1. Scheduling with Constraint Programming February 24/25, 2000

2. SMA HPC (c) NUS 15.094 Constraint Programming2 Today  Optimization  Scheduling  Assessment

3. SMA HPC (c) NUS 15.094 Constraint Programming3 Review Constraint programming is a framework for integrating three families of algorithms  Propagation algorithms  Branching algorithms  Exploration algorithms

4. SMA HPC (c) NUS 15.094 Constraint Programming4 S E N D + M O R E = M O N E Y S  {9} E  {4..7} N  {5..8} D  {2..8} M  {1} O  {0} R  {2..8} Y  {2..8} S  {9} E  {5..7} N  {6..8} D  {2..8} M  {1} O  {0} R  {2..8} Y  {2..8} E = 4 E  4 S  {9} E  {6..7} N  {7..8} D  {2..8} M  {1} O  {0} R  {2..8} Y  {2..8} E = 5 E  5 S  {9} E  {5} N  {6} D  {7} M  {1} O  {0} R  {8} Y  {2} E = 6 E  6

5. SMA HPC (c) NUS 15.094 Constraint Programming5 Using OPL Syntax enum Letter {S,E,N,D,M,O,R,Y}; var int l[Letter] in 0..9; solve { alldifferent(l); l[S] <> 0; l[M] <> 0; l[S]*1000 + l[E]*100 + l[N]*10 + l[D] + l[M]*1000 + l[O]*100 + l[R]*10 + l[E] = l[M]*10000 + l[O]*1000 + l[N]*100 + l[E]*10 + l[Y] }; search { forall(i in Letter ordered by increasing dsize(l[i])) tryall(v in 0..9) l[i] = v; };  All Solutions; Execution  Run

6. SMA HPC (c) NUS 15.094 Constraint Programming6 Today  Optimization  Scheduling  Assessment

7. SMA HPC (c) NUS 15.094 Constraint Programming7 Optimization  Modeling: define optimization function  Propagation algorithms: identify propagation algorithms for optimization function  Branching algorithms: identify branching algorithms that lead to good solutions early  Exploration algorithms: extend existing exploration algorithms to achieve optimization

8. SMA HPC (c) NUS 15.094 Constraint Programming8 Optimization: Example SEND + MOST = MONEY

9. SMA HPC (c) NUS 15.094 Constraint Programming9 SEND + MOST = MONEY Assign distinct digits to the letters S, E, N, D, M, O, T, Y such that S E N D + M O S T = M O N E Y holds and M O N E Y is maximal.

10. SMA HPC (c) NUS 15.094 Constraint Programming10 Modeling Formalize the problem as a constraint optimization problem:  Number of variables: n  Constraints: c 1,…,c m   n  Optimization constraints: d 1,…,d m   n  2  n Given a solution a, and an optimization constraint d i, the constraint d i (a)   n contains only those assignments b for which b is better than a.

11. SMA HPC (c) NUS 15.094 Constraint Programming11 A Model for MONEY  number of variables: 8  constraints: c 1 = {(S,E,N,D,M,O,T,Y)   8 | 0  S,…,Y  9 } c 2 = {(S,E,N,D,M,O,T,Y)   8 | 1000*S + 100*E + 10*N + D + 1000*M + 100*O + 10*S + T = 10000*M + 1000*O + 100*N + 10*E + Y}

12. SMA HPC (c) NUS 15.094 Constraint Programming12 A Model for MONEY (continued)  more constraints c 3 = {(S,E,N,D,M,O,T,Y)   8 | S  0 } c 4 = {(S,E,N,D,M,O,T,Y)   8 | M  0 } c 5 = {(S,E,N,D,M,O,T,Y)   8 | S…Y all different }  optimization constraint: d : (s,e,n,d,m,o,t,y)  {(S,E,N,D,M,O,T,Y)   8 | 10000*m + 1000*o + 100*n + 10*e + y < 10000*M + 1000*O + 100*N + 10*E + Y }

13. SMA HPC (c) NUS 15.094 Constraint Programming13 Propagation Algorithms Identify a propagation algorithm to implement the optimization constraints Example: SEND + MOST = MONEY d : (s,e,n,d,m,o,t,y)  {(S,E,N,D,M,O,T,Y))   8   8 | 10000*m + 1000*o + 100*n + 10*e + y < 10000*M + 1000*O + 100*N + 10*E + Y } Given a solution a, choose propagation algorithm for d(a).

