Production Scheduling and Rescheduling with ... - Semantic Scholar
Production Scheduling and Rescheduling with Genetic Algorithms Christian Bierwirth
Dirk C. Mattfeld
University of Bremen Department of Economics Box 330440 D-28334 Bremen, Germany [email protected]
University of Bremen Department of Economics Box 330440 D-28334 Bremen, Germany [email protected]
Abstract A general model for job shop scheduling is described which applies to static, dynamic and non-deterministic production environments. Next, a Genetic Algorithm is presented which solves the job shop scheduling problem. This algorithm is tested in a dynamic environment under different workload situations. Thereby, a highly efficient decoding procedure is proposed which strongly improves the quality of schedules. Finally, this technique is tested for scheduling and rescheduling in a non-deterministic environment. It is shown by experiment that conventional methods of production control are clearly outperformed at reasonable runtime costs. Keywords Genetic algorithm, permutation representation, tunable decoding, job shop scheduling problem, dynamic scheduling.
1 Introduction With the spread of automated manufacturing systems the optimization problem of assigning operations to a set of machines receives increasing attention (Parunak, 1992). From the viewpoint of combinatorial optimization the issue of finding an optimal production schedule is known to be NP -hard (Błazewicz et al., 1996). Therefore particular attention is paid to local search-based heuristics (Anderson et al., 1997) including Genetic Algorithms (GA). A good deal of previous research on GAs in scheduling concentrates on the static job shop problem (Mattfeld, 1996). However, production is an activity of sustained pursuit and therefore production scheduling is actually a dynamic task. In a manufacturing system all jobs have release times which define their date of arrival at the shop floor. Job releases are typically unforeseen events and, consequently, neither the release times of jobs nor their specific processing requirements are known in advance. Therefore, scheduling is a non-deterministic problem dealing with an open time horizon. The control of the jobs in a manufacturing system is referred to as rescheduling because unforeseen events frequently require schedule revisions. In practice predominantly priority-rule based control is used (Baker, 1974). The temporal decomposition of the non-deterministic problem into a series of dynamic, but deterministic
c 1999 by the Massachusetts Institute of Technology
Evolutionary Computation 7(1): 1-17
C. Bierwirth and D. Mattfeld
problems also opens this domain for optimization methods (Raman and Talbot, 1993). Since already small improvements of the schedule quality obtained by optimization often lead to considerable cost reductions, advanced optimization techniques are highly appreciated. The transfer of GA experiences to production control was first approached in Bierwirt et al. (1995) and has been taken up recently in Lin et al. (1997) by addressing the GA effectiveness. We think that the GA efficiency is of predominant importance. So far, however, the GA runtimes have hindered a thorough computational study as usually done for priority-rule based scheduling. This paper proposes an optimization method for the problem sketched above. In Section 2 we first review the static job shop problem. Next we extend the static scheduling problem to a dynamic one. In Section 3 we present a GA capable of solving the dynamic job shop problem. In Section 4 a highly efficient decoding procedure is presented and evaluated by experiment. In Section 5 the GA is applied to rescheduling and tested in two variants of which one again improves the GA efficiency. The results are compared with the outcome of priority-rule based scheduling before we conclude in Section 6.
2 Job Shop Scheduling 2.1 The Static Model The static model for job shop scheduling describes a simplified multistage production process. We consider a manufacturing system consisting of m machines M1 ; : : : Mm for which several assumptions are made. Machines cannot substitute each other and there is no preemption of a processing unit. In this ideal system no machine breakdowns occur and the passing times of jobs between machines are neglected. Furthermore, job-specific setup times of machines are not considered. A production program consists of n jobs J1 ; : : : Jn and each job consists of at most m operations. All operations have to be processed by dedicated machines but no job has to be processed twice by the same machine. The actual number of operations of job Ji (1 i n) is denoted by mi . Since the jobs do not necessarily have to pass every machine, mi m holds. Each job Ji defines a production recipe i given as an order of the machines it has to pass. The vector i points to the machines, hence for i (k ) = j (1 k mi ) machine Mj is the k-th machine that processes Ji . The k-th operation of job Ji processed on machine Mi (k) is denoted as oik . The required processing time of oik is known in advance and denoted by pik . A schedule of a production program can be seen as a table of starting times tik for the operations oik with respect to the technological machine orders of jobs. The completion time of an operation is given by its starting time plus its processing time (tik + pik ). The earliest possible starting time of an operation is determined by the maximum of the completion times of the predecessor operations of the same job and of the same machine. Hence the starting time for operation oik is calculated by
tik = max ti;k,1 + pi;k,1 ; thl + phl :
(1)
Here ohl refers to the operation of job Jh which precedes oik on its machine, i.e. i (k ) = h (l). For k = 1 we deal with the beginning operation of a job and therefore in this case the first argument in the max-function is set to zero. If oik is the first operation processed on a machine it has no machine predecessor and now the second argument in the max-function is set to zero. 2
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The completion time Ci of a job Ji is given by the completion time of its final operation i.e. Ci = ti;mi + pi;mi . As a measure of the performance of a schedule often the i minimization of Cmax = maxfC1 ; : : : Cn g is pursued. This objective function is optimized by reducing the time span needed in order to make the entire production program.
