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European Journal of Operational Research 186 (2008) 1212–1217

www.elsevier.com/locate/ejor

Short Communication

Single machine unbounded parallel-batch schedulingwith forbidden intervals q

Jinjiang Yuan *, Xianglai Qi, Lingfa Lu, Wenhua Li

Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People’s Republic of China

Received 4 October 2006; accepted 22 February 2007Available online 25 April 2007

Abstract

In this paper we consider the single machine parallel-batch scheduling with forbidden intervals. There are some forbid-den intervals in which the machine cannot be available. The jobs are processed in batches form in the remaining free time-slots without preemption, where the processing time of a batch is defined to be the maximum processing time of the jobs inthis batch. We show that, when the objective is bottleneck form, maximum lateness, or makespan with release dates ofjobs, the considered problem can be solved in polynomial time.� 2007 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Parallel-batch; Forbidden intervals; Compact schedule

1. Introduction and problem formulation

Let n jobs J1, J2, . . . , Jn and a single machine thatcan simultaneously process a batch of jobs be given.There are k specified forbidden intervals

½a1; b1Þ; ½a2; b2Þ; . . . ; ½ak; bkÞ;

which are occupied by machine maintenance (see[1,10,11]). The forbidden intervals can also be inter-preted as unavailability periods (see [12]) of the ma-chine. We assume in this paper that, ai and bi arepositive integers with ai < bi, i = 1, 2, . . . ,k, andbi < ai+1, i = 1, 2, . . . ,k� 1. We further definea0 = �1 and b0 = 0 for the convenience of the dis-

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2007.02.051

q Project partially supported by NSFC (10671183).* Corresponding author.

E-mail address: yuanjj@zzu.edu.cn (J. Yuan).

cussion. The jobs must be processed in the remain-ing k + 1 free time-slots (or intervals)

½b0; a1Þ; ½b1; a2Þ; . . . ; ½bk�1; akÞ; ½bk;1Þ

without preemption. Each job Jj has an integral pro-cessing time pj P 0, an integral release date rj P 0,and an integral due date dj P 0. The jobs are pro-cessed in parallel batches. The processing time of abatch is defined to be the largest processing timeof all jobs in the batch, and the release date of abatch is defined to be the largest release date of alljobs in the batch. Hence, if B is a batch that consistsof some jobs, then we define

pðBÞ ¼ maxfpj : J j 2 Bg; and

ðBÞ ¼ maxfrj : J j 2 Bg:

The completion time of all jobs in a batch is definedas the completion time of the batch. All jobs in the

.

Table 1Complexity of 1jp-batch; rjjfProblem Complexity Reference

1jp-batchjfmax Polynomial time [2]1jp-batchjRfj Pseudo-polynomial time [2]1jp-batchjLmax O(n2) [2]1jp-batchjRwjCj O(n logn) [2]1jp-batchjRUj O(n3) [2]1jp-batch; rjjCmax O(nlogn) [13]1jp-batchjRwjUj NP-hard [2]1jp-batchjRTj NP-hard [15]1jp-batch; rjjLmax NP-hard [14]1jp-batch; rjjRwjCj NP-hard [7]

J. Yuan et al. / European Journal of Operational Research 186 (2008) 1212–1217 1213

same batch start and complete simultaneously. Weuse S(B) = S(B, p) and C(B) = C(B, p) to denotethe starting time and completion time of batch B

in a schedule p. It is clear that the starting time ofa batch is no earlier than the maximum release dateof the jobs in it.

