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Rolling horizon algorithms for a single-machine dynamicscheduling problem with sequence-dependent setuptimesI. M. OVACIKT a & R. UZSOY aa School of Industrial Engineering , Purdue University , 1287 Grissom Hall, West Lafayette,IN, 47907-1287, USAPublished online: 07 May 2007.

To cite this article: I. M. OVACIKT & R. UZSOY (1994) Rolling horizon algorithms for a single-machine dynamic schedulingproblem with sequence-dependent setup times, International Journal of Production Research, 32:6, 1243-1263, DOI:10.1080/00207549408956998

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INT. J. PROD. RES., 1994, VOL. 32, No.6, 1243-1263

Rolling horizon algorithms for a single-machine dynamic schedulingproblem with sequence-dependent setup times

I. M. OVACIKt, and R. UZSOytt

We present a familyof rolling horizon heuristics 10 minimize maximum lateness ona single machine in the presence of sequence-dependent setup times. This problemoccurs as a subproblem in a decomposition procedure for more complicated jobshop scheduling problems. The procedures solve a sequence of subproblems tooptimality with a branch and bound algorithm and implements only part of thesolution obtained. The size and number of the subproblems are controlled byalgorithm parameters. Extensive computational experiments show that theseprocedures outperform myopic dispatching rules by an order of magnitude, both onaverage and in the worst case, in very reasonable computation times.

t. JntroductionThe effective control of material movement through manufacturing facilities is

becoming increasingly important in today's highly competitive global markets.Companies are under pressure to shorten lead times and meet customer due-dates tomaintain high levels of customer satisfaction. Effective management of work-in-processinventories (WIP) can also give companies significant cost advantages. Hence thedevelopment of scheduling procedures to achieve these advantages is of considerableeconomic significance. However, the proven intractability of job-shop schedulingproblems makes it difficult to develop efficient procedures that are applicable toproblems of realistic size. Most practical job-shop scheduling problems have beenaddressed using myopic dispatching rules (Bhaskaran and Pinedo 1991). While theserules are computationally efficient and easy to implement, they may result in poor long­term performance. In manufacturing environments with heavy competition forcapacity at key resources, scheduling procedures that take a global view of the shopshould result in substantial improvements in performance.

The research we describe in this 'paper is part of a larger effort to develop adecomposition methodology by scheduling complex dynamic job shops. Thesefacilities are characterized by the presence of different types of workcentres, some ofwhich have sequence-dependent setup times; reentrant product flows, where ajob mayreturn to a machine several times; and due-date related performance measures. Wefocus on the performance measure of maximum lateness (Lmax ), to capture manage­ments' concern with providing consistent levels of customer service. A workcentre mayconsist of a single machine, a number of parallel identical machines, or of a batchprocessing machine like a heat treatment oven, where a number of jobs are processedsimultaneously as a batch. These problems represent a considerable generalization of

Revision received April 1993.t School of Industrial Engineering, 1287 Grissom Hall, Purdue University, West Lafayette,

IN 47907-t287, USA.:I: To whom correspondence should be addressed.

002ll-7543/94 $\0-00 © 1994 Taylor & Francis Ltd.

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the classical job shop scheduling problem (Baker 1974), which assumes that there areno sequence-dependent setup times, that each job visits each workcentre exactly once,that each workcentre consists of a single machine and that the performance measure tobe minimized is makespan.

The obvious difficulty of these problems (Garey and Johnson 1979) has resulted intheir being largely ignored by researchers. However, decomposition methods thatexploit recent developments in information technology offer a promising avenue ofattack on these problems. In addition, decomposition methods allow us to exploit thespecial structure present in many industrial contexts, rendering these problems moreamenable to efficient, near-optimal solution procedures than the generic problems onwhich much past research has focused.

The decomposition method we propose proceeds in a manner similar to the ShiftingBottleneck approach of Adams et al.(1988) by decomposing the job shop into a numberof workcentres. These are scheduled in order of criticality until all workcentres havebeen scheduled and a feasible schedule achieved. A network representation of thescheduling problem is used to model the interactions between the workcentres so as toallow the solutions to the subproblems to be integrated into a solution to the job shopproblem (Uzsoy 1993).

An important aspect of the decomposition method we propose is that it takes aglobal view of the shop while developing a schedule. In the past, a major obstacle to thedevelopment and implementation of such methods has been the difficulty of obtainingreliable information on the current state of the shop. However, many companies haveimplemented sophisticated Shop Floor Information Systems (SFIS) which can trackWIP and machine status in real time. These systems provide real-time information onwhere cachjob is currently located, whether it is in process or in queue, what operationsit will require in the future and when it is due to the customer. They also provide statusinformation on machine breakdowns and setups. This information makes it possible fora scheduling system to take the state of the entire shop into account when developing aschedule, rather than only a subset of the shop such as most dispatching rules. Adescription of a commercially available SFIS is given in Consilium (1988).

In developing an effective decomposition method, there are two fundamental sets ofproblems. The first is that of deciding upon a decomposition that isolates the 'correct'subproblems as critical and ensures that their solution is meaningful relative to theoriginal problem, that is, that the solution to the subproblems should result in a feasibleschedule consistent with management goals. This requires a mechanism to integrate thesolutions of the subproblems, i.e. to model the interactions between subproblems andtheir effect on the solution to the overall problem. This can be thought of as amechanism to capture global information used in setting up the local subproblems'correctly'. Related to this issue is that of prioritizing the subproblems, deciding in whatorder to solve the subproblems so that the most constraining ones are solved first,leading to a better overall solution.

Once a set of subproblems has been formulated, the second issue is that of findingsolution procedures for the subproblems. These procedures must be fast enough to beused repeatedly without resulting in intolerable computational burden, and mustobtain high-quality solutions. Often the subproblems themselves are intractable, whichmakes this a challenging task. The inherent modularity of the decomposition methodallows us to apply different solution techniques to different subproblems, making itpossible to select the most appropriate procedure for each subproblem and exploit anyspecial structure the subproblems may have.

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Single-machine dynamic scheduling 1245

The main interaction between different subproblems in a scheduling problem is dueto jobs and machines becoming available at different times depending on schedulingdecisions made during the solution of the different subproblems. As a result, thesubproblems will be dynamic, where jobs arrive at machines or machines becomeavailable over time. Hence the scheduling algorithms for the subproblems must becapable of handling the dynamic versions of these problems.

Extensive computational experiments with a prototype decomposition methodhave shown that the decomposition method outperforms dispatching rules both onaverage and in the worst case (Ovacik and Uzsoy 1992). In these experiments, thedecomposition procedure used a simple dispatching heuristic combined with a localimprovement procedure to schedule the work centres. Our results indicate that thequality of the solution obtained for the subproblems has a significant effect on thequality of the solution obtained for the overall problem.

In this paper we present a class of procedures to minimize L m ax on a single machinewith sequence-dependent setup times. The objective is to use these procedures in adecomposition method to schedule complex job shops containing machines of thistype. The industrial application motivating this study was that of scheduling testsystems in the final test phase of semiconductor manufacturing (Ovacik and Uzsoy1992). In addition to the need for such procedures in decomposition method, thisproblem has not been studied extensively to date, giving it an interest in its own right.

