Single machine parallel batch scheduling subject to precedence constraints

Single Machine Parallel Batch Scheduling Subject toPrecedence Constraints

T.C.E. Cheng,1 C.T. Ng,1 J.J. Yuan,1,2 Z.H. Liu1,3

1 Department of Logistics, The Hong Kong Polytechnic University, Hung Hom,Kowloon, Hong Kong, People’s Republic of China

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

3 Department of Mathematics, East China University of Science and Technology,Shanghai 200237, People’s Republic of China

Received 31 October 2001; revised 24 June 2003; accepted 10 May 2004DOI 10.1002/nav.20035

Published online 30 July 2004 in Wiley InterScience (www.interscience.wiley.com).

Abstract: We consider the single machine parallel batch scheduling problems to minimizemakespan and total completion time, respectively, under precedence relations. The complexitiesof these two problems are reported as open in the literature. In this paper, we settle these openquestions by showing that both problems are strongly NP-hard, even when the precedencerelations are chains. When the processing times of jobs are directly agreeable or inverselyagreeable with the precedence relations, there is an O(n2) time algorithm to minimize themakespan. © 2004 Wiley Periodicals, Inc. Naval Research Logistics 51: 949–958, 2004.

Keywords: scheduling; single machine; parallel batching

1. INTRODUCTION

Let n jobs J1, J2, . . . , Jn and a single machine that can handle batch jobs at the same timebe given. There are precedence relations � between the jobs, i.e., Ji � Jj means Ji must becompleted before Jj starts; furthermore, Ji � Jj and Jj � Jk imply Ji � Jk. Especially, we saythe precedence relations � between the jobs are m chains if the set of all jobs can be partitionedinto m job subsets { J(i, j) : 1 � j � ni}, 1 � i � m, such that the precedence relations are

J�i,1� � J�i,2� � · · · � J�i,ni�, 1 � i � m.

Correspondence to: T. C. E. Cheng ([email protected])

© 2004 Wiley Periodicals, Inc.

More details on the formal definitions of precedence relations and chains can be found in [3].Each job Jj has a processing time pj. The jobs are processed in batches, where a batch is a subsetof jobs and we require that the batches form a partition of the set of all jobs. The processing timeof a batch is equal to the maximum processing time among the jobs in the batch. The completiontime of all the jobs in a batch is defined as the completion time of the batch. If Ji and Jj are twojobs such that Ji � Jj, we require that Jj is processed at or after the completion time of Ji, andso Ji and Jj cannot be processed in the same batch. All the jobs in the same batch startsimultaneously at the earliest possible starting time. Following [4] and [10], we call this modelthe parallel batch scheduling problem and denote it by

1�prec; p-batch�f,

where f is the objective function to be minimized, and “p-batch” means that the jobs containedin the same batch are processed in parallel, (i.e., concurrently) so that the processing time of abatch is equal to the maximum processing time among the jobs in the batch. The parallel batchscheduling problem is different from the serial batch scheduling problem [3, 6]. In serial batchscheduling, the processing time of a batch is equal to the sum of the processing times of the jobsin the batch.

For the problem 1�prec; p-batch�f, a feasible schedule is given by a batch sequence

BS � �B1, B2, . . . , BN�,

such that, for any two jobs Ji and Jj with Ji � Jj, if Ji � Bx and Jj � By, then x � y. Thecompletion time of any batch Bx is naturally defined by

�i�1

x

maxJj�Bi

pj.

The parallel batch scheduling is one of the important modern scheduling models that havereceived much attention in the literature. The fundamental model of the parallel batch sched-uling problem was first introduced by Lee, Uzsoy, and Martin-Vega in [11] with the restrictionthat the number of jobs in each batch is bounded by a number b, which is denoted by 1�p-batch;b � n�f. This bounded model is motivated by burn-in operations in semiconductor manufac-turing [11]. For example, a batch of integrated circuits (jobs) may be put inside an oven oflimited size to test for their thermal standing ability. The circuits are heated inside the oven untilall circuits are burned. The burn-in time of the circuits (job processing times) may be different.When a circuit is burned, it has to wait inside the oven until all circuits are burned. Therefore,the processing time of a batch of circuits is the processing time of the longest job in the batch.

