Research ArticleA Genetic Algorithm-Based Approach for Single-MachineScheduling with Learning Effect and Release Time

Der-Chiang Li1 Peng-Hsiang Hsu12 and Chih-Chieh Chang3

1 Department of Industrial and Information Management National Cheng Kung University 1 University Road Tainan Taiwan2Department of Business Administration Kang-Ning Junior College of Medical Care and Management Taipei Taiwan3 Research Center for Information Technology Innovation Academic Sinica Taipei Taiwan

Correspondence should be addressed to Peng-Hsiang Hsu r38991036mailnckuedutw

Received 24 August 2013 Revised 6 November 2013 Accepted 29 November 2013 Published 4 March 2014

Academic Editor Chin-Chia Wu

Copyright copy 2014 Der-Chiang Li 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 way to gain knowledge and experience of producing a product in a firm can be seen as new solution for reducing the unitcost in scheduling problems which is known as ldquolearning effectsrdquo In the scheduling of batch processing machines it is sometimesadvantageous to form a nonfull batch while in other situations it is a better strategy to wait for future job arrivals in order toincrease the fullness of the batch However research with learning effect and release times is relatively unexplored Motivated bythis observation we consider a single-machine problem with learning effect and release times where the objective is to minimizethe total completion times We develop a branch-and-bound algorithm and a genetic algorithm-based heuristic for this problemThe performances of the proposed algorithms are evaluated and compared via computational experiments which showed that ourapproach has superior ability in this scenario

1 Introduction

Learning effect has been considered as the most inten-sive phenomenon since it has been proposed in Biskup[1] The basic concept has been set to fix the processingtime of scheduling sequence job from the first job to thelast job which demonstrated that the processing time canbe improved after continuous learning Moreover someresearches also indicated that learning effect can be regardedas a controllable and important component affect process-ing time in scheduling problems (Vickson Nowicki andZdrzałka and Cheng et al [2ndash4]) Although there weremany researches focusing on this phenomenon all of themsometimes assumed that all jobs were allowed to processon machine in time However the release times must beconsidered in many real-world applications in order tomake this assumption valid For example products in asemiconductor wafer fabrication facilities undergo severalhundreds of manufacturing steps such as reentrant processflows sequence-dependent setups diversity of product mixand batch processing With such complexities it wouldbe a great challenge to meet the customersrsquo requirements

such as different priorities ready times and due dates Inthe presence of unequal ready times the application of anonfull batch would sometimes be advantageous In somecases it would be better to wait for new jobs to arrive toincrease the completeness of the batch (Monch et al [5])Following are some researches that focus on schedulingproblem by considering both learning effect and release times(Lee et al Eren Wu and Liu Toksarı Wu et al Ahmadizarand Hosseini Rudek Li and Hsu Kung et al and Yinet al [6ndash16]) In this paper we would like to discuss thesingle-machine total completion time problem with sum ofprocessing time-based learning and release times given thatit is a topic still to be studied and explored

Rinnooy Kan [17] showed that the same problem withoutlearning consideration was NP-hard unless the release timeswere identical Therefore we apply the branch-and-boundalgorithm and genetic algorithm search for the optimalsolution and near-optimal solutions The results show thatthe branch and bound algorithm has solved the instancesless than or equal to 24 jobs Moreover GA also shows goodperformance in computational experiments

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 249493 12 pageshttpdxdoiorg1011552014249493

2 Mathematical Problems in Engineering

The rest of the paper is organized as follows Somelearning effect works are described in Section 2 In Section 3the description of notations and the problem formulation aregiven Some dominance properties and two lower boundsare developed to enhance the search efficiency for theoptimal solution followed by descriptions of the geneticalgorithm and the branch-and-bound algorithms are shownin Section 4 The results of a computational experiment aregiven in Section 5 and the conclusions are given in the lastsection

2 Related Works

There were some related researches dealing with schedulingproblem via learning effect In Heizer and Render [18]authors verified that unit costs decrease while a firm gainsmore product knowledge and experience Cheng and Wang[19] introduced the framework of learning effect in a singlemachine More recently Biskup [20] provided reviews ofstate-of-the-art scheduling Wang et al [21] studied the time-dependent learning effect in scheduling recently and cameup with the same learning model as proposed by Kuo andYang [22] where the job processing time is a function oftotal normal processing time of the previously scheduled jobsCheng et al [23] brought up with a new learning model inwhich the actual processing time of a job depends on boththe jobrsquos scheduled position and the processing times of thejobs that are already processed The two models proposedby Biskup [1] and Koulamas and Kyparisis [24] were furthercombined by Wu and Lee Lee and Wu and Yin et al [25ndash27] Also a new experience-based learning effect modelwhich is based on the S-shaped learning curve where jobprocessing times would be dependent on the experience ofthe processor was introduced and analyzed by Janiak andRudek [28] S-J Yang and D-L Yang [29] investigated anew group learning effect model on scheduling problemaiming to minimize the total processing time Yin et al [30]brought a general learning effect model into the field ofscheduling which states that the actual processing time of ajob is a general function of both the total actual processingtimes of the jobs already processed and the jobrsquos scheduledposition They had shown that the problems of minimizingmakespan and the sum of the 119896th power of completion timecould be solved in polynomial time respectively A single-machine scheduling problem with a truncation learningeffect proposed by Wu et al [31] states that job processingtime depends on the processing times of the jobs alreadyprocessed and a controlled parameter They also showed thatpolynomial time can be used to solve some single-machinescheduling problems J-B Wang and M-Z Wang [32] alsocame up with a revised model based on general learningeffect and proved that some single-machine and flowshopscheduling problems can be solved using polynomial timeLu et al [33] applying different learning effect modelssimultaneously studied several single-machine schedulingproblems and concluded that under the proposed modelsthe scheduling problems of minimization of the makespanthe total completion time and the sum of the 119896th power of

job completion times can be solved in polynomial time J-B Wang and J-J Wang [34] studied learning effect modelwhere the actual processing time of a job is not only a non-increasing function of the total weighted normal processingtimes of the jobs already processed but also a nonincreasingfunction of the jobrsquos scheduled position where the weightis a position-dependent weight They also show that theirapproach can solve the problem in polynomial time Li et al[35] investigated several single-machine problems with atruncated sum of processing times based learning effect thatremain polynomially solvable Cheng et al [36] addressedtwo-machine flowshop scheduling with a truncated learningfunction while minimizing the makespan They applied abranch-and-bound and three heuristic algorithms to derivethe optimal and near-optimal solutionsWu [37] studied two-agent scheduling on a single machine involving the learningeffects and deteriorating jobs simultaneouslyThe objective isto minimize the total weighted completion time of the jobs ofthe first agent with the restriction that no tardy job is allowedfor the second agent

3 Notation and Problem Formulation

Before formulating the problem we first introduce somenotations that will be used throughout the paper

119899 the number of jobs119878 1198781015840 119878lowast 119878

1 the sequences of jobs

119869119894 119869119895 the job 119894 and job 119895

119901119894 the normal processing time of job 119894

119901[119894](119878) the normal processing time for the job sched-

uled in the 119894th position in 119878119901(119894) the 119894th job processing time when they are in a

nondecreasing order119901119895119903 the actual processing time of a given job 119895 if it is

scheduled in the 119903th position119903119894 the release time of job 119894

119903[119894](119878) the release time of a job scheduled in the 119894th

position in 119878119903(119894) the 119894th job release time when they are in a non-

decreasing order119886 the learning ratio where 119886 le 0119862119894(119878) 119862

119894(1198781015840) the completion time of job 119894 in 119878 and 119878

1015840119862119894(119878lowast) the completion time of job 119894 in optimal

schedule 119878lowast119862[119894](119878) 119862

[119894](1198781) the completion time of a job sched-

uled in the 119894th position in 119878 and 1198781

TC(119878) TC(1198781015840) the total completion times of sequen-ces 119878 and 119878

1015840120587 1205871015840 120587119888 the subsequences of jobs

The formulation of the problem is described as followsThere are 119899 jobs to be processed on a single machine Themachine can handle one job at a time and machine idle and

Mathematical Problems in Engineering 3

job preemption are not allowed Each job 119895 has a normalprocessing time 119901

119895and a release time 119903

119895 The general job

learning model is 119901119895119903

= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886 where 119901

119895119903is

the actual processing time of job 119895 scheduled in the 119903thposition and 119886 le 0 is a learning ratio The objective of thisproblem is to find an optimal schedule 119878

lowast that minimizesthe total completion time that is sum119899

119894=1119862119894(119878lowast) le sum

119899

119894=1119862119894(119878)

for any schedule 119878 Using the standard three-field notation ofGraham et al [38] our scheduling problem can be denotedas 1119901

119895119903= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886sum119862

4 The Branch-and-Bound andGenetic Algorithms

In this paper we will apply the branch-and-bound andgenetic algorithms search for the optimal solution and obtainnear-optimal solution respectively First in order to facilitatethe searching process and improve the branching procedurewe develop some adjacent pairwise interchange propertiesand two lower bounds to use in branch-and-bound algo-rithmThen the procedure of the genetic algorithms is givenat last

41 Dominance Properties Before presenting the adjacentpairwise interchange properties we provide two lemmaswhich will be used in the proofs of the properties in thesequel

Lemma 1 Let119891(119909) = 1+ 119886119909(1 minus 119909)119886minus1

minus (1 + 119909)119886 then119891(119909) ge

0 for 119886 le 0 and 119909 gt 0

Lemma 2 Let 119892(120582) = (120582 minus 1) + (1 + 120582119909)119886minus 120582(1 + 119909)

119886 then119892(120582) ge 0 for 120582 ge 1 119886 le 0 and 119909 gt 0

To fathom the searching tree we develop some dom-inance properties based on a pairwise interchange of twoadjacent jobs 119869

119894and 119869

119895 Let 119878 = (120587 119869

119894 119869119895 1205871015840) and 119878

1015840=

(120587 119869119895 119869119894 1205871015840) be two sequences in which 120587 and 120587

1015840 denotepartial sequences To show that 119878 dominates 1198781015840 it suffices toshow that 119862

119894(119878) + 119862

119895(119878) le 119862

119895(1198781015840) + 119862119894(1198781015840) and 119862

119895(119878) lt 119862

119894(1198781015840)

In addition let 119905 be the completion time of the last job insubsequence 120587 with (119903 minus 1) jobs

Property 1 If 119901119894lt 119901119895and max119903

119894 119903119895 le 119905 then 119878 dominates

1198781015840

Proof Since max119903119894 119903119895 le 119905 we have

119862119894(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119895(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

119862119895(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119894(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

(1)

After taking the difference of (1) we have

119862119894(1198781015840) minus 119862119895(119878) = (119901

119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(2)

On substituting120582 = 119901119895119901119894 119906 = (1+sum

119903minus1

119897=1119901[119897]) and119909 = (119901

119894(1+

sum119899

119897=1119901119897)) into (2) and simplifying it we obtain

119862119894(1198781015840) minus 119862119895(119878) = 119901

119894119906119886[(120582 minus 1) + (1 + 120582119909)

119886minus 120582(1 + 119909)

119886]

(3)

By Lemma 2 with 120582 gt 1 119886 le 0 and 119909 gt 0 we have 119862119894(1198781015840) minus

119862119895(119878) gt 0Moreover after taking the difference of total completion

times (TC) between sequences 119878 and 1198781015840 we have

TC (1198781015840) minus TC (119878)

= [119862119895(1198781015840) + 119862119894(1198781015840)] minus [119862

119894(119878) + 119862

119895(119878)]

= 2 (119901119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(4)