The heuristics we suggest operate on a rolling horizon basis. At any point in timewhen a scheduling decision is to be made, we solve a subproblem consisting of the jobson hand and a subset of the jobs that will arrive in the near future. Arrival times arecalculated from the network representation in the decomposition method, and so areknown a priori. We develop a branch and bound algorithm to solve the subproblemsoptimally. Although the computational burden of this procedure increases exponenti­ally with the number ofjobs, the restricted size of the subproblems in the rolling horizonprocedures allows us to use it effectively within this framework. The rolling horizonprocedures consistently yield better schedules than dispatching rules combined withlocal improvement procedures, demonstrating that the latter methods may performextremely poorly.

In the following section we state the problem of interest and review previous relatedwork. Section 3 describes the rolling horizon algorithms, while §4 describes the branchand bound procedure used to solve the subproblems. We present the design of ourcomputational experiments and their results in §§5 and 6, respectively, and concludethe paper with some directions for future research.

For the remainder of this paper, we shall use the notation of Lageweg et al. (1981) torefer to the problems studied in a concise manner. Thus the problem of interest,scheduling a single machine in the presence of sequence-dependent setup times andnon-simultaneous release times to minimize maximum lateness will be denoted asl/rj,s,)Lm.,. The notation is briefly described in the Appendix.

2. Problem description and previous related workThe problem of minimizing L m.. on a single machine without setup times has been

extensively examined. Cases with simultaneous release times (l//Lm.. and I/prec/Lmax)

are easy to solve using the Earliest Due Date rule and Lawler's Algorithm respectively(Baker 1974, Lawler 1973). However, the presence of non-simultaneous release timerenders the l/r)Lm a x problem NP-hard in the strong sense (Garey and Johnson 1979).Thus the problem addressed in this paper, l/rj,sjLm a " is NP-hard in the strong sense

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1246 I. M. Ovacik aL R. Uzsoy

Ieven without sequence-dependent setup times. Furthermore, the special case ofl/si}Lmu, where all jobs have a common duci date is equivalent to l/sulCm..' which isequivalent to the Travelling Salesman Probh~m (TSP) (Baker 1974), which is NP-hardin the strong sense. Thus it is unlikely thatl a polynomial-time procedure to obtainoptimal solutions exists. Research to date has focused on two main areas: developingimplicit enumeration algorithms to obtain optimal solutions, or using heuristics toefficiently obtain near-optimal solutions. In this paper we follow the latter approach.

The dynamic problem without sequence-dependent setup times, l/r}Lmax has beenstudied extensively. Baker and Su (1974). McMahon and Florian (1975) and Carlier(1982) present branch and bound algorithms. while Potts (1980), Carlier (1982) andHall and Shmoys (1992) analyse heuristics. It has been shown that this problem isequivalent to the problem of minimizing makespan (Cmu,) on a single machine in thepresence of delivery times qj = K-dj • where K ;;.max j{dj } (Lageweg et al. 1976). In thisproblem each job j requires qj units of time to reach its destination after completingprocessing on the machine. The objective is to minimize Cmu" where Cmu, denotes thetime the last job reaches its destination. We shall denote this problem by 1/rj, qj/Cmu,'This problem is also time-symmetric, in the sense that for any instance P of I/r j•q}Cmu"we can ercate another instance P' with release times rj =qj and delivery times qj=rj thathas the same optimal sequence (although in reverse) and Cmu, value as the originalproblem. These results motivate various aspects of our approach in this paper.

The problem of minimizing Lmu, with sequence-dependent setup times has not beenextensively examined to date. Monma and Potts (1989) present a dynamic program­ming algorithm and optimality properties for the case of batch setups, where setupsbetween jobs from the same batch are zero. Picard and Queyranne (1978) model arelated problem as a time-dependent travelling salesman problem and develop abranch and bound algorithm. Uzsoy et al. (1991) provide a branch and boundalgorithm for I/prec, su/Lmu,' For problems with more than fifteen operations,however, the computational burden of this algorithm increases rapidly. Uzsoy et al.(1992) develop dynamic programming procedures of the I/prec, su/Lmu, problem wherethe precedence constraints consist of a number of strings. Unal and Kiran (1992)consider the problem of determining whether a schedule in which all due dates can bemet exists in a situation without precedence constraints but with batch setups. Theyprovide a polynomial-time heuristic and an exact algorithm which runs in polynomialtime given a fixed upper bound on the number of setups.

Several authors have suggested heuristics for related problems. Zdrzalka (1992)considers the I /r j • pmll1/Lmu, problem where the jobs have sequence-independent setuptimes. He proves that this problem is N P-hard and presents a heuristic with a tightworst-case error bound. Uzsoy et al. (1992) analyse the performance of the myopicEarliest Due Date (EDD) dispatching rule, which gives priority to the available job withearliest due date, for the Ih. suiLmu, problem. Assuming that the setup times arebounded by the processing times, i.e. that Sij~Pj for all j, they develop tight worst-caseerror bounds for this heuristic. Sahni and Gonzalez (1976) show that unless P = NPthere can be no polynomial-time heuristic with a constant, data-independent worst­case error bound for the TSP with arbitrary intercity distances. Since the TSP is aspecial case of I/rj' sui LmDX> this indicates that efficient heuristics with data-dependentworst-case bounds are unlikely to exist for Ih,sulLmu,' Ovacik and Uzsoy (1992)combine the EDD heuristic with a local improvement procedure similar to that ofUzsoy et al. (1991). They show that the addition of the local improvement procedureresults in substantial improvements over the schedules obtained by the dispatching rule

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Single-machine dynamic scheduling 1247

alone. In addition, they show that EDD performs best out of a number of other myopicdispatching rules (Ovacik and Uzsoy 1992, Uzsoy et al. 1993).

The motivation for the rolling horizon approach followed in this paper is derivedfrom insights into the deficiencies of other techniques for related problems. While EDDis optimal for the static problem, when it is applied to the problem with nonsimul­taneous arrival times it may make poor decisions due to its myopic nature. An exampleof this is when a longjob with a large due date is scheduled just before a short job with avery tight due date arrives. The ability to predict future job arrivals over a certainforecast window in the future can alleviate this problem to some extent. However, whensequence-dependent setup times are also involved, simply having some visibility offuture events does not suffice. The complex interactions between setup times and duedates must be addressed explicitly in order to arrive at good decisions. This is clearlyachieved by a branch and bound procedure for the entire problem, taking into accountthe entire set of jobs. However, the computational burden of such a procedure increasesexponentially, rendering it impractical for problems of realistic size. In particular, theuse of such a technique in a decomposition procedure, where many single-machineproblems must be solved at each iteration, is impossible if the decomposition procedureis to have reasonable computational performance.