An extensive discussion of the unbounded version is provided in [2]. This unbounded modelcan be applied, for example, to situations where compositions need to be hardened in asufficiently large kiln and so the batch size is not restricted [2].

Recent developments of this topic can be found in the new book [3] and the Web site [4]. Inaddition, [5], [7], [12], and [13] present new complexity results on the parallel batch schedulingproblem subject to release dates.

For the model 1�prec; pi � p; p-batch�f, it is implied in [3] that there is an O(n2) timealgorithm for every regular objective function. To the best of our knowledge, this is the only

950 Naval Research Logistics, Vol. 51 (2004)

known result on the problem 1�prec; p-batch�f. By [4], even the complexities of the simplestproblems 1�prec; p-batch�Cmax and 1�prec; p-batch� ¥ Cj are open.

We show that both the problems 1�chains; p-batch�Cmax and 1�chains; p-batch� ¥ Cj arestrongly NP-hard. The problem 1�prec; p-batch�Cmax can be solved in O(n2) time for thefollowing two cases: (1) directly agreeable processing times, i.e., Ji � Jj implies pi � pj; (2)inversely agreeable processing times, i.e., Ji � Jj implies pi � pj.

2. NP-HARDNESS PROOFS

Our reduction will use the NP-complete vertex cover problem of graphs. Hence, we firstintroduce some graph theory terminology.

The graphs considered here are finite and simple. For a graph G, V � V(G) and E � E(G)denote its sets of vertices and edges, respectively. An edge e with end vertices u and v will bedenoted by e � uv � vu. For e � uv � E, we say that e is incident to u and u is incidentto e. A vertex subset S � V(G) is said to be a vertex cover of G if for every edge e � E(G)there is u � S such that e is incident to u. A path P of G is a sequence of vertices (v1, v2, . . . ,vk) such that vi � vj for i � j and vivi�1 � E(G) for 1 � i � k � 1. A path P of G canbe regarded as a subgraph of G. A Hamiltonian path of a graph G is a path P of G such thatV(P) � V(G). For two graphs G and H, the joint of G and H, denoted by G � H, is obtainedfrom G and H by joining each vertex in G with each vertex in H with edges. We refer the readerto reference [1] for the preliminaries of graph theory.

The vertex cover problem of graphs is defined as follows.

Vertex cover problem: For a given graph G and a positive integer k with k � �V(G)�� 1, is there a vertex cover S of G such that �S� � k?

By [8, 9], it is known that the vertex cover problem is NP-complete in the strong sense.For the vertex cover problem, if a Hamiltonian path of the considered graph is given as input,

the special vertex cover problem is called “restricted vertex cover problem” in this paper.

LEMMA 1: The restricted vertex cover problem of graphs is NP-complete in the strong sense.

PROOF: The restricted vertex cover problem of graphs is a subproblem of the vertex coverproblem, and so in the class NP. To prove the NP-completeness, we establish a polynomialreduction from the vertex cover problem of graphs to the restricted one.

Let an instance of the vertex cover problem be given, which inputs a graph G with V(G) �{v1, v2, . . . , vn} and an integer k with 1 � k � n � 1 and asks whether or not there is a vertexcover S of G such that �S� � k. We construct an instance of the restricted vertex cover problem(i.e., a graph H together with a Hamiltonian path P and an integer k� with 1 � k� � �V(H)�� 1) as follows.

Set H � G � Kn, where Kn is the complete graph with vertex set V(Kn) � {u1, u2, . . . ,un} such that V(Kn) � V(G) � A. Set P � (u1, v1, u2, v2, . . . , un, vn) and k� � n � k.

