Scheduling problems with multiple due windows assignmentand controllable processing times on a single machine

Dar-Li Yang a, Chien-Jung Lai b, Suh-Jenq Yang b,c,n

a Department of Information Management, National Formosa University, Taiwanb Department of Distribution Management, National Chin-Yi University of Technology, Taiwanc Department of Industrial Management, Nan Kai University of Technology, No. 568, Zhongzheng Rd., Caotun Township, Nantou County 54243, Taiwan

a r t i c l e i n f o

Article history:Received 29 August 2012Accepted 13 December 2013Available online 22 December 2013

Keywords:SchedulingMultiple due windowsControllable processing timeAging effect

a b s t r a c t

This paper deals with multiple due windows assignment scheduling problems and controllableprocessing times on a single machine. We assume that the actual processing time of a job can becontrolled by the introduction of additional resource and any due window is not allowed to containanother due window as a proper subset. The objective is to determine the optimal due window positionsand sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimalschedule to minimize a total cost function, which consists of the earliness, the tardiness, the processingtime compressions, and the due windows related costs. We show that for the case when the number ofjobs assigned to each due window is given in advance, the problem is polynomially solvable in Oðn3Þtime, where n is the total number of jobs; while if the number of jobs assigned to each due window isunknown, the problem can be optimally solved in Oðnmþ2Þ time, wherem is the number of due windows.Furthermore, we extend the problem by incorporating with the aging effect and prove that it remainspolynomially solvable.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Traditionally manufacturers have forecasted demand for theirproducts into the future and then have attempted to smooth outproduction to satisfy that forecasted demand. Unfortunately, thisapproach has a number of major drawbacks, such as large inven-tories, long production times, production obsolescence, inability tomeet delivery schedules, and high costs. On the other hand, Just-in-Time (JIT) is a pull system of production, in which products areproduced only as needed to meet the actual customer demand. It isbased on planned elimination of all waste and continuous improve-ment of productivity. To cope with intensified global competitionand escalating customer demand for superior service, JIT productionhas become a competitive strategy for world-class company.According to the principle of JIT production, a company producingorders (or jobs) early, as well as late, is discouraged. It incurs severalcosts when an order is produced early, for example, costs caused bythe extra investment in its finished products inventory, costsinvolved in extra storage facilities and product spoilage cost. Inpractice, order completions can be accepted without penalty withinan interval in time. This time interval is often called the due

window. Applications of the due window problem in real-lifesituations can readily be found. For example, the due windowmight reflect an assembly environment inwhich the components ofthe product should be ready at a time interval in order to avoidstaging delays, or a shop where several jobs constitute a singlecustomer’s order. It is clear that a wide due window increases thesupplier’s production and delivery flexibility. However, a large duewindow and delaying job completion reduce the supplier’s compe-titiveness and customer service level.

The classical JIT scheduling with due windows assignment aimsto determine the optimal due window locations and sizes and theoptimal schedule to minimize a total cost function. Sidney (1977)was among the pioneers, who studied a single machine schedulingproblem with due windows assignment. The objective was to finda schedule that minimizes the total costs of earliness and tardi-ness. He assumed that each job has its due window and no job’sdue window is allowed to contain the due window of another job.He proved that this problem is solvable in polynomial time anddeveloped an optimal procedure to determine the actual job starttimes to minimize the earliness and tardiness penalties. Limanet al. (1996) explored a variation of the problem where a commondue window size is given but the earliest due date is not. Limanet al. (1998) further generalized their result to the problem whereboth the earliest due date and the due window size are to bedetermined. They proposed an Oðn log nÞ algorithm to solve theproblem, where n is the total number of jobs. Recently, Mosheiov

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Int. J. Production Economics

0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ijpe.2013.12.021

n Corresponding author at: Department of Industrial Management, Nan KaiUniversity of Technology, No. 568, Zhongzheng Rd., Caotun Township, NantouCounty 54243, Taiwan. Fax: þ886 49 2565842.

E-mail address: sjyang@nkut.edu.tw (S.-J. Yang).

Int. J. Production Economics 150 (2014) 96–103

and Sarig (2008) extended the problem proposed by Liman et al.(1998) to the case of position-dependent job processing times.They assumed that the processing time of a job is a function of itsposition in a sequence. They provided an Oðn3Þ time solution forthis case. Janiak et al. (2009) considered various models of duewindow assignment scheduling problems on a single processorsuch that the objective function containing the maximum or totalearliness and tardiness and the due window parameters is mini-mized. They derived several properties of the solutions andconstructed polynomial time algorithms for the considered pro-blems. Wang and Wang (2011) studied a single machine commondue window scheduling problem simultaneously with the learningeffect and deteriorating jobs considerations. They introduced anOðn log nÞ time algorithm to solve the problem. Su and Tien (2011)considered a problem of scheduling jobs on a single machine tominimize the mean absolute deviation of the job completion timeabout a large common due window subject to the maximumtardiness constraint. They proposed a branch and bound algorithmand a heuristic to solve the problem under study. Cheng et al.(2012) explored a common due window assignment schedulingproblem with linear time-dependent deteriorating jobs and adeteriorating maintenance on a single machine setting. Theyshowed that the proposed model is polynomially solvable. Yinet al., 2013 considered a batch delivery single machine schedulingproblem where the jobs have an assignable common due window.They showed that the problem can be optimally solved in Oðn8Þtime under a reasonable assumption on the relationships amongthe cost parameters. For new trends in scheduling with duewindow, we refer the reader to Mor and Mosheiov (2012),Huang et al. (2013), Chen et al. (2013), Ji et al. (2013), and Yinet al. (2014).

