Constraint Programming

1
Constraint Programming

• Michael Trick
• (actually 75 Pascal Van Hentenryck, 20 Irv
  Lustig, 5 Trick)
  
• Carnegie Mellon

2
Outline

• Motivation
• An Overview of Constraint Programming
• Constraint Programming at Work
• Getting Started
• Sports Scheduling
• Manufacturing
• Perspectives

3
Combinatorial Optimization

• Many, many practical applications
• Resource allocation, scheduling, routing
• Properties
• Computationally difficult
• Technical and modeling expertise needed
• Experimental in nature
• Important () in practice
• Many solution techniques
• Integer programming
• Specialized methods
• Local search/metaheuristics
• Constraint programming

4
Constraint programming

• Began in 1980s from AI world
• Prolog III (Marseilles, France)
• CLP(R)
• CHIP (ECRC, Germany)
• Application areas
• Scheduling, sequencing, resource and personnel
  allocation, etc. etc.
  
• Active research area
• Specialized conferences (CP, CP/AI-OR, )
• Journal (Constraints)
• Companies

5
Constraint Programming

• Two main contributions
• A new approach to combinatorial optimization
• Orthogonal and complementary to standard OR
  methods
  
• Combinatorial versus numerical
• A new language for combinatorial optimization
• Rich language for constraints
• Language for search procedures
• Vertical extensions

6
The Tutorial

• Goal to provide an introduction
• What is constraint programming?
• What is it good for?
• How does it compare to integer programming?
• How easy is it to use?
• What is the underlying technology?

7
Constraint Programming

• Constraint programming by example
• Illustrate rich language
• Contrast with integer programming
• Illustrate some underlying technologies
• Disclaimers
• Cant cover all of CP
• I want to make you curious
• Language/system used
• Could use many choose OPL

8
Modeling in Constraint Programming

• A rich constraint language
• Arithmetic, higher-order, logical constraints
• Global constraints for natural substructures
• Specification of a search procedure
• Definition of search tree to explore
• Specification of search strategy

9
Comparison of CP/IP

• Branch and Prune
• Prune eliminate infeasible configurations
• Branch decompose into subproblems
• Prune
• Carefully examine constraints to reduce possible
  variable values
  
• Branch
• Use heuristics based on feasibility info
• Main focusconstraints and feasibility

• Branch and Bound
• Bound eliminate suboptimal solutions
• Branch decompose into subproblems
• Bound
• Use (linear) relaxation of problem ( cuts)
• Branch
• Use information from relaxation
• Main focus objective function and optimality

10
Illustrative artificial example

• Color a map of (part of) Europe Belgium,
  Denmark, France, Germany, Netherlands,
  Luxembourg
  
• No two adjacent countries same color
• Is four colors enough?

11
OPL example

• enum Country Belgium,Denmark,France,Germany,Nethe
  rlands,Luxembourg
  
• enum Colors blue,red,yellow,gray
• var Colors colorCountry
• solve
• colorFrance colorBelgium
• colorFrance colorLuxembourg
• colorFrance colorGermany
• colorLuxembourg colorGermany
• colorLuxembourg colorBelgium
• colorBelgium colorNetherlands
• colorBelgium colorGermany
• colorGermany colorNetherlands
• colorGermany colorDenmark

• Variables non-numeric
• Constraints are non-linear
• Looks nothing like IP!
• Perfectly legal CP

12
Constraint Programming

• Domain store
• For each variable what is the set of possible
  values?
  
• If empty for any variable, then infeasible
• If singleton for any variable, then solution

• Constraints
• Capture interesting and well studied
  substructures
  
• Need to
• Determine if constraint is feasible WRT the
  domain store
  
• Prune impossible values from the domains

13
Constraints

• Can have differing techniques to handle a
  constraint type
  
• 3x10y2z 4w 4
• x in 0,1, y in 0,1,2, z in 0,1,2, w in
  0,1
  
• Simple bound on sizes gives y in 0
• More complicated handling gives
• x in 0, y in 0, z in 0,2, w in 0,1

14
Constraint Solving

• General algorithm is
• Repeat
• select a constraint c
• if c is infeasible wrt domain store
• return infeasible
• else apply pruning algorithm of c
• Until no value can be removed

15
Branching

• Once the constraint solving is done, if the
  problem is not infeasible nor are the domains
  singletons, then apply the search method
  
• Choose a variable x with non-singleton domain
  (d1, d2, di)
  
• Foreach d in (d1, d2, di)
• add constraint xdi to problem and solve

16
Show OPL solving coloring problem
17
Strength of CP

• Since there is no need for a linear relaxation,
  the language can represent much more directly (no
  need for big-M IP formulations.

18
Examples of formulation abilities

• Facility location want a constraint that
  customer j can be assigned to warehouse i only if
  warehouse open. (yi1 if warehouse i open)
  
• IP xi,j is 1 if cust j assigned to i
• xi,j
• CPxj is the warehouse cust j assigned to (not
  a 0,1 variable)
  
• yxj 1

19
Similar example

• Routing type constraints.
• Let xi be the ith customer visited and di,j
  be distance from i to j
  
• sum (i in 1..n) dxi,xi1
• gives total distance traveled

20
Formulation strengths

• Logical requirements if A1 and Beither C3 or D1.
  
