Optim LettDOI 10.1007/s11590-012-0575-4

ORIGINAL PAPER

Scheduling jobs with release dates on parallel batchprocessing machines to minimize the makespan

L. L. Liu · C. T. Ng · T. C. E. Cheng

Received: 6 December 2011 / Accepted: 27 September 2012© Springer-Verlag Berlin Heidelberg 2012

Abstract Batch processing happens in many different industries, in which a numberof jobs are processed simultaneously as a batch. In this paper we develop two heuristicsfor the problem of scheduling jobs with release dates on parallel batch processingmachines to minimize the makespan and analyze their worst-case performance ratios.We also present a polynomial-time optimal algorithm for a special case of the problemwhere the jobs have equal processing times.

Keywords Scheduling · Release dates · Batch processing · Makespan

1 Introduction

Ikura and Gimple [5] first studied the deterministic batch scheduling problems. A batchprocessing machine is one that can process several jobs simultaneously as a batch. Theprocessing time of a batch is equal to the largest processing time of the jobs in thebatch. Once processing is begun on a batch, no job can be removed from or addedto the batch. Since the processing of the burn-in operations is very long comparedwith the other steps of the final testing stage, its efficient scheduling is very critical toinventory management, productivity improvement, and on-time delivery. More detailsabout the background of this model is given in [5].

In this paper we study the problem of scheduling jobs with release dates on parallelbatch processing machines to minimize the makespan. Specifically, there are n jobs

L. L. LiuSchool of Science, Shanghai Second Polytechnic University,No. 2360 Jinhai Road, Pudong Shanghai 201209, China

C. T. Ng (B) · T. C. E. ChengDepartment of Logistics and Maritime Studies, The Hong Kong Polytechnic University Hung Hom,Kowloon, Hong Konge-mail: lgtctng@polyu.edu.hk

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to be processed on m parallel batch processing machines. Each job j ( j = 1, . . . , n) isassociated with a release date r j , at which the job arrives and becomes available, anda processing time p j , which is the time needed to process the job. A batch processingmachine can process up to B jobs at the same time, where B < n. The processingtime of a batch is equal to the largest processing time of the jobs in the batch. If abatch contains exactly B jobs, we call it a full batch; otherwise, we call it a partialbatch. This type of batch processing is called parallel batch scheduling (p-batch).There is also serial batch scheduling (s-batch), where the processing time of a batchis equal to the sum of the processing times of the jobs in the batch. For both types ofbatch scheduling, the completion time of a job is defined as the completion time ofthe batch it belongs to. Extending the traditional three-field notation for schedulingproblems, Brucker et al. [2] introduced p-batch in the second field to denote parallelbatch scheduling problems. Following Brucker et al. [2], we denote the problem understudy as P|r j , p-batch|Cmax , where P and Cmax indicates parallel machines and themakespan, respectively.

For the single-machine case, considerable research has been carried out. Bruckeret al. [2] proved that the problem 1|r j , p-batch|Cmax is strongly NP-hard even ifB = 2. Lee and Uzsoy [6] developed polynomial-time algorithms for several specialcases of the problem and provided several heuristics for the general case. Liu andYu [10] showed that the problem is binary NP-hard even if the jobs are subject totwo distinct release dates. They presented a pseudopolynomial-time algorithm for thecase where there are a fixed number of distinct release dates. Independently, Deng etal. [4] presented a pseudopolynomial-time algorithm for the same special case of theproblem. Based on the pseudopolynomial-time algorithm they provided, Deng et al. [4]developed a polynomial-time approximation scheme for the problem. Yuan et al. [12]showed that the scheduling problem with family jobs and release dates on a singlebatch processing machine to minimize the makespan is strongly NP-hard and theydeveloped two dynamic programming algorithms and a heuristic with a performanceratio 2. For the scheduling problems with release dates and other objective functions ona single batch processing machine, Cheng et al. [3] proved that the scheduling problemwith release dates and deadlines on an unbounded batch processing machine is NP-hard and provided polynomial-time algorithms for several special cases. Liu et al. [11]showed that the problem of minimizing the total tardiness on a single unbounded batchprocessing machine is NP-hard and developed pseudopolynomial solvable problemson an unbounded batch processing machine. Liu et al. [9] presented a polynomial-time approximation scheme for the problem of scheduling jobs with release dates on asingle batch processing machine to minimize the total completion time and developeda fully polynomial-time approximation scheme for the case with an unbounded batchprocessing machine.