oi;m ,
2.2 The Dynamic Model We now consider an extension of the static model in which the n jobs are released at different points in time. The release time ri of job Ji determines the point in time at which Ji arrives in the shop floor. The starting time of the beginning operation of a job is now given by the maximum of the release time and the completion time of its machine predecessor operation
ohl
ti1 = max ri ; thl + phl :
(2)
The non-beginning operations of Ji are still scheduled according to equation (1). Whenever release times are taken into account for building a schedule, the job shop model is referred to as a dynamic one. Here, we still assume that all release times are known in advance. Therefore, we classify this type of job shop to be dynamic, but deterministic. In the dynamic model Cmax predominantly depends on the release time of the latest released job. Therefore it cannot serve as a proper measure of performance anymore. Instead, P the mean flow-time of jobs is used. It is calculated by F = n1 ni=1 Ci , ri . Minimizing F seeks to complete jobs as soon as possible. In practice short job flows are preferred because they reduce the work in progress inventory costs and increase the flexibility within the shop floor. 2.3 The Non-Deterministic Model Since production is an activity of sustained pursuit, scheduling of the production process is a non-deterministic problem. The production process is typically supposed to last beyond the completion of any of the known jobs. Therefore scheduling deals with an open time horizon. We follow Raman et al. (1993) to handle situations where jobs arrive non-deterministically in the manufacturing system. In their approach the non-deterministic scheduling problem is decomposed into a sequence of dynamic but deterministic scheduling problems. Let P0 be a dynamic job shop problem as introduced in Section 2.2. P0 arises from the release of a production program at time t0 = 0. At t0 the program consists of n jobs. Up to the next release of new jobs at time t1 > t0 problem P0 can be solved. This leads to a table of potential starting times for all operations involved in P0 .
To construct the new problem P1 we make a snapshot at the release time of new jobs t1 . The operations oik with potential starting times tik < t1 have already been implemented in the manufacturing system and are, therefore, now removed from the problem. We modify the jobs Ji of P0 which cover operations starting in the period from t0 to t1 . Then we remove the corresponding operations from the technological machine order and finally decrease m1 to the number of remaining operations of job Ji . If mi = 0 we remove job Ji completely. If a job has been modified but not removed from the production program we calculate a new release time for the modified job Ji by
n
o
ri = 1max t + pik j tik < t1 : km ik i
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Figure 1: Production scheduling on a rolling time basis. Figure 1 shows a snapshot of a manufacturing system at time t1 . The operations which have not been started before t1 appear in light grey shade. Sometimes the processing of an operation oik , starting shortly before a new job is released, overshoots t1 (tik < t1 < tik + pik ). In this situation Mi(k) is not available at time t1. In the charts this is depicted by black shadings on M2 and M3 . To model the situation of non-availability of machines we introduce initial setup times sj for the machines Mj (1 j m). The setup times sj denote the earliest point in time machine Mj is available again. It is calculated by
sj = max 1max t +p jt 0. This presentation focuses on the possible acceleration of the GA achieved by hybrid scheduling. As one would expect, Evolutionary Computation
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Figure 5: Influence of hybrid-decoding on GA generations. a continuous increase of the number of generations can be seen for increasing values. At = 0:5 less than the half of generations are needed compared to active scheduling.
Summarizing, a hybrid scheduler parameterized with = 0:5 yields an improved solution quality by saving a significant amount of runtime at the same time. If runtime is the limiting resource, one might think of using a smaller value in order to speed up the search. Particularly for large problems the expected loss of solution quality for < 0:5 seems to be bounded.
5 GA based Rescheduling The GA outlined in the previous section is now used to solve a series of dynamic problems resulting from a temporal decomposition in a non-deterministic environment (Raman and Talbot, 1993). Now the GA actually reschedules the production program as it is sketched in Figure 1. 5.1 Improving Efficiency by Biasing the Search The simplest approach to using the GA for rescheduling is to make a snapshot of the manufacturing system and to formulate a corresponding dynamic, but deterministic problem. Starting with a randomly generated population, the GA solves this problem and returns the best solution found. This solution is implemented in the manufacturing system as far as possible with respect to a forthcoming snapshot. In this manner the GA is restarted after each snapshot taken at the arrival of a new job. A more sophisticated approach proposed is to reuse some of the work already done (Fang et al., 1993; Bierwirth et al., 1995). Whenever a new job arrives at time t, a new problem is created like described above. Different from the previous approach, the population of the newly started GA is now obtained from the final population of the previous GA by adjusting it to the requirements of the new problem, see Section 2.3. In other words, the search process is given a bias by seeding the initial population instead of initializing it at random. The adjustment of the population requires two steps. 12
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1. The operations which have been started before t, including those which are still processed, are deleted in all permutations of the population. 2. The operations of new jobs are inserted at random positions in all permutations of the population respecting the job’s operation order. Furthermore, job release times and machine setup times are updated using Equations (3) and (4). The operations already started before time t are not considered any longer. Whenever the final operation of a job is eliminated in this way, the job itself vanishes completely. After all these modifications, the GA continues the optimization process. Now the initial population contains fragments of favorable solutions found in the previous GA run. We conjecture that these fragments resemble each other strongly, because, due to the termination criterion described in Section 3.3, the population of the previous problem is likely to have already converged regardless of the size of this problem. Therefore biasing the population will lead to a faster convergence which in turn will result in a considerable saving of runtime (Grefenstette, 1987). Regarding the quality achieved we see two possible impacts: The information transferred may concentrate the search on parts of the search space which are still promising in the current situation. On the other hand, the similarity of individuals in the biased population may hinder a thorough search. This issue is subject to a computational investigation. 