Batch Schedule Optimization

Transcription

1 Batch Schedule Optimization Steve Morrison, Ph.D. Chem. Eng Abstract: Batch schedule optimization is complex, but decomposing it to a simulation plus optimization problem, can make it easier to implement. Exploiting a commonality of approach of a fitness function, a costing function, and a sequencing function can enable an engineer to see derivations of standard optimization approaches that can potentially work well. Finally different classes of common optimization algorithms are briefly presented. Key words: batch schedule optimization, optimization, batch scheduling Introduction It is a paradox that batch process optimization is so complex, that it is done by hand instead of using computers. The problem can be complex with multiple levels of simultaneous optimization, special plant policies, and unusual types of constraints. Furthermore, there is considerable variation in batch process plants, so an ideal optimizer for one can simultaneously be overkill and too simplistic on another. However, process batch scheduling is not so intractable as it may first appear. Like all optimization problems, process batch scheduling has a calculation for a fitness function, and a set of inputs for the optimizer to manipulate. Like some other optimization problems, there are even promising search directions to try. First we will discuss optimization approaches in general, and then why a branch and bound or beam search is recommended for this type of problem. Problem Decomposition Both Morrison (1996) and Schuster and Framinan (2003) have decomposed scheduling problems into a simulation (or timetable) problem and an optimization (or sequencing) problem. This greatly simplifies later altering of the application to try differing optimization algorithms, and makes the application robust with respect to changes in the costing function or constraints. From this author s experience, the performance penalty in doing so is non-existent if it is welldesigned. A larger question is whether to use more programmer-friendly structures such as DateTime, many objects, etc. or to get closer to the metal with more integers, arrays, and raw linked lists. Having benchmarked both methods, the second approach in C# is about twice as fast as the first approach, when the first approach is well-written. This author expects that the difference would be smaller in languages such as C++ with no garbage collection. The Fitness Function 1

2 Once all the decisions are made, how good the schedule is can be calculated based on the multivalued objective function. There can be at least six different components, some of which are less obvious. Completion time or Due penalties: Consider the following example. A quantity of final product is due at the end of July, and there are three schedules: the first has it produced at the end of August, the second at the end of July, and the third at the end of June. All other things being equal, is the third schedule as much better than the second schedule, as the second is better than the first? - usually not. So either one would use due date constraints combined with completion time, or due penalties, which act as soft constraints. One can have a very late due penalty as well as a late due penalty. There can be start after constraints, but there can also be a start before penalty function if the campaign needs to start earlier than desired. Cost: changeover cost and buy vs. make decisions are one kind of cost. A second type is final product inventory holding cost and intermediate inventory holding cost, which can be combined by considering how soon one must buy the raw materials. A third kind of cost is using a more less expensive piece of equipment vs. a more expensive one. Reducing Variability: If the last campaign that was run on a set of equipment was considered successful, then all other things being equal, it is more desirable to run the next campaign on the same equipment. In process scheduling, the probability of a schedule not being significantly changed in the next six months is very low. The next two factors are a means of improving the adaptability of the schedule to new rush orders. Minimum Component Usage: When two pieces of equipment have the same cost and availability, the one with the fewest number of extra components (heaters, agitators, etc.) is a better choice, because this would in general leave the most versatile equipment free for unexpected rush orders. Maximize Gaps: If a piece of equipment could be free for alternating, that is not a flexible as have a piece of equipment free for consecutive weeks. Thus it is more desirable to use equipment that was most recently used, as long as it does not excessively delay the campaign. The real challenge of the fitness function is finding the equivalence between the chosen factors. For example, to save $10,000 in cost, what is the maximum time a scheduler would be permitted to delay a campaign? Is it better to have larger gaps for an additional $1,000 in cost, or $10,000? These tradeoffs should be decided by the user, but what is an intuitive way to display these options? One way is to have dial controllers, analogous to stereo volume settings. The user can set these, run a few schedules, and see what he or she likes better. A more rigorous way is to equate everything to a cost in dollars, and then simply pick the lowest cost. A more mathematically sophisticated enhancement of that is to make everything a cost, but compute the costs for each factor separately, and then define the cost function as the L2 norm, in other words, to minimize the sum of the square of the cost of each factor. 2