To the best of our knowledge, little research has been done on the problem ofminimizing the makespan on parallel batch processing machines with job releasedates except Li et al. [8], who presented a polynomial-time approximation scheme forthis problem.

We organize the rest of the paper as follows: In Sect. 2 we present two heuristics andanalyze their performance ratios. In Sect. 3 we propose a polynomial-time algorithmfor the special case where the jobs have equal processing times.

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2 Heuristics

We first develop an integer programming formulation for problem P|r j , p-batch|Cmax .Given a feasible solution, we denote the processing time, release date, and com-pletion time of the j-th batch on machine k as p jk , r jk , and y jk , respectively, forj = 1, . . . , n and k = 1, . . . , m. Then, we can formulate the problem considered inthis paper as the following integer programming problem, where xi jk takes the value1 if job i is assigned to the j-th batch on machine k and the value 0 otherwise fori = 1, . . . , n, j = 1, . . . , n, and k = 1, . . . , m, and Z denotes the makespan:

minZ

s.t.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

n∑

j=1

n∑

k=1xi jk = 1, i = 1, . . . , n;

n∑

i=1xi jk ≤ B, j = 1, . . . , n; k = 1, . . . , m;

pi xi jk ≤ p jk, i = 1, . . . , n; j = 1, . . . , n; k = 1, . . . , m;ri xi jk ≤ r jk, i = 1, . . . , n; j = 1, . . . , n; k = 1, . . . , m;y( j−1)k + p jk ≤ y jk, j = 1, . . . , n; k = 1, . . . , m;r jk + p jk ≤ y jk, j = 1, . . . , n; k = 1, . . . , m;ynk ≤ Z , k = 1, . . . , m;xi jk ∈ {0, 1}, i = 1, . . . , n; j = 1, . . . , n; k = 1, . . . , m;

where y0k = 0, p jk, r jk, y jk, Z ≥ 0, j = 1, . . . , n, and k = 1, . . . , m.We develop two heuristics for problem P|r j , p-batch|Cmax , which are based

on Algorithm FBLPT (Full Batch Largest Processing Time) [1] for problem1|p-batch|Cmax . Algorithm FBLPT first orders the jobs in non-increasing order oftheir processing times, then assigns adjacent B jobs as a batch from the beginning untilall the jobs have been assigned, and finally arranges the batches in any arbitrary order.We define available jobs as the jobs that have arrived but have not yet been scheduled.

The first heuristic GP(Greedy Processing) is based on the intuition that if there isa machine becoming free and there are jobs available, we should assign as many jobswith the largest possible processing times as possible to this machine. It generalizesthe following two algorithms: Algorithm GRLPT (Greedy Longest Processing Time)presented by Lee and Uzsoy [6] for problem 1|r j , p-batch|Cmax and Algorithm BLPTpresented by Lee et al. [7] for problem P|p-batch|Cmax .

Algorithm GRLPT

Whenever the machine becomes free, put as many unscheduled available jobs withthe largest possible processing times as possible into one batch and assign this batchto the machine.

Liu and Yu [10] proved that the performance ratio of Algorithm GRLPT is 2 andthat this bound is tight. We give an alternative proof of this result, which is simplerand more straightforward.

Theorem 1 The performance ratio of Algorithm GRLPT for problem 1|r j , p-batch|Cmax is 2.

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Fig. 1 Schedule obtained by algorithm GRLPT

Proof Let C∗max and Cmax be the optimal makespan and the makespan obtained by

Algorithm GRLPT, respectively; rmax be the last release date; and Bl be the batch thatstarts strictly before rmax and completes at or after time rmax . The processing time andstart time of batch Bl are denoted by pl and sl , respectively; the job set processed afterbatch Bl is denoted by S, which is arranged by Algorithm FBLPT according to theprocedures of Algorithm GRLPT; and the makespan obtained by applying AlgorithmFBLPT to S is denoted by Cmax (S). Then we consider two sub-cases.