Thus, given the computational impossibility of using an exact optimal procedureand the poor solution quality of myopic dispatching rules, we are motivated to seekintermediate methods which obtain higher-quality schedules than myopic dispatchingrules at the cost of additional computational effort. This leads us to the idea of rollinghorizon procedures (RHPs), where a dynamic scheduling problem is decomposed intoa series of smaller subproblems of the same type. The limited size ofthese subproblemsallows us to use exact methods for their solution, which would be impossible for theoverall problem. The solution to the overall problem is approximated by segments ofthe solutions of these subproblems. Thus we obtain a procedure that combines a degreeof forward visibility at each decision point with an optimization procedure thatexplicitly takes into account due dates and setup times, addressing both deficiencies ofdispatching rules described above. One extreme case of such a procedure, with noforward visibility, is a myopic dispatching rule. Another extreme, when forwardvisibility is perfect so that all jobs are considered in a single subproblem, yields an exactsolution procedure. This allows us to explicitly address the tradeoff between solutionquality and computation time through the choice of parameter values defining the sizeand number of the subproblems.

In a RH P, at each decision point a subproblem is solved using forecasts of futureevents that arc predicted to occur over a certain time period in the future called aforecast window. This yields decisions for a certain time period in the future. Only thedecisions related to the current decision point are implemented and decisions arerevised at the next decision point.

RHPs have been developed for a number of different problems (Morton 1981).However, there have been few efforts to apply them to dynamic scheduling problems.Glassey and Weng (1991) and Fowler et al. (1992) consider the problem of schedulingbatch processing machines in the presence of dynamic job arrivals. They assume theavailability of information on jobs that will arrive over a certain forecast window anduse this information to decide whether or not to start processing a batch at eachdecision epoch. Ovacik and Uzsoy (1993) use information on jobs that will becomeavailable over a given forecast window to make dispatching decisions in a job shopwith sequence-dependent setup times. Whenever a dispatching decision must be made,

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a subset of the jobs available over the forecast window is selected. An optimal scheduleis found for the resulting Ih,su/Lmu problem by complete enumeration, and the firstjob in this schedule is processed next on the machine. The encouraging results obtainedfor this approach motivate the work in this paper.

In this paper we present a family of rolling horizon algorithms for the I/rj,su/Lm ax

problem, which has not been addressed in the literature to date. We develop a branchand bound algorithm for the problem which we use to solve the subproblems in theRHPs. We study the effects of different forecast windows on the performance of ourprocedures, describing the tradeoff between computation time and solution quality.Our computational results show that the RHP obtains improvements of up to 58%over dispatching rules combined with local improvement methods. Solutions areobtained for problems with 100 jobs in 3 min of CPU time.

3. Rolling horizon proceduresIn this section we describe the problem under study and the RH Ps developed for its

solution. We are given n jobs, each job} with a known release time r j , a processing timePj' and a due date dj. We incur a setup time of sij when job} is processed immediatelyafter job i. We assume that the jobs are indexed in order of increasing release times, suchthat}>i implies rj~ri'

We define a decision point to be a point in time t when a decision as to which job(s)to schedule next needs to be made. The forecast window is the time period within whichwe can predict the arrival times of future jobs. Since arrival times of the jobs are givenby the decomposition method discussed in § I, the length of the forecast window is adecision variable rather than a system parameter. The set of jobs considered whilemaking a scheduling decision at a given point in time consists of the set J(t) of jobsalready available for processing and the set F(t) of those that will become availablewithin the forecast window.

Although it is important to take jobs that will arrive over the forecast window intoaccount while making the current decision, it is not necessarily to our advantage toconsider all jobs in the set J(t)vF(t). In the problem under study, the relative urgency ofa job is defined by its due date. If we consider jobs which are due far in the future, wemay make a poor decision due to considering jobs which could safely have beenprocessed later. Hence the selection of the set K(t) of candidate jobs considered at thecurrent decision point t becomes important. We define K(t) as the k jobs in J(t)vF(t)with the earliest due dates, where k = min {K, IJ(t)vF(t)l} and Kis a decision parameterdefining the maximum size of the candidate set K(t). This ensures that the k most urgentjobs in J(t)vF(t) are considered in the current decision.

This selection of candidate jobs follows naturally from insights into the timesymmetry of the Ilrj' q)Cma x problem, whose equivalence to the I/r)Lmax problem wasdiscussed in the previous section. It can be shown that similar relationships existbetween the problems with sequence-dependent setup times. Recall that for anyinstance of the I/r)Lma x problem, a corresponding instance P of the Ilr j,qjlC max

problem can be constructed, where the qj depend on the due dates of the jobs. Whenconstructing the set of jobs to be considered, the first consideration is the arrival timesof those jobs. lt is unlikely that a job arriving far into the future will affect the currentdecision. Hence we consider only jobs arriving over the forecast window. The selectionof a set of these jobs based on due dates is motivated by considering the time symmetricI/r j,qjlC m ax problem P' whose release times rj =qj and delivery times qj= rj.Consider aset G of jobs that become available in P' over some forecast window for this problem.

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Single-machine dynamic scheduling 1249

Index the c jobs in G such r'[ ,,:; r~":; ... ,,:; r;.Since for every job i in G, r;=qi =K -d" wehave d, ~d2~ ... ~dc' Hence choosing a set of jobs with consecutive arrival timesoccurring over a given forecast window in P' corresponds to selecting a set of jobs withconsecutive due dates in the original IjriLmax problem. Thus, selecting a set of jobswith consecutive due dates in the I jriL max problem corresponds to choosing a certainforecast window in the corresponding P'. Hence our process of selecting the jobs in K(t)based on both arrival times and due dates reflects the use of forecast windows in boththe problem of interest and its time-symmetric equivalent.

Since some jobs in K(t) may not be available at time t, each subproblem is aIh,siiLmax problem consisting of at most K jobs. We use a branch and boundprocedure to solve these subproblems to optimality. Although the computationalrequirements of this procedure grow exponentially as the number of jobs to bescheduled increases, the restricted size of the subproblems limits the computationaleffort required to solve a given subproblem, ensuring that the computational burden ofthe overall procedure does not show exponential growth.

Let us denote the current decision point by t and define S(t) to be the set of all jobsscheduled until that time. We can now state the RHP as follows.

3.1. Algorithm RHPStep 0 Let t=rt,S(t)={¢}.

Step I Determine the set K(t).

Step 2 Optimally schedule the jobs in K(t). Select the next I,I= min p., IK(t)l),jobs in theoptimal schedule to the subproblem and place them in the schedule, where). is adecision parameter denoting the maximum number of jobs that we schedule atany decision point. Let these jobs form the set L. Set t to the completion time ofthe last job scheduled in L, and S(t)= S(t)uL. If all jobs have been scheduled,stop. Else go to step I.

The procedure has three decision parameters: the length of the forecast window,which we shall denote by T, the maximum number of jobs considered for scheduling atany decision point (K), and the maximum number of jobs we schedule at each decisionpoint (A). The first two of these parameters have been discussed above. The parameter).determines the maximum number of jobs we schedule at each decision point. As ).increases, the number of decision points at which subproblems need to be solved, andtherefore the computational burden of the procedure, decreases.

When we set T=O and K=).= I, we obtain the EDD dispatching rule. If T=rm therelease time of the last job, and K =). = n, we obtain a single subproblem identical to theoriginal problem, and hence the RHP yields an optimal solution. By assigning differentvalues to the decision parameters we can define a range of solution procedures rangingfrom myopic dispatching rules to exact solution methods with increasing solutionquality and computational burden. This ability to specify the degree of precision andcomputational effort is useful when the procedure is to be used in a decompositionprocedure. Since the decomposition procedure schedules workcentres in order ofcriticality, we can use a more time consuming but more precise procedure for morecritical workcentres, and faster but less accurate methods for less critical ones.