On the other hand, scheduling problems with controllableprocessing times have received considerable attention by manyresearchers. In scheduling problems, the actual processing time ofa job can be controlled by the allocation of additional resources.The concept of controllable processing time is from the area ofproject planning and control. Comprehensive surveys on this lineof scheduling research can be found in Nowicki and Zdrzalka(1990), Chudzik et al. (2006), and Shabtay and Steiner (2007). Fornew trends in scheduling with controllable processing times, werefer the reader to Rudek and Rudek (2012), Shabtay et al. (2012),Yang et al. (2013), Oron (2014), and Kayvanfar et al. (2014).

Furthermore, scheduling problem with controllable processingtimes to include the due window aspect has received relativelylittle attention in the literature. Liman et al. (1997) investigatedsingle machine scheduling problems with a common due windowand controllable processing times. The objective was to find theoptimal common due window position and size, the optimal jobcompressions, and the optimal job sequence to minimize a totalcost function including earliness, tardiness, due window locationand size, and processing time reduction. They showed that theproblem can be formulated as an assignment problem and thuscan be solved in polynomial time. Wan (2007) studied a singlemachine common due window assignment scheduling problemwhere the job processing times are controllable with linear costsand the due window was variable. The objective was to find a jobsequence, a processing time for each job, and a position of thecommon due window to minimize the total costs of weightedearliness and tardiness and the processing time compression. Heproposed some properties of the optimal solution and provided apolynomial time algorithm to solve the problem.

Although the due window assignment in JIT scheduling hasbeen extensively studied in the literature, the multiple duewindows assignment scheduling problem with controllable pro-cessing times has never been explored. In practice, n jobs mayhave distinct m due windows, where 1rmrn. In this paper the

concept of single due window is extended to allow for a pre-determined number of due windows when scheduling a singlemachine. This extension allows for greater flexibility in modelingreal-life problems. For example, in an order picking operationprocess, the number of orders to be completed may be too great torealistically justify measurement from a single due window for asingle customer. By viewing the order picking operation process asbeing composed of several discrete segments, each group of orderscould be made uniform around its own due window. The ordersshould be ready at their due window in order to avoid stagingdelays. Moreover, a higher cost in the form of transship feegenerally accompanies a later due window. Another examplecan be found in an assembly process where many componentsconstitute a product. We consider single machine schedulingproblems with multiple due windows assignment and controllingprocessing times. Similar to Liman et al. (1997), the objective is todetermine the optimal due window positions and sizes, the set ofjobs assigned to each due window, the optimal job compressions,and the optimal job sequence such that the cost function contain-ing the earliness, the tardiness, the processing time compressions,and the due windows related costs is minimized. We also extendthe proposed problem by the introduction of the aging effect.The considered issues are relatively new in the scheduling theory.We show that the considered problems are polynomially solvableand propose two efficient algorithms to solve them.

This paper is divided into six sections. In Section 2 we formulatethe problem. In Section3 we provide some properties of the optimalschedule. In Section 4 we present the optimal solution for theproblem. We extend the problem to incorporate with the agingeffect in Section 5. We conclude the paper and suggest some topicsfor future research in the last section.

2. Notation and problem formulation

The following notation will be used throughout the paper andwe will introduce additional notation when needed.

n the total number of jobs;m the number of due windows, 1rmrn;Jj job j, j¼ 1;2;…;n;½r� the job scheduled in the rth position of a job sequence;di the due window starting time of the ith due window,

i¼ 1;2;…;m;wi the due window finishing time of the ith due window,

i¼ 1;2;…;m;Di the size of the ith due window, i¼ 1;2;…;m, i.e.,

Di ¼wi�di;Ii the set of jobs assigned to the ith due window,

i¼ 1;2;…;m;ni the number of jobs assigned to the ith due window,

i¼ 1;2;…;m, i.e., Iij j ¼ ni and n¼∑mi ¼ 1ni;

pj the normal processing time of job Jj, j¼ 1;2;…;n;xj the amount of reduction in processing time of job Jj,

j¼ 1;2;…;n;xj the upper bound on the amount of reduction in proces-

sing time of job Jj, j¼ 1;2;…;n;pj the actual processing time of job Jj, j¼ 1;2;…;n;Cj the completion time of job Jj, j¼ 1;2;…;n;Ej the earliness of job Jj ¼ max 0; di � Cj

� �, for JjA Ii,

j¼ 1;2;…;n and i¼ 1;2;…;m;Tj the tardiness for job Jj ¼ max 0;Cj �wi

� �, for JjA Ii,

j¼ 1;2;…;n and i¼ 1;2;…;m;Gj the unit cost of compression for job Jj, j¼ 1;2;…;n;α the per unit time earliness penalty, α40;β the per unit time tardiness penalty, β40;

D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103 97

γ the per unit time due window starting time penalty,γ40.

δ the per unit time due window size penalty, δ40.

