Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Operator assignment problem in aircraftassembly lines: a new planning approach
taking into account economic andergonomic constraints
Dmitry Arkhipov, Olga Battaıa, Julien Cegarra, Alexander Lazarev
May 12, 2018
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 1/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Overview
1 Context & Motivation
2 Operator assignment problem
3 Mathematical model
4 Numerical experiments
5 Conclusion
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 2/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Overview
1 Context & Motivation
2 Operator assignment problem
3 Mathematical model
4 Numerical experiments
5 Conclusion
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 3/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Considered problem
A brief problem formulation
There is an aircraft assembly line. How to schedule assembly tasksand assign them to operators optimally? How to satisfyprecedence, resource, time and ergonomic constraints?
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
RCPSP
Resource Constrained Project Scheduling Problem (RCPSP)
Considers resources of limited availability and activities of knowndurations and resource utilization, linked by precedence relations.The problem consists of finding a schedule of minimal duration byassigning a start time to each activity such that the precedencerelations and the resource availabilities are respected. Theobjective is to minimize the project makespan.
Complexity
The problem is NP-complete in a strong sense (Garey, Johnson1975).
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Industrial motivation
Aircraft companies
to reduce takt time;
to minimize the number of human errors;
to improve working conditions.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Challenges
Operational research
very high-dimensional instances;
the basic problem (RCPSP) is known to be NP-hard.
Ergonomics
scoring methods for long work cycles (in contrast to therepetitive environment for other assembly lines e.g.automotive);
consideration of cognitive and physical ergonomic factors.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Ergonomics
Physical ergonomics
Loaded body parts: neck, trunk, upper limbs, whole body.
Load types: static postures, movements, action forces, strains.
Load parameters: duration, force intensity, hand position,
Personal factors: age, sex, height.
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 8/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Ergonomics
Physical ergonomics evaluation methods
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Ergonomics
Cognitive ergonomics
Type of actions: motor vs cognitive.
Worker’s personal factors: skill, age, sex.
Learning, fatigue and motivation effects.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Operator assignment problem
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 11/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Operator assignment problem
Data
H – planning horizon (takt time);
N – set of tasks;
O – set of operators;
S – set of operator skills, each operator has only one.
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 12/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Operator assignment problem
Tasks
rj – release time;
pj – processing time;
ajx – amount of resource x ∈ R required to process task j ;
bjs – number of operators with skill s ∈ S required to processtask j .
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Operator assignment problem
Physical ergonomic risks
M – set of ergonomic risk evaluation methods;
ergmj – ergonomic score evaluated by method m ∈ M for onetime unit of task j ∈ N;
Umo – an upper bound on total ergonomic impact foroperator o evaluated by method m;
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 14/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Overview
1 Context & Motivation
2 Operator assignment problem
3 Mathematical model
4 Numerical experiments
5 Conclusion
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 15/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Constraint programming model for the aggregated demand
Decision variables
intervalj – interval variable associated to the execution of taskj ∈ N, i.e. intervalj = [Sj ,Cj);
Objective function
The objective is to find a schedule π∗ with the minimal makespani.e.
minπ
maxj∈N
(Sj(π) + pj). (1)
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Constraint programming model for the aggregated demand
Constraints
The task interval size has to be equal to the task processing time, i.e.
∀j ∈ N : |intervalj | = pj . (2)
Task processing intervals must satisfy the precedence relations with timelags, i.e.
∀eji ∈ E : Sj(π) + lji ≤ Si (π). (3)
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Constraint programming model for the aggregated demand
Resource capacity constraints
Resource capacity constraint:
F (x , t) =∑j∈N
ajx · f (intervalj , t), (4)
where f (intervalj , t) = 1 if t ∈ intervalj and f (intervalj , t) = 0otherwise.Then resource capacity constraint can be formulated as
∀x ∈ R, t : cx ≥ F (x , t). (5)
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 18/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
MIP model for Operator assignment problem
Decision variables
assignoj – binary variable equals to 1 if operator o ∈ Oassigned on task j ∈ N, otherwise assignoj = 0.
Objective function
The objective function is to minimize the highest ergonomicimpact calculated for each pair (m ∈ M, o ∈ O).
min maxm∈M,o∈O
∑j∈N
assignoj · ergmjso (6)
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 19/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
MIP model for Operator assignment problem
Constraints
For each task j ∈ N, the number of operators with skill s ∈ S hasto be equal to bjs
∀j ∈ N, s ∈ S :∑
o∈O:so=s
assignoj = bjs . (7)
The total ergonomic impact of the tasks assigned to the sameoperator o ∈ O measured by method m ∈ M has to be less thanthe defined critical level Umo , i.e.
∀m ∈ M, o ∈ O : Umo ≥∑j∈N
ergmjso · assignoj . (8)
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
MIP model for Operator assignment problem
Incompatibility constraints
Since the schedule of the tasks is known, the incompatible sets Eof tasks can be defined, i.e. the sets of the tasks e that cannot beperformed by the same operator.
∀e ∈ E , o ∈ O :∑j∈e
assignoj ≤ 1. (9)
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Overview
1 Context & Motivation
2 Operator assignment problem
3 Mathematical model
4 Numerical experiments
5 Conclusion
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 22/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Numerical experiments
Implementation
Software: IBM ILOG CPLEX 12.6Processor: Intel(R) Core(TM) i5-4670 3.40GHzRAM: 16 GB
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Numerical experiments
Instance 1
289 tasks;
7 operators with 3 skills;
3 ergonomic evaluation methods.
Optimal solution found in 18 minutes.
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 24/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Numerical experiments
Instance 2
447 tasks;
5 operators with 2 skills;
3 ergonomic evaluation methods.
Optimal solution found in 20 minutes.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Numerical experiments
Gantt chart for the optimal solution for instance 2
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 26/30
7th CIRP Conference on Assembly Technologies and Systems
Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Overview
1 Context & Motivation
2 Operator assignment problem
3 Mathematical model
4 Numerical experiments
5 Conclusion
D. Arkhipov, O. Battaıa, J. Cegarra, A. Lazarev 27/30
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Conclusion
Obtained results
Operator assignment problem for aircraft assembly line subjectto ergonomic constraints was considered;
Constraints programming and Integer linear programmingmodels were developed;
Optimal solutions were found for two industrial instances inreasonable time.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Conclusion
Future perspectives
Consideration of cognitive and physical ergonomic factorstogether.
Evaluation of impacts of sequences of tasks.
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Context & Motivation Operator assignment problem Mathematical model Numerical experiments Conclusion
Thanks!Questions?
O. Battaı[email protected]
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