3 What the Optimizer Can Change In process scheduling, the optimizer typically can only change up to four types of factor for a campaign: The site where the campaign will be produced. The master recipe used The choices of equipment, connections, and sometimes other resources used The order or priority of the campaign relative to other production campaigns and maintenance The choice of site is usually a global planning function. If it is a part of the schedule, it would typically be the outermost loop in a branch and bound. The choice of master recipe can change not only the batch size and duration, but even the number of pieces of equipment and other resources used. It should be the second outermost loop. The choices of equipment/resources and the priority of campaigns could be done in any order. However, if the choice of equipment or other resources can change the batch size, yield, batch cycle time, or batch duration (as likely is the case), then for performance reasons it is best to make the order/priority the innermost loop. Optimizer Lite One client wanted to roll simulation without optimization initially, to get the value of that before the optimizer was finished. However, it was seen that even a simple optimization function would greatly enhance simulation, if it could be done simply. Thus we came up with the idea of what came to be called Optimizer Lite, that would perform equipment selection optimization based on what was already allocated in the past, without looking at the future. This relieved the user of much of the responsibility of balancing availability vs. cost in equipment selection. However, it is far inferior to a true optimizer, as an ideal equipment choice with respect to one campaign, might cause a later campaign to be forced to either have a long delay or have no choice but a very expensive piece of equipment. Anchors The user might wish to anchor some choices, so that the scheduler can change only what is unanchored. In one approach, an optimization function could not override the user s choice, but in a second approach that was tried, the optimization could override the user s choice if another piece of equipment was available earlier. However, leaving the user no capability to fix the choice of equipment, perhaps for reasons not known to the scheduling software, has drawbacks. Search Directions Schedule metrics on a schedule can show which campaigns will complete later than their due dates, which are not using the least expensive pieces of equipment, and for some industries, which have cleanup costs that changing campaign order could reduce. Making campaign priority 3

4 the innermost loop, in other words, the most frequently changed item, enables the optimizer to run faster because that does not change campaign duration, batch size, or yield. Possible Solution Algorithms There are hundreds of possible algorithms for most complex scheduling algorithms, but not all of them would work well. In particular, linear programming, a powerful algorithm for continuous planning and scheduling, and MRP II, has severe shortcomings here due to the nature of constraints. Almost any problem that has equality, inequality, greater than or less than constraints is amenable to linear programming if the variables are all continuous. Unfortunately, in this type of scheduling most of the fields, if not binary, are at least discrete. For example, for ten equipment choices, the one chosen has a value of 1, and all of the others are 0. Linear programming has been extended handle some discrete variables using cutting plane algorithms, but this approximate method bogs down when the majority of variables in a large system are discrete. The potentially relevant ones can be put in four categories: randomized methods and deterministic iterative ones. Randomized Methods Randomized methods do not exploit any hints of search directions. One method is to apply GRASP (Greedy Randomized Search Procedure), which picks a random solution, for an initial guess. It locally tries to improve on the solution until there is no additional progress. Then it picks random guesses until one is not too much worse (say 10%) than the best solution so far. Then it locally tries to improve on that solution and repeats. See Feo et al. (1994) and Feo, Bard, and Holland (1995) for using GRASP on other types of problems. Tabu Search is another general purpose method that a schedule simulator can apply. After picking an initial solution and locally trying to improve upon it, it artificially sets the cost of part of that solution to be very high, a tabu. Then it continues to locally try to improve the solution, with the tabu forces a move away from that solution. Tabus eventually expire, after they have performed their purpose of diversification. See Barnes and Chambers (1991), Barnes and Laguna (1991) for examples applied to various jobshop scheduling problems. Genetic Algorithms work by taking pieces of different solutions and recombining them to try to find a better solution. This family of methods is analogous to taking different combinations of genes to try to produce better offspring. Statistics are kept so that the apparently less fit pieces are not tried so much. A gene could be a partial or complete set of equipment assignments for a campaign, or part of a sequence of campaigns. One problem with genetic algorithms for these kinds of problems is that the same equipment selection could be very good or very bad, depending on the equipment selection of the campaign that are before and after. See Dozier et al (1994), Green (1994), Morikawa et al (1994), and Murata and Ishibuchi (1994) for more on genetic algorithms. 4

