On the Asymptotic Optimality of the Spt Rule for ... - Stanford University

On the Asymptotic Optimality of the Spt Rule for the Flow Shop Average Completion Time Problem Author(s): Cathy H. Xia, J. George Shanthikumar, Peter W. Glynn Source: Operations Research, Vol. 48, No. 4 (Jul. - Aug., 2000), pp. 615-622 Published by: INFORMS Stable URL: http://www.jstor.org/stable/222880 Accessed: 20/07/2010 02:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=informs. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of 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

OPTIMALITY OFTHESPT RULEFORTHEFLOW ON THEASYMPTOTIC TIMEPROBLEM COMPLETION SHOPAVERAGE CATHYH. XIA IBM Thomas J. Watson Research Center, PO Box 704, Yorktovn Heights, NewvYork 10598 hhcx@,watson.ibm.com

J. GEORGESHANTHIKUMAR Department of IEOR and the Walter A. Haas School of Business, University of California at Berkeley, Berkeley, California 94720 [email protected]

PETERW. GLYNN Department of EES & OR, Stanford University, Stanford, California 94305 [email protected] edu (Received September 1997; revisions received October 1998, December 1998; accepted December 1998) Consider a flow shop with M machines in series, through which a set of jobs are to be processed. All jobs have the same routing, and they have to be processed in the same order on each of the machines. The objective is to determine such an order of the jobs, often referredto as a permutationschedule, so as to minimize the total completion time of all jobs on the final machine. We show that when the processing times are statistically exchangeable across machines and independentacross jobs, the ShortestProcessing Time first (SPT) scheduling rule, based on the total service requirementof each job on all M machines, is asymptoticallyoptimal as the total numberof jobs goes to infinity. This extends a recent result of Kaminsky and Simchi-Levi (1996), in which a crucial assumption is that the processing times on all M machines for all jobs must be i.i.d.. Our work provides an alternativeproof using martingales,which can also be carried out directly to show the asymptotic optimality of the weighted SPT rule for the Flow Shop Weighted Completion Time Problem.

1. INTRODUCTION

to be NP-hardeven in the two-machinecase, as pointed out by Garey et al. (1976). Research on completion time can be divided into two categories. One focuses on makespan, the time needed to complete processing all jobs, which takes a social point of view from the system side. The other concentrateson the average completion time of all jobs on the final machine, which focuses more on the individualjobs. Althoughthe latter capturesimportantreal-life managerialscheduling concerns that are not reflected in makespanrelated objectives, the analysis is often very complicatedand difficult. So far, most results in this area have focused on minimizing the makespan.For a complete survey of makespanrelatedwork see, for example, Pinedo (1995b). For average completion time related scheduling,most of the previous research has studied deterministicproblems using branch-and-boundstrategies and is often limited to a small number of jobs and machines. Examples include Krone and Steiglitz (1974) and Van de Velde (1990), along with many others. In this paper, we look at the stochastic case instead and search for an asymptotically optimal schedule. We adopt the notion of asymptotically optimal schedulingthat has been recently consideredby Shanthikumar and Xu (1997), and Xia (1999), in queueing control problems. The limiting performanceof tandem queues

Flow shop scheduling, as modeled by a tandem queue, is a rich classical topic motivated mainly by practical needs in the manufacturingcontext. With recent developmentsin communicationnetworks,the role of tandemqueue scheduling becomes increasinglysignificant.Researchhas concentrated on different issues such as buffering, service effort allocation, and completion time minimization.Results such as reversibility, duality, and the bowl-shaped phenomenon are all well known in the literature,as establishedby Muth (1979), Pinedo (1995a), and Hillier and Boling (1979). A more detailed literature review of these issues can be found in Buzacott and Shanthikumar(1993), Pinedo (1995b), and Weber (1992). In this paper,we restrictour attentionto the issue of completion time. Considera flow shop with M machines in series, throughwhich a set of jobs are to be processed.All jobs have the same routing,and they have to be processed in the same sequence on each of the machines. The objective is to determinea sequence of jobs, often referredto as a permutationschedule, that minimizes the average-or equivalently, the total-of the completion times of all jobs on the final machine. It is assumed that the buffer space between machines is unlimited and that the jobs are processed individually without preemption.This problem is well known

Subject classifications: Asymptotically optimal scheduling: flow shop average completion time problem. Flow shop scheduling: average completion time, asymptotic optimality. Tandem queueing: asymptotic optimality of SPT. Area of review: STOCHASTIC MODELS. Operations Research, (C 2000 INFORMS Vol. 48, No. 4, July-August 2000, pp. 615-622