5.2 Experimental Setup We run the decomposition by using both variants of the GA and compare the outcome in terms of solution quality and runtime. Jobs are generated in the same way as done in the experimental setup described in Section 4.2. This time we use slightly different utilization rates which correspond to reasonable workloads of a manufacturing system. Utilization rates of U 2 f0:65; 0:70g represent a relaxed situation of the manufacturing system. A moderate load is produced by U 2 f0:75; 0:80g whereas utilization rates of U 2 f0:85; 0:90g produce an excessive workload. Modeling the inter-arrival times by a Possion process as described in Section 4.2 can lead to extreme deviations of the workload in different phases of the simulation run. Therefore a large variance of both, the quality achieved as well as the runtime required can be observed. In order to alleviate this effect, a large number of job releases has to be simulated. Moreover, a large number of different simulations is needed in order to produce a sound overall mean. Consequently, for each U investigated we conduct 50 simulation scenarios consisting of 1000 jobs each. In a preliminary investigation we run the non-delay scheduler described in Section 3.2. Thereby the selection of an operation from the critical set (Step 2.N4) gives priority to the operation with the shortest imminent processing time (SPT). Jobs waiting in a queue may cause their dedicated successor machines to run idle. The SPT rule alleviates this risk by reducing the length of the queues in the fastest possible way. This in turn reduces the complexity of the manufacturing system. For this reason SPT dominates all other existing priority rules if the minimization of F is pursued. SPT runs are carried out for all utilization rates prescribed. Thereby the workload observed in the system is recorded. Keep in mind that the workload of the decomposed deterministic problem is simply the number of the operations involved. From Figure 6 we can easily recognize the six levels of workload corresponding to the Evolutionary Computation
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six utilization rates used. Consider the development of the workload in the early phase of the simulation. The workload continuously increases until a stationarity is reached (apart from U = 0:90) after approximately 200 job releases. Notice, that the difference between adjacent levels of stable workload increases with U . This observation indicates that a small additional increase of the workload may not be handled properly in a tense production situation anymore. Therefore, for U = 0:90, a stabilization of the workload cannot be observed within the 1000 jobs released. The figure depicts that jobs scheduled in the early phase of the simulation distort the F measure considerably. The same is true for the final phase of the simulation run. Consider the last problem constituted at job release 1000. Although it still consists of many operations, it must be solved entirely now. By taking these effects into account, we must doubt the results reported by Lin et al. (1997) for a similar study. The scenarios investigated by Lin et al. consist of 100 job releases — the leftmost 10% of job releases shown in Figure 6. In our computational study we discard job 1 to 200 as well as job 801 to 1000 from being evaluated in order to circumvent distortion effects. 5.3 Results and Discussion After we have carefully designed the experiment, we perform simulation runs for both GA variants described in Section 5.1. The randomly initialized GA is called “rand” and its seeded counterpart is called “bias” in Table 2. The mean flow-times of jobs achieved for the SPT rule and both GA variants are reported. For the GAs also the average number of generations and the average runtime of a simulation scenario are listed. The programs are run on a Pentium/200 Mhz computer. From Table 2 we see that the flow-times of jobs obtained by SPT increase strongly with the utilization rate. Under high utilization rates the large workload results in long flow-times of jobs. The prescribed average processing time P = 50 of jobs can serve as a simple lower bound for F . By subtracting this value from F we obtain the average waiting time of a job in the system which increases from 42 for U = 0:65 up to 150 units for U = 0:90. 14
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util. rate 0.65 0.70 0.75 0.80 0.85 0.90
Table 2: Results of non-deterministic scheduling. mean flow-time F generations runtime (sec) SPT rand bias rand bias rand bias 91.8 85.8 85.9 10.8 8.8 111.7 81.1 100.4 93.2 93.5 14.9 11.1 189.6 127.6 111.5 103.5 103.1 21.4 14.6 351.0 214.6 128.1 118.7 117.4 33.5 20.3 747.6 407.5 154.9 142.6 140.6 56.7 31.6 1876.4 953.0 200.4 184.3 179.4 103.8 52.4 5928.1 2705.5
The SPT rule is outperformed by both GA variants for all utilization rates tested. Thereby a reduction of 15% of the waiting time of the jobs is gained for all values of U . The solution quality produced by the GA variants differ only slightly from each other. As it can be taken from the lower curve in Figure 6, the dynamic problems solved under U = 0:65 consists of about 25 operations. For such small problems both versions of the GA come up with almost identical results. With an increasing U the problem size increases and now the biased GA dominates its rival by a few percent. Table 2 also reports the average number of generations performed in the GA runs until termination occurred. Recall the flexible termination criterion used. For U = 0:65 the biased GA uses slightly less generations than its randomly initialized counterpart. In contrast, for a heavily loaded system, the unbiased GA demands up to two times the generations needed when used on a biased basis. The larger the workload increases, the more advantageous the biasing of the initial GA population gets in terms of the generations spent. The advantage becomes even more clear by considering the runtime required for one simulation scenario. Note that the time needed by the hybrid scheduler increases at O(n2 ) with the problem size. For U = 0:90 the GA solves problems of an average size of 73.5 operations. The “rand” GA is clearly outperformed by its “bias” rival. Even under heavy load only 2705.5 seconds are spent on average by the biased GA in order to solve 1000 subsequent problems. Thus, the rescheduling of the system is performed competitively fast in 2.7 seconds. In order to derive the runtime of the SPT-scheduler recall that a priority rule just determines the order of operations in the scheduler. The same scheduler evaluates the fitness within the GA, this time driven by the order of operations in the permutation. Since the population consists of 100 permutations and the biased GA performs 52.4 generations on average for U = 0:90, we can easily examine that a priority rule based scheduler performs at least 5240 runs in a time span of 2.7 seconds. Thus we must admit that priority based scheduling requires a fraction of a second only. However, the results presented justify the additional time spent for the biased GA.
