Current trends in deterministic scheduling Chung-Yee Lee, Lei Lei and Michael Pinedo.

1. Current trends in deterministic schedulingChung-Yee Lee, Lei Lei and Michael Pinedo. Presented by Gökhan Metan

2. Outline • Brief introduction to Deterministic Scheduling • Recent developments in scheduling theory • Recent developments in search algorithms • Recent developments in scheduling practice • Conclusion

3. Introduction In deterministic scheduling, a set of jobs has to be processed by a set of machines while optimizing a certain performance measure. Recent developments focused on more practical issues Most practical problems are NP-hard models are extended! Consequence: Local Search Methods (especially the Meta-heuristics)

4. Notation  /  /  Machine (resource) configuration Processing restrictions & constraints Performance measure to be optimized

5. Recent developments in scheduling theory Trend: Results of classical algorithms Extend Models that are closely related to real problems. Two areas reviewed: Scheduling with 1-job-on-r-machine Machine scheduling with availability constraints

6. Recent developments in scheduling theory Scheduling with 1-job-on-r-machine Jobs may need to be processed simultaneously on several machines (r is a positive integer), or several jobs can be processed by a single processor simultaneously (0<r1). Machine scheduling with availability constraints Machines may not be available at all times, due to machine maintenance, or machines may only be available in given time windows.

7. Recent developments in scheduling theory Scheduling with 1-job-on-r-machine pattern Notation: /  /  : fix or nonfix =fix : each job can be processed simultaneously by several processors but these machines are prespecified (a fixed set of machines). =nonfix: each job can be processed simultaneously by several processors but the number of machines is fixed (not the machine set itself).

8. Recent developments in scheduling theory Scheduling with 1-job-on-r-machine pattern The machine set not fixed (=nonfix) Versions of Pm|nonfix|Cmax Pm|nonfix|wjCj Pm|nonfix|Lmax studied in the literature.

9. Recent developments in scheduling theory Scheduling with 1-job-on-r-machine pattern The machine set fixed (=fix) Versions of Pm|fix|Cmax Pm|fix|wjCj Pm|fix|Lmax Pm|fix|Tj Pm|fix|wjUj studied in the literature.

10. Recent developments in scheduling theory Machine scheduling with availability constraints Machines may not be available because of machine breakdown (stochastic) or preventive maintenance (deterministic) during the scheduling period. Machine scheduling with availability constraints in deterministic case is reviewed. Two cases are discussed in the literature: Resumable: If a job cannot be finished before the next down period of a machine and the job can continue after the machine has become available again. (=r-a) Nonresumable: If the job has to restart rather than continue. (=nr-a)

11. Recent developments in scheduling theory Machine scheduling with availability constraints Versions of 1|nr-a|  Cj F2|r-a|Cmax , F2|nr-a|Cmax 1|r-a|  Cj , 1|r-a| Lmax 1|r-a| wjCj 1|r-a|Uj studied in the literature.

12. Search Algorithms Most scheduling problems are so complex. Cannot be formulated as Mathematical Programs. Difficult to apply classical techniques (DP, B&B). Increasing popularity of “Search Techniques” other than classical operations research techniques.

13. Search Algorithms Search Techniques Constraint Guided Heuristic S.T. Neighbourhood S.T. Frequently used by OR and IE people. Often used by CS and AI people. Do not try to find the optimal, rather try to find good & feasible solutions. Based on the concept of local improvement. Programming effort required for the implementation is reasonable. Require more programming effort than Neighbourhood S.T. Structural knowledge needed about the problem is significantly less when compared with the MP approach.

14. Search Algorithms Definition:A heuristic is a technique which seeks good (i.e. near optimal) solutions as a reasonable computational cost without being able to guarantee either feasibility or optimality, or even in many cases to state how close to optimality a particular feasible solution is. (Reeves) NOTE: Many heuristics are problem specific, so that a method which works well for one problem cannot be used to solve a different one.

15. Search AlgorithmsNeighbourhood Search Techniques Four Important Design Concepts for Neighbourhood S.T. 1) The mapping of the data in a format suitable for the algorithm Data is a problem specific issue and it must be well described that how it should be represented in the search procedure. 2) The neighbourhood design The neighbourhood design specifies the set of all the neighbouring solutions of a given solution. (Note: For a more formal definition look to the “Definition-3” on page 2 of the notes provided.) It is a very critical decision which affects the solution quality and the computational efficiency of the algorithm.