615

0030-364X/00/4804-0615 $05.00 1526-5463 electronic ISSN

616 /

XIA, SHANTHIKUMAR, AND GLYNN

has been studied extensively by, for example, Glynn and Whitt( 1991). In this study,we combine ideas fromboth and apply them to the Flow Shop Average Completion Time Problem. We show that as the numberof jobs to be scheduled becomes larger and larger, which is often demanded in real-life applications,the Shortesttotal Processing Time (SPT) firstrule is asymptoticallyoptimal,providedthat the processing times are statistically exchangeable across machines and independentacrossjobs. It shouldbe notedthatthis researchwas motivatedby a recent resultof Kaminskyand Simchi-Levi(I1996b),in which it was firstpointed out thatthe SPT rule was asymptotically optimalwhen the processingtimes are i.i.d. acrossmachines and jobs. We provide a simplified proof using martingales and extendthe resultto a more generalrealm,wherethejobs need not be identical an assumptioncrucialto theirpaper. We show that for the SPT rule to be asymptoticallyoptimal, the jobs can have differentprocessing time distributions,as long as they are independentacrossjobs and the processing times of eachjob on all machinesare statisticallyexchangeable (not necessarily independent). In addition, our argument can be easily extendedto show the asymptoticoptimality of SWPT-Shortest Weighted total Process Time first rule-for the Flow Shop WeightedCompletionTime Problem. This extends anotherresult of Kaminskyand SimchiLevi (1996a) and simplifies their argument. The remainder of the paper is organized as follows. Section 2 introducesthe basic model. Section 3 reviews the well-known criticalpath approachusing an activitynetwork in determiningthe final completion time for each job. The total completion time for all jobs is then evaluated in ??4 and 5, where a simple lower bound and an upperbound are presented,respectively. In ?6 we show, using martingales, that when scheduling is based on the total service requirement on all machines for each job, the differencebetween the upper bound and the lower bound converges to zero under an exchangeability hypothesis. The main result is then presentedin ?7. 2. THE MODEL We considerthe traditionalFlow Shop Average Completion Time Problem,which is well known in the field of scheduling. In this problem,a flow shop consisting of M machines, each with unlimitedbufferspace, must sequentiallyprocess n jobs. Each machine can handle at most one job at a time, and each job can be processed on only one machine at a time without preemption.Job j, j 1,2,. .., n has a processing time tym on machine m, m 1,2,...,M, and the processing times must satisfy the following assumptions. Al. Processing times of different jobs are independent, i.e., ((tj,i,...,tjAd):j >1) is a sequence of independentrandom vectors.

ASSUMPTION

ASSUMPTION

A2. For each job, the processing times on the

M machines are exchangeable random variables, i.e., for any givenj ( > 1), tj , ...,tj,M are exchangeable.'

We restrictourattentionto the set ofpermutationscheduling policies, where the orderin which thejobs areprocessed on the first machine is maintainedthroughoutthe system; that is, once the order of processing the n jobs on the first machine is scheduled, all other machines process the jobs in afirst comefirst serve (FCFS) manner.It should also be noticed that permutationschedules are not necessarily optimal for the general flow shop scheduling problem where reorderingis allowed at each machine. For simplicity, we consider only the permutationschedules. The objective is to determine7r a schedule, or sequence of the jobs, such thatZj,(n), the total completiontimes of all jobs on the last machineM is minimized.Let Z4(n) denote the optimal objective function value. This paperprovides a proof of the asymptoticoptimality of the well-known rule in schedulingtheory, known as the Shortest Processing Time (SPT) firstpolicy. According to this policy, jobs are sequenced in increasingorder of their total service requirementson all M machines. Note that there may exist differentSPT schedules associated with the same total processing time sequence (when there is a tie in the total processing time). For the general flow shop problem(withoutAssumptionsAl andA2), these SPT schedules are not necessarily optimal. See Kaminsky and Simchi-Levi (1996b) for such a counterexample.Nevertheless, given AssumptionsAl, A2, and some finite moment conditionson the processingtimes, we show that SPT is asymptoticallyoptimal in the sense that M(n) Z*

1 a.s. as n -- oc,

where 7SPT(n)denotes the correspondingtotal completion time for processing the n jobs underthe SPT policy.

