Heuristics for a multiobjective car sequencing problem

1
Heuristics for a multi-objective car sequencing
problem

• Daniel Aloise 1
• Thiago Noronha 1
• Celso Ribeiro 1,2
• Caroline Rocha 2
• Sebastián Urrutia 1

ROADEF Challenge February 2005 Tours, France
1 Universidade Católica do Rio de Janeiro,
Brazil 2 Universidade Federal Fluminense, Brazil
2
Summary

• Problem statement
• Basic findings
• Construction heuristics
• Neighborhoods
• Local search
• Improvements heuristics for each problem
• EP-RAF-(ENP), EP-ENP-RAF, RAF-EP-(ENP)
• Implementation issues
• Numerical results
• Concluding remarks

3
Problem statement

• Problem find the sequence of cars that optimizes
  painting and mounting requirements.
• Three different problems are identified

EP-RAF-(ENP)

• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of paint color changes
• Minimize the number of violations of low
  priority ratio constraints

EP-ENP-RAF

• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low priority
  ratio constraints
• Minimize the number of paint color changes

RAF-EP-(ENP)

• Minimize the number of paint color changes
• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low
  priority ratio constraints

4
Problem statement

• Some notation
• Paint color changes PCC
• High priority ratio constraints HPRC
• Low priority ratio constraints LPRC
• Ratio constraint N/P at most N cars associated
  with this constraint in any sequence of P cars
• Number of cars n
• Number of constraints m

5
Basic findings

• Heuristics are very sensitive to initial
  solutions
• Effective quick construction heuristics are a
  must.
• Same algorithm behaves differently for each
  problem
• Specific heuristics for each problem.
• Weight structure strongly differentiates the
  three objectives
• Algorithms should handle one objective at a time.
• Specific algorithms for each objective of each
  problem.
• All objectives should be taken into account
  triggering strategies.

6
Basic findings
7
Basic findings

• Many neighborhood definitions exist
• Explore simple neighborhoods for local search.
• Use complex moves as perturbations.
• Time limit is restrictive
• Optimize move evaluations and local search.
• Use appropriate data structures.
• Optimal number of paint color changes can be
  exactly computed in polynomial time
• Initial solutions for problem RAF-EP-(ENP) will
  have the minimum number of paint color changes.

8
Construction heuristics

• Heuristic H5
• Starts with the sequence of cars from day D-1.
• At each iteration, a yet unselected car is
  considered for insertion into the partial
  solution.
• Best position (possibly in the middle) to
  schedule this car into the sequence of cars
  already scheduled is that with the smallest
  increase in the cost function.
• Insertions into positions corresponding to
  infeasible partial solutions are discarded.
• Obtains a solution minimizing PCC.
• Complexity O(m.n2)

9
Construction heuristics

• Heuristic H6
• Greedy strategy using the number of additional
  HPRC violations to define the next car to be
  placed in the end of the partial sequence.
• Ties are broken in favor of more equilibrated car
  distributions.
• Second tie breaking criterion based on the
  hardness of each constraint
• Harder constraints are those applied to more cars
  and that have smaller ratios.
• Cars with harder constraints are scheduled first.
  
• Complexity O(m.n2)

10
Neighborhoods

• Local search explores two different types of
  moves (neighborhoods) evaluated in time O(1)
• swap the positions of two cars are exchanged
• shift a car is moved from its current position
  to a new specific position

11
Local search

• Local search uses swap and shift moves.
• Quick local search only cars involved in
  violations.
• Full search too many cars involved in
  violations.
• For each car, select the best improving move.
• In case of ties, best moves are kept in a
  candidate list from which one of them is randomly
  selected.
• Better and same cost solutions are accepted.
• Move evaluations quickly performed in time O(m).
• Search stops when all cars have been investigated
  without improvement.