14. SMA HPC (c) NUS 15.094 Constraint Programming14 Branching Algorithms Identify a branching algorithm that finds good solutions early. Example: SEND + MOST = MONEY Idea: Naïve enumeration in the order M, O, N, E, Y. Try highest values first.

15. SMA HPC (c) NUS 15.094 Constraint Programming15 Exploration Algorithms Modify exploration such that for each solution a, corresponding optimization constraints are added. Two strategies:  branch-and-bound: continue as in original exploration  restart optimization: after each solution, start from the root.

16. SMA HPC (c) NUS 15.094 Constraint Programming16 Using OPL Syntax enum Letter {M,O,N,E,Y,S,D,T}; var int l[Letter] in 0..9; minimize l[M]*10000 + l[O]*1000 + l[N]*100 + l[E]*10 + l[Y] subject to { alldifferent(l); l[S] <> 0; l[M] <> 0; l[S]*1000 + l[E]*100 + l[N]*10 + l[D] + l[M]*1000 + l[O]*100 + l[R]*10 + l[E] = l[M]*10000 + l[O]*1000 + l[N]*100 + l[E]*10 + l[Y] }; search { forall(i in Letter) tryall(v in 0..9 ordered by decreasing v) l[i] = v; };  All Solutions; Execution  Run

17. SMA HPC (c) NUS 15.094 Constraint Programming17 Experiments with MONEY MONEY Search MONEY Optimization

18. SMA HPC (c) NUS 15.094 Constraint Programming18 Today  Optimization  Scheduling  Assessment

19. SMA HPC (c) NUS 15.094 Constraint Programming19 Scheduling  Scheduling Problems  Propagation Algorithms for Resource Constraints  Branching Algorithms  Other Constraints  Exploration Algorithms  Literature

20. SMA HPC (c) NUS 15.094 Constraint Programming20 Scheduling Problems Assign starting times (and sometimes durations) to tasks, subject to  resource constraints,  precedence constraints,  idiosyncratic and other constraints, and  an optimization function, typically to minimize overall schedule duration.

21. SMA HPC (c) NUS 15.094 Constraint Programming21 Example: Building a Bridge Show Problem  Resource constraints Example: a1 and a2 use excavator, cannot overlap in time  Precedence constraints Example: p1 requires a3, a3 must end before p1 starts Animate SolutionGantt Chart

22. SMA HPC (c) NUS 15.094 Constraint Programming22 Modeling  Indices: Task = {begin,a1,a2,a3,a4,a5,a6,p1,...,end}  Constants: duration of tasks duration = #[begin:0, a1:4, a2:2,...]#  Variables: represent each task with a finite domain variable representing its starting time. Activity a[t in Task](duration[t]) a[t].start is finite domain variable representing the starting time of a[t]

23. SMA HPC (c) NUS 15.094 Constraint Programming23 Precedence Constraints For each two tasks t1, t2, where t1 must precede t2, introduce a constraint a[t1].start + a[t1].duration  a[t2].start In OPL, just write a[t1] precedes a[t2]

24. SMA HPC (c) NUS 15.094 Constraint Programming24 Resource Constraints For each two tasks t1, t2 that require a unary resource r, we have the constraint a[t1].start + a[t1].duration  a[t2].start \/ a[t2].start + a[t2].duration  a[t1].start But: many constraints and weak propagation. Thus, introduce global resource constraints.

25. SMA HPC (c) NUS 15.094 Constraint Programming25 Scheduling  Scheduling Problems  Propagation Algorithms for Resource Constraints  Branching Algorithms  Other Constraints  Exploration Algorithms  Literature

26. SMA HPC (c) NUS 15.094 Constraint Programming26 Global Resource Constraints in OPL  Declare resource UnaryResource excavator;  Require resource a[a1] requires excavator; a[a2] requires excavator;...  All “ requires ” constraints on a resource together form a global resource constraint.

27. SMA HPC (c) NUS 15.094 Constraint Programming27 Propagation: Disjunctive Resource For all tasks t1, t2 using the same resource: a[t1].start + a[t1].duration  a[t2].start \/ a[t2].start + a[t2].duration  a[t1].start Weakest but most efficient form of propagation for unary resources.