There are n independent jobs J ¼ J1; J2;…; Jn� �

to be processedon a single machine. All the jobs are ready for processing at timezero and no preemption is allowed. The machine can handle atmost one job at a time. We consider the problem of multiple duewindows assignment scheduling with controllable processingtimes simultaneously. Under the proposed model, job Jj has anormal processing time pj, which can be controlled by an addi-tional resource xjZ0, j¼ 1;2;…;n. For each job Jj, its compressionhas an upper bound xj, j¼ 1;2;…;n. Then, the actual processingtime of job Jj is given by

pj ¼ pj�xj;0rxjrxjopj; j¼ 1;2;…;n ð1ÞThe condition xjopj is necessary to ensure that the actual

processing time of a job is greater than zero.In practice, n jobs may have distinct m due windows, where

1rmrn. We assume that the number of due windows m to beassigned to the jobs is given in advance, for 1rmrn. We furtherassume that any due window is not allowed to contain anotherdue window as a proper subset. Let diðZ0Þ and wiðZdiÞ denotethe starting time and finishing time of the ith due window,respectively, for i¼ 1;2;…;m, and Di ¼wi�di denotes the size ofthe ith due window. Note that both di and wi are decisionvariables. If m¼ 1, it means that all the jobs have one commondue window; if m¼ n, it indicates that there exists n distinct duewindows for n jobs. Let Ii denote the set of jobs assigned to the ithdue window, for i¼ 1;2; :::;m. Then the earliness and the tardinessof job Jj are Ej ¼ max f0; di�Cjg and Tj ¼ max f0;Cj�wig, respec-tively, for JjA Ii. Similar to Liman et al. (1997), we aim to determinethe set of the due window starting times d¼ fd1; d2; :::;dmg, the setof the due window sizes D¼ fD1;D2; :::;Dmg, the set of jobsassigned to each due window I¼ fI1; I2; :::; Img, the set of the jobcompressions x¼ fx1; x2; :::; xng, and the job sequence π such thatthe total cost function Zðd;D; I; x;πÞ

Zðd;D; I; x;πÞ ¼ ∑m

i ¼ 1∑jA Ii

ðαEjþβTjþγdiþδDiþGjxjÞ ð2Þ

is minimized.

3. Preliminary analysis

The following lemmas are useful to find the optimal solutionfor the problem under study.

Lemma 1. In an optimal schedule, there exists no idle time betweenconsecutive jobs on the machine and the first job starts at time zero.

Proof. The proof is obvious and omitted. □

Lemma 2. For any specified sequence π, there exists optimalcommon due windows with the property that the due windowstarting time di and finishing time wi coincide with some jobs’completion times, for i¼ 1;2;…;m.

Proof. When all the jobs have a common due window (i.e.,m¼ 1),Liman et al. (1998) and Mosheiov and Sarig (2008) showed thatthere exists an optimal common due window such that the duewindow starting time and finishing time coincide with some jobs’completion times. Because their proofs are independent of the jobdistribution on the time axis, the result can immediately begeneralized to the proposed problem. □

Let Ni ¼∑ik ¼ 1nk be the total number of jobs assigned to the

first i due windows, for i¼ 1;2;…;m, and N0 ¼ 0. Lemma

3 indicates that for an optimal sequence, the jobs assigned todifferent due windows are mutually disjoint, that is, there is anoptimal solution such that ni consecutive jobs (in positionsNi�1þ1 to Ni) in a job sequence π are assigned to the ith duewindow.

Lemma 3. For any given d, D, x, and π, there exists an optimal I suchthat Ii ¼ ðJ½Ni� 1 þ1�; J½Ni� 1 þ2�;…; J½Ni �Þ, for i¼ 1;2;…;m, where J½r� is thejob scheduled in position r in the job sequence π.

Proof. For any given d, D, x, and π, without loss of generality, weassume that in a schedule S1 ¼ ðπ1; Jk; Jl;π2Þ job Jk immediatelyprecedes job Jl, while in a schedule S2 ¼ ðπ1; Jl; Jk;π2Þ jobs Jk and Jlare mutually replaced, where π1 and π2 denote partial sequencesand jobs Jk and Jl are, respectively, scheduled in the rth positionand ðrþ1Þth position in the schedule S1. In addition, we assumethat for both schedules S1 and S2, Jk is early for the ðiþ1Þth duewindow and Jl is tardy for the ith due window, where JkA Iiþ1 andJlA Ii.Let CjðS1Þ and CjðS2Þ be the completion times of job Jj in

schedules S1 and S2, respectively. By definition, the completiontimes of jobs Jk and Jl in S1 are

CkðS1Þ ¼ ∑r�1

j ¼ 1p½j� þpk ð3Þ

and

ClðS1Þ ¼ ∑r�1

j ¼ 1p½r� þpkþpl ð4Þ

Similarly,

ClðS2Þ ¼ ∑r�1

j ¼ 1p½j� þpl ð5Þ

and

CkðS2Þ ¼ ∑r�1

j ¼ 1p½r� þplþpk ð6Þ

Then,

αðdiþ1�CkðS1ÞÞþβðClðS1Þ�wiÞ� �� αðdiþ1�CkðS2ÞÞþβðClðS2Þ�wiÞ

� �

¼ αplþβpk40 ð7Þ

Clearly,

α diþ1�CkðS1Þ� �þβ ClðS1Þ�wið Þ4α diþ1�CkðS2Þ

� �þβ ClðS2Þ�wið Þ

Similarly, the total cost decreases as repeating this interchangeargument for the jobs which assigned to the same due window arenot sequenced consecutively. Therefore, we conclude that for anygiven d, D, x, and π, the jobs assigned to the same due window arearranged consecutively. □

Furthermore, for the case that all the jobs have a common duewindow, Mosheiov and Sarig (2008) showed that if γ4 min β; δ

� �,

an optimal schedule exists in which the due window starts at timezero and if βo min γ; δ

� �, there exists an optimal schedule in

which the due window is reduced to a due date that starts at timezero. We assume δ4γ and βZδ throughout the remainder of thepaper. Given any real number x, the ceiling of x, denoted ⌈x⌉, is thesmallest integer greater than or equal to x. Lemma 4 provides theoptimal locations of common due windows.

