Computers & Industrial Engineering 61 (2011) 782–787

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Computers & Industrial Engineering

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

Deteriorating job scheduling to minimize the number of late jobs with setup times q

Wen-Chiung Lee a,⇑, Jian-Bang Lin a, Yau-Ren Shiau b

a Department of Statistics, Feng Chia University, Taichung, Taiwanb Department of Industrial Engineering and System Management, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o

Article history:Received 23 December 2010Received in revised form 19 May 2011Accepted 21 May 2011Available online 27 May 2011

Keywords:Deteriorating jobsNumber of late jobsSetup timeSingle-machine

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.05.010

q This manuscript was processed by Area Editor T.C⇑ Corresponding author.

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

a b s t r a c t

Deteriorating jobs scheduling problems have been widely studied recently. However, research on sched-uling problems with deteriorating jobs has rarely considered explicit setup times. With the currentemphasis on customer service and meeting the promised delivery dates, we consider a single-machinescheduling problem to minimize the number of late jobs with deteriorating jobs and setup times in thispaper. We derive some dominance properties, a lower bound, and an initial upper bound by using a heu-ristic algorithm to speed up the search process of the branch-and-bound algorithm. Computationalexperiments show that the algorithm can solve instances up to 1000 jobs in a reasonable amount of time.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In classical scheduling it is assumed that the job processingtimes are known and fixed throughout the entire process. How-ever, this assumption might not be appropriate to describe somereal-world environments. For instance, there are situations that ajob processed later consumes more time than the same job whenprocessed earlier, e.g., in scheduling maintenance jobs, cleaningassignments, medical treatments, or fire-fighting process, whereany delay in starting the process is penalized by incurring addi-tional time to accomplish the job. This is known as schedulingdeteriorating jobs in the literature. Alidaee and Womer (1999),Cheng, Ding, and Lin (2004) and Gawiejnowicz (2008) providedextensive reviews of different models and problems concerningdeteriorating jobs.

Recently, Wang, Ng, and Cheng (2008) considered some single-machine deteriorating jobs scheduling problems where the jobsare related by a series–parallel graph. They showed that polyno-mial algorithms exist for the problem with general linear deterio-rating jobs to minimize the makespan and for the problem withproportional linear deteriorating jobs to minimize the totalweighted completion time. Cheng, Wu, and Lee (2008), Wang(2009) and Sun (2009) considered the effects of deterioration andlearning at the same time. They provided the optimal solutionsfor some single-machine problems and some permutation flow-shop problems under certain agreeable conditions. He, Wu, andLee (2009) considered a single-machine problem to minimize the

ll rights reserved.

. Edwin Cheng.

total completion time with step-deteriorating jobs. They provideda branch-and-bound and a weight-combination search algorithmfor the problem. Li, Li, Sun, and Xu (2009) investigated a single ma-chine scheduling problem with deteriorating jobs. They showedthat the optimal schedule to minimize the total absolute differ-ences in completion times is V-shaped. Chung, Liu, Wu, and Lee(2009) studied a single-machine problem where the objective isto minimize the sum of the square of the job lateness. They pro-vided a branch-and-bound algorithm and two heuristics for theproposed problem. Lee, Wu, Chung, and Liu (2009) addressed amultiple-machine permutation flowshop problem to minimizethe total completion time where each machine has its own deteri-oration rate. Ji and Cheng (2010a, 2010b) showed that the single-machine makespan problem is NP-hard under the case that themachine has an unavailable period and the job is resumable. Mos-lehi and Jafari (2010) studied a single machine problem where theobjective is to minimize the number of tardy jobs. They provided abranch-and-bound and a heuristic algorithm for the problem. Ng,Wang, Cheng, and Liu (2010) addressed a two-machine problemwhere the objective is to minimize the makespan. Lee, Shiau, Chen,and Wu (2010) studied a two-machine flowshop problem withdeteriorating jobs and blocking to minimize the makespan. Huang,Wang, Wang, Gao, and Wang (2010) developed the optimal solu-tions for some single-machine problems with time-dependentdeterioration and exponential learning effect. Yin, Xu, and Wang(2011) provided the optimal solutions for some single-machineproblems where the effects of learning and deterioration are ex-pressed as a general function. Ng, Wang, Cheng, and Lam (2011)showed that the makespan and the total completion time prob-lems remain polynomially solvable on the no-wait and no-idlepermutation flowshop environment.