The computational burden of the RHP depends on the decision parameters K and)..The number of decision points at which subproblems have to be solved is defined by)..At each decision point we solve a branch and bound problem with at most K,,:;njobs,

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1250 I. M. Ovacik and R. Uzsoy

the worst case complexity of which is O(K!). The effort involved in developing the setK(t) at each decision point t is O(n log n), due to ordering the jobs in increasing orderof due dates. Hence in the worst case, when ).= I, the complexity of the algorithm isO(Il(K! +n log II)).Since Kis a parameter of the algorithm, not the problem, this leads to apolynomial-time complexity for this procedure which, by the results of Sahni andGonzalez (1976), implies that unless P= NP a data-independent worst-case bound forits performance does not exist. Hence its performance may be arbitrarily bad. Forrelatively small values of K, the nK! term in the complexity will dominate, resulting in theworst-case computational effort increasing linearly with II.

4. Branch and bound algorithmThe RH Ps described in the previous section require the solution of a I/rj,su/Lmax

problem at each decision point. In this section we present a branch and boundalgorithm to solve this problem optimally.

The key to-the branch and bound procedure is a tree of partial solutions each ofwhose nodes at level h represents a partial solution with h jobs. Associated with eachnode at level h is a lower bound LB which is the minimum Lma x value that can beobtained by any schedule whose first h jobs are scheduled as in the partial schedulecorresponding to the node.

We start by finding an initial upper bound UB to the optimal solution byconstructing a feasible solution to the problem using the EDD dispatching rule. A localsearch based on adjacent pairwise interchanges is applied to this schedule to ensurethat the initial solution is at least at a local minimum. This schedule becomes our initialincumbent solution and its Lma x value the initial UB. The incumbent solution and theUB are updated as better solutions are found throughout the course of the procedure.

We expand the tree by branching on a selected node S at level h.For each job i not inthe partial schedule of S, we add a new node Si to the tree. The first h jobs of the partialschedule of node Si are those of node S, and the (Il + I)st job is job i.

Whenever we branch on a specific node, we generate all possible nodes that can begenerated from that node. The new nodes generated inherit all characteristics of theirparent nodes. Therefore it is sufficient to keep track of only the active nodes, thosenodes that have not been branched on yet. By keeping these nodes in the order that weare going to select them, the problem of identifying which node to branch on reduces topicking the first node in an ordered list.

There arc two well-known methods for selecting the next node to branch on (Parkerand Rardin 1988). Depth-first search selects the last node that has been added to thetree, i.e. the node at the deepest level of the tree. It has the advantage of requiring theleast number of nodes to be kept active at any time, although it may end up processing alarge number of nodes to reach the optimal solution. On the other hand, best-boundsearch, which selects the active node with the lowest LB, minimizes the number of nodesprocessed, but its memory requirements can be prohibitive since many nodes are activeat any time. We adopt a hybrid of these two methods by generating all possible nodesthat can be generated when branching on a specific node and selecting the node withlowest LB among those at the deepest level of the tree.

We fathom a partial schedule in two different ways. A node is fathomed bycompletion of a solution when it represents a full schedule, since it cannot be expandedany further. If its objective function value is less than the UB, we have a solution that isbetter than any solution found so far, and we update the incumbent solution and theUB. A node is fathomed by bound ifits LB is greater than the current UB. This indicates

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Single-machine dynamic scheduling 1251

that expanding the tree from that node can only give us solutions inferior to what wealready have. If a new incumbent solution is found, all nodes with LBs larger than thenew UB are eliminated from the list of active nodes for the same reason.

The LBs we use are derived from the results of Potts (1980) and earlier (1982) for thel/r)Lmex problem. They show that for any subset S of the set N of jobs to be scheduled,

min {r/}+ Lp,-max {d,}JeS leS leS

(I)

is a lower bound on the optimal Lmax of the problem and is tightened by taking themaximum over all possible subsets S. This bound also applies to the lh,s;)Lmaxproblem since we can only do worse by inserting setup times into the schedule. Since foreachjobj scheduled, we incur a setup time of at least s minj = min'eN{siJ, we can tighten(I) by adding in the sum of the s min/s for all j in the subset S. Therefore,

min {r/} + L (s min,+ PI)-max {d/}leS leS leS

(2)

becomes a LB for I/rj , s;)Lmaxfor any subset S of N. The same bound applies when weare trying to find a LB for a partial schedule during the course of the branch and boundprocedure. For a partial schedule S' with operation h scheduled last and with makespanCmax(S') when N' is the set of all jobs remaining to be scheduled, the expression

min {r;}+ L (s min,+pil- max {d,}leS leP leP

(3)

where P is any subset of the set of unscheduled jobs, forms a LB on the minimum Lm..

value that can be obtained by completing the partial schedule S'. Note that the releasetime of any job i in the set P must be updated to r; = max {r" Cmax(S')} since ajobcannotstart before the completion time of the last job scheduled in the partial schedule S'.Since there are (2"- I) such subsets, where n is the number of jobs that remain to bescheduled, it is not feasible to check all subsets of N'. Therefore we consider only thoseP of size 1,2, and n. When P= N', that is if the subset consists of all unscheduled jobs, wecan tighten the LB further by using a better lower bound on the amount of setup timethat will be incurred. The minimum amount of setup time that we will incur can befound by solving a TSP problem where the intercity costs correspond to the sequence­dependent setup times between jobs. However, since TSP is NP-hard, solving thisproblem to optimality is computationally burdensome. Hence we opt for a lowerbound on the optimal value of the TSP obtained from the assignment problem which ispolynomially solvable (Balas and Toth 1985). The result is an expression of the form

min{ri}+SMIN+ L p=max{d,}leN' leN' leN'

(4)

where S MIN is a lower bound on the minimum amount of setup that will be incurred,which forms a LB on the Lmax of the partial solution S'.

Another lower bound on the Lmax value that can be achieved by completing apartial schedule S' is the Lmax of the partial schedule itself which we will denote asLm,,(S'). Since any schedule that we generate by expanding S' will contain S', its Lm"cannot be less than the Lmaxof the partial schedule S'. Therefore, for a partial scheduleS', a lower bound to the minimum Lmaxachievable is found by finding the maximum ofLmax(S'), expressions of the form (3) for all subsets P with I or 2jobs, and the expression(4).

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1252 I. M. Ovacik and R. Uzsoy

5. Experimental designTo evaluate the performance of the RHPs, we use two different algorithms as

benchmarks. The first of these is the EDD dispatching rule. Whenever the machine fallsidle, this rule myopically selects the available job with the earliest due date. This rulehas consistently shown itself to outperform other, more complex dispatching rules forthe performance measure of Lmax (Ovacik and Uzsoy 1992, 1993, Uzsoy et al. 1993). Inaddition, Uzsoy et al. (1992) have shown that if the setup times are bounded by theprocessing times, this rule has a tight worst-case error bound. The main weakness ofthis rule is that it ignores the setup times. To remedy this deficiency, we have augmentedit with a local search procedure that performs adjacent pairwise exchanges to improvethe EDD schedule. We shall refer to this procedure as the EDD-LI procedure. EDD-LIcan never perform worse than EDD, and we would expect it to yield improvedschedules at the expense of moderate increases in computation time.

