On the Asymptotic Optimality of Algorithms for the Flow Shop Problem withRelease Dates*

Hui Liu,1 Maurice Queyranne,2 David Simchi-Levi3

1 Verizon Laboratory, 40 Sylvan Road Mail Stop 55, Waltham, Massachusetts 02451

2 Faculty of Commerce and Business Administration, University of British Columbia,Vancouver, British Columbia, Canada

3 The Engineering Systems Division and the Department of Civil and Environmental Engineering, Massachusetts Instituteof Technology, 77 Massachusetts Avenue, Room 1-171, Cambridge, Massachusetts 02139

Received October 2001; revised February 2003, December 2003, October 2004; accepted 6 December 2004DOI 10.1002/nav.20066

Published online 26 January 2005 in Wiley InterScience (www.interscience.wiley.com).

Abstract: We consider the nonpermutation flow shop problem with release dates, with the objective of minimizing the sum ofthe weighted completion times on the final machine. Since the problem is NP-hard, we focus on the analysis of the performanceof several approximation algorithms, all of which are related to the classical Weighted Shortest Processing Time Among AvailableJobs heuristic. In particular, we perform a probabilistic analysis and prove that two online heuristics and one offline heuristic areasymptotically optimal. © 2005 Wiley Periodicals, Inc. Naval Research Logistics 52: 232–242, 2005.

Keywords: scheduling; online algorithms; probabilistic analysis; flow shop; release dates

1. INTRODUCTION

We consider the Flow Shop Weighted CompletionTime Problem with Release Dates. In this problem, a setof n jobs has to be sequentially processed on m machineswithout preemption. At any given time each machine canhandle at most one job and a job can be processed on onlyone machine.

Associated with each job j, is a release date rj, a weightwj, and a processing time pij of job j on machine i. Each jobis available for processing at its release date, but not before.The objective is to determine a (nonpermutation) scheduleof the jobs so that the sum of the weighted completion timesof all jobs on the final machine is as small as possible. Inflow shop environment, permutation schedules process thejobs on the second to the last machines according to theFirst In First Out discipline. This implies that the order (or

Permutation) in which the jobs go through the first machineis maintained throughout the shop. In nonpermutationschedules, there is no such restriction, and as a result, theorders in which jobs are processed on each machine can bedifferent.

Following the notation of Graham, Lawler, Lenstra, andRinnooy Kan [11], the model is best classified as F�rj� ¥wjCj. This problem is well known to be NP-hard (seeGarey, Johnson, and Sethi [8]), even in the two-machinecase when all jobs have equal weights and all jobs areavailable for processing at the same time.

As pointed out in Kaminsky and Simchi-Levi [16], themajority of the flow shop related research has focused onminimizing the makespan, that is, minimizing the time ittakes to complete processing all jobs on the final machine.This is due to the fact that individual job related objectives,such as total completion time, are very difficult to analyze.In fact, as Pinedo [21] points out on page 94, “makespanresults are already relatively hard to obtain.” Nevertheless,individual job related objectives capture important real-lifemanagerial scheduling concerns which are not reflected inthe makespan and similar objectives (see, for example,Morton and Pentico [20]). For a review of the literature on

* Research supported in part by the MIT-Singapore Alliance,The MIT Forum for Supply Chain Innovation and NSF ContractsDMI-9732795, DMI-0085683 and DMI-0245352, and ONR Con-tracts N00014-90-J-1649 and N00014-95-1-0232

Correspondence to: D. Simchi-Levi (dslevi@mit.edu)

© 2005 Wiley Periodicals, Inc.

flow shop problems, the reader is referred to Kaminsky andSimchi-Levi [16].

The flow shop problem with release dates has hardlyreceived any attention in the literature. The rare exception isthe work of Hall [12], who analyzed problem F2�rj�Cmax,that is, the two-machine flow shop problem with releasedates when the objective is to minimize the time it takes tofinish all jobs on the second machine. In this paper, Hallproposed a (1 � �) polynomial time approximation algo-rithm based on Johnson’s optimization algorithm for thetwo-machine flow shop problem without release dates. Inaddition, Bhaskaran and Pinedo [3] and Morton and Pentico[20] suggest a variety of dispatch rules as a way to solvereal-world industrial flow shop problems with or withoutrelease dates.

In contrast to the flow shop problem, much work has beendone on single and parallel machine scheduling problemswith release dates and when the objective is to minimize thetotal job (weighted) completion time. Specifically, in thepast several years we have seen a stream of papers focusingon the design and analysis of polynomial time approxima-tion algorithms for problems such as 1�rj� ¥ wjCj and P�rj�¥ wjCj. Many of these approximation algorithms are ob-tained by first constructing lower bounds and (nonfeasible)schedules based on linear programming and/or preemptiveschedule relaxations. These schedules are then convertedinto nonpreemptive, feasible solutions.

For instance, Goemans, Queyranne, Schulz, Skutella, andWang [10] considered the problem 1�rj� ¥ wjCj. They usedthe method of �-points to convert preemptive schedules intononpreemptive schedules. In a preemptive schedule, the�-point of each job j is defined as the first point in time atwhich an �-fraction of job j has been completed. To obtaina nonpreemptive schedule, they simply schedule the jobs inthe order of their �-points. Using this approach, they ob-tained an online randomized approximation algorithm withperformance ratio of 1.6853.

For this same problem, Anderson and Potts [2] used adifferent proof technique concentrating on the combinato-rial nature of the problem. They showed that a deterministiconline algorithm, Delayed SWPT has a competitive ratio of2. In this algorithm, if the machine is available at time t,choose from among the available jobs the one with thesmallest pj/wj. If pj � t, then process job j. If pj � t, themachine remains idle until time pj, or until another job isreleased and the condition is met before pj.

