Recent Developments in Integrated Methods for Optimization

Transcription

1 Recent Developments in Integrated Methods for Optimization John Hooker Carnegie Mellon University August 2012

2 Premise Constraint programming and mathematical programming can work together. There is underlying unity. Result: faster solution and more elegant models. August 2012 Slide 2

3 Premise Constraint programming and mathematical programming can work together. There is underlying unity. Result: faster solution and more elegant models. Math programming solvers constantly improve But they would reach a much higher level if this same effort were applied to integrated methods. Solvers are beginning to move in this direction. August 2012 Slide 3

4 Premise Constraint programming and mathematical programming can work together. There is underlying unity. Result: faster solution and more elegant models. Math programming solvers constantly improve But they would reach a much higher level if this same effort were applied to integrated methods. Solvers are beginning to move in this direction. The same integration principles apply more generally. Global optimization, exact/heuristic methods. August 2012 Slide 4

5 Outline What is constraint programming? Integrating OR and CP Constraint propagation + relaxation CP-based branch and price Decomposition Methods Software Some very recent work Caveat: This is a high-level overview. Don t worry about the technical details. August 2012 Slide 5

6 What Is Constraint Programming? Basic Idea Applications Examples CP and Math Programming Software

7 Basic Idea Each line of the model is both a constraint and a procedure. Constraint: often a high-level global constraint Different modeling paradigm than math programming Procedure: removes infeasible values from variable domains Filtering, domain consistency maintenance Passes reduced domains to next constraint (constraint propagation). August 2012 Slide 7

8 Early commercial successes Circuit design (Siemens) Container port scheduling (Hong Kong and Singapore) Real-time control (Siemens, Xerox) August 2012 Slide 8

9 Applications Job shop scheduling Assembly line smoothing and balancing Cellular frequency assignment Nurse scheduling Shift planning Maintenance planning Airline crew rostering and scheduling Airport gate allocation and stand planning August 2012 Slide 9

10 Applications Production scheduling chemicals aviation oil refining steel lumber photographic plates tires Transport scheduling (food, nuclear fuel) Warehouse management Course timetabling August 2012 Slide 10

11 Example: Employee scheduling Schedule four nurses in 8-hour shifts. A nurse works at most one shift a day, at least 5 days a week. Same schedule every week. No shift staffed by more than two different nurses in a week. A nurse cannot work different shifts on two consecutive days. A nurse who works shift 2 or 3 must do so at least two days in a row. August 2012 Slide 11

12 Two ways to view the problem Assign nurses to shifts Sun Mon Tue Wed Thu Fri Sat Shift 1 A B A A A A A Shift 2 C C C B B B B Shift 3 D D D D C C D Assign shifts to nurses Sun Mon Tue Wed Thu Fri Sat Nurse A Nurse B Nurse C Nurse D August 2012 Slide 12 0 = day off

13 Use both formulations in the same model! First, assign nurses to shifts. Let w sd = nurse assigned to shift s on day d ( ) alldiff w, w, w, all d 1d 2d 3d Schedule 3 different nurses on each day August 2012 Slide 13

14 Use both formulations in the same model! First, assign nurses to shifts. Let w sd = nurse assigned to shift s on day d ( w1 d w2d w3d ) d ( w A B C D ) alldiff,,, all cardinality (,,, ),(5,5,5,5),(6,6,6,6) Each nurse works at least 5 and at most 6 days a week August 2012 Slide 14

15 Use both formulations in the same model! First, assign nurses to shifts. Let w sd = nurse assigned to shift s on day d ( w1 d w2d w3d ) d ( w A B C D ) alldiff,,, all cardinality (,,, ),(5,5,5,5),(6,6,6,6) ( ) nvalues w,..., w 1,2, all s s,sun s,sat At least 1 and at most 2 nurses work any given shift. August 2012 Slide 15

