Computers & Operations Research 38 (2011) 1760–1765

Contents lists available at ScienceDirect

Computers & Operations Research

0305-05

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/caor

Single-machine scheduling with deteriorating jobs and setup times tominimize the maximum tardiness

T.C.E. Cheng a, Chou-Jung Hsu b, Yi-Chi Huang c, Wen-Chiung Lee c,n

a Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kongb Department of Industrial Engineering and Management, Nan Kai Institute of Technology, Taiwanc Department of Statistics, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o

Available online 21 February 2011

Keywords:

Single-machine scheduling

Deteriorating jobs

Setup time

Maximum tardiness

48/$ - see front matter & 2011 Elsevier Ltd. A

016/j.cor.2010.11.014

esponding author.

ail address: wclee@fcu.edu.tw (W.-C. Lee).

a b s t r a c t

In many realistic production situations, a job processed later consumes more time than the same job

when it is processed earlier. Production scheduling in such an environment is known as scheduling

with deteriorating jobs. However, research on scheduling problems with deteriorating jobs has rarely

considered explicit (separable) setup time (cost). In this paper, we consider a single-machine

scheduling problem with deteriorating jobs and setup times to minimize the maximum tardiness.

We provide a branch-and-bound algorithm to solve this problem. Computational experiments show

that the algorithm can solve instances up to 1000 jobs in reasonable time.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In many real-world scheduling environments, a job processedlater consumes more time than the same job when it is processedearlier. For example, in steel production, iron ingots need to bereheated before rolling if their temperatures drop below a thresh-old level while waiting to enter the rolling machine. In fire-fighting, the time and effort required to control a fire increase ifthere is a delay in starting the fire-fighting effort. In health care,more extensive medical treatment is necessary for a patient thatreceives late treatment because his/her health condition worsensover time [1–3]. Recently, Rachaniotis and Pappis [4] proposeseveral demand-covering models for the deployment of availablefire-fighting resources so that a forest fire is attacked within aspecified time limit. In these cases, accomplishing a task needsmore time as time passes.

Melnikov and Shafransky [5], Gupta and Gupta [1], andBrowne and Yechiali [6] were among the pioneers that introduceddeteriorating jobs to scheduling problems. Since then, manyscheduling models dealing with deteriorating jobs have beenproposed from a variety of perspectives. Alidaee and Womer [7]and Cheng et al. [8] provide reviews of different models andproblems concerning deteriorating jobs. Moreover, Gawiejno-wicz [9] presents a comprehensive discussion of different aspectsof time-dependent scheduling and its applications. Recently, Leeet al. [10] study the makespan problem in the two-machine

ll rights reserved.

flowshop. Low et al. [11] consider the single-machine makespanproblem with an availability constraint where jobs undergosimple linear deterioration, while Lee and Wu [12] study thesame problem in the multiple parallel-machine setting. Wanget al. [13] consider some single-machine scheduling problemswith deteriorating jobs where the jobs are related by a series–parallel graph. They show that polynomial algorithms exist forthe problem with general linear deteriorating jobs to minimizethe makespan and for the problem with proportional lineardeteriorating jobs to minimize the total weighted completiontime. Toksari and Guner [14] consider a parallel-machine ear-liness/tardiness scheduling problem with different penaltiesunder the effects of learning and deterioration. Lee et al. [15]study a two-machine flowshop problem with deteriorating jobsand blocking to minimize the makespan. Li et al. [16] investigate asingle-machine scheduling problem with deteriorating jobs. Theyshow that the optimal schedule to minimize the sum of absolutedifferences in completion times is V-shaped. Sun [17] and Wanget al. [18] study scheduling models in which deteriorating jobsand learning effect are both considered simultaneously. Theyprovide the optimal schedule for several single-machine pro-blems. Lee et al. [19] address a total completion time schedulingproblem in the multi-machine permutation flowshop where eachmachine has its own deterioration rate. Gawiejnowicz and Kono-nov [20] consider the problem of scheduling a set of independent,resumable, and proportionally deteriorating jobs on a singlemachine with multiple periods of machine non-availability.