Exciting results have also been obtained for problem P�rj�¥ wjCj. For example, Schulz and Skutella [23] developed arandomized online algorithm for P�rj� ¥ wjCj with a com-petitive ratio of 2. Bringing together the contributions fromfive different research groups, Afrati et al. [1] show thatthere exist polynomial time approximation schemes for P�rj�¥ wjCj, and Rm�rj� ¥ wjCj with a fixed number m of

unrelated parallel machines (where the processing time pij

of job j may depend on machine i). Thus, for any fixed � �0, these problems can be approximated (offline) to within afactor 1 � � of the optimum, in time polynomial in thenumber n of jobs and the inverse precision 1/�.

A departure from this line of research is presented inKaminsky and Simchi-Levi [18] and Chou, Queyranne, andSimchi-Levi [4], who focused on the asymptotic perfor-mance ratio of algorithms. In the latter work, problem 1�rj�¥ wjCj, which is a special case of their model is directlyrelated to the current paper. In this case, Chou, Queyranne,and Simchi-Levi [4] analyzed the asymptotic performanceof the following online heuristic referred to as the WeightedShortest Processing Time among Available jobs(WSPTA) heuristic. In this heuristic, every time we need toschedule a job, we consider all the jobs that have beenreleased but not yet processed and select the job with thelargest ratio of wj/pj to be processed next. If there is no jobavailable, the machine stays idle until at least one jobarrives.

In the past few years, flow shop scheduling problems withminsum objectives have attracted attention of several re-search groups.

Using probabilistic analysis approach, Kaminsky andSimchi-Levi [16, 17] and Xia, Shanthikumar, and Glynn[25] studied the flow shop problem F� � ¥ wjCj and itsspecial case, F� � ¥ Cj. These papers have focused oncharacterizing both the structure of the asymptotic optimalsolution and the effectiveness of the Weighted ShortestProcessing Time (WSPT) algorithm. In this algorithm jobsare sequenced in a decreasing order of the ratio of theirweighs to the total processing times on all of the machines.The most general result is obtained by Xia, Shanthikumar,and Glynn [25], who show that WSPT is asymptoticallyoptimal under some mild probabilistic assumptions on thedistributions of job processing times and weights. See Table1 for a summary of these results.

In the next section we discuss the model, the problem andthe main results.

2. MODEL, NOTATION, AND THE MAINRESULTS

In the Flow Shop Weighted Completion Time Problemwith Release Dates, a set of n jobs has to be sequentiallyprocessed on m machines with no preemption. The objec-tive is to determine a nonpermutation schedule so that thesum of the weighted completion times of all the jobs on thefinal machine is minimized. In our model, we assume thatthere is unlimited buffer space between any two machines.That is, there is no blocking of any machine because thebuffer of its subsequent machine is full.

Formally, let Cij be the completion time of job j onmachine i in a particular schedule. The objective value of

233Liu, Queyranne, and Simchi-Levi: Asymptotic Optimality of Algorithms

this flow shop problem is to minimize ¥j�1n wjCmj. We use

Z*m to denote the objective value of the optimal nonpermu-tation schedule of this problem. Similarly, given a heuristicH for this problem, we use Zm

H to denote the sum of theweighted completion times in the resulting schedule.

The processing times pij are drawn from an independent,identical, and bounded distribution defined on the interval(0, 1]. Similarly, the weights wj are independent and iden-tically distributed random variables defined on the interval(0, 1]. No assumption is made on the release dates exceptthat they are nonnegative.

The Single Machine Scheduling Problem with ReleaseDates, i.e., 1�rj� ¥ wjCj, plays an important role in ouranalysis. In this problem, there are n jobs, every job has arelease date rj, a weight wj, and a processing time pj. Theobjective is to schedule all the jobs without violating therelease date constraints so as to minimize the sum of theweighted completion times of all jobs. Given an instance ofthe problem, let Z*1 be its optimal solution and Z1

H is the costof the solution generated by some heuristic H. As pointedout in the introduction, problem 1�rj� ¥ wjCj is NP-com-plete.

Recently, Chou, Queyranne, and Simchi-Levi [4] provedthat the WSPTA heuristic, described in the Introduction, isasymptotically optimal for this problem. That is,

THEOREM 1: Consider any instance of 1�rj� ¥ wjCj

with bounded job processing times and weights. We have

limn3�

Z1WSPTA

n2 � limn3�

Z*1n2 � �, for some nonzero constant �.

Observe that no probabilistic assumptions are made in thestatement of the Theorem. A similar result is mentioned inPosner [22]. Unfortunately, no detail on either the algorithm

or the proof is provided in that paper. Gazmuri [9] analyzeda similar problem, 1�rj� ¥ Cj, under probabilistic assump-tions and showed the asymptotic optimality of some heu-ristics for various special cases.

In our analysis, we associate two specific single-machineproblems with every instance of the flow shop problem. Thefirst, referred to as problem PAverage, is constructed asfollows. Given an instance of the flow shop problem F�rj� ¥wjCj as described above, construct an instance of 1�rj� ¥wjCj with the same set of n jobs, each job j having the samerelease date rj and same weight wj, and a processing timepj

Average � (1/m) ¥i�1m pij. We let Z*1(PAverage) denote the

optimal objective value of this problem. The second singlemachine problem, referred to as problem PFirst, is identicalto PAverage, except that the processing time of each job j inthe single machine problem is now pj

First � p1j, its pro-cessing time on the first machine. We let Z*1(PFirst) denotethe optimal objective value of this single machine problem.