16 Remaining constraints are not easily expressed in this notation. So, assign shifts to nurses. Let y id = shift assigned to nurse i on day d alldiff (,, ) y y y, all d 1d 2d 3d Assign a different nurse to each shift on each day. Redundant, but speeds solution. August 2012 Slide 16

17 Remaining constraints are not easily expressed in this notation. So, assign shifts to nurses. Let y id = shift assigned to nurse i on day d alldiff ( y, y, y ) 1d 2d 3d ( y y P ) i,sun, all stretch,, (2,3),(2,2),(6,6),, all i,sat d i Shift 2 or 3 must be worked at least 2 days in a row. August 2012 Slide 17

18 Remaining constraints are not easily expressed in this notation. So, assign shifts to nurses. Let y id = shift assigned to nurse i on day d alldiff ( y, y, y ) 1d 2d 3d ( y y P ) i,sun, all stretch,, (2,3),(2,2),(6,6),, all i,sat d i Pattern constraint: P = {(s,0),(0,s) s = 1,2,3} Don t switch shifts without taking at least one day off. August 2012 Slide 18

19 Connect the w sd variables to the y id variables. Use channeling constraints: w y y w id sd d d = = i, all i, d s, all s, d Channeling constraints increase propagation and make the problem easier to solve. August 2012 Slide 19

20 The complete model is: ( w1 d w2d w3d ) d ( w A B C D ) alldiff,,, all cardinality (,,, ),(5,5,5,5),(6,6,6,6) ( ) nvalues w,..., w 1,2, all s alldiff s,sun ( y, y, y ) 1d 2d 3d s,sat ( y y P ) i,sun, all stretch,, (2,3),(2,2),(6,6),, all i,sat d i w y y w id sd d d = = i, all i, d s, all s, d August 2012 Slide 20

21 Example: Resource-Constrained Scheduling Use the cumulative scheduling constraint. Total resources consumed by jobs at any one time must not exceed L. ( t t p p c c C) cumulative (,, ),(,, ),(,, ), 1 n 1 n 1 n Job start times (variables) Job processing times Job resource requirements August 2012 Slide 21

22 Ship loading The problem Load 34 items on the ship in minimum time (min makespan) Each item i requires p i minutes and c i workers. Total of 8 workers available. From IBM OPL Studio manual August 2012 Slide 22

23 Precedence constraints 1 2, , , , ,30,31, August 2012 Slide 23

24 Use the cumulative scheduling constraint. min z s.t. z t + p, i = 1,,34 i i ( t t p p c c ) cumulative (,, ),(,, ),(,, ), t t + 3, t t + 3, etc. (precedence constraints) Note that there are no integer variables. The solver may treat the t i s as integers, but this is an implementation decision. August 2012 Slide 24

25 Edge finding for cumulative scheduling Consider a cumulative scheduling constraint: ( s s s p p p c c c C) cumulative (,, ),(,, ),(,, ), A feasible solution: August 2012 Slide 25

26 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } Because the total energy required exceeds the area between the earliest release time and the later deadline of jobs 1,2: ( ) e + e > C L E 3 {1,2} {1,2} {1,2,3} August 2012 Slide 26

27 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } Because the total energy required exceeds the area between the earliest release time and the later deadline of jobs 1,2: ( ) e + e > C L E 3 {1,2} {1,2} {1,2,3} Total energy required = August 2012 Slide 27

28 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } Because the total energy required exceeds the area between the earliest release time and the later deadline of jobs 1,2: ( ) e + e > C L E 3 {1,2} {1,2} {1,2,3} Total energy required = 22 Area available = August 2012 Slide 28

29 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } We can update the release time of job 3 to E {1,2} + e ( C c )( L E ) J 3 {1,2} {1,2} c 3 Energy available for jobs 1,2 if space is left for job 3 to start anytime = August 2012 Slide 29

30 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } We can update the release time of job 3 to E {1,2} + e ( C c )( L E ) J 3 {1,2} {1,2} c 3 Energy available for jobs 1,2 if space is left for job 3 to start anytime = 10 Excess energy required by jobs 1,2 = 4 August 2012 Slide