However, most studies assume the setup time is negligible orpart of the processing time. While this assumption simplifies theanalysis and/or reflects certain applications, it adversely affects

T.C.E. Cheng et al. / Computers & Operations Research 38 (2011) 1760–1765 1761

the solution quality in many applications that require an explicittreatment of the setup operation. There are many practicalapplications that call for a separate consideration of the setuptasks from the processing tasks. These applications can be foundin various shop types and situations, e.g., computer systems,paper bag factories, and the textile, container manufacturing,bottling, chemical, pharmaceutical, and food processing indus-tries. We refer the reader to the reviews by Allahverdi et al. [21],Yang and Liao [22], Cheng et al. [23], Potts and Kovalyov [24], andAllahverdi et al. [25]. To the best of our knowledge, research onscheduling problems with deteriorating jobs has rarely consid-ered explicit (separable) setup time (cost), except the followingstudies. Wang et al. [26] show that the single-machine groupscheduling problem to minimize the makespan or total comple-tion time is polynomially solvable under the model pij(t)¼aij�bijt

and si(t)¼si, where aij and bij are the basic processing time and thedeterioration rate of job j in group i, t is the starting time, and si isthe setup time of group i. Wu and Lee [27] show that the single-machine group scheduling problem to minimize the makespan ortotal completion time is polynomially solvable under the modelpij(t)¼aij+bt and si(t)¼di+gt, where aij is the basic processing timeof job j in group i, b is the common job deterioration rate, t is thestarting time, di is the basic setup time of group i, and g is thecommon deterioration rate of group setup time. Leung et al. [28]consider an identical parallel-machine scheduling problem wherethe jobs are processed in batches and the processing time of eachjob is a step function of its waiting time. They show that theproblem to minimize the total completion time is NP-hard in thestrong sense. Ji and Cheng [29] consider a batch schedulingproblem where the processing time of each job is a simple linearfunction of its waiting time. They show that the problem tominimize the makespan is strongly NP-hard. Pappis and Racha-niotis [30] consider the fire suppression problem where theobjective is to maximize the total value of the burnt arearemaining. They propose a branch-and-bound algorithm andheuristic algorithms to tackle this problem. Moreover, Pappisand Rachaniotis [31] provide a real-time synchronous heuristicalgorithm and test the efficiency of the heuristic using real data.

Wu et al. [32] point out that the late processing of a job mayrequire a longer setup or preparation time in the food processingand health care industries because food quality deteriorates or apatient’s condition worsens over time. In this paper, we consider asingle-machine scheduling problem where the job processingtimes and setup times are simple linear functions of their startingtimes. The objective is to minimize the maximum tardiness. Theremainder of this paper is organized into five sections. Weintroduce the notation and formulate the problem in Section 2.We provide a branch-and-bound algorithm to solve the problemin Section 3. We present the computational experiments to testthe performance of the algorithm and discuss the resultsin Section 4. We conclude the paper and suggest topics for futureresearch in the last section.

2. Problem formulation

There are n jobs to be processed on a single machine, each ofwhich belongs to one of M families. All the jobs are available attime t0, where t040. For each job j, there is a processing time pj, adue date dj, and a family code fj. When a job is processed first onthe machine or immediately after a job of another family, asequence-independent setup time is necessary. No setup isrequired between two jobs of the same family. During the setuptime, the machine is not available for processing. We assume thatthe actual job processing time of job j is a simple linear function of

its starting time t such that

pj ¼ ajt, j¼ 1,2,. . .,n,

where aj40 is the deterioration rate of job j’s processing time.Moreover, we assume that the actual setup time of a job from familyi is also a simple linear function of its starting time t such that

si ¼ yit, i¼ 1,2,. . .,M,

where yi40 is the deterioration rate of family i’s setup time. Undera schedule S, let CjðSÞ be the completion time of job j andTjðSÞ ¼maxf0,CjðSÞ�djg be the tardiness of job j. The objective is tofind a schedule such that Tmax ¼max1r jrnfTjðSÞg is minimized.

Let STsi,b denote the sequence-independent setup time sche-duling problem as in [25]. Using the traditional three-fieldnotation for scheduling problems, we denote the problem understudy as 1=STsi,b, pj ¼ ajt, si ¼ yit=Tmax.

3. A branch-and-bound algorithm

For an arbitrary number of families, Bruno and Downey [33]show that the classical single-machine scheduling problem withsequence-independent setup times to minimize the maximumlateness is NP-hard. Although the complexity of 1=STsi,b, pj ¼ ajt,si ¼ yit=Tmax is unknown, it is likely to be NP-hard. Thus weprovide a branch-and-bound algorithm to solve the problem.We first provide some dominance properties, followed by a lowerbound to speed up the search process. We then present the detailsof the branch-and-bound algorithm.

