Research ArticleSingle Machine Scheduling and Due Date Assignment withPast-Sequence-Dependent Setup Time and Position-DependentProcessing Time

Chuan-Li Zhao1 Chou-Jung Hsu2 and Hua-Feng Hsu2

1 College of Mathematics and Systems Sciences Shenyang Normal University Shenyang Liaoning 110034 China2Department of Industrial Management Nan Kai University of Technology Nantou 542 Taiwan

Correspondence should be addressed to Chou-Jung Hsu jrsheunkutedutw

Received 23 July 2014 Accepted 14 August 2014 Published 27 August 2014

Academic Editor Dehua Xu

Copyright copy 2014 Chuan-Li Zhao et alThis 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

This paper considers single machine scheduling and due date assignment with setup time The setup time is proportional to thelength of the already processed jobs that is the setup time is past-sequence-dependent (p-s-d) It is assumed that a jobrsquos processingtime depends on its position in a sequence The objective functions include total earliness the weighted number of tardy jobs andthe cost of due date assignment We analyze these problems with two different due date assignment methods We first consider themodel with job-dependent position effects For each case by converting the problem to a series of assignment problems we provedthat the problems can be solved in 119874 (119899

4) time For the model with job-independent position effects we proved that the problems

can be solved in 119874 (1198993) time by providing a dynamic programming algorithm

1 Introduction

In many realistic scheduling environments a jobrsquos processingtime may be depending on its position in the sequence [1]Two well-known special cases of this stream of research are(i) positional deterioration (aging effect) where the processingtime of a job increases as a function of its position ina processing sequence and (ii) learning effect where theprocessing time of a job decreases as a function of its positionin a processing sequence Biskup [2] andCheng andWang [3]independently introduced the learning concept to schedulingresearch Other studies include Mosheiov and Sidney [4]Mosheiov [5 6] Wu et al [7 8] and Yin et al [9 10]Biskup [11] presented an updated survey of the results onscheduling problems with the learning effect Mosheiov [6]first mentioned the aging effect in scheduling research Otherstudies include Mosheiov [12] Kuo and Yang [13] Janiak andRudek [14] Zhao and Tang [15] and Rustogi and Strusevich[16] among othersMoreover some studies consider schedul-ing problems with general position-dependent processingtime Mosheiov [17] considered a scheduling problem withgeneral position-dependent processing timeThe polynomial

algorithm is derived for makespan minimization on an m-machine proportionate flow shop Zhao et al [18] studiedscheduling and due date assignment problemThey provideda unifiedmodel for solving the single machine problems withrejection and position-dependent processing time Rustogiand Strusevich [19] presented a critical review of the knownresults for scheduling models with various positional effects

Koulamas and Kyparisis [20] first introduced a schedul-ing problem with past-sequence-dependent (p-s-d) setuptime They assumed that the job setup time is proportionalto the sum of processing time of all already scheduled jobsIt is proved that the standard single machine scheduling withp-s-d setup time can be solvable in polynomial time when theobjectives are the makespan the total completion time andthe total absolute differences in completion time respectivelyWang [21] studied the single machine scheduling problemswith time-dependent learning effect and p-s-d setup timeconsiderations He showed that the makespan minimizationproblem the total completion time minimization problemand the sum of the quadratic job completion time minimiza-tion problem can be solved in polynomial time respectivelyYin et al [22] considered a single machine scheduling model

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 620150 9 pageshttpdxdoiorg1011552014620150

2 The Scientific World Journal

with p-s-d setup time and a general learning effect Theyshowed that the single machine scheduling problems tominimize the makespan and the sum of the kth powerof completion time are polynomially solvable under theproposed model Hsu et al [23] presented a polynomial-time algorithm for an unrelated parallel machine schedulingproblem with setup time and learning effects to minimize thetotal completion time Lee [24] proposed a model with thedeteriorating jobs the learning effect and the p-s-d setuptime He provided the optimal schedules for some singlemachine problems Huang et al [25] considered some singlemachine scheduling problems with general time-dependentdeterioration position-dependent learning and p-s-d setuptimeThey proved that the makespan minimization problemthe total completion time minimization problem and thesum of the 120583th power of job completion time minimizationproblem can be solved by the SPT rule

Meeting due dates is one of themost important objectivesin scheduling (Gordon et al [26]) In some situations thetardiness penalties depend on whether the jobs are tardyrather than how late they are In these cases the number oftardy jobs should be minimized (Yin et al [27]) Kahlbacherand Cheng [28] considered scheduling problems tominimizecosts for earliness due date assignment and weighted num-ber of tardy jobsThey presented nearly a full classification forthe single and multiple machine models Shabtay and Steiner[29] studied two single machine scheduling problems Theobjectives are to minimize the sum of weighted earliness tar-diness and due date assignment penalties and minimize theweighted number of tardy jobs and due date assignment costsrespectively They proved that both problems are stronglyNP-hard and give polynomial solutions for some importantspecial cases Koulamas [30] considered the second problemof Shabtay and Steiner [29] He presented a faster algorithmfor a due date assignment problem with tardy jobs Gordonand Strusevich [31] addressed the problems of singlemachinescheduling and due date assignment problems in which ajobrsquos processing time depends on its position in a processingsequence The objective functions include the cost of thedue dates the total cost of discarded jobs that cannot becompleted by their due dates and the total earliness of thescheduled jobs They presented polynomial-time dynamicprogramming algorithms for solving problems with two duedate assignment methods provided that the processing timeof the jobs is positionally deteriorating Hsu et al [32]extended part of the objective functions proposed by Gordonand Strusevich [31] to the positional weighted earlinesspenalty and showed that the problems remain solvable inpolynomial time

2 Problem Formulation and Preliminaries

This paper studies the single machine scheduling problemswith simultaneous consideration of due date assignment p-s-d setup time and position-dependent processing time

The problem can be described as followsA set 119873 = 119869

1 1198692 119869119899 of 119899 jobs has to be scheduled on

a single machine All jobs are available for processing at time

zero and preemption is not permitted Each job 119869119895has a basic

processing time 119901119895 The actual processing time of job 119869

119895 if

scheduled in position 119903 of a sequence is given by

119901119860

119895= 119892 (119895 119903) 119901

119895 (1)

where 119892(119895 1) 119892(119895 2) 119892(119895 119899) represent an array of job-dependent positional factors

Each job 119869119895isin 119873 has to be assigned a due date 119889

119895 by which

it is desirable to complete that job Given a schedule denotethe completion time of job 119869

119895by 119862119895 Job 119869

119895is called tardy if

119862119895gt 119889119895 and it is called nontardy if 119862

119895le 119889119895 Let 119880

119895= 1 if job

119869119895is tardy and let 119880

119895= 0 if job 119869

119895is nontardy The earliness

of 119869119895is defined as 119864

119895= 119889119895minus 119862119895 provided that 119862

119895le 119889119895 In all

problems considered in this paper the jobs in set 119873 have tobe split into two subsets denoted by 119873

119864and 119873

119879 We refer to

the jobs in set 119873119864as ldquonontardyrdquo while the jobs in set 119873

119879are

termed ldquotardyrdquo A penalty 120573119895is paid for the tardy job 119869

119895isin 119873119879

Given a schedule 120587 = [119869[119894] 119869[2] 119869

[119899]] we assumed that the

p-s-d setup time of 119869[119895]

is given as Koulamas and Kyparisis[20] did as follows

119904[119895]

= 120575

119895minus1

sum

119894=1

119901119860

[119894] 119895 = 2 3 119899 119904

[1]= 0 (2)

where 120575 ge 0 is a normalizing constantThe purpose is to determine the optimal due dates and

the processing sequence such that the following function isminimized

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120593 (d) (3)

where 120587 is the sequence of jobs 120572 is the positive unit earlinesscost d is the vector of the assigned due dates and 120593(d)denotes the cost of assigning the due dates that depends ona specific rule chosen for due date assignmentWe denote theproblem as

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120593 (d) (4)

Most of the presented results hold for a general positionaleffect that is for any function 119892(119895 119903) that depends on bothposition 119903 and job 119869

119895 For each individual model there is a

particular rule that defines 119892(119895 119903) and explains how exactlythe value of 119901

119895changes for example

(i) Job-Dependent Learning Effect (Mosheiov and Sidney [4])The actual processing time of a job 119869