12
Other neighborhoods

• Four types of moves are explored as
  perturbations
• k-swap k pairs of cars have their positions
  exchanged

13
Other neighborhoods

• Four types of moves are explored as
  perturbations
• group swap two groups of cars painted with
  different colors are exchanged

14
Other neighborhoods

• Four types of moves are explored as
  perturbations
• inversion order of the cars in a group painted
  with the same color is reverted

15
Other neighborhoods

• Four types of moves are explored as
  perturbations
• reinsertion cars involved in violations are
  eliminated and greedily reinserted

16
Problem EP-RAF-(ENP)
EP-RAF-(ENP)

• Build initial solution H6
• Improve 1st objective ILS with restarts
• Make solution feasible for PCC
• Improve 2nd objective without deteriorating the
  1st VNS
• Improve 3rd objective without deteriorating the
  1st and 2nd ILS with restarts

• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of paint color changes
• Minimize the number of violations of low
  priority ratio constraints

17
Problem EP-RAF-(ENP)

• Optimization of the first objective HPRC
• Build initial solution H6
• Improvement Iterated Local Search (ILS) with
  restarts
• Only first objective is considered.
• Local search swap moves
• Intensification shift followed by swap moves
• Perturbations reinsertion moves
• Reinitializations H6 or reinsertions
• Stopping criterion number of reinitializations
  without improvement or given fraction of total
  time

18
Problem EP-RAF-(ENP)

• Optimization of the second objective PCC
• Repair heuristic to make solution feasible for
  PCC
• Improvement Variable Neighborhood Search (VNS)
• First and second objectives are considered.
• First objective does not deteriorate.
• Local search swap moves
• Shaking k-swap moves (kmax20)
• Intensification shift followed by swap moves
• Stopping criterion number of intensifications
  without improvement or given fraction of total
  time

19
Problem EP-RAF-(ENP)

• Optimization of the third objective LPRC
• Improvement Iterated Local Search (ILS) with
  restarts
• All three objectives are simultaneously
  considered.
• First and second objectives do not deteriorate.
• Local search swap moves
• Intensification shift followed by swap moves
• Perturbations inversion and group swap moves
• Reinitializations variant of H6 that do not
  deteriorate the first and second objectives
• Stopping criterion time limit

20
Problem EP-ENP-RAF
EP-ENP-RAF

• Build initial solution H6
• Improve 1st objective ILS with restarts
• Improve 2nd objective without deteriorating the
  1st VNS
• Make solution feasible for PCC
• Improve 3rd objective without deteriorating the
  1st and 2nd VNS

• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low priority
  ratio constraints
• Minimize the number of paint color changes

21
Problem RAF-EP-(ENP)
RAF-EP-(ENP)

• Build initial solution minimizing 1st objective
  PCC H5
• Improve 2nd objective without deteriorating the
  1st ILS with restarts
• Improve 3rd objective without deteriorating the
  1st and 2nd ILS with restarts

• Minimize the number of paint color changes
• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low
  priority ratio constraints

22
Implementation issues

• Same quality solutions (ties) encouraged,
  accepted, and explored to diversify the search.
• Neighbors that cannot improve the current
  solution are not investigated, for example
• To do not deteriorate PCC, a car inside (but not
  in the border of) a color group may only be
  exchanged with another car with the same color.
• Swap of two cars not involved in violations
  cannot improve the total number of violations.
• Only shift moves of isolated cars can reduce the
  number of paint color changes.

23
Implementation issues

• If the current solution was a local optimum,
  after a perturbation only moves of cars in the
  region affected by the perturbation can improve
  the solution.
• When a neighbor solution is being evaluated,
  first evaluate the first objective. If the latter
  deteriorates, then stop and discard this
  neighbor.

24
Implementation issues

• Codes in C compiled with version 3.2.2 of the
  gcc compiler with the optimization flag -O3.
• Extensive use of profiling for code optimization.
• Approximately 27000 lines of code.
• C library routines linked with flag -static
  -lstdc
• Computational experiments on a Pentium IV with
  1.8 GHz clock and 512 Mbytes of RAM memory.
• Time limit 600 seconds (10 runs)
• Schrages random number generator.