3.1. Dominance properties

In this subsection we derive some dominance rules that arehelpful in eliminating the dominated sequences.

Theorem 1. If jobs i and j are from the same family, aioaj, and

dirdj, then job i precedes job j in an optimal sequence.

Proof. Suppose that S and S0 are two job schedules and thedifference between S and S0 is a pairwise interchange of two jobsi and j from family u. That is, S¼ ðp,i,p0,j,p00Þ and S0 ¼ ðp,j,p0,i,p00Þ,where each of p, p0, and p00 denotes a partial sequence. Thetardiness of the jobs in the partial sequence p and p00 is the samein both sequences since jobs i and j are from the same family andthey are processed in the same order in both sequences. More-over, we have Ck(S)oCk(S0) for job kAp0 because aioaj. Thus, toshow S dominates S0, it suffices to show that maxfTiðSÞ,TjðSÞgrmaxfTjðS

0Þ,TiðS0Þg. We see that Ci(S)oCi(S

0) because job i is pro-cessed in a later position in S0 and Cj(S)¼Ci(S0) because they arefrom the same family and processed in the same position. Fromdirdj, we have

maxfTiðSÞ,TjðSÞg ¼maxf0,CiðSÞ�di,CjðSÞ�djg

rmaxf0,CiðS0Þ�digrmaxfTjðS

0Þ,TiðS0Þg:

Thus, S dominates S0 and the proof is completed.

To further expedite the search process, we provide a proposi-tion to determine the feasibility of a partial schedule. Assume that(pc,p) is a sequence of jobs where p is the scheduled part and pc isthe unscheduled part. Moreover, let S� ¼ ðp�,pÞ be a sequence inwhich the unscheduled jobs in pc are arranged as follows: Jobs inthe same family as the first job in p are scheduled last, if any, andthey are arranged in the earliest due date (EDD) order if there ismore than one job. For the other jobs, they are arranged family byfamily, where jobs in the same family are scheduled in the EDDorder, and the families are arranged in the EDD order of themaximum due dates of the families.

T.C.E. Cheng et al. / Computers & Operations Research 38 (2011) 1760–17651762

Theorem 2. If maxjApTjðS�ÞZmaxjAp�TjðS

�Þ, then sequence (p*,p)dominates sequences of type (pc,p).

Proof. Suppose that S is a sequence of the type (pc,p) andS� ¼ ðp�,pÞ. We have

max1r jrnTjðSÞ ¼maxfmaxjApTjðSÞ,maxjApc TjðSÞgZmaxjApTjðSÞ

ZmaxfmaxjApTjðS�Þ, maxjAp�TjðS

�Þg

¼maxjApTjðS�Þ

since maxjApTjðS�ÞZmaxjAp�TjðS

�Þ. Thus, sequence (p*,p) is opti-mal among sequences of type (pc,p).

In the following we provide two adjacency dominance proper-

ties. Suppose that S and S0 are two job schedules and the

difference between S and S0 is a pairwise interchange of two

adjacent jobs i and j. That is, S¼(p,i,j,p0) and S0 ¼ ðp,j,i,p0Þ, where pand p0 each denote a partial sequence. In addition, let t denote the

completion time of the last job in p.

Property 1. If (1) jobs i and j are from family u, (2) there is at least

one job in p that is from family u, (3) dirdj, and (4) diot(1+ai)(1+aj), then S dominates S0.

Proof. The tardiness of the jobs in partial sequences p and p0 isthe same in both sequences because jobs i and j are from the samefamily and they are processed in the same order in bothsequences. Thus, to show S dominates S0, it suffices to show thatmaxfTiðSÞ,TjðSÞgomaxfTjðS

0Þ,TiðS0Þg. To compare the maximum tar-

diness of jobs i and j in S and S0, we consider two cases. In the firstcase, the last job in p is from family u, so the completion times ofjobs i and j in S and in S0 are

CiðSÞ ¼ tð1þaiÞ,

CjðSÞ ¼ tð1þaiÞð1þajÞ,

CjðS0Þ ¼ tð1þajÞ;

and

CiðS0Þ ¼ tð1þaiÞð1þajÞ:

Since dirdj and diot(1+ai)(1+aj), we have

maxfTiðSÞ,TjðSÞgoTiðS0ÞrmaxfTjðS

0Þ,TiðS0Þg:

In the second case, the last job in p is not from family u, so the

completion times of jobs i and j in S and in S0 are

CiðSÞ ¼ tð1þyuÞð1þaiÞ,

CjðSÞ ¼ tð1þyuÞð1þaiÞð1þajÞ,

CjðS0Þ ¼ tð1þyuÞð1þajÞ;

and

CiðS0Þ ¼ tð1þyuÞð1þaiÞð1þajÞ:

Since dirdj and diot(1+ai)(1+aj), we have

maxfTiðSÞ,TjðSÞgoTiðS0ÞrmaxfTjðS

0Þ,TiðS0Þg:

In both cases, we have maxfTiðSÞ,TjðSÞgomaxfTjðS0Þ,TiðS

0Þg. So S

dominates S0.

Property 2. If (1) jobs i and j are from family u, (2) there is no job in

p that is from family u, (3) dirdj, and (4) diot(1+yu)(1+ai)(1+aj),then S dominates S0.

Proof. We omit the proof, which is similar to that of Property 1.

3.2. A lower bound

In this subsection we develop a lower bound for the branch-and-bound algorithm. First, we provide a lemma that is used inthe sequel.

Lemma 1. maxfa1�b2,a2�b1,0gZmaxfa2�b2,a1�b1,0g for 0oa1ra2

and 0ob1rb2.

We assume that PS is a partial schedule in which the order of thelast (n�k) jobs has been determined and k jobs (belonging to m

families) are yet to be scheduled. For these k unscheduled jobs, wearrange them in non-decreasing order of their deterioration rates, i.e.,a(1)ra(2)r?ra(k), and in non-decreasing order their due dates,i.e., d(1)rd(2)r?rd(k). Note that a(i) and d(i) are not necessarilyassociated with the same job. Moreover, for the m families associatedwith the k unscheduled jobs, we arrange them in non-increasingorder of their sizes, i.e., n(1)Zn(2)Z?Zn(m), and in non-decreasingorder of their setup times, i.e., y(1)ry(2)r?ry(m). By definition, thecompletion time of the first job is

C½1� ¼ t0ð1þyf½1� Þð1þa½1�ÞZt0ð1þyð1ÞÞð1það1ÞÞ;

where the subscript [ ] is used to signify the job position.Similarly, the completion time of the jth job is

C½j� ¼ t0ð1þyf½1� ÞYj

i ¼ 2

ð1þyf½i� Iðf½i�1�,f½i�ÞÞYj

i ¼ 1

ð1þa½i�Þ for 1r jrk,

where I(f[i�1],f[i])¼1 if f[i�1]a f[i] and I(f[i�1],f[i])¼0, otherwise. Wecan derive a lower bound on the completion time of the jth job byassuming that the first n(1) jobs with the smallest deteriorationrates form a family with setup time y(1), the next n(2) jobs with thesecond smallest deterioration rates form another family withsetup time y(2), and so on. Thus, we have

C½j�ZC�½j� ¼ t0

Ylj

i ¼ 1

ð1þyðiÞÞYj

i ¼ 1

ð1þaðiÞÞ for 1r jrk,

where lj is the smallest number such that jrn(1)+n(2)+?+n(lj).We can compute a lower bound on the completion times of thescheduled jobs accordingly. That is,

C½j�ZC�½j� ¼ C½k�Yj

i ¼ kþ2

ð1þyf½i� Iðf½i�,f½i�1�ÞÞYj

i ¼ kþ1

ð1þa½i�Þ for kþ1r jrn

if there is an unscheduled job from the same family of the (k+1)thjob or

C½j�ZC�½j� ¼ C½k�ð1þyf½kþ 1�ÞYj

i ¼ kþ2

ð1þyf½i� Iðf½i�,f½i�1�ÞÞ

�Yj

i ¼ kþ1

ð1þa½i�Þ for kþ1r jrn

if there is no unscheduled job from the same family of the (k+1)thjob. Thus, we have

TmaxðPSÞ ¼maxfmax1r jrkfC½j�ðPSÞ�d½j�g,maxkþ1r jrnfC½j�ðPSÞ�d½j�g,0g

Zmaxfmax1r jrkfC�½j��d½j�g,maxkþ1r jrnfC

�½j��d½j�g,0g

Zmaxfmax1r jrkfC�½j��dðjÞg,maxkþ1r jrnfC

�½j��d½j�g,0g:

The first inequality holds because C�½j� is a lower bound on the

actual job completion times while the second inequality isderived from Lemma 1 because C�

½j� is an increasing function of j.

