Using Constructive Search in Resource Scheduling

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Using Constructive Search in Resource Scheduling

• By Andrei Missine

2
Overview

• Brief Introduction to Constructive Search
• Running examples
• Basic Terms and Concepts in Resource Scheduling
• Some common propagators
• Disjunctive Constraint
• Edge Finding

3
Overview

• Scheduling Algorithms
• Branching by dynamic release dates
• Branching by assigning generalized precedence
  constraints
• Branching by assigning or delaying start times of
  activities
• Conclusion

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Brief Intro to Constructive Search

• Complete
• Builds and systematically traverses the search
  tree
• Root node of the tree contains assumptions only
  from the problem statement
• At each node a decision is made

• If a child backtracks to the parent the parent
  considers the next decision
• If there are no more decisions to consider at a
  node, backtrack
• Terminate when backtracking out of the root node

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Running Examples

• There will be two running examples, one for the
  unit-capacity resource and one for the
  capacitated resource.
• A unit capacity resource is a resource with a
  capacity of exactly 1 unit.
• A capacitated resource is a resource with a
  capacity that is greater than or equal to 1.

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Running Example - Unit Capacity Resource (1)

• The resource is the classroom
• The activities are the 3 classes
• Class A and class B can start at 1200 and must
  finish by 1400
• Class C can start at 1200 and must finish by
  1600
• Each class runs for an hour
• Find a schedule with the smallest makespan such
  that all classes are scheduled and there are no
  clashes

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Running Example - Capacitated Resource (2)

• A computer lab with 100 computers is the resource
• Classes are the activities
• Class A and B start at 1200 and must finish by
  1330
• Class C starts at 1200 and must finish by 1400
• Each class requires 50 computers for 1 hour
• Find a schedule with minimal makespan such that
  all classes can use the lab without going over
  the 100 computer limit

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Resource Scheduling Terms and Concepts

• Time windows an activity must execute in a
  given time window defined by the earliest start
  time and the latest end time
• Generalized precedence constraints impose a
  minimum delay between the start times of A and B

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Resource Scheduling Terms and Concepts

• EST - Earliest start time of an activity
• LST - Latest start time of an activity
• EET - Earliest end time of an activity
• LET - Latest end time of an activity

• Core execution interval - the time interval when
  the activity must execute, if any
• Makespan - the maximum of the latest end times of
  all activities

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Resource Scheduling Terms and Concepts

• Distance matrix - an n by n matrix where the
  entry (i, j) is the delay between the start time
  of activity i and the start time of activity j

• Active / Dominating schedule - a schedule such
  that all activities are scheduled as early as
  possible without violating any of the constraints

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Resource Scheduling Terms and Concepts

• Active / Dominating schedule - a schedule such
  that all activities are scheduled as early as
  possible without violating any of the constraints

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Common Propagators Disjunctive Constraint

• When two activities combined capacity
  requirement exceeds the resource capacity the two
  activities are in disjunction
• If start time of one activity is fixed then
  propagator can prune the other activitys start
  time domain.

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Common Propagators Edge-Finding (Baptiste et al,
2003)

• Edge-finding can be applied when resource usage
  of a subset of activities exceeds the resource
  capacity
• Considers activities one by one, trying to prune
  their start times.

• Pruning occurs when it is evident that given any
  feasible assignment of the remainder of the
  subset, the current domain of activity under
  consideration must be pruned.

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Common Propagators Edge-Finding

• This is done by considering core execution
  intervals and overall resource usage in an
  interval
• Can be costly - O(n2)

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Disjunctive Propagator versus Edge-Finding

• Edge-finding is overkill in unit-capacity
  resource problems
• Disjunctive propagator can be too weak on general
  capacitated resources

• Edge-finding is expensive - O(n2)
• Disjunctive propagator is cheap - O(1)
• Bottom line - use disjunctive whenever you can
  and edge finding only when necessary (e.g. for
  pre-processing)

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Propagators applied to the Examples

• For the disjunctive propagator, note that it will
  not deduce any extra info on the capacitated
  example
• Edge-finding is more powerful than disjunctive
  constraint

• For the disjunctive case and example 1, assume
  that class A is already scheduled to start at
  1200
• The propagator can then deduce that B and C must
  start at 1300 or later, and after a second
  iteration that C must start at 1400 or later

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Propagators applied to the Examples

• For edge-finding and example 2 it is possible to
  deduce that class C cannot start earlier than
  1300 even without any extra information
• The reasoning is because both A and B have a core
  execution interval 1230 - 1300)

• Thus C will not have any computers available if
  it starts any time before 1300
• Applying edge-finding to example 1 we note that
  edge finding will be able to deduce that C must
  start at 1400 or later, even without knowing any
  extra info

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The Branch-and-Bound idea

• Search for a solution
• Once a solution is found, remember it
• Keep searching
• If it becomes evident at a node that nothing
  below it leads to a solution better than the
  current, backtrack

• In the algorithms that follow the bound is on the
  makespan
• Lower-bound avoids exploring overly optimistic
  branches
• Upper-bound avoids exploring overly pessimistic
  branches
• Good bounds can greatly improve performance

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Branching by Dynamic Release Dates

• (Fest et al, 1998) introduce the idea of
  branching by selecting a set of activities to
  delay to fix broken resource utilization
  constraints
• Branches are formed by considering possible
  subsets to delay

• Overall structure of the algorithm
• Assign earliest start times to all activities
  such that precedence constraints are met
• Keep delaying activities until resource
  constraint is fixed
• Upon backtracking pick the next delaying
  alternative

