Post on 13-May-2023
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
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
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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
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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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
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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
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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