28. SMA HPC (c) NUS 15.094 Constraint Programming28 Propagation: Edge finding For a given task t and set of tasks S, all sharing the same unary resource, find out whether t can occur  before all tasks in S,  after all tasks in S,  between two tasks in S, and infer corresponding basic constraints.

29. SMA HPC (c) NUS 15.094 Constraint Programming29 Propagation: Task Intervals Notation: est(t): earliest starting time lct(t): latest completion time For two tasks t1 and t2 where est(t1)  est(t2) and lct(t1)  lct(t2), the task interval I(t1,t2) is defined: I(t1,t2) = { t | est(t1)  est(t) and lct(t)  lct(t2) } Perform edge finding on all task intervals.

30. SMA HPC (c) NUS 15.094 Constraint Programming30 Task Intervals: Example I(A,B) = {A,B,C} I(C,D) = {B,C,D} A B C D

31. SMA HPC (c) NUS 15.094 Constraint Programming31 Which Edge Finder? As usual, trade-off between run-time of propagation algorithm and strength of propagation. Edge finding can be expensive depending on what tasks are considered. Recent edge finders have complexity n 2 for one propagation step, where n is the number of tasks using the resource. Edge finding based on task intervals can be stronger than these, but typically have complexity n 3.

32. SMA HPC (c) NUS 15.094 Constraint Programming32 Specifying Propagation Algorithms OPL fixes propagation for unary resources to an (undisclosed) edge-finding algorithm. For discrete (non-unary) resources, the user can choose between default, disjunctive and edgeFinder when declaring a resource: DiscreteResource crane(3) using edgeFinder;

33. SMA HPC (c) NUS 15.094 Constraint Programming33 Demo: Propagation for Unary Resources Bridge

34. SMA HPC (c) NUS 15.094 Constraint Programming34 Scheduling  Scheduling Problems  Propagation Algorithms for Resource Constraints  Branching Algorithms  Other Constraints  Exploration Algorithms  Literature

35. SMA HPC (c) NUS 15.094 Constraint Programming35 Branching Algorithms: Serializers  Simple enumeration techniques such as first-fail are hopeless for scheduling.  Use unary resources to guide branching.  For two tasks t1, t2 sharing the same resource, use the constraints t1.start + t1.duration  t2.start and t2.start + t2.duration  t1.start for branching.

36. SMA HPC (c) NUS 15.094 Constraint Programming36 Which Tasks To Serialize?  Resource-oriented serialization: Serialize all tasks of one resource completely, before tasks of another resource are serialized.  Most-used-resource serialization  Global slack  Local slack  Task-oriented serialization: Choose two suitable tasks at a time, regardless of resources.  Slack-based task serialization (see later)

37. SMA HPC (c) NUS 15.094 Constraint Programming37 Most-used-resource Serialization “Most-used-resource” serialization: serialize first the resource that is used the most. Let T be a set of tasks running on resource r. demand(T) =  t  T t.duration Let S be the set of all tasks using resource r. demand(r) := demand(S), Serialize the resource r with maximal demand(r) first.

38. SMA HPC (c) NUS 15.094 Constraint Programming38 Slack-based Resource Serialization Let T be a set of tasks running on resource r. supply(T) = lct(T) - est(T) demand(T) =  t  T t.duration slack(T) = supply(T) - demand(T) Let S be the set of all tasks using resource r. slack(r) := slack(S), S is set of all task running on r. Global slack serialization: Serialize resource with smallest slack first

39. SMA HPC (c) NUS 15.094 Constraint Programming39 Local-Slack Resource Serialization Let I r be all task intervals on r. The local slack is defined as min {slack(I) | I  I r } Local slack serialization: Serialize resource with smallest local slack first. Use global slack for tie-breaking.

40. SMA HPC (c) NUS 15.094 Constraint Programming40 Which Tasks Are Serialized? Ideas:  Use edge finding to look for possible first tasks (and last tasks)  Choose tasks according to their est, lst ( lct, ect ).

41. SMA HPC (c) NUS 15.094 Constraint Programming41 Task-oriented Serialization Among all tasks using all resources, select a pair of tasks according to local/global slack criteria and other considerations, regardless what resources are serialized already. Provides more fine-grained control at the expense of runtime for finding candidate task pairs among all tasks.