We have selected these benchmarks due to the fact that it is extremely difficult toobtain optimal solutions, or even a reliable lower bound, on the optimal solution valuefor this problem. These two rules are, in our experience, representative of approachestaken to this problem in practice. One of our major results is that these rules oftenperform extremely poorly, indicating that the widespread reliance often placed ondispatching-based procedures may be misplaced for problems with sequence­dependent setup times.

We compare the dispatching rules discussed above to the RHP with differentcombinations of decision parameter values. We represent the forecast window in twodifferent ways: job- and time-based. If we assume that the n jobs to be scheduled areindexed by increasing release times and let S(I) be the set of jobs that have beenscheduled at time I, then using a job-based forecast window, we include the nextj jobswith release time greater than I in the forecast window. More formally, the forecastwindow will contain the jobs (s+ l,s+2, ... ,s+j) where job s is last job that hasarrived, i.e. the highest indexed job i with ';:$;I and j=min {Jl,n-s} where Jl is adecision parameter determining the maximum number of jobs we allow in the forecastwindow at any time. While, the job-based approach allows a fixed number of jobs in theforecast window, the time-based approach allows the jobs that will become availableover a fixed period of time to be in the window, i.e. all jobs i such that 'i:$; I + T where Tis the decision parameter denoting the length of the time-based forecast window. Forour experiments we use values of 1,2,3,4 for JI and 200,400,600 and 800 for T. Thesevalues for Tcorrespond to the expected processing and setup time for I, 2, 3 and 4 jobs,respectively. We also examine the two extreme cases where we have no visibility (Jl= T=0) and where we have visibility over the entire horizon (Jl= n, T = 'n). These enable usto examine the effects of having no forward visibility at all and perfect forward visibilityon the quality of the schedules generated.

For the parameter K, we use the values of 5 and 10. This parameter is the majorfactor determining the computational burden of the procedure by limiting the size ofthe largest subproblem solved. The choices of 5 and 10 represent a low and a high valuefor this parameter, allowing us to isolate its effect on the performance of the proceduresin the experiments.

For A, we use values of I, 2 and 3, corresponding to fixing the schedule of I, 2, and 3jobs at any decision point. As Adecreases, the number of subproblems solved, andtherefore the computational burden of the procedure, increases. By assigning a highervalue to A, i.e. by fixing a larger number of jobs at any decision point, we commitourselves to a schedule for a longer period of time, which prevents us from reacting toevents such as the arrival of an urgent job that may occur during that time.

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Single-machine dynamic scheduling 1253

We apply the different scheduling algorithms to randomly generated problems. Theprocessing and the setup times are taken from a uniform distribution in the interval[1,200]. Each job is assigned a release time uniformly distributed over an intervalbetween time 0 and an upper bound which is the product of a range parameter Randthe expected makes pan of the jobs. The range parameter R determines the time periodover which the jobs to be scheduled arrive. An R value of 0 corresponds to the staticproblem where all the jobs are available at time 0, and larger values of R correspond toless frequent arrivals over time. The expected makespan is the product of the number ofjobs and the expected setup and process time of a job which in this case is 200 minutes.For the computational experiments we use R values of 0'6, O'S, 1'0, 1'2, and 1·4corresponding to varying frequencies of job arrivals.

The due date d, of a job i with release time r i and processing time Pi is determined as

di = r i+2kpi

where k is an integer uniformly distributed over the interval [ - 1,4]. This way we alloweach job a multiple of its processing time to complete before it is due. The multiplicativefactor 2 serves to include an estimate of setup time in the due-date setting procedure.Since k can take on negative values, we may have jobs that are already tardy when theybecome available. This is often the case in industrial situations where a job may bedelayed in preceding stages of the manufacturing process. When the problem is solvedas a subproblem in a decomposition procedure, ajob may be tardy due to interactionswith other jobs and machines in the job shop problem the decomposition procedure isattempting to solve.

Values used Total

Release time range (R)

Number of jobs

Number of combinations

Problems/combination

Total number of problems

0'6,0'8, 1'0, 1-4

10,20, ... ,100

5

10

50

20

1000

Table 1. Randomly generated single machine problems.

Parameter and description

Ii Forecast window-job-based

T Forecast window-time-based

K Max. size of subproblems solved

,l Planning horizon

Values used

0,1,2,3,4,co

0,200,400,600, 800, co

5,10

1,2,3

# of comb.

6

6

2

3

Total

12

2

3

Total number of combinations 72

Table 2. Parameter values used for RH P.

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1254 I. M. Ovacik and R. Uzsoy

We examine problems of sizes ranging from 10 jobs through 100 jobs in 10 jobincrements. For each combination of range parameter R and problem size, werandomly generate 20 problems. Each of the 1,000 problems generated is solved usingthe EDD and EDD + LI procedures and the 72 different parameter combinations of theRHP procedure. For each problem, the L max is calculated and the CPU time to solvethe problem is measured. All algorithms are coded in C and run on a SUN SPARCworkstation. The design of the experiment is summarized in Tables I and 2.

6. ResultsTo evaluate the performance of the benchmarks and the RH Ps, we use the ratio of

the average solution value found by each procedure to the average of the best solutionsfound for a given problem class. A problem class is characterized by a release time rangeR and a problem size (number of jobs) n. We denote this ratio by r(R, n). We defineA VE(R, *), A VE(*, n), A VE(*, *) to be the average of r(R, n) over all values of n for fixedR, average of r(R, n)over all values of R for fixed n, and the average of all r(R, n) over allvalues of Rand n, respectively. MAX(R, *), MAX(*,n), and MAX(*, *) are definedsimilarly for the maximum values of r(R, n).

The first issue to be examined is the performance of EDD and EDD-LI relative tothe RHPs with time-based forecast windows. Table 3 shows the A VE(R, "), A VE(*, n),and A VE(*, *) values for the different algorithms. The columns marked xx denote theaverage results for all RHPs with the same K and Avalues. The columns marked 0 andC1J represent the results from the RH P with no knowledge and perfect knowledge of alljob arrival times, respectively.

These results show that EDD yields very poor solutions for this problem, being onaverage 184% worse than the best solution found, even though a number ofcomputational studies (Uzsoy el al. 1993) have shown that EDD performs better thanseveral other dispatching rules. This illustrates the difficulties of evaluating theperformance of dispatching rules against each other. While a given dispatching rulemay perform well relative to other dispatching rules, its performance relative to theoptimum may be extremely poor.

The addition of the local improvement procedure to the EDD rule leads todramatic improvements in performance. This is due to the fact that the localimprovement procedure in effect has perfect visibility of all jobs in the problem, thusremedying the poor decisions resulting from the myopic nature of EDD. However,these improved solutions obtained by EDD-LI are still on average 57%, worse than thebest solution obtained, indicating how unreliable procedures which guarantee onlylocal optimality can be. It is also interesting how much room for improvement remainsafter the improvements from EDD.