Case 1 Batch Bl is a partial batch. Then all the jobs in S must be released aftertime sl according to the algorithm and we have sl + Cmax (S) ≤ C∗

max . Sincepl ≤ C∗

max , it follows that (see Fig. 1)

Cmax = sl + pl + Cmax (S) ≤ 2C∗max .

Case 2 Batch Bl is a full batch. If the largest processing time of the jobs inS is no larger than the smallest processing time of the jobs in batch Bl , thenbatch Bl and S are a partial schedule obtained by applying Algorithm FBLPTto the job set Bl

⋃S. We have pl + Cmax (S) = Cmax (Bl

⋃S) ≤ C∗

max . Sincesl ≤ rmax ≤ C∗

max , we have Cmax = sl + pl + Cmax (S) ≤ 2C∗max . Otherwise, the

largest processing time in S is larger than the smallest processing time in batch Bl

and the jobs in S with larger processing times than the smallest processing time inbatch Bl must be released after time sl . Denote this job set as S1 and S2 = S − S1.Then we have Cmax (S) ≤ Cmax (S1) + Cmax (S2), sl + Cmax (S1) ≤ C∗

max , andpl + Cmax (S2) = Cmax (Bl

⋃S2) ≤ C∗

max . Thus, we have

Cmax = sl + pl + Cmax (S) ≤ sl + pl + Cmax (S1) + Cmax (S2) ≤ 2C∗max .

��

Algorithm BLPT

Whenever a machine becomes free, put as many unscheduled jobs with the largestpossible processing times as possible into one batch and assign this batch to themachine. The performance ratio of Algorithm BLPT for problem P|p-batch|Cmax is4/3 − 1/3m [7].

Inspired by Algorithm GRLPT and Algorithm BLPT, we develop the followingAlgorithm GP for problem P|r j , p-batch|Cmax .

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Table 1 Schedule obtained by Algorithm GP

Machine Start time Batches assigned

1 0 (3(m − 1))(1), (3(m − 1))(1)(2m − 3)(B−1)

2 δ (3(m − 1))(1), (2m − 3)(B), (m − 1)(B), (m − 1)(1)

3 2δ (3(m − 1))(1), (2m − 3)(1)(2m − 4)(B−1), (m − 1)(B)

.

.

....

.

.

.

m − 1 (m − 2)δ (3(m − 1))(1), (3(m − 1)/2)(B), ((3m − 1)/2)(B)

m (m − 1)δ (3(m − 1))(1), (3(m − 1)/2)(1)((3m − 1)/2)(B−1), ((3m − 1)/2)(B)

Algorithm GP

Whenever a machine becomes free, put as many unscheduled available jobs with thelargest possible processing times as possible into one batch and assign this batch tothe machine.

Theorem 2 The performance ratio of Algorithm GP is 10/3 − 1/3m.

Proof Let the batch starting strictly before time rmax and completing at or after timermax on machine i be batch Bi , the start time of batch Bi be si , the processing timeof batch Bi be pi (i = 1, . . . , m), and the batch Bl satisfy l = max1≤i≤m{si + pi }.Let S denote the job set that starts at or after time rmax and Cmax (S) be the makespanobtained by applying Algorithm GP to S. Assume that all the jobs in S are released attime zero. Then we have sl ≤ C∗

max , pl ≤ C∗max , and Cmax (S) ≤ (4/3 − 1/3m)C∗

max .It follows that

Cmax ≤ sl + pl + Cmax (S) ≤(

10

3− 1

3m

)

C∗max .

��Obviously, Algorithm GP is an on-line algorithm and we have no information about

the jobs before they arrive. We believe that the performance ratio 10/3 − 1/3m is nottight. The following example shows that the worst-case performance ratio of AlgorithmGP is at least 7/3 − 1/3(m − 1) for any given number of machines m.