42. SMA HPC (c) NUS 15.094 Constraint Programming42 Programming Branching Algorithms in OPL  Scheduling-specific try constructs:  tryRankFirst(u,a): activity a comes first  tryRankLast(u,a): activity a comes last  rank(u): serialize all tasks using u  Reflective functions for unary resources:  isRanked(Unary): 1 if the resource is ranked  nbPossibleFirst(Unary): number of activities that can come first  globalSlack(Unary): global slack of u  localSlack(Unary): local slack of u

43. SMA HPC (c) NUS 15.094 Constraint Programming43 Example: Task-oriented Serialization in OPL while not isRanked(tool) do select(r in Resources: not isRanked(tool[r])) select(t in tasks[r] : not isRanked(tool[r],a[t]) tryRankFirst(tool[r],a[t]);

44. SMA HPC (c) NUS 15.094 Constraint Programming44 Demo: Serialization Algorithms Bridge

45. SMA HPC (c) NUS 15.094 Constraint Programming45 Scheduling  Scheduling Problems  Propagation Algorithms for Resource Constraints  Branching Algorithms  Other Constraints  Exploration Algorithms  Literature

46. SMA HPC (c) NUS 15.094 Constraint Programming46 Other Scheduling Constraints  Discrete resources DiscreteResource crane(3); a requires(2) crane; a consumes(1) crane;  Reservoirs Reservoir plumbing(3,1); a requires(2) plumbing; a consumes(2) plumbing; a provides(2) plumbing; a produces(2) plumbing;

47. SMA HPC (c) NUS 15.094 Constraint Programming47 Exploration Algorithms  Usually: branch-and-bound using minimization of the starting time of an “ end ” task minimize a[end].start subject to {...}  Lower-bounding: Prove the non-existence of a solution with a[end].start  n, add constraint a[end].start > n ; increase n by .  Upper-bounding: Find solution with a[end].start = n, add constraint a[end].start<n ; decrease n by .

48. SMA HPC (c) NUS 15.094 Constraint Programming48 Literature  Van Hentenryck: The OPL optimization programming language, 1999  Constraint Programming Tutorial of Mozart, 2000 (www.mozart-oz.org)  Applegate, Cook: A computational study of the job-shop scheduling problem, 1991  Carlier, Pinson: An algorithm for solving the job- shop scheduling problem, 1989  Various papers by Laburthe, Caseau, Baptiste, Le Pape, Nuijten, see “Overview” paper

49. SMA HPC (c) NUS 15.094 Constraint Programming49 Today  Optimization  Scheduling  Assessment

50. SMA HPC (c) NUS 15.094 Constraint Programming50 Assessment: Don’t Use It! Don’t use constraint programming for:  Problems for which there are known efficient algorithms or heuristics. Example: Traveling salesman.  Problems for which integer programming works well. Example: Many discrete assignment problems.  Problems with weak constraints and a complex optimization function. Example: Timetabling problems.

51. SMA HPC (c) NUS 15.094 Constraint Programming51 Assessment: Do Use It! Use constraint programming for:  Problems for which integer programming does not work (linear models too large).  Problems for which there are no efficient solutions available.  Problems with tight constraints, where propagation can be employed. Example: ACC 97/98.  Problems for which strong branching algorithms exist. Example: Scheduling with unary resources.

52. SMA HPC (c) NUS 15.094 Constraint Programming52 Myths Debunked  A fad! Constraint programming has been used successfully in a number of application areas, most spectacularly in scheduling  Universal! More failure stories than success stories.  All new! Many ideas come from AI search and Operations Research. In particular, the important ideas for scheduling come from OR.  Artificial Intelligence! All quite earthly algorithms. Constraint programming systems provide framework for different kinds of algorithms to interact.

53. SMA HPC (c) NUS 15.094 Constraint Programming53 The Future  Constraint programming will be added in OR as a standard technique for solving combinatorial problems, along with local search.  Constraint programming techniques will be tightly integrated with integer programming and local search.  Look at the Workshop on Integration of AI and OR Techniques for Combinatorial Optimization Problems (www.uni-paderborn.de/CP-AI-OR)