In this paper, we analyze the asymptotic performance ofthe following three heuristics. Note all three heuristics con-struct a permutation schedule by determining the order ofthe jobs on the first machine. The jobs then go through theremaining m � 1 machines as early as feasible and on a firstcome first served basis. As a result, the job order on the firstmachine is maintained on all machines.

● Heuristic Firstm: In the m-machine flow shop prob-lem, when a job is finished on the first machine, weconsider all the jobs that have arrived and not yetprocessed: The next job to be processed is the onewith the largest wj/p1j ratio. That is, we rank theavailable jobs in nonincreasing order of their ratiosof weight to first-machine processing time. If there isno job available at that date, the first machine staysidle until the next job arrives. We denote the objec-tive value generated by this heuristic by Zm

First.

Table 1. Summary of asymptotic performance analysis.

Problem Reference Assumptionsa Schedule

F� ¥ Cj Kaminsky and Simchi-Levi [17] i.i.d. processing timeb SPTF� ¥ Cj Xia, Shanthikumar, and Glenn [25] Statistically exchangeable processing times across

machinesSPT

F� ¥ wjCj Kaminsky and Simchi-Levi [16] i.i.d. processing timeb i.i.d. weights OptimalF� ¥ wjCj Xia, Shanthikumar, and Glenn [25] Statistically exchangeable processing times across

machinesWSPT

1�rj� ¥ Cj Kaminsky and Simchi-Levi [18] Bounded processing times no probabilistic assumptions SPTA1�rj� ¥ wjCj Chou, Queyranne, and Simchi-Levi [4] Bounded processing times and weights no probabilistic

assumptionsWSPTA

Qm�rj� ¥ wjCj Chou, Queyranne, and Simchi-Levi [4] Bounded processing times and weights no probabilisticassumptions

WSPRc

a i.i.d. stands for independently, identically distributed.b In his Ph.D. thesis, Kaminsky [15] points out that these results hold for statistically exchangeable processing times across machines.c WSPR is the following algorithm: Whenever a machine becomes idle, start processing on it an available job, if any, with largest ratio ofits weight to the processing requirement, i.e., ignore the speed of the machines; otherwise, wait until the next job release date.

234 Naval Research Logistics, Vol. 52 (2005)

● Heuristic Averagem: This is identical to the Firstm

heuristic just described, except that the job selectedto be processed next on the first machine is one withlargest wj/pj

Average ratio, where as before, pjAverage

� (1/m) ¥i�1m pij is job j’s average processing time

over the m machines. We denote the objective valuegenerated by this heuristic by Zm

Average.

Observe that these two heuristics are online heuristics;decisions are made at any time using only informationpertaining to jobs which have been released by that time.The third heuristic is not an online algorithm:

● Heuristic Singlem: Consider an instance of the flowshop problem and construct the associated single-machine instance of problem PAverage. ApplyWSPTA to jobs in this single-machine instance andobtain a sequence of the n jobs. Then sequence alljobs in the original flow shop problem in this orderon each machine. Denote by Zm

Single the m-machineflow shop objective value of the resulting schedule.

Observe that WSPTA heuristic is an online heuristic forthe single-machine problem PAverage. The following exam-ple, however, shows that the flow shop heuristic Singlem isnot online.

EXAMPLE 1: Consider a 3-job, 2-machine instance withall processing times p1j � 1 and p2j � 5. With all wj �rj � j, job 1 is selected by WSPTA to be processed first atits release date 1. When it completes processing on the firstmachine at date 2 in the flow shop problem, the onlyavailable job is job 2, just released. However, since allpj

Average � 3, the WSPTA order on the single-machineinstance has job 3 processed immediately after job 1, be-cause this job has the highest wj/pj

Average ratio and isavailable for processing at the completion date 4 of job 1 onthe single-machine.

The main result of this paper is:

THEOREM 2: Consider a class of n-job instances of them-machine flow shop problem with data satisfying thefollowing probabilistic assumptions. (i) The processingtimes pij are independent and identically distributed randomvariables, defined on the interval (0, 1]; (ii) the weights wj

are independent and identically distributed random vari-ables, defined on the interval (0, 1]; and (iii) the releasedates rj are nonnegative random variables with rj � O(n).Then, with probability 1 we have

limn3�

ZmAverage

n2 � limn3�

ZmFirst

n2 � limn3�

ZmSingle

n2 � limn3�

Z*mn2 � �,

for some nonzero constant �.

REMARK 1: We can relax the third assumption and onlyrequire the release dates rj to be nonnegative random vari-ables, if we restate our main result in the following form:

limn3�

ZmAverage

Z*m� lim

n3�

ZmFirst

Z*m� lim

n3�

ZmSingle

Z*m� 1, w.p.1.

We observe that while the three heuristics analyzed inthis paper generate permutation schedules, i.e., schedules inwhich jobs go through all the machines using the sameorder, our analysis shows that the asymptotic optimality ofthese three heuristics holds with respect to nonpermutationschedules as well. This is true since Z*m is the optimalobjective value among all nonpermutation schedules. Thus,Theorem 2 states that these heuristics have the same asymp-totic performance ratio as that of the optimal scheduleamong all nonpermutation schedules.

The rest of this paper is organized as follows. In Section3, we provide an example of a simple and natural heuristicwhich is not asymptotically optimal for our problem. Thus,the example illustrates that Theorem 2 is truly the propertyof the heuristics analyzed in this paper, and does not holdfor any heuristic. In Section 4 we present two lower boundsthat are used in the proof of the main theorem. Section 5presents our general approach for proving Theorem 2. Thissection is followed by two sections in which we prove theoptimality of the three heuristics. In Section 8 we put ourasymptotic results in perspective by presenting a series ofnumerical experiments. Finally, Section 9 provides someconcluding remarks.