119895 if scheduled in position

119903 of a sequence is given by

119901119860

119895= 119901119895119903119886119895 (5)

where 119886119895le 0 is a job-dependent learning parameter (include

119886119895= 119886 as a special case ie 119901119860

119895= 119901119895119903119886 Biskup [2])

(ii) Job-Dependent Aging Effect (Zhao and Tang [15]) Theactual processing time of a job 119895 if scheduled in position 119903

of a sequence is given by

119901119860

119895= 119901119895119903119886119895 (6)

The Scientific World Journal 3

where 119886119895ge 0 is a job-dependent aging parameter (include

119886119895= 119886 as a special case ie 119901119860

119895= 119901119895119903119886 Moshieov [17])

(iii) Positional Exponential Deterioration (Wang [33]) Theactual processing time of a job 119895 if scheduled in position 119903

of a sequence is given by

119901119860

119895= 119901119895119886119903minus1 (7)

where 119886 ge 1 is a given positive constant representing a rate ofdeterioration which is common for all jobs

We study our problem with the two most frequently useddue date assignment methods

(i) The Common Due Date Assignment Method (usuallyreferred to as CON) Where all jobs are assigned the samedue date such that is 119889

119895= 119889 for 119895 = 1 2 119899 and 119889 ge 0 is a

decision variable

(ii) The Slack Due Date Assignment Method (usually referredto as SLK) Where all jobs are given an equal flow allowancethat reflects equal waiting time (ie equal slacks) such thatis 119889119895= 119901119860

119895+ 119902 for 119895 = 1 2 119899 and 119902 ge 0 is a decision

variable

We first provide some lemmas

Lemma 1 (Hardy et al [34]) Let there be two sequencesof numbers 119909

119894and 119910

119894(119894 = 1 2 119899) The sum sum

119899

119894=1119909119894119910119894

of products of the corresponding elements is the least if thesequences are monotonically ordered in the opposite sense

It is not difficult to see that the following property is validfor both the variants of our problem

Lemma 2 There exists an optimal schedule in which thefollowing properties hold (1) all the jobs are processed consec-utively without idle time and the first job starts at time 0 forboth the variants of the problem (2) all the nontardy jobs areprocessed before all the tardy jobs for both the variants of theproblem

3 The CON Due Date Assignment Method

In the CON model 119889119895= 119889 (119895 = 1 2 119899) We choose 120574119889 as

the cost function 120593(d) where 120574 is a positive constant Thusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889 (8)

The problem denotes

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ 120572 sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(9)

Kahlbacher and Cheng [28] provide an 119874(1198994) time

algorithm for the problem 1|119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 In this section we consider a generalization of the

basic model with p-s-d setup times and position-dependentprocessing times As a result of Lemma 2 we can restrict ourattention to those schedules without idle times and search forthe optimal schedule only among the schedules in which oneof the jobs is on time

Let 120587 = [119869[119894] 119869[2] 119869

[119899]] then

119862[1]

= 119892 ([1] 1) 119901[1]

119862[2]

= 119862[1]+ 119904[2]+ 119892 ([2] 2) 119901

[2]

= 119892 ([1] 1) 119901[1]+ 120575119892 ([1] 1) 119901

[1]+ 119892 ([2] 2) 119901

[2]

=

2

sum

119896=1

[1 + (2 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119895]

=

119895

sum

119896=1

[1 + (119895 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119899]

=

119899

sum

119896=1

[1 + (119899 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119899

sum

119895=1

119862[119895]

=

119899

sum

119895=1

(119899 minus 119895 + 1)(1 +120575 (119899 minus 119895)

2)119892 ([119895] 119895) 119901

[119895]

(10)

Note that we need only to consider the schedule in whichall the nontardy jobs are processed before all the tardy jobs

Lemma 3 For the problem

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(11)

if the number of jobs in119873119864is ℎ (denote |119873

119864| = ℎ) there exists

an optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that 119889 = 119862

[ℎ]

Proof Suppose 120587 = [119869[119894] 119869[2] 119869

[119899]] is an optimal schedule

Since jobs 119869[119894] 119869[2] 119869

[ℎ]are nontardy then 119889 ge 119862

[ℎ] Let

Δ = 119889 minus 119862[ℎ] Moving the due date Δ units of time to the

left such that 119889 = 119862[ℎ] the objective value will be decreasing

(120572119895 + 120574)Δ which is nonnegative This means we can find aschedule with 119889 = 119862

[ℎ]that is at least as good as 120587

As a consequence of Lemma 3 we consider the schedule120587 = [119869

[119894] 119869[2] 119869

[119899]] with 119889 = 119862

[ℎ]

4 The Scientific World Journal

Thus

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

= 120572

ℎ

sum

119895=1

119864[119895]+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119889

= 120572

ℎ

sum

119895=1

[119862[ℎ]

minus 119862[119895]] +

119899

sum

119895=ℎ+1

120573[119895]+ 120574119862[ℎ]

= (120572ℎ + 120574)(

ℎ

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895])

minus 120572

ℎ

sum

119895=1

(ℎ minus 119895 + 1) (1 +1

2120575 (ℎ minus 119895)) 119892 ([119895] 119895) 119901

[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(12)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(13)

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 From (12) we can formulate the problemwith objective (8) as the following assignment problemA1(h)which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(14)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903= 120572 (119903 minus 1) +

1

2120572120575 (ℎ minus 119903) (ℎ + 119903 minus 1)

+ [1 + 120575 (ℎ minus 119903)] 120574 119903 = 1 2 ℎ

(15)

Note that 119888119895119903is the cost of assigning job 119869

119895(119895 = 1 2 119899)

in the rth (119903 = 1 2 119899) position in the scheduleIn order to derive the optimal solution we have to solve

the above assignment problem A1(h) for any ℎ = 1 2 119899We summarize the results of the above analysis and presentthe following solution algorithm

Algorithm 4

Step 1 For ℎ = 0 (119889 = 0 all the jobs are tardy) calculate119865(0) = sum

119899

119895=1120573119895

Step 2 For ℎ from 1 to 119899 solve the assignment problem A1(h)and calculate the corresponding objective value 119865(ℎ)

Step 3 The optimal value of the function 119865 is equal tomin119865(ℎ) | ℎ = 0 1 2 119899

As a result we obtain the following theorem

Theorem 5 Problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119862119874119873|120572sum

119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

We demonstrate our approach using the following exam-ple

Example 6 Consider the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878

119901119904119889

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 ℎ = 3Theweights are120572 = 1 120574 = 2 and 120575 = 01The processing times are 119901

1= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are

119892 (119895 119903) =

[[[[[

[

2 1 3 2 4

1 3 2 2 3

2 3 1 4 3

1 2 3 1 3

2 1 2 3 4

]]]]]

]

1199081= 27 119908

2= 34 119908

3= 4

(16)

The Scientific World Journal 5

The assignment problem A1(3) is

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(17)

where

119888119895119903=

[[[[[

[

1199081119892 (1 1) 119901

11199082119892 (1 2) 119901

11199083119892 (1 3) 119901

112057311205731

1199081119892 (2 1) 119901

21199082119892 (2 2) 119901

21199083119892 (2 3) 119901

212057321205732

1199081119892 (3 1) 119901

31199082119892 (3 2) 119901

31199083119892 (3 3) 119901

312057331205733

1199081119892 (4 1) 119901

41199082119892 (4 2) 119901

41199083119892 (4 3) 119901

412057341205734

1199081119892 (5 1) 119901

51199082119892 (5 2) 119901

51199083119892 (5 3) 119901

512057351205735

]]]]]

]

=

[[[[[

[

378 238 84 10 10

162 612 48 8 8

27 51 20 4 4

54 136 24 5 5

54 34 8 6 6

]]]]]

]

(18)

The solution is

119909119895119903=

[[[[[

[

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

0 1 0 0 0

]]]]]

]

(19)

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 85

The total cost is 468

4 The SLK Due Date Assignment Method

In the SLKmodel 119889119895= 119901119860

119895+119902 (119895 = 1 2 119899)We choose 120574119902

as the cost function 120593(d) where 120574 is a positive constantThusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902 (20)

The problem denotes 1|119901119860119895

= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902

Similar to the CON model if the number of jobs in119873119864is

given we have the following solution

Lemma 7 For the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870

|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902 if |119873

119864| = ℎ there exists an

optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that the slack

time 119902 = 119862[ℎminus1]