6 Summary and Conclusion Throughout the last decade numerous GA approaches to production scheduling have been reported. These approaches differ strongly from each other with respect to the encoding and operators used, the constraints handled and the goals pursued. Despite these differences, all approaches have in common that domain knowledge is required in order to produce competitive schedules. Although highly desirable, the GA is not capable to conduct the search properly. In order to overcome this shortcoming many ways of incorporating domain knowledge Evolutionary Computation
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have been proposed. Probably the most successful approach is to introduce hill-climbing resulting in a genetic local search approach. Unfortunately, for most scheduling problems hill-climbing lacks an efficient estimate of promising neighborhood moves. For these cases the remedy of reducing the search space by means of an active scheduler has been proven successful. Despite searching in the set of active schedules only, the capabilities of the GA to conduct a proper search still vanish with an increasing problem size. This weakness of GA based scheduling can be alleviated by a further reduction of the search space. For this end we have proposed a tunable decoding procedure which proportions the size of a search space within useful bounds. Although this procedure may exclude optimal and even near optimal solutions from being searched, the positive effects of the search space reduction do prevail. Moreover, the GA runtime is reduced by approximately 50%. A temporal decomposition of a non-deterministic production environment opens this domain for GA based scheduling. One way to a further reduction of the computational effort is to fall back on information available at no cost. Based on this principle a rescheduling of the current problem is done whenever a new job is released. Thereby the initial population of the GA is built by modifying the final population of the preceding GA run. Again, this technique cuts down the GA runtime by about one half. In this paper we have dealt with the minimization of the mean flow-time of jobs, because for this objective the excellent performance of the SPT rule is well known. Against this simple but powerful rival, the GA approach produces far better results at a reasonable runtime.
References Anderson, E. J., Glass, C. A., and Potts, C. N. (1997). Machine scheduling. In Aarts, E. H. L. and Lenstra, J. K., editors, Local Search Algorithms in Combinatorial Optimization, pages 361–414, John Wiley, New York, New York. Baker, K. R. (1974). Introduction to Sequencing and Scheduling, John Wiley, New York, New York. Bierwirth, C., Kopfer, H., Mattfeld, D. C., and Rixen, I. (1995). Genetic algorithm based scheduling in a dynamic manufacturing environment. In Palaniswam, M., editor, Proceedings of the Second Conference on Evolutionary Computation, pages 439–443, IEEE Press, New York, New York. Bierwirth, C., Mattfeld, D., and Kopfer, H. (1996). On permutation representations for scheduling problems. In Voigt, H.-M., et al., editors, Proceedings of Parallel Problem Solving from Nature IV, pages 310–318, Springer Verlag, Berlin, Germany. Blanton, J. L. and Wainwright, R. L. (1993). Multiple vehicle routing with time and capacity constraints using genetic algorithms. In Forrest, S., editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 452–459, Morgan Kaufmann, San Mateo, California. Błazewicz, J., Domschke, W., and Pesch, E. (1996). The job shop scheduling problem: Conventional and new solution techniques. European Journal of Operational Research, 93:1–30. Della Croce, F. D., Tadei, R., and Volta, G. (1995). A genetic algorithm for the job shop problem. Computers and Operations Research, 22:15–24. Dorndorf, U. and Pesch, E. (1995). Evolution based learning in a job shop scheduling environment. Computers and Operations Research, 22:25–40. Fang, H.-L., Corne, D., and Ross, P. (1996). A genetic algorithm for job-shop problems with various schedule quality criteria. In Fogarty, T., editor, Proceedings of AISB Workshop, pages 39–49, Springer Verlag, Berlin, Germany.
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Fang, H.-L., Ross, P., and Corne, D. (1993). A promising genetic algorithm approach to job-shop scheduling, rescheduling, and open-shop scheduling problems. In Forrest, S., editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 375–382, Morgan Kaufmann, San Mateo, California. Giffler, B. and Thompson, G. (1960). Algorithms for solving production scheduling problems. Operations Research, 8:487–503. Grefenstette, J. (1987). Incorporating problem specific knowledge in genetic algorithms. In Davism, L., editor, Genetic Algorithms and Simulated Annealing, pages 42–60, Morgan Kaufmann, San Mateo, California. Holthaus, O. and Rajendran, C. (1997). Efficient dispatching rules for scheduling in a job shop. International Journal of Production Economics, 48:87–105. Lin, S., Goodman, E., and Punch, W. (1997). A genetic algorithm approach to dynamic job shop scheduling problems. In B¨ack, T., editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 481–489, Morgan Kaufmann, San Mateo, California. Mattfeld, D. (1996). Evolutionary Search and the Job Shop, Physica/Springer, Heidelberg, Germany. Nakano, R. and Yamada, T. (1991). Conventional genetic algorithm for job shop problems. In Belew, R. and Booker, L., editors, Proceedings of the Third International Conference on Genetic Algorithms, pages 474–479, Morgan Kaufmann, San Mateo, California. Parunak, H. V. D. (1992). Characterizing the manufacturing scheduling problem. Journal of Manufacturing Systems, 10:241–259. Raman, N. and Talbot, F. (1993). The job shop tardiness problem: A decomposition approach. European Journal of Operational Research, 69:187–199. Storer, R., Wu, S., and Vaccari, R. (1992). New search spaces for sequencing problems with application to job shop scheduling. Management Science, 38:1495–1509. Taillard, E. (1993). Benchmarks for basic scheduling problems. European Journal of Operational Research, 64:287–295. Yamada, T. and Nakano, R. (1992). A genetic algorithm applicable to large-scale job-shop problems. In M¨anner, R. and Manderick, B., editors, Proceedings of Parallel Problem Solving from Nature II, pages 281–290, Springer Verlag, Berlin, Germany.