By (3) it can be easily shown that (4) is nonnegative for 119901119894lt

119901119895 Therefore 119878 dominates 1198781015840

The proofs of Properties 2 to 5 are omitted since they aresimilar to that of Property 1

Property 2 If 119903119894le 119905 le 119903

119895le 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

and 119901119894lt 119901119895

then 119878 dominates 1198781015840

Property 3 If 119905 ge 119903119894and 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

Property 4 If 119905 le 119903119894le 119903119895 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

ge 119903119895 and

119901119894lt 119901119895 then 119878 dominates 1198781015840

Property 5 If 119905 le 119903119894and 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

4 Mathematical Problems in Engineering

In order to further determine the ordering of the remain-ing unscheduled jobs to further speed up the searchingprocess we provide the following property Assume that 119878 =

(120587 120587119888) is a sequence of jobs where 120587 is the scheduled part

containing 119896 jobs and 120587119888 is the unscheduled part Let 119878

1=

(120587 1205871015840) be the sequence in which the unscheduled jobs are

arranged in a nondecreasing order of job processing timesthat is 119901

(119896+1)le 119901(119896+2)

le sdot sdot sdot le 119901(119899)

Property 6 If 119862[119896](1198781) gt max

119895isin120587119888119903119895 then 119878

1= (120587 120587

1015840)

dominates sequences of the type (120587 120587119888) for any unscheduledsequence 120587119888

Proof Since 119862[119896](1198781) gt max

119895isin120587119888119903119895 it implies that all the

unscheduled jobs are ready to be processed on time 119862[119896](1198781)

To obtain the optimal subsequence let 1198781= (120587 120587

1015840) be the

sequence in which the unscheduled jobs are arranged innondecreasing order of jobs processing times

42 Lower Bounds In this subsection we develop two lowerbounds by using the following lemma from Hardy et al [39]

Lemma 3 Suppose that 119886119894and 119887

119894are two sequences of

numbers The sum sum119899

119894=1119886119894119887119894of products of the corresponding

elements is the least if the sequences are monotonic in theopposite sense

First let 119875119878 be a partial schedule in which the order ofthe first 119896 jobs has been determined and let 119878 be a completeschedule obtained from 119875119878 By definition the completiontime for the (119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119862[119896]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(5)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119862[119896]

(119878) +

119895

sum

119894=1

119901(119896+119894)

(1 +

119896

sum

119897=1

119901[119897]

+

119894minus1

sum

119897=1

119901(119896+119897)

)

119886

1 le 119895 le 119899 minus 119896

(6)

The first term on the right hand side of (6) is known anda lower bound of the total completion time for the partialsequence 119875119878 can be obtained by minimizing the secondterm Since the value of (1 + sum

119896

119897=1119901[119897]

+ sum119894minus1

119897=1119901[119896+119897]

)119886 is a

decreasing function of sum119894minus1119897=1

119901[119896+119897]

the total completion timeis minimized by sequencing the unscheduled jobs accordingto the shortest processing time (SPT) rule according toLemma 3 Consequently the first lower bound is

LB1=

119896

sum

119894=1

119862[119894](119878) +

119899minus119896

sum

119895=1

119862(119895) (7)

where 119862(119895)

= 119862[119896](119878) + sum

119895

119894=1119901(119896+119894)

(1 + sum119896

119897=1119901[119897]

+

sum119894minus1

119897=1119901(119899minus119896+119897minus1)

)119886 On the other hand this lower bound may

not be tight if the release time is long To overcome thissituation a second lower bound is established by takingaccount of the release time The completion time for the(119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119903[119896+1]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(8)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119903[119896+119895]

(119878) + 119901[119896+119895]

(1 +

119896

sum

119897=1

119901[119897]

+

119895minus1

sum

119897=1

119901[119896+119897]

)

119886

1 le 119895 le 119899 minus 119896

(9)

Note that 119862[119896+119895]

(119878) is greater than or equal to 1199031015840

(119896+119895) where

1199031015840

(119896+1)le 1199031015840

(119896+2)le sdot sdot sdot le 119903

1015840

(119899)denote the release times of

the unscheduled jobs arranged in a nondecreasing orderThesecond term on the right hand side of (9) is minimized bythe SPT rule since (1 +sum

119896

119897=1119901[119897]+sum119895minus1

119897=1119901[119896+119897]

)119886 is a decreasing

function of sum119895minus1119897=1

119901[119896+119897]

It follows that we have the followingsecond lower bound

LB2=

119896

sum

119894=1

119862[119894]

+

119899minus119896

sum

119895=1

119862(119895) (10)

where 119862(119895)

= 1199031015840

(119896+119895)(119878) +119901

1015840

(119896+1)(1+sum

119896

119897=1119901[119897]+sum119895minus1

119897=11199011015840

(119899minus119896+119897minus1))119886

Note that 1199011015840(119896+119895)

and 1199031015840

(119896+119895)do not necessarily come from the

same job In order tomake the lower bound tighter we choosethe maximum value from (7) and (10) as the lower bounds of119875119878 That is

LB = max LB1 LB2 (11)

43TheProcedure of Genetic Algorithms Agenetic algorithm(GA) is an optimization method that mimics natural pro-cesses GAwas invented by Holland [40] and themost widelyused to solve numerical optimization problems in a widevariety of application fields including biology economicsengineering business agriculture telecommunications andmanufacturing For example in Goldberg [41] authors usingGA in engineering design problems is reviewed in Gen andCheng [42] Soolaki et al [43] use a GA to solve an airlineboarding problem with linear programming models [44 45]and use genetic algorithms to optimize the parameters for thegiven test collections GAs start evolving by generating aninitial population of chromosomes Then a fitness functionis used to compute the relative fitness of each chromosomeof the population The selection crossover and mutationoperators are used in succession to create a new population

Mathematical Problems in Engineering 5

of chromosomes for the next generation This approach hasgained increasing popularity in solving many combinatorialoptimization problems in a wide variety of different disci-plines

431 Initial Settings In a GA every problem is presented bya code and each code is seen as a geneThe existing genes canbe combined and seen as a chromosome each of which is oneof the feasible solutions to a problem However traditionalrepresentation of GA does not work for scheduling problems(Etiler et al [46]) In dealing with this condition this studyadopts the same method that a structure can describe thejobs as a sequence in the problem To specify our approachseveral initial sequences are adopted In GA

1 jobs are placed

according to the shortest processing times (SPT) first ruleIn GA

2 jobs are arranged in earliest ready times (ERT) first

rule In GA3 jobs are arranged in a nondecreasing order on

the sum of job processing times and ready times Note thatbefore performing GA NEH algorithm (Nawaz et al [47]) isutilized to improve the quality of the solutions obtained fromthe previous rules to reduce many idle periods The processof GA

1 GA2 and GA

3are different initial sequences and use

the same selection crossover mutation operators populationsize and generations to obtain near-optimal solution Inaddition the fourth genetic algorithm denoted as GA

4 is

the best one among GA1 GA2 and GA

3 that is GA

4=

minGA1GA2GA3

In order to avoid rapidly observing a local optimum in asmall population or consume more waiting time in a largeone this study set a suitable population size as 60 (119873 =

60) in a preliminary trial It is also an important work toevaluate the fitness of selected chromosomes that each ofthe chromosomes is included or excluded from a feasiblesolution The main goal of this study is to minimize thetotal completion time Assume that 119878

119894(119905) is the 119894th string

in the 119894th generation and the total completion time of 119878119894(119905)

is sum119899

119895=1119862119895(119878119894(119905)) Then the fitness function of 119878

119894(119905) can be

represented as 119891(119878119894(119905)) Following are the calculations of the

strings in fitness function

119891 (119878119894(119905)) = max

1le119897le119873

119899

sum

119895=1

119862119895(119878119897(119905))

minus

119899

sum

119895=1

119862119895(119878119894(119905))

(12)

Moreover it is also crucial work to ensure that the probabilityof selection for a sequence with lower value of the objectivefunction is higher Thus the probability 119875(119878

119894(119905)) can be

written as follows

119875 (119878119894(119905)) =

119891 (119878119894(119905))

sum119899

119894=1119891 (119878119894(119905))

(13)

432 Operators There are a few operators that are used inthis study Following are the descriptions of those operatorscrossover mutation and selection

(a) CrossoverThis is an operator that exchanges some of thegenes of the selected parents with the main concept being

005 010 015 020 025 030000

001

001

002

002

003

Mea

n er

ror (

)

00040 00192 00108 00103 0020600171Pm

Figure 1 The performance of the genetic algorithms for various 119875119898

at (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899)

that the descendant can inherit the advantages of its parentsThis study applied the linear order crossover operator (LOX)proposed by Falkenauer and Bouffouix [48] and is one of thebetter performers among the others (Etiler et al [46]) Theprobability of crossover is set to 1

(b) Mutation The main object of mutation is to achieve foran overall optimal solution and to avoid a locally optimalone In this study the mutation rates (119875

119898) are set at 010

based on our preliminary experiment as shown in Figure 1For (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899) 100 sets of datawere randomly generated to evaluate the performance of theproposed algorithms with varying values of 119875

119898 The results

showed that the proposed algorithmshad the leastmean errorpercentage at 119875

119898= 010

(c) Selection This is a process that determines the proba-bility of each chromosome and is used to decide the betterchromosomes with the better fitness value The evolutionimplemented in our algorithm is based on the elitist list Wecopy the best offspring and use them to generate some ofthe next generation The rest of the offspring are generatedfrom the parent chromosomes by the roulette wheel selectionmethod which can maintain the variety of genes

433 Stopping Criteria In the preliminary experimentthe proposed GAs are terminated after 100 lowast 119899 genera-tions as shown in Figures 2 and 3 For (119899 119886 120579119873 119875

119898) =

(20 minus005 05 60 010) the above 100 sets of randomly gen-erated data were used to evaluate the performance of theproposed algorithms with varying values of 119892 The resultsshowed that the least mean error percentage of the proposedalgorithms would stabilize with reasonable CPU time rangeafter 119892 = 100119899

5 Computational Experiment

A computational experiment was conducted to evaluatethe efficiency of the branch-and-bound algorithm and

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

The rest of the paper is organized as follows Somelearning effect works are described in Section 2 In Section 3the description of notations and the problem formulation aregiven Some dominance properties and two lower boundsare developed to enhance the search efficiency for theoptimal solution followed by descriptions of the geneticalgorithm and the branch-and-bound algorithms are shownin Section 4 The results of a computational experiment aregiven in Section 5 and the conclusions are given in the lastsection

