Norman Sadeh, School of Computer Science, CMU Mark S.Fox

1. Variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem Norman Sadeh, School of Computer Science, CMU Mark S.Fox Dept. of Industrial Engineering, Univ. of Toronto Artificial Intelligence, 86(1996) p1-41

2. Contents 1. Introduction 2. Job shop CSP 3. Search Procedure 4. Some popular variable ordering heuristics 5. Some popular value ordering heuristics 6. New variable and value ordering heuristics 7. Empirical evaluation 8. Concluding remarks ? -> No!

3. About Authors • Fox and Sadeh • the most representatives in CSP and Scheduling area. • ISIS (1986) • MICRO-BOSS (1991) • CMU School of CS, Robotics Institute

4. 1. Introduction (1) • CSPs for design and scheduling • Generally entail large search spaces. • Only a very tiny fraction of all possible assignments participates in solution. • Variable and Value ordering • Reducing the effective size of the search space to be explored.

5. 1. Introduction (2) • Job Shop Scheduling Problem. • Large scale, tightly connected problem • Objective is to find a feasible solution as quickly as possible. (notes: in large scale problem in industry) • In this research, • Review generic variable and value ordering heuristics • Introduce new variable and value ordering heuristics with Probabilistic framework considering reliance of a variable, and measure of contention

6. 2. The job shop CSP (1) • Assumption on this study • Tree-like process routing (그림 참조) • J = { j1, j2, …., jn} • RES = {R1,R2,..,Rm} • Ol = {Ol1,…, Olnl } • Variables Ol • operation start time stli • resource requirements, Rlij

7. O14 O16 O11 O17 O12 O15 O13 O18 O21 O22 O23 Tree-like process routing (an example)

8. 2. The job shop CSP (2) Constraints • Precedence Constraints • stli + duli <=stlj (i.e. Oli Before Oli) • Capacity Constraints • Example : • 4 jobs, 4 resources, 2-3 operations • All jobs should be terminated by 15 time units.

9. Simple Job shop problem example P1 P2 O11 R1 O12 R2 O13 R3 j1 C1 C2 C4 C6 P3 C7 j2 O21 R1 O22 R2 C3 C5 C9 C8 P4 P5 j3 O31 R3 O32 R1 O33 R2 C10 P6 j4 O42 R2 O41 R4

10. 3. Search Procedure 1. If all operations have been scheduled then stop, else go on to 2. 2. Apply the consistency enforcing procedure. 3. If a deadend is detected then backtrack (i.e. select an alternative if there is one left and go back to 1, else stop and report that the problem is infeasible). else go on to step 4. 4. Select the next operation to be scheduled (variable ordering heuristic). 5. Select a promising reservation for that operation (value ordering heuristic). 6. Create a new search state by adding the new reservation assignment to the current partial schedule. Go back to 1.

11. Consistency Enforcing • Consistency with respect to precedence constraints • Propagate EST to downstream and LST to upstream • Forward consistency checks w.r.t. capacity constraints • difficult because of disjunctive nature • checks the remaining possible reservations, • and remove those reservations that would conflict with the new assignment.

12. Consistency Enforcing - Cont’d • Additional consistency checks w.r.t. capacity constraints • checks that no two unscheduled operations require overlapping resource/time intervals. • In example [lskki, efkki]and[lsklj, efklj] overlap --> oversubscribed interval • This enforces a higher level of are-consistency than forward checking.

13. Before Propagation j1 O11 3 R1 O12 4 R2 O13 5 R3 [0, ] [0, ] [0, 15] Downstream Propagation j1 O11 3 R1 O12 4 R2 O13 5 R3 [0, ] [3, ] [7, 15] Upstream Propagation j1 O11 3 R1 O12 4 R2 O13 5 R3 [0, 8] [3, 11] [7, 15] Consistency w. r. t. Precedence constraints

14. Forward consistency checks w.r.t. capacity constraints

15. Additional Consistency checks Detecting situations where two unscheduled operations requiring the same resource are in conflict

16. O11 R1 O12 R2 O13 R3 j1 [0,6] [3,9] [6,12] j2 O21 R1 O22 R2 [3,12] [0,9] j3 O31 R3 O32 R1 O33 R2 [0,6] [3,9] [6,12] j4 O42 R2 O41 R4 [0,9] [3,12] Job shop CSP after consistency labeling

17. Consistency Enforcing - Cont’d In Job shop Scheduling Problem • Consistence enforcement to Precedence constraint is sufficient to guarantee decomposability. • Backtracking can only occur as the result of capacity constraint violations.

18. 4. Popular variable ordering heuristics • Difficult variables first! • Fixed variable ordering heuristics • a unique variable ordering is determined prior to starting the search. • Require less computing! • Dynamic variable ordering heuristics • The ordering is dynamically revised in each state in order to account for earlier assignments. • More powerful!

19. 4.1 Variable ordering heuristics • DSR (Dynamic search reassignment) • DSR look for the the variable with the smallest number of remaining values. • BENCHMARK to determine whether it is worthwhile using a dynamic variable ordering heuristics. • Job shop scheduling is more difficult problem for which Dynamic variable ordering heuristic is justified.

20. 4.2 Variable ordering heuristics • MW (Minimum Width) • select a node which has minimal degree remaining after deleting all nodes that have already been selected. • MD (Minimum degree) • Ordering variables according to their degree in the initial constraint graph. • MC (Max Cardinality) Dynamic • picks the variable connected to the largest number of already instantiated variables.

21. 5. Popular value ordering heuristics • Least constraining values first! • ABT (Advised Backtracking) and MST(Min Spanning Tree) • Benchmark in this research.

22. 6. New variable and value ordering heuristics • General Consideration • A probabilistic model of the search space • a variable ordering heuristic based o measures of resource contention • a reservation ordering heuristic that attempts to minimize contention • Estimation the probability that a reservation survives contention • Estimating the probability that a job schedule survives contention

23. 6.1 General Consideration • This research aims at developing heuristics that efficiently reduce the expected number of search states that need to be explored

24. 6.2 A probabilistic model of the search space • The chance that a given value will be assigned to a variable • the chances that values assigned to different variables conflict with each other.

25. 6.3 a variable ordering heuristic • Calculate individual demand of each operation for each resources ==> Dli() (fig10-1, fig10-2) • Aggregate each demand for Resource Rj ==> DaggrRj() (fig11-1, fig11-2) • Compare each aggregated demand profile ==> select max contention resource Rj* • Select Operation Oli which has highest contribute to Rj*(fig12-1, fig12-2)

26. Fig10-1 R2’s demand profiles

27. Fig10-2 R2’s demand profiles

28. Fig11-1 Aggregate demand profiles

29. Fig11-2 Aggregate demand profiles

30. Fig12-1 ORR heuristic

31. Fig12-2 ORR heuristic

32. 6.4 A variable ordering heuristic • Estimate the Survivability for each reservation.==> (Fig 13) • Estimate the survivability for each job ==> Goodness. (Fig 14)

33. Fig 13 Survivability measures

34. Fig 14 Reservation Goodness

35. 7. Empirical Evaluation • Test data parameters • Range parameter : release date and due date distribution control • Bottleneck parameter : Number of bottleneck resources • Slack parameter

36. Comparison with other heuristics • Comparison competitors • DSR & ABT • DSR & FSS • ORR & ABT • ORR & FSS • Performance Measures • Search Efficiency : the number of operations to be scheduled over total number of search states explored • Number of experiment solved • Average CPU time

37. Table 1-1

38. Table 1-2

39. Table 1-3