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Dirk C. Mattfeld
University of Bremen Department of Economics Box 330440 D-28334 Bremen, Germany [email protected]
University of Bremen Department of Economics Box 330440 D-28334 Bremen, Germany [email protected]
Abstract A general model for job shop scheduling is described which applies to static, dynamic and non-deterministic production environments. Next, a Genetic Algorithm is presented which solves the job shop scheduling problem. This algorithm is tested in a dynamic environment under different workload situations. Thereby, a highly efficient decoding procedure is proposed which strongly improves the quality of schedules. Finally, this technique is tested for scheduling and rescheduling in a non-deterministic environment. It is shown by experiment that conventional methods of production control are clearly outperformed at reasonable runtime costs. Keywords Genetic algorithm, permutation representation, tunable decoding, job shop scheduling problem, dynamic scheduling.
1 Introduction With the spread of automated manufacturing systems the optimization problem of assigning operations to a set of machines receives increasing attention (Parunak, 1992). From the viewpoint of combinatorial optimization the issue of finding an optimal production schedule is known to be NP -hard (Błazewicz et al., 1996). Therefore particular attention is paid to local search-based heuristics (Anderson et al., 1997) including Genetic Algorithms (GA). A good deal of previous research on GAs in scheduling concentrates on the static job shop problem (Mattfeld, 1996). However, production is an activity of sustained pursuit and therefore production scheduling is actually a dynamic task. In a manufacturing system all jobs have release times which define their date of arrival at the shop floor. Job releases are typically unforeseen events and, consequently, neither the release times of jobs nor their specific processing requirements are known in advance. Therefore, scheduling is a non-deterministic problem dealing with an open time horizon. The control of the jobs in a manufacturing system is referred to as rescheduling because unforeseen events frequently require schedule revisions. In practice predominantly priority-rule based control is used (Baker, 1974). The temporal decomposition of the non-deterministic problem into a series of dynamic, but deterministic
c 1999 by the Massachusetts Institute of Technology
Evolutionary Computation 7(1): 1-17
C. Bierwirth and D. Mattfeld
problems also opens this domain for optimization methods (Raman and Talbot, 1993). Since already small improvements of the schedule quality obtained by optimization often lead to considerable cost reductions, advanced optimization techniques are highly appreciated. The transfer of GA experiences to production control was first approached in Bierwirt et al. (1995) and has been taken up recently in Lin et al. (1997) by addressing the GA effectiveness. We think that the GA efficiency is of predominant importance. So far, however, the GA runtimes have hindered a thorough computational study as usually done for priority-rule based scheduling. This paper proposes an optimization method for the problem sketched above. In Section 2 we first review the static job shop problem. Next we extend the static scheduling problem to a dynamic one. In Section 3 we present a GA capable of solving the dynamic job shop problem. In Section 4 a highly efficient decoding procedure is presented and evaluated by experiment. In Section 5 the GA is applied to rescheduling and tested in two variants of which one again improves the GA efficiency. The results are compared with the outcome of priority-rule based scheduling before we conclude in Section 6.
2 Job Shop Scheduling 2.1 The Static Model The static model for job shop scheduling describes a simplified multistage production process. We consider a manufacturing system consisting of m machines M1 ; : : : Mm for which several assumptions are made. Machines cannot substitute each other and there is no preemption of a processing unit. In this ideal system no machine breakdowns occur and the passing times of jobs between machines are neglected. Furthermore, job-specific setup times of machines are not considered. A production program consists of n jobs J1 ; : : : Jn and each job consists of at most m operations. All operations have to be processed by dedicated machines but no job has to be processed twice by the same machine. The actual number of operations of job Ji (1 i n) is denoted by mi . Since the jobs do not necessarily have to pass every machine, mi m holds. Each job Ji defines a production recipe i given as an order of the machines it has to pass. The vector i points to the machines, hence for i (k ) = j (1 k mi ) machine Mj is the k-th machine that processes Ji . The k-th operation of job Ji processed on machine Mi (k) is denoted as oik . The required processing time of oik is known in advance and denoted by pik . A schedule of a production program can be seen as a table of starting times tik for the operations oik with respect to the technological machine orders of jobs. The completion time of an operation is given by its starting time plus its processing time (tik + pik ). The earliest possible starting time of an operation is determined by the maximum of the completion times of the predecessor operations of the same job and of the same machine. Hence the starting time for operation oik is calculated by
tik = max ti;k,1 + pi;k,1 ; thl + phl :
(1)
Here ohl refers to the operation of job Jh which precedes oik on its machine, i.e. i (k ) = h (l). For k = 1 we deal with the beginning operation of a job and therefore in this case the first argument in the max-function is set to zero. If oik is the first operation processed on a machine it has no machine predecessor and now the second argument in the max-function is set to zero. 2
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The completion time Ci of a job Ji is given by the completion time of its final operation i.e. Ci = ti;mi + pi;mi . As a measure of the performance of a schedule often the i minimization of Cmax = maxfC1 ; : : : Cn g is pursued. This objective function is optimized by reducing the time span needed in order to make the entire production program.