2 Related Works

There were some related researches dealing with schedulingproblem via learning effect In Heizer and Render [18]authors verified that unit costs decrease while a firm gainsmore product knowledge and experience Cheng and Wang[19] introduced the framework of learning effect in a singlemachine More recently Biskup [20] provided reviews ofstate-of-the-art scheduling Wang et al [21] studied the time-dependent learning effect in scheduling recently and cameup with the same learning model as proposed by Kuo andYang [22] where the job processing time is a function oftotal normal processing time of the previously scheduled jobsCheng et al [23] brought up with a new learning model inwhich the actual processing time of a job depends on boththe jobrsquos scheduled position and the processing times of thejobs that are already processed The two models proposedby Biskup [1] and Koulamas and Kyparisis [24] were furthercombined by Wu and Lee Lee and Wu and Yin et al [25ndash27] Also a new experience-based learning effect modelwhich is based on the S-shaped learning curve where jobprocessing times would be dependent on the experience ofthe processor was introduced and analyzed by Janiak andRudek [28] S-J Yang and D-L Yang [29] investigated anew group learning effect model on scheduling problemaiming to minimize the total processing time Yin et al [30]brought a general learning effect model into the field ofscheduling which states that the actual processing time of ajob is a general function of both the total actual processingtimes of the jobs already processed and the jobrsquos scheduledposition They had shown that the problems of minimizingmakespan and the sum of the 119896th power of completion timecould be solved in polynomial time respectively A single-machine scheduling problem with a truncation learningeffect proposed by Wu et al [31] states that job processingtime depends on the processing times of the jobs alreadyprocessed and a controlled parameter They also showed thatpolynomial time can be used to solve some single-machinescheduling problems J-B Wang and M-Z Wang [32] alsocame up with a revised model based on general learningeffect and proved that some single-machine and flowshopscheduling problems can be solved using polynomial timeLu et al [33] applying different learning effect modelssimultaneously studied several single-machine schedulingproblems and concluded that under the proposed modelsthe scheduling problems of minimization of the makespanthe total completion time and the sum of the 119896th power of

job completion times can be solved in polynomial time J-B Wang and J-J Wang [34] studied learning effect modelwhere the actual processing time of a job is not only a non-increasing function of the total weighted normal processingtimes of the jobs already processed but also a nonincreasingfunction of the jobrsquos scheduled position where the weightis a position-dependent weight They also show that theirapproach can solve the problem in polynomial time Li et al[35] investigated several single-machine problems with atruncated sum of processing times based learning effect thatremain polynomially solvable Cheng et al [36] addressedtwo-machine flowshop scheduling with a truncated learningfunction while minimizing the makespan They applied abranch-and-bound and three heuristic algorithms to derivethe optimal and near-optimal solutionsWu [37] studied two-agent scheduling on a single machine involving the learningeffects and deteriorating jobs simultaneouslyThe objective isto minimize the total weighted completion time of the jobs ofthe first agent with the restriction that no tardy job is allowedfor the second agent

3 Notation and Problem Formulation

Before formulating the problem we first introduce somenotations that will be used throughout the paper

119899 the number of jobs119878 1198781015840 119878lowast 119878

1 the sequences of jobs

119869119894 119869119895 the job 119894 and job 119895

119901119894 the normal processing time of job 119894

119901[119894](119878) the normal processing time for the job sched-

uled in the 119894th position in 119878119901(119894) the 119894th job processing time when they are in a

nondecreasing order119901119895119903 the actual processing time of a given job 119895 if it is

scheduled in the 119903th position119903119894 the release time of job 119894

119903[119894](119878) the release time of a job scheduled in the 119894th

position in 119878119903(119894) the 119894th job release time when they are in a non-

decreasing order119886 the learning ratio where 119886 le 0119862119894(119878) 119862

119894(1198781015840) the completion time of job 119894 in 119878 and 119878

1015840119862119894(119878lowast) the completion time of job 119894 in optimal

schedule 119878lowast119862[119894](119878) 119862

[119894](1198781) the completion time of a job sched-

uled in the 119894th position in 119878 and 1198781

TC(119878) TC(1198781015840) the total completion times of sequen-ces 119878 and 119878

1015840120587 1205871015840 120587119888 the subsequences of jobs

The formulation of the problem is described as followsThere are 119899 jobs to be processed on a single machine Themachine can handle one job at a time and machine idle and

Mathematical Problems in Engineering 3

job preemption are not allowed Each job 119895 has a normalprocessing time 119901

119895and a release time 119903

119895 The general job

learning model is 119901119895119903

= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886 where 119901

119895119903is

the actual processing time of job 119895 scheduled in the 119903thposition and 119886 le 0 is a learning ratio The objective of thisproblem is to find an optimal schedule 119878

lowast that minimizesthe total completion time that is sum119899

119894=1119862119894(119878lowast) le sum

119899

119894=1119862119894(119878)

for any schedule 119878 Using the standard three-field notation ofGraham et al [38] our scheduling problem can be denotedas 1119901

119895119903= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886sum119862

4 The Branch-and-Bound andGenetic Algorithms

In this paper we will apply the branch-and-bound andgenetic algorithms search for the optimal solution and obtainnear-optimal solution respectively First in order to facilitatethe searching process and improve the branching procedurewe develop some adjacent pairwise interchange propertiesand two lower bounds to use in branch-and-bound algo-rithmThen the procedure of the genetic algorithms is givenat last

41 Dominance Properties Before presenting the adjacentpairwise interchange properties we provide two lemmaswhich will be used in the proofs of the properties in thesequel

Lemma 1 Let119891(119909) = 1+ 119886119909(1 minus 119909)119886minus1

minus (1 + 119909)119886 then119891(119909) ge

0 for 119886 le 0 and 119909 gt 0

Lemma 2 Let 119892(120582) = (120582 minus 1) + (1 + 120582119909)119886minus 120582(1 + 119909)

119886 then119892(120582) ge 0 for 120582 ge 1 119886 le 0 and 119909 gt 0

To fathom the searching tree we develop some dom-inance properties based on a pairwise interchange of twoadjacent jobs 119869

119894and 119869

119895 Let 119878 = (120587 119869

119894 119869119895 1205871015840) and 119878

1015840=

(120587 119869119895 119869119894 1205871015840) be two sequences in which 120587 and 120587

1015840 denotepartial sequences To show that 119878 dominates 1198781015840 it suffices toshow that 119862

119894(119878) + 119862

119895(119878) le 119862

119895(1198781015840) + 119862119894(1198781015840) and 119862

119895(119878) lt 119862

119894(1198781015840)

In addition let 119905 be the completion time of the last job insubsequence 120587 with (119903 minus 1) jobs

Property 1 If 119901119894lt 119901119895and max119903

119894 119903119895 le 119905 then 119878 dominates

1198781015840

Proof Since max119903119894 119903119895 le 119905 we have

119862119894(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119895(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

119862119895(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119894(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

(1)

After taking the difference of (1) we have

119862119894(1198781015840) minus 119862119895(119878) = (119901

119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(2)

On substituting120582 = 119901119895119901119894 119906 = (1+sum

119903minus1

119897=1119901[119897]) and119909 = (119901

119894(1+

sum119899

119897=1119901119897)) into (2) and simplifying it we obtain

119862119894(1198781015840) minus 119862119895(119878) = 119901

119894119906119886[(120582 minus 1) + (1 + 120582119909)

119886minus 120582(1 + 119909)

119886]

(3)

By Lemma 2 with 120582 gt 1 119886 le 0 and 119909 gt 0 we have 119862119894(1198781015840) minus

119862119895(119878) gt 0Moreover after taking the difference of total completion

times (TC) between sequences 119878 and 1198781015840 we have

TC (1198781015840) minus TC (119878)

= [119862119895(1198781015840) + 119862119894(1198781015840)] minus [119862

119894(119878) + 119862

119895(119878)]

= 2 (119901119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(4)

By (3) it can be easily shown that (4) is nonnegative for 119901119894lt

119901119895 Therefore 119878 dominates 1198781015840

The proofs of Properties 2 to 5 are omitted since they aresimilar to that of Property 1

Property 2 If 119903119894le 119905 le 119903

119895le 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

and 119901119894lt 119901119895

then 119878 dominates 1198781015840

Property 3 If 119905 ge 119903119894and 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

Property 4 If 119905 le 119903119894le 119903119895 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

ge 119903119895 and

119901119894lt 119901119895 then 119878 dominates 1198781015840

Property 5 If 119905 le 119903119894and 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

4 Mathematical Problems in Engineering

In order to further determine the ordering of the remain-ing unscheduled jobs to further speed up the searchingprocess we provide the following property Assume that 119878 =

(120587 120587119888) is a sequence of jobs where 120587 is the scheduled part

containing 119896 jobs and 120587119888 is the unscheduled part Let 119878

1=

(120587 1205871015840) be the sequence in which the unscheduled jobs are

arranged in a nondecreasing order of job processing timesthat is 119901

(119896+1)le 119901(119896+2)

le sdot sdot sdot le 119901(119899)

Property 6 If 119862[119896](1198781) gt max

119895isin120587119888119903119895 then 119878

1= (120587 120587

1015840)

dominates sequences of the type (120587 120587119888) for any unscheduledsequence 120587119888

Proof Since 119862[119896](1198781) gt max

119895isin120587119888119903119895 it implies that all the

unscheduled jobs are ready to be processed on time 119862[119896](1198781)

To obtain the optimal subsequence let 1198781= (120587 120587

1015840) be the

sequence in which the unscheduled jobs are arranged innondecreasing order of jobs processing times

42 Lower Bounds In this subsection we develop two lowerbounds by using the following lemma from Hardy et al [39]

Lemma 3 Suppose that 119886119894and 119887

119894are two sequences of

numbers The sum sum119899

119894=1119886119894119887119894of products of the corresponding

elements is the least if the sequences are monotonic in theopposite sense

First let 119875119878 be a partial schedule in which the order ofthe first 119896 jobs has been determined and let 119878 be a completeschedule obtained from 119875119878 By definition the completiontime for the (119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119862[119896]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(5)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119862[119896]

(119878) +

119895

sum

119894=1

119901(119896+119894)

(1 +

119896

sum

119897=1

119901[119897]

+

119894minus1

sum

119897=1

119901(119896+119897)

)

119886

1 le 119895 le 119899 minus 119896

(6)

The first term on the right hand side of (6) is known anda lower bound of the total completion time for the partialsequence 119875119878 can be obtained by minimizing the secondterm Since the value of (1 + sum

119896

119897=1119901[119897]

+ sum119894minus1

119897=1119901[119896+119897]

)119886 is a

decreasing function of sum119894minus1119897=1

119901[119896+119897]

the total completion timeis minimized by sequencing the unscheduled jobs accordingto the shortest processing time (SPT) rule according toLemma 3 Consequently the first lower bound is

LB1=

119896

sum

119894=1

119862[119894](119878) +

119899minus119896

sum

119895=1

119862(119895) (7)

where 119862(119895)

= 119862[119896](119878) + sum

119895

119894=1119901(119896+119894)

(1 + sum119896

119897=1119901[119897]

+

sum119894minus1

119897=1119901(119899minus119896+119897minus1)

)119886 On the other hand this lower bound may

not be tight if the release time is long To overcome thissituation a second lower bound is established by takingaccount of the release time The completion time for the(119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119903[119896+1]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(8)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119903[119896+119895]

(119878) + 119901[119896+119895]

(1 +

119896

sum

119897=1

119901[119897]

+

119895minus1

sum

119897=1

119901[119896+119897]

)

119886

1 le 119895 le 119899 minus 119896

(9)

Note that 119862[119896+119895]

(119878) is greater than or equal to 1199031015840

(119896+119895) where

1199031015840

(119896+1)le 1199031015840

(119896+2)le sdot sdot sdot le 119903

1015840

(119899)denote the release times of

the unscheduled jobs arranged in a nondecreasing orderThesecond term on the right hand side of (9) is minimized bythe SPT rule since (1 +sum

119896

119897=1119901[119897]+sum119895minus1

119897=1119901[119896+119897]

)119886 is a decreasing

function of sum119895minus1119897=1

119901[119896+119897]

It follows that we have the followingsecond lower bound

LB2=

119896

sum

119894=1

119862[119894]

+

119899minus119896

sum

119895=1

119862(119895) (10)

where 119862(119895)

= 1199031015840

(119896+119895)(119878) +119901

1015840

(119896+1)(1+sum

119896

119897=1119901[119897]+sum119895minus1

119897=11199011015840

(119899minus119896+119897minus1))119886

Note that 1199011015840(119896+119895)

and 1199031015840

(119896+119895)do not necessarily come from the

same job In order tomake the lower bound tighter we choosethe maximum value from (7) and (10) as the lower bounds of119875119878 That is