D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–10398

Lemma 4. For any given I, x, and π, there exists optimal duewindows such that di ¼ C ½ki � and wi ¼ C ½ki þhi �, where

ki ¼Ni�1þ⌈niðδ�γÞ

α⌉ ð8Þ

and

kiþhi ¼Ni�1þ⌈niðβ�δÞ

β⌉ ð9Þ

for i¼ 1;2;…;m.

Proof. When all the jobs share a common due window (i.e., m¼ 1and n1 ¼ n), consider an optimal schedule and an optimal duewindow such that C½k1 � ¼ d1 and C ½k1 þh1 � ¼w1. Using the smallperturbation technique (Mosheiov and Sarig, 2009), we have

k1 ¼ ⌈n1ðδ�γÞ

α⌉ ð10Þ

and

ðk1þh1Þ ¼ ⌈n1ðβ�δÞ

β⌉ ð11Þ

Since any due window is not allowed to contain another duewindow as a proper subset and both (10) and (11) are obtainedindependent of the job distribution on the time axis, the resultscan immediately be generalized to the proposed model. Conse-quently, we have (8) and (9). □

Consequently, by Lemmas 2–4, we obtain that for the ith duewindow, the first ðki�Ni�1�1Þ jobs in Ii are early and ðNi�ki�hiÞjobs in Ii are tardy.

4. Optimal solution

In this section we will find the set of jobs assigned to each duewindow, the optimal job compressions, and the optimal jobsequence to minimize the objective function.

4.1. Iij j ¼ ni known

This subsection assumes that the vector ðn1;n2;…;nmÞ of mpositive integers such that n¼∑m

i ¼ 1ni and Iij j ¼ ni, fori¼ 1;2;…;m, is externally specified. Note that while Iij j ¼ ni isassumed to be known in advance, the specific jobs in Ii areunknown and to be determined.

Using Lemmas 3 and 4 and substituting C ½j� ¼∑jr ¼ 1p½r� and

x½j� ¼ p½j� �p½j� into (2), we have

Zðd;D; I; x;πÞ ¼ ∑m

i ¼ 1∑jA Ii

ðαEjþβTjþγdiþδDiþGjxjÞ

¼ ∑m

i ¼ 1∑Ni

j ¼ Ni� 1 þ1ðαE½j� þβT ½j� þγdiþδDiþG½j�x½j�Þ

¼ ∑m

i ¼ 1niγC½ki � þniδðC½ki þhi� �C½ki �Þþ ∑

ki

j ¼ Ni� 1 þ1αðC½ki � �C½j�Þ

(

þ ∑Ni

j ¼ ki þhi þ1βðC ½j� �C ½ki þhi �Þþ ∑

Ni

j ¼ Ni� 1 þ1G½j�x½j�

)

¼ ∑n

r ¼ 1urp½r� þ ∑

n

r ¼ 1G½r�p½r� ð12Þ

where

ur ¼αðr�1�Ni�1Þþγðn�Ni�1Þ�G½r�; r¼Ni�1þ1;Ni�1þ2;…; ki;

δðNi�Ni�1Þþγðn�NiÞ�G½r�; r¼ kiþ1; kiþ2;…; kiþhi;

βðNi�rþ1Þþγðn�NiÞ�G½r�; r¼ kiþhiþ1; kiþhiþ2;…;Ni;

8><>:

ð13Þand i¼ 1;2;…;m. Since ∑n

r ¼ 1G½r�p½r� is a constant, for anysequence, the optimal processing time of a job in a position with

a negative position weight ur should be its normal processing timep½r�, and the processing time of a job in a position with a positiveweight ur should be its normal processing time p½r� minus theupper bound of its compression x½r�. If ur ¼ 0, then the optimalprocessing time of the job in this position may be any valuebetween p½r� �x½r�. Notice that if x½j� ¼ 0, it implies that no additionalresource is needed to compress the processing time of the job.Thus, after simplifying, we have the following result:

pn

½r� ¼p½r� if urr0;p½r� �x½r� if ur40; right:

(ð14Þ

where pn½r� denotes the optimal processing time of the job in

position r, for r¼ 1;2;…;n. As a result, the optimal job compres-sion xn½r� can be obtained by xn½r� ¼ p½r� �pn

½r�, for r¼ 1;2;…;n. There-fore, we have the following theorem.