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Branching by Dynamic Release Dates

• The resulting tree is rather bushy because there
  can be exponentially many different subsets to
  consider
• Authors present some ideas how to optimize the
  algorithm
• The overall performance is still rather slow

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Applying to the Examples

• Consider the unit-capacity case. At root 3
  possible delay subsets A, B, A, C and B, C
• Taking A, B results in backtracking since both
  are delayed to the 1300 - 1400 slot and cannot
  be scheduled

• Taking A, C produces the schedule B (1200), A
  (1300), C (1400)
• Taking B, C produces the schedule A (1200), B
  (1300), C (1400)

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Applying to Examples

• Now consider the capacitated case. At root the
  possible delay sets are A, B and C
• Delaying A or B leads to immediate
  backtracking (the interval 1300 - 1330 is too
  short for a 1 hour class)
• Delaying C results in the correct schedule
  where A and B start at 1200 and C starts at 1300

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Branching by Assigning generalized precedence
relationships

• (Brucker et al, 1998) suggest branching by making
  a decision at every node to assign either a
  concurrent or a disjunctive relationship between
  two free activities
• A leaf node in such a tree has all activities
  associated with some precedence relationship

• (Brucker et al, 1998) show that it is possible to
  tell whether or not a solution exists at such a
  leaf node, and if it does to find the active
  schedule
• Note that the branching factor is only 2
• This scheme was shown to be quite good by (Cesta
  et al, 2000)

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Applying to Examples

• The unit-capacity example is not interesting
  since all activities must be in disjunction
• For the capacitated example the root node
  contains no precedence relationships

• By following the algorithm, the following leafs
  form ( means concurrent, - means disjunctive)
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C
• A B, A C, B C

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Applying to Examples

• The last 4 have no solution since A and B cannot
  possibly be in disjunction (because their core
  execution intervals overlap)
• The first one has no solution also since ABC
  exceed the resource capacity

• A B, A C, B - C is a solution B starts at
  1200, A starts at 1215 and C starts at 1300
• Similarly for A B, A - C, B C (A and B are
  reversed)
• A B, A - C, B - C is also a solution A and B
  both start at 1200 and C starts at 1300

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Branching by Assigning or Delaying Start Times of
Activities

• (Dorndorf et al, 2000) present a branching scheme
  that picks a free activity at each node and
  either assigns it the earliest start time, or
  delays the start time by some amount
• Algorithm makes use of 4 consistency tests

• Precedence consistency test
• Resource consistency test
• Interval-based disjunctive consistency test
• Lag-based disjunctive consistency test
• These tests are applied at each node to further
  prune the search tree

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Branching by Assigning or Delaying Start Times of
Activities

• Selection of an activity from a subset of free
  activities is done as follows
• Find all activities such that either its earliest
  start time matches its precedence and resource
  feasible start time or it must precede some other
  free activity
• Heuristically pick an activity from this subset

• Delaying of the selected activity
• Look at all activities which share a resource
  with the current activity such that their end
  times are after the current activitys start time
• If there is at least one such activity pick the
  minimal end time of this activity as the delay
• Otherwise delay by one

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Branching by Assigning or Delaying Start Times of
Activities

• Note that again, the branching factor is only 2
• Unlike the two previous schemes, this scheme does
  not use lower bound on the makespan
• This scheme was shown to be quite good by (Cesta
  et al, 2000)

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Applying to Examples

• Consider unit-capacity case. Assume that the
  heuristics selects A
• It will first try assigning it to the earliest
  start time
• This will result in pruning B to 1300, 1400)
  and, consequently C to 1400, 1600)

• The algorithm then backtracks and will eventually
  find the solution B, A, C (after selecting B
  first) and no solutions starting with C

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Applying to Examples

• Now consider the general capacitated example
• The execution is very similar - selecting C as
  the first activity leads to backtracking and
  delaying C since it is impossible to have C start
  at 1200
• Selecting A or B will lead to the 3 solutions
  mentioned above

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Conclusion

• Propagators the tradeoff between speed versus
  pruning power
• Branch-and-bound and usefulness of good initial
  bounds
• The three branch-and-bound algorithms
• Disadvantages of Constructive Search on large
  problem instances

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References

• Baptiste Philippe, Claude Le Pape, Nuijten Wim.
  2003. Constraint-Based Scheduling Applying
  Constraint Programming to Scheduling Problems,
  Second Printing. Kluwer Academic Publishers.
• Brucker Peter, Knust Sigrid, Schoo Arno, Thiele
  Olaf. 1998. A branch and bound algorithm for
  the resource-constrained project scheduling
  problem. European Journal of Operations
  Research, 107 (1998) 272-288.

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References

• Cesta Amedeo, Oddi Angelo, Smith F. Stephen.
  2000. Iterative Flattening A Scalable Method
  for Solving Multi-Capacity Scheduling Problems.
  American Association for Artificial Intelligence,
  2000, pp 742-747.
• Fest Andreas, Möhring Rolf H., Stork Frederik,
  Uetz Marc. 1998. Resource-Constrained Project
  Scheduling with Time Windwos A Branching Scheme
  Based on Dynamic Release Dates. Technical Report
  596, Technische Universit at Berlin.
• Dorndorf Ulrich, Pesch Erwin, Phan-Huy Toàn.
  2000. A Time-Oriented Branch-and-Bound Algorithm
  for Resource-Constrained Project Scheduling with
  Generalised Precedence Constraints. Management
  Science, Vol. 46, No. 10, October 2000, pp.
  1365-1384.