3. PRELIMINARIES

In this section we demonstrate that a simple and naturalheuristic, the First Come First Served (FCFS), is not as-ymptotically optimal, even for the unweighted single-ma-chine problem, when the processing times are i.i.d. randomvariables defined on the interval (0, 1]. Thus, the asymptoticoptimality of the heuristics discussed in the paper is aproperty not necessarily shared by other reasonable heuris-tics.

The (FCFS) is arguably the simplest and most naturalheuristic for scheduling jobs that arrive over time: simplyprocess the jobs as soon as possible and in their order ofarrival. Thus FCFS, also known as First-In First-Out(FIFO), or Earliest Release Date (ERD), is an online, listscheduling heuristic with priority given to earliest releasedate. It is also probably the most widely used schedulingheuristic in industrial and service environments. The fol-lowing example shows that FCFS does not have asymptoticoptimality, even for unweighted single-machine instanceswith positive, bounded i.i.d. processing times.

235Liu, Queyranne, and Simchi-Levi: Asymptotic Optimality of Algorithms

EXAMPLE 2: Fix a parameter a, such that 0 � a � 1.Construct the n-job instance Ia(n) in the class with i.i.d.processing times pj drawn from the following distribution:pj � a2 with probability 1/(1 � a), and 1 otherwise [i.e.,with probability a/(1 � a)]. Thus E[ pj] � a. The jth jobin the instance is released at date rj � (1 � 1/j)a2, for j �1, . . . , n.

The FCFS heuristic schedules the jobs in their arrivalorder 1, 2, . . . , n. Using a well-known expression (e.g.,Smith [24]), the total completion time of the resultingschedule is zFCFS(Ia(n)) � ¥j�1

n CjFCFS � ¥j�1

n (n � j �1) pj, and therefore its expected value is

E�zFCFS�Ia�n � E� �j�1

n

CjFCFS� �

12

n�n � 1a.

On the other hand, the SPTA heuristic first schedules job 1and then, since all other jobs have been released by thecompletion of job 1, it schedules all “short” jobs from{2, . . . , n}, i.e., those with processing time equal to a2,followed by all remaining “long” jobs. The number S ofshort jobs, among jobs {2, . . . , n}, is a binomial randomvariable with parameters n � 1 (number of “trials”) and1/(1 � a) (probability of “success”). Thus E[S] � (n �1)/(1 � a) and E[S2] � Var[S] � E[S]2 � (n � 1)(a �n � 1)/(1 � a)2. The total completion time of the SPTAschedule is

zSPTA�Ia�n � �j�1

n

CjSPTA � np1 �

S�S � 1

2a2

� �n � 1 � S�Sa2 ��n � 1 � S�n � S

2,

where the first term, np1 accounts for the processing of job1; the second term, [S(S � 1)/ 2]a2 for the total completiontime thereafter of all short jobs; the third term, (n � 1 �S)(Sa2), for the additional delay imposed by all short jobson each one of the n � 1 � S remaining jobs; and the lastterm for the total completion time thereafter of these re-maining jobs. Rewriting, we obtain

zSPTA�Ia�n � �j�1

n

CjSPTA � np1 �

1 � a2

2S2

� �n �1

2� �1 � a2S ��n � 1n

2.

The expected total completion time is thus

E�zSPTA�Ia�n � E� �j�1

n

CjSPTA� � nE�p1 �

1 � a2

2E�S2

� �n �1

2� �1 � a2E�S ��n � 1n

2

�a

1 � a�an2 � �n � 1a � 2n � 1.

We now calculate the limiting ratio of the difference ofthese two expected values, we have

limn3�

E�zFCFS�Ia�n � E�zSPTA�Ia�n

n2 �1

2a �

a2

1 � a.

Use Z�(Ia(n)) to denote the optimal solution of this in-stance of the single-machine unweighted problem, we have

limn3�

E�zFCFS�Ia�n � E�z*�Ia�n

n2

� limn3�

E�zFCFS�Ia�n � E�zSPTA�Ia�n

n2 �1

2a �

a2

1 � a.

Because we have 0 � a � 1 as a nonzero parameter, thislimiting ratio is not 0. In fact, it reaches the maximum (3 �2�2)/2 when a � �8/ 2 � 1. Since zSPTA(Ia(n)) andz*(Ia(n)) does not converge in expectation; consequently,they do not converge with probability one.

From this example we can see that asymptotic optimalityis not a property shared by any (reasonable) heuristics, evenunder the restriction of bounded problem instances.

4. LOWER BOUNDS ON Z*m

In this section we present two lower bounds on Z*m, theobjective value of the optimal nonpermutation schedule ofproblem F�rj� ¥ wjCj. The first lower bound is constructedbased on the single machine problem, problem PFirst, andthe second lower bound is based on the preemptive versionof the single machine problem, problem PAverage.

To present the first lower bound on Z*m, consider aninstance of F�rj� ¥ wjCj and its associated single-machineproblem PFirst. It is easy to verify that the following ob-servation holds.

OBSERVATION 1: For every instance of the Flow ShopWeighted Completion Time Problem with Release Dates,we have

236 Naval Research Logistics, Vol. 52 (2005)

Z*m � Z*1�PFirst.

To construct the second lower bound on Z*m, we associateevery instance of F�rj� ¥ wjCj with an instance of the singlemachine problem PAverage. We relax the no-preemptionconstraint in problem PAverage and allow for preemption.We let Z*1

P(PAverage) denote the optimal solution of thispreemptive single machine problem, and we prove thatZ*1

P(PAverage) provides a lower bound for Z*m.

