Constraint Programming, Planning and Scheduling with Time and Resource Constraints

1
Constraint Programming, Planning and Scheduling
with Time and Resource Constraints

• Claude Le Pape - ILOG S.A.
• Disclaimer not (at all) a complete overview of
  the field!

2
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving

3
Constraint Programming

• Explicit problem definition
• Separation between problem definition and problem
  solving
• Systematic deduction of the consequences of made
  decisions (constraint propagation)
• Partial constraint propagation è development of
  heuristic search algorithms to generate and
  optimize solutions
• Incremental constraint propagation (with global
  fix point semantics)
• Localized definition of the constraint
  propagation process (each constraint propagates
  independently of other constraints) è many
  different types of constraints viewed as as many
  sub-problems

4
Constraint Programming
New constraints (decisions)
Problem definition
Decision-making (and retracting)
Initial constraints Dynamic changes
Constraint propagation
Deduced constraints Contradictions
5
Example Propagation
Constraints x - y 1y lt z x ! z
Variables x in 1..3y in 1..3z in 1..3
6
Example Decision-Making
Constraints x - y 1 y lt z x ! z
Variables x in 2..3y in 1..2z in 2..3
7
Incrementality Principle (1)
8
Incrementality Principle (2)
0
5
10
15
20
Masonry
Plumbing
Carpentry
R.
W.
Ceilings
Facade
Paint.
9
Incrementality Principle (3)

• "Plumber" "Roofer"
• Solution Order "Plumbing" and "Roofing"
• Heuristic choice
• For example, "Plumbing" before "Roofing"

10
Incrementality Principle (4)
11
Incrementality Principle (5)
0
5
10
15
20
Masonry
Plumbing
Carpentry
R.
W.
Ceilings
Facade
Paint.
12
Incrementality Principle (6)
0
5
10
15
20
Masonry
Plumbing
Carpentry
R.
Ceilings
Facade
Paint.
M.
13
Incrementality Principle (7)
0
5
10
15
20
Masonry
Plumbing
Carpentry
R.
Ceilings
Facade
Paint.
14
Incrementality Principle (8)

• Constraint propagation consists in
• incrementally
• updating characteristics of a partial problem
  solution
• when an additional constraint is added
• These characteristics may be
• indicators of contradictions
• domains of variables (relational propagation)
• the overall set of constraints (logic/algebraic
  propagation)

15
Incrementality Principle (9)

• Particular case arc-consistency
• A constraint propagation technique enforces
  arc-consistency if and only if when propagation
  stops the following statement holds
• for every constraint c(v1 ... vn)
• for every variable vi
• for every value vali in the domain of vi
• there are values val1 ... vali-1 vali1 ... valn
• in the domains of v1 ... vi-1 vi1 ... vn
• such that val1 ... vali-1 vali vali1 ... valn
  satisfy c
• Considering a constraint as a sub-problem,
  arc-consistency consists of removing all the
  values that do not belong to any solution of the
  sub-problem

16
Locality Principle (1)

• Each constraint (or constraint type) "includes"
  all the information necessary to enable its
  propagation and, in particular, to determine
  whether it is satisfied or not as soon as all its
  variables are instantiated
• The constraint propagation methods associated
  with a constraint (or constraint type) are a
  priori independent of the methods
  associated with other constraints

17
Locality Principle (2)

• Example temporal constraints and resource
  constraints
• User "Plumber" "Roofer"
• User "Moving" must end before 20
• Temporal constraint "Plumbing" must end before
  17
• Disjunctive resource constraint "Plumbing" must
  precede "Roofing"
• Temporal constraint "Moving" cannot end before 19

18
Locality Principle (3)
Temporal sub-problem
19
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving

20
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Scheduling Problems
• Propagation of Temporal Constraints
• Propagation of Resource Constraints
• Examples of Applications
• Cooperative Problem-Solving

21
Scheduling Problems

• "Scheduling is the allocation of resources over
  time to perform a collection of tasks" Baker 74
• Pure scheduling problems assignment of start and
  end times to activities (each activity requires
  given resources with given capacities)
• Pure resource allocation problems assignment of
  resources to activities (the start and end times
  of each activity are given)
• Joint scheduling and resource allocation
  problems assignment of resources and start and
  end times to activities

22
Scheduling Problems

• Transportation
• Traffic scheduling and control (aircrafts, buses,
  trains, trucks)
• Loading and unloading (aircrafts, ships, trucks)
• Crew rostering
• Production and maintenance
• Manpower planning and timetabling (shifts,
  courses, exams)
• Project or mission scheduling
• Network planning and routing
• Computer or printer job scheduling
• Mixture planning, rack configuration, textile
  cutting, ...