3. CRITICALPATH It is well known that an Activity Nenvork, composed of nodes and directedarcs, is helpful in calculatingcompletion times in a flow shop. We consider the model in which jobs are processed according to their nominal order 1, 2,... , n. Let node (j, m) be associatedwith the activity of processing job j on machinem. Activity node (j, m) is connectedwith its immediate consequent activities, i.e., node (j + 1,m) and node (j, m + 1) with a directedarc of length t1,,,. This representsthe fact that only when job j is completed on machinem, which shouldtaketime tj,rn,canthe serviceofjob j + 1 on machinem andthe service ofjob j on machinem+ 1 be initiated. Figure 1, for example, illustratesthe activity network graph for processingjobs 1,2, and 3 sequentially on machines 1 and 2. Let Fm,,,be the completion time of job j on machine m. A basic recursionfor the completiontimes is Fk+lni

max{Fkn,,Fk+?1n-1}

+ tk+1,m-

(1)

From (1), it is easy to establish the following lemma by induction.

XIA, SHANTHIKUMAR, AND GLYNN

Figure 1.

Activity network graph.

1,2,t

1,1

2,2

1. The completion time ofjob k on the last machine M is given by:

LEMMA

FkM=

I1
3

t/,I

1,...,n.

Lemma 1 is a well-known result; see, for example, Conway et al. (1967), Pinedo (1995b), and Glynn and Whitt (1991). Graphically,based on the activity network, this is equivalent to saying that the completion time of the last activity must be the directedpath with maximumlength from the startingnode to the finishing node, which is often identifiedas the Critical Path.

AllA2m.axAM~

2

+ . .+

4. A SIMPLELOWERBOUND FORTHETOTAL COMPLETION TIME

k

M-1

M-

(ti

+

l

tj

tlmli

2:

lb

m~ax I

bkE(tj'm - tj,l)

la

b=

M-1

Associated with the M-machine Flow Shop Average Completion Time problem introduced in ?2 is the following single machine Average Completion Time problem:n jobs are to be processed on a single machine where job j has J 1,.. ., n. The objective processing time Tj:= El is to determine a sequence of the jobs so as to minimize the average completion time of all jobs. Let Z*(n) be the correspondingoptimal total completion time. It turns out that the optimal total completion time of a single machine scheduling problem naturallygives a lower bound for the total completion time of the M-machine scheduling problem. This was first observed by Kaminsky and Simchi-Levi (1996b), and we state their result in the following lemma. 2. Consider the M-machine Flow Shop Average

Completion Time Problem and the Associated Single Machine Scheduling Problem. Suppose that the corresponding optimal total completion times are ZM(n) and Z*(n), respectively. We then have

I

O

j

jowl 1f

flP(tj,

mC

dxjm, m

To see this, we note that

1,21T).

j~ 1 Because the processing times are exchangeable, i.e., (tj, t)?- (ti, tj), we have P(tj,1IC dXjI, t,2 C dxy2|Tj) = P(tj,2

C

E[t

nil] =

-

2EF

(tin,

2EF

(tj i-

-)

du

U)I(tj, n > u) du

dxi1, tj, I C dXj2ITi). = 2jA

The results then follow immediately. El

t ni

E[tj,m-

> U]P(tj nl > u)

du

0C

1. Assume the conditions of Lemma 3. If 'V-measurable,and

THEOREM

on

is

42 R

P(tjnl >U) du

-2RE[tjn

where the exchange of the expectation and the integral in thirdequality is justified by Fubini's theorem. Examples satisfying condition (6) include cases where:

then,for 6 > 0, maxI

n

Ui

I (t~n(j),ni

tcrn(j)A

1l,12+6

as ni -- oc, Vm 2,..., distribution.

M, where

means convergencein

PROOF. We fix m =2 in this proof. The same argument works for the general case. Based on Lemma 3, tQn(j),2 - t(y,(j),I (1 O. 3. The tjms are new better than used in expectation (NBUE) with supjE[tj,m] < ??. Ournext theoremdeals with the almost sure convergence to zero of the right-handside of (4) as n goes to infinity, which requiresthe following strongercondition. THEOREM2.

(7)

supE[Itj, l lP]

for some p > 2, and (as, n > 1) is a sequence oJf i-measurable permutations, then max1?Al? n

2

Assume the condition of Lemma 3. If

EK-(ta,,(j),rn n

-tetj

-0

a.s. as

i

-

ox.

XIA, SHANTHIKUMAR, AND GLYNN

Again, we look only at m = 2 and extend the same argumentto general m. For 1< / < n, let 'I := V U(t6n(j),2- tan(j),: 1 j ). Then (Z=1 (tan(j),2 - tan(j) ): O < 1< n) is a martingalewith 1 n). This is obvious because, given respect to (*I: I 1,

PROOF.

