Download pdf - Parallel Machine Scheduling with Batch Setup Times

Parallel Machine Scheduling with Batch Setup TimesAuthor(s): T. C. E. Cheng and Z.-L. ChenSource: Operations Research, Vol. 42, No. 6 (Nov. - Dec., 1994), pp. 1171-1174Published by: INFORMSStable URL: http://www.jstor.org/stable/171994 .

Accessed: 09/05/2014 18:16

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.

http://www.jstor.org

This content downloaded from 169.229.32.138 on Fri, 9 May 2014 18:16:21 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=informs

http://www.jstor.org/stable/171994?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


CHENG AND CHEN / 1171

JACKSON, J. R. 1963. Jobshop-Like Queueing Systems. Mgmt. Sci. 10, 131-142.

SPEARMAN, M. L., AND M. A. ZAZANIs. 1992. Push and Pull Production Systems: Issues and Comparisons. Opns. Res. 40, 521-532.

SPEARMAN, M. L., W. J. Hopp AND D. L. WOODRUFF.

1989. A Hierarchical Control Architecture for CONWIP Production Systems. J. Manuf. and Opns. Mgmt. 3, 147-171.

SPEARMAN, M. L., D. L. WOODRUFF AND W. J. Hopp. 1990. CONWIP: A Pull Alternative to Kanban. Int. J. Prod. Res. 28, 879-894.

WEIN, L. M. 1990. Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Network With Controllable Inputs. Opns. Res. 38, 1065-1078.

WEIN, L. M. 1992. Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network With Controllable Inputs. Opns. Res. 40, S312-S334.

WEIN, L. M., AND P. B. CHEVALIER. 1992. A Broader View of the Job-Shop Scheduling Problem. Mgmt. Sci. 38, 1018-1033.

PARALLEL MACHINE SCHEDULING WITH BATCH SETUP TIMES

T. C. E. CHENG Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong

Z.-L. CHEN Shanghai Transportation Planning Institute, Shanghai, People's Republic of China

(Received September 1992; revisions received April, June 1993; accepted August 1993)

We consider a problem of scheduling several batches of jobs on two identical parallel machines to minimize the total completion time of jobs. A setup time is incurred whenever there is a switch from processing a job in one batch to a job in another batch. When the number of jobs is arbitrary, the computational complexity of the problem is posed as an open problem in the literature. We show in this note that the problem is binary NP-hard even when the setup times are sequence independent and all processing times are equal.

W e are given B batches of jobs. Each batch i for "Y 1 % i < B contains a set of Ni jobs: Ji = {Jil, Ji2, * ** JiNi. Let N = >fL1 Ni and J = UB 1 Ji. There are two identical parallel machines on which all the Njobs are to be processed. A batch of jobs cannot be processed by more than one machine simulta- neously. Each job Jii E J has a given processing time

pj E Z+, where Z+ is the set of positive integers. A setup time Si, E Z+ is incurred whenever a job from batch j is processed immediately after a job from batch i on the same machine. We let Sii = 0 for all 1 < i S B. Also, an initial setup time Soi E Z+ is incurred if a job from batchj is processed first on a machine. The setup times are called sequence independent if SJ = S, for 0 % i % B, where i 1 j; otherwise, the setup times are called sequence dependent.

Monma and Potts (1989) stated that when the number of batches B is fixed there exist pseudopolynomial algorithms for the maximum completion time, the maximum lateness, the total weighted completion

time, and the weighted number of late jobs problems. They also showed that when the number of batches B is arbitrary the maximum completion time, the maximum lateness, the number of late jobs, and the total weighted completion time problems are all NP-hard. But the computational complexity of the total completion-time problem with an arbitrary number of batches was posed as an open problem in their paper.

We will study this open problem in this note. Spe- cifically, we will consider a special case of the problem in which the setup times are sequence independent, i.e., Sij = Sj for all 0 S i S B with i X j, and all processing times are equal, i.e.,pij = p for all Ji1 E J. The objective is to find an optimal schedule to process the jobs on the two machines so that the total completion time f = -1--j ANi 1-i B Cij is mini- mized, where Ci1 is the completion time of job Jif. We call this special problem the two-parallel machine total completion time (P2TCT) problem and show that P2TCT is NP-hard by a reduction from PARTITION.

Subject classifications: Analysis of algorithms, computational complexity: NP-hardness. Production/scheduling: parallel machine scheduling. Area of review: MANUFACTURING, OPERATIONS AND SCHEDULING.



1172 / CHENG AND CHEN

PRELIMINARY RESULT

The following lemma identifies the structure of an optimal schedule for P2TCT.

Lemma 1. Given an optimal schedule, suppose that a machine is to process ni, 0 S ni > Ni jobs from batch i, 1 > i > B. Then, on this machine, the following properties hold:

i. the jobs from the same batch are processed consecutively;

ii. the batches are sequenced in nondecreasing order of Si/ni-

Proof. Lemma 1 follows immediately from Proposi- tion 1 in Dobson, Karmarkar and Rummel (1987) when pij = p for all Jij E J.

