A Hybrid Approach to Scheduling with Earliness and Tardiness Costs

1
A Hybrid Approach to Scheduling with Earliness
and Tardiness Costs

• Dai-Hanh
• Inf6101

2
Plan

• Introduction
• Problem definition
• Mixed Integer Programming technique
• Hybrid technique
• Experiment
• Conclusion

3
Introduction

• Optimization techniques based on linear
  relaxation perform well in reasoning about costs,
  providing an optimal solution of a linear
  relaxation of the problem
• In the constraint based scheduling, it is common
  to infer precedence constraints which state that
  one activity must start after the completion of
  another activity. The addition of constraints to
  the problem model can also be used as a branching
  rule
• A hybrid cp/lp model where the cp model contains
  the original scheduling problem plus the new
  resource capacities resulting from a resource
  breakdown and the lp model contains a linear
  representation of each activity and a cost
  function

4
Problem definition

• A set of jobs
• A set of resources
• ETSP

5
MIP technique

• Scheduling problem with MIP require a schedule
  horizon, T. For each activity I, it introduces T
  binary variable .The variable
  is set to 1 if the activity i start at time t.

6
Hybrid technique

• Probe
• CRS-ALL
• ProbePlus
• CRS-Root

7
Experiment
8
Conclusion

• The major drawback of the MIP approach is the
  size of the matrix which depends on T
• CRS-ALL will spend significantly more time than
  Probe at each search nodes
• The ProbePlus technique show that a simple
  addition to the Probe technique can lead to
  better problem solving
• The MIP technique out-performed the pure CP
  technique and the Probe approaches
• CRS can lead more performance than a linear
  relaxation
• For some problem, MIP technique produced slightly
  superior resultants than the CRS-based hybrid
  technique

9
Problem definitionJobs

• A set , J, of n jobs, m activities per job
• A(j) aj1, aj2,ajm
• Aji characterized by its start time S(aji) and
  its processing time pt(aji)
• Once an activity has started execution, it must
  execute for its entire duration without
  interruption.
• For each job, j ? J, there is a due date ddj and
  two cost factors, earliness, ecj, and tardiness,
  tcj.
• A cost
• Cj ecj( ddj - s(ajm) pt(ajm)) if s(ajm)
  pt(ajm) ddj
• Cj tcj(s(ajm) pt(ajm) - ddj ) if s(ajm)
  pt(ajm) gt ddj

10
Problem definitionResource

• A set, R, of k resource
• Rcap(aji,rh) is the capacity of resource rh ? R
  which is required by aji
• Cap(rh) is the capacity of resource rh ? R which
  is a maximum amount of the resource available.

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Problem definitionETSP

• The activities in a job are processed in the
  specified order.
• S(aji) pt(aji) S(aji1)
• For each resource, the resource capacity
  constraints are respected
• Rcap(aji, rh) cap(rh)
• Total cost of the problem, CJ, is minimized

12
Probe

• Procedure PROBE (node n)
• solve linear relaxation
• build resource profiles with relaxed optimal
  solution
• If no peaks then
• return optimal solution
• else
• heuristically identify activity pair at peak
  A,B
• return Probe(n(A?B)) \/ Probe(n(B?A))

13
CRS-ALL

• Procedure CRS-ALL (node n)
• solve CRS
• if CRS solution can extended to global solution
  then
• return global solution
• else
• use the cost of the CRS solution as a lower
  bound
• solve linear relaxation
• build resource profiles with relaxed optimal
  solution
• if no peaks then
• return relaxed optimal solution
• else
• heuristically identify activity pair at peak
  A, B
• return CRS-ALL(n(A?B)) \/ CRS-ALL(n(B?A))

14
ProbePlus

• Procedure ProbePLus(node n)
• solve linear relaxation
• assign relaxed optimal start time to cost
  relevant activities
• If CP goal finds a global solution then
• return global solution
• else
• retract start times assignment of cost relevant
  activities
• build resource profiles with relaxed optimal
  solution
• if no peaks then
• return relaxed optimal solution
• else
• heuristically identify activity pair at peak A,
  B
• return ProbePlus(n(A?B)) \/ ProbePlus(n(B?A))

15
CRS-ROOT

• Procedure CRS-ROOT(node n)
• solve CRS
• if CRS solution can be extended to a global
  solution then
• return global solution
• else
• use the cost of the CRS solution as a lower
  bound
• solve liner relaxation
• build resource profiles with relaxed optimal
  solution
• if no peaks then
• return relaxed optimal solution
• else
• heuristically identify activity pair peak A, B
• return ProbePlus(n(A?B)) \/ ProbePlus(n(B?A))

16
Solve linear relaxation

• A liner relaxation is obtained by using the
  mixed-integer programming
• smin(aji) Sji smax(aji)
• Sji pt(aji) Sji1
• Minimize
• Cj ecj( ddj - s(ajm) pt(ajm)) if s(ajm)
  pt(ajm) ddj
• Cj tcj(s(ajm) pt(ajm) - ddj ) if s(ajm)
  pt(ajm) gt ddj

17
Build resource profiles

• Using the relaxed optimal start time for each
  activity, we build resource profiles and identify
  a peak

18
CRS

• The CRS of an ETSP contains the resources and
  cost function of the full ETSP but only the cost
  relevant activities.
• CRS is itself an instance of an ETSP with each
  job containing one activity.
• If no solution exists for the subset of
  constraints in the CRS at nodes S, no solution
  exists in the global problem at node S