31 Edge finding for cumulative scheduling We can deduce that job 3 must finish after the others finish: 3 > { 1,2 } We can update the release time of job 3 to E {1,2} + e ( C c )( L E ) J 3 {1,2} {1,2} c 3 Energy available for jobs 1,2 if space is left for job 3 to start anytime = 10 Excess energy required by jobs 1,2 = 4 August 2012 Slide 31 4 Move up job 3 release time 4/2 = 2 units beyond E 10 {1,2} E 3

32 Edge finding for cumulative scheduling In general, if e ( ) J { k } > C LJ EJ { k } then k > J, and update E k to max J J e ( C c )( L E ) > 0 J k J J E J ej ( C ck )( LJ EJ ) + ck In general, if e ( ) J { k } > C LJ { k } EJ then k < J, and update L k to min J J e ( C c )( L E ) > 0 J k J J L J ej ( C ck )( LJ EJ ) ck August 2012 Slide 32

33 Edge finding for cumulative scheduling There is an O(n 2 ) algorithm that finds all applications of the edge finding rules. August 2012 Slide 33

34 Other propagation rules for cumulative scheduling Extended edge finding. Timetabling. Not-first/not-last rules. Energetic reasoning. August 2012 Slide 34

35 CP and Mathematical Programming Comparison CP Logic processing Inference (filtering, constraint propagation) High-level modeling (global constraints) Branching Constraint-based processing MP Numerical calculation Relaxation Atomistic modeling (linear inequalities) Branching Independence of model and algorithm August 2012 Slide 35

36 Software for Constraint Programming ECLiPSe (NICTA), open source Early CP (and hybrid) solver, still maintained CHIP (Cosytec), commercial State-of-the-art solver OPL CP Optimizer (IBM), commercial State-of-the-art solver, originally developed by ILOG Gecode (Schulte & Tack), free download State-of-the-art toolkit for building CP solvers Frontline MIP/CP solver (Frontline Systems), commercial Add-in for Excel spreadsheets G12 (NICTA), under development Major CP and hybrid system Google OR-tools (Google), open source Includes CP solver August 2012 Slide 36

37 Integrating OR and CP Complementary strengths How to integrate Constraint propagation + relaxation CP-based branch and price Decomposition methods

38 Complementary Strengths CP: Inference methods Modeling Exploits local structure Good at scheduling OR: Relaxation methods Duality theory More robust Good with continuous variables Let s bring them together! August 2012 Slide 38

39 How to Integrate Constraint propagation + relaxation Propagation reduces search space. Relaxation bounds prune the search CP-based column generation In branch-and-price methods CP accommodates complex constraints on columns Decomposition methods Distinguish master problem and subproblem MILP solves one, CP the other. Use CP-style modeling August 2012 Slide 39

40 Constraint propagation + relaxation Combine MIP-style bounding with CP-style propagation. At each node of search tree: Solve linear relaxation to get bound, as in MIP. Filter variable domains and propagate constraints, as in CP. August 2012 Slide 40

41 Search tree for an IP problem min 90x + 60x + 50x + 40x 7x + 5x + 4x + 3x 42 x + x + x + x 8 x x x x 1 i { 1,2 } + x 2 + x 2 + x , 3, x 1 3 CP Summer School June 2011 Slide 41 1 Knapsack cuts x 1 { } x 2 { } x 3 { } x 4 { } infeasible x 1 { 123} x 2 {0123} x 3 {0123} x 4 {0123} x = (2⅓,3,2⅔,0) value = 523⅓ x 1 {1,2} x 1 { 3} x 2 {012 } x 3 { 123} x 4 {0123} x = (3,2,2¾,0) value = 527½ x 1 { 3} x 3 {1,2} x 2 { 12 } x 3 { 12 } x 4 { 123} x = (3,2,2,1) value = 530 feasible solution x 1 = 3 x 3 = 3 x 1 { 3} x 2 {0123} x 3 {0123} x 4 {0123} x = (3,2.6,2,0) value = 526 x 2 = 3 x 2 {0,1,2} x 1 { 3} x 2 {012 } x 3 { 3} x 4 {012 } x = (3,1½,3,½) value = 530 backtrack due to bound x 1 { 3} x 2 { 3} x 3 {012 } x 4 {012 } x = (3,3,0,2) value = 530 feasible solution