+ 119878[ℎ]

Proof The proof is similar to that of Lemma 3Let 120587 = [119869

[1] 119869[2] 119869

[119899]] 119902 = 119862

[ℎminus1]+ 119878[ℎ] Thus

119902 =

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895 minus 1) 120575] 119892 ([119895] 119895) 119901[119895]

+ 120575

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

=

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

119864[119895]

= 119889[119895]minus 119862[119895]

= 119892 ([119895] 119895) 119901[119895]+ 119902 minus 119862

[119895]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

minus

119895

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

1 le 119895 le ℎ minus 1

(21)

Consequently

ℎminus1

sum

119895=1

119864[119895]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

(22)

Therefore

119865 (d 120587 ℎ) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

2 The Scientific World Journal

with p-s-d setup time and a general learning effect Theyshowed that the single machine scheduling problems tominimize the makespan and the sum of the kth powerof completion time are polynomially solvable under theproposed model Hsu et al [23] presented a polynomial-time algorithm for an unrelated parallel machine schedulingproblem with setup time and learning effects to minimize thetotal completion time Lee [24] proposed a model with thedeteriorating jobs the learning effect and the p-s-d setuptime He provided the optimal schedules for some singlemachine problems Huang et al [25] considered some singlemachine scheduling problems with general time-dependentdeterioration position-dependent learning and p-s-d setuptimeThey proved that the makespan minimization problemthe total completion time minimization problem and thesum of the 120583th power of job completion time minimizationproblem can be solved by the SPT rule

Meeting due dates is one of themost important objectivesin scheduling (Gordon et al [26]) In some situations thetardiness penalties depend on whether the jobs are tardyrather than how late they are In these cases the number oftardy jobs should be minimized (Yin et al [27]) Kahlbacherand Cheng [28] considered scheduling problems tominimizecosts for earliness due date assignment and weighted num-ber of tardy jobsThey presented nearly a full classification forthe single and multiple machine models Shabtay and Steiner[29] studied two single machine scheduling problems Theobjectives are to minimize the sum of weighted earliness tar-diness and due date assignment penalties and minimize theweighted number of tardy jobs and due date assignment costsrespectively They proved that both problems are stronglyNP-hard and give polynomial solutions for some importantspecial cases Koulamas [30] considered the second problemof Shabtay and Steiner [29] He presented a faster algorithmfor a due date assignment problem with tardy jobs Gordonand Strusevich [31] addressed the problems of singlemachinescheduling and due date assignment problems in which ajobrsquos processing time depends on its position in a processingsequence The objective functions include the cost of thedue dates the total cost of discarded jobs that cannot becompleted by their due dates and the total earliness of thescheduled jobs They presented polynomial-time dynamicprogramming algorithms for solving problems with two duedate assignment methods provided that the processing timeof the jobs is positionally deteriorating Hsu et al [32]extended part of the objective functions proposed by Gordonand Strusevich [31] to the positional weighted earlinesspenalty and showed that the problems remain solvable inpolynomial time

2 Problem Formulation and Preliminaries

This paper studies the single machine scheduling problemswith simultaneous consideration of due date assignment p-s-d setup time and position-dependent processing time

The problem can be described as followsA set 119873 = 119869

1 1198692 119869119899 of 119899 jobs has to be scheduled on

a single machine All jobs are available for processing at time

zero and preemption is not permitted Each job 119869119895has a basic

processing time 119901119895 The actual processing time of job 119869

119895 if

scheduled in position 119903 of a sequence is given by

119901119860

119895= 119892 (119895 119903) 119901

119895 (1)

where 119892(119895 1) 119892(119895 2) 119892(119895 119899) represent an array of job-dependent positional factors

Each job 119869119895isin 119873 has to be assigned a due date 119889

119895 by which

it is desirable to complete that job Given a schedule denotethe completion time of job 119869

119895by 119862119895 Job 119869

119895is called tardy if

119862119895gt 119889119895 and it is called nontardy if 119862

119895le 119889119895 Let 119880

119895= 1 if job

119869119895is tardy and let 119880

119895= 0 if job 119869

119895is nontardy The earliness

of 119869119895is defined as 119864

119895= 119889119895minus 119862119895 provided that 119862

119895le 119889119895 In all

problems considered in this paper the jobs in set 119873 have tobe split into two subsets denoted by 119873

119864and 119873

119879 We refer to

the jobs in set 119873119864as ldquonontardyrdquo while the jobs in set 119873

119879are

termed ldquotardyrdquo A penalty 120573119895is paid for the tardy job 119869

119895isin 119873119879

Given a schedule 120587 = [119869[119894] 119869[2] 119869

[119899]] we assumed that the

p-s-d setup time of 119869[119895]

is given as Koulamas and Kyparisis[20] did as follows

119904[119895]

= 120575

119895minus1

sum

119894=1

119901119860

[119894] 119895 = 2 3 119899 119904

[1]= 0 (2)

where 120575 ge 0 is a normalizing constantThe purpose is to determine the optimal due dates and

the processing sequence such that the following function isminimized

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120593 (d) (3)

where 120587 is the sequence of jobs 120572 is the positive unit earlinesscost d is the vector of the assigned due dates and 120593(d)denotes the cost of assigning the due dates that depends ona specific rule chosen for due date assignmentWe denote theproblem as

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120593 (d) (4)

Most of the presented results hold for a general positionaleffect that is for any function 119892(119895 119903) that depends on bothposition 119903 and job 119869

119895 For each individual model there is a

particular rule that defines 119892(119895 119903) and explains how exactlythe value of 119901

119895changes for example

(i) Job-Dependent Learning Effect (Mosheiov and Sidney [4])The actual processing time of a job 119869

119895 if scheduled in position

119903 of a sequence is given by

119901119860

119895= 119901119895119903119886119895 (5)

where 119886119895le 0 is a job-dependent learning parameter (include

119886119895= 119886 as a special case ie 119901119860

119895= 119901119895119903119886 Biskup [2])

(ii) Job-Dependent Aging Effect (Zhao and Tang [15]) Theactual processing time of a job 119895 if scheduled in position 119903

of a sequence is given by

119901119860

119895= 119901119895119903119886119895 (6)

The Scientific World Journal 3

where 119886119895ge 0 is a job-dependent aging parameter (include

119886119895= 119886 as a special case ie 119901119860

119895= 119901119895119903119886 Moshieov [17])

(iii) Positional Exponential Deterioration (Wang [33]) Theactual processing time of a job 119895 if scheduled in position 119903

of a sequence is given by

119901119860

119895= 119901119895119886119903minus1 (7)

where 119886 ge 1 is a given positive constant representing a rate ofdeterioration which is common for all jobs

We study our problem with the two most frequently useddue date assignment methods

(i) The Common Due Date Assignment Method (usuallyreferred to as CON) Where all jobs are assigned the samedue date such that is 119889

119895= 119889 for 119895 = 1 2 119899 and 119889 ge 0 is a

decision variable

(ii) The Slack Due Date Assignment Method (usually referredto as SLK) Where all jobs are given an equal flow allowancethat reflects equal waiting time (ie equal slacks) such thatis 119889119895= 119901119860

119895+ 119902 for 119895 = 1 2 119899 and 119902 ge 0 is a decision

variable

We first provide some lemmas

Lemma 1 (Hardy et al [34]) Let there be two sequencesof numbers 119909

119894and 119910

119894(119894 = 1 2 119899) The sum sum

119899

119894=1119909119894119910119894

of products of the corresponding elements is the least if thesequences are monotonically ordered in the opposite sense

It is not difficult to see that the following property is validfor both the variants of our problem

Lemma 2 There exists an optimal schedule in which thefollowing properties hold (1) all the jobs are processed consec-utively without idle time and the first job starts at time 0 forboth the variants of the problem (2) all the nontardy jobs areprocessed before all the tardy jobs for both the variants of theproblem

3 The CON Due Date Assignment Method

In the CON model 119889119895= 119889 (119895 = 1 2 119899) We choose 120574119889 as

the cost function 120593(d) where 120574 is a positive constant Thusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889 (8)

The problem denotes

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ 120572 sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(9)

Kahlbacher and Cheng [28] provide an 119874(1198994) time

algorithm for the problem 1|119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 In this section we consider a generalization of the

basic model with p-s-d setup times and position-dependentprocessing times As a result of Lemma 2 we can restrict ourattention to those schedules without idle times and search forthe optimal schedule only among the schedules in which oneof the jobs is on time