• Really painful in IP. Straightforward in CP
• ((A1) (B ((C3)\/(D1))

21
Global Constraints

• Recognize that some types of constraints come up
  often
  
• Create specialized routines to handle
• Strong pruning
• Efficient handling
• Extend system to include these

22
Global constraint alldifferent

• Most well known and studied constraint.
• alldifferent(x,y,z)
• states that x, y, and z take on different values.
  So x2, y1, z3 would be ok, but not x1, y3,
  z1.
  
• Clear uses in routing (xi is ith customer
  visited, alldifferentx says each customer
  visited at most once), very useful in many other
  situations.

23
Alldifferent feasibility and pruning

• Feasibility? Given domains, create
  domain/variable bipartite graph

x1
1
x2
2
3
x3
4
x4
5
x5
24
Alldifferent feasibility and pruning

• Pruning? Which edges are in no matching?

x1
1
Domain is sharply reduced
x2
2
3
x3
4
x4
5
x5
25
Global constraints

• Many different types of constraints have
  specialized routines
  
• distribute(card,value,base) the number of times
  valuei appears in base is cardi
  
• circuit(succ) the values in succ form a
  hamiltonian circuit (so if you follow the
  sequence 1, succ1, succsucc1 etc, you will
  get a loop through 1..n.

26
Global constraints

• Many others, and new ones being created all the
  time
  
• Strengthen and expand the language
• Make modeling easier and more natural
• System is faster at finding solutions
• Details hidden to user

27
Vertical language extensions

• Can add constraints and definitions to make
  modeling even more natural
  
• Ideas remain the same there are domains and
  constraints constraints check for feasibility
  and prune domains a search strategy guides the
  system in finding solutions

28
Scheduling

• Want concepts of jobs, machines, before,
  after, jobs requiring machines, and so on.
  
• Easy to extend

29
Example of scheduling
forall(j in Jobs) forall(t in 1..nbTasks-1
) taskj,t precedes taskj,t1
forall(j in Jobs) forall(t in Tasks)
taskj,t requires toolresourcej,t
30
Search Strategy

• Combined with model, search strategies are
  integral to constraint systems.
  
• Allow choice of branching variables or more
  powerful search strategies
  
• Can be key in solving problems
• Two steps
• Specify tree to search
• Specify how to explore the tree

31
Example of Search Strategies

• forall(s in Stores ordered by increasing
  regretdmax(costs))
  
• tryall(w in Warehouses ordered by increasing
  supplyCosts,w) suppliers w
  
• implements a maximum regret ordering (find a
  store with maximum regret then order the
  warehouses by increasing cost)

32
Example Problem

• Painting cars (from Magnanti and Sokel).
• Sequence cars to minimize paint changeover
• Cars cannot be sequenced too far out of order

33
Small example

• Small Example 10 cars in sequence. The order
  for assembly is 1, 2, ..., 10. A car must be
  painted within 3 positions of its assembly order.
  For instance, car 5 can be painted in positions
  2 through 8 inclusive. Cars 1, 5, and 9 are red
  2, 6, and 10 are blue 3 and 7 green and 4 and 8
  are yellow. Initial sequence 1, 2, ... 10
  corresponds to color pattern RBGYRBGYRB and has 9
  purgings. The sequence 2,1,5,3,7,4,8,6,10,9
  corresponds to color pattern BRRGGYYBBR and has 5
  purgings.
•  

34
Constraint Program

• int n
• int rnge
• int ncolor
• range Slots 1..n
• var Slots slot1..n
• var Slots revslot1..n
• int color1..n
• minimize
• sum (j in 1..n-1) (colorrevslotj
  colorrevslotj1)
  
• subject to
• forall (i in Slots)
• i-rngerange /
  
• alldifferent(slot) /must choose different
  slots /
  
• forall (i in Slots)
• revslotsloti i

35
Personal use

• Tremendous help in my work on sports scheduling
  much easier to formulate idiosyncratic
  constraints
  
• Very fast to create prototypes
• Competitive (at least!) to IP approaches

36
Result

• Formulation is much easier than IP formulation
• Gets good solutions much faster than IP
• Is competitive in proving optimality

37
Finding optimal solutions

• Constraint programs can find optimal solutions.
  Typically works by finding a feasible solution
  and adding a constraint that future solutions
  must be better than it. Repeat until infeasible
  the last solution found is optimal

38
Perspectives

• Many solution techniques
• Integer programming
• Constraint programming
• Local search
• Combinations
• Which to use?

39
Comparing IP and CP

• Complementary technologies
• Integer programming
• Objective function relaxations
• Constraint programming
• Feasibility domain reductions
• Might need to experiment with both
• CP particularly useful when IP formulation is
  hard or relaxation does not give much information

40
Combining Methods

• Local and Global Search
• Use CP/IP for very large neighborhood search
  (take a solution, remove large subset, find
  optimal completion)
  
• Combining CP and IP
• Use LP as constraint handler
• Use CP as subproblem solver in branch and price

41
Conclusions

• Constraint programming should become a part of
  every OR persons toolkit
  
• Combinations of CP and IP represent a big thing
  in future techniques
  
• Blurring of lines between optimization and
  heuristics
  
• This talk at http//mat.gsia.cmu.edu/INFORMS/cp.pp
  t