Therefore, a lower bound on the maximum tardiness of the partialsequence PS is

LB¼maxfmax1r jrkfC�½j��dðjÞg, maxkþ1r jrnfC

�½j��d½j�g,0g:

3.3. Details of the branch-and-bound algorithm

We adopt a depth-first search in the branching procedure.Throughout the branching procedure, we only need to store thelower bounds for at most n�1 active nodes. To tackle the problemunder study, we assign jobs in a backward manner starting from

Table 1Performance of the branch-and-bound algorithm with n¼10 and M¼10.

Property or lower bound

included

t R Branch-and-bound algorithm

CPU time Number of nodes

Mean Max Mean Max

Enumeration 0.50 0.75 3.1497 3.3438 3,628,800 3,628,800

Property 1 only 0.0008 0.0156 76.01 171

Property 2 only 0.0009 0.0156 82.10 195

Theorem 1 only 0.0008 0.0156 74.20 146

Theorem 2 only 0.0005 0.0156 16.17 53

Lower bound only 0.0009 0.0156 84.39 270

All included 0.0005 0.0156 16.08 42

Enumeration 0.75 0.50 3.2294 3.5000 3,628,800 3,628,800

Property 1 only 0.0006 0.0156 70.80 176

Property 2 only 0.0009 0.0156 88.09 225

Theorem 1 only 0.0006 0.0156 63.21 176

Theorem 2 only 0.0003 0.0156 11.49 53

Lower bound only 0.0008 0.0156 69.23 162

All included 0.0003 0.0156 11.41 45

T.C.E. Cheng et al. / Computers & Operations Research 38 (2011) 1760–1765 1763

the last position. In the searching tree, we choose a branch andsystematically work down the tree until we either eliminate itusing the results from Sections 3.1 and 3.2 or reach its final node.The branch-and-bound algorithm consists of the following steps.

Step 1. Set an initial sequence S¼(� ,� ,...,�) with maximumtardiness N.

Step 2. Apply Theorem 1 and Properties 1, 2 to eliminate thedominated partial sequences.

Step 3. For the non-dominated nodes, apply Theorem 2 todetermine the order of the unscheduled jobs.

Step 4. Compute a lower bound on the maximum tardiness ofthe unfathomed partial sequences or the maximumtardiness of the completed sequences.

Step 5. If the lower bound on the unfathomed partial sequenceis greater than or equal to the initial solution, eliminatethe node corresponding to the partial sequence and allthe nodes beyond it in the branch. If the value of thecompleted sequence is less than the initial solution,replace it as the new solution. Otherwise, eliminate it.

Step 6. Output S.

4. Computational experiments

We conducted computational experiments to evaluate theperformance of the branch-and-bound algorithm and report theresults in this section. We coded the algorithm in Fortran 90 usingCompaq Visual Fortran version 6.6 and performed the experi-ments on a personal computer with a 2.26 GHz 2 Duo CPU and3 GB RAM under Windows XP. We designed the experimentsaccording to Fisher’s framework [34]. For all the test instances, wegenerated the deterioration rates of the job processing times (aj)and of the family setup times (yi) from uniform distributions over0 and 1. For each job j, we generated the family code (fj) from adiscrete uniform distribution over the integers between 1 and M.Notice that the actual number of families might not be known inadvance. We generated the due dates from another uniformdistribution between T(1�t�R/2) and T(1�t+R/2), where R isthe due date range, t is the tardiness factor, and T is the productof 1 plus the job deterioration rates and 1 plus the setup timedeterioration rates, i.e., T ¼

Qnj ¼ 1ð1þajÞ

QMi ¼ 1ð1þyiÞ, which is an

estimate of the completion time of the last job. For eachcombination of the test parameters, we report the mean, standarddeviation, and maximum value of the execution times (in sec-onds), as well as the mean, standard deviation, and maximumnumber of nodes over the replications.

We conducted the computational experiments in three parts.In the first part, we studied the efficiency of the properties andthe lower bound. We fixed the number of jobs and the number offamilies at 10 and set the values of (t,R) at (0.50, 0.75) and (0.75,0.50). We randomly generated 100 instances and computed theoptimal solutions by full enumeration for comparison purposes.We tested the dominance properties and the lower bound one at atime. Table 1 gives the results. It is evident that all the dominanceproperties are effective in pruning the search tree, especiallyTheorem 2. The results show that the branch-and-bound algo-rithm incorporated with all the properties is the most efficient interms of execution time and number of nodes. So we adopted thisapproach in conducting the subsequent experiments.