oi;m ,
2.2 The Dynamic Model We now consider an extension of the static model in which the n jobs are released at different points in time. The release time ri of job Ji determines the point in time at which Ji arrives in the shop floor. The starting time of the beginning operation of a job is now given by the maximum of the release time and the completion time of its machine predecessor operation
ohl
ti1 = max ri ; thl + phl :
(2)
The non-beginning operations of Ji are still scheduled according to equation (1). Whenever release times are taken into account for building a schedule, the job shop model is referred to as a dynamic one. Here, we still assume that all release times are known in advance. Therefore, we classify this type of job shop to be dynamic, but deterministic. In the dynamic model Cmax predominantly depends on the release time of the latest released job. Therefore it cannot serve as a proper measure of performance anymore. Instead, P the mean flow-time of jobs is used. It is calculated by F = n1 ni=1 Ci , ri . Minimizing F seeks to complete jobs as soon as possible. In practice short job flows are preferred because they reduce the work in progress inventory costs and increase the flexibility within the shop floor. 2.3 The Non-Deterministic Model Since production is an activity of sustained pursuit, scheduling of the production process is a non-deterministic problem. The production process is typically supposed to last beyond the completion of any of the known jobs. Therefore scheduling deals with an open time horizon. We follow Raman et al. (1993) to handle situations where jobs arrive non-deterministically in the manufacturing system. In their approach the non-deterministic scheduling problem is decomposed into a sequence of dynamic but deterministic scheduling problems. Let P0 be a dynamic job shop problem as introduced in Section 2.2. P0 arises from the release of a production program at time t0 = 0. At t0 the program consists of n jobs. Up to the next release of new jobs at time t1 > t0 problem P0 can be solved. This leads to a table of potential starting times for all operations involved in P0 .
To construct the new problem P1 we make a snapshot at the release time of new jobs t1 . The operations oik with potential starting times tik < t1 have already been implemented in the manufacturing system and are, therefore, now removed from the problem. We modify the jobs Ji of P0 which cover operations starting in the period from t0 to t1 . Then we remove the corresponding operations from the technological machine order and finally decrease m1 to the number of remaining operations of job Ji . If mi = 0 we remove job Ji completely. If a job has been modified but not removed from the production program we calculate a new release time for the modified job Ji by
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sj = max 1max t +p jt 0. This presentation focuses on the possible acceleration of the GA achieved by hybrid scheduling. As one would expect, Evolutionary Computation
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Summarizing, a hybrid scheduler parameterized with = 0:5 yields an improved solution quality by saving a significant amount of runtime at the same time. If runtime is the limiting resource, one might think of using a smaller value in order to speed up the search. Particularly for large problems the expected loss of solution quality for < 0:5 seems to be bounded.
5 GA based Rescheduling The GA outlined in the previous section is now used to solve a series of dynamic problems resulting from a temporal decomposition in a non-deterministic environment (Raman and Talbot, 1993). Now the GA actually reschedules the production program as it is sketched in Figure 1. 5.1 Improving Efficiency by Biasing the Search The simplest approach to using the GA for rescheduling is to make a snapshot of the manufacturing system and to formulate a corresponding dynamic, but deterministic problem. Starting with a randomly generated population, the GA solves this problem and returns the best solution found. This solution is implemented in the manufacturing system as far as possible with respect to a forthcoming snapshot. In this manner the GA is restarted after each snapshot taken at the arrival of a new job. A more sophisticated approach proposed is to reuse some of the work already done (Fang et al., 1993; Bierwirth et al., 1995). Whenever a new job arrives at time t, a new problem is created like described above. Different from the previous approach, the population of the newly started GA is now obtained from the final population of the previous GA by adjusting it to the requirements of the new problem, see Section 2.3. In other words, the search process is given a bias by seeding the initial population instead of initializing it at random. The adjustment of the population requires two steps. 12
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1. The operations which have been started before t, including those which are still processed, are deleted in all permutations of the population. 2. The operations of new jobs are inserted at random positions in all permutations of the population respecting the job’s operation order. Furthermore, job release times and machine setup times are updated using Equations (3) and (4). The operations already started before time t are not considered any longer. Whenever the final operation of a job is eliminated in this way, the job itself vanishes completely. After all these modifications, the GA continues the optimization process. Now the initial population contains fragments of favorable solutions found in the previous GA run. We conjecture that these fragments resemble each other strongly, because, due to the termination criterion described in Section 3.3, the population of the previous problem is likely to have already converged regardless of the size of this problem. Therefore biasing the population will lead to a faster convergence which in turn will result in a considerable saving of runtime (Grefenstette, 1987). Regarding the quality achieved we see two possible impacts: The information transferred may concentrate the search on parts of the search space which are still promising in the current situation. On the other hand, the similarity of individuals in the biased population may hinder a thorough search. This issue is subject to a computational investigation. 