42 Search tree for an IP problem min 90x + 60x + 50x + 40x 7x + 5x + 4x + 3x 42 x + x + x + x 8 x x x x 1 i { 1,2 } + x 2 + x 2 + x , 3, x 1 3 CP Summer School June 2011 Slide 42 1 Knapsack cuts x 1 { } x 2 { } x 3 { } x 4 { } infeasible Prune tree due to propagation alone. x 1 { 123} x 2 {0123} x 3 {0123} x 4 {0123} x = (2⅓,3,2⅔,0) value = 523⅓ x 1 {1,2} x 1 { 3} x 2 {012 } x 3 { 123} x 4 {0123} x = (3,2,2¾,0) value = 527½ x 1 { 3} x 3 {1,2} x 2 { 12 } x 3 { 12 } x 4 { 123} x = (3,2,2,1) value = 530 feasible solution x 1 = 3 x 3 = 3 x 1 { 3} x 2 {0123} x 3 {0123} x 4 {0123} x = (3,2.6,2,0) value = 526 x 2 = 3 x 2 {0,1,2} x 1 { 3} x 2 {012 } x 3 { 3} x 4 {012 } x = (3,1½,3,½) value = 530 backtrack due to bound Filtered domains x 1 { 3} x 2 { 3} x 3 {012 } x 4 {012 } x = (3,3,0,2) value = 530 feasible solution Relaxation bound

43 Constraint propagation + relaxation Example: Relaxation of cumulative constraint Use discrete-time MILP model Total resource consumption at time t z = z 1 + c x c x,, all t t t j jt j j t p j j j t jt z C, all t t x 0 x 1, all j, t jt = 1, all j Add this to linear relaxation of problem while propagating the cumulative constraint. 0-1 variables x jt appear only in the relaxation = 1 when job j starts at time t August 2012 Slide 43

44 Constraint propagation + relaxation Example: Relaxation of cumulative constraint Or use valid inequalities in original variables t j Earliest release time in set J k k 1 t kr + ( k i + 1) p c p j J j j j j J C i = 1 i = 1 k 1 t kr ( k i + 1) p c j J ji j j J C i = 1 { } i i i where J = i 1,, i k is any subset of jobs. i August 2012 Slide 44

45 Constraint propagation + relaxation Example: Piecewise linear functions Use convex hull relaxation in original variables. Add these cuts to linear relaxation Reduced by bounds propagation August 2012 Slide 45 No need for 0-1 variables

46 Constraint propagation + relaxation Example: Piecewise linear functions Branch on original variables & tighten relaxation. Branch here August 2012 Slide 46 Solution value of x in current linear relaxation

47 Computational Advantage of Integrating CP and OR Using CP + relaxation from OR Problem Lesson timetabling (LP + red. cost var. fixing) Piecewise linear costs (LP relaxation) Flow shop scheduling, etc. (LP relaxation + cuts) Speedup 2 to 50 times faster than CP 2 to 120 times faster than MILP 4 to 150 times faster than MILP. Focacci, Lodi, Milano (1999) Refalo (1999), Yunes, Aron & Hooker (2010) Hooker & Osorio (1999) August 2012 Slide 47

48 Computational Advantage of Integrating CP and OR Using CP + relaxation from OR Problem Product configuration (LP relaxation + cuts) Automatic recording (Lagrangean relaxation) Stable set problem (semidefinite relaxation) Speedup 30 to 40 times faster than CP, MILP 1 to 10 times faster than CP, MILP Better than CP in less time Thorsteinsson & Ottosson (2001) Sellmann & Fahle (2001) Van Hoeve (2001) August 2012 Slide 48