Theorem 1. For a given job sequence, the optimal job compressionscan be determined as follows: The compression of the job in a non-positive weight position is zero; the compression of the job in apositive weight position is its maximum reduction in the processingtime. That is,

xn½r� ¼0 if urr0;x½r� if ur40; right:

(ð15Þ

Proof. The proof follows from the above analysis. □

In order to obtain the optimal job sequence, we formulate theproblem as an assignment problem. We may ignore the term of∑n

r ¼ 1G½r�p½r� in (12), as it represents a constant in the objectivefunction. Let yjr be a 0/1 variable such that yjr ¼ 1 if job Jj isscheduled in the rth position to be processed on the machine andyjr ¼ 0 otherwise. Then, we can formulate the sequencing problemas the following assignment problem:

minimize

∑n

j ¼ 1∑n

r ¼ 1ujrpjryjr ð16Þ

subject to

∑n

j ¼ 1yjr ¼ 1; r¼ 1;2;…;n ð17Þ

∑n

r ¼ 1yjr ¼ 1; j¼ 1;2;…;n ð18Þ

yjr ¼ 1 or 0; j¼ 1;2;…;n and r¼ 1;2;…;n ð19Þ

where

ujr ¼αðr�1�Ni�1Þþγðn�Ni�1Þ�Gj; j¼ 1;2;…;n; r¼Ni�1þ1;Ni�1þ2;…; ki;δðNi�Ni�1Þþγðn�NiÞ�Gj; j¼ 1;2;…;n; r¼ kiþ1; kiþ2;…; kiþhi;

βðNi�rþ1Þþγðn�NiÞ�Gj; j¼ 1;2;…;n; r¼ kiþhiþ1; kiþhiþ2;…;Ni;

8><>:

ð20Þand

pjr ¼pj; if ujrr0;pj�xj; if ujr40;

(ð21Þ

for i¼ 1;2;…;m.For a given vector ðn1;n2;…;nmÞ, the problem can be optimally

solved by the following algorithm.

Algorithm 1.

Step 1. By Lemma 4, calculate ki ¼Ni�1þ⌈niðδ�γÞ=α⌉ andkiþhi ¼Ni�1þ⌈niðβ�δÞ=β⌉, for i¼ 1;2;…;m, whereNi ¼∑i

k ¼ 1nk and N0 ¼ 0.

D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103 99

Step 2. Calculate the αjr and pjr values by using (20) and (21), forj¼ 1;2;…;n and r¼ 1;2;…;n.Step 3. Solve the assignment problem (16)-(19) to determinethe optimal job sequence.Step 4. Calculate the optimal job compressions by using (13)and (15).Step 5. Calculate the optimal job processing times by using (14).Step 6. Calculate the position of the due windowdi ¼ C½ki � ¼∑ki

r ¼ 1p½r� and wi ¼ C½ki þhi � ¼∑ki þhir ¼ 1 p½r�, for

i¼ 1;2;…;m.Step 7. Calculate the size of the due window Di ¼wi�di, fori¼ 1;2;…;m.

Theorem 2. For a given constant m, if Iij j ¼ ni is known in advance,for i¼ 1;2;…;m, the proposed model can be solved in Oðn3Þ time.

Proof.. In Algorithm 1, the time complexity of Step 2 is Oðn2Þ andStep 3 is Oðn3Þ time; Steps 1, 4, 5, 6, and 7 can be performed inlinear time. Thus the overall time complexity of the algorithm isOðn3Þ. □

The following example illustrates applying Algorithm 1 to findthe optimal solution of an 11 jobs instance. We solve the assign-ment problem of the example using the commercial softwarepackage LINGO version 11.0 on a personal computer.

Example 1. There are n¼ 11 jobs. The number of due windows ism¼ 3 and n1 ¼ 3, n2 ¼ 5, and n3 ¼ 3 are given. The set of jobparameters is presented in Table 1. The unit earliness, tardiness,due window starting time and due window size penalties areα¼ 3, β¼ 15, γ ¼ 5 and δ¼ 6, respectively.

Solution. By Lemma 4, we obtain that k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 5,k2þh2 ¼ 6, k3 ¼ 9, and k3þh3 ¼ 10. We solve the assignmentproblem (16)–(19) to obtain the optimal job sequence and thusobtain the optimal compressions and the actual processing timesof jobs. The results of the example are summarized in Table 2.From Table 2, we see that the optimal solution for the exampleconsists of the following: (i) the optimal job sequences for thethree due windows are I1 ¼ ðJ7; J1; J3Þ, I2 ¼ ðJ10; J5; J11; J9; J4Þ, andI3 ¼ ðJ6; J2; J8Þ, respectively; (ii) the actual processing times of jobsJ1, J2, J3, J4, J5, J7, J9, J10, and J11 are compressed by additionalresources, (iii) the first due window is starting at time 4.5 andfinishing at time 7.5; the second due window is starting at time31.0 and finishing at time 37.4; and the third due window isstarting at time 72.7 and finishing at time 84.7; (iv) the sizes of thedue windows are D1 ¼ 7:5�4:5¼ 3:0, D2 ¼ 37:4�31:0¼ 6:4, andD3 ¼ 84:7�72:7¼ 12:0, respectively; (v) job J10 is early and jobs J3,J4, J8, and J9 are tardy; (vi) the total cost is Z ¼ 3843:8.