W.-C. Lee et al. / Computers & Industrial Engineering 61 (2011) 782–787 783

However, most studies assumed that the setup time is negligi-ble or parts of the processing time. While this assumptionsimplifies the analysis and/or reflects certain applications, itadversely affects the solution quality for many applications whichrequire an explicit treatment of the setup operation. There aremany practical situations that need the separate considerationof the setup tasks from the processing tasks, e.g., the computersystem, the paper bag factory, the textile industry, the containerand the bottle manufacturing industry, the chemical industry,the pharmaceutical industry, and the food processing industry.Allahverdi, Gupta, and Aldowaisan (1999), Cheng, Gupta, andWang (2000), Potts and Kovalyov (2000), and Allahverdi, Ng,Cheng, and Kovalyov (2008) provided extensive reviews of differ-ent models and problems concerning setup time. Recently, Leung,Ng, and Cheng (2008) considered the scheduling problem on par-allel and identical machines where the jobs are processed inbatches and the processing time of each job is a step functionof its waiting time. They showed that the total completion timeproblem is NP-hard in the strong sense. Lee and Wu (2010) pro-vided the optimal schedule for the single machine makespanproblem under the group technology. Zhao and Tang (2010)considered some single-machine problems with deteriorating jobsand past-sequence-dependent setup times. Ji and Cheng (2010a,2010b) studied a batch scheduling problem and the processingtime of each job is a simple linear function of its waiting time.They showed that the makespan problem is strongly NP-hard.Cheng, Lee, and Wu (2010) considered the effects of deteriorationand learning at the same time. They provided the optimal solu-tions for some single-machine problems with the assumption thatthe setup time is past-sequence-dependent. Cheng, Lee, and Wu(2011) provided the optimal solutions for some single-machineproblems with the consideration of deterioration and past-sequence-dependent setup time. Although deteriorating jobsscheduling problems with the consideration of setup time hasstarted to draw the attention of the researchers, most of theresearch focuses on batch setup time, group setup time or past-sequence-dependent setup time. To the best of our knowledge,explicit family setup time is rarely considered in deterioratingjobs scheduling problems.

With the current emphasis on customer service and meetingthe promised delivery dates, we consider a single-machine sched-uling problem to minimize the number of late jobs where the jobprocessing times and setup times are simple linear functions oftheir starting times. The remainder of this paper is organized intofive sections. Notation and the problem formulation are presentedin Section 2. A branch-and-bound algorithm incorporated withsome dominance conditions, a lower bound, and an initial upperbound by using a heuristic algorithm is described in Sections 3.The computational experiments to test the performance of thealgorithm are conducted and provided in Section 4. The conclusionis given 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 t0 > 0. For job j, there is a processing time pj, a duedate dj, and a family code fj. When a job is processed first on themachine or immediately after a job of another family, a setup timeis necessary. No setup is required between jobs from the same fam-ily. During the setup time, the machine is not available for process-ing. It is assumed that the actual job processing time of job j is asimple linear function of its starting time t; that is pj = ajt,j = 1, 2, . . . , n, where aj > 0 is its deterioration rate. Moreover, theactual setup time of family i is also assumed to be a simple linearfunction of its starting time t as follows:

si ¼ hit ; i ¼ 1;2; . . . ;M;

where hi > 0 is its deterioration rate. Under a schedule S, let Cj(S) bethe completion time of job j, Uj(S) = 1 if Cj(S) > dj and 0 otherwise.The objective is to find a schedule S� such thatPn

j¼1UjðS�Þ 6Pn

j¼1UjðSÞ for any schedule S. Using the traditionalthree-field notation for scheduling problems, the problem understudy is denoted as 1/pj = ajt, si = hit/

PUj.

3. The branch-and-bound algorithm

For an arbitrary number of families, Bruno and Downey (1978)showed that the traditional single-machine problem with se-quence-independent setup times to minimize the number of latejobs is NP-hard. Thus, the problem under consideration is alsoNP-hard. In this paper, a branch-and-bound algorithm is developedto derive the optimal solution. In this section, we first providesome dominance properties, followed by a lower bound to speedup the search process. Finally, the descriptions of the branch-and-bound and the heuristic algorithms are presented.

