Temporal Constraint Propagation NonPreemptive Case

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Temporal Constraint Propagation(Non-Preemptive
Case)

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Outline

• Variables
• Relations between the variables
• Temporal constraints
• Time bounds
• Minimal and maximal distances between time points

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Variables (definition)

• Three variables
• start(A)
• end(A)
• duration(A)
• for each activity A

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Variables (implementation)

• Finite domain (bitvector)
• 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)

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Relation between the variables

• 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))

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Temporal constraints

• Simple precedences
• start(A) start(B)
• start(A) end(B)
• end(A) start(B)
• end(A) end(B)

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Temporal constraints

• Precedences with minimal delays
• start(A) delay start(B)
• start(A) delay end(B)
• end(A) delay start(B)
• end(A) delay end(B)

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Temporal constraints

• Precedences with fixed delays
• start(A) delay start(B)
• start(A) delay end(B)
• end(A) delay start(B)
• end(A) delay end(B)

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Temporal constraints

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

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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)

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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(n3) but not
  worth it in practice)

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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

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Minimal and maximal distances

• Matrix-based method
• Whenever dxy is modified, update dwz to max(dwz,
  dwx dxy dyz)

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Minimal and maximal distances

• Complexity
• O(n2) after each modification of the constraint
  network
• O(n3) to initialize the matrix