49 Computational Advantage of Integrating CP and OR Using CP + relaxation from OR Problem Structural design (nonlinear) (LP quasi-relaxation + logic cuts) Radiation therapy planning (Lagrangean relaxation) Speedup Up to 600 times faster than MILP. 10 times faster than CP, MILP Bollapragada, Ghattas & Hooker (2001) Cambazard, O Mahony and O Sullivan (2010) August 2012 Slide 49

50 Applications of Integrated CP and OR Using CP + relaxation from OR Application Orthogonal Latin squares Appa, Magos & Mourtos (2002) Truss structure design Bollapragada, Gattas & Hooker (2001) Chemical processing network design Vehicle routing Sadykov (2004) Grossmann et al. (1994), Hooker & Osorio (1999) Resource-constrained scheduling Demassy, Artiques & Michelon (2002), Berthold et al. (2010) Multiple machine scheduling Bockmayr & Pisaruk (2003) Shuttle transit routing Quadrifoglio, Dessouky & Ordóñez (2008) Boat party scheduling Hooker & Osorio (1999) August 2012 Slide 50

51 Applications of Integrated CP and OR Using CP + relaxation from OR Application Multidimensional knapsack problem Osorio & Glover (2001) Factory retrofit planning Sawaya & Grossmann (2005) Strip packing Sawaya & Grossmann (2005) Chemical process engineering Transport & production problems with piecewise linear costs Lee & Grossmann (2003), Vecchietti et al (2001), Swaya & Grossmann (2007) Refalo (1999), Ottosson et al. (1999) TSP with time windows Milano & van Hoeve (2002) Product configuration Milano & van Hoeve (2002) August 2012 Slide 51

52 Applications of Integrated CP and OR Using CP + relaxation from OR Application Network design Cronholm & Ajili (2004) Automatic digital recording Sellmann & Fahle (2001) Traveling tournament problem Resource-constrained shortest path problem Radiation therapy planning CP domain filtering Benoist, Laburthe & Rottembourg (2001) Gellermann, Sellmann & Wright (2005) Cambazard, O Mahony & O Sullivan (2010) Khemmoudj, Bennaceur & Nagih (2005) August 2012 Slide 52

53 CP-Based Branch and Price Same as traditional branch and price Except that CP generates the columns. Introduce an integer variable for each combinatorial possibility. Removes most or all of symmetry. Filter variable domains and propagate constraints, as in CP. At each node of MIP search tree: Solve linear relaxation using column generation. Generate improving columns with CP (pricing problem). Start with these columns at child nodes. August 2012 Slide 53

54 Airline Crew Scheduling Assign crew members to flights to minimize cost while covering the flights and observing complex work rules. Flight data Start time Finish time A roster is the sequence of flights assigned to a single crew member. The gap between two consecutive flights in a roster must be from 2 to 3 hours. Total flight time for a roster must be between 6 and 10 hours. The possible rosters are: (1,3,5), (1,4,6), (2,3,5), (2,4,6) August 2012 Slide 54

55 Airline Crew Scheduling There are 2 crew members, and the possible rosters are: (1,3,5), (1,4,6), (2,3,5), (2,4,6) The LP relaxation of the problem is: Cost of assigning crew member 1 to roster 2 = 1 if we assign crew member 1 to roster 2, = 0 otherwise. Each crew member is assigned to exactly 1 roster. Each flight is assigned at least 1 crew member. August 2012 Slide 55

56 Airline Crew Scheduling There are 2 crew members, and the possible rosters are: (1,3,5), (1,4,6), (2,3,5), (2,4,6) The LP relaxation of the problem is: Cost of assigning crew member 1 to roster 2 = 1 if we assign crew member 1 to roster 2, = 0 otherwise. Each crew member is assigned to exactly 1 roster. Each flight is assigned at least 1 crew member. August 2012 Slide 56 Rosters that cover flight 1.