5.2 Experimental Setup We run the decomposition by using both variants of the GA and compare the outcome in terms of solution quality and runtime. Jobs are generated in the same way as done in the experimental setup described in Section 4.2. This time we use slightly different utilization rates which correspond to reasonable workloads of a manufacturing system. Utilization rates of U 2 f0:65; 0:70g represent a relaxed situation of the manufacturing system. A moderate load is produced by U 2 f0:75; 0:80g whereas utilization rates of U 2 f0:85; 0:90g produce an excessive workload. Modeling the inter-arrival times by a Possion process as described in Section 4.2 can lead to extreme deviations of the workload in different phases of the simulation run. Therefore a large variance of both, the quality achieved as well as the runtime required can be observed. In order to alleviate this effect, a large number of job releases has to be simulated. Moreover, a large number of different simulations is needed in order to produce a sound overall mean. Consequently, for each U investigated we conduct 50 simulation scenarios consisting of 1000 jobs each. In a preliminary investigation we run the non-delay scheduler described in Section 3.2. Thereby the selection of an operation from the critical set (Step 2.N4) gives priority to the operation with the shortest imminent processing time (SPT). Jobs waiting in a queue may cause their dedicated successor machines to run idle. The SPT rule alleviates this risk by reducing the length of the queues in the fastest possible way. This in turn reduces the complexity of the manufacturing system. For this reason SPT dominates all other existing priority rules if the minimization of F is pursued. SPT runs are carried out for all utilization rates prescribed. Thereby the workload observed in the system is recorded. Keep in mind that the workload of the decomposed deterministic problem is simply the number of the operations involved. From Figure 6 we can easily recognize the six levels of workload corresponding to the Evolutionary Computation
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six utilization rates used. Consider the development of the workload in the early phase of the simulation. The workload continuously increases until a stationarity is reached (apart from U = 0:90) after approximately 200 job releases. Notice, that the difference between adjacent levels of stable workload increases with U . This observation indicates that a small additional increase of the workload may not be handled properly in a tense production situation anymore. Therefore, for U = 0:90, a stabilization of the workload cannot be observed within the 1000 jobs released. The figure depicts that jobs scheduled in the early phase of the simulation distort the F measure considerably. The same is true for the final phase of the simulation run. Consider the last problem constituted at job release 1000. Although it still consists of many operations, it must be solved entirely now. By taking these effects into account, we must doubt the results reported by Lin et al. (1997) for a similar study. The scenarios investigated by Lin et al. consist of 100 job releases — the leftmost 10% of job releases shown in Figure 6. In our computational study we discard job 1 to 200 as well as job 801 to 1000 from being evaluated in order to circumvent distortion effects. 5.3 Results and Discussion After we have carefully designed the experiment, we perform simulation runs for both GA variants described in Section 5.1. The randomly initialized GA is called “rand” and its seeded counterpart is called “bias” in Table 2. The mean flow-times of jobs achieved for the SPT rule and both GA variants are reported. For the GAs also the average number of generations and the average runtime of a simulation scenario are listed. The programs are run on a Pentium/200 Mhz computer. From Table 2 we see that the flow-times of jobs obtained by SPT increase strongly with the utilization rate. Under high utilization rates the large workload results in long flow-times of jobs. The prescribed average processing time P = 50 of jobs can serve as a simple lower bound for F . By subtracting this value from F we obtain the average waiting time of a job in the system which increases from 42 for U = 0:65 up to 150 units for U = 0:90. 14
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util. rate 0.65 0.70 0.75 0.80 0.85 0.90
Table 2: Results of non-deterministic scheduling. mean flow-time F generations runtime (sec) SPT rand bias rand bias rand bias 91.8 85.8 85.9 10.8 8.8 111.7 81.1 100.4 93.2 93.5 14.9 11.1 189.6 127.6 111.5 103.5 103.1 21.4 14.6 351.0 214.6 128.1 118.7 117.4 33.5 20.3 747.6 407.5 154.9 142.6 140.6 56.7 31.6 1876.4 953.0 200.4 184.3 179.4 103.8 52.4 5928.1 2705.5
The SPT rule is outperformed by both GA variants for all utilization rates tested. Thereby a reduction of 15% of the waiting time of the jobs is gained for all values of U . The solution quality produced by the GA variants differ only slightly from each other. As it can be taken from the lower curve in Figure 6, the dynamic problems solved under U = 0:65 consists of about 25 operations. For such small problems both versions of the GA come up with almost identical results. With an increasing U the problem size increases and now the biased GA dominates its rival by a few percent. Table 2 also reports the average number of generations performed in the GA runs until termination occurred. Recall the flexible termination criterion used. For U = 0:65 the biased GA uses slightly less generations than its randomly initialized counterpart. In contrast, for a heavily loaded system, the unbiased GA demands up to two times the generations needed when used on a biased basis. The larger the workload increases, the more advantageous the biasing of the initial GA population gets in terms of the generations spent. The advantage becomes even more clear by considering the runtime required for one simulation scenario. Note that the time needed by the hybrid scheduler increases at O(n2 ) with the problem size. For U = 0:90 the GA solves problems of an average size of 73.5 operations. The “rand” GA is clearly outperformed by its “bias” rival. Even under heavy load only 2705.5 seconds are spent on average by the biased GA in order to solve 1000 subsequent problems. Thus, the rescheduling of the system is performed competitively fast in 2.7 seconds. In order to derive the runtime of the SPT-scheduler recall that a priority rule just determines the order of operations in the scheduler. The same scheduler evaluates the fitness within the GA, this time driven by the order of operations in the permutation. Since the population consists of 100 permutations and the biased GA performs 52.4 generations on average for U = 0:90, we can easily examine that a priority rule based scheduler performs at least 5240 runs in a time span of 2.7 seconds. Thus we must admit that priority based scheduling requires a fraction of a second only. However, the results presented justify the additional time spent for the biased GA.