Consider an instance with B = m + 1 and n = 2m B = 2m(m + 1). Withoutloss of generality, we assume m is an odd number and δ is an arbitrary small positivenumber. r1 = 0, r2 = δ, r3 = 2δ, . . . , rm = (m − 1)δ, and rm+1 = rm+2 = · · · =rn = mδ.p1 = p2 = · · · = pm+1 = 3(m + 1), pm+2 = pm+3 = · · · = p3(m+1) =2m − 3, p3(m+1)+1 = · · · = p5(m+1) = 2m − 4, . . . , p(2m−5)(m+1)+1 = · · · =p(2m−3)(m+1) = m, and p(2m−3)(m+1)+1 = · · · = p2m(m+1) = m − 1. The makespanobtained by Algorithm GP is 7m−8+δ (see Table 1, in which the number in the bracketsdenote the processing time of the batch and the number in the brackets on the top righthand corner denotes the number of jobs in the batch, and different batches are separatedby commas). The optimal makespan of the instance is 3(m − 1) + mδ (see Table 2).

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Table 2 The optimal scheduleMachine Start time Batches assigned

1 mδ (3(m − 1))(B)

2 mδ (2m − 3)(B), (m)(B)

3 mδ (2m − 3)(B), (m)(B)

.

.

....

.

.

.

m − 1 mδ ((3m − 1)/2)(B), ((3m − 5)/2)(B)

m mδ (m − 1)(B), (m − 1)(B), (m − 1)(B)

Hence, the performance ratio of Algorithm GP is at least 7m−8+δ3(m−1)+mδ

→ 73 − 1

3(m−1)when δ → 0.

Considering the last release date rmax , we next present another algorithm for prob-lem P|r j , p-batch|Cmax .

Algorithm RP (Processing based on the largest Release date)

Apply Algorithm GP until there is a batch Bl that starts at or before time rmax andcompletes strictly after time rmax , then apply Algorithm BLPT to all the remainingjobs, together with the jobs in batch Bl at time rmax .

Theorem 3 The performance ratio of Algorithm RP is 7/3 − 1/3m and the bound istight.

Proof Let Cmax and C∗max denote the makespan obtained by Algorithm RP and the

optimal makespan, respectively. The job set processed after time rmax is denoted byS and the makespan obtained by applying Algorithm BLPT to job set S is denoted byCmax (S). Since rmax ≤ C∗

max and Cmax (S) ≤ (4/3 − 1/3m)C∗max ,

Cmax = rmax + Cmax (S) ≤(

7

3− 1

3m

)

C∗max .

The following instance shows that the bound is tight. Consider an instance ofproblem P|r j , B|Cmax with B = m + 1 and n = 2m B = (2m + 1)(m + 1) + 1.Without loss of generality, we assume that m is an even number and δ is an arbitrarysmall positive number. r1 = 0, r2 = δ, r3 = 2δ, . . . , rm = (m − 1)δ, rm+1 = rm+2 =· · · = r(2m+1)(m+1) = mδ, and r(2m+1)(m+1)+1 = 3m.p1 = p2 = · · · = p2(m+1) =2m −1, p2(m+1)+1 = · · · = p4(m+1) = 2m −2, p4(m+1)+1 = · · · = p6(m+1) = 2m −3, . . . , p(2m−4)(m+1)+1 = · · · = p(2m−2)(m+1) = m + 1, p(2m−2)(m+1)+1 = · · · =p(2m+1)(m+1) = m, and p(2m+1)(m+1)+1 = δ. Algorithm RP generates a schedule withmakespan 7m−1 (see Table 3). The optimal makespan, however, is 3m+(m+1)δ (seeTable 4.). Thus, the worst-case performance ratio of Algorithm RP is 7m−1

3m+(m+1)δ→

73 − 1

3m when δ → 0. ��For the special case where all the jobs have the same processing times, we denote it

as P|r j , p-batch, p j = p|Cmax and provide the following Algorithm FBERD (FullBatch Earliest Release Dates) to treat it.

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Table 3 Schedule obtained by Algorithm RP

Machine Start time Batches assigned Start time Batches assigned

1 0 (2m − 1)(1) 3m (2m − 1)(B), (m)(B), (m)(1)

2 δ (2m − 1)(1) 3m (2m − 1)(1)(2m − 2)(B−1), (m)(B)

3 2δ (2m − 1)(1) 3m (2m − 2)(B), (m + 1)(1)(m)(B−1)

.