The above construction takes a polynomial time. Moreover, P is clearly a Hamiltonian pathof H.

It is routine to check that S is a vertex cover of G with �S� � k if and only if S� � S � {u1,u2, . . . , un} is a vertex cover of H with �S�� k� � n � k. This means that the vertex coverproblem can be polynomially reduced to the restricted vertex cover problem. By the fact that the

951Cheng et al: Single Machine Parallel Batch Scheduling Subject to Precedence Constraints

vertex cover problem is strongly NP-complete, we conclude that the restricted vertex coverproblem is strongly NP-complete. �

THEOREM 2: The scheduling problem 1�chains; p-batch�Cmax is strongly NP-hard.

PROOF: The decision version of the considered scheduling problem asks, for a given instanceof the problem and a positive integer Y, whether there is a feasible batch sequence BS such thatCmax(BS) � Y. It can be easily seen that the decision problem is in NP. To prove the strongNP-completeness, we use the strongly NP-complete restricted vertex cover problem for thereduction.

Let an instance of the restricted vertex cover problem be given, which inputs a graph G withV(G) � {v1, v2, . . . , vn}, a Hamiltonian path P � (v1, v2, . . . , vn) of G and an integer kwith 1 � k � n � 1 and asks whether or not there is a vertex cover S of G such that �S� �k. Write m � �E(G)�. We construct an instance of the decision version of the schedulingproblem 1�chains; p-batch�Cmax as follows.

We have m(n � 1) � n jobs. Each vertex vi � V(G) corresponds to a vertex job Ji, 1 �i � n. Each edge vivj � E(G) with i � j corresponds to n � 1 edge jobs

J�1;i,j�, J�2;i,j�, J�3;i,j�, . . . , J�n�1;i,j�.

The processing time of each job is defined in the following way. For 1 � i � n, the processingtime pi of the vertex job Ji is n. For 1 � x � n � 1 and vivj � E(G) with i � j, the processingtime p( x;i, j) of the edge job J( x;i, j) is

p�x;i,j� � � n � 1, if either i � x or j � x � 1,n, otherwise.

[We should note that, for every edge vivj � E(G) with i � j, among the jobs J( x;i, j), 1 � x �n � 1, J(i;i, j) and J( j�1;i, j) are the only jobs with processing time n � 1.] The immediateprecedence relations between the jobs are defined by the following m � 1 chains:

J1 � J2 � · · · � Jn,

J�1;i,j� � J�2;i,j� � · · · � J�n�1;i,j�,

for vivj � E(G) with i � j. The threshold value Y is defined as Y � n2 � k. We ask whetherthere is a feasible batch sequence BS such that the makespan under BS is at most Y.

The above reduction takes a polynomial time. In the following, we will prove that the instanceof the restricted vertex cover problem has a vertex cover S � V(G) such that �S� � k if and onlyif the instance of the problem 1�chains; p-batch�Cmax has a feasible batch sequence withmakespan at most Y.

If the instance of the restricted vertex cover problem has a vertex cover S � V(G) such that�S� � k, then for each edge uv � E(G) at least one of u and v is in S. For the reason that P �(v1, v2, . . . , vn) is a Hamiltonian path of G, for each i with 1 � i � n � 1, either vi � Sor vi�1 � S. We define the batch sequence

BS � �B1, B2, . . . , Bn�


in the following way. For 1 � i � n, we set Ji � Bi. For vivj � E(G) with i � j, if {vi, vj}� S, we set

J�x;i,j� � Bx, when 1 � x � j � 2,

J�x;i,j� � Bx�1, when j � 1 � x � n � 1;

if vi � S and vj � S, we set

J�x;i,j� � Bx, for 1 � x � n � 1;


J�x;i,j� � Bx�1, for 1 � x � n � 1.

It is not hard to see that BS � (B1, B2, . . . , Bn) is a feasible schedule, and that theprocessing time of each batch is either n or n � 1.

Now for a job J, let �( J) be the vertex vx such that J � Bx.