57 Airline Crew Scheduling There are 2 crew members, and the possible rosters are: (1,3,5), (1,4,6), (2,3,5), (2,4,6) The LP relaxation of the problem is: Cost c 12 of assigning crew member 1 to roster 2 = 1 if we assign crew member 1 to roster 2, = 0 otherwise. Each crew member is assigned to exactly 1 roster. Each flight is assigned at least 1 crew member. August 2012 Slide 57 In a real problem, there can be millions of rosters.

58 Airline Crew Scheduling We start by solving the problem with a subset of the columns: Optimal dual solution u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6 August 2012 Slide 58

59 Airline Crew Scheduling We start by solving the problem with a subset of the columns: Dual variables u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6 August 2012 Slide 59

60 Airline Crew Scheduling We start by solving the problem with a subset of the columns: Dual variables u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6 The reduced cost of an excluded roster k for crew member i is c u v ik i j j in roster k We will formulate the pricing problem as a shortest path problem. August 2012 Slide 60

61 Pricing problem Crew member 1 Crew member 2 August 2012 Slide 61

62 Pricing problem Each s-t path corresponds to a roster, provided the flight time is within bounds. Crew member 1 Crew member 2 August 2012 Slide 62

63 Pricing problem Cost of flight 3 if it immediately follows flight 1, offset by dual multiplier for flight 1 Crew member 1 Crew member 2 August 2012 Slide 63

64 Pricing problem Cost of transferring from home to flight 1, offset by dual multiplier for crew member 1 Crew member 1 Dual multiplier omitted to break symmetry Crew member 2 August 2012 Slide 64

65 Pricing problem Length of a path is reduced cost of the corresponding roster. Crew member 1 Crew member 2 August 2012 Slide 65

66 Pricing problem 10 Arc lengths using dual solution of LP relaxation Crew member Crew member August 2012 Slide 66

67 Crew member 1 Reduced cost = 1 Add x 12 to problem. Crew member 2 Reduced cost = 2 Add x 23 to problem. August 2012 Slide Pricing problem Solution of shortest path problems After x 12 and x 23 are added to the problem, no remaining variable has negative reduced cost. 2 2

68 Pricing problem The shortest path problem cannot be solved by traditional shortest path algorithms, due to the bounds on total duration of flights. It can be solved by CP: Set of flights assigned to crew member i Path length Graph Path global constraint Setsum global constraint Path( X, z, G), all flights i min i j X i { } i T f s T max X flights, z < 0, all i i ( j j ) i Duration of flight j August 2012 Slide 68

69 Process Scheduling Assign batches to processing-unit/start-time combinations while observing time constraints. A schedule is the sequence of unit/start-time combinations assigned to a batch. The time gap between two consecutive processes in a schedule is bounded. Total time for a schedule is bounded. August 2012 Slide 69

70 Process Scheduling The integer programming model is = 1 if we assign batch i to schedule j = 1 if schedule j includes unit/start-time k min j ij x ij ij x = 1, all i ( u ) ij c x a x 1, all k ( v ) {0,1}, all i, j ij ij jk ij k i Cost of assigning batch i to schedule j Each batch is assigned to exactly 1 schedule. Each unit/start-time is assigned at most one batch. August 2012 Slide 70

71 Pricing problem Each s-t path corresponds to a schedule, provided the total processing time is within bounds. Batch 1 Unit/start-time combination Batch 2 August 2012 Slide 71 Everything else is the same as before, except network generation.