Let 120587 = [119869[119894] 119869[2] 119869

[119899]] then

119862[1]

= 119892 ([1] 1) 119901[1]

119862[2]

= 119862[1]+ 119904[2]+ 119892 ([2] 2) 119901

[2]

= 119892 ([1] 1) 119901[1]+ 120575119892 ([1] 1) 119901

[1]+ 119892 ([2] 2) 119901

[2]

=

2

sum

119896=1

[1 + (2 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119895]

=

119895

sum

119896=1

[1 + (119895 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119899]

=

119899

sum

119896=1

[1 + (119899 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119899

sum

119895=1

119862[119895]

=

119899

sum

119895=1

(119899 minus 119895 + 1)(1 +120575 (119899 minus 119895)

2)119892 ([119895] 119895) 119901

[119895]

(10)

Note that we need only to consider the schedule in whichall the nontardy jobs are processed before all the tardy jobs

Lemma 3 For the problem

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(11)

if the number of jobs in119873119864is ℎ (denote |119873

119864| = ℎ) there exists

an optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that 119889 = 119862

[ℎ]

Proof Suppose 120587 = [119869[119894] 119869[2] 119869

[119899]] is an optimal schedule

Since jobs 119869[119894] 119869[2] 119869

[ℎ]are nontardy then 119889 ge 119862

[ℎ] Let

Δ = 119889 minus 119862[ℎ] Moving the due date Δ units of time to the

left such that 119889 = 119862[ℎ] the objective value will be decreasing

(120572119895 + 120574)Δ which is nonnegative This means we can find aschedule with 119889 = 119862

[ℎ]that is at least as good as 120587

As a consequence of Lemma 3 we consider the schedule120587 = [119869

[119894] 119869[2] 119869

[119899]] with 119889 = 119862

[ℎ]

4 The Scientific World Journal

Thus

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

= 120572

ℎ

sum

119895=1

119864[119895]+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119889

= 120572

ℎ

sum

119895=1

[119862[ℎ]

minus 119862[119895]] +

119899

sum

119895=ℎ+1

120573[119895]+ 120574119862[ℎ]

= (120572ℎ + 120574)(

ℎ

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895])

minus 120572

ℎ

sum

119895=1

(ℎ minus 119895 + 1) (1 +1

2120575 (ℎ minus 119895)) 119892 ([119895] 119895) 119901

[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(12)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(13)

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 From (12) we can formulate the problemwith objective (8) as the following assignment problemA1(h)which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(14)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903= 120572 (119903 minus 1) +

1

2120572120575 (ℎ minus 119903) (ℎ + 119903 minus 1)

+ [1 + 120575 (ℎ minus 119903)] 120574 119903 = 1 2 ℎ

(15)

Note that 119888119895119903is the cost of assigning job 119869

119895(119895 = 1 2 119899)

in the rth (119903 = 1 2 119899) position in the scheduleIn order to derive the optimal solution we have to solve

the above assignment problem A1(h) for any ℎ = 1 2 119899We summarize the results of the above analysis and presentthe following solution algorithm

Algorithm 4

Step 1 For ℎ = 0 (119889 = 0 all the jobs are tardy) calculate119865(0) = sum

119899

119895=1120573119895

Step 2 For ℎ from 1 to 119899 solve the assignment problem A1(h)and calculate the corresponding objective value 119865(ℎ)

Step 3 The optimal value of the function 119865 is equal tomin119865(ℎ) | ℎ = 0 1 2 119899

As a result we obtain the following theorem

Theorem 5 Problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119862119874119873|120572sum

119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

We demonstrate our approach using the following exam-ple

Example 6 Consider the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878

119901119904119889

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 ℎ = 3Theweights are120572 = 1 120574 = 2 and 120575 = 01The processing times are 119901

1= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are

119892 (119895 119903) =

[[[[[

[

2 1 3 2 4

1 3 2 2 3

2 3 1 4 3

1 2 3 1 3

2 1 2 3 4

]]]]]

]

1199081= 27 119908

2= 34 119908

3= 4

(16)

The Scientific World Journal 5

The assignment problem A1(3) is

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(17)

where

119888119895119903=

[[[[[

[

1199081119892 (1 1) 119901

11199082119892 (1 2) 119901

11199083119892 (1 3) 119901

112057311205731

1199081119892 (2 1) 119901

21199082119892 (2 2) 119901

21199083119892 (2 3) 119901

212057321205732

1199081119892 (3 1) 119901

31199082119892 (3 2) 119901

31199083119892 (3 3) 119901

312057331205733

1199081119892 (4 1) 119901

41199082119892 (4 2) 119901

41199083119892 (4 3) 119901

412057341205734

1199081119892 (5 1) 119901

51199082119892 (5 2) 119901

51199083119892 (5 3) 119901

512057351205735

]]]]]

]

=

[[[[[

[

378 238 84 10 10

162 612 48 8 8

27 51 20 4 4

54 136 24 5 5

54 34 8 6 6

]]]]]

]

(18)

The solution is

119909119895119903=

[[[[[

[

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

0 1 0 0 0

]]]]]

]

(19)

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 85

The total cost is 468

4 The SLK Due Date Assignment Method

In the SLKmodel 119889119895= 119901119860

119895+119902 (119895 = 1 2 119899)We choose 120574119902

as the cost function 120593(d) where 120574 is a positive constantThusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902 (20)

The problem denotes 1|119901119860119895

= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902

Similar to the CON model if the number of jobs in119873119864is

given we have the following solution

Lemma 7 For the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870

|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902 if |119873

119864| = ℎ there exists an

optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that the slack

time 119902 = 119862[ℎminus1]

+ 119878[ℎ]

Proof The proof is similar to that of Lemma 3Let 120587 = [119869

[1] 119869[2] 119869

[119899]] 119902 = 119862

[ℎminus1]+ 119878[ℎ] Thus

119902 =

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895 minus 1) 120575] 119892 ([119895] 119895) 119901[119895]

+ 120575

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

=

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

119864[119895]

= 119889[119895]minus 119862[119895]

= 119892 ([119895] 119895) 119901[119895]+ 119902 minus 119862

[119895]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

minus

119895

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

1 le 119895 le ℎ minus 1

(21)

Consequently

ℎminus1

sum

119895=1

119864[119895]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

(22)

Therefore

119865 (d 120587 ℎ) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

The Scientific World Journal 3

where 119886119895ge 0 is a job-dependent aging parameter (include

119886119895= 119886 as a special case ie 119901119860

119895= 119901119895119903119886 Moshieov [17])

(iii) Positional Exponential Deterioration (Wang [33]) Theactual processing time of a job 119895 if scheduled in position 119903

of a sequence is given by

119901119860

119895= 119901119895119886119903minus1 (7)

where 119886 ge 1 is a given positive constant representing a rate ofdeterioration which is common for all jobs

We study our problem with the two most frequently useddue date assignment methods

(i) The Common Due Date Assignment Method (usuallyreferred to as CON) Where all jobs are assigned the samedue date such that is 119889

119895= 119889 for 119895 = 1 2 119899 and 119889 ge 0 is a

decision variable

(ii) The Slack Due Date Assignment Method (usually referredto as SLK) Where all jobs are given an equal flow allowancethat reflects equal waiting time (ie equal slacks) such thatis 119889119895= 119901119860

119895+ 119902 for 119895 = 1 2 119899 and 119902 ge 0 is a decision

variable

We first provide some lemmas

Lemma 1 (Hardy et al [34]) Let there be two sequencesof numbers 119909

119894and 119910

119894(119894 = 1 2 119899) The sum sum

119899

119894=1119909119894119910119894

of products of the corresponding elements is the least if thesequences are monotonically ordered in the opposite sense

It is not difficult to see that the following property is validfor both the variants of our problem

Lemma 2 There exists an optimal schedule in which thefollowing properties hold (1) all the jobs are processed consec-utively without idle time and the first job starts at time 0 forboth the variants of the problem (2) all the nontardy jobs areprocessed before all the tardy jobs for both the variants of theproblem

3 The CON Due Date Assignment Method

In the CON model 119889119895= 119889 (119895 = 1 2 119899) We choose 120574119889 as

the cost function 120593(d) where 120574 is a positive constant Thusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889 (8)

The problem denotes

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ 120572 sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(9)

