed model This company produces ten types of productsEach product is represented by a letter (119860 119861 119862 119863 119864 119865 119866119867 119869 and119870)The time horizon of APP decision is sixmonthsTables 1ndash3 represent costs of production and inventory hoursrequired to produce one ton for each product and forecastdemand for each product respectively The initial inventoryfor product 119860 is 105 tons 119864 is 333 tons 119867 is 025 ton and 119869

is 18 tonThe initial worker level is 3313workersThe cost ofregular worker per month is 500 $man the working hoursin onemonth are 140 hours 5357 $ (dollar) is overtime costsper worker per hour The costs associated with hiring andfiring are 774910 $ and 581182 $ per worker respectivelyHours of overtime allowed during the period are 60 hoursper period 119905

The authors select HS because it provides better resultsthan standard SA in terms of costs especially in the secondobjective 119885

2 while HS gives slower runtime than standard

SA

Table 1 Production and inventory costs in dollars

119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37

Table 2 Hours required to produce one ton of product

Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692

After we used the MSA to solve the case study it is clearfrom the results that the proposed MSA was effective andfaster than the standard SA and HS as shown in Table 4

Due to the fact that MSA provided better results than theSA and HS in terms of cost and time consequently MSAwas used to solve the model in the company as it is shownin Tables 5ndash7

Tables 5 and 6 contain the amount of the productionyield and inventory levels respectively to fulfill the forecast


Table 3 Forecast demand for all products

Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071

Table 4 Results for each algorithm

Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s

Table 5 Production yield

Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068

Table 6 Inventory levels

Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0

Table 7 The rate of workforce level

Workforce level P1 P2 P3 P4 P5 P6119882119905

1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0

demand while Table 7 shows the results in the first columnwhich represent the first period Number 1915 explains the

HSSAMSA

Z1 Z2 Z1 Z2 Z1 Z2 Z1 Z2 Z1 Z2 Z1 Z2 Z1 Z2 Z1 Z1 Z1 Z2

P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09

Figure 4 Optimal cost comparison MSA SA and HS

number of regular workers that should be hired during thatperiod Also number 0 refers to the fact that the companydoes not need to hire any worker in that period In additionthe result (119865

119905= 1400) represents the number of workers that

should be fired from the factory while number 31079 refers tothe hours of overtime that should be adopted in that period

62 Simulation Simulation is another criterion for analysingthe effectiveness of the proposed modified SA During sim-ulation based on the same preliminary data set for the casestudy the algorithm was used to resolve various problemsTo compare the performances of these algorithms SA MSAand HS 9 cases of problems have been used dependingon regular working hours (119860

119877) overtime hours (119860

119874) and

different number of products and periods The algorithmshave been implemented 1000 times for every problem Theaverage time and the total costs for each objective of MSASA and HS are depicted in Table 8 According to this tablethe MSA provided better results by minimizing the objectivefunction through utilising a different number of products andperiod Besides that we changed the regular 119860

119877and 119860

119874to

enlarge and change our problem where 119873 and 119875 representthe product number and period number respectively whileminimizing production costs and minimizing the workforcecosts are represented by 119885

1and 119885

2 respectively

From Figure 4 we can see that it is evident that themodified algorithm is superior even for complex and largeproblems in minimizing the total cost of production andworkforce levels As illustrated in Figure 5 the quality ofsolutions obtained by MSA is better in runtime than SA andHS

7 Conclusion

Simulated annealing provides a mechanism to escape localoptima by allowing hill-climbing moves in hopes of finding


Table 8 Optimal costs and time for each problem when 119860119877= 160 119860

119874= 100 hours respectively

SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180

Figure 5 Comparison time running for each algorithm

a global optimum However it has a disadvantage whichis the imperfections ability and unacceptable performanceespecially a large constrained APP issue Therefore this

algorithm works sequentially that the current solution willgenerate only one solution To improve its performance andlessen its insufficiencies to problem-solving a modified SA(MSA) is introduced that will expand the search space bygenerating 119873 + 1 solutions to create as many numbers aspossible of the neighbour statesThe study effectively explainsthe applicability of the MSA algorithm in solving the APPproblem in industries The results indicated that the use ofthe modified SA resolved numerous problems including thecomplex and large ones