LB = max LB1 LB2 (11)

43TheProcedure of Genetic Algorithms Agenetic algorithm(GA) is an optimization method that mimics natural pro-cesses GAwas invented by Holland [40] and themost widelyused to solve numerical optimization problems in a widevariety of application fields including biology economicsengineering business agriculture telecommunications andmanufacturing For example in Goldberg [41] authors usingGA in engineering design problems is reviewed in Gen andCheng [42] Soolaki et al [43] use a GA to solve an airlineboarding problem with linear programming models [44 45]and use genetic algorithms to optimize the parameters for thegiven test collections GAs start evolving by generating aninitial population of chromosomes Then a fitness functionis used to compute the relative fitness of each chromosomeof the population The selection crossover and mutationoperators are used in succession to create a new population

Mathematical Problems in Engineering 5

of chromosomes for the next generation This approach hasgained increasing popularity in solving many combinatorialoptimization problems in a wide variety of different disci-plines

431 Initial Settings In a GA every problem is presented bya code and each code is seen as a geneThe existing genes canbe combined and seen as a chromosome each of which is oneof the feasible solutions to a problem However traditionalrepresentation of GA does not work for scheduling problems(Etiler et al [46]) In dealing with this condition this studyadopts the same method that a structure can describe thejobs as a sequence in the problem To specify our approachseveral initial sequences are adopted In GA

1 jobs are placed

according to the shortest processing times (SPT) first ruleIn GA

2 jobs are arranged in earliest ready times (ERT) first

rule In GA3 jobs are arranged in a nondecreasing order on

the sum of job processing times and ready times Note thatbefore performing GA NEH algorithm (Nawaz et al [47]) isutilized to improve the quality of the solutions obtained fromthe previous rules to reduce many idle periods The processof GA

1 GA2 and GA

3are different initial sequences and use

the same selection crossover mutation operators populationsize and generations to obtain near-optimal solution Inaddition the fourth genetic algorithm denoted as GA

4 is

the best one among GA1 GA2 and GA

3 that is GA

4=

minGA1GA2GA3

In order to avoid rapidly observing a local optimum in asmall population or consume more waiting time in a largeone this study set a suitable population size as 60 (119873 =

60) in a preliminary trial It is also an important work toevaluate the fitness of selected chromosomes that each ofthe chromosomes is included or excluded from a feasiblesolution The main goal of this study is to minimize thetotal completion time Assume that 119878

119894(119905) is the 119894th string

in the 119894th generation and the total completion time of 119878119894(119905)

is sum119899

119895=1119862119895(119878119894(119905)) Then the fitness function of 119878

119894(119905) can be

represented as 119891(119878119894(119905)) Following are the calculations of the

strings in fitness function

119891 (119878119894(119905)) = max

1le119897le119873

119899

sum

119895=1

119862119895(119878119897(119905))

minus

119899

sum

119895=1

119862119895(119878119894(119905))

(12)

Moreover it is also crucial work to ensure that the probabilityof selection for a sequence with lower value of the objectivefunction is higher Thus the probability 119875(119878

119894(119905)) can be

written as follows

119875 (119878119894(119905)) =

119891 (119878119894(119905))

sum119899

119894=1119891 (119878119894(119905))

(13)

432 Operators There are a few operators that are used inthis study Following are the descriptions of those operatorscrossover mutation and selection

(a) CrossoverThis is an operator that exchanges some of thegenes of the selected parents with the main concept being

005 010 015 020 025 030000

001

001

002

002

003

Mea

n er

ror (

)

00040 00192 00108 00103 0020600171Pm

Figure 1 The performance of the genetic algorithms for various 119875119898

at (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899)

that the descendant can inherit the advantages of its parentsThis study applied the linear order crossover operator (LOX)proposed by Falkenauer and Bouffouix [48] and is one of thebetter performers among the others (Etiler et al [46]) Theprobability of crossover is set to 1

(b) Mutation The main object of mutation is to achieve foran overall optimal solution and to avoid a locally optimalone In this study the mutation rates (119875

119898) are set at 010

based on our preliminary experiment as shown in Figure 1For (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899) 100 sets of datawere randomly generated to evaluate the performance of theproposed algorithms with varying values of 119875

119898 The results

showed that the proposed algorithmshad the leastmean errorpercentage at 119875

119898= 010

(c) Selection This is a process that determines the proba-bility of each chromosome and is used to decide the betterchromosomes with the better fitness value The evolutionimplemented in our algorithm is based on the elitist list Wecopy the best offspring and use them to generate some ofthe next generation The rest of the offspring are generatedfrom the parent chromosomes by the roulette wheel selectionmethod which can maintain the variety of genes

433 Stopping Criteria In the preliminary experimentthe proposed GAs are terminated after 100 lowast 119899 genera-tions as shown in Figures 2 and 3 For (119899 119886 120579119873 119875

119898) =

(20 minus005 05 60 010) the above 100 sets of randomly gen-erated data were used to evaluate the performance of theproposed algorithms with varying values of 119892 The resultsshowed that the least mean error percentage of the proposedalgorithms would stabilize with reasonable CPU time rangeafter 119892 = 100119899

5 Computational Experiment

A computational experiment was conducted to evaluatethe efficiency of the branch-and-bound algorithm and

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

job preemption are not allowed Each job 119895 has a normalprocessing time 119901

119895and a release time 119903

119895 The general job

learning model is 119901119895119903

= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886 where 119901

119895119903is

the actual processing time of job 119895 scheduled in the 119903thposition and 119886 le 0 is a learning ratio The objective of thisproblem is to find an optimal schedule 119878

lowast that minimizesthe total completion time that is sum119899

119894=1119862119894(119878lowast) le sum

119899

119894=1119862119894(119878)

for any schedule 119878 Using the standard three-field notation ofGraham et al [38] our scheduling problem can be denotedas 1119901

119895119903= 119901119895(1 + sum

119903minus1

119897=1119901[119897])119886sum119862

4 The Branch-and-Bound andGenetic Algorithms

In this paper we will apply the branch-and-bound andgenetic algorithms search for the optimal solution and obtainnear-optimal solution respectively First in order to facilitatethe searching process and improve the branching procedurewe develop some adjacent pairwise interchange propertiesand two lower bounds to use in branch-and-bound algo-rithmThen the procedure of the genetic algorithms is givenat last

41 Dominance Properties Before presenting the adjacentpairwise interchange properties we provide two lemmaswhich will be used in the proofs of the properties in thesequel

Lemma 1 Let119891(119909) = 1+ 119886119909(1 minus 119909)119886minus1

minus (1 + 119909)119886 then119891(119909) ge

0 for 119886 le 0 and 119909 gt 0

Lemma 2 Let 119892(120582) = (120582 minus 1) + (1 + 120582119909)119886minus 120582(1 + 119909)

119886 then119892(120582) ge 0 for 120582 ge 1 119886 le 0 and 119909 gt 0

To fathom the searching tree we develop some dom-inance properties based on a pairwise interchange of twoadjacent jobs 119869

119894and 119869

119895 Let 119878 = (120587 119869

119894 119869119895 1205871015840) and 119878

1015840=

(120587 119869119895 119869119894 1205871015840) be two sequences in which 120587 and 120587

1015840 denotepartial sequences To show that 119878 dominates 1198781015840 it suffices toshow that 119862

119894(119878) + 119862

119895(119878) le 119862

119895(1198781015840) + 119862119894(1198781015840) and 119862

119895(119878) lt 119862

119894(1198781015840)

In addition let 119905 be the completion time of the last job insubsequence 120587 with (119903 minus 1) jobs

Property 1 If 119901119894lt 119901119895and max119903

119894 119903119895 le 119905 then 119878 dominates

1198781015840

Proof Since max119903119894 119903119895 le 119905 we have

119862119894(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119895(119878) = 119905 + 119901

119894(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

119862119895(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

119862119894(1198781015840) = 119905 + 119901

119895(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

(1)

After taking the difference of (1) we have

119862119894(1198781015840) minus 119862119895(119878) = (119901

119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(2)

On substituting120582 = 119901119895119901119894 119906 = (1+sum

119903minus1

119897=1119901[119897]) and119909 = (119901

119894(1+

sum119899

119897=1119901119897)) into (2) and simplifying it we obtain

119862119894(1198781015840) minus 119862119895(119878) = 119901

119894119906119886[(120582 minus 1) + (1 + 120582119909)

119886minus 120582(1 + 119909)

119886]

(3)

By Lemma 2 with 120582 gt 1 119886 le 0 and 119909 gt 0 we have 119862119894(1198781015840) minus

119862119895(119878) gt 0Moreover after taking the difference of total completion

times (TC) between sequences 119878 and 1198781015840 we have

TC (1198781015840) minus TC (119878)

= [119862119895(1198781015840) + 119862119894(1198781015840)] minus [119862

119894(119878) + 119862

119895(119878)]

= 2 (119901119895minus 119901119894)(1 +

119903minus1

sum

119897=1

119901[119897])

119886

+ 119901119894(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119895)

119886

minus 119901119895(1 +

119903minus1

sum

119897=1

119901[119897]

+ 119901119894)

119886

(4)

By (3) it can be easily shown that (4) is nonnegative for 119901119894lt

119901119895 Therefore 119878 dominates 1198781015840

The proofs of Properties 2 to 5 are omitted since they aresimilar to that of Property 1

Property 2 If 119903119894le 119905 le 119903

119895le 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

and 119901119894lt 119901119895

then 119878 dominates 1198781015840

Property 3 If 119905 ge 119903119894and 119905 + 119901

119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

Property 4 If 119905 le 119903119894le 119903119895 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

ge 119903119895 and

119901119894lt 119901119895 then 119878 dominates 1198781015840

Property 5 If 119905 le 119903119894and 119903119894+ 119901119894(1 + sum

119903minus1

119897=1119901[119897])119886

lt 119903119895 then 119878

dominates 1198781015840

4 Mathematical Problems in Engineering

In order to further determine the ordering of the remain-ing unscheduled jobs to further speed up the searchingprocess we provide the following property Assume that 119878 =

(120587 120587119888) is a sequence of jobs where 120587 is the scheduled part

containing 119896 jobs and 120587119888 is the unscheduled part Let 119878

1=

(120587 1205871015840) be the sequence in which the unscheduled jobs are

arranged in a nondecreasing order of job processing timesthat is 119901

(119896+1)le 119901(119896+2)

le sdot sdot sdot le 119901(119899)

Property 6 If 119862[119896](1198781) gt max

119895isin120587119888119903119895 then 119878

1= (120587 120587

1015840)

dominates sequences of the type (120587 120587119888) for any unscheduledsequence 120587119888

Proof Since 119862[119896](1198781) gt max

119895isin120587119888119903119895 it implies that all the

unscheduled jobs are ready to be processed on time 119862[119896](1198781)

To obtain the optimal subsequence let 1198781= (120587 120587

1015840) be the

sequence in which the unscheduled jobs are arranged innondecreasing order of jobs processing times

42 Lower Bounds In this subsection we develop two lowerbounds by using the following lemma from Hardy et al [39]

Lemma 3 Suppose that 119886119894and 119887

119894are two sequences of

numbers The sum sum119899

119894=1119886119894119887119894of products of the corresponding

elements is the least if the sequences are monotonic in theopposite sense

First let 119875119878 be a partial schedule in which the order ofthe first 119896 jobs has been determined and let 119878 be a completeschedule obtained from 119875119878 By definition the completiontime for the (119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119862[119896]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(5)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119862[119896]