Kahlbacher and Cheng [28] provide an 119874(1198994) time

algorithm for the problem 1|119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 In this section we consider a generalization of the

basic model with p-s-d setup times and position-dependentprocessing times As a result of Lemma 2 we can restrict ourattention to those schedules without idle times and search forthe optimal schedule only among the schedules in which oneof the jobs is on time

Let 120587 = [119869[119894] 119869[2] 119869

[119899]] then

119862[1]

= 119892 ([1] 1) 119901[1]

119862[2]

= 119862[1]+ 119904[2]+ 119892 ([2] 2) 119901

[2]

= 119892 ([1] 1) 119901[1]+ 120575119892 ([1] 1) 119901

[1]+ 119892 ([2] 2) 119901

[2]

=

2

sum

119896=1

[1 + (2 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119895]

=

119895

sum

119896=1

[1 + (119895 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119862[119899]

=

119899

sum

119896=1

[1 + (119899 minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

119899

sum

119895=1

119862[119895]

=

119899

sum

119895=1

(119899 minus 119895 + 1)(1 +120575 (119899 minus 119895)

2)119892 ([119895] 119895) 119901

[119895]

(10)

Note that we need only to consider the schedule in whichall the nontardy jobs are processed before all the tardy jobs

Lemma 3 For the problem

110038161003816100381610038161003816119901119860

119895= 119901119895119892 (119895 119903) 119878psd 119862119874119873

10038161003816100381610038161003816120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

(11)

if the number of jobs in119873119864is ℎ (denote |119873

119864| = ℎ) there exists

an optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that 119889 = 119862

[ℎ]

Proof Suppose 120587 = [119869[119894] 119869[2] 119869

[119899]] is an optimal schedule

Since jobs 119869[119894] 119869[2] 119869

[ℎ]are nontardy then 119889 ge 119862

[ℎ] Let

Δ = 119889 minus 119862[ℎ] Moving the due date Δ units of time to the

left such that 119889 = 119862[ℎ] the objective value will be decreasing

(120572119895 + 120574)Δ which is nonnegative This means we can find aschedule with 119889 = 119862

[ℎ]that is at least as good as 120587

As a consequence of Lemma 3 we consider the schedule120587 = [119869

[119894] 119869[2] 119869

[119899]] with 119889 = 119862

[ℎ]

4 The Scientific World Journal

Thus

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

= 120572

ℎ

sum

119895=1

119864[119895]+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119889

= 120572

ℎ

sum

119895=1

[119862[ℎ]

minus 119862[119895]] +

119899

sum

119895=ℎ+1

120573[119895]+ 120574119862[ℎ]

= (120572ℎ + 120574)(

ℎ

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895])

minus 120572

ℎ

sum

119895=1

(ℎ minus 119895 + 1) (1 +1

2120575 (ℎ minus 119895)) 119892 ([119895] 119895) 119901

[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(12)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(13)

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 From (12) we can formulate the problemwith objective (8) as the following assignment problemA1(h)which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(14)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903= 120572 (119903 minus 1) +

1

2120572120575 (ℎ minus 119903) (ℎ + 119903 minus 1)

+ [1 + 120575 (ℎ minus 119903)] 120574 119903 = 1 2 ℎ

(15)

Note that 119888119895119903is the cost of assigning job 119869

119895(119895 = 1 2 119899)

in the rth (119903 = 1 2 119899) position in the scheduleIn order to derive the optimal solution we have to solve

the above assignment problem A1(h) for any ℎ = 1 2 119899We summarize the results of the above analysis and presentthe following solution algorithm

Algorithm 4

Step 1 For ℎ = 0 (119889 = 0 all the jobs are tardy) calculate119865(0) = sum

119899

119895=1120573119895

Step 2 For ℎ from 1 to 119899 solve the assignment problem A1(h)and calculate the corresponding objective value 119865(ℎ)

Step 3 The optimal value of the function 119865 is equal tomin119865(ℎ) | ℎ = 0 1 2 119899

As a result we obtain the following theorem

Theorem 5 Problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119862119874119873|120572sum

119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

We demonstrate our approach using the following exam-ple

Example 6 Consider the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878

119901119904119889

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 ℎ = 3Theweights are120572 = 1 120574 = 2 and 120575 = 01The processing times are 119901

1= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are

119892 (119895 119903) =

[[[[[

[

2 1 3 2 4

1 3 2 2 3

2 3 1 4 3

1 2 3 1 3

2 1 2 3 4

]]]]]

]

1199081= 27 119908

2= 34 119908

3= 4

(16)

The Scientific World Journal 5

The assignment problem A1(3) is

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(17)

where

119888119895119903=

[[[[[

[

1199081119892 (1 1) 119901

11199082119892 (1 2) 119901

11199083119892 (1 3) 119901

112057311205731

1199081119892 (2 1) 119901

21199082119892 (2 2) 119901

21199083119892 (2 3) 119901

212057321205732

1199081119892 (3 1) 119901

31199082119892 (3 2) 119901

31199083119892 (3 3) 119901

312057331205733

1199081119892 (4 1) 119901

41199082119892 (4 2) 119901

41199083119892 (4 3) 119901

412057341205734

1199081119892 (5 1) 119901

51199082119892 (5 2) 119901

51199083119892 (5 3) 119901

512057351205735

]]]]]

]

=

[[[[[

[

378 238 84 10 10

162 612 48 8 8

27 51 20 4 4

54 136 24 5 5

54 34 8 6 6

]]]]]

]

(18)

The solution is

119909119895119903=

[[[[[

[

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

0 1 0 0 0

]]]]]

]

(19)

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 85

The total cost is 468

4 The SLK Due Date Assignment Method

In the SLKmodel 119889119895= 119901119860

119895+119902 (119895 = 1 2 119899)We choose 120574119902

as the cost function 120593(d) where 120574 is a positive constantThusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902 (20)

The problem denotes 1|119901119860119895

= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902

Similar to the CON model if the number of jobs in119873119864is

given we have the following solution

Lemma 7 For the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870

|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902 if |119873

119864| = ℎ there exists an

optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that the slack

time 119902 = 119862[ℎminus1]

+ 119878[ℎ]

Proof The proof is similar to that of Lemma 3Let 120587 = [119869

[1] 119869[2] 119869

[119899]] 119902 = 119862

[ℎminus1]+ 119878[ℎ] Thus

119902 =

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895 minus 1) 120575] 119892 ([119895] 119895) 119901[119895]

+ 120575

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

=

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

119864[119895]

= 119889[119895]minus 119862[119895]

= 119892 ([119895] 119895) 119901[119895]+ 119902 minus 119862

[119895]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

minus

119895

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

1 le 119895 le ℎ minus 1

(21)

Consequently

ℎminus1

sum

119895=1

119864[119895]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

(22)

Therefore

119865 (d 120587 ℎ) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

4 The Scientific World Journal

Thus

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119889

= 120572

ℎ

sum

119895=1

119864[119895]+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119889

= 120572

ℎ

sum

119895=1

[119862[ℎ]

minus 119862[119895]] +

119899

sum

119895=ℎ+1

120573[119895]+ 120574119862[ℎ]

= (120572ℎ + 120574)(

ℎ

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895])

minus 120572

ℎ

sum

119895=1

(ℎ minus 119895 + 1) (1 +1

2120575 (ℎ minus 119895)) 119892 ([119895] 119895) 119901

[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(12)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(13)

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 From (12) we can formulate the problemwith objective (8) as the following assignment problemA1(h)which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(14)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903= 120572 (119903 minus 1) +

1

2120572120575 (ℎ minus 119903) (ℎ + 119903 minus 1)

+ [1 + 120575 (ℎ minus 119903)] 120574 119903 = 1 2 ℎ

(15)

Note that 119888119895119903is the cost of assigning job 119869

119895(119895 = 1 2 119899)

in the rth (119903 = 1 2 119899) position in the scheduleIn order to derive the optimal solution we have to solve

the above assignment problem A1(h) for any ℎ = 1 2 119899We summarize the results of the above analysis and presentthe following solution algorithm

Algorithm 4

Step 1 For ℎ = 0 (119889 = 0 all the jobs are tardy) calculate119865(0) = sum

119899

119895=1120573119895

Step 2 For ℎ from 1 to 119899 solve the assignment problem A1(h)and calculate the corresponding objective value 119865(ℎ)