In the second part of the experiments, we studied the impactsof the number of families, the tardiness factor, and the rangefactor on the performance of the branch-and-bound algorithm.We fixed the number of jobs at 200. We used three differentnumbers of families and eight combinations of (t,R) values, i.e.,40, 100, and 200 for M, and (0.25, 0.25), (0.25, 0.50), (0.25, 0.75),

(0.50, 0.25), (0.50, 0.50), (0.50, 0.75), (0.75, 0.25), and (0.75, 0.50)for (t,R). For each case, we randomly generated 100 instances.Table 2 shows the results. It is evident that instances with greaternumbers of families are harder to solve. As regards the effect of(t,R), the cases with (t,R)¼(0.50, 0.75) and (0.75, 0.50) were thehardest to solve, whereas the other six cases were relatively easyto solve. The main reason is that Theorem 2 is less potent whenthe due date values are more variable and small. So we set thevalues of (t,R) at (0.50, 0.75) and (0.75, 0.50) in the next part ofthe experiments.

In this last part of the experiments, we tested the branch-and-bound algorithm with five different numbers of jobs (n¼200, 400,600, 800, and 1000). The number of families M took the values n,n/2, and n/5. For each situation, we randomly generated 100instances. Table 3 shows the results. The branch-and-boundalgorithm solved all of the instances up to 1000 jobs in reasonabletime. The worst case took about 72 min when n¼1000, M¼1000,and (t,R)¼(0.50, 0.75). To test the effects of the number of jobs,due dates, and number of families, we constructed a three-wayanalysis of variance (ANOVA) of the number of nodes in Table 3.The resulting F-value was 3.07 with a p-value of 0.0155, whichindicates that the number of jobs has a significant effect on thehardness of the problem. A closer look of Table 3 reveals that thenumber of nodes increases as the number of jobs increases.The test also shows that the F-value for the number of familieswas 6.36 with a p-value of 0.0018, which indicates that thenumber of families also has a significant effect on the hardnessof the problem. In fact, the problem is harder to solve as thenumber of families increases. The main reason is that Theorem 1,and Properties 1, 2 are less potent if there are more families.Finally, the due date factor is also statistically significant with anF-value of 9.46 and a p-value of 0.0021. The results show that thecase (t,R)¼(0.50, 0.75) was harder to solve than the case(t,R)¼(0.75, 0.50), especially for large numbers of families. Thisis due to the fact that Theorem 2 is less potent when due dates aremore variable and small, as explained earlier.

5. Conclusions

In this paper, we consider a single-machine scheduling pro-blem with deteriorating jobs and setup times to minimize themaximum tardiness. We develop some properties and a lower

Table 2Performance of the branch-and-bound algorithm with n¼200.