.

....

.

.

....

.

.

.

m − 1 (m − 2)δ (2m − 1)(1) 3m (3m/2)(B), ((3m/2) − 1)(1)((3m/2) − 2)(B−1)

m (m − 1)δ (2m − 1)(1) 3m (3m/2)(1)((3m)/2) − 1)(B−1), ((3m/2) − 1)(B)

Table 4 The optimal scheduleMachine Start time Batches assigned

1 mδ (2m − 1)(B), (m + 1)(B)

2 mδ (2m − 1)(B), (m + 1)(B)

3 mδ (2m − 2)(B), (m + 2)(B)

.

.

....

.

.

.

m − 1 mδ (3m/2)(B), (3m/2)(B)

m mδ (m)(B), (m)(B), (m)(B)

Table 5 Performance ratios ofthe heuristics to lower boundsand optimal values

GP/LB GP/OP RP/LB RP/OP

1.19 1 1.44 1.21

1.18 1.11 1.24 1.17

1.33 1.05 1.53 1.21

1.26 1.04 1.37 1.13

1.23 1.07 1.23 1.07

Algorithm FBERD

Step 1. Order the jobs in non-decreasing order of their release dates.Step 2. Assign the first n −n/BB jobs at time r j on one of the machines, wherej = n − n/BB. For the remaining jobs, if there are B jobs available, put theminto one batch and assign them to the first available machine; otherwise, wait untilthere are B jobs available.

Theorem 4 Algorithm FBERD yields an optimal solution to problem P|r j , p-batch,

p j = p|Cmax .

Proof Let S be the schedule generated by Algorithm FBERD and S∗ be an optimalschedule. We adjust schedule S∗ such that the first batch contains n − n/BB jobsand the other batches contain exactly B jobs without increasing the makespan. Theresulting schedule is denoted as S

′. Then S is also optimal since schedule S is almost

the same as S′

except that each batch starts at the earliest possible time. ��

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Table 6 Performance ratios of the heuristics to lower bounds for instances with processing times U[1,100]and release date ratio 0.5

n m = 3 m = 5GP RP GP RPMean Max Mean Max Mean Max Mean Max

B = 3

20 1.33 1.40 1.35 1.44 1.09 1.19 1.10 1.20

40 1.23 1.31 1.37 1.44 1.24 1.37 1.39 1.54

60 1.15 1.19 1.25 1.32 1.25 1.29 1.40 1.43

80 1.13 1.22 1.15 1.28 1.19 1.24 1.34 1.38

100 1.10 1.11 1.17 1.23 1.16 1.20 1.30 1.40

150 1.08 1.09 1.12 1.18 1.11 1.14 1.21 1.25

200 1.07 1.09 1.11 1.12 1.09 1.11 1.16 1.21

300 1.05 1.06 1.08 1.09 1.06 1.07 1.10 1.14

B = 5

20 1.21 1.39 1.11 1.18 1.17 1.31 1.11 1.21

40 1.31 1.42 1.38 1.49 1.22 1.46 1.09 1.13

60 1.30 1.36 1.42 1.45 1.22 1.36 1.30 1.41

80 1.23 1.26 1.39 1.46 1.30 1.39 1.48 1.55

100 1.19 1.24 1.31 1.43 1.30 1.34 1.44 1.50

150 1.14 1.16 1.23 1.32 1.21 1.23 1.36 1.44

200 1.11 1.14 1.17 1.20 1.17 1.20 1.26 1.30

300 1.09 1.10 1.13 1.15 1.11 1.13 1.19 1.21

3 Computational experiments

We conducted a series of computational experiments to evaluate the performance ofthe heuristic algorithms by generating random instances. Since the problem understudy is strongly NP-hard, it is difficult to obtain the optimal values for instances witha large number of jobs within a reasonable time. So for such instances, we comparethe makespan obtained by the heuristics with the largest value among the followinglower bounds for the optimal value.