Step 3 The optimal value of the function 119865 is equal tomin119865(ℎ) | ℎ = 0 1 2 119899

As a result we obtain the following theorem

Theorem 5 Problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119862119874119873|120572sum

119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

We demonstrate our approach using the following exam-ple

Example 6 Consider the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878

119901119904119889

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 ℎ = 3Theweights are120572 = 1 120574 = 2 and 120575 = 01The processing times are 119901

1= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are

119892 (119895 119903) =

[[[[[

[

2 1 3 2 4

1 3 2 2 3

2 3 1 4 3

1 2 3 1 3

2 1 2 3 4

]]]]]

]

1199081= 27 119908

2= 34 119908

3= 4

(16)

The Scientific World Journal 5

The assignment problem A1(3) is

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(17)

where

119888119895119903=

[[[[[

[

1199081119892 (1 1) 119901

11199082119892 (1 2) 119901

11199083119892 (1 3) 119901

112057311205731

1199081119892 (2 1) 119901

21199082119892 (2 2) 119901

21199083119892 (2 3) 119901

212057321205732

1199081119892 (3 1) 119901

31199082119892 (3 2) 119901

31199083119892 (3 3) 119901

312057331205733

1199081119892 (4 1) 119901

41199082119892 (4 2) 119901

41199083119892 (4 3) 119901

412057341205734

1199081119892 (5 1) 119901

51199082119892 (5 2) 119901

51199083119892 (5 3) 119901

512057351205735

]]]]]

]

=

[[[[[

[

378 238 84 10 10

162 612 48 8 8

27 51 20 4 4

54 136 24 5 5

54 34 8 6 6

]]]]]

]

(18)

The solution is

119909119895119903=

[[[[[

[

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

0 1 0 0 0

]]]]]

]

(19)

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 85

The total cost is 468

4 The SLK Due Date Assignment Method

In the SLKmodel 119889119895= 119901119860

119895+119902 (119895 = 1 2 119899)We choose 120574119902

as the cost function 120593(d) where 120574 is a positive constantThusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902 (20)

The problem denotes 1|119901119860119895

= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902

Similar to the CON model if the number of jobs in119873119864is

given we have the following solution

Lemma 7 For the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870

|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902 if |119873

119864| = ℎ there exists an

optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that the slack

time 119902 = 119862[ℎminus1]

+ 119878[ℎ]

Proof The proof is similar to that of Lemma 3Let 120587 = [119869

[1] 119869[2] 119869

[119899]] 119902 = 119862

[ℎminus1]+ 119878[ℎ] Thus

119902 =

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895 minus 1) 120575] 119892 ([119895] 119895) 119901[119895]

+ 120575

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

=

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

119864[119895]

= 119889[119895]minus 119862[119895]

= 119892 ([119895] 119895) 119901[119895]+ 119902 minus 119862

[119895]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

minus

119895

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

1 le 119895 le ℎ minus 1

(21)

Consequently

ℎminus1

sum

119895=1

119864[119895]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

(22)

Therefore

119865 (d 120587 ℎ) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

The Scientific World Journal 5

The assignment problem A1(3) is

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(17)

where

119888119895119903=

[[[[[

[

1199081119892 (1 1) 119901

11199082119892 (1 2) 119901

11199083119892 (1 3) 119901

112057311205731

1199081119892 (2 1) 119901

21199082119892 (2 2) 119901

21199083119892 (2 3) 119901

212057321205732

1199081119892 (3 1) 119901

31199082119892 (3 2) 119901

31199083119892 (3 3) 119901

312057331205733

1199081119892 (4 1) 119901

41199082119892 (4 2) 119901

41199083119892 (4 3) 119901

412057341205734

1199081119892 (5 1) 119901

51199082119892 (5 2) 119901

51199083119892 (5 3) 119901

512057351205735

]]]]]

]

=

[[[[[

[

378 238 84 10 10

162 612 48 8 8

27 51 20 4 4

54 136 24 5 5

54 34 8 6 6

]]]]]

]

(18)

The solution is

119909119895119903=

[[[[[

[

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

0 1 0 0 0

]]]]]

]

(19)

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 85

The total cost is 468

4 The SLK Due Date Assignment Method

In the SLKmodel 119889119895= 119901119860

119895+119902 (119895 = 1 2 119899)We choose 120574119902

as the cost function 120593(d) where 120574 is a positive constantThusit follows from function (3) that our problem is to minimizethe objective function

119865 (d 120587) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902 (20)

The problem denotes 1|119901119860119895

= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902

Similar to the CON model if the number of jobs in119873119864is

given we have the following solution

Lemma 7 For the problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870

|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119902 if |119873

119864| = ℎ there exists an

optimal schedule 120587 = [119869[1] 119869[2] 119869

[119899]] such that the slack

time 119902 = 119862[ℎminus1]

+ 119878[ℎ]

Proof The proof is similar to that of Lemma 3Let 120587 = [119869

[1] 119869[2] 119869

[119899]] 119902 = 119862

[ℎminus1]+ 119878[ℎ] Thus

119902 =

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895 minus 1) 120575] 119892 ([119895] 119895) 119901[119895]

+ 120575

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

=

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

119864[119895]

= 119889[119895]minus 119862[119895]

= 119892 ([119895] 119895) 119901[119895]+ 119902 minus 119862

[119895]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

minus

119895

sum

119896=1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

= 119892 ([119895] 119895) 119901[119895]+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

1 le 119895 le ℎ minus 1

(21)

Consequently

ℎminus1

sum

119895=1

119864[119895]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119896=119895+1

[1 + (ℎ minus 119896) 120575] 119892 ([119896] 119896) 119901[119896]

=

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

(22)

Therefore

119865 (d 120587 ℎ) = 120572 sum

119869119895isin119873119864

119864119895+ sum

119869119895isin119873119879

120573119895119880119895+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

6 The Scientific World Journal

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574119902

= 120572

ℎminus1

sum

119895=1

119892 ([119895] 119895) 119901[119895]

+ 120572

ℎminus1

sum

119895=2

(119895 minus 1) [1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

+

119899

sum

119895=ℎ+1

120573[119895]+ 120574

ℎminus1

sum

119895=1

[1 + (ℎ minus 119895) 120575] 119892 ([119895] 119895) 119901[119895]

=

ℎ

sum

119895=1

119908119895119892 ([119895] 119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(23)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(24)

Since the objective functions for CON and SLK due dateassignment methods have the same structure we have thefollowing solution

Let 119909119895119903

be a binary variable such that 119909119895119903

= 1 if job119869119895is scheduled in the rth position and 119909

119895119903= 0 otherwise

119895 119903 = 1 2 119899 If |119873119864| = ℎ then we can formulate

the problem with objective (20) as the following assignmentproblem A2(h) which can be solved in 119874(1198993) time

Min119899

sum

119895=1

119899

sum

119903=1

119888119895119903119909119895119903

st119899

sum

119903=1

119909119895119903= 1 119895 = 1 2 119899

119899

sum

119895=1

119909119895119903= 1 119903 = 1 2 119899

119909119895119903isin 0 1 119895 = 1 2 119899 119903 = 1 2 119899

(25)

where

119888119895119903=

119908119903119892 (119895 119903) 119901

119895for 119903 = 1 2 ℎ

120573119895

for 119903 = ℎ + 1 ℎ + 2 119899

119908119903

=

120572 + 120574 (1 + (ℎ minus 119903) 120575) for 119903 = 1

120572 + [120572 (119903 minus 1) + 120574] [1 + 120575 (ℎ minus 119903)] for 119903 = 2 ℎ minus 1

0 for 119903 = ℎ

(26)

In order to derive the optimal solution we have to solvethe above assignment problem A2(h) for any ℎ = 1 2 119899

As a result we obtain the following theorem

Theorem 8 The problem 1|119901119860

119895= 119901119895119892(119895 119903) 119878psd 119878119871119870|120572

sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 can be solved in 119874(1198994) time

5 Job-Independent Position Effects Case

In this section we explore the model with job-independentposition effects that is the actual processing time of job 119869

119895

if scheduled in position 119903 of a sequence is given by 119901119860119895

=

119901119895119892(119903) where 119892(1) 119892(2) 119892(119899) represent an array of job-

independent positional factors In Section 4 we have shownthat the general version (job-dependent position effects) canbe solved in119874(1198994) time In the following we present an119874(1198993)time dynamic programming algorithm for solving the specialversion with job-independent position effects The main ideathat will be used in the development of our algorithm issimilar to that of Shabtay et al [35]