6 Summary and Conclusion Throughout the last decade numerous GA approaches to production scheduling have been reported. These approaches differ strongly from each other with respect to the encoding and operators used, the constraints handled and the goals pursued. Despite these differences, all approaches have in common that domain knowledge is required in order to produce competitive schedules. Although highly desirable, the GA is not capable to conduct the search properly. In order to overcome this shortcoming many ways of incorporating domain knowledge Evolutionary Computation
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have been proposed. Probably the most successful approach is to introduce hill-climbing resulting in a genetic local search approach. Unfortunately, for most scheduling problems hill-climbing lacks an efficient estimate of promising neighborhood moves. For these cases the remedy of reducing the search space by means of an active scheduler has been proven successful. Despite searching in the set of active schedules only, the capabilities of the GA to conduct a proper search still vanish with an increasing problem size. This weakness of GA based scheduling can be alleviated by a further reduction of the search space. For this end we have proposed a tunable decoding procedure which proportions the size of a search space within useful bounds. Although this procedure may exclude optimal and even near optimal solutions from being searched, the positive effects of the search space reduction do prevail. Moreover, the GA runtime is reduced by approximately 50%. A temporal decomposition of a non-deterministic production environment opens this domain for GA based scheduling. One way to a further reduction of the computational effort is to fall back on information available at no cost. Based on this principle a rescheduling of the current problem is done whenever a new job is released. Thereby the initial population of the GA is built by modifying the final population of the preceding GA run. Again, this technique cuts down the GA runtime by about one half. In this paper we have dealt with the minimization of the mean flow-time of jobs, because for this objective the excellent performance of the SPT rule is well known. Against this simple but powerful rival, the GA approach produces far better results at a reasonable runtime.
References Anderson, E. J., Glass, C. A., and Potts, C. N. (1997). Machine scheduling. In Aarts, E. H. L. and Lenstra, J. K., editors, Local Search Algorithms in Combinatorial Optimization, pages 361–414, John Wiley, New York, New York. Baker, K. R. (1974). Introduction to Sequencing and Scheduling, John Wiley, New York, New York. Bierwirth, C., Kopfer, H., Mattfeld, D. C., and Rixen, I. (1995). Genetic algorithm based scheduling in a dynamic manufacturing environment. In Palaniswam, M., editor, Proceedings of the Second Conference on Evolutionary Computation, pages 439–443, IEEE Press, New York, New York. Bierwirth, C., Mattfeld, D., and Kopfer, H. (1996). On permutation representations for scheduling problems. In Voigt, H.-M., et al., editors, Proceedings of Parallel Problem Solving from Nature IV, pages 310–318, Springer Verlag, Berlin, Germany. Blanton, J. L. and Wainwright, R. L. (1993). Multiple vehicle routing with time and capacity constraints using genetic algorithms. In Forrest, S., editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 452–459, Morgan Kaufmann, San Mateo, California. Błazewicz, J., Domschke, W., and Pesch, E. (1996). The job shop scheduling problem: Conventional and new solution techniques. European Journal of Operational Research, 93:1–30. Della Croce, F. D., Tadei, R., and Volta, G. (1995). A genetic algorithm for the job shop problem. Computers and Operations Research, 22:15–24. Dorndorf, U. and Pesch, E. (1995). Evolution based learning in a job shop scheduling environment. Computers and Operations Research, 22:25–40. Fang, H.-L., Corne, D., and Ross, P. (1996). A genetic algorithm for job-shop problems with various schedule quality criteria. In Fogarty, T., editor, Proceedings of AISB Workshop, pages 39–49, Springer Verlag, Berlin, Germany.
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Fang, H.-L., Ross, P., and Corne, D. (1993). A promising genetic algorithm approach to job-shop scheduling, rescheduling, and open-shop scheduling problems. In Forrest, S., editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 375–382, Morgan Kaufmann, San Mateo, California. Giffler, B. and Thompson, G. (1960). Algorithms for solving production scheduling problems. Operations Research, 8:487–503. Grefenstette, J. (1987). Incorporating problem specific knowledge in genetic algorithms. In Davism, L., editor, Genetic Algorithms and Simulated Annealing, pages 42–60, Morgan Kaufmann, San Mateo, California. Holthaus, O. and Rajendran, C. (1997). Efficient dispatching rules for scheduling in a job shop. International Journal of Production Economics, 48:87–105. Lin, S., Goodman, E., and Punch, W. (1997). A genetic algorithm approach to dynamic job shop scheduling problems. In B¨ack, T., editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 481–489, Morgan Kaufmann, San Mateo, California. Mattfeld, D. (1996). Evolutionary Search and the Job Shop, Physica/Springer, Heidelberg, Germany. Nakano, R. and Yamada, T. (1991). Conventional genetic algorithm for job shop problems. In Belew, R. and Booker, L., editors, Proceedings of the Third International Conference on Genetic Algorithms, pages 474–479, Morgan Kaufmann, San Mateo, California. Parunak, H. V. D. (1992). Characterizing the manufacturing scheduling problem. Journal of Manufacturing Systems, 10:241–259. Raman, N. and Talbot, F. (1993). The job shop tardiness problem: A decomposition approach. European Journal of Operational Research, 69:187–199. Storer, R., Wu, S., and Vaccari, R. (1992). New search spaces for sequencing problems with application to job shop scheduling. Management Science, 38:1495–1509. Taillard, E. (1993). Benchmarks for basic scheduling problems. European Journal of Operational Research, 64:287–295. Yamada, T. and Nakano, R. (1992). A genetic algorithm applicable to large-scale job-shop problems. In M¨anner, R. and Manderick, B., editors, Proceedings of Parallel Problem Solving from Nature II, pages 281–290, Springer Verlag, Berlin, Germany.
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