APORS’97, Melbourne, 1-4 December 1997
Interactive Scheduling Peter. G. Higgins (1) and Andrew Wirth (2) 1. School of Mechanical and Manufacturing Engineering, Swinburne University of Technology, Hawthorn 3122, Australia,
[email protected] 2. Mechanical and Manufacturing Engineering, University of Melbourne, Parkville 3052, Australia,
[email protected] Abstract. The paper discusses the need for interactive human-computer scheduling. Modelling and solving most scheduling problems often involves making so many simplifying assumptions that the model solution may be inadequate to solve real problems. Interactive scheduling provides a means by which human abilities may be extended and scheduling heuristics applied to solve real problems.
Introduction It is common to make scheduling problems manageable by making assumptions that simplify the problem. While the abstracted problem may capture significant properties of the manufacturing domain, a scheduler may need to adapt its solution to the actual operating conditions. Interactive scheduling is seen to provide a means for modifying the solution to cater for factors that had been assumed away during problem simplification. In this paper we argue that interactive scheduling should extend beyond tinkering with Gantt charts built by standard heuristics. Instead, scheduling heuristics should be fitted within a human-computer decision-making architecture. The steps in our argument are: 1. 2. 3. 4.
A scheduler overcomes infeasibility by relaxing constraints Over-relaxation results in combinatorial complexity Light relaxation results in scheduling perplexity A scheduler in practice has to operate in an environment that is both complex and perplexing 5. The architecture for an Interactive Decision Support System needs to reflect the requirements of the manufacturing domain. Our argument depends on real data that are typical for an actual manufacturing situation. As the data are from an actual field study, specific attributes used in standard scheduling, for example, due dates and processing times, may not satisfy the specifications for data distribution used in standard data sets. While a standardised data set is useful for the comparison of novel heuristics with standard methods, this is not the subject of this study. To bring cogency to our argument we will present a case study from the printing industry. The company supplies printed forms on fan-fold. There are normally two operations in producing a form: printing followed by collating. Four printing presses operate in parallel.
© P. Higgins and A. Wirth 1997
Typically, the forms may be for bills or cheques. Customers use these forms on track-fed printers.
Tackling feasibility In planning manufacture a CONSTRAINTS DEFINE scheduler coordinates activities MANUFACTURE within the bounds set by Problem: no constraints. If there is only one feasible possible arrangement of schedule production that meets all RELAX CONSTRAINTS LIGHTLY HEAVILY constraints, then a scheduling Problem: Too Problem: Which problem does not exist. It is many feasible to loosen much more common, however, schedules for schedulers to find that either SIMPLIFY SELECTION MODEL USING GOALS there are no feasible schedules or there is only a few possibilities Problem: Still more Problem: conflicting than one feasible goals from which to choose. Some schedule constraints need to be relaxed SELECT USING EVALUATION where the manufacturing PERFORMANCE conditions cause infeasibility. A Figure 1 Constraints define the scheduling process scheduler has to decide which to relax. Where there is more than one feasible plan the problem is under constrained. A scheduler acting under these conditions has to decide which of the competing schedules will be applied. The planning process for producing a schedule described using constraints is shown in Figure 1. The over-constrained case is shown at the top. The problem becomes redefined when constraints are relaxed. Its redefinition depends upon the degree of relaxation. If the loosening of constraints results in many feasible schedules, then problem becomes finding the most suitable. If the constraints are only lightly relaxed to allow one, or only a few, feasible schedule, the scheduler has to decide which to relax to produce an appropriate schedule.
Complexity In the most relaxed description of our field study there is a single operation, printing, that is performed on one of four machines that are equal in capacity and capability. In the six-day field study, 59 jobs were processed. The processing times for these jobs are shown in Figure 2 and their distribution is shown in Figure 3. The arrival and due timesi are shown in Figure 4. i Due times were only given as a date. Tardiness was found to be very sensitive to due time.
Due times were assumed to be ex factory and a reasonable time for completion was assumed to be 4 p.m.
