Applications of Constraint Programming

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Applications of Constraint Programming

• The Data Error Detection Problem and the DDX
  Manpower Planning Problem

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Outline of Presentation

• Data Error Detection Problem
• Manpower Planning Problem for the DDX
• Some Results of Constraint Programming Applied to
  the Resource Constrained Project Scheduling
  Problem (RCPSP)

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Errors in Data Sets

• Large data files frequently contain missing,
  incorrect, and inconsistent data.
• Such errors can be detected by formulating a set
  of rules.
• First task is to form a set of rules for the EMF
  and check the set for consistency and
  redundancy.
• Second task is to use the set of rules to detect
  erroneous data and make the appropriate
  corrections which modify the data as little as
  possible and preserves the statistical
  distribution of correct data as much as possible.
• Our research focuses on the first one.

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Propositional Logic Formulation

• Due to Bruni and Sassano 2001
• An edit rule may be
• If marital status is married, age must be
  not less than 14
• The propositional logic formula for the above
  rule is
• (married_status married) /\ (age lt
  14)
• To get a formula which is satisfied by correct
  data, we need to negate the above formula.
• This negated formula is satisfiable if and only
  if there exists an assignment that makes the
  formula true otherwise the negated formula is
  unsatisfiable, which implies an inconsistency in
  the edit set.

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Complete Inconsistency

• Complete inconsistency occurs when every possible
  set of entries to the fields activates an edit.
• In order to restore consistency it is useful to
  know which are the offending edits, or the
  minimal unsatisfiable sub formula.
• The core is to solve a SAT and a MAX_SAT problem.

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Redundancy

• Some edits could be logically implied by others,
  thus being redundant. By removing these redundant
  edits, we can reduce the number of clauses in the
  system while maintaining the same power of
  inconsistency detection.
• An edit, e, can be checked by removing its
  clausal representation from the CNF formula,
  adding its negation to the formula, and testing
  if the resulting formula is unsatisfiable. If it
  is unsatisfiable, then e is redundant.
• The core is again to solve a SAT problem.

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SAT and MAX_SAT Problem

• Satisfiability (SAT) Problem (Gary and Johnson
  1979)
• Instance A set U of variables and a
  collection C of clauses over U.
• Question Is there a satisfying truth
  assignment for C?
• MAX_SAT Problem
• Instance A set U of variables and a
  collection C of clauses over U.
• Question What is the maximum number of
  clauses that can be satisfied?
• Both SAT and MAX_SAT are NP-complete and not
  easily expressed or solved with classical
  mathematical programming tools and concepts.
• Intense interest in exploring outside
  mathematical programming in areas such as
  Artificial Intelligence (AI) and Computer Science
  (CS) to find ideas to use to represent and solve
  satisfiability problems.

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Opportunities of Research

• Edit Generation. Attack the problem of
  generating a set of consistent and
  redundancy-free edit rules. This problem can be
  solved by encoding the formulation as a
  propositional logic formula and solving a
  sequence of satisfiability problems.
• Implementation. Design, implement, and test an
  application that executes the efficient set of
  edit rules for the types of personnel files
  handled by the Navy.

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Manpower Planning for DDX Ship

• The reduction in crew size associated with the
  new DDX class of ships, means that many of the
  routine and non routine tasks that are now
  performed by our forces at sea will need to be
  accomplished with fewer resources.
• Determining the correct crew size and a right mix
  of skills for the crew will require that we
  understand how to schedule the myriad of tasks
  that a ships crew must accomplish every day.
• Our Plan -- Formulate it as a Resource
  Constrained Project Scheduling Problem (RCPSP)
  with special objective function and resource type
  considerations.

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A Traditional Resource Constraint Project
Scheduling (RCPS) Problem

• Temporal time constraints (time windows)

starting time of task k in job j
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Additional Assumptions

• A set of renewable resources is required for
  carrying out the activities of the project.
• Each activity must be carried out without
  interruption (non preemption).
• Set-up time independent
• Objective is to find an optimal schedule to
  minimize the make span (completion time of the
  last project scheduled).