Following [2,8], we call this model the parallel-batch scheduling problem with forbidden intervalsand denote it by

1jp-batch; FB; rjjf ;where ‘‘p-batch’’ means parallel-batch, ‘‘FB’’ meansforbidden interval, and f is the objective function,which is a function of the job completion times Cj

under a given schedule, to be minimized. In this pa-per we will suppose that the objective function f(C),for C = (C1, C2, . . . , Cn) P 0, is of the followingforms:

MAX-form : f ðCÞ ¼ maxffiðCiÞ : 1 6 i 6 ng;SUM-from : f ðCÞ ¼ f1ðC1Þ þ f2ðC2Þ þ . . .þ fnðCnÞ:

We also suppose that the function f is regular [2],i.e., fi(Ci) is nondecreasing in Ci for each i, and that,for each given x P 0, fi(x) can be calculated in aconstant time.

For the problem 1jp-batch; FB; rjjf, a feasibleschedule p = (BS; ST) is given by a batch sequence

BS ¼ ðB1;B2; . . . ;BbÞtogether with a sequence ST = (s1, s2, . . . , sb) of thestarting times of the batches, i.e., si = S(Bi, p) for1 6 i 6 b, such that, (a) for each batch Bx,sx P r(Bx); (b) for 1 6 x 6 b�1, sx+1 P sx + p(Bx);(c) for each batch Bx, if sx 2 [bi�1, ai) for some i with1 6 i 6 k + 1, where b0 = 0 and ak+1 =1, thensx + p(Bx) 2 [bi�1, ai).

The batch sequence in an optimal schedule iscalled an optimal batch sequence.

For a nonnegative integer t, we define x(t) to bethe minimum positive integer x with t 2 [bx�1, ax).For two nonnegative integers t and p, we define

Sðt; pÞ ¼t; if t þ p 6 axðtÞ;

minfby : by þ p 6 ayþ1; xðtÞþ 1 6 y 6 kg; otherwise:

8><>:

S(t, p) can be interpreted as the earliest possiblestarting time of a batch of processing time p whenthis batch cannot start before time t. It can beobserved that each S(t, p) can be calculated inO(k) time, where k is the number of forbiddenintervals.

Suppose a batch sequence BS = (B1, B2, . . . , Bb)is given and we want to start the processing of eachbatch as early as possible. By noting that each Bi

cannot start before C(Bi�1) and r(Bi), the onlychoice is to define the starting time si of each batchBi recursively by the following formula

si ¼ S max si�1þ p Bi�1ð Þ; r Bið Þf g;p Bið Þð Þ; 16 i6 b;

where s0 = 0 and p(B0) = 0. The correspondingschedule p = (BS, ST) with ST = (s1, s2, . . . , sb) iscalled a compact schedule. Since the objective func-tion considered in this paper is always regular, it canbe observed that there is an optimal schedule for theproblem 1jp-batch; FB; rjjf which is a compactschedule.

The fundamental model of the parallel-batchingscheduling problem was first introduced by Leeet al. [9] with the restriction that the number of jobsin each batch is bounded by a number b, which isdenoted by 1jp-batch; b < njf. An extensive discus-sion of the unbounded version is provided in [3].Recent developments of this topic can be found in[2,4]. Lately, [5,14,15] presented some new complex-ity results on the parallel-batch scheduling problemsubject to release dates. [6] showed the strongNP-hardness of the parallel-batch scheduling prob-lem under the precedence constraints between jobs.The results in the following Table 1 are presented inthe literature.

By Scharbrodt et al. [16], even the problem1jFBjCmax is strongly NP-hard. This means that,when the number of jobs in each batch is boundedby a number b < n, all non-trivial scheduling prob-lem of 1jp-batch; FB; rjjf is strongly NP-hard.Hence, we assume in this paper that the capacityof each batch is unbounded.

If 1jp-batch; rjjf is NP-hard, then 1jp-batch; FB;rjjf is also NP-hard. Hence, we focus our attention

Table 2Complexity of 1jp-batch; FB; rjjfProblem Complexity Reference

1jp-batch;FBjLmax 6 0

O(n2k) Theorem 2.2.3

1jp-batch; FBjfmax Polynomial time Theorem 2.2.41jp-batch; FBjRfj Pseudo-polynomial

timeTheorems 2.3.1 and2.3.2

1jp-batch; FBjRUj O(n4k) Theorem 2.3.31jp-batch; FB;

rjjCmax

O(n2k) Theorem 3.2

1jp-batch;FBjRwjCj

Open –

1214 J. Yuan et al. / European Journal of Operational Research 186 (2008) 1212–1217

on the problem 1jp-batch; FB; rjjf for which 1jp-batch; rjjf can be solved in polynomial time orpseudo-polynomial time.