Based on the properties proved in Section 4 we have thefollowing solutions

For the CON model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎ] then

119865 (d 120587 ℎ) =ℎ

sum

119895=1

120572 (119895 minus 1) +1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(27)

where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 119895 = 1 2 ℎ

(28)

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

The Scientific World Journal 7

For the SLK model if |119873119864| = ℎ 120587 = [119869

[1] 119869[2] 119869

[119899]]

and 119889 = 119862[ℎminus1]

+ 119878[ℎ] then

119865 (d 120587 ℎ) = [120572 + 120574 (1 + (ℎ minus 119895) 120575)] 119892 (1) 119901[1]

+

ℎminus1

sum

119895=2

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)]

times 119892 (119895) 119901[119895]+

119899

sum

119895=ℎ+1

120573[119895]

=

ℎ

sum

119895=1

119908119895119892 (119895) 119901

[119895]+

119899

sum

119895=ℎ+1

120573[119895]

(29)

where

119908119895

=

120572 + 120574 (1 + (ℎ minus 119895) 120575) for 119895 = 1

120572 + [120572 (119895 minus 1) + 120574] [1 + 120575 (ℎ minus 119895)] for 119895 = 2 ℎ minus 1

0 for 119895 = ℎ

(30)

Using (27) and (29) and with any of the two previouslymentioned due date assignments methods let 119908

119895= 119908119895119892(119895)

the objective function can be formulated as 119865(d 120587) =

sum119899

119895=1119908119895119901120587(119895)

for the special case of119873119864= 119873 where no jobs are

tardy FromLemma 1 the optimal job sequence is obtained bymatching the largest 119908

119895value to the job with the smallest 119901

119895

value the second largest 119908119895value to the job with the second

smallest 119901119895value and so on The index of the 119908

119895matched

with119901119895specifies the position of job 119895 in the optimal sequence

For example first renumber the jobs in the LPT order suchthat 119901

1ge 1199012ge sdot sdot sdot ge 119901

119899 and then reorder the positional

weights such that 1199081198941le 1199081198942le sdot sdot sdot le 119908

119894119899(1198941 1198942 119894

119899is a

permutation of 1 2 119899) schedules job 119895 in the position119894119895(119895 = 1 2 119899)We now consider the due date assignment problem to

minimize the objective function (3) Since the objectivefunctions for all two due date assignment methods have thesame structure we provide a generic algorithm to solve theseproblems with two due date assignment methods If set119873

119864is

given (|119873119864| = ℎ) then we can reorder the positional weights

such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎThus an optimal job sequence

of119873119864is obtained in119874(ℎ log ℎ) timeHowever in order to find

the optimal solution for the due date assignment problem thecontribution of the total cost of the tardy jobs must be takeninto account Below we present a new dynamic programmingalgorithm For a given ℎ the idea of a dynamic programmingalgorithm to minimize the function (3) is as follows Wedefine the states of the form (119894 119903) where 119894 means that jobs1198691 1198692 119869

119894have been considered and 119903 (1 le 119903 le min119894 ℎ)

represents how many of these jobs have been sequenced asnontardy jobs A state (119894 119903) is associated with 119891(119894 119903) thesmallest value of the objective function in the class of partialschedules for processing 119894 jobs provided that 119903 of the thesejobs has been sequenced nontardy This method works byeither each job tardy or nontardy Next all 119891(119894 119903) values can

be calculated by applying the recursion for 119894 = 1 2 119899 and119903 ge max1 ℎ minus (119899 minus 119894) The condition is that 119903 ge ℎ minus (119899 minus 119894) isnecessary to ensure that we do not consider states that mightlead to a solution which has fewer than 119903 jobs in set 119873

119864

since |119873119864| = ℎ and there are 119903 jobs that have been sequenced

nontardy among the first 119894 jobs the remaining ℎminus 119903 nontardyjobs needed to be selected from the last 119899 minus 119894 jobs The formalstatement of the algorithm is below

Algorithm 9

Step 0 Renumber the jobs in the LPT order such that 1199011ge

1199012ge sdot sdot sdot ge 119901

119899

Step 1 Calculate positional weights 119908119895= 119908119895119892(119895) where

119908119895= 120572 (119895 minus 1) +

1

2120572120575 (ℎ minus 119895) (ℎ + 119895 minus 1)

+ [1 + 120575 (ℎ minus 119895)] 120574 (119895 = 1 2 ℎ)

(31)

for the CON model and 1199081= 120572 + 120574[1 + (ℎ minus 1)120575] 119908

119895= 120572 +

[120572(119895 minus 1) + 120574][1 + 120575(ℎ minus 119895)] (119895 = 2 ℎ minus 1) and 119908ℎ= 0

for the SLK model Reorder the positional weights such that1199081198941le 1199081198942le sdot sdot sdot le 119908

119894ℎ Initialize 119891(0 0) = 0 119891(119894 119903) = infin for

119903 gt 119894

Step 2 For 119894 from 1 to 119899 calculate

119891 (119894 0) = 119891 (119894 minus 1 0) + 120573119894 (32)

119891 (119894 119903) = min 119891 (119894 minus 1 119903) + 120573119894 119891 (119894 minus 1 119903 minus 1) + 119908

119894119903119901119894

max 1 ℎ minus (119899 minus 119894) le 119903 le min 119894 ℎ (33)

Step 3 Compute the optimal value of the function 119891lowast(ℎ) =

119891(119899 ℎ)

For a given ℎ value calculating all possible 119891(119899 ℎ) valuesusing the above recursion relation requires119874(119899ℎ) time Sincethe value of positional weights (and the order of positionalweights) can be altered by changing the ℎ value we mustrepeat the entire programming procedure for each ℎ =

0 1 2 119899 Thus the minimal objective value 119865lowast is givenby

119865lowast= minℎ=01119899

119891lowast(ℎ) (34)

Therefore the following statement holds

Theorem 10 Both the problems 1|119901119860

119895= 119901

119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889 and 1|119901

119860

119895= 119901119895119892(119903)

119878psd 119878119871119870|120572sum119869119895isin119873119864 119864119895+sum119869119895isin119873119879 120573119895119880119895+120574119889 can be solved in119874(1198993)

time

Example 11 Consider the problem 1|119901119860

119895= 119901119895119892(119903) 119878psd

119862119874119873|120572sum119869119895isin119873119864

119864119895+ sum119869119895isin119873119879

120573119895119880119895+ 120574119889

Let 119899 = 5 and ℎ = 3 The weights are 120572 = 1 120574 = 2 and120575 = 01

8 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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 The Scientific World Journal

The processing times are 1199011= 7 119901

2= 6 119901

3= 5 119901

4= 2

and 1199015= 1

The tardy penalties are 1205731= 10 120573

2= 8 120573

3= 4 120573

4= 5

and 1205735= 6

The positional effects are 119892(1) = 8 119892(2) = 7 119892(3) = 3119892(4) = 4 and 119892(5) = 7

Positional weights are1199081= 216119908

2= 238 and119908

3= 12

Positional weights are reordered such that 1199081198941le 1199081198942le

1199081198943 that is 119908

1198941= 1199083 1199081198942= 1199081 and 119908

1198943= 1199082

119891 (0 0) = 0

119894 = 1

119891 (1 0) = 119891 (0 0) + 1205731= 10

119891 (1 1) = 119891 (0 0) + 11990811989411199011= 84

119894 = 2

119891 (2 0) = 119891 (1 0) + 1205732= 18

119891 (2 1) = min 119891 (1 1) + 1205732 119891 (1 0) + 119908

11989411199012

= min 92 82 = 82

119891 (2 2) = 119891 (1 1) + 11990811989421199012= 2036

119894 = 3

119891 (3 0) = 119891 (2 0) + 1205733= 22

119891 (3 1) = min 119891 (2 1) + 1205733 119891 (2 0) + 119908

11989411199013

= min 86 78 = 78

119891 (3 2) = min 119891 (2 2) + 1205733 119891 (2 1) + 119908

11989421199013

= min 2076 190 = 190

119891 (3 3) = 119891 (2 2) + 11990811989431199013= 3226

119894 = 4

119891 (4 0) = 119891 (3 0) + 1205734= 27

119891 (4 2) = min 119891 (3 2) + 1205734 119891 (3 1) + 119908

11989421199014

= min 195 1212 = 1212

119891 (4 3) = min 119891 (3 3) + 1205734 119891 (3 2) + 119908

11989431199014

= min 3276 2376 = 2376

119894 = 5

119891 (5 0) = 119891 (4 0) + 1205735= 33

119891 (5 3) = min 119891 (4 3) + 1205735 119891 (4 2) + 119908

11989431199015

= min 2436 145 = 145

(35)