n M t R Branch-and-bound algorithm

CPU time Number of nodes

Mean SD Max Mean SD Max

200 40 0.25 0.25 0.0325 0.0127 0.1094 217.9100 69.8292 598

0.50 0.0333 0.0123 0.0781 229.8400 86.5607 598

0.75 0.0347 0.0090 0.0625 200.0000 0.0000 200

0.50 0.25 0.0345 0.0132 0.0781 221.8900 62.5787 399

0.50 0.0344 0.0155 0.1094 231.8400 88.1827 797

0.75 0.0534 0.0369 0.2031 331.0400 241.2627 1191

0.75 0.25 0.0284 0.0114 0.0781 217.9100 63.8444 598

0.50 0.0355 0.0284 0.2812 243.7200 127.9586 1191

100 0.25 0.25 0.0669 0.0219 0.1719 215.9200 61.1890 598

0.50 0.0839 0.0366 0.2500 255.6700 123.3930 796

0.75 0.0705 0.0090 0.0938 200.0000 0.0000 200

0.50 0.25 0.0659 0.0190 0.1250 221.8900 62.5787 399

0.50 0.0833 0.0501 0.5000 249.6900 160.4936 1588

0.75 0.1548 0.1526 1.1406 464.0700 543.8937 4355

0.75 0.25 0.0647 0.0206 0.1406 221.8900 62.5787 399

0.50 0.0917 0.0514 0.3906 285.4900 150.0737 992

200 0.25 0.25 0.1006 0.0332 0.2969 219.9000 66.3333 598

0.50 0.1195 0.0549 0.4844 241.7900 117.6280 996

0.75 0.1081 0.0101 0.1406 200.0000 0.0000 200

0.50 0.25 0.1044 0.0530 0.5625 219.8900 82.3925 796

0.50 0.1291 0.0854 0.7812 271.5500 194.3843 1785

0.75 0.4955 2.8995 29.1094 1042.6700 6422.2116 64,425

0.75 0.25 0.1031 0.0338 0.3125 217.9100 57.2371 399

0.50 0.1445 0.0784 0.5000 293.4700 168.2015 995

Table 3Performance of the branch-and-bound algorithm.

n M t R Branch-and-bound algorithm

CPU time Number of nodes

Mean SD Max Mean SD Max

200 40 0.50 0.75 0.0534 0.0369 0.2031 331.0400 241.2627 1191

0.75 0.50 0.0355 0.0284 0.2812 243.7200 127.9586 1191

100 0.50 0.75 0.1548 0.1526 1.1406 464.0700 543.8937 4355

0.75 0.50 0.0917 0.0514 0.3906 285.4900 150.0737 992

200 0.50 0.75 0.4955 2.8995 29.1094 1042.6700 6422.2116 64,425

0.75 0.50 0.1445 0.0784 0.5000 293.4700 168.2015 995

400 80 0.50 0.75 0.3089 0.2859 2.4688 738.5300 848.9785 7161

0.75 0.50 0.1922 0.1195 1.1406 459.8100 222.2005 1995

200 0.50 0.75 1.0098 1.2962 12.6719 953.7700 1540.1571 15,105

0.75 0.50 0.5652 0.2639 1.9219 535.6200 278.9592 1995

400 0.50 0.75 3.2989 13.6958 133.9531 2067.9700 9393.6808 92,313

0.75 0.50 0.8287 0.2789 1.8750 483.7800 172.8583 1197

600 120 0.50 0.75 0.9175 0.8652 6.7656 1132.3200 1355.6417 10,750

0.75 0.50 0.5475 0.2017 1.6250 713.7800 314.9452 2396

300 0.50 0.75 2.8275 2.0189 12.6094 1210.6700 1010.1317 6587

0.75 0.50 1.7936 0.9040 6.0469 833.5900 480.7702 2996

600 0.50 0.75 17.1744 84.8512 649.2500 5297.5400 27,534.3822 211,842

0.75 0.50 10.2902 75.8879 761.5000 3183.4200 24,444.1894 245,165

800 160 0.50 0.75 1.8628 1.3164 9.8125 1335.1000 1008.8867 7181

0.75 0.50 1.1936 0.4027 2.7188 967.7900 381.6694 2398

400 0.50 0.75 6.8769 7.3868 69.6250 1773.8900 2339.4585 22,306

0.75 0.50 4.2317 2.5106 20.9844 1167.4700 656.2393 3990

800 0.50 0.75 100.4819 517.0549 3676.0938 17,619.4400 93,694.3000 689,838

0.75 0.50 6.4583 2.8315 20.1250 1039.7000 447.1055 2398

1000 200 0.50 0.75 3.5320 2.2254 13.5938 1718.9800 1405.3844 7985

0.75 0.50 2.3156 1.6797 16.8281 1229.6300 774.8663 6983

500 0.50 0.75 30.3011 167.1812 1680.1875 6202.1100 38,600.3933 387,396

0.75 0.50 8.0716 5.0563 31.4688 1399.5400 827.7931 5994

1000 0.50 0.75 83.5794 449.0490 4320.8594 9186.9400 50,739.3739 487,420

0.75 0.50 13.0395 6.2746 42.6719 1389.6100 722.5549 4996

T.C.E. Cheng et al. / Computers & Operations Research 38 (2011) 1760–17651764

T.C.E. Cheng et al. / Computers & Operations Research 38 (2011) 1760–1765 1765

bound for this problem. Based on these results, we construct abranch-and-bound algorithm to solve the problem. The results ofthe computational experiments show that the branch-and-boundalgorithm can solve instances up to 1000 jobs in reasonable time.

The complexity of the problem is open and future researchshould address this issue. Another interesting topic for futurestudy is to consider the problem with batch setup times tominimize other due date related objective functions.

Acknowledgements

We are grateful to the Editor and two anonymous referees fortheir helpful comments on an earlier version of our paper.

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