Definition of Notations

119899 Number of products 119899 = 1 2 119873

119905 Number of periods in the planning horizon119905 = 1 2 119879

119888119899119905 Production cost per ton of product 119899 per period 119905

($ton)119894119899119905 Inventory carrying cost per ton of product 119899 perperiod 119905 ($ton)

ℎ119905 Hiring cost per worker in period 119905 ($ worker)

119891119905 Firing cost per worker in period 119905 ($ worker)

119900119905 Cost per man-hour of overtime labor per period 119905

($ worker)


119908119905 Cost of regular labor per period 119905 ($ worker)

119863119899119905 Forecasted demand for product 119899 per period

119905 (tons)119875119899119905 Production of product 119899 per period 119905 (tons)

119868119899119905 Inventory level of product 119899 per period 119905

(tons)119874119905 Man-hours of overtime labor per period 119905

119882119905 Workforce level per period 119905 (workers)

119867119905 Hired workers per period 119905 (workers)

119865119905 Fired workers per period 119905 (workers)

119867max Maximum hiring in each period119865max Maximum firing in each period119872119899 Hours required to produce one ton of

product 119899119860119877 Working regular hours per period 119905

119860119874 Working overtime hours which are allowed

during per period 119905

119870119899 Hours required to produce one ton for

product 119899 per worker

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] C C Holt F Modigliani and H A Simon ldquoA linear decisionrule for production and employment schedulingrdquoManagementScience vol 2 no 1 pp 1ndash30 1955

[2] G L Bergstrom andB E Smith ldquoMulti-itemproduction plann-ing-an extension of the hmms rulesrdquoManagement Science vol16 no 10 pp 614ndash629 1970

[3] G R Bitran and H H Yanasse ldquoDeterministic approximationsto stochastic production problemsrdquo Operations Research vol32 no 5 pp 999ndash1018 1984

[4] S-J Nam and R Logendran ldquoAggregate production planningmdasha survey of models and methodologiesrdquo European Journal ofOperational Research vol 61 no 3 pp 255ndash272 1992

[5] O S Silva Filho ldquoAn aggregate production planningmodel withdemandunder uncertaintyrdquoProduction PlanningampControl vol10 no 8 pp 745ndash756 1999

[6] R Y Fung J Tang and D Wang ldquoMultiproduct aggregateproduction planning with fuzzy demands and fuzzy capacitiesrdquoIEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 33 no 3 pp 302ndash313 2003

[7] S M J Mirzapour Al-E-Hashem HMalekly andM B Aryan-ezhad ldquoA multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supplychain under uncertaintyrdquo International Journal of ProductionEconomics vol 134 no 1 pp 28ndash42 2011

[8] R Zhang L Zhang Y Xiao and I Kaku ldquoThe activity-basedaggregate production planning with capacity expansion inmanufacturing systemsrdquo Computers and Industrial Engineeringvol 62 no 2 pp 491ndash503 2012

[9] A Jamalnia and A Feili ldquoA simulation testing and analysis ofaggregate production planning strategiesrdquo Production Planningand Control vol 24 no 6 pp 423ndash448 2013

[10] F Tonelli M Paolucci D Anghinolfi and P Taticchi ldquoProduc-tion planning of mixed-model assembly lines a heuristic mixed

integer programming based approachrdquo Production Planningand Control vol 24 no 1 pp 110ndash127 2013

[11] N Gholamian I Mahdavi and R Tavakkoli-MoghaddamldquoMulti-objectivemulti-productmulti-site aggregate productionplanning in a supply chain under uncertainty fuzzy multi-objective optimisationrdquo International Journal of Computer Inte-grated Manufacturing vol 29 no 2 pp 149ndash165 2016