72 Computational Advantage of Integrating CP and OR Using CP-based Branch and Price Problem Urban transit crew scheduling Speedup Schedules 210 trips, vs. 120 for traditional branch and price Yunes, Moura & de Souza (1999) Airline crew rostering Traveling tournament scheduling Incorporates complicated work rules First to solve 8-team instance Fahle et al. (2002) Easton, Nemhauser & Trick (2002) August 2012 Slide 72

73 Applications of Integrated CP and OR Using CP-based branch and price Application Airline crew rostering Junker et al. (1999), Kohl (2000), Chabrier (2000), Fahle et al. (2002), Sellmann et al. (2002) Aircraft scheduling Grönqvist (2003) Bus crew scheduling Yunes, Moura & de Souza (2005) Vehicle routing Rousseau, Gendreau & Pesant (2002) Network design Chabrier (2003) Employee timetabling Demassey, Pesant & Rousseau (2005) Physician scheduling Gendron, Lebbah & Pesant (2005) Radiation therapy planning Cambazard, O Mahony & O Sullivan (2010) Traveling tournament problem Easton, Nemhauser & Trick (2002), Trick & Yildiz (2007) August 2012 Slide 73

74 Decomposition Methods Can be applied to planning and scheduling problems. MILP solves planning (master) problem. CP solves scheduling subproblem. Link master and subproblem with logic-based Benders cuts. August 2012 Slide 74

75 Facility assignment and scheduling Assign jobs to facilities, and schedule them, to minimize makespan. Master problem assigns jobs, using MILP. Subproblem decomposes into a separate scheduling problem for each facility, solved by CP. Start time of job j min M M t + p, all j j x j r t d p, all j j j j x j j j ( j j = ij j = ij j = ) cumulative ( t x i),( p x i),( c x i), all i August 2012 Slide 75 Facility assigned to job j

76 Facility assignment and scheduling Subproblem for each facility i, given an assignment x from master min M M t + p, all j j x j r t d p, all j j j j x j j j ( t j x j = i pij x j = i cij x j = i ) cumulative ( ),( ),( ) Sample Benders cut (all release times the same): Min makespan on facility i in iteration k M M p (1 y ) + max d min d ik ik ik { } { } ik ij ij j j j J j J j J =1 if job j assigned to facility i (x j = i ) Deadline for job j Set of jobs assigned to facility i in iteration k

77 Facility assignment and scheduling The master problem is min M M Mik pij (1 yij ) + max{ d j} min { d j}, all i, k j J ik j Jik j J ik Relaxation of subproblem y ij {0,1} Benders cuts August 2012 Slide 77

78 Computational Advantage of Integrating CP and OR Using CP/MILP Benders methods Problem Min-cost planning & scheduing Min-cost planning & scheduling Polypropylene batch scheduling at BASF Speedup 20 to 1000 times faster than CP, MILP Solved some MIP-intractable instances in <1 sec Solved previously insoluble problem in 10 min Jain & Grossmann (2001), Thorsteinsson (2001) Yunes, Aron & Hooker (2010) Timpe (2002) August 2012 Slide 78

79 Computational Advantage of Integrating CP and OR Using CP/MILP Benders methods Problem Call center scheduling Min-cost, min-makespan planning & cumulative scheduling Min tardiness planning & cumulative scheduling Speedup Solved twice as many instances as traditional Benders times faster than CP, MILP times faster than CP, MILP Benoist, Gaudin, Rottembourg (2002) Hooker (2004) Hooker (2005) August 2012 Slide 79

80 Computational Advantage of Integrating CP and OR Using CP/MILP Benders methods Problem Sports scheduling Single-facility scheduling Speedup Several orders of magnitude speedup vs MILP Schedules several times as many jobs as MILP. Faster and/or more robust than CP. Rasmussen & Trick (2007) Coban & Hooker (2012) August 2012 Slide 80

81 Applications of Integrated CP and OR Using CP/MILP Benders methods Application Logic circuit verification Hooker & Yan (1995) Propositional satisfiability Planning & scheduling Hooker (2000), Hooker & Ottosson (2003) Jain & Grossmann (2001), Hooker (2000,2004,2005), Harjunkoski & Grossmann (2002), Chu & Xia (2005) Dispatch of automated vehicles Corréa, Langevin & Rousseau (2004) Steel production scheduling Harjunkoski & Grossmann (2001) Chemical batch scheduling Computer processor scheduling Timpe (2002), Maravelias & Grossmann (2004) Cambazard et al. (2004), Benini et al. (2005,2008), August 2012 Slide 81