CLAIM 1: For every job J with processing time n � 1, �( J) � S.

Let J be a job with processing time n � 1. Then there must be an edge vivj � E(G) withi � j, such that either J � J(i;i, j) or J � J( j�1;i, j). By the definition of vertex cover, at leastone of vi and vj is in S. We distinguish the following three cases.

CASE 1: J � J(i;i,i�1).

In this case, by the definition of the batch sequence BS, if vi�1 � S, then �( J) � vi�1 �S; if vi � S and vi�1 � S, then �( J) � vi � S.

CASE 2: J � J(i;i, j) and i � j � 2.

In this case, by the definition of the batch sequence BS, if vi � S, then �( J) � vi � S; ifvi � S, then by the fact vivi�1 � E(G), �( J) � vi�1 � S.

CASE 3: J � J( j�1;i, j) and i � j � 2.

In this case, again by the definition of the batch sequence BS, if vj � S, then �( J) � vj �S; if vj � S, then, by the fact vj�1vj � E(G), �( J) � vj�1 � S. This completes the proofof Claim 1.

Now by the result of Claim 1, for each vx � S, the processing time of Bx is n under the batchsequence BS; for each vy � S, the processing time of By is at most n � 1 under the batchsequence BS. Hence, the makespan under BS is at most n(n � �S�) � (n � 1)�S� � n2 � �S�� n2 � k � Y.

On the other hand, suppose that BS* � ( A1, A2, . . . , AN) is a feasible batch sequence suchthat the makespan under BS* is at most Y � n2 � k. By the fact that the processing time ofeach job is either n or n � 1, we know that the processing time of each batch Ai (1 � i � N)


is either n or n � 1. This means that N � n. Because J1 � J2 � . . . � Jn, there are at leastn different batches under any feasible batch sequence. Hence, we must have N � n. Set

S* � vx : the processing time of the batch Ax is n � 1.

By the fact that the makespan under BS* is at most Y � n2 � k, �S*� � k. Hence, we onlyneed to show that S* is a vertex cover of G in the following.

Let vivj with i � j be an edge of G. Then J(i;i, j) and J( j�1;i, j) are the only jobs withprocessing time n � 1 among the jobs J( x;i, j), 1 � i � n � 1. By the fact that the chain

J�1;i,j� � J�2;i,j� � J�3;i,j� � · · · � J�n�1;i,j�

contains n � 1 jobs, one of the following cases must occur: either J(i;i, j) � Ai and J( j�1;i, j) �Aj�1, or J(i;i, j) � Ai and J( j�1;i, j) � Aj, or J(i;i, j) � Ai�1 and J( j�1;i, j) � Aj. In the first case,vi � S*; in the second case, {vi, vj} � S*; and in the third case, vj � Aj. Hence, at least oneof vi and vj is in S*. This means that S* is a vertex cover of G with �S*� � k. The proof iscompleted. �

THEOREM 3: The scheduling problem 1�chains; p-batch� ¥ Cj is strongly NP-hard.

PROOF: The decision version of the considered scheduling problem asks, for a given instanceof the problem and a positive integer Y, whether there is a feasible batch sequence BS such that¥ Cj(BS) � Y. It can easily be seen that the decision problem is in NP. To prove the strongNP-completeness, we again use the strongly NP-complete restricted vertex cover problem forthe reduction.

Let an instance of the restricted vertex cover problem be given, which inputs a graph G withV(G) � {v1, v2, . . . , vn}, a Hamiltonian path P � (v1, v2, . . . , vn) of G and an integer kwith 1 � k � n � 1 and asks whether or not there is a vertex cover S of G such that �S� �k. Write m � �E(G)�. We construct an instance of the decision version of the schedulingproblem 1�chains; p-batch� ¥ Cj as follows.