[12] A Baykasoglu and T Gocken ldquoMulti-objective aggregate pro-duction planning with fuzzy parametersrdquoAdvances in Engineer-ing Software vol 41 no 9 pp 1124ndash1131 2010

[13] M Ramazanian andAModares ldquoApplication of particle swarmoptimization algorithm to aggregate production planningrdquoAsian Journal of Business Management Studies vol 2 no 2 pp44ndash54 2011

[14] D Wang and S-C Fang ldquoA genetics-based approach for aggre-gated production planning in a fuzzy environmentrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 27 no 5 pp 636ndash645 1997

[15] G M Kumar and A N Haq ldquoHybrid geneticmdashant colonyalgorithms for solving aggregate production planrdquo Journal ofAdvancedManufacturing Systems vol 4 no 1 pp 103ndash111 2005

[16] P Aungkulanon B Phruksaphanrat and P LuangpaiboonldquoHarmony search algorithmwith various evolutionary elementsfor fuzzy aggregate production planningrdquo in Intelligent Controland Innovative Computing S I Ao O Castillo and X HuangEds vol 110 of Lecture Notes in Electrical Engineering pp 189ndash201 Springer Berlin Germany 2012

[17] P Luangpaiboon and P Aungkulanon ldquoIntegrated approachesto enhance aggregate production planning with inventoryuncertainty based on improved harmony search algorithmrdquoProceedings of World Academy of Science Engineering andTechnology no 73 p 243 2013

[18] M Kaveh and V M Dalfard ldquoA simulated annealing algorithmfor aggregate production planning with considering of ancillarycostsrdquo International Journal of Mathematics in OperationalResearch vol 6 no 4 pp 474ndash490 2014

[19] S-C Wang and M-F Yeh ldquoA modified particle swarm opti-mization for aggregate production planningrdquo Expert Systemswith Applications vol 41 no 6 pp 3069ndash3077 2014

[20] R Ramezanian D Rahmani and F Barzinpour ldquoAn aggregateproduction planning model for two phase production systemssolving with genetic algorithm and tabu searchrdquo Expert Systemswith Applications vol 39 no 1 pp 1256ndash1263 2012

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

[22] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[23] V M Dalfard and G Mohammadi ldquoTwo meta-heuristic algo-rithms for solving multi-objective flexible job-shop schedulingwith parallel machine andmaintenance constraintsrdquoComputersand Mathematics with Applications vol 64 no 6 pp 2111ndash21172012

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


APP along with the other mathematical optimisation modelsindicated that linear programming has the most pervasiveacknowledgment Silva Filho [5] developed a stochasticoptimisationmodel with the limitations of production stockand workforce to explain a multiproduct and multiperiodAPP problem APP was described by Fung et al (2003) [6] asa method to ascertain inventory production and workforcelevels required to meet all market demands MirzapourAl-E-Hashem et al [7] formulated a robust multiproductmultiperiod multiobjective and multisite APP that handlescontradictions in relation to total expenses of supply chaincustomer assistance levels and efficiency of workers duringmedium-term planning in an uncertain environment

A mixed-integer linear programming model was pro-posed by Zhang et al [8] for APP problems related to theexpansion of production capacity Jamalnia and Feili [9]introduced a distinct hybrid event simulation and systemdynamics method to simulate and replicate APP problemsThe prime purpose of their study was to find out theefficiency of APP strategies with regard to the total profitTonelli et al [10] suggested an optimisation method toconfront aggregate planning difficulties of a hybrid modelproduction setting They assessed the planning problem of areal-world assembly manufacturing system and tackled themodel flexibility challenge A mixed-integer multiobjectivenonlinear programming model was proposed by Gholamianet al [11] for scheduling aggregate production in an SC underuncertainty demand The model resolves the problem witha fuzzy multiobjective optimisation approach The suggestedapproach changes the nonlinear model to a linear one andthen alters the fuzzy model to the parametric deterministicmodel