23
Variety of Scheduling Problems

• Activities
• Resources
• Constraints
• Optimization criteria
• Problem size
• Time available to make a decision

24
Activities

• Interval (block) activities
• Splittable activities (with interruption cost?)

A
A
A
A
time
25
Resources

• Unary resources
• one person
• one machine
• Discrete resources
• a group of people with the same capabilities
• State resources
• an oven with different possible temperatures
• Energetic resources
• a limited number of human-days each week
• Required, provided, consumed or produced by
  activities

26
Constraints

• Temporal constraints
• Fixed or variable durations
• Precedence constraints
• Minimal and maximal delays
• Resource constraints
• Fixed capacity
• Variable capacity (time versus capacity
  tradeoffs)
• Variable capacity over time
• Specific constraints

27
Capacity Constraints

• Periods during which a resource is not fully
  available
• Maintenance periods
• Vacations
• Forbidden states (at night)
• Periods during which some minimal amount must be
  required or provided

28
Optional/Variable Requirements

• Optional activities
• Resource alternatives
• Sub-contracts

29
Optional/Variable Requirements

• Time versus capacity tradeoffs
• 2 people during 3 days
• or 3 people during 2 days

30
Optional/Variable Requirements

• Variable requirement over time
• Example 8 person-days with either 2 or 3 people
  at any time

31
Transition Times

• Tool setup between two tasks on the same machine
• State change (oven temperature, color used in a
  painting shop)
• Cleaning

32
Percentage Constraints

• a of activity A on resource RA must be done
  before (or after)
• B starts
• B ends
• b of activity B on resource RB are done
• Often complements variable requirements over time

33
Synchronization Constraints

• When A executes, B requires (at least, at most)
  c1 units of resource R
• When A does not execute, B requires (at least, at
  most) c2 units of resource R

34
Optimization Criteria

• No optimization criterion
• A well-defined criterion
• Project makespan
• Number of activities performed within given
  delays
• Maximal or average tardiness or earliness
• Peak or average resource utilization
• A combination of well-defined criteria
• Preferences (soft constraints)
• Optimization versus robustness

35
Problem Size
unary resources discrete resources several types
of resources
36
Suitable Response Time

• Reactive train traffic control two seconds
• Predictive production and maintenance scheduling
  minutes to hours
• Reactive production and maintenance scheduling
  seconds
• Predictive timetabling minutes to hours
• Reactive computer or printer job scheduling
  small fraction of the average job duration

37
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Scheduling Problems
• Propagation of Temporal Constraints
• Propagation of Resource Constraints
• Examples of Applications
• Cooperative Problem-Solving

38
Variables (Definition)

• From now on, we focus on interval activities
• ? Three variables
• start(A)
• end(A)
• duration(A)
• for each activity A

39
Variables (Implementation)

• Finite domain (bit vector)
• The domain of each variable is a finite set
• Interval domain (pair of numbers)
• The domain of each variable is an interval
• startmin(A), startmax(A),
• endmin(A), endmax(A)
• durationmin(A), durationmax(A)

40
Basic Relation

• end(A) start(A) duration(A)
• endmin(A) max(endmin(A), startmin(A)
  durationmin(A))
• endmax(A) min(endmax(A), startmax(A)
  durationmax(A))
• startmin(A) max(startmin(A), endmin(A) -
  durationmax(A))
• startmax(A) min(startmax(A), endmax(A) -
  durationmin(A))
• durationmin(A) max(durationmin(A), endmin(A) -
  startmax(A))
• durationmax(A) min(durationmax(A), endmax(A) -
  startmin(A))

41
Temporal Constraints

• Simple precedence constraints
• start(A) start(B)
• start(A) end(B)
• end(A) start(B)
• end(A) end(B)

42
Temporal Constraints

• Precedence constraints with minimal delays
• start(A) delay start(B)
• start(A) delay end(B)
• end(A) delay start(B)
• end(A) delay end(B)

43
Temporal Constraints

• Precedence constraints with fixed delays
• start(A) delay start(B)
• start(A) delay end(B)
• end(A) delay start(B)
• end(A) delay end(B)

