A heuristic for a reallife car sequencing problem with multiple requirements

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A heuristic for a real-life car sequencing
problemwith multiple requirements

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

MIC2005 Vienna, Austria
1 Universidade Católica do Rio de Janeiro,
Brazil 2 Universidade Federal Fluminense, Brazil
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Summary

• Problem statement
• Basic findings
• Construction heuristics
• Neighborhoods
• Local search
• Other neighborhoods
• Improvement heuristics
• ROADEF challenge
• Implementation issues
• Numerical results

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Problem statement

• Scheduling in a car factory consists in
• Assigning a production day to each vehicle,
  according to production line capacities and
  delivery dates
• Scheduling the order of cars to be put on the
  production line for each day, while satisfying as
  many requirements as possible of the plant shops
  body shop, paint shop and assembly line.

X
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Problem statement

• Paint shop requirements
• The paint shop has to minimize the consumption of
  paint solvent used to wash spray guns each time
  the paint color is changed between two
  consecutive scheduled vehicles.
• Therefore, there is a requirement to group
  vehicles together by paint color.

Minimize the number of paint color changes (PCC)
in the sequence of scheduled vehicles.
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Problem statement
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Problem statement

• Assembly line requirements
• Vehicles that require special assembly operations
  have to be evenly distributed throughout the
  total processed cars.
• These cars may not exceed a given quota over any
  sequence of vehicles.
• This requirement is modeled by a ratio constraint
  N/P at most N cars in each consecutive sequence
  of P cars are associated with this constraint.

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Problem statement
P-1 cars
X _ _ ... _ _ X
Non-constrained car
Constrained car
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Problem statement

• Assembly line requirements (cont.)
• There are two classes of ratio constraints
• High priority level ratio constraints (HPRC) are
  due to car characteristics that require a heavy
  workload on the assembly line.
• Low priority level ratio constraints (LPRC)
  result from car characteristics that cause small
  inconvenience to production.

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Problem statement

• Cost function
• Weights are associated to the objectives
  according to their priorities
• Lexicographic formulation is handled as a
  single-objective problem

Solution cost P1 ? number of violations
of HPRC P2 ? number of violations of LPRC
P3 ? number of paint color changes
EP-ENP-RAF
P1 gtgt P2 gtgt P3
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Problem statement

• Problem find the sequence of cars that optimizes
  painting and assembling requirements.
• Three different lexicographic problems exist

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

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Notation

• 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

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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.

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Basic findings

• Four step approach

Construction heuristic
First objective optimization
Second objective optimization
Third objective optimization
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Basic findings
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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.

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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)

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Construction heuristics

• Heuristic H6
• Greedy strategy using the number of additional
  HPRC violations to define the next car to be
  placed at 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)

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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

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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.

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Other neighborhoods

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

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Other neighborhoods

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

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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

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Other neighborhoods

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

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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
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Variable Neighborhood Search
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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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ROADEF Challenge

• Real life problem proposed by Renault
• First phase
• Test set A provided by Renault (16 instances)
• Results evaluated for instances in test set A
• Best teams selected (52 candidates)
• Second phase
• Test set B provided by Renault (45 instances)
• Teams improved their codes using test set B
• Third and final phase
• Renault evaluated the algorithms using test set X
  of unknown instances (19 instances)
• Instances of the three types in each test set

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ROADEF Challenge
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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.

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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 (imposed by Renault).
• Schrages random number generator.

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Numerical results
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Numerical results
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Numerical results
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Numerical results
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Numerical results
average cost
running time (s)
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Numerical results
average cost
running time (s)
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Numerical results
average cost
running time (s)
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Numerical results
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Numerical results
Team A B. Estelllon, F. Gardi, K. Nouioua Team
PUC-UFF D. Aloise, T. Noronha, C. Ribeiro, C.
Rocha, S. Urrutia