3.1. Dominance properties

In this subsection, we provide two adjacent dominance proper-ties. Suppose that S and S0 are two job schedules and the differencebetween S and S0 is a pairwise interchange of two adjacent jobs iand j. That is, S = (p, i, j, p0) and S0 = (p, j, i, p0), where p and p0 eachdenote a partial sequence. In addition, let t denote the completiontime of the last job in p.

Property 1. If (1) jobs i and j are from family u, (2) there is no jobin p which is from family u, and (3) t(1 + hu)(1 + ai) 6 di < t(1 + hu)(1 + ai)(1 + aj) 6 dj, then S dominates S0.

Proof. The total numbers of late jobs in p are the same in bothsequences since jobs are processed in the same order in bothsequences. Since there is no job in p which is from family u, thecompletion times of jobs i and j in S and in S0 are

CiðSÞ ¼ tð1þ huÞð1þ aiÞ;CjðSÞ ¼ tð1þ huÞð1þ aiÞð1þ ajÞ;CjðS0Þ ¼ tð1þ huÞð1þ ajÞ;

and

CiðS0Þ ¼ tð1þ huÞð1þ aiÞð1þ ajÞ:

Since Cj(S) = Ci(S0), the total numbers of late jobs in p0 are thesame in both sequences since jobs are starting at the same timeand processed in the same order in both sequences. To show Sdominates S0, it suffices to show that Ui(S) + Uj(S) < Uj(S0) + Ui(S0).

Since t(1 + hu)(1 + ai) 6 di < t(1 + hu)(1 + ai)(1 + aj) 6 dj, we have

UiðSÞ þ UjðSÞ ¼ 0 < UjðS0Þ þ UiðS0Þ ¼ 1:

Thus, S dominates S0. h

Property 2. If (1) jobs i and j are from family u, (2) there is no jobin p which is from family u, (3) t(1 + hu)(1 + aj)(1 + ai) > di Pt(1 + hu)(1 + ai), and (4) dj < t(1 + hu)(1 + aj), then S dominates S0.

Proof. The proof is similar to that of Property 1. h

To further facilitate the search process, we provide a corollary todetermine the ordering of the remaining jobs of a partial schedule.Assume that (pc, p) is a sequence of jobs where p is the scheduled

784 W.-C. Lee et al. / Computers & Industrial Engineering 61 (2011) 782–787

part and pc is the unscheduled part. Moreover, let (p�, p) be a se-quence in which the unscheduled jobs in pc are arranged as fol-lows: jobs in the same family as the first job in p are scheduledlast, if any, and they are arranged in the earliest due date (EDD)order if there is more than one job. For the other jobs, they are ar-ranged family by family, where jobs in the same family are sched-uled in the EDD order, and the families are arranged in the EDDorder of the maximum due dates of the families.

Corollary 1. If there is no late jobs in p�, then sequence (p�, p) is theoptimal schedule among sequences of the types (pc, p).

3.2. Lower bound

In this subsection, we develop the lower bound for the branch-and-bound algorithm. We assume that PS is a partial schedule inwhich the order of the last (n � k) jobs has been determined andk jobs (belonging to m families) are yet to be scheduled. Withoutloss of generality, we assume that among the k unscheduled jobs,the deterioration rates are a(1) 6 a(2) 6 � � � 6 a(k) when they are ar-ranged in a non-decreasing order. Moreover, let dmax be the maxi-mal due date of the k unscheduled jobs. It is also assumed that thenumbers of jobs in each family are n(1) P n(2) P � � �P n(m) whenthey are arranged in a non-increasing order, and the setup timesare h(1) 6 h(2) 6 � � � 6 h(m) when they are arranged in a non-decreasing order. By definition, the completion time of the firstjob is

C ½1� ¼ ð1þ h½1�Þð1þ a½1�ÞP ð1þ hð1ÞÞð1þ að1ÞÞ:

Similarly, the completion time for the jth job is

C ½j� ¼ ð1þ h½1�ÞYj

i¼2

ð1þ h½i� � Iðf½i�1�–f½i�ÞÞYj

i¼1

ð1þ a½i�Þ for 1 6 j 6 k;

where I(f[i�1] – f[i]) = 1 if f[i�1] – f[i] and I(f[i�1] – f[i]) = 0 otherwise.The lower bound on the completion time for the jth job can be de-rived by assuming that the first n(1) jobs with the smallest deterio-ration rates form a family with setup time of h(1), and the second n(2)

jobs with the smallest deterioration rates form another family withsetup time of h(2), and so on. Thus, we have

C ½j� PYlj

i¼1

ð1þ h½i�Þ �Yj

i¼1

ð1þ aðiÞÞ for 1 6 j 6 k;

where lj is the smallest number such that j 6 nð1Þ þ nð2Þ þ � � � þ nðljÞ.The lower bound on the completion times of the scheduled jobscan be calculated accordingly. That is,

C ½j� P C ½k�Yj

i¼kþ1

ð1þ h½i� � Iðf½i�–f½i�1�ÞÞYj

i¼kþ1

ð1þ a½i�Þ for kþ 1 6 j 6 n

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

C ½j� P C ½k�ð1þ h½fkþ1 �ÞYj

i¼kþ1

ð1þ h½i� � Iðf½i�–f½i�1�ÞÞYj

i¼kþ1

ð1þ a½i�Þ

for kþ 1 6 j 6 n

if there is no unscheduled job from the same family of the (k + 1)thjob. Thus, the lower bound on the number of late jobs for partial se-quence PS is derived as

LB ¼Xk

j¼1

IfC ½j� � dmax > 0g þXn

j¼kþ1

IfC½j� � d½j� > 0g;

where the indicator function is defined as I{C[j] � dmax > 0} = 1 ifC[j] � dmax > 0 and I{C[j] � dmax > 0} = 0 otherwise.

3.3. The steps of the branch-and-bound algorithm

A depth-first search is adopted in the branching procedure. Thismethod has the advantage that it only requires very little storagespace and can be used for problems with a large number of jobs.In this paper, we assign jobs in a backward manner starting fromthe last position. In the searching tree, we choose a branch and sys-tematically work down the tree until we either eliminate it usingthe results from Sections 3.1 and 3.2 or reach its final node, whichis either used as a substitute for the initial solution or eliminated.The steps of the branch-and-bound algorithm are described asfollows:

Step 1. Perform the heuristic algorithm as described in the nextsubsection to obtain an initial incumbent solution S.Step 2. Utilize Properties 1–2 to eliminate the dominated partialsequences. For the non-dominated nodes, apply Corollary 1 tocheck whether the order of the unscheduled jobs can bedetermined.Step 3. Compute the lower bound on the number of late jobs ofthe unfathomed partial sequences or the number of late jobs ofthe completed sequences.Step 4. If the lower bound of the unfathomed partial sequence isgreater than or equal to the initial solution, eliminate the nodecorresponding to the partial sequence and all the nodes beyondit in the branch. If the value of the completed sequence is lessthan the initial solution, replace it as the new solution. Other-wise, eliminate it.Step 5. Output S.

3.4. Heuristic algorithm

French (1982) pointed out that apart from the choice of a searchstrategy and lower bounds, there are other ways in which we mayattempt to speed up the search process for the optimal solution.We may prime the procedure with a near-optimal schedule asthe first trial. In this section, a heuristic algorithm is proposed tofacilitate the search for the optimal solution. The steps of the algo-rithm are given as follows:

Step 1. Arrange jobs in the same family in the EDD order, sayF = {F1, F2, . . . , FM}

Step 2. For l M to 1 doChoose family Fi with the minimal number of late jobs fromF. If there is a tie of the number of late jobs, choose thefamily with maxj2Fi

fð1þ hiÞð1þ ajÞg. Place it on the lthfamily. Delete family Fi from F.endOutput sequence S;

Step 3. For i 1 to n doIf the ith job is a late job

For j 1 to i � 1 doCreate a new sequence Snew by interchanging jobs in the

ith and the jth positions from S.Replace S by Snew if the total number of late jobs in Snew is

less than that of Sendif

endOutput sequence S.