82 Applications of Integrated CP and OR Using CP/MILP Benders methods Application Location-allocation Fazel-Zarandi & Beck (2009) Traffic diversion Xia, Eremin & Wallace (2004) Transport network design Peterson & Trick (2009) Queuing design & control Terekhov, Beck & Brown (2007) Sports scheduling Rasmussen & Trick (2007), Cheung (2009) August 2012 Slide 82

83 Software for Integrated Methods ECLiPSe (NICTA), open source Exchanges information between ECLiPSEe solver, Xpress-MP IBM OPL Studio (IBM), commercial Combines CPLEX and ILOG CP Optimizer with script language Mosel (FICO), commercial Combines Xpress-MP, Xpress-Kalis with low-level modeling SIMPL (CMU), free download Full integration with high-level modeling (prototype) SCIP (ZIB), free download, commercial users asked to buy license Combines MILP and CP-based propagation G12 (NICTA), under development Converts generic model to one that invokes cooperating solvers BARON (global optimization), commercial Combines convexification and interval propagation August 2012 Slide 83

84 Some Very Recent Work Benders for scheduling Cutting planes from CP model BDDs as constraint store BDDs for relaxation bounds

85 Recent work Benders for Scheduling Joint work with Elvin Coban. Apply logic-based Benders to single-facility scheduling with long time horizons and many jobs. Decompose the problem by assigning jobs to segments of time horizon. Segmented problem Jobs cannot cross segment boundaries (e.g., weekends). Unsegmented problem Jobs can cross segment boundaries. August 2012 Slide 85

86 Recent work Benders for Scheduling Segmented instances, tight time windows August 2012 Slide 86

87 Recent work Benders for Scheduling Segmented instances, wide time windows August 2012 Slide 87

88 Recent work Benders for Scheduling Unsegmented instances CP solves it quickly (< 1 sec) or blows up, in which case Benders solves it in 6 seconds (average). So: try CP for 1 sec, then switch to Benders August 2012 Slide 88

89 Recent work Cutting Planes from CP Model Joint work with David Bergman. Polyhedral analysis of overlapping all-different constraints (equivalent to graph coloring). Used in many scheduling problems, sudoku puzzles, etc. etc. Derive cutting planes from CP alldiff formulation and map them into 0-1 model. Provides tighter bounds than all CPLEX cuts in a small fraction of the time (e.g., 1%). August 2012 Slide 89

90 Recent work BDDs as Constraint Store Joint work with Henrik Andersen, David Bergman, Andre Cire, Tarik Hadzic, Willem van Hoeve, Barry O Sullivan, Peter Tiedemann Replace variable domains in CP with relaxed binary decision diagrams (BDDs). BDDs have long been used for circuit design, configuration, etc. We use them to represent relaxation of feasible set. Replace domain filtering with BDD-based propagation. Reduces search tree for multiple alldiffs from 1 million nodes to 1 node, time speedup factor of 100. Speedups on other problems. Now being incorporated into Google CP solver. August 2012 Slide 90

91 Recent work BDDs for Relaxation Bounding Joint work with David Bergman, Andre Cire, Willem van Hoeve Replace LP relaxation with a relaxed binary decision diagram (BDD). Shortest path in BDD provides a lower bound on optimal value. For most instances of independent set problem, we get tighter bounds than full cutting plane technology in CPLEX. Bound is normally obtained in very small fraction of the time. August 2012 Slide 91

92 Further Reading J. N. Hooker, Integrated Methods for Optimization, 2 nd ed., Springer, M. Milano and P. van Hentenryck, eds., Hybrid Optimization: The Ten Years of CPAIOR, Springer, J. N. Hooker, Operations research methods in constraint programming, in F. Rossi, P. van Beek and T. Walsh, eds., Handbook of Constraint Programming, Elsevier, August 2012 Slide 92