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The slack times for jobs at the time of their arrival shows the time constraint on scheduling to meet the due date objective (Figure 5)ii. Note that 12 jobs (20%) are tardy on arrival. Distribution of Processing Times
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By comparing the makespan for a particular heuristic with the lower bound, found by dividing the total processing time, machine utilisation can be determined. For all the heuristics the machines are nearly completely utilised (Table 1)iii. Clearly, dynamic arrivals did not impede production. Utilisation therefore was not an issue. The scheduler was more likely to focus on minimising some measure of tardiness
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ii Slack time on arrival = d - p - ri = a - p where r is the arrival time, d is the due-date , p i i i i i i i
is the processing time and ai = di - ri is the total allowance for time in the shop. iii Utilisation is between 81% and 96% for the simple heuristics and 99.6% for the benchmark.
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to satisfy the delivery requirements of customers. The time to find the optimal schedule for 59 jobs such that the number tardy is a minimum would be inordinate. To overcome combinatorial complexity, a scheduler may apply an appropriate heuristic for minimising the performance objective. Table 2 shows that for this problem SPT gives the minimum number of tardy jobs, which is nearly as effective as the benchmark.iv However, the other heuristics, except LPT, only produced one extra tardy job. Where average tardiness is considered, Table 3 shows that the Rachamadugu & Morton (R&M) heuristic (Morton and Pentico, 1993) is the best performer and close to the benchmark. Note that EDD was only marginally poorer. The performance of these heuristics are reversed when maximum tardiness is regarded (Table 4) The scheduler in the field study had estimates of job arrivals and processing times. Actual arrival and processing times, obtained from observation and company records, were used by the simple model. While known processing times advantaged the simple model, the heuristics that were applied are oblivious to anticipated arrivals. In contrast, it was clear from our interviews of the scheduler as he worked that he considered anticipated arrivals when composing processing sequences. Significantly, he always underestimated processing times.
Table 1 Makespan Heuristic FCFS SPT LPT EDD SLACK R&M BENCHMARK
Objective Value 3.30E3 3.86E3 3.25E3 3.33E3 3.26E3 3.62E3 3.22E3
% Deviation 2.3 20.0 0.8 3.3 1.1 12.0 0.0
Table 2 Number of Tardy jobs Heuristic FCFS SPT LPT EDD SLACK R&M BENCHMARK
Objective Value 20.0 19.0 24.0 20.0 20.0 20.0 18.0
% Deviation 11.0 5.6 33.0 11.0 11.0 11.0 0.0
Table 3 Average Tardiness Heuristic FCFS SPT LPT EDD SLACK R&M BENCHMARK
Objective Value 351.0 347.0 635.0 291.0 316.0 265.0 243.0
% Deviation 45.0 43.0 161.0 20.0 30.0 8.9 0.0
Table 4 Maximum Tardiness Heuristic FCFS SPT LPT EDD SLACK R&M BENCHMARK
Objective Value 2.48E3 3.46E3 2.80E3 2.13E3 2.25E3 2.18E3 2.12E3
% Deviation 17.0 63.0 32.0 0.6 6.3 3.0 0.0
iv The benchmark value of 18 tardy jobs was found using Rachamadugu & Morton (R&M)
heuristic as a seed for general pairwise interchange.