44
Temporal Constraints

• Maximal delays
• start(A) start(B) delay
• start(A) end(B) delay
• end(A) start(B) delay
• end(A) end(B) delay

45
Propagation of Time Bounds

• var(A) delay var(B)
• varmin(B) max(varmin(B), varmin(A) delay)
• varmax(A) min(varmax(A), varmax(B) - delay)
• Complete propagation for bounded domains
• Contradiction found when the constraints conflict
• Best possible varmin(A) and varmax(A) found
  otherwise
• Incremental variant of an operations research
  algorithm for project scheduling (PERT networks)

46
Propagation of Time Bounds

• Complexity
• For a consistent network O(nm) where n is the
  number of activities and m the number of
  constraints if constraints are propagated in the
  first-in first-out order
• For an inconsistent network O(hn2) where h is
  the time horizon (can be reduced to O(nm) but
  not always worth it in practice)

47
Minimal and Maximal Distances

• x dxy y and y dyz z
• implies x (dxy dyz) z
• Useful to solve disjunctions of temporal
  constraints
• x - 5 y
• y 2 z
• z 4 x OR v 3 w

48
Minimal and Maximal Distances

• Matrix-based method
• Whenever dxy is modified, update dwz to max(dwz,
  dwx dxy dyz)
• Complexity
• O(n2) after each modification of the constraint
  network
• O(n3) to initialize the matrix

49
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Scheduling Problems
• Propagation of Temporal Constraints
• Propagation of Resource Constraints
• Examples of Applications
• Cooperative Problem-Solving

50
Unary Resources

• Main resources in 4/10 problems
• Example one individual machine or person

51
Discrete Resources

• Main resources in 5/10 problems
• Example group of identical machines

52
State Resources

• Main resources in 1/20 problems
• Example oven with different temperatures

53
Energetic Resources

• Main resources in 1/20 problems
• Example number of man-days per week

A
B
time
54
Constraint Propagation Models

• Explicit time-tables
• Disjunctive constraints
• Energetic reasoning
• Edge finding (task intervals)

55
Explicit Time-Tables (UD)

• WA(t) start(A) t lt end(A)
• "t, SA WA(t) capacity(A) capacity(t)

A
B
C
time
C
56
Explicit Time-Tables (S)

• WA(t) start(A) t lt end(A)
• "t, WA(t) implies state(A) state(t)

D
A
C
time
D
57
Explicit Time-Tables (E)

• aA(I) max(0, endmin(A) - start(I))
• bA(I) max(0, end(I) - startmax(A))
• d(I) end(I) - start(I)
• d(A) durationmin(A)
• WA(I) min(aA(I), bA(I), d(I), d(A))
• SA WA(I) capacity(A) capacity(I)

58
Explicit Time-Tables (E)

A
B
time
C
59
Explicit Time-Tables 2 Models

• Discrete array
• Sequential table or binary tree

B
D
A
C
time
60
Explicit Time-Tables Extensions

• Minimal capacity constraints (UDE)
• Default state (S)
• Constraints between time-tables or between
  time-tables and other variables (UDES)

61
Disjunctive Constraints (U)

• end(A) start(B)
• OR end(B) start(A)
• endmin(A) gt startmax(B)
• implies end(B) start(A)
• endmin(B) gt startmax(A)
• implies end(A) start(B)

62
Disjunctive Constraints (U)

• Optional activities
• Resource alternatives
• Sub-contracts
• Transition times
• Tool setups
• Color changes
• Cleaning

63
Disjunctive Constraints (U)

• end(A) ttime(A, B) start(B)
• OR end(B) ttime(B, A) start(A)
• OR duration(A) 0
• OR duration(B) 0
• OR capacity(A) 0
• OR capacity(B) 0

64
Energetic Reasoning (UD)

• aA(I) max(0, endmin(A) - start(I))
• bA(I) max(0, end(I) - startmax(A))
• d(I) end(I) - start(I)
• d(A) durationmin(A)
• WA(I) min(aA(I), bA(I), d(I), d(A))
• SA WA(I) capacity(A) capacity(I)

65
Energetic Reasoning (UD)

• Habographs Beck 92
• Intervals id .. jd) or b id .. b jd)
• Energetic resources Le Pape 94
• Intervals id .. (i1)d) or b id .. b
  (i1)d)
• Energetic reasoning rules Lopez 91
• Intervals startmin(A) .. endmax(B)) or
  startmin(A) .. x)