(119878) +

119895

sum

119894=1

119901(119896+119894)

(1 +

119896

sum

119897=1

119901[119897]

+

119894minus1

sum

119897=1

119901(119896+119897)

)

119886

1 le 119895 le 119899 minus 119896

(6)

The first term on the right hand side of (6) is known anda lower bound of the total completion time for the partialsequence 119875119878 can be obtained by minimizing the secondterm Since the value of (1 + sum

119896

119897=1119901[119897]

+ sum119894minus1

119897=1119901[119896+119897]

)119886 is a

decreasing function of sum119894minus1119897=1

119901[119896+119897]

the total completion timeis minimized by sequencing the unscheduled jobs accordingto the shortest processing time (SPT) rule according toLemma 3 Consequently the first lower bound is

LB1=

119896

sum

119894=1

119862[119894](119878) +

119899minus119896

sum

119895=1

119862(119895) (7)

where 119862(119895)

= 119862[119896](119878) + sum

119895

119894=1119901(119896+119894)

(1 + sum119896

119897=1119901[119897]

+

sum119894minus1

119897=1119901(119899minus119896+119897minus1)

)119886 On the other hand this lower bound may

not be tight if the release time is long To overcome thissituation a second lower bound is established by takingaccount of the release time The completion time for the(119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119903[119896+1]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(8)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119903[119896+119895]

(119878) + 119901[119896+119895]

(1 +

119896

sum

119897=1

119901[119897]

+

119895minus1

sum

119897=1

119901[119896+119897]

)

119886

1 le 119895 le 119899 minus 119896

(9)

Note that 119862[119896+119895]

(119878) is greater than or equal to 1199031015840

(119896+119895) where

1199031015840

(119896+1)le 1199031015840

(119896+2)le sdot sdot sdot le 119903

1015840

(119899)denote the release times of

the unscheduled jobs arranged in a nondecreasing orderThesecond term on the right hand side of (9) is minimized bythe SPT rule since (1 +sum

119896

119897=1119901[119897]+sum119895minus1

119897=1119901[119896+119897]

)119886 is a decreasing

function of sum119895minus1119897=1

119901[119896+119897]

It follows that we have the followingsecond lower bound

LB2=

119896

sum

119894=1

119862[119894]

+

119899minus119896

sum

119895=1

119862(119895) (10)

where 119862(119895)

= 1199031015840

(119896+119895)(119878) +119901

1015840

(119896+1)(1+sum

119896

119897=1119901[119897]+sum119895minus1

119897=11199011015840

(119899minus119896+119897minus1))119886

Note that 1199011015840(119896+119895)

and 1199031015840

(119896+119895)do not necessarily come from the

same job In order tomake the lower bound tighter we choosethe maximum value from (7) and (10) as the lower bounds of119875119878 That is

LB = max LB1 LB2 (11)

43TheProcedure of Genetic Algorithms Agenetic algorithm(GA) is an optimization method that mimics natural pro-cesses GAwas invented by Holland [40] and themost widelyused to solve numerical optimization problems in a widevariety of application fields including biology economicsengineering business agriculture telecommunications andmanufacturing For example in Goldberg [41] authors usingGA in engineering design problems is reviewed in Gen andCheng [42] Soolaki et al [43] use a GA to solve an airlineboarding problem with linear programming models [44 45]and use genetic algorithms to optimize the parameters for thegiven test collections GAs start evolving by generating aninitial population of chromosomes Then a fitness functionis used to compute the relative fitness of each chromosomeof the population The selection crossover and mutationoperators are used in succession to create a new population

Mathematical Problems in Engineering 5

of chromosomes for the next generation This approach hasgained increasing popularity in solving many combinatorialoptimization problems in a wide variety of different disci-plines

431 Initial Settings In a GA every problem is presented bya code and each code is seen as a geneThe existing genes canbe combined and seen as a chromosome each of which is oneof the feasible solutions to a problem However traditionalrepresentation of GA does not work for scheduling problems(Etiler et al [46]) In dealing with this condition this studyadopts the same method that a structure can describe thejobs as a sequence in the problem To specify our approachseveral initial sequences are adopted In GA

1 jobs are placed

according to the shortest processing times (SPT) first ruleIn GA

2 jobs are arranged in earliest ready times (ERT) first

rule In GA3 jobs are arranged in a nondecreasing order on

the sum of job processing times and ready times Note thatbefore performing GA NEH algorithm (Nawaz et al [47]) isutilized to improve the quality of the solutions obtained fromthe previous rules to reduce many idle periods The processof GA

1 GA2 and GA

3are different initial sequences and use

the same selection crossover mutation operators populationsize and generations to obtain near-optimal solution Inaddition the fourth genetic algorithm denoted as GA

4 is

the best one among GA1 GA2 and GA

3 that is GA

4=

minGA1GA2GA3

In order to avoid rapidly observing a local optimum in asmall population or consume more waiting time in a largeone this study set a suitable population size as 60 (119873 =

60) in a preliminary trial It is also an important work toevaluate the fitness of selected chromosomes that each ofthe chromosomes is included or excluded from a feasiblesolution The main goal of this study is to minimize thetotal completion time Assume that 119878

119894(119905) is the 119894th string

in the 119894th generation and the total completion time of 119878119894(119905)

is sum119899

119895=1119862119895(119878119894(119905)) Then the fitness function of 119878

119894(119905) can be

represented as 119891(119878119894(119905)) Following are the calculations of the

strings in fitness function

119891 (119878119894(119905)) = max

1le119897le119873

119899

sum

119895=1

119862119895(119878119897(119905))

minus

119899

sum

119895=1

119862119895(119878119894(119905))

(12)

Moreover it is also crucial work to ensure that the probabilityof selection for a sequence with lower value of the objectivefunction is higher Thus the probability 119875(119878

119894(119905)) can be

written as follows

119875 (119878119894(119905)) =

119891 (119878119894(119905))

sum119899

119894=1119891 (119878119894(119905))

(13)

432 Operators There are a few operators that are used inthis study Following are the descriptions of those operatorscrossover mutation and selection

(a) CrossoverThis is an operator that exchanges some of thegenes of the selected parents with the main concept being

005 010 015 020 025 030000

001

001

002

002

003

Mea

n er

ror (

)

00040 00192 00108 00103 0020600171Pm

Figure 1 The performance of the genetic algorithms for various 119875119898

at (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899)

that the descendant can inherit the advantages of its parentsThis study applied the linear order crossover operator (LOX)proposed by Falkenauer and Bouffouix [48] and is one of thebetter performers among the others (Etiler et al [46]) Theprobability of crossover is set to 1

(b) Mutation The main object of mutation is to achieve foran overall optimal solution and to avoid a locally optimalone In this study the mutation rates (119875

119898) are set at 010

based on our preliminary experiment as shown in Figure 1For (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899) 100 sets of datawere randomly generated to evaluate the performance of theproposed algorithms with varying values of 119875

119898 The results

showed that the proposed algorithmshad the leastmean errorpercentage at 119875

119898= 010

(c) Selection This is a process that determines the proba-bility of each chromosome and is used to decide the betterchromosomes with the better fitness value The evolutionimplemented in our algorithm is based on the elitist list Wecopy the best offspring and use them to generate some ofthe next generation The rest of the offspring are generatedfrom the parent chromosomes by the roulette wheel selectionmethod which can maintain the variety of genes

433 Stopping Criteria In the preliminary experimentthe proposed GAs are terminated after 100 lowast 119899 genera-tions as shown in Figures 2 and 3 For (119899 119886 120579119873 119875

119898) =

(20 minus005 05 60 010) the above 100 sets of randomly gen-erated data were used to evaluate the performance of theproposed algorithms with varying values of 119892 The resultsshowed that the least mean error percentage of the proposedalgorithms would stabilize with reasonable CPU time rangeafter 119892 = 100119899

5 Computational Experiment

A computational experiment was conducted to evaluatethe efficiency of the branch-and-bound algorithm and

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

In order to further determine the ordering of the remain-ing unscheduled jobs to further speed up the searchingprocess we provide the following property Assume that 119878 =

(120587 120587119888) is a sequence of jobs where 120587 is the scheduled part

containing 119896 jobs and 120587119888 is the unscheduled part Let 119878

1=

(120587 1205871015840) be the sequence in which the unscheduled jobs are

arranged in a nondecreasing order of job processing timesthat is 119901

(119896+1)le 119901(119896+2)

le sdot sdot sdot le 119901(119899)

Property 6 If 119862[119896](1198781) gt max

119895isin120587119888119903119895 then 119878

1= (120587 120587

1015840)

dominates sequences of the type (120587 120587119888) for any unscheduledsequence 120587119888

Proof Since 119862[119896](1198781) gt max

119895isin120587119888119903119895 it implies that all the

unscheduled jobs are ready to be processed on time 119862[119896](1198781)

To obtain the optimal subsequence let 1198781= (120587 120587

1015840) be the

sequence in which the unscheduled jobs are arranged innondecreasing order of jobs processing times

42 Lower Bounds In this subsection we develop two lowerbounds by using the following lemma from Hardy et al [39]

Lemma 3 Suppose that 119886119894and 119887

119894are two sequences of

numbers The sum sum119899

119894=1119886119894119887119894of products of the corresponding

elements is the least if the sequences are monotonic in theopposite sense

First let 119875119878 be a partial schedule in which the order ofthe first 119896 jobs has been determined and let 119878 be a completeschedule obtained from 119875119878 By definition the completiontime for the (119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119862[119896]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(5)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119862[119896]

(119878) +

119895

sum

119894=1

119901(119896+119894)

(1 +

119896

sum

119897=1

119901[119897]

+

119894minus1

sum

119897=1

119901(119896+119897)

)

119886

1 le 119895 le 119899 minus 119896

(6)

The first term on the right hand side of (6) is known anda lower bound of the total completion time for the partialsequence 119875119878 can be obtained by minimizing the secondterm Since the value of (1 + sum

119896

119897=1119901[119897]

+ sum119894minus1

119897=1119901[119896+119897]

)119886 is a

decreasing function of sum119894minus1119897=1

119901[119896+119897]

the total completion timeis minimized by sequencing the unscheduled jobs accordingto the shortest processing time (SPT) rule according toLemma 3 Consequently the first lower bound is

LB1=

119896

sum

119894=1

119862[119894](119878) +

119899minus119896

sum

119895=1

119862(119895) (7)

where 119862(119895)

= 119862[119896](119878) + sum

119895

119894=1119901(119896+119894)

(1 + sum119896

119897=1119901[119897]

+

sum119894minus1

119897=1119901(119899minus119896+119897minus1)

)119886 On the other hand this lower bound may

not be tight if the release time is long To overcome thissituation a second lower bound is established by takingaccount of the release time The completion time for the(119896 + 1)th job is

119862[119896+1]

(119878) = max 119862[119896]

(119878) 119903[119896+1]

+ 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

ge 119903[119896+1]

(119878) + 119901[119896+1]

(1 +

119896

sum

119897=1

119901[119897])

119886

(8)

Similarly the completion time for the (119896 + 119895)th job is

119862[119896+119895]

(119878) ge 119903[119896+119895]

(119878) + 119901[119896+119895]

(1 +

119896

sum

119897=1

119901[119897]

+

119895minus1

sum

119897=1

119901[119896+119897]

)

119886

1 le 119895 le 119899 minus 119896

(9)

Note that 119862[119896+119895]