In recent times with the influence of sophisticatedmodeling methods invented and increasing the number ofassumptions the APP problem has turned into a complicatedand large-scale problem In the research community thereis a tendency to resolve large and complex problems by theuse ofmodern heuristic optimisationmethodsThis ismainlydue to the time-consuming and unsuitability of classicaltechniques in many circumstances

A fuzzy multiobjective APP model was introduced byBaykasoglu and Gocken [12] along with a direct solutiontechnique based on ranking orders of fuzzy numbers andtabu search Ramazanian and Modares [13] presented agoal programming multiobjective model for a multiperiodmultistep and multiproduct APP problem in the cementsector The model was modified as a nonlinear programmingmodel with a single objective An extended objective functionmethod and a suggested PSO variant whose inertia weightwas established as a function were used to resolve it Acomparison of the simulation with GA and PSO confirmedthat PSO showsmore acceptable results thanGA To simulatethe process of human decision-making Wang and Fang [14]implemented a genetic algorithm (GA) based technique withfuzzy logic In place of finding the most optimal solutionthis procedure looks for a group of inaccurate solutions thatyields results within satisfactory levels Subsequently a finalsolution was selected by examining a convex combination ofthese solutions

Kumar andHaq [15] resolved anAPPproblemby utilisinggenetic algorithm (GA) ant colony algorithm (AGA) andhybrid genetic-ant colony algorithm (HGA) Based on theresults HGA and GA exhibited relatively good performance

Aungkulanon et al [16] considered various hybridisationtechniques of the harmony search algorithm to explain thefuzzy programming method for several objectives to theAPP problems To resolve the multiobjective APP problemLuangpaiboon and Aungkulanon [17] recommended hybridmetaheuristics of the hunting search (HuSIHSA) and firefly(FAIHSA) procedures based on the enhanced harmonysearch algorithm A mathematical model was suggested byKaveh and Dalfard [18] that took into consideration the timeand expenses for preventative upkeep of APP problems andused a simulated annealing algorithm to solve this In spiteof this they manage single objective only Wang and Yeh [19]introduced the concept of subparticles to the update rules ofPSO to relieve the lack of resources to change the PSO planso that the restrained integer linear programming model canbe implemented to solve real-world APP problems Howeverall these presented approaches are generality concentratedon the solution algorithm but not on a general model Themajority of models in the APP are relevant to single productsingle stage systems and even single objective they are notcompatible with production systems [20]

A simulated annealing algorithm (SAA) has not beenfunctional in solving multiobjective APP problems eventhough most applications made use of metaheuristic algo-rithms on APP In this paper we first suggest a conven-tional multiobjective linear programming model for APPThroughout the implementation of SA to the APP problemsit was observed that SA has an inadequacy with respect tobig APP problems that have plenty of decision variablesTo improve the SA efficiency for the production planningsystem a modified SA is introduced that can resolve APPproblems with multiobjective linear programming modelsince a large number of companies intend to fulfill more thanone objective while creating a response and flexibility Theobjective of themodel is to decrease the aggregate expenses ofproduction and workforce and simulation and real-life datawere used to verify the effectiveness of this method

The organisation of the paper is as follows Section 2gives an overview of harmony search Section 3 includessome essential theories of simulated annealing algorithmSection 4 introduces the scheme of modifying the simulatedannealing algorithm Section 5 explains the multiobjectivelinear programming model of the APP Section 6 discussesthe trials and comparisons for the altered SA with standardHS and SA Section 7 provides a few conclusions and offerssome discussions

2 Multiobjective Linear Programming Model

We proposed mathematical model for APP problem andassumed that an industrial companymanufacturing produces119899 types of products to fulfill market demand over planningtime horizon119879We considered two objective functions in thispaper tominimize production costs andminimizeworkforcecosts We will describe them below




min1198851=

119873

sum

119899=1

119879

sum

119905=1

119888119899119905119875119899119905

+ 119894119899119905119868119899119905 (1)


min1198852=

119879

sum

119905=1

119908119905119882119905+ ℎ119905119867119905+ 119891119905119865119905+ 119900119905119874119905 (2)


