2. Using Simulated Annealing and Evolution Strategy scheduling capital products with complex product structure By: Dongping SONG Supervisors: Dr. Chris Hicks and Prof. Chris F. Earl Department of MMM Engineering University of Newcastle, Oct., 2000.

3. Contents Introduction Problem formulation A discrete event-driven model Simulated Annealing Evolution Strategy Case studies Conclusions

4. Introduction Production scheduling -- the allocation of resources over time to perform a collection of tasks (Baker, 1974). Two important points in scheduling: Sequencing -- in which order to perform tasks on resources Timing -- when to start and complete tasks

5. Introduction The importance of sequencing has been well recognised, because the optimal schedule can only be characterised by the sequences for performance measures such as mean flow-time, percentage of jobs tardy, mean tardiness, etc. However, timing scheduling is necessary for performance measures such as earliness and tardiness costs, total discrepancy from the due dates, etc.

6. Introduction - effect of timing Example 1. One machine with three independent jobs. Schedule A: Schedule B: Schedule C: Due dates

7. Introduction - effect of timing Example 2. Two machines four jobs s.t. job 1 becomes job 4 and job 2 becomes job 3 after completion. Schedule A: Schedule B:

8. Complex product structure Introduction - a capital product

9. Gantt chart -- with 113 operations (13 assemblies). Waiting time Time period product

10. Resource chart -- work-load on 13 machines. Time period Work-load Machine

11. Introduction Constraints in our scheduling problem: –Operation precedence constraints –Resource capacity constraints –Due date constraints –Assembly co-ordination requirements Scheduling problem: to find optimal operation sequences and timings to meet above constraints and minimise total cost.

12. Problem formulation Notation: s i -- planned start time for operation i; N -- total operation number. Solution space of schedules := R N  {sequences on resources}. Solution space can be simplified to R N, because operations on the same resource have different start times ( i.e. timings imply sequences ).

13. Problem formulation The schedule problem can be formulated as a numerical optimisation problem. Find the optimal {s i, i=1,..,N} to minimise the total cost J(s) =  (Work-in-progress holding costs + product earliness costs + product tardiness costs)

14. Problem formulation Questions: (1) How to execute a schedule that is characterised by {s i } ? (2) How to evaluate the cost function for a given schedule ?

15. Discrete event-driven model Two types of events : –the start of an operation –the completion of an operation. Two constraints to trigger the start events : –Physical constraints : an event cannot occur before all preceding events are completed. –Planning constraints : an operation cannot be started before its planned start time s i.

16. Discrete event-driven model The evolution of the system for a given schedule {s i } can be described by: If a resource is idle, an operation will be processed as soon as the physical and planning constraints are satisfied. If there is a queue of operations ready for processing, the operation with the earliest s i will be processed first.

17. Simulated Annealing Neighbourhood of a solution -- by adding a random number to each s i. Outer loop -- cooling the temperature T until T=0. Inner loop -- perform Metropolis simulation with fixed T to find equilibrium state.

18. Simulated Annealing Adjust the solution : –shift the whole schedule (optional) –impose precedence constraints (optional) –make non-negative Evaluate cost function : –run the DED model

19. Simulated Annealing

20. Evolution Strategy Similarity of Genetic Algorithms and ES: –model organic evolution. –iterative scheme including “selection”, “crossover” and “mutation”. Difference of GA and ES: –GA uses binary or string representations, suitable for combinatorial optimisation problem. –ES uses continuous variable, suitable for numerical optimisation problem.

21. Evolution Strategy Crossover -- randomly copy elements from parents column by column.

22. Evolution Strategy Mutation -- add a random number from a Normal distribution to each element.

23. Evolution Strategy Adjust the solution : –shift the whole schedule (optional) –impose precedence constraints (optional) –make non-negative Evaluate cost function -- run the DED model. Selection -- choose a set of best offspring as parents for the next generation

24. Evolution Strategy

25. Case studies Characteristics of scheduling problems

26. Case study 1 Cost is reduced by 50% for SA and ES. MRP -- material requirement planning FIFO -- first in first out EDD -- earliest due date first SPT -- shortest processing time first

27. Case study 1 -- cost v.s cpu Cost CPU(s) SA Total cost v.s. CPU time for SA and ES ES

28. Case study 1 -- ES method Cost CPU(s) Maximum cost at each generation Maximum cost in all parents Minimum cost at each generation

29. Case study 2 Cost is reduced by 50% for SA and ES.

30. Case study 2 -- cost v.s. cpu Cost CPU(s) Total cost v.s CPU time for SA and ES SA ES

31. Case study 2 -- ES method Cost CPU(s) Maximum cost at each generation Maximum cost in all parents Minimum cost at each generation

32. Conclusions SA and ES can reduce total cost by 50% compared with MRP+dispatching rules. ES is generally better than SA in both cost and CPU time. ES is more robust to its initial parameter selection than SA.

33. Conclusions Suggestions for SA initial parameters: –T 0 and step-size is taken from [d/N, 20*d/N]; –Temperature cooling rate > 0.5 and step-size reduction factors > 0.70; –No-improvement number at inner loop > N/2; Suggestions for ES initial parameters: –Offspring population is from [N/2, 2*N]; –Parent population is 1/10 to 1/5 of offspring; –Initial standard deviation is from [d/N, 5*d/N].

34. Further work Compare our methods with GA (Pongcharoen, et al.) for the same cost function. Develop hybrid optimisation methods by combining SA, ES with Perturbation Analysis or heuristics. Extend to stochastic situations such as dynamic customer demand arrivals and processing uncertainties.

35. SA -- effect of parameters Initial temperature and temperature cooling factor T 0 =1 T 0 =40 T 0 =20 T 0 =10

36. ES -- effect of parameters Offspring Number(ON)/Generation Number(GN) and Standard deviation reduction factor Solid-line : ON/GN=80/250 dashed-line: ON/GN=100/400 dotted-line: ON/GN=160/250 dash-dotted: ON/GN=200/200