(119878) is greater than or equal to 1199031015840

(119896+119895) where

1199031015840

(119896+1)le 1199031015840

(119896+2)le sdot sdot sdot le 119903

1015840

(119899)denote the release times of

the unscheduled jobs arranged in a nondecreasing orderThesecond term on the right hand side of (9) is minimized bythe SPT rule since (1 +sum

119896

119897=1119901[119897]+sum119895minus1

119897=1119901[119896+119897]

)119886 is a decreasing

function of sum119895minus1119897=1

119901[119896+119897]

It follows that we have the followingsecond lower bound

LB2=

119896

sum

119894=1

119862[119894]

+

119899minus119896

sum

119895=1

119862(119895) (10)

where 119862(119895)

= 1199031015840

(119896+119895)(119878) +119901

1015840

(119896+1)(1+sum

119896

119897=1119901[119897]+sum119895minus1

119897=11199011015840

(119899minus119896+119897minus1))119886

Note that 1199011015840(119896+119895)

and 1199031015840

(119896+119895)do not necessarily come from the

same job In order tomake the lower bound tighter we choosethe maximum value from (7) and (10) as the lower bounds of119875119878 That is

LB = max LB1 LB2 (11)

43TheProcedure of Genetic Algorithms Agenetic algorithm(GA) is an optimization method that mimics natural pro-cesses GAwas invented by Holland [40] and themost widelyused to solve numerical optimization problems in a widevariety of application fields including biology economicsengineering business agriculture telecommunications andmanufacturing For example in Goldberg [41] authors usingGA in engineering design problems is reviewed in Gen andCheng [42] Soolaki et al [43] use a GA to solve an airlineboarding problem with linear programming models [44 45]and use genetic algorithms to optimize the parameters for thegiven test collections GAs start evolving by generating aninitial population of chromosomes Then a fitness functionis used to compute the relative fitness of each chromosomeof the population The selection crossover and mutationoperators are used in succession to create a new population

Mathematical Problems in Engineering 5

of chromosomes for the next generation This approach hasgained increasing popularity in solving many combinatorialoptimization problems in a wide variety of different disci-plines

431 Initial Settings In a GA every problem is presented bya code and each code is seen as a geneThe existing genes canbe combined and seen as a chromosome each of which is oneof the feasible solutions to a problem However traditionalrepresentation of GA does not work for scheduling problems(Etiler et al [46]) In dealing with this condition this studyadopts the same method that a structure can describe thejobs as a sequence in the problem To specify our approachseveral initial sequences are adopted In GA

1 jobs are placed

according to the shortest processing times (SPT) first ruleIn GA

2 jobs are arranged in earliest ready times (ERT) first

rule In GA3 jobs are arranged in a nondecreasing order on

the sum of job processing times and ready times Note thatbefore performing GA NEH algorithm (Nawaz et al [47]) isutilized to improve the quality of the solutions obtained fromthe previous rules to reduce many idle periods The processof GA

1 GA2 and GA

3are different initial sequences and use

the same selection crossover mutation operators populationsize and generations to obtain near-optimal solution Inaddition the fourth genetic algorithm denoted as GA

4 is

the best one among GA1 GA2 and GA

3 that is GA

4=

minGA1GA2GA3

In order to avoid rapidly observing a local optimum in asmall population or consume more waiting time in a largeone this study set a suitable population size as 60 (119873 =

60) in a preliminary trial It is also an important work toevaluate the fitness of selected chromosomes that each ofthe chromosomes is included or excluded from a feasiblesolution The main goal of this study is to minimize thetotal completion time Assume that 119878

119894(119905) is the 119894th string

in the 119894th generation and the total completion time of 119878119894(119905)

is sum119899

119895=1119862119895(119878119894(119905)) Then the fitness function of 119878

119894(119905) can be

represented as 119891(119878119894(119905)) Following are the calculations of the

strings in fitness function

119891 (119878119894(119905)) = max

1le119897le119873

119899

sum

119895=1

119862119895(119878119897(119905))

minus

119899

sum

119895=1

119862119895(119878119894(119905))

(12)

Moreover it is also crucial work to ensure that the probabilityof selection for a sequence with lower value of the objectivefunction is higher Thus the probability 119875(119878

119894(119905)) can be

written as follows

119875 (119878119894(119905)) =

119891 (119878119894(119905))

sum119899

119894=1119891 (119878119894(119905))

(13)

432 Operators There are a few operators that are used inthis study Following are the descriptions of those operatorscrossover mutation and selection

(a) CrossoverThis is an operator that exchanges some of thegenes of the selected parents with the main concept being

005 010 015 020 025 030000

001

001

002

002

003

Mea

n er

ror (

)

00040 00192 00108 00103 0020600171Pm

Figure 1 The performance of the genetic algorithms for various 119875119898

at (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899)

that the descendant can inherit the advantages of its parentsThis study applied the linear order crossover operator (LOX)proposed by Falkenauer and Bouffouix [48] and is one of thebetter performers among the others (Etiler et al [46]) Theprobability of crossover is set to 1

(b) Mutation The main object of mutation is to achieve foran overall optimal solution and to avoid a locally optimalone In this study the mutation rates (119875

119898) are set at 010

based on our preliminary experiment as shown in Figure 1For (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899) 100 sets of datawere randomly generated to evaluate the performance of theproposed algorithms with varying values of 119875

119898 The results

showed that the proposed algorithmshad the leastmean errorpercentage at 119875

119898= 010

(c) Selection This is a process that determines the proba-bility of each chromosome and is used to decide the betterchromosomes with the better fitness value The evolutionimplemented in our algorithm is based on the elitist list Wecopy the best offspring and use them to generate some ofthe next generation The rest of the offspring are generatedfrom the parent chromosomes by the roulette wheel selectionmethod which can maintain the variety of genes

433 Stopping Criteria In the preliminary experimentthe proposed GAs are terminated after 100 lowast 119899 genera-tions as shown in Figures 2 and 3 For (119899 119886 120579119873 119875

119898) =

(20 minus005 05 60 010) the above 100 sets of randomly gen-erated data were used to evaluate the performance of theproposed algorithms with varying values of 119892 The resultsshowed that the least mean error percentage of the proposedalgorithms would stabilize with reasonable CPU time rangeafter 119892 = 100119899

5 Computational Experiment

A computational experiment was conducted to evaluatethe efficiency of the branch-and-bound algorithm and

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

of chromosomes for the next generation This approach hasgained increasing popularity in solving many combinatorialoptimization problems in a wide variety of different disci-plines

431 Initial Settings In a GA every problem is presented bya code and each code is seen as a geneThe existing genes canbe combined and seen as a chromosome each of which is oneof the feasible solutions to a problem However traditionalrepresentation of GA does not work for scheduling problems(Etiler et al [46]) In dealing with this condition this studyadopts the same method that a structure can describe thejobs as a sequence in the problem To specify our approachseveral initial sequences are adopted In GA

1 jobs are placed

according to the shortest processing times (SPT) first ruleIn GA

2 jobs are arranged in earliest ready times (ERT) first

rule In GA3 jobs are arranged in a nondecreasing order on

the sum of job processing times and ready times Note thatbefore performing GA NEH algorithm (Nawaz et al [47]) isutilized to improve the quality of the solutions obtained fromthe previous rules to reduce many idle periods The processof GA

1 GA2 and GA

3are different initial sequences and use

the same selection crossover mutation operators populationsize and generations to obtain near-optimal solution Inaddition the fourth genetic algorithm denoted as GA

4 is

the best one among GA1 GA2 and GA

3 that is GA

4=

minGA1GA2GA3

In order to avoid rapidly observing a local optimum in asmall population or consume more waiting time in a largeone this study set a suitable population size as 60 (119873 =

60) in a preliminary trial It is also an important work toevaluate the fitness of selected chromosomes that each ofthe chromosomes is included or excluded from a feasiblesolution The main goal of this study is to minimize thetotal completion time Assume that 119878

119894(119905) is the 119894th string

in the 119894th generation and the total completion time of 119878119894(119905)

is sum119899

119895=1119862119895(119878119894(119905)) Then the fitness function of 119878

119894(119905) can be

represented as 119891(119878119894(119905)) Following are the calculations of the

strings in fitness function

119891 (119878119894(119905)) = max

1le119897le119873

119899

sum

119895=1

119862119895(119878119897(119905))

minus

119899

sum

119895=1

119862119895(119878119894(119905))

(12)

Moreover it is also crucial work to ensure that the probabilityof selection for a sequence with lower value of the objectivefunction is higher Thus the probability 119875(119878

119894(119905)) can be

written as follows

119875 (119878119894(119905)) =

119891 (119878119894(119905))

sum119899

119894=1119891 (119878119894(119905))

(13)

432 Operators There are a few operators that are used inthis study Following are the descriptions of those operatorscrossover mutation and selection

(a) CrossoverThis is an operator that exchanges some of thegenes of the selected parents with the main concept being

005 010 015 020 025 030000

001

001

002

002

003

Mea

n er

ror (

)

00040 00192 00108 00103 0020600171Pm

Figure 1 The performance of the genetic algorithms for various 119875119898

at (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899)

that the descendant can inherit the advantages of its parentsThis study applied the linear order crossover operator (LOX)proposed by Falkenauer and Bouffouix [48] and is one of thebetter performers among the others (Etiler et al [46]) Theprobability of crossover is set to 1

(b) Mutation The main object of mutation is to achieve foran overall optimal solution and to avoid a locally optimalone In this study the mutation rates (119875

119898) are set at 010

based on our preliminary experiment as shown in Figure 1For (119899 119886 120579119873 119892) = (20 minus005 05 60 100119899) 100 sets of datawere randomly generated to evaluate the performance of theproposed algorithms with varying values of 119875

119898 The results

showed that the proposed algorithmshad the leastmean errorpercentage at 119875

119898= 010

(c) Selection This is a process that determines the proba-bility of each chromosome and is used to decide the betterchromosomes with the better fitness value The evolutionimplemented in our algorithm is based on the elitist list Wecopy the best offspring and use them to generate some ofthe next generation The rest of the offspring are generatedfrom the parent chromosomes by the roulette wheel selectionmethod which can maintain the variety of genes

433 Stopping Criteria In the preliminary experimentthe proposed GAs are terminated after 100 lowast 119899 genera-tions as shown in Figures 2 and 3 For (119899 119886 120579119873 119875

119898) =

(20 minus005 05 60 010) the above 100 sets of randomly gen-erated data were used to evaluate the performance of theproposed algorithms with varying values of 119892 The resultsshowed that the least mean error percentage of the proposedalgorithms would stabilize with reasonable CPU time rangeafter 119892 = 100119899

5 Computational Experiment

A computational experiment was conducted to evaluatethe efficiency of the branch-and-bound algorithm and

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

00738 00510 00315 00258 00040

000

001

002

003

004

005

006

007

008

Mea

n er

ror (

)

40n 60n 80n 100n20n

g

Figure 2 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

01666 03260 04981 06800 08382

000

010

020

030

040

050

060

070

080

090

CPU

tim

es (s

)

g

40n 60n 80n 100n20n

Figure 3 The performance of the genetic algorithms for various 119892at (119899 119886 120579119873 119875

119898) = (20 minus005 05 60 010)

the accuracies of the genetic algorithmsThe algorithms werecoded in Fortran and run on Compaq Visual Fortran version66 on an Intel(R)Core(TM)2QuadCPU266GHzwith 4GBRAM on Windows Vista The experimental design followedReeves [49] design The job processing times were generatedfrom a uniform distribution over the integers between 1 and20 in every case while the release times were generated froma uniform distribution over the integers on (0 20119899120579) where 119899is the number of jobs Five different sets of problem instanceswere generated by giving 120579 the values 1119899 025 05 075and 1