We have m(n � 1) � n � M jobs, where M � (mn � m � n)(n2 � k). Each vertex vi �V(G) corresponds to a vertex job Ji, 1 � i � n. Each edge vivj � E(G) with i � j correspondsto n � 1 edge jobs

J�1;i,j�, J�2;i,j�, J�3;i,j�, . . . , J�n�1;i,j�.

In addition, we have M large jobs Jn�1, Jn�2, . . . , Jn�M. The processing time of each job isdefined in the following way. For 1 � i � n, the processing time pi of the vertex job Ji is n.For 1 � i � M, the processing time pn�i of the large job Jn�i is nM. For 1 � x � n � 1and vivj � E(G) with i � j, the processing time p( x;i, j) of the edge job J( x;i, j) is

p�x;i,j� � � n � 1, if either i � x or j � x � 1,n, otherwise.

The immediate precedence relations between jobs are defined by the following m � 1 chains:

J1 � J2 � · · · � Jn � Jn�1 � Jn�2 � · · · � Jn�M,


J�1;i,j� � J�2;i,j� � · · · � J�n�1;i,j�,

for vivj � E(G) with i � j. The threshold value Y is defined as

Y � �n2 � k � 1�M �1

2nM2�M � 1�.

We ask whether or not there is a feasible batch sequence BS such that ¥ Cj(BS) � Y.The above reduction takes a polynomial time. In the following, we will prove that the instance

of the restricted vertex cover problem has a vertex cover S � V(G) such that �S� � k if and onlyif the instance of the problem 1�chains; p-batch� ¥ Cj has a feasible batch sequence BS suchthat ¥ Cj(BS) � Y.

If the instance of the restricted vertex cover problem has a vertex cover S � V(G) such that�S� � k, we define the batch sequence

BS � �B1, B2, . . . , Bn�M�

in the following way. For 1 � i � n � M, we set Ji � Bi. For vivj � E(G) with i � j, if{vi, vj} � S, we set

J�x;i,j� � Bx, when 1 � x � j � 2,

J�x;i,j� � Bx�1, when j � 1 � x � n � 1;


J�x;i,j� � Bx, for 1 � x � n � 1;


J�x;i,j� � Bx�1, for 1 � x � n � 1.

It is not hard to see that BS � (B1, B2, . . . , Bn�M) is a feasible schedule. By the discussionof the if part of Theorem 2, the completion time of the batch Bi, 1 � i � n, is at most n2 �k. Then the completion time of the batch Bn�i � { Jn�i}, 1 � i � M, is at most (n2 � k) �inM. For the reason that the first n batches contain mn � m � n jobs, the total completion timeof jobs can be roughly estimated as

� Cj�BS� � �mn � m � n��n2 � k� � �i�1

M

�n2 � k � inM� � Y.

This implies that the instance of the scheduling problem has a required batch sequence.On the other hand, suppose that BS* � ( A1, A2, . . . , AN) is a feasible batch sequence such

that the total completion time of jobs under BS* is at most Y. By the fact that


J1 � J2 � · · · � Jn � Jn�1 � Jn�2 � · · · � Jn�M,

the n vertex jobs must be processed before the large jobs with each vertex job being processedin one batch. Because each vertex job has processing time n, the starting time of any batch thatcontains a large job must be at least n2. If there is an edge job J that is processed either in thesame batch as a large job or after a large job, then, by the fact that the large job has processingtime nM, the completion time of job J is at least n2 � nM. Because the completion time of thelarge job Jn�i is at least n2 � inM, the total completion time of the jobs under BS* is greaterthan

n2 � nM � �i�i

M

�n2 � inM� � Y � �n � k � 1�M � n2 � Y.

This contradicts our assumption. Hence, each edge job must be processed before every large job.Denote by � the maximum completion time of vertex jobs and edges jobs. If � � n2 � k �1, then the completion time of the large job Jn�i is at least n2 � k � 1 � inM. It follows thatthe total completion time of the jobs under BS* is greater than

�i�1

M

�n2 � k � 1 � inM� � Y.