5 A characteristic of many genetic algorithms is that they rapidly have a dominance of particular genes. A related approach, called Immune Algorithms, are said to be better than genetic algorithms by avoiding to quickly jumping to a solution. See Khoo and Alisantoso (2003) for more on this new approach. Deterministic Iterative Methods The most straightforward deterministic iterative method is simple enumeration of all the choices, but that is not practical for most of process scheduling problems. The deterministic methods mentioned here are similar to enumeration, except that they find shortcuts in making calculations, reasons to rigorously exclude certain branches, and guesses that certain branches are more promising than others. Expert Systems and Dispatching rules simply use rules of thumb to develop the schedule. The Optimizer Light function, that look at only past campaigns for equipment selection, is an example of this. Dynamic Programming applied to these problems involves storing intermediate results of calculations of parts of a potential schedule, and reusing these results in subsequent calculations. While impractical to do all of the optimization with dynamic programming, this approach is a helpful auxiliary method. For example, the cost (but not start time) of many choices of equipment selections can be stored and reused later. Sequential Methods work well when campaigns at the start of the schedule have little or not effect on campaigns toward the end of the schedule. Thus schedule optimization can be done on only say the first half of the schedule, then the schedule could be optimized on the middle half, and then on the last half. Note there needs to be significant overlap between the regions. A second benefit of sequential methods comes from the realization that having a guaranteed optimal solution on what to do six months from now is not very useful. Conditions will undoubtedly change by then. It is better to have as optimal solution as possible on what to do for the next week or so, and not to be so optimal after that. Branch and Bound and Beam Search Branch and Bound is similar to enumeration, except that it evaluates whether to go down a path or not. For example, if there are two or more equipment choices, and one piece of equipment costs a little more and takes longer, and no other campaign would have contention with this campaign for that piece of equipment, then every possible schedule that would use this piece of equipment in this campaign. When bounds are tight, branch and bound can be very powerful. See Rodrigues et al. (1993) for an example applied to flowshop scheduling. Yunpers (2003) solved the single-machine scheduling problem to minimize the total completion time with deadlines. Using the principle of optimality, if a particular choice cannot give a lower objective function without violating a deadline, then it can be eliminated. He rigorously solved up to 120 jobs. Morrison (1996) used a similar philosophy, except with multiple simultaneous bounds. For the batch digester scheduling problem, it was proved that any candidate that did not 5

6 have a lower cost that the fastest candidate, or was not faster than the lowest cost candidate, or did not improve the long-term order, could be excluded. In an example with 12 digesters and 40 batches, Morrison (1996) iterated through 40^12 potential iterations in less than one second. The reason for this time is that branch and bound determined that of these 40^12 potential iterations, it actually only had to iterate through 2,915 iterations. Applied to pharmaceutical plant scheduling, there are two separate places for branch and bound: campaign ordering and equipment selection. If a cost of the schedule is partially based on meeting campaign due dates, then all campaign ordering sequencing can be avoided except those which would potentially reduce a due date penalty. Changing an equipment selection should not be done except for three reasons: lower cost on the campaign, the campaign can complete sooner and there was a due date penalty using the old equipment choice, and the potential to improve the cost or due date penalty of a future campaign. Equipment choices that do not help in one of these ways do not help the overall fitness of the schedule. As long as the cost functions are accurate, branch and bound can give a guaranteed optimal solution. Branch and bound works by calculating one initial schedule, and then changing only the last choice of the last campaign. Only that one small part needs to be recalculated. Then it changes the next to last choice of the last campaign, and repeats. Then it goes back and changes the next to last campaign, and so forth. So the majority of calculations only a small fraction of a the time to do a full simulation. Even so, in many cases branch and bound does not perform fast enough because the bounds are too loose. While branch and bound only eliminate those branches where it is proven that the optimal solution is not there, beam search is a non-rigorous branch and bound method that also eliminates branches were the an optimal solution is less likely to be there. The number of choices to search, called the beam width, can vary. For batch process scheduling problems the beam width can be one value for campaign ordering, and a different value for equipment selection. Beam width does not need to be a constant number, but beam width can be defined as search the top 20% of the branches for a specific campaign. Both branch and bound and beam search usually start from the last campaign and work their way to the earliest campaign, because the iterations are more efficient that way. However, as previously mentioned getting the last campaigns optimal is least important, and getting the first campaigns optimal is the most important. For this type of problem, where the number of campaigns is fairly small, a reverse branch and bound starting with the earliest campaign will work better. There is additional housekeeping to perform the shifts of later campaigns, and there is significant performance decrease per iteration, but it would payoff for some problems because there are relatively few campaigns, and campaigns are an outer loop in the branch and bound. The equipment selection optimization would still be done last to first. Conclusion Despite any appearances to the contrary, optimization of batch process scheduling problems is not impossible. A number of algorithms are available, used either alone or in combination. Best first branch and bound for a rigorous solution, and its close cousin, beam search, are promising 6