Examining the performance of the RHPs, we see that the most significant factoraffecting solution quality is the parameter K, which defines the maximum size of thesubproblems. This effect can be seen clearly when we compare the performance of theEDD rule, which corresponds to K= I, A= I and T=O, with that of the RHPs with Kvalues of 5 and 10 and the same Tand ), values, corresponding to columns 3 and 12Table 3. As K goes from I to 5, there is a 151'6% improvement in solution quality.Increasing K to 10 yields a further improvement of 12'2%. The initial improvementindicates the benefit of solving the subproblems to optimality rather than using amyopic heuristic. The small improvement from K= 5 to K= 10 suggests the advantagesof using an optimal procedure myopically, without forward visibility, are limited.

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1<=5 1<=10

T=O T=xx T= co T=O T=xx T= co T=O T=xx T=ro T~O T=xx T= cc T=O T=xx T=ro T~O T=xx T=<:I)

EDD EDD-L1 ).=1 ).=2 ).=3 A=1 ).=2 A=3 ""S·AVE (Q-6, *) 3·380 (,853 1·411 1·419 ',419 1·463 1-481 1·487 1·532 1·556 1·578 1·182 1·095 1·029 1·206 1·123 1·054 1·251 1·159 1·091 ""'"AVE (0'8,*) 3·605 1·764 1·392 1-390 1·388 1·447 1·439 1·460 1·500 1·502 1·526 1·297 1'151 1'045 1·312 1'177 1·053 1·375 1'216 1·072 ;3AVE(I'O,*) 2-899 1·590 1·350 1'309 1·298 1·388 1·359 1·365 1-460 1·402 1·443 ',315 1·172 1·019 1·350 1·205 1·025 1-422 1·246 1·039 '"AVE(I'2, *) 2·328 1·423 ',323 1·242 1·220 1·399 1·270 1·300 1·483 1·314 1·355 1·317 1·173 1·010 1·395 1·188 1·020 1-469 1·228 1·039 ""..

AVE(I-4, *J 2·213 1·364 1·271 1·173 1·173 ',377 1·207 1·209 1·464 1·273 1·290 1·271 1·137 1·007 1·383 1·165 1·020 1·463 1·220 1·028 S·'"

AVE(*,IO) 1·984 1·307 1·296 1·125 1·089 1-364 1·155 1'112 1·405 \·195 1·119 1·296 1·090 1·000 1-364 1·116 1·000 1·405 1·161 1·000 "'-""AVE(',20) 2·235 1·388 1·280 1·242 1'218 1·331 1·256 1·269 1·406 1'295 1·330 1·273 1·155 1·022 1·317 1·168 1·028 1·407 1·208 1·031 ::s

'"AVE(',30) 2·606 1·541 1·359 1·285 1·281 1-417 1·319 1·343 1·526 1·399 1'430 1·328 1·155 1·020 1-379 1·182 1·050 1·478 1·238 1·057 sAVE(',40) 2·813 1·582 1·328 1·301 1·316 I·367 1-342 1·356 1·410 1·396 1·431 1·277 1·160 1·023 1·314 1·192 1·040 1·350 1·216 1·063

;:;.

AVE(',50) 2·794 1·593 1·339 1·312 1·280 1-419 1·367 1·377 1·471 1·425 1'466 1·293 1·167 1'025 1-337 1·\83 1·035 1·363 ',225 1·054 On

"AVE(',60) 3·029 1·656 1·382 1'345 1·358 1·430 1·390 1·416 1·506 1·453 1·507 1·292 1·138 1·026 1·315 1·170 1·043 1·403 1·221 1·069 "..

AVE(*,70) 3·176 1·709 1·377 1·376 1·374 1-434 1·420 1-427 1'534 1·500 1'499 1·249 1'152 1·026 \·313 ',175 1·033 1·402 1·228 1·062 it"AVE (*,80) 3'311 1·686 1·356 1·345 1·333 1·447 1·395 1·394 1·511 1-460 1·525 1·241 1·149 1·020 1·316 1·172 1·031 1·381 1·211 1·060 S·

AVE(',90) 3·290 1·767 1·404 1-355 1·372 1-457 1·407 1'468 1·532 1·490 1·540 1·275 1'141 1'029 1·306 1·171 1·040 1·351 1·215 1·067 ""AVE(',I00) 3·145 1·761 1'372 1·379 1-376 1-479 1·461 1-482 1·577 1·483 1·541 1·241 1'148 1·030 1·332 1·186 1·043 1'418 1·215 1'075

AVE(', *J 2·865 1·599 1·349 1·306 1·300 1'415 1·351 1·364 1·488 1-410 1-439 1·277 1·145 1·022 1'329 1·171 1·034 1·396 1·214 1·054

Table 3. AVE (a, *J, AVE(*,b), and AVEtO,*J values.

NV.V.

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<=S

I. M. Ovacik and R. Uzsoy

"35"__,"---',30 -----_ , _

1256

T=BOOT=600T=400T=200

""

1·25

1·20

1-15

1-10

l,OO.J-----+-----+------1------l-------lT=o

1·05

AVEr:)

Figure 1. Effect of length of forecast window (T) on RHP performance.

There are clear interactions between T, the length of the forecast window and K. Asshown in Fig. I, when K = 5, increasing Thas little effect on solution quality since thefuture information obtained cannot be taken into account in the subproblems. WhenA= I, increasing T from I to CIJ results in only 3·1 % improvement. However, whenK= 10, extending the forecast window results in a steady, significant improvement,reaching 20'2% as T increases to CIJ. This is due to the fact that when K is small, theamount offuture information taken into account in the current decision is limited. Thelarger K value allows more future oriented information to be considered, resulting insuperior solutions.

The effects of the forecast window become clear when we compare the RHPs withtime-base forecast windows to those with job-based forecast windows. Figure 2 plotsthe AVE(*, *) values for the two families of RHPs. It can be seen that the time-basedprocedures consistently outperform the job-based ones. When R is large, the time­based procedure considers fewer jobs than the job-based procedure, but the jobs itignores will be those arriving far into the future. When R is small, the time-basedprocedure may consider more jobs than the job-based procedure, allowing it to selectthe set K(t) from a larger set of candidates, hence capturing a 'better' set K(t). The job­based procedure, on the other hand, may ignore urgent jobs that arrive in the nearfuture, resulting in poor decisions. Since the time-based procedures are consistentlybetter than their job-based counterparts, we shall focus on the results of the time-basedprocedures for the rest of this paper.

The number of jobs fixed at each decision point, A, also affects solution quality. As).increases, solution quality degrades steadily, exhibiting a linear trend. This is illustratedin Fig. 3 for the cases where T = I and 4 and K = 5 and 10. This is because a procedurewith a low value of K uses little future information, resulting in poor schedules for thesubproblems. While for small Adecisions are revised frequently, as Aincreases we arecommitted to these poor decisions for a longer period of time, resulting in poorerperformance overall.

Although there are some exceptions, the performance of all procedures degradessomewhat as the number of jobs increases. However, the RHPs appear to performrather more consistently than EDD and EDD-L1, which exhibit a marked degradation

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Single-machine dynamic scheduling 1257

AVEr:)

1·05

1·00+--+--l-~--+--+-+--+--l----if----+--l

T=200 T=-eOO T=6()O T=800 T=200 T=400 r-soo T=800 T"'200 T=400 T=600 T=800~=1 1l=2 J.I=3 1l=4 ....=1 J.1=2 J.I=3 fJ=4 J.I=I 1.1=2 J.I=] J.1=4J..=l ),.:1 ),.::1 A.=I A.=2 A.=2 ).,;::2 A.=2 ).=3 )..:3 ).,;::) 11.=3

Figure 2. Performance of job- and time-based forecast windows on RHP performance.

