An Exact Approach to EarlyTardy Scheduling with Release Dates

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An Exact Approach to Early/Tardy Scheduling with
Release Dates

• J. Valente R. Alves
• presented by
• Gülay Samatli

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Introduction

• A single machine scheduling with
• Early and tardy costs
• Different release dates
• No unforced machine idle time (non-delay
  scheduling)
• JIT production

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Introduction

• No unforced machine idle time (non-delay
  scheduling)
• Machine idleness cost is higher than the cost
  incurred by completing the job early
• Machine is heavily loaded, it must be kept
  running in order to satisfy the demand
• Existence of rj is compatible with this
  assumption, if the forced idle time caused by rj
  is quite small
• If that is not the case, assumption is
  unrealistic

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Introduction

• Problem is to schedule
• set of n independent jobs without preemptions on
  a single machine
• Job Jj becomes available at rj
• Job Jj requires pj
• Job Jj should be completed on dj
• Ejmax(0, dj-Cj)
• Tjmax(0, Cj-dj)
• The objective min
• st no unforced machine idle time

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Introduction

• Strongly NP-hard problem
• In the paper, BB algorithm based on
  decomposition
• Lower bound procedures for each of these
  subproblems
• Dominance rules for equal release dates in order
  to eliminate dominated nodes from search tree
• Several BB are tested on a set of randomly
  generated problems with up to 30 jobs

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Decomposition of the problem
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Decomposition of the problem

• Motivation for decomposition
• P1 P2 have a simpler structure
• P2(weighted tardiness with release dates) has a
  lower bounding procedure
• Given no unforced idle time, P1 is symmetrical to
  P2
• P1 is also NP-hard, given its symmetry to P2
• Solving P1 P2 would not yield a direct
  solution to P
• So it is developed efficient lower bounding
  procedures for P1 P2 to obtain a lower bound
  for P

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Decomposition of the problem
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Lower bound procedures for P1P2

• S partial schedule for P1 or P2
• U set of unscheduled jobs
• Objective obtain a lower bound on the minimum
  cost of scheduling the jobs in U after the
  partial schedule S
• time at which the last job in U to be
  scheduled will be completed (seq indp)
• (V1) (V2) optimal objective value of P1 P2
  on set U

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Lower bound procedures for P1P2

• Prop5 Given P2 on the set of unscheduled jobs U,
  let be a new problem in which the release
  dates of all jobs in U are set equal to .
  Then,
• Prop6 Given P2 on U, let
• be a lower bound for the weighted completion
  time problem 1rj on U and starting
  at .Then,

• Prop3 Given P1 on the set of unscheduled jobs U,
  let be a new problem in which the release
  dates of all jobs in U are set equal to .
  Then,
• Prop4 Given P1 on U, let
• be a lower bound for the weighted completion
  time problem 1rj on U and
  starting at .Then,

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Lower bound procedures for P

• Theorem7Let t be the current time and rmax be
  the largest release date. If trmax, we have

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Dominance rules

• used to reduce the number of nodes
• developed for the problem with identical release
  dates
• two dominance rules
• adjacent pairs of jobs
• non-adjacent pairs of jobs( it can be used only
  pi pj)

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Implementation of the BB algorithm

• The decision theory local search heuristic is
  used to generate an initial sequence for upper
  bound
• Then, dominance rules are applied to improve
• Forward-sequencing branching rule is applied
• Depth-first strategy is used to search the tree
• To discard the node
• Adjacent rule to two jobs most recently added
• Non-adjacent rule
• Lower bound

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Computational results

• 180 instances were generated for each (var,n)
  combinations
• BB was used to solve the optimality the
  instances with up to 30 jobs

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Computational results

• For smaller instances, E2T2 E1T2 provide a
  noticeable increase in the lower bound value over
  E1T1
• Improvement of weighted completion time lower
  bound is higher for tardiness subproblem
• The relative improvement over E1T1 decreases with
  n and variability

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Computational results

• Lower bound performance is poor
• The performance
• is better when pj and penalty var. is low
• improves as n increases

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Computational results

• The performance is good when are at their
  lowest value
• The performance deteriorates as increase
  (expect )

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Computational results

• The performance of upper bound is good (results
  are 1-2 above the optimum)
• Procedure generates an optimal schedule
• 2/3 of 15 20 job instances
• 1/3 of 25 30 job instances

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Computational results

• E1T1N algorithm provides the best results
• The non-adjacent rule usually leads to lower
  runtimes, even for instances with high pj and
  penalty variability

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Conclusion

• Lower bounds are far below the optimum
• BB is still able to find optimal solutions for
  problems with up to 30 jobs
• The use of non-adjacent dominance rule is
  recommended

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Comment and questions