For the branch-and-bound algorithm the average and themaximum numbers of nodes as well as the average and themaximum execution times (in seconds) were recorded Forthe three genetic algorithms the mean and the maximum

0

200

400

600

800

1000

1200

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005

minus010

minus015

minus020

120579

1n

Figure 4Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 12

error percentages were recorded where the error percentagewas calculated as

(GA119894minus TClowast)

TClowastlowast 100 (14)

where GA119894(119894 = 1 2 3 4) is the total completion time

obtained from the genetic algorithmand TClowast is the totalcompletion time of the optimal schedule The computationaltimes of the heuristic algorithmswere not recorded since theywere finished within a second

In the computational experiment four different numbersof jobs (119899 = 12 16 20 and 24) four different values of learn-ing effect (119886 = minus005 minus010 minus015 andminus020) and five differ-ent values of generation parameter of release times (120579 = 1119899025 05 075 and 1) were tested in the branch-and-boundalgorithmAs a consequence 80 experimental situationswereexamined A set of 20 instances were randomly generatedfor each situation and a total of 1600 problems were testedThe algorithms were set to skip to the next set of data if thenumber of nodes exceeded 108 The results are presented inTable 1 and Figures 4 5 6 and 7 Figures 4ndash7 showed theaverage number of nodes for various 120579 and 119886 at job size 1216 20 and 24 respectively The average number of nodesdecreased as the value of 120579 increased when 119899 was greaterthan 16 This was the direct result of the efficiency of LB

1

and LB2 As 120579 increased the frequency of applications of LB

2

would increase Consequently it would yield longer releasetimes in those cases and the properties were more powerfulMoreover LB

1is more efficient than LB

2 Table 1 and Figures

4ndash7 also showed whether in job size the algorithms had theleast mean number of nodes at 120579 = 1119899 It was due to thefact that with 120579 = 1119899 the release time was relatively shortand the completion time would readily exceed the release

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

10000

20000

30000

40000

50000

60000

025 05 075 1

Aver

age n

umbe

r of n

odes

minus005minus010

minus015

minus020

120579

1n

Figure 5The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 16

time In those cases Property 6 was applied more frequentlyconversely the completion time would not easily exceed therelease time when the values of 120579 increased Moreover thenumber of nodes increased exponentially as the number ofjobs increased which was typical of an NP-hard problem Asillustrated in Table 1 when 119899 = 24 there were five cases inwhich the branch-and-bound algorithm could solve all theproblems optimally larger than 10

8 nodes The branch-and-bound algorithm had the worst performance when (119899 119886 120579) =

(24 minus005 025)with 87times107 nodes and 5234 secondsWith

120579 fixed at 1119899 the decrease of the completion time would berelatively small at the beginning when the learning effectwassmall (eg 119886 = minus005) In other words the completion timewould easily exceed the release time which would expeditethe timing of invoking Property 6 and consequently theaverage number of nodes would be smaller With 120579 = 025 as119899 increased the corresponding least average number of nodeswould occur at greater values of learning effect

The performance of the proposed GA algorithms out ofthe 80 evaluations and a total of 1600 problems was testedThe number of times that each of the objective functions ofthe GA

1 GA2 and GA

3algorithms had the smallest mean

error percentage was 45 41 and 49 respectively In additionin Table 1 and Figures 8 9 and 10 their performanceswere not affected with the learning rate the generationparameter 120579 of release times or the number of jobs Noneof the three genetic algorithms had absolutely dominantperformance in terms of mean error percentage Howeverthe combined algorithm GA

4strikingly outperformed each

of the three algorithms in terms of the maximummean error

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

5000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 6Theperformance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 20

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

40000000

45000000

50000000

025 05 075 1

Aver

age n

umbe

r of n

odes

120579

1n

minus005minus010

minus015

minus020

Figure 7The performance of the branch-and-bound algorithms forvarious 120579 and 119886 at 119899 = 24

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table1Th

eperform

ance

oftheb

ranch-

and-bo

undandgenetic

algorithm

s

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

12

minus005

111989942

385

00031

00156

2000858

17151

000

00000

00004

0108013

000

00000

00025

549

1781

00187

00624

2001055

21091

00142

02839

000

00000

00000

00000

00050

473

1189

00140

00312

20000

00000

00000

00000

00000

00000

00000

00000

00075

414

1013

00125

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0569

2427

00164

00624

20000

00000

00000

00000

00000

00000

00000

00000

00

minus010

111989962

488

00023

00156

2001646

20568

01617

20568

000

00000

00000

00000

00025

1047

4318

00312

01092

20000

4400878

000

4400878

00283

05665

000

00000

00050

480

1321

00140

004

6820

000

00000

00006

8413

676

00019

00373

000

00000

00075

786

5291

00203

01404

20000

00000

00000

00000

00000

00000

00000

00000

0010

0396

1382

00125

004

6820

000

00000

00000

00000

00000

00000

00000

00000

00

minus015

111989921

7100016

00156

2000206

04130

00000

00000

00360

04130

00000

00000

025

817

4363

00226

00936

20000

00000

00000

00000

0001228

24567

000

00000

00050

493

1969

00148

00624

20000

00000

00000

00000

00000

00000

00000

00000

00075

286

803

000

9400312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0417

1259

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

111989947

336

00023

00156

2001295

25896

01295

25896

000

03000

69000

00000

00025

1091

6689

00289

01560

2000106

02119

000

00000

00000

00000

00000

00000

00050

337

948

00109

00312

2000314

06284

000

00000

00000

00000

00000

00000

00075

452

1121

00117

00312

20000

00000

00000

00000

00000

00000

00000

00000

0010

0319

1387

00086

00312

20000

00000

00000

00000

00000

00000

00000

00000

00

16

minus005

111989974

365

00078

00312

2002066

34518

02202

27743

01114

12163

006

0812

163

025

39325

105847

20108

58188

2004522

53071

01479

17548

00621

04629

00309

03126

050

18779

107488

08697

44616

2000052

01035

00935

16983

00849

16983

000

00000

00075

22967

290574

09259

102025

20000

00000

00000

00000

00000

00000

00000

00000

0010

05144

34260

02535

15288

20000

4100811

00115

02306

000

00000

00000

00000

00

minus010

111989991

323

00086

00312

2001813

34958

02368

27957

01906

19030

000

00000

00025

53237

291654

25826

127297

2003151

25393

02824

21606

01918

21606

01271

21606

050

15022

193968

06778

81277

2000616

10305

00154

03088

00154

03088

00000

00000

075

15144

197196

06474

80029

20000

00000

00000

00000

00000

0600120

000

00000

0010

03403

9875

01794

04992

20000

00000

00000

00000

00000

00000

00000

00000

00

minus015

1119899118

1368

00101

01092

2002462

45183

00431

06619

00103

02068

000

00000

00025

52185

4846

6222994

190165

2000762

08319

03212

37507

00969

10191

00418

08319

050

17808

101503

08221

46020

2000740

07668

00356

06773

00574

06773

00339

06773

075

3763

16453

01997

08112

20000

00000

00000

00000

00000

6701343

000

00000

0010

03882

14411

01911

07020

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899200

828

00172

00624

2001832

26781

03397

1766

003644

22420

000

00000

00025

25881

149925

11224

53664

2000140

02772

00299

04614

000

9801922

000

0200034

050

11829

48284

05819

23556

20000

00000

0000072

01431

000

00000

00000

00000

00075

2687

12382

01427

05928

20000

00000

00000

00000

00000

00000

00000

00000

0010

02047

5258

01076

02340

20000

00000

00000

00000

00000

00000

00000

00000

00

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table1Con

tinued

119899a

120579

Branch-a

nd-bou

ndalgorithm

GA

1GA

2GA

3GA

4Nod

eCP

Utim

eOF

Errorp

ercentage

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

Mean

Max

20

minus005

111989982

278

00156

00624

2016

268

74028

03538

69562

00113

02265

000

00000

00025

1319111

4407360

1059941

3224697

2007803

29510

05928

32815

08700

42205

02480

15709

050

963361

9983475

618895

5687017

2000377

040

1601608

1746

402224

15708

00033

00510

075

323776

2518001

229485

1831606

2000022

004

46000

00000

0000027

004

46000

00000

0010

041188

138955

33642

1110

7220

00017

00338

00017

00338

000

00000

00000

00000

00

minus010

11198991345

14123

01716

17317

2006707

34958

04690

28787

04353

34958

00000

00000

025

2134707

16978059

1543318

12044214

2003845

23175

01848

06798

03536

21979

004

6002909

050

202434

1320363

156266

1024927

2000000

00000

000

4500786

00000

00000

00000

00000

075

67733

450297

55645

3452

2920

000

00000

00000

00000

00000

00000

00000

00000

0010

030928

238389

26192

185483

2000023

004

62000

00000

0000023

004

62000

00000

00

minus015

1119899293

2224

004

2902964

2001670

13088

01845

29345

01835

29345

000

00000

00025

1539138

7544

291

1039052

4970

186

20046

8529738

01876

10582

02690

19134

00544

09353

050

235214

2737579

16906

61920840

20000

4600923

00032

006

4100073

01284

000

00000

00075

56901

281822

45716

225576

20000

00000

00000

00000

00000

00000

00000

00000

0010

017992

51529

15163

36973

20000

00000

00000

00000

00000

00000

00000

00000

00

minus020

1119899698

4825

00991

06240

2002436

15781

04239

404

2602224

15781

01927

15781

025

434800

664

668017

2750735

40004590

2001596

14118

00549

02968

02765

21869

00203

02249

050

107201

675811

80878

476895

20000

00000

00000

00000

0000058

01155

000

00000

00075

31529

163854

25990

110918

20000

00000

00000

00000

00000

00000

00000

00000

0010

015118

63845

13198

49453

20000

00000

0000016

00324

00016

00318

000

00000

00

24

minus005

1119899478

2082

01248

04836

2004707

39003

02017

20990

01368

18118

00959

18118

025

43205517

875044

4452338760

1060

43535

1304991

21418

06032

19032

046

0720265

02283

13231

050

11780220

66191783

13054869

65948027

1900758

10456

00313

02363

01334

10456

00161

02363

075

1216867

4261994

1562529

4708125

2000056

01126

00198

03359

00174

01871

000

00000

0010

0888475

5439067

1144057

69244

1720

000

00000

00000

00000

00000

00000

00000

00000

00

minus010

11198991074

12050

02496

26364

2007378

42545

03868

24023

02577

18340

00541

06057

025

24953166

5300

6630

28547095

6344

0156

1104324

22382

05843

29043

06883

25478

03452

22382

050

3116322

31612798

3661781

38159717

2000137

0118

400087

00883

00075

01028

000

00000

00075

513685

2435858

672926

2964331

20000

00000

0000015

00306

00112

01371

000

00000

0010

0270431

1462391

35260

91960620

20000

00000

0000052

01037

00039

00771

000

00000

00

minus015

1119899674

2613

01693

06084

2007280

78202

03115

27017

02512

34255

00284

05683

025

23417410

61951876

253876

736371360

415

09250

33429

02948

12902

02232

17855

00499

04354

050

4118014

36622393

4749373

39276216

2000114

0119

300053

01058

000

00000

00000

00000

00075

661765

5958864

874869

7920327

2000073

01458

00075

01509

000

4300867

000

00000

0010

0837932

11995859

968626

13244172

2000000

00000

00000

00000

00000

00000

00000

00000

minus020

11198993126

15554

07082

31980

2001942

32396

04290

33146

03196

32396

01669

32396

025

9333827

72575949

9993

219

80682031

1500589

02822

006

4607914

00844

07331

00204

02057

050

384242

2271989

508384

2808486

2000038

00760

00130

02596

00130

02596

00000

00000

075

570054

4053320

7017

325311990

20000

00000

00000

00000

00000

00000

00000

00000

0010

0233727

1202819

2915

7413260

0620

000

00000

00000

00000

00000

00000

00000

00000

00NoteldquoO

Frdquodeno

testhe

numbero

finstances

in20

setsof

datathatcanbe

solved

inlessthan

108no

desb

yusingtheb

ranch-

and-bo

undmetho

d

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

000

005

010

015

020

025

030

GA1 GA2 GA3 GA4GA

n16

n20n12

n24

Mea

n er

ror (

)