The idea of Step 2 of the algorithm is to process jobs in the samefamily successively to reduce the setup cost. However, there mightbe some tardy jobs in later processed families in the sequence ob-tained from Step 2. Thus, the purpose of Step 3 is to move late jobsto early positions to reduce the total number of late jobs.

Table 1Performance of the branch-and-bound algorithm with n = 10.

Property or lower bound included M R Branch-and-bound algorithm

CPU time Number of nodes

mean max mean Max

Enumeration 5 0.8 0.4 6.109 6.421 36,28,800 36,28,800Property 1 only 0.006 0.062 1097.83 15,088Property 2 only 0.005 0.062 1036.54 15,256Corollary 1 only 0.005 0.046 959.74 15,400Lower bound only 0.004 0.046 777.75 11,786All included 0.005 0.062 691.60 11,786

Enumeration 0.9 0.2 6.064 6.343 36,28,800 36,28,800Property 1 only 0.013 0.187 3331.73 52,292Property 2 only 0.014 0.203 3278.26 52,274Corollary 1 only 0.013 0.187 3251.65 52,381Lower bound only 0.006 0.093 951.33 22,689All included 0.007 0.125 928.31 22,684

Enumeration 10 0.8 0.4 6.530 7.093 36,28,800 36,28,800Property 1 only 0.005 0.062 648.02 16,019Property 2 only 0.004 0.062 637.27 15,914Corollary 1 only 0.003 0.046 499.88 15,999Lower bound only 0.004 0.062 537.17 15,947All included 0.004 0.078 403.61 15,927

Enumeration 0.9 0.2 6.186 6.531 36,28,800 36,28,800Property 1 only 0.009 0.078 1719.00 19,972Property 2 only 0.008 0.078 1725.56 20,650Corollary 1 only 0.008 0.062 1682.38 21,454Lower bound only 0.006 0.062 970.02 17,070All included 0.006 0.078 926.91 17,070

Table 2Performance of the branch-and-bound algorithm with n = 200.

M s R With the heuristic algorithm Without the heuristic algorithm

CPU time Number of nodes CPU time Number of nodes

mean Max mean max mean max mean max

20 0.7 0.1 0.077 0.109 207.96 399 0.091 0.109 399.00 3990.2 0.077 0.093 203.98 399 0.091 0.109 399.00 3990.4 0.077 0.093 200.00 200 0.086 0.109 397.01 399

0.8 0.1 0.081 0.156 211.94 598 0.121 0.156 595.04 7960.2 0.095 0.125 341.28 597 0.100 0.125 402.96 5970.4 0.082 0.109 229.85 399 0.096 0.109 399.00 399

0.9 0.1 0.141 0.718 624.62 4379 0.144 0.828 684.10 51700.2 0.108 0.203 408.39 992 0.129 0.203 602.92 992

40 0.7 0.1 0.105 0.140 203.98 399 0.134 0.203 363.17 7960.2 0.107 0.125 200.00 200 0.135 0.171 361.19 3990.4 0.105 0.125 200.00 200 0.128 0.171 309.45 399

0.8 0.1 0.691 6.515 2098.30 19,105 0.823 6.421 2615.06 19,3030.2 0.123 0.421 255.71 1195 0.172 0.453 442.69 13930.4 0.116 0.187 237.81 399 0.148 0.187 357.21 399

0.9 0.1 1.965 10.437 6139.50 34,066 2.100 10.328 6717.99 34,2630.2 1.510 7.015 4761.20 25,721 1.959 35.640 7023.53 190,713

60 0.7 0.1 0.133 0.218 207.96 399 0.152 0.250 263.68 3990.2 0.132 0.171 203.98 399 0.139 0.218 227.86 3990.4 0.131 0.203 203.98 399 0.133 0.203 205.97 399

0.8 0.1 0.189 5.120 325.37 11,543 0.302 5.187 585.89 11,7410.2 0.133 0.187 201.99 399 0.207 0.796 390.92 19810.4 0.140 0.281 221.88 597 0.147 0.265 249.74 597

0.9 0.1 1.153 5.843 2444.70 12,538 1.680 6.171 3675.01 12,7360.2 0.414 10.453 802.50 20,896 0.567 10.500 1159.84 21,291