16. Search AlgorithmsNeighbourhood Search Techniques Job-1 Job-1 Job-2 Job-2 Job-3 Job-3 Job-(n-1) Job-(n-1) Job-n Job-n For any solution S, neighbourhood of S, N(S), includes (n-1) different alternative neighbouring solutions. (ex: for n= 20, 19 alternative solutions.) For any solution S, neighbourhood of S, N(S), includes (n-1)! different alternative neighbouring solutions. (ex: for n=20, 19!=121,645,100,408,832,000 alternative solutions)

17. Search AlgorithmsNeighbourhood Search Techniques 3) The search process within the neighbourhood How to select a neighbouring solution as the new solution? Different strategies (simple or complicated) can be employed. (e.g. select any neighbouring solution at random, or select the one that improves the objective much) 4) The acceptance-rejection criterion How to decide a temporarily seleted neighbouring solution to be the new solution or not? Depends on the user specified decision parameters. (e.g. cooling parameter in SA, tabu length in TS)

18. Search AlgorithmsSimulated Annealing(SA) General Properties SA algorithm is based on the analogy between the annealing (cooling of material) process of solids and the problem of solving combinatorial optimization problems. Employes probabilistic approaches such as the probability of accepting a worse solution than the current solution. Has a memoryless property, that is, it does not keep track of the information gained during the search process. Simpler to implement (in terms of programming effort) than TS & GA. Note: For a detailed information refer to the notes provided.

19. Search AlgorithmsTabu Search(TS) General Properties The philosophy of TS is to derive and exploit a collection of principles of intelligent problem solving. TS is based on the selected concepts that unite the fields of AI and optimization. One of the main components of TS is its use of adaptive memory(attributive memory, explicit memory etc.) (Memory-based strategies are employed!) Not a random search technique! A bad choice can yield more information than a good random choice. In a system that uses memory, a bad choice based on strategy can provide useful clues about how the strategy may profitably be changed. (Glover) Note: For a detailed information refer to the notes provided.

20. Search AlgorithmsGenetic Algorithms(GA) General Properties GA is the meta-heuristic technique that originates from the analogy between the representation of a complex structure by means of a vector of components, and the idea, familiar to biologist, of genetic structure of a chromosome. Employes probabilistic approaches such as the probability of mutation, cross-over etc. Has a memoryless property, that is, it does not keep track of the information gained during the search process. Works on a population of solutions, not a single current solution.

21. NOTE (A mistake in the paper): On page 15, 3rd paragraph of section-2.3, 1st line. Search Algorithms Genetic Algorithms(GA)

22. Search Algorithms SA, TS & GA A Comparision: What is going on in the literature? Among the modern heuristic techniques, SA was the first one that appears in the literature, but TS is the most popular one. SA used for TSP and job shop scheduling problem with the objective of makespan (Jm // Cmax). TS used for TSP, single machine, parallel machine, flow shop, flexible flow shop and job shop problems with the objective of makespan, total weighted completion time and total weighted tardiness. A number of hybrid approaches, that is combining the SA & TS philosophies, exist in the literature, as well.

23. Search Algorithms SA, TS & GA A Comparision: What is going on in the literature? Cont’d GA is used for TSP, job shop problem (Jm // Cmax). Also, machine learning techniques from computer science (AI) is combined with GA for a job shop problem. (Lee, Piramuthu and Tsai, 1995.) Recently, there exists a number of applitions and implementations of GA in the real world problems.

24. Search Algorithms Constraint-guided Heuristic Search General Properties Constraints of the problem are incorporated into the system as rules. Technique focuses on partial solutions of the problem and tries to extend these partial solutions until a complete feasible solution is obtained. Technique first attempts to satisfy the most strict constraints at the beginning of the search and the least strict ones at the end.

25. Recent Developments in Scheduling Practice Recent popular areas in Scheduling Practice: Flexible-resourse scheduling Scheduling variable-speed machines Scheduling with finite capacity input and output buffers Scheduling of machine and material handling operations Integrating scheduling with batching and lot-sizing Our scope: “machine scheduling with material handling operations”

26. Recent Developments in Scheduling Practice Differences from classical machine scheduling problems: Two types of resources are now involved: Machines Material handling transporters Either resource could become a bottleneck (if not properly scheduled). Transportation operations are not instantaneous, depends on the sequence in which material movement (between machines) is executed.

27. Recent Developments in Scheduling Practice Issues that should be addressed simultaneously: Sequencing that specifies the order in which jobs are processed at machining centers. Scheduling that makes time-phased routing and dispatching of transporters for job pick-up and delivery. Facility layout and flowpath design that makes efficient operations possible. Extended Notation: (K) /  /  : K denotes the number of transporters.