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The Effects of Constraint Relaxation on Model Suitability Heavy Relaxation We formed the simple model by heavily relaxing constraints. This model ignores set-up times: one major and two minor.v The major set-up is associated with the depth of the paper. It takes 40 minutes to change the cylinders on a press, which is significant when compared to the median processing time of 100 minutes. Each minor set-up takes about 10 minutes to complete. Finding heuristics for minimising either the number of tardy jobs or average tardiness for jobs that include set-ups is problematic. Table 5. The However, there is a much more significant constraint than these. relationship The presses differ in the number of colours (one, two, four and six) between number of they can print (Table 5). This manufacturing environment, while colours and the number of parallel not being extraordinary, is a member of the difficult class of an machines. identical parallel-machine problem (French, 1982). The number of parallel machines a job “sees” depends upon the number of colours Colours Machines used in its production. Over recent years there have been some 4 theoretical advances in allocating and sequencing jobs on parallel 1 2 3 machines with major and minor set-ups (Tang, 1990; So, 1990; 2 Rajgopal and Bidanda, 1991; Wittrock, 1990). While these 3 4 2 advances move forward the understanding of this class of problem, 1 they apply to situations that are far simpler than this field study (in 5 6 1 which the level of difficulty is nothing unusual). Light Relaxation Classical OR approaches scheduling complexity by minimising information. Jobs are described by a few defining attributes. Characteristics of the real problem are ignored: set-up times are assumed to be predictable; measures of performance not simply directed towards profit maximisation are disregarded. Scheduling heuristics are also minimalist. They use a few job attributes, at most. The scheduler in the field study only lightly relaxed constraints sufficiently to obtain a feasible schedule. He followed the right-hand path shown in Figure 1. The most obvious constraint to relax is due date. He does not take due date as an inviolate constraint. The extent he relaxes it depends upon a subjective judgment based upon context. The value he places on a due date is an outcome of the interplay between interested persons and groups. For some customers the due date is rigid. Others may not be too concerned if they receive their orders a day or two late. Yet again, these very same customers may have jobs in the system with due dates that are atypically firm. His attitude towards the customer also has an affect. Major players within the company (e.g., the manager and sales representatives) affect his perception of a customer. Their disposition towards a customer depends upon whether they are regular, new, slow to pay, belligerent when jobs are late, etc. Even where a v The production records did not separate set up and processing times. For the simple model
production time was taken as processing time.
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customer may not be unduly concerned about a late delivery, a sales representative may see a late delivery as a threat to his or her reputation and therefore may try to influence scheduler. Under light relaxation the problem is not one of too many feasible schedules from which to select. Which constraints should be relaxed to meet the goals is the question that arises. During the pilot study there was one very long job for which it was not permissible to relax the due date constraint. It also had to be produced on the six-colour machine. The machine broke down during production. To meet the due-date constraint the scheduler changed machine constraints. He removed parts from other machines and placed them on the six-colour machine so that it could operate. The details are insightful: A3 required four blankets suitable for B size jobs. Damaged blankets had to be replaced by blankets from other machines. In taking one from A4, jobs had to be re-ordered so B size jobs were not run. By the time the press transfers to B size, there would be spare blankets available. Two came from A1 as it was not processing B jobs. One was taken from A2 leaving it capable of producing three colours. Later on more blankets were damaged and two more blankets were taken from A2 and its capability reduced to one colour. vi
To keep one machine operating, each other machine’s capability was reduced. A machine’s capability may also be increased. To meet capacity demands, the scheduler may increase a machine’s processing speed beyond its normal limits. As well as being aware of this possibility, the scheduler needs to appreciate the repercussions of the action; shorter life, lower quality, etc. Thus, in violating machine constraints, the scheduler has to recognise the conditions and requirements for violation to be permissible. Hence, the scheduler needs to have a deep understanding of machine functions. The rules that the scheduler applies may be intricate. Consider the change of colours between jobs. Washing the colour applicators may or may not be necessary. The decision is complex. Take as an example a three-colour job. Which machine should the scheduler place it on? The choice affects set-up conditions for both it and the jobs that follow. If the job was on the six-colour machine and the job that follows requires completely different colours, then if the three unused applicators are clean or set to inks of the required colours then the job could proceed immediately without waiting while the applicators are cleaned. If instead the threecolour job was placed on the four-colour machine, then the six-colour machine would be available for jobs requiring five or six colours. The scheduler’s choice will depend,inter alia, on the current sequences on both machines, the possible sequences on each, the set up costs on each when the job is loaded and the total set up cost for each over a selected period. As the company’s strategic niche in the market is the provision of quick turnaround, the scheduler tries to have cylinders for the most common depths immediately available. He can then process a premium job without delay. This rule is not simple. The obvious configuration is for the cylinder size with the greatest demand from such jobs being on the six-colour press to allow processing of all jobs of that size. This may also decrease the time to wash the applicators. When the likelihood of “quick-turnaround” jobs for the other cylinder sizes is vi A1, A2, A3 and A4 are presses that are can print 2, 4, 6 and 1 colours, respectively. B and
C are cylinder sizes.