The main results of this paper are presented inthe following Table 2.

2. Jobs with common release dates

In this section, we assume that all jobs arereleased at time zero. Hence, the problem underconsideration is denoted by 1jp-batch; FBjf. In Sec-tion 2.1, we first give two basic lemmas.

2.1. Two basic lemmas

Since the objective function f is regular, we havethe following result about the optimal solutionswhich is similar to Lemma 1 in Brucker et al. [3].

Lemma 2.1.1. For every regular objective function f,

there is an optimal batch sequence BS = (B1, B2, . . . ,

Bb) for the problem 1jp-batch; FBjf such that, forevery two batches Bx and By with x < y.

max pi : J i 2 Bxf g < min pj : J j 2 By

� �:

Proof. Let p = (BS; ST) be an optimal schedule forthe problem 1jp-batch; FBjf, where BS = (B1,B2, . . . , Bb) and ST = (s1, s2, . . . , sb). Suppose thatthere are two batches Bx and By in BS with x < ysuch that max{pi: Ji 2 Bx} P min{pj: Jj 2 By}.Then there is Jj 2 By such that pj 6 max{pi: Ji 2 Bx}.By shifting Jj from By to Bx, we obtain a new batchsequence BS� ¼ ðB�1;B�2; . . . ;B�bÞ. Note that, ifBy ¼ fJ jg;B�y will not appear in BS*. By settings�z ¼ sz for each batch B�z in BS*, we obtain a newschedule p* = (BS*, ST*). It can be observed thatCj(p*) < Cj(p) and Ci(p*) 6 Ci(p) for i5j. Since f isregular, p* is still an optimal schedule. A finite num-

ber of repetitions of this procedure yields an optimalschedule of the required form. h

We refer to a schedule which satisfies the prop-erty in Lemma 2.1.1 an SPT-batch schedule. ByLemma 2.1.1, we only need to find an optimalSPT-batch schedule.

In the remaining part of this section, we alwaysassume that the jobs are renumbered such that

p1 6 p2 6 � � � 6 pn:

Then in an SPT-batch schedule, every batch Bx is ofthe form

Bx ¼ J i; J iþ1; . . . ; J j

� �for some job indices i and j with i 6 j.

By Lemma 2.1.1, we also have

Lemma 2.1.2. For every regular objective function f,

there is an optimal schedule which is of SPT-batchand compact.

2.2. 1jp-batch; FBjfmax

To simplify the discussion, we assume thatfmax(p) P 0 for every feasible schedule p.

The decision version of the problem 1jp-batch;FBjfmax, denoted by

1jp � batch; FBjfmax 6 Y ;

asks whether there is a feasible schedule p such thatfmax(p) 6 Y, i.e.,

fiðCiðpÞÞ 6 Y ; for each job J i:

Note that in any schedule, the completion time ofevery job cannot exceed D = bk + R16i6npi. Hence,the value of Y can be chosen as an integer in theinterval [0, D], where D = max16i6nfi(D). By the bin-ary search method for the value of Y 2 [0,D], we canobtain the following observation.

Observation 2.2.1. If, for each integer Y 2 [0,D], thedecision problem 1jp-batch; FBjfmax 6 Y can besolved in O(F(n, k)) time, then the problem 1jp-batch; FBjfmax can be solved in O(F(n, k) logD)time.