119875119899119905

+ 119868119899(119905minus1)

minus 119868119899119905

= 119863119899119905 forall119899 forall119905


= 0 forall119905

119874119905minus 119860119874

times 119882119905le 0 forall119905

119873

sum

119899=1

119872119899119875119899119905


119867119905le 119867max

119865119905le 119865max

119867119905 119865119905119882119905are integer 119875

119899119905 119868119899119905 119874119905ge 0

(3)






119894= 119909119894119895 119894 =


HM

=

[[[[[[[

[

11990911

11990912

11990913


119891 (1199091)

11990921

11990922

11990923


119891 (1199092)

119909HMS1

119909HMS2

119909HMS3


119891 (119909HMS119899

)

]]]]]]]

]

(4)







119909new119894119895

= 119909119894119895 119909119894119895

isin 1199091119895 1199092119895 119909HMS

119895

(5)




119909new119894119895

= 119909119894119895plusmn 1199033sdot BW (6)






119909new119894119895

= 119897119894119895+ (119906119894119895minus 119897119894119895) 1199034 (7)


3 1199034isin (0 1)


if 119891 (1199091015840new

) lt 119891 (119909worst

) then 119909worst

= 119909new (8)




Start



r1 lt HMCR

Yes

xnewij = xij

No

No

r2 lt PAR

Yes

xnewij = xnew

ij plusmn r3 BW

No

Yes



Yes

End













NoP lt Z

YesYes

k = k + 1


No

Yes

x = x998400

Nok = m


NoStopping criteria

YesEnd

Start



f(x998400) le f(x)

f(x) = f(x998400)

Tk+1 = 120572T 120572 = 095




119909119899 119909119899+1



119899 119899 + 1



119891(119909119899) le 119891(119909

119899+1) where 119891(119909


solution)



119894for each 119909

119894

(ii) If 119891(1199091015840

119894) le 119891(119909

119894) then 119909

119894= 1199091015840

119894and 119891(119909

119894) = 119891(119909

1015840

119894)

(iii) Else if 119901 = 119890(minus(|119891(119909

1015840

119894)minus119891(119909

119894)|)119879) and 119875 lt 119885 then 119909

119894= 1199091015840

119894

119891(119909) = 119891(1199091015840




= 120572119879 120572 =







NoP lt Z

Yes

k = k + 1


No

Yes

xi = x0i

Nok = m



Yes

End

Start




f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095






SA


119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37


Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692






Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












min1198851=

119873

sum

119899=1

119879

sum

119905=1

119888119899119905119875119899119905

+ 119894119899119905119868119899119905 (1)


min1198852=

119879

sum

119905=1

119908119905119882119905+ ℎ119905119867119905+ 119891119905119865119905+ 119900119905119874119905 (2)


119875119899119905

+ 119868119899(119905minus1)

minus 119868119899119905

= 119863119899119905 forall119899 forall119905


= 0 forall119905

119874119905minus 119860119874

times 119882119905le 0 forall119905

119873

sum

119899=1

119872119899119875119899119905


119867119905le 119867max

119865119905le 119865max

119867119905 119865119905119882119905are integer 119875

119899119905 119868119899119905 119874119905ge 0

(3)






119894= 119909119894119895 119894 =


HM

=

[[[[[[[

[

11990911

11990912

11990913


119891 (1199091)

11990921

11990922

11990923


119891 (1199092)

119909HMS1

119909HMS2

119909HMS3


119891 (119909HMS119899

)

]]]]]]]

]

(4)







119909new119894119895

= 119909119894119895 119909119894119895

isin 1199091119895 1199092119895 119909HMS

119895

(5)




119909new119894119895

= 119909119894119895plusmn 1199033sdot BW (6)