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considered, along with some estimate of the number of colours to be printed, the allocation of cylinders to machines becomes more problematic. The obligation to process existing jobs, efficiently and on time, tempers the scheduler’s desire to provide quick turnaround for jobs that may never exist. On the arrival of a “hot” job the scheduler has to decide whether to pre-empt the current job. These, and other, constraints make the scheduling process beyond the capabilities of heuristics developed for single-stage parallel processors. Environmental conditions may affect operations. The ink’s drying time may depend upon the density of the colour, the ambient temperature and the relative humidity. This in turn affect processing speed and, consequently, processing time. Quality demands vary between customers. Some customers expect exceptional quality. To achieve such quality, the scheduler may need to allocate the job to a particular machine with an especially good operator. While good operators enjoy the challenge of high quality, they also find it stressful. The scheduler therefore tries to mix lower quality with high quality work so the operator is not overloaded with exacting work. This forms part of the scheduler’s overall objective. Rodammer and White (1988) noted in trying to minimise operating stresses human schedulers may also apply measures that improve schedule stability, reduce confusion, and placate a demanding customer. Schedulers also take into account a broad-range of factors covering issues such as labour allocation (e.g., absenteeism, skill distribution, overtime and extra shifts), availability of tooling and raw materials, use of subcontractors, and alternative job routes.
Perplexity Rather than being defined by combinatorial complexity, scheduling in practice is often characterised by perplexity. Schedulers have to satisfy many stated and unstated conflicting goals, using hard and soft information that is possibly incomplete, ambiguous, biased, outdated, and erroneous. Goals may be neither clear nor explicit. They pursue goals without fully articulating them by following practices they believe to be good, from years of experience within the industry. From their extensive study of real job shops, McKay, Safayeni and Buzacott (1988) found that job shops are seldom stable for more than half an hour. Something is always happening unexpectedly. Manufacturing goals are not simple, but affected by hidden agenda. Processing and set-up times are not reliable. Information on processing requirements are inaccurate and incomplete. New jobs usually arrive before previously scheduled jobs in the system have been processed. The new arrivals may make the prevailing plan irrelevant. The state of the shop may restrict choices available for amending the schedule. For example it may be impractical to alter the place of some jobs in a queue. Often those jobs for which processing is imminent have already placed calls on resources and materials. Reversing these calls may be difficult. Reallocating materials that have been earmarked for one job to another, can cause difficulty in tracking materials. If the materials for an operation consist of parts produced by preceding operations, associating materials and jobs can be quite a perplexing activity. Changes also may be restrained to limit chaos and confusion at the shop floor due to the chopping and changing the order of work. Under these circumstances, if all jobs in the queue are not available for revision, the justification for applying a particular heuristic becomes questionable.
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The scheduler in the field study did not adhere to classical assumptions (Buxey, 1989) in the strategies they pursue. He: • assigns priorities to jobs • splits operations between machines and overlaps operations to speed up work (see Figure 6) • interrupts operations to run more urgent jobs • renegotiates due dates with customers to spread the work load, and • uses machines in non-standard ways to increase short-term capacity. Currie: 16611, 16785, 16612
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Figure 6 Complex factors present — pre-emption, overlapping, constraint on operator availability Under conditions where the factors that can affect the schedule are myriad and subjective, algorithmic dispatching rules alone ill-equip schedulers for carrying out their responsibilities. Where goals are multiple and ill-defined, scheduling activity cannot be based solely on procedures directed towards obtaining near-optimal performance of a single criterion.
Performance under Perplexity The performance of the scheduler acting under perplexity was compared to the simple model. While the simplicity of the model using simple measures of performance give it an unfair advantage, it provides some benchmark on which to judge the performance of the scheduler. To be able to equate the scheduler’s performance with the model the data had to be adjusted. Over the six-day period the time associated with each day was modified to reflect actual machine availability: time lost due to machine breakdown and operator unavailability were deducted. vii Arrival times had to be proportionally adjusted to the length of the day. vii Including these times, the utilisation was 50%.