Define

di ¼ max Ci : Ciis an integer with f iðCiÞ 6 Yf g;for each job J i:

Clearly, each di can be calculated by the binarysearch method for the value of Ci 2 [0,D] in O(logD)

J. Yuan et al. / European Journal of Operational Research 186 (2008) 1212–1217 1215

time. This means that, for any given Y, we candetermine all values of di, 1 6 i 6 n, in O(n logD)time.

The above discussion means the following result.

Theorem 2.2.2. By using O(n logD) time, the problem

1jp-batch; FBjfmax 6 Y is polynomially reduced to the

problem 1jp-batch; FBjLmax 6 0.

We now focus our attention to the problem 1jp-batch; FBjLmax 6 0.

First consider, for a given j with 1 6 j 6 n, thejobs in Jj ¼ J 1; J 2; . . . ; J j

� �to be processed in the

machine with forbidden intervals. If p is an SPT-batch schedule for the jobs in J such thatLmax(p) 6 0, we say that p is Jj-feasible. Define,for a given j with 1 6 j 6 n,

F ðjÞ ¼ min CjðpÞ : p is a Jj-feasible schedule� �

:

If there is no Jj-feasible schedule, we defineF(j) =1. We further define F(0) = 0 as an initialcondition. If p is a Jj-feasible SPT-batch compactschedule such that the last batch is of the form{Ji+1, . . . , Jj}, then F(j) = pj + S(F(i), pj), sinceS(F(i), pj) is the earliest possible starting time ofthe last batch.

Hence, F(j) can be calculated by the followingdynamic programming recursion:

F ðjÞ ¼ pjþmin SðF ðiÞ;pjÞþ dði; jÞ : 06 i6 j� 1� �

;

where

dði; jÞ ¼0; if S F ðiÞ; pj

� �þ pj 6 dl

for iþ 1 6 l 6 j;

1; otherwise:

8><>:

The dynamic programming function has n + 1states. Each recursion runs only O(nk) time, sincethe calculation of each S(t, p) needs O(k) time.Hence, all F(j) can be calculated in O(n2k) time.

The problem 1jp-batch; FBjLmax 6 0 has a solu-tion if and only if F(n) <1. We thus have

Theorem 2.2.3. The problem 1jp-batch; FBjLmax 6 0

can be solved in O(n2k) time.

Combining Observation 2.2.1, Theorems 2.2.2and 2.2.3, we obtain

Theorem 2.2.4. The problem 1jp-batch; FBjfmax can

be solved in O(n(nk + logD)logD) time.

2.3. 1jp-batch; FBjRfj

We will show in this subsection that the problem1jp-batch; FBjRfj can be solved in pseudo-polyno-mial time for every regular function Rfj. We willuse two different approaches.

First algorithm for 1jp-batch; FBjRfj:Let Pðt; jÞ be the problem 1jp-batch; FBjR16i6jfi

with jobs J1, J2, . . . , Jj under the restriction thatthe last batch is completed at time t, where 1 6 j 6 n

and t is a positive integer. Let F(t, j) be the minimumobjective value for SPT-batch schedules of the prob-lem Pðt; jÞ. If the problem Pðt; jÞ is infeasible, wedefine F(t, j) =1. We further define F(t,0) = 0 forany t as initial condition. We consider the followingtwo cases.

Case 1. There is a forbidden interval [ax, bx) suchthat [t�pj, t) \ [ax, bx) is not empty. Then there is nospaces to process the last batch without overlappingthe forbidden interval [ax, bx), and so the problemPðt; jÞ is infeasible. In this case, we have F(t, j) =1.

