Operator assignment problem in aircraft assembly lines: a new … · Context & MotivationOperator assignment problemMathematical modelNumerical experimentsConclusion Overview 1 Context

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

7th CIRP Conference on Assembly Technologies and Systems


Overview

1 Context & Motivation

2 Operator assignment problem

3 Mathematical model

4 Numerical experiments

5 Conclusion




Overview





5 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?




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).




Industrial motivation

Aircraft companies

to reduce takt time;

to minimize the number of human errors;

to improve working conditions.




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.




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.




Ergonomics

Physical ergonomics evaluation methods




Ergonomics

Cognitive ergonomics

Type of actions: motor vs cognitive.

Worker’s personal factors: skill, age, sex.

Learning, fatigue and motivation effects.




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.





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 .





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;




Overview





5 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)





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)





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)




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)





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)





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)




Overview





5 Conclusion




Numerical experiments

Implementation

Software: IBM ILOG CPLEX 12.6Processor: Intel(R) Core(TM) i5-4670 3.40GHzRAM: 16 GB





Instance 1

289 tasks;

7 operators with 3 skills;

3 ergonomic evaluation methods.

Optimal solution found in 18 minutes.





Instance 2

447 tasks;

5 operators with 2 skills;

3 ergonomic evaluation methods.

Optimal solution found in 20 minutes.





Gantt chart for the optimal solution for instance 2




Overview





5 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.




Conclusion

Future perspectives

Consideration of cognitive and physical ergonomic factorstogether.

Evaluation of impacts of sequences of tasks.




Thanks!Questions?

O. Battaı[email protected]



Documents

Operator assignment problem in aircraft assembly lines: a new … · Context & MotivationOperator assignment problemMathematical modelNumerical experimentsConclusion Overview 1 Context