THEOREM 3: For every instance of the Flow ShopWeighted Completion Time Problem with Release Datesand its associated single machine problem PAverage, wehave

Z*m � Z*1P�PAverage.

A similar lower bound is proven in Eastman, Even, andIsaacs [7] for F� ¥ wjCj. To extend the lower bound to ourmodel, which includes release dates, it is sufficient to intro-duce the following lemma whose proof is based on the workof Chou, Queyranne, and Simchi-Levi [4] for the uniformparallel machine scheduling problem.

LEMMA 4: Every feasible schedule of the flow shopproblem with completion time Cmj of job j, j � 1, 2, . . . ,n, on the last machine, machine m, has a correspondingpreemptive schedule for problem PAverage, where job j, j �1, 2, . . . , n, completes processing at time Cj, such thatCj � Cmj for all j.

Here we provide a sketch of the proof, the interestedreader is referred to Liu [19] for details. Construct a pre-emptive schedule for the single machine problem PAverage

from a feasible schedule of the flow shop problem as fol-lows. Partition the time interval [minj{rj}, maxj{Cmj}] intosubintervals where the boundaries of the subintervals are theset of release dates and the completion times of the jobs. Inthe single machine problem, process the same jobs in eachsubinterval with the job processed on the last machine first(if exists), and the rest of the jobs in an arbitrary order, eachfor a duration equals to 1/m of the time being processed inthe flow shop. This generates a feasible schedule because bythe choice of the boundaries, the release date are respected.And since all jobs in PAverage have processing time 1/mtimes of the total processing time in the flow shop problem,we have Cj � Cmj for all j.

Consequently, the optimal solution to the preemptiveversion of problem PAverage provides a lower bound on Z*m.On the other hand, it is relatively easy to construct examplesin which the optimal solution of the nonpreemptive versionof PAverage does not provides a lower bound on Z*m.

5. THE GENERAL APPROACH

In this section we present our approach for proving theasymptotic optimality of the three heuristics introduced inSection 2, i.e., heuristic Averagem, Firstm, and Singlem.Since all three heuristics generate permutation schedules, inthis section, we focus on feasible permutation schedulesonly. We show that these permutation schedules generateasymptotically optimal schedules for the flow shop problemeven when we consider nonpermutation schedules.

Our approach for proving the main result is based onusing a recursive equation that specifies the completion timeof every job in a given schedule, similar to the recursionequation applied by Xia, Shanthikumar, and Glynn [25] forthe flow shop problem without release dates.

To present our approach, consider any permutation heu-ristic H for the flow shop problem. Without loss of gener-ality we index the jobs according to their starting time onthe first machine (or the completion time on the last ma-chine).

LEMMA 5: For any given permutation schedule gener-ated by heuristic H for the flow shop weighted completiontime problem with release dates, we have

�j�1

n

wjCmj � �j�1

n

wjmax1�x�j

�rx � �k�x

j

p1k � �j�1

n

wj max1��1�· · ·��m�1�j

� � �k��1

�2

�p2k � p1k � · · · � �k��m�1

j

�pmk � p1k� m max

j�1,2,...,n

p1j �j�1

n

wj.

PROOF: To prove the lemma, we first observe that, givena schedule generated by H, the following recursion allowsone to calculate the completion time of each job on anymachine,

Ci,j�1 � max Cij, Ci�1,j�1� � pi,j�1 i � 2, 3, . . . , m,

C1,j�1 � max rj�1, C1j� � p1,j�1.

Applying the recursion multiple times, we get that

Cmj � max1�x��1�· · ·��m�1�j

�rx � �k�x

�1

p1k � �k��1

�2

p2k � · · · � �k��m�1

j

pmk.

237Liu, Queyranne, and Simchi-Levi: Asymptotic Optimality of Algorithms

Rearranging, we have

Cmj � max1�x��1�· · ·��m�1�j

� �rx � �k�x

j

p1k � �k��1

�2

�p2k � p1k � · · · � �k��m�1

j

�pmk � p1k

� �i�1

m�1

p1,�i � max1�x�j

�rx � �k�x

j

p1k � max1��1�· · ·��m�1�j

� � �k��1

�2

�p2k � p1k � · · · � �k��m�1

j

�pmk � p1k� m max

j�1,2,...,n

p1j.

Calculating the cost associated with heuristic H using theabove expression of Cmj proves the lemma. �

Observe that the upper bound on Z*m developed byLemma 5 has three components. The first term, i.e.,

�j�1

n

wjmax1�x�j

�rx � �k�x

j

p1k,

can be interpreted as the cost associated with a singlemachine scheduling problem. The second term satisfies

�j�1

n

wj max1��1�· · ·��m�1�j

� �k��1

�2

�p2k � p1k � · · ·

� �k��m�1

j

�pmk � p1k � �j�1

n

wj �i�2

m

max1�a�b�n

�k�a

b

�pik � p1k

� �i�2

m

max1�a�b�n

�k�a

b

�pik � p1k �j�1

n

wj.

This last quantity was analyzed by Xia, Shanthikumar, andGlynn [25]. They prove that under the assumptions de-scribed in Theorem 2 on the distributions of processingtimes and weights, we have for every i, i � 2, 3, . . . , m,almost surely that

limn3�

1

nmax

1�a�b�n�

k�a

b

�pik � p1k � 0. (1)

Finally, the last term, i.e.,

maxj�1,2,...,n

p1j �j�1

n

wj grows as n.