Case 2. [t�pj, t) is included in a free time interval[bx�1, ax). If p is an optimal SPT-batch schedule forthe problem Pðt; jÞ, then the last batch in p is of theform {Ji, . . . , Jj} for some i with 1 6 i 6 j. Since Jj iscompleted in time t, the completion time t* of Ji�1

belongs to [0, t�pj]. Hence, we have

F ðt; jÞ ¼Xi6l6j

flðtÞþminfF ðt�; i� 1Þ : 06 t� 6 t� pjg:

By summing the above discussion, F(t, j) can becalculated by the following dynamic programmingrecursion:

If [t�pj, t) \ [ax, bx) for some x with 0 6 x 6 k,then F(t, j) =1. Otherwise

F ðt; jÞ ¼ min16i6j

Xi6l6j

flðtÞ þ min06t�6t�pj

F t�; i� 1ð Þ !

:

The value of t belongs to [0,bk + Rpj]. WriteM = bk + Rpj. The dynamic programming functionhas nM states. Each recursion runs only O(npmax)time. Hence, all F(t, j) can be calculated inO(n2pmaxM) time. The minimum objective value ofthe problem is given by min{F(t, n): 0 6 t 6M}.We thus have

Theorem 2.3.1. The problem 1jp-batch; FBjRfi can

be solved in O(n2pmaxM) time.

Second algorithm for 1jp-batch; FBjRfi:Let N = max{fj(bk + Ripi): 1 6 j 6 n}. Then

fj(Cj(p)) 6 N in any compact SPT-batch schedule p.

1216 J. Yuan et al. / European Journal of Operational Research 186 (2008) 1212–1217

Let QðjÞ be the problem 1jp-batch; FBjR16i6jfi

with jobs J1, J2, . . . , Jj, where 1 6 j 6 n. Let H(t, j)be the minimum completion time of job Jj in a com-pact SPT-batch schedule for QðjÞ under the restric-tion that objective value of the schedule is exactly t.If the problem QðjÞ has no compact SPT-batch sche-dule with objective value t, we define H(t, j) =1.We further define H(0,0) = 0 and H(t,0) =1 forany t > 0 as initial conditions.

If p is a compact SPT-batch schedule for theproblem QðjÞ such that the objective value is t andthe completion time of Jj is minimum, then the lastbatch in p is of the form {Ji, . . . , Jj} for some i with1 6 i 6 j, and therefore contributes Ri6l6jfl(H(t, j))to the objective value. Furthermore, the schedule prestricted in the jobs J1, J2, . . . , Ji�1 is a compactSPT-batch schedule for the problem Qði� 1Þ suchthat the objective value is t* = t�Ri6l6jfl(H(t, j)).Since p is a compact SPT-batch schedule, the com-pletion time of Jj is calculated by

Hðt; jÞ ¼ pj þ SðHðt�; i� 1Þ; pjÞ;where the value of t* satisfies the condition

t� þXi6l6j

flðpj þ SðHðt�; i� 1Þ; pjÞ ¼ t:

By summing the above discussion, H(t, j) can becalculated by the following dynamic programmingrecursion:

Hðt; jÞ ¼ pj þ min16i6j

mint�2T

SðHðt�; i� 1Þ; pjÞ;where

T ¼ t� : t� þXi6l6j

flðpj þ SðHðt�; i� 1Þ; pjÞ ¼ t

( ):

The value of t belongs to [0,nN]. The dynamic pro-gramming function has n2N states. In the calculationof each recursion, the value of t* also has at most nN

choices, and for each t*, we run O(k) time to calculateS(H(t*, i � 1), pj), and consequently, run O(nk) timeto check whether t� 2T or not. Hence, each recur-sion runs only O(n2kN) time. Therefore, all H(t, j)can be calculated in O(n4kN2) time. The minimumobjective value of the problem is given by

minft 2 ½0; nN � : Hðt; nÞ <1g:We thus have

Theorem 2.3.2. The problem 1jp-batch; FBjRfj can

be solved in O(n4kN2) time.

When the objective function is RUj, we haveN = 1 in the above discussion. Hence, we have

Theorem 2.3.3. The problem 1jp-batch; FBjRUj can

be solved in O(n4k) time.