Therefore 119891(3) = 119891(5 3) = 145

The optimal sequence is 120587lowast = [1198694 1198695 1198693 1198691 1198692] 119889lowast = 419

The total cost is 145

6 Conclusions

Scheduling problems involving position-dependent process-ing time have received increasing attention in recent yearsIn this paper we considered single machine schedulingand due date assignment with setup time in which a jobrsquosprocessing time depends on its position in a sequence Thesetup time is past-sequence-dependent (p-s-d)The objectivefunctions include total earliness the weighted number oftardy jobs and the cost of due date assignment The due dateassignment methods used in this problem include commondue date (CON) and equal slack (SLK) We have presentedan 119874(119899

4) time algorithm for the general case and an 119874(119899

3)

time dynamic programming algorithm for the special casesIn the paper the model with position-dependent effectsis considered However in some other situations a jobrsquosprocessing time may be time-dependent or both position-dependent and time-dependent Therefore it is worthwhilefor future research to investigate the model in which a jobrsquosprocessing time depends both on its position in a sequenceand its start time It is also interesting for future researchto investigate the model in the context of other schedulingsettings including multimachine and job-shop scheduling

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and three anonymousreferees for their helpful comments on the earlier version ofthe paper This paper was supported in part by the Ministryof Science and Technology of Taiwan under Grant no MOST103-2221-E-252-004

References

[1] A Bachman and A Janiak ldquoScheduling jobs with position-dependent processing timesrdquo Journal of the OperationalResearch Society vol 55 no 3 pp 257ndash264 2004

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

[3] T C E Cheng and G Wang ldquoSingle machine scheduling withlearning effect considerationsrdquo Annals of Operations Researchvol 98 pp 273ndash290 (2001) 2000

[4] G Mosheiov and J B Sidney ldquoScheduling with general job-dependent learning curvesrdquo European Journal of OperationalResearch vol 147 no 3 pp 665ndash670 2003

[5] G Mosheiov ldquoScheduling problems with a learning effectrdquoEuropean Journal of Operational Research vol 132 no 3 pp687ndash693 2001

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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

The Scientific World Journal 9

[6] G Mosheiov ldquoParallel machine scheduling with a learningeffectrdquo Journal of the Operational Research Society vol 52 no10 pp 1165ndash1169 2001

[7] C Wu W Lee and T Chen ldquoHeuristic algorithms for solvingthe maximum lateness scheduling problem with learning con-siderationsrdquo Computers and Industrial Engineering vol 52 no1 pp 124ndash132 2007

[8] C Wu P Hsu J C Chen and N S Wang ldquoGenetic algorithmfor minimizing the total weighted completion time schedulingproblem with learning and release timesrdquo Computers amp Opera-tions Research vol 38 no 7 pp 1025ndash1034 2011

[9] Y Q Yin W H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014

[10] Y Q Yin T C E Cheng and C C Wu ldquoScheduling withtime-dependent processing timesrdquo Mathematical Problems inEngineering vol 2014 Article ID 201421 2 pages 2014

[11] 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

[12] G Mosheiov ldquoA note on scheduling deteriorating jobsrdquo Math-ematical and Computer Modelling vol 41 no 8-9 pp 883ndash8862005

[13] W-H Kuo and D-L Yang ldquoMinimizing the makespan in asingle-machine scheduling problem with the cyclic process ofan aging effectrdquo Journal of the Operational Research Society vol59 no 3 pp 416ndash420 2008

[14] A Janiak and R Rudek ldquoScheduling jobs under an aging effectrdquoJournal of the Operational Research Society vol 61 no 6 pp1041ndash1048 2010

[15] C L Zhao and H Y Tang ldquoSingle machine scheduling withgeneral job-dependent aging effect and maintenance activitiesto minimize makespanrdquo Applied Mathematical Modelling vol34 no 3 pp 837ndash841 2010

[16] K Rustogi and V A Strusevich ldquoSingle machine schedulingwith general positional deterioration and rate-modifyingmain-tenancerdquo Omega vol 40 no 6 pp 791ndash804 2012

[17] G Mosheiov ldquoProportionate flowshops with general position-dependent processing timesrdquo Information Processing Lettersvol 111 no 4 pp 174ndash177 2011

[18] C Zhao Y Yin T C E Cheng and C Wu ldquoSingle-machine scheduling anddue date assignmentwith rejection andposition-dependent processing timesrdquo Journal of Industrial andManagement Optimization vol 10 no 3 pp 691ndash700 2014

[19] K Rustogi and V A Strusevich ldquoSimple matching vs linearassignment in scheduling models with positional effects acritical reviewrdquo European Journal of Operational Research vol222 no 3 pp 393ndash407 2012

[20] C Koulamas and G J Kyparisis ldquoSingle-machine schedulingproblems with past-sequence-dependent setup timesrdquo Euro-pean Journal of Operational Research vol 187 no 3 pp 1045ndash1049 2008

[21] J Wang ldquoSingle-machine scheduling with past-sequence-dependent setup times and time-dependent learning effectrdquoComputers and Industrial Engineering vol 55 no 3 pp 584ndash591 2008

[22] Y Yin D Xu and J Wang ldquoSome single-machine schedul-ing problems with past-sequence-dependent setup times anda general learning effectrdquo International Journal of AdvancedManufacturing Technology vol 48 no 9ndash12 pp 1123ndash1132 2010

[23] C-J Hsu W-H Kuo and D-L Yang ldquoUnrelated parallelmachine scheduling with past-sequence-dependent setup timeand learning effectsrdquo Applied Mathematical Modelling vol 35no 3 pp 1492ndash1496 2011

[24] W Lee ldquoSingle-machine scheduling with past-sequence-dependent setup times and general effects of deterioration andlearningrdquo Optimization Letters vol 8 no 1 pp 135ndash144 2014

[25] X Huang G Li Y Huo and P Ji ldquoSingle machine schedul-ing with general time-dependent deterioration position-dependent learning and past-sequence-dependent setup timesrdquoOptimization Letters vol 7 no 8 pp 1793ndash1804 2013

[26] V Gordon J Proth and C Chu ldquoA survey of the state-of-the-art of common due date assignment and scheduling researchrdquoEuropean Journal of Operational Research vol 139 no 1 pp 1ndash25 2002

[27] Y Yin M Liu T C E Cheng C Wu and S Cheng ldquoFoursingle-machine scheduling problems involving due date deter-mination decisionsrdquo Information Sciences An InternationalJournal vol 251 pp 164ndash181 2013

[28] H G Kahlbacher and T C E Cheng ldquoParallel machinescheduling to minimize costs for earliness and number of tardyjobsrdquo Discrete Applied Mathematics vol 47 no 2 pp 139ndash1641993

[29] D Shabtay and G Steiner ldquoTwo due date assignment problemsin scheduling a single machinerdquo Operations Research Lettersvol 34 no 6 pp 683ndash691 2006

[30] C Koulamas ldquoA faster algorithm for a due date assignmentproblem with tardy jobsrdquo Operations Research Letters vol 38no 2 pp 127ndash128 2010

[31] V S Gordon andVA Strusevich ldquoMachine scheduling and duedate assignment with positionally dependent processing timesrdquoEuropean Journal of Operational Research vol 198 pp 57ndash622009

[32] C Hsu S Yang and D Yang ldquoTwo due date assignmentproblems with position-dependent processing time on a single-machinerdquo Computers and Industrial Engineering vol 60 no 4pp 796ndash800 2011

[33] J Wang ldquoFlow shop scheduling jobs with position-dependentprocessing timesrdquo Journal of Applied Mathematics amp Comput-ing vol 18 no 1-2 pp 383ndash391 2005

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

[35] D Shabtay N Gaspar and L Yedidsion ldquoA bicriteria approachto scheduling a singlemachine with job rejection and positionalpenaltiesrdquo Journal of Combinatorial Optimization vol 23 no 4pp 395ndash424 2012

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