66
Energetic Reasoning (U)

• I startmin(A) .. endmax(B))
• d(I) lt SCltgtA,BWC(I) d(A) d(B)
• implies end(B) start(A)
• A1..8 - 2 - 3..10
• B0..3 - 2 - 2..5
• C2..4 - 1 - 3..5
• I 1 .. 5)
• 4 lt 1 2 2 implies 2 start(A)

67
Edge Finding (U)

• Basic idea
• Prove that an activity A executes before (or
  after) a set of other activities W
• Notations
• smin(W) minBÎW startmin(B)
• emax(W) maxBÎW endmax(B)
• dmin(W) SBÎW durationmin(B)

68
Edge Finding (U)

• emax(W) lt smin(W A) dmin(W A)
• implies smin(W') dmin(W') start(A)
• for every W' included in W
• A0..11 - 6 - 6..17
• B1..7 - 4 - 5..11
• C1..8 - 3 - 4..11
• W B C
• 11 lt 0 13 implies 1 7 start(A)

69
Edge Finding (U)

• Jackson's Preemptive Schedule Pinson 88
  Carlier Pinson 90/94
• O(n2) O(nlog(n))
• Iterative algorithm Nuijten 94
• O(n2) with no specific data structure
• Task intervals Caseau Laburthe 94
• O(n3)
• Incremental
• More deductions è more precise time-bounds

70
Edge Finding (U)

• "Not-first" deduction rule
• emax(W) lt smin(A) dmin(W A)
• implies minBÎW(smin(B) dmin(B)) start(A)
• Nuijten 94 Caseau Laburthe 94
• Baptiste Le Pape 96 Complete application in
  O(n2) time and O(n) space

71
Edge Finding (U)

• "Not-first" deduction rule with knapsack
• D smin(A) dmin(W A) - emax(W) gt 0
• implies minW' f(W')(smin(W') dmin(W'))
  start(A)
• with f(W') defined as W W' OR D dmin(W')
• Complete application in exponential time
  (includes the NP-complete knapsack problem)
• Partial application Caseau Laburthe 95

72
Comparison

• Theoretical results
• Unique fix point semantics
• Modeling power
• Time and space complexity
• Pruning power
• Experimental results

73
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Scheduling Problems
• Propagation of Temporal Constraints
• Propagation of Resource Constraints
• Examples of Applications
• Cooperative Problem-Solving

74
MSS Problem Definition (1)

• Molding Shop Scheduling
• 2 types of cast-iron (standard and special)
• For each cast-iron type
• available quantity every hour
• 3 molding machines
• For each machine
• shifts
• breaks (lunch, maintenance)
• periods during which the machine can manufacture
  only some types of parts

75
MSS Problem Definition (2)

• 30 types of parts
• For each part type
• cast-iron type
• one or more possible machines
• For each (part, machine) pair
• production speed
• cast-iron consumption
• minimal and maximal batch size
• maximal break duration (one break per batch
  allowed)
• minimal production time before and after the break

76
MSS Problem Definition (3)

• 200 orders for parts
• For each order
• part type
• number of parts
• due-date
• Optimization criterion
• relaxation of due-dates
• closer due-dates are more important than further
  due-dates

77
MSS Problem Representation (1)

• Part type Discrete resource
• Produced by activities
• Due-dates è minimal production constraints
• Machine State resource
• Part type produced at time t state at time t
• Default state 0 out of shifts and during breaks
• Cast-iron type Energetic resource

78
MSS Problem Representation (2)

• Minimal and maximal number of (ordered) batches
  for each (part, machine) pair
• Batch 3 Interval activities
• BB before the break, B for the break, AB after
  the break
• end(BB) start(B) and end(B) start(AB)
• duration(BB) not in 1, minimal-duration-before-br
  eak - 1
• duration(AB) not in 1, minimal-duration-after-bre
  ak - 1
• duration(B) maximal-break-duration
• batch-duration duration(BB) duration(AB)
• batch-duration maximal-batch-size /
  production-rate
• batch-duration not in 1, (minimal-batch-size -
  1) / production-rate