L B1: In any feasible schedule, the earliest possible completion time of job j isr j + p j . Hence we have L B1 = max1≤ j≤n{r j + p j }.L B2: Order the jobs in increasing order of their release dates r j . Let C F BL PT

max ( j, n)

be the makespan obtained by applying Algorithm FBLPT to jobs j through n,assuming that they are released at time zero. Algorithm FBLPT is optimal forproblem 1|p-batch|Cmax [2]. Then we have

L B2 = max1≤ j≤n

{

r j + C F BL PTmax ( j, n)

m

}

.

L B3: Order the jobs in increasing order of their release dates r j . Let C BL PTmax ( j, n)

be the makespan obtained by applying Algorithm BLPT to jobs j through n,

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Scheduling jobs with release dates

Table 7 Performance ratios of the heuristics to lower bounds for instances with processing times U[1,100]and release date ratio 1

n m = 3 m = 5GP RP GP RPMean Max Mean Max Mean Max Mean Max

B = 3

20 1.09 1.23 1.19 1.28 1.08 1.19 1.16 1.25

40 1.10 1.17 1.16 1.34 1.06 1.16 1.11 1.27

60 1.09 1.16 1.15 1.27 1.06 1.13 1.12 1.22

80 1.10 1.14 1.17 1.25 1.05 1.12 1.13 1.24

100 1.11 1.13 1.14 1.17 1.07 1.12 1.13 1.23

150 1.12 1.16 1.15 1.22 1.08 1.10 1.13 1.18

200 1.12 1.13 1.14 1.17 1.10 1.15 1.14 1.19

300 1.11 1.15 1.14 1.18 1.10 1.11 1.13 1.15

B = 5

20 1.14 1.23 1.10 1.24 1.05 1.16 1.09 1.19

40 1.12 1.28 1.16 1.25 1.09 1.25 1.07 1.15

60 1.09 1.15 1.17 1.25 1.07 1.15 1.14 1.24

80 1.13 1.19 1.19 1.24 1.05 1.08 1.14 1.24

100 1.10 1.12 1.17 1.23 1.06 1.12 1.13 1.26

150 1.13 1.18 1.19 1.25 1.06 1.11 1.16 1.21

200 1.13 1.15 1.18 1.22 1.09 1.12 1.14 1.18

300 1.14 1.17 1.17 1.22 1.10 1.15 1.15 1.19

assuming that they are released at time zero. Algorithm BLPT is a 4/3 − 1/3mapproximation algorithm for problem P|p-batch|Cmax [7]. Thus we have

L B3 = max1≤ j≤n

{

r j + C BL PTmax ( j, n)

4/3 − 1/3m

}

.

There are five parameters that may influence the performance of the heuristics:number of machines (m), machine capacity (B), number of jobs (n), processing times(p j , 1 ≤ j ≤ n), and release dates (r j , 1 ≤ j ≤ n). We generated two levelsof processing times from two discrete uniform distributions U[1,100] and U[1,20],respectively. To generate the release dates, we first computed the makespan C BL PT

maxobtained by applying Algorithm BLPT to all the jobs by assuming that they are releasedat time zero, then generated the release dates from a discrete uniform distribution rang-ing from 0 to F R · C BL PT

max , where FR is the frequency of the release dates, and wecall it the release date ratio. Thus we generated all the instances by assigning dif-ferent values to the five parameters, whereby there were two different values of m(3 and 5), two different values of B (3 and 5), eight different values of n (20, 40,60, 80, 100, 150, 200, and 300), two different levels of processing time distribu-tion (U[1,100] and U[1,20]), and two different values of the release date ratio (0.5