119909new119894119895

= 119897119894119895+ (119906119894119895minus 119897119894119895) 1199034 (7)


3 1199034isin (0 1)


if 119891 (1199091015840new

) lt 119891 (119909worst

) then 119909worst

= 119909new (8)




Start



r1 lt HMCR

Yes

xnewij = xij

No

No

r2 lt PAR

Yes

xnewij = xnew

ij plusmn r3 BW

No

Yes



Yes

End













NoP lt Z

YesYes

k = k + 1


No

Yes

x = x998400

Nok = m


NoStopping criteria

YesEnd

Start



f(x998400) le f(x)

f(x) = f(x998400)

Tk+1 = 120572T 120572 = 095




119909119899 119909119899+1



119899 119899 + 1



119891(119909119899) le 119891(119909

119899+1) where 119891(119909


solution)



119894for each 119909

119894

(ii) If 119891(1199091015840

119894) le 119891(119909

119894) then 119909

119894= 1199091015840

119894and 119891(119909

119894) = 119891(119909

1015840

119894)

(iii) Else if 119901 = 119890(minus(|119891(119909

1015840

119894)minus119891(119909

119894)|)119879) and 119875 lt 119885 then 119909

119894= 1199091015840

119894

119891(119909) = 119891(1199091015840




= 120572119879 120572 =







NoP lt Z

Yes

k = k + 1


No

Yes

xi = x0i

Nok = m



Yes

End

Start




f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095






SA


119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37


Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692






Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start



r1 lt HMCR

Yes

xnewij = xij

No

No

r2 lt PAR

Yes

xnewij = xnew

ij plusmn r3 BW

No

Yes



Yes

End













NoP lt Z

YesYes

k = k + 1


No

Yes

x = x998400

Nok = m


NoStopping criteria

YesEnd

Start



f(x998400) le f(x)

f(x) = f(x998400)

Tk+1 = 120572T 120572 = 095




119909119899 119909119899+1



119899 119899 + 1



119891(119909119899) le 119891(119909

119899+1) where 119891(119909


solution)



119894for each 119909

119894

(ii) If 119891(1199091015840

119894) le 119891(119909

119894) then 119909

119894= 1199091015840

119894and 119891(119909

119894) = 119891(119909

1015840

119894)

(iii) Else if 119901 = 119890(minus(|119891(119909

1015840

119894)minus119891(119909

119894)|)119879) and 119875 lt 119885 then 119909

119894= 1199091015840

119894

119891(119909) = 119891(1199091015840




= 120572119879 120572 =







NoP lt Z

Yes

k = k + 1


No

Yes

xi = x0i

Nok = m



Yes

End

Start




f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095






SA


119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37


Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692






Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












NoP lt Z

YesYes

k = k + 1


No

Yes

x = x998400

Nok = m


NoStopping criteria

YesEnd

Start



f(x998400) le f(x)

f(x) = f(x998400)

Tk+1 = 120572T 120572 = 095




119909119899 119909119899+1



119899 119899 + 1



119891(119909119899) le 119891(119909

119899+1) where 119891(119909


solution)



119894for each 119909

119894

(ii) If 119891(1199091015840

119894) le 119891(119909

119894) then 119909

119894= 1199091015840

119894and 119891(119909

119894) = 119891(119909

1015840

119894)

(iii) Else if 119901 = 119890(minus(|119891(119909

1015840

119894)minus119891(119909

119894)|)119879) and 119875 lt 119885 then 119909

119894= 1199091015840

119894

119891(119909) = 119891(1199091015840




= 120572119879 120572 =







NoP lt Z

Yes

k = k + 1


No

Yes

xi = x0i

Nok = m



Yes

End

Start




f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095






SA


119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37


Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692






Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












NoP lt Z

Yes

k = k + 1


No

Yes

xi = x0i

Nok = m



Yes

End

Start




f(x998400i ) le f(xi)

f(xi) = f(x998400i )