79
MSS Problem Representation (3)

• Resource constraints
• BB and AB require the machine in the "part type"
  state
• BB and AB require the cast-iron at the given
  consumption rate
• B requires the machine in the state 0
• AB produces the batch of parts (batch-duration
  production-rate units of the part type)
• Redundant constraints
• From start(BB) to end(AB), the state of the
  machine is either 0 or "part-type"
• Counter of the minimal and maximal production of
  each part type each day

80
MSS Search Procedure

• Select a batch with unbound start time or unbound
  duration
• If the batch is optional
• either confirm or cancel the batch
• Else if the start time is unbound
• either schedule (ASAP) or postpone the batch
• Else (the duration is unbound)
• instantiate the duration
• Iterate

81
MSS Results

• 2200 lines of ILOG SCHEDULER source code
• 1400 lines for representing the problem
• 600 lines for reading the data
• 200 lines for solving the problem
• Executable
• 880K on an IBM RS600
• 420K on a SUN-4
• CPU time (for 300 non-zero activities and 40
  resources)
• In most cases 10 seconds
• When more search is stopped and restarted with
  relaxed due-dates

82
CSS Problem Definition

• Construction Site Scheduling
• Resource-constrained project scheduling problem
  with
• 1 unary resource and 3 discrete resources
• 15 interval activities and 15 interruptible
  activities
• 15 activities with time-versus-capacity tradeoffs
• 10 activities with variable requirements over
  time
• temporal constraints
• periods during which a resource is not or not
  fully available
• percentage constraints
• synchronization constraints
• 40 preference constraints (10 levels of
  preferences)

83
CSS Search Procedure

• Select preferences
• Impose the selected preferences as constraints
• Search for a solution within a limited number of
  backtracks (blend of preemptive job-shop
  scheduling and resource-constrained project
  scheduling)
• Save the solution if one is found
• Iterate with a different set of preferences

84
CSS Results

• 2200 lines of CLAIRE source code ( 2200 lines
  library)
• 1100 lines for representing the problem
• 300 lines for reading the data
• 800 lines for solving the problem
• 3000 lines for the graphical interface
• Executable
• 1130K on a PC under OS2 (with graphical
  interface)
• CPU time (for 30 activities and 4 resources)
• "Good" schedules obtained in 5 to 10 iterations
• 15 seconds per iteration

85
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving

86
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving
• Introduction
• Problem Decomposition
• Constraint Propagation
• Search

87
Problem Solving

• Two main other classes of techniques
• Linear Programming and Mixed Integer Programming
• Local Search

88
Mixed Integer Programming (1)

• Explicit problem definition
• Linear constraints
• Real (floating point) and integer variables
• One optimization criterion
• Separation between problem definition and problem
  solving
• The continuous relaxation (when fractional values
  are allowed for integer variables) is solvable in
  polynomial time
• Bound on optimization criterion
• Guidance for solving the MIP problem by
  branch-and-bound (rounding, cuts, )

89
Mixed Integer Programming (2)
90
Mixed Integer Programming (3)
91
Mixed Integer Programming (4)
92
Mixed Integer Programming (5)
93
Mixed Integer Programming (6)
94
Mixed Integer Programming (7)
95
Mixed Integer Programming (8)
96
Mixed Integer Programming (9)
97
MIP/CP Most Important Points

• MIP
• Linear constraints
• One relaxation of the global problem
• Guidance from the optimal solution of the relaxed
  problem
• Cuts
• Scheduling example hard to deal with resource
  constraints

• CP
• Any constraints
• Linked relaxations of multiple sub-problems
• Guidance from the possible solutions of the
  relaxed sub-problems
• Redundant constraints
• Scheduling example hard to deal with  sum 
  cost functions

98
MIP/CP Most Important Points
Temporal sub-problem
99
Local Search

• Operators to move from solutions to other
  (neighbor) solutions (or from populations of
  solutions to populations of solutions)
• Search control strategy (meta-heuristic)
• Many different types simulated annealing, tabu
  search, genetic algorithms,

100
Main Issues (1)

• Mixed Integer Programming
• Is the continuous relaxation a good approximation
  of the convex envelope of the solutions?
• Can the formulation of the problem be iteratively
  modified to make the optimal solution of the
  continuous relaxation converge toward the optimal
  solution of the initial problem?

101
Main Issues (2)

• Constraint Programming
• Are the critical constraints propagating well?
• Is a tight bound on the optimization criterion
  efficiently translated (by propagation) into
  constraints that effectively guide the search
  toward a solution?