This contradicts our assumption again. Hence, we must have � � n2 � k.The above discussion means that there is a feasible batch sequence BS� for the vertex jobs and

edge jobs such that the makespan � is at most n2 � k. By the discussion of the only if part ofTheorem 2, there is a vertex cover S* of G such that �S*� � k. This completes the proof. �

3. DIRECTLY AGREEABLE AND INVERSELY AGREEABLE PROCESSINGTIMES

If Ji � Jj implies pi � pj, we say that the processing times of the jobs are directly agreeablewith the precedence relations. If Ji � Jj implies pi � pj, we say that the processing times ofthe jobs are inversely agreeable with the precedence relations. The corresponding problems willbe denoted by

1�prec; p-batch; directly agreeable�f

and

1�prec; p-batch; inversely agreeable�f,

respectively.

THEOREM 4: For the scheduling problem 1�prec; p-batch; directly agreeable�Cmax, set

F � Ji : there is no successor of Ji.


Then there is an optimal batch sequence for the problem such that F is the last batch.

PROOF: Let BS � (B1, B2, . . . , BN) be an optimal batch sequence for the problem 1�prec;p-batch; directly agreeable�Cmax such that �BN� is as large as possible. Then BN � F.

Suppose, for the sake of contradiction, that BN � F. Let x be the maximum index such thatBx � (F�BN) is not empty, then x � N � 1. If Bx � F, then

BS� � �B1, B2, . . . , Bx�1, Bx�1, . . . , BN�1, Bx � BN�

is a feasible batch sequence such that Cmax(BS�) � Cmax(BS). This contradicts our assumptionthat BS is an optimal batch sequence.

Now suppose that Bx�F is not empty. Define a new batch sequence BS� by

BS� � �B1, B2, . . . , Bx�1, Bx�F, Bx�1, . . . , BN�1, BN � �Bx � F��.

For every job Ji � Bx�F, by the definition of F, there must be Jj � F such that Ji � Jj. Fromthe maximality of x, we deduce that Jj � BN. Because the processing times are directlyagreeable, Ji � Jj implies that pi � pj. Hence,

maxpi : Ji � Bx�F � maxpj : Jj � BN.

Let max{ pi : Ji � Bx�F} � a, max{ pj : Jj � BN} � b, and max{ pj : Jj � Bx � F} � c.Then a � b. Thus,

Cmax�BS� � Cmax�BS�� maxpi : Ji � Bx � maxpi : Ji � BN � maxpi : Ji � Bx�F

� maxpi : Ji � BN � �Bx � F� � maxa, c � b � a � maxb, c

� max0, c � a � max0, c � b � 0.

This implies that BS� is still an optimal batch sequence. But this contradicts the choice that BSis an optimal batch sequence such that �BN� is as large as possible. The result follows. �

THEOREM 5: For the scheduling problem 1�prec; p-batch; inversely agreeable�Cmax, set

E � Ji : there is no predecessor of Ji.

Then there is an optimal batch sequence for the problem such that E is the first batch.

PROOF: Similar to Theorem 4. �

By Theorem 4 and Theorem 5, we can establish the following batching rules.

Batching Rule for 1�prec; p-batch; directly agreeable�Cmax: At each point, let thenext last batch to be formed consist of all the unbatched jobs without any unbatchedsuccessor.


Batching Rule for 1�prec; p-batch; inversely aggreeable�Cmax: At each point, let thenext first batch to be formed consist of all the unbatched jobs without any unbatchedpredecessor.

The above discussions show that the scheduling problem 1�prec; p-batch�Cmax can be solvedin O(n2) time in the case that the processing times of jobs are either directly agreeable orinversely agreeable with the precedence relations.

ACKNOWLEDGMENTS

This research was supported in part by The Hong Kong Polytechnic University under a grantfrom the Area of Strategic Development in China Business Services. The third author wassupported in part by the National Natural Science Foundation of China under Grant No.10371112.

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Single machine parallel batch scheduling subject to precedence constraints