W.-C. Lee et al. / Computers & Industrial Engineering 61 (2011) 782–787 785

4. Computational experiment

In order to evaluate the performance of the branch-and-boundalgorithm, a computational experiment was conducted in this sec-tion. The algorithm was coded in Fortran 90 and run on Compaq Vi-

sual Fortran version 6.6 in a personal computer with Dual CPU2.0 GHz, 1.99 GHz and 1.99 GB RAM on Windows XP. The experi-ment was designed according to Fisher’s framework (1976). Forall the test instances, the deterioration rates of the job processingtimes (aj) and of the family setup times (hi) were generated from

786 W.-C. Lee et al. / Computers & Industrial Engineering 61 (2011) 782–787

uniform distributions over 0 and 1. For each job j, the family code(fj) was generated from a discrete uniform distribution over theintegers between 1 and M. Notice that the actual number of fami-lies might not be known in advance. The due dates were generatedfrom another uniform distribution between T(1 � s � R/2) andT(1 � s + R/2), where R is the due date range, s is the tardiness fac-tor, and T is the product of 1 plus the job deterioration rates and 1plus the setup time deterioration rates, which is an estimation ofthe completion time of the last job where t0 = 1. That is,T ¼

Qnj¼1ð1þ ajÞ

QMi¼1ð1þ hiÞ. For the branch-and-bound algorithm,

the mean, the standard deviation and the maximum executiontimes (in seconds) as well as the mean, the standard deviationand the maximum numbers of nodes were reported.

The computational experiment was divided into three parts. Inthe first part, we studied the efficiency of the properties and thelower bound. We fixed the number of jobs at 10, the number offamilies at 5 and 10, and the values of (s, R) at (0.8, 0.4) and(0.9, 0.2). For each situation, we generated 100 instances randomly.The number of nodes and the execution time of the enumerationmethod was used as the comparison base, and we included thedominance properties or the lower bound one each time to thebranch-and-bound algorithm. The results were given in Table 1.It was seen that all the dominance properties and the lower boundare useful in cutting the search tree. It was also observed thatthe branch-and-bound algorithm with all the properties and the

Table 3Performance of the branch-and-bound algorithm for large job-sized problems.

n M s R CPU time

mean SD

500 50 0.8 0.2 1.007 0.2000.4 0.755 0.067

0.9 0.1 2.842 7.2440.2 1.153 0.457

100 0.8 0.2 1.227 0.3060.4 1.372 1.196

0.9 0.1 54.578 54.1430.2 31.73 55.888

150 0.8 0.2 1.519 0.1630.4 1.523 0.173

0.9 0.1 5.415 19.4960.2 1.819 0.877

750 75 0.8 0.2 2.475 0.6170.4 1.869 0.127

0.9 0.1 8.455 28.4540.2 9.065 42.362

150 0.8 0.2 3.302 0.6730.4 3.433 0.672

0.9 0.1 212.852 244.7490.2 71.855 197.418

225 0.8 0.2 4.598 0.2450.4 4.683 0.175

0.9 0.1 6.962 13.2830.2 4.892 1.177

1000 100 0.8 0.2 5.197 1.1450.4 4.031 0.248

0.9 0.1 56.062 166.4560.2 45.802 193.189

200 0.8 0.2 7.453 0.9250.4 7.630 1.378

0.9 0.1 420.822 669.1680.2 122.568 459.725

300 0.8 0.2 10.718 1.1050.4 10.329 0.283

0.9 0.1 10.636 1.6260.2 10.893 2.529

lower bound included is the most efficient in terms of the execu-tion time and the number of nodes. Thus, it was used in lateranalysis.

The second part of the experiment is to study the impacts of thenumber of families, the tardiness factor and the range factor, aswell as the heuristic algorithm on the performance of thebranch-and-bound algorithm. We fixed the number of jobs at200 and the numbers of groups at 20, 40 and 60. We chose (s, R)values at (0.7, 0.1), (0.7, 0.2), (0.7, 0.4), (0.8, 0.1), (0.8, 0.2),(0.8, 0.4), (0.9, 0.1) and (0.9, 0.2). For each case, we generated 100instances randomly. The results for the branch-and-bound algo-rithm with or without the heuristic algorithm were presented inTable 2. It was observed that problems are harder to solve whenthe value of s increases for a fixed value of R or when the valueof R decreases for a fixed value of s. The main reason is that theproperties are less powerful when the values of the due date areless variable or small. Thus, we will choose (s, R) at (0.8, 0.2),(0.8, 0.4), (0.9, 0.1) and (0.9, 0.2) in subsequent experiments sincethey are difficult cases. It was also seen that instances are easierto solve with the solution of the heuristic algorithm as an initialupper bound. It was seen in Table 2 that the average number ofnodes for the branch-and-bound algorithm without an initial solu-tion is 7023.53 when (M, s, R) = (40, 0.9, 0.2). With the help of theheuristic algorithm, it reduced to 4761.2, a reduction of more than30%. Thus, the branch-and-bound algorithm with the solution of