28. Recent Developments in Scheduling Practice Properties of the general model: n jobs to be processed, m machining centers. All jobs are ready at time zero, each with its own route and processing specifications. The deliveries of jobs between machining centers are performed by K, K1, identical transporters. Transporters travel on a shared network & collisions must be avoided. Operations of the transporters (loading, unloading, moving jobs) are non-instantaneous & non-preemptive.

29. Recent Developments in Scheduling Practice Properties of the general model: (Cont’d) Neither a machine nor a transporter can hold more than 1 job at any time. The problem is finding a simultaneous feasible schedule for job sequencing and time-phased dispatching and routing of transporters while optimizing a given objective. Three sub-categories: 1- Robotic cell scheduling 2- Scheduling of AGVs 3- Cyclic scheduling of hoists subject to time-window constraints

30. Recent Developments in Scheduling Practice 1- Robotic Cell Scheduling: Typically arises in cellular manufacturing systems. Cell MPS Material handling robot Flexible machines Because of material handling cost, buffers between machines are limited (zero or finite). Cell performance depends on the sequence of robot moves and the order in which jobs are loaded into the cell.

31. Recent Developments in Scheduling Practice The no-buffer case: The finite buffer case: No buffer between machines. Finite (but non-zero) input and output buffers at machine centers. Several polynomial algorithms are available for 2-machine cells (m=2). 2-machine scheduling with finite input buffer but no output buffer for each machine is considered and B&B procedure is employed (King, Hodgson and Chafee, 1993.) For 3-machine cells (m=3), efficient algorithms are reported. (Hall, Kamoun, Wan, 1994.)

32. Recent Developments in Scheduling Practice 2- Scheduling of AGVs: Typically arises in flexible manufacturing systems. Such an FMS contains NC machining centers, limited input & output buffers, and a material flow network. A major constraint that must be satisfied by any AGV schedule is to avoid trafic collisions of AGVs during their operations. There are two types of AGV flowpaths commonly used in practice: Unidirectional flowpath () Bi-directional flowpath ()

33. Recent Developments in Scheduling Practice Advantages & disadvantages of bi-directional flowpath designs over the unidirectional networks: Disadvantages: Requires higher control. Implementation cost is higher. Advantages: Have greater potential to improve productivity. Require fewer AGVs. Reduce AGV travel time. Most analytical approaches that guarantee the optimal AGV schedule wih respect to certain objective functions are limited to special cases.

34. Recent Developments in Scheduling Practice AGV dispatching rules: Work center-initiated rules A work center selects an AGV for a delivery operation whenever it finishes an operation. Vehicle-initiated rules An AGV selects a pick up whenever it becomes idle. Pull-based vehicle-initiated rules Push-based vehicle-initiated rules

35. Recent Developments in Scheduling Practice Pull-based vehicle-initiated rules 1)Selects the one with highest need. Work Center 2) A job is selected to be sent to the work center. Work Center AGV Work Center Work Center Work Center Set of candidate Jobs Highest need for replenishment!

36. Recent Developments in Scheduling Practice Push-based vehicle-initiated rules 2) selects the work center which the job should be sent Work Center 1) A job is selected to be moved. Work Center AGV Work Center Work Center Work Center

37. Recent Developments in Scheduling Practice 3- Hoist Scheduling: Typically deals with the scheduling of multiple hoists in a flexible flow shop. The most distinct feature is that the job processing time at each machine is strictly limited by a lower and an upper bound. This means that any hoist schedule that causes a hoist not to pick up a job within this time window is infeasible. Subject to nwt (jobs are not allowed to wait in process) and collision free constraints. Most of the real life problems are NP hard in the strong sense and even they preclude a formal mathematical formulation.

38. Conclusion Future research directions: Nonpreemptive case with different job release times and different machine available time windows. In machine scheduling with availability constraints problem, it may be interesting to study on the semi-resumable case where some extra setup time may be required when a job restarts. Studies comparing neighbourhood search techniques with constraint-guided search techniques. Studies that integrates the facility layout and flowpath design problems with the machine and transporter scheduling. Transporter scheduling with dynamic job arrivals.

39. References C.-Y. Lee, L. Lei, M. Pinedo, Current trends in deterministic scheduling, Annals of Operations Research 70(1997)1-41. C. R. REEVES, Modern Heuristic Techniques for Combinatorial Problems, New York, Toronto, GB: John Wiley & Sons INC., 1993. F. Glover and M. Laguna, Tabu Search, Massachusetts, USA: Kluwer Academic Publishers, 1997.