AVEr:)

::~:::t ~~5'T=800.K'=10. T::200

'-25 •

1·20

"'5

IC'=IO. T:800110L-_--~',05

1·00+-----------+-----------1l.=.

Figure 3. Effect of number of jobs fixed at each decision point (A) on RHP performance.

in performance with increasing problem size. This indicates another benefit of theRHPs, that their performance relative to the other procedures improves as problemsize increases. Similar conclusions can be drawn for the effect of the range parameter Ron the performance of EDD and EDD-LI. Both these procedures show decliningperformance as R decreases. This is due to the fact that with a small R, the number ofavailable jobs for the dispatching rule to choose from is high, and thus a myopic choiceignoring setup times is more likely to be a poor one.

To evaluate the robustness ofthe algorithms we use the MAX (R, *),MAX (*, n), andMAX (*, *) values shown in Table 4. All the RHPs outperform EDD and EDD-LIsignificantly in the worst case. The worst of the RHPs outperforms EDD-LI by 38'6%in the worst case, and the best by 104'2%. This indicates a major strength of the RHPs,that even when they do not yield the best solution they are unlikely to deviate from it

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NV.00

1(=5 1(= 10

T=O T=xx T=ro T=O T=xx T=ro T=O T=xx T= ro T~O T=xx T=ro T=O T=xx T=ro T=O T=xx T=ro

EDD EDD-L1 ).=1 i.=2 <=3 i.= 1 i.=2 <=3

MAX(0'6,*) 3·767 2']18 1·503 1·521 1·515 ],586 1·629 1·631 ],665 1·732 ],731 1·275 1·231 ],046 1·313 1·254 1·089 1·390 1·409 1·131 -MAX (0'8, *) 5·068 2·065 ]'493 1-497 1·491 1·589 ],631 1·624 1·662 1·677 1·687 1-425 1·306 1·076 I·589 1-300 1·074 1·662 1·451 1'129MAX(I·O, *) 3·697 1·809 1'456 1-446 1·435 1·531 1·553 '·569 ',615 ],565 1·628 1·422 1·404 1·034 ]'468 1·401 ',042 1·595 I-488 ]·080 ~MAX(\'2, *) 2·734 1·627 1·442 1·399 1-375 1·484 "437 1·507 1·624 1·474 1·546 1'442 1·363 1·029 1-460 1·405 1·047 1·616 ]-438 1·067 0MAX(]'4,*) 2·640 1·540 1·366 1·326 1·394 1·534 1·411 \·406 ],720 1'536 1·485 1·366 \·317 1·016 1·534 1·371 1·067 1·720 1·571 1·081 '"'"MAX(*,IO) 2·389 1·455 1-425 1·233 1·233 1·589 1'263 1·242 1·662 1-451 1·263 1-425 1-193 1-000 1'589 1·216 1·000 1·000

n1·662 1·451 '"MAX(*,20) 2·728 1·556 1·360 1·374 1·373 1·453 1·388 1·387 1·605 1·472 1·501 1·375 1·278 1·076 1'453 ],320 1-074 ],605 1-438 1·067 '"MAX(*,30) 3·057 1·734 1·442 1·422 1·368 1·534 1'443 1·439 1·720 ',536 ],528 1·442 1·404 1·041 1·534 1·401 1-067 1·720 1'571 \·073 ;::

"-MAX (*,40) 3-431 1·819 1·454 ]'428 1·413 1-490 1'492 1-486 1·489 1·574 1·531 1·383 1·317 ]·049 1·339 1·371 1·061 1·489 1·408 1·082 ?:'MAX(*,50) 3·471 ],869 1-398 1·443 1·385 ',531 1·513 1-474 1·554 1·586 1·628 1·376 1·355 1·045 1·435 ],320 1·057 1·417 1·488 1·107MAX(*,60) 3·625 ',991 1·463 ',498 1·465 1·560 ',594 1·560 1-614 1·710 1·710 ',422 1·317 1·046 1'444 1-318 1·067 ]'527 1'443 1·131 c::MAX(*,70) 4·135 2'112 1·495 1·516 1·504 1·553 1-621 ],612 ],665 1·732 ],731 1·339 1·277 1·053 1-410 1·355 1·063 1·536 1-434 ]']19

N

'"'"MAX(*,80) 4·662 1·995 1·460 1·468 1·462 1'523 1·562 '-536 1-609 1·654 1·656 ],342 1·363 1·032 1·455 1-405 ],051 1'521 1·399 1-105 '"MAX (*,90) 4·814 2·065 1·503 1·521 1'515 1·547 1·573 1·564 1·654 1·664 1·687 1·362 1·349 ],055 1·434 ',342 1·071 1-477 1·365 1·106MAX (*,100) 5-068 2·118 1·493 1-519 1·506 1-586 1·631 1·631 ]·647 1·677 1·675 1·380 1'363 1·059 1'460 1·356 1-089 1·616 1-376 ],129

MAX(*, *) 5·068 - 2-118 1·503 1·521 1·515 ],589 1·631 1·631 1·720 1·732 1·731 1-442 1·404 1·076 1·589 1·405 1·089 1·720 1'571 1·131

Table 4. MAX (a,*), MAX (*,b), and MAX (*, *) values.

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Single-machine dynamic scheduling 1259

drastically. Dispatching rules, on the other hand, may yield extremely poor solutions,as the results for EDD show.

Summarizing our results on solution quality, several conclusions emerge. The firstis that dispatching rules can yield extremely poor solutions in the presence of sequence­dependent setup times. Even the inclusion of a local improvement procedure does notremedy these defects. The RH Ps with appropriate choices of parameters consistentlyyield better solutions than EDD and EDD-LI both on average and in the worst case.The RHPs with time-based forecast windows consistently outperform their job-basedcounterparts. The performance of both job-based and time-based procedures isaffected by the algorithm parameters in the same way. However, solution quality is notthe only attribute to be considered when selecting for a problem. The computationaleffort required by the algorithm is also an important factor which must often be tradedoff against solution quality. We shall first discuss the computational burden of thedifferent procedures studied, and then address the issue of the quality/time tradeoff.

The computational effort required by the RHPs is heavily affected by the choice ofthe parameters K, A. and T. The average CPU times for the RHPs are shown in Table 5,and the maximum times in Table 6. The effect of K is particularly significant, whichfollows from the discussion of the complexity of the RHPs in § 3. As K increases from 5to 10 there is an order of magnitude increase in both average and maximum CPU time.This is due to the exponential worst-case complexity of the branch and boundalgorithm used to solve the subproblems. The effects of ). and Tare weaker, but stillsignificant. As Tincreases, the number of jobs considered in a given subproblem, andthus computation time, increases. As A. increases, the number of subproblems solved

T

K Ie 0 200 400 600 800 00

1 0·73 0·78 0·85 0·91 0·95 1·065 2 0·65 0·63 0·66 0·68 0·70 0·75

3 0·63 0·63 0·61 0·62 0·64 0·67

1 10·82 12-29 11·67 12-93 17·99 25'1710 2 5·25 7·25 6·63 7·31 7·87 10·98

3 4·17 4·66 4·94 5·44 7·11 9-40

Table 5. Average CPU times (s/problem).