102
Main Issues (3)

• Local Search
• Is the neighborhood topology consistent with the
  optimization criterion?
• Are the operators connecting (in a few steps)
  good solutions with better solutions, without
  downgrading the solution too much in intermediate
  steps?

103
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving
• Introduction
• Problem Decomposition
• Constraint Propagation
• Search

104
Problem Decomposition
Complete model
105
Sequential Decomposition

• Solve sub-problem 1
• Impose the solution of sub-problem 1 and solve
  sub-problem 2
• Infer characteristics of  good  solutions of
  sub-problem 1
• Iterate

106
Column Generation

• Generate candidate solution components
• Find the optimal combination of these components
• Infer characteristics of  good  solution
  components to add
• Iterate

107
Example Inventory Management
108
Example

• Stock 4
• st(o1) 0, et(o1) 35, rq(o1) 2
• st(o2) 5, et(o2) 30, rq(o2) 3
• st(o3) 32, et(o3) 87, rq(o3) 5

109
Maintenance Constraints

• No resource is used more than Utime time units
  without maintenance
• No more than Mnumber resources are in maintenance
  at the same time
• Cost of each maintenance Cmaint

110
Mixed Integer Programming (1)
crane allocation problem
select maintenance intervals
core problem
maintenance schedule
select cranes
maintenance problem
99 cost
1 cost
Polynomial
NP-hard
80 vars 40 constrs
1000 vars 630 constrs
6000 vars 4000 constrs
45000 vars 32000 constrs
111
Mixed Integer Programming (2)
crane allocation
maintenance scheduling
80 vars 40 constrs
1000 vars 630 constrs
6000 vars 4000 constrs
45000 vars 32000 constrs
1 node
30 nodes
1200 nodes
gt 20 000 nodes
gt 20 000 nodes
112
Hybrid Algorithms (1)
LINEAR PROGRAMMING Solve the (core) resource
allocation problem
HEURISTIC ALGORITHM Select resource items for
each order
CONSTRAINT PROGRAMMING Solve the resulting
maintenance scheduling problem
On failure, tighten the resource allocation
problem and iterate
113
Hybrid Algorithms (2)

• Tightness of the crane allocation problem
• Extend the duration of all tasks (add the
  maintenance duration)
• Extend the duration of the conflict task
• Final result 4.5 above the lower bound 87 927
  919

114
Hybrid Algorithms (3)

• Other hybrid algorithms, including column
  generation, provided  good  solutions
• Some instances remain  hard 
• Details available in Caseau Kökény 98,
  Baptiste et al. 98

115
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving
• Introduction
• Problem Decomposition
• Constraint Propagation
• Search

116
LP as a Global Constraint

• Collection of linear constraints (general or of a
  particular form) treated as a global constraint
• Detect conflicts
• Find variables with fixed values
• Find bounds of variables
• Infer additional constraints (cuts)

117
Objective Functions

• Minimize COST F(V1, V2, , Vn)
• Apply linear programming (or an operations
  research algorithm) to a relaxation of the
  problem
• Global lower bound at the root of the search tree
• Lower bound LB at each node of the search tree
• Characteristics of solutions with COST lt LB D
  inferred from the optimal solution of the relaxed
  problem and  reduced costs 

118
Outline

• Constraint Programming
• Constraint-Based Scheduling
• Cooperative Problem-Solving
• Introduction
• Problem Decomposition
• Constraint Propagation
• Search

119
Example Vehicle Routing

• Efficient local search algorithms
• No specific constraint
• Incremental computation of tour length and cost
  from a solution to its neighbors
• Constraint programming for specific constraints
• Use constraints to test and reject  bad 
  neighbors
• Enlarge the neighborhood if there is no valid
  neighbor
• Find the valid neighbor with the smallest cost
• References
• Pesant Gendreau 96, Shaw 98, Kilby, Prosser
  Shaw 00
• Caseau Laburthe 99, Caseau et al. 01

120
Examples Scheduling

• Forget and extend
• Random fragments Baptiste, Le Pape Nuijten 95,
  Nuijten Le Pape 98
• Chosen fragments Caseau et al. 01
• Minimal perturbation
• El Sakkout Wallace 00

121
Conclusion (1)

• Already a significant number of applications
• Credo hybrid problem-solving techniques will
  enable the resolution of problems that are still
  open today
• More efficient exact resolution
• Better lower and upper bounds in given CPU time
• More robust algorithms