25
Numerical results Set A

• Best average results for all EP-ENP-RAF instances
  in test set A
• Improved results for the second objective LPRC

26
Numerical results Set A

• Best average results for four out of eigth
  EP-RAF-(ENP) instances in test set A
• Improved results for the second objective PCC

27
Numerical results Set A

• Best average results for two out of four
  RAF-EP-(ENP) instances in test set A
• Results are very close for the other four
  instances

28
Numerical results Set B
29
Numerical results Set B
30
Numerical results Set B
31
Problem EP-ENP-RAF
EP-ENP-RAF

• Build initial solution H6
• Improve 1st objective ILS with restarts
• Improve 2nd objective without deteriorating the
  1st VNS
• Make solution feasible for PCC
• Improve 3rd objective without deteriorating the
  1st and 2nd VNS

• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low priority
  ratio constraints
• Minimize the number of paint color changes

32
Problem EP-ENP-RAF

• Optimization of the first objective HPRC
• Build initial solution H6
• Improvement Iterated Local Search (ILS) with
  restarts
• Only first objective is considered.
• Local search swap moves
• Intensification shift followed by swap moves
• Perturbations reinsertion moves
• Reinitializations H6 or reinsertions
• Stopping criterion number of reinitializations
  without improvement or given fraction of total
  time

33
Problem EP-ENP-RAF

• Optimization of the second objective LPRC
• Improvement Variable Neighborhood Search (VNS)
• First and second objectives are considered.
• First objective does not deteriorate.
• Local search swap moves
• Shaking reinsertion and k-swap moves
• Intensification shift followed by swap moves
• Stopping criterion number of intensifications
  without improvement or given fraction of total
  time

34
Problem EP-ENP-RAF

• Optimization of the third objective PCC
• Repair heuristics to make solution feasible for
  PCC
• Antecipatory analysis build good solution for
  PCC
• Swap moves to find feasible solution for PCC
• Shift moves to ensure feasibility solution may
  deteriorate
• Improvement Variable Neighborhood Search (VNS)
• All three objectives are simultaneously
  considered.
• First and second objectives do not deteriorate.
• Local search swap moves
• Shaking reinsertion and k-swap moves
• Intensification shift followed by swap moves
• Stopping criterion time limit

35
Problem RAF-EP-(ENP)
RAF-EP-(ENP)

• Build initial solution minimizing 1st objective
  PCC H5
• Improve 2nd objective without deteriorating the
  1st ILS with restarts
• Improve 3rd objective without deteriorating the
  1st and 2nd ILS with restarts

• Minimize the number of paint color changes
• Minimize the number of violations of high
  priority ratio constraints
• Minimize the number of violations of low
  priority ratio constraints

36
Problem RAF-EP-(ENP)

• Optimization of the second objective HPRC
• Improvement Iterated Local Search (ILS) with
  restarts
• First and second objectives are considered.
• First objective does not deteriorate.
• Local search swap moves
• Intensification shift followed by swap moves
• Perturbations group swap and inversion moves
• Reinitializations H5
• Stopping criterion same solution hit many times
  after given fraction of total time

37
Problem RAF-EP-(ENP)

• Optimization of the third objective LPRC
• Improvement Iterated Local Search (ILS) with
  restarts
• All three objectives are simultaneously
  considered.
• First and second objectives do not deteriorate.
• Local search swap moves
• Intensification shift followed by swap moves
• Perturbations inversion and group swap moves
• Reinitializations variant of H6 that do not
  deteriorate the first and second objectives
• Stopping criterion time limit

38
Iterated Local Search
procedure ILS while stopping criterion not
satisfied do s0 ? BuildRandomizedInitialSolu
tion() s ? LocalSearch(s0) repeat
s ? Perturbation(s) s ?
LocalSearch(s) s ? AcceptanceCriterion(
s,s) until reinitialization criterion
satisfied end-while end
39
Variable Neighborhood Search