Tk+1 = 120572T 120572 = 095






SA


119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1198881198991199053285 385 451 1006 801 487 449 1007 496 739

119894119899119905

38 47 536 35511 35415 246 3775 37666 58666 37


Product 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

Hours 92 525 69 64 50 121 42 607 172 692






Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Period 119860 119861 119862 119863 119864 119865 119866 119867 119869 119870

1 30491 539 3406 100 6064 231 17 12 31 0742 16641 509 7081 152 4827 265 33 2 18 113 12364 354 700 138 4968 148 74 17 23 0474 7825 408 650 77 4299 25 87 25 29 0765 9144 275 439 56 3247 15 215 24 21 236 6529 379 6191 50 6529 124 291 13 27 071


Algorithm 1198851

1198852

TimeHS 7043102 5737053 157 sSA 7159657 5918976 17 sMSA 6993111 5814778 11 s


Product P1 P2 P3 P4 P5 P6119860 294482 16641 12364 7825 9144 6529119861 539 509 354 408 275 379119862 3406 7081 700 6502 439 6191119863 100 152 138 77 56 50119864 144071 4827 4968 4299 3247 6529119865 23101 26501 14801 25 145671 117534119866 17 33 74 87 215 291119867 119 2 17 25 369 07119869 1258 18 23 29 21 27119870 0741 11669 04614 0722 30498 0068


Product P1 P2 P3 P4 P5 P6119860 3650167 365017 365017 36502 36502 36502119861 129 259 363 415 415 527119862 0 0 0 0 0 0119863 0 0 0 0 0 0119864 01612 01612 01612 01612 01612 01612119865 0 0 0 0 02308 0119866 0 0 0 0 0 0119867 01434 01434 01434 11434 01946 01263119869 0 0 0 0 0 0119870 01 06 006 002 077 0



1915 1831 1602 1296 1190 1191119867119905

0 0 0 0 0 1119865119905

1400 84 229 306 106 0119874119905

31079 0 0 0 0 0


HSSAMSA


P=6

P=6

P=12

P=18

P=6

P=12

P=18

P=12

P=18

N = 15 N = 25 N = 35

0

500000000

1E + 09

15E + 09

2E + 09

25E + 09







119874) and


119877and 119860

119874to


1and 119885

2 respectively


7 Conclusion





SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












SA HS MSA

119873 = 15 119875 = 6

1198851

9623121904 9585047304 93276091811198852

4474374222 4468666941 4432744638Time 48048 98436 06864

119873 = 15 119875 = 12

1198851

2500537229 247445920 24179694721198852

6109619847 6109619847 6109619847Time 84552 174097 07956

119873 = 15 119875 = 18

1198851

3645638404 3636604939 35084383311198852

4883802824 4883802824 4883802824Time 131352 443042 04212

119873 = 25 119875 = 6

1198851

1788719071 1764678125 1732631611198852

3798854597 3798854597 3798854597Time 50856 14821 03276

119873 = 25 119875 = 12

1198851

2728110509 2712766913 26092657721198852

5482170295 4780504377 4572669502Time 8081 68999 04056

119873 = 25 119875 = 18

1198851

4890209322 486493908 46774941411198852

5547808678 5055586502 5016375384Time 169105 1351904 04524

119873 = 35 119875 = 6

1198851

2097361519 2088245585 20262962221198852

4289075951 4139691552 4139691552Time 8346 1833 03276

119873 = 35 119875 = 12

1198851

4000608781 3997316399 38490624051198852

6251650151 5471269441 4690962702Time 114817 1260956 0421

119873 = 35 119875 = 18

1198851

7693953235 7671725819 74063503051198852

8781557215 8501321868 8501321868Time 163801 165478 04683

HSSAMSA

P=6

P=12

P=18

P=6

P=12

P=18

P=6

P=12

P=18

N = 15 N = 25 N = 35

020406080

100120140160180












($ worker)
















Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Competing Interests


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article enhanced simulated annealing for solving...

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