4.2. Iij j ¼ ni unknown

In this subsection, we consider that the vector ðn1;n2;…;nmÞ isunknown. d, D, I, x, and π are decision variables. Following Yangand Yang (2010b), we denote by Pðn;mÞ ¼ ðn1;n2;…;nmÞ theallocation vector of number of jobs assigned to each due window,where niZ1 is the number of jobs assigned to the ith due windowand n¼∑m

i ¼ 1ni. Note that the number of jobs assigned to a duewindow may be 1;2;…;n�mþ1. So if we know the numbers of

jobs on the first m�1 due windows, the number of jobs processedon the last due window is then determined uniquely due to thefact that n¼∑m

i ¼ 1ni. Therefore, the upper bound on Pðn;mÞ vectoris ðn�mþ1Þm�1rnm�1.

Based on the analysis in the previous section, if ðn1;n2;…;nmÞ isgiven, then we can obtain the optimal job sequence and theoptimal job compressions by using (16)–(19) and (13) and (15),respectively. Consequently, we can determine the actual proces-sing time of jobs, the optimal positions of the common duewindows, and the size of each due window.

Theorem 3. For a given constant m, if Iij j ¼ ni is unknown, fori¼ 1;2;…;m, the proposed model can be solved in Oðnmþ2Þ time.

Proof. From Algorithm 1, for a given vector ðn1;n2;…;nmÞ, theproposed problem can be solved in Oðn3Þ time. In addition, theupper bound on Pðn;mÞ vector is ðn�mþ1Þm�1rnm�1. Therefore,if Iij j ¼ ni is unknown, for i¼ 1;2; :::;m, the proposed problem canoptimally be solved in Oðnmþ2Þ time. □

Clearly, for a given vector Pðn;mÞ ¼ ðn1;n2;…;nmÞ, we candetermine the local optimal solution of the problem by Algorithm1. The global optimal solution for the problem is the one withthe minimum total cost for all the possible vectorsPðn;mÞ ¼ ðn1;n2;…;nmÞ.

Example 2. There are n¼ 7 jobs. The number of due windows ism¼ 3. The set of job parameters is presented in Table 3. The unitearliness, tardiness, due window starting time and due windowsize penalties are α¼ 3, β¼ 15, γ ¼ 5 and δ¼ 6, respectively.

Solution. For given the number of jobs on each due window, i.e.,ðn1;n2;n3Þ, we solve the corresponding problem using Algorithm 1.Table 4 shows the optimal position of due windows and the totalcost Z for all the possible vectors n1;n2;n3ð Þ. From Table 4, we seethat the minimum total cost for this example is obtained whenn1;n2;n3ð Þ ¼ ð3;2;2Þ. The optimal solution for the example isshowed in Table 5. From Table 5, we know that: (i) the optimaljob sequences for the three due windows are I1 ¼ ðJ3; J1; J7Þ,I2 ¼ ðJ2; J4Þ, and I3 ¼ ðJ6; J5Þ, respectively; (ii) the actual processingtimes of jobs J1, J2, J3, J4, and J7 are compressed by additionalresources, (iii) the first due window is starting at time 4.5 andfinishing at time 7.5; the second due window is starting at time24.0 and finishing at time 34.3; and the third due window isstarting at time 51.3 and finishing at time 63.3; (iv) the sizes of thedue windows are D1 ¼ 7:5�4:5¼ 3:0, D2 ¼ 34:3�24:0¼ 10:3, andD3 ¼ 63:3�51:3¼ 12:0, respectively; (v) job J7 is tardy; (vi) thetotal cost is Z ¼ 1548:4.

Table 1Job parameters for Example 1.

Job J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11

pj 8 16 7 14 12 17 9 15 11 13 10xj 5 4 2.5 3.7 3.2 1.5 4.5 3.5 3 2.8 3.6Gj 21 12 23 14 35 26 17 28 39 20 31

Table 2The optimal solution for Example 1.

r 1 2 3 4 5 6 7 8 9 10 11

J½r� J7 J1 J3 J10 J5 J11 J9 J4 J6 J2 J8xn½r� 4.5 5.0 2.5 2.8 3.2 3.6 3.0 3.7 0.0 4.0 0.0

p½r� 4.5 3.0 4.5 10.2 8.8 6.4 8.0 10.3 17.0 12.0 15.0C½r� 4.5 7.5 12.0 22.2 31.0 37.4 45.4 55.7 72.7 84.7 99.7

Table 3Job parameters for Example 2.

Job J1 J2 J3 J4 J5 J6 J7

pj 8 16 7 14 12 17 9xj 5 4 2.5 3.7 3.2 1.5 4.5Gj 21 12 23 14 35 26 17

D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103100

5. Extension

In this section, the proposed model is extended by theintroduction of the effect of aging. The aging effect occurs whenthe production facility becomes less efficiency (e.g., wearing ortiredness) and the production rate deteriorates (Janiak and Rudek,2006). Under the aging effect environment, the later a given job isscheduled in the sequence, the longer its processing time is.Several recent papers have studied scheduling problems withthe aging effect in different machine environments, includingKuo and Yang (2008), Janiak and Rudek (2010), Zhao and Tang(2010), and Yang and Yang (2010a, 2010b, 2010c). We will showthat the problem with incorporating the aging effect remainspolynomially solvable. In this model, if job Jj is scheduled inthe rth position in a job sequence, its actual processing time isgiven by

pj ¼ pjra�xj;0rxjrxjopjn

a; j¼ 1;2;…;n ð22Þ

where aZ0 is the common aging factor of jobs.Clearly, Lemmas 1–4 still hold when the aging effect is examined.

