A heuristic for a real life car sequencing problem with multiple requirements
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A heuristic for a real-life car sequencing problem with multiple requirements. EURO VNS Tenerife, Spain November 2005. Daniel Aloise 1 Thiago Noronha 1 Celso Ribeiro 1,2 Caroline Rocha 2 Sebastián Urrutia 1. 1 Universidade Católica do Rio de Janeiro, Brazil

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A heuristic for a real-life car sequencing problem with multiple requirements

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A heuristic for a real life car sequencing problem with multiple requirements

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

EURO VNS

Tenerife, Spain

November 2005

Daniel Aloise 1

Thiago Noronha 1

Celso Ribeiro 1,2

Caroline Rocha 2

Sebastián Urrutia1

1Universidade Católica do Rio de Janeiro, Brazil

2Universidade Federal Fluminense, Brazil


Summary

Summary

  • Problem statement

  • Basic findings

  • Construction heuristics

  • Neighborhoods

  • Local search

  • Other neighborhoods

  • Improvement heuristics

  • ROADEF challenge

  • Implementation issues

  • Numerical results

Heuristics for a multi-objective car sequencing problem


Problem statement

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

Heuristics for a multi-objective car sequencing problem


Problem statement1

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.

Heuristics for a multi-objective car sequencing problem


Problem statement2

7 washes!

2 washes!

Color batches have an upper bound on the batch size.

HARD CONSTRAINT

Problem statement

Heuristics for a multi-objective car sequencing problem


Problem statement3

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.

Heuristics for a multi-objective car sequencing problem


Problem statement4

There must be no more than 3 constrained cars in any consecutive sequence of 5 vehicles .

N/P = 3/5

It means that 2 constrained cars must be separated by at least P-1 consecutive non-constrained vehicles .

N/P = 1/P

Problem statement

P-1 cars

X _ _ ... _ _ X

Non-constrained car

Constrained car

Heuristics for a multi-objective car sequencing problem


Problem statement5

Minimize the number of violations of ratio constraints.

SOFT CONSTRAINTS

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.

Heuristics for a multi-objective car sequencing problem


Problem statement6

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 >> P2 >> P3

Heuristics for a multi-objective car sequencing problem


Problem statement7

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

Heuristics for a multi-objective car sequencing problem


Notation

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

Heuristics for a multi-objective car sequencing problem


Basic findings

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.

Heuristics for a multi-objective car sequencing problem


Basic findings1

Basic findings

  • Four step approach:

Construction heuristic

First objective optimization

Second objective optimization

Third objective optimization

Heuristics for a multi-objective car sequencing problem


Basic findings2

Basic findings

Heuristics for a multi-objective car sequencing problem


Basic findings3

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.

Heuristics for a multi-objective car sequencing problem


Construction heuristics

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)

Heuristics for a multi-objective car sequencing problem


Construction heuristics1

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)

Heuristics for a multi-objective car sequencing problem


Neighborhoods

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

Heuristics for a multi-objective car sequencing problem


Local search

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.

Heuristics for a multi-objective car sequencing problem


Other neighborhoods

Other neighborhoods

  • Four types of moves are explored as perturbations:

    • k-swap: k pairs of cars have their positions exchanged

Heuristics for a multi-objective car sequencing problem


Other neighborhoods1

Other neighborhoods

  • Four types of moves are explored as perturbations:

    • group swap: two groups of cars painted with different colors are exchanged

Heuristics for a multi-objective car sequencing problem


Other neighborhoods2

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

Heuristics for a multi-objective car sequencing problem


Other neighborhoods3

Other neighborhoods

  • Four types of moves are explored as perturbations:

    • reinsertion: cars involved in violations are eliminated and greedily reinserted

Heuristics for a multi-objective car sequencing problem


Iterated local search

Iterated Local Search

procedureILS

whilestopping criterion not satisfieddo

s0 BuildRandomizedInitialSolution()

s*  LocalSearch(s0)

repeat

s’  Perturbation(s*)

s’  LocalSearch(s’)

s*  AcceptanceCriterion(s*,s’)

until reinitialization criterion satisfied

end-while

end

Heuristics for a multi-objective car sequencing problem


Variable neighborhood search

procedureVNS

s*  BuildInitialSolution()

Select neighborhoods Nk, k = 1,...,kmax

while stopping criterion not satisfied do

k  1

while k  kmaxdo

s’  Shaking(s*, Nk)

s”  LocalSearch(s’)

s*  AcceptanceCriterion(s*,s”)

if s* = s” then k  1

else k  k + 1

end-while

end-while

end

Variable Neighborhood Search

Heuristics for a multi-objective car sequencing problem


Problem ep raf enp

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

Heuristics for a multi-objective car sequencing problem


Problem ep raf enp1

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

Heuristics for a multi-objective car sequencing problem


Problem ep raf enp2

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 (kmax=20)

    • Intensification: shift followed by swap moves

    • Stopping criterion: number of intensifications without improvement or given fraction of total time

Heuristics for a multi-objective car sequencing problem


Problem ep raf enp3

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

Heuristics for a multi-objective car sequencing problem


Problem ep enp raf

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

Heuristics for a multi-objective car sequencing problem


Problem ep enp raf1

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

Heuristics for a multi-objective car sequencing problem


Problem ep enp raf2

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

Heuristics for a multi-objective car sequencing problem


Problem ep enp raf3

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

Heuristics for a multi-objective car sequencing problem


Problem raf ep enp

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

Heuristics for a multi-objective car sequencing problem


Problem raf ep enp1

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

Heuristics for a multi-objective car sequencing problem


Problem raf ep enp2

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

Heuristics for a multi-objective car sequencing problem


Roadef challenge

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

Heuristics for a multi-objective car sequencing problem


Roadef challenge1

ROADEF Challenge

Heuristics for a multi-objective car sequencing problem


Implementation issues

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.

Heuristics for a multi-objective car sequencing problem


Implementation issues1

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).

  • Schrage’s random number generator.

Heuristics for a multi-objective car sequencing problem


Numerical results

Numerical results

Heuristics for a multi-objective car sequencing problem


Numerical results1

Numerical results

Heuristics for a multi-objective car sequencing problem


Numerical results2

Numerical results

Heuristics for a multi-objective car sequencing problem


Numerical results3

Numerical results

Heuristics for a multi-objective car sequencing problem


Numerical results4

Numerical results

average cost

running time (s)

Heuristics for a multi-objective car sequencing problem


Numerical results5

Numerical results

average cost

running time (s)

Heuristics for a multi-objective car sequencing problem


Numerical results6

Numerical results

average cost

running time (s)

Heuristics for a multi-objective car sequencing problem


Numerical results7

Numerical results

Heuristics for a multi-objective car sequencing problem


Numerical results8

Numerical results

Team A: B. Estelllon, F. Gardi, K. Nouioua

Team PUC-UFF: D. Aloise, T. Noronha, C. Ribeiro, C. Rocha, S. Urrutia

Heuristics for a multi-objective car sequencing problem


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