3. 1jp-batch; FB; rjjCmax

Let J be the job system under consideration. Ifthere are two jobs Ji and Jj such that ri 6 rj andpi 6 pj, then we can delete job Ji from the job sys-tem, since we can obtain an optimal schedule forJ from an optimal schedule for J n fJ ig by addingJi to the batch containing Jj. We can sort the jobs byERD order in O(nlogn) time, and then, by usingadditional O(n) time, we iteratively compare twoconsecutive jobs to decide if any job should bedeleted. Hence, by using at most O(n logn) time,we can obtain an equivalent and simplified job sys-tem in which any two jobs Ji and Jj have differentrelease dates and different processing times, and fur-thermore ri < rj implies that pi > pj. Now, the jobs inthe job system can be renumbered such that

r1 < r2 < � � � < rn

and

p1 > p2 > � � � > pn:

Lemma 3.1. There is an optimal batch sequence

BS = (B1, B2, . . . , Bb) for the problem 1jp-batch; FB;

rjjCmax such that, for every two batches Bx and By

with x < y,

minfpi : J i 2 Bxg > maxfpj : J j 2 Byg:

Proof. Let BS = (B1,B2, . . . ,Bb) be an optimal batchsequence such that the property of Lemma 3.1 doesnot hold. Then there are two batches Bx and By withx < y and a job Ji 2 Bx such that pi 6 max{pj:Jj 2 By}. We obtain a new batch BS* by shifting Ji

from Bx to By. Clearly, BS* is still an optimal batchsequence.

Continuing this procedure, we eventually obtainan optimal batch sequence with the requiredproperty. h

We refer to a batch sequence which satisfies theproperty in Lemma 3.1 an LPT-batch sequence.By Lemma 3.1, we only need to find an optimalLPT-batch sequence. Clearly, in an LPT-batch sche-dule, every batch Bx is of the form

Bx ¼ fJ i; J iþ1; . . . ; J jÞ

for some job indices i and j with i 6 j.

J. Yuan et al. / European Journal of Operational Research 186 (2008) 1212–1217 1217

Recall that S(t, p) is the earliest possible startingtime of a batch of processing time p when this batchcannot start before time t. Note that each batch B

cannot start before time r(B).Let PðjÞ be the problem 1jp-batch; FB; rjjCmax

with jobs J1, J2, . . . , Jj. Let F(j) be the minimumobjective value for LPT-batch schedules of theproblem PðjÞ. We further define F(0) = 0 as the ini-tial condition. In an optimal compact LPT-batchschedule p for the problem PðjÞ, the last batch inp is of the form {Ji, . . . , Jj} for some i with 1 6 i 6 j.Since the last batch cannot start before both F(i� 1)and rj, its starting time is given by S(max{F(i� 1),rj}, pi). Hence, F(j) = pi + S(max{F(i � 1), rj}, pi).

Hence, F(j) (1 6 j 6 n) can be calculated by thefollowing dynamic programming recursion:

F ðjÞ ¼ min16i6jðpi þ SðmaxfF ði� 1Þ; rjg; piÞÞ:

The dynamic programming function has n + 1states. Each recursion runs only O(nk) time, sincethe calculation of each S(t, p) needs O(k) time.Hence, all F(j) can be calculated in O(n2k) time.

The optimal objective value of the problem 1jp-batch; FB; rjjCmax is given by F(n). We thus have

Theorem 3.2. The problem 1jp-batch; FB; rjjCmax

can be solved in O(n2k) time.

4. Conclusions

In this paper we study the single machine paral-lel-batch scheduling with forbidden intervals. Weshow that 1jp-batch; FBjfmax can be solved in poly-nomial time, 1jp-batch; FBjRfj can be solved inpseudo-polynomial time, 1jp-batch; FBjRUj can besolved in O(n4k) time, and 1jp-batch; FB; rjjCmax

can be solved in O(n2k) time. For the furtherresearch, it is interesting to resolve the complexityof the problem 1jp-batch; FBjRwjCj.

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