Thus,

THEOREM 6: Consider any permutation heuristic H forthe flow shop problem and index the jobs according to theirstarting times on the first machine. Under the assumptions(i) the processing times pij are independent and identicallydistributed random variables, defined on the interval (0, 1];(ii) the weights wj are independent and identically distrib-uted random variables, defined on the interval (0, 1]; and(iii) the release dates rj are nonnegative random variableswith rj � O(n), we have, with probability 1,

limn3�

Z*mn2 � lim

n3�

ZmH

n2 � limn3�

¥j�1n wjmax1�x�j rx � ¥k�x

j p1k�

n2 .

6. ANALYSIS OF HEURISTICS FirstmAND Averagem

In this section, we prove the asymptotic optimality ofheuristic Firstm and we provide some insight on the anal-ysis of heuristic Averagem.

To analyze the Firstm heuristic, consider single machineproblem PFirst and its optimal solution Z*1(PFirst). Obser-vation 1 implies that Z*1(PFirst) provides a lower bound onthe optimal solution of the flow shop problem, that is, alower bound on Z*m. In addition, Theorem 1 implies that theWSPTA heuristic is asymptotically optimal for problemPFirst. Of course in this case, the heuristic works as follows.At the completion time of any job on the single machine,consider all jobs released but not yet processed and selectthe one with the largest ratio wj/p1j to be scheduled next.

Consider problem PFirst and index the jobs according tothe order generated by the WSPTA heuristic when appliedto this problem. Evidently, the completion time Cj of eachjob must satisfy Cj � max1�x�j{rx � ¥k�x

j p1k}. Thus,Theorem 1 implies

limn3�

Z*1�PFirst

n2 � limn3�

1

n2 �j�1

n

wjmax1�x�j

�rx � �k�x

j

p1k.

Finally, observe that the sequence generated by theWSPTA heuristic when applied to problem PFirst is thesame as the sequence generated by heuristic Firstm whenapplied to the flow shop problem. Hence, using Theorem 6,we have

238 Naval Research Logistics, Vol. 52 (2005)

limn3�

1

n2 �ZmFirst � Z*m � lim

n3�

1

n2 �ZmFirst � Z*1�PFirst

� limn3�

1

n2 �j�1

n

wj�max1�x�j

�rx � �k�x

j

p1k � max1�x�j

�rx � �k�x

j

p1k�� 0.

That is, heuristic Firstm is asymptotically optimal for theflow shop problem.

The proof of the asymptotic optimality of heuristic Av-eragem uses a similar framework to that in the proof ofheuristic Firstm. The only difference is that in case ofAveragem we apply a lower bound based on a singlemachine problem where jobs are scheduled in order accord-ing to the heuristic Averagem. The interested reader isreferred to Liu [19], who makes use of results found inHoeffding [14] and Coffman and Lueker [5].

7. ANALYSIS OF HEURISTIC Singlem

In this section, we prove the asymptotical optimality ofheuristic Singlem. The proof is similar, but not identical tothe previous proofs, and we perform this proof in two steps.

In the first step, we prove that for a given instance of theflow shop problem, under the assumptions stated in Theo-rem 2, with probability 1 we have

limn3�

ZmSingle

n2 � limn3�

Z*1�PAverage

n2 � �,

for some nonzero constant �.

For this purpose, observe that the sequence generated byWSPTA when applied to problem PAverage is identical tothe one used by heuristic Singlem. This, together with theresult of Theorem 1 and Theorem 3, implies that

limn3�

1

n2 �ZmSingle � Z*1�PAverage � lim

n3�

1

n2 �j�1

n

wj

� �max1�x�j

�rx � �k�x

j

p1k � max1�x�j

�rx � �k�x

j

pkAverage�

� limn3�

1

mn ��i�1

m

max1�a�b�n

��k�a

b

�p1k � pik� 1

n �j�1

n

wj

� limn3�

1

n �j�1

n

wj

1

m �i�2

m �limn3�

1

nmax

1�a�b�n��

k�a

b

�p1k � pik � 0.

This last equality is obtained again by using Eq. (1).In the second step of this proof, we show that there is a

strong connection between the optimal objective values ofthe two single-machine problems that we constructed forevery instance of the flow shop problem. In particular, weprove that optimal schedules for problem PFirst andPAverage have the same asymptotic performance ratio. Thatis,

LEMMA 7: Under the assumptions of Theorem 2, for agiven instance of the flow shop problem and its correspond-ing single machine problem PAverage and PFirst, we havealmost surely,

limn3�

Z*1�PFirst

n2 � limn3�

Z*1�PAverage

n2 � �,

for some nonzero constant �.

The proof of this lemma can be found in the Appendix.Since Z*1(PFirst) provides a lower bound on the optimalnonpermutation solution for the flow shop problem, Lemma7, combined with the previous result, indicates that

limn3�

1

n2 �ZmSingle � Z*m � lim

n3�

1

n2 �ZmSingle � Z*1�PFirst

� limn3�

1

n2 �ZmSingle � Z*1�PAverage � 0.

Thus, heuristic Singlem is asymptotically optimal among allnonpermutation schedules.

This completes the proof of Theorem 2.

8. A NUMERICAL STUDY

The analysis provided in the previous sections indicatesthat heuristics Averagem, Firstm, and Singlem are asymp-totically optimal. The difficulty, of course, is that the rate ofconvergence of the heuristics to the optimal solution may bequite slow. To show the effectiveness of the algorithms, weconducted a numerical study.

Ideally, we would like to report the performance of thethree heuristics relative to the optimal solution. Since theFlow Shop Weighted Completion Time Problem with Re-lease Dates is strongly NP-hard, even for problems with twomachines and equal weights, finding the optimal solutionusing branch and bound or other technique may take pro-hibitively long time. As a result, we measure the perfor-mance of each heuristic relative to the best of the two lowerbounds which we introduce in the following paragraph.

