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

2. 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

3. X 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.
Qual a tradução para body shop e paint shop???? X
Heuristics for a multi-objective car sequencing problem

4. 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

5. Color batches have an upper bound on the batch size.
Problem statement
7 washes!
2 washes!
Color batches have an upper bound on the batch size.
HARD CONSTRAINT
Heuristics for a multi-objective car sequencing problem

6. 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

7. Problem statement
N/P = 3/5
There must be no more than 3 constrained cars in any consecutive sequence of 5 vehicles .
N/P = 1/P
It means that 2 constrained cars must be separated by at least P-1 consecutive non-constrained vehicles .
P-1 cars
X _ _ _ _ X
Constrained car
Non-constrained car
Heuristics for a multi-objective car sequencing problem

8. 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.
Minimize the number of violations of ratio constraints.
SOFT CONSTRAINTS
Heuristics for a multi-objective car sequencing problem

9. 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

10. 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

11. 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

12. 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

13. Basic findings Four step approach: Construction heuristic
First objective optimization
Second objective optimization
Third objective optimization
Heuristics for a multi-objective car sequencing problem

14. Basic findings H6 H5 ILS VNS Algorithms: EP-RAF-(ENP) EP-ENP-RAF
RAF-EP-(ENP)
Construction H6 H5
First objective
ILS
Second objective
VNS
Third objective
Heuristics for a multi-objective car sequencing problem

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. Iterated Local Search procedure ILS
while stopping criterion not satisfied do
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

25. Variable Neighborhood Search
procedure VNS
s*  BuildInitialSolution()
Select neighborhoods Nk, k = 1,...,kmax
while stopping criterion not satisfied do
k  1
while k  kmax do
s’  Shaking(s*, Nk)
s”  LocalSearch(s’)
s*  AcceptanceCriterion(s*,s”)
if s* = s” then k  1
else k  k + 1
end-while
end
Heuristics for a multi-objective car sequencing problem

26. Problem 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

27. 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

28. 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

29. 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

30. Problem 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. ROADEF Challenge ID Instances EP-RAF-ENP X4 023_S49_J2 X5 024_S49_J2
034_VP_S51_J1_J2_J3 X8
034_VU_S51_J1_J2_J3 X9
039_CH1_S49_J1
X10
039_CH3_S49_J1
X11
048_CH1_S50_J4
X12
048_CH2_S49_J5
X13
064_CH1_S49_J1
X14
064_CH2_S49_J4
X15
655_CH1_S51_J2_J3_J4
X16
655_CH2_S52_J1_J2_S01_J1 ID
Instances
EP-ENP-RAF X1
025_S49_J1 X2
028_CH1_S50_J4 X3
028_CH2_S51_J1 ID
Instances
RAF-EP-(ENP)
X17
022_S49_J2
X18
035_CH1_S50_J4*
X19
035_CH2_S50_J4*
Heuristics for a multi-objective car sequencing problem

39. 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

40. Implementation issues
Codes in C++ compiled with version of the gcc compiler with the optimization flag -O3.
Extensive use of profiling for code optimization.
Approximately 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

41. Our average results (5 runs) Best average results (5 runs)
Numerical results
Instances
EP-ENP-RAF
Our average results (5 runs)
Best average results (5 runs)
HPRC
LPRC
PCC
Cost X1
0.0
160.0
602.2
407.6 X2
36.0
341.4
95.4 X3
3.0
Team A
PUC-UFF
PUC-UFF+
Heuristics for a multi-objective car sequencing problem

42. Our average results (5 runs) Best average results (5 runs)
Numerical results
Instances
EP-RAF-(ENP)
Our average results (5 runs)
Best average results (5 runs)
HPRC
LPRC
PCC
Cost X4
0.0
77.2
193.0
66.0
192.4 X5
12.0
352.8
6.0
337.0 X6
111.8
98.4
110.2 X7
643.6
58.0
794.8
55.2 X8
8.0
35.8
87.0 X9
479.6
69.0
239.0
Team A
Team C
Team E
PUC-UFF
Heuristics for a multi-objective car sequencing problem

43. Our average results (5 runs) Best average results (5 runs)
Numerical results
Instances
EP-RAF-(ENP)
Our average results (5 runs)
Best average results (5 runs)
HPRC
LPRC
PCC
Cost
X10
0.0
2162.6
231.0
30.0
X11
1016.0
196.0
1005.6
X12
31.0
1128.4
79.0
1116.2
76.8
X13
61.0
81.4
190.8
29.8
187.2
X14
37.0
X15
X16
153.0
34.0
Team B
Team D
Team A
PUC-UFF+
Heuristics for a multi-objective car sequencing problem

44. Our average results (5 runs) Best average results (5 runs)
Numerical results
Instances
RAF-EP-(ENP)
Our average results (5 runs)
Best average results (5 runs)
HPRC
LPRC
PCC
Cost
X17
2.0
3.0
12.0
X18
10.0
0.0
5.0
X19
56.0
6.0
PUC-UFF+
Heuristics for a multi-objective car sequencing problem

45. Numerical results average cost running time (s)
Heuristics for a multi-objective car sequencing problem

46. Numerical results average cost running time (s)
Heuristics for a multi-objective car sequencing problem

47. Numerical results average cost running time (s)
Heuristics for a multi-objective car sequencing problem

48. Numerical results
Heuristics for a multi-objective car sequencing problem

49. 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