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Notation
amount of resource k used at t for a schedule S
E the edge set of activity on node graph,
which connects pairs of nodes having
temporal relations
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A General Model Formulation
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An Illustration of RCPSP
1. The current illustration is not feasible for
unary resources. 2. Our task is to find a
feasible solution while optimizing an objective
function.
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Where is the Gap?

• The manpower planning problem faced by the Navy
  is different from the traditional RCPSP. It has
  combined characteristics of both RCPSP and a
  generalized assignment problem (GAP).
• The assignment has to be made which is usually
  the case in GAP but rare in RCPSP.
• The objective is to minimize the number of
  sailors on site (at work), which is different
  from minimizing make-span in traditional RCPSP.
  Also note that this is not simply minimizing the
  number of sailors, but the attendance of
  sailors since one sailor might be assigned
  several times for different tasks.
• On the other hand, it is significantly different
  from GAP because the assignment needs to be made
  while satisfying the temporal constraints among
  tasks (time windows).

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Introduction to Constraint Programming (CP)

• CP is the study of computational systems based on
  constraints.
• Earliest ideas leading to CP may be found in AI
  area of CSP (Constraint Satisfaction Problem),
  dating back to 60s and 70s.
• In CP, the word programming refers to its roots
  in the field of programming language. Other
  programming paradigms are procedural programming,
  OOP, functional programming, logic programming,
  etc. (Hentenryck, 1999)
• In Mathematical Programming, the word
  programming roots in Dantzig 1963, which is
  associated with a specific mathematical problem.
• Constraint Programming represents one of the
  closest approaches computer science has yet made
  to the Holy Grail of programming the user states
  the problem, the computer solves it. (E. Freuder)

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An Illustration for Domain Reduction

• Constraint Propagation
• When a variables domain is modified, the
  effects of this modification are then
  communicated to any constraint that interacts
  with that variable. The domain reduction
  algorithms modifies the domains of all the
  variables in that constraint, given the
  modification of one of the variables in that
  constraint.

01234 56789
1 3
01234
Y2X
Ylt10
(X modulo 2) 1
Y2X
01234 56789
02468
2 6
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Experimental Study of Constraint Programming
Approach to Solve RCPSP

• A RCPSP with unary resource, i.e. for each type
  of resource there is only one unit available
  throughout the process of the project.
• Implementing constraint programming with
  Optimization Programming Language (OPL).
• Branching and Bound approach to solve the same
  problem with Excel Solver as comparison.
• Our study indicates that the constraint
  programming approach outperforms the B B
  approach significantly.

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Effects of Number of Types of Resources
Note 1. Time in seconds. 2. Performed
on AMD Athlon 1GHz with 256 MB Memory.
3. Set the base case to include 9 jobs, 3 types
of resources and 18 temporal
constraints.
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Note The base case has 9 tasks, 3 types of
resources, 8 min-delay relations and no due date
constraints. Resource 3 has two units and others
have one unit each.
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Note The base case has 9 tasks, 3 types of
resources, 10 precedence relations and no due
date constraints.
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Note The base case has 9 tasks, 3 types of
resources, 10 precedence relations and 8
min-delay relations.
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Integer Programming Approach(Unary Resource)
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Comparison of CP and BB
Note 1. Time in seconds. 2. Performed
on AMD Athlon 1GHz, 256 MB memory
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Conclusion

• The declarative nature of Constraint Programming
  reduces the effort to express and formulate the
  models for SAT, MAX_SAT and RCPSP.
• The novelty of specifying the search procedure in
  OPL separately from expressing the model will
  play an invaluable part in solving these NP-hard
  combinatorial problems.
• For large instances of these problems, we might
  need to develop hybrid integer constraint
  programming model to take advantages of both
  approaches.

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Questions? Comments?
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Thank You!