239Liu, Queyranne, and Simchi-Levi: Asymptotic Optimality of Algorithms

Recall the two lower bounds introduced in Section 4,namely, Z*1

P(PAverage) and Z*1(PFirst). The first lowerbound requires finding the optimal solution of the singlemachine problem when preemption is allowed and the sec-ond lower bound is the optimal solution when preemption isnot allowed. Since both of these single machine problemsare strongly NP-hard, in the computation study, we focus ona lower bound on Z*1

P(PAverage) and Z*1(PFirst) that iseasily obtained.

Specifically, consider the single machine weighted com-pletion time problem with release dates with n jobs eachwith an associated processing time pj, weight wj, and re-lease date rj. Dyer and Wolsey [6] proposed the followinglinear programming formulation, whose optimal solutionprovides a lower bound on the optimal solution for thesingle machine problem when preemption is allowed. Con-sequently, it also provides a lower bound on the optimalsolution of the single machine problem when no preemptionis allowed. In this formulation, it is required that jobs haveintegral processing times and release dates, T is the leastupper bound on the completion time of the last job, andyjt � 1 if job j is being processed in the time period [t �1, t) and yjt � 0 otherwise.

minimize �j�1

n

wjCj subject to �j�1

n

yjt � 1,

t � 1, 2, . . . , T, �t�1

T

yjt � pj, j � 1, 2, . . . , n,

pj

2�

1

pj�t�1

T �t �1

2�yjt � Cj, j � 1, 2, . . . , n,

yjt � 0, j � 1, 2, . . . , n, t � rj � 1, . . . , T.

Dyer and Wolsey [6] (see also Hall, Schulz, Shmoys, andWein [13]) proved that the following sequence is optimalfor this linear program. At the completion time and therelease date of any job, consider all the jobs which havebeen released by that time but not yet completed, and selectthe job with the largest wj/pj to be processed next. If no jobis available, the machine is idle until at least one job arrives.

Thus, we use this linear program to find lower bounds onZ*1

P(PAverage) and Z*1(PFirst). Let ZLP(PAverage) andZLP(PFirst) be the corresponding lower bounds, and letLB � max{ZLP(PAverage), ZLP(PFirst)}. Evidently, LBprovides lower bound on Z*m. As a result, in this study, wemeasure the performance of the heuristics relative to LB.

We randomly generated different instances of the flowshop problem. The number of machines in our study was 2,5, and 10 while the number of jobs was 100, 500, and 1000.For each combination of n and m, 10 problems are ran-domly generated.

For each problem, two cases are considered for job pro-cessing times. In the first case, pij, j � 1, 2, . . . , n, i �1, 2, . . . , m, are generated independently from DiscreteUniform [1, 10] distribution. In the second case, pij aregenerated independently using the following equation:min{10, expo(0.5)�10 � 1}. This method guarantees thatthe processing times are bounded i.i.d. integers. We call thismethod discrete exponential. In all problem instances, jobweights wj, j � 1, 2, . . . , n, are generated independentlyfrom Discrete Uniform [1, 10] distribution. To generate therj, we associate with each job j a random variable Irj, whichwe referred to as job j’s interarrival times. Interarrival timeis the time between the release dates of two successivelyreleased jobs. Thus, if we let r1 � 0 and index the jobsaccording to the order they are released, we have Irj � rj �rj�1, j � 2, 3, . . . , n. In all of the problem instances, Irj’sare generated independently from a Discrete Uniform [1,10] distribution. We summarize the problem parameters inTable 2.

Tables 3 and 4 report the average ratios between theheuristic objective value and the objective value of the bestlower bound over 10 problem instances generated, for everycombination of n and m. To try and identify the best lower

Table 2. List of parameter values.

Parameter Value or distribution

Machine number m 2, 5, 10Job number n 100, 500, 1000Processing times pij i.i.d. D.U. [1, 10] or i.i.d. discrete

exponentialJob weights wj i.i.d. D.U. [1, 10]Interarrival times Irj i.i.d. D.U. [1, 10]

Table 3. Summary of computational results for discrete uniform processing times.

Job no.

m � 2 m � 5 m � 10

100 500 1000 100 500 1000 100 500 1000

ZmAveragem/LB 1.0474 1.0118 1.0072 1.1635 1.0446 1.0273 1.3572 1.1002 1.0574

ZmFirstm/LB 1.0488 1.0118 1.0071 1.1640 1.0047 1.0267 1.3642 1.1007 1.0562

ZmSinglem/LB 1.0555 1.0159 1.0091 1.1776 1.0484 1.0299 1.3690 1.1046 1.0585

ZLP(PAverage)/ZLP(PFirst) 0.9979 0.9988 0.9995 0.9941 0.9984 0.9992 0.9923 0.9987 0.9992

240 Naval Research Logistics, Vol. 52 (2005)

bounds, we also report the average relative performanceratios of the two lower bounds, i.e., ZLP(PAverage)/ZLP(PFirst), over the 10 problems instances for every com-bination of n and m.

Tables 3 and 4 reveal that, regardless of the job process-ing time distributions, the performance of each heuristicimproves as the number of jobs increases, precisely what thetheoretical analysis suggests. The tables also indicate thatthe performance of the heuristics decreases as the number ofmachines increases. This suggests that the rate of conver-gence of the heuristic to the optimal solution depends notonly on the number of jobs but also on the number ofmachines. Two possible explanations exist. First, it could bethe case that the lower bounds are worse for a larger numberof machines. Second, it is possible that the performance ofthe heuristics improves as the number of machines decrease.That is, the more machines we have, the larger the numberof jobs that we need in order to achieve a similar perfor-mance of the quality of the solution.