Figure 8 The performance of the genetic algorithms for various 119899

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

Mea

n er

ror (

)

minus005

minus010 minus020

minus015

Figure 9 The performance of the genetic algorithms for various 119886

percentage among all the 80 scenarios with varying learningrates numbers of jobs and values of 120579 The maximummean error percentages of GA

1 GA2 GA3 and GA

4were

16268 06032 08700 and 03452 respectively Themaximum mean error percentage of GA

1was more than

four times that of GA4 The combined algorithm GA

4also

clearly outperformed each of the three algorithms in terms

000

002

004

006

008

010

012

014

016

GA1 GA2 GA3 GA4GA

02505

0751

1n

Mea

n er

ror (

)

Figure 10The performance of the genetic algorithms for various 120579

of the maximum error percentage The worst cases of thecorresponding algorithms were 78202 69562 42205and 32396 respectively The worst case error of GA

1was

more than twice that of GA4Thus we would recommend the

combined algorithm GA4

6 Conclusions

In this paper we address a single-machine total completiontime problem with the sum of processing times basedlearning effect and release timesThe objective is to minimizethe total completion time The problem without learningconsideration is NP-hard one Thus we develop a branch-and-bound algorithm incorporatingwith several dominancesand two lower bounds to derive optimal solution and geneticalgorithms that was proposed to obtain near-optimal solu-tions respectively

The branch-and-bound algorithm performs well to solvethe instances of less than or equal to 24 jobs in a reasonableamount of time In the different varying parameter theproposed genetic algorithms are quite accurate for small sizeproblems and the mean error percentage of the proposedgenetic algorithms are less than 035 Another interestingtopic for future study is to consider and study the problem inthe multimachine environments or multicriteria cases

Conflict of Interests

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

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

References

[1] D Biskup ldquoSingle-machine scheduling with learning consider-ationsrdquo European Journal of Operational Research vol 115 no 1pp 173ndash178 1999

[2] R G Vickson ldquoTwo single-machine sequencing problemsinvolving controllable job processing timesrdquo American Instituteof Industrial Engineers Transactions vol 12 no 3 pp 258ndash2621980

[3] E Nowicki and S Zdrzałka ldquoA survey of results for sequencingproblems with controllable processing timesrdquo Discrete AppliedMathematics vol 26 no 2-3 pp 271ndash287 1990

[4] T C E Cheng Z L Chen and L I Chung-Lun ldquoParallel-machine scheduling with controllable processing timesrdquo IIETransactions vol 28 no 2 pp 177ndash180 1996

[5] L Monch H Balasubramanian J W Fowler and M E PfundldquoHeuristic scheduling of jobs on parallel batch machines withincompatible job families and unequal ready timesrdquo Computersand Operations Research vol 32 no 11 pp 2731ndash2750 2005

[6] W-C Lee C-C Wu and M-F Liu ldquoA single-machine bi-criterion learning scheduling problem with release timesrdquoExpert Systems with Applications vol 36 no 7 pp 10295ndash103032009

[7] W-C Lee C-CWu and P-HHsu ldquoA single-machine learningeffect scheduling problem with release timesrdquo Omega vol 38no 1-2 pp 3ndash11 2010

[8] T Eren ldquoMinimizing the total weighted completion time ona single machine scheduling with release dates and a learningeffectrdquo Applied Mathematics and Computation vol 208 no 2pp 355ndash358 2009

[9] C-C Wu and C-L Liu ldquoMinimizing the makespan on a singlemachine with learning and unequal release timesrdquo Computersand Industrial Engineering vol 59 no 3 pp 419ndash424 2010

[10] M D Toksarı ldquoA branch and bound algorithm for minimizingmakespan on a singlemachinewith unequal release times underlearning effect and deteriorating jobsrdquo Computers amp OperationsResearch vol 38 no 9 pp 1361ndash1365 2011

[11] C-C Wu P-H Hsu J-C Chen and N-S Wang ldquoGeneticalgorithm for minimizing the total weighted completion timescheduling problem with learning and release timesrdquo Comput-ers amp Operations Research vol 38 no 7 pp 1025ndash1034 2011

[12] F Ahmadizar and L Hosseini ldquoBi-criteria single machinescheduling with a time-dependent learning effect and releasetimesrdquo Applied Mathematical Modelling vol 36 no 12 pp6203ndash6214 2012

[13] R Rudek ldquoComputational complexity of the single processormakespan minimization problem with release dates and job-dependent learningrdquo Journal of the Operational Research Soci-ety 2013

[14] D-C Li and P-H Hsu ldquoCompetitive two-agent schedulingwith learning effect and release times on a single machinerdquoMathematical Problems in Engineering vol 2013 Article ID754826 9 pages 2013

[15] J-Y Kung Y-P Chao K-I Lee C-C Kang and W-CLin ldquoTwo-agent single-machine scheduling of jobs with time-dependent processing times and ready timesrdquo MathematicalProblems in Engineering vol 2013 Article ID 806325 13 pages2013

[16] Y Yin W-H Wu W-H Wu and C-C Wu ldquoA branch-and-bound algorithm for a single machine sequencing tominimize the total tardiness with arbitrary release dates and

position-dependent learning effectsrdquo Information Sciences AnInternational Journal vol 256 pp 91ndash108 2014

[17] A H G Rinnooy Kan Machine Scheduling Problems Classifi-cations Complexity and Computations Martinus Nijhoff TheHague The Netherlands 1976

[18] J Heizer and B RenderOperations Management Prentice HallEnglewood Cliffs NJ USA 5th edition 1999

[19] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 no 1ndash4 pp 273ndash290 2000

[20] D Biskup ldquoA state-of-the-art review on scheduling with learn-ing effectsrdquo European Journal of Operational Research vol 188no 2 pp 315ndash329 2008

[21] J-B Wang C T Ng T C E Cheng and L L Liu ldquoSingle-machine scheduling with a time-dependent learning effectrdquoInternational Journal of Production Economics vol 111 no 2 pp802ndash811 2008

[22] W-H Kuo and D-L Yang ldquoMinimizing the total completiontime in a single-machine scheduling problem with a time-dependent learning effectrdquo European Journal of OperationalResearch vol 174 no 2 pp 1184ndash1190 2006

[23] T C E Cheng C-C Wu and W-C Lee ldquoSome schedul-ing problems with sum-of-processing-times-based and job-position-based learning effectsrdquo Information Sciences vol 178no 11 pp 2476ndash2487 2008

[24] C Koulamas and G J Kyparisis ldquoSingle-machine and two-machine flowshop scheduling with general learning functionsrdquoEuropean Journal of Operational Research vol 178 no 2 pp402ndash407 2007

[25] C-C Wu and W-C Lee ldquoSingle-machine and flowshopschedulingwith a general learning effectmodelrdquoComputers andIndustrial Engineering vol 56 no 4 pp 1553ndash1558 2009

[26] W-C Lee and C-C Wu ldquoSome single-machine and 119898-machine flowshop scheduling problems with learning consid-erationsrdquo Information Sciences vol 179 no 22 pp 3885ndash38922009

[27] Y Yin D Xu K Sun and H Li ldquoSome scheduling problemswith general position-dependent and time-dependent learningeffectsrdquo Information Sciences vol 179 no 14 pp 2416ndash24252009

[28] A Janiak and R Rudek ldquoExperience-based approach toscheduling problems with the learning effectrdquo IEEE Transac-tions on SystemsMan and Cybernetics A vol 39 no 2 pp 344ndash357 2009

[29] S-J Yang and D-L Yang ldquoA note on single-machine groupscheduling problems with position-based learning effectrdquoApplied Mathematical Modelling vol 34 no 12 pp 4306ndash43082010

[30] Y Yin D Xu and J Wang ldquoSingle-machine schedulingwith a general sum-of-actual-processing-times-based and job-position-based learning effectrdquo Applied Mathematical Mod-elling vol 34 no 11 pp 3623ndash3630 2010

[31] C-C Wu Y Yin and S-R Cheng ldquoSome single-machinescheduling problems with a truncation learning effectrdquo Com-puters and Industrial Engineering vol 60 no 4 pp 790ndash7952011

[32] J-B Wang andM-Z Wang ldquoA revision of Machine schedulingproblems with a general learning effectrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 330ndash336 2011

[33] Y-Y Lu C-M Wei and J-B Wang ldquoSeveral single-machinescheduling problems with general learning effectsrdquo AppliedMathematical Modelling vol 36 no 11 pp 5650ndash5656 2012

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[34] J-B Wang and J-J Wang ldquoScheduling jobs with a generallearning effect modelrdquo Applied Mathematical Modelling vol 37no 4 pp 2364ndash2373 2013

[35] L Li S W Yang Y B Wu Y Huo and P Ji ldquoSingle machinescheduling jobs with a truncated sum-of-processing-times-based learning effectrdquo The International Journal of AdvancedManufacturing Technology vol 67 no 1ndash4 pp 261ndash267 2013

[36] T C E Cheng C-C Wu J-C Chen W-H Wu and S-RCheng ldquoTwo-machine flowshop scheduling with a truncatedlearning function to minimize the makespanrdquo InternationalJournal of Production Economics vol 141 no 1 pp 79ndash86 2013

[37] W-H Wu ldquoA two-agent single-machine scheduling problemwith learning and deteriorating considerationsrdquo MathematicalProblems in Engineering vol 2013 Article ID 648082 18 pages2013

[38] R L Graham E L Lawler J K Lenstra and A H GRinnooy Kan ldquoOptimization and approximation in determin-istic sequencing and scheduling a surveyrdquo Annals of DiscreteMathematics vol 5 pp 287ndash326 1979

[39] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press London UK 1967

[40] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[41] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[42] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1996

[43] M Soolaki I Mahdavi N Mahdavi-Amiri R Hassanzadehand A Aghajani ldquoA new linear programming approach andgenetic algorithm for solving airline boarding problemrdquoAppliedMathematical Modelling vol 36 no 9 pp 4060ndash4072 2012

[44] S Fujita ldquoRetrieval parameter optimization using genetic algo-rithmsrdquo Information Processing and Management vol 45 no 6pp 664ndash682 2009

[45] W Fan M D Gordon and P Pathak ldquoA generic rankingfunction discovery framework by genetic programming forinformation retrievalrdquo Information Processing andManagementvol 40 no 4 pp 587ndash602 2004

[46] O Etiler B Toklu M Atak and J Wilson ldquoA genetic algorithmfor flow shop scheduling problemsrdquo Journal of the OperationalResearch Society vol 55 no 8 pp 830ndash835 2004

[47] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOMEGA vol 11 no 1 pp 91ndash95 1983

[48] E Falkenauer and S Bouffouix ldquoA genetic algorithm for jobshoprdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 824ndash829 April 1991

[49] C Reeves ldquoHeuristics for scheduling a single machine subjectto unequal job release timesrdquo European Journal of OperationalResearch vol 80 no 2 pp 397ndash403 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of