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Under these conditions, machine utilisation for both the model and actual production reflects time lost time waiting for jobs. Some jobs were so tardy on arrival, that the average tardiness would change little no matter what schedule was followed. Therefore, all jobs that were tardy at the beginning of scheduling period were set to zero. After making these adjustments, machine utilisation of 86% for actual production compared favourably with the simple model results of 81% for SPT (which minimises the number of tardy jobs) and 86% for R&M (which minimises average tardiness). For the period of the study, there were 25 tardy jobs (taken at 4 p.m. or 20 by end of the day) as compared to 19 for the best heuristic operating on the simple model. Note that tardiness was unavoidable for 12 jobs. An average tardiness of 1560 minutes for the actual schedule was quite inferior to the simple model, which ranged from 265 minutes for the best heuristic to 635 minutes for the worst.
IDSS Architecture From the discussion it is clear that an interactive scheduling system that uses a simple model of the domain and applying simple heuristics are inappropriate where conditions are perplexing. The schedule would have to be so heavily amended that the final schedule would bear little resemblance to the original schedule. Nevertheless. the authors do not propose that classical OR methods should be abandoned. Instead they accept Morton and Pentico’s (1993) challenge that ‘All useful approaches should be pursued’. While the characteristics of many jobs in the field study forced the scheduler to consider them individually, a significant number of jobs have like characteristics. Jobs with like characteristics were grouped and then treated as a single entity when allocated to a machine. Within some groups the scheduler did care what way they were ordered, as all jobs were considered equal. If he was able to compare the performance of different sequences within a group, the overall scheduling performance may improve. Higgins (1996) put forward an Scheduling Rules Knowledge-Based architecture for an Intelligent Decision Adviser Support System (IDSS) in which the human is central to the decision-making GANTT process. A computer displays all available JOBS CHART detail on jobs to the human scheduler. Timing at SCREENS resources Using this information and other domain Performance Unassigned prediction knowledge the scheduler seeks patterns in Sequence Job attributes the data on which they draw inferences Machine 1 Sequence about possible scheduling strategies. The Job attributes scheduler is therefore able to handle Machine n perplexity by applying methods that he HUMAN DECISION MAKING Sequence Context Setting Job attributes naturally uses, but finds hard to represent Pattern Recognition as algorithms (Kempf, Le Pape, Smith, and Fox, 1991). Where jobs can be grouped the scheduler can apply various Figure 7 Architecture of IDSS for scheduling OR heuristics to see how they affect
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performance. A knowledge-based adviser can alert the scheduler when he attempts to violate constraints. Where the constraints are soft, the scheduler may violate them and suffer a penalty. However, the system does not allow hard constraints to be violated.
Conclusion A system based on Higgins’s IDSS architecture is unlike other interactive systems. The scheduler actively participates in the decision-making process and does not merely alter a Gantt chart produced by a computer. The Gantt chart instead is a product of the humancomputer decision-making process.
References Buxey, G. (1989). Production scheduling: practice and theory. European Journal of Operational Research, 39, 17-31. French, S. (1982). Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop, Ellis Horwood, Chichester Higgins, P. G., (1996). Interaction in hybrid intelligent scheduling. The International Journal of Human Factors in Manufacturing, 6(3), 185-203. Kempf, K., Le Pape, C., Smith, S. F., and Fox, B. R., (1991). Issues in the Design of AI-Based Schedulers: A Workshop Report. AI Magazine, 37-46. McKay, K. N., Safayeni, F. R., and Buzacott, J. A. (1988). Job-Shop Scheduling Theory: What is Relevant? Interfaces, 18 (4), 84-90. Morton, T. E., and Pentico, D. W. (1993). Heuristic Scheduling Systems: With Applications to Production Systems and Project Management, John Wiley & Sons, New York. Rajgopal, J., and Bidanda, B., (1991). On scheduling parallel machines with two setup classes.International Journal of Production Research, 29(12), 2443-2458. Rodammer, F. A., and White, K. P. (1988). A recent survey of production scheduling. IEEE Transactions on Systems, Man and Cybernetics, 18(6), 841-851. Tang, C. S., (1990). Scheduling batches on parallel machines with major and minor set-ups.European Journal of Operational Research, 46 28-37. So, K. C., (1990). Some heuristics for scheduling jobs on parallel machines with setups.Management Science, 36(4), 467-475. Wittrock, R. J., (1990). Scheduling Parallel Machines with major and minor setup times.The International Journal of Flexible Manufacturing Systems, 2, 329-341.
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