7 methods due to their scalability and speed in iterating in the solution space. Dynamic programming and tabu search are an excellent auxiliary methods to use in conjunction with branch and bound and beam search. References 1. Barnes, J.W. and J.B. Chambers. Solving the Job Shop Scheduling Problem Using Tabu Search ORP Barnes, J.W. and M. Laguna. Solving the Multiple-Machine Weighted Flow Time Problem Using Tabu Search ORP Barnes, J.W. and M. Laguna. A Review and Synthesis of Tabu Search Applications to Production Scheduling Problems ORP Dozier, G., J. Bowen, and D. Bahler. Solving Small and Large Scale Constraint Satisfaction Problems Using a Heuristic-Based Microgenetic Algorithm. Proceedings of the First IEEE Conference on Evolutionary Computing June 27-29, 1994 Orlando, Florida p Feo, T.A., et al. A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set. Operations Research vol.42 no.5 September-October 1994 p Feo, T.A., J.F. Bard, S.D. Holland. Facility-Wide Planning and Scheduling of Printed Wiring Board Assembly. Operations Research vol.43 no p Greene, F. A Method for Utilizing Diploid/Dominance in Genetic Search. Proceedings of the First IEEE Conference on Evolutionary Computing vol.i June 27-29, 1994 Orlando, Florida p Khoo, L.P. and D. Alisantoso, Line Balancing of PCB Assembly Line Using Immune Algorithms. Engineering with Computers. Vol.19 no p Morikawa, K. T. Furuhashi, and Y. Uchikawa. Evolution of CIM System with Genetic Algorithms. Proceedings of the First IEEE Conference on Evolutionary Computation. vol.ii June 1994 p Morrison, S.M., Edgar, T.F., and Burns, H.A. Dynamic Scheduling. American Institute of Chemical Engineers. 11/1991 Meeting. 11. Morrison, S.M.. Scheduling Methods for Batch Digesters, Batch Process Plants, and Shops with the Conveyor Algorithm. Dissertation University of Texas at Austin, May Morrison, S.M. Quantifying the Benefits of Batch Scheduling. 2004a To be published. 13. Morrison. S.M. Focusing on the Primary Purpose in Batch Scheduling. 2004b. To be published. 7

8 14. Morrison, S.M. Gathering (and Reducing) Requirements for Batch Scheduling. 2004c. To be published. 15. Morrison, S.M. Architecture Alternatives for Batch Scheduling. 2004d. To be published. 16. Morrison, S.M. Types of Batch Schedule Simulation. 2004e. To be published. 17. Murata, T. and H. Ishibuchi. Performance Evaluation of Genetic Algorithms for Flowshop Scheduling Problems. Proceedings of the First IEEE Conference on Evolutionary Computing vol.ii June 27-29, 1994 Orlando, Florida p Rodrigues, M.T.M. et al. Flow Shop Scheduling : A Resource based Branch and Bound Perspective. International Federation of Automatic Control, 1993, Schuster, C. J. and Framinan, J.M. Approximate Procedure for No-Wait jobshop Scheduling. Operations Research Letters. vol.31 no.4 July p Yunpers, P. An Improved Branch and Bound Algorithm for Single Machine Scheduling with Deadlines to Minimize Total Weighted Completion Time. Operations Research Letters. vol.31 no.6 November p