T

K Ie 0 200 400 600 800 00

1 3·02 2-97 2-99 2·80 2-86 2-915 2 2-45 2·78 2-68 2·69 2-68 2·68

3 2-92 2·96 2-85 2·88 2·52 2-43

1 170·84 151·21 122·02 120·34 184·31 171·3910 2 73-68 103-30 84-41 81·75 83-05 85·69

3 50'76 56·67 51·48 56·58 79·74 80·69

Table 6. Maximum CPU times (s/problern).

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1. M. Ovacik and R. Uzsoy

CPU

10 20 30 40 50 60 70 80 90 100

Number of jobs

Figure 4. Effect of problem size on average and maximum CPU time (s) for K =5, T =800and A=2.

1'50

1'45 1l

...0

\1·35

1·30

AVEj",' 1-25

,.",

1015

HO

1'05

"00

10 15CPU (se<:ondsl

20 2. 30

I(, A. .5,1 Q 5.2 • 5,3 0 10.1 .. 10,2 6 10,3

Figure 5. Tradeoff between CPU time and RHP performance.

decreases, reducing computation time. The effects of the range parameter R and thenumber ofjobs are more marked than for solution quality. As R increases, compuiationtime decreases rapidly since fewer jobs are available in the forecast window. Neither theaverage nor the maximum computation time increase exponentially with number ofjobs, as shown in Fig. 4 for a representative RHP with K=5, T=800, ).=2. This isconsistent with our analysis of the complexity of the RHPs in § 3.

The tradeoff between solution time and quality is illustrated in Fig. 5. The verticalaxis represents A VE(*, *), and the horizontal axis is the average computation timerequired by the procedure. Each point corresponds to an RHP with a specific set ofparameter values. There are a number of procedures which are dominated, in the sensethat another procedure which obtains a better solution faster exists. Once we discard

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these points, we have a set of procedures that form the efficient frontier. We can seediminishing returns on CPU time. Getting within 3'4% of the best solution on averagerequires an average of approximately II s. Improving this to 2'2% requires approxi­mately 25 s. The choice of procedure to use depends on the purpose for which thesolution will be used. If we are trying to make a real-time dispatching decision, then asolution time of II s may be acceptable. On the other hand, if we seek a procedure to beused repeatedly in a decomposition procedure which is itself being used in a real-timeenvironment, we may seek a faster, slightly less accurate procedure.

7. Conclusions and future directionsIn this paper, we present a family of procedures for single machine problems with

nonsimultaneous arrival times and sequence-dependent setup times where theperformance measure to be minimized is Lmax. This work is significant because itaddresses a problem which has not been extensively studied in the literature to date.However, our main motivation stems from the fact that these problems arise assubproblems of a decomposition procedure we have developed to schedule complexjob shops. The decomposition procedure works by dividing an intractable job shopproblem into smaller, more tractable subproblems, developing solutions for thesubproblems, and assembling these into a schedule for the job shop. The effectiveimplementation of this procedure in real-world environments requires fast proceduresto obtain high-quality solutions to the subproblems.

The rolling horizon procedures we present address the dynamic schedulingproblem by solving a series of smaller subproblems to optimality. The size and numberof the subproblems is determined by algorithm parameters, such as the forecastwindow length and the maximum size of the subproblems. These parameters allow usto describe the tradeoff between solution time and quality explicitly and select the mostappropriate parameter settings for the application at hand. With appropriateparameter settings, these procedures outperform the best available myopic dispatchingrule by an order of magnitude, and yield solutions that are on average 60% better than adispatching rule combined with local search. The maximum solution time for problemsof up to 100jobs is of the order of 3 min. Thus these procedures represent a substantialimprovement over heuristics commonly used in practice for these problems. Anotherimportant insight from this paper is that dispatching rules, even when combined withlocal improvement procedures, are capable of producing very poor solutions,indicating that the widespread reliance on these methods in practice may be misplacedin certain circumstances.

We have also developed a branch and bound algorithm to solve this problem tooptimality. While the computational burden of this procedure becomes prohibitive asproblem size increases, the limited size of the subproblems allows us to use it effectivelyin the RHPs.

The most important direction for future research is to implement these proceduresin the decomposition procedure which motivated their development. Considerablecomputational experimentation will be required to determine what parameter settingsare appropriate for their use in this environment. In earlier experiments (Ovacik andUzsoy 1992)we noted that adding a local improvement procedure to a myopic heuristicused to solve the subproblems significantly improved the performance of thedecomposition procedure. We conjecture that the higher quality solutions obtainedusing the RH Ps will improve its performance even further.

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1262 I. M. Ovacik and R. Uzsoy

There are a number of issues to explore to further improve the efficiency of theRH Ps. Empirically, problems where arrival times are distributed over a wide intervalare easier to solve. For problems which do not have this characteristic, we may be ableto exploit the time-symmetry or the related makespan problem with delivery times. Hthe due dates are such that the time-symmetric problem has its arrival widelydistributed, then we may obtain considerable computational savings by applying theRH P to this problem. Another aspect is that very often the subproblems arising in thedecomposition methods have precedence constraints between jobs, which could reducecomputation time if exploited appropriately.

In summary, rolling horizon procedures provide a promising avenue or attack in abroad family or complex dynamic scheduling problems. When combined with anintelligent exploitation or the structure or the problems at hand, they can yield highquality solutions in very reasonable computation times. For this reason they form anatural building block or decomposition methods for more complex schedulingproblems, and have considerable theoretical and practical interest in their own right.Research is in progress on exploiting these characteristics in such a decompositionmethod.

AcknowledgmentsThis research was partially supported by the National Science Foundation under

Grant No. DDM-9J07591 and the Purdue Research Foundation.

AppendixIn order to be able to refer to the problems under study in a concise manner, we shall

use the notation of Lageweg et al. (1981), extended to include sequence-dependent setuptimes. This notation consists of three fields alPly. The first field represents the type ofshop (single machine (a = I), parallel identical machines (a = P), etc). The second field isused to represent problem characteristics such as precedence constraints, dynamic jobarrivals, batch processing machines or special processing time structures. The last fielddenotes the measure of performance to be optimized. Thus, for example, I/ri,sulLm..represents the problem or minimizing maximum lateness on a single machine whereeach job j is available at time rj and there arc sequence-dependent setup times. Someexamples of the notation are as follows:

JIILmax : minimize Lmax on a single machine with all jobs availablesimultaneously,

I/s,)Lmax: IllLmax with sequence-dependent setup times,I/r)Lmax: minimize L max on a single machine withjobj available at time ri,I/r)Cmax: minimize Cmax on a single machine withjobj available at time ri'

IIri , prec]Lmax: IhiL max with precedence constraints,IIri, pmtn]Lmax : IIr)L max where preemption of jobs is allowed,

I/r j , prec, sui L max : JIri' prec/Lmax with sequence-dependent setup times.

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