Similar to the above analysis, if the vector ðn1;n2;…;nmÞ is given inadvance, using Lemmas 3 and 4 and substituting C ½j� ¼∑j

r ¼ 1p½r� and

x½j� ¼ p½j�ja�p½j� into (2), we obtain

Zðd;D; I; x;πÞ ¼ ∑m

i ¼ 1∑jA Ii

ðαEjþβTjþγdiþδDiþGjxjÞ

¼ ∑m

i ¼ 1∑Ni

j ¼ Ni� 1 þ1ðαE½j� þβT ½j� þγdiþδDiþG½j�x½j�Þ

¼ ∑m

i ¼ 1niγC½ki � þniδðC½ki þhi � �C½ki �Þþ ∑

ki

j ¼ Ni� 1 þ1αðC½ki � �C½j�Þ

(

þ ∑Ni

j ¼ ki þhi þ1βðC½j� �C½ki þhi �Þþ ∑

Ni

j ¼ Ni� 1 þ1G½j�x½j�

)

¼ ∑n

r ¼ 1θrp½r�r

aþ ∑n

r ¼ 1ðG½r� �θrÞx½r� ð23Þ

where

θr ¼αðr�1�Ni�1Þþγðn�Ni�1Þ; r¼Ni�1þ1;Ni�1þ2;…; ki;

δðNi�Ni�1Þþγðn�NiÞ; r¼ kiþ1; kiþ2;…; kiþhi;

βðNi�rþ1Þþγðn�NiÞ; r¼ kiþhiþ1; kiþhiþ2;…;Ni;

8><>:

ð24Þand i¼ 1;2;…;m. Recall that Ni ¼∑i

k ¼ 1nk denotes the total numberof jobs assigned to the first i due windows, for i¼ 1;2;…;m, andN0 ¼ 0.

From (23), for any job sequence, we can find that the optimalcompression of a job in a position with a negative G½r� �θr shouldbe its maximum reduction in the processing time, and the optimalcompression of a job in a position with a positive G½r� �θr should be0. If G½r� �θr ¼ 0, then the optimal compression of the job in thisposition may be any value between 0 and x½j�. After simplifying, fora given job sequence, the optimal job compressions can bedetermined as follows:

xn½r� ¼x½r� if G½r� �θro0;0 if G½r� �θrZ0;

(ð25Þ

for r¼ 1;2;…;n. Then, we can formulate the sequencing problemas the following assignment problem:

minimize

∑n

j ¼ 1∑n

r ¼ 1λjryjr ð26Þ

subject to

∑n

j ¼ 1yjr ¼ 1; r¼ 1;2;…;n ð27Þ

∑n

r ¼ 1yjr ¼ 1; j¼ 1;2;…;n ð28Þ

yjr ¼ 1 or 0; j¼ 1;2;…;n; and r¼ 1;2;…;n ð29Þwhere

λjr ¼θrpjr

a if Gj�θrZ0;

θrpjraþðGj�θrÞxj if Gj�θro0

(ð30Þ

For a given vector ðn1;n2;…;nmÞ, we propose a polynomial timealgorithm to solve the problem.

Algorithm 2.

Step 1. By Lemma 4, calculate ki ¼Ni�1þ⌈niðδ�γÞ=α⌉ andkiþhi ¼Ni�1þ⌈niðβ�δÞ=β⌉, for i¼ 1;2;…;m, whereNi ¼∑i

k ¼ 1nk and N0 ¼ 0.Step 2. Calculate the λjr values by using (30), for j¼ 1;2;…;nand r¼ 1;2;…;n.Step 3. Solve the assignment problem (26)-(29) to determinethe optimal job sequence.

Table 4The optimal due window positions and total cost for all the possible vectorsn1 ;n2 ;n3ð Þ of Example 2.

n1 ;n2 ;n3ð Þ Optimal position of due windows Total cost Z

ð1;1;5Þ k1 ¼ 1, k1þh1 ¼ 1, k2 ¼ 2, k2þh2 ¼ 2, k3 ¼ 4,k3þh3 ¼ 5

1913.8

ð1;2;4Þ k1 ¼ 1, k1þh1 ¼ 1, k2 ¼ 2, k2þh2 ¼ 3, k3 ¼ 5,k3þh3 ¼ 6

1741.0

ð1;3;3Þ k1 ¼ 1, k1þh1 ¼ 1, k2 ¼ 2, k2þh2 ¼ 3, k3 ¼ 5,k3þh3 ¼ 6

1687.3

ð1;4;2Þ k1 ¼ 1, k1þh1 ¼ 1, k2 ¼ 3, k2þh2 ¼ 4, k3 ¼ 5,k3þh3 ¼ 7

1702.3

ð1;5;1Þ k1 ¼ 1, k1þh1 ¼ 1, k2 ¼ 3, k2þh2 ¼ 4, k3 ¼ 7,k3þh3 ¼ 7

1783.7

ð2;1;4Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 3, k2þh2 ¼ 3, k3 ¼ 5,k3þh3 ¼ 6

1738.0

ð2;2;3Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 3, k2þh2 ¼ 4, k3 ¼ 5,k3þh3 ¼ 6

1637.3

ð2;3;2Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 3, k2þh2 ¼ 4, k3 ¼ 6,k3þh3 ¼ 7