122
Conclusion (2)

• Main difficulties
• Identification of promising combinations of
  techniques
• Implementation and test of several of these
  combinations
• Guidelines and tools are needed
• Modeling
• Combination of sub-models
• Testing and interpretation of test results

123
References (1)

• Disclaimer not (at all) a complete bibliography!
• K. R. Baker.
• Introduction to Sequencing and Scheduling.
• John Wiley and Sons, 1974.
• Ph. Baptiste, C. Le Pape and W. Nuijten.
• Constraint-Based Optimization and Approximation
  for Job Shop Scheduling.
• IJCAI'95 Workshop on Intelligent Manufacturing
  Systems, 1995.
• Ph. Baptiste and C. Le Pape.
• Edge-Finding Constraint Propagation Algorithms
  for Disjunctive and Cumulative Scheduling.
• 15th Workshop of the U.K. Planning Special
  Interest Group, 1996.
• Ph. Baptiste, Y. Caseau, T. Kökény, C. Le Pape
  and R. Rodosek.
• Creating and Evaluating Hybrid Algorithms for
  Inventory Management Problems.
• 4th National Meeting on Practical Approaches to
  NP-Complete Problems, 1998.
• H. Beck.
• Constraint Monitoring in TOSCA.
• AAAI Spring Symposium on Practical Approaches to
  Planning and Scheduling, 1992.

124
References (2)

• J. Carlier and E. Pinson.
• A Practical Use of Jackson's Preemptive Schedule
  for Solving the Job-Shop Problem.
• Annals of Operations Research, 26269-287, 1990.
• J. Carlier and E. Pinson.
• Adjustment of Heads and Tails for the Job-Shop
  Problem.
• European Journal of Operational Research,
  78(2)146-161, 1994.
• Y Caseau and F. Laburthe.
• Improved CLP Scheduling with Task Intervals.
• 11th International Conference on Logic
  Programming, 1994.
• Y. Caseau and F. Laburthe.
• Disjunctive Scheduling with Task Intervals.
• Technical Report, Ecole Normale Supérieure,
  1995.
• Y. Caseau and T. Kökény.
• An Inventory Management Problem.
• Constraints, 3(4)363-373, 1998.
• Y. Caseau and F. Laburthe.
• Heuristics for Large Constrained Vehicle Routing
  Problems.
• Journal of Heuristics, 5281-303, 1999.

125
References (3)

• Y. Caseau, F. Laburthe, C. Le Pape and B.
  Rottembourg.
• Combining Local and Global Search in a
  Constraint Programming Environment.
• Knowledge Engineering Review, to appear.
• H. El Sakkout and M. Wallace.
• Probe Backtrack Search for Minimal Perturbation
  in Dynamic Scheduling.
• Constraints, 5(4)359-388, 2000.
• P. Kilby, P. Prosser and P. Shaw.
• A Comparison of Traditional and Constraint-based
  Heuristic Methods on Vehicle Routing Problems
  with Side Constraints.
• Constraints, 5(4)389-414, 2000.
• C. Le Pape.
• Implementation of Resource Constraints in ILOG
  SCHEDULE A Library for the Development of
  Constraint-Based Scheduling Systems.
• Intelligent Systems Engineering, 3(2)55-66,
  1994.
• P. Lopez.
• Approche énergétique pour l'ordonnancement de
  tâches sous contraintes de temps et de
  ressources.
• Thèse de l'Université Paul Sabatier, 1991.

126
References (4)

• W. Nuijten.
• Time and Resource Constrained Scheduling A
  Constraint Satisfaction Approach.
• PhD Thesis, Eindhoven University of Technology,
  1994.
• W. Nuijten and C. Le Pape.
• Constraint-Based Job Shop Scheduling with ILOG
  Scheduler.
• Journal of Heuristics, 3271-286, 1998.
• G. Pesant and M. Gendreau.
• A View of Local Search in Constraint
  Programming.
• 2nd International Conference on Principles and
  Practice of Constraint Programming, 1996.
• E. Pinson.
• Le problème de job-shop.
• Thèse de l'Université Paris VI, 1988.
• P. Shaw.
• Using Constraint Programming and Local Search
  Methods to Solve Vehicle Routing Problems.
• 4th International Conference on Principles and
  Practice of Constraint Programming, 1998.