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Table 8 Performance ratios of the heuristics to lower bounds for instances with processing times U[1,20]and release date ratio 0.5

n m = 3 m = 5GP RP GP RPMean Max Mean Max Mean Max Mean Max

B = 3

20 1.36 1.48 1.37 1.45 1.15 1.34 1.09 1.27

40 1.20 1.29 1.34 1.44 1.22 1.42 1.38 1.55

60 1.13 1.18 1.22 1.28 1.24 1.30 1.39 1.45

80 1.13 1.18 1.24 1.31 1.18 1.26 1.34 1.40

100 1.09 1.12 1.16 1.20 1.14 1.17 1.25 1.38

150 1.07 1.08 1.12 1.17 1.10 1.12 1.19 1.24

200 1.06 1.07 1.10 1.14 1.08 1.09 1.14 1.20

300 1.05 1.06 1.07 1.08 1.06 1.07 1.10 1.13

B = 5

20 1.22 1.38 1.09 1.16 1.14 1.22 1.09 1.21

40 1.31 1.47 1.45 1.55 1.21 1.48 1.10 1.20

60 1.30 1.39 1.41 1.45 1.20 1.27 1.29 1.38

80 1.23 1.26 1.39 1.45 1.30 1.41 1.45 1.54

100 1.17 1.22 1.29 1.42 1.24 1.34 1.35 1.41

150 1.13 1.17 1.20 1.28 1.18 1.21 1.33 1.39

200 1.10 1.13 1.14 1.18 1.15 1.17 1.23 1.27

300 1.08 1.10 1.11 1.15 1.11 1.13 1.17 1.21

and 1). There are totally 128 combinations of these five parameters. For each com-bination, we applied the two heuristics to ten randomly generated instances, whichresulted in a total of 1,280 instances. We coded all the algorithms and the lowerbound in Matlab 6.0 and ran the computational experiments on a Pentium 4/2.0 GHzpersonal computer.

We first compare the makespan obtained by the heuristics with the optimal val-ues obtained by integer programming and with the lower bound, respectively, forinstances with eight jobs, in which B = 2, m = 2, the processing time distributionis U[1,10], and the release date ratio is 0.5. Table 5 reports the computational results,where GP/LB (RP/LB) and GP/OP (RP/OP) denote the ratios of the makespan gen-erated by Algorithm GP (RP) to the lower bound and the optimal value, respectively.The results indicate that the differences in the ratios between the heuristics to thelower bound and to the optimal value are about 0.2, revealing the effectiveness of thelower bounds.

Naturally, it is convenient to compare the makespan obtained by the heuristics withthe lower bounds for instances with a large number of jobs. Tables 6, 7, 8, 9 reportthe computational results instances with a large number of jobs. For the ten instancesof each combination, they show the average and maximum ratios obtained by theheuristics to the largest lower bounds on the optimal values. In general, both algorithmsperform well since almost all the ratios are between 1 and 1.5, and Algorithm GP

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Table 9 Performance ratios of the heuristics to lower bounds for instances with processing times U[1,20]and release date ratio 1

n m = 3 m = 5GP RP GP RPMean Max Mean Max Mean Max Mean Max

B = 3

20 1.13 1.26 1.16 1.29 1.09 1.23 1.08 1.19

40 1.07 1.18 1.17 1.32 1.05 1.14 1.12 1.26

60 1.08 1.15 1.16 1.22 1.02 1.05 1.11 1.19

80 1.11 1.15 1.16 1.19 1.06 1.12 1.13 1.26

100 1.11 1.15 1.15 1.21 1.07 1.13 1.15 1.24

150 1.10 1.13 1.15 1.16 1.08 1.12 1.14 1.18

200 1.11 1.13 1.13 1.16 1.08 1.10 1.12 1.16

300 1.12 1.14 1.14 1.17 1.11 1.14 1.14 1.19

B = 5

20 1.10 1.19 1.12 1.38 1.04 1.12 1.08 1.19

40 1.11 1.32 1.21 1.55 1.03 1.08 1.08 1.18

60 1.11 1.20 1.22 1.40 1.09 1.16 1.10 1.20

80 1.10 1.16 1.16 1.22 1.05 1.11 1.11 1.24

100 1.11 1.16 1.18 1.24 1.05 1.10 1.13 1.20

150 1.12 1.16 1.17 1.28 1.07 1.12 1.17 1.21

200 1.13 1.16 1.16 1.21 1.09 1.15 1.15 1.21

300 1.14 1.15 1.16 1.19 1.11 1.13 1.15 1.17

performs much better than Algorithm RP with respect to the average ratio and themaximum ratio because the latter delays some jobs to the largest release date.