1604.8

ð2;4;1Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 4, k2þh2 ¼ 5, k3 ¼ 7,k3þh3 ¼ 7

1651.2

ð3;1;3Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 4, k2þh2 ¼ 4, k3 ¼ 5,k3þh3 ¼ 6

1624.8

ð3;2;2Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 4, k2þh2 ¼ 5, k3 ¼ 6,k3þh3 ¼ 7

1548.4 a

ð3;3;1Þ k1 ¼ 1, k1þh1 ¼ 2, k2 ¼ 4, k2þh2 ¼ 5, k3 ¼ 7,k3þh3 ¼ 7

1569.7

ð4;1;2Þ k1 ¼ 2, k1þh1 ¼ 3, k2 ¼ 5, k2þh2 ¼ 5, k3 ¼ 6,k3þh3 ¼ 7

1587.3

ð4;2;1Þ k1 ¼ 2, k1þh1 ¼ 3, k2 ¼ 5, k2þh2 ¼ 6, k3 ¼ 7,k3þh3 ¼ 7

1562.3

ð5;1;1Þ k1 ¼ 2, k1þh1 ¼ 3, k2 ¼ 6, k2þh2 ¼ 6, k3 ¼ 7,k3þh3 ¼ 7

1667.3

a The minimum total cost.

Table 5The optimal solution for Example 2.

r 1 2 3 4 5 6 7

J½r� J3 J1 J7 J2 J4 J6 J5xn½r� 2.5 5.0 4.5 4.0 3.7 0.0 0.0

p½r� 4.5 3.0 4.5 12.0 10.3 17.0 12.0C ½r� 4.5 7.5 12.0 24.0 34.3 51.3 63.3

D.-L. Yang et al. / Int. J. Production Economics 150 (2014) 96–103 101

Step 4. Calculate the optimal job compressions by using (25).Step 5. Calculate the optimal job processing times by using (22).Step 6. Calculate the position of the due windowdi ¼ C½ki � ¼∑ki

r ¼ 1p½r� and wi ¼ C½ki þhi � ¼∑ki þhir ¼ 1 p½r�, for

i¼ 1;2;…;m.Step 7. Calculate the size of the due window Di ¼wi�di, fori¼ 1;2;…;m.

Clearly, the overall time complexity of Algorithm 2 is Oðn3Þ andthus the following theorem holds.

Theorem 4. For a given constant m, if Iij j ¼ ni is known in advance,for i¼ 1;2;…;m, the proposed model with the aging effect can besolved in Oðn3Þ time.

Example 3 illustrates applying Algorithm 2 to find the optimalsolution of an 11 job instance.

Example 3. The same data in Example 1 is used and the commonaging factor is a¼ 0:3.

Solution. By Lemma 4, we determine that k1 ¼ 1, k1þh1 ¼ 2,k2 ¼ 5, k2þh2 ¼ 6, k3 ¼ 9, and k3þh3 ¼ 10. We solve the assign-ment problem (26)–(29) to obtain the optimal job sequence andthus get the optimal compressions and the actual processing timesof jobs. The results of the example are listed in Table 6. FromTable 6, we obtain that the optimal solution for the exampleconsists of the following: (i) the optimal job sequences for thethree due windows are I1 ¼ ðJ4; J7; J1Þ, I2 ¼ ðJ10; J9; J11; J3; J5Þ, andI3 ¼ ðJ6; J2; J8Þ, respectively; (ii) the actual processing times of jobsJ1, J2, J3, J4, J7, J9, J10, and J11 are compressed by additionalresources; (iii) the first due window is starting at time 10.3 andfinishing at time 16.88; the second due window is starting at time54.73 and finishing at time 68.25; and the third due window isstarting at time 133.55 and finishing at time 161.47; (iv) the sizesof the due windows are D1 ¼ 16:88�10:3¼ 6:58,D2 ¼ 68:25�54:73¼ 13:52, and D3 ¼ 161:47�133:55¼ 27:92,respectively; (v) job J10 is early and jobs J1, J3, J5, and J8 are tardy;(vi) the total cost is Z ¼ 6411:93.

In view of the analysis in the section 4.2, we have the followingtheorem.

Theorem 5. For a given constant m, if Iij j ¼ ni is unknown, fori¼ 1;2;…;m, the proposed model with the aging effect can be solvedin Oðnmþ2Þ time.

6. Conclusions

We considered single machine scheduling problem with multi-ple due windows assignment and controllable processing timeswhere the objective is to determine the optimal due windowlocations and sizes, the set of jobs assigned to each due window,the optimal job compressions, and the optimal job sequence tominimize the total penalty based on job earliness, tardiness, duewindow, and processing time compression. We assumed that theactual processing time of a job is a function of its normal

processing time and an amount of reduction. We showed thatthe problem is polynomially solvable and presented an efficientalgorithm to solve it. We also extended the proposed model by theintroduction of the aging effect. We proved that even the intro-duction of the effect of aging, the problem remains polynomiallysolvable. Further research might be to consider the problem withother models of resource allocation or in multi-machine settings.

Acknowledgments

The authors thank the Editor and anonymous reviewers fortheir helpful comments and suggestions on an earlier version ofthe paper. This research was supported in part by the NationalScience Council of Taiwan, Republic of China, under grant numberNSC 102–2221-E-252-010-MY2.

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