/

619

Hence,

P (1 max

,X)

P)

Under condition (7), we know that for p > 2, Z@( ta (j ),2 -

(

0 < < n)

tan(j),) :I

< ?? 1,IP]

sup E[ t1,2 -t, j]?

is a submartingale,providedthat

3 (tan (j),2

E(

-

tan(), 1)

)

This implies, using the Burkholder-Gundysquare inequality for martingales(see Burkholder1972), that

< o,

(8) E [Z(tj,2-tjl)

for 1 -j-n. In order to verify (8), note that under condition (7) we have

Therefore,V's> 0, p (P

oo>E

) = ?E nE Itan(j),2

Itj,2-tj,

tE

-

max

Z(

(

Zoo

E(Itan (j),

2 -ta,(j)

an(j),2

-

>?n

tn(j)l)

tan(j),l)

n

=E j=1

(9)

] =Q(nIp)

tn

P

9)

IIIP) 00

= ,-P E

So, it follows that E( t6n(j),2 -t6n((),l lP)< 00 for 1 j- n. Note that x IP is a convex function in x; therefore, by Jensen's inequality,

Z(tan(j), 2 -tan()l)

cn

max1Sln

P)

< oo-

Consequently,the Borel-Cantelli lemma establishes that is W-measurable,then

if

E(

n-PO(np12)

n=1

IE=I (tan(),2 -tan(j), I

n

E

E( tan(j),

(j), I IP oc. Thus the claim in Lemma 4 follows. I

1 Ta j). Therefore,

>

I

Tn(j)

FromTheorem3 it then follows immediatelythat,as n 4ZM7(n)- Z1M(n)

Ej=

n2n2

Ek%

ZM (n)

N1>

n

It then follows that

and call the correspondingperformanceZMPT(n),then MZ(n) I T.>_ } where Nn =E= where {Uj: j? 1} is a seLet Yzj-I{fTl}J{U I}, quence of independent uniform random variables, independent of the processing time processes. Note that P( Yj 1) =y for all js, therefore the Yjs are i.i.d. r.v.s. Hence, based on the SLLN,

0 0 undercondition (5),

-

|

oc,

To see this, let c be the upperboundon the variancesimplied by (5), and let b be the assumed lower bound on the means of the t,1 , s. Then, for 0 < ? < 1, Chebyshev'sinequality implies that

b (_E[tjnj]= =

>

P(t1,,J

P(tjn

>

u)du

u) d u +

P(tj

n?

>

u) dti

XIA, SHANTHIKUMAR, AND GLYNN

) P(ti

e+ (




> E) +

P(tj,m

E)+ (I

+

j

2

du

C)E.

Hence, P(ti, n >

E)b-

(1 + c)e

By choosing E sufficiently small, we find the requiredpositive lower bound on P(tj,m > c). Examples of distributions satisfying the lower bound propertyare: 1. the tjms are uniformlybounded away from zero, and 2. ti m exp(uJ) with supj1Hi< oc. We then conclude with the following main result.

/

621

8. CONCLUSION In this paper we have provided a martingaleapproachthat proves the asymptotic optimality of the SPT rule for the Flow Shop Average Completion Time Problem when the processing times on the machines are i.i.d., or more generally, statisticallyexchangeableacross machines and independentacrossjobs. The same argumentcan also be carried out to show the asymptotic optimality of the SWPT rule for the Flow Shop Weighted Completion Time Problem. This extends the recent results of Kaminsky and SimchiLevi (1996a, 1996b).

END NOTES RandomvariablesXl,.. .,Xk are said to be exchangeable if ...* Xak (k) )for any deterministicper(XI,.. ., Xk)-(X~k(1), mutationSk of {1,. .., k}.

For the M-machineflow shop total completion time scheduling problem, the SPT rule is asymptotically optimal in the sense that (i) under the conditions of Lemma 3 and (5), THEOREM 4.

ZSPT(n)-ZM(n)

=,

0

as

n

o,

ACKNOWLEDGMENTS This work was supportedin partby the U.S. Army Research Office under contract/grantDAAG 55-97-1-0377.

n2

(ii) under the conditions of Lemma 3 and (7), - Z(n

ZSPT(n)

)-

a.s. as n

oo.

n2

If, in addition, all the processing times tjms are uniformly

boundedaway from zero, then

ZM IM

) ,1

a.s. as n

-0.

ZM(n )

The same argumentcan be applied to the Flow Shop Weighted Completion Time Problem. In this problem, the processing times of job j, j 1,... ,n, are given an associated weight o)j, with 0