One might suspect that heuristic Averagem should outper-form heuristic Firstm since it ranks the jobs by taking intoaccount the processing time of jobs across all the machines,while heuristic Firstm ranks the jobs based only on processingtimes on the first machine. Based on our limited numericalstudy, it is appropriate to point out that there is no strongevidence to support this speculation as long as the job process-ing times have the same distribution on all the machines. Infact, there is no indication that any heuristic outperforms theother two heuristics. This observation also holds when wecompare the two lower bounds; i.e., it is not possible to say thatone lower bound dominates the other one.

On the other hand, if the distribution of the job processingtime is machine dependent, then one might suspect thatheuristic Averagem will outperform heuristic Firstm. Toaddress this issue, we conducted a limited numerical studyin which job processing times on different machines havedifferent distributions. In these experiments, we fixed thedistributions of p1j, Irj, and wj and varied the distributionsof the job processing times on the other machines. Ourlimited results show that if the job processing times on somemachine other than the first one are stochastically largerthan the job processing time on the first machine, thenheuristic Averagem outperforms heuristic Firstm in all ofthe instances tested, although the magnitude depends on theprocessing time distributions. These preliminary results

suggest that heuristic Averagem is more robust than heu-ristic Firstm when processing time distributions on differentmachines are different.

9. CONCLUDING REMARKS

In this paper, we provide a probabilistic analysis of threeonline heuristics for the Flow Shop Weighted CompletionTime Problem with Release Dates. These three heuristicsare closely related to the classical Weighted Shortest Pro-cessing Time among Available Jobs heuristic. We provethat when the jobs processing times and weights are i.i.d.random variables, the three heuristics are asymptoticallyoptimal. Although these three heuristics generate permuta-tion schedules, the asymptotic optimality is with respect toall nonpermutation and permutation schedules. In fact, ouranalysis shows that these three heuristics have the sameasymptotic performance ratio as the optimal scheduleamong all nonpermutation schedules. An empirical analysisis conducted to test the effectiveness of these heuristics onmoderate size problems.

The reader observes that the i.i.d. assumption is onlyrequired in two places: (i) for the validity of Eq. (1), and (ii)the proof of the asymptotic optimality of heuristicAveragem. It is appropriate to point out that these resultsalso hold under the more general assumption that job pro-cessing times are statistically exchangeable across ma-chines. Indeed, Xia, Shantikumar, and Glynn [25] provedEq. (1) under this more general assumption, and it is easy toverify that the asymptotic optimality of heuristic Averagem

holds in this case as well. Thus, Theorem 2 also holds whenthe job processing times are statistically exchangeable.

Finally, it is appropriate to point out that the two lowerbounds presented in Section 4 are also valid for the openshop weighted completion time problem with release dates.In an open shop, each job has to be processed on m ma-chines with no restriction with regard to machine routing.This implies that the three heuristics designed for the flowshop problem are also feasible with respect to the open shopproblem. Thus, the theoretical analysis and the computa-tional results that are presented in this paper apply also tothe open shop problem. That is, the online heuristicsAveragem, Firstm, and Singlem are also asymptoticallyoptimal for the open shop weighted completion time prob-lem with release dates.

Table 4. Summary of computational results for discrete exponential processing times.

Job no.

m � 2 m � 5 m � 10

100 500 1000 100 500 1000 100 500 1000

ZmAveragem/LB 1.0364 1.0089 1.0041 1.1584 1.0316 1.0150 1.3113 1.0683 1.0347

ZmFirstm/LB 1.0349 1.0085 1.0040 1.1592 1.0312 1.0147 1.3051 1.0679 1.0344

ZmSinglem/LB 1.0419 1.0098 1.0046 1.1689 1.0331 1.0157 1.3113 1.0692 1.0355

ZLP(PAverage)/ZLP(PFirst) 0.9973 0.9994 0.9997 0.9958 0.9991 0.9996 0.9973 0.9992 0.9996

241Liu, Queyranne, and Simchi-Levi: Asymptotic Optimality of Algorithms

APPENDIX: PROOF OF LEMMA 7

To prove this lemma, we first prove that

limn3�

Z*1�PAverage

n2 � limn3�

Z*1�PFirst

n2 .

For this purpose, we process the single machine problem PAverage using theoptimal schedule for problem PFirst. We index the jobs according to theirappearance in this order. The completion time of job j in problem PAverage

can be expressed as

max1�x�j

�rx � �k�x

j

pkAverage,

and the completion time of job j in the optimal schedule of problem PFirst

can be characterized as

max1�x�j

�rx � �k�x

j

p1k.

Thus, the difference in the objective values is

limn3�

Z*1�PAverage � Z*1�PFirst

n2 � limn3�

1

n2 �j�1

n

wj� �max1�x�j

�rx � �k�x

j

pkAverage

� max1�x�j

�rx � �k�x

j

p1k�� limn3�

1

n �j�1

n

� wj

1

m �i�2

m �limn3�

1

nmax

1�a�b�n��

k�a

b

�pik � p1k� 0.

The last equation is obtained again by using Eq. (1). To prove theinequality in the reversed direction, i.e., to prove

limn3�

Z*1�PFirst

n2 � limn3�

Z*1�PAverage

n2 ,

we process the jobs in problem PFirst according to the optimal schedule ofproblem PAverage. By similar arguments, we can show this inequalityholds. Thus, we have proved the Lemma 7. �

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