NP-HARDNESS RESULT

Now, we will show that P2TCT is binary NP-hard by showing that its associated decision problem is NP- complete by a reduction from PARTITION, a known NP-complete problem (Garey and Johnson 1979).

The decision problems of PARTITION and P2TCT are stated as follows.

PARTITION. Given t positive integers a1, a2, ...,

at such that -t=1 ai = 2A is even, does there exist a subset U C T = {1, 2, ..., t} such that >LEu ai = >ie\u ai - A?

P2TCT. Given two identical parallel machines, B batches of jobs, a set of Ni jobs: Ji = {Jil, Ji2, . . .*

JiNi} a setup time Si for each batch i, a processing timep for each jobJij, 1 S i S B, 1 > j S Ni, and a threshold Y, does there exist a schedule such that

f = y- El --j --Ni l -_ Si --B C ij < Y?

Given any instance I of PARTITION, we can construct in polynomial time a corresponding instance I' of P2TCT as: B = t; Ni = ai, Si = 2Aai, 1 < i S B; p = 1;

Y=2A > aiaj-2A3+A2+A. 1 <_i -j _<t

The following result holds for instance I'.

Lemma 2. An optimal schedule for instance I' is one in which no more than one batch is processed by both machines and all the other batches are processed on only one of the machines.

Proof. Suppose that an optimal schedule is given in which more than one batches are processed on both machines. Without loss of generality, let 1, 2, ... , kg k ? 2, denote these "split" batches. For each 1 < i k, let n' and nj (nJ + n 2 = Ni) be the number of

jobs from batch i that are processed on machines one and two, respectively. Also, let k + 1, k + 2, ... ., m represent the batches that are processed on machine one only and m + 1, m + 2, ... , B represent the batches that are processed on machine two only. Thus, on machine one, there are n' jobs from batch i, 1 < i < k, and Nj jobs from batchj, k + 1 S j m; on machine two, there are n 2 jobs from batch i, 1 S i S k, and Nj jobs from batch j, m + 1 < j S B.

Assume without loss of generality that Sl/nl S2/nl ... % Sknl. Since for each 1 S i S k, Si = 2Aai, n1 + n 2 =.Ni so SJ1n2l ? S2/n2 ... 2S/n. It is easy n1 +n 11 22 kk es to verify that Sk+l/Nk+l = Sk+2/Nk+2 = SB/NB <

min{51/nl, Sk/nk2}. Since, by Lemma 1, jobs from the same batch on the same machine will be processed consecutively, the schedule can be expressed in terms of the processing orders of the batches on machines one and two, which are (k + 1, k + 2, ..., m, 1, 2, ..., k) and (m + 1, m + 2 ..., B, k, k - 1, ..., 1), respectively. The schedule is illustrated in Figure 1, where Di denotes the set of re, jobs from batch i that are processed on machinej (i = 1, k;j = 1, 2), H' represents the set of jobs processed between DI and Sk on machine one and t' its completion time, and H2 represents the set of jobs processed between D 2 and S1 on machine two and t2 its completion time.

In the following, we show that a better schedule can be created by combining a split batch, thus reducing the number of split batches. There are two cases to consider.

Case 1 (t1 ? t2)

Construct a new schedule by putting D 1 immediately after Dk. Since t' ? t2, the completion times of jobs in D 1 in the new schedule will be no greater than those in the original schedule. The new schedule will also save the setup time Sk on machine one and create no additional setups. Let L be the number of jobs in H2 U D2; evidently, L < 2A. While the completion time of each of the jobs in H2 U D12 will increase by nk, there is a net decrease in total completion time by at least Sknk - Lnk > 2Anl(ak - 1) > 0. Thus, the new schedule which has one fewer split batch is better than the original schedule, a contradiction.

Case 2 (t1 < t2)

Construct a new schedule by putting D 2 immediately after D1. Arguing in a way analogous to Case 1, we see that the new schedule which has one fewer split batch is better than the original schedule, a contradiction.

We can repeat the above process until an optimal schedule with no more than one split batch is ob- tained. This completes the proof.



CHENG AND CHEN / 1173

Figure 1. A schedule with a batch split over both machines.

We now begin to show the NP-completeness of P2TCT.

Theorem. The decision version of P2TCT is binary NP-complete.

Proof

i. The proof that P2TCT E NP is trivial, and we omit it.

ii. PARTITION is polynomially reducible to P2TCT. To prove part ii, it suffices to show that there exists a solution to instance I' if and only if there exists a solution to instance I.

If Part

Suppose that there exists a subset U C T = {1, 2, ... , t} such that Lieu ai = >ie7\u ai = A. Without loss of generality, let U = {1, 2, ... , k} and T\U = {k + 1, k + 2, ... , t}. We now construct a schedule V for instance I' as follows: Machine one processes batches 1, 2,..., k and machine two processes batches k + 1, k + 2, ..., t; batches are processed in an arbitrary order. Then the total completion time of schedule V is given by

k k N Ni+-+Nk

f (V)=I Si I Nj + I i i=l j=i i=l

B B Nk+1+ +NB

+ 2 Si ( Nj + i i=k+l j=l i-1

+2A I ai ( aj + 2 ai - 1 +a )

i=k1 \= 2i=1 \i=k 1

2A( aIai +?2 aj + ai) a2+

1 i=kl k+= 1 2 i=k+l

t ~

=2A I aiaj - + aiaj-2+

2 .i=1S i=l ik+l

= 2A I aiaj - 2A3+A2 +A = Y, 1 <i]j t

which implies that V is a solution to instance I'.

