47 Constraint Programming for Scheduling

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47 Constraint Programming for Scheduling

• Cathleen Blackmon
• CS 599 Scheduling Algorithms
• Winter 2005

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Constraint Programming

• An approach for formulating and solving discrete
  variable constraint satisfaction or constrained
  optimization problems
• Systematically employs deductive reasoning
• Reduce Search Space
• Allow for a Wide Variety of Constraints
• Extends Logic Programming
• More Powerful Search Strategies using
  Problem-Specific Knowledge

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The Constraint Satisfaction Problem - CSP

• Given
• A Set of Discrete Variables
• Associated Finite Domains
• Set of Constraints (involving the variables)
• Find
• A Solution that Satisfies all the Constraints

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General Algorithmfor CSP (Constraint
Satisfaction Problem)
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Constraint Propagation and Domain Reduction
Start

• Discrete Variables
• Boolean
• Integer
• Symbolic
• Set Elements
• Subsets of Sets

• Constraints
• Mathematical
• Disjunctive
• Relational
• Explicit
• Special Purpose

• If solution is found algorithm ends
• If no solution found, check for inconsistency

• Inconsistency The domain of at least one
  variable has become empty.

• If inconsistency not proven,
• Then, search
• Using some strategy for branching

• Branching
• Divides the problem into subproblems
• Mutually exclusive
• Collectively exhaustive
• Select a branch
• Propagate all constraints again

• If inconsistency is proven (a failure), then
  examine the search tree to see if all subproblems
  have been explored.

• If all branches have been fathomed (explored)
• Then problem inconsistency has been proven

• If not, then backtrack to the previous stage and
  branch to a different subproblem

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Define variables Initialize variable
domains Initialize constraint store

• Domains reduced systematically using logic-based
  filtering algorithms

• Then, the reduced domains are propagated to the
  constraints that use those variables

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Propagate constraints
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Solution found ?
Yes
End1
No
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5
Inconsistency proven ?
All branches fathomed ?
Yes
Yes
End2
6
7
No
No
Branch
Backtrack
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Constraint Propagation - Example
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Constraint Propagation Logic
Step1
C1 0, 1, 2, 3, 4, 5, 6
S1 0, 1, 2, 3, 4, 5, 6
C2 0, 1, 2, 3, 4, 5, 6
S2 0, 1, 2, 3, 4, 5, 6
C1 s1 3
C2 s2 2
C1 0, 1, 2, 3, 4, 5, 6
S1 0, 1, 2, 3, 4, 5, 6
C2 0, 1, 2, 3, 4, 5, 6
S2 0, 1, 2, 3, 4
Step2
C1 lt 5
Step3
C1 3, 4
either (C1 lt s2) or (C2 lt s1)
S1 0, 1
C1 s1 3
Step4
C2 5, 6
s2 3, 4
C2 s2 2
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Branching

• Strategies
• Depth-first strategy - typical
• More sophisticated look-ahead strategies
• What to branch?
• First select variable whose domain is not yet
  bound (not yet reduced to a single value)
• Selection of variable based on heuristics
• Then instantiate variable again using
  heuristics
• Choice points ? The points at which the search
  strategy chooses to advance along the search tree

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Constraint Optimization Problem - COP

• Given Objective Z to be minimized
• When a first solution to the original CSP is
  found
• Calculate its objective value (Z)
• Add the constraint Z lt Z to the constraint store
  (to form a new CSP)
• Repeat the CSP algorithm on this new problem
• Repeat until inconsistency is found
• The last found solution is the optimum

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Selected CP Applications

• Job Shop Scheduling
• Single-Machine Sequencing
• Parallel Machine Scheduling
• Timetabling
• Vehicle Dispatching

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Research

• Hybrid methods for CP and IP search methodologies
• CPs strengths
• Flexibility in modeling
• Rich set of operators
• Domain reduction (combinatorial problems)
• IPs strengths
• Specialized search techniques
• I.E., Relaxation, cutting planes, duals

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Modeling Scheduling Problems w/ CP

• Variable Indexing
• Constraints
• Inequalities
• Boolean variables
• Integer variables
• Logical constraints
• Global constraints

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CP vs. IP Approaches

• IP
• Relies on the mathematical structure of the
  problem
• Objective-centric
• Generate-and-test strategy
• Constraints ? The fewer the better

• CP
• Constraints search strategy determined by the
  researcher
• Relies less on math. struct of prob
• - more on domain knowledge
• Constraint-centric
• Constant reevaluation of logical conclusions
  resulting from interactions of constraints
  reducing domain of feasible solns.
• Constraints ? The more the better

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Choosing the CP Methodology

• CP is most appropriate for
• Pure integer and Boolean decision variable
  problems
• (Not effective for floating-point)
• Combinatorial optimization problems with large
  number of logical, global and disjunctive
  constraints
• Large num interrelated constraints with few
  variables in each constraint
• Better constraint propagation and domain
  reduction
• CP algorithms perform better
• Optimal mapping of one ordered set to another
  relation btwn terms expressed in mathematical
  terms significant to the objective

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• Questions?