Number of nodes

max mean SD max

1.375 834.33 235.81 9991.109 514.97 85.55 999

47.375 2604.85 7024.88 44,4112.171 953.68 501.53 1996

2.796 559.88 177.74 149812.921 674.06 1098.36 11,419

139.656 28837.92 28548.01 68,364180.078 17659.42 31407.54 98,803

2.375 519.96 98.28 9992.578 514.97 85.55 999

170.562 1992.01 7489.03 65,8697.984 624.71 342.76 2994

3.640 1139.48 376.89 14992.843 757.49 74.90 1499

208.406 4067.05 13904.16 101,115322.671 4314.84 20487.4 152,048

9.187 772.47 166.80 22486.125 817.41 215.43 1499

636.843 53689.14 61740.73 144,557887.468 18425.75 50516.78 218,709

6.671 757.49 74.90 14995.109 764.98 105.38 1499

101.687 1079.55 2169.55 16,47913.609 787.45 195.57 2248

7.973 1439.56 498.38 19996.075 1009.99 99.90 1999

814.577 15731.75 46938.84 237,5271134.506 12318.38 51590.35 284,716

13.710 1059.94 238.44 199913.246 1079.92 272.38 1999

1885.698 61269.61 97151.06 262,7382879.423 17593.2 65444.98 402,598

16.495 265.61 1039.96 199910.947 134.86 1009.99 199926.578 269.59 1019.97 299731.155 327.55 1039.96 2998

W.-C. Lee et al. / Computers & Industrial Engineering 61 (2011) 782–787 787

the heuristic algorithm as an initial upper bound is adopted in sub-sequently experiments.

In this last part of the experiments, we tested the branch-and-bound algorithm with three different numbers of jobs (n = 500,750, and 1000). The number of families M took the values 0.1n,0.2n, and 0.3n. For each situation, we randomly generated 100 in-stances and presented the results in Table 3. It was seen that thebranch-and-bound algorithm can solve the instances of up to1000 jobs in a reasonable amount of time. The worst case tookabout 48 min when n = 1000, M = 200, and (s, R) = (0.9, 0.2). To testthe effects of the number of jobs, due dates, and number of fami-lies, we constructed a three-way analysis of variance (ANOVA) onthe number of nodes in Table 3. The resulting F-value on the num-ber of jobs was 9.61 with a p-value of less than 0.0001, which indi-cates that the number of jobs has a significant effect on thehardness of the problem. A closer look of Table 3 revealed thatthe number of nodes increases as the number of jobs increasessince this is an NP-hard problem. The test also showed that theF-value on the number of families was 113.47 with a p-value of lessthan 0.0001, which indicated that the number of families also has asignificant effect on the hardness of the problem. Finally, the teston the due date factor is also statistically significant with an F-va-lue of 86.12 and a p-value of less than 0.0001. The results showedthat the case (s, R) = (0.9, 0.2) is the hardest case to solve. This isdue to the fact that the properties are less potent when due datesare less variable and small, as explained earlier.

5. Conclusions

In this paper we considered a single-machine scheduling prob-lem with deteriorating jobs and setup times to minimize the num-ber of late jobs. We derived some dominance conditions, a lowerbound, and an initial upper bound by using a heuristic algorithmfor the problem to minimize the total number of late jobs. Thecomputational experiment showed that the proposed branch-and-bound algorithm can solve the instances of up to 1000 jobsin a reasonable amount of time.

Acknowledgements

The authors are grateful to the editor and the referees, whoseconstructive comments have led to a substantial improvement inthe presentation of the paper.

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