Next we examine the results related to the five job parameters. From the four tableswe notice that the patterns of the performance of the heuristics are very different forthe cases of smaller release date ratio and larger release date ratio, so we discuss thecomputational results according to these two cases.

If the jobs are released frequently, i.e., the release ratio is small, then from Tables 6and 8, we see that if the number of jobs increases, or number of machines decreases, ormachine capacity decreases, or processing time variance decreases, the performanceof both algorithms improves. The reason is that if the number of jobs increases, orprocessing time variance decreases, or machine capacity decreases, there are morejobs with close processing times available at a time and both algorithms will assignunscheduled jobs with close processing times into one batch. If the number of machinesdecreases, the workload on each machine can be balanced much more than the casewith a large number of machines.

If the jobs are released less frequently, the above mentioned pattern does not existand there are no significant differences in the performance ratios of Algorithms GPand RP if the number of jobs, number of machines, machine capacity, or processingtime variance changes.

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With respect to the parameter of release dates, if the jobs are released less frequently,both algorithms perform better than the case with a small ratio of release dates. How-ever, this performance gap becomes smaller when the number of jobs increases.

4 Conclusions

In this paper we consider the problem of scheduling jobs with release dates on parallelbatch processing machines. Since this problem is strongly NP-hard, we provide twoefficient heuristic algorithms and evaluate their performance by exploring their per-formance ratios and conducting computational experiments. We also identify a specialcase of the problem that can be solved in polynomial time.

A challenging topic for future research is to develop a better performance ratiofor the on-line Algorithm GP. To develop more efficient heuristics than the heuristicspresented in this paper is also worth further studying.

Acknowledgments We thank two anonymous referees for their helpful comments on an earlier ver-sion of our paper. This research was supported in part by the Logistics Research Centre of The HongKong Polytechnic University, and the NSFC under Grant No. 71072157. The first author was also sup-ported by Institute of Applied Mathematics, “Dawn” Academy, Shanghai Second Polytechnic UniversityNo. A30XK121y03.

References

1. Bartholdi, J.J.: unpublished manuscript (1988)2. Brucker, P., Gladky, A., Hoogevreen, H., Kovalyov, M.Y., Potts, C.N., Tautenhahn, T., Van de Velde,

S.: Scheduling a batching machine. J Schedul 1, 31–54 (1998)3. Cheng, T.C.E., Liu, Z.H., Yu, W.C.: Scheduling jobs with release dates and deadlines on a batch

processing machine. IIE Trans. 33, 685–690 (2001)4. Deng, X.T., Poon, C.K., Zhang, Y.Z.: Approximation algorithms in batch processing. J. Comb. Optim.

7, 247–257 (2003)5. Ikura, Y., Gimple, M.: Efficient scheduling algorithms for a single batch processing machine. Oper.

Res. Lett. 5, 61–65 (1986)6. Lee, C.Y., Uzsoy, R.: Minimizing makespan on a single batch processing machine with dynamic job

arrivals. Int. J. Prod. Res. 37, 219–236 (1999)7. Lee, C.Y., Uzsoy, R., Martin Vega, L.A.: Efficient algorithm for scheduling semiconductor burn-in

operations. Oper. Res. 40, 764–775 (1992)8. Li, S.G., Li, G.J., Zhang, S.Q.: Minimizing makespan with release times on identical parallel batching

machines. Discret. Appl. Math. 148, 127–134 (2005)9. Liu, Z.H., Cheng, T.C.E.: Approximation schemes for minimizing total (weighted) completion time

with release dates on a batch machine. Theor. Comput. Sci. 347, 288–298 (2005)10. Liu, Z.H., Yu, W.C.: Scheduling one batch processor subject to job release dates. Discret. Appl. Math.

105, 129–136 (2000)11. Liu, Z.H., Yuan, J.J., Cheng, T.C.E.: On scheduling an unbounded batch machine. Oper. Res. Lett. 31,

42–48 (2003)12. Yuan, J.J., Liu, Z.H., Ng, C.T., Cheng, T.C.E.: The unbounded single machine parallel batch scheduling

problem with family jobs and release dates to minimize makespan. Theor. Comput. Sci. 320, 199–212(2004)

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