Only If Part

Suppose that there does not exist a subset U C T ={1, 2, . .., t} such that Xi,u ai = Xi,7\u ai = A. Given an optimal schedule Wfor instance I', by Lemma 2 there are two cases to consider.

Case 1. There is no split batch in schedule W. Without loss of generality, suppose that in schedule

W batches 1, 2, . . ., m are processed on machine one and batches m + 1, m + 2, ... , B are processed on machine two. Since S1/Ni = 2A for all 1 < i < B by Lemma 1 we can suppose that in schedule W all jobs from the same batch are processed consecutively on the same machine and the processing orders of the batches on machines one and two are (1, 2, ... , m) and (m + 1, m + 2, .. , B), respectively. Then, similar to the pre- ceding case, the total completion time of W is

f(W) = 2A(2 aia1 - aj ? ai)

Sine +~

lS

i)2S +i=l+ i)=]+A.

Snethere exists no solution to instance I,

m t

fa1 ( a, <A2 i=l i=m+l

and

~~~~m 2 t 2 '

(z a1) + (I+ ai > 2A2.

2= i=1m+l+

Thus,f (W) > Y, implying that W is not a solution to instance I'.

Case 2. There is one split batch in schedule W. Without loss of generality, suppose that batch k

is the split batch with n1 jobs processed on machine one and Nk - n1 jobs processed on machine two, batches 1, 2, ... , k - 1 are processed on machine one only and batches k + 1, k + 2, ... , B are processed on machine two only. By Lemmas 1 and 2, the processing orders of the batches on machines one



1174 / CHENG AND CHEN

andtwoare(1, 2, ..., k - 1, k) and (k + 1, k + 2, . .. , B, k), respectively. Then the total completion time of W is

k-1 k-1 k

PMW)= 2 Si 2 Nj) + nj 2 Si i=1 j=l i=1

N * *+Nk-I +nl

+ i=l

B B B + 2 Si(2 Nj) + (Nk -nl) 2 Si

i=k+l j=1 i=k

Nk +- * *+NB--nl

+ (1) i=l1

=2A( aiaj + aiaj) 1 <i-<j-k- 1 k+ 1-i <j<-t

k t

+ 2A(n, 2 ai + (ak - nl) 2 ai) i=l i=k

1 k-1 2 t

2'

2 -

Eai +nl + Eai -ni +A.

Since there exists no solution to instance I,

k-1 t

2 ai , ai <A2 (2) i=1 i=k

and

k t

ai 2 ai <A2. (3) i=1 i=k+l

Also, since n1 < Nk = ak and

If >ik=l a1 ? >t=k at, then from (1), (2) and (4), we have

=()>2A( aia1 + ajai)A2+ k-k-t

a + n a 2 ai nA 2+A, i=k

k-1lijS- 2 iSj2

=2A ana -2Aai >a 4+A2+A

_i=l a t i=l i=ka,te fo 1,() n 4,w

>2A aia1-2A3 +A2+A=Y. (5) 1 SiSjkt

On the other hand, if jk=1 ai < >Li=k a , then from (1), (3) and (4), we obtain

f (W) > 2A( 2 aiaj + aiaj 1 _i_<j_k- I k+1 <i<_j_t

k\

+ ak 2 ai +A2+A

= 2A( aiaj + 2 aiaj) +A2 +A 1<_i_<j_k k+1 _i<j_<t

k t

= 2A aiaj - 2A >ai > ai +A2+A 1 _<i <j _t i=1 i=k+l

> 2A 2 aiaj - 2A3+A2 +A =Y. (6) 1 <i <j t

From (5) and (6), we see that schedule W is not a solution to instance I'.

Combining Cases 1 and 2, we establish the correctness of the "only if" part. This completes the proof.

Note that we assumed in the above proof that the setup times are sequence independent and all processing times are equal, thus the general problem with an arbitrary number of batches, sequence-dependent setup times, and variable processing times is also NP-hard.

CONCLUSION

We have shown that the parallel machine scheduling problem with an arbitrary number of batches and independent batch setup times to minimize the total completion time is binary NP-hard by a reduction from PARTITION.

REFERENCES

DODSON, G., U. S. KARMARKAR AND J. L. RUMMEL. 1987. Batching to Minimize Flow Times on One Machine. Mgmt. Sci. 33, 784-799.

GAREY, M. R., AND D. S. JOHNSON. 1979. Computers and Intractability: A Guide to the Theory of NP- Completeness. W. H. Freeman, New York.

MONMA, C. L., AND C. N. Porrs. 1989. On the Complex- ity of Scheduling With Batch Setup Times. Opns. Res. 37, 798-804.

